microsoft word 3-王剑飞.doc 14-18 advances in systems science and applications (2010), vol.10, no.1 issn 1078-6236 international institute for general systems studies, inc. extensions of liouville’s theorem jianfei wang applied science college, harbin university of science and technology, harbin 150080, china email: jianfei1965@sohu.com abstract in this paper, we first generalize liouville’s theorem into the general forms based on power series representations for analytic functions. second, in simply connected domains harmonic functions can be identified as real parts of analytic functions. observing the relations between analytic functions and harmonic functions, we extend liouville’s theorem to harmonic functions by the harnack’s inequality. the generalized liouville’s theorems obtained in this paper will help us to further study the properties of entire functions and harmonic functions. keywords liouville’s theorem entire function extension harmonic function harnack’s inequality analytic 1. introduction if f (z) is analytic on the whole complex plane, then it is said to be an entire function. for entire functions, there exists a beautiful theorem, known as liouville’s theorem. it gives many important properties of entire functions, these properties have been widely applied to complex analysis. and it enables us to prove the fundamental theorem of algebra. therefore, it is necessary for us to further discuss liouville’s theorem. in this paper, we first derive two extension forms of liouville’s theorem by simplifying some conditions of classical liouville’s theorem. meanwhile we get two results for analytic functions. second, in simply connected domains harmonic functions can be identified as real parts of analytic functions. and there are many consequences for analytic functions. some of these are the infinite differentiability of analytic functions, liouville’s theorem, and the maximum modulus theorem. hence we think that these results have analogues for harmonic functions. in terms of the ideas, we extend liouville’s theorem to harmonic functions. the generalized liouville’s theorems provide theoretical basis for us to further study the properties of entire functions and harmonic functions. 2. inequalities and lemmas we list some useful inequalities and lemmas before giving to the generalized liouville’s theorems. theorem 2.1 (liouville’s theorem) [1] the only bounded entire functions are the constant functions. lemma 2.1 [2] let ϕ be harmonic on a simply connected domain d . then there is an analytic function f such that re fϕ = on d . lemma 2.2 (mean-value theorem for harmonic function) [3] let ϕ be harmonic in a domain containing the dish z r≤ . then 2 0 1(0) ( ) 2 i tre dt π ϕ ϕ π = ∫ (1) lemma 2.3 (poisson integral formula) [2] let ϕ be harmonic in a domain containing the dish z r≤ . then for 0 r r≤ < , we have advances in systems science and applications (2010), vol.10, no.1 15 ( ) ( ) ( ) 2 2 2 2 202 2 cos i t i rer rre dt r r rr t πθ ϕ ϕ π θ − = + − −∫ (2) proof: using lemma 2.1, we have re fϕ = here f is analytic on a simply connected domain d . (assuming the domain d includes the circle :rc z r= as well as its interior) applying cauchy integral formula, we obtain ( ) ( ) ( )1 2 rc f f z d z r i z ζ ζ π ζ = < −∫ (3) for fixed z , with z r< , the function ( ) 2 f z r z ζ ζ− is an analytic function of ζ inside and on rc . hence by cauchy theorem ( ) 2 1 0 2 rc f z d i r z ζ ζ π ζ = −∫ (4) we add it to equation (eq.) (3): ( ) ( )2 1 1 2 rc zf z f d i z r z ζ ζ π ζ ζ ⎛ ⎞ = +⎜ ⎟⎜ ⎟− −⎝ ⎠ ∫ ( )( ) ( ) 22 2 1 2 rc r z f d i z r z ζ ζ π ζ ζ − = − −∫ (5) if we parameterize rc by , 0 2 ,i tre tζ π= ≤ ≤ eq. (5) becomes ( ) ( )( ) ( ) 22 2 20 1 2 i t i t i t i t r z f z f re r e dt re z r re z π π − = − −∫ (6) ( ) ( )( ) 22 2 02 i t i t i t f rer z dt re z re z π π − − = − −∫ (7) ( )22 2 202 i t i t f rer z dt re z π π − = − ∫ (8) writing z in the polar form iz re θ= , we have ( ) ( )2 2 2 202 i t i i t i f rer rf re dt re re πθ θπ − = − ∫ (9) ( ) ( ) 2 2 2 2 202 2 cos i tf rer r dt r r rr t π π θ − = + − −∫ (10) finally, by taking the real part of this equation, we arrive at poisson integral formula ( ) ( ) ( ) 2 2 2 2 20 , , 2 2 cos r tr rr dt r r rr t π ϕ ϕ θ π θ − = + − −∫ (11) or, equivalently, ( ) ( ) ( ) 2 2 2 2 202 2 cos i t i rer rre dt r r rr t πθ ϕ ϕ π θ − = + − −∫ (12) poisson integral formula expresses the values of a harmonic function in a region is wang: extensions of liouville’s theorem 16 completely determined by its values on the boundary. using poisson integral formula and mean value theorem for harmonic function, we may derive the following important inequality. theorem 2.2 (harnack’s inequality) [2] let ϕ be harmonic and nonnegative in a domain containing the dish z r≤ . then for 0 r r≤ < , we have ( ) ( ) ( )0 0ir r r rre r r r r θϕ ϕ ϕ− + ≤ ≤ + − (13) proof: applying lemma 2.3 to harmonic functionϕ . then for 0 r r≤ < , ( ) ( ) ( ) 2 2 2 2 202 2 cos i t i rer rre dt r r rr t πθ ϕ ϕ π θ − = + − −∫ (14) observing that ( ) ( ) ( )2 22 2 2 cosr r r r rr t r rθ− ≤ + − − ≤ + (15) since ϕ is nonnegative, we have ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 20 0 02 cos i t i t i tre re re dt dt dt r r rr tr r r r π π πϕ ϕ ϕ θ ≤ ≤ + − −+ −∫ ∫ ∫ (16) hence ( ) ( ) ( )2 2 0 0 1 1 2 2 i t i i tr r r rre dt re re dt r r r r π πθϕ ϕ ϕ π π − + ⋅ ≤ ≤ ⋅ + −∫ ∫ (17) finally, by lemma 2.2, the proof is completed. 3. the generalized liouville’s theorems in this section, we consider the extension problem of classical liouville’s theorem. this is the main work of this paper. firstly, we obtain two extension forms of liouville’s theorem by reducing the condition of classical liouville’s theorem. these theorems are stated below. theorem 3.1 i f f is an entire function, and suppose there are a nonnegative integer n , and two positive constants ,r m such that ( ) nf z m z≤ when z r≥ , then f is a polynomial with ( )deg f n≤ or constant. proof: the case 0n = will be treated first. in this case, ( )f z m≤ when z r≥ . since f is an entire function, it must be continuous on the closed domain z r≤ .under such circumstances it is known from calculus that the function must be bounded there. in other words, there exists a positive constant g such that ( )f z g≤ when z r≤ . by taking { }max , 0n m g= > , we have ( )f z n≤ (18) when z < +∞ . thus from theorem 2.1 there follows f is constant. this completes the proof. then the general case 1n ≥ will be dealt with. since f is an entire function, it must have a maclaurin series representation, namely ( )2 0 1 2( ) n nf z c c z c z c z z= + + + + + < +∞ (19) where ( ) ( ) ( ) ( ) 1 0 1 0; 0,1, 2, ! 2 n n nz r f f c d r n n i ζ ζ π ζ += = = > =∫ (20) for an arbitrary integer 1p ≥ and a large enough 0r r> , we consider that advances in systems science and applications (2010), vol.10, no.1 17 ( ) 0 1 1 2n p n pz r f c d i ζ ζ π ζ+ + += = ∫ (21) notice that ( ) 0 1 1 2n p n pz r f c ds ζ π ζ + + += ≤ ∫ 0 1 1 2 n n pz r m d s ζ π ζ + += ≤ ∫ 01 0 0 1 2 2 p p m mr r r π π += ⋅ ⋅ = (22) we can get ( )0 1,2,n pc p+ = = . thus ( )2 0 1 2( ) n nf z c c z c z c z z= + + + + < +∞ (23) it says that f is a polynomial with ( )deg f n≤ . by modifying the condition of theorem 3.1, we easily obtain the following results. corollary 3.1 i f f is an entire function, and suppose there are a nonnegative integer n , and three positive constants ,r m and n such that ( ) nf z n m z≤ + when z r≥ , then f is a polynomial with ( )deg f n≤ or constant. corollary 3.2 i f f is an entire function, and there is a positive integer n such that ( )lim 0nz f z k z→∞ = > , then f is a polynomial with ( )deg f n≤ . theorem 3.2 let f be analytic in the extended complex plane. then f is constant. proof: since f is analytic in the extended complex plane, it must have a maclaurin series representation, namely ( ) 0 ( ) n n n f z c z z ∞ = = < +∞∑ (24) meanwhile, z = ∞ is a removable singularity of f . hence ( )f z has a finite limit as z approaches 0z . recalling (24), the conclusion is obtained. secondly, in simply connected domains harmonic functions can be identified as real parts of analytic functions. based on the relations between analytic functions and harmonic functions, harmonic function in 2r is considered [4] in generalizing liouville’s theorem. we state liouville’s theorem for harmonic functions as follows. theorem 3.3 (liouville’s theorem for harmonic functions) [5] let ϕ be harmonic in the whole real plane 2r and bounded from above or below there. thenϕ is constant. proof: assume that ϕ is bounded from above there, namely, there exists a constant m such that mϕ ≤ for any 2z r∈ . clearly, mψ ϕ= − is harmonic and nonnegative in the whole real plane 2r . using theorem 2.2, for 0 r r≤ < < +∞ , we have ( ) ( ) ( )0 0ir r r rre r r r r θψ ψ ψ− + ≤ ≤ + − (25) letting r →+∞ , deduce that ( ) ( )0ire θψ ψ= (26) where [ )0,r∈ +∞ . wang: extensions of liouville’s theorem 18 notice that r is an arbitrary nonnegative real number hence mψ ϕ= − is constant. it follows immediately that ϕ is constant. 4. conclusions summing up, we first transform some useful inequalities and lemmas to generalize liouville’s theorem. then using power series representations for analytic functions [6] , we derive two extension forms of liouville’s theorem by simplifying some conditions of classical liouville’s theorem. finally, observing the relations between analytic functions and harmonic functions, we extend liouville’s theorem to harmonic functions by harnack’s inequality. in this course, we know that the generalized liouville’s theorems will help us to further study the properties of entire functions. meanwhile, we can get some important conclusions for harmonic functions [7] based on the consequences for analytic functions in the future. references [1] yuquan zhong. functions of complex variable. higher education press, beijing, 2004: 127-130. [2] saff, e. b. fundamental of complex analysis with applications to engineering and science. china machine press, beijing, 2004: 221-226. [3] jiarong yu. functions of complex variable. higher education press, beijing, 2000: 156-157. [4] weixing dai, shaobo zhou. attraction and stability for neutral stochastic differential delay equations. advances in systems science and applications, 2008, 8(2): 212-219. [5] zhenhua jiao. a note on liouville’s theorem. journal of hangzhou dianzi university, 2006, 26(2): 96-98. [6] jiancong chen, zhihong guan and hu chen. fuzzy association analysis of complex system in correlation. advances in systems science and applications, 2008, 8(2): 251-257. [7] elias zafiris. categorical modeling of natural complex systems. part i: functorial process of representation. advances in systems science and applications, 2008, 8(2): 187-200. advances in systems science and application (2016) vol.16 no.2 94-102 driver drowsiness detection system meriem boumehed1, belal alshaqaqi2, abdullah salem baquhaizel2 and mohamed el amine ouis2 1preparatory school in science and technology 2university of sciences and technology of oran mohamed boudiaf (usto-mb) oran, algeria abstract drowsiness and fatigue of drivers are amongst the significant causes of road accidents. every year, they increase the amounts of deaths and fatalities injuries globally. in this paper, a module for advanced driver as-sistance system (adas) is presented to reduce the number of accidents due to drivers fatigue and hence increase the transportation safety; this system deals with automatic driver drowsiness detection based on visual information and artificial intelligence. we proposed an algorithm to locate, track, and analyze both the drivers face and eyes to measure percentage of eye dosure, a scientifically supported measure of drowsiness associated with slow eye closure. keywords drowsiness detection, adas, face detection and tracking, eyes detection and tracking 1 introduction currently, transport systems are an essential part of human activities. we all can be victim of drowsiness while driving, simply after a night’s sleep too short, altered physical condition or during long journeys. the sensation of sleep reduces the driver’s level of vigilance producing dangerous situations and increases the probability of an occurrence of accidents. driver drowsiness and fatigue are among the important causes of road accidents. every year, they increase the number of deaths and fatalities injuries globally. in this context, it is important to use new technologies to design and build systems that are able to monitor drivers and to measure their level of attention during the entire process of driving. in this paper, a module for adas (advanced driver assistance system) is presented in order to reduce the number of accidents caused by driver fatigue and thus improve road safety. this system treats the automatic detection of driver drowsiness based on visual information and artificial intelligence. we proposed an algorithm to locate, track and analyze both the driver face and eyes to measure percentage of eye closure. the remainder of this paper is organized as follows, section 2 presents the related works, section 3 presents the proposed system and the implementation of each block of the system, the experimental results are shown in section 4 and in the last section conclusions and perspectives are presented. advances in systems science and application (2016) vol.6 no.2 95 2 related works some efforts have been reported in the literature on the development of the notintrusive monitoring drowsiness systems based on the vision. malla et al.[1] develop a light-insensitive system. they used the haar algorithm to detect objects[2] and face classifier implemented by[3] in opencv[4] libraries. eye regions are derived from the facial region with anthropometric factors. then, they detect the eyelid to measure the level of eye closure. vitabile et al. implement a system to detect symptoms of driver drowsiness based on an infrared camera[5]. by exploiting the phenomenon of bright pupils, an algorithm for detecting and tracking the driver’s eyes has been developed. when drowsiness is detected, the system warns the driver with an alarm message. bhowmick et kumar use the otsu thresholding to extract face region[6,7]. the localization of the eye is done by locating facial landmarks such as eyebrow and possible face center. morphological operation and k-means is used for accurate eye segmentation. then a set of shape features are calculated and trained using non-linear svm to get the status of the eye. hong et al. define a system for detecting the eye states in real time to identify the driver drowsiness state[8]. the face region is detected based on the optimized jones and viola method[2]. the eye area is obtained by an horizontal projection. finally, a new complexity function with a dynamic threshold to identify the eye state. tian et qin build a system that checks the driver eye states. their system uses the cb and cr components of the ycbcr color space[9]. this system locates the face with a vertical projection function, and the eyes with a horizontal projection function. once the eyes are located the system calculates the eyes states using a function of complexity. under the light of what has been mentioned above, the identification of the driver drowsy state given by the perclos is generally passed by the following stages: 1) face detection, 2) eyes location, 3) face and eyes tracking, 4) identification of the eyes states, 5) calculation of perclos and identification the driver state. 3 the proposed system in this section, we discuss our presented system which detects driver drowsiness. the overall flowchart of our system is shown in fig 1. 96 meriem b., belal a., abdullah salem b, etc: driver drowsiness detection system a. face detection the symmetry is one of the most important facial features.we modeled the symmetry in a digital image by a one-dimensional signal (vector accumulator) with a size equal the width of the image, which gives us the value corresponding to the position of the vertical axis of symmetry of objects in the image. the traditional principle to calculate the signal of symmetry is for each two white pixels which are on the same line we increment the value in the medium between these two pixels in the accumulating vector. (the algorithm is applied on an edge image, we called a white pixel: the pixel with value 1). we introduce improvements on the calculation algorithm of symmetry into an image to adapt it to the detection of face, by applying a set of rules to provide a better calculation of symmetry of the face. instead of computing the symmetry between two white pixels in the image, it is calculated between two windows (z1 and z2) (fig.2). fig. 1 flowchart of the proposed system for each window z1, we sweep the window z2 in the area determined by the parameters s min, s max, and h. we increment the signal of symmetry between advances in systems science and application (2016) vol.6 no.2 97 these two windows if the sum of white pixels is located between two thresholds s1 (maximum) and s2 (minimum). fig. 2 the new method to improve the calculation of the symmetry in the image (a) original image (b) edge detection (c) symmetry signal (d) localization of the maximum of symmetry (e) region of interest roi (f) result fig. 3 face detection using symmetry then we extract the vertical region of the image contours (region of interest roi) corresponding to the maximum index of the obtained signal of symmetry. next, we take a rectangle with an estimated size of face (because the camera is fixed and the driver moves in a limited zone so we can estimate the size of the face using the camera focal length after the step of camera calibration) and we 98 meriem b., belal a., abdullah salem b, etc: driver drowsiness detection system scan the roi by searching the region that contains the maximum energy corresponding to the face (fig.3). we propose a checking on two axes: the position variance of the face detected according to time; i.e., in several successive images, it is necessary that the variance of the positions of the detected face is limited; because the speed of movement of the face is limited of some pixels from a frame to another frame which follows. b. eyes localization since the eyes are always in a defined area in the face (facial anthropometric properties), we limit our research in the area between the forehead and the mouth (eye region of interest ’eroi’) (fig.4.a). we benefit from the symmetrical characteristic of the eyes to detect them in the face. first, we sweep vertically the eroi by a rectangular mask with an estimated height of height of the eye and a width equal to the width of the face, and we calculate the symmetry. the eye area corresponds to the position which has a high measurement of symmetry. then, in this obtained region, we calculate the symmetry again in both left and right sides. the highest value corresponds to the center of the eye. the result is shown in fig.4.b. (a) eroi (b) result fig. 4 eyes localization using symmetry c. tracking the tracking is done by template matching using the sad algorithm (sum of absolute differences). sad(x, y) = n∑ i=1 m∑ i=1 |i(x+ i, y + j)−m(i, j)| (1) we proposed to make a regular update of the reference model m to adjust it every time when light conditions changes while driving, by making a tracking advances in systems science and application (2016) vol.6 no.2 99 test: tracking = { good if sad ≤ th bad if sad > th (2) d. eyes states the determination of the eye state is to classify the eye into two states: open or closed. we use the hough transform for circles [10] (htc) on the image of the eye to detect the iris. for that, we apply the htc to the edge image of the eye to detect the circles with defined rays, and we take at the end the circle which has the highest value in the accumulator of hough for all the rays. then, we apply the logical ’and’ logic between edges image and complete circle obtained by the htc by measuring the intersection level between them “s”. finally, the eye state “stateeye” is defined by testing the value “s” by a threshold: stateeye { open if s ≤ th closed if s ≥ th (3) the results are shown in fig.5. (a) and (b) edge detection (c) and (d) eyes states results fig. 5 eyes states using htc e. driver state we determine the driver state by measuring perclos. if the driver closed his eyes in at least 5 successive frames several times over a period of up to 5 seconds, 100 meriem b., belal a., abdullah salem b, etc: driver drowsiness detection system it is considered drowsy. 4 experimental results to validate our system (fig.6), we test on several drivers in the car with real driving conditions. we use an ir camera with infrared lighting system operates automatically under the conditions of reduced luminosity and night even in total darkness. fig. 6 our system installed in the car based on ir camera the results of the eye states are illustrated in table 1. table 1 results obtained from the system driver frames number false eyes sates false rate open closed d1/day 420 17 0 4% d2/day 430 15 0 3.50% d3/day 245 7 1 3.20% d1/night 200 3 1 2% d2/night 200 1 0 0.50% d3/night 200 6 3 4.50% according to the obtained results, our system can determine the eye states advances in systems science and application (2016) vol.6 no.2 101 with a high rate of correct decision. 5 conclusion and perspectives in this paper, we presented the conception and implementation of a system for detecting driver drowsiness based on vision that aims to warn the driver if he is in drowsy state. this system is able to determine the driver state under real day and night conditions using ir camera. face and eyes detection are implemented based on symmetry. hough transform for circles is used for the decision of the eyes states. the results are satisfactory with an opportunity for improvement in face detection using other techniques concerning the calculation of symmetry. moreover, we will implement our algorithm on a dsp (digital signal processor) to create an autonomous system working in real time. references [1] a. malla, p. davidson, p. bones, r. green and r. jones. (2010), “automated video-based measurement of eye closure for detecting behavioral microsleep”, in 32nd annual international conference of the ieee, buenos aires, argentina. [2] p. viola and m. jones. (2001), “rapid object detection using a boosted cascade of simple features”, in proceedings of the ieee computer society conference on computer vision and pattern recognition. [3] r. lienhart and j. maydt. (2002), “an extended set of haar-like features for rapid object detection”, in proceedings of the ieee international conference on image processing. [4] open source computer vision library: reference manual, available at: http://opencvlibrary.sourceforge.net [5] s. vitabile, a. paola and f. sorbello. (2010), “bright pupil detection in an embedded, real-time drowsiness monitoring system”, in 24th ieee international conference on advanced information networking and applications. [6] b. bhowmick and c. kumar. (2009), “detection and classification of eye state in ir camera for driver drowsiness identification”, in proceeding of the ieee international conference on signal and image processing applications. [7] n. otsu. (1979), “a threshold selection method from gray-level histograms”, ieee transactions on systems,man and cybernatics, pp.62-66. 102 meriem b., belal a., abdullah salem b, etc: driver drowsiness detection system [8] t. hong, h. qin and q. sun. (2007), “an improved real time eye state identification system in driver drowsiness detection”, in proceeding of the ieee international conference on control and automation, guangzhou, china. [9] z. tian et h. qin. (2005), “real-time driver’s eye state detection”, in proceedings of the ieee international conference on vehicular electronics and safety. [10] m. s. nixon and a. s. aguado. (2008), feature extraction and image processing(2nded.), jordan hill, oxford ox2 8dp, uk. corresponding author belal alshaqaqi can be contacted at: alshaqaqi belal@hotmail.fr microsoft word 7 s. x. mei j. l. xie--3d numerical simulation in combustion space of an oxy-fuel glass furnace.doc advances in systems science and applications (2011), vol.11, no.3-4 257-263 issn 1078-6236 international institute for general systems studies, inc 3d numerical simulation in combustion space of an oxy-fuel glass furnace s. x. mei and j. l. xie school of materials science and engineering, wuhan university of technology, hubei, wuhan 430070, china abstract in glass manufacturing, oxy-fuel combustion is a new technology. in design of oxy-fuel glass furnace, the numerical simulation and virtual reality techniques have become necessary tools. in this paper, the numerical simulation in the combustion space of a designed oxy-fuel glass furnace was carried out with the results displayed by the virtual reality technology. the gas phase was expressed with k-ε two-equation model; the combustion was described with non-premixed model; the radiation was expressed with discrete ordinates radiation model. the results of simulation agree well with the related reference, showing that the flame from each burner is individual and similar, and most fuel streams combust adequately with high temperature except that from the burner 1, which is suggested to be moved to other place. what we learned from the simulation results can give direct insight in the combustion space of the furnace, and can be used to guide the design of the oxy-fuel glass furnace. keywords glass furnace; oxy-fuel; combustion space; virtual reality; numerical simulation 1.introduction in glass manufacturing, the oxy-fuel furnaces, using pure oxygen instead of air as the primary oxidant, began to rapidly instead of the conventional air-fuel furnaces since 1990 because of many benefits, such as glass quality improvement, fuel reduction, productivity increase, emissions reduction (nox, so2, particulates), and expansion of the existing furnace [1, 2, 3]. the oxy-fuel technology is different from the air-fuel technology, to meet higher quality of the produced glass, reduce energy consumption, and prolong furnace life, it is important to obtain a rational flow field in the combustion space of the furnace. in the past，most designs of the glass furnaces were based on empirical rules or traditional methods, being difficult to obtain information such as temperatures and pressures in the combustion space of the furnace. and now with the computational fluid dynamics (cfd) modelling has been widely used to predict and estimate the flow field of the glass furnace [4, 5, 6], it is more scientific to design and optimize the oxy-fuel glass furnaces using mathematical simulation. in this paper, an oxy-fuel glass furnace was designed, and the numerical simulation in the combustion space of the furnace was carried out. by using the virtual reality technology for the simulation results, the fields of temperature, velocity and species concentration were displayed clearly. what we learned from the simulation results can give direct “insight” in the furnace, and can be used to guide the design of the oxy-fuel glass manufacturing technology. 2.geometrical model fig. 1 shows schematically the configurations of the combustion space of the glass furnace. there are seven pairs of staggered burners, and two exhaust ports which are between the first pairs of burners. fig. 2 shows the mesh. structural hexahedral grid was used in the whole combustion space with mesh refined around the burners. 258 mei：3d numerical simulation in combustion space of an oxy-fuel glass furnace fig.1. configurations of the combustion space of the glass furnace fig.2. meshes 3. mathematical model there occur fluid flow, heat transfer, and combustion phenomena inside the combustion space. to deal with turbulence, thermal radiation, and combustion, some submodels are as follows. 3.1 turbulence model in eularian system we solve the fluid phase continuity and momentum equations using the k-ε model, which is widely used in engineering [7, 8, 9]. the general form of the governing equations for the gas phase is given as follow: ( ) ,/ [ ( / )] /j j j j pv x x x s sϕ ϕ ϕρ ϕ ϕ∂ ∂ = ∂ γ ⋅ ∂ ∂ ∂ + + (1) where ρ is the fluid density, φ is the general different variable, γφ is the effective viscosity, sφ is the source term of the gas phase, sp,φ, the sourse term from the interaction with the discrete phase. 3.2 combustion model combustion is modeled by the non-premixed modeling, which involves the solution of transport equations for one or two conserved scalars (the mixture fractions f). equations for individual species are not solved. instead, species concentrations are derived from the predicted mixture fraction fields. interaction of turbulence and chemistry is accounted for with an assumed-shape probability density function (pdf). the mean (density-averaged) mixture fraction equation is: ( ) / ( ) [( / ) ]t t mf t vf f s sρ ρ μ σ ′∂ ∂ +∇⋅ = ∇ ⋅ ⋅∇ + + (2) where the source term sm is due solely to transfer of mass into the gas phase from liquid fuel droplets or reacting particles (e.g., coal), and s ′ is any other source term. 3.3 radiation model radiation heat transfer from surface to surface and radiation absorption and emission by h2o and co2 in the combustion atmosphere is modeled with the discrete ordinates (do) radiation model [10], which solves the radiative transfer equation for a finite number of discrete solid angles, each associated with a vector direction s fixed in the global cartesian system (x,y,z). the do model transforms equation (7) into a transport equation for radiation intensity in advances in systems science and applications (2011), vol.11, no.3-4 259 the spatial coordinates (x,y,z). the do model solves for as many transport equations as there are directions s . the solution method is identical to that used for the fluid flow and energy equations. ( ) ( ) ( ) ( ) ( ) 42 4 0 , / , / [ , , ] / 4s sdi r s ds a i r s an t i r s s s d π σ σ π σ π′ ′ ′+ + = ⋅ + ⋅ φ ω∫ (3) where r is the position vector, s is the direction vector, s ′ is the scattering direction vector, s is the path length, a is the absorption coefficient, n is the refractive index, σs is the scattering coefficient, σ is the stefan-boltzmann constant (5.672 * 10-8 w/m2-k4), i is the radiation intensity, which depends on position ( r ) and direction ( s ), t is the local temperature,φis the phase function, and ω′ is the solid angle. 4. boundary conditions and numerical solution method the numerical procedure was based on a well-known finite volume method. to solve the elliptic form of differential equations, appropriate boundary conditions are required. (i) the fuel is natural gas with the distribution listed in table 1, the oxidizer is pure oxy. at the inlet, all velocities and temperature were specified. (ii) at the outlet, the pressure was at an ambient atmosphere. (iii) at an impenetrable wall of the furnace, the wall temperatures on different region were specified, and the usual non-slip conditions were applied. to account for the wall effect in the nearby regions, the wall-function was introduced to link velocities in the near-wall region. the flow field equations with boundary conditions were solved numerically. the process was repeated until convergence was achieved table 1 fuel distribution ratio (%) no. 1# 2# 3# 4# 5# 6# 7# distribution ratio 6 13 9.5 7.5 6 4.5 3.5 no. 1’# 2’# 3’# 4’# 5’# 6’# 7’# distribution ratio 6 13 9.5 7.5 6 4.5 3.5 5. results and discussions to visualize the flame envelope, concentration of co (carbon monoxide) can be used as an indicator since the flame is rich in soot, which is formed in fuel-rich regions. fig. 3 displays the iso-surface of co concentration of 0.04. we can see the “flame” shape from each burner. fig. 4 shows the temperature contours through the burner plane, and fig. 5 shows the temperature contours at the vertical middle slice of burner 2. fig.3. iso-surface of co concentration of 0.04 figs. 3-5 show that not only the “flame” shape, but also the temperature field agree well with the related reference [11], indicating the reliability of the simulation results. from figs. 3 and 4 we can see that the length and width of each “flame” from burners 2-7 correspond with the fuel 260 mei：3d numerical simulation in combustion space of an oxy-fuel glass furnace distribution ratio listed in table 1, while the length of the “flame” from burner 1 is the shortest, indicating the incomplete combustion of the fuel. fig.4. temperature contours through the burner plane fig.5. temperature contour at the vertical middle slice of burner 2 fig. 6 shows the velocity vector on the vertical middle slice of burner 2, fig.7 displays the streamlines from burners 1-7. from fig. 6 we can see that the gas stream runs straightly from the inlet to the opposed wall, and then rebounds when meets the wall, resulting in a wider sized reversed flows under the crown. from fig.7 it can be observed that there is large backflow under the crown in the whole space, being beneficial for decreasing the crown heat duty. fig.6. velocity vector at the vertical middle slice of burner 2 fig.7. streamlines from burners 1-7 fig.8. temperature contours at the bottom surface figs. 8-10 show the temperature iso-lines at the bottom surface, the cross middle slice of burners and the crown, respectively. from fig 9 it can be observed that the crest “flame” advances in systems science and applications (2011), vol.11, no.3-4 261 temperature is around the burner 2, resulting in the corresponding local high temperature region not only on the bottom (see fig. 8) but also on the crown (see fig. 10). except the high temperature region, in other regions the temperature on the bottom and on the crown is uniform. fig.9. temperature iso-lines at the cross middle slices of burners fig.10. temperature iso-lines at the crown to know the temperature field along the length direction (x direction) in the whole combustion space, we created a series of vertical slices, of which the average temperature values are shown in the curve of fig. 11. it can be observed that there are 13 temperature crests corresponding with 13 burners, except the burner 1, indicating the incomplete combustion process there, which should be improved. the maximum of the temperature crest is near the burner 2, around which the expected hot-spot located, indicating the rationality of the results. fig.11. average temperatures at the vertical slices along x direction the above analyses results show that the combustion condition from the burner 1 is quite bad. it is necessary to find the reason. fig. 12 displays the velocity vector through the burner plane, fig. 13 shows the velocity vector at the vertical middle slice of exhaust port, and fig. 14 displays the co2 concentration through the burner plane. from figs. 12-14 it can be known that since the burner 1 is near the outlet 1, and the fuel distribution ratio of burner 1 is much less than that of the neighbor burners (2# and 2’# ), when large flue gas pass by the burner 1 (see fig. 14), and run to the outlets finally, the oxy is insufficient around the burner 1, resulting in the incomplete combustion. to improve this condition, it is suggested to move the burner 1 to other place. 262 mei：3d numerical simulation in combustion space of an oxy-fuel glass furnace fig.12. velocity vector through the burner plane fig.13. velocity vector at the vertical middle slice of exhaust ports fig.14. co2 concentration through the burner plane 6. conclusion in this paper, the numerical simulation in the combustion space of a designed oxy-fuel glass furnace was carried out. the virtual reality techniques was used to display the simulation results which are agree well with the related reference. the results show that the “flame” from each burner is individual and similar on the whole, and most fuel streams combust adequately with high temperature except that from the burner 1. the maximum flame temperature in the oxy-flames is higher than that in the air-fired furnace, and the maximum average temperature along the length direction is near the second pairs of burners, corresponding to the predicted region of hot-spot. there are back flows occurring near the crown, being beneficial for decreasing the crown heat duty. to obtain more rational flow field the placement of burner 1 will be discussed in future works. acknowledgements the authors are grateful for the supports provided by the national key technology r&d program (2006baf02a26). references [1] simpson, neil g., wilcox richard, et al., oxy-fuel technologies for boosting and 100% conversions of cross fired furnaces. glass technology: european journal of glass science and technology part a. 48(4) (2007) 168-175. [2] habel michael, lievre kevin, inskip julian, et al., advanced cleanfire® hri oxy-fuel boosting application lowers emissions and reduces fuel consumption. ceramic engineering and science proceedings: 68th conference on glass problems a collection of papers presented at the 68th conference on glass problems. 29(1) (2008) 203-211. advances in systems science and applications (2011), vol.11, no.3-4 263 [3] kobayashi h, evenson e., and xue y., development of an advanced batch/cullet preheater for oxy-fuel fired glass furnaces. ceramic engineering and science proceedings: 68th conference on glass problems a collection of papers presented at the 68th conference on glass problems. 29(1) (2008) 137-148. [4] s.l. chang, c.q. zhou, and b. golchert, eulerian approach for multiphase flow simulation in a glass melter. applied thermal engineering. 25 (2005) 3083–3103. [5] vishal sardeshpande, u.n. gaitonde, and rangan banerjee, energy conversion and management, 48(10) (2007) 2718-2738. [6] a. abbassi, and kh. khoshmanesh, numerical simulation and experimental analysis of an industrial glass melting furnace. applied thermal engineering. 28(5-6) (2008) 450-459. [7] u. shah, c. zhang, j. zhu, et al., validation of a numerical model for the simulation of an electrostatic powder coating process. international journal of multiphase flow. 33(5) (2007) 557-573. [8] d.m. hargreaves, and n.g. wright, on the use of the k–ε model in commercial cfd software to model the neutral atmospheric boundary layer. journal of wind engineering and industrial aerodynamics. 95(5) (2007) 355-369. [9] fatih üneş, investigation of density flow in dam reservoirs using a three-dimensional mathematical model including coriolis effect. computers & fluids. 37(9) (2008) 1170-1192. [10] jian c q, dutta a, mukhopadhyay a, et al., explicit coulpling between combustion space and glass tank simulation for complete furnace analysis. presented at the 1st balkan conference on glass science and technology , volos , greece ,2000. [11] lankhorst a. m., bauer r. a., coupled combustion modeling and glass tank modeling in oxy and air-fired glass melting furnaces. the 4th int sem on mathematical simulation in glass melting ,1997. advances in systems science and applications (2013) vol.13 no.3 227-232 thoughts on the general systems theory michael v. tokarev corporation axis ltd. miami, fl, s.-petersburg, russia abstract the most common definition of systems is formulated as “a combination of elements”, or even as “any object is a system”. many authors have presented the formal definitions of an aggregation of two multitudes: elements and relationships. but a single, universally accepted systems definition still does not exist. in this article, why this happens, what matters in definitions do not have the answers yet, how “general systems theory” metascience and “systems engineering” particularistic science relate, the paradoxes of general systems theory, and much more will be discussed. keywords systems’ definition, general systems theory 1 introduction known classic of general systems theory v.n. sadowski brings dozens of existing systems’ definitions [1]. other authors add new definitions, but a single, universally accepted systems definition still does not exist. why this happens, what matters in definitions do not have the answers yet, how “general systems theory” metascience and “systems engineering” particularistic science relate, the paradoxes of general systems theory [1], and much more will be discussed in this article. these issues arise primarily in the course of the analysis of existing definitions and their research by v.n. sadowski in the book “foundations of general systems theory”. from our point of view as long as it is the best and most comprehensive study to date of the existing state of things in terms of the general system theory (gst). the purpose of this paper is to formulate the most pressing issues in the context of system definitions and attempt to identify ways to address them. 2 existing definitions and issues arising from them. entropy the most common definition of systems is formulated as “a combination of elements”, or even as “any object is a system”. many authors have presented the formal definitions of an aggregation of two multitudes: elements and relationships [2]. our task is not to refute or criticize existing definitions. we’re just going to try to formulate questions to the definitions and try to find answers to those for which it is possible within this article. in the analysis of any definition there is the first important question, the answer to which is connected to all systems definitions and classifications. this michael v. tokarev: thoughts on the general systems theory 228 question will be worded as follows: “does a system really exist?” not many researchers ask this question, but those who are wondering, as a rule, respond to it positively. it seems to us that the question can’t be answered at all. why? let’s think constructively. were there systems before the mankind? if we answer in the affirmative (systems have always existed), what is the point in trying to determine them? they already exist, someone created them, and therefore defined? and what about those systems that the man himself creates? and what about those systems, which include both elements: those existed prior humans and artificial elements? this question is based on the idea that if we say that the system is there, so we know what we say, and this system has already been described by someone and you can explore it as a system. obviously, prior to mankind, to be exact, even to the end of xix century, no one knew what the system in the modern sense were, did not describe them. so, we can only speak of the objects or entities existence, but not the system. if the answer to the existence of systems prior humans is negative ( system concept was invented by a man, and prior him the concept did not exist), then we are just coming to terms of kant transcendental idealism, according to which the person is not doing anything, but only “reflects”. it seems to us that the answer to this fundamental question must be sought in the following. a man created the concept of a system. he “adjusts” the existing reality to his concept. in fact we find the objects, try to assign them the system properties (which we have invented ourselves) and state that systems exist. it’s fine if scientists understand that examine just a model, not the system. the researchers come to paradoxical conclusions: for example, the type (genus, family) of animals-is also a system. obviously, from the point of view of nature of it is not so, because the concept of genus, species, family also came up with people. certainly, nature was not busy with splitting all living beings into the genera and species. in fact, these systems are completely virtual. otherwise we’ll have to admit a creator, who invented not only flora and fauna, but also classified them. thus, if we wave away the non-scientific idea of a universal creator, we are forced to admit that the system did not exist before humans and does not exist now. here we see the fundamental paradox of the system science: everybody talks about a non-existing, and not only talks, but also uses in their practice and research. of course, aside from all of this there are artificial systems that people originally created as such. but in this case, and the conceptual apparatus is immediately objective. in particular, if we build a car as a system of interacting elements that has to deliver goods from point a to point b, we define the system on the basis of interacting elements, goals and functions. 229 advances in systems science and applications (2013) vol.13 no.3 thus, solving this major issue, we split at least (and fundamentally) natural and artificial systems, knowing that the latter are systems a priori. now we continue the argument about natural systems (which exist only in our minds?). usually, natural systems are classified as animate (living) and inanimate (nonliving) ones. if every living individual can be somehow dragged into the system concept, the inanimate objects can be appealed to only in the form of models, as we do not know enough about them, about the hierarchy of these systems, and possibly even the dimension in which they exist. for example, a stone lying in the mountains. is it a system? from the point of view of the most general definition (“any object is a system”) a stone is undoubtedly the system. from the point of view of the extension of the concept (for example, an observer or function), the stone can’t be considered a system, until we started to study it and / or until it is comes into operation. of course, we can think of the functions of internal heat, molecules and atoms inside it, the crystal structure, etc. but is it a system while it is lying, and it is not being studied? in that case, why bother to talk about such an object as a system? only because of a common definition of the system itself? no less obvious the application of the notion “system” to objects of fauna. is a particular bird a system? from a biological point of view (only because it is more convenient to be studied by a man)-of course. and from the point of view of other fauna? possible-it is also a system. a bird feeds, feed others, in addition it gets a lot more besides input actions. we may say that here the interpretations of definitions of the hierarchy, morphology and interaction of elements, the micro and macro levels are performed. one point confuses us: the bird itself does not know it and, moreover, does not take any concerted action to be in the system. just feeds, breeds and dies. from this point of view, it is like the inanimate nature, stone, which also does not “know” that it is the system. there is only one answer to all these questions: a human called it a system. he himself invented system. and each person (the researcher), has its own system. in fact, this system is some kind of a model for our understanding of a particular entity. the model itself, in turn, is also a system. and when the researchers begins to study the model, he/she has to disengage himself/herself from the “system” (more precisely, from the subject). no researcher argues with this. do we do with an object, which exists, not realizing that it a system? but the most paradoxical is that the person giving the definition of the system, tries to simplify the study of all the systems to unify their properties, suggesting that in the future it may be possible by studying one (the one that he understands) system to project (extrapolate) the system properties on the other system he understands less. if this does not work directly, it makes changes to the definimichael v. tokarev: thoughts on the general systems theory 230 tions, complicating them by adding new elements and features. such “fitting” of definitions for a research specific needs leads to the greater separation of new definitions from classical ones, and, as a consequence, to greater entropy in the system definitions themselves. discuss the entropy and the second law of thermodynamics, which, from our point of view, is applied indiscriminately to almost all known systems. interpretation of second law of thermodynamics is unambiguous: “entropy of an isolated system may increase or remain unchanged. reduction of entropy in an isolated system is impossible” [3]. what do many systems researchers do? they apply the second law of thermodynamics and the concept of entropy itself to any of the studied systems. in this case, few people pay attention to the paradox that, for example, self-organizing system does not increase the entropy inside themselves, but reduce, not desintegrate, but are being created, and organize themselves. this error has been long known to physicists and many other scientists. the concept of an isolated system in the definition of thermodynamics means no system interchange with the environment, not only by means of substance (closed system), but also by energy. the gross error is also the application of the concept of systems entropy to the systems which are not in equilibrium (initially entropy is a measure of the thermodynamic system in equilibrium). another mistake made by researchers who follow the fashion-application of the laws of thermodynamics to any system, including ones of not physical, and certainly not of a thermodynamic nature. for example, how does a perfect system of geometric axioms relates to thermodynamics and how we can apply the principles of entropy change to it? however, despite these considerations, all systems eventually die (entropy increases). why? is there a common cause of death of all the systems? and may the root cause of this common cause be the determination, which is man-made? meaning that the man gives the definition of the system in which it ends its existence (and this definition may include this course latently, possible only in the context of the investigator). most researchers believe that the system is, by definition, hierarchical, that the system itself may enter into other systems as an element (subsystem), and each of the elements of the system is the system on its own level of consideration. however, if at the intuitive level, this can be imagined, at the level of universal definition systems this property may seem controversial. why? if we assume that the system as a term was invented by people (not nature and not the creator), then considering a particular system in a large number of cases, the researcher does not appeal to the macro or micro systems, even not assuming that there is a hierarchy. as an example, study the alphabet. with this study, it is possible to ignore the fact that a particular alphabet is a subset of 231 advances in systems science and applications (2013) vol.13 no.3 all human alphabets. but at the micro level, in some context, we are absolutely not interested in each letter as a system (for example, lettering or placing them side by side in constructing words, etc.). one can argue that in this case a systematic approach does not apply. but in course of the alphabet model study we can look at it from the point of view of other system properties, in particular the main-integrity (agregation of letters has properties not possessed by each letter separately). perhaps we have no questions only to this particular property system-a property of integrity (the interaction of elements of the system leads to system properties / functions that each element separately unable to perform or “the object properties can’t be reduced to the sum of the properties of its constituent elements and non-deducibility of the last properties of the whole”. 3 system paradoxes the urgency of our questions is confirmed by systemic paradoxes in v.n. sadowski works [1]. and in particular, the first paradox of hierarchy, which is the following: for a complete description of the element as a “system element” there must be a full description of the system that can’t be described fully until each element is described. therefore, the question of the legality of a systematic approach analysis of micro and macro-level systems remains a question, the answer to which is possible in our view only in specific applications, with significant restrictions in the context, or the use of artificial systems with known properties, as-built for specific purposes. let’s consider a simple example. a man, as a biological system, consists of elements (subsystems): circulatory, digestive system, musculoskeletal, respiratory, etc. if we try to describe the circulatory system, as a separate, outside of the body, we can miss important features of nutrients carried by the blood, oxygen, and various chemical elements necessary for metabolism, etc. for a complete description of the circulatory system, as part of the body, we need to have a complete description of the body (including the blood system). thus, for the study of any element of a complex system, we have to build a model that is different from the object due to substantial simplification. considering the provision of oxygen, we can abstract away from the digestive system, which significantly affects feedback and the circulatory system, which in some cases may be affected so much that the work of the respiratory system will be blocked. even more interesting example is when a person studies a community of people, to which he /she belongs, and his/her decisions can affect the outcome of the michael v. tokarev: thoughts on the general systems theory 232 system. in this case, self-knowledge is not just difficult, it is impossible. on the other hand, if we consider a vehicle as the system, studying any of its units (subsystem) is simplified by the fact that creating a car we base its structure on system properties. the rest of the paradoxes is formulated similarly based on the fact that it is impossible to explore a part, not knowing to the whole and vice versa. here we do not just agree with v.n. sadowski, but get confirmation of these issues. namely: “the attempt to interpret paradoxes considered static, applied to the system knowledge, taken out of its development, inevitably leads to the conclusion that the system thinking is impossible.” 4 how to answer these questions in what direction should one look for the answers to these questions? the first way is, of course, a sharp decline in the number of systems, consideration of only artificial systems, which were originally created as a system, not disseminating research results to other systems. the second method, proposed by v.n. sadowski, is to explore the hierarchical systems by fixing some elements/connections/properties. as is done in the systems of equations, where the number of unknowns by more than 1 greater than the number of equations. solving such systems of equations, the researcher captures part of the unknown variables and then solves the solvable system of equations. unfortunately, this method, as well as the method of successive approximations, proposed v.n. sadovski, does not allow to solve systemic paradoxes efficiently, in real time, with the right qualities in the study of systems. references [1] sadowski v.n. (1974), foundations of general systems theory, moscow, nauka. [2] mesarovic m.d. and yasuhiko takahara. (1975), general system theory: mathematical foundations, system research center, cleveland, ohio. [3] osipov a.i. and uvarov a.v. (2004), “entropy and its role in science”, journal.issep.rssi.ru, vol.8, no.1. corresponding author author can be contacted at: mtokarev@axisconsulting.ru advances in systems science and applications (2012), vol.12, no.2 133-140 research on the 3-dimensional grab design system based on case-based reasoning yuantao sun1 and ran li2 1college of mechanical engineering, tongji university, shanghai, 200439,china 2changjiang three gorges navigation administratioiyichang,443133, china abstract with the increasing intense competition of mechanical manufacturing market, companies must shorten products design cycles, improve products’ quality, and reduce the costs of products. this paper takes into account the characteristics of grab product design and put forward the method the integrate three-dimensional design and case-based reasoning technology for the grab design, and propose the procedure of grab bucket design based on case-based reasoning and apply the method to grab bucket design field,. in the paper, the 3-d grab case library and explore intelligent reasoning module are established. and case-modified technology through solidworks and its secondary development are achieved. at last, a set of three-dimensional grab design system is developed, which can realize grab virtual assembly, kinematics analysis, interference check, finite element analysis and creating engineering drawings so that it makes the applications of intelligence design possibility. keywords digital manufacturing, case-based reasoning, virtual design, intelligent design, grab 1 introduction the grab is a main handling device which is used for loading and unloading staple scattered material. but all countries have not a uniform design rules for grab bucket because of the diversity of the material shape, the type of material or load capacity and working environment. for example in germany, various manufacturers have their own reference data and design rules. and in china, there is no similar national standard. the study on the grab design also is less[1]. in the actual design process, the experience occupies a large proportion. in general, designers use the method such as a simple analogy, size, and zoom in or out according to the reference drawings to revise and design new grab products based on the similar types of grab products drawings. but the method is very difficult to avoid interference between the parts of grab and guarantee the products have reasonable dredging rate and so on. with the increasing intense competition of mechanical manufacturing market, companies must shorten products design cycles, improve products’ quality, and reduce the costs of products. to improve the efficiency and quality of grab design, 134 yuantao sun:research on the 3-dimensional grab design system based on case-based· · · the paper focus on the method how to integrate both of intelligent design and three-dimensional grab modeling . in common, the intelligence design method is the process that choosing the design satisfied solution through the rbr (rule-base reasoning) way base on record the product design knowledge and discipline in rule form, through the rbr (rule-base reasoning) way to choose the design satisfied solution. intelligent design is method which record product design knowledge and law based on the usual rule form, and make the design plan which meet the requirements through the rule-based reasoning (rule-based reasoning, rbr). however in the design field, a wealth of experience and fragmented knowledge are difficult to be summarized in the form of rules. those problems limit such system application for the mechanical design field in a narrow range. these problems restrict the application of this kinds system in mechanical design field. recent years, with the intelligence technology studying, case-based design (case-based design, cbd) method has shown haven a good ability to resolve these issues. along with the studying of the intelligence technology, cbd has show its ability on these problem solving. case-based reasoning (case-base reasoning, cbr) is a form of artificial intelligence from machine imitation to machine thinking[2-3]. the characteristics of grab product design are took account in the paper. three-dimensional design and case-based reasoning technology are integrated in the grab bucket design field. the procedure of grab bucket design based on case-based reasoning is put forward in the paper. the design system establishes the 3-d grab case library and explore intelligent reasoning module. and achieve case-modified technology through solidworks and its secondary development. at last, a set of three-dimensional grab buckets design system is developed, that it makes grab intelligence design come true. 2 the principal of grab design based on case reasoning because the foundation of cbr is that similar problems have similar solutions, the case-based design method is very close to the actual design process[4]. faced to new design requirements, the system first select the case which is most close to design requirements from the past design cases library ,and then simulate to get design program which meet current requirements. the results case also can be modified again as a reference case that means the system has a self-learning ability. the advantages are: (1) no need to build the rule or model, but only collect the past cases to establish the cbr system; (2) only need determine the related case characters. cbr system’s case can be improved and enlarged during the using, as long as some examples are added in the system; (3) no need to reasoning from the beginning, only through a completed program which can generate the solution quick; (4) easy to maintain. the increasing new cases advances in systems science and applications (2012), vol.12, no.2 135 not only achieve the purpose of the study, but also reflect the customer demand character. 2.1 design workflow of grab design based on cbr design workflow is shown as fig.1. designer input the parameter of grab first. then the system search the cases library through the key words and find out a number of similar cases and identify and select the most similar one as design reference .the user can get the guidance information to modify the case and establish the three-dimensional geometric model for the product through human-computer interface after a series of steps such as virtual assembly model, kinematic analysis, interference checking, structural analysis, mechanical steps finally, to generate two-dimensional drawings, at the same time, new product design is completed as a new three-dimensional cases which is stored in the cases library. fig.1 design flow of grab based on cbr so that the grab intelligence design depend on the following key technologies: case expression retrieval cases modify case modification. 136 yuantao sun:research on the 3-dimensional grab design system based on case-based· · · 2.2 case expression because the number of cases and the degree of perfection of case base is the cornerstone of using cbr technology for compute-aided design .the establishment of case library should take the content of the case, the expression of the case and the organization of the case base etc. into consideration. it is considered that the software solidworks has the functions of three-dimensional solid modeling, so that the software and its secondary development are used to the established geometric model can display three-dimensional appearance of entities in actual time[5]. through the operation of the model the entitle shapes modification of the model can be concisely completed and the same times. the physical changes slice product model, interference detects, motion simulate, finite elements analyze and so on can be concisely completed. in view of the characteristics of grab, in order to facilitate search and matching cases, the paper put forward the method to define cases that the cases =cases three-dimensional graphic + case feature the data. feature data refers to the factors that has a decisive influence on the product design, and can be extracted from the cases. the following is workflow. at first, the three-dimensional graphic of the cases are set up in solidworks, then feature data of the cases are input into the cases library. in practical applications, regarding to different storage requirements of case’s three-dimensional graphic library and feature data, different storage management functions are established in system respectively so that they can be integrate into the same interface to be operated. so that a cases consist of a set of three-dimensional graphics include the related sub-components hierarchy and a number of feature date of the case. the method makes the case query and retrieval more easier. fig.2 shows the three-dimensional graphics, the structure and the expression of feature data in the design system respectively. fig.2 case expression of grab in order to make the model more practical, both hands of modeling methods and modeling sequence should be take into; on the one hand, it should be easily modified ,that means the main size of upon -bearing beam , under-bearing beam, advances in systems science and applications (2012), vol.12, no.2 137 stay bar, bucket body can be stretched and reduced in the corresponding parts through changing the numerical size, and they are easily be matched each other also ); on the other hand, it is easy to establish model, and the model’s level and structure coincide with the actual as far as possible. the system use the software solidworks, through top-down design methods to establish the model of grab’s every part. case features are summarized from the case. the description of the case feature is the core of case representation. in the system, the case features are classified three types to be expressed according to different case objects. 1. every complete set of grab product is a case object. the type of grab as one of case feature will be placed as the first important. in general, there are several types of grab: double-cables two-jaw grab, four-cable double-jaw grab, four-cable grab and four cables multiple jaws and so on. there are also several special type grab such as hydraulic grab .and then features such as lifting capacity, material characteristics and work occasions will be took into account as the design requirements. 2 .the main part of grab such as upon -bearing beam, under-bearing beam, stay bar, bucket body also are deal as cases .the feature are expressed according to the range of lifting capacity, shaping feature, mating dimension. 3. the case feature of grab accessory such as pin is geometrical dimensions and processing technology. these features are input into product lib as database. then the features will be modified by the product data management module and matched with the corresponding three-dimensional drawings .so that the corresponding three-dimensional drawings will be easily got through the feature database retrieval. 2.3 case retrieval the key to case-based reasoning achieved successfully is the case retrieval and matching .the function of the case retrieval and matching is that the most similar cases will be retrieved and used as the template of new design program. to cases which be retrieved, there are two requirements: 1) the cases should be less as possible. 2) the cases are most similar with new product. so that the retrieval algorithm is the core of cases retrieval. at present there are three kind of retrieval algorithm: nearest neighbor method, induction indexing method, knowledge inducting method. because the feature of grab can be classed according to grab parameter such as dimension, type of grab. the system uses a knowledge-guided strategy and the recent adjacent strategy. the nearest neighbor method and the knowledge inducting method are used in the system .those grab parameter are define as feature key word and are given weight value .after calculating case similarity according to the weight, the most similarity case is selected and modified. 138 yuantao sun:research on the 3-dimensional grab design system based on case-based· · · the following equation is similarity formula sim (l, k) = 1− √√√√ n∑ i=1 wi × ( fki − fli fli )2 where sim means the similarity between objective design and the case, the more value of sim is large ,the more similarity is close. fli means design requirement feature item l includes attribute variable i. fli means the case feature item k includes attribute variable i . if fki and fli is same, then fki − fli value is 0, otherwise is 1).wi is weight of the attribute variable i. for example, when the case is searched , considering the design requirement and the using condition for the grab ,its feature array is (the grab type, lifting capacity ,material ,bucket cubage),the weight array is (0.4,0.2,0.3,0.1). 2.4 the case modification base the knowledge it is a key problem to propose the advice about the modification through case reasoning. after the system concluded the experts’ design experience on grab, it establishes knowledge base. the knowledge base has concluded the main parts’ design experience and data on different types of grabs, modify suggest on overall geometric parameters and the related reasoning calculate formula. if there is a difference between the case base is the grab capacity when a new grab is designed., you can propose the suggestion that modify the grab’s width in a certain range firstly according to the ratio between the two grab capacity in the case base and then the modified modules are called to the grab three-dimensional model can be modified. because the application of the interface with solidworks and the top down design theory, the function can be applied that the grab three-dimensional model could automatically update as the parameter changes. topdown design means the ability to design the relevant sub-components in the assembly environment, not only the relevance between the size parameters, but also realize the automatic and total relevance between the geometry appearances and parts. the user can design some other parts on the case that the assembly layout diagram is already, and make sure that the assembly layout diagram and the parts’ size are totally automatic relevance, so to make the system’s modified modules in three-dimensional model more convenient. as long as the key sizes are modified on the matching position, the relevant parts can be automatically modified and updated. 3 application to design a double -cable double-jaw grab, whose load capacity is 12t, grab material is sand, bucket capacity is 1.6t/ m3, after retrieval the similar grab advances in systems science and applications (2012), vol.12, no.2 139 bucket case from the three dimensional case library. in the case library, one which loads capacity is 10t, grab material is sand, bucket capacity is 6m3 can be got. considering the difference with the objective product, keeping the same bucket area, the bucket wide is modified from 2.05m to 2.46m, and then it can satisfy the capacity requirement. the new capacity is 7.2m3. after the accordingly adjustment for other components, the 12t three dimensional model and drawing can be got shown as fig.3 and fig.4 respectively. fig.3 grab model fig.4 grab part drawing 4 conclusion the intelligent t design process of the grab based on the cbr and three-dimensional entity’s model technology is introduced in this paper. the basic structure of implementation, the expression s of cases and their retrieval method are also described. an intelligent t design system is developed using solidworks and its secondary development. the application of the system has shortened the design cycle and improved the design efficiency. at present improving the knowledge 140 yuantao sun:research on the 3-dimensional grab design system based on case-based· · · library is still in progress. acknowledgements the paper is sponsored by “the open research projects supported by the project fund of the hubei province key laboratory of mechanical transmission and manufacturing engineering wuhan university of science and technology”. the sponsor project series number is 2007a22. references [1] li ran. (2005), research and development of 3-dimensional grab bucket design system based on cbr, degree description of wuhan university of technology, pp.2-5. [2] dubois.d, hullermeier, e and prade.h. (2006), “fuzzy methods for casebased recommendation and decision support”, journal of intelligent information systems, vol.27, no.2, pp.95-115. [3] hamza.h, belaid.yand, belaid.a. (2007), “case-based reasoning research and development”, proceedings 7th international conference on case-based reasoning, pp.404-418. [4] lee sangjae, kim kyoung-jae. (2009), “using case-based reasoning for the design of controls for internet-based information systems”, expert systems with applications, vol.36, no.3, pp.5582-5591. [5] ding yufeng, wei zhongling. (2006), “research on parametric process planning technology based on three-dimensional part model”, journal of wuhan university of technology, vol.28, no.s1, pp.502-506. microsoft word 4-李国东.doc advances in systems science and applications (2010), vol.10, no.1 19-24 issn 1078-6236 international institute for general systems studies, inc. analysis b-scan image by cnn and polyfit* guodong li1,2 and wenxia xu3,4 1school of mathematics and physics, north china electric power university, beijing 102206, china 2industrial systems engineering, university of regina, wascana parkway, regina, sask. s4s 0a2, canada 3dep of electric engineering, chengdu university of information technology, chengdu 610225, china 4xinjiang weather modification office, urumqi 830002, china email: lgdzhy@ncepu.edu.cn, xwxqiuye@live.cn abstract in this paper, we have be processed ultrasound with cell neural network(cnn). first we detected the edge from b-scan images, and then we analysis the data that from the edge of image. some interesting result has been finding. there are some regular between the number and the patient’s b-scan image. keywords b-scan image edge detection liver damages 10-degree polynomials fitting 1. introduction in the last two decades, medical image processing technology has been developed rapidly. it can help doctor to diagnose patients' disease more accurately. medical image processing with computer has attracted much attention recently [1]. a lot of methods for dealing with medical images have appeared [2],[3]. in chua's articles (see [4] or [5]), many important and interesting cnns are described. one of them is the edge detection cnn, which can detect the edge in gray-scale images. in a recent conference paper, a robustness theorem for designing edge detection cnns is set up [6][7][8]. detecting edges of gray-scale images may be required for a variety of purposes, such as machine vision, image analysis, and image pick-up and so on. edge detection is one of the most important steps for image recognition since there is a direct relationship between edge and object recognition. a lot of scene information can be interpreted from the edges. in natural images, most edges are associated with abrupt changes in intensity distribution and can be approximately modelled as step edges. pathological changes of chronic hepatitis patient include liver fibrosis, heterotrichosis and liver cancer, etc; however, the three states of liver disease are difficult to distinguish from b-scan image by naked eyes. digital analysis of the b-scan images will help doctors to diagnose the stage of the damages of patients' livers. 2. edge detection cnn in a chua’s exposition [1], many important and interesting cnns are described. one of them is edgegray cnn, which can detect the edges in gray-scale images. the template of the standard edge gray cnn has the form。 the standard m×n cnn architecture is composed of cells jic , . the dynamics of each cell is given via the following equation [1]: jiljki r rk r rk lkljki r rk r rk lkjiji zubyaxx ,,,,,,, +++−= ++−= −=++−= −= ∑ ∑∑ ∑ (1) * this work is supported by meteorology bureau of xinjiang uighur autonomy, science and technical item (project no. 201012). li: analysis b-scan image by cnn and polyfit 20 ( )11 2 1 ,,, −−+= ++++++ ljkiljkiljki xxy njmi nmljki ,2,1;,2,1 ],1[],1[),( == ×∈++ (2) where jijijiji zuyx ,,,, ,,, represent state, output, input, and threshold respectively; lklk ba ., , are the elements of the a-template and the b-template respectively. figure 1 the dynamic routes of the cnn the standard edge cnn's local rules and template are listed as follows: local ruler jiu , → )(, ∞jiy (1) white → white, independent of neighbours. (2) black → white, if all nearest neighbours are black. (3) black →black, if at least one nearest neighbours is white. (4) grey →black, if the laplacian operator zu ji >∇ , 2 . (5) grey →white, if the laplacian operator zu ji <∇ , 2 . (6) grey →0, if the laplacian operator 0, 2 =∇ jiu . the standard cnn template has been generalized by the following theorem. theorem 1[6]: let the positions of cnn template parameters be described by (3).then the cnn can perform the local ruler of the detection of edges in gray-scale images, if the following parameter inequalities hold: cbzzcz 6 .2 8 .1 −<<− 000 00 000 1,1 zz ccc cbc ccc baa −= ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ −−− −− −−− = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = (3) where a > 1, 1,1 >> cb ∑ ∑ ≠ −∈ ++−δ∇ )0,0(),( }1,0,1{),( ,,, 2 lk lk ljkijiji ucbuu where： advances in systems science and applications (2010), vol.10, no.1 21 ⎪⎩ ⎪ ⎨ ⎧ − >δ =δ otherwise u if c ub ji jilk 1 g )( , ,, (4) 3. segmentation of the b scan image figures 3 show the ultrasound b-scan images (ubsis) of six people's livers. each original ubsi is an rgb image with 720×576 pixels. we chose the same eath part in the ultrasound b-scan images as that shown in fig.2, which is on the left of the cholecyst and up of the portal vein. expert doctors analyze this region when they diagnose sufferer according to b-scan image. so we cut this region and then processed it with edge gray detected cnn. clinical diagnoses for the six ubsis are listed as follows. figure 3(1) is obtained from a healthy hepatitis b virus (hbv) carrier. figure 3(2) is the ubsi of a normal liver without hbv infection. the ubsi of a young (under 20 years old) chronic hepatitis b patient without liver cirrhosis is shown in figure 3 (3) and that of an old one (over 50 years old) is given in figure 3 (4). figure 3 (5) and figure 3 (6) are the ubsis of a patient with ascites and a patient with marked liver cirrhosis, respectively. 4. application of the edge detection cnn in this section, we shall use edge detection cnn to process the part of b-scan image. the template parameters of the cnns are given in table 1. table 1 template parameters of the cd cnn a b c z g 4 16 2 0.4 0.4 the edge detection cnn can be used to process rgb images. an rgb image is usually represented by an m×n×3 (for the above ubsis, m = 576, n = 720) data matrix p where m, n stand for the rows and the columns of the pixels in the image, and 3 represents 3 color planes ---red, green and blue, each color plane with 256 levels of intensity denoted as (r, g, b) = ( ) ( ) ( )( ),3:,:,~,2:,:,~,1:,:,~ ppp . such an image is called an rgb 24-bit map. in order to use cnn to process rgb images, we use a transform *pp → (5) to change the 256 levels of intensity of each color plane into the levels of intensity in [-1, 1]. consequently, the lighten pixels in the original rgb image p correspond to smaller values in the transformed image *p and vice versa. in the following discussions, we always assume that the levels of intensity of input rgb images 1p and 2p have been transformed via formula (5); the color planes of processed images have been changed into the rgb forms when the processed images are shown as color pictures. firstly the edge detection cnn given in table 1 is used to process the part of the ubsis shown in figure 3. the processing results are demonstrated in figure 4. it is difficult to give a correct judgement for non medical researchers. however, we arrange the pixels of the three color planes p(:,:,1), p(:,:,2), p(:,:,3), of each processed image (the output ( )"" , ∞jix , not the output ( )"" , ∞jiy ) in row-wise packing scheme [5]. then wavelet transform is applied to process the images in figure 4. li: analysis b-scan image by cnn and polyfit 22 figure 2 the selected part of the b scan image the part in the white rectangle is chosen for edge cnn analysis. patient liver no1(a) patient liver no2(a) patient liver no3(a) patient liver no4(a) patient liver no5(a) patient liver no6(a) figure 3 segmentation of patient liver patient liver no1(b) patient liver no2(b) patient liver no3(b) patient liver no4(b) patient liver no5(b) patient liver no6(b) figure 4 the result of the b scan image processed by edge detect cnn. 5. introduction of polynomial fitting given data ( )ii yx , (i=0,1,…，m)，we shall find a polynomial p(x) such that the error’s square sum is the least, i.e. ( )[ ] min 0 2 =−∑ = m i ii yxp suppose that φ is the set of polynomials of degree ( )mnn ≤ . let ( ) ∑ = φ∈= n k k kn xaxp 0 and advances in systems science and applications (2010), vol.10, no.1 23 ( )[ ] min 0 2 00 2 =⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −=−= ∑ ∑∑ = == m i n k i k ik m i ii yxayxpi (6) it is obvious that ∑ ∑ = = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −= m i n k i k ik yxai 0 2 0 is a function of ,,,, 10 naaa . so the problem is to find the extreme value of ( )naaaii ,,, 10= . by the necessary condition of extreme points, we have: 02 0 0 =⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −= ∂ ∂ ∑ ∑ = = m i j i n k i k ik j xyxa a i nj ,1,0= (7) i.e. njyxax m i i j i n k k m i kj i ,1,0 00 0 ==⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∑∑ ∑ == = + (8) in matrix form, (8) becomes ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ + ∑ ∑ ∑ ∑∑∑ ∑∑∑ ∑∑ = = = == + = = + == == m i i n i m i ii m i i nm i n i m i n i m i n i m i n i m i i m i i m i n i m i i yx yx y a a a xxx xxx xxm 0 0 0 1 0 0 2 0 1 0 0 1 0 2 0 00 1 (9) it is easy to show that the coefficient matrix of (9) is positive definite, therefore (9) has a unique solution. solving (9), we obtain ka (k=0, 1,…，n)，and thus the polynomial ( ) ∑ = = n k k kn xaxp 0 　 (10) it is easy to show that (10) satisfies, i.e. (10) is the fitting polynomial. figure 3 and figure 4 are 6 livers’ original b-scan images and resulting images processed by cnn, respectively. each original b-scan image is an rgb image with 720×576 pixels. in this paper we deal with these b-scan images by edcnn. then by polynomial fitting (10-degree), we find that the coefficients of the polynomials are related to the damage of the patients’ livers. 6. concluding remarks we have detected 6 b-scan images by cnn. then using polynomial fitting, we obtain the following data as shown in table 2. data of normal liver is no. 1 in table 2, from 2 to 5 are b-hepatitis in table 2. from the tables 2, it is easy to see that the value of a1 is closely related to the condition of the liver. most values of a1 are smaller than 7, and few are smaller than 12. therefore, according statistic regular[9], we can conclude that if a1 is larger than 7, the liver is normal; and if a1 is smaller than 7, the liver is chb. we can use this way to diagnose the chb. li: analysis b-scan image by cnn and polyfit 24 table 2 data of edge b-scan analysis by cnn a10 e-47 a9 e-41 a8 e-35 a7 e-29 a6 e-24 a5 e-19 a4 e-13 a3 e-09 a2 e-04 a1 e-00 a0 e+5 1 -2.1535 4.9041 -4.7809 2.6041 -8.6754 18.181 -2.3715 18.313 -7.4913 12.252 1.8006 2 1.1509 -2.2140 1.7396 -0.7036 1.4633 -1.0387 -0.1564 3.6640 -2.5314 5.3752 2.0242 3 -0.3927 0.9139 -0.9437 0.5649 -2.1400 5.2406 -0.8123 7.4907 -3.6249 6.5730 1.9970 4 -0.8249 1.7516 -1.6135 0.8469 -2.7937 5.9910 -0.8262 6.8948 -3.0492 5.1082 2.0678 5 1.3496 -2.4470 1.7724 -0.6260 0.9187 0.6274 -0.4292 6.0397 -3.5273 6.9943 1.9646 6 0.5490 -0.8967 0.5185 -0.7841 -0.4708 2.6723 -0.5914 6.5870 -3.5138 6.6962 1.9830 references [1] b. li. evole and application of medicine image in technology of computer image processed. journal of medicine engineer, 1993, 10(4): 360-363. [2] s. r. xia. medicine image searchers technology base on charpter fuse and feedback correlative. spaceflight medicine and medicine engineer, 2004, vol. 12, 429-433. [3] z. l. tian. two dimension wave transform and application in processed of medicine image. journal of medicine treatment appliance in china, 2004, 28(6): 409-411. [4] l.o.chua. cnn: a version of complexity. int. j. bifurcation and chaos, 1997, 7(10): 2219-2425. [5] l.o.chua, t.roska. cellular neural networks and visual computing. cambridge: cambridge university press, 2000, 89-94 [6] g.li, l.min, h.zang. design for robustness edgegrey detection cnn, international conference on communication, circuits and systems and chaos, 2004, (2): 1161-1165. [7] zhao, x., liao, x., jia, x "the neural networks of regression analysis and its application", advances in systems science and applications, 2006, 6(1): 7-15. [8] jian, j.g., zhang, f.k., chen, d.y. analysis of absolute stability for a class of neural networks. advance in systems science and application, 2005, 5(1): 59-64. [9] fanling kong, zhiguo zhang, guizhi wang. a martingale method of optimization design in investment and effectiveness model. advances in systems science and applications, 2007, 7(1): 1-6. advances in systems science and applications (2014) vol.14 no.3 278-285 the new form of mixed economy with rationing: agent based approach valery l. makarov and albert r. bakhtizin central economics and mathematics institute of russian academy of sciences, moscow abstract in nowdays, there is a number of mixed economy types in different country. however, any one type has its advantages and insufficient. in this paper, a new form of mixed economy with rationing based on agent is proposed, and the results of computational simulation are investigated. keywords mixed economy types, agent based approach 1 introduction. there is sizable number of mixed economy’ types. generally speaking every existing economy is mixed one in a sense that it uses various economic mechanisms. every country tries to find optimal combination of the mechanisms. the most famous mechanisms are market, rationing, planning, direct distribution, gifts, some others. in the paper we explore the rationing as a way to implement a variety of incentives for people’ activity. we assume that the population is divided to the six social clusters with different incentives to work. the first social cluster (business oriented people) has standard incentive: getting profit. other clusters’ members (state and military servants, scientists, art workers, priests, etc.) try to lift on the social stairs, getting higher rank. having a position of a certain rank a person obtains the possibility to consume goods according to fixed norms. so, we have mixture of the two mechanisms: market and rationing. mathematical economists start to study this kind of models long ago. see, tobin james [1]. the literature was growing since that. we indicate a few. in howard david h. one can find some results about dependence of a market equilibrium on quantity constraints for a consumer [2]. in makarov v. l, vasil’ev v. a. there are number of proofs of the equilibrium’ existence for various mixed economies [3]. it is important to mentioned that in the all papers quantities norms (standards. rates, quotas) are given. the only task related to changing the norms is comparative static’s problem. for example, what happened to equilibrium prices, when some norm is increasing. no studies of mechanisms, seeking the norms. in a reality the norms are generating by a society. much depends on a type of the society. under totalitarian regime the norms are given by a dictator. other regimes produce a number of way how to generate the norms, starting from a decision of relatively small groups (elite) to taking into account the opinion of everybody (referendum). advances in systems science and applications (2014) vol.14 no.3 279 a number of papers attempt to reveal the mechanism for formation of social standards [4]. it is clear, that mathematical modeling is not an effective method to study the problem. the agent based approach looks promising. there are number of papers where the agent based models are successfully used for generating of so called social norms. see, for example, epstein joshua m. [5]. epstein proposes a simple agent-based model where agents learn behavioral rules (i.e., accept certain standards formed in the society). agents are placed in order in a certain loop. agents interact with each other. each agent has a fixed position in the loop and is endowed with two characteristics. the first is “quota”, a binary variable which depends on the values of the quotas of the neighboring agents. the neighbors taken into consideration are situated within the radius of vision, which is the second characteristic of the agent. the simulations demonstrated that after certain rules have been established in the society, the majority of agents subconsciously keep following them. fent, thomas described an agent-based model developed for analyzing emergence, stability, and change of social standards within a certain population of artificial agents [6]. the information on social standards is kept within a social network, which unites the agents of the model. each agent has an inner group (i.e., a set of agents with the same social standards) and an outer group. accordingly, the agents of the model receive utility from following social rules of their inner group and from refusing the rules of outer group. the model was applied to explaining the emergence of temporal social rules, prevalent in certain groups. the quantities’ norms are the special type of social ones. the social norms play role of constraints in behavior. for example, taboo is a typical social norm. the quantities’ norms look as prices. one can buy or not by the price, one can use the quantities’ norm or not. prices can be set by market (people) or by a state agency. so with norms. its can be generated by a society (people) or by an agency. essential difference between prices and norms lies on the speed of changes. in the described below model we consider that the norms change once in a time unit period. 2 mixed economy with rationing based on agent we can mentioned the recent paper with the some rules of generating the quantities’ norms (tips). savarimuthu et al. describe an agent-based model with a micro-level mechanism for the emergence of social standards [7]. the approach differs from the common macro-level mechanism, which is implemented in most papers on social standards. agents have memory of events and radius of vision, i.e. a parameter which determines the square of the surface where the agent can observe events, related to violating or following social standards. the pro280 valery l. makarov:the new form of mixed economy with rationing: agent based ... posed algorithm for the emergence of social standards is applied to a visit to a restaurant where agents make decisions about tips. computational simulations revealed that the time for emergence of social standards directly depends on the length of memory (the number of recent events memorized by the agent) and on the value of the radius of vision. we constructed the agent based model, where agents are people of the six social clusters mentioned above. moreover people in the clusters are divided to three ranks; high, middle and low ones with exclusion of the first social cluster. the first cluster (business oriented people) has objective to achieve maximal profit. the other people destinations are to get high rank. the quantities’ norms associated with the clusters and the ranks in. an agent consume goods from two sources: from market and rationing spheres. in a simple case we consider that an agent used only one source: market, if he/she belongs to the first cluster, or rationing, otherwise. now we describe the first version of the agent based model, where each social cluster produces one “good”, and the norms relate to the first good only. other “goods” we interpret as development’s levels of the clusters. so the production function of the first cluster looks as a1 (t) = n1(t) α1(t) · k1(t− 1)β 1(t) · a2(t)γ2 · a3(t)γ3 · a4(t)γ4 · a5(t)γ5 · a6(t)γ6 (1) where a1 (t) is a production of the first cluster in the period t; aj (t) is the level of the cluster’ j development in the period t; n1 (t) is the quantity of agents, populated cluster 1; k1 (t− 1) is accumulated capital (production funds) to the beginning of the period t. other clusters’ production functions are about the same type. so there is a strong dependence of all clusters on each other. it is impossible to develop one cluster with no development of others. the consumption of an agent ci is defined by his/her budget, if he belongs to the first cluster and is equal to the existing at the given period norm of a rank & cluster for others. so a macro-path {a (t) , c (t)} of the agent-based model, which is the outcome of simulations, depends on the norms, generated by the mechanism, we try to design. the comparison of the simulated trajectories can be done by standard way, using well know criteria. the simple way to generate the norms we used is the following. the initial norms for all clusters and ranks are given. further agents make influence on the advances in systems science and applications (2014) vol.14 no.3 281 norms according to the formula. csr(t) = { csr(t− 1) · ex(n) · fx(n)(r) · ox(n)(t− 1); s = x csr(t− 1)/(ex(n) · fx(n)(r) · ox(n)(t− 1)); s ̸= x (2) where ex(n) the level of n-th agent’s influence on the total level of consumption f social cluster x is calculated as: ex(n) = 1 + µ · (1/n) (3) with n the number of agents in the society and µ a coefficient, which determines the power of the agent. by default µ equals 10 for agents of the first (high) rank, 3 for the second and 1 for the third rank. this implies that the influence is an order higher for the agents of the first (high) rank if compared to the third rank. it is important to mentioned that the state of agents is changing over time. the age of the agent increases and the life expectancy decreases in each moment of time. with a certain probability, estimated on the basis of statistics, the agent can have a child. upon reaching the age 18, the child with a high probability enters the cluster of parents. agents can migrate from one cluster to another according to some rules. we produced a number of scenarios, how agents influence to the norms, divided to the two blocks. the first block deals with assessing the change in the socio-cluster society due to the change in the mechanism for the formation of the norms. the second estimates the impact of various factors on the processes for the formation of the norms. the first block contains three scenarios to make calculations. the first one is the baseline scenario for the development of the socio-economic system in our model. the second calculation deals with a slight modification in the work of the model. while the first calculation studied the totality of agents, in which each agent was capable to directly influence the consumption quotas for all social clusters, in the second calculation agents elect representatives of their clusters proportional to the number of agents in the cluster. in other words, we supplement the model with a special (new or additional) cluster. the only function of the agents in this cluster is to regulate social quotas. the agents of the supplementary cluster change quotas according to the formula analogous to presented above. the only difference is that the agents of the supplementary cluster have higher level of influence in changing quotas. moreover, the very change of the quota in each cluster happens stochastically: the quota is changed with probability 1 for the cluster represented by the agent. for other social clusters the quotas are changed with probability 0.5. this implies that the government elected by the agents fully 282 valery l. makarov:the new form of mixed economy with rationing: agent based ... fig.1 the results of computational simulations (x coordinates is years, y coordinates is the volume of production) supports its electorate but does not hurt others strongly. the third scenario is similar to the second. the difference is higher values of probability for the decrease of the quotas for the social clusters, which agents are not represented in the regulating body. fig.1 demonstrates the results of calculations with respect to the estimated values of production in all the three scenarios. we see a justification of the assumption about the need for equal rights of all clusters for successful development of the society. when the regulating body strengthens the policy aimed at discriminating certain social clusters, it is immediately reflected on the volume of production in the whole economic system. note that a more democratic principle of change of social quotas (when each agent has a direct influence on parameters) is slightly more efficient. the second block of calculations quantitatively assesses the impact of various factors on the processes for formation of social quotas. namely, it distinguishes between two mechanisms of formation: by each agent or by a selected group. first, we studied the dependency of the values for social quotas on the initial number of agents in each social cluster. in this case the size of the first social cluster was 50% of the society, and the sizes of other clusters were 10%. in the second case the sizes of all the 6 clusters were equal (16(.6)%). note that the initial level of social quotas was the same in both cases. the results for the 8th year of the model are presented in table 1 (all values are normalized relative to the initial values, i.e., all quotas equal to 1 in the zero year). the results demonstrate that in the first case (when the first social cluster initially includes 50% of the society), quotas of the first social cluster increased advances in systems science and applications (2014) vol.14 no.3 283 table 1 the impact of the initial number of agents in clusters social cluster experiment-1.1 (each agent can change quota) experiment-1.2 (quotas are changed only by representatives of a special cluster) experiment-2.1 (each agent can change quota) experiment-2.2 (quotas are changed only by representatives of a special cluster) first 1.053 1.042 1.001 0.996 second 0.962 0.944 1.011 1.002 third 0.930 0.981 0.983 1.010 fourth 0.954 0.956 1.035 0.987 fifth 0.936 0.948 0.996 1.003 sixth 0.933 0.967 1.022 0.989 standard deviation (for all clusters) 0.03060 0.02568 0.01479 0.00713 table 2 the impact of the number of ranks social cluster ranks are present ranks are absent first 0.981 0.995 second 0.943 0.981 third 1.034 0.975 fourth 0.967 1.036 fifth 1.045 0.983 sixth 0.954 1.055 standard deviation (for all clusters) 0.03478 0.02759 more than quotas of other clusters. note that the standard deviation of the values for all clusters is larger than in case of change only by the representatives of a special cluster. we conducted experiments no.2.1 and 2.2 several times. each time we obtained a certain (different) set of values for social quotas. consequently, even making a society more balanced (i.e., with the equal number of agents in social clusters), in the end we inevitably have a differentiation in consumption. we assessed the dependency of the values of social quotas on the presence/absence of ranks in the society. (with the absence of ranks the agents have equal abilities to change social quotas). similarly to previous estimations we conducted calculations for 8 model years (see results in table 2; all values are normalized relative to the initial values, i.e., all quotas equal to 1 in the zero year). we see that quotas change faster in the presence of ranks. in other words the society with the differentiation s more inclined to an additional stratification in income. 284 valery l. makarov:the new form of mixed economy with rationing: agent based ... the final experiment deals with estimating the dependency of social quotas on their initial values. in initializing the model we increased the quotas for the first cluster, decreased for the sixth cluster, and left unchanged for other clusters. our purpose was to study the impact of initial misbalance in social quotas on the values of social quotas in the long run (15 years). the results (table 3) demonstrate a clear long run tendency: the discrepancy in social quotas increases. this implies that no movement towards balance can happen without an exogenous disturbance. table 3 the impact initial values of social quotas social initial values of each agent only representatives cluster social quotas changes quotas of a special cluster change quotas first 1.500 1.544 1.525 second 1.000 0.820 0.833 third 1.000 0.811 0.824 fourth 1.000 0.810 0.829 fifth 1.000 0.809 0.826 sixth 0.500 0.397 0.406 3 conclusion to sum up, the results of computational simulations revealed that: (1) successful development of a society with social clusters requires equal rights of social clusters; (2) democratic principle of change in quotas (when each individual has a direct influence on the values of quotas) is a more efficient economic mechanism; (3) social quotas change faster for the dominant social cluster than for other clusters; (4) even trying to make a society more balanced (i.e, with equal number of individuals in social clusters) at the end we inevitably face differentiation in production and consumption; (5) a society with differentiation is more inclined to additional stratification in the level of consumption. references [1] tobin james. (1952), “a survey of the theory of rationing”, econometrica, vol.20 (1952), pp.521-553. [2] howard david h. (1977), “rationing, quantity constraints, and consumption theory”, econometrica, vol.45, pp.399-412. advances in systems science and applications (2014) vol.14 no.3 285 [3] makarov v. l, vasil’ev v. a. (1989), “equilibrium, rationing an stability”, matecon, vol.25 no.4, pp.4-95. [4] neumann martin. (2010), “norm internalisation in human and artificial intelligence”, journal of artificial societies and social simulation, vol.13 no.1, 2010. publishers. [5] epstein joshua m. (2000), “learning to be thoughtless: social norms and individual computation, the brookings institution and santa fe institute, center on social and economic dynamics”, working paper, no.6, january 2000. [6] fent, thomas. (2006), “collective social dynamics and social norms”, mpra paper, no.2841. [7] bastin tony roy savarimuthu, stephen cranefield, maryam a. purvis and martin k. purvis. (2010), “obligation norm identification in agent societies”, journal of artificial societies and social simulation, vol.13 no.4. this work is supported by russian humanitarian scientific fund (grants no.1402-00431 and no.15-02-00276). corresponding author albert r. bakhtizin can be contacted at: albert.bakhtizin@gmail.com advances in systems science and applications (2013) vol. 13 no. 3 218-226 integrated mechanisms of organizational behavior control v.n. burkov, m.v. goubko, n.a. korgin and d.a. novikov institute of control sciences, russian academy of sciences, moscow, russia abstract problems of control mechanisms integration are formulated and discussed in the framework of mechanism design for organizational behavior control. unified schemes for control mechanisms description and design are proposed. an example of the integrated production cycle optimization mechanism is considered. keywords mechanism design, management theory, organizational behavior, integrated control mechanisms. 1 introduction according to general control methodology [1], control is the activity of subjects controlling other subjects or objects. formal models of control in organizational systems are studied in the framework of mechanism design (see surveys in [2] and [3]), theory of contracts [4, 5] and collective choice theory [6, 7], involving results of game theory [10, 11] and operations research (see textbooks [8, 9]), and are applied in microeconomics and management theory. a set of control mechanisms may be considered as a “kit”, which contains elementary blocks, intended for solving of typical problems of planning, organizing, motivating and controlling. but in practice one faces not typical but real complex problems of organizational control (according to [12] controlled complex systems nowadays are usually decentralized, hierarchical and networked, hence, heterogeneity approach in control models would surely be in demand). hence the technique of complexing and integrating different control mechanisms is required. below an attempt of integrated control mechanisms construction is taken for the problem of production cycle optimization. 2 control mechanisms suppose that a man is engaged in a control loop. then there is a need to control him or her – a controlled object which • is subjectified and acts according to individual interests and preferences (the principle of rationality); • appears not completely known to a corresponding control subject (the principle of information asymmetry); • may not reveal true information and not perform the expected actions. in the sequel, such controlled objects are said to be economic agents or, shortly, agents; the corresponding control subject is referred to as a principal. 219 advances in systems science and applications (2013) vol. 13 no. 3 the “principal – agent” terminology seems convenient and is traditionally used to describe control mechanisms. a control mechanism represents a certain set of rules and procedures involved by the principal for making decisions that influence on the behavior of active economic agents (in particular, on information revealed and actions chosen by them), hence the principal has to solve the problems of control mechanism design. in fig. 1 the typical aggregated scheme of a control mechanism structure is provided in a operational form. fig. 1 typical aggregated scheme of a control mechanism subjects of control are: norms and restrictions of agents’ activity (institutional control), preferences of the agents (motivational control, including incentive problems) and awareness of the agents (informational control). sequence of moves in an organizational system (being common for all stages of a control cycle) under a fixed control mechanism is shown in fig. 1: stage i: the principal makes the first move, i.e., informs the agent of a control mechanism (“rules of play”) in a general form. for instance, this could be the relation between the amount of resource allocated and the reported need in the resource; alternatively, rules of play may be specified by the relationship between the reward and results achieved (the state of the agent). stage ii: the agent reports information on uncertain parameters to the principal (e.g., submits a claim for a resource or reports information on his or her preferences). stage iii: the principal informs the agent of control mechanism parameters. for instance, he or she assigns a plan as the result of the agent’s activity expected by the principal. stage iv: the agent chooses an action, and the result of activity is formed. stage v: the principal receives information on the agent’s action. stage vi: the principal informs the agent of his or her own action according to the control mechanism (e.g., the amount of resource allocated, the reward and so v.n. burkov: integrated mechanisms of organizational behavior control 220 on). stages i-iii (see fig. 1) correspond to the planning cycle, while stages iii-vi are related to the implementation cycle. the lower rectangle in fig. 1 represents the model of an economic agent. when solving design problems for sophisticated (complex) control mechanisms, it appears reasonable to describe the mechanisms using the notation similar to that of idef0 standard [13] and the input-output schemes in control science (see fig. 2). fig. 2 the general input-output scheme of a control mechanism within the framework of such representation, the six general stages discussed above (see fig. 1) could be formulated in the following way: a. the external conditions and constraints imposed on the principal during the process of making management decisions. b. the parameters of a control mechanism, being defined by the principal (stages i and iii). c. the input information from (on) agent (agents), being necessary to make a management decision (stages ii and iv). d. the result of the mechanism applied, i.e., the management decision made by the principal (stages iii and vi). now let’s consider the technique of integrating several mechanisms, which follow the unified description (fig. 1 and fig. 2), in order to solve certain control problem (the production cycle optimization problem is used as an example). 221 advances in systems science and applications (2013) vol. 13 no. 3 3 integrated mechanism of production cycle optimization production cycle optimization (pco) appears an important factor in improving the efficiency of a manufacturing process and in reducing the demand for circulating assets. pco represents an integrated mechanism, since the underlying process includes four primary stages as follows. stage 1. data acquisition regarding capabilities of pco (such information is supplied by units, i.e., shop floors and offices). stage 2. pco planning to ensure the required rate of optimization (what operations should be optimized and at what rates). stage 3. incentive scheme design (what rewards should be assigned to the units for reducing operation time of a manufacturing process). stage 4. implementation of pco plan and providing the rewards to the units. a simplified description of the pco-mechanism is shown in fig. 3. fig. 3 stages of production cycle optimization according to the stages mentioned, one would observe that (at least) four basic mechanisms are necessary: • stage 1 (data acquisition) – the mechanism for counter plans, which motivates the units to report higher capability estimates for pco; • stage 2 (pco planning) – the coordinated planning mechanism, which guarantees the required rate of pco in the sense of the minimum total costs to motivate such optimization; • stage 3 (incentive scheme design) – the incentive mechanism for a unit as the result of a certain optimization rate achieved; the mechanism must ensure truth-telling of all units; • stage 4 (implementation of pco plan) – the mechanism of predictive selfcontrol, which serves for well-timed informing of possible frustration of the plans. the mechanisms for counter plans ensure better estimates of feasible work time optimization (reported by the units) via coordinating the rewards for tense plans, the penalties caused by plan non-fulfillment and the rewards for plan overfulfillment. when an agent is paid merely for fulfillment (or overfulfillment) of the plan assigned by a principal, the agent is not interested in having a high (“tight”) plan. the reason is performing it would require additional efforts (costs) of the agent. for instance, an agent may inform the principal of his or her preferences, eo ipso reporting his or her estimate of the plan (referred to as a “counter plan”). v.n. burkov: integrated mechanisms of organizational behavior control 222 within the framework of incentive scheme for counter plans, an agent is given rewards for reporting counter plans that better meet the principal’s interests (yet, are “tighter” for the agent). the planning mechanism enables solving an optimization problem of defining the planned rates of operation time optimization to minimize the total optimization rate; this mechanism is also used to evaluate a reward norm for a unit optimization rate. for example, resource allocation mechanism is intended to distribute a specific resource based on claims of control objects (agents) for a desired quantity of the resource. note this is done under the conditions of deficiency (the resource is limited), while a control subject (a principal) has no information on optimal quantity for every agent. the mechanism ensures truth-telling of the claims submitted by the agents. modifying the mechanism parameters (priorities of the agents, prices, rates, etc) allows the principal to minimize his or her losses caused by the gap between actual distribution and its optimal counterpart (the latter could be achieved if the principal knew exact quantities of the resource needed by the agents). another example is a transfer price mechanism, which is involved to perform mutual payments within a company (in the system of internal accounting) or between companies that enter the same corporation. in the case of vertical integration, a transfer price is a tool to distribute profits between participants of a technological chain. horizontal integration being considered, a transfer price provides a tool of coordinating interests between the enterprises (units – agents) and a principal. the latter acquires from the former information on the price and quantity of a product they are ready to supply; in other words, the principal finds the relationship between an optimal output (or a plan) and a transfer price. based on the acquired information, the principal establishes a transfer price of the product such that the total output of the agents equals the required one. under a sufficiently great number of the agents, the mechanism of transfer prices ensures, first, truth-telling (since the agents benefit from reporting the actual dependence between the optimal output and the transfer price) and, second, minimum manufacturing costs for the products. the corresponding incentive mechanism could be represented by a proportional unified incentive scheme; here the income (bonus) of a unit equals the product of the reward norm and the planned rate of optimization. collective incentive scheme is designed for situations when a principal turns out unable to separately observe the action of every agent (the principal merely knows a certain aggregated rate, e.g., the result of collective activity). imagine that the principal can evaluate the minimum costs to-be-incurred by the agents to achieve the required result of collective activity. in this case, the efficient 223 advances in systems science and applications (2013) vol. 13 no. 3 incentive scheme takes the following form. the minimum costs of each agent are compensated (provided that the result of collective activity agrees with the requirements of the principal). moreover, sometimes the principal has no costs related to observing individual actions of every agent; thus, the principal’s workload to acquire and process information is substantially reduced. unified incentives mechanism is employed in situations when a principal has to motivate large groups of agents, to involve “democratic” management methods and to decrease the amount of processed data. under unified incentives, the relationship between the reward and labor intensity of the agents (alternatively, the results attained by them) is identical for all agents. in several cases, the described unification leads to no loss of efficiency, while the wages fund is spent in an optimal way. yet, unified control may be inefficient, when non-consideration of individual features of the agents results in inefficient spending of financial resources. the mechanism of predictive self-control allows a principal to obtain well-timed information on possible deviations from the planned rates of pco; this is done by coordinated assignment of the penalties for plan correction (they depend on the moment a certain unit reports of such correction) and the penalties for plan non-fulfillment. mechanism of predictive self-control is intended for well-timed informing a principal of possible deviations (from a plan) in the agents’ activity. the earlier the principal gets aware of possible deviations from the plan (e.g., in due dates, financial investments, etc), the more efficient and well-timed would be his or her decision (e.g., additional measures to eliminate deviations and reduce losses, or plan correction); note the agents report deviations. the matter is that the penalties of the agents (in the case of plan correction) depend on the moment the agents report of the correction (they are smaller if the report is early); moreover, these penalties are less than in the case of plan non-fulfillment. detailed description of these mechanisms may be found in [3]. fig. 4 shows a block diagram of the integrated mechanism of pco. the algorithm of pco-optimization mechanism operation: (1). each unit informs the principal of an optimization rate of the corresponding operations in a manufacturing process (depending on the reward norm). (2). the principal determines a minimum reward norm such that the duration of the manufacturing process is reduced to a required rate. the principal defines an optimization rate of the corresponding operations, using a rank-order tournament in the case of multiple-valued solution. (3). the units implement the tasks regarding production cycle optimization. (4). the principal computes the rewards of the units and pays them to the latter. v.n. burkov: integrated mechanisms of organizational behavior control 224 fig. 4 the block diagram of pco-optimization mechanism 4 conclusions the theory of control in organizations traditionally studies a certain system of nested control problems (solutions to “special” problems are widely used to solve more “general” ones) [3]. today, there are two common ways to describe an organizational system model (as well as to formulate and solve the corresponding control problems). they are referred to as the “bottom-top” approach and the “top-bottom” approach. according to the first (“bottom-top”) approach, particular problems are solved first; using the obtained solutions to the particular problems, general ones are treated then. for instance, a particular problem could be that of incentive scheme design. suppose this problem has been solved for any possible staff of the organization. next, one may pose the problem of staff optimization, i.e., choosing a certain staff to maximize the efficiency (under a proper optimal incentive scheme). an advantage of this approach lies in its constructivity. a shortcoming lies in high complexity due to the large number of possible solutions of the upper-level problem, each requiring the solution of the corresponding set of particular subproblems. the second (“top-bottom”) approach eliminates this shortcoming; this approach states that the upper-level problems must be solved first, while their solutions serve as constraints for the particular lower-level problems. in fact, we doubt whether (e.g., when creating a new department) a manager of a large-scale company would first think over the details of regulations describing the lowestlevel employees’ interactions. quite the contrary, he would delegate this task to 225 advances in systems science and applications (2013) vol. 13 no. 3 the head of the department (providing him with necessary resources and authorities). construction of an efficient control system for an organization requires combining both approaches in theory and in applications. as well, both approaches may be applied when designing certain integrated control mechanism. the challenge is to develop a simple and efficient technique of integration. references [1] novikov, d. and rusjaeva, e. (2012), “foundations of control methodology”, advances in systems science and application, vol. 12, no. 3, pp. 33-52. [2] burkov, v., goubko, m., korgin, n., and novikov, d. (2013), “mechanisms of organizational behavior control”, advances in systems science and application, vol. 13, no. 1, pp. 1-21. [3] novikov, d. (2013), theory of control in organizations, new york. nova science publishers. [4] bolton, p. and dewatripont, m. (2005), contract theory, cambridge, mit press. [5] salanie, b.m. (2005), the economics of contracts. 2nd edition, massachusetts, mit press. [6] moulin, h. (1995), cooperative microeconomics: a game-theoretic introduction, princeton, princeton university press. [7] nitzan, s. (2010), collective preference and choice, cambridge, cambridge university press. [8] camerer, c. (2003), behavioral game theory: experiments in strategic interactions, princeton, princeton university press. [9] myerson, r.b. (1991), game theory: analysis of conflict, london, harvard university press. [10] hillier, f. and lieberman, g. (2005), introduction to operations research (8th ed.), boston, mcgraw-hill. [11] taha, h. (2011), operations research: an introduction (9th ed.), new york, prentice hall. v.n. burkov: integrated mechanisms of organizational behavior control 226 [12] forrest, j., novikov, d. (2012), “modern trends in control theory: networks, hierarchies and interdisciplinarity”, advances in systems science and application, vol.12, no.3, pp.1-13. [13] idef0 overview at idef.com, http://www.idef.com/idef0.htm, last reviewed on july 7, 2011. corresponding author author can be contacted at: novikov@ipu.ru advances in systems science and applications (2013) vol.13 no.4 392-400 stability results for an inverse parabolic problem j. damirchi1 and a. shidfar2 1faculty of mathematics, statistics and computer science, department of mathematics, semnan university, semnan, iran. 2department of mathematics, iran university of science and technology, narmak, tehran 16844, iran. abstract in this paper, a one dimensional inverse parabolic problem in a quarter plane will be considered. the unknown function in a boundary is estimated from an over specified condition at a fixed location inside the region by solving an illposed integral equation. the tikhonov regularization method of the 1st order is applied in order to stabilize the solution of the ill-posed problem. the solution of the inverse problem is defined by minimization of the tikhonov functional. some analytical results for regularization parameter determination and stability of solution of inverse problem are derived. keywords tikhonov regularization method, inverse parabolic problem, regularization parameter, stable solution, ill-posed problem 1 introduction inverse parabolic problems play a crucial role in applied mathematics, physics and engineering science. they arise for example, in the study of heat conduction processes, diffusion, control theory [1-10]. in recent years, a lot of attention has been devoted to the study of inverse parabolic problems. hence, the last 20 years have seen growing attention paid in the literature to the development, analysis, and implementation of accurate methods for the solution of inverse parabolic problems, i.e., the determination of unknown boundary condition g(t) in the parabolic partial differential equation. in this paper, we investigate an inverse parabolic problem in quarter plane to obtain an unknown function g(t) from over specified data p(t) at a fixed location inside the body. this inverse problem can be reduced to the operator integral equation ag = g. it is well known that the operator equation ag = g is an ill-posed problem, when a solution is unstable with respect to small variations in input data. since in the usual case where only a measured or computation approximation gδ is available, some kind of regularization methods is required in order to obtain a reasonable stable approximation gδ to g [6, 11-15]. in order to obtain stable solution for this ill-posed problem the tikhonov regularization method is applied for operator equation ag = g for retrieving solution in a stable manner. applying general result in the theory of tikhonov regularization method for ill-posed inverse problem which consists in solving the unconstrained minimization problem, we can find stable solution. since the regularization paadvances in systems science and applications (2013) vol.13 no.4 393 rameter play an important role in applying the tikhonov regularization method to the operator equation, we obtain this parameter based on the error in input data directly, which can be one of the advantage of our method for selecting regularization parameter with respect to other methods [16-19]. the organization of this paper is as follows, in forcecoming section, mathematical formulation for this inverse parabolic problem in a quarter plane is introduced. in section 3, thikhonov regularization method is stated and we use this method to construct a stable solution for this ill-posed problem. some theoretical results will be proved about the solution of this ill-posed problem and the choice of regularization parameter. it will be shown that the solution of thikhonov regularization method is stable under small errors in input data and the existence of this stable solution is shown. we conclude this article with a brief conclusive discussion in section 4. finally, some references are given at the end of this paper. 2 mathematical formulation in this section, we consider the following inverse parabolic problem in a quarter plane: ut = uxx, 0 < x, 0 < t < t (1) u(x, 0) = f(x), 0 < x (2) u(0, t) = g(t), 0 < t < t, (3) |u(x, t)| ≤ m, 0 < x, 0 < t < t (4) where f(x) and g(t) are piecewise known continuous functions and m is a positive number.the problem consist of using an overspecified data p(t), which is given by p(t) = u(1, t), 0 < t < t (5) to determine the unknown function g(t). for the forward problem (1)-(4), the unique bounded solution u(x, t) is given by [20], u(x, t) = x√ 4π ∫ t 0 g(τ)√ (t− τ)3 e − x2 4(t−τ)dτ + 1√ 4πt ∫ ∞ 0 (e− (x−ξ)2 4t − e− (x+ξ)2 4t )f(ξ)dξ (6) due to the setting x = 1, and using (5) we obtain ∫ t 0 e − 1 4(t−τ)√ (t− τ)3 g(τ)dτ = 2 √ πp(t)− 1√ t ∫ ∞ 0 (e− (x−ξ)2 4t − e− (x+ξ)2 4t )f(ξ)dξ (7) 394 j. damirchi: stability results for an inverse parabolic problem which is written in the form∫ t 0 h(t− τ)g(τ)dτ = 2 √ πp(t)− 1√ t ∫ ∞ 0 (e− (x−ξ)2 4t − e− (x+ξ)2 4t )f(ξ)dξ where the kernel h(t, τ) = 1√ (t−τ)3 e − 1 4(t−τ) is a continuous function on [0, t ] × [0, t ]. now, we formulate (7) in term of an operator integral equation : (ag)(t) = g(t), 0 < t < t (8) where the operator a is defined by (ag)(t) = ∫ t 0 e − 1 4(t−τ)√ (t− τ)3 g(τ)dτ, 0 < t < t and g(t) = 2 √ πp(t)− 1√ t ∫ ∞ 0 (e− (x−ξ)2 4t − e− (x+ξ)2 4t )f(ξ)dξ, 0 < t < t the above integral equation of the first kind (7) cannot be reduce into an integral equation of the second kind by differentiation and the problem is inherently ill-posed. for 1 ≤ p < ∞, a is a compact linear operator in lp[0, t ]. zero is not an eigenvalue of a and is the only point in the spectrum of a, thus a−1 exist and is unbounded, so if g(t) on 0 ≤ t ≤ t is in the range of a, g(t) is uniquely determined from g = a−1g. in practice, with g(t) obtained from measurement, small error in g(t) lead to enormous errors in g(t) because a−1 is unbounded. in order to regularize the problem, we use tikhonov regularization method for finding approximate stable solution for ill-posed integral equation (8). in the next section we describe tikhonov regularization method. 3 tikhonov regularization method for ill-posed integral equation it is well known that the tikhonov regularization method is one of the useful tools for solving an ill-posed problem of the form (7)[12-13, 19]. since in practical purpose, the input data are non smooth, we apply tikhonov regularization method to construct stable solution for solving ill-posed equation (8). based on the main concept of tikhonov regularization method, we introduce smoothing functional mα[g,g] which is called also thikhonov functional and stabilizing functional ω(g) as follows: mα[g,g] = ∥ag −g∥2l2[0,t ] + αω(g) advances in systems science and applications (2013) vol.13 no.4 395 α is regularization parameter and ω(g) is a stabilizer of the 1th order constant coefficient which is defined as: ω(g) = ∫ t 0 (g2(τ) + g′2(τ))dτ we construct a regularize solution for the integral equation (8) by using the following minimization problem which is state by the following theorem: theorem 1. for every function g ∈ l2[0, t ], and any positive α, there exists an element gα ∈ w 1 2 , such that smoothing functional mα[g,g] attain it greatest lower bound. infmα[g,g] = mα[gα, g] proof. we have mα[g,g] = ∫ t 0 ( ∫ t 0 e − 1 4(t−τ)√ (t− τ)3 g(τ)dτ −g(t))2dt+ α ∫ t 0 (g2(τ) + g′2(τ))dτ a condition for a minimum of this functional is vanishing of its first variation, this is written in the form 1 2 d dε mα[g + εη,g]|ε=0 = ∫ t 0 [ ∫ t τ ∫ t 0 h(t− s)h(t− τ)g(s)dsdt− ∫ t τ h(t− τ)dt − α(g′′(τ)− g(τ))]η(τ)dτ + g′(τ)η(τ)|t0 =0 (9) here, η(τ) is an arbitrary variation of the function g(τ) such that both g(τ) and g(τ) + εη(τ) belong to the class of admissible function. condition (9) will be satisfied if∫ t τ ∫ t 0 h(t− s)h(t− τ)g(s)dsdt− ∫ t τ h(t− τ)dt = α(g′′(τ)− g(τ)) (10) and g′(0) = g′(t ) = 0 (11) the equation (10) is called euler-lagrange equation, therefore the minimizer gα(t) for tikhonov functional is determined by the solution of euler-lagrange equation corresponding to functional mα[g,g]. the solution of (10) with boundary condition (11) is unique by classical theorems in ode. now, we can assume that the regularized solution of the above minimization problem, gα as an regularizing operator r(g,α) such that gα = r(g,α) where 396 j. damirchi: stability results for an inverse parabolic problem α = α(δ,gα) in accordance with the error in the initial data g and δ, measuring the error in data, we select α in a suitable way such that r(g,α) is a regularizing operator for the equation (8), and gα = r(gα, α(δ)) can be taken as an approximate stable solution for ill-posed problem (8). theorem 2. if gt (t) ∈ c[0, t ] be the exact solution of the original problem (8) associated with the exact right hand member g = gt ; that, (agt )(t) = gt (t). then, for every positive number ε, there exists a positive number δ(ε) such that for every gδ ∈ l2[0, t ] the inequality ∥gt (t)−gδ(t)∥l2[0,t ] ≤ δ < δ(ε) implies the inequality ∥gα(δ)(t)− gt (t)∥c[0,t ] ≤ ε where gα = r(gα(δ), α(δ)) be the solution of (8) associated with perturbed data gδ for all α satisfying α(δ) = δγ , 0 ≤ γ < 2. proof. since gα is a minimizer of functional mα, we have mα(δ)[gα, gδ] ≤ mα(δ)[gt , gδ] therefore, ∥agα −gδ∥2 ≤ mα(δ)[gα, gδ] ≤ ∫ t 0 (agt (t)−gδ(t)) 2dt+ α(δ) ∫ t 0 (g2t (τ) + g′2t (τ))dτ = ∫ t 0 (gt (t)−gδ(t)) 2dt+ δγ ∫ t 0 (g2t (τ) + g′2t (τ))dτ ≤ δ2 + δγ ∫ t 0 (g2t (τ) + g′2t (τ))dτ ≤ δγ(1 + ∫ t 0 (g2t (τ) + g′2t (τ))dτ) = δγn with n = 1 + ∫ t 0 (g2t (τ) + g′2t (τ))dτ . consequently, the elements gt (t) and gα(δ)(t) belong to the compact subset e of element g of c[0, t ] such that e = {g(t)| ∥g∥2w 1 2 ≤ n} since e is compact in c[0, t ], and the operator a is continuous, the mapping a : e → ae is continuous and one to one, therefore the inverse mapping a−1 : ae → e is also continuous. this means that, ∀ε > 0, ∃η(ε), ∥gt −gα∥ ≤ η(ε), agt = gt , agα = gα advances in systems science and applications (2013) vol.13 no.4 397 then ∥gt − gα∥c[0,t ] ≤ ε on the other hand we have ∥gt −gα∥2l2 = ∫ t 0 (agα −gt (t)) 2dt < δ2 an so, ∥gt (t)− gα(δ)(t)∥c[0,t ] = ∥a−1agt −a−1agα(δ)∥ ≤ ∥a−1∥∥agt −agα(δ)∥ on the other hand ∥agt −agα(δ)∥l2 ≤ ∥agt −gδ∥+ ∥agα(δ) −gδ∥ ≤ ∥gt −gδ∥+ ∥agα(δ) −gδ∥ ≤ δ + δ γ 2 √ n ≤ δ γ 2 (1 + √ n) therefore, ∥gt − gα(δ)∥ ≤ ∥a−1∥δ γ 2 (1 + √ n) the above results show that δ(ε) should be chosen in the form δ(ε) ≤ [ ε ∥a−1∥(1 + √ n) ] 2 γ such that the theorem is satisfied. the above theorem shows that when we construct regularizing operator by minimizing the smoothing functional mα[g,g], the regularization parameter α can be obtain according to the error in the right hand member. by the next theorem we will show that g depends continuously on the initial data f, p. theorem 3. if the exact data ft (x) and pt (t) satisfy (8) and the approximate data fδ(x) and pδ(t) also satisfy (8), then inequalities ∥ft (x)− fδ(x)∥l2[0,∞] < δ and ∥pt (t)− pδ(t)∥l2[0,t ] < δ imply that ∥gt (t)−gδ(t)∥l2[0,t ] < δd, d = (9 √ 2πt + 12π) 1 2 398 j. damirchi: stability results for an inverse parabolic problem proof. by using of cauchy-schwartz inequality, we have ∥gt (t)−gδ(t)∥2 =∥2 √ π(pt (t)− pδ(t))− 1√ t ∫ ∞ 0 (e− (1−x)2 4t − e− (1+x)2 4t )(ft (x)− fδ(x))dx∥2 ≤3[ ∫ t 0 4π(pt (t)− pδ(t)) 2dt+ ∫ t 0 1 t ( ∫ ∞ 0 e− (1−x)2 4t (ft (x)− fδ(x))dx) 2dt + ∫ t 0 1 t ( ∫ ∞ 0 e− (1+x)2 4t (ft (x)− fδ(x))dx) 2dt] ≤3(4δ2π + ∫ t 0 1 t ( ∫ ∞ 0 (e− (1−x)2 4t )2dx)( ∫ ∞ 0 (ft (x)− fδ(x)) 2dx)dt + ∫ t 0 1 t ( ∫ ∞ 0 (e− (1+x)2 4t )2dx)( ∫ ∞ 0 (ft (x)− fδ(x)) 2dx)dt) ≤3(4δ2π + δ2 ∫ t 0 1 t ∫ ∞ 0 e− (1−x)2 2t dxdt+ δ2 ∫ t 0 1 t ∫ ∞ 0 e− x2 2t dxdt) ≤3(4πδ2 + 3 √ 2πtδ2) =δ2(12π + 9 √ 2πt ) theorem (3) and (4) show that by choosing α in such a way that the choice for α is consistent with the accuracy δ of the initial data, then the element gα = r(gα, α) obtained with the aid of the regularizing operator r(g,α), can be taken as approximate stable solution of equation (8). we summarize the above results by the following theorem: theorem 4. if gt (t) is the exact solution of equation (8) with exact data ft (x) and pt (t), then for every ε > 0 and the approximate data fδ(x) and pδ(t) also satisfy (8), there exist δ(ε) and α(δ) such that inequalities ∥ft (x)−fδ(x)∥l2[0,∞] < δ and ∥pt (t)− pδ(t)∥l2[0,∞] < δ imply that inequality ∥gt (t)−gα(t)∥ < ε where gα(δ) = r(gδ, α(δ)). 4 conclusion in this paper, we have introduced tikhonov regularization method and we have shown why it is important to use it in order to solve ill-posed problems. we have shown that the tikhonov regularization technique is introduced to treat the instability of obtaining a stable solution. the choice of regularization parameter and stability of solution are proved. it is very interesting to extend these results for higher dimensional problem and for nonstandard heat equation. advances in systems science and applications (2013) vol.13 no.4 399 references [1] ramm a g. (2005), inverse problems, springer, new york. [2] ramm a g. (2001), “an inverse problem for the heat equation”, j. math. anal. appl, vol.264, pp.691-697. [3] cannon j r, lin y, wang s. (1992), “determination of source parameter in parabolic equations”, meccanica, vol.27, pp.85-94. [4] chen q, liu j j. (2006), “solving an inverse parabolic problem by optimization from final measurement data”, j. comput. appl. math, vol.193, pp.183-203. [5] dehghan m, tatari m. (2006), “determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions”, math. comput. modelling, vol.44, pp.1160-1168. [6] zheng g h, wei t. (2011), “spectral regularization method for solving a time-fractional inverse diffusion problem”, appl. math. comp, vol.38, pp.317-336. [7] isakov v. (1998), inverse problems for partial differential equations, springer, new york. [8] alifanov o m. (1994), inverse heat transfer problem, springer, berlin heidelberg. 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(1984), the one-dimensional heat equation, addison-wesley, california. corresponding author j. damirchi can be contacted at: damirchi.javad@gmail.com adv syst sci appl 2018; 4; 12-19 published online at http://ijassa.ipu.ru/ severity of breast masses prediction in mammograms based on optimized naive bayes diagnostic system abeer s. desuky faculty of science, al-azhar university, cairo, egypt e-mail: abeerdesuky@azhar.edu.eg abstract. mammography is the most effective tool for breast masses screening. it is a special ct scan technique used only to detect breast tumors early and accurately. detecting tumors in its early stage has improved the survival rate for breast cancer patients. computer aided diagnostic systems help the physicians to detect breast cells abnormalities earlier than other traditional procedures. in this paper, an improved naive bayes classifier based on chicken swarm optimization algorithm (cso-nbc) is analyzed on mammographic mass dataset. the main aim of this research is to increase physician's ability to determine the severity of a mammographic mass lesion from the bi-rads features and the patient's age using the bio inspired chicken swarm optimization (cso) algorithm for naive bayes classifier (nbc) improvement. the dataset is preprocessed and divided to train the cso-nbc system and test it by 5-folds cross validation technique. the performance of our proposed classification system is compared with papers' results of other researchers to show the efficiency of our system in predicting severity of breast tumors with the highest accuracy. keywords: breast masses mammography, naive bayes, chicken swarm optimization, computer aided diagnosis. 1. introduction cancer is one of the top leading causes of death. it is caused by uncontrolled growth of cells which invade and spread around the body, often resulting in death. cancer is the second leading cause of death globally, and was responsible for 8.8 million deaths in 2015 [1]. globally, nearly 1 in 6 deaths is due to cancer according to the who (world health organization). per the us national cancer institute, 60 percent of the world’s new cancer cases happen in asia, africa, central and south america, and 70 percent of global cancer deaths occur in those same regions as well. moreover, breast cancer death-rate was 571 000 equivalently 6.5 percent of all deaths in 2015 [2]. because of this fact, early detection and true diagnosis is an important issue and plays a key role to reduce mortality of this disease. mammography is an efficient imaging appliance for early breast cells abnormality detection. mammography exists in two types: first type, screening mammography which is breast x ray used to test the changes in breast area for early detection of breast tumors in women with no symptoms of cancer. it can also detect tiny calcium deposits (micro calcification) that is one of cancer manifestations. the second type, diagnostic mammography which is a breast x ray to check the signs of breast cancer after detecting a mass or other symptom. symptoms include breast size/shape change, skin thickening, pain or nipple discharge [3, 4]. significant improvements can be made in the lives of breast cancer patients by detecting cancer early and avoiding delays in care, computer aided diagnosis (cad) can help the 13 a. s. desuky copyright ©2018 assa. adv. in systems science and appl. (2018) physicians to do this task faster and more accurate than traditional procedures. during the last decade with development of machine learning approaches in diagnostic systems, breast cancer detection has improved. the aim of using machine learning approaches is to minimize mistakes that may occur by specialists in diagnosis [5]. different methods have been proposed to detect and classify masses in mammogram images. ramani and vanitha [4] used weighted histogram algorithms to select features from mammogram data then classified the selected features using random forest, naive bayes and ann algorithms. another method proposed in [6] based on missing values imputation and three models svm with polynomial kernel, ann with pruning parameters and dt with chi-squared interaction detection were derived for prediction. bio-inspired swarm algorithms can be used to enhance effectiveness of cad systems. this paper presents the bio inspired chicken swarm optimization (cso) algorithm to improve naive bayesian classifier (nbc) for the severity of breast masses prediction in mammograms. 2. the proposed method 2.1. chicken swarm optimization chicken swarm optimization (cso) algorithm proposed by meng et al. [7], is a bio-inspired swarm algorithm which mimics the behavior of chicken swarms in nature. each chicken swarm contains several groups of one rooster with two or more hens and many chicks move in a hierarchy groups and searching for food which has an effect on the swarm movements. dividing the chicken swarm into groups and identifying the chickens into roosters (r), hens (h) and chicks (c) mainly depends on the fitness values of the chickens. the chickens with best fitness values would be the roosters. chicks would be the chickens with worst fitness values while the others would be the hens (mothers). these statuses are updated according to the change in the fitness values only every several (g) time steps. there are three basic movements in the swarm as proposed in [7]: first movement is rooster’s movement which has the better fitness values since it can search for food in a wider range than the rest of chicks with less fitness values, the movement of the rooster is defined in eq. 1 and eq. 2: (1) where x is the selected rooster, is a gaussian distribution with mean 0 and standard deviation , and f is the fitness value of the corresponding rooster x. (2) where k is rooster index, i is the related position and ε is the smallest integer number in the computer used to avoid zero-division-error. second movement is hen’s; hens follow the roosters to search for food and their movement is defined in eq. 3 – eq. 5: (3) (4) (5) severity of breast masses prediction in mammograms 24 copyright ©2018 assa. adv. in systems science and appl. (2018) where r1 and r2 are the indices of the rooster and the randomly chosen chicken (rooster or hen), r1 ≠ r2, and rand is a uniform random value between [0, 1]. third movement is the movements of the chicks which follow their mothers, defined in eq. 6: (6) where xm,j is the position of the chick’s mother, and l is a randomly chosen parameter between 0 and 2. 2.2. naive bayesian classifier the bayesian classifier represents a statistical classification algorithm as well as a supervised learning method for classification based on bayes theorem [8]. assuming a probabilistic model, in bayes theorem, posterior probability of a data sample x with unknown class label is calculated as in eq. 7. (7) where each data sample is represented by an n-dimensional feature vector, x = (x1, x2.... xn) and h is some hypothesis, such that the data sample x belongs to a specified class label l1, l2.... lm. the nb classifier will predict that the given data sample x belongs to the class label li that have the higher posterior probability, i.e. for p (li | x) is the maximum [9]. for 1 ≤ j ≤ m (8) 2.3. the dataset researchers proposed several (cad) computer aided diagnostic systems in the last few years. these systems help doctors in deciding to perform short term follow-up examination or proceed a breast biopsy on a suspicious lesion seen in a mammogram instead. table 1. mammographic mass dataset features feature type values and labels missing values bi-rads ordinal (nonpredictive) 0 assessment incomplete 1 negative (non-predictive) 2 benign findings 3 probably benign 4 suspicious abnormality 5 highly suggestive of malignancy 2 age integer 18 96 (patient's age in years) 5 shape nominal 1 round 2 oval 3 lobular 4 irregular 31 margin nominal 1 circumscribed 2 microlobulated 3 obscured 4 ill-defined 5 spiculated 48 density ordinal 1 high 2 iso 3 low 76 15 a. s. desuky copyright ©2018 assa. adv. in systems science and appl. (2018) 4 fat-containing severity binominal (goal field) 0 benign 1 malignant 0 bi-rads (breast imaging reporting and data system) is developed to standardize nomenclature between radiologist and doctors. bi-rads assessment is done based on shape, age, margin density. bi-rads features and the patient's age in mammographic data set can be used to predict the severity of a mammographic mass lesion. this data set contains the patient's age, a bi-rads assessment, and three bi-rads features together with the class feature (severity) for 445 malignant and 516 benign masses that have been collected and identified on full field digital mammograms at the institute of radiology of the university erlangen-nuremberg between 2003 and 2006. table 1 shows that each instance has bi-rads features, age, shape, margin, density and severity features. mammographic data set includes 162 missing values [6, 10]. 2.4. naive bayes based on chicken swarm optimization traditional nbc is a statistical classifier with mutual independency among the features [4]. in the medical domain, all the symptoms do not contribute equally in diagnosing a specific disease. for example, in heart disease domain, the bmi feature is having less impact than the prior-stroke feature in predicting the probability of being heart patient [9]. in our proposed work called csonbc we assign weights to each feature according to their efficiency in prediction. we use cso technique to learn the features weight in nbc. no information or assumptions are needed about the weight in nbc, the mechanism of cso can help us get automatically the optimal weights. fig. 1 cso-nb classification technique we use the weight to increase the classification performance of the nbc. a high weight will be assigned to features that have more impact in prediction and low weight are assigned to features that have less impact. so, our main aim is to get optimum weight configuration that gives the highest accuracy for cso-nbc. the major steps of the working technique are shown in figure 1 and described as follows: severity of breast masses prediction in mammograms 26 copyright ©2018 assa. adv. in systems science and appl. (2018) first step: the mammographic mass data is preprocessed divided into training and testing sets then, the nbc applied on it firstly to measure the nb classification performance. second step: an initial generation for weight individuals (chickens) is formed in a random manner. each chicken swarm is a vector of all weight variables ranging from -10 to 10. for the weight individuals, the generation size n should be determined first. third step: the total generation cost of each chicken is calculated. fourth step: generate new rooster around the global best swarm by searching for the best fitness index and the worst fitness index of chicken swarm in same iteration and store their indexes; then, cso-nbc replaces the worst index with the best index. fifth step: third and fourth steps are repeated till get the best individual weight which is the one getting the highest classification accuracy. last step: the test instances can be classified under the learned weight by cso-nbc and the performance measures can be determined. 3. experimental results the effectiveness of any diagnostic system is evaluated by its ability to give the maximum accurate classification result [11]. per the real nature for any given case and the prediction from diagnostic system, there are only four possible outcomes, true negatives (tn) as well as true positives (tp) correspond to a correct diagnosis; that is, cases are successfully labeled as uninfected and infected patients, respectively; false positives (fp) refer to uninfected patient being classified as infected; false negatives (fn) are infected patients incorrectly classified as uninfected. the most popular performance metrics is accuracy which can be calculated as: n n n (9) to evaluate our system (cso-nbc), besides the classical accuracy, the standard metrics of sensitivity, g-mean (geometric mean), and f-measure have been used. n (10) (11) 2 2 n (12) a diagnosis system should have a high accuracy, sensitivity, g-mean and f-measure [4,12]. he experiments were implemented on a computer with intel core™ i7 processor and 6gb ram running microsoft windows 7 professional and the algorithm is coded in matlab 15. first, we replaced all the missing features values substitute it with the average in the mammographic mass dataset then, applied information gain (ig) feature selection technique [13], which is used to evaluate the importance of a feature by measuring the information gain (entropy) with regard to the class to remove useless features and improve the classification performance. the instances are randomly divided into training and testing sets in a 5-folds cross validation manner. the used parameters in evolution were: 100 for population size, 1000 generations (iterations) and time steps (g=10). tables 2, 3, 4 and 5 show the performance evaluation: classification accuracy, sensitivity, gmean and f-measure respectively of the nb classifier and the proposed technique using the full 17 a. s. desuky copyright ©2018 assa. adv. in systems science and appl. (2018) mammographic mass dataset features and the subset of selected features after applying information gain technique and removing “density” feature. the results show a perceivable improvement in classification performance for cso-nbc and show that with ig feature subset selection our proposed technique enhanced the classification performance against the enhancement with nb classifier. table 2 classification accuracy table 3 sensitivity nb cso-nbc full data 70.16 89.52 reduced data 87.57 82.94 table 4 g-mean nb cso-nbc full data 78.81 82.63 reduced data 85.11 87.37 table 5 f-measure nb cso-nbc full data 77.93 85.80 reduced data 86.48 87.36 table 6 comparison of algorithms applied on mammographic mass data method accuracy proposed cso-nbc 87.20 nb [5] 82.49 svm [6] 81.25 bagging svm-smo [9] 82.00 bagging dt [9] 83.40 the experiments also showed table 6 -that the proposed algorithm cso-nbc outperforms algorithms proposed by other researchers applied on the mammographic mass data. 4. conclusion mammography is used to help in the early detection (diagnosis) of breast diseases. the computer aided diagnostic systems can help physicians in detection (diagnosis) of nb cso-nbc full data 78.87 83.88 reduced data 85.42 87.20 severity of breast masses prediction in mammograms 28 copyright ©2018 assa. adv. in systems science and appl. (2018) abnormalities early than traditional procedures. many algorithms and diagnostic systems have been developed recently. the objective is not to replace medical researchers and professionals, but to increase their abilities to take decisions about the disease. this paper proposed a new automated mammogram classification system cso-nbc based on the improved naive bayes classifier using chicken swarm optimization algorithm. the mammographic mass data is preprocessed by replacing all the missing features values and applying information gain (ig) feature selection technique to remove the useless features. the dataset then divided to training and testing sets using 5-folds cross validation technique. the experiments showed the efficiency of our proposed classification system in predicting severity of breast tumors since it performs better than the traditional nb classifier also, better than algorithms proposed by other researchers for the same mammographic mass data. references [1]world health organization (2018, may) cancer [online]. available: http://www.who.int/cancer/en/ [2] world atlas (2018, may) explore the world [online]. available: http://www.worldatlas.com [3] sickles, e. a., wolverton, d. e., & dee, k. e. (2002). performance parameters for screening and diagnostic mammography: specialist and general radiologists. radiology, 224(3), 861869. [4] ramani, r. & suthanthira vanitha, n. (2014). computer aided detection of tumors in mammograms, int.j. image, graphics and signal processing, 4, 54-59. [5] güzel, c., kaya, m. & yıldız, o. (2013). breast cancer diagnosis based on naïve bayes machine learning classifier with knn missing data imputation, awerprocedia information technology & computer science, 4, 401-407, [online]. available: www.awer-center.org/pitcs [6] mokhtar, s. a., elsayad, a. m. (2013). predicting the severity of breast masses with data mining methods, international journal of computer science issues, 10(2). [7] meng, x., liu, y., gao, x. & zhang, h. (2014). a new bio-inspired algorithm: chicken swarm optimization. in: tan y., shi y., coello c.a.c. (eds.) advances in swarm intelligence. icsi 2014. lecture notes in computer science, vol 8794. springer, cham, https://doi.org/10.1007/978-3-319-11857-4_10 [8] ozer, p. (2008). data mining algorithms for classification, b.sc thesis, redbound university nijimegan. [9] jeyarani, d. s., anushya, g., raja, r. & pethalakshmi, a. (2013). a comparative study of decision tree and naive bayesian classifiers on medical datasets, international journal of computer applications (0975 – 8887), international conference on computing and information technology (ic2it-2013), 5-7. [10] luo, s.-t., cheng, b.-w. (2012). diagnosing breast masses in digital mammography using feature selection and ensemble methods, journal of medical systems, 36(2), 569-77. https://doi.org/10.1007/s10916-010-9518-8. [11] el bakrawy, l. m. & desuky, a. s. (2015). a hybrid classification algorithm and its application on four real-world data sets, international journal of computer science and information security (ijcsis), 13(10): 93-97. [12] han, j. & kamber, m. (2001). data mining concepts & techniques. san francisco, ca, usa: morgan kaufmann publishers inc. 19 a. s. desuky copyright ©2018 assa. adv. in systems science and appl. (2018) [13] kumar, c. s. & sree, r. j. (2014). application of ranking based attribute selection filters to perform automated evaluation of descriptive answers through sequential minimal optimization models, ictact journal on soft computing, 5(1), 860-868. advances in systems science and application (2015) vol.15 no.3 267-278 modern changes of the climatic conditions and rhythmicity of the long-term oscillations of the mode parameters of water bodies as the integral index of the climate fluctuation(based on the urals example) nadezhda s. rasskazova1, aleksandr v. bobylev1, michail n. bubin2 and alexander v. malaev3 1south ural state university russian federation 454080, chelyabinsk lenin v.76. 2yurginski technological institute (branch) of the national research tomski polytechnic university 652055, russian federation, kemerovo region, yurga, st. leningradski, 26. 3 chelyabinsk state pedagogical university, russian federation, 454074, chelyabinsk, st.bazhova 48 abstract the mechanism of how global factors of terrestrial and extraterrestrial origins influence the natural objects is very complicated and has not been fully studied yet. the majority of scientists think that the lunar and solar tidal forces produce an effect on the water surface of the sea, making the ocean currents pulse under certain rhythms. their pulsation causes changes in the global moisture and heat transfer on the earth. as a result, there appear complex rhythmic oscillations of the planets climate and inland waters. this article represents an analysis of the reasons for the modern climate fluctuations, including those at the local level (the urals region). keywords rhythmicity of the long-term river flow oscillations; dependence on cosmo and geophysical factors; prevailing atmospheric circulation type c (longitudinal); rossby waves 1 introduction determining the rhythms of natural phenomena and their reasons represents one of quite important tasks of the modern science. rhythmicity is the major parameter of the long-term oscillations of the natural processes in time and space. the major rhythms that determine the development character of the natural phenomena are the cosmic cycles. the cosmic rhythms of various durations and different origins permeate through and regulate every development process of the earth. the authors have found out that the reasons for the modern climate fluctuations in the urals, along with the major rhythms, are the longitudinal atmospheric circulation type c, which prevails in the zauralie area, and the rossby waves. to determine the major rhythms, the authors used fourier analysis and the difference integral curve method. to reveal the connection between the long-term oscillations of the hydrological characteristics and the cosmoand 268 nadezhda s. rasskazova, aleksandr v. bobylev, michail n. bubin and alexander... geophysical factors which represent a reflection of the climate fluctuations, cluster analysis was used. the accuracy of the research results was estimated using the mathematical statistics methods. 2 material & methods there are two essentially different rhythm categories in nature: the limited set of the cosmic rhythms and the unlimited number of the derivative interaction rhythms or environment rhythms. unlike the interaction rhythms which appear in particular environments and in limited territories, the cosmic rhythms reveal themselves, with amazing stability, in every geocomponent and geosphere. according to their duration, rhythms are divided into centuries-old and interdecadal ones. among the centuries-old rhythms, those with durations of 120, 300, 600, and 1,200 to 2,000 years are the most often found in nature. but it is not only natural phenomena that obey the rhythmicity laws. l.n. gumilev [1] noticed that even ethno-social processes taking place on earth are also subject to rhythms, and e.v. maximov [2] found that the passionary shocks described by gumilev [1] , which lead to disappearance of some nations and birth of some others, are connected with the cosmic rhythms, and in particular with one of the most important cosmic rhythms with the duration of about 1,800 years. on the shores of the picturesque lake zurich there are two ancient terraces: high cliffs where, in the rock strata, once can clearly distinguish the strata of different eras. in these stratified rocks, scientists found some very clear evidence of the 1,800-year rhythm. the same rhythm was also determined in the sequence of the oozy deposits, the glacier movements, humidification oscillations, and after all in the climate fluctuations. of course, we cannot observe any rhythms of such durations in that short life we live. however, there are a whole series of space-generated interdecadal rhythms the effects of which can be felt during our lifetime: 80 to 90, 28 to 35, 21 to 22, 15, 11 to 13, 5 to 8, and 2 to 4 years. these are the interdecadal rhythms. the interconnection between the terrestrial and cosmic phenomena becomes the most evidential if based on the example of the rhythms occurring in the physical and geographical processes recorded by means of instrumental observations which are introduced in table 1. today, the overwhelming majority of researchers have no doubt that rhythmicity is an integral natural law of the earths geosphere, and that its rhythmical fluctuations are caused by the cosmic rhythms. the cosmic rhythms give rise to rhythms in the geosphere, similar to themselves, while derivative rhythms produced by their interference cause accidental fluctuations. rhythmical oscillations have been found in the solar activity, the earths magnetic field, atmospheric precipitations, air and water temperature, tides, and advances in systems science and application (2015) vol.15 no.3 269 many other natural phenomena. they are conditioned by three groups of factors: astronomical, geophysical and circulating. table 1 average annual indices reflecting the connection between the long-term oscillations of the annual river flow and the cosmoand geophysical factors dendro-gram number index factors represented by the index symbol unit of measure 175 relative numbers of sunspots (wolf numbers) solar activity w n/a 195 troposphere-effective indices of solar activity (according to loginov) t 186,187 19-year constituent of the lunar and solar tide-generating force potential at 56◦ and 60◦ north polar tide f19 cm2/sec2 188 average annual values of the day length fluctuations earth axis nutation △t of 0.00005 sec 176,177 longitudinal and latitudinal coordinates of the icelandic low activity of pressure formations φ,λ degree 178 soi (southern oscillation index) intensity of ocean currents isoi mb 179 nao (north atlantic oscillation index) inao mb 185 longitudinal circulation indices in the northern atlantic (january) rn mb 189,190 water temperature anomalies in the northern atlantic in smed squares d and e △t◦ degree 180,181 annual number of days with cyclones in the 4th and 8th vitels zones global atmospheric circulation z4, z8 days192 recurrence of the western type a.c. w 193 recurrence of the eastern type a.c. e 194 recurrence of the longitudinal type a.c. c 184 annual anomalies of the surface air temperature in the arctic underlying surface humidification long-term river flow oscillations in the urals t degree 182,183 aridity index dm (etp,atp) dm n/a 197-199,301-317 humidification coefficient chumid n/a q m3/sec 270 nadezhda s. rasskazova, aleksandr v. bobylev, michail n. bubin and alexander... the oscillations of the hydrological parameters of water bodies (river flows and levels) are taken as the integral index of the climate fluctuations. the mechanism of how global factors of terrestrial and extraterrestrial origins influence the natural objects is very complicated and has not been fully studied yet. the majority of scientists think that the lunar and solar tidal forces produce an effect on the water surface of the sea, making the ocean currents pulse under certain rhythms. their pulsation causes changes in the global moisture and heat transfer on the earth. as a result, there appear complex rhythmic oscillations of the planets climate and inland waters. the fluctuations of the solar activity, in its turn, is connected with the gravitational effect of the planets in the solar system and with the effect produced on it by other space bodies: comets, asteroids and meteorites coming to us from outer space and breaking the abovementioned rhythms. in particular, the average rhythm of meteoric impacts on the earth is about 80 years[3]. as for the nature of various cycles, scientific literature still introduces quite contradictory hypotheses. the coincidence in the durations of the cycles of specific phenomena is normally considered to be the major argument for the existence of some connections between the external factors and the oscillations of the hydrometeorological elements. but this kind of approach is one-sided and insufficient to make statements on the existence of those connections and to reveal their nature. 3 results so we spread our research both in the time and space aspects. as it was said above, we used the fourier analysis and difference integral curves to determine the rhythms, and cluster analysis was used to reveal the connection between the natural phenomena and cosmic rhythms. the research resulted in the map (fig. 1) that the author made using the cartographic method. according to the studies that have been performed [4] the full 80-year solar rhythm can be observed in the series with execution lengths of over 100 years, such the wolf numbers, the nao (north atlantic oscillations) indices, the soi (el niño southern oscillation indices), and others. it means that all the rhythms are the reflection of the solar rhythms and have the cosmic origin. but unfortunately, their accuracy is so low that determining such a rhythm requires series of 250 years (3 x 80) and more. the map (fig. 1) shows that the rhythm of 80-90 years (the full solar rhythm), the rhythm of 28-35 years (the brückner cycles), as well as others, are actually cosmic rhythms which should be considered as inherent features, not only of the entire planet but also of the urals in particular. the duration of every rhythm gets longer southwards; the rhythms of 28-35 advances in systems science and application (2015) vol.15 no.3 271 and 80-90 years are no exception. in the north, the duration of the brckner cycle in the flow of the rivers in the mountain taiga areas is 28 years (7%-14% of the total dispersion of the characteristic), while in the south it increases up to 34-35 years. fig. 1 schematic map of the major rhythms in the long-term oscillations of the river flow in the basin of the tobol river. as for the 80-to-90-year rhythm, it increases up 90 (3 x 30) years southwards. the difference integral curves also prove the presence of the abovementioned rhythms in the flow of the rivers in the urals. the difference integral curves, an example of which is represented on fig. 2 below, show that there are rhythm 272 nadezhda s. rasskazova, aleksandr v. bobylev, michail n. bubin and alexander... elements with durations of about 35 and 80 years. as it was said above, those rhythms are of global character and are typical of the entire geosphere. fig. 2 difference integral curves of the long-term oscillations of the river flow in the basin of the tobol river. rhythms of 30 to 35 years have been found in the geological deposits, in the growth of the annual rings, which is related to the humidification dynamics, and in the spectrum of the air haziness caused by volcanic eruptions. in the hydrosphere, the 30-year periods have been found in the oscillations of the water level of the caspian sea. at the same time, we have to mention that the long-term river flow osciallations are inert to the cosmoand geophysical factors. the delays of the long-term flow oscillations are caused by the peaks of the solar activity and the large-scale processes taking place in the atmosphere and hydrosphere. as for the modern climate fluctuations, the authors have determined three reasons for the climate fluctuations occurring in the southern urals this years: (1)the prevailing longitudinal atmospheric circulation type in the zauralie area is c; (2)the 35-year rhythmic oscillations (the brückner cycles) that depend on the cosmic rhythms; (3)rossby waves. 4 discussion 1. the longitudinal air mass transport in the zauralie area is the one that prevails, for the ural mountains are located longitudinally and the west siberian plain is open to the invention of the arctic air masses, and it takes them about 6 hours to reach the southern urals. if the process is directed southwards, it causes sharp temperature falls, and if it goes northwards, then in wintertime it gives rise advances in systems science and application (2015) vol.15 no.3 273 to warm air masses with large amount of precipitations. in the flat land it leads to thaws and snowdrifts and to heavy snowfalls, avalanches and mudflows in the mountains. lately, the frequency of the longitudinal processes has been growing, which leads to weather anomalies: extremely high and low air temperatures, heavy rains and snowfalls, and longer droughty periods. as for the territory of the region, in summertime low air pressure prevails there. arctic air masses come there from the barents sea and kara sea, while tropical air masses travel from the south, from kazakhstan and central asia. when continental tropical air comes, hot and dry weather sets in. western winds from the atlantic ocean bring humid and changeable weather. representative for this region is the period comprising several a.c. epochs. to explore the effect produced by the cosmogeophysical factors on the climate fluctuations in the region, we used the values of the water discharge and river level variations. a period when the longitudinal circulation prevails (c epoch) was chosen as the period of the study. the research was carried out using the cluster analysis. the analysis results were represented in the form of a dendrogram (fig. 3). the optimal connectivity level was taken as 0.3 (according to the connectivity criterion [4]), where 8 clusters are determined. the first cluster at the level of 0.3 is formed by the rivers of the forest and forest-steppe zones of the territory that was explored, with the probability of the connection between the objects (rivers) of 85% to 99%. the second cluster consists of two sub-clusters combined at the level of 0.3. the first sub-cluster is represented by a group of rivers of the urals forest-steppe and mountain taiga zones situated at approximately the same latitude. they become combined at the level of 0.7, with the connection probability of 94% to 97%. the second sub-cluster is represented by the rivers of the zauralie and preduralie areas (forest and forest-steppe zones; r=0.8, p=99,9%), which proves the synchronism of the long-term oscillations of the river flows in the basins of the rivers kama and tobol during the c epoch. at the level of 0.6, this sub-cluster is joined by the index of the water temperature anomalies in the north atlantic (smed square d, the southwestern part of the north atlantic), which proves the interrelation of the processes occurring in the hydrosphere and atmosphere. and with the probability of the connection between the objects of 88% to 93%, we can suppose that there are also some other factors producing their effects on the long-term river flow oscillations. the next cluster also consists of two sub-clusters. the first sub-cluster comprises three factors: tide-generating force index (at φ=60◦). wolf numbers and the index of the numbers of cyclones in the zauralie area (z8) (r=0.7, p=94% 95%). their combination proves the close interconnection between the cosmic 274 nadezhda s. rasskazova, aleksandr v. bobylev, michail n. bubin and alexander... factors and the processes occurring in the atmosphere and hydrosphere. in the second sub-cluster, we observe a correlation between the tide-generating force index (at φ=56◦) and the air aridity index at (r=0.7, p=99%), which proves that the abovementioned conclusion is right. the sub-clusters get combined at the level of 0.3. the connection probabilities between the objects in the sub-clusters exceed 95%, except for the connection between the indices z8 and dm (p=83%) and between z8 and f19 (at φ=56◦) (p=85%). fig. 3 dendrogram of the connection between the river flow and the cosmoand geophysical factors during the c epoch the c epoch also reveals a considerable connection between the intensity indices of the ocean currents, the soi and nao indices (r=0.6, p=97%). at the level of 0.6, we see the indices of the number of cyclones (z4) at etp (the preduralie area) combine with those of the process recurrence of the w form (p=85%). at the level of 0.3, they get reunited with the soi and nao indices. the connection probability between the soi and nao indices and the recurrence index of the western form of the atmospheric circulation is quite considerable (94% 95%), which proves the fact of their close interconnection. but the accuracy of the connection between the soi and nao indices and z4 is not high (p<80%). the recurrence value of the eastern form processes of the atmospheric circulation (e) correlates with the index of the water temperature anomalies in the northeastern part of the north atlantic, which proves yet again that the there is advances in systems science and application (2015) vol.15 no.3 275 a connection between the processes in the atmosphere and hydrosphere. in the c epoch, the recurrence of the processes of this form depends on the position of the longitudinal center of the icelandic low (λicl.low), which is proved by the combination of those indices at the level of 0.5 with the connection probability of 94%. the latitudinal coordinate of the icelandic low (φicl.low) correlates with the longitudinal circulation index in the north atlantic and the troposphereeffective index of the solar activity (the connection probabilities of 93% and 97% relatively). this allows representing the interrelation chain as follows: solar activity →...→ sea level oscillations →...→ large-scle atmospheric processes →...→ annual river flow moreover, in the dendrograms of the other periods under study, we found stable and reliable connections within the cosmoand geophysical factors themselves and also between them and the long-term oscillations of the annual river flow: (1)the index of the number of cyclones in the zauralie area (z8) correlates with the solar activity (w); (2)the latitudinal value of the center of the icelandic low (φicl.low) correlates with the nao index and the recurrence of the w form processes; (3)the recurrence of the c form processes correlates with the long-term changeability of the annual river flow of all the natural zones, except for the rivers situated in the area of the stovas critical parallels (59◦ 62◦), where the major influence is produced by the w form. hence, according to the results of the studies, the chain of the interrelations between the cosmoand geophysical factors and the long-term river flow oscillations, represented in the most detailed way, looks as follows: solar activity → large-scale atmospheric processes → ocean currents → number of cyclones (z) and anticyclones (az) → air humidification (aridity indices) → annual river flow. 2. along with the average climate conditions and the abovementioned climatic cycles, the peculiar thing about the climatic regime is its variations. the modern period is characterized by a global temperature trend. it is subject to interannual and decadal (with characteristic rhythmicity of about a decade) variations. if we consider the period of instrumental observations only, we can state that finding the planetary average appears more reasonable. but if we excluded, from our consideration, the 19th century instrumental observations as the least reliable, the modern 100-year trend would totally disappear. instead, we could only speak of the temperature growth that started in the 1980s and in the first decade of the third millennium and consider it an example of the positive temperature 276 nadezhda s. rasskazova, aleksandr v. bobylev, michail n. bubin and alexander... anomaly of the 1940s warming type. at the same time, it would be reasonable to distinguish changeability from changes. changeability is understood as variations around some average condition, but changes is what characterizes a trend, a transition to a brand new state, with the changes occurring both to the average regime itself and the changeability condition. this differentiation is methodologically convenient, but the physical value of this approach is doubtful, for a lot of things depend on the time scale taken into consideration: the same phenomenon may act both as changeability and changes if viewed from different time scales. as appears on fig. 1, the major rhythms in the zauralie area are those of about 13.8, 19 and 35 years. in 2014, we could observe the 35-year brckner cycle (it may vary from 28 to 35 years; it is closer to 35 years in the zauralie area). the authors have determined this rhythm as one of the major ones. a similar abnormally cold summer was observed in the southern urals in 1980. according to the scientists forecast, the next anomalous summer is expected in 2050, and climatologists even promise us a minor ice age then. hence, the rhythms of nature represent a result of the joint effect produced on it by the space and geophysical factors. 3. not so long ago, scientists found a correlation between the anomalous weather phenomena[5]: the hardest frosts[6] in the past 20 years in the usa (50 below zero c◦) and the thaws at the beginning of the year in russia. the reason for the climate changeability on the opposite sides of the earth is the rossby waves. they may occur in different seasons (k. nyullis, press secretary of the world meteorological organization). last summer, during a two-month period, there was a blocking anticyclone “hanging” over russia. such blocking anticyclones are typical representatives of a natural phenomenon called rossby solitons (spatial solitary waves). rossby waves are air waves which are generated in the atmosphere at moderate altitudes and are moving from the east towards the west. they participate in the formation of cyclones and anticyclones. the atmospheric rossby waves appear in the atmospheres of planets as a result of the shift in the vortices due to the effect of the coriolis forces during latitudinal variations of the area. such waves were first determined in the earths atmosphere in 1939 by carl-gustaf arvid rossby. such formations are not unique; they are found at different levels of the organization of matter, and in particular, the rossby soliton is exactly what determines the spiral structure of the galaxies. the most striking and best-known rossby soliton example is jupiters great red spot which has been observed for more than 300 years [7]. in theory, these waves had been known since the late 19th century. back in 1893, the academy of sciences of vienna published an article by max margules, where the author stated that in a rotating liquid or gaseous environment there advances in systems science and application (2015) vol.15 no.3 277 appear drift waves caused by the global rotation of the environment where they are developing. a little later, in the works by the german geophysicist and oceanologist bernhard haurwitz, those waves were considered as an additional element of the earths ocean. but it was in 1939 when the swedish geophysicist carl-gustaf rossby became the first to draw special attention to the extremely important role played by those waves in the global atmospheric circulation. since then those drift waves, as well as the global vortices they cause, have been named after him. the oceanic rossby waves cause fluctuations of the height of the sea surface. it had been difficult to determine the wave lengths before satellite altimetry was invented. the observations made with the nasa/snes topex/poseidon satellite proved the existence of the oceanic rossby waves. the satellite observations showed the majestic progress of rossby waves in every ocean basin, at low and middle latitudes in particular. the waves may live for months or even years, so that they cross the pacific ocean. rossby waves have also been considered as an important mechanism to monitor the warming of europas ocean. most probably, rossby waves also participate in the generation of magnetic fields in nature and are capable of causing local fluctuations of the condition of the lower troposphere. 5 conclusions according to our studies, the following phenomena accumulated in the zauralie area in summer 2014: a) recurrence of the brückner cycle with the duration of 34 years (1980 2014); b) intensification of the longitudinal shift that prevails in the zauralie area and is caused by el niño (rossby waves are the “conductor” of el niño); c) the shift being blocked in the zauralie area by a rossby wave; d) “hanging” of the weather conditions caused by el niño [8]. thus, the rhythms observed in nature are the results of the effect produced on it by the cosmogeophysical factors. in particular, “interference” in this interaction between the geophysical factors (rossby wave) and celestial bodies in the solar system and outside it is able to “disturb” that interaction and lead both to local (zauralie) and global climate fluctuation, which has been observed more than once in the history of our planet. references [1] gumilev l.n. (1992), ethnogenesis and the biosphere of earth, oscow. [2] maksimov e.v.. (1995) rhythms on earth and in space, publishing house of saint petersburg state university. [3] rasskazova n.s. (2013), “rhythmic changeability of the regime characteristics of water bodies as the integral index of the climate fluctuation (based on 278 nadezhda s. rasskazova, aleksandr v. bobylev, michail n. bubin and alexander... the urals example)”, climate change and industrial city ecology, pp. 49-51. http://www.forum-ecology.ru/content/forum/catalog 2013/194 catalog ef 2013.pdf [4] bubin m.n. and rasskazova n.s. (2013), “rhythmicity of the long-term river flow oscillations as the integral index of the climate fluctuation (based on the urals example) ”(monograph by m.n. bubin, n.s. rasskazova of the institute of technologies in yurga), tomsk nrpu publishing house, 278 pages. [5] (2013), access mode : “the el nino phenomenon” [electronic resource]. http://nature.web.ru/db/msg.html?mid=1158162 [6] (2014), access mode: “anomalous frosts” [electronic resource]. http://okoplanet.su/pogoda/pogodaday/225659-ssha-anomalnye-holoda.html [7] rossby soliton, (2010), access mode:“jupiters great red spot and anomalous summer in russia” [electronic resource]. http://okoplanet.su/pogoda/listpogoda/48596-soliton-rossbi-bolshoe-krasnoe-pyatnoyupitera-i.html [8] rasskazova n.s., (2009), “el nino-southern oscillation signal depression and its possible reasons”,science & technology. theses of the reports of the 29th russian school dedicated to the academician v.p. makeevs 85th anniversary, miass, pp. 83. corresponding author nadezhda s. rasskazova can be contacted at: yal05@mail.ru advances in systems science and application (2016) vol.16 no.2 81-93 an artificial volition architecture for autonomous robotics t. e. raptis1 and k. karamanos2 1division of applied technologies, national centre for science and research “demokritos”, athens, 15341 greece. 2physics department, university of athens, zografou, gr-15784 abstract we introduce a new computational architecture capable of exhibiting an archetypal type of volition in a simplified modularized version of m. minskys hive-mind. in this model, three relatively independent computational cores which themselves can also be whole multi-agent systems are engaged in an endless interaction each one representing the internal “imaginative” world, the external world interface and the arbitrator or internal observer. volition then is expected to occur as the result of an endless antagonism for control between the internal and the external world models. keywords a.i., robotics, volition, free will. 1 introduction history of a. i. is marked with a division between two main abstractions, the one of “connectionists” that try to imitate directly the real brain functions closer to the spirit of old wieners cybernetics paradigm and “symbolists” who advocate the old belief that mind is algorithmic, first introduced by alan turing. both schools have had various successes in diverse fields but when it comes to the inner psychological or subjective experiences they face a major obstacle which is the correct objective definition and experimental verification of such abstract concepts as volition (exercise of will), consciousness and/or self-awareness. these problems also arose in the early phase of development of cognitive sciences and still constitute a major set of debatable issues at the heart of this research. questions concerning the possibility of externally verifying ones own awareness of a self and a personal identity have given rise to the argument of “philosophical zombies” against functionalist interpretations of the identity and awareness problem.although the particular architecture proposed seems to be unique, at least to our knowledge, there are several parts that may be comprised into the proposed structure and have already been proposed as separate items and implemented into various robotic platforms. a crucial property for self-aware agents is that of self-observation which requires a constant scan of both the history of the external agent behavior and the internal loop activity and their correlation. recent work from the cornell group[1-3] has revealed the possibility of an advanced reverse engineering of 82 t. e. raptis and k. karamanos:an artificial volition architecture for autonomous robotics complex environmental signals and accurate modeling of external reality. such models can then be contrasted with the internal personality model and affect future planning. furthermore, the agent can effectively randomize its own behavior to avoid stereotypical pre-programmed behavior by an internal mechanism which optimizes the antagonistic need to preserve the personality model and the environmental pressures by an adaptation protocol which affects the probabilities/weights of certain actions. attempts to introduce a “self-model” compatible with the idea of an “inner world” have been superficially carried out based on a 3 -dimensional mental space together with the so called “big five model” of psychology. previous work by the waseda group[4-6] on a 3-d mental space has shown promising results in control of a robotic face (emotional agent). this way we can build “idiosyncratic agents” with a certain predefined set of preferences. a second important step in the construction of intelligent agents is the previous work of luc steels on “fluid construction grammars” which led to the development of the talkingheads project. in this, a number of bots were sustained inside the internet moving in a network of host computers which were equipped with steerable cameras. they were then able to analyze optical signals and classify objects while developing a primitive linguistic structure through which they were communicating their experience into one another. the same model can also be utilized in an advanced volitional agent who would become capable of classifying both internal and external signals and thus reaching at level 3.1 or 3.3 of tables 1(a) (b) of the next section. it could also be used for an agent applying an artificial linguistic structure to purely internal variables. the present proposal attempts to provide an example of an advanced architecture that could encompass all previous developments and unify their separate approaches in a unique frame thus extending their capabilities towards an advanced agent design with truly inherent volitional attitude that would arise as a result of the internal dynamics of its major components. the presentation is based on a high level description of the whole architecture ignoring the details of the software implementation which in principle could be numerous. it is intended in giving a correct understanding of the foundations of an alternative volitional theory that could possibly be applicable in other fields as cognitive sciences and human psychology. in section 2, we lay the foundations of our model while in section 3 we describe their possible implementation in more detail using a top-down approach. in section 4 we conclude and also discuss the significance of learning as an additional concept of a higher level that was not absolutely essential in the previous development. advances in systems science and application (2016) vol.16 no.2 83 2 foundations of artificial volition adopting a practical approach, we choose to concentrate in the necessary and sufficient conditions for a behavioral evaluation of the existence of volitional attributes of an agent. present state-of-the -art in cognitive sciences allows one to write a generic test that can be applied to an arbitrary set of agents in order to discriminate between several levels of awareness as proposed in [1]. this is summarized in the table 1(a) below. table 1 (a) the awareness level awareness level discriminating question categorization 0 is it animate? alive/dead 0.1 “does the system move or act on its own, i.e., without obvious prompting by external forces?” autonomous/nonautonomous 0.2 “is the systems spontaneous behavior modified by events/conditions in the environment?” 1 ”does the system appear to be trying to approach or avoid any object or occurrence of an event in its environment” 1.1 ”does the system have different sets of goals active during different environmental or bodily conditions?” modal value-driven automaton/“pacman ghost” 2 “does the system develop new adaptive approach or avoidance patterns over time?” 2.1 “can the system engage in a task that requires working memory (e.g. delayed non-match-to-sample)?” 2.2 “can the system engage in a task that requires longterm memory?” 2.3 “can the system engage in a behavior(e.g. gameplaying, navigation) that requires evaluation of multiple possibilities without action?” most animals 3 “does the organism send and selectively respond to social cues? ” 3.1 “can the agent pick up and move around objects in its environment?” 3.2 “does the system communicate using language that has syntax as well as semantics?” chimpanzee some observations are due at this point. nowadays, robotic arms and even autonomous robots exist that are capable of fulfilling 3.1 although they are no more than modal automata thus we should move this question into level 1. secondly, the overall description of level 3 seems incomplete in that it does not explicitly shows true volitional acts in the absence of competition with a truly intelligent environment which includes also other intelligent agents either artificial or human. we thus propose the following modified table 1(b) for level 3.0. 84 t. e. raptis and k. karamanos:an artificial volition architecture for autonomous robotics table 2 (b) the modified awareness level 3 “does the organism send and selectively respond to social cues?” 3.1 “does the agent attempts to escape captivity enforced by an external agent or situation?” 3.2 “is the agent capable of enforcing a certain task to other less intelligent agents?” 3.3 “does the system communicate using language that has syntax as well as semantics?” we can now proceed to a separate examination of the three different levels. in fact we propose that the above hierarchy is not strictly necessary and that certain properties may be intermixed at least in artificial agents. that is to say that we can in principle separate between different architectures or implementations of systems that could emulate several of the characteristics pertaining at different levels without strictly obeying the above hierarchy. in particular, we would like to make a crucial separation between consciousness as a purely subjective state and volition as a more primordial level necessary for its existence. in fact, it is not in principle possible to directly assert the existence of an internal “feeling” by the agent.it is only possible to assert the resistive actions taken by the agent against external obstacles or environmental “laws” that are not preprogrammed. next we concentrate on a mechanism capable of exhibiting volitional effects at the level of 3.1 without necessarily reproducing all of the characteristics of the previous levels. specifically, we seek for the minimal behavioral test that could verify the ability of an agent to exhibit an element of “free will” in the sense of a) either randomizing its own behavior in order to cope with a contradiction or a conflict between its own self model and its obtained world model, or b) undertaking evasive actions against an enforced captivity or restrain by another agent. we thus concentrate on the fact that the presence of any kind of “free will” is definitely asserted only in an antagonistic situation where the agents will is exerted against another agent as a resistive force. this assumption does not exclude a cooperative behavior which would only occur under a state of agreement or symbiosis between different agents. to explain the significance of the levels 3.1 and 3.2 we may also add a quote from a dialogue between c. zvosil and d. greenberger[8] where they criticize the incomplete nature of turings test with respect to free will. “· · · assume an artificial world and/or an artificial creature. assume further a super selection rule which should not be broken by any circumstances; eg eating from the tree of genesis. this creature should be termed intelligent if it breaks its super selection rule. in this approach it is evident that uncontrollability is a trade-off for intelligence and that it might be impossible to create a machine which is both intelligent and a advances in systems science and application (2016) vol.16 no.2 85 reliable server.” we choose to call this principle, a “non serviam” principle from the latin expression for denial of service. we will next show that it is possible to extract from the above general argumentation a full computing architecture that can fulfill the above principle. the significance of such a construct is that a) if it is possible than it will probably be realized in the future yielding new problems for security and reliability of services (friendly ai problem) and b) it seems to be more closely associated with human behavior than other mechanistic approaches. at first we will use a metaphorical example to lay down the description of the principles of the model. the basic obstacle in present state-of-the art is that the mind is still represented much like an operating system which waits for input from the external environment and then reacts to it according to a prescribed set of functions. what is clearly missing from all these approaches is the notion of a purely internal world with autonomous life independent of any external activity. such is the case of imagination but also of dreaming. the only way we can achieve a machine with a “dreaming state” is to endow such a construct with an internal autonomous and endless loop of which the variables are equally sensed by an “internal observer” (io) as well as the external response signals albeit not always on an equal footing. we thus present this general idea with the following schematic. in fig.1, the three modules should be interpreted as relatively independent computational engines of different characteristics and purpose. while the inner world (iw) is represented by an autonomous activity, like a huge dynamical system, the io module is more like an operating system which needs to be fed with inputs from both subsystems having the environmental variables fed by sensory systems and also form the interface of the machine with the outside world. each module may act as an independent agent in a continuous dialogue with the other two thus forming a tri-dialogical system. the dynamics envisioned behind this scheme can be described with the aid of fig.2. the io module acts like an independent agent “trapped” between the activities of the iw core which is fed to it through appropriate internal sensory inputs and those provided by the external interface. the io module then attempts an endless evaluation and arbitration task between the conflicting activities in order to control the appropriate responses that are also imposed by the third survival task (st) module which plays the role of an autonomic nervous system. in what follows we give precise meaning and some possible technical methods to implement the above logic. we of course assume the existence of an appropriate environmental interface for both input and response signals (eg. a full robotic body). 86 t. e. raptis and k. karamanos:an artificial volition architecture for autonomous robotics fig. 1 a tri-dialogical system fig. 2 a dynamic response process 1. the iw-module is a specially constructed closed computational loop which is to have access to some or all the external inputs not for processing but in the form of direct perturbations of its own internal variables. it is also prewired in a certain way with the io module in a way that allows for alterations of its internal structure -not just its variables!-imposed by the io module. (practically this means that the io module may alter even the form of the “equations of motion” of the internal dynamics). this is supposedly an endless internal activity. at advances in systems science and application (2016) vol.16 no.2 87 this point one may ask how we can guarantee that this will indeed be an endless computation without solving the halting problem! this will be further clarified into the next paragraph that fully describes the dynamics with the environment mediated by the io activity. at the moment it is important to accept the iw module as an isolated “dreaming” machine or a kind of “subconscious” in the overall architecture. 2. the st-interface module contains all such attributes that represent predetermined protocols for the survival of the agent in an arbitrary environment including input signal processing and automatic response circuitry. it is thus close to a model of the autonomic nervous system. 3. the io is by necessity the part of the machine responsible for the categorization and evaluation of the activities of both the internal and external environmental variables. it thus holds and updates a primitive self-model made by the subsequent observations of each ones activity. on the other hand, it also plays the role of an arbitrator between possible conflicting demands of the two separate iw and st agents in case of conflicting demands posed to the external response modules. in order for this scheme to exhibit full volitional attributes there is one more crucial step in the definition of the io module. we thus propose to introduce an “egotistic” attitude to the agent in the following sense. the fundamental property of “ego” is not to be just a self-model in the form of a data structure but also to be “demanding”. this is also evident from early childhood psychology. the only way we can give a precise technical meaning to this proposition is to introduce a certain kind of internal hyper-tasks attributed to the io module beyond the simple self-identification task. we thus attempt to derive such a hyper-task from a generalized functional control problem. we assume that the simplest hyper-task is the attempt by the machine to create a “higher self” model into which all or as many as possible of the external variables have been “internalized” or in fact “enslaved”. this is to be understood here in a somewhat more broad manner than a “master-slave” configuration is often realized in engineering. this is quite a broad concept that can extend even to systems with continuous variables. we interpret this internalization process as follows. assume a composite system, e.g. a neural network, which has several processing nodes and several other input nodes (fig.3). the processing nodes are to be considered as “immediately active” in the sense that they carry a certain processing capacity while the sensory nodes form an interface for some in principle unknown external nodes that represent the activity of the environment. thus all nodes f1, f2, · · · , fn represented by circles are “known” to the system while the nodes e1, e2, · · · , em represented by red squares may not even have a functional expression due to their extreme complexity. in a sense, although the agent may 88 t. e. raptis and k. karamanos:an artificial volition architecture for autonomous robotics possess a primitive notion of identity it still considers the rest of its world as an extension that can be manipulated. the systems dynamics executes the hypertask of trying to minimize a generalized distance between the functional forms of the external world source nodes and the internal nodes. thus the system should start to transform the external world continuously until it would be able to bring all or most of the external variables to a state in which they conform to the internal world dynamics. while this in actual reality is an open and endless task (considering the rest of the universe as the “environment”!) it is still an accurate representation of the origin of the agents volition. in fact, we propose that this is the only mathematically acceptable expression of an agents “free will”. an interesting consequence of the above type of dynamics that we can predict is that cooperation with such an agent would be possible through “enticement” which presupposes a degree of access or knowledge to its own internal variables that make up its “character” or self-model without sacrificing an amount of fluidity in it. conclusively, we may end this section with the following propositions. fig. 3 the internalization process of artificial volitional attributes proposition 1. a sufficient condition for the appearance of volitional attributes in a network of parallel turing machines is the existence of conflicting dynamics between computational tasks of different purpose that result in the formation of a master-slave configuration inside the network. proposition 2. a necessary condition for ascertaining the volitional attribute of an agent is the testability of its capacity to interpret restrictions as such even in the absence of any knowledge about their origin (“suspicious agent”) and seek more freedom in the transformation of a given environment. 3 possible implementation and dynamics a more concise and detailed description of the system units, their interconnection and dynamics follows. the design philosophy explained in the previous section advances in systems science and application (2016) vol.16 no.2 89 is further elaborated on the technical details of a possible implementation. design of the “inner world” model can be better described with the aid of the analog computing paradigm which is here simplified on a system of differential equations. these can be seen as an average over a multitude of microscopic degrees of freedom that could be realized by neuromorphic circuitry. we believe this to be true for natural biological circuits although we do not make strict use of neural implementations as this is unimportant to our purpose. let then x1,· · · ,xnin and x1,· · · ,xnout be a set of internal and external variables defined as follows : the internal variables form the basis of the inner loop dynamics while they are coupled to the set of external variables. the external variables are coupled to environmental signals and they originate at the st module where a first processing of the system inputs takes place. system response is due to both the internal dynamics of the st module as well as the supervising inputs from the io module which mediates between the iw module and the st module. the inner loop attempts to bring the set of external variables under its own control. this of course is an always incomplete task in the sense that the external variables obey an in principle unknown, non-stationary dynamics which may also have different underlying laws from a previous instant to the next. the philosophy behind this control scenario is that the inner loop attempts not just to identify the external variables dynamics but it tries to “enslave” them in order to follow its own internal differential equations. that is to say, the inner loop attempts to internalize the external world. it is possible to borrow from nature the additional well known biological principle of antagonistic signals. for this, we would have to define pairs of antagonistic variables. one can utilize for example the well known model of competitive lotka-volterra equations to simulate the inner loop dynamics. the choice of the system is somewhat arbitrary and has been chosen for having stable attractors. one could in principle try other choices or a direct neural implementation. even a spiking network could have been used but at the cost of an increased complexity. in fact, we may assume the continuous version of a pde system or equivalently the corresponding pseudo-spectral kernel by which the iw module builds the subsequent configurations of its own internal state. assuming an appropriate sampler interface between the io and the iw modules, the io “sees” snapshots of the iw activity the same way it sees the external world through the additional interface of the st module. moreover, the inner loop module should be coupled to a set of supervising signals from the io module that affect a matrix of coefficients for the ode/pde system that defines the iw loop dynamics. the inner loop dynamics may be made to obey a 1st order hyper-task. this 90 t. e. raptis and k. karamanos:an artificial volition architecture for autonomous robotics is described by an internal update process of the system coefficient matrix or of the corresponding kernel parameters towards the production of more coherent and symmetric patterns of activity (artistic agent). this is also influenced by the perturbations from the external variables so that an antagonistic dynamics between the inner and the outer world models is established. the criteria of what constitutes a like pattern are prewired into the particular agent and form an integral part of its “character”. (martian spiders for example may have a very peculiar idea of what constitutes a beautiful face!) the io module is primarily responsible for a 2nd order hyper-task of reducing the functional distance between the iw and the outer world as it appears to the interface variables provided by the st module. this 2nd order hyper-task is eventually linked to the 1st order one through the attempt to bring the environmental dynamics close to fulfilling the directive of the iw module of producing more symmetric and coherent patterns. in principle there is no limit in the hierarchy of hyper-tasks that could be implemented and could also be said to stand for the agents “talents” in a metaphorical sense. the io module should contain a reverse engineering engine which separately examines time-series of both the internal and the external variables and their history which are recorded into a dedicated part of memory (long term memory). it should then extract mathematical models of both and attempt a possible match by proposing certain changes into both the internal parameters and the external assumed environmental parameters. the supervising signals towards the iw module can then be directly implemented into the underlying system of differential equations while the supervising signals towards the st module must be appropriately translated into actions through an interpreter that extracts from them the discrete signals towards the drivers of the external devices in a manner that can best allow the modification of the assumed external dynamical functions. such a modification of the external world may also obey certain directives like the overall entropy reduction in the surrounding space. other scenarios can also be tested with more complicated directives. in fact, the io module can be updated to include a more general planner under certain learning strategies. for the moment we may ignore this advanced capability as it is not inside our objectives to test learning strategies of which are many in the trade. the io module is in addition responsible for continuously building and refining a self model by projecting the present state of match or distance between the iw and the external world model into the mental space representation which may affect the decision making and subsequent planning. in total, the io module is responsible for all the abstract representations of both the self and the external world. the io module can also incorporate a fcg engine for artificial linguistics in order to fulfill the final level of self-expression. advances in systems science and application (2016) vol.16 no.2 91 the st module must contain certain instructions for preprogrammed tasks like step generators for locomotion, or other necessary movements, auto-charging in case of a power source present, “food” hunting if abscent etc. in general it may be characterized as the equivalent of the autonomic nervous system. on the other hand, the st module must interpret the supervising signals from the io module in order to feed appropriately the drivers of the external peripheral devices (legs, arms, sensors) so that the combined motion will affect the environment in a way fit to the higher task of enforcing the desired structure to the external world dynamics. building the appropriate interpreter for the drivers is a non-trivial task as it requires also an amount of computational geometry in order to take into account the exact actual details of the environment and the robotic devices and the possible ways of their interaction. at this level, it is necessary to include a system of somatosensory perception which entails the robot with the capability of describing itself in both shape and movement inside a certain environment. 4 discussion and conclusions the above general scheme is here presented as having certain advantages over other existing approaches due to its holistic nature that attempts to incorporate the most fundamental elements of what human beings intuitively know about themselves. to our opinion these advantages include. • deeper understanding of the foundational principles behind awareness in artificial and natural systems. • enhanced capabilities of operation of autonomous systems in real time. • practical approach to the problem of measuring various forms of awareness in situ. • prediction of possible malignant applications of true self-aware software or hardware and experimentation with counter-measures (“asimov laws” onr report[9]). • practical investigation of the problem of friendly ai (hal9000 problem). in principle there seems to be no restriction in the kind of hyper-tasks that could be implemented. what has not been examined in detail here due to space restrictions is the role of learning processes in a direct interaction with the io layer. it seems reasonable to assume that a learning machine with the structure described above could also discover its own set of hyper-tasks or modify previously programmed ones unless this is strictly forbidden via some dedicated censorship circuitry (freudian viewpoint). with respect to the last two points above it deserves to mention that there 92 t. e. raptis and k. karamanos:an artificial volition architecture for autonomous robotics exists indeed a possibility that a learning machine could also have been imprinted or even develop on its own! a “seek and destroy” attitude. in fact, it is quite possible that a “skynet” scenario is in principle technically feasible and could become reality in the next 20 years or so as already predicted by de garis[10] and kurzweil[11]. the architecture presented above to our opinion justifies such fears and shows the necessity towards more research in the direction of friendly ai as well as the need to ask for demilitarization of robotics. acknowledgements: the first author would like to express his gratitude to the members of the computational applications group of d.a.t.-ncsrd for helpful discussions and support in the preparation of this document. references [1] bongard j. and lipson h. (2005) “active coevolutionary learning of deterministic finite automata”, journal of machine learning research, 6(10):1651-1678. [2] bongard j., zykov v. and lipson h. (2006), “resilient machines through continuous self-modeling”, science, 314(5802): 1118-1121. [3] bongard j. and lipson h. (2007), “automated reverse engineering of nonlinear dynamical systems”, proceedings of the national academy of science, vol. 104, no. 24, pp. 9943-9948. [4] miwa, h., umetsu t., takanishi a. and takanobu h. (2001), “robot personality based on the equations of emotion defined in the 3d mental space”, proceedings of the 2001 ieee international conference on robotics and automation, pp.2602-2607. [5] miwa, h., umetsu t., takanishi a. and takanobu h. (2001), “human-like robot head that has personality based on equations of emotion”, preprints of the sixth symposium on theory of machines and mechanisms,pp.1-8. [6] miwa, h., umetsu t., takanishi a. and takanobu h. (2000), “robot personalization based on the mental dynamics”, proceedings of the 2000 ieee/rsj international conference on intelligent robots and systems,pp.8-14. [7] chadderdon and g. l. (2008), “assessing machine volition: an ordinal scale for rating artificial and natural systems”, adaptive behavior 16, pp.246-263. [8] svozil, k. (1993), randomness and undecidability in physics, world scientific. advances in systems science and application (2016) vol.16 no.2 93 [9] lin p., bekey g. and abney k. (2008), autonomous military robotics: risk, ethics, and design, ethics & emerging technologies group, california polytechnic state university. [10] de garis h. (2005), the artilect war, etc publication. [11] kurzweil r. (1990), the age of intelligent machines, mit press. corresponding author t. e. raptis can be contacted at: rtheo@dat.demokritos.gr advances in systems science and applications (2014) vol.14 no.2 170-182 air materiel supply intelligent collaborative decision-making mode based on ontology and multi-agent xiong li1, chang-xin liu1, fang bing1 and xiongyi li2 1school of management, shanghai university, china 2graziadio school of business and management, pepperdine university, usa abstract to solve the complexity problem of each node and the diversity problem of decision-influencing factors in air materiel supply chain, an air materiel supply intelligent collaborative decision-making mode based on ontology theory and multi-agent has been proposed. to better organize the collaborative knowledge utilized by agents and facilitate agents’ adaptive collaborative decision-making ability, an ontology-based approach is presented in this paper. the knowledge is separated into shared ontology and private ontology to ensure both the agent communicative interoperability and the privacy of strategic knowledge. then, the collaborative mode and action planning of agents are analyzed, and the architecture of intelligent collaborative decision-making system of two-echelon air materiel supply chain has been designed. thus a platform for the consultations and coordination of each agent has been provided, and an effective decisionmaking method has been proposed to decision-makers of air materiel supply. keywords intelligent collaborative decision, ontology, multi-agent, action planning 1 introduction the decision-making mode of the air materiel supplying is actually one of the most important subjects in the field of air materiel management and engineering. in the decision-making process, the airline and air force mainly use the best analytical model available for the aircraft spares provisioning problem, which includes the order, transportation, storage and consumption of air materiel[1-3].in supply chain network, the complexity of each node and the diversity of decisioninfluencing factors are indispensable attributes, however, when making decisions, decision makers only consider a certain part or some key factors in the decision process. usually, the actual decision authority and processes are distributed among the members in the supply chain, who are primarily concerned with optimizing their own objectives. as a result, making appropriate decisions to attain optimal performance in air materiel supply chain is a very challenging problem. traditionally, contractual agreements and complex accounting schemes are used to ensure that the supply-chain works effectively during daily operations. and the centralized or hierarchical decision-making process in these supply chains readvances in systems science and applications (2014) vol.14 no.2 171 sults in losses of efficiency in a competitive and dynamic market environment. members in a supply chain have to coordinate their individual decision-making closely and cooperatively to achieve optimal supply chain performance with balanced services level and inventory level, the each node of supply systems can no longer be viewed in isolation, they must be managed in the context of the total business and the associated key linkages of the business, and the collaboration is an attractive strategy in air materiel supply chain network. due to the complexity of each node and the diversity of decision-influencing factors in the field of air materiel supply, it is worth to research the air materiel supply intelligent collaborative decision-making system to aid managers to make decisions. decision support systems (dss) is the area of the information systems (is) discipline that is focused on supporting and improving managerial decision-making[4]. the history of dss reveals the evolution of a number of sub-groupings of research and practice. the major dss sub-fields include personal decision support systems (pdss), group support systems (gss), organization decision support systems(odss), negotiation support systems (nss), intelligent decision support systems (idss),distributed intelligent decision support systems (didss),knowledge management-based dss (kmdss), distributed intelligent cooperative decision support systems(dicdss) [5-8],etc. with the development of decision support systems, it has been applied in many areas, such as the application to support the temporal and spatial distributed decision-making process in supply chain collaborative planning[9]. a software agent is a program that performs a specific task on behalf of a user, independently or with little guidance, it is characterized with environment awareness, ongoing execution, autonomy, adaptiveness, mobility, intelligence, anthropomorphism, reproduction, independence, collaboration, distribution, etc. the software agent is crucial factor as dss components to build intelligent collaborative decision support systems characterized by cooperating agents, either human or non-human actors[9-11]. ontology is a knowledge representation method with a philosophical concept as the branch of metaphysics which has been widely used in science and technology. from computer specialists perspective, ontology means a vocabulary and a set of terms and relations that define, with the needed accuracy, a set of entities enabling the definition of classes, hierarchies, and other relations among them [12], it has been established as a powerful paradigm to enable knowledge sharing, and became the foundation for many multi-agent system applications as a means to achieve semantic interoperability among heterogeneous agents systems. many intelligent cooperative decision support systems based on ontology and multi-agent has been proposed by researchers[13-15]. for example, chang-shing lee, etc. present an ontology-based intelligent decision support agent to apply to project monitoring and control to reduce the human efforts and 172 xiong li:air materiel supply intelligent collaborative decision-making mode based on ... the costs of the project [13]. in the study, the natural language processing agent, the fuzzy inference agent, the performance decision support agent, the capability maturity model integration ontology and project personal ontology are designed. through the collaborative work of the agents and ontologies, the ontology-based intelligent decision support agent can work effectively for project monitoring and control of capability maturity model integration. compared with the above articles, this paper has the following characteristics. at the first, the collaborative mode of agents based on ontologies has been designed, and according to the collaborative mode, the collaborative decision-making problem in agents located on the different nodes of supply chain network or between managements and agents can be resolved. secondly, the approach of dynamic description logics(ddl) has been applied to planning the action sequences of agents in the instance of air material supply decisions. at last, the structure of air materiel supply intelligent collaborative decision support system based on the collaborative mode of ontologies and agents is designed. the details of this article are as follows. the paper proposes a framework for building decision support systems using software agent technology and ontology to support organizations characterized by physically distributed, enterprise-wide, heterogeneous information systems. in section 2, the structure and interoperability mode of ontology is presented, the multi-agents collaborative mode based on ontology is also analyzed, and a platform for the consultations and coordination of each agent has been provided. section 3 analyzes the action planning of agent. in section 4, the architecture of intelligent collaborative decision-making system of two-echelon air materiel supply chain is designed, and an effective decision-making method is presented to decision-makers of air materiel supply. section5 presents conclusions of this paper. 2 the multi-agents’ collaborative mode based on ontology 2.1 ontology based intelligent agent applications an agent is defined as a software entity that is situated in some environment, and is capable of autonomous action in that environment in order to meet its design objectives[16]. a generic agent has a set of goals, certain capabilities to perform tasks and some knowledge about its environment. to achieve its goals, an agent needs to use its knowledge to reason about its environment and the behaviors of other agents, to generate plans and to execute these plans. a multiagent system(mas) consists of a group of agents, interacting with one another to collectively achieve their goals. by absorbing other agents’ knowledge and capabilities, agents can overcome their inherent bounds of intelligence. air materiel supply intelligent collaborative decision-making system is the intelligent system based on air materiel supply knowledge. in the air materiel supply chain network, advances in systems science and applications (2014) vol.14 no.2 173 managers get consensus about some decision problems through the collaboration of intelligent agents or the intelligent agents and managers. agent technology can provide flexible, distributed, and intelligent solutions for air materiel supply management. however, each agent is an independent entity, one agent needs to exchange the domain knowledge, related concepts with other agents in the internet, and the software agents need achieve the collaborative protocols regulating the set of rules that govern the interaction of agent. ontology as an explicit specification of a conceptualization, and the agreements about the conceptual frameworks for modeling domain knowledge, it is more than just a vocabulary and taxonomy of terms. it provides a set of well-founded constructs that can be leveraged to build meaningful higher level knowledge and relationships between terms[17]. by describing a set of concepts and the relationships between them, ontology can construct both the hierarchical architecture of the knowledge and the descriptive logics of regulations and activities. according to the ontology structure, inference rules can be defined to guide agents collaborative behaviors to adapt to various collaborative environments. as a novel knowledge organization concept, ontology is widely applied in the domain of information sharing, supply collaborative management, multi-agent systems, dsss(decision support systems) and rule-based reasoning systems to enable interoperable decision knowledge structures for knowledge sharing and utilization[18-20]. to effect the intelligent agent applications in the collaborative environment, ontology can help agents in representing and storing domain knowledge, enabling a semantic interoperable environment, reaching mutual understanding, reasoning and querying the knowledge repositories , maintaining a secure system access, etc. in air materiel supply chain network, the members embrace mainly air materiel manufacturers, maintenance contractors, airlines, pooling providers of air materiel, etc. the pooling provider of air materiel is the air materiel warehouse or airline, manufacturer, maintenance contractor in essence which provides service of air materiel supply for the airlines, and the pooling provider charges the airlines in some calculation for the air materiel supply service in accordance with the agreement. in the decision-making analysis process of production, order and storage, generally the air materiel manufacturers, maintenance contractors and pooling providers are considered as the analysis object. due to supply members needing to exchange the concept, domain knowledge and the agents’ related activity specification in the collaborative decision process of air materiel manufacturers, maintenance contractors and pooling providers, two aspects have to be considered. the first is to solve the ontology interoperability problem in agent communication. the second is to build sophisticated private ontology and shared ontology to facilitate collaborative behavior deployment[15]. the private ontology abstracts the knowledge in supply chain collaborative decision which config174 xiong li:air materiel supply intelligent collaborative decision-making mode based on ... ures software agents computing and inference parameters. the shared ontology is developed in accordance with some criteria to ensure mutual understanding between agents developed by different air materiel supply chain members, and it is structured as the types of schemas: concept and agentaction, etc. the structure and interoperability mode of ontology are displayed by fig.1. the contents of shared ontology developed by pooling providers embrace the air materiel category, air materiel name, payment method, order quantity, ordering price, shared price, unit cost of transportation , unit cost of inventory, unit cost of missing parts, agent name, agent address, aid (agent identifier), etc. the contents of private ontology developed by pooling providers comprise order cycle, manufacturer’s credibility, transportation methods, ordering price concessions degree, the way of ordering price concessions, agent negotiation strategies, etc. the contents of shared ontology developed by manufacturers embrace the air materiel category, air materiel name, sale price, delivery time, terms of service, unit operating costs, agent name, agent address, aid, etc. the contents of private ontology developed by manufacturers comprise pooling providers credibility, ordering price concessions degree, the way of ordering price concessions, agent negotiation strategies, etc. fig.1 the structure and interoperability mode of ontology 2.2 the multi-agent collaborative mode in the collaborative decision process of air materiel supply chain members, the members exchange the information and define the software agents collaborative mode through the ontology interoperability. the software agents solve the decision problem by interacting together. there is a set of intelligent agents defined advances in systems science and applications (2014) vol.14 no.2 175 as ordering decision agent, available inventory agent, manufacturer select agent, manufacturer production decisions agent, transport planning agent, etc. in the air materiel supply chain network. when the pooling providers make ordering decision, the ordering decision agent needs to choose adaptive agents from defined agents according to the task, and carry out manufacturer select agent to select possible manufacturers. then the pooling provider sends information of order request to the possible manufacturers, exchanges shared ontology with them. according to the shared ontology exchanged and respective private ontology, the related parameters are calculated collaboratively by the ordering decision agent, available inventory agent and manufacturer production decision agent. at last, the order quantity and order price, etc. are negotiated repeatedly to determine the final order scheme by the ordering decision agent, manufacturer production decision agent and transport planning agent. the multi-agents collaborative mode is displayed by fig.2. fig.2 the multi-agent collaborative mode 3 agents action plan an agent is a tuple, and concrete agents have the same architecture. the components of task agent is defined as follows: w=<act, belief, desire, intent>, act is the act ability component, and represents available behaviours, as well as the situations in which these plans are applicable. its action derive from collaborative mode of agent. beliefs comprise information known by the agent, and regularly updated as a result of perception. desire represents situations that the agent reacts to by adopting plans, corresponding to desired states. intention structures comprise the set of partially instantiated plans currently adopted by the agent[21-23]. when the members of air materiel supply chain make ordering decision, the 176 xiong li:air materiel supply intelligent collaborative decision-making mode based on ... ordering decision agent embrace the order information of a certain air materiel. the content of its belief as follows: belief = {av i(avi), equal(avi, “agent5“), prc(prc), equal(prc, “agent7“), mpd(mpd), equal(mpd, “agent8“), m(m), equal(m, “materiel6“), p (pj), equal(pj , “producer − 1, p roducer − 2, ..., p roducer − n“), ma1(m, a1), equal(a1, v1), ..., man(m, an), equal(an, vn), ..., pb1(p1, b11), equal(b11, x1), ..., pbn(pj , bjn), equal(bjn, xn), ...} concept avi represents the collection of available inventory agent. concept prc represents the collection of manufacturer select agent. concept mpd represents the collection of manufacturer production decisions agent. concept tp represents the collection of transport planning agent. concept m represents the collection of air materiel category. concept p represents the collection of manufacturer.mak(m,ak) represents the kth attribute of an item air materiel named m is ak . relation of equal(a,vk) represents that the value of a is vk.pbk(pj ,bjk) represents the kth attribute of a manufacturer named pj is bjk. relation of equal (bjk, xk) represents that the value of bjk is xk. the collection of belief represents that it knows an available inventory agent named agent5, an manufacturer select agent named agent7, an manufacturer production decision agent named agent8, an item air materiel named materiel6 and n attributes of the air materiel, a set of manufacturers named producer-j and n attributes of every manufacturer. the attributes of the air materiel may be the index of price, performance specifications and transport conditions, etc. in the decision process, ordering decision agent receives the ordering instructions from upper layer agent to formulate an order scheme of an air materiel. according to collaborative mode of agent, the ordering decision agent sets the desire as follows: {ai(m, q0) ∧hr (q0) ∧ pc (m, p) ∧hr (p) ∧oq (m, q) ∧hr (q) ∧op (m,ω)∧ hr (ω) ∧ap (m,ϕ) ∧hr(ϕ) ∧ ep (m, e) ∧hr (x)} . relation of ai(m,q0) represents the available inventory of air materiel named m is q0. relation of pc(m, p) represents manufacturer of air materiel named m is p. relation of oq(m, q) represents order quantity of air materiel named m is q. relation op(m, ω) represents the collaboration price of air materiel named m advances in systems science and applications (2014) vol.14 no.2 177 is ω. relation ap(m, ϕ) represents profit distribution parameters of air materiel named m is ϕ.relation ep(m, e) represents optimal expected profit is e in the collaborative ordering decision process. concept hr (x) represents returning to the upper layer agent. ordering decision agent determines the order quantity, collaboration price, profit distribution parameters and optimal expected profit, then the order scheme is returned to upper layer agent. after the objective of agents determined, it searches the act ability base according to the collaborative mode, and plan the sequence action for achieving the goals. the act of ordering decision agent comprises the action as follows: requestavailableinventory(availableinventory agent5,materiel, attribute1, ..., attributen, availablequantity)= < { ai(availableinventory agent5), k(availableinventory agent5), m (materiel), k(materiel), ma1(materiel, attribute1) , k(attribute1),..., man (materiel, aattributen) , k(attributen), ai(materiel, availablequantity)} , { k(availablequantity), ai(materie1, availablequantity)} > requestproducerschoose(producerschoose agent7, producer-j, attributej1, ..., attributejn, producer)= < { pc(producerschoose agent7), k(producerschoose agent7), p (producer-j), k(producer-j), pb1(producer-1, attribute11) , k(attribute11),..., pbn (materiel, aattributejn) , k(attributejn),..., pc(materiel, producer-j)} , { k(producer-j), ai(materie1, producer-j)} > requestdecisionsplan(manufacturerproductiondecision agent8, quantity, price, assigning parameters, optimalexpectedprofit)= < {mpd(manufacturerproductiondecision), k(manufacturerproductiondecision), m (materiel), k(materiel), p (producer-j), k(producer-j), ai (materiel, availablequantity) , k(availablequantity), pc (materiel, producer-j) , k(producer-j), (oq(materiel,quantity)∧op(materiel, price)∧ap(materiel, assigning parameters) ∧ep(materiel, optimalexpectedprofit))} , { k(quantity), oq(materiel,quantity), k(price), op(materiel, price), k(assigning parameters), ap(materiel, assigning parameters), 178 xiong li:air materiel supply intelligent collaborative decision-making mode based on ... k(optimalexpectedprofit), ep(materiel, optimalexpectedprofit) } > returnordersplan(producer, quantity, price, assigning parameters, optimalexpectedprofit)= < {mpd(manufacturerproductiondecision), k(manufacturerproductiondecision), m (materiel), k(materiel), pc(materiel, producer-j), k(producer-j), oq(materiel,quantity), k(quantity), op(materiel, price), k(price), ap(materiel, assigning parameters), k(assigning parameters), ep(materiel, optimalexpectedprofit), k(optimalexpectedprofit), -(hr(producer-j)∧hr(quantity)∧hr(price)∧hr(assigning parameters)∧ hr(optimalexpectedprofit))}, { hr (producer-j), hr(quantity), hr (price), hr (assigning parameters), hr(optimalexpectedprofit) } > the concept k represents the value of an element has been determined. the requestavailableinventory indicates if the attribute of surplus stock, repairing parts and parts waiting for repair of an air materiel and an available inventory agent are known, then the number of available stock can be obtained through the available inventory agent. the requestproducerschoose indicates if the attribute of quality, order price, reliability, maintainability of an air materiel produced by different manufacturers and a manufacturer select agent are known, then the manufacturers can be obtained through the manufacturer select agent. the requestdecisionsplan indicates if the number of available inventory and manufacturer of an air materiel and a manufacturer production decisions agent are known, then order quantity, collaborative price, profit distribution parameters and optimal expected profit can be determined through the manufacturer production decisions agent. the appropriate ordering scheme of an air materiel can be formulate. the returnordersplan indicates if manufacturer, the number of available inventory, order quantity, collaborative price, profit distribution parameters and optimal expected profit are known, then it can be considered as an order scheme returned to upper layer agent. the action achieving decision objective as follows: requestavailableinventory (avi,m, a1, . . . , an, q0) requestproducerschoose (prc, pj , aj1, . . . , ajn, p) requestdecisionsplan (mpd, q, ω, ϕ, e) returnordersplan (p, q, ω, ϕ, e) the sequence action represents firstly the available stock and manufacturer can be obtained through available inventory agent and manufacturer selection agent, advances in systems science and applications (2014) vol.14 no.2 179 then the request of ordering is sent to manufacturer production decision agent for obtaining the collaborative price, profit distribution parameters and optimal expected profit of an air materiel, at last the order scheme is returned to upper layer agent. 4 the structure of the intelligence collaborative decision-making support system owing to the distribution of the geographical location of the air materiel supply chain members, the heterogeneity of the network and internal business data resources, and the dynamic of decision support activities, the problem of mistakes and misunderstandings among the supply chain members are easy caused. to facilitate the collaborative decision support activities in the air materiel supply chain members, it is essential to integrate and filter the information of inventory, distribution, production and logistics, and to provide a shared domain knowledge structure to enable the members to reach mutual understanding with each other. at the same time, collaborative mechanism of the members and collaborative mode of agent need to be designed, and the interaction between man and computer is realized based on web. fig.3 the system architecture there are many types of decision agents owned by the members of air materiel supply chain. in the every decisions process, each decision agent calls on the corresponding agent in supply chain network according to the decision tasks, and they complete the decision task through the collaborative mode. the system 180 xiong li:air materiel supply intelligent collaborative decision-making mode based on ... architecture shows as fig.3. it mainly comprises the interface, task management module, internal database and management system of each node in air materiel supply chain, data warehouse and management system, the agents integrated base, model base, knowledge base, method base, problem base, case base, shared ontology base, private ontology base, the management systems, etc. when the members of air materiel supply chain submit decision problems to the system through interface, the task management module calls on the corresponding agent to resolve and deal with the decision problems, and the shared ontologies are exchanged to guarantee the mutual understanding of agents through the interface agent and ontology agent. then the collaborative decision-making model is established, and decision problems are solved by calculation and reasoning through calling on the database, model base, knowledge base, method base and case base. the subsequent collaborative processes between the members involve iterative exchanges of proposals and counter-proposals until the final result is achieved. 5 conclusions in this paper, an intelligence collaborative decision-making mode based on the ontology and multi-agent 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[23] guoyin jiang, bin hu, youtian wang.(2010), “agent-based simulation of competitive and collaborative mechanisms for mobile service chains”, information sciences, vol.180, no.2, pp.225-240. corresponding author author can be contacted at: liuchangxin5128@msn.com. мягкая вероятность ошибки при передаче информации advances in systems science and application(2016) vol.16 no.4 43-52 concordance of private and public interests: dynamic graph representation, identification and simulation modeling a.v. antonenko, o.i. gorbaneva, g.a. ougolnitsky department of applied mathematics and computer science, southern federal university, rostov-on-don, russia annotation a universal approach to the description of dynamics of complex systems, identification of their models, and finding the solutions by means of computer simulation is proposed (on the example of cppi-models). the dynamic graph balance models permit to reflect in a convenient visual form the main matter-energetic processes of real-world systems dynamics. computer simulation is proposed to use both for solving of complex game theoretic problems and for their identification. an idea of representative scenarios is developed in this frame. a special computer software should be developed for the implementation of the method. key words: computer simulation, concordance of private and public interests, dynamic graph balance models, identification. 1 introduction in the seminal paper [7] a static model of joint consideration of private and public interests was proposed. they proved that if each agent's payoff function is a convolution by minimum of the private and public parts then a pareto optimal nash equilibrium exists in the agents' game in normal form. an investigation of the models of concordance of private and public interests (cppi-models) was continued by the authors [8,9]. the conditions of system compatibility in cppi-models based on the notion of price of anarchy [1] were studied, and different mechanisms of control providing the system compatibility were analyzed. dynamic versions of cppi-models were also built [3,4]. this paper makes a contribution to dynamic graph representation, identification and simulation of cppi-models as instruments of the applied systems analysis. first, there is a number of mathematical models which permit to describe the state of complex systems including explicit or implicit consideration of their dynamics. some examples are markov chains, finite automates, petri nets, queuing systems. in this paper we develop a technique of dynamic graph balanced cppi-models [15]. second, the standard methods of econometrics [6] and theory of identification [14] are based on long time series of reliable data which are often absent in real applications. we propose to solve the problems of structural and numerical identification by means of building a special computer software. third, computer simulation [13] is an appropriate method of solving complex dynamic problems. we specify this method for cppi-models with different information structure and introduce the idea of representative scenarios. the rest of the paper is organized as follows. in section 2 dynamic graph balanced cppimodels are described. section 3 gives an idea of computer simulation with cppi-models with different information structure based on a small number of representative scenarios. section 4 is concerned with computer simulation support of the identification of cppi-models. section 5 concludes. 44 a.v. antonenko, o.i. gorbaneva, g.a. ougolnitsky: concordance of private and public interests: dynamic graph … 2 dynamic graph balanced cppi-models the description of the state of a complex system in the moment of time t taking into consideration its structure by means of dynamic multidigraphs [15] includes the following elements. 1. a set of vertices y(t)=(y1(t),...,yn(t)) where n(t) is a number of vertices in the moment t. decompose the set y(t) onto two non-intersecting subsets: t y(t) = y1(t)  y2(t),y1(t)  y2(t) =  (it is possible that y2(t) = ). let’s name the vertices from the subset y1(t) compartments and denote them by squares and the vertices from the subset y2(t) transformers and denote by circles. 2. a set of arcs z(t)={zij k (t)}, 1 i, j  n(t), 1  k  n, where zij k (t) is the arc from the vertex yi to the vertex yj (in particular the loop if i=j) on which a resource k can move in the moment t; n is a total number of resources within the system. 3. a set of state variables of the compartments x(t) = {xi k (t)}, 1 i n(t), 1  k  n, where xi k (t) is a value of the resource k in the compartment yiy1 in the moment t. then xi(t) is a state vector of the compartment yi in the moment t (the collection of all its resources). 4. a set of state variables of the arcs f(t) = {fij k (t)}, 1 i, j  n(t), 1  k  n, where fij k (t) is a weight of the arc zij k (t), i.e. a number of the resource k moved during the time [t,t+1] from the vertex yi to the vertex yj, i j, or a quantity of increase (decrease) of the resource k in the compartment yi during the same time, i=j (∆t=1). in each considered situation (problem) the set f(t) can be split into two non-intersecting subsets: t f(t)=f1(t)  f2(t), f1(t)  f2(t) =  (in particular it is possible that f2(t) = ). variables from the set f1(t) are called regulated (they change in the strength of given rules) and variables from the subset f2(t) are called regulators (they can change arbitrarily in the admissible set). 5. a set of limitations on the compartments capacity x = {xi k }, 1 in(t), 1  k  n, where xi k is a maximal number of the resource k which can be stored in the compartment yi. 6. a set of limitations on the carrying capacity of the arcs f = {fij k }, 1 i, j n(t), 1  k  n, where fij k is a maximal number of the resource k which can be moved from the vertex yi to the vertex yj, ij, or produced (destructed) in the compartment yi, i=j, during the time unit. thus, the extended state of a complex system is a set s(t) = < y(t), z(t), x(t), f(t), x, f>. to avoid a consideration of digraphs with multiple arcs let’s map to each vertex yiy1 the only value xi(t) and to each arc zijz the only weight aij(t). then a dynamic structure of the system consists of the separate “scalar” structures each of which represent a certain aspect of matter and energy interactions within the system. the partition of a set of vertices of the dynamic digraph onto two parts permits to describe the principal matter-energetic processes in the realworld systems such as 1) movement (transfer, exchange) of resource between the compartments; 2) production/destruction of the resource in the compartments; 3) transformation of the resource. the respective models can be called dynamic graph balanced ones. describe the processes by such models. 1. a movement of the resource k between two compartments yi and yj in the moment t can be performed if the arc zij k (t) exists (fig.1). fig.1 a movement of the resource k between compartments yi and yj advances in systems science and application(2016) vol.16 no.4 45 assume that in the moment t the stocks of the resource k in the compartments yi and yj are equal to xi k (t) and xj k (t) respectively and the state variable of the arc zij k (t) is fij k (t). then ).()()1(),()()1( tftxtxtftxtx k ij k j k j k ij k i k i  (1) 2. production/destruction of the resource k in the compartment yi in the moment t is possible if the loop zii k (t) exists (fig. 2). fig.2 production/destruction of the resource k in the compartment yi the case fii k (t) > 0 corresponds to the production and the case fii k (t) < 0 to the destruction of the resource k. the result is ).()()1( tftxtx k ii k i k i  (2) 3. a transformation of some resource into other one is possible if a vertex-transformer from the set y2 exists. it is the most complicated class of processes which contains a number of subclasses. the subclasses can be classified by different criterions such as а) a simple transformation (resource k into resource l); b) a composite transformation (one resource into several ones, many resources into one or many to many); or a) a unit transformation (within one compartment); b) a binary transformation (between two compartments); c) a multiple transformation (between several compartments). consider the case bb as an example. assume that initial stocks of the resource are xi k (t), xi l (t), xj l (t). the transformation satisfies the equations ),()()1(),()()1( ),()()1( tftxtxtftxtx tftxtx l pj l j l j l ip l i l i k ip k i k i   (3) where yi, yj are compartments, yp is a transformer. now consider as a more detailed example a known predator-prey model[12] ,, 21222 2 21111 1 xxx dt dx xxx dt dx   (4) where x1(t), x2(t) are biomasses of the prey and predator respectively in the moment t; ε1, ε2 are coefficients of the natural increase of the populations; γ1, γ2 are coefficients of the predatorprey interaction. a representation of the model (4) by means of the dynamical digraph is shown in fig. 3. fig.3 a representation of the predator-prey model by means of the dynamical hierarchical digraph 46 a.v. antonenko, o.i. gorbaneva, g.a. ougolnitsky: concordance of private and public interests: dynamic graph … the loops z11 1 and z22 2 describe an increase of the prey biomass (resource 1) and a decrease of the predator biomass (resource 2) in the compartments y1 and y2 respectively, and the transformation y3 describes a simple binary transformation of the prey biomass into the predator biomass. in the general case a natural dynamics of the system resources is represented by a balance equation for each compartment and each resource: ,1),(,,1 ),()()()1( )()( nktnlji tftftxtx tsy k jl k tsy ij k j k j jlji      (5) the relation (5) added by initial data represents in fact a simulation model describing the system dynamics with consideration of its structure. the equation (5) considers both a passive regulation of the system (due to change of fjl k , fmj k  f1 in the strength of given relations) and an active one (due to choice of fjp k , fqj k  f2). the active regulation could additionally change the sets y and z. it is natural to name the changes of the sets x and f resource ones and the changes of the sets y and z structural ones. the totality of resource and structural changes determines the dynamics of the system s. an extended state of the system s(t) is also changed by external impacts. the system environment can be represented by a vertex y0 with the state vector x0(t) = (x0 1 (t), ... , x0 n (t)). respectively the set of arcs z(t) is added by elements of the type z0i k (t), zi0 k (t) and the set of state variables of the arcs f(t) by elements f0i k (t), fi0 k (t), 1 ≤ i ≤ n(t), 1 ≤ k ≤ n. an influence of the environment is considered in (5) without loss of generality with the condition that y0 can belong to the sets sj + , sj . besides, an external impact can change the sets y(t), z(t). if there are several sources of impact then it is necessary to introduce several external vertices y01 ,..., y0m , respective arcs and state variables. consider as an example the predator-prey model with man-made impact ,, 221222 2 121111 1 xxxx dt dx xxxx dt dx   (6) where in comparison with the model (4) the characteristics of man-made exploitation of the community are added, namely an intensity of use λ and methods of use α, β. a representation of the model (6) by means of the dynamical hierarchical digraph is given in fig. 4. in comparison with the fig.3 to the compartments y1 ("preys") and y2 ("predators") the compartment y0 reflecting the community environment (a source of exploitation) and arcs z10 1 , z20 2 with state variables αλx1, βλx2 are added [15]. fig.4 modeling of exploitation in the predator-prey system by means of the dynamical digraph advances in systems science and application(2016) vol.16 no.4 47 now consider cppi-model written in a discrete form:    ni ijj max; (7)    ni t i t i ss ;1;0 (8) max;)]()([ 0   t t tt i t i t iii xcsurpj (9) ;0 t i t i ru  (10)     ni t ii tt ubxx ;1 (11) .,...,1,0;);,(1 ttniuxgrr t i t i t i t i  (12) in this dynamic model each agent shares his resource ir between a production of a public good )( iu and a private activity )( ii ur  . respectively, his current payoff is a sum of the private gain )( iii urp  and the share in the consumption of the public good )(xcsi . his integral payoff is given by the formula (9), where ipc, are continuous increasing concave functions, .0)0()0(  ipc the utilitarian social welfare function (7) is also introduced. if it is associated with a principal then a choice of the variables is s. t. (8) is considered as an economic control of the principal. the equations of dynamics are given by formulas (11)-(12), where x is a state vector, and the function of its dynamics is linear for simplicity. the implementation of the model dynamics (11)-(12) can be given by the following algorithm: given ;,,,, 0000 niryzx ii  ttotfor 0: );(:{ 11 t i t ii t i t i urpyy       ni t ii tt ubxx ;: 1 );(: 1 ttt xczz   })()(: 1 tt i t i tt i t i t ii t i zsyxcsurpr   . this algorithm can be represented by a dynamic graph balanced model as follows (fig. 5). notice that the equation (12) is specified by means of this model. the relations for j and ij are omitted for simplicity, they can be represented similarly. in fact, in fig. 5 a general relation )(:1 t ijk t jk t jk xx  is presented graphically as fig.5 a balance relation 48 a.v. antonenko, o.i. gorbaneva, g.a. ougolnitsky: concordance of private and public interests: dynamic graph … by default, it is supposed that t ijk t ijk  )( (a linear transformation), and t ik t ijk x (a transference). fig.6 cppi-model presented by the dynamic graph balanced model 3 computer simulation with cppi-models schematically, a process of simulation modeling with a cppi-model can be represented as follo ws. each scenario mj ,...,1 includes a set of the principal's control variables t t n i t jis 11)( }{  and a s et of the agents' control variables t t n i t jiu 11)( }{  . these two sequences generate the respective syste m trajectory t t t jx 1)( }{  and payoffs )(1)( ,}{ j n iji jj  . however, it is important to consider an information structure of the hierarchical game (7)(12). to develop a classification of information structures in the hierarchical differential games with many followers three attributes characterizing the principal's strategy can be used: 1) absence/presence of a feedback of the principal's strategy on the state of a controlled dynamic system. this attribute has two basic values: open-loop strategies (ol) which depend only on the moment of time t , and closed-loop strategies (cl) which depend on the game position ))(,( txt [2]; 2) absence/presence of a feedback of the leader's strategy on the followers' strategies. in the first case we deal with a stackelberg game, and games of the second type we propose to call germeier games [10,11]; 3) methods of hierarchical control. here we differentiate compulsion, when the principal influences the followers' sets of feasible strategies, and impulsion, when the principal influences the followers' payoff functionals [15]. advances in systems science and application(2016) vol.16 no.4 49 in turn, the followers can choose one of the three modes of behavior: (a) isolation, when the followers act independently and come to a nash equilibrium; (b) cooperation, when they pool resources and combine efforts to maximize the summarized payoff functional; (c) collaboration, when the followers voluntarily maximize the principal's payoff functional. notice that in the case of cppi-models cooperation and collaboration coincide because    ni ijj . to explain the proposed classification we use the following two tables. table 1 basic information structures in the hierarchical games [11] without a feedback on the followers' controls with a feedback on the followers' controls without a feedback on the system state t1 t2 with a feedback on the system state x1 x2 table 2 maximal guaranteed payoffs of the principal for different information structures principal followers inaction (s-const) impulsion t1 , x1 (st) t2 , x2 (ger) isolation (ne) 0 nej stimp nej  gerimp nej  cooperation (c) 0 cj stimp cj  gerimp cj  in table 1 the types of leader's strategies using the denotations proposed in [11] are shown. the table 2 should be explained in more details. in hierarchical differential games the principle of optimality is a maximal guaranteed strategy of the principal with consideration of an optimal reaction of the followers. the respective maximal guaranteed payoffs of the principal for the enumerated information structures are collected in the table 2. in the case of isolation it is supposed that the optimal reaction of the followers is their nash equilibria set ne. in the case of cooperation the optimal reaction of the followers is the set c of points of maximum of their summary payoff functional. in this paper we consider only a case of impulsion, when the principal chooses a vector of strategies ),...,( 1 nsss  in the modes t1 , x1 (stackelberg games) or t2 , x2 (germeier games). the strategies can be ol ( t1 , t2 ) or cl ( x1 , x2 ).in the degenerate case of inaction s is constant (no control). thus, in the case of inaction ),(inf0 ujj neu ne   ),(inf0 ujj cu c   )(inf0 max ujj uu  . in the case of impulsion for stackelberg games we have ),(infsup )( usjj sneuss stimp ne    , ),(infsup )( usjj scuss stimp c    , and for germeier games )),~((infsup )~(~~ usjj sneuss gerimp ne     , )),~((infsup )(~~ usjj scuss gerimp c     , where usussuss  ~ :},:~{ ~  . now we can describe an approach to the implementation of the characterized information structures in the simulation mode. in the case t1 strategies have the form )(),( tuts . a scenario mj ,...,1 represents a pair of discrete control trajectories t t n i t jis 11)( }{  , t t n i t jiu 11)( }{  for which a discrete phase trajectory t t t jx 1)( }{  and the payoffs )(1)( ,}{ j n iji jj  are calculated. 50 a.v. antonenko, o.i. gorbaneva, g.a. ougolnitsky: concordance of private and public interests: dynamic graph … in the case of information structure t2 strategies have the form )()),(,( tututs . asarule, impulsion is implemented by a mechanism of reward and punishment of the type       ,),( ,)(),( ))(,( otherwisets ututs tuts p r r (13) where ru is a set of the agents' strategies encouraged by the principal, )(),( tsts pr strategies of reward and punishment respectively ),(),(( i r iii p ii usjusj  . in many situations it is possible to think that pprr stssts  )(,)( . for each agent's control trajectory t t n i t jiu 11)( }{  belonging to the scenario mj ,...,1 , at each step tt ,..,1 the condition (13) is checked, and the respective value t jx )( and the respective summands of payoffs are calculated. in the case of information structure x1 strategies have the form ))(,()),(,( txtutxts , and rules of the type            ,)(, ... ,)(, ,)(, ))(,( )( 2 )( 2 1 )( 1 )( l m j m lj lj j xtxs xtxs xtxs txts            ,)(, ... ,)(, ,)(, ))(,( )( 2 )( 2 1 )( 1 )( f l j l fj fj j xtxu xtxu xtxu txtu mj ,...,1 , (14) where  f r f k f l fl r l k l m l xxxxxxxxxx ,...,,... 11 , should be applied step by step for all scenarios t t t j t t t j us 1)(1)( }{,}{  , mj ,...,1 . the implementation of the game x2 is more complicated and is omitted here. it is important to note the following thing. in the majority of organizational and socioeconomic systems a number of scenarios reflecting qualitatively different control strategies is quite small. in fact, for a qualitative differentiation of control strategies it is sufficient to study "strong", "moderate", and "weak" types of them. for example, if an effort is measured on a scale [0,1] then the values 0, 1/2, 1 representatively reflect the mentioned strategies. if it is not sufficient due to a reason then the values 1/4, 3/4 can be considered additionally, and so on. such scenarios can be called representative ones. therefore, in the set of representative scenarios the complete enumeration becomes a practically implementable procedure. 4 computer simulation support of the identification of cppi-models the problems of identification of mathematical models are solved by econometrics [6] and theory of identification [14]. but these theories have at least two essential shortages. first, their implementation requires to use long time series of reliable data of observations. but such time series are often hardly available and even absent (for example, in social processes such as corruption). second, the standard methods solve only the problem of numerical identification, i.e. determination of the numerical values of model parameters. in the same time, in applied systems analysis a problem of structural identification is more important (which classes of functions should be used in the model). we propose a unified computer-based approach to solving the problems of structural and numerical identification (cppi-models are used as an example). in mathematical modeling it is well known that to provide adequacy one must build hierarchical gradually complicated sequences of models. first, the simplest model is built which reflects only the most essential features of a modeled object. as a rule, the model contains only a few parameters and allows for an analytic investigation. after that investigation is made and some results are received, it becomes clear in which direction the model is bounded and which properties of the real system it does not describe. then the model is perfected in the required advances in systems science and application(2016) vol.16 no.4 51 direction, and the procedure repeats. in the limit, the respective sequence of gradually complicated models can describe the system's behavior with any required accuracy. this methodology is used for both structural and numerical investigation of mathematical models. as for the structural identification, we always start from linear functions. specific features of the models should always be considered. in cppi-models the functions c and ip are increasing, thus we use linear functions )0(  abaxy . after that, it is natural to explore power functions. in cppi-models the functions c and ip are concave, thus functions )1(  baxy b are used. again, it is natural to start from 2/1b ; if it is necessary then other values 1b can be analyzed. at last, other classes of continuous increasing concave functions can be taken if necessary. in the numerical identification a feasible range of values is determined for each parameter (for example, maxmin aaa  ). using the idea of representative scenarios, the values 2/)(,, maxminmaxmin aaaa  are examined first. if it is not sufficient, a dichotomy of the segments ],2/)[(],2/)(,[ maxmaxminmaxminmin aaaaaa  is made, and so on. the idea is implemented by developing a computer software providing to a user the possibility of choice of different classes of functions and values of their parameters. the essence of the software consists in a modification of genetic algorithms for solving problems with several input parameters. the genetic algorithm decreases a number of required calculations on some orders in comparison with known numerical methods. the numerical methods give the value of a result with almost any given accuracy, but in the case of more than four input parameters too many calculations are required. in turn, genetic algorithms don't guarantee such an accuracy but practically don't depend on the number of model parameters. in the same time, the accuracy of the order 10 -6 can be achieved which is more than sufficient for cppi-models. a possible development of the software includes usage of parallel calculations. though genetic algorithms can decreases the number of calculations on some orders there are still hundreds of thousands and even millions of them, therefore paralleling seems a logical way. each stream will contain its own genetic algorithm and input data set. the solutions of each stream are cumulated for further processing. another promising direction of the software development is to seek a solution not in the domain of parameters values but in the domain of types of the solved problem. a modification of the genetic algorithm of another type is required for this. at last, the software can be developed by adding a possibility of the formation of arbitrary kinds of problems. the respective tool is parsing, i.e. a decomposition of a given term on more fine formulas which form a new term available for processing. thus, the described approach assumes an active usage of the methodology of applied systems analysis based on simulation modeling. if a precise analytical solution of an optimization problem is troubled then it can be investigated qualitatively on the base of scenario method using the idea of representative scenarios determined by means of systems analysis. therefore, simulation modeling serves as a universal numerical method of solving complex mathematical problems. 5 conclusion in this paper a universal approach to the description of dynamics of complex systems, identification of their models, and finding the solutions by means of computer simulation is proposed (on the example of cppi-models). the dynamic graph balance models permit to reflect in a convenient visual form the main matter-energetic processes of real-world systems dynamics. this technique is applied to cppi-models. computer simulation is proposed to use both for 52 a.v. antonenko, o.i. gorbaneva, g.a. ougolnitsky: concordance of private and public interests: dynamic graph … solving of complex game theoretic problems and for their identification. an idea of representative scenarios is developed in this frame. a special computer software should be developed for the implementation of the method. references [1] n. nisan, t. roughgarden, e. tardos and v. vazirani(2007), algorithmic game theory, cambridge university press. [2] basar t. and olsder g.j. (1999), dynamic noncooperative game theory, siam. [3] chistyakov a.e., nikitina a.v., ougolnitsky g.a. et al. (2015), "a differential game model of preventing fish kills in shallow waterbodies", game theory and applications. vol. 17. [4] dyachenko v.k., ougolnitsky g.a. and tarasenko l.v. (2014), "computer simulation of social partnership in the system of continuing professional education", advances in systems science and applications, vol.14, no.4, pp.27-37. [5] dockner e., jorgensen s., long n.v. and sorger g. (2000). differential games in economics and management science. cambridge university press. [6] dougherty c. (2011) introduction to econometrics. oxford university press. [7] germeier yu. b., vatel i.a. (1975). "equilibrium situations in games with a hierarchical structure of the vector of criteria", lecture notes in computer science, 27, 460-465. [8] gorbaneva o.i. and ougolnitsky g.a. (2013), "purpose and non-purpose resource use models in two-level control systems", advances in systems science and applications, vol.14, no.4, pp.378-390. [9] gorbaneva o.i. and ougolnitsky g.a. (2015) , "system compatibility, price of anarchy and control mechanisms in the models of concordance of private and public interests", advances in systems science and applications, vol.15, no.1, pp. 45-59. [10] gorelov m.a. and kononenko a.f. (2015). "dynamic models of conflicts. iii. hierarchical games", automation and remote control, vol.76, no.2, pp. 264-277. [11] kononenko a.f. (1977), " game-theory analysis of a two-level hierarchical control system", ussr computational mathematics and mathematical physics, vol.14, no.5, pp. 72-81. [12] kot m. (2001), elements of mathematical ecology. cambridge university press. [13] law a.m. and kelton w.d. (2000) simulation modeling and analysis. mcgraw-hill. [14] ljung l. (1999) system identification: theory for the user. prentice hall. [15] ougolnitsky g.a. (2011) sustainable management. nova science publishers. corresponding author guennady a. ougolnitsky can be contacted at: ougoln@gmail.com. microsoft word 3-hu rong.doc issn 1078-6236 international institute for general systems studies, inc. existence, uniqueness and asymptotic properties of neutrastochastic functional differential equations with markovian switching hu rong 1,2, hu shigeng 1 1department of mathematics, huazhong university of science and technology, wuhan 430074. china 2department of mathematics in science institute, wuhan university of technology, wuhan 430070. china abstract this paper considers the existence and uniqueness of solution to neutral stochastic functional differential equation with markovian switching with local lipschitz condition but neither the linear growth condition. and we discuss the asymptotic properties of this solution including moment boundedness and moment average boundedness in time. a one-dimension nonlinear example is discussed to illustrate the theory. keywords moment boundedness lyapunov function stochastic functional di fferential equations markovian switching generalized ito formula 1. introduction and preliminaries many practical systems may experience abrupt changes in their structure and parameters caused by phenomena such as component failures or repairs, changing subsystem interconnections, and abrupt environmental disturbances. the hybrid systems driven by continuous-time markov chains have recently been developed to cope with such situation, which have therefore received a great deal of attention, and have played a more and more important role in recent years. stochastic functional differential equations with markovian switching have been studied by many authors, and we here mention [8-12], in which they mainly discuss the asymptotic property of the solution, including the stability and moment boundedness and so on with the linear growth condition. kolmanovskii [1] studied the neutral stochastic differential delay equations with markovian switching, and discussed the existence and uniqueness of the solution of the equation and the moment asymptotic boundedness and moment exponential stability. mao [2] discussed the almost surely asymptotic stability of nsdde. in this paper, we will mainly consider neutral stochastic functional differential equations with markovian switching and discuss the existence and uniqueness of a global solution without the linear growth condition, and asymptotic properties including moment boundedness and moment average boundedness in time of the this global solution. 42-54 advances in systems science and applications (2011), vol. 11, no. 1-2 consider the neutral stochastic functional differential equations with markovian switching of the form: [ ( ) ( , ( ))] ( ( ), , ( )) ( ( ), , ( )) ( ),t t td x t u x r t f x t x r t dt g x t x r t dw t   (1) where ( ) ( ), [ ,0]tx x t       which is regarded as in ([ ,0]; ), ( )( 0)nc r r t t  is a right-continuous markovian chain on the probability space taking values in a finite state space  1, 2, ,s n  . moreover : ([ ,0]; ) ,n n nf r c r s r    : ([ ,0]; ) ,n n n mg r c r s r     : ([ ,0]; ) .n n mu c r s r    let  },}{,, 0 pff tt  be a complete probability space with a filtration 0}{ ttf satisfying the usual conditions (i.e. it is right continuous and 0f contains all p-null sets). let ( )( 0)w t t  be an m-dimensional brownian motion defined on this space. let 0 and )];0,([ nrc  denote the family of continuous functions  from ]0,[  to nr with the norm |)(|sup|||| 0    . if a is a vector or matrix, its transpose is denoted by ta . let ( ) 0r t t  , be a right-continuous markovian chain on the probability space taking values in a finite state space },,2,1{ ns  with generator nnij  )( given by jio jio ij ii itrjtrp   ),( ),(1{})(|)({   where 0 . here 0ij is the transition rate from i to j if ji  while    ij ijii  . we assume that the markovian chain )(r is independent of the brownian motion )(w . for any 2( , ) ( ; )nv x i c r s r  , define an operator lv from ([ ,0]; )n nr c r s   to r by advances in systems science and applications (2011), vol.11, no.1-2 43 1 ( , , ) ( ( , ), ) ( , , ) [ ( , , ) ( ( , ), ) ( , , )] 2 t x xxlv x i v x u i i f x i trace g x i v x u i i g x i         1 ( ( , ), ), n ij j v x u i j     (2) where 2 1 ( , ) ( , ) ( , ) ( , ) , , , ( , ) .x xx n i j n n v x i v x i v x i v x i v x i x x x x                    if x(t) is a solution to eq.(1) and let ( ) ( ) ( , ( ))tz t x t u x r t  (as is the following), then by the generalized ito  formula, we have 0 ( ( ), ( )) ( (0), (0)) ( ( ), ( )) t ev z t r t ev z r e lv z s r s ds   , where ( ( ), ( )) ( ( ), , ( ))tlv z t r t lv x t x r t . in this paper the following assumptions are imposed as standing hypothesis. assumption 1.1 both f and g are locally lipschitz continuous. assumption 1.2 for each ,i s there is constant (0,1)i  such that 0 ( , ) ( , ) ( ) ( ) ( ),iu i u i d               (3) where  is a probability measure and those , ([ ,0]; ).nc r    assume moreover that u(0,i)=0, f(0,0,i)=0, g(0,0,i)=0. in general, these assumptions will only guarantee a unique maximal local solution to eq.(1) for any given initial data ([ ,0]; )nc r   and 0(0)r i s  . however, the additional conditions imposed in it, we will guarantee that this maximal local solution is in fact a unique global solution, which is denoted by 0( , , ),x t i and this solution has properties 0limsup ( , , ) , p p x e x t i k   * 00 1 limsup ( , , ) , t p p x e x t i ds k t      (4) where 0  and 0p  are proper parameters, pk and * pk  are positive constants independent of  and 0i . 44 rong: existence, uniqueness and asymptotic properties of neutral…… for the convenience of reference, several elementary inequalities are given in the following which will be used frequently. for any , nx y r , , , 0. x y x y                  (5) 1 1( ) (1 ) , 1,0 1.p p p p px y x y p          (6) ( ) ,0 1.p p px y x y p     (7) 2 2 2 ( ) ,0 1. 1 x y x y          (8) before we state our main results, let us cite several useful lemmas. lemma 1.3 for any ( ) ( ; ), , 0,nh x c r r b  when ,x  ( ) ( ),h x o x  then sup[ ( ) ] . nx r h x b x      in this paper, when we use the notation ( )o x  , it is always under the condition x  . in addition, throughout this paper, const represents a positive constant, whose precise value or expression is not important. ( )i x const always implies that ( )( )ni x x r is bounded above. note that the notation ( )o x  includes the continuity. hence lemma 1.3 can be rewritten as ( ) .b x o x const     in this paper, let 2( , ) ( ) ( ) p t n iv x i x q x x r  . ( )n n iq r i s  are positive definite matrices and 0p  . clearly, we have 2 2( , ) , p pp p i iq x v x i q x  (9) here min ( )i iq q . by (2), we have 12( , , ) ( ) [2 ( , , ) ( , , ) ( , , )] 2 p t t t i i i p lv x i z q z z q f x i g x i q g x i     advances in systems science and applications (2011), vol.11, no.1-2 45 2 22 2 ( 2) ( ) [ ( , , )] ( ) , 2 p p t t t i i ij i j p p z q z z q g x i z q z    (10) where ( , )z x u i  . lemma 1.4 let i be the last term of (10), then we have , p pi m z (11) where 2 2max ( ) 0. p p p i ii i j i ij jm q q    proof clearly, (11) is obtained directly. we only need to prove that 0pm  . we may suppose 1 2 .nq q q   noting that 1 1q q , then 2 2 22 2 11 1 1 11 1 1 1 1 1 1 1 1 0. p p pp p p j j j j j j j m q q q q q                lemma 1.5 assume 1p  , let x(t) be a solution of eq.(1) with 0x  , we have limsup ( ) (1 ) limsup ( ) , p pp t t e x t e z t       (12) where  1( ) ( ) ( , ( )), max .t i n iz t x t u x r t      proof by (3) and (6), we have 01( ) (1 ) ( ) ( ) p p pe x t e x t d              1 0 (1 ) sup ( ) sup ( ) . p pp s t s t e z s e x s             this implies 1 0 0 sup ( ) (1 ) sup ( ) sup ( ) . p p p pp s t s t s t e x s e z s e x s                   so, we can get sup ( ) p s t e x s     . then limsup ( ) (1 ) limsup ( ) p pp t t e x t e z t       . 2. a basic lemma the following lemma plays a key role in this paper. 46 rong: existence, uniqueness and asymptotic properties of neutral…… lemma 2.1 under assumptions 1.1 and 1.2, let 1,p  if there exist constants 00, , , , , 0( ,1 ),i j i ija k k i s j m       positive definite matrices iq and probability measures j , such that ( , , ) ( ( , ), )lv x i v x u i i     0 0 ( ( ) ( ) ),j jp i i ij j j a x k k d e x               (13) then for any initial data ([ ,0], )nc r   and 0(0)r i s  there exists a unique global solution 0( , , )x t i to eq.(1) and this solution satisfies (4). proof for any given initial data ([ ,0]; )nc r   and 0i s , write 0( , , ) ( )x t i x t  , we will divide the whole proof into three steps. step 1 let us first show the existence of the global solution x(t). under assumption 1.1 and 1.2, eq.(1) admits a unique maximal local solution ( )( )x t t    , where  is the explosion time. let ( ) ( ) ( , ( ))tz t x t u x r t  , define the stopping time  inf 0 : ( ( ), ( )) , ( )k t v z t r t k k n      . since  is bounded, when k is large enough, ( ( ), ( ))v z r k   for      , thus, 0k  . if    , when t  , z(t) may explode. hence,  : ( ( ), ( )) , ( )t v z t r t k k n        shows that k  . thus, we may assume 0 ( )k k n    . obviously, k is increasing and ( ) . .k k a s     . if we could show , . .a s   , then . .a s   . thus it need only, for any 0t  , ( ) 0kp t   as k  . fix 0t  . now we prove that ( ) 0kp t   as k  . first note that if k   , then by the continuity of x(t) and the right continuity of r(t), ( ( ), ( ))k kv z r k   . hence, by (13), we have advances in systems science and applications (2011), vol.11, no.1-2 47 ( ) ( ( ), ( )) ( ) ( ( ), ( ))k k k k k kkp t v z r p t ev z t r t           0 0 ( (0), ) ( ( ), ( )) kt ev z i lv z s r s ds     0( (0), )ev z i + 0 00 [ ( ) ( ) ( ) ] k j j t r rj j j e k k x s d x s ds                     0 0 0 0 ( (0), ) ( ) ( ) ( ) k kj j t t j j j ev z i k t k e d x s ds x s ds                     0 0 0 0( (0) ( (0), ), ) ( ) : ,j j t j v u i i k t k d k                 where the index r represents r(t), max (0 )j i ijk k j m    , and tk is a positive constant independent of k. so we can get 1( ) 0( ).k tp t k k k     that shows that x(t) is a global solution to eq.(1). step 2 let us now show inequality (4). by (13), we obtain that 0 ( ( ), ( )) ( (0), (0)) [ ( ( ), ( ))] t se ev z t r t ev z r e l e v z s r s ds    0 00 ( (0), (0)) [ ( ) ( ) ( ) ]j j t s r rj j j ev z r e e k k x s d e x s ds                     01 ( ) 0 0 0 1( (0) ( (0), ), ) ( 1) ( ) : ,jt t j j v u i i k e k e d c ke                         where 1c is a positive constant independent of t and 1 0k k   is a positive constant independent of  and 0i . hence, we have limsup ( ( ), ( )) . t ev z t r t k   then the required assertion (4) follows from (9) and (12). step 3 finally, using (13), we obtain that 0 ( ) t p a e x s ds    0 00 ( ( ), ( )) [ ( ) ( ) ( ) ]j j r rj j j t lv z s r s k k x s d x s de s                     0 0 0 0 2 0( (0) ( (0), ), ) ( ) : ,j j j v u i i k t k d c k t                   48 rong: existence, uniqueness and asymptotic properties of neutral…… where min i ia a and 2c is a positive constant independent of t. the assertion (4) follows directly. the proof is therefore complete. denote the left side of (13) by  and establish the inequality 0 ( ( ) ( ) ) ,j j ij j j k d e x i              (14) where ( ). p p ii a x o x      (15) by lemma 2.1, we have ( ) . 2 p pia x o x const      this together with (15) yields . 2 pia i x const    substituting this into (14) shows that the condition (13) are required. to get (14) and (15), some conditions imposing on the coefficients f and g. these conditions are considered in the next section. 3. main results recall  to denote the left hand of (13). if p>2, by (9) (10) and (11) 1 12 2( ) ( , , ) ( ) ( , ) ( , , ) p p t t t i i i ip z q z x q f x i p z q z u i q f x i      2 22 2 1 2 3 4 ( 1) ( , , ) [ ] : . 2 p pp p i p i p p q z g x i m q z i i i i         (16) we firstly list the following conditions that we will need: (h1) there exist , , 0,i ia   positive-definite matrices iq and a probability measure  , such that 02 2 2 ( , , ) ( ) ( ) ( ).t i i ix q f x i a x d o x                  (h2) there exist 0, , 0i ir r   and a probability measure  , such that 01 1 1 ( , , ) ( ) ( ) ( ).i if x i r x r d o x                (h3) there exist 0, , 0,i i    positive-definite matrices iq and a probability measure  , such that 01 1 1 ( , , ) ( ) ( ) ( ).i ig x i x d o x                  advances in systems science and applications (2011), vol.11, no.1-2 49 we can now state our main result in this paper. theorem 4.1 under assumptions 1.1 and 1.2, if the conditions (h1)-(h3) hold, 2 , 2 3p    and 2 2 2 2 1 ( )( 2) [ 1 ( 1)(1 ) p p i i i i i i ip p i i r r p p a q p                  2( 1)( ) (1 sgn( 2 ))], 2 i i i p q         (17) then for any initial data ([ ,0], )nc r   and 0(0)r i s  there exists a unique global solution 0( , , )x t i to eq.(1) and this solution satisfies (4). proof let 0( ) ( , , )x t x t i and  be sufficiently small. now we estimate 1 4i i respectively. first, by the condition (h1), the inequalities (5) and (7), we can have 01 2 2 222 1 [ ] p p pp i i ii a p q x x d              0 01 2 2 2 222 [ ][ ( )] p p pp i i iq x d d op x                     01 22 ( 2) ( 2) [ p p p p p i i i i p x q a x ap d p                        0 0( 2) ( 2) ( ) ( ) p p p p i p x d o d o x p                           0 02 ( 2) ( ) ( 2) ( ) ( ) ( )].i p p p i p s d s d p                          (18) next, by the condition (h2), the inequalities (5) and (7), we obtain 01 112 2 [( 2) ] 1 p p pp i i p i q p x p d p            01 1 1 [ ( ) ( ) ( )]i ir x r d o x              0 2 ( 1) ( 1) [( 2) ( 2) 1 p p p p i i i p xp q p r x p r d p p                      50 rong: existence, uniqueness and asymptotic properties of neutral…… 0 01 ( 1) ( 1) ( ) ( ) p p p pp i i p x p r d o d o x p                            0 01 ( 1) ( ) ( 1) ( ) ( ) ( )]. p p p i i p s p r d s d p                         (19) then by the condition (h3) and the inequalities (5), (7) and (8), we can get 02 222 3 ( 1) [ ] 2 p p pp i i p p i q x d             0 2 222 22 2 2 [ ( )] 1 ii i i dx o x             2 22 2 2 022 ( 2) (2 2)( 1) [ 2 2 p ppp pi i i i i i p xp p q x d v p                       2 22 0 0 2 2(2 2) ( 2) ( ) ( ) 1 2 p p p pi i p x d o d o x p                             2 22 2 0 0 ( 2) ( ) (2 2) ( ) ( ) ( )], 1 2 p pp i i i p s d s d p                          (20) where , (0,1)i v  are constants. it is easy to see that 012 4 [ ][(1 ) ]. p p pp p i i ii m q x d             (21) then substituting (18)-(21) into (16), we can get  whose form is similar to (14), where 1 22 ( 2) ( 2) ( 2) ( 2) { p p i i i i i e p p e q a ai p p p                    2 2 1( 1) ( 1) ( 1) ( 1) [( 2) 1 p ip p i ii i i q p e e p e p r p r p p p                       2 2 2 2 2 1 2 ( 1) ( 2) ]} [ 2 1 p pppp i i i i i i i i i i i i p p p r p re x q e v                     advances in systems science and applications (2011), vol.11, no.1-2 51 2 2 2( 2) (2 2) (2 2) ( 2) ] ( ) ( ). 2 1 2 p p pi i e p e p x o x o x p p                        if 2  , then we have 12 ( ), p p p i ii p q a x o x       where 2 2( 2) ( 2) ( 2) ( 2 ] { ) [1 p p i i i ii i e p p e a a e p p                      1( 1) ( 1) ( 1) ( 1) [( 2) 1 i p i i i q p e e p p r p r p p p                  1( 2) ]} : ( ).p i i i ip r p re a     by (17), we have (0) 0ia  . since  is sufficiently small, we get 0ia  . therefore, the form of i is similar to (15). if 2  , then we can get 12 ( ), p p p i ii p q a x o x       where 2 2 2 2 21 ( 2) ( 2) [ 2 1 p p i i i i i i i i i i i p e p a a q e v p                        2 ( 2) ( 2) ] : ( , ). 1 i i i e p a v p              choosing that ( )i i i i     and by (17), we get ( , ) 0ia v  . then we also have 0ia  , and the form of i is similar to (15). thus, by lemma 2.1, we can get that for any initial data  and 0i , there exists a unique global solution 0( , , )x t i to eq.(1) and this solution satisfies (4). theorem 4.2 under assumptions 1.1 and 1.2, if the conditions (h1)-(h3) hold, 2 , 2 3p    and 2 1 2 21 ( )( 2) [ 1 ( 1)(1 ) p p i i i i i i i i i i i r r p p a q p                     52 rong: existence, uniqueness and asymptotic properties of neutral…… 2( 1)( ) (1 sgn( 2 ))], 2 i i i p q         (22) then for any initial data ([ ,0], )nc r   and 0(0)r i s  there exists a unique global solution 0( , , )x t i to eq.(1) and this solution satisfies (4). the proof is mostly the same as the one we provided previously, only when we estimate the 1 3i i , we use the inequality (6) not (7). 4. one-dimension nonlinear example let us discuss a one-dimension nonlinear neutral stochastic functional differential with markovian switching to illustrate our theory. 04 2 4 2 1 [ ( ) 0.1 ( 1)] [ ( ) ( ) ( ) ] ( ) ( ).rd x t x t b x t cx t d x t d dt x t dw t             let 3, , 3, 1, 0.1, 0, 0i i ip q e b          , then 0 05 5 55 3 4 1 1 1 4 ( , , ) ( ) ( ), 5 5 t i i ix q f x i b x cx dx x d b x d o x                  04 4 4 1 ( , , ) ( ),if x i b x d o x       2 ( , , ) .g x i x  now, 1 4 , , , 1, , 0. 5 5i i i i i i i i ia b r b r          by theorem 4.1, when 166 25ib  , we can conclude that there 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[12] c. yuan, x. mao, asymptotic stability in distribution of stochastic differential equations with markovian switching. stochastic processes appl., 2003, 103: 277-291. 54 rong: existence, uniqueness and asymptotic properties of neutral…… advances in systems science and application (2015) vol.15 no.1 45-59 system compatibility: price of anarchy and control mechanisms in the models of concordance of private and public interests olga i. gorbaneva and guennady a. ougolnitsky southern federal university, russia abstract the problem of system compatibility is considered. its solution ensures maximization of the social welfare by consideration of individual interests of the agents. the conditions of system compatibility and respective control mechanisms are analyzed for the models of concordance of common and private interests in the agents’ resource allocation. keywords concordance of interests; control mechanisms; hierarchical game theory; system compatibility 1 introduction a problem of concordance of interests in the active systems may be considered in two aspects. first, it is well known that an egoistic behavior of independent active agents often implies a less social welfare than in the case of their coordinated actions. the quantitative side of this problem is named “inefficiency of equilibria”[1] and can be characterized by the price of anarchy index introduced by papadimitriou [2].second, the agents can allocate their resources between private and public interests. in the seminal paper by germeier and vatel [3],it is shown that if payoff functions of all agents are convolutions by minimum of the functions of public and private interests then in the respective game there is a pareto-optimal nash equilibrium (i.e. the price of anarchy is equal to the ideal value of one). we continue to investigate models of that type in literature [4,5] and in the present paper. mathematical methods of solution of the static problems of concordance of interests of active agents are developed in the theory of incentives [6], the information theory of hierarchical systems [7,8], the theory of control in organizations [9,10], mechanism design [1]. it should be noticed that the mechanism design investigates another setting of the problem: how to motivate agents to report the true information about their type (the problem of strategy-proofness, or incentive compatibility). in the present paper a notion of system compatibility is introduced. the system compatibility means that individually optimal controls of agents form the globally optimal vector of controls for a social welfare function. this setting is close to the problem of meta-game synthesis in the theory of active systems [11]. conditions of the system compatibility are studied for the models of concordance of private and public interests (cppi-models). as the conditions are quite re46 olga i. gorbaneva and guennady a. ougolnitsky: system compatibility,price of anarchy and... strictive, the control mechanisms directed to ensure the system compatibility are proposed. in the framework of the authors’ concept [12], the administrative and economic mechanisms with/without a feedback are constructed. the rest of the paper is organized as follows. in the section 2 the principal notions such as price of anarchy, system compatibility, control mechanisms, and cppi-models are introduced. economic mechanisms without feedback and with it as well as algorithms of their implementation are considered in the sections 3 and 4 respectively. administrative control mechanisms are discussed in the section 5. section 6 concludes. 2 principal notions. let’s consider a set n = {1, 2, ..., n} of active agents maximizing their payoff functions gi(u1, ..., un) → max (1) s.t. ui ∈ ui, i ∈ n (2) let the solution of the game (1)-(2) be a nash equilibrium une ∈ ne,u = (u1, ..., un). introduce a utilitarian social welfare function. g0(u) = ∑ j∈n gj(u). let umax be a solution of the maximization problem g0(u) → max, u ∈ u = u1 × ...× un (3) denote gmax 0 = g0(u max) . definition 1. the model(1)-(3)is system compatible if ∀une ∈ ne g0(u ne) = gmax 0 . a quantitative measure of the system compatibility is the price of anarchy [1-2] pa = min une∈ne g0(u ne) gmax 0 (4) it is evident that a model is system compatible if pa = 1 . the system compatibility is a rare phenomenon, and it is worthwhile to use control mechanisms for its achievement[9]. suppose that maximization of the social welfare (3) is the objective of a specific agent (center, principal, mechanism designer and so on) who has an ability of impact on the sets of feasible controls and/or payoff functions of the other agents to provide this objective. denote the first possibility as ui = ui(qi) , and the second one gi = gi(pi, ui), where q, p are vectors of the principal’s administrative and economic controls respectively. in the context of literature[12] we can differentiate the following control mechanisms (methods of control). advances in systems science and application (2015) vol.15 no.1 47 table 1 control mechanisms principal’s impact without a feedback (γ1) with a feedback (γ2) on the sets of feasible controls of the agents(administrative one,or compulsion) qi = const qi = qi(u) on the agents’ payoff functions(economic one,or impulsion) pi = const pi = pi(u) thus, the principal can exert influence on the sets of feasible controls of the agents (administrative control mechanism, or compulsion) or on the agents’ payoff functions (economic control mechanism, or impulsion). both mechanisms can include not or include a feedback on control. in the first case a hierarchical game of the type γ1(stackelberg game) holds, meanwhile the second case generates a hierarchical game of the type γ2(germeier game). so, four types of control mechanisms are possible (table 1). notice that now or in dependence of the used control mechanism. definition 2. a control mechanism q(p) in the model (1)-(3) is system compatible if gmax 0 = g0(u ne(q)) or gmax 0 = g0(u ne(p)) respectively. for definiteness let’s specify the model (1)-(3) in the form gi(u) = pi(ri − ui) + sic(u) → max, 0 ≤ ui ≤ ri, i ∈ n (5) g0(u) = ∑ j∈n pj(rj − uj) + c(u) → max, ∑ j∈n sj = { 1, ∃i : si > 0, 0, otherwise. (6) here, ri > 0 is a resource of the i-th agent; ui is a part of the resource assigned for production of the public payoff described by a function c(u); si is a share of the i-th agent in the public payoff; pi(ri − ui) is a function of the i-th agent’s private interest. the functions pi, c are supposed to be continuously differentiable and concave in all arguments. so, each agent shares his resource between public and private interests according to the ratio ui and ri − ui respectively. thus, the model (5)-(6) describes a concordance of the private and public interests in resource allocation. our investigation of models of the type (5)-(6) (cppimodels) develops the approach by germeier and vatel and burkov and opoitsev [3,11]. economic control mechanisms in the model (5)-(6) are implemented by the principal’s choice of the values si. to use administrative mechanisms one should suppose additionally that the principal can bound feasible controls of the agents: q̃i ≤ ui ≤ q̄i, i ∈ n (7) 48 olga i. gorbaneva and guennady a. ougolnitsky: system compatibility,price of anarchy and... the control mechanisms from the table 1 can be specified for the cppi-model (5)-(7) as follows (table 2). table 2 mechanisms of system compatibility in cppi-models principal’s impact without a feedback (γ1) with a feedback (γ2) on the sets of feasible controls of the agents(administrative one,or compulsion) q̃i ≤ ui ≤ q̄i, q̃i, q̄i = const q̃i(u) ≤ ui ≤ q̄i(u) on the agents’ payoff functions(economic one,or impulsion) si = const si = si(u) 3 economic control mechanisms without a feedback. assume that in the model (5)-(6) ∀i si = const .using the first order conditions we get that the internal system compatibility in the model (5)-(6) holds only if ∂c ∂ui = 0, i ∈ n (8) let’s notice that it is true also for models with partly coincident interests in more general form [5] gi(u) = pi(ui) + sic(u) → max, ui ∈ ui, i ∈ n (9) so, the following statement is true. theorem 1. suppose that ∃i ∈ n : ∂c/∂ui ̸= 0. then for system compatibility in the model (5)-(6) it is necessary that ∀i ∈ n ui = 0 ∨ ui = ri . in other words, the system compatibility in the model (5)-(6) is possible only if all agents are pure individualists (ui = 0) or pure collectivists (ui = ri) . example 1 (linear cppi-model). consider a linear specification of the model (5)-(6): gi(u) = ki(ri − ui) + sik ∑ j∈n uj → max, 0 ≤ ui ≤ ri, i ∈ n (10) g0(u) = ∑ j∈n kj(rj − uj) +k ∑ j∈n uj → max; s : 0 ≤ si ≤ 1, ∑ j∈n sj = { 1, ∃i : s > 0, 0, otherwise. (11) advances in systems science and application (2015) vol.15 no.1 49 here ki > 0,k > 0 are given constants which characterize efficiencies of the functions of private and public payoffs respectively. the first order conditions for the agents and the principal give respectively: ∂gi ∂ui = ksi − ki { ≥ 0, si ≥ ki k ⇒ une i = ri; < 0, si < ki k ⇒ une i = 0; ∂g0 ∂ui = k − ki { ≥ 0, ki ≤ k ⇒ umax i = ri; < 0, ki > k ⇒ umax i = 0. thus, the system compatibility is possible and holds on the bounds of the segments of feasible controls. two partitions of the set n can be defined: n = i0 ∪ c0, i0 ∩ c0 = ∅ : i0 = {i ∈ n : ki > k ⇒ umax i = 0}(immanent individualists); c0 = {i ∈ n : ki ≤ k ⇒ umax i = ri}(immanent collectivists). n = i(s) ∪ c(s), i(s) ∩ c(s) = ∅ : i(s) = {i ∈ n : si < ki k ⇒ une i = 0}(controlled individualists); c(s) = {i ∈ n : si ≥ ki k ⇒ une i = ri}(controlled collectivists). the immanent partition is determined by the objective properties of the model (10)-(11), meanwhile the controlled partition results from the optimal reaction of the agents on the choice of a vector s = (s1, ..., sn) by the principal. notice that i ∈ i0 ⇒ ki/k > 1 ⇒ ∀s ∈ s si ≤ ki/k ⇒ i ∈ i(s), i.e. ∀s ∈ s i0 ⊂ i(s) . the inverse statement is wrong: let ki ≤ k ⇒ i ∈ c0 but si = 0 ⇒ i ∈ i(s). similarly, it is simple to show that ∀s ∈ s c(s) ⊂ c0. it is also clear that if ∃s ∈ s : i(s) = i0, c(s) = c0 then the model is system compatible. to find her optimal control s∗ the principal should solve a discrete optimization problem g0(s) = ∑ j∈i(s) kjrj +k ∑ j∈c(s) rj → max (12) s : 0 ≤ si ≤ 1, i ∈ n, ∑ j∈n sj = { 1, ∃i : si > 0, 0, otherwise. (13) if c0 = ∅ then the model is system compatible for ∀s ∈ s .in this case n = i(s) = i0, g max 0 = gi0 = ∑ j∈n kjrj (individualistic society). 50 olga i. gorbaneva and guennady a. ougolnitsky: system compatibility,price of anarchy and... suppose that c0 ̸= ∅ if ∑ i∈c0 ki ≤ k then let for i ∈ c0, s ∗ i = ki k + εi so that∑ i∈c0 s∗i = 1. then (s∗) = c0 , and the model is system compatible. particularly, if i0 = ∅ then n = c0 = c(s∗), gmax 0 = gc0 = k ∑ j∈c0 rj (collectivistic society). in general case i0 ̸= ∅, c0 ̸= ∅, ∑ j∈c0 kj > k. here |c0| > 1. subject to ∀s ∈ s , i0 ⊂ i(s) it is impossible to put any i ∈ i0 into c(s), thus the problem (12)-(13) is reduced to the construction of a set c(s) ⊂ c0 such that g0(s) = ∑ j∈i(s)=c0\c(s) kjrj +k ∑ j∈c(s) rj → max, ∑ j∈c(s) kj ≤ k, s ∈ s this problem is still being solved. example 2 (power-linear cppi-model). consider the following specification of (5)-(6): gi(u) = ki √ ri − ui + sik ∑ j∈n uj → max, 0 ≤ ui ≤ ri, i ∈ n (14) g0(u) = ∑ j∈n kj √ rj − uj +k ∑ j∈n uj → max, 0 ≤ si ≤ 1, ∑ j∈n sj = { 1,∃i : s > 0, 0, otherwise. (15) the first order conditions for the agents and the principal give respectively: 0 = ∂gi ∂ui = ksi − ki 2 √ ri − ui ⇒ une i = { ri − k2i 4k2s2i , si ≥ ki 2k √ ri ; 0, otherwise; 0 = ∂g0 ∂ui = k − ki 2 √ ri − ui ⇒ umax i = { ri − ki 4k2 , ki ≤ 2k √ ri; 0, otherwise. thus, the system compatibility in the model (14)-(15) holds only if ∀i ∈ n , ki ≥ 2k √ ri (16) if this condition holds, then n = i(s) = i0, gmax 0 = gi0 = ∑ j∈n kj √ rj for any s ∈ s. if (16) doesn’t hold, then the problem of system compatibility can be formulated in a weaker form of maximization the price of anarchy (4) by a mechanism of economic control. the following problem of discrete optimization arises g0(s) = ∑ j∈i(s) kj √ rj + ∑ j∈c′(s) [krj + kj 2 2ksj − k2j 4ks2j ] → max (17) advances in systems science and application (2015) vol.15 no.1 51 s.t. (13), where i(s) = {i ∈ n : si ≤ ki 2k √ ri ⇒ une i = 0}, c ′(s) = {i ∈ n : si > ki 2k √ ri ⇒ une i = ri − k2i 4k2s2i } this problem is reduced to the following one: ∑ j∈c′(s) ( 2k2j sj − k2j s2j ) + λ( ∑ j∈c′(s) sj − 1) → max (18) the foc gives: −2k2j 2s2j + k2j 2s3j + λ = 0, s.t. s ∈ s. multiplication by sj gives the system s3j − 2k2j sj λ + 2k2j λ = 0∑ i∈c′(s) si = 1 to find sj and λ the following should be done: 1) to express sj by λ from the first equation; 2) to substitute the expression into the second equation and find λ; 3) to substitute the value of λ back to the expression from 1) and find the respective si. given λ the first equation can be solved analytically by cartan method (the solution is omitted due to its awkwardness) or numerically. as only real solutions such as 0 ≤ si ≤ 1 are feasible, the following conclusions can be received: (1) a solution exists only if λ < 0. rewriting the equation as s3j = 2k2j λ (sj − 1), we get λ < 0 due to positive left part and negative terms in the numerator and in the brackets; (2) when λ < 0 the only real solution si of the equation exists, and 0 ≤ si ≤ 1. actually, lets find the derivative of the expression f (sj) = s3j − 2k2j sj λ + 2k2j λ : f ′ (sj) = 3s2j − 2k2j λ > 0. therefore, the function f(si) increases in r , and if a root of the equation f(si) = 0 exists then it is unique.now let’s prove that the root exists and satisfies 0 ≤ si ≤ 1. f (0) = 2k2j λ < 0, f (1) = 1 > 0. subject to continuousness of f(si) the property is proved. 52 olga i. gorbaneva and guennady a. ougolnitsky: system compatibility,price of anarchy and... (3) sj increases in λ. let’s find the derivative ∂sj ∂λ of an implicit function: 3s2j · s′j − 2k2jλ·s′j−2k2j sj λ2 − 2k2j λ2 = 0, or s′j = 2k2j (1−sj) λ(3s2jλ−2kj) < 0 that implies. (4) ∑ i∈c′ si increases in λ. therefore, it is possible to choose such λ that ∑ i∈c′(s) si = 1. thus, a problem with two variables is reduced to a problem with one variable. the problem can be solved by the method of dichotomy in which the left bound of a segment is equal to λ1 : ∑ i∈c′(s) si > 1 (for example,λ1 = −ε √ b2 − 4ac , and the right bound is a big enough λ2 : ∑ i∈c′(s) si < 1. 4 economic control mechanisms with a feedback. now assume that in the model (5)-(6) si = si(ui) or even si = si(u). according to foc, the internal system compatibility is possible only if ∂si(u) ∂ui c(u) = [1− si(u)] ∂c(u) ∂ui , i ∈ n (19) this condition is less restrictive than (8) when si = const. both empirical and theoretical approaches can be used in the following analysis. in the context of empirical approach widely spread in practical activity methods of resource allocated are investigated. for example, consider a natural method of proportional allocation si(u) =  ui∑ j∈n uj , ∃m : um > 0, 0, otherwise. (20) in this case (19) takes the form ∑ j ̸=i uj [ ∂c(u) ∂ui ∑ j∈n uj − c(u)] = 0, i ∈ n and therefore the following statement is evident. theorem 2. the mechanism of proportional allocation (20) is system compatible in the cppi-models with linear function of public payoff c(u) and any functions of private payoffs. example 3. suppose that gi(u) = ki √ ri − ui+sik ∑ j∈n uj , where is determined advances in systems science and application (2015) vol.15 no.1 53 by (20). then ∂gi ∂ui = ∂g0 ∂ui = k− ki 2 √ ri−ui , une i = umax i = { ri − k2i 4k2 , ki ≤ 2k √ ri, 0, otherwise; gmax 0 = ∑ j∈i kj √ rj+k ∑ j∈c′ rji = {i ∈ n : ki > 2k √ ri}, c ′ = {i ∈ n : ki ≤ 2k √ ri}. theoretical approach to building of system compatible economic impulsion mechanisms is based on germeier’s theorem for games of the type γ2 (see appendix). let’s apply this theorem to the linear model (5)-(6). we obtain sdi is arbitrary (g0 does not depend on s ), spi ≡ 0; li = kiri; ei = {ui = 0}; di = {(si, ui) : si > kiui k ∑ j∈n uj , n∑ i=1 si = 1}; k2 = gi0 = ∑ j∈n kjrj to find k1 we must solve an optimization problem g0(u) = ∑ j∈i(s) kj(rj − uj) +k ∑ j∈c(s) uj → max s.t. kiui k ∑ j∈n uj < si ≤ 1, ∑ j∈n sj = { 1, ∃i : si > 0, 0, otherwise, , 0 ≤ ui ≤ ri, i ∈ n from the first order condition ∂g0 ∂ui = k − ki ⇒ u∗i = { ri, ki ≤ k (set c0), 0, otherwise (set i0). it is proved that it is possible to find si from the set di: si∈c0 = kiri k ∑ j /∈i0 rj + εi, n∑ i=1 εi = 1− ∑ i/∈i0 kiri k ∑ j /∈i0 rj ; si∈i0 = 0. therefore, like in γ1 formulation (example 1), we obtain g0(u ∗) = ∑ j∈i0 kjrj +k ∑ j∈c0 rj 54 olga i. gorbaneva and guennady a. ougolnitsky: system compatibility,price of anarchy and... three cases are possible: 1. ∀i ∈ n , ki > k ⇒ u∗i = 0, k1 = gmax 0 = gi0 = ∑ j∈n kjrj , in this case k2 = k1, and gi = kiri = li . 2. ∀i ∈ n , ki ≤ k ⇒ u∗i = ri, k1 = gmax 0 = gc0 = k ∑ j∈n rj , that is k2 ≥ k1, and gi = kri > li, hence, condition si > kiui k ∑ j∈n uj is satisfied. in this case n = c0 = c(s), i = ∅, gmax 0 = gc0 = k ∑ j∈n rj , s∗, is any allocation satisfying (13) (“collectivistic” society). 3. ∃l,m : kl > k, km ≤ k. here the solution of the problem (12)-(13) is given by the following algorithm: to assign s∗i =  kiri k ∑ j /∈i0 rj + εi, ki ≤ k; 0, ki > k. then u∗i = { ri, ki ≤ k, 0, ki > k, and gmax 0 = k ∑ j∈i0 rj + ∑ j /∈i0 kjrj . in this case c0 = c(s) ̸= ∅, i0 = i(s) ̸= ∅ (≪mixed≪ society). in all cases 1-3 there is a system compatibility of the model (12)-(13) and economic impulsion mechanism s∗. gi(u) = ki √ ri − ui + sik ∑ j∈n uj → max, 0 ≤ ui ≤ ri, i ∈ n g0(u) = ∑ j∈n kj √ rj − uj +k ∑ j∈n uj → max, 0 ≤ si ≤ 1, ∑ j∈n sj = { 1,∃i : s > 0, 0, otherwise. we obtain sdi is arbitrary (g0 does not depend on s ),spi ≡ 0; li = ki √ ri; ei = {ui = 0}; di = {(si, ui) : si > kiui k( √ ri + √ ri − ui) ∑ j∈n uj }; k2 = gi0 = ∑ j∈n kj √ rj . to find k1 we must solve an optimization problem g0(u) = ∑ j∈n kj √ rj − uj +k ∑ j∈n uj → max ki( √ ri − √ ri − ui) k ∑ j∈n uj < si ≤ 1, ∑ j∈n sj = { 1, ∃i : si > 0, 0, otherwise, , 0 ≤ ui ≤ ri, i ∈ n advances in systems science and application (2015) vol.15 no.1 55 from the first order condition u∗i = { ri − ki 2 4k2 , ri ≤ 2k √ ri; 0, otherwise. it is proved that it is possible to find si from the set di: s∗i =  ki( √ ri− ki 2k ) k ∑ j /∈i0 ( rj− kj 2 4k2 ) + εi, ui > 0; 0, ui = 0. where n∑ i=1 εi = 1− ∑ i/∈i0 ki( √ ri− ki 2k ) k ∑ j /∈i0 ( rj− kj 2 4k2 ) ; therefore k1 = ∑ j∈i0 kj √ rj + ∑ j /∈i0 ( kri + k2i 4k ) > k2. the following cases are possible. 1. ∀i ∈ n , ki > 2k √ ri ⇒ u∗i = 0, k1 = gmax 0 = gi0 = ∑ j∈n kj √ rj , in this case k2 = k1, and gi = ki √ ri = li. 2. ∀i ∈ n , ki ≤ 2k √ ri ⇒ u∗i = ri, k1 = gmax 0 = ∑ j∈n ( kri + k2i 4k ) , i.e. k2 ≤ k1, li = ki √ ri; ei = {ui = 0}; di = { (si, ui) : si > kiui k( √ ri+ √ ri−ui) ∑ j∈n uj } ; and gi = kri + k2i 4k > li, hence, condition si > kiui k( √ ri+ √ ri−ui) ∑ j∈n uj is satisfied. in this case n = c ′, i0 = ∅, ∑ j∈n ( kri + k2i 4k ) , s∗ is any allocation satisfying (13) (“collectivistic” society). 3. ∃l,m : kl > 2k √ rl, km ≤ ki > 2k √ rm. here the solution of the problem (12)-(13) is given by the following algorithm: to assign s∗i =  ki( √ ri− ki 2k ) k ∑ j /∈i0 ( rj− kj 2 4k2 ) i , ki ≤ 2k √ ri; 0, ki > k. then u∗i = { ri − ki 2 4k2 , ri ≤ 2k √ ri; 0, otherwise. and gmax 0 = ∑ j∈i0 kj √ rj + ∑ j /∈i0 ( kri + k2i 4k ) . in this case c0 = c(s) ̸= ∅, i0 = i(s) ̸= ∅ (≪mixed≫ society). in all cases 1-3 there is a system compatibility of the model (12)-(13) and economical impulsion mechanism s∗. 5 administrative control mechanisms. suppose that the principal can bound the agents’ sets of feasible controls. consider the case of administrative control without a feedback. then the model (5)-(6) takes the form gi(q̃i, q̄i, u) = pi(ri − ui) + sic(u) → max, q̃i ≤ ui ≤ q̄i, si ∈ [0, 1]; (21) 56 olga i. gorbaneva and guennady a. ougolnitsky: system compatibility,price of anarchy and... g0(q̃, q̄, u) = ∑ j∈n pj(rj − uj) + c(u) → max, 0 ≤ q̃i ≤ q̄i ≤ ri, i ∈ n. (22) it is clear that if there are no restrictions then the problem (22) has a trivial solution q̃i = q̄i = umax i , i ∈ n . therefore the principals control cost should be considered. then (22) takes the form g0(q̃, q̄, u) = ∑ j∈n pj(rj − uj) + c(u)− c(q̃, q̄) → max, 0 ≤ q̃i ≤ q̄i ≤ ri, i ∈ n, where c(q̃, q̄) is a continuously differentiable and convex in all arguments compulsion cost function. let’s give a simple example. example 4. assume that gi(q̃i, q̄i, u) = ki(ri − ui) + sik ∑ j∈n uj → max, q̃i ≤ ui ≤ q̄i, si ∈ [0, 1]; g0(q̃, q̄, u) = ∑ j∈n kj(rj − uj) +k ∑ j∈n uj − ∑ j∈n (m̃j q̃j + m̄j q̄j) → max, 0 ≤ q̃i ≤ q̄i ≤ ri, i ∈ n where k, ki, m̃i, m̄i are known positive constants. the foc give ∂gi ∂ui = sik − ki ⇒ une i = { ri, ki ≤ sik, 0, ki > sik; ∂g0 ∂ui = k − ki ⇒ umax i = { ri, ki ≤ k, 0, ki > k; ∂g0 ∂q̃i = −m̃i, ∂g0 ∂q̄i = −m̄i ⇒ q̃max i = q̄max i = 0 notice that if ki > k then ki > sik, therefore umax i = une i = 0, and compulsion is not required (the model is system compatible). otherwise two cases should be differentiated: (a) ki ≤ sik ⇒ umax i = une i = ri, and compulsion is not required again; (b) sik < ki ≤ k ⇒ une i = 0, umax i = ri . in this case the principal’s payoffs with and without compulsion should be compared. denote by m the set of agents for whom sik < ki ≤ k. if compulsion holds we have g0(r, 0, r) = k ∑ j∈m rj − ∑ j∈m m̃jrj , and if not then g0(0, 0, 0) = ∑ j∈m kjrj . thus, compulsion is rational if ∑ j∈m (k − m̃j − kj)rj > 0 holds. advances in systems science and application (2015) vol.15 no.1 57 6 conclusion. in the present paper the problem of system compatibility was analyzed. its solution ensures maximization of the social welfare by considering individual interests of the agents. this setting is close to the problem of mechanism design but distinct because in the latter problem it is required to motivate agents to report the true information about their types instead of the direct maximization of the social welfare. the conditions of system compatibility are quite restrictive, therefore to provide them it is worthwhile to construct control mechanisms. in the framework of authors’ concept, the mechanisms are classified by two attributes: direction of impact (sets of feasible controls of the agents or their payoff functions) and presence or absence of feedback in the control system. the first attribute differentiates administrative and economic control mechanisms (methods of compulsion and impulsion respectively), while the second one leads to hierarchical games of the types γ1 and γ2. the conditions and mechanisms of system compatibility are analyzed in the class of models of concordance of the private and public interests in resource allocation (cppi-models). some preliminary results about system compatibility of the cppi-models are obtained. thus, system compatibility in cppi-models when si = const is reachable only if all agents are pure individualists (all resources are assigned for private interests) or pure collectivists (all resources are assigned for public interest). the exact dichotomous partition is built by specific algorithms of discrete optimization. economic mechanisms with a feedback simplify the achievement of system compatibility. administrative mechanisms of system compatibility are under development. the research perspectives include: investigation of the system compatibility for more general classes of models; considering of corruption (an additional feedback on bribe); analysis of dynamic settings, including phase constraints (requirements of sustainable development), investigation of the conditions of time consistence. appendix (germeier theorem). assume that payoff functions of both players m1(x1, x2),m2(x1, x2) are continuous on compact sets of feasible controls x1, x2. introduce the punishment function xp1 (x2) such that m2(x p 1 , x2) = min x1∈x1 m2(x1, x2), and the dominant strategy of the player 1 xd1 (x2), which satisfies the condition m1(x d 1 , x2) = max x1∈x1 m(x1, x2). introduce also the following values and sets: l2 = max x2∈x2 m2(x p 1 (x2), x2); e2 = {x2 ∈ x2 : m2(x p 1 (x2), x2) = l2}; d2 = {(x1, x2) ∈ x1 ×x2 : m2(x1, x2) > l2}; 58 olga i. gorbaneva and guennady a. ougolnitsky: system compatibility,price of anarchy and... k1 = sup (x1,x2)∈d2 m1(x1, x2) ≤ m1(x ε 1, x ε 2) + ε (d2 = ∅ ⇒ k1 = −∞); k2 = min x2∈e2 max x1∈x1 m1(x1, x2). then the guaranteed payoff of the player 1 (leader) in the game γ2 (in which the first player knows the choice of the second player) is equal to w1 = max(k1,k2), and the respective -optimal guaranteeing strategy has the form x̃ε1(x2) =  xε1, x2 = xε2, k1 > k2, xd1 (x2), x2 ∈ d2, k1 ≤ k2, xp1 (x2), otherwise, where xε1 and xε2 are described above. acknowledgements the work is supported by the russian foundation for basic research, project # 15-01-00432. references [1] n. nisan, t. roughgarden, e. tardos and v. vazirani. (2007), algorithmic game theory, cambridge university press. [2] papadimitriou c.h. (2001), “algorithms, games, and the internet”, proc. 33th symposium theory of computing, pp.749-753. [3] germeier yu.b. and vatel i.a. (1975), “equilibrium situations in games with a hierarchical structure of the vector of criteria” ,lecture notes in computer science, vol.27, pp.460-465. [4] gorbaneva o.i. and ougolnitsky g.a. (2013), “purpose and non-purpose resource use models in two-level control systems”, advances in systems science and application ,vol.13,no.4,pp.378-390. [5] ougolnitsky g.a. (2011), “games with differently directed interests”, contributions to game theory and management, vol.4,pp.327-338.[collected papers presented on the fourth international conference game theory and management / editors l. petrosyan, n. zenkevich.] [6] laffont j.-j. and martimort d. (2002), the theory of incentives: the principal-agent model, princeton university press. [7] kononenko a.f. (1974), “game-theory analysis of a two-level hierarchical control system”, ussr computational mathematics and mathematical physics,vol.14,no.5,pp.72-81. advances in systems science and application (2015) vol.15 no.1 59 [8] kukushkin n.s. (1994), “a condition for existence of nash equilibrium in games with public and private objectives”, games and economic behavior,vol.7,pp.177-192. [9] d. novikov. (2013), mechanism design and management: mathematical methods for smart organizations, nova science publishers. [10] novikov d. (2013), theory of control in organizations, nova science publishers. [11] burkov v.n. and opoitsev v.i. (1974), “metagame approach to the control in hierarchical systems”, automation and remote control, vol 35, no.1, pp.93-103. [12] ougolnitsky g. (2011), sustainable management, nova science publishers. corresponding author olga i. gorbaneva, ph.d. can be contacted at: gorbaneva@mail.ru guennady a. ougolnitsky, ph.d. can be contacted at:ougoln@gmail.com advances in systems science and applications (2012) vol.12 no.4 373-387 predictive inferences for a future number of failures coming from underlying models under parametric uncertainty konstantin n. nechval1, nicholas a. nechval2, maris purgailis2, uldis rozevskis2, vladimir f. strelchonok3 and max moldovan4 1applied mathematics department, transport and telecommunication institute lomonosov street 1, lv-1019, riga, latvia 2statistics department, evf research institute, university of latvia raina blvd 19, lv-1050, riga, latvia 3informatics department, baltic international academy lomonosov street 4, lv-1019, riga, latvia 4australian institute of health innovation, university of new south wales, level 1 agsm building, sydney nsw 2052, australia abstract in this paper, we present an accurate procedure to obtain prediction limits for the number of failures that will be observed in a future inspection of a sample of units, based only on the results of the first in-service inspection of the same sample. the failure-time of such units is modeled with a two-parameter weibull distribution indexed by scale and shape parameters β and δ, respectively. it will be noted that in the literature only the case is considered when the scale parameter β is unknown, but the shape parameter δ is known. as a rule, in practice the weibull shape parameter δ is not known. instead it is estimated subjectively or from relevant data. thus its value is uncertain. this δ uncertainty may contribute greater uncertainty to the construction of prediction limits for a future number of failures. in this paper, we consider the case when both parameters β and δ,are unknown. in literature, for this situation, usually a bayesian approach is used. bayesian methods are not considered here. we note, however, that although subjective bayesian prediction has a clear personal probability interpretation, it is not generally clear how this should be applied to non-personal prediction or decisions. objective bayesian methods, on the other hand, do not have clear probability interpretations in finite samples. the technique proposed here for constructing prediction limits emphasizes pivotal quantities relevant for obtaining ancillary statistics. and represents a special case of the method of invariant embedding of sample statistics into a performance index. two versions of prediction limits for a future number of failures are given. keywords weibull distribution, parametric uncertainty, future number of failures, prediction limits 374 konstantin n. nechval:predictive inferences for a future number of failures coming from... 1 introduction this paper extends the results of nelson [1]. nelson’s prediction limits were motivated by the following application. nuclear power plants contain large heat exchangers that transfer energy from the reactor to steam turbines. such exchangers typically have 10,000 to 20,000 stainless steel tubes that conduct the flow of steam. due to stress and corrosion, the tubes develop cracks over time. cracks are detected during planned inspections. the cracked tubes are subsequently plugged to remove them from service. to develop efficient inspection and plugging strategies, plant management can use a prediction of the added number of tubes that will need plugging by a specified future time. nelson presents simple prediction limits for the number of failures that will be observed in a future inspection of a sample of units. the past data consist of the cumulative number of failures in a previous inspection of the same sample of units. life of such units is modeled with a weibull distribution with a given shape parameter value. prediction of an unobserved random variable is a fundamental problem in statistics. hahn and nelson [2], patel [3], and hahn and meeker [4] provided surveys of methods for statistical prediction for a variety of situations on this topic. in the areas of reliability and life-testing, this problem translates to obtaining prediction intervals for lifetime distributions. nordman and meeker [5] compared probability ratio, simplified probability ratio and likelihood ratio methods proposed by nelson [1], assuming known the weibull shape parameter δ. in this paper, we use a frequentist procedure, which is called ‘within-sample prediction of future order statistics’, when the time-to-failure follows the twoparameter weibull distribution indexed by scale and shape parameters β and δ. we consider the case when both parameters β and δ are unknown. the technique proposed here for constructing prediction limits emphasizes pivotal quantities relevant for obtaining ancillary statistics and represent a special case of the method of invariant embedding of sample statistics into a performance index applicable whenever the statistical problem is invariant under a group of transformations, which acts transitively on the parameter space (nechval et al. [6-7]). conceptually, it is useful to distinguish between “new-sample” prediction, “within-sample” prediction, and “new-within-sample” prediction. some mathematical preliminaries for the within-sample prediction are given below. 2 mathematical preliminaries for within-sample prediction theorem 1 let x1 ≤ . . . ≤ xk be the first k ordered observations (order statistics) in a sample of size m from a continuous distribution with some probability density functio fθ(x) and distribution function fθ(x), where θ is a parameter (in general, vector). then the joint probability density function of x1 ≤ . . . ≤ xk advances in systems science and applications (2012) vol.12 no.4 375 and the lth order statistics xl(1 < k < l < m) is given by gθ(x1, . . . , xk, xl) = gθ(x1, . . . , xk)gθ(xl|xk), (1) where gθ(x1, . . . , xk) = m! (m− k)! πk i=1fθ(xi)[1− fθ(xk)] m−k, (2) gθ(xl|xk) = (m− k)! (l − k − 1)!(m− l)! [ fθ(xl)− fθ(xk) 1− fθ(xk) ]l−k−1[1− fθ(xl)− fθ(xk) 1− fθ(xk) ]m−l fθ(xl) 1− fθ(xk) = (m− k)! (l − k − 1)!(m− l)! l−k−1x j=0 � l − k − 1 j � (−1)j [ 1− fθ(xl) 1− fθ(xk) ]m−l+j fθ(xl) 1− fθ(xk) = (m− k)! (l − k − 1)!(m− l)! m−lx j=0 � m− l j � (−1)j [ fθ(xl)− fθ(xk) 1− fθ(xk) ]l−k−1+j fθ(xl) 1− fθ(xk) (3) represents the conditional probability density function of xl given xk = xk. proof.the joint density of x1 ≤ . . . ≤ xk and xl is given by gθ(x1, . . . , xk, xl) = (m)! (l − k − 1)!(m− l)! ky i=1 fθ(xi)[fθ(xl)− fθ(xk)] l−k−1fθ(xl) [1− fθ(xl)] m−l = gθ(x1, . . . , xk)gθ(xl|xk). (4) it follows from (4) that gθ(xl|x1, . . . , xk) = gθ(x1, . . . , xk, xl) gθ(x1, . . . , xk) = gθ(xl|xk), (5) i.e., the conditional distribution of xl l, given xi = xi for all i = 1, . . . , k , is the same as the conditional distribution of xl , given only xk = xk,which is given by (3). this ends the proof. corollary 1.1.the conditional probability distribution function of xl given xk = xk is pθ(xl ≤ xl|xk = xk) =1− (m− k)! (l − k − 1)!(m− l)! l−k−1x j=0 � l − k − 1 j � (−1)j m− l + 1 + j h 1− fθ(xl) 1− fθ(xk) im−l+1+j = (m− k)! (l − k − 1)!(m− l)! m−lx j=0 � m− l j � (−1)j l − k + j hfθ(xl)− fθ(xk) 1− fθ(xk) il−k+j . (6) 376 konstantin n. nechval:predictive inferences for a future number of failures coming from... corollary 1.2. let x1 ≤ . . . ≤ xk be the first k order statistics in a sample of size m from the two-parameter weibull distribution with the probability density function fθ(x) = δ β ( x β )δ−1 exp[−( x β )δ] (x > 0), (7) where θ = (β, σ),β > 0 and σ > 0 are the scale and shape parameters, respectively. then the conditional probability distribution function of xl given xk = xk is pθ{xl ≤ xl|xk = xk} = 1− (m− k)! (l − k − 1)!(m− l)! l−k−1x j=0 � l − k − 1 j � (−1)j m− l + 1 + j h exp � − xδl − xδk βδ �im−l+1+j . (8) theorem 2 if in (8) the scale parameter is unknown, then the predictive probability distribution function of xl based on (xk, δ) is given by pδ n�xl xk �δ ≤ � xl xk �δ } = 1− m! (l − k − 1)!(m− l)! × � l − k − 1 j � (−1)j m− l + 1 + j � πk−1 s=0 h� xl xk �δ − 1)(m− l + 1 + j) + (m− k + 1 + s) i�−1 . (9) proof.we reduce (8) to pθ n�xl xk �δ ≤ �xl xk �δ | �xk β �δ = �xk β �δo =1− (m− k)! (l − k − 1)!(m− l)! l−k−1x j=0 � l − k − 1 j � (−1)j m− l + 1 + jh exp(−ω[νδ − 1]) im−l+1+j =pδ{v δ ≤ νδ|w = ω}, (10) where v = xl/xk is the ancillary statistic whose distribution does not depend on the parameter β. since xk does not depend on v , w = (xk/β) δ is the pivotal quantity, whose distribution is known and does not depend on the parameters β and δ, we eliminate the parameter from the problem as pδ{xl ≤ xl} = z ∞ 0 pθ{xl ≤ xl|xk = xk}gθ(xk)dxk, (11) where gθ(xk) = m! (k − 1)!(m− k)! f k−1 θ (xk) h 1− fθ(xk) im−k fθ(xk), xk ∈ (0,∞), (12) advances in systems science and applications (2012) vol.12 no.4 377 represents the probability density function of the kth order statisticxk k. indeed, it follows from (12) that gθ(xk)dxk = m! (k − 1)!(m− k)! h 1− exp � − �xk β �δ�ik−1 exp � − �xk β �δ(m−k) � exp � − �x β �δ� d �x β �δ = m! (k − 1)!(m− k)! [1− e−ω]k−1e−ω(m−k+1)dω = g(ω)dω. (13) it follows from (10) and (13) that pδ{v δ ≤ νδ} = z ∞ 0 pδ{v δ ≤ νδ|w = ω}g(ω)dω =1− (m)! (l − k − 1)!(m− l)! l−k−1x j=0 � l − k − 1 j � (−1)j m− l + 1 + j � πk−1 s=0 � (νδ − 1)(m− l + 1 + j) + (m− k + 1 + s) ��−1 . (14) now (9) follows from (14). this ends the proof. corollary 2.1.if the parameter δ = 1 , i.e. we deal with the exponential distribution, then the predictive probability distribution function of xl based on xk is given by p n�xl xk � ≤ � xl xk �o = 1− m! (l − k − 1)!(m− l)! × l−k−1x j=0 � l − k − 1 j � (−1)j m− l + 1 + j � πk−1 s=0 h� xl xk − 1 � (m− l + 1 + j) + (m− k + 1 + s) i�−1 . (15) theorem 3 let x1 ≤ . . . ,≤ xk be the first k ordered observations from a sample of size m from the two-parameter weibull distribution (7). then the joint probability density function of the pivotal quantities w2 = δóδ , w3 = � óβ β �δ̂ , (16) conditional on fixed zk = (zi, . . . , zk),where zi = (xi/óβ)δ̂, i = 1, . . . , k,are ancillary statistics, any k − 2 of which form a functionally independent set,óβ and óδ are the estimators of β and δ,based on the first k ordered observations (x1 ≤ . . . ≤ xk) from a sample of size m from the two-parameter weibull distribution (7), such that w2 and w3 are the pivotal quantities (in particular, the 378 konstantin n. nechval:predictive inferences for a future number of failures coming from... maximum likelihood estimators of β and δ, óβ = �h kx i=1 xδ̂i + (m− k)xδ̂k i /k �1/δ̂ (17) and óδ = �� kx i=1 xδ̂i lnxi+(m−k)xδ̂k lnxk �� kx i=1 xδ̂i +(m−k)xδ̂k �−1 − 1 k kx i=1 lnxi �−1 (18) respectively, lead to the pivotal quantities w2 and w3 )is given by f(ω2, ω3|z(k)) =ϑ•(z(k))ωk−1 2 ky i=1 zω2 i ωkω2−1 3 exp � − ωω2 3 � kx i=1 zω2 i + (m− k)zω2 k �� =ϑ•(z(k))ωk−2 2 ky i=1 zω2 i ω ω2(k−1) 3 exp � − ωω2 3 � kx i=1 zω2 i + (m− k)zω2 k �� ω2ω ω2−1 3 =f(ω2|z(k))f(ω3|ω2, z (k)), ω3 ∈ (0,∞), (19) where ϑ•(z(k)) = h z ∞ 0 γ(k)ωk−2 2 ky i=1 zω2 i � kx i=1 zω2 i + (m− k)zω2 k �−k dω2 i−1 (20) is the normalizing constant, f(ω2|z(k)) = ϑ(z(k))ωk−2 2 ky i=1 zω2 i � kx i=1 zω2 i + (m− k)zω2 k �−k , ω2 ∈ (0,∞), (21) ϑ(z(k)) = h z ∞ 0 ωk−2 2 ky i=1 zω2 i � kx i=1 zω2 i + (m− k)zω2 k �−k dω2 i−1 , (22) f(ω3, ω2|z(k)) = hpk i=1 z ω2 i + (m− k)zω2 k ik γ(k) ω w2(k−1) 3 × exp � − ωw2 3 h ( kx i=1 zω2 i + (m− k)zω2 k i� ω2ω ω2−1 3 , ω3 ∈ (0,∞). (23) proof.the joint density x1 ≤ . . . ≤ xk is given by fθ(x1, . . . , xk) = m! (m− k)! ky i=1 δ β ( xi β )δ−1 exp(−( xi β )δ) exp(−(m− k)(( xk β )δ). (24) advances in systems science and applications (2012) vol.12 no.4 379 using óβ and óδ (the maximum likelihood estimators of β and δ obtained from solution of (17) and (18)) and the invariant embedding technique [8-14], we transform (24) as follows: fθ(x1, . . . , xk)dóβdóδ = m! (m− k)! ky i=1 x−1 i δk ky i=1 �xi β �δ exp � − kx i=1 �xi β �δ − (m− k) �xk β �δ� dóβdóδ =− m! (m− k)! óβóδk ky i=1 x−1 i �δóδ �k−2 ky i=1 �xióβ �δ̂( δ δ̂ )� óβ β �δ̂( δ δ̂ )(k−1) × exp � − � óβ β �δ̂( δ δ̂ ) � kx i=1 �xióβ �δ̂( δ δ̂ ) + (m− k) �xkóβ �δ̂( δ δ̂ )���óδ( δóδ ) β � óβ β �δ̂( δ δ̂ )−1 dóβ��− δóδ2dóδ � =− m! (m− k)! óβóδk ky i=1 x−1 i ωk−2 2 ky i=1 zω2 i ω ω2(r−1) 3 exp � − ωω2 3 h kx i=1 zω2 i + (m− k)zω2 k i� d(ωω2 3 )dω2 =− m! (m− k)! óβóδk ky i=1 x−1 i ωk−2 2 ky i=1 zω2 i ω ω2(k−1) 3 exp � − ωω2 3 h kx i=1 zω2 i + (m− k)zω2 k i� ω2ω ω2−1 3 dω2dω3. (25) normalizing (25), we obtain (19). this ends the proof. it will be noted that more general case of distributions indexed by location and scale parameters has been considered in [15]. theorem 4 if in (8) both parameters β and δ are unknown, then the predictive probability distribution function of xl based on (xk, óδ) and conditional on fixed z(k) is given by p n�xl xk �δ̂ ≤ � xl xk �δ̂|z(k)o =1− m! (l − k − 1)!(m− l)! × z ∞ 0 l−k−1x j=0 � l − k − 1 j � (−1)j m− l + 1 + j� k−1y s=0 h��� xl xk �δ̂�ω2 − 1 � (m− l + 1 + j) + (m− k + 1 + s) i�−1 f � ω2|z(k) � dω2. (26) 380 konstantin n. nechval:predictive inferences for a future number of failures coming from... proof. we reduce (9) to pδ n�xl xk �δ̂( δ δ̂ ) ≤ � xl xk �δ̂( δ δ̂ ) o = 1− m! (l − k − 1)!(m− l)! × l−k−1x j=0 � l − k − 1 j � (−1)j m− l + 1 + j � k−1y s=0 h�� xl xk �δ̂( δ δ̂ ) − 1 � (m− l + 1 + j) + (m− k + 1 + s) i�−1 =1− m! (l − k − 1)!(m− l)! × l−k−1x j=0 � l − k − 1 j � (−1)j m− l + 1 + j � k−1y s=0 h (νω2 2 − 1)(m− l + 1 + j) + (m− k + 1 + s) i�−1 =p ¦ v w2 2 ≤ νω2 2 © , (27) where v2 = (xl/xk) δ̂ is the ancillary statistic whose distribution does not depend on the parameters β and δ. since the pivotal quantity w2, whose distribution is given by (21), does not depend on v2, it follows from (21) and (27) that p ¦ v2 ≤ ν2|z(k) © = z ∞ 0 p ¦ v w2 2 ≤ νω2 2 © f � ω2|z(k) � dω2, (28) where the unknown parameters β and δ are eliminated from the problem. now (26) follows from (28). this ends the proof. 3 prediction limits for a future number of failures consider the situation in which m units start service at time 0 and are observed until a time tc when the available weibull failure data are to be analyzed. failure times are recorded for the k units that fail in the interval [0, tc]. then the data consist of the k smallest-order statistics x1 ≤ . . . ≤ xk ≤ tc and the information that the other m?k units will have failed after tc.with time (or type i) censored data, tc is prescribed and k is random. with failure (or type ii) censored data, k is prescribed and tc = xk is random. the problem of interest is to use the information obtained up to tc to construct the weibull within-sample prediction limits (lower and upper) for the number of units that will fail in the time interval [tc, tω].for example, this tω could be the end of a warranty period. consider the situation when tc = xk. under conditions of theorem 4, the lower prediction limit for the number of units that will fail in the time interval [tc, tω] is given by llower = lmax − k, (29) advances in systems science and applications (2012) vol.12 no.4 381 where lmax = max k<l≤m argp �¦ xl > tω|zk © ≤ α � (30) the upper prediction limit for the number of units that will fail in the time interval [tc, tω] is given by lupper = lmin − k − 1, (31) where lmin = min k<l≤m argp �¦ xl > tω|zk © ≥ 1− α � (32) in the above case, where both parameters β and δ are unknown, the prediction limits (lower and upper) for the number of units that will fail in the time interval [tc, tω] are based on (xk, óδ) and conditional on fixed z(k). if l, which satisfies (30), does not exist then lmax = k and the lower prediction limit for the number of units that will fail in the time interval [tc, tω] is given by llower = lmax − k = 0. (33) if l, which satisfies (32), does not exist then lmin = m+1 and upper prediction limit for the number of units that will fail in the time interval [tc, tω] is given by lupper = lmin − k − 1 = m− k, (34) 4 second version of prediction limits for a future number of failures in this section, we wish to show how to obtain the second version of prediction limits for a future number of failures. the methodology is based on the following results. theorem 5 let x1 ≤ . . . ≤ xk be the first k ordered observations from a sample of size m from the two-parameter weibull distribution (7). then the joint probability density function of the pivotal quantities w1 = � óβ β �δ , w3 = δóδ , (35) conditional on fixed z(k) = (zi, . . . , zk), where zi = (xi/óβ)δ̂, i = 1, . . . , k are ancillary statistics, any k−2 of which form a functionally independent set,óβ andóδ are, for instance, the maximum likelihood estimators for β and δ based on the first k ordered observations (x1 ≤ . . . ≤ xk) from a sample of size m from the 382 konstantin n. nechval:predictive inferences for a future number of failures coming from... two-parameter weibull distribution (7), which can be found from solution of (17) and (18), is given by f(ω1, ω2|z(k)) = ϑ•(z(k))ωk−2 2 ky i=1 zω2 i ωk−1 1 exp(−ω1[ kx i=1 zω2 i + (m− k)zω2 k ]) = f(ω2|z(k))f(ω1|ω2, z (k)), ω1 ∈ (0,∞), ω2 ∈ (0,∞), (36) where ϑ•(z(k)) = � z ∞ 0 γ(k)ωk−2 2 ky i=1 zω2 i � kx i=1 zω2 i + (m− k)zω2 k �−k dω2 �−1 (37) is the normalizing constant, f(ω2|z(k)) is given by (21), f(ω1, ω2|z(k)) = �pk i=1 z ω2 i + (m− k)zω2 k �k γ(k) ωk−1 1 exp � − ω1 � kx i=1 zω2 i + (m− k)zω2 k ) �� ω1 ∈ (0,∞), (38) proof. the joint density of x1 ≤ . . . ≤ xk is given by fθ(x1, . . . , xk) = m! (m− k)! ky i=1 δ β ( xi β )δ−1 exp(−( xi β )δ) exp(−(m− k)( xk β )δ). (39) using the invariant embedding technique [8-14], we transform (39) to fθ(x1, . . . , xk)dóβdóδ = m! (m− k)! ky i=1 x−1 i δk ky i=1 �xi β �δ exp � − kx i=1 �xi β �δ − (m− k) �xk β �δ� dóβdóδ =− m! (m− k)! óβóδk ky i=1 x−1 i �δóδ �k−2 ky i=1 �xióδ �δ̂( δ δ̂ )� óβ β �δ(k−1) × exp � − � óβ β �δ h kx i=1 ( xióδ ) δ̂( δ δ̂ ) + (m− k) �xkóδ �δ̂( δ δ̂ ) i�� δ β � óβ β �δ(k−1) dóβ�(− δóδ2 )dóδ2 =− m! (m− k)! óβóδk ky i=1 x−1 i ωk−2 2 ky i=1 x−1 i zω2 i ωk−1 1 exp � − ω1 h kx i=1 zω2 i + (m− k)zω2 k i� dω1ω2. (40) normalizing (40), we obtain (36). this ends the proof. corollary 5.1. if the parameter δ is known then w1 ∼ f(ω1) = kk γ(k) ωk−1 1 exp(−ω1k), ω1 ∈ (0,∞). (41) advances in systems science and applications (2012) vol.12 no.4 383 theorem 6 if in (8) the scale parameter β is unknown, then the predictive probability distribution function of xl based on (óβ, δ) and conditional on fixed xk is given by pδ{xl ≤ xl|xk = xk} =1− (m− k)! (l − k − 1)!(m− l)! l−k−1x j=0 � l − k − 1 j � (−1)j m− l + 1 + j h 1− (m− l + 1 + j) xδl − xδk k óβδ �−k (42) proof.we reduce (8) to pθ{xl ≤ xl|xk = xk} =1− (m− k)! (l − k − 1)!(m− l)! l−k−1x j=0 � l − k − 1 j � (−1)j m− l + 1 + j h exp � − � óβ β �δ xδl − xδkóβδ �im−l+1+j =1− (m− k)! (l − k − 1)!(m− l)! l−k−1x j=0 � l − k − 1 j � (−1)j m− l + 1 + j h exp � − ω1 xδl − xδkóβδ �im−l+1+j . (43) now, we eliminate the unknown parameter β from the problem and find (42) as pδ{xl ≤ xl|xk = xk} = z ∞ 0 pθ{xl ≤ xl|xk = xk}f(ω1)dω1. (44) this ends the proof. corollary 6.1.if the parameter δ = 1, i.e. we deal with the exponential distribution, then the predictive probability distribution function of xl based on óβ and conditional on fixed xk is given by xk p{xl ≤ xl|xk = xk} = 1− 1 b(l − k,m− l + 1) l−k−1x j=0 � l − k − 1 j � (−1)j m− l + 1 + j h 1 + (m− l + 1 + j) xl − xk k óβ i−k , (45) where k óβ = kx i=1 xi + (m− k)xk. (46) theorem 7 if in (8) both parameters β and δ are unknown, then the predictive probability distribution function of xl based on (wideparenβ,wideparenδ) and 384 konstantin n. nechval:predictive inferences for a future number of failures coming from... conditional on fixed xk and z(k) is given by pθ{xl ≤ xl|xk = xk; z (k)} = 1− (m− k)! (l − k − 1)!(m− l)! × z ∞ 0 l−k−1x j=0 � l − k − 1 j � (−1)j m− l + 1 + j h 1 + (m− l + 1 + j) ��xlóβ �δ̂ω2 − �xkóβ �δ̂ω2 �� kx i=1 zω2 i + (m− k) zω2 i �−1i−k × f(ω2|z(k))dω2. (47) proof. we reduce (8) to pθ{xl ≤ xl|xk =xk} = 1− (m− k)! (l − k − 1)!(m− l)! l−k−1x j=0 � l − k − 1 j � (−1)j m− l + 1 + j � exp � − � óβ β �δh�xlóβ �δ̂( δ δ̄ ) − �xkóβ �δ̂( δ δ̄ ) i��m−l+1+j =1− (m− k)! (l − k − 1)!(m− l)! l−k−1x j=0 � l − k − 1 j � (−1)j m− l + 1 + j exp h� − ω1 ��xlóβ �δ̂ω2 − �xkóβ �δ̂ω2 ��im−l+1+j . (48) now, we eliminate the unknown parameters β and δ from the problem and find (47) as pθ{xl ≤ xl|xk = xk; z (k)} = z ∞ 0 z ∞ 0 pθ{xl ≤ xl|xk = xk}f(ω1, ω2|z(k))dω1dω2 = z ∞ 0 z ∞ 0 pθ{xl ≤ xl|xk = xk}f(ω1|ω2, z (k))f(ω2|z(k))dω1dω2. (49) this ends the proof. under conditions of theorem 7, the lower prediction limit for the number of units that will fail in the time interval [tc, tω] is given by llower = lmax − k, (50) where lmax = max k<l≤m arg � p{xl > tω|xk = xk; z (k))} ≤ α � , (51) advances in systems science and applications (2012) vol.12 no.4 385 the upper prediction limit for the number of units that will fail in the time interval [tc, tω] is given by lupper = lmin − k − 1, (52) lmin = min k<l≤m arg � p{xl > tω|xk = xk; z (k))} ≥ 1− α � , (53) in the above case, when both parameters β and δ are unknown, the prediction limits (lower and upper) for the number of units that will fail in the time interval [tc, tω] are based on (óβ, óδ) and conditional on fixed xk,z (k). if l, which satisfies (51), does not exist then lmax = k and the lower prediction limit for the number of units that will fail in the time interval [tc, tω] is given by llower = 0. if l, which satisfies (53), does not exist then lmin = m+ 1 and upper prediction limit for the number of units that will fail in the time interval [tc, tω] is given by lupper = m− k. 5 numerical example for the sake of simplicity, but without loss of generality, we consider (for illustration) the special case of theorem 2 where m = 40 items simultaneously tested have life times, which follow the weibull distribution with δ = 1 . in other words, we deal with the exponential distribution. two items have failed by the inspection at times, x1 = 45 and x2 = 100 hours. let us assume that the situation takes place when tc = xk = 100 hours, where k = 2. suppose, say, tω = 450 hours. taking into account (15), we find the lower prediction limit for the number of units that will fail in the time interval [tc, tω] as llower = lmax − k = 3− 2 = 1, (54) lmax = max k<l≤m arg � p{xl > tω} ≤ α � = 3, α = 0.05 (55) p{xl > tω} = m! (l − k − 1)!(m− l)! l−k−1x j=0 � l − k − 1 j � (−1)j m− l + 1 + j� k−1y s=0 h� tω xk − 1 � (m− l + 1 + j) + (m− k + 1 + s) i�−1 , (56) the upper prediction limit for the number of units that will fail in the time interval [tc, tω] is given by lupper = lmin − k − 1 = 17− 2− 1 = 14, (57) lmin = min k<l≤m arg � p{xl > tω|xk = xk; z (k))} ≥ 1− α � = 17. (58) 386 konstantin n. nechval:predictive inferences for a future number of failures coming from... it will be noted that when both parameters β and δ are unknown, the lower and upper prediction limits for the number of units that will fail in the time interval [tc, tω] can be found either from (29) and (31), which are based on (xk, óδ),or from (50) and (52), which are based on (óβ, óδ). conclusion and future work the methodology described here can be extended in several different directions to handle various problems that arise in practice. we have illustrated the prediction method for log-location-scale distributions (such as the weibull or exponential distributions). application to other distributions could follow directly. acknowledgements this research was supported in part by grant no. 06.1936, grant no. 07.2036, grant no. 09.1014, and grant no. 09.1544 from the latvian council of science and the national institute of mathematics and informatics of latvia. references [1] nelson w. (2000), “weibull prediction of a future number of failures”, quality reliability engineering international, vol.16, pp.23-26. [2] hahn g.j and nelson w. (1973), “a survey of prediction intervals and their applications”, journal of quality technology, vol.5, pp.178-188. [3] patel j.k. (1989), “prediction intervals-a review”, communications in statistics. theory and methods, vol.18, pp.2393-2465. [4] hahn g.j and meeker w.q. (1991), statistical intervals: a guide for practitioners, new york: wiley. [5] nordman d.j and meeker w.q. (2002), “weibull prediction for a future number of failures”, technometrics, vol.44, pp.15-23. [6] nechval n.a, nechval k.n and vasermanis e.k. (2003), “effective state estimation of stochastic systems”, kybernetes (an international journal of systems & cybernetics), vol.32, pp.666-678 1218-1224. [7] nechval n.a, purgailis m, berzins g, cikste k, krasts j, and nechval k.n. (2010), “invariant embedding technique and its applications for improvement or optimization of statistical decisions”, in: k. al-begain, d. fiems and w. knottenbelt (eds.), analytical and stochastic modeling techniques and applications, lncs, vol.6148, berlin, heidelberg: springer-verlag, pp.306320. advances in systems science and applications (2012) vol.12 no.4 387 [8] nechval n.a, nechval k.n and purgailis m. (2011), ‘prediction of future values of random quantities based on previously observed data”, engineering letters, vol.9, pp.346-359. [9] nechval n.a and purgailis m. (2010), “improved state estimation of stochastic systems via a new technique of invariant embedding”, in: chris myers (ed.) stochastic control, publisher: sciyo, croatia, india, pp.167-193. [10] nechval n. a, nechval k. n, purgailis m, strelchonok v. f. (2011), “planning inspections in the case of damage tolerance approach to service of fatigued aircraft structures”, international journal of performability engineering, vol.7, pp.279-290. [11] nechval n.a, purgailis m, nechval k.n, and strelchonok v.f. (2012), “optimal predictive inferences for future order statistics via a specific loss function”, iaeng international journal of applied mathematics, math.1180, springer, berlin, vol.42, pp.40-51. [12] nechval n.a, purgailis m, nechval k.n, and bruna i. (2012), “optimal inventory control under parametric uncertainty via cumulative customer demand”, in:lecture notes in engineering and computer science: proceedings of the world congress on engineering, wce vol.i, pp.4-6 july, london, u.k, pp.6-11. [13] nechval n.a, purgailis m, nechval k.n and bruna i. (2012), “optimal prediction intervals for future order statistics from extreme value distributions”, in: lecture notes in engineering and computer science: proceedings of the world congress on engineering 2012, wce 2012, vol.iii, pp.4-6 july, london, u.k, pp.1340-1345. [14] nechval n.a and purgailis m. (2012), “stochastic control and improvement of statistical decisions in revenue optimization systems”, in:stochastic control, ivan ganchev ivanov (ed.), croatia, india, publisher: sciyo, pp.151-176. [15] paramonov yu.m (1992), methods of mathematical statistics in problems on the estimation and maintenance of fatigue life of aircraft structures, (in russian), riga: riiga. corresponding author konstantin n. nechval can be contacted at:e-mail: konstan@tsi.lv мягкая вероятность ошибки при передаче информации advances in systems science and application(2016) vol.16 no.4 1-12 analysis method of structural equation modeling a. v. burkov 1 , e. a. murzina 2 1 applied statistics and informatics department, mari state university, lenina 1, yoshkar-ola, 424002, mari el, russia 2 department of management and law, volga state university of technology, lenina 3, yoshkar-ola, 424002, mari el, russia annotation the goal of this article is to define a set of factors which most strongly influence a probability of finding (opening) self-employment by graduates of high schools of the united states of america. different methods for reducing the number of independent variables are reviewed and compared. first, it is the correlation analysis. this method was used for the assessment of these factors’ influence on self-employment of graduates of high schools of the united states of america. secondly, it is the classical factor analysis, and finally it is the confirmatory factor analysis with application of structural equations. these methods were used to decrease the quantity of the variables influencing self-employment. confirmatory factor analysis with application of structural equations was used for the confirmation of the received results. as the data base the information of the national science foundation of the united states of america was used. moreover in this article the factors which have the most significant influence on selfemployment of american graduates are described. keywords: correlation analysis, factor analysis, structural equations, higher education, labor market. 1 introduction in the modern world education is one of the most important prerequisites of successful development of society. sustained economic growth of any country and its competitiveness in the times of globalization of world economy are impossible without highly educated and highly skilled laborers. when considering the tasks facing higher education and statistics of education, the development of a methodological basis of statistical analysis of higher education expert labor market and the employment of such specialists for jobs in their degree field seems potentially productive and extremely important. it is important for national and international studies. the experience of the usa is interesting from the point of view that if in the russian federation the introduction of a two-stage system of higher education is still developing, in the usa the similar experience of training specialists is already obtained. to our mind, special attention should be paid to the criteria of education quality which is used in the usa. the usa is a recognized leader among other countries with market economy, it is a country with a highly developed private sector. a considerable share of concentration of medium-sized and small business is in many respects caused by the mentality: in the usa the choice of speciality is influenced by possibilities of employment, but the prestige of a business also influences the education one’s gets. 2 a. v. burkov, e. a. murzina : analysis method of structural equation modeling. the organization of business by graduates from american higher education institutions is influenced by the whole set of factors. taking into consideration everything said before, at the first stage of our research it is necessary to define factors influencing the basis of a new business organized by american university graduates. 2 research methodology the methods used in the research to find out factors which most strongly influence a probability of finding (opening) self-employment by graduates of high schools in the usa are the following: correlation analysis, eigenvalue method, factor analysis. to prove the results the method of structural equations was used. the data base was the information of the national science foundation of the usa (http://www.nsf.org). 3 findings and discussion our goal is to define a set of factors which most strongly influence a probability of finding (opening) self-employment by graduates of high schools in the usa. we used the database on experts with higher education of the national science foundation of the usa. there are 447 parameters for each graduate. for the construction of a regression model we cannot use all of 447 parameters from the database, therefore, it is necessary to make a reduction of data which is carried out in several stages. on the basis of expert estimation, the parameters which have no influence on the process of self-employment have been removed from the database. for example: ”taking courses during last (reference) week“, ”living in the usa during last (reference) week“ or ”reason for working less than 35 hours a week“. at the given stage the number of parameters was reduced from 447 to 224. using the correlation analysis (pair correlation), from 224 parameters those which most strongly influence the probability of self-employment have been selected. as a result, 36 parameters have been selected with the module of pair correlation coefficients of 0,1 or more (table 1). the parameters with high autocorrelation have been removed from the selected 36 parameters: for example, parameters ”age group [5 year intervals]” and ”year, date of birth [recoded for public use]” as they correlate with the “age” parameter (the value of correlation is 0,99). moreover, some variables can be grouped together: for example, ”employer size“ or ”type of educational institution [employer]” (table 1). hence, such parameters cannot be used as independent variables and should be removed. after the removal of highly correlated parameters and parameters that can be grouped together only 18 parameters were left. it is noteworthy that at first there were 100 000 cases in the database, but after the removal of cases with missing values, there were 569 cases left in the database. to check the received representative sampling we compared histograms of distribution of the variable "age", both for the general set (fig.1) and for sampling (fig.2). so, the histogram of distribution of the variable “age” for the general set does not considerably differ from the same histogram for sampling, it is possible to make a conclusion about the representative sampling. on the basis of the remained 18 parameters the 3 and 2 factor models were received using factor analysis. the kaiser criterion (kaiser, 1960) based on eigenvalues was used to find out the number of factors (table2) and the cattell’s scree test (cattell, 1966). advances in systems science and application(2016) vol.16 no.4 3 table 1 correlation ot the parameters name descriptions included group correlations age age x 0,11 agegr age group [5 year intervals] x 0,11 biryrp year, date of birth [recoded for public use] -0,11 emsec sm employer sector [summary code] x 0,31 emsec pb employer sector [recoded for public use] x 0,29 hdacy r academic year of highest degree -0,14 dgryr year of highest degree x -0,14 hdacy 3 year of highest degree [3 year intervals] x -0,14 hday5 year of highest degree [5 year intervals] x -0,14 hsyr year of receiving high school diploma x -0,11 acdrg type of degree valid during the week of oct. 1 x -0,12 actrd t activity, research, development, and teaching x -0,14 acttc h activity, teaching x -0,16 emed employer is an educational institution x -0,25 emsize employer size x -0,49 edtp type of educational institution [employer] x 0,33 fptind full-time/part-time status including all jobs during the reference week x -0,13 nedtp type of a non-educational institution [employer] x -0,57 nrfam reason for working outside the highest degree field: family-related reasons x 0,10 nrocn a reason for working outside the highest degree field: a desired job is not available x -0,10 waacc work activities on principal job: accounting, finance, contracts x 0,16 waprs m summarized primary work activity x 0,15 wasvc work activities in the principal job: professional services x 0,13 wasal e work activities in the principal job: sales, purchasing, marketing x 0,15 watea work activities in the principal job: teaching x -0,14 newbu s new business x 0,25 baacy r academic year of first bachelor degree -0,11 bayr year of first bachelor’s degree x -0,11 mrdac yr academic year of most recent degree -0,14 mryr year of most recent degree x -0,14 mr3yr year of most recent degree [3-year intervals] x -0,14 mr5yr year of most recent degree [5-year intervals] x -0,14 d2ayr degree award date based on academic year -0,12 d2yr year of second highest degree x -0,12 4 a. v. burkov, e. a. murzina : analysis method of structural equation modeling. d3ayr degree award date based on academic year -0,10 d3yr year of third highest degree x -0,10 histogram (spreadsheet в workbook1.stw 225v*100402c) age = 100402*5*normal(x; 46,9476; 11,8435) 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 age 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 n o o f o b s fig. 1 histogram of distribution of the variable "age" for the general set histogram (spreadsheet в workbook1.stw 23v*569c) age = 569*5*normal(x; 49,7856; 10,4602) 20 25 30 35 40 45 50 55 60 65 70 75 80 age 0 20 40 60 80 100 120 n o o f o b s fig.2 histogram of distribution of the variable "age" for sampling table 2 eigenvalues of factors eigenvalues extraction: principal components eigen value % total variance cumulative eigenvalue cumulati ve % 1 6,3724 88 35,40271 6,37249 35,40271 2 2,9252 16 16,25120 9,29770 51,65391 3 1,3341 97 7,41221 10,63190 59,06611 4 1,1181 76 6,21209 11,75008 65,27820 5 1,0623 68 5,90204 12,81244 71,18025 the scree plot of eigenvalues is given in fig.3. from the table of eigenvalues it is clear, that the greatest share of variance is being described by the first two factors, the same fact is confirmed by the plot, it has excesses on the second and third points (fig.3). hence, it is most logical to consider the 2 and 3 factor models. of the factor analysis methods we selected the principal component analysis with various variants of rotation of axes for the 3 factor and 2 factor models. for rotation of axes the following methods were used: unrotated, varimax raw, varimax normalized, biquartimax raw, advances in systems science and application(2016) vol.16 no.4 5 biquartimax normalized, quartimax raw, quartimax normalized, equamax raw and equamax normalized. plot of eigenvalues number of eigenvalues 0 1 2 3 4 5 6 7 8 v a lu e fig.3 the scree plot of eigenvalues as the criterion of of the rotation of axes method, the values of factor loadings were used. the method with the greatest values of factor loadings is considered the best. factor loading is considered to be high if its value is 0,5 or more. the variable joined in the factor which has the greatest factor loading. moreover, the best model should have a minimal correlation between factors. after analyzing models with various variants of rotation of axes, we came to the conclusion, that the optimal method is varimax normalized, both for the 3 factor and 2 factor models. factor loadings for the given models are given in table 3 and table 4 (the most significant loadings are in bold type). relying on factor loadings, it is possible to assign variables to those factors. let’s have a closer look at the received models and describe them in detail. the received 3 factor model describes dependence of probability of self-employment on the following 3 factors: ”experience“, ”the attitude to education and science“ and ”business characteristics“. the factor «experience» describes work experience of a graduate and is linear approximation of the following characteristics of a high school graduate: ”age“, ”year of highest degree“, ”year of receiving a high school diploma“, ”year of first bachelor’s degree“, ”year of most recent degree“, ”year of second highest degree“, ”year of third highest degree“. it is noteworthy that factor loadings of variables for this factor are higher than 0,9, that shows good approximation of variables by the given factor. the factor ”the attitude to education and science“ shows the attitude to education and science, how much the activity of a graduate is connected with education and science, it is linear approximation of the following characteristics: ”activity, research, development, and teaching“, ”activity, teaching“, ”employer is an educational institution“, ”work activities in the principal job: teaching“. factor loadings for this factor are not so unequivocal if compared with the previous ones, but all their values are high and not lower than 0,6. table 3 factor loadings for the 3 factor model factor loadings (varimax normalized) clusters of loadings are marked; they determine the oblique factors for hierarchical analysis factor 1 factor 2 factor 3 age 0,95299 6 0,06146 8 0,02065 4 dgryr 0,92170 1 0,082739 0,06004 9 table 4 factor loadings for the 2 factor model factor loadings (varimax normalized) clusters of loadings are marked; they determine the oblique factors for hierarchical analysis factor 1 factor 2 age 0,954366 0,012021 dgryr 0,916927 0,135735 6 a. v. burkov, e. a. murzina : analysis method of structural equation modeling. hsyr 0,95689 6 0,06113 0 0,02092 9 actrd t 0,044594 0,65320 6 0,280236 acttc h 0,04986 3 0,90217 4 0,01144 5 emed 0,02353 2 0,80138 2 0,049644 fptind 0,11708 2 0,16849 9 0,36340 0 nrfam 0,057464 0,16323 3 0,52586 1 nrocn a 0,09272 4 0,06485 9 0,28800 0 waac c 0,01964 4 0,346718 0,52497 4 wasv c 0,03913 7 0,049291 0,27536 7 wasal e 0,003421 0,279992 0,58086 5 wate a 0,03008 7 0,82129 8 0,14496 0 newb us 0,077575 0,102705 0,36437 0 bayr 0,97749 3 0,05961 8 0,03345 3 mryr 0,92306 0 0,084538 0,05954 2 d2yr 0,95271 9 0,02518 6 0,04798 4 d3yr 0,95504 6 0,07871 4 0,07030 6 hsyr 0,958248 0,011453 actrdt 0,022482 0,700040 acttch 0,105162 0,874567 emed 0,068952 0,790894 fptind 0,149831 0,079367 nrfam 0,014094 0,046620 nrocn a 0,114510 0,004039 waacc 0,031704 0,453740 wasvc 0,053400 0,110096 wasal e 0,016311 0,399886 watea 0,088993 0,767415 newbu s 0,060479 0,176327 bayr 0,979465 0,006301 mryr 0,918140 0,137439 d2yr 0,953615 0,029312 d3yr 0,960593 0,017869 finally, the third factor «business characteristics» describes a new business where a graduate is employed, it is linearization of characteristics: ”full-time/part-time status including all jobs during the reference week“, ”reason for working outside the highest degree field: family-related reasons“, ”reason for working outside the highest degree field: a desired job is not available“, ”work activities in the principal job: accounting, finance, contracts“, ”work activities in the principal job: professional services“, ”work activities in the principal job: sales, purchasing, marketing“, ”new business“. factor loadings for the variables of this factor are not so significant, they all exceed two times the values of factor loadings for other factors and for some variables do not exceed 0,15. the received 2 factor model describes the dependence of probability of self-employment on the following 2 factors: «experience and environment conditions» and «business characteristics». the first factor «experience and environment conditions» describes work experience of a graduate and work conditions. it is linear approximation of the following characteristics: “age“, “year of highest degree“, “year of receiving a high school diploma“, ”full-time/part-time status including all jobs during the reference week“, ”reason for working outside the highest degree field: a desired job is not available“, ”year of first bachelor’s degree“, ”year of most recent degree“, ”year of second highest degree“ and ”year of third highest degree“. practically all factor loadings of variables for the given factor have high value of more than 0,9, except for factor loadings for parameters ”full-time/part-time status including all jobs during the reference week“ and ”reason for working outside the highest degree field: a desired job ia not available“, the value of their factor loadings does not exceed 0,15. this fact shows the low influence of these parameters on this factor. the second factor ”business characteristics“, as well as the third factor in the 3 factor models characterizes the business in which a graduate is employed. the given factor is linearization of the following parameters: ”activity, research, development, and teaching“, ”activity, advances in systems science and application(2016) vol.16 no.4 7 teaching“, ”employer is an educational institution”, ”reason for working outside the highest degree field: family-related reasons“, ”work activities in the principal job: accounting, finance, contracts“, ”work activities in the principal job: professional services“, ”work activities in the principal job: sales, purchasing, marketing“, ”work activities in the principal job: teaching“, ”new business“. the situation with factor loadings for this factor is similar to the first factor of the given model. factor loadings change within 0,8-0,04, and the parameter ”reason for working outside the highest degree field: family-related reasons” has the least influence on the given factor. below there are given tables of correlations between factors (table 5 and table 6), both for the 3 factor and the 2 factor models. table 5 correlations between factors for the 3 factor model correlations between oblique factors (clusters of variables with unique loadings) fact or 1 facto r 2 facto r 3 fac tor 1 1,00 0000 0,033 756 0,088 356 fac tor 2 0,03 3756 1,000 000 0,182457 fac tor 3 0,08 8356 0,182457 1,000 000 table 6 correlations between factors for the 2 factor model correlations between oblique factors (clusters of variables with unique loadings) factor 1 factor 2 factor 1 1,000000 0,022367 factor 2 0,022367 1,000000 the level of correlation between factors in the both models is low enough, this confirms their importance. in the 3 factor model the correlation between factors is 0,03, 0,08 and -0,18 accordingly, and in the 2 factor model the correlation between factors is -0,02. if to consider the correlation between probability of self-employment and the received factors (table 7 and table 8), we can come to the following conclusions. first, taking into consideration the 3 factor model it is obvious that the third factor has the greatest influence on probability of self-employment, it is followed by the first factor and then the second one. secondly, with the growth of the first or third factors, the value of probability decreases, and with the growth of the second factor, the probability increases. thirdly, in the 2 factor model the both factors have equally negative influence on probability of self-employment. table 7 correlation between probability of self-employment and the received factors for the 3 factor model correlations marked correlations are significant at p <, 05000 n=569 (casewise deletion of missing data) factor 1 factor 2 factor 3 selfemp l -0,20 0,11 -0,37 table 8 correlation between probability of self-employment and the received factors for the 2 factor model correlations marked correlations are significant at p <, 05000 n=569 (casewise deletion of missing data) factor 1 factor 2 selfempl -0,22 -0,22 if to compare the 3 factor and the 2 factor models, it is possible to draw the following conclusion: though the correlation between factors in the 2 factor model is less than in the 3 factor model, from the point of view of the explanation of factor value, the 3 factor model is better. we will verify the results of the factor analysis using structural equations. reliability of the received factors was verified by the confirming factor analysis using structural equations. using the structural equations for both the 3 factor and the 2 factor models, we employed the method of maximum likelihood estimation together with the method of least squares, with absence of correlation between factors and the residuals since low correlation between factors was given above (table 5 and table 6). the data for the analysis was the matrix 8 a. v. burkov, e. a. murzina : analysis method of structural equation modeling. of correlations between parameters. the results of the analysis of the constructed structural equations were the following. table 9 the confirming factor analysis using structural equations for the 3 factor model model estimates para meter estim ate stan dard erro r t statistic p rob. l evel (experience)-1> [age] 0,980 0,00 7 1,409422e+02 0 ,000 (experience)-2> [dgryr] 1,000 0,00 0 8,086041e +03 0 ,000 (experience)-3> [hsyr] 0,980 0,00 7 1,416687e +02 0 ,000 (experience)-4> [bayr] 0,985 0,00 5 1,880929e +02 0 ,000 (experience)-5> [mryr] 1,000 0,00 0 8,033911e +03 0 ,000 (experience)-6> [d2yr] 0,988 0,00 4 2,397669e +02 0 ,000 (experience)-7> [d3yr] 0,977 0,00 8 1,200892e +02 0 ,000 (teaching)-8> [actrdt] 0,814 0,08 2 9,930843e +00 0 ,000 (teaching)-9> [acttch] 1,000 0,00 0 1,475717e +09 0 ,000 (teaching)-10> [emed] 0,930 0,03 3 2,829592e +01 0 ,000 (teaching)-11> [watea] 0,951 0,02 3 4,060509e +01 0 ,000 (business)-12> [fptind] 0,013 0,27 1 4,958209e -02 0 ,960 (business)-13> [nrfam] 0,085 0,27 0 3,138192e -01 0 ,754 (business)-14> [nrocna] 0,193 0,26 3 -7,332709e-01 0 ,463 (business)-15> [waacc] 0,813 0,27 5 2,961953e +00 0 ,003 (business)-16> [wasvc] 0,169 0,26 5 6,393056e -01 0 ,523 (business)-17> [wasale] 0,806 0,27 3 2,948677e +00 0 ,003 (business)-18> [newbus] 0,272 0,25 5 1,066804e +00 0 ,286 (delta1)-19 (delta1) 0,040 0,01 4 2,915270e +00 0 ,004 (delta2)-20 (delta2) 0,000 0,00 0 2,041707e -01 0 ,838 (delta3)-21 (delta3) 0,040 0,01 4 2,915235e +00 0 ,004 (delta4)-22 (delta4) 0,030 0,01 0 2,913469e +00 0 ,004 (delta5)-23 (delta5) 0,000 0,00 0 3,872706e -01 0 ,699 (delta6)-24 (delta6) 0,024 0,00 8 2,912069e +00 0 ,004 (delta7)-25 (delta7) 0,046 0,01 6 2,916457e +00 0 ,004 (delta8)-26 (delta8) 0,338 0,13 3 2,533464e +00 0 ,011 (delta9)-27 (delta9) 0,000 0,00 0 (delta10)-28 (delta10) 0,135 0,06 1 2,217215e +00 0 ,027 (delta11)-29 0,097 0,04 2,168875e 0 advances in systems science and application(2016) vol.16 no.4 9 (delta11) 5 +00 ,030 (delta12)-30 (delta12) 1,000 0,00 7 1,368791e +02 0 ,000 (delta13)-31 (delta13) 0,993 0,04 6 2,173207e +01 0 ,000 (delta14)-32 (delta14) 0,963 0,10 1 9,498401e +00 0 ,000 (delta15)-33 (delta15) 0,338 0,44 7 7,571400e -01 0 ,449 (delta16)-34 (delta16) 0,971 0,09 0 1,082963e +01 0 ,000 (delta17)-35 (delta17) 0,351 0,44 0 7,979269e -01 0 ,425 (delta18)-36 (delta18) 0,926 0,13 8 6,699364e +00 0 ,000 for the 3 factor model the results are described in table 9 (the significant facts are in bold type), and the normal probability plot of residuals is given in fig.4. from table 9 it is obvious that the ways for the factors “experience” and “the attitude to education and science” are significant as they have high value of t-statistics and low probability. hence, it is possible to conclude that the considered model precisely describes the set forth above factors. however, we can observe that the factor “business characteristics” is not so well described by the given model, not all ways for this factor have high t-statistics and low probability. for additional verification of the importance of the model the normal probability plot of residuals (fig.4) has been constructed. from the plot it is obvious that the residuals of model are situated close to the straight line of normal distribution that confirms the importance of the constructed model. on the whole we can conclude that the results received, using the diagram of ways for the 3 factor model, coincide with the results of the factor analysis for this model which proves their correctness. table 10 the confirming factor analysis using structural equations for the 2 factor model model estimates param eter estima te stan dard erro r t statistic p rob. l evel (experience)-1> [age] -1,000 0,00 0 2,083354e+05 0 ,000 (experience)-2> [dgryr] 0,980 0,01 0 1,020890e +02 0 ,000 (experience)-3> [hsyr] 1,000 0,00 0 3,205600e +12 0 ,000 (experience)-4> [fptind] 0,254 0,22 7 1,117756e +00 0 ,264 (experience)-5> [nrocna] 0,136 0,23 8 5,715618e -01 0 ,568 (experience)-6> [bayr] 0,997 0,00 2 6,361259e +02 0 ,000 (experience)-7> [mryr] 0,980 0,01 0 1,000551e +02 0 ,000 (experience)-8> [d2yr] 0,995 0,00 3 3,952639e +02 0 ,000 (experience)-9> [d3yr] 0,997 0,00 1 6,970720e +02 0 ,000 (business)-10> [actrdt] 0,814 0,08 2 9,930843e +00 0 ,000 (business)-11> [acttch] 1,000 0,00 0 3,545235e +02 0 ,000 (business)-12> [emed] 0,930 0,03 3 2,829592e +01 0 ,000 (business)-13> [nrfam] 0,027 0,24 2 1,124848e -01 0 ,910 (business)-14> -0,588 0,15 0 10 a. v. burkov, e. a. murzina : analysis method of structural equation modeling. [waacc] 9 3,700352e+00 ,000 (business)-15> [wasvc] -0,164 0,23 6 -6,967788e-01 0 ,486 (business)-16> [wasale] -0,530 0,17 4 3,040265e+00 0 ,002 (business)-17> [watea] 0,951 0,02 3 4,060509e +01 0 ,000 (business)-18> [newbus] -0,346 0,21 4 1,617683e+00 0 ,106 (delta1)-19 (delta1) 0,000 0,00 0 2,061573e +00 0 ,039 (delta2)-20 (delta2) 0,040 0,01 9 2,103603e +00 0 ,035 (delta3)-21 (delta3) 0,000 0,00 0 (delta4)-22 (delta4) 0,936 0,11 5 8,127443e +00 0 ,000 (delta5)-23 (delta5) 0,981 0,06 5 1,515202e +01 0 ,000 (delta6)-24 (delta6) 0,006 0,00 3 2,068245e +00 0 ,039 (delta7)-25 (delta7) 0,040 0,01 9 2,104467e +00 0 ,035 (delta8)-26 (delta8) 0,010 0,00 5 2,072333e +00 0 ,038 (delta9)-27 (delta9) 0,006 0,00 3 2,067659e +00 0 ,039 (delta10)-28 (delta10) 0,338 0,13 3 2,533464e +00 0 ,011 (delta11)-29 (delta11) 0,000 0,00 0 (delta12)-30 (delta12) 0,135 0,06 1 2,217215e +00 0 ,027 (delta13)-31 (delta13) 0,999 0,01 3 7,562196e +01 0 ,000 (delta14)-32 (delta14) 0,655 0,18 7 3,508443e +00 0 ,000 (delta15)-33 (delta15) 0,973 0,07 8 1,253796e +01 0 ,000 (delta16)-34 (delta16) 0,719 0,18 5 3,888715e +00 0 ,000 (delta17)-35 (delta17) 0,097 0,04 5 2,168875e +00 0 ,030 (delta18)-36 (delta18) 0,881 0,14 8 5,966712e +00 0 ,000 let’s now consider the 2 factor model. the parameters of the 2 factor model are given in table 10 and the normal probability plot of residuals is given in fig.5. as well as in the factor analysis, the factor “experience and environment conditions” is well enough described by the diagram of ways, practically all the ways have high t-statistics and low probability. the exceptions are only the ways for the parameters “full-time/part-time status including all jobs during the reference week” and “reason for working outside the highest degree field: a desired job is not available”. the second factor is not so well described by the given model. the ways for the parameters “reason for working outside the highest degree field: family-related reasons”, “work activities in the principal job: professional services” and “new business” have low tstatistics and high probability. the normal probability plot of residuals for this model is practically similar to the normal probability plot of residuals for the 3 factor model this proves that the model conforms to real data. on the whole, the model confirms the results of the factor analysis. considering all the results described above we can draw the conclusion that the received structure of correlation between the parameters and the factors which include them coincides with the structure of correlation of the factor analysis for the 3 factor and the 2 factor models. hence, advances in systems science and application(2016) vol.16 no.4 11 the results of the confirming factor analysis using structural equations, on the whole, confirm the results of the factor analysis. normal probability plot normalized residuals -3,0 -2,5 -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 value -3 -2 -1 0 1 2 3 e x p e c te d n o rm a l v a lu e fig. 4 the normal probability plot of residuals for the 3 factor model normal probability plot normalized residuals -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 value -3 -2 -1 0 1 2 3 e x p e c te d n o rm a l v a lu e fig. 5 the normal probability plot of residuals for the 2 factor model further on, on the basis of the received factors the regression models showing dependence of probability of self-employment on given factors will be constructed. for it various methods are to be used to find out the best regression model for binary responses. references [1] brown m.b., and forsythe a.b(1974). 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(2015) , "the system of parameters efficiency of financial supervision". advances in systems science and application vol.15 no.4, pp.384-391. corresponding author elena a. murzina can be contacted at: elena.murzina@gmail.com. advances in systems science and applications (2011), vol. 11, no. 1-2 149-162 monotone hybrid methods for a finite family of nonexpasive multi-valued maps and equilibrium problems ∗ suthep suantai and watcharaporn cholamjiak department of mathematics, faculty of science, chiang mai university, chiang mai 50200, thailand centre of excellence in mathematics, che, si ayutthaya rd., bangkok 10400, thailand email: scmti005@chiangmai.ac.th, c-wchp007@hotmail.com abstract in this paper, we introduce a new monotone hybrid iterative scheme for finding a common element of the set of common fixed points of a finite family of nonexpansive multivalued maps and the set of the solutions of the equilibrium problem in a hilbert space. moreover, we also introduce a new iterative scheme for finding a common fixed point of a finite family of nonexpansive multi-valued maps in a banach space. strong convergence theorem of the proposed iteration is established. keywords nonexpansive multi-valued map monotone hybrid method, 1. introduction letd be a nonempty convex subset of a banach spacee. let f be a bifunction fromd×d to r, where r is the set of all real number. the equilibrium problem for f is to find x ∈ d such that f(x, y) ≥ 0 for all y ∈ d. the set of such solutions is denoted by ep (f). the set d is called proximinal if for each x ∈ e, there exists an element y ∈ d such that ‖x−y‖ = d(x,d), where d(x,d) = inf{‖x − z‖ : z ∈ d}. let cb(d),k(d) and p (d) denote the families of nonempty closed bounded subsets, nonempty compact subsets, and nonempty proximinal bounded subsets of d, respectively. the hausdorff metric on cb(d) is defined by h(a,b) = max { sup x∈a d(x,b), sup y∈b d(y,a) } for a,b ∈ cb(d). a single-valued map t : d → d is called nonexpansive if ‖tx− ty‖ ≤ ‖x − y‖ for all x, y ∈ d. a multi-valued map t : d → cb(d) is said to be nonexpansive if h(tx, ty) ≤ ‖x−y‖ for all x, y ∈ d. an element p ∈ d is called a fixed point of t : d → d (respectively, t : d → cb(d)) if p = tp (respectively, p ∈ tp). the set of fixed points of t is denoted by f (t ). the mapping t : d → cb(d) is called quasi-nonexpansive[18] if f (t ) 6= ∅ and h(tx, tp) ≤ ‖x − p‖ for all x ∈ d and all p ∈ f (t ). it is clear that every nonexpansive multi-valued map t with f (t ) 6= ∅ is quasi-nonexpansive. but there exist quasi-nonexpansive mappings that are not nonexpansive, see [17]. the mapping t : d → cb(d) is called hemicompact if, for any sequence {xn} in d such that d(xn, txn) → 0 as n → ∞, there exists a subsequence {xnk } of {xn} such that xnk → ∗this research is supported by the centre of excellence in mathematics, the commission on higher education, thailand, the thailand research fund and the graduate school of chiang mai university for the financial support. issn 1078-6236 international institute for general systems studies, inc. 1 2 1 2 150 suantai:monotone hybrid methods for a finite family of nonexpasive. . . . . . p ∈ d. we note that if d is compact, then every multi-valued mapping t : d → cb(d) is hemicompact. a mapping t : d → cb(d) is said to satisfy condition (i) if there is a nondecreasing function f : [0,∞)→ [0,∞) with f(0) = 0, f(r) > 0 for r ∈ (0,∞) such that d(x, tx) ≥ f(d(x, f (t ))) for all x ∈ d. a family {ti : d → cb(d), i = 1, 2, ..., n} is said to satisfy condition (ii) if there is a nondecreasing function f : [0,∞)→ [0,∞) with f(0) = 0, f(r) > 0 for r ∈ (0,∞) such that d(x, tix) ≥ f(d(x, n⋂ i=1 f (ti))) for all i = 1, 2, ..., n and x ∈ d. in 1953, mann [10] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping t in a hilbert space h: xn+1 = αnxn + (1− αn)txn, ∀n ∈ n, (1) where the initial point x0 is taken in c arbitrarily and {αn} is a sequence in [0,1]. however, we note that mann’s iteration process (1) has only weak convergence, in general; for instance, see [1, 7, 15]. in 2003, nakajo and takahashi [12] introduced the method which is the so-called cq method to modify the process (1) so that strong convergence is guaranteed have recently been made. they also proved a strong convergence theorem for a nonexpansive mapping in a hilbert space. recently, tada and takahashi [20] proposed a new iteration for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping t in a hilbert space h . in 2005, sastry and babu [16] proved that the mann and ishikawa iteration schemes for multi-valued map t with a fixed point p converge to a fixed point q of t under certain conditions. they also claimed that the fixed point q may be different from p. more precisely, they proved the following result for nonexpansive multi-valued map with compact domain. in 2007, panyanak [13] extended the above result of sastry and babu [16] to uniformly convex banach spaces but the domain of t remains compact. later, song and wang [19] noted that there was a gap in the proofs of theorem 3.1(see [13]) and theorem 5 (see [16]). they further solved/revised the gap and also gave the affirmative answer to panyanak [13] question using the following ishikawa iteration scheme. in the main results, domain of t is still compact, which is a strong condition (see [19], theorem 1) and t satisfies condition(i) (see [19], theorem 1). in 2009, shahzad and zegeye [17] extended and improved the results of panyanak [13], sastry and babu [16] and song and wang [19] to quasi-nonexpansive multi-valued maps. they also relaxed compactness of the domain of t and constructed an iteration scheme which removes advances in systems science and applications (2011), vol. 11, no. 1-2 151 the restriction of t namely tp = {p} for any p ∈ f (t ). the results provided an affirmative answer to panyanak [13] question in a more general setting. in the main results, t satisfies condition(i)(see [17], theorem 2.3) and t is hemicompact and continuous (see [17], theorem 2.5). question: how can we modify iteration process for a nonexpansive multi-valued map t which the domain of t is not necessary to be compact to obtain strong convergence theorems for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of t ? in the recent years, the problem of finding a common element of the set of solutions of equilibrium problems and the set of fixed points in the framework of hilbert spaces and banach spaces have been intensively studied by many authors, for instance, see [2, 3, 4, 5, 6, 8, 14, 20] and the references cited theorem. in this paper, we introduce a monotone hybrid iterative scheme for finding a common element of the set of a common fixed points of a finite family of nonexpasive multi-valued maps and the set of solutions of an equilibrium problem in a hilbert space. let d be nonempty, closed and convex subset of a hilbert space h and αi n ∈ (0, 1) for all i = 0, 1, ...,m with∑m i=0 α i n = 1, ∀n ≥ 0 and rn ∈ (0,∞). for an initial point x0 ∈ d = c0, compute the sequence {xn} by the iterative process f(un, y) + 1 rn 〈y − un, un − xn〉 ≥ 0, ∀y ∈ d, yn = ∑m i=0 α i nz i n, z i n ∈ tiun, ∀i = 1, 2, ...,m, z0n = un, cn+1 = {z ∈ cn : ‖yn − z‖ ≤ ‖xn − z‖}, xn+1 = pcn+1x0, n ≥ 0, (2) where ti is a nonexpansive multi-valued map for all i = 1, 2, ...,m. 2. preliminaries the following lemmas give some characterizations and a useful property of the metric projection pd in a hilbert space. let h be a real hilbert space with inner product 〈·, ·〉 and norm ‖ · ‖. let d be a closed and convex subset of h . for every point x ∈ h , there exists a unique nearest point in d, denoted by pdx, such that ‖x− pdx‖ ≤ ‖x− y‖, ∀y ∈ d. pd is called the metric projection of h onto d. we know that pd is a nonexpansive mapping of h onto d. lemma 2.1. [11] let d be a closed and convex subset of a real hilbert space h and let pd be the metric projection from h onto d. given x ∈ h and z ∈ d. then z = pdx if and only if the following holds: 〈x− z, y − z〉 ≤ 0, ∀y ∈ d. lemma 2.2. [12] let d be a nonempty, closed and convex subset of a real hilbert space h and pd : h → d be the metric projection from h onto d. then the following inequality holds: ‖y − pdx‖2 + ‖x− pdx‖2 ≤ ‖x− y‖2, ∀x ∈ h, ∀y ∈ d. 152 suantai:monotone hybrid methods for a finite family of nonexpasive. . . . . . lemma 2.3. [11] let h be a real hilbert space. then the following equations hold: (i) ‖x− y‖2 = ‖x‖2 − ‖y‖2 − 2〈x− y, y〉, ∀x, y ∈ h; (ii) ‖tx+ (1− t)y‖2 = t‖x‖2 + (1− t)‖y‖2 − t(1− t)‖x− y‖2, ∀t ∈ [0, 1] and x, y ∈ h . by using lemma 2.3, we obtain the following lemma. lemma 2.4. let h be a real hilbert space. then for each m ∈ n ‖ m∑ i=1 tixi‖2 = m∑ i=1 ti‖xi‖2 − m∑ i=1,i 6=j titj‖xi − xj‖2, xi ∈ h and ti, tj ∈ [0, 1] for all i, j = 1, 2, ...,m with ∑m i=1 ti = 1. lemma 2.5. [9]letd be a nonempty, closed and convex subset of a real hilbert spaceh . given x, y, z ∈ h and also given a ∈ r, the set {v ∈ d : ‖y − v‖2 ≤ ‖x− v‖2 + 〈z, v〉+ a} is convex and closed. for solving the equilibrium problem, we assume the bifunction f : d × d → r satisfies the following conditions: (a1) f(x, x) = 0 for all x ∈ d; (a2) f is monotone, i.e., f(x, y) + f(y, x) ≤ 0 for all x, y ∈ d; (a3) for each x, y, z ∈ d, lim supt↓0 f(tz + (1− t)x, y) ≤ f(x, y); (a4) f(x, ·) is convex and lower semicontinuous for each x ∈ d. lemma 2.6. [2] let d be a nonempty, closed and convex subset of a real hilbert space h . let f be a bifunction from d ×d to r satisfying (a1)-(a4) and let r > 0 and x ∈ h . then, there exists z ∈ d such that f(z, y) + 1 r 〈y − z, z − x〉 ≥ 0, for all y ∈ d. lemma 2.7. [6] for r > 0, x ∈ h , defined a mapping tr : h → 2d as follows: tr(x) = { z ∈ d : f(z, y) + 1 r 〈y − z, z − x〉 ≥ 0, for all y ∈ d } . then the followings hold: (1) tr is single-value; (2) tr is firmly nonexpansive, i.e., for any x, y ∈ h , ‖trx− try‖2 ≤ 〈trx− try, x− y〉; (3) f (tr) = ep (f); (4) ep (f) is closed and convex. advances in systems science and applications (2011), vol. 11, no. 1-2 153 lemma 2.8. let d be a closed and convex subset of a real hilbert space h . let t : d → cb(d) be a nonexpansive multi-valued map with f (t ) 6= ∅ and tp = {p} for each p ∈ f (t ). then f (t ) is a closed and convex subset of d. proof. first, we will show that f (t ) is closed. let {xn} be a sequence in f (t ) such that xn → x as n→∞. we have d(x, tx) ≤ d(x, xn) + d(xn, tx) ≤ d(x, xn) +h(txn, tx) ≤ 2d(x, xn). it follows that d(x, tx) = 0, so x ∈ f (t ). next, we show that f (t ) is convex. let p = tp1 + (1− t)p2 where p1, p2 ∈ f (t ) and t ∈ (0, 1) . let z ∈ tp, by lemma 2.3, we have ‖p− z‖2 = ‖t(z − p1) + (1− t)(z − p2)‖2 = t‖z − p1‖2 + (1− t)‖z − p2‖2 − t(1− t)‖p1 − p2‖2 = td(z, tp1) 2 + (1− t)d(z, tp2) 2 − t(1− t)‖p1 − p2‖2 ≤ th(tp, tp1) 2 + (1− t)h(tp, tp2) 2 − t(1− t)‖p1 − p2‖2 ≤ t‖p− p1‖2 + (1− t)‖p− p2‖2 − t(1− t)‖p1 − p2‖2 = t(1− t)2‖p1 − p2‖2 + (1− t)t2‖p1 − p2‖2 − t(1− t)‖p1 − p2‖2 = 0, hence p = z. therefore p ∈ f (t ). lemma 2.9. [21] let p > 1, r > 0 be two fixed numbers. then a banach space e is uniformly convex if and only if there exists a continuous, strictly increasing, and convex function g : [0,∞)→ [0,∞) with g(0) = 0 such that ‖λx+ (1− λ)y‖p ≤ λ‖x‖p + (1− λ)‖y‖p − ωp(λ)g(‖x− y‖), for all x, y ∈ br(0) = {x ∈ e : ‖x‖ ≤ r} and λ ∈ [0, 1] where ωp(λ) = λ(1−λ)p+λp(1−λ). by using lemma 2.9, we can prove the following lemma by induction. lemma 2.10. let e be a uniformly convex banach space and br(0) = { x ∈ e : ‖x‖ ≤ r} be a closed ball of e. then there exists a continuous strictly increasing convex function g : [0,∞)→ [0,∞) with g(0) = 0 such that ‖ m∑ i=1 αixi‖2 ≤ m∑ i=1 αi‖xi‖2 − α1α2g(‖x1 − x2‖), for all m ∈ n, xi ∈ br(0) and αi ∈ [0, 1], i = 1, 2, ...,m with ∑m i=1 αi = 1. by interchanging the roles of vectors xi in lemma 2.10 and summing the inequalities together we obtain the following lemma. 154 suantai:monotone hybrid methods for a finite family of nonexpasive. . . . . . lemma 2.11. let e be a uniformly convex banach space and br(0) = {x ∈ e : ‖x‖ ≤ r} be a closed ball of e. then there exists a continuous strictly increasing convex function g : [0,∞)→ [0,∞) with g(0) = 0 such that for each j ∈ {1, 2, ...,m}, ‖ m∑ i=1 αixi‖2 ≤ m∑ i=1 αi‖xi‖2 − αj m− 1 ( m∑ i=1 αig(‖xj − xi‖) ) , for all m ∈ n, xi ∈ br(0) and αi ∈ [0, 1] for all i = 1, 2, ...,m with ∑m i=1 αi = 1. proof. let j ∈ {1, 2, ...,m} be fixed. by lemma 2.10, there is a continuous strictly increasing convex function g : [0,∞)→ [0,∞) with g(0) = 0 such that ‖α1x1 + α2x2 + α3x3 + ...+ αmxm‖2 ≤ m∑ i=1 αi‖xi‖2 − αjα1g(‖xj − x1‖) ‖α1x1 + α3x3 + α4x4 + ...+ α2x2‖2 ≤ m∑ i=1 αi‖xi‖2 − αjα2g(‖xj − x2‖) ... ‖α1x1 + α4x4 + α5x5 + ...+ α3x3‖2 ≤ m∑ i=1 αi‖xi‖2 − αjαj−1g(‖xj − xj−1‖) ‖α1x1 + α4x4 + α5x5 + ...+ α3x3‖2 ≤ m∑ i=1 αi‖xi‖2 − αjαj+1g(‖xj − xj+1‖) ... ‖α1x1 + αmxm + α2x2 + ...+ αm−1xm−1‖2 ≤ m∑ i=1 αi‖xi‖2 − αjαmg(‖xj − xm‖). by summing up above inequalities, we obtain ‖ m∑ i=1 αixi‖2 ≤ m∑ i=1 αi‖xi‖2 − αj m− 1 ( m∑ i=1 αig(‖xj − xi‖) ) . 3. main result first, we prove a strong convergence theorem for a finite family of nonexpansive multivalued mappings which satisfies the condition (ii) in a uniformly convex banach space. theorem 3.1. let d be a nonempty, closed and convex subset of a uniformly convex banach spacee. let ti : d → cb(d) be a nonexpansive multi-valued map for all i = 1, 2, ...,m with⋂m i=1 f (ti) 6= ∅ and tip = {p} for each p ∈ ⋂m i=1 f (ti). assume that {ti : i = 1, 2, ...,m} satisfies the condition (ii) for all i = 1, 2, ...,m and αi n ∈ (0, 1) with 0 < lim infn→∞ α i n ≤ advances in systems science and applications (2011), vol. 11, no. 1-2 155 lim supn→∞ α i n < 1 for all i = 0, 1, 2, ...,m. let x0 ∈ d and let {xn} be the sequence in d generated by iteration process: xn+1 = m∑ i=0 αi nz i n, (3) where z0n = xn, zin ∈ tixn for all i = 1, 2, ...,m and ∑m i=0 α i n = 1. then {xn} converges strongly to a common fixed point of ti, i = 1, 2, ...,m. proof. let p ∈ ⋂m i=1 f (ti). by the nonexpansiveness of ti, we have ‖xn+1 − p‖ ≤ m∑ i=0 αi n‖zin − p‖ = α0 n‖xn − p‖+ m∑ i=1 αi nd(zin, tip) ≤ α0 n‖xn − p‖+ m∑ i=1 αi nh(tixn, tip) ≤ ‖xn − p‖, (4) which implies that limn→∞ ‖xn − p‖ exists. for each i = 1, 2, ...,m, we have ‖zin − p‖ = d(zin, tip) ≤ h(tixn, tip) ≤ ‖xn − p‖. it follows that {‖zin − p‖} is bounded for all i = 1, 2, ...,m. put r = max1≤i≤m{supn ‖zin− p‖}. by lemma 2.11, there is a continuous strictly increasing convex function g : [0,∞)→ [0,∞) with g(0) = 0 such that ‖xn+1 − p‖2 ≤ m∑ i=0 αi n‖zin − p‖2 − α0 n m m∑ i=1 αi ng ( ‖zin − xn‖ ) = α0 n‖xn − p‖2 + m∑ i=1 αi nd(zin, tip) 2 − α0 n m m∑ i=1 αi ng ( ‖zin − xn‖ ) ≤ α0 n‖xn − p‖2 + m∑ i=1 αi nh(tixn, tip) 2 − α0 n m m∑ i=1 αi ng ( ‖zin − xn‖ ) ≤ ‖xn − p‖2 − α0 n m m∑ i=1 αi ng ( ‖zin − xn‖ ) . it follows that α0 n m m∑ i=1 αi ng ( ‖zin − xn‖ ) ≤ ‖xn − p‖2 − ‖xn+1 − p‖2. this implies that g ( ‖zin − xn‖ ) → 0 as n → ∞ for all i = 1, 2, ...,m. since g is continuous strictly increasing with g(0) = 0, we can conclude that ‖zin − xn‖ → 0 as n → ∞ for all i = 1, 2, ...,m. also d(xn, tixn) ≤ ‖zin − xn‖ → 0 as n → ∞ for all i = 1, 2, ...,m. since that {ti}mi=1 satisfies the condition (ii), we have d(xn, ⋂m i=1 f (ti))→ 0. thus there is a 156 suantai:monotone hybrid methods for a finite family of nonexpasive. . . . . . subsequence {xnk } of {xn} such that ‖xnk − pk‖ < 1 2k for some {pk} ⊂ ⋂m i=1 f (ti) and all k. from (4), we obtain ‖xnk+1 − pk‖ ≤ ‖xnk − pk‖ < 1 2k . next, we shall show that {pk} is cauchy sequence in d. notice that ‖pk+1 − pk‖ ≤ ‖pk+1 − xnk+1 ‖+ ‖xnk+1 − pk‖ < 1 2k+1 + 1 2k < 1 2k−1 . this implies that {pk} is cauchy sequence in d and thus converges to q ∈ d. since d(pk, tiq) ≤ h(tiq, tipk) ≤ ‖q − pk‖ for all i = 1, 2, ...,m and pk → q as n→∞, it follows that d(q, tiq) = 0 for all i = 1, 2, ...,m and thus q ∈ ⋂m i=1 f (ti) and {xnk } converges strongly to q. since limn→∞ ‖xn − q‖ exists, it follows that {xn} converges strongly to q. this completes the proof. note that in theorem 3.1 in order to have strong convergence of the iterative sequence {xn} defined by (3), we need to assume that {ti}mi=1 satisfy the condition (ii). in the following theorem, we introduce a new monotone hybrid iterative scheme (2) for finding a common element of the set of a common fixed points of a family of nonexpasive multi-valued maps and the set of solutions of an equilibrium problem in a hilbert space, and we prove strong convergence of the sequence {xn} defined by (2) without the condition (ii). theorem 3.2. let d be a nonempty, closed and convex subset of a real hilbert space h . let f be a bifunction from d × d to r satisfying (a1)-(a4) and let ti : d → cb(d) be nonexpansive multi-valued maps for all i = 1, 2, ...,m with ⋂m i=1 f (ti) ∩ ep (f) 6= ∅ and tip = {p} for each p ∈ ⋂m i=1 f (ti). assume that αi n ∈ (0, 1) with 0 < lim infn→∞ α i n ≤ lim supn→∞ α i n < 1 for all i = 0, 1, 2, ...,m and rn ∈ (0,∞) with lim infn→∞ rn > 0. then the sequence {xn} generated by (2) converges strongly to p⋂m i=1 f (ti)∩ep (f)x0. proof. we split the proof into six steps. step 1. show that pcn+1x0 is well defined for every x0 ∈ d. by lemma 2.8, we obtain that ⋂m i=1 f (ti) is a closed and convex subset of d. since ep (f) is also closed and convex, then ⋂m i=1 f (ti) ∩ ep (f) is a closed and convex subset of d. from the definition of cn+1, it follows from lemma 2.5 that cn+1 is closed and convex for each n ≥ 0. let v ∈ ⋂m i=1 f (ti) ∩ ep (f). from un = trnxn, we have ‖un − v‖ = ‖trnxn − trnv‖ ≤ ‖xn − v‖, (5) for every n ≥ 0. from this, we have ‖yn − v‖ = ‖ m∑ i=0 αi nz i n − v‖ ≤ m∑ i=0 αi n‖zin − v‖ = α0 n‖un − v‖+ m∑ i=1 αi nd(zin, tiv) ≤ α0 n‖un − v‖+ m∑ i=1 αi nh(tiun, tiv) ≤ ‖un − v‖ ≤ ‖xn − v‖. (6) advances in systems science and applications (2011), vol. 11, no. 1-2 157 so, we have v ∈ cn+1, thus ⋂m i=1 f (ti)∩ep (f) ⊂ cn+1. therefore pcn+1x0 is well defined. step 2. show that limn→∞ ‖xn − x0‖ exists. since ⋂m i=1 f (ti) ∩ ep (f) is a nonempty, closed and convex subset of h , there exists a unique v ∈ ⋂m i=1 f (ti) ∩ ep (f) such that v = p⋂m i=1 f (ti)∩ep (f)x0. from xn = pcnx0, cn+1 ⊂ cn and xn+1 ∈ cn, ∀n ≥ 0, we get ‖xn − x0‖ ≤ ‖xn+1 − x0‖, ∀n ≥ 0. on the other hand, as ⋂m i=1 f (ti) ∩ ep (f) ⊂ cn, we obtain ‖xn − x0‖ ≤ ‖v − x0‖, ∀n ≥ 0. it follows that the sequence {xn} is bounded and nondecreasing. therefore limn→∞ ‖xn−x0‖ exists. step 3. show that xn → w ∈ d as n→∞. for m > n, by the definition of cn, we see that xm = pcmx0 ∈ cm ⊂ cn. by lemma 2.2, we get ‖xm − xn‖2 ≤ ‖xm − x0‖2 − ‖xn − x0‖2. from step 2, we obtain that {xn} is cauchy. hence, there exists w ∈ d such that xn → w as n→∞. step 4. show that ‖zin − xn‖ → 0 as n→∞ for every i = 1, 2, ..,m. from xn+1 ∈ cn+1, we have ‖xn − yn‖ ≤ ‖xn − xn+1‖+ ‖xn+1 − yn‖ ≤ 2‖xn − xn+1‖ → 0 (7) 158 suantai:monotone hybrid methods for a finite family of nonexpasive. . . . . . as n→∞. for v ∈ ⋂m i=1 f (ti) ∩ ep (f), by lemma 2.4 we have ‖yn − v‖2 = ‖ m∑ i=0 αi nz i n − v‖2 ≤ m∑ i=0 αi n‖zin − v‖2 − m∑ i=1 α0 nα i n‖zin − un‖2 ≤ α0 n‖un − v‖2 + m∑ i=1 αi nd(zin, tiv)2 − m∑ i=1 α0 nα i n‖zin − un‖2 ≤ α0 n‖un − v‖2 + m∑ i=1 αi nh(tiun, tiv)2 − m∑ i=1 α0 nα i n‖zin − un‖2 ≤ ‖un − v‖2 − m∑ i=1 α0 nα i n‖zin − un‖2 ≤ ‖xn − v‖2 − m∑ i=1 α0 nα i n‖zin − un‖2. this implies that m∑ i=1 α0 nα i n‖zin − un‖2 ≤ ‖xn − v‖2 − ‖yn − v‖2 ≤ m‖xn − yn‖, where m = supn≥0{‖xn − v‖+ ‖yn − v‖}. by our assumptions and (7), we obtain ‖zin − un‖ → 0 as n→∞, ∀i = 1, 2, ...,m. (8) from lemma 2.7, we obtain ‖un − v‖2 = ‖trnxn − trnv‖2 ≤ 〈trnxn − trnv, xn − v〉 = 〈un − v, xn − v〉 = 1 2 { ‖un − v‖2 + ‖xn − v‖2 − ‖xn − un‖2 } , hence ‖un − v‖2 ≤ ‖xn − v‖2 − ‖xn − un‖2. advances in systems science and applications (2011), vol. 11, no. 1-2 159 therefore, by lemma 2.4, we get ‖yn − v‖2 = ‖ m∑ i=0 αi nz i n − v‖2 ≤ m∑ i=0 αi n‖zin − v‖2 ≤ α0 n‖un − v‖2 + m∑ i=1 αi nd(zin, tiv)2 ≤ α0 n‖un − v‖2 + m∑ i=1 αi nh(tiun, tiv)2 ≤ ‖un − v‖2 ≤ ‖xn − v‖2 − ‖xn − un‖2. it follows that ‖xn − un‖2 ≤ ‖xn − v‖2 − ‖yn − v‖2 ≤ m‖xn − yn‖, where m = supn≥0{‖xn − v‖+ ‖yn − v‖}. from (7), we obtain ‖xn − un‖ → 0 as n→∞. (9) from (8) and (9), we have ‖xn − zin‖ ≤ ‖xn − un‖+ ‖un − zin‖ → 0 as n→∞. (10) step 5. show that w ∈ ⋂m i=1 f (ti) ∩ ep (f). from (9) and lim infn→∞ rn > 0, we get ‖xn − un rn ‖ = 1 rn ‖xn − un‖ → 0 , n→∞. (11) from xn → w as n → ∞ and (9), we obtain also that un → w. we shall show that w ∈ ep (f). by un = trnxn, we get f(un, y) + 1 rn 〈y − un, un − xn〉 ≥ 0, ∀y ∈ d. from the monotonicity of f , we have 1 rn 〈y − un, un − xn〉 ≥ f(y, un), ∀y ∈ d, hence 〈y − un, un − xn rn 〉 ≥ f(y, un), ∀y ∈ d. from (11) and condition (a4), we have 0 ≥ f(y, w), ∀y ∈ d. 160 suantai:monotone hybrid methods for a finite family of nonexpasive. . . . . . for t with 0 < t ≤ 1 and y ∈ d, let yt = ty+ (1− t)w. since y, w ∈ d and d is convex, then yt ∈ d and hence f(yt, w) ≤ 0. so, we have 0 = f(yt, yt) ≤ tf(yt, y) + (1− t)f(yt, w) ≤ tf(yt, y). dividing by t, we obtain f(yt, y) ≥ 0, ∀y ∈ d. letting t ↓ 0 and from (a3), we get f(w, y) ≥ 0, ∀y ∈ d. therefore, we obtain w ∈ ep (f). next, we will show that w ∈ ⋂m i=1 f (ti). for each i = 1, 2, ...,m, we have d(w, tiw) ≤ ‖w − xn‖+ ‖xn − zin‖+ d(zin, tiw) ≤ ‖w − xn‖+ ‖xn − zin‖+h(tiun, tiw) ≤ ‖w − xn‖+ ‖xn − zin‖+ ‖un − w‖. it follows from step 4 that d(w, tiw) = 0 and thus w ∈ f (ti) for all i = 1, 2, ...,m. step 6. show that w = p⋂m i=1 f (ti)∩ep (f)x0. since xn = pcnx0, by lemma 2.1, we have 〈z − xn, x0 − xn〉 ≤ 0 for all z ∈ cn. since w ∈ ⋂m i=1 f (ti) ∩ ep (f) ⊂ cn, we get 〈z − w, x0 − w〉 ≤ 0 for all z ∈ ⋂m i=1 f (ti)∩ep (f). again by lemma 2.1, we obtain thatw = p⋂m i=1 f (ti)∩ep (f)x0. this completes the proof. corollary 3.3. let d be a nonempty, closed and convex subset of a real hilbert space h . let f be a bifunction from d × d to r satisfying (a1)-(a4) and let t : d → cb(d) be a nonexpansive multi-valued map for all i = 1, 2, ...,m with f (t ) ∩ ep (f) 6= ∅ and tp = {p} for each p ∈ f (t ). assume that βn ∈ (0, 1) with 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1 and rn ∈ (0,∞) with lim infn→∞ rn > 0. for an initial point x0 ∈ d = c0, compute the sequence {xn} by the iterative process f(un, y) + 1 rn 〈y − un, un − xn〉 ≥ 0, ∀y ∈ d, yn = βnun + (1− βn)zn, zn ∈ tun, cn+1 = {z ∈ cn : ‖yn − z‖ ≤ ‖xn − z‖}, xn+1 = pcn+1x0, n ≥ 0, then the sequence {xn} converges strongly to pf (t )∩ep (f)x0. proof. putting t1 = t and ti = i for i = 2, 3, ...,m, where i : d → cb(d) such that id = {d} for all d ∈ d in theorem 3.2, we obtain the desired result directly from theorem 3.2. advances in systems science and applications (2011), vol. 11, no. 1-2 161 corollary 3.4. let d be a nonempty, closed and convex subset of a real hilbert space h . let t : d → cb(d) be a nonexpansive multi-valued map for all i = 1, 2, ...,m with f (t ) ∩ ep (f) 6= ∅ and tp = {p} for each p ∈ f (t ). assume that βn ∈ (0, 1) with 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1. for an initial point x0 ∈ d = c0, compute the sequence {xn} by the iterative process yn = βnxn + (1− βn)zn, zn ∈ txn, cn+1 = {z ∈ cn : ‖yn − z‖ ≤ ‖xn − z‖}, xn+1 = pcn+1x0, n ≥ 0, then the sequence {xn} converges strongly to pf (t )x0. proof. putting f(x, y) = 0 for all x, y ∈ d in corollary 3.3, we obtain the desired result directly from corollary 3.3. the main result of this paper holds true under the assumption that tp = {p} for all p ∈ f (t ). this condition was introduced by shahzad and zegeye [17]. the following example gives an example of a nonexpansive multi-valued map t which satisfies the property that tp = {p} for all p ∈ f (t ) and tx is not a singleton for all x /∈ f (t ). example. consider d = [0, 1]× [0, 1] with the usual norm. define t : d → cb(d) by t (x, y) =  {(x, 0)}, x 6= 0, y = 0 {(0, y)}, x = 0, y 6= 0 {(x, 0), (0, y)}, x, y 6= 0 {(0, 0)}, x, y = 0. open problem: can we drop the condition that tp = {p} for all p ∈ f (t ) in the main result of this paper ? references [1] bauschke,h. h., matoušková,e., reich,s. 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(1991). inequality in banach spacees with applications. nonlinear. anal. 16 (1991) 1127-1138. advances in systems science and application (2015) vol.15 no.3 279-298 simd acceleration of spmv kernel on multi-core cpu architecture j.saira banu and dr m.rajasekhara babu school of computing science and engineering, vit university, vellore, india. abstract as the field of sparse matrix vector multiplication (spmv) matures and its breadth of application increases, the need for parallel implementation becomes necessary. spmv is proved to be a bottleneck due to its irregular access patterns. various storage formats for sparse matrices have been proposed to solve this issue. the quad tree-compressed row storage (qcsr) format shows good performance improvement over other formats such as compressed row storage (csr) and blocked csr (bcsr) for spmv. this paper extends qcsr format to exploit the single instruction multiple data (simd) registers which are available in current processors. programming with simd registers in a single core processor achieves parallelism with reduced power and without any additional hardware requirement, as in the case of graphics processing unit (gpu) computing. to program effectively in simd units intels streaming simd extension (sse) instructions are used. in this paper computational performance of qcsr-spmv is determined for over a collection of 10 benchmark matrices on simd units of x86 architecture. experimental results demonstrate qcsr-simd achieves significant average speedup of 2.0 x compared to csr-simd. keywords openmp; gpu; sparse matrix; spmv; csr; qcsr; simd; sse; performance; programming model; optimization; parallel computing. 1 introduction there are multiple ways of accomplishing parallel computing in single machine to cloud environment. in single machine, parallel computing is achieved through thread level parallelism with open multiprocessing (openmp), message passing method with message passing interface (mpi) and through simd technique with sse instruction. increasing clock speed has been ceased recently which gave a way for multi or many cores cpu. the other alternative method to achieve many core computing is through gpu. in gpu, parallelization is achieved with the expense of new hardware. parallel computing in cloud environment is realized by distributing the computing task to various computing resources and gathering the result. in cloud computing allocating task is a major issue. guiyi wei, et al addressed this issue of allocating the task to various computing resources in the cloud[1]. it involves allocation matrix and expense matrix. if a matrix used in an algorithm is dense then there is no issue but algorithm involving sparse matrix is a concern one. 280 j.saira banu and m.rajasekhara babu:simd acceleration of spmv kernel on ... a lot of work is being done in the field of sparse matrices and spmv, to improve the efficiency. emphasis has been laid on the parallel implementation of spmv in the past decade. sparse matrix structures are present in various applications, and efficient method for optimizing the performance of these applications is a critical issue. spmv is an operation found in many computational science applications. improving the speedup of spmv boosts the performance of these applications. extensive research has been done in improving the performance of spmv using thread level and distributed memory parallelism. samuel williams et al. examined spmv on various multicore design namely amd quad core, intel quad core[2], etc. shengfei liu, et al. worked on spmv with compressed sparse row (csr) and blocked csr (bcsr) format[3]. they have implemented multithreaded spmv using openmp. partitioning the matrices in to sub matrices and hybrid programming model with mpi and openmp are the future methods proposed in the paper to increase the performance of the kernel. dakuan discussed spmv with mpi[4]. gerald schubert et al. examined the performance of spmv kernel with hybrid programming (openmp and mpi) model[5]. this paper does not explore the load balancing issues. m.krotiewski and m.dabrowski presented a parallel implementation of spmv on multicores with wellknown optimization such as matrix reordering, blocking and prefetching[6]. with the rise of high performance computing, most of the focus is now on gpu computing. in literature, different formats of storage have been developed to either improve the space efficiency, or the time of access of non-zero elements for various operations. some formats are specific to the execution environment such as cpu or gpu. tomas oberhuber et al. proposed a new format such as new row grouped csr which performed well in gpu devices[7]. jilin zhang et al. have implemented spmv using a new type of storage format, quad tree csr (qcsr) format using gpus[8]. this format outperforms the blocked csr (bcsr) format. recently vectorization capability using sse instructions of the current processor proved to be better technology to improve the performance of many applications. susana ladra, et al. exploited simd instructions in current processors to improve the classical string algorithms[9]. kai zeng et al. proved the usage of simd technique in computed tomography ct reconstruction is efficient rather than gpu implementation[10]. image processing applications such as feature detection, stereo vision class model estimation, and object detection achieved better performance on the compressed sparse extended ( csx ) simd architecture compared with gpu[11]. s.j.pennycook et al. explored the use of simd registers for molecular dynamics problem set[12]. to obtain better performance on numerical kernels, martin kong et al. proposed a 3-step framework such as data locality, multicore parallelism and simd advances in systems science and application (2015) vol.15 no.3 281 execution of programs[13]. libo huang et al. presents a dynamic vectorization method to address the constraints in current simd engines such as register variation in the processors requires changing of vector operands and aligned memory accesses[14]. nasersedaghati proposed an extension to the instruction available in instruction set architecture (isa) to overcome the disadvantage of simd for stencil computation application[15]. neil g. dickson et al. found that explicit vectorization on the cpu gave 9x-12x speedup over the original cpu version, and was also found to be 2x faster than the fully optimized gpu version that uses explicit memory coalescing. these papers signifies the importance of single core optimization on cpu[16]. ji-lin zhang et al. have implemented spmv using csr format with simd methodology and thread-level parallelizing[17]. they found out that these methods gave a speed up of around 2.11 over that of the non-optimized method. kai zhang et al. improved upon this and broke the performance bottlenecks of the simd processors like the low utilization of simd processors and the memory bandwidth[18]. their method showed a good speedup over the normal csr vector kernel. nazligoharian et al. demonstrated that csr is the best storage format for information retrieval and query processing among various storage format of sparse matrix[19].these paper explored simd in spmv and its importance. in this paper, we investigate the impact of simd acceleration on spmv kernel. for this, spmv kernel with csr and qcsr format has been accelerated, i.e., simd is applied to all suitable operations and implemented in haswell processor using sse instructions. the main contribution of this paper are as follows. 1. made an extensive study on strorage formats of sparse matrix structure and analyzed the space complexity for benchmark matrices from different applications. 2. simd acceleration is presented for all the data-parallel kernnels of spmv with csr and qcsr format. 3. implementation and optimization is performed with sse instructions. 4. simd implementation of csr outperforms the na?ve and thread level parallelism. 5. with simd optimization,spmv with qcsr format gives 2 fold of speedup compared to csr-simd. the rest of the paper is organized as follows: section 2 deals with architecture specifications, section 3 describes the spmv algorithm with its storage format analysis; section 4 explores about simd optimization of spmv kernel with csr and qcsr storage formats; section 5 deals with the results and observations; section 6 talks about the conclusion. 282 j.saira banu and m.rajasekhara babu:simd acceleration of spmv kernel on ... 2 architecture specification since the beginning of the processor, its performance has been steadily increasing. developments in semiconductor capability are the main reason for this remarkable change. advancement in architectural features like instructionlevel parallelism, data-level parallelism (dlp) thread-level parallelism (tlp) and memory-level parallelism (mlp) brought remarkable improvement in performance. here, in this paper dlp is achieved through simd, and tlp is done through openmp. 2.1 dlp-simd simd is one of the classifications of the flynns taxonomy of computer architec− ture[20]. it exploits dlp by performing the same operation on multiple data points simultaneously. simd architecture consists of multiple processing elements that perform identical task concurrently on the data elements as shown in fig.1. fig. 1 simd architecture. sse instructions are now available in current generation processors to program effectively with simd registers. these instructions are used effectively to vectorise the code and to modify the data layout. automatic vectorization has certain limitations such as handling loops of irregular access patterns, pointers, etc. since automatic vectorization has restricted applicability, programmers write vectorised code using intrinsic which is directly expanded to machine instructions. programming using intrinsic are tied to a specific instruction set. with the help of these sse instructions, simd programming is performed in high level programming languages such as c. compared to gpu, there is no overhead incurred, like moving data from device to device or thread processing. advances in systems science and application (2015) vol.15 no.3 283 it can also be combined with thread level parallelism technique to increase the speed up further. sse is included from pentium iii processors by adding 128 bit registers and the instructions that can operate on them. advanced intel processors supports avx which includes 256 bit register with extra simd instructions in addition to sse instruction set. sse instruction set includes integer and floating point arithmetic instructions, comparison, shuffling, data type conversion, bitwise operations, minimum, maximum, conditional copies, crc 32 and population count. currently in many applications such as gaming, graphics, physics, and mathematics, simd instructions are used to perform shuffling, scalar product, checksum calculations and complex operations. there are various versions of sse such as ssesse4.2 which are capable of processing 2 double precision floating point numbers or 4 single precision 32 bit floating point numbers. sse intrinsic is used to improve the performance of an application which has fine grain parallelism in it. when dealing with sparse matrices, one of the most important restrictions is that the average width of non-zero elements per row must be greater than the simd width of the computer device. else, the simd registers are only partially utilized, and do not give a considerable increase in the efficiency. 2.2 tlp openmp thread level parallelism is achieved when a single process is divided into various subparts, and each part is carried out by a different thread, so at the end, it could be combined to give the result. openmp is an api that provides shared memory multi-processing programming using high level languages such as c, c++ and fortran. there are various methods to perform thread level parallelism for spmv-csr kernel. parallelism can be done row wise or column wise or block wise. here, we have performed row wise parallelism, where each row is assigned to a thread. the accesses to the data are independent and hence can easily be parallelized. one advantage of using the row wise partitioning is that individual threads operate on different parts of the final resultant array unlike that of the column wise partitioning, where all threads have to write to all parts of the array. 3 spmv algorithm spmv forms a basic computational kernel in many scientific and industrial applications. modern high performance computer systems (hpc) such as multicores, gpu,simd computation using coprocessor and special xmm registers rely on spmv computation for numerous task. spmv is usually represented as y = a ∗x where a represents a sparse matrix and y and x represents a vector. sparse matrix is a matrix which has more number of zero elements than the non-zero values. storing a sparse matrix as it is in a memory is a space overhead and in some applications it doesnt fit in to the available memory. to effectively reduce the storage space many data structures have been proposed in the literature. 284 j.saira banu and m.rajasekhara babu:simd acceleration of spmv kernel on ... these data structure stores only the non-zero elements of the matrix and thereby reduces the computation of spmv algorithm. this is important in hpc community since all modern computer system is equipped with small storage devices. the compression of sparse matrix using efficient storage format influence the performance of spmv operation.for the remainder of the paper, we use the following notation: n : total number of non-zero values n : order of the matrix s( ): storage space occupied nr : number of rows 3.1 storage formats sparse matrices are stored in computer memory with specific formats that give high performance of spmv operation. few of the popular formats of storing sparse matrices with space and time complexity are discussed here.equations 1-5 represents the space complexity of various storage formats. 3.1.1 the coordinate format (coo) this is one of the most general purpose formats. it consists of three 1-dimensional arrays one to store the non-zero values, one to store the corresponding column index, and the other to store the corresponding row index [21]. fig. 2 example matrix. for example, the coo format for the matrix shown in fig.2 is represented as : data = [1 1 1 1 1] col = [1 1 2 2 4] row = [1 2 2 3 3]. a time complexity : the time complexity to convert this format to/from the compressed row storage (csr) format is o (n+n) b space complexity : the space complexity of this format excluding the data array is given by s(coo) = 2 ·n · s(n) (1) advances in systems science and application (2015) vol.15 no.3 285 3.1.2 the compressed row storage format (csr) this is one of the most efficient and hence most popular storage formats. it consists of three 1-dimensional arrays c one to store the non-zero value, one to store the corresponding column index, and the other to store the row pointer value, that gives the total non-zero values in each row [21] for the example in figure 2: csr format is given as: data = [1 1 1 1 1] col = [1 1 2 2 4] row-ptr = [0 1 3 5 5] a. time complexity : the time complexity to convert this format to/from the coordinate (coo) format is o (n+n). b. space complexity : the space complexity of this format excluding the data array is given by s(csr) = n · s(n) + n · s(n) (2) 3.1.3 the quadtree storage format quadtree is a recursive tree data structure given by ivan simecek [22]. the matrix is recursively divided into four quadrants until each block is equal to a predefined size (called density). these blocks or nodes can be of three types: a. empty: the entire node is made up of zeroes. b. mixed: the node consist of a mixture of zeroes and non-zero elements. c. full: the entire node id made up of non-zero elements. empty nodes are ignored, and the mixed and full nodes are stored in any format that is found to be appropriate for the application.the matrix in fig.1, is divided in to four quadrants and each quadrant is checked for the node types. here the second quadrant is empty and it is ignored and all other quadrants are further retrieved through the indices for its operation as shown in fig.3. the advantages of this format are: a. easy conversion from popular formats. b. easy modifications to the data. c. the recursive style of programming leads to better performance due to better cache memory utilization. a. time complexity: the entire algorithm can be broken down into smaller functions. the time complexity of each such function is as given below: 286 j.saira banu and m.rajasekhara babu:simd acceleration of spmv kernel on ... (1) to find the total non-zero values in each quadrant: o (average per row. (y2 y1 + 1)). (2) to check the given quadrant is empty or not:o (log2 average per row. (y2 y1 + 1)). (3) for converting the matrix into quadtree format, the complexity depends on the density. b. space complexity: the space complexity completely depends on the density chosen and the creation of leaves. 3.1.4 the qcsr (quadtree compressed row storage) format the qcsr format is a combination of the quadtree format and the csr format(jilin zhang, etal, 2013)[8]. after dividing the matrix into various nodes using the quadtree logic, the mixed and full nodes are stored in the csr format. though it is found to have a space overhead, the implementation of spmv is much faster in this format. a. time complexity : the time complexity of this format is the same as that of the quadtree format, in addition to which each quadrant is converted into the csr format. b. space complexity : this format gives an overhead in space.the maximum overhead over the csr format is given by: soh(qcsr) = (2sr + sl)xo(4d− 1) (3) where, sr is the space occupied by the index pointer to the region, sl is the region length and d is the maximum depth of the tree. fig. 3 splitting of matrix. 3.1.5 minimal quadtree format: this format is developed by i.simecek et al.[23]. it is a derivative of the quadtree format and consists of a bit stream of 1s and 0s. the matrix is recursively divided advances in systems science and application (2015) vol.15 no.3 287 into blocks as in quadtree format, but here, each block is represented by a single bit0 if all the elements in the block are zeroes, 1 otherwise. the example given in fig. 1, can be divided into 4 blocks if the density is 2, and can be represented as 4 bits1011 for only the second block is an empty block, and the others have non-zero elements. a. time complexity : the total time complexity for this conversion is o(n(n+ √ n)) · log2avg per row b. space complexity : the minimal size of the mqt format is given by: s(mqtmin) = 4 · (n 3 + log4( n2 n )) (4) the maximal size of the mqt format is given by: s(mqtmax) = 4 · (1 3 + log4( n2 n )) (5) as minimization of memory is an important criterion for spmv operation, we compared thespace complexity of various formats for the benchmark matrices obtained from university of florida database [24]. table 1 shows the properties of benchmark matrices used. table 2 depicts the space occupied for these matrices in coo, csr, qcsr and minqcsr format. table 1 overview of the benchmark matrices benchmark matrices number of rows number of columns number of nonzero elements bcsstk07 420 420 4140 adder dcop 08 1813 1813 11242 mhd4800b 4800 4800 16160 meg4 5860 5860 26324 gemat11 4929 4929 33185 cell1 b 7055 7055 34855 ex12 3973 3973 42092 sina 5743 5743 102265 na5 5832 5832 155731 from table 2, it is evident that minimal qcsr, which is just a series of bits is well suited for storing the structure of matrices and hence not suitable for spmv operation. csr is the next format that has the least space complexity and proved to be well utilized in spmv operation. qcsr format has the space overhead but that is not accounted for spmv operation. hence, among all the 288 j.saira banu and m.rajasekhara babu:simd acceleration of spmv kernel on ... formats that have been developed so far, the csr format is the most efficient and effective format. a lot of improvements have been done on this format to improve its efficiency. in this paper, we analyse the csr format along with the qcsr format for spmv computation in simd architecture. table 2 the space complexity of various formats for benchmark matrices benchmark matrices original coo csr qcsr minqcsr bcsstk07 391kb 97kb 76kb 118kb 394b adder dcop 08 6.35mb 326kb 176kb 265kb 999b mhd4800b 44mb 436kb 263kb 416kb 810b meg4 65.7mb 425kb 386kb 854kb 739b gemat11 46.5mb 648kb 506kb 1.2mb 367b cell1 b 95.2mb 587kb 386kb 854kb 739b ex12 30.4mb 1.13mb 635kb 845kb 349b sina 63.6mb 2.95mb 1.52mb 3.19mb 46b na5 65.9mb 4.54mb 2.3mb 4.25mb 384b 3.2 spmv csr spmv kernel acts as a core kernel of many iterative algorithms and scientific applications. it is one of the time consuming kernel in these methods. optimizing this kernel plays a vital role in improving the performance of these applications. the performance of spmv kernel is based on the storage format used. csr format has been proved to be the best format with space efficiency. algorithm 3.1 and 3.2 shows spmv with csr format in naive and thread level parallelism. algorithm 3.1 spmv with csrnaive inputs ptr pointer array data data array column column index array x vector for multiplication output: z result array 1. begin 2. for each item i from 0 nr do 3. z[i] = 0 4. for each item j in ptr[i] to ptr[i+1] -1 do 5. temp = column [j] 6. z[i] = z[i]+ (data[j] * x[temp]) 7. end for 8. end for advances in systems science and application (2015) vol.15 no.3 289 9. end algorithm 3.2 spmv with csr-thread level parallelism inputs ptr pointer array data data array column column index array x vector for multiplication output: z result array 1. begin 2. ♯ pragma omp parallel num threads(4) 3. ♯ pragma omp parallel for private (k, j, i) schedule(static, 10) 4. for i = 0 to n 1 do 5. z[i] = 0 6. for j = ptr[i] to ptr[i+1]-1 do 7. temp = colj 8. z[i] = z[i] + (val[j] * x[temp]) 9. end for 10. end for 11. end 3.3 spmv qcsr as elucidated in section 3.1, the qcsr format is a combination of the quadtree format and the csr format. the given matrix is first divided into various quadrants using the quadtree format, and the mixed and full nodes are converted into the csr format as shown in algorithm 3.3 and 3.4 [8]. here, sr stands for start row, er for end row, sc for start column, ec for end column, a[][] is the input matrix, and density is the size of the quadrant. qcsr format compared to csr format has space overhead. this overhead is due to storage of block and intermediate node information. this overhead accounts for transformation and representation of matrix and not in spmv kernel. this implies that qcsr is suitable for spmv kernel with all general cases. algorithm 3.3 matrix conversion to qcsr format inputs sr start row sc start column 290 j.saira banu and m.rajasekhara babu:simd acceleration of spmv kernel on ... er end row ec end column 1. begin 2. int mr=(sr+er)/2; 3. int mc=(sc+ec)/2; 4. int r=er-sr; 5. int c=ec-sc; 6. if( r≥ density && c ≥ density ) 7. { 8. if( !isempty(sr,mr,sc,mc) ) 9. trans(sr,mr,sc,mc); 10. if( !isempty(sr,mr,(mc+1),ec) ) 11. trans(sr,mr,(mc+1),ec); 12. if( !isempty((mr+1),er,sc,mc) ) 13. trans((mr+1),er,sc,mc); 14. if( !isempty((mr+1),er,(mc+1),ec) ) 15. trans((mr+1),er,(mc+1),ec); 16. } 17. else if(c>density) 18. { 19. trans(sr,er,sc,mc); 20. trans(sr,er,(mc+1),ec); 21. } 22. else if(r>density) 23. { 24. trans(sr,mr,sc,ec); 25. trans((mr+1),er,sc,ec); 26. } 27. else 28. { 29. if( !isempty(sr, er, sc, ec)) 30. convert the quadrant into csr format 31. } 32. end algorithm 3.4 isempty(sr,er,sc,ec) inputs sr start row sc start column advances in systems science and application (2015) vol.15 no.3 291 er end row ec end column 1. begin 2. int isempty() 3. for all elements i from sr to er 4. { 5. for all elements j from sc to ec 6. { 7. if( a[i][j] != 0) 8. return 0 9. } 10. } 11. return 1 12. end 4 simd optimization 4.1 spmv ccsr simd optimization is performed with the help of sse 4.2 instructions. here, nr is the total number of rows in the matrix, val is the array used to store the nonzero elements, col is the array to store the column indices, row is the row pointer array, and x is the vector for multiplication. algorithm 3.5 gives simd optimization of spmv kernel with csr format. in vectorization using simd technique, four values are operated simultaneously, as the size of the simd register is 4. the non-zero elements were loaded into the data register, and the corresponding vector elements in the x register. here data and x are the special xmm registers available in the current generation processors. these values are multiplied simultaneously, hence decreasing the total number of operations performed. algorithm 3.5 spmv with csr-simd inputs nr c number of rows val c data array row c pointer array x vector for multiplication output: z result array 1. begin 292 j.saira banu and m.rajasekhara babu:simd acceleration of spmv kernel on ... 2. for every element i from 0 to nr-1 do 3. for every element j in row[i] to row[i+1]-1 do 4. data = {val[j] , val[j+1], val[j+2], val[j+3]} 5. x = {x[col[j]] , x[col[j+1]], xcol[j+2]], x[col[j+3]]} 6. y = mm mul ps(data, x) 7. zi = sum{y} 8. j = j + 4 9. end for 10. end for 11. end 4.2 spmv -qcsr qcsr format is an improvisation on the quad tree format, in which each of the blocks is stored in the csr format. we have performed sparse matrix vector multiplication using this format. one condition that has to be met to maximize efficiency is that, the total non-zero values in each block should be greater than or equal to the size of the simd register. if not, the remaining bits in the simd register must be padded with zeros which results in time overhead. qcsr with sse instructions is specified in algorithm 3.6. the following instructions are used to vectorize the spmv implementation of qcsr format. 1. mm loadu ps(array) : this is used for loading the registers with the elements in the array. 2. mm mul ps(reg1, reg2) : this is used to perform vectorized multiplication of the elements in the two mentioned registers. the result is stored in reg1. 3. mm hadd ps(reg1, reg2) : this is used to perform vectorized addition of the elements in the two mentioned registers. the result is stored in reg1. 4. mm storeu ps(ptr, reg) : this is used to store the results in the register to the pointer. algorithm 3.6 qcsr with simd optimization each block in qcsr format consists of the following components: startrow: the row index of the first row in the block. startcolumn: the column index of the first column in the block. endrow: the row index of the last row in the block. endcolumn: the column index of the last column in the block. structure csr: csr representation of the block, which contains advances in systems science and application (2015) vol.15 no.3 293 nz: the total non-zeroes values in the block data[n]: data array for the block. ptr[n]: row-pointer array for the block col[n]: column indices array for the block multiply (each block) i: to keep track of the number of rows in the block. k: to keep track of the number of non-zero elements in each row. t: to keep track of the ptr arrays index. temparr: to store the padded data values. data1, vect1, temp, res1: simd registers. res: to store the multiplication result. 1. begin 2. for i from startrow to endrow, do 3. for k from csr.ptr[t] to csr.ptr[t+1], do 4. q= csr.ptr[t+1] c csr.ptr[t]; 5. a = minimum ( q-k, 4) 6. for j from 0 to a, do 7. temparr[j]= csr.data[k+j]; 8. if j < 4 9. pad the remaining places with 0. 10. end if 11. data1= mm loadu ps ( temparr ) 12. vect1= mm loadu ps ( v[col[k]], v[col[k+1]], v[col[k+2]], v[col[k+3]] ) 13. temp= mm mul ps (data1 , vect1) 14. res1= mm hadd ps (temp , temp ) 15. res1= mm hadd ps (res1 , res1) 16. mm storeu ps ( c , res1) 17. res[i] = res[i] + c 18. end for 19. t++ 20. k=k+4 21. end for 22. end 294 j.saira banu and m.rajasekhara babu:simd acceleration of spmv kernel on ... 5 results and discussion in this section we analyse the performance improvement of csr-spmv and qcsrspmv format using instructions included in sse 4.1/4.2 extension. to carry out experiments, we have used intel(r) coretm i5-4200u cpu@ 1.60 ghz (2 cores) with 4 gb ram. visual studio 2010 compiler was used for compilation. the width of the simd register in the test system is 4. in this experiment, the density size has been chosen to be 16 such that the nonzero elements in each block are equal to or greater than the width of simd register. speed-up is a performance index, which is a ratio of the time taken to implement a program sequentially to the time taken to do the same in parallel. we first performed spmv using the csr format both sequentially and in parallel using openmp. in parallel implementation, we have considered the block size to be 10. each thread is given a single row and multiplication of these rows is performed in parallel. one disadvantage of this method is, if the total non-zero elements in one of the rows are much greater than that in the other rows, then stalling of the other threads occurs. the remaining threads have to be idle until the thread with that particular row finishes execution. next, we implemented the same using simd vectorization and compared this with that of sequential execution. the simd registers are of size 128 bits, and hence 4 32-bit integers can be loaded in it at a given time. each row is considered separately, and the nonzero elements in it are loaded into the register, 4 at a time. if the number of non-zero elements left is less than 4, it is padded with zeroes. this overcomes the disadvantage of stalling that occurs in thread level parallelism. another overhead that this method avoids is the time taken for the creation of threads. the efficiency of the execution of spmv using simd completely depends on the arrangement of the non-zero elements in the matrix. fig. 4 shows results obtained when implementing spmv using csr format with simd and openmp techniques. from the graph shown in fig. 4, we determine that the simd vectorization technique gives better results when compared to that of sequential and parallel execution. hence, we implemented simd version of spmv for various benchmark matrices using the qcsr format using this technique of simd vectorization. in this method of implementation, each quadrant is considered separately and is treated as a separate matrix. csr spmv using vectorization, as discussed above, is then carried out for each quadrant. here, the size of the quadrant depends on the density chosen. since the density is usually in powers of two and much smaller than the original matrix size, the probability that the total non-zero elements in each row in the quadrant is less than 4 is quite high. hence the number of times the simd registers have to be loaded and padded is reduced. these results were compared with the advances in systems science and application (2015) vol.15 no.3 295 results obtained from implementing spmv with regular csr format with simd vectorization. fig. 5 shows the graph of the speed-up obtained by the qcsr format. the graph clearly shows that the qcsr format when implemented with simd is more efficient than that of the csr format with simd. fig. 4 computing time for optimization schemes using csr. fig. 5 computing time for optimization schemes using csr. 6 conclusions this paper explores the use of the simd technique provided in current processors, to accelerate the performance of spmv using qcsr format. as simd units are readily available with the current generation processors, and do not incur ad296 j.saira banu and m.rajasekhara babu:simd acceleration of spmv kernel on ... ditional processing cost unlike 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[24] t.davis (1997), “the university of florida sparse matrix collection” [eb/ol], http://www.cise.ufl.edu/research/sparse/matrices. corresponding author j.saira banu can be contacted at: jsairabanu@vit.ac.in microsoft word 13-杨冰陈喜等.doc issn 1078-6236 international institute for general systems studies, inc. real-time 3d smoking simulation based on computational fluid dynamics and gpu computation for virtual surgery� bing yang, xi chen, cheng yang, jiaxiang guo, jie han, jiayu tang and zhiyong yuan school of computer, wuhan university, wuhan 430072, china email: zhiyongyuan@whu.edu.cn (* corresponding author) abstract in this paper a smoking simulation model based on computational fluid dynamics (cfd) for virtual surgery is proposed. navier-stokes equations are used to model smoke as a constant and incompressible fluid. in order to realize the visual effects of real-time 3d smoking, we utilize gpu (graphics processing unit) and cuda (compute unified device architecture) computing technology to perform the massive parallel computation by executing multithreading concurrently. the rendering of the smoke is implemented by the ray-tracing algorithm based on hardware. the visual effects of the real-time dynamic smoking simulation are given by the paper. the results show that the methods applied to the smoking simulation are elucidating and practical. keywords virtual surgery cfd gpu 3d smoking simulation 1. introduction virtual surgery training system based on virtual reality can be used to improve the skill level of the surgeons and to reduce the incidence of complications in surgical practice. the ultimate advantage of this simulation system is to bridge the gap between the basic training of surgeon and the practical operation on patients, which means a real configurable training environment without the limitation of repeated training. however, the current surgical training systems have not yet reached the necessary level of sufficient reality, and the application of the simulation technology to the field of surgery has not yet been widely accepted [1]. the real simulation of smoke is an important research which has not yet been soundly solved. by far, there are few researches referring to smoke simulation in the virtual surgical simulation. considering that smoke is a kind of fluid, a fluid dynamics method is adopted to solve the problems. usually, navier-stokes equations are used to represent the accurate mathematic equations of fluid dynamics [2], but this method is only applicable under simple conditions. with the * the research work described in this paper was supported by the national natural science foundation of china (grant no. 61070079) and the fundamental research funds for the central universities of china (grant no. 6081006). ∗ ∗ advances in systems science and applications (2011), vol.11, no.1-2 173-179 development of computer technology, it has been possible to compute the equations by numerical calculation and as a result, a new subject, computational fluid dynamics, has been born. in 1997, foster and metaxas [3, 4] adopted incompressible navier-stokes equations to a coarse mesh and solved nonlinearity partial differential equations by explicit differential equations, but it was too slow to show real-time animations. in 1999, stam proposed a fluid solution with unconditionally stable and simulated the flow of the fluid with a random time step [5]. because the simulation process of virtual surgery requires great visual effect and excellent quality of real-time, on the premise of great visual effect, the solution to the smoke model built by navier-stokes equations should meet the need of high real-time. therefore, this paper implements a rapid solution to the three-dimensional smoke simulation model by cuda+gpu technology. cuda is a development environment for gpu computation. it is a new software and hardware structure based on c programming language. the code which runs in the display core can be wrote directly by c language and it is unnecessary to learn the special commands or structures of the display core [6]. it can perform large-scale parallel computation by executing multithreading concurrently, which offers a significant opportunity to meet the need of real-time simulation. therefore, this paper accomplishes a rapid solution to navier-stokes equations by cuda technology so as to meet the challenge of real-time. compared with cpu, the result of this paper represents great advantages. to meet the need of real-time quality and real visual effect of the smoke simulation during the surgical training process, this paper adopts a real-time smoke synthesis and simulation model based on computational fluid dynamics. navier-stokes equations are adopted to model smoke as a constant, incompressible fluid with stable temperature. in order to simulate the smoke with a real-time effect, the gauss-seidel iterative method is employed to obtain diffusive items of the density field governing equations. successive over relaxation method is used to derive the amended velocity field which meets the mass conservation condition. and then, ray tracing algorithm is used to implement smoke rendering [7]. for the purpose of getting real-time smoke, the solutions to navier-stokes equations based on gpu are performed by cuda. 2. modeling of smoking simulation 2.1 smoking model smoke state is determined by its density, temperature and velocity. supposing that smoke is incompressible fluids with a stable temperature, then the smoking model can be expressed by incompressible navier-stokes governing equations: continuity equation and momentum equation [8]. 0=⋅∇ v (1) 174 yang: real-time 3d smoking simulation based on computational fluid dynamics…… fvvvv t v +∇+∇⋅−= ∂ ∂ 2)( (2) skv t +∇+∇⋅−= ∂ ∂ ρρρ 2)( (3) equation (1) is the conservative differential form of incompressible smoke navier-stokes continuity equation. where v⋅∇ is called the divergence of velocity (divergence). in the 3d space, divergence is defined as ),,()( z w y v x uvdivv ∂ ∂ ∂ ∂ ∂ ∂ ==⋅∇ . equation (2) is a simplified incompressible navier-stokes smoking velocity momentum equation, while ignoring the pressure gradient item. v is velocity vector, ν represents the kinematic viscosity coefficient and f corresponds to the vector of external body forces. equation (3) is the smoking density momentum equation, in which ρ is the density of the smoke, k is the diffusion constant and s is the scalar field of the amount of substance injected. 2.2 solving the smoking model in 3d space fluid model, we consider smoke fluid to be stored in a discrete grid of a cube. the density field is divided into a n×n×n grid. each cell in it has a number which stands for the density of the cell. if the number is 0, it means that there is no fluid particle in this cell. the larger the number is, the greater the density is. the density field is shown in fig. 1, in which a) shows the three-dimensional fluid mesh and b) is the density distribution of each layer. a) 3d fluid grid b) distribution of density field in each layer of fluid gird fig. 1 density field in each layer of the 3d fluid gird, the outermost ring is the boundary of grid. there are five state types for the boundary: no slip, free slip, inflow, outflow and periodic. in this paper, we employ free slip for the smoking simulation, which means that the density values on boundary are set to 0. the structure of velocity field is similar to the density field. each cell in the grid is a velocity vector, which includes 3 velocity component values at position (x, y, z) of a cell. fdm advances in systems science and applications (2011), vol.11, no.1-2 175 method is used to solve the governing equations. for convenience, density field is solved first, and then the velocity field. solution by parts is used to resolve governing equation of the density field. the detailed steps are as follows: when solving the equation, two of the three items of the right side of equation are set to zero. therefore, the solving process can be divided into 3 steps: solving the adding source item, solving the diffusion item by g-s relaxation method; solving the advection item using tracing backwards method to get the solution. at this point, the calculation of the density field is completed. governing equation of smoke simulation velocity field is exactly the same format with the density field equation. therefore we only need to modify the density field equation and apply the solution by parts to solve the velocity field. for the time being, we complete the calculation of velocity field and velocity field to be amended. the sequences of solving the smoking model governing equation are solving velocity field equations, solving to be amended velocity field which meets mass conservation conditions, solving density field governing equation, repeating till the end of calculation. 3. 3d smoking simulation based on gpu computing considering that smoke is a constant and incompressible fluid with stable temperature, the smoking model can be expressed by incompressible navier-stokes governing equations. the algorithms for the smoking model are highly data-parallel and therefore a good match for the cuda based on gpu programming model to improve the reality of the simulation. 3.1 cuda+gpu model in cuda+gpu architecture, the main function is executed in cpu host and the parallel running data is transferred to the gpu device to implement the parallel computation. all of the tasks executed by cuda are finished by a series of threads in the device allocated by the user. an array of threads is specified as a block. all the blocks are distributed in a parallel running gird. the programming frame based on cuda+gpu is shown in fig. 2 fig. 2 cuda+gpu programming architecture 3.2 3d smoking simulation model based on cuda+gpu 176 yang: real-time 3d smoking simulation based on computational fluid dynamics…… in 3d fluid model, the smoke is considered to be saved in a discrete mesh of a square. at first, the cuda api functions transfer the data of velocity field and density field into the dram of the device and then the corresponding computation of resolving the velocity field and density field is executed in the device. in the end, the result obtained in the device is transferred to the host so as to implement the rendering. the whole frame is shown in fig. 3. fig. 3 simulation model of 3d smoking based on cuda+gpu considering that the bandwidth between the device and device memory is much higher than that between device memory and host memory, it is highly necessary to minimize the data transmission between the host and the device. in our test, the velocity field and the density field are initialized in the host firstly, and then they are sent to the device. at time t, the simulation model is resolved in device and as a result the velocity field and the density field of time t+1 can be got. for the reason that the final rendering is implemented in the host, the computation result in the device should be written back into the host. considering that only the data of density filed is used when rendering, it just need to write the density filed to the host. meanwhile, the velocity field and the density filed at time t+1 exchange with the data of time t to prepare for the next iteration. in the detailed running process, the 3d mesh is corresponding to the parallel running grid in the device, the slice of each layer in the fluid mesh is corresponding to the thread block in the device and the gird in the slice to the thread in the device. the final solving of the smoking model is implemented by the allocated thread. 4. smoking simulation results the rendering of the smoke is implemented by the ray-tracing algorithm based on hardware [7]. g-s iterative method is applied to solve the diffusive item of smoking density governing equation. sor method is utilized to solve possion equation and the velocity field to be amended which meets the mass conservation condition [8]. the ray tracing method is used to render the advances in systems science and applications (2011), vol.11, no.1-2 177 smoke. cuda+gpu technology is employed to solve the navier-stokes equations based on gpu to obtain a visual effect of real-time smoking. in the following, the smoking simulation based on cuda+gpu is implemented, and the results are displayed in opengl. the hardware configuration of our experimental platform is as follows: dell precision t3500, intel(r) xeon(r) w3503 2.40ghz cpu, 4gb memory, nvidia tesla c1060. the software configuration is: windows vista, visual c++.net 2005. the size of the fluid mesh in the investigation is 323232 ×× . we forced our codes to execute 1000 times in order to get an objective running time of the smoking model. a frame of dynamic smoking simulation results is shown in fig. 4. besides , the efficiency of the whole smoking simulation for virtual surgery (including the spatial positioning of the surgical instruments) is tested and the real-time smoking simulation result for virtual surgery is shown in fig. 5. fig. 4 a frame of dynamical smoking simulation fig. 5 real-time smoking s imulation for virtual surgery the performance of the smoking simulation system implemented on the cpu and cuda+gpu are compared in table 1. we get 36.0 speedup for the mesh. the result listed in table 1 shows that significant increase of the performance is achieved and the frame per second of the whole system reaches to 56.2, which meets the requirement of real-time simulation for virtual surgery. 178 yang: real-time 3d smoking simulation based on computational fluid dynamics…… table 1 performance comparison on the cpu and the cuda+gpu data processing location total running time (ms) average running time (ms) speedup ratio frame per second (fps) cpu 193384 193.384 4.7 cuda+gpu 5370 5.370 36.0 56.2 5. conclusion this paper investigates the methods for building the real-time dynamical smoking model based on cfd and employs the incompressible navier-stokes equations as the governing model to the smoke to be constant and incompressible. in order to develop a realistic simulation of smoking, we apply g-s iterative method to solve the diffusive item of smoking density governing equation. sor method is utilized to solve possion equation and the velocity field to be amended which meets the mass conservation condition. the ray tracing method is used to render the smoke and cuda technology is employed to solve the navier-stokes equations based on gpu to obtain a real-time smoking visual effect. the results show that the smoking simulation model proposed is practical and elucidating. references [1] montgomery k., bruyns c., wildermuth s. et al. surgical simulator for operative hysteroscopy. ieee visualization 2001, washington dc, pp.14-17, 2001. [2] jiyuan tu, guan heng yeoh, chaoqun liu. computational fluid dynamics: a practical approach. 1st ed., oxford: butterworth-heinemann, 2007. [3] foster n., metaxas d. controlling fluid animation. proceedings of the cgi_97, belgium: ieee computer society, pp.178-188, 1997. [4] foster n., metaxas d. modeling the motion of a hot, turbulent gas. proceedings of the 24th annual conference on computer graphics and interactive techniques. new york, pp. 181-188, 1997. [5] stam j. stable fluids. siggraph 99 conference proceedings, annual conference series, los angeles, pp. 121-128, august 1999. [6] nvidia: cuda programming guide ver. 2.0. [7] ronald fedkiw, jos stam, henrikwann jensen. visual simulation of smoke. in siggraph 2001 conference proceedings, annual conference series, pp.15-22, august 2001. [8] zhiyong yuan, qian yin, shikun feng, jun hu. real-time dynamic bleeding synthesis and simulation of endoscopic images. geomatics and information science of wuhan university, 34(3): 317-320, 2009. (in chinese) advances in systems science and applications (2011), vol.11, no.1-2 179 advances in systems science and application (2016) vol.16 no.1 1-18 cybernetics 2.0 dmitry a. novikov institute of control sciences, moscow, russia moscow institute of physics and technology, moscow, russia abstract the evolution of cybernetics (from n. wiener to the present day) is briefly considered. a new development stage of cybernetics (the so-called cybernetics 2.0) is discussed as a science on general regularities of systems organization and control. the author substantiates the topicality of elaborating a new branch of cybernetics, i.e., organization theory (o3) which studies an organization as a property, process and system. keywords cybernetics; organization; system; control. 1 cybernetics of n. wiener this section is intended to consider in brief the history of cybernetics and describe “classical” cybernetics. let us call it “cybernetics 1.0”. cybernetics is the science of general regularities of control and information transmission processes in different systems, whether machines, animals or society. cybernetics studies the concepts of control and communication in living organisms, machines and organizations including self-organization. it focuses on how a (digital, mechanical or biological) system processes information, responds to it and changes or being changed for better functioning (including control and communication). cybernetics is an interdisciplinary science. it originated “at the junction” of mathematics, logic, semiotics, physiology, biology and sociology. among its inherent features, we mention analysis and revelation of general principles and approaches in scientific cognition. control theory, communication theory, operations research and others represent most weighty theories within cybernetics 1.0. in ancient greece, the term “cybernetics” denoted the art of a municipal governor (e.g., in platos laws). a. ampere (1834) related cybernetics to political sciences: the book [2] defined cybernetics (“the science of civil government”) as a science of current policy and practical governance in a state or society. b. trentowsky (1843, see [3]) viewed cybernetics as “the art of how to govern a nation.” in its tektology (1925, see [4]), a. bogdanov examined common organizational principles for all types of systems. in fact, he anticipated many results of n. wiener and l. bertalanffy, as the both were not familiar with bogdanovs works. 2 dmitry a. novikov: cybernetics 2.0 the modern (and classical!) interpretation of the term “cybernetics” as “the scientific study of control and communication in the animal and the machine” was pioneered by norbert wiener in 1948, see the monograph [5]. two years later, wiener also added society as the third object of cybernetics[6]. among other classics, we mention william ashby [7, 8] (1956) and stafford beer [9] (1959), who made their emphasis on the biological and “economic” aspects of cybernetics, respectively. therefore, cybernetics 1.0 (or simply cybernetics) can be defined as “the science ofcontrol and data processing in animals, machines and society.” an alternative is the definition of cybernetics (with capital c, to distinguish it from cybernetics whenever confuse may occur) as “the science of general regularities of control and data processing in animals, machines and society.” the second definition differs from its first counterpart in the words “general regularities,” which is crucial and will be repeatedly underlined and used below. in the former case, the matter concerns “the umbrella brand,” i.e., the “integrated” results of all sciences dealing with problems of control and data processing in animals, machines and society. the latter case covers partial “intersection” of these results (see fig.1 figuratively speaking, the central rode of the “umbrella.”), i.e., usage of common results for all component sciences. furthermore, we will adhere to this approach over and over again for discrimination between the corresponding umbrella brand and the common results of all component sciences in the context of different categories such as interdisciplinarity, systems analysis, organization theory, etc. cybernetics today (disciplines included in cybernetics in the descending order of their “grades” of membership, see fig.1, with year of birth if available): –control theory (1868-the papers [10, 11] published by j. maxwell and i. vyshnegradsky); –mathematical theory of communication and information (1948-k. shannons works[12, 13]); –general systems theory, systems engineering and systems analysis (1968-the book [14] and 1956-the book [15]); –optimization (including linear and nonlinear programming; dynamic programming; optimal control; fuzzy optimization; discrete optimization, genetic algorithms, and so on); –operations research (graph theory, game theory and statistical decisions, etc.); –artificial intelligence (1956-the dartmouth summer research project on artificial intelligence); –data analysis and decision-making ; –robotics and others (purely mathematical and applied sciences and scientifadvances in systems science and application (2016) vol.16 no.1 3 ic directions, in an arbitrary order) including systems engineering, recognition, artificial neural networks and neural computers, ergatic systems, fuzzy systems (rough sets, grey systems, etc), mathematical logic, identification theory, algorithm theory, scheduling theory and queuing theory, mathematical linguistics, programming theory, synergetics and all that jazz. fig. 1 the composition and structure of cybernetics in its components, cybernetics intersects considerably with many other sciences, in the first place, with such metasciences as general systems theory and systems analysis and informatics (see below and [16]). there exist a few classical monographs and textbooks on cybernetics with its “own” results; here we refer to [1, 6, 7, 9, 14, 17–23]. on the other hand, textbooks on cybernetics (mostly published in the former ussr) include many of the above-mentioned directions (par excellence, control in technical systems and informatics)-see [24–28]. the prefix “cyber” induces new terms on a regular basis, viz., cybersystem, cyberspace, cyberthreat, cybersecurity, etc. in a broader view of things, this prefix embraces all connected with automation, computers, virtual reality, internet and so on. alongside with general cybernetics, there exist special (“sectoral”) types of cybernetics[27]. a most natural approach (which follows from wieners extended 4 dmitry a. novikov: cybernetics 2.0 definition) is to separate out technical cybernetics, biological cybernetics and socioeconomic cybernetics besides theoretical cybernetics (i.e., cybernetics). it is possible to compile a more complete list of special types of cybernetics (see references in [16]): physical cybernetics (to be more precise, “cybernetical physics”, see [29, 30]), social cybernetics, educational cybernetics, quantum cybernetics (quantum systems control, quantum computing), etc. as standing apart, we mention a branch of biological cybernetics known as cybernetic brain modeling integrated with artificial intelligence, neural and cognitive sciences. a romantic idea to create a cybernetic (computer-aided) brain at least partially resembling a natural brain stimulated the founding fathers of cybernetics (see the works of w. ashby[8], g. walter[31], m. arbib[32], f. george[33], k. steinbuch[34] and others) and their followers (for a modern overview, we refer to [35]). 2 cybernetics of cybernetics and other types of cybernetics in addition to wieners classical cybernetics, the last 50+ years yielded other types of cybernetics declaring their connection with the former and endeavoring to develop it further. no doubt, the most striking phenomenon was the appearance of second-order cybernetics (cybernetics of cybernetics, metacybernetics, new cybernetics; here “order” corresponds to “reflexion rank”). cybernetics of cybernetic systems is associated with the names of m. mead, g. bateson and h. foerster and puts its emphasis on the role of subject/observer performing control[36–40]. the central concept of second-order cybernetics is an observer as a subject refining the subject from the object (indeed, any system is a “model” generated from reality for a certain cognitive purpose and from some point of view/abstraction). h. foerster noted that “a brain is required to write a theory of a brain. from this follows that a theory of the brain, that has any aspirations for completeness, has to account for the writing of this theory. and even more fascinating, the writer of this theory has to account for her or himself. translated into the domain of cybernetics; the cybernetician, by entering his own domain, has to account for his or her own activity. cybernetics then becomes cybernetics of cybernetics, or second-order cybernetics.”[38]. in contrast to wieners cybernetics, second-order cybernetics possesses the conceptual-philosophical character (for a mathematician or engineer, it is demonstrative that all publications on second-order cybernetics contain no formal models, algorithms, etc.). in fact, this type of cybernetics “transmits” the complementarity principle (with insufficient grounds) from physics to all other sciences, phenomena and processes. moreover, a series of works postulated that any system must have positive feedback loops amplifying positive control actions (e.g., advances in systems science and application (2016) vol.16 no.1 5 see [41]). but any expert in control theory knows the potential danger of such loops for system stability! the “biological” stage in second-order cybernetics is associated with the names of h. maturana and f. varela[42–44] and their notion of autopoiesis (self-generation and self-development of systems). f. varela underlined that “first-order cybernetics is the cybernetics of observed systems; second-order cybernetics is the cybernetics of observing systems.” the latter focuses on feedback of a controlled system and an observer. therefore, the key terms of second-order cybernetics are recursiveness, selfregulation, reflexion, autopoeisis. for a good survey of this direction, we refer to [45, 46]. however, the historical picture has appeared much more colorful and diverse, not confining to the second order. some authors adopt the terms “third-order cybernetics” (social autopoeisis; second-order cybernetics considering autoreflexion) and “fourth-order cybernetics” (third-order cybernetics considering observers system of values), but they are conceptual and still have no generally accepted meanings (e.g., see a discussion in [47–53]). for instance, v. lepsky wrote: “third-order cybernetics can be formed basing on the thesis “from observing systems to self-developing systems.” in this case, control is gradually transformed into a wide spectrum of support processes for system self-development, namely, social control, stimulation, maintenance, modeling, organization, “assembly/disassembly” of subjects and others.”[54]. we point out other directions (see table 1): homeostatics (yu. gorsky and his scientific school), a science studying contradictions control for the sake of maintaining the permanency of processes, functions, development trajectories, etc.[55]; neo-cybernetics (b. sokolov and r. yusupov), an interdisciplinary science which elaborates a methodology of stating and solving analysis and synthesis problems of intelligent control processes and systems for complex arbitrary-nature objects[56, 57]; neo-cybernetics (s. krylov)[58]; control methodology (d. novikov)[59]; new cybernetics, post-cybernetics (g. tesler), a fundamental science about general laws and models of informational interaction and influence in processes and phenomena running in animate, inanimate and artificial nature[60]. interestingly, k. kolin had proposed almost a same definition to informatics 20 years before g. tesler, see [61]; evergetics (v. vittikh), a value-oriented science about control processes in a society, which focuses on problem situations for a group of heterogeneous actors 6 dmitry a. novikov: cybernetics 2.0 with different viewpoints, interests and value preferences[62]. in other words, evergetics can be defined as third-order cybernetics for interacting control subjects. according to vittikhs fair remark, in everyday social life control processes will be realized by the “tandem” of common and professional control experts (theoreticians): the former face concrete problem situations in daily routine and acquire conventional knowledge (in the sense of h. poincare) on the situation and define directions of its control, whereas the latter create necessary methods and means for their activity. involvement of “common” people into social control processes is an important development trend of control science. subject-oriented control in noosphere, the so-called hi-hume cybernetics (v. kharitonov and a. alekseev), a science mostly considering subjectness and subjectivity of control[63]. table 1 different types of cybernetics type main authors period cybernetics n. wiener, w. ashby, s. beer the 1948-1950s second-order cybernetics m. mead, g. bateson, h.foerster the 1960-1970s autopoiesis h. maturana, f. varela the 1970s homeostatics yu. gorsky the 1980s conceptual cybernetics of third and fourth orders v. kenny, r. mancilla, s.umpleby the 1990-2010s neo-cybernetics b. sokolov, r. yusupov the 2000s neo-cybernetics s. krylov the 2000s third-order cybernetics v. lepsky the 2000s new cybernetics, post-cybernetics g. tesler the 2000s control methodology d. novikov the 2000s evergetics v. vittikh the 2010s subject-oriented control in noosphere (hi-hume cybernetics) v. kharitonov, a. alekseev the 2010s it is possible to introduce the notion of “fifth-order cybernetics”[16] as fourthorder cybernetics considering the mutual reflexion of control subjects[46] making coordinated decisions, etc. note that all types of cybernetics in table 1 are conceptual, i.e., absorbed by cybernetics. the observed variety of the approaches claiming (explicitly or implicitly) to be a new mainstream in classical cybernetics development seems natural, as reflecting the evolution of cybernetics. with the lapse of time, certain approaches will be further developed, others will stop growing. of course, it is extremely desirable to obtain a general picture with integration, generalization and joint positioning of all existing approaches or most of them. advances in systems science and application (2016) vol.16 no.1 7 3 cybernetics 2.0 the history of cybernetics and its state-of-the-art, as well as the development trends and prospects of several components of cybernetics (mainly, control theory see also [64]) is briefly considered in [16]. what are the prospects of cybernetics? to answer this question, let us address the primary source-the initial definition of cybernetics as the science of control and communication. its interrelation with control seems more or less clear. at the first glance, this is also the case for communication: by the joint effort of scientists (including n. wiener), the mathematical theory of communication and information appeared in the 1940s (quantitative models of information and communication channels capacity, coding theory, etc.). but take a broader view of communication. both in the paper [65] and in the original book [5], n. wiener explicitly or implicitly mentioned interrelation or intercommunication or interaction-reasonability and causality (cause-effect relations). really, in feedback control systems, control-effect is defined by its cause, i.e., the state of a controlled system (plant); conversely, control supplied to the input of a plant is induced by its cause, i.e., the state of a controller, and so on. no doubt, the channels and methods of communication are important but secondary whenever the matter concerns universal regularities for animals, machines and society. a much broader view of communication implies interpreting communication as intercommunication, e.g., between elements of a plant, between a controller and a plant, etc. including different types of impacts and interactions (material, informational and other ones). “intercommunication” is a more general category than “communication.” in the general systems context, intercommunication corresponds to the category of organization (see its definition and discussion below). therefore, a simple correction (replacing “communication” with “organization” in wieners definition of cybernetics) yields a more general and modern definition of cybernetics: “the science of systems organization and their control.” we call it cybernetics 2.0. making such substitution, we get distanced from informatics. consider the soundness and consequences of this distancing. cybernetics and informatics.nowadays, cybernetics and informatics form independent interdisciplinary fundamental sciences[61]. according to a figurative expression of b. sokolov and r. yusupov[56], informatics and cybernetics are “siamese twins.” yet, in nature siamese twins represent pathology for instance, the definition of informatics as the “union” of general laws of informatics and control would induce a megascience without concrete content, subsisting at conceptual level exclusively. 8 dmitry a. novikov: cybernetics 2.0 cybernetics and informatics have a strong intersection (including the level of common scientific base-statistical information theory). their accents much differ. the fundamental ideas of cybernetics are wieners “control and communication in the animal and the machine,” whereas the fundamental ideas of informatics are formalization (theory) and computerization (practice). accordingly, in the mathematical sense cybernetics bases on control theory and information theory, whereas informatics proceeds from theory of algorithms and formal systems. note, that this distinction partly elucidates why some sciences often related to informatics or computer sciences have not been mentioned: theory of formal languages and grammars, “true” artificial intelligence (knowledge engineering, reasoning formalization, behavior planning, etc. instead of artificial neural networks as a modern empirical engineering science), automata theory, computational complexity theory, and so on. the subject of modern informatics (or even the “umbrella brands” of informational sciences) covering information science, computer science and computational science[66] are informational processes. indeed, on the one hand, information processing arises everywhere (!), not only in control and/or organizing. on the other hand, informational processes and corresponding information and communication technology are integrated into control processes so that their discrimination seems almost impossible. a close cooperation of informatics and cybernetics at partial operational level will be continued and even extended in future. organization and organization theory.according to the definition provided by merriam-webster dictionary, an organization is: 1. the condition or manner of being organized; 2. the act or process of organizing or of being organized; 3. an administrative and functional structure (as a business or a political party); also, the personnel of such a structure. well use the notion “organization” mostly in its second and first meanings, i.e., as a process and a result of this process. the third meaning (an organizational system) as a class of controlled objects appears in theory of control in organizational systems[67, 68]. at descriptive (phenomenological) and explanatory levels[69], “system organization” reflects how and why exactly so, respectively, a system is organized (organization as a property). at normative level, “system organization” reflects how it must be organized (requirements to the property of organization) and how it should be organized (requirements to the process of organization). note that nowadays also exists “theory of organizations” (“organizational theory”) a branch of management science, both in its subject (organizational systems) and methods used. unfortunately, numerous textbooks (and just a few advances in systems science and application (2016) vol.16 no.1 9 monographs!) give only descriptive generalizations on the property and process of organization in their introductions, with most attention then switched to organizational systems, viz., management of organizations (for instance, see the classical textbook [70]). a scientific branch responsible for the posed questions (organization theory, or o3 (organization as a property, process and system, by analogy to c3 control, computation, communication[16, 64]) has almost not been developed to-date. yet, this branch obviously has a close connection and partial intersection with general systems theory and systems analysis (mostly focused on descriptive level problems and a little bit dealing with normative level ones), as well as with methodology (as the general science of activity organization[59, 69]). creating a full-fledged organization theory is a topical problem of cybernetics! consider the correlation of the two basic categories in the definition of cybernetics 2.0 (“organization” and “control”). control is “an element, function of different organized systems (biological, social, technical ones) preserving their definite structure, maintaining activity mode, implementing a program, a goal of activity.” control is “an impact on a controlled system, intended for ensuring its necessary behavior”[68]. consequently, the categories of organization and control do intersect, but do not coincide. the former fits system design and the latter fits system functioning (a conditional analogy: organization corresponds to deism (the creator of a system does not interfere in its functioning), while control corresponds to teism (the opposite picture)); they are jointly realized during system implementation and adaptation, see fig.2. in other words, organization (strategic loop) “foregoes” control (tactical loop). fig. 2 organization and control the domains in fig.2 have the following content (as examples): i. design (construction) of systems (including their stuff, structure and functions)organization but not control (despite that theory of control in organizational systems suggests stuff control and structure control). 10 dmitry a. novikov: cybernetics 2.0 ii. joint design of a system and a controlled object. adaptation. control mechanisms adjustment. iii. functioning of controllers in technical systems-control but not organization. on the one part, control process calls for organization (organization as a stage in fayols management cycle and a function of organizational control, see [67]). on the other part, organization process (e.g., system life cycle) might and should be controlled. organization and control can have a “hierarchical” correlation. generally speaking, the correlation of organization and control is far from trivial and requires further perception. for instance, in multi-agent systems decentralized control (choosing the laws and rules of autonomous agents interaction) can be treated as organization. another example is the bible as a tool of organization[71] (a system of norms making common knowledge and implementing institutional control of a society). following the complication of systems created by mankind, the process and property of organization will attract more and more attention. indeed, control of standard objects (e.g., controller design for technical and/or production systems) gradually becomes a handicraft rather than a science; modern challenges highlight standardization of activity organization technologies, creation of new activity technologies, etc. (activity systems engineering). a fruitful combination of organization and control within cybernetics 2.0 would give a substantiated and efficient answer to the primary question of activity systems engineering: how should control systems for them be constructed? actually, this is a “reflexive” question related to second-order and even higher-order cybernetics. mankind has to learn to design and implement control systems for complex systems (high-technology manufacturing, product life cycle, organizations, regions, etc.), similarly to the existing achievements in technical systems engineering. cybernetics is important from general educational viewpoint, since it forms the integral modern scientific world outlook. cybernetics 2.0.we have defined cybernetics 2.0 as the science of (general regularities in) systems organization and their control. a close connection between cybernetics and general systems theory and systems analysis[16], as well as the growing role of technologies leads to a worthy hypothesis. cybernetics 2.0 includes cybernetics (wieners cybernetics and higher-order cybernetics), cybernetics, and general systems theory and systems analysis with results in the following forms: general laws, regularities and principles studied within metasciences-cybernetics and systems analysis; advances in systems science and application (2016) vol.16 no.1 11 a set of results obtained by sciences-components (“umbrella brands”-cybernetics and systems studies uniting appropriate sciences); design principles of corresponding technologies. keywords for cybernetics 2.0 are control, organization and system. similarly to cybernetics in its common sense, cybernetics 2.0 has a conceptual core (cybernetics 2.0 with capital c). at conceptual level, cybernetics 2.0 is composed of control philosophy (including general laws, regularities and principles of control), control methodology, organization theory (including general laws, regularities and principles of (a) complex systems functioning and (b) development and choice of general technologies), as illustrated by fig.3. basic sciences for cybernetics 2.0 are control theory, general systems theory and systems analysis, as well as systems engineering-see fig.3. complementary sciences for cybernetics 2.0 are informatics, optimization, operations research and artificial intelligence-see fig.3. fig. 3 the composition and structure of cybernetics 2.0 12 dmitry a. novikov: cybernetics 2.0 the general architecture of cybernetics 2.0 (see fig.3) admits projection to different application domains and branches of subject-oriented sciences depending on a class of posed problems (technical, biological, social, etc.). 4 the prospects of cybernetics 2.0 further development of cybernetics has several alternative scenarios as follows: –the negativistic scenario (the prevailing opinion is that “cybernetics does not exist” and it gradually falls into oblivion); –the “umbrella” scenario (owing to past endeavors, cybernetics is considered as a “mechanistic” (non-emergent) union, and its further development is forecasted using the aggregate of trends displayed by the basic and complementary sciences under the “umbrella brand” of cybernetics); –the “philosophical” scenario (the framework of new results in cybernetics 2.0 includes conceptual considerations only-the development of conceptual level); –the subject-oriented (sectoral) scenario (the basic results of cybernetics are obtained at the junction of sectoral applications); –the constructive-optimistic (desired) scenario (the balanced development of the basic, complementary and “conceptual” sciences is the case, accompanied by the convergence and interdisciplinary translation of their common results, with subsequent generation of conceptual level generalizations (realization of wieners dream “to understand the region as a whole”). the development of cybernetics 2.0 in the conditions of intensified sciences differentiation provides the following: –for scientists specialized in cybernetics proper and the representatives of adjacent sciences: the general picture of a wide subject domain (and a common language of its description), the positioning of their results and promotion in new theoretical and applied fields; –for potential users of applied results (authorities, business structures): (1) confidence in the uniform positions of researchers; (2) more efficient solution of control problems for different objects based on new fundamental results and associated applied results. main challenges are control in social and living systems. several classes of control problems seem topical, namely: –network-centric systems (including military applications, networked and cloud production); –informational control and cybersafety; –life cycle control of complex organization-technical systems; –activity systems engineering. among promising application domains, we mention living systems, social systems, microsystems, energetics and transport. advances in systems science and application (2016) vol.16 no.1 13 there exists a series of global challenges to cybernetics 2.0 (i.e., observed phenomena going beyond cybernetics 1.0), see [16]: 1)the scientific tower of babel (interdisciplinarity, differentiation of sciences; in the first place, in the context of cybernetics-sciences of control and adjacent sciences); 2)centralization collapse (decentralization and networkism, including systems of systems, distributed optimization, emergent intelligence, multi-agent systems, and so on); 3) strategic behavior (in all manifestations, including interests inconsistency, goal-setting, reflexion and so on); 4) complexity damnation (including all aspects of complexity and nonlinearity (figuratively, in this sense cybernetics 2.0 has to include nonlinear automatic control theory studying nonlinear decentralized objects with nonlinear observers, etc.) of modern systems, as well as dimensionality damnation-big data and big control[72]). thus, the main tasks of cybernetics 2.0 are developing the basic and complementary sciences, responding to the stated global challenges, as well as advancing in appropriate application domains. and here are the main tasks of cybernetics 2.0: 1) ensuring the interdisciplinarity of investigations (with respect to the basic and complementary sciences, as illustrated by fig.3); 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(2015), “big data and big control”, advances in systems studies and applications, vol.15, no.1, pp.21-36. corresponding author dmitry a. novikov can be contacted at: novikov@ipu.ru advances in systems science and applications (2012) vol.12 no.4 347-352 a universal nonlinear control law for the synchronization of arbitrary 3-d continuous-time quadratic systems zeraoulia elhadj1 and j.c.sprott2 1department of mathematics, university of tébessa, 12002, algeria. 2department of physics, university of wisconsin, madison, wi 53706, usa abstract in this letter we present a universal nonlinear control law for the synchronization of arbitrary 3-d continuous-time quadratic systems. this control law does not require any type of conditions on the considered systems. keywords synchronization, universal nonlinear control law, chaos. pacs numbers: 05.45.-a, 05.45.gg 1 introduction several methods have been successfully applied to chaos synchronization. for example, in [1] a method is introduced to synchronize two identical chaotic systems with different initial conditions. an adaptive control approach is presented in [2], a backstepping design was presented in [3], an active control method is presented in [4-6], and a nonlinear control scheme was given in [7-9]. consequently, there are many applications of chaos synchronization in physical, chemical, and ecological systems, and in secure communications as shown in [1-2,10-13]. in this letter, we apply nonlinear control theory to synchronize two arbitrary 3-d continuous-time quadratic systems. the proposed control law does not need any conditions on the considered systems, and hence it is a universal synchronization approach for general 3-d continuous-time quadratic systems. in other words, the present letter is concerned with synchronization of nonlinear systems in the framework of nonlinear observers. the investigation is restricted to a pair of quadratic three dimensional systems, for which a control feedback can be chosen in such a way that global asymptotic stability of the error system can be established in the framework of classical lyapunov theory. this restriction is justified by the importance of this type of systems in real applications [14] which is certainly a useful result. 2 synchronization using a universal nonlinear control law in this section, we consider two arbitrary 3-d continuous-time quadratic systems. the one with variables x1, y1, and z1 will be controlled to be the new system given by 348 zeraoulia elhadj:a universal nonlinear control law for the synchronization of arbitrary...  x′1 = a0 + a1x1 + a2y1 + a3z1 + f1(x1, y1, z1) y′1 = b0 + b1x1 + b2y1 + b3z1 + f2(x1, y1, z1) z′1 = c0 + c1x1 + c2y1 + c3z1 + f3(x1, y1, z1) (1) where  f1(x1, y1, z1) = a4x 2 1 + a5y 2 1 + a6z 2 1 + a7x1y1 + a8x1z1 + a9y1z1 f2(x1, y1, z1) = b4x 2 1 + b5y 2 1 + b6z 2 1 + b7x1y1 + b8x1z1 + b9y1z1 f3(x1, y1, z1) = c4x 2 1 + c5y 2 1 + c6z 2 1 + c7x1y1 + c8x1z1 + c9y1z1 (2) and the one with variables x2, y2, and z2 as the response system  x′2 = d0 + d1x2 + d2y2 + d3z2 + g1(x2, y2, z2) + u1(t) y′2 = r0 + r1x2 + r2y2 + r3z2 + g2(x2, y2, z2) + u2(t) z′2 = s0 + s1x2 + s2y2 + s3z2 + g3(x2, y2, z2) + u3(t) (3) where  g1(x2, y2, z2) = d4x 2 2 + d5y 2 2 + d6z 2 2 + d7x2y2 + d8x2z2 + d9y2z2 g2(x2, y2, z2) = r4x 2 2 + r5y 2 2 + r6z 2 2 + r7x2y2 + r8x2z2 + r9y2z2 g3(x2, y2, z2) = s4x 2 2 + s5y 2 2 + s6z 2 2 + s7x2y2 + s8x2z2 + s9y2z2 (4) here (ai, bi, ci)0≤i≤9 ⊂ r30 and (di, ri, si)0≤i≤9 ⊂ r30 are bifurcation parameters, and u1(t), u2(t), u3(t) are the unknown (to be determined) nonlinear controller such that two systems (1) and (3) can be synchronized. first, let us define the following quantities depending on the above two systems in which we can proceed with our proposed method:  ξ1 = a1 + d1 + a4(x1 + x2) + d4(x1 + x2) + a7y1 + a8z1 + d7y2 + d8z2 ξ2 = a2 + d2 + a5(y1 + y2) + d5(y1 + y2) + a9z1 + d9z2 ξ3 = a3 + d3 + a6(z1 + z2) + d6(z1 + z2) ξ4 = η1 + η2 + η3 ξ5 = b1 + r1 + b4(x1 + x2) + r4(x1 + x2) + b7y1 + b8z1 + r7y2 + r8z2 (5) and advances in systems science and applications (2012) vol.12 no.4 349  ξ6 = b2 + r2 + b5(y1 + y2) + r5(y1 + y2) + b9z1 + r9z2 ξ7 = b3 + r3 + b6(z1 + z2) ξ8 = η4 + η5 + η6 ξ9 = c1 + s1 + c4(x1 + x2) + s4(x1 + x2) + c7y1 + c8z1 + s7y2 + s8z2 ξ10 = c2 + s2 + c5(y1 + y2) + s5(y1 + y2) + c9z1 + s9z2 ξ11 = c3 + s3 + c6(z1 + z2) + s6(z1 + z2) ξ12 = η7 + η8 + η9 (6) where  η1 = d4x 2 1 + d7x1y2 + d8x1z2 + d1x1 − a4x2 − a7x2y1 − a8x2z1 η2 = −a1x2 + d5y 2 1 + d9y1z2 + d2y1 − a5y 2 2 − a9y2z1 − a2y2 η3 = d6z 2 1 + d3z1 − a6z 2 2 − a3z2 − a0 + d0 η4 = r4x 2 1 + r7x1y2 + r8x1z2 + r1x1 − b4x 2 2 − b7x2y1 − b8x2z1 η5 = −b1x2 + r5y 2 2 + r9y1z2 + r2y1 − b5y 2 2 − b9y2z1 − b2y2 η6 = r6z 2 1 + r3z1 − b6z 2 2 − b3z2 − b0 + r0 η7 = s4x 2 1 + s7x1y2 + s8x1z2 + s1x1 − c4x 2 2 − c7x2y1 − c8x2z1 η8 = −c1x2 + s5y 2 1 + s9y1z2 + s2y1 − c5y 2 2 − c9y2z1 − c2y2 η9 = s6z 2 1 + s3z1 − c6z 2 2 − c3z2 − c0 + s0 (7) the above quantities comes from the formulation of the problem as the system in (8) below. now let the error states be e1 = x2−x1, e2 = y2−y1, and e3 = z2−z1. then the error system is given by e′1 = ξ1e1 + ξ2e2 + ξ3e3 + ξ4 + u1(t) e′2 = ξ5e1 + ξ6e2 + ξ7e3 + ξ8 + u2(t) e′3 = ξ9e1 + ξ10e2 + ξ11e3 + ξ12 + u3(t) (8) we propose the following universal control law for the system (3): u1 = −(ξ1 + 1)e1 − (ξ2 + ξ5)e2 − ξ4 u2 = −(ξ6 + 1)e2 − (ξ7 + ξ10)e3 − ξ8 u3 = −(ξ3 + ξ9)e1 − (ξ11 + 1)e3 − ξ12 (9) then the two 3-d continuous-time quadratic systems (1) and (3) approach synchronization for any initial condition. indeed, the error system (8) becomes 350 zeraoulia elhadj:a universal nonlinear control law for the synchronization of arbitrary...  e′1 = −e1 − ξ5e2 + ξ3e3 e′2 = ξ5e1 − e2 − ξ10e3 e′3 = −ξ3e1 + ξ10e2 − e3 (10) and if we consider the lyapunov function v = e21+e22+e23 2 , then it is easy to verify the asymptotic stability of the error system (10) by lyapunov stability theory since we have dv dt = −e21 − e22 − e23 < 0 for all (ai, bi, ci)0≤i≤9 ⊂ r30, (di, ri, si)0≤i≤9 ⊂ r30 and for all initial conditions. in particular, if the two systems (1) and (3) are chaotic, then the control law (9) guarantees also their synchronization for any initial condition. a practical example of this situation can be found in [8]. on the other hand, any 3-d continuoustime quadratic chaotic system can be stabilized (controlled) to a stable 3-d continuous-time quadratic system that converges to an equilibrium point (to a 3-d continuous-time quadratic system that converges to a periodic solution). furthermore, any 3-d continuous-time quadratic system can be chaotified to a chaotic 3-d continuous-time quadratic system. 3 example the most known example of 3-d quadratic systems, is the original lorenz system given by:  x′1 = a1x1 − a1y1 y′1 = b1x1 − y1 − x1z1 z′1 = −c3z1 + x1y1 (11) to apply the above method, we consider the one with variables x2, y2, and z2 as the response system  x′2 = d1x2 − d1y2 y′2 = r1x1 − y2 − x2z2 + u2(t) z′2 = −s3z2 + x1y2 + u3(t) (12) thus, the universal control law for the system (12) is given by: u1(t) = −(ξ1 + 1)e1 − (ξ2 + ξ5)e2 − ξ4 u2(t) = e2 − ξ8 u3(t) = −ξ9e1 − (ξ11 + 1)e3 − ξ12 (13) advances in systems science and applications (2012) vol.12 no.4 351 where  ξ1 = a1 + d1, ξ2 = −a1 − d1, ξ4 = d1x1 − a1x2 − d1y1 + a1y2 ξ5 = −z1 − z2, ξ6 = −2, ξ8 = −x1z2 + z1x2 − y1 + y2 ξ9 = y1 + y2, ξ11 = −c3 − s3, ξ12 = x1y2 − x2y1 − s3z1 + c3z2 (14) in particular, if the two systems (11) and (12) are chaotic, then the control law (13) guarantees their synchronization for any initial condition. 4 conclusion we have presented a universal nonlinear control law (without any conditions) for the synchronization of arbitrary 3-d continuous-time quadratic systems. this universal law (9) can be considered either as a stabilization, or a control, or as a chaotification approach for the system under 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(2005), “global chaos synchronization of new chaotic systems via nonlinear control”, chaos, solitons & fractals, vol.23, pp.1245-1251. [8] j. h park. (2005), “chaos synchronization of a chaotic system via nonlinear control”, chaos, solitons & fractals, vol.25, pp.579-584. [9] l. huang, r. feng and m. wang. (2004), “synchronization of chaotic systems via nonlinear control”, phys. lett. a, vol.320, pp.271-275. 352 zeraoulia elhadj:a universal nonlinear control law for the synchronization of arbitrary... [10] a. pikovsky, m. rosenblum and j. kurths. (2001), synchronization, a universal concept in nonlinear science, cambridge university press, cambridge. [11] e. mosekilde, y. mastrenko and d. postnov. (2002), chaotic synchronization: applications for living systems, world scienctific, singapore. [12] h. k. chen. (2005), “synchronization of two different chaotic systems: a new system and each of the dynamical systems lorenz”, chen and lu, chaos, solitons & fractals, vol.25, pp.1049-1056. [13] j. lu, x. wu and j. l. (2002), “synchronization of a unified chaotic system and the application in secure communication”, phys. lett. a, vol.305, pp.365370. [14] g. chen. (1999), controlling chaos and bifurcations in engineering systems, crc press, boca raton, fl. corresponding author zeraoulia elhadj can be contacted at:zeraoulia@mail.univ-tebessa.dz, and zelhadj12@yahoo.fr. advances in systems science and application (2016) vol.16 no.1 85-94 design of full adder using subthreshold dtpt logic kishore sanapala and r.sakthivel school of electronics engineering, vit university, vellore, tamilnadu, india. abstract as technology scaling has enabled small, design of digital circuits optimal for subthreshold region is becoming an active area for ultra low power applications.this paper presents the design of full adder using subthreshold dynamic threshold pass transistor (sub-dtpt) logic to achieve low power with acceptable performance. the simulations are carried out in cadence 90nm technology for vdd=0.2v and 1.2v. from the simulations the power delay product (pdp) of the proposed design is found to be extremely low in subthreshold region and is reduced by more than 80% when compared with the earlier reports. in strong inversion region the proposed adder is achieved more than 50% savings in delay when compared with the existing designs. keywords full adder; pdp; subthreshold region; sub-dtpt logic; ultra low power. 1 introduction increase in the usage of modern portable battery operated devices such as wearable electronics, cellular phones, ipods and remote sensors are leading to the demand of ultra low power applications. from [? ? ? ? ? ? ? ], it is noted that digital sub-threshold circuit design became the assured method for achieving the ultra-low power applications with acceptable performance. the sub-threshold voltage is basically the supply voltage (vdd) less than the threshold voltage (vth) of the transistor, circuits operating in this region consider the sub-threshold leakage current of the device for the necessary computations. exponential reduction of power at the cost of reduced performance is the impact with the subthreshold region of operation. this impacts being positive, there has been interest for the digital computations which uses the subthreshold leakage current achieving in ultra low power consumptions in portable computing devices. the formerly parasitic sub-threshold leakage current is exploited by the operation in the sub-threshold or weak-inversion region, and this exploited leakage current is considered as the primary operation current[? ]. these primary currents are considered to be much weaker comparatively to the standard strong-inversion currents; this will increase the time for charging or discharging the capacitive nodes which will limit the operation frequency of the circuit. in order to achieve ultra low power benefits, these leakage currents expected to drive the logic should be minimized in the device off state. for an mos transistor, when the gate to source voltage (vgs) of the transistor 86 kishore sanapala and r.sakthivel: design of full adder using subthreshold dtpt logic is biased under the threshold voltage (vth), the subthreshold or weak inversion region of operation occurs. vth is independent of the drain bias and the channel in case of a long channel device. the case differs when comes to submicron channel lengths and results in the effect of drain induced barrier lowering (dibl)[? ]. the drain current in the subthreshold region of operation is [? ] ids = i0e (vgs − ηvds − vth)/ηvt (1− e −v ds vt ) (1) where i0 = µ0c0x w l (n− 1)vth 2 (2) and the parameters vt thermal voltage (kt/q = 26mv at 3000k) η dibl coefficient n subthreshold swing coefficient (n= 1 + cdep/cox) µ0 zero bias mobility cdep depletion capacitance cox oxide capacitance w effective width of the channel l length of the channel vds drain to source voltage as device behavior depends on many parameters, in this study of dtpt logic, bulk terminal voltage is the key variable choice. varying the bulk terminal potential yields to further perceive into the device behavior to changes at the drain/ source and gate terminals. in the standard cmos configuration the bulk of the nmos is tied to the ground and the bulk of the pmos is tied to the vdd for an inverter. it is found that the drain current increases, when the bulk potential raise above the ground and the vdd to below threshold for the nmos and pmos devices respectively[? ]. one solution to increase/decrease subthreshold currents in on/off states is to use dtmos configuration[? ], where the bulk terminal is tied to its gate as shown in fig.1 (a). using dtmos pass gate along with the augmented device tends to a new configuration as shown in fig.1 (b) called dtpt[? ]. this paper presents the full adder design using sub-dtpt logic for the first time in the literature. the remaining of this paper is organized as follows. section 2 presents the brief description about existing full adder designs reviewed in comparison with the proposed design. the proposed sub-dtpt full adder design is described in section 3. simulation results and performance parameters advances in systems science and application (2016) vol.16 no.1 87 of the comparisons made with respect to different adders are presented in section 4. finally some conclusions are summarized in section 5. 2 existing full adder designs the adder is the basic building block in many of the vlsi systems such as digital signal processors (dsp) and microprocessors. since adder is the core module in many of the arithmetic operations, enhancing the performance of this module would lead to enhance the overall system performance. thus the design of full adder with lower pdp becomes the engineers interest for implementing the modern vlsi systems. number of adder circuits was designed using different logic styles and circuit techniques to reduce the power consumption and delay. the functionality of the adders compared in table1 is similar but differ in the design methodologies. each logic style tends to favor one of performance aspects like power, delay, and area but at the expense of the other. the different full adders are considered for performance comparison in conventional strong inversion region is discussed briefly in the following. the standard complementary metal oxide semiconductor (cmos) adder is the more robust and most conventional design[? ? ? ]. it is designed using the regular cmos structure with pull up and pull down networks. another conventional complementary pass transistor logic (cpl) with swing restoration logic is proposed in[? ? ]. it produces the complementary output of the many intermediate switching nodes. voltage degradation is the main issue in the performance of cpl design. this was improved with the transmission gate full adder (tgfa) designs[? ? ]. transmission gate (tg) is constructed by connecting the pmos and nmos transistors in parallel. but the driving capability of the tgfa design is less when cascaded. this results in performance degradation. later many hybrid logic styles which use more than one logic style in implementation are proposed. a 14 transistor hybrid adder was proposed in [? ? ] and zhang et al. proposed hybrid pass logic with static cmos output drive (hspc) adder [? ]. in this logic the xor and xnor functions are generated concurrently by using pass transistors and implemented in cmos module to produce full swing outputs. chiou-kou tung et al. proposed another hybrid fa core which uses the mirror type static cmos logic to improve output driving capability[? ]. sumeer goel et al. proposed a new hybrid adder which also targets high speed applications and is noise immune[? ]. mariano aguirre et al. proposed a hybrid double pass transistor logic (dpl) and swing restored cpl (srcpl) full adders[? ]. multiplexing of the boolean functions xnor/xor and or/and is used as internal logic style for implementing this full adder architecture. the different full adders considered for performance comparison in subthresh88 kishore sanapala and r.sakthivel: design of full adder using subthreshold dtpt logic old region (for vdd= 200mv) is discussed briefly in the following. direct synthesis fa [? ] is implemented based on karnaugh map driven digital design and is realized using standard nand and xor gates. the other three different adders proposed in [? ] are min3 stacked fa, min3 mirrored fa, and min3 ijcnn fa. these three adders are based on the stacked minority 3 element[? ], mirror minority 3 element[? ], cmos inverter and dynamically reconfigurable ijcnn element[? ? ]. min3 mirrored adder is suitable only for low performance applications because of its large power dissipation. min3 stacked and ijcnn adder architectures are applicable for low noise margin systems. 3 proposed sub-dtpt full adder many circuit techniques are introduced to operate the digital circuit in subthreshold region to meet the ultra low power requirement. in all the subthreshold digital circuits, power supply less than the threshold voltage of the transistors is used to power the circuit. in the subthreshold region, there is no conducting inversion channels, so therefore, the transistors behave in different manner as compared to when they are operated in a strong inversion region. therefore, there will be a change in the circuit properties, such as noise margin, tolerance to temperature and process variations. thus there are noted favorable changes, such as increased transconductance gain, near-ideal static noise margin, sensitivity of subthreshold circuit to power supply, temperature and process variations [? ? ]. this increase in the sensitivity may lead to improper functioning of the subthreshold circuits. thus to make sure proper functioning of the subthreshold circuits, moalemi et al.[? ] and lindert et al.[? ] proposed a logic family called sub-dtpt logic which gives lesser pdp when compared other subthreshold logic families reviewed in [? ]. (a) (b) fig. 1 (a) standard dtmos (b) dtmos with pass gate sub-dtpt logic uses the dynamic threshold transistors whose gates are tied to the substrates. to mitigate the drop problem, restoration has been commonly used; this will assist the pass-gate pull-up at the output. a new idea in pass-gate advances in systems science and application (2016) vol.16 no.1 89 logic is to use dynamic threshold mos (dtmos)[? ] for restoration, but the combination of the two will help scale the voltage even further. to provide a new symmetric design to the circuit, augmented circuit styles need to be customized for pass-gate logic, which makes it to add a secondary auxiliary device. the dtmos pass gate with the augmented device is known to be sub-dtpt and is shown in fig.1(b) and the standard dtmos is shown in fig.1(a). fig. 2 subthreshold dtpt nand gate fig. 3 full adder using sub-dtpt nand gates the design of nand gate using the sub-dtpt logic is as shown in the fig.2. 90 kishore sanapala and r.sakthivel: design of full adder using subthreshold dtpt logic the combinational circuit full adder is designed in sub-dtpt logic, since the full adders are the basic digital blocks in many of the arithmetic units. because of the universal characteristics, of nand/nor gates, engineers prefer these universal gates in the design of combinational digital circuits. in the proposed design, the full adder is realized using only nand gates. since nor gate has more delay because of the series stack of pmos transistors in series. the full adder using subthreshold dtpt nand gate is shown in fig.3. 4 results and discussion the proposed sub-dtpt full adder is simulated using cadence virtuoso tool in 90nm technology for the supply voltages of 1.2v (strong inversion region) and 200mv (subthreshold region). the obtained simulation waveforms are shown in fig.4. the performance parameters power, delay and power delay product (pdp) are measured for the proposed design and also compared with the existing designs. the comparisons for the proposed adder with different existing full adders in conventional strong inversion region and subthreshold region are shown in table 1 and table 2 respectively. the longest delay is considered as the cell delay which is obtained by considering the 50% of the input voltage swing to the 50% of output voltage swing. the average power is measured using the predefined calculator functions available in the cadence virtuoso tool. from the comparisons in the table 1, it is noted that in strong inversion region, the proposed design achieved major s avings in terms of delay and is 36.25% less than the fa-srpl, 58.26% less than the fa-dpl designs proposed in [? ], approximate 25% less than the sumeer goel design[? ] and chiou-kou tung table 1 results of the simulations for full adders in 90nm technology with vdd=1.2v design reference power(µw) delay(ns) pdp(fj) c-cmos [? ? ] 1.5799 0.1274 0.20127 cpl [? ? ? ] 1.7683 0.0791 0.13987 tgfa(16t) [? ? ] 1.7459 0.3258 0.5688 tgfa(20t) [? ? ] 1.7796 0.2348 0.41785 14t hybrid [? ? ] 3.3328 0.3377 1.1254 hspc-hybrid [? ] 1.576 0.2301 0.3626 mirror type-hybrid [? ? ] 7.707 0.1406 1.0836 fa-hyb [? ] 6.21 0.143 0.888 fa-dpl [? ] 7.34 0.254 1.864 fa-srpl [? ] 7.4 0.167 1.235 proposed [present] 12.3704 0.10646 1.31695 advances in systems science and application (2016) vol.16 no.1 91 (a) (b) fig. 4 (a) simulation response of the proposed full adder design for vdd=1.2v; (b) simulation response of the proposed full adder design for vdd=0.2v. et al.[? ], more than 53% less than the hybrid hspc adder design and 20t tgfa design, more than 65% savings than the vesterback et al.[? ] adder and 16t tgfa and also 16.4% less than the conventional cmos adder. it is also noted that the power delay product (pdp) of the proposed adder is 29.2% less than the fa-dpl design. table 2 results of the simulations for full adders in 90nm technology with vdd=0.2v design reference power(pw) delay(ns) pdp(aj) direct synthesis [? ? ] 854.1 263.5 225.3 c-cmos [? ? ] 931.6 162.4 151.3 min3-stacked [? ? ] 191.8 4767.7 914.5 min3-mirrored [? ? ] 1160 159.1 184.6 min3-ijcnn [? ? ] 5206 173.7 904.1 proposed [present] 2293 69.25 158.79 from the comparisons in table2 it is noted that proposed adder has achieved major savings in terms of pdp and is more than 82% lesser than the min3ijcnn and min3 stacked adder designs, 29.5% lesser than the direct synthesis adder design and also 13.98% lesser than the min3 mirrored adder design. from the above result analysis, it is observed that the delay of the sub-dtpt design is extremely low but the power is found to be very large. this increase in power is due to the excessive gate current caused by forward biasing the source body junctions. 5 conclusions in this paper, the design of full adder using dynamic threshold pass transistor logic is presented. the simulations are done using cadence virtuoso 90nm 92 kishore sanapala and r.sakthivel: design of full adder using subthreshold dtpt logic technology for supply voltages of 1.2v and 0.2v. at 1.2v, the results of the proposed design are compared with many logic styles like conventional cmos, cpl, tgfa, hybrid, srpl and dpl. at 0.2v, the results of the proposed logic is compared with different logic designs proposed earlier like standard cmos and min-3 based adders. from the simulations, it is found that in subthreshold region of operation the proposed design achieves major savings in the pdp compared with the previous reports. the proposed logic is useful in the implementation of the chip which overcomes the issues like power, delay, dibl, and process sensitivity. further improvements can be done for the proposed design in reducing the area overhead. references [1] k.roy and h.soeleman (1999), “ultra low power subthreshold digital logic circuits”, proc. ieee conf. low power electronics and design, pp. 94-96. 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[30] p.bhattacharyya, bijoy.k and s.ghosh et al (2014), “performance analysis of a low power high speed hybrid 1-bit full addert circuit”, ieee trans. very large scale integration systems, vol. 23, pp. 2001-2008. corresponding author kishore sanapala can be contacted at: kishorelendi@gmail.com study of external and internal factors affecting enterprise’s stability catherine g. zinovieva margarita v. kuznetsova tatyana v. dorfman pavel v. limarev juliya a. limareva nosov magnitogorsk state technical university, magnitogorsk, russia abstract. the paper analyzes the factors of enterprise’s stability meaning system’s ability to maintain its qualitative certainty via self-organization targeted to overcome the environment. the authors prove that enterprise’s stability is characterized as system’s activeness and adaptiveness and is formed under the influence of a set of external/internal environmental factors. the first directly depend on enterprise’s operations arrangement, the second are external to that arrangement and are outside of enterprise’s influence zone. so, the most significant direct influence factors are consumers, competitors, suppliers, contact audiences. indirect influence factors do not act on an enterprise directly but cause the environment to change which may affect an enterprise. this article covers on such factors like economic, state and political, scientific and technological, legal and socio-demographic, etc. finally the authors come to the strong conclusion: one of the most important ways to enterprise’s stability is the training of organizational culture of staff, featuring its philosophy, its basic operational principles (company’s relationships with suppliers, consumers and competitors), management style, attitude to staff, etc. keywords: enterprise’s stability, external environment factors, internal environment factors, direct and indirect influence factors, consumers, competitors, suppliers, contact audiences, activeness, system’s adaptiveness, innovational activity, pre-adaptive elements, human potential. 1. introduction an enterprise is one of complex systems functioning as the unity of stability and unsteadiness. synergetic approach understands system’s complexity as the ability for self-organization, typical, first of all, for nonequilibrium systems. stability of an enterprise may (temporarily) act as equilibrium. in equilibrium state, each element exercises its function, stably and independently from other elements. meantime the stability of an enterprise is ensured through management, i.e., the ability of entrepreneur/manager to maintain steadiness of functions. but, as an enterprise is an open system, equilibrium may not be a moment of its existence. own existence of an enterprise is an open nonequilibrium system capable for self-organization. in that case, the stability mechanism acts otherwise: the system behaves as a whole but the level of elements freedom decreases and coherent interaction of elements, forces, goals, motives, wishes occurs. in such a system, fluctuations are not suppressed but intensified, thus causing the growing role of actions coordination and efforts cooperation inside the system. the objective of this paper is to prove that stability of an enterprise is characterized as system’s activeness and adaptiveness and is formed under the influence of a set of external/internal environmental factors. in the economic literature, self-organization is equivalent to self-adjustment borrowed from engineering. but j. m. keynes in his day showed that self-adjustment idea does not reflect the specifics of the contemporary economy and causes fatal consequences in practice. keynes offered to add to self-organization the relative organization on the macroeconomic level. the stability mechanism of a separate enterprise is the same. so, stability of an enterprise is system’s ability to maintain its qualitative certainty via selforganization targeted to overcome the environment [8]. 2. choosing methodology. to assess the level of stability of an enterprise and build its stable development strategy, the factors affecting operational stability should be analyzed. in the study of those factors we are based on the following methodological provisions: – stability as a complex dynamic feature is not just subject to the influence of a great number of factors but is formed and maintained by their whole interacting aggregate; – the influence of any factor depends on the development of other factors and the aggregate of subjective conditions created in the economic system. being closely interrelated, stability factors often are oppositely directed affecting enterprise’s operation: some are positive, some – negative. the negative effect of some factors is able to reduce or even eliminate the positive effect of others, and vice versa. 3. main part. the availability of a great number of various factors makes it necessary to group them. various characteristics may become the classification basis: – by place of existence – external and internal; – by nature of effect – objective and subjective; – by level of influence – basic and secondary; – by structure – simple and complex; – by time of effect – permanent and temporary, etc. we hold the opinion that enterprise’s stability is formed under the influence of a set of factors of internal and external environment. the first directly depend on enterprise’s operations arrangement, the second are external to that arrangement and are outside of enterprise’s influence zone. external factors of enterprise’s stability are the conditions which may not as a rule be changed but should be accounted for by the subject of an enterprise as they affect the state of its affairs, i.e., the factors outside an enterprise. external factors are interrelated: change of a one may cause change of others and therefore their effect on enterprise’s stability is correlated. among the total aggregate of external factors of enterprise’s stability, foreign economic trade, production, market, information and innovational infrastructure factors may be specified. external factors affecting the level of enterprise’s stability are logically divided into two groups: – direct influence factors; – indirect influence factors. direct influence means the straight impact of environment on an enterprise. indirect influence is the environmental impact on an enterprise causing changes of its operating conditions. external factors of direct influence include consumers, competitors, suppliers, contact audiences – those are the factors manifesting through the activity of enterprise’s subjects, although having objective reasoning. an enterprise in the course of its activities continuously interacts with various elements and processes, so a rather significant change of activities always causes the change of an enterprise itself. the reasons like growth of raw materials prices and loan interests, loss of buyers, etc. may change enterprise’s structure, its actual form and size. the most important factor is consumers forming the sales market. there is a widespread point of view that the only true purpose of a business is to create consumers. that means the following: enterprise’s survival and justification of existence depends on its ability to find consumers of its activity and meet their demands. in the scientific literature, there are various approaches to consumers classification. to study and analyze the consumers, their aggregate may be grouped as follows: – direct consumers; – manufacturers receiving products/services for production purposes; – intermediaries buying products/services for resale; – governmental organizations buying products/services for own use. regarding the relationships between an enterprise and its products buyer, in the course of business an entrepreneur does not have to enter into relationships with investors on calling for capital (having own capital), does not have to enter into relationships with employees (doing all production activities independently), but there has to be a relationship with the consumer/buyer of products/services. in any case, an enterprise has to cooperate with the consumer as the need for production process arrangement and its length depend on the consumer: it is the consumer which determines whether the production process of a business will further exist. the consumer, from producer’s point of view, is its main partner. 4. discussion however, in the contemporary world producer’s dependency on the consumer changed greatly. as noted by j. baudrillard, the idea of primary needs is a myth [3, p. 54]. the things are not exhausted with what they serve for, but are imposed prestige and signs of social position. political economy traditionally explains the reasoning of production by the existing needs. as opined by j. baudrillard, it camouflages the internal feasibility of production order [3, p. 64]. in fact, producers manage the needs via advertising. the real power lies in the real spheres of decision making, management of needs, manipulation of signs and people [3, pp. 99-100]. thus, that external factor of stability is gradually getting the internal nature. the main stability factors include the competition of business entities. in many cases, not consumers but competitors determine which kind of products may be sold and at what price. consumers are not the only object of competition between businesses. the latter may compete for labor resources, materials, capital and the right to use some technical novelties. internal factors like working conditions, labor compensation, nature of relationships between leaders and subordinates are depending on competitive response. competitive success is ensured to those which can find new needs, arrange production of new goods and introduce new technologies, thus making competition a tool of economic play, making businesses review their strategies. competition affects the quantity and quality of products made, removes inefficient enterprises from production, assists in rational use of resources, prevents from producer’s (monopolistic) dictatorship over consumers. for growth and prosperity, enterprise’s subject needs capital suppliers. those may be banks, individuals, governmental institutions engaged in loans granting. the wish of investors to invest their capital may be a certain criterion of enterprise’s success as the higher its performance statements, the higher the probability to get the required funds on good terms and conditions. the significance of factor suppliers of material resources is characterized by the fact that in an enterprise the portion of material resources in production cost-price is 60-80% and higher. contact audiences are the external forces directly affecting enterprise’s decision making due to various kinds of interests related to its business. contact audiences may be characterized as follows: – governmental authorities on supervision and regulation of business; – mass media (advertising agencies, newspapers, magazines, radio and tv); – public organizations, trade unions, civil public opinion groups, etc.; – local contact audiences: communities, religious organizations, etc. governmental institutions play an important role in regulating the activities of enterprises. enterprise’s legal status reasons a certain taxation procedure. various governmental regulators are authorized to set forth the procedure of financial accounting keeping, issue licenses, set the standards and operating conditions, etc. indirect influence factors do not affect an enterprise straightly, but cause the changes in the external environment which may affect an enterprise. they are economic, governmental and political, scientific and technological, legal, socio-demographic and others. indirect influence environment is usually more complex than that of direct influence. top management, taking incomplete information as the basis and trying to prognosticate possible consequences for an enterprise often has to rely upon the suggestions about such environment. for efficient interaction with the external environment, the leader has to continuously analyze the dynamics of such environment (changing structure of factors) taking into account the following: – structure of factors is rather complex and there is plenty of factors; – the level of each factor’s impact on business structure is different; – some factors are continuous, others – short-term; – external environment changes are mobile, chaotic and rather violent, making hard to monitor them. to ensure survival and achievement of the goals to be sought, enterprise’s subject should be able to efficiently response and adapt to the changing environment. enterprise’s stability is system’s activeness and adaptiveness. adaptiveness is the representation in the reflecting system of the specifics of the reflected one, or system’s response to environment’s influence. adaptiveness in the general sense is characterized as passive adaptation of a system to the external environment [2]. it is oriented at the compensation of negative consequences caused by external factors. enterprise’s adaptiveness is characterized by its ability to change the functioning method in compliance with the environment’s changes and may not be considered absolutely passive as it is exercised through conscious activity of individuals – employees and owners of an enterprise. nevertheless, it is relative activeness as it is forced by external factors. activeness suggests feasible response of a system directed to the external environment and related to the elimination of negative external factors by acting on the elements of the environment producing them [7]. enterprise’s activeness is the implementation of measures aimed at decrease of external environment factors’ dependency. however, if during the adaptation the dependency is decreased by elimination of the influence of factors, active behavior suggests preventive measures of an enterprise aimed to eliminate the external environment factors. w.r. ashby [14] worded the basic principle of systems stability – law of requisite variety, according to which only a variety may destroy a variety so the variety of system’s states should not be lower than external environment’s variety. n.v. chepurnykh notes: if an enterprise is capable to affect the external environment’s factors, it may be used for reducing environment’s variety [12, p. 76]. we opine that through adaptiveness an enterprise reduces the variety of the external environment by eliminating its factors. enterprise’s active behavior contributes to the growth of the variety of states of an enterprise itself. enterprise’s adaptiveness as a way of functioning in compliance with the external environment changes regarding direct factors of the environment is characterized by the level of control of enterprise’s top management over its liabilities. in economic literature on business matters some four groups of enterprise’s stability factors are specified: 1) innovations and investments; 2) resources and their use (material/financial, human, technical/technological facilities, information); 3) marketing quality and level of use; 4) organizational culture of a business. it should be noted that the above groups include dozens of specific factors selectively acting in each business structure. an enterprise may increase the variety of its states and keep ahead of the environment’s variety via production optimization, introduction of new technologies and kinds of products. enterprise’s effect on the environment is achieved by establishment and development of competitive advantages causing the consumer market structure to change ensuring reliability of raw materials and semi-finished products supply and sales. according to m. porter’s theory of five competitive forces [18], achievement of competitive advantages by an enterprise is ensured by the developed technology, innovative equipment, intellectual resources and other elements. enterprise’s stability may be ensured either via keeping traditional business forms or via innovations. innovation notion relates to a new product or service, their production method, novelties in organizational, financial, scientific and research and other spheres. any improvement ensuring costs saving or creating any conditions for such saving may be deemed an innovation. the innovational process combines science, technology and management covering the aggregate of production, exchange, consumption relationships. under institutional evolutionary theory, a number of models was developed explaining company’s development as the result of its innovative activity. j. schumpeter specified seven directions of innovative changes on the microlevel: products, technologies, sale markets, raw materials, semi-finished products, organizational structure principles. through innovations, an enterprise increases the variety of products range, resources used, production and management methods. it was found that between innovative growth and relative quality of products there is a certain correlation, while enterprises offering a great number of new products compared to its competitors the probability to offer more useful products from consumer’s point of view is higher. intensive development of production implies opportunity to increase production output without extra resources as distinct from extensive development which is oriented to call for extra production resources [13]. intensive use of resources means the introduction of new technologies and production optimization. an essential consequence of the intensification process is growing products quality while cutting cost-price, which makes them more attractive for consumers. therefore, an enterprise becomes less dependent on certain buyers’ groups. the use of new saving technologies enables to increase production output using less resources and makes production more flexible and not dependent on a particular kind of resource. meantime, cutting cost-price makes price increase on raw materials and semi-finished goods a less substantial factor. increasing innovations quantity and quality, intensifying and diversifying production, an enterprise may affect the environment’s variety via increasing the variety of own states: product differentiation and product line expansion, opportunities for consumption of larger volume of interchangeable resources, products quality improvement, increasing number of buyers and markets. staff is the central factor of any enterprise. this is because an individual, his/her norms of behavior and values greatly determine the achievement of enterprise’s goals. to ensure staff’s normal working conditions, it is required to take into consideration individual skills of employees, develop their creative potential and apply it, accounting for socio-psychological and physiological needs of people, improve professional training thus ensuring social and psychological stability of the staff team. the account for those aspects should be the cornerstone of enterprise’s potential. the model of keeping enterprise’s stability suggests the availability of two parameters at least: business efficiency and free acts of the leader and the staff. freedom means growing role of choice of each element functioning as acts of certain subjects. systems methodology worded so-called law of varieties exchange: in social systems, dropping variety on the macrolevel (economic life of the society) is accompanied by growing variety on the microlevel (in our case, we may speak of an enterprise) [11, p. 125]. 5. result. this, the ability of top management to manage human resources rapidly taking into consideration the growing role of employee’s personality, knowing his/her motivations, ability to create them and align with enterprise’s goals may become a reserve of enterprise’s stability. transferring the center of gravity to an individual, a manager creates the conditions for a new level of system’s existence where an individual acts not as a cog in the management system machine in general but as an independent element. from the point of view of system management methodology, organizational activity should be aimed at the development of active, constructive consciousness, accounting for free choice alternatives of production parties. as opined by i. novik, systems analysis of alternatives suggests three main methods: 1) analysis of compliance level of living program and its implementation tool; 2) pros and contras analysis enabling to assess objective/subjective conditions in the state of instability; 3) estimating consequences of alternatives [11, pp. 122-123]. to solve the strategic tasks, the information on the environment is required. information flows as a means of control ensure enterprise’s links with markets, consumers, scientific and engineering novelties, thus achieving enterprise’s work optimization. meantime, the information collected on the organizational environment should possess a number of characteristics: reliability (assuredness in non-distortion of the information received), timeliness (no delays in getting the information required), trustworthiness, exactness (quality characteristic describing object’s condition), sufficiency (information about an object should cover all of object’s state), necessity (only necessary knowledge about an object is permitted), usefulness (effect of its use should exceed the costs on its receipt), regularity, etc. information’s impact on sustainable development may open new opportunities by solving the following tasks: – growing requirements to information quality and content; – defining enterprise’s need for information by kinds and forms; – defining basic directions on collecting, processing and storing initial data; – planning actions on collecting and exchanging information between enterprise’s divisions; ensuring feedback; – creating database used in the course of development and implementation of marketing programs, planning and control at an enterprise; – creating system for meeting the technical needs in connection with collection, processing and storage of information; automation and computerizing of administrative and management work; – development of protection and security of information. the information system’s efficiency which determines stable operation of an enterprise is determined by flexibility, i.e., the ability to timely describe emerging problems and give possible solution variants, thus checking for potential opportunities of an enterprise, tasks and goals, alternative solutions, possible risks and effect. thus, information forms manager’s ability to analyze and choose alternatives, i.e., contributing to the creation of internal variety for keeping enterprise’s stability. it also should be noted that the decisive role in enterprise’s dynamics is played by the subject – the manager. the abilities ensuring system’s self-organization in crisis conditions depend on so-called pre-adaptive elements. (pre-adaptation is the emergence of some or other useful characteristics in changing systems before they become actually useful [4, p. 79]). on the personal level it means cognitive complexity of thinking, high variation of human behavior and motivation. t.i. zaslavskaya [7] opines that the most important and decisive feature ensuring social integrity is human potential. a subject (a manager) should be correctly motivated, possess professional knowledge, morals, ability to think progressively. 6. conclusion. thus, one of the most important ways to enterprise’s stability is the training or organizational culture of employees [5], characterizing enterprise’s philosophy, its basic principles (interrelation of enterprise’s subject with suppliers, consumers and competitors), management style, of enterprise’s subject with suppliers, consumers and competitors), management style, attitude to employees, etc. stability is one of the sides of dynamic existence of an enterprise, the other side is its development. references 1. abalyan a.s. building the system of enterprise’s dynamic stability in modernization conditions // scientific review. – 2013. – no. 6. – p. 178-183. 2. agafonov v.a. principles of building structural model of strategy // strategic planning and development of enterprises. moscow: central economic and mathematical institute of ras, 2001. p. 21-22. 3. baudrillard j. for a critique of the political economy of the sign. moscow: biblion – russkaya kniga, 2003. 272 p. 4. vasilyeva l.n. theory of elites (synergetic approach) // public sciences and contemporaneity. 2005. no. 4. p. 75-85. 5. votchel l.m. philosophical analysis of ontological reasons of business: phd thesis, author’s abstract; magnitogorsk state university. magnitogorsk, 2000. 156 p. 6. golovanov p.v. improving financial and economic stability of industrial enterprises // scientific review. – 2013. – no. 7. – p. 158-160. 7. zaslavskaya t.i. business stratum of the russian society: notion, structure, identification // economic and social changes: public opinion monitoring. 1994. no. 5. p. 7-15. 8. zinovieva ye.g. enterprise’s existence: dialectics of stability and unsteadiness: phd thesis, author’s abstract; magnitogorsk state university. magnitogorsk, 2006. 138 p. 9. zinovieva ye.g., usmanova ye.g. enterprise’s stability factors // scientific review. – 2014. – no. 6. – p. 402-409. 10. kaminskiy m.a. economic stability of construction sector enterprises // scientific review. – 2013. – no. 9. – p. 692-694. 11. novik i. systematicity of optimal choice // public sciences and contemporaneity. 1995. no. 1. p. 118-126. 12. chepurnykh n.v. economy and ecology: development, catastrophes. moscow: nauka, 1996. 271 p. 13. sheremet a.d. methodology of financial analysis. moscow: infra-m, 1995. 176 p. 14. ashby w.r. principles of self-organization // principles of self-organization. moscow: mir, 1966. p. 314-343. 15. aldrich h., zimmer c. entrepreneurship through social networks / sexton p., smilor r. (eds) the art and science of entrepreneurship. – cambridge (ms), 1986. 16. carland j.w. diffierentioning entrepreneurs and small business owners: a conceptualization // academy of management rev. – 1984. – no. 9. – p. 358. 17. kirtzner i. competition and entrepreneurship. chicago: the university of chicago press, 1973. 18. porter m. competitive advantage. n.y.: macmillan, 1985. advances in systems science and applications (2012) vol.12 no.1 89-95 rapid manufacturing metallic parts via selective laser melting ruidi li1, yusheng shi1, jinhui liu2, mingzhang du3 and zhan xie3 1state key laboratory of material processing and die & mould technology, huazhong university of science and technology, wuhan 430074, china 2heilongjiang institute of science and technology, harbin, 150027, china 3shichuan petroleum perforating materials ltd, longchang, 642177, china abstract 316l stainless steel parts were manufactured via selective laser melting in this work. the surface morphology, microstructure, density and mechanical property were characterized. it is found that the surface contains a little amount of oxide and splash with balling effect. microstructure in low magnification shows features of scan molten tracks and molten pools; microstructure in high magnification depicts very fine crystal under rapid cooling. the tensile strength of as-received samples is 652.12 mpa, with a density of 95.6%. therefore the gas atomized 316l stainless steel powders could be used in manufacturing high quality parts with complex shapes via selective laser melting method. keywords selective laser melting, metallic parts, 316l stainless steel, powder 1 introduction selective laser melting (slm) is a newly developed rapid manufacturing technology, which can directly manufacturing intricate metal parts according to a three dimensional model from metal powders[1-4]. this forming technique is based on means of adding layers to build net shape components. each layer represents a slice geometry graph of the objective part in two-dimensional pattern. owing to its unlimited flexibility of geometry and complexity, slm is capable of manufacturing short run components which are not easily made through other forming method. moreover, metal parts made by slm need no or very little postprocessing procedure, and it is differ from selective laser sintering (sls). during slm process, metallic powders are fully melted because of a higher laser energy input. while in sls process, the forming mechanism is melting high polymer powder to binder un-melted metal powder. thus sls needs very trivial procedures such as post treatment to degrease to remove the binder. so it is urgent to develop and optimize slm technique with an aim to improve its practicability and universality in advanced manufacturing field. at present, slm technology is faced with many problems such low density, balling phenomenon, delaminating, warp and crack etc. that is due to the complex heat transfer mode under rapid moving gauss heat source, inducing acute temperature variation[5-6]. metal powders are melted in an extremely short time 90 ruidi li:rapid manufacturing metallic parts via selective laser melting coupled with multi-lines and multi-layers process, resulting in a very complex physicochemical process and multi modes heat and mass transfer. under this circumstance, the densification is especially important for creating high quality parts. in addition, the as prepared microstructure is particular comparing with other microstructure obtained from conventional forming method. therefore, investigation into a series of special phenomenon during slm is essential. in this paper, the densification, microstructure, balling effect, mechanical property of 316l stainless steel part was presented and the corresponding mechanisms were also addressed. 2 experimental procedures 2.1 powder materials 316l stainless steel powders (99% purity) were used in this experiment. these powders were prepared through water atomization in institute of powder metallurgy, central south university (csu). the morphology of the starting powder was examined by a quanta 200 scanning electron microscope (sem). it can be seen that these powders show sphere shape and bimodal-size feature, which can induce a high loose density and it is favorable for slm experiment. fig.1 sem figure showing morphology of 316l stainless steel powders 2.2 slm forming the forming process was carried on in the hrpm-iislm system which was developed by huazhong univ. of sci.&tech. (hust). the shaping space of the equipment was 250mm(l)×250mm(w)×200mm(h). this slm equipment contained a 100w continuous wave fiber laser. the building chamber of this system could also be vacuumed and protected by inert gas. forming process of slm was followed the listed procedures. (i) build 3d-cad model and transfer it into stl file. slice this 3d-cad model into horizontal layers according layer thickness and input it into slm system (ii) a quality of powders is dropped, then a roller spread a powder layer (iii)high energy laser scan the powder bed, thus the powders in scanned zone advances in systems science and applications (2012) vol.12 no.1 91 are completely melted (iv) the working platform descend a layer thickness (v) repeat procedure (b)-(d) until an integrated part is formed scan speed of 50∼100mm/s, laser power of 90∼100w, scan interval of 0.05∼0.2mm and layer thickness of 0.02∼0.1mm were selected for slm process. using above procedures and processing parameters, samples of slm-parts can be made as shown in fig.2. (a) (b) fig.2 samples of as fabricated slm parts 2.3 characterization at laser, densities of slm parts were measured through archimedes laser. surface morphologies and microstructure of as-received samples were analyzed by a quanta 200 sem. the element composition of the laser processed material was measured by energy dispersive x-ray spectrometer analysis. 3 experimental procedures 3.1 surface morphology surface characteristic could represent melting and solidification feature of slm process. fig.3 shows the surface morphologies. from the low magnification of fig.3a, it can be seen that the surface is relatively smooth in addition to some white flakes and splash. edx analysis resulted that the white flakes on the surface were oxygen rich, which is formed because of oxidation. it should be noted that the oxidation is disadvantageous for slm technique, due to a worsened wetting ability caused by oxide. in next layer forming process, the melted liquid could not wet the previous layer easily. therefore, it should be cautious to control oxygen content in atmosphere to prevent oxidation. moreover, it also should control oxygen contents in powder materials. fig.3b is the high magnification of splash form fig.3a, showing the detailed characteristic of splash. it exhibits spherical feature, with a large number of spheres on the top surface. the forma92 ruidi li:rapid manufacturing metallic parts via selective laser melting tion mechanism of balling effect during solidification process can be explained as follow. in liquid metal solidifying process, the surface energy tends to reach a lower value which is according to the lowest energy principle. thus under surface tension action, the many spheres are generated. (a) (b) fig.3 sem figure showing top surface morphologies of as prepared part (a) (b) fig.4 metallograph showing low magnification of microstructure. a: top view; b: side view 3.2 microstructure slm-produced microstructure is very different from microstructure obtained by conventional forming method. hence investigation on its microstructure is requisite. fig.4 shows the metallograph of top view and side view respectively, overall it can be seen that the microstructure in low magnification is dense with few pores. it is widely accepted that pores in a metal part are detrimental to mechanical property, so a high density is a significant index of a metal part. in this work the final density of 316l stainless is 95.6%, which is measured by displacement method. consequently this dense structure facilitates mechanical property. fig.4a describes the microstructure from top view. it can be seen that scan tracks are overlapped, and formed into a dense layer. fig.4b shows the microstructure advances in systems science and applications (2012) vol.12 no.1 93 from side view. it indicates that molten pools are accumulated tightly with a small amount of pores. fig.5 illustrates the microstructure in high magnifica(a) (b) fig.5 sem figures showing microstructure in high magnification of slmproduced material tion of slm-produced material. it is obvious that the grains are very fine and the grains grow along multiple directions. the extremely fine size is caused by the significantly rapid cooling rate during solidification process. a rapid solidification could inhibit grain growth, accordingly forming a large number of fine grains. the grains during molten pool solidification tend to grow along the easy growth direction. for cubic crystal, the growth directions are < 110 >. when one of the easy growth directions coincides with the heat flow direction, growth conditions are optimum. for this reason, among the randomly oriented grains, those grains have one of their < 100 > crystallographic axis most aligned with the heat flow direction will be favored. however, slm is a multi-line based technology in every cad slice layer, the moving heat resource controlled by a complex scan path can easily induce a very complicated heat diffusion process, thereby a varied heat transfer direction. this reason leads to a variable growth direction in order to accommodate heat flow direction. thus the as-received microstructures are formed. here, it should be pointed out that the slm produced parts contain fine microstructure, which is in favor of mechanical property. 3.3 tensile strength evaluation mechanical property is an important factor which ultimately determines its practical application. thus the tensile strength experiment was conducted with an eye to evaluate its mechanical property. a tensile test specimen was made by wire-electrode cutting from bulk slm fabricated samples. fig.6 expresses the stress-strain curves of extension test of slm-prepared 316l stainless steel materials. by calculation from above data, the tensile strength value of 652.12 mpa is obtained. to an extent, this mechanical property is equivalent to forgeable piece. therefore, this slm-produced parts can be used as many practical requirement in metal components. 94 ruidi li:rapid manufacturing metallic parts via selective laser melting fig.6 stress-strain curves from extension test 4 conclusions based on the investigation on selective laser melting 316l stainless steel powders, the following conclusion can be drawn. the gas atomized 316l stainless steel powders with spherical shape could be used as slm material for rapid manufacturing metallic parts. the corresponding properties and characteristic for evaluation slm-produced material are addressed. the surface morphology shows multi-lined feature, with a little amount of oxide and splash with balling effect. moreover, the microstructure in low magnification is composted of scan molten tracks and molten pools, which reflect the accumulative characteristic of rapid prototype technique. the microstructure in high magnification indicates the intrinsic nature of rapid cooling rate, resulting in extraordinary fine crystal. finally, the density of 95.6% is measured by drainage, possessing of tensile strength of 652.12 mpa. overall, slm technology in manufacturing 316l stainless steel parts shows its excellent feature in both flexible formability and high mechanical property, which can meet demand for practical applications. acknowledgements the authors would like to give thanks to the national high-tech program (863) of china (2007aa03z115), independent fund of state key lab. of material processing and die & mould technology of huazhong university of sci & technol and open fund of state key lab. of powder metallurgy of central south university of china (2008112022) the authors also thank for hua yan, bin hua and the analytic and testing center of huazhong university of science & technology for their assistance. advances in systems science and applications (2012) vol.12 no.1 95 references [1] yevko v, park cb, zak g,benhabib b. (1998), “cladding formation in laserbeam fusion of metal powder”, rapid prototyping, vol.4, pp.168-184. [2] mumtaz k.a, erasenthiran p, hopkinson n. (2008), “high density selective laser melting of waspaloy”, j. mater. process. technol, vol.195, pp.77-87. [3] asgharzadeh h, simchi a. (2005), “effect of sintering atmosphere and carbon content on the densification and microstructure of laser-sintered m2 high-speed steel powder”, mater. sci. eng. a, vol.403, pp.290-298. [4] simchi a, pohl h. (2004), “direct laser sintering of iron-graphite powder mixture”, mater. sci. eng. a, vol.383, pp.191-200. [5] simchi a. (2006), “direct laser sintering of metal powders: mechanism, kinetics and microstructural features”, mater. sci. eng. a, vol.428, pp.148158. [6] shen yf, gu dd, wu p. (2008), “development of porous 316l stainless steel with controllable microcellular features using selective laser melting”, mater. sci. technol, vol.24, pp.1501-1505. corresponding author author can be contacted at: shiyusheng@263.net advances in systems science and application (2016) vol.16 no.2 15-38 intrusion detection model using pca and ensemble of classifiers sumaiya thaseen1 and ch.aswani kumar2 1school of computing science and engineering, vit university, chennai, tamil nadu. 2school of information technology and engineering, vit university, vellore. abstract most of the intrusion detection systems examine all network features to identify intrusions with different classification approaches. the major challenges for any intrusion detection model is to achieve maximum accuracy with minimal false alarms. while many ensemble techniques are present to improve the accuracy of intrusion detection models, building an ensemble that can be generically applied for any network traffic is still a difficult task. in this paper, we propose a hybrid model for intrusion detection integrating base classifiers such as svm, linear discriminant and quadratic discriminant analysis.the aim of this paper is to identify the class label by constructing an individual classifier for each of the attack type and merging the results of every classifier. the resultant decision of the class label is obtained using weighted majority voting approach. we analyzed the performance of the model on two different data sets such as nsl-kdd and unsw-nb datasets. the experimental results indicate that the ensemble produces high accuracy in comparison to the base classifiers. as there is a huge class imbalance problem in network traffic, it is also observed that rather than relying on a single classifier, predicting the class label by weighted majority voting of svm, linear and quadratic discriminant classifier is an optimal solution which is proposed in this paper. keywords accuracy; intrusion; linear discriminant; quadratic discriminant; support vector machine. 1 introduction malicious intruders in the network are increasing day by day due to the rapid development of internet. the intruders can access, manipulate and disable the systems connected on the internet. intrusion detection systems (ids) are designed to discover the unauthorized access to computers in the network. intrusion detection systems are classified in two categories signature detection and anomaly detection. signature detection is used to identify attacks based on the known pattern of attacks. anomaly detection compares unknown profiles with known profiles and then identifies the unknown traffic profile as an attack. anomaly detection techniques have high false positive rates. many machine learning techniques have been used by researchers to overcome the disadvantages of anomaly detection models. several intelligent approaches such as svms [1], 16 sumaiya thaseen and ch.aswani kumar: intrusion detection model using pca and ... anns [2], petri nets and data mining approaches[3, 4] have been used to build an ids. there are also many ensemble methods of machine learning which are more efficient than individual techniques that can reduce the false alarm rate and increase the classification accuracy. the different ensemble methods are bagging, boosting and stacking. bagging and boosting are mostly used to implement intrusion detection models as the stacking technique requires more time. there are two major limitations of existing approaches. the first limitation is even though there are many sophisticated detection techniques only few focus on feature representation for normal traffic and attack traffic which is a major issue to enhance performance of the classifier. the second issue is the computation time involved in integrating multiple techniques which may degrade the efficiency of on-line detection. the contribution of this research is to to build a desirable ids model with high accuracy using machine learning ensemble techniques with a feature reduction technique to identify the suitable features in ids. ensemble is preferred because the aggregation of multiple classifier predictions improves the accuracy of ids. in this paper we specify a manager as a combination of classifiers that will be generated depending on the class labels in each dataset. for instance, the manager will have 5 different classifiers in the svm, mnb and ldc ensemble for nsl-kdd data set as there are five class labels and the manager will have 9 different classifiers in the svm, ldc and qdc for unswnb dataset. thus the manager has a variable number of classifiers generated dynamically according to the number of class labels available in the dataset. thus the ensemble model utilizes a individual classifier for every class type and is an integration of base classifiers svm, linear discriminant and quadratic discriminant with the resultant class label predicted by weighted majority voting ensemble which is deployed to classify the different kinds of network attacks. a similar ensemble of classifiers was already developed [5] using four dif-ferent base classifiers namely linear discriminant classifier (ldc), quadratic discriminant classifier (qdc), k-nearest neighbor (knn) and back propagation and tested on four datasets namely hearth, diabetes, iris and transfusion. the uniqueness of the proposed model over earlier developed ensemble techniques are 1) the model can be evaluated on any real time intrusion detection datasets and the managers can be extended based on the number of class labels in the samples. 2) any combination of base classifiers can replace the existing techniques to improve the model. the rest of this paper is summarized as follows. section 2 provides a discussion on various developed intrusion detection models. section 3 provides the background of various techniques utilized in the model such as svm, linear disadvances in systems science and application (2016) vol.16 no.2 17 criminant , quadratic discriminant classifier and weighted majority approach. section 4 gives the overview of proposed intrusion detection model. experimental results and discussions are discussed in section 5. section 6 finally concludes the paper. 2 related work in this section we will discuss the various intrusion detection models developed using machine learning approaches, models developed by integrating classifiers and the ensemble techniques developed for intrusion detection. various artificial intelligence methods have been developed for intrusion detection models such as fuzzy logic [6], k-nearest neighbors [7], support vector machines [1], artificial neural networks [2], naïve bayes networks [8], decision trees [9], and genetic algorithms [10]. sumaiya et al. developed an intrusion detection model by using pca as the dimensionality reduction technique and svm as the classifier [11, 12]. the kernel parameters of svm are optimized by considering the variance of samples available in the same and different classes. this model provided a better classification accuracy. ajith et al. built a light weight ids using genetic programming approaches [13]. the experimental results proved that the accuracy was better in comparison to traditional intrusion detection models. kuanga et al. developed a hybrid kpca svm with ga model for intrusion detection [14]. the authors used kpca to extract the primary features of intrusion detection dataset. svm multilayer model is employed as the classifier to identify the attack. chebrolu et al. evaluated the performance of two feature selection techniques such as bayesian networks (bn) and classification and regression trees (cart) and also an integration of cart and bn [15]. results illustrate that feature selection is very effective in the development of real world intrusion detection models. sandhya et al. developed a hybrid intrusion detection model by combining decision trees and svm and also an ensemble of other base classifiers [3]. this hybrid model maximized accuracy and minimized complexity and results illustrated that the proposed model provided an accurate ids. perin et al. built a three layer multi classifier intrusion detection model to increase the overall accuracy [16]. the performances were analyzed from a vari-ety of combination techniques such as fuzzy k-nn classifier, na¨ıve bayes classifier and back propagation neural network classifier and the decision obtained from multiple classifiers are combined into a single result. the results proved that the detection performance is better than deploying a single classifier when using a full feature set or partial feature set. chandra and yao developed an ensemble based neural network wherein the outputs are combined in a form that resulted in a sig-nificant improvement in the generalization performance [17]. srinivas et al. built 18 sumaiya thaseen and ch.aswani kumar: intrusion detection model using pca and ... an ensemble model of svm, artificial neural network (ann) and multivariate adaptive regression splines (mars) and analyzed the performance which was superior to individual approaches with respect to classification accuracy [18]. syarif et al. improved the accuracy and false positive rate of intrusion detec-tion by constructing an ensemble of bagging, boosting and stacking [19]. the base classifiers for these ensemble models were na¨ıve bayes, decision tree, rule induction and nearest neighbor. their results indicated that the accuracy for known intrusions was more than 99% but novel intrusions were identified with accuracy levels of 60%. thus bagging and boosting did not improve the accuracy significantly whereas stacking decreased the false positive rate by 46%. these ensembles increase the execution time and hence are not practical to be implemented in an ids. bahri et al built a ensemble method called greedyboost and compared with adaboost and c4.5 [20]. however the base classifier details are not specified in the paper but the results indicated that greedyboost scores a higher precision and recall in comparison to probe, u2r and r2l attacks present in kdd’99 dataset. bukhtoyarov et al designed neural network classifiers by applying a probabilistic approach to the network intrusion detection. genetic programming based ensembling (gpen) was deployed to design neural network ensembles [21]. they also analyzed with the kddcup 1999 dataset and classified the attacks. cordeiro and pappa utilized the particle swarm optimization (pso) by weighing the classifications obtained from different classifiers [22]. the four classification algorithms used were knn, näıve bayes, rocchio and svm. they used datasets of users of video social network for classification. their results outperformed the single classifiers. the motivation for selecting algorithms in the ensemble is due to the fact that an ensemble based on the four expert algorithms: linear discriminant classifier (ldc), quadratic discriminant classifier (qdc), k-nearest neighbor (knn) and back propagation. the ensemble is obtained by integrating the experts opinion with a weight coefficient assigned by weighted majority voting is already tested on four widely used datasets of hearth, diabetes, iris and transfusion. this ensemble resulted in better accuracy in comparison to simple majority voting approach, mean, maximum, minimum and median combiner [23]. thus many hybrid models using ensemble of techniques for intrusion detection have been developed but the major issue in ensemble approaches is the models were built and tested only on kdd datasets and thus a generic model that can deploy and test for any real time datasets is the necessity for the current scenario. the proposed model aims to overcome the issues in the existing ensemble approach and also with the advantage of developing modular structures that can have interchangeable positions. another advantage of our proposed ensemble deadvances in systems science and application (2016) vol.16 no.2 19 signs is that the algorithms can be replaced anytime with a more precise one. hence the approach aims to improve intrusion detection accuracy using simple techniques in ensemble learning integrated with pca as a dimensionality reduction technique. 3 background 3.1 preprocessing data preprocessing is very essential for huge data such as network traffic. reduction of redundant data and normalization are essential to be performed in preprocessing to build a balanced set of data. normalization is the process of transforming the data within a small specified range. the different normalization techniques are min-max normalization, z-score normalization and normalization by decimal scaling. we select z-score technique because it considers the mean and standard deviation of the attribute. d1 = b−mean (f) std (f) (1) where , mean(f)= sum of all attribute values of f std(f) = standard deviation of all values of f. 3.2 dimensionality reduction dimensionality reduction transforms the data in the high dimensional space to a lower dimension. pca performs a linear mapping of the data to a lower dimension such that the maximum variance for the data is obtained. the procedure begins with the construction of correlation matrix and computation of eigen vectors. the eigen vectors that correspond to the highest eigen values will be deployed to reconstruct the variance of the original data. transformation thus the original space is reduced to the space obtained by a few eigen vectors. the advantages of using dimensionality reduction are as follows: • time and storage space is reduced. • improves the performance of the machine learning model. • visualization of data is easier as the dimensions are reduced to 2d or 3d. 3.3 datasets we have analyzed the knowledge discovery and data mining 1999 standard dataset such as nsl-kdd data set which is widely used as intrusion detection benchmark datasets. the nsl-kdd data set contains roughly 33,300 samples. this dataset is chosen because of the following benefits [24]: 1) no redundant records in the training set. 2) due to the reduction in the number of records, the complete data 20 sumaiya thaseen and ch.aswani kumar: intrusion detection model using pca and ... can be used for both training and testing. each packet in the dataset can be classified in any one of the classes namely normal, dos, u2r, r2l and probe. all the class labels except normal indicate the different attacks in the dataset. the other dataset analyzed in our model is unsw-nb dataset obtained from the cyber range lab of the australian center for cyber security (accs). this dataset contains nearly 1,56,000 samples falling in one of the nine classifica-tion categories namely normal, analysis, backdoor, reconnaissance, exploits, fuzzers, generic, dos and shellcode. this dataset has the advantages of con-taining current attacks in the network domain. 3.4 support vector machine support vector machines are widely used for classification and regression problems. svm is preferred over other techniques due to the low generalization error and less over fitting issues that arise from the training data set. there exist a possibility of high generalization error or overfitting if the model doesnt scale well on instances not available in the training set. svm is very effective on data samples that are separable in a linear fashion. the objective is to identify the hyperplane h that can split the instances into two categories such that samples in one class fall entirely on one side of h. as we can determine unlimited number of candidate hyperplanes, svm selects only the hyperplane that maximizes distance to the closest data samples in either class. this is known as margin maximization. the major features of svm are: • deals with very large data sets efficiently. • multiclass classification can be done with any number of class labels. • high dimensional data in both sparse and dense formats are supported. • expensive computing not required. • used in many applications like e-commerce, text classification, bioinformatics, banking and other areas. there are many real time applications where such a hyperplane does not exist. in such cases, svm utilizes a function to transform the data into a different feature space such that there is a possibility of separation. the function that performs such a transformation is called as kernel function. kernels play a major role in svm. the different kernel functions widely used along with svm are [25] as given below: i) linear kernel: k (xi, xj) = xixj j) polynomial kernel : k ( x, x ′ ) = ( xx ′ + 1 )d k) rbf kernel: k ( x, x ′ ) = exp ( −γ∥ x− x ′ ∥2 ) l) sigmoid kernel: k (xi, xj) = tanh ( yxi txj + r ) k (xi, xj) = tanh ( yxi txj + r ) advances in systems science and application (2016) vol.16 no.2 21 svm can be extended to multi-class classification, a set of binary classifiers are trained one for each class depending on the data set and its respective class labels.ie. if we train the nsl-kdd data set, then let i=15 be a index in the set s=(normal, probe, dos, u2r and r2l) and let bi denote the matching binary classifier for the target set s. similarly if we train the unsw-nb dataset, then let i= 1· · · 9 be a index in the set s=(normal, analysis, backdoor, reconnaissance, exploits, fuzzers, generic, dos and shellcode) and let bi denote the matching binary classifier for the target set s. thus the observations are classified using one-versus-all approach in both the datasets. to distinguish among the binary classifiers, we deploy manager to denote one set of classification. svm produces the best results when the rbf kernel function is utilized. experimental results show that the performance of svm classifiers will differ with the selection of rbf function. therefore in this paper we train the svm manager with five different rbf values= [ 5, 2, 1, 0.5 ,0.1] for both datasets to ensure that the svm algorithm is utilized maximally. this approach will ensure greater diversity of managers in ensemble classifier as the accuracy will vary for each binary classifier according to the selected rbf values in the vector. construct a set of binary classifiers f 1 , f 2· · · f n for 1· · ·n classes each trained to differentiate one class from the rest. a multi class categorization can be obtained by combining them according to the maximal output before applying the sgn function. argmax gk(x) where gk(x) = n∑ i=1 yiai kk(x, xi) + bk where k = 1 · · · n. (2) wherein gk (x) returns a signed real value which is the distance from the hyper plane to the point x. this value is referred as the confidence value. the higher the value, the more is the confident that the point x belongs to positive class. hence we need to assign x to the class having highest confidence value. given normal data χ = {x1, x2, ...xm} ∈ rd and let r be the radius of the hypersphere and c ∈ rd which is the center. the optimization problem can be solved by determining the minimum enclosing hypersphere. minimize r2 subject to ∥ ϕ (xj)− c ∥2 ≤ r2, j = 1, ...m (3) l (c, r, α) = r2 + m∑ j=1 αj{∥ ϕ (xi − c) ∥2 − r2} (4) 22 sumaiya thaseen and ch.aswani kumar: intrusion detection model using pca and ... setting the derivatives δl (c, r, α) δc = c m∑ j=1 αj (ϕ (xj)− c) = 0 (5) we can obtain the following equation,∑m j=1 αj = 1 and c = ∑m j=1 αjϕ (xj) hence the equation (4) becomes, l (c, r, α) = m∑ j=1 αjk (xj , xj)− m∑ i,j=1 αiαjk (xi, xj) (6) which is the dual form of equation (4). the dual form of α can be obtained by solving the optimization problem, maximizing, w (α) = m∑ i=1 αik (xi, xi)− m∑ i,j=1 αiαjk (xi, xj) (7) subject to∑m i=1 = αi = 1 and αi ≥ 0 , i = 1 to m it should be noted that lagrange multiplier can be non-zero only if the inequality constraint is an equality for the solution. the complementarity conditions are satisfied by the optimal solutions α,(c, γ) αi{∥ ϕ (xi)− c ∥2 − r2}, i = 1...m (8) hence it implies that the training samples x i lie on the surface of the optimal hypersphere corresponding to αi > 0. the decision function becomes, f (x) = sgn ( r2 − ∥ ϕ (x)− c ∥2 ) this implies, = sgn(r2 − ϕ(x).ϕ(x)− 2 m∑ i=1 αiϕ(x).ϕ(xi) + m∑ i,j=1 αiαj(ϕ(xi).ϕ(xj))) = sgn(r2 − k(x, x)− 2 m∑ i=1 ϕ(xi)k(x, xi) + m∑ i,j=1 ϕiϕjk(xi, xj)))) (9) thus the aim of obtaining minimum enclosing hypersphere containing all training samples is satisfied. advances in systems science and application (2016) vol.16 no.2 23 3.5 linear discriminant classifier discriminant analysis is a classification problem where more than two groups of populations are known a priori and one or more samples from the population are classified according to the characteristics measured. the assumption is that the population πi has a probability density function of x which has a mean vector ui and variance-covariance matrix σ (similar for all populations). it is specified as f (x | πi) = 1 | σ |1/2(2π)p/2 exp [ −1 2 (x− ui) ′ σ−1 (x− ui) ] (10) we classify to the population for which p i f(x— i ) is the highest. lda is used when the variance-covariance matrix is not dependent on the population from the available data. in such cases the decision rule is based on the linear score function which is a function of the means for each of our g population ui and also the variance ccovariance matrix. the linear score function is as follows: si l (x) = −1 2 ui ′ σ−1ui ′ + ui ′ σ−1x+ logpi = d̂i0 + p∑ j=1 d̂ijxj + logpi (11) where di0 = −1 2ui ′ σ−1uiui ′ σ−1 dij = jth element of ui ′ σ−1 the far left hand expression represents a linear regression with intercept di0 and regression coefficients dij . di l (x) = −1 2ui ′ σ−1ui ′ + ui ′ σ−1x = d̂i0 + ∑p j=1 d̂ijxj given a sample unit with measurements x1,x2 · xp, the sample unit is classified into the population that has the highest linear score. this is comparable to the population that has the highest membership of posterior probability. linear score has to be calculated for each class of population and then the assignment of the sample to the population with highest score. but as this function utilizes unknown parameters ui and σ these parameters have to be determined from the data. hence discriminant analysis requires estimation of the following prior probabilities: pi = pr (πi) ; i = 1, 2, ...g population means: these can be determined by sample vectors. ui = e (x | πi) ; i = 1, 2, ...g variance-covariance matrix: this can be determined using the pooled variancecovariance matrix. 24 sumaiya thaseen and ch.aswani kumar: intrusion detection model using pca and ... σ = var (x | πi) ; i = 1, 2, ...g (12) generally these parameters are determined from training set for which the population membership is already calculated. conditional density function parameters population means: i can be determined by replacing in the sample means xi variance-covariance matrix : let s i represent the sample variance-covariance matrix for the population i. thus the variance-covariance matrix σ can be determined by substituting the pooled variance-covariance into the linear score as given below: sp = ∑g i=1 (ni − 1)si∑g i=1 (ni − 1) (13) to obtain the linear score, ˆsil (x) = −1 2xisp −1xi + ui ′ σ−1x+ logpi = d̂i0 + ∑p j=1 d̂ijxj + logpi where, d̂i0 = −1 2xisp −1xi and dij = jth element of xisp −1 this is a function of the sample mean vectors, the variance-covariance matrix and prior probabilities for different populations. thus the expression looks similar to linear regression formula with a term for intercept and a linear combination of response variables with the natural log of the probabilities. thus the decision rule is to classify the sample item into the population that has the highest calculated linear score. the steps to identify the class label is as follows step 1: delete one observation from the sample. step 2: compute the discriminant function using the remaining observations. step 3: calculate the discriminant function from step 2 to identify the class label of the observation removed from sample in step 1. steps 1-3 are repeated for all the samples. calculate the misclassified observations. 3.6 quadratic discriminant analysis qda is very similar to lda where in the assumption is that the each class measurements are distributed normally. but in qda there is no such assumption that the covariance of each of the class is identical. if the normality assumption holds true, then the best possible test for a hypothesis that the given measurement from a given class is named as the likelihood test. assume that there are only 2 groups (yϵ{0, 1}) and the means of each class are specified as uy = 0 ,uy = 1 and the covariances are defined as σy=0 and σy=1 . then the likelihood ratio will be specified as, advances in systems science and application (2016) vol.16 no.2 25 likelihoodratio = exp ( −1 2 (x− uy=1) t ∑−1 y=1 (x− uy=1) )√ 2π | σy=1 | −1 exp ( −1 2 (x− uy=0) )t ∑−1 y=0 √ 2π | σy=0 ∥ −1 < t (14) for a specific threshold ’t’. the sample estimates of the mean vector and variance-covariance matrices will substitute the population quantities in the formula. 3.7 ensemble approach using wma the basic idea of majority voting is that the votes are initialized to each managers opinion. the opinion with the highest votes is selected as the final result. littlestone and warmuth [37] have specified that the number of errors can be reduced in an ensemble model by introducing weights to the majority voting technique. we utilize voting approach in this paper as given in [37]. each manager is initialized with a weight obtained from managers accuracy in classifying the sample. as each manager based on the dataset contains different number of binary classifiers bi, we need to consider each manager’s opinion for every class i separately. thus we can divide the manager’s opinion into two categories: • managers which classify given sample as an object of class i (output value 1) • managers which assert that the given observation fits to some other class than i. (output value 0 ). the voting approach is repeated for each sample x and for each binary classifier inside the manager. this results in an ensemble manager one for each class. we define a set of weight coefficients w as a 15 element vector for nsl-kdd dataset where each element j represents weight for jth manager in ensemble ie. w = (w1, w2...w15) and similarly w as a 27 element vector for unsw-nb dataset. ie. w = (w1, w2...w27). to obtain the final decision function, we consider the weights used in the voting approach. for every single observation x, we obtain fifteen output values (y1, y2...y15) for nsl-kdd dataset and twenty seven output values (y1, y2...y27) for unsw-nb dataset, one output value per manager. each value can be a positive or negative,i.e yj = {−1, 1} where value 1 correspond to managers output 1 and negative value -1 represents managers output 0. the final decision is evaluated by the equation given below y = sgn ( n∑ i=1 wiyi ) (15) 26 sumaiya thaseen and ch.aswani kumar: intrusion detection model using pca and ... where n= 1· · · 15 for nsl-kdd dataset and n=1· · · 27 for unsw-nb dataset. each coefficient wi is multiplied with output from ith manager yi and the final decision is determined by the sign of the sum of weight coefficients for all managers. 4 proposed model the proposed model is an integrated intrusion detection system combining pca as the dimensionality reduction technique and a hybrid model of base and ensemble classifiers. in stage 1, the raw data is sent to a preprocessing unit for performing normalization by z-score technique and the noisy attributes are removed using dimensionality reduction. the resultant subset is then fed to the stage 2 which is the ensemble layer for classification. three classifiers are deployed in stage 2 namely svm. linear and quadratic discriminant classifiers. thus the ensembled approach is a merger of different supervised classifiers. a 5-fold cross validation is performed to split the data into training and testing sets. the class label is obtained by majority voting from the three classifier results. fig 1 depicts the proposed intrusion detection model. in the next subsection we discuss the approach for obtaining optimal subset using pca and ensemble approach for classification of network traffic label. algorithm for obtaining the optimal feature subset using pca: input(training set, test set) output(optimal training set, optimal test set) step 1: determine the size of training and test data step 2: scale the training and test data step 3: subtract the mean for each row m = ∑n k=1 xk n (16) wherein x specifies the individual elements and ‘n’ denotes the no. of samples. step 4: determine the covariance matrix c = xixit n (17) where x represents the matrix after subtracting the mean and xt is the transpose matrix and n is the total number of elements. step 5: determine the eigenvectors and eigenvalues of the covariance matrix. σv = λv (18) step 6: obtain a feature vector = ( eig1, eig2...eigp ) where eig1 is principal component and p ≤ n. select ’m’ such eigen vectors that match to the largest advances in systems science and application (2016) vol.16 no.2 27 ‘m’ eigenvalues in the set. algorithm for obtaining the class label using hybrid model given: classifier m1,(svm)m2,(ldc),m3(qdc), ensemble(wma) input: optimal attribute dataset d output: class label step1: initialize all the weights in d. wi = 1/n, where n is the total number of elements. step2: for every sample data di fit the svm classifier to (xt, yt) using weights wi for each class label k = 1...k obtain the hypothesis xs ← argmin (λ | f (xj) | +(1− λ)) max k(xi,xj)√ k(xi,xj)k(xi,xj) d ← d ∪ {xs} label(d) l← l ∪ {s} step 3: for every sample data di i) compute the sample estimates π̂m,ûm,σ̂ ii) make two transformations: sphere the data points based on factoring σ̂ and project to the subspace by the centroids. thus a transformation of aϵrp(k−1) iii) given any data, xϵrp transform to x̄ = axϵrk−1 x̄ = axϵrk−1 and classify according to class m = 1...k for which 1 2∥ x̃− ũm ∥2 − log π̂m is lowest where ũj = aûj step 4: for every sample data di, perform qda by repeating the process in step 3 but with different and for each class and compute the class label. step 5: the class label is predicted by weighted majority voting from results of steps (2), (3)and (4) given in equation (15) step 6: determine the performance of the model and analyze the accuracy and time complexity of the different algorithms. 4.1 class label prediction using weighted majority approach (wma) in this paper, we first deploy svm, ldc and qdc classifiers individually to result in a good generalization performance. after passing through each of the individual classifiers, a majority voting technique is adopted to predict the resultant class label from the different classifier labels. the major advantage of using ensemble approach is that the performance is improved because the approach selects only the class label which are correctly identified by all the classification techniques. the efficiency of existing approaches is compared by deploying a validation set to obtain weights for each classifier. all managers assign initial weight with value 1 and all the managers weights 28 sumaiya thaseen and ch.aswani kumar: intrusion detection model using pca and ... are combined using wmv. the predicted class label is then compared with the target label present in the validation set. if the manager has made a mistake in identifying the class label then the weights will be subtracted by a learning factor β. learning factor is user defined and the values range between 0 and 1. this process is repeated for every sample in the dataset. after each test in which a mistake arises the sum of the weights is at most ‘u’ times the sum of the weights before the test wstart for specific u < 1. if the initial eight is wbegin and the final weight is wfin , then wbeginu f ≥ wfin must be true where ‘f’ is the number of faults. f ≤ log (wbegin −wfin) log (1/u) (19) then semble approach overcomes the difference in misclassification. the overall complexity of the model is o(n4 *m) where n is the number of attributes and m is the number of training instances. ldc performs computation in o(n), qdc performs computation in o(n2 ) and svm performs computation in o(nm) time. fig. 1 proposed intrusion detection model advances in systems science and application (2016) vol.16 no.2 29 5 experiment setup and results the experimental study was conducted with nsl-kdd and unsw-nb datasets as discussed in the section 3.2. the experiments utilized matlab-2013b installed on windows 7 ultimate 64-bit machine. a 5-fold cross validation is performed on all the classifiers for splitting the dataset into 5 non repeated subsets for training, validation and testing. in this paper we compare the efficiency of the proposed algorithms with the metrics namely accuracy rate and elapsed time. • accuracy: number of test instances correctly classified by the model. • elapsed time : time to complete the detection by a classification approach. an important advantage of intergrating complementary classifiers is to improve accuracy and generalization performance. we assume the accuracy of each classifier separately. the ensemble approach is compared by considering the average score from each classifier. we analyzed the efficiency of each classifier in the base and ensemble. thus the managers accuracy is defined by ei = ∑n i=1ai n (20) where ‘n’ is the number of classes. 5.1 study 1: nsl-kdd dataset the initial step is to perform preprocessing on the data and dimensionality reduction using pca to remove the noisy attributes in the traffic that do not contribute for classification. table 1 shows the features retrieved after dimensionality reduction on the nsl-kdd dataset. the symbolic features and the features with less variance are removed from the dataset. experimental results for each manager separately for svm, ldc, qdc and ensemble managers on the nslkdd datasets are specified in tables 2-5 respectively. the results show that by deploying different classifiers for each class, higher accuracy is achieved for all classes namely normal, dos, probe, u2r and r2l which is highlighted in each table. it is also shown that in comparison to base classifiers and ensemble managers, the latter results in high accuracy above 99% for all the class labels. time required for classification for each of the managers is presented in table 6. the time consumption for the ensemble wma is relatively higher in comparison to base classifiers but it can be neglected as the accuracy is high. figs 2-5 represent the accuracies of svm, ldc, qdc and ensemble wma managers respectively. it is evident that wma produces highest accuracy in comparison to all base classifiers though individually svm, ldc and qdc produces optimal results for a single or two class label only. 30 sumaiya thaseen and ch.aswani kumar: intrusion detection model using pca and ... table 1 features selected by pca in nsl-kdd dataset 22 features selected after dimensionality reduction in nslkdd data set. service, dst bytes, dst host diff srv rate, flag, dst host serror rate, dst host srv count, same srv rate , dst host same srv rate, serror rate, src bytes, dst host srv diff host rate, host, dst host rerror rate duration, srv diff host rate, dst host srv rerror rate er ror rateprotocol type, srv rerror rate, is guest login, srv count, num compromised. table 2 accuracy obtained using svm manager with different rbf kernel in nsl-kdd dataset manager normal probe dos u2r r2l svm 1 (rbf:5) 92.9 98.08 74.41 90.08 87.5 svm 2 (rbf:2) 91.93 97.79 51.42 84.1 100 svm 3 (rbf: 1) 90.39 99.8 85.71 85.2 88.88 svm 4 (rbf:0.5) 89.73 97.78 71.82 100 83.3 svm5 (rbf:0.2) 99.54 97.94 56.25 99.2 89.47 table 3 accuracy obtained by linear discriminant analysis in nsl-kdd dataset manager normal dos probe u2r r2l ldc 1 98.87 99.83 82.05 96.99 99.12 ldc 2 93.22 97.79 94.12 86.77 99.65 ldc 3 96.58 98.56 100 99.45 98.56 ldc 4 99.58 99.6 86.48 100 81.81 ldc 5 70.4 93.92 71.84 55.86 97.42 table 4 accuracy obtained by quadratic discriminant analysis in nsl-kdd dataset manager normal probe dos u2r r2l qdc 1 92.898 97.94 82.75 83.81 89.83 qdc 2 91.73 98.28 86.29 86.81 93.32 qdc 3 87.17 97.55 67.92 91.71 92.21 qdc 4 90.59 97.92 94.05 87.44 92.45 qdc 5 89.83 97.8 78.98 86.02 95.31 advances in systems science and application (2016) vol.16 no.2 31 table 5 accuracy obtained by wma managers in nsl-kdd dataset manager normal probe dos u2r r2l wma 1 98.93 99.47 94.61 99.45 99.11 wma 2 78.77 94.47 94.59 82.78 99.6 wma 3 78.89 94.31 88.1 82.36 84.85 fig. 2 accuracy of svm managers in kdd dataset fig. 3 accuracy of ldc managers in nsl-kdd dataset fig. 4 accuracy of qdc managers in nsl-kdd dataset fig. 5 accuracy of wma managers in nsl-kdd dataset 32 sumaiya thaseen and ch.aswani kumar: intrusion detection model using pca and ... table 6 elapsed time for managers in nsl-kdd managers elapsed time (secs) svm 31.96 s ldc 63.95 s qdc 66.09 s wma 53.36 s 5.2 study ii: unsw-nb dataset after preprocessing on the dataset, dimensionality reduction using pca is performed and the results are shown in table 7. experimental results for each manager separately for svm, ldc, qdc and ensemble manager on the unsw-nb datasets are specified in tables 811 respectively. the results show that by deploying unique classifiers for each class higher accuracies are obtained for all the class labels namely normal, analysis, backdoor, exploits, reconnaissance, fuzzers, exploits, dos and shellcode respectively. the highest accuracy for all the class labels are obtained using ensemble manager higher than the base classifiers as weighted majority approach reduces the misclassification rate. time required for classification for each of the managers is presented in table 12. the time consumption for the ensemble wma is relatively higher in comparison to base classifiers but as the dataset is huge the time span is relatively huge in comparison to the nsl-kdd dataset. figs 6-9 represent the graphical accuracies using svm, ldc, qdc and wma managers for the unsw-nb dataset respectively. it is inferred that svm performs poorly for attacks namely reconnaissance, dos and analysis with accuracy of 35%, 29% and 40% respectively. ldc performs poorly for attacks such as shellcode, reconnaissance, dos and fuzzers with accuracy of 40%,36%,38% and 28% respectively. qdc performs poorly for backdoor, analysis, exploits, shellcode and reconnaissance with accuracy of 34%,33%,30% and 33% respectively whereas wma results in an average accuracy for reconnaissance,exploits, shellcode and fuzzers more than 50% which is a considerable increase in comparison to base classifiers and other class labels also produce higher accuracy. table 7 attributes retrieved after dimensionality reduction using pca in unswnb dataset. 21 attributes selected after pca id, dur, service, state, spkts, dpkts,sbytes,dbytes, rate, sttl, dttl, sload, dload, sloss, dloss, sinpkt, dinpkt, sjit, djit, swin, stcpb, dtcpb, dwin, tcprtt, synack, ackdat, smean, dmean, trans depth, response body len, ct srv src, ct state ttl, ct dst ltm, ct src dport ltm, ct dst sport ltm, is-2 login, ct 2 cmd, ctflw 4 mthd, ct src ltm, ct srv dst, is sm ips ports. advances in systems science and application (2016) vol.16 no.2 33 table 8 experimental results using svm manager in unsw-nb dataset manager normal analysis backdoor reconna issance exploits fuzzers generic dos shellcode svm 1 99.94 66.21 59.14 90.28 86.76 91.35 99.61 91.35 91.26 svm 2 51.66 60.94 50.81 71.89 64.86 98.33 84.06 76.66 86.66 svm 3 69.14 93.79 45.81 48.21 56.17 99.72 86.89 56 78.87 svm 4 99.03 39.28 45.95 60.79 87.76 53.05 98.4 70.2 27.83 svm 5 93.76 43.33 65.51 25.97 47.81 57.88 72.64 58.62 43.13 svm 6 82.34 34.76 62 30.39 52.22 63.57 87.37 80.76 73.22 svm 7 99.55 31.33 47.36 33.55 75.49 42.67 99.26 47.42 42 svm 8 96.19 32.15 47.07 33.4 52.54 33.84 98.83 28.95 34.18 svm 9 99.65 79.63 63.73 70.78 87.38 96.28 99.65 93.97 87.51 table 9 experimental results using ldc manager in unsw-nb dataset manager normal analysis backdoor reconnaissance exploits fuzzers generic dos shellcode ldc 1 87.91 53.49 54.15 75.07 67.88 63.91 99.09 71.95 39.61 ldc 2 86.41 50.22 52.95 36.07 84.34 55.66 94.71 54.16 84.94 ldc 3 98.67 75.14 62.59 36.84 65.45 56.08 92.36 38.67 50 ldc 4 97.59 80.11 92.3 32.98 65.79 42.87 99.26 46.25 41.37 ldc 5 96.59 89.2 90.44 50.89 84.26 47.55 63.43 35.26 36.74 ldc 6 94.22 76.56 85.66 36 75.44 94.59 80.82 48.21 66.99 ldc 7 87.56 44.7 59.53 37.39 55.4 28.41 99.57 38.39 43.27 table 10 experimental results using qdc manager in unsw-nb dataset manager normal analysis backdoor reconnaissance exploits fuzzers generic dos shellcode qdc 1 82.11 46.41 77.41 26.38 48.33 50 65.21 67.1 67.16 qdc 2 97.86 28.67 25.03 35.8 60.34 57.06 80.68 75.49 58.76 qdc 3 86.85 32.98 34.03 33.42 83.7 60.34 59.06 85.68 50 qdc 4 91.73 43.43 52.76 36.34 46.27 51.37 94.03 77.36 30.17 qdc 5 84.21 83.33 63.37 42.19 51.76 45.83 80.94 74.15 42.23 qdc 6 81.24 35.82 81.51 24.4 52.8 53.95 87.59 81.92 41.95 qdc 7 88.39 40.52 78.94 49.49 38.58 54.98 87.58 68.85 64.19 qdc 8 87.48 44.45 58.95 35.43 54.54 66.31 81.54 91.66 50.63 qdc 9 88.25 46.7 63.85 36.72 62.33 62.83 96.43 82.48 53.56 table 11 experimental results using wma manager in unsw-nb dataset manager normal analysis backdoor reconnaissance exploits fuzzers generic dos shellcode wma1 99.99 78.41 86.71 65.97 76.66 91.35 91.85 58.94 86.66 wma1 98.51 51.95 64.71 71.89 68.84 98.33 90.88 50.46 91.26 wma 3 88.93 52 62.09 48.21 71.14 99.72 81.45 90.75 78.87 wma 4 99.41 50.8 100 60.79 73.97 53.05 97.65 ‘100 57.83 wma 5 93.2 67.32 84.61 90.28 59.9 57.88 90.19 67.67 53.13 wma 6 54 100 50 60.39 46.84 63.57 84.32 68.8 73.22 wma 7 95.25 55.89 57.09 63.55 62.67 42.67 99.15 58.8 62 wma 8 81.71 74.72 50.22 63.4 88.21 63.84 91.4 69.34 64.18 wma 9 99.21 58.67 65.62 70.78 87.92 96.28 84.13 64.96 87.51 table 12 elapsed time for managers in unsw-nb data set managers elapsed time (secs) svm 464.82 s ldc 485.56 s qdc 622.50 s wma 751.61 s 34 sumaiya thaseen and ch.aswani kumar: intrusion detection model using pca and ... fig. 6 proposed intrusion detection model fig. 7 proposed intrusion detection model fig. 8 proposed intrusion detection model advances in systems science and application (2016) vol.16 no.2 35 fig. 9 proposed intrusion detection model 5.3 discussion in this paper, we have analyzed the accuracy of the base and ensemble classifier managers. it is to be noted that the ensemble managers accuracy is relatively high in comparison to the base classifiers and managers accuracies are above 99% for nsl-kdd dataset and the range is from 88-100% for unsw-nb dataset. the variation is due to the imbalance of samples in the dataset for the classes exploits, shellcode and reconnaissance. the reason for the high accuracy in both the datasets are 1) proper selection of training data 2) well-selected parameters in wma technique namely the learning factor and weight assignment wi for each sample. 3) misclassification error is reduced due to weight assignment to each sample. there were poor results obtained for ldc and qdc manager only in the unsw-nb dataset. but svm performs well for nsl-kdd dataset and in specific for class labels normal, fuzzers, generic and dos in the unsw-nb dataset. svm produced poor accuracy of 85% for dos and qdc produced poor accuracy of 91% and 95% for u2r and r2l attacks in the nsl-kdd dataset. the accuracy was as low as 49.49% for reconnaissance class label because the test data contains samples that were not utilized while training the classifiers. thus in comparison to wma, the base classifiers svm, ldc and qdc perform poorly and also with increase in elapsed time. the constraint on the weights generated by wma also is a major factor in improving the accuracy of the intrusion detection model. the weights can lie between 0 and 1, the higher the value the more is the possibility of correctly predicted. thus this model can be extended to include other ensemble techniques but it should consider the time complexity as many ensemble approaches complete the task in extremely long time. 36 sumaiya thaseen and ch.aswani kumar: intrusion detection model using pca and ... 6 conclusions the classification accuracy can be improved with minimal elapsed time by integrating opinions from multiple managers into single using an ensemble approach. we have deployed weighted majority voting to integrate results from different managers. the three base classifiers namely svm, ldc and qdc were experimentally compared using two different datasets namely nsl-kdd and unswnb. thus wma results in good accuracy for both the datasets. the accuracy improvement is of 1% in comparison to base classifiers. thus the success of the model is due to the generated weights in wma which were tuned by the learning factor. thus the integration of base classifiers in to an ensemble manager with wma will be very well utilized in intrusion detection as it has been tested on recent datasets with modern day attacks. the other improvement in our approach is instead of relying on binary classification techniques, we have utilized multi class classification for the base managers. in future we can integrate optimization techniques with wma to tune the parameters for generating weights. references [1] khan l., awad m. and thruaisingham b. (2007), “a new intrusion detection system using support vector machines and hierarchical clustering”, the vldb journal, vol. 16, pp. 507-521. [2] wang g., hao j., ma j. and huang l. (2010), “a new approach to intrusion detection using artificial neural networks and fuzzy clustering, expert systems with applications”, vol. 37, pp. 6225-6232. [3] sandhya peddabachigiri, ajith abraham, crina grosan and johnson thomas. (2007), “modeling intrusion detection system using hybrid intelligent systems”, journal of network and computer applications, vol. 30, pp. 114-132. [4] lee w, stolfo s and mok k. (1999), “a data mining framework for building intrusion detection model”, in: proc. of ieee symposium on security and privacy, pp. 120-32. [5] a. kausar, m. ishtiaq, m.a.jaffar and a.m.mirza. (2010), “optimization of ensemble based decision using pso”, in: proceedings of the world congress on engineering, vol. 10. [6] tsang c.h., kwong s. and wang h. 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[20] e. bahri, n.harbi and h.n.huu. (2011), “approach based ensemble methods for better and faster intrusion detection”, in: computational intelligence in security for information systems, pp. 17-24. [21] v.bukhtoyarov and v.zhukov. (2014), “ensemble-distributed approach in classification problem solution for intrusion detection systems”, in: intelligent data engineering and automated learning-ideal 2014, pp. 662-668. [22] cordeiro junior, z. and g.l.pappa. (2011), “a pso algorithm for improving multi-view classification”, in: 2011 ieee congress on evolutionary computation ( cec ), pp. 925-932. [23] a.kausar, m.ishtiaq, m.a.jaffar and a.m.mirza. (2010), “optimization of ensemble based decision using pco”, in proceedings of the world congress on engineering, wce, vol. 10. [24] http://nsl.cs.unb.ca/nsl-kdd/ [25] s.j.horng, m.y.su, y.h.chen, t.w.kao, r.j.chen, j.l.lai and c.d.perkasa. (2011), “a novel intrusion detection system based on hierarchical clustering and support vector machines”, expert systems and applications, vol. 38, no. 1, pp. 306-313. corresponding author sumaiya thaseen can be contacted at: sumaiyathaseen@gmail.com advances in systems science and applications (2012) vol.12 no.4 327-346 what are natural systems, actually? alan rayner bath bio art, 81 dovers park, bathford, bath ba1 7ud, uk abstract to this day, the applicability of abstract, definitive logic and mathematics to natural systems is rarely challenged or even questioned. consequently we find ourselves predominantly living, working and researching in a way that contradicts how we naturally are in the world as it naturally is. this seems unwise, to put it mildly. in such circumstances can it be any surprise when we are drawn into needless conflict and misunderstanding, unable to work out what it means to live in an ecologically sustainable way, and prone to inflict profound psychological, social and environmental harm on ourselves and our natural neighbourhood? a way out from this predicament is offered by what has been called ‘natural inclusionality’, which has the potential radically to transform human understanding of natural systems and our place within them. keywords boundary perceptions, evolutionary process, flow-geometry, intangibility, natural inclusion, place-time, receptivity, space perceptions 1 introduction: abstract and natural perceptions of space and boundaries by imposing non-existent rigid structure onto naturally continuous space and dynamic boundaries, and enshrining this within three dimensional planes set at right-angles to one another, abstract mathematical logic engenders profound paradox[1-3]. natural space cannot be confined within a box-frame, nor can it be cut into discrete segments that can be moved around independently and/or relative to one another. natural space is infinite at all scales i.e. dynamically distinguishable into different localities, but not divisible into separable, discretely packaged units or quantities. by the same token, natural boundaries cannot cut, they can only dynamically configure space. to cut space, as abstract models require, they would have to be reduced to zero thickness, and hence be nowhere. this paradox of treating space and boundaries as rigidly definable fabric or structure is laid bare by the following statements: “when a smaller box s is situated, relatively at rest, inside the hollow space of a larger box s, then the hollow space of s is a part of the hollow space of s, and the same “space”, which contains both of them, belongs to each of the boxes. when s is in motion with respect to s, however, the concept is less simple. one is then inclined to think that s encloses always the same space, but a variable part of the space s. it then becomes necessary to apportion to each box its particular space, not thought of as bounded, and to assume that these two spaces are in motion with respect to each other”[4]. 328 alan rayner:what are natural systems, actually “space is another framework we impose upon the world . . . here the mind may affirm because it lays down its own laws; but let us clearly understand that while these laws are imposed on our science, which otherwise could not exist, they are not imposed on nature. . . .euclidian geometry is . . . the simplest, . . . just as the polynomial of the first degree is simpler than a polynomial of the second degree. . . . the space revealed to us by our senses is absolutely different from the space of geometry” poincaré[5]. a moment’s contemplation reveals how utterly unrepresentative of the space and boundaries of our natural experience this treatment is. all we have to do is ask ‘what needs to be present for natural form to be distinguishable?’ or, more concretely, ‘why is our direct experience of walking into a brick wall different from that of walking through an open doorway?’ or, ‘what makes it possible to paint a picture?’ it then becomes apparent that the only way of answering these questions is to acknowledge the occurrence of at least two kinds of natural presence: a receptive context or medium which provides freedom for local movement and/or expression, and local formative content, which informs or configures that context. the former is necessarily spacious, the latter necessarily cohesive. moreover, for form to be and become distinguishable, each of these presences must naturally include the other. spacious presence alone would be formless void, and formative presence alone would have no shape or size. they are necessarily distinct, but mutually inclusive presences. they can neither be abstracted from one another as independent entities, nor be homogenised into ‘oneness’. the only way in which this necessity can be fulfilled is for one of these presences, natural space, ultimately to be everywhere, continuous, intangible (i.e. frictionless) and immobile, and for the other ultimately to be somewhere, distinctive, tangible and continually in motion. natural space and figural boundaries are hence, respectively, continuous and dynamically distinct energetic interfacings between the insides and outsides of all natural forms as flow-forms[1-3]. in summary, natural systems are hence actually radically different in their dynamic, evolutionary organization from the self-contained abstract scientific models that are widely used to represent and simulate their behaviour, both conceptually and practically. this difference arises from the preand/or postimposition of rigidly definitive structure in abstract models onto the spatial and dynamic continuity of natural flow-geometry. such imposition is embedded in the foundations of propositional and dialectic logics and both classical and modern mathematics[1-3]. whereas it has predictive utility in unchanging or repetitive systems, it cannot be expected to account adequately for the dynamics of evolutionary systems, where it has the potential to give rise to serious and damaging misunderstanding and miscalculation. in the latter systems a radically different, more natural approach is needed in which conventional discontinuous treatments advances in systems science and applications (2012) vol.12 no.4 329 of space and boundaries are replaced by continuous ones. this more natural approach is available through the fluid boundary logic of what has been called ‘natural inclusionality’[2-3]. here, i discuss the benefits of ‘naturalizing’ systems science using this approach to understand all forms as flow-forms energetic configurations of space in space, displaying varying degrees of deformability, permeability and connectivity depending on context. 2 natural inclusionality conceptually, natural inclusionality is simply a way of understanding all natural evolutionary form or organization as flow-form, an energetic configuration of space in space. implicit within this simple description are, however, two radical innovations in thought: (1). the recognition that natural boundaries are intrinsically energetic ‘dynamic interfacings’ between distinct localities, not the ‘inert limits’ of discrete objects[1,6]. (2). the recognition that natural boundaries can only be dynamic through the inclusion of space as infinite, intangible, frictionless presence[1-3]. correspondingly, the dynamic origin of all natural form as an energetic configuration of space can be understood in terms of an evolutionary process of natural inclusion: the co-creative, fluid-dynamic transformation of all through all in receptive spatial context[2-3,7]. this understanding differs radically from the conception of the evolutionary origin of biological species by ‘natural selection, or the preservation of favoured races in the struggle for life’[8]. it similarly challenges any notion of cosmological origins and expansion from an isolated locality. such notions of ‘something out of nowhere’ are an abstract product of theoretically dividing or unifying space and form[9]. natural inclusionality therefore calls for what amounts to a paradigmatic transformation a radical de-framing and reframing of abstract perceptions of space and boundaries that have imposed non-existent definition onto naturally continuous systems for millennia. such a transformation has profound implications for every aspect of our philosophical understanding of and relationship with nature, including human nature: logical, mathematical, scientific, artistic, theological, linguistic, educational, social, psychological and political. the need for such a transformation was recognised by polanyi[10] when he stated that: “for once men have been made to realize the crippling mutilations imposed by an objectivist framework once the veil of ambiguities covering up these mutilations has been definitely dissolved many fresh minds will turn to the task of reinterpreting the world as it is, and as it then once more will be seen to be”. to summarize in general terms, natural inclusionality is a kind of awareness that helps us to appreciate our selves and other tangible forms as dynamic in330 alan rayner:what are natural systems, actually habitants of nature, not discrete subjects and objects rigidly set apart from one another. this awareness comes with recognizing that natural space is a limitless intangible presence everywhere, which permeates throughout and beyond all tangible expressions of energy, whether in the form of radiation or massy bodies. natural space cannot be cut and can neither resist nor be resisted by nor be removed from the presence and movement of tangible forms. far from being just empty distance between, outside or occupied by discrete material objects or structures as is assumed by abstract logic natural space is a receptive presence, vital for movement and communication. as natural dynamic inclusions of space, all forms are variably fluid flow-forms. their boundaries are energetic configurations of space, not exclusions from space. when they move, they do not move through space; instead space permeates through them. with this awareness comes an appreciation of self-identity as an inclusion of neighbourhood a fluid inclusion, not a rigid exclusion of others’ identities. our understanding of physical reality is such as to bring profound compassion for ourselves and other life forms, and is a source of deep inspiration and creativity. it calls for an expansion of conventional theoretical reasoning to include more fluid, artistic and poetic forms of expression. in more technical and philosophical terms, natural inclusionality is a new philosophy and fluid boundary logic of self-identity and ecological and evolutionary diversity and sustainability. it is intended to supersede the abstract rationality that has dominated human thought for millennia, based on definitive logic that can only apply to inert material systems that are unknown to exist anywhere in nature. whereas abstract rationality treats space as empty distance between, occupied by or outside completely definable tangible material structures or objects with discrete boundary limits, natural inclusionality recognizes space as a limitless, indivisible, receptive (non-resistive) ‘intangible presence’ vital for movement and communication. this allows all form to be understood as flow-form, distinctive but dynamically continuous, not singularly discrete. the simple move from regarding intangible space and tangible boundaries as mutually exclusive sources of discontinuity and discrete definition to mutually inclusive sources of continuity and dynamic distinction enables self-identity to be understood as a dynamic inclusion of neighborhood. intangible space is included throughout and beyond all tangible figural forms as configurations of energy, whether as massy bodies or mass-less electromagnetic radiation. 3 the relationship between natural inclusionality and ecological sustainability the natural inclusional perception of living systems as flow-forms that receive, retain and pass on energy in the process of growing, living and dying is inconsistent advances in systems science and applications (2012) vol.12 no.4 331 with any model of them as independent entities. to be entirely self-contained is to be an inert, hermetically closed structure with no capacity for take up or loss of energy between inner world and outer world. the nearest any life forms actually get to this condition is when they form survival capsules such as spores, seeds, pupae and cysts that carry them through periods of scarcity. this, by contrast with the darwinian perception of survival by competitive exclusion of others, is what real biological ‘survival’ or ‘preservation’ entails. in such a dormant condition they are incapable of any active growth or relationship with others. but no sooner is any activity resumed that can support growth, so too is any life form’s capacity to lose as well as take up energy through its necessarily permeable bodily boundaries and those of others in its vicinity. it is therefore clear that the availability of sources of energy is the principal influence that governs the growth, organization and function of all natural forms of organic life as variably open systems. any activity or pattern of development in which energy loss through permeable boundaries persistently exceeds energy acquisition will result in unsustainable deficit. on the other hand, any pattern of development that permanently prevents energy loss also prevents energy gain. for any living system to sustain itself, its primary need is therefore to be able to attune its activities and development to correspond with energy availability and hence with the local conditions of its habitat. this availability varies, both in amount and rate of supply, due to seasonal and climatic fluctuations, and where and in what form it is located. it also changes due to the growth, death and decomposition of the systems themselves, which respectively deplete and replenish supplies as they come under one another’s simultaneous mutual influence. real life does not, therefore, inhabit an even playing field of energy, space and time. instead it continually both changes and responds to changes in the contextual circumstances of its natural neighbourhood in an improvisational process of autocatalytic flow, which gives rise to evolutionary and ecological complexity and succession[1,6]. through this process of ‘natural inclusion’ an opening is made dynamically for an extraordinary diversity and complexity of interdependent forms and patterns of life to co-evolve over myriad nested temporal and spatial scales. the breathtaking variety that we can find in a crumb of soil, a patch of chalk grassland, a coral reef and a tropical forest comes into being under the guidance of no more and no less than the responses and contributions of its membership to natural energy flow in a natural ‘sustainability of the fitting’[11-13]. fig.1 illustrates the general principles arising from observations of how living systems (except modern human cultures) attune their patterns of growth and development to variable availabilities of energy sources. as natural inclusional energetic inner-outer interfacings of continuous space, the boundaries of real organisms, populations and communities do not remain constant throughout their 332 alan rayner:what are natural systems, actually life span, but fluidly vary in permeability, deformability and contiguity (connectivity)[6,14]. they change in dynamic relationship with the availability of energy predominantly assimilated from sunlight into organic compounds via the process of photosynthesis, and rendered into chemical form (adenosine triphosphate) via the oxidative-reductive reactions of respiration as a form of combustion. moreover, these changes themselves entail alterations in boundary chemistry induced by and involving shifts in availability and production of oxidizing and reducing power[6,15]. fig.1 the interplay between boundary-proliferating (‘differentiation’) and boundary-condensing (‘integration’) processes in energy-rich (stippled) and energy-restricted circumstances. this interplay enables energy to be assimilated (allowing regeneration and proliferation of boundaries), conserved (by conversion of boundaries into relatively impermeable form), explored for (through internal distribution of energy) and recycled (via redistribution/reconfiguration of boundaries) in spatial capsules, channels, branches and networks of life forms in dynamic attunement with their natural neighbourhood. thin lines indicate relatively more permeable boundaries, thick lines relatively impermeable boundaries and dotted lines degenerating boundaries[6]. the ecological and evolutionary sustainability of natural life forms, from the cells and tissues in a human body to the trees in a forest correspondingly depend upon close harmonization with (as distinct from unilateral adaptation to) advances in systems science and applications (2012) vol.12 no.4 333 fig.2 ‘fungal foraging’[6,16]. the mycelium of the wood-decaying fungus, hypholoma fasciculare, finds an ‘oasis in a desert’, by fluid-dynamically spreading and narrowing its energetic focus. the fungus has been inoculated into a tray full of soil on a block of wood (‘starter’ food source), with an uncolonized wood block (‘bait’ food source) placed some distance away from it. distinct stages are shown in the radial spreading of the fungal colony from the inoculated wood block, followed by the redistribution and directional focusing of its energy following upon contact with the bait. as indicated in fig.2, similar fluid dynamic patterns of gathering in, conservation of, exploration for and redistribution of energy supplies within variably connective channels and capsules of receptive space are found throughout the living world, from subcellular to ecosystem scales of organization. 334 alan rayner:what are natural systems, actually the diversity, complementary nature and changeability of all within their neighbourhood, to which they themselves contribute. when energy supplies become scarce, sustainable living systems pool and redistribute internal resources within integrated structures and survival capsules they do not compete to proliferate faster on the dwindling supplies than their neighbours. when supplies are abundant they proliferate and differentiate. moreover, as is beautifully illustrated by the exploratory patterns of some kinds of fungi, this ability to attune their capacity to differentiate and integrate activity in dynamic relationship with energy availability allows life forms to locate and sustain supplies in heterogeneous habitats with extraordinary efficiency. as illustrated in fig.2, they do this through a combination of all round exploration and directional focus. sustainability, not supremacy, is therefore the path of evolutionary and ecological continuity. natural energy flow is variably fluid, circulatory and redistributive along pressure gradients from higher concentration (relative ‘abundance’) to lower concentration (relative ‘scarcity’), as illustrated, for example by atmospheric and ocean currents. the primary need for all life forms is not to seek competitive advantage through the unilateral accumulation of energy ‘wealth’ at the expense of their neighbourhood, but to sustain themselves and their offspring as variable channels for natural energy flow. they are more like members of a relay team than a set of autonomous individuals striving to be first past the post. to succeed in this they have to be open to the energetic influence of their neighbourhood at the same time as sustaining the distinctiveness but not discreteness (or separateness) of their inner worlds from their outer worlds through their dynamic boundaries. any ecological or evolutionary or management model that treats an individual or group as a discrete, autonomous object or subject with the set objective of promulgating and preserving its self at all costs as sole survivor of a war of attrition is therefore partial and unsustainable in a changeable world of natural energy flow. unfortunately, just such models are implicit in the objectivistic framing of natural energy flow that continues to underpin our strategic planning for a desirable future and perceptions of what it means to be sustainable. we confuse sustainability with self-preservation, just as darwin[8] did when describing ‘natural selection’ as the preservation of favoured races in the struggle for life. why? 4 unsustainable logic: self-dislocation from natural neighbourhood notions of adversarial ‘competition’ and coercive ‘co-operation’, which respectively underlie individualistic ‘capitalism’ and collectivistic ‘socialism’, are predicated upon definitive logic that is incompatible with the cumulative energetic transformation of an evolving system[2]. it is presupposed that individual or group advances in systems science and applications (2012) vol.12 no.4 335 entities can be defined independently from their spatial context and correspondingly that their ‘future’ can be fully defined by present or ‘initial conditions’. as recognized by bateson[17], this narrows the focus of perception and purpose at the outset of enquiry into nature instead of in the process of discovery (cf. fig.2) and can give rise to the familiar idea that undesirable present ‘means’ can be justified by desirable future ‘ends’. human beings may be cognitively and culturally predisposed to make this presupposition through a combination of our inter-related capacities for categorization, sociality, abstract thought, tool and language use and awareness of mortality[12-14,18]. on the other hand, the imagination that comes alongside these capacities offers the creative potential to escape the restrictions imposed by purposive abstract objectivity through what is actually the more comprehensive worldview of natural inclusionality[2-3,12]. as terrestrial, omnivorous, bipedal primates unable to digest cellulose but equipped with binocular vision and opposable thumbs that enable us to catch and grasp, we are predisposed to view the geometry of our natural neighborhood in an overly definitive way. we are prone to see the world in terms of what it can do for us and to us as detached observers or abstracted ‘exhabitants’, not how we are inextricably involved in it as natural inhabitants. we perceive ‘boundaries’ as the limits of definable ‘objects’ and ‘space’ as ‘nothing’ a gap or absence outside and between these objects[1]. as discussed earlier, this perception of space and boundaries as definitively discontinuous is incompatible with the comprehension of continuity and change[23,19]. if two adjacent locations in space and/or time are distinguished by a boundary, which one does the boundary belong to? if it belongs to both of them, how can the mutual exclusivity of definitive logic be satisfied, and where do both cease to be both and become either one or the other? if it belongs to neither, then where does one location end and the other begin and what really comes between them? in the case of a curved boundary, does it belong to whatever lies within it or to whatever lies without it? if two distinct locations are both contained within a larger location, are they mutually exclusive or coexistent? upon such dilemmas rests the whole gamut of alternative propositional (either/or) and dialectical/transcendental logics (both/and in mutual opposition) that have been in conflict for millennia and continue to be so[20]. so too do the ‘holons’ as ‘janus-faced’ entities combining individual and collective aspects, and ‘holarchies’ as nested arrays of holons, of koestler[21] in his ‘open hierarchical systems theory’[22-23]. that it is nonetheless possible to avoid this perception is, however, evident from the indigenous cultures that sustain a much stronger sense of inclusion in nature, aided by the preservation of oral, aural and nomadic traditions [24-25]. 336 alan rayner:what are natural systems, actually according to walker[26], “cross-cultural views of the self define individuality in terms of boundaries, locus of control and inclusiveness versus exclusiveness, or that which is intrinsic versus that which is extrinsic to the self [27-28]. cultures that emphasize firm boundaries and high personal control tend to view the self as exclusionary or ‘self contained’. fluid boundary, strong field control cultures, view the self as “ensembled,” meaning that the self is inclusive of other individuals. while ‘self contained’ individualism is indigenous to the united states and to the european countries from which its dominant ethnic groups draw their roots, ‘ensembled’ individualism is far more prevalent as a percentage of all known cultures[29]. ensembled individualism is also indigenous to aboriginal, native american, senoi and other cultures that are widely known to use dreams for social purposes.” the perception of completely definable objects separated by intervals of space as ‘gaps of nothingness’ sets the scene for the hard line logic of abstract rationality to become established in the foundations of our mathematical, scientific, theological, linguistic, governmental and economic endeavors. it also profoundly affects our perceptions of ‘self’ and ‘self-interest’. the definitive supposition that one thing is not another thing, and, specifically, that ‘one self cannot be another self’ leads to what c.s. lewis[30] called ‘the philosophy of hell’, in which ‘to be means to be in competition’. it is easy to see that this detached perception of nature and human nature in unnatural opposition could lead to profound human conflict and jealous possessiveness. with the continuous presence of space throughout and beyond all form erased from consideration, ‘subjective self’ and ‘objective other’ are brought into fear-full confrontation. priorities are inverted from seeking sustainable relationship with others in a natural ‘communion of diversity’, to seeking cancerous dominion over other as the only certain route to ‘self-preservation’[25]. sustaining ‘ego’ becomes the focus of attention at the expense of the natural neighbourhood upon which individual self-identity actually depends to sustain itself. love and trust of others break down into xenophobia and avarice. can this abstraction actually be intellectually justified as a means of representation consistent with sensory experience (i.e. evidence) and that makes consistent sense? in a word, no, it cannot, because energy/matter cannot physically be cut away from space[2,18,31-32]. nonetheless, the dissociation of matter from space is embedded in the numerical and geometrical foundations of classical and modern mathematics. here it may be recalled that euclidean geometry is the abstract geometry of zerodimensional (size-less) numerical points, one-dimensional (breadth-less) lines, two-dimensional (depthless) planes and three-dimensional solids (self-contained volumes). its figures are used to represent definitive tangible structure and yet advances in systems science and applications (2012) vol.12 no.4 337 can only actually represent the intangible presence in the core of tangible form because it is impossible to reach zero without removing the tangible presence. the same applies to the so-called ‘non-euclidean’, riemannian and lobachevskian geometries of curved surfaces. the scientifically inconvenient truth is hence that abstract euclidian and noneuclidean points, lines and planes/curved surfaces can consist only of intangible presence, not tangible presence! by the same token, it is impossible to drive or rotate a solid body from or around a solid fixed centre. the central ‘still’ point, axis or plane of symmetry of any bodily form can only consist of intangible presence, with correspondingly zero pressure. in effect, conventional mathematics and its discontinuous underpinning logic thereby treat ‘1’, as a ‘unit of tangible presence’, as if it is ‘0’, a vanishing point of intangible presence. they literally attempt to construct ‘one thing from nothing’ and then to sum an infinite number of these one things up into an infinite ‘whole’ as a ‘one’ that is also ‘many’, whilst discounting the very presence that truly is infinite, at all scales. this difficulty can only be resolved realistically by accepting that in nature, tangible and intangible presences are distinct but mutually inclusive. this is the point recognized by the fluid geometry of natural inclusionality. here, space and boundaries are regarded as mutually inclusive sources of continuity and dynamic distinction with variable connectivity, not mutually exclusive sources of discontinuity and discrete definition, as in euclidean and non-euclidean geometries. so far, the only mathematical formulation explicitly to accept and incorporate this natural inclusion of non-local space in and throughout local figural form is the ‘transfigural mathematics’ introduced in 1985 by lere shakunle[32-34]. natural inclusionality effectively transforms the fixed frameworks of euclidean and non-euclidean geometries into fluid framings of omnipresent, non-local intangible space everywhere, within (intra-), throughout (trans-), between (inter-) and beyond (extra-) local tangible energetic form[32]. this opens the possibility of a dynamic, co-creative, mutually inclusive relationship between internally and externally situated non-resistive (and hence receptive) intangible spatial presence and locally situated, tangible energetic presence. 5 variable connectivity: the sustainable self-cultivation of life all that may therefore be needed to unlock our imagination and the world of real, live organisms and communities from the unnatural confinement imposed by abstract rationality is the simple understanding that space cannot be cut, occupied, confined or excluded. space is a continuous presence throughout and beyond the boundaries of natural figures. by the same token, these boundaries are energetic interfacings between inner and outer realms, not fixed limits. this simple move from regarding space and boundaries as sources of discontinuity and 338 alan rayner:what are natural systems, actually discrete definition to sources of continuity and dynamic distinction is the ecological and evolutionary point of departure of ‘natural inclusionality’ from objective rationality. the underlying logic of natural inclusionality can be described as ‘the understanding of all form as flow-form, an energetic configuration of space throughout figure and figure in space’, such that space, as a receptive (non-resistive) presence, is not assumed to be discontinuous (i.e. to stop at discrete boundary limits)[12,26]. correspondingly, we can recognize the impossibility of defining or measuring anything in absolute numerical terms anywhere, because all form has both a ‘figural’, energetic inner-outer interfacing or dynamic boundary, which makes it distinct, and a ‘transfigural’ (this term was first conceived by lere shakunle in 1985) ‘through the figure’ spatial reach that cannot be sliced or limited. the continuous space throughout and beyond the figure pools it within the co-creative, influential neighbourhood of all others: local ‘self’ as an ‘including middle’ finds identity in its non-local neighbourhood as neighbourhood finds identity through its local ‘self’. without spatial continuity, figures are rendered into lifeless bodies, integral or fractional numbers and idealized geometric points, lines and solids. with space included, we can escape the confinement and inconsistencies of the ‘excluded middle’, discrete boundary logic of ‘one opposed to other’ that has held human imagination to ransom for millennia. this enables us to move on to a more natural and comprehensive form of reasoning in the fluid boundary logic of each in the other’s mutual influence. the real meanings of ‘zero’ and ‘infinity’ as qualities of space and sources of creativity, not abstract quantities of material, are brought into our natural accounting systems, not excluded by abstract definition. the following simple exercise might help illustrate the difference between the hard-line, space-cutting view of discontinuous models and fluid-line understanding of natural inclusionality. draw an outline of two figures using a dotted line on a plain sheet of paper. the ‘paper’ infinitely stretched would represent what in the transfigural geometry developed by lere shakunle is called ‘omni-space’[32,34]. the space within each figure represents ‘intra-space’, the space between figures ‘inter-space’, the space beyond the figures ‘extra-space’ and the space transcending the figures’ permeable and dynamic boundaries ‘trans-space’. you can see how the continuous non-local space everywhere (omni-space’) is locally configured into distinctive, but not discrete regions. in the way that you have drawn them, the figures are not contiguous (connected), and so their ‘intra-spaces’ can only communicate through the ‘inter-space’ and ‘trans-space’ between and permeating their boundaries as energetic interfacings and restraining influences (not restrictive material definitions or external forces see later). nonetheless, they advances in systems science and applications (2012) vol.12 no.4 339 inhabit the same limitless pool of omni-space everywhere. if you were now to draw the figures closer together, so that their boundaries first connect and then coalesce at one or more points, their intra-space now becomes continuous (cf. fig.3). on the other hand, if you were to take a pair of scissors and cut around the dotted lines, the figures will drop out of their spatial context as discontinuous individual entities. this ‘dropping out’ of context is what discontinuous models of reality effectively do they treat boundaries as cut-out zones between discrete inner realms and outer realms, instead of dynamic relational interfacings through which these realms remain continuous through trans-space. fig.3 distinct but not discrete figures of space in space (redrawn by philip tattersall from original pencil sketch by alan rayner, 2010). fig.3 illustrates the dynamic relationships between figural flow-forms as energetic configurations of space throughout figure and figure in space. it also serves to distinguish the natural inclusional dynamic relationship between distinct but not discrete flow-forms both from reductive schemas that cut off inner from outer spatial realms and from connective and holistic schemas where individual dynamic locality is eschewed from a seamless, purely figural whole or ‘unity’. since the cartoons can only represent an instantaneous ‘slice’ through the figures, the dotted lines shouldn’t be taken to represent ‘sieves’ but more the seething ‘fluid mosaic’ that constitutes real biological membranes. a very simple example of what is represented in the cartoon can also be seen between surface-tense droplets 340 alan rayner:what are natural systems, actually of water condensing on a surface. as they expand and come into proximity their tensely curved inner-outer interfacings first touch and then coalesce in a visible rush as each flows reciprocally into the other and the tension of their boundaries is released. fig.4 stages (from top left clockwise) in fusion between the protoplasm-filled cellular tubes (hyphae) within the mycelium of the basidiomycete fungus, phanerochaete velutina. the tubes are internally partitioned into distinct compartments by septa, which have a door-like pore in their middle. as fusion occurs (third picture in the sequence) the cell walls and membranes around initially distinct tubes coalesce, so that their intracellular cytoplasm, which in its turn contains membrane bound organelles (nuclei and mitochondria) becomes continuous. a visible recoil can occur in the receptive hypha when the tubes coalesce. (photographed by dr a.m. ainsworth). a living illustration of the process of figural boundaries coming into proximity, contiguity and conjugation occurs during the process of hyphal fusion that is found in many fungi[16] and is shown in fig.4. here some fundamental differences between rationalistic and natural inclusional perceptions of connectivity and continuity emerge: (1). in rationalistic thought, continuity is equated with ‘connectedness’ because space is regarded as void, a source of discontinuity or disruptive gap between and around ‘things’ as discrete objects. hence the only way of deriving continuity in this ‘whole way of thinking’, is either by totally excluding space and boundaries from form as a continuous line or network of width-less threads, or by totally conflating space with form in a seamless [distinction-less] whole. such exclusion or conflation is neither consistent with evidence/experience nor does it advances in systems science and applications (2012) vol.12 no.4 341 make consistent sense. (2). in natural inclusional thought, space is a continuous omnipresence that cannot be cut, occupied, confined or excluded, and form is dynamically continuous through its energetic inclusion of space throughout figure and figure in space. distinction and difference are hence accommodated in a natural fluid continuum, without contradiction. local identity is recognised as a dynamic inclusion of non-local space in which all forms are pooled together (but not merged into complete unity) in natural communion as flow-forms. (3). correspondingly, the treatment of continuity by objective rationality as the same as connectedness as exemplified in conventional calculus, where continuity is approximated by connecting infinitesimal discontinuous units is an idealized abstraction that is physically impossible. the very idea of complete ‘whole units’ existing anywhere, at any scale in nature as an energetically open, fluid system does not make sense. the fluidly variable connectivity of natural inclusionality arises from the coming together (contiguity/interconnectivity), fusion (confluence/intra-connectivity) and dissociation (individuation/differentiation) of energetic paths, corridors or channels of included space in labyrinthine branching systems and networks (i.e. as shown in fig. 2), not the ‘ties that bind all into a web of one’[1,31,35]. 6 flow-networking the space-including processes of regeneration, degeneration, differentiation and integration illustrated in figs 1-4 are very different from the purely tangible connectedness of modern network theory. they inform us about how energy is assimilated, located, conserved and redistributed in real-world sustainable systems, as distinct from abstract mathematical models. this is the understanding that i suggest we need to incorporate into systems science. as we do this, there are a number of principles that we need to remember. rather than being formed by stringing together a given set of initially independent entities, flow-networks grow into place through a combination of selfdifferentiating (boundary-maximizing) and self-integrating (boundary-minimizing) processes that configure and reconfigure space in dynamic correspondence with energy availability. for example, fungal mycelia form when a spore germinates by first swelling symmetrically as it takes in water and nutrients across its bounding cell wall and membrane. the resulting structure then becomes polarized, hence breaking spherical symmetry and increasing surface area to volume ratio, through the emergence of a germ-tube or ‘hypha’ with a parabolic growing tip. as this tube elongates, its growth accelerates exponentially, as the absorptive surface increases, before attaining a more or less constant rate of extension, whence branches begin to emerge, each with their own parabolic growing tips. a den342 alan rayner:what are natural systems, actually dritic (tree-like) system of hyphal branches develops, which radiates out in all directions. eventually, in many fungi, as resources are depleted by the growing system, some of the branches begin to fuse or anastomose with one another, so converting the inner part of the system into a network of labyrinthine channels (as in fig.4). during this process, the branches open up their external boundaries to one another, so that the inter-space initially between them becomes the continuum of intra-space within them. in other words, they ‘let go’ of their individuated self-identity ‘agenda’ in the process of coalescing by self-integration (cf. fig.2). within the integrated system, the branches do not disappear, but retain their form as connective channels of intra-space. the nodes in this system are the places from which the branches originally arose, rather than the loci of initially discrete entities. the branch-identities are the connective channels in the system, not the ‘knots’ or local centres through which network transactions are administratively controlled. at no stage in the evolution of the system have these identities been fully dislocated from one another or the limitless pool of common space in which they are immersed and of which they are dynamic inclusions. by growing into place, these dynamic systems exhibit indeterminacy, the potential for indefinite expansion and transformation within boundaries that vary in their deformability, permeability and connectivity depending on contextual circumstances. this contrasts with the determinacy assumed by many to apply to creatures like our individual selves, sentenced to death within a fixed frame of bodily space and time and so bustling through life as if there were no place else to care for, notwithstanding the continuum of our social space. such indeterminacy brings scope for continual improvisation, discovery and learning through co-creative evolutionary play that is not fixed on a pre-determined course, but eases its own passage through a process of autocatalytic flow in which the flow of current lowers resistance to subsequent flow: sheep, wildebeest, ants and humans all exhibit this phenomenon as they create paths by following in one another’s wake[6]. some fungal mycelia making their way through ancient forest in this fashion are thought to cover up to square kilometres of ground and to be thousands of years old. by connecting their internal space in parallel rather than purely in series (as applies to dendritic systems, lacking anastomoses/cross links), flow-form networks greatly increase their conductivity and consequent capacity to store (i.e. ‘memorize’) and supply power at or to localities on their boundaries (cf figs 1, 3). in fungi, this increased capacity is what allows mycelial systems literally to ‘mushroom’ as well as to produce survival structures such as sclerotia (of which ‘ergots’ are a well known example) and rapidly extending cable-like aggregations known as ‘rhizomorphs’ because of their root-like appearance and growth. mycelial systems that lack or lose the ability to form anastomoses are prone to advances in systems science and applications (2012) vol.12 no.4 343 become dysfunctional and degenerate, proliferating numerous branches from local nodal sites in a way that looks very similar to some unrealistic ‘maps’ that have been made of the internet using purely abstractive analytical techniques. local, well connected centers in flow-form networks drain resources from the system, and inhibit its expansion. in fungi, fruit bodies and storage structures may form at such centers. in human organizations they have the potential to develop into exploitative growths and megalithic power structures. degenerative processes in flow-form networks are vital as a means of preventing retention of power by core components of the system. for example, ‘fairy rings’, consisting of an annulus of spreading mycelium, result from the degeneration of the colony centre and release of its resources to supply the growing margin. in the absence of such degeneration, expansion of the system stalls. death is vital to the possibility of continuing life: it feeds life and opens up new possibilities for reconfiguration it does not annihilate life in the way that the rationalistic view of space as an absence of presence may lead us to believe. the ability of flow-form networks to differentiate, integrate and degenerate, by varying the dynamic properties of their boundaries in tune with their circumstances and avoiding the wastage implicit in rationalistic ‘cost-cutting’, allows them to produce extraordinarily efficient organizations in highly heterogeneous situations. in fungi inhabiting the forest floor, for example, this ability allows them to make connections between local sources of nutrients in decaying wood, leaf litter and roots, to form an underground communicative infrastructure, which brings the lives and deaths of the trees into a common circulation . so, altogether, these living flow networks are far more sensitively attuned to the ever-reconfiguring space that their channels embody, than the inflexible meshwork entrapments our current abstractions represent. how do these principles translate into management praxis? i have just two general suggestions: (1). being alive to any life-form’s unique situation, the way it attunes with its neighbourhood, the complex relationships that such attunement entails, and the ease with which these relationships can be destroyed by insensitive intervention. (2). value one’s own learning experience, be prepared to share this with others and value others’ unique experience, rather than simply following or desiring some ‘one size fits all’ doctrine, fad or short term ‘fix’. i have the feeling that these suggestions might sound rather obvious and lacking any absolute, clear, fixed, authoritarian direction. they might seem like not much more than we might gather about life’s patterns and uncertainties from our everyday experience as relational human beings good neighbours using all our sentient faculties. i do hope so! 344 alan rayner:what are natural systems, actually references [1] rayner a.d.m. (2004), “inclusionality and the role of place, space and dynamic boundaries in evolutionary processes”, philosophica, vol.73, pp.51-70. [2] rayner a.d. (2011), “space cannot be cut: why self-identity naturally includes neighbourhood”, integrative psychological and behavioural science, vol.45, pp.161-184. [3] rayner a.d.m. 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(2012), “place-time: the flow-geometry of space”, available from http://www.bestthinking.c om/article/permalink/1798?tab=article&title=place-time-the-flow-geometr y-of-space. [10] polanyi m. (1958), personal knowledge: towards a post-critical philosophy, london, routledge and kegan paul, pp.381. [11] rayner a.d.m. (2008), “natural communion: poems and paintings about our human inclusion in the evolutionary flow of place-time”, available from http://www.inclusional-research.org/furtherreading/naturalcommunion.pdf. [12] rayner a.d.m. (2010), “inclusionality and sustainability attuning with the currency of natural energy flow and how this contrasts with abstract economic rationality”, environmental economics, vol.1, pp.98-108. advances in systems science and applications (2012) vol.12 no.4 345 [13] elstrup o. (2009), “the ways of humans: modelling the fundamentals of psychology and social relations”, integrative psychological and behavioural science, vol.43, pp.267-300. [14] elstrup o. 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(2005), the fall. winchester, new york: o books. [26] walker e.m. (2003), “the confusion of dreams between selves and the other: non-linear continuities in the social dreaming experience”, in w.g. lawrence (ed.), london: karnac books, experiences in social dreaming, pp.215-227. [27] heelas p & lock a. (1981), indigenous psychologies: the anthropology of the self, london: academic press. [28] sampson e. (1988), “indigenous psychologies of the individual and their role in personal and societal functioning”, american psychologist, vol.43, pp.15-22. [29] sampson e. (2000), “reinterpreting individualism and collectivism: their religious roots and monologic versus dialogic person-other relationship”, american psychologist, vol.55, pp.1425-1432. [30] lewis c.s. (1942), the screwtape letters, geoffrey bles. [31] tesson k.j.a. (2006), dynamic networks: an interdisciplinary study of network organization in biological and human organizations, phd thesis, university of bath. [32] shakunle l.o. & rayner a.d.m. (2009), “transfigural foundations for a new physics of natural diversity the variable inclusion of gravitational space in electromagnetic flow-form”, journal of transfigural mathematics, vol.1, no.2, pp.109-122. [33] shakunle l.o. (1994), spiral geometry. the principles (with discourse), hitit verlag, berlin, germany. [34] shakunle l.o & rayner a.d.m. (2008), “superchannel inside and beyond superstring: the natural inclusion of one in all iii”, transfigural mathematics, vol.1, no.3, pp.9-55, pp.59-69. [35] barabási a-l. (2002), linked: the new science of networks, perseus publishing. corresponding author alan rayner can be contacted at:alan@admrayner.plus.com advances in systems science and applications (2011), vol. 11, no. 1-2 27-41 existence results for semilinear fractional functional differential equations with state-dependent delay∗ yong-kui chang1, m.mallika arjunan2, g. m. n’guér ékata3 and v. kavitha4 1department of mathematics, lanzhou jiaotong university, lanzhou, gansu 7300070, p.r. china 2department of mathematics, karunya university, karunya nagar, coimbatore641 114, tamil nadu, india 3 department of mathematics, morgan state university, 1700 e. cold spring lane, baltimore, m.d. 21251, usa 4 department of mathematics, karunya university, karunya nagar, coimbatore641 114, tamil nadu, india email: lzchangyk@163.com, arjunphd07@yahoo.co.in, gaston.n’guerekata@morgan.edu, kavi velubagyam@yahoo.co.in abstract according to theories for α-resolvent family (sα(t))t≥0 and fixed point methods, this paper is mainly concerned with existence of mild solutions to a semilinear fractional functional differential equation with state-dependent delay in a complex banach spacex. some sufficient conditions are established without the compactness of (sα(t))t≥0. keywords fractional differential equations α-resolvent family state-dependent delay fixed point 1. introduction in this paper, we establish the existence of mild solutions to the following fractional func∗this research is supported by nnsf of china (10901075), the key project of chinese ministry of education (210226), the scientific research fund of gansu provincial education department (0804-08) and “qing lan” talent engineering funds (ql-05-16a) by lanzhou jiaotong university. issn 1078-6236 international institute for general systems studies, inc. 28 chang:existence results for semilinear fractional functional . . . . . . tional differential equation with state-dependent delay dαx(t) = ax(t) + f(t, xρ(t,xt)), t ∈ j = [0, b], (1) x0 = ϕ ∈ b, (2) where b > 0, 0 < α < 1 and a : d(a) ⊂ x → x is the infinitesimal generator of an α-resolvent family (sα(t))t≥0 defined on a complex banach space x. the function xs : (−∞, 0] → x, xs(θ) = x(s + θ), belongs to some phase space b that will be defined later ( see section 2), f : j ×b → x, ρ : j ×b → (−∞, b] are appropriate functions and ϕ belongs to the phase space b with ϕ(0) = 0. the fractional derivative dα is understood here in the riemann-liouville sense. the theory of functional differential equations has emerged as an important branch in nonlinear analysis. it is worth mentioning that several important practical problems have lead to investigations of functional differential equations of various types ( see the books of hale et al. [11], wu [31], and the references therein). on the other hand, functional differential equations with state-dependent delay appear frequently in applications as model of equations and for this reason, the study of this type of equation has gained great attention in the last decades, we refer to [4, 5, 9, 14, 15, 22] and the references therein. differential equations of fractional order play a very important role in describing some real world problems. for example some problems in physics, mechanics and other fields can be described with the help of fractional differential equations, see [2, 7, 17, 24, 25] and references therein. the theory of differential equations of fractional order has recently received much attention and now constitutes a significant branch in differential equations. lots of research papers and monographs have appeared devoted to fractional differential equations, for example see [1, 3, 6, 8, 18, 19, 20, 21, 26, 27, 28, 30, 33, 34] and the references therein. motivated by the above mentioned works, the purpose of this paper is to investigate the existence results of mild solutions to a semilinear fractional functional differential equation with state-dependent delay described in the general abstract form (1)-(2). the main technique is based upon the α-resolvent family (sα(t))t≥0 combined with suitable fixed point theorems. this paper is organized as follows. in section 2, we introduce notations, definitions and some lemmas which are used in the sequel. in section 3, we prove the existence of mild solutions advances in systems science and applications (2011), vol. 11, no. 1-2 29 for the problem (1)-(2). 2. preliminaries from now on, we set j = [0, b]. we denote by x a complex banach space with norm ‖ · ‖,c(j,x) the space of all x-valued continuous functions on j , endowed with the topology of uniform convergence with norm ‖x‖∞ := sup t∈j ‖x(t)‖. and l(x) the banach space of all linear and bounded operators on x. moreover, br(z0,z) denotes the closed ball with center at z0 and radius r > 0 in z. definition 2.1 [25] assume that f ∈ cm(r+,x). if α ∈ (m− 1,m), where m ∈ n, then the riemann-liouville fractional derivative of order α ∈ (m− 1,m) is the expression dα t f(t) = dm dtm ∫ t 0 gm−α(t− s)f(s)ds, where for β > 0 gβ(t) =  tβ−1 γ(β) for t > 0, 0 for t ≥ 0. definition 2.2 [25] let α > 0 and f : r+ → r be in l1(r+,x). then the riemann-liouville integral is given by: iαf(t) = 1 γ(α) ∫ t 0 (t− s)α−1f(s)ds. recall that the laplace transform of a function f ∈ l1(r+,x) is defined by: f̂(λ) = ∫ ∞ 0 e−λtf(t)dt, re(λ) > ω, if the integral is absolutely convergent for re(λ) > ω. 30 chang:existence results for semilinear fractional functional . . . . . . definition 2.3 [26] let a be a closed and linear operator with domain d(a) defined on a banach space x and α > 0. let ρ(a) be the resolvent set of a. we call a the generator of an α-resolvent family if there exists ω ≥ 0 and a strongly continuous function sα : r+ → l(x) satisfying sα(0) = i such that {λα : re(λ) > ω} ⊂ ρ(a) and (λα −a)−1x = ∫ ∞ 0 e−λtsα(t)xdt, re(λ) > ω, x ∈ x. in this case, sα(t) is called the α-resolvent family generated by a. for construction of solution by using α-resolvent family, we refer to [26] and the references therein. we also refer to [23, 29] for more information about resolvent or solution operator. remark 2.1 [26] note that if a is the generator of an α-resolvent family (sα(t))t≥0 then the laplace transform of sα(t) is ŝα(λ) = (λα −a)−1. in this paper, we will employ the axiomatic definition of the phase space b introduced by hale and kato in [12] and follow the terminology used in [16]. thus, (b, ‖ · ‖b) will be a seminormed linear space of functions mapping (−∞, 0] to x, and satisfying the following axioms: (a1) if x : (−∞, b) → x with b > 0, is continuous on [0, b] and x0 ∈ b, then for every t ∈ [0, b) the following conditions hold: (i) xt ∈ b; (ii) there exists a positive constant h such that ||x(t)|| ≤ h‖xt‖b ; (iii) there exist two functions k(·),m(·) : r+ → [1,+∞) independent of x(t) with k continuous and m locally bounded such that ‖xt‖b ≤ k(t) sup{||x(s)|| : 0 ≤ s ≤ t}+m(t)‖x0‖b. denote kb = sup{k(t) : t ∈ [0, b]} and mb = sup{m(t) : t ∈ [0, b]}. (a2) for the function x(·) in (a1), xt is a b-valued continuous function on [0, b]. (a3) the space b is complete. advances in systems science and applications (2011), vol. 11, no. 1-2 31 an example of phase space b satisfying (a1) − (a3) is the following space c0 g (see [31], pp.44), where g : [−∞, 0]→ [0,∞) is a given continuous nondecreasing function such that: (i) g(0) = 1 and g(−∞) =∞. (ii) the function g(t) = sup { g(t+s) g(t) : −∞ < s ≤ −t } is locally bounded for t ≥ 0. let c0 g = {φ : (−∞, 0]→ x;φ is continuous and lim s→−∞ |φ(s)| g(s) = 0} then c0 g , together with the following norm: ‖φ‖c0 g = sup |φ(s)| g(s) , satisfies axioms (a1)− (a3). the next lemma is a consequence of the phase space axioms and is proved in [14]. lemma 2.1 ([14]) let ϕ ∈ b and i = (γ, 0] be such that ϕt ∈ b for every t ∈ i . assume that there exists a locally bounded function jϕ : i → [0,∞) such that ‖ϕt‖b ≤ jϕ(t)‖ϕ‖b for every t ∈ i . if x : (−∞, b]→ r is continuous on j and x0 = ϕ, then ‖xs‖b ≤ (mb + jϕ(max{γ,−|s|})‖ϕ‖b +kb‖x‖max{0,s}, for s ∈ (γ, b], where we denotedkb = sup t∈j k(t) andmb = sup t∈j m(t), ‖x‖max{0,s} = sup {‖x(θ)‖, θ ∈ [0,max{0, s}]}. to conclude the current section, we recall the following well-known results. theorem 2.2 [10, theorem 6.5.4]. let d be a closed convex subset of a banach space x and assume that 0 ∈ d. let γ : d → d be a completely continuous map. then, either the set {x ∈ d : x = λγ(x), 0 < λ < 1} is unbounded or the map γ has a fixed point in d. lemma 2.3 ([13, 32]) suppose b ≥ 0, α > 0 and a(t) is a nonnegative function locally integrable on 0 ≤ t < t ( for some t ≤ +∞), and suppose u(t) is nonnegative and locally integrable on 0 ≤ t < t with u(t) ≤ a(t) + b ∫ t 0 (t− s)α−1u(s)ds 32 chang:existence results for semilinear fractional functional . . . . . . on this interval; then u(t) ≤ a(t) + ∫ t 0 [ ∞∑ n=1 (bγ(α))n γ(nα) (t− s)nα−1a(s) ] ds. 3. existence results in this section, we present and prove the existence results for the fractional differential problem (1)-(2). first, we present its mild solution. definition 3.1 a function x : (−∞, b] → x is called a mild solution of (1)-(2) if x0 = φ, xρ(s,xs) ∈ b for each s ∈ j and x(t) = ∫ t 0 sα(t− s)f(s, xρ(s,xs))ds, for each t ∈ j. (3) we are now in a position to state and prove our existence result for the problem (1)-(2). for the study of this, we first list the following hypotheses: (h1) there exist m > 0 and δ > 0 such that ‖sα(t)‖l(x) ≤meδt, t ∈ j . (h2) the function f : j → b → x is completely continuous and there exists a continuous function µ : j → (0,+∞) such that ‖f(t, ψ)‖ ≤ µ(t)‖ψ‖b, (t, ψ) ∈ j × b. (h3) the function t → ϕt is well defined and continuous from the set r(ρ−) = {ρ(s, ψ) : (s, ψ) ∈ j × b, ρ(s, ψ) ≤ 0} into b. moreover, there exists a continuous and bounded function jϕ : r(ρ−)→ (0,∞) such that ‖ϕt‖b ≤ jϕ(t)‖ϕ‖b for every t ∈ r(ρ−). remark 3.1 for more details on the hypothesis (h3), we refer to [14]. theorem 3.1 assume that the hypotheses (h1)-(h3) hold, then the problem (1)-(2) has at least one mild solution on (−∞, b]. advances in systems science and applications (2011), vol. 11, no. 1-2 33 proof. let y = {u ∈ c(j,x) : u(0) = ϕ(0) = 0} endowed with the uniform convergence topology and n : y → y be the operator defined by nx(t) = ∫ t 0 sα(t− s)f(s, xρ(s,xs))ds, for each t ∈ j. (4) where x : (−∞, b] → x is such that x0 = ϕ and x = x on j . from axiom (a1) and our assumption on ϕ, we infer that nx(·) is well defined and continuous. let ϕ : (−∞, b] → x be the extension of ϕ to (−∞, b] such that ϕ(0) = ϕ(0) = 0 on j and jϕ = sup{jϕ : s ∈ r(ρ−)}. we will prove that n(·) is completely continuous from br(0, y ) to br(0, y ). step 1: n is continuous on br(0, y ). let (xn)n∈n be a sequence in br(0, y ) and x ∈ br(0, y ) such that xn → x in y . from the axiom (a1), it is easy to see that (xn)s → xs uniformly for s ∈ (−∞, b] as n → ∞. by (h2) we have ‖f(s, xn ρ(s,(xn)s) )− f(s, x ρ(s,(x)s) )‖ ≤ ‖f(s, xn ρ(s,(xn)s) )− f(s, x ρ(s,(xn)s) )‖+ ‖f(s, x ρ(s,(xn)s) )− f(s, x ρ(s,(x)s) )‖, which implies that f(s, xn ρ(s,(xn)s) ) → f(s, x ρ(s,(x)s) ) as n → ∞ for each x ∈ j . by axiom (a1), lemma 2.1 and dominated convergence theorem, we obtain ‖n(xn)−n(x)‖ = sup t∈j ∥∥∥∥∫ t 0 sα(t− s) [ f(s, xn ρ(s,(xn)s) )− f(s, x ρ(s,(x)s) ) ] ds ∥∥∥∥ → 0 as n→∞. thus, n(·) is continuous. step 2: n maps bounded sets into bounded sets. let µ∗ = sup 0≤τ≤b µ(τ). if x ∈ br(0, y ), from lemma 2.1, follows that ‖xρ(t,xt)‖b ≤ r ∗ = (mb + j ϕ )‖ϕ‖b +kbr. (5) and so ‖n(x)(t)‖ = ∥∥∥∥∫ t 0 sα(t− s)f(s, xρ(s,xs))ds ∥∥∥∥ 34 chang:existence results for semilinear fractional functional . . . . . . ≤ ∫ t 0 ‖sα(t− s)‖l(x)‖f(s, xρ(s,xs))‖ds ≤m ∫ t 0 eδ(t−s)µ(s)‖xρ(s,xs)‖bds ≤mµ∗r∗ ∫ t 0 eδ(t−s)ds ≤mµ∗r∗ eδb δ . this implies that ‖n(x)‖∞ ≤mµ∗r∗ eδb δ = l. step 3: n maps bounded sets into equicontinuous sets. let t1, t2 ∈ j with t1 > t2 and x ∈ br(0, y ). then ‖n(x)(t1)−n(x)(t2)‖ = ∥∥∥∥∫ t1 t2 sα(t1 − s)f(s, xρ(s,xs))ds + ∫ t2 0 [ sα(t1 − s)− sα(t2 − s) ] f(s, xρ(s,xs))ds ∥∥∥∥ ≤ i1 + i2, where i1 = ∫ t1 t2 ‖sα(t1 − s)f(s, xρ(s,xs))‖ds, i2 = ∫ t2 0 ∥∥∥[sα(t1 − s)− sα(t2 − s) ] f(s, xρ(s,xs)) ∥∥∥ ds. here i1 and i2 tend to 0 as t1 → t2 independently of x ∈ br(0, y ). in fact, i1 = ∫ t1 t2 ‖sα(t1 − s)f(s, xρ(s,xs))‖ds ≤ ∫ t1 t2 ‖sα(t1 − s)‖l(x)‖f(s, xρ(s,xs))‖ds ≤m ∫ t1 t2 eδ(t1−s)µ(s)‖xρ(s,xs)‖bds ≤mµ∗r∗ ∫ t1 t2 eδ(t1−s)ds = mµ∗r∗ [eδ(t1−t2) − 1 δ ] . advances in systems science and applications (2011), vol. 11, no. 1-2 35 hence lim t1→t2 i1 = 0. and i2 = ∫ t2 0 ∥∥∥[sα(t1 − s)− sα(t2 − s) ] f(s, xρ(s,xs)) ∥∥∥ ds → 0, since f is compact and sα is strongly continuous, ∥∥∥[sα(t1 − s)− sα(t2 − s) ] f(s, xρ(s,xs)) ∥∥∥ → 0 as t1 → t2 uniformly for x ∈ br(0, y ). we conclude that lim t1→t2 i2 = 0. step 4: the operator n maps br(0, y ) into a relatively compact set in x. indeed from the strong continuity of sα(·) and (h2), the set { sα(t− s)f(s, xρ(s,xs)), t, s ∈ [0, b], x ∈ br(0, y )} is relatively compact in x. moreover, for x ∈ br(0, y ), using the mean value theorem for the bochner integral, we obtain nx(t) ∈ tconv{sα(t− s)f(s, xρ(s,xs)) : s ∈ [0, b], x ∈ br(0, y )}, for all t ∈ [0, b]. consequently the set {nx(t) : x ∈ br(ϕ|j , y )} is relatively compact in x, for every t ∈ [0, b]. step 5: a priori bounds. set λ = {x ∈ y such that x = λn(x) for some 0 < λ < 1}. let x ∈ λ. then for each t ∈ [0, b], we have ‖x(t)‖ ≤ λ ∫ t 0 ‖sα(t− s)‖l(x)‖f(s, xρ(s,xs))‖ds ≤m ∫ t 0 eδ(t−s)µ(s)‖xρ(s,xs)‖bds ≤mµ∗ ∫ t 0 eδ(t−s) [ (mb + j ϕ )‖ϕ‖b +kb‖x‖max{0,s} ] ds ≤mµ∗ ∫ t 0 eδ(t−s) [ (mb + j ϕ )‖ϕ‖b +kb‖x‖s ] ds ≤mµ∗(mb + j ϕ )‖ϕ‖b eδb δ +mµ∗kb ∫ t 0 eδ(t−s)(t− s)α−1(t− s)1−α‖x‖sds ≤ θ1 + θ2 ∫ t 0 (t− s)α−1‖x‖sds, where θ1 = mµ∗(mb + j ϕ )‖ϕ‖b eδb δ θ2 = mµ∗kbe δbb1−α. 36 chang:existence results for semilinear fractional functional . . . . . . in view of lemma 2.3, we have for all t ∈ j , ‖x(t)‖ ≤ θ1 [ 1 + ∫ t 0 ∞∑ n=1 (θ2γ(α))n γ(nα) (t− s)nα−1 ] ds ≤ θ1 [ 1 + ∞∑ n=1 (θ2γ(α))nbnα nαγ(nα) ] = θ1 [ 1 + ∞∑ n=1 [θ2γ(α)bα]n γ(nα+ 1) ] ≤ θ1λα[θ2γ(α)bα], where λα[θ2γ(α)bα] = ∞∑ n=0 [θ2γ(α)bα]n γ(nα+ 1) is the mittag-leffer function. this implies that ‖x‖∞ ≤ θ1λα[θ2γ(α)bα]. hence combining step 1–step 5 and using the theorem 2.2, we obtain that n has a fixed point which is a mild solution of (1)-(2) on (−∞, b]. next, we give an existence result when the nonlinearity f has a sublinear growth with its state variable. let us list the following condition: (h2∗) the function f : j → b → x is completely continuous such that there exist a continuous function µ : j → (0,+∞) and a continuous nondecreasing function w : [0,+∞) → (0,+∞) satisfying ‖f(t, ψ)‖ ≤ µ(t)w (‖ψ‖b), (t, ψ) ∈ j × b, lim inf ξ→+∞ w (ξ) ξ = γ < +∞. theorem 3.2 assume that the hypotheses (h1), (h2∗) and (h3) are satisfied. then the problem ( 1)-(2) admits at least one mild solution on (−∞, b] provided that meδbµ∗kb δ γ < 1,where µ∗ = sup 0≤τ≤b µ(τ). (6) proof. let n be the operator defined by (4). we shall complete the proof by schauder’s fixed point theorem. let r∗ be defined as (5). we claim that there exists a positive number r such that nbr(0, y ) ⊆ br(0, y ). if it is not true, then for each r > 0, there exists xr(·) ∈ br(0, y ), but nxr /∈ br(0, y ), that is, advances in systems science and applications (2011), vol. 11, no. 1-2 37 ‖n(xr)(t)‖ > r for some t(r) ∈ j , where t(r) denotes t depending on r. however, on the other hand, we have from (h1), (h2∗) that r < ‖n(xr)(t)‖ = ∥∥∥∥∫ t 0 sα (t− s) f ( s, xρ(s,xrs) ) ds ∥∥∥∥ ≤ m ∫ t 0 eδ(t−s)µ (s)w (r∗) ds ≤ m eδb δ µ∗w (r∗) . dividing both sides by r and taking the lower limit, we get meδbµ∗kb δ γ ≥ 1, where contradicts (6). hence for some positive r, nbr(0, y ) ⊆ br(0, y ). just the same as the proof in theorem 3.1, we can show that n is continuous on br(0, y ) and n maps br(0, y ) into a relatively compact set in x. next we prove that the family {nx : x ∈ br(0, y )} is an equicontinuous family of functions. let t1, t2 ∈ j with t1 > t2 and x ∈ br(0, y ). then ‖n(x)(t1)−n(x)(t2)‖ = ∥∥∥∥∫ t1 t2 sα(t1 − s)f(s, xρ(s,xs))ds + ∫ t2 0 [ sα(t1 − s)− sα(t2 − s) ] f(s, xρ(s,xs))ds ∥∥∥∥ ≤ i1 + i2. where i1 = ∫ t1 t2 ‖sα(t1 − s)f(s, xρ(s,xs))‖ds, i2 = ∫ t2 0 ∥∥∥[sα(t1 − s)− sα(t2 − s) ] f(s, xρ(s,xs)) ∥∥∥ ds. here i1 and i2 tend to 0 as t1 → t2 independently of x ∈ br(0, y ). indeed, i1 = ∫ t1 t2 ‖sα(t1 − s)f(s, xρ(s,xs))‖ds 38 chang:existence results for semilinear fractional functional . . . . . . ≤ ∫ t1 t2 ‖sα(t1 − s)‖l(x)‖f(s, xρ(s,xs))‖ds ≤m ∫ t1 t2 eδ(t1−s)µ(s)w (‖xρ(s,xs)‖b)ds ≤mµ∗w (r∗) ∫ t1 t2 eδ(t1−s)ds = mµ∗w (r∗) [ eδ(t1−t2) − 1 δ ] . hence lim t1→t2 i1 = 0. and i2 = ∫ t2 0 ∥∥∥[sα(t1 − s)− sα(t2 − s) ] f(s, xρ(s,xs)) ∥∥∥ ds → 0, since f is compact and sα is strongly continuous, ∥∥∥[sα(t1 − s)− sα(t2 − s) ] f(s, xρ(s,xs)) ∥∥∥ → 0 as t1 → t2 uniformly for x ∈ br(0, y ). we deduce that lim t1→t2 i2 = 0. thus, by the arzela-ascoli theorem n is a completely continuous operator. in view of schauder’s fixed point theorem, we deduce that n has a fixed point which is a mild solution of (1)-(2) on (−∞, b]. this finishes the proof. according to theorem 3.2, we can easily obtain the following consequence for the sublinear growth case. 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[34] y. zhong, j. feng and j. li, existence and uniqueness for fractional neutral differential equations with infinite delay, nonlinear anal., 71 (2009), 3249-3256. introduction preliminaries existence results advances in systems science and application (2015) vol.15 no.2 99-118 towards a quantum model for meditation françois dubois1 and christian miquel2 1 dept. of mathematics, conservatoire national des arts et métiers, paris, france. 2 association for the development of mindfulness and the center for mindfulness in medicine, health care, and society. abstract we study the meditative states of human beings from the conceptual framework provided by the fractaquantum hypothesis : analogously to an atom, man can from his “quiet” base state explores various states of higher energy as loving or mystical state. we then look what energy states are explored during meditation: is it the “hyperfine” structure of his base state? is there a love ecstatic state? a very high energy structure mystical state? on one hand we illustrate these hypothesis from the experience of a large part of mystical traditions such as hinduism or buddhism and on another hand from contemporary cognitive sciences. in addition, quantum mechanics indicates that any interaction between energy levels is mediated by a boson of exchange. so we aim to identify the nature of this boson linking the various human being energy levels. keywords fractaquantum hypothesis. 1 fundamental state and fractaquantum hypothesis this work is placed in the context of the fractaquantum hypothesis: the quantum approach is applicable to all indivisible scales in nature, regardless of their size [1]. in addition, the underlying question is that of a quantum model for humans, knowing that a quantum model is not interested in specific individuals, but rather to what they have in common, and makes them “indistinguishable” through this approach. we are guided by the atomic analogy. it is known that an atom is, from the quantum point of view, correctly described by its energy levels. these can be calculated as the eigenvalues of the hamiltonian operator describing the interaction between the atom and the electromagnetic environment outside. thus, the atom has energy levels as proposed by niels bohr [2] and exactly calculable (at least for the hydrogen atom) with the quantum mechanics methods; we refer e.g. the book of cohen-tannoudgi et al [3]. the fundamental level | 0 >, a first excited level | 1 >, a second noted as | 2 >, etc. when the atom passes from the level | 0 > to the level | 1 >, it absorbs the energy gap corresponding to an absorbing wave of frequency ν according to the relation h ν = ε1 − ε0 , (1) where h is planck’s constant. when the atom moves from level | 1 > to the basic level | 0 >, it forwards a ν frequency wave according to the (1) equation. 100 françois dubois and christian miquel: towards a quantum model for meditation in this quantum description, the initial analogy recalled here is located in the human being excited levels. we admit that everyday life, with joys, sorrows, desires and frustrations, is in fact a sum of a fundamental state modulations. this base state could be defined in a first approach, implicitly compared to all states of greater excitement. it is the “current state” of the profane or everyday life, made of low excitations, unstable discontinuous changes according to the every moment exchange of energy made with the outside world, without the usual subject being necessarily aware. these can be closer to what christophe andré [4] calls “states of mind” these mixtures of feelings, thoughts and emotions match each time to a state of temporary power of my relationship to the world, changing during each new configuration, thus being replaced by another state, with both fluctuations and matching micro energy jumps. we call this fundamental state of weakness and disorganized excitement of everyday life, the “daily agitated state”, noted | 0 >, as we will later see. this basic structure is referred to an atom (known technically as molecular orbital) and physics tells us that a “fine structure” (see e.g. the book [3]) exists for each level. the difference of energy between sublevels that forms this fine structure is much smaller than the difference ε1 − ε0 of the relation (1). still in the same analogy, we propose that a succession of energy states | 0, j > and levels ε̃j actually compose the fundamental level | 0 >. as for the atom, it is a “fine structure” even though we cannot yet offer any explicit representation. but as evoked earlier, different feelings, anger, sadness, desires, etc. are all “understate of excitation” of the base state. we know that anger requires energy then falls, such as desires and their acomplishments, or sadness and fear. a real spectroscopist work starts here, classifying energy gaps of these states. following descartes, spinoza was the first to go this route, since his whole ethics is to show how ethics invites to deviate from the resentmentfull sad passions towards joy thus corresponding to the maximum intensity of being, as presented by damasio [5]. the wheel of emotions of plutchik shows how the eight basic emotions (joy, fear, disgust, anger, sadness, surprise, trust, anticipation) can each pass through three degrees or intensity[6]. moreover, descartes opened the way with his treatise on the passions, where he quoted six basic emotions: admiration, hate, love, desire, joy, sadness. from irritationton anger then rage just as aprehension, from right to fear, or in a positive way from serenity to joy then extasy there is allways a fluctuation going along with possible macro-jumps of energy. the lauri nummenmaa et al. finnish study of the aalto university recently deepened and nuanced this classification of emotions, showing in spinoza’s way, that they affect more or less subjective and energy body according to their nature[7]. more than seven hundred volunteers from several countries were asked advances in systems science and application (2015) vol.15 no.2 101 to show on a human silhouette the parts of their bodies that were superactivated, or otherwise impaired, while feeling one of the seven emotions, including the five emotions currently recognized as primary by researchers: joy, sadness, anger, fear, disgust. the results show that the body activation is “very low” to “low” when passing from depression to sadness, contempt or shame, every time with a lower limbs sub-activation and a slight activation of the upper body. conversely when going from disgust to fear and anger, the body activation becomes stronger, mainly in the upper areas of the body. both emotions of happiness and especially love match the strongest activations, this time of the entire body. let’observe how energy, atomic levels energy differences, and by extension man’s excitation levels are of the same nature as an engine’s mechanical energy. this physical concept is well understood since sadi carnot and josiah gibbs pioneering works during the 19th century[8, 9]. 2 from base state to love and mystical states if we now focus on excited states above the base state, we assume that the first excited state | 1 > is love. like the equation (1) it “gives energy”, places the individual in a very specific state. we all have experienced it and know that after a while, passion time, one falls from the state of love to the natural state. it has been described by all cultures over the course of time. as we write these lines in trouville, we believe the christian bobin books (we refer e. g. to the two fundamental la part manquante or le très-bas [10]), to alain de botton, swiss-born english writer who develops a philosophy of love [11], to pierre de ronsart : “mignonne allons voir si la rose?” [12]. to the disappointment in love, staying in the love state | 1 >, while the other is no longer in the relationship, we refer to the works of poets, popular singers and e.g. to “l’écharpe” of de maurice fanon [13] or to the famous french song “mon amant de saint jean” [13]. our idea is not to stop the quantum description to the two previous states, the fundamental state and the being in love. doesn’t plato, in the σνµπóσιoν (the sumpósion), invite us to see the desire, or eros to see the being in love as a springboard in the quest for the beauty, the good and the true, higher states of contemplation, involving most unconscient couples? we propose that the mystical state would be considered “second excited state” of man and noted | 2 >. the mystical state experience is more rare than the being in love. a very poetic description is given by teresa of avila [14] or john of the cross [15] in the christian culture, among the sufi poets, or closer to us by karlfried graf dürkheim [16] or jiddu krishnamurti [17]. we believe that the mystic state is as natural as the being in love. a more “excited” state: the mystics often speak about being “delighted” transported out of them, the body sometimes trembling, approaching the trance phenomena, due to such an excitement. most of the time 102 françois dubois and christian miquel: towards a quantum model for meditation these experiences are fleeting and fragile as these state are difficult to reconcile with the agitation and violence of everyday life. a priori, this state is limited in time and john of the cross speaks of “la noche oscura” to describe the shock of returning to the “fundamental” state. we think possible to exceed the mystical state, beyond the level | 2 > to state | 3 >? what happens then to the over-excitement we were talking about? will it dilute to an appeased base state no longer crossed by any micro-excitation, or will it immerse in a higher energy state? is it only an excitement modulation experience, not such as a fine structure, but rather as “hyperfine” structure as in quantum physics. in addition to their fine structure, atoms have a hyperfine structure involving the coupling the proton magnetic moment with the electron orbital moment and spin of the electron with other quantum effects and an additional coupling being the contact between the electron and the “frontier” of proton, see cohentannoudji et al. op cit.. these famous couplings are widely studied transitions by atomic physicists. should we now speak of a hyperfine structure of mystical excitement, or a new comparable quantum jump towards a state of excitement or a peaceful energy stage higher level | 3 >? to go forward, we propose no longer to focus on ecstatic mystics experiences, but to consider the already mapped meditation experiences, for over two thousand years, with the multiple mindmaps that a regularly trained spirit can experience and more with the recent development of neuroscience, meditation is itself becomes the subject of numerous scientific researches. 3 the different stages and experiences of meditation what is meditation? according to francisco varela [18], who identified the common structures between buddhist meditations, sufi, and orthodox prayers, we always find the same movement of mobilization and attention refocusing, done in three steps: first suspending the attention focus to the world watched as an external object that constantly comes, thus leading to focus (samatha in buddhism) calmly, back to oneself and especially on breath. second, this suspension allows to change the sight and flow of attention, re-directing it towards internal feelings, to all the body or mental micro-excitations that occur at any moment while observing (vipassana) according to michel bitbol [19], their variations as small as unpermanents without being caught by them. by maintaining this self re-directed attention a new state is created marking a real break from the usual excitement state level. according to varela and bitbol, it is then characterized by the third step as being inherent to any meditation practice: attention becomes just open and available as a single “host state”. this is, according to them and to the experiences of people engaged in meditation, a open standby state, obadvances in systems science and application (2015) vol.15 no.2 103 ject and knowledge free and independent of any excitation trace. this relative vacuum time can be very brief, lasting a few seconds or hundredths of seconds, several minutes if not more, time for new objects or stimuli to inevitably come in the field of consciousness. according to bitbol who incorporates the terms of romano [20], the most confusing and sometimes scaring for beginners meditation is to approach this terra incognita of the internal events without knowing in advance what will happen, to accept this short vacuum time that will leave emerge, to getin a “immersed in the seem”, rather than in a “taking reflective distance” (see [19], op. cit.). what is happening now, during meditation? everything? and nothing! thoughts, emotions, pleasant or painful sensations arise, drowsiness or mental agitation threatens. the main thing is to learn not to get caught by them, gradually discovering that there are several possible stages in meditation, each one with interesting correlations to brain. we can see these thoughts as “virtual bosons”, as a classical concept for quantum field theory calculus. if there is communication or exchange between two partners two entities, a interaction boson is transmitted and received (see e.g. [21]). in the case of non-communication, such as the thought of a subject without explicit communication, the boson is emitted and reabsorbed by the same entity, which is characteristic of the virtual bosons, as described e.g. in literature [22]. 4 a first type of meditation experience, exploration of the base states in the first stage, meditation opens to the discovery of internal micro-sensations which are not usually under attention. still, away from the stress of the outside world, indeed attention drops unconsciously from the level of ordinary consciousness to a succession of moods and multiple excitations, a state of mindfulness to inner events. this is not an external posture of the observer, analytically decomposing internal sensations as many objects to observe, but it is using intuitive ability to self-presence, falling below the subject-object cutoff as written by michel bitbol [23]. we learn to see in the field of consciousness, without getting caught by them, internal micro-physical sensations, emotions and partial impulses, that come and go, fleeting as bosons, without leaving trace. this ability to carefully observe the emergence, deployment and loss of multiple emotions or sensations that pass through our bodies every moment without being drawn by them corresponds to a change in the normal work of the brain. current neuroscientists studies show that during full conscience mindfulness meditation, rational areas of thought (broca’s and wernicke’s) are less active, while internal perceptions areas of our own body are instead stimulated and overactive: the fronto-limbic cortex and fronto-parietal cortex that promotes introceptive percep104 françois dubois and christian miquel: towards a quantum model for meditation tion of internal sensations, the anterior cingulate cortex, which helps strengthen body sensations and pain responses as presented in summary of various neuroscience studies about the effects of meditation on the brain including katya rubia’s, of the university of london [24], the left prefrontal cortex that promotes stress management and emotional well-being, particularly in response to negative emotions such as sadness. the more you meditate and become “expert”, the more changes occur in the brain, which reshapes, with, among others, a decrease in the amygdala which is the place of the automatic stress reactions, but also a sensible thickening of the gray matter for a better control of the cortex, an elongation of telomeres responsible for maintaining cell life (stress instead shortening the telomeres and the matching duration of cells life), and a better capacity for sustained attention focus. when the meditator is able to install a certain stability in his mind, we make the assumption that this very fine meditative observation of his internal energy states corresponds, according to the quantum physics model, to a lowering of the daily agitated state, to what we call a “quiet state”, in which it is possible to become aware of the fine and even hyperfine structures man’s base state and micro-sensations and excitations which allways succeed in him. similar observations occur in quantum physics and common buddhist meditation: there is no streaming, but discrete and ephemeral appearances, followed by disappearances and new energy emergencies, whenever discontinued. from the restless state to the exploration of micro-movements and fine or hyperfine structures of the base state, involves attention gymnastics, which gradually through the meditations, instills a four steps new cycle and cognitive mechanism leading different areas of the brain power to work together (see antoine lutz [25]), gradually calming and fixing attention as shown in a irm study of wendy hasenkamp et al. from atlanta university [26]. mental vagrancy which usually occurs by “default” brain operation in automatic mode at the sensory and motor cortices, the meditation, awareness of this mental wandering is followed by a conscious awarness in the cingulate cortex, somatosensory cortex and anterior insula, which allows to pause for a moment the always prevailing elsewhere mind flows. this awareness allows attention to detach the object vagrancy, mobilizing this time the dorsolateral prefrontal cortex and anterior parietal regions. it can then refocus, sustained and quiet, in the prefrontal cortex, with the setting of a broad and open attentive presence to what is happening. 5 a second type of meditative experience, parallel with the being in love if learning meditation is a deepening of man’s finest base state, certain types of meditation help in many traditions, exploring new, more excited states, which would probably refer to love states previously presented, offering all the features advances in systems science and application (2015) vol.15 no.2 105 of the | 1 > step of love or | 2 > step of mystic experience. to support our hypothesis we take examples drawing on two often opposed major eastern traditions: hinduism and buddhism. in hinduism, the bhakti yoga meditation, or path of devotion, is famous since it was themed for the first time in the bhagavad gita [27], between the 5th and 1st century bc. primarily oriented towards vishnu or krishna this bhakti allows to develop and grow, in everyday life and in meditation, a state of love transport with one’s heart divinity. the bhakti implies in fact a complete abandonment to the divinity, installing a person to person relationship to the god, the devotee receiving or discovering an energy and intensity of love far from with the usual feelings of the secular world. as one might expect, in the most advanced stages, such a loving devotion naturally extends towards a mystical state, since it is then to unite and merge into the infinite love of the god, devotees of vishnu or krishna sometimes being remarked in the streets by real love and energy transport while they chant and meditate, seeking to perpetuate this state of ecstasy in love before it fatally falls as this dark night described by st john of the cross. in pathological cases, and specially in the bengali bhakti as noted by esnoul [28], these second fervor states may exceed the meditation framework and overflow crowds of jubilant devotees phenomenons with nervous or functional disorders caused by extreme emotional stimuli. in buddhism, the compassion and kindness loving meditations highlighted by matthieu ricard [29] for tibetan buddhism, are to develop a feeling of infinite love, without limit, extending the feeling of kindness, empathy and compassionate love, even to people we do not like or comply. during these meditations cultivating compassion, meditator goes through strong and sometimes unpleasant thrills leading through into mental intensity and higher energy states, with strong activations in certain areas of brain, measured by neurociences. during the kindness love meditation, richard davidson and antoine lutz [30] (from cern) have registered a higher activation in brain regions that manage empathy for the suffering of others: the anterior insula and the anterior cingulate cortex, but also the three areas that allow us to put ourselves in the place of others (medial prefrontal cortex, superior temporal sulcus, temporoparietal junction). experienced meditators feel even more compassion and suffering of others, with additional activation of the somatosensory cortex that makes them part of their own body. according to matthieu ricard [29], other studies about this loving kindness meditation testify logically from the decreased activity of the amygdala, which manages aggression and fear, while areas related to empathy as the insula were more active, increase in size, creating more neural connections and thus reshaping the brain. the feeling of kindness love goes even further than a simple 106 françois dubois and christian miquel: towards a quantum model for meditation increasing empathy, according to studies reported by matthieu ricard, particularly by promoting and developing oxytocin, this maternal love peptide produced by hypothalamus. 6 a third type of meditative experience, approaching the mystical state if these meditations based on love seem to scale up from the base state to the love state level | 1 >, even to the mystical state | 2 >, are there non devotional meditative practices which would specifically lift to the so called mystic level | 2 >? it seems that this is the case of advanced dhyana samadhi states of concentration and contemplation, which have been cataloged in all meditation traditions of the two great hindu and buddhist currents. for over two thousand years, these states of consciousness were indeed widely listed and presented, sometimes in hinduism as the accession to new energy strata and higher conscience, sometimes in buddhism as stages of consciousness, obtaining advanced powers to keep beware of without stopping. in hinduism: the yoga sutras, between 200 and +500, codified and compiled for probably several centuries experiences and stages developed by yogis during their yoga exercises and meditation. in buddhism: the abhidharma, between first and 5th century, is still an impressive collection of texts and reviews about pyschological and philosophical experiences experienced by the buddha and buddhist meditators. michel bitbol, in his research on consciousness, builds among others on this corpus. let’s present now the first advanced states of consciousness found in these various traditions, stopping for the convenience on the terms and stages described by buddhism (see e.g.[31]), and more specifically in the dhyanas or jhanas, more or less ultimate absorbing states which can reach after long mediations and samadhi concentrations. the first dhyana state that seems to be the very definition of the mystical state given at the beginning of this article is first the experimentation of a pure inner energy or virya excitement state, excitement feeling arising in the depths of oneself. this burst of excitement and energy can be experienced with weak signs as heat, tingling, creepy, or cause, as with the mystics, most violent movements that literally carry the meditator, almost out of himself, giving the impression of being transported untill almost levitating or feel of emerging from his own body in a state of priti passadhi rapture and pure joy which then shows access to the second state. this state of joy can almost naturally lead to a third pure happiness state free of body sensation or excitement, again with a mental experience level break, and the access to a fourth prasrabdhbi equanimity state, of serenity and immutable peace. which can then lead to be completely absorbed into new dhyana or jhana more subtle states according to the texts: the fifth stage, dilution or absorption in an unlimited space; sixth, absorption in a state of unlimited consciousness; seven advances in systems science and application (2015) vol.15 no.2 107 and eight, the absorption in other areas be empty or all of any form, perfectly silent, beyond perceptions and non-perceptions. hindu texts and the different stages of yoga present variations and terms inversions, distinguishing such as mircea eliade [32] after concentration dhyana, and the more and more refined samadhi: first, an enstasis or standard limits liberation in samadhi with support, still maintaining the support of the meditation object. thus leading to miraculous yogic powers [33], and finally when exceeded to the ultimate samadhi without any support, which is a perfect and sudden enlightenment, a most deep inside enstasis, in a higher state of consciousness, “a total and saturated by direct intuition be”. we refer to mircea eliade [34] and to zimmer’s description [35] about these states of samadhi, according to the great vedanta philosopher cankara at 8th century. it is not surprising that these absorption states reached through meditation find the states evidenced by the mystics. in her spiritual journey, st. teresa and reached in the fifth home of her soul a pure attention succeeding an infinite bliss, which is according to bitbol [19] , the third dhyana buddhist, removing the awareness of body sensations to become a pure attention, just after a feeling of joy and bliss. hatching such states requires, ability to focus attention, then let go with abandon, are essential as observed too by jeanne guyon, 13th century mystical: “surrender is the key to the the inner”[36]. regardless the difference stages and terms in yoga or buddhism, favoring samadhi or dhyana. the main thing is to realize that these two traditions mapped these mystical level “two” states quantum of our assumptions for a long while, each time with the same degree of change, a break level, an extension and intensification of energy awareness in both aspects physiological and cognitive. physiological studies in neuroscience allow, again, to aproach indirectly and by comparaison the energy transformations that probably occur in the brain, when such statements are reached in meditation for a short or long period. measures that were conducted on trained meditators [37] show indeed both increased alpha and theta waves of deep relaxation and rem (rapid eye movement) sleep, but also the increase of gamma waves thus reflecting the activation and mobilization of neural resources allowing mental effort and the emergence of a global intuitive flash. as they allow, according chaskalson [37], to daily accumulate sound, visual and cognitive data, to suddenly understand by example in a flash: it is a train, the same waves that cause these flashes of intuitive understanding known by the great scientists, artists and meditators. this is the “aha!”, the “eureka” that last a few milliseconds, which suddenly allows to see the reality otherwise. according chaskalson, even without speaking of the states of samadhi or dhyana we have presented, the mental meditation training allows monks to have the same type of illumination and intuitive flash, with a broad and comprehensive vision that can 108 françois dubois and christian miquel: towards a quantum model for meditation last up to five minutes. how to explain or account for neuroscience? studies by lutz in his research laboratory at the university of wisconsin [25], on expert meditators who have accumulated more than 10,000 hours of meditation, show that stress areas located in the amygdala are much less mobilized, even with an objective stress (sending aggressive sounds), then providing them an attentional and emotional stability. in other studies on more experienced meditators, totaling more than 50,000 hours of meditation, antoine lutz also shows that synchronization of gamma waves is actually much sharper than in average individuals[38], which indicates levels of “increased” consciousness with more integration of the different brain areas begining to work in synergy. 7 paradoxes of mystical states: third state, or return to the base states? if the existence of mystical states corresponding to a “jump” into a higher energy level seems to be demonstrated that these become mystical states: they can exceed that in a later stage we would call a quantum state level | 3 >, or they eventually return to the base state? hinduism and yoga seem to militate in favor of the possibility of reaching a final status level | 3 >. in hinduism, mystical peak condition naturally to contemplation, to the union or merger in a higher power or divine, which goes beyond feelings states absorption described above. this was already revealed in the most ancient hindu texts, bitbol reminder that the chandogya upanishad already stated that six centuries bc “dhyana meditation is more than consciousness” (see [19]). for hindus, the ultimate experience is the effect of the fusion and resorption in the primordial ocean, conceived on the model of the wave or the water drop (atman or ultimate principle in itself), which returns blend into the primordial ocean the absolute (brahman). cankara, large non-dualistic philosopher who belongs to the hindu tradition vedanta to 8th century ad, will be part of the same line when his distinguished according zimmer commentator different samadhi with and without support, to finally end with a oceanic fusion in the absolute: “in the first type of samadhi savikalpa with mindfulness the subject-object duality, the oscillating vitality of consciousness assumes the form of brahman, but remains conscious of itself, has a beneficial ecstasy, nirvikalpa samadhi, absorption without consciousness is immersed in the self, without distinction about objects, such as waves vanish in the water” (see zimmer [35]). but this merger in the infinite seems quantum level | 2 > or | 3 >, is it not the same resorption time in a base state, naturally peaceful and informal, which refers to the paradox of a new level | 0 > ? this is what seems to indicate for which buddhism different stages of dhyana absorption are not an end in itself: they simply allow to abandon the game categories and normal mental projectionadvances in systems science and application (2015) vol.15 no.2 109 s, to evade excitations which continue to occur without one is now assigned or forced to react by simply return to a peaceful state originally, where we show the world just leaves appear here and now, in a perpetual present. nirvana simply means extinction, blow and quench the thirst and the projection of his desires, to let appear the phenomena as they are, in their “like this being” (just as well, tathagata, the epithet enlightened buddha, literally meaning: and went, and came, or more exactly according bitbol: standing in the well). this is what allows bitbol to write: “the state of absorption once pushed into its ends, leading to a territory which is not a stranger, and that seems unusual because he has been stripped of its grid cadastral: what is, as it is” (see [19]). in this sense, it is just an awakening to what is presented to each moment, as if pondering managed to reverse in the original and fundamental quantum field which emerge discrete events or appearances, which immediately effaced, as strange bosons enigmatic. to speak of this final state, the tibetan tradition of talk “open presence” and the zen tradition back to an originally awake and calm the conscience, conscience hishiryo of non-thought, before any thought of where the boundary between the self and the world continues to find itself in what is sometimes mistakenly called the “nature of buddha” (nature of what is just, well). our thinking leads to a strange paradox: how to reconcile the hindu metaphor the drop of water that melts into an infinite ocean of energy, with the buddhist image of the empty mirror dust, which has nothing to reflect? maybe he should accept the aporia as the place and the reverse of the same coin, and that the ultimate mystical state is ultimately nothing more than a return to the base state “appeased” rights, ie the base state within the structure hyperfine state quiet day. that is the common point between the “semi-silent buddhist and theological verb: they look like each other to prepare the mind to meet the unprecedented and the unnamed themselves, evoking in a case like a proto-type, and the painting in the other case under the guise of an over-kind” (see [19]). compared to our quantum assumptions, this would imply that the quantum leap inducing a change of consciousness is thinner in what is called metaphorically an ocean of energy, would be correlative or followed by a return of consciousness ordinary that we named the base state appeased by letting go and extinction apprehensions and tensions of the ego. 8 transition from one state to another and quantum physics at the end of this presentation, meditative states seem to respond well to different intensity scales, and to a lesser extent the affective and emotional states of the human spirit can, within the scope of increased excitement, split from a low state to an average state, then exacerbated such as frustration, anger and rage. but if, for the emotional states of plutchik, we understand that the external stimulus 110 françois dubois and christian miquel: towards a quantum model for meditation makes suddenly switch from one state to another, how will the transition occur with a much stronger meditative state | 0 > to | 1 >, | 2 > or even | 3 > while the meditator seeks to abstract from any external stimulus? we will try to approach this issue from three aspects: the relationships dynamics, bosons, and finally heat exchanges, which each help to determine whether the changes produced by meditation fully meet the requirements of quantum physics. 9 what dynamics of relationships, what breaking levels? in this second part, it is therefore to ask how the transition from one stage to another can be explained. we know that during meditation, thoughts come and go, such as interaction bosons, light, photon to the electromagnetic field. are thoughts bosons that occasionally emerge from the quantum field of a selfrefocused and soothed consciousness? this hypothesis probably deserves further developments! one can also imagine that they are the result of a measurement made by our conscience, the trace of a reduction of the wave packet, the trace of the interaction between the macroscopic observer which is our conscience and microscopic phenomena of our body, specially our brain. let’s recall that we have identified three different scales in energy levels structuring the human psyche. we start from the everyday agitated state, the ordinary mental state of everyday social exchanges. with a high energy input, one goes to the love state then beyond to the mystical state that allows to shift to a contemplation state. the “daily hectic” state itself is made of myriad of sub-states, fine structure of the energy system. of these, joy, anger, etc. as many levels of excitations in the daily regular. in this new spectral system fine structure, the “quiet” state is fundamental for us. the“quiet” state is the reference state of the meditator who gradually descends towards “hyperfine” sub-levels to lead to a new appeased base state, we may call “vacuum state”, in analogy with the vacuum of a photon free electromagnetic field. so we have three interlocked structures for the psychic structure, with levels of energy exchanges inside each similar subsystem, but at very different levels between subsystems. as always in quantum mechanics, the relationship between energy levels is carried by discrete jumps, transfers with a definite energy, such as the photo-electric effect introduced with the relation (1) at the beginning of this contribution. in this case, the interaction mediator, the “boson” as called by the physicists is the photon, unbreakable grain light provided in 1905 by albert einstein [39]. to extend the analogy between the atomic system and human psyche, to develop our attempt of spectroscopy of the human psyche, we must now search this boson of interaction between energy levels of the psyche. it is known that looking for a boson is still a “big deal” in physics. the understanding of the weak interaction with the weinberg-salam model [40] allowed to hypothesize the “intermediate advances in systems science and application (2015) vol.15 no.2 111 boson”, highlighted at cern in 1984 [41]. more recently, the unification of the electro-weak interaction with the strong interaction that binds protons and neutrons in the atomic nucleus [42] led to the discovery of a boson planned in 1964 by robert brout, françois englert, peter higgs [43] and probably many others! we here suggest the book of gilles cohen tannoudji and michel spiro [44] to the reader. 10 a boson for meditation in quantum physics, the change in energy level for the system is possible if a quantum of energy is exchanged with the outside world. a photon, elementary particle of light, allows the transition between energy e0 and energy e1 if (and only if!) its frequency ν is exactly compatibe with the two previous energies through the equation (1). reciprocally, when the atom descends from e1 level to e0 level, it emits a ν frequency photon. the photon is the interaction boson, which mediates the exchange between the atom and the outside world. the question now, as part of a quantum model for humans, is to understand what these energy grains are, these interaction bosons that allow for example to move from the ordinary state to the being in love. this transition generally occurs abruptly; this is the famous “lightning strike”, whose name recalls the exchange of light within the atomic system. butthe matter is also to understand what are these micro-energy exchanges that allow the mediator to explore the hyperfine structure of the human spectrum, and to gradually descend towards the base level, called “vacuum” in the case of the system obtained by quantifying the entire electromagnetic field. in the case of meditation, reversing the outside world look towards the inner being allows the progressive exploration. but it’s probably a simple “quiet state” overall support framework avoiding the strongest disturbances of everyday life. the precise shift from an energy level to the following one should happen (if the quantum framework of this model is correct) throught a very low energy expense of the subject. the outside world disturbance (better: a disturbance from the outside world) results in a still possible energy contribution moving the meditator from an energy level to a more agitated level. what is the nature of this energy input? let’s first recall that it must exacly be on a tuned frequency, as the photon, through a compatibility relationship similar to the (1) relation. if the outside world sends detuned energy grains, even on much larger frequencies, the system state does not change. this is a great discovery of albert einstein in 1905 while he made the assumption of photon. considering the very complex structure of human being, we believe that we should not look for this energy exchange in pure electromagnetism as for the atoms, but probably rather in a still to highlight assembly of biomolecular structures ultimately organizing trade within the human psyche. 112 françois dubois and christian miquel: towards a quantum model for meditation we can hypothesize in line with théodule ribot [45] that a “lightning strike” results of the resonance between an unconscious rememberance buried in the memory and a glimpse of present, with a previous incubation and updating work on the buried memory. so this exchange of information, whom the subject is generally unaware, causes the “lightning strike”, the shift from of the “daily status” to the being in love. is the thought making this exchange of information? it seems clear that thinking requires an energy expenditure (see e.g. the work of giuseppe vitiello [46]) and results in the exchange of information within the bio-physical body system. but it is likely that the resonant equation harmony (1) is not reached and that thought does not allow such a transition. it seems instead that, if one refers to the assumption made for the “lightning strike” during the appearance of the being in love, that the transition between two states escapes the consciousness of the subject. no one chooses to fall in love and neither not to be in love. we should probably seek the exchange of energy between the states of consciousness, hyperfine energy levels similar to those the atom’s in a non conscious process. of course, this process is stimulated by the thought that controls the aware mental state; but the transition itself is beyond thought. we can assume it is produced by an exchange boson. boson which needs to be highlighted. boson allowing the transition between two states of consciousness. boson unconsciously emitted by the body bio-physical system. universal boson which would not depend on the individual: everything could then be explained by reference to a biological level to a neurotransmitter being an interaction boson, not that far from what happens when you’re in love, which cause and match the change of states we are interested in. the lighting of what is happening in the state changes during meditation helps to bring other complementary lightings, surprisingly finding what happens in the love state. changing of state to leave the daily agitated state, requires at least three conditions: first, as seen with varela, a suspension of the link engaged in to the world and a redirection of sight to the pre-reflective life then the setting of that we have called a quiet state, especially with the techniques of concentration on the breath that allow you to install a minimum of stability within discontinuous appearance / disappearance of various excitations and events; finally, a necessary control of attention, which tirelessly leads intentional thought ready to escape to outside, and inside, to turn into a simple and intuitive attentive presence, open and available at what is happening at every moment. this important and even decisive role of attention as a third factor is confirmed by studies in neuroscience that highlight the importance of the prefrontal cortex mobilization, being, among others, the seat of the thought control by the attention. but what is interesting is that these three factors are necessary conditions but not sufficient for a significant change of state in meditation. neither introspective sight advances in systems science and application (2015) vol.15 no.2 113 redirection nor installing a state of tranquility or opening a wide attention as a wellcoming and 360 degrees acceptance as stated bitbol, is sufficient to explain or cause a change of state, the daily agitated state to a mystical state, or emptyness or wellcoming: that change will never be determined and causally provoqued, most of the time it appears suddenly and discontinuously, even brutally, as in the state of love. in soto zen, it is said that you can just promote the enlightenment emergence, but cause it in no case: it will happen spontaneously, naturally, automatically and unconsciously as liked to say deshimaru [47], without any intervention of man. under a neurotransmitter biological influence, a sudden compatibility ofrelationship or, to resume zen terminology, when mysteriously body, mind and breath finally get synchronized and reunited in a “fair” posture, in synch, then, along with the “letting go”, a sudden illumination may occur. rinzai zen is even more radical, by not advocating a progressive illumination that can be prepared as in the soto zen, but a sudden illumination. pai-chang huai-hai, t’chan eighth century chinese master (the chinese t’chan is the original form that will create the japanese zen), was thus responding to a disciple who asked him how to reach the issue: “it can only be reached by the sudden illumination” [48]. hence the further development of koans, these seemingly absurd riddles that teachers offer their disciple. when the student finally drops his intellectual efforts to understand, a sudden illumination, satory, may occur, which leads him to another level of reality or consciousness. “satori is a spiritual experience; it describes the sudden click of buddhist enlightenment. if it is difficult to describe to the uninitiated, the intellectual approach is easily accessible. just imagine archimedes in his bathtub discovering the same name famous thrust: “eureka!”. he experiences a cognitive process in which violence is like an illumination. suddenly the incomprehensible lights in front of the mind flash. this is a sudden process whose instantaneous contrasts with the heaviness of a verbal explanation”, as suggested in the abc-book “ombres nippones” on the website kichigai.com. in the hindu and vedanta tradition, access to samadhi and ultimate levels of consciousness can only, in the same way, occur suddenly, with a sharp break of ontological level of consciousness, as noted by mircea eliade [32] (page 70): “the level break india aims to achieve is in the samaddhi. this enstasis is in fact a rapture, since it is experienced without being provoqued. the enstasis is equivalent to a reintegration of different modes towards the real mode: primary non-duality, before the bipartition of the real into subject-object, undifferentiated fullness (with feeling of) unit and bliss. there is a return to the origin, but enriched by dimensions of freedom and trans-awareness (or consciousness).” 114 françois dubois and christian miquel: towards a quantum model for meditation 11 state changes and thermal changes during meditation, the subject navigates among the hyperfine structures of the quiet state. we want to establish statistics relating to the occupation of each of the energy levels. but we can not talk about temperature because a priori we study a single subject. however, one can imagine averaging up the time the subject spends in each state. this approach is a little daring because there is a real state change dynamic over time. some tried to describe it with thermostatic statistical tools, which implies invariance in time throughout the experiment. however, let’s try continue this analogy. the nj population of the εj level of statistical physics now represents the time θj spent in the ej state of energy. then we can through the partition function “z” (see e.g. the book of bernard diu et al. [49]) defining a “temperature” of the meditator, modulo this transformation by considering the population of nj states during a time θj : z = ∑ j exp ( − εj kt ) , nj = 1 z exp ( − εj kt ) . (2) we then observe on the equation (2) that the psychic temperature is negative (!) if the subject spends more time in the excited state than in the base state. in meditations, it would also implies that when going from daily agitated state to quiet state, there should be a decrease in temperature and a regulation of vital functions. or this is exactly what we see as neuroscience studies on the subject show that meditation causes almost mechanically a decrease in the release of stress cortisol, a regulation of blood pressure and heart rythm, an elevation of immune defenses [38], and for some types of meditation, a decrease in skin temperature as presented by manocha et al. [50]. other tibetan meditation or yoga can also focus on increasing the temperature of the skin to fight against the cold. 12 conclusion in conclusion, if we take the fractaquantum hypothesis in its very maximum, the analogy observed between the atomic system and the psychic system can continue: just as the photon allows moving from one atomic system state to another, there should be a boson (still to identify!) which would be the direct cause of the transition between two states of mind. this boson could be chemical or electromagnetic or purely physical or biophysical. without going to such a prediction, the fractaquantum hypothesis can minimally and by analogy notice the existence of a same mechanism explaining the process and state changes to quantum level and psychological level particularly in meditation. meditation, when it reaches a stable and peaceful state of attention would thus reach or experience a contentfree underlying quantum field, which advances in systems science and application (2015) vol.15 no.2 115 manifests itself locally and occasionally by small bursts or quanta of energy, fragments of thoughts, emotions or fleeting sensations appearing then disappearing immediately, which may ultimately lead to a state of joy and soothed presence. as if those changes and modulations tiny lived during meditation, were musical notes reflecting a fundamental quantum field, quantum song. acknowledgments the english translation is a collaboration with claire couratier. the authors are happy to thank her! references [1] f. dubois. 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(2010), skin temperature changes of a mental silence orientated form of meditation compared to rest”, journal of the international society of life information sciences, vol. 28, issue 1, p. 23-31. corresponding author françois dubois can be contacted at: francois.dubois@cnam.fr advances in systems science and applications (2012) vol.12 no.1 38-45 the vibration response analysis on a 6-ups pkm based on fem jinquan li and zhenxing luan school of automation, beijing university of posts and telecommunications, beijing 100876, china abstract in this paper, a finite element full-scale entity model for 6-ups pkm and its accessories were established based on substructure method and were further confirmed through a modal experiment and a finite element modal analysis. the displacements on response curves of the nodes along the virtual x, y and z axes of pkm were obtained through a random vibration response analysis on pkm performed with a finite element method. based on these resultant curves, the law of vibration response on the pkm was investigated, and the resultant parameters clearly demonstrated the vibration response characteristic of pkm and its response scope magnitude, which can provide an important theoretical base for the optimal structure design of pkm. keywords pkm, random vibration response, stewart platform, fem, dynamics 1 introduction the bkx–i parallel kinematic machine (pkm)[1] is a typical using of stewart platform in machine. it is a 6 -ups pkm, shown as fig.1. fig.1 bkx-i pkm the pkm’s rigid body dynamics model can be built by nearly all classical mechanics methods, such as newton-euler method, lagrange method, virtual displacement principal, kane equation[2-3], etc. so, these methods based on dynamic can be chosen proper and simply according to different type machines advances in systems science and applications (2012), vol.12, no.1 39 which are extensively used in control system. but, the telescopic shafts in pkm shows the flexible characters that the rigid parallel kinematic structures don’t have during the high speed working. the shaft’s elastic deformation, dynamic stress and elastic vibration can cause the whole machine’s impact, noise and fatigue[4]. the pkm’s dynamic system is actual a multi-elastic system, having the characters of mechanism-structure couple, time-varying, nonlinearity, etc. so, during the period of the pkm’s design and optimization, the finite element method is usually used in the modal analysis and the response analysis to reflect the machines’ dynamic characters and the convenience of solution accurately[5]. the machine structure’s random vibration response analysis is one of spectral analysis in the dynamic analysis. it is mainly used in the probability statistics of structure to random vibration response analysis. it is also a relationship curve of power spectral destiny and frequency which reflects the intensity of load and the frequency in time changing. 2 the machine’s finite element solid modeling method the bkx-i pkm’s geometric structure is a complex spatial structure. directly establish its finite element model by using the software of ansys would be complex. in this paper, the geometry solid model including 14 substructures was established by 3d software pro/e. the substructures included the machine frame, motional platform, 6 telescopic shafts and 6 crosses of universal joint which the universal joint’s other accessories belonged to the machine frame and telescopic shaft. the main idea for making substructures was that it could make every moving structure as the substructure while the machine was working. the motional platform is a substructure because it is the pkm’s actuator and also an individual moving part. the machine’s frame is an entirety and holds stationary relative the earth when the machine is working. the universal joint’s upper part was fixed the frame firmly by all kinds of connection, so they could be ranged as a substructure. inside the telescopic shaft assembly, although it’s lower slide bar and upper swing bar had the tendency of motion relatively, both of them and the universal joint’s lower part which connected with them were supposed as a substructure in order to simplify the structure because their connections and power transmission were obtained by the ball screw. the universal joint’s cross as the main member of connecting the machine frame and telescopic shaft is always an individual active structure in any time, so it also was a substructure. after establishing the geometry solid model, it could be transferred to finite element model by the interface between the pro/e and ansys. during the transferring, the relationship between the original pro/e substructures would disappear. the couple configuration of every joint face needs to be made again in ansys to simulate spherical hinge, universal joint and other structures. the 40 jinquan li:the vibration response analysis on a 6-ups pkm based on fem couple configuration was that it equalized the entire contact node’s displacement dof which between the two contact face, and each other’s rotational dof need not to be constrained to make the contact node have the tendency of rotation relatively. after coupling, it could use the solid45 unit to meshing every solid model freely. then the pkm’s finite element solid model including 170 thousand solid units was established, shown as fig.2. fig.2 bkx-i pkm fem model 3 the confirmation and the modal analysis on the finite element model of pkm the modal experiment and finite element theoretic modal analysis on the machine were carried out respectively. then, the comparison of results was made to confirm the validity of finite element model. during the experiment modal analysis the method of single point exciting and multi-point receiving[6] was used. in the experiment, the machine was fixed on the foundation firmly first, which its position and posture were adjusted to as the same as finite element model’s. then the excitation with the hammer were made, while the 4 acceleration sensors picked up and recorded the system’s single exciting signal and multi-response signals at the same time. the experiment statistic’s acquisition and disposal were made by the software of dasp. after a series of signals disposing like analog digital conversion and faster fourier transformation, the system’s transfer function, amplitude-frequency were obtained which can reflect the functional relationship of machine’s dynamic characteristics and then the machine’s natural frequency could be got by normalizing the modal parameter. the boundary condition of finite element theoretic modal analysis was that the machine basement’s dof of displacement was zero. then the block-lanczos method was used during the finite element modal analysis. after that, all of the machine’s 10 order natural frequency, vibration shape and its animation were advances in systems science and applications (2012), vol.12, no.1 41 obtained in ansys. the comparable result between the machine’s natural frequency got from the finite element modal analysis and the machine’s natural frequency got from the experiment was shown in the table 1. from the table we could get that the proper finite element model could be build with the error less than 10% after many times of simplifying the structures and choosing proper finite element unit. table 1 the results comparison between fem’s analysis and experimental modal analysis modal fem actually order calculation value error 1 25.836 23.357 9.6% 2 49.436 45.472 8.0% 3 67.988 63.434 6.7% 4 76.475 72.365 5.4% 4 the theoretic basis of response analysis on pkm when n order natural frequencies of pkm are ω1, ω2, . . . ωn and the corresponding vibration shapes are x1,x2, . . .xn , which natural frequencies are arranged according to sort ascending and the ω1 and x1 are the pkm’s basic frequency and basic vibration shape, the relationship of them can be expressed as follows: [k]xi = ωi 2[m ]xi (i = 1, 2, . . . n) (1) because the integer stiffness matrix [k] and the integer mass matrix [m] are real symmetric positive definite matrix which the vibration shape vectorsx1,x2, . . .xn are a group of basis of n-dimensional space vector, the node displacement vector {ϕ (t)} can be expressed as follows: {ϕ (t)} = q1x1 + q2x2 + . . .+ qnxn = [x]{q} (2) here, q1, q2, . . . qn is the coordinate of {ϕ (t)} in the coordinate system which the vibration shape vectors are basis and they are the time function. so, {q} = {q1 q2 . . . qn}t · [x] is the vibration shape matrix of system which [x] = [x1 x2 . . . xn]. the dynamic equation of machine in the entire coordinate system can be obtained as follows: [m ]{ϕ̈}+ [c]{ϕ̇}+ [k]{ϕ} = {f} (3) 42 jinquan li:the vibration response analysis on a 6-ups pkm based on fem the eq.(2) is substituted into eq.(3) and the equation can be obtained as follows: [m][x]{q̈}+ [c][x]{q̇}+ [k][x]{q} = {f (t)} (4) the equation is left multiplicity by [x]t and the equation can be obtained as follows after collate: mq̈+cq̇+kq = f(t) (5) here, m–main mass matrix of system. m = [x]t [m][x] = diag(m1,m2, . . . ,mn) k–main stiffness matrix of system. k = [x]t [k][x] = diag(k1, k2, . . . , kn) here, ωi 2 = ki/mi c–main damp matrix of system c = [x]t [c][x] = diag(c1, c2, . . . , cn) and ci = 2ξiωimi. here, the ξi is the damp ratio corresponding to the i order vibration shape. f (t) –main active load vector. f (t) = [x]t {f (t)}. expand the eq.(5) and every equation is divided by m̄i , the equation can be obtained as follows after collate: q̈i + 2ξiωiq̇i + ωi 2qi = f i (t) /mi (i = 1, 2, . . . , n) (6) this is a typical single freedom vibration equation belong to linear constant coefficient quadratic differential equation. the chief response of structure system in modal space can be got after solving the equation. qi = eξiωit ( qi0 cosωdit+ q̇i0 + ξiωiqi0 ωdi sinωdit ) + 1 ωdi ∫ t 0 fi (τ) mi e−ξiωi(t−τ) sinωdi (t− τ) dτ (7) here,ωdi = ωi √ 1− ξi 2 {q}0 = [x]t [m ]{ϕ0} {q̇}0 = [x]t [m ]{ϕ̇0} last, when qi are replaced by eq.(7) in eq.(2), the relationship equation between the displacement response in the original physics coordinate system and the excitation could be obtained. advances in systems science and applications (2012), vol.12, no.1 43 5 the random vibration response analysis based on fem the random vibration analysis based on the fem was solved on the base of modal analysis after extending the modal. in ansys, the nodes’ dof of machine base plane were fixed. after extending the modal, the random displacement vibration loads of 1 × 10−5 m amplitude were imposed on servo motors which located on the 6 telescopic shafts and the main axis motors which located on the center of motional platform along the machine’s virtual x and y axes. according to the machine structure damper rate which got from the modal experiment, it could definite the damper rate as 2%. finally, random vibration analysis was implemented to the pkm’s general structure to get the curve of the machine structure node’s displacement response when the random vibration frequency was between zero and 200hz. fig.3 the random vibration response curve of motional platform centre node along the x axis the displacement response curve of the node centre on the motional platform along the virtual x axis was shown as fig3, on the y axis curve shown as fig 4, on the z axis curve shown as fig5. for saving the words, other nodes’ curves were not shown. in fig3, fig4 and fig5, the horizontal axis was frequency which its unit was hz, the longitudinal axis was displacement which its unit was meter. 6 the analysis and conclusions of calculation results some conclusions were got from the pkm’s random vibration response curves: first, all of the virtual x, y, z axes had a great response around the 26hz of the machine’s first order natural frequency. it was the resonance response which didn’t happen in the second and third order natural frequency. at the same time, 44 jinquan li:the vibration response analysis on a 6-ups pkm based on fem fig.4 the random vibration response curve of motional platform centre node along the y axis fig.5 the random vibration response curve of motional platform centre node along the z axis a smaller resonance happened in the fourth order natural frequency. from the modal shape of machine and its animation, the result could be made that the pkm’s first order modal shape was the machine’s whole modal shape, the second and third order modal shape were machine’s local modal shape. the vibration response excitation which loaded on the centre of motional platform didn’t excite the machine’s second and third order response. the result of machine’s vibration analysis and the result of machine’s modal analysis were closely fit each other. second, the motional platform centre’s maximum resonance amplitude was 4.5× 10−6 m which was half of the loaded maximum resonance amplitude. comparing to the vibration excitation amplitude, the vibration response value was advances in systems science and applications (2012), vol.12, no.1 45 less than 2 orders of magnitude when the pkm didn’t resonate, so its influence could be neglected during the actually manufacturing progress. third, the motional platform centre resonated in some low-order natural frequency along the virtual z axis, but compared with the biggest random vibration amplitude, its values were less than 4 orders of magnitude in vibration magnitude. so it could be nearly neglected. fourth, the result, which the random response along the x and y axis’ was greater than that along the z axis’, was just like the rigidity which along the machine x and y axis’ was far lower than that along the z axis’. the result of machine’s vibration analysis and the result of machine’s rigidity analysis[6] were closely fit each other. acknowledgements the project is supported by national postdoctoral fund.(china 2005038083). references [1] jinquan lin, hongsheng ding, tie fu, siqin pang. (2003), “the parallel machine tool’s history, status quo and prospect”, machine tool & hydraulics, no.3, pp.3-8. [2] zhao-cai du, yue-qing yu, li-ying su. (2006), “effects of inertia parameters of moving platform on dynamic characteristic of flexible parallel mechanism and optimal design”, optic and precision engineer, vol.14, pp.1009-1016. [3] shan-zeng liu, yue-qing yu, zhao-cai du, jian-xin yang. (2007), “recent development and current status of parallel manipulators”, modular machine tool & automatic manufacturing technique, no.7, pp.4-10. [4] ju f, choo y s, cui f s. (2006), “dynamic response of tower crane induced by the pendulum motion of the payload”, int.j.solids structures, vol.43, no.2, pp.376-389. [5] jin-quan li, tie fu, zhen-xing luan. (2008), “the dynamic analysis on a 6ups pkm based on fem. optic and precision engineer”, journal of beijing university of posts and telecommunications, vol.31, no.5, pp.40-43. [6] jin-quan li, hong-sheng ding, tie fu, si-qin pang, xie dian-huang. (2004), “the experimental modal analysis of the bkx-i model pkm”, manufacturing technology and machine tool, no.10, pp.45-48. advances in systems science and applications (2014) vol.14 no.3 244-253 symbolic dynamics applied to velocity time-series in wind farms k. karamanos1, i.s. mistakidis2, s.i. mistakidis3 and i.e. sarris4, 1 complex systems group, institute of nuclear and particle physics, ncsr demokritos, gr-15310, aghia paraskevi, attiki, greece. 2 department of physical sciences and applications, hellenic army academy, vari, attiki, greece. 3 zentrum für optische quantentechnologien, universität hamburg, luruper chaussee 149, 22761 hamburg, germany. 4 department of energy technology, technological & educational institute of athens, ag. spyridona 17, 12210 athens, greece. abstract the development of standards for wind farms, presupposes the correct description of wind potential and this can be done with the field measurements of wind flow by cup anemometers. the utilization of new concepts, coming from the world of cybernetics of nonlinear science and complex systems could open the road to uncover information hidden in both the mean polar velocity and the mean angle time-series. in particular, with the use of block entropies, it is shown that we can achieve a better and deeper understanding of the phenomenon of filtered turbulence, produced by time-series of the average wind velocity logged every ten minutes. the present analysis allows in principle a characterization of the experimental time-series in terms of the complexity for selected stationary windows of the signal, as well as the underlying mechanisms of the filtered turbulence. keywords wind farm, anemometer, wind velocity measurements, filtered turbulence, symbolic dynamics, block-entropies. 1 introduction. in the context of energetic problems of modern societies, a proposed solution has been the use of alternative energy sources. one particular realization of this idea is for instance the installation and usage of wind farms for electricity. a cluster of wind turbines in the same site used to produce energy is called wind farm. investment in a wind farm consults temporal measurements of the wind potential on the prospective site by using suitably located towers. the wind velocity, the pressure and temperature, are collected in the measurement tower by cup anemometers and meteorological instruments [1]. the height of the tower is up to the hub height of the planned wind turbines, and the data are recorded frequently, i.e. contains the averaged quantities every ten minutes, and for at several months. these data, in principle, may allow the developer to decide if the investment in a wind farm is economically feasible for the selected site. understanding and quantify aspects of their behaviour and nature is one of advances in systems science and applications (2014) vol.14 no.3 245 the most important and challenging problem of modern engineering [2]. the long recordings of the averaged wind velocity time-series from the anemometer, are a kind of filtered turbulent temporal data set with intermediate properties that may be analysed. the most important problem in this framework is the prediction of the production of electric energy in wind farms [2]. the problem of prediction is exactly, what connects these studies with cybernetics and nonlinear science. 0 1 2 3 4 5 6 7 8 9 10 0 20 months m /s 0 1 2 3 4 5 6 7 8 9 10 0 5 months m /s 0 1 2 3 4 5 6 7 8 9 10 0 300 months d eg re es 0 1 2 3 4 5 6 7 8 9 10 0 150 months d eg re es (c) (a) (d) (b) fig.1 time-series of the velocity field between 23 march 2005 and 16 december 2005. the collected data have been obtained from the mountains of peloponnesus, greece. shown are the distributions over time (obtained from the anemometer) of (a) the mean polar velocity in m/s, (b) the standard deviation of the mean velocity, the distribution of (c) the mean angles (degrees) and (d) the standard deviation of the angles. 0 1 2 3 4 5 6 7 8 9 10 0 10 20 30 months m /s 0 1 2 3 4 5 6 7 8 9 10 0 200 400 months d eg re es w 6 w 12 w 10 w 1 w 5 (a) (b) w 11 w 8w 7 w 3 w 2 w 4 w 9 fig.2 segments of experimental time-series (a) for the mean value of the velocity and (b) the mean angle of the anemometer depicted in fig.1. the vertical lines indicate the months during the measurements. 246 k. karamanos : symbolic dynamics applied to velocity time-series in wind farms the development of standards for wind farms presupposes the correct description of wind potential and this can be done only with the field measurements of wind force by anemometers. herein, we utilize new concepts coming from the world of nonlinear physics, to quantify and understand the relative complexity of wind signals of filtered or mild turbulence. although field measurements of filtered turbulence by mechanical purpose anemometers presuppose enormous and violent averagings on the molecular nature of turbulence, also averagings in time (months, epochs), not to mention the inertial phenomena in the measuring apparatus. we intend to show that a coherent and self-consistent mathematical description of turbulence within the arsenal of nonlinear physics is not only possible, but also beneficial for physicists and engineers. in this paper, one set of wind speed data from one measurement tower situated in the mountains of the region achaia, peloponnesus, greece, in the form of polar velocity and angle was analyzed. the data set consists of some thousands of wind speed values, recorded over every ten minutes by a cup anemometer, covers the period march 2005 to december 2005. more specifically, the wind polar speed and angle was measured using cup anemometers and the 38500 recordings logged into a digital anemograph logging equipment system. the complexity and variability of the data is analyzed here by the use of nonlinear techniques, and the information hidden in both the mean polar velocity and the mean polar angle time series are discussed. more precisely, the temporal evolution of nonlinear characteristics is studied by applying a recently proposed technique [4-5]. the original continuous time, of the mean polar velocity/angle data are projected to symbolic sequence and a block entropy analysis by the novelty of lumping follows [4-5]. the paper is articulated as follows : in sec 2, we recall basic facts about symbolic sequences, and the block entropy analysis by lumping. sec 3, will be devoted to the application of the entropy analysis by lumping to anemometer recordings. finally, in the last sec 4, we draw the main conclusions and discuss future plans. 2 symbolic dynamics a way to examine transient phenomena is to analyze the original time series (anemometer recordings) into a sequence of distinct time windows (epochs). the basic aim is to discover a clear difference of dynamical characteristics as time evolves by employing techniques from the toolbox of symbolic dynamics. in particular here we employ the notion of block entropy analysis by lumping [4-10]. towards this direction, within a stationary time window, the block entropy serves as a measure of "complexity" of the signal. the lower the value of entropy, the more "ordered" it is. in the following, in order to proceed with the analysis of the experimental data we will briefly review the concepts of symbolic dynamics and advances in systems science and applications (2014) vol.14 no.3 247 0 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 n h (n ) / n 0 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 n h (n ) / n w 1 w 2 w 3 w 4 w 5 w 6 w 7 w 8 w 9 w 10 w 11 w 12 (a) (b) fig.3 block-entropy per letter as a function of the word length for the various stationary windows shown in figs.2 (a) and (b). we observe a reduction of the block-entropy per letter which can be interpreted as a sign of complexity reduction of the respective time-window of the signal. the notion of block entropies [3-18]. we restrict ourselves to the simplest possible coarse graining of the recording. this is given by choosing a threshold c and assigning the symbols "1" and "0" to the signal, depending on whether it is above or below the threshold (binary partition or bipartition). in this way, each stationary time window of the original aiolic time-series for a given threshold is transformed into symbolic sequences, which contains "linguistic" or "symbolic dynamics" characteristics. more specifically, the block entropies, depending on the word-frequency distribution, are of special interest, extending shannon’s classical definition of the entropy of a single state to the entropy of a succession of states. thus, each entropy takes a large (small) value if there are many (few) kinds of patterns, i.e it decreases while the organization of patterns is increasing. in this manner we can argue that the block entropy constitutes a measure the complexity of a stationary signal. in particular, we estimate the block entropy by lumping [4-5]. lumping is the reading of the symbolic sequence by "taking portions", as opposed to gliding [1017] where one has essentially a "moving frame". in general, the basic novelty of the analysis by lumping is that, unlike the fourier transform or the conventional entropy by gliding, it gives results that can be related to algorithmic aspects of the sequences. it is useful to transform the initial raw data of the anemometer recording into symbolic sequences taking values in the alphabet {0, 1}, according to the rules ai = 1 if a(ti) > e[a(ti)] and ai = 0, if a(ti) < e[a(ti)]. here the quantities a(ti) denote the values of the measured mean velocity/polar angle at 248 k. karamanos : symbolic dynamics applied to velocity time-series in wind farms 0 2 4 6 8 10 12 0.5 1 1.5 2 2.5 3 3.5 n h (n ) w 2 w 6 w 12 fig.4 the observed scaling of the block entropy h(n) (eq. 6) as a function of the word length n. the experimental values show a linear best fit for small word lengths with a very good precision. the slope of the line gives the kolmogorovsinai entropy which in 1d coincides with the lyapunov exponent. this scaling is consistent with the corresponding theoretical predictions, see nicolis and gaspard [13]. time ti and e[a(ti)] =< a(ti) > is the mean value in the particular time windows. so, let us consider a subsequence of length n , selected out of a very long (theoretically infinite) symbolic sequence. we stipulate that this subsequence is to be read in terms of distinct "blocks" of length n, i.e : · · ·a1...an︸ ︷︷ ︸ b1 an+1...a2n︸ ︷︷ ︸ b2 ... ajn+1...a(j+1)n︸ ︷︷ ︸ bj+1 · · · . (1) we call this reading procedure "lumping" and we shall implement it in the following to the experimental time-series of the filtered turbulence. it is also useful to mention the following quantities which give the information content of the sequence : – the dynamical (shannon-like) block-entropy for blocks of length n has the form : h(n) = ∑ (a1,...,an) p(n)(a1, ..., an) ln p (n)(a1, ..., an), (2) where the probability of occurrence of a block a1, ..., an, denoted p(n)(a1, ..., an), is defined by the fraction (when it exists) in the statistical limit as follows : no of blocksa1, ..., anencounteredwhen lumping total no of blockswhen lumping , (3) advances in systems science and applications (2014) vol.14 no.3 249 table 1 the kolmogorov-sinai (ks) entropy h, the estimated error δh and the corresponding percentage h/ln(2) (see fig.5) in respect to the maximum value of the ks entropy for the different windows (see figs. 2(a),(b)) window no. h δh h/ ln 2(%) w1 0.2192 0.0350 31.6 w2 0.2784 0.0163 40.2 w3 0.2588 0.0407 37.3 w4 0.2040 0.0128 29.4 w5 0.2301 0.0152 33.2 w6 0.2065 0.0194 29.8 w7 0.2611 0.0576 37.7 w8 0.1879 0.0414 27.1 w9 0.1781 0.0611 25.7 w10 0.1245 0.0398 18.0 w11 0.3140 0.0535 45.3 w12 0.1277 0.0201 18.4 starting from the beginning of the sequence. however, the associate entropy per letter reads : h(n) = h(n) n . (4) – on the other hand, the entropy of the source (a topological invariant), defined in the limit (if it exists) reads : h = lim n→∞ h(n), (5) which is the discrete analog of metric or the kolmogorov-sinai entropy. therefore, in order to determine the abundance of long blocks one is led to examine the scaling properties of h(n) as a function of n (see also fig.4). 3 results in terms of symbolic dynamics. to begin with, in fig. 3 we depict the block entropy by lumping per letter as a function of the word length for the selected time windows that we present in fig. 2. we note that a complete absence of structure in the signal, would lead to an horizontal line in the block entropy diagram. as one can observe this is not the present case. however, one important conjecture, due essentially to ebeling and nicolis [13-14] states that the most general (asymptotic) scaling of the block entropies takes the form h(n) = y + nh+ gnµ0(lnn)µ1 , (6) 250 k. karamanos : symbolic dynamics applied to velocity time-series in wind farms 1 2 3 4 5 6 0 10 20 30 40 50 # of window h / ln (2 ) (% ) 1 2 3 4 5 6 0 10 20 30 40 50 # of window h / ln (2 ) (% ) (a) (b) fig.5 the normalized kolmogorov-sinai entropy (taken as the normalized slope of the linear part of the block entropy h(n)), for the respective time-windows of (a) the mean velocities (w1-w6), and (b) the polar angles (w7-w12). where y, h and g are constants and µ0 and µ1 are the corresponding non-scaling constant exponents. in the following, we attempt to examine the behavior of eq.(6) for each of the twelve stationary windows under study, which are depicted in fig. 2. hence, in fig. 4 we present the typical variation of the block entropy by lumping h(n) as a function of the word length n for three representative windows. this study reveals that if we restrict ourselves to the first five values of h(n), a linear scaling is observed with a great precision. in this manner, we next perform a least square method for this region and we estimate the slope h. note, that the associated correlation coefficients r, are close to 1 with a precision better than 10−4. working similarly for the rest of the 9 time windows, we conclude that for n < 6 the same behavior is observed, i.e. the equation for the scaling of the block entropy by lumping, is transformed to the remarkably simple linear relation h(n) = y + nh. (7) this means that g = 0 , for n < 6. in fig 2(a),2(b) we isolate 6 time windows for the mean velocity and 6 time windows for the polar angle respectively, which present a good overall stationary behavior according to our tests. their corresponding ks entropy is given in table 1. the ks entropy (complexity) of the whole window w6 (mean velocity) is of the order of 30%. this shows that the underlying mechanism of filtered turbulence has an underlying organized molecular basis, as it does not correspond to a completely random process (bernoulli shift). the maximum value of ks entropy for the mean velocity is about 41% for the window (w2), while the minimum value is about 30% (see also fig.5). a close inspection of the w2 window, shows that it has a structure more reminiscent of advances in systems science and applications (2014) vol.14 no.3 251 noise and this is confirmed by its high value of ks entropy. the ks entropy of the whole window for the mean angle (w12) is about 19% and it is quite low. the highest value is achieved in w11 which is of the order of 46%. we note that when g = 0 and h > 0, long words are penalized exponentially. we focus on the quantity h, namely the kolmogorov-sinai entropy defined as the slope of eq. (7). we notice that for a one-dimensional process the kolmogorov-sinai entropy coincides with its lyapunov exponent. the lyapunov exponent under these conditions gives a measure of the chaoticity (or dynamical randomness) of the signal. for a two-letter alphabet, the kolmogorov-sinai entropy h takes values from zero to ln 2 (see the discussion in sec 2), so that one can normalize dividing by ln 2 and obtain the respective percentage. hence, it is important to note that the linear part of the scaling helps for a classification and categorization of the recordings. the question which arises naturally, is whether this is an independent algorithmic law of nature. this seems to be an open problem for the moment. however, our results strongly support this hypothesis. remark : we restrict ourselves to the region n < 6, because the maximum statistical accuracy for the block entropies by lumping is of the order of lnl, where l is the total number of points (the size of the window). in our case l, is of the order of 2800, so that n < 8 and due to the underestimation of the higher entropies, we have enough statistical precision for n < 6. 4 conclusions. in this paper, we made an attempt to apply techniques and methods from cybernetics, nonlinear physics and general systems to the temporal unfolding of the phenomenon of filtered turbulence in wind farms as measured by cup anemometers. the tremendous molecular averaging implied by this simplistic procedure, has been equilibrated by the high specialization and the novelty of the techniques used in general systems. block entropy analysis by lumping, as introduced by karamanos et al. [4-5], is for the first time used for the understanding and categorization of time series of filtered turbulence in anemometer recording. a first linear region has been revealed, as it has been already happened for em preseismic precursors [6], cardiac signals of coronary patients [9], and dna strands in oligonucleotide basis acgt [8]. hence, it is important to note that the linear part of the scaling helps us for a complete classification and categorization of the recordings. as it has been already pointed out, the question which arises naturally, is whether this is an independent algorithmic law of nature. this seems to be an open problem for the moment. however, our results strongly support this hypothesis. future projects include the enhancement of data and recordings, the crosschecking of many complexity measures, the month-to-month monitoring of the 252 k. karamanos: symbolic dynamics applied to velocity time-series in wind farms dynamics and the application of prediction techniques, with special use to wind parks. references [1] probst, o. and cárdenas, d. (2010), “state of the art and trends in wind resource assessment”, energies, 3, 1087-1141. [2] huang, z. and chalabi, z.s. (1995), “use of time-series analysis to model and forecast wind speed”, journal of wind engineering and industrial aerodynamics, vol.56, no.2-3, pp.311-322. [3] nicolis, g., rao, g., rao, j. and nicolis, c. (1988), generation of spatially asymmetric, information-rich structures in far from equilibrium systems, in: christiansen. p. l., and parmentier, r. d. (eds.). (1989), “structure, coherence, and chaos in dynamical systems”, r. d., manchester university press. [4] karamanos, k., and nicolis, g. (1999), “symbolic dynamics and entropy analysis of feigenbaum limit sets”, chaos, solitons and fractals, vol.10, no.7, pp.1135-1150. [5] karamanos, k. (2001), “entropy analysis of substitutive sequences revisited”, journal of physics a: mathematical and general, vol.34, no.43, pp.9231. [6] karamanos, k., peratzakis a., kapiris p., nikolopoulos s., kopanas j. and eftaxias k. (2005), “extracting preseismic electromagnetic signatures in terms of symbolic dynamics”, nonlinear processes in geophysics (npg), vol.12, pp.835-848. [7] karamanos, k., dakopoulos, d., aloupis., k., peratzakis, a., athanasopoulou, l., nikolopoulos, s., kapiris, p. and eftaxias, k. (2006), “preseismic electromagnetic signals in terms of complexity”, physical review e, vol.74, no.1, 016104, pp.21. [8] karamanos, k., kotsireas, i.s., peratzakis, a., and eftaxias, k. (2006), “statistical compressibility analysis of dna sequences by generalized entropy-like quantities: towards algorithmic laws for biology?”, wseas transactions on systems, vol.5, no.11, pp.2503-2509. [9] karamanos, k., nikolopoulos, s., hizanidis, k., manis, g., alexandridi, a. and nikolakeas, s. (2006), “block entropy analysis of heart rate variability signals”, int. j. bif. chaos, vol.16, no.7, pp.2093-2101. [10] khinchin, a.i. (1949), mathematical foundations of statistical mechanics, dover publications. [11] mistakidis, i., karamanos, k., and mistakidis, s. (2013), “statistical versus optimal partitioning for block entropies”, kybernetes, vol.42, no.1, pp.35-54. advances in systems science and applications (2014) vol.14 no.3 253 [12] nicolis, g. (1995), introduction to nonlinear science, cambridge university press. [13] nicolis, g., and gaspard, p. (1994), “toward a probabilistic approach to complex systems”, chaos, solitons and fractals, vol.4, no.1, pp.41-57. [14] ebeling, w., and nicolis, g. (1992), “word frequency and entropy of symbolic sequences: a dynamical perspective”, chaos solitons and fractals, vol.2, pp.635-650. [15] steuer, r., molgedey, l., ebeling, w. and jimenez-montano, m.a. (2001), “entropy and optimal partition for data analysis”, eur. phys. j. b, vol.19, pp.265-269. [16] nicolis, g., nicolis, c., and nicolis, j. s. (1989), “chaotic dynamics, markov partitions, and zipf’s law”, journal of statistical physics, vol.54, no.3, pp.915-924. [17] nicolis, j. (1991), “chaos and information processing: a heuristic outline”, world scientific. [18] nikolopoulos, s., kapiris, p., karamanos, k., and eftaxias, k. (2004), “a unified approach of catastrophic events”, natural hazards and earth system science, vol.4, no.5/6, pp.615-631. [19] sammis, c. g., and sornette, d. (2002), “positive feedback, memory, and the predictability of earthquakes”, proceedings of the national academy of sciences of the united states of america, vol.99(suppl 1), pp.2501-2508. corresponding author author can be contacted at: sarris@teiath.gr. advances in systems science and applications (2017) vol.17 no.1 40 control cost in adaptive systems with an identifier under disturbance description uncertainties alexander bunich v.a. trapeznikov institute of control sciences of the russian academy of sciences, 117997, profsoyuznaya street, 65, moscow, russia; e-mail: bunfone@ipu.ru abstract. a problem of the criterion optimization of a linear-quadratic system with a stationary disturbance and set transfer function on control is considered. to solve the problem, a regularized method of substitution of estimates of the spectral density of the disturbance into the control cost functional is proposed. using the non-regularized method of substitution is shown to be able to lead to deflate the control cost. key words: linear-quadratic system, control cost, spectral smoothing of disturbances, substitution method. 1. substitution method and control cost a natural way of synthesis automatic systems with incomplete initial description is a reduction to a completely defined analog with substitution in the control law of corresponding empirical estimates instead of unknown parameters, calculated by an identifier. the identifier, as a sensor of parametric plant disturbances, may also be used to solve auxiliary problems, for instance, to predict slow faults, what demonstrates a universality of systems with an identifier. however, a justification of the substitution method, whose name was taken from the statistical estimation theory [1], requires considerable efforts. in the first turn, the optimization abilities are limited by two-block structures with separation of processes of forming the controls (identifier and controller with tuned parameters). generically, restricting the class of strategies increases the control cost in comparison to the optimal (dual [2]) strategy. besides that, the bayesian scheme of recalculating a posteriori parameters distributions over past observations requires setting a priori distributions of these parameters. finally, it is necessary to modify this scheme for problems with infinite control horizon [3]. the forced substitution of the initial problem statement of the synthesis with an asymptotic version implies considerable losses (a detailing of the term of ''asymptotic'' is needed: under increasing a sample, normalized risks may converge almost surely, in probability, in mean). formally, the synthesis problem (of dual control) for a plant with a parametric uncertainty relates to the stochastic optimal control of a completely defined plant with partially observed state ),( tt xx  , extended by partial inclusion of unknown plant parameters  , and involves calculating a priori distributions of the parameter )/( t u ydp  with respect to the observations t t yyy ,...,( 1 ), where the sub-index u underlines the dependence of these distributions on selecting a control strategy )},({ 1tt t uyu (in problems with a scalar criterion, the class of selectors is enough). if the strategy selection does not influence the rate of studying (passive feed-back), then the filtering algorithm admits 41 a. bunich: control cost in adaptive systems with an identifier under disturbance description uncertainties decomposition with emphasizing the parameters estimation in a functionally autonomous block (the identifier). computational problems of the dual control are supplemented with inadequate assumption on known a priori distribution of the parameters, characterizing the so called ''a priori hardness''. a natural way to overcome this hardness is the asymptotic statement of the synthesis problem. losses of the substitution are considerable. firstly, there are not distinguished asymptotically optimal systems with different transient quality, so the problem statement needs detailing. secondly, since a problem with infinite control horizon is considered, then a new constraint on the admissible strategies appears, which enables the closed-loop system stability, having no finite analogs. in systems with an identifier, investigating the stability is implemented within the method of ''frozen coefficients''. correct using the heuristic principle of ''frozen coefficients'', excluding effects of the parametric resonance, is possible under artificial decreasing the rate of controller retune with respect to the identification rate). asymptotically optimal strategies may be constructed without an identification by use of the direct method, as well as in systems with an identifier, when for a closed-loop plant conditions of the consistent estimation are violated (for instance, the gauss-markov conditions for the least squares method (lsm) [4]). let us be distracted from a more detailed discussion of the place and role of the identification in control systems. we will be restricted by more weak purpose of the criterion optimization [5] (the criterion optimization problem is always solvable, involving cases, when the lower bound of the control cost in the class of admissible controllers is not achieved, and the problem of the argument optimization is unsolvable). let us adopt the following assumptions. the plant is linear time-invariant, all variables are scalar and measured without disturbances; the transfer function of the plant on control is known; the disturbance 0}{  ttvv is centered stationary gaussian random process with the spectral density of an unknown order ,ks for which a consistent estimate ,...}2,1,{*  nkss n is known by use of the input/output observations  0},{ ttt yu , where k is a cone of spectral densities of stationary random processes of the autoregression/moving average with the 1l -metrics. the discrete-time ,...1,0t plant is described by the equation ttt vubya  )()( , 1)0( a with known coefficients of co-prime operator polynomials in the backward one time step shift operator . the initial data are disturbance independent random values with finite second moments. the controller (output feed-back) tt yu )()(   , 1)0(  is admissible, if the characteristic polynomial of the closed-loop system bag   is stable (the schur one). the control cost is the output variance in the steady-state mode. it turns out, that under accounting even the simplifications listed the criterion optimization is rather complex and the ''natural'' substitution method may decrease the control cost. let n be the completion of .k the cone n includes spectral densities of any stationary disturbances (indeed, k contains polynomials, in particular, convolutions of any density from n with the fejer kernels of any orders, forming the approximative unit in 1l ). thus, k may be advances in systems science and applications (2017) vol.17 no.1 42 considered as a class of parametric approximations of gaussian stationary random processes with absolutely continuous spectrum. on n , a concave functional ,||inf)( 20    dbfwi y f     is defined, where 0 yw is the transfer function from the disturbance to the output of the system with some fixed admissible controller,  is the ring of rational functions without poles in the closed disk }.1|{| z due to the completeness of the affine parameterization of control systems by the functional parameter f [6] kssi ),( is the control cost, that is the lower bound of the control cost over the class of all admissible controllers. the substitution method to determine the control cost )(si recommends the estimate *).(si the recommendation is justified, if the cost functional is uniformly continuous: in this case, to small errors of estimating the spectral density, small errors of the estimation by use of the substitution method correspond. however, the cost functional meets only the more weak condition of the semi-continuity from below [7]. indeed, let us continue uniformly the continuous cost functional from k on .n from the definition of the functional cost and the runge theorem it follows that 0)( si for any densities taking zero values on some arcs of the circumference 1|| z , and the set of such densities is dens in ,n so .0)( si however, in contrast to the last said, the cost is strictly positive for any density ks and the assumption on the uniform continuity is a delusion. in a number of cases, the designer possesses information on the spectral make-up of the disturbance in the form of the inclusion s with an a priori given class k . this information may be set by the inclusions ns . the paper is organized as follows. in section 2, a class (of degenerated problems) is emphasized. since the degeneracy is equivalent to the disturbance singularity, the control cost is zero. in section 3, an example of decreasing the control cost under using the substitution method (a consistent estimate of the spectral density of a regular disturbance) is presented. in section 4, a formula for the control cost is presented, and a procedure of the regularization of calculating the cost with using substitution of the metrics in the space of spectral densities of disturbances is proposed. required reference information was presented in section 1. 2. degenerate problems synthesis problems with zero cost will be referred as degenerate. let us resemble that the factorization method assumes the process to be regular, that is control system performance under the conditions of stationary inflow of new (renovating the disturbance) information from outside. a stationary process is regular if and only if its spectral density exists and has no ''deep zeros'' in the sense of convergence of the logarithmic integral: .ln 1ls this a rather weak constraint does not even guarantee the fourier series convergence for sln on a set of a positive measure (an example of a.n. kolmogorov), in computational practice it is conventionally substituted by considerably more strict one of existing the spectrum (absolute convergence of this series). correspondingly, the complex problem of the density s factorization is reduced to the separation for sln . a technical way of reducing the density factorization to separating its logarithm is complicated by the fact that the separation operator or ''natural projecting'' p ,  kj n k k kj n nk k eceсp            1 || , 43 a. bunich: control cost in adaptive systems with an identifier under disturbance description uncertainties is not bounded in the 1l -norm. the high prefilter sensitivity to variations of the spectral make-up of the disturbances indicates that 1l -norm is not an adequate closeness measure; and possible way of regularization of synthesis of wiener controllers is substitution of the metrics. from the category point of view, the regularity property is not typical: spectral densities meeting the regularity criterion forms in the space of all spectral densities the set of the first baire category. the regularity (singularity) property is preserved under transforming the process by a stable filter. the factorization solves the inverse problem: determining the transfer function of the prefilter via the spectral density of the disturbance. these well known results admit a generalization to multivariate processes of a constant rank (with substitution in the criterion ss det ). for more general classes of stationary processes, effectively verified regularity criteria are not known, while determining the prefilter via the spectral density is a complex problem. a general theory of singular processes perhaps does not exist, however enough simple way is available to transfer from singular processes to regular ones by adding (mixing) the white noise of a small power (what is interpreted by ''measurement noise''). to pursue the purpose, one determines a prefilter that factorizes the density s with small 0 , reducing the case of singular disturbance to regular one (non-formal question of selecting  is solved experimentally). under a fixed spectral density s , the control cost does not increase monotonically under as ,0 while solving the argument optimization problem is extremely non-regular and for the limit system (as )0 does not exist (let us resemble that for a singular disturbance the criterion optimization problem has zero solution). forming control has a dual function: directing (minimizing the action risk) and studying (minimizing future risks due to an uncertainty non-removed) [6]. the adaptive systems with an identifier (asi) are emphasized with their ability to parallelize these functions, what is seen from the structure asi scheme with a functionally autonomous block, the identifier, calculating in the real time plant parameters estimates and controlling another block, the tuned controller, constructed at the stage of the synthesis of the main control system loop. justification of the identification (indirect) approach to the synthesis problem is reduced to the fact, that if the control cost does not depend on transients in the identifier, then substitution of unknown plant parameters with their consistent estimates will not enlarge the control cost. 3.decreasing the control cost let us show, by use of an example of a minimum-phase plant, that the substitution method may decrease the control cost [7]. in this example, the control cost coincides with the error variance of the optimal one step ahead prediction. the example idea is to approximate a density with the ''deep zero'' by spectral densities of regular processes. for a positive density ks and a small number 0 , we will set s ( )()  s under ),(   and )exp()( 2 s under ).,(   the functions 1)( ls  are bounded from zero and so ,)( rs  where r is the class of spectral densities of regular processes. in accordance to the construction, ss l1  as ,0 however advances in systems science and applications (2017) vol.17 no.1 44     dsln as 0 , and by virtue of the szegő formula [8] ,0)( 0 2     where )(2  is the error variance of the optimal one step ahead prediction for the process with the spectral density s , coinciding, due to the minimum-phase plant property, with the control cost. thus, for some family of regular processes the following conditions hold: , 1 ss l  rs  (1a) ),(0)(lim 0 sisi     (1b) hence, the control cost takes as small as needed values in any neighborhood of the point .s one may prove that the latter affirmation is valid also without the constraints of minimum-phase and under any delays in the measurement. lemma on decreasing the control cost. for any spectral density ks , there exists a family of densities from k , meeting conditions (1). the lemma name is explained by the inequality 0)(inflim   ikv < ),(si (2) where v is any neighborhood of the point .ks due to this inequality, 1l -metrics is not an adequate measure of closeness of disturbances in the spectral make-up, what is expressed by the incorrectness of the criterion optimization problem, possible decreasing the control cost, defined by the substitution method, in comparison to its magnitude under exactly known spectral distribution density. the problem regularization is possible by using additional information, or by introducing a new metrics in the class of spectral densities of regular disturbances. 4. calculating the cost and solution regularization the control cost )(si has a simple geometric interpretation: it is the square of the distance from the point 201 lhwb yi  to 2h subspace. its calculation enables one to regularize the criterion optimization problem by the metrics substitution: in the new metrics, the cost functional is slowly changed under small variations of the spectral make-up of disturbances. to calculate the control cost, let us introduce the canonical (interior-exterior factorization of the polynomial eibbb  , factorization *)2( 1hh of the spectral density ,s the distance )( from the point 2l to 2h subspace. let us fix some admissible (support) controller and designate 00, uy ww the transfer functions of the control system with the support controller. lemma. the control cost is defined by the formula )()( 012 hwbsi yi  . (3) indeed, the distance d from the point hwy 0 to the 2hbi subspace is equal to 2 0 2 | || |inf fbhw iyhf   , and due to the unimodular property of ib , the equality holds )( 0122 hwbd yi  . (4) but hbe is an exterior function, so the linear manifold hbe is dense in 2h , bh is dense in в 2bh , and 2d ,| || |inf 2 2whmw (5) 45 a. bunich: control cost in adaptive systems with an identifier under disturbance description uncertainties where m 0 yw .ib the value 2 2| || |wh is the control cost of the control system with the spectral density s of the disturbance, and due to the completeness of the class m , 2d is the control cost, what was to be proven. remark. as a support regulator, one may select the h -optimal regulator. on the code r of spectral densities of regular processes, let us introduce a new metrics ,| || |),( 22121 hhss  where 2,1h are solutions of the problem of the factorization of the densities rs 2,1 with norming 0)0( h . it is easily to verify, that the mapping introduced as is a metric: ),0[: rr . let ssn  r , for solutions hhn , of the problem of the factorization of corresponding densities , 2 hh h n  but  )(|||| ssssss nnn ,)(|| sshh nn  (6) where from due to the cauchy inequality, we will obtain: chhss nn 2 2 2 1 | || || || |  (7) with a large enough constant c . the limit transfer as n proves the following implication: ssn  r . 1 ss l n however, the inverse affirmation is mistaken. let, for instance, the density s is a singular process density, while ns is its polynomial approximation, then .rs the new metrics regularizes the criterion optimization problem, and in the new metrics on the class of spectral densities of regular disturbances the functional )(2/1 si is a lipschitz one. let us underline, that the metrics introduced does not permit a continuation to more wide classes of densities, in contrast to the 1l -metrics, under definition of which the regularity condition is not pre-assumed. for rather poor classes of  , the regularization is not required. example. let  admit embedding in the simplex interior (parametric uncertainty). the continuity follows from the concave property. references [1] borovkov a.a., mathematical statistics, nauka publ., moscow, 1984. (in russian) [2] feldbaum a.a., foundations of the theory of optimal automatic systems, nauka publ., moscow, 1966. (in russian) [3] fomin v.n., methods of control of linear discrete-time plants, lgu publ., leningrad, 1985. (in russian) [4] kogan m.m. and neymark yu.i., “the identifiability of locally-optimal adaptive control laws under indirect observations”, automation and remote control, vol. 51, no. 1, 1990, pp. 53-62. [5] tsypkin ya.z., “the optimality in problems and methods of the advanced control theory”, vestnik an sssr, no. 9, 1982, pp. 116-121. [6] fomin v.n., fradkov a.l., and yakubovich v.a., adaptive control of dynamic plants, nauka publ., moscow, 1981. (in russian) [7] bunich a.l., “degenerate linear-quadratic problem of discrete plant control under uncertainty”, automation and remote control, vol. 72, no. 11, 2011, pp. 2351-2363. [8] hoffman k., banach spaces of analytical functions, prentice hall, englewood cliffs, n.j., 1962. advances in systems science and application (2015) vol.15 no.1 90-98 the birth triangle: a new approach to study the birth of systems j.-f. vautier french society of systems science (afscet), paris, france abstract this paper focus on the birth of systems. it starts with a physical model which provides some necessary conditions to the birth of a specific system: the fire. next, a more general model: the birth triangle is presented. similarly to the fire triangle, in the birth triangle two objects interact together when there is an activator. the notion of environment, in which the two objects are included, is also introduced. different examples of application of this model from physics to human organizations are examined: conglomerates of matters, molecules, cells and markets. the first main interest of the birth triangle is to propose ways to induce the birth of systems and also to stop their development. the second main interest of this model is to propose a new framework for a triggering condition and thus a way to revise the question of causality of different factors. indeed, with this model, it is the triangle which is causal (the set of the necessary factors and their conjunction). this new vision permits to avoid some quarrels to know which factor is the most responsible for something and some debates on which is the first inducer of an effect. keywords fire triangle, birth, system, field 1 introduction a lot of studies deal with the behavior of systems, the occurrence of some specific properties like for example emergences (properties of systems which cannot be observed in the elements [1]) or the collapse of systems [2]. considering the systemic methods, the same remarks can be done: a lot of methods aim to represent the behavior and/or the collapse of systems. the system dynamics, for example, belongs to this kind of methods [3]. but there are few studies about the birth of systems. then, this article will be focused on this topic. moreover, a lot of studies deal with causes [4-6]. some methods try in particular to find variables that most influence the others [7-9]: the control variables. indeed, the question is often: what are the variables which induce variations of the other ones? then, this article will propose a specific answer to this question related to the factors which cause the birth of systems. finally, this work starts with a physical model which provides some necessary conditions to the birth of fire (which is considered as a system). this model is the fire triangle. it was adapted to get a more general one: the birth triangle. then, the birth triangle will be firstly described. next, examples of application of this model from physics to human organizations will be provided. finally, a advances in systems science and application (2015) vol.15 no.1 91 discussion will be presented. 2 the birth triangle the fire triangle it is proposed to study the question of the birth of systems with a metaphor: the birth of fire. the common representation used to explain this birth is called the fire triangle. in a few words, this representation indicates that the birth of a fire depends on the conjunction and the amount of three factors which have to be in the same place at the same time: something to burn: a fuel, another factor which is necessary for the chemical reaction with the fuel: an oxidizing agent (usually oxygen), an activating energy which is necessary to induce the reaction between the two previous factors. the different components of the birth triangle similarly to the fire triangle, in the birth triangle two objects interact together when there is an activator. the notion of environment, in which the two objects are included, is also introduced here. fig.1 the birth triangle in this model, objects are generally “physical” and may have different sizes: two molecules (of fuel and oxygen), two people, two towns, two countries· · · there may be also virtual objects like the cells of conway’s game of life [10]. but objects are not qualities or emotions like gladness, sadness· · · . the activator induces a movement of one object closer to the other one in such a way that a system (i.e. two objects grouped and interacting together) may exist after (this movement). the activator is a field which may be a: 92 j.-f. vautier: the birth triangle: a new approach to study the birth of systems ... pushing field. object 2 moves closer to object 1 due to an internal energy (in object 2) or the two objects are pushed toward each other due to an external energy. this field is related to: · intrinsic pushing energies: if the objects are living beings, it means that only object 2 moves closer to object 1 (a prey tries to run away when the predator attacks). then, when a predator attacks a prey, a system of pursuit may be created; and/or · extrinsic pushing energies: heat (for example the birth of fire), mechanical energy (for example to make a mayonnaise) or pulling field (in the meaning of attraction). the two objects are pulled toward each other due to an internal energy (in the two objects) or due to an external energy. this field is related to: · intrinsic pulling energies of the two objects (leading to a mutual attraction). for example, two boxers move closer to constitute a pair after the ringing of bell; and or · extrinsic pulling energies: introduction of new entities in the environment like a catalyst. in a few words, energies are what permit the existence of the fields (which induce a movement of one object closer to another one). according to this general definition, energies are not only physical ones. they may correspond also for example to wishes of people. moreover, for living beings or human organizations, this latest kind of energy may appear to create a movement after an event, an accident, an information, a chance meeting. for example, the murder of archduke franz ferdinand of austria in 1914 was an event which leads to the start of the first world war. the wishes of fighting appeared (even if they were “fed” by grievances existing previously). the european peoples constituted in 1914 a system of belligerents; the environment supports the movement of moving one object closer to the other one. it means that there are also some characteristics of the environment that maintain together the two objects (to be able to interact to build a system afterwards): a small space inside a bottle or a bowl a place, a town which “maintain” people together. it means also that there are not obstacles in the environment between the objects. this may result from the removal of these obstacles, previously present in the environment, removal which may be caused by action of an entity (a human for example) or not (the natural removal of the link between a fruit and its tree). in the birth triangle, only the interactions between object 1 and object 2 are considered. object 1 is the entity on which the focus is. object 2 may consist of advances in systems science and application (2015) vol.15 no.1 93 several other entities which interact with object 1. moreover the question of the shape or the form of the set of these entities which interact in order to build the system is not considered in this article. 3 examples of birth of systems: from the physics to the human organizations conglomerates of matter in this case, the physical forces of gravitation are only taken into account. they attract the objects toward each other. when an apple falls down on the ground, this phenomenon can be described with the birth triangle: object 1: the earth; object 2: an apple; environment: the space around these two objects; activator: a pulling field related to gravitational forces in the same way, two magnets attract (pulling field) or repel (pushing field) each other according to their nature, orientation and distance from one to each other i.e. according to the field related to electromagnetic forces. “life bricks” let us consider now the chemical domain (the fire belongs to this domain). s. miller succeeded to produce some molecules: the amino acids which are often considered as the “life bricks”[11]. object 1: methane (ch4); object 2: ammonia (nh3) and hydrogen (h2); environment: aqueous solution in a bowl; activator: a pushing field (electric shock). cells concerning the process of fecundation: object 1: an ovule; object 2: a spermatozoid; environment: liquid solution in a small space; activator: for in vitro fecundation, there is pushing field resulting from an electric stimulation or mechanical push (a sting with a pipette). for natural fecundation, there is a pulling field. ovule and spermatozoids move closer toward each other. markets let us consider a market as a set of economic actors which sell and buy different kind of products: object 1: a company; 94 j.-f. vautier: the birth triangle: a new approach to study the birth of systems ... object 2: its clients; environment: a place to support the interactions between these two objects i.e. the transfer of products between a selling company and a buying actor; activator: a pulling field due to, for example, the occurrence of new offers of products from the company and/or new demands from the clients. 4 discussion several points are presented. first of all, this discussion deals with some ways to induce the birth of a system or to stop its development. next, the focus is on one characteristic of the objects: the fertile soil. finally, the notion of causality is questioned. ways to induce the birth of a system or to stop its development according to the birth triangle, if there is a birth of a system then a pushing or a pulling field exists. it means that if these kinds of fields are identified somewhere, then a system may exist if there are an environment and sufficient fertile soil in the objects. it is a way to look for the existence of some potential new systems. according to this model, what should be done, in practice, to induce the birth of systems? the way is often to: change the environment, for example, by introducing new entities (catalysts), or energy (e.g. heat), or removing distances, obstacles· · · between the objects in the environment and/or increase in object 1 and 2 the potentiality of interaction with the other one. for example, let us consider: object 1: a baker who also sells some kinds of cakes; object 2: the clients for cakes; environment: the location of the bakery; activator: a pulling field. then to induce the purchase of the cakes, it could be important to work, respectively, on: the quality of the cakes, their colors, their shapes· · · ; the needs of cakes from the clients (a way to increase them may be an advertising campaign for example); all information to locate the bakery· · · in order to increase the foot traffic to the storefront; if there are not enough clients, a special event may be organized with promotions and low costs or prepared like during the days of christmas or epiphany. this example shows that this birth triangle is in fact already used in marketing· · · even if people did not know it. on the other hands, the birth triangle may be also advances in systems science and application (2015) vol.15 no.1 95 a way to observe a marketing mix strategy with a new framework of examination. moreover, a hypothesis is proposed: it is possible to use also the birth triangle to stop the development of a system. it means that the different factors of the birth triangle could be pointed out during the development of the system. if we consider, for example, the problem of irregular migrants, a lot of countries try to prevent these migrants from crossings their borders. according to a birth triangle view, these actions deal with the environment in putting obstacles between the objects. nevertheless, some people propose also other approaches, for example trying to decrease the pushing field by improving the standard of living of people in their country with investments· · · then, in this way, the birth triangle, as a generalization of the fire triangle, may propose some ways to stop the development of systems, in particular, in decreasing the level of the different factors to remove the conjunction. importance of the “fertile soil” in the objects to create a system from interactions between object 1 and object 2, there must be already some “fertile soil” in the objects. for physical and chemical domains the fertile soil is in fact a basic property of the objects (the mass, the atomic composition· · · ) which will not change when the system will be made for human organizations like for example a market, it is different. its birth needs: sufficient “fertile soil” in the selling company i.e. some people who can answer the demands and understand the wishes, the needs expressed by the potential clients, who can provide some specific products for clients and this fertile soil may exist, in the company, in a domain but not in another one. in particular, it consists of top managers who think that their company has to answer the demands, has to go in this direction and who can provide resources and decide the development of teams in specific domains otherwise the response of the company will be certainly: “it is not my business”; sufficient “fertile soil” in the clients i.e. some people who can provide “demands” to the potential selling company, top managers of a buying company who think they may send these demands to this selling company. it concerns also the capacity of changing its buying habits by purchasing new products. an example: a seed may fall down from the tree to the ground but it will grow only if there is sufficient fertile soil in the ground (object 2: water). if the seed is too young (and then not ready to sprout) or if there is a stone (environment) or no possibility for water to enter the seed (the activator: a pulling field), there will be not sufficient conditions to the birth of the plant. besides, the seed may stay sometimes a long time in this state if there is not a chance meeting with a suitable environment 96 j.-f. vautier: the birth triangle: a new approach to study the birth of systems ... another example: if a problem occurs, for example a shortage of a product (like during the oil crisis of 1973), it may induce a pushing field from the clients of a company which sells this kind of products. but a system will appear only if there are sufficient fertile soils in object 1 (enough products to sell) and object 2 (enough money) and an environment suitable for development (without obstacles). then, this view, from the birth triangle, contrasts with another view which indicates that a problem, an accident, an event would be the triggering factor of modifications of the organizations. according to the birth triangle, an event, like the murder of archduke franz ferdinand of austria in 1914, may induce some changes in the objects (concerning for example the wishes of people to fight together) which may create, in this case, a pulling field but the birth of a system needs also some fertile soil in the objects (for example the resources necessary for the fighting). finally, the question may be even more complex for certain kinds of human organizations. indeed, the necessary level of fertile soil in object 1 may depend on the environment. for example, to own a research department or not in its company may be a weakness or not. that depends often on the other companies of this market or those which might enter this market. then, let us note that this birth triangle proposes some conditions to the birth of systems (factors and their conjunction). for physical process, the birth of a system is more predictable than for human organizations since, in this latest case, we cannot know often precisely the fertile soil of object 1 or 2 which is necessary. a causality based on the conjunction of necessary conditions the birth triangle proposes a new framework for a triggering condition and thus a way to revise the question of causality of different factors. with this model, it is the triangle which is causal (the set of the necessary factors and their conjunction). furthermore, we are not in a cumulative vision in which the value of a factor can compensate the value of another one (it is, for instance, the propriety which underlines the possibility to calculate a mean value). we enter in a more complex reality in which no component can compensate the lack of another one. for example, in order to create a market, the needs for the product are important but the capacity of the clients to pay for the product is naturally important too· · · this new vision permits often to avoid some quarrels to know which factor is the most responsible for something and some debates on which is the first inducer of an effect. 5 conclusion let us note that, in the birth triangle, the efficiency of each factor depends on the other factors and a suitable conjunction between the three factors is necessary to advances in systems science and application (2015) vol.15 no.1 97 induce the birth of a system. and to end this article, here are some perspectives of works for the future: the size of systems does not seem to be linked to a pushing or pulling field.is it true the more the systems are large the more the objects seem to be modified to induce the birth of systems and also after when the systems work. does a relationship exist really between the size of systems and the amount and the nature of the modifications? is there a possibility to combine some birth triangles together? for example, may an object be represented as a triangle? in a causal analysis, could it be possible to use the birth triangle to help to identify and to go back to the root causes of an event, a problem? if we consider a fire, there is an auto-activation of the reaction until sufficient fuel or oxygen are present in the environment of this system i.e. until new elements can support the combustion regularly. in other words, it could be represented as a continuous birth of the system. is this kind of functioning may be observed in a few or in a lot of systems? finally, it could be certainly interesting to better define the limits of this formalization. for example, it seems to be important that the two objects are different to build a system, like in a musical chord, the necessary difference between two notes. but is there a minimal difference? references [1] l. von bertalanffy. 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(1994), “structural analysis of a man-machine system (samms): presentation of the method”, xiith congress of the international ergonomics association (iea) toronto vol. 4, pp. 59-60. [8] a. dassens and r. launay. (2008), “systemic study of the risks analysis: presentation of a general approach”, systemic study of the risks analysis, ag 1585, editions t.i. a. dassens and r. launay. (2008), “etude systémique de l’analyse de risques :the original is in french.) [9] m. godet. (1994), from anticipation to action: a handbook of strategic prospective , unesco publishing. [10] m. gardner. (1970), “mathematical games. the fantastic combinations of john conway’s new solitaire game ‘life’ ”, scientific american,no.223,pp.120-123. [11] s. miller. (1953), “a production of amino acids under possible primitive earth conditions”, science, vol.117, pp.528-529. corresponding author j.-f. vautier can be contacted at:jean-francois.vautier@cegetel.net advances in systems science and applications (2014) vol.14 no.2 129-143 adaptive cfar tests for detection and recognition of target signals in radar clutter konstantin n. nechval1 and nicholas a. nechval2 1applied mathematics department, transport and telecommunication institute lomonosov street 1, lv-1019, riga, latvia 2statistics department, evf research institute, university of latvia raina blvd 19, lv-1050, riga, latvia abstract in this paper, adaptive cfar tests are described which allow one to classify radar clutter into one of several major categories, including bird, weather, and target classes. these tests do not require the arbitrary selection of priors as in the bayesian classifier. the decision rule of the recognition techniques is in the form of associating the p-dimensional vector of observations on the object with one of the m specific classes. when there is the possibility that the object does not belong to any of the m classes, then this object is to be classified as belonging to one of the m classes or to class m+ 1 whose distribution is unspecified. the tests are invariant to intensity changes in the clutter background and achieve a fixed probability of a false alarm. the results obtained in this paper agree with the simulation results, which confirm the validity of the theoretical predictions of performance of the suggested adaptive cfar tests. keywords radar clutter, target signal, detection, recognition, adaptive cfar tests 1 introduction modern air traffic control radar systems rely heavily on automatic target detection and tracking to maximize air traffic safety. moving target indicator and moving target detector algorithms achieve good target detection performance through the suppression of most or all forms of radar clutter. unfortunately, real-time information on airborne hazards to aircraft, such as birds and storm systems, is also suppressed. the ability to classify clutter and hence identify these hazards can thus contribute significantly to air traffic safety. the process of classification can be formalized as follows. the unprocessed radar data are passed through a feature extractor, which transforms the available data samples into a set of separable features. these features are derived from the reflection coefficients computed using the multisegment version of burgs formula[1]. the aforementioned coefficients (that contain all spectral information, including the mean doppler shift) are then transformed and grouped to satisfy the requirements for multivariate gaussian behaviour. only information that is different from class to class is maintained, and in such a form that a reliable decision, based on a discriminant function derived from the above features, may 130 konstantin n. nechval: adaptive cfar tests for detection and recognition of target... be made. the classification problem consists in the following. there are m classes (populations), the elements (objects) of which are characterized by p measurements (features). next, suppose that we are investigating a certain object on the basis of the corresponding p measurements. we postulate that this object can be regarded as a random drawing from one of the m populations but we do not know from which one. we suppose that m samples are available, each sample being drawn from a different class. the elements of these samples are realizations of p-dimensional normal random variables with unknown parameters. after a sample of p-dimensional vectors of observations on the object is drawn from a class known a priori to be one of the above set of m classes, the problem is to infer from which class the sample has been drawn. the decision rule should be in the form of associating the sample of observations on the object with one of the m samples and declaring that the object has come from the same class as the sample with which it is associated. when there is the possibility that the object does not belong to any of the m classes, then this object is to be classified as belonging to one of the m classes or to class m+ 1 whose distribution is unspecified. stehwien and haykin [2] solved the problem of statistical classification of radar clutter in a bayesian framework. in this paper, the problem is treated in a nonbayesian setting. a classification technique is described which allows one to classify radar clutter into one of several major categories, including bird, weather, and target classes. this technique is based on applying the theory of generalized maximum likelihood ratio testing for composite hypotheses. the unknown parameters are then estimated using maximum likelihood estimators. this approach does not require the arbitrary selection of priors as in the bayesian classifier. yet the generalized likelihood ratio test (glrt) is widely preferred because of its nice asymptotic (large sample size) properties such as consistency, unbiasedness, and constant false alarm rate (cfar). it is also called the uniformly most powerful invariant (umpi) test since it exhibits the ump property among the class of tests that are invariant to a natural set of transformations. the asymptotic performance of the glrt becomes equivalent to the test with perfectly known parameters. the main feature of the proposed classification technique is the class elimination rule. when certain conditions are met, the decision is taken to eliminate specific class from further considerations, and the classification process is continued with a reduced number of classes. the class elimination rule is based on the generalized likelihood ratio. the outline of the paper is as follows. a problem of signal detection in clutter is considered in section 2. section 3 is devoted to a problem of target signal recognition. advances in systems science and applications (2014) vol.14 no.2 131 2 signal detection in clutter the problem of detecting the unknown deterministic signal s in the presence of a clutter process, which is incompletely specified, can be viewed as a binary hypothesis-testing problem. the decision is based on a sample of observation vectors xi = (xi1, ..., xip) ′, i = 1(1)n, each of which is composed of clutter wi = (wi1, ..., wip) ′ under the hypothesish0 and a signal s = (s1, ..., sp) ′ added to clutter wi under the alternativeh1, where n > p. the two hypotheses that the detector must distinguish are given by h0 : x = w(clutteralone) (1) h1 : x = w + cs ′ (signalpresent) (2) where x = (x1, ...,xn) ′ (3) w = (w1, ...,wn) ′ (4) are n > p random matrices, and c = (1, ..., 1) ′ (5) is a column vector of n units. it is assumed that wi, i = 1(1)n, are independent and normally distributed with common mean 0 and covariance matrix (positive definite) q, i.e. wi ∼ np(0,q), ∀i = 1(1)n. (6) thus, for fixed n, the problem is to construct a test, which consists of testing the null hypothesis h0 : xi ∼ np (0,q) , ∀i = 1(1)n. (7) versus the alternative h1 : xi ∼ np (s,q) , ∀i = 1(1)n. (8) where the parameters q and s are unknown. remark 1. characterization of the multivariate normality is given by the following theorem. theorem 1 (characterization of the multivariate normality). let xi, i = 1(1)n, be n independent p-multivariate random variables (n ≥ p+2) with common mean bfa and covariance matrix (positive definite) bfq. let zk, k = p+2, , n, be defined by zk = k − (p+ 1) p k − 1 k (xk − x̄k−1) ′s−1 k−1 (xk − x̄k−1) 132 konstantin n. nechval: adaptive cfar tests for detection and recognition of target... = k − (p+ 1) p ( |sk| |sk−1| − 1 ) , k = p+ 2,...,n, (9) where x̄k−1 = k−1∑ i=1 xi/(k − 1), (10) sk−1 = k−1∑ i=1 (xi − x̄k−1)(xi − x̄k−1) ′, (11) then the xi (i = 1, . . . , n) arenp (a,q) if and only if zp+2, ..., zn are independently distributed according to the centralf distribution with p and 1, 2, ...n− (p+ 1) degrees of freedom, respectively. proof. the proof is similar to that of the characterization theorems [3-4] and so it is omitted here. 2.1 goodness-of-fit testing for the multivariate normality the results of theorem 1 can be used to obtain test for the hypothesis of the form h0 : xi follows np (a,q) versus ha : xi does not follow np (a,q) , ∀i = 1(1)n the general strategy is to apply the probability integral transforms [5] of zk, ∀k = p + 2 (1)n, to obtain a set of i.i.d. u(0, 1) random variables under h0. under ha this set of random variables will, in general, not be i.i.d. u(0, 1). any statistic, which measures a distance from uniformity in the transformed sample (say, a kolmogorov-smirnov statistic), can be used as a test statistic. 2.2 gmlr statistic and its distribution one of the possible statistics for testing h0 versus h1 is given by the generalized maximum likelihood ratio (gmlr) gmlr=max θ∈θ1 lh1(x; θ) / max θ∈θ0 lh0(x; θ), (12) where q = (s,q) ,q0 = {(s,q) : s = 0,q ∈ qp},q1 = q − q0,q = {(s,q) : s ∈ rp,q ∈ qp}, qp denotes the set of p × p positive definite matrices. under h0, the joint likelihood for x based on (7) is lh0(x; θ)= (2π)−np/2|q|−n/2 exp ( − n∑ i=1 x′iq −1xi/2 ) ., (13) under h1, the joint likelihood for x based on (8) is lh1(x; θ)= (2π)−np/2|q|−n/2 exp ( − n∑ i=1 (xi − s)′q−1(xi − s)/2 ) . (14) advances in systems science and applications (2014) vol.14 no.2 133 it can be shown that gmlr= ∣∣∣q̂0 ∣∣∣n/2∣∣∣q̂1 ∣∣∣−n/2 , (15) where q̂0=x ′x/n, (16) q̂1= (x ′ − ŝc′)(x ′ − ŝc′)′/n, (17) and ŝ=x ′c/n (18) are the well-known maximum likelihood estimators of the unknown parameters q and s under the hypotheses h0 and h1, respectively. it can be shown, after some algebra, that (15) is equivalent finally to the statistic y = t ′ 1t −1 2 t1/n, (19) where t1 = x′c,t2 = x′x.it is known that(t1,t2) is a complete sufficient statistic for the parameter q = (s,q) thus, the problem has been reduced to consideration of the sufficient statistic (t1,t2). it can be shown that under h0 , the result (19) is a q-free statistic y which has the property that its distribution does not depend on the actual covariance matrixq. this is given by the following theorem. theorem 2 (pdf of the gmlr statistic y). under h0, the statistic y is subject to a noncentral beta-distribution with the probability density function (pdf) fh1(y;n, q) = [ b (p 2 , n−p 2 )]−1 y (p 2 ) −1 (1− y) (n−p 2 ) −1 ×e−q/2 1f1 (n 2 ; p 2 ; qy 2 ) , 0 <y< 1, (20) where f1(a; b;x) is the confluent hypergeometric function [6], q = n ( s′q−1s ) (21) is a noncentrality parameter representing the generalized signal-to-noise ratio (gsnr). under h0, when q = 0, (20) reduces to a standard beta-function density of the form fh0(y;n) = [ b ( p 2 , n− p 2 )]−1 y ( p2 )−1 (1− y) (n−p 2 )−1 , 0 <y< 1. (22) proof. the proof is given by nechval [7] and so it is omitted here. it is clear that the statistic y is equivalent to the statistic v = [(n− p)/p] y/(1− y) = [n(n− p)/p] ( ŝ′ [ ĝ1 ]−1 ŝ ) , (23) 134 konstantin n. nechval: adaptive cfar tests for detection and recognition of target... where ĝ1 = nq̂1 = (x ′ − ŝc′)(x ′ − ŝc′)′ = n∑ i=1 (xi − ŝ)(xi − ŝ)′. (24) here the following theorem clearly holds. theorem 3 (pdf of the gmlr statistic v). under h1, the statistic v is subject to a noncentral f -distribution with p and n−p degrees of freedom, the probability density function of which is fh1(v;n, q) = [ b ( p 2 , n− p 2 )]−1 ( p n−p )p/2 vp/2−1( 1 + p n−pv )n/2 × e−q/2 1f1 ( n 2 ; p 2 ; q 2 ( p n− p v ( 1 + p n− p v )−1 )) , 0 < v < ∞, (25) where q is a noncentrality parameter given by (21). under h0, when q = 0, (25) reduces to a standard f -distribution with p and n− p degrees of freedom, fh0(v;n) = [ b (p 2 , n−p 2 )]−1 ( p n−p )p/2 vp/2−1( 1 + p n−pv )n/2 ,0 < v < ∞, (26) proof. the proof follows by applying theorem 2 and being straightforward it is omitted. 2.3 test for signal detection based on the gmlr statistic the test of h0 versus h1, based on the gmlr statistic v, is given by v { > h, then h1 (signalpresent), ≤ h, then h0 (clutteralone), (27) and can be written in the form of a decision rule u(v) over {v : v ∈ (0,∞)} , u(v) = { 1, v > h (h1), 0, v ≤ h (h0), (28) where h > 0 is a threshold of the test which is uniquely determined for a prescribed level of significance α so that sup θ∈θ0 eθ {u(v)} = α. (29) advances in systems science and applications (2014) vol.14 no.2 135 for fixed n, in terms of the probability density function (26), tables of the central f -distribution permit one to choose h to achieve the desired test size (false alarm probability pfa ), pfa = α = ∞∫ h fh0(v;n)dv. (30) furthermore, once h is chosen, tables of the noncentral f -distribution permit one to evaluate, in terms of the probability density function (25), the power (detection probability pd) of the test, pd = γ= ∞∫ h fh1(v;n, q)dv. (31) the probability of a miss is given by β = 1− γ. (32) it follows from (30) that the gmlr test is invariant to intensity changes in the clutter background and achieves a fixed probability of a false alarm, i.e. the resulting analyses indicate that the test has the property of a constant false alarm rate (cfar). also, no learning process is necessary in order to achieve the cfar. thus, operating in accordance to the local clutter situation, the test is adaptive. when the parameter q = (s,q) is unknown, it is well known that no the uniformly most powerful (ump) test exists for testing h0 versus h1 [8]. however, some hypothesis testing problems that do not admit ump decision rules (tests) nevertheless exhibit certain natural invariance properties [8-9]. these properties suggest restricting attention to a limited class of decision rules, viz., the invariant decision rules. it is then sometimes possible to derive decision rules that are ump within this limited class. in this sense, invariance is a concept of fundamental importance in hypothesis testing. the following theorem shows that the test (27) is umpi for a natural group of transformations on the space of observations. theorem 4 (umpi test). for testing the hypothesis h0(1) versus the alternative h1(2), the cfar test given by (27) is uniformly most powerful invariant (umpi). proof. the proof is similar to that of nechval [10] and so it is omitted here. a robustness property of the v-test can be studied in the following set-up. let x = (x1, ...,xn) ′ be an n × p random matrix with a pdf φ,let cnp be the class of pdf s on rnp with respect to lebesque measure dx, and let h be the set of nonincreasing convex functions from [0,∞) into [0,∞). we assume n ≥ p + 1. 136 konstantin n. nechval: adaptive cfar tests for detection and recognition of target... for s ∈ rp and q ∈ qp define a broad class of pdfs on rnp as follows cnp(s,q) =  f ∈ cnp : f(x; s,q) = |q|−n/2η ( n∑ i=1 (xi − s)′q−1(xi − s) ) , η ∈ h  (33) in this model, it can be considered the testing problem h0 : ϕ ∈ cnp(0, q), q ∈ qp (34) versus h1 : ϕ ∈ cnp(s,q), s ̸= 0, q ∈ qp (35) and shown that v-test is umpi. clearly if (x1, ...,xn) is a random sample of xi ∼ np(s,q), i = 1(1)n, or x ∼ nnp(cs ′, in ⊗q), the pdf φ of x belongs to cnp (s,q) . further if f (x; s,q)) belongs to cnp (s,q), then g∗(x; s,q) = ∞∫ 0 f(x; s, rq)dg∗(r) (36) also belongs to cnp (s,q) where g∗ is a distribution function on (0,∞), and so cnp (s,q) contains the (np-dimensional) multivariate t-distribution, the multivariate cauchy distribution, the contaminated normal distribution, etc. [10-12]. here the following theorem holds. theorem 5 (robustness property). for problem (34)-(35), the cfar v-test is umpi and the null distribution of v is f -distribution with d.f.s p and n− p, i.e., the cfar test is still umpi in a broad class of distributions given by (33), and the null distribution under any member of the class is the same as that under normality. proof. the proof is similar to that of nechval [10] and so it is omitted here. 2.4 risk minimization for fixed n, in terms of the above probability density functions in (25) and (26), the probability of making the first type of wrong decision (false alarm probability) is found by α(h;n) = ∞∫ h fh0(v;n)dv (37) and the probability of making the second type of wrong decision (the probability of a miss) by β(h;n, q)= h∫ 0 fh1(v;n, q)dv. (38) advances in systems science and applications (2014) vol.14 no.2 137 any value of s will result in a value for q that is greater than zero. as the value of s increases, the value of q will also increase. a good detector is certainty expected to minimize α and β in some manner. for example, the neyman-pearson criterion defines optimality to be that of maximizing 1− β subject to the constraint that α ≤ α0, where α0 is a fixed constant between zero and unity. for this criterion, the optimum threshold can be found from (30). in general the structure of an optimum detector depends on the signal (or the signal-to-noise ratio). let us assume that a noncentrality parameter q representing the generalized signal-to-noise ratio (gsnr) is given. if we let wα and wβ be the unit weight (cost) of the probability of making the first type of wrong decision (α) and the probability of making the second type of wrong decision (β), respectively, then the optimal threshold of test, h∗, can be found by solving the following optimization problem (with respect to h): minimize r(h;n, q) = wαα(h;n) + wββ(h;n, q) (39) subject to h ∈ (0, 1), (40) where r (h;n, q) is a risk representing the weighted sum of the false alarm risk and the miss risk. it can be shown that h∗ satisfies the equation wαfh0(h ∗;n) = wβfh1(h ∗;n, q). (41) generally, the miss risk is more important that the false alarm risk, so that wa ≤ wb. if the sample size of observations, n is not bounded above, then the optimal value n∗ of n can be found as n∗ = inf n : ( α(h∗;n)+β(h∗;n,q) ≤ ϑ, h∗= arg min h∈(0,1) r(h;n, q) ) , (42) where ϑ is a preassigned value of the sum of the false alarm risk and the miss risk. 3 target signal recongnition suppose that the hypothesis h0: (clutter alone) is rejected. then the target (signal in clutter) classification problem using the target identity information consists in the following. let the target signal belong to one of m classes and each class has equal a priori probability. there is available a sample of radar measurements of size n from each class. the elements of the sample from the jth class are realizations of p-dimensional random variables si (j) ∼ np (s (j) ,q (j)) , i = 1 (1)n, 138 konstantin n. nechval: adaptive cfar tests for detection and recognition of target... with unknown parameters s (j) and q (j) for each j ∈ {1, . . . ,m}. we are investigating a detected target signal on the basis of the corresponding sample of size n of p-dimensional radar measurements ri = (ri1, ..., rip) ′ , i = 1 (1)n, where ri ∼ np (s,q) . we postulate that this target signal can be regarded as a random drawing from one of the m classes but we do not know from which one. the problem is to classify a detected target signal as belonging to one of the m specified classes. when there is the possibility that a target signal does not belong to any of the m above classes, it is desirable to recognize this case. let ri and si (j) be the ith observation of the target and jth class variable, respectively. it is assumed that all observation vectors,ri = (ri1, ..., rip) ′ , si (j) = (si1 (j) , ..., sip (j)) ′ , i = 1 (1)n, are independent of each other, where n is a number of paired observations. let xi (j) = ri − si (j) , i = 1 (1)n, be paired comparisons leading to a series of vector differences. thus, for classification of a detected target signal as belonging to the jth class, it can be obtained and used a sample of n independent observation vectors x (j) = (x1 (j) , ...,xn (j)) , j ∈ {1, . . . ,m} . it is assumed that under h0 (j),xi (j) ∼ np (0,q+q (j)) , ∀i = 1 (1)n, where q+q (j) is a positive definite covariance matrix. under h1 (j), xi (j) ∼ np (a (j) ,q+q (j)),∀i = 1 (1)n, where a (j) = (a1 (j) , ..., ap (j)) ′ ̸= (0, ..., 0)′ is a mean vector. for fixed n, the problem is to construct a test which consists of testing the null hypothesis h0 (j) : xi (j) ∼ np (0,q+q (j)) , ∀i = 1 (1)n, versus the alternative h1 (j) : xi (j) ∼ np (a (j) ,q+q (j)) ,∀i = 1 (1)n, where the parameters a (j) ,q and q (j) are unknown. the cfar test of h0(j) versus h1 (j) (j ∈ {1, ...,m}) is based on the statistic given by (23), v(j) = [n(n− p)/p] ( â′(j) [ ĝ1(j) ]−1 â(j) ) , (43) where ĝ1(j) = (x ′(j)− â(j)c′)(x ′(j)− â(j)c′)′ = n∑ i=1 (xi(j)− â(j))(xi(j)− â(j))′. (44) the test of h0 versus h1, based on the gmlr statistic v(j), is given by v(j) { > h(j), then h1(j) (targetdoesnotbelongtoclassj), ≤ h(j), then h0(j) (targetbelongstoclassj), (45) where h (j) > 0 is a threshold of the test which is uniquely determined for a prescribed level of significance α(j) so that sup θ(j)∈θ0(j) eθ(j) {u(v(j))} = α(j), (46) advances in systems science and applications (2014) vol.14 no.2 139 where θ (j) = (a (j) ,q+q (j)) ,θ0 (j) = {(a (j) ,q+q (j)) : a (j) = 0, (q + q (j)) ∈ qp} , u(v(j)) = { 1, v(j) > h(j) (h1(j)), 0, v(j) ≤ h(j) (h0(j)). (47) thus, if v(j) > h(j) then the jth target class is eliminated from further consideration. if (m− 1) target classes are so eliminated, then the remaining class (say, kth) is the one to which a detected target signal being classified belongs. if all the target classes are eliminated from further consideration, we decide that a detected target signal belongs to the (m+1)th class whose distribution is unspecified. if the set of target classes not yet eliminated has more than one element, then we declare that a detected target signal belongs to the class j∗ if j∗ = arg max j∈d (h(j)− v(j)), (48) where d is the set of target classes not yet eliminated by the above test. now consider the situation in which a detected target signal s is related to the true target signal of the jth class, s(j), by s = us(j) = u(s1(j), ...sp(j)) ′, j ∈ {1, ...,m} (49) where ν is a scalar amplitude parameter. it is assumed that the target signal vectors s(j), j = 1(1)m, are known. the generalized maximum likelihood ratio statistics for this recognition problem are given by max υ {max q lh1(j)(x; υ,q)} / max q lh0(j)(x;q), (50) where lh0(j)(x;q)= (2π)−np/2|q|−n/2 exp ( − n∑ i=1 x′iq −1xi/2 ) , (51) lh1(j)(x; υ,q)= (2π)−np/2|q|−n/2 exp ( − n∑ i=1 (xi − υs(j))′q−1(xi − υs(j))/2 ) (52) are the likelihood functions under h0(j) and h1(j), j ∈ {1, . . . ,m} , respectively, and max υ {max q lh1(j)(x; υ,q)} = max υ 1 (2π)np/2 ∣∣∣⌢q1(j) ∣∣∣n/2 exp ( −np 2 ) , (53) 140 konstantin n. nechval: adaptive cfar tests for detection and recognition of target... max q lh0(j)(x;q) = 1 (2π)np/2 ∣∣∣⌢q0 ∣∣∣n/2 exp ( −np 2 ) . (54) the well-known maximum likelihood estimates (mles) of the unknown covariance matrix q under the respective hypotheses, h0(j) and h1(j), are given by ⌢ q0 = 1 n n∑ i=1 xix ′ i = 1 n xx ′, (55) ⌢ q1(j) = 1 n n∑ i=1 (xi − υs(j))(xi − υs(j))′ = 1 n (x − υs(j)c′)(x − υs(j)c′)′. (56) after several algebraic manipulations, (28) reduces to the following clutter-adaptive test of detection of the jth target signal, j ∈ {1, . . . ,m} :{ > h(j), then h1(j), ≤ h(j), then h0(j), (57) where z(j) = [s′(j)(xx ′)−1xc] 2 [s′(j)(xx ′)−1s(j)][1− c′x ′(xx ′)−1xc] , (58) h(j) > 0 is a threshold of the test which is uniquely determined for a prescribed level of significance α(j) so that the probability of a false alarm is equal to α(j). theorem 6 (pdf of the gmlr statistic v(j)). the probability density function of v(j) under hypothesis h1(j) is given as follows: fh1(j)(v(j);n, q(j)) = 1∫ 0 f(v(j);g, n, q(j))f(g;n)dg, (59) where f(g;n) = γ ( n−1 2 ) γ (n−p 2 ) γ ( p−1 2 )(1− g) p−3 2 g n−p−2 2 , (60) for 0 ≤ g ≤ 1, and f(v(j); g, n, q(j)) = γ ( n−p+1 2 ) exp ( − q(j)g 2 ) γ (n−p 2 ) γ ( 1 2 ) advances in systems science and applications (2014) vol.14 no.2 141 ×[1− v(j)] n−p−2 2 [v(j)]− 1 2 1f1 ( n− p+ 1 2 ; 1 2 ; q(j)gv(j) 2 ) (61) for 0 < v(j) < 1. in (61) 1f1 (a; b;x) is the confluent hypergeometric function, and q(j) is the generalized signal-to-noise ratio (gsnr) defined by q(j) = gsnr = nυ2s′(j)q−1s(j). (62) under hypothesis h0(j), no signal is present. thus, if one sets q(j) = 0 in (61), fh0(j)(v(j);n) = γ ( n−p+1 2 ) γ (n−p 2 ) γ ( 1 2 ) [1− v(j)] n−p−2 2 [v(j)]− 1 2 , 0 < v(j) < 1. (63) proof. the proof is similar to that of theorem 2 and so it is omitted here. finally, in terms of the above probability density functions in (59) and (63) the probability of false alarm is given by pfa(j) = 1∫ h(j) fh0(j)(v(j);n)dv(j) (64) and the probability of detection of the jth target signal is pd(j) = 1∫ h(j) fh1(j)(v(j);n, q(j))dv(j). (65) thus, if v(j) < h(j) then the jth target class is eliminated from further consideration. if (m− 1) target classes are so eliminated, then the remaining class (say, kth) is the one to which a detected target being classified belongs. if all the target classes are eliminated from further consideration, we decide that we deal with a clutter alone. if the set of target classes not yet eliminated has more than one element, then we declare that a detected target belongs to the class j∗ if j∗ = arg max j∈d (v(j)− h(j)), (66) where d is the set of target classes not yet eliminated by the above test. 142 konstantin n. nechval: adaptive cfar tests for detection and recognition of target... 4 conclusion the main idea of this paper is to find a test statistic whose distribution, under the null hypothesis, does not depend on unknown (nuisance) parameters. this allows one to eliminate the unknown parameters of noisy processes, which are changing with time and position, from the problem. the authors hope that this work will stimulate further investigation using the approach on specific applications to see whether obtained results with it are feasible for realistic applications. acknowledgments this research was supported in part by grant no.06.1936 and grant no.01.0031 from the latvian council of science and the national institute of mathematics and informatics of latvia. this support is gratefully acknowledged. references [1] kay, s.m. and makhoul, j. (1983), “on the statistics of the estimated reflection coefficients of an autoregressive process”, ieee trans. on assp, vol. 31, pp.1447-1455. [2] stehwien, w. and haykin, s. (1989), “a statistical radar clutter classifier”, proceedings of the ieee national radar conference, dallas, texas, pp.164169. [3] nechval, n.a. and nechval, k.n. (1998), “characterization theorems for selecting the type of underlying distribution”, abstracts of communications of the 7th vilnius conference on probability theory and mathematical statistics & the 22nd european meeting of statisticians, tev, vilnius, lithuania, pp.352-353. [4] nechval, n.a., nechval, k.n., and vasermanis, e.k. (2000), “technique of testing for two-phase regressions”, proceedings of the second international conference on simulation, gaming, training and business process reengineering in operations, riga, latvia, pp.129-133. [5] nechval, n.a. (1988), “a general method for constructing automated procedures for testing quickest detection of a change in quality control”, computers in industry, vol. 10, pp.177-183. [6] abramowitz, m. and stegun, i.a. (1964), handbook of mathematical functions, national bureau of standards, new york. advances in systems science and applications (2014) vol.14 no.2 143 [7] nechval, n.a. (1992), “radar cfar thresholding in clutter under detection of airborne birds”, proceedings of the 21st meeting of bird strike committee europe, bsce, jerusalem, israel, pp.127-140. [8] lehmann, e.l. (1959), testing statistical hypotheses, john wiley, new york. [9] nechval, n.a. (1997), “adaptive cfar tests for detection of a signal in noise and deflection criterion”, digital signal processing for communication systems (edited by t. wysocki, h. razavi, b. honary), kluwer academic publishers, boston·dordrecht·london, pp.177-186. [10] nechval, n.a. (1997), “umpi test for adaptive signal detection”, signal processing, sensor fusion, and target recognition vi (edited by i. kadar), proc. spie, vol. 3068. orlando, florida usa, paper no.3068-73, 12 pages. [11] nechval, n.a. and nechval, k.n. (1999), “cfar test for moving window detection of a signal in noise”, proceedings of the 5th international symposium on dsp for communication systems, curtin university of technology, perth-scarborough, australia, pp.134-141. [12] nechval, n.a., nechval, k.n., and purgailis, m. (2011), “statistical pattern recognition principles”, international encyclopedia of statistical science (edited by miodrag lovric), springer-verlag, berlin, heidelberg, part 19, pp.1453-1457. corresponding author konstantin n. nechval can be contacted at: konstan@tsi.lv. microsoft word 1-robert vallee.doc advances in systems science and applications (2010), vol.10, no.1 1-5 issn 1078-6236 international institute for general systems studies, inc. internal time of a dynamical system robert vallée professor emeritus université paris-nord, president of wosc email: r.vallee@afscet.asso.fr abstract we have a dynamical system. the evolution of its state, x(t) at instant t, is given by a differential equation dx(t)/dt = f(x(t),t), independent of the environment. we propose to introduce a time s, or internal time, different from time t, or reference time. for this purpose we consider duration, or time elapsed between two instants. reference duration, between instants t1 and t2 is obviously given by dr(t1,t2) = t2-t1. any duration, for example internal duration di(t1,t2), must satisfy certain conditions. once we have an internal duration di(t1,t2), we can generate an internal time s = di(t0,t). the choice of di(t1,t2) depends upon the “weight” of reference duration t2-t1, seen from the internal point of view, or equivalently that of infinitesimal reference duration dt between t and t+dt. we propose that the internal duration corresponding to reference duration dt is equal to (d(x(t)/dt)2 dt. in a way (dx(t)/dt)2 is an index of the “importance” of instant t. as an example, we consider an “explosive-implosive” dynamical system described by a certain evolution equation. the corresponding internal time varies from -∞ to +∞ while reference time varies from 0 to +∞. interpretations (physiology, cosmology) are given. keywords explosion-implosion internal duration cosmological time 1. introduction we want to give a definition of the internal time, or intrinsic time of a dynamical system evolving independently of its environment. we proposed this definition for the first time in 1996. we developed it mainly in an article (vallée, 2005) which the present text reproduces partly. the notion of internal timeis opposed to that of external time, or reference time, taken for granted and which is used in the evolution equation. the basic idea is that the internal time does not elapse if the state of the system does not change, a conception close to that of aristotle for whom time ceases to be known when the “soul” does not vary. so if x(t), belonging to a finite dimensional linear space, is the state of the system at reference instant t, any real positive and increasing function, null for argument 0, of a norm of dx(t)/dt, is a measure of the intensity of change of the system at instant t .we make the most simple choice, that of the square of the euclidian norm (or scalar square) (dx(t)/dt)2 which may be seen as an index of “importance” of instant t. so we consider that the internal duration corresponding to reference duration dt (between t and t+dt) is equal to (dx(t)/dt)2 dt and we define (vallée, 1996, 2001) the internal duration di(tl , t2) of interval (t1,t2), whose reference duration is t2 –t1, by di(t1,t2) = ∫t1,t2 (dx(t)/dt)2 dt (1) so if (dx(t)/dt)2 is equal to 0 on the interval, the internal duration is 0, and if (dx(t)/dt)2 is equal to1, the internal duration is equal to the reference duration. in short, the higher the values of (dx(t)/dt)2 on the interval, the greater the internal duration. we can now, define the internal time s(t) by s(t) = di(t0,t) = ∫t0, t (dx(s)/ds)2 ds (2) where t0 is any reference instant. so s(t) if determined up to an arbitrary additive constant. of course we have di(t1,t2)=s(t2)-s(t1). (3) robert: internal time of a dynamical system 2 it is interesting to verify if equation (3) is consistent with the axioms that a « time » must satisfy. if f(a,b) is a duration attached to interval (a,b), we must have, with a≤b≤c, f(a,b) + f(b,c) ≡ f(a,c), f(a,b) >0 for b> a, f(a,a) ≡ 0 (4) f (a,b) increasing with b and decreasing with a. with the hypothesis that f is differentiable, it is easy to solve functional equation (4). we have f(a+da,b) + f(b,c+dc) ≡ f(a+da, c+dc) replacing f(a+da,b) by f(a,b) + ∂f(a,b)/∂a da and the like for f(b,c+dc) and f(a+da, c+dc), we obtain ∂f(a,b)/∂a ≡ ∂f(a,c)/∂a so ∂f(a,b)/∂a is independent of b. consequently, by integration with respect to a, we have f(a,b) = f(a) +g(b) but since f(a,a) ≡ 0 ≡ f(a) + g(a) we have g = -f so f(a,b) = g(b) – g(a) which is consistent with (3). 2. examples we call explosion the evolution of a system whose state vector has a modulus starting with value 0 at t = 0 then increasing with t, and such that the modulus of its speed vector starts with value +∞ at t = 0. the first instants of the evolution of the system have an exceptional importance since (dx(t)/dt)2 tends to +∞ when t tends to 0. we have here an idealisation as well as in the case of what we call implosion where x(t) decreases with t and attains value 0 at the final instant while (dx(t)/dt)2 tends to +∞ system may be explosive at the beginning and implosive at the end, then we say that we have an explosion-implosion. for the sake of simplicity we shall suppose now that x(t) is a mere scalar. we start with an explosion-implosion (vallée, 1996, 2001) defined by the differential equation x (t)/dt = q/p sgn(p-t) (q2 x2 (t))1/2 / x(t), x(0) = 0, p et q > 0, 0≤ t ≤ p (5) where sgn(p-t) is the sign of p-t. the solution of this equation is given by function x(t) = q/p (p2 (p-t)2)1/2 (6) whose graph is an half-ellipse (great axis 2p, small axis 2q). we say that we have an elliptic explosion-implosion. when t varies from 0 to 2p, x(t) increases from 0 to q, then decreases from q to 0,with a speed of infinite absolute value at t = 0 and t = 2p.the square of the speed of evolution is given by (dx(t)/dt)2 = q2/p2 (p-t)2/ t(2p-t) = q2/2p (1/t 2/p + 1/2p-t) which shows that the “importance” of instant t is infinite at the beginning (t = 0) and at the end (t=2p). if we integrate (dx(t)/dt)2 from t1 to t2 we obtain the internal duration of the reference time interval (t1,t2) di(t1,t2) = q2/2p (log (t2/t1) 2(t2-t1) /p -log (2p-t2 /2p-t1)) (7) now, choosing t1 = p and t2 = t we have an internal time s(t) defined by s(t) = di(p,t) = q2/2p (log t 2t/p +2 -log (2p-t)) (8) of course constant q2/2p may be supressed, then we have another internal time σ(t) = q2/2p (log t – 2t/p – log(2p-t) we see that when the reference time t varies from 0 to 2p, generating a finite reference duration of the evolution equal to 2p, the internal time svaries from -∞ to +∞generatingan infinite internal duration of the evolution. the initial instant 0 is pushed back to -∞ and the final instant 2p is pushed forward to +∞.the internal duration of any interval (0, t) is infinite as well as the internal duration of any interval (t, 2p). we consider now the differential equation advances in systems science and applications (2010), vol.10, no.1 3 dx(t)/dt = q/p (q2 + x2(t))1/ 2 / x(t) , x(0) = 0, p et q >0, 0≤τ< + ∞ (9) its solution is given by function x(t) = q/p ((p+t)2 – p2)1/2 (10) whose graph is the right part of an half-hyperbola (great axis 2p,“small axis” 2q).we say that we have an hyperbolic explosion : x(t) increases from 0 to +∞ with an infinite speed at instant 0 and, for the great values of t, x(t) behaves like (t-p) a/p. the square of the speed is given by (dx(t)/dt)2 = q2/p2 (p+t)2 / t(t+2p) = q2/2p (1/t + 2/p -1/2p+t) it shows that the « importance » of instant t is infinite at t = 0. calculations, similar to those for the elliptic case, give the internal duration of interval (t1,t2) and consequently an internal time such as s(t) = q2 /2p (log t + 2t/p – log (2p+t) (11) when t varies from 0 to +∞, s(t) varies from -∞ to +∞, the initial instant 0 being pushed back to −∞. any interval (0, t), of reference duration t, has an infinite internal duration. moreover s(t) behaves as q2 /2p log t for small values of t and as q2/p2 t for great values. an intermediary case, which we call parabolic explosion (vallée, 1996, 2001), is obtained when p and q tend to infinity while q2/p keeps a constant value 2h. starting indifferently from equation (6) or (9), we obtain dx(t)/dt = 2h / x(t), x(0) = 0, h>0 (12) the solution is given by function x(t) = 2 (ht)1/2 (13) whose graph is an half-parabola of parameter h. we see that xt) increases from 0 to +∞ with an infinite initial speed. then we have (dx(t)/dt)2 = h/t the “importance” of instant t is infinite at t = 0 and tends to 0 when t tends to +∞. the calculation of internal duration generates an internal time s(t) = h log t, (14) the initial instant 0 is pushed back to -∞ and any interval (0,t) has an infinite internal duration. 3. interpretations in the first interpretation we consider an elliptic explosion-implosion as the evolution of a living being whose birth may be compared to a kind of explosion and the end of life as an involution, or a kind of implosion, more or less quick. in our model the implosive part is symmetrical with the explosive one. this is not very realistic, since it seems that the implosive part must be shorter. nevertheless if we consider only the qualitative aspect of the conclusions we can say that, from the internal time point of view, the initial instant (conception) is pushed back to ∞and the final instant (death) is pushed forward to +∞ (vallée,1991,1996). the first part of this qualitative conclusion is in accordance with the natural feeling of a human being (if we consider this case) of having no beginning. we can also consider the case of a parabolic explosion limited at the instant of death. the initial instant is pushed back to −∞ and the internal time is proportional to the logarithm of the elapsed reference time. it elapses slowly at the beginning and more and more quickly near the end. this is close to the ideas of lecomte du noüy (lecomte du noüy, 1936). for him the physiological duration of an astronomical time interval (reference time interval in our terminology), of given length, is proportional to the speed of healing of wounds. this speed varying roughly as the inverse of age, a logarithmic physiological time is generated. but a remark seems necessary in order to avoid apparent paradoxes: the internal time of a conscious being may be different from the perceived internal time. the second interpretation concerns internal duration in cosmology. the cases of elliptic explosion-implosion, parabolic or hyperbolic explosion have common traits with certain robert: internal time of a dynamical system 4 cosmological models with primordial explosion followed by final implosion or with primordial explosion only. generally speaking, the differential equation giving the evolution of the universe, whose state at instant t is described by the cosmological scale factor r(t), is according to lemaître, friedmann, robertson (berry, 1989) (dr(t)/dt)2 = 8πg/3ρ(t) r2(t) kc2 + λ/3 r2(t), r(0) = 0 (15) where g is the gravitational constant, c the speed of light, k the index of curvature (k = -1, we have a space with negative curvature; k = 0, we have a flat space; k = +1, the space has a positive curvature (then r(t) may be considered as the radius of the universe), λ the cosmic constant or cosmic repulsion term, ρ(t) the density of matter or its material equivalent in case of pure radiation. in the material case ρ(t) = a/r3(t) and in the case of pure radiation it is equal to b/r4(t), a and b being constants. equation (5), corresponding to an elliptic explosion-implosion, gives if we consider the square of its two members (dx(t)/dt)2 = q4/p2 / x2(t) q2/p2 if we substitute r(t) to x(t), the above equation takes one of the possible forms of (15) if ρ(t) = b/r4(t), k = +1, λ= 0 . we have a case of pure radiation with positive curvature and null cosmic constant. more precisely q = c p and p = 1/c2 (b 8πg/3)1/ 2. then (dr(t)/dt)2 = 8πg/3 b/r2(t) – c2 (16) the internal time of this system, which we propose to call generalized cosmological time (vallée, 1995, 1996, 2005) is then given, according to (8), by s(t)= c2p/2 (log t 2t/p – log (2p-t) (17) the initial reference instant t = 0 (“big bang”) is pushed back to -∞ and the final referenceinstant t = 2p (“big crunch”) is pushed forward to +∞. but classically a cosmological model with pure radiation is accepted mainly as an approximation valid when the density of matter (a/r3(t)) is negligible compared to the (equivalent) density of matter of pure radiation (b/r4(t)). this happens when r(t) is small, so when t is close to 0. in that case –kc2 is negligible as well as λ/3r2 and it is not even necessary to suppose that λ = 0 . we then have (dr(t)/dt)2 = 8πg/3 b/r2(t) (18) this equation is that of the radiation-dominated era at the beginning of the universe, or that of an evolution with pure radiation, null cosmic constant and flat universe. it corresponds to a parabolic explosion whose differential equation, after taking the square of its two members, gives (dx(t)/dt) = 4h2 / x2(t) we have just to substitute r(t) to x(t) and (b 2πg/3)1/ 2 to h. according to (14), the internal time of this universe is s(t) = (b 2πg/3)1/2 log t (19) the initial instant t = 0 (“big bang”) is pushed back to -∞. we recognize here what milne has called cosmological time (milne, 1948). references [1] berry, m. principles of cosmology and gravitation. institute of physics publishing, bristol & philadelphia, 1989. [2] lecomte du noüy. p. le temps et la vie. gallimard, paris, 1936. [3] milne, e. a. kinematic relativiy. clarendon press, oxford, 1948. [4] vallée, r. perception, memorisation and multidimensional time, kybernetes, 1991, vol.20, 15-28. [5] vallée, r. cognition et système. essai d’épistémo-praxéologie. l’interdisciplinaire, limonest (france), 1995. advances in systems science and applications (2010), vol.10, no.1 5 [6] vallée, r. temps propre d’un système dynamique, cas d’un système explosif-implosif, in actes du 3ème congrès international de systémique, (e. pessa, m.p. penna, dirs), edizioni kappa, rome, pp.967-970, 1996. [7] vallée, r. time and dynamical systems. systems science vol.27, 97-100, 2001. [8] vallée, r. time and systems, kybernetes, vol.34, 9-10, 1563-1569, 2005. adv syst sci appl 2016; 16(2); 70-80 published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/351 copyright ©2016 assa. adv. in systems science and appl. (2016) improved prediction of post-operative life expectancy after thoracic surgery abeer s. desuky, lamiaa m. el bakrawy al-azhar university, cairo, egypt e-mail: abeerdesuky@yahoo.com, lamiaabak@yahoo.com abstract: monitoring health outcomes is essential to enhance quality initiatives, healthcare management and consumer education. thoracic surgery is the data collected for patients who underwent major lung resections for primary lung cancer. the application of machine learning techniques for predicting post-operative life expectancy in the lung cancer patients is an area with little research and few concrete recommendations. in order to use machine learning techniques effectively, attribute ranking and selection is an integral component to successful health outcome prediction. in this paper, we present three attribute ranking and selection methods to improve algorithms performance for health outcomes research. two papers results for other researchers are used in comparison to show the efficiency of our proposed attribute ranking and selection methods. keywords: attribute ranking, machine learning, prediction, thoracic surgery. 1. introduction integrating computer applications into the medical field have directly affected the productivity and accuracy of doctors nowadays. measuring health outcomes is one of these applications. clearly, there is a growing role for health outcomes in the purchasing and management of healthcare. these days cancer is one of the major causes of death in the most countries. currently, lung cancer is the most frequent augury for thoracic surgery [1]. researchers applied different strategies, such as examination in early stage, to identify the type of cancer before the emergence of symptoms. furthermore, new methods for the early prediction of cancer therapy outcome have been developed [2].with the raise of new techniques in the field of medicine, massive datasets of cancer have been collected and now available to researchers in the medical field. however, the most challenging task is predicting a disease outcome accurately. so, the current research efforts examine the use of machine learning techniques for discover and identify models and relationships between them, from large datasets, the data is analyzed to extract useful information that supports disease augury, and to improve models that predict patient’s health more accurately [3,4]. huge datasets usually lead to crumble the performance and accuracy of the machine learning systems. datasets with high dimensional attributes have more processing complexity with longer computational time for prediction. attribute ranking and selection is a solution to complex datasets [5]. several attribute and selection methods have been presented in the machine learning domain. the main aim of these methods is to remove attributes that can be irrelevant, misleading, or redundant which increase search space size resulting in difficulty to process data further thus not contributing to the learning process. attribute and ranking selection is the process of choosing best attributes from all the attributes that are useful to discriminate classes [6, 7]. mailto:abeerdesuky@yahoo.com mailto:lamiaabak@yahoo.com 71 a. s. desuky, l.m. el bakrawy copyright ©2016 assa. adv. in systems science and appl. (2016) the rest of this paper is organized as follows: brief introduction on machine learning algorithms and attribute ranking and selection methods have been applied to disease prognosis and prediction are introduced in sections (2 and 3). the details of the proposed methods and the data set are presented in section (4). section (5) shows experimental results. conclusions are discussed in section (6). 2. related work the most major operation that performed on lung cancer patients is the thoracic surgery. survival rate is very critical factor for sawbones to decide on which patient surgery would be performed. selection of the appropriate patient for surgery is one of the common clinical decision challenges in thoracic surgery, bearing in mind risk and benefits for a patient, both in short-term (e.g. post-operative complications, including death-rate in the first month) and long-term perspective (e.g. survival for 1-5 years) [8]. a variety of different machine learning algorithms and attribute ranking and selection methods have been applied to disease prognosis and prediction in the last decades. a comprehensive search was performed relevant to the use of machine learning algorithms in cancer receptivity, recurrence and survival prediction [2]. k. kourou et al. [2] presented predictive models based on various supervised machine learning techniques including support vector machines, bayesian networks, artificial neural networks, and decision trees as an aim to model cancer risk or patient outcomes. in their work, maciej zieba et al.[9], used boosted svm for predicting post-operative life expectancy. in their research, they applied oracle-based approach for extracting decision rules from the boosted svm in order to solve imbalanced data problems. sindhu et al. [1] used six classification approaches-naive bayes, j48, part, oner, decision stump and random forest-toanalyse thoracic surgery data and they found that random forest gives the best classification accuracy with all split percentages. another paper [10] have analyzed and compared the performance of four machine learning techniques (naïve bayes, simple logistic regression, multilayer perceptron and j48) with their boosted versions by different metrics. their results indicate that boosted simple logistic regression technique is generally better or at least competitive against the rest of four machine learning techniques with 84.53% prediction accuracy. 3. machine learning machine learning is a branch of artificial intelligence which utilizes statistical, optimization and probabilistic techniques that allows computers to “learn” from past examples and to detect hard-to-discern patterns from large, noisy or complex data sets. these techniques have become a popular tool in medical diagnosis, which can find and identify models and relationships between them from large, noisy or complex datasets[3]. the inputs are the information about the patient's age, gender, past medical history, past medical procedures, family medical history and current symptoms , while labels are the illnesses. in some cases, these inputs are missed because some tests haven't been applied to the patient, so we do not apply machine learning techniques unless we confirm that the patient will give us valuable information. if the medical diagnosis is wrong, decision may lead to a wrong or no treatment, so machine learning is extremely used to diagnose and detect cancer [4]. more recently, it has been widely applied in the field of cancer prediction and prognosis which are differ from cancer detection and diagnosis. there are three types of cancer prediction and prognosis: one of them is prediction of cancer receptivity. in this type, one is trying to predict the probability of cancer progression before occurrence of the disease. second type is the prediction of cancer recurrence by trying to predict the probability of redeveloping cancer after treatment and after a period of time during which the cancer cannot improved prediction of post-operative life expectancy after thoracic surgery 72 copyright ©2016 assa. adv. in systems science and appl. (2016) be detected. third type is the prediction of cancer survivability by trying to predict an outcome which usually refers to life expectancy, survivability, progression and tumor-drug sensitivity. these days, different types of cancer such as prostate, brain, cervical, esophageal, leukemia, head, neck, breast, and thoracic are appear to be compatible with machine learning prediction. the thoracic datasets is concerned with classification problem related to the post-operative life expectancy in the lung cancer patients[2,3,4] in order to improve machine learning techniques when the datasets have a large number of features or attributes, attribute ranking and selection is used to identify the most relevant attributes and remove the redundant and irrelevant attributes from the dataset. attribute ranking and selection algorithms can be divided into wrapper and filter methods. the wrapper methods select attributes based on an estimation of the accuracy according to target learning algorithm. after applying the learning algorithm, wrapper searches the feature space by removing some attributes and testing the effectiveness of attribute removing on the prediction metrics. the attribute which make important difference in learning process should be selected as high quality attribute, while filters methods estimate the quality of selected attributes independently from the learning algorithm. it depends on the statistical correlation between the set of attributes and the target attribute, since the value of correlation identify the importance of target attribute [6,11]. by using filtering methods attributes can be ranked independently, then according to the ranking result optimal subset of attributes can be selected [12]. 4. the proposed method 4.1 dataset description table 1. characteristic of dataset features. name description characteristics dgn diagnosis specific combination of icd-10 codes for primary and secondary as well multiple tumors if any nominal pre4 forced vital capacity fvc numeric pre5 volume that has been exhaled at the end of the first second of forced expiration fev1 numeric pre6 performance status zubrod scale nominal pre7 pain before surgery binary pre8 haemoptysis before surgery binary pre9 dyspnoea before surgery binary pre10 cough before surgery binary pre11 weakness before surgery binary pre14 t in clinical tnm size of the original tumor, from oc11 (smallest) to oc14 (largest) nominal pre17 type 2 dm diabetes mellitus binary pre19 mi up to 6 months binary pre25 pad peripheral arterial diseases binary pre30 smoking binary pre32 asthma binary age age at surgery numeric risk1y 1 year survival period t value if died binary 73 a. s. desuky, l.m. el bakrawy copyright ©2016 assa. adv. in systems science and appl. (2016) thoracic surgery data is dedicated mainly to elicit surgical risk for real-life clinical lung cancer patients. the data was collected retrospectively by mareklubicz et al. [13] at wroclaw thoracic surgery centre for consecutive patients –ages from 21 to 87 years old who underwent major lung resections for primary lung cancer in the years 2007–2011. the centre is associated with the department of thoracic surgery of the medical university of wroclaw and lower-silesian centre for pulmonary diseases, poland, while the research database constitutes a part of the national lung cancer registry, administered by the institute of tuberculosis and pulmonary diseases in warsaw, poland. the dataset includes 470 instances (70 true and 400 false) and 16 attributes with no missing values and binary valued class (death within one year after surgery – survival). 4.2 research methodology in this work, version 3.7.12 of weka (waikato environment for knowledge analysis) toolkit [14] has been used for analysis. it is the product of the university of waikato (new zealand) and it is licensed under the gnu general public license. weka is a popular suite of machine learning software written in java, also it provides access to sql database and process the result retrieved by a database query. we have run our experiments on a system with a 2.30 ghz intel(r) coretmi5 processor and 512 mb of ram running microsoft windows 7 professional (sp2). cross-validation (10 folds) has been used in this study to validate the results. in this model, the dataset is partitioned into complementary 10 equal sized subsets. the analysis is performed on 9 subsets (training) and validating the analysis on one subset (testing). ten rounds of cross-validation are performed and in each round another subset 2 through 10 used as testing dataset. the validation results are averaged over the ten rounds in the final phase. researchers in the machine learning field have proposed numerous attribute ranking and attribute selection methods. the main aim of these methods is to eliminate redundant or irrelevant attributes from the original set of attributes. in our work, we use the attribute ranking methods (information gain (ig) attribute evaluation, symmetrical uncertainty (su) attribute evaluation and relief-f (rf) attribute evaluation) information gain (ig) attribute evaluation [15] is used to evaluate the importance of an attribute by measuring the information gain with regard to the class. the bases of ig depend on entropy which measure the randomness of the system. information gain can be calculated by the following equation: attribute)| h(class-h(class) = attribute) ig(class, (1) where h is the entropy which stands for the greek alphabet eta. symmetrical uncertainty (su) attribute evaluation [16]is used to evaluate the importance of an attribute by measuring the symmetrical uncertainty with respect to the class. symmetrical uncertainty compensates for the inherent bias in information gain. symmetrical uncertainty is given by the following equation: e))h(attribut-h(class) /(attribute) ig(class,*2 = attribute) su(class, (2) relief-f (rf) attribute evaluation is used to rank the quality of features depending on how well their values differ from the cases that are close to each other. it is sensible to predict that a valuable feature should have different values between cases belong to different classes and have the same value for cases from the same class [17]. the aim of this paper is to analyze the effect of number of attributes on accuracy of machine learning techniques to solve the problem for prediction of the post-operative life https://en.wikipedia.org/wiki/gnu_general_public_license https://en.wikipedia.org/wiki/machine_learning https://en.wikipedia.org/wiki/java_(programming_language) https://en.wikipedia.org/wiki/complement_(set_theory) improved prediction of post-operative life expectancy after thoracic surgery 74 copyright ©2016 assa. adv. in systems science and appl. (2016) expectancy in the lung cancer patients. reducing the number of attributes and increasing the accuracy is required to minimize the computational time of prediction techniques. in this study, we used information gain, symmetrical uncertainty and relief-f as attribute ranking methods to reduce the number of attributes (from 16 to 13 attributes), then we examined the quality of techniques naïve bayes, simple logistic regression, j48, multilayer perceptron, and svm after applying the three ranking methods for prediction of post-operative life expectancy after thoracic surgery. the quality of the proposed methods is evaluated by comparing the performance of naïve bayes, simple logistic regression, j48 and multilayer perceptron techniques with and without using attribute ranking methods as first step. also, our proposed` methods is compared to boosted naïve bayes , boosted simple logistic regression, boosted j48, boosted multilayer perceptron and boosted svm. 5. experimental results performances of the methods were analyzed by using six metricsaccuracy, f measure, roc curve, gmean, tnr and tpr [9, 10]. accuracy is the percentage of observations that were correctly predicted by the method. it was used to evaluate the performance of each algorithm. n)+tn/(p+tp =accuracy (3) table 2. shows the confusion matrix which clarifies the prediction tendencies tp (true positive), tn (true negative), fp (false positive) and fn (false negative) of considered machine learning technique. table 2. confusion matrix predicted outcome p n actual value p tp fn n fp tn accuracy is not a reliable metric for the real performance of a machine learning technique, because it will yield misleading results if the data set is imbalanced (i.e. when the number of samples in different classes vary greatly). since thoracic surgery data is imbalanced data with 70 true and 400 false instances we used f measure (f1 score), roc curve, gmean, tnr and tpr. where, f measure was used to test the accuracy depending on harmonic mean of precision & recall. fn)+fp+2tp/(2tp = measure f (4) while, roc curve was also used as an effective method to evaluate the performance of predicted models by plotting the true positives against the false positives and area under the roc curve is used for predicting accuracy of models. the gmean (geometric mean) is a widely used quality rate and is defined as equation: tnr tpr=gmean  (5) where tnr (specificity or true negative rate) is described by: fp) + tn/(tn = tnr (6) and tpr (sensitivity or true positive rate) and described by the equation: fn) + tp tp/( = tpr (7) 75 a. s. desuky, l.m. el bakrawy copyright ©2016 assa. adv. in systems science and appl. (2016) table 3. shows the accuracy of naïve bayes, simple logistic regression, j48 and multilayer perceptron techniques with and without using attribute ranking methods. also, it shows the accuracy of boosted naïve bayes, boosted simple logistic regression, boosted j48, boosted multilayer perceptron for prediction of post-operative life expectancy after thoracic surgery. results show that using ig and su as ranking methods before applying naïve bayes gives the better accuracy than applying naïve bayes without using ranking methods and with boosted naïve bayes. also, in the case of applying simple logistic, using the three ranking methods gives better accuracy than applying simple logistic without using ranking methods and with boosted simple logistic. similarity, in the case of applying multilayer perceptron, using the three ranking methods gives better accuracy than applying multilayer perceptron without using ranking methods and with boosted multilayer perceptron. but in the case of applying j48 without using ranking methods gives better accuracy than applying j48 with using ranking methods and applying j48 with using ranking methods gives better accuracy than boosted j48. table 3. also shows that the simple logistic technique applied with the three ranking methods gives the best accuracy. table 3. prediction accuracy comparison of machine learning techniques using thoracic surgery data set ml techniques method accuracy naïve bayes original [10] 77.74 boosted [10] 78.32 su 82.12 rf 77.74 ig 82.13 simple logistic original [10] 84.55 boosted [10] 84.53 su 84.68 rf 84.68 ig 84.68 multilayer perceptron original [10] 80.91 boosted [10] 80.70 su 81.27 rf 81.28 ig 81.28 j48 original [10] 84.64 boosted [10] 79.34 su 84.46 rf 84.47 ig 84.47 table 4. shows the f measure and roc curve of naïve bayes, simple logistic regression, j48 and multilayer perceptron techniques with and without using attribute ranking methods. also, it shows the f measure, roc curve of boosted naïve bayes, boosted simple logistic regression, boosted j48, boosted multilayer perceptron for prediction of post-operative life expectancy after thoracic surgery. results show that applying naïve bayes without using ranking methods gives the better f measure than using the three ranking methods before applying naïve bayes and with boosted naïve bayes, but, boosted naïve bayes gives the best roc curve. in the case of applying simple logistic, it gives the same results for the f measure in all methods, but using the three ranking methods gives the best roc curve. in the case of applying multilayer perceptron, using su and ig ranking methods gives the best f measure and the best roc curve. in the case of applying j48, boosted j48 gives the best f measure but applying j48 with and without using ranking methods gives better roc curve than boosted j48. improved prediction of post-operative life expectancy after thoracic surgery 76 copyright ©2016 assa. adv. in systems science and appl. (2016) table 4. prediction measures comparison of machine learning techniques using thoracic surgery data set ml techniques method f measure roc naïve bayes original [10] 0.13 0.68 boosted [10] 0.12 0.60 rf 0.06 0.66 su 0.06 0.66 ig 0.07 0.67 simple logistic original [10] 0.00 0.53 boosted [10] 0.00 0.61 rf 0.00 0.50 su 0.00 0.50 ig 0.00 0.50 multilayer perceptron original [10] 0.22 0.60 boosted [10] 0.18 0.56 rf 0.20 0.58 su 0.24 0.55 ig 0.24 0.55 j48 original [10] 0.00 0.50 boosted [10] 0.18 0.61 rf 0.02 0.50 su 0.00 0.50 ig 0.00 0.51 table 5. shows the tpr, tnr, and gmean for support vector machine after applying the three attribute ranking and selection methods and boosted support vector machine. the rf gives the best prediction quality where it has the higher gmean value. it shows also that the proposed methods give better gmean and tnr than boosted svm but boosted svm gives better tpr. table 5. performance evaluation of boosted svm vs. svm with ranking methods method tpr tnr gmean boosted svm(bsi) [9] 60.00 72.00 65.73 svm (ig) 44.30 99.80 66.49 svm (su) 44.30 99.80 66.49 svm (rf) 51.40 99.80 71.62 6. conclusion in this study, the quality of three attribute ranking and selection methods has been evaluated to improve the prediction for life expectancy of lung cancer patients after thoracic surgery. five machine learning techniques before and after applying the attribute ranking and selection methods have been compared with their boosted versions. the results show that boosting is not always the better choice where attribute ranking and selection can perform better in improving prediction accuracy. other attribute selection and machine learning techniques can be introduced in the future work to gain a better prediction model performance of the dataset. references [1] v. sindhu, s. a. s. prabha, s. veni , and m. hemalatha, “thoracic surgery analysis using data mining techniques” , international journal of computer technology & applications , vol. 5, pp 578-586, may, 2014 [2] konstantina kourou, themis p. exarchos, konstantinos p. exarchos, michalis v. karamouzis, dimitrios i. fotiadisa, “machine learning applications in cancer prognosis and prediction”, computational and structural biotechnology journal, vol 13, pp 8-17, 2015. http://www.sciencedirect.com/science/article/pii/s2001037014000464 http://www.sciencedirect.com/science/article/pii/s2001037014000464 http://www.sciencedirect.com/science/article/pii/s2001037014000464 http://www.sciencedirect.com/science/article/pii/s2001037014000464 http://www.sciencedirect.com/science/article/pii/s2001037014000464 http://www.sciencedirect.com/science/article/pii/s2001037014000464 http://www.sciencedirect.com/science/article/pii/s2001037014000464 http://www.sciencedirect.com/science/journal/20010370 77 a. s. desuky, l.m. el bakrawy copyright ©2016 assa. adv. in systems science and appl. (2016) [3] kwetishe joro danjumal, “performance evaluation of machine learning algorithms in post-operative life expectancy in the lung cancer patients”, international journal of computer science issues, vol. 12, no. 2, pp. 189-199, 2015 [4] joseph a. cruz, david s. wishart, “applications of machine learning in cancer prediction and prognosis”, cancer informatics, vol. 2, pp. 59–77, 2006 [5] mehdi naseriparsa, amir-masoudbidgoli,tourajvaraee, “a hybrid feature selection method to improve performance of a group of classification algorithms”, international journal of computer applications, vol. 69, no 17, pp 28-35, 2013 [6] pinar yildirim, “filter based feature selection methods for prediction of risks in hepatitis disease”, international journal of machine learning and computing, vol. 5, no. 4, pp. 258-263, august 2015 [7] samina khalid, tehmina khalil, shamila nasreen, “a survey of feature selection and feature extraction techniques in machine learning”, science and information conference (sai), pp. 372-378, 2014 [8] koklu murat, humar kahramanli, and novruz allahverdi. “applications of rule based classification techniques for thoracic surgery”, tiim international conference, 27-29 may 2015, italy, pp 1991-1998. [9] m. zięba, j .m. tomczak, m. lubicz, and j. świątek, “boosted svm for extracting rules from imbalanced data in application to prediction of the post-operative life expectancy in the lung cancer patients” , applied soft computing, vol. 14, pp 99-108, jan. 2014. [10] md. ahasan uddin harun and md. nure alam, “predicting outcome of thoracic surgery by data mining techniques”, ijarcsse, vol. 5, no. 1, pp 7-10, 2015 [11] mark a. hall, “correlation-based feature selection for discrete and numeric class machine learning”, international conference on machine learning, pp 359-366, 2000 [12] vishal gupta, et al., “performance of various feature selection techniques under loaded networks”, international journal of computer applications, vol. 78, no. 2, september 2013 [13] uci machine learning repository. url: https://archive.ics.uci.edu [14] machine learning group at the university of waikato. url: http://www.cs.waikato.ac.nz/ml/ [15] c. sunil kumar and r.j. rama sree, “application of ranking based attribute selection filters to perform automated evaluation of descriptive answers through sequential minimal optimization models”, ictact journal on soft computing: special issue on distributed intelligent systems and applications, vol. 5, no. 1, october 2014 [16] jasmina novaković, perica strbac, dusan bulatović , “toward optimal feature selection using ranking methods and classification algorithms”, yugoslav journal of operations research, vol. 21, no. 1, pp. 119-135, 2011 [17] niket kumar choudhary, yogita shinde, rajeswari kannan, vaithiyanathan venkatraman, “impact of attribute selection on the accuracy of multilayer perceptron”, ijitkmi, vol. 7, no. 2, pp. 32-36, 2014 http://ieeexplore.ieee.org/search/searchresult.jsp?searchwithin=%22authors%22:.qt.khalil,%20t..qt.&newsearch=true http://ieeexplore.ieee.org/search/searchresult.jsp?searchwithin=%22authors%22:.qt.nasreen,%20s..qt.&newsearch=true https://archive.ics.uci.edu/ http://www.cs.waikato.ac.nz/ml/ advances in systems science and applications (2012) vol.12 no.1 27-37 study on the horizontal subgrade reaction of expressway protective guard pillar lu yang1, daiheng chen2 and bo xiao1 1shenyang university of technology school of civil engineering and architecture ,shenyang,110023 2tokyo university of science engineering department , tokyo ,102-0073 abstract it is quite necessary to describe the relationship between the horizontal subgrade reaction and the horizontal displacement with a simple form. this study provides the analysis of the horizontal subgrade reaction of protective guard pillar in the conflict between automobiles and protective guard pillars. making use of the relationship between soil load and horizontal load, we can build the theory of assessment of the coefficient of horizontal subgrade reaction, and then simulate the relationship between soil and column with spring in the help of the theory. in this paper, we use mohr-coulomb yield criterion and integrate abaqus finite element numerical simulations, to find that it is the difference between the greatest horizontal stress components σx and the smallest horizontal stress components σy that causes the fact that horizontal load leads to the plastic yielding of protective guard pillar and earth coupling model soil. and also, the paper provides the theoretical formula of coefficient of horizontal subgrade reaction and the ultimate bearing capacity of horizontal subgrade reaction, then build the theoretical model of coefficient of horizontal subgrade reaction when soil is under the action of horizontal force, to offer design and analysis reference for practical projects. keywords protective guard pillar, level of load, abaqus, level reaction site, mohr-coulomb yield criterion, numerical simulation 1 introduction the accident that automobiles collide guardrails is one of the main forms of road traffic accidents, and their collision has become main research direction of traffic safety[1-3]. in the analysis of building structures, spring is usually exerted in the foundation to simulate the relationship between soil and structures, so as to predict the coefficient of soil subgrade reaction in the system of building structures, and then we can ascertain spring coefficient. usually, the prediction method of coefficient of soil subgrade reaction refers only to semi-infinite space soil or beam on elastic foundation[4-5], so it’s necessary to revise the existing prediction method of coefficient of soil subgrade reaction, and ascertain its failure mechanism. this research makes use of the relationship between soil load and horizontal load, taking their elastic plastic behavior into consideration[6-7], and builds the prediction method of coefficient of soil subgrade reaction which is applied to buiding structure system, expecting that the research result is helpful 28 lu yang:study on the horizontal subgrade reaction of expressway protective guard pillar for the selection of coefficient of engineering soil subgrade reaction. take the columns of semi-rigid guardrail system as object of study, and research each component’s characteristics of deformation and energy-absorbing in the impact of shock. because of the fact that the column is rammed and buried into soil,and the nonlinear characteristics of soil’s height, in-depth study of the relationship between surrounding soil and column and also the characteristics of deformation and energy-absorbing of column is of great importance. this paper integrates finite element numerical simulations, providing the analysis of horizontal subgrade reaction and limiting uniform force,and finally simulates the relationship between soil and column with a series of horizontal springs, to build the theoretical model of coefficient of horizontal subgrade reaction when the soil is under the action of horizontal force;offers design and analysis references for practical projects. 2 previous research for coefficient of subgrade reaction ks piles’ horizontal resistance is represented by coefficient of subgrade reaction ks. coefficient of subgrade reaction ks is proposed by kinds of proposals. usually, the derived relational expression kh0 = α · e0 · d−3/4 acts as the coefficient of subgrade reaction when the benchmark displacement is 10mm,and we can represent the coefficient of subgrade reaction with the nonlinear curve of second order kh = kh0/y 1/2 of the relative displacement y between piles and the surrounding subgrade.among them,:determinate number=80 (for cohesive soil ,when e0 is derived by n,is 60). gold proposed that relation curve of horizontal load pmax and displacement could be represented approximately by three crease line, among them, y is the relative displacement of piles and surrounding subgrade, 1(y < 0.6mm),1/4(0.6mm < y < 10mm), 1/12(y > 10mm) acts as the slope of each line[8]. as can be seen from the above, kh’s character is that its value reduces gradually with deformation. seed and others analyzed many horizontal subgrade response results, and proposed that stress reduction factor should be decreased with depth, and show bandwidth viriation with depth[9]. the deeper, the wider. also, they suggested that when analyzing and designing, we could use average curve. the most important data of pile detail design includes coefficient of subgrade reaction ks that responses deformation behavior. in existing d sees the formula (2). esign, ks’s precise value is not as precise as other soil parameters (such as: undrained cohesion, internal friction angle, etc.). usually, we can estimate the relationship of horizontal subgrade reaction with young’s modulus es: non-sticky soil kh = 3×es/d, sticky soil kh = 1.6×es/d, d is pile diameter. in evaluating pile’s coefficient of horizontal subgrade reaction, its horizontal advances in systems science and applications (2012) vol.12 no.1 29 load’s horizontal displacement and the effect of horizontal subgrade reaction are pretty important, only in reasonable way to build appropriate design value, and it’s essential to operate the experiment of loading pile load horizontally or conduct the simulation of correct value. the traditional method is to conduct the experiment of pile’s destructive load, and it’s rather difficult in loading condition or test technology[10-11]. finite element method is the most appropriate method at this stage, through the material parameters obtained from laboratory testing, the mechanical behavior of making use of simulation analysis to determine the pile-soil interaction is worthy of wide concern. 3 numerical simulation analysis analyzing and calculating the factors that affect pile’s horizontal subgrade reaction and displacement correctly is of the utmost importance.this study simulates the contact problem of pile-soil interaction with abaqus finite element software, and makes use of the one by one contact algorithm in abaqus, and then build contact couple between the side of pile and the soil; for pile shaft, we can use elastic plastic model, and for soil, we can simulate with mohr-coulomb model; taking the effect of initial earth stress into consideration, we introduce soil lateral pressure coefficient to achive balance in the stage of ‘geostatic’, and this is very effective for simulation of geotechnical issues. 3.1 the establishment of finite element model of column soil in the geometric model, use large body to simulate semi-infinite space, and in the simulate calculation , the radius of the soil is much lager than the radius of the pile’s cross-section. as the picture 1 shown. in this model, the soil is built by eight-node entity reduced element, and others are four node shell element. the whole mode system includes 89158 nodes and 76656 elements, and material models are all elastic plastic materials that consider isotropic hardening. the main parameters that can be revised in desigh are as follows(see table 1). 3.2 interpretation of result mohr-coulomb damage and strength criteria is widely applied to geotechnical engineering. the constitutive model that is used in abaqus is the classical expansion of mohr-coulomb yield criterion[12]. at some point the role of the shear stress equal to the shear strength, the damage occurred, and the role of shear strength in the face of the linear relationship between normal stress; allow the material isotropic hardening or softening, however, the shape of the flow of the model potential function in the meridional plane is hyperbola, and there is no cusp in the π plane, potential function is therefore completely smooth, ensuring the uniqueness of the plastic flow direction. fig.2 is the curve of path and displcement for the soil in front of column at 30 lu yang:study on the horizontal subgrade reaction of expressway protective guard pillar fig.1 mesh model different loading time on the depth of the soil. as can be seen from figure, the displacement of the soil in front of the column is gradually increasing with the load; and is of inverse relationship with the buried depth. because of the role of soil bound, the region that column and soil doesn’t saperate, has the characteristic that displacement reduces with buried depth; at about 1.05 meters in depth accurs the phenomenon that column saperate from the soil. kh is the ratio of the distribution force destiny q and relative displacement y,that is q/y; q is obtained form the distribution stress in the surface of the pile, and here we should firstly study the stress σx of the point in front of pillar. fig.3 is the curve of the relationship between the path in depth and σx (σx = s11) , as can be seen, according to the horizontal load, s11 is negative value, and its absolute value is roughly linear and gradually become larger. fig.4 is the distribution force of the soil in front of pillar in depth, as can be table 1 model of material parameters pile length pile length 2.2 m buried in soil 1.5m elastic modulus 2.1e12 pile diameter 0.14m of column pa elastic coefficient of modulus of 5.0e+6pa soil lateral 0.538 soil pressure soil poisson’s ratio 0.35m friction angle 20 soil cohesion 3000pa expansion angle 0.1 advances in systems science and applications (2012) vol.12 no.1 31 seen, with the development of deformation, the latter phenomenon of spin-off accurs, so we can predict the theoretical starting point of the spin-off, and the following spin-off broadens gradually. 4 limiting distribution force qcr based on the study on limiting distribution force qcr, we can draw the qualitative conclusions as follows: fig.2 the depth way-deflection curve (1) the limiting distribution force qcr that varies with the the soil density γ, this function as shown in fig.5, and the performance is not obvious. especially in a lot of literatures, qcr is in proportional relationship with proportion γ, here the relationship qcr∞rgx isn’t performed very well. the relationship is as stated above, σx is the middle stress, and yield stress is related to σy and σz. that is to say, usually, when we consider coulomb earth pressure, we take it as plane strain issue, σz is middle stress, and coulomb limiting stress is determined by the difference of σy and earth pressure σx∞rgx and believe that σy∞rgx. the performance of the relationship is: the movement of the soil pile, is in threedimensional stress state in soil that in front of the pile. coulomb limiting stress is determined by the difference of σy and earth pressure σx. consequently, the function qcr∞rgx can’t exist. (2) about the relationship with cohesion c, according to the research and analysis into numerical simulations, we can gain a deeper knowledge: no matter big or small the depth x is, qcr will become bigger with the cohesion c. consequently, the expression of qcr is: 32 lu yang:study on the horizontal subgrade reaction of expressway protective guard pillar fig.3 depth way-load direction curve of pressure and resistance fig.4 depth way-distributed force relational graph qcr = qcro + cc × x (1) qcr can be ascertained from the following expression qcro ∼= 2 cosϕ 1− sinϕ cb (2) meanwhile, the relationship between coefficient cc and cohesion c is ascertained.for previous research on the construction of pile, coefficient cc is not dependent on cohesion, but has founction relationship with friction angle ϕ in 3kprg form and density γ. advances in systems science and applications (2012) vol.12 no.1 33 fig.5 r= 1800 and r= 900situation q/e0b and v/b relations fig.6 in logarithm graph q/e0b and v/b relations based on the research above, the approximate expression of limiting force qcr is: qcr ∼= 2 cosϕ 1− sinϕ b(c+ 0.24k1k2rg 1 + sinϕ 1− sinϕ x b ) (3) 34 lu yang:study on the horizontal subgrade reaction of expressway protective guard pillar k1 and k2 are the modified coefficient of soil proportion γ and cohesion c. k1 = 0.3 + 0.8× r0 r , r0 = 1800kg/m3 (4) k2 = 0.5 + 0.5× c c0 , c0 = 3000pa (5) 5 the proposal of the relationship of horizontal subgrade distribution force q and horizontal displacement v the purpose of this study is to propose the analysis of the simple model of horizontal subgrade reaction in the conflict between automotives and protective guard pillars. therefore, based on the statement above, it’s quite essential to describe the relationship between the horizontal subgrade distribution force q and the horizontal displacement v with a simple form. this paper researched the logarithmic chart that is shown in fig.6, and express the relationship between q and q approximately in the chart with three lines. in the logarithmic chart, firstly, for initial elastic deformation, can be expressed approximately as the line whose slope λ = 1; secondly, the latter non-elastic deformation can be expressed approximately as the line whose slope λ = 1/2; at last, if horizontal subgrade distribution force q reaches limiting force qcr, then it can be expressed approximately as the line whose slope λ = 0 . from the fig.6 we can see: in the early stages of non-elastic deformation, horizontal subgrade distribution force q’s limiting resistance is qcro in the place where x = 0, sees the formula (2). however, we can see from the chart, when the depth x is big, the starting point transit form the line whose slop λ = 1 to the line whose slope λ = 1/2, when we calculate horizontal subgrade reaction qcr, we can use the following expression : qre = p2 × [qcro + 0.1(qcr − qcro)] (6) qcr can be ascertained by it. therefore, q can be ascertained by the following expression : q =  p1 × c1e0v (v ≤ vre) cxe0 √ vb (vre ≤ v ≤ vcr) p2 × qcr (vcr ≤ v) (7) p1 and p2 in equation (6) and (7) modified coefficients when considering the friction between soil and pillar. when the friction coefficients µ = 0, p1 = p2 = 1; when µ ̸= 0, the calculation of p1 and p2 will be stated later. the unknown quantities vre, cx and vcr in equation (7) can be ascertained as follows: firstly, when v reaches vre, q reaches qre from the first expression in equation advances in systems science and applications (2012) vol.12 no.1 35 (7) we can obtain: vre = qre p1c1e0 (8) and also, when v = vre from the first and second expressions in equation (7)we can obtain: cx = qre e0 √ bvre (9) so when v = vcr, from the second and third expression in equation (7)we can obtain: vcr = ( p2qcrcxe0 )2 b (10) the predicting value according to the expression (3) is quite closed to the abaqus numerical simulations, therefore, the theoretical model of the horizontal resistance of the protective guard pillar in subgrade is feasible. 6 conclusion as is shown in the research result, with the help of the abaqus software, we built the coupling model of pillar and soil under the action of horizontal load, and analyzed soil’s mechanical behavior when pillar is under the action of horizontal load, and also, proposed theoretical expression about the coefficient of horizontal subgrade reaction. the simulation results fit the trend of the literature[13] experimental data very well, indicating that abaqus has a great ability in dealing with highly nonlinear issues of the interaction between pillar and soil. (1) along the depth of the pillar, the coefficient of horizontal subgrade reaction is not certain, in the region more nearer to the surface, the value is relatively smaller, and this is the result that the horizontal resistance is small for the region near the surface . (2) in pillar’s slewing deformation, the coefficient of horizontal subgrade reaction is not certain too. it will become smaller with the development of the deformation. this is because the effects of plastic deformation become greater gradually. (3) according to the argument above, using mohr-coulomb yield criterion, we can find that soil plastic yielding damage is caused by the difference between σx and σy. because of the action of horizontal load, σx that is s11) is negative value, and its absolute value is roughly linear and gradually become larger, and σy (that is s22)’s absolute value gradually become smaller. the greatest stress component is σx, and the smallest stress component is not σz (that is s33) which in gravity direction ,but is σy nd the cause of the entire yielding is the difference between σx and σy. (4) proposed the theoretical expressions of the limiting distribution force qcr 36 lu yang:study on the horizontal subgrade reaction of expressway protective guard pillar and horizontal subgrade reaction q and horizontal displacement v . acknowledgements supported by key laboratory of geological hazards in three gorges reservoir of ministry of education area, china three gorges universityunder grant no.2008kdz10. especially thanks to professor chen from tokyo university of science for his instruction and help when the author worked at tokyo university of science engineering department during mar.2008-mar.2009. references [1] nathaniel r. seckinger, a.m.asce, akram abu-odeh. (2005), “performance of guardrail systems encased in pavement mow strips”, journal of transportation engineering , asce / november, pp.851-860. [2] atahan a.o, and ross h. e, jr. (2004), “computer simulation of recycled content guardrail post impacts”, transp. eng., pp.733-741. [3] tabiei a and jin w. (2000), “roadmap for crashworthiness finite element simulation of roadside safety structures”, finite elem. anal. design, pp.145157. [4] yuanshun bi huanren zhang, “tilted bored pile the foundation system of economy and safe”, the user convention of germany bao e mechanic equipment ltd. [5] xunshan zhan, jianzhi wang, guanxiong wang. (2001), “mechanics parameter of gao xiong jie yun red-line soil layer”, proceeding of 9th geotechnical engineering proseminar (paper no. a031), august 30-31 and september 1. [6] mizuue oozutsuchiya maneka. (2002), “the analysis of horizontal subgrade reaction”, proceeding of 37th subgrade proseminar, pp.1477-1478. [7] kaimi seiyanakai teruokido hiraki. (2002), “the actual locale experiment of the horizontal load”, model proceeding of 37th subgrade proseminar, pp.1467-1468. [8] seed h.b and peacock w.h. (1971), “test procedures for measuring soil liqufaction characteristics”, journal of the soil mechanics and foundations division, asce, vol.97 no.sm8, proc., paper 8330, pp.1099-1119. [9] torinami shinsuke, tomi kougi, “the non-linear analysis of horizontal subgrade reaction and pile foundation”, proceeding of japan civil engineering proseminar(11), issn, pp.61-64. advances in systems science and applications (2012) vol.12 no.1 37 [10] kanaako naoru, tomi kougi, kokuhu taiyou. (2004), “experiment of lateral resistance of pile foundation and horizontal subgrade reaction”, proceeding of subgrade proseminar, vol.jgs39, pp.1515-1516. [11] cishun chen, weixhao xu, xinjie lv. (2000), “subgrade reaction coefficient of layer soil system”, journal of china mechanic academy. vol.16, no.1, pp.69-80. [12] jinchang wang, yekai chen. (2006), the usage of abaqus in civil engineering, zhengjiang university press. [13] suzuki yasuji. (2002), “the horizontal load experiment of subgrade reaction model”, proceeding of 37th subgrade proseminar, pp.1469-1470. advances in systems science and application (2015) vol.15 no.3 220-232 labour market model — focused on final products a.oinarov1, s. baizakov2, n.baizakov3 and jeffrey forrest4 1 the board of jsc “kazakhstan public-private partnership” 2 economic research institute, str. temirkazyk 65, astana, 0100000, kazakhstan; 3 association for “leadership in education development”, 36 saraishyk street, apt 105, astana, 0100000, kazakhstan; 4 school of business, slippery rock university, slippery rock, pa 16057, usa. abstract this paper studied history of development of commodity and cash flows balancing models. keynesian model , monetary model and mundell-fleming model are investigated, the issues of mutual consistency of indicators of economic growth is studied. at the same time , the mathematical formulation of market equilibrium of levels of production, employment, incomes and prices is gives. a new model of labour market focusing economy on the final product is established. keywords labour market; gross output; labour-intensity of products; interindustry balance; final product 1 market imbalance theory is an instrument to identify macroeconomic imbalances this section studied history of development of commodity and cash flows balancing models. there are assumptions that adequately reflected development of market forces of economic development. relationship between indicators of economic growth in both sectors of the economy was established. if keynesian model developed a balance between real economy and financial economics by using only one indicator, then monetary model used two, and mundell-fleming model used three indicators of economic growth. it also studied the issues of mutual consistency of indicators of economic growth. historically, analysis of production repeatability since 40-ies of the last century was conducted based on keynesian model, which assesses the market equilibrium between commodity and cash flows. using this model as an anti-crisis tool to eliminate the consequences of the great depression of the us economy in the 1930s is associated with the name of the us president f. roosevelt. circuit diagram of the replenishment cycle of this model is shown in figure 1. according to keynesian model we may admit the constancy of prices for goods and services, and consider investment growth and increase in other consumption costs as an impetus for economic development by the state. as prices for goods and services are constant, commodity and cash flows balancing based on this model is carried out due to one indicator nominal gdp. in other words, economic growth indicator is nominal gdp, which serves as an indicator of real advances in systems science and application (2015) vol.15 no.3 221 economic growth, as prices are constant, and volumes of real gdp and nominal gdp are the same because of constant prices. fig.1 principle circuit diagram of the replenishment cycle of keynesian model fig.2 principle circuit diagram of the replenishment cycle of monetary model in the mid-80s of the last century a model of economic management has 222 a.oinarov, s. baizakov, n.baizakov and jeffrey forrest: labour market model — focused... changed. keynesian model was replaced by more progressive model a model of monetary policy. the reason of progressivity of monetary model was that it considered variability of prices of goods and services that was clear in real life and which is a key advantage of the market economy.conditions of goods and services are rapidly renewed because of the influence of market competition of goods and services, and there will be their new prices, and prices of previous goods and services will change due to scientific and technological progress. monetary model in production repeatability analysis started to be actively used since the 1980s. its appearance on practice of economic management of the country is associated with the name of the president of r. reagan. monetary model means constant money velocity and it uses two gdp indicators: nominal and real. a real gdp is defined in prices of basic year and characterizes the dynamics of market forces of real economy. a nominal gdp is defined in prices of the year and characterizes the dynamics of market forces of monetary and financial system. gdp deflator connects their balancing between each other. but later it is turned out that a weak point of the model of monetary policy is assuming the constancy of money velocity, which is considered as a basis for monetary model. fig.3 principle circuit diagram of the replenishment cycle of monetary model advances in systems science and application (2015) vol.15 no.3 223 multinational companies and companies with developed foreign trade relations started to see the narrowness of this model during their economic analysis. in this regard, since the 1990s of the last century fleming mundell model was used, which overcame limitations of the monetary model. the advantage of fleming mundell model is that it admits that money velocity is dynamic. however, this model again assumes the constancy of prices for goods and services. production repeatability analysis according to fleming mundell model (since 1990) considers dynamics of exchange rate, which serves as a reliable development instrument of international relations of the world countries. it uses three gdp indicators: nominal, real and gdp based on purchasing power parity; figure 3 shows a principle circuit diagram based on fleming mundell model. firstly, mundell-fleming model assumes that prices are constant, and secondly, it links three gdps to each other. to ensure conceptual relationships of three gdps and stop assuming that prices for goods and services are constant, market equilibrium of levels of production, employment, incomes and prices was established. fig.4 principle circuit diagram of the replenishment cycle of kazakhstan model of double regulation 224 a.oinarov, s. baizakov, n.baizakov and jeffrey forrest: labour market model — focused... a new model of market equilibrium of levels of production, employment, incomes and prices in employment terms summarizes keynesian, monetary and fleming mundell models, and do not contradict their structure principles, but develops them. 2 economic content of kazakhstan model of double regulation this section gives mathematical formulation of market equilibrium of levels of production, employment, incomes and prices.. a new model of labour market focusing economy on the final product is established, which is shown in the example of the republic of kazakhstan. it explains why the market forces of labour and capital are developing according to the laws of free competition despite wishes of entrepreneurs to increase competitiveness of their companies. an article of american scientists was published in the column ”new world order” in magazine ”foreign affairs” about ”labour, capital and ideas in economy of power laws”, which states that the future belongs not to those who provide cheap labour or owns capital (they will be driven out by automation), but to those who are willing to introduce innovations and create products, services and business models based on them. new technologies do not only integrate existing sources of labour and capital, but also generate products that replace both labour and capital. according to the authors of the article, innovations in the nearest future, will lead to great changes, associated with distribution of incomes, as profit sources will be represented by high technology and globalization, in which this process is usually demonstrated as power laws. in this case, an employee in the usa will receive the same salary as employees working in the industrial sector in china or india. this is good news for developing countries, and problem for the united states and europe, where there are no possibilities of reducing production costs and monetary growth.can kazakhstan take advantage of the situation in the new world order? economic research institute (kazakhstan) applies a system of techniques to do forecasting and analysis. innovative model of the labour market plays an important role in it, which is focused on the final result. top modules of balance labour model are coefficients of direct labour intensity of products. to measure labour intensity we can use the unit of man-hours, man-years or jobs, and for their calculation we can use the table ”input output”, which are produced in kazakhstan every year: ti = li xi , i = 1, n (1) where ti coefficients of direct labour intensity of products; li number of people employed in the economy; xi gross output. advances in systems science and application (2015) vol.15 no.3 225 to measure labour intensity we can use the unit of man-hours, man-years or jobs. full coefficients of labour intensity per unit of production are calculated by coefficients of direct labour intensity: tj = n∑ i=1 aijti + tj , i, j = 1, n (2) where aij input-output coefficient i type of economic activity for issue j type of economic activity; ti, tj full labour intensity coefficients for production of yi or yj component of final product y ; if we consider that coefficients of direct t = (t1, t2, . . . , tn) and full t = (t1, t2, . . . , tn) labour intensities are raw vectors, then the ratio (2) can be written in matrix form: t = ta+ t (3) or t = t(e −a)−1 (4) where e is identity matrix, a is matrix coefficient, representing production process technology of this year. (e −a)−1 input coefficient matrix of production processes represented by full material costs in the table of interindustry balance. labour intensity turns into full material costs in accordance with duality principle of koopmans and kantorovich. if we introduce a notation = (e −a)−1 then formula (4) can be rewritten as follows: t = tb (5) as it is seen from the formula, the only link between direct and full labour intensity indicators of products is still input coefficient matrix of production processes. it means that coefficients of of full labour intensity of each goods and services provided are very important, which reflect the influence of production processes. if we consider l as a number of jobs in all sectors of national economy and apply module (1), we will get the following formula to calculate a number of people employed in the economy: l = n∑ j=1 lj = n∑ j=1 tjxj = tx (6) where x is gross output of goods and services. if in formula (1) we marked a number of people employed in the economy by the components of raw vector li, then we can have the record shown in formula 226 a.oinarov, s. baizakov, n.baizakov and jeffrey forrest: labour market model — focused... (6) by representing the same number of people employed in the economy as that of components of raw vector j. if we multiply both sides of (5) by y-final vector of production, we can have the following model of the labour market:= tby . and if we take into account the formula (6), then balancing model will have a complete form, where the basic law of labour market and capital is reflected: tx = ty (7) on the other hand, a similar assessment on the level of national economy of the country represents the average value formed by the ratio of nominal gdp to gross output: c = t t = y x left side (7) is equal to ty where t according to formula (5) is full labour intensity of products, nd y is the final product. and the cost of the final product is considered to be equal to nominal gdp. right side (7) is equal to tx where t according to the module (1) is direct labour intensity of products, nd is gross output of goods and services provided. based on this simple formula the yield of gdp (cdp (t)) is estimated as per turnover unit x(t). in fact, the gross output value x(t), which was correctly identified by famous american scientist harrington emerson in his main work “twelve principles of productivity” (1911), at business level it is ”total expenditure = material costs + labour costs + capital costs. or: total expenditure =qp+tw+tr; where compensation for intermediate materials, and other current material resources, used during production of goods is qp; remuneration of labour costs of employees is tw; remuneration of fixed capital is tr. thus, there is relationship between the final product and overall costs to create it x. so, by putting in place of t in the formula (7) its value from the formula (1) we will have t = l/x. then we will have l = ty . it means that a number of people employed in the economy is equal to multiplication of full labour intensity of products by nominal gdp. and full labour intensity of products is t , which in turn, is determined by multiplication of direct labour intensity of products t to full costs matrix.b = (e −a)−1. multiplicationt x according to (6) alsomeans number of working time fund used or number of jobs on economic activity, which are required to produce gross output (x), necessarry for production of the final product (y ). on the one hand, development model of the labour market (7) defines the advances in systems science and application (2015) vol.15 no.3 227 marginal valuation of the ratio of direct and full labour intensity of products for every type of economic activity, on the other a similar assessment at the level of national economy of the country represents the average value, formed by the ratio of nominal gdp to gross output: t/t = y/x (8) a number of people employed in the economy of state is similar to the total number of employees in all economic activities. therefore, new interpretation of the main situation of the market in labour terms is calculated by the model (7), which is completed by equation (8). in this case, the share of direct labour intensity of goods and services in their full labour intensity is equal to the level of production of the final product (y ) in the gross output (x). as a rule, destructive forces of laws of market equilibrium appear in macroeconomic imbalances, which are easily detected by analytical tools built on the basis of the table of interindustry balance. thus, according to the above formula (5), full material costs matrix allows to turn actual labour used during production of gross output (x) into abstract labour. however, this abstract labour, expressed in full labour intensity of products is needed to evaluate the effectiveness of production of a particular amount of the final product (y ). as defined in formula (1), direct labour intensity is expressed by the ratio of a number of people employed in every economic activity to the total volume of gross output, issued in this sector. multiplication of raw vector of direct labour intensity of a particular type of human activity by inverse matrix of material costs expresses a column vector of full labour intensity of production of the final product (y)). figure 5 shows a piece of values of direct and full labour intensity of products for certain types of economic activity of kazakhstan for 2012. as we can see from the diagram, the first indicator reflects the specific amount of jobs (man-years, man-hours, man-days) required to produce 1 million tenge worth of gross output. for example, in agriculture, hunting and forestry, it amounted for 965 jobs, and in production area of crude oil and natural gas only 10. figure 5 also shows that full labour intensity is greater in number than direct labour intensity. this is due to the fact that the first indicator reflects abstract labour that expresses the social form of unit cost of the final product produced, and the second represents economy of a particular type of human activity. here, the “black box”, which allows you to transfer direct labour intensity (t) into full (t ) is represented by inverse matrix of interindustry balance (b), t = t ∗ b. it also makes it possible to determine rapid development of technological improvement of production, since it is its dynamics (τ) that is a factor defining the rapidness of this process. that is why the advantages of new kazakh model, focused on 228 a.oinarov, s. baizakov, n.baizakov and jeffrey forrest: labour market model — focused... the final result, are provided on the basis of analytical calculations, conducted using reverse (τ)-matrix. the results are shown in table 1. fig.5 diagrams of the values of direct and full labour intensity, defined on the basis of the table of interindustry balance of kazakhstan for 2012. the first three columns show the components of final demand, translated into jobs, and the fourth column reflects total results. thus, we see that to cover final consumption, to have gross savings, and to attract export of the national economy we will require 6903.22, 1907.75 and 2227.89 thousand jobs, respectively. in general, the final demand (fourth column) can be satisfied if kazakhstan will have more than 11 million jobs. since supply of the country’s economy, according to the data of interindustry balance of 2012, is amounted to just tx = 8507000 jobs in total, then to cover the final demand of the economy additional capacity will be fully required, which must be repaid by import financing of foreign goods (the fifth column). in general, as we can see from the sixth column, kazakhstan’s jobs defined on the basis of labour intensity of product in calculation equal to 1 million tenge worth gross output ( by economic activity) significantly deviated from the jobs defined on the basis of full labour intensity (y ). however, their ratio (direct labour intensity 0,177 jobs, full 0.298) in terms of the national economy is equal to the ratio of scientific and technological excellence of real sector of the country, because we have the equation: c = t t = l x + l y = y x (9) advances in systems science and application (2015) vol.15 no.3 229 index value of the level of scientific and technological excellence of real sector of the country for kazakhstan is 0.587. this equation (9) of market equilibrium of levels of production, employment, incomes and prices in man terms is based on the balance of aggregate demand and aggregate supply. final demand in the economy of the country is generally determined by total costs of final consumption of households (y 1), general government costs (y 2), non-profit organizations (y 3), changes in inventories (y 4), purchase minus retirement of valuables (y 5), gross saving of fixed capital (y 6) and export (y 7). final demand = y 1 + y 2 + y 3 + y 4 + y 5 + y 6 + y 7 an aggregate supply, which is used to cover it, is defined as total of final demand and import with a minus sign (−y 8): final product = ∑ y i− y 8 thus, market equilibrium of levels of production, employment, incomes and prices is carried out by all agents of production, exchange, distribution and consumption. for example, to measure performance of the economy in working time we have the results of final demand in kazakhstan: 11,005,000 people. (8.505 million of which is covered by human resources within the country, and 2.5 million beyond the country). volumes of aggregate demand and aggregate supply are balanced, and they are equal to the work of 11,005,000 people, while direct labour intensity of products for production of 1 million tenge worth gross products in kazakhstan in 2012 amounted to 177 people, and full labour intensity 298 people. this equilibrium in formula (9) is defined in current prices and it expresses a market equilibrium. to assess the level of economic development of the country comparison base in comparable prices or quantum indexes should be determined based on this level. since nominal gdp (ngdp) is monetary terms of the final product and is defined in prices of this year, then, multiplying both sides of a well-known equation of gdp deflator: ngdp = b∗rgdp by purchasing power (pp), real value of the final product (q) can be defined, which corresponds to the balance of supply and demand in the country: q = pp ∗ngdp = c ∗rgdp, where c = pp ∗ pb = rgdp x ∗ ngdp rgdp = rgdp x (10) the main indicator of equation (10) is an indicator of scientific and technological progress (stp) c. its multiplication by real gdp is identical to multiplication of nominal gdp with purchasing power. it turns out that stp’s contribution helped to divide gdp deflator into two indicators. one of them 230 a.oinarov, s. baizakov, n.baizakov and jeffrey forrest: labour market model — focused... is purchasing power, the other is coefficient of scientific technological progress itself. as a result, the formula (10) balances the final product both in terms of money and terms of commodity. it is developed by generalizing and developing the equation of exchange of monetarists, which do not contradict its structure principles, but there are variables that are also represented by purchasing power and an stp index at real gdp. it means that the cost of the final product is equal to the nominal value of the product, i.e.q = pp*ngdp. therefore, multiplying both sides of equation ngdp = pb ∗ rgdp by purchasing power, a balance between nominal and real gdp can be set with weighing coefficient pp and c, respectively: pp ∗ngdp = c ∗ rgdp (11) it explains the viability of creative formula of scientific and technological progress: ċ c = ẏ y − ẋ x (12) rapid change of stp coefficient in equation (12) depends on difference in speed of the final product and gross output. let us assume that principle of using direct and full labour intensity to determine technical and technological excellence of the national economy is applied in the countries all over the world, which have developed foreign economic relations. then value of the dollar, euro, kazakhstani tenge and other currencies will be established according to the level of economic development within the state itself and its foreign relations. in this case, assessment principle of stp level deviations between the countries can be preserved, and real exchange rates of the national currencies will be set in accordance with rules of sdr design. such an approach can serve as an effective tool for building a methodology for sustainable development of the global economy: measured economy is managed economy. in conclusion, it can be summarized that it is possible to construct an analytical model for managing a market economy. it can be built in labour terms using the principle of double regulation and the table “input-output”. thus, balances of commodity and cash flows on economic activity in labour terms are formed with imbalances: ti ∗xi − ti ∗ yi = ±θi and at the level of the national economy of the state commodity and cash flows are always balanced: ∑ ti ∗xi − ∑ ti ∗ yi = 0 where components of final demand are yi defined, and their total value is the final demand, which corresponds to the final product of the country. advances in systems science and application (2015) vol.15 no.3 231 t a b le 1 b al an ci n g m o d el of th e la b o u r m a rk et o f k a za k h st a n a cc o rd in g to d a ta o f th e in te ri n d u st ry b al a n ce of 20 12 (p ie ce ). 232 a.oinarov, s. baizakov, n.baizakov and jeffrey forrest: labour market model — focused... these balance equations fully meet the conditions of globalization of the world economy and can be used to ensure market equilibrium of the levels of production, employment, incomes and prices anywhere in the world. the construction of an analytical model of market forces for development of real sector and financial system based on parity of their autonomous and parallel operation is the final result of this study. corresponding author s.baizakov can be contacted at: baizakov37@mail.ru advances in systems science and applications (2014) vol.14 no.3 294-301 estimates of the relative deviation for the equity home bias yirong ying1, ye ying1 and jeffrey forrest2 1 college of economics, shanghai university, shanghai, 200444, china 2 department of mathematics, slippery rock university, slippery rock, pa16057, usa, abstract there are a variety of explanations of equity home bias, the underlying difference being that different factors are emphasized, such as trade costs that impede international diversification, and the possibility that terms of trade responses to supply shocks provide international risk sharing. we found that with the ratio of world consumption of home vs. foreign goods and the ratio of world demand for home vs. foreign goods used for physical investment, we have three different estimates of the relative deviation about terms of trade, and then figured out three expressions representing these estimates. what we concluded helps to uncover the puzzle of the equity home bias. keywords international equity and bond portfolios, capital flows, current account 1 introduction though international capital flows have rallied with the liberalization of the global capital market two decades ago, equity home bias still exits in all industrial economies. abundant researches have been contributed to solve this puzzle. there are two major explanations of the continuously sizable equity home bias. the first one emphasizes on transaction costs and information barriers in cross-border financial transactions and suggests that international risk sharing is insufficient. the second one emphasized on the possibility that terms of trade changes in response to supply shocks may provide international insurance against these shocks, so that even a portfolio with home bias delivers efficient international risk sharing. seen from these very different opinions, where we start the work can make a big difference to what we conclude about the equity home bias. much published research made use of statistical methods and empirical analysis to interpret the home bias under capital account liberalization by transaction costs, information barriers and financial openness (recently from behavioural finance, cultural factors and legal system perspectives). ferreira and miguel, bekaert and wang and mondria and wu attributes much of home bias to information, familiarity and capital market openness [1-3]. another class of literature tried to decide home bias determinants through characterization of steady-state equilibrium portfolios in a general equilibrium model. engel and matsumoto analysed international equity portfolio choices in a advances in systems science and applications (2014) vol.14 no.3 295 model with money, sticky prices and trade in bonds [4-5]. under price stickiness, the short-run output is fixed, so that a positive productivity shock leads to a fall in employment and labor income, but an increase in profits. ownership of local equity is thus an effective hedge against labor income risk. heathcote and perri investigated the importance of physical investment for equity portfolios for the first time [6]. the hp model only generates realistic equity home bias if the terms of trade respond strongly to total factor productivity shocks. since the empirical evidence concerning the response of the terms of trade to technology shocks is mixed, it is important that our model does not require strong terms of trade effects of productivity shocksnevertheless, there is sizable equity home bias. the main contribution of this paper is that with the ratio of world consumption of home vs. foreign goods and the ratio of world demand for home vs. foreign goods used for physical investment, we have three different estimates of the relative deviation about terms of trade, and then figured out three expressions representing these estimates. 2 model we consider two symmetric countries, home (h) and foreign (f ), each with a representative household. country i = h,f produces one good using labor and capital. goods and financial assets (stocks and bonds) are traded in perfectly competitive markets. country i is inhabited by a representative household who lives in periods t = 0, 1, 2.... the household has the following life-time utility function: e0 ∞∑ t=0 βt ( c1−σ i,t 1− σ − l1+ω i,t 1 + ω ) (1) with w > 0. ci,t is country i’s aggregate consumption in period t and li,t is labor input. like much of the macroeconomics and finance literature, we take the coefficient of relative risk aversion to be greater than one: σ > 1. ci,t is a composite good given by: ci,t = [ a1/φ ( cii,t )(φ−1)/φ + (1− a)1/φ ( cij,t )(φ−1)/φ ]φ/(φ−1) , j ̸= i (2) where ci j,t is country i’s consumption of the good produced by country j at time t. ϕ > 0 is the elasticity of substitution between the two goods. in the (symmetric) deterministic steady state, a is the share of consumption spending devoted to the local good. we assume a preference bias for local goods, 1/2 < a < 1. the welfare-based consumer price index that corresponds to these preferences is: pi,t = [ a(pi,t) 1−φ + (1− a)(pj,t) 1−φ ]1/(1−φ) , j ̸= i (3) 296 yirong ying: estimates of the relative deviation for the equity home bias where pi,t is the price of good i. likewise, the associated investment price index is: p i i,t = [ ai(pi,t) 1−φi + (1− ai)(pj,t) 1−φi ]1/(1−φi) , j ̸= i (4) 3 relative deviation estimation coeurdacier, kollman and martin (2010) introduced a concept in their research on equity home bias, relative deviation. ∧ zt = (zt − z)/z denotes the relative deviation of a variable zt from its steady state value z. here, variables without a time subscript refer to the steady state. chh,t + cfh,t = p−φ h,t [ ach,tp φ h,t + (1− a)cf,tp φ f,t ] , cff,t + chf,t = p−φ f,t [ acf,tp φ f,t + (1− a)ch,tp φ h,t ] coeurdacier et al. (2010) derived from the above first-oder conditions that: yc,t ≡ chh,t + cfh,t cff,t + chf,t = q−ϕ t ωa [( pf,t ph,t )ϕ cf,t ch,t ] , with ωz(x) ≡ 1 + x ( 1−z z ) x+ ( 1−z z ) (5) where yc,t is the ratio of world consumption of home goods over world consumption of foreign goods, while qt ≡ ph,t/pf,t denotes the country h terms of trade. the ratio of world demand for home vs. foreign goods used for physical investment yi,t ≡ ihh,t+ifh,t iff,t+ihf,t can similarly be expressed as: yi,t ≡ q−ϕi t ωai (p i f,t p i h,t )ϕi if,t ih,t  (6) coeurdacier et al. (2010) found a zero-order portfolio such that the ratio of home to foreign marginal utilities of aggregate consumption (c−σ h,t/c −σ f,t ) is equated to the consumption-based real exchange rate (rert ≡ ph,t/pf,t), up to the following first-order condition: −σ ( ∧ ch,t− ∧ cf,t ) = ∧ rert (7) which is a linearized version of a risk sharing condition that holds under complete markets. it follows from the definition of home and foreign cpi indices (see equation 3) that: ∧ rert = ∧ ph,t− ∧ pf,t = (2a− 1) ∧ qt (8) advances in systems science and applications (2014) vol.14 no.3 297 due to consumption home bias (a > 1/2), an improvement of the home terms of trade leads to an appreciation of the home real exchange rate. lemma 1 if α, β are non-zero constants and α+β ̸= 0, then we have for continues functions f1 and f2 that ∧ (αf1(qt) + βf2(qt)) = αf1(q) αf1(q) + βf2(q) ∧ f1(qt) + βf2(q) αf1(q) + βf2(q) ∧ f2(qt) (9) proof : ∧ (αf1(qt) + βf2(qt)) = αf1(qt) + βf2(qt)− (αf1(q) + βf2(q)) αf1(q) + βf2(q) = α (f1(qt)− f1(q)) + β (f2(qt)− f2(q)) αf1(q) + βf2(q) = αf1(q) · ∧ f1(qt)+βf2(q) · ∧ f2(qt) αf1(q) + βf2(q) lemma 2 for continues functions f1 and f1, we have that ∧ (f1(qt) · f2(qt)) = ∧ f1(qt) + ∧ f2(qt) + ∧ f1(qt) · ∧ f2(qt) (10) proof : ∧ (f1(qt) · f2(qt)) = f1(qt)f2(qt)− f1(q)f2(q) f1(q)f2(q) = (f1(qt)− f1(q)) f2(qt) + f1(q)f2(qt)− f1(q)f2(q) f1(q)f2(q) = f1(qt)− f1(q) f1(q) · f2(qt) f2(q) + f2(qt)− f2(q) f2(q) = ∧ f1(qt) ( 1 + ∧ f2(qt) ) + ∧ f2(qt) = ∧ f1(qt) + ∧ f2(qt) + ∧ f1(qt) · ∧ f2(qt) lemma 3 for continues functions f1 and f1, we have that ∧( f1(qt) f2(qt) ) = ∧ f1(qt)− ∧ f2(qt) 1 + ∧ f2(qt) (11) 298 yirong ying: estimates of the relative deviation for the equity home bias proof : ∧( f1(qt) f2(qt) ) = f1(qt) f2(qt) − f1(q) f2(q) f1(q) f2(q) = f1(qt)f2(q)− f1(q)f2(qt) f1(q)f2(qt) = (f1(qt)− f1(q)) f2(q) + f1(q)f2(q)− f1(q)f2(qt) f1(q)f2(qt) = ∧ f1(qt) · f2(q) f2(qt) − f2(qt)− f2(q) f2(q) · f2(q) f2(qt) = ∧ f1(qt)− ∧ f2(qt) 1 + ∧ f2(qt) in order to improve the expression of equation 8, we need to modify equation 9, 10 and 11 and present as follows: lemma 4 for continues functions f1 and f1, we have that ∧ (αf1(qt) + βf2(qt)) = α ∧ f1(qt) + β ∧ f2(qt) α+ β (12) lemma 5 for continues functions f1 and f2, we have that ∧ (f1(qt) · f2(qt)) = ∧ f1(qt) + ∧ f2(qt) (13) lemma 6 for continues functions f1 and f1, we have that ∧( f1(qt) f2(qt) ) = ∧ f1(qt)− ∧ f2(qt) (14) obviously, lemma 4 is unique when f1(q) equals to f2(q) as is implied by ex-ante symmetry of two countries in our model, and lemma 5 and lemma 6 is one special case when ignoring second-order relative deviations and ∧ f2(qt) in the dominator, respectively. theorem 1 when equation 5 holds, then the relative world consumption demand for the home good obeys ∧ yc,t = − [ (1− (2a− 1)2)φ+ (2a− 1)2 1 σ ] ∧ qt (15.a) advances in systems science and applications (2014) vol.14 no.3 299 proof:since ∧ yc,t = ∧ q−ϕ t ωa [( pf,t ph,t )ϕ cf,t ch,t ] = ∧ q−ϕ t ωa(x), we derive from lemma 4-6 that: ∧ ωa(x) = ∧( 1 + x ( 1−a a ) x+ ( 1−a a ) ) = ∧( 1 + x ( 1− a a )) − ∧( x+ ( 1− a a )) = (1− a) ∧ x−a ∧ x = (1− 2a) ∧ x then, ∧ yc,t = ∧ q−ϕ t + ∧ ωa(x) = −ϕ ∧ qt+(1− 2a) ( −ϕ+ 1 σ ) (2a− 1) ∧ qt = − [ (1− (2a− 1)2)φ+ (2a− 1)2 1 σ ] ∧ qt and we can similarly prove for expression 2 and 3. theorem 2 when equation 5 holds, then the relative world consumption demand for the home good obeys ∧ yc,t = − [ (1− (2a− 1)2 a )φ+ (2a− 1)2 a 1 σ ] ∧ qt (15.b) theorem 3 when equation 5 holds, then the relative world consumption demand for the home good obeys ∧ yc,t = − [ (1− (2a− 1)2 1− a )φ+ (2a− 1)2 1− a 1 σ ] ∧ qt (15.c) note that λ > 0(as 1/2 < a < 1 implies 0 < 1 − (2a− 1)2). thus, an improvement in the home terms of trade lowers worldwide relative consumption of the home good. introduce the following signs, whose expressions are illustrated in fig.1: λ11 = 1− (2a− 1)2, λ12 = (2a− 1)2, λ21 = 1− (2a− 1)2 a λ22 = (2a− 1)2 a , λ31 = 1− (2a− 1)2 1− a , λ32 = (2a− 1)2 1− a 300 yirong ying: estimates of the relative deviation for the equity home bias fig.1 home terms of trade lowers worldwide relative consumption of the home good. acknowledges this research was supported by national natural science foundation of china (71171128) and research fund of program foundation of education ministry of advances in systems science and applications (2014) vol.14 no.3 301 china (10yja790233). references [1] ferreira, m.a. and a.f. migue. (2007). “the determinants of domestic and foreign bond bias”. working paper serious, may. [2] bekaert g. and x. s. wang. (2009), “home bias revisited”. nber working paper series, february. [3] mondria j. and t. wu. (2010), “the puzzling evolution of the home bias, information processing and financial openness”. journal of economic dynamics and control, vol.34, pp.875-896. [4] engel, c. and a. matsumoto. (2006), “portfolio choice in a monetary openeconomy dsge model”. nber working papers, may, pp.12214. [5] coeurdacier n., r. kollman and p. martin.(2010), “international portfolios, capital accumulation and foreign assets dynamics”. journal of international economics, vol.80, pp.100-112. [6] heathcote j. and f. perri. (2007), “the international diversification puzzle is not as bad as you think”. nber working papers, october, pp. 13483. corresponding author yirong ying and wenjun lv can be contacted at: yrying@staff.shu.edu.cn advances in systems science and application (2016) vol.16 no.1 47-61 development of a biodegradable additive from brown algae of the russian far east for the production of plastic packaging tatyanav. chadova, lyudmila o. korshenko, elena s. smertina, alexey e. nekrasov and natalya v. berlova far eastern federal university. vladivostok, russian; vladivostok branch of the russian customs academy, vladivostok. russian abstract the article presents the development of a biodegradable additive from brown algae of the russian far east for the packaging polymer industry. we consider a possibility of using cellulose derived from algae of the primorsky krai, as an additive to polymeric packaging products, which impartsthe properties of biodegradability to the polymer. a way to produce the biodegradable additive for plastic packaging is to reduce the duration of the technological process, increase the yield and quality of the end product and expand the resource base through the use of cheap and non-traditional local raw materials. keywords biodecomposition; biodegradation, brown algae; biodegradable packaging, polymeric materials. 1 introduction russia annually produces about 70 million tonnes of municipal solid waste (msw), the major share of which (more than 50%) refers to packages. up to 10 thousand hectares of land is annually alienated for msw landfills and dumps. this land also includes fertile lands, withdrawn from the agricultural use. currently, in the russian federation measures to implement new investment projects in the sphere of solid waste management are developed and being taken[1]. only 3% of this amount are recycled, and the rest are incinerated or disposed of in landfills. however, incineration is an expensive process that also leads to the formation of the highly toxic (such as furans and dioxins) compounds. time needed for degradation of packaging materials in natural conditions may range from a few decades to hundreds of years, the use of biodegradable materials in the packaging industry will lead to a significant reduction of this period. therefore, the topic of the development of biodegradable materials is relevant and is of practical and scientific interest. polymeric materials have become part of our lives and have replaced wood, metal, glass, ceramics. a large number of scientific papers and reviews[2-20] are dedicated to the studies on the degradation of polymers. according to the international organization for standardization, biodegradable plastics are polymers the degradation of which takes place under the influence of microorganisms, bac48 tatyanav. chadova, lyudmila o. korshenko, etc.: development of a biodegradable... teria and fungi. the use of such plastics minimizes the adverse effects on the environment ecology. the degradation rate depends on a number of factors such as the type of a polymer, the type and concentration of degrading materials, moisture, temperature and a series of other factors. the corresponding public opinion and the legislative ways of control and regulation contribute to the accelerated proliferation of the production technologies of these packaging materials. in many countries, the government prohibits the use of plastic bags and disposable plastic utensils made of traditional types of plastic, and one has to pay for plastic bags in supermarkets, while paper bags are free. in recent years, the production of packaging materials is the leading sector of the economy and is growing worldwide, including in russia. based on this, one of the relevant directions is the production of environmentally friendly biodegradable packaging. the development of new highly efficient technologies, the creation of products based on plant raw materials, their further assimilation and standardization become more popular due to the wide spectrum of biological effects. particular attention is paid to the need to use local plant raw materials with sufficiently large natural reserves of natural or cultivated natural sources that are promising in terms of their introduction into production. one of the ways to produce biodegradable materials is to create composite materials based on natural biodegradable polymers such as cellulose, chitosan, gelatin, polypeptides, casein, and others. this research work aims to study the possibility of using local plant raw material brown algae of the primorsky krai in order to create a new biodegradable polymeric composition, which along with the high molecular basis comprises organic fillers (starch, cellulose) that should serve as a nutrient medium for microorganisms. the advantage of biogenic packaging made using local algae lies precisely in the high speed of growth some species can grow up to a meter during twelve hours. 2 main part at the department of commodity research and examination of goods of the far eastern federal university, the work was undertaken to develop a method for producing cellulose from brown algae of the primorsky krai. the main objective of the development was to provide an efficient method for producing cellulose from sea algae, mostly of the fucus genus, the use of which as a raw material makes it possible to clean coastal areas littered with the cast ashore algae and improve their ecological condition. today, modern biotechnology is aimed at the production of both traditional and new types of substances and materials. in addition, at this point, renewable plant raw materials can become a solution. such technology can be successfully advances in systems science and application (2016) vol.16 no.1 49 used for the further development of the industry in the far east. today, many physical and technical characteristics of biodegradable polymers are not inferior to the characteristics of conventional plastics. biodegradable polymers are degradable in natural conditions under the influence of such environmental factors as light, temperature, humidity and with the participation of living microorganisms as well. simultaneously, high-molecular substances degrade into low molecular ones, which are safe for the environment. another positive characteristic of biopolymers is that they decompose in a short time from several months to several years. thus, the creation of compositions that along with the high molecular basis include organic fillers (starch, cellulose, amylose, amylopectin, dextrin, etc.), which are a nutrient medium for microorganisms is a promising trend in the packaging industry. by the general structure, the majority of algae cells are similar to the cells of green plant such as corn or tomatoes; they have rigid cell wall, which consists primarily of cellulose, hemicellulose and pectins. cellulose pulp, alkyl ketones and compounds containing carbonyl groups are the catalysts of photo and biodegradation of the films based on polyethylene, polypropylene, or polyethylene terephthalate. photo and biodegradation of such films begins after 8-12 weeks, the remnants of the film completely disappear during harrowing and ploughing, thus loosening the soil. the main commercial algae is brown algae, so this paper puts more emphasis on the representatives of this division. brown algae (phaeophycophyta) are spread exclusively in the seas; their appearance resembles the one of higher plants. representatives of brown algae have a yellowish-brown colour of thalli, which is due to the presence of chlorophyll and carotenoids, as well as the brown pigment from the group of xanthophylls-fucoxanthine. their vegetative body (thallus) is of complex internal and external structure. the thallus is composed of several types of tissues with different functions. the chemical composition of brown algae is highly dependent on the species, season and habitat. according to the literature, brown algae contains from 2.5 to 17% of algal cellulose, which, due to certain differences from the usual, is called eucellulose[21]. extraction of cellulose from natural materials is based on the action of reagents that solve or destruct non-cellulosic components contained in plant tissues (proteins, fats, waxes, resins, lignin and polysaccharides cellulose companions). extraction methods depend on the type of plant material and designation of cellulose. the main method is alkaline pulping processing of plant materials with diluted sodium hydroxide solution. in the course of the research, we obtained the optimal technological parameters of the cellulose extraction process, thus allowing us to increase the yield of the final product. 50 tatyanav. chadova, lyudmila o. korshenko, etc.: development of a biodegradable... the authors propose a new method for producing cellulose from non-wood cellulose-containing raw materials, which comprises the milling of dried raw material, delignification with dilute acid by heating followed by separation of the solid phase, which is treated with the aqueous solution of sodium hydroxide under heating, and then the desired product is filtered off, washed with water and dried. an important advantage of the method proposed by the authors is the possibility to improve the sanitary condition of the coastline sections cluttered with deposits of cast ashore dry algae and improve the environmental situation in general. the study found that raw algal cellulose fibres are very small, in the shape of irregular flakes and dissolve in the schweitzer reagent. thus, sea algae is an ideal raw material for bio-packaging material. 3 methodology the method consists in the purification of large extraneous impurities of dried brown algae, mostly of the fucus genus, followed by refinement through grinding and sifting. 4 to 6% solution of sulfuric acid (2so4) is added to the sifted fine fraction with the 1: 45-55 hydromodule, and is heated at the boiling point of the mixture (100 to 105◦c) for 5-10 minutes under reflux with continuous mechanical stirring. the reacted and cooled down mixture is filtered, and the mass precipitated on the filter is washed with hot water (60 to 70◦c) until neutral and dried to constant weight. then, 5-10% solution of the koh alkali is added to the resulting solid residue, followed by hot distilled water (from 60 to 70◦c) in equal amounts calculated to provide a hydro module of 1/50-60 at an alkali solution concentration of 2.5-5.0%, and it is heated at the boiling point of the mixture for 5-10 minutes with continuous mechanical stirring. the cooled mixture is filtered with it being rinsed with water as well. after complete filtration, the solid residue in the filter, which is the desired product, is washed and dried in natural conditions. after it is dried, it is in the form of brown powder, which goes into the reactions characteristics for cellulose: it is dissolved in schweizer’s reagent and forms a gelatinous cellulose mass if diluted acid is added to the resulting solution. lignin-like residual content and other organic substances in the resulting product amounts to no more than 5-8 wt.%. the proposed method differs from the known methods[22-25], as local sea brown algae is used as the non-wood cellulose raw material; at the same time, the delignification is carried out by heating in a 4-6% solution of sulfuric acid (s/l = 1/45-55) at the boiling point temperature of the mixture for 5-10 min; the alkaline treatment of the obtained solid residue is carried out after washing with water by heating in a 2.5-5.0% solution of potassium hydroxide koh (s/l = 1/50-60) at the boiling point for 5-10 min. advances in systems science and application (2016) vol.16 no.1 51 technological parameters of the proposed method ensure efficient processing of specific algal raw material with the product being based on cellulose, containing lignin residues (lignin-like substances) and trace content of other organic substances. the conducted field tests showed that the resulting cellulosic semi-finished product acts as an effective filler of composite polymeric materials that serves as a nutrient medium for microorganisms, destroying high molecular basis, and can be successfully used in the production of various types of packages and containers with limited lifetime in particular. for a more complete utilization of the algal mass and higher profitability the author suggest using acidic and alkaline filtrates for obtaining a valuable nutrient of the fertilizer, for this purpose, it is necessary to mix and separate them for the formation of the flocculated settling fraction, which is washed with water until neutral and dried. discussion.there are certain developments that envisage the use of different plant fillers as natural bio additives: flax waste, wheat stalks, corn stalks, wastes of sawmill and wood chemical industries, food production waste, etc. today polymer blends are widely used (most often polyethylene) with corn and starch, biodegradable polymers with natural additives cellulose, soy flour, spent grains. the combination of a synthetic polymer and a natural material can give a new set of properties. the authors studied the technical level in the sphere of development of the biodegradable additives into polymeric materials of a wide application range; a patent search is conducted, which revealed that many scientists, both in russia and europe mainly work in the field of biodegradable materials that are used as packaging materials[26-46]. from the perspective of rational environmental management as a natural supplement, it is feasible to consider the resources that one or another region is rich with. the fauna of the far east is a rich source of unique plants. algae make up the bulk of plant organisms in bodies of water and are the most productive plants. large amounts of sea weed and algae are cast ashore in the promorsky krai; these plants contain fiber plant cellulose, which, as the authors suggest, can be used as a natural filler for biodegradable composition. it is the study on the degradation of algae that a large number of scientific papers and reviews are dedicated to [47-57]. by its general structure, most algae cells have similar cells to the cells of green plants such as corn or tomatoes. the studies have found that algae (cyanobacteria) are capable of synthesizing cellulose, which can be used as a biodegradable additive. the far east is rich with brown algae, so in our work we focused on representatives of this division. the chemical composition of brown algae is highly dependent on the species, 52 tatyanav. chadova, lyudmila o. korshenko, etc.: development of a biodegradable... season and habitat. it is found that brown algae contain up to 17% of algal cellulose, eu-cellulose, which is structurally similar to cellulose of higher plants, and thus it is a potential source[57]. the use of dry brown algae, which can be considered as non-wood cellulose waste, allows expanding the raw material base for producing cellulose, reduce the consumption of more valuable wood and get cheaper products due to the use of affordable renewable non-conventional raw materials. each plant contains cellulose, but not every plant is suitable for industrial extraction of cellulose. when choosing raw materials, the content of fibre is important, as well as structural features of the constituent cellular tissues, the possibility of using industrial processing methods, the quality of the product obtained as a result of this treatment, the prevalence of plant raw materials, cost of collection, storage. despite the proven similarity of the algal cellulose with the cellulose of higher plants in structure and properties, with the use of dry brown algae as cellulosecontaining raw materials, the known cellulose production methods are ineffective and cannot exhibit sufficiently convincing results. the reason for this is that algal fibre contains lignin-like substances that are difficult to remove as they have strong bonds with it, as well as due to the residual amount of alginic acid, prone to certain conditions, to gel formation, and in some cases chlorophyll formation. there is a known method for manufacturing cellulose powder from various kinds of lignocellulosic materials derived from wood semi-finished products in the course of their processing on the cellulose and paper plants, straw of herbaceous plants, waste paper (rf patent. no. 2478664, published on 2013.04.10) which includes the destruction of said materials via exposure to a lewis acid solutions of low concentration and an organic solvent with stirring, followed by washing and drying of the desired product[24]. if there is a common idea consisting in the use of lewis acids, this method includes unspecified number of variants, each of which requires an independent selection of optimum conditions for obtaining powdered cellulose: concentration values and the type of lewis acid in various organic solvents, liquid module, temperature and a destructive treatment of the lignocellulosic material, the intensity of stirring and the conditions of drying of the end product, depending on the used raw material. furthermore, cellulose production in the known manner on an industrial scale involves the use of substantial quantities of organic solvent, requiring certain safety precautions during handling and disposal. when this method is used for dry brown algae as a cellulose-containing raw material, it is ineffective and does not provide the production of the necessary target product. there is a method of cellulose production from paddy straw (rf patent no. 2418122, published on 2011.05.10) comprising two pulping stages, the first of advances in systems science and application (2016) vol.16 no.1 53 which is carried out in an alkaline medium with subsequent separation of the cellulose product, and the second-in an acidic environment, with a mixture of peracetic acid, acetic acid and hydrogen peroxide in the presence of a stabilizer, in this case, mixtures of organ ophosphonates, wherein second pulping stage is carried out in the presence of ozone at a rate of 2-4 g/hr[25]. the disadvantage of this method is the difficulty and cost of technology that are due to the need to conduct the process in the presence of ozone and due to the use of hydrogen peroxide, which is subject to rapid decomposition, and organic stabilizers. furthermore, the known method is also ineffective when dry brown algae is used as raw cellulose-containing material. there is a method of cellulose production from straw (rf patent no. 2423570, published on 2011.07.10) comprising impregnation in the reactor and maceration of chopped straw with the aqueous solution of sodium hydroxide with a concentration of 20-30g/l in na2o units at a temperature of 30-80◦c at a ratio of the solution weight to the weight of dry chopped straw of 7:1[24]. the impregnated chopped straw is maintained at a predetermined temperature for 30 minutes, and then the liquid phase is withdrawn. heated water is added to the mass, the mass temperature is raised to 96◦c and the pulping is carried out at this temperature for 2 h 30 min. when used as a dry cellulosic raw material of brown algae containing non-hydrolysable substances strongly bonded with fibre and similar in elemental composition to lignins of higher plants, the known method is inefficient and does not provide a marketable product. the closest to the claimed method is a methodof producing cellulose from nonwood plant raw material with the content of the native cellulose being not more than 50%, described in the rf patent no. 2448118, publ. on 2012.04.20[27]. according to the conventional method, the raw material is washed with water at 40-70◦c and atmospheric pressure for 0.5-4 hours, and processing is carried out with an aqueous solution of nitric acid with a concentration of 2-8% at 90-95◦c for 4-20h followed by separation of the solid phase, which is treated with the aqueous solution of sodium hydroxide with a concentration of 1-4% at 60-95◦c for 1-6 hours. a disadvantage of this method is the long duration of the technological process, in addition, with the cellulose extraction from dry brown algae, the fibre of which contains lignin-like components and alginic acids that are hard to separate, it is possible to use the method with positive results and it does not provide an acceptable yield and quality of the end product due to the complexity of removing the tightly bound substances contained in algal cellulose and, on the one hand, and the partial hydrolysis of the fibre itself after prolonged treatment, on the other hand. result.algae are a non-food biomass, the use of which does not pose a threat to the production of packaging and food safety. algae grow 20-30 times faster 54 tatyanav. chadova, lyudmila o. korshenko, etc.: development of a biodegradable... than land plants (some species can double their mass several times a day). the lack of a hard shell and almost no lignin makes the processing technology easier and more efficient than biomass processing of any ground material. it should be noted that the intensification of research in the field of polymers is important not only for further successful development of the market of biodegradable plastic packaging, but is one of the most promising ways to solve the global ecological problem related to the environmental pollution with polymeric materials wastes. the proposal to use local raw materials as raw materials base can solve the problem of non-waste production in the industry. among the possible priority areas we can note the development of ocean resources and the ecology, new technologies and materials. today, modern biotechnology is aimed at the production of both traditional and new types of substances and materials. moreover, at this point, renewable plant raw materials can become a solution to this problem, and the proposed technology for producing biodegradable additives for plastic packaging can be successfully used for the further industrial development in the far east region of russia. the development presented in the article is an invention (patent no. 2556115 russian federation, ipc s08b15/02 (2006.01), appl. dated 15.04.14, published on 07.10.15), which relates to algae processing methods, in particular, of the sea brown algae cast ashore; this method can be used to produce fillers for synthetic polymersthat provide biodegradation of polymer compositions and are required for manufacturing materials with a controlled lifetime and to producecellulose semi-finished products used as raw materials for the chemical industry, in the manufacture of paper and cardboard. cellulose is not a product intended for direct consumption: cellulose obtained after a single technological cycle serves as a raw material for other processing cycle; varying degrees of cellulose purification determine its various applications. an important advantage of the proposed method is the possibility to improve the sanitary condition of the coastline sectionscluttered with deposits of cast ashore dry algae and improve the environmental situation in general. 4 conclusions 1. the technical result of the proposed method for producing the biodegradable additive from the far east brown sea algae for plastic packaging is to reduce the duration of the technological process, increase the yield and quality of the end product and expand the resource base through the use of cheap and nontraditional local raw materials. 2. in the course of the research, we obtained the optimal technological parameters of the cellulose extraction process, thus allowing us to increase the yield of advances in systems science and application (2016) vol.16 no.1 55 the final product. 3. technological parameters of the proposed method ensure efficient processing of specific algal raw material with the end product being based on cellulose, containing lignin residues (lignin-like substances) and trace content of other organic substances. 4. the resulting cellulosic semi-finished product acts as an effective filler for composite polymeric materials that serves as a nutrient medium for microorganisms, destroying high molecular basis, and can be successfully used in the production of various types of packages and containers with limited lifetime in particular. 5. during the study, the authors found that raw algal cellulose fibres are very small, in shape of irregular flakes and dissolve in the schweitzer reagent. thus, sea algae is an ideal raw material for bio-packaging material. 6. for a more complete utilization of the algal mass and higher profitability, we propose to use acidic and alkaline filtrates for obtaining a valuable nutrient of the fertilizer, for this purpose, it is necessary to mix and separate them for the formation of the flocculated settling fraction, which is washed with water until neutral and dried. 7. the novelty of the technological solution of cellulose production from brown sea algae is confirmed by rf patent no. 2556115 “the process for producing cellulose from brown sea algae”. acknowledgement the authors are deeply grateful to their colleagues victor e. vaskov,olesya a. zdor, margarita d. boyarova, who are not the authors, but with whose assistance the study was conducted. references [1] the committee on environment and natural resources. 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(2004), “vegetable fiber: structure, proper-ties, applications”. publishing house: university, forestry journal, no. 4.c, pp.122-129. corresponding author tatiana can be contacted at: yal05@mail.ru microsoft word 17 y. g. jiang--the application of norm functional multi-dimensional space theory in dynamic design.doc advances in systems science and applications (2011), vol.11, no.3-4 331-338 issn 1078-6236 international institute for general systems studies, inc the application of norm functional multi-dimensional space theory in dynamic design y. g. jiang machine-electrical engineering department, chengdu electromechanical college, chengdu, 610031, china abstract the dynamic design for space over three-dimension is a tough problem in mechanical design, because there are no proper cognition and analysis methods for multi-dimensional space. so, more abstract and comprehensive methods are in need. normed function space theory is a fully-fledged mathematics theory for multi-dimensional space, which is a powerful aid in analysis and representation. by using arbitrary function, dynamic design problem can be solved, such as the description of multi-variable system, relation modeling and optimization. variable systems with totally different physical meanings may have an identical norm function structure, which makes the abstraction and analysis of complex problem easier. the design is the modeling based on physical essence and the analysis and calculation of model, instead of traditional geometry analysis. the method is not analogy but precise calculation, which can produce an optimum solution in norm function. centrifugal governing system is a typical mechanical power system. according to dynamic requirement, the energy arbitrary function is set up and solved. that completes the mechanical structure design. the example shows, the arbitrary function design is workable and digitized. keywords multi-dimensional space route planning dynamic design norm function space 1.introduction along with the development of science and technology, the major task in mechanical design is the dynamic design of multi-variable system. the dimensions of multi-variable dynamic space are over three, at least four including time, without visual geometry. the major task of dynamic design is the representation and analysis of ultra three-dimension space. nowadays, the mechanical design is basically in analogy and static design stage [1-2]. it’s difficult to represent and analyze dynamic space without scientific method to express multi-dimension space. in mechanical structure design, the following problems cannot be solved at the same time: the representation of motion position and performance in mechanical drive system, the elastodynamic characteristic of mechanism motion, the time-sharing occupation of space, the spatial encounter and capture of moving object, the influence of different motion parameter on mechanical and motion characteristics, the structure of high-speed mechanism and dynamic strength and so on. these involve position, tangent line, normal line, time, velocity, accelerated velocity, which can be concluded to study function and its derivative. norm function space theory is a mathematic theory to describe and analyze multi-dimension space, which establish the relation between three-dimension and infinite space. the normal three-dimension space is a fully-fledged mathematic method, with direct geometry view, which helps the comprehension of the concepts in high-dimension, even infinite dimension space [3-4]. dynamic design problem, such as the description, the representation and the overall optimization of multi-dimension space in mechanical design, can be analyzed with norm function space theory. so, mechanical design can be transferred from geometry analogy to precise calculation, from three-dimension to multi-dimension. the content is expanded by the project is funded by the sichuan educational commission in the area of nature science (contract no. 2004a163). 332 jiang: the application of norm functional multi-dimensional space theory in dynamic design changing the method of design, thus complete the fullness and scope in mechanical design. 2.the arbitrary function problem of n-dimension design space 2.1 the expansion and composition of design space mechanical design has developed from static to dynamic design, from three-dimension to multidimension, from mechanism to inter-discipline of mechanism, electricity, magnetism, liquid and light, from geometry analogy to precise calculation, from sequential to parallel design. the emphasis is transferred from structure design to life process of product. the continual development of mechanical design will expand dimension and content in design space further. the structure parameter and the function or differential equations set which expresses physical process are foundation to form overall performance. the multi-dimension space of mechanical design consists of structure parameter or function sequential set. normally, the spatial variable is represented by vector: [ ]tn21 x,,x,x=x the sub-spaces are collections of parameters with identical characteristics. the design space can be divided into sub-space according to following functions: geometry shape, shape restrict, geometry factor; requirements on tolerance and surface technology; statics performance; motion performance; dynamics performance; process performance; and other performances concerned with geometry topology. all these performances are represented with functions and derivatives, such as the design of contour curve which need to consider its function and derivative, with nth continual derivatives to ensure smoothness of contour. the mechanical motion design hopes for motion equation with continual derivatives over 3 to ensure steady movement. 2.1.2 the arbitrary function problem of design space arbitrary function is more abstract and comprehensive than higher mathematics and linear algebra. to study space composed of different functions, the property and trend can be analyzed with abstract operator. normally, the problems of arbitrary function are the representation and analysis of multi-dimension function space and to choose one set optimized function which meets the requirement from functions group. the design process is to choose representation parameter according to preset dynamic characteristics from the system, to derive the solution which meets the design requirement, to choose one set optimized function, such as contour, motion, strength function and so on, in order to meet the standard. for example, a design with, in certain range, least deflection in vibrating frequency and lightest mass is a reverse problem of dynamics. that is, to preset requirements of the system on frequency, stress, strain and displacement, which determine the material and geometry characteristics in mechanical system. the problem is to solve the extremum of design arbitrary function. so, dynamic design problem is also arbitrary function problem: to represent and analyze multi-dimension space in mechanical design and to solve the extremum of arbitrary function. the first arbitrary function problem of dynamic design is to study n-dimensional variable and establish a space which can express n-dimensional variable. the second arbitrary function problem of dynamic design is to study the variable and its change rate at the same time. the key problem is to establish a norm function space which can express the function and its n th derivative at the same time. the third arbitrary function problem of dynamic design is to choose optimized solution which meets dynamic requirements and proper operator to solve and design the extremum of arbitrary function. 3.the representation and simplification of n-dimension design space with arbitrary function 3.1.1 the representation of design space advances in systems science and applications (2011), vol.11, no.3-4 333 traditional mechanical design space is three-dimensional euclid space, which is represented by distance and angle, and can be analyzed continuously and with extremes. for multi-dimension, such as linear space, normally there is no need to represent but only introduce norm into it, and make it a normed space. with norm, the concepts of open set, closed set, convergence and continuation can be introduced correspondingly. in order to solve the first arbitrary function problem of dynamic design, a normed space should be defined. definition: supposing x is the vector space in number field k (real number field r or complex number field c), if for every x∈x , appointed one real number ║x║ which called norm of x, it meets following norm axioms. (1) homogeneity： xx αα = ； (2) triangle inequality： yxyx ++ ≤ ； (3) positive definiteness： 0≥x 00 =⇔= xx as above, x， xy∈ ，if k∈α , x is called a nomed vector space in k， normed space for short. when norm must be expressed definitely, it is marked as (x，║•║）. if k＝r (or c)，the normed space in k is real (or complex) . the way to define norm diversifies. norm can be defined according to its physical meaning. furthermore, the distance of three-dimensional space, the displacement, velocity and accelerated velocity of motion can also be defined into norm separately, which can also be combined together for norm. the same function can be bestowed different norm. for example, c[a，b] is defined all continuous derivable functions in [a，b], which forms linear space x. for every x∈x , we define norm: )( bta txmaxx ≤≤ = or )()( txmaxtxmaxx b≤t≤ab≤t≤a += or dttxx a∫= b )( they can all be normed space. n-dimensional euclid space set distance as norm: so it is also a normed space. norm is a representation of multi-dimension. the definition of norm is artificial, which makes its physical meaning diversified. some norm has such a complicated physical meaning that it is unfathomed by using three-dimension. multi-dimension can express much richer physical meaning than three. norm can express n variables at the same time, which can represent and analyze variables from several design spaces systematically and comprehensively at the same time. 3.1.2 the simplification of n-dimension design space with norm norm analysis is highly comprehensive and abstract. it is not focused on specific structure and physical meaning of function, but on the function system and its commonality. mechanical design space analyzed with arbitrary function can unify geometry, statics, kinematics and dynamics design into one norm structure. the diversification of norm definitions makes the study comprehensive and flexible. the normed space with distance as its norm is a traditional geometry space. and the one with the linear combination of displacement, velocity and accelerated velocity can show the difference among different moving modes easily and clearly. normed space with linear combination of 2 1 1 2 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ∑ = n i ix ξ 334 jiang: the application of norm functional multi-dimensional space theory in dynamic design contour function and n th derivative as norm can also express the smoothness of curve and the continuity of processing easily. define for any )(tu mc∈ ，set norm: p ma p p a pm uu 1 , ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂= ∑ ≤ { }∞<∈= pm, m u:)t(cus is a normed space. its completion is a banach space, called sobolev space. the second arbitrary function problem of dynamic design can be solved by sobolev space definition, that is, to study variable and its change rate at the same time. though two functions with equal norm║u║m,p may not be the same, the norm of identical functions must be equal. so, accompanied by other conditions, norm can be used to study function. distance, motion and energy space all involves function and its derivatives. sobolev space is concerned with motion: suppose the motion space is composed of displacement, velocity, acceleration velocity and accelerating acceleration velocity, represented with u ，u ′，u ′′ ，u ′′′ respectively, )(tu mc∈ , set norm: ( )2 2 2 2 1 1 1 1 1 1 22 2 2 2 , u dt u dt u dt u dt p pa m p p a m u u ξ ξ ξ ξ ξ ξ ξ ξ ≤ ⎛ ⎞ ′ ′′ ′′′= ∂ = + + +⎜ ⎟⎜ ⎟ ⎝ ⎠ ∑ ∫ ∫ ∫ ∫ ║u║m,p norm can express for variables u ，u ′，u ′′ ，u ′′′ at the same time. space variables in dynamics are: u ，u′，u ′′ ， um ′′ , in which m is mass. space variables of contour curve are ( )tc0 ， ( )tc1 ， ( )tc 2 ，…， ( )tc n , which can also be defined into norm with similar method. they share the same form, all belonging to sobolev space. sobolev space, which combines parameter and its derivative to define norm, includes the main content in dynamic design by unified structure of norm function. this highly abstract and comprehensive structure makes the analysis of mechanical design space more simple and comprehensive. 4.example for dynamic design analyzed with arbitrary function 4.1.1 flexible and digital characteristics to define norm with several parameters or only one, no matter which way, the analysis of arbitrary function directly with norm is discrete and fussy. if to analyze directly with calculus, it’s also discrete and fussy for function space because calculus can only study one function. operators in arbitrary function are all effective methods to analyze arbitrary function. the optimization of arbitrary function, effective theoretical method, is an expansion of differential calculus. gateaux differential is equivalent to vatiation in variational calculus, or direction derivative in differential calculus. fréchet differential is equivalent to gradient in differential calculus or complete differential or gradient. by using gateaux and fréchet differential, the extreme value of arbitrary function in linear space can be solved easily, and so can the necessary condition for local extreme. if x is a linear space and f is real value arbitrary function in x, with gateaux or fréchet differentiable, the necessary condition for extreme in x∈0x is, for every x∈h , ( ) 0; =hxfδ . that is, the extremum of arbitrary function can be solved by vatiation, which is the commonest and simplest method in analysis and calculation. at present, the main task for dynamic design is to solve kinematics and dynamics problems, such as vibration frequency, periodic and inertial force, which involves mainly in analytical mechanics. the major theory of analytical mechanics is deduced according to the necessary condition for arbitrary function extreme, such as hamilton’s principle and lagrange equation. advances in systems science and applications (2011), vol.11, no.3-4 335 a x y z d b c m ω θ φ m m hamilton’s principle focuses on energy arbitrary function composed of general displacement and its derivative, time and so on. if the constraint is ideal and complete, and the conservative force is drive force, the real motion is the motion which makes get exrtreme value. ( ) ( ) ( ) ( )[ ] ttqv-t,qq,ttt,qqlqs t t t t d,d, 2 1 2 1 ∫∫ ′=′= if the function which makes arbitrary function s(q) get extremum is existing and exclusive, according to arbitrary function differential theory, the necessary condition equivalent to arbitrary function extremum is: 0 q l q l t aa = ∂ ∂ −⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′∂ ∂ d d this is called lagrange equations set [5-6]. the solution can be very easy and simple only by partial differential or differential. for example: to design a centrifugal governor shown in fig.1, a slide c with mass m can slide along a vertical axis, and the whole system can also rotate on this axis. point a is fixed, point b and d hinge two particles with mass m. the requirement: when the system acquires stable angular velocity ω, the vibration periodic is t, and determine the length of lever a. solution: (1) list laplace equation: the angle between plane abcd and axis x isφ , θ = dac∠ , degree of freedom f=2. to choose θ , φ is generalized coordinates, constraint is ideal and stable. the cartesian coordinate of particle m: φcosθ sinax = θθ cossinay = θcosaz = lagrange function: ( ) ( ) ( ) ( )2 22 2m ml θ,φ,θ ,φ 2 aθ aφ sinθ 4a θ sinθ 2mga cosθ 2mga cosθ 2 2 ⎡ ⎤ ⎡ ⎤′ ′ ′ ′ ′= + + + +⎣ ⎦ ⎣ ⎦ to solve partial derivative: fig.1 the vibration periodic design of centrifugal 336 jiang: the application of norm functional multi-dimensional space theory in dynamic design to solve partial derivative and simplification: θsinθma4θma2)φθφ,θ,(l θ 222 ′+′=′′ ′∂ ∂ lagrange equation: 2 2 2 2 2 2(2ma θ 4ma θ sin θ) (2mφ 4mθ )a cosθ (2m 2m)ga sin θ t d d ′ ′ ′ ′⎡ ⎤+ = + +⎣ ⎦ ( ) ( ) ( )2 2 2 gm 2m sin θ θ mφ 2mθ cosθm+m sinθ a ⎡ ⎤′′ ′ ′+ = −⎢ ⎥⎣ ⎦ （2） to solve relatively balanced condition and micro-vibration periodic, the system rotates on vertical axis at even angular velocity ω, ωφ =′ , on balanced position: ( ) 0 a gmmθcosmω2 =⎥⎦ ⎤ ⎢⎣ ⎡ +− to simplify, then: 2 2 gm 2mθ sin θ mω cosθ (m m) sinθ a ⎡ ⎤′+ = − +⎢ ⎥⎣ ⎦ at balanced position: a gmmθmω s 2 )(cos += solve angle sθ , then put it into lagrange equation: [ ] θsin)θcosθ(cosmωθ)θsinm2m( s 22 −=′′+ by using formula: ) 2 θθsin() 2 θθsin(2θcosθcos ss s −+ −=− define sθθφ −= , take micro-vibration approximation: s s θ 2 θθ = + sss θsinφ) 2 φsin(θsin2θcosθcos −=−=− define and set the micro-vibration approximation: [ ] ss 2 s 2 θsin)φθsin(mωφ)θsinm2m( −=′′+ φ θsinm2m θsinmωφ s 2 s 22 + − =′′ s 2 s 22 2 θsinm2m θsinmωω + = micro-vibration frequency: s 2 s 2 θsin m m21 θsinωω + = the micro-vibration periodic near the balanced position is: advances in systems science and applications (2011), vol.11, no.3-4 337 s 2 s 2 θsin θsin m m21 ω 2π ω π2t + == solve: 2 1 2 s m m 2 ωtθsin − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ −⎟ ⎠ ⎞ ⎜ ⎝ ⎛= π set: 0.6=t s， 3 m m = ， sω π10= solve: by the angle sθ at balanced position: 0435.0 θcosω g m m1 a s 2 = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = （m） by simple dynamic design example above, we know that according to the expression of micro-vibration periodic, the structure of a centrifugal governor which meets dynamics requirements can be produced. the step is that based on the physical characteristics of system, to describe and establish arbitrary function rule by choosing motion parameters, and to deduce function solution that is in accordance with the design condition. it is different from tradition. this design is not confined to size or three-dimension any more, but dealing with the physical essence of system directly, choosing parameters flexibly, establishing arbitrary function rule and solving it. compared with traditional die forming and hard jig assemblies, mpf/mpp can perform in dieless and jigless manufacturing. the flexibility characteristic can be completely shown by using active elements in the eg of the mpf/mpp tooling. for example, using mppf tooling, the deformation path can be changed to gain the best loading for raising the formability limit through increases in the contact area between the punches and the sheet, which changes the loading condition. various outputs from a 3d surface model can be digitally used to adjust the eg using a computer control system to configure the surfaces of tools. by means of the flexible and digital characteristics of mpf/mpp, larger and more complex panel parts can be formed and assembled incrementally and continuously with the dt/jt system. 5.conclusion to describe multi-dimension in mechanical design with arbitrary function theory, the selection of parameter is not geometry size, the object is not geometrical body, the representation is not limited to three-dimension figure any more, but to build model just according to the system’s physical essence, and with diverse expression, to express different kinds of parameter at the same time. the design is not limited to the arrangement of three-dimensional figure, but to analyze and compare parameters form different design field with space theory. by using normed function, the convergence and extreme problems of function in banach space can be discussed and designed. by utilizing hilbert space, the optimization problem can be discussed. and by using differential operator, dynamic space function, derived function and optimized solution of parameter can be solved. to sum up, arbitrary function theory can solve the difficulty in dynamic design, and facilitate dynamic design. acknowledgements =sθsin 0.4082， 094924.θs = a gmmθmω s 2 )()cos( += 338 jiang: the application of norm functional multi-dimensional space theory in dynamic design the project is funded by the sichuan educational commission in the area of nature science (contract no. 2004a163). references [1] a. illarramendi1, j. l. azpeitia1, r. bueno1，new control techniques based on state space observers for improving the precision and dynamic behaviour of machine tools， annals of cirp，54/(1) (2005)： 632635. [2] y.altintas.，k. eykorkmaz. feedrate optimization for spline interpolation in high speed machine tools[j]． cirp, 52（1）(2003)：288-305． [3] francois isnard , gordon dodds. dynamic positioning ofclosed chain robot mechanisms in virtual reality environments. international conference on intelligent robots and systems , victoria , b. c. , canada ,1998. [4] luokang,xiongnan the structure identification of link gear based on connected matrix aggregtion operation, sichuan industry technology college scientific journal, 19 (2) (2000) 35∶ ～37. [5] wang shuang. functional analysis and optimization theory [m]. beijing: beijing aerospace university press, 2004. [6] isnard f , dodds g ,claude vallée. efficient multi arm closed chain dynamics computation for visualisation . international conference on intelligent robots and sys2 tems , grenoble , france ,1997. мягкая вероятность ошибки при передаче информации advances in systems science and application(2016) vol.16 no.4 81-88 the asymmetry of the present-day social and demographic conditions of the north vitali s.belozerov, natalia a. shchitova ,vasiliyv.chichikhin north-caucasus federal university, stavropol, russia annotation the integral analysis of the current socio-demographic conditions in the north caucasus, the specific features of natural population development, migration and their influence on the dynamics of the ethnic structure are presented in the article. substantial regional disparities in the demographic development of the territories have been found. the regional differences in the reproduction and the ethnic composition of the population of the mountain and plain land regions have been shown. the four types of regions with their specific way of population generation have been identified. special attention is paid to the processes connected with the transformation of the ethnic structure of the population. the increasing number of the areas rapidly changing their ethnic composition, the growth of multi-ethnicity in the plain land regions, derussification and ethnic homogenization in the mountain areas have been noted. a significant asymmetry of the north caucasus regions in terms of their social and economic development proving the territorial disparities between the republics in the south-east and the relatively developed plain regions in the north-west has been revealed. key words: the north caucasus, demographic conditions, migration of the population, ethnic structure, interregional disparities, social and economic processes.. 1 introduction the north caucasus is one of the most peculiar regions in russia in terms of social and demographic issues. here, quite developed, self-sustaining provincial and falling behind areas adjoin each other on a relatively small territory. some of them are rather politically stable and characterized by a rapid economic growth, while other regions are inert and inactive and still others are seen to have growing social inequality and deteriorating economic conditions. among the factors, influencing the social and economic development of the north caucasus region in the first place is its geopolitical and cross-border position to have gained pace after the collapse of the soviet union. in fact, it is the southern outpost of russia, providing access to the south caucasus, the black sea and the caspian sea. it is located between two highly differing in terms of socio-cultural, and ethno-demographic processes territories slavic, primarily orthodox (central russia, belarus and ukraine) and the caucasus, being intervened in ethnic and religious terms (north caucasian republics and south caucasus states). the north caucasus is one of the most important and at the same time complex and unstable ethnic and political elements in the structure of the russian nationhood. the main causes of the political precariousness arise from the ethnic heterogeneity of the geo-space, high multiethnic, overlapping and intersection of ethnic settlement areas, the contradictions in the political and economic purposes of different ethnic, cultural and social groups [5]. hence, the conditions there are quite loaded and controversial and resolved not only through negotiations but sometimes 82 vitali s.belozerov, natalia a. shchitova ,vasiliyv.chichikhin: the asymmetry of the present-day social and … with force applied. often ill-conceived administrative and territorial structure itself proved to be a detonator of the conflict. being observed in the last decades growth in the ethnic identification takes place along with the boost in religious consciousness. for many centuries, the north caucasus is the contact zone of the peoples belonging to two major world religions christianity and islam, the representatives of other religions being present as well. however, one cannot ignore the fact of non-traditional religious movements in the area and their destructive influence, in particular, the islamic fundamentalist movement (e.g. wahhabism). in general, it is no secret that the socio-economic and political situation in the north caucasus is deteriorated by a number of factors preventing strengthening of the unity of the russian nation and harm onisation of interethnic relations. it is impossible to understand the nature of these problems without a thorough analysis of the social and demographic processes, which largely influence not only the quality of human potential, but also the quality of life of the population. 2 sources of information and research methods in our study the main accent was made on finding answers to the following questions: 1. what is the demographic and migration situation in the north caucasus as a whole and in its certain regions? 2. how inhomogeneous is the social and demographic extent of the north caucasus, what are the reasons for this heterogeneity? 3. how is the ethnic structure of the population of the north caucasus transformed under the influence of the demographic and migration processes? 4. how do ethnic and demographic processes affect the population’s standard of living and socio-economic situation in the north caucasus? we used the following sources of information in our study: population current record data (mainly demographic and migration records) available on the official website of the federal state statistics service rosstat and its regional offices in the north caucasus; data collected during the census of 1989 (the all-union 1989 census), 2002 and 2010 (all-russia national population census of 2002, national population census of 2010 ) on the ethnic structure of the north caucasus’ regions’ inhabitants; design and analysis data of the rating agencies ria-rating and expert ra on the level of social and economic development of the north caucasus regions; results of the previous complex and industry (demographic, migration, ethnic, social and economic) researches of the north caucasus conducted, including the authors of the article [1][8]. the main research methods used are quantitative ones. they are conventionally applied for this kind of work. the focus is made on the multilateral statistical analysis of demographic, migration, ethnic and socio-economic processes, ratios and indicators of birth rate, death rate, natural population increase, net migration, the share of the various ethnic groups in the general population being calculated. the main qualitative methods used in this research are aimed at identifying the territorial disparities of the social and demographic processes in the north caucasus combined with the analysis of the temporal trends over the last 15 20 years. those methods are widely applied in social and economic geography. the method of multi-immensity is based on the classic geographical papers and it is understood as the manipulation of scale ranks of the territory within the particular object of the research to find the spatial patterns of events and processes. multi-immensity of this study includes the following territorial “steps”: federal (comparison of advances in systems science and application(2016) vol.16 no.4 83 the north caucasus with the processes of the russian federation as a whole), regional (the subjects of the russian federation in the north caucasus), intra-regional (municipal districts). the typological method deals with the development of such groupings of geographical features used to identify the qualitative differences. we have proposed a typology of the north caucasus regions according to the nature of the population. this method allowed us to conduct the zoning of the north caucasus territory in terms of socio-economic development and living standards. in general, finding the territorial disparities of the processes was carried out through the intensity vectors in the directions of north south and west east. the main spatial heterogeneity is the asymmetry of the mountain land (south or more precisely south-east) and plainl and (north-west) parts of the north caucasus. the mountain part of the republic includes dagestan, chechnya, ingushetia, north ossetia, kabardino-balkaria, karachay-cherkessia and the plain part includes rostov, krasnodar and stavropol regions, and the republic of adygea. 3 main results and their discussion specifics features of the processes in the mountainous and plain land parts of the north caucasus. demographic processes. in the 60 – 80s of the 20 th century, the plain land and mountainous parts of the north caucasus “changed” their features of the demographic conditions. in the plain land part, the high level of natural growth inherent in it fell to the population replacement level since the second half of the 19 th century and currently to the restricted population replacement level. in the mountainous area, on the contrary, the restricted population replacement level of 1960s turned into the expanded one and even population explosion” was observed there. at the present time, the north caucasus differs greatly for the better in terms of birth and death rate from the rest of russia and the region can be quite clearly divided into “relatively successful south” (mountainous area) and “disadvantaged north” (the plain land).the range of the crude birth rate (cbr) is from 25 29 to 10 11 ‰. in all the republics, the cbr is quite high. for example, in chechnya it is 25 29 ‰, in ingushetia 21 27 ‰, in dagestan about 19 ‰.when moving to the north-west, the birth rate falls and reaches its minimum in rostov region (10 11 ‰.). at the same time, since the 2000s, the slow growth of birth rate is quite obvious in all regions of the north caucasian plains (from 9 10 and 11 12 ‰) being a consequence (as will be proved below) of change in the ethnic and demographic structure and increase in the share of non-slavic, more demographically active population. however, it should be noted that in most regions of the north caucasus, as well as in the whole country, in the 1990s early 2000s the birth rate was significantly lower than it is today. the particular negative peak at the border of the centuries was due to the widespread protracted demographic crisis caused by the serious political and socio-economic disturbances. the second most important demographic process is population mortality. in rostov region, its indicators are the highest and even slightly higher than the national average ones, they are slightly lower in krasnodar and stavropol regions. in the republics, the crude death rate (cdr) is not high. as is known, the value of cdr is highly dependent on the age structure of the population and life expectancy. in mountainous republics, the age structure of the population is much younger than in that in the plain land regions. for example, in stavropol region, the share of young people is 18%, older people is 22%, and in chechnya, these figures are 35 and 9% respectively, in ingushetia they are 31% and 10%, in dagestan they are 27% and 11% respectively. at the same time, the republics show the highest average life expectancy: in ingushetia it is almost 81 years old, in north ossetia 79 years, in dagestan, kabardino-balkaria and karachay-cherkessia it is 78 years (in plain land regions it is lower than 70 years). as a result, the north caucasus republics show the highest natural population growth in russia: in chechnya it is 19.9 ‰, in ingushetia – 17.7 ‰, in dagestan – 13.3 ‰. in the plain land regions, the natural population growth is slightly positive or even negative. 84 vitali s.belozerov, natalia a. shchitova ,vasiliyv.chichikhin: the asymmetry of the present-day social and … thus, the modern period in the north caucasus is characterrised by the increase in the interregional disparities in demographic development the south shows an impressive multi-ethnic human potential increase, and the north the mono-ethnic slavic community of nations is aging, weakened and eroded. migration processes. it is obvious that the demographic processes govern the vector of the interregional migration. from the 1960s the population of the mountain are as accumulating a powerful demographic potential was actively drawing upon the labour deficient plain territories. plain land areas are, on the one hand, a powerful accumulator of the north caucasian migration flows, and on the other hand, they are a kind of “a corridor of the migration winds” between the north caucasian republics and the rest of the country. being at the forefront of the caucasus problems, krasnodar, rostov and especially stavropol region appeared to be the russian base in the caucasus and at the same time the buffer against the acuteness of the ethno-political and socio-economic crisis. the location next to the hot spots of chechnya, dagestan, and the zone of the osset-ingush conflict, etc. and what is more as acute hearths of inter-ethnic tensions in the countries of the south caucasus (georgia-south ossetia, azerbaijani-armenian, georgian-abkhazian) resulted in a mass influx of the forced migrants to stavropol, don and kuban regions. gradually the stressful migration factors have lost their significance. the current quiet geopolitical conditions have exacerbated the impact of economic and demographic factors on the migration process. the inter-regional migration growth in the plain land regions will remain positive due to the high demographic potential in the republics. the north caucasian peoples have high demographic potential growth rates and are actively advancing on the plain land, adjacent to the caucasus territories. the population and young people in the first place, will be leaving to choose new places of residence in the economically developed regions. at the same time, the ukrainian instability hearth is heading toward the renewed stress migration flow in the regions of the plain land part of the north caucasus. territorial heterogeneity of demographic and migration processes in the north caucasus can be divided into four types of the regions with their specific features of population formation: – regions, where natural population growth exceeds the total one (chechnya) and has a steady migration outflow; – regions, where the excess of the natural and total growth alternate (dagestan, ingushetia, north ossetia, kabardino-balkaria, karachay-cherkessia) and at certain periods they either accept or give a large number of migrants; – regions, where natural growth is negative or very low, and the total one is almost always positive or higher than the natural growth (stavropol and krasnodar regions), the population growth is achieved due to the migrants. here we can also observe the growth of the intraregional disparities in demographic development in the form of the formation of areas of high positive and high negative population growth being the consequence of the transformation of the population ethnic structure. – regions, where the total population growth is negative or slightly positive (rostov region, adygea). migration there does not overlap the natural population decline. ethnic processes. the ethnic structure of the north caucasus population is largely dependent upon the correlation of the three groups of ethnic groups to be the russians, the title nations of the national entities (the adygei, karachai, cherkesses, kabardin, balkar, ossetian, ingush, chechen and the peoples of dagestan), and a number of non-indigenous peoples (the greeks, armenians, azerbaijani, germans, meskhetian turks, etc.). the dynamics of the ethnic structure is dependent upon the balance of the natural and mechanical population growth. from the beginning of colonisation to the present time, two trends are observed: at an early stage, it is “slavyanization” and at present, it is strengthening of the caucasian characteristics. advances in systems science and application(2016) vol.16 no.4 85 for nearly two centuries, the two main ethnic groups – the slavic people (mostly russians) and the title nations of the republics had been changing roles in the course of their interaction. initially, the main ethnic group to ensure the inclusion of the mountain peoples in the russian economy and social sphere were russians. currently, the north caucasian peoples having high demographic growth rates are actively “attacking” the plain land part of the north caucasus, which is seen not only in expanding the settlement territory but also in active participation in the economic activity [6]. at the same time, the expansion of the title ethnic groups of the republics originally was seen in the rural districts and the agricultural sector, and now the expansion takes place in the cities and in the new sectors of the economy (including business, management, etc.) and more young people are found to be engaged in the educational sphere. simultaneously, the title ethnic groups are concentrated in “their” original regions, whereas the “aliens” are pushed to leave. the main factor of ethnic homogenisation of there publics’ population is not an ethnic incompatibility and hatred, but growing competition for becoming scarce job and study placements, comfortable living conditions, water and land. these processes often turn into covertly or overtly confrontational and they are accompanied by the change in the settlement of the ethnic groups [5]. these changes are particularly well seen in the geography of the displacement of the russian ethnos, having gone through several stages in its evolution in the republics of the region. at the initial stage, they chose the administrative, metropolitan and industrial centers as their place of residence, as a rule. at the next stage, there occurs a slowdown in the growth of the russian population and the outflow of the russians from the rural areas. later the inclusion of the title ethnic groups of the republics in the urbanisation process in the conditions of expanded reproduction was accompanied by their active resettlement in the rural areas by means of “pushing out” and displacement of the russian population living there. the third stage is characterised by the sustained reduction in the absolute and relative indicators of the russians in both rural and urban areas. the outflow of the russians was seen throughout all the settlements and especially the cities, including the capital ones in the conditions of the deep economic crisis and the lack of effective national policy, high ethnic tensions, actively propagating xenophobia and actual civil insecurity of the russian and russianspeaking population. the structure of the non-indigenous population of the north caucasus is changing rapidly. the active outflow of some nations (particularly the germans emigrating to germany) combined with the increasing inflow of other nations (the armenians) and the emergence of the new nations (the azerbaijanis, meskhetian turks). some of them, first of all, the armenians and the greeks, and in recent years meskhetian turks have formed the compact settlement areas. especially noticeable is the increase in the proportion of the armenian ethnos. if in the national republics the armenian population was declining (except adygea republic), in the steppe areas of the north caucasus the influx of the armenians has increased dramatically, primarily in stavropol territory and kuban [3][4]. new in their resettlement was the settlement, along with the cities and suburbs, in the rural areas. the indicator of the concentration of the armenian population in areas traditionally inhabited regions of the steppe ciscaucasia dropped significantly. a new feature about their resettlement was the settlement in the cities, suburbs, and rural areas. the indicator of the armenian population concentration in the traditionally inhabited areas has significantly reduced. contrasting zones of socio-demographic area of the north caucasus. the course of the modern demographic, migration and ethnic processes in the north caucasus depends on both political and geographical changes and socio-economic conditions in the caucasus as a whole in the post-soviet years. the active migration of migrants from the crisis regions resulted in a more stable transformation of the ethnic structure of both receiving and giving territories has changed the trends of their social and economic development. the north caucasus regions occupying the peripheral position in the russian sociogeographical space, form a rather complex discrete conglomerate, its parts being contrasted with 86 vitali s.belozerov, natalia a. shchitova ,vasiliyv.chichikhin: the asymmetry of the present-day social and … each other on the main socio-economic characteristics [10], the inter-regional disparities in terms of welfare, development of social and cultural infrastructure and other indicators have reached here enormous values. the comparative analysis of the socio-economic situation in the north caucasus regions show a close correlation between the parameters of ethno-demographic and socio-economic processes [16]. the discrepancy between the level of the socio-economic well-being and demographic prosperity of the population is becoming a source of the key contradictions adding to the instability of the regional development. the crisis of the early 1990s led to the considerable deterioration of the population’s living conditions in all areas of the north caucasus, but the economic boom of the 2000s manifested itself in many ways and led to a significant socioeconomic differentiation of the regions and to the formation of depression areas and areas of socio-economic growth. currently, the north caucasus can be divided into three contrasting zones, differing in the nature and pace of socio-economic development and the living standards of the population [15]. the western zone is represented by krasnodar territory and rostov region being stable leaders, developing in the most dynamic way in economic and social sphere. however, the falling migration growth there does not compensate for the natural population decline. the central zone includes one region – stavropol territory, where a living standard is going up due to the economic growth, unemployment reduction, income increase and decrease in the proportion of the poor people. demographic and migration conditions are relatively favorable there. the western and central zones are characterised by multi-ethnicity strengthened through active, sometimes a point settling of the title peoples of the north caucasus republics of the south caucasus states, “pushing out” the russians from the old russian regions. many traditionally “rural” ethnic groups (dargin, chechens) having settled in the eastern agricultural areas are gradually shifting to the west and to the cities [2]. the influx of dargin, chechen, karachay, cherkess in large cities and towns is growing due to educational migration, which is becoming a real channel of social mobility and transformation of the ethnic villagers into townspeople [1][14]. all this increases the inter-ethnic tensions, forms ahostile attitude of the local population to the ethnic migrants and raises domestic aggression. the southern zone–the north caucasus republics can be divided into two subtypes. the first one the least affected in the 1990s early 2000s karachay-cherkessia, north ossetia and kabardino-balkaria, which currently are having deep economic and social problems, but having several high indicators in the social sphere (e.g. housing, the number of students, health indicators). the second subtype of the southern zone includes chechnya, ingushetia characterised by the low levels of socio-economic development and a large number of unresolved issues negatively affecting the quality of life in general. these republics are the outsiders as regards the difficulties in the labour market, observed for a long time, despite the efforts of the federal and regional authorities. the highest unemployment rate has been recorded in ingushetia: almost half of its inhabitants do not have a permanent place of employment. in two regions chechnya and dagestan the unemployment rate is over 10% [16]. it is believed that these figures are not reliable as the population of these republics have shadow employment [9], existing at least, in two forms. the first form of employment is non-incorporated entrepreneurs engaged in farm households and producing products for sale. the second form is work for private individuals [13]. in this subtype, the russian population in chechnya is declining fast: from 1989 till 2010 it has fallen 11 times, in ingushetia the figure is almost 8 times, in dagestan it is 1.6 times. there has been a rapid reduction in long-time territorial expansion of the russians in the caucasus. the most dramatic was the fate of nearly 300-thousand russian population of the chechen republic from which the forced exodus has exceeded 90%. the most striking change in the ethnic structure of the population is observed in the “old russian” regions of chechnya, ingushetia and advances in systems science and application(2016) vol.16 no.4 87 dagestan. over the last 50 years, not only the dominance the dominance of the russian population but also the prevalence has been lost and in the 1990s, these trends increased significantly and in a number of “old russian” areas almost complete exodus of the russian population is observed [6]. 4 conclusions 1. the north caucasus having a high population size is gradually losing its demographic advantages. the territories with both natural population decline and a negative migration balance are expanding. in the plain land regions, the falling migration gain does not compensate for the natural population decline. 2. multi-ethnicity and derussification of the migration growth has considerably increased. the area of the positive migratory growth of the russians and armenians is falling rapidly. the migration of the rural ethnic groups into the cities is growing and the ethnic structure of those groups is being transformed. 3. the regions of the north caucasus, being at the crossroads of inter-regional and crosscountry migration flows fulfil the function of integration of different nations. here, numerous ethnic cultures of numerous language groups interact (slavic, armenian, iranian, greek, german, nakh-dagestani, abkhazian-circassian, turkic, etc.). at the same time, russian population in all regions of the north caucasus is rapidly going down. 4. strengthening of multi-ethnicity in the plain land regions of the north caucasus owing to the active settlement of the title peoples of the north caucasus republics of the south caucasus states, “pushing out” of the russians from the old russian regions, enhances inter-ethnic tensions, forms the hostile attitude of the local population to the ethnic migrants and growth of domestic aggression. 5. the russians outflow from the republics prevents from the rapid revival of the industrial economy sectors. areas of the russians’ mass exodus are a risk zone for the integrity of the russian state and indicators of ethnic tensions proving the propagation of nationalist sentiments and actions, extremism and other extreme forms of ethnic tensions manifestations. the most negative result of this situation is the russian center influence weakening and the spread of political, economic and religious expansion of the muslim countries. 6. the impact of migration and ethno-demographic processes on the development and socioeconomic stability in the north caucasus regions is evident. the discrepancy between demographic and socio-economic well-being becomes a source of political instability in the republics. however, per capita income is falling behind the population growth, despite some economic recovery. high birth and life expectancy rates raise dependency burden on the economically active population, whose structure is now dominated by the unemployed. the location of regions with high birth rate and younger age structure of the population in one place requires the adequate response from the federal authorities. the socio-economic policy is to be set taking into account the regional conditions and specific measures aimed at eliminating contrasts in socio-economic development of the population are to be proposed taking into account the ethnodemographic characteristics. acknowledgment this researchers was supported by the russian foundation for basic research № 16-06-00179 "development and approbation of system of geo information monitoring of ethno-demographic processes (on the example of regions of the north caucasus)". references 88 vitali s.belozerov, natalia a. shchitova ,vasiliyv.chichikhin: the asymmetry of the present-day social and … [1] belozerov v.s., panin a.n., prikhodko r.a. and chihichin v.v. 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"comparative analysis of the socio-economic development of the north caucasus regions". science. innovation. technologies. no.1, pp.161 174. corresponding author vitali s.belozerov can be contacted at: yal05@mail.ru. advances in systems science and applications (2013) vol.13 no.3 233-248 ways of fusing different types of information and how systemic yoyo model is applied in complex systems evaluation and estimation xiaojun duan1 and yi lin2 1department of mathematics and systems science national university of defense technology changsha 410073, pr china 2department of mathematics slippery rock university slippery rock, pa 16057 usa abstract continuing the works in literature[1], we show in this paper how different types of information can be fused together consistently in order to produce accurate evaluations and estimations for complex systems. the theoretical part of this presentation is based on the standard statistical reasoning, while the ending part constructs three case studies in order to validate the main thinking logic and results obtained in literature[1] and in this paper. it is shown that (1) for linear systems, when fusing data of different types, the weights placed on the data have profound effects on the outcomes and the achieved precisions, meaning that in this case, the unique optimal weight matrix is determined by the precisions of the data (gauss-markov theorem of linear models); (2) for nonlinear models, when fusing heterogeneous sets of data with varied scales of precision, the structure of the weight matrix is no longer uniquely determined by the precisionsbut also related to the degree of model nonlinearity, indicating that the classical gauss-markov theorem of linear models no longer holds true. at the same time, a specific method of determining the optimal weighting factor and the relevant computational method for estimating the parameters are established. combined with the process of conserved information applied in systems evaluation, we provide three case studies, including (1) how to quantitatively measure prior knowledge and observational data so that prior knowledge can be considered in obtaining much improved optimal systems evaluations; (2) how to excavate new sources of observational data of processes so that the established models can be validated jointly using process data collected under different test environments and the directly measured information of the specific indices of concern in order to improve the quality of systems evaluation and estimation and to obtain model validation results of better accuracy; and (3) how to more effectively fuse prior knowledge and heterogeneous sets of data. all of these case studies further witness the epistemological validity of the information conservation existing in the systemic recognition process beneath the systems model description, prior knowledge, and observational data, and their transformational relationship, as obtained in literature[1]. xiaojun duan, yi lin:ways of fusing different types of information and how systemic... 234 because other than establishing the theory, particular procedures are provided, conclusions of this work can be directly employed in system evaluations and estimations and related works. this work shows how systemic thinking can be practically applied to benefit the efforts of system evaluation and model estimations involved in various engineering projects. keywords systemic yoyo model, data fusion, linear/nonlinear model, parameter estimation, gauss-markov theorem 1 introduction in literature[1], after clarifying the relationship between observational quantities and the target indices to be measured, the systemic yoyo model[2] is established for system evaluations and estimations (fig.1), where the form of the model, observational data, and prior knowledge are the main sources of information useful for the evaluation, estimation, and prediction of the performance index of the system to be measured. fig.1 the systemic yoyo model for complex system evaluation from analyzing the characteristics of and connections between the three main sources of information, model descriptions, prior knowledge, and observational data, for system evaluations and estimations, the following conservation law of information for system analysis is obtained. aeim ×beid × ceip = a (1) where a, b, and c are constants, im stands for the information content described by the model, id the information content of the autoptic test data, and ip the information content of the prior knowledge. the constant a should somehow depict the minimum amount of information required to satisfy the given precision (in the estimate of the model or parameters), where the precision is 235 advances in systems science and applications (2013) vol.13 no.3 given in terms of the model accuracy and parameter estimation precision. for the detailed expressions of im , id, and ip , see [1]. with this law of conservation is established, duan and lin use it to investigate the evolution direction of the process of a system evaluation and estimation. continuing what is obtained in literature[1], in this paper, we show that for linear systems, when different types of data are available for systems evaluation and estimation, then the analysis outcomes and precisions achieved are greatly determined by the weights placed on the data. more specifically, the gauss-markov theorem, established on the method of least squares method, holds true. however, when nonlinear systems are involved, the classical gauss-markov theorem of linear models no longer holds. that is, the structure of the weight matrix is not uniquely determined by the precisions of the available data. to this end, we provide a specific method of determining the optimal weighting factor and the relevant computational method for estimating the parameters. after this theoretical exploration, combined with the conservation law of information of system evaluations and estimations, we construct three case studies to show (1)how to quantitatively measure prior knowledge and observational data so that better evaluation and estimation results can be obtained using prior knowledge; (2)how to excavate the available observational data of processes so that the process information collected under different test environments and directly measured data of the specific indices can be employed jointly to fine-tune the model, leading to improved system evaluations, estimations, and more accurate test results, and (3)how to make prior knowledge and heterogeneous sets of data work effectively together. these case studies further verify the validity of the conservation of information of the model information, prior knowledge, and observational data in the recognition process of systems and the evolutionary relationship between three main sources of information. this paper is organized as follows: section 2 looks at various ways one can fuse information in his analysis of complex systems. section 3 focuses on three specific cases studies. and, the paper is concluded by section 4. 2 ways information fusion takes place in processes of system evaluation with the requirement of precision given, we can optimize the process of a system evaluation. that is such a problem as how to obtain the optimal evaluation results when multiple types of models and multiple kinds of data are available. this end can be analyzed by placing various weights on the multiple kinds of data and prior information. xiaojun duan, yi lin:ways of fusing different types of information and how systemic... 236 when dealing with information by combining heterogeneous data, the most typical case is the fusion of such information that are of different types and various precisions. when observational information is expressed by using a parametric model, the problem of how to fuse the information together can be transformed into that of estimating the parameters of some regression models. here, by different types of information, we mean such information that is of different functional relationships with the parameters to be estimated so that their various orders of derivatives are also different. if in our treatment we have to deal with different types of information of varying precision, different weightings placed on these information will have direct effect on the estimations of the parameters. sometimes, the effects can also be quite significant. hence, how to place weights on the information that are of heterogeneous types and varied precisions becomes a key technique for obtaining high accuracies in the parameter estimations. as for the parameter estimations of linear regression models, gauss-markov theorem provides the most optimal weighting method for observational data of different precisions [3]. as for nonlinear regression models, current publications have assumed that all observational data have the same scale of precision. that is, the random errors in these data are identically independently distributed [3]. to this end, our work in this section theoretically shows that when jointly dealing with heterogeneous sets of data of varied scales of precision, the structure of the weight matrix is no longer uniquely determined by the precision of the observational data; instead, it also has something to do with the degree of nonlinearity of the model, which can be measured by different orders of derivative functions. that is, the classical gauss-markov theorem of linear models does not hold true anymore. in the following, we will specifically address the problem of how to determine the most optimal weighting factor, while providing the relevant computational method for estimating the parameters. 2.1 the optimally weighted information fusion of linear system evaluation according to the research on the mean square errors of parameter estimations of linear systems, it is readily to show that the estimate corresponding to the optimal weighting factor is the bayesian estimation, which is gauss-markov theorem[1,6]. for the problem of estimating parameters βp×1, assume that there are the following two types of observational information:{ ym×1 = xm×pβp×1 + εm×1 eε = 0;cov(ε, ε) = σ2 1im×m (2) and { β̃k×1 = zk×pβp×1 + ηk×1 η ∼ n(0, σ2 2ik×k) (3) 237 advances in systems science and applications (2013) vol.13 no.3 satisfying eεηt = 0. if we treat model (2) as the direct observational data, while (3) the prior information on the parameters, then the bayesian estimation of the parameters is the solution of the following extremum problem: min β∈rp σ−2 1 ∥ y −xβ ∥22 +σ−2 2 ∥ β̃ − zβ ∥22 (4) that is given as follows: β̂b = (σ−2 1 xtx + σ−2 2 ztz)−1(σ−2 1 xty + σ−2 2 zt β̃) (5) therefore, the following conclusions can be shown readily[3-5]: (1)eβ̂b = β; that is the bayesian estimate is unbiased; and (2)mse(β̂b) = tr(σ−2 1 xtx + σ−2 2 ztz)−1 < mse(β̂ls) = σ2 1tr(x tx)−1 that is, by making use of appropriate prior knowledge, one can always improve the estimation precision of the parameters, where tr(ak×k) = ∑k i=1 ai,i is the trace of the matrix. when fusing these data, the weights placed on the data have profound effects on the outcomes and the achieved precisions. these conclusions indicate that when fusing observations of different precisions, the unique optimal weight matrix is determined by the precisions of the data, which in essence is still the gaussmarkov theorem of linear models established on the least squares method. 2.2 the optimally weighted information fusion of nonlinear system evaluation for nonlinear models, we show in theory that when fusing heterogeneous sets of data with varied scales of precision, the structure of the weight matrix is no longer uniquely determined by the precisions. that is, the classical gaussmarkov theorem of linear models no longer holds true. at the same time, we establish a method for determining the optimal weighting factor and the relevant computational method for estimating the parameters. for the sake of convenience, for the parameter θ that is to be estimated, we assume that we have the prior information (3) for a linear model, where the error satisfies eεηt = 0. so, the weighting problem of fusing these two types of observational data can be reduced to the following minimization problem: min ρ∈r1,ρ>0 min β∈rp ∥ y − f(x,β) ∥22 +ρ ∥ β̃ − zβ ∥22 (6) then, we have the following result: theorem1. denote s(β) =∥ y −f(β) ∥22 +ρ ∥ β̃−zβ ∥22, s(β̂) = minβ s(β), c =∑m i=1 ḟ 2 i +ρ ∑k i=1 z 2 i , d = ∑m i=1 ḟif̈i, ξ = ∑m i=1 ḟiεi+ρ ∑k i=1 ziηi, a = σ2 1 ∑m i=1 ḟ 2 i + xiaojun duan, yi lin:ways of fusing different types of information and how systemic... 238 ρ2σ2 2 ∑k i=1 z 2 i , then under the assumed conditions (i) and (ii), the following estimation holds true: β̂ − β =c−1ξ + c−2 m∑ i=1 f̈iεiξ − 3 2 c−3dξ2 + c−3 m∑ i,j=1 f̈if̈jεiεjξ − 9 2 c−4d m∑ i=1 f̈iεiξ 2 + 9 2 c−5d2ξ3 (7) with the bias and mean square error approximated as follows: e(β̂ − β) = −1 2σ 2 1c −2d− 3 2c −3dρ(ρσ2 2 − σ2 1) ∑k i=1 z 2 i mse(β̂) =c−2a+ 6c−4d2σ4 1 + 3c−4aσ2 1 m∑ i=1 f̈2 i + 135 4 c−6d2a2 − 36c−5ad2σ2 1 (8) where the assumed conditions (i) and (ii) are given below: (i) the derivative of f(t, β) with respect to the parameter β exists and is continuous, and lim m→+∞ 1 m m∑ i=1 = ( df(ti, β) dβ )2 = ω1(β) > 0 (9) (ii) the second order derivative of f(t, β) with respect to β exists and is continuous, and lim m→+∞ 1 m m∑ i=1 = ( d2f(ti, β) dβ2 )2 = ω2(β) (10) proof. taking the series expansion of ṡ(β̂) about the true value of the parameter β produces ṡ(β̂) = ṡ(β) + s̈(β)(β̂ − β) + 2−1... s (β)(β̂ − β)2 notice that ṡ(β̂) = 0, ṡ(β) = −2( ∑m i=1 ḟiεi + ρ ∑m i=1 ziηi), s̈(β) = 2c − 2 ∑m i=1 f̈iεi, ... s (β) = 6d − 2 ∑m i=1 ... f iεi, so, ignoring all terms of third or higher order derivatives produces β̂ − β = c−1 { ξ + m∑ i=1 f̈iεi(β̂ − β)− 3 2 d(β̂ − β)2 + 1 2 m∑ i=1 ... f iεi(β̂ − β)2 } = c−1ξ + c−2 m∑ i=1 f̈iεiξ − 3 2 c−3dξ2 + c−3 m∑ i,j=1 f̈if̈jεiεjξ − 9 2 c−4d m∑ i=1 f̈iεiξ 2 + 9 2 c−5d2ξ3 239 advances in systems science and applications (2013) vol.13 no.3 that is equ.(7) holds true. because the expected values of normal variables to odd powers are zero and ε and η are independent, we obtain the first equation in equ. (8) by ignoring the error terms of the fourth and higher orders and then calculating the expected value such that e(β̂−β)2 = c−2eξ2+3c−4e( m∑ i=1 f̈iεi) 2ξ2+ 45 4 c−6d2eξ4−12c−5d m∑ i=1 f̈iεiξ 3 (11) because eξ2 = σ2 1 m∑ i=1 ḟ2i + ρ2σ2 2 k∑ i=1 z2i =̂a, eξ4 = 3σ4 1( m∑ i=1 ḟ2i ) 2 + 6ρ2σ2 1σ 2 2 m∑ i=1 ḟ2i k∑ i=1 z2i + 3ρ4σ4 2( m∑ i=1 z2i ) 2 = 3a2, e( m∑ i=1 f̈iεi) 2ξ2 = 2d2σ4 1 + σ4 1 m∑ i=1 ḟ2 i m∑ i=1 f̈2 i + ρ2σ2 1σ 2 2 m∑ i=1 f̈2 i k∑ i=1 z2i = 2d2σ4 1 + σ2 1a m∑ i=1 f̈2 i , e m∑ i=1 f̈iεiξ 3 = 3dσ4 1 m∑ i=1 ḟ2i + 3ρ2σ2 1σ 2 2d k∑ i=1 z2i = 3daσ2 1, substituting these equations into equ.(11) leads to e(β̂−β)2 = c−2a+6c−4d2σ4 1 +3c−4aσ2 1 m∑ i=1 f̈2i + 135 4 c−6d2a2−36c−5ad2σ2 1 that is the second equation in equ.(8). qed. theorem2. for mse(β̂)(ρ) in theorem 1, the solution to the following minimization problem min ρ mse(β̂)(ρ) (12) exists, satisfying min ρ mse(β̂)(ρ) < minρmse(β̂)( σ2 1 σ2 2 ). proof. because lim ρ→+∞ mse(β̂)(ρ) = σ2 2( ∑k i=1 z 2 i ) −1, mse(β̂)(ρ) is infinitely difxiaojun duan, yi lin:ways of fusing different types of information and how systemic... 240 ferentiable on [0,+∞), mse(β̂)(ρ) has its minimum value on [0,+∞). because d dρ mse(β̂)(ρ) = ( k∑ i=1 z2i )(−2c−3a+ 2ρc−2σ2 2 − 24c−5d2σ4 1 − 12c−5σ2 1 m∑ i=1 f̈2 i + 6ρc−4σ2 1σ 2 2 m∑ i=1 f̈2 i − 405 2 c−7d2a2 + 135ρc−6d2aσ2 2 + 180c−6ad2σ2 1 − 72ρc−5d2σ2 1σ 2 2) (13) we have d dρ mse(β̂)(0) = −( k∑ i=1 z2i )(2( m∑ i=1 ḟ2 i ) −2σ2 1 + 93 2 ( m∑ i=1 ḟ2 i ) −5d2σ4 1+ 12( m∑ i=1 ḟ2 i ) −5 m∑ i=1 ḟ2 i m∑ i=1 f̈2 i σ 4 1) < 0 also, because lim ρ→+∞ d dρmse(β̂)(ρ) = 0, and when ρ → +∞, each term starting from the third in equ.(13) is a higher order infinitesimal when compared to the previous two terms; and when ρ > σ2 1σ −2 2 , the sum of the previous two terms is greater than zero there is ρ0 > 0 so that d dρmse(β̂)(ρ) > 0 when ρ ∈ [ρ0,+∞). therefore, the solution of min ρ mse(β̂)(ρ) satisfies ρ̂ ∈ (0, ρ0). and because d dρ mse(β̂)( σ2 1 σ2 2 ) = σ4 1 2c5 k∑ i=1 z2i (33d 2 − 12( m∑ i=1 ḟ2i + σ2 1 σ2 2 k∑ i=1 z2i ) m∑ i=1 f̈2i ) ̸= 0, ρ = σ2 1σ −2 2 is not a solution of min ρ mse(β̂)(ρ). hence, theorem 2 is proven. qed. remarks: (1) theorem 2 indicates that for nonlinear models, because their least squares estimates are generally biased, when fusing observational data of varied scales of precision, the weight matrix obtained by using gauss-markov theorem, which is derived out of the least squares estimation for linear models, is no longer optimal. the optimal weights can be obtained by solving the minimization problem in equ.(12). (2) if the prior knowledge equ. (3) is a nonlinear model and the first and second order derivatives of the model are the same as those of nonlinear function f(x,β), then the optimal weight can be approximated by ρ = σ2 1σ −2 2 . 241 advances in systems science and applications (2013) vol.13 no.3 2.3 the parameter estimation of heterogeneous data fusion when solving problem (6), one can simply follow the following iterative method: step 1: for an initial weight value ρ0 = σ2 1σ −2 2 , solve the following minimization problem min β∈r1 ∥ y − f(x,β) ∥2 +ρ0 ∥ β̃ − zβ ∥2 (14) to obtain its solution β̂(1); step 2: calculate the mean square error mse(β̂(1))(β̂(1), β) of the estimated parameter at β̂(1); step 3: solve the minimization problem min ρ>0 mse(β̂(1))(β̂(1), β) to obtain ρ1; and step 4: repeat steps 1-4 with the initial value ρ0 replaced by ρ1 until the estimated value of the parameter becomes stable. fig.2 the three samples and their relationships to the information flow of the yoyo model 3 case studies in this section, we will use case studies to illustrate the systemic yoyo model and the evolutionary process naturally existing in system evaluations and estimations. fig.2 shows the connection of these examples and their individual relationships with the information flow of the systemic yoyo model. we will mainly consider the scenario of supplementing prior information. because of the shortage of observational data, the convergence of the system evaluation and estimation process is very slow or becomes stagnant after converging to a certain degree. however, after additional prior information becomes available, the stagnated process will continue to converge; and the speed of the resumed convergence is dependent on the quality of the newly supplied prior information and how consistent it is with the observational data. we will look at three examples to respectively illustrate the following: (1) how to quantitatively measure both the prior and data information; (2) how to xiaojun duan, yi lin:ways of fusing different types of information and how systemic... 242 excavate a new source of observational process data so that the quality of the ultimate system evaluation and estimation is improved; and (3) how to fuse prior information with heterogeneous sets of data effectively. fig.3 the relationship between the accuracy of post fusion parameter estimation and sample size (the left represents posterior fusion estimation accuracy (measured by information); the right the posterior fusion accuracy improvement (measured by gain in information)) table 1 the relation between the fisher information gain and the increase in sample size increase of sample size 1 2 3 4 5 6 7 informa(1)without prior 29.2893 12.9757 7.7350 5.2786 3.8965 3.0284 2.4411 (2)fuse prior,w = 0.4 9.3127 6.0794 4.3675 3.3328 2.6511 2.1741 1.8249 tion gain (3)fuse prior,w = 1 3.8965 3.0284 2.4411 2.0220 1.7106 1.4716 1.2836 increase of sample size 8 9 10 11 12 13 14 informa(1)without prior 2.0220 1.7106 1.4716 1.2836 1.1325 1.0089 0.9062 (2)fuse prior,w = 0.4 1.5601 1.3537 1.1892 1.0555 0.9451 0.8527 0.7744 tion gain (3)fuse prior,w = 1 1.1325 1.0089 0.9062 0.8199 0.7464 0.6833 0.6287 increase of sample size 15 16 17 18 19 informa(1)without prior 0.8199 0.7464 0.6833 0.6287 0.5809 (2)fuse prior,w = 0.4 0.7075 0.6496 0.5992 0.5551 0.5161 tion gain (3)fuse prior,w = 1 0.5809 0.5389 0.5017 0.4686 0.4390 example1. let us look at the information measurement of the prior and observational data. take the prior parameter variance to be sigma0 = 50 and the sample variance sigma1 = 100. let us vary the sample size from 1 to 20 and consider three scenarios: no prior information is fused, and prior information is fused with the 243 advances in systems science and applications (2013) vol.13 no.3 consistency weights w = 1 and w = 0.4, respectively. figure 3 shows the relationship between the accuracy of the post-fusion parameter estimation and the sample size. the fisher information gain is shown in table 1. evidently, when the threshold of fusion accuracy is fixed at 1.5, one needs to repeat his test ten times if he does not fuse any additional prior information; if he fuses additional prior information and sets its weight at w = 0.4 (that means the consistency between the additional prior information and the test data is measured by weight 0.4), then he needs to repeat the test 9 times; and if he fuses additional prior information with weight set at w = 1 (that means the additional prior information has very good consistency with the test data), then he only needs to repeat the test 6 times. this result indicates that with correct fusion of prior information, the same requirement of precision can be met with a fewer number of times the test is repeated. example2. in this case study, we will see how we can speed up the convergence of our recognition of the underlying system by making use of process information and data. the precision evaluation of active homing radar [6] is a complex recognition process of systems. the impact error of clustered warhead missiles with active homing radar is mainly composed of the measurement error of the radar navigation system, the instrumental error of the inertial navigation system (ins), the method error of the terminal guidance, the error of the distribution, and some random error. let us take the impact error of an active homing radar system [6] as our system evaluation performance index and compare the outcomes of the following two methods: one is to fuse the different observational information, directly observed index data, and some indirectly observed data; and the other the point estimate method [7] for the impact error. the main difference here lies in that the later, more traditional method uses only the final impact point error information so that the evaluation outcomes are strongly influenced by the random error of the impact points, while the former method validates the procedure error model with systematic error and the characteristic of the random error by making using of all the observational information. that is how the former method provides more robust evaluation results. in order to avoid analyzing a heterogeneous population created by different testing states, we will transform these different testing states to standard full range process testing states so that the consequent analysis will become manageable. the impact errors of different testing states are denoted as follows: △lradar stands for the assessed impact deviation caused by the error of the radar measurement in the standard overall operational test; ∆l̃radar the measured impact xiaojun duan, yi lin:ways of fusing different types of information and how systemic... 244 deviation caused by the error of radar measurement in a substitute test. for a missile with active homing radar, its warhead-target relative positional x-, y-, and z-errors are mainly caused by its radar’s measurement error. what an actual radar measures includes the range r, azimuth angle a, and elevation angle e. these measurements satisfy the following transformational connection with the measured warhead-target relative x-, y-, and z-positions: x = r cose cosa y = r sine z = r cose sina (15) that is, the performance index (that is to be evaluated), the relative positional error of warhead and target, is a function of the observational quantities (range r, azimuth angle a, and elevation angle e). for details, see fig.4. fig.4 the relationship between radar measurements r, a, and e, and the warhead-target relative x-, y-, and z-positions according to the gaussian law of error propagation, combined with equ. (15), we can obtain the transfer relation from the x-, y-, and z-errors to the r-, a-, and e-errors below: △x △y △z  =  ∂x ∂r ∂x ∂a ∂x ∂e ∂y ∂r ∂y ∂a ∂y ∂e ∂z ∂r ∂z ∂a ∂z ∂e  △r △a △e  =  cose cosa −r cose sina −r sine cosa sine 0 r cose cose sina r cose cosa −r sine sina  △r △a △e  (16) where △x, △y, and △z stand for the warhead-target relative positional errors of the standard testing state, and △r, △a, and △e the errors in the radar measurements r, a, and e of the whole testing state. 245 advances in systems science and applications (2013) vol.13 no.3 firstly, we use the observational data of actual tests to obtain the errors △̃r, △̃a, and △̃e of radar measurements. secondly, we analyze the iterative process of the errors in the measured range and angles. in the following, we use a monopulse radar system as our example to specifically analyze the influencing factors on the errors in radar measured range and angles. based on the analysis on the sources of errors in radar measurements r, a, and e [6], and the law of error synthesis, we can obtain the main errors in radar measured ranges and angles as follows: u2 angle = △2 rader +△2 target +△2 enviroment + ... = △2 thermanoise +△2 phaseunbalance +△2 angularglint+ △2 dynamiclag +△2 clutterinterference + ... u2 distance = △2 rader +△2 target +△2 enviroment + ... = △2 thermanoise +△2 angularglint +△2 dynamiclag+ △2 clutterinterference + ... if we look at radar measured ranges, we see that the systematic error is mainly the dynamic lag error. the time-dependent random error mainly includes the error of clutter interference, thermal noise, and distance glint. now, let us consider the methods of computation for the impact errors of active homing radar systems, as mentioned earlier, under two different testing states: one is to fuse different observational information, including the direct observational index data and indirect observational data(the concrete fusion model refers to subsection 2.2 and 2.3), and the other the point estimate method of impact errors [7]. our simulated radar measurements are the range r, azimuth angle a, and elevation angle e. other than analyzing the single point measurements at the impact moments [7], we also provide a method on how to combine observed process quantities. we respectively model the errors in radar measured range and angle signals, and the speed and acceleration of the actual measured ranges and angles for the two testing states: the simulated substitute tests and the whole process tests. assume that the random error term includes the independent errors of clutter interference, thermal noise, and distance glint. for the systematic deviation, we mainly consider the error caused by dynamic lag. by comparing the point estimate method of impact points and that of combining process information that is used to calculate the impact point error caused by radar errors, the outcomes are listed in table 2. the point estimate method does not employ the process data of the active homing radar. instead, it only applies the observational data of the last moments to calculate the impact point deviation caused by radar errors. when the sample size is small, the outcome xiaojun duan, yi lin:ways of fusing different types of information and how systemic... 246 table 2 the impact error estimate comparisons between the two methods considered (significance level α= 0.01) unit (meter) cross impact cross impact longitudinal longitudinal error point error confidence impact error impact error estimation interval point estimation confidence interval real impact error -12.93 [-15.72,-10.13] 18.78 [18.09,19.47]for standard overall test substitute test -13.96 [-49.43,21.51] 20.23 [-29.56,70.02]point conversion (traditional method) substitute test -12.56 [-15.33,-9.80] 18.33 [17.61,19.05] conversion method fusing with indirect observational information of this method is greatly affected by random factors. on the other hand, the method developed in this research righteously employs the physical background information and observed process data so that interval estimates can be directly produced from the estimation of the parameters. comparing to the point estimate method, our method reduces the effect of random errors and makes the system evaluation and estimation process converge more quickly. example3. let us now look at how to place optimal weights for nonlinear models. assume f(t, β) = 1 + (5 + tβ)0.1, y(t) = f(t, β + ε(t), ε(t) iid∼n(0, 0.012), β̃ = β + η, η ∼ n(0, 0.052) let t = 0.01 ∗ j, j = 1, ..., 100, and the true value of β is 8. let us generate 50 observational data {yi(t)}1001 , β̃i, i=1,...,50. then, when ρ = 0.012/0.052, let us solve the minimization problem (6) 50 times, producing the mean square error 0.0394. and when ρ = 2.33∗0.012/0.052, the mean square error reaches the minimum value 0.0372. however, in theory, the mean square error is 0.0361. this end indicates that for a nonlinear system, the optimal choice of weights for fusing information not only depends on observational precisions but also the model curvature involved. all the three examples above verify the transformational relationship of system model information, prior knowledge, and observational data in the recognition process of the underlying system, where the model information can be 247 advances in systems science and applications (2013) vol.13 no.3 strengthened through the usage of process data, and when the observational data is insufficient, the shortage in information can be made up by supplementing additional prior knowledge. 4 summary continuing literature[1], in this paper we studied how to fuse heterogeneous sets of data together consistently so that better results can be obtained for system evaluations and estimations. it is shown that if the types of data considered are few and the available observational data is insufficient, one can consider obtaining process information and additional prior knowledge. it is because the limited amount of observational data could make the process of system evaluation and estimation converge extremely slowly or stop converging completely after reaching a certain degree. with process information or additional prior knowledge added, the convergence will continue. the speed of the resumed convergence is dependent on how the process information is applied, how good quality the prior knowledge is and how consistent the newly adopted prior information is with the available observational data. this end has been well illustrated by the case studies considered in section 3. when there is only a small sample available for a specific system evaluation and estimation, this work provides the theoretical guideline for how to excavate other sources of information and how newly adopted information should be fused with what is available. acknowledgements this work is supported by the natural science foundation of china (60974124), the program for new century excellent talents in university and the projectsponsored by srf for rocs, sem in china, the key lab open foundation for space flight dynamics technique (sfdlxz-2010-004). references [1] xj duan, y. lin. (2011), “conservation law of information and its application in evaluation and estimation of complex systems”, kybernetes: the international journal of cybernetics, systems, and management science, vol.40 no.1/2, pp.262-274 [2] y. lin. (2008), systemic yoyos: some impacts of the second dimension, taylor and francis, new york. [3] s. s. mao. (1999), bayesian statistics. beijing: china statistics publishing house. xiaojun duan, yi lin:ways of fusing different types of information and how systemic... 248 [4] d. m. bates and d. g. watts. (1997), nonlinear regressive analysis and its application, translator: bocheng wei. beijing: china statistics publishing house. [5] x. p. zhang, j. h. zhang, and h. w. xie (2003), “a few discussion of samples, a prior information and bayesian statistical decision”, acta electronica sinica, vol.31 no.4, pp.536-538. [6] d. c. wang, j. h. ding, w. d. chen (2006), radar measurement technique in precise tracking, beijing: publishing house of electronics industry. [7] g. wang, x. j. duan, z. m. wang (2009), “conversion method of impact dispersion in substitute equivalent tests vased on error propagation”, defence science journal, vol.59 no.1, pp.15-21. corresponding author xiaojun duan can be contracted at: xjduan@nudt.edu.cn advances in systems science and application (2016) vol.16 no.2 1-14 evaluation of a germ stability of a differentiable mapping defined by a mathematical model abdykappar ashimov1 and yuriy borovskiy2 1academician of national academy of science of republic of kazakhstan. 2candidate of physical and mathematical sciences. abstract the paper presents a set of algorithms to assess: the set of singular points of a differentiable mapping defined by some mathematical model; the stability of the germ of this mapping in its singular point, and (in the case of its stability) the form of such a germ in cases of corank 1 and all the possible relations of dimensions of the domain and mapping image. the stability of these germs in all singular points of the mapping under study is a necessary condition for the stability of the mapping in its domain. the implementation of the developed algorithms can be used to verify the model under study by investigating mappings defined by it. keywords germ of smooth mapping; germ stability; singular point; algorithm; mathematical model. 1 introduction as is known, in natural science and economics to study properties of the objects under investigation are widely used mathematical models related to different classes. important stage of accepting a model to use it in practice is verification and validation v&v[1, 2]. successful testing of the model using v&v methods gives reason to carry on the practice the conclusions drawn on the basis of calculations of the model. note that the currently used v&v methods contain no estimates of conditions for preserving qualitative properties of the considered models under arbitrarily small changes in these models. that is, these methods permit transfer of initial model (under its arbitrarily small changes) to the model with totally different properties[3]. necessary conditions for preserving qualitative properties of model under its small changes can be conditions of the weak structural stability of the model, being dynamical system and stability conditions of mappings, defined by the model under study[4–6]. the study of authors ashimov et al on the basis of the robinson theorem on sufficient conditions for the weak structural stability and an algorithm for constructing a symbolic set provide convenient for practice numerical algorithms for the evaluation of weak structural stability of dynamical systems[4, 7–9]. such system stability is required for small sensitivity of its phase portrait to small disturbances in model. these papers also provide the application examples of these algorithms to macroeconomic models. 2 abdykappar ashimov and yuriy borovskiy: evaluation of a germ stability ... in literature [8] authors propose v&v method, based on stability estimation of the defined by the model differentiable mappings in the sense of definition used in the theory of singularities of differentiable mappings[5, 6]. literature [5] provide sufficient theoretical conditions for the stability of differentiable mapping when the mapping is an immersion, submersion or submersion with a fold. proposed in [8] on the basis of these theoretical propositions the numerical algorithms permit to evaluate the set of singular points of the studied mapping, and also its stability (and instability) when the mapping belongs to the specified class of mappings. if using these algorithms the defined by model mappings are evaluated as stable, the appropriate model can be considered as more adequate to describe corresponding phenomena of reality. however, the proposed in [8] algorithms are insufficient to assess the stability of these differentiable mappings in general case (if they do not belong to classes of immersions, submersions or submersions with a fold). based on the fact that a required condition for the stability of a differentiable mapping is the stability of its germs in all the singular points of the mapping, the present work provides (based on theoretical propositions of arnold et al[6]) an algorithm to estimate the stability of such germs. also this paper proposes three algorithms to evaluate the form of stable germ with corank 1 for the cases of all possible ratios of dimensions of the domain and the image of the studied mapping. the use of the developed algorithms in the framework of v&v mathematical model would enable to evaluate the conditions of the low sensitivity of qualitative properties of the models to its small perturbations. 2 algorithms to evaluate a set of singular points of differentiable mapping and germ stability of the mapping in its singular point as noted above, the evaluation of stability of the given by model mapping f can be replaced by the evaluation of stability of germs of f in all singular points of this mapping. ashimov et al[8] proposed an algorithm to estimate the set of singular points of the differentiable mapping in the parallelepiped d f : d0 → e (1) (where dim(d) = m, d0 the set of internal points d) using the set of parallelepipeds d̃ with arbitrarily small size, covering the estimated set. for the convenience of the reader, this algorithm (as well as an illustration of its application), we present below. designate the vector of arguments of the mapping (1) through p = (p1, ..., pm) ∈ d, and respective image of the point p vector of the model solutions designate through y = y(p) = (y1, ..., yn) ∈ e, ( d ⊂ rm and e ⊂ rn some domains). in this case the jacobian matrix with dimension v ∗ n for the mapping (1) in the advances in systems science and application (2016) vol.16 no.2 3 point p would be written as follows: j(p) = ( ∂yi ∂pj (p) ) i=1,...,n; j=1,...,m (2) evaluation of the jacobian matrix (2) in some point p ∈ d, derived using the numerical differentiation, we also designate through j(p). designate total quantity of maximal order minors in j(p) through l. estimate of determinant value of such a minor of order min(v, n) in j(p) for p ∈ d designate through |mi(p)|, i = 1, ..., l. algorithm 1 to estimate a set of singular points of mapping (1). 1) parallelepiped d is divided into sufficiently great number of (elementary) parallelepipeds dk with the same size, and a grid p composed of n points is determined, which are vertices of chosen parallelepipeds: p = {pj : j = 1, ..., n}. 2)values of all j(pj) matrix elements are computed for j = 1, ..., n . 3) for i = 1, ..., l =; are computed determinants |mi(pj)|. 4) for every i = 1, ..., j = 1, ..., l the set d(i) is determined in the following way. d(i) is a union of all (closed) parallelepipeds dk with the property: not all values of |mi(pj)| in vertices dk have the same sign. 5) the set d̃ = ∩l i=1d(i) is found. 6) if set d̃ is empty, then stop. 7) if not, the steps 1 5 of this algorithm are performed with replacement of domain d by d̃ and decrease of parallelepipeds size, participating in subdividing d̃ until the diameter dk is not less than some given forward number . the abovementioned algorithms (as well as provided in [8] algorithms for evaluating points of a fold) were tested based on the whitney mapping, defined by relations y1 = x31 + x1x2, y2 = x2,where (x1, x2) ∈ r2. it is known that singular points of this mapping form parabola 3x21 + x2 = 0 and all points of this parabola, except the origin of coordinates, are fold points; origin of coordinates the point of cusp[6]. fig.1 presents the derived by the mentioned algorithms points estimates of the whitney mappings fold as the set of vertices of parallelepipeds p̃ covering the fold. note that the distance from every point in the set p̃ to the fold does not exceed the diameter of elementary square from the set , namely, the number 0.05 √ 2. here are further required to construct an algorithm to evaluate the stability of the germ of a differentiable mapping notations and the theoretical propositions from [6]: definitions of a stable and infinitesimally v stable germ, as well as theorems, enabling to obtain sufficient conditions for the stability of the germ by conditions of its infinitesimally v stability. then the conditions from the definition of infinitesimally v stable germ we have rewritten equivalently as condition 6. the proposed further algorithm 2 permits to evaluate condition 4 abdykappar ashimov and yuriy borovskiy: evaluation of a germ stability ... 6, that is, the infinitesimal v stability, and therefore the stability of the germ under study. fig. 1 estimate of the whitney mappings fold. here are further required to construct an algorithm to evaluate the stability of the germ of a differentiable mapping notations and the theoretical propositions from [6]: definitions of a stable and infinitesimally v -stable germ, as well as theorems, enabling to obtain sufficient conditions for the stability of the germ by conditions of its infinitesimally v -stability. then the conditions from the definition of infinitesimally v -stable germ we have rewritten equivalently as condition 6. the proposed further algorithm 2 permits to evaluate condition 6, that is, the infinitesimal v -stability, and therefore the stability of the germ under study. consider a germ of a differentiable mapping f : rm → rn in some of its singular point x0 ∈ rm . we denote also the mapping f : d → rn, specifying the germ, determined in a certain parallelepiped d ⊂ rm centered at point x0 . such a germ is called stable if for any other arbitrarily close (in the corresponding topology) to it mapping f there exist diffeomorphisms in some neighborhoods of points of the domain x0 and image f (x0) , during application of which the mapping f converted into f . the formal definition of the (left − right, differentiable) advances in systems science and application (2016) vol.16 no.2 5 stability of germ f is provided in the monograph[4]. definition 1. a germ f is called stable, if for arbitrarily small neighborhood u of point x0 exists such a neighborhood e of the mapping f , that for any mapping f from e is found the point x ∈ u such that the germ f in x (left-right, differentiable) equivalent to the germ f in x0 . topology in the set of all differentiable germs in point x0 is given by the set of neighborhoods of the form[6]: e = {f : sup |α|≤k,x∈u,i=1,...,n | ∂ |α| ∂αx f i(x)− ∂|α| ∂αx f i 0(x)|} (3) here f i coordinate function of the germ f = (f 1, ..., fn), (i = 1, ..., n); f0 fixed germ the center of neighborhood e; u some neighborhood of point x0 ; x = (x1, ..., xn), α = (α1, ..., αm); αj integral non-negative number; |α| = |α1 + ...+ αm| ; k arbitrarily large number; ∂|α| ∂αx = ∂|α| ∂α1x1...∂αmxm . further we introduce the following notations. let ∂f / ∂xj a germ of the mapping, defined by partial derivatives of coordinate functions of the mapping f : ∂f / ∂xj = ( ∂f 1/ ∂xj , ..., ∂fn/ ∂xj ) ; j = 1, ...,m. basis germs of mappings rm → rn we call constant germs er, (r = 1, ..., n) which have r-th coordinate function identically equal to 1, and the rest coordinate functions zero: e1 = (1, 0, ..., 0), ..., en = (0, 0, ..., 1). let ax0 algebra of all germs of differentiable functions rm → r in point x0, (ax0) n ax0 module of all germs of differentiable mappings rm → rn in point x0. designate trough submodule in (ax0) n generated by the following m+ n2 germs of mappings: ∂f / ∂xj ,where j = 1, ...,m and f ier,where i, r = 1, ..., n. definition 2. [5]a germ of the mapping f in point x0 is called infinitesimally v −stable, if factor module (ax0) n/b is generated above r images of basis germs e1, e2, ..., en. note that from infinitesimal v -stability of a germ follows its stability. this fact follows from the following theorems 3 and 4, provided in arnold et al[6]. theorem 3. infinitesimal v -stability of a germ is equivalent to its infinitesimal stability. theorem 4. (mather theorem (local version)) infinitesimally stable germ is stable. using facts: submodule b is a set of all linear combinations of germs of mappings ∂f / ∂xj and f ier with coefficients from ax0 and fulfillment of condition of definition 2 is equal to the fact that any element of the factor module (ax0) n/b of the form g+b where g some germ from (ax0) n can be written as the sum of some linear combination of germs e1, e2, ..., en with numerical coefficients and submodule b reformulate the condition of infinitesimal 6 abdykappar ashimov and yuriy borovskiy: evaluation of a germ stability ... v -stability of the germ f from definition 2 by the following equivalent way. condition 5. any germ g from (ax0) n can be presented as the sum of some linear combination with numerical coefficients of constant germs e1, e2, ..., en and linear combination with coefficients from ax0 of the mentioned germs ∂f / ∂xj , f ier. since for the case x0 = 0 any germ g from (ax0) n can be written as the sum of constant germ (resulting from linear combination with numerical coefficients of germs e1, e2, ..., en) and the germ taking zero value in point x0, and the last germ is written as the sum of linear combinations of functions of the form xjer (xj-j-th coordinate of vector x) with coefficients from ax0 , then in condition 5, the term “any germ g”, can be replaced by “any germ of the form xjer where j = 1, ...,m; r = 1, ..., n”. moreover, for sufficiently small neighborhood x0=0 coefficients of germs, generating submodule can be considered in the rough (with precision till infinitely small of high-order infinitesimality) polynomial germs qk(x) from ax0 , of order not exceeding k, where k sufficiently large fixed number: qk(x) = ∑ α:|α|≤k cαx α (4) here xα = (x1)α1 ...(xm)αm , cα coefficient of the polynomial. therefore, for the case x0=0 condition 5 infinitesimally v -stable germ f can equivalently be rewritten in the following way. condition 6. any germ of the mapping of the form xjer where j = 1, ...,m; r = 1, ..., n (with precision till infinitely small of high-order infinitesimality) can be written as linear combination of germs ∂f / ∂xj , f ier, with polynomial coefficients as (4). from theorem 4 results that condition 6 is sufficient for the stability of germ f in its singular point the origin of coordinates. here are based on an assessment of condition 6 enlarged algorithm to evaluate the stability of the germ in a singular point x0 of the mapping f (defined in dε parallelepiped centered at x0 , where ε is a diameter of the parallelepiped) defined by some model. estimate of singular point x0 can be obtained by algorithm 1 and selecting one point from the obtained set p̃ . algorithm 2 to estimate the stability of germ of the differentiable mapping 1) appropriate parallel shift and normalizing coordinate system in rm yields that singular point x0 corresponds with the origin of coordinates, and parallelepiped dε is close to cube. here ε sufficiently small given number. 2) dividing every edge dε into sufficiently large even integer n of equal segments, yields a grid pε composed of nm points. 3) determine grid functions f i, i = 1, ..., n, corresponding to coordinate functions of the mapping f and determined in grid junctions p ∈ pε. advances in systems science and application (2016) vol.16 no.2 7 4) using numerical differentiation (with step less than step of the grid pε) mine grid functions f i j , i = 1, ..., n, j = 1, ...,m,determined in grid functions pε and being estimates of partial derivatives ∂f / ∂xj in points p ∈ pε. 5) setting grid mapping linear combination of germs by its coordinate grid functions determined in pε (here i = 1, ..., n ): yi = ri(x, {αj : j = 1, ...,m; |αj | ≤ k}, {αi,r : r = 1, ..., n; |αi,r| ≤ k}) = m∑ j=1 qj(x)f i j + n∑ r=1 qi,r(x)f r (5) here qj and qi,r some polynomials of degrees, not exceeding k with arbitrary coefficients αj and αi,r respectively. k sufficiently large fixed number. 6) using numerical differentiation (with step less than step of the grid pε ) determine grid functions f i β, i = 1, ..., n, β = (β1, ..., βm), |β| ≤ k, determined in grid junctions pε and being estimates of partial derivatives ∂|β| ∂βx f i.(for the cases |β|=0 and |β|=1 these functions were determined above, correspondingly, in steps 3 and 4 of the algoritm). 7) setting grid functions ri β, i = 1, ..., n, β = (β1, ..., βm), |β| ≤ k , determined in grid functions pε and being estimates of partial derivatives ∂|β| ∂βx ri using the derived f i β values and values of partial derivatives of polynomials: ∂|β| ∂βx qj , ∂|β| ∂βx qi,r in the mentioned grid functions. 8) for every r = 1, ..., n and j = 1, ...,m the steps 9, 10, 11 are performed. 9) for i = 1, ..., n determine functions m i r,j ({αj : j = 1, ...,m; |αj | ≤ k}, {αi,r : r = 1, ..., n; |αi,r| ≤ k}) = sup x∈pε;β:|β|≤k ∣∣∣∣∣ri β(x)− ∂|β| ∂βx ( xjeir )∣∣∣∣∣ (6) here for β = 0, ∂|β| ∂βx ( xjeir ) = xj , if i = r and j = j, if not, ∂ ∂βx ( xjei i ) = 0. for cases |β| > 1 ∂|β| ∂βx ( xjei i ) ≡ 0. 10) finding estimate of deviation of the germ xjer from approximating it linear combination. m = 1 ε inf {αj :j=1,...,m;|αj |≤k},{αi,r:r,i=1,...,n;|αi,r|≤k} ( sup i∈{1,...,n} m i r,j ) (7) 11) if m > δ, where δ sufficiently small given forward number, then go to step 13. 8 abdykappar ashimov and yuriy borovskiy: evaluation of a germ stability ... 12) the germ f is assessed as stable. stop. 13) two-fold decrease of ε and all of the edges of parallelepiped dε . 14) if ε > ε0 , where ε0 sufficiently small given forward number, then go to step 2. 15) question about f germ stability left unanswered. stop. 3 algorithms to estimate the form of stable germ of corank 1 of the differentiable mapping after evaluation by the algorithm 2 of the studied germ f in its singular point x0 as a stable, there is a question on the evaluation of its form in accordance with [6] classification of genotypes of stable germs. in [6] there are three morin theorems determining the form of stable germ in the case where the germ has corank 1; the question on classification of stable germs with greater corank is still open. here corank of germ f its singular point x0 is determined as the difference between the number min(m,n) and the rank of the jacobian matrix of germ f in x0 . this section presents these morin theorems and developed algorithms enabling for respective estimation of the form of the germ f , and to evaluate some of the other characteristics of the germ under study. let there be a differentiable mapping f : rm → rn defined in some neighborhood of its singular point 0 ∈ rm, f (0) = 0 ∈ rn. next, consider the options of relations of dimensions values m and n. 2.1. let the dimension of the image and the full domain of the germ f match: n = m. the morin theorem[6], which describes these stable germs is as follows. theorem 7. a stable germ f : (rn, 0) → (rn, 0), of corank 1 left-right equivalent to the germ (i.e., driven by a diffeomorphic changes of coordinates in the spaces of domain and the image to the form) ỹ1 = (x̃1)k+x̃2(x̃1)k−2+...+x̃k−1x̃1), ỹ2 = x̃2, ... ỹn = x̃n, (8) here ỹ = (ỹ1, ỹ2, ..., ỹn) and x̃ = (x̃1, x̃2, ..., x̃n) new coordinates correspondingly in spaces of image and domain of the germ f . k integer (indicator of germ), 2 ≤ k ≤ n+ 1. we present an enlarged algorithm for assessing the axis oỹ for mapping (8), and the k order value of genotype specified in this theorem of the mapping f . it is based on the obvious remark that in the space of images rn there is the only direction (defined by the axis oỹ), satisfying the following property: the derivative of the mapping coordinate f corresponding to ỹ1 at the origin of coordinates advances in systems science and application (2016) vol.16 no.2 9 o in any direction in domain of f is zero. for any other axis in space of images the corresponding derivative at point o in some direction is different from zero. algorithm 3 1) finding by algorithm 1 estimate p of the set of singular points of the mapping f : d → rn, d ⊂ rn. 2) if p is empty, then stop. there are no singular points of the mapping f . 3) choosing the singular point x0 ∈ p and estimating the stability of the germ f in x0 by algorithm 2. 4) if the germ f is not assessed as stable, then stop. 5) computing the jacobian estimate of the mapping f in point x0 and its rank r = rank(j(x0)). 6) if r ̸= n− 1 then stop. corank f in singular point x0 is greater than one. 7) setting the shifted mapping y = f (x−x0)+f (x0) = f (x), for which x = 0 is the investigated singular point and f (0) = 0 . 8) finding the estimate of axis oỹ1. 8.1) for arbitrary unit vector ỹ ( |ỹ| = 1 ) setting coordinate function of n variables a = f̄ỹ(x) = prỹf̄ (x) projection of vector f̄ (x) to vector ỹ. 8.2) setting function of n variables s = g(ỹ) = ∑n i=1 ( ∂f̄ỹ(x) ∂xi )2 x=0 , where ∂f̄ỹ(x) ∂xi appropriate difference derivative. 8.3) finding the estimate of constrained minimum s0 of function s = g(ỹ) under constraint |ỹ| = 1. 8.4) if s0 ≈ 0 then the detected ỹ is an estimate of axis unit vector ỹ1, if not, then stop. in this case recalculation of steps 1-8 of the algorithm is possible with decreased values of grid steps and increments. 9) finding the estimate of value of indicator k of the germ f . 9.1) for arbitrary unit vector x ( |x| = 1 ) setting the constraint of coordinate function f̄ỹ1(x) on axis, determined by this vector: c = fx(t) = f̄ỹ1(tx). 9.2) assignment k = 2. 9.3) finding the estimate of maximum value of module of k-th derivative of function c = fx(t) at zero: m = max x,|x|=1 |f (k) x (t)|t=0|, where the estimate in the form of corresponding difference derivative is used as a derivative. 9.4) if m > ε, where ε sufficiently small number, then output of k value and stop. 9.5) increasing k value by one: k := k + 1. 9.6) if k ≤ n+ 1, then go to step 9.3. 9.7) indicator k of point x0 is not defined. stop. 2.2 consider now the case, when the germ image dimension f̄ is strongly greater than the domain dimension: n > m. take the morin theorem[5], describ10 abdykappar ashimov and yuriy borovskiy: evaluation of a germ stability ... ing the mentioned stable germs for this case. theorem 8. in the case n > m stable germ f̄ : (rm, 0) → (rn, 0), of corank 1 is right-left equivalent the germ (that is, reduced by diffeomorphic substitution of coordinates in spaces of domain and image to the form) ỹ1 = (x̃1)k + x̃2(x̃1)k−2 + · · ·+ x̃k−1x̃1, ỹ2 = x̃k(x̃1)k−1 + x̃k+1(x̃1)k−2 + · · ·+ x̃2k−2x̃1, · · · ỹt = x̃(t−1)k−t(x̃1)k−1 + x̃k+1(x̃1)k−2 + · · ·+ x̃tk−tx̃1, ỹt+1 = x̃2, · · · ỹn = x̃m. (9) here k ≥ 2 − integer; t = n −m + 1; t(k − 1) ≤ m; ỹ = (ỹ1, ỹ2, . . . , ỹn) and x̃ = (x̃1, x̃2, . . . , x̃n) − new coordinates correspondingly in spaces of image and domain of the germ f . we present an enlarged algorithm for assessing the plane ∏ spanned by the axes ỹ1, ỹ2, . . . , ỹt for mapping (9), as well as the value of k genotype order of the mapping f specified in this theorem. in the construction of this algorithm is used the following remark. for any axis of the plane ∏ the coordinate derivative of the mapping f corresponding to the axis at the origin o in any direction in space of the inverse images is zero. for any axis that is not in from the space of images the derivative at point o in some direction is different from zero. algorithm 4. 1) finding by algorithm 1 estimate p of the set of singular points of the mapping f : d → rn, d ⊂ rm. 2) if p is empty, then stop. there are no singular points of the mapping f . 3) choosing the singular point x0 ∈ p and estimating the stability of the germ f in x0 by algorithm 2. 4) if the germ f is not assessed as stable, then stop. 5) computing the jacobian estimate of the mapping f in point x0 and its rank r = rank(j(x0)). 6) if r ̸= m − 1 then stop. corank f in inverse image in singular point x0 is greater than one. 7) setting the shifted mapping y = f (x−x0)+f (x0) = f̄ (x), for which x = 0 is the investigated singular point and f̄ (0) = 0. 8) finding the estimate of plane ∏ , spanned by the axes ỹ1, ỹ2, ...,ỹt (where t = n−m+ 1) for the mapping (9). 8.1) for arbitrary unit vector ỹ ( |ỹ| = 1 ) setting coordinate function n of variables a = f̄ỹ(x) = prỹf̄ (x) projection of vector f̄ (x) to vector ỹ. 8.2) setting function of n variables s = g(ỹ) = ∑n i=1 ( ∂f̄ỹ(x) ∂xi )2 x=0 , where ∂f̄ỹ(x) ∂xi advances in systems science and application (2016) vol.16 no.2 11 appropriate difference derivative. 8.3) assignment k = 1, where k enumerator of coordinate axes of the required plane ∏ . 8.4) finding the estimate of constrained minimum s0 of function s = g(ỹ) under constraints:|ỹ| = 1, ỹ · ỹi = 0,where i = 1, . . . , k − 1 numbers of the derived at the previous steps coordinate axes of the plane ∏ , · sign of scalar product (when k = 1 this constraint is not used). 8.5) if s0 ≈ 0 then the detected value ỹ is an estimate of axis unit vector ỹk, if not, then stop. in this case recalculation of steps 1-8 of the algorithm is possible with decreased values of grid steps and increments. 8.6) increasing k by one: k := k + 1. 8.7) if k ≤ t, then go to step 8.4. 9) finding the estimate of value of indicator k of the germ f . 9.1) for arbitrary unit vector x̃ ∈ rm (|x̃| = 1) and arbitrary unit vector ỹ ∈ rn (|ỹ| = 1) setting the constraint of coordinate function f̃ỹ(x) on axis, determined by this vector: c = fx,y(t) = f̃y(tx). 9.2) assignment k = 2. 9.3) finding the estimate of maximum value of module of k-th derivative of function c = fx,y(t) at zero: m = max y∈π,|y|=1,x,|x|=1 ∣∣∣f (k) x,y (t)|t=0 ∣∣∣, where the estimate in the form of corresponding difference derivative is used as a derivative. 9.4) if m > ε, where ε sufficiently small number, then output of k value and stop. 9.5) increasing k value by one: k := k + 1. 9.6) if t(k − 1) ≤ m, then go to step 9.3. 9.7) indicator k of point x0 is not defined. stop. 2.3 now consider the rest case, when the dimension of f̄ germ image is less than the dimension of domain: n < m. we present the morin theorem[6] for this case, describing the mentioned stable germs of corank 1. theorem 9. in the case n < m the stable germ f̄ : (rm, 0) → (rn, 0) of corank 1, the corank of the second differential the genotype of which at zero does not exceed 1, right-left is equivalent to the germ (i.e., by diffeomorphic substitutions of coordinates in spaces of domain and image reduced to the form). ỹ1 = (x̃1)k + x̃2(x̃1)k−2 + · · ·+ x̃k−1x̃1 ± (x̃n+1)2 ± (x̃n+2)2 ± · · · ± (x̃m)2, ỹ2 = x̃2, · · · ỹn = x̃n. (10) here ỹ = (ỹ1, ỹ2, . . . , ỹn) and x̃ = (x̃1, x̃2, . . . , x̃n) − new coordinates correspondingly in spaces of image and domain of the germ f . k − integer (genotype order),2 ≤ k ≤ n+ 1. 12 abdykappar ashimov and yuriy borovskiy: evaluation of a germ stability ... below is presented an enlarged algorithm for assessing the axis oỹ1 for the mapping (10), and k order value of the genotype of the specified in this theorem mapping f . the first part of the algorithm (steps 1-8) is similar to the corresponding steps of algorithm 3. algorithm 5. 1) finding the estimate p of the set of singular points of the mapping f : d → rm, d ⊂ rn using algorithm 1. 2) if p is empty, then stop. there are no singular points of the mapping f . 3) choosing the singular point x0 ∈ p and estimating the stability of the germ f in x0 by algorithm 2. 4) if the germ f is not assessed as stable, then stop. 5) computing the jacobian estimate of the mapping f in point x0 and its rank r = rank(j(x0)). 6) if r ̸= m− 1 then stop. corank f in singular point x0 is greater than one. 7) setting the shifted mapping y = f (x−x0)+f (x0) = f̄ (x) , for which x = 0 is the investigated singular point and f̄ (0) = 0 . 8) finding the estimate of axis oỹ1 . 8.1) for arbitrary unit vector ỹ ∈ rm(|ỹ| = 1) setting coordinate function n of variables a = f̄ỹ(x) = prỹf̄ (x) − projection of vector f̄ (x) to vector ỹ. 8.2) setting function of m variables s = g(ỹ) = ∑n i=1 ( ∂f̄ỹ(x) ∂xi )2 x=0 , where ∂f̄ỹ(x) ∂xi − appropriate difference derivative. 8.3) finding the estimate of constrained minimum s0 of function s = g(ỹ) under constraint |ỹ| = 1 . 8.4) if s0 ≈ 0 then the derived value is an estimate of axis unit vector oỹ1 , if not, then stop. in this case recalculation of steps 1-8 of the algorithm is possible with decreased values of grid steps and increments. 9) finding the estimate of value of indicator k of the germ f . 9.1) for arbitrary unit vector x(|x| = 1) setting the constraint of coordinate function f̄ỹ1(x) on axis, determined by this vector: c = fx(t) = f̄ỹ1(tx) . 9.2) assignment i = 1. i enumerator of axes, corresponding to morse singularity of function f̄ỹ1(x). 9.3) finding the estimate of maximum value of module of 2nd derivative of function c = fx(t) at zero: m = max x,|x|=1;xx̃j=0,j<i ∣∣∣f (2) x (t)|t=0 ∣∣∣ and corresponding vector x̃i , where the estimate in the form of corresponding difference derivative is used as a derivative:f (2) x (t)|t=0 = (fx(h) + fx(−h))/h2. 9.4) if m < ε where ε sufficiently small number, then stop. in this case the corank of the second differential of the studied germs genotype is estimated by the number greater than one. 9.5) increasing the enumerator by one: i := i+ 1 . advances in systems science and application (2016) vol.16 no.2 13 9.6) if i ≤ m− n , then go to step 9.3. 9.7) assignment k = 2 . 9.8) finding the estimate of maximum value of module of k-th derivative of function c = fx(t) at zero:m = max x,|x|=1,xx̃j=0,j<m−n ∣∣∣f (k) x (t)|t=0 ∣∣∣ , where the estimate in the form of corresponding difference derivative is used as a derivative. 9.9) if m > ε , where ε sufficiently small number, then output of the required k value and stop. 9.10) increasing k value by one: k := k + 1. 9.11) if k ≤ n+ 1 , then go to step 9.8. 9.12) characteristics k of point x0 is not defined. stop. the use of algorithms 2, 3 or 4 enables to evaluate the axis oỹ1 (or the plane∏ for the case n > m ) in the space of the mapping image under study, with the following property. the coordinate derivative of the mapping f corresponding to this axis (or any axis of the mentioned plane for the case n > m ) at the origin (corresponding to a singular point x0 ) in any direction in space of the inverse images is zero. detection of the specified axis (or plane) may be important to study the behavior of the germ (in the singular point) of the mapping defined by the studied model. coordinate function corresponding to this axis gives an example of the local invariant of model, in the sense that in some neighborhood of the singular point under study the derived function is constant (with accuracy up to infinitesimals of highest order) with respect to all exogenous parameters of the model involved in the construction of the initial mapping. if the studied point is estimated (using algorithm 1) as a non-singular (regular) for this mapping, then space of images does not contain the axis with the above properties and the mapping has no local invariant in the neighborhood of this regular point. 4 conclusion this paper presents a set of algorithms that allow: to estimate the set of singular points of a differentiable mapping created by mathematical model. to estimate the stability of germ of this mapping in its singular point. in the case of such a stability of germ of corank 1 to estimate k indicator of its genotype and the corresponding axis in the domain of the germ, which determines the local invariant of the mapping under study. the developed algorithms can be used to evaluate the possibility of transferring obtained on the basis of model results into practice. 14 abdykappar ashimov and yuriy borovskiy: evaluation of a germ stability ... references [1] o. balci. (1998), “verification, validation and testing”, in handbook of simulation: principles, methodology, advances, applications, and practice, john wiley & sons, new york, pp. 335-393. [2] r.c. kennedy, x. xiang, g.r. madey and t.f. cosimano. (2005), “verification and validation of scientific and economic models”, agent 2005 conference proceedings, chikago, pp. 177-192. [3] v.i. arnold (1988), geometrical methods in the theory of ordinary differential equations, springer-verlag, new york. [4] c. robinson (1980), “structural stability on manifolds with boundary”, journal of differential equations,no. 37. pp. 1-11. [5] m.golubitsky and v. gueillemin. (1973),stable mappings and their singularities,springer-verlag, new york, heidelberg, berlin. [6] v.i. arnold, s.m. gusein-zade and a.n. varchenko. (1985), “singularities of differential maps”, birkhauser, boston, basel, stuttgard, vol. 1. [7] a. ashimov et al (2013), macroeconomic analysis and parametrical control of a national economy, springer, new york. [8] a. ashimov et al. (2014), “the theory of parametric control of macroeconomic systems and its applications (i)”, advances in systems science and application, vol. 14, no. 1, pp. 1-21. [9] e.i. petrenko (2006), “development and realization of the algorithms for constructing the symbolic set”, differential equations and control processes , no. 3, pp. 55-96. corresponding author abdykappar ashimov can be contacted at: ashimov37@mail.ru мягкая вероятность ошибки при передаче информации advances in systems science and application(2016) vol.16 no.4 22-28 system of criteria and indicators for the development of resource-based multiclusters dmitry l. napolskikh, tatyana v., yalyalieva, nina i. larionova, elena a. murzina volga state university of technology annotation the paper presents the system of criteria and indicators of development of resource-based multiclusters able to be used for the purposes of government control of performance efficiency of the territory on which they are located. the limitations of the paper are general adaptation mechanisms of a region economic system within the frames of innovation development that would significantly enrich the theoretical aspect of the study. an indisputable advantage of the suggested system of the parameters for the purposes of governmental control of the resource-based multiclusters efficiency is its versatility, as it is adapted to any kind of regional economic system. key words: governmental control, economy clustering, natural resource-based multicluster, efficiency of governmental cluster policy. 1 introduction in the modern conditions of global economy, the competitiveness of the cluster residents is interpreted as a concentrated expression of the production, scientific, educational and technical advantages implemented in innovation technology, goods and services [1]. the multi-subject composition of the multicluster can be classified according to the subject composition of the “triple spiral” model, i.e., it appears possible to single out «state and municipalities», «science and education» and «business [2]. the aggregate social and economic effect produced through implementation of governmental cluster policy, which is achieved by applying structural modifications to the economy, can be used as the efficiency indicator [3]. the formation of resource-based multiclusters at the regional level is based not only on building an industry infrastructure but also on the formation of a network interaction structure between innovation businesses and the government. in this case, the methods used to assess the natural resource-based multiclusters can be applied to control the efficiency of the use of the state resources [4]. the practical importance of this approach lies mainly in the opportunities to formulate and implement large national-and regional-scale investment and innovation projects. issues of government regulation of the cluster have been studied in works by [5, 6]. in this regard, studying theoretical approaches to improving government control of the formation of natural resource-based multiclusters at the regional level becomes an important scientific task. 2 data and method advances in systems science and application(2016) vol.16 no.4 23 most international studies on the problem of efficiency of the resource-based multiclusters are devoted to comparative analysis of transaction costs of economic activity at the regional level. in the modern economy system the structure of transaction costs are determined by a wide range of technological and ecological factors [7, 8]. minimization of transaction costs as a main function of government control allows us to consider costs of interactions between economic agents in the multicluster. an multicluster performance efficiency for economic agents can be defined as the ratio of the benefits of transaction costs reduction in multicluster and the institution’s maintenance costs and/or institutional constraints loss: where: me = multicluster efficiency tc = transaction costs imc = institution maintenance costs icl = institutional constraints loss efficiency of informal institutions of control of the formation of resource-based multiclusters can be evaluated using the following formula: where: uie = informal institution efficiency lgc = losses associated with government control iic= informal interactions costs icl = informal constraints loss if the informal institution is of a “grey” (illegal) nature, its efficiency can be evaluated using the following formula: where: gie = grey institution efficiency cla = losses associated with activity legitimization lgc = losses associated with government control iic = informal interaction costs icl = losses associated with informal constraints rial = losses associated with risks of illegal activities in the context of the network analysis of the of the formation of resource-based multiclusters at the regional level, it is expedient to use the instruments of the mathematical graph theory. considering the development criteria of the resource-based multiclusters, we can single out such parameters as the density of the institutional environment and its innovation conductivity, which produce effect on the diffusion speed of the management and production innovations within the cluster. let us make an assumption that any economical agent of the multicluster may be connected with other economical subjects through identifiable bilateral relations, both direct connections and indirect ones operating through intermediaries. the multicluster density can be characterized as the ratio of the stable formal connections between the multicluster organizations and businesses to their total number. the multicluster density can be represented with the formula below: where: d = multicluster density sxi = number of the stable formal connections between the ith muticluster agent with the other ones 24 dmitry l. napolskikh, tatyana v., yalyalieva, nina i. larionova, et al.: system of criteria and indicators… n = total number of the multicluster agents considering the multicluster density, it is required to single out the institutional environment integrity parameters related to the cluster. the multicluster integrity can be characterized as the ratio of the sum of the number of the indirect connections of each multicluster agent with the other ones to their total number: where: i = multicluster integrity uxi = number of the multicluster agents connected with indirect non-formal connections with the ith cluster agent n = total number of the multicluster agents the multicluster complementarity can be characterized as the average number of the primary communication channels between each multicluster agent and the other ones: where: c= multicluster complementarity axi = number of the primary communication channels between each multi cluster agent and the other ones n = total number of the multicluster agents the multicluster conductivity is considered as the average length of the bilateral interaction chains between the cluster organizations and businesses which are not directly interconnected but build communication channels for innovation diffusion, technology transfer, as well as for informational and sociocultural interactions between the cluster agents. the multicluster conductivity can be represented with the formula below. where: φ = multicluster conductivity xixj = communication channel between the ith and the jth multicluster agents which are not directly interconnected lxixj = aggregate length of the bilateral interaction chains between the multicluster agents building the communication channel. 3 analysis and results to assess the impact of innovation on economic development of the regions of the russian authors applied regression analysis. as a final indicator of innovative development of regional economic systems used the "share of innovative goods, works and services in their total volume". the analysis revealed that in the russian regions is no direct correlation between the share of innovative products and factors, which in accordance with the theory of cluster should define an innovative vector of territory development. the results of the correlation analysis are presented in table 1. using the territorial method to identify and assess the development of the resource-based multiclusters at the regional level allows assessing the synergetic effect produced by the interaction between primary sector and innovation businesses. a local resource sphere is created in the territory of this cluster, which is intended for providing innovation technologies, outside the region as well. advances in systems science and application(2016) vol.16 no.4 25 table 1 the results of correlation analysis of innovative development of the russian regions indicators of innovative development of the russian regions № 1.1 № 1.2 № 1.3 № 1.4 № 2.1 № 2.2 № 2.3 № 2.4 № 3 1.1. the proportion of organizations implementing technological innovation 1,00 1.2. the proportion of organizations implementing organizational innovation 0,69 1,00 1.3. the proportion of organizations implementing marketing innovation 0,35 0,48 1,00 1.4. the proportion of small innovative businesses, implementing technological innovation 0,22 0,12 0,26 1,00 2.1. number of advanced production technologies 0,27 0,27 0,26 0,21 1,00 2.2. the number of used advanced production technologies 0,28 0,28 0,29 0,25 0,74 1,00 2.3. the number of organizations engaged in research and development 0,28 0,22 0,24 0,19 0,88 0,74 1,00 2.4. the number of issued patents for inventions and utility models 0,26 0,18 0,20 0,14 0,74 0,67 0,96 1,00 3. the proportion of innovative goods, works and services in the total volume 0,00 0,03 0,07 0,05 0,06 0,15 0,03 0,01 1,00 competitive advantage of multicluster are the result of the synthesis of competitiveness factors, formed at previous stages of regional economic development. the main factors of competitiveness at different evolution stages of territorial economic systems are presented in table 2. table 2 the main factors of competitiveness at the evolution stages of territorial economic systems the evolution stages of territorial economic systems the main factors of competitiveness industrial agglomeration of industry transportation and logistics advantages, reducing uncertainty and transaction costs on the basis of geographical concentration, rapid response to competitors' innovations innovative industrial zones staffing and infrastructural benefits of innovative development, reducing uncertainty and transaction costs with the use of formal institutions and on the basis of an explicit contract with the participants of cooperation local innovation networks information benefits, reducing uncertainty and transaction costs with informal institutions, social capital formation, diffusion of management innovations multiclusters innovative advantages of joint activities in the network of scientific and technical cooperation, formation of the institutional environment of innovative development, a partnership with the government and the local community the current russian model for multicluster relations between the state, business and society is characterized under the three main segments of the institutional socio-economic interactions: «white sphere». it combines the formal institutions of the legislative and administrative regulation within the framework of multicluster, which include: registration, licensing, arbitration proceedings, auctions, etc. 26 dmitry l. napolskikh, tatyana v., yalyalieva, nina i. larionova, et al.: system of criteria and indicators…  «gray sphere». it includes informal institutions, shadow rent from business, political "bargaining" with the regional management, etc.  «black sphere». it includes informal practices of corrupt interactions, interaction with the criminal world, raider grabs businesses, etc. a variety of business and government in the formation and development of clusters is given space with two axes shown in figure 1. fig. 1. models and institutional spheres of interaction between business, government and society in the framework of multicluster it was found out in the course of the research that l resource-based multiclusters do not only broaden and develop but, in time, they can also become narrower and disintegrate. this kind of dynamics and flexibility of resource-based multiclusters represents their distinctive feature and requires permanent governmental control. in time, efficient active clusters become objects of big governmental investments. the economic institutions uniting into a multicluster on the basis of vertical and horizontal integration build a unified distribution sphere innovation technologies and the essential condition for the efficient transformation from innovations to competitive advantages is establishing a network of stable connections between all the cluster agents. formation of resource-based multiclusters at the regional level may, on the whole, be regarded as a response to excessive transaction costs [9,10]. at this point, multicluster integration processes are characterized by aggregation and consolidation of enterprises. this implicitly proves the desire for diversification and economic control of enterprise risks associated with imperfect institutional environment and excessive transaction costs. comprehensive mechanism for efficiency control of transaction costs which is used in economy to reduce them involves development of specification and protection of property rights, standardization of measurements, accounting and reporting, maintenance of monetary system, improvement of law enforcement effectiveness and efficiency, as well as implementation of measures aimed at eliminating unnecessary administrative burdens and infrastructure markets of various transactions. the basic requirements of institutional changes to improving effectiveness of natural resource-based multiclusters include recognition of the critical role of the government control of gray sphere white sphere the degree of involvement of residents in the process of territorial development the level of institutional development of the business environment contradi ctions model affiliates model policymaking model corporate model advances in systems science and application(2016) vol.16 no.4 27 economic development; government’s commitment to economic development; accounting of institutional transformation costs; review of efficiency of the current control. 4 discussion from the position of the institutional theory, the functioning sphere of the multicluster institutions represents an environment that is commonly called institutional. the study of the institutional environment of a territory as an evolving endogenous factor was initiated by the representatives of the school of economics of washington university in the 1970s [11]. douglass north uses the institutional environment term to define the institutional limits which exist at the macro level and determine the possible conditions of contractual agreements between individuals [12]. oliver williamson defines the institutional environment as an established system of the informal “rules of the game” which build the sociocultural context of economic activity [13]. in the context of this research, the institutional environment of natural resource-based multiclusters is interpreted as the aggregate of the institutional connections which surround and fill the regional economic system and produce their effect on it. on the other hand, the development degree of the governmental control mechanisms applied to the cluster policy efficiency at the regional level is determined by the activities of various groups of interests within the multicluster, while the effect they produce on the environment depends on the level of the institutional control of the interrelations inside those groups. thus, we can speak about a network aspect of functioning of the natural resource-based multiclusters which sets trajectories of interactions between economic entities. network analysis for the purposes of government control of the formation of resource-based multiclusters at the regional level allows to: · identify the influence of informal relationships between economic agents within a multicluster on competitive ability and efficiency of a multicluster as a whole. · evaluate structural consequences for the economic system of a multicluster due to change in equilibrium of institutional environment of a territorial unit. · identify the optimal organizational structure of communication channels in the resourcebased multiclusters at the regional level. 5 conclusion studying the formation of resource-based multiclusters in the modern practices of the economic development allows determining the formation trends of an efficient governmental control system at the regional level. the research shows that the formation of the governmental control mechanisms is based on a general study of the transaction costs of economic activity. the resource-based multicluster have their potential both for generating fundamental and applied knowledge and for managing innovation projects. the efficiency parameters suggested by the author for the modern cluster policy implementation for the governmental control purposes, such as density, integrity, complementarity and conductivity of multicluster will allow controlling the development efficiency of the resource-based multiclusters. the paper offers the method for calculating performance efficiency of an cluster institution for economic agents, as well as efficiency of informal institutions. the article formulates recommendations for implementing control principles to improve government regulation of multiclusters. the results proposed in the study can find application in the sphere of government control under conditions of the natural resource-driven economy. thus, the proposed efficiency indicators of multicluster effectiveness were used by government authorities of the republic of mari el (russia) while developing the programs for development of the region’s economy. the 28 dmitry l. napolskikh, tatyana v., yalyalieva, nina i. larionova, et al.: system of criteria and indicators… mechanisms for government regulation considered in this article were used by the city administration of yoshkar-ola in the process of formation of a local natural resource-based innovation cluster. it is planned to implement future research findings into the practice of public administration in russia through long-term programs of collaboration between the volga state university of technology and government agencies of the mari el republic. acknowledgment this researchers was supported by the russian foundation for basic research. project № 16-3600126 mol_a "development of mathematical methods to assess the efficiency of formation of innovative multiclasters at the sub-national level." references [1] porter, m.e., (2009). "clusters and economic policy: aligning public policy with the new economics of competition". institute for strategy and ompetitiveness. [2] enright, m., (1996). "regional clusters and economic development: a research agenda. in: business networks: prospects for regional development",walter, d.g. (ed.)., berlin, isbn10: 3110151073, pp: 190-213. [3] feldman, m.p., (1994). "the geography of innovation". springer science and business media,dordrecht, isbn-10: 0792326989, pp: 154. [4] dritsaki, c. and a. adamopoulos, (2005). "a causal relationship and macroeconomic activity: empirical results from european union". am. j. applied sci, no.2, pp.504-507. [5] liu, c.c. and c.y. chen, (2004). "a computable general equilibrium model of southern region in taiwan: the impact of the tainan science-based industrial park". am. j. applied sci. [6] kim, y.d., s. yoon and h.g. kim, (2014). "an economic perspective and policy implication for social enterprise ". am. j. applied sci., no.11, pp. 406-413. [7] larionova, n. i., napolskikh, d. l. and. yalyalieva, t.v. (2015) "institutional aspects of state control the effectiveness of innovation clusters development", actual problems of economics. no.5, pp. 20-24. [8] larionova, n. i., napolskikh, d. l. and. yalyalieva, t.v. (2014) "ensuring efficiency control of institutional environment of the cluster", american journal of applied sciences, vol.11, no.9, pp. 1594-1597. [9] larionova, n. i., napolskikh, d. l., yalyalieva, t.v., and shebashev, v.r. e. (2014) "governmental control of the formation efficiency of educational clusters at the regional level", american journal of applied sciences, vol.11, no.9, pp. 1594-1597. [10] larionova, n. i.,.napolskikh, d. l and. yalyalieva, t.v. (2015) " theoretical approaches to improving government control systems for educational clusters development", actual problems of economics. no. 4, pp. 285-288. [11] coase, r.h., (1992). "the institutional structure of production". am. economic rev., no.82, pp.713-719. [12] north, d., (1994). "economic performance through time". am. economic rev., no.84, pp. 360-361. [13] williamson, o., (2000). "the new institutional economics: taking stock, looking ahead". j. economic literature, vol.38, pp.595-613. corresponding author dmitry l. napolskikh can be contacted at: yal05@mail.ru. advances in systems science and applications (2012) vol.12 no.1 46-53 a method for determining importance degree of customer requirements in software quality function deployment lixiong gong1, mingzhong yang1, shunshenga guo and qi wang2 1school of mechanical and electrical engineering, wuhan university of technology,wuhan 430070,china 2vic, monash university, victoria 3800 ,australia abstract a method based on fuzzy analytic hierarchy process was provided to determine customer requirements in this study, and the importance degree was analyzed by using trapezoid fuzzy function. the method overcame the impact on subjective judgments & preference, and caused decision-making more reasonable. finally, the effectiveness and practicality of the method was verified by the example of fetching customer requirement in the process of software project development. keywords analytic hierarchical process, customer requirements, quality function deployment, house of quality 1 introduction software is a product that integrating knowledge and procedure. software engineers must focus on how to consider the maximum requirements of users in programming. therefore, the software must be designed to collect and analyze customer requirements, and be oriented to maximize the value of customers in the whole software development. however, what are customer requirements for a complex problem must be investigated and analyzed. data existing software programming showed that more than 50% unsuccessful projects of software development occurred in the wrong requirement analysis. quality function deployment (qfd) is a widely used customer-driven quality, design and manufacturing management tool. it is becoming a methodology of modern design theory applied for new products design and old products improvement[1-3]. customer requirements were mapped to the corresponding technical characteristics of products design that will be around customer requirements by the house of quality (hoq) in qfd[4]. but the success rate of traditional qfd is subject to restrictions because of various shortcomings, such as: fussy process, too large matrix, complicated manual calculation, unreasonable score mechanism. for the determination of the house of quality problems, the non-linear programming method that determined customer demands and technical requirements of the relationship was provided by large number of experimental data in quotation[5], and the method of multi-feature map based on taguchi theory was used to obtain the above-mentioned relationship in quotation[6]. however, advances in systems science and applications (2012), vol.12, no.1 47 these methods must rely on a large number of relevant experimental data. consequently, the experimental time and cost will be fundamental obstacles. with the development of decision-making and information science, fuzzy analytic hierarchy process (fahp) is more and more widely applied in various fields[7-8]. it decreased subjective judgmental errors account of the fuzzy factor, so the final set of index weight are more realistic. based on this, the algorithm model based on fahp was provided to determine the importance degree in this study, and applied to get customer requirements in software programming. 2 structure of software quality function deployment methodology 2.1 house of quality the core of qfd is the demand for conversion, the house of quality, that is qfd matrix, is a quality function deployment plans that associate with customer requirements and technical characteristics of products. software qfd run through the whole process of software development, which is derived from manufacture qfd and originated in japan[9]. the house of quality of software qfd is similar to the traditional qfd, as is shown in fig.1: fig.1 structure of hoq the house of quality can converted customer requirements to the quality characteristics. it is composed of six matrices, as is shown in fig.1: (1) whats matrix, said customer demand; (2) plan matrix, said the evaluation of whats matrix. (3) hows matrix, the demands for what to do; (4) the relationship matrix, the relationship between whats and the hows; (5) the matrix of hows inner relationship; (6) technology matrix, said that the evaluation of the technical cost: comparison of competitiveness or feasibility. the conversion of “what needs” to “how to do” was completed after building house of quality. 2.2 process of qfd implementation in general, the implementation of software qfd consists of two basic processes: the extraction of customer requirements and waterfall decomposition of customer requirements. information of customer requirements through face-to-face, telephone, e-mail, network, on-site investigation, after-sales service, were collected, 48 lixiong gong:a method for determining importance degree of customer requirements . . . classified, organized and analyzed to form well-organized user requirements and weighted importance degree. then, software engineers disassembled customer requirements and constructed hoq according to technical feasibility, practicability, economy and other aspects. the importance degree of qfd indexes were determined after completed above process. 3 model of determining importance degree of customer requirements in software qfd based on fahp 3.1 algorithm of fahp it is reasonable decision-making for fahp because of overcoming shortcomings of human subjective judgments, choices and preferences. this study was used trapezoidal fuzzy number to score the weight because it is more accordant with actual states and more extensive application, although trigonometric number, logarithmic trigonometric number, normal distribution function can be used while scoring. fahp algorithm steps are as follows: step 1: hierarchical structure construction. put the goal of the desired problem on the top layer of the hierarchical structure, and then put the evaluation criteria on the second layer of the hierarchical structure. further, the third layer is the sub-indexes of evaluation criteria. finally, the candidate alternatives lay in the bottom layer. hierarchical structure is shown in fig.2. fig.2 hierarchical structure step 2: constructing the fuzzy judgment matrix in this study, the fuzzy judgment matrix x is the matrix of the combination of each candidate alternative and evaluation criteria, and the fuzzy judgment matrix x is represented by trapezoid fuzzy numbers such as 1, 3, 5, 7 and 9; the advances in systems science and applications (2012), vol.12, no.1 49 values are shown in tab.1. the judgment matrix is as follows: x =  x11 x12 . . . x1n x21 x22 . . . x2n ... ... . . . ... xn1 xn2 . . . xnn  where xij = (aij , bij , cij , dij , ) , and aij , bij , cij , dij denote four values of trapezoid fuzzy numbers respectively. table 1 membership of trapezoid fuzzy numbers meaning of scale of value of relative importance relative importance fuzzy numbers equal importance 1 ( 1, 1, 1, 1 ) weak importance 3 ( 2, 2.5, 3.5, 4 ) strong importance 5 ( 4, 4.5, 5.5, 6 ) demonstrated importance 7 ( 6, 6.5, 7.5, 8 ) absolute importance 9 ( 9, 9, 9, 9 ) step 3: testing the consistency of the matrix and calculating fuzzy weights testing the consistency of fuzzy matrix after clarifying above fuzzy matrix, algorithm of testing the consistency of ahp is described in quotation[10]. return to step 2 to re-structure trapezoidal fuzzy judgment matrix and calculate it until the consistency is passed, and then, calculating fuzzy weights, fuzzy weighting formula is defined as follows: dx̄ = n∑ j=1 xij ⊗  n∑ i=1 n∑ j=1 xij −1 (1) where x̄ is judgment matrix, xij is the element of judgment matrix. step 4: single sorting weight of same hierarchy for any of two trapezoidal fuzzy numbers m and n, the possibility of m ≥ n is following results. theorem: suppose m = (r1, r2, r3, r4), n = (s1, s2, s3, s4) are two trapezoidal fuzzy numbers, then the possibility degree is v (m ≥ n) =  1 r3 ≥ s2 r4−s1 (r4−r3)+(s2−s1) r3 ≤ s2, r4 ≥ s1 0 r4 ≤ s1 (2) 50 lixiong gong:a method for determining importance degree of customer requirements . . . where r1 > 0, r2 > 0, r3 > 0, r4 > 0, s1 > 0, s2 > 0, s3 > 0, s4 > 0 approach to sorting the weights of hierarchical indexes is as follows: (1) calculating possibility degree of dx̄1 ≥ dx̄2 , . . . , dx̄1 ≥ dx̄n .calculating possibility degree of dx̄1 ≥ dx̄2 , . . . , dx̄1 ≥ dx̄n according to formula (2), that is v ( dx̄1 ≥ dx̄2 ) , v ( dx̄1 ≥ dx̄3 ) , . . . , v ( dx̄1 ≥ dx̄n ) . (2) calculating the possibility value that x̄1 is greater than other matrices. d ( x̄1 = minv ( dx̄1 ≥ dx̄2 , dx̄3 , . . . , dx̄n )) and analog,d ( x̄2 ) ,d ( x̄3 ) ,. . . d ( x̄n ) can also be calculated. (3) normalizing the matrix and getting the weight vectors ( wx̄1 ,wx̄2 . . .wx̄n ) . step 5: sorting the general hierarchy the sorting general hierarchy means that weights of all schemes are ordered in the layer of candidate alternative. the general sorting value is the weight of the scheme multiply the value of single sorting of the same hierarchy. 3.2 application in accordance with steps of fahp, calculated as follows: step 1: constructing trapezoidal fuzzy judgment matrix, testing consistency and completing the single-sort in the same hierarchy. (1) constructing trapezoidal fuzzy judgment matrix c̄ in c layer and calculating weight. the trapezoidal fuzzy judgment matrix c̄ is as follows: c̄ = [ c1 c2 ] = [ (1, 1, 1, 1) (1, 1.5, 2.5, 3) (0.333, 0.4, 0.6667, 1) (1, 1, 1, 1) ] (2) calculating synthetically fuzzy values. dc1 = 2∑ j=1 x1j ⊗  2∑ i=1 2∑ j xij  −1 = (0.3334, 0.4838, 0.8974, 1.2) dc2 = (0.2223, 0.2709, 0.4237, 0.6) (3) single sorting in c layer. single sorting in c layer using step 4 of section 3.1, then d (c1) = minv (dc1 ≥ (dc2) = 1 d (c2) = minv (dc2 ≥ (dc1) = 0.8251. the weight can be concluded while normalizing the matrix, it is shown as follows: (wc1 ,wc2) = (0.5479, 0.5421) , and then advances in systems science and applications (2012), vol.12, no.1 51 (wr1 ,wr2 ,wr3) = (0.5270, 0.4730, 0.0007) , (ws1 ,ws2) = (0.5479, 0.4521) . fig.3 hierarchical structure of software qfd development step 2: sorting the general hierarchy. vr1 = wr1 ×wc1 = 0.5479× 0.5270 = 0.2883, vr2 = 0.2592, vr3 = 0.0004, vs1 = 0.2477, vs2 = 0.2044. therefore, the general sorting and respective weight is as follows: r1(0.2883), r2(0.2592), s1(0.2477), s2(0.2044), r3(0.0004). as can be seen from the result the most important customer requirements are robust procedure & low faults, user-friendly & easy to operation and low prices. 4 conclusion it is very important to determine the importance degree, relationship degree and competitive forces in hoq. in this study, fahp method is used to determine the importance degree of customer requirements of the effective factors in the model of software qfd. humans are often uncertain in assigning the evaluation scores in conventional ahp. fahp can capture the vagueness of human thinking style 52 lixiong gong:a method for determining importance degree of customer requirements . . . and effectively solve multi-criteria decision making problems. by using fahp and appropriate calculations, there are extremely accurate for fetching customer requirements and the quality characteristics. the application shows that fahp can access to design new products, software development and evaluation of customer satisfaction that provide a new thinking for exactly getting customer requirements. acknowledgements the project is supported by the hubei international cooperation key projects, china (no. 2007ca008). references [1] hauser j r, clausing d. (1988), “the house of quality”, harvard business review, vol.66, no.3, pp.63-73. [2] sireliy y, kauffmann p, ozan e. (2007), “integration of kano’s model into qfd for multiple product design”, ieee transactions on engineering management, vol.54, no.2, pp.380-390. [3] zheng l y, chin k s. (2007), “qfd based optimal process quality planning”, international journal of advanced manufacturing technology, vol.26, no.7, pp.831-841. [4] karsak e e. (2004), “fuzzy multiple objective programming framework to prioritize design requirements in quality function deployment”, computers & industrial engineering, vol.47, no.2-3, pp.149-163. [5] dawson d, asking g. (1999), “optimal new product design using quality function deployment with empirical value functions”, quality and reliability engineering international. vol.15, no.1, pp.17-32. [6] kumar p, barua p b, gaindhar j l. (2000), “quality optimization (multicharacteristics) through taguchis technique and utility concept”, quality and reliability engineering international, vol.16, no.2, pp.475-485. [7] han shilian, li xuhong, liu xinwang. (2004), “stmulti-person and multicriteria evaluation and selection of logistic centers with fuzzy analytic hierarchical process method”, system engineering theory & practice,vol.24, no.7, pp.128-134. [8] hu yaoguang, fan yushun. (2006), “decision model based on fahp for selection of enterprise core business systems”, computer integrated manufacturing systems, vol.12, no.2, pp.215-219. advances in systems science and applications (2012), vol.12, no.1 53 [9] haag s. (1996), “quality function deployment: usage in software development”, in communications of the acm, vol.39, no.1, pp.41-49. [10] sang song, lin yan, ji zhuoshang. (2002), “an improved ahp method for mcdm in ship type’s demonstration”, journal of dalian university of technology, vol.42, no.2, pp.204-207. advances in systems science and application (2015) vol.15 no.3 254-266 optimal estimation and precision analysis of measuring data fusion model xuanying zhou, jiongqi wang, zhengming wang and zhangming he college of science, national university of defense technology, changsha, hunan, china. abstract data fusion is an effective method to improve the data processing accuracy, and the fusion weight has a great influence to the accuracy of the fusion estimation. in this paper, we study the problem of optimal weight and parameter estimation for the linear fusion model with unequal precision measuring data. the properties of the unbiased estimation are discussed for linear model, and then the optimal estimation for the fusion model with unequal precision measuring data is given. it is proved that there exists the optimal fusion weight and it is unique. besides, the accuracy of the optimal estimation for the multivariate linear model is analyzed and some conclusions suitable for practical application are obtained, which can provide the theory foundation for the experiment design and the data selection. finally, two simulations are offered to validate the theories conclusions in this paper. keywords data fusion; regression model; fusion weight; parameter estimation; precision analysis. 1 introduction the main purpose of data fusion is to improve the accuracy of measuring data, to establish a proper processing model, and to give an effective and reliable fusion algorithm [1, 2]. fusion with different types and unequal precision data is the most typical situation in the data fusion processing [3-5]. after the data are modeled in a parametric model, the data fusion problem can be transferred into the parameter estimation problem [6,7]. to evaluate the performance of the parameter estimation result, we need an evaluation standard, i.e., evaluation criterion or optimal criterion. such criteria include minimum mean-square error (mse) criterion, maximum likelihood (ml) criterion, maximum a posteriori (map) criterion, best linear unbiased estimation (blue) criterion and least squares estimation (lse) criterion, etc. whether the estimated parameter satisfies the need of the application depends on the estimation criteria as well as the data accuracy and the model properties [8]. obviously, the selection of the evaluation criterion is affected by the characteristic of the estimation parameter, the demand of the estimation accuracy and the complexity of the estimation algorithm. specifically, the parameters, estimated according to lse criterion, will lead to the minimal norm of the obseradvances in systems science and application (2015) vol.15 no.3 255 vation residual, i.e. the differencebetween the observed value and the calculated value. usually, lse does not involve the dynamic and statistical information of the parameter to be estimated. therefore, lse is easy implemented but with low estimation accuracy. nevertheless, when we are short of the error information about the measuring data, lse also can provide us with an acceptable solution. mse criterion is the best in terms of that mse has the minimal mean square error. however, this method needs some statistical prior information, such as the first and the second moment of the data and parameter. the map and the ml estimation are both related to the conditional probability density functions, and the estimation is hard to be obtained except for some special cases. therefore, in the actual application, the efficient and reliable data fusion algorithm for data fusion should be selected according to the specific situation. although most of the fusion systems are nonlinear, they can be linearized into some linear regression model when proper base functions are selected or the nonlinear iterative means are adopted. that is to say, nonlinear fusion problem can be approximated to process with the linear fusion problem. furthermore, when the parameters and the measuring data are with the normal distribution, some optimal estimation methods, such as blue, mse and lse, are equivalence to each other [10]. therefore, the minimal linear variance criterion is usually applied for the actual application. following the introduction in section 1, the structure of the paper is organized as follows. the form of unbiased estimation for linear model is given and the estimation characters are discussed in section 2. in section 3, the optimal weight and parameter estimation of unequal-precision data fusion are researched. besides, it is proved that there exists the optimal fusion weight and it is unique, and the accuracy of multivariable optimal estimation for linear fusion model is analyzed. section 4 provides two numerical examples to validate the proposed theory and method. finally, the paper is concluded in section 5. 2 unbiased estimate of linear model consider the linear measuring regression model as follow: y = xβ + ε, ε ∼ (0, σ2i) (1) where y = ym×1 is the measuring data, x = xm×n is the design matrix and rank(x) = n, β = βn×1 is the estimated parameter vector, and ε = εm×1 is the measuring random error vector with the zero expectation and diagonal covariance, i.e. ε ∼ (0, σ2i). for the parameter estimation problem in model (1), the lse β̂ls = ( xtx )−1 xty has some good properties as follows: property 1: β̂ls is the umvue (uniform minimum variance unbiased esti256 xuanying zhou, jiongqi wang and zhengming wang, zhangming he: optimal... mation) of the parameter β, and moreover, ∀c ∈ rn, ctβ̂ls is the linear unbiased estimation of the parameter ctβ. property 2: assuming e ∥∥∥xβ̂ls −xβ ∥∥∥2 = (xβ̂ls −xβ)t(xβ̂ls−xβ) and then e ∥∥∥xβ̂ls −xβ ∥∥∥2 = nσ2. property 3: assuming cov(β̂ls) = e(β̂ls − β)(β̂ls − β)t, mse(β̂ls) = e(β̂ls − β)t(β̂ls − β) and then cov(β̂ls) = σ2s−1,mse(β̂ls) = σ2 n∑ i=1 λ−1 i , where s = xtx and λi, i = 1, · · · , n are all eigenvalues of the matrix s; remark 1: property 1 shows that in the actual engineering application, different constant vector c can be chosen for estimating some components or the linear combination of the parameter β. remark 2: property 2 shows that the estimation precision of xβ̂ls is directly proportional to n. that is to say, the more the number of the estimated parameters, the lower estimation precision will be obtained. therefore, the proper base function and parameter model should be chosen when modeling the measuring data to let the parameter number as few as possible. the sparse parameter modeling methods are often used in the real application [11, 12]. remark 3:property 3 shows that when the model (1) is multi-collinearity, i.e. the matrix xtx has some extremely small eigenvalues λi, the large mse(β̂ls) or mse(ctβ̂ls) will lead to the bad estimation accuracy. the regularizing methods, a series of biased estimation methods, were proposed to handle the multi-collinear problem in application [13-15]. by choosing the proper regularizing factor µ and the regularizing matrix d with full column rank to solve the following optimization problem min β∈rn,µ>0 ∥y −xβ∥22 + µ ∥dβ∥22 (2) the solution of (2) can be easily calculated as follows β̂r = (xtx + µdtd)−1xty (3) compared to lse, the regularizing estimation has properties as follows: property 4: eβ̂r = (xtx + µdtd)−1xtxβ. i.e. the regularizing parameter estimation is biased; property 5: there exists µ,d, and make mse(β̂r) < mse(β̂ls), i.e. the regularizing estimation can better than lse by choosing some proper regularization parameter and matrix. in the linear measuring regression model, ε ∼ (0, σ2i) means the measures are irrelevant and the precision are equal. in actual, if the measures are relevant and have unequal precision, the model (1) can be transferred to y = xβ + ε, ε ∼ (0, σ2g) (4) advances in systems science and application (2015) vol.15 no.3 257 where σ2 is known or unknown, and g is a known positive definite matrix. for model (4), its umvue is the weighted least squares estimation (wlse)β̃wls =( xtg−1x )−1 xtg−1y , and mse(β̃wls) = tr ( xtg−1x )−1 . in real application, in order to get the lse for the linear fusion model, the measuring data should be parametric modeling to make it satisfy the model (1) or (4). actually, a typical application of model (4) is the unequal precision data fusion processing problem with several kinds of measuring equipment. although the measuring equations of different equipment are non-linear, the proper basis function can be chosen or the nonlinear iterative means can be adopted to linearize the measuring equations. therefore, the lse or wlse can be an important theory foundation for the measuring data fusion. no matter the parameters estimated by model (1) or (4), the statistic properties of the measuring random error, including the correlation, meaning, variance, and covariance and so on, need to be estimated at first. σ2 in the model (1) or (4) reflects the accuracy of the measuring data. therefore, as the base of unequal precision data fusion, the estimation of the parameter σ2 is very important. besides, when the estimation performance of the lse (wlse) is worse, the information of the parameter σ2 is also needed to be used in order to build the biased estimation of the parameter β. there is the property about the estimation of σ2: property 6:assume the observation error in model (1) satisfy the normal distribution, i.e. ε ∼ n(0, σ2i), then σ̂2 = rss/(m− n),where rss = m∑ i=1 µ2 i =∥∥∥y −xβ̂ls ∥∥∥2 is the measuring residual square sum, µi = yi−xiβ̂ls , i = 1, · · · ,m is the ith residual between of the measuring data and the calculated value and eσ̂2 = σ2, mse(σ̂2) = 2σ4 / (m− n). remark 4:property 6 shows that the parameters σ2 can be estimated by the residual if the precision of the actual measuring data is unknown. the estimation value and its accuracy of the parameter σ2 are related to the number of the measuring data as well as the dimensional of the estimated parameter. as a result, in the actual application, the estimation variance can be decreased by increasing the sampling number of the measuring data. 3 optimal fusion estimation of linear model with unequal precision data in many measuring processing problem, like trajectory tracking, the unequal precision data fusion processing often need to be considered. obviously, the weighting methods for different measuring data have a great influence to the accuracy of the fusion estimation. 258 xuanying zhou, jiongqi wang and zhengming wang, zhangming he: optimal... 3.1 optimal estimation in unequal precision linear fusion model considering s kinds of unequal precision linear data fusion model: y1 = x1β + ε1, ε1 ∼ (0, σ2 1im1), · · · · · · · · · ys = xsβ + εs, εs ∼ (0, σ2 s ims), eεiεj t = omi×mj , i, j = 1, · · · , s, i ̸= j (5) the definitions of the parameter in (5) are same to that in model (1), and the optimal estimation can be given by the follow theorem. theorem 1: for model (5), ∀c ∈ rn, the uniformly minimum variance estimation of ctβ is ctβ̃f where β̃f = ( s∑ i=1 σ−2 i s∑ i=1 σ−2 i xt i xi) −1( s∑ i=1 σ−2 i s∑ i=1 σ−2 i xt i yi) (6) proof : assuming t = 1 /√ s∑ i=1 σ−2 i , then model (6) can be rewritten as:  tσ−1 1 y1 = tσ−1 1 x1β + tσ−1 1 ε1, tσ −1 1 ε1 ∼ (0, t2im1), · · · · · · · · · tσ−1 s ys = tσ−1 s xsβ + tσ−1 s εs, tσ −1 s εs ∼ (0, t2ims), (7) assuming y = [tσ−1 1 y t 1 , · · · , tσ−1 s y t s ]t, x = [tσ−1 1 xt 1 , · · · , tσ−1 s xt s ] t, ε = [tσ−1 1 εt1 , · · · , tσ−1 s εts ] t, combining with (6) and (7), the fusion model (5) can be written as follows: y = xβ + ε, ε ∼ (0, t2im),m = s∑ i=1 mi (8) using the lse, the theorem can be proved that ∀c ∈ rn, the uniformly minimum variance estimation of ctβ is ctβ̃f , where β̃f = (xtx)−1xty = ( s∑ i=1 t2σ−2 i xt i xi) −1( s∑ i=1 t2σ−2 i xt i yi) (9) the proof is completed. 3.2 optimal weight for the linear fusion model the purpose of data fusion is to find the optimal weight ρi and then to optimize the fusion problem and obtain the optimal parameter estimation. theorem 1 advances in systems science and application (2015) vol.15 no.3 259 above shows that the optimal weight of the data fusion model with unequal precision measuring data is ρi = σ−1 i /√ s∑ i=1 σ−2 i = tσ−1 i , and moreover, it satisfies s∑ i=1 ρ2i = 1. this indicates that the optimal only related to the data accuracy σ−1 i . for convenient, the weight method for two types of unequal-precision linear observed data is discussed firstly: y1 = x1β + ε1, ε1 ∼ (0, σ2 1im1), y2 = x2β + ε2, ε2 ∼ (0, σ2 2im2), eε1ε t 2 = om1×m2 (10) considering the following optimization problem: arg β min 2∑ i=1 ρ2i ∥yi −xiβ∥2 2∑ i=1 ρ2i = 1 (11) and the solution can be easily obtained as follow: β̂(ρ) = ( 2∑ i=1 ρ2ix t i xi) −1( 2∑ i=1 ρ2ix t i yi) (12) theorem 2: under the assumption of model (10), the solution of the optimization problem arg ρ mine ∥∥∥β̂(ρ)− β ∥∥∥2 = arg ρ minmse(β̂(ρ)) is: ρi = σ−1 i /√ σ−2 1 + σ−2 2 , i = 1, 2 (13) proof : by calculating e ∥∥∥β̂(ρ)− β ∥∥∥2 = tr( 2∑ i=1 ρ2ix t i xi) −2( 2∑ i=1 ρ4iσ 2 ix t i xi) assuming a = xt 1 x1, b=xt 2 x2, and then f(ρ1, ρ2) = (ρ21a+ ρ22b)−1(ρ41σ 2 1a+ ρ42σ 2 2b)(ρ21a+ ρ22b)−1 = σ2 2 ( ρ21 ρ22 a+b )−1 (ρ41 ρ42 σ2 1 σ2 2 a+b )( ρ21 ρ22 a+b )−1 (14) as both of a and b are real symmetric positive definite matrices, and can be similarity diagonalized simultaneously, that is to say existing an invertible matrix 260 xuanying zhou, jiongqi wang and zhengming wang, zhangming he: optimal... p , let a = ptλp, b=ptp , where λ = diag(λ1, · · · , λn), therefore f(ρ1, ρ2) = σ2 2p t ( ρ21 ρ22 λ + i )−1 (ρ41 ρ42 σ2 1 σ2 2 λ + i )( ρ21 ρ22 λ + i )−1 (p−1)t = σ2 2p tdiag [( ρ41 ρ42 σ2 1 σ2 2 λi + 1 )( ρ21 ρ22 λi + 1 )−2 ] (p−1)t (15) assuming that t = ρ21 ρ22 , σ = σ1 σ2 , λi = a, g(t) = (t2σ2a+ 1) / (ta+ 1)2, let dg(t) dt = 2a(tσ2−1) (ta+1)3 = 0 and then t = σ−2. besides, d2g(t) dt2 |t=σ−2 = 2a(σ2 + a) (ta+ 1)4 > 0 (16) noticed that ρ21 + ρ22 = 1, and the function g(t) gets the minimum value at ρi = σ−1 i /√ σ−2 1 + σ−2 2 , i = 1, 2, then e ∥∥∥β̂(ρ)− β ∥∥∥2 = tr( 2∑ i=1 ρ2ix t i xi) −2( 2∑ i=1 ρ4iσ 2 ix t i xi) = tr [ σ2 2p tdiag [( ρ41 ρ42 σ2 1 σ2 2 λi + 1 )( ρ21 ρ22 λi + 1 )−2 ] (p−1) t ] = σ2 2 n∑ i=1 t2σ2λi+1 (tλi+1)2 (17) since the each item in equation (17) gets the minimum at ρi = σ−1 i /√ σ−2 1 + σ−2 2 , i = 1, 2, therefore, the solution of arg ρ mine ∥∥∥β̂(ρ)− β ∥∥∥2 is ρi = σ−1 i /√ σ−2 1 + σ−2 2 , i = 1, 2. the proof is complete. remark 5: the conclusion of theorem 2 has important application value. in the actual problem, unequal precision data are usually measured, so the data weight has an important effect to the data fusion accuracy. theorem 2 shows that the unique optimal fusion weigh depends on the data precision in the linear fusion model with unequal precision measuring data. actually, this is the gauss-markov theorem applies to the lse for the linear model. however, the gauss-markov theorem shows the optimal estimation only can be given when the data accuracy is known, and while theorem 2 shows the optimal estimation can be obtained by solving the optimal problem arg ρ mine ∥∥∥β̂(ρ)− β ∥∥∥2 when the data precision is unknown. that is to say, for model (10), ρi = σ−1 i /√ σ−2 1 + σ−2 2 , i = 1, 2 is the necessary and sufficient condition if β̂(ρ) = ( 2∑ i=1 ρ2ix t i xi) −1( 2∑ i=1 ρ2ix t i yi) is the advances in systems science and application (2015) vol.15 no.3 261 uniformly minimum variance solution (optimal solution) for the parameter β. from theorem 2, for the linear fusion model (5), assuming ρ = [ρ1, · · · , ρs], the parameter estimation and optimal weight determination can be handled by the following two-step minimal problem (1)  arg β min s∑ i=1 ρ2i ∥yi −xiβ∥2 s∑ i=1 ρ2i = 1 (18) (2) arg ρi mine ∥∥∥β̃(ρ)− β ∥∥∥2 (19) 3.3 accuracy analysis of optimal fusion estimation for convenient, consider the parameter estimation accuracy problem of two kinds of unequal precision data fusion model. suppose that β̂(i), i = 1, 2 are the estimation by measuring yi, i = 1, 2, separately, β̂(1, 2) is the traditional joint estimation of these two kinds of measuring data, and β̂f is the optimal fusion estimation, i.e. β̂(1) = (xt 1 x1) −1xt 1 y1 β̂(2) = (xt 2 x2) −1xt 2 y2 β̂(1, 2) = (xt 1 x1 +xt 2 x2) −1(xt 1 y1 +xt 2 y2) β̂f = (σ−2 1 xt 1 x1 + σ−2 2 xt 2 x2) −1(σ−2 1 xt 1 y1 + σ−2 2 xt 2 y2) (20) then the follow conclusions can be drawn. theorem 3: for the different estimation for the parameter β there are: (1)e ∥∥∥β̂f − β ∥∥∥2 ≤ min{e ∥∥∥β̂(1)− β ∥∥∥2, e ∥∥∥β̂(2)− β ∥∥∥2, e ∥∥∥β̂(1, 2)− β ∥∥∥2} (21) (2) e ∥∥∥β̂(1, 2)− β ∥∥∥2 < max{e ∥∥∥β̂(1)− β ∥∥∥2, e ∥∥∥β̂(2)− β ∥∥∥2} (22) (3)if σ2 2/σ 2 1 ≤ 2, and then e ∥∥∥β̂(1, 2)− β ∥∥∥2 ≤ min{e ∥∥∥β̂(1)− β ∥∥∥2, e ∥∥∥β̂(2)− β ∥∥∥2} (23) proof : (1) from the lse properties: e ∥∥∥β̂(1)− β ∥∥∥2=σ2 1tr(x t 1 x1) −1,e ∥∥∥β̂(2)− β ∥∥∥2 = σ2 2tr(x t 2 x2) −1 e ∥∥∥β̂(1, 2)− β ∥∥∥2= tr(σ2 1x t 1 x1+σ2 2x t 2 x2)(x t 1 x1+xt 2 x2) −2 e ∥∥∥β̂f − β ∥∥∥2 = tr(σ−2 1 xt 1 x1+σ−2 2 xt 2 x2) −1 (24) 262 xuanying zhou, jiongqi wang and zhengming wang, zhangming he: optimal... obviously, e ∥∥∥β̂f − β ∥∥∥2 ≤ min{e ∥∥∥β̂(1)− β ∥∥∥2,e∥∥∥β̂(2)− β ∥∥∥2} and furthermore, from theorem 2, e ∥∥∥β̂f − β ∥∥∥2 ≤ e ∥∥∥β̂(1, 2)− β ∥∥∥2. (2) assuming e ∥∥∥β̂(1)− β ∥∥∥2 ≤ e ∥∥∥β̂(2)− β ∥∥∥2 and a = xt 1 x1, b = xt 2 x2, e ∥∥∥β̂(1, 2)− β ∥∥∥2 ≤ e ∥∥∥β̂(2)− β ∥∥∥2 need to be proved, the follow inequality need to be proved first: tr(σ2 1a+σ2 2b)(a+b)−2 ≤ σ2 2tr(b −1) (25) that is tr[(σ2 1a+σ2 2b)− (a+b)σ2 2b −1(a+b)] ≤ 0 (26) noticed that σ2 1a+σ2 2b−(a+b)σ2 2b −1(a+b) =a(σ2 1a −1−2σ2 2a −1−σ2 2b −1)a < 0. then equation (26) is right, and thus equation (22) is proved. (3) following the symbol and assumption in (2), if want to prove e ∥∥∥β̂(1, 2)− β ∥∥∥2 ≤ e ∥∥∥β̂(1)− β ∥∥∥2, i.e. to prove tr(σ2 1a+σ2 2b)(a+b)−2 ≤ σ2 1tr(a −1) that is tr[(σ2 1a+σ2 2b)− (a+b)σ2 1a −1(a+b)] ≤ 0 (27) noticed that when σ2 2/σ 2 1 ≤ 2, and then σ2 1a+σ2 2b − (a+b)σ2 1a −1(a+b) =(σ2 2 − 2σ2 1)b −1 − σ2 1ba−1b < 0, then equation (27) as well as equation (23) is proved. the proof is complete. obviously, the conclusion in theorem 3 can also be adapted to s kinds of unequal precision data fusion processing problem. it has the great effect to experiment designed and data selection scheme optimization problem in the actual application. equation (21) shows that the accuracy of several sensors optimal fusion estimation is the best comparing to the any single or any combination sensors joint estimation. and equation (22) shows that the precision of several sensors joint (traditional weighted scheme) estimation is better than the worst single sensors estimation, but the estimation precision of several sensors joint can better than the best single sensors if each sensors measuring accuracy reaches some certain conditions. 4 numerical examples in this section, two calculation examples are given to validate the proposed theory and algorithm for optimal weight and parameter estimation of unequal-precision data fusion. example 1: fusion processing of static measuring data advances in systems science and application (2015) vol.15 no.3 263 assuming two unequal-precision equipment measure the physical signal β. suppose the real value of the signal β is 10, and randomly create 100 highprecision measuring data (the root mean square error, rms, is 3), 100 mediumprecision measuring data (the rms is 4) and 100 low-precision measuring data (the rms is 6), and simulate 100 times. the estimation of β and its variance is get. seen in the table 1 as follow (the root variance is come from the 100 simulate data statistic) the true value of the physical quantity is β = 10. we have 100 groups of data. in each group, there are 100 high-precision data samples (the standard deviation of is 3), 100 medium-precision data samples (the standard deviation of is 4) and 100 low-precision data samples (the standard deviation of is 6). the parameter estimated and its mse is shown in table 1 below. (the estimated variance is obtained based on statistics of 100 groups of observed data.) table 1 parameter estimate result in different weighted methods hhhhhh result method highprecision only mediumprecision only lowprecision only traditional weighted jointestimation with highand medium-precision data traditional weighted joint estimation with highand low-precision data optimal fusion estimation truth parameter 10 10 10 10 10 10 estimated value 9.975 10.087 10.114 9.983 10.052 9.995 mean square error 0.087 0.131 0.154 0.071 0.116 0.045 in the linear measuring data fusion processing, the unique fusion weight depends on the measuring data accuracy. solve the minimum optimization problem (18) and (19), and get the mse of the estimated parameter is smallest, 0.045. for the highand medium-precision data fusion, the precision of two data satisfies σ2 2/σ 2 1 < 2, therefore, the mse of the parameter with the traditional joint weighted method is 0.071, which is better than that only with the high-precision data, 0.087. for the highand low-precision data fusion, the precision of two data does not dissatisfy σ2 3/σ 2 1 < 2, the mse of the estimated parameter with the traditional weight joint method is 0.116, which is worse than that only with the high-precision data, 0.087, while better than the mse, 0.154, which only use the low-precision data. example 2: fusion processing of dynamic tracking data assuming that gps and bds are tracking and measuring a dynamic target, simultaneity, and the measuring data are the single point positioning data. suppose (x(t), y(t), z(t))t is the position of the target orbit at time t, the positioning accuracy of gps in every direction is 1m, and that of bds is 3m. simulate 80 groups of measuring data by the theoretical orbit, including t = 0.05× j, j = 1, · · · , 600 gps and bds positioning data in each group. in the tracking period, the orbit data is model by the cubic spline function of the optimal node according 264 xuanying zhou, jiongqi wang and zhengming wang, zhangming he: optimal... to the reference [6]. the spline coefficient is estimated first and then the orbit (x̂(k)(tj), ŷ (k)(tj), ẑ (k)(tj),̂ ẋ (k)(tj),̂ ẏ (k)(tj),̂ ż (k)(tj))(ρ), k = 1, 2, · · · , 80, j = 1, 2, · · · 600 can be calculated sequentially. assume that, mser(ρ21, ρ 2 2) = 1 80 1 600 80∑ k=1 600∑ j=1 √ (x(tj)− x̂(k)(tj)) 2 + (y(tj)− ŷ(k)(tj)) 2 + (z(tj)− ẑ(k)(tj)) 2 (ρ21, ρ 2 2) (28) where (x(tj), y(tj), z(tj)) t is the theoretical orbit and table 2 gives the estimation precision for the orbit position with different weighted methods. table 2 parameter estimate result in different weighted methods xxxxxxxxxxx method estimation result weighted only use gps data only use bds data traditional weighted estimation optimal fusion estimation weighted method (1, 0) (0, 1) (1/2, 1/2) (9/10, 1/10) mse 1.512 3.247 1.243 0.716 5 conclusion data fusion is one effective method to improve the precision of data processing. this paper researches the optimal weight and parameter estimation of unequal precision linear data fusion. for the linear fusion model, the optimal weight depends on the precision of the measuring data only, and it is consistent with the classical gauss-markov theory. furthermore, when the data precision in the fusion model is unknown, the parameter estimation and the optimal weight can be obtained by minimizing the mean square error. the parameter estimation precision of multivariate linear fusion model is given in this paper as well as some conclusions, which can be used in the practical engineering application. the accuracy of the optimal fusion estimation with multi-measuring data is better than that of only using any single measuring data and the traditional weighted estimation. the estimation precision with the traditional weighted method is better than that only with the low-precision data, and if the data precision of each measuring data satisfies some certain condition, the precision with the traditional weighted method can better than that only with the high-precision data. and these conclusions can be the support to experiment design and data selected scheme. note that the optimal fusion weight of the unequal precision discussed in this paper is the linear fusion model. the nonlinear model needs to be considered further, although the theory and method in this paper is the basic of nonlinear advances in systems science and application (2015) vol.15 no.3 265 problem. besides, the optimal fusion weight of the multi-structure linear regression model is obtained under certain evaluation criterion. different evaluation criterion leads to different optimal fusion weight. the evaluation criterion corresponding to the minimal mse of parameter estimation is used in the paper, i.e. arg ρ mine ∥∥∥β̂(ρ)− β ∥∥∥2. nevertheless, this criterion has some limitations, e.g. the contribution of each parameter to the problem is not distinguished. certainly, other evaluation criteria should be considered for 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(2013), parameter estimation and inverse problems , academic press. corresponding author xuanying zhou can be contacted at: julia chow07@163.com advances in systems science and application (2015) vol.15 no.3 202-219 a novel multi-attribute decision making methodology and application yong liu1 and yi lin 2 1 school of business, jiangnan university, wuxi, 214122, china; 2 mathematics department, slippery rock university of usa, pennsylvania, usa. abstract in the multi-attribute decision making problems, how to effectively extract the decision making rules and rank the schemes are much more important research contents. however, the acquisitions of the decision making rules often are ignored. in view of this, with respect to the problems that there exist a lot of preference information and fuzzy information in the real decision making information system, a novel decision making methodology based on dominance intuitionistic fuzzy rough set is constructed in the paper, and then it is applied to audit risk assessment and risk judgment. based on the analysis of model and example, the result shows that the proposed model can well realize the extraction of the decision making rules and the ranking of the schemes, and effectively deal with the intuitionistic fuzzy information system with preference information. keywords preference information; decision making rule; dominance distance index 1 introduction as a useful mathematical tool to deal with knowledge with inaccuracy, uncertainty and fuzziness, rough set theory was initially proposed by pawlak [1]. it has been widely applied in many fields, such as knowledge discovery, data mining, decision analysis, and pattern recognition[2–4]. the classical rough set theory conducts data reasoning on the basis of equivalence relationship, while it is difficult to satisfy the harsh conditions of equivalence relationship in the practical applications. at the same time, the binary relationship existing on the field of discourse is often a fuzzy relationship and a similarity relationship instead of an equivalence relationship. in view of this, based on the idea and method of the fuzzy set theory[5] put forward the fuzzy rough sets theory. because this new theory can well describe the uncertainty of various types of knowledge and more objectively reflect the physical world, it has been rapidly becoming a research focus of rough set theory, leading to its rapid development. as a result of simultaneously considering the positive, negative and hesitancy degrees for an object to belong to a set, intuitionistic fuzzy sets possess stronger ability of information expression and well describe and portray delicate ambiguities of the nature of the objective world when compared with the traditional fuzzy sets[6, 7]. therefore, intuitionistic fuzzy sets and rough sets are first proadvances in systems science and application (2015) vol.15 no.3 203 posed to hybrid, leading to the construction of the intuitionistic fuzzy rough set model [8]. due to the important theoretical value and application implications, intuitionistic fuzzy rough set theory has soon become a hot academic research area. currently, most of the related research on intuitionistic fuzzy rough set lies in the aspects of constructing different models and exploring their relevant properties. for the related researches on constructing different models and exploring their properties, the relationship between intuitionistic fuzzy set theory and rough set theory firstly is revealed, and then they employ intuitionistic fuzzy set to define approximation operators in the intuitionistic fuzzy approximation space. by making use of the cut set of intuitionistic fuzzy sets, the upper and lower approximation operators of intuitionistic fuzzy rough set and the axiomatic method of the approximation operators based on general binary intuitionistic fuzzy relationship are respectively constructed [8–10], and then it is well known that the upper and lower approximation sets of intuitionistic fuzzy rough sets are intuitionistic fuzzy sets by making the proof [11]. based on intuitionistic fuzzy residual implication and intuitionistic fuzzy relationship, an intuitionistic fuzzy rough set model is established[12]. however, it is difficult to apply this model to deal with an information system with noise data. with the intuitionistic fuzzy triangle model t = min, intuitionistic fuzzy t-conorms s= max, and intuitionistic fuzzy inverse operator n, the approximation operators of the intuitionistic fuzzy rough sets is defined, and the intuitionistic fuzzy rough set models based on the general intuitionistic fuzzy logic operators are developed[13–15]. by using the thought of intuitionistic fuzzy set and rough set, an improved intuitionistic fuzzy rough set model based on hamming distance and establish the models such properties as interval, symmetry, complete similarity and complete dissimilarity are proposed[16], while the novel intuitionistic fuzzy rough set based on general intuitionistic fuzzy information systems is constructed in order to expand the model and its application[17]. the interval-valued intuitionistic fuzzy rough set based on the thought of implication is established, and then the related properties of the models are developed[18, 19]. by combining interval intuitionistic fuzzy set and rough set, the interval intuitionistic fuzzy rough set models based on interval intuitionistic fuzzy relationship are constructed[19–21]. by using interval-valued intuitionistic fuzzy compatibility relationship, the interval-valued intuitionistic fuzzy rough set model based on the concept of double universes and relevant properties is constructed, and then it is applied into the decision making[22, 23]. for the related literature on attribute reduction, a genetic algorithm is proposed to reduce attributes by making use of the characteristics of intuitionistic fuzzy information systems[24], while the kind of attribute reduction algorithm of intuitionistic fuzzy rough set based on mutual information by combining in204 s. baizakov, n.baizakov and jeffrey forrest: a novel multi-attribute decision... formation entropy theory and intuitionistic fuzzy rough set is constructed and designed[25]. by using the intuitionistic fuzzy distance formula, an intuitionistic fuzzy rough model and an attribute reduction algorithm is constructed[26–28]. as discussed above, although there exist many related literatures on the intuitionistic fuzzy rough set model to construct models and discuss their theories from different levels, there are few intuitionistic fuzzy rough set models in the existing literatures those can be applied into the multi-attribute decision making and effectively deal with the real problems. in the multi-attribute group decision making problems, the dominated and dominating relationship for the attributes should be considered, and the subjective preferences of attribute value from decision-makers are based on some methods to aggregate, so that the acquisition of the decision rules and the ranking of the schemes can be made. however, it is difficult for the group decision making methods based on rough set[29–38] to deal with multi-attribute decision making problems with noise data, preference information and fuzzy information. in view of this, with respect to the shortcoming of the size comparison of the intuitionistic fuzzy numbers, the concept of the intuitionistic fuzzy dominance distance index is defined, and then it is used to construct a novel intuitionistic fuzzy rough set model, finally an example illustrates the effectiveness and applicability of the proposed model. 2 dominance distance index intuitionistic fuzzy sets proposed by atanassov[6, 7] are an expansion and further development of the traditional fuzzy sets. as a result of taking into account to the membership and non-membership information by adding a new attribute parameter: non-membership function in intuitionistic fuzzy sets, it provides additional options for describing the properties of things and possesses a stronger capability of dealing with uncertainty information. that is, intuitionistic fuzzy sets can well describe and portray delicate ambiguities of the nature of the objective world. 2.1 intuitionistic fuzzy dominance distance index definition 1. (intuitionistic fuzzy sets[6, 7]. suppose thatx = {x1, x2, ..., xn} is a nonempty, finite set of objects with xi(i = 1, 2, ..., n) being the ith object. then the set a = {< x, µa(x), υa(x) > |x ∈ x} of triplets is called an intuitionistic fuzzy set, where µa(x) and υa(x) are respectively known as the membership and non-membership for the object to belong to , that is, µa(x) : x → [0, 1], x ∈ x → µa(x) ∈ [0, 1] (1) υa(x) : x → [0, 1], x ∈ x → υa(x) ∈ [0, 1] (2) satisfying 0 ≤ µa(x) + υa(x) ≤ 1 , for any x ∈ x. and πa(x) = 1 − µa(x) − υa(x), x ∈ x stands for the degree of hesitation or uncertainty for the object to advances in systems science and application (2015) vol.15 no.3 205 belong to . so the intuitionistic fuzzy number is denoted as α =< µa(x), υa(x) >. definition 2. (intuitionistic fuzzy sets[6, 7]. for any intuitionistic fuzzy number α =< µ, ν >, the score function s(α) of this number is defined as follows: s(α) = µ− ν, s(α) ∈ [−1, 1] (3) the larger s(α) is, the greater the intuitionistic fuzzy number α =< µ, ν > is. for example, assume that both intuitionistic fuzzy numbers are respectively α1 =< 0.8, 0.1 > and α1 =< 0.9, 0.1 >, because s(α1) = 0.7 and s(α2) = 0.8, then s(α1) < s(α2). so we can regard the intuitionistic fuzzy number α2 as being greater than α2. definition 3. (intuitionistic fuzzy sets[6, 7]. for any intuitionistic fuzzy number α =< µ, ν >, the accuracy function h(α) of this number is defined as follows: h(α) = µ+ ν 2 (4) the larger h(α) is, the greater the intuitionistic fuzzy number α =< µ, ν > is. for example, for intuitionistic fuzzy numbers α3 =< 0.7, 0.2 > and α4 =< 0.4, 0.2 >, according to definition 4, the accuracy functions of these intuitionistic fuzzy numbers are respectively h(α3) = 0.45 and h(α3) = 0.3. therefore,α3 > α4. according to the definition 2 and 3, based on the score function s(α) and precision function h(α), the intuitionistic fuzzy numbers are compared. for any both intuitionistic fuzzy numbers αi =< µi, υi > and αk =< µk, υk >, if s(αi) ≥ s(αk), then αi ≥ αk; if s(αi) = s(αk) and h(αi) ≥ h(αk), then αi ≥ αk. for example, for intuitionistic fuzzy numbers α5 =< 0.8, 0.1 > and α6 =< 0.6, 0.3 >, due to s(α5) = 0.7, s(α6) = 0.3, therefore α5 > α6; while for the intuitionistic fuzzy numbers α7 =< 0.7, 0.3 > and α8 =< 0.5, 0.1 >, due to s(α7) = 0.4, s(α8) = 0.4, h(α7) = 0.5, h(α8) = 0.3, therefore α7 > α8. however, there exist the shortcomings that how much their uncertainty allotted to the membership and non-membership for two intuitionistic fuzzy numbers, so that the size of the intuitionistic fuzzy numbers cannot be exactly determined, for example, the intuitionistic fuzzy numbers α7, α8. in order to determine the size relationship of the intuitionistic fuzzy numbers, a novel method should be proposed. definition 4. for any given two intuitionistic fuzzy numbers αi =< µi, υi > and αk =< µk, υk >, then the dominance distance index ifdd(xi, xk) of the intuitionistic fuzzy numbers αi and αk can be defined as follows: ifdd(xi, xk) =  1 µi ≥ µk, υi ≤ υk 0 µi < µk, υi > υk 1 2 + 1 4 µi−υi−µk+υk µ(ω) other (5) 206 s. baizakov, n.baizakov and jeffrey forrest: a novel multi-attribute decision... where, µ(ω) stands for the background measure, and it is generally taken as µ(ω) = max {1− υ1, 1− υ2, ..., 1− υn}−min {µ1, µ2, ..., µn}, while ifdd(xi, xk) expresses the degree of intuitionistic fuzzy numbers vi better than the intuitionistic fuzzy number vk. based on the definition 4, intuitionistic fuzzy numbers can be compared. for example, suppose that the background measure µ(ω) is 1, for the intuitionistic fuzzy numbers α9 =< 0.8, 0.1 > and α10 =< 0.5, 0.3 >, such that ifdd(xi, xk) = 0.55, therefore, α11is greater than α12 by the possibility of 55%. 2.2 dominance relationship based on dominance distance index definition 5. assume that an ordered 4-tuple s = (u,a, v, f) is known as an preference information system, if u = {u1, u2, ..., un} is a finite and nonempty set, known as the universe; a = c ∪ d is a finite, nonempty attribute set, while c = {a1, a2, ..., am} and d are respectively the condition attribute set and the decision attribute set; v = ∪va stands for the value domain of the information system s, where va is the value of u with respect to the attribute a ∈ a; f : u ×a → v is an information function. if va is an intuitionistic fuzzy number αa =< µa, υa >, then the information system s is called as the intuitionistic fuzzy preference information system, and denoted as ifs. definition 6. suppose that there exist an intuitionistic fuzzy preference information system ifs = (u,a, v, f), for ∀a ∈ p ⊆ a, xi, xk ∈ u, f(xi, a) = αia =< µia, υia >∈ v , f(xk, a) = αka =< µka, υka >∈ v , λ ∈ (0.5, 1], if ifdd(xi, xk) ≥ λ, then the dominance relationship between the objects xi and xk with respect to the α attribute or the attribute set p can be called as a dominance relationship based on the dominance distance index with threshold value λ, written as (xi, xk) ∈ r≥λ a , (xi, xk) ∈ r≥λ p . property 1.suppose that there exist an intuitionistic fuzzy preference information system ifs = (u,a, v, f), ∀a ∈ p ⊆ a, ∀xi, xk ∈ u, f(xi, a) = αia =< µia, υia >∈ v, f(xk, a) = αka =< µka, υka >∈ v , it holds true: 0 ≤ ifdda(xi, xk) ≤ 1 (6) 0 ≤ ifddp (xi, xk) ≤ 1 (7) proof. according to the definition 4, (1)if µia ≥ µka, υia ≤ υka, then ifdda(xi, xk) = 1; (2)if µia < µka, υia > υka, then ifdda(xi, xk) = 0; (3)if µia ≥ µka, υia ≥ υka or µia ≤ µka, υia ≤ υka, because of the background value measure µ(ω) = max {1− υ1, 1− υ2, ..., 1− υn}−min {µ1, µ2, ..., µn}, then 2(min i (µia)−max i (1−υia)) ≤ µia−υia−µka+υka ≤ 2(max i (1−υia)−min i (µia)), so that −2 ≤ µia−υia−µka+υka max i (1−υia)−min i (µia) ≤ 2, and then there exists advances in systems science and application (2015) vol.15 no.3 207 0 ≤ 1 2 + 1 4 µia−υia−µka+υka max i (1−υia)−min i (µia) ≤ 1, therefore, 0 ≤ ifdda(xi, xk) ≤ 1. in conclusion, 0 ≤ ifdda(xi, xk) ≤ 1 can be obtained. let the weight vector of attributes is w = (w1, w2, ..., wp), and it satisfies wt > 0 and p∑ t=1 wt = 1, and then there exists ifddp (xi, xk) = p∑ t=1 wtifddat(xi, xk), therefore, 0 ≤ ifddp (xi, xk) ≤ 1. property 2. for ∀a ∈ p ⊆ a, xi, xk ∈ u , f(xi, a) = αia =< µia, υia >∈ v and f(xk, a) = αka =< µka, υka >∈ v , λ ∈ (0.5, 1], if (xk, xs) ∈ r≥λ a , the following holds true: ifdda(xi, xk) ≤ ifdda(xi, xs). proof. it suffices to show ifdda(xi, xk)− ifdda(xi, xs) ≤ 0. for ifdda(xi, xk), there exist three cases:ifdda(xi, xk) = 1. ifdda(xi, xk) = 1 2 + 1 4 µia−υia−µka+υka max i (1−υia)−min i (µia) and ifdda(xi, xk) = 0. for ifdda(xi, xs), there also are three cases. for (xk, xs) ∈ r≥ a , according to the dominance relationship r≥λ a , that is, ifdda(xk, xs) ≥ λ by analyzing the situation, there exit the following 6 cases. (1) when the object xi is definitely better than the objects xk and xs, that is αi > αk and αi > αs, thus ifdda(xi, xs) = 1, ifdda(xi, xk) = 1, therefore, ifdda(xi, xs)− ifdda(xi, xk) = 0. (2)when the objectxi is definitely better than the object xs, and not necessarily better than the object xk, ifdda(xi, xs) = 1 and ifdda(xi, xk) = 1 2 + 1 4 µia−υia−µka+υka max i (1−υia)−min i (µia) can be acquired. therefore, the following holds true: ifdda(xi, xs) − ifdda(xi, xk) = 1 2 − 1 4 µia−υia−µka+υka max i (1−υia)−min i (µia) . according to definition 4, there exits −2 ≤ µia−υia−µka+υka max i (1−υia)−min i (µia) ≤ 2, that is, ifdda(xi, xk) − ifdda(xi, xs) ≤ 0. (3) when the object xi is definitely better than the objects xs and must be inferior to the object xk, ifdda(xi, xs) = 1 and ifdda(xi, xk) = 0 can be obtained. thus, there exists ifdda(xi, xs) − ifdda(xi, xk) = 1 that is, ifdda(xi, xk)− ifdda(xi, xs) ≤ 0. (4) when the object xi may be superior to the object xs and may be superior to the object xk, ifdda(xi, xs) = 1 2+ 1 4 µia−υia−µsa+υsa max i (1−υia)−min i (µia) and ifdda(xi, xk) = 1 2+ 1 4 µia−υia−µka+υka max i (1−υia)−min i (µia) can be acquired. thus, ifdda(xi, xs)−ifdda(xi, xk) = 1 4 µka−υka−µsa+υsa max i (1−υia)−min i (µia) due to ifdda(xk, xs) ≥ 1 2 , thus µka − υka − µsa + υsa ≥ 0, therefore, it then follows that ifdda(xi, xs)− ifdda(xi, xk) ≥ 0. when the object xi may be superior to the object xs and must be inferior to the object xk, ifdda(xi, xs) = 1 2+ 1 4 µia−υia−µsa+υsa max i (1−υia)−min i (µia) and ifdda(xi, xk) = 0 can be obtained. thus ifdda(xi, xs)− ifdda(xi, xk) = 1 2 + 1 4 µia−υia−µsa+υsa max i (1−υia)−min i (µia) . 208 s. baizakov, n.baizakov and jeffrey forrest: a novel multi-attribute decision... from the fact that −2 ≤ µia−υia−µsa+υsa max i (1−υia)−min i (µia) ≤ 2, it follows that ifdda(xi, xs)− ifdda(xi, xk) ≥ 0. when the object xi must be inferior to the objects xs and xk, there exist ifdda(xi, xs) = 0 and ifdda(xi, xk) = 0, and then ifdda(xi, xs) − ifdda(xi, xk) = 0 can be concluded. in conclusion, ifdda(xi, xs)−ifdda(xi, xk) ≥ 0 therefore, ifdda(xi, xk) ≤ ifdda(xi, xs). qed property 3. the dominance relationship based on dominance distance index satisfies the condition of transitivity. proof. it suffices to show that for ∀a ∈ p ⊆ a, xi, xk ∈ u, if ifdda(xi, xk) ≥ 1 2 and ifdda(xk, xs) ≥ 1 2 , then ifdda(xi, xs) ≥ 1 2 . when ifdda(xi, xk) ≥ 1 2 , there exists µia − υia − µka + υka ≥ 0 . accordingly there are two cases which are respectively certainly better and perhaps superior for the object xi than the object xk. that is, if µi ≥ µk, υi ≤ υk, then ifdda(xi, xk) = 1. if ifdda(xi, xk) = 1 2 + 1 4 µia−υia−µka+υka max i (1−υia)−min i (µia) and µia − υia − µka + υka ≥ 0 then ifdda(xi, xk) ≥ 1 2 . when ifdda(xk, xs) ≥ 1 2 similarly there are also two cases which are respectively and certainly better and perhaps superior for the object xk than the object xs. thus there exist the following 4 cases where the object xi is better than the object xs. (1)if ifdda(xi, xk) = 1 and ifdda(xk, xs) = 1, then µi ≥ µk, υi ≤ υk and µk ≥ µs, υk ≤ υs, and then µi ≥ µs, υi ≤ υs. therefore ifdda(xi, xs) = 1. (2)if ifdda(xi, xk) = 1, ifdda(xk, xs) = 1 2 + 1 4 µka−υka−µsa+υsa max i (1−υia)−min i (µia) and µka−υka−µsa+υsa ≥ 0, then µia−υia−µsa+υsa ≥ 0. according to definition 4, ifdda(xk, xs) ≥ 1 2 can be obtained. (3)ifdda(xk, xs) = 1 2+ 1 4 µka−υka−µsa+υsa max i (1−υia)−min i (µia) and µia−υia−µka+υka ≥ 0 and ifdda(xk, xs) = 1 2 + 1 4 µka−υka−µsa+υsa max i (1−υia)−min i (µia) and µka − υka − µsa + υsa ≥ 0, such that µia − υia − µsa + υsa ≥ 0 and according to the definition of the dominance distance index, ifdda(xk, xs) ≥ 1 2 can be obtained. (4) if ifdda(xi, xk) = 1 2 + 1 4 µia−υia−µka+υka max i (1−υia)−min i (µia) , µia−υia−µka+υka ≥ 0 and ifdda(xk, xs) = 1, then µia − υia − µsa + υsa ≥ 0. according to the definition of the dominance distance index, therefore, ifdda(xk, xs) ≥ 1 2 . in conclusion, ifdda(xk, xs) ≥ 1 2 can be always obtained. qed. property 4. for ∀a ∈ p ⊆ a, xi, xk ∈ u , αia =< µia, υia >∈ v , αka =< µka, υka >∈ v ,then the following hold true: ifdda(xi, xk)+ifdda(xk, xi) = 1. proof. according to equation (5), (1)if µia ≥ µka, υia ≤ υka, then ifdda(xi, xk) = 1 and ifdda(xk, xi) = 0, therefore ifdda(xi, xk) + ifdda(xk, xi) = 1 can be obtained. advances in systems science and application (2015) vol.15 no.3 209 (2) when µia < µka, υia > υka, it follows that ifdda(xi, xk) = 0 and gdda(xk, xi) = 1. thus, we have gdda(xi, xk) +gdda(xk, xi) = 1. (3) when the other conditions, ifdda(xi, xk) = 1 2 + 1 4 µia−υia−µka+υka µa(ω) and ifdda(xk, xi) = 1 2 + 1 4 µka−υka−µia+υia µa(ω) can be acquired. thus, ifdda(xi, xk) + ifdda(xk, xi) = 1 can be obtained. qed. 3 dominance intuitionistic fuzzy variable precision rough set model in order to deal with preference attributes, with respect to the information system with preference value[39? –41], by introducing dominance relationship into the rough set mode and taking advantage of it to substitute for the indistinguishable relationship constructed the dominance rough set model, and then acquired the decision making rules. based on the thought of dominance rough set model, the dominance relationship based on intuitionistic fuzzy dominance distance index can be used to substitute the equivalence relationship of rough set, and then the dominance intuitionistic fuzzy rough set model is constructed. 3.1 the construction of model definition 7. suppose that there exist an intuitionistic fuzzy preference information system ifs = (u,a, v, f), for p ⊆ a, cl≥t ⊆ d, a threshold value λ ∈ (0.5, 1] and β ∈ (0.5, 1], if d+ p (x) = {y ∈ u |ifdda(x) ≥ λ, ∀a ∈ p } and d− p (x) = {y ∈ u |ifdda(x) < λ, ∀a ∈ p } respectively stand for the λ−p dominating set and dominated set with respect to x, and then the lower approximation and upper approximation of the decision making class cl≥t are respectively defined as follows: aprλ p (cl≥t ) = ∪{x ∈ u : d+ p (x) ⊆ cl≥t } (8) aprλp (cl≥t ) = ∪{x ∈ u : d− p (x) ∩ cl≥t ̸= ∅} (9) and thus, [ aprλ p (cl≥t ), apr λ p (cl≥t ) ] is called as the dominance intuitionistic fuzzy rough set. the λ−p -lower approximation aprλ p (cl≥t ) of (cl≥t ) can be called as the positive domain of the dominance intuitionistic fuzzy rough set and interpreted as the set of the union of all condition classes with confidence threshold value , where the classified objects definitely belong to the upward union (cl≥t ). accordingly, the λ − p -upper approximation aprλ p (cl≥t ) of (cl≥t ) can be interpreted as the union of all classes with confidence threshold value λ, where the classified objects possibly belong to the upward union (cl≥t ). accordingly, based on the lower approximation and upper approximation of the intuitionistic fuzzy rough set, the λ boundary domain and classification quality of the set (cl≥t ) can be defined as follows: bndλp (cl≥t ) = aprλp (cl≥t )− aprλ p (cl≥t ) (10) 210 s. baizakov, n.baizakov and jeffrey forrest: a novel multi-attribute decision... αλ p (cl≥t ) = |aprλ p (cl≥t )| |aprλp (cl≥t )| (11) analogously, the λ -lower and λ -upper approximations of can be respectively defined as follows: aprλ p (cl≤t ) = ∪{x ∈ u : d− p (x) ⊆ cl≤t } (12) aprλp (cl≤t ) = ∪{x ∈ u : d+ p (x) ∩ cl≤t ̸= ∅} (13) and thus, [ aprλ p (cl≤t ), apr λ p (cl≤t ) ] stands for the dominance intuitionistic fuzzy rough set. accordingly, based on the lower approximation and upper approximation of the intuitionistic fuzzy rough set, the λ boundary domain and classification quality of the set (cl≥t ) can be defined as follows: bndλp (cl≤t ) = aprλp (cl≤t )− aprλ p (cl≤t ) (14) αλ p (cl≤t ) = |aprλ p (cl≤t )| |aprλp (cl≤t )| (15) 3.2 attribute reduction and preference rules definition 8. suppose that that there exist an intuitionistic fuzzy preference information system ifs = (u,a, v, f), for p ⊆ a and the given threshold value λ ∈ (0.5, 1], the classification quality of cl can be defined as follows: γλp (cl) = |u − ((∪bnd(cl≥t )) ∪ (∪bnd(cl≤t )))| |u | (16) the classification quality γλp (cl) of cl stands for the ratio of the relation between all the correctly classified objects and all the objects with respect to the attribute set p in the information system. for every minimal subset p ⊆ c, the attribute set p satisfying γλp (cl) = γλc(cl) s called as the reduction of c with respect to cl and denoted byredcl(p ). the preferential decision rule is one kind of dependence form between condition preference attribute and decision preference attribute. based on dominance relationship with the dominance distance index, the rough approximation is acquired, and then the preferential decision rule can be induced and shown as follows: for the given threshold value λ, d≥ -decision rules can take on the following form: if f(x, q1) ≥ rq1 ∧ f(x, q2) ≥ rq2... ∧ f(x, qp) ≥ rqpthenx ∈ cl≥t ; for the given threshold value λ, d≤ -decision rules can take on the following advances in systems science and application (2015) vol.15 no.3 211 form: if f(x, q1) ≤ rq1 ∧ f(x, q2) ≤ rq2... ∧ f(x, qp) ≤ rqpthenx ∈ cl≥t ; where {q1, q2, ...qp} ⊆ c, iff(x, q1) ≤ rq1 ∧ f(x, q2) ≤ rq2... ∧ f(x, qp) ≤ rqpthenx ∈ cl≥t , t ∈ {1, 2, ..., l}. 3.3 comprehensive dominance degree definition 9. suppose that the intuitionistic fuzzy preference information system ifs = (u,a, v, f), for ∀aj ∈ p ⊆ a, xi, xk ∈ u , f(xi, aj) = αij =< µij , υij >∈ v , f(xk, aj) = αkj =< µkj , υkj >∈ v the following is called as the dominance degree with equal weight with respect to the attribute set p for the object xi over the object xk based on the dominance distance index: ifddp (xi, xk) = 1 |p | ∑ ∀a∈p ifdda(xi, xk) (17) where |p | is the cardinality of the attribute set p . in the situation of real-life decision making, due to the fact that different decision makers may very well weigh the attributes differently, the results, obtained on the basis of this uniform treatment of equal weights, can be obviously expected to be inconsistent with the reality. therefore, there is a need to study the situation that the attributes are given different weights. definition 10. suppose that the intuitionistic fuzzy preference information system ifs = (u,a, v, f), for ∀aj ∈ p ⊆ a, xi, xk ∈ u , f(xi, aj) = αij =< µij , υij >∈ v , f(xk, aj) = αkj =< µkj , υkj >∈ v , let the weight vector of the attribute be w = (w1, ..., wt, ..., w|p |), satisfying wt > 0 and |p |∑ t=1 wt = 1. then ifddp (xi, xk) = |p |∑ t=1 wtifdda(xi, xk) (18) is referred to as the different weight dominance degree with respect to the attribute set p for the object xi over the object xk based on the dominance distance index. definition 11. suppose that the intuitionistic fuzzy preference information system is ifs = (u,a, v, f), for ∀xi ∈ u , p ⊆ c, the comprehensive dominance degree of the object xi in all the objects based on the dominance distance index is defined as follows: ifddp (xi) = 1 |u | − 1 ∑ i ̸=k ifddp (xi, xk) = 1 |u | − 1 ∑ i̸=k |p |∑ t=1 wtifdda(xi, xk) (19) 212 s. baizakov, n.baizakov and jeffrey forrest: a novel multi-attribute decision... 3.4 the decision steps based on dominance intuitionistic fuzzy rough set to sum up, the decision steps are following: step1: by adjusting the given threshold value , based on the condition attribute set, the different classifications can be determined. step2: based on the decision attribute set, all classifications can be determined. step3: according to all classifications based on the condition attribute set and the decision attribute set, the lower approximation and the upper approximation of the decision classifications can be acquired, and then the attribute reduction can be obtained, so that t the decision rules can be extracted. step4: by the classification results based on the condition attribute set or the decision attribute set, based on the expressions (18) and (19), the comprehensive dominance degrees of the objects are calculated, and then the ranking of the objects can be determine. 4 case analysis information system security design is that by testing the legal compliance, the confidentiality, the usability, and the reliability of the audiles, the security risk of information system is assessed, and then some audit opinions and suggestions of information system security can be given. while the risk guidance information system security audit based on the risk identification and evaluation is that through the comprehensive analysis of the audiles operation environment and information system running environment, the factors affected the system security are extracted, and then according to the risk assessment, the implementation audit scope and key target are determined, so that the substantive tests are implemented. therefore, the evaluation and judgment of audit risk possess are much more important in the information system security audit process. currently, few scholars take advantage of rough set to make the audit risk judgment of the information system security, and the evaluation is mainly based on the knowledge and experience of audit personnel in the practice process, however, it lacks the scientific and rationality, which has effect on the implementation of information system security audit, and ultimately affects the audit results. in recent years, the audit risk assessment and risk judgment expert system is established to effectively decrease audit judgment deviation and reduce the audit risk. however, due to the complexity and uncertainty of the objective world, as well as the limitation of human ability to understand, there always exist a variety of preference information and fuzzy information in the audit risk assessment and risk judgment expert system, while it is difficult for the existing representation and processing methods of the information system security audit based on the expert experience to acquire exactly knowledge. in view of this, based on the proposed model in advances in systems science and application (2015) vol.15 no.3 213 this paper, the attribute reduction is used to extract the decision rule of the audit risk assessment of the information system security and acquire the key factors and bottleneck factors of affecting the audit risk assessment of the information system security, and then the ranking of the different information system security can be determined, so that the proposed model can provide for audit personals a much reasonable and effective audit risk assessment method and tool. according to the actual audit cases, the related data can be collected and shown in the table 1. in the table 1, there exist 10 audited objects denoted as u = {x1, x2, ..., x10}, and the five attributes c = {a1, a2, a3, a4, a5} which respectively good system environment, good system control, reliable financial data, reliable audit software and standard operation. for each condition attribute, based on the audit results and their own professional quality, its values can be obtained by the comprehensive judgment from the information system audit experts, for example, f(x2, a1) =< 0.5, 0.4 > stands for the fact that 50% experts think that the system environment of the audited object x2 is good, while 40% experts think that it is bad and 10% experts hesitate. the decision attribute set is denoted as d = {d}, and it indicates whether the audit risk of information system security is acceptable. for example, f(x2, d) = 1 expresses that experts regard that the audit risk of the audited object x2 is acceptable. table 1 the audit risk assessment intuitionistic fuzzy information system of the information system security u a1 a2 a3 a4 a5 d u1 < 0.2, 0.6 > < 0.1, 0.7 > < 0.4, 0.4 > < 0.5, 0.4 > < 0.4, 0.4 > 0 u2 < 0.5, 0.4 > < 0.3, 0.6 > < 0.3, 0.6 > < 0.5, 0.2 > < 0.5, 0.4 > 0 u3 < 0.2, 0.6 > < 0.1, 0.8 > < 0.2, 0.7 > < 0.3, 0.6 > < 0.3, 0.6 > 0 u4 < 0.7, 0.1 > < 0.6, 0.4 > < 0.7, 0.2 > < 0.8, 0.2 > < 0.7, 0.2 > 1 u5 < 0.3, 0.6 > < 0.2, 0.7 > < 0.2, 0.7 > < 0.2, 0.6 > < 0.3, 0.7 > 0 u6 < 0.6, 0.3 > < 0.6, 0.4 > < 0.6, 0.3 > < 0.7, 0.2 > < 0.5, 0.4 > 1 u7 < 0.2, 0.6 > < 0.2, 0.6 > < 0.5, 0.4 > < 0.5, 0.4 > < 0.2, 0.6 > 1 u8 < 0.1, 0.6 > < 0.2, 0.6 > < 0.4, 0.5 > < 0.3, 0.6 > < 0.2, 0.7 > 0 u9 < 0.7, 0.2 > < 0.6, 0.4 > < 0.8, 0.1 > < 0.6, 0.3 > < 0.8, 0.2 > 1 u10 < 0.6, 0.2 > < 0.6, 0.2 > < 0.8, 0.2 > < 0.4, 0.5 > < 0.4, 0.5 > 1 step1: according to the condition attribute set c, when λ = 0.55, based on the dominance distance index, the universe can be divided into the following classifications: u/c = {x1, x2, x3, x4} where x1 = {x1, x3, x5} , x2 = {x7, x8} , x3 = {x2, x6, x9, x10} , x4 = {x4} and 214 s. baizakov, n.baizakov and jeffrey forrest: a novel multi-attribute decision... x1 < x2 < x3 < x4. step2:according to the decision attribute set d, the universe can be divided into the following classifications: u/d = {cl1, cl2}. where cl1 = {x1, x2, x3, x5, x7, x8}, u/d = {cl1, cl2}. step3: according to the classifications based on the condition attribute set and the decision attribute set, the lower approximation aprλ p (cl≥1 ) of (cl≥1 ) and the upper approximation aprλ p (cl≥1 ).the lower approximation aprλ p (cl≤2 ) of (cl≤2 ) and the upper approximation aprλ p (cl≤2 ) respectively are: aprλ p (cl≤1 ) = {x1, x3, x5, x7, x8} , aprλp (cl≤1 ) = {x1, x2, x3, x5, x6, x7, x8, x9, x10} aprλ p (cl≥2 ) = {x4} , aprλp (cl≥2 ) = {x2, x4, x6, x9, x10} γλ,βp (cl) = | {x1, x3, x5, x7, x8} ∪ {x4} | | {x1, x2, x3, x4, x5, x6, x7, x8, x9, x10} | = 6 10 = 0.6 when λ = 0.55, the reduction {a1, a2} based on the genetic algorithm can be acquired, therefore, it is well known that the factors of good system environment and good system control should be considered in the audit risk assessment of information system. according to the reduction {a1, a2}, the probabilistic decision rules can be generated and shown in the table 2. table 2 the probabilistic decision rules based on the reduction {a1, a2} with λ = 0.55. rules support number confidence a1 6< 0.3, 0.6 > and a2 6< 0.2, 0.7 > 100%−−−→ d = 0 2 100% a1 6< 0.2, 0.6 > and a2 6< 0.2, 0.6 > 100%−−−→ d = 0 3 100% a1 >< 0.6, 0.3 > and a2 >< 0.6, 0.4 > 100%−−−→ d = 1 4 100% step4: because there are four classifications based on the condition attribute set, and it satisfies x1 < x2 < x3 < x4, that is, {x1, x3, x5} < {x7, x8} < {x2, x6, x9, x10} < x4. for the object x4, due to the fact that there only exists the object x4 in the x4, so its comprehensive dominance degree no longer needs to calculate; for the classes {x1, x3, x5} , {x7, x8} and {x2, x6, x9, x10}, their comprehensive dominance degrees should be computed. according to the attribute dependency based on the proposed the model, the attribute dependency degree of each attribute can be acquired and then they are standardized, thus the weight vector is obtained as follows w = (w1, w2, w3, w4, w5) = (0.2125, 0.2024, 0.1964, 0.1998, 0.1889). for {x1, x3, x5}, based on the formula (18) and (19), ifdda(x1) = 0.5204, advances in systems science and application (2015) vol.15 no.3 215 ifdda(x3) = 0.1745 and ifdda(x5) = 0.3035 can be calculated, therefore, x1 > x5 > x3. for {x7, x8}, because the both objects are in the classification, it is necessary to only calculate the dominance degree with weights, and then it will exist ifdda(x7) = ifdda(x7, x8) = 0.8167 and ifdda(x8) = ifdda(x8, x7) = 0.1833, therefore, x7 > x8. for {x2, x6, x9, x10}, based on the formula (18) and (19), ifdda(x2) = 0.1469, ifdda(x6) = 0.4039, ifdda(x9) = 0.5743 and ifdda(x10) = 0.3563 can be calculated, therefore, x2 < x10 < x6 < x9. to sum up, the ranking of the audited objects is x3 < x5 < x1 < x7 < x8 < x2 < x10 < x6 < x9 < x4. based on the above calculation and analysis, the proposed model possesses certain fault-tolerant ability by adjusting parameter λ, and it can well do with the intuitionistic fuzzy information system with preference information and realize the extraction of group decision rules and the ranking of schemes, so that it can well deal with the real multi-attribute decision making problems with preference information and fuzzy information. 5 conclusion in order to achieve the law of mining the real decision information system and extract decision rules, with respect to the intuitionistic fuzzy preference information system, we construct the dominance intuitionistic fuzzy rough set based on dominance distance index, and then exploit it to extract decision-making rules and determine the ranking of decision schemes. the results show that the hybrid model can well 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(2001), “an algorithm for induction of decision rules consistent with dominance principle”, lecture notes in artificial intelligence, vol. 2005, pp. 304-313. corresponding author yong liu can be contacted at: clly1985528@163.com adv sist sci appl 2017; 17(2); 43-51 published online in http://ijassa.ipu.ru/ojs/ijassa/article/view/258 copyright ©0000 assa. adv. in systems science and appl. (0000) reconfigurable architecture for image feature detection rajesh nandalike 1 , saroja devi hande 2 1) department of electronics and communication engineering e-mail: nrajesh7@gmail.com 2) department of computer science and engineering nitte meenakshi institute of technology, bengaluru, india e-mail: hsarojadevi@gmail.com abstract: hardware-based developments are useful for variety of image based applications such as highly challenging video surveillance. fpga comprises of combination of the hardware attributes of an asic, supporting reconfigurability, reduced time-to-market and real-time performance. hardware-based feature detection is a promising solution that exploits inherent parallelism in algorithms to accomplish significant improvement in speed. the efficient usage of resources still remains a challenge that specifically determines the cost of hardware, which can be addressed using our approach. in this paper, we have presented the hardware architecture for feature detection part of the scale invariant feature transform (sift) algorithm for frames from an hd-720p video. the proposed architecture is designed using xilinx system generator (sysgen) tool and implemented on genesys2 kintex-7 fpga development board. keywords: field programmable gate array (fpga), scale-invariant feature transform (sift), feature detection, keypoint detection, feature point, image feature. 1. introduction feature detection and matching are fundamental functionalities in many image applications, video surveillance and also computer vision. they are computationally intensive, requiring considerable amount of resources in terms of time, power, memory and silicon. scale invariant feature transform (sift) algorithm proposed by lowe is one of the potent algorithms in image matching and object recognition [1,2]. high resource utilization and computational complexity make sift algorithm challenging to cope up with the realtime performance for software implementation. feature-based identification is the prevailing object identification strategy. it employs one of the feature extraction algorithms to extract the important features of the image. the development of numerous feature extraction algorithms has been taking place during last few years. canny and sobel’s edge detectors, binary robust independent elementary feature (brief) and harris corner detectors are local feature extraction algorithms that are employed to extract features from an image in object recognition systems [3,4]. every algorithm attempted to enhance the uniqueness and robustness across image transformation operations. this paper proposes hardware architecture for feature detection on fpga, so that the chip acts as a standalone system for feature detection. based on the analysis of feature detection algorithms, five steps must be performed for feature detection and matching, visualization, pre-processing, feature detection, feature descriptor building, feature matching and postprocessing. sift algorithm is used for feature detection in this paper. the approach aims at suitable algorithmic selection and testing using matlab. different algorithms are tested and the corresponding hardware implementation architectures 44 rajesh n., saroja devi h. copyright ©2017 assa. adv. in systems science and appl. (2017) are investigated. feature detection techniques are implemented on fpga, which are directly adaptable by robotic units and air-borne vehicles. 2. scale invariant feature transform (sift) sift is the most competent approach to identify and describe invariant features of an image [5]. the features obtained are invariant to image scaling, rotation and partially invariant to change in illumination. it is an algorithm where image data is converted to scaleinvariant coordinates corresponding to its local features. sift consists of four major stages:  scale-space peak selection: in the first stage, the identification of feasible interest points in an image with their position and scale is accomplished. this is utilized effectively to find out the interest points that are stable, by constructing the gaussian pyramid and searching for the interest points in a series of difference-of-gaussian (dog) images.  key point localization: in the second stage, the feature points are restricted to sub-pixel precision and are waived if they are ambiguous.  orientation assignment: in the third stage, orientations are allotted to each and every feature point position depending on the image gradient directions. the orientation, scale and location to each feature point empowers sift to construct an accepted perspective for the feature point that is invariable to identical transformations.  key point descriptor: in the fourth stage, image feature descriptor is built for each feature point. the sift algorithm forms a depiction for each feature point depending on a patch of pixels in its neighborhood. the patch that has been formerly centered about the feature point's location will be rotated depending on the dominant orientation and scaled to suitable size. the sift descriptor produces 128 feature vectors for each feature point [69]. 3. feature detection the properties of features found in an image make them relevant for identifying different images of a similar scene. detecting good features is a complex problem. the following section gives a discussion on the properties of the ideal local feature. the desired properties from a feature depend on the actual application. a feature is a point-of-interest in an image. it is a slice of information which is suitable for determining the computational assignment identified with respect to a particular application. the properties of a good feature are: they are consistent over several images of the same scene, insensitive to noise, invariant towards certain transformations [10]. sift uses scale-space extrema as candidate features. this work concentrates on the scale-space extrema detection with focus on dedicated hardware implementation. the gaussian kernel is well-defined in 2d as, (1) in equation (1), σ determines the width of kernel and is often referred as the inner scale. it is the standard deviation while the σ 2 is the variance. the term 1/(2πσ 2 ) is the normalization constant and it makes the integral over the exponential function unity. this constant in the equation makes it a normalized kernel with the integral unity for every σ. this means that increasing the σ value effectively decreases the height of kernel while increasing its width. reconfigurable architecture for image feature detection 45 copyright ©2017 assa. adv. in systems science and appl. (2017) the procedure of sift feature point detection comprises of developing difference-ofgaussian (dog) pyramid of the image. from the developed dog pyramid, maxima and minima known as scale-space extrema (also known as feature points or interest points) are recognized. the procedure of sift feature identification is given in fig. 1. fig. 1. computational flow of feature detection 3.1. dog pyramid construction the sift algorithm focuses on the image locations that display immense neighborhood transforms in their visual appearances. the dog pyramid is built with the objective of identifying the feature points. the input image i(x, y) is convolved with a gaussian kernel k(x, y; σ), where σ is the size of gaussian kernel. the product is gaussian-filtered image symbolized by equation (2). (2) where conv2(•) symbolizes the 2-d convolution procedure and gaussian kernel is represented by equation (3). (3) the dog image is the difference between the two gaussian filtered images over successive scales, as depicted by equation (4). (4) where k is a multiplicative factor. the convolution of image with the variable size gaussian kernel produces a mass of blurred images with the quantity of blur dependent on scale aspect. the size of gaussian kernel is varied using a constant multiplicative factor k. 3.2. stable key-point detection after the creation of dog image pyramid [11], the local maxima and minima [12,15] is found by comparing every pixel with its 26 neighborhood in the 3x3 regions. these local maxima and minima feature points are shown in fig. 7. when feature point candidates have been created, the low contrast and solid edge response points must be eliminated to make it sturdy against disturbance. this approach gives rise to a lot of feature points depending on the image size. quality of feature points is more important than the quantity for reliable object recognition. the features that have greater probability to be found in the other version of the image exhibit better quality. to check the stability of feature points, the local extrema are compared with a minimum threshold value. the extrema that have relatively higher minima or maxima value pass the threshold test and are considered, while the weak feature points having low contrast are discarded, resulting in less but more stable candidates. 4. sysgen model for feature detection the architecture developed for each of the stage feature detection and matching is first implemented in software, then feature detection based on sift algorithm is implemented in 46 rajesh n., saroja devi h. copyright ©2017 assa. adv. in systems science and appl. (2017) sysgen model based environment [13]. this was carried on to deal with the issues concerned with that of synchronization and data type matching. upon obtaining a simulated design, hardware co-simulation of the same carried on. this was intended to perform functional verification also taking into account the issues concerned with routing delays, latency and timing constraints. implementation of feature detection on fpga is followed on a ‘design module’ basis [14]. initially, the input image which is a 2-d matrix is converted into a 1-d vector. this 1-d vector from matlab workspace environment is passed as an input to the gaussian filter blocks of sysgen to perform 2-d convolution operation between input image and gaussian kernels. the add/sub blocks are used to find the difference between two gaussian images will be taken. the results then passed through a 3x3 window generator subsystem and maxima and minima blocks. finally using and and or gates strong features of the image will be detected. then finally the resultant of the sysgen model will be a 1-d vector taken onto the matlab workspace. this 1-d vector is then converted back to its 2d form using a matlab program to obtain the strong key-points. the overall sysgen model of feature detection is as shown in fig. 2. the model shown can be categorized into the following major subsystems: subsystem 1: 3x3 window generator. subsystem 2: maxima and minima block. subsystem 3: 5x5 gaussian filter. fig. 2. sysgen model for feature detection 4.1. subsystem 1: sysgen model for 3x3 window generator this subsystem implements the 3x3 window on every dog images. the design comprises of six delays and two virtex line buffers to form a 3x3 window on every dog images. the virtex line buffers have a depth equal to the number of columns in an input image.the sysgen model of 3x3 window generator is as shown in the fig. 3. reconfigurable architecture for image feature detection 47 copyright ©2017 assa. adv. in systems science and appl. (2017) fig. 3. sysgen model for window generator 4.2. subsystem 2: sysgen model for maxima and minima blocks the implementation of the maxima and minima blocks is carried out in this subsystem. these blocks are used for finding the maximum value among the 26 neighbors and employs ‘compare and select’ process. the maxima and minima are found by comparing every pixel with its 26 neighborhood in the 3x3 regions. these local maxima and minima points are treated as candidate feature points. the design comprises of 26 relational blocks with comparison is set to a > b for maxima block and a < b for minima block, to find the maximum and minimum value among the 26 neighbors. the sysgen model of maxima and minima block is as shown in the fig. 4. fig. 4. sysgen model for maxima and minima block 4.3. subsystem 3: sysgen model for 5x5 gaussian filter the 1-d vector from matlab workspace environment is passed as an input to the convolution blocks of sysgen to perform 2-d convolution operation between input image and gaussian kernels. the resultant of the sysgen model will be a 1-d vector taken onto the matlab workspace. this 1-d vector is to be then converted back to its 2-d form using a matlab program to obtain the gaussian filtered image. the overall sysgen model of 5x5 gaussian filter is as shown in fig. 5. convolution block subsystem gives 2-d convolution of input image with the corresponding gaussian kernels. this design comprises of five multiplier blocks, concatenated using delay blocks and virtex line buffers with a depth equal to the number of columns in an input image. the design of a 2-d convolution block is shown in fig. 5. 48 rajesh n., saroja devi h. copyright ©2017 assa. adv. in systems science and appl. (2017) fig. 5. sysgen model for gaussian filter 5. results and discussions the experimental result of feature detection for the above proposed architecture is presented. the performance can be considered based on three parameters: accuracy, speed and hardware utilization. there is a trade-off present between these parameters; higher accuracy consumes more hardware resources and processing time. in this work, the preference is given to accomplish high accuracy in real-time restraint. processing time: the processing time is anticipated based on the number of clock cycles necessary to accomplish every task and the operational frequency for that particular module [16]. the processing time is anticipated as number of clock cycles per task divided by operational frequency in hertz. sift module processing time: the maximum operating frequency of the anticipated design is 85.390mhz. the sift feature extraction module takes 1280x720 clock cycles to examine the input image and identify the feature points. feature detection: a high definition image is selected in the process of analysing the performance of architecture. the hd image shown in fig. 6 is selected. at this stage the input image was convolved with the gaussian kernel. (a) (b) fig. 6. (a) input image (b) gaussian smoothened image reconfigurable architecture for image feature detection 49 copyright ©2017 assa. adv. in systems science and appl. (2017) the next step is to take difference of gaussian to get the minima and maxima points. it gives the co-ordinates of the pixel. the feature points of the image are obtained. the feature points for different values of size of the gaussian kernel or standard deviation (σ) are shown in the fig. 7. increase in the σ value decreases the height of the kernel while the width of the kernel increases. the above analysis is carried out for different 200 images. the results of synthesis show that the proposed architecture requires less resources and achieves the real-time processing requirements. speed can be further increased by using higher frequency. the system is able to achieve maximum frequency of 100mhz. furthermore it is also possible to increase the frequency beyond 100mhz by introducing more pipeline stages at the expense of more resources. however increasing the frequency directly impacts the power requirements. in applications that require lower frame rate a lower clock can be used to get more efficiency in terms of power and resources. (a) (b) (c) (d) fig. 7. feature points using gaussian kernels of different variances: (a) σ = 5 (b) σ = 5.62 (c) σ = 5.93 (d) σ = 6.25 the hardware utilized the feature detection module in the proposed architecture is given in table 1. the fpga essentially consists of hardware resources such as memory, slice registers, slice luts, lut flip flop pairs and dsp blocks [11,12]. the results are reported from the synthesis reports generated by xilinx ise environment. table 1. utilized hardware for the feature detection device utilization summary (estimated values) logic utilization used available utilization number of slices 19487 25350 7.68% number of slices lut 5345 101400 5.27% number of slices registers 8427 202800 4.15% lut as flip flop pairs 6959 101400 6.86% 50 rajesh n., saroja devi h. copyright ©2017 assa. adv. in systems science and appl. (2017) number of brams 34 350 10.46% number of iobs 33 400 8.25% the input image size of 1280x720 is considered with five gaussian levels and a kernel size of 5. the effect of changing number of octaves is negligible since the resources are time shared among the octaves. the maximum operating frequency of hardware is 100 mhz and minimum period is 10ns. the total time to extract the number feature points does not depend on the number of feature points. for this design, we have achieved 108 frames per second for hd-720p video which is much above the real-time processing requirement. this gives an overview of synthesis results for the target fpga xilinx genesys 2 kintex – 7 xc7k325t (package: fbg676, speed grade: -1). xilinx vivado (version 2014.4) has been used for the synthesis of design. table 1 recapitulates the hardware resources used in the implementation the of the feature detection module. conclusion the hardware architecture for feature detection based on scale invariant feature transform (sift) is proposed in this paper. the design of computationally effective hardware architecture for feature recognition and coordinating system on a solitary fpga chip is projected. this system has the capability to detect sift features, extract the descriptors for the detected features and complete features matching for two images taken at separate standpoints, rotation, scaling and change in illumination. the real time feature identification and matching for a series of images are accomplished. the architecture is implemented on a xilinx genesys 2 kintex 7 fpga. the results presented shows that the hardware is reliable and supports real-time applications on an hd image size of up to 1280x720. the maximum operating frequency of hardware is 100 mhz and minimum period is 10ns. it is possible to use this feature extraction hardware in realtime high definition video applications. references [1] lowe, d. g. (2004). distinctive image features from scale-invariant keypoints, int. j. computer vision, 60(2), 91-110. [2] lowe, d. g. (1999). object recognition from local scale-invariant features, proc. of the seventh ieee int. conf. on computer vision, kerkyra, greece, 1150-1157. [3] jian wu, j. cui, z., sheng, v. s., zhao, p., su, d. & gong, s. (2013). a comparative study of sift and its variants, measurement science review, 13(3). https://doi.org/10.2478/msr-2013-0021 [4] harris, c. (1988). a combined corner and edge detector, proc. 4th alvey vision conf., manchester, uk, 147-152. [5] wang, j., zhong, s., yan, l. & cao, z. (2014). an embedded system-on-chip architecture for real-time visual detection and matching, ieee transactions on circuits and systems for video technology, 24(3), 525-538. [6] mishra, p., nidhi a.i., kishore, j.k., nandini, s. & iffat, u. (2014) embedded hardware architectures for scale and rotation invariant feature detection, proc. of ieee int. conf. electronics, computing and communication technologies (ieee conecct), bangalore, india. https://doi.org/10.2478/msr-2013-0021 reconfigurable architecture for image feature detection 51 copyright ©2017 assa. adv. in systems science and appl. (2017) [7] alhwarin, f., wang, c., risti-durrant, d. & graser, a. (2008). improved siftfeatures matching for object recognition, proc. of bcs int. academic conf., london, uk, 179-190. [8] wang, z., xiao, h., he, w., wen, f. & yuan, k. (2013) real-time sift-based object recognition system, in proc. ieee int. conf. on mechatronics and automation (icma), takamatsu, japan, 1361-1366. [9] cheung, w. & hamarneh, g. (2009). n-sift: n-dimensional scale invariant feature transform, ieee trans. on image processing, 18(9), 2012-2021. [10] tuytelaars, t. & mikolajczyk, k. (2007). local invariant feature detectors: a survey, foundations and trends in computer graphics and vision, 3(3), 177-280. [11] raut, n.p. & gokhale, a.v. (2013). fpga implementation for image processing algorithms using xilinx system generator, iosr j. of vlsi and signal processing (iosr-jvsp), 2(4), 26-36. [12] swaraj, d. & madhumati, g.l. (2014). fpga implementation of sift algorithm using xilinx system generator, int. j. emerging trends in electrical and electronics, 10(10), 80-85. [13] qasaimeh, m., sagahyroon, a. & shanableh, t. (2014). a parallel hardware architecture for scale invariant feature transform (sift), int. conf. multimedia computing and systems (icmcs), marrakech, morocco, https://doi.org/10.1109/icmcs.2014.6911251 . [14] bonato, v., marques, e. & constantinides, g.a. (2008). a parallel hardware architecture for scale and rotation invariant feature detection, ieee trans. circuits and systems for video technology, 18(12), 1703-1712. [15] rajesh, n., kulkarni, r.r., sarojadevi, h. (2014). hardware architecture for scale and rotation invariant feature detection for image registration, int. conf. emerging research in computing, information, communications and applications, 239-244. [16] zhong, s., wang, j., yan, l., kang, l., cao, z. (2013). a real-time embedded architecture for sift, j. systems architecture, 59, 16–29. https://doi.org/10.1109/icmcs.2014.6911251 microsoft word 9 gang xu, yutang dai, jianlei cui--micro-manufacturing technology using duv laser and nc.doc 270-279 advances in systems science and applications (2011), vol.11, no.3-4 issn 1078-6236 international institute for general systems studies, inc micro-manufacturing technology using duv laser and nc* gang xu, yutang dai and jianlei cui key laboratory of fiber optic sensing technology, wuhan university of technology, wuhan 430070, china abstract combining deep uv laser with nc technology, complex 3d structures can be generated. for fused silica, the ablation depth and surface roughness are investigated after 157nm laser processing. under laser fluence of ~4j/cm2 and spot size of ~35μm, the ablation rate of fused silica is about 75nm/pulse for 157nm laser. a better ablation quality could be achieved under the repetition rate of ~10hz. assisted by numerical control, three-dimensional micro-structures are fabricated using 157nm laser technique. keywords duv laser, micro-manufacturing, numerical control, fused silica 1.introduction moems (micro-optical-electro-mechanical system) is a very dynamic research direction developed from mems technology in recent years, which is a new type of micro-optical system resulting from the combination of micro-mechanical, micro-electronics and micro-optics. moems device has smaller size, lighter weight, so has lower power consumption, smaller inertia, shorter response times, etc[1,2,3,4]. but the forecast advantages are not fully reflected, in large part due to the poor level of micro-mechanical processing[5]. currently, the majority of such photonics chips or mems parts are planar devices based on silicon or silica-on-silicon substrates but the manufacturing procedures are still largely reliant on combinations of conventional exposure and etching steps. in particular, three-dimensional structuring of most mems parts is usually reliant on liga process[6]. there are so many constraints using abovementioned process that developments of more cost-effective process is needed. among various new manufacturing methods, laser micro-processing is a promising choice. one of the main attractions of laser micro-processing is that lasers offer great flexibility in the rapid prototyping and evaluation of different designs. many different process routes can also be tried out with the same laser tool in a relatively short time so the developmental cycle is also much faster than with many conventional techniques. sio2 has low thermal expansion coefficient, permeability to gases and to ionic contaminants, excellent chemical inertness, good electric insulation, and has more attractive mechanical properties than ordinary glass [7,8,9], it has become one of the most important materials for fabricating optical lens and moems devices, such as micro fluidic chip, soc (system on chip) and mcm(multi-chip module). however, fused silica is relatively transparent for visible light and ultraviolet radiation, which can not be effectively absorbed, so the issue of how to process it efficiently is of crucial importance. due to development of special optical component, three dimension micromachining with deep-ultraviolet laser (duv, mainly 157nm laser) has become feasible technically. the photon energy of 157 nm laser is up to 7.9-ev [10], can directly hit the ionic bond of sio2 by photo-chemical effect. besides, 157nm laser have the benefit of being compatible with mask projection systems, so that a precision laser-ablation processing can be performed on glass surfaces. the main process steps of laser direct ablation with the sample are shown in fig. 1. * national natural science foundation of china (grant no.50775169 and no.60537050) advances in systems science and applications (2011), vol.11, no.3-4 271 1 2 3 4 laser beam is irradiated on substrate through a mask photonic energy is absorbed and ionic bonding is destroyed material particles spray out and redeposit around micro structure is shaped fig.1. the ablation steps of 157nm laser process in this study, preliminary research on micro-ablation of 157nm duv laser was performed. the main aim of those researches is to investigate the micro-processing performance of 157nm laser.influence of laser process parameters on surface qulity was investigated. 3d micro structures produced by duv laser and nc process, are demonstrated. 2.duv laser micromachining system a dual-laser system (m2000: dual laser processing tool) manufactured by exitech ltd of england was used. the system is adaptive to wide variety of laser processing requirements, based around uv laser materials processing. two laser sources are fitted to the m2000. in one case the beam from a high repetition rate uv dpss laser operating at 355nm illuminates an aperture. this is then imaged onto the workpiece using a high resolution projection lens. in the other case an f2 laser is fitted which operates at a wavelength of 157nm. the beam from this is imaged onto the workpiece using a high resolution reflective lens. fig. 2 illustrates schematic of dual-laser precision micro-ablation system.. the 157nm laser source is the model m-100 manufactured by tui laser of germany. the pulse duration is 20ns, and the pulse repetition rate is 100hz. the output of the 157nm laser source is 1.5w, and the laser pulse energy is 25mj. due to the high absorption of 157nm radiation in the atmosphere, at atmospheric pressure the photon penetration of o2 is only ~50μm, the entire beam path from the laser output coupler to the sample was purged with high purity nitrogen. the purge gas escaped from the bottom of the schwarzschild lens onto the sample. the purging also helps prevent contamination of the optics by deposited films from airborne organic contaminants that are photo-dissociated by 157nm radiation. 272 xu: micro-manufacturing technology using duv laser and nc the workpiece mounted on the worktable can be moved by 4-axis (x, y, z and r) nc stages. motions of the stages and the firing of the laser are coordinated by the aerotech unidex 500 motion controller. providing up to four axes of synchronized servo control and four axes of stepper control, the unidex 500 supports the motion requirements of today’s most demanding machines. the parameters of the motion control stages are shown in table 1, and fig. 3 shows the schematic of the motion system. fig.2. schematic of optical system of duel laser process tool fig.3. schematic of the motion control stages table 1 the parameters of the aerotech motion control stages axis description travel drive resolution maximum feedrate typical feedrate x x workpiece 300mm ac linear motor 0.5um 60000mm/min 10000mm/min y y workpiece 300mm ac linear motor 0.5um 60000mm/min 10000mm/min z focus 10mm ac brushless motor 0.1um 500mm/min 250mm/min r workpiece rotation ±45° ac brushless motor 0.8mdeg 5rpm 1rpm a attenuator 0-1 stepper motor 0.01 1/sec 0.25/sec advances in systems science and applications (2011), vol.11, no.3-4 273 3.laser micro-ablation experiments 3.1 ablation rates fig. 4 shows vuv–uv transmission spectra of fused silica at room temperature. fused silica is relatively transparent when laser wavelength is larger than 180nm. but the transparency decrease dramatically as the wavelength is lower than 175nm. this means that 157nm laser beam can be better absorbed by fused silica. not only is the range of materials which can be machined enlarged by using 157nm radiation, but the higher machining quality can be achieved. in general, the ablation rate is largely affected by absorption coefficients α of materials and laser fluences f. in the 157nm laser ablation, the material removal is predominantly photon-chemical process. wavelength（nm） tr a n sm is si o n （ % ） 100 80 40 60 20 0 140 150 170160 200190180 fig.4. uv transmission of fused silica for incident laser fluences just above an ablative threshold value ft, the etch depth per pulse t is approximately: ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = tf ft ln1 α (1) for a laser pulse duration τ giving a temperature rise δt to the vaporization point of a material with density ρ, specific heat cp and reflectivity r of the surface at the laser wavelength, the ablation threshold is given by: ( )r ct f p t − δ = 1 1 ρ α (2) for photon absorption, the penetration depth of 1/α is dependent on laser wavelength, bandgap energy of materials and so on. the higher the photon absorption, the smaller the ablation threshold and etch depth per pulse (eqs. 1 and 2). as for fused silica ablated by 157nm laser, the penetration depth is about 59nm, and the ablation threshold of ft is about 1.0j/cm2. through calculation, the ablation depth is about 81nm per pulse as fluence of 157nm laser was 4j/cm2. in this study, we also measured laser ablation depth under conditions with different laser pulses and constant energy (f=4j/cm2), as shown in fig. 5. the 157nm laser spot size was about 35μm×35μm. as the number of pulses is smaller than 100, the ablation rates is 73-77nm per pulse, generally close to the calculation result. but the ablation rate reduces gradually to about 50nm per pulse as the number of pulses increase to 600[11]. that is result from energy reduction when the ablated material far from the focusing plane, even though the energy is still constant on the focusing plane. 274 xu: micro-manufacturing technology using duv laser and nc fig.5. relationship between ablation depth and number of laser pulses (f=4j/cm 2 ) 3.2 micromachining of silica glasses 157nm laser beam can be largely absorbed (α=170,000 cm-1) by fused silica, it can thus be used for 3d micro-structuring tool for silica materials. fig. 6 shows channels formed by a 20μm×20μm f2 laser beam scanning along the horizontal direction in fused silica. the size of each channel is ~120μm×20μm. the ablating conditions are: repetition rate for each column is 8hz, 12hz and 16hz, respectively; scanning velocity for each row is 0.18mm/min, 0.23mm/min and 0.28mm/min, respectively; laser fluence is ~4j/cm2, the scanning cycles are three loops. the plume dynamics in laser ablation of silica glasses generates non-volatile products, so after ablation, some of re-deposited particles scattered onto and over nearby the ablated surface, as shown in fig. 6(a). (a) (b) fig.6. the images of the channels ablated on silica glasses using 157nm laser (a) ablation debris deposited around the channels, immediately after ablation (b) the sem image after ultrasonic cleaning in hf solution advances in systems science and applications (2011), vol.11, no.3-4 275 submicron order glass particulates will collect and partially sinter on surfaces to form coatings around small structures, and growing thicker and broader with the volume of material removed by the laser. since there is no strong bonding between the re-deposited particles, the debris tends to be rather weak in structure and can be easily removed by etching process using vol. <10% hf acid. the sem image shown in fig. 6(b) is the result after 5 minutes ultrasonic cleaning in 5% hf solution. it is seen that, the channels produced by 157nm laser is relatively regular and precise. using a profiler, the depth of each channels were measured. fig. 7 shows the depth of the channels under different ablating conditions. the measured results is very close to the theoretical value. the depth of the channel can be calculated by: l f t nh v ⋅ ⋅ ⋅ = (3) l: length of laser spot along scanning direction v: shift velocity of the workpiece f: pulse repetition frequency t: etch depth per pulse n: scanning loops fig.7. the depth of the channels under different frequency and velocity fig.8. the relationship between roughness and laser parameters (frequency, velocity) 276 xu: micro-manufacturing technology using duv laser and nc we also investigated ablated surface roughness of fused silica under different process parameters. like shown in fig. 6, the short channels (4×4 array) were produced using a quadrate beam, the spot size ~35μm×25μm, the laser fluence ~4j/cm2. we changed laser pulse repetition rate from 5hz to 30hz, and the shift velocity of worktable from 0.05mm/min to 0.30mm/min. fig. 8 shows the ablated bottom roughness ra of channels for the different scanning velocity and repletion rate. it is obvious that, there exists a strong dependence of the surface quality on the laser process parameters. on the whole, when laser repletion rate increases, the averaged roughness of the channel bottom would become larger, i.e. the surface quality become worse. and, slower laser scanning velocity such as v=0.05mm/min is not recommended, because the averaged roughness shows higher value. using high velocity (v=0.2 mm/min or 0.30mm/min) and low pulse repetition rate (f=5hz), the ablated surface roughness is very low. but take into account the machining efficiency, the repletion rate of 10hz is better. from the trend of ablated roughness, the influence of the heat produced during the laser process and the cleaning process with hf solution can not be ignored. in fact, when the laser ablation was performed under higher repetition rate and lower scanning speed, the accumulation of heat become bigger at same point, so that thermo-stress would formed at bottom of the channel. after ultrasonic cleaning with hf solution, cracks resulting from the thermo-stress would be produced, like the upper-right channel shown in fig. 6. this causes the increase of bottom roughness. 3.3 micromachining assisted by numerical control nc programming is an important part of laser processing, which affects the processing quality and production efficiency[12]. in this paper, the nc programming system is alphacam which programmed by licom system ltd of england. based upon geometric graphics information, this system can automatically generate and optimize the tool-path. as the user input the process parameters on the basis of the tool-path, the system then generate laser processing nc code automatically, includes the laser fire on/off control. the flow chart of laser micro-manufacturing is shown in fig. 9. trajectory optimization is aimed at generating the reasonable laser processing tool-path, getting the shortest processing path and the least number of laser switching, and then improving the efficiency of laser processing. typically, the processing parameters include laser fluence, beam size, pulse repetition frequency, scanning velocity (for processing with scanning mode) and total pulse numbers (usually for machining at a constant position). creating toolpath determining the optimal processing parameters step and repeat mode direct writing mode generating nc code actual machining drawing 3d structure fig.9. the flow chart of laser processing advances in systems science and applications (2011), vol.11, no.3-4 277 as examples of laser nc micromachining, fig. 10 shows two microstructures fabricated by 157nm laser. fig. 10(a) is a chinese character “田”. the process tool-path was decided as from line 1 to line 6. the process parameters were: a quadrate beam with size of ~35μm×25μm, the laser fluence ~4j/cm2; scanning velocity 0.15mm/min, pulse repetition rate 12hz. fig. 10(b) is a special shaped channel ablated into a smf-28 fiber. laser scanning was performed along a path as line1 to 2, and so on. laser spot is about 10μm×10μm, and scanning velocity is 0.12mm/min. on the whole, the micromachining quality of those microstructures is considerably good. (a) (b) fig.10. the nc process tool-path and ablated images of 157nm laser micro-fabrication (a) on a silica chip (b) on a silica fiber 4. discussion fused silica has a ~9.3ev bandgap energy, equivalent to 133nm wavelength. practical lasers are not yet available at the bandgap energy (9.3ev). at 157 nm wavelength, fused silica is only weakly absorbing, α=10cm-1[10], but from our experiments and literature [10,11,13,14,15], the 157nm lasers nevertheless provide sufficiently strong interaction that smooth etched surfaces can be precisely excised, free of micro-cracks and with little debris. the inverse of the slope provides an effective single-photon absorption coefficient of αeff =170,000cm-1 — a 17,000-fold 278 xu: micro-manufacturing technology using duv laser and nc increase that attests to the importance of defect formation during such single-pulse interactions. this is due to, the 157nm output of the f2 laser, which strongly couples energy into glass via defects, dopants or near-bandedge states, provides one approach for micromachining glass. in general, germanium and other elements are doped into silica fibers or glasses. fused silica has ~8-ev bandgap energy, and bonding energy of si-o is 4.5ev [10]. the dopants lower the bandgap and provide copious new defect centers. and then, induced by irradiation of 157nm laser, many defects such as nbohcs, odcs and e’ centers were also created. the possible processes for photoinduced defect formation in the fused silica upon f2 laser irradiation can be given by: ( ) ( ) ( ) hvsi o si si o si e nbohc ≡ − − ≡ ⎯⎯→ ≡ • + • − ≡ ′ ⅰ 2 1 2 ( )( ) 2 (e ) hv si o si si si o odcs si si si ≡ − − ≡ ⎯⎯→≡ − ≡ + ≡ − ≡ ⎯⎯→ ≡ • ′ ⅱ the photoexcited states of si–o–si rings would result in dissociation into pairs of nbohcs and e’ centers [process (i)]or formation of odcs accompanied by interstitial oxygen molecules [process (ii)]. the photoinduced odc defects were also excited by f2 laser, which led to the formation of e’ centers. all of them would cause the ablation threshold was lowered, sometimes to 70%. due to the large photon energy of 7.9-ev, it is easy to break the bonding of silica glasses for 157nm laser, only by one-photon-absorption processes. in fact, when fused silica exposed to 157nm laser irradiation, ion (si+, o+) emission phenomenon and changes of ion intensities have been observed [16]. that is also evidence of photo-chemical interaction. unlike material removal by 193nm and 248nm excimer laser, which are mainly by photo-thermal effect [17], few of heat generated during ablation of 157nm laser [18,19]. however, there is still evidence of weak thermal effect in 157nm laser ablation. for example, slight color-difference resulted from heat on glass surface can be observed, as shown in fig. 6 and fig. 10. after all, the 157nm laser ablation is one-photon-absorption process for silica glasses or fibers, accompanying with weak thermal effect only. 5. conclusions a new micro-manufacturing tool -157nm duv laser is demonstrated in this paper. for 157nm laser ablation of fused silica, the material removal is predominantly photo-chemical reaction induced by one-photon absorption process. experiments and analysis show that: (1) the ablation rate of fused silica is about 75nm/pulse under laser fluence of ~4j/cm2 and spot size smaller than 35μm. but when the number of pulses increases, the ablation rate would drop down gradually. (2) take into account the machining efficiency, a better ablation quality can be achieved under the repetition rate of ~10hz. (3) 157nm duv laser processing has great potential for precision microfabrication of hard materials for practical applications. acknowledgements this work was financially supported by the national natural science foundation of china (grant no.50775169 and no.60537050). references [1] m. edward motamedi, ming c.wu and kristorfer s.j.pister. micro-opto-electro-mechanical devices and on-chip optical processing. opt. eng., vol. advances in systems science and applications (2011), vol.11, no.3-4 279 36 (1997) 1282-1297. [2] w. zhang and j. xiong. micro-optical-electromechanical-system,moems. china machine press, (2006). [3] ming c. wu. micromachining for optical and optoelectronic systems.vol.85, no.11(1997) 18333-1856. 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[11] y. dai, d. jiang and g. xu. microstructuring of photonic materials by deep-ultraviolet laser. proceedings of spie, 7284(2009)72840p1-6. [12] ryoichi kuwano, tsuyoshi tokunaga, yukitoshi otani and norihiro umeda. beam shaping optics for yag laser processing fabricated by computerized numerical control lathe. optical review vol. 12, no. 6 (2005) 476–479. [13] h. deng, y. rao, z. ran, x. liao, w. liu. photonic crystal fiber based fabry-perot sensor fabricated by using 157nm laser micromachining. acta optica sinica. 28(2)(2008)255-258. [14] y. dai, w. li and d. jiang. micro ablation of optical fibers using vacuum ultraviolet laser. proc. int. conf. integr. commer. micro nanosyst, sanya china. (2007)1353-1356. [15] j. greuters and n.h. rizvi. laser micromachining of optical materials with a 157-nm fluorine laser, proc. spie, belgium, 4941(2003) 77-83. [16] s.r. john, j.a. leraas, s.c. langford and j.t. dickinson. laser induced ion emission from wide bandgap materials. applied surface science. 253(2007) 6283–6288. [17] y. liao, y. chen, c. chao and y. liu. surface morphology and sub-surface damaged layer of various glasses machined by 193nm arf excimer laser. proceedings of spie, 5717(2005)110-117. [18] a. pusel, p. hess, et al. photochemical processing (157nm) of semiconductor surfaces without heating. surface science, 74(11)(1999)433-435. [19] vaidynathan a, walker t, et al. comparison of keldysh and perturbation formulas for one-photon absorption. j. physics review, 20b(5)(1979)3526–3527. advances in systems science and applications (2012) vol.12 no.1 54-66 the application of semantic web technology in manufacturing grid h.j.zhang1,2 and y.f.hu1,2 1the school of mechanical and electronic engineering, wuhan university of technology, wuhan 430070,china 2hubei digital manufacturing key laboratory, wuhan university of technology, wuhan 430070, china abstract concerning the knowledge-intensive environment, the paper introduces semantic web(sw) technology to manufacturing grid (mgrid). resources and services in mgrid are described in the well-defined meaning, so that computers can understand resource and service information to interoperate seamlessly. in the paper, a semantic-aware mgrid architecture (samga) based on ws-resource framework (wsrf) is first presented, in which the stateless web services are semantically described using owl-s and the stateful resource descriptions are semantically enhanced using owl. then a semantic matching algorithm is proposed for samga. finally, samga has been applied to the magnetic bearing resource and service sharing platform (mbrssp). keywords manufacturing grid, semantic web technology, semantic matching algorithm, architecture 1 introduction through the network, manufacturing grid (mgrid)[1] makes all kinds of enterprises and resources geographically distributed connected and form a virtual organization (vo), which is centralized in logic but distributed in physics. in this vo, all resources can be shared and all enterprises can be employed to work collaboratively towards to a common target of dealing with a distributed manufacturing task or problem. the kernel of mgrid is to share manufacturing resources and use services provided by manufacturing resources pellucidly. however, mgrid resources are far more diverse and complex than those of grid computing and there isn’t a unified format for describing manufacturing resources and services. there exist lack of relationships and semantics. moreover, manufacturing service composition is notoriously complex and challenging. these problems seriously hinder the development and application of mgrid. even though web services description language (wsdl) was used extensively in web service and grid at present, regrettably, it couldn’t address these problems due to lack of semantic parts. tangmuarunkit argued that existing resource description and resource selection in the grid was highly constrained[2]. due to the complexity of some manufacturing processes, some manufacturadvances in systems science and applications (2012) vol.12 no.1 55 ing tasks in mgrid should be decomposed into several subtasks, which cannot be further decomposed and can be executed by a single resource service. extensive studies have been conducted related to web service composition problem in service-oriented system and distributed system. existing research efforts for web service composition have been undertaken in two orthogonal directions: 1) manual composition, and 2) automated composition. manual composition is a time-consuming task, and it is supported by major it enterprises such as ibm and microsoft. in this approach, the web service developer selects the outsourced web services that are relevant to their composition intents and programs the interaction logic of the component services with a low level programming language such as bpel[3]. for the automated composition, the information must be understandable by computers, so that they can perform more of the tedious work involved in finding, sharing, and finally combining information in mgrid. the mutual understanding between computers is also important for the optimal composition of mgrid resources. the appearance of sw[4] brings a new hope of solving the above issues. sw is an evolving extension of the world wide web in which the semantics of information and services on the web is defined, making it possible for the web to understand and satisfy the requests of people and machines to use the web content. many researches have applied sw technologies to grid. all resources and services in grid are adequately described in the well-defined meaning that is machine-processable, which is favor of enabling computers and people to work in cooperation. in the grid system, “semantics” plays an important role in the following aspects: • more effectively discovering and managing dynamic resources in virtual organizations, such as the description and processing of semantic service, resource classification, event notification, origin tracking. the problem of heterogeneity of information in the distributed environment would be resolved at the bottom of the grid architecture; • linking and cooperating with the information stored and available within a grid automatically, so that it can support the knowledge-intensive applications at the top of the architecture. the rest of the paper is organized as follows. in section 2, some related work is summarized. in section 3, samga is presented. section 4 introduces a semantic matching algorithm which is especially designed for samga. section 5 provides the resource and service of magnetic bearing resource and service sharing platform based on samga. the conclusion is given in section 6. 56 h.j.zhang:the application of semantic web technology in manufacturing grid 2 related work early in 2001, a number of researchers were increasingly conscious of the necessity of combination of semantic web and grid. this was firstly captured by david de roure in the uk e-science program[5]. global grid forum (ggf) also established semantic grid research group (sem-grg) to realize the added value of emerging web technologies and approaches, in particular semantic web and web 2.0, for grid users and developers. there are some representative research and projects abroad, such as mygrid, combechem, geodise etc; in china, there is the national grand fundamental research 973 program: study on basic theory, model and method of semantic grid, which would mainly resolve three issues: standard resource organization, semantic interconnection and intelligent aggregation[6]. above researches are mostly applied in the environment of computing grid or data grid. there are few researches of semantic web technology for manufacturing. for instance, yang et al. presented a semantic web services approach for automated integration of manufacturing systems and services[7]. lemaignan et al. presented a proposal for a manufacturing upper ontology in order to draft a common semantic net in manufacturing domain, and applied the ontology in the automatic cost estimation and semantic-aware multi-agent system for manufacturing[8]. he et al. proposed an ontology-based manufacturing resource discovery architecture, and gave the discovery algorithm of manufacturing resources based on semantic extending and qos[9]. li et al. used ontology to describe resources and services in manufacturing grid, and extend uddi to support service publishing and retrieval[10]. it has been realized that sw have an extremely broad development prospect, but the application in manufacturing just begin, especially in mgrid. most researches have focused on the description of resource and service of mgrid, not on the semantic-enabled mgrid architecture and semantic matching algorithm. therefore, this paper proposes a new semantic-aware architecture and a semantic matching algorithm for mgrid. 3 semantic-aware mgrid architecture based on concepts and technologies of wsrf and sw, the semantic-aware mgrid architecture (samga) is designed as shown in fig.1. for the purpose of the pellucid interoperation, sw technology should be applied upon grid middleware. it is important to note that the “semantics” permeates the full vertical extent of architecture and is not just a semantic (or knowledge) layer on top: it is semantics in, on and for the grid. the function of every layer as follows: advances in systems science and applications (2012) vol.12 no.1 57 fig.1 the semaritic-aware mgrid architecture (1) fabric layer: its basic function is that local resources would be encapsulated into global resources which could be shared for the mgrid application layer. at the endpoint of physical resources node, the interfaces of lifetime management, state management and notification would be provided; web services standardize various functional operations provided by resources node and offer the standard 58 h.j.zhang:the application of semantic web technology in manufacturing grid interfaces of access services. web services would be described by using service upper ontology owl-s and manufacturing domain ontologies which are written in standard ontology language owl. a flexible and high-quality “ws-resource” would be constructed by combining standard web services with dynamic manufacturing resources. ws-resource is the extended grid service according with wsrf specification, hiding the heterogeneity of manufacturing resources and being rich in “semantics”. (2) mgrid core middleware layer: mgrid core middleware is essentially a semantic enhanced web service container, because the sw standards provide the web service container with the capabilities of processing semantic information, such as services composition monitoring, batch job processing and service level agreement (sla) management. it is the basic running environment of grid service, which will be deployed in each grid node in advance. it is recommended that mgrid core middleware is developed on globus toolkit 4.0.5 core. (3) mgrid public service layer: this is a service set which provides the common operations for mgrid applications, including information service, data service, task management and semantic service. the semantic service includes three components: • to provide the domain ontology library (knowledge workers who are a little familiar with the knowledge of manufacturing can easily use protg[11] to establish and maintain ontology library) and the general ontology library (e.g., wordnet [12], cyc[13], sumo[14]); • to provide the api of ontology libraries (e.g., jena framework is open source and grown out of work with the hp labs semantic web programme, which provides an owl api and sparql query engine[15]); • to provide the ontology reasoner (e.g., racer[16], pellet[17], fact++[18]), owl-s based matching engine[19] and owl & owl-s editor[20]. (4) manufacturing service layer: on the basis of the mgrid public service layer, the intelligent toolkits for the manufacturing applications are developed, for example, capp expert system based on artificial intelligence. (5) mgrid application layer: in virtue of the domain dependent programming model (e.g. corba, com, javabean etc.) and man-machine interaction mechanism, the man-machine interfaces (e.g. visual component portlet, servlet, etc.)are provided to mgrid users for mgrid applications. both owl-s and the web service modeling ontology (wsmo) [21] can describe services semantically. owl-s uses owl in combination with wsdl. wsmo uses f-logic to perform inferences with services, and its xml-based format gives external agents access to the service features. the paper adopts owl-s as mgrid services description language (see fig. 5). advances in systems science and applications (2012) vol.12 no.1 59 4 procedures of mgrid resource & service publication and discovery based on sw 4.1 mgrid resource & service publication elenius et al. developed an owl-s plug-in for protégé as owl-s editor [20]. the owl-s editor provides a graphical user interface to create and modify an owl-s description including all three parts: serviceprofile, servicemodel and servicegrounding. publishing sw services is more complex than web services. however, it can contribute to automated discovery, negotiation, composition, execution, and monitoring of web services. first of all, rsp develop wsdl documents of web services. then with the help of owl-s editor, rsp fill in serviceprofile information and use service–model to define service executable process. at last, rsp import the above wsdl documents into owl-s files. in order to describe mgrid services formally as much as possible, rsp can enter some keywords and browse ontolgoies that include the keywords, as well as synonyms, superclass or subclass of these ontologies, when entering iopes. information. thus rsp select the right ontology names as the iopes terms. 4.2 mgrid resource & service discover the procedures of semantic-aware service discovery are shown in fig.3. rsc just need to enter input and (or) output term(s) via the man-machine interaction interface. in the same way of resource & service publication, after that ontology reasoner performed reasoning and expansion of ontology, rsc select the right ontology names. the selected ontology names and those related terms (e.g., synonyms, superclass or subclass) are sent to mds4 as keywords. mds4 connect to all mgiis and return the service profile information to owl-s based matching engine. matching engine would extract i/o information from mds4 and then start to perform the matching algorithm (detailed in section 5) to compare i/o information between returned services and rsc request. service query interface will show the matching results (including service name and address). thus rsc connect the selected mgris to schedule this service node. 5 semantic matching algorithm the matching algorithm is the key to the implementation of the sw technology in mgrid. the algorithm is the main process of the owl-s based matching engine in the core middleware layer. the principles of the semantic matching algorithm, which are based on the semantic ontology library-wordnet, are as follows: 60 h.j.zhang:the application of semantic web technology in manufacturing grid fig.2 semaritic-aware mgrid service dicovery model (1) the smaller the semantic distance, the greater the similarity. the semantic distance between one concept and itself is 0, and their similarity is 1. when the semantic distance between concepts is infinite, their similarity is 0. in the hierarchical structure tree (see fig.3), the semantic distance refers to the length of the shortest path between two concepts. as shown in fig.3, the semantic distance of o31 to o36 is denoted by l1 +l2, where l1 refers to the distance of o31 to o11, l2 refers to the distance of o36 to o11 , and o11 is the nearest common concept of o31 and o36. (2)the more the semantic overlap, the greater the similarity. the semantic overlap refers to the level of the same meaning that both concepts involve. as shown in fig.3, the semantic overlap of o31 to o36 is denoted by l, where l refers to the distance of o11 to o0, and o0 is the top concept in the hierarchical structure tree. generally speaking, the larger the semantic overlap , the smaller the difference between the meanings of both concepts. for example, the similarity between o31 to o32 is larger than that between o11 to o12. (3) the more detailed the classification, the lower the similarity. there are fine and coarse classifications of concepts in the semantic dictionary. hence, the concept density isn’t a fixed value. there must consider the concept density in the algorithm . (4) the similarity is asymmetric. generally speaking, the similarity from o1 to o2 is unequal to that from o2 to o1. for example, the similarity from “lathe” to “machine tool” is larger than that from “machine tool” to “lathe”. because machine tool includes lathe, grinder, shaper etc. therefore, we introduce the vector curves, such as the broken blue curves with arrow in the fig.3. according to the above principles, the similarity of resource functionality from concept o1 to concept o2 is defined sim(o1 → o2) as follows: advances in systems science and applications (2012) vol.12 no.1 61 sim(o1 → o2) = l l+ α(o1, o2) l1 ρ(o) + [1− α(o1, o2)] l2 ρ(o) where α(o1, o2) = { l+l1 2l+l1+l2 , l1 ≤ l2 l+l2 2l+l1+l2 , l1 > l2 is the adjusting parameter of the asymmetry of similarity. the concept density [22] ρ(o) = m−1∑ i=0 nhypi 0.20 descendants0 ,where nhyp refers to the mean number of hyponyms per node, h refers to the height of the subhierarchy, and m refers to the number of senses in the hierarchical structure tree. there is an example of how to cacluate the similarity in the field of machine tool service. the goal of calculating the similarity is to match two resource names. the resource name stands for the resource functionality. the hierarchial structure tree in the wordnet is shown in fig.4. from the fig.4, we can obtain the data in the table 1. table 1 the similarity calculation o1 → o2 l l1 l2 m desc− α(o1, o2) nhpy ρ(o) sim(o1 endanto → o2) lathe→ 9 2 0 2 5 0.45000 1.15091 0.43018 0.81138 machine tool grinder→ 9 1 0 2 5 0.47368 1.15091 0.43018 0.89099 machine tool shaper→ 9 1 0 2 5 0.47368 1.15091 0.43018 0.89099 machine tool machine tool 9 0 2 2 5 0.45000 1.15091 0.43018 0.77874→lathe machine tool 9 0 1 2 5 0.47368 1.15091 0.43018 0.88033→grinder machine tool 9 0 1 2 5 0.47368 1.15091 0.43018 0.88033→shaper lathe→ 10 1 1 2 5 0.45000 1.00000 0.66667 0.86957 milling machine lathe→ 9 2 1 2 5 0.47619 1.15091 0.43018 0.72396 grinder in the table 1, it’s easy to see that sim(lathe → tool) > sim(lathe → grinder) and sim(lathe → machine tool)> sim(machine tool → lathe). the first inequation shows when a lathe is required, the resource node of machine tool is much better for the searching result than that of grinder. the second inequation verifies the asymmetry of similarity. we set a threshold for the similarity in the mgrid 62 h.j.zhang:the application of semantic web technology in manufacturing grid sysytem. if sim(o1 → o2) is no less than the value of threshold, we argue that o1 is similar enough to o2. fig.3 the hierarchical structure tree fig.4 an example of hierarchical structure tree in wordnet 6 application a magnetic bearing is a new type of high performance bearing. however, each type of magnetic bearings must be designed and manufactured according to the concrete objects. a large number of resources are needed in the development of magnetic bearing. in order to realize the sharing and collaborative work of all needed resources, according to samga, the magnetic bearing resource and service sharing platform (mbrssp) is developed. resource & service providers (rsps) can publish the own resources through the manufacturing resources publication center, as shown in fig.4. the publication center provides the semantic template for resource and service publication and automatically generates the semantic documents. resource & service demanders (rsds) can search the needed resources at the resource optimal-allocation center as shown in fig.5; rsds can advances in systems science and applications (2012) vol.12 no.1 63 also reserve all kinds of resources for mgrid tasks through the co-reservation system, as shown in fig.6. the semantic matching can provide the better searching result for both the optimal-allocation and the co-reservation. fig.5manufacturing resource & servicepublication center based on owl&owls fig.6 mgrid resource & service optimal-allocation center 64 h.j.zhang:the application of semantic web technology in manufacturing grid 7 conclusion this paper presented a semantic-aware mgrid architecture that exploits semantic web technologies to solve manufacturing problems. the semantic environment at the bottom of samga can adequately support the knowledge-intensive application of mgrid on the top layer. with the support of samga, a new information model suitable for the semantic and grid environment is designed, which can realize the fuzzy matching. furthermore, the information model is also called the pull model, which avoids a large number of redundant information. fig.7 mgrid resource & service co-reservation system acknowledgements this paper is supported by the national natural science foundation key project of china: digit manufacturing basic theories and key techniques under network environment (no.50335020), and the hubei digital manufacturing key laboratory opening fund project: research on resource service search and optimalselection theories and experiments in manufacturing grid system (no.sz0621). references [1] fei tao, yefa hu and zude zhou. (2008), “study on manufacturing grid and its executing platform”, international journal of manufacturing technology and management, vol.14, no.1-2, pp.35-51. [2] hongsuda tangmuarunkit, stefan decker and carl kesselman. (2003), “ontology based resource matching in the grid-the grid meets the semantic web”, proceedings of the 2nd inter. semantic web conference, pp.706-721. advances in systems science and applications (2012) vol.12 no.1 65 [3] matjaz b. juric, benny mathew, poornachandra sarang. (2004), business process execution language for web services, birmingham.u.k.: packt. [4] http://www.semanticweb.org [5] http://www.intsci.ac.cn/ai/sg.html [6] zhonghhua yang, robert gay, chunyan miao, et al. (2005), “automating integration of manufacturing systems and services: a semantic web services approach”, 31st annual conference of ieee industrial electronics society, pp.2255-2260. [7] severin lemaignan, ali siadat, jean yves dantan, et al. (2006), “mason: a proposal for an ontology of manufacturing domain”, ieee workshop on distributed intelligent systems: collective intelligence and its applications, pp.195-200. [8] yu’an he, tao yu, lilan liu, et al. (2006), “research on manufacturing resource discovery based on ontology and qos in manufacturing grid”, 2006 international conference on cyberworlds, pp.209–215. [9] jing li, xiangxu meng, shijun liu, ran yuan. (2007), “ontology-based resource description in manufacturing grid”, computer supported cooperative work in design, pp.646–650. [10] http://protege.stanford.edu [11] http://www.cyc.com [12] http://wordnet.princeton.edu [13] http://www.ontologyportal.org [14] http://jena.sourceforge.net [15] http://www.racer-systems.com [16] http://pellet.owldl.com [17] http://owl.man.ac.uk/factplusplus [18] naveen srinivasan, massimo paolucci, katia svcara. (2004), “adding owls to uddi. implementation and throughput”, proceedings of the 1st international workshop on semantic web services and web process composition.san diego, california. 66 h.j.zhang:the application of semantic web technology in manufacturing grid [19] daniel elenius, grit denker, david martin, et al. (2005), “the owl-s editor-a development tool for semantic web services”, in proceedings of the second european semantic web conference. [20] http://www.wsmo.org [21] zhizhong liu, huaimin wang, bing zhou. (2007), “a two layered p2p model for semantic service discovery”, journal of software, vol.18, no.8, pp.1922-1932 [22] eneko agirre, german rigau. (1995), “a proposal for word sense disambiguation using conceptual distance” (submited), international conference of recent advances in natural language processing. corresponding author y. f. hu can be contacted at: haijun@whut.edu.cn advances in systems science and application (2016) vol.16 no.1 72-84 the development system of independent (commercial) sector of the cultural complex of the region (on the example of the khanty-mansiysk autonomous okrug-ugra) yulia s. rod1, lina s. khromtsova1, svetlana a. yesipova2 and anna i. panenko2 1institute of management and economics, yugra state university, khanty-mansiysk, russia. 2yugra state university, khanty-mansiysk, russia. abstract the article presents the results of a study of the current state of independent (commercial) cultural complex of khanty-mansiysk autonomous okrug ugra, including analysis of statistical indicators of development of independent (commercial) cultural complex, analysis of russian and international experience of development, the sociological study of satisfaction of residents, visitors and entrepreneurs the terms of service, business practices, existing in the region in the field of culture. the aim of the study was to develop proposals for the development of independent (commercial) cultural complex of the region and its interaction with the public sector. as a result of study a system of statistical indicators that reflects the status of independent (commercial) sector of the cultural complex of ugra; the technique of “mapping” of the independent (commercial) sector of the cultural complex of ugrahave been developed. a set of activities aimed at the development of the cultural environment of the region through creation of a favorable investment climate and cooperation between the two sectors of the cultural complex of yugra state (municipal) and independent (commercial); support of innovative projects in the field of culture and the creation of “creative clusters”; the development of public-private partnership in the sphere of culture of ugra was formulated. keywords independent (commercial) sector of culture; the technique of “mapping”; systems of creative clusters; public-private partnership 1 introduction the composition of the commercial (or independent) sector of the cultural complex of khanty-mansiysk autonomous okrug-ugra (hereinafter khmao-ugra) includes cinema circuits, art galleries, organizations of show business, publishing and bookselling network and other cultural institutions, the main purpose of which is to obtain profits. state and independent sectors represent two parts of a consistent regional cultural situation. there are no insurmountable walls between them. the same advances in systems science and application (2016) vol.16 no.1 73 creative professionals sometimes work in both sectors. there are examples of cooperation between state and independent culture organizations. many culture organizations have recently had independent status. finally, new, experimental directions, which then become the constituency of the regional cultural situation and work out also in the state organizations of culture, often develop particularly in the independent sector. thus, the independent sector is an innovative resource and creative reserve of the regional culture. but its value is not limited. independent cultural institutions create their own jobs and independent cultural products and services. they make a significant contribution to the development of the regional environment, enable raising of cultural diversity in the region. they are often more flexible than public sector organization, perceive new trends more quickly, develop new technologies, respond to social needs and therefore occupy niches which are not occupied by public organizations, due to some reasons. although region authorities are not able to influence this sector directly in the same way that they affect the state organizations however, they can conduct district policy in relation to the independent sector, aimed at stimulating its growth, the maintenance of diversity and the use of its resources for further cultural development of khmaougra. a serious obstacle to the development of region policy in relation to the independent sector in the field of culture is the lack of reliable statistical data on the number, composition and activities of independent culture organizations. the role and importance of entrepreneurship development in the field of culture do not cause doubts, as evidenced by the analysis of methods and forms of state support in various countries. for instance, the policy of support to small and medium-sized enterprises in austria and germany[1,2] aims at support young entrepreneurs, venture enterprises. sectorial projects in the creative industries including the sphere of culture have a priority. in china, the government directs most of its efforts at reducing administrative barriers, while providing substantial fiscal benefits to micro and small enterprises. large-scale industrial development policy of spatial concentration in the form of free trade zones develops also[3,4]. direct support of art and culture from the public sector takes the form of subsidies, grants and awards. the allocation of funds differs among european countries in accordance with their cultural priorities. in addition, autonomous regions and municipalities make significant contributions to the culture at the local level in some countries, such as germany and poland. public financial support for culture is distributed through foundations, arts councils in some countries. lottery funds have a special role for culture in some countries (e.g. italy). 74 yulia s. rod et al: the development system of independent (commercial) sector of ... indirect support for culture is carried out in reduction of incomes associated with payment of national or local taxes. there is a general trend in many european countries towards the introduction of legal measures in the field of taxation for donations or sponsorship in the field of culture (transnational public-privatepartnership concep). there are government programs aimed at supporting organizations working in the field of culture in the russian federation and some of its constituents. thus,vgrants to support innovative projects in the field of contemporary art; grants to music organizations established by the russian federation subjects and municipal entities, as well as independent music collectives; grants to theatres under the jurisdiction of subjects of the russian federation and municipal entities and independent theatre ensembles; and other forms of support are provided in the framework of implementation of the federal target program “culture of russia (2012-2018)”. 2 materials and methods this research was carried out in the framework of the state contract for a comprehensive study of independent (commercial) cultural complex of khmao-ugra. analysis of statistical data in the field of development of (independent) commercial cultural sector in the region, analysis of russian and international experience of state support of subjects of the cultural sector, analysis of “mapping” technique and creation of “creative clusters”, sociological research have been used as the main methods of research.a large-scale survey of residents and guests of khmao-ugra (using questionnaires), as well as a survey of representatives of independent (commercial) sector of the cultural complex of the region (interviews) to identify issues and trends of its development included the sociological study held during the work. 2099 people out of 22 municipalities of khmao-ugra have been interviewed in the survey. on average, 95 people (56% of women and 44% of men) have been interviewed in each municipality.60% of people are employed and 40% do not work out of musters of respondents, almost two-thirds have children (62%). 3 the main part the state and municipal sector culture of ugra was represented by a multidisciplinary network of cultural institutions on all types of cultural activities, consisting of 466 cultural institutions in 2014. 231 public library, 112 organizations of cultural and leisure type 3 crafts establishments, 8 theatres, 34 museums, 3 parks of culture and leisure (urban gardens), 5 concert organization and 1 independent company, 1 institution of cinema and film distribution, 2 institutions for the protection of monuments of history and advances in systems science and application (2016) vol.16 no.1 75 culture and 5 other institutions act on the territory of autonomous district. 3 educational organization conducting educational activity on educational programs of secondary vocational education, 58 children’s music, art, choreographic school and art schools, which operate on the territory of municipal formations conduct educational activities in the field of culture on the territory of ugra. we conducted a grouping of small and medium enterprises of non-state (commercial) sector of culture of khanty-mansiysk autonomous okrug-ugra on activities according to statistics provided by the territorial body of federal state statistics service for khmao-ugra, budget institution of khmao-ugra “museum of geology of oil and gas”, budget institution “state library of ugra”(table 1). table 1 the number of small and medium enterprises of private (commercial) sector of culture in khmao-ugra line of activity as of rate of growth %01.01.2010 01.01.2015 manufacture of products of national art crafts 25 26 104 additional education of children (in culture) 18 22 122 film production 7 12 171 distribution of a motion picture 8 6 75 screening 9 11 122 activity in the field of art 10 12 120 activity in the field of creation work of art 2 3 150 activity in the field of art literary executive works 6 7 117 the activities of the organization and the production of theatrical and opera performances, concerts and other stage performances 15 18 120 the activities of actors, directors, composers, artists and other representatives of creative professions, acting on an individual basis 4 5 125 the activity of concert and theatre halls 1 10 1000 the activity of fairs and amusement parks 12 17 142 the activity of dance halls, discos, schools of dances 11 11 100 other entertainment activities 7 8 114 activities of libraries, archives, institutions of club type 18 20 112 the activities of museums and protection of historical sites and buildings 4 5 125 total: 157 192 122,3 thus, there are 192 subjects of small and average business of private (commercial) sector of culture on 01.01.2015 in the region. over the past 5 years the rate of growth of such institutions was 22.3%. the share of small and medium 76 yulia s. rod et al: the development system of independent (commercial) sector of ... enterprises of private (commercial) sector of culture in khmao-ugra is less than 1% of all business entities in the region. participants of the mass survey were asked to answer a series of questions, with the aim to study satisfaction level of the residents of khmao-ugra by level, quantity and range of services in the field of culture: 1.“what cultural facilities do you visit?” the most popular, according to respondents, are the palaces of culture, clubs (fig. 1) fig. 1 the level of demand for culture facilities. 2.“how often do you visit the culture facilities?” fig. 2 the frequency of culture facilities visits. half of the respondents visit cultural institutions at least 1-2 times per month (fig. 2). on answer option “other” respondents were asked for their own answers, the main ones are almost every day (7%) and do not attend at all (5%). 3.“please, specify the reasons why do not you attend or visit the culture faciladvances in systems science and application (2016) vol.16 no.1 77 ities rarely”. fig. 3 the reasons for rear visits of culture facilities. respondents cited excessive employment and “high cost” of services as the main reasons for rear visits (fig. 3). 4.“please rate the level of satisfaction with the quality of services provided by the region cultural institutions” fig. 4 satisfaction with a quality of services. two-thirds of respondents rated the level of satisfaction with the activities of cultural institutions as positive (fig. 4) 5. “please rate the level of professionalism of the region cultural institutions” more than half of respondents rated the level of professionalism as high (52,3%), a quarter of respondents rated the level as average (24.6 percent) and 14% as low. 6. “what the new culture facilities would you like to see in your community?” the respondents were offered the following answer options: amusement park; entertainment center with a skating rink; theme parks; entertainment nightclub; -imax cinema, planetarium, aquarium, dolphinarium to the last question. thus the summary of questionnaire survey of residents of the region the following conclusions have been made: 78 yulia s. rod et al: the development system of independent (commercial) sector of ... virtually all types of cultural objects are represented in the region clubs (house of culture) (49%) are the most popular among inhabitants of the region; half of the respondents visit cultural institutions at least 1-2 times a month; the range of services offered satisfied about 80% of the population; two-thirds of respondents satisfied with the level of satisfaction with the quality of services provided by the region cultural institutions (67%); more than half of respondents rated the level of professionalism of employees of cultural institutions of khmao-ugra as high (52.3%); 70% of citizens rated the level of satisfaction with the activities of cultural institutions positively. a survey of hotel and other tourist cities in khanty-mansiysk, surgut, nizhnevartovsk has been conducted to identify the degree of satisfaction the guests of ugra of the level, quality and range of services in the sphere of culture. 78 people have been interviewed just in 13 hotels in these cities. among these cultural facilities that attract visitors of our region were named as follows: archeopark, temple complexes, center of national cultures, biathlon center, the ethnographic museum under the open sky torum maa and the other. more than 23% of respondents indicated “cinemas”, 16% “temple complexes” and 11% “cultural-leisure centers” as cultural objects that attract them in the cities of ugra. preferences of region visitors in terms of cultural events were as follows: about one third of respondents are attracted to concerts and performances (concerts, cfi), 27% of respondents named sport events (biathlon) and 19.2% are attracted to exhibitions and nearly 18% festivals (the spirit of fire, rescue and save). 87.2% of all respondents answered positively on the survey question “are you satisfied with the range of services offered in the field of culture”, while 12.8% indicated that they were not satisfied by reason of the fact that there are not enough facilities incities ”where you can go with children”, there are no circuses, zoos, leisure centers. guests are most satisfied with the variety of services available in khanty-mansiysk (93.5% of the respondents), because of the large number of cultural events and the presence of a more diverse culture. guests were also asked to rate the quality of services offered in the field of culture in khmao-ugra on a five point scale. over half of respondents rated the quality of services offered in the field of culture in our region as “excellent”, indicating that all activities (especially in khanty-mansiysk) are always organized at “the highest level”. respondents who rated the quality of services as “good” and “satisfactory”, outlined the following reasons: -there are not enough places for young people; -there are not enough advertising activities and culture facilities; advances in systems science and application (2016) vol.16 no.1 79 -the lack of information in the hotel; -there are not enough facilities (museums, galleries), working in the evening. guests of our region would like to see an amusement park, new cinemas, theater, virtual museum, the circus, the museum of modern art, leisure centers, zoo as new culture facilities. meetings were held with entrepreneurs in khanty-mansiysk, surgut, nizhnevartovsk and 2 municipalities: surgut and nizhnevartovsk in the study of the independent (commercial) culture sector. analysis of the results of the interviews showed in general, underdeveloped commercial sector of culture in khmao ugra. major obstacles to business development in the field of culture, according to leaders of organizations in the industry, were as follows: the problem of underreporting of income; significant rents; the high cost of advertising; the complex of problems connected with competition and the “overflow” of clients with low incomes from one organization to another; the problem of lack of qualified personnel. 4 conclusions and suggestions 1.we developed a map, which gives an idea about the distribution of institutions and organizations independent (commercial) sector of the cultural complex of ugra based on the russian[5] and abroad[6] mapping data obtained in sociological research of management and administrative bodies of municipalities of the region, as well as information obtained from the territorial body of federal state statistics service of the khanty-mansiysk autonomous okrug ugra; budgetary institution of the khanty-mansiysk autonomous okrug ugra “museum of geology of oil and gas”; budgetary institution “state library of ugra” (fig.5). according to the results of the mapping of the organizations of small and medium-sized business of the cultural sector of khmao ugra, you can make a general conclusion, according to which commercial (independent) enterprises are not the leaders in the field of region culture. the sector is most complete and diverse developed in surgut, nizhnevartovsk, khanty-mansiysk, nefteyugansk, beloyarsk, megion and raduzhny. almost all activities are widely represented here. outsiders are berezovsky, sovetsky and kondinsky areas, as there are nocommercial enterprises in the field of culture even in region centers. 2.on the basis of domestic experience of creation of special institutions for the development of culture is currently possible to develop the following infrastructure requirements as the basis for investment development of independent (commercial) cultural sector in the region: 1) the creation of national centres of cultural development, whose activities should be focused on the disparities levelling in the quality of provision and diversity of the spectrum of cultural services for the population as a major, and 80 yulia s. rod et al: the development system of independent (commercial) sector of ... small towns of the region, as well as ensuring maximum involvement of local people in joint cultural and creative activity: fig. 5 the distribution of institutions and organizations of independent (commercial) sector of the cultural complex of ugra. 2) the creation of centres of international cultural exchange, whose activity should be aimed at community involvement in the process of intercultural integration and orientation processes of local cultural services (including support of private cultural initiatives) in the system of cultural “subregions; 3) the creation of centres for monitoring the state of activity of the organizations of the cultural sector, engaged in the maintenance and development of infrastructure to ensure the safety of these values and guaranteeing access to citizens, and replication of successful projects in the field of culture, the creation and promotion of network projects, maintaining information networks (including mapping), the promotion of small businesses. the activities of all the above centres must respond to the needs of regions, and the characteristics and priorities of the centres should be developed by the authorities of subjects of the russian federation in the field of culture. 3.a model of interaction between these sectors, including the expansion of the organizational structure of the department of culture of khmao-ugra has been developed with the aim of developing the cultural environment of the region through creation of a favorable investment climate and cooperation between the two sectors of the cultural complex of ugra state and independent (commercial) advances in systems science and application (2016) vol.16 no.1 81 sectors (fig. 6). the basic premise of creating additional structural unit (department for cooperation with the independent (commercial) sector of culture)is the lack of reliable statistical data on the number, composition and activities of independent organizations culture. the tasks of the structural unit will include: -the collection and analysis of information about independent (commercial) organizations; -consultation of creative communities; -providing information about the presence and functioning of an independent (commercial) organizations. -providing information about the forms of state support for the subjects of independent (commercial) culture sector. 4.there is a sector of activity in the independent sector that lies on the boundary of the sphere of culture and sometimes goes beyond this boundary. however, creative and cultural resources are actively used in many cases, and their products are becoming an integral part of ugra cultural environment. therefore, the following proposal relates to attract investors to the region for the construction of entertainment (shopping) centers, as well as increase the level of satisfaction of residents and guests of the region cultural sphere. these sites offer visitors not only a wide selection of products, but also services of cultural complex associated with providing of innovative resources: theme parks, amusement parks for adults and children, indoor aquarium, terrarium, zoo, cinemas, ice rinks. in addition to the shopping center you can create such cultural facilities for residents and visitors to the region as museum of local celebrities, the museum of entertaining science,museum for children, a miniature park and other. 5.the introduction of additional forms of support: subsidies from the budget of the khanty-mansiysk autonomous okrug ugra on the implementation of innovative projects in cultural sphere; the competition for the award of the government of khmao-ugra for the implementation of innovative educational projects in the field of culture; the competition for the grant of the government of kmao-ugra; on the implementation of innovative projects in the field of theater and concerts; the competition for the grant of the government of khmao-ugra on the development of innovative museum technology are suggested with the aim of further development of the cultural environment of the region and the increase of innovative activity in the sphere of culture and art. the use of regional tax concessions for entities engaged in entrepreneurial activity in the sphere of culture and art should be also provided for in the legislation of khmao-ugra. 6.it is possible to acknowledge the tremendous role of creative clusters in the development of the commercial sector of culture: the activities of the centres are wholly aimed at providing services in the field of culture, organization of leisure 82 yulia s. rod et al: the development system of independent (commercial) sector of ... and creative development of the areas population of all ages, and it is conducted on a commercial basis. we have proposed a model of the circuit of the creative cluster on the area of municipal formation of surgut, because it is a major hub of the region (table 2). table 2 “galleries”+g2:j16 educational projects industrial interactive platform oil and gas lecture studios debating club wood and paper ethnic cinema halls mass cinema creative workshop fine art art films dance documentary film the design and simulation of techniques handicraft experimental theatre platform radio journalism “virtual museums of the world” animated cartoonsphotostudios interactive platform of street culture project “wall” (graffiti) libraries branch “booksurfing” the club of sport achievements clubs of city-quests and sport orienteering board games club rooms for forums, festivals, presentations, private exhibitions related services hire of sports equipment centres of coworking timeclub hippodrome amusement park the formation of creative clusters should occur with the mandatory support of state or regional authorities, which may create favorable conditions for interaction between representatives of business and creative environment. 7.propositions for the development of public-private partnership in the sphere of culture of the khanty-mansiysk autonomous okrug ugra is based on the study of russian[7] and international experience implementing projects in the field of culture on the basis of the mechanism of state-private partnership[? ], the analysis of existing region programs and activities in the field of culture and legislation of khmao-ugra on participation in public-private partnerships, the study of satisfaction of residents and visitors to the region level, quality and range of services in the field of culture, the study views of the representatives of independent (commercial) sector of the cultural complex of ugra. in order to improve the security of the population and guests of municipalities cultural institutions (museums, libraries, cinemas, circuses, zoos), the construction of new objects of culture, reconstruction and equipment of existing cultural institutions and cultural heritage the following activities are offered: -conclusion of contracts of rent of objects of cultural significance of khmaoadvances in systems science and application (2016) vol.16 no.1 83 ugra with commercial organizations providing funding for the maintenance and repair of buildings; equipped with modern technology. so renting for long term rent movie theaters will allow to implement their modernization with modern equipment. renting museums to specialized business entities will reduce the financial costs of the region budget associated with these facilities. renting individual objects of cultural heritage of khmao-ugra to business entities will allow the financial cost of their reconstruction at the expense of investors, providing for the latter a significant reduction of the rent. -conclusion of service contracts (outsourcing) of separate functions for maintenance of buildings and premises of cultural institutions (museums, theatres, libraries); -the conclusion of concession agreements in creation (construction, reconstruction) of objects of infrastructure of cultural heritage. thus, the use of concession agreements is possible to perform repair and restoration work on cultural heritage sites of regional importance within the framework of the implementation of the state program “development of culture and tourism in the khanty-mansi autonomous district-ugra on 2014 2020”. -conclusion of investment agreements with business entities for the construction of new cultural facilities that are absent in the municipalities of khmaougra at the time of the study (the circus; zoo; folk arts and crafts enterprise, producing handicrafts in industrial scale). the use of the above forms of public-private partnership in the sphere of culture of khanty-mansiysk autonomous okrug-ugra will attract private investment in solving problems of infrastructural support of the cultural sphere of the region. references [1] 2014 sba fact sheet. (2014), austria-european comission, 2. [2] 2014 sba fact sheet. (2014), germany-european comission, 2. [3] liu, x. and lim, h. (ed.) (2008), “sme development in china: a policy perspective on sme industrial clustering”, sme in asia and globalization, eria research project report. [4] tang. (2013), facilitating sme: the chinese perspective, mekong forum, khom kaen, thailand. [5] kuzovnikova l. (2010), “historical and cultural resources mapping as a stage creative industries development”, in l.e. vostryakov (ed.) ecology of culture, moscow: dashkovik. 84 yulia s. rod et al: the development system of independent (commercial) sector of ... [6] brown j. (2003), presentation at “creative industries mapping” seminar, november 10-13, petrozavodsk. available at: http://www.cpolicy.ru/doc.plx?id=64 [7] evmenov a.d. and golubev g.m. (2012), “introduction prospects of private-public partnership concept in cultural sphere of the russian federation”, russian entrepreneurship, vol.202, no.4, pp.17-22. corresponding author yulia s. rod can be contacted at:sci.publ@gmail.com microsoft word 9-shang shaoqiang.doc issn 1078-6236 international institute for general systems studies, inc. dentability and convexity shang shaoqiang1 , suyalatu wulede2 1department of mathematics, harbin institute of technology , harbin 150001, china 2college of mathematics science, inner mongolia normal university, huhhot 010022, china email: yizhitu@163.com, suyila@imnu.edu.cn abstract in this paper, we introduced the notion of weak denting point of ( )u x (respectively, uniformly dentable) and described the characterization of reflexive very convex (respectively, uniformly convex) spaces by using the notion of weak denting point (respectively, uniformly dentable), and studied the properties of them. keywords dentability convexity denting point banach space 1. introduction and preliminaries throughout this paper, x will denote a real banach space and x will denote its conjugate space . set ( ) { : , 1}, ( ) { : , 1}, { : ( ), ( ) 1 }.xs x x x x x u x x x x x s f f s x f x x          for a convex set ,c x ext c will denote the extreme point of c . for ( )f s x and 0 ,  set ( , )f f  will denote the slice{ : ( ) : ( ) 1 }.x x u x f x    the weak topology of x is denoted by ( , )x x  and the weak  topology of x is denoted by ( , ).x x  let d be a subset of .x d is said to be dentable if for any 0  there is a x d  such that ( \ ( )) ,x co d b x   where ( ) { : , }.b x x x x x x       a point x d is said to be denting point of d if for any 0 ,  we have ( \ ( )) .x co d b x m.a.rieffel [1] first introduced the notion of dentability and proved that x has the radon-nikodym property whenever every bounded subset of x is dentable. h.b.maynard[2] improved the result of m.a.rieffel and proved that x has the radon-nikodym property if and only if x is dentable. it is known that there is a close connection between the extreme point and the denting point. for example, if m is a compact convex set in banach space ,x then x is extreme point of m if and only if x is denting point of .m in 1993, congxin wu and yongjin li [3] introduced the notion of strongly convex banach spaces and proved that if x is reflexive banach space, then x is dentable if and only if every point of ( )s x is denting point of ( ).u x this is only a result about describing the straight relations between dentability and convexity. * this research has been financially supported by the national natural science foundation of china (no.11061022). ∗ 110-122 advances in systems science and applications (2011), vol. 11,no.1-2 definition 1 let b be a bounded subset of .x a point 0x b is said to be weakly exposed point of b if there is a function ( )x s x  such that 0( ) sup{ ( ): }x x x x x b   and for any sequence 0{ } , ( ) ( ) , ( )n nx b x x x x n    imply that 0 , ( ).w nx x n  definition 2 a point 0 ( )x u x is said to be very extreme point of ( )u x if for 0  there exists a weak neighborhood v of 0 in weak topology ( , )x x  such that for any ( )f s x does not exist , ( )a b u x v  satisfying that ( ) ,{ (1 ) } ( ) ,f a b ta t b u x v      0 1 ( ) , 2 x a b  where [ 0 ,1 ].t  definition 3 let d be a subset of .x d is said to be weak dentable if for any weak neig -hborhood v of 0 in weak topology ( , ),x x  there is a vx d such that ( \( )).w v vx co d x v  a point x d is said to be weak denting point of d if for any weak neighborhood v of 0 in weak topology ( , ),x x  we have ( \ ( )) .wx co d x v  definition 4 let d be a subset of .x a point x d  is said to be weak  denting point of d if for any weak  neighborhood v of 0 in weak  topology ( , ),x x  we have ( \ ( )) .wx co d x v    definition5 ( )u x is said to be uniformly dentable. if for any 0, ( )x s x   there exists 0  such that inf{ ( ) ( ( ( ) \ ( ))) , } .xx x x co u x b x x s       definition 6 [4] a space x is said to be uniformly convex if and only if for any 0  there is a 0  such that for , ( ) ,x y s x if 1 , 2 x y   then .x y   definition 7 [5] a space x is said to be very convex if and only if for any ( ) ,x s x { } ( )nx s x and for some xx s  there holds ( ) 1, ( ),nx x n   then 0,( ).w nx x n  definition 8 [3] a space x is said to be strongly convex if and only if for any ( ) ,x s x { } ( )nx s x and for some xx s  there holds ( ) 1, ( ),nx x n   then 0 , ( ).nx x n  lemma 1 [6] x is uniformly convex if and only if for any 0, ( ),f s x   there is a 0  and some compact set c with dim 1c  such that ( , ) { : ( , ) }.f f y x d y c    lemma 2 let x be a strictly convex space. if 01 0{ } ( ), ( ) , ,n n xx u x x s x x s        0 0( ) ( ) , ( ),nx x x x n   then there exists a net 1{ } { }n nx x     such that .wx x   proof if 01 0 0 0{ } ( ), ( ) , , ( ) ( ) , ( ),n n x nx u x x s x x s x x x x n            we may assume that n mx x  for all .n m ( )u x is weak  compact set, so there exists 0 ( )x u x  such that 0x  is accumulation point of 1{ }n nx   about weak  topology ( , ).x x  we construct a family of sets 0 0 { : x x u u  is weak  neighborhood of 0x  in weak  topology ( , )}x x  and advances in systems science and applications (2011), vol.11, no.1-2 111 define a order by inclusive relation, i.e., 0 0x x u u  if and only if 0 0 , x x u u  thus we obtain a ordered set . we also construct another family of sets 0 0 1{ { } :n nx x u x u      is weak  neighborhood of 0x  in weak  topology ( , )},x x  then by zermelo axiom, there is a mapping f such that 0 0 1 1( { } ) { } .n n n nx x f u x u x         put 0 1( { } ) ,n nx x f u x       then 1{ } { }n nx x       is a net in x and .wx x   denoting 0 0| ( ) | ,r x x then 1.r otherwise, 0 1 ,r  we consider a neighborhood 0 0 0 1 { :| ( ) ( )| (1 )} 2 z z x x x r     of point 0 .x because 0| ( )| 1,nx x  there is an integer n such that for all n n there holds inequality 0 1 | ( )| , 2n r x x   hence 0 1 : | ( ) | 2n n r x x x        is finite set, furthermore, there exists a weak  neighborhood v in weak  topology ( , )x x  such that 0 1 :| ( ) | 2n n r x x x v          because of ( , )x x  is hausdorff topology. by 0 ,wx x   we know that there is a 0 such that for 0  there holds 0 0 0 1 { :| ( ) ( )| (1 )} . 2 x z z x x x r v         one hand, by 0 0 0 1 { :| ( ) ( )| (1 )}, 2 x z z x x x r        we have 0 1 | ( )| . 2 r x x    on the other hand, by 0 0 0 1 { } { }, { :| ( ) ( )| (1 )} 2nx x x z z x x x r v             and 0 1 :| ( ) | , 2n n r x x x v          we have 0 1 | ( )| , 2 r x x    which leads to a contradiction, this shows that 0 0( ) 1,x x  noticing that 0( ) 1x x  and the hypothesis that xis strictly convex, we have 0 ,x x  but 0x x  is impossible because of 0 0 0, ( ) ( ), ( )w nx x x x x x n       and 1{ } { } ,n nx x       which leads to 0 .wx x   2. main results and proof theorem 1 let x be a reflexive banach space. then x is very convex if and only if every point of ( )s x is weak denting point of ( ).u x proof we divide the proof into three parts. firstly, we will prove that if every point of ( )s x is weak denting point of ( ),u x then x is strictly convex. suppose that x is not strictly convex, then there exist three different point 0 1 2, ,x x x in )(xs such that 0 1 2 1 1 . 2 2 x x x  we choose a scalar 0l  and define an linear functional 0f on a subspace 0 1 2{ : , }x x x r      of x such that 0 1 2( ) ( ) ( )f x x l l          112 shang: dentability and convexity for some 0.  since 0x is finite dimensional subspace of ,x then there exists a real number 0m  such that 0|| || .f m by hahn-banach theorem, there exists ( )f s x such that || ||f m and 0( ) ( )f y f y whenever 0 .y x therefore we have 1 0 1 0 1 2( ) ( ) ( 0 ) ( ) 0( ) ,f x f x f x x l l l           2 0 2 0 1 2( ) ( ) (0 ) 0( ) ( ) ,f x f x f x x l l l           0 1 2 0 1 2 1 1 1 1 1 1 ( ) ( ) ( ) . 2 2 2 2 2 2 f x f x x f x x l l l                    we consider a weak neighborhood 0 0: ( ) ( ) | 2 x v x f x f x         of 0x in weak topology ( , ),x x  where v denotes the weak neighborhood of 0 in weak topology ( , ),x x  then 1 2 0, ,x x x v  hence 1 2 0, ( ) \ ( ) ,x x u x x v  it follows that  0 0( \ ( )).wx co u x x v  this contradicts that 0x is weak denting point of ( ).u x secondly, we will prove the necessity of theorem 1. by hahn-banach theorem, we know that for ( )x s x  there exists a ( )f s x such that ( ) 1.f x  if ( ) , ( ) 1, ( ) ,n nx u x f x n   then || || 1, ( ) .nx n  let , || || n n n x y x  then ( ) 1.nf y  because x is very convex (which implies strictly convex), we ( ) ( ( ) \{ })f x f u x x and , ( ).w ny x n  on the other hand, | ( ) | ,n n n nf x x f x y f y x       hence | ( ) ( )| 0, ( ). nf x f x n   this shows that x is weakly exposed point of ( )u x and f is corresponding weakly exposing function. if ( ) \ ( )y u x x v  (where v is the weak neighborhood of 0 in weak topology ( , ),x x  then there exists scalar 0m  such that ( ) ( ) ,f x f y m  hence    ( ) sup ( ): ( )\( ) sup ( ): ( ( )\( )) ,wf x m f y y u x x v f y y co u x x v       this shows that ( ( ) \ ( )) ,wx co u x x v  hence x is weak denting point of ( ).u x thirdly, we will prove the sufficiency of theorem 1. suppose that 1( ),{ } ( )n nx s x x s x    and ( ) 1, ( )nf x n  for some .xf s by the reflexivity of ,x we know that ( )u x is weak sequential compact, hence there exists a subsequence 1 1{ } { } kn k n nx x    such that , ( ) . k w nx x k  because ( )u x is closed convex set, we have ( ) ( ) ( ) , w u x u x u x  hence 1.x  on the other hand, ( ) 1.x f x   this shows that 1.x  by ( ) ( ) 1f x f x  and the fact that x is strictly convex, we have .x x furthermore, we can deduce that , ( ) ,w nx x n  this shows that x is very convex. assume the contrary,i.e., there exist weak neighborhood x v of x in weak topology ( , ),x x  and a subsequence 1 1{ } { } in i n nx x    such that . inx x v  by the assumption that every point of ( )s x is weak denting point of ( ) ,u x we have ( ( ) \ ( )).wx co u x x v  hence there is a function ( , )g x x x    which separates x and ( ( ) \ ( )) ,wco u x x v i.e., there is advances in systems science and applications (2011), vol.11, no.1-2 113 a scalar 0r  such that ( ) sup ( ( ( ) \ ( ))).wg x r g co u x x v   evidently, ( ( )\( )) , i w nx co u x x v  thus ( ) ( ) . ing x g x r  on the other hand, by the reflexivity of ,x we know that there exist a subsequence 1 1{ } { } i il n l n ix x    such that 1{ } il n lx   converges weakly to x . which contradicts that ( ) ( ) . ing x g x r  . theorem 2 x is uniformly convex if and only if ( )u x is uniformly dentable and reflexive. proof we divide the proof into three parts. firstly, we will prove that x is uniformly convex if and only if for any 0 , ( ) ,x s x    there exists 00 , ( )x s x   such that 0( ) 1,x x  then  0( , ) : , .f x x x x x x      suppose that x is uniformly convex. by lemma 1 we know that for any 0, ( ),x s x    there exist a scalar 0 and a compact setc with dim 1c  such that ( , ) :f x x x   ( , ) .d x c  let ( , ) ,x f x  then ( ) 1 .x x    we select a 00 , ( )x s x   such that 0( ) 1x x  because of uniform convexity implies reflexivity, then 0( ) ( ) ,x x x x    it follows that 0 0( ) 1 . 2 2 2 x x x x       by the assumption that x is uniformly convex, we have 0 .x x   this shows that  0( , ) : , .f x x x x x x      conversely, suppose that for any 0 , ( ) ,x s x    there exists 00, ( )x s x   such that 0( ) 1,x x  then  0( , ) : , .f x x x x x x      by lemma 1 we know that x is uniformly convex. secondly, we will prove the necessity of theorem 2. suppose that x is uniformly convex. evidently, x is reflexive. let ( ) , .xx s x x s  by we have proved above, there exist 0  such that  ( , ) : , ,f x y y x y x      i.e., if ( ) ( ) ,x x x y    then .y x   hence, for ( ) \ ( ) ,y u x b x we have .y x   (where ( ) { : , }b x y y x y x     ). we take , 2   then ( ) ( ) .x x x y      therefore, inf{ ( ) ( ( ( ) \ ( )))} .x x x co u x b x      this shows that ( )u x is uniformly dentable. thirdly, we will prove the sufficiency of theorem 2. by the reflexivity of ,x we know that for ( )x s x  there exists 0 ( )x s x such that 0( ) 1.x x  hence, there exists 0 such that 0 0{ ( ) ( ( ( ) \ ( )))}x x x co u x b x    because of ( )u x is uniformly dentable. we take   and let ( , ) ,x f x  then ( ) 1 ,x x    114 shang: dentability and convexity i.e., 0( ) ( ) .x x x x      by the inequality 0 0{ ( ) ( ( ( ) \ ( )))} ,x x x co u x b x    we have 0 0( ) { : , }.x b x x x x x x      this shows that  0( , ) : , .f x y y x y x      by we have proved above, we know that x is uniformly convex. theorem 3 if x  is strictly convex space, then weak  denting points of ( )u x are dense in ( ).s x proof firstly, we will prove that if for ( )x s x  there exists 0 ( )x s x such that 0( ) 1,x x  then x is weak  denting points of ( ).u x for any weak  neighborhood v of 0 in weak  topology ( , ),x x  there exists a scalar 0r such that for any ( ) \ ( )y u x x v    there holds inequality 0 0( ) ( ) .x x y x r   assume the contrary,i.e., 0 0( ) sup ( ( )\( )),x x x u x x v    it is obvious that there exists ( ) \ ( )ny u x x v    such that 0 0( ) ( ) 1,( ).ny x x x n    by lemma 2, we know that there is a 1{ } { }n nx y       such that ,wx x   which contradicts to that ( ) \ ( ).ny u x x v    thus  0 0( ) sup ( ) : ( ) \ ( )x x r y x y u x x v         0sup ( ) : ( ( ) \ ( ) )y x y co u x x v       0sup ( ) : ( ( ) \ ( ) ) ,wy x y co u x x v       hence 0 ( ( ) \ ( )).wx co u x x v     this shows that x is weak denting points of ( ).u x secondly, we will prove that for any ( )x s x  and 0  there is a ( )y s x  which attains its norm on ( )s x such that .y x    let ( ).x s x  for 0,  take a  0,1 such that     25 1 0 . 4 5 1        we consider the norm neighborhood 1 : 4 z z x          of x  in norm topology ( , ) ,x   then 5 1 3 1 . 4 4 z      thus z z x x x x x x z z z z z z                        1 4 4 4 1 1 4 5 1 5 1 5 1                      25 1 . 4 5 1        by bishp-phelp theorem, we know that there exist 0 0, ( )z x z s x   with  0 0 0z z z  such that 0 1 : . 4 z z z x            taking 0 0 , z y z     then  0 1y z  and ,y x    this shows that weak  denting points of ( )u x are dense in ( ).s x advances in systems science and applications (2011), vol.11, no.1-2 115 theorem4 let d be a bounded closed convex set of x .then d is weak dentable if and only if for any weak neighborhood v of 0 in weak topology ( , ),x x  there is a slice  ( , , ) : , ( ) sup ( )s x d y y d x y x d       and 0 ( , , )x s x d such that 0( , , ) .s x d x v   where 0  . proof sufficiency. for any 1 2, , , nf f f x  and 0 ,  let   1 : ( ) | . n i i v x f x     then, by the conditions given here, we know that there exist a slice 0 ( , , )x s x d and 0 ( , , )x s x d such that 0( , , ) ,s x d x v   it follows that 0( ) sup ( ) .x x x d    noticing that  0\ ( ) \ ( , , ). : , ( ) sup ( )d v x d s x d y y d x y x d         and  : , ( ) sup ( )y y d x y x d     is weak closed convex set of weak topology ( , ),x x  we know that  0( \ ( )) : , ( ) sup ( ) ,wco d x v y y d x y x d       it follows that 0 0( \ ( )) ,wx co d x v  this shows that d is weak dentable. necessity. suppose that d is weak dentable. for any 1 2, , , nf f f x  and 0 ,  let 1 : ( ) | . 2 n i i v x f x          then, there a is 0x x such that 0 0( \ ( )) .wx co d x v  hence there is a function x x  which separates 0x and 0( \ ( )) ,wco d x v i.e., there is a scalar 0r  such that 0 0( ) sup ( ( \ ( ))).wx x r x co d x v    let 0sup ( ) ( ) ,x d x x r     then 0( ) sup ( ) sup ( ) ,x x x d r x d        this shows that 0 ( , , ) .x s x d furthermore, for any ( , , ) ,y s x d we can deduce 0 0( ) sup ( ) sup ( ) sup ( ) ( ) ( ) ,x y x d x d x d x x r x x r             which leads to 0 .y x v  otherwise, 0 ,y x v  furthermore 0 0\( ) ( \( )).wy d x v co d x v    by the inequality 0 0( ) sup ( ( \ ( ))),wx x r x co d x v    we have 0( ) ( ).x x r x y   a contradiction. theorem5 let x be a reflexive banach space. if every point of ( )s x is weak denting point of ( ) ,u x then every point of ( )s x is weakly exposed point of ( ).u x proof suppose that 0 ( )x s x is weak denting point of ( ) ,u x by hahn-banach theorem we know that there is a function 0 ( )x s x  such that 0 0( ) 1.x x  we will prove that if 0x attains its norm at another point 0 ( ) ,y s x then 0 0 .x y otherwise, 0 0 ,x y it is obvious that    0 0 0 0 0 1 1 1 1, 2 2 x y x y x     thus  0 0 1 1. 2 y x  by hahn-banach theorem we know that there is a function 0 ( )y s x  such that    0 0 0 0 .y y y x  we consider a weak neighborhood 0 0 0 0 0 0 0 | ( ) ( ) |1 : ( ( )) 2 4 y y y x v x y x y x            of point 0 0 1 ( ) , 2 y x 116 shang: dentability and convexity it is clear that 0 0, ,x y v thus  0 0, \ ,x y u x v furthermore  0 0 1 ( ) ( \ ), 2 wy x co u x v  this shows that  0 0 1 2 y x is not weak denting point of ( ) ,u x which contradicts to the hypothesis that every point of ( )s x is weak denting point of ( ).u x if 1 0 0 0{ } ( ) , ( ) ( ) 1, ( ) ,n n nx u x x x x x n        then, by the hypothesis that x is reflexive space we know that there exist 1 1{ } { } kn k n nx x    and ( )x u x such that ,( ). k w nx x k  hence 0 0 0 0( ) ( ) ( ) 1, ( ) , knx x x x x x k      it follows that 1,x  this shows that ( ).x s x by the proof above, we have 0 ,x x thus 0 , ( ) k w nx x k   and we can deduce that 0 , ( ).w nx x n  if 1{ }n nx   does not converges weakly to 0 ,x then there exist a weak neighborhood 1v of 0 in weak topology ( , )x x  and a subsequence 1{ } in ix   of 1{ }n nx   such that 1 0 1{ } ( ) . in ix x v     noticing that 0x is weak denting point of ( )u x , we know that 0 0 1( ( ) \ ( )).wx co u x x v  hence there is a function x x  which separates 0 1( ( ) \ ( ))wco u x x v and 0 ,x i.e., there is a scalar 0r  such that  0 0 1sup ( ( ( ) \ ( ))).wx x r x co u x x v    noticing that 1 0 1{ } ( ) , in ix x v    we have 0 1 0 1( ) \ ( ) ( ( ) \ ( )). i w nx u x x v co u x x v    by the inequality 0 0 1( ) sup ( ( ( ) \ ( ))),wx x r x co u x x v    we have 0( ) ( ). inx x r x x   on the other hand, by we have proved above, there exists a subsequence 1{ } jn jx   of 1{ } in ix   such that 1{ } jn jx   converges weakly to 0 ,x which leads to    0 0 .x x r x x   a contradiction. hence 0 ( )x s x is weakly exposed point of ( ).u x theorem6 let x be a separable reflexive banach space. if 0 ( )x u x is extreme point of ( ) ,u x then 0 ( )x s x is weak denting point of ( ).u x proof suppose that 0 ( )x u x is extreme point of ( ).u x if 0 ( )x s x is not weak denting point of ( ),u x then there exists a weak neighborhood v of 0 in weak topology ( , )x x  such that 0 0( ( ) \ ( )).wx co u x x v  noticing that 0( ( )\( )) ( )wco u x x v u x  and the assumption that x is reflexive banach space, we know that 0( ( ) \ ( ))wco u x x v is weak compact set. by krein-milman theorem, we have 0 0( ( ( ) \ ( ))) ( ( ) \ ( )),w w wco ext co u x x v co u x x v   hence 0 0( ( ) \ ( )) ( ) \ ( ) . wwext co u x x v u x x v   in fact, by the assumption that x is separable banach space, we know that weak topology ( , )x x  is metrizable space, hence, for any 0( ( ) \ ( )) ,wx ext co u x x v  there exists a sequence 1{ (1 ) }n n n n nt x t y    such that  (1 ) , .w n n n nt x t y x n    where   00,1 , , ( ) \ ( ).n n nt x y u x x v   by the reflexivity of ,x we know that there exists a sequence    in n such that 0 0, , i i w w n nx x y y  0, ( ). i w nt t i  it is obvious that  0 0 0 00,1 , , ( )\( ) w t x y u x x v   and (1 ) , i i i i w n n n nt x t y x   advances in systems science and applications (2011), vol.11, no.1-2 117 thus  0 0 0 01 .x t x t y   case (i): if 0 0,t  then 0 0( ) \ ( ) ; w x y u x x v   case (ii): if 0 1,t  then 0 0( ) \ ( ) ; w x x u x x v   case (iii): if  0 0,1 ,t  then, by 0 0 0, ( ) \ ( ) w x y u x x u  0( ( ) \ ( ))wco u x x v  and 0( ( ) \ ( )) ,wx ext co u x x v  we have 0 0 0( ) \ ( ) . w x x y u x x v    combining case (i),(ii) and (iii), we have 0 0( ( ) \ ( )) ( ) \ ( ) . wwext co u x x v u x x v   because   0 0( ( ( ) \ ( ))) ( ( ) \ ( )) ,w wext u x co u x x v ext co u x x v   we have 0 0 0( ( ) \ ( )) ( ) \ ( ) . wwx ext co u x x v u x x v    this is a contradiction. theorem7 if 0 ( )x s x is weak denting point of ( )u x , then 0x is very extreme point of ( )u x . proof suppose that 0 ( )x s x is not very extreme point of ( ),u x then there exist 0 0 ,  0 ( )f s x  such that for any neighborhood v of 0 in weak topology ( , ),x x  there exists , ( )a b u x v  satisfying that     0 0 1 ( ),| | , 1 ( ) , 2 x a b f a b ta t b u x v        where  0,1 .t  we consider a weak neighborhood 0 0: | ( ) | 6 v x f x       of 0 in weak topology ( , )x x  and a family of sets  v (where v is balanced convex neighborhood of 0 in weak topology ( , ),x x  then weak topology ( , )x x  has a neighborhood base  v v  of 0. for any ,v v  there exist , ( ) ( )a b u x u v     satisfying that 0 0 1 ( ), ( ) , 2 x a b f a b        { (1 ) } ( ) ,ta t b u x v v       where [0,1].t let ,a a a b b b            (where , ( ) , ,a b u x a b v v          ), then 0 0 0 0 0 1 1 1 ( ) ( ) , ( ) | ( ( )) | , 2 2 2 2 x a b a b f a x f a b                  0 0 0 0 0 0 0 0( ) ( ) | ( ) | . 2 6 3 f a x f a x f a a               noticing that ( )u x is convex set and v v  is balanced convex set, we know that 1 1 1 ( ) ( ) , ( ) , ( ) . 2 2 2 a b u x a b v v a b v v                    let 0 0 0: ( ) . 3 v x f x       it is obvious that 0 0 .a x v   similarly, 0 0 .b x v   hence 118 shang: dentability and convexity 0, ( ) \ ( ) ,a b u x x v     furthermore, 0 0 1 ( ) ( ( ) \ ( )). 2 a b co u x x v     on the other hand, 0 1 1 ( ) ( ) 2 2 a b x a b          and 1 ( 2 a  ) ,b v v    so we have 0 1 ( ) . 2 a b x v v       because  v v  is neighborhood base of 0 in weak topology ( , ),x x  from the definition of weak closure about weak topology ( , ),x x  we know that 0 0 0( ( ) \ ( )).wx co u x x v  which contradicts that 0 ( )x s x is weak denting point of ( ).u x theorem8 let x be a reflexive separable banach space. x x is a banach space with the norm  ( , ) max , .x y x y if 1 2,x x are weak denting points of ( ) ,u x then 1 2( , )x x is weak denting point of ( ) ( ) ( ).u x u x u x x   . proof suppose that 1 2,x x are weak denting points of ( ) .u x by theorem 7 we know that 1 2,x x are very extreme points of ( ).u x it is obvious that 1 2,x x are extreme points of ( ).u x we will prove that 1 2( , )x x is extreme point of ( ) ( ).u x u x if 1 2 1 2( , ), ( , ) ( ) ( )y y z z u x u x  and 1 2 1 2 1 2 1 1 2 2 1 1 1 1 1 1 ( , ) ( , ) ( , ) ( , ), 2 2 2 2 2 2 x x y y z z y z y z     then  1 1 1 2 2 2 1 1 ( ), . 2 2 x y z x y z    hence 1 1 1 2 2 2, .x y z x y z    this shows that 1 2 1 2 1 2( , ) ( , ) ( , ).x x y y z z  therefore 1 2( , )x x is extreme point of ( ) ( )u x u x . by theorem 6, 1 2( , )x x is weak denting point of ( ) ( ) ( ).u x u x u x x   theorem9 let x be a very convex space and x x is a banach space with the norm  ( , ) max , .x y x y if 1 1, ( ) ,x y s x then 1 1( , )x y is weak denting point of ( ).u x x if 1 11, 1,x y  then 1 1( , )x y is not weak denting point of ( ).u x x proof firstly, we will prove that ( , ( ) ) ( , ) ( , ).x x x x x x x x        for 0, ( ) ,f x x     we consider a eighborhood{( , ) : ( , ) }x y f x y  of (0,0) in weak topology ( ,( ) )x x x x   and define 1 2( ) ( ,0), ( ) (0, )f x f x f x f x  for any ,x x then ( )if x f x  and ( 1,2)if i  are linear functions. this shows that *, ( 1,2).if x i  we construct a set 1 2: ( ) : ( ) , 2 2 g x f x x f x                then g is open set in ( , ) ( , ).x x x x   for any ( , ) ,x y g we have 1 2( , ) ( ,0) (0, ) ( ) ( ) , 2 2 f x y f x f y f x f y          hence {( , ) : | ( , ) | }.g x y f x y   this shows that topology ( , ) ( , )x x x x   is smaller than topology ( , ( ) ).x x x x   for * 1 20, , ,f f x   we also consider the above open setg in topology and define advances in systems science and applications (2011), vol.11, no.1-2 119 1 2( , ) ( ) , ( , ) ( )f x y f x g x y f x  for ( , ) ,x y x x  then 1 1( , ) . ( , ) ,f x y f x f x y  2 2( , ) . ( , )g x y f x f x y  and ( , ) , ( , )f x y g x y are linear functions. this shows that , ( ) ,f g x x   we consider a neighborhood ( , ): ( , ) ( , ): ( , ) 2 2 x y f x y x y g x y               of (0,0) in weak topology ( ,( ) ),x x x x   then ( , ) : ( , ) ( , ) : ( , ) . 2 2 x y f x y x y g x y g                this shows that topology ( ,( ) )x x x x   is smaller than topology ( , ) ( , ).x x x x   this completes the proof that ( ,( ) )x x x x   ( , ) ( , )x x x x    . secondly, we will prove that if 1 1, ( ),x y s x then 1 1( , )x y is weak denting point of ( ).u x x if 1 1, ( ) ,x y s x then, by hahn banach theorem we know that there exists 1 ( )f s x such that 1 1( ) 1.f x  we choose ( )nx u x with 1,nx  such that  1 1 1( ) ( ) 1, .nf x f x n   by the assumption that x is very convex space, we have 1, ( ) .w nx x n  it follows that for any weak neighborhood v of 0 in weak topology ( , ),x x  there exists 1 0r  such that for any 1( ) \ ( )x u x x v  there holds 1 1 1 1( ) 1 ( ) .f x f x r   otherwise, there exists 1( ) \ ( )nx u x x v  such that 1( ) 1,( ),nf x n  which leads to 1,( ).w nx x n  this contradicts that 1( )\( )nx u x x v  .similarly, there exists 2 ( )f s x such that 2 1( ) 1f y  and for any weak neighborhood v of 0 in weak topology ( , ),x x  there exists 2 0r  such that for any   1\ ( )y u x y v  there holds 2 1 2 2( ) 1 ( ) .f y f y r   for any ( , ) ,x y x x  let 1 2( , ) ( ) ( ) ,f x y f x f y  then f is linear function on ,x x and 1 2 1 2( , ) ( ) ( ) . .f x y f x f y f x f y    2 2 1 22 f f  2 2 x y  2 2 2 2 1 2 1 24 max , 4 ( , )f f x y f f x y    . this shows that ( ) .f x x   we consider any weak neighborhood w of (0,0) in weak topology ( ,( ) )x x x x   and notice that 1 1, ( ) ,x y s x then, by ( ,( ) )x x x x   ( , ) ( , ),x x x x    we know that there exists a weak neighborhood v of 0 in weak topology ( , )x x  such that 1 1 1 1( ) ( ) ( , ) .x v y v x y w     for 1 1( , ) ( ( ) ( ) ) \ (( ) ( )),x y u x u x x v y v     we have 1( ) \ ( )x u x x v  or 1( ) \ ( ).x u x y v  without loss of generality, we may assume 120 shang: dentability and convexity that  1( ) \ ,x u x x v  then 1 1 1 1 1 2 1 2 1( , ) ( , ) ( ( ) ( )) ( ( ) ( )) .f x y f x y f x f x f y f y k      thus 1 1 1( , )f x y k   1 1sup ( , ) : ( , ) ( ) ( ) \ ( ) ( )f x y x y u x u x x v y v      1 1sup ( , ) : ( , ) ( ) ( ) \ (( , ) )f x y x y u x u x x y w     1 1sup ( , ) : ( , ) ( ( ) ( ) \ (( , ) ))f x y x y co u x u x x y w     1 1sup ( , ) : ( , ) ( ( ) ( ) \ (( , ) )) .wf x y x y co u x u x x y w    hence 1 1 1 1( , ) ( ( ) ( ) \ (( , ) )).wx y co u x u x x y w   obviously, ( ) ( ) ( ),u x x u x u x   which leads to 1 1 1 1( , ) ( ( ) \ (( , ) )) ,wx y co u x x x y w   this shows that 1 1( , )x y is weak denting point of ( )u x x . thirdly, we will prove that if 1 11, 1,x y  then 1 1( , )x y is not weak denting point of ( ).u x x case (i). if 1 1( ), , (0,1),x s x y a a   then, there exist 1 1 2 2 2 2 1 , 1 3 3 a a y y y y a a                   such that 1 1 1 1 2 2 (1 ) 1, (1 ) 1. 3 3 y y y y a a y y y y a a                 by hahn-banach theorem, there exists *f x such that 1 2 2 2 2 ( ) 1, ( ) 1 , ( ) 1 , 3 3 a a f y f y f y a a        hence 1 1 2 2 2 2 ( ) , ( ) 3 3 a a f y y f y y a a       and 0 2 2 . 6 a a    we consider a weak neighborhood 1 0{ : | ( )| }x f x y   of 1y in weak topology ( , ) ,x x  then 1 0{ : | ( )| }x f x y   can be written by 1 0 1{ : | ( )| } ,x f x y y v     where v  is a weak neighborhood of 0 in weak topology ( , ).x x  evidently, 1, ,y y y v    it follows that 1 1 1 1( , ),( , ) ( ( ) ( ) \ ( )( )).x y x y u x u x x v y v       by ( ,( ) ) ( , ) ( , ),x x x x x x x x        we know that there exists a weak neighborhood v of 0 in weak topology ( , )x x  such that 1 1 1 1( , ) ( ) ( ) .x y v x v y v      noticing that 1 1 1 1 1 1 ( , ) ( , ) ( , ) 2 2 x y x y x y   and the equality ( ) ( ) ( ) ,u x x u x u x   we have 1 1 1 1 1 1( , ) ( ( ) ( ) \ ( ) ( )) ( ( ) \ (( , ) )) ,x y u x u x x v y v co u x x x y v         furthermore, 1 1 1 1( , ) ( ( ) \ (( , ) )).wx y co u x x x y v   hence, 1 1( , )x y is not weak denting point o ( ).u x x case (ii). if 1 10, ( ),y x s x  let 1 ( ) , . 2 z u x z  by hahn-banach theorem, there exists *f x such that ( ) 1.f z  we consider a neighborhood 1 : ( ) 2 v x f x        of 0 in weak advances in systems science and applications (2011), vol.11, no.1-2 121 topology ( , ),x x  then , .z z v  which leads to 1 1 1( , ) , ( , ) ( ( ) ( ) \ ( ) (0 )).x z x z u x u x x v v       similar to case (i), there exists a v such that 1 1( ,0) ( ) (0 ).x x v v     noticing that 1( ,0)x  1 1 1 1 ( , ) ( , ), 2 2 x z x z  we have 1 1 1( ,0) ( ( )\( ) (0 )) ( ( )\(( ,0) )).w wx co u x x x v v co u x x x v         this shows that 1( ,0)x is not weak denting point of ( ).u x x . theorem10 let x be a strongly convex space. x x is a banach space with the norm  ( , ) max , .x y x y if 1 1, ( ) ,x y s x then 1 1( , )x y is denting point of ( ).u x x if 1 11, 1,x y  then 1 1( , )x y is not denting point of ( ).u x x proof case (i). 1 1, ( ).x y s x using a similar method to that used in the proof of theorem 9, we can prove that 1 1( , )x y is denting point of ( ).u x x case (ii). 1 11, 1.x y  by the assumption that x is strongly convex space, we know that x is very convex space. by theorem 9, we know that 1 1( , )x y is not weak denting point of ( ) ,u x x so there exists a weak neighborhood v of (0,0) in weak topology ( , ),x x  such that 1 1 1 1( , ) ( ( ) \ ( , ) ).wx y co u x x x y v   it is obvious that there exists a scalar 0  such that 1 1 1 1( , ) ( , ) .b x y x y v   hence 1 1 1 1 1 1( , ) ( ( )\ ( , )) ( ( )\ ( , )).wx y co u x x b x y co u x x b x y     this shows that 1 1( , )x y is not denting point of ( ).u x x references [1] m.a.rieffel. dentable subsets of banach spaces with applications to a radon-nikodym theorem, funct.anal.proc.(conf. irvine, calif., 1966), acad.press.london. thompson.wash -ington, d.c. 71-77. 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[9] jinghong pan, xuemou wu. a pansystem topology approach grayscale image processing. advances in systems science and applications, 2003, 2: 151-156. 122 shang: dentability and convexity мягкая вероятность ошибки при передаче информации advances in systems science and application(2016) vol.16 no.4 89-99 modeling the outbreak of an infectious disease on a heterogeneous network m. hajizadeh 1 , f. r. vishkaie 2 , f. bakouie 3 , s. gharibzadeh 3 1 faculty of biomedical engineering, amirkabir university of technology (tehran polytechnic) tehran, iran 2 faculty of electrical engineering, shahid beheshti university, tehran, iran 3 institute for cognitive and brain sciences, shahid beheshti university, tehran, iran annotation the outbreak of infectious diseases is a global public health threat for the international community. modelling the propagation of epidemics in a society is one of the important fields in epidemiology science. it is essential to know the number of infected cases for estimating and controlling the spread of disease in the affected countries. in this study, we used complex network theory to model the spread mechanism of epidemic disease in social networks. we modeled a social complex network by graph theory. individuals are considered as nodes and acquaintances between them are considered as links. disease virus can transmit along the links between nodes (people) according to different situations. in this work, we proposed a dynamic model for simulating the outbreak of infectious disease on a social network based on the susceptible, exposed, infected and recovered (seir) dynamical categories. it has been tried to study the heterogeneity on the network by considering two key factors in the epidemic propagation: 1) the communications weights between individuals in the network 2) different body resistances of people based on age. the proposed dynamic model was applied on a real social network which was constructed in our previous research. we compared the proposed model with two different dynamic models. finally, the simulations were compared with the reported data of infected cases of sars outbreak in hong kong in 2003. the results indicated some similarity between our proposed model and the real reported data. based on the results, it could be concluded that considering communications weights and body resistances of people captures the dynamic of disease spread in a proper way. key words: social network, infectious disease, heterogeneity, body resistance, communications weights. 1 introduction complex network theory could help to understand the spread mechanism of epidemic disease in social networks [1, 2]. graph theory is a powerful tool for modeling complex networks [3-6]. in graph theory, individuals are considered as nodes and acquaintances between individuals are considered as links [7]. virus transmission can occur along the existing links according to different situations. some studies indicated that human interactions are important for the spread of infectious disease [8]. in the real-word, social networks links strengths aren’t equal and some interactions have a high probability of infection transmission than others, i.e. the weight of links shows the amount of communications between individuals [7]. prevalence of infectious diseases such as influenza, meningitis, pertussis, yellow fever and etc., since the beginning of history has been a major concern of humanity. some of them are 90 zhaobin li, jian liu, zhuo zhang, et al.: a new decision method for multi-criteria decision making with numerical values… widely spread in the last decade: severe acute respiratory syndrome (sars) in 2002, methicillin-resistant staphylococcus aureus (mrsa) in 2005 and ebola in 2014. several types of research have studied the spread of infectious disease in social networks [913]. some of them, considered the complex network approach in their researches. christian l. althaus presented mathematical modeling that was appropriate for estimate the outbreak of ebola in west africa in 2014. he applied seir model in the simulations [14]. abdulrahman et al. presented and analyzed a model for controlling the spread of ebola virus disease (evd) in a population by slir dynamic model. they defined 12 parameters in their model such as rate of public enlightenment, enhanced personal hygiene due to public enlightenment, number of quarantined individuals and availability of isolation centers. their simulations showed that improved personal hygiene and quarantining of infectious individuals are enough to control the spread of evd [15]. small and tse modeled transmission of sars in hong kong with complex small world network. they used seir dynamic model. they supposed different probabilities of transition between the states of disease in the network. in their model, random nodes are isolated in order to control the spread of infection [16]. colizza et al. used real airline networks data for their study because they believed that people’s transportations are causes of sars disease propagation. they considered the population of each city is classified into seven different compartments, and hospitalized as well as infectious individuals are able to transmit the infection. they concluded that their models are fit for the forecast and analysis of emerging disease spreading at the global level [17]. some scientists have attempted to produce their database through questionnaires [18]. previously, we recorded the communications between 100 persons during one week and constructed a sample of a social network consisting human communications. it was shown that the constructed social encounters network has a small-world topology. in the next step, the common cold outbreak was modeled with sir dynamic model [1]. in this study, we used our previous social network for modeling sars propagation by seir model. in a disease spread network, there are some sources of heterogeneity. for example, individuals’ characteristics against the disease and the level of interaction between people are some bases of inhomogeneity. in this paper, we tried to apply this heterogeneity factors in the simulations. for this purpose, we proposed a dynamic model with two important features: 1) communications levels between individuals which is reflected in the connections’ weights of the network 2) different body resistances based on ages of people which is used as different thresholds for each node (people). our suggested model considers cumulative weights of all links between each node and its infected neighbors. we named this model as cumulative weight model (cw). this model was compared with two different dynamics. the first one is abramson and kuperman’s model (ak) that it is based on the number of infected neighbors for each node [19]. this model doesn’t consider the amount of communication between persons. the second model is rajabi vishkaie et al. model (r) that it’s based on the largest connection weight between one node and its infected neighbors [1]. at the end, the simulation results of these models were compared with reported data of infected cases of sars outbreak in hong kong in 2003 [20]. 2 methods dataset: previously, for data gathering, we used the questionnaire distribution method in a small social network. in order to construct the network structure, 15 participants were asked in the questionnaires to record their encounters in a week. moreover, they were asked to record their contact time weekly. based on the level of contact between the persons, we assigned different amounts for edge weight in integer values from 0 to 9: more time of weekly contacts receives more scores. advances in systems science and application(2016) vol.16 no.4 91 the questionnaires were distributed among the author’s acquaintances. these participants were asked to distribute the questionnaires between their own acquaintances (exactly those people that the participants mentioned in their own questionnaires). in this way, the network was expanded. all the participants were asked to record their encounters with their acquaintances; also, we requested participants to record the encounters between their acquaintances. there were 15 participants. the number of all people that they recorded was 100. the produced network (database network) had 326 links between persons [1]. in this study, since we used our previous database, the run time must be 7 days but it isn’t long enough to see the propagation of disease. hence, the run time must be a multiple of 7. therefore, the run time was chosen 98 days. (we considered the run time long enough to see the changes.) 2.1. basic definitions in disease transmission the dynamic of the disease spread governed by seir model (that ‘s’ stands for susceptible, ’e’ shows exposed persons, ‘i’ shows infected person, and ‘r’ indicates recovered or deceased person). it has reported that sars has incubation period between 2 to 7 days (or longer) [21]. in our model, we supposed the incubation period is 7 days ( ). this means that when the sars’s virus goes inside a person body (s)he goes to exposed state (e) while (s)he doesn’t have disease symptoms in this period (first week). after this incubation period, the person goes to infected state (i) which has disease symptoms such as fever, cough etc. here, we assumed the symptoms period is about 7 days. since cdc (centers for disease control and prevention) recommends that persons with sars limit their interactions outside the home until 10 days after their respiratory symptoms have gotten better [22], we considered infection period ( ) as 17 (10+7) days which each infected person could carry the virus. after the infected period (s)he goes to recovered state (r) in which (s)he can’t transmit the infection anymore (see fig.1). fig.1 schematic of state transition by seir dynamic model for sars disease, this dynamic model has four state for a society that has infectious disease: s (susceptible), e (exposed), i (infected) and r (recovered) in our network, we assumed that at the start time of the simulations, sars viruses infected 1% of the population and before this time, nobody is sick. we defined two time-based vectors: where state(t) is a vector that indicates the conditions of each node (individuals) and n is the number of network’s nodes. we also defined probability vector as: where (t) is a vector that i ndicates the likelihood of getting illness for the node i at time t. generally, if the interaction between two people increases, the probability of getting illness ( (t)) will increase, too. the states of nodes in time step t+1 (state (t+1)) will change according to: (t+1) = 0 (s) ; 1 (e) ; 2 (i) 3 (r) http://www.cdc.gov/ 92 zhaobin li, jian liu, zhuo zhang, et al.: a new decision method for multi-criteria decision making with numerical values… one of the important factors in the outbreak of epidemic disease is the probability of virus transmission from each infected nodes to its neighbor nodes (q). we assumed that q is between 0 and 1 with a normal distribution. another factor in propagation is the threshold (th). we considered it as body resistance against the disease. if a person has more body resistance, (s)he will have a higher threshold. in this study, at first, in some simulations we considered th as a fixed parameter between 0 and 1 to all the individuals for obtaining results in different situations. since the disease mortality depends on age [23], so body resistance against the virus may be different according to the people’ age. hence, in some simulations, we assigned different th to the network nodes based on the ages. 2.2. transmission dynamics we applied three different models for the dynamic of the disease spread: 1) ak model. this model is proposed by abramson et al. and it was used for modeling infectious disease [19]. this model has expressed based on the number of the infected neighbors. where is the number of the infected neighbors for the node i at time t, and q is the probability of transmission from each infected neighbor to node i and it is a random quantity between 0 and 1. (t) is the probability of becoming ill for node i at time t. in simulations if (t) is more than th, node i will go to the e state. (th is the body resistance against the disease for each node.) 2) r model. this model is proposed by rajabi et al. and it was used for modeling common cold outbreak [1]. where is the weight of the link between the node i and one of its infected neighbors. if the node i has several infected neighbors, simulations will check all of the links and if it get a connection which has a (t) more than th, the contagion transmits to the node i and it will go to the e state. (th is the body resistance against the disease for each node.) in fact, one dangerous connection (a link that its weight is more than probability *10) is sufficient to transmit the infection and the other connections will be ignored. 3) cw model. the amount of interactions between persons (w) and the number of infected neighbors have the main roles in disease spread. as the number of infected neighbors increase, the probability of getting illness will be increased too. also, the probability of transmission from each infected neighbor to a susceptible node (q) is significant. in order to consider these factors and the heterogeneities on the network, we changed eq. (4) (ak model) to eq. (6). this equation comprised cumulative weights of links to each node and its infected neighbors: where (t) is the probability of becoming ill for i node at time t. is the sum of the edges weights between the node i and its infected neighbors at time t. if (t) is more than th, node i will go to the e state. (th is the body resistance against the disease for each node.) 3 results the diagrams of the susceptible, exposed, infected and recovered persons for three models is illustrated in fig.2. in these three diagrams, the total number of susceptible, exposed, infected and recovered persons is equal to 100 (number of network’s vertices) at each time. for all 3 advances in systems science and application(2016) vol.16 no.4 93 dynamics, it was indicated that the number of exposed people is 1 from the first day until 7 th day. during this period, the number of infected people is 0. fig.2 number of susceptible, exposed, infected and recovered persons for (a) ak model (eq. (4)) with mean of the q=0.4 and th=0.5 (b) r model (eq. (5)) with th=0.1 (c) cw model (eq. (6)) with mean of the q=0.4 and th=0.5 the results of simulation for ak model with the mean of q=0.4 and th=0.5 are shown in fig.2.a. as it is shown in this figure, the number of infected persons is increased from 7 th day to 40 days from the beginning of the simulation, and then it will be decreased till the 46 th day. after that, it will be increased till 48 th day and then it will be decreased till the 77 th day. the maximum number of infected persons is 55 which is reached in the 48 th day. the number of susceptible persons becomes 0 at the 52 nd day and it will be remained 0 until the end of run time. therefore, all the people (100 persons in the social network) became sick in ak model. simulation results for r model with th=0.1 are illustrated in fig.2.b. as it is shown in this diagram, the number of infected persons is increased from 7 th day till 32 nd day of simulation cycle and then it will be decreased till the 49th day. the maximum of the infected subject is 24 in the 32 nd day. the number of susceptible individuals remained in 75 from 25 th day till the end of simulation cycle. in this model, 25 persons in the social network became infected. the results of implementation for cw model with the mean of q=0.4 and th=0.5 are indicated in fig.2.c. as it is shown in this figure, the number of infected people is increased from 7 th day till 40 th day, and then it will be decreased till 41 st day quickly. again it will increase till 42 nd day, after that it will be decreased till the 62 th day. the maximum number of infected people is 81 which is reached in the 40 th day of simulation cycle. the number of susceptible people became 2 in the 38 th day and it will remain in 2 until the end of the simulation, i.e.in cw model 98 persons became sick. with giving different quantities to the q and th for each dynamic model, the number of susceptible people will change. the results are shown in fig.3. 94 zhaobin li, jian liu, zhuo zhang, et al.: a new decision method for multi-criteria decision making with numerical values… fig.3 (a) number of susceptible persons for (a) ak model (eq. (4)) when th<q : th=0.2 & q=0.4 and when th>q : th=0.8 & q=0.4 (b) r model (eq. (5)) when th<0.5 : th=0.2 and when th>0.5 : th=0.8 (c) cw model (eq. (6)) when th<q : th=0.2 & q=0.4 and when th>q : th=0.8 & q=0.4 in this figure when th q in ak model any node won’t be infected although in cw model 26 persons became ill. also, when th q in ak and cw model after 34 days from beginning of simulation number of susceptible people became 0. it means, in this case, all the susceptible individuals will be sick (see fig.3.a and fig.3.c). in fig.3.b for r model when th=0 all the people in the network will be infected, since the number of susceptible people becomes 0 at 33 rd day. in addition, when 0 th 0.5 only 25 persons became infected and it remained 25 from 25 th day till the end of the simulation. moreover, when th 0.5 any person won’t be infected. in the real world, old people and kids have lower body resistance against the disease. in order to have a more plausible model, we consider different body resistances (th) in our simulations. we assign different quantities to the th based on the persons’ ages in the network. old people and kids were given the lower th and younger people were given higher th. the simulation results for three models in which th quantities are assigned based on persons’ ages is shown in fig4. the diagram of the number of susceptible, exposed, infected and recovered people for ak model in which th quantities are based on individuals ages is shown in fig.4.a. as you see the number of infected individuals is increased from 7 th day till 40 th day, and then it will be decreased till the 69 th day. the maximum number of infected individuals is 75 which is reached in the 40 th day of simulation cycle. the number of susceptible individuals became 0 in the 45 th day and it will remain in 0 until the end of the simulation. therefore, all the people in this social network became sick. for r model as it is shown in fig.4.b, the number of infected individuals is increased from 7 th day till 56 th day of simulation cycle and then it will be decreased till the 65 th day. the maximum of the number of infected individuals is 8 from the 40 th day till 48 th day. the number of susceptible individuals remained in 89 from 41 st day till the end of run time. thus, in this model, just 11 persons in the social network became ill. the simulation results for cw model in which th quantities are assigned based on ages and with the mean of q=0.7 is shown in fig.4.c. as it is shown in this figure, the number of infected people is increased from 7 th day till 40 th day, and then it will be decreased till the 44 th day. again it is increased till 48 th day and after that, it will be decreased till the 70 th day of the simulation period. the maximum number of infected people is 73 which is reached in the 40 th day of simulation cycle. the number of susceptible individuals became 1 in the 46 th day and it will remain in 1 until the end of the simulation. so, 99 persons became infected. advances in systems science and application(2016) vol.16 no.4 95 fig.4 number of susceptible, exposed, infected and recovered persons giving th quantity based on age for (a) ak model (eq. (4)) with mean of the q=0.7 (b) r model (eq. (5)) (c) cw model (eq.(6)) with mean of the q=0.7 in order to have a better comparison with real data from sars infection data for hong kong since 15 february 2003 [20], the number of infected people in fig.4 for each model is depicted in fig.5. fig.5 number of infected persons (a) ak model (eq. (4)) with mean of q=0.7 and giving th quantity based on age (b) r model (eq. (5)) and giving th quantity based on age (c) cw mode l (eq. (6)) with mean of q=0.7 giving th quantity based on age (d) daily reported sars infection data for hong kong since 15 february 2003 all these diagrams have the same trend of propagation of disease. ak and cw model have higher amplitude for the number of infected people. in addition cw model has a higher slope than ak and r model. 4 discussion 96 zhaobin li, jian liu, zhuo zhang, et al.: a new decision method for multi-criteria decision making with numerical values… in this study, we examined three different dynamic models for simulating the outbreak of an infectious disease (here the sars was modeled) on a social network. in all the simulations, at the beginning of the disease spread, 1% of people will receive disease virus, (in this study the number of nodes is 100 persons) therefore just one node will go to exposed status. according to fig.2, in the first 7 days, there are no changes in the number of infected people. in fact, the number of infected people depends on the changing status of the first exposed person in the network. the incubation period is supposed 7 days, and the contagion couldn’t transmit from the first exposed person to others in this period. the ak model (eq. (4)) considers these factors: the number of infected neighbors and the probability of infection transmission from an infected person to its susceptible neighbors (q). this probability depends on the transmission rate. in the transmission of epidemic diseases, another important factor is the weights of communications between persons [24, 25]. there are different strategies to assign a weight to a link based on the amount of communication between two nodes [26, 27]. previously, we allocated different amounts for links weights in integer values from 0 to 10 based on the level of communication between the people and we applied this factor (w) in r model (eq. (5)) [1]. according to r model, having an infected neighbor with strong communication weight is enough to become sick. in our data, most of the links have the weight equal to 1. for these links, the probability of becoming sick is equal to 0.1 (eq. (5)) and it is not more than th (th= 0.1). so this dynamic doesn’t let the communication links with weight 1 to transmit the infection. this is the reason for the r model’s result wherein a few people have been ill (see fig.2.b). in the real-word social networks, the communications between individuals aren’t equal. in order to consider heterogeneity on the network, we changed eq. (4) (ak model) to eq. (6) (cw model). in the proposed model the communication weights of all the infected neighbors are considered. the diagram trend of infected people in the cw model is similar to the ak because both of them consider q. in both of these models, almost all the people became infected. besides, the cw model shows more fluctuations in disease propagation than the ak and r model. it may be as a result of considering the summation of communication weights. with giving different amounts to the q and th the below results were achieved from simulations (see fig.3). according to fig.3.a in ak model we observe: if th ≥ q: any node can’t be infected. if th < q: all nodes will be infected. these former statements can be obtained from the following equations: at the beginning of the disease spread just one node will be sick. so some nodes have just one infected neighbor and the remaining ones does not have any infected neighbors. so we have or → =1, 1>1-q > 1-th →th> 1 →th > → =1-q > 1-th →th> 1 →th > therefore, node i doesn’t become infected. it means that if th ≥ q at the beginning of the spread of disease, the infection does not transmit from node i to any node. in r model, we observe that (see fig.3.b): if th ≥ 0.5: no node will be infected. if th < 0.5: some persons will be infected. if th = 0: all the nodes will be infected. in this model as th decreases, the probability of becoming infected will increase. in our data, the weights of links are between 0 and 10. in the beginning, there is only one infected node which its maximum weight of connected links is 5. according to eq. (5) advances in systems science and application(2016) vol.16 no.4 97 ( ), if th ≥ 0.5, the infection can’t transmit from first infected node to its neighbors. in cw model, we observe that (see fig.3.c): if th ≥ q: some persons will be infected. if th < q: all the nodes will be infected. in an overview, when th≥ q in ak model no one became sick, but in cw model some persons became sick. as cw model considers the weight of links and so it is more impressible than ak model. in addition to aforementioned factors, the characteristics of individuals are not the same and people may have different body resistance against the disease, i.e. each person may have a particular threshold of becoming sick (th). for example, infection risks may be related to age. previously hethcote suggested that in realistic infectious disease models it would be beneficial to include the age of individuals [28]. here we allocated different quantities for the resistance body from 0 to 1 based on the persons’ age (see fig.4 and fig.5). for the r model, the number of infected individuals in fig.4 are fewer than it in fig.2, since in our network data most of the people are young and they have high th (compare fig.2 with fig.4). and as mentioned before, the r model depends on th very much. comparing the number of infected persons in our simulated models with real data suggests that the ak and the cw showed more similarity to real result (see fig.5). however, the cw results have more fluctuations similar to real data which is as a result of considering both q and w. 5 conclusions and future prospect modeling transmission of infectious diseases in a society is one the most important field in epidemiology science. here, we have attempted to model the transmission of sars disease in a small population with three different models. based on our results, weights of communications between individuals and the thresholds of body resistances of people are important factors for disease spread. the cw model which has all these factors is more similar to the reported data of infected cases of sars outbreak in hong kong in 2003 [20]. based on these results it may be concluded that the cw captures the dynamic of disease spread in a proper way. as suggestions for the future prospect, it will be useful to model the vaccination effect of the person who has more connections with other people in a society, since in other studies it has expressed that the prevalence of the disease depends on the degree distribution [2]. also, it has shown that quarantining infected persons can reduce the infectious in a society [16]. considering this state in simulated models will be interesting. we believe the model can be applied to another contagious diseases such as ebola, influenza, aids and etc., because these diseases have some similar features. for example, ebola and sars have the analogous incubation period. acknowledgment we want to thank dr.sajad jafari for useful guidance and improvements. references [1] f. r. vishkaie, f. bakouie, and s. gharibzadeh(2014), "common cold outbreaks: a network theory approach," communications in nonlinear science and numerical simulation, vol. 19, pp. 3994-4002. 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[26] c. kamp, m. moslonka-lefebvre, and s. alizon(2013), "epidemic spread on weighted networks," plos comput biol, vol. 9. [27] j. stehlé, n. voirin, a. barrat, c. cattuto, l. isella, j.-f. pinton, et al.(2011), "highresolution measurements of face-to-face contact patterns in a primary school," plos one, vol. 6. [28] h. w. hethcote(2000), "the mathematics of infectious diseases," siam review, vol. 42, pp. 599-653. corresponding author f. bakouie can be contacted at: f_bakouie@sbu.ac.ir. mailto:f_bakouie@sbu.ac.ir advances in systems science and applications (2014) vol.14 no.1 22-35 optimizing investment decisions for a set of projects under uncetainty of future returns konstantin n. nechval1 and nicholas a. nechval2 1applied mathematics department, transport and telecommunication institute lomonosov street 1, lv-1019, riga, latvia 2statistics department, evf research institute, university of latvia raina blvd 19, lv-1050, riga, latvia abstract project portfolio selection is a crucial decision in many organizations, which must make informed decisions on investment, where the appropriate distribution of investment is complex, due to varying levels of risk, resource requirements, and interaction among the proposed projects. in this paper, we present a mathematical model of investment for a set of projects under uncertainty of future returns. the model addresses the problem of an investor with access to a limited pool of capital, who makes decisions on investments. the problem is to decide how much to invest in each project so as to maximize the total expected return by the end of the horizon in relation to a given utility function. we discuss optimal investment decisions for the cases where the return from investment is a random variable. the single-period and multi period cases of investment decisions are considered. the paper presents closed form solutions for commonly adopted utility functions. keywords projects, future return, uncertainty, investment decisions, optimization 1 introduction project portfolio investment is the periodic activity involved in investing a portfolio of projects, that meets an organizations stated objectives without exceeding available resources or violating other constraints. some of the issues that have to be addressed in this process are the organization’s objectives and priorities, financial benefits, intangible benefits, availability of resources, and risk level of the project portfolio [1]. difficulties associated with project portfolio investment result from several factors: (i) there are multiple and often-conflicting objectives, (ii) some of the objectives may be qualitative, (iii) uncertainty and risk can affect projects, (iv) the project portfolio may need to be balanced in terms of important factors, such as risk and time to completion, (v) some projects may be interdependent, and (vi) the number of feasible portfolios is often enormous. in addition to these difficulties, due to resource limitations there are usually constraints such as finance, work force, and facilities or equipment, to be considered. as some researchers have noted [2], the major reason why some projects are selected but not completed is that resource limitations are not always formally advances in systems science and applications (2014) vol.14 no.1 23 included in the project selection process. in cases where resource limitations are at fault for a failed project, a selection model that incorporated resource limitations could have aided the decision maker in avoiding such mistakes [1]. portfolio selection becomes more complex when resource availability and consumption are not uniform over time. there are many different techniques that can be used to estimate, evaluate, and choose project portfolios [3-4]. some of these techniques are not widely used because they address only some of the above issues, they are too complex and require too much input data, they may be too difficult for decision makers to understand and use, or they may not be used in the form of an organized process [5]. among all of the techniques that are available, optimization techniques are the most fundamental quantitative tool for project portfolio selection and address most of the important issues [6]. however, they have largely failed to gain user acceptance [7], and few modeling approaches, from a variety of optimization approaches that have been developed, are being utilized as aids to decision making in this area [8]. according to hess “management science has failed altogether to implement project selection models; we have proposed more and more sophistication with less and less practical impact” [9]. one of the major reasons for the failure of traditional optimization techniques is that they prescribe solutions to portfolio selection problems without allowing for the judgment, experience and insight of the decision-maker [7]. a literature review we conducted in this field clearly showed that, although there are many different methods for project evaluation and portfolio selection that have their own advantages, no single technique addresses all of the issues that should be considered in project portfolio selection [10]. among published methodologies for project portfolio selection, there has been little progress towards achieving an integrated framework that: (a) simultaneously considers all the different criteria in determining the most suitable project portfolio, (b) takes advantage of the best characteristics of existing methods by decomposing the process into a flexible and logical series of activities and applying the most appropriate technique(s) at each stage. well-known pragmatic difficulties that make project selection challenging include the following: (1) success uncertainty (unknown rewards): whether (or how well) a project will succeed, both technically and in the market, may be uncertain. (2) changing opportunities (randomly arriving opportunities): the opportunity of funding a project may be uncertain. new projects are continually being proposed throughout the year, stimulated by changes in technology and market opportunities that might help the company to achieve its goal. (3) need for quick funding decisions (on-line decision): the organizations proposing projects lack a comprehensive view of all projects activities company-wide. it is essential 24 konstantin n. nechval: optimizing investment decisions for a set of projects... for top management to give the organization prompt accept/revise/reject decisions, since such feedback helps to coordinate their activities. (4) combinatorial complexity: the costs and benefits from different projects may interact. 2 problem statement the problem we will examine can be formulated as follows. let w0 denote the initial wealth (measured in monetary units) of the investor and assume that there are m projects (opportunities) for investments which belong to a set of m risk categories, with corresponding random rates of return r1, r2, ..., rm, among which the investor can allocate his wealth. the investor can also invest in a project of a riskless category offering a sure rate of return s. if we denote by u1, ..., um the corresponding amounts to invest in the projects belonging to the set of m risk categories and by (w0 − u1 −λ− um) the amount of investment in the project of the riskless category, the final wealth is given by w1 = s(w0 − u1 − · · · − um) + m∑ j=1 rjuj , (1) or equivalently w1 = sw0 + m∑ j=1 (rj − s)uj . (2) the objective is to maximize over u1, ..., um, e{g(ω1)} (3) where g is a known utility function for the investor, w1 is a random variable. we assume that the given expected value is well defined and finite for all w0, ui, and that g is concave and twice continuously differentiable. we will not impose constraints on u1, ..., um. this is necessary in order to obtain the results in convenient form. a few additional assumptions will be made later. if a given amount w0 of the initial wealth is available for investment within n time periods, we are interested in determining decision rules on how much to invest in each project so as to maximize the total expected return by the end of the horizon in relation to a given utility function. 3 investment decisions for the single case let us consider the preceding problem for every value of initial wealth and denote by µ∗ j = µ∗ j (ω0), j = 1, ...,m, the optimal amounts to invest in projects belonging to the set of m risk categories when the initial wealth is w0. we say that the investment portfolio {µ1 j (ω0), ..., µ m j (ω0)} is partially separated if u∗j (w0) = cjh(w0), j = 1,...,m (4) advances in systems science and applications (2014) vol.14 no.1 25 where cj , j = 1, ...,m, are fixed constants and h(w0) is a function of w0 (which is the same for all j). when partial separation holds, the ratios of amounts invested in projects belonging to the set of m risk categories are fixed and independent of the initial wealth; that is, u∗j (w0) u∗k(w0) = cj ck , forj, k ∈ {1, ...,m}, ck ̸= 0 (5) actually, in the cases we will examine, when partial separation holds, the investment portfolio {µ1 j (ω0), ..., µ m j (ω0)} will be shown to consist of affine (linear plus constant) functions of w0 that have the form u∗j (w0) = cj [a+ bsw0], j = 1,...,m (6) where a and b are constants characterizing the utility function g. in the special case where a=0 in (6), we say that the optimal investment portfolio is completely separated in the sense that the ratios of the amounts invested in projects belonging to both the set of m risk categories and the riskless category are fixed and independent of initial wealth. here the following theorem holds. theorem 1. if the utility function satisfies − g′(w1) g′′(w1) = a+ bw1, forallw1 (7) where g′ and g′′ denote the first and second derivatives of g, respectively, and a and b are some scalars, then the optimal investment portfolio is given by (6). furthermore, if g(w0) is the optimal value of the problem, i.e., g(w0) = max u1,...,um e{g(w1)} (8) then we have −g′(w0) g′′(w0) = a s + bw0, forallw0 (9) proof. let us assume that an optimal investment portfolio exists and is of the form u∗j (w0) = cj(w0)[a+ bsw0], j = 1,...,m (10) where cj(w0), j = 1, ...,m, are some differentiable functions. we will prove that dcj(w0)/dw0 = 0 for all w0 and hence the functions cj must be constant. we have for every w0, by the optimality of µ∗ j (ω0) for j = 1, ...,m, de{g(w1)} duj = e { g′ [ sw0 + m∑ k=1 (rk − s)ck(w0)(a+ bsw0) ] (rj − s) } = 0 26 konstantin n. nechval: optimizing investment decisions for a set of projects... j = 1(1)m (11) differentiating the m equations in (11) with respect to w0 yields e   (r1 − s)2 · · · (r1 − s)(rm − s) · · · · · · · · · · · · (rm − s)(r1 − s) · · · (rm − s)2  g′′(w1)(a+ bsw0) ×  dc1(w0)/dw0 ... dcm(w0)/dw0  = −  e { g′′(w1)(r1 − s)s [ 1 + m∑ k=1 (rk − s)ck(w0)b ]} ... e { g′′(w1)(rm − s)s [ 1 + m∑ k=1 (rk − s)ck(w0)b ]}  = −  e { g′′(w1)(r1 − s)s [ 1 + m∑ j=1 (rj − s)cj(w0)b ]} ... e { g′′(w1)(rm − s)s [ 1 + m∑ j=1 (rj − s)cj(w0)b ]}  . (12) using relation (7), we have g′′(w1) = − g′(w1) a+ b [ sw0 + m∑ j=1 (rj − s)cj(w0)(a+ bsw0) ] = − g′(w1) (a+ bsw0) [ 1 + m∑ j=1 (rj − s)cj(w0)b ] . (13) substituting in (12) and using (11), we have that the right side of (12) is the zero vector. the matrix on the left in (12), except for degenerate cases, can be shown to be nonsingular. assuming that it is indeed nonsingular, we obtain dcj(w0) dw0 = 0, j = 1,...,m (14) advances in systems science and applications (2014) vol.14 no.1 27 and cj(w0) = cj , where cj are some constants, thus proving (6). we now turn our attention to proving relation (9). we have g(w0) = e{g(w1)} = e g s 1 + m∑ j=1 (rj − s)cjb w0 + m∑ j=1 (rj − s)cja  (15) and hence g′(w0) = e g′(w1)s 1 + m∑ j=1 (rj − s)cjb  (16) g′′(w0) = e g′′(w1)s 2 1 + m∑ j=1 (rj − s)cjb 2 (17) the last relation after some calculation and using (13) yields g′′(w0) = − e { g′(w1)s [ 1 + m∑ j=1 (rj − s)cjb ]} s a+ bsw0 (18) by combining (16) and (18), we obtain the desired result: −g′(w0) g′′(w0) = a s + bw0 (19) this ends the proof. it can be shown that the following utility functions satisfy (19): exponential: −e−w/a, forb = 0 (20) logarithmic ln(ω + a), forb = 1 (21) power [1/(b− 1)](a+ bω)1−(1/b), otherwise (22) naturally in our problem only concave utility functions from this class are admissible. furthermore, if a utility function that is not defined over the whole real line is used, the problem should be formulated in a way that ensures that all possible values of the resulting final wealth are within the domain of definition of the utility function. 28 konstantin n. nechval: optimizing investment decisions for a set of projects... 4 investment decisions for the multiperiod case it is now easy to extend the one-period result of the preceding analysis to the multiperiod case. we will assume that the current wealth can be used to invest amounts in projects at the beginning of each of n consecutive time periods. if we denote: wυ the wealth of the investor at the beginning of the υth period, uj(υ) the amount to invest in project belonging to the jth risk category at the beginning of the υth period, rj(υ) the rate of return of project which belongs to the jth risk category at the beginning of the υth period, sυ the rate of return of project which belongs to the riskless category at the beginning of the υth period then we have (in accordance with the single-period model) the system equation wν+1 = sνwν + m∑ j=1 (rj(ν) − sν)uj(ν),v = 0, 1, ..., n − 1 (23) we assume that the vectors rυ = (r1(υ), ..., rm(υ)), υ = 0, ..., n − 1, are independent with given probability distributions that result in finite expected values throughout the following analysis. the objective is to maximize e{g(wn )}, the expected utility of the terminal wealth wn , where we assume that g satisfies for all w − g′(w) g′′(w) = a+ bw (24) applying the dynamic programming algorithm to this problem [11-12] , we have gn (ωn ) = g(ωn ) (25) gν(wν) = max u1(ν),...,um(ν) e gν+1 sνwν + m∑ j=1 (rj(ν) − sν)uj(ν)  , ν = 0, 1, ..., n−1 (26) from the solution of the one-period problem we have that the optimal policy at the beginning of period n − 1 is of the form u∗n−1(wn−1) = cn−1[a+ bsn−1wn−1], (27) where cn−1 is an appropriate m-dimensional vector, µ∗ n−1 = [µ∗ 1(n−1), ..., µ ∗ m(n−1)] ′. furthermore, we have −g′ n−1(w) g′′ n−1(w) = a sn−1 + bw. (28) advances in systems science and applications (2014) vol.14 no.1 29 hence, applying the result of this section in (26) for the next to the last period, we obtain the optimal policy u∗n−2(wn−2) = cn−2 ( a sn−1 + bsn−2wn−2 ) (29) where cn−2 is again an appropriate m-dimensional vector. proceedings similarly, we have for the υth period u∗ν(wν) = cν ( a sn−1 · · · sν+1 + bsνwν ) (30) where cυ, υ = 0, 1, ..., n − 1, are m-dimensional vectors that depend on the probability distributions of the rates of return rj(υ) of projects belonging to the risk categories and are determined by optimization of the expected value of the optimal cost-to-go functions gυ. these functions satisfy −g′ ν(w) g′′ ν(w) = a sn−1 · · · sν + bw, ν = 0, 1, ..., n − 1 (31) thus one can see that the investor, when faced with the opportunity to reuse sequentially his wealth, uses a policy similar to that of the single-period case. 5 multiperiod project portfolio investment with markowitz mean-variance optimization to our knowledge, no analytical or efficient numerical method for finding the optimal multiperiod portfolio policy for the constrained mean-variance formulation of markowitz has been reported in the literature. this section presents an optimal solution to the constrained mean-variance formulation of the multiperiod project portfolio investment problem. we consider a portfolio with (m + 1) risky projects, with random rates of returns. let w0 be an initial wealth of an investor at time 0. the investor can allocate his wealth among the (m + 1) projects. the wealth can be reallocated among the (m + 1) projects at the beginning of each of the following (t − 1) consecutive time periods. the rates of return of the risky projects at time period τ within the planning horizon are denoted by a vector rτ = [rτ(0), rτ(1), ..., rτ(m)], where rτ(j) is the random return for project j at time period τ . it is assumed in this paper that vectors rτ , = 0, 1, ..., t − 1, are statistically independent and return rτ has a mean e{rτ} = [e{rτ(0)}, e{rτ(1)}, ..., e{rτ(m)}]′ and a covariance cov{rτ} =  στ(00) · · · στ(0m) ... . . . ... στ(0m) · · · στ(mm)  (32) 30 konstantin n. nechval: optimizing investment decisions for a set of projects... which can be found from the data of observations. let wτ be the wealth of the investor at the beginning of the τth period, and let uτ(j), j ∈ {1, ...,m}, be the amount invested in the jth risky project at the beginning of the τth time period. the amount investigated in the 0th risky project at the beginning of the τth time period is equal to uτ(0) = wτ − m∑ j=1 uτ(j). (33) an investor is seeking a best multiperiod investment strategy, (uτ(0), uτ(1), ..., uτ(m)) for τ = 0, 1, 2, ..., t − 1, such that either (i) the expected value of the terminal wealth wt , e{wt }, is maximized if the variance of the terminal wealth, v ar{wt }, is not greater than a preassigned risk level v∗, or (ii) the variance of the terminal wealth, v ar{wt }, is minimized if the expected terminal wealth, v ar{wt }, is not smaller than a preassigned level e∗. mathematically, a meanvariance formulation for multiperiod project portfolio investment can be posed as one of the following two forms: (i) maximize v ar{wt } (34) subject to v ar{wt } ≤ ν∗ (35) m∑ j=1 uτ(j) ≤ e{wτ} (36) µτ(j) ≥ 0, j = 1, ...,m, τ = 0, 1, 2, .., t − 1 (37) with wτ+1 = m∑ j=1 rτ(j)uτ(j) + wτ − m∑ j=1 uτ(j)  rτ(0) = rτ(0)wτ +∆′ τuτ (38) and (ii) minimize v ar{wt } (39) subject to e{ωt } ≥ e∗ (40) m∑ j=1 uτ(j) ≤ e{wτ} (41) µτ(i) ≥ 0, j = 1, ...,m, τ = 0, 1, 2, .., t − 1 (42) advances in systems science and applications (2014) vol.14 no.1 31 with wτ+1 = m∑ j=1 rτ(j)uτ(j) + wτ − m∑ j=1 uτ(j)  rτ(0) = rτ(0)wτ +∆′ τuτ , (43) where ∆τ = [∆τ(1),∆τ(2), ...,∆τ(m)] ′ = [rτ(1) − rτ(0), rτ(2) − rτ(0), ...,rτ(m) − rτ(0)] ′ (44) formulation (i) or (ii) enables an investor to specify a risk level he can afford when he is seeking to maximize his expected terminal wealth or specify an expected terminal wealth he would like to achieve when he is seeking to minimize the corresponding risk. a strategy of multiperiod project portfolio investment is an investment sequence, ut = [u0, u1, u2, ..., un−1] (45) where uτ = [uτ1 , uτ2 , ..., uτm ] ′, ∀τ = 0(1)t − 1 (46) more specifically, ut is a feedback strategy and uτ maps the wealth at the beginning of the τth period, wτ , into a project portfolio decision in the τth period, i.e., uτ = uτ (wτ ). a multiperiod project portfolio investment strategy, u∗ t , is said to be efficient if there exists no other portfolio one, ut , such that e{wt ;ut } ≥ e{wt ;u ∗ t } and v ar{wt ;ut } ≤ v ar{wt ;u ∗ t } with at least one equality strictly. by varying the value of v∗ in (i) or the value of e∗ in (ii), the set of efficient multiperiod project portfolio investment strategies can be generated. an equivalent formulation to either (i) or (ii) in generating efficient multiperiod portfolio strategies is (iii) maximize e{wt } − ϑv ar{wt } (47) subject to m∑ j=1 uτ(j) ≤ e{wτ} (48) uτj ≥ 0, j = 1, ...,m, τ = 0, 1, 2, .., t − 1 (49) with wτ+1 = m∑ j=1 rτ(j)uτ(j) + wτ − m∑ j=1 uτ(j)  rτ(0) = rτ(0)wτ +∆′ τuτ (50) 32 konstantin n. nechval: optimizing investment decisions for a set of projects... table 1 the observations of returns for three risky projects. year(t) project 0 1 2 1 1.3000 1.2250 1.1490 2 1.1030 1.2900 1.2600 3 1.2160 1.2160 1.4190 4 0.9540 0.7280 0.9220 5 0.9290 1.1440 1.1690 6 1.0560 1.1070 0.9650 7 1.0380 1.3210 1.1330 8 1.0890 1.3050 1.7320 9 1.0900 1.1950 1.0210 10 1.0830 1.3900 1.1310 11 1.0350 0.9280 1.0060 12 1.1760 1.7150 1.9080 where ϑ ∈ [0,∞). it will be noted that if u∗ t } solves (iii), then u∗ t } solves (i) with v∗ = v arwt ;u∗ t , and solves (ii) with e∗ = ewt ;u∗ t . note that ϑ is equal to ∂e{wt }/∂v ar{wt } at the optimal solution of (iii). problem formulation (iii) is preferable to be adopted in investment situations where an investor is able to specify his desirable trade-off between the expected terminal wealth and the associated risk. 6 numerical example consider the case of a stationary multiperiod process with t = 2. an investor has one unit of wealth at the very beginning of the planning horizon, i.e., w0 = 1. the investor is trying to find the best allocation of his wealth among three risky projects, 0, 1, and 2 in order to maximize e{w2} while keeping his risk not exceeding 0.07; that is, v∗ = 0.07. the observations of returns for risky projects, 0, 1, and 2 are given in table 1. at first consider the case of a stationary multiperiod process with t = 1. the problem is to maximize e{ω1} (51) subject to v ar{ω1} ≤ ν∗ = 0.07 (52) 2∑ j=1 u0(j) ≤ e{w0} = w0 = 1 (53) advances in systems science and applications (2014) vol.14 no.1 33 u0(j) ≥ 0, j = 1, 2, ... (54) with w1 = 2∑ j=1 r0(j)u0(j) + w0 − 2∑ j=1 u0(j)  r0(0) = r0(0)w0 +∆′ 0u0. (55) using (55) and table 1, it can be shown that e{w1} = e{r0(0)}w0 + e{∆′ 0}u0, (56) where e{r0(0)} = 1.0891, e{∆′ 0} = [0.1246, 0.1455] (57) v ar{ω1} = [ω0, u ′ 0]cov{[r0(0),△′ 0] ′}[ω0, u ′ 0] ′ (58) with cov{[r0(0),∆′ 0] ′} =  0.0099 0.0015 0.0021 0.0015 0.0407 0.0374 0.0021 0.0374 0.0723  (59) using solver software (ms excel), we obtain from maximization of (51) a percent investment (on w0 = 1) in three risky projects, 0, 1, and 2, at period τ = 0 as follows: µ∗ 0(0) = 0%, µ∗ 0(1) = 26.92%, and µ∗ 0(2) = 73.08%. the corresponding expected terminal wealth and the risk level are given by e{w1} = 1.229 and v ar{w1} = 0.07, respectively. now consider the case of a stationary multiperiod process with t = 2. the problem is to maximize e{w2} (60) subject to v ar{ω2} ≤ ν∗ = 0.07 (61) 2∑ j=1 u0(j) ≤ e{w0} = w0 = 1 (62) 2∑ j=1 u1(j) ≤ e{w1} (63) uτ(j) ≥ 0, j = 1, 2, τ = 0, 1 (64) with wτ+1 = m∑ j=1 rτ(j)uτ(j) + wτ − m∑ j=1 uτ(j)  rτ(0) = rτ(0)wτ +∆′ τuτ (65) 34 konstantin n. nechval: optimizing investment decisions for a set of projects... using (65) and table 1, it can be shown that e{w2} = e{r1(0)r0(0)}w0 + e{r1(0)∆′ 0}u0 + e{∆′ 1}u1 (66) where e{r1(0)r0(0)} = 1.1960 (67) e{r1(0)∆′ 0} = [0.1371, 0.1605] (68) e{∆′ 1} = [0.1246, 0.1455] (69) v ar{w2} = [w0, u ′ 0, u ′ 1]× cov{[r1(0)r0(0), r1(0)∆′ 0,∆ ′ 1] ′}[w0, u ′ 0, u ′ 1] (70) where cov{[r1(0)r0(0), r1(0)∆′ 0,∆ ′ 1] ′} =  0.0491 0.0028 0.0053 0.0022 0.0037 0.0028 0.0496 0.0490 0.0447 0.0427 0.0053 0.0490 0.0942 0.0426 0.0823 0.0022 0.0447 0.0426 0.0407 0.0374 0.0037 0.0427 0.0823 0.0374 0.0723  (71) using solver software (ms excel), we obtain from maximization of (60) a percent investment (on w0 = 1) in three risky projects, 0, 1, and 2, at period τ = 0 as follows: µ∗ 0(0) = 100%, µ∗ 0(1) = 0%, and µ∗ 0(2) = 0%; and a percent investment (on e{w1} = 1.0891) in three risky projects, 0, 1, and 2, at period τ = 1 as follows: µ∗ 1(0) = 39.80%, µ∗ 1(1) = 47.24% and µ∗ 1(2) = 12.96%. the corresponding expected terminal wealth and the risk level are given by e{w2} = 1.2807 and v ar{w2} = 0.07, respectively. 7 conclusion the problem considered in this paper is to decide how much of our available resources to invest in each project so as to maximize the total expected return by the end of the horizon in relation to a given utility function. we formulate the problem in terms of dynamic programming, which allows one to obtain optimal investment decisions for a set of projects under uncertainty of future returns in a simple form. the derived optimal multiperiod project portfolio investment strategy provides investors with the best strategy to follow in a dynamic investment environment. acknowledgments this research was supported, in part, by the latvian council of science and the national institute of mathematics and informatics of latvia under grant no. 06.1936 and grant no. 01.0031. this support is gratefully acknowledged. advances in systems science and applications (2014) vol.14 no.1 35 references [1] schniederjans, m. and santhanam, r. 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(1996), “portfolio selection techniques: a review and a suggested integrated approach”, innovation research center, working paper 46, school of business, mcmaster university, hamilton. corresponding author konstantin n. nechval can be contacted at: konstan@tsi.lv advances in systems science and application (2015) vol.15 no.1 72-89 trying to evaluate human dignity in a social group antonio caselles spanish society of general systems abstract this study attempts to make progress in the way an instrument can be created, by consensus, to monitor the development of human beings quality of life in all aspects. based on various recent studies into human values, quality of life and subjective well-being, and on the universal declaration of human rights, this study takes human dignity as the supreme value, and development, freedom and equality (with solidarity, justice and peace as the subsidiary values) as the subsidiary values. all these values were disaggregated hierarchically by considering the literature on this matter to obtain measurable variables. they were converted into indices and geometric averages were used to aggregate the considered variables into each level. obviously, this is an initial attempt, and the definitive instrument will need more in-depth studies and ampler consensuses. keywords human values; subjective well-being; quality of life; universal declaration of human rights. 1 introduction from immemorial times, and without going into much detail, philosophers in particular, and human beings in general, have attempted to determine and explain what they need to survive and to obtain satisfactory conditions of life, both individually and also in their social group. in the present-day, and taking advantage of new technologies, we can aspire to find a procedure to give a numerical value to these conditions in a given social group. that is, to devise a system that evaluates human organizations, and not just individuals (or perhaps it does). to this end, it would be necessary to first determine the partial objectives to be reached by asking: what is well? do objectively right social conduct norms exist? if so, how can their degree of implantation and effectiveness be evaluated? evidently this it is the field of ethics. the aim is to find what is good. yet “good” seems a relative thing (“one man’s meat is another man’s poison”, “it never rains to everyone’s taste”, are well-known sayings). for this reason, what the common good actually is must be determined by consensus, and must represent the common desires or preferences in a social group. the larger social group is humanity as a whole (at the moment), and we have the united nations (un), which issued the universal declaration of human rights (udhr) in 1948. this is the first manifestation of the existence of a global consensus on “what is well” that we know. in the udhr, “human dignity” is determined implicitly as the upper rank value, and its subordinated values are also determined [1]. psychology studies the so-called “subjective well-being”, which we could consider advances in systems science and application (2015) vol.15 no.1 73 the equivalent to happiness. some authors, e.g., khaneman and krueger, have attempted to evaluate a group’s happiness by averaging the happiness of their component individuals (determined through questionnaires)[2]. other authors, e.g., parra-luna, have proposed their own scale of values and suggested a way to evaluate it[3]. regarding the philosophical treatment of udhr, the book of moncho is most interesting[1]. from the abundant literature on the matter we have selected, more or less rightly, these two traditional approaches based on (a) human dignity, and (b) subjective happiness, since economic well-being is considered in the udhr to be a subsidiary objective of human dignity. therefore, first we set out to analyze the human dignity concept from the udhr perspective by hierarchically disaggregating its components until we obtain the directly measurable components; second, we do this to suggest a way to evaluate all these lower level components; third, we do this to confer a value to the degree of respect to human dignity in a given social group (city, country, region, world, etc.) from these values by means of mathematical formulas and with a staggered aggregation process. finally, we analyze the happiness or subjective well-being concept, and attempt to find aspects or factors not contemplated in the udhr that can affect human dignity; e.g., environmental caring and sustainability. we attempt to integrate them into this hierarchy as a suggestion for future consensus. 2 human dignity from the udhr analysis and following moncho [1], we deduce that human dignity is considered here to be the supreme value or the upper rank value, and development, freedom and equality to be immediate the subordinated values. solidarity, justice and peace are subordinated to equality (which acts as justification for them). in order to establish common criteria, we attempt to summarize definitions for these seven basic concepts: dignity: the equivalent to being a “person”; that is to say, subject of operations, and not a “thing”; that is to say, an object or usable instrument. this definition assumes that self-conscience and reason exist in a person. development: survival and self-fulfillment options, which include: life/ health, social progress (education, culture, etc.) and standard of life (economic resources, comforts, etc.). freedom: no restrictions to self-fulfillment would be the total freedom which, obviously in a group, must be limited by the dignity of the other group members. equality: non discrimination to face opportunities and rights, and obviously with the limits determined by the social group’s resources. 74 antonio caselles: trying to evaluate human dignity in a social group solidarity: considered synonymous of brotherhood; that is to say, mutual aid. justice: mechanisms of prevention, protection and compensation for individuals or groups to face possible damage or benefits. peace: absence of violence, coercion and fear. next we attempt to not only identify articles on the udhr which explicitly mention the diverse components or ingredients of human dignity existing, but to also locate other possible components in the literature that are not explicitly mentioned in the present udhr, but could perhaps be included in the future according to our criterion. 2.1 development for this aspect, the udhr includes: health, education, sufficient rent and free time. health: includes feeding, dress, house and health care (article 25-1). education: refers to knowledge and aptitudes, and emphasizes the following subsidiary values: ◦ in the state (article 26-1) • free-of-charge and obligatory nature of elementary and fundamental education. • generalization for technical and professional education. • free access through merits to higher education. ◦ like objective (article 26-2) • respect human rights and fundamental liberties. • understanding, tolerance and friendship. • peace-keeping. • values that do not explicitly appear in the udhr, like education objectives (perhaps they are implicit in the three previous ones), but frequently appear in the literature, like desirable: courage, love, joy, calm, prudence, respect of opinions and other people’s customs, communication and cooperation, self-control, self-knowledge, self-acceptance, self-esteem, flexibility, loyalty, integrity/honesty (word), self-discipline, honor the elderly and parents, empathy, generosity, efficiency (yield rate, rapidity, success, social recognition), exploration, reliability, environmental respect, pleasure, responsibility (in case of failure), and to have objectives that go beyond ones own person (see for instance: [4-11]). advances in systems science and application (2015) vol.15 no.1 75 sufficient income: enough to ensure health and family well-being (article 25-1). free time: includes rest, leisure and paid vacations (article 24). sustainability and environment care do not appear explicitly in the udhr. 2.2 freedom it includes the following as subsidiary values: opinion and publication by any means (article 19). choice of partner (in marriage) (article 16). pacific meeting (article 20). pacific association (article 20). choice of work (article 23). choice of asylum (not for common or anti-un crimes) (article 14). displacement (in the territory of a state) (article 13-1). to leave any country and to return to ones own country (article 13-2). trade unions (to endow and/or to affiliate) (article 23-4). choice of childrens type of education (article 26-3). thought, conscience and religion (aims and values) (article 18). ◦ to change ones own religion or beliefs. ◦ to show ones own religion or beliefs (individual and collectively, publicly and privately, education, cult and observance). access to public functions (article 21). legislation: indirectly by means of electing legislators and governors, or directly (article 21-1). vote (article 21-3). the following do not appear explicitly: right to strike, freedom to hire (market), sexual freedom (with consent and fidelity). 2.3 equality the udhr understands equality to be equality in rights and liberties (article 2) and facing the law (article 7). subsidiary values of equality are solidarity, justice and peace because equality justifies solidarity as both justice and peace [1]. 76 antonio caselles: trying to evaluate human dignity in a social group solidarity ◦ non discrimination: by race, color, sex, language, opinion, national or social origin, economic position, birth, etc. (article 2). ◦ same rights: economic, social, cultural and social security (article 22). it clarifies by stating that it refers to rights that are indispensable to people’s dignity, them free developing their personality, and always limited by the resources of the state itself and international cooperation. • right to work: under equitable, satisfactory conditions (article 23-1). • the same wage for the same work (article 23-2). • security and social welfare: insurances for disease, unemployment, widowhood, disability, old age and involuntary loss of means of subsistence (articles 23-1, 25-1 and 26). ◦ the solidarity mechanisms that are not explicit in the udhr: • urgent palliative for ensuring survival. • reintegration mechanisms: for workers and delinquents. • welfare aid for people with difficulties. justice ◦ protection by law (article 7). ◦ appeal to courts (article 8). ◦ presumption of innocence and guarantees of defense (article 11-1). ◦ non retroactivity of laws (article 11-2). ◦ right of property (article 17). ◦ protection of author rights (article 27-2). ◦ protection of human rights (article 28). ◦ duties to the community (article 29-1). ◦ respect of others rights and liberties (article 29-2). ◦ no opposition to the general un principles (article 29-3). ◦ no contradiction (article 30). ◦ effectiveness and efficiency of justice are not explicitly considered. peace: no coercion, no violence, no fear, no misery. ◦ prohibition of slavery and servitude (article 4). advances in systems science and application (2015) vol.15 no.1 77 ◦ prohibition of torture and cruel and/or degrading treatment (article 5). ◦ right to legal personality (article 6). ◦ prohibition of arbitrariness: in cases of detention, prison and exile (article 9). ◦ right to being heard publicly and with justice by an impartial court (article 10). ◦ right to having a nationality and being able to change it (article 15). ◦ right to participating in the scientific progress and in artistic and cultural activities (article 27-1). 3 subjective well-being, happiness or satisfaction with one’s own life for our purpose, these three concepts are considered synonymous. as our objective in this section is to find the factors that influence subjective well-being, and this objective agrees with that in the recent publication of vijayamohanan and asalatha[5], for details we recommend reading this publication. from the literature review that these authors did, we emphasize the following factors that determine human happiness: (a) material conditions and consumption (income, unemployment, inequality, inflation, free time, etc.). (b) satisfactory family life (partner, children, relatives, etc.). (c) personal and family health. (d) satisfaction in the workplace. (e) ones own character or personality. (f) environmental, socio-demographic or institutional factors (community life, friends, liberties, activities, social control, religion/values, etc.). according to these authors, or those mentioned by them, the main factor that influences individual happiness is one’s own character or personality (determined by genetic and environmental factors), followed by health. money seems to have less influence than what people think, mainly from a minimum. sex and age seem to have little influence and depend on certain aspects. influence of marriage is different for men than it is for women. the united nations development program distinguishes between well-being and happiness. the well-being components are health, work and standard of life. the components of happiness are: life with a purpose, receiving respectful 78 antonio caselles: trying to evaluate human dignity in a social group treatment and having a network of social support[12]. the evaluation of these components is made by means of interviews, which determine the percentage of people who state having these components. in order to measure the degree of personal happiness kahneman and krueger propose a scale of adjectives (happy, enjoying, pleasant, depressed, angry and frustrated) and a questionnaire to determine the distribution of the time a person spends between pleasant and disagreeable situations[2]. it distinguishes 19 possible situations in daily life; e.g. intimate relationships, meeting people after work, eating supper, relaxing, eating, exercising, etc.. later it proposes a formula to average the happiness of the individuals in a social group (sum of the degree-of-satisfaction and time-in-the-situation products of different individuals in various situations). as observed, this approach does not attempt to analyze the causes of happiness, but to evaluate happiness as the result. perhaps it would be necessary to ask with questionnaires because of each answer, and to thus attempt to reach the corresponding cause. the literature includes several questionnaires that have been devised to determine an individual’s level of satisfaction with his/her life. a very popular one is the oxford questionnaire[9], although it has received severe critics[13]. to conclude our literature review of happiness causes, we state that we have not found anything new that is objectively measurable to be incorporated into calculations at the human dignity level. therefore, if what is more influential seems to indicate respect to a person’s happiness is his/her own character, and the other conditions have already been considered between the factors that most influence his/her human dignity, we agree with most of the opinions voiced in the literature that the best way to determine a person’s degree of happiness is to directly ask him/her, and with more or less disaggregation according to the questionnaire used. 4 other scales to measure the value of a society or group of people among the more recent studies on this subject, we selected the following for them possibly leading, more or less quickly, to a numerical global evaluation of a society or a group of people which are, at the same time, a source of ideas or suggestions for our purpose. an approach to the subject in more detail is found in the work of parra-luna[3]. this author distinguishes nine groups of values: health, wealth, security, knowledge, freedom, justice, conservation of the environment, quality of activities, prestige. each group is disaggregated into its respective components to obtain 84 measurable lower level ones. the proposed averaging formula is arithmetic. thus, for example, the health value is disaggregated as so: (a) life expectancy, which includes that of 1-year-old children, mortality at the age of 1 year and advances in systems science and application (2015) vol.15 no.1 79 mortality at the age of 50; (b) quality of life, which includes days not worked due to disease or an accident; (c) sanitary means available, which includes the people for 10000 inhabitants and hospital beds for 10000 inhabitants. all the lowest level components are assumed to be measurable and registered in official statistics. dolan et al. propose a detailed hierarchy of the factors that influence subjective well-being or happiness based on the literature review they did[7]. they distinguish seven groups of factors: income, personal characteristics, socially developed characteristics, distribution of available time, attitudes to and beliefs in others, relations with others, and characteristics of an ample social environment. these factors are composed of sub factors; for example, in the factor distribution of available time, they distinguish: hours worked, hours spent traveling from home to work, taking care of others, voluntary community service, physical exercise and religious activities. all the sub factors are assumed to be measurable and are recorded in statistics or can be determined by surveys. they propose aggregation by means of a linear model with an uncertainty term (y = a + b1 · x1 + b2 · x2 + . . . + ε). as mentioned earlier, kahneman and krueger propose a procedure to measure the happiness of a country based on individual surveys, distributed by what each individual does in his/her own time and how he/she feels like in all the considered situations[2]. schwartz puts forward a hierarchy of basic values to motivate action and interrelations (some are hardly compatible) that are common to all cultures (justified), between which there would be differences only for priorities and each value’s relative importance[6]. these values are aggregated into 10 higher ranking values: self-direction, stimulation, hedonism, success, power/prestige, security, conformity (with social norms), tradition (acceptance of one’s own culture and customs), benevolence (group aid) and universalism (well-being for everyone and for nature). the proposed measurement method is based on surveys conducted by means of validated questionnaires. the priorities of the diverse values in each individual and each culture are calculated. the evaluation at the group level is assumed based on averages. no higher level value to group these 10 basic values is considered musek proposes a hierarchy that begins with two macro categories: dionysian values and apollonian values[10]. dionysian values are classified into hedonistic (sensual and heath related) and values of power (profit, success, etc.). apollonian values are classified into moral values (traditional, social, etc.) and satisfaction values (cognitive, cultural, etc.). singular values are fitted to this scheme. nevertheless, the author observes that this hierarchy can change with country and with each individual’s age and time. 80 antonio caselles: trying to evaluate human dignity in a social group maslow proposes a hierarchy of values (motivating priorities), which is not altogether justified and receives considerable feedback[11]. for him, level 1 (physiological) is occupied by: breathing, food, water, sex, dreaming, homeostasis and excretion; level 2 (security) is occupied by: corporal security, and those of work, resources, morality, family, health and property; level 3 (love/belonging) by: friendship, family and sexual intimacy; level 4 (esteem) by: self-esteem, confidence, success, respect of others and to be respected by others; level 5 (self-fulfillment) by: morality, creativity, spontaneity, problem solving, lack of prejudices and acceptance of facts. 5 the development concept considered by the undp the development concept that the un uses attempts to include all the aspects implied in: (a) a prolonged, healthy, creative life; (b) knowledge acquisition; (c) a decent standard of life; (d) political freedom; (e) human rights; (f) to interact freely with others[12]. in our opinion, its amplitude is total. however, a constructive process is followed there, which tends to produce an adequate index for measuring development. the un began by defining the human development index (the hdi, in 1990), whose calculation was specified in detail by anand and sen[14]. for this calculation, the following variables were used: life expectancy at birth (years), adult literacy rate (%), combined gross enrolment ratio (%), and gross domestic product (gdp ) per capita (ppp us$). the hdi measures health by life expectancy when born, wealth by gdp and education by the percentage of people with a degree or registered in regulated studies. in order to certainly complete the hdi, other indices were created, among them the hpi2, which is a poverty index applied to developed countries. this index is based on four variables: probability of not surviving to the age of 60; the long-term unemployment rate; the proportion of adults who lack functional aptitudes; the proportion of the population that lives below the poverty threshold. later, hdi − d (that corrects the hdi by considering inequality)[12], hybrid hdi (that uses a geometric average instead of an arithmetic one in order to be more sensitive to minor differences in lower valued factors), the multidimensional poverty index (mpi) (that analyzes deprivations in hdi components), and the gender inequality index (gii) appeared. in the undp , today the idea seems to develop and perfect these indices in order to obtain a suitable, efficient specification and measurement to monitor human development in diverse countries. we suggest restricting the development concept to the components that are deduced from the udhr, that is, health, education, sufficient rent and free time, to which we would add what concerns sustainability and the environment. thus development, along with freedom and equality (which includes solidarity, justice and peace), as described by the udhr, would be the components of the supreme advances in systems science and application (2015) vol.15 no.1 81 value: human dignity. 6 attempting to suggest a proposal for future consensus let us now consider the scale that the udhr implicitly developed and which we have attempted to make explicit based on the philosophical approach of moncho [1] (see heading 2.). it is now necessary to associate all the values specified in udhr articles with a measurable variable that is registered, or can be registered, in the official statistics of a given country or region. this scale also needs the incorporation of new values which, in 1948, the date when the udhr came into being, were not as important as they are now. attempting to find the best way to measure the concepts specified in the various udhr articles and incorporating new values into it can imply remarkable work, which also requires a consensus. however we dare, as an attempt, to suggest a preliminary proposal which, if it contains something interesting, will have to be improved in subsequent approaches. following the implicit scheme in the udhr, considering the statistics that the undp uses and stressing that our suggestion is merely a preliminary attempt[12], we propose the following procedure to evaluate the degree of human dignity in a social group (world, country, region, municipality, company, etc.). 1. specify the implicit hierarchy of values in the udhr and complement it by considering the statistics and procedures that the undp habitually uses. 2. evaluate all the first-level variables (input or basic) as they appear in the statistics or by means of scores provided by experts. 3. transform all these values into indices with the formula habitually used in the undp ; that is: (present value minimum value)/(maximum value minimum value). 4. calculate the indices of the second-level variables (those that depend solely on first-level variables) by using the geometric average (as the undp suggests). indices with a negative sense will be entered as (1-index) in the corresponding formula. 5. calculate the indices of the third-level variables (those depending on the secondor firstlevel variables) by also using the geometric average. 6. continue in this way until the human dignity index is calculated. this procedure is simple and quite feasible in a computer. in order to complicate it, a weight or importance measure may be assigned to each index. for geometric averages, each weight will appear as an exponent of its corresponding index. for instance, idignity = (idevelopment) a · (ifreedom)b · (iequality)c 82 antonio caselles: trying to evaluate human dignity in a social group where a, b, c are higher than zero, and a+ b+ c = 1. this calculation can be facilitated using logarithms. as a first approach, the list of variables (turned into indices) can be the following: they are numbered between 0 and 142 (both inclusive). when their codified name begins with a y , it is understood that they have a negative sense and that their respective aggregation formula will consider (1− y · ··). i000 human dignity i001 development i002 health (disaggregated according to [4]) i003 resources i004 cost in health per capita (ppa in us$) i005 doctors (for every 10000 inhabitants) i006 hospital beds (for every 10000 inhabitants) y007 risk factors i008 children not immunized against: i009 diphtheria, pertussis and tetanus (% of children aged 1 year) i010 measles (% of children aged 1 year) i011 incidence of hiv i012 young people (% aged 15-24 years) (women) (men) i013 adults (% aged 15-49 years) (total) y014 mortality i015 infantile (for every 1000 live born babies) i016 children aged under 5 (for every 1000 live born babies) i017 adults (for every 1000 inhabitants) (women) (men) i018 rates of death by non transmissible diseases, standardized for age (for every 100000 inhabitants) i019 education (disaggregated according to [4]) i020 achievements in education i021 literacy rates for adults (% of 15-year-olds or older) i022 population who at least finished secondary education (% of 25-year-olds old and older) i023 access to education i024 rate of enrollment in primary education (% of the population of primary education ages) (gross) (net) i025 rate of enrollment in secondary education (% of the population of secondary education ages) (gross) (net) i026 rate of enrollment in tertiary education (% of the population of tertiary education ages) (gross) i027 efficiency of primary education y028 dropout rate, all levels (% of the cohort in primary education) y029 repetition rate, all levels (% of the total enrollment in primary education in advances in systems science and application (2015) vol.15 no.1 83 the previous year) i030 quality of primary education i031 student-teacher ratio (number of students per teacher) i032 teachers trained in primary education (%) i033 income/work/standard of living (disaggregated according to literature[4]) i034 jobs-population ratio (% of the population aged 15-64 years) i035 formal jobs i036 (% of all jobs) i037 womens rate/mens rate ratio y038 vulnerable jobs i039 (% of all jobs) i040 womens rate/mens rate ratio y041 people who work and live with less than us$ 1.25 per day (% of all jobs) y042 unemployment rate per levels of education (% of the labor force with the indicated level of education) i043 primary education or less i044 secondary education or better y045 infantile work (% of children aged 5-14 years) i046 obligatory and paid maternity (in days) i047 free time y048 working hours per year (average per worker) i049 days of paid vacations per year (average per worker) i050 stainability and non vulnerability (disaggregated according to [4]) i051 fitted net saving (% of gross net income) y052 ecological footprint of consumption (hectares per capita) i053 proportion of the total provision of primary energy y054 fossil fuels (%) i055 renewable sources (%) y056 emissions of carbon dioxide per capita (tons) i057 protected area (% of the terrestrial area) y058 population that lives on degraded terrain (%) y059 population with no access to improved services i060 water (%) i061 sewerage (%) y062 deaths caused by intra-domiciliary, atmospheric and water contamination (per 1 million people) y063 population affected by natural disasters (annual average, per 1 million people) i064 freedom i065 of choice i066 of partner (score) 84 antonio caselles: trying to evaluate human dignity in a social group i067 of work (score) i068 of asylum (score) i069 of displacement in a country (score) i070 to leave and return to a country (score) i071 of childrens type of education (score) i072 of religion or beliefs (score) i073 of access to public functions (score) i074 of opinion and publication i075 freedom of press index y076 number of jailed journalists i077 of pacific meetings (score) i078 of pacific associations (to create and to become a member, including trade unions) (score) i079 of pacific manifestations (including religion or beliefs) (score) i080 of legislation (directly or by choosing legislators) i081 to vote (score) y082 % corruption/bribe victims i083 degree of democratic decentralization (score) i084 % of political participation i085 equality i086 solidarity i087 non discrimination (race, color, sex, language, opinion, national or social origin, economic position, birth, etc.) i088 seats in parliament (score) i089 population who have at least completed secondary education (score) i090 rate of participation in the labor force (score) i091 same rights (economic, social, cultural, equal wage for the same work, fair wage)(everything limited by the resources of the state and international cooperation) (score) i092 social security i093 disease (score) i094 unemployment (score) i095 widowhood (score) i096 disability (score) i097 old age (score) i098 non voluntary loss of means of subsistence (score) i099 attending to people who have difficulties surviving (not in udhr) (score) i100 justice i101 protection by law i102 physical (score) i103 of property (score) advances in systems science and application (2015) vol.15 no.1 85 i104 of author rights (score) i105 of human rights (score) i106 obligations to others and the community (score) i107 resources to courts (score) i108 presumption of innocence (score) i109 guarantees of defense (score) i110 non retroactive laws (score) i111 non contradictory laws (score) i112 effectiveness and efficiency in application of laws i113 hearing (score) i114 publicity (score) i115 impartiality (score) i116 non abuse (score) i117 no errors (score) i118 celerity (score) i119 peace i120 no coercion i121 no slavery (score) i122 no servitude (score) i123 right to legal personality (score) i124 right to nationality and its change (score) i125 right to participate in scientific progress (score) i126 right to participate in artistic and/or cultural activities (score) i127 non violence i128 no torture (score) i129 no cruel and/or degrading treatment (score) y130 rate of homicides y131 rate of robberies y132 rate of assaults i133 no fear (disaggregated according to [4]) y134 selling/purchasing arms y135 refugees y136 displaced internally y137 civil war i138 victims i139 intensity i140 no misery y141 % undernourished y142 insufficiency average 86 antonio caselles: trying to evaluate human dignity in a social group 7 conclusions and discussion different approaches have been observed in the literature as to not only the identity of the superior values in a human group, but also its hierarchy and composition. the values detected there have been checked against the subjective well-being components and the result is that well-being depends mostly on genetic factors and personality, while the other factors are assumed in other previously considered values. it is observed that values and their priorities can change according to country and time. accordingly, if the intention is to design a scale of values that is useful for any human group, it is necessary to reach a global consensus and to update it whenever necessary. the undp is immersed in constructing an index that measures the general value of both a human group and partial indices that measure different aspects of this value. this process comes across several difficulties, such as those that derive from the nonexistence or inaccuracy of necessary data and the nonexistence of a consensus about the definition and structure of diverse partial values. in order to advance in determining a definition and a structure between the partial values and the total value of a human group, we based our work on the udhr (a first universal consensus) and on the philosophical studies, which are also based on the udhr. with this information and the suggestions that we found in the undp reports and in studies by different authors, we drew a list of indices, structured by levels, and a general formula to aggregate lower level indices into higher level ones. in this way, the upper rank/higher level index would be the human dignity index (idig) (level 6). like the indices of level 5 (with whose aggregation the idig would be calculated), we would obtain the human development index (idev ), the human freedom index (ifre) and the human equality index (iequ). from this stage onward, we would suppress the adjective “human” for being redundant and obvious. level 4 indices, subsidiaries of the idev , would include the index of health (ihea), the index of education (iedu), the index of income (iinc), the index of free time (ifrt ) and the index of sustainability (isus). we would continue in the same way until we reach the indices of level 1, which would be calculated from the statistical data or from the scores provided by experts. as formulas to calculate the indices of level 1 and to aggregate indices, we agree with the habitual ones employed in the undp reports (present-minimum)/(maximum-minimum) and the geometric average (perhaps weighed), respectively. all the indices must be aggregated in a positive sense. the indices in a negative sense (in the list in heading 6, those that begin with a y) would enter the respective aggregation formulas as the 1-index. despite being obvious, we insist that this proposal (based on the consulted literature) has to be considered a germ or an attempt, must be the object of a more advances in systems science and application (2015) vol.15 no.1 87 detailed or refined study, and has to be submitted to consensus; for instance, the definition of each index, deduced from its assigned components, ingredients or dimensions, and also the weight assigned to each component when averaging. another question to consider in future studies is that which refers to crosssectional indices. we assumed that each component, ingredient or dimension appears once in the structure; that is to say, the considered structure is hierarchical or tree-shaped. however, it is possible that a given component can be considered as a starting point to pertain to more than one branch. in this case, we chose the branch that was more concordant with the concept being dealt with; for example, free medical aid could be considered within the health concept and also on the development branch. however, we considered that it was more likely a solidarity subject than (although also) a health subject. there are also some concepts, such as poverty or gender inequality, that do not appear in the list of heading 6, but include components that either appear on several branches of the tree or do not appear on any. thus in the multidimensional poverty index (mpi)[12], 10 components enter: 2 in health (nutrition and infantile mortality), 2 in education (school enrollment and training years) and 6 in standard of life (goods, floor, electricity, water, sewerage and fuel to cook). both the health ones are on different branches (y141 and i015, respectively). in education, training years do not appear on the list and school enrollment is disaggregated into three levels (i024, i025, and i026). we see that the six standard of life ones do not appear on the list. since many of the items on the list were obtained from tables in the undp report[12] and were assumed to be a first approach, we observed that perhaps it would be advisable to better select the components of the list so that the cross-sectional indices, such as the mpi, are based solely on the components on the list. a similar situation occurs with the gender inequality index (gii); the ideal situation would be that all the components of the cross-sectional indices are included on the list, but this ideal situation is perhaps not attainable if simplicity is preferred. in this last case, the cross-sectional indices would be independent indices of the human dignity index, which we are attempting to design herein. we sincerely hope that somebody finds new ideas in this paper which contribute to or are useful for better monitoring the progress of humanity on our planet, and we encourage anyone who feels motivated by this matter to continue with this type of work. references [1] j.r. moncho. (2003), theory of superior values (in spainish:teoŕia de los valores superiors), campgrafic, valencia, spain. [2] d. kahneman and a.b. krueger. (2006), “developments in the measuremen88 antonio caselles: trying to evaluate human dignity in a social group t of subjective well-being”, the journal of economic perspectives, no.20, pp.3-24. [3] f. parra-luna. (2013), “axiological systems theory: a general model of society”, triplec, no.6-1, pp.1-23. [4] s. roth. (2013), “common values? fifty-two cases of value semantics copying on corporate websites”, human systems management, no.32, pp.249265. [5] p.n. vijayamohanan and b.p. asalatha. (2013), “objectivizing the subjective: measuring subjective wellbeing”, munich personal repec archive, no.45005, http://mpra.ub.uni-muenchen.de/45005/. [6] s.h. schwartz. (2012), “an overview of the schwartz theory of basic values”, online readings in psychology and culture, no.2-1, http://dx.doi.org/10.9707/2307-0919.1116. [7] p. dolan, t. peasgood and m. white. (2008), “do we really know what makes us happy? a review of the economic literature on the factors associated with subjective well-being”, journal of economic psychology, no.29, pp.94-122. [8] w.a. kritsonis. (2007), “ways of knowing through the realms of meaning”, national forum press, houston, tx. [9] p. hills and m. argyle. (2002), “the oxford happiness questionnaire: a compact scale for the measurement of psychological well-being”, personality and individual differences, no.33, pp.1073-1082. [10] j. musek. (1994), “values and value orientations in the background of european cultural traditions”, anthropos (international issue), ljubljana. [11] m. ferguson and abraham maslow. (1980), millennium: glimpses into the 21st century, k. dychtwald and a. villoldo.(eds.), j.p. tarcher, los angeles, ca. [12] united nations development program. (2010), human development report 2010, new york, usa. [13] t.b. kashdan. (2004), ”the assessment of subjective well-being (issues raised by the oxford happiness questionnaire)”, personality and individual differences, no.36, pp.1225-1232. advances in systems science and application (2015) vol.15 no.1 89 [14] s. anand and a. sen. (1994), “human development index: methodology and measurement”, human development report office, new york.vol.5,no.2,pp.1433-1434. corresponding author antonio caselles can be contacted at: antonio.caselles@uv.es advances in systems science and applications (2012) vol.12 no.2 113-121 sup-pixel edge detection technology and its application in non-contact measurement of precise parts bo lei, hong lu and chenghuo shang school of mechanical and electronic engineering, wuhan university of technology, wuhan, china abstract the precision of traditional methods of edge detection of precise parts with linear ccd is determined by the size of ccd cells. the precision can only reach µm at high cost. a sup-pixel edge detection method of ccd based on least square method and derivative operator method is proposed to improve the measurement precision. the image gradient obtained using derivative operator method. point a with the max gradient is then regarded as the point of the edge, and then the pixels nearby edge are interpolated linearly. for example, a high-speed data acquisition system is designed using high accuracy linear ccd tcd1501d and high speed a/d converter tlc5510. fifo memory cy7c460a is used to store the converted data. using the sample pulse of ccd as the system control clock, a high speed data acquisition and storing system with simple circuit is designed in this paper. experimental results show that the resolution µ of the system design is 0.036 pixels and the measurement precision is improved by one order of magnitude over traditional methods of edge detection. it reaches sub-pixel accuracy. keywords edge detection, sub-pixel, linear ccd 1 introduction with the growth of heavy industries and the demand for high quality production, many industries require some form of technology for controlling their production quality. one common quality control (qc) technique employs linear ccd cameras for precise parts measurement. the edge detection and its localization in the image are, in particular, used for dimensional inspection and for localizing objects in industrial applications[1]. the precision is determined by the size of ccd cells used for measurement by linear ccd. people try to improve the precision of the measurement system in this way, for example, improving the manufacturing process to decrease the size of ccd cells, and improving the precision of system by amplifying measured parts with optical system. but the precision can only reach µm at high cost. in order to obtain accurate edge measurements, it is necessary to determine the location of an edge to a greater resolution than the spacing between the pixels of the image sensor, that is to say, at sub-pixel resolution[2]. ohtani[3] summarizes the common sub-pixel edge detection techniques and divide them into first derivative algorithm, second derivative algorithm, template 114 bo lei:sup-pixel edge detection technology and its application in non-contact... matching, edge fitting, and statistical approaches. linearity interpolation, which is a mature theory, is widely used in many fields, such as image and signal processing. but the interpolation result is too smooth, and it would loose some edge of images information[4]. therefore, a sub-pixel measurement method of ccd is proposed to so that the edge of static images is detected using derivative operator method in digital image processing, and the pixels nearby edge are then interpolated linearly. 2 design of measurement system 2.1 design of optical system the optical system is as shown (see fig.1). after the amplified by the optical system, the shade of part is imaged on the photosensitive cells of ccd. fig.1 optical system 1-laser 2-test part 3-objective lens 4-imaging objective lens 5-linear ccd 2.2 system composition the system is composed of illumination unit, test part, optical imaging unit, linear ccd, signal acquisition unit, fifo memory and pc computer (see fig.2). we choose laser as illumination unit. and linear ccd is a toshiba ccd linear image sensor (tcd1501d). the parallel beam irradiates the edge of the test part, it passes through the optical amplification unit and images on the advances in systems science and applications (2012) vol.12 no.2 115 photosensitive surface of ccd. after being sampled and a/d conversion by the signal acquisition unit, the output signals of ccd are stored at fifo memory (cy7c460a). then the digital signals are transmitted to pc computer. 2.3 composition of hardware 2.3.1 linear ccd tcd1501d the tcd1501d which includes sample-and-hold circuits is a high sensitive and low dark current 5000 elements ccd image sensor. the sensor is designed for facsimile, image scanner and ocr. the device is operated by 5v (pulse), and 12v power supply[5]. the features of tcd1501d are shown as table 1 below. table 1 the features of tcd1501d image sensor elements size 7µm by 7µm on 7µm centers photo sensor region high sensitive and low voltage dark signal pn photodiode clock 2 phase (5v) internal circuit s/h circuit package 22pin 2.3.2 analog-to-digital converters tlc5510 the tlc5510 is cmos, 8-bit, 20msps analog-to-digital converters (adcs) that utilize a semiflash architecture. the tlc5510 operates with a single 5-v supply and typically consume only 130 mw of power. included is an internal sample-and-hold circuit, parallel outputs with highimpedance mode, and internal reference resistors. the semiflash architecture reduces power consumption and die size compared to flash converters. by implementing the conversion in a 2-step process, the number of comparators is significantly reduced. the latency of the data output valid is 2.5 clocks. the tlc5510 uses the three internal reference resistors to create a standard, 2v, full-scale conversion range using vdda[6]. so its peripheral circuit is simplified. 2.3.3 fifo memory cy7c460a the cy7c460a is 8k words by 9-bit wide first-in first-out (fifo) memory. each fifo memory is organized such that the data is read in the same sequential order that it was written. full and empty flags are provided to prevent overrun and underrun. three additional pins are also provided to facilitate unlimited expansion in width, depth, or both. the depth expansion technique steers the control signals from one device to another by passing tokens. the read and write operations may be asynchronous; each can occur at a rate of up to 50 mhz. the write operation occurs when the write (w) signal is 116 bo lei:sup-pixel edge detection technology and its application in non-contact... fig.2 system composition low. read occurs when read (r) goes low. the nine data outputs go to the high-impedance state when r is high[7]. 2.3.4 single-chip computer at89s52 the at89s52 is a low-power, high-performance cmos 8-bit microcontroller with 8k bytes of in-system programmable flash memory. the at89s52 provides the following standard features: 8k bytes of flash, 256 bytes of ram, 32 i/o lines, watchdog timer, two data pointers, three 16-bit timer/counters, a six-vector twolevel interrupt architecture, a full duplex serial port, on-chip oscillator, and clock circuitry[8]. 2.4 design of hardware system after being sampled and a/d conversion the output video signals of ccd are transmitted to pc computer. and it is transformed into digital images by pc computer. the key parts of the signal acquisition unit are high-speed analog-todigital converter (tlc5510), fifo memory (cy7c460a) and simple chip computer (at89s52). its principle is shown (see fig.3). advances in systems science and applications (2012) vol.12 no.2 117 fig.3 block diagram of hardware system through differential amplifier and low-pass filter, the output signals of ccd (opt) are sent into a/d converts tlc5510. under the control of the a/d convert controlling clock (clk) that generated by the sample pulse of ccd (sp), the simulated output signal of ccd (opt) are converted into 8-bit digital signals that is corresponding to its simulated amplitude. if enabling port (oe) captures the low level signal, the converted signals are sent to the 8-bit input data ports (a0a0) of fifo memory (cy7c460a). and under the control of write clock (w), which is generated by the sample pulse of ccd (sp), the converted signals are stored in fifo in turn. relative address methods are used in fifo memory, and the address coding generator can be omitted, so the circuits are simplified. upon power-up, the fifo must be reset with a master reset (mr) cycle. this causes the fifo to enter the empty condition signified by the empty flag (ef) being low, and both the half full (hf), and full flags (ff) being high. so while enabling port (oe) of tlc5510 captures the low level signal, the reset 118 bo lei:sup-pixel edge detection technology and its application in non-contact... port (mr) of fifo memory should capture low level to reset the fifo .the shift pulse (sh) of the ccd is sent to the trunk line (into) of the simple chip computer (at89s52). we can get the beginning time and end time of the output signals of ccd by (into). so the signals of 5000 elements of ccd are stored into fifo memory in turn. then the at89s52 reads data which stored into the fifo with bus mode and sends them to pc computer one by the serial ports. the 74ls373 is used for improving the stability of reading data. 3 analysis of the measurement signal of ccd when the test part is imaged on the photosensitive cells of ccd, the ccd converts the optical signals of the test part into discrete voltage signals. the value of each discrete voltage signal corresponds to the light intensity of the photosensitive cells, and the order of output signal is corresponding to the position of the cell. after preprocessing the discrete output voltage signal of ccd, such as high pass filter, differential amplifier and low-pass filter. the edge of one-dimensional image signal of ccd is corresponding to the physical boundary of the test part. the low voltage signals of the ccd output correspond to the shadow of the test part and the high voltage signals is corresponding to the bright area where the test part cant shade the light. the ideal edge signal is a step signal (see fig.4(a)). but the edge signal actually is a gradually-changed signal which decreases gradually(see fig.4(b)), and the actual edge point is located at this area. fig.4 edge signal of one dimension image 4 sub-pixel edge detection algorithm we define that the resolution equals the numbers of cells x when the ccd sensor gets valid signal, and the signal of ccd is valid if the signal to noise ratio (snr) s>1. the image snr is signal square to noise square ratio. and if we take s=1, the valid signal f(x) is the least local variance of all pixel h(x). f (x) = h(x) (1) advances in systems science and applications (2012) vol.12 no.2 119 we can get the image gradient by derivative operator method. then the point a with the max gradient is regarded as the point of the edge. and we get several points nearby point a. the number of points is m. we can get the polynomial curve equation (2) and the max residue by polynomial curve fitting. according to error theory and data processing method [9], and take confidence probability p= 95.44%, can get (3): g(x) = k ∗ x+ c (2) and h(x) = 2δ (3) we take ∆g(x) = 2δ, therefore obtain from (2) and (3): ∆x = 2δ/|k| (4) which leads to: µ = 2δ/|k| (5) the magnification of optical system is β, and it leads to µ = 2δ/(|k| ∗ β) (6) but the output of ccd is discrete signals, so the actual edge point p is between the point p and the point p’. the point p’ is nearest point of the point p. we take h(xa) = ξ ·h(xp ) + (1− ξ) ·h(xp ) (7) and the linear factor ξ ∈ [0, 1],then obtain the position of actual edge point xa is: xa = (h(xa)− c)/k (8) 5 application fig.5 one of the pictures of the experiment data according to the algorithm mentioned above, we measured the location a specific edge with the system. the number of measure times is 5. the magnification of optical system β=20, and the polynomial curve equation under the condition of 120 bo lei:sup-pixel edge detection technology and its application in non-contact... the number of points m=7. one of data of the experiment is shown (see fig.5), we get the results of its polynomial curve fitting (see fig.6).the experimental results show that the performance of this algorithm is better than the traditional method in precision of measurement. fig.6 results of polynomial curve fitting table 2 the features of tcd1501d order ξ=0.3(pixel) ξ=0.5(pixel) ξ=0.7(pixel) µ(pixel) 1 1048.6 1048.3 1048.1 0.021 2 1048.7 1048.3 1048.0 0.034 3 1049.2 1048.9 1048.6 0.031 4 1049.2 1048.7 1048.2 0.047 5 1049.1 1048.7 1048.2 0.048 statistical results average 1049.0 1048.6 1048.2 0.036 standard deviation 0.288 0.268 0.228 0.011 then we get the results under the condition of linear factor ξ=0.3, ξ=0.5 and ξ=0.7 . the results are shown as table 2 blow. the data of the table means the location of the pixel. there are no abnormal data in this table. with the comparison analysis on actual location of the specific edge, the result is best under the condition of ξ=0.7. the repetitive error (2δ) is 0.45, which is lest than a pixel, and the resolution µ of the system is 0.036pixel in theory. thus it reaches sub-pixel accuracy. advances in systems science and applications (2012) vol.12 no.2 121 6 conclusions in this paper, a sub-pixel measurement method of ccd was presented, which detect edge of static images with derivative operator method in digital image processing, and then the pixels nearby edge are interpolated linearly. the technique is to improve the precision of measurement at low cost. at the same time the resolution of the measurement system is calculated in theory. finally the results of experiment verify that the system can extract the accurate sub-pixel edge position. acknowledgements this work was supported by hubei province science foundation (no.2008cdb274) and wuhan high-tech development project foundation (no.200812121559). references [1] o. faugeras. (1993), three dimensional computer vision, cambridge, the mit press:cambridge. [2] a.j. tabatai and o.r. mitchell. (1984), “edge location to subpixel values in digital imagery”, ieee trans. pattern analysis machine intell, pp.188-201. [3] k. ohtani, m. baba. (2001), “a fast edge location measurement with subpixel accuracy”, ieee proceedings instrumentation and measurement technology, pp.2087-2092. [4] zhang xiong, bi duyan, yang baoqiang. (2007), “an edge preserved image interpolation method”, journal of air force engineer university, vol.8, no.3, pp.78-80. [5] toshiba. (1996), the toshiba ccd linear image sensor tcd1501d. [6] li cai, wang an. (2003), “application of 8 bit high speed a/d converter tlc5510”, foreign electronic devices and components, pp.59-61. [7] cypress. (2002), datasheet of cy7c460a. [8] atmel company. (2002), datasheet of at89s52. [9] fei yetai. (2000), error theory and data processing, china machine press. beijing, pp.72-82. advances in systems science and applications (2014) vol.14 no.1 1-21 the theory of parametric control of macroeconomic systems and its applications(i) a. ashimov, zh. adilov, r. alshanov, yu. borovskiy and b. sultanov kazakh national technical university named after k. satpayev, 22 satpaev street, almaty, 05x0013, kazakhstan abstract this work consists of three parts and presents the recent results of development of the theory of parametric control of macroeconomic systems and some its applications for solving a number of concrete problems. keywords mathematical model, structural stability, parametrical identification, parametric control introduction development of adequate methods on the basis of mathematical models for macroeconomic analysis and of evaluating optimal values of parameters (economic policy tools) for macroeconomic systems control on the level of national economies, regional economic unions and world economic system is urgent problem, sharply necessity in solving which was emphasized by the latest global crisis. nowadays, mathematical models of corresponding macroeconomic systems without comprehensive testing for possibility of their application are widely used for macroeconomic analysis (including scenario analysis) and evaluating optimal parameter values of economic policy for controlof macroeconomic systems evolution [1-15]. this paper is devoted to the development of the theory of macroeconomic analysis and evaluating optimal parameter values of economic policy for control of macroeconomic systems on the basis of the corresponding mathematical models, tested for possibility of their application, and it consists of three parts. the first part describes the components of the parametric control theory and its algorithmic foundations. the second part describes mathematical foundations of the parametric control theory, and the third part-the developed theory applications for solving a number of applied problems on the basis of some mathematical models of macroeconomic systems. part 1. components of the parametric control theory and its algorithmic foundations 1.1 components of the parametric control theory of macroeconomic systems given the following facts: solution of either continuous or discrete dynamical system [which can include both controllable parameter vectors-state policy tools (µ), and uncontrollable 2 a. ashimov: the theory of parametric control of macroeconomic systems and ... parameter vectors (a)] depends on initial condition vectors and parameters (coefficients) of this system; solution of static system (for instance, static model of small open economy) depends on parameters (coefficients) of this system; for judging by the study results of dynamical system about an object, described by it, an existence of structural stability (or robustness) of this system is required [16]; for judging by the study results of (static or dynamic) model about an object, described by it, an existence of stability of mapping, defined by this model, is required [17]; and also a condition of macroeconomic model (presented by one of dynamic or static system) stability is required at small perturbations of the initial statistical data for parametric identification of the model (input parameters) and the following components of the parametric control theory are proposed [18-19]. 1. the methods for forming the set (library) of macroeconomic mathematical models. these methods are oriented towards the description of various specific socio-economic situations. 2. the methods for estimating the conditions for robustness (structural stability) of the dynamical mathematical models, the methods for estimating the stability indicators and the methods for estimating stability of mappings, set by models of national economic system from the library (without parametric control). 3. the methods for adjusting the structural instable dynamical mathematical model to obtain its structural stability (methods for attenuation of structural instability). choosing (or synthesizing) the algorithms for attenuation of structural instability for the mathematical model of macroeconomic system. 4. the methods for choosing and synthesizing the laws of parametric control of macroeconomic system based on its dynamical mathematical models. the methods for setting and solving the parametric control problems in terms of corresponding mathematical programming problems on the basis of static mathematical models of macroeconomic systems. 5. the methods for estimating the robustness (structural stability) of dynamical mathematical model. the methods for estimating the stability and the methods for estimating the stability of mappings, set by models of macroeconomic systems (with parametric control). 6. the methods for adjusting the constraints on parametric control of macroeconomic system in the case of structural instability of its mathematical model with parametric control. specification of constraints on the parametric control of macroeconomic system. 7. the methods for studying the effects of uncontrollable parameters and advances in systems science and applications (2014) vol.14 no.1 3 functions (uncontrollable factors) on the results ofsolving of variational calculus problems of synthesis and choice (among given finite algorithms set) of parametric control laws. study of bifurcation points of extremals of variational calculus problems of choosing optimal laws of parametric control. the methods for studying the effects of uncontrollable factors variance on the solution results of mathematical programming problems based on static mathematical models. 8. approach for choosing recommendations on evaluating political rules in the frame of implementing the laws of parametric control of macroeconomic system on the base of the analysis of dependences of optimal criteria values of corresponding parametric control problems on uncontrollable factor values. this paper presents the general results of component-specific development of the parametric control theory (its mathematical and algorithmic foundations). within the framework of the methods for forming the set (library) of macroeconomic mathematical models, it is proposed an algorithm for parametric identification of large-scale macroeconomic models, which uses jointly two identification criteria. within the framework of the methods for examining mathematical models stability, it is proposed numerical algorithms for stability indicators estimation and numerical algorithms for estimation stability of mappings, set by model (in terms of the theory of differentiated mappings singularities); within the framework of the methods for examining the weak structural stability, it is described the proposed numerical algorithm based on the robinson theorem about sufficient conditions of weak structural stability of dynamical mathematical models. within the framework of the methods for choosing and synthesizing parametric control of national economy, based on continuous and discrete non-autonomous dynamical systems, as well as discrete dynamical systems with additive noise, there are formulated and proved the corresponding theorems about conditions for the existence of solutions of variational calculus problems on synthesis and choice (among given finite algorithms set) of optimal parametric control laws. within the framework of the methods for studying the effects of uncontrollable factors variance on the solution results of variational calculus problems on synthesis and choice of optimal parametric control laws, there are formulated and proved the theorems about conditions for the continuous dependence of optimal criteria values of variational calculus problems on uncontrollable parameters (uncontrollable function values). within the framework of studying the bifurcations of extremals of the variational calculus problem of choosing the optimal parametric control laws, there are formulated and proved the theorems about sufficient conditions for the existence of appropriately defined bifurcational point of extremals of the variational 4 a. ashimov: the theory of parametric control of macroeconomic systems and ... calculus problem; there is proposed an approach for choosing the recommendations on evaluating political rules within the framework of implementation of appropriate economic tools for adjusting national economy on the base of analyzing dependences of optimal criteria values of corresponding parametric control problems on uncontrollable factors values. 1.2 algorithm for the parametric identification of large-scale macroeconomic models the following algorithm for the parametric identification of large-scale macroeconomic models is proposed within the framework of elaborating the 1st component of the parametric control theory [19]. the parametric identification problem for discrete dynamical macroeconomic model is finding the estimates of unknown values of its parameters (to which belong unknown values of exogenous functions of model and unknown initial values of its dynamical equations), at those one can obtain the minimum of the objective, characterizing the deviations of output variables values of the model from corresponding observed values (of known statistical data for the period t = t1, t1 + 1, ..., t2). this problem comes to finding the minimum of the multi variable function (parameters) in some closed domain d of euclidean space with constraints, overlaying both on endogenous variables values of the model (e constraints)and on initial parameter values (f constraints). in the case of large number of dimensions n of this domain, the standard methods for finding function extrema are often ineffective because of presence of several local minima of the objective. below we present the algorithm, allowing for features of the parametric identification problems for macroeconomic models and allowing passing over the mentioned problem of “local extrema”. constraints e are formed by economic meaning of endogenous variables of model (for example, by their non-negativity). domain of type d = ∏ni i−1[a i, bi], where [ai, bi] is segments of possible values of the parameter pi; i = 1, ..., n was considered as a range, defined by f constraints for evaluating possible values of exogenous parameters.herewith, parameter estimates, for which had observed values, were searched either within [ai, bi] segments with centers at corresponding observed values (in the case of one such value) or within some segments, covering observed values (in the case of several such values). other [ai, bi] segments for search of the parameters were chosen using indirect estimates of their possible values.nedler-mead algorithm of directed search was used in calculating experiments for finding the minimal values for continuous function k: d → r of several variables [20]. use of this algorithm for initial point p1 ∈ d can be interpreted in terms of (converged to local minimum p0 = argmink of criterion k) sequencep1, p2, ..., where k(pj + 1) ≤ k(pj), pj ∈ d, j = 1, 2,... we will assume that point can be found accurately enough, when we describe the following algorithm. advances in systems science and applications (2014) vol.14 no.1 5 for solving the parametric identification problem of model in question on the base of obvious assumption about divergence (in general case) of minimum points of two different functions, two criteria of the following type were proposed: ka(p ) = √√√√ 1 nµ(t2 − t1 + 1) t2∑ t=t1 na∑ i=1 αi ( yi(t)− yi∗(t) yi∗(t) )2 kb(p ) = √√√√ 1 nβ(t2 − t1 + 1) t2∑ t=t1 nβ∑ i=1 αi ( yi(t)− yi∗(t) yi∗(t) )2 here {t1, ..., t2} is identification period;yi(t), yi ∗ (t)-correspondingly computed and observed values of model output variables, ka(p )-subsidiary criterion, kb(p )-basic criterion; nb > na; αi > 0 and βi > 0 are some weight coefficients, values of which are defined during solution of parametric identification problem for dynamical system; ∑na i=1 αi = nα, ∑nb i=1 βi = nβ. the minimization problems based on the model of corresponding criterion (ka and kb) in the domain d, we will call the problem a and the problem b.theaggregate algorithm for solving the parametric identification problem of model was chosen in terms of the following steps: 1. for some vector of initial values of parameter p1 ∈ d, solve problems a and b simultaneously. then, find the minimum points pa0 and pb0 of criteria ka and kb, respectively. 2. if kb(pb0) < ε for some sufficiently small number ε, then the model parametric identification problem is solved. 3. otherwise, choose the point pb0 as the initial point p1, solve problem a and, choosing the point pa0 as the initial point p1, solve problem b. go to step 2. after sufficiently large number of iterations of stages 2 and 3, initial values of the parameters might leave neighborhoods of the non-global minima in one criterion with help of the other and thereby solve the parametric identification problem. the following methods for evaluating the stability indicators and the structural stability of mathematical models are proposed within the framework of elaborating the 2 component of the parametric control theory. 1.3 methods for evaluating the stability of mathematical models of macroeconomic systems 1.3.1 methods for evaluating the weak structural stability of dynamical models the methods of analysis of the robustness (structural stability) of mathematical model of national economic system are based on: fundamental results on dynamical systems theory in the plane; 6 a. ashimov: the theory of parametric control of macroeconomic systems and ... methods of verification of mathematical models belonging to certain classes of structurally stable systems (classes of morse-smale systems, ω-robust systems, -systems, systems with weak structural stability). at present, the theory of parametric control of market economic development has available a number of theorems about structural stability of specific mathematical models (the model of the neoclassical theory of optimal growth; model of national economic system taking into consideration the influence of the share of public expenses and of the interest rate of governmental loans on economic growth; model of national economic systems taking into consideration the influence of international trade and currency exchanges on economic growth; and others) formulated and proved on the basis of the aforementioned fundamental results. along with analysis of the structural stability of specific mathematical models (both with and without parametric control), based on results of the theory of dynamical systems, one can consider approaches to the analysis of structural stability of mathematical models of national economic system by means of computer simulations. we shall consider below the construction of a computational algorithm for estimating the structural stability of mathematical models of national economic system on the basis of robinson theorem (theorem a) on weak structural stability [21]. theorem.let n ′ be some manifold, and n a compact subset in n ′ such that the closure of the interior of n is n. let some vector field be given in a neighborhood of the set n in n ′. this field defines the c1-flux f in this neighborhood. let r(f,n) denote the chain-recurrent set of the flux f on n. let r(f,n) be contained in the interior of n. let it have a hyperbolic structure. moreover, let the flux f upon r(f,n) also satisfy the transversability conditions of stable and unstable manifolds. then the flux f on n is weakly structurally stable. in particular if r(f,n)an empty set, then the flux f is weakly structurally stable on n. a similar result is also correct for the discrete-time dynamical system (cascade) specified by the homeomorphism (with image) f : n → n ′. therefore, one can estimate the weak structural stability of the flux (or cascade) f via numerical algorithms based on this theorem by means of numerical estimation of the chain-recurrent set r(f,n) for some compact region n of the phase space of the considered dynamical system. let us further propose an algorithm of localization of the chain-recurrent set for a compact subset of the phase space of the dynamical system described by a system of ordinary differential (or difference) equations and algebraic system. the proposed algorithm is based on the algorithm of construction of the symbolic image [22]. a directed graph (symbolic image), being a discretization of the advances in systems science and applications (2014) vol.14 no.1 7 shift mapping along the trajectories defined by this dynamical system, is used for computer simulation of the chain-recurrent subset. suppose an estimate of the chain-recurrent set r(f,n) of some dynamical system in the compact set n of its phase space has been found. for a specific mathematical model of the economic system, one can consider, for instance, some parallelepiped of its phase space including all possible trajectories of the economic system evolution for the considered time interval as the compact set n. the localization algorithm for the chain-recurrent set consists of the following: 1. define the mapping f defined on n and given by the shift along the trajectories of the dynamical system for the fixed time interval. 2. construct the partition c of the compact set n into cells ni. assign the directed graph g with graph nodes corresponding to the cells and branches between the cells ni and nj corresponding to the conditions of the intersection of the image of one cell f(ni) with another cell nj . 3. find all recurrent nodes (nodes belonging to cycles) of the graph g. if the set of such nodes is empty, then r(f,n) is empty, and the process of its localization ceases. one can draw a conclusion about the weak structural stability of the dynamical system. 4. the cells corresponding to the recurrent nodes of the graph g are partitioned into cells of lower size, from which a new directed graph g is constructed (see item 2 of the algorithm). 5. go to item 3. items 3, 4, 5 must be repeated until the diameters of the partition cells become less than some given number ε. the last set of cells is the estimate of the chain-recurrent set r(f,n). the method of estimating the chain-recurrent set for a compact subset of the phase space of a dynamical system developed here allows one, in the case in which the obtained chain-recurrent set r(f,n) is empty, to draw a conclusion about the weak structural stability of the dynamical system. in the case that the considered discrete-time dynamical system is a priori the semi-cascade f, one should verify the invertibility of the mapping f defined on n (since in this case, the semi-cascade defined by f is the cascade) before applying robinson’s theorem a for estimating its weak structural stability. let us give a numerical algorithm for estimating the invertibility of the differentiable mapping f : n → n ′, where some closed neighborhood of the discrete-time trajectory {f ′(x0), t = 0, ..., t} in the phase space of the dynamical system is used as n. suppose that n contains a continuous curve l,which sequentially connects the points {f ′(x0), t = 0, ..., t}. one can choose as such curve a piecewise linear curve with nodes at the points of the above mentioned discrete-time trajectory of the semi-cascade. 8 a. ashimov: the theory of parametric control of macroeconomic systems and ... an invertibility test for the mapping f : n → n ′ can be implemented in the following two stages: 1. an invertibility test for the restriction of the mapping f : n → n ′ to the curve l, namely, f : l → f(l). this test reduces to the ascertainment of the fact that the curve f(l) does not have points of self-crossing, that is, (x1 ̸= x2) ⇒ (f(x1) ̸= (f(x2)). for instance, one can determine the absence of self-crossing points by means of testing monotonicity of the limitation of the mapping f onto l along any coordinate of the phase space of the semi-cascade f. let us choose sufficiently large set of points like xi = (x1i , x 2 i , ..., x n i ) ∈ l, yi = f(xi), yi = (y1i , y 2 i , ..., y n i ) and coordinate number of these points (j). if for all xji , i = 1, ..., n at xji1 < xji2 theine quality yji1 < yji2 is met (or at xji1 < xji2 the inequality yji1 > yji2 is met), then f : l → f(l) mapping is estimatedas invertible. 2. an invertibility test for the mapping f in neighborhoods of the points of curve l (local invertibility). based on the inverse function theorem, such a test can be carried out as follows: for a sufficiently large number of chosen points x ∈ l one can estimate the jacobians of the mapping f using the difference derivations: j(x) = det( ∂fi∂xj (x)), x, j = 1, ..., n. here i, j are the coordinates of the vectors, and n is the dimension of the phase space of the dynamical system. if all the obtained estimates of jacobians are nonzero and have the same sign, one can conclude that j(x) = 0 for all x ∈ l and, hence, that the mapping f is invertible in some neighborhood of each point x ∈ l. an aggregate algorithm for estimating the weak structural stability of the discrete-time dynamical system (semi-cascade defined by the mapping f) with phase space n ′ ∈ r′′ defined by the continuously differentiable mapping f can be formulated as follows: 1. find the discrete-time trajectory {f ′(x0), t = 0, ..., t} and curve l in a closed neighborhood n which is required to estimate the weak structural stability of the dynamical system. 2. test the invertibility of the mapping f in a neighborhood of the curve l using the algorithm described above. 3. estimate (localize) the chain-recurrent set r(f,n). by virtue of the evident inclusion r(f,n1) ⊆ r(f,n2) for n1 ⊂ n2 ⊂ n ′, one can use any parallelepiped belonging to and containing l as the compact set n. 4. if r(f,n) = φ, draw a conclusion about the weak structural stability of the considered dynamical system in n. this aggregate algorithm can be also applied for estimating the weak structural stability of a continuous-time dynamical system (the flux f), if the trajectory l = {f ′(x0), 0 ≤ t ≤ t} of the dynamical system is considered as the curve l. in this case, item 2 of the aggregate algorithm is omitted. the mapping f t for some fixed t(t > 0) can be accepted as the mapping f in item 3. advances in systems science and applications (2014) vol.14 no.1 9 1.3.2 methods for evaluating the weak structural stability of dynamical models by definition of orlov [18], the mathematical model of an economic system in general view is some mapping f : a → b transferring values of initial (exogenous) data p ∈ a to solutions (values of endogenous variables) y ∈ b. after constructing a mathematical model of some real-life phenomena or process and defining some actual values of the point p by known measured data or solving the parametric identification problem, the question about adequacy of the analyzed model arises. the condition of model stability relative to admissible perturbations of the initial data [16] is a one of the conditions of the model adequacy. in case of such stability, small perturbations of the model’s initial data results in small changes of its solution. in the mentioned monograph, the definitions of the basic stability indicators are introduced (these definitions are presented below). monograph, however, does not propose any algorithm for computing the considered indicators of the mathematical model stability. below we present the developed algorithms for evaluating the mathematical model stability indicators which characterize stability of solutions of the mathematical model relative to initial data perturbations. at that, all of the model parameters and variables must be made dimensionless beforehand. let x = (x1, x2, ..., xk) be some vector of values of the model exogenous parameters for the time interval t ∈ {0, ..., t}. let x0 = (x1 0 , x 2 0 , ..., x k 0 ) denote the respective vector of base values for the same time interval. the vector that incorporates the values of parameters and initial values of the variables of differential (or difference) equations is considered as the vector x. the vector of measured statistical data used for finding the model equation coefficients is considered as vector x for the econometric models. let p = (p1, p2, ..., pk) be a vector of the normalized input data of the mathematical model where pi = xi xi 0 , i = 0, ..., k. the vector p0 = (1, 1, ..., 1). let be a space of the normalized input data vectors which includes all admissible sets p, a ⊂ rk is a metric space with the euclidean metric defined by the space rk, p0 ∈ a. let y = y (p) = (y 1, y 2, ..., y k) be a selected vector of the values of endogenous variables for some chosen interval (or moment) of time obtained for the selected values of p. the vector that incorporates the values of some selected set of the model endogenous variables for the aforesaid interval (or moment) of time is considered as vector y for the dynamical models. the vector of coefficients of the model equations or vector of values of some selected set of the model en10 a. ashimov: the theory of parametric control of macroeconomic systems and ... dogenous variables for the aforesaid interval (or moment) of time is considered as vector y for the econometric models. in particular, with p = p0, introduce the notation y0 = y0(p) = (y 1 0 , y 2 0 , ..., y k 0 ). the normalized vector of values of the endogenous variables for the moment of time t1 is denoted by y = y(p) = (y 1 y 1 0 , y 2 y 2 0 , ..., y n y n 0 ); y0 = y(p0) = (1, 1, ..., 1). let b ⊂ rn be a region which contains all possible output values y for p ∈ a with the euclidean metric of space rn, y0 ∈ b. the considered model defines the mapping f of set a into set b. for the selected point p ∈ a and number α > 0, let uα(p) denote the intersection of a neighborhood of the point p with radius α with set a: uα(p) = {p1 ∈ a : ρ(p1, p) ≤ α} here and below, ρ(., .) denotes the euclidean distance between two points of the euclidean space. for some subset b1 ⊂ b, let d(b1) denote the diameter of set v1, that is d(b1) = sup(ρ(y1, y2) : y1, y2 ∈ b1) definition1.1. the number β(p, α) = d(f(uα(p))) is defined as the stability indicator of the econometric model at the point for ¿0. algorithm1.1 for evaluating the model stability indicator β(p, α) by the monte carlo method is as follows: 2. define the vector of normalized input data p = (p1, p2, ..., pk), number α > 0, and set uα(p). 3. generate a set of sufficiently large number m of pseudo-random points (p1, p2, ..., pm ) uniformly distributed in β(p, α). for this purpose, consecutively generate the coordinates pij(i = 1, ..., k; j = 1, ...,m) of the point pj in numerical segments [pi − α, pi + α] covering uα(p) using a generator of uniformly distributed pseudo-random numbers. if the inequality k∑ i=1 (pij − pi)2 ≤ α2 holds (i.e. xj ∈ uα(p)), this point is added to the created set. 4. for each point pj of the set, define point yj = f(pj), j = 1, ...,m , by simulation. 5. evaluate β = max(ρ(yi, yj) : i, j = 1, ...,m). 6. stop. with α = 0.01, the obtained number β/2 characterizes the (maximum) percentage change of values of the model output variables under the perturbed input data by 1%. definition1.2. the number β(x) = inf 0≤α≤α0 β(p, α) is called the absolute stability indicator of the econometric model at point x ∈ a. here, α0 is the maximal advances in systems science and applications (2014) vol.14 no.1 11 admissible relative deviation of values of the model input data. algorithm 1.2 for evaluating the absolute stability indicator β(p) of the econometric model is as follows: for the selected value α0 and numbers j = 0, 1, 2, consecutively find (by algorithm 1.1) numbers βj = β(p, α0/2 j), and then evaluate the number β(p) = inf j=0,1,2,... βj if β(p) turns out to be less than some a priori given small number (i.e. β(p) is considered to be approximately zero), then the mapping f defined by the analyzed model is evaluated at point p continuously depending on the input values. definition1.3. the number γ = sup p∈a β(p) is called the maximal absolute stability indicator of the model for region a. algorithm1.3 for evaluating the maximal absolute stability indicator of the model by the monte carlo method is as follows: 1. generate the set of sufficiently large number m of pseudo-random points (p1, p2, ..., pm ) uniformly distributed in . 2. for each point pj in the set and chosen ,α0 > 0 find the numbers β(pj) by algorithm 1.2. 3. determine the number γ = max j=1,...,m β(pj). 4. stop. if the number turns out to be less than some a priori given small number ε(i.e. is considered to be approximately zero), then the mapping f defined by the analyzed model is evaluated in set a continuously depending on the input values. the developed algorithms were applied for evaluating econometric model of a small open economy and computable general equilibrium model of economic branches. 1.3.3 methods for evaluating the stability of mappings, defined by models, in terms of the theory of differentiated mappings singularities this section describes the methods for evaluating in sense of definition the stability of smooth mappings f : d → e, defined by statical or dynamical model [17]. as domains (d) of the mappings in question are used corresponding domains of possible values of uncontrollable, controllable parameters, coefficients of the model econometric equations in question, and also domain of possible values of observed (statistical) data, used for functions building, which set models econometric equations of the model. range of values (e) of the mapping f contains 12 a. ashimov: the theory of parametric control of macroeconomic systems and ... set of possible values of endogenous variables of the model. existence of such stability property indicates preservation of qualitative properties of mapping, using which the model is described, at small variances of this mapping. when real economic phenomena are described adequately using the mathematical model, stability (or instability) of the mapping, presented by the model, may indicate stability (or instability) of corresponding dependencies of possible values of economic indicators on external (controllable or uncontrollable) factors at small variances of these dependencies. instability of mapping, set by the model, may also indicate inadequacy of the model in question [17]. an algorithm for estimating critical point set of the mappings in question is presented within the framework of this study of the given mappings stability. this algorithm, in particular, allow sestimating the maximality of jacobian matrix rank of mapping at all points of its domain, i.e. to check whether either the investigated map is immersion or submersion. for the immersion case, an algorithm for estimating injectivity of the mapping is proposed. there are formulated and proved the statements, which allow to estimate the stability (and in some case ratios of the image dimension to the counter image dimension-instability) of the mapping in question, when conditions of immersion and injectivity or condition of submersion are satisfied for the mapping in question. there are presented the statements about stability conditions for the mapping in question in the case if this mapping is submersion with fold [23]. there is presented the algorithm, which allows to estimate the mapping as submersion with fold and estimate the stability of the mapping in question in this case. 1.3.3.1 algorithm for estimating the critical point set of the mappings, set by the model hereinafter, we will imply the mapping, defined by the mathematical model when we use the mapping f : d → e (1) let’s denote the mapping arguments vector (1) through p = (p1, ..., pn) ∈ d, and corresponding p point image the model solutions vector denote through y = y(p) = y(p1, ..., pυ) ∈ e, (d ⊂ rn and e ⊂ rυ are some regions). in this case, jacobian matrix with size vn for the mapping (1) at the point p will be written in the following form: j(p) = ( ∂yi ∂pj (p) ) i=1,...,υ;j=1,2,...,n (2) we will also denote the jacobian matrix estimate (2) at some point p ∈ d, obtained by numerical differentiation from (1), through j(p). advances in systems science and applications (2014) vol.14 no.1 13 within the framework of the solution of f mapping stability studying problem, we will present an algorithm, which allows to estimate the j(p) matrix rank maximality for p ∈ d, that is the algorithm for condition estimate rank((j(p))) = min(υ, n), p ∈ d (3) when this condition is satisfied, the mapping f will noth ave any critical point in the domain d [17]. for (3) condition estimate to find any nonsingular minor (of the matrix j(p) of order min(υ, n)) for each p ∈ d is enough, taking into account that total amount of maximal-order minors in j is l = cn υ = υ! n!(υ−n)! if n < υ and l = cυ n = n! υ!(n−υ)! , if n ≥ υ. we will denote the determinant value estimate of such minor of order min(υ, n)) in j(p) for p ∈ d through |mi(p)|, i = 1, ..., l. algorithm1.4. the aggregate algorithm for estimating condition(1.3)for estimation of mapping critical point set. 1) domain d divides to sufficiently large amount of (elementary) parallelepipeds dk of the same size, and define the net p from n points, those are vertices of chosen parallelepipeds: p = {pj : j = 1, ..., n}. 2) compute the values of all j(pj) matrix elements for j = 1, ..., n . 3) for i = 1, ..., l, j = 1, ..., n , compute the determinants |mi(pj)|. 4) for each i = 1, ..., l the set d(i) is defined in the following way. d(i) isa sumof all (closed) parallelepipeds dk, which have property that not all values of |mi(pj)| at dk vertices have the same sign. 5) find the set d̃ = ∩l i=1d(i). 6) if the set d̃ is empty, then condition (3) is evaluated as satisfied. stop. 7) otherwise,the steps 1) 6) of present algorithm are performed substituting domain d by d̃ and diminishing sizes of parallelepipeds, participating in partitioning of d̃. sufficiently large amount iteration of the steps of the presented above algorithm allow either to estimate satisfaction of the condition (3) or to obtain the estimate (using set d̃)the set of critical points of f mapping. 1.3.3.2 algorithm for estimating the nonlocal injectivity of mapping,set by model in this section it assumes that n < υ (d domain dimension of mapping (1) is less than erange of values dimension). this section presents an algorithm for estimating conditions of nonlocal injectivity (absence of non-close points in d, having equal images at mapping f) of mapping (1), set by model in domain d. satisfaction of mentioned nonlocal injectivity condition and condition (3), ensuring local injectivity in neighborhood of each point means an existence of inverseto fmapping, determined in set f (d). fix sufficiently small number ε > 0. for each point pj ∈ p through d(pj) 14 a. ashimov: the theory of parametric control of macroeconomic systems and ... denote set of all points pk ∈ p which have |pj − pk| > ε. here | · | is magnitude of vector. algorithm1.5 for estimating the condition of the nonlocal injectivity is presented in terms of the following steps. 1) compute the numbers mj = min pk∈d(pj) |f (pk)− f (pj)| for each point pj ∈ p . 2) compute m == minpj∈pmj and determine the points pj , pk ∈ p such that |f (pk)− f (pj)| = m. 3) repeat steps 1) and 2) of this algorithm substituting the net p by net p1 being the vertices of less size parallelepipeds and containing all points of the net p, being distant from one of pj , pk points by the distance not exceeding 2ε. there are two possible cases given sufficiently large amount of iterations of steps of the presented above algorithm. a) the sequence of obtained values are diminishing about proportional to the p1 net step. in this case,f mapping is estimated as non-injective (in other words, f (d) set is estimated as self-crossing). b) condition m > ε is met for all nets in question with sufficiently small step. in this case, f mapping is estimated as injective (in other words, f (d) set is estimated as non-self-crossing). 1.3.3.3 estimating the stability of mappings set by the model this section presents propositions about sufficient conditions for f mapping stability in open domain d0 = d \ γ(d), where γ(d) is boundary of set d, within the framework of determining stable mapping [17]. it also presents an aggregate algorithm for estimating mapping f : d → e, as stable submersion with fold. according to stability theorem of mazer [17], mapping f is stable in manifold d0, if it is infinitesimally stable in d0. condition of infinitesimal stability of mapping h(p) in k(y) is formulated in terms of solubility in relative to mappings and of the following homologous equation [17]. µ(p) = −j(p)h(p) + k(f (p)) (4) here µ(p) is an arbitrary infinitesimal deformation of f mapping, which is presented in terms of smooth correspondence to each point p ∈ d0 of tangent vector to manifold e atpoint f (p); h(p) is the smooth vector field in d0; k(y) is the smooth vector field in e. consider two possible cases. 1. let n < υ let condition (3) be satisfied, that is f mapping is the immersion advances in systems science and applications (2014) vol.14 no.1 15 in d. according to the theorem about inverse function (when f mapping injectivity condition is satisfied, which checks by the algorithm described in item 2) in set f (d0) the smooth mapping f−1 : f (d0) → d0, inverse to f is defined. if in equation (4) input h(p) = 0, and k(y) = µ(f−1(y)), then this equation become an identity µ(p) = 0+µ(f−1(f (p))). this means that the mentioned mappings, h(p) and k(y), are solutions of homologous equation (4), that is this equation is soluble. consequently, the first statement of the following proposition is true. proposition 1.3.1. let given n < υ for all points of chosen set d,condition (3)for injective mapping(1.1)is satisfied.then mapping(1) is stable in domain d0 = d \ γ(d). if condition(3) is not satisfied for any point p ∈ d0, then mapping(1) is not stable in its domain. the second statement of the proposition 1.3.1 results from the sentences 2.4 and 3.12 [23].formulate them. statement 2.4. let x is compact and dimy ≥ 2dimx+1. mapping f : x → y is stable, if and only if f isone-to-one immersion. statement3.12. let x is compact manifold and dimy = 2dimx. mapping f : x → y is stable, if and only if it is an immersion with normal crossings. from these sentences results that when condition dimy ≥ 2dimx is satisfied, the immersion property is a requirement for stability, that is when (3)is not satisfied, mapping f cannot be stable. note that although in formulations of mentioned sentences x manifold compactness condition is used, proofs of immersion properties for stable mappings given dimy ≥ 2dimx + 1(or dimy = 2dimx) are based on the theorems 5.6 and 5.7 [23], which do not require x manifold compactness. proposition 1.3.1 is fully proved. 2. let n ≥ µ let condition (3) be satisfied, that is f mapping is a submersion in d0. test, in this case, the solubility of homologous equation (4). consider first the case, when some v-order minor determinant mi(p) of jacobian matrix has constant sign for p ∈ d0, that is, |mi(p)| ≥ ε ≥ 0 in d0. let, for determinacy, such anon singular minor mi(p) consists of first six columns of matrix j(p). for arbitrary deformation µ(p), written in terms of v-dimensional column vector in equation (4) determine h(p) in the following way. first v coordinates of n-dimensional column vector h(p) set using column −(mi(p)) −1µ(p), and all other coordinates of vector h(p) assume to be zero. if put k(y) = 0, then equation (4) becomes the identity: µ(p) = −j(p)h(p) + 0 = mi(p)(mi(p)) −1µ(p) (5) consider the general case of satisfaction of condition (3). let for each point p ∈ d (and some its neighborhood) be found its nonsingular minormi(p). choose from such neighborhoods uj finite domain cover: d ⊂ u s j=1uj . make subject to this cover the partition of unity into domains d in terms of s smooth functions 16 a. ashimov: the theory of parametric control of macroeconomic systems and ... φj(p) ≥ 0 [23], where φj(p) = 0 at p ∈ rn\uj(j = 1, ..., s) and ∑s j=1 φj(p) = 1 for p ∈ d. consider arbitrary deformation µ(p) in equation (4). then, for each neighborhood uj and its corresponding nonsingular minor mi(p), make (according to the method proposed in the previous paragraph) the vector field hj(p) in uj , which is the solution of equation (4) in uj at k(y) = 0. determine the vector field h(p) in d0 using the formula: h(p) = s∑ j=1 φj(p)hj(p) thus, from (4) (where h(p) should be substitute hj(p)) results that the vector fields h(p) and k(y) = 0 are the solutions of (4): −j(p)h(p) + k(f (p)) = −j(p) s∑ j=1 φj(p)hj(p) + 0 = s∑ j=1 φj(p)[−j(p)hj(p)] = s∑ j=1 φj(p)µ(p) = µ(p) s∑ j=1 φj(p) = µ(p) this means that the equation (4) is soluble. therefore, the following proposition is true. proposition 1.3.2. let given n ≥ υ for chosen set d condition(3)be satisfied for mapping (1).then mapping(1) is stable in domain d0 = d \ γ(d) 3. now consider the case, when given n ≥ υ condition (3) is not satisfied for some points of domain d0 (that is, the case, when domain d0 contains the critical points of mapping f). denote by s1(f ) set of points of domain d0, in which jacobian matrix of mapping frank is less by unit than the maximal one, that is s1(f ) = {p ∈ d : rank(j(p) = υ − 1} it is known that when the additional condition (j1f ◃▹ s1, where j1f is 1stream of mapping f, s1 is submanifold in the space of 1-streams j1(d0, e) consisting of streams with 1 co-rank, ◃▹ is the sign of transversality), is satisfied, the set s1(f ) is a submanifold in d0 with dimension υ − 1 [23]. definition. let the mapping f : d0 → e satisfy condition j1f ◃▹ s1. point p ∈ s1(f ) is called as fold point, if sum of tangent space to s1(f ) and kernel of tangent mapping df at this point has dimension n, that is, if tps1(f ) +ker(df )p = tpd (6) since sum of dimensions of summands of lhs is equal to the dimension of rhs (υ − 1) + (n − υ + 1) = n, then condition (6) is equivalent to that these advances in systems science and applications (2014) vol.14 no.1 17 summands have the only common point origin of coordinates and cos∠(tps1(f ),ker(df )p) ̸= 1 (7) definition. mapping f : d0 → e is called a submersion with folds, if eachits singular pointis a fold point. in this case submanyfold s1(f ) is called as a fold. it is known that if f : d0 → e is submersion with fold, then f mapping constraint on s1(f ) fold is immersion. the following theorem is valid [23]. consequence 1.3.4. if f : d0 → e is submersion with fold and f |s1(f ) is injective, then mapping f : d0 → e is stable. we present an algorithm for estimating the satisfaction of these conditions of this consequence. algorithm 1.6. aggregate algorithm for estimating mapping f : d0 → e as stable submersion with fold. let given n ≥ υ, after use of algorithm 1.4, the estimate of singular points set of the mapping fin terms of non-empty set d̃ ⊂ d and set of p̃ vertices of elementary parallelepipeds, containing d̃, be found. algorithm steps for testing transversality condition (j1f ◃▹ s1) is not presented here because of unhandiness. we will assume that condition j1f ◃▹ s1 is estimated as satisfied. 1. for each primary parallelepipeddk ⊂ d̃, satisfaction of condition rank(j(p)) = υ − 1 is estimated in the following way. let {pkj }2 n j=1 be set of parallelepiped vertices dk; {|mi(p k j )|}li=1 is set of values of υ − 1-order minor determinants of jacobian matrix j(pkj ) at point pkj ; l = cυ−1 n n = n!n (υ−1)!(n−υ+1)! . if for chosen dk one can find such number i, that all determinants |mi(p k j )| for j = 1, ..., 2n have the same sign, then rank(j(p)) is estimated by number υ − 1 in parallelepiped dk. if rank(j(p)) is estimated by number υ − 1 for all dk ⊂ d̃, then set d̃ is considered to be the estimate of submanifold s1(f ). otherwise, if one can find such parallelepiped dk ⊂ d̃, that for each chosen i = 1, ..., l numbers |mi(p k j )| for j = 1, ..., 2n have different signs, then subset d̃ is not estimated as a fold. stop. 2 the next steps 3, 4, 5 are performed for each point of the net p̃ . 3. for p ∈ p̃ basis vectors (e1, ..., eυ−1) of tangent space tps1(f ) are estimated in the following way. choose m (where m ≫ υ−1) close (except this point itself) to p points of the net p̃ : {p1, ..., pm}. identify the following set of m vectors {fi = pi − p : i = 1, ...,m}. here the points pi and p are consider edasradiusvectors. linear envelope of arbitrary set of n-dimensional vectors (e1, ..., eυ−1) denote by t = t (e1, ..., eυ−1). the distance from point fi up to the plane t 18 a. ashimov: the theory of parametric control of macroeconomic systems and ... denote by d(fi, t ). sum of squares of distances from points fi to t denote by s(e1, ..., eυ−1) = m∑ i=1 (d(fi, t )) 2 (8) coordinates of required e1, ..., eυ−1 vectors are determined by the least squares method from the condition of s(e1, ..., eυ−1) function minimum. 4. (g1, ..., gυ−n+1) basic vectors of ker(df )p kernel for p ∈ p̃ are estimated in the following way. 4.1. since mapping matrix (df )p with theoretical rank υ − 1 is estimated by numerically found jacobian matrix j(p) , all υ − 1-order minors of which have determinants close to zero, we firstly determine the row close to linear combination of other rows of the matrix j(p). in case, if matrix rank is less by unit than the number of its rows, according to the theorem about principal minor, one of matrix rows is a linear combination of its other rows and its elimination does not change the kernel of the linear operator, appropriate to this matrix. let {j1, j2, ..., jυ} be the set of all normalized (the elements of each row divide to the magnitudeof this row, if magnitude of any row is zero, then the problem in item 4.1 is solved) row of j(p) matrix. let p i, (i = 1, ..., υ) be linear envelope of all rows of mentioned set, except the row j i, which is considered as plane in the space rn. let mi be the distance from the point j i ∈ rn to plane p i : mi = d(j i, p i). mi value can be found by finding minimum of function of υ − 1 variable (α1, ..., αi−1, αi+1, αυ): di(α1, ..., αi−1, ..., αi+1, αυ) = |j2 − υ∑ j=1,j ̸=i αij j |2 choose number i relevant to minimal number mi. denote the matrix j(p) of size (υ − 1)× n with removed i-row by j̃(p). 4.2. solve the linear homogeneous system from (υ − 1) equation with n unknowns and with matrix of system j̃(p) by gauss method. herewith, basis of its decision space (g1, ..., gυ−n+1) is found. 5. for p ∈ p̃ estimate angle cosine (cosφp) between the planes tps1(f ) and ker(df )p in rn in the following way. let (e1, ..., eυ−1) and (g1, ..., gυ−n+1) be the found above estimates of bases of these planes; (α1, ..., αυ−1) and (β1, ..., βυ−n+1) be variables sets. let the vector e = ∑υ−1 i=1 αiei be arbitrary vector of plane tps1(f ), vector g = ∑υ−1 i=1 βigi be arbitrary vector of plane ker(df )p. define function f (expressing angle cosine between the vectors e and g) on n variables in terms of (α1, ..., αi−1, β1, ..., βn−υ+1) y = f(α1, ..., αi−1, β1, ..., βn−υ+1) = fg |f ||g| advances in systems science and applications (2014) vol.14 no.1 19 its maximal value takes as the value of required φp. 6. choose small number ε > 0. if for all p ∈ p̃ condition cosφp < 1− ε is met, then based on the definition, d̃ set is the fold estimate and mapping f : d0 → e is estimated as the submersion with fold. otherwise, if p ∈ p̃ is found, for which cosφp ≥ 1 − ε, then d̃ set is not estimated as fold. stop. 7. estimate the injectivity of the mapping f on fold d̃. since f mapping constraint on fold is immersion then local injectivity of this constraint is guaranteed. nonlocal injectivity is estimated by algorithm 1.5, in which one should substitute the net p by the net p̃ . number ε, used in this algorithm, should exceed doubled diameter of elementary parallelepipeds of d̃ set. two cases are possible given sufficiently large amount of step iterations of algorithm 1.5. a) the sequence of obtained m values are diminishing about proportional to the p̃1 net step. in this case, f mapping constraint on the fold d̃ is estimated as non-injective (in other words, f (d̃) set is estimated as self-crossing). additional study of f (d̃) self-crossing points for normality is required. b) condition m > ε is met for all nets in question with sufficiently small step. in this case, f mapping constraint on the fold d̃ is estimated as injective (in other words, f (d̃) set is estimated as non-self-crossing). based on the consequence 1.3.4, the mapping f : d0 → e, in this case, is estimated as stable. references [1] r.a. mundell. (2000), “a reconsideration of the twentieth century”, the american economic review, vol.90, no.3. [2] j.a. frenkel, a. razin. (1987), “the mundell-fleming model: a quarter century later”, national bureau of economic research, working paper , cambridge, massachusetts, no.2321. [3] c. erceg, l. guerrieri, c. gust. (2005), “sigma: a new open economy model for policy analysis”, board of governors of the federal reserve system, international finance discussion papers, july, no.835. [4] j. andrés, p. burriel, á estrada. (2006), “bemod: a dsge model for the spanish economy and the rest of the euro area”, documentos de trabajo banco de españa, no.0631. [5] n. stähler, c. thomas. (2011), “fimod-a dsge model for fiscal policy simulations”, deutsche bundesbank discussion paper. series 1: economic studies, no.06. [6] s. paltsev, j.m. reilly, h.d. jacoby, r.s. eckaus, j. mcfarland, m. sarofim, m. asadoorian, m.h. babiker. (2005), “the mit e20 a. ashimov: the theory of parametric control of macroeconomic systems and ... missions prediction and policy analysis (eppa) model: version 4”, http://globalchange.mit.edu/research/publications/697, report no.125. [7] p.d. dixon, m.t. rimmer. (1998), “forecasting and policy analysis with a dynamic cge model of australia”, preliminary working paper,no.op-90, http://www.monash.edu.au/policy/elecpapr/op-90.htm. [8] v.l. makarov, a.r. bahtizin, and s.s. sulakshin. (2007), “application of computable models in state administration”, moscow: nauchniy expert, russian. [9] g. gandolfo. (2014), “international trade theory and policy”, springer heidelberg new york dordrecht london. [10] k. farmer, m. schelnast. (2013), “growth and international trade. an introduction to the overlapping generations approach”, springer heidelberg new york dordrecht london. [11] r.c. (2004), “fair, estimating how the macroeconomy works”, cambridge, massachusetts: harvard university press, london, england. [12] s. murchison, a. rennison. (2006), “to tem: the bank of canada’s new quarterly projection model”, bank of canada technical report, no.97. [13] m. adolfson, s. laséen, j. lindé and l.svensson. (2011), “optimal monetary policy in an operational medium sized dsge model”, journal of money, credit and banking, blackwell publishing, vol.43(7), pp.1287-1331. [14] d. acemoğlu. (2008), “introduction to modern economic growth”, princeton university press. [15] f.j. andré, m.a. cardenete, and c. romero. (2010), “designing public policies. an approach based on multi-criteria analysis and computable general equilibrium modeling”, lecture notes in economics and mathematical systems, 1st ed, vol. 642, springer. [16] v.i. arnold. (1988), “geometrical methods in the theory of ordinary differential equations”, springer-verlag, new york. [17] v.i. arnold, s.m. gusein-zade, a.n. varchenko. (1985), “singularities of differential maps. volume i”, basel, stuttgard: birkhauser, boston. [18] a.i. orlov, econometrics. (2002), “a textbook”, ekzamen, moskow, russian. advances in systems science and applications (2014) vol.14 no.1 21 [19] a.a. ashimov, b.t. sultanov, zh.m. adilov, yu.v. borovskiy, d.a. novikov, r.a. alshanov, as.a. ashimov. (2013), “macroeconomic analysis and parametrical control of a national economy”, springer, new york. [20] j.a. nelder, r. mead. (1965), “a simplex method for function minimization”, the computer journal, no.7, pp.308-313. [21] robinson c. (1980), “structural stability on manifolds with boundary”, journal of differential equations, no.37, pp.1-11. [22] e.i. petrenko. (2006), “development and realization of the algorithms for constructing the symbolic set”, differential equations and control processes (electronic journal), no.3, pp.55-96. [23] . golubitsky, gueilleminv. (1973), “stable mappings and their singularities”, new york, heidelberg, berlin: springer-verlag. corresponding author a. ashimov can be contacted at: ashimov37@mail.ru advances in systems science and applications (2014) vol.14 no.2 183-189 competitive equilibrium of a sequence of incomplete markets with a continuum of agents guo-sheng zhang the school of economics, trade and event management, beijing international studies university, beijing, china, 100024 abstract by introducing the large economical ideas to multi-period financial market, we have constructed the multi-period economy with incomplete market and a continuum of agents. the competitive equilibrium has been proposed and the existence has been claimed. our equilibrium definition is a development compared to that described by radner, and our conclusion for equilibrium existence has generalized the related results obtained by aumann and zhang, if only the future contracts and goods are traded on security-spot markets. keywords perfect competition, equilibrium, incomplete markets, correspondence integral 1 introduction consider a sequence of markets at successive dates, no one of which is complete in the arrow debreu sense, i.e., at every date and for every commodity there will be some future dates and some events at those date for which the spot goods and future contracts contingent on those events are traded. for such economy, radner had first proposed the concept of common expectations that require traders to associate the same future prices to same future exogenous events[1]. an equilibrium is a set of prices at the first date, a set of common price expectations for the future, and a consistent set of individual plans for agents such that, given the current prices and price expectations, each individual agents plan is optimal for him, subject to an appropriate sequence of budget constraints. radner’s common expectation is a foundation for modern incomplete market theory. but a basic assumption of such model is that the current prices and price expectations for future contracts be not affected by a single agents action, which needs the market be perfect competition. otherwise a change in an individuals offer to buy or sell can easily upset the prevailing prices, so that the equilibrium will never be achieved, and price system is meaningless. as early as 1964, aumann had suggested that the most natural mathematical model for a commodity market with such perfect competition is one in which there are a continuum of traders (like the continuum of points on a line)[2]. for decades, aumann’s large economy has always been a vigorous field of economics. a deficiency of modern incomplete market theory is the lack of introduction of aumann’s large economy ideas. as a novelty, zhang first discussed security-spot markets with a measurable space of agents[3-4], where aumann’s ideas have been 184 guo-sheng zhang: competitive equilibrium of a sequence of incomplete markets with ... applied to security-spot market, particularly, to financial markets. but these research works are primary: the model is two-periods, and the existing discussion needs to proceed technically. in this paper, by combining aumann’s ideas with radner’s model, we discuss the economy with a sequence of incomplete markets and a continuum of agents. first, the concept of a competitive equilibrium for such economy is proposed, which has developed radner’s definition. then, we claim the existence of equilibrium. the sufficient conditions are all used by aumann and radner[5-6]. for the security-spot market with goods and future contracts, our conclusion is the generalization of related results proved by aumann and zhang[4,5]. 2 a multiperiod model for a sequence of incomplete markets with a continuum of agents consider an economy extending through a finite sequence of elementary dates 1, 2, ..., t , in an environment with a finite set s of alternative states. the set of events observable at date t will be represented by partition φt of s . it is assumed the sequence of partition, φt is monotonous, no decreasing in fineness, that is, φt+1 is as fine as φt 1 . also, take φ1 = {s}. for each date, there is a finite set of commodities, numbered 1, 2, ..., l. trade contract (such as future contract) at date t in event a, denoted by θhtu(a,b), specifies the number of units of commodity h that the trader will receive from the market at date u ≥ t in event b(θhtu(a,b)) < 0 means the delivery to market; u > t means a future trade and u = t means a spot trade). for each pair of dates t and u such that u > t , and each commodity h, there is a given family f h tu of events, which is either empty or is a partition of s. in the latter case, φu must be as fine as f h tu. assume that . assume further that if f h tu is not empty and t ≤ v ≤ u, f h vu is as fine as f h tu. in other word, if at date t, one can buy a contract for receipt at date u contingent on event b, then at a later date v , one can do the same. a portfolio plan, which was described as a trade plan by radner, is an array (θhtu(a,b)), one for each combination(h,t,u,a,b) such that for a in φt , b in f h tu, b ⊆ a, t ≤ u. the security price paid at date t in event a for receipt of commodity h at date u in event b will be denoted by phtu(a,b) .when t = u, phuu(a) is spot price of commodity h. an array p = {phtu(a,b)} will be called a commodity price system. in the situation just described, there is for each event pair (t,a), with a ∈ φt, a market in contracts for current and future receipt, with cost to be made currently in units of account. to simplify the notation, let denote the set of all pair ( t,a) such that, t = 1, 2, ..., t , a ∈ φt. endow order for m = (t,a) and n = (u,b) in 1φ is said to be as fine as partition φ′ if, for every a’ in φ ,either a ⊂ a′ or a’ ∩ a = ϕ advances in systems science and applications (2014) vol.14 no.2 185 m : m ≤ n if and only if t ≤ u, a ⊇ b. we assume short sales have low bound l, −l ∈ r++. then all of the portfolio plans can be denoted as z = ×m∈mzm with zm denoting the vector space of all arrays of number θhtu(a,b) ≥ l; h = 1, 2, ..., l; u = t, t+ 1, ..., t ; b ⊆ a,b ∈ f h tu for each m = (t, a) ∈ m . thus a portfolio θ = (θm)m∈m,θm ∈ zm, is a point in z. for m = (t, a), n = (u,b), if u ≥ t and b ⊆ a, a ∈ φt, b ∈ f h tu, we denote θhmn = θhtu(a,b); or θmn = (θ1mn, θ 2 mn, · · · θlmn) = 0. the payment at m for portfolio plan θ = (θm)m∈m , θm = (θmn)n≥m ∈ zm,given the commodity price system p = (pm)m∈m , is the inner product pm · θm. let {a,f, u} be the measure space of agents. for any agent a ∈ a, we denote his preference as ≺a, which is a binary relation on (×m∈mrl +)× (×m∈mrl +). we assume the ≺a is complete, transitive, reflexive and satisfies a) closeness: the set {(x, y) ∈ (×m∈mrl +) × (×m∈mrl +)|x≺ay} is closed in (×m∈mrl +)× (×m∈mrl +) and b) monotony: if x,y are two points in ×m∈mrl + , such that x < y, then x≺ay. the set of all preference relations on ×m∈mrl + satisfying all of these assumptions is denoted by β. we endow β with the topology of closed convergence on (×m∈mrl +) × (×m∈mrl +). for each a ∈ a, we require ε = (≺a, ea) : a → β × (×m∈mrl +) is measurable and ea is integrable, where ea ∈ ×m∈mrl + is the real endowment of agent α. we call ε large sequence of security-spot market. each agent must select a consumption plan xa = (xam)m∈m ∈ ×m∈mrl +, and a portfolio plan θa = (θam)m∈m ∈ z within his budget set. an pair (xa, θa) is called an assign if (xa, θa) : a → (×m∈mrl +)× z is integrable (we endow z with bore -field). give p = (pm)m∈m , agent a′s budget set is defined as xa(p) = {(x, θ) : x = (xm)m∈m ∈ ×m∈mrl +, θ = (θm)m∈m ∈ z, such that for each m ∈ m,pm · θm ≤ 0, xm ≤ eam + ∑ j≤m θjm} the agent a′s demand correspondence is then defined as ξa(p) = {(x, θ) ∈ xa(p)|∀(x′, θ′) ∈ xa(p), x≻ax ′} definition 2.1 an equilibrium of the economy ε is an assign (xa, θa) of consumption-portfolio plan and a price system p such that a · e, a ∈ a, (xa, θa) ∈ ξa(p) (1)∫ a θadu = 0, ∫ a xadu = ∫ a eadu (2) where ∫ a θadu = 0 means ∫ a θamdu = 0 for each m ∈ m . this definition has obviously generalized radner’s definition for pure exchange 186 guo-sheng zhang: competitive equilibrium of a sequence of incomplete markets with ... economy, that is, the finite market participants is replaced by infinite market participants. note that (1) implies a.e., a ∈ a, (xa, θa) is the best consumptionportfolio in his budget set and (2) implies the market is clear at each dataevent pair. our definition is also a development for the equilibrium described by debreu[7], which only considered spot-market with one-period. but compared with general incomplete market as that geankoplos proposed[8], we only consider future contracts as securities here. equilibrium existence our main result can be described as the following. theorem 3.1 if {a,f, u} is atomless2 and endowment satisfies ∫ a eam ≫ 03, for any m ∈ m , then economy ε has equilibrium. for each m ∈ m , let p = ×m∈mpm, pm be the set of all nonnegative vectors in zm whose coordinates sum to 1.denote the excess demand correspondence as ξ(p) = ∫ a ξa(p)du− ∫ a (ea, 0)du 4, here (ea, 0) ∈ (×m∈m )rl +×z. we will give some properties of ξ(p) restricted on . for the reason of shortening this paper, some simple proof processes similar to that used by debreu are omitted here[9]. proposition 3.1 under conditions of theorem 3.1, ξ(p) is non-empty, compact, lower bounded and upper hemicontinuous at every p ≫ 0 in p . proof. we only prove ξ(p) is upper hemicontinuous. we first prove xa(p) is continuous at each p ≫ 0 in p .the graph of correspondence xa(p) is obviously closed in p× (×m∈mrl +)×z. thus, xa(p) is upper hemicontinuous on p . to show that xa(p) is lower hemicontinuous at any point po ≫ 0 in p , we consider a pt sequence in p converging to po(t → ∞) and a point (xo, θo) ∈ xa(p o).denote xo = (xom)m∈m , θo = (θom)m∈m , θom ∈ zm . we will discuss the problem under two cases. i) pom · θom < 0 for any m ∈ m . because of ptm → pom, we have ptm · θom < 0 for large enough t. so (xo, θo) ∈ xa(p t), which satisfies the condition appearing in the definition of lower hemicontinuity. ii) ∃m1,m2 · · · ,mg ∈ m , such that pomj · θomj = 0, j = 1, 2, . . . , gand pom · θom < 0, for m ̸= m1,m2 . . . ,mg. for any j, we select a point θ ′ mj ∈ zmj satisfying pomj · θ′mj < 0. this is possible because −l > 0.thus pomj ·θ′mj < pomj ·θomj , and for large enough t, the hyperplane {θmj ∈ zmj |ptmj · θmj = 0} intersects the straight line through θ′mj and θomj in a unique point θ t mj . 2{a,f, u} is called atomless if, for any b ∈ f , u(b) > 0, there is c ∈ f such that 0 < u(c) < u(b). 3for x ∈ l, x ≫ 0 means all of its coordinates are strictly positive. 4for integrals of correspondences and their properties, see definition of part , d. of [7]. advances in systems science and applications (2014) vol.14 no.2 187 define θt as θt = (θtm)m∈m , θtm =  θom, m ̸= m1,m2, · · · ,mg θ t m, m = mj and θ t mj is between θ′mj and θomj 0, others. it is easily checked that θtm → θom(t → ∞) for any m ∈ m , and ptm · θtm ≤ 0. let xom = (xohm ), h = 1, 2, . . . , l, eam = (eahm ), h = 1, 2, . . . , l for each m ∈ m . for any h, if xohm < eahm + ∑ j≤m θohjm, we can take xthm such that xth < eahm + ∑ j≤m θthjm, and xthm → xohm (t → ∞). this is possible because xohm < eahm + ∑ j≤m θthjm for large enough t. if xohm = eahm + ∑ j≤m θohjm, then we take xthm = eahm + ∑ j≤m θthjm, which also implies xthm → xohm . therefore, we can find xt = (xtm)m∈m → xo such that (xt, θtm) satisfies the condition for lower hemicontinuity of xa(p). note that xa(p) is nonempty ((0, 0) ∈ xa(p)) and compact, according to debreu[10],ξ(p) is upper hemicontinuous at each p ∈ p , p ≫ 0 . # proposition 3.2 under the conditions of theorem 3.1, ξ(p) satisfies boundary condition, that is, if pt ≫ 0 in converges to p0 in ∂p, then d(0, ξ(pt)) → (t → ∞). proof. by the similar discussion to debreu (1982, p. 729), the proposition holds if only for a.e.,a ∈ a d(0, ξ(pt)) → ∞. suppose that the conclusion does not hold. then there is a subsequence (pt ′ ) such that d(0, ξa(p t′)) is bounded. for each t′, one can select (ct ′ , θt ′ ) ∈ ξa(p t′) in such a way that sequence (ct ′ , θt ′ ) is bounded. therefore, one can extract from (pt ′ , ct ′ , θt ′ ) a sequence (pt ′′ , ct ′′ , θt ′′ ) converging to (po, co, θo). by proposition 3.1 we have (co, θo) ∈ ξa(p o). since po ∈ ∂p , there is mo ∈ m such that some coordinates of pomo is zero. without loss of generality, we suppose the first coordinate of pomo is zero. then by replacing the first coordinate of θomo with large one, we can get θ′ = (θ′m)m∈m ∈ z, satisfy in pom · θ′m ≤ 0 and eau + ∑ j≤u θ′ju > eau + ∑ j≤u θoju. thus, we can obtain x′ = (x′m)m∈m , such that (x′, θ′) ∈ xa(p o) and x′ > x0. this contradicts the monotony of preference ≺a.# proposition 3.3 under conditions of theorem 3.1, walras law holds, that is, for any p ∈ p , p ≫ 0 and (x, θ) ∈ ∫ a ξa(p)du, we have (pm · θm)m∈m = 0 and xm = ∫ a eamdu+ ∑ j≤m ∫ a θajmdu. proof. for any a ∈ a, (x, θ) ∈ ξa(p), by the monotony of preference ≺a, it is easily proven that (pm · θam)m∈m = 0 and xam = eam+ ∑ j≤m θajm. by integrating the two sides of (pm · θam)m∈m = 0 and xam = eam + ∑ j≤m θajm, the conclusion holds. # 188 guo-sheng zhang: competitive equilibrium of a sequence of incomplete markets with ... proposition 3.4 under conditions of theorem 3.1, is convex-valued. noticing the fact that ξa(p) is convex-valued for any a ∈ a, the proposition is the direct corollary of theorem 3 of part i, d.ii of [9]. proof of theorem 3.1. for the real number b > 0, let b·pm = {b·p : p ∈ pm}. notice that if we replace p by p′ : p′ = ×m∈m (bm · pm), all of the conclusions of proposition 3.1-proposition 3.4 are true. assume the number of dimensions of pm is vm, and ∑ m∈m νm = ν. let p ′ m = νm ν · pm, p′ = ×m∈mp ′ m, and e = {p ∈ p′|p ≫ 0, there is(x, θ) ∈ ξ(p), such that ∑ n≥m,n,m∈m l∑ h=1 θhmn ≤ 0}. since ξ(p) is bounded below, when p varies in e, the θ with (x, θ) ∈ ξ(p) remains bounded. so does χ. so, by the boundary conditions, there cant be in e a sequence pt converging to p0 in ∂p ′. consequently, the distance from p ∈ e to ∂p ′ is bounded below by a strictly positive real number. thus, there is a closed convex cone c with vertex 0 in ×m∈mrm, such that e ⊂ intc and c\0 ⊂ int(×m∈mr+ m), where rm denotes the vector space of all arrays of real number ghmn ∈ r for any n = (u,b) ≥ m,b ∈ f h mn and h = 1, 2, ..., l, and r+ m is the positive cone of rm. let ξ′(p) = {θ ∈ z : thereis(x, θ) ∈ ξ(p)}. if we restrict ξ′(p) on p ′, it is easily proved that ξ′(p) is convex-valued, bounded below and satisfies walras law. we further claim ξ′(p) is upper hemicontinuous at each p ≫ 0, p ∈ p′. suppose pt in p′ converge to po ∈ p′, po ≫ 0 and θt ∈ ξ′(pt) converge to θo, which is a portfolio plan. by the definition of ξ′(p) , there is (xt, θt) ∈ ξ(pt). let u be a compact neighborhood of p0 contained in the relative interior of p′. then when t large enough, (xt, θt) ∈ ξ(pt) is uniformly bounded. thus we can take a subsequence (xt ′ , θt ′ ) → (xo, θo) with xo ∈ ×m∈mr+ m. by proposition 3.1, we have (xo, θo) ∈ ξ(po), which yields θo ∈ ξ′(po). this shows that ξ′(p) is upper hemicontinuous at any po ∈ p′, po ≫ 0. according to debreu[9], there is a point p∗ ∈ c ∩p′ such that ξ′(p∗)∩co ̸= ϕ. let θ∗ ∈ ξ′(p∗) ∩ co. by walras’ law, the point ( 1 v , 1 v , · · · , 1 v ) in p′ belongs to e, hence to c. therefore, ∑ n≤m,n,m∈m l∑ h=1 θ∗hmn ≤ 0. consequently,p∗ ∈ e, hence p∗ ∈ intc. moreover, by another application of the walras’ law, we has p∗ · θ∗ = 0. this equality together with p∗ ∈ intc and θ∗ ∈ co implies θ∗ ∈ 0. then (x∗, θ∗) is a equilibrium for price p ∗.# references [1] aumann, r. j. (1964), “space cannot be cut: why self-identity naturally includes neighbourhood”, integrated psychological behaviour, vol.32, pp.3950. advances in systems science and applications (2014) vol.14 no.2 189 [2] aumann, r. j. (1966), “existence of competitive equilibria in market with a continuum of traders”, econometrica, vol.34, pp.1-17. [3] yun, x. (2007), “optimal portfolio on optimal control”, advances in system science and applications, vol.7, no.1, pp.66-71. [4] arrow, k., debreu, g. (1954), “existence of an equilibrium for a competitive economy”, econometrica, vol.22, pp.265-290. [5] debreu, g. (1982), “existence of competitive equilibrium”, handbook of mathematical economics, vol. ii, chapter 15, pp.697-743, north holland, new york.http://www.calphysics.org/articles/chown2007.html [6] radner, r. (1972), “existence of equilibrium of plans, prices, and price expectations in a sequence of market”, econometrica, vol.40, no.2, pp.289-302. [7] hildenbrand, w. (1974), core and equilibria of a large economy, princeton university press. [8] geankoplos, j. (1990), “an introduction to general equilibrium with incomplete asset market”, journal of mathematical economics, vol.19, pp.1-38. [9] jifeng, z. and xuemou, wu. (2002), “pansystems or principles to societyeconomy systems”, advances in system science and applications, vol.2, no.1, pp.24-31. [10] hart, o. d. (1975), “on the optimality of equilibrium when the market structure is incomplete”, journal of economic theory, vol.11, pp.418-443. [11] bingan, j. etc.(2007), “equilibrium of manufactures’ r&d decision-making in defense procurement”, advances in system science and applications, vol.7, no.1, pp.117-120. [12] guosheng zhang (1998), on large finacial market, mathematics in economy (in chinese), vol.15, no.3, pp.1-6. [13] guosheng zhang (2000), “spot-financial market with large characteristics”, systems engineering-theory & paractice (in chinese), vol.20, no.3, pp.3945. [14] guosheng zhang (2007), “competitive equilibrium of large security-spot market with incomplete asset structure”, journal of systems science and complexity, vol.20, no.3, pp.386-396. corresponding author author can be contacted at: zhangguosheng@bisu.edu.cn. advances in systems science and application (2015) vol.15 no.3 233-241 computational study of the particles interaction distance under the influence of steady magnetic field n. k. lampropoulos1, e. g. karvelas 1,2 and i. e. sarris1 1 department of energy technology, technological & educational institute of athens, ag. spyridona 17, 12210 athens, greece. 2 department of civil engineering, university of thessaly, volos, greece. abstract a computational method for the estimation of the particles’ maximum interaction distance, when these are under the influence of a steady magnetic field, is presented. the computational model was developed for the simulation of the particles and the forces that are exerted on them. the chains of particles are formed under the influence of these forces. also, the mean value of cogglomeration is estimated with simulations, which have different number of particles under a steady concentration and a uniform magnetic field. keywords interaction distance, magnetic driving, aggregations 1 introduction for the treatment of serious diseases, drug is inserted into the human body through a blood vessel and through the blood circulation reaches the infected area. as a result of the above mentioned injection, the drug circulates all over the human body and causes damages in areas that are not infected. in order to overcome this difficulty, we attach drug to particles and drive them to the infected area. for the particles’ guidance to the targeted area, a magnetic resonance imaging (mri) device is needed. this method was presented for the first time in the 1970’s [1, 2]. it minimizes the side effects, as the drug is driven only to the infected area. the above mentioned method is applied with even better results on small blood vessels with low blood flow rates [3]. other factors that affect the efficiency of the proposed method is the size, the material of the particles and the magnetic intensity. the smaller a particle is, the smaller the magnetic response will be. as a result, the magnetic driving to the infected area is non-feasible [4,5]. to overcome this problem, magnetic particles are used in order to form cogglomerates [6]. it is proven that the total magnetic moment of cogglomerates is higher than this of the isolated particles, and therefore, clusters are more magnetically responsive [7]. consequently, the magnetic driving is feasible. when the aggregates reach the targeted area, break into isolated particles. this is feasible with the usage of superparamagnetic particles, which lose their magnetism when there is no magnetic field [6]. it is verified that the computational methods can simulate with accuracy or with a small discrepancy the experimental data [8]. interaction distance of par234 n. k. lampropoulos, e. g. karvelas and i. e. sarri computational study of the particles... ticles plays an important role in the accuracy of the results, when these are compared with the experimental data. the minimum interaction distance is chosen, in order to export accurate results in minimum time, because the magnetic moments that are exerted on a particle are calculated from particles that are located inside the minimum interaction distance. the aim of this study is to define the minimum particle’s interaction distance under a constant magnetic field. the forces acting on a particle are described in section 2 and the results in section 3. finally, discussion is presented in section 4. 2 numerical model for the propulsion model of the particles, four major forces are considered, i.e. the magnetic force from mris main magnet static field, as well as the magnetic field gradient force from the special propulsion gradient coils. the static field caters for the aggregation of nanoparticles, while the magnetic gradient navigates the agglomerations. moreover, the contact forces among the aggregated nanoparticles and the wall is used. the stokes drag force for each particle is considered, while only spherical particles are used in the calculation process. finally, gravitational forces due to gravity and the force due to buoyancy are added. the motion of particles is given by the newton equations. mi ∂ui ∂t = fmag i + f nc i + f tc i + f hydro i + f boy i +w i (1) ii ∂ωi ∂t = mdrag i +m con i + tmag i (2) where the index i stands for the particle i. all quantities indicated in (1) and (2) are presented in table 1 and depicted in figure 1. the bold variables represent vector quantities. the numerical model for the forces is given in [8]. fig. 1 forces and moments of a particle. advances in systems science and application (2015) vol.15 no.3 235 fig. 2 domain of particle close magnetic field inside the mri bore. 2.1 magnetic field in the mri bore the magnetic field b in the mri bore is given by b = b0 + g̃+b1 (3) and is depicted in figure 2. b0 is the mri superconducting magnet field that is constant and uniform, g̃ is the gradient field and b1 is the time dependent radio frequency field [9]. the steady magnetic field b0 is used for the cogglomeration forming and the gradient magnetic field g̃ for the navigation of the cogglomerates into the desired area. 2.2 particle’s interaction distance the magnetic interaction force, f , in the parallel direction between two spheres is inversely proportional to the fourth power of the separation distance [10,11]: f ∝ mimj (h+ ai + aj)4 (4) where, mi and mj are the magnetic moments of the center of each sphere and h is the nearest distance between the surfaces of two spheres with radii ai and aj . although, a weak interaction is being observed for particles that are more than five radii apart, as time goes by, the interaction force is getting stronger, because the particles are getting faster closer to each other [8]. this study is trying to define the minimum interaction distance di of the particle, as is depicted in figure ??, in which the mean length of aggregations remains stable. to estimate the minimum particle’s interaction distance, five series of simulations were performed with 100, 200, 300, 400, 500 particles, respectively. for each series, fifteen simulations have been performed. each time, the particle’s interaction distance varied from 5 to 20 radius. 236 n. k. lampropoulos, e. g. karvelas and i. e. sarri computational study of the particles... table 1 forces and moments simulation quantities quantity description vi velocity ωi rotational velocity t time ii mass moment of inertia matrix mi mass fmag i total applied magnetic force f nc i normal contact force f tc i tangential contact force f boy i buoyancy force w i weight force f drag i drag force mdrag i drag moments m con i contact moments tmag i torque due to the magnetic field 3 numerical method the openfoam platform was used in order to calculate the flow field and the uncoupled equations of particles’ motion [12]. the simulation process reads as follows: firstly, the fluid flow is found using the pressure correction method. upon finding the flow field (pressure, velocity) the motion of particles is evaluated by the lagrangian method by solving eq. (1) and (2) along the trajectory of each particle. the equations are solved in time using the euler time marching method. the stability of the algorithm is guaranteed with a time step of 10−6s. table 2 simulation parameters flow domain and computational grid of case 1 (3d) no concentr. (mg/ml) particles volume (m3) dimension in x-dir (m) dimension in y-dir (m) dimension in z-dir (m) 1 1.125 100 7.303× 10−11 4.18× 10−4 4.18× 10−4 4.18× 10−4 2 1.125 200 1.381× 10−10 5.17× 10−4 5.17× 10−4 5.17× 10−4 3 1.125 300 2.018× 10−10 5.94× 10−4 5.83× 10−4 5.83× 10−4 4 1.125 400 2.686× 10−10 6.6× 10−3 6.38× 10−3 6.38× 10−5 5 1.125 500 3.489× 10−10 7.04× 10−4 7.04× 10−4 7.04× 10−4 the spacing of the computational grid is equal to 2 ∗ diameter of the particles that were simulated. the summary of the domain parameters for the simulated cases is tabulated in table 2. advances in systems science and application (2015) vol.15 no.3 237 4 results (a) 0.1ms (b) 1ms (c) 3ms (d) 5ms fig. 3 snapshots of aggregation process of 500 particles under a uniform magnetic field of b0 = 0.4t . simulations with 100, 200, 300, 400, 500 particles were performed. for each number of particles, fifteen simulations were tested. each time the particle’s interaction distance varied from 5 to 20 radius with an increment of 1. the concentration and the magnetic field of the domain were 1.125 kg/m3 and 0.4t , respectively. the diameter of each particle was 11 um and the relative magnetic permeability (µr) of the fe3o4 particles was 12.3. the young modulus (y ) and the poisson ratio (ν) of the material was 109 m−2pa and 0.5, respectively. the tangential stiffness (γ) was 10 nsm−1 and the coefficient of friction (µ) was 0.5 . the density of the fluidic environment (ρf ) and particle (ρp) was 1000 and 1087 kg/m3. from figure 4 is depicted that the mean length of the chains remains steady, 238 n. k. lampropoulos, e. g. karvelas and i. e. sarri computational study of the particles... when the particle’s interaction distance is more than 11 radius. the mean length and std deviation of aggregations are steady above 11 radii. on the other hand, when the particle’s interaction distance is between 5 and 10 radii, the results show discrepancies from the experimental data. these discrepancies appeared due to truncation error, which is minimized when the particle’s interaction distance is more than 11 radii. apparently, the forces that are exerted on each particle are eliminated more than 11 radii when a magnetic field is applied. the computational platform does not estimate all the forces that are exerted on the particle from each neighbour, due to the existence of a certain interaction distance each time. for the simulations of 200 to 500 particles, the mean length of aggregations that is estimated from the computational platform is in a good agreement with the experimental data, as is depicted in figure 5. the simulations that were performed with 100 particles show a small discrepancy from the experimental data. this difference occurs due to the small numbers of particles that were used in the simulation, in comparison with the measurements that were conducted using a number of particles in the order of 106 . fig. 4 mean length of aggregations with different particle’s interaction distance. 5 discussion after all the above mentioned simulations, it is observed that under the influence of a steady magnetic field of 0.4 t and concentration of 1.125 kg/m3 the results are steady, when the particles interaction distance is above 11 diameters, as is depicted in figure 4. in figure 5, is observed that the computational platform estimates accurately the experimental data [6]. although, in the simulations of 100 particles the computational platform shows discrepancies with the experimental data, in these with more particles the proposed method is in agreement advances in systems science and application (2015) vol.15 no.3 239 with the experimental data. the small number of particles is the cause of these discrepancies. in figure 3, we present some snapshots of the aggregation process. in time of 0.1ms, the particles are isolated, as they have not form chains. from 1ms to 3ms, chains are observed, but some particles are still isolated. they have, thus, the tendency to form aggregations either with other particles that are isolated or with the chains that are already formed. in time of 5ms, the aggregation process has ended. as a result, the chains are the longest possible under the current magnetic field. small number of particles is still isolated and they will not form aggregations as time goes by, because there is no force exerted on them. fig. 5 mean length of aggregations with different number of particles. fig. 6 mean length of aggregations in each ms in figure 6, the aggregation process is presented in accordance with time. the 240 n. k. lampropoulos, e. g. karvelas and i. e. sarri computational study of the particles... mean length of the aggregation is estimated for each ms of the process till its completion. the aggregation process is not linear, because the particles react with each other. as a result, the size of the chains is constantly changing. for tests of 100 particles, the aggregation process shows discrepancies with the other simulations, due to the small number of particles that has been simulated. factors, such as size, the material of the particles and the magnetic intensity of the field are important for the interaction distance. higher intensity of the magnetic field creates stronger forces between particles. as a result, the particles interact in greater distances. therefore, the mean length of aggregates tends to be larger. bigger particles with the same magnetic intensity become into stronger magnets, due to bigger magnetic volume. as a result, smaller particles are attracted by bigger ones. the material of the particles plays an important role in the particles’ interaction distance. particles with high material magnetic susceptibility attract particles that are located in greater distances than those which have low. as a result of all the above, the interaction distance becomes greater. 6 conclusion in this work, a parametric study for the particles’ interaction distance is presented. after the simulations, it was observed that there is no interaction between the particles, which have a distance above 11 diameters. the computational method simulated the experimental results with a minimal discrepancy. acknowledgments the work is funded by the nanother program (magnetic nanoparticles for targeted mri therapy) through the operational program cooperation 2011 of gsrt, greece. discussions with dr klinakis from brfaa, greece, prof. zergioti from ntua, greece and the people from future intelligence ltd. are also acknowledged. references [1] a. senyei, k. widder, and c. czerlinski, (1978), “magnetic guidance of drug carrying microspheres,” appl. phys., vol. 49, pp. 3578–3583. [2] k. widder, a. senyei, and g. scarpelli, (1978), “magnetic microspheres : a model system of site specific drug delivery in vivo.” proc. soc. exp. biol. med., vol. 158, pp. 141–146. [3] b. yellen, z. forbes, and d. halverson, (2005), “targeted drug delivery to magnetic implants for therapeutic applications,” magn. magn. mater, vol. 293, pp. 647–54. advances in systems science and application (2015) vol.15 no.3 241 [4] q. pankhurst, j. connolly, s. jones, and j. dobson, (2003), “applications of magnetic nanoparticles in biomedicine,” physiscs d:applied physics, vol. 36, pp. 166–181. [5] o. petracic, (2010), “superparamagnetic nanoparticle ensembles,” superlattices and microstructures, vol. 47, p. 569. [6] j.-b. mathieu and s. mantel, (2009), “aggregation of magnetic microparticles in the context of targeted therapies actuated by a magnetic resonance imaging system,” j. appl. phys., vol. 106, p. 044904. [7] p. babinec, a. krafcik, m. babincova, and j. rosenecker, (2010), “dynamics of magnetic particles in cylindrical halbach array: implicationsfor magnetic cell separation and drug targeting,” med. biol. eng. comput., vol. 48, pp. 745–753. [8] n. k. lampropoulos, e. g. karvelas, and i. e. sarris, (2014), “computational modeling of an mri guided drug delivery system based on magnetic nanoparticle aggregations for the navigation of paramagnetic nanocapsules”, under review. [9] c. l. epstein and f. w. wehrli. (2005), “magnetic resonance imaging,” elsevier encyclopedia on mathematical physics, [online]. available: http://www.math.upenn.edu. [10] t. fujita and m. mamiya, (1987), “interaction forces between nonmagnetic particels in the magnetized magnetic fluid,” j. of magnetism and magnetic materials, vol. 65, pp. 207–210. [11] a. mehdizadeh, r. mei, j. f. klausner, and n. rahmatian, (2010), “interaction forces between soft magnetic particles in uniform and non-uniform magnetic fields,” acta mechanica sin., vol. 26, pp. 921–929. [12] h. g. weller, g. tabor, h. jasak, and c. fureby, (1998) “a tensorial approach to computational continuum mechanics using object-oriented techniques, computers in physics,” vol. 12, no. 6, pp. 620–631. corresponding author e. g. karvelas can be contacted at: karvelas@uth.gr microsoft word 2-yirong ying.doc 6-13 advances in systems science and applications (2010), vol.10, no.1 issn 1078-6236 international institute for general systems studies, inc. dynamics of price model with nonlinear demand function∗ yirong ying1, ke chen1 and jeffrey forrest2 1college of international business and management, shanghai university, shanghai 200444, china 2department of mathematics, slippery rock university, slippery rock, pa16057, usa abstract in this paper we derive the dynamic price model by introducing a general nonlinear form of the demand function into the traditional cobweb model, and establish three propositions about the existence and structure stability of limit cycles by applying the qualitative theory of ordinary differential equations. the dynamic characteristics under the specific nonlinear demand function in six individual situations are discussed and illustrated by the petroleum oil daily price data from january 2 to october 27, 2009. keywords demand function limit cycle dynamic characteristics of price petroleum oil price 1. introduction the cobweb theory proposed by ragnar frisch and jan tinbergen belongs to the category dynamic equilibrium analysis. it investigates the change of commodity prices in the supply-demand equilibrium and the inherent stability problems. the early cobweb model assumes that producers have the same expectations, only to produce one commodity or participate in one market, and to adjust the supply by the price of the last period. besides, the demand and supply functions are of a linear form of the prices, and the market is clear at every period (namely, the overall balance of supply and demand), etc. recently, the researchers continue to relax these assumptions or add the new ones in order to make the situation studied closer to the real market, and apply the new methods to analyze the complex performances and equilibrium conditions of the price in the nonlinear dynamic cobweb model. for instant, chiarella (1988) introduces a general nonlinear supply function into the traditional cobweb model with adaptive expectations, and finds that the dynamics of the model is driven by a chaotic type of single-hump map; hommes (1994) studies the dynamics of the price-quantities model derived from the cobweb model with adaptive expectations and nonlinear supply and demand curves, also find the chaotic dynamic behavior even if both the supply and demand functions are monotone; brock & hommes (1997) analyze the nonlinear dynamic equilibrium process by allowing the suppliers with heterogeneous expectations to switch between the naïve and rational ones freely, but restricting the linear function formation of supply and demand in the cobweb model, and dominate that a bifurcation route will change to the chaos and strange attractors when the intensity of choice to switch prediction strategies increases; goeree & hommes (2000) develop the brock & hommes’s evolutionary cobweb model by expanding the linear supply and demand to the nonlinear ones, while reaching similar conclusions. particularly, with adding other more realistic hypotheses into the traditional cobweb model, some recent literatures show that the improved models also generate chaotic dynamic behaviors. chiarella et al. (2006) discuss the cobweb model with boundedly rational heterogeneous producers (i.e., risk-averse producers), and conclude that each dimension of heterogeneity enriches the cobweb dynamics with respect to the case of homogeneous producers; choudhary & orszag (2008) study the traditional cobweb model in one market with local externalities which imply that the ∗ this work has been supported partly by the research fund of subject construction for reading material of financial economics. advances in systems science and applications (2010), vol.10, no.1 7 firms must forecast both price and quantities, and obtain the evidence of clusters of firms whose output behavior is correlated as equilibrium is reached; dieci & westerhoff (2009) consider that the producers face two kinds of commodity markets and tend to enter the more profitable one in the recent past, and find such a switching process is a further source of nonlinearity for the dynamics of prices. in addition, the more widely the new research methods are applied, the more deeply the chaotic behaviors of dynamic cobweb model are researched, including the model of stability and convergence, unstable conditions and the possibility of internal changes, and the evolutionary steps of the prediction strategy choice. for instance, ying yi-rong (1996) applies the ordinary differential equation qualitative theory, especially the “limit cycle” theory, to explore the existence and uniqueness of stable limit cycles, and the magnitude and period of price shocks in the nonlinear cobweb model. onozaki et al. (2000) use the classical homoclinic point theorem to discuss the nonlinear cobweb model with adaptive production adjustment, and find the observable chaos (strange attractors) as well as topological chaos (saddle point) associated with homoclinic points. (for the mathematical and physical mechanisms underlying the appearance of chaos, please consult with (lin and ouyang, 1998; lin, 2008)). chiarella et al. (2006) construct the geometric decay processes (gdp) of supply quantities to study the dynamic features of the nonlinear cobweb model while the supply function contains the limited memory or infinite memory. li et al. (2008) introduce the gaussian white noise into the demand function, and transform the equilibrium price model into a quasi-integrable hamiltonian system, then take the random averaging method to discuss the first-pass damage of this system and give the conditional reliability function and the probability distribution function. in short, there is a diversification trend in the study of dynamic cobweb model due to the complexity of economic and social activities. in fact, chaos is everywhere so that different assumptions and research methods are introduced and developed to explore the different sides of the complex phenomena. for enriching the studies of nonlinear cobweb models, this paper focuses on the dynamic equilibrium of price in a single commodity market with a general nonlinear formation of demand function basing on ying yi-rong’s earlier study, and derives three propositions about the existence and structure stability of limit cycles from this nonlinear cobweb model. furthermore, we discuss the dynamic characteristics under the specific nonlinear demand function in six individual situations, and find that there is the specific demand function in a shorter term illustrated by the petroleum oil daily price data. but there exist many other nonlinear forms of the demand function which are hard to determinate because of the chaotic behaviors existing in the real market. in the next section we will construct the general nonlinear demand function and the nonlinear dynamic cobweb model. the third section focuses on the existence and stability of the limit cycles under different conditions; and the fourth section expands the conclusions developed in section 3 by discussing the dynamic price equilibrium with the special nonlinear demand function form. the final section concludes the presentation of this work. 2. modeling 2.1 nonlinear demand function ying yi-rong (1996) assumes that the current demand is a nonlinear function of the price and its growth rate. that is dt dppcpccaptd )()( 2 210 ++++=α (1) where ( )d t is demand of the goods at time t; p is price of the goods at time t; 210 ,,,, cccaα are parameters. denote 2( )f p ap bp c= + + ，then ying: dynamics of price model with nonlinear demand function 8 ( ) ( ) dpd t ap f p dt α= + + (2) in eq. (2), ( )d t is made up of the simple linear impacts of price ( apα + ) and the nonlinear impact ( ( ) dpf p dt ) that explains the consuming attitude respect to the growth rate of price. this assumption makes the model closer to the real market than the simple linear demand function in the traditional cobweb model. in accordance with microeconomics views, besides the price of the commodity itself, the other important factors, generally including income, consumption preferences, and other commodity prices, can directly generate either positive or negative impact on the commodity demands. furthermore, these factors also have some unascertained impact on the consumer price sensitivity and pricing behavior of suppliers. the uncertainty involved eventually leads to the proposed nonlinear demand function. for example, with the growth in income, the consumer increases the demand but reduces the degree of consuming price sensitivity at the same time. so rational price hikes by the supplier might not necessarily lead to declining demand. in addition, in some commodity markets, the short-term change in consumption preferences may greatly improve or reduce the degree of consuming price sensitivity with respect to the price increases, which directly cause volatile fluctuations in demand. therefore, given the consuming attitude that is mainly influenced by the sensitivity of consumers to price changes, ( )f p in eq. (2) may take any function formation. if still denoted by ( )f p , then eq. (2) can be considered as the general form of any nonlinear demand function. 2.2 the nonlinear cobweb equilibrium model basing on ying yi-rong’s study (1996), in general, one can assume that the supply at time t is still a linear function of the price. the general form is as follows: ( )s t bpβ= + ， 0b > (3) where ( )s t stands for the supply of the goods at time t with ,bβ being the parameters. assumed that the suppliers do pricing all the time to make the rate of change in price proportional to the amount of shortage, which is caused by the stock fallen below a certain threshold level. that is, ( ( ) )dp q t q dt λ= − − ， 0λ > (4) where ( )q t is stock of the goods at time t; q is the threshold level of stock set by suppliers;λ the proportion of the stock shortage. let 0 ( ) (0) ( ( ) ( )) t q t q s t d t dt= + −∫ with (0)q being the initial level of ( )q t . then 0 [ (0) ( ( ) ( )) ] tdp q q s t d t dt dt λ= − − + −∫ ， 0λ > (5) substituting eqs. (2), (3) into eq. (5), one can easily obtain the following second order differential equation 2 2 ( ) ( ) ( )d p dpf p b a p dt dt λ λ λ α β− + − = − ， 0λ > (6) when p p= , then ( ) ( )s t d t= , 0dp dt = . so one can get p b a α β− = − （b a≠ ）, where p is the equilibrium price. let ( )p p t p= − and substitute it into eq. (6), we have advances in systems science and applications (2010), vol.10, no.1 9 2 2 ( ) ( ) 0d p dpf p p b a p dt dt λ λ− + + − = (7) without loss of generality, we still denote ( )f p p+ by ( )f p , meanwhile let t τ λα = , 0 b a λμ = − > − , we can get 2 2 ( ) 0d p dpf p p d d μ τ τ + + = (8) let p x= , dy x dτ = − . then eq. (8) can be rewritten as the rayleigh equations by using lienard transformation. that is ( )dx y f x d dy x d τ τ ⎧ = −⎪⎪ ⎨ ⎪ = − ⎪⎩ (9) where 0 ( ) ( ) x f x f p dpμ= ∫ . 3. analysis according to the qualitative theory of ordinary differential equations, there are three conclusions in terms of eq. (9). proposition 1 if (0) 0f > , the singular point (0,0) is an unstable focus or unstable node of eq. (9); if (0) 0f < , the singular point (0,0) is a stable focus or stable node of eq. (9). proof. the characteristic equation of the linear part in eq. (9) at point (0,0) is 2 (0) 1 0fλ μ λ+ + = therefore the proposition 1 must be true according to the qualitative theory of ordinary differential equations. proposition 2 if the divergence of eq. (9) is constant within a simply connected domain area g, then there does not exist any limit cycle in eq.(9) so that all closed orbits are located within the area g. proof. let us compute the divergence of eq. (9) as follows: (9)| ( ) ( ( )) 2 0div x y f x x y ∂ ∂ = + − = > ∂ ∂ hence proposition 2 can be shown to hold true by applying theorem 1.10 derived by ye yan-qian (1984). proposition 3 if eq. (9) satisfies the following two conditions: (a) ( ) 0xf x < , when 0x ≠ and x is sufficiently small; (b) there exist a positive constant m and constants k and k ′ with k k ′> such that ( )f x k≥ ，when x m> ; ( )f x k ′≤ ，when x m< − .then eq. (9) has the stable limit cycle. proof. the function in eq. (9) shows that there exist ( )g x x≡ . then the three conditions in theorem 5.1 in the monograph of ye yan-qian (1984) can be satisfied. thus considering the conclusions of proposition 1, proposition 3 must be true. ying: dynamics of price model with nonlinear demand function 10 4. applications 4.1 case 1: constpf =)( let 0( )f p c= , 0c is a none-zero constant. substituting ( )f p into eq. (8) leads to 2 02 0d p dpc p d d μ τ τ + + = (10) (1) if 2 0( ) 4 0cμδ = − > , then 0 2( )c b a> − , and 1 2 1 2 p t p tp a e a e= + , where 1 2,a a are any constants, 1 2,p p the two distinct real roots of the characteristic equation of eq. (10). when 1 0p > and 1 2p p> , the price keeps on moving further and further away from the equilibrium price p (for the convenience of drawing the figure, assume 0p = ); when 1 0p < and 2 0p < , the price gradually moves toward the equilibrium price. figure 1 and 2 show the two kinds of price movements at some specific values 0 15c = and 0.2μ=− . in the same fashion, we can produce figure 3 – 6. 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 -200 -150 -100 -50 0 50 100 150 200 0 1 2 3 4 5 6 7 8 9 10 -20 -15 -10 -5 0 5 10 15 20 figure 1 (0) 20, (0) 30p p′= ± = ± figure 2 (0) 20, (0) 30p p′= ± = ± (2) if 2 0( ) 4 0cμδ = − = , then 0 2 1 2( ) c tp a a t e μ−= + , where 1 2p p= . when 0 0c > , after reaching the equilibrium price, the price continues to move further and further away from it, see figure 3 ( 0 10c = , 0.2μ =− ) for more details. when 0 0c < , the price firstly experiences larger fluctuations in the vicinity of the equilibrium price, then moves closer to it, figure 4 ( 0 10c = , 0.2μ =− ). 3 3.5 4 4.5 5 5.5 -30 -20 -10 0 10 20 30 40 50 60 70 0 1 2 3 4 5 6 -5 0 5 10 15 20 25 figure 3 (0) 20, (0) 10p p′= − = − figure 4 (0) 20, (0) 10p p′= = (3) if 2 0( ) 4 0cμδ = − < , then 0 2( )c b a< − , and 0 2 cos( ) c p ae t μ ω ψ−= + , where 0 1 2, 2 cp p iμ ω= − ± , ψ stands for the initial phase. when 0 0c > , the price contains a period of oscillations with an increasing amplitude around the equilibrium price, and then moves away advances in systems science and applications (2010), vol.10, no.1 11 from it, figure 5. however, when 0 0c < , the price contains a period of oscillations with a decreasing amplitude around the equilibrium price, and finally close in to it, figure 6. 0 50 100 150 -100 -80 -60 -40 -20 0 20 40 60 80 0 50 100 150 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 figure 5 0 2.5c = ， 0.2μ = − ， (0) 2, (0) 0p p′= = figure 6 0 2.5c = − ， 0.2μ = − ， (0) 2, (0) 0p p′= = 4.2 case 2: the short-term fluctuations of petroleum oil price according to the u.s. government's official energy statistics, the daily data of petroleum oil prices (fob, except weekends and holidays) obviously has a growing uptrend during january 2 to october 27 in 2009 (see figure 7). what’s shown is that petroleum oil equilibrium price points were continuously broken within the interval of the 207 said days due to changes in supply and demand, and gradually went upward from the initial equilibrium price point with the short-term irregular fluctuations or a random walk. however, in a smaller time interval, the price fluctuations show the similar characteristics of dynamic changes as in figure 1– 6. for example, the data set is simply divided into two stages at the price point of june 11, 2009. in stage 1, although the petroleum oil price irregularly moves without a clear direction in the first 40 days, it goes upward and moves far away from the initial price after that; eventually it reaches the highest price on june 11. it has the similar characteristics as described in figure 3. in the stage 2, before october 10, the petroleum oil price seems to keep moving around a certain equilibrium price with decreasing amplitude. it is similar in principle to what is shown in figure 6. but it is not easy to find whether its movement within the rest of the time period would have the similar performance as those in the previous figures. 0 20 40 60 80 100 120 140 160 180 200 35 40 45 50 55 60 65 70 75 80 stage 1 stage 2 data source： http://tonto.eia.doe.gov/dnav/pet/hist/rwtcd.htm； figure 7 01/02/2009-10/27/2009 petroleum oil price trend. some evidences can be found from this real market experience. generally speaking, if these factors, such as price, income, preferences, and the prices of other commodities, change regularly without a significant fluctuation, or no war, no pestilence, no earthquakes, etc. to occur, both suppliers and consumers’ price sensitivity may vary little within a shorter term or a sufficiently shorter interval. that is, the price sensitivity can be assumed to be constant theoretically so that ( )f p in eq. (9) can be roughly seen as a constant in a short time period. ying: dynamics of price model with nonlinear demand function 12 however, if those factors change chaotically, or one or some of the events mentioned above suddenly occur, then the price sensitivity is easily changed drastically even within a sufficiently shorter interval. so ( )f p may be a linear or even nonlinear function instead of a constant. therefore, the price movements of the petroleum oil always stand for a combined effect of two individual performances even in a sufficiently small interval. one is similar to what are described in the cases of figure 1– 6, another represents a complex process. given 10 days as a sufficiently short interval for the petroleum oil price data, then there are 198 samples by sampling randomly 10 consecutive days. all the price movement may be roughly divided into two categories with ( )f p being constant or non-constant. so, let us simply choose two special samples for easy comparative analysis. additionally, assume the lag one of the initial price in the sample is the equilibrium price under the adaptive expectations, which is estimated by smoothing spline interpolation method (with the smoothing coefficient being 0.5). figure 8 shows that during april 27 to may 8, the fitting curve by fourier interpolation method goes up to cross the equilibrium price line on april 28, and keeps on moving further upward. if excluding the irregular change, and contrasting with figure 3, this curve implies that there exists a constant ( )f p greater than zero in the petroleum oil price dynamics of this sample. whereas the fitting curve obtained in the same way in figure 9 is not similar to any of those in figure 1– 6 during the 10th to 24th of february. it has a period of almost six trading days of oscillation with the same amplitude underneath the equilibrium price line, and hovers over the line in the last four days. it can be inferred that ( )f p in figure 9 is no longer a constant but a nonlinear function. unfortunately, it is not easy to find its specific functional expression. because it is more difficult to determine the existence and stability conditions of limit cycles according to proposition 3, more in depth study will be needed. 0 1 2 3 4 5 6 7 8 9 49 50 51 52 53 54 55 56 57 58 59 0 1 2 3 4 5 6 7 8 9 34 35 36 37 38 39 40 figure 8 petroleum oil price trend in 04/27-05/08/2009 figure 9 petroleum oil price trend in 02/10-02/24/2009 5. conclusions based on the studies of ying yi-rong (1996), the hypothesis of a linear demand function in the traditional cobweb model is extended to a nonlinear form. then for the given nonlinear demand function of the general form ( ) ( ) dpd t ap f p dt α= + + , of which ( )f p could be any function, the nonlinear dynamic price model is derived and transformed into the rayleigh equations, and three propositions on the limit cycles are proved to hold true by applying the qualitative theory of ordinary differential equations. on this basis, two specific cases are discussed. one is the study of the dynamics of the nonlinear price model with a constant ( )f p , where six situations of price fluctuations are graphed. another is the analysis of the petroleum oil price movement from january 2 to october 27, 2009; it is found that there may appear two kinds of situations where ( )f p is either a constant or non-constant in a sufficiently short term by contrasting to what are shown in figure 1– 6. for non-constant ( )f p , it might be caused by significant changes in price, income, preferences, and prices of other commodities, or by the advances in systems science and applications (2010), vol.10, no.1 13 appearance of one or more uncontrollable events. so, ( )f p might be a nonlinear function (to this end, please consult with (lin, 2008), where it is shown that the study of economics should be mainly about interactions of nonlinear forces). but its specific functional expression is difficult to find, constituting an open problem for future studies. references [1] carl chiarella. the cobweb model: its instability and the onset of chaos. economic modelling. 1988, 24(4): 377-384. [2] cars h. hommes. dynamics of the cobweb model with adaptive expectations and nonlinear supply and demand. journal of economic behavior & organization. 1994, 24 (3):315-335. [3] brock, w.r., hommes, c. rational route to randomness. econometrica. 1997, 65(5):1059 –1096. [4] jacob k. goeree, cars h. hommes. heterogeneous beliefs and the non-linear cobweb model. journal of economic dynamics and control, 2000, 24(5-7):761-798. [5] carl chiarella, xue-zhong he, hing hung, peiyuan zhu. an analysis of the cobweb model with boundedly rational heterogeneous producers. journal of economic behavior & organization, 2006, 61(4):750-768. [6] m. ali choudhary, j. michael orszag. a cobweb model with local externalities. journal of economic dynamics and control, 2008, 32(3):821-847. [7] roberto dieci, frank westerhoff. stability analysis of a cobweb model with market interactions. applied mathematics and computation, 2009, 215(6): 2011-2023. [8] lin lin, ying yi-rong, dang xin-yi. the significance, methodology and application about infinite. xi’an: press of northwestern university, 1996. (in chinese) [9] yi lin. systemic yoyos: some impacts of the second dimension. auerbach publications, an imprint of taylor and francis, new york, 2008. [10] yi lin and s. c. ouyang. invisible tao and realistic nonlinearity proposition. kybernetes: the international journal of cybernetics, systems and management science, vol. 27, nos. 6 – 7 (1998), pp. 809 – 822. [11] tamotsu onozaki, gernot sieg, masanori yokoo. complex dynamics in a cobweb model with adaptive production adjustment. journal of economic behavior & organization, 2000, 41(2):101-115. [12] jiaorui li, wei xu, wenxian xie, zhengzheng ren. research on nonlinear stochastic dynamical price model. chaos, solitons and fractal, 2008 (37):1391-1396. [13] yan-qian ye. limit cycle. shanghai: press of science and technology, 1984. (in chinese) microsoft word 5 peng zhao, buyun sheng--process planning service mechanism based on saas mode.doc 240-248 advances in systems science and applications (2011), vol.11, no.3-4 issn 1078-6236 international institute for general systems studies, inc. process planning service mechanism based on saas mode peng zhao and buyun sheng hubei digital manufacturing key laboratory, wuhan university of technology, wuhan 430070, p.r.china abstract software-as-a-service (saas) is a new mode that the software is deployed as a series of services on the internet. the customers could choose the necessary function services or the better services to meet their normal and individual requirements. for process planning is a highly individual work, process planning service mechanism based on saas mode was proposed and researched. the service-oriented architecture of capp was established. process planning services were designed to service components, which could communicate each other with messages. the thought of process service bus (psb) was put forward and applied to manage the process planning services needed in real time. on this basis, a case was analyzed to show how to apply the service mechanism to gears process planning, the feasibility and validity of the process planning service mechanism were demonstrated by the case. keywords process planning software-as-a-service service-oriented architecture process service bus 1. introduction with the development of china's manufacturing informatization technology in the last decade, the technology informatization （ cad/cam ） and management informatization (pdm/erp) have made significant achievements and brought certain economic and social benefits to the country and enterprises. however, in actual production circumstance, process planning is considering as the connecting bridge between product design and manufacture, whose informatization degree lags far behind the actual application requirements[1]. the development and commercialization of capp are difficult, even become a bottleneck of the application of cims, which can be ascribed to the complicated and changeable characters of process planning project. not only does process planning exist in almost all production activities including product design, production schema, shop fabrication, stock management, quality management, sales management, cost management and other business activities, but also be related to parts’ types, product structures, technology and equipments and so on. furthermore, it is influenced by personnel practical experiences and the constraints of production management system, any change of these factors may lead to revise the process planning project. for the actual state and conditions of different enterprises are different, there are various different process planning requirements. further speaking, the individual process planning service of the capp system can become a decisive factor for the development of capp systems. a flexible multi-user service mechanism is needed to meet the enterprises' individual requirements. in detail，the system architecture, the service management and the serving model of capp must be unified, complete and agile. ref.[2-5] gave some researches on capp system architecture using components technology. the related capp system architectures are classified as traditional system architecture, which is need to be revise wholly when system updating or modifying are happened because the connections among components are tight coupling and complex and the communication standards are different. ref.[6-9] constructed capp systems with web services. although these studies were still based on traditional system architecture, they provided us one new thinking way: forming system with services, just like web services. with the emergence and application of service-oriented architecture (soa), it became a hot point doing research on capp based on soa. because the interface between the modules in this advances in systems science and applications (2011), vol.11, no.3-4 241 architecture is unified and standard, different manufacturers provide integrated solutions for better compatibility between each other, and the architecture is easy to be extended while the business expanding. the attendant problem is about service management, namely，how do the service "components" run in the service-oriented architecture? ref.[10,11] adopted the thought of enterprise service bus(esb) when building system architectures and which can be used for reference. esb shows that the services are distributed and are connected via bus-like method, which supports message communication and data dynamical interaction based on a widely accepted and open standard. process service bus (psb) is proposed to express process planning service management clearly. so far, it is still not enough to meet the multi-user individual service requirements, given that the service thought claims to choose and configure services according to the users' practical requirements. traditional capp systems are set up in the users' computers as a whole software, and most functions of the software are not be needed or are not suitable for the different applying environments. so a better serving model is needed. ref.[12-16] described one serving model named as application service provider(asp) and ref.[17,18] discussed the applications of asp. however, asp is just a primary serving model. ref.[19-22] used an advanced serving model, software-as-a-service(saas), which is more comprehensive and sophisticated. it is summarized that soa,psb and saas are coincident, trying to provide customers with flexible, diverse and personalized services, rather than a fixed, simple software. in the paper, process planning service mechanism based on saas mode is proposed and be researched detailedly. 2. saas mode saas is a mode of software service by renting software or on-line services [23]. the core of saas is deploying software into services and accessing services via network. software vendors provide services such as maintenance, daily operation and technical support service, which conveniences users to focus on their own business, regardless of the purchasing related hardware and software, and the cumbersome system maintenance and upgrade. this service mode has much difference from the traditional software services mode, especially on the ways of paying, deploying, serving, using and upgrading [24]. saas maturity model is assorted into four grades according to the data structure and deployment for customers[25] (see fig.1). fig.1. saas maturity model on the first grade, application service vendors offer individuals a customization sample, which includes independent database, virtual content, website, etc. what the software provides is not a service but an application to each customer. software vendors haven’t consider the various 242 zhao: process planning service mechanism based on saas mode needs of customers as a whole, thus the personal demands wouldn’t be better satisfied. the ideal saas model should be the fourth grade: the clients connect into the server cluster, and a layer used to balance the loads of clients is added between clients and servers, which also can deal with the customization, expansibility and the efficiency of multi-tenant. so it is regarded to meet the demands of customers excellently and solve the load problem by coordinating the multi-tenants. on the one hand, the processes of domestic commercial capp have fixed in the procedure, which results that the system can not respond to the part’s process changes to meet individual customer requirements opportunely. on the other hand, most of systems, designed by one common process planning management model, couldn’t meet individual customer requirements, so the workload of software development increases while the software life cycle shortens. capp lags behind the use of modern information-based manufacturing requirements due to the particularity and complexity of process planning. saas model provides a good solution for the development of the capp system. in this model, customers will consider that each service is provided as their needs. 3. process planning service mechanism in the saas model, to provide customers with individual services, the capp system must be studied in the aspect of process planning service mechanism, including system architecture, design & realization of process planning services and service management mechanism and other relative problems. 3.1 system architecture capp should not only be general but also adaptable to different requirements of individual users, which determines the system itself should be flexible. according to the saas model, software vendors provide process planning services to users, and users access to services after registering and searching. it should be emphasized that the provided software must be in the form of services. when service-oriented architecture (soa) is introduced to capp system, independent, standard and loosely coupled services could be provided because these services in soa architecture can be configurable, reconfigurable, scalability. in accordance with the thinking of soa[26], capp system is divided into multi-level services, business could be completed by calling services. users can invoke services dynamically and configure on-demand system and choose the right interface style, business processes and management through customizing services. the architecture is divided into four layers: the user layer, business layer, services layer and data layer (see fig.2). (1) user layer. system users are classified into enterprise users and individual users. users login in system with web browser or other network equipments directly after identity authentication. (2) business layer. in accordance with process planning task, process management modules invoke appropriate services through psb. for process planning is multi-threaded task, a number of process management modules may be needed.，but there is only one process management center that harmonizes and controls their work process. (3) services layer. this layer contains four categories of basic services: process planning service, process service, process rules service and basic system service (such as data service， logical service, agency service). each type of services contains many fine granularity services. these services can be searched and used after services registration by uddi[27], short for universal description discovery and integration. the whole system is based on the soa architecture and the services are independent, standards, loosely coupled and call or visit each other by process controlling , information mapping. (4) database layer. it supports the system data provision and contains process knowledge national natural science fund project of china (contract no.50620130441) scientific&technological project of wuhan city (contract no.200810321153) advances in systems science and applications (2011), vol.11, no.3-4 243 database, process files database, process resources database and process cases database. users can customize the type and structure of the database and add, modify or delete them without having any impact on other users’. fig.2. the framework of capp system 3.2 design & realization of process planning services generally, a service describes the input and output parameters, right, accessing control lists, security and the quality of service (priority levels, delivery, transaction characteristics, recovery semantics, etc.) and services levels, including response time, availability rate. under an ideal circumstance, process planning services could be designed in accordance with the existing definition of the domain. if the existing capp systems are expected to be reused, process planning services could be got by packaging components interface, traditional message format or apis. the common service design method is that classifying functional services wholly with top-down manner, these coarse granularity services could be further classified fine granularity services or atomic services. according to the analysis of the process planning field, seven categories services are abstracted: cad parts information service, process decision service, process edited service, manufacturing resources service, process data service, process knowledge service and output of process service. messages transfer among the seven services (see fig.3). fig.3. message passing among services 244 zhao: process planning service mechanism based on saas mode fig.4. the interfaces of process decision service in the service-oriented architecture, it is necessary to ensure that services are loosely coupled and can be dynamically discovered and binded to other services in order to deal with changes in business processes or interactive partners flexibly. so process planning services are packaged into service-oriented component models， which could afford one kind uniform invoking way for different interfaces[7].the component declaration should be took to regard service as component like as following： ＜component name=" processordercomponent "＞ ＜implementation.java class="processdecision.processorder "/＞ ＜reference name="addservice"＞processdivivdecomponent ＜/reference＞ ＜/component＞ 3.3 process service management all functions provided as services, only is there a reasonable mechanism for service management, users’ requirements would be met. psb is proposed to manage the services in the process planning system. it is an architecture mode, supporting virtual communication, interaction among services and services management. it affords the connection between the service vendors and requesters, even the services are not perfectly matched. the services participants don’t interact directly, but through psb, which supplies management functions and virtualizations, including the location and identity, interactive protocol, quality of service (qos), and many other virtualizations, to achieve and expand the core services definition. fig.5. the basic psb mode advances in systems science and applications (2011), vol.11, no.3-4 245 in the basic psb mode, the message flows link the various communication participants into the bus (see fig.5). some participants invoke the provided services while others just are interested in the published services information. the endpoint that services and psb interact is named service interactive point (sip), which could be web services endpoint, webspere message queue, or remote method invocation (rmi) agents. the service registering table captures the following metadata: the requirements and functions of sip, the hoped interaction ways (synchronous or asynchronous, http or jms, etc.)，the requirements of qos (the first safe choice, etc.) and other information about supporting other sip. process planning services put messages on the bus or take away from the bus. sip will deal with these dynamic messages and ensure the qos of the hosting interactions. 4. application the case is about the application in a gear enterprise, which has an annual output of 1.5 million middle-module gears and 50 thousand spline shafts and other gear boxes. in accordance with the requirements of the function of the system，the coarsegranularity process planning services were： gear aided design service，gear intensity checking service，process edit service， typical process management service， users management service，gear routing service，etc. fine-granularity services were defined based on detail function requirements. interfaces and function of each service were designed. take gears routing service as an example, the correlation between gears routing service and other services was described and the interfaces could be defined (see fig.6) fig.6 the interfaces of gears routing service it was designed with java language and packaged into service component，the following java code described the interface definition and function realization of gear routing service： package services.gearrouting; public interface gearroutingservice { string getgearinfo(string info); string getgearrouting( string info ); string sendgearprocess( string routing ); } package services. gearrouting; import org.osoa.sca.annotations.*;@service(gearproecess;.class) public class gearrouting implements gearroutingservice { public string getgearinfo(string info){…} 246 zhao: process planning service mechanism based on saas mode public string getgearrouting( string info ){…} public string sendgearprocess( string outing ){..} } psb was realized by model driven architecture (mda). in practical application, the core services were retrieved and matched from the local service or network database. the so-called process planning system was in essence composed dynamically with scalability, agility and availability. fig.7 shows the dynamical correlation of services on the psb. fig.7 the dynamical correlation of services on the psb more efficient and robust process planning services were needed to be searched to meet the process planning requirements in a high extent. the user interface was different with different practical requirements. the interface of gear process planning system was designed as web pages(see fig.8). fig.8. interface of gear process planning system 5. conclusion this paper researches on process planning service mechanism, including how to compose advances in systems science and applications (2011), vol.11, no.3-4 247 process planning service-oriented architecture based saas model, design & realization of process planning services packaged as service-oriented component and process service management with psb. capp system based 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(2006) [25] she w, duan zm, liu yp. research on e-commerce platform based on soa, china computer & network,11,2008, 38-41 [26] wang xh, yao sj, jiao zy, et al. research on uddi-based web service discovery, computer and modernization,65,2009,31-34 [27] rao ph, zhang yh, liu diet. soa design and implementation based-on service component architecture, computer systems & applications,.8,2008 advances in systems science and applications (2012) vol.12 no.2 103-112 the stability of milling of thin-walled workpiece tongyue wang1,2, ning he1, liang li1 and dahu liu1 1nanjing university of aeronautics and astronautics, nanjing, 210016 2huaiyin institute of technology, huaian, 223003 abstract in order to control the cutting chatter in machining of thin-walled workpieces, the dynamic milling model of thin-walled workpieces is analyzed and built based on the analysis of degrees in two perpendicular directions of tool-workpiece system. in high speed milling of 2a12 aluminum alloy, the compensation method based on the modification of inertia effect was proposed and accurate cutting force coefficients were obtained. modal parameters of tool-workpiece system were acquired via modal analysis tests. the stable lobe for high speed milling of 2a12 aluminum alloy thin-walled workpieces and limit cutting axial-depth at different cutting radial-depth were obtained. the results were verified with cutting tests. the method can be used in milling of thin-walled workpieces to select cutting parameters properly. all these work lay a reliable foundation to the further studies on the cutting chatter rules of thin-walled workpieces. keywords thin-walled workpiece, cutting chatter, dynamic milling model, inertia modification, stable lobe, limit cutting depth 1 introduction with the structural properties of light weight, high strength et al, thin-walled workpiece has been used widely in many fields such as aeronautics & astronautics, mold & die manufacturing. because of the inherent poor rigidity, complicated structures, large metal allowance and bad processing properties, the milling of thin-walled components is difficult for cutting deformation and vibration. a noted previous research on end milling of thin-walled structures was carried out in ref.[1]. a flexible thin-walled rectangular plate was clamped on three edges (cccf) was assumed, the effect of the deflection on the chip load and cutting geometry was not considered in the force calculation. sutherland and devor[2] presented an improved model to take into account the effect of the deflection on the chip load. a dynamic model for milling of a very flexible cantilever plate with rigid end mill by neglecting the time varying structural properties and the changes in the immersion boundaries was built by altintas et al.[3]. budak and altintas went forward one-step by considering the milling of a flexible cantilever plate with slender end mills that incorporate with the mechanistic force model and finite element methods[4]. lim et al. developed a mechanistic force model for predicting the machining errors caused by tool deflection[5]. budak considered the dynamic model of milling of thin-walled workpiece as a mdof(multi-degree 104 tongyue wang: the stability of milling of thin-walled workpiece of freedom) system in[6]. duncan et al. proved the frf (frequency response function) of the tip and error for measuring force coefficients had influenced the prediction accuracy of stability limit[7]. although these models are very useful to analyze the milling of thin-walled structures, the machining of thin-walled workpiece is limited due to the non-liner dependency between the forces and the continuously changing tool immersion angle and chip thickness. in this paper, considering the degrees in two perpendicular directions of toolworkpiece is analyzed and built. the authors proposed the compensation method based on the modification of inertia effect. theoretical analysis to high speed milling stability of thin-walled workpiece is carried out via high speed milling tests and modal analysis tests. the stable lobe and limit cutting axial-depth at different cutting radial-depth for high speed milling of 2a12 aluminum alloy thin-walled workpieces are obtained. 2 dynamic milling model of thin-walled workpiece milling of thin-walled workpiece is usually expressed by the dynamic milling model as shown in figure 1, the cutter and the workpiece are modeled as twodegree-of freedom structures, respectively. after the first revolution, the cutter starts leaving a wavy surface behind because of the bending vibration of the cutter in the normal direction, which is the direction of radial cutting force. when the second revolution starts, the wavy surface causes the tool-workpiece system fluctuating. hence, the resulting dynamic chip thickness is no longer constant. the general dynamic chip thickness can be divided into two parts, one is the intended static chip thickness and the other is the dynamic chip thickness produced owing to vibrations at the present time and one spindle revolution period before. it can be expressed as follows: h(ϕj) = [fz sinϕj + (υj,0 − υj)]g(ϕj) (1) where fz is the feed rate per tooth (mm/rev-tooth),ϕj is the instantaneous angular immersion of tooth j, and (υj,0 − υj) are dynamic offsets of the cutter at the previous and present tooth periods, respectively. the function g(ϕj) is a unit step function, which is used to decide the cutting status of the cutter tooth. if g(ϕj) = 1, the cutter is in cutting, and g(ϕj) = 0, the cutter is out of cutting. henceforth, the static component of the chip thickness fz sinϕj can be dropped from the above equation because it does not contribute to the regeneration dynamic chip thickness. then, the chip thickness can be written in terms of the fixed coordinate system x and y (see fig.1) as follows: h(ϕj) = [∆x sinϕj +∆y cosϕj ]g(ϕj) (2) advances in systems science and applications (2012) vol.12 no.2 105 where ∆x = x− x0 and ∆y = y − y0. here (x, y) and (x0, y0) represent the dynamic offsets of the cutter at the present and previous tooth periods, respectively. fig.1 dynamic milling model of thin-walled workpieces the tangential and radial cutting forces acting on the tooth are proportional to the chip thickness and the axial depth of cut ftj = ktah(ϕj), frj = krftj (3) where kt is the tangential milling force coefficient which is experimentally determined for a tool-workpiece material pair,kr is the ratio of radial force coefficient to tangential force coefficient, a is the cutting width or axial depth of cut. resolving the cutting forces in the x and y directions and summing the cutting forces contributed by all teeth, the milling forces formulate can be built. the dynamic milling forces can be expressed in matrix form: ( fx fy ) = 1 2 akt ( αxx αxy αyx αyy )( ∆x ∆y ) (4) where time-varying directional force coefficients are given by: 106 tongyue wang: the stability of milling of thin-walled workpiece αxx = n−1∑ j=1 −gj [sin 2ϕj + kr(1− cos 2ϕj)] αxy = n−1∑ j=1 −gj [(1 + cos 2ϕj) + kr sin 2ϕj ] αxy = n−1∑ j=1 −gj [(1− cos 2ϕj)− kr sin 2ϕj ] αyy = n−1∑ j=1 −gj [sin 2ϕj − kr(1 + cos 2ϕj)] (5) as the tool rotates, the directional force coefficients vary with time, they can be expanded into fourier series. take the average component of the fourier series expansion, the directional force coefficients are written as: αxx = 1 2 [cos 2ϕ− 2krϕ+ kr sin 2ϕ] ϕex ϕst αxy = 1 2 [− sin 2ϕ− 2krϕ+ kr cos 2ϕ] ϕex ϕst αyx = 1 2 [− sin 2ϕ+ 2krϕ+ kr cos 2ϕ] ϕex ϕst αyy = 1 2 [cos 2ϕ− 2krϕ− kr sin 2ϕ] ϕex ϕst (6) where n is the cutter tooth number, ϕst and ϕex are the entry and exit angles of a tooth, respectively. the transfer function matrix [φ(iw)] of the milling system is: [φ(iw)] = ( φxx(iw) φxy(iw) φyx(iw) φyy(iw) ) (7) where φxx(iw) and φyy(iw) are the direct transfer functions in the x and y directions, and φxy(iw) and φyx(iw) are the cross transfer functions. considering the vibration at the present and previous tooth period, system equation at the chatter frequency wc in the frequency domain can be written as: [f ]eiwct = 1 2 akt[1− eiwct][a0][φ(iwc)][f ]eiwct (8) where a0 is the directional milling matrix. advances in systems science and applications (2012) vol.12 no.2 107 3 principle of dynamic force coefficient measurement and inertia modification a force measuring chain is functionally illustrated in fig.2. in milling, the cutting force produced between the cutter and workpiece acts on the workpiece and is transmitted to the dynamometer. the cutting force leads to a deformation of the piezozlectric sensors of the dynamometer, which produce electric charges in correspondence with the deformation of the sensors. the electric charges are processed into a data file which reflects the size of cutting force following several steps of amplifier and filter. the validity of the acquired force data depends not only on the precision of the hardware devices used and their parameter settings, but also the dynamic effect of the dynamometer itself, particularly when measuring forces in high speed milling. in high speed milling, the tooth passing frequency and/or the high harmonic frequency components induced from the impact effect of the milling cutter are often in the neighborhood of the natural frequencies of the dynamometer. so the vibration of the structure including the dynamometer is exaggerated in the output signals and the measured force signals are damaged. fig.2 diagram of force measuring principle the structure including the dynamometer and workpiece can be treated as typical spring-damper-mass model. to obtain the accurate cutting force data in high speed milling, compensation of the inertia forces is necessary to improve the measurement results. it can be modified as follows: → f= → f0 − → fi (9) where → f is the modified cutting force (n), → f0 is the directly measured cutting force (n), → fi is the inertia force (n). the inertia force → fi can be written as → fi= (me +mw)a , me is the equivalent mass of the dynamometer (kg), mw is the mass of workpiece (kg), and a is the measured acceleration(m/s2). following the quick mechanistic method of calibrating and the above inertia modification approach, a set of high speed milling experiments are conducted at following cutting conditions: workpiece: 2a12 aluminum alloy 108 tongyue wang: the stability of milling of thin-walled workpiece equipment: mikron ucp 710 high speed machining center, kistler 9265b dynamometer, kd1001a acceleration sensor tool: yg813 carbide tipped tool, two flutes, diameter=20mm test parameters: slotting, axial depth of cut=1mm, radial depth of cut=3mm, spindle revolution=10000 rev/min, feed rate per tooth changes from 0.01mm/z to 0.055mm/z at 0.005mm/z increment after treating the experimental results, the values of force coefficients can be worked out: kt=4252mpa,kr=0.858. 4 stability analysis of milling of thin-walled workpiece[3,4,6] 4.1 the limit axial depth of cut after solving the characteristic equation (8) of dynamic milling system, the eigenvalue is given as: λ = − 1 2a0 (a1 ± √ a12 − 4a0) (10) where { a0 = φxx(iwc)φyy(iwc)(αxxαyy − αxyαyx) a1 = αxxφxx(iwc) + αyyφyy(iwc) (11) in [3,4,6] , altintas and budak presented the solution of stability is in detail and provided the limit depth of cut and spindle speed, i.e. stability lobes as: alim = −2πλr nkt (1 +k2) (12) n = − 60wc n(ε+ 2kπ) (13) where κ = sinwct 1−coswct , λr is the real part of the eigenvalue, ψ = arctanκ is the phase shift of the eigenvalue, ε = π − 2ψ is the phase shift between the current chatter mark and the previous chatter mark. 4.2 end milling with a flexible cutter a above mentioned yg813 carbide tipped tool with 2 flutes, 20mm diameter is used in end milling of aluminum alloy 2a12. the gage distance is 150mm from the collet. the transfer function of the cutter attached to the spindle is measured in both feed and normal directions with an impact hammer instrumented with a pcd208c02 piezoelectric force transducer and a b&w22100 acceleration sensor. the modal parameters such as modal mass, modal dampness etc are identified from modal analysis software uteklma and are given in table 1. later the modal parameters are used to simulate the stability lobes for milling advances in systems science and applications (2012) vol.12 no.2 109 of 2a12 alloy. four kinds of radial depth of cut are adopted. the results are given in fig.3. the region above the curve is unstable cutting region and under the curve is stable cutting region. table 1 identified modal parameters direction modal mass m(kg) dampness c(n ∗ s/m) stiffness k(n/m) nature frequency ωc(hz) tangential 1.37 533.1 1.67e+7 556 radial 1.32 510.0 1.57e+7 550 (a) ae = 1mm (b) ae = 2mm (c) ae = 3mm (d) ae = 4mm fig.3 stability lobe 4.3 analysis of experimental results of varying axial depth of cut two sets of varying axial depth of cut experiments are carried out to investigate the influence of axial depth of cut to cutting stability. the principle scheme is given in fig.4. the 2a12 aluminum alloy workpiece with 30 inclined angles is 110 tongyue wang: the stability of milling of thin-walled workpiece adopted in order to increase or decrease the axial depth of cut during the milling period. the above mentioned yg813 carbide tipped tool with 2 flutes, 12mm diameter is used with the 150mm gage distance measured from the collet. the milling stability is studied by analyzing the radial cutting force, which influences the stability most. during the milling period, the 3mm radial depth of cut, 10000 rev/min spindle revolution, 0.01mm/z feed rate per tooth are kept unchanged. the axial depth of cut changes from 0.1mm to 3.98mm and from 3.98mm to 0.1mm, respectively. the radial cutting force results measured via the kistler 9265b dynamometer are shown in fig.5. although the varying way is different, the radial cutting force fluctuated severely at about 2mm axial depth of cut together. the limit axial depth of cut seems to be about 2mm at 3mm radial depth of cut, 10000 rev/min spindle revolution. obviously, the measured results are in very close agreement with the results of stability lobe (see fig.3(c)). the stability lobes presented a reasonable range for selecting the cutting parameters. fig.4 diagram of varying axial depth of cut test principlee (a) axial depth of cut, 0.1mm to 3.98mm (b) axial depth of cut, 3.98mm to 0.1mm fig 5 radial cutting force of varying cutting-depth test advances in systems science and applications (2012) vol.12 no.2 111 5 conclusions on the basis of analysis of degrees in two perpendicular directions of tool-workpiece system, the dynamic milling model of thin-walled workpieces is analyzed and built. the compensation method based on the modification of inertia effect is proposed and accurate cutting force coefficients are obtained through high speed milling test. modal parameters of tool-workpiece system are acquired via modal analysis tests. the modal parameters and cutting force coefficients are used to simulate the stability lobe at 4 cutting radial depth of cut for high speed milling of 2a12 aluminum alloy thin-walled workpieces. the results are verified with varying axial depth of cut tests. all these work lay a reliable foundation to the further studies on the high speed milling stability of thin-walled workpiece. acknowledgements the authors greatly appreciate the financial support provided by the national natural science foundation of china (no.10477008), the foundation of education department of jiangsu province (no.07kjb460008) and the research foundation of dml-hyit(hgdml-0801). references [1] w.a. kline, r.e. devor, j.r. lindberg. (1982), the prediction of cutting forces in end milling with application to cornering cuts, international journal of machine tool design and research, vol.22, no.1, pp.7-22. [2] j.w. sutherland, r.e. devor. (1986), an improved method for cutting force and surface error prediction in flexible end milling systems, asme journal of engineering for industry,vol. vol.108, no.b-4, pp.269-279. [3] y. altintas, e. budak. (1995), analytical prediction of stability lobes in milling, annals of the cirp, vol.44, no.1, pp.357-362. [4] e. budak and y. altintas. (1998), analytical prediction of chatter stability in milling. part ii:application of the general formulation to common milling systems. trans, asme journal of dynamci systems, measurement and control, vol.120, no.1, pp.22-36. [5] e.m. lim, h. feng, c. menq, z. lin. (1995), the prediction of dimensional error for sculptured surface productions using the ball-end milling process. part 1: chip geometry analysis and cutting force prediction, international journal of machine tools manufacture, vol.35, no.8, pp.1149-1169. 112 tongyue wang: the stability of milling of thin-walled workpiece [6] e. budak. (2006), analytical models for high performance milling. part i: cutting forces, structural deformations and tolerance integrity, international journal of machine tools and manufacture, vol.46, no.12-13, pp.1478-1488. [7] g.s. duncan, m. kurd, t.l. schmitz. (2006), uncertainty propagation for selected analytical milling stability limit analyses, transactions of namri/sme, vol.34, no.1, pp.17-24. advances in systems science and applications (2012) vol.12 no.1 96-102 dmu-oriented design based on cases and the research on kbe system yiming qian, xu chen, bin zhang and jinjia zhou school of mechanical engineering, chongqing university, chongqing 400044, p.r.china abstract by analyzing the technology composition of digital mock-up (dmu) and knowledge based engineering (kbe) system, the dmu-oriented design methods based on cases is proposed, which is significant for the development of model design methodology. the design flow used in the kbe system is developed based on the product cases and neural network technology, and the kbe system for motorcycle design is described in detail. keywords knowledge acquisition, dmu, kbe, intelligent design 1 dmu&kbe dmu (digital mock-up) technology is the technology of product development and design growing with the development of computer, computer graphics knowledge and other emerging technologies[1-3]. there is no consistent definition so far. the main model of dmu is a dendriform relation model based on the hierarchical structure of products. it describes the assembly information, functional information, movement relation information and cooperative relation information of the whole product, and also expresses design parameter and engineering semantic constraint of products parts and describes the design information about each stage of the life circle of the whole product. fig.1 the framework of kbe advances in systems science and applications (2012) vol.12 no.1 97 kbe (knowledge-based engineering) is a computer integrated disposal technology, which provides the best solutions to engineering problems and tasks through the driving and reproduction of knowledge[4]. the four cores of kbe technology are knowledge system, knowledge acquisition, product modeling and analysis technique. knowledge system is mainly used to denote and process engineering design knowledge, facing to the engineering design personnel and embodying the intelligent level of system. knowledge acquisition technique is primarily applied in engineering knowledge acquisition, including automatic acquisition and manual acquisition, embodying the knowledge of experts in all fields, making the whole design system improve engineering design and analysis ability step by step, thereby making the system achieve the goal of kbe system of engineering design. the framework of kbe is presented in fig.1. 2 case-based design case-based design (cbd) method is a reasoning method based on case applying in the design filed. its design idea originates from human thinking modes. confronted with the new design requirements, the similar design conditions, which happened before, usually emerge into the designers mind firstly, and according to them, the designer identifies a new design scheme associated with the standardized design rules[5-6]. the contents of case consist of the description of design problems, the design rationales, the evaluation of design scheme and the final design scheme. at present, the application of artificial neural network in reusing case experience and knowledge of parts is the relative advanced technology. the design process of case-based and nn-based (neural network-based) kbe system is summarized in fig.2. the whole process involves four key technologies: case knowledge reuse, case filtering, case supplement and modification and case base maintenance. the whole process involves four key technologies: case knowledge reuse, case filtering, case supplement and modification and case base maintenance. nn(neural network) indicates the given conception or knowledge by interlinking a great deal of nerve cells and connecting weight distribution. in the engineering that uses artificial network to acquire case knowledge, these cases which are systematically filtered and have appropriate similarity and corresponding results should be provided by the special adaptive algorithm of nn to learn the samples and constant modifying connecting weight distribution to meet the demands in order to make nn acquire the same exporting cases as much as possible under the condition of the identical input. this is the ideal process for nn to learn automatically. on this occasion, the case experience and knowledge will be transformed into the joint strength of each nerve cell in nn. when the error between output case and filtration one comes to a certain precision, nn finishes 98 yiming qian:dmu-oriented design based on cases and the research on kbe system the process of learning, and stores the acquired nn into case base simultaneously and establishes the corresponding relation with the relevant case. fig.2 the design flow of case-based and nn-based kbe the structure-function model of motorcycle parts is the base of establishing artificial nn, and the artificial nn of cases can be set up according to the structurefunction model of parts. we should analyze the cases structure after choosing the motorcycle parts cases. firstly, the designer should analyze the relationship between the structure parameters of the parts and its function, and confirm what main structure parameter affects its function. secondly, the designer presents the exact numerical range of every key parameter and function parameter, and the two parameters have different function relation in different count range, which shows the relationship between structure and function. we can adjust the function relation to make the output function and the case function similar as much as possible. for instance, the modal parameter of frame structure is the primary parameter that affects the comfort of four-wheel motorcycle. when designing the frame, the main structure parameters that have influence on modal parameter are the height and width between the front and the back upright pole of the frame, the length and width between the bottom and the top horizontal pipe of frame, the whole height of frame, the height of mid upright pole, the length of frame tail, the assembly angle and the materials of frame, displayed in fig. 3. according to the design experience data that enterprise accumulates step by step, the fitting function relation of module parameter of multiple parameters can be advances in systems science and applications (2012) vol.12 no.1 99 established to make the structure parameter based on cases acquire the function parameters that are close to the cases. this is the artificial nn for system to establish the motorcycle parts and the process of finish studying. then we describe the design requirements, inputting the description into the learned nn, a integrated case experience scheme is output, and the final design scheme will be formed through modifying it according to the cad system; or, we directly call the cases combined with cad system to modify, at last, an outcome that meets the design requirements will be got. the application of nn and the directly call of cases is a side-by-side route, proceeding simultaneously. this is the design flow case-based and nn-based kbe system. fig.3 frame of four-wheel motorcycle searching cases includes the following procedures: distributing index, looking up the relative cases and choosing the best cases. the index of cases is to determine when we can use the cases in future and it indicates the parts deserved to learn in cases. when establishing the index, its relativity, the generality and the feasibility should be taken into account. the filtration of cases can be chosen with regard to the similarity of cases, and the calculation of the similarity of cases will be listed later. 2.1 the calculation of similarity of property when two properties are identical, the similarity is 1. the calculation formula of two properties is as follows: sim(ti, si) = 1− △d ti (1) in the formula, sim is the similarity function; ti is to show the certain property in design requirements of the object case; si is to show the corresponding property in the source case;△d = |ti − si|. 2.2 the definition of attribute weights the setting of property can be designed according to the requirements of nn, and the user can configure the corresponding weights to the every property according to the different focus on design requirements. 100 yiming qian:dmu-oriented design based on cases and the research on kbe system 2.3 the calculation of the whole similarity of product sim(t, s) = n∑ i=1 wi × sim(ti, si) (2) n∑ i=1 wi = 1 in the formula, sim is the whole similarity function, t is to show the object case; s is to show the case in the case base; n is to show the number of property of each case; wi is to show the weight of the property of i. the modification and supplement of case is the difficulty of cbd method, and it includes identifying the difference between case and problem, finding out the parts that require modification and reservation. according to the characteristics of case, the types of modifications are as follows: (1) direct change. when the cases fit the design requirements completely, it only requires altering the relative parameter of the previous case. (2) change with modification. do partial correction and make use of the relative knowledge in this field. (3) scheme change based on frame. store the new cases into case base after finishing designing new product 3 the development of kbe system and its application example in the motorcycle design with the powerful secondary development function of ug, we can successfully develop kbe of knowledge-oriented motorcycle based on ug, integrating it with pdm software (team center), multi-cae soft wares and database system to realize the intellectualization and being knowledgeable. the system consist mainly of four modules: the assembly-oriented design guide, the management of parts base, the analysis of man-machine engineering and the interface with pdm system. 3.1 the assembly-oriented design guide the process of this kbe module is, first, entering the assembly-oriented design guide, configuring the basic performance parameter of the designed vehicle model. kbe program forces the designer to design and assemble parts from system-class parts. because the system-class basic part of the whole motorcycle is frame, so the frame is the first part that is to be designed and assembled. fig.5 shows the main interface of assembly design guide of frame of a four-wheeled atv motorcycle. after the design parameters are completely defined in the design platformthe management system of parts base makes use of cases inquiry system to inquire advances in systems science and applications (2012) vol.12 no.1 101 the storage position of the parts model in accordance with the requirements in the parts storeroom according to the design knowledge in the knowledge storeroom and these parameters, and it will be showed on the interface for the choice of designer according to similarity. after choosing the frame, the designers can establish the new frame according to the procedure based on cases and nn. after designing the new frame, we store it named by the number of file into the parts storeroom, and at the same time register the storage position of the file into the registration table of parts. the designer can enter the design of the rest parts fixed on the frame after finishing designing frame, and the process is similar to the design of frame. fig.4 interface for frame design of oriented-assembly 3.2 the management of parts storeroom and the analysis of man-machine engineering the cases base of kbe system is to establish the parts base and the management system of parts base under the condition of three-dimension cad circumstance. these new established parts are stored into the right position of parts storeroom by the management system and registered its position information into registration table. the designers must simulate and analyze every function of the vehicle after accomplishing the assembly design of the vehicle. the following are kbe system and the integration interfaces of cae analysis software: adams, msc. patran, msc. nastran, anasys and so on. 3.3 the interface to the pdm system the main function of this module is to achieve the integration between kbe system and pdm system. the knowledge of design procedure of extracting dmu mainly include the scheme management procedure of dmu and the procedure of development assembly of the motorcycle dmu, storing the relative information with procedure into pdm system and achieving the reuse of the procedure 102 yiming qian:dmu-oriented design based on cases and the research on kbe system knowledge and experience. 4 conclusion this paper has discussed the composition of dmu and kbe technology, presented the products design procedure of kbe system based on cases and nn. then, the kbe of motorcycle design based on ug that integrates with pdm and cae is developed. the integration of dmu, pdm, cae and kbe makes the analytical technology of product, the experiences of experts and the analysis results reused in the engineering design, improving the design efficiency and quality of products greatly, and it is an important direction of intelligent development of product design. acknowledgements project of nature sustentation fund for chongqing science committee (cstc, 2007bb0116). references [1] guo gang. (2004), digital design and management of new products, chongqing: chongqing university press. [2] wu meiping, liao wenhe. (2008), “application and research of development management knowledge of virtual products in dmu development of helicopter”, science and technology of mechanical, vol.27, no.5, pp.633-639. [3] chen xu, huang zehao, guo gang. (2005), “technology research of dmu of motorcycle products development”, modern manufacture engineering, no.3, pp.69-71. [4] li zhi, jing xianlong, jia huaiyu. (2006), “expression and reuse technology of knowledge for products design”, shanghai transportation university journal, vol.40, no.7, pp.1183-1186. [5] gu jianguang, zhang weihua, xie hongyu. (2008), “knowledge acquisition technology of product design by integrating cases with experiences”, computer integration manufacture system, vol.14, no.3, pp.418-424. [6] dai rong, he yuling, he xiansong. (2009), “integration application research of cases reasoning and regulation reasoning of motorcycle by intelligent designing”, computer integration manufacture system, vol.15, no.3, pp.411423. analysis and studying of cascading failures in gene networks advances in systems science and applications (2011), vol. 11, no. 1-2 163-172 issn 1078-6236 international institute for general systems studies, inc. analysis and study of cascading failures in gene network * wang shudong 1 , shi songtao 2 , ge yanru 3 , sun longxiao 1 , xu dashun 4 , meng dazhi 1,2 1 college of information science and engineering, shandong university of science and technology, qingdao, shandong 266510, china 2 school of software engineering; college of applied science, beijing university of technology, beijing 100124, china 3 department of control science and engineering, huazhong university of science and technology, wuhan, hubei 430074, china 4 department of mathematics, southern illinois university carbondale, makanida 62958, usa email: wangshd2008@yahoo.com.cn, dzhmeng07@yahoo.com.cn abstract genome forms gene networks in terms of complicated interactions to realize its functions. further research for gene networks can help to comprehend and predict many unknown functions of genome. in this work, cascading failure models of weighted gene networks are built and the robustness of the models is also analyzed and discussed. based on the data of normal and lung adenocarcinoma stages, by simulating and analysing the cascading failures of the two gene network models, that cascading failures occur more likely in the networks for adenocarcinoma experimental groups than the one of normal control group is discovered. in the numerical experiments, we notice that nine genes of experimental group and eight genes of control group are of very strong destructibility for the robustness of experimental and control networks respectively. the failures of these genes can lead to the collapse or paralysis of the whole network. therefore, we conclude that these genes might play important roles in keeping normal level or developing lung adenocarcinoma of organisms. when applying the methods of modeling and analysing cascading failures in gene network to other diseases’ data, biomedical scientists can be enlightened for understanding the mechanism of diseases and predicting the functions of significant genes. keywords systems biology gene network cascading failure network statistics 1. introduction research on the biological functions of genome is a major issue in life science [1-3] . the study for gene networks describing the complicated genetic interactions in genome is an important way to understand biological functions [4-9] . so far, many methods have been proposed to build and analyze gene networks [4] . amy hin yan tong et al. [5] investigated the correspondence between the dense local neighborhoods in gene regulatory network and biological functions of yeasts. mark kittisopikul and gürol m. süel [6] studied the biological significance of feed-forward loop motifs in gene network of escherichia coli and discovered that most * this work was supported by the national natural science foundation of china (grant nos. 60874036, 60503002) and sdust research fund. mailto:wangshd2008@yahoo.com.cn http://www.pnas.org/search?author1=mark+kittisopikul&sortspec=date&submit=submit http://www.pnas.org/search?author1=g%c3%bcrol+m.+s%c3%bcel&sortspec=date&submit=submit 164 wang: analysis and study of cascading failures in gene network feed-forward loops have two kinds of regulatory functions. hallinan j. s. et al. [7] analyzed the relationships among the motifs, feedback loops and their dynamical features in gene regulatory network. bowers et al. [8] proposed a computational approach-logic analysis of phylogenetic profiles to identify detailed relationships among genes or proteins on the basis of genomic data, which was applied into 4873 distinct orthologous protein families of 67 fully sequenced organisms, and identified 750, 000 triplets previously unknown logic regulatory relationships. shudong wang et al. [9] constructed a logical network with 16 active genes of shoot in different external stimuli, and analyzed the dynamics of the logical network. now, the theoretical models of cascading failures and their mechanisms, prevention and control for various actual complex networks have been relatively deeply studied [10-17] . for instance, r. kinney et al. [16] analyzed the cascading failures in the north american power grid. the results show that deliberate attacks can lead to a substantial decline in the transmission efficiencies of power grid, while random failures have nearly no influence. ashley g. smart et al. [17] investigated the relationships between structure and robustness in the metabolic networks of escherichia coli, methanosarcina barkeri, staphylococcus aureus, and saccharomyces cerevisiae using a cascading failure model based on a topological flux balance criterion and found that metabolic networks are exceptionally robust compared to appropriate null models. but the reports are rare about cascading failures in gene network. this work investigates the influences of cascading failures on gene networks. based on the documental data, by comparing cascading failures of gene networks for normal and lung adenocarcinoma groups, we discover control networks are quite robuster than lung adenocarcinoma experimental ones. this indicates the change from normal organisms into lung adenocarcinoma ones may result from dysfunctions or gene mutations (considered as failures) of some genes. through numerical experiments, we notice that failures of some genes in experimental and control groups can lead to collapse or paralysis of the whole network. these genes might play important roles in keeping normal level or developing lung adenocarcinoma in organisms. this paper is organized as follows: the work background and existing methods for studying gene networks are introduced in the first part. the cascading failure model and its algorithm used in this work are presented in detail in the second part. the methods and main results of numerical experiments of cascading failure model are described based on two data sets of (normal) control group and (lung adenocarcinoma) experimental group in the third part. the obtained results are analyzed and discussed, and the possible corresponding biological significances are also pointed out in the fourth part. 2. the model and algorithm of cascading failure in this research, we consider cascading failures of complex gene networks, so we treat genes no different from nodes of complex networks. we use the capacity-load in cascading failure model. let  wevg ,, be a complex (directed or undirected) gene network with node-set  1,2, ,v n  , edge-set e and weight-set w . suppose ijw is the weight from node i to j in complex gene network g . then the edge-length from node i to j is defined as the reciprocal value ijw 1 of ijw . if 0ijw , then the edge-length from node i to j is  . the greater the weight between two nodes is, the lesser the edge-length is; the lesser the weight is, the greater the edge-length is, and vice versa. the shortest paths from node i to j are these paths corresponding to the smallest sum of edge-length in all the paths from node i to j . obviously, the shortest paths from node i to j are not always unique. suppose there exist app:ds:have app:ds:describe app:ds:always advances in systems science and applications (2011), vol. 11, no. 1-2 165 p shortest paths from node i to j . then the load of any shortest path r is defined as p 1 of the product of all the weights in r , i.e.   p w rji ij , . the load jl of node j is defined as the sum of the loads of all the shortest paths passing through node j . the capacity jc of node j is proportional to its initial load 0 jl , i.e.   01 jj lc  , 1,2, ,j n  , where constant 0  is a tolerance factor. if the load of a node is greater than its capacity, then it is called a failure node. after deleting node i , and causing is failure nodes (including node i ), then is is defined as the size of cascading failure of node i and n s d i i  as the size-ratio of cascading failure. if cfi td  , then the network breaks down, otherwise, the network doesn’t have failure. this is a criterion of network failure, where cft is the threshold of network failure. let       cfi cfi td td isign ,0 ,1 )(1 . then the percentage of failure nodes of the network n isign p n i   1 )(1 ; the largest size-ratio of cascading failure  max max , 1,2, ,ir d i n   ; the average size-ratio of cascading failure 1 n i i d r n   . let 1, 2( ) 0, i i d d sign i d d     ( d is a variable parameter). then the cumulative probability of size-ratio of cascading failure n isign ddp n i   1' )(2 )( , which indicates the probability of size-ratio id of cascading failure greater than d . obviously, maxr , r and '( )p d d are the important parameters measuring the robustness or fragility of network. based on the above mentioned definitions and symbols, we present the algorithm （cfa）of cascading failure model as follows: ① input the weight matrix of complex gene network  , ,g v e w . ② calculate initial load 0 jl of node j and its capacity   01 jj lc  , nj ,,2,1  . 1i . ③ delete node i and its incident edges in the network. ④ calculate the load of every node in the present network and compare the capacity with the load of every node. if the load is lesser than the capacity for every node in the present network, then go to ⑤, otherwise, delete every node and its incident edges whose load is greater than its capacity, go to ④. ⑤ if the size-ratio of cascading failure after deleting node i is greater than or equal to the threshold cft of network failure, then the network breaks down. ⑥ 1 ii . if ni  , then go to ③. ⑦ calculate the largest size-ratio of cascading failure maxr , average size-ratio of cascading 166 wang: analysis and study of cascading failures in gene network failure r , and the cumulative probability of size-ratio of cascading failure '( )p d d . 3. the methods and r esults of numerical experiments 3.1 data sources data used in this work are from the results of lung adenocarcinoma network studied by yuanyuan zhang et al. (the detailed data sources can be seen in). we use the mutual information network and directed (or 1-order logic) network for control group (abbreviated as n) and lung adenocarcinoma experimental group (abbreviated as ac). in the mutual information network (directed weighted network), mutual information value (u value) is the weight of networks. we denote the weight matrices of mutual information (directed-weighted) network for control and experimental groups by nm ( nl ) and acm ( acl ), respectively. table 1 the list of  ddp ' along with the change of the network threshold nett in control and experimental networks respectively. when 34.0d ,  ddp ' is equal to zero nett 0.52 0.5 0.45 0.4 d ac n ac n ac n ac n 0.01 1 1 1 1 1 1 1 1 0.02 1 1 1 1 1 0.0794 1 0.0714 0.03 1 0.1053 1 0.0870 0.1351 0.0794 0.1667 0.0714 0.04 0.2069 0.1053 0.1875 0.0870 0.1351 0.0794 0.1667 0.0714 0.05 0.2069 0.1053 0.1875 0.0870 0.1351 0.0794 0.1667 0.0714 0.06 0.2069 0.1053 0.1875 0.0870 0.1351 0.0794 0.1667 0.0714 0.07 0.1724 0.1053 0.1563 0.0870 0.1351 0.0794 0.1667 0.0714 0.08 0.1724 0.1053 0.1563 0.0870 0.1351 0.0794 0.1667 0.0714 0.09 0.1724 0.1053 0.1563 0.0870 0.1351 0.0794 0.1667 0.0714 0.1 0.1724 0.1053 0.1563 0.0870 0.1351 0.0794 0.1667 0.0714 0.11 0.1724 0.1053 0.1563 0.0870 0.1351 0.0794 0.1667 0.0714 0.12 0.1724 0.1053 0.1563 0.0870 0.1351 0.0794 0.1667 0.0714 0.13 0.1724 0.1053 0.1563 0.0870 0.1351 0.0794 0.1667 0.0714 0.14 0.1724 0.1053 0.1563 0.0870 0.1351 0.0794 0.1667 0.0714 0.15 0.1724 0.1053 0.1563 0.0870 0.1351 0 0.1667 0.0714 0.16 0.1724 0.1053 0.1563 0.0870 0.1351 0 0.1667 0.0286 0.17 0.1724 0.1053 0.1563 0.0870 0.1351 0 0.1667 0.0286 0.18 0.1724 0.1053 0.1563 0 0.1351 0 0.1667 0 0.19 0.1724 0.1053 0.1563 0 0.1351 0 0.1667 0 0.2 0.1724 0.1053 0.1563 0 0.1351 0 0.1667 0 0.21 0.1724 0.1053 0.1563 0 0.1351 0 0.1667 0 0.22 0.1724 0.0790 0.1563 0 0.1351 0 0.1667 0 0.23 0.1724 0.0790 0.1563 0 0.1351 0 0.1667 0 0.24 0.1724 0 0.1563 0 0.1351 0 0.1667 0 0.25 0.1724 0 0.1563 0 0.1351 0 0.1667 0 0.26 0.1724 0 0.1563 0 0.1351 0 0.1667 0 0.27 0.1724 0 0.1563 0 0.1351 0 0.1667 0 advances in systems science and applications (2011), vol. 11, no. 1-2 167 0.28 0.1724 0 0.1563 0 0.1351 0 0.1667 0 0.29 0.1724 0 0.0625 0 0.1351 0 0.1667 0 0.3 0.1724 0 0.0625 0 0 0 0.1667 0 0.31 0.1724 0 0.0625 0 0 0 0 0 0.32 0.069 0 0 0 0 0 0 0 0.33 0.069 0 0 0 0 0 0 0 0.34 0.069 0 0 0 0 0 0 0 3.2 the methods and results 3.2.1 the results of mutual information gene network in order to highlight the characteristics of the network structure, we analyze the changes of '( )p d d along with the network threshold nett . when taking different network thresholds, we can obtain the mutual information networks with different coarse granularities. the corresponding weight matrices nm and acm are the inputs in the above cfa algorithm. obviously, the greater the network threshold nett is, the coarser the granularity is, the more the lost information is and the computational complexity is relatively low; on the contrary, the lesser the network threshold nett is, the finer the granularity is, the less the lost information is, but the computational complexity is relatively high. the detailed data and the changing curves are in table 1 and fig. 1. from table 1 and fig. 1, it is obvious that '( )p d d of experimental network is clearly higher than that of control one under any network threshold. this indicates the ratio of nodes in experimental network which can result in cascading failures is much greater than the one of control group. with the increasing of d ,  ddp ' in control network reduces to zero earlier than in experimental one. in other words, taking certain appropriate d , control network has no failure while experimental network has more failures. moreover, with the increasing of network threshold nett , the platform value of  ddp ' of control network is 0.0714, 0.0794, 0.0870 and 0.1053 respectively, showing gradually increasing tendency. this indicates that with the decreasing of the numbers of nodes and edges, cascading failures are more likely to occur in the gene networks, namely: the robustness goes worse. the genes resulting in the cascading failures of control and experimental groups under all four network thresholds are nras, pik3ca, mapk9, top2a and fgf1, ret, wt1, tcl1a, hrk, respectively. the detailed situations can be seen in table 2 and 3. fig. 1 taking the network threshold nett as 0.4, 0.45, 0.5, 0.52 respectively, the changing curves of )( ' ddp  along with d in control and experimental networks. table 2 the list of the relative greater size-ratio *d of cascading failure of control network under different network thresholds nett , where * denotes the gene. the following presentation is similar. 0.52 0.5 0.45 0.4 app:ds:clearly app:ds:appropriate app:ds:situation app:ds:relative 168 wang: analysis and study of cascading failures in gene network gene *d (%) gene *d (%) gene *d (%) gene *d (%) nras 23.68 nras 17.39 nras 14.2857 nras 15.71 pik3ca 23.68 pik3ca 17.39 pik3ca 14.2857 pik3ca 17.14 mapk9 21.05 mapk9 17.39 mapk9 14.2857 mapk9 17.14 top2a 23.68 top2a 17.39 rbl1 14.2857 rbl1 15.71 top2a 14.2857 top2a 15.71 table 3 the list of the relative greater size-ratio *d of cascading failure of experimental network under different network thresholds nett . 0.52 0.5 0.45 0.4 gene *d (%) gene *d (%) gene *d (%) gene *d (%) fgf1 34.48 fgf1 31.25 fgf1 29.73 fgf1 30.95 ret 31.03 ret 28.125 ret 29.73 fgf2 30.95 wt1 31.03 wt1 28.125 wt1 29.73 hspb2 30.95 tcl1a 34.48 tcl1a 31.25 tcl1a 29.73 ret 30.95 hrk 31.03 hrk 28.125 hrk 29.73 wt1 30.95 tcl1a 30.95 hrk 30.95 3.2.2 the results of directed gene network to comprehensively measure the robustness and fragility of directed weighted gene network, we analyze the situations of r , maxr and '( )p d d with the changes of network thresholds nett (table 4 and table 5). from table 4 and 5, we discover that r , maxr and '( )p d d of experimental network are clearly greater than the ones of control network under any network threshold. this shows that cascading failures occur in experimental network more easily than in control one. the genes resulting in cascading failures of control and experimental networks under five network thresholds are bad, ing1, raf1, traf3 and esr2, hspb2, nov, tal1 respectively. the detailed situations can be seen in table 6 and 7. app:ds:relative app:ds:situation app:ds:clearly app:ds:situation advances in systems science and applications (2011), vol. 11, no. 1-2 169 fig. 2 taking the network threshold nett as 0.1, 0.125, 0.15, 0.175, 0.2 respectively, the changing curves of )( ' ddp  along with d in control and experimental networks. table 4 the list of r , maxr along with the change of network threshold nett in control and experimental networks respectively. nett stage no. of nodes no. of edges r maxr 0.100 ac 60 487 0.1355 0.2167 n 98 1124 0.0858 0.1531 0.125 ac 60 392 0.1385 0.2000 n 95 887 0.0756 0.1158 0.150 ac 59 338 0.1390 0.2034 n 90 700 0.0635 0.1000 0.175 ac 58 285 0.1281 0.1897 n 86 560 0.0686 0.1163 0.200 ac 58 240 0.0888 0.1552 n 77 446 0.0727 0.1169 table 5 the list of '( )p d d along with the change of network threshold nett in control and experimental networks respectively. when 22.0d ,  ddp ' is equal to zero. nett 0.1 0.125 0.15 0.175 0.2 d ac n ac n ac n ac n ac n 0 1 1 1 1 1 1 1 1 1 1 0.01 1 1 1 1 1 1 1 1 1 1 0.02 0.5167 0.3469 0.4833 0.2947 0.4237 0.2333 0.3621 0.2326 0.3448 0.2597 0.03 0.5167 0.2959 0.4833 0.2947 0.4237 0.2000 0.3621 0.1977 0.3448 0.2208 0.04 0.5167 0.2449 0.4833 0.2316 0.4237 0.1889 0.3621 0.1744 0.2931 0.1818 0.05 0.5167 0.2449 0.4833 0.2211 0.4237 0.1556 0.3621 0.1512 0.2931 0.1818 170 wang: analysis and study of cascading failures in gene network 0.06 0.5167 0.2347 0.4667 0.2211 0.4068 0.1222 0.3448 0.1512 0.2759 0.1558 0.07 0.4833 0.2143 0.4667 0.2000 0.3898 0.0889 0.3103 0.1163 0.2069 0.1299 0.08 0.4833 0.2143 0.4667 0.1684 0.3898 0.0778 0.3103 0.1163 0.2069 0.1169 0.09 0.4500 0.2143 0.4167 0.1158 0.3390 0.0333 0.2414 0.0814 0.1552 0.1169 0.10 0.4500 0.1939 0.4167 0.0632 0.3390 0.0333 0.2414 0.0233 0.1552 0.0909 0.11 0.3667 0.1837 0.3833 0.0105 0.3051 0 0.2241 0.0116 0.0517 0.0390 0.12 0.3000 0.1122 0.3000 0 0.2712 0 0.2241 0 0.0517 0 0.13 0.3000 0.0510 0.3000 0 0.2712 0 0.2069 0 0.0517 0 0.14 0.2333 0.0102 0.2333 0 0.2542 0 0.1552 0 0.0517 0 0.15 0.2333 0.0102 0.2333 0 0.2542 0 0.1552 0 0.0517 0 0.16 0.1333 0 0.1833 0 0.1864 0 0.0690 0 0 0 0.17 0.0833 0 0.0833 0 0.0339 0 0.0690 0 0 0 0.18 0.0833 0 0.0833 0 0.0339 0 0.0517 0 0 0 0.19 0.0500 0 0.0333 0 0.0169 0 0 0 0 0 0.20 0.0500 0 0.0333 0 0.0169 0 0 0 0 0 0.21 0.0333 0 0 0 0 0 0 0 0 0 0.22 0 0 0 0 0 0 0 0 0 0 table 6 the list of the relative greater size-ratio *d of cascading failure of control network under different network thresholds nett , where * denotes the gene. 0.1 0.125 0.15 0.175 0.2 gene *d (%) gene *d (%) gene *d (%) gene *d (%) gene *d (%) apc 15.31 ing1 11.58 elk1 10 traf3 11.63 bad 11.69 akt1 13.27 apc 10.53 raf1 10 bad 10.47 atf2 11.69 axl 13.27 bad 10.53 traf3 10 fas 9.30 ing1 11.69 fosl2 13.27 mll 10.53 bad 8.89 hck 9.30 apc 10.39 grb2 13.27 pml 10.53 hck 8.89 ing1 9.30 hck 10.39 bad 12.24 raf1 10.53 ing1 8.89 nras 9.30 mll 10.39 ing1 12.24 akt1 9.47 mll 8.89 raf1 9.30 traf3 10.39 nras 12.24 axl 9.47 akt1 7.78 akt1 8.14 cxcl2 9.09 sell 12.24 elk1 9.47 axl 8.14 raf1 9.09 tp53 12.24 grb2 9.47 cxcl2 8.14 sfrs3 7.79 traf3 12.24 nras 9.47 mcc 11.22 bcl2 8.42 mll 11.22 cxcl2 8.42 notch1 11.22 mapk9 8.42 mapk3 11.22 traf3 8.42 raf1 11.22 aven 8.42 rara 11.22 supt4h1 11.22 elk1 10.2 hck 9.18 app:ds:relative advances in systems science and applications (2011), vol. 11, no. 1-2 171 mapk9 9.18 table 7 the list of the relative greater size-ratio *d of cascading failure of the experimental network under different network thresholds nett , where * denotes the gene. 0.1 0.125 0.15 0.175 0.2 gene *d (%) gene *d (%) gene *d (%) gene *d (%) gene *d (%) gli2 21.67 fes 20.00 hspb2 20.34 nov 18.97 extl3 15.52 tp63 21.67 ros1 20.00 nov 18.64 wnt3 18.97 nov 15.52 ret 20.00 hspb2 18.33 erg 16.95 tcl1a 18.97 ros1 15.52 fes 18.33 wnt3 18.33 esr2 16.95 extl3 17.24 esr2 10.34 tal1 18.33 tcl1a 18.33 extl3 16.95 esr2 15.52 gli2 10.34 esr2 16.67 esr2 16.67 fes 16.95 hspb2 15.52 hspb2 10.34 extl3 16.67 gli2 16.67 cxcl3 16.95 tal1 15.52 il1a 10.34 nov 16.67 cxcl3 16.67 tal1 16.95 tp63 15.52 tal1 10.34 e2f1 15.00 il1a 16.67 wnt3 16.95 hrk 15.52 tcl1a 10.34 cxcl3 15.00 nov 16.67 tp63 16.95 hspb2 15.00 tal1 16.67 hrk 16.95 ros1 15.00 wnt3 15.00 tcl1a 15.00 4. conclusion and analysis in this research, we analyze and investigate the cascading failures in control and experimental networks. through numerical experiments, we discover: under all the network thresholds, cascading failures occur in experimental networks more easily than in control ones for undirected and directed weighted gene networks. this indicates that the normal organisms are quite robust while diseased organisms are more fragile. in table1, we notice that: with the increasing of network thresholds, the platform values of  ddp ' of control network are gradually increasing. this shows that with the decreasing of the numbers of nodes and edges, the robustness goes worse. in other words, with the increasing of the numbers of nodes and edges, the robustness goes better. this indicates the intrinsic reason of organisms functioning normally and stably maybe is that most genes play their own roles in organisms. in the process of numerical experiments, we notice that failures of genes bad, ing1, raf1, traf3, nras, pik3ca, mapk9, top2a of control group and esr2, hspb2, nov, tal1, fgf1, ret, wt1, tcl1a, hrk of experimental group under all the network thresholds result in collapse or paralysis of the whole network. this provides some useful reference informations for the normal or dieased organisms. for example, activation of gene bad may induce apoptosis in human lung adenocarcinoma cells [18] . in other words, the failure of gene bad leads to the defunctionalization of inducing the apoptosis of human lung adenocarcinoma cells and the organism might suffer from lung adenocarcinoma. gene mapk9 may enhance the stability of tumor suppressor p53 and its failure can reduce the stability of p53. thus the organism might develop into cancer. references app:ds:relative app:ds:normally app:ds:enhance app:ds:stability app:ds:stability 172 wang: analysis and study of cascading failures in gene network [1] motoki shiga, ichigaku takigawa, hiroshi mamitsuka. annotating gene function by combining expression data with a modular gene network. bioinformatics, 2007, 23(13):i468-i478. [2] marco sardiello, michela palmieri, alberto di ronza, et al.. a gene network regulating lysosomal biogenesis and function. science, 2009, 325(5939):473-477. [3] roded sharan1, trey ideker. modeling cellular machinery through biological network comparison. nature biotechnology, 2006, 24: 427-433. [4] thomas schlitt, alvis brazma. present approaches to gene regulatory network modelling. bmc bioinformatics, 2007, 8(suppl 6) s9:1-22. [5] amy hin yan tong, guillaume lesage, gary d. bader, et al.. global mapping of the yeast genetic interaction network. science, 2004, 303(5659):808-813. [6] mark kittisopikul, gürol m. süel. biological role of noise encoded in a genetic network motif. pnas, 2010, 107 (30): 13300-13305. [7] hallinan, j.s., jackway, p.t. network motifs, feedback loops and the dynamics of genetic regulatory networks. computational intelligence in bioinformatics and computational biology, 2005. cibcb '05, 1-7. [8] bowers p m, cokus s j, eisenberg d, et a1.. use of logic relationships to decipher protein network organization. science, 2004, 306(5706):2246-2249. [9] shudong wang,yan chen, qingyun wang, eryan li, yansen su, dazhi meng. analysis for gene networks based on logical relationships. journal of systems science and complexity, 2010, 23(5): 999-1011. [10] asavathiratham c. the influence model: a tractable representation for the dynamics of networked markov chains. elec. eng. and comp. sci. dept., mit, 2000. [11] dobson i, chen j, thorp j s, et al.. examining criticality of blackouts in power system models with cascading events. proceedings of 35th hawaii international conference on system sciences, 2002, 63-72. [12] moreno y, gómez j b, pacheco a f. instability of scale-free networks under node-breaking avalanches. europhys. lett., 2002, 58(4):630-636. [13] moreno y, pastor-satorras r,vázquez a, vespignani a. critical load and congestion instabilities in scale-free networks. europhys. lett., 2003, 62:630-636. [14] crucitti p, latora v, marchiori m. model for cascading failures in complex networks. phys. rev. e, 2004, 69, 045104:1-4. [15] xiaofan wang, xiang li, guanrong chen. the theory and application of complex networks. tsinghua press, beijing, 2006.4. [16] r.kinney, p.crucitti, r.albert, v.latora. modeling cascading failures in the north american power grid. eur. phys. j. b, 2005, 46:101-107. [17] ashley g. smart, luis a. n. amaral, julio m. ottino. cascading failure and robustness in metabolic networks. pnas, 2008, 105(36):13223-13228. [18] chen-tzu kuo, ming-jen hsu, bing-chang chen, et al.. denbinobin induces apoptosis in human lung adenocarcinoma cells via akt inactivation, bad activation, and mitochondrial dysfunction. toxicol lett, 2008, 177(1):48-58. http://www.nature.com/nbt/journal/v24/n4/full/nbt1196.html#a1#a1 http://www.pnas.org/search?author1=mark+kittisopikul&sortspec=date&submit=submit http://www.pnas.org/search?author1=g%c3%bcrol+m.+s%c3%bcel&sortspec=date&submit=submit http://ieeexplore.ieee.org/xpl/mostrecentissue.jsp?punumber=10629 http://ieeexplore.ieee.org/xpl/mostrecentissue.jsp?punumber=10629 http://ieeexplore.ieee.org/xpl/mostrecentissue.jsp?punumber=10629 advances in systems science and applications (2012) vol.12 no.3 272-282 improved airline seat inventory control policies under parametric uncertainty of customer demand models nicholas a. nechval1, konstantin n. nechval2 and maris purgailis1 1university of latvia, evf research institute, statistics department, raina blvd 19, lv-1050 riga, latvia 2transport and telecommunication institute, applied mathematics department, lomonosov street 1, lv-1019 riga, latvia abstract most models, which are used for solving airline seat inventory control problems, are developed in the literature under the assumptions that the parameter values of the models are known with certainty. when these models are applied to solve real-world problems, the parameters are estimated and then treated as if they were the true values. the risk associated with using estimates rather than the true parameters is called estimation risk and is often ignored. when data are limited and/or unreliable, estimation risk may be significant, and failure to incorporate it into the model design may lead to serious errors. in this paper, we consider the static and dynamic problems of airline seat inventory control under parametric uncertainty, which are invariant with respect to a certain group of transformations. since common practice for airlines is to charge several different fares for a common pool of seats, this paper presents the policies that have been used to address the problem of when to refuse booking requests for a given fare level to save the seat for a potential request at a higher fare level. in this paper, we present the innovative technologies for constructing the static and dynamic policies of the airline seat inventory control.on the basis of the ‘unbiasedness performance index’. the idea of prediction of a future cumulative customer demand for the seats on a flight via the order statistics from the underlying distribution, introduced in the paper, allows one to use the invariant embedding technique in order to eliminate the unknown parameters from the problem and to use the previous and current sample data as completely as possible. the proposed unbiased static and dynamic policies are more efficient as compared with the policies, where the unknown parameters of the airline customer demand models are estimated and then treated as if they were the true values. an illustrative example is given. keywords airlines, demand, uncertainty, airline booking, optimization 1 introduction passenger reservations systems have evolved from low level inventory control processes to major strategic information systems. today, airlines and other transportation companies view revenue management systems and related information advances in systems science and applications (2012) vol.12 no.3 273 technologies as critical determinants of future success. indeed, expectations of revenue gains that are possible with expanded revenue management capabilities are now driving the acquisition of new information technology. each advance in information technology creates an opportunity for more comprehensive reservations control and greater integration with other important transportation planning functions. the airline seat inventory control problem lies at the heart of airline revenue management. it is common practice for airlines to sell a pool of identical seats at different prices according to different booking classes to improve revenues in a very competitive market. in other words, airlines sell the same seat at different prices according to different types of travelers (first class, business and economy) and other conditions. the question then arises whether to offer seats at a relatively low price at a given time with a given number of seats remaining or to wait for the possible arrival of a higher paying customer. assigning seats in the same compartment to different fare classes of passengers in order to improve revenues is a major problem of airline seat inventory control. this problem has been considered in numerous papers. for details, the reader is referred to a review of yield management, as well as perishable asset revenue management, by weatherford et al. [1], and a review of relevant mathematical models by belobaba [2]. this paper deals with the airline seat inventory control problem when customers for different fare levels are booked into a common seating pool in the aircraft. the following assumptions are made: (1) single-leg flight: bookings are made on the basis of a single departure and landing; no allowance is made for the possibility that bookings may be part of larger trip itineraries, (2) independent demands: the demands for different fare classes are stochastically independent, (3) low before high demands: the lowest fare reservations requests arrive first, followed by the next lowest, etc., (4) no cancellations: cancellations, no-shows and overbooking are not considered, (5) nested classes: any fare class can be booked into seats not taken by bookings in lower fare classes, (6) fare classes: the business and economy fare classes are considered. the first purpose of this paper is to present the innovative information technologies for constructing the static and dynamic policies of the airline seat inventory control on the basis of the ‘unbiasedness performance index’. the static and dynamic policies (unbiased) are more efficient (from the point of view of airline revenue management) as compared with the policies, where the unknown parameters of the airline customer demand models are estimated and then treated as if they were the true values. the second purpose of this paper is to introduce the idea of prediction of a future cumulative customer demand for the seats on a flight via the order statistics from the underlying distribution, where only the functional form of the 274 nicholas a. nechval:improved airline seat inventory control policies under parametric... distribution is specified, but some or all of its parameters are unspecified. this idea allows one to use the technique of invariant embedding of sample statistics in a performance index in order to eliminate the unknown parameters from the problem [3-4]. the technique represents a simple and computationally attractive statistical method based on the constructive use of the invariance principle in mathematical statistics. unlike the bayesian approach, an invariant embedding technique is independent of the choice of priors, i.e., subjectivity of investigator is eliminated from the problem. it allows one to find the improved invariant statistical decision rules, which have smaller risk than any of the well-known traditional statistical decision rules, and to use the previous and current sample data as completely as possible. 2 state-of-the-art and progress beyond airline seat inventory control is a very profitable tool in the airline industry. a major problem of airline seat inventory control is to sell the same seat at different prices according to different types of travelers (first class, business and economy) and other conditions in order to improve revenues. this problem has been considered in numerous papers. littlewood [5] was the first to propose a solution method of the seat inventory control problem for a single leg flight with two fare classes. the idea of his scheme is to equate the marginal revenues in each of the two fare classes. he suggests closing down the low fare class when the certain revenue from selling low fare seat is exceeded by the expected revenue of selling the same seat at the higher fare. that is, low fare booking requests should be accepted as long as c2 ≥ c1pr{y1 > µ1}, (1) where c1 and c2 are the high and low fare levels respectively, y1 denotes the demand for the high fare (or business) class,µ1 is the number of seats to protect for the high fare class and pr{y1 > µ1} is the probability of selling more than µ1 protected seats to high fare class customers. the smallest value of µ1 that satisfies the above condition is the number of seats to protect for the high fare class, and is known as the protection level of the high fare class customers. the concept of determining a protection level for the high fare class can also be seen as setting a booking limit, a maximum number of bookings, for the low fare class. both concepts restrict the number of bookings for the low fare class in order to accept bookings for the high fare class. it should be remarked that there is no protection level for the low fare (or economy) class;µ2 is the booking limit, or number of seats available, for the low fare class; the low fare class is open as long as the number of bookings in this class remains less than this limit. thus,is the booking limit, or number of seats available, for the low fare class; the low fare class is open as long as the number advances in systems science and applications (2012) vol.12 no.3 275 of bookings in this class remains less than this limit. thus, µ1+µ2 is the booking limit or number of seats available, for the high fare class at time. the high fare class is open as long as the number of bookings in this and low classes remain less than this limit. richter [6] gave a marginal analysis, which proved that (1) gives an optimal allocation (assuming certain continuity conditions). optimal policies for more than two classes have been presented independently by curry [7], wollmer [8], and brumelle & mcgill [9]. 3 airline booking policies which are used in practice 3.1 static airline booking policy under complete information it will be noted that (1) represents the static policy of airline seat inventory control (or airline booking) under complete information. if fθ , the probability distribution function of y1 with the parameter θ (in general, vector), is continuous and strictly increasing, the definition (1) of µ1 is equivalent to µ1 = arg ( f̄θ(µ1) = γ ) (2) where γ = c1/c2, (3) f̄θ(τj) = 1− fθ(τj). (4) 3.2 static airline booking policy under parametric uncertainty in practice, under parametric uncertainty, i.e. when the parameter θ is unknown, the performance index, f̄θ(µ1) = γ, (5) is usually used to construct the static policy given by µ1 = arg ( f̄θ̂(µ1) = γ ) , (6) where θ̂ represents the maximum likelihood estimator of θ . the performance index (5) is named as ‘maximum likelihood performance index’. the static policy (6) based on (5) is named as ‘static maximum likelihood airline booking policy’. 3.3 dynamic airline booking policy under parametric uncertainty the static policy of airline booking is optimal as long as no change in the probability distributions of the customer demand is foreseen. however, information on the actual customer demand process can reduce the uncertainty associated with the estimates of demand. hence, repetitive use of a static policy over the booking period, based on the most recent demand and capacity information, is the general way to proceed. 276 nicholas a. nechval:improved airline seat inventory control policies under parametric... 4 improved airline booking policies proposed in the paper 4.1 static unbiased airline booking policy under parametric uncertainty this policy is based on the performance index, eθ{f̄θ(µ1)} = γ, (7) which takes into account (2) and the previous data of cumulative customer demand y1 for the seats on a flight. it allows one to construct the static airline booking policy given by µ (µnb) 1 = arg ( eθ{f̄θ(µ1)} = γ ) , (8) where µ1 ≡ µ1(θ̂),θ̂ represents either the maximum likelihood estimator of θ or sufficient statistic s for θ, i.e., µ1 ≡ µ1(s) the performance index (7) is named as ‘unbiasedness performance index’.the static policy (8), which is based on (7), is named as ‘static unbiased airline booking policy’. the relative bias of the static airline booking policy is given by γ(µ1) = ∣∣eθ{f̄θ(µ1)} − γ ∣∣ γ 100% (9) 4.2 dynamic airline booking policy under complete information in this section, we consider a flight for a single departure date with m predefined reading dates at which the dynamic policy is to be updated, i.e., the booking period before departure is divided intom readings periods:(0, τ1], (τ1, τ2], . . . , (τm−1, τm] determined by them reading dates:τ1, τ2, . . . , τm. these reading dates are indexed in increasing order:0 < τ1 < τ2 < . . . < τm, where (τm−1, τm] denotes the reading period immediately preceding departure, and τm] is at departure. typically, the reading periods that are closer to departure cover much shorter periods of time than those further from departure. for example, the reading period immediately preceding departure may cover 1 day whereas the reading period 1-month from departure may cover 1 week. let us suppose that the cumulative passenger demand for the high fare class at the kth reading date (time τk, 1 ≤ k ≤ m) is y1k representing the kth order statistic from the underlying distribution with the probability distribution function gθ(y1k), where θ is a parameter (in general, vector). in other words,y1k represents the number of seats sold for the customers of the high fare class at the kth reading date. we assume that the cumulative passenger demands for the high and low fare classes are stochastically independent. each booking of a seat of the high fare class generates average revenue of c1. each booking of a seat of the low fare class generates average revenue of c2 , where c2 < c1.let µ1k be an individual protection level for the high fare class at time τk (the kth advances in systems science and applications (2012) vol.12 no.3 277 reading date).this many seats are protected for the high fare class from the low fare class. there is no protection level for the low fare class;µ2k is the booking limit for the low fare class at time τk; the low fare class is open as long as the number of bookings in this class remains less than this limit. thus,µ1k + µ2k is the booking limit for the high fare class at time τk the high fare class is open as long as the number of bookings in this and low classes remain less than this limit. the maximum number of seats that may be booked by fare classes in the next at time τk prior to flight departure is the number of unsold seats µ◦ k. under the complete information, the dynamic airline booking policy is given by µ1k = arg ( ḡθ(µ1k|y1k) = γ ) , k = 1, 2, . . . ,m− 1, (10) where ḡθ(µ1k|y1k) = 1−gθ(µ1k|y1k), (11) gθ(µ1k|y1k) represents the conditional probability distribution function of the mth order statistic y1m . the number of unsold seats protected for the high fare class from the low fare class in the next at time τk prior to flight departure is the number of unsold seats,µ◦ 1k,which is given by µ◦ 1k = min(µ◦ k, µ1k − y1k). (12) 4.3 dynamic unbiased airline booking policy under parametric uncertainty under the parametric uncertainty, the dynamic unbiased airline booking policy is given by µunb 1k = arg ( eθ{ḡθ(µ1k|y1k)} = γ ) , k = 1, 2, . . . ,m− 1, (13) where µ1k ≡ µ1k(θ̂),θ̂ represents either the maximum likelihood estimator of θ or sufficient statistic s for θ, i.e.,µ1k ≡ µ1k(s).the number of unsold seats protected for the high fare class from the low fare class in the next at time τk prior to flight departure is the number of unsold seats µ◦ 1k,which is given by µ ◦(unb) 1k = min(µ◦ k, µ unb 1k − y1k). (14) 5 mathematical preliminaries theorem 1 let x1 ≤ . . . ≤ xk be the first k ordered observations (order statistics) in a sample of size m from a continuous distribution with some probability density function fθ(x) and distribution function fθ(x) where θ is a parameter (in general, vector). then the joint probability density function of x1 ≤ . . . ≤ xk and the lth order statistics xl(1 ≤ k ≤ l ≤ m) is given by gθ(x1, . . . , xk, xl) = gθ(x1, . . . , xk)gθ(xl|xk), (15) 278 nicholas a. nechval:improved airline seat inventory control policies under parametric... where gθ(x1, . . . , xk) = m! (m− k)! k∏ i=1 fθ(xi) [ 1− fθ(xk) ]m−k , (16) gθ(xl|xk) = (m− k)! (l − k − 1)!(m− l)! [fθ(xl)− fθ(xk) 1− fθ(xk) ]l−k−1[ 1− fθ(xl)− fθ(xk) 1− fθ(xk) ]m−l fθ(xl) 1− fθ(xk) = (m− k)! (l − k − 1)!(m− l)! k−l−1∑ j=0 ( l − k − 1 j ) (−1)j [ 1− fθ(xl) 1− fθ(xk) ]m−l+j fθ(xl) 1− fθ(xk) = (m− k)! (l − k − 1)!(m− l)! m−l∑ j=0 ( m− l j ) (−1)j [fθ(xl)− fθ(xk) 1− fθ(xk) ]l−k−1+j fθ(xl) 1− fθ(xk) (17) represents the conditional probability density function of xl given xk = xk. proof. the joint density of x1 ≤ . . . ≤ xk and xl is given by gθ(x1, . . . , xk, xl) = m! (l − k − 1)!(m− l)! k∏ i=1 fθ(xi) [ fθ(xl)− fθ(xk) ]l−k−1 fθ(xl) [ 1− fθ(xl) ]m−l =gθ(x1, . . . , xk)gθ(xl|xk). (18) it follows from (4) that gθ(xl|x1, . . . , xk) = gθ(x1, . . . , xk, xl) gθ(x1, . . . , xk) = gθ(xl|xk), (19) i.e., the conditional distribution of xl, given xi = xi for all i = 1, . . . , k, is the same as the conditional distribution of xl, given only xk = xk, which is given by (17). this ends the proof. corollary 1.1. the conditional probability distribution function of xl given xk = xk is pθ{xl ≤ xl|xk = xk} =1− (m− k)! (l − k − 1)!(m− l)! l−k−1∑ j=0 ( l − k − 1 j ) (−1)j m− l + 1 + j [ 1− fθ(xl) 1− fθ(xk) ]m−l+1+j = (m− k)! (l − k − 1)!(m− l)! m−k∑ j=0 ( m− l j )[fθ(xl)− fθ(xk) 1− fθ(xk) ]l−k+j . (20) advances in systems science and applications (2012) vol.12 no.3 279 corollary 1.2.let x1 ≤ . . . ≤ xk be the first k order statistics in a sample of size m from the two-parameter weibull distribution with the probability density function fθ(x) = δ β (x β )δ−1 exp [ −( x β )δ ] (x > 0), (21) where θ = (β, δ), β > 0 and δ > 0 are the scale and shape parameters, respectively. then the conditional probability distribution function of xl given xk = xk is pθ{xl ≤ xl|xk = xk} = 1− (m− k)! (l − k − 1)!(m− l)! l−k−1∑ j=0 ( l − k − 1 j ) (−1)j m− l + 1 + j [ exp(− xδl − xδk βδ ) ]m−l+1+j . (22) theorem 2 if in (22) the scale parameter β is unknown, then the predictive probability distribution function of xl based on (xk, δ) is given by pδ {(xl xk )δ ≤ ( xl xk )δ ) } = 1− m! (l − k − 1)!(m− l)! × l−k−1∑ j=0 ( l − k − 1 j ) (−1)j m− l + 1 + j ( πk−1 s=0 [(( xl xk )δ − 1 ) (m− j +1+ j) + (m− k+1+ s) ])−1 . (23) proof.we reduce (22) to pθ {(xl xk )δ ≤ ( xl xk )δ | (xk β )δ = (xk β )δ} = 1− (m− k)! (l − k − 1)!(m− l)! l−k−1∑ j=0 ( l − k − 1 j ) × (−1)j m− l + 1 + j [ exp ( − ω[νδ − 1] )]m−l+1+j =pδ{v δ ≤ νδ|w = ω} (24) where v = xl/xk is the ancillary statistic whose distribution does not depend on the parameter β. since xk does not depend on v ,w = (xk/β) δ is the pivotal quantity, whose distribution is known and does not depend on the parameters β and δ, we eliminate the parameter β from the problem as pδ{xl ≤ xl} = ∫ ∞ 0 pθ{xl ≤ xl|xk = xk}gθ(xk)dxk, (25) where gθ(xk) = m! (k − 1)!(m− k)! f k−1 θ (xk) [ 1− fθ(xk) ]m−k fθ(xk), xk ∈ (0,∞), (26) 280 nicholas a. nechval:improved airline seat inventory control policies under parametric... represents the probability density function of the kth order statistic xk.indeed, it follows from (26) that gθ(xk)dxk = m! (k − 1)!(m− k)! [ 1− exp ( − (xk β )δ)]k−1 exp ( − (xk β )δ(m−k) ) exp ( − (x β )δ) d (x β )δ = m! (k − 1)!(m− k)! [ 1− e−ω ]k−1 e−ω(m−k+1)dω = g(ω)dω. (27) it follows from (24) and (27) that pδ{v δ ≤ νδ} = ∫ ∞ 0 pδ{v δ ≤ νδ|w = ω}g(ω)d(ω) =1− m! (l − k − 1)!(m− l)! l−k−1∑ j=0 ( l − k − 1 j ) (−1)j m− l + 1 + j( πk−1 s=0 [ (νδ − 1)(m− l + 1 + j) + (m− k + 1 + s) ])−1 . (28) now (23) follows from (28). this ends the proof. corollary 2.1.if the parameter δ = 1 , i.e., we deal with the exponential distribution, then the predictive probability distribution function of xl based on xk is given by p {(xl xk ≤ xlxk )} =1− m! (l − k − 1)!(m− l)! × l−k−1∑ j=0 ( l − k − 1 j ) (−1)j m− l + 1 + j( πk−1 s=0 [( xl xk − 1 ) (m− l + 1 + j) + (m− k + 1 + s) ])−1 . (29) 6 illustrative example of airline booking policies let x1, . . . , xn be the random sample of the previous independent observations of the cumulative customer demand for the high fare class, which follow the exponential distribution with the probability density function (21)(δ = 1) , where the parameter β is unknown. then the static policies of airline booking under parametric uncertainty are given as follows. advances in systems science and applications (2012) vol.12 no.3 281 the static maximum likelihood airline booking policy follows from (6): µml 1 = ln γ−s/n, (30) where s = ∑n i=1xi is the sufficient statistic for β,with v = s/β ∼ f(ν) = 1 γ(n) νn−1 exp(−ν), ν ≥ 0, (31) and the relative bias, r(µml 1 ) = ∣∣eθ{f̄θ(µ ml 1 )} − γ ∣∣ γ 100% = 1 + (1 + ln γ−1/n)−1 − γ γ 100%. (32) if, say,n = 1 and γ = 0.4, then rrb(µ ml 1 ) = 30%.thus, in this example the static maximum likelihood airline booking policy has the relative bias equal to 30%.it follows that the protection level for customers of the high fare class will be determined incorrectly. this may lead to serious loss. the static unbiased airline booking policyfollows from (8): µunb 1 = [γ−1/n − 1]s, (33) where the relative bias r(µunb 1 ) = 0. the dynamic unbiased airline booking policyfollows from (13) and (29): µunb 1k = arg ( m! (m− k − 1)! m−k−1∑ j=0 ( m− k − 1 j ) (−1)j 1 + j ( πk−1 s=0 [(µ1k y1k − 1 ) (1 + j) + (m− k + 1 + s) ])−1 = γ ) , k = 1, 2 . . . ,m− 1, (34) µ ◦(unb) 1k = min ( µ◦ k, µ unb 1k ,−y1k ) . (35) 7 conclusion the methodology, which is developed in this paper for the use in the airline industry under parametric uncertainty of airline customer demand models, may be found to be useful in other industries such as hotels, car rental companies, shipping companies, etc. while the details of problems considered in this project can change significantly from one industry to the next, the focus is always on making better demand decisions and not manually with guess work and intuition but rather scientifically with models and technology, all implemented with disciplined processes and systems. 282 nicholas a. nechval:improved airline seat inventory control policies under parametric... acknowledgements this research was supported in part by grant no. 06.1936, grant no. 09.1014, and grant no. 09.1544 from the latvian council of science and the national institute of mathematics and informatics of latvia. references [1] weatherford l.r., bodily s.e. and pfeifer p.e. (1993), “modeling the customer arrival process and comparing decision rules in perishable asset revenue management situation”, transportation science, vol.27, pp.239-251. [2] belobaba p.p. (1987), “airline yield management: an overview of seat inventory control”, transportation science, vol.21, pp.66-73. [3] nechval n.a., berzins g., purgailis m., nechval k.n. (2008), “improved estimation of state of stochastic systems via invariant embedding technique”, wseas transactions on mathematics, vol.7, pp.141-159. [4] nechval n.a., nechval k.n., purgailis m., rozevskis u. (2011), “improvement of inventory control under parametric uncertainty and constraints”, in: dobnikar, a., lotric, u., ster, b. (eds.), adaptive and natural computing algorithms, lncs, vol.6594, part ii, pp.136-146, springer, heidelberg. [5] littlewood k. (1972), “forecasting and control of passenger bookings”, in: proceedings of the 12th agifors symposium, pp.95-117, american airlines, new york. [6] i richter h. (1982), “the differential revenue method to determine optimal seat allotments by fare type”, in: proceedings of the xxii agifors symposium, pp.339-362, american airlines, new york. [7] curry, r.e. (1990), “optimal airline beat allocation with fare classes nested by origins and destinations”, transportation science, vol.24, pp.193-203. [8] wollmer, r.d. (1992), “an airline seat management model for a single leg route when lower fare classes book first”, operations research, vol.40, pp.2637. [9] brumelle, s.l. and mcgill, j.i. (1993), “airline seat allocation with multiple nested fare classes”, operations research, vol.41, pp.127-137. corresponding author nicholas nechval can be contacted at:jnechval@junik.lv мягкая вероятность ошибки при передаче информации advances in systems science and application(2016) vol.16 no.4 29-42 a new decision method for multi-criteria decision making with numerical values based on criteria reduction zhaobin li 1 , jian liu 2 , zhuo zhang 1 , si-feng liu 1 1 school of economics and management, nanjing university of aeronautics and astronautics, nanjing, china, 210016 2 school of economics and management, nanjing university of science and technology, nanjing, china, 210094 annotation the work is contribution a new decision method to address the challenge (large number of criteria) in multi-criteria decision making (mcdm) problems with numerical values. this new method involves criteria reduction based on the rough set theory and the relation of criteria values (tolerance and advantage relations). using this method and building a discrenibility matrix for numerical value mcdm problems, find useful criteria and avoid useless criteria. then, we find a new way to obtain the weights based on the discernibility matrix when criteria weights of alternatives are completely unknown. later, we also propose a new method to rank the alternatives according to weighted combinatorial advantage values (wcav). finally, we use a realistic voting example to demonstrate the proposed method. key words: multi-criteria decision making, criteria reduction, discernibility matrix, obtain weight, relation. 1 introduction with the development of information technology, most decision makers (dms) face the problem of how to make a wise decision when there are massive data in the decision table. how can we filter information is a potential application and development area for mcdm/maut in an internet or mobile environment that was proposed by wallenius et al. on management science, 2008 [1]. undoubtedly, we need to find out the useful data that really affect the decision making also take to human subjective. the related problems have been one of the most popular research topics in decision making science since 2008 [2-4]. massive data of mcdm problems contain three situations that are large number of criteria, large 4umber of alternatives or both. in this paper, we focus on the problems which are large number of criteria in the decision table. extracting useful information from large quantity of uncertain problems has become an important research field in computer science-attribute reduction [5-6]. rough set theory has been recognized as one of the most powerful techniques to deal with uncertainty problems since its appearance in 1982 [7-8]. the original rough set approach validated to be very useful in dealing with discrete problems. rough set theory [9-10] is based on equivalence relation and captures useful information from a great deal of information through attribute reduction is a basic research method. attribute (criteria) reduction find useful information by using smaller criteria set b a which is to make the criteria set a describes replaced and described by b. in this paper, we want to find out the useful criteria set b from a, and use b to make a decision will address the problem proposed by wallenius et al. on management science, 2008. 30 zhaobin li, jian liu, zhuo zhang, et al.: a new decision method for multi-criteria decision making with numerical values… obtaining criteria weights is also an important research topic of mcdm problems. for uncertain of criteria weights problems, there are several methods to obtain them, such as obtaining criteria weights approach of subject [11-12], objective obtaining criteria weights approach[13-14], subjective and objective obtaining criteria weights approach [15], the owa operator weights [16] and feedback model [17], etc. as for the uncertain criteria weights problems, most approaches obtain the criteria weights according to their deviation of criteria values to facilitate the ranking of alternatives. generally speaking, the bigger the deviation of criteria values are, the larger the weight will be [18]. in reality, we find that some criteria values change lager than others but these criteria only have a little influence on the result, while some tiny changes of few criteria would lead to different consequences for some mcdm problems. according to the traditional methods of mcdm problems, we need several procedures, such as unifying criteria and obtaining criteria weights and information fusion as well as ranking and selecting the most desirable alternative(s) [18-19]. in order to rank alternatives, we need to compare expectations of combinatorial criteria values [20-21]. but we can not filter off the absolute disparity through unifying criteria, so different criteria are incomparable even if we unify them into the same meaning. in this paper, we think the criteria can be compared only on the same criteria. some errors would be produce in the processing of uniform criteria. in order to filter off these errors, we propose through comparing the weighted combinatorial advantage values (wcav) [22] to rank alternatives in this paper. this paper is organized as follows: in section 2, for numerical value mcdm problems, there are a large number of criteria in the decision table. we propose a criteria reduction technique based on tolerance and advantage relation and rough set theory to find out useful criteria, respectively. in section 3, we propose a new way of obtaining criteria weights according to the discernibility matrix and using weighted combinatorial advantage value (wcav) instead of traditional methods to rank those alternatives. in section 4, we validate the method a useful and effective tool for mcdm problems. finally, section 5 discusses the conclusion and future work. 2 a large number criteria mcdm problems and criteria reduction in this section, we will give one type of numerical value mcdm problems that involve a large number of criteria in decision tables. the primary goal of mcdm problems is to rank the alternatives and select the most desirable one(s). to achieve this goal, several processing steps are needed to compare the alternatives based on multiple criteria. in each processing step, specific algorithms and operations are involved. the problems concerned in this paper are how to find out the useful criteria and obtain weights of useful criteria and to rank alternatives or select the most desirable alternative(s). 2.1 addressing numerical value mcdm problems example mcdm problem (supplier choice): the commercial aircraft corporation of china, ltd. (cacc) builds huge commercial aircrafts to serve commercial airlines in china. to build aircrafts the company needs to buy and use some key parts from international or domestic suppliers. therefore, the cacc must make a scientific decision to choose the most desirable supplier that relate to the success of commercial aircraft program. there are lots of complicated factors that affect decision makers (dms) to make decision and they need to combine all information for every supplier and analyze them as well as select the most desirable supplier(s) [22]. suppose that there are five international suppliers in the first round competing for the cacc demand of some key parts of the huge commercial aircrafts, and these five suppliers are represented by 1 2 3 4 5{ , , , , }a a a a aa . suppose we invite 100 experts to make judgments for the sake of obtaining the degrees to which alternative ai satisfies and does not satisfy criteria cj (i=1, app:ds:comparability advances in systems science and application(2016) vol.16 no.4 31 2, 3, 4, 5; j=1, 2, ……, m). there are two kinds of poll results “yes” or “no” to the question whether alternative ai satisfies criteria cj. then we should choose the most desirable choice based on the results in table 1. table 1 the result of “yes” answers from 100 experts u c1 c2 …… cm …… a1 a2 a3 a4 a5 55 61 55 65 89 50 62 55 65 85 …… …… …… …… …… 58 60 70 65 55 …… …… …… …… …… 2.2.1 the challenge of numerical value mcdm problems obviously, there are a large number of criteria in decision table 1. the first problem is how to make a wise decision within limited time when the dms face a large number of criteria? what kind of decision support do dms want under large number of criteria environment? to address the problem, we need to find out useful criteria that really affect the decision results. this is a significant potential research area for mcdm problems. 2.2.2 the principles and methods of criteria reduction for numerical value mcdm problems we often face a question whether we can remove some criteria from an information table while preserving its basic properties, that is, whether a table contains some superfluous criteria. through criteria reduction we want to find out useful information that really affects the processing of making decision. this work is to our best knowledge the first one that applies rough set theory [7-10] to do criteria reduction. in addition, the rough set theory contains many attributes reduction [5-6] techniques, such as the reduction of attributes based on similarity relation [23], advantage relation or disadvantage relation [22] and automatic threshold estimation [24] etc. in this paper, we apply rough set theory to find out useful information from the decision table by using criteria reduction based on tolerance relation and advantage relation of criteria value. now, let’s use the concrete example shown in table 1 to illustrate our idea in using tolerance relation or advantage relation and rough set theory to find out useful information. generally speaking, there are three different type criteria  the criteria of cost type (the smaller of criteria values, the better)  the criteria of benefit type (the larger of criteria values, the better)  the criteria of middle type (at a special point of criteria values, the better) obviously, all the criteria belong to benefit type in table 1. there are two different criteria values of alternatives a1 and a2 on criterion c1 are f(a1, c1) and f(a2, c1), such as f(a1, c1)=55, f(a2, c1)=61. obviously, both of them have different criteria values on c1, i.e., f(a1, c1)< f(a2, c1). for benefit criteria, if f(a1, c1)is smaller than f(a2, c1), it means that the alternative a2 locates at a more advantage position than a1 on criterion c1 when we want to compare them. in other words, alternative a2 is better than a1 on criteria c1. in this paper, we use 2 1 1a a c to indicate this advantage relation of them. at this situation, c1 is a useful criterion when we compare these two alternatives a1 and a2. two criteria values of alternatives a2 and a4 on criterion c1 are f(a2, c1) and f(a4, c1), i.e., f(a2, c1)=61 and f(a4, c1)=65. obviously, both of them have different criteria values on c1, i.e., f(a2, c1)< f(a4, c1). for benefit criteria, if f(a2, c1)is smaller than f(a4, c1), it means that the alternative a2 locates at a more disadvantage position than a4 on criterion c1 when we compare them. at the same time, the alternative a4 locates at a more advantage position than a2 on criterion c1. in this paper, we use 2 4 1a a c to indicate this disadvantage relation of them. thus, c1 is still a useful criterion when we compare these two alternatives a2 and a4. 32 zhaobin li, jian liu, zhuo zhang, et al.: a new decision method for multi-criteria decision making with numerical values… two criteria values of alternatives a1 and a3 on criterion c1 are f(a1, c1) and f(a3, c1), obvious f(a1, c1)=55 and f(a3, c1)=55. both of them have the same criteria value on c1, i.e., f(a1, c1)= f(a3, c1). for benefit or cost criteria, if two alternatives have the same criteria value on a criterion, it means that these two alternatives locate at the same position on this criterion when we compare them. that means the alternative a1 and a3 locate at the same position on criterion c1. according to attribute reduction based on tolerance relation and rough set theory, if two alternatives have the same values on the same criteria, it means that these two criteria values have a tolerance relation for numerical value mcdm problems. when two alternatives have different values on the same criterion, it means that a tolerance relation does not exist. in this paper, we use 1 3 1a a c to indicate this tolerance relation of them. thus, c1 is a useless criterion when we compare these two alternatives a1 and a3, so we can remove it. as the previous discussion, there are three different relations of criteria values for numerical value mcdm problems. these three relations are advantage relation, disadvantage relation and equivalence relation. definition 1. suppose that {a1, a2, … , an} indicates a set of n alternatives, and{c1, c2, … , cm} is a set of m criteria for numerical value mcdm problems. f(ai, cj) and f(ak, cj) indicate the possible outcome of alternatives ai and ak on criterion cj, three relations of the criteria values are defined as follows: ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) i k j i j k j i k j i j k j i k j i j k j a a c f a c f a c a a c f a c f a c a a c f a c f a c        (1) our idea regarding how to find out useful information: as the previous discussion, when we want to compare two alternatives, we do not need to consider those criteria with the same value on the same criteria. that means we do not need to consider those criteria that two alternatives locate at the same position on these criteria. however, we need to consider those criteria that two alternative locate at different positions on these criteria. from the previous discussion, criterion c1 is useless to compare alternatives a1 and a3, but it is needed when we want to know which is better between alternatives a1 and a2. that means the same criteria may play different roles for different alternatives in the decision table. thus, if we want to find out the useful criteria that we need, we need to make a comparison on every pair of alternatives in the decision table. to check every pair of alternatives separately, we need to construct a discernibility matrix [25] and find out useful criteria for all the alternatives in the decision table. the data in the discernibility matrix indicate a set of criteria that must be considered when we want to compare two corresponding alternatives. we think this criteria reduction method reflect and convey the below information.  from the perspective of the alternative: every alternative is chosen as the most desirable alternative(s) in decision making. thus, it is necessary to find out these kinds of desirable criteria that locate in a comparative advantage position as the useful criteria in decision making for every alternative and the larger these criteria’s weights are the better. thus, the alternative could have its weighted combinatorial advantage in decision making.  from the perspective of the competitors: those criteria that locate at a less advantage position could be chosen as the useful criteria in decision making for their opponents and the heavier of those kinds of criteria are, the better. thus, the competitors could have their own weighted combinatorial advantage in decision making and they will have more chance to be selected as the best choice in decision making. from the perspective of the decision makers: they want to make a scientific and reasonable decision within limited time. so they need to remove those useless criteria that locate in the same position when they compare two different alternatives. 2.2 building the discernibility matrix advances in systems science and application(2016) vol.16 no.4 33 as we proposed in section 2.1.2, need to construct three discernibility matrixes as criteria reduction based on tolerance relation, advantage relation and disadvantage relation and rough set theory in order to find out useful criteria from the decision table. the discernibility matrix for numerical values mcdm problems as follows: definition 2. suppose that {a1, a2, … , an} indicates a set of n alternatives, and{c1, c2, … , cm} is a set of m criteria for numerical value mcdm problems. f(ai, cj) and f(ak, cj) indicate the possible outcome of alternatives ai and ak on criterion cj, m is a discernibility matrix for numerical value mcdm problems based on tolerance relation defined as follows: 1 1 11 1 1 n n n n nn a a a m m m a m m           , where { : ( , ) ( , )}j i j k j ik c f a c f a c m else      c (2) and ikm denotes those criteria of two alternatives ai and ak have the different criteria values on the same criteria in set c. in other words, ikm is a set of criteria that contains all the criteria of two alternatives ai and ak have different criteria values in set c. obviously, if there is i k ja a c , there will be k i ja a c when ( , ) ( , )i j k jf a c f a c . so there is mik=mki in the discernibility matrix. thus, the discernibility matrix is a symmetric matrix. definition 3. suppose that {a1, a2, … , an} indicates a set of n alternatives for numerical value or interval number mcdm problems, {c1, c2, … , cm} is a set of m criteria. for the numerical value mcdm problems, m is a discernibility matrix for numerical value mcdm problems based on advantage relation defined as follows: 1 1 11 1 1 n n n n nn a a a m m m a m m            , where, { :j i k j ik c a a c m else     c (3) and ikm is a set of criteria which contains those criteria that the alternative ai locates at a more advantage position than ak on those criteria. definition 4. suppose that {a1, a2, … , an} indicates a set of n alternatives for numerical value or interval number mcdm problems, {c1, c2, … , cm} is a set of m criteria. for the numerical value mcdm problems, m is a discernibility matrix for numerical value mcdm problems based on disadvantage relation defined as follows: 1 1 11 1 1 n n n n nn a a a m m m a m m            , where, { :j i k j ik c a a c m else     c (4) and ikm is a set of criteria which contains those criteria that the alternative ai locates at a more disadvantage position than ak on those criteria. there is tm m between these two discernibility matrixes. so we will get the same useful criteria by using the discernibility matrix based on advantage relation, the discernibility matrix based on disadvantage relation or both. thus, we will just use the discernibility matrixes based on advantage and tolerance relations to find out the useful criteria avoid the useless criteria in this paper. we use the relative discernibility function of discernibility matrix by using boolean reasoning techniques [6-7, 25-26]. we can get the useful criteria for all the alternatives. 3 the method for a large number of criteria mcdm problems 34 zhaobin li, jian liu, zhuo zhang, et al.: a new decision method for multi-criteria decision making with numerical values… according to the traditional methods of mcdm problems, we need several procedures, such as unifying criteria and obtaining criteria weights and information fusion as well as ranking and selecting the most desirable alternative(s)[18]. in this paper, we apply the method of criteria reduction based on tolerance and advantage relations and rough set theory to a large number criteria mcdm problems. so, we will change the traditional procedure of mcdm problems. 3.1 resolution procedure for the a large number of criteria mcdm problems to solve the above problems, a new resolution procedure is proposed, as shown in fig. 1. a brief description of the resolution procedure is given below. fig.1 the resolution procedure for a large number of criteria mcdm problems first, by using eq. (2)-(4), we construct a discernibility matrix to find out the useful criteria for numerical value mcdm problems. second, by using eq. (5) and (6), we find a way to obtain criteria weights for useful criteria and all criteria in the decision table, respectively. third, by using eq. (7)-(11), we find a new algorithm to rank alternatives of numerical value of mcdm problems. finally, we validate this new resolution procedure a feasibility and validity method for mcdm problems. all equations (eq. (5)-(11)) and details (sections 4.1 and 4.2) will be provided in the next sections. 3.2 obtaining criteria weights of the useful criteria the challenge of obtaining criteria weights: there are several methods to obtain the criteria weights. according to the traditional opinion, the most popular one in the existing methods is to obtain the criteria weights based on the deviation of criteria values [18]. but we think this method has two places that deserve further consideration.  first, for some mcdm problems, some criteria values change lager than others but these criteria only have a little influence on the result, while some tiny changes of few criteria would lead to different consequences.  second, building the discernibility matrix and finding out useful criteria depend on whether the criteria values are the same or not, but do not depend on whether the deviation of criteria values larger or smaller. large decision table find out useful criteria rank and select alternatives validate the method using eq. (2)-(4) using eq. (5)-(6) using eq. (7)-(10) sections 4.1 and 4.2 information aggregation using eq. (11) obtain criteria weights for the useful criteria advances in systems science and application(2016) vol.16 no.4 35 how to find a scientific and reasonable method and obtain the criteria weights is becoming a very important research area in mcmd fields. based on the previous representing, we propose a new way to obtain the criteria weights of mcmd problems. our idea about how to obtain criteria weights: in the discernibility matrix, ( , 1,2, , )ikm i k n indicates a useful set which contains those criteria that two alternatives ai and ak have different criteria values. for the dms, ( , 1,2, , )ikm i k n contains all the criteria that we must compare if we want to know which is the better between the alternatives ai and ak in table 2. the times of the criteria appear in the discernibility matrix mean how many times we need to consider it. the times of criteria appear in the advantage discernibility matrix means how many alternatives locate at a more advantage position than others on these criteria. every alternative wants to become the best one in decision making. thus, these alternatives want those criteria with big weights in advantage matrix. at the same time, the times of these criteria appear in the disadvantage discernibility matrix means how many alternatives locate at a disadvantage position on these criteria. every alternative hopes its competitors having big weights in disadvantage matrix. thus, the criteria weights should have a proportional relation with the times it appears in the discernibility matrix. definition 5. suppose that 1 2{ , , , }na a a is a set of n alternatives of mcdm problems, 1 2 '{ , , , }mc c c indicates a set of m’ useful criteria，and 1 2 '{ , , , }m   is a set corresponding to all useful criteria weights. ( 1,2, ')j j m  for useful criteria are listed as follows: ' 1 | | | | j j m j j c c     (5) where, | |jc indicates the times of criteria cj appearing in the discernibility matrix, ' 1 | | m j j c   indicates the total times of all useful criteria appearing in the discernibility matrix, and 'm indicates how many useful criteria in table 2. definition 6. suppose that 1 2{ , , , }na a a is a set of n alternatives of mcdm problems, 1 2{ , , , }mc c c indicates a set of m criteria o，and 1 2{ , , , }m   is a set corresponding to all criteria weights. ( 1,2, , )j j m  for all criteria are listed as follows: 1 | | | | j j m j j c c     (6) where, | |jc indicates the times of criteria cj appearing in the discernibility matrix, 1 | | m j j c   indicates the total times of all criteria appearing in the discernibility matrix, and m indicates how many criteria in the decision table. as previous representing in section 2.2.2, we also use the same reasons to explain through eq. (5) and (6) to obtain the criteria weights is reasonable and scientific. as the previous discussion of the risk preference assumptions, we get different advantage orders for the same two interval numbers under a special situation. different dms will get different discernibility matrixes for the interval number mcdm problems. thus, we will get different criteria weights for different types of dms in the same decision table. 3.3 ranking and selecting the most desirable alternative(s) for numerical value mcdm problems the challenge of ranking alternatives: there are several existing techniques to rank alternatives in mcdm, such as gower plots and decision balls method [21], theseus method [20], information fusion [18], weight restrictions (reza farzipoor saen 2009) and rational research method [27] etc. we must make all criteria the same meaning if we want to compare them. but it would produce some errors in this step. furthermore, uniting criteria and comparing the 36 zhaobin li, jian liu, zhuo zhang, et al.: a new decision method for multi-criteria decision making with numerical values… weighted combinatorial expectation of alternatives still has another problem. we can not filter off the absolute disparity through unifying criteria, so different criteria are incomparable even if we unify them into the same meaning. our idea of how to rank and select alternatives: we think the criteria only on the same criteria can be compared. through criteria reduction and obtaining criteria weights we don’t need to consider the deviation between two criteria values, because we only consider whether they are the same or not and find out the useful criteria as well as obtain the criteria weights. for the dms, they just need to know the criteria that locate at an advantage or a disadvantage in decision making. they do not need to consider these useless criteria. in this paper, we propose through comparing the weighted combinatorial advantage values (wcav) of alternatives to rank alternatives and select the most desirable alternative(s) [22]. for benefit type criteria, criteria values of two alternatives ai and ak on criterion cj are f(ai, cj) and f(ak, cj). if f(ai, cj)is larger than f(ak, cj) it means that the alternative ai locates at a more advantage position than a2 on criterion cj. in other words, alternative ai is better than ak on criteria cj. undoubtedly, if alternative ai is better than ak on criteria cj, it also means that alternative ak locates at a more disadvantage position than ai on criterion cj. in this paper, we use i k ja a c to indicate the relation. if there is i k ja a c , it means that alternative ak locates at a more advantage position than ai on criterion cj, respectively, there has f(ai, cj)< f(ak, cj). two criteria values of two alternatives on the same criterion. if two alternatives have the same criteria values, it means that these locate at the same position on this criterion in decision making. we use i k ja a cav to express an advantage value (av) between decision alternative ai and ak on criterion cj. so we get the advantage value as follows: 1 0 1 i k j i k j a a c i k j i k j a a c av a a c a a c       (7) 1 2a awav represents a weighted advantage value (wav) of criteria between ai and ak for useful criteria as follows: 1 2 '/ 1 / 2 / 'i k i k i k i k ma a a a c a a c a a c mwav av av av         (8) 1 2a awav represents a weighted advantage value (wav) criteria between ai and ak for all criteria as follows: 1 2/ 1 / 2 /i k i k i k i k ma a a a c a a c a a c mwav av av av         (9) comparing every pair of all alternatives, we construct the weighted advantage relation matrix (warm), so we get the wadm for all the alternatives of mcdm problems as follows: 1 1 1 2 1 2 1 2 2 2 1 2 n n n n n n a a a a a a a a a a a a a a a a a a wav wav wav wav wav wav warm wav wav wav               (10) for the wav there are the following characteristics: 1. 0 i k k ia a a awav wav  2. 0 i ka a i kwav a a   3. 0 i ka a i kwav a a  4. i k k ia a a awav wav  kawcav represents a weighted combinatorial advantage value (wcav) of alternatives for alternative ak in decision tables, we can get the wcav as follows: 1 1k k ia a a i k wcav wav n     (11) app:ds:comparability advances in systems science and application(2016) vol.16 no.4 37 the larger the wcav of ak, the better alternative ak. therefore, all the alternatives can be ranked according to the wcav, respectively. thus, the best alternative can be selected. as the previous discussion of the risk preference assumptions, we get different advantage values for the same two intuitionistic fuzzy sets under a special situation. thus, we will get different weighted combinatorial advantage values for different types of dms in the same decision table. 4 applying the method to numerical value mcdm problems in this section, an example for numerical value mcdm problem is used to illustrate the feasibility and validity of the proposed method. in order to demonstrate the method of criteria reduction based on tolerance relation and rough set theory，we proposed an effective tool for mcdm problems. suppose that the dms have seven criteria of 1 2 3 4 5 6 7{ , , , , , , }c c c c c c cc to make a decision, such as quality c1, competitive c2, price c3, design plan c4, delivery time c5, safety index c6, and sale service c7. suppose we invite 100 experts to make their judgement and voting for all the competitors based on seven criteria, there are two kinds of poll results “yes” or “no”. which is the best one in table 2? table 2 the result of “yes” from 100 experts u c1 c2 c3 c4 c5 c6 c7 a1 45 50 75 20 50 40 48 a2 61 62 65 54 45 50 50 a3 45 55 30 54 45 45 70 a4 65 65 30 65 70 45 65 a5 89 85 65 65 65 50 50 as denoted in fig. 1, we need several steps (find useful criteria, obtain criteria weights, rank alternatives, etc.) to select the most desirable alternative(s). we can build two different discernibility matrixes based on tolerance and advantage relations to find out the useful criteria, respectively. first, we will build the discernibility matrix based on tolerance relation and rough set and validate the method in section 4.1. second, we will build the discernibility matrix based on advantage relation and rough set and validate it in section 4.3. as we have denoted in fig. 1 that we need. 4.1 the procedures for numerical value mcdm problems based on tolerance relation step 1 find out the useful criteria. according to criteria of table 2 by using eq. (2), we construct the discernibility matrix based on tolerance relation and find out the useful criteria as follows: 2 3 4 5 6 7 1 2 3 6 7 1 2 4 5 1 2 3 5 7 1 2 3 5 6 7 c c c c c c c c c c c c c c c m c c c c c c c c c c c                     c c c c c as the previous representing, we know discernibility matrix is a symmetric matrix. in this paper we just give the upper triangular matrix of it. if we want to compare alternatives a1 and a2, we must compare those criteria that m12 contains. as we proposed in section 2, c is a criterion set which contains all the criteria in table 2. thus, we need to compare all criteria in table 2 if we want to know which is better between alternatives a1 and a2. 38 zhaobin li, jian liu, zhuo zhang, et al.: a new decision method for multi-criteria decision making with numerical values… we use a relative discernibility function of discernibility matrix by using boolean reasoning techniques[27-30]. we get 1 2 4 5{ , , , }c c c c a group of useful criteria. if we want to compare alternatives these four criteria are needed. step 2 obtain criteria weights of the useful criteria. by using eq. (5) and the useful criteria of 1 2 4 5{ , , , }c c c c , we can get the criteria weights as follows: 1 2 4 5 1 5 2 1 , , , . 4 18 9 4        step 3 rank and select the most desirable alternative(s). by using eq. (7), (8) and (10) as well as the useful criteria of 1 2 4 5{ , , , }c c c c , we construct the warm as follows: 0 0.5 0.25 1 1 0.5 0 19 36 1 1 0.75 19 36 0 1 1 1 1 1 0 5 18 1 1 1 5 18 0 warm                        by using eq. (11), we get the wcav for all alternatives as follows: 1 2 3 4 5 0.6875, 0.2431, 0.4444, 0.6806, 0.8194.a a a a awcav wcav wcav wcav wcav        therefore, the ranking order of all the alternatives is 5 4 2 3 1 0.2778 1 0.5278 0.75 a a a a a . thus, the alternative a5 is the best choice in table 2. 4.1.2 validating the method for numerical value mcdm problems based on tolerance relation in section 4.1.1, we got the alternatives order for numerical value mcdm problems of cacc by using the new method based on the rough set and tolerance relation. several algorithms are needed in it, such as finding out useful criteria through criteria reduction and making a decision using useful criteria. in this section, we will validate this method that we proposed an effective and useful tool of mcdm problems. to address this problem, we use all the criteria in table 2 to make a decision. if we get the same ranking order and the most desirable alternative as we have got in section 4.1 by using the useful criteria, it means that our method is correct. using all criteria in table 2 of mcdm problems to make a decision, by using the eq. (6) to obtain the criterion weights as follows: 1 2 3 4 5 6 7 9 61, 10 61, 8 61, 8 61, 9 61, 8 61, 9 61.c c c c c c c             by using eq. (7), (9), (10) and all the criteria，we get the warm as follows: 0 27 61 18 61 45 61 45 61 27 61 0 26 61 29 61 36 61 18 61 26 61 0 36 61 43 61 45 61 29 61 36 61 0 17 61 45 61 36 61 43 61 17 61 0 warm                        by using eq. (11), we get the wcav for all alternatives as follows: 1 2 3 4 5 0.5533, 0.0492, 0.3566, 0.3811, 0.5779.a a a a awcav wcav wcav wcav wcav        therefore, the ranking order of all the alternatives is 5 4 2 3 1 0.2787 0.4754 0.4262 0.2951 a a a a a . obviously, we get the same ranking order and the most desirable choice in two different situations. so, finding out and using the useful criteria to make decision is an effective and useful tool for numerical value mcdm problems, especially, when there are large number of criteria in decision tables. in section 4.1, we validate the new method for criteria reduction based on rough set and tolerance relation. through this method, we can find out the useful criteria and avoid the useless criteria. just use these useful criteria to make decision. thus, we find a useful method for advances in systems science and application(2016) vol.16 no.4 39 criteria reduction based on tolerance relation to make decision when there have large number of criteria of numerical value mcdm problems. 4.2 the procedures for numerical value mcdm problems based on advantage relation step 1 find out the useful criteria. according to criteria of table 2 by using eq.(3), we construct the discernibility matrix based on advantage relation and find out the useful criteria as follows: 3 5 3 5 3 3 1 2 4 6 7 1 2 3 6 3 6 2 4 6 7 7 7 7 1 2 4 5 6 7 1 2 4 5 7 1 2 4 5 5 7 1 2 4 5 6 7 1 2 4 5 1 2 3 4 5 6 1 2 3 6 c c c c c c c c c c c c c c c c c m c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c                       as the previous representing, we know that the discernibility matrixes based on advantage and disadvantage relations have a relation of transposed, it means that they are transposed matrix each other. in this paper we just use the advantage discernibilty matrix to find out the useful criteria and make a decision. as the discussion in section 2.2, we know that if we want to compare alternatives a1 and a2, alternatives a1 locates at a more advantage position than a2 on criteria c3 and c5 in table 2. at the same time, alternatives a2 locates at a more advantage position than a1 on criteria c1, c2, c4, c6, and c7 in table 2. we use a relative discernibility function of discernibility matrix by using boolean reasoning techniques [24-26, 27-30]. we get 1 2 3 4 5 7{ , , , , , }c c c c c c a group of useful criteria. if we want to compare alternatives these four criteria are needed. step 2 obtain criteria weights of the useful criteria. by using eq. (5) and the useful criteria of 1 2 3 4 5 7{ , , , , , }c c c c c c , we can get the criteria weights as follows: 1 2 3 4 5 7 9 10 8 8 9 9 , , , , , . 53 53 53 53 53 53            step 3 rank and select the most desirable alternative(s). by using (7), (8) and (10) as well as the useful criteria of 1 2 3 4 5 7{ , , , , , }c c c c c c , we construct the warm as follows: 0 19 53 10 53 37 53 37 53 19 53 0 18 53 37 53 36 53 10 53 18 53 0 27 53 35 53 37 53 37 53 27 53 0 9 53 37 53 36 53 35 53 9 53 0 warm                        by using eq. (11), we get the wcav for all alternatives as follows: 1 2 3 4 5 0.4858, 0.1698, 0.3302, 0.2480, 0.5519.a a a a awcav wcav wcav wcav wcav        therefore, the ranking order of all the alternatives is 5 4 2 3 1 0.1698 0.6981 0.3396 0.1887 a a a a a . thus, the alternative a5 is the best choice. 4.2.2 validating the model for numerical value mcdm problems in this section, we will validate this method based on the rough set and advantage relation an effective and useful tool of mcdm problems. to address this problem, we still use all the criteria in table 2 to make a decision. if we get the same ranking order and the most desirable alternative as we have got in section 4.1 by using the useful criteria, it means that our model is effectively. using all criteria in table 2 of mcdm problems to make a decision, by using the eq. (6) to obtain the criterion weights as follows: 1 2 3 4 5 6 7 9 10 8 8 9 8 9 , , , , , . 61 61 61 61 61 61 61              40 zhaobin li, jian liu, zhuo zhang, et al.: a new decision method for multi-criteria decision making with numerical values… by using eq. (7), (9), (10) and all criteria，we get the warm as follows: 0 27 61 18 61 45 61 45 61 27 61 0 26 61 29 61 36 61 18 61 26 61 0 27 61 43 61 45 61 29 61 27 61 0 17 61 45 61 36 61 43 61 17 61 0 warm                        by using eq. (11), we get the wcav for all alternatives as follows: 1 2 3 4 5 0.5533, 0.0492, 0.3197, 0.3343, 0.5779.a a a a awcav wcav wcav wcav wcav        therefore, the ranking order of all the alternatives is 5 4 2 3 1 0.2787 0.4754 0.4262 0.2951 a a a a a . obviously, we get the same ranking order and the most desirable choice in two different situations. so, finding out and using the useful criteria to make a decision is an effective and useful tool for numerical value mcdm problems, especially, when there are large number of criteria in decision tables. thus, we find a useful model to make a decision when dms face large number of criteria of numerical value mcdm problems. if we can get the same ranking order, it means that this model is a useful tool. in fact, if we get the same the most desirable alternative(s), we can say this model is still a useful tool for mcdm problems. because the decision makers are always concerned with finding the best alternative(s), so getting the most desirable alternative(s) seems to be more important than the other ranks. in this section, we use two different methods to address the “large decision table” (e.g. a large number of criteria) challenge in multiple criteria decision making. this new method involves criteria reduction based on rough set and criteria value relation (tolerance and advantage relations). we also validate the new method. thus, we can say that we find an effective method to address the problem “there may be not one of having insufficient information, but rather one of having too much or an unknown quality of information how we filter information for mcdm in an internet or mobile environment.” 4.3 validating the model using maximizing deviation method in this paragraph, we will compare the results from our method with the one based on the maximum deviation method to obtain the criteria weights. by using the maximizing deviation method for interval numbers proposed in [18], we need to take all the criteria into consideration in table 2. if we use all the criteria in table 4 to make a decision, the ranking order of all the alternatives is also 5 4 2 3 1a a a a a . alternative a5 is the best choice. we get the same best choice and ranking order of all alternatives. 5 conclusion this paper mainly focuses upon how to make a wise and reasonable decision within limited time when dms face a large number of criteria in mcdm problems. in this paper, our solution mainly focuses on four aspects. first, according to numerical value mcdm problems, we proposed a method of criteria reduction based on tolerance relation and then build a discernibility matrix to find out useful criteria. just use useful criteria to make a decision. second, through the idea of building the discernibility matrix, we proposed a new method to obtain criteria weights. third, a new method based on weighted combinatorial advantage criteria value and criteria reduction, we give the model of advantage matrix of criteria. finally, we compare the ranking result by using useful criteria and all criteria in the table to make a decision. we validated the method of finding out the useful criteria through criteria reduction a useful and advances in systems science and application(2016) vol.16 no.4 41 scientific method for mcdm problems. our work is an underline research field of mcdm problems with large number of criteria. in future work, we will study the criteria reduction algorithms to find the useful criteria when the criteria are dependent. acknowledgment this research is supported by the national natural science foundation of china (no. 71671092; 71301075), national natural science foundation of jiangsu province, china (no. bk20130770), international postdoctoral exchange fellowship program of china (no. 20140072), postdoctoral science foundation funded project of jiangsu province, china (no. 1501040a). we thank the department editor and anonymous reviewers for their helpful comments. references [1] j. wallenius, j. s. dyer, p. c. fishburn, r. e. steuer, s. zionts and k. deb(2008), "multiple criteria decision making, multicriteria utility theory: recent accomplishments and what lies ahead", management science, vol.54, no. 7, pp. 1336-1349. 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[30] r. w. swiniarskia and a. skowronb(2006), "rough set methods in feature selection and recognition", pattern recognition letters, vol. 24, no. 6, pp. 833-849. corresponding author jian liu can be contacted at: jianlau@njust.edu.cn. http://www.sciencedirect.com/science/journal/00200255 http://www.sciencedirect.com/science/article/pii/s0167865502001964 http://www.sciencedirect.com/science/article/pii/s0167865502001964 http://www.sciencedirect.com/science/article/pii/s0167865502001964 http://www.sciencedirect.com/science/article/pii/s0167865502001964 http://www.sciencedirect.com/science/journal/01678655 microsoft word 4 jiang jianhua, sheng buyun, yang mingzhong--research on owa based multi-source heterogeneous data fusion.do 232-239 advances in systems science and applications (2011), vol.11, no.3-4 issn 1078-6236 international institute for general systems studies, inc. research on owa based multi-source heterogeneous data fusion jianhua jiang, buyun sheng and mingzhong yang school of mechanical and electronic engineering, wuhan university of technology, wuhan 430070, china abstract for the problem of multi-source heterogeneous data fusion, an architecture model of multi-source heterogeneous data fusion was designed. in the process of data fusion, triangular fuzzy number (tfn) was used to give uniform express of multi-data description in quantity, the ordered weight average (owa) was used to deal with the preference of decision-maker and design the algorithm of data fusion. at last, the feasibility was validated by an example. keywords data fusion; triangular fuzzy number; ordered weight average 1. introduction data fusion is a process of multi-source data cooperating as to reduce data redundancy and capture comprehensive information from data. it has become a research focus in data processing, object identification, situation assessment and intelligent decision-making domains. in general view, the data for data fusion come from multi-sensors[1], the data type is almost numerical and the methods are mainly from statistics and artificial intelligent, and some literatures have researched the multi-source data fusion[2,3,4]. in fact, except for numerical value, there are many other expression ways of data such as language and symbols, and the multi-ways of data expression will led to the ambiguity, difference and heterogeneity of data. on the other hand, in order to make an ultimate decision, a decision-maker must integrate a wide range of heterogeneous data and information. therefore, in allusion to the features of heterogeneous data, this paper set focus on the method of multi-source heterogeneous data fusion and its application. 2. multi-source heterogeneous data fusion methods and its architecture 2.1 multi-source heterogeneous data fusion methods the methods for data fusion in decision level mainly include weight average[5], d-s evidence theory[6] and voting[7]. (1) weight average method the method uses formula ∑ iji tw to compute decision support value, where iw denotes weight of data source i and ijt denotes support value of data source i to decision j . it evaluates every decision according the support value, it is easy to operate and gives full consideration to the importance of data sources. but it is difficult to eliminate the impact of subjective factors in determining the weights. (2) d-s evidence theory it defines the space including all possible results of the object to be identified as a framework set d , its subset marked d2 and for da ⊆∀ : ]1,0[2: →dm where 0)( =φm , 1)(2 =∑ ⊆ da am , φ is empty set, and m is called basic probability assignment function (bpaf) which actually refers to assign the trust value to subsets of d on advances in systems science and applications (2011), vol.11, no.3-4 233 issn 1078-6236 international institute for general systems studies, inc basis of the available evidence. in practice, many bpafs should be combined for different evidences of a problem may led to different im , it can be realized by the following formal: )()( 1 i aa i amkam i ∑∏×= =∩ − （ ni ≤≤1 ） where )( i a i amk i ∑∏= ≠∩ φ and dai ⊆ d-s evidence theory is based on bpaf and able to compute uncertainty caused by unknown factors. but it requests all the items in d mutually exclusive and is difficult to calculate when there are too many bpafs. (3) voting it treats every data source as a voter and chooses a decision by comparing the votes obtained, the votes is defined as follow: ))(()( iji asupfasup = where ia denotes decision i ( ni ≤≤1 ), )( iasup denotes the total votes it obtained, )( ij asup denotes the support value of data source j to decision i and its value is 0 or 1, and f may be defined as sum. it is difficulty to find bpaf for d-s evidence theory and distinguish two decisions have same votes, this paper uses owa to fusion multi-source data in full considering the preference of decision-makers. 2.2 multi-source heterogeneous data fusion architecture literature [2] proposes an architecture for multi-source data fusion as shown in figure 1, it takes into account user requirements and source credibility, and uses proximity knowledge base, knowledge of reasonableness and voting method to resolve data conflicts. figure 1 schematic of data fusion under the guidance of the above model, an architecture model of multi-source heterogeneous data fusion in decision level was designed as shown in figure 2. the fusion engine is made up of four parts of data warehouse, decision support value computing, owa operator weight vector computing and data conversion and sorting. figure 2 multi-source data fusion architecture (1) data warehouse is used to integrate multi-source data and eliminate data heterogeneity and differences by operation of data selection, feature extraction and statistical analysis. 234 jiang:research on owa based multi-source heterogeneous data fusion (2) decision support computing module obtains related data from data warehouse according to decision-making attributes and computes the support value ijs of data source i to decision j . (3) owa operator weight vector computing module computes the weight of data source according to the fuzzy semantic parameters provided by decision-maker, and these parameters reflect the preferences of decision-makers to the data source. (4) data conversion and sorting calculates ' ijs in combining with data source credibility (or importance degree) and owa weight, and sorts ' ijs according its value. at last, the final decision value can be calculated by ' ijs and iw . 3. algorithm of data fusion 3.1. data type and its features data can be expressed in quantity and quality, and numbers for quantity while language variables for quality[8]. according to difference of data description, data type can be divided into quantitative and qualitative two types, this paper focus on 4 types of data description, as shown in table 1. table 1 data type type data description notes random variable random variable submits to certain distribution quantitative two value type data value either 1 or 0, or true or false degree type the degree general described by 7 or 9 standard qualitative terminology based description method depends on vocabulary space in case of large samples, random variable submits to normal distribution marked ),(~ 2σμx , where μ denotes expectation and σ denotes standard variance, and 9974.0)33( =+<<− σμσμ xp . two-value type data is used to give answer for affirming or denying, if agreed the value is 1, otherwise 0, it can also be described by true or false. degree type data is generally described by degree adverbs such as good, a little good and so on, the grading standard widely used is 7 or 9. terminology based data is described by terms from a pre-defined vocabulary space and the space depends on specific circumstance. 3.2 computing of tfn based support value in allusion to the ambiguity in data description, tfn is used to compute the decision support value. (1) transform method for random data supposed that: σ30 −= ux σ6 0' xxx − = if support value increases with the increased of random variable x and ]3,3[ σμσμ +− is divided into n intervals, support value can be defined as: ⎪ ⎪ ⎩ ⎪⎪ ⎨ ⎧ +> + + ≤<+ + −≤ = σ σσ σμ 3,)1,1,1( )1(66,)1,,( 3,)0,0,0( )( 00 ' ux x n ixx n i n ix n i x xs (1) advances in systems science and applications (2011), vol.11, no.3-4 235 issn 1078-6236 international institute for general systems studies, inc if support value decreases with the increased of random variable x , support value defined as: )()1,1,1(' )( xss x −= (2) transform method for two value type data supposed that two-value type data described by set {1, 0}, the number of answer 1 and 0 is n and m respectively. if support value is defined by number of value 1, support value can be defined as: )/,/,/()( mnnmnnmnnxs +++= (2) (3) transform method for degree type data if data described by the 7 grading standard, the support value can be defined according to table 2. table 2 fuzzy quantification of degree type data (4) transform method for terminology based data if the vocabulary space w includes n terminologies, and they are ordered by their contribution value for decision from low to high as },...,,{ 110 −= nwwww , the support value defined as: ))1/(),1/(),1/(()( −−−= nininiws i (3) 3.3 computing of owa operator[9] weight vector supposed that rrf n →: and a n dimension vector ),...,,( 21 nwwww = related to f makes: ∑ == n i iin bwaaaf 121 ),...,,( (4) where ]1,0[∈iw , ni ≤≤1 and 11 =∑ = n i iw . ib is the ith largest element of ia . then f is called n dimension owa operator and ),...,,( 21 nwwww = can be represented by a function as: )/)1(()/( nifnifwi −−= (5) where ni ,...,2,1= , f is named fuzzy semantic operator (fso) and defined as: ⎪ ⎩ ⎪ ⎨ ⎧ < ≤≤−− < = xb bxaabax ax xf ,1 ,)/()( ,0 )( (6) where ]1,0[,, ∈bax . in addition, owa also defines a measure operator for reflecting preference of decision-makers named )(worness and it is defined as: inverse proportion direct proportion )(xs very high very low (0.00,0.00,0.17) high low (0.00,0.17,0.34) a little high a little low (0.17,0.34,0.50) common common (0.34,0.50,0.67) a little low a little high (0.50,0.67,0.84) low high (0.67,0.84,1.00) very low very high (0.84,1.00,1.00) 236 jiang:research on owa based multi-source heterogeneous data fusion ∑ = −== n i iwin n wornessc 1 )(1)( (7) 3.4 algorithm of data fusion supposing there are n decisions ),...,,( 21 naaaa = , m data sources ),...,,( 21 mssss = and ip is the credibility or importance of is , the algorithm of data fusion is described as: step1: compute support value of data sources to decisions access data from data warehouse and transform it to support value of every decision according to the method of section 3.1. the support value marked as: ),,( ijijijij cbas = where ijs is support value of data source i to decision j , ),,( jiijij cba is tfn expression of ijs and 10 ≤≤≤≤ ijijij cba . step2: compute owa operator weight vector considering the preference of decision-makers, choose fuzzy semantic parameters for formula 6. in most cases, fuzzy semantic parameters is defined as “the majority”, “at least half” and “as much as possible”, their parameters are )8.0,3.0( , )5.0,0( and )1,5.0( respectively. according to the parameters, the fso )(xf can be determined. with fso, formula 5 and 7, owa weight vector ),...,,( 21 nwwww = and )(worness can be calculated. step3: transform ijs according to ip and ijs in order to use owa weight vector, ijs must be transformed and sorted, the transform method as follow: supposed that: ijiij sps ×=min_ ijiijiij spsps ×−+=max_ (8) ijin i i averageij sp p ns ∑ = = 1 _ define: if 5.0≤c 0)( =ch ， cm c 2)( = ， cl c 21)( −= if 5.0≥c 12)( −= ch c ， cm c 22)( −= ， 0)( =cl ijs can be transformed into ' ijs by following formula: min_)(_)(max_)( ' ijcaverageijcijcij slsmshs ++= (9) step 4: compute final support value for every decision the final support value of decisions can be computed by the following formula: ∑ == m i ijij bws 1 nj ,...,2,1= (10) where ijb denotes the ith largesse element of ),...,,( '' 2 ' 1 njjj sss . step5: make a decision according to the final support value of decisions. 4. an example this paper takes steam turbine product development decision-making as example. supposing there are 5 kinds of product named 1a , 2a , 3a , 4a and 5a , the data can be collected include market demand, product feedback, product parameters, historical data, failure statistics, advances in systems science and applications (2011), vol.11, no.3-4 237 issn 1078-6236 international institute for general systems studies, inc expert advice and so on. based on these data sources, products can be compared from 6 aspects, they are assessment of market demand 1a , average failures per year 2a ( 8.0,5.3 == σμ ）, longest average no failure time 3a (month as unit and 53.2,28.12 == σμ ), economy evaluation 4a , customer evaluation 5a and expert advice 6a , and the preliminary results are shown in table 3. table 3 support value of product and data source credibility (1) in table 3, 1a and 4a are degree-type data and transformed by table 2, 2a and 3a are random data and transformed by formula 1 (supposing n=15), 5a is two-value type data(the value in table 3 is the ratio of positive evaluation) and transformed by formula 2, 6a is terminology based data and transformed by formula 3. the transformed results are shown in table 4. (2) if “the majority” is selected as fso, the parameters a and b in formula 6 are 0.3 and 0.8 respectively. according to formula 5 and 6, the owa weight vector can be calculated as: )0,27.0,33.0,33.0,067.0,0(=w put every iw into formula 7, the value of c can be calculated and 37.0=c . (3) according to formula 8 and 9, the data in table in table 4 can be transformed as shown in table 5. (4) for every column data in table 5, sorted them from high to low according the second value and computed final support value by formula 10, the result is shown in table 6. (5) from table 6, product 3a gets the highest support value, so it should be developed full table 4 disposed attribute values ia ip a1 a2 a3 a4 a5 1a 0.5 (0.67,0.84,1.00) (0.34,0.5,0.67） (0.84,1.00,1.00) (0.17,0.34,0.50) (0.00,0.17,0.34) 2a 0.9 (0.13,0.14,0.20） (0.00,0.00,0.00) (0.53,0.55,0.60) (0.80,0.85,0.86) (0.02,0.23,0.27) 3a 0.9 (0.20,0.24,0.26) (0.53,0.56,0.6) (0.66,0.68,0.73) (0.13,0.16,0.20) (0.80,0.83,0.86) 4a 0.3 (0.34,0.50,0.67） (0.84,1.00,1.00) (0.17,0.34,0.50) (0.50,0.67,0.84) (0.67,0.84,1.00) 5a 0.6 (0.65,0.65,0.65) (0.73,0.73,0.73) (0.48,0.48,0.48) (0.54,0.54,0.54) (0.67,0.67,0.67) 6a 0.4 (0.33,0.33,0.33) (1.00,1.00,1.00) (0.66,0.66,0.66) (0.00,0.00,0.00) (0.33,0.33,0.33) ia ip a1 a2 a3 a4 a5 1a 0.5 large common very large a little small very small 2a 0.9 5.25 8.64 3.28 1.81 4.76 3a 0.9 8.36 13.28 15.14 7.08 17.31 4a 0.3 commo n very good a little bad a little good good 5a 0.6 0.65 0.73 0.48 0.54 0.67 6a 0.4 improve fully recommen d as usual out of market improve 238 jiang:research on owa based multi-source heterogeneous data fusion table 5 results of data transformed ia a1 a2 a3 a4 a5 1a (0.50,0.62,0.75) (0.25,0.37,0.50) (0.63,0.75,0.75) (0.13,0.25,0.37) (0.00,0.13,0.25) 2a (0.18,0.19,0.27） (0.00,0.00,0.00) (0.71,0.74,0.81) (1.07,1.14,1.16) (0.27,0.31,0.36) 3a (0.27,0.32,0.35) (0.71,0.75,0.81) (0.89,0.92,0.98) (0.18,0.22,0.27) (1.07.1.12,1.16) 4a (0.15,0.22,0.30） (0.38,0.45,0.45) (0.07,0.15,0.22) (0.22,0.30,0.38) (0.30,0.38,0.45) 5a (0.58,0.58,0.58) (0.65,0.65,0.65) (0.43,0.43,0.43) (0.48,0.48,0.48) (0.60,0.60,0.60) 6a (0.20,0.20，0.20) (0.60,0.60,0.60) (0.39,0.39,0.39) (0.00,0.00,0.00) (0.20,0.20,0.20) table 6 final support values a1 a2 a3 a4 a5 value (0.23,0.27,0.31) (0.43,0.49,0.53) (0.52,0.54,0.56) (0.20,0.27,0.35) (0.28,0.32,0.36) 5. conclusion in this paper, an architecture model for multi-source heterogeneous data fusion was constructed, tfn based data processing for multi-data description in quantity was researched and owa operator based data fusion algorithm was designed, the practical application indicates that the algorithm designed is effective. the study of this paper presents a feasible option for constructing intelligent decision support system and has certain reference value for similar data processing and fusion. acknowledgements this paper is supported by national natural science fund project of china (contract no.50620130441), scientific and technological project of wuhanc city (contact no. 200810321153) and youth science and technology chenguang project of wuhan city (contact no. 200750731289). references [1] d. l. hall, j. llinas. hand book of multi-sensor data fusion. beijing: electronic industry press, 2008. 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[7] s. kang. two-phase identification algorithm based on fuzzy set and voting for intelligent multi-sensor data fusion, lecture notes in computer science, vol. 4252 (2006) 769~776. [8] f.herrera, l.martinez. an approach for combing linguistic and numerical information based on the 2-tuple fuzzy linguistic representation model in decision-making. international advances in systems science and applications (2011), vol.11, no.3-4 239 issn 1078-6236 international institute for general systems studies, inc journal of uncertainty, fuzziness and knowledge-based systems, 8(5) (2000)539~562. [9] z.zhang, c.zhang. result fusion in multi-expert systems based on owa operator. proceedings of the 23rd computer science conference, (2000) 234~240. advances in systems science and applications (2012) vol.12 no.3 283-296 central limit theorems for the single point catalytic super-brownian motion with immigration xu yang and mei zhang laboratory of mathematics and complex systems, school of mathematical sciences, beijing normal university, beijing 100875, people’s republic of china abstract we establish the central limit theorems for the single point catalytic superbrownian motion with deterministic immigration and single point catalytic superbrownian motion immigration on the schwartz space and a weighted sobolev space. for the catalytic immigration case, the weak convergence depends on the branching rates of both immigration part and non-immigration part. keywords super-brownian motion, catalytic, conditional log-laplace functional, immigration, central limit theorem 1 introduction and main results catalytic super-brownian motion is the superprocess with brownian motion as underlying spatial motion,whose branching occurs only in the presence of some catalysts. dawson and fleischmann [1] considered the case of single point catalytic super-brownian motion, namely, the underlying particles move as independent brownian motions in r. the life time of each particle is exponentially distributed. when it dies, it splits according to critical branching only if they pass 0, which is called the (single) catalyst point. fleischmann and xiong [2], yang and zhang [3] and li and wang [4] proved the large deviation, moderate deviation and central limit theorem for the single point catalytic super-brownian motion, respectively. if we suppose the situation where there are additional particles added, we need to consider the process with immigration. superprocesses with immigration are studied by many authors; see e.g. [5-10]. in the present paper,first we will prove the central limit theorem for the single point catalytic super-brownian motion with deterministic immigration on schwartz space, and then extend the result to a weighted sobolev space. we shall also investigate the processes with catalytic immigration. in this case, the weak convergence depends not only on the branching rate ϱ of non-immigration part, but also on the branching rate ϱ0 of immigration part. 1.1 notations and preliminaries first we introduce some notations. let p ≥ 2, hp(x) = (1 + x2)− p 2 and cp(r) denote the set of all real-valued continuous functions φ on r such that φ(x)/hp(x) has a finite limit as |x| → ∞. equipped with the norm 9φ9p := sup{|φ(x)|/hp(x) : x ∈ r}, cp(r) is a banach space. let c+ p (r) denote all the positive functions 284 xu yang:central limit theorems for the single point catalytic... of cp(r) and mp(r) be the set of all measures µ on r such that ⟨µ, hp⟩ < ∞. suppose that 9µ9p = ⟨µ, hp⟩. let c∞(r) be the set of bounded infinitely differentiable functions on r with bounded derivatives. let s (r) ⊂ c∞(r) denote the schwartz space of rapidly decreasing functions on r and s+(r) be the collection of non-negative elements of s (r). that is, each f ∈ s (r) is infinitely differentiable and for each non-negative integer k and each non-negative integer α we have lim |x|→∞ xk dα dxα f(x) = 0. now we introduce some basic results in the following subsection, which can be found in [11]. we define the hilbertian norms {q0, q1, q2, · · · } on s (r) by qn(f) 2 = n∑ k=0 ∫ r (1 + x2)n(f (k)(x))2dx. the hermite polynomials on r are given by gk(x) = (−1)kex 2 dk dxk e−x2 , k = 0, 1, 2, · · · . based on those we define the hermite functions hk(x) = 1 4 √ π √ 2kk! e−x2/2gk(x), k = 0, 1, 2 · · · . then hk ∈ s (r) and {hk : k ≥ 0} is a complete orthonormal system in l2(r). let ⟨·, ·⟩ denote the inner product of l2(r). for f ∈ s (r) we write f = ∞∑ k=0 ⟨f, hk⟩hk and define ∥f∥2n = ∞∑ k=0 (2k + 1)2n⟨f, hk⟩2 (1) for n = 0,±1,±2, · · · . let hn(r) be the completion of s (r) with respect to ∥ · ∥n. by approximation we can extend ⟨·, ·⟩ to a bilinear form between h−n(r) and hn(r). let ⟨·, ·⟩n denote the inner product of hn(r). for g, f ∈ hn(r) we have ⟨g, f⟩n = ∞∑ k=0 (2k + 1)2n⟨g, hk⟩⟨f, hk⟩ = ⟨πng, f⟩, where πng = ∞∑ k=0 (2k + 1)2n⟨g, hk⟩hk ∈ hn(r). then h−n(r) and hn(r) are dual spaces with the duality ⟨·, ·⟩. advances in systems science and applications (2012) vol.12 no.3 285 lemma 1.1 for every n ≥ 0 there is a constant c(n) > 0 such that qn(f) ≤ c(n)∥f∥n and ∥f∥n ≤ c(n)q2n(f), f ∈ s (r). the sequence of norms defined by (1) induces a topology on the set h∞ := ∞∩ k=0 hn(r), which is compatible with the metric ρ defined by ρ(f, g) = ∞∑ k=0 ∥f − g∥k 2k(1 + ∥f − g∥k) . then (s (r), ρ) (written as s (r) for simplicity) is a nuclear space. this implies s ′(r) = ∞∪ n=0 h−n(r) ⊃ · · · ⊃ h−2(r) ⊃ h−1(r) ⊃ h0(r) ⊃ h1(r) ⊃ h2(r) ⊃ · · · ⊃ ∞∩ n=0 hn(r) = s (r). a subset b of the nuclear space s (r) is said to be bounded if it is bounded in each norm ∥ · ∥n, that is, sup x∈b ∥x∥n < ∞ for each n ≥ 0. for each bounded set b ⊂ s (r) we define the semi-norm pb on s ′(r) by pb(f) = sup{|f(x)| : x ∈ b}, f ∈ s ′(r). we endow s ′(r) with the topology generated by the collection of semi-norms {pb : b ⊂ s (r) is bounded}, which is called the strong topology. then s ′(r) is a nuclear space. 1.2 models for a process x taking its value in mp(r), let pr,ν denote its conditional law given xr = ν. suppose that p is the heat kernel in r with constant ς > 0: 1√ 2πςt exp { − a2 2ςt } , t > 0, a ∈ r · · · . (2) for φ ∈ c+ p (r) fixed, u(t, z; ϱ) denotes the unique non-negative solution to the log-laplace equation u(t, z; ϱ) = ptφ(z)− ϱ ∫ t 0 pt−r(z)u 2(r, 0; ϱ)dr, t ≥ 0, z ∈ r (3) 286 xu yang:central limit theorems for the single point catalytic... and ω(t, z; ϱ, ϱ0) denotes the unique non-negative solution to the log-laplace equation ω(t, z; ϱ, ϱ0) = ∫ t 0 pt−r [u(r, ·; ϱ)] (x)dr − ϱ0 ∫ t 0 pt−r(z)ω 2(r, 0; ϱ, ϱ0)dr, t ≥ 0, z ∈ r · · · . (4) where ϱ and ϱ0 are positive constants and {pt : t ≥ 0} denotes the brownian semigroup corresponding to (2). let a be the generator of {pt : t ≥ 0}. in the following we suppose µ, η ∈ mp(r). we say ξµ,ϱ = {ξµ,ϱt : t ≥ 0} is a single point catalytic super-brownian motion, if ξµ,ϱ0 = µ, and for r ≥ 0 and ν ∈ mp(r), − logpr,ν exp {−⟨ξt, φ⟩} = ⟨ν, u(t− r, ·; ϱ)⟩, 0 ≤ r ≤ t, φ ∈ c+ p (r) · · · . (5) suppose z = {zt : t ≥ 0} is the single point catalytic super-brownian motion with z0 = µ, deterministic immigration controlled by η and log-laplace functional given by − logpr,νe −⟨zt,φ⟩ =⟨ν, u(t− r, ·; ϱ)⟩+ ∫ t r ⟨η, u(t− s, ·; ϱ)⟩ds, 0 ≤ r ≤ t, φ ∈ c+ p (r) · · · . (6) now we suppose that xξ = {xξ t : t ≥ 0} is the single point catalytic superbrownian motion with single point catalytic immigration determined by ξη,ϱ0 . let p0,ν,η denote its conditional law given xξ 0 = ν and ξ0 = η. by theorem 3.2 of [12] we have the log-laplace functional of xξ: − logp0,ν,η exp { −⟨xξ t , φ⟩ } =− logp0,ν,η [ p0,ν,η exp { −⟨xξ t , φ⟩ } ∣∣∣∣ {σ(ξs : 0 ≤ s ≤ t)} ] =− logp0,η exp { −⟨ν, u(t, ·; ϱ)⟩ − ∫ t 0 ⟨ξs, u(t− s, ·; ϱ)⟩ds } =⟨ν, u(t, ·)⟩+ ⟨η, ω(t, ·; ϱ, ϱ0)⟩, µ, η ∈ mp(r), t ≥ 0, φ ∈ c+ p (r) · · · . (7) in the following, we breviate u(t, ·; ϱ) and ω(t, ·; ϱ, ϱ0) by u(t, ·) and ω(t, ·), respectively. we construct the superprocesses z, ξη,ϱ0 and xξ on the same probability space (ω,f ,p). advances in systems science and applications (2012) vol.12 no.3 287 1.3 main results let {z(k) t : t ≥ 0} be the single point catalyst super-brownian motion with deterministic immigration characterized by(3) and (6), with ϱ replaced by 1/k, let w (k) t = k 1 2 ( z (k) t − µpt − ∫ t 0 ηpsds ) . theorem 1.2 as k → ∞, the sequence {w (k) t : t ≥ 0} converges weakly to the gaussian process {wt : t ≥ 0} in c([0,∞),s ′(r)) with w0 = 0 and laplace functional given by p exp { − ⟨wt, f⟩ } =exp {∫ t 0 ⟨µ, pt−r(·)⟩p 2 r f(0)dr + ∫ t 0 ds ∫ s 0 ⟨η, ps−r(·)⟩p 2 r f(0)dr } , where f ∈ s+(r). remark 1.3 if η ≡ 0 in theorem ??, the result can be found in [4]. theorem 1.4 for any n ≥ 3, c([0,∞),s ′(r)) can be replaced by c([0,∞), h−n(r)) in theorem ??. similarly, we can establish the central limit theorem for the single point catalytic super-brownian motion with immigration controlled by another single point catalytic super-brownian motion. the weak convergence depends on ρ and ρ0, which are the branching rates of non-immigration and immigration parts. to specify the effects of ϱ and ϱ0 on the convergence of xξ, we suppose that ϱ = γ1k −1, ϱ0 = γ2k −β(β > 0), where k ∈ n, γ1, γ2 are positive constants. let {ξ(k)t : t ≥ 0} be defined by (3) and (5) with ϱ0 replaced by γ2k −β(β > 0). let {x(k) t : t ≥ 0} be defined accordingly by (7), (3) and(4) with ϱ, ϱ0 replaced by γ1k −1, γ2k −β(β > 0), respectively. define y k t = kα ( x (k) t − µpt − tηpt ) (8) theorem 1.5 as k → ∞, the sequence {y (k) t : t ≥ 0} converges weakly to the gaussian process {yt : t ≥ 0} in c([0,∞),s ′(r)) with y0 = 0 and laplace functional given by p exp { − ⟨yt, f⟩ } = exp {∫ t 0 fs(f)ds } , f ∈ s+(r) (9) 288 xu yang:central limit theorems for the single point catalytic... where fs(f) is determined by the following table: β α fs(f) (0, 1) 1 2β γ2⟨η, pt−s(·)⟩ (spsf(0)) 2 1 1 2 γ1⟨µ, pt−s(·)⟩ (psf(0)) 2 + γ1 ∫ s 0 ⟨η, ps−r(·)⟩ (prf(0)) 2 dr +γ2⟨η, pt−s(·)⟩ (spsf(0)) 2 (1,∞) 1 2 γ1⟨µ, pt−s(·)⟩ (psf(0)) 2 + γ1 ∫ s 0 ⟨η, ps−r(·)⟩ (prf(0)) 2 dr theorem 1.6 for any n ≥ 3, c([0,∞),s ′(r)) can be replaced by c([0,∞), h−n(r)) in theorem ??. since the proofs are similar, we only show the case β = 1 of theorems ??–??. 2 proofs of of theorem ?? and theorem ?? (case β = 1) in the proof of theorem ??, we need the following proposition. proposition 2.1 as k → ∞, the finite dimensional distributions of {y (k) t : t ≥ 0} converge weakly to a s ′(r)-valued gaussian process {yt : t ≥ 0} with y0 = 0 and laplace functional determined by (9). proof. to simplify the notations, we consider the two dimensional distributions of y (k). first we calculate the laplace transform of (⟨y (k) t1 , f⟩, ⟨y (k) t2 , f⟩). by the markov property, (7), (3), (8), [10] and [12], for f ∈ c+ p (r), t2 > t1 > 0, we get the laplace transform of (⟨y (k) t1 , f⟩, ⟨y (k) t2 , f⟩): logp0,µ,η exp { − θ1⟨y (k) t1 , f⟩ − θ2⟨y (k) t2 , f⟩ } = logp0,µ,η [ p0,µ,η ( exp { − ⟨x(k) t1 , θ1k 1 2 f + v(k)(t2 − t1, ·; θ2)⟩ − ∫ t2 t1 ⟨ξ(k)s , v(k)(t2 − s, ·; θ2)⟩ds }∣∣∣∣ {σ(ξ(k)s : s ≤ t2) })] + k 1 2 ( ⟨µpt1 , θ1f⟩+ ⟨µpt2 , θ2f⟩+ t1⟨ηpt1 , θ1f⟩+ t2⟨ηpt2 , θ2f⟩ ) = logp0,η exp { − ⟨µ, u(k)(t1, ·; θ1, θ2)− ∫ t1 0 ⟨ξ(k)s , u(k)(t1 − s, ·; θ1, θ2)⟩ds − ∫ t2 t1 ⟨ξ(k)s , v(k)(t2 − s, ·; θ2)⟩ds } + ⟨µpt1 , k 1 2 θ1f⟩ + ⟨µpt2 , k 1 2 θ2f⟩+ t1⟨ηpt1 , k 1 2 θ1f⟩+ t2⟨ηpt2 , k 1 2 θ2f⟩. advances in systems science and applications (2012) vol.12 no.3 289 using markov property, (3) and (8), we get logp0,ν,η exp { − θ1⟨y (k) t1 , f⟩ − θ2⟨y (k) t2 , f⟩ } = logp0,η exp { − ⟨µ, u(k)(t1, ·; θ1, θ2)− ∫ t1 0 ⟨ξ(k)s , u(k)(t1 − s, ·; θ1, θ2)⟩ds − ⟨ξ(k)t1 , ω(k)(t2 − t1, ·; θ2)⟩ } + ⟨µpt1 , k 1 2 θ1f⟩+ ⟨µpt2 , k 1 2 θ2f⟩+ t1⟨ηpt1 , k 1 2 θ1f⟩+ t2⟨ηpt2 , k 1 2 θ2f⟩ =k−1γ1 ∫ t2−t1 0 ⟨µ, pt2−r(·)⟩ [ v(k)(r, 0; θ2) ]2 dr + k−1γ1 ∫ t1 0 ⟨µ, pt1−r(·)⟩ [ u(k)(r, 0; θ1, θ2) ]2 dr + k−1γ2 ∫ t2−t1 0 ⟨η, pt2−l(·)⟩[ω(k)(l, 0; θ1, θ2)] 2dl + k−1γ2 ∫ t1 0 ⟨η, pt1−r(·)⟩[s(k)(r, 0; θ1, θ2)]2dr + k−1γ1 ∫ t2−t1 0 dl ∫ l 0 ⟨η, pt2−r(·)⟩[v(k)(r, 0; θ2)]2dr + k−1γ1t1 ∫ t2−t1 0 ⟨η, pt2−r(·)⟩[v(k)(r, 0; θ2)]2dr + k−1γ1 ∫ t1 0 dl ∫ l 0 ⟨η, pt1−l(·)⟩[u(k)(r, 0; θ1, θ2)]2dr, where θ1, θ2 ≥ 0, v(k)(·, ·; θ2) is the non-negative solution to v(k)(r, x; θ2) = k 1 2 θ2prf(x)− γ1k −1 ∫ r 0 pr−l(x) [ v(k)(l, 0; θ2) ]2 dl, u(k)(·, ·; θ1, θ2) is the non-negative solution to u(k)(r, x; θ1, θ2) = pr [ k 1 2 θ1f + v(k)(t2 − t1, ·; θ2) ] (x) − γ1k −1 ∫ r 0 pr−l(x) [ u(k)(l, 0; θ1, θ2) ]2 dl, ω(k)(·, ·; θ2) is the non-negative solution to ω(k)(r, x; θ2) = ∫ r 0 pr−l [ v(k)(l, ·; θ2) ] (x)dl − γ2k −1 ∫ r 0 pr−l(x) [ ω(k)(l, 0; θ2) ]2 dl, 290 xu yang:central limit theorems for the single point catalytic... and s(k)(·, ·; θ1, θ2) is the non-negative solution to s(k)(r, x; θ1, θ2) = pr [ ω(k)(t2 − t1, ·; θ2) ] (x) + ∫ r 0 pr−l [ u(k)(l, ·; θ1, θ2) ] (x)dl −γ2k−1 ∫ r 0 pr−l(x) [ s(k)(l, 0; θ1, θ2) ]2 dl. it is easy to show that k−1/2v(k), k−1/2u(k), k−1/2ω(k) and k−1/2s(k) are all convergent as k → ∞. then lim k→∞ logp0,µ,η exp { − θ1⟨y (k) t1 , f⟩ − θ2⟨y (k) t2 , f⟩ } =γ1θ 2 1 ∫ t1 0 ⟨µ, pt1−r(·)⟩ [ prf(0) ]2 dr + γ1θ 2 2 ∫ t2 0 ⟨µ, pt2−r(·)⟩ [ prf(0) ]2 dr + 2γ1θ1θ2 ∫ t1 0 ⟨µ, pt1−r(·)⟩prf(0)pt2−t1+rf(0)dr + γ1θ 2 1 ∫ t1 0 dl ∫ l 0 ⟨η, pt1−r(·)⟩ [ prf(0) ]2 dr + γ1θ 2 2 ∫ t2 0 dl ∫ l 0 ⟨η, pt2−r(·)⟩ [ prf(0) ]2 dr + 2γ1θ1θ2 ∫ t1 0 dl ∫ l 0 ⟨η, pt1−r(·)⟩prf(0)pt2−t1+rf(0)dr + γ2θ 2 1 ∫ t1 0 ⟨η, pt1−r(·)⟩ [ rprf(0) ]2 dr + γ2θ 2 2 ∫ t2 0 ⟨η, pt2−r(·)⟩ [ rprf(0) ]2 dr + 2γ2θ1θ2 ∫ t1 0 dl ∫ l 0 ⟨η, pt1−r(·)⟩r(t2 − t1 + r)prf(0)pt2−t1+rf(0)dr. recalling (9), we can obtain the result by the method of [12] page 110. 2 in the following, for fixed interval i := [0, t ], t > 0, we introduce the banach space ci p (r) of all continuous maps u of i into cp(r) equipped with the norm ∥u∥ip := sup{9u(t)9p : t ∈ i}. the proofs of the following lemmas ??, ?? and ?? are essentially similar to those of [1, lemma 2.5.2], [1, lemma 2.6.2] and [1, lemma 3.2.1], so we only present the results and omit the proofs here: lemma 2.2 there are two positive constants ε1, ε2, such that for |θ| < ε1, there is a unique solution u = uθ ∈ ci p (r) to the following equation u(t, x) = θptf(x)− ∫ t 0 pt−r(x) [u(r, 0)] 2 dr, 0 ≤ t ≤ t, x ∈ r satisfying ∥u∥ip < ε2. advances in systems science and applications (2012) vol.12 no.3 291 set v(t, x) := θptf(x)− u(t, x), 0 ≤ t ≤ t, x ∈ r, |θ| < ε1. let v(n) denote the nth derivative of v with respect to θ, taken at θ = 0. put ∥sf∥t := sup { |ptf(0)| : t ∈ (0, t ] } , f ∈ cp(r), t > 0. lemma 2.3 for each f ∈ cp(r), 0 ≤ t ≤ t, n ≥ 2, v(2)(t, x) = 2 ∫ t 0 pt−r(x) [prf(0)] 2 dr, x ∈ r and there is a constant cn such that 9v(n)(t)9p ≤ n!cn∥sf∥ntα(t)n−1, where α(t) = t 1 2 + t p+1 2 . lemma 2.4 for each k ≥ 1, n ≥ 2, f ∈ cp(r), p0,µ,η [∣∣∣⟨y (k) t , f⟩ ∣∣∣2] = 2 ∫ t 0 ⟨µ, pt−s(·)⟩(s2 + 1) [psf(0)] 2 ds +2 ∫ t 0 ds ∫ s 0 ⟨η, pt−r(·)⟩ [prf(0)] 2 dr, and there exists a constant cn such that∣∣∣p0,µ,η [ ⟨y (k) t , f⟩ ]n∣∣∣ ≤ cnt n 4 ∥sf∥n1 n−1∑ i=1 ( 9 µ 9p + 9 η 9p )i . by the proofs of proposition ?? and [1, lemma 3.2.1], for all t2 > t1 > 0 and φ,ψ ∈ cp(r) we have p0,µ,η [ ⟨y (k) t1 , φ⟩+ ⟨y (k) t2 , ψ⟩ ] = 0 (10) and (−1)np0,µ,η [ ⟨y (k) t1 , φ⟩+ ⟨y (k) t2 , ψ⟩ ]n =⟨kµ, ū(n)k (t1, ·, φ, ψ)⟩+ ⟨kη, s̄(n)k (t1, ·, φ, ψ)⟩+ ∑ 2≤j≤n−2 ( n− 1 j )[ ⟨kµ, ū(n−j) k (t1, ·, φ, ψ)⟩+ ⟨kη, s̄(n−j) k (t1, ·, φ, ψ)⟩ ] (−1)jp0,µ,η [ ⟨y (k) t1 , φ⟩+ ⟨y (k) t2 , ψ⟩ ]j , (11) 292 xu yang:central limit theorems for the single point catalytic... where n ≥ 2, v̄k(·, ·, ψ) is the non-negative solution to v̄k(r, x, ψ) = k− 1 2prψ(x)− ∫ r 0 pr−l(x) [ v̄k(l, 0, ψ) ]2 dl, ūk(·, ·, φ, ψ) is the non-negative solution to ūk(r, x, φ, ψ) = pr [ k− 1 2φ+ v̄k(t2 − t1, ·, ψ) ] (x)− ∫ r 0 pr−l(x) [ ūk(l, 0, φ, ψ) ]2 dl, ω̄k(·, ·, ψ) is the non-negative solution to ω̄k(r, x, ψ) = ∫ r 0 pr−l [ v̄k(l, ·, ψ) ] (x)dl − ∫ r 0 pr−l(x) [ ω̄k(l, 0, ψ) ]2 dl, and s̄k(·, ·, φ, ψ) is the non-negative solution to s̄k(r, x, φ, ψ) = pr [ ω̄k(t2 − t1, ·, ψ) ] (x) + ∫ r 0 pr−l [ ūk(l, ·, φ, ψ) ] (x)dl − ∫ r 0 pr−l(x) [ s̄k(l, 0, φ, ψ) ]2 dl. then by (10), (11) and calculations, we get the following estimate. lemma 2.5 there exists a positive constant c0 such that p0,µ,η [ ⟨y (k) t , phf⟩ − ⟨y (k) t+h, f⟩ ]6 ≤ c0h 3 2 ∥sf∥61, 0 ≤ t < t+h ≤ 1, k ≥ 1, f ∈ s (r). similarly to [1, lemma 3.2.2], we have the sixth moment of the increments of the process {y (k) t : t ≥ 0}: lemma 2.6 there is a positive constant c0 such that p0,µ,η [ ⟨y (k) t − y (k) s , f⟩ ]6 ≤ c0(t− s) 3 2 , 0 ≤ s < t ≤ 1, k ≥ 1, f ∈ s+(r). proof. applying the elementary inequality |x+ y|n ≤ 2n−1(|x|n + |y|n), x, y ∈ r, n ≥ 0, there are positive constants d1, d2, d3 such that p0,µ,η [ ⟨y (k) t − y (k) s , f⟩ ]6 ≤ d1 ( p0,µ,η [ ⟨y (k) s , pt−sf − f⟩ ]6 +p0,µ,η [ ⟨y (k) s pt−s − y (k) t , f⟩ ]6) ≤ d2 ( ∥s(pt−sf − f)∥61 + (t− s) 3 2 ∥sf∥61 ) ≤ d3 ( ∥pt−sf − f∥6∞ + (t− s) 3 2 ∥f∥6∞ ) advances in systems science and applications (2012) vol.12 no.3 293 by lemmas ?? and ??. since phf−f h −→ ς 2f ′′ = af as h → 0, h 7→ phf−f h is strongly continuous on h ∈ [0, 1]. this implies that there exists a positive constant d4 such that ∥phf − f∥∞ ≤ hd4 for h ∈ [0, 1]. let c0 = d3(d 6 4 + ∥f∥6∞). we finish the proof. 2 lemma ?? and kolmogorov’s criterion lead to lemma 2.7 for any f ∈ s (r), the sequence {⟨y (k) t , f⟩ : t ≥ 0; k ≥ 1} is tight in c([0,∞),r). proof of theorem ??. by lemma ?? and proposition ??, the result follows from [13, theorem 6.15]. 2 proposition 2.8 for each k ≥ 1 and f ∈ cp(r), m (k) t (f) := ⟨y (k) t , f⟩− ∫ t 0 ⟨y (k) s , af⟩ds− ∫ t 0 ( ⟨ξ(k)r , k 1 2 f⟩ − ⟨η, prk 1 2 f⟩ ) dr, t ≥ 0 is a continuous martingale. proof. fix k ≥ 1. by the markov property of {x(k) t : t ≥ 0}, for t > s ≥ 0, p0,µ,η ( ⟨x(k) t , f⟩ − ∫ t s ⟨x(k) r , af⟩dr ∣∣fs ) =⟨x(k) s , pt−sf⟩+ (t− s)⟨ξ(k)s , pt−sf⟩ − ∫ t s ( ⟨x(k) s , pr−saf⟩+ (r − s)⟨ξ(k)s , pr−saf⟩ ) dr. then by [14, proposition 1.5], we have p0,µ,η ( ⟨x(k) t , f⟩ − ∫ t 0 ⟨x(k) r , af⟩dr ∣∣fs ) =⟨x(k) s , f⟩ − ∫ s 0 ⟨x(k) r , af⟩dr + ∫ t−s 0 ⟨ξ(k)s , prf⟩dr and ⟨µ, ptf⟩+ t⟨η, ptf⟩ − ∫ t 0 ( ⟨µ, praf⟩+ r⟨η, praf⟩ ) dr = ⟨µ, f⟩+ ∫ t 0 ⟨η, prf⟩dr. recalling y (k) t = k 1 2 (x (k) t − µpt − tηpt), we have p0,µ,η [ m (k) t (f) ∣∣fs ] =m (k) s (f). 2 294 xu yang:central limit theorems for the single point catalytic... lemma 2.9 there is a locally bounded function t→ c(t) on [0,∞) such that sup k≥1 p0,µ,η { sup 0≤t≤t ∣∣∣⟨y (k) t , f⟩ ∣∣∣2} ≤ c(t )∥f∥22, t ≥ 0, f ∈ h2(r). proof. we only consider f ∈ s (r). for each k ≥ 1, p0,µ,η { sup 0≤t≤t ∣∣∣⟨y (k) t , f⟩ ∣∣∣2} ≤ 3 p0,µ,η { sup 0≤t≤t m (k) t (f)2 } + 3 p0,µ,η {[∫ t 0 ∣∣∣⟨y (k) s , af⟩ ∣∣∣ ds]2} +3 p0,η {[∫ t 0 ∣∣∣⟨ξ(k)s , k 1 2 f⟩ − ⟨η, psk 1 2 f⟩ ∣∣∣ ds]2} . using hölder inequality, fubini theorem and (5), we have p0,η {[∫ t 0 ∣∣∣⟨ξ(k)s , k 1 2 f⟩ − ⟨η, psk 1 2 f⟩ ∣∣∣ ds]2} ≤2 t ∫ t 0 ds ∫ s 0 ⟨η, ps−r(·)⟩ (prf(0)) 2 dr. by proposition ??, m (k) t (f) is a continuous martingale. then by doob’s martingale inequality, hölder inequality and fubini theorem, we have p0,µ,η { sup 0≤t≤t m (k) t (f)2 } ≤ 12 p0,µ,η {∣∣∣⟨y (k) t , f⟩ ∣∣∣2}+ 12 t {∫ t 0 p0,µ,η [∣∣∣⟨y (k) s , af⟩ ∣∣∣2] ds} +24 t ∫ t 0 ds ∫ s 0 ⟨η, ps−r(·)⟩ (prf(0)) 2 dr. by lemma ??, there is a constant c such that [prf(0)] 2 ≤ 1√ 2πςr q0(f) 2 ≤ c√ 2πςr ∥f∥20. it is easy to show that ⟨µ, ps−r(·)⟩ ≤ 1√ 2πς(s−r) (9µ 9p +µ([− √ 2pπςs, √ 2pπςs]) ) . then by lemmas ?? and ??, there is a locally bounded function t → c1(t) on advances in systems science and applications (2012) vol.12 no.3 295 [0,∞) such that p0,µ,η [∣∣∣⟨y (k) s , f⟩ ∣∣∣2] ≤ c2(s)∥f∥20 and p0,µ,η [∣∣∣⟨y (k) s , af⟩ ∣∣∣2] ≤ c2(s)∥f∥22 for any all s ≥ 0. this finishes the proof. 2 proof of theorem ??. using [13, corollary 6.16], the result follows from proposition ??, lemmas ?? and ??. 2 acknowledgements mathematics subject classifications (2000):supported by nsfc (no.11071021),primary 60j80; secondary 60g20, 60j68. references [1] dawson, d.a, fleischmann, k. 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[8] li z. h, shiga t. (1995), “measure-valued branching diffusions: immigraiton, excursions and limit theorems”, math. kyoto univ, vol.35, pp.233274. [9] zhang, m. (2004), “large deviation for super-brownian motion with immigration”, j. appl. prob, vol.41, pp.187-201. [10] zhang, m. (2005), “functional central limit theorem for the super-brownian motion with super-brownian immigration”, j. theoret. probab, vol.18, pp.665-685. 296 xu yang:central limit theorems for the single point catalytic... [11] li, z.h. (2011), measure-valued branching markov process, springer, berlin. [12] iscoe i. (1986), “a weighted occupation time for a class of measure-valued critical branching brownian motion”, probab.th. rel. fields, vol.71, pp.85116. [13] walsh, j.b. (1986), “an introduction to stochastic partial differential equations”, in: ecole d’eté de probabilités de saint-flour xiv -1984, pp.265-439, lecture notes math, pp.1180, springer, berlin. [14] ethier, s.n, kurtz, t.g. (1986), markov processes: characterization and convergenc, wiley, new york. corresponding author xu yang can be contacted at:xuyang@mail.bnu.edu.cn мягкая вероятность ошибки при передаче информации advances in systems science and application(2016) vol.16 no.4 62-70 method of parametric structural synthesis of the machine-assembly departments anatoliy serduk, aleksandr sergeev, aleksandr rusyaev, sergey kamenev orenburg state university, orenburg, russia annotation reconstruction and technical re-equipment of enterprises is preferable to the creation of a similar capacity as a result of new construction. stringent requirements apply to projects of modernization of production, design methods used, duration and quality of the project works. traditional methods of designing high-tech assembly areas, based on average calculations are inefficient and weakly correlated with the automated production process. important research task is to develop a method of automated design of assembly areas, based on computer modeling of technological processes and automated synthesis of structure and parameters of the production equipment. the article describes the formalized representation of the production process machining industries, including operations of separation and assembly. developed algorithmic, mathematical and information support of the process simulation of the machining industry, including operations of separation and assembly. implemented the lexicographic algorithm generates the best options for production jobs. these options are processed statistically. developed a method of structural-parametric synthesis of machining industries. the method takes into account changes in production item due to the simulation of a sample optimal replacement jobs that have the highest probability. the use of the developed method will ensure the optimality of the design solutions according to the criteria of efficiency of functioning of the machining industries. keywords: machine-assembly plant, structural-parametric synthesis, lexicographic algorithm, simulation plant 1 introduction a reorganization of existing enterprises is most efficient way to adjust for challenging market environment. return of the reorganization investment is estimated to be 3 times faster comparing to establishment of similar facilities. in the conditions of the formation of a new technological order the tasks of the reconstruction and re-equipment are important for vast majority of machine-building enterprises in the country especially in adopting of new product ramping. highly competitive product markets require very efficient organization of manufacturing facilities saving material and human resources. level of manufacturing automation is most crucial to achieve production efficiency and, therefore, production of the competitive products. in this regard, modernization of manufacturing projects, design styles, timing and quality of a design work are subjected to very demanding requirements. the traditional averaged calculations methods that used for design of the high-tech machineassembly plant (map) are ineffective with low correlation to the automated manufacturing process. for this reason computer modeling of manufacturing processes is necessary to improve the methods of a computer-aided design of the map. on the basis of this, the improvement of computer-aided design of the map based approaches is important scientific challenge that would provide great benefit to the country advances in systems science and application(2016) vol.16 no.4 63 economy. solution of the specified problem is possible with development of computer modeling of manufacturing processes that provides the automated structural-parametric synthesis of the map. the developed method will provide the optimality of accepted design solutions according to criteria of operational efficiency of the machine-assembly lines. 2 review of works on the subject of research a great contribution to the theory of design and optimization of manufacturing processes in the machining production made the researches of [1]-[8] and other authors. the successful solutions for design and optimization of manufacturing processes are offered by the companies who develops the special software such as the dassault systems, siemens plm software, autodesk, anylogic company, mstu «stankin», sprut-technology. in accordance to numerous analyses in the field of a machining production design, systems of operating scheduling, systems of simulation techniques which implemented on the machinebuilding enterprises the design-automation systems cannot be developed without connection with the operating scheduling regardless of production seriality. therefore the leading world companies provide the set of solutions for production design including the package to compose production schedules which are used in the maintenance process of created system. the domestic developments in the field of operating scheduling such as «fobos», «sprut-okp» aren’t the tools for manufacturing processes modeling that makes difficulties for getting of statistics on the production assortment. the solutions offered by foreign vendors which are leaders in the field of modeling (tecnomatix, delmia) has a wide set of tools but their manufacturing application for domestic enterprises involves the additional costs on adaptation to production standards, purchasing of unused modules and the development of specific subsystems. therefore, the development of automated system for design of the machine-assembly departments that include the module for composition of production schedules has a great importance. the mathematical model was based on the method for automated construction of sequence diagrams which was developed in orenburg state university and described at the works of [5] [9-12]. 3 simulation algorithm for the working process of the map that takes into account the separation and assembly operations the technological process of manufacturing of the product named «drill pipe» was used as model object for development of the product structure. this technological process contains the following operations: sawcut the pipe in a predetermined size, cutoff the workpieces for manufacturing of the nipple and coupling, machining of the pipe, nipple and coupling, welding the nipple and coupling with the pipe, machining of the drill pipe. the product structure in the form of a bi-directional graph (fig. 1) is proposed to account partitioning and assemblage operations. on this figure left to right shown the direction of workpieces flow and right to left – the information about workpieces required for pipe assembly. in the block specified the nodes which are connected with graph according to the technological process of manufacturing and composition of the product. at the first blocks the nodes wherein specified the composition of the product are equal to zero, at the last blocks – on the place of node that indicate the following manufacturing stage, assigned the value «–1», which fix the machining completion. the operations of partitioning and assemblage of the products introduce significant changes to the work of modeling subsystem. thus, for example, without the data of operations the user by itself calculates and sets the number of workpieces that start up to manufacturing. in our 64 anatoliy serduk, aleksandr sergeev, aleksandr rusyaev, et al: method of parametric structural synthesis of the machine... modeling case it is necessary to automatically provide the compliance between the number of assembling products, number of assembly units and amount of a raw stock for workpieces production. number of machined workpieces in a batch start up depends on whether the product is unit or workpiece and is described by the expression:     .. .. , , prodprod prodprod vfromtakenzifzdevedv zincludevifzv z (1) where vprod. – number of products to be produced, pcs.; z – number of assembly units with the same name belonging to the product or number of workpieces produced from raw stock, pcs. fig. 1 bi-directional graph of the product development with account the blanking and assembling operations the amount of a raw stock (pcs.) required for manufacturing of given number of workpieces can be determined by expression:      residuewithoutzondividedvifzv residuewithoutzondividedtdonvifzv z prodprod prod .. . ,div ',1div (2) thus, this expression takes into account the situation where, for example, the one bar can be sawn to 8 workpieces for nipple. but if required to make ten such workpieces then it is necessary to put the two bars on sawing operation. for the selection of an acceptable variant of the work order (wo) it is necessary to analyze the product composition and distribute the productive time of assembly units (au) between the equipment groups: , 100 ∑, iji i i ft fdf f    at i=1.. g, j=1..n, (3) where fi – fund of productive time of the equipment group i, min; тi,j∑ – total productive time of the workpieces machining in composition of product j on the equipment group i, min; g – number of the equipment groups, pcs; n – assortment of the delivered products, pcs. then the total productive time of the workpieces machining in composition of product j on advances in systems science and application(2016) vol.16 no.4 65 the equipment group i determines by expression: jji prod n j prodji vтt . 1 .∑, ∑ ,   , (4) where тprod.i,j – productive time of product j on the equipment group i with account of product composition (min), determined by equation: ∑ 1 ,,, kol k jauprod zтт kjiji   , (5) where тau i,j,k – productive time of assembly unit k of product j on the equipment group i, min; kol – number of au in product j, pcs. the developed structure for describing of the products allows keeping and using the information about assembly units and raw stock in the modeling process. based on equations above the algorithm for modeling of manufacturing process was developed (fig. 2), the corresponding software was accomplished and estimation of adequacy was performed. the input data are the capabilities of primary and auxiliary equipment and geometrical position of process modules. the output data are the actual duration of work cycle, coefficient of equipment utilization, productivity, utilization by the equipment groups and machine tools, sequence diagram for work of map. fig.2 simulation algorithm for the working process of the map that takes into account the separation and assembly operations 66 anatoliy serduk, aleksandr sergeev, aleksandr rusyaev, et al: method of parametric structural synthesis of the machine... 4 synthesis algorithm for operating schedule based on the use of the truncated lexicographic algorithm in the conditions of multiproduct manufacture the multiple passes of manufacturing process on the different variants of wo for collection of statistical data about the productive efficiency, in the dependence of content of the batch run are required. as it is well known, the brute force approach of all possible variants of the batch run has a factorial dependency and cannot be used for real life problems even on the modern highly productive computers. the formation of work orders relates to problems in combinatorial optimization and may be reduced to statement of knapsack problem. let exists the set of items consisting of the m names and l parameters. then the characteristics of items will defined the by vector ][ia  , which determines the values of each parameter of the item: ],...,,[][ ,2,1, liii aaaia   , where  mi ,...,2,1∈ . the number of items of the each name n[i] may be the any integer in the range  maxm0,1,...,∈ n[i] , where mmax – maximum number of the identical items. within a predetermined range of constraints  ][],...,2[],1[ minminminmin lssss   and  ][],...,2[],1[ maxmaxmaxmax lssss   it is required to find the combinations of packing which meets the specified constraints. in the mathematical statement the conditions of possibility of the packing for given combination of items are described by the following expressions:              ][≤≤][ ... ]2[≤≤]2[ ]1[≤≤]1[ max 1 min max2 1 min max1 1 min ∑ ∑ ∑ lsnals snas snas mlm m i mm m i mm m i (6) when the condition (6) meets the constraints for combination k then number of items in the knapsack ][∑ kn will be ∑ 1 ∑ ][][ m i inkn   . subject to the introduced mathematical support the algorithm for generation of the work orders variants based on the branch and bound method was developed if in a production flow of the processing of the parts from different groups without their interleaving is found a situation when the some machine tools are not active due to the specific of a manufacturing cycle then it may be concluded that the given situation involves the abrupt reduction of the productive efficiency. therefore, attempt to optimize the operating schedule through interleaving the parts of various groups in a sequence can be engaged. solution of this problem was realized with truncated lexicographic algorithm (fig. 3). the number of steps in this algorithm was reduced at the expense of the selection from the sum of all rearrangements of the sequence run the rearrangements with the predefined interval only. for statistical treatment of the modeling results for sampling of the wo the advanced algorithm was developed that allows to receive the frequency histograms and cumulative probability plots for the various factors of map efficiency. comparison of modeling results of the work orders with and without optimization reveals higher efficiency for optimization with the sequence run. to account the sensibility of the map to alterations of the manufactured product assortment the technique for selection of representative batches based on the estimated probability of higher efficiency of the manufacturing system was realized (fig. 4). selection of the representative batches that allowed to use in the developed computer model of the map in the algorithm of advances in systems science and application(2016) vol.16 no.4 67 structural-parametric synthesis of the production equipment is subject to the wide assortment of produced objects. fig.3 synthesis algorithm for operating schedule based on the use of the truncated lexicographic algorithm fig.4 statistical analysis of the work orders 68 anatoliy serduk, aleksandr sergeev, aleksandr rusyaev, et al: method of parametric structural synthesis of the machine... 5 the development of the method of structural-parametric synthesis for machine-assembly plants the next stage is the development of the method of structural-parametric synthesis for machineassembly plants. the introduced method uses the three strategies based on using of genetic algorithms, hooke-jeeves method and alternating-variable descent method. in the process of computing experiments the structural-parametric synthesis method is found to be most effective if based on the genetic algorithms. it may be explained by the fact that the two other methods use the same principle of optimization, i.e. here it matches the optimal value of the one parameter when the other parameters are constant. the advantage of the genetic algorithm is the variations of all synthesized values on every iteration step that as the result allows to get the uniform set of parameters that provide the maximum efficiency of the machineassembly departments. the other distinctive feature of the algorithm is that algorithm during the execution takes into account all used efficiency indexes: usage and duration of the actual equipment work. this approach allows to avoid the procedure of criteria convolution so as to proceed for the complex efficiency index. meanwhile, the parameters selection is performed with regard to extreme value from all efficiency indices. for solution integrity in the problem of structural-parametric synthesis of the machineassembly departments the algorithm for synthesis of coordinate arrangement of equipment was introduced. in the capacity of synthesis method the genetic algorithm based on the chromosome conception wherein the values of genes not modified but their locations are changing is implemented. this algorithm was realized in the software «komponovka 1.3». on the basis of computer model of the manufacturing system the method of structuralparametric synthesis of the machine-assembly departments was developed (fig. 5) that take into account the partitioning and assemblage operations of the produced objects. this method takes into account the change of manufacturing assortment at the expense of modeling the sampling of optimal wo which has a maximum probability of production starting. fig. 5 method of structural-parametric synthesis of the map on the framework of introduced algorithms the program «plan3d» was developed which is possible to transfer the generated model of map to cad-system kompas and automatically advances in systems science and application(2016) vol.16 no.4 69 construct the layouts of designed machine-assemble department in 2d and 3d formats. on the layout the table with the selected parameters of manufacturing equipment is also displayed (fig. 6). fig. 6 3d model of equipment arrangement for machine-assembly department the design data that formed as a result of automated structural-parametric synthesis are easy integrated into uniform information space of enterprise and allow to make changes via tools of cad-system kompas. 5 conclusions in this work it was determined that the modern approach to design the machine-assemble departments is a detailed analysis of the design solutions related to selection of structure and parameters of manufacturing equipment via computer modeling of map with the running on multiple manufacturing processes. the formalized presentation for functioning process of the machine-assemble department was performed. this presentation which describes the dynamics of assembly units manufacturing is based on the diagrammatic work method and may be used for most categories of the machinebuilding products. the algorithm for synthesis of optimal operating schedules which allows reducing a turnaround time by 10-15% was developed. the method for designing of machine-assembly departments which distinctive feature is the usage of operating scheduling procedures and computer modeling of the map functioning was developed. it was proved that the inclusion the procedures of an operating scheduling into method for design allows to increase the map efficiency on 23%. the program realization of a developed algorithms and procedures was designed, tested and approved. the developed software allows to form the complex of design solutions realized as 3d model of the equipment arrangement and nomenclature of the equipment parameters on the base of computer simulation of functioning and analysis of structure and parameters of the map. 70 anatoliy serduk, aleksandr sergeev, aleksandr rusyaev, et al: method of parametric structural synthesis of the machine... references [1] g. n. melnikov and v. p. voronenko(1990), designing of the machine-assembly departments, machine building, moscow,. 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"structuralparametric synthesis of the flexible manufacturing systems with application of the genetic algorithms", the study, orenburg,. corresponding author anatoliy serduk can be contacted at: yal05@mail.ru. advances in systems science and applications (2014), vol.14, no.2 158-169 teaching chaos with a pendulum to greek secondary school students constantine skordoulis, vasilis tolias, dimitris stavrou, kostas karamanos and aristotelis gkiolmas department of education, national and kapodistrian university of athens, athens, greece abstract in this paper we report on how a commercial “chaos pendulum system” was used to teach aspects of chaos theory to greek upper secondary school students. the principal aim of this project was to investigate to what extent students can develop an understanding of the chaotic behaviour using representations in the phase space. the didactical methodology used was that of the “physics workshop” where students follow a discovery type laboratory course with detailed lab worksheets and are guided to the study of chaotic motion starting from observations of the oscillation of a simple physical pendulum. interesting conclusions concerning students’ ability in using the technology associated with the microcomputer based laboratory (mbl), the development of a qualitative understanding of the chaotic behaviour using the representation in the phase space and in interpreting graphs can be drawn from the analysis of their recorded answers. 1 why teaching chaos? the study of non-linear systems has become a major area of research in science as well as in philosophy. this research has led to a shift in the emphasis concerning fundamental scientific c philosophical concepts such as ‘determinism’, ‘predictability’, ‘causality’, and ‘chance’. there are systems, the chaotic pendulum is one of them, which are very ‘sensitive’ to small changes in the initial conditions when the process is running. subtle changes of the initial parameters lead to a large deviation in the path followed so that detailed prediction of the system’s behavior is not possible. despite the irregular behavior of the system in the representation of the movement in the phase space a characteristic structure emerges indicating a kind of order. a number of non-linear systems develop complex self-organizing structures. self-organizing structures as well as structures that appears in the phase space could be described by the model of fractals. in this sense, chaos, self-organisation and fractals are intimately linked. from an educational point of view this new domain has an important contribution to scientific literacy by developing a more adequate worldview and a more advances in systems science and applications (2014), vol.14, no.2 159 relevant view concerning the nature of science[1] . more specifically, teaching nonlinearity to students makes them understand that the mechanistic worldview has been challenged, the physical world is not governed by simple deterministic laws leading to complete predictability, that causality and chance are an integral part of physical reality. a considerable amount of research work in science education has been done on issues of non-linear systems in the last years[1-4] . nevertheless there is a deficiency in empirical investigations concerning the possibility to teach and learn of aspects of the chaos theory using representations in the phase space. the principal aim of this research is to investigate to what extent students can develop an understanding of the chaotic behavior using representations in the phase space. for this task our approach adopts the mbl methodology, which allows a direct connection between the real experiment and the abstract graphical representation, based on the ‘workshop physics’ project using the pasco chaotic pendulum system. in the following paragraphs a short description of the mbl methodology, the ‘workshop physics’ project and the pasco chaotic pendulum system is given. 2 microcomputer based laboratory c (mbl) methodology the development of the laboratories using mbl-tools has been inspired by the pedagogical approaches applied in ‘realtime physics’[5] and ‘tools for scientific thinking’[6] and by research in physics education[7-9] . for some decades sensors interfaced with a computer have been used in most physics research laboratories. an attachment of a sensor to a computer creates a very powerful system for collection, analysis and display of experimental data. today several systems, specially developed for schools and undergraduate courses, are commercially available. in a microcomputer based laboratory, students do real experiments using different sensors connected to a computer via an interface. experimental data from real-world experiment are immediately presented on the computer screen in a graphical format and can immediately be analysed. one of the main educational advantages of using mbl is the real-time display of experimental results and graphs thus facilitating direct connection between the real experiment and the abstract representation. because data are quickly taken and displayed, students can easily examine the consequences of a large number of changes in experimental conditions during a short period of time. brasell (1987)[10] has shown that even a very short delay in the presentation of experimental data is detrimental for student comprehension. 160 constantine skordoulis:teaching chaos with a pendulum to greek secondary school students 3 the ‘workshop physics’ course the ‘workshop physics’ course has developed since 1986 at dickinson college by p. laws (2004) and is based on mbl methodology. the major objective of workshop physics courses is to help students understand the basis of knowledge in physics as an interplay between observations, experiments, definitions and mathematical description. the main idea of the course is to engage students in laboratory activities without any previous lecturing. in this discovery type laboratory course the students gain real-world experience using hands-on experiments. students following “workshop physics” courses learn to use computer tools to collect data and to develop mathematical models of phenomena. the workshop physics curriculum is centred on a workbook-style activity guide, that consists of twenty-eight units[11] . these units are based on activities that include predictions, qualitative observations, explanations, equation derivations, mathematical model building, quantitative experiments and problem solving. students in our sample class did activities selected from unit 15 on “oscillations, determinism and chaos”. this unit includes a series of observations and experiments on a physical pendulum system that becomes increasingly complex and eventually appears to undergo chaotic behaviour. the major advantage of the workshop physics for us is the ‘step by step’ approach used in unit 15 where, starting from observations of the motion of the physical pendulum, students can gradually built up their knowledge up to a point of understanding chaotic behaviour. 4 the chaotic pendulum system the chaotic pendulum system by pasco is one of the chaotic pendulums that are commercially available. a detailed comparison of the chaotic pendulums available has been published[12]. the pasco chaos pendulum can be used in two different configurations: a horizontal and a vertical. both configurations have been analysed and modelled in the relevant literature. we have used the horizontal configuration, shown in fig.1, a brief description of which follows in the next paragraphs. the oscillator consists of an aluminum disk connected to two springs. a mass attached on the edge of the aluminum disk makes the oscillator non-linear. the frequency of the sinusoidal driver can be varied to investigate the progression from predictable motion to chaotic motion. magnetic damping can be also adjusted to change the character of the chaotic motion. the angular position and velocity of the disk are recorded as a function of time using a rotary motion sensor connected to a computer via an interface. data are recorded and displayed advances in systems science and applications (2014), vol.14, no.2 161 fig.1 the pasco chaos pendulum with the horizontal configuration using the data studio software. the chaotic behaviour of the driven non-linear pendulum is explored by graphing its motion in phase space and by making a poincare plot. these plots are compared to the motion of the pendulum when it is not chaotic. a real time phase space diagram is made by graphing the angular velocity versus the displacement angle of the oscillation, while the poincare plot is also graphed in real time and can be superimposed on the phase plot. this is achieved by recording the point on the phase plot once every cycle of the driver arm as the driver arm blocks a photogate. the variables of the experiment (driving frequency, displacement amplitude and magnetic damping) can be varied so as to cause the regular motion to become chaotic.students, using the data studio software, can view the following graphs: 1. angular displacement (θ) vs. time (t) 2. angular velocity (ω) vs. time (t) 3. phase space: angular velocity (ω) vs. angular displacement (θ) 4. poincare plot: angular velocity (ω) vs. angular displacement (θ) plotted only once per period of the driving force. the phase space and the poincare plot are particularly useful for recognizing chaotic oscillations. when the motion is chaotic, the graphs do not repeat. 5 didactical activities our sample class consists of six gifted final year secondary school students, with high scores in maths and physics, selected from a private school based in athens and worked in two teams of three persons during three lab sessions of three hours each (a total of nine hours of lab instruction) in asel where the apparatus is based. 162 constantine skordoulis:teaching chaos with a pendulum to greek secondary school students the students had not any previous knowledge either of the concepts or the graphical representations (eg phase space, poincare mapping) associated with chaotic motion and they were not lectured on them before commencing on the lab course. in accordance with the greek upper secondary school science curriculum the students had a fairly good knowledge of simple harmonic motion, rotational dynamics and electromagnetism and so it was rather easy for them to understand the operation of the dc motor driver, the rotational sensor and the interface during a demonstration lab period conducted by the teacher. the guidelines of the activities selected from unit 15 on “oscillations, determinism and chaos” of ‘workshop physics’ were modified to meet the tasks and the aim of the present study. the students had to make observations, discuss the phenomena, give explanations and draw graphics, following the instructions in a worksheet designed by our group to meet the needs of the present research. the performance of the students, both individually and as a team, has been evaluated from the questions completed in the laboratory worksheets and from the discussions among them and with the teacher. during the lab period the teacher was giving the necessary information concerning the operation of the devices, analyzing the instructions for doing the experiment when necessary, and intervening only when the discussion among students had reached a dead end. below a detailed description of the procedure followed is given highlighting the ‘step by step’ approach and the students’ performance in every step. activity one: observation of the oscillation of the physical pendulum. in the first activity students made a series of observations, at different oscillation angles, of the motion of the aluminium disk mounted on the low friction bearing with and without various quantities of mass attached to its edge. this disk was part of the physical pendulum system. the students keep a record of their observations and discuss about the oscillation of the physical pendulum (fig.2). activity two: time series and phase space plot of the physical pendulum. next the rotary motion sensor was attached to a universal laboratory interface. the students using the data studio software could create a plot of the angular displacement of the pendulum as a function of time in real time (time series plot) as well as a plot of the phase space (angular velocity as a function of angular displacement phase plot). the students observe the time series graph, the phase plot graph and the experiment in real time. it has to be emphasised that the students had no difficulty in understanding advances in systems science and applications (2014), vol.14, no.2 163 fig.2 observation of the oscillation of the physical pendulum the phase space graph. they started from the graph of the time series, which is very familiar to them and they can easily connect it with the motion of the disk (experiment in the real world). then they correspond specific points of the time series graph to the graph of the phase space (see fig.3) and connect these with the oscillation in the real experiment. fig. 3 time series and phase space plot of the physical pendulum activity three: determination of the natural frequencies of the system. students were asked to modify the physical pendulum so that a string with 164 constantine skordoulis:teaching chaos with a pendulum to greek secondary school students the two springs attached and the driver motor are coupled to the aluminium disk with the edge mass in place (see fig.4). fig.4 determination of the natural frequencies of the system then with the motor turned off the students explored the natural frequencies of the system with and without the spring coupling and with and without the edge mass in place. the students using data studio software create the time series graph ω(t) and take a direct reading of the natural frequency of the system in the various cases. then students activate the driver motor and they set the driver frequency very close to the natural frequencies of the physical pendulum system thus creating a situation “near resonance”. the shape of the phase space graph is elliptic (see fig.5). fig.5 the shape of the phase space graph “at resonance” the amplitude in the time series graph remains constant (within experimental error) while the shape of the graph in the phase space is a circle. the students had some difficulty in understanding the shape of the phase space graph and the teacher had to talk about the phase difference between θ(t) and advances in systems science and applications (2014), vol.14, no.2 165 ω(t) in the corresponding time series graphs at resonance. at this point the poincare plot was introduced. a photogate was placed in the driver arm so as the plot can be produced. the students could not understand the physical significance and could not make any connection with the actual experiment despite the efforts of the teacher. activity four: exploring conditions for chaotic motion-sensitivity to initial conditions students were asked to place the damping magnet fairly close to the metal pendulum disk, activate the driver motion and set the driver frequency in a position that is different than any of the natural frequencies of the system and see if they can achieve a situation in which there is an irregular pattern in the time series graph (angular position vs. time). a typical pattern is shown in fig.6 for the time series graph and the corresponding phase space plot. fig.6 the time series graph and the corresponding phase space an important observation in this sample activity was to have each participant attempt to repeat a pattern like that shown in fig.6 by setting up the same initial conditions for the pendulum. all attempts failed after only a few seconds of run time. the students realize that this system is so sensitive to the initial values of angular position and angular velocity that it is impossible to recreate the initial conditions accurately enough to repeat a pattern on a graph. in this activity students initially show serious difficulties to explain the final form of the graph in the phase space and connect it directly with the motion of the device. nevertheless, they managed to give a qualitative explanation of the 166 constantine skordoulis:teaching chaos with a pendulum to greek secondary school students shape of the phase space graph by connecting it to the time series graph and then connect the motion of the pendulum with the time series graph. more specifically, students connect the random number of peaks on the waveform in the time series graph with the number of oscillations of the pendulum. in the next step, they relate the random number of peaks on the waveform in the time series graph with the irregular way of shaping the graph in the phase space and by extension to the motion of the pendulum. concerning the poincar plot, students can make use of the experimental apparatus and produce the plot, they can use the shape of the plot as a criterion to identify chaotic motion but they cannot relate it to phase space plot and/or understand the physical meaning of it despite the intervention of the teacher. in order to clarify if and to what extent students had develop an understanding of the phenomena discussed during this laboratory period, the same students have been asked six months later to repeat the course. in this second phase, the main emphasis was on students’ predictions about the process of the experiment and the kind of graphs expected. after the experiment was carried out students had to compare their predictions and their graphs with the graphs produced by the software based on the date collected from the apparatus and discuss the possible agreement or differences. the results of this second phase showed that students were able to make predictions about the behaviour of the system which are very successful, and their graphs in most cases are similar to the graphs drawn by the software. their explanations were also very satisfactory. in the following fig.7 and 8, the graphs drawn by the two groups of students are presented concerning their prediction about phase space plots for the oscillation of the physical pendulum and chaotic motion. fig.7 phase space plots for the oscillation of the physical pendulum and chaoticmotion by group a advances in systems science and applications (2014), vol.14, no.2 167 fig.8 phase space plots for the oscillation of the physical pendulum and chaoticmotion by group b 6 discussion and conclusions the results presented here are based mainly on students’ worksheets and the discussion, the teacher had with the students during and after the lab period. in sum, we can say, that despite the difficulties students have, they can go through these activities and develop up to a point a qualitative understanding of the chaotic behavior using the representation in the phase space. the ‘step by step’ approach, ie starting from observing the motion of the simple physical pendulum and ending with the study of the complex motion of the chaotic pendulum system, in combination with the simultaneous presence of the real experiment and the corresponding graphs facilitate students’ understanding. the students had the opportunity to discuss the graphs of simple motions, which are up to a point familiar to them, and connect them with the real experiments. they connect the time series graph with the phase space graph and by extension with the motion of the pendulum. based on the gained knowledge they could make also connections between the real experiment and the graphs in the case of the chaotic motion and give up to a point explanations of the observed complex behavior and the corresponding graphs. interventions of the teacher are necessary and concern mainly the clarification of the graphs in the phase space. this study gave insights about difficulties and opportunities of teaching a new field of physics using the mbl methodology. the results presented here are stem from a preliminary explorative study. thus they need further elaboration and should be tested in a broader sample. a further investigation based on the results presented here is intended. this work also contributes in the field of pendulum 168 constantine skordoulis:teaching chaos with a pendulum to greek secondary school students studies[13] by revealing aspects of the educational dynamic of the pendulum. apart from using the simple pendulum in cross disciplinary teaching giving students an understanding of the way that science has developed in relation to other forms of cultural life, the physical pendulum can be utilised to promote knowledge about the nature of science where causality and chance are characteristics of an integrated scientific worldview. references [1] komorek, m., stavrou, d., & duit, r. (2003), “non-linear physics in upper physics classes: educational reconstruction as a frame for development and research in a study of teaching and learning basic ideas of nonlinearity. in: d. psillos, p. kariotoglou, v. tselfes, e. hatzikraniotis, g. fassoulopoulos, & m. kallery (eds.)”, science education research in the knowledge based society, pp.269-276. [2] adams, h.m. & russ, j.c. (1992), “chaos in the classroom: exposing gifted elementary school children to chaos and fractals”, journal of science education and technology 1, pp.91-209. [3] chacn, r., batres, y. & cuadros, f.(1992), “teaching deterministic chaos through music”, physics education, vol.27, pp.151-154. [4] duit, r., komorek, m. & wilbers j. (1997), “studies on educational reconstruction of chaos theory”, research in science education. vol.27, pp.339357. [5] sokoloff, d., thornton, r. and laws, p. (1998), “realtime physics”, active learning laboratories. [6] thornton, r. (1989), “tools for scientific thinking: learning physical concepts with real-time laboratory measurements tools in redish, e. and risley, j. (eds), proc. conf”, computers in physics instruction, pp.177-189. [7] thornton, r. (1997), “learning physics concepts in the introductory course: microcomputer-based labs and interactive lecture demonstrations in wilson, j. (ed.)”, conference on the introductory physics course, pp.6986. [8] laws, p.w., (1997a), “a new order for mechanics”, conference on the introductory physics course, pp.125-136. [9] mcdermott, l. (1997), “how research can guide us in improving the introductory course”, proceedings of the conference on introductory physics course, pp.33-45. advances in systems science and applications (2014), vol.14, no.2 169 [10] brasell, h., (1987), “the effect of real-time laboratory graphing on learning graphic representation of distance and velocity”, journal of research in science teaching, vol.24, pp.385. [11] laws, p.w., (1997b), “workshop physics activity guide, new york: wiley laws, p.w., (2004), a unit on oscillations, determinism and chaos”, american journal of physics, vol.72, no.4, pp.446. [12] blackburn, j. a. & baker g. l. (1998), “a comparison of commercial chaotic pendulums”, american journal of physics , vol.66, no.9, pp.821. [13] matthews, m. r., (2001), “how pendulum studies can promote knowledge of the nature of science”, journal of science education and technology, vol.10, no.4, pp.359. [14] komorek, m. & duit, r., (2004), “the teaching experiment as a powerful method to develop and evaluate teaching and learning sequences in the domain of non-linear systems”, international journal of science education, vol.26, pp.619-633. corresponding author constantine skordoulis can be contacted at: kkaraman@ulb.ac.be. advances in systems science and applications (2017) vol.17 no.1 1 the concept of intelligent tutoring for enterprise staff as a component of integrated manufacturing control system development rustem a. sabitov, gulnara s. smirnova 1 , bulat. r. sirazetdinov, natalya. u. elizarova, alexander. v. eponeshnikov kazan national research technical university named after a. n. tupolev — kai, kazan, russian federation abstract. modern industrial manufacturing has high requirements for employees. workers must continually improve their competence. you must constantly test, train and retrain specialists of enterprises. in the light of manufacturing control problems and cognitive issues of identification, verification and other problems for processes of identification, modeling, and control, intelligent tutoring system for specialists training can be considered as an important part of manufacturing intelligent control and planning system. the development and a pilot implementation of adaptive approaches to the universal computer system of individual training, functioning on the basis of subject and meta knowledge bases and meta knowledge bases with the latest advances in information and communication technologies, and computing and artificial intelligence is a very important part of intelligent control and planning for manufacturing system. staff must continuously learn novel the techniques and use of the learning system in the educational process for solving practical problems in individual courses within their existing production competence and future qualification requirements. to improve the system functionality and its compliance with the modern educational process needs it is necessary to take into account the comments and suggestions from users. this system also can be used for more productive generation of ideas to improve the system and eliminate bottlenecks. actually an intelligent tutoring system helps to solve the broader problems, i.e. a learner understands and sees not only one option for solving the a problem task and working on a predetermined pattern, but begins to perceive the subject areas of the enterprise as a part of the whole system. key words: learning management systems, intelligent tutoring system, model identification, manufacturing control, cognitive issues of identification, positive-formed formulas. 1. introduction modern industrial manufacturing has high requirements for employees. workers must continually improve their competence. you must constantly test, train and retrain specialists of enterprises. an adaptive program of development of new competencies is needed to perform advanced enterprise plans. they should be formed depending on the program and production plans, as well as based on the analysis of skills of learners themselves. in the light of manufacturing control problems and cognitive issues of identification, verification and other problems for processes of identification, modeling, and control, intelligent tutoring system for specialists training can be considered as an important part of manufacturing intelligent control and planning system [1]. the proposed approach can be considered as a component of the global problem of forecasting and management of the development of productive forces and production relations of society depends on the individual capabilities of each worker, and as a component of the inverse problem of advanced training of specialists, who is "needed" for the economy. training of highly skilled professionals that are competitive in the labor market, as well as competent and able to work effectively in the specialty, is a prerequisite for the successful development of any country. the recently formed lack of trained engineers in high-tech industries reflects the prevailing tension between the modern engineering complexity and the actual training level of technical specialists with higher education. 1 corresponding author. e-mail: seyl@mail.ru 2 r. s. sabitov et.al.: the concept of intelligent tutoring for enterprise staff as a component of integrated manufacturing control system development today it is a large and growing in scale problem of many businesses. there is a lack of education: instead of bringing innovative ideas to the production, investments are needed for further training and retraining of young professionals. universities are not always able to train a sufficient quantity of decent level specialists in information technologies, data mining, machinebuilding design and technological specialties, integrated logistics, production planning and modeling, analysis and design of business processes, etc. [2]. the essence of the proposed approach in this paper is quite simple and can use well-known scientific and technical solutions. as necessary conditions for the successful introduction of the results and further development of a new training system are the rational use of modern and advanced information technologies, taking into account recent advances of cognitive science and developing of original high-tech methods for solving problems in the modern theory of intelligent control, the theory of “c cubed” (control, computation and communication), etc. 2. statement of the problem and the main results teaching many technical and mathematical subjects is impossible without learning problem solving. it is necessary not only to demonstrate the examples of reasoning in solving various problems, but also to offer the student a tool for problem solving and self-check of the results. comparison of the numerical value of the response task obtained by a student with a reference value is not sufficient to control the level of the student’s assimilation of the problem-solving technique. especially if he or she is asked to choose an answer from a fixed set of options. first, the student should find a solution in the general form (to obtain a formula), and only then perform the calculations. thus, we need to be able to compare two formulas up to equivalence transformations (in the general case is algorithmically undecidable), but first it is necessary to get the formula from the user (in a machine-readable format). second, to trace the student’s reasoning, you need to know at least what formulas and in what sequence he or she used (because we cannot count on the analysis of natural reasoning). that is, the task of getting the formula from the user and comparing it with the existing needs to be repeated many times. modern information technologies training mainly provides students with electronic versions of educational hypertext material, possibly using multimedia tools, as well as remote access to these information resources and remote interaction with the teacher and other students, including real time. the achievements of artificial intelligences are still used in a much lesser extent. this conclusion applies to the learning management systems and learning content (lcms), both commercial and open source systems (e.g., moodle). content of this work is the development and pilot implementation of adaptive approaches to the universal computer system of individual training, functioning on the basis of subject knowledge and meta knowledge bases with the latest advances in information and communication technologies, computing and artificial intelligence as an important part of intelligent control and planning manufacturing system. the work includes the presentation of the basic theoretical foundations, detailed solutions of typical examples; explanations of practical applications of the mathematical results, as well as challenges for independent solutions. it is not simple tests that require calculate a numeric value, or select one of several options, but it must provide control of the correct course of solving the problem, review and updating of the results. we are talking about the formation of the individual learning program material based on real opportunities to learn using the level of training dynamic selection and feedback. the entire process can be analyzed and logged, while training the next lesson; you will have more efficiency using the accumulated experience of the system. 3. brief description of the algorithm of the module "universal tasks solver on the basis of positive-formed formulas" the main components of the pilot software package [3] is the module "universal tasks solver on the basis of positiveformed formulas". the basis of knowledge representation here is a advances in systems science and applications (2017) vol.17 no.1 3 classic first-order language l of positively constructed formulas. positively formed formula the formula is derived from n concluding statements (n ≥ 1) using standard quantifiers and logical connectives &, ⸀, and which do not limit (in its syntax) representability of various properties. moreover, the presentation of knowledge in the form of language l using only standard existential and universal quantifiers, logical connectives conjunction and disjunction is not explicitly recorded and assumed in conjunctive and disjunctive branching-formulas [4]. output is to focus on the method of contradiction. inference rule in the calculus j positively constructed formulas denoted as ω. thus, an algorithm for finding solutions using calculus j, based on the language l can be represented as follows:  have a starting base b (which is given), i.e. many well-known facts;  have the relations of computability in the knowledge base that can be used to form the basis of questions to b;  by the operation ω find answers to some questions and expand the facts base b;  if the answer is made by the target issue, i.e. refutation of contradictions, we come to a successful conclusion;  if the answer to the question is made with disjunctive subformula branching, the fact base is split by the number of bases corresponding to the number of elements in the disjunctive formula;  in case of splitting database operations using ω again is searched for the new database response and so on, to successfully complete location solutions or not. the algorithm of the universal problem solvers in this case can be summarized as follows:  the raw data (given, goal) enter the solver from the graphical user interface (interactive equation editor);  the raw initial data base is formed by the facts;  loop solver associated with the knowledge base (which contains the ratio of computability) and selects the required ratio of computability;  on the basis of computability relations the solver generates a design according to the structure formulas, applies the rule ω and given the potential ramifications of the facts base (raw data), through all the branches in the original conjunctive branching;  solver adds facts to the base;  the process continues as long as the option is not found/solutions in the form of a refutation of the contradictions, otherwise the message about the absence of the possibility of solving the problem with this facts and knowledge database. schematically, the process of finding a solution to the problem can be represented by the following relationships between the graphical user interface, solver of problems, the original data and knowledge base (fig. 1-2) 4 r. s. sabitov et.al.: the concept of intelligent tutoring for enterprise staff as a component of integrated manufacturing control system development fig. 1. process of finding a solution to the problem 4. principles of formation of the formula editor for interactive intelligent tutoring system computer software formula editors are of great importance in teaching technical subjects. they are needed when you create training material, as well as directly in the learning process. it offers ideas for the construction of the formula editor you would really want to use. on the basis of these ideas a working formula editor has already been created. this equation editor is used in the interactive learning system of intellectual training of engineers [7]. these ideas are as follows. form formula by typing the name of the keyboard characters, operators, etc., that you want to insert. moreover, it made directly in the text of the formula. the system then replaces the typed text to the desired mark, operator or member. enough variants of the name that you use it (integral matrix, matrix, etc.). also, the system tries to guess what you want, with the first letter typed, and offers options for items to choose from. as you type, a list of options varies. this approach works well in integrated development environments used by programmers. here the system guesses what the programmer is going to write on, and offers options for continuing or replacement text. this feature is known as code completion. our experiments showed that the use of this approach for formulas reduces the formula typing time compared with the traditional interface to enter a formula using the mouse. revealed the following features. beginners do not need to study the menu and the editor of formulas to find the right icon. they already know names of the needed elements. the student, typing formula, can continue to think about the formula itself. he or she doesn’t need to transfer attention to finding the right button, the window or the icon. inserting the name of the element is made directly in the text of the formula, and not in a separate window. interface to enter a formula should contain the necessary elements to enter and control the formula using the mouse as in traditional systems, since it also has certain advantages. advances in systems science and applications (2017) vol.17 no.1 5 fig. 2. algorithm for finding the solution of the problem allow you to specify the semantics of typed formulas. that is, allow the user to specify exactly what he got – a superscript or degree opening bracket arithmetic expression or a list of function parameters, vector, matrix, etc. automatic guessing what is typed by the user, based on the context surrounding the formula. that is, if a function is assumed in calculations of v, but there is a variable with the same name, then the bracket for the symbol v rather marks the beginning of the list of its parameters, than anything else. also in the opposite direction, it is possible to use semantic information from a typed formula to automatically generate a list of user-entered or used variables, functions, and other objects. this is true for both research and training systems. for example, to request the user to provide the meaning of an introduction of a variable already present in the context of the problem being solved. a formula may be managed fully with the keyboard, waiting for the user’s next action. for example, adding new parameters of a function by pressing the comma. inserting new rows and columns by pressing the enter key and the space bar, respectively. add to a user the possibility to undo and redo actions and also the opportunity to return to any of the formulas in the timeline, copy from an old version of the desired piece and insert it into the current version. further some of the implementation details are provided. 6 r. s. sabitov et.al.: the concept of intelligent tutoring for enterprise staff as a component of integrated manufacturing control system development it is proposed to move away from the standard approach abstraction and unification of structures and methods. for a software implementation does not use the formula editor arithmetic data types (adt) and / or parse trees formats for mathematical formulas (mathml, etc.) and reflects the perception of the structure of the formula written by the user. for example, a formula does not seem tree operators /, operations on arguments (adt), and an ordered sequence of its constituent elements (variables, operators, etc.), see fig. 3. fig. 3. representation of formulas in the computer memory fig. 4. presentation exponentiation operation in the computer memory exponentiation is represented not as a transaction with two arguments (mathml, etc.). it is a characteristic of an element (variable, function, etc.) the extent to which he was elevated (fig. 4). division operation will then be presented in two ways (fig. 5). this approach simplifies the system. system behavior when navigating and editing a formula corresponds to the user's expectations, and not abstract structures in computer memory. it then becomes easier to predict what the user wants to do next, but has not yet made, and to help the user to perform these actions. the program realizes the possibility of undo and redo actions. changes are made through the design pattern (command). in this case a formula itself does not change (fig. 5). if you want to make changes to the formula, the program creates a special object (command object) with two methods – "apply changes" and "undo the changes." this object allows you to make changes to the formula these objects are stored in a special stack. if necessary, they in turn are popped from the stack, used for undo and placed in another stack to repeat actions. the command object, which inserts a new element into the formula, when recalling the method "apply changes", it inserts the same object as the first call. objects that removes an element after canceling of the removal return to the formula the object that has been deleted, and do not generate a new identical object. this allows you to insert, delete, and modify the formula to implement elements directly via pointers to the object in memory (since reached the immutability of pointers), and not only use the location element in the formula with respect to its root / start. application of direct pointers to mutable object simplifies writing and empowering online and accelerates the work itself online. advances in systems science and applications (2017) vol.17 no.1 7 fig. 5. substitution the division operation in the computer memory implementation of these ideas has already shown tremendous speed gain in typing a set of formulas and invaluable assistance to concentrate on the essence of the case that the user is working on. actually, the same formula editor can be used for scientific research and for writing articles. together with our colleagues, we continue to develop it. thus, the role of an editor of mathematical formulas is twofold: on the one hand, it should allow a student to be comfortable enough to express their thoughts, and on the other – should allow effective handling of user’s input. in doing so, it would be desirable to reduce the possibility of typing errors. moreover, the editor’s usage should not assume that a user has special knowledge, representation of formulas should be familiar and intuitive. these requirements are met by the flexible editor. formulas in the editor are presented in a familiar graphical form, as they would be printed in a textbook. the editor uses the concept of a template formula – a sort of a "canonical" representation in which this type of formula is written (which makes them easy to compare). a user selects a pattern corresponding to the desired formula, and then fills it with slots of necessary constants, variables and functions. symbols of constants, variables and functions are presented by graphic primitives, roaming through the mechanism of drag-n-drop. in addition, the editor allows you to further adjust the difficulty of a task, varying the amount of "designer details" available to the user. templates in the system are generated automatically from text formulas. in a formula (generated by parsing the text representation) variables, functions, constants are indicated. they are not inserted in the template automatically, instead of that cells for substitution variables, functions and constants are put in their places. other elements (operators, etc.) are inserted into the template unchanged. 6. conclusions the development and a pilot implementation of adaptive approaches to the universal computer system of individual training, functioning on the basis of subject and meta knowledge bases with the latest advances in information and communication technologies, computing and artificial intelligence is a very important part of intelligent control and planning for 8 r. s. sabitov et.al.: the concept of intelligent tutoring for enterprise staff as a component of integrated manufacturing control system development manufacturing system. staff must continuously learn novel techniques and use a learning system in the educational process for solving practical problems in individual courses within their existing production competence and future qualification requirements. to improve the system functionality and its compliance with the modern educational process needs it is necessary to take into account comments and suggestions from users [5-6]. staff can also use a complex learning system to have a holistic picture for more productive generation of ideas which can be used for improving the production system and eliminating bottlenecks. actually, an intelligent tutoring system helps to solve the broader problems, i.e. a learner understands and sees not only one option for solving a task and working on a predetermined pattern, but begins to perceive the subject areas of the enterprise as a part of the whole system. this happens largely because using the latest achievements of artificial intelligence, telecommunications and information technologies it is possible to study fast and find efficient solutions of a various options with the learner’s active participation. this also significantly simplifies the process of domain knowledge modeling. a similar effect can also be achieved with conventional training methods, i.e. using a constant direct contact between a teacher and a learner. obviously, to achieve a similar effect in the case of traditional teaching methods in average much more time and money is required than with the learning process of the developed software system. here the learning system actually allows you to simulate the process of individual learning. the effect of the individual approach in this case is very significant and convenient. you can continue with your work at any time that is not practicable at the standard training scheme. in the case of a group of traditional forms of education it is extremely difficult for a teacher to give sufficient time for an individual approach for each group member. the system can be used very effectively in groups where there are different levels of competence of learners. weaker learners will not hinder the work of strong learners. all motivated learners have an opportunity to improve their performance through their individual work. references [1] vassilyev, s.n., sabitov r.a., degtyarev, g.l., kozlov, v.v, malivanov, n.n., sabitov, sh.r. and sirazetdinov, r.t. adaptive approach to developing advanced distributed elearning management system for manufacturing. preprints of the 13th ifac symposium on information control problems in manufacturing, moscow, russia, pp. 2198-2203, 2009 [2] vasiliev s.n. and sabitov r.a. the knowledge economy and the intelligent management. proceedings of the x international conference chetaev, kazan, 2012. [3] vassilyev, s.n., sabitov sh.r., sirazetdinov b.r., smirnova, g.s., sukonnova a.a. architecture and functions tracking intelligent tutoring system "volga" №4, knrtu-kai named after a.n. tupolev, russia, kazan, 2012 [4] vasiliev s. n., zherlov a. k., fedosov e. a., b. e. fedunov (2000) intelligent control of dynamic systems. – moscow: physical and mathematical literature [5] smirnova, g.s., sabitov, r.a., sabitov, sh.r., korobkova, e.a, kislov, a.s. programmethodical complex of operations management №4, knrtu-kai named after a.n. tupolev, russia, kazan, 2012 [6] smirnova, g.s., sabitov, r.a., elizarova, n.y., sh.r. sabitov operational management system of enterprise production processes "1c: mes: cloudy production management. journal “automation in industry”, 2014, № 8, russia, moscow [7] sirazetdinov b.r., smirnova, g.s., korobkova e.a. principles of formation equation editor for interactive intelligent tutoring system for training engineers in aerospace disciplines. international scientific-practical conference "modern technologies and materials – the key point in revival of domestic aircraft building", kazan, 2010 advances in systems science and applications (2011), vol. 11, no. 1-2 1-26 time-optimal control of infinite variables parabolic systems with time lags given in integral form g. m. bahaa1 and m. m. tharwat 2 1 department of mathematics, faculty of science, taibah university, al-madinah al-munawarah, saudi arabia 2 departmentofmathematics,universitycollege,ummal-qurauniversity,makkah,saudiarabia emall: bahaa−gm@hotmail.com, zahraa26@yahoo.com abstract in this paper, the time-optimal control problem for second order parabolic system and also for (n× n)– parabolic systems with infinite number of variables involving constant time lags appearing in integral form in both the state equation and in the boundary condition is presented. some specific properties of the optimal control are discussed. keywords time-optimal control (n × n) parabolic systems operator with an infinite number of 1. introduction distributed parameters systems with delays can be used to describe many phenomena in the real world. as is well known, heat conduction, properties of elastic-plastic material, fluid dynamics, diffusion-reaction processes, the transmission of the signals at a certain distance by using electric long lines, etc., all lie within this area. the object that we are studying (temperature, displacement, concentration, velocity, etc.) is usually referred to as the state. the time-optimal control problems of distributed second order parabolic systems with finite number of variables involving time lags appearing in the boundary condition have been widely discussed in many papers and monographs. a fundamental study of such problems is given by (wang, 1975) and was next developed by (knowles, 1978) and (wong, 1987). it was also intensively investigated by (kowalewski, 1988; 1990a; 1990b; 1993; 1998; 1999; 2009), (kowalewski and duda, 1992 ), (kowalewski and krakowiak, 1994; 2000; 2006; 2008), (kotarski, 1997), (kotarski & el-saify and bahaa, 2002b), (kotarski and bahaa, 2007) and (elsaify, 2005; 2006) in which linear quadratic problem for parabolic systems with time delays given in the different form (constant time delays, time-varying delays, time delays given in the integral form, etc.) were presented. the necessary and sufficient conditions of optimality for systems consists of only one equation and for (n × n) systems governed by different types of partial differential equations defined on spaces of functions of infinitely many variables and also for infinite order systems are discussed for example in ( gali, i. m. & el-saify, h. a. 1982; 1983), (el-saify & bahaa, 2001; 2003), (el-saify, h. a., serag, h. m, & bahaa, g. m. 2000), (el-saify, 2005; 2006), (kowalewski, 2009) and (kowalewski and krakowiak, 2008) in which the argument of (lions, 1971 and lions & magenes, 1972) were used. variables time lag.variables time lag issn 1078-6236 international institute for general systems studies, inc. 2 bahaa:time-optimal control of infinite variables parabolic systems. . . . . . making use of the dubovitskii-milyutin theorem in (kotarski, el-saify & bahaa, 2002a,b), (bahaa, 2003; 2005a,b; 2008) and (bahaa and kotarski, 2008), the necessary and sufficient conditions of optimality for similar systems governed by second order operator with an infinite number of variables and also for infinite order systems were investigated. the interest in the study of this class of operators is stimulated by problems in quantum field theory. in particular, the papers of (kowalewski & krakowiak, 2006, 2008), the time-optimal boundary control problem for a second order distributed parabolic systems with finite number of variables in which constant time lags appear in integral form in both the state equation and the boundary condition is presented. some particular properties of optimal control are discussed. in this paper we recall the problem in a more general formulation. we consider the timeoptimal distributed and boundary control problem for second order parabolic system and also for (n×n) –second order parabolic systems with infinite number of variables involving constant time lags appearing in integral form in both the state equation and in the boundary condition simultaneously. such an infinite variables parabolic systems can be treated as a generalization of the mathematical model for a plasma control process. the quadratic performance functional defined over a fixed time horizon are taken and some constraints are imposed on the boundary control. following a line of the lions scheme (lions, 1971) and (lions & magenes, 1972), necessary and sufficient optimality conditions for the neumann and dirichlet problem applied to the above systems were derived. the optimal control is characterized by the adjoint equations. this paper is organized as follows. in section 2, we introduce sobolev spaces with infinite number of variables. in section 3, we formulate the mixed neumann problem for infinite variables parabolic systems involving time lags. in section 4, the time-distributed control problem for this case is formulated, then we give the necessary and sufficient conditions for the time control to be an optimal. in section 5, we concluded and generalized our results. 2. sobolev spaces with infinite number of variables this section covers the basic notations, definitions and properties, which are necessary to present this work (berezanskii, 1975), ( gali & el-saify 1982; 1983), (el-saify & serag & bahaa, 2000) and (el-saify & bahaa, 2001). let (pk(t)) ∞ k=1 be a sequence of weights, fixed in all that follows, such that; 0 < pk(t) ∈ c∞(r1), ∫ r1 pk(t)dt = 1, with respect to it we introduce on the region r∞ = r1 × r1 × . . . , the measure dρ(x) by setting, dρ(x) = p1(x1)dx1 ⊗ p2(x2)dx2 ⊗ . . . , (r∞ 3 x = (xk) ∞ k=1, xk ∈ r1). on r∞ we construct the space l2(r∞, dρ(x)) with respect to this measure i.e., l2(r∞, dρ(x)) is the space of quadratic integrable functions on r∞. we shall often set l2(r∞, dρ(x)) = l2(r∞). advances in systems science and applications (2011), vol. 11, no. 1-2 3 it is classical result that l2(r∞) is a hilbert space for the scalar product (φ, ψ)l2(r∞) = ∫ r∞ φ(x)ψ(x)dρ(x). we next consider a sobolev space in the case of an unbounded region. for functions which are ` = 1, 2, . . . times continuously differentiable up to the boundary γ of r∞ ( γ is meant to be the boundary of the support of the measure dρ(x)) and which vanish in a neighborhood of ∞, we introduce the scalar product (φ, ψ)w `(r∞) = ∑ |α|≤` (dαφ,dαψ)l2(r∞), where dα is defined by dα = ∂|α| (∂x1)α1(∂x2)α2 · · · , |α| = ∞∑ i=1 αi, and the differentiation is taken in the sense of generalized functions on r∞, and after the completion, we obtain the sobolev space w `(r∞). so in short, sobolev space w 1(r∞) is defined by : w 1(r∞) = {φ|φ,dφ ∈ l2(r∞)}. as in the case of a bounded region, the space w 1(r∞) form the space with positive norm ||.||w 1(r∞). we can construct the space w−1(r∞) = (w 1(r∞))∗ with negative norm ||.||w−1(r∞) with respect to the space w 0(r∞) = l2(r∞) with zero norm ||.||l2(r∞), then we have the following equipped, w 1(r∞) ⊆ l2(r∞) ⊆w−1(r∞), ||φ||w 1(r∞) ≥ ||φ||l2(r∞) ≥ ||φ||w -1(r∞). letl2(0, t ;w 1(r∞)) be the space of square integrable measurable functions t→ φ(t) of ]0, t [→ w 1(r∞), where the variable t denotes the “ time ”; t ∈]0, t [, t <∞. this space is a hilbert space with respect to the scalar product (φ, ψ)l2(0,t ;w 1(r∞)) = ∫ t 0 (φ(t), ψ(t))w 1(r∞)dt, and its dual is the spacel2(0, t ;w−1(r∞)), analogously, we can define the spacesl2(0, t ;l2(r∞)) which we shall denote by l2(q). let ω ⊂ r∞ is a bounded, open set with boundary γ, which is ac∞ manifold of dimension (n − 1). locally, ω is totally on one side of γ and denote by w 1(ω,r∞, dρ(x)) (briefly w 1(ω,r∞)) the sobolev space of vector function y(x) defined on ω. the construction of the cartesian product of n-times to the above hilbert spaces can be construct, for example (w 1(ω,r∞))n = w 1(ω,r∞)×w 1(ω,r∞)× · · · ×w 1(ω,r∞)︸ ︷︷ ︸ n−times = n∏ i=1 (w 1(ω,r∞))i, 4 bahaa:time-optimal control of infinite variables parabolic systems. . . . . . with norm defined by: ||φ||(w 1(ω,r∞))n = n∑ i=1 ||φi||w 1(ω,r∞), where φ = (φ1, φ2, ..., φn) = (φi) n i=1 is a vector function and φi ∈w 1(ω,r∞). finally, we have the following chain: (l2(0, t ;w 1(ω,r∞)))n ⊆ (l2(q))n ⊆ (l2(0, t ;w−1(ω,r∞)))n, where (l2(0, t ;w−1(ω,r∞)))n are the dual spaces of (l2(0, t ;w 1(ω,r∞)))n. the spaces considered in this paper are assumed to be real. 3. existence and uniqueness of solutions consider now the distributed-parameter system described by the following parabolic delay equation: ∂y ∂t +a(t)y + ∫ b a c(x, t)y(x, t− h) dh = u, x ∈ ω, t ∈ (0, t ), h ∈ (a, b), (1) y(x, t′) = φ0(x, t′), x ∈ ω, t′ ∈ [−b, 0), (2) y(x, 0) = y0(x), x ∈ ω, (3) ∂y(x, t) ∂ηa = ∫ b a d(x, t)y(x, t− h) dh+ v, x ∈ γ, t ∈ (0, t ), h ∈ (a, b), (4) y(x, t′) = ψ0(x, t′), x ∈ γ, t′ ∈ [−b, 0), (5) where ω and γ have the same properties as in section 2. we have y ≡ y(x, t;u), u ≡ u(x, t), v ≡ v(x, t), q ≡ ω× (0, t ), q ≡ ω× [0, t ], q0 ≡ ω× [−b, 0) σ ≡ γ× (0, t ), σ0 ≡ γ× [−b, 0), t is a specified positive number representing a time horizon, c is a given real c∞ function defined on q, d is a given real c∞ function defined on σ, h is a time lag such that h ∈ (a, b) and a > 0, φ0 and ψ0 are initial functions defined on q0 and σ0, respectively. the parabolic operator ∂ ∂t + a(t) in the state equation (1) is a second order parabolic operator with infinite number of variables anda(t) (berezanskii, 1975), (gali & el-saify, 1982; 1983) and (kotarski & el-saify & bahaa, 2002b ) is given by: a(t)y(x) = ( − ∞∑ k=1 1√ pk(xk, t) ∂2 ∂x2 k √ pk(xk, t) + q(x, t) ) y(x) = − ∞∑ k=1 d2 ky(x) + q(x, t)y(x), (6) advances in systems science and applications (2011), vol. 11, no. 1-2 5 where dky(x) = 1√ pk(xk, t) ∂ ∂xk √ pk(xk, t)y(x), (7) and q(x, t) is a real-valued function in x which is a bounded and measurable on ω ⊂ r∞, such that q(x, t) ≥ ξ0 > 1, ξ0 is a constant. the operatora(t) is a bounded second order self-adjoint elliptic partial differential operator with an infinite number of variables maps w 1(ω,r∞) onto w−1(ω,r∞). for this operator we define the bilinear form as follows: definition 3.1. for each t ∈ (0, t ), we define a family of bilinear forms on w 1(ω,r∞) by: π(t; y, φ) = (a(t)y, φ)l2(ω,r∞), y, φ ∈w 1(ω,r∞), (8) where a(t) maps w 1(ω,r∞) onto w−1(ω,r∞) and takes the above form. then π(t; y, φ) = ( a(t)y, φ ) l2(ω,r∞) = ( − ∞∑ k=1 d2 ky(x) + q(x, t)y(x), φ(x) ) l2(ω,r∞) = ∫ ω ∞∑ k=1 dky(x)dkφ(x) dρ(x) + ∫ ω q(x, t)y(x)φ(x) dρ(x). lemma 3.1. the bilinear form π(t; y, φ) is coercive on w 1(ω,r∞), that is π(t; y, y) ≥ λ ||y||2w 1(ω,r∞), λ > 0. (9) proof. it is well known that the ellipticity ofa(t) is sufficient for the coerciveness of π(t; y, φ) on w 1(ω,r∞). π(t;φ, ψ) = ∫ ω ∞∑ k=1 dkφ(x)dkψ(x) dρ+ ∫ ω q(x, t)φ(x)ψ(x) dρ. then π(t; y, y) = ∫ ω ∞∑ k=1 |dky(x)|2 dρ(x) + ∫ ω q(x, t)|y(x)|2 dρ(x) ≥ ∞∑ k=1 ||dky(x)||2l2(ω,r∞) + ξ0||y(x)||2l2(ω,r∞) = ||y(x)||2w 1(ω,r∞) + ξ0||y(x)||2l2(ω,r∞) ≥ ||y(x)||2w 1(ω,r∞) = λ||y||2w 1(ω,r∞), λ > 0. 6 bahaa:time-optimal control of infinite variables parabolic systems. . . . . . also we have: ∀y, φ ∈w 1(ω,r∞) the function t→ π(t; y, φ) is continuously differentiable in (0, t ) and π(t; y, φ) = π(t;φ, y) } (10) equations (1)–(5) constitute a neumann problem. then the left-hand side of the boundary condition (4) may be written in the following form: ∂y(u) ∂ηa = ∞∑ k=1 (dky(u)) cos(n, xk) = g(x, t), (11) where ∂ ∂ηa is a normal derivative at γ, directed towards the exterior of ω, cos(n, xk) is the k − th direction cosine of n, with n being the normal at γ exterior to ω, and g(x, t) = ∫ b a d(x, t)y(x, t− h) dh+ v(x, t), x ∈ γ, t ∈ (0, t ), h ∈ (a, b). (12) first we shall prove sufficient conditions for the existence of a unique solution of the mixed initial boundary value problem (1)–(5) for the cases where the control u or v belong to l2(q) or l2(σ) respectively. to this purpose, for any pair of real numbers r, s ≥ 0, we introduction the sobolev space w r,s(q) (lions and magenes, 1972, vol. 2, p. 6) defined by w r,s(q) = l2 (0, t ;w r(ω,r∞)) ∩w s ( 0, t ;l2(ω,r∞) ) (13) which is a hilbert space normed by(∫ t 0 ||y(t)||2w r(ω,r∞)dt+ ||y||2w s(0,t ;l2(ω,r∞)) )1/2 , (14) where w s ( 0, t ;l2(ω,r∞) ) denotes the sobolev space of order s of functions defined on (0, t ) and taking values in l2(ω,r∞). the existence of a unique solution for the mixed initial-boundary value problem (1)–(5) on the cylinder q can be proved using a constructive method, i.e., first, solving (1)–(5) on the sub-cylinder q1 and in turn on q2, and so on, until the procedure covers the whole cylinder q. in this way, the solution in the previous step determines the next one. for simplicity, we introduce the following notation: ej , ((j − 1)a, ja), qj = ω× ej , σj = γ× ej , j = 1, 2, . . . . (15) case 1: u ∈ l2(q) using theorem 6.1 of lions & magenes (1972, vol. 2, p. 33), we can prove the following lemma. advances in systems science and applications (2011), vol. 11, no. 1-2 7 lemma 3.2. let u ∈ l2(q), (16) fj(x, t) ∈ l2(qj), (17) where fj = u(x, t)− ∫ b a c(x, t)yj−1(x, t− h) dh, yj−1(·, (j − 1)a) ∈w 1(ω,r∞), (18) gj ∈w 1 2 , 1 4 (σj), (19) where gj(x, t) = ∫ b a d(x, t)yj−1(x, t− h) dh+ v(x, t). then, there exists a unique solution yj ∈ w 2,1(qj) for the mixed initial-boundary value problem (1), (4) and (18). proof. we observe that for j = 1, y0|q0(x, t−h) = φ0(x, t−h) and y0|σ0(x, t−h) = ψ0(x, t− h). then the assumptions (17)–(19) are fulfilled if we assume that φ0 ∈ w 2,1(q0), y0 ∈ w 1(ω,r∞), v ∈ w 1 2 , 1 4 (σ) and ψ0 ∈ w 1 2 , 1 4 (σ0). these assumptions are sufficient to ensure the existence of a unique solution y1 ∈ w 2,1(q1). in order to extend the result to q2, we have to prove that y1(·, a) ∈w 1(ω,r∞), g2 ∈w 1 2 , 1 4 (σ2) and f2 ∈ l2(q2). really, from theorem 3.1, p.19 of lions & magenes vol.1, y1 ∈ w 2,1(q1) implies that the mapping t → y1(·, t) is continuous from [0, a] → w 1(ω,r∞). thus y1(·, a) ∈ w 1(ω,r∞). then using the trace theorem of lions & magenes (1972, vol. 2, p. 9) we can verify that y1 ∈ w 2,1((q1) implies that y1 → y1|σ1 is a linear, continuous mapping of w 2,1(q1)→ w 1 2 , 1 4 (σ1). assuming that d is a c∞ function and v ∈ w 1 2 , 1 4 (σ), the condition g2 ∈ w 1 2 , 1 4 (σ2) is fulfilled. also it is easy to notice that the assumption (17) follows from the fact that y1 ∈ w 2,1(q1) and u ∈ l2(q). then, there exists a unique solution y2 ∈ w 2,1(q2). finally, we can extend our result to any qj , j = 3, 4 . . .. theorem 3.3. let y0, φ0, ψ0, v and u be given with y0 ∈ w 1(ω,r∞), φ0 ∈ w 2,1(q0), v ∈ w 1 2 , 1 4 (σ), ψ0 ∈w 1 2 , 1 4 (σ0) and u ∈ l2(q). then, there exists a unique solution y ∈w 2,1(q) for the mixed initial-boundary value problem (1)–(5). moreover, y(·, ja) ∈w 1(ω,r∞) for j = 1, 2, . . .. case 2: v ∈ l2(σ) using theorem 15.2 of lions & magenes (1972, vol. 2, p. 81), we can prove the following lemma. lemma 3.4. let u ∈w− 1 2 ,− 1 4 (q), v ∈ l2(σ) (20) fj ∈w− 1 2 ,− 1 4 (qj), (21) yj−1(·, (j − 1)a) ∈w 1 2 (ω,r∞), (22) 8 bahaa:time-optimal control of infinite variables parabolic systems. . . . . . gj ∈ l2(σj). (23) then, there exists a unique solution yj ∈ w 3 2 , 3 4 (qj) for the mixed initial-boundary value problem (1), (4) and (22). proof. for j = 1, the assumptions (21)–(23) are fulfilled if we assume that φ0 ∈w 3 2 , 3 4 (q0), y0 ∈ w 1 2 (ω,r∞) and ψ0 ∈ l2(σ0). these assumptions are sufficient to ensure the existence of a unique solution y1 ∈ w 3 2 , 3 4 (q1). in order to extend the result to q2, we have to prove that y1(·, a) ∈ w 1 2 (ω,r∞), y1|σ1 ∈ l2(σ1) and f2 ∈ w− 1 2 ,− 1 4 (q2). first using theorem 3.1 of lions & magenes (1972, vol. 1, p. 19) we can prove that y1 ∈ w 3 2 , 3 4 (q1) implies that the mapping t → y1(·, t) is continuous from [0, a] → w 3 4 (ω,r∞) ⊂ w 1 2 (ω,r∞). hence y1(·, a) ∈ w 1 2 (ω,r∞). again, from trace theorem of lions & magenes (1972, vol. 2, p. 9), we can verify that y1 ∈ w 3 2 , 3 4 (q1) implies that y1 → y1|σ1 is a linear, continuous mapping of w 3 2 , 3 4 (q1) → w 1, 1 2 (σ1). thus y1|σ1 ∈ l2(σ1). moreover, it is worth mentioning that the assumption (21) follows from the fact that y1 ∈ w 3 2 , 3 4 (q1) and u ∈ w− 1 2 ,− 1 4 (q). then, there exists a unique solution y2 ∈ w 3 2 , 3 4 (q2). finally, we can extend our result to any qj , j = 3, 4 . . .. theorem 3.5. let y0, φ0, ψ0, v and u be given with y0 ∈w 1 2 (ω,r∞), φ0 ∈w 3 2 , 3 4 (q0), ψ0 ∈ l2(σ0), v ∈ l2(σ) and u ∈ w− 1 2 ,− 1 4 (q). then, there exists a unique solution y ∈ w 3 2 , 3 4 (q) for the mixed initial-boundary value problem (1)–(5). moreover, y(·, ja) ∈ w 1 2 (ω,r∞) for j = 1, 2, . . .. now we shall verify the existence of a unique solution for the problem (1), (2), (3) and (5) with the dirichlet boundary condition involving a time lag y(x, t) = g(x, t) (24) where g is given by the formula (12). making use of the results of lions & magenes (1972, vol. 2, p. 33 and p. 81) we can prove the following lemmas and theorems. case 3: u ∈ l2(q) lemma 3.6. let u ∈ l2(q), (25) fj ∈ l2(qj), (26) yj−1(·, (j − 1)a) ∈w 1(ω,r∞), (27) gj ∈w 3 2 , 3 4 (σj), (28) and the following compatibility relation is fulfilled yj−1(x, (j − 1)a) = gj(x, (j − 1)a), on γ. (29) then, there exists a unique solution yj ∈ w 2,1(qj) for the mixed initial-boundary value problem (1), (24) and (27). advances in systems science and applications (2011), vol. 11, no. 1-2 9 proof. for j = 1, the assumptions (26)–(28) can be satisfied if we assume that φ0 ∈ w 2,1(q0), v ∈ w 3 2 , 3 4 (σ) and ψ0 ∈ w 3 2 , 3 4 (σ0). these assumptions are sufficient to ensure the existence of a unique solution y1 ∈ w 2,1(q1) if y0 ∈ w 1(ω,r∞) and the following compatibility relation is satisfied y0(x, 0) = g1(x, 0), on γ. (30) in order to extend the result to q2, we have to prove that y1 ∈ w 2,1(q1) and it is necessary to impose the compatibility relation y1(x, a) = g2(x, a), on γ (31) and it is sufficient to verify that f2 ∈ l2(q2), (32) y1(·, a) ∈w 1(ω,r∞), (33) g2 ∈w 3 2 , 3 4 (σ2). (34) first using the solution in the previous step and the condition (25) we can prove immediately the condition (32). to verify (33), we use the fact that y1 ∈ w 2,1(q1) implies that the mapping t → y1(·, t) is continuous from [0, a] → w 1(ω,r∞) (by theorem 3.1 of lions & magenes (1972, vol. 1, p. 19)), hence y1(·, a) ∈w 1(ω,r∞). from the trace theorem of lions & magenes (1972, vol. 2, p. 9) y1 ∈w 2,1((q1) implies that y1 → y1|σ1 is a linear, continuous mapping of w 2,1(q1) → w 3 2 , 3 4 (σ1). assuming that d is a c∞ function and v ∈ w 3 2 , 3 4 (σ), the condition (34) is fulfilled. then, there exists a unique solution y2 ∈ w 2,1(q2). finally, we can extend our result to any qj , j = 3, 4 . . .. theorem 3.7. let y0, φ0, ψ0, v and u be given with y0 ∈ w 1(ω,r∞), φ0 ∈ w 2,1(q0), v ∈ w 3 2 , 3 4 (σ), ψ0 ∈ w 3 2 , 3 4 (σ0), u ∈ l2(q) and the compatibility relation (29) is fulfilled. then, there exists a unique solution y ∈w 2,1(q) for the mixed initial-boundary value problem (1), (2), (3) (5) and (24) with y(·, ja) ∈w 1(ω,r∞) for j = 1, 2, . . .. case 4: v ∈ l2(σ) lemma 3.8. let u ∈w− 3 2 ,− 3 4 (q), v ∈ l2(σ) (35) fj ∈w− 3 2 ,− 3 4 (qj), (36) yj−1(·, (j − 1)a) ∈w− 1 2 (ω,r∞), (37) gj ∈ l2(σj). (38) then, there exists a unique solution yj ∈ w 1 2 , 1 4 (qj) for the mixed initial-boundary value problem (1), (24) and (37). 10 bahaa:time-optimal control of infinite variables parabolic systems. . . . . . proof. we observe that for j = 1, the assumptions (36)–(38) are satisfied if we assume that φ0 ∈ w 1 2 , 1 4 (q0), y0 ∈ w− 1 2 (ω,r∞) and ψ0 ∈ l2(σ0). these assumptions are sufficient to ensure the existence of a unique solution y1 ∈ w 1 2 , 1 4 (q1). next for j = 2, using the solution in the first step, it is sufficient to verify that f2 ∈ w− 3 2 ,− 3 4 (q2), y1(·, a) ∈ w− 1 2 (ω,r∞) and y1|σ1 ∈ l2(σ1). then it is worth mentioning that the condition f2 ∈ w− 3 2 ,− 3 4 (q2) follows from the fact that y1 ∈ w 1 2 , 1 4 (q1) and u ∈ w− 3 2 ,− 3 4 (q). since y1 ∈ w 1 2 , 1 4 (q1) implies that the mapping t → y1(·, t) is continuous from [0, a] → w 1 4 (ω,r∞) (by theorem 3.1 of lions & magenes (1972, vol. 1, p. 19)), hence y1(·, a) ∈ w 1 4 (ω,r∞) ⊂ l2(ω,r∞) ⊂ w− 1 4 (ω,r∞) ⊂ w− 1 2 (ω,r∞). we shall prove that y1|σ1 ∈ l2(σ1). we must notice that for proving y1|σ1 ∈ l2(σ1) we cannot use the trace theorem of lions & magenes (1972, vol. 2, p. 9), since y1 ∈w 1 2 , 1 4 (q1). it is worth mentioning that this difficulty can be avoided by using the condition y1|σ1 = g1 ∈ l2(σ1). this implies that y1|σ1 ∈ l2(σ1). then, there exists a unique solution y2 ∈w 1 2 , 1 4 (q2). finally, we can extend our result to any qj , j = 3, 4 . . .. theorem 3.9. let y0, φ0, ψ0, v and u be given with y0 ∈ w− 1 2 (ω,r∞), φ0 ∈ w 1 2 , 1 4 (q0), ψ0 ∈ l2(σ0), v ∈ l2(σ) and u ∈ w− 3 2 ,− 3 4 (q). then, there exists a unique solution y ∈ w 1 2 , 1 4 (q) for the mixed initial-boundary value problem (1), (2), (3), (5) and (24). moreover, y(·, ja) ∈w− 1 2 (ω,r∞) for j = 1, 2, . . .. 4. optimal distributed control now, we shall restrict our considerations to the case of the distribute control for the neumann problem. therefore, we shall formulate the minimum-time problem for (1)–(5) in the context of the theorem 3.3, i.e., u ∈ u = {u ∈ l2(q) : |u(x, t)| ≤ 1}. (39) we shall define the reachable set h such that h = {y ∈ l2(ω,r∞) : ||y − zd||l2(ω,r∞) ≤ ε} (40) where zd ∈ l2(ω,r∞) and ε > 0. solving the stated minimum-time problem is equivalent to hitting the target set h in minimum time, that is, minimizing the time t, for which y(t;u) ∈ h and u ∈ u . moreover, we assume that there exists a t > 0 and u ∈ u with y(t ;u) ∈ h (41) then we have the following theorem theorem 4.1. if the assumption (41) holds, then the seth is reached in minimum time t∗ by an admissible control u∗ ∈ u . moreover∫ ω [zd − y(t∗;u∗)] [y(t∗;u)− y(t∗;u∗)] dρ ≤ 0, ∀u ∈ u. (42) advances in systems science and applications (2011), vol. 11, no. 1-2 11 proof. let us define the following set t∗ := inf{t : y(t;u) ∈ h for some u ∈ u} (43) the minimum is well defined, as (41) guarantees that this set is nonempty. by definition, we can choose tn ↓ t∗ and admissible controls {un} such that y(tn;un) ∈ h, n = 1, 2, 3, . . . . (44) each un is defined on ω × (0, tn) ⊃ ω × (0, t∗). to simplify the notation, we denote the restriction of un to ω× (0, t∗) again by un. the set of admissible controls then forms a weakly compact, convex set in l2(ω×(0, t∗)), and so we can extract a weakly convergent subset {um}, which converges weakly to some admissible control u∗. consequently, theorem 3.3 implies that y(t;u) ∈ w 1(ω,r∞) ⊂ l2(ω,r∞) for each u ∈ l2(q) and t > 0. then using theorem 1.2 of (lions, 1971, p. 102) and theorem 3.3 it is easy to verify that the mapping u → y(t∗;u) from l2(ω × (0, t∗)) into l2(ω,r∞), is continuous. since any continuous linear mapping between banach spaces is also weakly continuous (dunford and schwartz, 1958), theorem v. 3.15, the affine mapping u → y(t∗;u) must also be weakly continuous. hence, y(t∗;um)→ y(t∗;u∗) weakly in l2(ω,r∞). (45) moreover, dy(u) dt ∈ l2 ( [0, t∗];l2(ω,r∞) ) , (46) for each u ∈ u , by definition of w 2,1(ω× (0, t∗)) and ||y(tm;um)− y(t∗;um)||l2(ω,r∞) = ∣∣∣∣∣∣∣∣∫ tm t∗ ẏ(σ;um) dσ ∣∣∣∣∣∣∣∣ l2(ω,r∞) (47) ≤ √ tm − t∗ (∫ tm t∗ ||ẏ(σ;um)||2l2(ω,r∞) dσ )1/2 .(48) applying theorem 1.2 of (lions, 1971) and theorem 3.3 again, the set {ẏ(um)} must be bounded in l2(0, t∗;l2(ω,r∞)), and so ||y(tm;um)− y(t∗;um)||l2(ω,r∞) ≤m √ tm − t∗. (49) combining (45) and (49) shows that y(tm;um)− y(t∗;u∗) = (y(tm;um)− y(t∗;um)) + (y(t∗;um)− y(t∗;u∗)), (50) converges weakly to zero in l2(ω,r∞), and therefore y(t∗;u∗) ∈ h as h is closed and convex, hence weakly closed. this shows that h is reached in time t∗ by an admissible control accordingly, t∗ must be the minimum time and u∗ an optimal control. we shall now prove the second part of our theorem. indeed, from theorem 3.1 (lions and magenes, 1972, vol. 1, p. 19) y(u) ∈ w 2,1(q) implies that the mapping t → y(t;u) is 12 bahaa:time-optimal control of infinite variables parabolic systems. . . . . . continuous from [0, t ] → w 1(ω,r∞) ⊂ l2(ω,r∞) is continuous for each fixed u, and so y(t∗;u) 6∈ inth , for any u ∈ u , by the minimality of t∗. from our earlier remarks, the set a(t∗) = {y(t∗;ux) : ux ∈ u}, (51) is weakly compact and convex in l2(ω,r∞). applying theorem 21.11 of (choquet, 1969) to the setsa(t∗) and h shows that there exists a nontrivial hyperplane z ∈ l2(ω,r∞) separating these sets, that is, ∫ ω zy(t∗;u) dρ ≤ ∫ ω zy(t∗;u∗) dρ ≤ ∫ ω zy dρ (52) for all u ∈ u and y ∈ l2(ω,r∞) with ||y − zd||l2(ω,r∞) ≤ ε. from the second inequality in (52), z must support the seth at y(t∗;u∗). sincel2(ω,r∞) is a hilbert space, z must be of the form z = µ(zd − y(t∗;u∗)) for some µ > 0. (53) subsequently, dividing (52) by µ gives the desired result (42). we shall apply theorem 4.1 to the control problem of (1)–(5). to simplify (42), we introduce the adjoint equation, and for every u ∈ u we define the adjoint variable p = p(u) = p(x, t;u) as the solution of the following system −∂p(u) ∂t +a∗(t)p(u) + ∫ b a c(x, t+h)p(x, t+h;u) dh = 0, x ∈ ω, t ∈ (0, t∗− b), (54) −∂p(u) ∂t +a∗(t)p(u) + ∫ t∗−t a c(x, t+ h)p(x, t+ h;u) dh = 0, x ∈ ω, t ∈ (t∗− b, t∗− a), (55) −∂p(u) ∂t +a∗(t)p(u) = 0, x ∈ ω, t ∈ (t∗ − a, t∗), (56) p(x, t∗;u) = zd(x)− y(x, t∗;u), x ∈ ω, (57) ∂p(u) ∂ηa∗ (x, t) = ∫ b a d(x, t+ h)p(x, t+ h;u) dh, x ∈ γ, t ∈ (0, t∗ − b), (58) ∂p(u) ∂ηa∗ (x, t) = ∫ t∗−t a d(x, t+ h)p(x, t+ h;u) dh, x ∈ γ, t ∈ (t∗ − b, t∗ − a), (59) ∂p(u) ∂ηa∗ (x, t) = 0, x ∈ γ, t ∈ (t∗ − a, t∗), (60) where ∂p(u) ∂ηa∗ (x, t) = ∞∑ k=1 (dkp(u)) cos(n, xk), (61) a∗(t)p(u) = ( − ∞∑ k=1 d2 k + q(x, t) ) p(u). (62) advances in systems science and applications (2011), vol. 11, no. 1-2 13 remark 4.2. if t∗ < b, then we consider (55) and (59) on ω×(0, t∗−a) and γ×(0, t∗−a), respectively. the existence of a unique solution to the problem (54)–(60) on the cylinder ω× (0, t∗) can be proved using a constructive method. it is easy to notice that for given zd and u, the problem (54)–(60) can be solved backwards in time starting from t = t∗, i.e., first, solving (54)–(60) on the sub-cylinder qk and in turn onqk−1, and so on, until the procedure covers the whole cylinder ω × (0, t∗). for this purpose, we may apply theorem 3.3 (with an obvious change of variables). hence, using theorem 3.3, the following result can be proved. theorem 4.3. let the hypothesis of theorem 3.3 be satisfied. then for given zd ∈ l2(ω,r∞) and any u ∈ l2(q), there exists a unique solution p(u) ∈ w 2,1(ω × (0, t∗)) for the adjoint problem (54)–(60). now, we have the main result. theorem 4.4. if the assumptions concerning system (1)–(5) and controllability condition (41) are satisfied, then the time-optimal control u∗ exists and is characterized by the following condition ∫ t∗ 0 ∫ ω p(u∗)(u− u∗) dρ dt ≤ 0, ∀u ∈ u, (63) where p(u∗) is the solution of the adjoint system (54)–(60). proof. we simplify the left-hand side of the inequality (42) using the adjoint equation (54)– (60). for this purpose, setting u = u∗ in (54)–(60), multiplying both sides of (54), (55) and (56) by y(u) − y(u∗), then integrating over ω × (0, t∗ − b), ω × (t∗ − b, t∗ − a) and ω× (t∗− a, t∗) respectively and then adding both sides of (54), (55) and (56), we get∫ t∗ 0 ∫ ω ( −∂p(u ∗) ∂t +a∗(t)p(u∗) ) (y(u)− y(u∗)) dρ dt + ∫ t∗−b 0 ∫ ω (∫ b a c(x, t+ h)p(x, t+ h;u∗) dh ) × (y(x, t;u)− y(x, t;u∗)) dρ dt + ∫ t∗−a t∗−b ∫ ω (∫ t∗−t a c(x, t+ h)p(x, t+ h;u∗) dh ) (y(x, t;u)− y(x, t;u∗)) dρ dt =− ∫ ω p(x, t∗;u∗)(y(x, t∗;u)− y(x, t∗;u∗)) dρ+ ∫ t∗ 0 ∫ ω p(u∗) ∂ ∂t (y(u)− y(u∗)) dρ dt + ∫ t∗ 0 ∫ ω a∗p(u∗)(y(u)− y(u∗)) dρ dt + ∫ t∗−b 0 ∫ ω ∫ b a c(x, t+ h)p(x, t+ h;u∗) (y(x, t;u)− y(x, t;u∗)) dh dρ dt + ∫ t∗−a t∗−b ∫ ω ∫ t∗−t a c(x, t+ h)p(x, t+ h;u∗) (y(x, t;u)− y(x, t;u∗)) dh dρ dt = 0. (64) 14 bahaa:time-optimal control of infinite variables parabolic systems. . . . . . then, applying (57), the equation (64) can be expressed as ∫ ω (zd − y(t∗;u∗))(y(x, t∗;u)− y(x, t∗;u∗)) dρ = ∫ t∗ 0 ∫ ω p(u∗) ∂ ∂t (y(u)− y(u∗)) dρ dt+ ∫ t∗ 0 ∫ ω a∗p(u∗)(y(u)− y(u∗)) dρ dt + ∫ b a ∫ ω ∫ t∗−b 0 c(x, t+ h)p(x, t+ h;u∗) (y(x, t;u)− y(x, t;u∗)) dt dρ dh + ∫ t∗−t a ∫ ω ∫ t∗−a t∗−b c(x, t+ h)p(x, t+ h;u∗) (y(x, t;u)− y(x, t;u∗)) dt dρ dh. (65) using (1), the first integral on the right-hand side of (65) can be rewritten as ∫ t∗ 0 ∫ ω p(u∗) ∂ ∂t (y(u)− y(u∗)) dρ dt =− ∫ t∗ 0 ∫ ω p(u∗)a(y(u)− y(u∗)) dρ dt − ∫ t∗ 0 ∫ ω p(x, t;u∗) (∫ b a c(x, t)(y(x, t− h;u)− y(x, t− h;u∗)) dh ) dρ dt + ∫ t∗ 0 ∫ ω p(x, t;u∗)(u− u∗) dρ dt advances in systems science and applications (2011), vol. 11, no. 1-2 15 =− ∫ t∗ 0 ∫ ω p(u∗)a(y(u)− y(u∗)) dρ dt − ∫ t∗ 0 ∫ ω ∫ b a p(x, t;u∗)c(x, t)(y(x, t− h;u)− y(x, t− h;u∗)) dh dρ dt + ∫ t∗ 0 ∫ ω p(x, t;u∗)(u− u∗) dρ dt =− ∫ t∗ 0 ∫ ω p(u∗)a(y(u)− y(u∗)) dρ dt − ∫ b a ∫ ω ∫ t∗ 0 p(x, t;u∗)c(x, t)(y(x, t− h;u)− y(x, t− h;u∗)) dt dρ dh + ∫ t∗ 0 ∫ ω p(x, t;u∗)(u− u∗) dρ dt =− ∫ t∗ 0 ∫ ω p(u∗)a(y(u)− y(u∗)) dρ dt − ∫ b a ∫ ω ∫ t∗−h −h p(x, t′ + h;u∗)c(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dρ dh + ∫ t∗ 0 ∫ ω p(x, t;u∗)(u− u∗) dρ dt =− ∫ t∗ 0 ∫ ω p(u∗)a(y(u)− y(u∗)) dρ dt − ∫ b a ∫ ω ∫ 0 −h p(x, t′ + h;u∗)c(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dρ dh − ∫ b a ∫ ω ∫ t∗−b 0 p(x, t′ + h;u∗)c(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dρ dh − ∫ b a ∫ ω ∫ t∗−h t∗−b p(x, t′ + h;u∗)c(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dρ dh + ∫ t∗ 0 ∫ ω p(x, t;u∗)(u− u∗) dρ dt =− ∫ t∗ 0 ∫ ω p(u∗)a(y(u)− y(u∗)) dρ dt − ∫ b a ∫ ω ∫ 0 −h p(x, t′ + h;u∗)c(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dρ dh − ∫ b a ∫ ω ∫ t∗−b 0 p(x, t′ + h;u∗)c(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dρ dh − ∫ t∗−t a ∫ ω ∫ t∗−a t∗−b p(x, t′ + h;u∗)c(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dρ dh + ∫ t∗ 0 ∫ ω p(x, t;u∗)(u− u∗) dρ dt. (66) the second integral on the right-hand side of (65), in view of green formula, can be 16 bahaa:time-optimal control of infinite variables parabolic systems. . . . . . expressed as ∫ t∗ 0 ∫ ω a∗p(u∗)(y(u)− y(u∗)) dρ dt = ∫ t∗ 0 ∫ ω p(u∗)a(y(u)− y(u∗)) dρ dt + ∫ t∗ 0 ∫ γ p(u∗) ( ∂y(u) ∂ηa − ∂y(u∗) ∂ηa ) dγ dt− ∫ t∗ 0 ∫ γ ∂p(u∗) ∂ηa∗ (y(u)− y(u∗)) dγ dt. (67) using the boundary condition (4), the second component on the right-hand side of (67) can be written as ∫ t∗ 0 ∫ γ p(u∗) ( ∂y(u) ∂ηa − ∂y(u∗) ∂ηa ) dγ dt = ∫ t∗ 0 ∫ γ p(x, t;u∗) (∫ b a d(x, t)(y(x, t− h;u)− y(x, t− h;u∗)) dh ) dγ dt = ∫ t∗ 0 ∫ γ ∫ b a p(x, t;u∗)d(x, t)(y(x, t− h;u)− y(x, t− h;u∗)) dh dγ dt = ∫ b a ∫ γ ∫ t∗ 0 p(x, t;u∗)d(x, t)(y(x, t− h;u)− y(x, t− h;u∗)) dt dγ dh = ∫ b a ∫ γ ∫ t∗−h −h p(x, t′ + h;u∗)d(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dγ dh = ∫ b a ∫ γ ∫ 0 −h p(x, t′ + h;u∗)d(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dγ dh + ∫ b a ∫ γ ∫ t∗−b 0 p(x, t′ + h;u∗)d(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dγ dh + ∫ b a ∫ γ ∫ t∗−h t∗−b p(x, t′ + h;u∗)d(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dγ dh = ∫ b a ∫ γ ∫ 0 −h p(x, t′ + h;u∗)d(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dγ dh + ∫ b a ∫ γ ∫ t∗−b 0 p(x, t′ + h;u∗)d(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dγ dh + ∫ t∗−t a ∫ γ ∫ t∗−a t∗−b p(x, t′ + h;u∗)d(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dγ dh. (68) the last component in (67) can be rewritten as ∫ t∗ 0 ∫ γ ∂p(u∗) ∂ηa∗ (y(u)− y(u∗)) dγ dt = ∫ t∗−b 0 ∫ γ ∂p(u∗) ∂ηa∗ (y(u)− y(u∗)) dγ dt + ∫ t∗−a t∗−b ∫ γ ∂p(u∗) ∂ηa∗ (y(u)− y(u∗)) dγ dt+ ∫ t∗ t∗−a ∫ γ ∂p(u∗) ∂ηa∗ (y(u)− y(u∗)) dγ dt. (69) advances in systems science and applications (2011), vol. 11, no. 1-2 17 substituting (68) and (69) into (67) and then (66) and (67) into (65), we obtain∫ ω (zd − y(t∗;u∗))(y(x, t∗;u)− y(x, t∗;u∗)) dρ =− ∫ t∗ 0 ∫ ω p(u∗)a(y(u)− y(u∗)) dρ dt − ∫ b a ∫ ω ∫ 0 −h c(x, t+ h)p(x, t+ h;u∗)(y(x, t;u)− y(x, t;u∗)) dt dρ dh − ∫ b a ∫ ω ∫ t∗−b 0 c(x, t+ h)p(x, t+ h;u∗)(y(x, t;u)− y(x, t;u∗)) dt dρ dh − ∫ t∗−t a ∫ ω ∫ t∗−a t∗−b c(x, t+ h)p(x, t+ h;u∗)(y(x, t;u)− y(x, t;u∗)) dt dρ dh + ∫ t∗ 0 ∫ ω p(u∗)a(y(u)− y(u∗)) dρ dt + ∫ b a ∫ γ ∫ 0 −h d(x, t+ h)p(x, t+ h;u∗)(y(x, t;u)− y(x, t;u∗)) dt dγ dh + ∫ b a ∫ γ ∫ t∗−b 0 d(x, t+ h)p(x, t+ h;u∗)(y(x, t;u)− y(x, t;u∗)) dt dγ dh + ∫ t∗−t a ∫ γ ∫ t∗−a t∗−b d(x, t+ h)p(x, t+ h;u∗)(y(x, t;u)− y(x, t;u∗)) dt dγ dh + ∫ t∗ 0 ∫ ω p(x, t;u∗)(u− u∗) dρ dt − ∫ t∗−b 0 ∫ γ ∂p(u∗) ∂ηa∗ (y(u)− y(u∗)) dγ dt − ∫ t∗−a t∗−b ∫ γ ∂p(u∗) ∂ηa∗ (y(u)− y(u∗)) dγ dt− ∫ t∗ t∗−a ∫ γ ∂p(u∗) ∂ηa∗ (y(u)− y(u∗)) dγ dt + ∫ b a ∫ ω ∫ t∗−b 0 c(x, t+ h)p(x, t+ h;u∗)(y(x, t;u)− y(x, t;u∗)) dt dρ dh + ∫ t∗−t a ∫ ω ∫ t∗−a t∗−b c(x, t+ h)p(x, t+ h;u∗)(y(x, t;u)− y(x, t;u∗)) dt dρ dh. (70) then, using the fact that y(x, t;u) = y(x, t;u∗) = φ0(x, t) for x ∈ ω and t ∈ [−b, 0), and y(x, t;u) = y(x, t;u∗) = ψ0(x, t) for x ∈ γ and t ∈ [−b, 0), we obtain∫ ω (zd − y(t∗;u∗))(y(x, t∗;u)− y(x, t∗;u∗)) dρ = ∫ t∗ 0 ∫ ω p(x, t;u∗)(u− u∗) dρ dt, (71) then, substituting (71) into (42), we get (63) and this finishes proof of the theorem. 5. generalization time-optimal control problem presented her can be extended to certain different two cases. case 1: time-optimal control problem for (2 × 2) coupled system of parabolic equations with infinite number of variables, in which time lags appear in integral form in both the state equation 18 bahaa:time-optimal control of infinite variables parabolic systems. . . . . . and the boundary condition. case 2: time-optimal control problem for (n×n) coupled system of parabolic equations with infinite number of variables, in which time lags appear in integral form in both the state equation and the boundary condition. 5.1 time-optimal control problem for (2 × 2) coupled system of parabolic equations with infinite number of variables. we can extend the discussions to study the time-optimal control problem for 2×2 coupled system of parabolic equations with infinite number of variables, in which time lags appear in integral form in both the state equation and the boundary condition. consider now the distributed-parameter system described by the following (2× 2) coupled system of parabolic equations with infinite number of variables, for i = 1, 2, ∂yi ∂t + a(t)yi + ∫ b a ci(x, t)y(x, t− h) dh = ui, x ∈ ω, t ∈ (0, t ), h ∈ (a, b), (72) yi(x, t ′) = φi,0(x, t′), x ∈ ω, t′ ∈ [−b, 0), (73) yi(x, 0) = yi,0(x), x ∈ ω, (74) ∂yi(x, t) ∂ηa = ∫ b a di(x, t)yi(x, t− h) dh+ vi, x ∈ γ, t ∈ (0, t ), h ∈ (a, b), (75) yi(x, t ′) = ψi,0(x, t′), x ∈ γ, t′ ∈ [−b, 0), (76) where a(t)yi(x) = ( − ∞∑ k=1 d2 k + q(x, t) ) yi(x) + 2∑ j=1 aijyj(x) ∀ i = 1, 2, (77) aij = { 1, i ≥ j; −1, i < j. (78) it is easy to see that a(t) is (2× 2) matrix which takes the form a(t) =  − ∞∑ k=1 d2 k + q + 1 −1 1 − ∞∑ k=1 d2 k + q + 1  2×2 . (79) also we have yi ≡ yi(x, t;u), ui ≡ ui(x, t), vi ≡ vi(x, t), u ≡ (u1, u2), ci and di, i = 1, 2, are real c∞ functions defined on q and σ, respectively, φi,0 and ψi,0, i = 1, 2, are initial functions defined on q0 and σ0, respectively. now we discuss the case of the distributed control for the neumann problem. then, as section 3, for u = (u1, u2) ∈ (l2(q))2, we can obtain the following results. advances in systems science and applications (2011), vol. 11, no. 1-2 19 lemma 5.1. let u ∈ ( l2(q) )2 , (80) fj = (f1,j , f2,j) ∈ ( l2(qj) )2 , (81) where fi,j(x, t) = ui(x, t)− ∫ b a ci(x, t)yi,j−1(x, t− h) dh, i = 1, 2, yj−1(·, (j − 1)a) = (y1,j−1(·, (j − 1)a), y2,j−1(·, (j − 1)a)) ∈ ( w 1(ω,r∞) )2 , (82) gj = (g1,j , g2,j) ∈ ( w 1 2 , 1 4 (σj) )2 , (83) where gi,j(x, t) = ∫ b a di(x, t)yi,j−1(x, t− h) dh+ vi(x, t), i = 1, 2. then, there exists a unique solution yj ∈ ( w 2,1(qj) )2 for the mixed initial-boundary value problem (72), (75) and (82). theorem 5.2. let yi,0, φi,0, ψi,0, v and u be given with yi,0 ∈w 1(ω,r∞), φi,0 ∈w 2,1(q0), vi ∈ w 1 2 , 1 4 (σ), ψi,0 ∈ w 1 2 , 1 4 (σ0) and ui ∈ l2(q), i = 1, 2. then, there exists a unique solution y ∈ ( w 2,1(q) )2 for the mixed initial-boundary value problem (72)–(76). moreover, y(·, ja) ∈ ( w 1(ω,r∞) )2 for j = 1, 2, . . .. now, we shall formulate the minimum-time problem for (72)–(76) in the context of the theorem 5.2, i.e., u ∈ u = {u ∈ ( l2(q) )2 : |ui(x, t)| ≤ 1, i = 1, 2}. (84) we shall define the reachable set h such that h = {y ∈ ( l2(ω,r∞) )2 : 2∑ i=1 ||yi − zi,d||l2(ω,r∞) ≤ ε} (85) where zi,d ∈ l2(ω,r∞), i = 1, 2, and ε > 0. solving the stated minimum-time problem is equivalent to hitting the target set h in minimum time, that is, minimizing the time t, for which y(t;u) ∈ h and u ∈ u. moreover, we assume that there exists a t > 0 and u ∈ u with y(t ;u) ∈ h (86) then, as section 4, we can prove the following theorem theorem 5.3. if the assumption (86) holds, then the set h is reached in minimum time t∗ by an admissible control u∗ ∈ u. moreover 2∑ i=1 ∫ ω [zi,d − yi(t∗;u∗)] [yi(t ∗;u)− yi(t∗;u∗)] dρ ≤ 0, ∀u ∈ u. (87) 20 bahaa:time-optimal control of infinite variables parabolic systems. . . . . . now, we introduce the adjoint equation, and for every u ∈ u we define the adjoint variable p = p(u) = p(x, t;u) as the solution of the following (2× 2) system −∂pi(u) ∂t +a∗(t)pi(u)+ ∫ b a ci(x, t+h)pi(x, t+h;u) dh = 0, x ∈ ω, t ∈ (0, t∗−b), (88) −∂pi(u) ∂t +a∗(t)pi(u)+ ∫ t∗−t a ci(x, t+h)pi(x, t+h;u) dh = 0, x ∈ ω, t ∈ (t∗−b, t∗−a), (89) −∂pi(u) ∂t + a∗(t)pi(u) = 0, x ∈ ω, t ∈ (t∗ − a, t∗), (90) pi(x, t ∗;u) = zi,d(x)− yi(x, t∗;u), x ∈ ω, (91) ∂pi(u) ∂ηa∗ (x, t) = ∫ b a di(x, t+ h)pi(x, t+ h;u) dh, x ∈ γ, t ∈ (0, t∗ − b), (92) ∂pi(u) ∂ηa∗ (x, t) = ∫ t∗−t a di(x, t+ h)pi(x, t+ h;u) dh, x ∈ γ, t ∈ (t∗ − b, t∗ − a), (93) ∂pi(u) ∂ηa∗ (x, t) = 0, x ∈ γ, t ∈ (t∗ − a, t∗), (94) where ∂pi(u) ∂ηa∗ (x, t) = ∞∑ k=1 (dkpi(u)) cos(n, xk), (95) a∗(t)pi(u) = ( − ∞∑ k=1 d2 k + q(x, t) ) pi(u) + 2∑ j=1 ajipj(u), (96) and aji is the transpose of aij . hence, as the proof of theorem 4.4 in section 4, we can prove the following theorem. theorem 5.4. if the assumptions concerning system (72)–(76) and controllability condition (86) are satisfied, then the time-optimal control u∗ = (u∗1, u ∗ 2) exists and is characterized by the following condition 2∑ i=1 ∫ t∗ 0 ∫ ω pi(u ∗)(ui − u∗i ) dρ dt ≤ 0, ∀u ∈ u, (97) where p(u∗) is the solution of the adjoint system (88)–(94). 5.2 time-optimal control problem for (n × n )coupled system of parabolic equations with infinite number of variables. we can extend the discussions to study the time-optimal control problem for n×n coupled system of parabolic equations with infinite number of variables, in which time lags appear in integral form in both the state equation and the boundary condition. advances in systems science and applications (2011), vol. 11, no. 1-2 21 consider now the distributed-parameter system described by the following (n×n) coupled system of parabolic equations with infinite number of variables, for i = 1, 2, . . . , n, ∂yi ∂t +a(t)yi + ∫ b a ci(x, t)y(x, t− h) dh = ui, x ∈ ω, t ∈ (0, t ), h ∈ (a, b), (98) yi(x, t ′) = φi,0(x, t′), x ∈ ω, t′ ∈ [−b, 0), (99) yi(x, 0) = yi,0(x), x ∈ ω, (100) ∂yi(x, t) ∂ηa = ∫ b a di(x, t)yi(x, t− h) dh+ vi, x ∈ γ, t ∈ (0, t ), h ∈ (a, b), (101) yi(x, t ′) = ψi,0(x, t′), x ∈ γ, t′ ∈ [−b, 0), (102) where a(t)yi(x) = ( − ∞∑ k=1 d2 k + q(x, t) ) yi(x) + n∑ j=1 aijyj(x) ∀ i = 1, 2, . . . , n, (103) aij = { 1, i ≥ j; −1, i < j. (104) it is easy to see thata(t) is (n× n) matrix which takes the form a(t) =  − ∞∑ k=1 d2 k + q + 1 −1 · · · −1 1 − ∞∑ k=1 d2 k + q + 1 · · · −1 ... ... ... ... 1 1 · · · − ∞∑ k=1 d2 k + q + 1  n×n . (105) also we have yi ≡ yi(x, t;u), ui ≡ ui(x, t), vi ≡ vi(x, t), u ≡ (u1, u2, . . . , un), ci and di, i = 1, 2, . . . , n, are real c∞ functions defined on q and σ, respectively, φi,0 and ψi,0, i = 1, 2, . . . , n, are initial functions defined on q0 and σ0, respectively. now we discuss the case of the distributed control for the neumann problem. then, as section 3, for u = (u1, u2, . . . , un) ∈ ( l2(q) )n, we can obtain the following results. lemma 5.5. let u ∈ ( l2(q) )n , (106) fj = (f1,j , f2,j , . . . , fn,j) ∈ ( l2(qj) )n , (107) where fi,j(x, t) = ui(x, t)− ∫ b a ci(x, t)yi,j−1(x, t− h) dh, i = 1, 2, . . . , n, 22 bahaa:time-optimal control of infinite variables parabolic systems. . . . . . yj−1(·, (j − 1)a) = (y1,j−1(·, (j − 1)a), y2,j−1(·, (j − 1)a), . . . , yn,j−1(·, (j − 1)a)) ∈ ( w 1(ω,r∞) )n , (108) gj = (g1,j , g2,j , . . . , gn,j) ∈ ( w 1 2 , 1 4 (σj) )n , (109) where gi,j(x, t) = ∫ b a di(x, t)yi,j−1(x, t− h) dh+ vi(x, t), i = 1, 2, . . . n. then, there exists a unique solution yj ∈ ( w 2,1(qj) )n for the mixed initial-boundary value problem (98), (101) and (108). theorem 5.6. let yi,0, φi,0, ψi,0, v and u be given with yi,0 ∈w 1(ω,r∞), φi,0 ∈w 2,1(q0), vi ∈ w 1 2 , 1 4 (σ), ψi,0 ∈ w 1 2 , 1 4 (σ0) and ui ∈ l2(q), i = 1, 2, . . . , n. then, there exists a unique solution y ∈ ( w 2,1(q) )n for the mixed initial-boundary value problem (98)–(102). moreover, y(·, ja) ∈ ( w 1(ω,r∞) )n for j = 1, 2, . . .. now, we shall formulate the minimum-time problem for (98)–(102) in the context of the theorem 5.6, i.e., u ∈ u = {u ∈ ( l2(q) )n : |ui(x, t)| ≤ 1, i = 1, 2, . . . , n}. (110) we shall define the reachable seth such that h = {y ∈ ( l2(ω,r∞) )n : n∑ i=1 ||yi − zi,d||l2(ω,r∞) ≤ ε} (111) where zi,d ∈ l2(ω,r∞), i = 1, 2, . . . , n, and ε > 0. solving the stated minimum-time problem is equivalent to hitting the target seth in minimum time, that is, minimizing the time t, for which y(t;u) ∈ h and u ∈ u . moreover, we assume that there exists a t > 0 and u ∈ u with y(t ;u) ∈h (112) then, as section 4, we can prove the following theorem theorem 5.7. if the assumption (112) holds, then the seth is reached in minimum time t∗ by an admissible control u∗ ∈ u . moreover n∑ i=1 ∫ ω [zi,d − yi(t∗;u∗)] [yi(t ∗;u)− yi(t∗;u∗)] dρ ≤ 0, ∀u ∈ u . (113) now, we introduce the adjoint equation, and for every u ∈ u we define the adjoint variable p = p(u) = p(x, t;u) as the solution of the following (n× n) system, i = 1, 2, . . . , n, −∂pi(u) ∂t +a∗(t)pi(u) + ∫ b a ci(x, t+ h)pi(x, t+ h;u) dh = 0, x ∈ ω, t ∈ (0, t∗ − b), (114) advances in systems science and applications (2011), vol. 11, no. 1-2 23 −∂pi(u) ∂t +a∗(t)pi(u)+ ∫ t∗−t a ci(x, t+h)pi(x, t+h;u) dh = 0, x ∈ ω, t ∈ (t∗−b, t∗−a), (115) −∂pi(u) ∂t +a∗(t)pi(u) = 0, x ∈ ω, t ∈ (t∗ − a, t∗), (116) pi(x, t ∗;u) = zi,d(x)− yi(x, t∗;u), x ∈ ω, (117) ∂pi(u) ∂ηa∗ (x, t) = ∫ b a di(x, t+ h)pi(x, t+ h;u) dh, x ∈ γ, t ∈ (0, t∗ − b), (118) ∂pi(u) ∂ηa∗ (x, t) = ∫ t∗−t a di(x, t+ h)pi(x, t+ h;u) dh, x ∈ γ, t ∈ (t∗ − b, t∗ − a), (119) ∂pi(u) ∂ηa∗ (x, t) = 0, x ∈ γ, t ∈ (t∗ − a, t∗), (120) where ∂pi(u) ∂ηa∗ (x, t) = ∞∑ k=1 (dkpi(u)) cos(n, xk), (121) a∗(t)pi(u) = ( − ∞∑ k=1 d2 k + q(x, t) ) pi(u) + n∑ j=1 ajipj(u), (122) and aji is the transpose of aij . hence, as the proof of theorem 4.4 in section 4, we can prove the following theorem. theorem 5.8. if the assumptions concerning system (98)–(102) and controllability condition (112) are satisfied, then the time-optimal control u∗ = (u∗1, u ∗ 2, . . . , u ∗ n) exists and is characterized by the following condition n∑ i=1 ∫ t∗ 0 ∫ ω pi(u ∗)(ui − u∗i ) dρ dt ≤ 0, ∀u ∈ u , (123) where p(u∗) is the solution of the adjoint system (114)–(120). 6. conclusions and perspectives the results presented in the paper can be treated as a generalization of the results obtained by knowles (1978), and kowalewski and krakowiak (2006; 2008) onto the case of time optimal distributed and boundary control of second order infinite variables parabolic systems with deviating arguments appearing in the integral form both in state equations and in boundary conditions. we considered a different type of control, namely, the control function defined in the distributed and boundary of the spatial domain. sufficient conditions for the existence of a unique solution of such parabolic equations with neumann boundary conditions are proved (lemmas 3.2; 3.4; 3.6; 3.8; 5.1 and 5.5) and (theorems 3.3; 3.5; 3.7; 3.9; 5.2 and 5.6). the optimal control is characterized by using the adjoint equations (theorems 4.3; 4.4; 5.4 and 5.8). the conditions (41; 86 and; 112) plays a fundamental role in controllability problems 24 bahaa:time-optimal control of infinite variables parabolic systems. . . . . . for time-delay parabolic systems. with regard to the controllability assumption (41; 86 and 112), we can investigate the exact controllability problem for the parabolic system (1)-(5). in this paper, we considered the time-optimal distributed and boundary control problem for infinite variables parabolic systems with non-homogeneous neumann and dirichlet boundary conditions. we can also consider an analogous minimum time problem for hyperbolic systems with non-homogeneous neumann and dirichlet boundary conditions. finally, we can consider the time-optimal control problem for discrete time delay distributed and boundary parameter systems. the ideas mentioned above will be developed in forthcoming papers. references [1] bahaa, g. m. 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(1987). optimal control computation for parabolic systems with boundary conditions involving time delays. j. optim. theory appl., 53, 475-57. advances in systems science and applications (2013) vol.13 no.4 379-391 purpose and non-purpose resource use models in two-level control systems olga i. gorbaneva and guennady a. ougolnitsky southern federal university, russia abstract in the paper a two-level control system consisting of one element in top level and one element in bottom level is considered. both levels have purpose use and non-purpose use interests. the model of resource allocation between purpose and non-purpose use interests for different classes of payoff functions is investigated. the model is built as a two-person game where the stackelberg equilibrium is found. analytical and numerical results are presented. keywords resource allocation, purpose use interests, non-purpose use interests, stackelberg equilibrium 1 introduction many problems of the social-economic development are solved due to the federal financing. the financing has different forms (grants, subsidies, assignments, credits) and is always strictly purpose oriented, i.e. the allocated resources should be spent for the pre-scribed needs only. there are legislative sanctions for the non-purpose use of the federal financing. nevertheless, the non-purpose use of federal resources is widely spread and may be considered as a variety of the opportunistic behavior meeting the private interests of active agents[1]. the non-purpose resource use is closely connected with corruption, especially with so-called “returns” when federal resources in some programs are assigned in exchange of a bribe and only partly satisfy the prescribed social destination being mostly used in the private interests of bribe-givers. it is natural to consider the problem of non-purpose resource use from the point of view of the concordance of interests in hierarchical control systems. this permits to use the mathematical formalism of the hierarchical game theory [2], the theory of incentives [3] and the theory of organizational systems [4-5]. in the same time, namely the models of resource allocation in the hierarchical systems with respect of their non-purpose use are scantily known and are analyzed by the authors’ methodology [6]. in this paper the emphasis is on the dependence of the distribution of resources between the purpose and non-purpose use on different classes of payoff functions characterizing the common (social) interest and the private interests of the resource distributor and resource recipient. in section 2 the structure of investigation is presented. sections 3 and 4 include analytical and numerical results respectively. section 5 concludes. 380 olga i. gorbaneva: purpose and non-purpose resource use models in ... 2 structure of investigation let us consider a two-level control system which consists of one element on the top level a1 (resource distributor-she) and one element on the bottom level a2 (resource recipient-he). without loss of generality we can equate to one the number of resources on the top level. the distributor delegates a part of her resources to the recipient for the purpose use, and the other part she keeps for the non-purpose use. in turn, the bottom level divides the received resource between the purpose use and his private non-purpose use. both levels have their shares in the purpose-use income and have their private payoff functions (fig.1). fig.1 the structure of the modeled system the model is built as a hierarchical two-person game in which a stackelberg equilibrium is sought [2]. the payoff functions of both players include two terms: a non-purpose income and the respective share of the purpose-use income. so, the payoff functions are: g1(u1, u2) = a1(1− u1) + b1(u1, u2)c(u1, u2) → max u1 , g2(u1, u2) = a2(u1, 1− u2) + b2(u1, u2)c(u1, u2) → max u2 subject to 0 ≤ ui ≤ 1. and the following conditions for the functions a, b and c: ai ≥ 0, ∂ai ∂ui ≤ 0, ∂ai ∂uj ̸=i ≥ 0,bi ≥ 0, ∂c ∂ui ≥ 0, i = 1, 2. here the subscript 1 relates to the parameters of the top level (the leader), and the subscript 2 relates to the parameters of the bottom level (the follower); ui a part of resources assigned by the i-th level for the purpose use (respectively the part 1ui leaves for the private non-purpose use); advances in systems science and applications (2013) vol.13 no.4 381 gi the i-th level payoff function; ai the i-th level function of his/her private interest; bi a share of the purpose-use income received by the i-th level; c a function of the purpose-use income of the whole system (society, organization). as functions a and c power, exponential and logarithmic functions of the variables u1 and u2 are considered which are cumulative ones, i.e. a1=a1(1-u1), a2=a2(u1 (1-u2)), c=c(u1u2). in this case the share u1u2 of resources is assigned for the purpose use. the relations a1=a1(1-u1), a2=a2(u1 (1-u2)) reflect the system’s hierarchical structure. the non-purpose income of the top level does not depend on the part of resources assigned by the bottom level for the purpose use, but the non-purpose income of the bottom level depends on the share of resources received by him from the top level. the following distributions of the purpose-use income b are considered: (1) uniform one, in particular, for n = 2 bi = 1 2 , i = 1, 2. (2) proportional one b1 = u1 u1 + u2 , b2 = u2 u1 + u2 . in this paper it is assumed that b1 + b2 = 1. the strategy of a player i is a part ui of his/her resources as-signed for the purpose use. the top-level player moves first, i.e. she chooses a value u1 and informs about it the bottom-level player who chooses his optimal reaction u2. the aim of investigation is study of the influence of the relations between functions a1, a2, b1, b2, c to the solution of the game (stackelberg equilibrium). the following types of non-purpose use functions were used: power function with an exponent smaller than one (a(x) = axα, 0 < α < 1, a > 0), linear function (a(x) = ax), power function with an exponent greater than one (a(x) = axk, k > 1, a > 0); exponential function (a(x) = a(1− e−λx), λ > 0, a > 0); logarithmic function (a(x) = a log2(1 + x), a > 0). almost all of the functions satisfy the conditions ∂a/∂x ≥ 0, ∂2a/∂x2≤0 (except the second condition for the function a (x) = axk, k > 1). similarly, the following types of purpose use functions were used: power function with an exponent smaller than one (c (x) = cxα, 0 < α < 1, c > 0); 382 olga i. gorbaneva: purpose and non-purpose resource use models in ... linear function (c(x) = cx); power function with an exponent greater than one (c(x) = cxk, k > 1, c > 0); exponential function (c(x) = c(1− e−λx), λ > 0, c > 0); logarithmic function (c(x) = c log2(1 + x), c > 0). thirteen of the possible twenty five combinations of the functions a and c are studied analytically, namely: (1) combinations of the one-type functions (both functions a and c are power, exponential, or logarithmic ones); (2) combinations of any non-purpose use function with linear purpose-use function; (3) combinations of any purpose-use function with non-purpose use linear function. six of other twelve cases are investigated numerically. 3 analytical investigation of the different classes of models first, let’s consider the following parameterization: a1(u1, u2) = a1(1− u1), a2(u1, u2) = a2u1(1−u2), c(u1, u2) = clog2(1 + u1u2), b1 = b, b2 = 1− b. in this case the payoff functions are g1(u1, u2) = a1(1− u1) + bclog2(1 + u1u2) (1) g2(u1, u2) = a2u1(1−u2) + (1− b)clog2(1 + u1u2) (2) subject to 0 ≤ ui ≤ 1, i = 1, 2. omitting the calculations, consider each branch of the stackelberg equilibrium separately: i. u = (0; 0) , if a2 > (1− b) c/ln2 or a1 > bc/ln2 (fig.2), i.e. for one of the players the non-purpose resource use is much more profitable than the purposeuse activity; therefore, it is disadvantageous for him/her to assign resources for the purpose-use activity, and in this case it is also disadvantageous for the other player. the payoffs in this case are equal to: g1 = a1, g2 = 0. ii. u = (1; 1) , if a2 < (1− b) c/ln2 and a1 < bc/ln2 (fig.2), i.e. for both players the purpose-use activity is much more profitable, and each of them assigns all resources for it. the payoffs are equal to: g1 = bc, g2 = (1− b) c. iii. u = (bc/ (a1ln2)− 1; 1), if the conditions bc/2ln2 < a1 < bc/ln2 and a2 < a1 (1− b) /b are satisfied (fig.2), i.e. for the top-level player it is profitable to assign only a part of her resources for the purpose use because her incomes from both activities are comparable, meanwhile for the bottom-level player it is profitable to assign all his resources to the purpose-use activity. the payoffs are advances in systems science and applications (2013) vol.13 no.4 383 fig.2 equilibrium outcomes in the game(1) (2) equal to: g1 = 2a1 − bc ln 2 + bclog2 ( bc a1 ln 2 ) , g2 = (1− b)clog2 ( bc a1 ln 2 ) iv. u = ((1− b) c/ (a2ln2)− 1; 1), if (1− b) c/2ln2 < a2 < (1− b) c/ln2 and a2 > a1 (1− b) /b (fig.2), i.e. for both players it is profitable to assign only a part of their resources for the purpose-use activity because their incomes from both types of activities are comparable. but the leader gives to the follower exactly the fixed number of resources which he planned to allocate for the purpose use, therefore compel-ling him to assign all his resources for the purpose use. the players’ payoffs are equal to g1 = 2a1 − a1(1− b)c a2 ln 2 + bclog2 ( (1− b)c a2 ln 2 ) , g2 = (1− b)clog2 ( (1− b)c a2 ln 2 ) . second, consider the following parameterization: a1(u1, u2) = a1(1− u1) k, a2(u1, u2) = a2(u1(1−u2)) k, c(u1, u2) = c(u1u2), b1 = b, b2 = 1− b. 384 olga i. gorbaneva: purpose and non-purpose resource use models in ... then the payoff functions have the form g1(u1, u2) = a1(1− u1) k + bc(u1u2) → max u1 (3) g2(u1, u2) = a2(u1(1−u2)) k + (1− b)c(u1u2) → max u2 (4) subject to 0 ≤ ui ≤ 1, i = 1, 2. the stackelberg equilibrium outcomes are the following : ū = { (1; 1), (a1 < bc)& (a2 < (1− b)c) (0; 0), (a1 > bc) ∨ (a2 > (1− b)c) consider the cases separately (fig.3): fig.3 equilibrium outcomes in the game(3) (4) i. u = (0; 0), if a2 > (1− b) c or a1 > bc, i.e. for one of the players the nonpurpose resource use is much more profitable than the purpose-use one, therefore it is disadvantageous for her/him to finance the purpose-use activity. the payoffs are: g1 = a1, g2 = 0. ii. u = (1; 1), if a2 < (1− b) c and a1 < bc, i.e. for both players the pur-poseuse activity is much more profitable, and each of them assigns all resources for it. the payoffs are equal to: g1 = bc, g2 = (1− b) c. when even one of the functions of purpose or non-purpose re-source use is power with an exponent greater than one (and the other function is the same or linear) then it is advantageous for both players to allocate resources or only to the purpose use (altruistic strategy), or only to the non-purpose use (egoistic strategy). advances in systems science and applications (2013) vol.13 no.4 385 third, consider the following case: a1(u1, u2) = a1(1− e−λ(1−u1)), a2(u1, u2) = a2(1− e−λu1(1−u2)), c(u1, u2) = c(1− e−λu1u2), b1 = b, b2 = 1− b. then the payoff functions have the form g1(u1, u2) = a1(1− e−λ(1−u1)) + bc(1− e−λu1u2) (5) g2(u1, u2) = a2(1− e−λu1(1−u2)) + (1− b)c(1− e−λu1u2) (6) subject to 0 ≤ ui ≤ 1, i = 1, 2. let’s consider all stackelberg outcomes separately (fig.4): fig.4 equilibrium outcomes in the game(5) (6) i. u = (0; 0), if ( a1 > bceλ ) &(a1 > (1− b) c) or a1 > bceλ √ a2 (1−b)c , i.e. for one of the players the non-purpose resource use is much more profitable than the purpose-use one, therefore it is disadvantageous for her/him to finance the purpose-use activity. the payoffs are: g1 = a1(1− e−λ), g2 = 0. ii. u = (1; 1), if a1 < bce−λ and a2 < (1− b) ce−λ, i.e. for both players the purpose-use activity is much more profitable, and each of them assigns all resources for it. the payoffs are equal to:g1 = bc(1− e−λ), g2 = (1− b)c(1− e−λ). iii. u = ( 1; 12 − 1 2λ ln a2 (1−b)c ) , if a1 < b 2 √ a2c 1−be −λ 2 and (1− b) ce−λ < a2 < (1− b) ceλ , i.e. for the top-level player it is profitable to assign all her resources to the purpose-use activity, meanwhile for the bottom-level player it is advantageous to divide his resources. payoffs are the following: 386 olga i. gorbaneva: purpose and non-purpose resource use models in ... g1 = bc ( 1− e −λ ( 1 2 − 1 2λ ln a2 (1−b)c )) = b ( 1− e− λ 2 )√ a2c (1− b) , g2 = (1− b) ( 1− e− λ 2 )√ a2c (1− b) . iv. u = ( 1 2 − 1 2λ ln a1 bc ; 1 ) , if bce−λ < a1 < bceλ and a2 < (1 − b) √ bc a1 ce λ 2 , i.e. the situation is opposite to the previous one. the payoffs are: g1 = a1 − a1e −λ 2 √ bc a1 + bc− e− λ 2 √ bca1, g2 = (1− b)c ( 1− e− λ 2 √ a1 bc ) . v. u = ( 2 3 − 2 3λ ln 2a1 b √ 1−b a2c ; λ−ln 2a1a2 b(1−b)c 2 ( λ−ln 2a1 b √ 1−b a2c ) ) , if the conditions b 2 √ a2c 1−be −λ 2 < a1 < b 2 √ a2c 1−be λ and (1− b) √ 2a1 bc e −λ 2 < a2 < bc2 2a1(1−b)e λ are satisfied, i.e. for both players it is profitable to divide their resources. the payoffs are omitted due to their tediousness. at last, consider the following case: a1(u1, u2) = a1(1− u1), a2(u1, u2) = a2 (u1(1−u2)) , c(u1, u2) = c(u1u2) α, b1 = b, b2 = 1− b. then the payoff functions have the form g1(u1, u2) = a1(1− u1) + bc(u1u2) α → max u1 (7) g2(u1, u2) = a2 (u1(1−u2)) + (1− b)c(u1u2) α → max u2 (8) the stackelberg equilibrium has the form (fig.5): ū =  (1; 1), (a1 < bcα)&(a2 < (1− b)cα),( 1−α √ αbc a1 ; 1 ) , (a1 > bcα)&(a2 < a1),( 1−α √ (1−b)cα a2 ; 1 ) , (a2 > (1− b)cα)&(a2 > a1). in this case the egoistic strategy is disadvantageous for both players. besides, the leader is always able to compel the follower to assign all his resources to the advances in systems science and applications (2013) vol.13 no.4 387 fig.5 equilibrium outcomes in the game(7) (8) purpose use. the payoffs are: g1 = a1 − a1 ( α(1− b) a2 ) 1 1−α + b ( αα(1− b)αc a2α ) 1 1−α , g2 = ( αα(1− b)c a2α ) 1 1−α . when the purpose-use function is linear, and the function of non-purpose use is power with exponent smaller than one, the altruistic strategy is profitable for the bottom-level player. the egoistic strategy is disadvantageous for both players. the thirteen analyzed cases are grouped by the structure of equilibrium outcomes of the game: i. one outcome when both functions of purpose and non-purpose use are power with exponent smaller than one. in this case it is profitable for both players to divide their resources between purpose and non-purpose use. ii. two outcomes (0; 0) and (1; 1) (fig.3) when: • the function of non-purpose use is power with exponent smaller than one, and the function of purpose use is linear; • both functions are linear or power with exponent greater than one, in any combination. iii. three outcomes (fig.5) when the non-purpose resource use function is linear, and the purpose-use function is power with exponent smaller than one. in this the altruistic strategy is profitable even for one player. iv. four outcomes (fig.2) in cases when one of the functions is linear, and the 388 olga i. gorbaneva: purpose and non-purpose resource use models in ... other is logarithmic. v. five outcomes (fig.4) when • both functions are linear or exponential in any combination except the case when they are both linear. • both functions are logarithmic. 4 numerical analysis let’s consider an example of the numerical analysis for the following parameterization: a1(u1, u2) = a1(1− u1) α, a2(u1, u2) = a2(u1 (1− u2)) α, c(u1, u2) = c(1− e−λu1u2), b1 = b, b2 = 1− b. in this case the game has the form g1(u1, u2) = a1(1− u1) α + bc(1− e−λu1u2) → max u1 (9) g2(u1, u2) = a2(u1 (1− u2)) α + (1− b)c(1− e−λu1u2) → max u2 (10) to find an optimal strategy of the bottom-level player let’s calculate the derivative of the function g2 with respect to the variable u2 and equate it to zero: ∂g2 ∂u2 (u1, u2) = − a2αu1 α (1− u2) 1−α + λu1(1− b)ce−λu1u2 = 0 (11) let’s prove that the method of bisection is applicable for the solution of (11). note that the second derivative of the function g2 with respect to the variable u2 is negative ∂2g2 ∂u22 (u1, u2) = a2α(1− α)u1 α (1− u2) 2−α − λ2u1 2(1− b)ce−λu1u2 < 0, and therefore the function ∂g2/∂u2 is monotone. now let’s calculate the signs of ∂g2/∂u2 in the ends of the segment [0; 1]. ∂g2 ∂u2 (u1, 0) = −a2αu1 α + λu1(1− b)c (12) ∂g2 ∂u2 (u1, u2) → u2→1− −a2αu1 α 0+ + λu1(1− b)ce−λu1u2 → u2→1− −∞ (13) if (12) is positive then the equation can be solved by bisection, and the solution will be the point of maximum due to the negativity of the second derivative. if (12) is negative then the method of bisection is not applicable but the left side of the equation is monotone and therefore it is negative in the segment [0; 1], so the function g2 de-creases and the point of maximum is u2 = 0. thus, advances in systems science and applications (2013) vol.13 no.4 389 u2 ∗ = { 0, −a2αu1 α + λu1(1− b)c < 0, ∈ (0; 1), −a2αu1 α + λu1(1− b)c > 0. the top-level player can use the information to impel the bottom-level player to choose a non-zero strategy u2 > 0. it is necessary for this to assure the condition −a2αu1 α+λu1 (1− b) c > 0. solving the inequality with respect to u1 we rewrite the condition as u1 > 1−α √ a2α λ(1−b)c . the top-level player can satisfy the condition only if 1−α √ a2α λ(1−b)c < 1 , or a2 < ( λ(1−b)c α )1−α . if the top-level player cannot choose the strategy then the bottom-level player chooses u2 = 0. in this case g1(u1, 0) = a1(1− u1) α. as far as the function g1 decreases with respect to u1 we receive u1 = 0. let’s summarize: i. if a2 > ( λ(1−b)c α )1−α then the leader cannot influence to the follower and u2 = 0, therefore u1 = 0. this case takes place when the effect from non-purpose activity on the bottom level is essentially greater than the effect from his purposeuse activity. ii. if a2 < ( λ(1−b)c α )1−α then the leader can impel the follower to assign a part of his resources for the purpose-use activity by choosing u1 > 1−α √ a2α λ(1−b)c . this case takes place when the effect from purpose-use activity on the bottom level is essentially greater than the effect from his non-purpose use activity. the case of parameterization a1(u1, u2) = a1log2(2− u1), a2(u1, u2) = a2log2(1 + u1(1− u2)), c(u1, u2) = c(u1u2) α, b1 = b, b2 = 1− b. is considered similarly. the payoff functions have the form g1(u1, u2) = a1log2(2− u1) + bc(u1u2) α, (14) g2(u1, u2) = a2log2(1 + u1(1− u2)) + (1− b)c(u1u2) α (15) the results of analysis are presented in fig.6. if a1 < αbcln2 and a2 > α (1− b) cln2 then u1 = 1−α √ (1−b)cα ln 2 a2 , u2 = 1. the payoffs of the players are: g1 ( 1−α √ (1−b)cα ln 2 a2 , 1 ) = a1log2 ( 2− 1−α √ (1−b)cα ln 2 a2 ) + bc ( 1−α √ (1−b)cα ln 2 a2 )α g2  1−α √ (1− b)cα ln 2 a2 , 1  = (1− b)c ( (1− b)cα ln 2 a2 ) α 1−α . 390 olga i. gorbaneva: purpose and non-purpose resource use models in ... fig.6 equilibrium outcomes in the game(14) (15) if a1 < αbcln2 and a2 < α (1− b) cln2 then the altruistic strategy is advantageous for both players: u1 = 1, u2 = 1. the payoffs are:g1 (1, 1) = bc, g2 (1, 1) = (1− b) c. if a1 > αbcln2 then it is profitable for the top-level player to allocate for the purpose use a part of her resource u1 ∈ ( 0;min { 1−α √ (1−b)cα ln 2 a2 ; 1 }) . 5 conclusion in this paper the problem of non-purpose resource use is considered from the point of view of analysis and design of the control mechanisms providing the concordance of interests in the hierarchical (two-level) systems. interests of the agents are described by their payoff functions including two terms: profit from the purpose and non-purpose resource use respectively. different classes of the payoff functions are studied. the top-level control agent (resource distributor) is treated as leader, and the bottom-level agent (resource recipient) as follower what results in the concept of stackelberg equilibrium. the analytical and numerical investigation allows for the following conclusions. • when both purpose and non-purpose interests functions are power (k < 1) then it is advantageous for both players to invest a part of their resources to the purpose use and the other part to the non-purpose use; • when even one of the payoff functions is power (k > 1), and the other is also power (k > 1) or linear then it is advantageous for both players or assign the resources only for the purpose use (the altruistic strategy), or only for non-purpose use (the egoistic strategy); • in other cases the following situations are possible: advances in systems science and applications (2013) vol.13 no.4 391 a) when the payoff from the non-purpose activity of a player is much greater than the payoff from the purpose activity then the egoistic strategy is advantageous; b) when the payoff of non-purpose activity of both players is much smaller than the payoff from the purpose activity then the strategy of pure altruism is advantageous; c) when the payoffs of the purpose and non-purpose activities are comparable then it is profitable to divide the resources between the purpose and non-purpose interests. acknowledgements the work is supported by rfbr (projects 12-01-00017, 12-01-31287) and by southern federal university. references [1] williamson o.e. (1985), the economic institutions of capitalism, free press. [2] basar t, olsder g.y. (1999), dynamic noncooperative game theory, siam, philadelphia, pp.536. [3] laffont j.j, martimort d. (2002), the theory of incentives: the principalagent model, princeton university press, pp.421. [4] prof. d. novikov (2013),mechanism design and management: mathematical meth-ods for smart organizations, n.y: nova science publishers, pp.163. [5] novikov d. (2013), theory of control in organizations, n.y: nova science publishers, pp.341. [6] ougolnitsky g. (2011), sustainable management, n.y: nova science publishers, pp.285. corresponding author guennady a. ougolnitsky can be contacted at: ougoln@gmail.com microsoft word 14 haoming dong, dingfang chen shaojun zou--development framework for gantry crane training system based on v advances in systems science and applications (2011), vol.11, no.3-4 309-314 issn 1078-6236 international institute for general systems studies, inc development framework for gantry crane training system based on virtual reality haoming dong1,2 , dingfang chen1 and shaojun zou2 1wuhan university of technology , wuhan 430063, china； 2wuhan supervise and test institute of especial equipment, , wuhan 430018, china abstract with the rapid development of science and technology, and the improvement of production safety and management standards, the use of virtual reality technology to gantry crane operator teaching simulation to train high quality personnel has become a pressing need. this article points out that virtual reality technology will be applied to gantry crane operator teaching simulation, a virtual simulation on the gantry crane operator teaching system and the composition of achieving. this article focuses on the virtual simulation platform for the establishment and adopted by key technologies, introduces the 3d modeling, kinetic analysis of crane, and the control hardware system, and analyzes and forecasts the application of the simulation system. keywords virtual reality, gantry crane, simulation training system 1.introduction in recent years, the national production safety situation is not optimistic. the gantry crane accidents have occurred occasionally. the main reason is that some gantry crane operators are lack of expertise and operating skills, poor quality of the safety operation, sometimes even wrong operation and so on. at present, the gantry crane as the category of special equipment in china has established a very sound safety training management system for special operations, but the teaching process and method is much lagged. in the teaching process, as a result of outdated teaching materials and methods, the actual operation conditions is limited, leading to the problem that gantry crane operators did not really acquire advanced theoretical knowledge. the practice of operating capacity is not strong, resulting in ineffective training and mining a large number of potential causes of accidents. therefore, it is a pressing task to change the status quo of operating personnel training of the gantry crane, and to improve the quality of personnel in professional and operational skills to ensure the security. implementation of the system can fundamentally change the gantry crane operation training status, and improve the training effectiveness and economize resources, and improve the operating personnel quality and safety skills, and improve the existing examination procedures, and eliminate safety incidents occurring due to human factors, and provide technical support for scientific management to special equipment for operating personnel. 2.virtual reality virtual reality (vr) is the modern hi-tech which can generate realistic visual, hearing, touch the specific scope of the integration of virtual environments with the computer technology. with the necessary equipment, users can interact with the virtual environmental object and exert the impact. in this way, the users can get the immersive experience of the real environment [1]. vr technology is that users can immerse in an artificial virtual environment and then design and complete the mission by interaction with the computers fully through the virtual reality software and its external devices. see its concept in figure 1. 310 dong: development framework for gantry crane training system based on virtual reality it combines advanced information technologies (such as network computing, graphics and image processing, multimedia technology, the new sensors, simulation, etc.), and even psychology, bionics, arts and other fields of research. the purpose is to enable people to build an immersion combined with the actual situation, the efficient multi-dimensional interactions and more harmonious human-computer environment. fig. 1 i3 of vr 3.research of simulated system on gantry crane the system is developed on osg system. it is easy to develop all types of vr system. the simulated system of gantry crane is composed of several parts, including visual system, human-computer system, hardware system. see the function of these parts in figure 2. fig.2 module of function 3.1 creation of visual system the system creates the visual system of gantry crane for the trained personnel. the visual scene includes the exterior and the crane. the factory, material, device and operators constitute the visual scene. and the scene is created by the technology of image processing, such as romance, texture, shade, lighting and so on. these technologies can immerse the operator in the scene. [3] the main body of crane provides suppositional operation platform, and with organic anastomosis of true driver's cage together. the trainee can see operation action's looking at scenery effect in the true cab. see the scenes in figure 3~ 4. advances in systems science and applications (2011), vol.11, no.3-4 311 fig.3 workshop scene fig.4 harbor scene 3.2 the research of human-computer interaction system in order to obtain the real interactive training effect, the sway simulation testing based on physical property and virtual interactive study are to be carried out for the training and assessment system of gantry crane. they mainly includes: (1) the research of elasto-dynamics system is to simulate the carrying sway and simulated failures and accidents of gantry crane under the circumstances of different carrying capacity, different braking time, and different running speed and so on. [2] (2) interactive simulation technology based on the collision detection technology is to achieve the real-time simulation of the gantry crane carrying objects and the collision avoidance of obstacles. and, the stress analysis of wirerope is the key to solve the problem of carrying sway and collision detection during the carrying process. operating condition of gantry machine can be divided into the following parts: the start (i.e. to accelerate movement), the uniform speed movement, the brake (i.e. to decelerate movement). the system does not obtain acceleration during the uniform speed movement, that is, it is easy to carry out real-time simulation since there is no horizontal force attached to a crane and non-sway problem. as for the stage of startup or braking, it will not affect the forces of each other by taking into account the fact that the car and the cart are rigidly connected with the interactive movement. in order to facilitate the analysis, we only take into account the forces on the wire rope during the running of the car when we analyze. when the car starts running or parking brake, the goods will make a sway which will foist additional horizontal forces on the crane structure. the emergence of such a dynamic load has a certain impact on the operation of the crane and will 312 dong: development framework for gantry crane training system based on virtual reality produce sway problems. when the dynamic load calculation of the car braking operating is similar to the operation condition of the startup, the additional horizontal force generated is opposite to it. therefore, the mechanical condition of different car models and dynamic load calculations can be simplified for the car to start, namely the force analysis and the calculation of dynamic load during the acceleration. see the fig.5. figure 5 force analysis during the movement of the car the dynamics analysis of large cart can be gradually worked out through the establishment of the above model and we can conduct calculating simulation. this system enables operator's to interact with the three-dimensional scene well. the operator can be trained to move cars or carts, high or low lift hook, and obtain all kinds of safety device skill in various scenes so that the operator acquires preliminary understanding of the basic principles and fundamental of the gantry crane, and experiences essentials to operate the gantry crane and taste the moving mode of all kinds of safety devices,. 3.3 the research of hardware control system based on the present operating room of the gantry crane, the research of the system utilize the sensor, the data communication and acquisition techniques to design and develop a set of controlling and demanding system in accordance to a general-purpose gantry crane control mode (controller mode), through which each sensor can collect control signal, and connect the main correspondence line through the interface i/o. the hardware control system collect various types of control lever, button signal and control signal to put on processing through the transferring relevant dynamic link storehouse. the transferring procedure is as follows: hplx = pciclose(hplx) public hplx as long public addr as boolean public dwvendorid as integer public dwdeviceid as integer public fuseint as boolean apart from the signal acquisition system, the hardware control system also includes the driver's cab structural design, reasonable layout of the control box, and display devices, sensors, power supply, as well as the location of signal alignment, and so on. on the basis of the above, we focus on the timeliness and anti-jamming of the operation signal. through the research and integration of the above systems, we can enable the trainees to interact directly with the control system. the operating movement signal is gathered by the control system and carried on the real-time processing, then the control visual system responses and gives feedback to the trainee via display device of the visual system. meanwhile the advances in systems science and applications (2011), vol.11, no.3-4 313 operating movement signal is simultaneously delivered to the system of training and assessment, which enables the experts system to make judgments on the operating movements which provide the fundamental basis for the final appraisal training. these four modules are organically linked together to provide users with realistic training environment. 4.the application and merits of virtual reality in training the system is a set of simulation operating system used in the hands-on training, skills training, safety education and practice of the gantry crane. it mainly promotes or improved the effects of training and assessment of the gantry crane operator in following aspects. (1) the use of the immersion of virtual reality to improve the understanding of the structure and working principle of the hoisting machinery. 2）the use of the interactivity of virtual reality to enhance the actual ability of operating personnel. 3）accident reappearance and case analysis the trainees may observe and analyze their own hands-on training process through on-site reappearance during the operation training process, especially when there occur glaring errors, such as a collision, hitting obstacles, gnawing rails, and safety devices movements. the use of the reappearance effect can help to determine and analyze the reasons for the error. it is suitable for students without any experience of operating gantry crane to carry out hands-on training, and some green hands who have qualified to be operators to conduct the actual practice. compare with the previous methods of training and assessment, the simulation operating system of the gantry crane based on the virtual reality technology will have a powerful functions. the advantages are as follows in terms of conducting training, assessment, education of security techniques: 1）be able to set a variety of conditions conveniently, efficiency, economy and high security compared by calculating, the training in terms of the present system can consumes only about one kilowatt in energy and occupies an area of within 10 square. while the previous real vehicle training consumes over 20 kilowatts in terms of energy and occupies over 200 square. if 3,000 persons will be trained each year in our city and 40 hours will be spent on each person per training hours, the present system only in terms of energy consumption will be able to save: （20-1）×3000×40=2280000 kilowatts hour in terms of the current electric charge of 0.5 yuan/ kwh, the total savings on electricity amount to 1.14 million yuan. 2）good reproducibility the factors concerning the working and operating conditions of the gantry crane is hard to manage, so the reproducibility of the real vehicle test is poor. while gantry crane simulator based on virtual reality technology can conveniently carry out the data acquisition, model selection and the environment settings of simulation model with a good reproducibility. 3）high security high-speeding and maximal driving, as well as very dangerous safety experiment can be safely carried out with the utility of the simulation operating system of gantry crane, which cannot achieve in the real vehicle experiment. 5.conclusion the virtual reality technology is an integrated multi-disciplinary technology. its basic philosophy is to use the method of modern science and technology to create artificially a virtual space in which people can achieve interaction such as watching, listening, and moving and so on, just like in the real environment. the users may enter this environment through the computer and can control and interact with the object in the system with timeliness and interactiveness in the three-dimensional environments as its chief features. 314 dong: development framework for gantry crane training system based on virtual reality the application of the virtual reality to the gantry crane operating personnel training is to design the idea of virtual scenes and conceive it to be viewable and feasible; to achieve the vivid spot effects; to provide simultaneously all kinds of direct-viewing and natural sensation interactive method such as the sensations of hearing, seeing, touching, etc. through the computer user’s connection; to utilize the computer simulation trainer to train the operating personnel to the full while shorten the training cycle, and reduce the training expense, thus enhance training quality and efficiency of gantry crane operator. developed gantry crane virtual training system which based on virtual reality technology achieves good results in the process of actual use. acknowledgements the project is funded by general administration quality supervision, inspection and quarantine of people’s public of china (project no. 20070k0229), and wuhan science and technology bureau (project no. 200711021381) references [1] shao-jun zou. on management of hoisting and conveying machinery operators. hoisting and conveying machinery. 2008 vol 03 [2] zhonghua lu, chenlin shen, zhe liu. research on distributed multi-screen overhead crane simulator. 2008 3rd international conference on pervasive computing and applications. icpca08 [3] guoqian wei, zhengyan zhang. research on the scene displaying scheme of the training simulator for crane. proceedings of 2008 ieee 9th international conference on computer aided industrial design & conceptual design vol.1. 2008-11-01 [4] jian li. the development and application of container crane training simulator. master dissertation of shanghai maritime university [5] de segura, j.d.g., peral, r. using virtual reality for gesture and vocal interface validation in industrial environments. 17th international conference on artificial reality and telexistence. icat 2007 , page: 294-5 [6] guoqian wei, qiuhua tang．research on the framework of the crane's virtual design system and its key technologies[c]．2006 ieee international conference on robio： 1293-1298 advances in systems science and applications (2012) vol.12 no.3 238-257 foundations of control methodology dmitry novikov1,2 and elena rusjaeva1 1institute of control sciences ,russian academy of science, moscow, russia 2moscow institute of physics and technology, moscow, russia abstract control methodology is defined as the theory of control activity organization. methodology of control activity, its characteristics, logical and temporal structures are described. philosophical foundations of control methodology are introduced. keywords control, methodology, control activity, control philosophy 1 introduction methodology is the theory of organization of an activity[1-2]. such definition uniquely determinates the subject of methodology which is organization of an activity (an activity is a purposeful human action). methodology being treated as the theory of organization of an activity, one should naturally consider the notion of an “organization”. according to the definition provided by merriamwebster dictionary and[3], an organization is: 1) the condition or manner of being organized; 2) the act or process of organizing or of being organized; 3) an administrative and functional structure (as a business or a political party); also, the personnel of such a structure. thus, an organization may be considered as the property of being organized (the first meaning) and the process of organizing including the result of this process (the second meaning). the third meaning is an organizational system[3]. let us classify an activity based on its ultimate goal (play-learning-labor[2]). in this case, one distinguishes among: -the methodology of play activity (in the first place, play of children); -the methodology of learning activity; -the methodology of labor (professional) activity. next, professional (or practical) activity can be subdivided into: -practical activity (in the fields of material and immaterial production). in the above sense, most of people are engaged in practical professional activity; -specific forms of professional activity such as philosophy, science, art, and religion. accordingly, we separate out philosophic activity, scientific activity, art activity, and religious activity. in scientific literature, one can find relatively full coverage of the methodologies of scientific activity (research methodology), practical activity, educational activity, as well as the basics of the methodologies of art activity and play activity[1-2]. advances in systems science and applications (2012) vol.12 no.3 239 control activity is the primary subject of this paper. control activity represents a type of practical activity, see fig.1. control methodology is the theory of organization of control activity1[4]. within the framework of general methodology approaches[2], it is possible to construct methodologies of other types of practical activity (learning activity, medical activity, etc.) by analogy with control methodology. methodology considers organization of an activity. organizing an activity means arranging it as an integral system with clearly defined characteristics, a logical structure and the accompanying process of its realization, the temporal structure. the corresponding reasoning lies in the pair of the dialectic categories “historical (temporal)” and “logical”. fig.1 types of activity: a classification the logical structure includes the following components of control activity: subject, object, topic, forms, means, methods, and result. the following characteristics of activity are external with respect to this structure: features, principles, conditions, and norms. the process of activity implementation is usually considered within the framework of a project realized in a time sequence by phases, stages and steps. furthermore, this sequence is common for all kinds of activity[2]. the completeness 1alternative approaches to the definition of methodology take place, as well. for instance, methodology is considered as the theory of methods (generally, methods of scientific research). following such idea, one can understand control “methodology” as epistemological foundations of control science. another approach lies in treating methodology as the theory of methods of practical activity. as a branch of cybernetics, the corresponding control “methodology” studies general methods (ways of implementation) of control, e.g., disturbance-based control or deviation-based control. in addition, there exist narrower “branchwise” interpretations, e.g., “project management methodology” as the whole set of general rules of efficient project management. another example consists in “quality management methodology”. and so on. and these are different control “methodologies”! 240 dmitry novikov:foundations of control methodology of an activity cycle (a project) is defined by the following three phases: -design phase,which yields the models of activity of a control subject and a controlled system, as well as and the plan of their implementation; -technological phase,which yields implementation of control actions; -reflexive phase,which yields an estimate of the results of control activity and indicates the necessity of its further correction or “launching” of a new project (i.e., designing a new control system). therefore, it is possible to suggest the following “textitscheme of control methodology”[4]: 1.the characteristics of control activity (features and principles); 2.the logical structure of control activity including subject, object, topic, forms, means, methods, and result of control activity; 3.the temporal structure of control activity (phases, stages, and steps). let us outline the structure and logic of the paper. first the general scheme of any activity is discussed (see section 2). in section 3 characteristics, logical and temporal structures of control activity are presented. philosophical foundations of control methodology are given in section 4. 2 control activity let us consider the basic structural (procedural[1,5]) components of any activity of some subject, see fig.2 (for convenience, the margins of a subject (individual or collective one) are marked by the dotted rectangle. the chain “need → motive→ goal→ tasks→ technology→ action→ result,” highlighted by thick arrows in fig.2, corresponds to a single “cycle” of activity. the goal is decomposed with respect to conditions, norms and principles of activity into a set of tasks. next, taking into account the chosen technology (that is, a system of conditions, forms, methods and means to solve tasks), a certain action is chosen; note that technology includes content and forms, methods and means. the above-mentioned action leads (under the influence of an environment) to a certain result of activity). a particular position within the activity structure is occupied by those components referred to as either self-regulation (in the case of an individual subject) or control (in the case of a collective subject), see fig.4. self-regulation represents a closed control loop. during the process of self-regulation the subject modifies the components of his activity based on the assessment of the achieved results (see the thin arrow in fig.2). thus, we have discussed primary characteristics of activity and the corresponding structural components. now, let us proceed to control issues. starting the discussion about control, one should precisely formulate what control is; therefore, below we give a series of common definitions, paying special attention to the purposefullness of control: advances in systems science and applications (2012) vol.12 no.3 241 fig.2 structural components of activity control is “the process of checking to make certain that rules or standards being applied” (macmillan dictionary). control is “the act or activity of looking after and making decisions about something” (merriam-webster dictionary). control is “an influence on a controlled system with the aim of providing the required behavior of the latter[3]”. there exist numerous alternative definitions, which consider control as a certain element, a function, an action, a process, a result, an alternative, and so on. we would not intend to state another definition; instead, let us merely emphasize that control being implemented by a subject2 should be considered as an activity. such approach, when control is meant as a type of practical activity3 (control activity, management activity-see above) puts many things into place; in fact, it explains “versatile character” of control and balances different approaches to this 2this eliminates from consideration situations when control is implemented by a technical system (activity is inherent to human beings only). hence, control methodology, as the theory of organization of control activity, studies exclusively (!) situations when control is performed by a human being or by a group of people. furthermore, choosing between two remaining alternatives (a controlled system comprises people or represents a techniqal system), we will be mainly focused on the first alternative as the most complicated one. in addition, we emphasize that the activity of a researcher designing a control system is not control activity but scientific activity. similarly, the activity of an engineer designing a technical system is not control activity but practical (engineering) activity. 3at first glance, interpreting control as a sort of practical activity seems a bit surprising. the reader knows that control is traditionally seen as something lofty and very general; however, activity of any manager is organized similarly (satisfies the same general laws) to that of any practitioner, e.g., a teacher, a doctor, an engineer. moreover, sometimes “control” (or management activity) and “organization” are considered together. 242 dmitry novikov:foundations of control methodology notion. let us clarify the last statement. if control is considered as activity of a control subject (principal), then implementing this activity turns out to be a function of a control system; moreover, the control process corresponds to the process of activity, a control action corresponds to its result, etc.[3]. in other words, if a principal and controlled system both represent subjects (see fig.3), then control is activity (of principals) regarding organization of activity (of controlled subjects). therefore, control methodology is the theory of organization of control activity, i.e., the activity of subjects controlling other subjects or objects. one can further increase the level of reflexion (who organizers whose activity). on the one hand, in a multilevel control system the activity of a top manager may be considered as activity regarding organization of activities of his subordinates; in turn, their activity consists in organization of activity of their subordinates, and so on. on the other hand, an army of consultants represent experts in organization of management activity (first of all, the matter applies to management consulting). such consultants regularly operate the term “control methodology” (sometimes, incorrectly and inappropriately). let us take the general formulation of a control problem for a certain system. assume there exist a control subject (a principal) and a controlled system or control object (in terminology of automatic control theory). the state of the controlled system depends on external disturbances, actions of the control subject (control actions) and, probably, on actions of the system, see fig.3 (if the control object appears active). a problem of the control subject consists in performing control actions (see the thick line in fig.3) to ensure a required state of the control object. this is done using information on external disturbances (see the dashed line in fig.3). the so-called subject-object (input-output) structure of a control system is illustrated by fig.3. this is the basic structure used in control theory to study control problems for systems of different nature. fig.3 structure of a control system the primary input-output structure of a control system illustrated by fig.3 advances in systems science and applications (2012) vol.12 no.3 243 bases on the scheme of activity presented by fig.2. the point is that both the control subject and control object carry out the corresponding activity. combining the structure of both sorts of activity according to fig.2, one obtains the structure of control activity illustrated by fig.4. note the following. from agents point of view, the principal is a part of an external environment (numbers of actions in fig.2 and fig.4 coincide), which exerts an influence for a definite purpose (double arrows (1)-(4) and (6) in fig.2), see fig.4. some components of environmental influence may even have a random (nondeterministic) character, and be beyond the principals control. along with actions of the controlled system, these actions exert an impact on the outcome (the state) of the con-trolled system (double arrow (5) in fig.2); see also external disturbances in fig.4. the structure given by fig.4 may be augmented by adding new hierarchical levels. the principles used to describe control in multilevel systems remain unchanged. however, multilevel systems have specifics distinguishing them from a serial combination of two-level “blocks”. fig.4 structural components of control activity for a controlled system, a criterion of operating efficiency depends on its state and (in some cases) on the control actions. the most important feature is whose viewpoint serves a reference in efficiency analysis. suppose that one knows the relationship between the state of a controlled system and control actions applied. hence, it seems possible to consider operating efficiency of a controlled system as a certain function of control actions. such function is referred to as a control 244 dmitry novikov:foundations of control methodology efficiency criterion. consequently, any control problem4 could be formally stated as follows. find feasible control actions ensuring the maximal efficiency (such controls are said optimal). to succeed, one should solve an optimization problem, notably, choose an optimal control (optimal controls). we have provided the general scheme of any human activity. a cycle of activity terminates with achieving a certain result. thus, the efficiency of activity is assessed in the sense of estimating the corresponding result. the presence of a measurable result (otherwise, control makes no sense!) allows for estimating the level of goal attainment as an anticipated, expected result of activity. the efficiency of activity is a degree of conformity between the result and goals of the subject performing the activity. exerting an impact on the components of activity (controlling them), one may influence on the result and efficiency of the activity. control is the activity of a control subject with respect to a controlled system. when a control subject coincides with a controlled system, the matter concerns self-regulation. the result of activity performed by a control subject is defined by his state and the state (the result of activity) of a controlled system. hence, the efficiency of control activity (control efficiency) represents a degree of conformity between the result of operation of a controlled system and goals of a control subject. evaluation of controls ensuring maximal efficiency is the scope of optimization. in fact, optimization consists in finding the best (optimal) alternatives in a set of feasible alternatives under given conditions. let us emphasize the relevance of every word in this statement. using the term “the best alternatives”, we assume there is a certain criterion (several criteria) and a way (ways) to compare alternatives. it is of crucial importance to account for the existing conditions and constraints; varying them leads to optimality of other alternatives under the same criterion (criteria). we have earlier discussed control efficiency. the efficiency being measured, control aims for efficiency optimization (in fact, maximization) under given constraints and conditions. in fact, optimization consists in finding the best (optimal) alternatives in a set of feasible alternatives under given conditions. given the general structure of control activity, lets procedd to its characteristics, logical and temporary structures. 4a problem is something requiring execution or solution; a goal of activity specified in certain conditions. in this book, the term control problem has two meanings. the first (wide) one is searching for an optimal control within the framework of the general model (efficiency maximization as the goal of control activity). the second (narrow) meaning consists in searching for an optimal control of a certain type (e.g., resource allocation problems, operative control problems, etc.). advances in systems science and applications (2012) vol.12 no.3 245 3 characteristics, logical and temporal structures of control activity characteristics, logical and temporal structures of control activity are presented in a summarised form in tables 1-3 correpondingly (see details in[4]). table 1 characteristics of control activity characteristics organization of control activity 1.the personalized nature of control activity; 2.independent goal-setting by a control subject (principal); 3.the mediated outcome of control activity; 4.the creative character of control activity; features of activity 5.the necessity of modeling (predicting, forecasting the behavior of a controlled system under specific control actions); 6.the responsibility of a control subject for the process and result of his her activity and activity of subjects and/or objects controlled by him/her; 7.development and adaptation. principles of activity principles of hierarchy; unification; purposefulness; openness; efficiency; responsibility; non-interference;social and state control; development; completeness and prediction; regulation and resource provision; feedback; adequacy; well-timed control; predictive reflection; adaptivity; rational centralization; democratic control; coordination; ethics . conditions of activity motivational, personnel-related, material and technical, methodical, organizational, financial, regulatory and legal, and informational conditions. norms: 1)general; universal ethical,legal and other norms. 2) specific norms of managerial ethics, organizational culture. table 2 the logical structure of control activity structural components organization of control activity active subject control subject (individual or collective). the object of activity control object and/or controlled subject (individual or collective). 246 dmitry novikov:foundations of control methodology structural components organization of control activity the subject of activity elements of a controlled system, components of activity of a controlled subject. for instance, in the case of organizational systems: staff of the system; structure of the system; constraints and norms of activity of participants; goals and preferences of participants; awareness of participants; the sequence of function-ing. the result of activity state of a control object, result of activity of a controlled subject; consumed resources. the forms of activity organization individual and collective control; unified and personalized control. projectand process-based management; reflectory (situational) and forward-looking control. hierarchical control, distributed control, and network control. the functions of activity in the case of organizational control: planning, organizing, motivating, and controlling. tasks in the case of organizational control: monitoring and analysis of the actual state of a controlled system, forecasting the evolution of the system, goal-setting, planning and distributing the resources, motivation (incentives), control and operative management, analysis and improvement of activity. the methods of activity in the case of organizational systems: staff control; structure control; institutional control (normative control, i.e., control of constraints and norms of activity); motivational control (economic control, i.e., control of preferences); informational control (socio-psychological control, i.e., control of information being available to controlled subjects at the moment of decision making). the means of activity in the case of organizational control: orders, directives, instructions, plans, strategies, policies, norms, standards, procedures, regulations concerning activity organization. advances in systems science and applications (2012) vol.12 no.3 247 structural components organization of control activity mechanisms in the case of organizational control: mechanisms of active expertise, mechanisms of active expertise, transfer pricing mechanisms, mechanisms of contract renegotiation, mechanisms of cost-benefit analysis, mechanisms of institutional control, mechanisms of informational control, integrated rating mechanisms (mechanisms of data aggregation), rankorder tournaments (tenders), multi-channel mechanisms, mechanisms of assignment, mechanisms of exchange, mechanisms of predictive self-control, mechanisms of production cycle optimization, incentive mechanisms for cost reduction, resource allocation mechanisms (including costs and incomes), mechanisms of self-financing, mechanisms of structure choice, mechanisms of staff choice, mechanisms of joint financing, mechanisms of consent, incentive mechanisms, insurance mechanisms, etc. table 3 organizing the process (temporal structure) of control activity temporal structure a control activity cycle phases stages steps 1. design phase 1.1. conceptual stage 1.1.1. identifying contradictions a contradiction between the actual (or forecasted) state of a controlled system and its desired state. 1.1.2.stating a problem a control problem as the need for exerting an impact on activity (state) of a controlled system; such need must be recognized by a control subject. 1.1.3.defining the goal of control. identifying contradictions defining the goals of control as a desired state (result of activity) of a controlled system (in the narrow sense, as a way of organizing of controlled subjects activity). 1.1.4.choosing criteria criteria for describing/assessing the state (result of activity) of a controlled system. control efficiency criteria. 248 dmitry novikov:foundations of control methodology temporal structure a control activity cycle phases stages steps 1.2. modeling stage 1.2.1. constructing a model constructing a model of a controlled system (taking into account its active property if necessary). studying the dependence of the controlled systems state (result of controlled subjects activity) on control actions and the state of an external environment. 1.2.2. optimization solving the problem of optimal control synthesis (for the constructed model of a controlled system). analyzing stability and adequacy of solutions. 1.3. the stage of control planning 1.3.1. decomposing formulating control problems as the goals for specific subproblems ensuring a definite overall goal of control (within the framework of existing constraints). 1.3.2.aggregation coordinating the results of solution of specific control problems, assessing the feasibility of joint application of different methods, means, forms and mechanisms of control. 1.3.3. analyzing the conditions (available resources) analyzing the influence of conditions (resource constraints) on the efficiency of control activity, including resources decomposition by methods, forms, means of control, etc. 1.3.4. making up the program of control identifying the controlled system. choosing conditions, methods, means, forms and mechanisms of control. solving the problem of optimal control synthesis. advances in systems science and applications (2012) vol.12 no.3 249 temporal structure a control activity cycle phases stages steps 1.4.the stage of technological preparations for control 1.4.1. technological preparations detailed elaboration and preparation of necessary conditions, methods, means and forms of control. 2. implem entation phase 2.1. organizing stage implementing conditions, methods, means, forms and mechanisms of control. resources allocation. distributing functions and tasks among elements of a controlled system. 2.2. motivating stage implementing the mechanisms of non-financial and financial incentives of controlled subjects. 2.3. monitoring stage organizing the system of permanent assessment of the activity performed by a controlled subject and/or an external environment. 2.4. the stage of operational management well-timed correction of conditions and mechanisms of control based on monitoring results. 3. reflexiv -e phase the stage of accounting and controlling acquiring information on the results of activity performed by a control subject and a controlled system, results assessment (comparison with posed goals). the stage of activity analysis (results analysis) reflexion as a way of control subjects recognition of his/her activity integrity, as well as of the goals, content, forms, and means of such activity. analyzing the obtained results (taking into consideration resources consumed). the stage of decisions correction in the case of cyclic (repetitive) activity, “local” modification of its content and parameters based on analysis of achieved results. 250 dmitry novikov:foundations of control methodology temporal structure a control activity cycle phases stages steps the stage of activity improvement systematic reviewing of the whole organizational structure of control activity (in particular, efficiency criteria adopted, as well as methods, forms, means and mechanisms of control). 4 philosophical foundations of control methodology a foundation is a sufficient condition of something (entity, cognition, an idea or activity). from the control theory point of view cybernetics and systems analysis are remarkable for occupying the interdisciplinary or overdisciplinary position and may be treated as applied dialectics. within the framework of these approaches, control activity is a complex system intended for preparing, substantiating and implementing solutions to complex problems of different character (e.g., political, social, economic, technical problems, etc.)[6-11]. by comparing the conceptions adopted by different scientific disciplines (viz., philosophy, psychology, sociology and systems analysis or systems engineering), one would easily choose the general structure of activity (see fig.2). but the fundamental foundations for control methodology are given by philosophy. philosophy studies activity as a universal way of human existence. accordingly, humans represent active creatures. human activity covers material-practical, intelligent and spiritual operations, external and internal processes. activity is the behavior of mind just exactly as the behavior of arms, whereas human activity makes up cognition process similarly to human behavior. activity enables an individual to reveal his/her particular place in the world and to assert himself/herself as a social being. having reached a certain level of epistemological maturity, scientists perform “reflexion” by formulating general laws in corresponding scientific fields, i.e., create metasciences. on the other part, any “mature” science becomes the subject of philosophical research. for instance, the philosophy of physics appeared at the junction of the 19th century and the 20th century as the result of such processes[12]. originated in the 1850s, research in the field of control theory5 led to the appearance of other metasciences, i.e., cybernetics[6-7,11] (in the 1950s) and systems analysis[8-10] (later). moreover, cybernetics quickly became the subject of philosophical investigations (e.g., see[11,13]) conducted by “fathers” of cybernetics 5following the established tradition, we will occasionally call control science by control theory (yet, keeping in mind that the name is narrower than the subject). advances in systems science and applications (2012) vol.12 no.3 251 and professional philosophers. the 20th century was accompanied with rapid progress of management science[14-16] as a branch of control theory studying practical control in organizational systems. by the beginning of the 2000s, management science engendered management philosophy. books and papers entitled “management philosophy” appeared exactly at that times (for instance, see[13,15,17]); as a rule, their authors represented professional philosophers. generally speaking, one may acknowledge the long-felt need for more precise mutual positioning of philosophy and control. consider fig.5 illustrating different connections between the categories of philosophy and control; they are treated in the maximal possible interpretation (philosophy includes ontology, epistemology, logic, axiology, ethics, aesthetics, etc.; control is viewed as a science and a type of practical activity). we believe that the three domains shaded in fig.5 are the major ones. control philosophy (as a branch of philosophy). historically (and similarly to the subjects of most modern sciences), control problems analysis was first the prerogative of philosophy. r. descartes was used to say, “philosophy is like a tree whose roots are metaphysics and then the trunk is physics. the branches coming out of the trunk are all the other sciences”. historical and philosophical analysis implies that first control theorists were exactly philosophers. confucius, lao-tzu, socrates, platon, aristotle, n. machiavelli, t. hobbes, i. kant, g. hegel, k. marx, m. weber, a. bogdanovcthis is a short list of philosophers that laid down the foundations of modern control theory for the development and perfection of managerial practice. presently, concrete control problems are no more the subject of philosophical analysis. philosophy (as a form of social consciousness, the theory of general principles of entity and cognition, human attitude to the reality, as the science of universe laws of natural development) studies general prob-lems and laws separated out by experts in certain sciences. by analogy to the notions of “historical philosophy”, “cultural philosophy”, “legal philosophy”, etc. (see philosophical encyclopedias), one can define control philosophy as a branch of philosophy connected with comprehension and interpretation of control processes and control cognition, studying the essence and role of control. such meaning of the term “control philosophy” (see the dashed-line contour in fig.5) has rich internal structure and covers epistemological research of control science, the analysis of logical, ontological, ethical and other foundations (both for control science and management science). the basic goals of research in control philosophy are as follows: 1.identifying the content of control as a science and practical activity, analyzing their subject and place in the system of scientific knowledge; 2.performing the ideological, methodological and logical-epistemological anal252 dmitry novikov:foundations of control methodology fig.5 philosophy and control ysis of primary notions, results, techniques, functions and theories in control science; 3.translating philosophical laws to enrich the content of control laws; 4.involving the achievements of control theory and practice to enrich the content of philosophical categories and laws; 5.substantiating the feasibility and conditions of using common approaches to control problems in systems of interdisciplinary nature, constructing uniform control theory; 6.performing methodological analysis of control with application to different areas of human activity and different classes of control objects; 7.substantiating philosophically the key directions in control theory and practice; 8.systematizing and classifying theories of control; 9.identifying and systematizing axiological dominants in control theory and practice; 10.developing the integrated conceptual framework of control science (including the terminology of all embedded theories). let us formulate a series of “questions” determining perspective directions of research in control philosophy (according to experts in control theory, these issues lie “in the plane” of control philosophy). • what would general laws and regularities studied by philosophy gain for control theory and practice? which modern directions of philosophical research advances in systems science and applications (2012) vol.12 no.3 253 can find (alternatively, have already found) applications in control science (structuralism, post-structuralism, hermeneutics, etc.)? what are the manifestation and influence of general scientific meaningfulness and interdependency of adopted terminology? • what are the epistemological specifics of control science? are there general approaches to the statement and solution of control problems? how does control science position itself in the general system of sciences? what is the epistemological status of a researcher in control theory and practice? • how are basic categories of philosophy (a language, ordinary consciousness, ethics, a law, philosophy, a science, art, a religion, a political ideology, etc.) correlated with that of control science (control, an activity, an organization, decision making)? how is the latter group of categories correlated with other categories (such as a human being, nature, a society, production)? • which laws (features) of control science formation as a metascience can be identified in historical retrospective and at the modern stage of its development? what is the connection between control theory and practice (again, in historical retrospective and in future perspective)? • how does philosophy (as the “quintessence of culture”) affect the formation of “organizational culture” in control theory and practice? what is the interrelation between universal principles, laws and features of development of particular organizational, social and cultural formations in control theory and practice? cybernetics (as a branch of control science, studying its most general theoretical laws). for many scientific disciplines, there exists a range of problems related to their foundations and traditionally referred to as the philosophy of a corresponding science. control science follows this tradition, as well. foundations of control science also include general laws of efficient control (representing the subject of cybernetics). nowadays, one often faces the opinion that cybernetics has become old-fashioned as a scientific discipline and no more pretends to the role of certain universal control science. this is true, but only in part. as a matter of fact, in the middle of the 1940s cybernetics appeared the theory of “control and communication in the animal and the machine” (see the pioneering monograph[11]). furthermore, it originated even as the theory of general laws of control. triumphal advancements of cybernetics during the 1950-1960s (e.g., technical cybernetics, economic cybernetics, biological cybernetics, etc., and well as their close connections to operations research, mathematical theory of control; plus intensive implementation of results in designing new and upgrading existing technical and information systems) created the illusion of the universal character of cybernetics and inevitability of its rapid development in future. however, the evolvement of cybernetics slowed down in the early 1970s. this “integral” science branched out into a set of 254 dmitry novikov:foundations of control methodology partial directions and “mingled with details”; indeed, the number of subbranches grew and all of them showed independent development (almost without identification and systematization of general laws). curiously enough, the only bearers of canonical cybernetic traditions were philosophers, whereas experts in control theory lost their confidence in ample opportunities of cybernetics. things can’t carry on as they are. on the one hand, philosophers vitally need knowledge of the subject (actually, the generalized knowledge). in this context, v. ilin mentioned that “philosophy represents second-rank reflexion; it provides theoretical grounds to other ways of spiritual production. the empirical base of philosophy consists in specific reflections of different types of cognition; philosophy covers not the reality itself, but the treatment of reality in figurative and category-logical forms” (see references in[18]). on the other hand, experts in control theory need “to see the wood for the trees”. hence, one can hypothesize that cybernetics must and would play the role of control philosophy in its second meaning (as a branch of control theory, studying its most general laws). here the emphasis should be made on constructive development of control philosophy, i.e., on formation of its content through obtaining concrete results (probably, first partial results and then general ones). management “philosophy”. a detailed analysis of modern textbooks on management science, sociology and psychology of management separates out the following categories6 used to describe managerial practice (see fig.6). management “philosophy” tops the pyramid demonstrated in fig.6. it reflects the maximally abstracted level of description and consideration of solving the problems of managerial practice. there are intensive discussions regarding the comprehension of management “philosophy”, its subject and main content. for instance, the following opinions are quoted in[18]: -“possibly, management philosophy is the pragmatism, where an essential characteristic of a human being lies in actions, purposeful activity. cognizing exactly the laws of human activity must form the object of management philosophy” (l. bessonova); -“management philosophy considers axiological, epistemological, and methodological foundations of human activity in control processes” (v. diev), and so on. the examples of inhomogeneous definitions could be continued. many authors of textbooks on management science adopt the term “personal management philosophy” (similarly to the existence of numerous opinions regarding necessary qualities of a good leader, there are many different management philosophies). in other words, sometimes management “philosophy” is commonly treated as an6note that the corresponding terms are generally not defined explicitly and addressed somewhat inadvertently (in management science). advances in systems science and applications (2012) vol.12 no.3 255 alyzing the set of qualities of an efficient manager and his/her decisions leading to a success. fig.6 levels and categories of managerial practice description fig.7 control philosophy, cybernetics and management “philosophy” almost all authors agree with the following. management “philosophy” is a system of ideas, views and beliefs of managers about human nature and society, control problems and ethical principles of their behavior (this system forms mostly empirically). yet, we believe such definition appears eclectic and not operational. our approach is to understand management “philosophy” (“the top of 256 dmitry novikov:foundations of control methodology management”) as a branch of control science dealing with generalization of laws of successful managerial practice. we have briefly analyzed the correlation of control philosophy (as a branch of philosophy studying general problems of control theory and practice), cybernetics (as a branch of control science generalizing the methods and results of solving theoretical problems of control) and management (as a branch of control science generalizing the experience of successful managerial practice), see fig.7. 5 conclusion the paper has endeavored to systematize control methodology (as the theory of organizing of control activity). philosophical foundations of the methodology of control activity, its characteristics, as well as the logical and temporal structures were described. a series of “questions” determining perspective directions of research in control philosophy were posed. authors hope that a unified approach (based on control methodology) to the consideration and research of control activity will help to interconnect and develop parallely mathematical control theory and control philosophy. references [1] novikov, a., novikov, d. (2013), research methodology: from philosophy of science to research design, leiden: crc press. [2] novikov, a., novikov, d. (2007), methodology, moscow: sinteg. (in russian). [3] novikov, d. (2013), theory of control in organizations, n.y.: nova science publishers. [4] novikov, d. (2013), control methodology, n.y.: nova science publishers. [5] leontjev, a. (1978), activity, consciousness and personality, prentice: prentice-hall. [6] ashby, w. (1956), an introduction to cybernetics, london: chapman and hall. [7] beer, s. (1995), brain of the firm, 2nd ed. ny: wiley. [8] bertalanffy, l. (1968), general system theory: foundations, development, applications, n.y.: george braziller. [9] mesarovic, m., mako, d. and takahara, y. (1970), theory of hierarchical multilevel systems, new york: academic. advances in systems science and applications (2012) vol.12 no.3 257 [10] peregudov, f. and tarasenko, f. (1993), introduction to systems analysis, oh: columbus, glencoe/mcgraw-hill. [11] wiener, n. (1965), cybernetics: or the control and communication in the animal and the machine, 2nd ed, massachusetts: the mit press. [12] heisenberg, w. (1962), physics and philosophy: the revolution in modern science, new york: harper & row publishers, inc. [13] rosenberg, a. (2000), a philosophy of science: a contemporary introduction, london: routledge. [14] ackoff, r. (1999), classic writings on management, 2nd ed, ny: wiley. [15] drucker, p. (2006), the effective ecutive: the definitive guide to getting the right things done, n.y.: collins business. [16] mintzberg, h. (1983), structure in fives: designing effective organizations, englewood cliffs, nj: prentice-hall. [17] kirkeby, o. (2000), management philosophy: a radical-normative perspective, heidelberg: springer. [18] novikov, d. and rusjaeva, e. (2012), “philosophy and conbtrol”, philosophy problems, no.5, pp.19-26. (in russian). corresponding author author can be contacted at: novikov@ipu.ru. advances in systems science and applications (2014) vol.14 no.3 254-278 mathematical models of informational and strategic reflexion: a survey novikov d.a. and chkhartishvili a.g. institute of control sciences, moscow abstract the paper is dedicated to a survey (in the framework of game theory and theory of collective behavior) of modern approaches to mathematical modeling of reflexive games and reflexive processes in control. keywords mathematical models, informational and strategic reflexion 1 introduction. 1.1 reflexion a fundamental property of human entity lies in the following. in addition to natural (“objective”) reality, there exists its image in human minds. furthermore, an inevitable gap (mismatch) takes place between the latter and the former. in the sequel, the described image will be called a part of reflexive reality. traditionally, purposeful study of this phenomenon relates to the term “reflexion”. the term reflexion (from latin reflex ‘bent back’; was first suggested by j. locke) means [1]: • principle of human thinking, guiding humans towards comprehension and perception of ones own forms and premises; • subjective consideration of a knowledge, critical analysis of its content and cognition methods; • the activity of self-actualization, revealing the internal structure and specifics of spiritual world of a human. to elucidate the whole essence of reflexion, let us consider the case of a single subject. he/she possesses certain beliefs about natural reality; however, a subject may perform reflexion (construct images) with respect to these beliefs (thus, generating new beliefs). generally, this process is infinite and results in formation of reflexive reality. the reflexion of a subject with respect to his/her own beliefs of reality, principles of his/her activity, etc., is said to be self-reflexion or reflexion of the first kind. we emphasize that most social research works concentrate on self-reflexion. in philosophy, self-reflexion represents the process of individuals thinking about beliefs in his/her own mind [2]. reflexion of the second kind takes place with respect to other subjects (includes beliefs of a subject about possible beliefs, decision principles and self-reflexion of other subjects). 1.2 reflexion and control control is an element, a function of organized systems of different nature (biological, social, technical, etc.), preserving their definite structure, sustaining their advances in systems science and applications (2014) vol.14 no.3 255 mode of activity and implementing the program or goal of their activity; control is a purposeful impact exerted on a controlled system to ensure its required behavior [3]. assume there is a control subject (a principal) and a controlled system (control object-in terminology of technical systems-or a controlled subject). the state of a controlled system depends on external disturbances, control actions applied by a principal and possibly on actions performed by the controlled system (if the latter represents an active subject), see fig. 1. the principals problem lies in choosing control actions (see the thick line in fig.1) to ensure the required behavior of a controlled system taking into account information on external disturbances (see the dashed line in fig.1). the so-called input-output structure of a control system (fig.1) is typical for control theory dealing with control problems in systems of different nature. the presence of feedback (see the double line in fig.1) which provides a principal with information on the state of a controlled system is the key (but not compulsory!) property of a control system. some researchers interpret feedback as reflexion (as an image of the controlled systems state in the “mind” of a control subject). this forms the first aspect of interrelation between control and reflexion. fig.1 the structure of a control system a series of scientific directions investigate the interaction and activity of a control subject and controlled system. control science (or control theory in the terminology of corresponding experts) mostly focuses on the interaction between a control subject and controlled system. control methodology [4] is the theory of organizing of control activity, i.e., the activity performed by a control subject. we emphasize that activity can be mentioned only with respect to active subjects (e.g., a human being, a group, a collective). in the case of passive (e.g., technical) systems, the term “functioning” is used instead. in the sequel, we believe that a control subject and controlled system appear active (otherwise, there is a clear provision for the opposite). hence, each of them may perform (at least) 256 novikov d.a. and chkhartishvili a.g.:mathematical models of informational and ... self-reflexion, constructing “images” of the process, organization principles and results of his/her own activity. this is the second aspect of interrelation between control and reflexion. searching for optimal control (i.e., the most efficient admissible control) requires control subjects ability of predicting controlled systems response to certain control actions. one of prerequisites is a model of a controlled system. generally speaking, a model is an image of a certain system; an analog (a scheme, a structure or a sign system) of a certain fragment of the natural or social reality, a “substitute” for the original in cognition process and practice. a model can be considered as an image of a controlled system in the mind of a control subject. modeling (as a process of “reflecting”, i.e., constructing this image) can be viewed as reflexion. furthermore, a controlled system may predict and assess the activity performed by a control subject. and so, we obtain the third aspect of interrelation between control and reflexion. the fourth aspect lies in the following. a control subject or controlled system performs reflexion with respect to external subjects and objects, phenomena or processes, their properties and laws of activity/functioning. for instance, the matter concerns an external environment (for a control subject), an external environment and/or other elements of a controlled system (for a fixed element of a controlled system). indeed, suppose that a controlled system includes several active agents; each of them may perform reflexion with respect to the others. exactly this aspect-mutual reflexion of controlled subjects-is discussed in game-theoretical models. of crucial importance here is that the process and/or result of reflexion can be controlled, i.e., can represent a component of controlled systems activity, being modified by a control subject for a definite goal. precisely this relationship between control and reflexion enables informational control and reflexive control, considered below. 1.3 game theory formal (mathematical) models of human behavior have been constructed and studied for over last 150 years. gradually, these models find wider application in control theory, economics, psychology, sociology, etc., as well as in practical problems. in the sequel, we will understand a game as the interaction of subjects with noncoinciding interests. still, an alternative interpretation treats a game as a type of unproductive activity whose motive consists not in the corresponding results, but in the process of activity itself (see [2, 5], where the notion of a game is assigned a broader sense). game theory represents a branch of applied mathematics, which analyzes models of decision making in the conditions of noncoinciding interests of opponents (players); each player strives for influencing the situation in his/her favor [6-7]. advances in systems science and applications (2014) vol.14 no.3 257 in what follows, a decision-maker (a player) is called an agent. the major task of game theory is describing the interaction among several agents with noncoinciding interests, where the results of agents activity (payoff, utility, etc.) generally depend on actions of all agents. such description yields a forecast of a rational and “stable” outcome of the game-the so-called game solution (equilibrium). describing a game means specifying the following parameters: a set of agents; preferences of agents (relationships between payoffs and actions). each agent is supposed to strive for maxi-mizing his/her payoff (and so, the behavior of each agent appears purposeful); a set of feasible actions of agents; awareness of agents (information on essential parameters, being available to agents at the moment of their choice); sequence of moves (the sequence of obtaining information and choosing actions). the above parameters define a game; unfortunately, they are insufficient for forecasting its outcome, i.e., a solution (or an equilibrium) of the game-the set of rational and stable actions of agents. nowadays, game theory suggests no universal concept of equilibria. by adopting different assumptions regarding principles of agents decision making, one can construct different solutions. thus, designing an equilibrium concept forms a basic problem for any game-theoretic research; this book does not represent an exception, as well. reflexive games are defined as a direct interaction among agents, where they make decisions based on hierarchies of their beliefs. in other words, awareness of agents is extremely important. 1.4 the role of awareness. common knowledge in game theory, psychology, distributed systems and other fields of science (see the overviews in [8-9]), one should consider not only agents beliefs about essential parameters, but also their beliefs about the beliefs of other agents, etc. the set of such beliefs is called the hierarchy of beliefs. we will model it using the tree of awareness structure of a reflexive game (see below). in other words, situations of interactive decision making (modeled in game theory) require that each agent “forecasts” opponents behavior prior to his/her choice. and so, each agent should possess definite beliefs about the view of the game by his/her opponents. on the other hand, opponents should do the same. consequently, the uncertainty regarding the game to-be-played generates an infinite hierarchy of beliefs of game participants. a special case of awareness concerns common knowledge when beliefs of all orders coincide. a rigorous definition of common knowledge was introduced in [10]. notably, common knowledge is a fact with the following properties: 1 ) all agents know it; 258 novikov d.a. and chkhartishvili a.g.:mathematical models of informational and ... 2 ) all agents know 1; 3 ) all agents know 2 and so on-ad infinitum. the formal model of common knowledge was originally proposed in [11]. later on, many investigators refined and redeveloped it see surveys and references in [1,12-18]. the present paper is almost completely dedicated to models of agents awareness in game theory (viz., hierarchies of beliefs and common knowledge). thus, we give several references demonstrating the role of common knowledge in different fields of science-philosophy, psychology, etc. (see also the overview in [19]). in philosophy, common knowledge has been studied in convention analysis [10, 20]. in psychology, one would face the notion of discourse (from latin discursus ‘argument’). it means human thinking in words, being mediated by past experience; discourse acts as the process of connected logical reasoning, where a next idea stems from the previous one. the importance of common knowledge in discourse comprehension has been explored in [19, 21]. mutual awareness of agents turns out significant in distributed computer systems [13, 15, 22], artificial intelligence [23-24] and other fields. game theory often assumes that all1 parameters of a game are a common knowledge. such assumption corresponds to the objective description of a game and enables addressing the nash equilibrium2 concept [25] as a forecasted outcome of a noncooperative game (a game, where agents do not agree about coalitions, data exchange, joint actions, redistribution of payoffs, etc.). thus, the assumption regarding common knowledge allows claiming that all agents know which game they play and that their beliefs about the game coincide. generally, each agent may possess individual beliefs about parameters of a game. and so, each belief corresponds to a subjective description of the game [6] (see also modern models of awareness in [26-29]). consequently, agents participate in the game, having no objective views of it or interpreting this game in different ways (rules, goals, the roles and awareness of opponents, etc.). unfortunately, still no universal approaches have been proposed for equilibria design under insufficient common knowledge. on the other part, within the “reflexive tradition” of the humanities, the surrounding world of each agent includes the rest agents; moreover, beliefs about other agents get reflected during the process of reflexion (in particular, variations of beliefs may result from nonidentical awareness). however, researchers have not succeeded in deriving constructive formal outcomes in this field to date. 1if the initial model incorporates uncertain factors, specific procedures of uncertainty elimination are involved to obtain a deterministic model. 2an agents action vector is a nash equilibrium if none of them benefits by unilateral deviation from it (provided that the rest agents choose the corresponding components of the nash equilibrium). a more rigorous definition could be found below. advances in systems science and applications (2014) vol.14 no.3 259 hence, an urgent problem lies in designing and analyzing mathematical models of games, where agents awareness is not a common knowledge and agents make decisions based on hierarchies of their beliefs. such class of games is called reflexive games [30-32]. we will provide a formal definition later. the term “reflexive games” was introduced by v. lefebvre in 1965, see [33]. however, the cited work and his other publications [34-37] represented qualitative discussions of reflexion effects in interaction among subjects (actually, no general concept of solution was suggested for this class of games). similar remarks apply to [38-41], where a series of special cases of players awareness was studied. the monograph [32] concentrated on systematical treatment of reflexive games and an endeavor of constructing a uniform equilibrium concept for these games. according to game theory and reflexive models of decision making, it seems reasonable to distinguish between strategic reflexion and informational reflexion. informational reflexion is the process and result of agents thinking about (a) the values of uncertain parameters and (b) what his/her opponents (other agents) know about these values. here the “game” component actually disappears-an agent makes no decisions. strategic reflexion is the process and result of agents thinking about which decision making principles his/her opponents (other agents) employ under the awareness assigned by him/her via informational reflexion. therefore, informational reflexion often relates to insufficient mutual awareness, and its result serves for decision making (including informational reflexion). strategic reflexion takes place even in the case of complete awareness, precessing agents choice of an action. in other words, informational and strategic reflexion can be studied independently, but the both occur in the case of incomplete or insufficient awareness. 1.5 general approaches to the description of information and strategic reflexion according to [12, 42-43], there are two different approaches to the description of awareness structures, viz., syntactic and semantic ones. recall that syntactics means syntax of sign systems, i.e., the structure of sign combinations and rules of their formation, “translation”and interpretation irrespective of their values and functions of sign systems. semantics studies sign systems as tools of meaning expression; here the basic subject lies in interpretations of signs and sign combinations. foundations of these approaches were laid in mathematical logic [44-45]. within the framework of syntactic approach, an hierarchy of beliefs is described explicitly. suppose that beliefs are defined by a probability distribution. then hierarchies of beliefs (at a certain level) correspond to distributions on the product of the set of states of nature and distributions reflecting beliefs of preceding levels [46]. an alternative is using “logic formulas”-rules of transforming elements of an initial set based on logic operations and operators such as “player i believes the probability of event . . . is not smaller than α” [43, 47]. a knowledge is modeled 260 novikov d.a. and chkhartishvili a.g.:mathematical models of informational and ... by propositions (formulas) constructed according to certain syntactic rules. according to semantic approach, beliefs of agents are defined by probability distributions on the set of states of nature. hierarchies of beliefs get generated only by virtue of these distributions. in the elementary (deterministic) case, a knowledge represents the set θ of feasible values of an uncertain parameter and different partitions {pi}i∈n of this set. an element of the partition pi containing θ ∈ θ forms the knowledge of agent i, namely, the set of values of the uncertain parameter, being indistinguishable for this agent under a known fact θ [11-12]. the correspondence (or “equivalence”) between syntactic and semantic approaches was established in [18, 42] and other works. we also cite experimental research on hierarchies of beliefs [48-50]; see the surveys in [51-52]. the above overview points at two existing “extremes”. the first one lies in common knowledge. here j. harsanyis merits [53] are (a) reducing all information on an agent (determining the latters behavior) to a single characteristic-agent‘s type-and (b) constructing a bayes-nash equilibrium by hypothesizing that the probability distribution of types is a common knowledge. the second “extreme” relates to infinite hierarchy of compatible or incompatible beliefs. (for an example, see the structure discussed in [46]. on the one hand, it describes all possible bayesian games and all possible hierarchies of beliefs. on the other hand, it appears very general and, consequently, very cumbersome, thus interfering with constructive statement and solution of specific problems). most research on awareness seeks to answer the following question. when does an hierarchy of agents beliefs describe a common knowledge and/or reflect adequately their awareness? [19, 54]. the dependence of game solutions on a finite hierarchy of compatible or incompatible beliefs of agents (the whole range between the above “extremes”) has been studied in [1, 32]. 1.6 theory of collective behavior traditionally, game-theoretic models and/or models of collective decision making utilize one of two assumptions regarding mutual awareness of agents [51]. the first one implies that all essential information and decision principles adopted by agents are known to all agents, all agents know this fact and so on (such reasoning could be infinite). actually, this is the concept of a common knowledge, which serves, e.g., in constructing a nash equilibrium. the second assumption claims that each agent (according to his/her awareness) follows a certain procedure of individual decision making and has “almost no idea” of the knowledge and behavior of the rest agents. the first approach appears canonical in game theory, while the second approach has become popular in models of collective behavior. yet, a variety of intermediate situations exists between these “extreme cases”. imagine that informational reflexion takes no place-a common knowledge on essential external parameters is observed. let an agent have performed an advances in systems science and applications (2014) vol.14 no.3 261 act of strategic reflexion, i.e., an attempt to predict the behavior of other agents (not their awareness but decision principles). this agent chooses his/her actions using the forecast (we believe he/she possesses reflexion rank 1). another agent (with reflexion rank 2) possibly knows about the existence of agents having reflexion rank 1. consequently, such agent endeavors to predict their behavior, as well. again, this line of reasoning could be infinite. a series of questions arises immediately. how does the behavior of a collective of agents depend on their distribution by reflexion rank (the number of agents with a specific rank in a collective)? suppose that the shares of reflexing agents can be controlled. what are the optimal values of these shares? here optimality is “measured” in terms of some criterion defined on the set of agents actions. classic game-theoretic models proceed from the following. in a normal form game, agents choose nash equilibrium actions. however, investigations in the field of experimental economics indicate this not always the case (e.g., see [55] and the overview [56]). the divergence between actual behavior and theoretical expectations has several explanations: limited cognitive capabilities of agents [57] (decentralized evaluation of a nash equilibrium represents a cumbersome computational problem [58]). furthermore, sometimes nash equilibria provide no adequate description to the real behavior of agents in experimental single stage games (agents have not enough time for “correcting” their wrong beliefs about essential parameters of a game [59]). for instance, d. bernheims concept of rationalizable strategies requires unlimited rationality from agents (their high cognitive capabilities); agents full confidence in that all the opponents would evaluate a nash equilibrium; incomplete awareness; the presence of several equilibria. therefore, there exist at least two foundations (“theoretical” and “experimental” ones) for considering models of collective behavior of agents with different reflexion ranks. in contrast to game theory, the theory of collective behavior analyzes the behavior dynamics of rational agents under rather weak assumptions regarding their awareness. for instance, far from always agents need a common knowledge about the set of agents, sets of feasible actions and goal functions of opponents. alternatively, agents may not predict the behavior of their opponents (as in game theory). moreover, making decisions, agents may “know nothing about the existence of” specific agents or possess aggregated information about them. the most widespread model of collective behavior dynamics is the model of indicator behavior (see references in [51]). the essence of the model consists in the following. suppose that at instant t each agent observes the actions of all 262 novikov d.a. and chkhartishvili a.g.:mathematical models of informational and ... agents {xt−1 i }i∈n that have been chosen at the preceding instant t− 1, t = 1, 2, . . . the initial action vector x0 = (x01, . . . , x 0 n) is assumed known. each agent can evaluate his/her current goal -an action maximizing his/her goal function provided that at a current instant all agents choose the same actions as at the previous instant: wi({xt−1 −i }) = argmax y∈ℜ1 fi(y, x t−1 −1 ), t = 1, 2, . . . , i ∈ n (1) according to the hypothesis of indicator behavior, at each instant an agent makes a “step” from his/her previous action to the current goal: xti = xt−1 i + γti [wi(x t−1 i )− xt−1 i ], i ∈ n, t = 1, 2, . . . (2) where γti ∈ [0; 1] designate “the values of steps”. for convenience, such collective behavior can be called “optimization behavior” (thus, we emphasize its difference from play behavior). the approaches adopted by the theory of collective behavior and game theory agree in the following sense. the both study the behavior of rational agents, while game equilibria generally represent equilibria for dynamic procedures of collective behavior. for instance, the nash equilibrium specifies an equilibrium for the dynamics (2) of collective behavior. to make the picture complete, note one more aspect, as well. the theory of collective behavior proposes another approach (going beyond the scope of this book), namely, evolutionary game theory [60]. this science studies the behavior of large homogeneous groups (populations) of individuals in typical repeated conflicts; each strategy is applied by a set of players, whereas a corresponding goal function characterizes the success of specific strategies (instead of specific participants of such interaction). thus, game theory often employs maximal assumptions regarding agents awareness (e.g., the hypothesis of existing common knowledge), while the theory of collective behavior involves the minimal assumptions. the intermediate position belongs to reflexive models. and so, let us discuss the role of (informational and strategic) reflexion in decision making by agents. 2 mathematical models of informational and strategic reflexion 2.1 reflexion in game theory and models of collective behavior: the structure of problem domain game theory and the theory of collective behavior analyze interaction models for rational agents. approaches and results of these theories can be considered at three interconnected epistemological levels (that correspond to different functions of modeling [2])-see fig.2 [1]: -phenomenological level, where a model aims at describing and/or explaining advances in systems science and applications (2014) vol.14 no.3 263 fig.2 descriptive and normative models of informational and strategic reflexion table 1 modeling of informational and strategic reflexion: a comparison of approaches parameter informational reflexion strategic reflexion parameter informational reflexion strategic reflexion model of a “game” awareness structure reflexive structure equilibrium information equilibrium reflexive equilibrium control information control reflexive control the behavior of a system (a collective of agents); predictive level (the aim is forecasting the system behavior); normative level (the aim is ensuring a required system behavior). in game theory, a common scheme consists in (1) describing the “model of a game” (phenomenological level), (2) choosing an equilibrium concept defining the stable outcome of a game (predictive level) and (3) stating a certain control problem-find values of controlled “game parameters” implementing a required equilibrium (normative level). an interested reader would find the corresponding illustration in fig.2. taking into account informational reflexion leads to the necessity of constructing and analyzing awareness structures. this enables defining an informational equilibrium, as well as posing and solving informational control problems-see fig.2. taking into account strategic reflexion generates a similar chain marked by heavy lines in fig.2: “models of strategic reflexion” “reflexive structure” 264 novikov d.a. and chkhartishvili a.g.:mathematical models of informational and ... “reflexive equilibrium” “reflexive control”. a comparison of approaches to modeling of informational and strategic reflexion is given by table 1. 2.2 awareness structure and informational equilibrium consider the set of agents: n = {1, 2, ..., n}. denote by θ ∈ θ the uncertain parameter (we believe that the set θ is a common knowledge for all agents). the awareness structure ii of agent i includes the following elements. first, the belief of agent i about the parameter θ ; denote it by θi, θi ∈ θ. second, the beliefs of agent i about the beliefs of the other agents about the parameter θ; denote them by θij , θij ∈ θ, j ∈ n . third, the beliefs of agent i about the beliefs of agent j about the belief of agent k; denote them by θijk, θijk ∈ θ, j, k ∈ n . and so on (evidently, this reasoning is generally infinite). in the sequel, we employ the term “awareness structure”, which is a synonym of “informational structure” and “hierarchy of beliefs”. therefore, the awareness structure ii of agent i is specified by the set of values θij1...jl , where l runs over the set of nonnegative integer numbers, j1, ..., jl ∈ n , while θi1...il ∈ θ. the awareness structure i of the whole game is defined in a similar manner; in particular, the set of the values θi1...il is employed, with l running over the set of nonnegative integer numbers, j1, . . . , jl ∈ n , and θij1...jl ∈ θ. we emphasize that the agents are not aware of the whole structure i; each of them knows only a substructure ii. thus, an awareness structure is an infinite n-tree; the corresponding nodes of the tree describe specific awareness of real agents from the set n , and also phantom agents (complex reflexions of real agents in the mind of their opponents). a reflexive game γi is a game defined by the following tuple: γi = {n, (xi)i∈n , fi(·)i∈n , i} (3) where n stands for a set of real agents, xi means a set of feasible actions of agent i, fi(·) : θ × x ′ → ℜ1 is his/her goal function (i ∈ n); θ indicates a set of feasible values of the uncertain parameter and i designates the awareness structure. therefore, a reflexive game generalizes the notion of a normal-form game (det -ermined by the tuple {n, (xi)i∈n , fi(·)i∈n , i} ) to the case when agents’ awaren -ess is reflected by an hierarchy of their beliefs (i.e., the awareness structure i). within the framework of the accepted definition, a “classical” normal-form game is a special case of a reflexive game (a game under a common knowledge among the agents). consider the “extreme” case when the state of nature appears a common knowledge; for a reflexive game, the solution concept (proposed in this book based on an informational equilibrium, see below) turns out equivalent to advances in systems science and applications (2014) vol.14 no.3 265 the nash equilibrium concept. to proceed and formulate a series of definitions and properties, we introduce the following notation:∑ + stands for a set of finite sequences of indexes belonging to n ;∑ is the sum of ∑ + and the empty sequence; |σ| indicates the number of indexes in the sequence σ ∈ ∑ (for the empty sequence, it equals zero); this parameter is known as the length of an index sequence. imagine θi represents the belief of agent i about the uncertain parameter, while θii means the belief of agent i about his/her own belief. it seems then natural that θii = θi. in other words, agent i is well-informed on his/her own beliefs. moreover, he/she assumes that the rest agents possess the same property. formally, this means that the axiom of self-awareness is accepted: ∀i ∈ n,∀τ, σ ∈ n : θτiiσ = θtiσ. in particular, being aware of θτ for all τ ∈ ∑ + such that |τ | = γ, an agent may explicitly evaluate θτ for all τ ∈ ∑ + with |τ | < γ. in addition to the awareness structures ii(i ∈ n), one may also analyze the awareness structures iij (i.e., the awareness of agent j according to the belief of agent i), iijk, and so on. let us identify the awareness structure with the agent being characterized by it. in this case, one may claim that n real agents (i − agents, where i ∈ n) having the awareness structures ii also play with phantom agents (τ − agents, where τ ∈ ∑ + , |τ | ≥ 2) having the awareness structures iτ = {θτσ}, σ ∈ ∑ . it should be emphasized that phantom agents exist merely in the minds of real agents; still, they have an impact on their actions; these aspects will be discussed below. assume that the awareness structure i of a game is given; this means that the awareness structures are also defined for all (real and phantom) agents. within the framework of the hypothesis of rational behavior, the choice of an action xτ performed by a τ − agent is described by his/her awareness structure iτ . hence, the mentioned structure being available, one may model agents reasoning and evaluate his/her action. on the other hand, while choosing his/her action, the agent models actions of the rest agents (i.e., performs reflexion). therefore, estimating the game outcome, we should account for the actions of real and phantom agents. a set of actions x∗τ , τ ∈ ∑ + , is called an informational equilibrium, if the following conditions are met: 1. the awareness structure i possesses finite complexity v [30]; 2. ∀λ, µ ∈ σ : iλi = iµi ⇒ xλi ∗ = xµi ∗; 3. ∀i ∈ n, ∀σ ∈ σ: x∗σi ∈ arg max xi∈xi fi(θσi, x ∗ σi1, ..., x ∗ σi,i−1, xi, x ∗ σi,i+1..., x ∗ σi,n) (4) 266 novikov d.a. and chkhartishvili a.g.:mathematical models of informational and ... here condition 1 claims that a reflexive game involves a finite number of real and phantom agents (what happens when this assumption is rejected, is discussed in [61]). condition 2 expresses the requirement that the agents with an identical awareness choose identical actions. finally, condition 3 reflects rational behavior of agentsceach agent strives for maximizing the individual goal function via a proper choice of his/her action. for this, an agent substitutes actions of the opponents into his/her goal function; the actions are rational in the view of the considered agent (according to the available beliefs of the rest agents). the “classical” concept of a nash equilibrium is remarkable for its self-sustained nature. notably, assume that a repeated game takes place and all agents (except agent i) choose the same equilibrium actions. then agent i benefits nothing by deviating from his/her equilibrium action; evidently, this feature is directly related to the following. beliefs of all agents about reality are adequate, i.e., the state of nature appears a common knowledge. generally speaking, the situation may change in the case of an informational equilibrium. indeed, after a single play of the game some agents (or even all of them) may observe an unexpected outcome due to an inconsistent belief about the state of nature (or due to an inadequate awareness of opponents beliefs). anyway, the self-sustained nature of the equilibrium is violated; actions of agents may change as the game is repeated. informational equilibrium is stable [62], if each agent (real or phantom) observes exactly the expected result (in this case agents awareness does not change). some models of awareness dynamics are considered in [63]. 2.3 informational control the model of informational control (purposeful impact on agents awareness to form informational structure which leads to the desired informational equilibrium) includes an agent (or several agents) and a principal. each agent is characterized by the cycle “awareness of the agent → action of the agent → result observed by the agent → awareness of the agent”. generally speaking, these components vary for different agents. at the same time, the cycle could be viewed common for the whole controlled subsystem (i.e., for the complete set of agents). this feature is indicated by the word “agent(s)” in fig.3. the interaction between an agent (agents) and the principal is characterized by the following elements: an informational impact of the principal, which forms a certain awareness of an agent (agents). it seems possible to study the principals influence on the outcome observed by an agent (agents), see the chain “principal → observed outcome” in fig.3; an actual outcome of the agents action (or agents actions), which has an impact on the preferences of the principal. advances in systems science and applications (2014) vol.14 no.3 267 fig.3 the model of informational control implementing informational control, the principal (as usual) strives to maximize his/her utility. assume the principal can form any awareness structure from a certain feasible set. the problem of informational control may be posed as follows. find an awareness structure from the set of feasible structures, which maximizes the principals utility in a corresponding informational equilibrium (perhaps, taking into account the principals costs to form such an awareness structure). define the following objects: the set ψx(i) ⊆ x ′ of the action vectors of real agents, representing equilibria under the awareness structure i; and the set ψi(x) of awareness structures, making the action vector x of real agents an equilibrium (solution to the inverse problem). let us give a formal statement to the control problem. assume that the goal function of the principal, φ (x, i), is defined on a set of real agents actions and awareness structures. next, suppose that the principal can form any awareness structure from a certain set ℑ′. under the awareness structure i ∈ ℑ′, the action vector of real agents is an element of the set of equilibrium vectors φx (i). we emphasize that the set ψx(i) may be empty; in the case of a missed equilibrium, the principal cannot predict the outcome of a game. to avoid this problem, introduce the set of feasible structures leading to the non-empty set of equilibria: ℑ = {i ∈ ℑ′|ψx (i) ̸= ∅}. imagine that, under the specified awareness structure i ∈ ℑ, the set of equilibrium vectors ψx(i) includes (at least) two elements. as a rule, one of the following assumptions is then adopted [3]: 1) the hypothesis of benevolence(hb), which implies that agents always choose 268 novikov d.a. and chkhartishvili a.g.:mathematical models of informational and ... the equilibrium desired by the principal; 2) the principle of maximal guaranteed result(pmgr), i.e., the principal expects the worst-case equilibrium of the game. using either the hb or the pmgr, one has the problem of informational control in two settings as follows: max x∈ψx(i) φ(x, i) −−→ i∈ℑ max; (5) min x∈ψx(i) φ(x, i) −−→ i∈ℑ max; (6) naturally, if for any i ∈ ℑ the set ψx(i) consists of a single element, formulas (1) and (2) coincide. in the sequel, the problem (5) (alternatively, (6)) is called the informational control problem in the form of the goal function. now, provide an alternative formulation to the problem of informational control (being independent from the goal function of the principal). assume that the principal wants agents choose an action vector x ∈ x ′. the question arises, “for which vectors and by which awareness structure i would the principal achieve this?” in other words, the second possible formulation of the informational control problem is to find the following components. first, the attainability set, viz, the one composed of the vectors x ∈ x ′ such that for each of them the set of awareness structures ψi(x) ∩ ℑ is nonempty (7) or consists of a single element. (8) second, the corresponding feasible awareness structures i ∈ ψi(x)∩ℑ, meeting the above property for each vector x. note that the condition (7) “corresponds” to the hb, while the one of (8) “corresponds” to the pmgr. the problem (7) (alternatively, (8)) will be referred to as the problem of informational control in the form of the attainability set. once again, we underline that the second formulation of the problem does not depend on the goal function of the principal. it merely reflects the possibility of bringing the system to a certain state by informational control. methods and examples of problems (5)-(8) solution are described in [1, 64]. particular case is the concordant informational control ; here agents are informed about the fact of control implementation by a principal, and still they trust messages of the principal. evidently, implementing such control requires specific conditions [65]. advances in systems science and applications (2014) vol.14 no.3 269 2.4 reflexion structure and reflexion equilibrium publications on strategic reflexion models, the so-called level k models, appeared in the mid-1990s [66-68]. in 2004, they were generalized by the cognitive hierarchies model(chm) [48]. the survey [56] identified four basic approaches to the construction and study of strategic reflexion theoretical models within the framework of game theory and experimental economics (see also experimental results in [48, 69-72] ). we cite fundamental works only (references to later research can be found in [56]). notably, the four basic approaches are: the level k approach [73]; the approach of quantal best response equilibria [74]; the quantal level k approach [50]; the approach of cognitive hierarchies [48]. all of this approaches are mainly generalized by the following model. the hypothesis of indicator behavior implies that choosing his/her actions by the procedure (2), an agent does not ponder over that the rest agents act similarly. otherwise, an agent would perform reflexion and (making decisions at a time instant ) seek for the best response to the actions of the rest agents, forecasted according to (2). in this case, the state of goal is no more defined by formula (1). instead, we obtain wi(x t −i) = argmax y∈ℜ1 fi(y, x t −i) (9) here xt−i satisfies (1). we will believe that a reflexing agent of rank 1 considers the rest agents as non-reflexing. similarly, it is possible to consider agents with higher reflexion ranks (the term “an agent of reflexion rank k” possesses many synonyms (a step k player, a level k player, a k-level player, a smart k-player, etc c see [75-76] and the survey [51]). for this, define ℵ = {n0, n1, . . . , nm} as a partition of the agents set n , where ni is the set of agents with reflexion rank i, i = 0, m , and m specifies the maximal reflexion rank, ni = |ni| , i ∈ n, m∑ i=0 ni = n. we will call ℵ a reflexive partition [77]. suppose that an agent with reflexion rank k exactly knows the sets (shares) of the agents with ranks k′ < k − 1. moreover, assume that he/she considers all agents as having reflexion rank k − 1. in other words, this agent does not concede the existence of agents with the same (or even higher) reflexion rank than his/her rank. in addition, the agent in question may incorrectly estimate the sets of agents possessing reflexion ranks k − 1, k, .... consider a given initial action vector of the agents. let us study the following dynamic reflexive model of their decision making. the corresponding expressions for the one-step “game” model represent a special case, when decisions are made 270 novikov d.a. and chkhartishvili a.g.:mathematical models of informational and ... one-time under γ1i ≡ 1, i ∈ n . reflexion rank 0 take agents with reflexion rank 0 (belonging to the set n0. assume that they choose actions, thinking that the rest agents act similarly to the previous period. formula (1) yields xti = xt−1 i + γti [wi(x t−1 −i )− xt−1 i ], i ∈ n0, t = 1, 2, . . . (10) n the case n0 = n (no reflexing agents), all agents observe the real trajectory (x0, . . . , xt, . . . ) of the agents action vectors, see (10). reflexion rank 1 x1tj = x1t−1 j + γtj [wj(x t −j)− x1t−1 j ], j ∈ n1 (11) for agent j ∈ n1, the forecasted trajectory is defined by (x0, . . . , (x1tj , x t −j), . . .); however, actually the trajectory (x0, . . . , (x1tj∈n1 , xti∈n0 ), . . .) is realized. this means that the real trajectory may differ from the forecasted trajectories of agents with reflexion ranks 0 and 1 [1]. reflexion rank 2 suppose that each agent j with reflexion rank 2 (j ∈ n2) exactly knows the set n0; moreover, he/she considers all agents from the set n1 ∪ n2\{j} as having reflexion rank 1. in the general case of several agents with reflexion rank 2, this agent wrongly assigns rank 1 to them. consequently, he/she can “forecast” the behavior of the opponents. therefore, his/her choice is the best response to the expected outcome: x2tj = x2t−1 j + γtj [wj(x t−1 i∈n0 , x1tl∈n1∪n2\{j})− x2t−1 j ], j ∈ n2 (12) for agent j ∈ n2, the forecasted trajectory is given by (x0, . . . , (x2tj , x1 t l∈n1∪n2\{j}, xti∈n0 ), . . .), while actually the trajectory (x0, . . . , (x2tj∈n2 , x1tl∈n1 , xtl∈n0 ), . . .) is realized. reflexion rank k(k ≤ m) the behavior of agents with reflexion rank k is described by analogy to the three cases above (reflexion ranks 0, 1 and 2). this is done on the basis of the following awareness structure of the agents. for agent j with reflexion rank k, denote by ℵjk the subjective reflexive partition (the beliefs of the agent about the partitions of all agents): ℵjk = (n0, n1, . . . , nk−2, nk−1 ∪nk ∪ . . . ∪nm\{j}︸ ︷︷ ︸ k , {j}, ∅, . . . , ∅︸ ︷︷ ︸ m−k−1 ), j ∈ nk (13) an agent with reflexion rank chooses actions by the procedure xktj = xkt−1 j +γtj [wj(x t l∈n0 , x1tl∈n1 , . . . , x[k−1]tl∈nk−1∪nk∪...∪nm\{j})−xkt−1 j ], j ∈ nk (14) advances in systems science and applications (2014) vol.14 no.3 271 in the “static” case, this agent selects the action xk∗j (ℵjk) = argmax y∈ℜ1 fj(y, x 1 l∈n0 , x11l∈n1 , . . . , x[k−1]1l∈nk−1∪nk∪...∪nm\{j}), j ∈ nk. (15) therefore, a reflexive structure represents the set of subjective reflexive partitions of all agents. assume that agents beliefs about the reflexion ranks of each other satisfy (13). then the awareness structure is uniquely defined by the reflexive partition ℵ. the vector of agents actions x∗(ℵ) = {xk∗j (ℵjk)}j∈nk, k=0,m (16) is said to be a reflexive equilibrium of the game γℵ = {n,fi(·)i∈n ,ℵ} [51, 77]. in other words, a reflexive equilibrium forms the set of agents actions being the best responses to opponents actions (according to an existing reflexive structure). by virtue of the assumptions regarding the existence and uniqueness of best responses, a reflexive equilibrium always exists. furthermore, a reflexive equilibrium seems rather exotic. generally, the actions of agents are not the best responses to opponents actions. detailed classification of strategic reflexion models is given in [51, 77]. the described general model of reflexive collective behavior would hardly lead to general analytical derivations. nevertheless, it may provide a basis for developing particular analytical models or general simulation models (e.g., according to the classification suggested in [78]). such models serve for describing and forecasting collective behavior (human beings, mobile robots, program agents) in various situations. for instance, we refer an interested reader to [1] for reflexive simulation models of evacuation, reflexive models of transport flows and other numerous examples from different applications. by proper variation of reflexive partitions, one can change the actions of agents, i.e., perform reflexive control [1, 77]. consider reflexive partition as a control parameter. it is possible to formulate controllability problem, as follows. under a given set ℑ of feasible reflexive partitions, find the set of agents action vectors x(ℑ) = ∪ ℵ∈ℑ x(ℵ) that can be realized by reflexive control. the inverse problem lies in obtaining the “minimal” set of feasible reflexive partitions (in a certain sense), allowing to realize a given agents action vector. now, let us address the control problem. suppose that the preferences of a control subject (a principal) are described by his/her real-valued goal function f0(q(x∗)) defined on the set of aggregated outcomes (q : ℜn → ℜ1), i.e., f0(·) : ℜ1 → ℜ1. using the expression (16), the efficiency of the reflexive partition ℵ can be characterized by k(ℵ) = f0(q(x∗(ℵ))). consequently, the problem of reflexive control (in terms of reflexive partitions) 272 novikov d.a. and chkhartishvili a.g.:mathematical models of informational and ... can be formally stated as [1, 77] k(ℵ) → max ℵ∈ℑ . (17) let km be the maximal value of the efficiency criterion in the problem (17) under a fixed maximal reflexion rank m. the problem of the maximal rational rank of reflexion (a rank being pointless to exceed for the principal in the sense of controllability or/and efficiency of reflexive control) is to find: m∗ = min{m|m ∈ arg max w=0, 1, 2, ... kw}. to proceed, we discuss conformity of subjective reflexive partitions of the agents. suppose that each agent observes merely the aggregated outcome. trajectories forecasted by the agents may differ from the real trajectory (see the general reflexive model of collective behavior). this motivates the agents to doubt the correctness of their subjective reflexive partitions. imagine that the agents observe just the aggregated outcome of the game (in addition to their own actions). by analogy to the condition of stable informational control (see above), one can introduce the condition of stable reflexive partition. notably, require that the aggregated outcome for the real trajectory coincides with the forecasted aggregated outcomes for all agents. stability of reflexive partitions is closely associated with learning in games. observing the behavior of opponents (which differs from the forecasted behavior), agents may modify their beliefs about the reflexion ranks of the opponents or pass to higher levels of reflexion. under a fixed reflexive partition ℵ ∈ ℑ, we have realization of the action vector (16). and the aggregated outcome q(x∗(ℵ)) is realized. according to agent with reflexion rank , the following vector is realized x̃jk(ℵjk) =(xl∈n0 , x1l∈n1 , x2l∈n2 , . . . , x[k − 1]l∈nk−1∪nk∪...∪nm\{j}, xkj), j ∈ nk, k = 0, m the condition of stable reflexive partition ℵ ∈ ℑ takes the form q(x̃jk(ℵjk)) = q(x ∗ (ℵ)), j ∈ nk, k = 0, m. the problem of reflexive control (ℵ) can be stated on the set of stable reflexive controls (if nonempty). in practice, this means that the principal forms an optimal partition of the agents into reflexion ranks. in such partition, the agents do not doubt the correctness of their beliefs about reflexion ranks of the opponents (based on observing the results of the “game”). 3 conclusion mathematical models of informational and/or reflexive structures and equilibria (and models of corresponding control problems) allow the following: • from the decision theory viewpoint, extending the class of collective behavior advances in systems science and applications (2014) vol.14 no.3 273 models for intelligent agents performing a joint activity under incomplete awareness and missed common knowledge; • from the descriptive viewpoint, enlarging the set of outcomes that can be “explained” (within the framework of the model) as stable results of agents interaction; accordingly, extending the controllability domain (for control problems); • from the normative viewpoint, posing/solving the problems of collective behavior by choosing a proper structure of agents awareness. numerous applied models of informational or/and reflexive control in economic, social and organizational systems, military problems and other fields are described in [1, 32, 64]. as strategic objectives of future investigations, we mention integration of informational reflexion models with strategic reflexion ones. in other words, it seems promising to construct a language for uniform joint description of informational and reflexive structures. references [1] novikov d.a., ckhartishvili a.g. 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(2010), “≪cognitive games≫: a linear impulse model”, automation and remote control, vol.71, no.4, pp.718-730. corresponding author d.a. novikov can be contacted at: novikov@ipu.ru. advances in systems science and applications (2012) vol.12 no.1 1-11 simulation based design for refrigerator development k. fukuyo graduate school of innovation and technology management, yamaguchi university, tokiwadai 2-16-1, ube, yamaguchi,7558611,japan abstract the simulation based design (sbd) is one of the new approaches to shorten development periods, reduce cost of development and avoid failures. this approach emphasizes an in-depth analytical understanding of target products or services in the upstream of a design process by using the simulation technology. this short communication presents some cases of simulation in the development of refrigerator as examples of bsd approach. keywords function s-rough sets, structure of function s-rough sets, relation theorem, function transfer rough law mining 1 introduction manufactures of refrigerators are forced to shorten development periods, reduce cost of development and avoid the failures and incidents. for those purposes, they are now trying to introduce new technologies and processes such as cad/cam/cae, digital engineering, or concurrent engineering instead of traditional ones. the simulation based design (sbd) is one of the new approaches to meet those purposes. according to bossak[1], its concept is “simulation of the entire life cycle of the product, from concept development to detailed design, prototyping, testing, manufacturing operations, maintenance and disposal” and it consists of “modeling methods and computational tools, virtual reality environment, an infrastructure for collaborative engineering and integration technologies and tools.” the concept of sbd is almost the same as those of the analysis-leads or analysis led design (ald, the terminology of isi and yamaguchi univ.) and the simulation driven product developmenttm (sdpd, the terminology of fluent). either word emphasizes a radical front-loading of design, i.e., an in-depth consideration in the upstream of a design process. an ordinary design process consists of the requirement definition, conceptual design, project design, and detail design phases. manufacturing and testing start almost as soon as the detailed design starts. some researchers argue that a sufficient analytical understanding of a target product/service is required in the upstream of design, i.e., the requirement definition and conceptual design phases and it results in decrease in the cost of development[2]. in the other words, a poor understanding of target products/services will bring some serious problems in the downstream of the design process, cause many trials and errors in the manufac2 k. fukuyo:the application of semantic web technology in ... turing and testing process, and significantly increase cost and time to market. therefore, the necessity for the analytical understanding in the upstream of a problem-solving process has been mentioned since a long time ago[3-4]. nowadays, a progress of information technology allows manufacturers to use computer simulations as a tool for the analytical understanding of target products/services; hence new approaches of design such as sbd, ald, and sdpd tm are proposed. 2 simulation methods in refrigerator development concept of sbd has become important among the refrigerator manufacturers, although sbd is not perfectly deployed in the design and manufacturing process. in this industry two types of simulation are mainly used for design process. one is cycle simulation, which is the system simulation of a refrigerating cycle, and the other is computational fluid dynamics (cfd). this short communication treats the latter. (a) dynamic and changing. process knowledge should be updated along with the emergence of new techniques and new methods. cfd simulation applied for designing refrigerators are as follows: cortella et al. applied a fem (finite element method) code to carry out unsteady 2-dimensional simulation of a refrigerated display cabinet[5]. chang et al. applied a cfd code with a k−ε turbulent flow model to refrigerators equipped with freezing cabinets at the top[6]. afonso and matos used a commercial cfd code, fluent tm to study the effect of radiation shields around a condenser and compressor[7]. gupta et al. used the cfd model to study effects of various operating and design parameters on the refrigerator performance[8]. haldar et al. applied their developed code to simulate free convection inside a freezer[9]. the author of this short communication has also carried out cfd analyses in the upstream of the refrigerator development. some cfd analyses carried out by the author are presented in the following sections as examples of bsd. 3 realization of thermal uniformity inside refrigerators fig.1 shows a household refrigerator with a conventional air-supply system. the fan supplies the air cooled by an evaporator to the fresh food cabinet, the vegetable cabinet and the freezing cabinet. in the fresh food cabinet, the cooled air is supplied from openings in the rear panel. temperature of the supplied air is set at a sufficiently low temperature to cool the freezing cabinet. thus the flow rate of the supplied air to the fresh food cabinet is suppressed to prevent freezing and drying foods, and increasing the heat gain from outside the refrigerator. however, the suppression of the flow rate causes rises in temperature in the door pockets and the upper shelf. besides this problem, the conventional air-supply system can not increase cooling rate because of the suppression of the flow rate. advances in systems science and applications (2012) vol.12 no.1 3 to meet the requirements for keeping food fresh, a new air-supply system for improving the thermal uniformity and the cooling rate inside the fresh food cabinet was developed by adding a blower and jet slots to a conventional cooled air supply system[10]. in the design process of this new air-supply system, cfd is applied to study air distribution inside refrigerators. fig.1 cross section of the refrigerator 3.1 design of the air-supply system fig.2 shows the schematics of the fresh food cabinets with a conventional air supply-system and our developed system. the volume of each cabinet is 0.24 m3. each air-supply system has ten openings and six grills. the openings supply cooled air at 0.17 m3 min−1 and -10◦c to keep the cabinet temperature below 5◦c, and the grills returns the air to an evaporator. the flow rate of the supplied air to the cabinet is suppressed to prevent freezing and drying foods, and increasing the heat gain. a thermal sensor is set on the rear panel of each cabinet. when the sensor detects that the surrounding temperature is lower than the preset one, air-supply is stopped. system a is the conventional air-supply system. the temperatures in the door pockets and the upper shelf will rise, because of the distances between those places and the supply openings. system b is the new air-supply system. one blower, three jet slots and two return grills are added to system a to improve the thermal uniformity and the cooling rate. the flow rate of the air jetted by the additional blower is 0.3 m3 min−1. the total opening area of the jet slots is one sixth of that of the supply openings. the velocity of the jets is thus ten times higher than that of the cooled air. the jet stirs the air inside the cabinet and improves thermal uniformity. the jet also improves the heat transfer on the surface of foods and cools it rapidly. 4 k. fukuyo:the application of semantic web technology in ... the blower collects the air inside the cabinet through the grills and jets air from the slots. the air does not pass through the evaporator. the air is thus at about the same temperature as the average temperature inside a cabinet and does not freeze foods. (a) (a)system a (b) (b)system b fig.2 layouts of air supply systems it takes much time to repeat the measurements for studying the air distribution inside the cabinet with the new system. in order to cut down the development period, we applied cfd simulations to the study on the air distribution inside the cabinet. 3.2 simulation results fig.3 shows the simulated thermal distribution for each air-supply system. regarding system a, there are high temperature regions above the top shelf and advances in systems science and applications (2012) vol.12 no.1 5 the top door pocket, and there is also a low temperature region in the center of each shelf. in system b, the air is kept at about 2◦c in most of the area, because the jetted air sufficiently stirs the air inside the cabinet. the heat gains of system b increases by 5% compared to system a because of the decrease in temperature of the regions that are not sufficiently cooled by system a. however, the standard deviation of the temperature for the fresh-food cabinet with system b is about half of that of system a. simulation results are validated by comparing with the measurement after the (a) (a)system a (b) (b)system b fig.3 thermal distributions in the central cross section simulation. the difference between the temperatures at the points indicated by circles in fig. 3 is 0.5◦c in the simulation and it is 0.3◦c in the measurement. it is concluded from these simulation results that system b improves thermal uniformity. after these simulations, the project design and detail design were carried out and the refrigerators with this new air-supply system were released 6 k. fukuyo:the application of semantic web technology in ... to the market. 4 thermal-bridge problem in order to decrease heat exchange between the inside and outside of refrigerators, insulating walls are applied. however, supporting elements of the insulating walls (frames and reinforcements) often act as thermal bridges and increase heat gain or loss. boughton et al. have shown that heat loss through thermal bridges accounted for about 30% of a refrigerators overall heat loss[11]. applying cfd and a heat-flow visualization technique to the upstream stage of the refrigerator design will help to know and solve the thermal-bridge problems. below is an example of the thermal-bridge problems[12]. 4.1 thermal-bridge problem around gaskets fig.5 shows a cross section of the region where the insulating wall and door of a refrigerator contact through a gasket. the wall and door consist of polyurethane foam insulators and reinforcements. the foam of the insulators is filled with isobutane. the wall and door are 100.0-mm-thick, and are separated by a 4.0mm-wide gasket. the left side of the insulation is the inside of the refrigerator and the right side is the outside. the inside and outside air are at temperatures of 273 and 303k respectively. the heat transfer coefficients on both surfaces are set to 10.0 w m−2 k. the upper side of the wall and lower side of the door are set to be adiabatic boundaries. two cases of steady-state simulations were carried out. the region shown in fig.4 is divided into ten thousand of 1.0 mm2 control volumes. the inner reinforcements were made of steel sheets in case (a) and of abs resin in case (b). the thermal properties of the material used in this region are listed in table 1. 4.2 simulation results fig.5 (a) and (b) shows the distributions of heat-flow intensity in cases (a) and (b) respectively. the respective heat flow intensities are 1.8 mw m−3 and 64 kw m−3 at the marked points in fig.5 (a) and (b); this indicates that the steel inner reinforcements conduct heat about 28 times as large as abs reinforcements. 5 radical change in the layout of cabinets the design for environment (dfe), i.e., environmentally conscious design is an important issue for manufacturers. they pay attention to the energy efficiency of their products to decrease co2 emission resulting from their energy consumption. in the field of the household refrigerators, manufacturers develop high performance insulation walls, compressors, heat exchangers, and control devices to improve the energy efficiency of their products. moreover, they try to change advances in systems science and applications (2012) vol.12 no.1 7 fig.4 cross-sectional view of the region where an insulating wall is connected with an insulating door. unit: mm the refrigerants to natural ones to keep the ozone layer. however, the improvement of energy efficiency will reach the limit if the shapes of the refrigerators are fixed. manufacturers should try to change the shape of the refrigerators to improve the energy efficiency. the current mainstream refrigerators shown in fig.1 are called “bottom freezer type” because of the location of the freezing cabinet. the layout of each cabinet is decided based on the usability or ergonomics. this layout has not changed for many years. however, there are two following problems in this type of refrigerator from the view point of thermal engineering. • nearness of the compressor and freezing cabinet causes temperature rise in the freezing cabinet because of heat gain from the compressor. • placing the compressor in the limited space decreases the heat transfer from the compressor. these problems decrease refrigerators’ energy efficiency. one of the solutions of these problems is change in the shape of the refrigerators. the effects of change in the layout of cabinets were studied by cfd[13]. table 1 conditions of the heat gain calculation conductivity density specific heat [w m1 k] [kg m−3] [j kg−1 k−1 ] air 0.026 1.205 1006 gasket 0.16 1230 1.42 inner reinforcements (steel) 43 7880 473 (abs resin) 0.22 1040 1000 insulator 0.03 100 836 outer reinforcement (steel) 43 7800 473 8 k. fukuyo:the application of semantic web technology in ... fig.5 distributions of heat-flow intensities for the region shown in fig.4 fig.6 layouts of cabinets and compressors 5.1 alternative layouts of cabinets some alternative shapes of refrigerators have been already proposed as shown in fig.6. the “top compressor type” refrigerators are the ones that have the compressor at the top. they increase the heat transfer from the compressor. the “separate type” refrigerators are the ones that the compressors are separated from the refrigerators’ cabinets. this type of refrigerator not only increases the heat transfer from the compressor but also prevent the freezing cabinet from gaining the compressor’s heat. the heat gains of the cabinets of the ordinary and alternative types from the ambient air were calculated by cfd on the conditions listed in table 2. 5.2 simulation results the result is shown in fig.7. the heat gains of the alternative types are smaller than that of the current one. this tells that the energy efficiency of the alternative advances in systems science and applications (2012) vol.12 no.1 9 types will be improved. fig.7 heat gain of cabinets from the ambient air table 2 thermal property of materials fresh food & vegetable cabinets freezing cabinet size width [m] 0.65 0.65 depth [m] 0.65 0.65 height [m] 1.35 0.45 insulation wall thickness [m] 0.04 0.06 thickness [m] 0.04 0.06 conductivity[w m1 k−1] 0.02 0.02 heat transfer inside[w m−2 k−1] 10.0 10.0 outside[w m−2 k−1] 5.0 temperature cabinet [◦c] 5.0 -18.0 ambient [◦c] 30.0 compressor [◦c] 50.0 however, there are some problems in the serviceability or usability dimension. the top compressor type doesn’t allow users to put something on the ceiling and the location of the compressor makes it difficult to maintain the compressor. moreover, the separate type needs an excess area and time to set it up. this indicates that it is not only the engineering performance that decides the value of a product/service. a design of a product/service should be carried out in view of performance, features, reliability, conformance, durability, serviceability, aesthetics, and perceived quality [14]. this is why an in-depth consideration by 10 k. fukuyo:the application of semantic web technology in ... using simulation is required in the upstream of a design process. 6 conclusion three examples of sbd in the refrigerator development are presented in this short communication. as shown in each example, by using cfd simulations, we can study the performance of target products analytically in the upstream of the design process, avoid to repeat the measurements and test in the downstream, and cut down the development period and cost. acknowledgements the research is supported by national natural science foundation of china(no. 11171065). references [1] m.a. bossak. (1998), “simulation based design”, technology, pp.8-11. [2] l.r. jenkinson, j.f. marchman iii. (2003), aircraft design project, butterworth-heinemann . [3] d.w. ver planck, b.r. teare, jr. (1954), “engineering analysis”, an introduction to professional method, john wiley & sons. [4] g. polya. (1957), how to solve it, princeton university press. [5] g. cortella, m. manzan, c. comini. (2005), “cfd simulation of refrigerated display cabinets”, international journal of refrigeration, no.24, pp.250-260. [6] w.r. chang, j.y. lin, h.c. hsu. (2001), “air flow simulation and energy estimation for household refrigerators/freezers”, proceedings of 52th annual international appliance technical conference, columbus, vol.17, no.6, pp.750-771. [7] c. afonso, j. matos. (2006), “the effect of radiation shields around the air condenser and compressor of a refrigerator on the temperature distribution inside it”, international journal of refrigeration, no.29, pp.1144-1151. [8] j.k. gupta, m.r. gopal, s. chakraborty. (2007), “modeling of a domestic frost-free refrigerator”, international journal of refrigeration, no.30, pp. 311-322. [9] s.c. haldar, k. manohar, g.s. kochhar. (2008), “conjugate conductioncconvection analysis of empty freezers”, industrial management and data systems, no.49, pp.783-790. advances in systems science and applications (2012) vol.12 no.1 11 [10] fukuyo k. et al. (2003), thermal uniformity and rapid cooling inside refrigerators, international journal of refrigeration, vol.26, no.2, pp.249-255. [11] b.e. boughton, a.m. clausing, t.a. newell. (1996), “an investigation of household refrigerator cabinet thermal loads”, hvac&r research, no.2, pp.135-148. [12] k. fukuyo. (2003), “heat flow visualization for thermal bridge problems”, international journal of refrigeration, no.26, pp.614-617. [13] k. fukuyo, et al. (2005), “conflicts between eco-design and usability of refrigerators”, 4th international symposium on environmentally conscious design and inverse manufacturing ecodesign, vol.12-14, no.12, pp.3d-11s. [14] d.a. garvin. (1987), “competing on the eight dimensions of quality”, harvard business review, vol.65, pp. 101-109. advances in systems science and applications (2012) vol.12 no.4 353-372 ecosystem food-webs as dynamic systems: educating undergraduate teachers in conceptualizing aspects of food-webs’ systemic nature and comportment kostas karamanos1, aristotelis gkiolmas1, anthimos chalkidis1, constantine skordoulis1, maria papaconstantinou2 and dimitrios stavrou3 1athens science & education laboratory, faculty of primary education, university of athens, greece. 2faculty of informatics, ionian university, corfu, greece. 3faculty of primary education, university of crete, greece. abstract the current research aimed at attempting to familiarize greek prospective primary school teachers with notions of the dynamic systems nature of ecosystem food-webs and evaluate the results. the sample consisted of 85 undergraduate students of the faculty of primary education, of the university of athens, greece, who had chosen the optional topic of the autumn semester: “environmental science: the laboratory approach”. students were initially given specifically designed pre-test worksheets, to find out their initial knowledge and pre-instructional ideas about the structure and the properties of food-webs as systems. following this, instructional-teaching took place in four stages. in the first stage of the instruction process, students received real data concerning pellets of barn owls (tyto alba) gathered from two areas of the united states, and were asked to complete given worksheets. at this stage, students’ discussions were recorded, hence trying to extract answers to issues such as: the food-webs’ ability to reflect biodiversity; their mental representation of a food-web as a system; and what would happen should a “node” of the food-web-network disappear. the second stage involved students interacting with computer software in two phases. in the first, students were asked to reconstruct, on a computer screen, the barn owl food-web, given the arrows and the food-web elements. in the second stage, students used the system dynamics modeler of the netlogo (version 4.0.4) simulation environment, as an interactive tool for learning. in the third stage, after familiarizing themselves with the modeler, using the simple predator-prey ecosystem “wolf sheep predation (docked)” model, students interacted with a variation of the modeler, created in order to learn things about the stability and instability of food-webs as systems and, as a result, of ecosystems. in the fourth-final-stage, following the instruction, post-test worksheets were handed out to students and stratified samples of them were interviewed on further ideas, predictions and possible extensions of their knowledge. 354 kostas karamanos:ecosystem food-webs as dynamic systems... the results gathered helped to reach encouraging conclusions related to the fact that students seemed to have conceptualized food-webs as dynamic systems. keywords food-webs, dynamical systems, stability, teaching, netlogo 1 introduction in the recent scientific literature, the study of ecological food-webs is important in understanding the systemic behavior of ecosystem populations, in revealing aspects of their comportment as complex systems, as well as in making predictions about their future states. typical questions in this latter field of research are: whether an ecological system will become more stable or unstable in the case that one population becomes extinct, or whether the interconnectedness in the food-web is increased or decreased. other problems concerning the systemic nature of food-webs are: (i) the degree of relation if any between the biodiversity and the connectivity in a food-web, (ii) the relative importance of the abundance of a population in the food-web as compared to its biomass content. the relationship between the systemic structure of food-webs and the properties of ecological systems as “complex systems” has been thoroughly researched. it is stated, for instance, that when a hypothetical food-web is formed by adding edges at random to a set of n “nodes”, a connectivity avalanche occurs when the number of edges is approximately n/2, which means that critical phenomena arise[1]. the critical phenomena have the form of a phase-change, where a connected sub-web is clearly formed, and the food-web looks more “connected”. such relations between the systemic structure, food-web behavior and the complexity of ecological systems, are the basis of this research. the broader research project, carried out in the faculty of primary education of the university of athens, greece, aimed at familiarizing prospective primary school teachers with the concepts of complex systems, specifically ecological systems. similar researches have been carried out in many european and north american educational systems[2], due to its outmost importance in developing complexity-thinking abilities in future educators, especially with respect to non-deterministic thinking, holistic attitudes, absence of pre-existing ideas about a central-control in all ecological phenomena and non mechanistic, non-clockwise treatment of all procedures in nature[3-4]. the inherent relation between systemic-thinking and “complexity-thinking” is among the core ideas in this research[5-6]. a person who has developed the ability to conceive a system as whole and is additionally able to conceptualize the relations between these parts, is also capable of understanding complex system properties such as emergence and feedback loops[6]. the well known property of complex systems “the whole is greater than the sum of the parts” is implicit in the systemic treatment of natural entities. advances in systems science and applications (2012) vol.12 no.4 355 food-webs are a clear case of the direct relation between aspects of ecological complexity and the systemic nature of ecosystems. in a wide known definition given by asher[7]: “. . . ‘complexity’ is the multiplicity of interconnected relationships and levels.” the existence of this property of multiplicity in ecological food-webs, when systemically treated, is obvious. asher declares that all the characteristics so often attributed to complex systems (and, therefore, complex ecological systems), such as emergence and non-linearity, are consequences of the fundamental properties identified in his definition[8]. another famous statement about complexity is that: “. . . complex interactions result from multiple pathways linking organisms with abiotic resources”[9]. once more these pathways become revealed in detail, within the ontological, systemic representations of food-webs in various ecosystems. a broader link between ecological complexity and the systemic nature of foodwebs is the notion of “connectivity”, i.e. the number, form, complicatedness and scale of connections between organisms (but also between abiotic components) in an ecosystem[8]. the greater and broader the connectivity of a food-web, the more properties of complex systems is found in it. in fig.1, connectivity is depicted as one of the aspects of ecological complexity[8]. fig.1 connectivity as one of the aspects of ecological complexity. connectivity and complexity increases from top to bottom. lastly, there is the concept of ‘stability’. the majority of ecologists believe that as food-web complexity increases, the stability of the ecosystem as a whole increases[10-12], in the sense that no populations or species tend to become extinct or to have an unrestrained increase. 356 kostas karamanos:ecosystem food-webs as dynamic systems... 2 method 2.1 the objectives the main scopes of the current research were: (i) to test and improve the ability of prospective primary school teachers (undergraduate students of the faculty of primary education) to create representations and then construct (on paper and on screen) relatively simple food-webs, given the species that participate in these webs. (ii) to help students conceptualize the main properties of food-webs as systems. for example, the property that not all populations have the same importance in the food-web, or that biodiversity and multiplicity of food-webs depend on geographic latitude (they increase as one moves from the polar to the equatorial zone). (iii) to increase students’ abstractive and comparative thinking, by providing them with a scientific/theoretical food-web, and then asking them to construct one similar to it based on the data of food-web-findings from certain areas. (iv) enticing students to examine the relationship between food-web interconnectedness and ecosystem stability. lateral objectives of this research were also to: help students understand and derive information from systemic diagrams; depict graphical representations of physical concepts, such as stability and instability, and to verbally express opinions about systemic properties of ecological systems. 2.2 research instruments in addition to pre-test and post-test worksheets, with each one lasting one and a half teaching hour1, undergraduate students were given data in form of tables, diagrams of food-webs, and worksheets. during each of the four teaching sessions (each lasting two hours) students had to complete a worksheet containing both open and closed questions. recorded classroom discussions were instigated so as to reach unanimous conclusions. computer interaction was a composite part of each teaching session. specifically: (i) “microsoft word”, with regards to its “drawing” tools, and (ii) “netlogo version 4.0.4.”[13] as well as the system dynamics’ modeler, built-in in netlogo. 2.3 sample and the settings the sample consisted of 85 undergraduate students (66 women and 19 men) from the faculty of primary education of the university of athens, greece, who 1pre-test and post-test sheets, as well as the teaching sequences worksheets, are available, upon request of the reviewers. advances in systems science and applications (2012) vol.12 no.4 357 had chosen the optional subject “science and environment: the laboratory approach”. students were grouped with respect to the following parameters: a. year of studies (a1freshmen to a4senior) b. selected biology as an exam subject for their university entry national exams (b1 implies ‘yes’ and b2 implies ‘no’). c. students’ orientation in the final year of high-school, namely: (i) science, (ii) humanities and law and (iii) technology, economics and informatics (denoted as: c1, c2 and c3, respectively). d. at high-school or university, did the student select optional topics relevant to computers? this parameter implies a pre-existing familiarity with computers (d1 implies ‘yes’ and d2 implies ‘no’). the sample of 85 students was divided into five groups; four of 18 students and one of 13 students. the first author of the paper met up with each group once a week, and the students worked either alone (e.g. when completing worksheets) or in groups of two or three, when working on a computer. in the interviewing phase, each student underwent a recorded interview, lasting about an hour. 3 the instruction process and the delivered material the overall teaching sequence started with a very short interview with each one of the 85 undergraduate students, in an effort to group them according to the parameter values mentioned above. their basic knowledge of ecosystems and food-webs, if any, were also questioned. each group of students was then asked to complete a pre-test worksheet2, with questions, tasks, and interactions with the computer. each group received four weekly teaching sessions (i to iv), each lasting two teaching hours (two 50 minute-sessions). teaching session i: students are given real data from a table[14], depicting findings regarding the pellets of the barn owl (tyto alba) from two different regions of the united states of america: northwestern and southwestern. the table with the pellet findings is given in fig.2. the barn owl (tyto alba) is a very important bird species for the study of food-webs and biodiversity, as it is a “high-level” predator and it has the property of emitting ‘pellets’, where the spinal cord of its preys remains intact. therefore, by studying the pellets of the barn owl, one can more or less study the biodiversity and the abundance of the species living in a given area. 2both the pre-test and the post-test sheets, as well as the teaching sequences worksheets, are available, upon request of the reviewers. 358 kostas karamanos:ecosystem food-webs as dynamic systems... fig.2 the pellet findings for tyto alba in northwest and southwest u.s.a.[14] given the above table, students were asked to answer certain questions (more details follow in the results section). teaching session ii: in the second teaching session, which took place one week later, students were given a worksheet with an actual representation of the food-web of the barn owl (fig.3), as derived from scientific literature[15]. the worksheet contained questions asking to correlate the theoretical foodweb of the barn owl with the actual two food-webs that stem from the table in fig.2. advances in systems science and applications (2012) vol.12 no.4 359 fig.3 the food-web of the barn owl, as given in the scientific literature[15]. students attempt to represent, on paper, two food-webs (one for each region), where the names of prey and predators must correspond to the given pelletfindings table. this involved having to exclude or add species to fig.3. following this, students answered questions regarding the complexity of the food-web as a system, in relation to geographical latitude, the relative importance of species and their biomasses in the food-web system. teaching session iii: in the third teaching session, students were given a simpler table of the barnowl-pellet findings, containing only those species which have a significant change in their abundance between the northwestern and the southwestern states. students were asked to work in groups of two or three and to open an existing word (*.doc) file named “exercise in the food-web of the barn owl.” a screenshot of this word file is depicted in fig.4. students were asked to cooperate in moving the pre-existing arrows of the word file and the pre-existing shapes, to form, on the screen, a simplified foodweb of the owl, once for northwest and once for southwest. each of the two files was uniquely named. they were then asked to answer a series of questions on the worksheet, the most significant being: ‘question number 9. a) which one of the two food-webs that you have represented on screen is more 360 kostas karamanos:ecosystem food-webs as dynamic systems... fig.4 a screenshot of the microsoft word file exercise in the food-web of the barn owl. the objects (animals, plants and arrows) are movable. complex? the one for the northern or the one for the southern united states? justify our answer. b) in which of the two food-webs does the species of the barn owl (tyto alba) have more possibilities of becoming extinct? why?’ teaching session iv in the final teaching session, the students were asked once again to work in groups. this time, they interacted with the netlogo version 4.0.4 programming and the simulation environment[13]. specifically, they worked with the “wolf sheep predation (docked)” model[16]. this model is accompanied with a “system dynamics modeler” file, created in java platform[16]. both these files in the model are depicted in fig.5 and 6 respectively. students, after trying to interpret the terms in the “system dynamics modeler” screen, focused on interacting with the sliders in the “aggregate model” section (right part of the interface screen) of the “wolf sheep predation (docked)” model. this section is nothing more than the graphical representation of a simple one-predator-and-one-prey population system, obeying to two simple lotkavolterra equations. students were guided by questions in their worksheet, to try and “destabilize” this system, by finding certain positions of the three sliders (‘predatorefficiency’, ‘predationrate’ and ‘wolves-death-rate’) where the wolves or the sheep become extinct, and hence, the other population becomes extinct too or deviates exponentially. the effort was to guide them to the conclusion that this is relatively easy, when there is such a simple food chain. advances in systems science and applications (2012) vol.12 no.4 361 fig.5 a screenshot of the netlogo ”wolf sheep predation (docked)” model[16], with both sections of the interface screen (the ”agent model” and the ”aggregate model”) activated. fig.6 a screenshot of the system dynamics modeler screen, belonging to the netlogo wolf sheep predation (docked) model[16] in the next step of this session, students were confronted by a variation of this simple prey-and-predator system, where, instead of one there are two species of prey. the idea behind this concept is depicted in fig.7 and in fig.8. in fig.7, the simple food chain of the initial “wolf sheep predation (docked)” model, is explained. in fig.8, the new simple food-web is shown. this food-web was constructed, by creating a very simple variation of the “system dynamics modeler” 362 kostas karamanos:ecosystem food-webs as dynamic systems... that was named: “system modeler food-web without grass”. in fig.9, below the new “system modeler food-web without grass” screen, fig.7 the food chain corresponding to the initial wolf sheep predation (docked) model, in the aggregate model part. fig.8 the simple food-web corresponding to the new “system modeler food-web without grass” model. which the students had in front of them in the computers, is depicted. and in fig.10, the interaction / interface screen of the same model (“sysfig.9 a screenshot of the “system modeler food-web without grass” model (“diagram” window from “system dynamics modeler”) tem modeler food-web without grass”) is shown, where the students were again advances in systems science and applications (2012) vol.12 no.4 363 asked to “destabilize” the system, by making one or two of the three populations extinct, with the use of the sliders. the concept behind this variation is that, in this case, it was more difficult fig.10 a screenshot of the interface of the “system modeler food-web without grass”. to “destabilize” the eco-system. twenty days following the final teaching session, a post-test was delivered to all groups of students3, containing identical questions to the initial pre-test worksheet. following the teaching session, a phase of one-hour interviews with a “stratified” sample of 17 students, begun. 4 results and discussion initially, students were handed a diagram, similar to fig.3, containing only the names and the images of the species involved in the barn owl food-web, without the predation connections between them. they were then asked to hand-draw the food-web, as they thought it should be and provide an explanation behind their food-web diagram. a. 72 out of the 85 students (percentage 84.7% of the sample) drew the web with arrows pointing, as they should, from the prey to the predator. b. 57 out of the 85 (percentage 67.1%) used in their argumentation on why they drew lines like that, the rationale “who is eaten by whom”, instead of “who eats who”.both these findings imply that the students have conceptualized a “bottom to top” perspective of ecosystems as systems, instead of the “top to bottom” perspective, the former being considered as the appropriate systemic treatment[17]. at a later stage, the students were asked to create on paper, a barn-owl-foodweb for the northern and sothern states of america. here, the systemic-thinking 3both the pre-test and the post-test sheets, as well as the teaching sequences worksheets, are available, upon request of the reviewers. 364 kostas karamanos:ecosystem food-webs as dynamic systems... skill examined was to correlate the theoretically existing elements (“stocks”) and relations (“flows”) in a system, with the actual ones, even if the latter have sometimes different names. the drawing of a student, alex, for the northern and for the southern united states, are depicted below, in fig.11. the drawings of alex show good correlation between given data and theory, fig.11 the food-web of the barn owl, as drawn by a student, alex. northwest u.s.a. (left) and southwest u.s.a. (right) even though this specific student reveals a “misconception”, by creating a more complex food-web in the north instead of in the south. due to the discrepancies between the theoretical food-web and actual barn-owlpellet table data, with regards to the type of species present, a “scoring system” was established for the drawings in the worksheets: whenever the student includes one of the eight species in his drawing, one (1) point is given. whenever the student misses one species, or includes a wrong one, zero (0) points are given. the actual scores and the corresponding numbers of students, are depicted in table 1 below. by calculating the average and the standard deviation of the scores of the students, we find: mean value = 4.36 sd = 1.85 the resulting mean value and standard deviation for the sample reveal a satisfactory ability to correlate to systems, achieved by the teaching sequence, which is an encouraging outcome. advances in systems science and applications (2012) vol.12 no.4 365 table 1 the scores of the n=85 students, in correlating the parts (“stocks”) of the theoretical and of the actual (real-data) barn-owl-food-web. number of students score 4 0 3 1 5 2 12 3 21 4 13 5 19 6 6 7 2 8 in concluding the results concerning objective (i), some statistical analysis was performed. frank draper [18], based on earlier work of barry richmond[19] proposes seven skills of systems thinking of students and the associated levels of activity[18]. each skill has also levels of acquainting it within it. for the current work, only the following skills and the specific sub-levels were considered relevant, ordered in increasing achievement difficulty: table 2 some of the proposed skills of systems’ thinking of students and the associated levels of activity[18] as they were selected for the current work. skill 1 (structural thinking), level ii identifying stocks and flows in phenomena skill 3 (generic thinking), level ii tabincellcrecognizing and using stockand-flow generic structures skill 4 (operational thinking), level ii building paper-and-pencil stock-and-flow diagrams skill 5 (scientific thinking), level i manipulating and modifying preconstructed computer models skill 6 (closed-loop thinking), level i identifying simple internal causal relations in the pre-test and post-test work sheets, with questions and interactions with the computer, there was effort to measure all these skills, based on a marine food-web, as retrieved by[20]. 366 kostas karamanos:ecosystem food-webs as dynamic systems... fig.12 the marine food-web delivered to the students in the pre-test and posttest worksheets[20]. by semantically-analyzing and content-analyzing students’ answers in both pre and post test, each was graded from 0 to 5 according to the skills demonstrated by their answers. following this, the non-parametric wilcoxon signed-rank test was carried out. there were statistically significant shifts of systemic-thinking skills in all levels of the scale 0-5. the results are briefly depicted in table 3 below. table 3 results of the non-parametric wilcoxon signed-rank test for the sample of n=85 students. grade significance level 0 p= .018 (¡0.05) 1 p= .014 (¡0.05) 2 p= .021 (¡0.05) 3 p= .023 (¡0.05) 4 p= .019 (¡0.05) 5 p= .011 (¡0.05) the findings are again considered encouraging, for the impact of the chosen teaching method. objective (ii) referring to the “relative importance” of each population (“stock”) in the foodwebas-a-system, the students had to use the table of food-web-pellets, together with simple calculators, to answer specific questions : q1: which species of prey contributes more to the barn owl’s food in each area of u.s.a. (northern and southern)? advances in systems science and applications (2012) vol.12 no.4 367 q2: is this the same species for both areas? q3: if all the species of shrew (blarina, cryptotis and sorex) become extinct, which one of the two areas’ food-webs will be affected more? or will they be equally affected? justify your answer. q4: if the only species of vole (microtus) becomes extinct, which one of the two areas’ food-webs will be affected more? or will they be equally affected? justify your answer. we briefly present the “correct” answers to q1-q4 and the corresponding ratio of students that found them: q1: the mice. (correct answers: 76 out of 85. percentage 89.4%) q2: yes, it was the same. (correct answers: 71 out of 85. percentage 83.5%) q3: the south will be affected much more. (correct answers: 40 out of 85.percentage 47.1%) q4: the two areas will be almost equally affected. (correct answers: 32 out of 85.percentage 37.6%) in the following fig.13, the percentages of correct answers, with respect to each question, are presented as a bar chart. fig.13 the percentages of students’ correct answers, with respect to each question. what can be observed is that within the same food-web as a system, the students are adequately capable of finding the prevailing “stock” (population), but this capability is reduced, when having to think comparatively between two food-webs/ systems. this has also been noticed by other researchers[21]. the answers obtained, regarding the question outlined below are shown in table 4. ‘the diversity of species and the complexity of the food-webs increases, de368 kostas karamanos:ecosystem food-webs as dynamic systems... creases or does something else, as one moves from the poles to the equator? explain your answer.’ from the n=85 undergraduate students, the results were: table 4 results of the non-parametric wilcoxon signed-rank test for the sample of n=85 students. “increases” “decreases” “do not know/ other answers i cannot answer” pre-test 50 14 5 16 post-test 73 6 r.2 4 in fig.14, the findings of table 4 are expressed in the form of pie charts. it is easily concluded, that following the teaching sequence, students seem to have conceptualized that as the geographical latitude decreases in absolute values, the diversity and the connectedness of food-webs as systems increases. fig.14 comparative pie-charts, depicting the relative (%) answers of the students to the question referring to biodiversity and the geographical latitude, in the preand post-test. objective (iv). after interacting with the word file “exercise in the food-web of the barn owl” (fig.4), students were confronted with two questions in their worksheets: q1: which one of the food-webs is more complex? the one of the north or the one of the south? justify your answers. q2: in which one of the two webs, the barn owl faces greater dangers of becoming extinct? it was considered as encouraging that in both questions the students selected the correct statement, which means: in q1 : 63 of the 85 students (74.11%) answered “the food-web of the south”. in q1 : 55 of the 85 students (64.71%) answered “in the food-web of the north”. advances in systems science and applications (2012) vol.12 no.4 369 table 5 the answers of the students referring to stability as related to food-web complexity. “i agree” “i disagree” “do not know/ other answers i cannot answer” pre-test 46 21 11 7 post-test 68 11 5 1 furthermore, both in the pre-test and the post-test, there was the question:‘do you agree or disagree with the following statement: “the more complex the foodweb is in an ecosystem, within an area, the more stable the ecosystem is, i.e. the more difficult it is for species populations to become extinct.” justify your selection.’ from the n=85 undergraduate students, the results were: it is apparent that the opinion of students, after interacting with the software and with the barn-owl-pellet data, lies closer to that which prevails in science nowadays. additionally, students’ ability to correlate food-web interconnectedness with ecological stability improved through their interaction with the “aggregate model” of the netlogo “wolf sheep predation (docked)” model. following attempts to ‘destabilize’ the system by moving the sliders in fig.5, students were asked to fill-in the cloze question in the worksheet: the system becomes unstable, i.e. one population is extinct, in n1 =. . . cases or in n1=. . . combinations of the three slider values [here we expect the same integer number]. these values are . . . . . . . . . the same was repeated for the systemmodeler food-web without grass (fig.10) this new system becomes unstable, i.e. one population is extinct, in n2 =. . . cases or in n2=. . . . combinations of the two slider values [here we expect the same integer number]. these values are. . . . . . . in the first case n1 varied from 1 to 6, and in the second case n2 varied from 0 to 2. the exact results are depicted in table 6. table 6 the findings for the n=85 students, as regards the (integer) number of cases they succeeded in destabilizing the system of populations. integer number mean value standard deviation n1 3,418 1,082 n2 0,760 0,112 the drawn here is that the more “complex” the food-web the more difficult it is to “bring it to instability”. 370 kostas karamanos:ecosystem food-webs as dynamic systems... 5 conclusions the overall teaching sequence used, including drawing of shapes, interaction with simple computer software, oral instruction, as well as discussion and personal interviews with the undergraduate students, seemed to have had adequate results in their conceptualizing the systemic properties and comportment of food-webs. the students within the evaluation phase in each worksheet, as well as during the post-test stage, gave answers and descriptions regarding: food-web structure, relation of food-web complexity with ecosystem stability and the dynamic system properties of the food-web, which were close enough to the scientifically accepted ones.the research is still continuing with interviews and new samples of students are being used for instruction. references [1] green d.g, sadedin s. & leishman t. g. 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[online] available at: http://www.bemidjistate.edu/academics/departments/science/k12-scie nce-units/barn-owl-project-grade7-biology.pdf. [accessed 13 august 2010] [16] wilensky u. (2005), netlogo wolf sheep predation (docked) model. available at: http://ccl.northwestern.edu/netlogo/models/wolfsheeppredation (docked). center for connected learning and computer-based modeling, northwestern university, evanston, il. [17] magntorn o, & helldén, g. (2007), “reading nature from a ‘bottom-up’ perspective”, journal of biological education, vol.41, no.2, pp.68-75. [18] draper f. (1993), “a proposed sequence for developing systems thinking in a grades 4-12 curriculum”, system dynamics review, vol.9, no.2, pp.207-214. 372 kostas karamanos:ecosystem food-webs as dynamic systems... [19] richmond b.( 1993), “systems thinking: critical thinking skills for the 1990s and beyond”, system dynamics review, vol.9, no.2, pp.113-133. [20] pauly d. (1998) “fishing down marine food-webs’ as an integrative concept”, proceedings of the expo’98 conference on ocean food-webs and economic productivity, lisbon, portugal, 1-3 july 1998.[online]. available at: http://cordis.europa.eu/inco/fp5/icons/pauly1.gif. [accessed 13august 2010]. [21] gallegos l, jerezano m e, & flores f. (1994), “preconceptions and relations used by children in the construction of food chains”, journal of research in science teaching, vol.31, no.3, pp.259-272. corresponding author aristotelis gkiolmas can be contacted at: agkiolm@primedu.uoa.gr advances in systems science and applications (2014) vol.14 no.3 286-293 empirical study on location choice of foreign banks in china tong xue school of international economics and trade, bisu, beijing, china abstract using the panel data from 2006 to 2010, this paper analyses the determinants of location choice of foreign banks in china with the amount of assets of foreign banks in different regions as dependant variables. the empirical results show that market opportunity and following clients factors are both important determinants for foreign banks location choice. meanwhile the presence of foreign banks tends to be more in financial centers of china, holding everything else being equal. however, the non-performing loan ratio of commercial banks of one region is not important factor for foreign banks to consider for now. keywords foreign banks, china, location choice. 1 introduction with the complete liberalization of banking industry at the end of 2006, foreign banks accelerated the pace of entering into china. by 2010, 185 banks from 45 countries and regions set up 216 representative offices in china; 37 banks from 14 countries and regions were locally incorporated which maintained 223 branches. in addition, there were 90 foreign bank branches established by 74 banks from 25 countries and regions. and the total assets of foreign banking institutions in china increased 29.13 percent year-on-year to rmb1.74 trillion (china banking regulatory commission (cbrc), 2010). according to the authors statistics, these foreign banks maintained business branches in 34 cities in china. shanghai, beijing, shenzhen, guangzhou and tianjin have the largest number of business branches respectively, in which shanghai is the top one and has 102 business branches, accounting for 30% of the number of business branches of foreign banks in china. it can be seen that there is an agglomeration effect in the location choice made by foreign banks in china, i.e. the offices of foreign banks concentrate in those more developed and first-deregulated cities, such as shanghai, beijing, etc. therefore, what factors determine the location choice of foreign banks in china? and which regions can attract foreign banks to do more business? these are interesting problems worth concerning. among foreign literatures, in addition to the studies on the location choices among various host countries, there are some researches on the internal location choices in a host country, such as literature[1] and [2]. in china, some researchers made some descriptive analysis on the features of location choices of foreign banks in china (for example [3-5]) using panel data, zhang and yang made the empirical analysis on the determinants of location of foreign banks in 16 cities in advances in systems science and applications (2014) vol.14 no.3 287 china for the first time[6]. li examined the determinants in 24 cities on the basis of zhang and yang[6-7]. he and yeung[8] examined the locational distribution in 32 cities with conditional logit models. existing studies, however, all took the number of institutions but not the amount of assets of foreign banks as the dependent variables, while the number of institutions cannot accurately reflect the business development status of foreign banks. this paper aims to contribute to the existing literature by using the amount of assets of foreign banks as the dependent variable to investigate the determinants of location choice of foreign banks in china. 2 hypothesis development on the basis of previous studies and taking into account the availability of data, this paper empirically examines four factors that affect the location distribution of foreign banks in china. 2.1 market opportunity existing literatures prove that market opportunity is one of the important factors that affect the location choice of foreign banks. the market opportunity is generally measured from two aspects, economic development level and size of financial sector. the higher the level of economic development, (which is usually measured by gdp, gdp per capita or the growth rate of gdp), and the larger the financial markets, the demand of households and businesses for financial services is greater, which can bring more profit opportunity for foreign banks. many studies have proved it. for example, buch and yamori proved that from the perspective of national level, the entry of foreign banks is positively related with the gdp or gdp per capita of host countries[9-10]. the further study by focarelli and pozzolo showed that profit opportunities resulting from a high expected economic growth and the prospect of competing with relatively less efficient banks appear to be a key factor affecting the expansion abroad[11]. goldberg and grosse found that foreign bank presence in various states in the u.s., as measured by assets or offices, is positively related with the size of the states banking market[2]. studies on location distribution of foreign banks in china have the identical results. zhang and yang proved that disposable income per capita is positively related with the office number of that city[6]. li proved that the office number of foreign banks is significantly positively related with both disposable income per capita and credit aggregates[7]. therefore, based on the above discussion, the following are inferred: h1: foreign bank presence is drawn to the economically developed areas. h2: foreign bank presence is drawn to the areas with large banking market sizes. 288 anne talkington: an extension of a logistic model for microbial kinetics 2.2 “follow-the-customer” factor theories on multinational banking indicated that one of the motives of multinational banks to go abroad is to follow the customers. numerous empirical studies also proved that foreign banks presence is significantly correlated with the foreign trade or fdi volume between the home and host countries. meanwhile, some studies showed the foreign trade or fdi volumes of one area play important role in location choice of foreign banks within one country. for example, goldberg and grosse proved that the amount of fdi in the state is a significant determinant for attracting foreign bank assets, while the foreign trade volume is not[2]. studies on china made by zhang and yang and li also proved that number of foreign banks’ offices in one city is positively related with the foreign trade volume, while the amount of fdi is not a significant determinant[6-7]. therefore, the following is speculated: h3: foreign bank presence is drawn to the areas with more foreign trade or fdi volume. 2.3 institutional factors institutional factors are also important for location distribution of foreign banks. numerous studies found that foreign banks tend to enter the financial centers of one country, which result in agglomeration effect. setting up offices in financial center makes it convenient for foreign banks to establish good relationships with regulatory authorities and other financial institutions. it is also convenient to gain funds and information with low cost and to receive specialized services from other companies. studies made by bagchi-sen and o huallachin all showed foreign banks in the u.s. concentrated in the financial centers[1,12]. he and yeung showed that foreign banks agglomerate in beijing, shanghai, and cities hosting the regional branches of the people’s bank of china (central bank of china)[8]. based on the above discussion, it is reasonable to deduce the following: h4: foreign bank presence agglomerates in financial centers in china. 2.4 risk factor in consideration of risk control, foreign banks usually make the regional risk assessments for location choice. therefore, financial risk status of one region is also one of factors affecting the location distribution. empirical study made by li used weighted average nonperforming loan ratios of 24 cities as independent variable. but the coefficient was not significant[7]. this paper tries to study whether the asset distribution of foreign banks is related with the risk status of one region and thus the following hypothesis is proposed: h5: foreign bank presence is negatively related with the financial risk level of one region. advances in systems science and applications (2014) vol.14 no.3 289 3 methodology and data sources 3.1 dependent variable as mentioned before, different from the previous studies on china, this paper uses the amount of assets of foreign banks but not the number of offices in different regions as the dependent variable. however, due to the availability of data, the amount of assets of foreign banks in different provinces, autonomous regions, or municipalities but not in different cities is used, which is denoted as fbasset (in natural logarithmic form). the data is from financial performance report of each region from 2006 to 2010. since foreign banks did not enter some provinces until recently, this paper selects 20 among 31 provinces, autonomous regions, or municipalities which have foreign bank presence at least after 20081. 3.2 explanatory variables based on the previous studies and the availability of data, definitions of explanatory variables are as follows: (1) the natural logarithmic form of gdp and the growth rate of gdp are introduced to measure the level of economic development, which are denoted as gdp and gdpgrowth respectively and the data are from china statistical yearbook. (2) the ratio of total banking loan or total amount of banking assets to gdp are introduced to measure the banking market sizes of one region. the reason is that compared with provinces and autonomous regions, municipalities are much smaller with respect to area and population. then it is biased to use the absolute value of total banking loan or banking assets to measure the banking market sizes. two variables are denoted as loan and asset respectively and the data are from financial performance report. (3) to test the “follow-the-customer” hypothesis (h3), the natural logarithmic form of volume of foreign trade (trade) and the natural logarithmic form of realized amount of foreign direct investment (fdi) are included as explanatory variables. the data are from china statistical yearbook. (4) dummy variable (center) is introduced to test whether foreign bank presence agglomerates in financial centers in china (h4), which represents the locations of regional branches of the central bank, assigning a value of “1” for beijing, shanghai, tianjin, chongqin, guangdong, liaoning, shandong, sichuan, jiangsu, hubei and shanxi, and zero for other provinces. (5) the nonperforming loan ratios of commercial banks of each region (npl) are introduced to measure the financial risk status. the data are from distribution of npls of commercial banks by region , annual report of china banking reg1twenty provinces, autonomous regions, or municipalities include beijing, shanghai, tianjin, chongqin, guangdong, fujian, liaoning, shandong, sichuan, jiangsu, hubei, hunan, jiangxi, anhui, shanxi, guangxi, yunnan, heilongjiang, zhejiang, hainan. 290 anne talkington: an extension of a logistic model for microbial kinetics ulatory commission of each year. in summary, regression model based on panel data is as follows: fbassetit =cit + β1oppotunityit + β2followit+ β3centeri + β4nplit + εit where i =1, ...n, t = 1, ..., t 4 regression results first, the correlation test of all variables except dummy variable is made and the result is shown in table 1. table 1 correlation coefficients among variables fbasset gdp gdpgrowth loan asset trade fdi npl fbasset 1.000 gdp 0.564 1.000 (0.000) gdpgrowth -0.060 -0.050 1.000 (0.552) (0.615) loan 0.529 -0.006 -0.335 1.000 (0.000) (0.952) (0.001) asset 0.540 0.010 -0.326 0.926 1.000 (0.000) (0.919) (0.001) (0.000) trade 0.831 0.789 -0.158 0.374 0.372 1.000 (0.000) (0.000) (0.114) (0.000) (0.000) fdi 0.713 0.790 0.072 0.121 0.127 0.866 1.000 (0.000) (0.000) (0.471) (0.228) (0.204) (0.000) npl -0.217 -0.427 0.234 -0.476 -0.479 -0.335 -0.354 1.000 (0.043) (0.000) (0.029) (0.000) (0.000) (0.002) (0.001) notes: p values are in parentheses. as the table 1 shows, among the explanatory variables, loan and asset are highly correlated and the correlation coefficient is 0.926. trade and fdi are also highly correlated with the coefficient being 0.866. to mitigate the multicollinearity issue of estimates, the interaction of loan and asset, and trade and fdi are introduced in the model, to show the effect of banking market sizes and “follow-the-customer” factor on the location choice respectively. besides, gdp is also highly correlated with trade and fdi. so the significance of gdp and trade*fdi is tested separately. advances in systems science and applications (2014) vol.14 no.3 291 table 2 regression results on location choice of foreign banks in china (20062010) variable (1) (2) gdpgrowth 0.0630* (1.66) gdp 1.5105*** (5.37) asset*loan 0.1550*** 0.1112** (2.97) (2.13) fdi*trade 0.1054*** (5.45) center 1.7155*** 2.0640*** (3.57) (3.26) npl -0.0143 0.0099 (-0.76) (0.51) adjusted r2 0.5830 0.4990 number of observation 82 89 hausman test(p value) 2.05 6.18 (0.73) (0.10) estimation metho random effects random effects notes: (1) the sample period is from 2006 to 2010 and the number of cross-section is 20. the actual sample is less than 100 because some data are default or wrong. (2) the constants are left out in the results. (3) *significant at the 10% level; ** significant at the 5% level; *** significant at the 1% level, t values are in parentheses. (4) hausman-test statistic shows whether the random effects method is suitable. the empirical results of table 2 show that the asset distribution of foreign banks in china is significantly positively related with gdp or gdp growth rate of that region, demonstrating that the level of economic development is one of important determinants for foreign banks presence. thus, h1 is supported. in addition, the disposable income per capita is also introduced into the model instead of gdp and the result is similar. this is slightly different from the results of zhang and yang and li which showed the number of offices of foreign banks is positively related with the disposable income per capita but not gdp[6-7]. both of results (1) and (2) verify that the foreign bank presence is significantly positively related with the total banking loan or total amount of banking assets, proving foreign bank presence is drawn to the areas with large banking market sizes (h2). this is consistent with li[7]. the coefficient of trade*fdi is significantly positive, proving that foreign bank presence is drawn to the areas with more foreign trade or fdi volume (h3). foreign banks may follow their customers and enter the area with more international trade or fdi, providing financial services to those customers. this result is consistent with other studies on china. 292 anne talkington: an extension of a logistic model for microbial kinetics the coefficient of center is significantly positive and thus h4 is supported very well, showing that holding everything else being equal, foreign banks tend to operate in financial centers to obtain the advantages in funds, information and others. this result is consistent with he and yeung[8]. h5 is not proved since the coefficient of npl is positive or negative but not significant, showing that the foreign bank presence is not significantly related with the nonperforming loan ratios of commercial banks, which is consistent with li[7]. this indicates that foreign banks do not take the nonperforming loan ratios as an important factor when making location choice. one of reasons may be that foreign banks do not consider the nonperforming loan ratios of commercial banks equivalently with the financial risk status of that region. 5 conclusion based on the previous studies and using the panel data from 2006 to 2010, this paper analyses the determinants of location choice of foreign banks in china with the amount of assets of foreign banks in different regions as dependant variables. market opportunity, “follow-the-customer”, institution factor and risk factor are all included. the empirical results show that when foreign banks make the location choice, market opportunity is one of the important factors so that the economically developed regions with large banking sizes are more attractive to foreign banks. on the other hand, foreign bank also consider “follow-the-customer” strategy so that foreign bank presence is drawn to the areas with more foreign trade or fdi volume. meanwhile the presence of foreign banks tends to be more in financial centers of china, holding everything else being equal. however, the non-performing loan ratio of commercial banks of one region is not important factor for foreign banks to consider for now. acknowledgements this research is supported by funding project for academic human resources development in institutions of higher learning under the jurisdiction of beijing municipality (phr(ihlb)). references [1] bagchi-sen, s. (1991), “the location of foreign direct investment in finance, insurance, and real estate in the united states”, geografiska annaler, vol.73b, pp.187-197. [2] goldberg, l.g., and r. grosse. (1994), “location choice of foreign banks in the united states”, journal of economic business, no.46, pp.367-379. advances in systems science and applications (2014) vol.14 no.3 293 [3] zheng, bohong., jianzhong tang. (2001), “study on location of multinational banks in china”, world regional studies, vol.10, no.4, pp.21-28. [4] xie, shouhong, mingfeng wang. (2004), “a study on the location of foreign financial institutions in china”, human geography, vol.19, no.3, pp.50-55. [5] bai,yongping, fajun ji. (2010), “dynamic development and spatial distribution of foreign-funded banks in china”, journal of northwest normal university (natural science) , no.1, pp.108-113. [6] zhang, hongjun, chaojun yang. (2007), “empirical study on location choice and motivation of foreign banks in china”, journal of financial research, no.9, pp.160-172. [7] li, aixi. (2009), “behavior and empirical study on location choice of foreign banks in china”, journal of international trade, no.3, pp.118-124. [8] he, canfei, and godfrey yeung. (2010), “the locational distribution of foreign banks in china: a disaggregated analysis”, regional studies, vol.45, no.6, pp.733-754. [9] buch, c.m. (2000), “why do banks go abroad? evidence from german data”, financial markets, institutions & instruments, vol.9, no.1, pp.33-67. [10] nobuyoshi yamori. (1998), “a note on the location choice of multinational banks: the case of japanese financial institutions”, journal of banking & finance, vol.22, no.1, pp.109-120. [11] focarelli d. and a.f. pozzolo. (2005), “where do banks expand abroad? an empirical analysis”, journal of business, no.78, pp.2435-2463. [12] o huallachin b. (1994), “foreign banking in the american urban system of financial organization”, economic geography, vol.70, no.3, pp.206-228. corresponding author tong xue can be contacted at: xuetong@bisu.edu.cn. a new modified block iterative algorithm for a system of equilibrium problems and a fixed point set of uniformly φ-asymptotically nonexpansive mappings∗ siwaporn saewan and poom kumam department of mathematics, faculty of science, king mongkut’s university of technology thonburi (kmutt),bangmod, bangkok 10140, thailand email: 52501406@st.kmutt.ac.th, poom.kum@kmutt.ac.th abstract in this paper, we construct a new modified block hybrid projection algorithm for finding a common element of the set of common fixed points of an infinite family of closed and uniformly quasi -φasymptotically nonexpansive mappings, the set of the variational inequality for an α-inverse-strongly monotone operator, the set of solutions of a system of equilibrium problems. moreover, we obtain a strong convergence theorem for the sequences generated by this process in the framework banach spaces. the results presented in this paper improve and generalize some well-known results in the literature. keywords modified block iterative algorithm inverse-strongly monotone operator variational inequality a system of equilibrium problem uniformly quasi-φ-asympto tically nonexpansive mapping. 1. introduction let c be a nonempty closed convex subset of a real banach space e with ‖ · ‖ and e∗ the dual space of e and a : c → e∗ be an operator. the classical variational inequality problem for an operator a is to find x∗ ∈ c such that 〈ax∗, y − x∗〉 ≥ 0, ∀y ∈ c. (1) the set of solution of (1) is denote by v i(a,c). recall that let a : c → e∗ be a mapping. then a is called (i) monotone if 〈ax−ay, x− y〉 ≥ 0, ∀x, y ∈ c, (ii) α−inverse-strongly monotone if there exists a constant α > 0 such that 〈ax−ay, x− y〉 ≥ α‖x− y‖2, ∀x, y ∈ c. such a problem is connected with the convex minimization problem, the complementary problem, the problem of finding a point x∗ ∈ e satisfying ax∗ = 0. ∗this work was completed with the support by the national research university project of thailand’s office of the higher education commission (under nru-csec project no.54000267). issn 1078-6236 international institute for general systems studies, inc. advances in systems science and applications (2011), vol. 11, no. 1-2 123-148 quasilet {fi}i∈γ : c × c −→ r be a bifunction, {ϕi}i∈γ : c −→ r be a real-valued function, where γ is an arbitrary index set. the system of equilibrium problems, is to find x ∈ c such that fi(x, y) ≥ 0, i ∈ γ, ∀y ∈ c. (2) if γ is a singleton, then problem (2) reduces to the equilibrium problem, is to find x ∈ c such that f(x, y) ≥ 0, ∀y ∈ c. (3) the above formulation (3) was shown in [5] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, variational inequality problems, vector equilibrium problems, nash equilibria in noncooperative games. in addition, there are several other problems, for example, the complementarity problem, fixed point problem and optimization problem, which can also be written in the form of an ep (f). in other words, the ep (f) is an unifying model for several problems arising in physics, engineering, science, optimization, economics, etc. in the last two decades, many papers have appeared in the literature on the existence of solutions of ep (f); see, for example [5, 13] and references therein. some solution methods have been proposed to solve the ep (f); see, for example, [5, 13, 15, 16, 21, 24, 30, 29, 28, 35, 46] and references therein. for each p > 1, the generalized duality mapping jp : e → 2e ∗ is defined by jp(x) = {x∗ ∈ e∗ : 〈x, x∗〉 = ‖x‖p, ‖x∗‖ = ‖x‖p−1} for all x ∈ e. in particular, j = j2 is called the normalized duality mapping. if e is a hilbert space, then j = i , where i is the identity mapping. consider the functional defined by φ(x, y) = ‖x‖2 − 2〈x, jy〉+ ‖y‖2, ∀x, y ∈ e. (4) as well know that if c is a nonempty closed convex subset of a hilbert space h and pc : h → c is the metric projection of h onto c, then pc is nonexpansive. this fact actually characterizes hilbert spaces and consequently, it is not available in more general banach spaces. it is obvious from the definition of function φ that (‖x‖ − ‖y‖)2 ≤ φ(x, y) ≤ (‖x‖+ ‖y‖)2, ∀x, y ∈ e. (5) if e is a hilbert space, then φ(x, y) = ‖x − y‖2, for all x, y ∈ e. on the author hand, the generalized projection (alber [2]) πc : e → c is a map that assigns to an arbitrary point x ∈ e the minimum point of the functional φ(x, y), that is, πcx = x̄, where x̄ is the solution to the minimization problem φ(x̄, x) = inf y∈c φ(y, x), (6) existence and uniqueness of the operator πc follows from the properties of the functional φ(x, y) and strict monotonicity of the mapping j (see, for example, [1, 2, 12, 17, 37]). remark 1.1. if e is a reflexive, strictly convex and smooth banach space, then for x, y ∈ e, φ(x, y) = 0 if and only if x = y. it is sufficient to show that if φ(x, y) = 0 then x = y. from ( 4 ), we have ‖x‖ = ‖y‖. this implies that 〈x, jy〉 = ‖x‖2 = ‖jy‖2. from the definition of j, one has jx = jy. therefore, we have x = y; see [12, 37] for more details. 124 saewan:a new modified block iterative algorithm for a system of. . . . . . let c be a closed convex subset of e, a mapping t : c → c is said to be l-lipschitz continuous if ‖tx− ty‖ ≤ l‖x− y‖,∀x, y ∈ c and a mapping t is said to be nonexpansive if ‖tx − ty‖ ≤ ‖x − y‖, ∀x, y ∈ c. a point x ∈ c is a fixed point of t provided tx = x. denote by f (t ) the set of fixed points of t ; that is, f (t ) = {x ∈ c : tx = x}. recall that a point p in c is said to be an asymptotic fixed point of t [31] if c contains a sequence {xn} which converges weakly to p such that limn→∞ ‖xn − txn‖ = 0. the set of asymptotic fixed points of t will be denoted by f̃ (t ). a mapping t from c into itself is said to be relatively nonexpansive [25, 36, 45] if f̃ (t ) = f (t ) and φ(p, tx) ≤ φ(p, x) for all x ∈ c and p ∈ f (t ). the asymptotic behavior of a relatively nonexpansive mapping was studied in [6, 7, 8]. t is said to be φ-nonexpansive, if φ(tx, ty) ≤ φ(x, y) for x, y ∈ c. t is said to be relatively quasi-nonexpansive if f (t ) 6= ∅ and φ(p, tx) ≤ φ(p, x) for all x ∈ c and p ∈ f (t ). t is said to be quasi-φ-asymptotically nonexpansive if f (t ) 6= ∅ and there exists a real sequence {kn} ⊂ [1,∞) with kn → 1 such that φ(p, tnx) ≤ knφ(p, x) for all n ≥ 1 x ∈ c and p ∈ f (t ). we note that the class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings [6, 7, 8, 23, 33] which requires the strong restriction: f (t ) = f̃ (t ). a mapping t is said to be closed if for any sequence {xn} ⊂ c with xn → x and txn → y, then tx = y. it is easy to know that each relatively nonexpansive mapping is closed. definition 1.2. ([9]) (1) let {ti}∞i=1 : c → c be a sequence of mapping. {ti}∞i=1 is said to be a family of uniformly quasi-φ-asymptotically nonexpansive mappings, if ∩∞i=1f (ti) 6= ∅, and there exists a sequence {kn} ⊂ [1,∞) with kn → 1 such that for each i ≥ 1 φ(p, tni x) ≤ knφ(p, x), ∀p ∈ ∩∞i=1f (ti), x ∈ c, ∀n ≥ 1. (7) (2) a mapping t : c → c is said to be uniformly l-lipschitz continuous, if there exists a constant l > 0 such that ‖tnx− tny‖ ≤ l‖x− y‖, ∀x, y ∈ c. (8) remark 1.3. it is easy to see that anα−inverse-strongly monotone is monotone and 1 α -lipschitz continuous. in 2004, matsushita and takahashi [22] introduced the following iteration: a sequence {xn} defined by xn+1 = πcj −1(αnjxn + (1− αn)jtxn), (9) where the initial guess element x0 ∈ c is arbitrary, {αn} is a real sequence in [0, 1], t is a relatively nonexpansive mapping and πc denotes the generalized projection from e onto a closed convex subset c of e. they proved that the sequence {xn} converges weakly to a fixed point of t . in 2005, matsushita and takahashi [23] proposed the following hybrid iteration method (it is also called the cq method) with generalized projection for relatively nonexpansive mapping advances in systems science and applications (2011), vol. 11, no. 1-2 125 t in a banach space e: x0 ∈ c chosen arbitrarily, yn = j−1(αnjxn + (1− αn)jtxn), cn = {z ∈ c : φ(z, yn) ≤ φ(z, xn)}, qn = {z ∈ c : 〈xn − z, jx0 − jxn〉 ≥ 0}, xn+1 = πcn∩qnx0. (10) they proved that {xn} converges strongly to πf (t )x0, where πf (t ) is the generalized projection from c onto f (t ). in 2008, iiduka and takahashi [14] introduced the following iterative scheme for finding a solution of the variational inequality problem for an inversestrongly monotone operator a in a 2-uniformly convex and uniformly smooth banach space e : x1 = x ∈ c and xn+1 = πcj −1(jxn − λnaxn), (11) for every n = 1, 2, 3, . . ., where πc is the generalized metric projection frome ontoc, j is the duality mapping from e into e∗ and {λn} is a sequence of positive real numbers. they proved that the sequence {xn} generated by (11) converges weakly to some element of v i(a,c). takahashi and zembayashi [39, 40], studied the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in the framework of banach spaces. in 2009, wattanawitoon and kumam [41] using the idea of takahashi and zembayashi [39] extend the notion from relatively nonexpansive mappings or φ-nonexpansive mappings to two relatively quasi-nonexpansive mappings and also proved some strong convergence theorems to approximate a common fixed point of relatively quasi-nonexpansive mappings and the set of solutions of an equilibrium problen in the framework of banach spaces. cholamjiak [10], proved the following iteration: zn = πcj −1(jxn − λnaxn), yn = j−1(αnjxn + βnjtxn + γnjszn), un ∈ c such that f(un, y) + 1 rn 〈y − un, jun − jyn〉 ≥ 0, ∀y ∈ c, cn+1 = {z ∈ cn : φ(z, un) ≤ φ(z, xn), xn+1 = πcn+1x0, (12) where j is the duality mapping on e. assume that αn, βn and γn are sequence in [0, 1]. then {xn} converges strongly to q = πfx0, where f := f (t ) ∩ f (s) ∩ ep (f) ∩ v i(a,c). in 2010, saewan et al. [33] introduced a new hybrid projection iterative scheme which is difference from the algorithm (12) of cholamjiak in [10, theorem 3.1] for two relatively quasinonexpansive mappings in a banach space. motivated by the results of takahashi and zembayashi [40], cholumjiak and suantai [11] proved the following strong convergence theorem by the hybrid iterative scheme for approximation of common fixed point of countable families of relatively quasi-nonexpansive mappings in a uniformly convex and uniformly smooth banach space: x0 ∈ e, x1 = πc1x0, c1 = c yn,i = j−1(αnjxn + (1− αn)jtxn, ) un,i = t fmrm,nt fm−1 rm−1,n · · ·t f1 r1,nyn,i cn+1 = {z ∈ cn : supi>1 φ(z, jun,i) ≤ φ(w, jxn)}, xn+1 = πcn+1x0, n ≥ 1. (13) 126 saewan:a new modified block iterative algorithm for a system of. . . . . . then, they proved that under certain appropriate conditions imposed on {αn}, and {rn,i}, the sequence {xn} converges strongly to πcn+1x0. we note that the block iterative method is a method which often used by many authors to solve the convex feasibility problem (see, [18, 20], etc.). in 2008, plubtieng and ungchittrakool [27] established strong convergence theorems of block iterative methods for a finite family of relatively nonexpansive mappings in a banach space by using the hybrid method in mathematical programming. chang et al. [9] proposed the modified block iterative algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasi-φasymptotically nonexpansive mapping, they obtain the strong convergence theorems in a banach space. in 2010, saewan and kumam [34] obtain the following result for the set of solutions of the generalized equilibrium problems and the set of common fixed points of an infinite family of closed and uniformly quasi-φ-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex banach space e with kadec-klee property. very recently, qin, cho and kang [28] purposed the problem of approximating a common fixed point of two asymptotically quasi-φ-nonexpansive mappings based on hybrid projection methods. strong convergence theorems are established in a real banach space. h. zegeye, e. u. ofoedu and n. shahzad [46] introduced an iterative process which converges strongly to a common element of set of common fixed points of countably infinite family of closed relatively quasinonexpansive mappings, the solution set of generalized equilibrium problem and the solution set of the variational inequality problem for a α-inverse strongly monotone mapping in banach spaces. motivated and inspired by the work of chang et al. [9], qin et al. [30], takahashi and zembayashi [39], wattanawitoon and kumam [41], zegeye [44] and saewan and kumam [34], we introduce a new modified block hybrid projection algorithm for finding a common element of the set of the variational inequality for an α-inverse-strongly monotone operator, the set of solutions of the system of equilibrium problems and the set of common fixed points of an infinite family of closed and uniformly quasi-φ-asymptotically nonexpansive mappings in a 2-uniformly convex and uniformly smooth banach space. the results presented in this paper improve and generalize some well-known results in the literature. 2. preliminaries a banach space e is said to be strictly convex if ‖x+y 2 ‖ < 1 for all x, y ∈ e with ‖x‖ = ‖y‖ = 1 and x 6= y. let u = {x ∈ e : ‖x‖ = 1} be the unit sphere of e. then a banach space e is said to be smooth if the limit lim t→0 ‖x+ ty‖ − ‖x‖ t exists for each x, y ∈ u. it is also said to be uniformly smooth if the limit is attained uniformly for x, y ∈ u . let e be a banach space. the modulus of convexity of e is the function δ : [0, 2]→ [0, 1] defined by δ(ε) = inf{1− ‖x+ y 2 ‖ : x, y ∈ e, ‖x‖ = ‖y‖ = 1, ‖x− y‖ ≥ ε}. advances in systems science and applications (2011), vol. 11, no. 1-2 127 a banach space e is uniformly convex if and only if δ(ε) > 0 for all ε ∈ (0, 2]. let p be a fixed real number with p ≥ 2. a banach space e is said to be p-uniformly convex if there exists a constant c > 0 such that δ(ε) ≥ cεp for all ε ∈ [0, 2]; see [3, 38] for more details. observe that every p-uniformly convex is uniformly convex. one should note that no a banach space is p-uniformly convex for 1 < p < 2. it is well known that a hilbert space is 2-uniformly convex, uniformly smooth. it is also known that if e is uniformly smooth, then j is uniformly norm-to-norm continuous on each bounded subset of e. remark 2.1. the following basic properties can be found in cioranescu [12]. (i) if e is a uniformly smooth banach space, then j is uniformly continuous on each bounded subset of e. (ii) if e is a reflexive and strictly convex banach space, then j−1 is norm-weak∗-continuous. (iii) if e is a smooth, strictly convex, and reflexive banach space, then the normalized duality mapping j : e → 2e ∗ is single-valued, one-to-one, and onto. (iv) a banach space e is uniformly smooth if and only if e∗ is uniformly convex. (v) each uniformly convex banach space e has the kadec-klee property, that is, for any sequence {xn} ⊂ e, if xn ⇀ x ∈ e and ‖xn‖ → ‖x‖, then xn → x. we also need the following lemmas for the proof of our main results. lemma 2.2. (beauzamy [4] and xu[42]). if e be a 2-uniformly convex banach space. then for all x, y ∈ e we have ‖x− y‖ ≤ 2 c2 ‖jx− jy‖, where j is the normalized duality mapping of e and 0 < c ≤ 1. the best constant 1 c in lemma is called the p-uniformly convex constant of e. lemma 2.3. (beauzamy [4] and zalinescu [43]). ife be a p-uniformly convex banach space and let p be a given real number with p ≥ 2. then for all x, y ∈ e, jx ∈ jp(x) and jy ∈ jp(y) 〈x− y, jx − jy〉 ≥ cp 2p−2p ‖x− y‖p, where jp is the generalized duality mapping of e and 1 c is the p-uniformly convexity constant of e. lemma 2.4. (kamimura and takahashi [17]). let e be a uniformly convex and smooth banach space and let {xn} and {yn} be two sequences of e. if φ(xn, yn)→ 0 and either {xn} or {yn} is bounded, then ‖xn − yn‖ → 0. lemma 2.5. (alber [2]). let c be a nonempty closed convex subset of a smooth banach space e and x ∈ e. then x0 = πcx if and only if 〈x0 − y, jx− jx0〉 ≥ 0, ∀y ∈ c. lemma 2.6. (alber [2, lemma 2.4]). let e be a reflexive, strictly convex and smooth banach space, let c be a nonempty closed convex subset of e and let x ∈ e. then φ(y,πcx) + φ(πcx, x) ≤ φ(y, x), ∀y ∈ c. 128 saewan:a new modified block iterative algorithm for a system of. . . . . . let e be a reflexive, strictly convex, smooth banach space and j is the duality mapping from e into e∗. then j−1 is also single value, one-to-one, surjective, and it is the duality mapping from e∗ into e. we make use of the following mapping v studied in alber [2] v (x, x∗) = ‖x‖2 − 2〈x, x∗〉+ ‖x∗‖2, (14) for all x ∈ e and x∗ ∈ e∗, that is, v (x, x∗) = φ(x, j−1(x∗)). lemma 2.7. (alber [2]). let e be a reflexive, strictly convex smooth banach space and let v be as in (14) . then v (x, x∗) + 2〈j−1(x∗)− x, y∗〉 ≤ v (x, x∗ + y∗), for all x ∈ e and x∗, y∗ ∈ e∗. a set valued mapping b : e ⇒ e∗ with graph g(b) = {(x, x∗) : x∗ ∈ bx}, domain d(b) = {x ∈ e : bx 6= ∅}, and rang r(b) = ∪{bx : x ∈ d(b)}. b is said to be monotone if 〈x − y, x∗ − y∗〉 ≥ 0 whenever (x, x∗) ∈ g(b), (y, y∗) ∈ g(b). we denote a set valued operator b form e to e∗ by b ⊂ e × e∗. a monotone b is said to be maximal if its graph is not property contained in the graph of any other monotone operator. ifb is maximal monotone, then the solution set b−10 is closed and convex. let e be a reflexive, strictly convex and smooth banach space, it is knows that b is a maximal monotone if and only if r(j + rb) = e∗ for all r > 0. define the resolvent of b by jrx = xr. in other words, jr = (j + rb)−1j for all r > 0. jr is a single-valued mapping from e to d(b). also, b−1(0) = f (jr) for all r > 0, where f (jr) is the set of all fixed points of jr. define, for r > 0, the yosida approximation of b by trx = (jx − jjrx)/r for all x ∈ c. we know that trx ∈ b(jrx) for all r > 0 and x ∈ e. let a be an inverse-strongly monotone mapping of c into e∗ which is said to be hemicontinuous if for all x, y ∈ c, the mapping f of [0, 1] intoe∗, defined by f (t) = a(tx+(1−t)y), is continuous with respect to the weak∗ topology of e∗. we define by nc(v) the normal cone for c at a point v ∈ c, that is, nc(v) = {x∗ ∈ e∗ : 〈v − y, x∗〉 ≥ 0, ∀y ∈ c}. (15) lemma 2.8. (rockafellar [32]). letc be a nonempty, closed convex subset of a banach spacee and a is a monotone, hemicontinuous operator of c into e∗. let b ⊂ e × e∗ be an operator defined as follows: bv = { av +nc(v), v ∈ c; ∅, otherwise. (16) then b is maximal monotone and b−10 = v i(a,c). lemma 2.9. (chang et al.[9]) . let e be a uniformly convex banach space, r > 0 be a positive number andbr(0) be a closed ball ofe. then, for any given sequence {xi}∞i=1 ⊂ br(0) and for any given sequence {λi}∞i=1 of positive number with ∑∞ n=1 λn = 1, there exists a continuous, strictly increasing, and convex function g : [0, 2r) → [0,∞) with g(0) = 0 such that, for any positive integer i, j with i < j, ‖ ∞∑ n=1 λnxn‖2 ≤ ∞∑ n=1 λn‖xn‖2 − λiλjg(‖xi − xj‖). (17) advances in systems science and applications (2011), vol. 11, no. 1-2 129 lemma 2.10. (chang et al.[9]) . let e be a real uniformly smooth and strictly convex banach space, and c be a nonempty closed convex subset of e. let t : c → c be a closed and quasi-φ-asymptotically nonexpansive mapping with a sequence {kn} ⊂ [1,∞), kn → 1. then f (t ) is a closed convex subset of c. for solving the equilibrium problem for a bifunction f : c × c → r, let us assume that f satisfies the following conditions: (a1) f(x, x) = 0 for all x ∈ c; (a2) f is monotone, i.e., f(x, y) + f(y, x) ≤ 0 for all x, y ∈ c; (a3) for each x, y, z ∈ c, lim t↓0 f(tz + (1− t)x, y) ≤ f(x, y); (a4) for each x ∈ c, y 7→ f(x, y) is convex and lower semi-continuous. for example, let a be a continuous and monotone operator of c into e∗ and define f(x, y) = 〈ax, y − x〉,∀x, y ∈ c. then, f satisfies (a1)-(a4). the following result is in blum and oettli [5]. lemma 2.11. (blum and oettli [5]). letc be a closed convex subset of a smooth, strictly convex and reflexive banach space e, let f be a bifunction from c×c to r satisfying (a1)-(a4), and let r > 0 and x ∈ e. then, there exists z ∈ c such that f(z, y) + 1 r 〈y − z, jz − jx〉 ≥ 0, ∀y ∈ c. lemma 2.12. (takahashi and zembayashi [39]). let c be a closed convex subset of a uniformly smooth, strictly convex and reflexive banach space e and let f be a bifunction from c × c to r satisfying conditions (a1)-(a4). for all r > 0 and x ∈ e, define a mapping t fr : e → c as follows: t fr x = {z ∈ c : f(z, y) + 1 r 〈y − z, jz − jx〉 ≥ 0, ∀y ∈ c}. then the following hold: (1) t fr is single-valued; (2) t fr is a firmly nonexpansive-type mapping [19], that is, for all x, y ∈ e, 〈t fr x− t fr y, jt fr x− jt fr y〉 ≤ 〈t fr x− t fr y, jx− jy〉; (3) f (t fr ) = ep (f); (4) ep (f) is closed and convex. lemma 2.13. (takahashi and zembayashi [39]). let c be a closed convex subset of a smooth, strictly convex, and reflexive banach space e, let f be a bifunction from c ×c to r satisfying (a1)-(a4) and let r > 0. then, for x ∈ e and q ∈ f (t fr ), φ(q, t fr x) + φ(t fr x, x) ≤ φ(q, x). 130 saewan:a new modified block iterative algorithm for a system of. . . . . . 3. main results in this section, we prove the new convergence theorems for finding the set of solutions of system of generalized mixed equilibrium problems, the common fixed point set of a family of closed and uniformly quasi-φ-asymptotically nonexpansive mappings, and the solution set of variational inequalities for an α-inverse strongly monotone mapping in a 2-uniformly convex and uniformly smooth banach space. theorem 3.1. let c be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth banach space e. for each j = 1, 2, ...,m let fj be a bifunction from c × c to r which satisfies conditions (a1)-(a4). let a be an α-inverse-strongly monotone mapping of c into e∗ satisfying ‖ay‖ ≤ ‖ay −au‖, ∀y ∈ c and u ∈ v i(a,c) 6= ∅. let {si}∞i=1 : c → c be an infinite family of closed uniformly li-lipschitz continuous and uniformly quasiφ-asymptotically nonexpansive mappings with a sequence {kn} ⊂ [1,∞), kn → 1 such that f := (∩∞i=1f (si))∩ (∩mj=1ep (fj))(∩v i(a,c)) is a nonempty and bounded subset in c. for an initial point x0 ∈ e with x1 = πc1x0 and c1 = c, we define the sequence {xn} as follows:  vn = πcj −1(jxn − λnaxn), zn = j−1(αn,0jxn + ∑∞ i=1 αn,ijs n i vn), yn = j−1(βnjxn + (1− βn)jzn), un = t fmrm,nt fm−1 rm−1,n ...t f2 r2,nt f1 r1,nyn, cn+1 = {z ∈ cn : φ(z, un) ≤ φ(z, xn) + θn}, xn+1 = πcn+1x0, ∀n ≥ 1, (18) where j is the duality mapping on e, θn = supq∈f (kn − 1)φ(q, xn), for each i ≥ 0, {αn,i} and {βn} are sequences in [0, 1], {rj,n} ⊂ [d,∞) for some d > 0 and {λn} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, where 1 c is the 2-uniformly convexity constant of e. if∑∞ i=0 αn,i = 1 for all n ≥ 0, lim infn−→∞ βn(1− βn) > 0 and lim infn−→∞ αn,0αn,i > 0 for all i ≥ 1, then {xn} converges strongly to p ∈ f , where p = πfx0. proof . we first show that cn+1 is closed and convex for each n ≥ 0. clearly c1 = c is closed and convex. suppose that cn is closed and convex for each n ∈ n. since for any z ∈ cn, we known that φ(z, un) ≤ φ(z, xn) + θn is equivalent to 2〈z, jxn − jun〉 ≤ ‖xn‖2 − ‖un‖2 + θn. hence, cn+1 is closed and convex. next, we show that f ⊂ cn for all n ≥ 0. since by the convexity of ‖ · ‖2, property of φ, lemma 2.9 and by uniformly quasi-φ-asymptotically nonexpansive of sn for each q ∈ f ⊂ cn, advances in systems science and applications (2011), vol. 11, no. 1-2 131 we have φ(q, un) = φ(q, t fmrm,n t fm−1 rm−1,n ...t f2r2,nt f1 r1,nyn) ≤ φ(q, yn) = φ(q, j−1(βnjxn + (1− βn)jzn) = ‖q‖2 − 2〈q, βnjxn + (1− βn)jzn〉+ ‖βnjxn + (1− βn)jzn‖2 ≤ ‖q‖2 − 2βn〈q, jxn〉 − 2(1− βn)〈q, jzn〉+ βn‖xn‖2 + (1− βn)‖zn‖2 = βnφ(q, xn) + (1− βn)φ(q, zn), (19) and φ(q, zn) = φ(q, j−1(αn,0jxn + ∑∞ i=1 αn,ijs n i vn)) = ‖q‖2 − 2〈q, αn,0jxn + ∑∞ i=1 αn,ijs n i vn〉+ ‖αn,0jxn + ∑∞ i=1 αn,ijs n i vn‖2 = ‖q‖2 − 2αn,0〈q, jxn〉 − 2 ∑∞ i=1 αn,i〈q, jsni vn〉+ ‖αn,0jxn + ∑∞ i=1 αn,ijs n i vn‖2 ≤ ‖q‖2 − 2αn,0〈q, jxn〉 − 2 ∑∞ i=1 αn,i〈q, jsni vn〉+ αn,0‖jxn‖2 + ∑∞ i=1 αn,i‖jsni vn‖2 −αn,0αn,jg‖jvn − jsnj vn‖ = ‖q‖2 − 2αn,0〈q, jxn〉+ αn,0‖jxn‖2 − 2 ∑∞ i=1 αn,i〈q, jsni vn〉 + ∑∞ i=1 αn,i‖jsni vn‖2 − αn,0αn,jg‖jvn − jsnj vn‖ = αn,0φ(q, xn) + ∑∞ i=1 αn,iφ(q, sni vn)− αn,0αn,jg‖jvn − jsnj vn‖ ≤ αn,0φ(q, xn) + ∑∞ i=1 αn,iknφ(q, vn)− αn,0αn,jg‖jvn − jsnj vn‖. (20) it follows from lemma 2.7, that φ(q, vn) = φ(q,πcj −1(jxn − λnaxn)) ≤ φ(q, j−1(jxn − λnaxn)) = v (q, jxn − λnaxn) ≤ v (q, (jxn − λnaxn) + λnaxn)− 2〈j−1(jxn − λnaxn)− q, λnaxn〉 = v (q, jxn)− 2λn〈j−1(jxn − λnaxn)− q, axn〉 = φ(q, xn)− 2λn〈xn − q,axn〉+ 2〈j−1(jxn − λnaxn)− xn,−λnaxn〉. (21) since q ∈ v i(a,c) and a is an α-inverse-strongly monotone mapping, we have −2λn〈xn − q, axn〉 = −2λn〈xn − q,axn −aq〉 − 2λn〈xn − q, aq〉 ≤ −2λn〈xn − q,axn −aq〉 ≤ −2αλn‖axn −aq‖2. (22) by lemma 2.2 and ‖axn‖ ≤ ‖axn −aq‖, ∀q ∈ v i(a,c), we also have 2〈j−1(jxn − λnaxn)− xn,−λnaxn〉 = 2〈j−1(jxn − λnaxn)− j−1(jxn),−λnaxn〉 ≤ 2‖j−1(jxn − λnaxn)− j−1(jxn)‖‖λnaxn‖ ≤ 4 c2 ‖jj−1(jxn − λnaxn)− jj−1(jxn)‖‖λnaxn‖ = 4 c2 ‖jxn − λnaxn − jxn‖‖λnaxn‖ = 4 c2 ‖λnaxn‖2 = 4 c2 λ2 n‖axn‖2 ≤ 4 c2 λ2 n‖axn −aq‖2. (23) 132 saewan:a new modified block iterative algorithm for a system of. . . . . . substituting (22) and (23) into (21), we have φ(q, vn) ≤ φ(q, xn)− 2αλn‖axn −aq‖2 + 4 c2 λ2 n‖axn −aq‖2 = φ(q, xn) + 2λn( 2 c2 λn − α)‖axn −aq‖2 ≤ φ(q, xn). (24) substituting (24) into (20), we also have φ(q, zn) ≤ αn,0φ(q, xn) + ∑∞ i=1 αn,iknφ(q, xn)− αn,0αn,jg‖jvn − jsnj vn‖ ≤ αn,0knφ(q, xn) + ∑∞ i=1 αn,iknφ(q, xn)− αn,0αn,jg‖jvn − jsnj vn‖ = knφ(q, xn)− αn,0αn,jg‖jvn − jsnj vn‖ ≤ φ(q, xn) + supq∈f (kn − 1)φ(q, xn)− αn,0αn,jg‖jvn − jsnj vn‖ = φ(q, xn) + θn − αn,0αn,jg‖jvn − jsnj vn‖ ≤ φ(q, xn) + θn. (25) and substituting (25) into (19), we obtain φ(q, un) ≤ φ(q, xn) + θn. (26) thus, this show that q ∈ cn+1 implies that f ⊂ cn+1 and hence, f ⊂ cn for all n ≥ 0. this implies that the sequence {xn} is well defined. from definition of cn+1 that xn = πcnx0 and xn+1 = πcn+1x0,∈ cn+1 ⊂ cn we have φ(xn, x0) ≤ φ(xn+1, x0), ∀n ≥ 0. (27) form lemma 2.6, it follows that φ(xn, x0) = φ(πcnx0, x0) ≤ φ(q, x0)− φ(q, xn) ≤ φ(q, x0), ∀q ∈ f. (28) by (27) and (28), then {φ(xn, x0)} are nondecreasing and bounded. so, we obtain that lim n→∞ φ(xn, x0) exists. in particular, by (5), the sequence {(‖xn‖− ‖x0‖)2} is bounded. this implies {xn} is also bounded. we denote m := sup n≥0 {‖xn‖} <∞. (29) moreover, by the definition of θn and (29), it follows that θn −→ 0 as n −→∞. (30) next, we show that {xn} is a cauchy sequence in c. since xm = πcmx0 ∈ cm ⊂ cn, for m > n, by lemma 2.6, we have φ(xm, xn) = φ(xm,πcnx0) ≤ φ(xm, x0)− φ(πcnx0, x0) = φ(xm, x0)− φ(xn, x0). advances in systems science and applications (2011), vol. 11, no. 1-2 133 since limn−→∞ φ(xn, x0) exists and we taking m,n→∞ then, we get φ(xm, xn)→ 0. from lemma 2.4, we have limn→∞ ‖xm − xn‖ = 0. thus {xn} is a cauchy sequence and by the completeness of e and there exist a point p ∈ c such that xn → p as n→∞. now, we claim that ‖jun−jxn‖ → 0, as n→∞. by definition of xn = πcnx0, we have φ(xn+1, xn) = φ(xn+1,πcnx0) ≤ φ(xn+1, x0)− φ(πcnx0, x0) = φ(xn+1, x0)− φ(xn, x0). since lim n→∞ φ(xn, x0) exists, we also have lim n→∞ φ(xn+1, xn) = 0. (31) again form lemma 2.4, that lim n→∞ ‖xn+1 − xn‖ = 0. (32) from j is uniformly norm-to-norm continuous on bounded subsets of e, we obtain lim n→∞ ‖jxn+1 − jxn‖ = 0. (33) since xn+1 = πcn+1x0 ∈ cn+1 ⊂ cn and the definition of cn+1, we have φ(xn+1, un) ≤ φ(xn+1, xn) + θn. by (30) and (31), that lim n→∞ φ(xn+1, un) = 0. (34) applying lemma 2.4, we have lim n→∞ ‖xn+1 − un‖ = 0. (35) since ‖un − xn‖ = ‖un − xn+1 + xn+1 − xn‖ ≤ ‖un − xn+1‖+ ‖xn+1 − xn‖ it follows from (32) and (35), that lim n→∞ ‖un − xn‖ = 0. (36) since j is uniformly norm-to-norm continuous on bounded subsets of e, we also have lim n→∞ ‖jun − jxn‖ = 0. (37) next, we will show that xn → p ∈ f := ∩mj=1ep (fj) ∩ (∩∞i=1f (si)) ∩ v i(a,c). (i) we show that xn → p ∈ ∩∞i=1f (si). since xn+1 = πcn+1x0 ∈ cn+1 ⊂ cn, it follow from (25), we have φ(xn+1, zn) ≤ φ(xn+1, xn) + θn, by (30) and (31), we get lim n→∞ φ(xn+1, zn) = 0 (38) 134 saewan:a new modified block iterative algorithm for a system of. . . . . . again form lemma 2.4, that lim n→∞ ‖xn+1 − zn‖ = 0. (39) since ‖zn − xn‖ ≤ ‖zn − xn+1‖+ ‖xn+1 − xn‖ from (32) and (39), we have lim n→∞ ‖zn − xn‖ = 0. (40) by using the triangle inequality, we obtain ‖xn+1 − zn‖ ≤ ‖xn+1 − xn‖+ ‖xn − zn‖. (41) by (32) and (40), we get lim n→∞ ‖xn+1 − zn‖ = 0. (42) since j is uniformly norm-to-norm continuous, we obtain lim n→∞ ‖jxn+1 − jzn‖ = 0. (43) from (67), we note that ‖jxn+1 − jzn‖ = ‖jxn+1 − (αn,0jxn + ∑∞ i=1 αn,ijs n i vn)‖ = ‖αn,0jxn+1 − αn,0jxn + ∑∞ i=1 αn,ijxn+1 − ∑∞ i=1 αn,ijs n i vn‖ = ‖αn,0(jxn+1 − jxn) + ∑∞ i=1 αn,i(jxn+1 − jsni vn)‖ = ‖ ∑∞ i=1 αn,i(jxn+1 − jsni vn)− αn,0(jxn − jxn+1)‖ ≥ ∑∞ i=1 αn,i‖jxn+1 − jsni vn‖ − αn,0‖jxn − jxn+1‖, and hence ‖jxn+1 − jsni vn‖ ≤ 1∑∞ i=1 αn,i (‖jxn+1 − jzn‖+ αn,0‖jxn − jxn+1‖). (44) from (33), (43) and lim inf n→∞ ∑∞ i=1 αn,i > 0, we get lim n→∞ ‖jxn+1 − jsni vn‖ = 0. (45) since j−1 is uniformly norm-to-norm continuous on bounded sets, we have lim n→∞ ‖xn+1 − sni vn‖ = 0. (46) using the triangle inequality, that ‖xn − sni vn‖ = ‖xn − xn+1 + xn+1 − sni vn‖ ≤ ‖xn − xn+1‖+ ‖xn+1 − sni vn‖. from (32) and (46), we have lim n→∞ ‖xn − sni vn‖ = 0. (47) advances in systems science and applications (2011), vol. 11, no. 1-2 135 on the other hand, we observe that φ(q, xn)− φ(q, un) + θn = ‖xn‖2 − ‖un‖2 − 2〈q, jxn − jun〉+ θn. it follows from θn −→ 0, ‖xn − un‖ −→ 0 and ‖jxn − jun‖ −→ 0, that φ(q, xn)− φ(q, un) + θn −→ 0 as n −→∞. (48) from (19), (20) and (24), we compute φ(q, un) ≤ φ(q, yn) ≤ βnφ(q, xn) + (1− βn)φ(q, zn) ≤ βnφ(q, xn) + (1− βn)[αn,0φ(q, xn) + ∑∞ i=1 αn,iknφ(q, vn) −αn,0αn,jg‖jvn − jsnj vn‖] = βnφ(q, xn) + (1− βn)αn,0φ(q, xn) + (1− βn) ∑∞ i=1 αn,iknφ(q, vn) −(1− βn)αn,0αn,jg‖jvn − jsnj vn‖ ≤ βnφ(q, xn) + (1− βn)αn,0φ(q, xn) + (1− βn) ∑∞ i=1 αn,iknφ(q, vn) ≤ βnφ(q, xn) + (1− βn)αn,0φ(q, xn) + (1− βn) ∑∞ i=1 αn,ikn[φ(q, xn)− 2λn(α− 2 c2 λn)‖axn −aq‖2] ≤ βnφ(q, xn) + (1− βn)αn,0knφ(q, xn) + (1− βn) ∑∞ i=1 αn,iknφ(q, xn) −(1− βn) ∑∞ i=1 αn,ikn2λn(α− 2 c2 λn)‖axn −aq‖2 = βnknφ(q, xn) + (1− βn)knφ(q, xn)− (1− βn) ∑∞ i=1 αn,ikn2λn(α− 2 c2 λn)‖axn −aq‖2 ≤ knφ(q, xn)− (1− βn) ∑∞ i=1 αn,ikn2λn(α− 2 c2 λn)‖axn −aq‖2] ≤ φ(q, xn) + supq∈f (kn − 1)φ(q, xn)− (1− βn) ∑∞ i=1 αn,ikn2λn(α− 2 c2 λn)‖axn −aq‖2 ≤ φ(q, xn) + θn − (1− βn) ∑∞ i=1 αn,ikn2λn(α− 2 c2 λn)‖axn −aq‖2 and hence 2a(α− 2b c2 )‖axn −aq‖2 ≤ 2λn(α− 2 c2 λn)‖axn −aq‖2 ≤ 1 (1−βn) ∑∞ i=1 αn,ikn (φ(q, xn)− φ(q, un) + θn). (49) from (48), {λn} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, lim infn−→∞(1− βn) > 0 and lim infn−→∞ αn,0αn,i > 0, for i ≥ 0 and kn −→ 1 as n −→∞, we obtain that lim n→∞ ‖axn −aq‖ = 0. (50) from lemma 2.6, lemma 2.7 and (23), we compute φ(xn, vn) = φ(xn,πcj −1(jxn − λnaxn)) ≤ φ(xn, j −1(jxn − λnaxn)) = v (xn, jxn − λnaxn) ≤ v (xn, (jxn − λnaxn) + λnaxn)− 2〈j−1(jxn − λnaxn)− xn, λnaxn〉 = φ(xn, xn) + 2〈j−1(jxn − λnaxn)− xn,−λnaxn〉 = 2〈j−1(jxn − λnaxn)− xn,−λnaxn〉 ≤ 4λ2n c2 ‖axn −aq‖2 ≤ 4b2 c2 ‖axn −aq‖2. 136 saewan:a new modified block iterative algorithm for a system of. . . . . . applying lemma 2.4 and (50) that lim n→∞ ‖xn − vn‖ = 0 (51) and we also obtain lim n→∞ ‖jxn − jvn‖ = 0 (52) from sni is continuous, for any i ≥ 1 lim n→∞ ‖sni xn − sni vn‖ = 0. (53) again by the triangle inequality, we get ‖xn − sni xn‖ ≤ ‖xn − sni vn‖+ ‖sni vn − sni xn‖. from (47) and (53), we have lim n→∞ ‖xn − sni xn‖ = 0, ∀i ≥ 1. (54) since j is uniformly continuous on any bounded subset of e, we obtain limn→∞ ‖jxn − jsni xn‖ = 0, ∀i ≥ 1. (55) since xn −→ p and j is uniformly continuous, it yields jxn −→ jp. hence, from (55), we get jsni xn −→ jp, ∀i ≥ 1. (56) since j−1 : e∗ −→ e is norm-weake*-continuous, we have sni xn ⇀ p, for each i ≥ 1. (57) on the other hand, for each i ≥ 1, we have |‖sni xn‖ − ‖p‖| = |‖j(sni xn)‖ − ‖jp‖| ≤ ||j(sni xn)− jp||. by (56), we obtain ‖sni xn‖ −→ ‖p‖ for each i ≥ 1. since e is uniformly convex banach spaces then e has the kadec-klee property, we get sni xn −→ p for each i ≥ 1. by the assumption that ∀i ≥ 1, si is uniformly li-lipschitz continuous, hence we have. ‖sn+1 i xn − sni xn‖ ≤ ‖sn+1 i xn − sn+1 i xn+1‖+ ‖sn+1 i xn+1 − xn+1‖+ ‖xn+1 − xn‖+ ‖xn − sni xn‖ ≤ (li + 1)‖xn+1 − xn‖+ ‖sn+1 i xn+1 − xn+1‖+ ‖xn − sni xn‖. (58) by (32) and (54), it follows that ‖sn+1 i xn − sni xn‖ → 0. from sni xn −→ p, we have sn+1 i xn → p, that is sisni xn → p. in view of closeness of si, we have sip = p, for all i ≥ 1. this imply that p ∈ ∩∞i=1f (si). advances in systems science and applications (2011), vol. 11, no. 1-2 137 (ii) we show that xn → p ∈ ∩mj=1ep (fj). applying (19) and (25), we get φ(p, yn) ≤ φ(p, xn)+θn. from lemma 2.13 and un = ωm n yn, when ωj n = t qj rj,nt qj−1 rj−1,n ...t q2 r2,nt q1 r1,n , j = 1, 2, 3, ...,m, ω0 n = i , for p ∈ f , we observe that φ(p, un) = φ(p,ωm n yn) ≤ φ(p,ωm−1 n yn) ... ≤ φ(p,ωj nyn) ... ≤ φ(p, yn) ≤ φ(p, xn) + θn ∀j = 1, 2, 3, ...,m. (59) follows from lemma 2.13, that φ(un,ω j nyn) ≤ φ(p,ωj nyn)− φ(p, un) ≤ φ(p, xn)− φ(p, un) + θn = ‖p‖2 − 2〈p, jxn〉+ ‖xn‖2 − (‖p‖2 − 2〈p, jun〉+ ‖un‖2) + θn = ‖xn‖2 − ‖un‖2 − 2〈p, jxn − jun〉+ θn ≤ ‖xn − un‖(‖xn + un‖) + 2‖p‖‖jxn − jun‖+ θn. (60) from (36), (37), θn → 0 as n→∞ and lemma 2.4, we get lim n→∞ ‖un − ωj nyn‖ = 0∀j = 1, 2, 3, ...,m. (61) by using triangle inequality, we have ‖xn − ωj nyn‖ ≤ ‖xn − un‖+ ‖un − ωj nyn‖. from (36) and (61), we have lim n→∞ ‖xn − ωj nyn‖ = 0 ∀j = 1, 2, 3, ...,m. (62) again by using triangle inequality, we have ‖ωj nyn − ωj−1 n yn‖ ≤ ‖ωj nyn − xn‖+ ‖xn − ωj−1 n yn‖. from (62),we also have lim n→∞ ‖ωj nyn − ωj−1 n yn‖ = 0 ∀j = 1, 2, 3, ...,m. (63) since j is uniformly norm-to-norm continuous, we obtain lim n→∞ ‖jωj nyn − jωj−1 n yn‖ = 0 ∀j = 1, 2, 3, ...,m. from rj,n > 0 we have ‖jωj nyn−jωj−1 n yn‖ rj,n → 0 as n→∞ ∀j = 1, 2, 3, ...,m, and fj(ω j nyn, y) + 1 rj,n 〈y − ωj nyn, jωj nyn − jωj−1 n yn〉 ≥ 0, ∀y ∈ c. 138 saewan:a new modified block iterative algorithm for a system of. . . . . . by (a2), that ‖y − ωj nyn‖‖jωj nyn−jωj−1 n yn‖ rn ≥ 1 rj,n 〈y − ωj nyn, jωj nyn − jωj−1 n yn〉 ≥ −fj(ωj nyn, y) ≥ fj(y,ω j nyn), ∀y ∈ c, and ωj nyn → p we get f(y, p) ≤ 0 for all y ∈ c. for 0 < t < 1, define yt = ty + (1 − t)p. then yt ∈ c which imply that fj(yt, p) ≤ 0. from (a1), we obtain that 0 = fj(yt, yt) ≤ tfj(yt, y) + (1− t)fj(yt, p) ≤ tfj(yt, y). thus fj(yt, y) ≥ 0. from (a3), we have fj(p, y) ≥ 0 for all y ∈ c and j = 1, 2, 3, ...,m. hence p ∈ ep (fj) ∀j = 1, 2, 3, ...,m. this imply that p ∈ ∩mj=1ep (fj). (iii) we show that xn → p ∈ v i(a,c). indeed, define b ⊂ e × e∗ by bv = { av +nc(v), v ∈ c; ∅, v /∈ c. (64) by lemma 2.8, b is maximal monotone and b−10 = v i(a,c). let (v, w) ∈ g(b). since w ∈ bv = av +nc(v), we get w −av ∈ nc(v). from vn ∈ c, we have 〈v − vn, w −av〉 ≥ 0. (65) on the other hand, since vn = πcj −1(jxn − λnaxn). then by lemma 2.5, we have 〈v − vn, jvn − (jxn − λnaxn)〉 ≥ 0, and thus 〈v − vn, jxn−jvnλn −axn〉 ≤ 0. (66) it follows from (65), (66) and a is monotone and 1 α -lipschitz continuous, that 〈v − vn, w〉 ≥ 〈v − vn, av〉 ≥ 〈v − vn, av〉+ 〈v − vn, jxn−jvnλn −axn〉 = 〈v − vn, av −axn〉+ 〈v − zvn, jxn−jvnλn 〉 = 〈v − vn, av −avn〉+ 〈v − vn, avn −axn〉+ 〈v − vn, jxn−jvnλn 〉 ≥ −‖v − vn‖‖vn−xn‖α − ‖v − vn‖‖jxn−jvn‖a ≥ −h(‖vn−xn‖α + ‖jxn−jvn‖ a ), where h = supn≥1 ‖v − vn‖. take the limit as n → ∞, (51) and (52), we obtain 〈v − p, w〉 ≥ 0. by the maximality of b we have p ∈ b−10, that is p ∈ v i(a,c). finally, we show that p = πfx0. from xn = πcnx0, we have 〈jx0 − jxn, xn − z〉 ≥ 0, ∀z ∈ cn. since f ⊂ cn, we also have 〈jx0 − jxn, xn − y〉 ≥ 0, ∀y ∈ f. advances in systems science and applications (2011), vol. 11, no. 1-2 139 taking limit n→∞, we obtain 〈jx0 − jp, p− y〉 ≥ 0, ∀y ∈ f. by lemma 2.5, we can conclude that p = πfx0 and xn → p as n → ∞. this completes the proof. if si = s for each i ∈ n, then theorem 3.1 is reduced to the following corollary. corollary 3.2. let c be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth banach space e. for each j = 1, 2, ...,m let fj be a bifunction from c × c to r which satisfies conditions (a1)-(a4). let a be an α-inverse-strongly monotone mapping of c into e∗ satisfying ‖ay‖ ≤ ‖ay − au‖, ∀y ∈ c and u ∈ v i(a,c) 6= ∅. let s : c → c be a closed l-lipschitz continuous and quasi-φ-asymptotically nonexpansive mappings with a sequence {kn} ⊂ [1,∞), kn → 1 such that f := (f (s)) ∩ (∩mj=1ep (fj)) ∩ (v i(a,c)) is a nonempty and bounded subset in c. for an initial point x0 ∈ e with x1 = πc1x0 and c1 = c, we define the sequence {xn} as follows: vn = πcj −1(jxn − λnaxn), zn = j−1(αnjxn + (1− αn)jsnvn), yn = j−1(βnjxn + (1− βn)jzn), un = t fmrm,nt fm−1 rm−1,n ...t f2 r2,nt f1 r1,nyn, cn+1 = {z ∈ cn : φ(z, un) ≤ φ(z, xn) + θn}, xn+1 = πcn+1x0, ∀n ≥ 1, (67) where j is the duality mapping one, θn = supq∈f (kn−1)φ(q, xn), {αn}, {βn} are sequences in [0, 1], {rj,n} ⊂ [d,∞) for some d > 0 and {λn} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, where 1 c is the 2-uniformly convexity constant of e. if lim infn−→∞(1 − βn) > 0 and lim infn−→∞ αn(1− αn) > 0, then {xn} converges strongly to p ∈ f , where p = πfx0. for a special case that i = 1, 2, we can obtain the following results on a pair of quasi-φasymptotically nonexpansive mappings immediately from theorem 3.1. corollary 3.3. let c be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth banach space e. for each j = 1, 2, ...,m let fj be a bifunction from c × c to r which satisfies conditions (a1)-(a4). let a be an α-inverse-strongly monotone mapping of c into e∗ satisfying ‖ay‖ ≤ ‖ay − au‖, ∀y ∈ c and u ∈ v i(a,c) 6= ∅. let s, t : c → c be two closed quasi-φ-asymptotically nonexpansive mappings and ls , lt -lipschitz continuous, respectively with a sequence {kn} ⊂ [1,∞), kn → 1 such that f := f (s) ∩ f (t ) ∩ (∩mj=1ep (fj)) ∩ v i(a,c) is a nonempty and bounded subset in c. for an initial point x0 ∈ e with x1 = πc1x0 and c1 = c, we define the sequence {xn} as follows: vn = πcj −1(jxn − λnaxn), zn = j−1(αnjxn + βnjs nvn + γnjt nvn), yn = j−1(δnjxn + (1− δn)jzn), un = t fmrm,nt fm−1 rm−1,n ...t f2 r2,nt f1 r1,nyn, cn+1 = {z ∈ cn : φ(z, un) ≤ φ(z, xn) + θn}, xn+1 = πcn+1x0, ∀n ≥ 0, (68) 140 saewan:a new modified block iterative algorithm for a system of. . . . . . where j is the duality mapping on e, θn = supq∈f (kn − 1)φ(q, xn), {αn}, {βn}, {γn} and {δn} are sequences in [0, 1], {rj,n} ⊂ [d,∞) for some d > 0 and {λn} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, where 1 c is the 2-uniformly convexity constant ofe. if αn+βn+γn = 1 for all n ≥ 0 and lim infn−→∞ αnβn > 0, lim infn−→∞ αnγn > 0, lim infn−→∞ βnγn > 0 and lim infn−→∞ δn(1− δn) > 0, then {xn} converges strongly to p ∈ f , where p = πfx0. corollary 3.4. let c be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth banach spacee. for each j = 1, 2, ...,m let fj be a bifunction fromc×c to r which satisfies conditions (a1)-(a4). let a be an α-inverse-strongly monotone mapping of c into e∗ satisfying ‖ay‖ ≤ ‖ay−au‖, ∀y ∈ c and u ∈ v i(a,c) 6= ∅. let {si}∞i=1 : c → c be an infinite family of closed quasi-φnonexpansive mappings such that f := ∩∞i=1f (si)∩ (∩mj=1ep (fj)) ∩ v i(a,c) 6= ∅. for an initial point x0 ∈ e with x1 = πc1x0 and c1 = c, we define the sequence {xn} as follows: vn = πcj −1(jxn − λnaxn), zn = j−1(αn,0jxn + ∑∞ i=1 αn,ijsivn), yn = j−1(βnjxn + (1− βn)jzn), un = t fmrm,nt fm−1 rm−1,n ...t f2 r2,nt f1 r1,nyn, cn+1 = {z ∈ cn : φ(z, un) ≤ φ(z, xn), xn+1 = πcn+1x0, ∀n ≥ 0, (69) where j is the duality mapping on e, {αn,i} and {βn} is sequence in [0, 1], {rj,n} ⊂ [d,∞) for some d > 0 and {λn} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, where 1 c is the 2uniformly convexity constant of e. if ∑∞ i=0 αn,i = 1 for all n ≥ 0, lim infn−→∞(1− βn) > 0 and lim infn−→∞ αn,0αn,i > 0 for all i ≥ 1, then {xn} converges strongly to p ∈ f , where p = πfx0. proof since {si}∞i=1 : c −→ c is an infinite family of closed quasi-φ-nonexpansive mappings, it is an infinite family of closed and uniformly quasi-φ-asymptotically nonexpansive mappings with sequence kn = 1. hence the conditions appearing in theorem 3.1 f is a bounded subset in c and for each i ≥ 1, si is uniformly li-lipschitz continuous are of no use here. by virtue of the closeness of mapping si for each i ≥ 1, it yields that p ∈ f (si) for each i ≥ 1, that is, p ∈ ∩∞i=1f (si). therefore all conditions in theorem 3.1 are satisfied. the conclusion of corollary 3.4 is obtained from theorem 3.1 immediately. corollary 3.5. [44, theorem 3.2] let c be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth banach spacee. let f be a bifunction fromc×c to r satisfying (a1)-(a4). leta be an α-inverse-strongly monotone mapping of c into e∗ satisfying ‖ay‖ ≤ ‖ay − au‖, ∀y ∈ c and u ∈ v i(a,c) 6= ∅. let {si}ni=1 : c → c be a finite family of closed quasi-φnonexpansive mappings such that f := ∩ni=1f (si)∩ep (f)∩v i(a,c) 6= ∅. for an initial point x0 ∈ e with x1 = πc1x0 and c1 = c, we define the sequence {xn} as follows:  zn = πcj −1(jxn − λnaxn), yn = j−1(α0jxn + ∑n i=1 αijsizn), f(un, y) + 1 rn 〈y − un, jun − jyn〉 ≥ 0, ∀y ∈ c, cn+1 = {z ∈ cn : φ(z, un) ≤ φ(z, xn), xn+1 = πcn+1x0, ∀n ≥ 0, (70) advances in systems science and applications (2011), vol. 11, no. 1-2 141 where j is the duality mapping on e, {αn,i} is sequence in [0, 1], {rn} ⊂ [d,∞) for some d > 0 and {λn} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, where 1 c is the 2-uniformly convexity constant of e. if αi ∈ (0, 1) such that ∑n i=0 αi = 1, then {xn} converges strongly to p ∈ f , where p = πfx0. corollary 3.6. let c be a nonempty closed and convex subset of a uniformly convex and uniformly smooth banach space e. let f be a bifunction from c × c to r satisfying (a1)-(a4). let {si}∞i=1 : c → c be an infinite family of closed and uniformly quasi-φ-asymptotically nonexpansive mappings with a sequence {kn} ⊂ [1,∞), kn → 1 and uniformly li-lipschitz continuous such that f := ∩∞i=1f (si) ∩ ep (f) is a nonempty and bounded subset in c. for an initial point x0 ∈ e with x1 = πc1x0 and c1 = c, we define the sequence {xn} as follows: yn = j−1(αn,0jxn + ∑∞ i=1 αn,ijs n i xn), f(un, y) + 1 rn 〈y − un, jun − jyn〉 ≥ 0, ∀y ∈ c, cn+1 = {z ∈ cn : φ(z, un) ≤ φ(z, xn) + θn}, xn+1 = πcn+1x0, ∀n ≥ 0, (71) where j is the duality mapping one, θn = supq∈f (kn−1)φ(q, xn), {αn,i} is sequence in [0, 1], {rn} ⊂ [a,∞) for some a > 0. if ∑∞ i=0 αn,i = 1 for all n ≥ 0 and lim infn−→∞ αn,0αn,i > 0 for all i ≥ 1, then {xn} converges strongly to p ∈ f , where p = πfx0. 4. deduced to hilbert spaces if e = h , a hilbert space, then e is 2-uniformly convex (we can choose c = 1) and uniformly smooth real banach space and closed relatively quasi-nonexpansive map reduces to closed quasi-nonexpansive map. moreover, j = i , identity operator on h and πc = pc , projection mapping from h into c. thus, the following corollaries hold. theorem 4.1. let c be a nonempty closed and convex subset of a hilbert space h . for each j = 1, 2, ...,m let fj be a bifunction from c × c to r which satisfies conditions (a1)(a4), bj : c −→ e∗ be a continuous and monotone mapping and ϕj : c → r be a lower semicontinuous and convex function. let a be an α-inverse-strongly monotone mapping of c into h satisfying ‖ay‖ ≤ ‖ay −au‖, ∀y ∈ c and u ∈ v i(a,c) 6= ∅. let {si}∞i=1 : c → c be an infinite family of closed and uniformly quasi-φ-asymptotically nonexpansive mappings with a sequence {kn} ⊂ [1,∞), kn → 1 and uniformly li-lipschitz continuous such that f := ∩∞i=1f (si)∩ (∩mj=1gmep (fj , bj , ϕj))∩ v i(a,c) is a nonempty and bounded subset in c. for an initial point x0 ∈ h with x1 = pc1x0 and c1 = c, we define the sequence {xn} as follows:  zn = pc(xn − λnaxn), yn = αn,0xn + ∑∞ i=1 αn,is n i zn, un = t fmrm,nt fm−1 rm−1,n ...t f2 r2,nt f1 r1,nyn, cn+1 = {z ∈ cn : ‖z − un‖ ≤ ‖z − xn‖+ θn}, xn+1 = pcn+1x0, ∀n ≥ 0, (72) where θn = supq∈f (kn − 1)‖q − xn‖, {αn,i} is sequence in [0, 1], {rj,n} ⊂ [a,∞) for some a > 0 and {λn} ⊂ [a, b] for some a, b with 0 < a < b < α/2. if ∑∞ i=0 αn,i = 1 for all n ≥ 0 142 saewan:a new modified block iterative algorithm for a system of. . . . . . and lim infn−→∞ αn,0αn,i > 0 for all i ≥ 1, then {xn} converges strongly to p ∈ f , where p = πfx0. remark 4.2. theorem 4.1 improve and extend the corollary 3.7 in zegeye [44] in the aspect for the mappings, we extend the mappings from a finite family of closed relatively quasinonexpansive mappings to more general an infinite family of closed and uniformly quasi-φasymptotically nonexpansive mappings. 5. applications 5.1 zero points of an inverse-strongly monotone operator next, we consider the problem of finding a zero point of an inverse-strongly monotone operator of e into e∗. assume that a satisfies the conditions: (c1) a is α-inverse-strongly monotone, (c2) a−10 = {u ∈ e : au = 0} 6= ∅. theorem 5.1. let c be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth banach space e. for each j = 1, 2, ...,m let fj be a bifunction from c × c to r which satisfies conditions (a1)-(a4), bj : c −→ e∗ be a continuous and monotone mapping and ϕj : c → r be a lower semicontinuous and convex function. let a be an operator of e into e∗ satisfying (c1) and (c2). let {si}∞i=1 : c → c be an infinite family of closed uniformly li-lipschitz continuous and uniformly quasi-φ-asymptotically nonexpansive mappings with a sequence {kn} ⊂ [1,∞), kn → 1 such that f := ∩∞i=1f (si) ∩ (∩mj=1gmep (fj , bj , ϕj)) ∩a−10 is a nonempty and bounded subset in c. for an initial point x0 ∈ e with x1 = πc1x0 and c1 = c, we define the sequence {xn} as follows: zn = j−1(αn,0jxn + ∑∞ i=1 αn,ijs n i vn), yn = j−1(βnjxn + (1− βn)jzn), un = t fmrm,nt fm−1 rm−1,n ...t f2 r2,nt f1 r1,nyn, cn+1 = {z ∈ cn : φ(z, un) ≤ φ(z, xn) + θn}, xn+1 = πcn+1x0, ∀n ≥ 0, (73) where j is the duality mapping on e, θn = supq∈f (kn − 1)φ(q, xn), for each i ≥ 0, {αn,i} and {βn} are sequences in [0, 1], {rj,n} ⊂ [d,∞) for some d > 0 and {λn} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, where 1 c is the 2-uniformly convexity constant of e. if∑∞ i=0 αn,i = 1 for all n ≥ 0, lim infn−→∞(1− βn) > 0 and lim infn−→∞ αn,0αn,i > 0 for all i ≥ 1, then {xn} converges strongly to p ∈ f , where p = πfx0. proof. setting c = e in corollary 3.4, we also get πe = i. we also have v i(a,c) = v i(a,e) = {x ∈ e : ax = 0} 6= ∅ and then the condition ‖ay‖ ≤ ‖ay − au‖ holds for all y ∈ e and u ∈ a−10. so, we obtain the result. advances in systems science and applications (2011), vol. 11, no. 1-2 143 5.2 complementarity problems let k be a nonempty, closed convex cone in e. we define the polar k∗ of k as follows: k∗ = {y∗ ∈ e∗ : 〈x, y∗〉 ≥ 0, ∀x ∈ k}. (74) if a : k −→ e∗ is an operator, then an element u ∈ k is called a solution of the complementarity problem ([37]) if au ∈ k∗, and 〈u,au〉 = 0. (75) the set of solutions of the complementarity problem is denoted by cp (a,k). theorem 5.2. let k be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth banach space e. for each j = 1, 2, ...,m let fj be a bifunction from c × c to r which satisfies conditions (a1)-(a4), bj : c −→ e∗ be a continuous and monotone mapping and ϕj : c → r be a lower semicontinuous and convex function. let a be an αinverse-strongly monotone mapping of k into e∗ satisfying ‖ay‖ ≤ ‖ay − au‖, ∀y ∈ k and u ∈ cp (a,k) 6= ∅. let {si}∞i=1 : k → k be an infinite family of closed uniformly li-lipschitz continuous and uniformly quasi-φ-asymptotically nonexpansive mappings with a sequence {kn} ⊂ [1,∞), kn → 1 such that f := ∩∞i=1f (si) ∩ (∩mj=1gmep (fj , bj , ϕj)) ∩ cp (a,k) is a nonempty and bounded subset in k. for an initial point x0 ∈ e with x1 = πk1x0 and k1 = k, we define the sequence {xn} as follows: vn = πkj −1(jxn − λnaxn), zn = j−1(αn,0jxn + ∑∞ i=1 αn,ijs n i vn), yn = j−1(βnjxn + (1− βn)jzn), un = t fmrm,nt fm−1 rm−1,n ...t f2 r2,nt f1 r1,nyn, kn+1 = {z ∈ kn : φ(z, un) ≤ φ(z, xn) + θn}, xn+1 = πkn+1x0, ∀n ≥ 0, (76) where j is the duality mapping on e, θn = supq∈f (kn − 1)φ(q, xn), for each i ≥ 0, {αn,i} and {βn} are sequences in [0, 1], {rj,n} ⊂ [d,∞) for some d > 0 and {λn} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, where 1 c is the 2-uniformly convexity constant of e. if∑∞ i=0 αn,i = 1 for all n ≥ 0, lim infn−→∞(1− βn) > 0 and lim infn−→∞ αn,0αn,i > 0 for all i ≥ 1, then {xn} converges strongly to p ∈ f , where p = πfx0. proof . as in the proof of takahashi in [37, lemma 7.11], we get that v i(a,k) = cp (a,k). so, we obtain the result. references [1] y. i. alber, s. reich, an iterative method for solving a class of nonlinear operator equations in banach spaces, panamer. math. j. 4 (1994) 39–54. 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[47] s. zhang, generalized mixed equilibrium problem in banach spaces, appl. math. mech. -engl. ed., 30 (2009) 1105–1112. 148 saewan:a new modified block iterative algorithm for a system of. . . . . . microsoft word 6-liu wei.doc issn 1078-6236 international institute for general systems studies, inc. structural analysis of leukemia related gene network liu wei1, zhang yuanyuan2, xu xiuzhu2 , yang meixi2, xu dashun 3 and wang shudong 2 1department of mathematics and information, ludong university, yantai, shandong 264025, china 2. college of information science and engineering, shandong university of science and technology, qingdao, shandong 266510, china 3. department of mathematics, southern illinois university carbondale, makanida 62958, usa email: wangshd2008@yahoo.com.cn ，liuyiwei1030@yahoo.com.cn abstract genome forms gene networks in the complicated interactive ways and further study on the cancer-related gene networks can help to understand and predict many unknown biological functions of cancer-related genome, and then obtain the useful informations about molecular mechanism of the formation of cancer. in this research, using the method of mutual information, we construct gene networks corresponding to normal control group and diseased experimental group of acute lymphoblastic leukemia (abbreviated as all). through contrasting the structure difference of two kinds of networks, we find 23 structural key genes of all with significant degree-difference, where 21 genes are confirmed to be closely related to the formation of all. the match ratio between the prediction by model and literature is up to 91.3%. according to the effectiveness of the method in this work, we can predict that the remaining two genes cxcl1 and tacstd2 are closely related to all. the significant differences of the network structures between control and experimental groups will enlighten the biomedical scientists that the reason of normal organism suffering from leukemia maybe is the great changes of some genes in vivo. the finding of structural key genes of all will help the biomedical scientist to further research the pathogenesis of all. keywords systems biology gene network mutual information leukemia 1. introduction leukemia is one of the top 10 human cancers. now many biomedical scientists are engaging in discovering the oncogenes and tumor-suppressor genes of leukemia, hope to know well the pathogenesis of leukemia, and find out the effective methods to treat leukemia. at present, besides doing clinical experiments directly, the main research ways to discover the pathogenesis of leukemia are based on the gene expression profiles data of dna chip. for instance, the method of cluster analysis was used to analyze the pattern of genes expressed in leukemic blasts from 360 pediatric all patients. distinct expression profiles identified each of the prognostically important leukemia subtypes, including t-all, e2a-pbx1, bcr-abl, *supported by national natural science foundation of china (grant nos. 60874036, 60503002) and sdust research fund. ∗ advances in systems science and applications (2011), vol.11, no.1-2 71-81 tel-aml1, mll rearrangement (yeoh et al., 2002)[1]. the method of pattern recognition was utilized to analyze the gene expression profiles data of patients with leukemia(yakovlev et al., 2002) [2]. the method of multivariate statistics was used to identify genes modulated by adhesion of human precursor b leukemia cells that regulate proliferation and apoptosis, highlighting new pathways that might provide insights into future therapy aiming at targeting apoptosis of leukemia cells (astier et al., 2003) [3]. the analysis of neural network internal structure allowed the identification of specific phenotype markers and the extraction of peculiar associations among genes and physiological states. at the same time, the neural network outputs provided assignment to multiple classes, such as different pathological conditions or tissue samples, for previously unseen instances (silvio et al., 2003)[4]. the method of graphical gaussian models (ggms) was used to describe gene association networks and to detect conditionally dependent genes (korbinian strimmer et al., 2005)[5]. a robust gene selection approach based on a hybrid between genetic algorithm and support vector machine was formalized and the major goal of this hybridization was to exploit fully their respective merits for identification of key feature genes (or molecular signatures) for a complex biological phenotype (shaoqi rao et al., 2005)[6]. the data-mining methods were applied to biomedical research. the results and how these results would affect the diagnosis and treatment of all in the future were discussed (morton, geoffrey, 2010)[7]. these above-mentioned methods can compensate for the deficiency of clinical trials, and they are of important significance to the prediction, diagnosis and treatment of cancer. systems biology is a newly interdiscipline after genomics, proteomics. now, forward modeling and reverse modeling are two main methods to study systems biology. modeling and analysis method of complex network is one of the important ways of reverse network modeling and has been widely applied to the study of complex biological system. the complex network models of biological system mainly include genetic networks, protein interaction networks, metabolic networks, signal networks and cellular networks, etc. harald lahm et al. researched the complex network of family of endogenous agglutinin in 2004[8], which would help to understand certain types of malignant phenotypes. adriano v. werhli et al. constructed raf signal transduction network using gaussian and bayesian model respectively based on the data of system expression profiles in 2006[9]. theodore j. perkins et al. analyzed the gene regulatory network of drosophila using the method of reverse network modeling, which can explain the activation of genes[10]. shudong wang et al. constructed the logical network of arabidopsis genes under different external stimuli using reverse network modeling of information entropy, simulated and analyzed the dynamical behaviours of the obtained logic network [11]. in this work, based on the gene expression profiles of control group and experimental groups of all, we construct the mutual information network of healthy marrow, b_all and t_all respectively. through contrasting the changes of network structures under different groups, we discover the significant difference in the networks for control and experimental groups. furthermore, we find 23 structural key genes of all. the paper is organised as follows. we introduce the related background in the first part, discuss the analysis method of complex network in the second one, construct the mutual information gene network and give the main results in the third part, and analyze the biological significances of the experimental results combining with the gene functions in the fourth part. 2. analysis method of complex network complex network can show the complex relationships among large numbers of elements more clearly, and better explain the mutual influence relationships between the structures and functions. therefore, further study on complex network of the genes of diseases can help to understand pathogenesis of diseases. biological systems are composed of interactions and mutual regulations of many elements, such as genes, proteins, protein complexes and 72 liu: structural analysis of leukemia related gene network transcription factors, etc. if these elements and the interactions and mutual regulations of elements can be simplified as nodes and edges respectively, then the complicated biological systems can be abstracted to complex networks, for example, undirected network, directed network, and weighted network etc. now, the analysis method of complex network is mainly studying the overall (average path length, clustering coefficient, assortativity coefficient, degree distribution, etc.) and local (community structure, motif etc.) properties of network. in this work, we will analyze the structure of complex life systems to obtain further understanding its function, which is of important reference values for life scientists to predict and cure disease. let  ,g v e be a complex network with node-set  1,2, ,v n  and edge-set e . the statistics of complex network used in the work are as follows: (1) average degree ( k ) the degree of a node is defined as the number of nodes adjacent to it. the average degree is the average of the degrees of all nodes in the network, denoted by k . (2) average path length ( l ) the shortest path length is defined as the distance between any two nodes in the network. the average path length is the average of all of the shortest path lengths in the network, denoted by l . (3) average clustering coefficient (c ) the clustering coefficient of node i in the network is defined as the number of triangles dividing by the number of triples connected with node i. it reflects the tightness of the network connection. the average clustering coefficient is the average of the clustering coefficients of all nodes in the network, denoted by c . (4) modularity (q ) modularity is defined as 2( )i ij i i i q q e a    where ije denotes the proportion of the edges connecting two different communities to all the edges of network, and ia is the sum of each element of thi row which denotes the proportion of the edges connecting with thi community to all the edges of network. modularity is the probability measure of community structure of network, and is mainly used to judge the quality of community structure. (5) average coreness ( k ) k -core of a graph is defined as the remaining subgraph after repeatedly removing those nodes whose degrees are less than or equal to k . the coreness of a node denotes the depth of the node in the core. the average coreness is the average of the corenesses of the graph, denoted by k . (6) average betweenness centrality (b ) the betweenness centrality of a node is defined as the number of the shortest paths through the node in a network. it is an important index which can reflect the topological properties of the network. the average betweenness centrality is the average of the betweenness centralities of all nodes in the network, denoted byb . (7) non-isolated node proportion ( i ) the isolated node of a network is defined as the node that is not adjacent to any other nodes. it usually doesn't work in the network. the non-isolated node proportion is the number of non-isolated node dividing the number of all nodes. it can reflect the intensive extent of the nodes associated in the network. advances in systems science and applications (2011), vol.11, no.1-2 73 3. construction and analysis of gene network of mutual information 3.1 establishment of working database all the data in this research are from e13159 sample library in gpl570 of ncbi. e13159 sample library is composed of more than 2000 samples. we extract 40 samples from normal samples, b_all samples and t_all samples respectively. for description convenience, we refer to normal samples as the control group, b_all samples and t_all samples as two experimental groups to establish three original databases which contain more than 20,000 genes sample data of human genome. it is impossible to establish a gene network of mutual information with more than 20,000 genes using the present computer from the view of computational complexity. therefore, we should process the original database firstly. according to 301 cancer-related genes we have mastered [12], we select 801 probes corresponding to these genes. if there are several probes corresponding to a gene, then we choose the probe that has the highest expression profiles corresponding to the gene. through the above processing, we get 286 genes as the research object. because we focus on gene network structure, the genes whose expression levels are almost 0 or 1 in all of the samples have no contribution to the differences of network structure. hence, we delete the genes whose the proportions of data expression profiles of samples being 1 are greater than 90% or less than 15% in each library. at last we get 58, 73 and 85 genes from the normal group, the b_all group and the t_all group respectively to form the corresponding working databases. next we will construct the gene networks of mutual information based on the 3 working databases. 3.2 selection of threshold 3.2.1 comparison of network statistics firstly, we should discretize p_value of the working database appropriately. specific methods are as follows: divide the range of p_value [0, 1] into 20 parts, and mark with 1, 2, …, 20, respectively. this kind of discretization has finer granularity than 0-1 discretization, and it lose less information. secondly, we can get a complete connected weighted network of all genes in each library by use of the mutual information formula, and the mutual information value is denoted by the weight. in order to highlight the specificity of network structure and obtain useful biological informations, the mutual information values of the complete connected network are need to be coarse-grained to seek threshold. as the distribution interval of mutual information value in each database is different, all the mutual information values of the 3 working databases are normalized, so that different databases can be comparable. finally, we analyze 7 statistics (average degree, average path length, average clustering coefficient, modularity, average coreness, average betweenness centrality, non-isolated node proportion) with the variation of thresholds between 0.1 and 0.9 in steps of 0.01, as shown in fig. 1. 74 liu: structural analysis of leukemia related gene network fig. 1. the change situation of 7 network statistics with the increasing of threshold, where the abscissa and vertical axis denotes the threshold and the corresponding statistics respectively. red, green, blue c urve denotes the change curve of normal group, b_all group and t_all group respectively. 3.2.2 construction of mutual information network we discover that when the threshold is taken as 0.53, 7 statistics of the networks for control and two experimental groups will be separated obviously by studying the statistics curves changing with the increasing of threshold in fig. 1. so we take 0.53 as the threshold to construct the mutual information gene networks of 3 working databases, as shown in fig. 2: (a) normal (b) b-all (c) t-all fig. 2. when the threshold is taken as 0.53, the mutual information gene networks of 68 coordinate positioning genes of 3 working databases, where (a)，(b)，(c) denotes normal, b_all and t_all network, respectively. obviously, the relationships of many nodes in the three gene networks change in quantity, such as most of isolated nodes in the normal mutual information gene network are no longer isolated in b-all or t-all network, especially the changes of t-all network are significantly. at the same time, many non-isolated nodes in the normal network become isolated in b-all or t-all network, the connectivity of nodes in the network also changes significantly. 3.3 selection of structural key genes a: some genes are isolated in the normal and t-all network, but high connectivity in b-all network. because there are 41 non-isolated nodes in the b_all network, take the genes whose degree changes in quantity are more than 20 as the significant genes called the structural key genes. at last, we discover that 8 genes ar, erbb4, esr2, evi-1, pdgfrb, fat1, cxcl1 and cxcl2 satisfy the condition. their degrees in 3 gene networks are showed as table 1. table 1 the gene list of type a advances in systems science and applications (2011), vol.11, no.1-2 75 b: some genes are isolated nodes in normal network and b-all network, but high connectivity in the t-all network. because there are 38 non-isolated nodes in the b_all network, take the genes whose degree changes in quantity are more than 19 as the structural key genes. at last, we find 8 genes fer, fes, lta, wnt3, tcl1a, btk, msh4 and ptk2 genes satisfy the condition, their degrees in 3 gene networks are showed as table 2. table 2 the gene list of type b gene normal b_all t_all fer* 0 0 34 fes* 0 0 27 lta* 0 0 32 wnt3* 0 0 26 tcl1a* 0 0 31 msh4* 0 0 25 ptk2* 0 0 24 btk* 0 0 23 c: the degrees of some genes have changed more significantly in the networks of experimental groups than in the one of normal control group. take the genes whose degree changes in quantity are more than 18 as the structural key genes. at last, we find 7 genes cdh1, kdr, igf-1, fosl1, cttn, mos and tacstd2 satisfy the condition, their degrees in 3 gene networks are showed as table 3. table 3 the gene list of type c gene normal b_all t_all cdh1* 0 31 27 tacstd2 0 30 27 fosl1* 0 21 21 cttn* 0 20 20 kdr* 20 0 0 igf1* 20 0 0 mos* 19 0 0 4. conclusion and analysis we discover some genes whose degrees change significantly between the networks for control group and experimental groups. for example, some genes are very active (with greater degree) in the normal control group, and abnormal silence (isolated nodes) in the experimental groups, while some others are very active in the experimental groups, and abnormal silence in the gene normal b_all t_all ar* 0 26 0 erbb4* 0 27 0 esr2* 0 26 0 evi1* 0 33 0 fat1* 0 28 0 pdgfrb* 0 35 0 cxcl1 0 21 0 cxcl2* 0 25 0 76 liu: structural analysis of leukemia related gene network normal control group. that is to say, these genes have great contribute to structure changes of mutual information gene networks for control and experimental groups, so we define those genes whose degrees change significantly between control group and experimental groups as “the structural key gene”. now, we discuss these genes in three conditions as follows: a: considering genes ar, erbb4, esr2, evi-1, pdgfrb, fat1, cxcl1 and cxcl2, they are isolated nodes in normal and t-all gene network, but with greater degree in the b-all network. because their degrees in the b-all network significantly increase, we can speculate that their functions or the life processes involved in should be to promote cell proliferation and division, and they should promote the development and deterioration of cancer forward. the specific functions of the 8 genes as follows: yun cai et al. investigated the methylation status of the androgen receptor gene (ar) in leukemia cell lines. results showed the presence of both methylated and unmethylated cpg islands of the ar promotor in leukemia cell lines. in the normal blood samples, only unmethylated bands were observed[13]. the over-expression of erbb4 gene (leukemia virus oncogene, homologous chromosome 4) is related with the occurrence and development of tumor. in leukemias, the downstream pathways of erbb family signaling are frequently activated, and introduction of activated egfr into hematopoietic cell lines induced proliferation, survival and abrogation of cytokine dependency of the cells. however, the activation mechanisms of the erbb family signaling are largely unknown[14]. otabek imamov et al. demonstrated the novel role for esr2 in regulating the differentiation of pluripotent hematopoietic progenitor cells, and suggested that the esr2(er beta)-mouse should be a potential model for myeloid and lymphoid leukemia, and that esr2 agonists might have clinical value in the treatment of leukemia if the esr2 is not itself mutated in this disease. the human esr2 gene has been mapped to chromosome 14q22[15]. evi-1 gene is frequently overexpressed in leukemias having 3q26 abnormalities such as t(3;3)(q21;q26) and inv(3)(q21 q26), and subjects to structural alteration in t(3;21)(q26;q22). ogawa s et al. presented another case of structural alteration of evi-1 gene in a case of inv(3)(q21 q26), in which evi-1 is truncated and a shorter form of evi-1 protein is expressed upon rearrangement of the gene. their result also supports an idea that evi-1 is a relevant oncogene whose overexpression or structural changes might play a crucial role in development of human leukemias[16]. pdgfrb is constitutively activated by gene fusion with different partners in myeloproliferative disorders with peculiar clinical characteristics. translocation t(5;12)(q33;p13), resulting in an etv6/pdgfrb gene fusion, is a recurrent chromosomal abnormality associated with chronic myelomonocytic leukemia (cmml). an analogous translocation was also found in four cell lines with features of pre-b acute lymphoblastic leukemia (all) [17]. thomas dunwell et al. demonstrated fat1 gene is significantly more methylated in b-all compared to t-all[18]. gill d. et al. demonstrated the prolonged survival of b-cll in vitro without additional stromal cell support. furthermore, novel cytokines ccl2 and cxcl2 appear to prolong the survival of b-cll cells in vitro. this culture system and these chemokines may allow us to gain insight into factors modulating b-cll survival and potentially could lead to targeted therapy as well as serve as an appropriate model to test new therapies[19]. cxcl1 is expressed by macrophages, neutrophils and epithelial cells, and has neutrophil chemoattractant activity. cxcl1 plays a role in spinal cord development by inhibiting the migration of oligodendrocyte precursors and is involved in the processes of angiogenesis, inflammation, wound healing, and tumorigenesis. based on the specific functions of the above genes, we discover that they are closely related to b_all, maybe they are the oncogenes of b_all. b: considering genes fer, fes, lta, wnt3, tcl1a, btk, msh4 and ptk2, they are isolated nodes in normal and b-all network, but with greater degree in the t-all network. because their degrees in the t-all network significantly increase, we can speculate that their functions or the life process involved in should be to promote cell proliferation and division, and advances in systems science and applications (2011), vol.11, no.1-2 77 they should promote the development and deterioration of cancer forward. the specific functions of the 8 genes as follows: expression of fer gene in a wide range of cell types indicates a general role in intracellular signalling or differentiation processes. j. groffen et al. found that the negative regulation to be the main cause for dysfunctioning of the fer promoter in a t-cell leukemia cell line[20]. the human fes proto-oncogene is expressed as a transcript of about 3.0 kb in both normal and leukemic myeloid cells. tesch h. et al. detected truncated fes transcripts of about 0.9 kb in a panel of human lymphoma and lymphoid leukemia cell lines, but not in normal untransformed hematopoietic cells[21]. zhou mx, et al. discovered that the partially purified lta (tnf-alpha) obtained from the eu-1 cell line also suppressed the proliferation of tnf-sensitive primary leukemic cells, and this inhibitory activity was abolished by an anti-tnf-alpha specific antibody. the results demonstrated that tnf-alpha is an inhibitor of in vitro proliferation of bcp-all cells from most patients[22]. transfection with a wnt3 plasmid resulted in a small increase in reporter gene activity, which was augmented by the transfection with the lrp6 coreceptor. recent microarray analyses have demonstrated that the wnt3 gene is overexpressed in cll, compared with normal b and t cells[23]. in t-cell prolymphocytic leukemia (t-pll), chromosomal imbalances affecting the long arm of chromosome 22 are regarded as typical chromosomal aberrations secondary to a tcrad-tcl1a fusion due to inv(14) or t(14;14) [24]. bruton's tyrosine kinase (btk) deficiency results in a differentiation block at the pre-b cell stage. likewise, acute lymphoblastic leukemia cells are typically arrested at early stages of b cell development. janet d. rowley et al. identified kinase-deficient splice variants of btk throughout all leukemia subtypes[25]. msh4 gene belongs to the human dna mismatch repair system (mmr).the principal function of mmr is to edit replication and to reverse dna polymerase errors. inactivation of mmr greatly increases spontaneous mutation rates. h hirai et al. suggested that disruption of mmr may play an important role in the development of human lymphoid leukemias[26]. focal adhesion kinase (fak) is constitutively activated and tyrosine phosphorylated in bcr/abl-transformed hematopoietic cells. yi le et al. suggested that fak is critical for leukemogenesis and might be a potential target for leukemia therapy[27]. based on the specific functions of the above genes, we discover that they are closely related to t_all, maybe they are the oncogenes of t_all. c: considering genes cdh1, kdr, igf-1, fosl1, cttn, mos and tacstd2, their degrees change more significantly in the networks for experimental groups than those for control group. the specific functions of the 7 genes as follows: e-cadherin, the gene product of cdh1, plays a key role in cell–cell adhesion. decrease or loss of e-cadherin expression accompanied by cdh1 promoter methylation has been reported in many human cancers. john r. et al. found that all normal donor samples expressed e-cadherin mrna, whereas both samples of acute myelogenous leukemia and chronic lymphocytic leukemia had a significant reduction or absence of expression. however, normal blast counterparts expressed only a low level of e-cadherin surface protein[28]. kdr (flt) gene plays an important role in new angiogenesis，and it is related with tumor grade, growth and prognosis. kai neben et al. showed that unique gene expression patterns can be correlated with flt3-itd and flt3-tkd. this might lead to the identification of further pathogenetic relevant candidate genes particularly in aml with normal karyotype[29]. igf-1 is important in blood formation and regulation and has been shown to stimulate the growth of both myeloid and lymphoid cells in culture. since infants who develop leukemia are likely to have had at least one transforming event occur in utero, julie a. ross et al. hypothesized that high levels of igf-1 may both produce a larger baby and contribute to leukemogenesis[30]. fosl1(fra-1) gene, which is a tax1-inducible fos-related gene, was isolated and tax1 or serum-responsive cis elements were analyzed to obtain further insight into the mechanism of tax1 action. tax1 of human t-cell leukemia virus type 1 stimulates the expression of several cellular immediate-early genes[31]. patients with lymph node metastasis apt to emerge the amplification of cttn (ems1) gene, 78 liu: structural analysis of leukemia related gene network accompanied the phenomenon of ems1 over-expression, tumor cell invasion and metastasis increases simultaneously. the mutation of ems1 maybe one of the reasons of all[32]. cytogenetic analysis of an infant with down syndrome with concomitant acute myelogenous leukemia revealed a unique t(8;16)(q22;q24). in situ chromosomal hybridization was used to demonstrate that the protooncogene mos was translocated from chromosome 8 to chromosome 16. mark j. et al. reported that the transposition of mos in association with acute leukemia[33]. tacstd2 is considered as a cancer-associated antigen, and the antibody of its extracellular domain can reduce the invasiveness of tumor cell. based on the specific functions of the above genes, we discover that they are closely related to all, maybe they are the oncogenes of all. therefore, comparing the 23 structural key genes of all predicted by model with the functions of these genes showed in literatures, we discover that 21 genes (marked with “*” in table 1, 2 and 3) are confirmed to be closely related to the formation of all. the match ratio between the prediction by model and literature is up to 91.3%. hence, we can predict that the remaining two genes cxcl1 and tacstd2 are also closely related to all based on the effectiveness of the method in this method. the authenticity and reliability of the above speculations, as well as their pathogenic mechanism of all are needed to be further validated by biological experiments. in this research, we use 7 statistics parameters to characterize the network structure from different angles. from the results of the numerical experiments, we can see that the 7 parameters can 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[32] donald pasqualea et al.. clonal evolution with +11q 13, t(1;7) and t(1;4) at relapse in a patient with ph positive acute lymphocytic leukemia (all) treated with single agent front line imatinib followed by dasatinib. hematology, 2007, 12(6): 505 509. 80 liu: structural analysis of leukemia related gene network [33] mark j. et al.. translocation of the mos gene in a rare t(8;16) associated with acute myeloblastic leukemia and down syndrome. cancer genetics and cytogenetics, 1989, 37(2): 221-227. advances in systems science and applications (2011), vol.11, no.1-2 81 advances in systems science and applications (2013) vol.13 no.3 276-298 using simulations of netlogo as a tool for introducing greek high-school students to eco-systemic thinking aristotelis gkiolmas1, kostas karamanos1, anthimos chalkidis1, constantine skordoulis1, maria papaconstantinou2 and dimitrios stavrou3 1department of primary education, university of athens, greece. 2department of informatics, ionian university, corfu, greece. 3department of primary education, university of crete, greece. abstract in this paper, the effectiveness of the netlogo programming environment is investigated, as regards assisting students (of various levels of achievement) of the greek higher secondary education, to understand how some simple ecosystems are structured and to model the systemic behaviour of such ecosystems by conceptualising their complexity features. this paper is part of a wider research on teaching ecosystem complexity to high-school students, with the use of information and communication technologies (ict’s). specific models from the netlogo models’ library were used and students of the 2nd class of the greek lyceum (aged between 16 and 17) participated in the investigation. apart from oral instruction, the students were asked to run the netlogo simulations and do specific things with the models, answering simultaneously questions on worksheets provided to them. the studying and the evaluation of the worksheets by the researchers, as well as the postinstructional evaluation of the students, both oral (by means of cassettes) and written, through the use of an evaluation sheet, gave research findings that proved to be encouraging in that: a) students developed a greater understanding of the complex/systemic behaviour of ecosystems and b) they were capable, to a certain extent, of analyzing the systemic relations within simple ecosystems and built analogous relations in other, also simple, ecosystems. keywords eco-systemic thinking, education, netlogo, classroom, complexity. 1 introduction the aim of the current research was to investigate whether high-school students’ ability to achieve systemic thinking, and in particular eco-systemic thinking, could be slightly enforced and enhanced with the use of a specific software. systems-thinking is defined in many ways, and one way of defining it, is as: “the ability to understand and interpret complex systems” [1]. ecosystems are considered complex systems with respect to three essential properties of them [2]: (i) sustained diversity and individuality of components, (ii) localized interactions among these components and (iii) an autonomous process that selects from these components based on the results of local interactions a subset for repli277 advances in systems science and applications (2013) vol.13 no.3 cation or enhancement. taking these properties into consideration, it becomes apparent that eco-systems’ thinking is a perfectly valid expression of systemic thinking. many attempts, on the other hand, have been made so far in the literature to define what “modeling” of systems and especially of complex systems [4]. even in the case that the famous alternative definition of the (so called) complex “adaptive” systems (cas’s) (i.e. complex systems that “learn” as time evolves and modify their comportment) given by holland (1995)[5] is used, attributing to such systems four basic properties: aggregation, nonlinearity, diversity and flows, it becomes again obvious that ecosystems are indeed complex (adaptive) systems, which “learn” as time evolves, and thus, eco-systemic thinking is a form of systemic thinking and eco-systemic modeling an aspect of systems’ modeling. ecosystems are also, in a variety of other aspects “complex systems” [6] and can, therefore, be treated through their complexity characteristics, which also reveal systemic behavior, since the dynamics within complex systems have all the characteristics of system dynamics [7]. in this respect, the current research examines whether students, of greek upper secondary education, can understand the systemic features of a simple modelecosystem, consisting of only one prey and one predator. the definition of what “learning about the systemic nature of ecosystems” actually means, is given as: “the main goal of having students learn about systems is not to have them talk about systems in abstract terms, but to enhance their ability (and inclination) to attend to various aspects of particular systems, in attempting to understand or deal with the whole” [8], and is employed in this research. in his classic paper comparing the way of thinking of “novices” and an “experts” in complex systems, jacobson provides a table, namely table 1 in which he juxtaposes the mental models and beliefs of complex systems experts and novices [3]. some of these mental models and beliefs are directly pertinent to system-thinking. these mental models and beliefs, taken from table 1 of jacobson, are presented in the table below, together with the system-thinking ability that was treated as related to each one: (see tab.1 ) the current research emphasizes on the fourth column of the above table and attributes a score for each one of the system-thinking abilities achieved by the students, as it progresses. this is a measure of the students’ improvement in systems’ thinking as will be shown in the “results” section of the paper. a separate model of treating knowledge and learning in complex systems, very popular in the field of science education, is the model of “sbf structure, behavior and function”, developed mainly by cindy hmelo-silver and her associates [9-11]. in the sbf treatment and evaluation of knowledge about complex systems, the learning subject is considered to have adequate knowledge about a (usually complex) system if he/she can describe in exact terms the system’s: aristotelis gkiolmas: using simulations of netlogo as a tool for introducing greek ... 278 table 1 juxtaposition of the complex systems novices’ and experts’ mental models and beliefs, those related to systems’ thinking, as taken by jacobson [3] category of component belief component beliefs associated with clockwork mental model component beliefs associated with complex systems mental model associated system-thinking ability 1. understanding phenomena reductive (e.g., step-wise sequences, isolated parts) nonreductive: whole-is-greaterthan-the-parts seeing only the stocks vs. seeing all the stocks and the flows and the arrows and the faucet controls 4. action effects small actions → small effects small action → big effect seeing a proportional increase/decrease in the stock and the flow vs. seeing a nonproportional increase/decrease in the stock and the flow 6. complex actions from complex rules from simple rules seeing a stock as affecting only very close stocks by the neighbouring flows vs. seeing a stock as affecting even very distant stocks by arrays of flows 7. final causes or purposefulness of natural phenomena teleological nonteleological or stochastic not observing the arrows, i.e. the “closed” feedback loops, seeing only “open” procedures vs. observing the arrows, i.e. the “closed” feedback loops, and not seeing only “open” procedures 8. ontology static structures events equilibration processes seeing the stocks as reaching a final, constant size and the flows as stopping vs. seeing the stocks as reaching a final constant size but the flows as never stopping table 2 the s-f elements of the modeled wolves-sheep-grass ecosystem and their respective system dynamics’ depictions properties and aspects of the modeled ecosystem as system, related to the sbf conceptual model. elements of this property and aspect system dynamics expression of this element structure s1) the wolves s2) the sheep s3) the grass s1) a box (“stock”) called “wolves” or “wolves’ population” or equivalent s2) a box (“stock”) called “sheep” or “sheep’s population” or equivalent s3) a box (“stock”) called “grass” or “amount of grass” or equivalent function f1) wolves eat sheep f2) sheep eat or do not eat grass f3) wolves die if they do not find something to eat for some time f4) sheep die if they do not find something to eat for some time (in the case: “grass plays a role”) f5) sheep die due to predation by wolves f6) grass is reduced when eaten by sheep (in the case: “grass plays a role”) f7) new wolves are added due to wolves’ births f8) new sheep are added, due to sheep’s births f9) grass is replaced with a certain rate (in the case: “grass plays a role”) f1) an arrow showing the predation or something equivalent f2) an arrow showing the eating of grass (loss of grass) or something equivalent f3) something depicting the deaths of wolves f4) something depicting the deaths of sheep f5) something depicting the deaths of sheep due to predation f6) an arrow depicting the loss of grass f7) something depicting the addition of wolves f8) something depicting the addition of sheep f9) something describing the replacement of grass 279 advances in systems science and applications (2013) vol.13 no.3 (i) structure (i.e. the parts that constitute the system), (ii) behavior (i.e. the algorithms that underlie the system and make it perform the way it performs and (iii) function (i.e. what actions the system really carries out, in the observable level). for the scope and the needs of the current research, the aspect of “behavior” of the system and the relative characteristics of systems dynamics were considered highly difficult to achieve for high-school students, so emphasis was given only to the aspects of “structure” (“s”) and “function” (“f”), as well as their corresponding system dynamics’ expressions. in the table 2 [9], the s and f characteristics of the modeled simple ecosystem of wolf predating sheep (and, optionally, sheep eating grass) are presented, together with the corresponding system dynamic representation(s) for each characteristic, that the student is expected to find and express. table 2 will also be used in the analysis of the results section of the research. 2 method 2.1 the objectives a number of methods and tools have been developed, in order to reveal students’ and teachers’ conceptions of simple dynamic systems and ecosystems [12]. a variety among these methods of revealing and improving the understanding of “systems”, have used agent-based systems and especially netlogo as a tool [1316], and furthermore, some of them used the environment of netlogo to see how a student changes from “expert” to “novice” in complex systems [10, 17]. there is clear evidence that even laypeople that have completed secondary education cannot easily define the structure or understand even the simplest simulations of ecosystem dynamics [18]. on the other hand, netlogo educational researchers have demonstrated that this modeling environment helps students, to a certain extent, to create mathematical models of a system, in the form of equations, or create dynamical/graphical models of a system [19-21]. therefore, the strategy of the current research, which makes it slightly different than all the aforementioned researches, focused on six didactical objectives, which stem mainly from authentic system-thinking abilities related to finding relations in the system and representing it, whereas the other researches seek for learners’ mental models regarding the systems’ nature or the learners abilities to model the system. in this context with the help of the teaching sequence carried out in this research, there are as already said six objectives. these objectives are that the students should: • understand, through computer simulation, the meaning of the entities of a modeled ecosystem. for example: a “stock” (a box) called “wolves” corresponds to the wolves’ population, and a “faucet control” corresponds to “wolves’ birth rate”. aristotelis gkiolmas: using simulations of netlogo as a tool for introducing greek ... 280 • be able to conceptualise the effects that parameter changes on the gui (graphical user interface) of netlogo have on the modeled system. • be in a position to reproduce, on paper, a similar model of an ecosystem, using the same set of symbols as the ones they were faced with, originally. the only things altered are the names of the constituents of the model. • develop the ability of constructing a similar ecosystem dynamics’ model, given only the parts of it. • in their oral and graphical description of the dynamics of the system, be a little more closer on the side of the “expert”, compared to the one of the “novice”, after interacting with the netlogo models, than before interacting with them. the comparison between “expert” and “novice” descriptions, refers to table 1 of jacobson and especially its last column on the right (“associated system thinking abilities”) [3]. • in their oral and graphical description of the dynamics of the system they should get a higher score (as explained in the results) regarding the sbf description of the system [9-10], after the interaction with the netlogo models. a secondary objective, apart from these six, was to be able to realise that while intervening on an (eco)system, the parts of it that are affected are much more than the ones that one initially sees, a skill that relates even to their active citizenship virtues. 2.2 the instrument as a working environment for teaching students, the programming language/environment netlogo was used (version 4.0.4). netlogo is a programming tool suitable for simulating [22], studying and understanding of complex systems. it is a modern variation of the logo programming language, simulating the function of multiagentbased systems. many researchers so far have used netlogo in educating people about understanding the nature of complex systems [23-25]. also specific use of this programming and modeling environment has been done to enhance students modeling abilities and skills, in domains such as graphical modeling or mathematical modeling [18, 20]. each agent of netlogo (typically called a “turtle”) follows a simple set of rules, defined by the writer of the code. this set of rules “guide” the agent in its motion among certain pixels of the screen, named now “patches”. the code additionally defines the action which the turtle should perform on the patch when it meets it, such as changing its colour. the agents act, to a certain extent, independently from each other and yet the “turtle’s” choice of which “patch” to move to and what task to perform on it, is also determined by the “status” of the patch it arrives at the degree of influence being usually determined by the programmer. netlogo provides an extensive “models’ library”, for various complex systems. the two simulations used for our teaching sequence come directly from this 281 advances in systems science and applications (2013) vol.13 no.3 models library and are: the wolf sheep predation model and the “wolf sheep predation (docked)” model [26-27], which is a variation of the first, concerning more the dynamics and the differential equations (mathematics) of the system. both models describe a simple predator (wolf) and prey (sheep) ecosystem, the first of the two giving also the option of adding a third population to the system (the grass). in both these simulations, there are two kinds of agents: the “wolves” and the “sheep”. [the “grass”, which can be also optionally activated, only in the first of the two models, as mentioned above, is not an agent. it is a netlogo “patch” (pixel) property]. the agents are interconnected by relationships of a simple prey-and-predator nature. both kinds of agent follow also a rate of reproduction, controlled by the user of the simulation, and they also die when they run out of energy, apart for the sheep deaths due to predation. the first model, “wolf sheep predation” is used mainly to familiarize the student with the multi-agent simulation, in the sense that the students interact with it, change its parameters and see the results both on the simulation screen but also on the graph screen which, in turn, depicts the time-evolution of the populations. the focus of interest for the teaching sequence is the second simulation, called the “wolf sheep predation (docked)” [27]. when opening the corresponding file, the student is faced with two screens. the one is the typical netlogo console, depicted in fig.1. in the left part of this screen the typical agent-model (simulation) of netlogo for one-prey and one-predator simulation exists. it is called “agent model” and the students may interact with it, as in the previous model, to visualise the time-evolution of the populations when altering parameters of the system. the right part of the screen is called the aggregate model and its evolution is only graphically depicted. this part of the screen describes the evolution of the two populations, based strictly on the mathematical treatment, and provided, in turn, by a form of the famous set of the two lotka-volterra equations [28-29]. epistemologically and didactically, the difference between the two parts of the screen is that on the left part (the simulation/agent screen) randomness plays an important role (“sheep die when and if they encounter wolves”) where as on the right part of the screen (the aggregate model screen) randomness and stochastic behaviour play no role, since the rate of dying of both populations are strictly determined by “rates” (i.e. differential equations, mathematics). apart from this two-sided screen, another, very important for this research, part of the second simulation is the screen called “system dynamics modeler”, created in a java environment, which is shown in fig.2. this screen is exactly the one that focus is given on, being directly related to the “aggregate model”. it represents the dynamic system-model of the simple prey-predator system, depicting additionally its interrelations, aristotelis gkiolmas: using simulations of netlogo as a tool for introducing greek ... 282 fig.1 a screenshot of the netlogo model: “wolf sheep predation (docked)” feedback loops and flows/stocks. 2.3 the sample and the settings this teaching sequence was part of a wider research, concerning the teaching and learning of ecosystem complexity at the upper-secondary-education student level, with the use of computers. for this research, a sample of 10 voluntarily participating students was used. the students belonged to two different senior secondary schools, and were at the 2nd class of the greek lyceum (ages between 16 and 17). their orientation was either the technical or science education, therefore guaranteeing a satisfactory background in mathematics, physics and biology. the two schools were of neighbouring areas of athens, greece, and the two groups had similar socioeconomic status, gender mix and school-grade achievement. the students were grouped according to their school achievement in the previous year of the high-school to three categories: –low achievement students (la). their overall grade of graduation from the first class of the senior high school was less than 14 (in the greek scale of 0 to 20, where 20 is the best possible grade). –medium achievement students (ma). their overall grade of graduation from the first class of the senior high school was between 14 and 18.5. (from 18.5 283 advances in systems science and applications (2013) vol.13 no.3 fig.2 a screenshot of the netlogo system dynamics modeler of the model: “wolf sheep predation (docked)”. up to 20, the grades characterization is “excellent” in greece). –high achievement students (ha). their overall grade of graduation from the first class of the senior high school exceeded 18.5. there are two things to notice: a. the choice of the grade score of 14 as a limit is not arbitrary. in the greek high school system very few students fall below 10 in the scale of 0 to 20 and also if one gets a grade of 14, he/she is considered as adequately having followed the class in the respective year of studies, even if he has lost a severe amount of teaching hours. b. in this particular research the parameterisation with respect to the school achievement (la, ma and ha) was not taken into consideration in detail, but only as an indicative index. in further researches of this research group, samples of students are examined with respect to their systemic thinking and systemic modeling abilities and in these researches, the school achievement plays an important role as a parameter. each one of the two groups of the ten students, was taught separately by the first of the authors, for sixteen teaching hours, and he provided each student with 4 worksheets one for each quartet of teaching hours which they aristotelis gkiolmas: using simulations of netlogo as a tool for introducing greek ... 284 completed during the process of the instruction. the students worked with computers, in groups of two or three, depending on computer availability. there was never a student sitting alone in a computer screen. in addition to the printed worksheets, students answers and the classroom discussion were tape-recorded. 3 the instruction process and the delivered material first teaching session (4 didactic hours) the first two teaching hours (of the first quartet of hours) were dedicated in familiarizing students with the netlogo environment, particularly with prey-andpredator simulations. for this purpose, the “wolf sheep predation” model was used. within this part of the first teaching session (the 2 teaching hours), the students were asked to handle each button and “slider” in the netlogo console making trials with it and watching the model evolving on the screen with time and try to learn the button’s or slider’s function. after each question on the worksheet, there followed a class discussion, mediated by the instructor. the following are characteristic questions from the first-2-hour worksheet: • question number 3: run the simulation at a low speed (“lower”) and try to find out what the sliders “initial-number-sheep”, “initial-number-wolves”, “wolf gain from food”, “sheep-reproduce” and “wolf-reproduce” actually do. write your answers down. • question number 4: ‘now let us discuss and reach a common conclusion about the role of each of the sliders’. in the rest of the first teaching session (two more teaching hours), through some other questions, the students are introduced visually and verbally to the concept of “(eco)system instability”, since some population may occasionally become extinct. second teaching session (4 didactic hours) during the second teaching session, the simulation “wolf sheep predation (docked)” was introduced. the participants were asked first to investigate the “agent model” section of the netlogo screen, which resembles the previous simulation. they interacted with this section, choosing various values and combinations of values for the sliders, and noticing the model’s evolution with time, as well as its final outcome, both in the simulation area and on the graph. their attention was then driven to the right part of the screen, the “aggregate model”, being now asked to find the meaning and the role of the sliders, noticing that the outcome of the user’s interaction with the system is shown only graphically. simultaneously, the students were given on their computers’ screens a typed presentation about the lotka-volterra equations. the students were then encouraged to discuss their findings about the sliders’ role with the class. next, the 285 advances in systems science and applications (2013) vol.13 no.3 students were prompted, by the worksheet, to write down or discuss verbally the relation and differences between the agent and the aggregate model, observing the effects live, by pushing the “compare” and “step compare” buttons. third teaching session (4 didactic hours) in the third teaching session (4-hour), the class was introduced to the system dynamics modeler. the aim was to conceptualize the direct connection of this model with the aggregate models’ parameters and the way in which this relation is established. the students were asked what the flows, stocks, arrows and faucet controls actually mean in the modeler, and how they can interfere with them (through the mouse of their pc). towards the end of the worksheet, a first low-level direct objective was achieved, to see what the hypothetical effect would be on the system dynamics modeler, when they altered the position of a slider in the aggregate model worksheet. question number 6: fill in the gaps below and create similar sentences: “when i increase ...... in the simulation, in essence i make the box (stock) ...... in size”. “when i reduce ...... in the simulation, in essence i make the arrow ...... in width.”, or “let less flow pass through the faucet control ......”. we also asked for verbal answers, which we tape-recorded. later in the same session, the students were given the opportunity to think and reconsider question 6. the issue is whether only one thing would be affected in the modeler if one changed something in the netlogo screen, or more things would be affected. afterwards, the notion of “systemic thinking” and especially “ecosystemic thinking” was introduced, and its importance to the conservation of the earth, as a whole. at the end of this third 4-hour session, a paper was given to the students, reading: “we see the parts of an ecosystem as interrelated and not as separate entities (this is called “reductionist thinking”). it is easy to see that human interference constitutes one of these parts and can affect many more parts than those that we initially estimated!!” fourth teaching session (4 didactic hours) in the last session, further skills of systemic thinking and modeling were developed, through the system dynamics modeler. mainly, it consisted of two stages: at the first stage, students were given a piece of paper other than the worksheet, which was identical to the model depicted in the modeler, but the terms the words had been removed (refer to fig.3). here the students were asked to fill in the following words or phrases: water deposits in an area / average rainfall rate in the area / birth rate in the area / water returning to the sea, to the lakes or to the under-surface horizon / consumption of water in cubic meters per person / population increase in the area / water increase in the area / peoples’ deaths in the area / population of the area aristotelis gkiolmas: using simulations of netlogo as a tool for introducing greek ... 286 fig.3 the system dynamics modeler of the netlogo model: “wolf sheep predation (docked)”, without the terms in it.. / rate of deaths in the area. they did this filling of gaps in the way they think proper. at the second stage of this fourth session, students were given the elements of the system dynamics modeler, as shown at fig.4. they were allowed to use each part/element of the figure freely, in order to fig.4 the elements (stocks, flows etc) of the dynamic model of the system. create a description of the following (eco)-system: “the amount of solar energy entering the baikal lake in russia, in relation to the seaweeds growing and dying in this lake, and the photosynthesis they perform, engaging this energy.” a crucial question was posed to the students after the second and before the third teaching session. it was related to the skills of “novice’s” vs. “expert’s” description of the system as given by jacobson [3], as well as to the sbf understanding of the system as given by hmelo-silver et. al. [9-10]). the question was this: 287 advances in systems science and applications (2013) vol.13 no.3 overall question: “you are asked now to write an essay (around one page or one page and a half) about what you do see in that system of the populations. describe, among other things, what this ecosystem consists of, what each part of the ecosystem does, and what the ecosystem does as a whole. when is a population extinct? when is it endangered? when does a population fall and when does it augment tremendously? what parameters affect each population and how does a population affects (or is affected by) another or others? you are also encouraged a lot to make drawings about the relations of the populations, using “boxes”, various kinds of “arrows”, “faucets”. write things or comments, or titles (legends) on or in every part of your figure that you thing it is appropriate to write. write and draw a lot!” the same question was also handled to the students after the completion of the fourth and final teaching session of the whole sequence, and the results were compared. 4 results and discussion despite the limitation of the small size of the sample, which does not allow to draw general conclusions, and makes this more of a case study, this research depicted that the use of the netlogo as a teaching environment, together with the system dynamics modeler, could potentially help high-school students in understanding simple (eco)system structures, and to acquire or slightly improve their skills on representing and even building models. triangulating the results with the ones of using other modeling tools than the system dynamics modeler of netlogo, such as the stagecast creator (sc) [30], used for primary school students, the oral answers of the students were used in the present survey as an encouraging feedback with respect to the use of the software as a means of understanding the model. at first the results of this research are presented in an indicative form. the answer of a girl, christine, of low school achievement, is quoted, to the tape-recorded discussion, following the aforementioned question number 6, in the third teaching session: ‘yes, by varying the sliders positions in the aggregate model, i can better see what every shape in the system dynamics modeler means.’ in question number 7, of the same worksheet, where they were asked to reconsider their answer in number 6, they are prompted to see the wider interconnectedness of the factors in this simple prey-and-predator population system. this is not at all easy for high-school students, as was shown in the research of barman, griffiths and okebukola, recorded as “misconception no. 2”[31]. there, students were found rather to believe that the change in one population affects only its closest pray and itself. in the answer to question 7, a student, john, is aristotelis gkiolmas: using simulations of netlogo as a tool for introducing greek ... 288 quoted, who is of a medium school achievement: ‘it seems that one motion of the slider can change the whole system. so if we touch the “predator-efficiency”, we increase the number of births of wolves, thus we increase predation-rate of them which, in turn, increases the number of dying sheep. as a result, the number of wolves is increased in one hand, but the amount of sheep (food) is decreased, so the wolves will have problems. many such changes are done.’ the concept investigated and aimed at improving here is a concept of what white calls “naive ecology”[32]. the relations in this simple prey-and-predator system are strictly causal (‘the wolf eats the sheep thus sheep die’, or ‘the wolves have no sheep to eat, thus they gradually become extinct’) and the learners are asked to follow such causal relations, as far as they can, relatively to the initial point of focus, which could be everything (e.g. ‘wolves die’). white’s research also depicted that the persons cannot follow such a causal string of facts in ecology too far from the point of origin. the students’ answers are also close enough to the eco-systemic, holistic conclusion, before this was finally distributed to them, written on paper, therefore their stands and attitudes seem to have been adequately investigated and, possibly, changed. katherine, a student with school achievement belonging to the lower part of the sample, wrote: ‘this indicates that whatever we touch in an ecosystem, this also affects the other factors too, and, therefore, we must never change something that needs not be touched, because this would affect many more factors than the ones we initially estimated!’ the one of the two higher system-thinking objectives, which is to represent a very similar system but with differently named parts or terms, was satisfactorily achieved, as is represented by the fig.5 given by denis, who belonged to the medium area of the sample as regards school grades achievement. this skill here refers mainly to the systems-based inquiry (s-bi) protocol used by sweeney and sterman, in ecosystems, and especially its first two parts: part i: systemic scenarios: participants consider system dynamics in [six] simple scenarios. these scenarios emphasize feedback dynamics. part ii: homology challenges: participants imagine [six] related but different systemic scenarios. this requires participants to use homological reasoning.’ [12]. homological reasoning, as well as analogical system-thinking, both in their very simple expressions, was cultivated, to a certain extend, to all of the students, as denis’s drawing (who reflects an average of the sample) reveals. passing to an ecological phenomenon other than “predation”, which is “water consumption”, the students appear to be 100% correct in adopting the terminology and conceptualising the analogies. 289 advances in systems science and applications (2013) vol.13 no.3 fig.5 the denis’ first sheet. finally the other, even higher, system-thinking and system-constructing objective, aimed at, was to construct a model again similar to the initial having only the elements of it. such modeling tasks are not easy, as researchers have shown [17, 33]. hogan used only textbook-based teaching and asked the learners to construct simple food webs, in the sense of only putting the arrows in them (‘who eats what’), giving the reasoning for their choices. the results were poor. jensen and brehmen asked a sample of post-secondary-education individuals to study a predator-and-prey model very close to the one we chose. it is depicted in fig.6. the task here was to establish population equilibrium in this system. the learners tried this on the computer (changing a numeric parameter, i.e. the “foxes”) and what they relied on, was either mathematical approach or negative feedback-loops. the results were again poor (less than 50% did it) and as one of the main reasons for this low performance, the authors give: ‘one possible reason why the rabbit-fox problem is difficult may be that it is not easily decomposed into subsystems or homomorphs. since they are so closely intertwined, it is not feasible to split the system into a rabbit system and a fox system. therefore it is not possible to simplify it. this may contribute to the difficulties in forming a clear-cut model of the system.’ [18]. aristotelis gkiolmas: using simulations of netlogo as a tool for introducing greek ... 290 fig.6 the figure of jensen and brehmer fig.7 the sheet as completed by george (at left) and christine (at right). therefore, one important competency in understanding an (eco)system, is decomposing it to its parts, which is what the half of our last activity is about. the 291 advances in systems science and applications (2013) vol.13 no.3 other half of the activity is attempting, with these same parts, to build a similar ecosystem. as a measure of the results of this last activity on the sample, we present two drawings: figure 7 depicts the model made by the highest-school-achievement student of the whole sample (george), as well as the one made by the lowestschool-achievement student of the whole sample (christine). it can be noticed that, even though the “better student” has made a more flexible shape, and he adds even subtle-unneeded details, whereas the “weaker student” sticks more to the original and omits certain terms, the basic representation of this alternative ecosystem (“lake baikal and the seaweeds”) is essentially achieved in both drawings. as regards the “numerical” representations of the results, each one concerning the six didactic objectives mentioned in section 2.1 of the current paper, the results were: first objective. the 7 out of the 10 students corresponded totally correctly the elements of the system dynamics model with the entities it describes. it seems to have been an easy task, especially with the aid of the netlogo model. second objective. 8 out of 10 students gave all the correct answers when writing what is affected in the modeled system by each slider’s and each button’s variation. again the proportion of successful completion of the task is high, combined with the use of the netlogo models. third objective. once again 7 out of the 10 learners constructed on paper a quite similar model of an ecosystem, with only the names of the terms altered. high performance also appeared here, with the help of the netlogo screen. fourth objective. here as was expected there was relative failure. only 3 out of 10 students succeeded in building correctly on paper, the dynamics’ model of a system “from the scratch”, given only the elements of it. it is a high systemic-thinking ability and, even with netlogo, it cannot be easily enhanced. fifth objective. as regards the table 1 of the classic paper of jacobson a score was attributed to each of the 10 students, as regards their performance of the last right column of table 1. the question of reference is of course the “overall question”, mentioned before in the paper, in which the students’ answers are scrutinized with respect to their content. each time a student acts as a “novice” on the modeled (complex) ecosystem, in his/her writings, descriptions or in his/her shapes of the system’s dynamics he/she gets a “0”, for the corresponding aspect mentioned in aristotelis gkiolmas: using simulations of netlogo as a tool for introducing greek ... 292 the last column on the right of table 1. on the contrary, each time the student acts as an “expert” on the modeled ecosystem he/she gets an 1 for this aspect. since there are five aspects of systemic thinking, it is obvious that the maximum score each student could take is “5” and the minimum score is “0”. for the whole sample of 10 students the respective maximum score is “50” and the minimum score is “0”. the results and the scores of this attributing system are shown in table 3, both before and after the study of the netlogo’s system dynamics’ modeler (the “overall question” was delivered to them twice). interpreting the table 3 the (preand post-interaction with the model) scores achieved by the 10 students, as regards the jacobson [3] model of mental modes, component beliefs and the associated with them system-thinking abilities. category of component belief and associated system-thinking ability overall score achieved by the 10 students before interacting with the two netlogo models overall score achieved by the 10 students after interacting with the two netlogo models 1. understanding phenomena 42 44 4. action effects 14 17 6. complex actions 20 39 7. final causes or purposefulness of natural phenomena 19 38 8. ontology 12 15 findings of table 3, one can see that: – the study of the system dynamic modeler of netlogo and the interaction with netlogo model “wolf sheep predation (docked)” was very helpful for the students , as regards “complex actions” and “final causes or purposefulness” of systems, which in systemic terms means it helped them to “see very distant interactions within elements of the system” and to “see closed loops within the system and not only open processes”. especially the first of these two aspects, is in close proximity, with what sharona t. levy and uri wilensky define as a “mid-level” which means that the students understand better a system if they can describe it by means of the actions of a group of neighbouring and interacting parts [34], not by the actions of one single part alone and not by the action of all the parts as a whole. – in the domain of “understanding phenomena”, which in systemic terms means: “seeing most the compartments and the flows in the system”, the students performance was significantly good from the beginning, and the interaction with the system dynamic modeler of netlogo and the model itself, failed to improve it significantly. – in the domains of “action effects”, and “ontology” which in systemic terms mean respectively: “seeing beyond proportionality between stocks’ size and flows’ rates in the system”, and “seeing the size of stocks as dynamic, even when it looks 293 advances in systems science and applications (2013) vol.13 no.3 constant”, the students performance was poor in the beginning, and the interaction with the system dynamic modeler of netlogo and the model itself, seems to have failed also to make it better. sixth objective. once more, the question that was scrutinized is the “overall question” and the drawings or the writings of students in it, about the system, both before and after their study of the system dynamics modeler and the “wolf-sheep predation (docked)” model of netlogo. analyzing again the content of the writings and the drawings of the students in this question, a scoring system was established: – for each one of the three “structure” elements of the system that the student managed to find, he got a score “1”. consequently the score for each student would range from “0” to “3”, and the score for the whole sample of 10 students would range from “0” to “30”. – for each one of the nine “function” elements of the system that the student managed to find, he got a score “1”. consequently the score for each student would range from “0” to “9”, and the score for the whole sample of 10 students would range from “0” to “90”. – additionally, it was decided that for each “faucet control” or “rate” that the student would describe, for the corresponding “function” element, he would get “0.5” point more, since this reveals a slightly deeper understanding of the relative “function” aspect of the system. thus, the score for each student here would range from “0” to “4.5”, and the score for the whole sample of 10 students would range from “0” to “45”. the cumulative results for sbf treatment of system dynamics are shown in table 4. the results show that the interaction with the system dynamics’ modeler of table 4 the (preand post-interaction with the model) scores achieved by the 10 students, as regards the sbf model [9] of knowledge of , learning about and description of the system and the associated with them system dynamics’ expressions (“behavior” is excluded). aspect of the systems description according to the sbf model of knowledge and learning (“behavior” is excluded, as said) “structure” overall score achieved by the 10 students before interacting with the two netlogo models overall score achieved by the 10 students after interacting with the two netlogo models “structure” 21 28 “function” 33 68 “function” with “faucet controls” 10 41 netlogo, as well as with the model “wolf-sheep predation (docked)”: – helped the students conceptualise better to a certain extent the elements that constitute the system (the “structure”) . it should be stressed here that aristotelis gkiolmas: using simulations of netlogo as a tool for introducing greek ... 294 the element mostly missed before interaction was “grass”, which is an expected result. – helped the students much more to conceptualise the “functional” aspects of the system, which are mainly: (i) the “flows” (inflows, outflows) and (ii) the “interaction arrows” between system elements. this is a gain for system dynamics’ knowledge and students’ systemic thinking. – proved extremely helpful in making students conceptualise and describe the “faucet controls” which are actually the “rates”, within the system. this is an aspect of the system description and understanding of its “function”, that was very poor before the interaction with the netlogo models and the dynamics’ modeler, and improved significantly afterwards. 5 conclusions taking into consideration the restrictions of the current research (small sample size, absence of pre-test and post-test evaluations) and the imposed settings on the teaching sequence (few hours of teaching, students working out of their classroom schedule) , it is, nevertheless, argued, that there are significant conclusions from the research. at first, it is concluded that the combined use of a netlogo model and the system dynamics modeler associated with it can make students improve their understanding and their system-analysing abilities of a simple modeled ecosystem. secondly, it is concluded that upon interaction with netlogo models and system dynamics’ modeler, a student who is a “novice” in conceiving and describing systems and relations within systems, can slightly move towards the direction of being an “expert” on it, or at least on some of the system’s aspects. a third conclusion is that low-level system thinking abilities exist at large even before working with netlogo model, and remain there after interacting with it. the same is valid, on the opposite view, for high-level thinking abilities. they are mostly absent and remain absent after the interaction with the netlogo models and the system dynamics modeler. the abilities that are mostly enhanced by netlogo, are the medium-level system-thinking, system-analysing and systemrepresenting abilities. finally, a core concept of this research and a conclusion of it, is that a learner who is weak in: (i) seeing detailed structures in a system, (ii) seeing loops, (iii) predicting time-evolutions in it, (iv) observing large-reaching interactions among the elements of the system and (v) making generalizations about the system behavior will definitely benefit somehow, if he/she is taught about the system, not only with the graphic representation of the system dynamics but also with the study of (and interaction with) an agent-based simulation (such as a netlogo 295 advances in systems science and applications (2013) vol.13 no.3 model) of the system. questions regarding the time-evolution of the system, questions regarding extreme events (“what if a stock becomes too large or too small”? “what if a stock vanishes”? “what if a flow diminishes in thickness”?) and questions regarding random events (“stochasticity”) (“what if wolves find no sheep to eat”?) may not be easily answered for the novice, by the graphic or mathematical representation of the system dynamics, but are 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(2006), “thinking like a wolf, a sheep or a firefly: learning biology through constructing and testing computational theoriescan embodied modeling approach”, cognition and instruction, vol.24, no.2, pp.171-209. corresponding author author can be contacted at: agkiolm@primedu.uoa.gr 82-92 advances in systems science and applications (2011), vol. 11, no. 1-2 existence and nonexistence of positive solutions for a kirchhoff-type equation with inhomogeneous strong allee effect ruyun ma and guowei dai department of mathematics, northwest normal university, lanzhou 730070, p.r. china email: mary@nwnu.edu.cn, daiguowei@uwnu.edu.cn abstract in this paper, we deal with the nonlocal semilinear elliptic equation with inhomogeneous strong allee effect { −m (∫ ω 1 2 |∇u| 2 dx ) ∆u = λf(x, u) in ω, u = 0 on ∂ω, where the nonlocal coefficient m (∫ ω 1 2 |∇u| 2 dx ) is a continuous function of ∫ ω 1 2 |∇u| 2 dx. by means of variational approach, we prove that the equation has at least two positive solutions for largeλ under suitable hypotheses about nonlinearity. we also prove some nonexistence results. in particular, we shall give a positive answer to the conjecture by liu, wang and shi’s of [1]. keywords nonlocal differential equation variational method positive solutions inhomogeneous strong allee effect. 1. introduction in this paper we study the following problem{ −m (∫ ω 1 2 |∇u| 2 dx ) ∆u = λf(x, u) in ω, u = 0 on ∂ω, (1) where ω is a smooth bounded domain in rn for n ≥ 1, the nonlocal coefficient m(t) is a continuous function of t = ∫ ω 1 2 |∇u| 2 dx. we shall give a positive answer to a conjecture by liu, wang and shi’s of [1]. the problem (1) is a generalization of a model introduced by kirchhoff[2]. more precisely, kirchhoff proposed a model given by the equation ρ ∂2u ∂t2 − ( ρ0 h + e 2l ∫ l 0 ∣∣∣∣∂u∂x ∣∣∣∣2 dx ) ∂2u ∂x2 = 0, (2) where ρ, ρ0, h, e, l are constants, which extends the classical d’alembert’s wave equation, by considering the effect of the changing in the length of the string during the vibration. a distinguishing feature of equation (2) is that the equation contains a nonlocal coefficient ρ0 h + e 2l ∫ l 0 ∣∣∂u ∂x ∣∣2 dx which depends on the average 1 2l ∫ l 0 ∣∣∂u ∂x ∣∣2 dx of the kinetic energy 1 2 ∣∣∂u ∂x ∣∣2 on [0, l], and hence the equation is no longer a pointwise identity. the equation{ − ( a+ b ∫ ω |∇u| 2 dx ) ∆u = f(x, u) in ω, u = 0 on ∂ω, (3) issn 1078-6236 international institute for general systems studies, inc. advances in systems science and applications (2011), vol. 11, no. 1-2 83 is related to the stationary analogue of the equation (2). equation (3) received much attention only after lions [3] proposed an abstract framework to the problem. some important and interesting results can be found, for example, in [4-15]. in the context of population biology, the nonlinear function f(x, u) ≡ ug(x, u) represents a density dependent growth if g(x, u) is a function depending on the population density u. while traditionally g(x, u) is assumed to be declining to reflect the crowding effect of the increasing population, allee suggested that physiological and demographic precesses often possess an optimal density, with the response decreasing as either higher or lower densities. such growth pattern is called an allee effect. if the growth rate per capita is negative when u is small, we call it a strong allee effect; if the growth rate per capita is small than the maximum but still positive for small u, we call it a weak allee effect (for detail, see [16] or [17]). under the special case of equation (3) with a = 1, b = 0 and f(x, u) satisfies inhomogeneous strong allee effect growth pattern, liu, wang and shi[16] prove that the equation{ −∆u = λf(x, u) in ω, u = 0 on ∂ω (4) has at least two positive solutions for large λ if ∫ c(x) 0 f(x, s) ds > 0 for x in an open subset of ω, where c(x) ∈ c1(ω) such that f(x, c(x)) = 0(see the assumption of (f2)). they also prove some nonexistence results. in particular, they conjecture that the nonexistence holds if∫ c(x) 0 f(x, s) ds ≤ 0 for any x ∈ ω (see remark 1.7 of [16]). motivated by above, we generalize existence and nonexistence results for the semilinear elliptic equation (4) to the case of nonlocal semilinear elliptic equation (1). more precisely, if f(x, u) satisfies inhomogeneous strong allee effect growth pattern and the nonlocal coefficient m(t) satisfies some suitable conditions, we establish the existence of at least two positive solutions for the nonlocal problem (1) withλ large enough. we also prove some nonexistence results for the nonlocal problem (1). in particular, we shall give a positive answer to the conjecture by liu, wang and shi. to the best of our knowledge, this is the first paper that discusses the nonlocal semilinear elliptic equation with inhomogeneous strong allee effect via variational method. we point out the nonlocal coefficient m(t) raises some of the essential difficulties. for example, the way of proving the geometry condition of mountain pass theorem in [16] can not be used here because the functional of (1) is notc2 function under our assumptions. in order to overcome this difficulty, we divided ω into b1 and b2 by comparing the value of c(x) with b, then use poincaré inequality to prove it(see lemma 3.3). this paper is organized as follows. in section 2, we present our main resuts and some necessary preliminary lemmas. in sections 3, we use variational method and sub-supersolution method to prove the main results. in section 4, we prove the conjecture of liu, wang and shi’s and give some examples which satisfy our hypotheses. 2. main resuts and preliminaries in this section, we give our main results and some necessary preliminary lemmas which will be used in the following proof. for simplicity we write x = h1 0 (ω) with the norm ‖u‖ = 84 ma:existence and nonexistence of positive solutions for a kirchhoff-type . . . . . .(∫ 1 0 |∇u| 2 dx ) 1 2 . hereafter, f(x, t) and m(t) are always supposed to verify the following assumptions: (f1) f(x, u) ∈ c(ω× r+) and f(x, ·) ∈ c1(r+) for any x ∈ ω; (f2) there exist b(x) ∈ c(ω), c(x) ∈ c1(ω) such that 0 < b(x) < c(x) and f(x, 0) = f(x, b(x)) = f(x, c(x)) = 0 for any x ∈ ω; (f3) for almost all x ∈ ω, f(x, s) < 0 for any s ∈ (0, b(x)) ∪ (c(x),+∞) and f(x, s) > 0 for any s ∈ (b(x), c(x)). remark 2.1. note that the weak maximum principle (theorem 8.1 of [18]) and strong maximum principle (theorem 8.1 of [18]) also hold for the nonlocal problem (1) because m(t) satisfies the assumption (m ). (m ) ∃m0 > 0 such that m(t) ≥ m0. definition 2.1. we say that u ∈ x is a weak solution of (1), if m (∫ ω 1 2 |∇u|2 dx )∫ ω ∇u∇ϕdx = λ ∫ ω f(x, u)ϕdx for any ϕ ∈ x. define φ(u) = m̂ (∫ ω 1 2 |∇u|2 dx ) ,ψ(u) = ∫ ω f (x, u) dx, where m̂(t) = ∫ t 0 m(s) ds, f (x, u) = ∫ u 0 f(x, t) dt. we redefine f(x, u), such that f(x, u) ≡ 0 when u ∈ (−∞, 0) ∪ (c(x),∞), but it does not change the solution set of (1) by the weak maximum principle, since all the solution of (1) satisfies 0 ≤ u(x) ≤ c(x). then the energy functional iλ(u) = φ(u) − λψ(u) : x → r associated with problem (1) is well defined. then it is easy to see that iλ ∈ c1 (x,r) is weakly lower semi-continuous and u ∈ x is a weak solution of (1) if and only if u is a critical point of iλ. from the regularity assumptions on f(x, u), any critical point u of iλ(·) is a classical solution of (1) (see [19, 20]), and from the strong maximum principle and the definition of modified f(x, u) above, u is either zero or satisfies 0 < u(x) < c(x) for any x ∈ ω. moreover, we have i ′λ(u)v = m (∫ ω 1 2 |∇u|2 dx )∫ ω ∇u∇v dx− λ ∫ ω f(x, u)v dx = φ′(u)v − λψ′(u), for any v ∈ x. from (m ) and lemma 4.1 of [21] we can easily see that φ′ is of (s+) type, i.e. if un ⇀ u in x and lim n→+∞ (φ′(un)− φ′(u), un − u) ≤ 0, then un → u in x . it is clear that ψ′ is weak-strong continuous (or see lemma 1.2 of [1]). so i ′λ is of (s+) type. advances in systems science and applications (2011), vol. 11, no. 1-2 85 our main existence result is as follows: theorem 2.1. if m(t) satisfies (m ) and f(x, u) satisfies (f1)–(f3), and ω1 is an open subset of ω such that ∫ c(x) 0 f(x, s) ds > 0 (5) for x ∈ ω1, then for λ large enough, (1) has at least two positive solutions, and (1) has no solution for small λ. in order to prove our main existence result we need the following lemma: lemma 2.1 (see [1].) suppose that f satisfies (f1)–(f3). if u(x) is an integrable function in ω, and there is a measurable subset ω0 of ω with positive measure, such that∫ c(x) 0 f(x, s) ds > 0 in ω0 and ∫ c(x) 0 f(x, s) ds ≤ 0 in ω \ ω0, then ∫ u(x) 0 f(x, s) ds ≤ ∫ c(x) 0 f(x, s) ds in ω0 and ∫ u(x) 0 f(x, s) ds ≤ 0 in ω \ ω0, now we turn to the nonexistence of the positive solutions of (1) when (5) does not hold for any x ∈ ω. we define c = maxx∈ω c(x), f(u) = maxx∈ω f(x, u). our main nonexistence result is theorem 2.2. if ∫ c 0 f(u) du ≤ 0, then (1) has no positive solution for any λ > 0. in order to prove our main nonexistence result, we recall a theorem in [22] for (1) with the special case of m(t) ≡ 1 and f(x, u) ≡ f(u). in fact, the theorem also holds for the nonlocal problem (1) with f(x, u) ≡ f(u). because the proof is similar to the proof of[22], we omit it here (for detail, see the proof of theorem 1 in [22]). let us assume that f : r→ r is a c1 function and let the following conditions hold: there exist 0 ≤ s0 < s1 < s2, such that f(si) = 0, i = 1, 2, f(s0) ≤ 0, f(s) < 0, s0 < s < s1, f(s) > 0, s1 < s < s2 (6) and let ∫ s2 s0 f(s) ds ≤ 0. (7) we have the following lemma. lemma 2.2. assume that f satisfies (6) and (7). let ω be a bounded domain with smooth boundary. if (1) with f(x, u) ≡ f(u) has a positive solution u, then u can not satisfy{ umax = maxx∈ω u(x) ∈ (s1, s2), u(x) > 0, x ∈ ω. (8) 86 ma:existence and nonexistence of positive solutions for a kirchhoff-type . . . . . . remark 2.2. note that our assumptions (f1)–(f3) are weaker than (f1)–(f4) of[1] even in the case ofm(t) ≡ 1. in fact, from (f1)–(f3), we can easily see that there exist a positive constant β such that f(x, s) ≤ βs for any s ≥ 0 and a.e. x ∈ ω, i.e., the condition (f4) of [1]. we do not need the conditions of b(x) ∈ c1,α(ω)(0 < α < 1) and f(·, u) ∈ c1,α(ω) for any u ≥ 0 because we do not need energy functional of (1) is ac2 function in x in our proof. remark 2.3. the condition of f(x, ·) ∈ c1(r+) for any x ∈ ω can be relaxed to f(x, ·) is locally lipschitz in r+ for any x ∈ ω. in fact, lemma 2.2 also holds when f : r → r is a locally lipschitz function because the symmetry results of [23] holds under this weaker condition. 3. proof of theorem 2.1 and 2.2 in this section we will prove theorem 2.1 and 2.2. lemma 3.1. if m(t) satisfies (m ), and f(x, u) satisfies (f1)–(f3) and (5), then for λ large enough, iλ(·) has a global minimum point u1 such that iλ(u1) < 0. proof. since ∫ c(x) 0 f(x, s) ds > 0 in ω1, then there exists a measurable set ω0 ⊂ ω with positive measure, such that ∫ c(x) 0 f(x, s) ds > 0 in ω0, and ∫ c(x) 0 f(x, s) ds ≤ 0 in ω \ ω0. from (m ) and the definition of m̂(t), we have m̂(t) ≥ m0t. in view of lemma 2.1, we have iλ(u) = m̂ (∫ ω 1 2 |∇u|2 dx ) − λ ∫ ω f (x, u) dx ≥ m0 ∫ ω 1 2 |∇u|2 dx− λ ∫ ω (∫ u(x) 0 f(x, s) ds ) dx ≥ m0 ∫ ω 1 2 |∇u|2 dx− λ ∫ ω0 (∫ u(x) 0 f(x, s) ds ) dx− λ ∫ ω\ω0 (∫ u(x) 0 f(x, s) ds ) dx ≥ m0 ∫ ω 1 2 |∇u|2 dx− λ ∫ ω0 (∫ c(x) 0 f(x, s) ds ) dx ≥ m0 ∫ ω 1 2 |∇u|2 dx− λ ∫ ω0 a1 dx = m0 2 ‖u‖2 − λ|ω0|a1 → +∞, as ‖u‖ → +∞, (9) where a1 = maxx∈ω0 |f (x, c(x))|. since iλ is weakly lower semi-continuous, iλ has a minimum point u1 in x . next we shall prove iλ(u1) < 0, thus u1 is a positive solution of (1). in fact, we only need to verify that when λ is large there exists a u0 ∈ x , such that iλ(u0) < 0 = iλ(0). we define u0(x) = 0 in ω \ ω1ε, and u0(x) = c(x) in ω1 and properly in ω1ε \ ω1 such that u0 ∈ x , where ω1ε = {x ∈ ω : dist(x,ω1) ≤ ε}. using the similar method with[1], we have iλ(u0) ≤ m̂ (∫ ω 1 2 |∇u0|2 dx ) − λ ∫ ω1 f (x, c(x)) dx− λ [−a1 (|ω1ε| − |ω1|)] dx. (10) advances in systems science and applications (2011), vol. 11, no. 1-2 87 since ∫ c(cx) 0 f(x, s) ds > 0 when x ∈ ω1 and ∫ c(x) 0 f(x, s) ds is continuous, then there must exists an open subset ω2 with ω2 ⊂ ω1 and δ > 0, such that |ω2| > 0 and ∫ c(x) 0 f(x, s) ds ≥ δ for x ∈ ω2. choose ε small enough, such that δ|ω2| + a(|ω1| − |ω1ε|) > 0. again using the similar method with [1], we have iλ(u0) ≤ m̂ (∫ ω 1 2 |∇u0|2 dx ) − λ [δ|ω2|+a1(|ω1| − |ω1ε|)] . therefore when λ large enough, iλ(u0) < 0, and consequently when λ is large enough, (1) has a positive solution u1(x) satisfying iλ(u1) = infu∈x iλ(u) < 0. next we shall use mountain pass theorem to prove that (1) has another positive solution u2. first we prove iλ(u) satisfies palais-smale condition. definition 3.1. we say that iλ satisfies (p.s.) condition in x , if any sequence {un} ⊂ x such that {iλ(un)} is bounded and i ′λ(un) → 0 as n → +∞, has a convergent subsequence, where (p.s.) means palais-smale. lemma 3.2. if m(t) satisfies (m ), f satisfies (f1)–(f3) and (5), then iλ satisfies (p.s.) condition. proof. suppose that {un} ⊂ x , |iλ(un)| ≤ c0 and i ′λ(un) → 0 as n → +∞. in view of (9), we have c0 ≥ iλ(un) ≥ m0 2 ‖un‖2 − λ|ω0|a. hence, {‖un‖} is bounded. without loss of generality, we assume that un ⇀ u, then i ′(un)(un− u)→ 0. therefore, we have un → u by the (s+) property of i ′λ. lemma 3.3. ifm(t) satisfies (m ), f satisfies (f1)–(f3), then there exist ρ > 0 and γ > 0 such that iλ(u) ≥ γ for every u ∈ x with ‖u‖ = ρ. proof. we define b = minx∈ω. for any u(x) ∈ x , we also define b1 = {x ∈ ω : u(x) < b}, b2 = {x ∈ ω : u(x) ≥ b}. it is well known that the embedding of x ↪→ lp(ω) is continuous when 2 < p ≤ 2∗, where 2∗ is the critical exponent. from poincaré inequality, we have b|b2| 1 p ≤ (∫ b2 up dx ) 1 p ≤ c1 (∫ b2 |∇u|2 dx ) 1 2 ≤ c1 (∫ ω |∇u|2 dx ) 1 2 = c1‖u‖, 88 ma:existence and nonexistence of positive solutions for a kirchhoff-type . . . . . . where c1 is the embedding constant of x ↪→ lp(ω). thus, we have iλ(u) = m̂ (∫ ω 1 2 |∇u|2 dx ) − λ ∫ ω f (x, u) dx ≥ m0 2 ‖u‖2 − λ ∫ b1 f (x, u) dx− λ ∫ b2 f (x, u) dx ≥ m0 2 ‖u‖2 − λ ∫ b2 f (x, u) dx ≥ m0 2 ‖u‖2 − λa2|b2| ≥ m0 2 ‖u‖2 − λa2 ( c1 b )p ‖u‖p = ‖u‖2 ( m0 2 − λa2 ( c1 b )p ‖u‖p−2 ) , where a2 = max(x,s)∈b2×[b,c] |f (x, s)|. therefore, there exist m0b p 2λa2c p 1 > ρ > 0 such that iλ(u) ≥ ρ2 ( m0 2 − λa2 ( c1 b )p ρp−2 ) := γ > 0 for every ‖u‖ = ρ and fixed λ. proof of theorem 2.1 concluded. first let us show that iλ satisfies the conditions of mountain pass theorem (see theorem 2.10 of [24]). by lemma 3.2, iλ satisfies (p.s.) condition in x . by lemma 3.3, for fixed λ > 0, there exist min { ‖u0‖, m0b p 2λa2c p 1 } > ρ > 0, γ > 0 such that iλ(u) ≥ γ > 0 for every ‖u‖ = ρ, where u0 comes from (10). on the other hand, since iλ(0) = 0 and from the proof lemma 3.1, there exists u0 ∈ x such that iλ(u0) < 0 and ‖u0‖ > ρ. so from mountain pass theorem, iλ has another critical point u2 such that iλ(u2) ≥ γ > 0 > iλ(u1). therefore, u2 is another positive solution of (1). finally we show that (1) has no positive solution when λ is small. we assume (1) has a positive solution u, let (λ1, ϕ1(x)) be the principal eigen-pair of the problem{ −∆φ = λφ in ω, u = 0 on ∂ω, (11) such that ϕ1(x) > 0 in ω. we rewrite (1) as the following form{ −∆u = λ f(x,u) m( ∫ ω 1 2 |∇u|2 dx) in ω, u = 0 on ∂ω. (12) multiplying (11) by u, multiplying (12) by ϕ1, subtracting and integrating in ω, we obtain 0 = ∫ ω [ λ1uϕ1 − λϕ1 f(x, u) m (t) ] dx = ∫ ω uϕ1 m (t) [ m (t) λ1 − λ f(x, u) u ] dx, (13) where t = ∫ ω 1 2 |∇u| 2 dx. if λ < m0λ1 β , then by remark 2.2, we have m (t) λ1 − λ f(x, u) u ≥ m0λ1 − λ f(x, u) u > m0λ1 − λβ > 0. advances in systems science and applications (2011), vol. 11, no. 1-2 89 that is contrary to (13). so for small λ, (1) has no positive solution. proof of theorem 2.2. the proof is similar to the proof of [1]. for the sake of completeness, we include it here. if there exists a positive solution (λ, u∗) for (1), then u∗ is a subsolution of { m (∫ ω 1 2 |∇u| 2 dx ) ∆u+ λf(u) = 0 in ω, u = 0 on ∂ω, (14) since m (∫ ω 1 2 |∇u∗| 2 dx ) ∆u∗ + λf(u∗) ≥ m (∫ ω 1 2 |∇u∗| 2 dx ) ∆u∗ + λf(x, u∗). and c is supersolution of (13). so by the standard comparison arguments, (13) has a positive solution u such that u∗ ≤ u ≤ c. but if we let s0 = 0, s1 = b and s2 = c, f satisfies (6) and (7), then by lemma 2.2, (13) has no positive solution. this is a contradiction. so (1) has no positive solution if ∫ c 0 f(u) du ≤ 0. 4. proof of a conjecture and some examples in this section we will prove the conjecture of liu, wang and shi’s and give some typical consequences of theorem 2.1 to theorem 2.2. in [1], liu, wang and shi conjecture that the nonexistence holds with a weaker condition:∫ c(x) 0 f(x, s) ds ≤ 0 for any x ∈ ω. (15) in fact, as we will see in the following proposition, the condition (15) is more strong than∫ c 0 f(s) ds ≤ 0. therefore, by theorem 2.2, the conjecture is right. proposition 4.1. if f(x, u) satisfies (f1)–(f3) and ∫ c(x) 0 f(x, s) ds ≤ 0 for any x ∈ ω, we have ∫ c 0 f(s) ds ≤ 0. proof. from (f1)–(f3), we can easily see that f(x, s) ≤ 0 when s ∈ [c(x), c]. thus, we have ∫ c c(x) f(x, s) ds ≤ 0. then, for any x ∈ ω, we have 0 ≥ ∫ c(x) 0 f(x, s) ds = ∫ c 0 f(x, s) ds− ∫ c c(x) f(x, s) ds ≥ ∫ c 0 f(x, s) ds. in particular, ∫ c 0 f(s) ds ≤ 0. now, we give some examples which satisfy our hypotheses. example 4.1. let m(t) = a + bt with t = ∫ ω 1 2 |∇u| 2 dx, here a, b are two positive constants and f(x, u) = u(u − b(x))(c(x) − u) with b(x) ∈ c(ω), c(x) ∈ c1(ω) such that 0 < b(x) < c(x) for any x ∈ ω. it is clear that m(t) and f(x, u) verify our assumptions (m) and (f1)–(f3). example 4.2. we consider a special case of example 4.1:{ ∆u+ λu(u− b(x))(c(x)− u) = 0 in ω, u = 0 on ∂ω, (16) 90 ma:existence and nonexistence of positive solutions for a kirchhoff-type . . . . . . where b(x) ∈ c(ω), c(x) ∈ c1(ω) such that 0 < b(x) < c(x) for any x ∈ ω. we have known that f(x, u) satisfies (f1)–(f3) from example 4.1. moreover, we have∫ c(x) 0 f(x, s) ds = ∫ c(x) 0 s(s− b(x))(c(x)− s) ds = 1 12 [c(x)]3(c(x)− 2b(x)). then by theorem 2.1, if there exists an open subset ω1 ⊂ ω, such that c(x) > 2b(x) in ω1, then (16) has at least two positive solutions for large λ. if c(x) ≡ 1 for all x ∈ ω, we obtain∫ 1 0 f(s) ds = ∫ 1 0 max x∈ω s(s− b(x))(1− s) ds = ∫ 1 0 max x∈ω [s2 − s3 + b(x)(s2 − s)] ds = 1 12 − b 6 , since s2 − s ≤ 0 for s ∈ [0, 1]. then by theorem 2.2, if b = minx∈ω b(x) ≥ 1 2 , then (16) has no positive solution for any λ > 0. example 4.3. let m(t) ≡ 1 and f(x, s) = s(s − 1)(c(x) − s) with 3 2 ≤ c(x) for any x ∈ ω. we can easily obtain∫ c 0 f(s) ds = ∫ c 0 max x∈ω s(s− 1)(c(x)− s) ds = ∫ c 0 (c(x)s2 − s3 + s2 − c(x)s) ds = c3 3 − c4 4 + ∫ c 0 max x∈ω c(x)(s2 − s) ds = c3 3 − c4 4 + max x∈ω c(x) ( c3 3 − c2 2 ) ds = c3 3 − c4 4 + c ( c3 3 − c2 2 ) ds = c3 12 [c− 2]. so ∫ c 0 f(s) ds ≤ 0 if and only if c ≤ 2. on the other hand,∫ c(x) 0 f(x, s) ds = ∫ c 0 s(s− 1)(c(x)− s) ds− ∫ c c(x) s(s− 1)(c(x)− s) ds ≥ ∫ c 0 s(s− 1)(c(x)− s) ds = −c 4 4 + 1 + c(x) 3 c3 − c(x) 2 c2. advances in systems science and applications (2011), vol. 11, no. 1-2 91 if ∫ c(x) 0 f(x, s) ds ≤ 0 for any x ∈ ω, we have 0 ≥ −c 4 4 + 1 + c(x) 3 c3 − c(x) 2 c2 ⇒ 4(1 + c(x))c− 6c(x) ≤ 3c2. in particular, we have 4(1 + c)c− 6c ≤ 3c2 ⇒ c ≤ 2. however, it is clear that∫ c 0 f(s) ds ≤ 0 ; ∫ c(x) 0 f(x, s) ds ≤ 0 for any x ∈ ω. therefore, the condition “ ∫ c(x) 0 f(x, s) ds ≤ 0 for any x ∈ ω” is more strong than the condition “ ∫ c 0 f(s) ds ≤ 0” in this example, which verifies proposition 4.1 by a concrete example. remark 4.1. in [25], dancer and yan proved when c(x) ≡ 1 and {x ∈ ω : b(x) < 1/2} is of positive measure, then (16) may have many positive solutions of local minimum type. the results of example 4.2 shows that the condition ∫ 1 0 f(s) ds ≤ 0 is optimal for the nonexistence of positive solution of (16). however, we do not know whether ∫ c 0 f(s) ds ≤ 0 is optimal for the nonexistence of positive solution of (1). references [1] g.liu, ,y. wang, & j. shi, existence and nonexistence of positive solutions of semilinear elliptic equation with inhomogeneous strong allee effect, appl. math. mech. -engl. ed. 30(11), 1461-1468 (2009) [2] g.kirchhoff, mechanik, teubner, leipzig. 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[7] f. j. s. a. corrêa, & g.m. figueiredo, on a elliptic equation of p-kirchhoff type via variational methods. bull. austral. math. soc. vol. 74 (2006) 263-277. 92 ma:existence and nonexistence of positive solutions for a kirchhoff-type . . . . . . [8] p.d’ancona, , & s. spagnolo, global solvability for the degenerate kirchhoff equation with real analytic data. invent. math. 108 (1992) 247-262. [9] m.chipot,, & b. lovat, some remarks on nonlocal elliptic and parabolic problems. nonlinear anal. 30 (1997) 4619-4627. [10] m.dreher, the kirchhoff equation for the p-laplacian. rend. semin. mat. univ. politec. torino 64 (2006) 217-238. [11] m.dreher, the ware equation for the p-laplacian. hokkaido math. j. 36 (2007) 21-52. [12] g.ai, & r. hao, existence of solutions for a p(x)-kirchhoff-type equation. j. math. anal. appl. 359 (2009) 275-284. [13] dai, g., & liu, d. infinitely many positive solutions for a p(x)-kirchhoff-type equation. j. math. anal. appl. 359 (2009) 704-710. 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[21] g.dai, nonsmooth version of fountain theorem and its application to a dirichlet-type differential inclusion problem. nonlinear analysis 72 (2010) 1454-1461 [22] e. n.dancer, & k. schmitt, on positive solution of semilinear elliptic equations. proc. amer. math.soc. 101 (1987), no.3, 445-452. [23] b. gidas,, w. ni, w. & nirenberg, l. symmetry and related properties via the maximum principle. comm. math. phys. 68 (1979), 209-243. [24] m. willem, minimax theorems. birkhäuser, boston. [25] e. n.dancer,, & s. yan, construction of various types of solutions for an elliptic problem. calculus variations and partial differential equations 20(1)(2004) , 93-118. microsoft word 4-li xiaogen.doc issn 1078-6236 international institute for general systems studies, inc. the key techniques study of constructing the roaming modelin virtual environment li xiaogen 1,2, wang zong-min1, wang an-ming2 ,yang xiang-bing3 1 department of water conservancy and environment , zhengzhou university, zheng zhou 450001, p.r china 2 north china institute of water conservancy and hydroelectric power,zhengzhou, 450011 p.r.china 3programme and construction commission of linzhou city,linzhou henan 456550, p.r.china email: lixiaogen@ncwu.edu.cn abstract taking the digitization of zhengzhou university campus as an example, the 3-d model of every buildings of zhengzhou university were constructed with vr technology and 3d image processing technology. the paper has studied the questions that need to be solved in the construction process of digitizing campus and in the model roaming process, this solution has been successfully used in the scientific research item of digitizing zhengzhou university. the study results can be summarized as follows：⑴the texture mapping technology make the 3d model roaming more smoother;⑵the construction technologies of many trees models make the data volume worse;⑶the 3d digitization of zhengzhou university campus provides a new idea to construct 3d virtual large scene model. keywords virtual reality 3d module the roaming virtual reality-geographical information system (vrgis) the texture mapping technology 1. introduction nowadays, virtual reality-geographical information system which was the aim of establishing digital globe in the future, was a hot spot but difficult issue in the field of geographical information system and it was a high level to implement immersed roaming in the established virtual environment. digital campus project, which was the demonstration project carried out by education ministry of henan province, has realized the virtualization of campus. in the process of constructing the campus model a few difficult questions were solved. the used ideas which were less in the references about virtual reality were a few new thoughts to construct the 3d virtual large scene which was the bigger data volume model in the future[1][2].  this study was supported by innovative prominent talents project foundation for henan universities in 2005,henan innovation project for university prominent research talents in 2005(haipurt)(2005kycx015), important science & technology foundation of henan province. this support is greatly appreciated. ∗ advances in systems science and applications (2011), vol.11, no.1-2 55-60 2. the basic principles of establishing virtual campus according to the basic principle that entity positioning in actual world is decided by both eyes of a person, virtual scenes which are composed of transverse wave and longitudinal wave are projected onto a screen with divided projectors. the transverse wave and longitudinal wave are separately projected by two groups of projectors. at this moment, beholders should wear special glasses which is made up of two layers of membrane. the two layers of glasses could filtrate transverse waves and longitudinal waves respectively[3][4]. because the structure of both glasses are contrary, they could receive a set of transverse wave and a set of longitudinal wave respectively. that is to say, people’s eyes could respectively receive a set of transverse wave and a set of longitudinal wave. so, two groups of projector suit both eyes of a person respectively. in this way, beholders could see distinct virtual geographical environment in the screen[5][6]. in this virtual environment, we adopt the creator of multigen corporation to construct the model of campus. the scenes are rendered by vega presented by multigen corporation. finally, steering wheel and remote controlling pole are used as interactive equipments to simulate roaming of driving vehicles or overlook the whole campus from the sky by air. the established virtual environment are presented as follows: fig. 1, fig. 2, fig. 3, fig. 4. fig. 1 comprehensive administration building fig. 2 students’ dormitory building on the second phase 56 li: the key techniques study of constructing the roaming modelin virtual environment fig. 3 cafeteria and rest house in the living areas fig. 4 flank greenbelt of electric engineering department on the second phase 3. a few key techniques 3.1 texture patching pictures digital camera will be used to collect surface texture pictures to show actual scenes. rgb or rgba format graphics file are recommended in vega. when exporting texture files, attention should be focused on length and width of these graphics. the pixel of length and width of the graphics texture files should be limited within power of two. the strict limitation guarantees the normal display of patching picture texture. as for large area of lawn and roof, multi-surface patching picture should be adopted. mipmap bilinear and mipmap trilinear should be adopted as the ways of filtration so as to avoid twinkle of graphics texture effectively[7]. 3.2 processing of trees as for trees along the roads, it is a difficult technique problem to display them quickly and effectively while the speed of roaming is not decreased. this problem is settled effectively with the adoption of billboard and interactive patching pictures. in addition, large quantities of trees could be produced by applying the instance technique provided by creator. trees of the same kind could share the same texture and memory is used only once, which greatly economizes resources of the system and speed up displaying on the screen[8]. 3.3 combination of building units if the number of building is small (which means the number of files which need to be combined is small)，we could produce the final document by applying the methods of duplicating and paste with one of the files as dominance. meanwhile, we could integrate the advances in systems science and applications (2011), vol.11, no.1-2 57 actual scenes by copying other building models into the present documents in the way of paste, placing them in the proper position and adjusting the levels of documents in the catalogue frame work. if single building units at different stages are processed, it is not proper to apply the above stated methods, while single document is dominant. in this way, single building units are introduced with the method of external reference so that graphics texture files of building models and independence of three-dimension model file could be maintained and the actual scenes could be integrated[9]. 3.4 roaming of sight to make the users interact with the virtual environment naturally and simultaneously, we could manipulate joystick like steering wheel to roam in the campus. we could realize this aim with the application of c++ procedure and dynamic model which is based on vega. on the basis of analysis of vega structure and operational principle, we could upgrade the immersed sense of roaming system by applying directinput and controlling poll of force feedback. users could sense that they are immersed in roaming by controlling the automobiles in the virtual scenes with steering wheel manipulated. there are ten bull type trigger buttons on the input equipment which separately symbolize “forward” “backward” “turn on” “turn off”. the other two buttons are used to simulate steering wheel and accelerator. ( as shown in fig.5 and fig. 6). fig. 5 virtual roaming environment fig. 6 steering wheel 3.5 accomplishment of collision test the phenomenon of perforation should be avoided in virtual environment. tools for collision joints are not directly provided in creator, while perforation could be solved by collision test. collision test could be carried out among entities in the virtual scene where collisions probably happen, which is main characteristic of video rendering software. it could be described 58 li: the key techniques study of constructing the roaming modelin virtual environment as evaluation of intersection point in mathematics. in collision test and intersection test, vector which is composed of certain number of directed line segments will be considered. there are generally seven methods to define vector entity. these methods confine the position of the entities and determine the minimum test results. they are as follows: z, hat, tripod, los, bump, xyzpr, volume. each method has its own applicatory field. for example: the method of “z” is fit to be used in calculation of elevation. the method of “los” is fit to be used in tests of eyesight direction. if collisions happen, collision responses will be made according to the information which is stored in gathered intersection test results[10]. to carry out collision test, we intercalate anti-collision for every building by applying c++ procedure in vega environment, that is to set up a category which prevents perforation. effect of mountain climbing could be achieved by applying the method of “z” to calculate the collision between dynamic objects and ground. the method of “bump” could be applied to calculate collision between dynamic objects and buildings and trees in the virtual scene. 4. conclusion through the above study as follow the conclusion can be summarized. the texture mapping technology make the 3d model roaming more smoother, which can make the observers’ eyes more comfy. the construction technologies of many trees models make the data volume worse, which make the computer run faster. the 3d digitization of zhengzhou university campus provides a new thought to construct 3d virtual large scene model which has bigger data volume in the future. 5. future work it is not the best choice to apply multigen creator to building modeling in large-area real scene. its superiority lies in modeling of geometric objects. while the best way is to patch picture by modeling with autocad or 3dsmax software and then switch them into creator to revise. in addition, authenticity, applicability and interest could be promoted by further improvement of dynamic modeling to support coordinated roaming of individuals. references [1] li xiaode, peng bin, virtual reality technique development summarize[j], innovation forum,,10-12, 2004. advances in systems science and applications (2011), vol.11, no.1-2 59 [2] yu li, wang cheng. digital campus three dimension simulation system based on the virtual reality [j], computer simulation newspaper. 2004:99-102. [3] wang yujiang, the virtual reality discuss based on multigen creator and vega[j], mapping information and engineering newspaper, 2003:14-16. [4] zhenjun wu, jianhui deng, hong min, “ gis-based management and analysis system for landslide monitoring information” (in chinese), rock and soil mechanics, 11, p1739-1743, 2004. [5] ashis k. saha, ravi p. gupta, irene sarkar, manoj k. arora, elmar csaplovics, an approach for gis-based statistical landslide susceptibility zonation-with a case study in the himalayas, landslides 2, p61–69, 2005. [6] guirong zhang , kunlong yin, “ web database construction of geo-hazard information system in zhejiang province based on webgis“(in chinese), tile chinese journal of geological hazard and control,3, p114~118, 2005. [7] gui-rong zhang, kunlong yin , liu li-ling etc, “a real time regional geological hazard warning system in terms of webgis and rainfall “(in chinese), rock and soil mechanics,3,p1312~1317, 2005. [8] yiping wu, kunlong yin, “the landslide data management information system” (in chinese), hydrogeology and engineering geology, 1, p 14-16, 2007. [9] daan liu, xiaojia liu, “the development of geologic engineering monitoring information system” (in chinese), journal of engineering geology, 4, p351-356, 2007. [10] xinli hu, huiming tang, “ frame design of space prediction and eevaluation system for the slope geological hazard based on gis” (in chinese), geological science and technology information, 21, p99-103, 2002. 60 li: the key techniques study of constructing the roaming modelin virtual environment advances in systems science and applications (2013) vol.13 no.2 144-166 fashion erp systems and supply chain coordination wing-yan li, tsan-ming choi and pui-sze chow business division, institute of textiles and clothing, the hong kong polytechnic university, hung hom, kowloon, hong kong abstract the enterprise resource planning (erp) system is an enterprise information system designed to integrate and optimize the business processes and transactions in a corporation. an erp system enables an organization to integrate all the primary business processes in order to enhance efficiency and achieve a competitive edge. it also facilitates the smooth flow of functional information and practices across the supply chain. tal apparel limited is chosen to be the case study target and its cooperation with j.c. penney under vendor-managed inventory (vmi) scheme is also being investigated. we also investigate the two latest recently implemented modules in tal called fashion planning workbench (fpw) and supply chain order (sco), which relate closely to the company’s production planning and order arrangement. based on the case study results, and the literature review, we identify the critical successful factors in coordinating supply chains with erp systems which include: (i) assessing the fitness between organization and the target erp system, (ii) managing the cost and time of erp implementation (iii) arranging extensive education, training and rewards to employees, (iv) ensuring the internal processes are working properly and fully integrated before it is implemented in managing the supply chain, (v) building up a trusting and collaborative relationship and (vi) ensuring information visibility. finally, future research opportunities are explored. keywords erp systems, coordination, fashion planning workbench, supply chain order, fashion industry 1 introduction in fashion industry, requirements to reduce the product cost while meeting quality requirements and improving the time-to-market lead to many operational challenges and magnify competition among companies. fashion companies are hence encouraged to use advanced technologies to improve operational efficiency. the enterprise resource planning (erp) system is an enterprise information system designed to integrate and optimize the business processes and transactions in a corporation. an erp system enables an organization to integrate all the primary business processes in order to enhance efficiency and maintain a competitive position. it also facilitates the smooth flow of functional information and practices across the supply chain (sc). erp platform is specifically necessary in the fashion industry. this is because the fashion industry and its supply chains face a demand-driven market and it becomes upmost important to obtain the latest 145 advances in systems science and applications (2013) vol.13 no.2 market information and share the information among the channel members in order to facilitate the operation [1]. there is a considerable amount of literature reviewing the erp system implementation [2-21]. in this paper, we will focus on erp system implementation and its modular functionalities in a manufacturing company and how it can help achieve supply chain coordination. owing to data availability, tal apparel limited is chosen to be the case study target. the company has recently adopted two modules, namely: fashion planning workbench (fpw) and supply chain order (sco). these two latest applied modules concern about production planning and order arrangement, which are critical for fashion manufacturers to schedule their production, even the production cycle, and to be responsive to customer orders. with the case study and literature, we aim at investigating the following four research perspectives: 1. how do erp systems benefit fashion supply chain coordination and internal operations? 2. what are the possible issues that fashion companies may encounter in supply chain coordination and erp system implementation? 3. what are the critical successful factors for achieving supply chain coordination with erp system implementation? 4. what are the trends of erp system implementation and coordination? the organization of the rest of the paper is as follows. we conduct in section 2 a comprehensive review of the literature on the apparel supply chain characteristics, supply chain coordination, erp system in fashion industry and the challenges and successful factors in erp system implementation. then, we introduce the case study research methodology, review and discuss the system implementation project of tal in section 3. afterwards, we explore the above perspectives and discuss the finding from the literature and the case in section 4. finally, we have a conclusion and further research direction in section 5. 2 literature review 2.1 apparel supply chain characteristic in apparel supply chain, there are several characteristics. (i) as it is marketoriented, the brand-owners are usually the controller and have the power to influent other units in the supply chain [22]. (ii) it is a subcontracting supply chain in which the apparel retailers are usually big companies that place orders with manufacturers whereas those manufacturers make the product under a maketo-order production structure in order to lower the risk of building up obsolete inventories [23]. (iii) demand is uncertain and unpredictable. most products of the apparel industry are seasonal with short product life cycle while the demands are driven by ever-changing market trends, so it requires quick response and shorter lead-time [24] (iv) information flow is complicated as a buyer may connect wingyan li:fashion erp systems and supply chain coordination 146 to a number of manufacturers or the other way round [25], as work process in fashion industries are often highly interdependent and time sensitive. accordingly, the business value is co-created by all partners who provide different services [26]. 2.2 supply chain coordination there is a vast amount of literature about supply chain coordination. according to [27], coordination is a central lever of supply chain management. and it plays a key role in focusing on the innovation, flexibility, and speed that serve as the sources of competitive advantage necessary for survival in global competition [24,28]. simatupang and sridharan (2004)[29] state that two or more independent companies work jointly to plan to execute sc operations is with greater success than when acting in isolation. but in reality, the objectives and interests of different members along a supply chain are usually diversified and conflicting. therefore supply chain coordination becomes vitally important to achieve the all-level consensus, through which different members along a supply chain can react to market requirements in highly congruous ways [30]. under the coordination scenario in a two-echelon supply chain, achieving a win-win improvement situation can provide improved business success for both parties [31] and becomes a strategic response to the challenges that arise in the market [32]. the inter-firm coordination processes are characterized by effective communication, information exchange, partnering and performance monitoring [33]. and structures of coordination are related to the levels of information sharing and integration of physical flow from the perspective of operations management [34]. supply chain partners are working for joint planning, joint product development, mutual information exchange and integrated information systems, cross coordination on several levels in the companies on the network, long-term cooperation and fair sharing of risks and benefits [35]. lack of coordination occurs when decision makers have incomplete information or have incentives that are not compatible with system-wide objectives [36]; and it may result in poor performance of sc. the consequences of lack of coordination are: inaccurate forecasts, low capacity utilization, excessive inventory, inadequate customer service, low inventory turns, high inventory costs, slow time to market, as well as poor performance in order fulfillment response, quality, customer focus and customer satisfaction [37]. 2.3 erp systems in fashion industry in fashion industry, the product life-cycle is relatively short whilst the demand is uncertain. the emergence of fast fashion further leads to a more dynamic market with more frequently changing merchandising and purchasing decisions, and hence erp systems can play a crucial role [38]. many textiles and clothing 147 advances in systems science and applications (2013) vol.13 no.2 companies have employed the company-wide centralized erp system to improve information sharing and operations [39]. in the literature, the importance of erp system for fashion companies has been explored. au and ho (2002)[40] conduct a review study on how erp system enhances supply chain management and comment that erp system can support the fashion supply chains by allowing the implementation of e-commerce business models. erp system can also provide companies with flexibility and help solve many operational problems such as achieving more coordinated decisions on material resource planning (mrp) and capacity scheduling [41]. it allows effective planning of all resources in an organization. fin (2006)[42] investigates the relationship between electronic data interchange (edi) in apparel industry and the three performance levels: operational, financial and strategic, and finds that edi can help in reduction of lead time from several weeks to 3days. according to hodge (2002)[43], an erp system can be defined as an information system for identifying and planning the enterprise-wide resources needed to take, make, ship, and account for customer order. in general, erp systems used in textile and clothing industries are modular in structure and they can be tailored to the needs of individual companies, both large and small, with variable numbers and features of the modules. functionalities commonly covered in an erp system include finance, sales and distribution, production planning and scheduling, manufacturing and quality management, inventory management, global enterprise management, and customer relationship management[44]. for example, choi et al.(2012) [45] examine the implementation of the order-to-cash module in levi strauss & co. in chinahong kong, showing that a modular design allows erp implementation in different processes to fit the companys requirement with flexibility. 3 case study tal apparel ltd & j.c. penney 3.1 methodology in this paper, we follow the relevant literature and employ a case study methodology, supplemented by literature review, to investigate four perspectives and propose future research directions. this research focuses on a single, in-depth case study, based on the publicly available data from the company’s website, and several published articles on the target company. according to benbasat et al. (1987)[46], case study methodology examines a phenomenon in its natural setting, employing multiple methods of data collection to gather information. it provides a systematic way of looking at events, collecting data, analyzing information, and reporting the results. as a result, a sharpened understanding of why the instance happened as it did and what might become important to look at more extensively in future research is investigated and discussed [47]. this approach is deemed as wingyan li:fashion erp systems and supply chain coordination 148 appropriate for this piece of research because the topic is exploratory in nature and a number of insights and critical issues can be identified by examining the target case. 3.2 overview of target company founded in 1947 with headquarter in hong kong, tal apparel ltd (tal) is one of the world leaders in production of innovative clothes that combine style, comfort and functionality. it specializes in the manufacture of quality men’s and women’s garment for the world’s leading brands and creates one in every six dress shirts sold in the united states. (tal official website)[48] j. c. penney co.ltd. was founded in 1902 in kemmerer, wyoming by james cash penney and william henry mcmanus. today j.c. penney offers a range of family apparel, jewelry, shoes, accessories, and home furnishing products through a chain of department stores and its company website. headquartered in plano, tx, the company operates in the united states and puerto rico, with a total of 1102 department stores in 49 states. it receives merchandise from over 2,500 suppliers, both domestic and foreign. the company also operates buying and quality assurance inspection offices in 15 countries. (j.c. penney annual report 2011)[49]. 3.3 erp implementation in tal in 1990s, tal was one of those first companies that use a shop-floor system to manage apparel bundles on the factory floor. each bundle has a barcode and whenever a sewer finishes an operation, she scans the bundle and it’s tracked with the shop-floor systems. the system, named the apparel manufacturing management information systems (ammis), is still in use, though its database has been upgraded [50]. tal is known for using information technology to improve shop floor efficiency and process flow to differentiate itself from the competitors. lawson m3 enterprise management systems implementation in early 2002, tal implemented lawson m3 enterprise management systems (m3 stands for make, move and maintain) to manage its internal business process. tal invested us$ 12 to 15 million to develop and perfect the system in around 5 years starting from 2000 [51]. this system is provided by lawson software which is a global provider of enterprise software, services and support to customers primarily in three sectors: services, trade and manufacturing/distribution. lawson’s solutions include enterprise performance management, human capital management, supply chain management, enterprise resource planning, customer relationship management, manufacturing resource planning, enterprise asset management and industry-tailored application [52]. tal implemented m3 enterprise management systems with the following five modules: customer sales & service, enterprise asset management, supply chain management, manufacturing operation and finance management. table 1 summarizes their corre149 advances in systems science and applications (2013) vol.13 no.2 sponding functions and benefits (adapted from the infor official website). tal has been using the lawson m3 enterprise software for supply chain management, order management, capacity planning, production planning, inventory management, procurement, vendor-managed inventory, material requirements planning, and finance. then tal spent two years with lawson’s research and development team to co-develop two more modules in 2006, which are: fashion planning workbench (fpw) and supply chain order (sco). fashion planning workbench (fpw) in the past, tal used excel spreadsheets to plan the mediumand long-term capacity for individual factories, which took hours to download data from the erp system and is susceptible to human error. now the workbench tool allows executives in the company to see order capacity across the group and handle multi site planning in a more efficient manner. it takes only 30 minutes and that’s with 200,000 manufacturing orders in the system the company has 10 factories located in seven countries, and it can swap short orders from one factory to another one according to individual factory loads. with the workbench tool, tal can arrange six-month production planning [53]. supply chain order (sco) in traditional material requirements planning (mrp), demands from multiple customer orders were satisfied at an aggregate level. it did not tell which order was specifying the demand, and it lacked visibility. when tal needed to make changes in a customer order like size breakdown, it had to go though three more steps to make sure the changes were done. however in tal’s business where most things are made-to-order, materials are highly specific to particular orders, and often need to be uniquely linked, rather than grouped together as a whole, so tracking and having visibility to customers level is crucial. the sco module improves the way tal plans and manages order chains by linking orders to supply chain data. specifically, if tal changes the quantity of a customer’s order, all other related material requirement orders are taken care of. tal implemented sco at 10 of tal’s production sites, which help to save 80 percent of the time spent for revising related orders already in the system, while reducing inventory levels by approximately 30 percent. in short, fpw and sco help tal to smooth the production workflow and even out the production cycle, and a flat production cycle results in $3 millions saving a year. [54-55]. wingyan li:fashion erp systems and supply chain coordination 150 table 1 m3 enterprise management systems (adapted from infor official website)[56] modules functions and benefits customer sales & service • provides support for market development, sales and after-sales service processes • integrates web, mobile and traditional order management tools • supports sales channels using the same availability, pricing, and fulfillment controls enterprise asset management • integrates with other operational, planning and execution applications • synchronizes maintenance with production plan • increases asset reliability and improves operational capacity and output • enhances scheduling • reduces the need for emergency repair and maintenance cost supply chain management • controls and optimizes information, material and financial flows • drives efficiencies from planning to procurement, from warehouse to work-in-progress • provides functionality based on open standards • helps to capitalize on new mobile and data collection manufacturing operations • simplifies complex challenges with the capability to manage traceability, change control, and enhance manufacturing environments • provides a single source of product data and improves information availability • brings control and transparency to the manufacturing processes finance management • records operational transactions from other modules • handles accounting, budgeting, and helps consolidate all required reports and documents • coordinates multiple units in terms of the financial requirements • detects critical trends effectively and in a timely manner continuous upgrade of the erp system in tal afterwards, tal continued to launch its system upgrade in hong kong, mainland china, thailand, malaysia, indonesia and vietnam in 2009 and the upgrade project was complete in january 2010, on time and on budget. the upgraded system helps tal improve productivity across its global operations, by helping users personalize their information workspace and bring together enterprise applications, business intelligence, desktop tools, and group collaboration. thus it helps tal to work in a more productive and dynamic manner [52]. 151 advances in systems science and applications (2013) vol.13 no.2 3.4 vmi scheme implementation between tal and j.c. penney traditionally, information systems are designed such that they are not accessible to exterior companies. there are organizational boundaries isolated a company’s system from the vendors and customers. useful information like sales data, inventory level and order status cannot reach it partners along the supply chain effectively. this would result in longer lead time and lower flexibility. under vmi scheme with tal, j.c. penney uses edi over a third-party network. there is both centralized and non-centralized ordering, the stores place orders directly to supplier, and many items are handled through automated replenishment system. edi can also be used to transmit pos-data. tal collects point-of-sales data of j.c. penney’s 1,102 stores. the data is fed into tal’s demand forecasting model yielding model stock of stock keeping units (skus) for individual stores. tal then decides how many shirts to be made in which color and size[57]. this entire program is designed and operated by tal. the model stock and sku information is now dynamically linked to tal’s factory floor, so the factories promptly know the detailed orders and then deliver the shirts directly to individual stores, using the technology of x-docking. although forecasts are never completely accurate, tal’s collaborative information and fast manufacturing capabilities enable the company to react quickly to changes. if necessary, tal can yield shirts ready for shipment in just four hours. according to [58], j.c. jenney is doing something beyond vmi called supplier managed availability (sma), which is an extension of vmi. there are no minimum or maximum inventory level requirements placed on the supplier whereas the supplier evaluates solely on product availability in the store when the product is needed. depending on the urgency of the need at a particular store, the supplier may choose to ship the product much more rapidly by air (express).tal’s coordinated power makes the chain operating in a more efficient mode. under vmi scheme, both the manufacturer and the retailer are benefited. j.c. penney can respond more quickly to consumer demand and it has shown sales increases by 20-25% and improvement in inventory turnover by 30% [59]. this form of collaboration completely eliminates warehouse inventory thus bypassing warehouse handling, improved fill rates, and increased customer satisfaction. the direct-ship processes have helped j.c. penney save roughly $30 million of the average monthly inventory expenses [60]. on the other hand, the main advantage of the vmi scheme for the manufacturer is the mitigation of demand uncertainty; it monitors the pos data and determines the most cost-effective way to meet the projected availability requirements, enhancing tal’s scheduling and planning, allowing a specified service level to be maintained at minimal inventory and production cost. tal claims that the vmi program can result in costs savings of up to 15% through reduced wingyan li:fashion erp systems and supply chain coordination 152 inventory and operational costs. (tal official website)[48]. close collaboration with j.c. jenney allows tal to design products for it. when tal’s design teams in new york and dallas come up with a new style, within a month its factories can churn out 100,000 new shirts and offer for sale in 50 penney stores [36]. as tal participates in the whole process including design, production, marketing and distribution, it can be more flexible and accurate to response to the consumer market, resulting in greater sales and fewer closeouts. 4 discussions 4.1 how do erp systems benefit fashion supply chain coordination and internal operation? allowing information accessibility among supply chain partners erp system allows integration and coordination, supports information flow and activities both within a company and along supply chain, provides reliable services to make information flow smoothly amongst all partners and ensures that any partner can get information timely and accurately when in need [25]. under the vendor managed inventory scheme between tal and j.c. jenney, tal (the manufacturer) can access to j.c. penney (the retailer)’s system to obtain point-of-sale data so that tal can have a clearer picture about the real market demand. this successfully reduces the forrester’s bullwhip effect and results in just-in-time inventory and delivery, quick response, collaborative planning and forecasting in their supply chain. also the two members maintain the updated single forecasting process in the system by sharing information related to future demand, resulting in substantial cost reduction. developing a successful synchronization strategy in the case of tal and j.c. penney, sophisticated skills in the deployment and use of technologies such as edi, advanced planning and scheduling (aps), forecasting software, and enterprise resource planning (erp) help the manufacturer in synchronizing with its retailers for information sharing, inventory and logistics management. information sharing helps supply chain partners build up trust and responsibility interdependence. the stronger the responsibility interdependence between supply chain partners, the more they need to become involved in sharing private information, learning how to open another’s viewpoint, as well as resolving shared problems [29]. thus, it develops a differential compatible advantage and creates higher switching costs for retailers and substantial entry barriers for competitors. in addition, most commodity manufacturers do not have the sophistication in a complex partnership between supplier and customer [61]. identifying mediumto long-term production planning tal has been using the lawson m3 enterprise management system in its factories in seven countries to help minimize production peaks and valleys. to achieve 153 advances in systems science and applications (2013) vol.13 no.2 that, it needs to match production capacities across all factories with orders that it expects six to nine months in advance. tal balances production load across its multiple sites and assists with planning at the style/color level. this gives tal the ability to better plan capacity across the company’s various production sites and simulate load based on possible operational and business changes [53]. thus, tal is better positioned to manage seasonal peaks and valleys for production. achieving a higher visibility in material requirements planning erp system helps tal plan and manage order chains by linking orders with supply chain data. this helps the company managing issues related to changes along the supply chain as demand and supply information are all in one single view. this allows improvement of delivery accuracy, reduction in inventories, and manual work elimination. with more immediate access to comprehensive integrated business information, tal has cut its reporting time by 50 percent. besides, improved demand management with relevant information consolidated to a single database also allows the company for better customer service using real-time information, and increase customer responsiveness [50]. 4.2 what are the possible issues that fashion companies may encounter in supply chain coordination and erp system implementation? a trade-off between autonomy and control collaboration implies visibility of internal activities and metrics by external parties. the adoption of an integrated approach throughout the supply chain requires a trade-off between autonomy and control between each supply partner relationship [62]. in the case of tal and j.c. penney, j.c. penney shares packing, shipping, inventory and product movement with tal and let tal control its inventory level. it is actually giving an important function to supply chain partners when outsourcing the inventory management. many companies in virtual integration may not be willing to allow partners to view their systems and processes as their ability to perform will become more transparent, which may create pressure on themselves, especially for small companies. therefore, this becomes an organizational challenge to reach an acceptable balance among autonomy and control between supply chain partners. conflict in inter-organization relationships with effective coordination, partners should work together to reach a mutual goal. however, such approach will be difficult for many firms to take on board as they have focused on squeezing margins with their suppliers rather than cooperating with them [63]. conflict in inter-organization relationships refers to the disagreements that occur in cooperation relationship or the incompatibility of activities, shared resources and goals between partners [64]. and this is impediment to effective inter-organization information sharing. if the supply chain partners do not cooperate with each others, this would affect the accuracy of wingyan li:fashion erp systems and supply chain coordination 154 information being shared, for example, the required lead time provided by the manufacturers and the expected due date by the retailers. power asymmetry according to hingley (2005)[65], inter-organizational relationships emphasize the necessity for symmetry and mutuality and that symmetric dependence structures foster longer-term relationships while asymmetric relationships are associated with less stability and more conflict. under vmi, we can observe a significant shift in power from the retailer to the manufacturer since it is now the manufacturer who manages the inventory. in the case of tal and j.c. penney, they are giant manufacturer and retailer respectively with high bargaining power; this leads them to become inter-dependant and work together under fair terms. tal is willing to design and operate entire program for vmi and j.c. penney is willing to share pos data and design work with tal. but for other manufacturers in the apparel industry that are small or medium in scale and lack resources and bargaining power in the supply chain, the cost involved in implementing new it systems is high. those small manufacturers will be worried about an unequal distribution of costs and benefits between the various members of the supply chain [26]. these will obviously hinder smaller firms to share their information. operation reengineering shifts the management power implementing an erp system may force the reengineering of key business processes or developing new business processes to support the organizations goals [66]. and redesigned processes require corresponding realignment in organizational control to sustain the effectiveness of the reengineering efforts. and this kind of alignment may change the functional areas and many social systems within the organization. the resulting changes may significantly affect organizational structures, policies, processes, and employees. according to burca (2005)[63], implementation of erp system changes the career prospects and aspirations of many people in the company. power was perceived to move strongly in the direction of people who had embraced the it agenda. as a result, management may feel that there is a change, and want to resist it. this would definitely affect erp implementation. 4.3 what are the critical successful factors for achieving supply chain coordination with erp system implementation? assessing the fit between organization and the target erp system according to swan et al. (1999)[67], organizational misfits of erp exist due to the conflicting interests of the user organization and the erp vendor. in order to have successful erp implementation, erp implementation managers as well as top management of the organization should be able to assess the fit between their organization and the target erp system before its adoption, identify gaps between the erp generic functionality and the specific organizational require155 advances in systems science and applications (2013) vol.13 no.2 ment, and then decide how these gaps will be handled [68]. after erp system implementation, the impact of erp and process adaptations should be well measured and managed so as to minimize the potential business disruptions and user resistance. in the case of tal’s erp implementation, tal spent two years with lawson’s research and development team co-developing new erp modules (the fashion planning workbench and supply chain order systems) to ensure the modules fit tal and work efficiently. managing the cost and time of erp implementation erp systems come in modular fashion and do not have to be implemented entirely at once as the length and cost of implementation is affected to a great extent by the number of modules being implemented and the scope of the implementation [69]. the adopting company is suggested to follow a phase-in approach in which few modules are implemented at a time. besides, the implementation costs increase with the degree of customization. therefore, the company should have a thoughtful plan in order to balance customization and cost, keep the project plan aggressive, but achievable and flexible. in the case of tal’s erp system implementation, tal first adopted 5 modules in 2002; after the system had been run smoothly for few years, it added two more new modules in 2006 and eventually upgraded the erp systems in 2009 on time and on budget. gradual implementation and modification of erp system can also lower user resistance and the risk that may encounter. arranging extensive education, training and rewards to employees education and training is probably the most widely recognized critical success factor for erp system implementation, because user understanding is essential. erp implementation requires a critical mass of knowledge to enable people to solve problems within the framework of the system. if employees do not understand how the system works, they will invent their own processes using those parts of the system they are able to manipulate [70]. top management must allocate adequate expense on education and end-user training and incorporate such expense as part of the erp budget. according to volwer (1999)[71], it is suggested that reserving 10-15% of the total erp implementation budget for training will give an organization an 80% chance of implementation success. once the selected employees are trained after investing a huge sum of money, it is a challenge to retain them; especially the market is seeking for skilled erp system consultants. employees could double or triple their salaries by accepting other positions [69]. it is recommended the company should provide attractive retention strategies such as bonus programs, company perks, salary increases, continual training and education, and appeals to company loyalty. ensuring internal processes work properly under vmi, the manufacturer receives sales data from the retailer via edi and wingyan li:fashion erp systems and supply chain coordination 156 the information will be transmitted to the manufacturer’s erp system for inventory planning, scheduling and production. the manufacturer should ensure its internal processes integrated properly so that the orders can link up with other supply chain data. norris et al. (2001)[72] have stressed that before an organization can participate in e-supply chain management (e-scm) it needs to ensure that its internal processes are fully integrated. the assertion is that processes that span the supply chain cannot work if the internal processes do not work correctly in the first place. building up trusting and collaborative relationships all parties involved must view collaboration as a strategic asset and an operational priority in order to foster trust among trading partners. organizations wishing to extend their processes will have to develop more trusting and collaborative relationships with their business partners. all parties need to recognize that success for one part of the supply chain means success for all and all improvements made to the operation of the supply chain will ultimately benefit all member firms [73]. smooth communication is the foundation for supply chain management. frequent communication contributes to faster problem resolution, trust, and relationship building as well as quicker decision-making and those are resulted from having access to up-to-date information. moreover, willingness to share information enhances the quality and relevance of the information that is shared. for example, sharing actual sales data combined with rolling forecasts helps improving mutual supply chain decision making. in addition, willingness to share future product strategies and technology plans rather than simply sharing forecasting data allows more cooperation and integration. ensuring information visibility good supply chain management is essential for a successful company. supply chain management aims at going beyond the boundary of a single company. this also applies the corresponding use of information. in order to have efficient and effective supply chain operations, information visibility, such as those on inventory level and transportation schedule, is critical. regarding information visibility across the supply chain, it should be managed with strict policies, disciplines and monitoring [72]. allowing all partners in the supply chain to view and dynamically manage both demand and capacity data raises opportunities for the simultaneous improvement in customer service levels and reduction in overall inventory levels and associated costs [74]. partners in virtual integration need to be willing to share information with each other in order for the end-to-end process to work correctly and organizations also need to understand the implications of integration across the entire supply chain [75]. according to kelle and akbulut (2005)[76], if one of the supply chain partners forces its optimal policy on the 157 advances in systems science and applications (2013) vol.13 no.2 other partner, the total operating cost of the system can be much higher than the case with a coordinated ordering/setup and shipment policy. so the potential for cost reduction in policy coordination motivates sharing of information, too. 4.4 what are the trends of erp system implementation and coordination? intensive cooperation between the organization and its erp vendor competing in a dynamic environment and meeting global challenges requires agility. successful companies must be able to respond quickly and cost-effectively to change. the change could be of any type, for example shifting in customer demands and supply chain partners, modifications of a business model, business process or business expansion [77]. organizations need to convert their business into responsive, demand-driven, profit-making enterprises by optimizing their operations, so a company with erp system would be more active in cooperating with its erp vendor to create new modules that can increase operational effectiveness and market responsiveness. this phenomenon also appears in the case of tal, where tal actually teams up with its erp vendor to co-create erp modules for effective operation and coordination. the need of extended information system applications and technology from the case of tal we can see the erp system should be monitored and upgraded continually to enhance its functions and efficiency, so as to increase the company’s competitiveness. according to sleeper (2004)[78], only 35% of the organizations are satisfied with the erp they use at the moment, and the main reason for dissatisfaction is that the software does not map well with the business goals. a major problem with existing erps is the “misfit” between delivered functionality and needed functionality, described as a gap between the processes the erp supports and the processes the organizations work by. thus, there is an increasing interest among vendors to improve future erp-systems [77], and further look into how to identify and present business requirements for the future erp. this creates the need for extended information system applications and technology (e.g. erp ii, soa, web 2.0 or software as a service saas, etc.). staff (2006)[79] describes mysap as the next generation of erps, and says that this software is fast, flexible, and an efficient foundation offering organizations new functions, greater productivity, and integrated analytical insights into business processes which increase the flexibility to design the software so that it meets customer requirements. this is described as role-specific development aiming at increasing employee productivity by supporting them with a tool tailored to their specific work tasks. amongst companies that are relatively satisfied with their existing erp operations, many of them are now considering extension of the functionalities provided by the original erp systems (i.e. erp ii) toward e-business, supply chain management (scm), customer relationship management, supplier relationship wingyan li:fashion erp systems and supply chain coordination 158 management, business intelligence, and manufacturing execution systems, etc. according to tenkorang et al. (2011)[80], erp ii aggregates and manages the data surrounding all the transactions of an enterprise as accurately as possible in real time. it also facilitates opening up of the system to make information available to trading partners in the supply chain. 5 conclusion and future research opportunities through the study of the talj.c. penney case and the reviewed literature, we have demonstrated how the erp system can improve business performance and organization in tal and how the system helps coordination between the retailer and the manufacturer. erp allows useful information to be integrated and flowed effectively within the organization and across the supply chain, which is important to uncertain and demand-driven fashion industry. the latest implemented modules – fashion planning workbench (fpw) and supply chain order (sco) – help tal to smooth the production workflow and even out the production cycle. besides, information sharing between tal and j.c. penney benefit in (i) maintaining optimal stock level, (ii) reducing inventory and operational cost, (iii) enhancing manufacturers’ scheduling and planning and (iv) building up close relationship between manufacturer and retailer. combining the findings from the case study and the literature review, below is the summary of what we have discussed in this paper. 5.1 the benefit of erp systems in fashion supply chain coordination and internal operations four major benefits in fashion supply chain coordination and internal operation are identified: (i) it allows information accessible among supply chain partners, providing reliable services to make information flow smoothly between all partners and get information from partners timely and accurately when partner needs. (ii) it develops a successful synchronization strategy, creating a differential compatible advantage and creates higher switching costs for supply chain partners. (iii) it helps the manufacturers to identify mediumto long-term production planning. (iv) it helps achieving a higher visibility in material requirements planning by linking orders with supply chain data. 5.2 the possible issues that fashion companies may encounter in supply chain coordination and erp system implementation from the case of tal and j.c. penney, we reveal the possible issues in supply chain coordination and erp system implementation, namely: (i) the trade-off between autonomy and control, (ii) conflict in inter-organization relationships, (iii) power asymmetry between the supply chain partners, and (iv) the management power shifted though operation reengineering. 159 advances in systems science and applications (2013) vol.13 no.2 5.3 the critical successful factors for achieving supply chain coordination with erp system implementation in order to have a successful erp systems implementation and make sure it can work effectively in supply chain coordination, the following six factors should be considered: the company should (i) assess the fitness between organization and the target erp system, (ii) manage the cost and time of erp implementation, (iii) arrange extensive education, training and rewards to employees, (iv) ensure the internal processes are working properly and fully integrated before it spans the supply chain. (v) the supply chain partners should build up a trusting and collaborative relationship and have better communication. (vi) supply chain participants should ensure information visibility, allowing all partners in the supply chain to view and dynamically manage both demand and capacity data. 5.4 the trends of erp system implementation and coordination according to the case study and the literature review, we have observed two trends of erp system implementation and coordination. (i) intense cooperation between organization and its erp vendors: companies with erp system installed would be more active in cooperating with their erp vendors to create new modules that can increase operational effectiveness and market responsiveness. (ii) the need for extended information system applications and technology: there are many companies which have implemented erp systems and are relatively satisfied with their operations are now considering the extension of the functionalities provided by the original erp systems. 5.5 the challenges and further research opportunity in erp system implementation mismatch between existing and new erp system as discussed in section 4, there are some new versions of erp systems developed in order to handle the problem of mis-fit between erp system and organizational functionality. however according to kumar and van hillegerberg (2000)[12], this creates new problems in the migration between different versions, and either it could be that the new version is not backward compatible or it could be that the customer organization has made modifications of the erp, and these modifications do not automatically adjust to the new version of the erp. thus, there is a need to examine further how companies should extend their erp systems and the factors that should be considered when extending erp systems. small and medium-sized firms hardly develop a costly erp system companies that invest in technology can improve overall efficiency and effectiveness and have a competitive advantage. but the implementation cost of those technology systems is high, only large companies are able to invest in these big projects to improve their performance whilst small and medium-sized firms hardly develop such a system on their own. besides, not many small companies are taking a proactive approach to implementing erp or even sharing information. wingyan li:fashion erp systems and supply chain coordination 160 the usual reason is they tend to rely on the industrial giants to offer help (e.g. teaming up with big suppliers such as tal can help enhance their own supply chains). then it may come to another problem mentioned by burca (2005)[63]: those small companies may have a distrust of passing confidential information to customers and suppliers as this may indicate how well they are performing. further study in this area may indicate how small and medium-sized firms can put practices in place to ensure that collaboration takes place. references [1] fernie, j. 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(2011). “enterprise resource planning (erp): a review literature report”, proceedings of the world congress on engineering and computer science, vol.2, pp.19-21. corresponding author author can be contracted at jason.choi@polyu.edu.hk. microsoft word 11 x.h.yang，c.h. li--design and simulation for conjugate cam mechanism.doc 286-293 advances in systems science and applications (2011), vol.11, no.3-4 issn 1078-6236 international institute for general systems studies, inc design and simulation for conjugate cam mechanism x.h. yang and c.h. li school of mechanical engineering, shandong university of technology, zibo , 255049, china abstract fuzzy synthetically appraising theory is put forward and to be used in the optimal selection of the law-of-cam motion. and its model is also established. an example has been given to illustrate the validity of this method. by compiling program about the optimal design and the simulation system, the motion and process simulation of the conjugate cams mechanism are realized. so the system can improve the efficiency, reduce the cost and solve the precision problem of high speed and over-loading about cam segmentation. keywords conjugate cam mechanism fuzzy evaluation simulation software 1.introduction the conjugate cam mechanism has many excellences, including high speeding and overloading, exact and effective control, steady capability and so on. it is a part of intermission and displacement mechanism, atc equipment and walking equipment, which is widely used in wrapping machine, agricultural machine, printed machine and wiring assemble line and so on. meanwhile, there exist a lot of faults. for example, there are complicated design processes, uneasy to ensure design quality, difficult to manufacture, to depend on import product for domestic need of high speeding and over-loading conjugate cam [1-4]. the cam’s dynamic capability can be effectively improved by fuzzy synthetically appraising theory and optimal selection the cam’s contour line. some problems appeared during the process of design can be earlier solved by combining with advanced virtual design and manufacture technology which can shorten the cycle of design, improve the precision of design and effectively enhance the conjugate cam’s design and the manufacture level. 2.fuzzy synthetically selection of the-law-cam mation it is important to select the-law-follower in the process of conjugate cam design. according to some fuzzy factors including the cam’s working condition, economy, precision, and kinematics and dynamics characteristics of the motion law, meanwhile, synthetically considering a few relevant factors influenced the law-of-follower, the motion capability of conjugate cam can be effectively improved by fuzzy synthetically appraising theory. during the process of confirming the motion law, both loading and speed must be satisfied. so, we regard the two factors as sub-factors of the basic requirements[5-8]. other factors include working conditions, economics, and precision and so on. its secondary appraising rational tree, such as fig. 1: the process to choose the optimal motion law of the cam’s followers by fuzzy synthetically appraising as follows: a. the motion law-of-follower deduced by positive sequence is known as spare set. it can be denoted by the following set fundatin of shan dong and sdut ( no y2008f02、2005kjm04) { }1 11 12 1, , , mv v v v= ⋅⋅⋅ advances in systems science and applications (2011), vol.11, no.3-4 287 fig.1. secondary appraising tree of connection b. the necessarily considered factors such as the maximum speed mv , the maximum acceleration ma , extracted the root-mean-square of speed rmsa , and move-loading torque characteristic value ( )m av and so on, are known as key factors, denoted by iu { }1 11 12 1, , , nu u u u= ⋅⋅⋅ c. as a result of different design requirement and the use spot, the importance of key factors is different. so, a fuzzy subset is established by analysing every key factor. { } ( )1 11 12 13 1 1, , , ,0 1, 1,2, ,n iw w w w w w i n= ⋅⋅⋅ ≤ ≤ = ⋅⋅⋅ in the set, iw1 is important degree coefficient of iu , which is known as weight coefficient. d. according to ( )1,2, ,iu i n= ⋅⋅⋅ , satisfactory subject degree of every element in the spare set is confirmed., forming the following matrix: 111 112 121 122 1 1 ~ 1 1 1 2 1 ... ... ,0 1 ... ... ... ij n n nm r r r r r r r r r ⎡ ⎤ ⎢ ⎥ ⎢ ⎥= ≤ ≤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ it denotes the subject degree from element jv to factor iju in the spare set. e. the choice of motion law ( ) ( )1m1211 1nm1n21n1 12m122121 11m112111 11211 11 b,...,b,b r...rr ............ r...rr r...rr ,...,, = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅= ⋅= nwww rwb hereinto, ( )1 1 1 1 1, 2,..., , n k j jk j b w r k m = = ⋅ =∑ jv corresponding to 1max kb , jv is the best choice.the flow chart of synthetically choice is denoted as fig. 2. whole requirement precision economyworking condition basic requirement speedload 288 yang: design and simulation for conjugate cam mechanism f. example ①requirement of use: middle speed, light-load, low noise. condition choice condition analysis, pick-up the spare set pick-up the weight a>− compute ija aa/a ij >− pick-up the parameters of characteristic curves compute ijr ( ) rd/f,f0.1 ijimax >− rab1 ×= whether to choose the other requirements 1bb = pick-up the weight c>− compute c cc/cij >− rcb2 ×= pick-up the secondary weight ( )21, www =>− 2211 b*wb*b +=×= wbw for i=0 to 5 extract ( )ibmax , output the relevant motion law end y n fig.2 the flow chart of the cam’s followers’ motion law confirmed by fuzzy synthetically theory advances in systems science and applications (2011), vol.11, no.3-4 289 ②spare set: { }1 11 12 13 14 15, , , ,v v v v v v= = {sine, amending constant speed, amending trapezoid, amending sine, equal addition and minus} ③characteristic values of every curve are regarded as key factors: ( ){ }1 , , , , ,m m rms m mm u v a av a jτ= because of the higher speed, ma and mj should be lower. so, its weight coefficient should be a little bigger; because of the light-load, weight coefficient of mv and ( )m av can be a little lower; because of the lower noise, weight coefficient of mτ should be proper bigger. synthetically considering various requirements, the right weight modulus can be taken as: { } { } 1 11 12 13 14 15 16, , , , , 0.2,1,0.1,0.5,0.6,0.8 w w w w w w w= = after computing every percentage of their weights: { }0.0625,0.3125,0.03125,0.15625,0.1875,0.25w = ④characteristic values of various motion law, such as tab.1: tab.1 characteristic values of various motion law subject degree can be formed with ( )max0.1 /ij i ijr f f d= + − ( ) ( )max min / 1 0.1i id f f= − − . the matrix can be formed: ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 082.011.057.0 083.088.01.01 85.0145.084.01.0 15.0113.092.01.0 166.08.01.05.0 1.04.01.011.0 r1 ⑤adopting synthetic appraise model of weighted average, namely: 1 , 1, 2,..., m j j ij j b w r i m = = ⋅ =∑ that: ( )1 0.511,0.298,0.746,0.779,0.456b w r= ⋅ = hereinto, the amending sine curve is the best choice which is corresponding to max 0.779,jb = the following curve is amending trapezoid curve (corresponding to 0.746jb = ), sequenc e number curve’s name mv ma ( )m am rmsa mj mτ 1 2 3 4 5 amending equal speed equal addition and minus amending trapezoid sine the mending sine 1.275 2.0 2.0 2.0 1.76 8.013 4.0 4.888 6.238 5.528 5.671 8.0 8.048 8.126 5.435 4.001 4.0 4.232 4.443 3.908 201.4 ∞ 61.43 39.48 69.47 63.3 — 26.71 44.14 34.17 290 yang: design and simulation for conjugate cam mechanism sine, equal addition and minus, amending constant speed. known by experience, amending sine curve among several curves has the strongest use, the better capability/performance. the method’s accuracy and feasibility are validated. the amending trapezoid is in the second place. 3.motion simulation for conjugate cam because contour curve of cam is formed by multiple curves, the correct contour curve of cam will not be gained if the input cam’s parameters and motion law aren’t proper (fig. 3). in order to avoid manufacturing the unqualified cam, the cam’s rationality and accuracy must be detected by motion simulation after gaining the relevant datum of designing cam model. we pick up the above data of the design model and change them into date format which can be recognized by graph software autocad and ug. the cam’s accurate contour graph and threedimensional dynamic simulation model can be automatically built in the graph environment. in the whole process of design, no only the relevant datum can be disposed of; but the visualization of designing results can be achieved by motion simulation. the designer can amend design parameters at any moment. so, the optimal design scheme can be gained, the ratio of waster during the practical process can be reduced, meanwhile, the cost can be reduced too. fig.3. the error cam’s contour curve the flow of simulation analysis and numerical controllable programming module is mainly divided into two parts: first, implement graph simulation in autocad2006 by compiling data interface files (camshape.scr) on the platform of vc++6.0, we can extremely intuitively judge whether contour is distortion or comes to a point. meanwhile, we can compile the interface files that numerical controllable line incision machine tool needs by autolisp language. fig.4. the flow chart of simulation and program model second, implement three-dimensional graph simulation and numerical controllable programming in ug nx3.0 by compiling data interface files (camshape.scr) on the platform of advances in systems science and applications (2011), vol.11, no.3-4 291 vc++6.0, simulate the processing movement of the cutting tool and cutting situation of the workpieces by using the ug’s formidable processing module. examine whether there occurs over-cutting or interference collision in order to validate the accuracy and the rationality of the procedure in front of the post-processing and make the prompt revision, thus guarantee accuracy of numerical controllable programming of cam’s contour. its flow chart of simulation and program model is denoted as fig. 4. the motion simulation is denoted as fig. 5. fig.5 the three-dimensional motion simulation of conjugate cam 4.machining simulation for conjugate cam for the regular cam tested by motion simulation model, we pick up the data which program needs from the graphics software and send data files to the automatic program model, and meanwhile, we choose the type of program, then, the numerical control line incision 3b or machining g programming can be realized according to practical needs. and the accuracy can be detected by machining simulation. if there were some problems, the program would be compiled over again. by the procedure of drawing data in autocad, the cam’s counter data can be gained and changed into 3 b line cutting codes such as fig. 6. and the knife track codes such as fig. 7; the machining simulation is denoted as fig. 8. fig.6. the 3 b line cutting codes 292 yang: design and simulation for conjugate cam mechanism fig.7. the knife track codes at last, by the program’s transferring model, send the correct program to the cam machining tool and complete the cam’s machining. to do so, the design, simulation and manufacture are integrated; it can share the datum and enhance the work efficiency. fig.8. the machining simulation of conjugate cam 5.conclusions according to some fuzzy factors including the cam’s working condition, economy, precision, and kinematics and dynamics characteristics of the motion law, the fuzzy optimal design for the conjugate cam is realised by applying the fuzzy synthetically appraising theory. and its model is also established. meanwhile, program the procedure of the fuzzy optimal design and simulation system. by combining existing advanced graphic simulation software, the design, simulation and manufacture are integrated. therefore, 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[8] long-long wu, shiau-huei wu, hong-sen yan. simplified graphical determination of diskcam curvature. mechanism and machine theory. 34(1999)1023-1036. advances in systems science and application (2015) vol.15 no.4 384-391 the system of parameters efficiency of financial supervision tatiana v. yalyalieva and elena a. murzina volga state university of technology, department of management and law, yoshkar-ola, russia abstract the solution for the problem of monitoring the effectiveness of the process depends on how timely and adequate state control bodies will react to changes in the external environment and the needs of society. the modern system of government performance management control is another step for russia on its way to enter the european and world space. the presented results of the study can be applied to the government’s management and control. parameters of the effectiveness of controls offered in this article have been used by state bodies of the republic of mari el, russia, while working on the program of innovative development of the region. it is planned to implement the results of future research in the russian practice of governance based on a long-term program of cooperation volga state university of technology and government bodies of the republic of mari el, russia. the article suggests methods for assessing the efficiency of management aimed at public economic supervision, both in a municipality and in a region as a whole. based on the data obtained from the method, a mechanism has been developed to improve the management efficiency. keywords governmental control; economic institutions; legal regulation; effectiveness of institutions. 1 introduction governmental control is performed by officials representing the authorized bodies, within their competences, by collecting explanations, checking the accounting and reporting information, inspecting the premises and territories used to gain income (profit), and by using other instruments provided by the law. one of the most significant areas of governmental control is financial control. financial control is traditionally considered as a part of the governmental control system which covers the sphere of formation, distribution and use of the monetary funds. financial control is control of the legality and appropriateness of the activities performed in the sphere of formation, distribution and use of the monetary funds of the state and municipalities, which is aimed at efficient social and economic development of the country and its regions. representing a constituent part of the economic management, financial control ensures the interconnection between companies and the governmental bodies that have the corresponding rights and authorities. advances in systems science and application (2015) vol.15 no.4 385 calculation of the payments additionally charged as resulting from the controlling activity: components represented in the standardized manner. based on the calculation results, we build a trend line to find the expected amount of the payments in the future. as a result of this work, we have carried out a comprehensive study of the theoretical bases of the governmental control. formation of a certain type of market structures, may, on the whole, be regarded as a response to excessive transaction costs. at this point, integration processes in russia are characterized by aggregation and consolidation of enterprises. this implicitly proves the desire for diversification and economic control of enterprise risks associated with imperfect institutional environment and excessive transaction costs. it should be noted that control over efficiency of costs or any other category may be an extremely complicated and expensive procedure. an example would be the control which the government exercises over taxpayers since it has to maintain the costly tax-collection apparatus. 2 research methodology the calculation method is suitable for the remaining efficiency types, as it allows calculating the total efficiency indicator for the structural components, which are multidirectional indicators expressed in different measures, so the total indicator allows bringing all of them to a unified scale and making their convolution. however, the normalization procedure requires either a data array for several municipalities or information available for a specific period. some issues related to the improvement of the governmental control efficiency were also examined in the works by shemyakina [1]. however, we find it possible to continue studying the subject. certain theses were already mentioned in some of the works by the authors of this article [2–4]. we are going to proceed with the analysis here. the suggested system contains three major blocks of criteria: 1) general efficiency assessment of the controlling activity; 2) assessment of the parameters related to organization and implementation of field audits; 3) assessment of the parameters related to organization and implementation of office audits. to make the analysis and develop appropriate efficiency assessment methods for the work done by governmental bodies, we used the information provided by the governmental bodies of the mari el republic, and by the taxation authorities in particular. the efficiency was analyzed by the major parameters of their activities. here are the most popular coefficients and criteria, which can be divided into three group of parameters: a) general assessment of the controlling activity performed by a taxation authority; b) assessment of the organization and implementation of office tax audits; c) assessment of the organization and implementation of field tax audits. this system of parameters includes: 386tatiana v. yalyalieva and elena a. murzina:the system of parameters efficiency of financial ... a. dynamics of the payments additionally charged as resulting from the controlling activity: ∆a = at + 1−at, where a stands for the amount of the payments additionally charged as resulting from the controlling activity, and t is one year. this value represents the comparison of the amounts of the payments additionally charged for several reported periods, and any positive dynamics would mean improvement of the controlling activity efficiency. b. specific share of the payments additionally charged as resulting from the controlling activity in the total amount of the incoming payments: sadd = a p × 100, where a stands for the amount of the payments additionally charged as resulting from the controlling activity, and p is the total amount of the incoming payments. this value is calculated as the ratio of the amount of the additionally charged payments to the total amount of the payments charged. c. specific share of the payments recovered as resulting from the controlling activity in the total amount of the additionally charged payments: srec = r a × 100, where a stands for the amount of the payments additionally charged as resulting from the controlling activity, and r is the total amount of the payments recovered to the budget system of russia as resulting from the tax audit. this value is calculated as the ratio of the amount of the payments recovered as resulting from the controlling activity to the amount of the payments additionally charged as resulting from the tax audits. it shows what share of the payments additionally charged came to the budget system. d. general effectiveness coefficient: ceff = r c × 100, where r stands for the amount of the payments recovered to the budget system of russia as resulting from the tax audit, and c is the costs of the activities performed by the taxation authority. this value is calculated as the ratio of the amount recovered from the payments additionally charged as resulting from the tax audit to the amount of the costs advances in systems science and application (2015) vol.15 no.4 387 of the activities performed by the taxation authority. it shows the effectiveness of the costs required for the recovery. e. legality coefficient of the extra charges resulting from tax audits: clegal = a−d a × 100, where d stands for the deduction amount of the payments additionally charged as resulting from the decisions of judicial and superior bodies, and a is the amount of the payments additionally charged as resulting from the controlling activity. this value represents the judicial acknowledgement of the legality of the fines and penalties imposed. the ideal value is 100%. 3 findings and discussion in scientific and journalistic literature, governmental control is defined in the broad sense as a special method of ensuring legality. its key tasks are ensuring that all the taxes and duties stipulated by the law are duly paid to the budgets of various levels and preventing tax evasion. the study of the institutional environment of a territory as an evolving endogenous factor was initiated by the representatives of the school of economics of washington university in the 1970s coase [5]. douglass north uses the institutional environment term to define the institutional limits which exist at the macrolevel and determine the possible conditions of contractual agreements between individuals north [6]. the aggregate social and economic effect produced through implementation of governmental programs, which is achieved by applying structural modifications to the economy, can be used as the efficiency indicator feldman [7]. the multi-subject composition of the parties of the contract relations within a cluster can be classified according to the subject composition of the triple spiral model, i.e. it appears possible to single out the state (state and self-government bodies), business and universities enright [8]. oliver williamson defines the institutional environment as an established system of the informal rules of the game which build the sociocultural context of economic activity williamson [9]. when analyzing the efficiency for the control purposes, it is required to consider the main principles that the financing of the state expenses is based on (including those when allotting the funds intended implementation of the target programs). in this case, the methods used to assess the educational clusters can be applied to control the efficiency of the use of the state resources dritsaki and adamopoulos [10]. in the modern conditions, the competitiveness of the subfederal cluster territory is interpreted as a concentrated expression of the production, scientific and technical, and organizational institutional advantages implemented in high technology goods and services porter [11]. 388tatiana v. yalyalieva and elena a. murzina:the system of parameters efficiency of financial ... in the context of this research, the institutional environment is interpreted as the aggregate of the institutions and institutional connections which surround and fill the subfederal economic system and produce their effect on it. the institutional environment creates favorable conditions for the formation of an interaction network between the agents and governmental bodies e.a.murzina[2] , t.v. yalyalieva [3]. at the same time, while changing in the course of time, the institutional aspects of the governmental control mechanisms form the limits of the economic behavior of innovation companies, thus establishing the sociocultural norms taking effect on the behavior of the agents of economy napolskikh [12]. in other words, governmental control means checking on how the laws are observed and elimination of errors and violations [13]. for instance, nikolaos, petros, eleni, evangelos and dimitrios, in their article, looks at governmental control from the position of the necessity to observe the financial discipline as a precondition for individuals and legal entities to properly fulfill their obligations to the state and considers fiscal supervision to be the major function of the taxation control system [14]. larionova, yalyalieva, napolskikh, shebashev [15] representatives of another school of economists and financiers introduce their grounds as follows: “governmental control is a system of measures aimed at supervising the legality, appropriateness and effectiveness of the activities related to the formation of the monetary funds of the state at all the levels of management” [15]. some issues related to the improvement of the governmental control efficiency were also examined in the works by shemyakina [1]. however, we find it possible to continue studying the subject. certain theses were already mentioned in some of the works by the authors of this article [2–4]. we are going to proceed with the analysis here. integration trends in the russian economy are mainly associated with transition processes and enterprises’ adaptation to the imperfect market conditions. the natural reaction of a rational economic agent to excessive transaction costs would be trying to independently reduce them. but as a result, the reproduction system may fall into an institutional trap. since it is impossible to rely on natural tendencies of transaction costs to decrease, the need arises for the government active interference in the process of their ‘natural’ reduction. as a subject of government control, dual nature of transaction costs, especially transformation processes with a high share of informal sector, as well as significant difference in conditions of functioning of various types of markets require a differentiated approach to transaction costs reduction in the national economy. comprehensive mechanism for efficiency control of transaction costs which is used in economy to reduce them involves development of ideology, specification and protection of property rights, standardization of measurements, accounting and reporting, maintenance of monetary system, improvement of law enforceadvances in systems science and application (2015) vol.15 no.4 389 ment effectiveness and efficiency, as well as implementation of measures aimed at eliminating unnecessary administrative burdens and infrastructure markets of various transactions. 4 concluding remarks an indisputable advantage of the suggested system of the parameters for the purposes of governmental control of the efficiency is its versatility, as it is adapted to any kind of system and is valid in the conditions of establishing an uninterrupted regional and national system resources. hence, we suggest assessment methods for efficiency of governmental control of regional economy aimed at economic supervision over the regional activities. based on these methods, we develop a mechanism to improve the resources management efficiency and find the essential ways to improve it. this mechanism is expected to represent a complex of measures aimed at changing the factors producing the biggest effect on the efficiency of governmental control of regional economy. for every factor and efficiency type, we have to suggest ways for efficiency improvement, and those ways have to consist in specific measures. as seen from the data we obtained, the recovery level of the payments resulting from the controlling activity is extremely low; it did not exceed 43% in the period of the analysis. in 2012 it was even less than 30%.it means that only one third of the amount additionally charged reached the budget system of the country. at the regional level, this value is several points lower in the reported period than the average value for the country. however, within the period we can observe a growing trend for this value, which represents a positive result of the work done by the taxation authorities. the recovery value, as estimated per governmental body official, grew by 35% within the period. totally for the entire country, it amounted to 978,000 rubles in 2012, 952,000 rubles in 2013, and 1,106,000 rubles in 2014. the main reasons for the decrease of the value of the payments additionally charged as resulting from the field audits, if compared to the same value of the previous year, are reduction of the number of the audits performed, which, in turn, resulted from the reduction of the number of the officials who actually performed the field audits, and selection of the subjects to be included in the audit plan, basing on the recovery possibility factor applied to the payments additionally charged as resulting from the audits. thanks. this researchers was supported by the ministry of economic development and trade of the mari el republic (agreement no. 126/8 of october 8, 2013) and volga state university of technology, state-financed scientific-research work no 20 “ efficiency of state control in the sphere of economic activity of the 390tatiana v. yalyalieva and elena a. murzina:the system of parameters efficiency of financial ... region ”, no 9 “development institutions of local markets and the formation of an innovative cluster”. references [1] blumenthal, o. i., shemyakina, m.s. 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[11] porter, m.e., (2009), “clusters and economic policy: aligning public policy with the new economics of competition”, nstitute for strategy and competitiveness. advances in systems science and application (2015) vol.15 no.4 391 [12] larionova.n.i., napolskikh d. l, yalyalieva , and shebashev v.r e.(2014), “governmental control of the formation efficiency of educational clusters at the regional level”, american journal of applied sciences, vol.11, no.9, pp.1594-1597. [13] koteeswaran, s., p. visu and j. janet (2012), “a review on clustering and outlier analysis techniques in datamining”. am. j. applied sci, vol.9, pp.254258. [14] nikolaos, e., k. petros, p. eleni, p. evangelos and v. dimitrios, (2013), “the impact of the greek economic crisis on the greek construction companies”. back to basics: the statistical cost accounting model. am. j. econom. bus. admin, vol.5, pp.168-174. [15] larionova n.i., yalyalieva t.v.and napolskikh d.l. (2014), “ensuring efficiency control of institutional environment of the cluster”. published online vol.11, no.9, http://www.thescipub.com/ajas.toc [16] kim, y.d., s. yoon and h.g. kim (2014),“ an economic perspective and policy implication for social enterprise”. am. j. applied sci, vol.11, pp.406413. [17] liu, c.c. and c.y. chen (2004), “a computable general equilibrium model of southern region in taiwan: the impact of the tainan science-based industrial park”. am. j. applied sci, vol.1, pp.220-224. [18] murzina, e.a., larionova, n, i. and yalyalieva t.v. (2015), “supervising the efficiency of governmental control of regional economy: legal aspects and economic essence”. mediterranean journal of social sciences, vol 6, pp.370-375 corresponding author tatiana can be contacted at:yal05@mail.ru advances in systems science and applications (2012) vol.12 no.3 297-304 an improvement on kim’s inequality for pricing asian option yirong ying1, sheng xu1 and jeffrey forrest2 1college of management, shanghai university, shanghai, 200444, china 2department of mathematics, slippery rock university, slippery rock, pa16057, usa abstract in his study of a simple one-dimensional parabolic pde, kim[11] obtained an inequality that the price of asian options satisfies. this paper aims to improve this particular kim’s inequality. our inequality has an advantage over kim’s, for our estimation has an upper bound as r vanishes. therefore, our newly established estimation can help us to determine the solution error of a second order partial differential equation with variable coefficients. keywords asian option, parabolic pde, error estimation 1 introduction as a new financial product, asian options, also known as average options, can be seen as innovative european options. the commonality with the european options is that investors are only allowed to exercise their option contracts on the maturity dates, while the difference is that investors of asian options decide whether or not to exercise their option contracts based on the price level of the average share price during the contract term. because the value of european options on the maturity date has nothing to do with the price path and depends only on the maturity date of the stock price, it is difficult to prevent speculators from manipulating the maturity price and consequently from arbitrage. on the other hand, because asian options are associated with the price path, they can be applied to ease market behaviors. they represent the exotic options most actively traded in financial derivatives market today. their difference with the usual-sense stock options is the implementation of price limits so that the exercise price is the average price of the stock price of the secondary market over a period of six months prior to exercise. their difference with the standard options is that the options’ maturity-date benefits are determined not by the prevailing market price of the underlying asset, but by the average price of the underlying asset over some time period during the option contract period. this time period is known as an average period, over which either the arithmetic or geometric mean is applied. asian options can be well designed to avoid stock price manipulation and damage of company interests caused by insider trading. so, they are welcomed by both investors and issuers. additionally, compared to the standard options, asian options also possess such advantages as lower prices and can be applied to hedge the risk of a specified time period. along with baskets, spreads, as well as 298 yirong ying:an improvement on kim’s inequality for pricing asian option other strategies, asian options have been extensively studied by many scholars. even so, few of published works deal with the problem of pricing general asian options. in terms of the pricing of asian options, [1-4] represent some of the most recent results, while dewynne et al. provide a simplified means of pricing asian options using parabolic pde[5]. a progressive solution is established by using laplace transforms and power series expansions. equations of kolmogorov type have also turned out to be relevant in option pricing in the setting of certain models for stochastic volatility and in the pricing of asian options. frentz et al. numerically solve the cauchy problem of a general class of second order degenerate parabolic differential operators of kolmogorov type with variable coefficients by using posteriori error estimates and an algorithm developed for adaptive weak approximation of stochastic differential equations[6]. on the basis of these works, we show how to apply the relevant results in the context of mathematical finance and option pricing. the approach outlined in this paper circumvents many of the difficulties confronted by any deterministic approach based on, for example, a finite-difference discretization of the partial differential equation. meanwhile, frentz[7] also analyze the second order pde operators arising in the pricing of asian options, while proving the optimal interior regularity. min dai presents a lattice algorithm for pricing both europeanand american-style moving average barrier options (mabos)[8]. they develop a finite-dimensional partial differential equation (pde) model for discretely monitoring mabos and solve it numerically by using a forward shooting grid method. however, their modeling pde for continuously monitored mabos is of infinite dimensions and cannot be solved directly by using existing numerical methods. recently, bayraktar et al. construct a sequence of functions based on the value of asian option[9]. as a result, each term of the sequence is the unique classical solution of a parabolic pde so that they provide the relevant numerical approximation. deelstra et al. obtain an approximation formula by using comonotonic bounds[10], leading to four different approximations: the upper, the improved upper, the lower, and the intermediary bounds. in this way, they improve the traditional hybrid moment matching method. their methods have the advantage that they can be applied in other frameworks, e.g., in lévy settings, as well. these results can be adapted to deal with options written in a foreign currency, such as compo and quanto options. kim studies a simple one-dimensional parabolic pde that the price of asian options satisfies[11]. the result indicates that the generalized solution is a classical solution and cannot be obtained. additionally, kim’s inequality is shown to be unbounded as the parameter r vanishes. by improving kim’s inequality, our estimation has an upper bound as r vanishes. advances in systems science and applications (2012) vol.12 no.3 299 2 main results kim (2009)[11] considers the following parabolic type equation ut + 1 2 [x− e− ∫ t 0 dv(s)q(t)]2σ2uxx = 0 (1) along with the boundary condition u(t, x) = max(x−k1, 0) (2) where v(t) stands for dividend yield, σ the volatility of the underlying asset, and q(t) the trading strategy. first let us introduce a lemma about gauss estimation. suppose that g(x) is a continuous function in the interval [−r,r] satisfying 1 2 ≤ g(x) ≤ 3 2 , x ∈ [−r,r] denote q := {(t, x) ∈ r2 : 0 < t < 2, |x| < r}, ω := {(t, x) ∈ q : t > g(x)},σ := {(t, x) ∈ q : t = g(x)}. lemma1[11] let ω and σ be defined as above and a(t, x) a function satisfying{ lu := ut − a(t, x)uxx = 0, (t, x) ∈ ω u = 0, (t, x) ∈ σ assume that u ∈ c1,2 loc (ω) ∩ c(ω̄) and a(t, x) satisfy 0 ≤ a(t, x) ≤ 1, ∀(t, x) ∈ ω then, we have the following estimate: |u|0;ω′ ≤ ( 16√ 2π )r−1e− r2 32 |u|0;ω ,ω′ := {(t, x) ∈ ω : |x| < r 2 } (3) proposition 1 if |x| < r, then∫ e ϕ(2, x− y)dy < 2 ∫ 2r 0 ϕ(2, y)dy + 1√ 2πr e−2r2 (4) where e = ∪ j∈z ((4j + 1)r, (4j + 3)r). proof: first we prove that the following equality holds true:∫ e ϕ(2, x− y)dy = ∞∑ j=1 ∫ (4j−1)r+x (4j−3)r+x ϕ(2, y)dy + ∞∑ j=0 ∫ (4j+3)r−x (4j+1)r−x ϕ(2, y)dy 300 yirong ying:an improvement on kim’s inequality for pricing asian option in fact,∫ e ϕ(2, x− y)dy = ∫ ∞∪ j=1 ((−4j+1)r,(−4j+3)r) ϕ(2, x− y)dy + ∫ ∞∪ j=0 ((4j+1)r,(4j+3)r) ϕ(2, x− y)dy = ∞∑ j=1 ∫ (−4j+3)r (−4j+1)r ϕ(2, x− y)dy + ∞∑ j=0 ∫ (4j+3)r (4j+1)r ϕ(2, x− y)dy = ∞∑ j=1 ∫ x−(−4j+3)r x−(−4j+1)r −ϕ(2, y)dy + ∞∑ j=0 ∫ x−(4j+3)r x−(4j+1)r ϕ(2, y)dy ∵ ϕ(2, y) = 1√ 8π e− y2 8 is an even function, ∴ ∫ e ϕ(2, x− y)dy = ∞∑ j=1 ∫ (−4j+3)r−x (−4j+1)r−x ϕ(2, y)dy + ∞∑ j=0 ∫ (4j+3)r−x (4j+1)r−x ϕ(2, y)dy = ∞∑ j=1 ∫ (4j−1)r+x (4j−3)r+x −ϕ(2, y)dy + ∞∑ j=0 ∫ (4j+3)r−x (4j+1)r−x ϕ(2, y)dy secondly, we prove that when −r < x < 0, the following inequality holds true ∞∑ j=1 ∫ (4j−1)r+x (4j−3)r+x ϕ(2, y)dy ≤ ∫ 2r 0 ϕ(2, y)dy + 1 2 √ 2πr e−2r2 (5) in fact, when −r < x < 0, ∞∑ j=1 ∫ (4j−1)r+x (4j−3)r+x ϕ(2, y)dy < ∞∑ j=0 ∫ (4j+2)r 4jr ϕ(2, y)dy = ∫ 2r 0 ϕ(2, y)dy + ∞∑ j=1 ∫ (4j+2)r 4jr ϕ(2, y)dy ∵ ∞∑ j=1 ∫ (4j+2)r 4jr ϕ(2, y)dy < ∫ ∞ 4r ϕ(2, y)dy = ∫ ∞ 4r 1√ 8π e− y2 8 dy < 1√ 8π ∫ ∞ 4r y 4r e− y2 8 dy = 1 2 √ 2πr e−2r2 advances in systems science and applications (2012) vol.12 no.3 301 ∴ ∞∑ j=1 ∫ (4j−1)r+x (4j−3)r+x ϕ(2, y)dy ≤ ∫ 2r 0 ϕ(2, y)dy + 1 2 √ 2πr e−2r2 thirdly, we prove that as 0 < x < r, we have ∞∑ j=0 ∫ (4j+3)r−x (4j+1)r−x ϕ(2, y)dy ≤ ∫ 2r 0 ϕ(2, y)dy + 1 2 √ 2πr e−2r2 (6) in practice, as 0 < x < r, ∞∑ j=0 ∫ (4j+3)r−x (4j+1)r−x ϕ(2, y)dy ≤ ∞∑ j=0 ∫ (4j+2)r 4jr ϕ(2, y)dy = ∫ 2r 0 ϕ(2, y)dy + ∞∑ j=1 ∫ (4j+2)r 4jr ϕ(2, y)dy ≤ ∫ 2r 0 ϕ(2, y)dy + 1 2 √ 2πr e−2r2 put (5) and (6) together, the proof is completed. qed. proposition 2, if |x| < r, then∫ e ϕ(2, x− y)dy < √ 1− e−r2 + 1√ 2πr e−2r2 (7) proof: for arbitrary r > 0, we have inequality ∫ 2r 0 ϕ(2, y)dy < 1 2 √ 1− e−r2 . in fact, (∫ 2r 0 ϕ(2, y)dy )2 = ∫ 2r 0 ∫ 2r 0 1 8π e− x2+y2 8 dxdy ≤ ∫∫ d 1 8π e− x2+y2 8 dxdy (8) = 1 4 (1− e−r2 ) where d = {(x, y)|x2 + y2 ≤ 8r2, x ≥ 0, y ≥ 0}. hence ∫ 2r 0 ϕ(2, y)dy < 1 2 √ 1− e−r2 . by substituting the above inequality in proposition 1, inequality (7) is obtained at once. qed. proposition 3 if k > 1, r2 < 2ln2 16k−9 , then∫ (4j+2)r 4jr ϕ(2, y)dy ≤ ∫ 8jr (8j−4)r ϕ(2, √ ky)dy (9) 302 yirong ying:an improvement on kim’s inequality for pricing asian option proof: after changing the variable t = y+4r 2 , we have∫ 8jr (8j−4)r ϕ(2, √ ky)dy = 1√ 8π ∫ 8jr (8j−4)r e− ky2 8 dy = 1√ 8π ∫ (4j+2)r 4jr eln2− k(y−2r)2 2 dy as r2 < 2ln2 16k−9 , 4jr < y < (4j + 2)r, j = 1, 2, · · · , the following inequality holds true:ln2− k(y−2r)2 2 + y2 8 ≥ 0. hence 1√ 8π ∫ (4j+2)r 4jr e− y2 8 dy ≤ 1√ 8π ∫ (4j+2)r 4jr eln2− k(y−2r)2 2 dy. that is ∫ (4j+2)r 4jr ϕ(2, y)dy ≤ ∫ 8jr (8j−4)r ϕ(2, √ ky)dy. qed. proposition 4 if k > 1, r2 < 2ln2 16k−9 , then∫ e ϕ(2, x− y)dy < 2 ∫ 2r 0 ϕ(2, y)dy + 1√ 2πkr e−2kr2 (10) proof: combining the results of proposition 2 and proposition 3 leads to∫ e ϕ(2, x− y)dy ≤ 2 ∫ 2r 0 ϕ(2, y)dy + 2 ∞∑ j=1 ∫ (4j+2)r 4jr ϕ(2, y)dy ≤ 2 ∫ 2r 0 ϕ(2, y)dy + 2 ∞∑ j=1 ∫ 8jr (8j−4)r ϕ(2, √ ky)dy ≤ 2 ∫ 2r 0 ϕ(2, y)dy + 2 ∫ ∞ 4r ϕ(2, √ ky)dy ≤ 2 ∫ 2r 0 ϕ(2, y)dy + 1√ 2π ∫ ∞ 4r y 4r e− ky2 8 dy = 2 ∫ 2r 0 ϕ(2, y)dy + 1√ 2πkr e−2kr2 .qed. proposition 5 if k > 1, r2 < 2ln2 16k−9 , then ∫ e ϕ(2, x − y)dy < √ 1− e−r2 + 1√ 2πkr e−2kr2 . proof: by substituting inequality (8) in proposition 4, one knows that inequality (10) is true. qed. 3 conclusions with the increasing globalization of investment in recent years, a variety of asian options has obtained a wide range of applications and development. for example, quanto option represents a contingent claim that the option’s benefit depends on the prices of financial derivatives in a foreign currency, while the actual transaction of the option is paid in local currency. quanto options, as a kind of option advances in systems science and applications (2012) vol.12 no.3 303 designed for avoiding different types of risks involved in international trades and investments, represent a financial derivative instrument. they can be used as a hedge in international trades against fluctuations in exchange rates. therefore, with the rapid development of international trades and the globalization of the financial industry, this option will definitely attract more attention. the revenue function of quanto option stands for a joint concern of foreign asset prices and exchange rates, creating many alternative investing and hedging opportunities. additionally, other than asian options, power options represent yet another new class of options. they alter the pricing structure of assets and greatly increase the sensitivity of price over time. power-asian options are a unification of the two, making it possible for investors to better avoid risks. an example is the moving average call option. it has been often used to design a poison pill, a business strategy used to increase the likelihood of negative results over positive ones against a party that attempts a takeover. the moving average calls, issued to existing shareholders, would be triggered by the event of a hostile takeover. the french investment bank compagnie financiere indosuez and the french construction company bouygues have successfully issued such options/warrants to protect themselves against potentially unfriendly investors. kim’s inequality is very useful for our work[12]. however, kim’s estimation is possibly unbounded as r vanishes. our inequality, developed in this paper, has an advantage over kims inequality, because our estimation is bounded on the top as r vanishes. therefore, our work in this paper may help us to look for the solution error of a second order partial differential equation with variable coefficients. acknowledgements this research is supported by the research fund of program foundation of ministry of education of china (10yja790233). references [1] xu chenglong, zhou jing, ren xuemin. (2007), “arbitrage analysis of a class of deposit product with option style”, journal of tong ji university (natural science), vol.35, no.7, pp.994-997. [2] wu zhen, wang guangchen. (2007), “a black-scholes formula for option pricing with dividends and optimal investment problems under partial information”, journal of system science and mathematic sciences, vol.27, no.5, pp.676-683. [3] wang xiaotian. (2010), “scaling and long-range dependence in option pricing i: pricing european option with transaction costs under the fractional 304 yirong ying:an improvement on kim’s inequality for pricing asian option black-scholes model”, physica a: statistical mechanics and its applications, vol.389, no.3, pp.438-444. [4] chen caisheng, li gang, zhou jidong, wang wenchu. (2008), mathematical physics equations, science press. [5] dewynne, shaw. (2008), “differential equations and asymptotic solutions for arithmetic asian options: ‘black-scholes formulae’ for asian rate calls”, european journal of applied mathematics, vol.19, pp.353-391. [6] marie frentz, kaj nyström. (2010), “adaptive stochastic weak approximation of degenerate parabolic equations of kolmogorov type”, journal of computational and applied mathematics, no.234, pp.146-164. [7] marie frentz, kaj nyström, andrea pascucci, sergio polidoro. (2010), “optimal regularity in the obstacle problem for kolmogorov operators related to american asian options”, math. ann, vol.347, pp.805-838. [8] min dai, peifanli, jine.zhang. (2010), “a lattice algorithm for pricing moving average barrier options”, journal of economic dynamics & control, no.34, pp.542-554. [9] erhan bayraktar, hao xing. (2011), “pricing asian options for jump diffusion”, mathematical finance, pp.117-143. [10] griselda deelstra, alexandre petkovic, michèle vanmaele. (2010), “pricing and hedging asian basket spread options”, journal of computational and applied mathematics, no.233, pp.2814-2830. [11] seick kim. (2009), “on a degenerate parabolic equation arising in pricing of asian options”, journal of mathematical analysis and applications, vol.351, pp.326-333. [12] yi-rong ying, yu-yuan tong, jeffrey forrest. (2010), “pricing asian options: approach of decomposition and estimation”, advances in systems science and applications, vol.10, no.2, pp.241-247. corresponding author author can be contacted at:jeffrey.forrest@sru.edu microsoft word 15 li shuiping, li xiaotian，shi lugang--simulation of the flow field of cement mixer based on numerical metho advances in systems science and applications (2011), vol.11, no.3-4 315-321 issn 1078-6236 international institute for general systems studies, inc simulation of the flow field of cement mixer based on numerical methods li shuiping, li xiaotian, shi lugang school of mechanical and electronical engineering, wuhan university of technology, wuhan 430070, p.r.china abstract different kinds of research methods of the flow field in the cement mixer were introduced and compared, theory bases of the numerical simulation method of the flow field in the cement mixer were described, whole of thoughts of realization this method were analyzed. and then, the geometrical model and mathematical mode of the cement mixer were created by fluent, which was based on cfd theory; and the characteristic of flow field in the cement mixer was simulated by mixture model, simple algorithm and standard k-ε turbulence model. the calculation results showed that the distribution of flow field in cement mixer could be simulated preferably by the numerical technology, which supplied the valuable theoretical guidance for designing of the cement mixer. keywords cement mixer numerical simulation flow field 1.introduction cement slurry is frequently needed to grout the cracks of the buildings when reinforcing the constructions. currently, most cement slurry is produced by the common mechanical mixer which makes grout by vane, scraper or spiral device. and the mixer is composed by gear, mixing shaft and impeller. it will easily produce the dead angle and precipitation at the bottom of mixing tank in the mixing process due to the blade is fixed on the shaft. apart from that, it has the problems which are big energy waste, low efficiency and can not prepare for different water-cement ratio cement slurry fast. therefore, the development of a new type of high-performance cement slurry mixing equipment has become rather important. currently,cement mixer commonly can be divided into three categories:colloidal mills mixer, blades mixer, water-jet mixer. although these devices are generally able to achieve the desired technical effect, the mixing efficiency and cost is another matter. colloidal mills mixer is an advanced equipment, mixing, high efficiency, high concentration slurry can be stirred, but the structure is more complex and costly. the structure of blades mixer is simple, the cost is inexpensive, and widely used in engineering, but the efficiency is low. water-jet mixer is a low-cost equipment, have the characteristics of high speed stirring, but because the mixing uniformity is not high, to some extent these limit its wider use. so how to provide advanced cement mixing equipment is an important issue that engineering and technical personnel in the country have to faced. for the design of cement mixer, traditional theory and experimental measurements usually used in analysis of the flow field. study found that in the mixer there are various of slurry flow forms at the same time, combined with the interaction of cement powder and water, make the two-phase flow field very complex and difficult to carry out a detailed theoretical analysis. also the complex flow characteristics and opacity of the slurry limit the application of experimental methods. in process of today's research about fluid mechanical design, the cfd method to simulate the flow fluid inside the machine has become an important technical means when the software of cfd have emerged. in engineering, numerical simulation has been more widely 316 li: simulation of the flow field of cement mixer based on numerical methods used, it can be replaced experiment to cut down the cost in some extent, in addition, numerical simulation can provide a lot of information about flow field, it can give reliable basis for the optimal design of fluid machinery for designers. comprehending of the cement slurry flow characteristics and the mechanism of mixing is a basic prerequisite for the structural design [1]. the main purpose of this paper is to simulate and analyze the flow field of the equipment based on the numerical simulation, and then study the mixed performance. 2.research methods of the mixer flow 2.1 introduction of research methods of the flow field 2.1.1 theoretical analysis method on the basis of the relevant experimental data and the law of conservation of energy and so on, it will establish the velocity field distribution of fluid flow, the stress field distribution of fluid flow , mass conservation law and energy conversion law by the mathematical analysis, thus construct the mathematical model of fluid flow[2] .as a result, it can forecast the intensity and characteristics of the fluid flow of mixer as well as its function and influence on mass transfer and energy exchange. 2.1.2 experimental methods (1) visualization of fluid method visualization of fluid method which can monitor the trajectory of fluid is a basic method of studying flow field. there are anisotropic thin (particulate) method, laser-induced fluorescence method and dye tracer method [3]. anisotropic thin (particulate) act refers to adding the anisotropic particles (such as titanium dioxide, mica, aluminum stearate, etc.) which have the reflective capacity to the fluid so that the anisotropic particles flow with the fluid, and the strong reflective capacity of particulate makes the flow trajectory of fluid can be visible. in addition, we are able to study the movement law of two-phase fluid and the migration law of material according to the selective dispersion of some particles in fluid (for example, water phase is chosen for mica, while hydrophobic organic phase are chosen for aluminum and mica). (2) video measurement using advanced camera equipments (such as magnetic resonance fluid imager, laser doppler veloeimetry, particle image vefocimetry etc.) to film the track of fluid flow and then obtain it’s boundary conditions as well as the related parameters [3]. piv technique is non-contact measurement technology which has many advantages such as high measurement accuracy, wide range of speed measurement speed, having simple principle and small influence from outside .consequently, it is able to depict the whole velocity vector of the total velocity field, so the flow field information of one profile can be analyzed at once. 2.1.3 numerical simulation method numerical simulation is that adopting numerical method and logical way to reflect changes prototype process and the rules of movement in the computer. mathematical model, different from the physical model, is often made from a single equation, a group of equation or groups formulas. we can get the various locations of the basic physical quantities (such as speed, pressure, temperature, concentration, etc.) and their changes in the very complex flow by numerical simulation. in addition, it can optimize the design of structural by combining with cad. 2.2comparison of the research methods of the mixer’s internal flow field advances in systems science and applications (2011), vol.11, no.3-4 317 table 1 comparison of the research methods numerical simulation method combined with traditional theoretical analysis method and experimental measurement method form a perfect system of studying the problems of the mixing equipment’s internal fluid flow. from table 1, we can we see clearly that these methods have their own advantages and disadvantages. both traditional theoretical analysis method and experimental test method can not obtain the characteristics of flow field mainly due to the following two points: first, the visualization methods such as staining piv and tracer method can not be used to measure the flow field which has high concentration and poor visibility; second, at present, all of the contact velocimeter can not test the velocity vector of cementing slurry flow field accurately. therefore, the slurry flow field can only be studied by numerical simulation. 3. numerical simulation method of the internal flow field of mixer 3.1 theoretical basis of numerical simulation the theoretical basis of numerical simulation is computational fluid dynamics. fluid flow is dominated by physical conservation laws which relates to the principal basic laws including mass conservation law, momentum conservation law, energy conversion and conservation law, components conversion and balance law，and so on[2,5]. these laws of the basic equations can be expressed as follows: ( ) ( ) ( )div u div grad s t φ ρφ ρ φ φ∂ + = γ + ∂ (1) expand the form of: ( ) ( ) ( ) ( )u v w t x y z ρφ ρ φ ρ φ ρ φ∂ ∂ ∂ ∂ + + + ∂ ∂ ∂ ∂ ( ) ( ) ( ) s x x y y z z φ φ φ∂ ∂ ∂ ∂ ∂ ∂ = γ + γ + γ + ∂ ∂ ∂ ∂ ∂ ∂ (2) whereφ is the general dependent variable, and so can represent the solution of variables of u v w t、 、 、 ; γ is the generalized diffusion coefficient; s is the generalized source term. fheoretical nnalysis method experimental method numerical simulation m ain a dvantage the results whose all kinds of influencing factors are clearly visible have universality. it is the theoretical basis which is used to guide the experimental research and verify the new numerical calculation method. the experimental results which are true and believable,are the basis of theoretical analysis and numerical method. it is easy to choose different physical parameters to make the various effective and sensitive tests because you are not restricted by physical model and mathematical model. m ain d isadvantages we are often required to abstract and simplify the object of calculation. for the non-linear situation, only the analytic results of simple flows can be given. experiments are often limited by model size, flow disturbances and measuring accuracy, in addition ,they will also encountered lots of problem such as the enormous cost of funding and manpower,material resources and the long cycle. it strongly rely on the mathematical model,and numerical processing method will lead to false results. 318 li: simulation of the flow field of cement mixer based on numerical methods in the macro category of macro continuous mechanics, two-phase flow analysis can be divided into two types: (1) local homogeneous flow analysis; (2) two-phase slip flow analysis. cementing slurry flow in stirred tank belongs to two-phase slip flow which mainly has three types of analysis model: euler-lagrange model, eulerian-eulerian model that is a mixture of the two-fluid model and mixture model. euler-lagrange model centralizes the advantages of macroscopic pseudo-fluid model and micro-dynamics model, that is, take into account not only the solid phase and liquid-phase interactions, and can describe the collision between particles, but it needs to record the spatial location and movement of every particle in the flow field , therefore a larger computer memory is necessary. currently, there are certain difficulties in the simulation of high concentration of solid-liquid two-phase flow [4, 5]. at present, multi-fluid model is given a wide range of applications. in a multi-fluid model, the continuous (liquid) phase movements is described by euler equations, assuming that the dispersed (particle) phase is the mutual penetration and continuous quasi-fluid, its movement is also described by euler equation, and then solve the momentum, energy and quality equations of each phase. however, the computational complexity of multi-fluid model results in poor stability, so it confronts many difficulties in the practical application. mixture model is a simplified multiphase flow model, which is used to simulate each phase of the multiphase flow at different speeds, however, the local equilibrium under short spatial scale is assumed. for mixture model, there is a strong coupling between different phases. when there exist a wide range of particle distribution, mixture model can achieve good results as perfect multiphase flow model. due to space limitations, the control equations of two-phase flow will not be listed here. 3.2 the process of numerical simulation of the internal flow field at present, the numerical simulation of mixing process is on the basis of macro flow field. first of all, it obtains the velocity field distribution law of macro-flow field and then joins tracer when the flow field is stable. through the calculation of tracer concentration changes with space and time to simulate the mixing process. this method applies to the flow field of relatively low concentration. in spite of that, there are great difficulties in the simulation of high concentration slurry flow field. in this paper, cementing slurry two-phase flow field was simulated by mixture model, and then the mixing effects were assessed according to the results of simulation. the aim of numerical simulation is to obtain the useful data used to instruct the optimal design of mixer, such as mixing time and mixing power number. the basic flow of numerical simulation is shown in figure1. in the process of numerical simulation, we can create the geometric model according to the structure of mixer. generally, we should deal with the physical mode reasonably during the establishment of geometric model since because it is usually rather complex [6, 8]. in order to reduce the unnecessary computation time, the physical model need to be simplified as much as possible, such as the cover of the mixer which has little impact on the effect of stirring, therefore, it ought to be neglected during the geometric model establishment; for those structures which have greater impact on the effect of stirring, not only to retain but also to establish them accurately. the establishment and dispersion of the mathematical model which directly affects the accuracy and precision of simulation are the core parts of numerical analysis. the key of numerical simulation is to establish the mathematical model which can reflect the essence of engineering problems. for different fluids, we can choose different suitable mathematical models, such as rng k-ε two-equation turbulence model or two-realizable k-ε model may be the best choice in a flow field with a cyclone or strong bending wall [9]. we discretize the regional of calculation before numerical calculation, that is, divide the space of calculation into many sub-regions, and to identify nodes in each region in order to generate grid. and then, the control equations are discretize in the discrete grids, as a result, partial differential equations turn into the algebraic equations of each node for numerical solution. algebraic equations have a advances in systems science and applications (2011), vol.11, no.3-4 319 variety of discrete formats including quick format which can provide higher calculation accuracy when we study swirling flow by structured grid. so the quick format may be the best choice during the numerical simulation of the internal flow field in mixer. the physical model of mixer establishing geometric model mesh generation establishing mathematical model algorithm optimization numerical simulation visual data mining analyze， compare,optim ize judgment of convergence yes no figure 1 the basic flow of numerical simulation apart from a few simple questions, the discrete equations can not be directly solved, because certain adjustments of the dispersion equations are necessary, and we ought to deal with the order and way of unknown variables (speed, pressure, temperature, etc.) especially in order to solve them easily. simple algorithm is widely used in engineering. before carrying out the computer simulation, setting the model boundary conditions is required because the boundary conditions which include the inlet velocity, boundary conditions, outlet pressure, the concentration of slurry, are necessary for solving the discretized equations. setting appropriate initial value and sub-relaxation factor appropriate before the iterative solution in favor of iterative calculations converge faster and more easily. once convergence of iterative solution achieves prescriptive convergence precision, the data we need can be obtained through the user interface. 4. process of numerical simulation of flow field of mixer by fluent 4.1 tools of numerical simulation of flow field on the basis of the theory and method above, we can achieve the numerical simulation of flow field by gambit and fluent soft wares [10]. fluent is specific cfd software which is used to simulate and analyze a complex geometry in the field of fluid flow and heat exchange while gambit is professional pre-processing software for computational fluid dynamics, its main features include three aspects: geometric modeling, mesh generation and assignment of boundaries. among them, mesh generation is the most important function. it generates grid file containing boundary information. fluent can realize numerical simulation of flow field of mixer effectively in connection with its pre-processing software gambit. 320 li: simulation of the flow field of cement mixer based on numerical methods 4.2 the establishment of geometric model mesh generation is the necessary condition to establish discrete control equations, because the quality of the mesh has a direct impact on the accuracy of simulation results. through analyzing the physical model of mixer, this paper used gambit to establish the mesh model on the basis of geometric model of mixer, as shown in figure 2. figure 2 the mesh model of mixer to reduce quantity of mesh cells and to improve precision and convergence, we used structured mesh in the nozzles. for the internal structure with complex geometric shape, the flow field changes rapidly, it is inconvenient to adopt structured mesh, so a structured-unstructured mixed type mesh was employed in this computation. 4.3 solve the discrete mathematical model mesh model file will be optimized after being imported in fluent. we discreted control equations of mesh nodes by numerical discretization of quick format in this paper. numerical algorithm is one of the methods for solving the discrete control equations of grid model, and simple algorithm is quite applicable. 4.4 preliminary analysis of simulation results to a large extent, the effect of mixing depends on the internal structure of flow field, so the existence of turbulence which has complex flow field is an important prerequisite for enhancing the effect of mixing. figure 3 velocity vectors colored by radial velocity (m/s) after analyzing the velocity vector distribution chart, it can be seen that the mixer produced a strong vortex under the action of the spiral blade combined the nozzles. the radial velocity is advances in systems science and applications (2011), vol.11, no.3-4 321 shown in figure 3. 5. future research at present, the literatures about the numerical simulation of cement mixer are rare. although the numerical simulation method has its unique advantages, there are many aspects need to be further improved, which mainly include the numerical simulation accuracy and convergence rate. numerical simulation accuracy of multiphase flow mainly depends on the discrete mathematical model, therefore how to establish mathematical model in accordance with the mechanical structure of mixer is one of the difficulties of numerical simulation. low–order dispersion and high-order dispersion has its own excellent shortcomings, so the way to solve the discrete control equations is a very key technology. the purpose of numerical simulation of flow field is obtaining the useful data to optimize the mechanical structure, thus making further analysis of the simulation results is one of the future research directions. acknowledgements this research is funded by the national key technology r&d program of china (no. 2006baj03a09-08). references [1] zhao jianjun，yuan shouqi，liu houlin, huang zhongfu and tan mingao. simulation of solid-liquid two-phase turbulent flow in double-channel pump based on mixture model. journal of transactions of the csae, 24 (1)( 2008) 7-10. [2] wang fujun. analysis of computational fluid dynamics—principle and application of cfd software. beijing: tsinghua university press , 2006. [3] zhan hanhui, chenghao, liu jianwen and zhan xuehui. principle of secondary flow. changsha:central south university press, 2005. [4] anders darelius, anders rasmuson, berend vanwachem, ingela niklasson and staffan folestad. cfd simulation of the high shear mixing process using kinetic theory of granular flowand frictional stress models. journal of chemical engineering science, 63(8)( 2008) 2188-2197. [5] zhou lixing. dynamics of multiphase turbulent reacting fluid flows, beijing: national defense industry press, 2002. [6] wu bo,yan hongzhi,duan yiqun. study on 3-d turbulence numerical simulation and wear characteristics of slurry pump. journal of china mechanical engineering. 20(6)( 2009) 719-780. [7] huang si, wang guo-yu. a 3d numerical simulation of solid-liquid turbulent flow with high solid-concentration in a centrifugal pump. journal of coal mining machinery. vol.11(2005) 53-57. [8] k.h. javed, t. mahmud and j.m. zhu. numerical simulation of turbulent batch mixing in a vessel agitated by a rushton turbine. journal of chemical engineering and processing, 45(2)( 2006) 99–112. [9] benjamin coesnon, mourad heniche, christophe devals, franc¸ois bertrand and philippe a. tanguy. a fast and robust fictitious domain method for modeling viscous flows in complex mixers: the example of propellant make-down. international journal for numerical methods in fluids, 58(4)( 2008) 427–449. [10] fluent inc. fluent user guide.2003. advances in systems science and application (2016) vol.16 no.3 1-10 on mutual coincidence of return on capital measuring principles by adam smith and karl marx (a critique of “capital in the twenty-first century” by thomas piketty) s.baizakov1,y.utembayev2,a.r. oinarov3 and j.forrest4 1 scientific supervisor, jsc "economic research institute"; astana, kazakhstan; 2 independent expert, astana, kazakhstan; 3 oinarovazamat, chairman of the board of the jsc "kazakhstan center for public-private partnership", astana, kazakhstan; 4 jeffrey forrest, school of business, slippery rock university, philadelphia, usa annotation in his book “capital in the xxi century” by thomas piketty justified the two basic laws of capitalism. on the basis of these economic laws thomas piketty concluded destabilizing role of accumulated national capital. based on his critical analysis this paper studied alternative ways of identification and knowledge of the objective laws’ system that would become the tools of detection imbalances in the economy and assessing the impact of the regulatory impacts on the development of a market economy. develop an appropriate model for the analysis of regulatory impacts. keywords capital, profitability, scientific and technological potential, law, macroeconomics, balance, effect methods thomas piketty, author of the bestselling book “capital in the twenty-first century”, has paid serious attention to the methods of measuring economic growth, including the method of measurement of nominal gdp, defining its by multiplication of return on capital by its volume. he correctly noted that “the concepts of inflation and growth are not always very well defined. the decomposition of the nominal growth (the only kind that can be observed with the naked eye, as it were) into a real component and inflation component is in part arbitrary and has been the source of numerous controversies” [1]. speaking about the effects of accumulation and return on capital’s decrease, he notes, “...on the basis of historical experience, the most likely outcome is that the volume effect will outweigh the price effect, which means that the accumulation effect will outweigh the decrease in the return on capital.”[1]. at the same time, he pays attention to that in agriculture there is a reverse process: “... capital (such as farmland in the case in point), it is inevitable that beyond a certain point, the price effect will outweigh the volume effect... there is no better illustration of the maxim “too much capital kills the return on capital” than the relative value of land and land rents in the new world and the world” [1]. in general in the piketty’s book, the principle of two-dimensional measurement 2 s.baizakov, y.utembayev, a.r.oinarov, j.forrest:on mutual coincidence ... of economic growth was successfully used to analyze the price effect, as a form of the capital’s cost in its form of goods, and volume effect, as a form of the capital’s cost of in its form of money. the same principle is used for two-dimensional measurement of the two fundamental economic laws of capitalism (in the terminology of the author). however, deeper analysis shows that these laws have no any relation to the fact that the “the volume effect is will outweigh the price effect, which means that the accumulation effect will outweigh the decrease in the return on capital”. moreover it hasn’t related to“...farmland in the case in point it is inevitable that beyond a certain point, the price effect will outweigh the volume effect”. pikettydoes his findings on the basis of historical retrospective in some combination of it with the dynamics of economic laws. the reason for these inconsistencies and inaccuracies has often derived from an outdated theoretical basis of establishment of analytical toolsthat are being used now in the practice of market economies institutions and management agents’ regulatory impact assessment. in our opinion, the main factor that has a destabilizing effect on the sustainability of the market economy is the outdated persistence of a one-dimensional measure of return on capital, which is recommended by adam smith as a tool to assess the true value of goods and money[2]. so, up to this day the principle of a one-dimensional measure of return on capital by adam smith has existed in the theoretical basis of current models of balanced economic growth. as an example, the traditional formula of the gdp deflator (inflation) pb, can be taken for, which has the known form as follows: pb = ngdp rgdp (a) where ngdp-is nominal gdp, which by its content is the final product in money terms. the theoretical basis of the formula (a) is the adam smith’s principle of a one-dimensional measure of return on capital. according to this formula, if nominal gdp (ngdp) grows faster than real gdp (rgdp) in the long term, what is happening in some developing countries nowadays, then the gdp deflator (pb) as an indicator of inflation, fueled by the national currencies devaluation may tend to infinity (pb −→ ∞). this is not the only example, which defines the adam smith’s principle of one-sidedness of return on capital’s measurement and its limitation as a balanced growth’s theoretical base models. thus, in the equation of monetarism, by replacing pb*rgdp onto ngdp, we have got: ngdp = v ∗m where m money supply, v velocity of money. advances in systems science and application (2016) vol.16 no.3 3 since v (velocity of circulation) according to the monetarism model is a constant, and taken m (money supply) tending to infinity, then nominal gdp (ngdp) may also tend to infinity. even though thomas piketty properly finds out the formula (a)’s limit to assess the balanced economic growth, he himself falls into the formula’s trap. for instance, in his analysis of the real growth rate’s measurement tools thomaspikettyhas failed to go beyond the limits of the adam smith’s principle of a one-dimensional measure of return on capital. his artistic success in the establishment of two economic laws of national income and the national capital, could not spread further towards the correct breakdown of “nominal growth into real and inflationary components”, and towards the elimination of errors in evaluation methods for growth rateand inflation. in short, thomas piketty diagnosedproperly the market economy disease, but was not able to determine the treatment methods. identification and knowledge of the market economy development laws is needed to make key management decisions and adjustments of previously taken ones. otherwise, inaccurate directions and contradictory readings of analytical tools result in the destabilizing and social inequalities that are listed in the thomas piketty’sbook. as already mentioned, the theoretical basis of all actual current market equilibrium and balanced growth’s models, including piketty’s economic equilibrium model, up to this day is founded on the adam smith’s one-sided principle of return on capital’s measurement. adam smith’s one-sided approach consists in reducing the economic growth’stwo-dimensional,by working time and national currency, measurement system into the one-dimensional system measured by money. for adam smith, “the annual cost of the product” is identified with “the cost that is newly established during the year[3]. more precisely, he was correct that the value of the annual product can be reduced to the same components of the newly created during the year value. adam smith and his modern followers are right in establishing the value of the final product by deducting material resources spent costs from the “cost of the ready product”. from a mathematical point of view these transactions by a.smith and his followers are perfect. however, economy development indicators, both by the adam smith’s theory and by the karl marx’s theory, are measured in the first place by working time. by both theories the price of the working time is the base measure for productive forces of labour and capital . thus, adam smith specifically states that “labour measures the value, not only of that part of price which resolves itself into labour, but of that which resolves itself into rent, and of that which resolves itself into profit”[2]. then he gives a specific example:“in the price of corn, for example, one part pays the rent of the landlord, another pays the wages or maintenance of the labourers and labouring cattle employed in producing it, and the third pays the 4 s.baizakov, y.utembayev, a.r.oinarov, j.forrest:on mutual coincidence ... profit of the farmer. these three parts seem either immediately or ultimately to make up the whole price of corn. a fourth part, it may perhaps be thought is necessary for replacing the stock of the farmer, or for compensating the wear and tear of his labouring cattle, and other instruments of husbandry. but it must be considered, that the price of any instrument of husbandry, such as a labouring horse, is itself made up of the same time parts; the rent of the land upon which he is reared, the labourof tending and rearing him, and the profits of the farmer, who advances both the rent of this land, and the wages of this labour. though the price of the corn, therefore, may pay the price as well as the maintenance of the horse, the whole price still resolves itself, either immediately or ultimately, into the same three parts of rent, labour, and profit.”[ibid, p.105]. this approach of adam smith is static, and focused on the short term and for a momentary effect when the scientific and technological potential of the country has no time to be updated, or when its impact on the real economy can be ignored. karl marx’s approach is focused on long-term development and the market economy sustainable development. but his approach does not replace the adam smith’s principle of return on capital, but summarizes and complements it. a commodity, according to marx, is in development, its dividing onto “the goods and money is the law of the expression of the product as a commodity” [3]. it is this law thomas piketty did not consider by partly remaining within the framework of the adam smith’s principle of a one-dimensional measurement. he failed to take into account that goods and money were developing and transforming into capital, which became a form of its product or its form of money. in the third volume of “capital” which explores the development and selfdevelopment of capital in general karl marx introduces two-dimensional measurement of capital in its form of money and capital in the form of its goods. these measures or economic growth measurement “rulers” are being presented by working time in form of man-hours and money in form of the national currency of each country. marx’s summary with respect to the adam smith’s one-dimensional approach is the following: “he does not distinguish the dual nature of labour itself, i.e.does not distinguish between labour,as a cost of wage creating value, and labour, as a specific and useful labour creating commodities (use-value). the total amount of goods produced in a year i.e. an entire annual product is a product of useful labour active during the past year; all these products exist only due to the fact that publicly employedlabourwas consumed in diversified extensive system of various kinds of usefullabour. that is the only reason the cost of the means of production has been consumed during manufacturing but kept in the total cost of manufactured commodities, preserved as reappearing in a new kind. consequently, the entire annual product is a result of the useful labour expended during advances in systems science and application (2016) vol.16 no.3 5 the year. but only a part of the annual cost of the product has being created newly; this part is the newly created cost for the year, which embodies the sum of labour,spent during the year” [3]. our studies done based oneconomic science latest achievements, in particular the nobel prize leonid kantorovich and tjallingkoopmans’ duality principle, show that adam smith is partly right in saying that “the cost of the annual product”,according to that principle, can be reduced to “the newly created cost for the year.” the adam smith’s principle of return on capital allows to estimate the cost of capital in its form of money; while karl marx is right in saying that the total costs of producing a particular product, as capital, in the form of its goods are consumed in the production of the final product “capital plus income” [4]. both of these indicators are not just theoretical, but also have a crucial practical significance. in accordance with the principle of duality, they form the basis for the twodimensional measurement of the balanced economic growth. thus, the newly created value for the year by adam smith is an exact expression of the cost of capital in its form of money. the practical implementation of this concept is found in the nominal gdp, which is the cost of the final product. and the cost of the annual product by karl marx is an exact expression of the cost of capital in its form of goods. the practical implementation of this concept is in real gdp, which is to represent the volume of actually produced final product. the conclusion is that to eliminate errors in the measurement of inflation and real growth,which is possible by adding the adam smith’s one-dimensional approach with the karl marx’s approach based on assessmentof the labour’ total costs, that are objectively necessary for the production of the given structure’s final product amount. therefore, the rationale of the new market equilibrium equation can start from that point where thomaspikettystopped his theoretical research of two-dimensional measurement: of the national income ĺcby money, and of the national capital ĺc by working time, man-years. in this case, the duality theory allows you to take advantage of the twodimensional measurement criterion. firstly, the cost of the working time is measured by estimating the product’s direct and full labour-intensity of economic activities. second, the cost of the final product (nominal gdp) and resources total costs for the production of nominal gdp are set in monetary terms. according to the duality principle, solutions for interlinked problems identified on the basis of the industries intersectional balance report will meet following criteria [5-7]: l = t ∗x = t ∗ y (b) where t is direct line and tproduct fulllabour-intensity, l the entire fund of working time, in man-years, yfinal product value (nominal gdp) and x6 s.baizakov, y.utembayev, a.r.oinarov, j.forrest:on mutual coincidence ... resources full costs for the nominal gdp production. fourthly, we introduce the extension transformation below: the tool for solving the contradictory problem is extension transformation. by using certain transformations, an unfeasible problem can be transformed a feasible one. we only introduce the general concept and the basic types of transformation. in the formula (b), each component of the full labour-intensity tof the final product value y is determined by the scalar multiplication of the product’s direct labour-intensity components t by components of each column of the full costs technology matrixb = (e a)-1, which serves as a carrier of scientific and technological progress and, therefore, the true value of goods and services. at the national level we have the equalities t = l / x and t = l / y. from here we denote the level of scientific and technological potential (stp) as c and get as follows: c = t t = y x (c) stp ratio of the country c with its growth (+,-) is determined either by the magnitude of the margin, which is correlated with changes in the prices of goods and services, or by the amount of rent which changingrate is correlated with the intensity of productive resources utilization. in any case, the effect of volume, which represents the level of scientific and technological potential of the country, is measured by the difference between the growth in labour productivity, defined by value of the final product and used in its resources production, as they have both utilized the very same equal fund of working time: c c = y/l y/l − x/l x/l . the new equation of market equilibrium derives from the basis of adding the adam smith’sprinciple of return on capital to the principle based on the ratio of direct and fulllabour-intensity, as determined by the karl marx’s theory of capital’s labour value. thus, on one hand, the new equation ofthe balanced economic growth of nominal value of the final product (ngdp = y) defined by the formula: ngdp = c ∗x. in this formula the level of national income is measured by the value of nominal gdp. it is determined by multiplying the level of scientific and technological potential (stp) of the country c by a volume of domestic capital x. here, the scientific and technological potential (stp) of the country c represents coefficient of national capital efficiency. this equation is the analog of the first fundamental advances in systems science and application (2016) vol.16 no.3 7 law of market economy development, defined on the basis of the principle of duality. on the other hand, multiplying both sides of the gdp deflator equation by the purchasing power of the national currency (not to be mixed up with purchasing power parity) pp, we have got as follows: pp ∗ngdp = pp ∗ pb ∗rgdp hence we have a qualitatively new equation of balanced economic growth, which determines the actual volume of the final product rq: rq = pp ∗ngdp = pp ∗ pb ∗rgdp (d) where ngdp, as well as previously, represents nominal gdp, which determines the cost of the final product, and its multiplication by the level of the true value of money represents the real final product. since the pp * pb is equal to c = ngdp / xas by definition the value of money pp = rgdp / x, while the gdp deflator pb = ngdp / rgdp, then the purchasing power of money (not to be mixed up with the parity of purchasing power of money) is defined by formula: pp = c pb , where the value of the ccoefficient, which represents the level of scientific and technological potential of the country, is determined by the ratio of direct labourintensity to its full labour-intensity (t / t). this ratio is determined due to the principle of duality solution for interlinked problems, based on the data of given reports on the industries intersectional balance of the country; and it is equal to the ratio of the value of the final product cost used in the country to the total consumption of resources used for its production (y / x). to change economic policy, aimed at establishing a real growth index of prices of goods and services, and the level of depreciation of the national currency, is doable due to the system of economic laws and regulations, which are derived from combining the adam smith’s principle of one-dimensional measure of return on capital to the karl marx’sprinciple of two-dimensional measurement of return on labourand capital. the main ones, which are caused by the definition of the true cost of price indices of goods and services by the purchasing power of money, are as follows: 1) the law on determining overall impact of the takenincentives on innovative investmentsinto the economy and on scientific and technological improvement of the production process, including the strength of competition c(t): c(t) = ngdp/x(t) ̸= 1, 8 s.baizakov, y.utembayev, a.r.oinarov, j.forrest:on mutual coincidence ... though on the basis of return on equity ratio by adam smith an stp coefficient: c (t) = pp ∗ pb = 1,because by definition, the value of money pp = rgdp/ngdp , and the gdp deflator pb = ngdp/rgdp . 2) the basic law on determiningreal purchasing power of money, as the ratio of the scientific and technological potential of the country to the gdp deflator pp(t): pp (t) = c (t) /pb (t) . 3) the law on determining prices of goods and services pc(t) = 1/pp(t): pc (t) = 1/pp (t) = ngdp (t) /(c (t) ∗rgdp (t)). 4) leading law on determining real growth of the final product on the basis of nominal gdp growthrq(t): rq (t) = pp (t) ∗ngdp (t) . 5) the control law on determining real growth of the economy, on the basis of real gdp growth rq(t): rq (t) = c (t) ∗rgdp (t) . 6) the law on defininga gdp deflator, the general price deflation of goods and services pb(t): pb (t) = c (t) /pp (t) = ngdp (t) /rgdp (t) . 7) the law on determining net benefits of promoting scientific and technological advances and competition (±△c (t)): ±△c (t)% = (c (t) /c (t− 1))%− 100%. since any incentives on scientific and technological improvements have certain expenses of costs of labour, capital and materials then their return is associated with risks, and return on these funds can be a value greater than 100 and less than 100. however, where the competitive environment is developed there the contribution of science and technology potential into the development of the real economy, as a rule, would be positive. the exceptions are those developed countries where sources of reduction of material resources through scientific and technological advances have been exhausted; and those countries trying to maintainthe efficiency of their economies at the expense of reducing their national currenciespurchasing value. however, these attempts are easily detected by a tool developed with the help of a system of laws to promote competition in the market economy. advances in systems science and application (2016) vol.16 no.3 9 the path to sustainable economic growth goes through the open labour market, focused on the final product production. thus, last five six years the actual “restoration” of the old world monetary and financial system by transferring private risk to the global (system-wide) level, and by the public debt exponential increase in a number of leading countries and the money supply, makes the issue of transition to a new world economic order more difficult. the problem of transition from the current crisis and unstable state of the world economy to a sustainable growth mode should be investigated in the system, as shown above, in the unity of the macroeconomic, sectoral and microeconomic aspects of market economy development based on feedbacks that exist between the various regulatedagents, and of patterns of the long-term economic development countries of the world. this approach significantly expands the understanding of the causes of the global financial crisis, allowing to realise mechanisms of modern economy reproduction and to develop reliable tools for its sustainable development. thus, the history of economic thought tells that objective laws of market economy can justify new, adequate to the new world economic order, models of economic governance, by which it will be able to ensure the unity of the three product types: nominal, real and final. by and large, the core to ensure the unity of three gdp growth rates is the final product, the real value of which is determined by the equation (d) under the influence of the real purchasing power of the national currency at nominal gdp and the rate of scientific and technological potentialat the real gdp. according to this model of balanced economic growth the procedure of targeting nominal gdp is implemented instead of inflation targeting’monetary policy.this is the tate-shafarevich group: xe/k . here it is as a subscript: ax . references [1] piketty and thomas(2014),capital in the twenty-first century,the belknap press of harvard university press. [2] smith and adam(2007),an inquiry intothe nature and causes of the wealth of nations,moscow eksmo. [3] marx and karl(1978),capital,critique of political economy:the process of circulation of capital,vol.iv, 648 p. [4] marx and karl(1975),capital,critique of political economy:the process of capitalist production as a whole, vol.iii, part 1. [5] y. hasanli, s. bayzakov, v. valiyev and g. sarsembaeva. (2011), “modeling 10 s.baizakov, y.utembayev, a.r.oinarov, j.forrest:on mutual coincidence ... of the multiplicative effects of opening of the work places on the bases of ‘intersectoral labor balance’ (on example of azerbaijan and kazakhstan)”, ecomodazkz_full pape eng. [6] s. baizakov(2014), “model of the labour market, based on the final product”, science and innovation, no.12, pp.31-34. [7] s. baizakov(2015), “declaration on market laws of economic development of world countries”, veo of russia, no.195, pp.143-156. corresponding author sailau baizakov can be contacted at: baizakov37@mail.ru advances in systems science and applications (2013) vol.13 no.3 198-217 currency wars and a possible self-defense (i): how currency wars take place jeffrey forrest1, zack hopkins2 and sifeng liu3 1department of mathematics, slippery rock university, slippery rock, pa 16127, usa 211934 route 89, wattsburg, pa 16442, usa 3school of economics and management, nanjing university of aeronautics and astronautics, nanjing 210016, pr china abstract going along with the common knowledge that money can easily destroy a person, a family, and a healthy business enterprise, this work investigates the nature of currency wars in two parts. the first part shows, by making use of the systemic yoyo model and bernanke-gertler model of fundamental value of capital, how money can be purposefully and strategically employed as a weapon of mass destruction. based on how a currency war could be potentially raged against a nation, the second part of this work makes use of the results of feedback systems to develop a self-defense mechanism that could conceivably protect the nation under siege. keywords systemic yoyo, foreign investment, financial crisis, financial liberalization, speculative attack 1 introduction the main focus of this work is on financial crises that are caused and/or created purposefully by certain group(s) of people in order to acquire economic and political gains. here, financial crises include all those crises with respect to either respectively or jointly currency, credit, bank, debt, and the markets of stocks, bond, and financial derivatives, etc. they generally mean severe fluctuations and chaos that appear in the financial area of a nation, interfering very negatively with the operation of the real economy. each of these crises is accompanied by sudden deterioration of all or most financial indexes, stock market crash, capital flight, credit destruction, extremely tight money supply, rising interest rate, bank runs, bankruptcy of a large number of financial institutions, major decrease in the official reserves, inability to repay the interest and principal of maturing debt, currency devaluation both internally and externally, etc. because the modern-day national economies have been closely intertwined with each other to a high degree of globalization, when a nation or geographic region suffers from economic and financial crises, in terms of their damaging effects and adverse impacts, the crises tend to possess international and global characteristics. since the 1920s, there had appeared many large scale financial crises. the 199 advances in systems science and applications (2013) vol.13 no.3 most noteworthy of these crises include 1) the global stock market crisis that started on october 28, 1929, in the new york stock exchange and spread to other nations quickly, leading to a worldwide financial and economic crisis until 1933; 2) six us dollar crises one after another during the time period from 1960 to 1973; 3) a bank bankruptcy wave that started with the failure of the united states national bank in san diego and quickly spread to many other western countries during 1973-1975; 4) the debt crisis that broke out in august 1982 and quickly involved over 50 developing countries from around the world; 5) the international stock market crisis that started with the plummet of the dow jones industrial average on october 19, 1987; the consequent chaos of the wall street momentarily spread throughout all the major stock exchanges of the western world; 6) the british pound crisis of 1992; 7) mexican financial crisis of 1994; 8) the southeast asian financial crisis of 1997; 9) the russian currency, financial crisis of 1998; 10) the brazilian currency, financial crisis of 1999; 11) the argentinean currency, financial crisis of 2001. this work makes use of the systemic yoyo model and bernanke-gertler model of fundamental value of capital to show that when a large amount of foreign investments gathers in one place over either a long period or a short period of time and then leaves suddenly and massively, that local economy has to suffer through a positive bubble, caused by the increased money supply as a consequence of the foreign investments, and then a following negative, disastrous bubble, caused by the sudden dry-out of the money supply. because a large number of economic activities are either unexpectedly delayed or totally impossible to complete, the local investors are actually unable to continue to collect their originally expected dividends for many time periods to come. in other words, foreign investments can be employed as a weapon of mass destruction, if they leave strategically and suddenly, no matter whether they come quickly in a short period of time or slowly over a relatively longer period of time. with the conclusion that money can be employed as a weapon of mass destruction is established, this work then is turned to design a self-defense mechanism for the purpose of eliminating or greatly reducing the disastrous aftermath of such currency warfare. this work is organized as follows: in the first part, with relevant concepts systematically gathered in section 2, recent speculative attacks and currency crises jeffrey forrest: currency wars and a possible self-defense (i): how currency wars ... 200 are compared in section 3. by focusing on the analysis of the fundamental value of capitals, section 4 shows that how potentially currency wars can be launched purposefully. in the second part we present a possible way to build up a national defense against possible currency wars. then a few final remarks conclude this presentation. 2 the basic concepts and systemic intuition the general concept of financial crises includes the following four classes of crises: currency, bank, foreign debt, and systems. here, by currency crisis it represents such a situation that due to purposeful and targeted speculative activitiesagainsta nation, the nation’s currency suffers from drastic devaluation, or the nation’s government is forced to drastically increase its interest rate or spend a large amount of the foreign reserves to defend its currency. by bank crisis, it means actual and/or potential bank runs or such a scenario that a number of banks stop repaying their debts because of their falling into bankruptcy or the government is forced to interfere by providing large amounts of support. by foreign debt crisis it implies such a case that a nation can no longer repay its foreign debts on time, no matter whether the debtors are governments or private individuals. by systematic financial crisis, it stands for the destructive effects on the real economy due to severe damages of the financial infrastructure so that the efficiency of the financial markets is greatly affected. the connotation of systematic financial crises might overlap with those of other kinds of crises, while currency and bank crises might not lead to severe damages to a nations payment system. so, neither currency crises nor bank crises can be identified with systematic financial crises. in many circumstances, the specific definition of financial crises simply means currency crises. the early investigation of currency crises can be traced back to at least [1]. paul krugman treats a currency crisis as an internationalbalance-ofpayments crisis. he believes that in order to prevent their currencies to devaluate, countries with either a fixed exchange rate system or pegged exchange rate system would pay the price of either spending their international reserves or increasing inflation due to their raising domestic interest rates; when the governments give up on the fixed exchange rate system or the pegged exchange rate system, their currencies experience drastic drops in value, leading to currency crises. currency crises are generally indicated by the collapse of the fixed exchange rate system or forced adjustment to the system, such as official devaluation of the local currency, expanded floating range of the exchange rate, drastic decrease in international reserves, noticeable rise in the interest rate of local currency, etc. according to the literature, there are four main criteria for judging currency crises: (1) sudden and large scale changes in exchange rate; (2) the weighted average of exchange rate and foreign reserves fluctuates widely; (3) the weighted 201 advances in systems science and applications (2013) vol.13 no.3 average of exchange rate, foreign reserves, and interest rate vibrates wildly; (4) import drops drastically. the second criterion was established by kaminsky, lizondo and reinhard who believe that a currency crisis represents such a scenario that is caused by either devaluation of the nation’s currency or drastic drop in the nation’s international reserve or both as a consequence of an attack on a nation’s currency [2]. because of this reason, currency crises can be verified afterward by using the index named exchange market pressure (emp). this index stands for a weighted average between the monthly percentage change in the exchange rate of the local currency and that of the international reserves.because this index increases with the devaluation of the local currency and the loss of international reserve, major increases in this index indicate a strong pressure to sell off the local currency. because this second criterion possesses very practical operationality, it has been employed most widely among these four criteria. most of the financial crises that occurred since the 1990s have brought along with them clear characteristics of currency crises; and another outstanding feature of recent currency crises is that they have been accompanied by bank crises. such scenarios are referred to as twin crises. since the 1980s, along with the gradual strengthening of globalization of the capital markets, international flows of capital have been developed unprecedentedly with ever increasing magnitude, velocity, and accompanying dangers and risks. large amounts of short-term capitals that are not under the watch of any national government and international financial organization have been moving freely in the international financial markets for pursuing profit opportunities by making use of various new financial tools, trading platforms,and advanced trading technologies. these new characteristics of the international capital have led to frequent occurrence of turmoil in the international financial markets; speculative attacks happen often with ever increasing strength and duration of impact. the asian currency crises that started in 1997 further indicate that the currency crises caused by speculative attacks can also possibly develop into full scale financial crises and deepening social crises of large magnitude. the so-called international speculative capital or hot money represents such capital that is frequently moved within and between various markets in pursuit of short-term, high levels of profits without any particular fields of investment focus. speculative capital tends to be short term even though there are exceptions to this rule of thumb. one of the modern characteristics of international speculative capitals is their camouflage. at the same time, these capitals can also go along with the market cycles by pursuing midand long term investments. additionally, not all short-term capitals are speculative. for example, the short-term capitals involved in the financial intermediation and settlement of international trades, short-term interbank funds, banks’ short-term positions for allocation, etc., are jeffrey forrest: currency wars and a possible self-defense (i): how currency wars ... 202 not speculative in nature. along with the expansion in the size, circulation speed, and coverage of the international capital markets have international speculative capitals grown. based on their predictions on the changes in exchange rate, interest rate, security prices, gold price, or the prices of certain commodities, speculative capitals could be suddenly involved in large scale in both long and short trades in a short period of time. by substantially altering the composites of their portfolios and by affecting the confidence of the holders of other assets, these speculative capitals cause severe instability in the market prices so that short-term high profit opportunities could be created. such market behaviors that disturb the market prices and appear suddenly are known as speculative attacks. as limited by the constraint of pursuing after quick gains, international speculative capitals generally choose to attack such economic sectors and geographic areas that can hold large amount of capitals and allow fast movements of funds with expected high returns, and few financial regulations. these economic fields include (foreign) currencies, futures, options, precious metals, real estates, etc. from what has happened empirically in the past, it seems that international speculative capitals have been fond of attacking the currency of one nation or the currencies of several nations at the same time. most often seen speculative attacks are those that assault either fixed exchange rate systems or regulated exchange rate systems. speaking generally, when a nation either employs a fixed exchange rate system or pegs a target exchange rate, the speculators often make their judgment that as long as the official parity or the target exchange rate does not conflict with the fundamental conditions and states of the economy, the official exchange rate will be maintained. however, if the speculators believe that the fundamental states of the current economy could not sustain the prevalent level of exchange rate for long, they would launch a speculative attack in order to speed up the dissolution of the fixed exchange rate system so that opportunities of quick profits would be generated. under the conditions of either fixed or pegged exchange rate system, as long as there appears either a domestic inflation or recession accompanied with sustained current account deficits, the governmental promise on the fixed exchange rate would lose its reliability. it is because in these situations there is a heavy pressure to devaluate the local currency; in order to maintain the promised exchange rate, the government will be forced to mobilize and spend its international reserve. even with the support of the international financial markets, the fundamental imbalances existing in the economic states still cannot be corrected, which can most likely delay the occurrence of the devaluation of the local currency, although the devaluation will sooner or later happen inevitably. if speculators predicted this forthcoming event, they would mobilize their capitals ahead of 203 advances in systems science and applications (2013) vol.13 no.3 time and launch their speculative attack in order to position themselves for quick profits. by making use of the spot and forward transactions, futures contracts, options trades, and swaps of various financial tools, speculators carry out their multi-dimensional speculative strategy by positioning their capitals at the same time on the markets of foreign exchanges, securities, and all different forms of financial derivatives. because of the fixed exchange rate or the promise that the government would maintain the rate fixed, the risk to the speculators is actually quite low, because the direction along which the exchange rate would move is clear. to say the least, even if the prediction is incorrect, the worst is that the exchange rate parity did not change so that the most the speculators could lose is their minimal amounts of trade costs. hence, once a speculative wave is started, the magnitude in general is large, leading to the expected consequences, as a self-fulfilling prophecy of a humongous scale. 3 recent speculative attacks and currency crises during the time period of bretton woods system after world war ii, the strength and power of private capitals grew drastically; relevant speculative activities evolved with increasing levels of energy. their attacks on various national currencies from around the world were mostly successful and amplified with evergrowing vigor and intensity. the most typical are the british pound crisis of the late 1967, french franc crisis of august 1969, and the u.s. dollar crisis of 1971-1973. if we say that the root problem for bretton woods system to eventually collapse were the defects of the system itself, then the direct triggering factor for the system’s collapse would be the speculative attack of the international short-term hot money on the u.s. dollar-the base currency. when bretton woods system was over, the world was in a wave of deregulation, strengthening the market mechanism, promoting economic and financial liberalization. correspondingly, the international financial markets become further liberalized and global. along with the application of modern technology of communication and computer networks, financial derivatives and methods of trading are developed in abundance. all these political, societal, and technological advances provided the space for international capitals to grow and to be mobilized unprecedentedly. with their greatly increased speed of mobility, international capitals have launched frequent speculative attacks. among the most typical are the attacks on the pegged exchange rate system employed by some countries of latin america in the early 1980s, the mexican peso crisis of 1994, and financial crises of eastern asia during 1997-1998. the following provides a list of recent speculative attacks. for more details please consult with wang and hu [3]. case 1: the attacks on the exchange rate system of latin american countries in the early 1980s jeffrey forrest: currency wars and a possible self-defense (i): how currency wars ... 204 in 1978, chile, uruguay, and argentina decided to employ a crawling peg exchange rate system. each of these national governments established its plan to gradually depreciate its local currency against u.s. dollar. however, in their implementations, their rates of inflation were much higher than that in the u.s.a., while their degrees of depreciation were much smaller than the difference of the u.s. inflation rates. therefore, the over-evaluations of their local currencies made the deficits of their current accounts rise. during 1981-1982, the interest rate in the international financial markets reached an historical high, making the burdens of foreign debts and the deficits of the current accounts of these three countries difficult to sustain, so that it became inevitable for the local currencies to devaluate while departing from the targeted exchange rates. under this background, these countries respectively experienced speculative attacks, corresponding currency devaluations and the consequent capital flights, and crises of domestic financial institutions’ runs. case 2: mexican peso crisis of 1994 in 1982 after having suffered from its debt crisis, under the supervision of the imf mexico implemented a comprehensive policy for economic adjustment and reform, while tightening its economy and dramatically reducing its fiscal deficit. in 1987, mexico re-fixed its exchange rate between mexican peso and u.s. dollar. in january 1989, mexico started to employ a crawling peg exchange rate system, which was changed to a moving target regional exchange rate system in december 1992, while gradually expanding the floating range for peso. this series of measures of economic reform achieved a certain degree of success; the national economy steadily recovered. however, in 1994, mexican economy once again stalled while accompanied by political instability. therefore, the expectation and rumor for peso to depreciate grew; capitals fled one after another. interventions of the central bank made the market interest rate rise drastically, while the national foreign reserves were depleted quickly. on december 30, mexican government eventually had to allow peso to depreciate. however, the new exchange rate established after the depreciation immediately suffered from speculative attacks so that mexican government had to implement a floating exchange rate system. after then, the domestic economic conditions and the political situation made foreign investors extremely nervous, causing continued capital flight, banks subjected to runs, and the economy falling into crises. in the newly adopted floating exchange rate system, peso continued to depreciate; until the end of 1995, peso had reached consecutive historical lows one after another. case 3: the speculative attack on the joint floating mechanism of european monetary system (1992-1993) the national currencies within the european monetary system of the nations that were members of the european community had followed a joint floating ex205 advances in systems science and applications (2013) vol.13 no.3 change rate mechanism. this mechanismhad led to the creation of the european currency unit (ecu) and established the statutory central exchange rates between the individual national currencies and the european currency unit. therefore, between these member nations a system of fixed exchange rates was employed, while externally a joint floating rate system was implemented. in the early 1990s, the member nations of the european community experienced varied degrees of economic turmoil. uniformities in their respective macroeconomic states, such as inflation rates, unemployment rates, fiscal deficits, economic growth rates, etc., started to be broken. over time, it became clear that some member nations could no longer maintain their statutory exchange rates with ecu, providing opportunities for the international speculative capitals. in the late 1992, an initial round of speculative attacks appeared. among the first group of victims of the attacks were finnish marks and swedish krona. at the time, neither finland nor sweden was a member of the european community. however, they all hoped to join so that they voluntarily pegged their own currencies with ecu. under the speculative attacks, finland quickly gave up its fixed exchange rate and drastically depreciated its currency on september 8. on the contrary, swedish government decided to protect its krona by raising its short-term interest rate to 500% annually. that eventually defeated the speculative attacks. at the same time, british pound and italian lira also suffered from continued attacks. on september 11, european monetary system agreed for lira to depreciate 7%. although german central bank spent around 24 billion marks to support lira, three days later lira was still forced out of the european monetary system. by this time, the bank of england had lost several billions of u.s. dollars to protect its pound. even so, on september 16, the british pound was still forced to float freely. although french franc also suffered from speculative attacks, through joint interventions with germany and by greatly increasing the interest rate, the value of franc recovered. this crisis of european monetary system started in 1992 and lasted until 1993, during which speculative attacks often occurred. at the end of 1992, portuguese currency escudo depreciated; spanish currency pesetas was devalued once again, while swedish krona and norwegian krone started to float. in the early part of 1993, ireland’s pound depreciated, portuguese escudo depreciated another time, and spanish pesetas experienced its third round of depreciation. on the other side, french franc and danish kroner successfully stood against the sporadic speculative attacks. case 4: 1997 currency crises of east asia (1997-1998) in the 1980s and early 1990s, nations in southeast asia sped up their steps toward financial liberalization by completely opening up their domestic financial markets to attract maximal scales of foreign investments. such fast-speed jeffrey forrest: currency wars and a possible self-defense (i): how currency wars ... 206 liberalization led to drastic economic growth, which was known as “southeast asia miracle”. however, after entering the mid-1990s, the rising labor costs were translated into the decreased international competitiveness in their products so that deficits began to appear in the current accounts of some southeast asian countries. because these countries did not in a timely basis upgrade their industrial structures in order to keep pace with the increasing competitiveness of their products, the continued influx of the foreign capitals together with domestic investments led to the formation of economic bubbles and overheated sector of real estates. for example, in 1996, thailand’s balance of foreign debts had reached over 90 billion u.s. dollars with more than 40 billion dollars of shortand mid-term foreign debts, both of which surpassed the corresponding levels of foreign reserves at the start of 1997. additionally, because of the overheated investments, particularly the overheated investments in real estates, the bad debts of thai financial institutions had amounted to more than 30 billion u.s. dollars in early 1997. therefore, the public and foreign investors started to worry about the economic conditions and financial order in thailand, which inevitably helped to consolidate the expectation for baht to depreciate. at the same time, international speculators were also building up their monetary energy and preparing to launch their large scale attacks. on february 14, thai currency baht depreciated 5% against u.s. dollars, making the covered speculative attacks public. after then, baht suffered from ever increasing pressure to depreciation further; and interventions of bank of thailand were quickly exhausting the national foreign reserves. after mid-may, speculative capitals launched their new rounds of attacks, creating an eleven-year new low for the exchange rate of baht. several nations in southeast asia jointly intervened in the foreign exchange markets by buying in baht, while bank of thailand sacrificed 5 billion u.s. dollars of foreign reserves and once again raised its short-term interest rate. however, all of these still could notrebuild the public confidence and drive off the speculative attacks. eventually on july 2, bank of thailand was forced to allow baht to float freely in the exchange markets, causing baht to depreciate 20% against u.s. dollar on that single day. subsequently, the speculative attacks quickly spread over to the neighboring nations and regions so that philippines, malaysia, indonesia, singapore, hong kong, and taiwan were all affected. other than hong kong, the local currencies of these countries and regions all depreciated against u.s. dollar with different scales. at the same time, all these nations and regions except singapore and taiwan fell into deep financial and economic crises. after october of the same year, the crises spread over to south korea, causing south korean currency won to depreciate deeply against u.s. dollar and making the economy of south korea fall deeply into an economic crisis. case 5: russian currency crisis (1998) 207 advances in systems science and applications (2013) vol.13 no.3 at the early stage of russian economic transition, large amounts of international capitals entered russia. as of july 1, 1997, the accumulated foreign investments totaled 18 billion u.s. dollars, about 10 billion dollars of which were short-term capitals and invested in the securities markets. although the economic growth of the real economy was nearly zero, the stock market rallied rapidly in the first half of 1997; bubble expansion appeared in the prices of financial assets. when the currency crises of southeast asia broke out and started to spread toward the regions of northeast asia, the probability for russia to experience a similar currency crisis was very big, considering the market long-term expectation of instability for russian economy. starting in november 1997, speculators launched their attack on ruble with many others followed. during the end of 1997 and early part of 1998, russian government and central bank sacrificed foreign reserves to purchase ruble, while expanding rubles floating range and increasing the interest rate from 21% to 35%. although the situation was temporarily stabilized, large amounts of foreign reserves were lost. however, after may, another wave of speculation started, causing rubles exchange rate to fluctuate severely again. though russian government raised the interest rate to 150% and sought for international assistance in order to counter the attack, the situation was never successfully under control. on august 17, russian government had to expand ruble floating exchange corridor; that is, allowing ruble to depreciate. however, unexpectedly, the situation continued to worsen. eventually on september 2 without a choice the exchange corridor was abandoned. that meant that russian government gave up its over-two-year old managed target floating mechanism of ruble. case 6: brazilian currency crisis (1999) the currency crises that appeared in east asia and russia increased the market expectation for brazilian currency to depreciate, causing large amounts of capital leaving brazil since september 1998. at the end of 1998, brazilian congress did not pass the bills to increase the benefit taxes of civil servants and those of retired civil servants as contained in the fiscal adjustment plan, while the 1998 deficits of the fiscal balance, trades, and current accounts exceeded the expected levels. so, along with the decreasing market confidence, speculative attacks were triggered, forcing brazilian central bank to allow its real to float freely against u.s. dollar. all the speculative attacks that were launched since the 1980s have shown a good number of new characteristics, including amplifying scales and forces, enhancing dimensionality in terms of their strategies of attack, covering larger geographic areas, increasing publicity of planned attacks, etc. in particular, the main reasons for the amplifying scales and forces of the speculative attacks include: 1.the size of the international speculative capital has been increasing conjeffrey forrest: currency wars and a possible self-defense (i): how currency wars ... 208 stantly. since the 1980s, the economic growth of the main industrialized nations has been slowed, while the reform for financial liberalization of various countries has been strengthened. that caused large amounts of capitals toenter into the international financial markets to look for new opportunities. such continuous accumulation of international speculative capitals, together with the multiplying effect of international credit, makes the available speculative capitals increase multiple times. based on the relevant estimates of the international monetary fund, there were at least 7,200 billion u.s. dollars of speculative capitals currently floating around the international financial markets. that amount is equivalent to 20% of the world gdp; and each day, over 1,200 billion u.s. dollars of speculative capitals are looking for various profit opportunities. that is nearly one hundred times more than the amount of trades of real consumable goods. 2.the capitals available for speculation purposes have shown the tendency to collaborate and to take actions jointly. along with the rapid development of communication technology and the expanding availability of internet, a 24/7 operational system for the globalized exchange markets has appeared so that capitals can be transferred from one exchange market to another instantly. that makes the once stranglers and disbanded international speculators develop into powerful speculation assemblages. as a third force different from various national currency authorities and international financial organizations, they have constituted a major thread to the stability of each nation’s exchange system and the normal operation of the international currency system. 3.the introduction of financial derivatives provided a leveraged platform of trading for the speculators. since the 1970s, financial innovations have led to the profligate development of new financial derivatives and their trades. because of the characteristic of high leverages of the financial derivatives, speculators can trade such financial products that are valued tens or even hundreds of times of the little amounts of capital they mobilized. that enables a hedge fund to embark on trades worthseveral hundreds of billions of u.s. dollars with a small amount of capital, affecting the entire international financial markets. since the 1990s, the strategies of speculative attacks have been further developed. the traditional speculator would simply use the price differences between spot and future trades, while the current strategies of speculative attacks have been quite complicated. by utilizing all kinds of available financial tools, the speculator gets involved in various markets of the traditional tools and derivatives. it can be said that current speculative attacks make use of the inherent linkages among the prices of various financial products, traded in different markets, to make comprehensive and profitable arrangement of capitals. traditional speculative attacks have been isolated and scattered in different geographic regions. however, along with the recent deepening globalization 209 advances in systems science and applications (2013) vol.13 no.3 of the international financial markets, and along with the further unification of regional economies, speculative attacks have also showed clear regional characteristics. no matter whether it is the latin american currency crisis of the early 1980s, the mexican crisis of 1994, the european monetary system crisis of 1992, or the southeast sian crisis of 1997, each started with the initial shocks in such a market that contained the most concentrated imbalances. then, the initial shocks spread over to other neighboring markets, displaying a dynamic process of mutual influences. traditional speculations used to be hidden or semi-public arbitrage activities. however, since the 1990s, along with the deregulation and financial liberalization of the international financial markets and the widening use of information and network technologies, originally speculative activities have been gradually evolved into open and purposeful attacks on specified currencies. such openness and publicity could be and have been strengthening the market expectation of depreciation of the target currency. for example, at the end of 1997, george soros published an article to openly declare that he and associates would launch another attack on hong kong dollars. after that, he announced in newspapers that the next currency crisis would appear in russia, which was later shown to be accurate. soon after then, soros commented that brazilian currency was evaluated too high so that brazilian real would be his next target of attack, which was indeed what happened next. before we start to develop and present our main results, let us first look at the systemic intuition that lies underneath this work. when von bertalanffy pointed out that the fundamental character of living things is its organization [4], the customary investigation of individual parts and processes cannot provide a complete explanation of the phenomenon of life, this holistic view of nature and social events has spread over all corners of science and technology [5]. accompanying this realization of the holistic nature, in the past 80 some years, studies in systems science and systems thinking have brought forward brand new understandings and discoveries to some of the major unsettled problems in the conventional science [6-7]. similar to how numbers are theoretically abstracted, systems can also be proposed out of any and every object, event, and process of concern. for instance, behind collections of objects, say, apples, there is a set of numbers such as 0 (apples), 1 (apple), 2 (apples), 3 (apples), ...; and behind each organization, such a regional economy, there is an abstract, theoretical system within which the relevant whole, component parts, and the related interconnectedness are emphasized. and, it is because of these interconnected whole and parts that the totality is known as an economy. in other words, when internal structures can be ignored, numbers can be very useful; otherwise the world consists of dominantly systems jeffrey forrest: currency wars and a possible self-defense (i): how currency wars ... 210 (or structures or organizations). historically speaking, on top of numbers and quantities has traditional science been developed; and along with systemhood comes the systems science. that jointly gives rise of a 2-dimensional spectrum of knowledge, where the classical science, which is classified by the thinghood it studies, constitutes the first dimension, and the systems science, which investigates structures and organizations, forms the genuine second dimension [8]. the importance of the systems science, the second dimension of knowledge, cannot be in any way over-emphasized. for example, when there are difficulties in studying dynamics in an n-dimensional space, one can conveniently get help from a higher-dimensional space. in particular, when a one-dimensional flow is stopped by a blockage located over a fixed interval, the movement of the flow has to cease. however, if the flow is located in a two-dimensional space, instead of being completely stopped, the 1-dimensional blockage would only create a local (minor) irregularity in the otherwise linear movement of the flow (that is how nonlinearity appears [9]. additionally, if one desires to peek into the internal structure of the 1-dimensional blockage, he can simply take advantage of the second dimension by looking into the blockage from either above or below the blockage. that is, when an extra dimension is available, science will gain additional strength in terms of solving more problems that have been challenging the very survival of the mankind since the start of history. additional to the afore-described strong promise of systems science, on the basis of the blown-up theory [10], the systemic yoyo model, fig.1, is formally introduced by lin for each and every system [11], be they tangible or intangible, physical or intellectual. (a) (b) (c) fig.1 the eddy motion model of the general system in particular, on the basis of the blown-up theory and the discussion on whether or not the world can be seen from the viewpoint of systems [12-13], the concepts of black holes, big bangs, and converging and diverging eddy motions are coined together in the model shown in fig.1. in other words, each system or object 211 advances in systems science and applications (2013) vol.13 no.3 considered in a study is a multi-dimensional entity that spins about its either visible or invisible axis. if we fathom such a spinning entity in our 3-dimensional space, we will have a structure as shown in fig.1(a). the side of black hole sucks in all things, such as materials, information, energy, profit, etc. after funneling through the short narrow neck, all things are spit out in the form of a big bang. some of the materials, spit out from the end of big bang, never return to the other side and some will (fig.1(b)). for the sake of convenience of communication, such a structure as shown in fig.1(a), is referred to as a (chinese) yoyo due to its general shape. more specifically, what this systemic model says is that each physical or intellectual entity in the universe, be it a tangible or intangible object, a living being, an organization, a culture, a civilization, etc., can all be seen as a kind of realization of a certain multi-dimensional spinning yoyo with either an invisible or visible spin field around it. it stays in a constant spinning motion as depicted in fig.1(a). if it does stop its spinning, it will no longer exist as an identifiable system. what fig.1(c) shows is that due to the interactions between the eddy field, which spins perpendicularly to the axis of spin, of the model, and the meridian field, which rotates parallel to axis of spin, all the materials that actually return to the black-hole side travel along a spiral trajectory. to show this yoyo model can indeed, as expected, play the role of intuition and playground for systems researchers, literature [5, 9] have successfully applied it to investigate newtonian physics of motion, the concept of energy, economics, finance, history, foundations of mathematics, small-probability disastrous weather forecasting, civilization, business organizations, the mind, among others. now, what is important to our work in hand here, which constitutes the necessary intuition for our reasoning, is that the systemic yoyo model implies that each economy is a system so that it can be investigated as a pool of rotational fluids of information, knowledge and money. and when the world economy is concerned with, we really have a theoretical ocean of rotational pools (of fluids) that interact with other. with time, some regional pools are destroyed while some stronger, more powerful spin fields are formed. in other words, the so-called speculative attacks, as described earlier, are natural phenomena that appear along with the globalization of the world economy. so, one natural question is how such attacks appear and how their damaging effect can be maintained at the theoretical height first and then at the level of real life practice. in the rest of this work, we will address this question by using some of the recent results of systems research. 4 one possible form of currency wars according to literature[14], the fundamental value of a particular capital is equal to the present value of the dividends the capital is expected to generate throughout the indefinite future. symbolically, the fundamental value qt of a depreciable jeffrey forrest: currency wars and a possible self-defense (i): how currency wars ... 212 capital in period t is given by qt =et( ∞∑ i=0 [ (1− δ)id(t+1+i)∏i j=0r q t+1+i ]) =et( dt+1 rq t+1 + (1− δ)1dt+2 rq t+1r q t+2 + (1− δ)2dt+3 rq t+1r q t+2r q t+3 + ...) (1) where et stands for the mathematical expectation as of period t ,δ the rate of physical depreciation of the capital, dt+i the dividends and rq t+1 the relevant stochastic gross discount rate at t for dividends received in period t+ 1. then, we can rewrite equ. (1) as follows: qt = et( dt+1 + (1− δ)qt+1 rq t+1 ) (2) because of various reasons, such as fads, the market price st of the capital differs persistently from the capitals fundamental value qt. when st ̸= qt, we say, as in literature [14], there is a bubble. however, to be more specific, when st > qt, we say that there is a positive bubble in the market place; and a negative bubble, when st < qt. in the realistic market place, asset prices mostly like deviate from the fundamental values due to various reasons, such as liquidity trading or to waves of alternating optimism or pessimism. if a bubble exists at period t with probability p to persist into the next period, then by using the mathematical expectations the difference between the market price and the fundamental value of the capital in period (t+ 1) satisfies the following: p(st+1 −qt+1) + (1− p) · 0 = a[(st −qt)r q t+1] + (1− a) · 0 (3) it means that the mathematically expected (st+1 −qt+1) value with probability p for st+1 ̸= qt+1 to happen is equal to the expected growth of the t-period difference (st −qt)r q t+1 with probability a(> p) for st −qt ̸= 0 . that is, what is expected is a more severe “bubbl”, since a/p > 1 . so, if we assume a/p < 1 , it means that the bubble in period (t+ 1) is expected to be less severe than in period t. from equ.(3), it follows that we know the following expression is true: p · st+1 −qt+1 rq t+1 = a(st −qt) (4) now by taking the mathematical expected value for in period t, we have the following: et( st+1 −qt+1 rq t+1 ) = p( st+1 −qt+1 rq t+1 ) + (1− p)· = a(st −qt) (5) 213 advances in systems science and applications (2013) vol.13 no.3 equ.(2) implies that qt =et[ dt+1 + (1− δ)st+1 − (1− δ)st+1 + (1− δ)qt rq t+1 ] =et[ dt+1 + (1− δ)st+1 rq t+1 − (1− δ) st+1 −qt+1 rq t+1 ] =et[ dt+1 + (1− δ)st+1 rq t+1 ]− (1− δ)et st+1 −qt+1 rq t+1 =et[ dt+1 + (1− δ)st+1 rq t+1 ]− (1− δ)a(st −qt) therefore, we have qt + (1− δ)a(st −qt) =et[ dt+1 + (1− δ)st+1 rq t+1 ] which is the equivalent to: st[qt + (1− δ)a(st −qt)] st =et[ dt+1 + (1− δ)st+1 rq t+1 ] by isolating the factor in the numerator and cross multiplying the rest from the left hand side onto the right hand side produce the following: st =et [dt+1 + (1− δ)st+1]st rq t+1[qt + (1− δ)a(st −qt)] =et [dt+1 + (1− δ)st+1]st rq t+1[a(1− δ)st − a(1− δ)qt] +qt =et [dt+1 + (1− δ)st+1]st rq t+1[bst + (1− b)qt] , where b = a(1− δ) =et dt+1 + (1− δ)st+1 rq t+1[b+ (1− b)qt st ] = et dt+1 + (1− δ)st+1 rs t+1 therefore, we have derived at st = et dt+1 + (1− δ)st+1 rs t+1 (6) where the return on stocks rs t+1 is related to the fundamental return of the capital rq t+1 as follows: rs t+1 = rq t+1[b+ (1− b) qt st ] (7) jeffrey forrest: currency wars and a possible self-defense (i): how currency wars ... 214 if st > qt , then rs t+1 < rq t+1 , meaning that the expected stock return st = et dt+1+(1−δ)st+1 rs t+1 is less than the fundamental return of st = et dt+1+(1−δ)st+1 rq t+1 . from equ.(3), it follows that st −qt = p a × st+1 −qt+1 rq t+1 (8) which implies that at period t+ 1, there is a positive bubble if and only if there is a positive bubble at period t. because 0 < p < a < 1, if st+1 > qt+1 (assuming a positive bubble), then from 0 < p a < 1 , equ.(8) implies that the current bubble st −qt is smaller than the fundamental return of the bubble in period t+ 1. in other words, the bubble in period t+ 1 gets more severe than when in period t. now, if st+1 < qt+1 with 0 < p < a < 1, then equ.(8) implies that the underpricing st+1 of the asset qt+1 in period t+ 1 is more severe than that in period t. moreover, this equation also indicates that at period t+ 1 there is a negative bubble if and only if there is a negative bubble at period t. these two conclusions evidently contradict the efficient market hypothesis, because these analogies indicate that if in period t the market over prices the asset, then the overpricing will continue forever; and if in period t the market underprices the asset, then the underpricing will also continue forever. that is, neither positive nor negative bubble will ever crash. so there are two possibilities: (1) the model in equ.(8) does not hold true in general, or (2) the efficient market hypothesis is not ever true. evidence appears to show that the efficient market hypothesis holds true at least occasionally and also bubbles, both positive and negative, do burst [15-16]. so, the model in equ.(8) needs to be modified in order to describe the more realistic market situation better. if in equ.(8) we assume 0 < p < a < 1, then a similar analysis as above indicates that the phenomenon of overpricing or under pricing disappears over time. and, if 0 < p = a < 1 is assumed, then the model in equ.(8) implies that the existing underpricing or overpricing stays fundamentally stable. if 0 < p < a < 1 and a ≈ 0 are assumed, then equ.(8) implies that p a ≈ +∞ so that the fundamental return st+1−qt+1 rq t+1 of the asset approaches 0, meaning that the existing bubble gradually disappears with time. next, let us focus on the analysis of equs.(1) and (2). in particular, assume that in the (t+ i)-th period foreign investments are suddenly increased drastically due to expected appearance of activities from the current weak economic state and weak local currency. so for this period, δ(= the physical depreciation rate of capital) would increase due to the increased amount of money supply, which in turn pushes up the inflation. so (1− δ)i would decrease drastically if the influx of foreign investments is large. at the same time, rq t+i (= the stochas215 advances in systems science and applications (2013) vol.13 no.3 tic gross discount rate of the (t+ i)-th period) would also increase, because of the increased inflation while the dividends dt+i would generally decrease due to the reason that everybody would like to reinvest much of the available capital back into the maket in order to capture the rising book values, including stocks, real estate, and others with of impressive increasing prices. that is, the present value of the return of the (t+ i)-th period (1−δ)idt+i rq t+1...r q t+i would drop from the level of expectation. in reality, the investor would hold onto the increased book value by receiving less tangible returns, hoping that the book value would continue to rise drastically. due to the wide and conveniently availability of capital, caused by the increased money supply, the local economic activities pick up too in large quantities, while the interest rate also goes up due to the fact that the central bank, in order to control the inflation, revises the interest rate and attempts to limit the money supply. at this very moment of financially prosperity, assume that a huge amount of foreign investments suddenly leave in the (t+ j)-th period, where j > i, because of the much higher prices in assets, in capital investments, etc., for them to take profit in order to move their capitals to other regions to capture new economic opportunities. so, in the (t+ j)-th period, when a huge amount of foreign investment leaves, most of the economic activities that got started because of the foreign investments become stalled and/or negatively impacted. therefore, a good portion of the local investments is forced to be retained with the interrupted economic activities. that is, the investors of the local investments can no longer receive their expected dividends, and at the same time a large amount of the investments evaporates totally if many of the stalled activities can no longer be continued until their expected completion, or are indefinitely delayed. that of course costs additional local capital to do the cleanup of what is left behind, unfinished, and unusable ruins. in particular, in equs. (1) and (2), the dividends dt+j of the (t+ i)-th period decreases drastically and the remaining future dividends, if they still come fortunately as expected, would have to be used to bail out (to finish up) some of the other potentially possibly profitable projects. that is, right before the large amount of foreign investments leaves suddenly and strategically, the local economy is more active than ever before. hence, the stochastic gross discount rate rq t+i of the (t+ i)-th period would be much lower than rq t+j of the (t+ j)-th period, because much higher returns on earlier investments are optimistically expected. so, the present ratio: (1−δ)idt+i rq t+1...r q t+i...r q t+j would in reality be very close to zero. summarizing what is just analyzed above, one can see that when a large amount of foreign investments gathers in one place over either a long period or a short period of time and then leaves suddenly and massively, that local ejeffrey forrest: currency wars and a possible self-defense (i): how currency wars ... 216 conomy has to suffer through a positive bubble, caused by the increased money supply as a consequence of the foreign investments, and then a following negative, disastrous bubble, caused by the sudden dry-out of the money supply. and due to a large number of economic activities that are either unexpectedly delayed or totally impossible to complete, the local investors are actually unable to continue to collect their originally expected dividends for many periods to come. that is, foreign investments can be employed as a weapon of mass destruction, if they leave strategically and suddenly, no matter whether they come quickly in a short period of time or slowly over a relatively longer period of time. acknowledges the authors would like to give thanks to the national high-tech program (863) of china (2007aa03z115), independent fund of state key lab. of material processing and die & mould technology of huazhong university of sci & technol and open fund of state key lab. of powder metallurgy of central south university of china (2008112022). the authors also thank for hua yan, bin hua and the analytic and testing center of huazhong university of science & technology for their assistance. references [1] krugman p. (1979), “a model of balance-of-payments crises”, journal of money. credit and banking, vol.11, pp.311-25. [2] kaminsky g, lizondo s. and reinhart c. m. (1998), “leading indicators of currency crises”, imf staff papers. palgrave macmillan, vol.45, no.1, pp.1-48, march. [3] wang r. x, hu g. h. (2005), international finance, wuhan. hubei: press of wuhan university of science and technology. [4] von bertalanffy l. (1924), einfuhrung in spengler’s werk, literaturblatt kolnische zeitung. [5] lin y. and forrest b. (2011), systemic structure behind human organizations: from civilizations to individuals, new york: springer. [6] lin y. (1999), general systems theory: a mathematical approach, new york: plenum and kluwer academic publishers. [7] klir g. (1985), architecture of systems problem solving, new york. ny: plenum press. 217 advances in systems science and applications (2013) vol.13 no.3 [8] klir g. (2001), facets of systems science, new york: springer. [9] lin y. (2008), systemic yoyos: some impacts of the second dimension, new york: auerbach publications, am imprint of taylor and francis. [10] wu y. and lin y. (2002), beyond nonstructural quantitative analysis: blown-ups, spinning currents and modern science, river edge nj: world scientific. [11] lin y. (2007), “systemic yoyo model and applications in newton’s, kepler’s laws, etc”, kybernetes: the international journal of cybernetics, systems and management science, vol.36 no.3-4 pp.484-516. [12] lin y. (1988), “can the world be studied in the viewpoint of systems”, mathl. comput. modeling, vol.11, pp.738-742. [13] lin y, ma y. and port. r. (1990), “several epistemological problems related to the concept of systems”, math. comput. modeling, vol.14, pp.52-57. [14] bernanke b., gertler m. (1999), “monetary policy and asset price volatility”, economic review. federal reserve bank of kansas city, fourth quarter, pp.1751. [15] beechey m, gruen d. and vickrey j. (2000), “the efficient markets hypothesis: a survey”, reserve bank of australia in its series rba research discussion papers numbered rdp. [16] smith v. l, suchanek g. l. and williams a. w. (1988), “bubbles. crashes. and endogenous expectations in experimental spot asset markets”, econometrica, vol.56, no.5 pp.1119-1151. corresponding author author can be contracted at jeffrey.forrest@sru.edu microsoft word 16 y. sun, j. luo, g.f. mi, x.j.wang,x.lin--numerical simulation and defect analysis in the casting of the nod advances in systems science and applications (2011), vol.11, no.3-4 322-330 issn 1078-6236 international institute for general systems studies, inc numerical simulation and defect analysis in the casting of the nodular cast iron truck rear axle y. sun1, j. luo1, g.f. mi2, x.j. wang1 and x. lin3 1state key laboratory of mechanical transmission, chongqing university, chongqing 400030, china 2 college of material science and engineering，henan polytechnic university, jiaozuo 454000, china 3 state key laboratory of solidification processing, northwestern polytechnical university, xi’an 710072, china abstract the paper analyzes the reasons of some typical defects such as shrinkage porosity (macro or micro), shrinkage cavity, cold shut, segregation and hot crack which occurring the truck rear axle casing using nodular cast iron during the casting process. a three-dimensional (3d) cad engineering model is created by the pro/e software, one 3d fdm (finite difference methods) numerical solidification model and z-cast simulation software are used to study the casting solidification of the rear axle casing using nodular cast iron. based on the simulation research results, the structure design of the truck rear axle casing is improved and the casting technique is developed, these simulations and experiments show that these optimization methods are contribute to reduce the casting defect and improve the product quality. keywords pouring system, mold filling, solidification, simulation, casting technique 1.introduction as an important component of truck, the rear axle casing need to withstand the bending stress, torque stress and fatigue during truck transport (takeda 1994), so the rear axle casing will have a good comprehensive mechanical properties and behaviors (such as, high strength and good toughness) by a good casting (pinkaew 2007). the general manufacture technique of a truck rear axle is casting technology, and the general material for manufacturing the truck rear axle is nodular cast iron which is of some good casting characteristics and behaviors (wang, 2003). but as a result of graphite expansion action and effect, which can lead to the fluid feeding channel is blocked and lose the feeding ability in later period of casting solidification, so many kinds of defects will present in the casting, such as cavity, shrinkage and crack. on the other hand, the structure of pouring system and casting technology can affect the stability on casting mold filling and solidification process, when the casting mold filling is lack of physical stability, it can lead to splashing, spatter, gas evolved and molten metal oxidized etc., the formation of casting defects includes sand burning, pore, oxidation and slag (wei, 1996). these above defects reduce casting strength, toughness and quality of the truck rear axle, which is very dangerous to a truck transport. many experimental studies on casting technology are carried out in order to improve casting quality of a truck rear axle and to guarantee the safety of truck transport (pinkaew 2007; huang 2004; gui 2004). in previous research works, these issues are usually dealt with according to personal intuition and experience( wei,1996; liu, 2005), the cost of this traditional research method is very high and the time is more. now the cad/cam/cae techniques are employed to cope with optimization design in casting process (wu, 2005; luo, 2006), which is a more efficiently and low-cost method and can save a plenty of time (luo, 2007; mi, 2007). the paper is supported by the natural science foundation project of cq cstc2008bb3303 and the program for new century excellent talents in university (ncet-08-0607). 323 sun: numerical simulation and defect analysis in the casting of the nodular cast iron truck rear axle the paper will study the casting process of the truck rear axle casing by the numerical simulation method, and then improve the casting technology and casting structure in order to improve the defects of he truck rear axle casing. 2.analysis of casting defects 2.1 structure and material of the truck rear axle casing fig.1 shows that structure of the truck rear axle casing, which is a symmetrical structure, the middle part of the truck rear axle casing is a rotator with an internal flange, the two waist parts are a smoothing transitional surface from the rotator, and there has a internal cavity structure with some small sidesteps. the truck rear axle casing has some groove, sidesteps and flanges, the internal mould cavity structure of the rear axle casing is a main structure forms, which is made up of complex geometrical structure. the basic size of the rear axle casing is 1559mm×448mm×196mm, the thinnest wall of the rear axle casing is 10mm, the thickest part of the rear axle casing is 20mm, and the average thickness of the rear axle casing is 15mm. fig.1. structure of the rear axle casing the nodular cast iron is used to manufacture the truck rear axle casing. the type of material is qt450-10 (trademark in china.), which is ferrite type nodular cast iron. this kind of nodular cast iron includes a high c and si element contents, the content of p and s element are in a low level, the nodular cast iron has a good fluidity and casting behaviors. so the casting defects such as misrun, cold-shut is difficult to appear in the casting workpiece at an appropriate temperature and reasonable pouring system condition, the casting workpiece has a good mechanical properties (strength and toughness ). 2.2 casting defects of the truck rear axle casing fig.2 shows an old pouring system of the truck rear axle casing, which is used in the actual casting process at some factory. the pouring system locates at the legs of the truck rear axle casing, and has two casting ladles and two risers. this structure design of pouring system is of compact and simple characteristics, it is advantageous to a good mold filling and save molten metal on the casting process. the shortcoming of this pouring system in fig.2 is as following by the experimental studies and theory analysis: (1) there are two casting ladles. because the two casting ladles must work simultaneously, so this pouring system makes the casting technology complex and casting quality controlling difficult. (2) there are some differences of casting temperature and beginning time between the two casting ladles, so the casting temperature field is not uniformly at the two sides of the truck rear axle casing, which disturbs the direction of heat transfer, temperature gradient and solidification process, so the casting workpiece is likely to present many defects (such as shrinkage) at the later period of casting solidification. (3) this pouring system increases the instability of mould filling when the molten metal liquid is poured into two independent casting ladles at same time. it is very possible that the sand advances in systems science and applications (2011), vol.11, no.3-4 324 burning, splashing, spatter, gas evolved, slag and molten metal oxidized take place in the casting process, which has a bad effect on the casting quality. fig.2. the old pouring system (a) slag (b) sand burning fig.3. the casting defects fig.3 shows an actual typical appearance casting defects of the rear axle casing, such as slag and sand burning, by adopting this pouring system. 3.new pouring system design and technology optimization 3.1 new pouring system design based on actual experiments and above analysis, the pro/e software is used to establish the three-dimensional (3d) cad engineering model for the truck rear axle casing with a new pouring system, the cad model is one precondition for the structure design and numerical simulation of new truck rear axle casing. by improving the pouring system and considering many important parameters (such as casting shrinkage rate, machining allowance etc.), a 3d engineering model of the rear axle casing with a new pouring system is designed as shown in fig.4. the cad model is saved as stl format by pro/e software, in order that the 3d model can be input fdm numerical simulation software for meshing and simulation. 325 sun: numerical simulation and defect analysis in the casting of the nodular cast iron truck rear axle fig.4. the new pouring system and 3d solid model 3.2 new structure analysis 3.2.1 model of casting process simulation the z-cast fdm simulation software is used to simulate casting mold filling and solidification process in order to check the pouring system reliability, optimize the structure design and guide the casting technology. the simulation model is shown in fig.4 which is input from the pro/e model. the continuity equation, momentum conservation (navier-stokes) equation and energy (enthalpy) conservation equation are respectively as following, 0= ∂ ∂ + ∂ ∂ + ∂ ∂ z w y v x u (1) ug x p z uw y uv x uu t u x 2)( ∇++ ∂ ∂ −= ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ μρρ vg y p z vw y vv x vu t v y 2)( ∇++ ∂ ∂ −= ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ μρρ (2) wg z p z ww y wv x wu t w z 2)( ∇++ ∂ ∂ −= ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ μρρ s z tk zy tk yx tk x z tcw y tcv x tcu t tc + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ =⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ )()()( ρρρρ (3) where , ,u v w is the velocity on three direction. , ,x y z is direction factors. g is the gravity, ρ is density, f is the rate of volume, k is the heat conduction ratio, s is the inner heat source. the details of mathematical model please refer to the user guide manual of z-cast. the 3d simulation model is meshed using one uniform method (about 5mm per grid), the meshing numbers of casting solid model is 353×168×90. 3.2.2 results analysis of mold filling simulation fig.5, fig.6 and fig.7 are the temperature field when the casting mold filling states is 30%, 60% and 100% respectively. the 30% filling state (in fig.5) shows that the internal temperature gradient of casting workpiece is small, the temperature at the inlet of ingate is little higher than that of at the bottom of flange. the 60% filling state (in fig.6) shows the casting temperature of the flange body ( in middle of the rear axle casing) is 1300 . ℃ fig.5. the temperature field about the 30% casting mold filling states advances in systems science and applications (2011), vol.11, no.3-4 326 the temperature of two risers and two edges of the rear axle casing are higher than that of the flange body. the molten metal flows into the casting mould steadily through the ingate and long pouring channel, which lies on one side of the casting workpiece. the liquid level of the casting mould rises also steadily in the mold filling process, so the bottom of casting workpiece has little defects about cold shut and feeding insufficiency in this way. fig.6. the temperature field about the 60% casting mold filling states it shows obviously that the casting workpiece establishes a good temperature gradient when the mold filling percents is 100% in fig.7. the temperature of the risers is higher than that of the casting body. this kind of temperature distribution is helpful to bring into full filling and overflow, and reduce the existences of internal shrinkages defect in the casting workpiece. fig.7. the temperature field about the 100%casting mold filling states 3.2.3 results analysis of casting solidification simulation fig.8. the temperature field about solidification time is 900 second 327 sun: numerical simulation and defect analysis in the casting of the nodular cast iron truck rear axle fig.9. the temperature field about solidification time is 1200 second at the 1200s solidification final stage (in fig.9), it shows obviously that solidification speed of the ingate is faster than that of casting body, the temperature of the ingate is lower that that of casting body, so the pouring channel is closed. it makes the risers can not play the role of feeding and overflow at the end period of solidification, and then the expanded solid graphite make metal liquid attempt to flows back into pouring channel from the casting body, so many defects ( such as hot crack and shrinkage cavity) take place at this solidification stage. in addition, there has one interface between the ingate and casting mould, the temperature at combining site where the fluid enter casting mould by the ingate is higher than other parts, it causes the solidification time is long at the combining site, many defects also present at this site, the reason is that the structure of inner channel near this combining site is very complex. so the pouring system , riser and structure of the ingate need to improve. 3.3 optimization a new improvement pouring system is designed, the position of the casting ladles and risers are optimized, and one new structure of the ingate at the combining site (between pouring channel and casting workpiece) is proposed, at the aid of the cad technique( pro/e software) and many numerical simulations (such as z-cast, procast and fluent softwares ) for mold filling and solidification process. for avoiding early solidification of molten metal in the ingate, the height of the ingate increases from 12mm to 24mm, and the length of the ingate improve from 75mm to 50mm considering the feeding ability of the risers and sand burning ratio. correspondingly the width of the horizontal runner changes from 22mm to 24mm, the diameter of the sprue bottom part which is located in the middle of the runner adds from 41mm to 43mm, and the position of the riser is also change. the temperature field of the new improvement system is shown in fig.10 and fig.11 respectively when the solidification time is 900 second and 1200 second after the mold filling process is finished. fig.10. the temperature field about solidification time is 900 second (improved) advances in systems science and applications (2011), vol.11, no.3-4 328 it shows obviously that the temperatures at the ingate and the bottom of riser are higher than that of casting body, so the pouring channel is unblocked and feeding at the final stage of casting solidification, this good temperature distribution in fig.11 is also helpful to reduce many internal defects in casting workpiece. compared fig.9 with fig.11, the numerical simulation results show that the improvement system is of availability. fig.11. the temperature field about solidification time is 1200 second (improved) on the other hand, the truck rear axle casing has a complicated internal structure. in order to improve the casting quality, the internal structure of the truck rear axle casing is also optimized as shown in fig.12. cooperated with the optimization of pouring system, the casting technology is improved to suit for the new optimization structure simultaneously. the results of numerical simulation and experiences show that the truck rear axle casing has a good casting profile and high mechanical properties, and the casting production meet the requirement of design and quality. fig.12. the optimized internal structure and pouring system of casting workpiece 4. discusses compared to fig.2, the runner of the new pouring system (in fig. 4) employ a new smooth circle transition structure, which can ensure that molten metal liquid flow steadily from the ingate into the casting workpiece with an uniform speed and adequate quantity, this kind of mold filling method can prevent the fluid form a turbulence state, and avoid the air mixture enter the casting, reduce the defects such as metal oxide, shrinkage porosity (macro or micro). the new pouring system is also of great benefit to the casting behaviors and improve gases release from molten metal, it can make molten metal liquid feeding and reduce the appearance of slag, shrinkage and hot cracks. at the same time, it is very convenient that workers operate the pouring system, in one word, the integrated methods including the new structure, casting 329 sun: numerical simulation and defect analysis in the casting of the nodular cast iron truck rear axle technology and pouring system are useful to establish effectively an sequential solidification procedure from mid part to two edge part of the truck rear axle casing. 5. conclusion the mold filling and casting solidification process of the truck rear axle casing using nodular cast iron are studied. the pouring system and structure of rear axle casing are improved, and the casting technique is developed. the integrated methods including the new casting structure, casting technology and pouring system are useful to establish a great directional and sequential solidification procedure from middle part to two edge part of the truck rear axle casing. the optimization pouring system reduces the defects in casting workpiece. the simulation research results and experiments show that these optimization methods are contributed to reduce the casting defect and improve the product quality. acknowledgements the authors gratefully acknowledge the support of a grant from the natural science foundation project of cq cstc 2008bb3303 and cq cstc 2009ba3026, the ph.d. programs foundation of ministry of education of china (no.20070611030 ), the outstanding young scholar program of natural science foundation of hubei province (no. 2006abb027), the program for new century excellent talents in university (ncet-08-0607), changjiang scholars and innovative research team in university (irt 0763), the project of the state key laboratory of mechanical transmission in chongqing university and the fund of the state key laboratory of solidification processing in northwestern polytechnical university. the paper is also supported by the exchange study foundation program of chongqing university of the project 211 tertiary system. references [1] 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[14] gui q.s. casting technology of back bridge housing, foundry technology, 25(5) (2004) 337-339. microsoft word 6 bing zhang, ying wang, dejun chen--a research on technology project credit evaluation model based on ahp and advances in systems science and applications (2011), vol.11, no.3-4 249-256 issn 1078-6236 international institute for general systems studies, inc a research on technology project credit evaluation model based on ahp and fcem bing zhang, ying wang and dejun chen school of information engineering, wuhan university of technology, wuhan 430070, p.r.china abstract this paper analyzes the basic requirements of the technology project credit evaluation, and presents a technology project credit evaluation model based on analytic hierarchy process (ahp) and fuzzy comprehensive evaluation method (fcem). combining with the actual situation of one scientific research center, a technology project credit evaluation index system is established, and its weight of each evaluation index is determined by ahp, and then an evaluation results is analyzed and evaluated through fcem, also it’s valuable theoretical foundation for the management of the technology project. keywords ahp; fcem; project credit evaluation 1. introduction the technology project credit evaluation is an important part of technology project process management. in the technology project concluding, it is necessary to evaluate the technology project implementation process, which is not only the summary of project implementation process, but also the archive of credit of undertakers, which provide important historical basis for the future project application approval procedures. currently, the technology project credit evaluation is based on subjective qualitative assessment, there is no reasonable technology project credit evaluation index system, or the evaluation indexes are lack of scientific weight distribution. to solve this problem, this paper presents a comprehensive evaluation method based on ahp and fcem. combining with the actual situation of one scientific research center, this paper proposes a technology project credit evaluation index system, which evaluate the credit of project stakeholders from the contract compliance, reporting significant matters, and implementation within the stipulated time those three aspects, and determines the weight of each index by ahp and checks the consistency, then taking one credit evaluation results in a project as example, calculates the technology project credit situation through fuzzy analysis and quantitative assessment to validate this model. 2. technology project credit evaluation index system technology project credit evaluation system should be operated from the multi-level, multi-angle, which should be able to fully reflect the technology project's credit rating commitment, combined with an actual situation of r & d center, based on ahp, a three-level technology project credit evaluation index system is proposed , as shown in figure 1. figure 1 show that, this system is made up of 3 respects of contract compliance, reporting significant events and implementation within the stipulated time, and has 8 indexes; of course, it can be adjusted according to actual situation. explain the specific content of each index as follows: (1) completion condition of assessment indicators: refers to the completion condition of the content stipulated in the contract. (2) rate of progress is the completion situation of the project progress according to the contract rules. (3) reporting significant events is to account for the significant issues to the virtual 250 zhang: a research on technology project credit evaluation model based on ahp and fcem coordination center faithfully and timely. u 1u 2u 3u 11u 12u 31u 32u 32u 34u 35u figure 1 technology project credit evaluation index system (4) submit research plan: refers to submit their work outline within the specified time, such as it is finished after being noticed in two months. (5) submit contract: refers to hand over contract within the specified time, such as it is finished after being noticed in three months. (6) submit the sheet about the execution situation: refers to submit it in scheduled time, such as finishing it on 15 of the first month of each quarter. (7) submit the acceptance of applications: refers to submit it before the deadline specified in the contract. (8) submit archive data: refers to submit it within the specified time after project acceptance, such as 1 month after the inspection (30 days) for submission. 3 the establishment of technology project credit evaluation model based on the ahp and fcem ahp is used first to define the weight of each level index, and then use fcem for project credit evaluation. the model is established in following steps: 1.tablishment of factors set from the project credit evaluation index system in figure 1 can be seen, there are two levels of evaluation indices, and now the first level is defined },,{ 321 uuuu = ; the second level is },{ 12111 uuu = ， },,,,{ 35343332313 uuuuuu = 。 2. establishment of reviews set according to the actual situation of the r&d center mentioned before, reviews are set to 4 levels, namely, n=4, },,,{ 4321 vvvvv = represent {excellent, good, medium, poor}. reviews level can be adjusted and determined according to the specific situations. 3. determine weights set of evaluation indexes by ahp using the ahp to determine weights set of evaluation indexes can be divided into the following steps: (1) construct judgment matrix judgment matrix represents the relative importance between two elements in the same level to some element in the upper level, the evaluation about the importance of indexes at all levels are a subjective process, based on expert evaluation results or the result of the questionnaire. use advances in systems science and applications (2011), vol.11, no.3-4 251 issn 1078-6236 international institute for general systems studies, inc ),...2,1,(, njibb ji = to represent the indexes. ijb expressed the value of the importance that ib relative to jb , construct the judgment matrix p through 1-9 ratio scaling, and the matrix have reciprocity and basic consistency, that is, 0>ijb ， 1=iib ， 1=∗ jiij bb .   ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = nnnn n n bbb bbb bbb p 21 22221 11211                                       (1)                      table 1 1-9 description of proportion quotients ijb meaning explanation 1 equal importance both have the same importance 3 weak importance ib important than jb slightly 5 strong importance ib important than jb obvious 7 very strong importance ib more important than jb 9 absolute importance ib absolute important than jb obvious 2、4、6、8 between the various levels above the importance is between the adjacent levels 1、1/2、… 、1/9 reverse comparison the importance of jb relative to ib with experience and knowledge related to credit evaluation and project management, construct judgment matrix of each level as 1p 2p 3p . in order to facilitate analysis and more intuitive, we graphically shows the matrix, as shown in table 2, 3, 4. table 2 judgment matrix 1p table 3 judgment matrix 2p table 4 judgment matrix 3p 3p 31u 32u 33u 34u 35u 31u 1 1 1 1/2 2 2p 11u 12u 11u 1 2 12u 1/2 1 1p 1u 2u 3u 1u 1 2 1/2 2u 1/2 1 1/3 3u 2 3 1 252 zhang: a research on technology project credit evaluation model based on ahp and fcem 32u 1 1 1 1/2 2 33u 1 1 1 1/2 2 34u 2 2 2 1 3 35u 1/2 1/2 1/2 1/3 1 (2) calculate the weight vector of each level and make consistency check first, calculate the maximized eigenvalue and eigenvector of judgment matrix. generally, it is to use geometric averaging (root method) or normative column average (sum method) to calculate the approximate eigenvectors [2], and then calculate the maximized eigenvalue. geometric average method: calculate the product of each element of each row, then calculate the nth root of each product; and then normalized the obtained vector. vector obtained above is the approximate eigenvectors, if the consistency check is passed, the vector is the relative weight vector of each index. calculating the maximized eigenvalue and eigenvector using geometric averaging mean is as follows: ① calculate the geometric average of all elements of each row of the judgment matrix. based on n n j iji bw ∏ = = 1 ， ( )ni ,,2,1= ，so ( )tnwwww ,,, 21= 。 ② normalize w , ∑ = = n i iii www 1 ， ni ,,2,1= , then t ni wwww ),,,( 2= which is approximate eigenvectors, and its value of each element is the weight value of each index. ③ calculate the largest eigenvalue maxλ ( )∑ = = n i i i nw pw 1 maxλ (2) in the formula 2, vector ( )ipw is the first i component of pw . then check on the consistency of judging matrix. matrix consistency test as follows: ① calculate the inconsistent level (ci) of judgment matrix. 1 max − − = n n ci λ (3) in the formula 3, maxλ is the maximized eigenvalue of )1( >nn order matrix. ② calculate the random consistency level (ri) of judgment matrix, which only determined by the order of the judgment matrix. note that, when 20 ≤< n , there is no inconsistency issue, matrix does not need be tested. standards of random consistency level shown in table 5: ③ calculate the consistency ratio of judgment matrix (cr). ri cicr = (4) advances in systems science and applications (2011), vol.11, no.3-4 253 issn 1078-6236 international institute for general systems studies, inc table 5 random consistency level ri n 1 2 3 4 5 6 7 8 9 10 ri 0 0 0.58 0.9 1.12 1.24 1.32 1.41 1.45 1.49 the method to determine the consistency of judgment matrix is [3]: when .10<cr , judgment matrix has satisfactory consistency, pass the consistency check; otherwise, the matrix does not pass the consistency check, degree of consistency is unacceptable, judgment matrix needs to be adjusted to your satisfaction. now, we use the judgment matrix 1p as example to explain the calculation. ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = 132 31121 2121 1p find the geometric mean ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ 8171.1 5503.0 1 normalization ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ 5396.0 1634.0 297.0 =w so, wp1 = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ∗ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ 5396.0 1634.0 297.0 132 31121 2121 = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ 6238.1 4918.0 8936.0 from the formula 2, we can get 0093.3 5396.0 6238.1 1634.0 4918.0 297.0 8936.0 3 1 max =⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ++=λ then according to the formula 3, ci =0.00465，checking the table 5, when n=3, ri=0.58, and finally, calculating according to the formula 4, =cr 0.008<0.1, so the judgment matrix has satisfactory consistency, the corresponding weight vector is ( )5396.01634.0297.01 =a . similarly, the cr of judgment matrix 3p equals 0.0022 <0.1, pass the consistency check, the weight vector is ( )0977.03485.01846.01846.01846.03 =a . because the order of judgment matrix 2p is 2, so there is no necessary to check consistency, the corresponding weight vector is )3333.06667.0(2 =a . 1. build the fuzzy evaluation matrix evaluate the object based on the reviews setv . build a fuzzy mapping from u tov , and determine the fuzzy evaluation matrix formed by ),,2,1;,,2,1)(10( mjnirr ijij ==≤≤ , which is the grade of membership that iu to jv . ⎟⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = nmnn m m rrr rrr rrr r 21 22221 11211 ppr ijij = ; p is the total number of experts; ijp is the number of experts who evaluate the index iu by the review jv . 254 zhang: a research on technology project credit evaluation model based on ahp and fcem to create a fuzzy evaluation matrix, firstly select a number of experts to evaluate the credit status of a project, and get the statistical data of the evaluation result, and then create fuzzy evaluation vectors of each single index, finally construct fuzzy evaluation matrix. construct all fuzzy evaluation vectors of the initial single indexes; so the underlying fuzzy evaluation matrix is determined, and then build the upper fuzzy evaluation matrix based on sublevel fuzzy performance matrix. 2. calculate the result of fuzzy comprehensive evaluation the weight of each evaluation index has been determined by ahp, using one fuzzy subset of u ）（ naaaa ,,, 21= represent weight set, namely ia is the quantitative index of iu . after getting r and a , the result rab = can be obtained, b is the fuzzy subset of v , is a kind of fuzzy operator, and generally consists of 4 models, here, is the weighted average type. ( ) ),,,(,,, 21 21 22221 11211 21 m nmnn m m n bbb rrr rrr rrr aaaarb = ⎟⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎜ ⎜ ⎝ ⎛ == 1 1 =∑ = m i ib , according to the principle of maximum degree of membership, if ib is the maximum number ofb , corresponds to i , then the object evaluation result level is iv . as the project credit evaluation model is a multi-layer model, the evaluation results of indexes in previous layer can be gotten from fuzzy evaluate matrix and weight vectors of the indexes in this layer, and then set these evaluation results as fuzzy evaluation matrixes of the previous layer, combining with weight vectors in the previous layer, the results of the project is gotten. another point to note is that not all the initial indexes are at the bottom, and some initial indexes are in the middle layers, such as "reporting significant events", when conducting the initial fuzzy evaluation, only calculate the initial indexes in the bottom, then put the fuzzy evaluation vector of the single initial index in the middle layer and the initial evaluation results of the bottom layer together, so get the fuzzy evaluation matrix of the middle layer. 3. fuzzy comprehensive evaluation of the results the evaluation results is a fuzzy result, do not give a quantitative evaluation result or a score, generally based on the principle of maximum degree of membership [3], if the maximum number of b is ib , corresponds to i , then the evaluation result level of the object is iv . usually, in order to give a quantitative result for fuzzy evaluation result, we can assign a weight value in to each review, so there will be a weight value set ( )tmnnnn 21= . the total score of evaluation result is ∑ = == m i iibnbns 1 . note that the determination of the weight value of reviews is a precise and seriously problem, it should be able to respond the actual situation well. general the weight value of reviews can be determined by the vote of experts or the historical data analysis, preferably both. 4 validation of technology project credit evaluation module based on ahp and fcem the algorithm of technology project credit evaluation module is given above, and the consistency of judgment matrix is verified. now we will show a practical example to verify this module. table 6 shows a statistical result of project evaluation. advances in systems science and applications (2011), vol.11, no.3-4 255 issn 1078-6236 international institute for general systems studies, inc table 6 a result of project evaluation result index excellent good fair poor completion of assessment indicators 1 6 0 0 contract fulfillment the progress 2 4 1 0 reporting significant events 6 0 1 0 submitting their work plan 3 4 0 0 submitting the contract 2 4 1 0 the implementation of the report 4 3 0 0 acceptance of applications 1 5 1 0 the implementation of the stipulated time archive information 2 3 1 1 combining data in the table above, according to step 4, in the bottom layer, the fuzzy evaluation matrix 2r , 3r ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = 0717472 007671 2r ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = 71717372 0717571 007374 0717472 007473 3r weight vector 2a 3a have been obtained in step 3, according to rab = : ( )00476.07619.01905.0 0717472 007671 )3333.06667.0(222 =⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ×== rab similarly, ( )014.00901.05809.0315.03 =b . the result of "the fulfillment of the contract" and "implementation within the stipulated time" can be seen from 2b 3b . according to the principle of maximum degree of membership, the result levels of the two indicators are all good. then, put 2b 3b and the single index evaluation vector of "reporting on major issues" ( )01429.008571.0 together to get the fuzzy evaluation matrix 1r , 1a has been determined in step 3, from rab = , the result of this technology project credit evaluation is 1b . 256 zhang: a research on technology project credit evaluation model based on ahp and fcem ( ) ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ×== 014.00901.05809.0315.0 01429.008571.0 00476.07619.01905.0 5396.01634.0297.0111 rab ( )0076.00861.05397.03666.0= according to the principle of maximum membership degree, the credit of this project is good. according to the actual situation of this r&d center, the weight value of each review is given, excellent 90, good 80, medium 70, and poor 50, then according to step 6: 553.820076.0500861.0705397.080366.090 =×+×+×+×=s from the calculation process and the fuzzy comprehensive evaluation result, the technology project credit evaluation model based on ahp and fcem is a scientific and reasonable, mainly fit reality. on condition that passing the consistency test of judgment matrix, the specific indexes of technology project credit evaluation can be adjusted flexibly, so that the model has the flexibility and scalability. 5 conclusion this paper introduces the relevant theory of the ahp and fcem, presents a technology project credit evaluation model based on ahp and fcem, establishes a credit evaluation index system of the technology project, and checks the consistency of the judgment matrix of each layer, calculates the weight vector of each layer, algorithm steps is given in detail, and finally the model and algorithm is validated by a practical examples. this model provides a scientific basis for the technology project credit evaluation. references [1] arquero a, alvarez m, martinez e. decision management making by ahp (analytical hierarchy process) trought gis data [j]. latin america transactions, ieee (revista ieee america latina),2009,7(1):101-106. [2] xuning peng, feng dai. information systems risk evaluation based on the ahp-fuzzy algorithm [c]. networking and digital society, 2009. icnds '09. international conference on 30-31 may 2009:178-180. [3] ohnishi s, yamanoi t, imai h. a kind of fuzzy weights representation for inner dependence ahp [c]. fuzzy systems (fuzz), 2010 ieee international conference on 1823 july 2010:1-5. advances in systems science and applications (2012) vol.12 no.4 305-326 a development plan for the new millennium yi lin1,2,tianxiang yao3,xiangdong li4 and sifeng liu1 1college of economics and management, nanjing university of aeronautics and astronautics,nanjing, 210016, pr china 2department of mathematics, slippery rock university, slippery rock, pa 16057, usa 3college of economics and management, nanjing university of information science and technology, 219 ningliu road, nanjing 210044, pr china 4school of business, jiangsu teachers university of technology, 1801 zhongwu road, changzhou city, jiangsu province, pr china , 213001 abstract this paper points out the importance of systems science as the second dimension of knowledge and introduces the yoyo model as the intuition and playground commonly useful for systems thinking and reasoning. by revealing how traditional science abstracts numbers and quantities out of objects of rich internal structures, this work shows the wide range of applicability of systems research in not only new territories created by the research but also the studies of all the age-old quests of the mankind. by detailing a careful design, this paper establishes the idea and purpose of creating educational programs at specifically undergraduate levels in order to stimulate further development of systems science. keywords yoyo model, social organization, systems thinking, curriculum design, calculus 1 introduction the breadth and diversity the vast amount of the literature of systems science and engineering covers point to a golden opportunity for the next stage development of systems movement. by comparing the origin where numbers are from and the places systems are seen, this paper establishes another reason for the 2-dimensional landscape of knowledge. by pointing to the advantage of the additional dimension, it is argued that because of the maturing development of systems research, some age-old problems that have challenged the mankind for thousands of years can now be hopefully resolved. after introducing the systemic yoyo model as the common intuition and playground for general systems thinking and reasoning, this paper looks at how even at the ground root level, the traditional science has intelligently dealt with mathematical entities without involving the internal structures of objects it studies and the impacts of the environment. that is, to successfully employ numbers and quantities, only thinghood the traditional science captures. however, the natural world consists mainly of structures and organizations, that is, systems. hence, systems researchers should 306 yi lin:a development plan for the new millennium start working on improving almost all, if not all, the basic concepts and elementary procedures of the traditional science so that problems of systemhood, which are more realistic than those problems addressed by using the traditional science, can be effectively addressed. to this end, considering the scope and magnitude modern science covers and embodies, improving the basic concepts and elementary procedures of such science from the angle of systemhood requires a steady supply of a huge amount of manpower. the number of current scholars who are interested in systems research is quite limited. it implies a severe shortage of manpower within the community of systems researchers. to help resolve this problem, this paper lays out a particular plan to establish educational programs worldwide that parallels that of mathematics education. by doing so, a steady job market for systems researchers will appear, while major advances in systems science can be achieved at the same time. as what george klir[1] did, let us look at the currently available knowledge as 2-dimensional. along the first dimension, the traditional science dimension, subjects that are investigated are compartmented based on their thinghood; along the second dimension, the systems science dimension, subjects are classified according to their systemhood. with this conceptual 2-dimensionality in place, it is argued in this paper that by taking the responsibility of reshaping the traditional science, from its very bottom up, as part of the future works of systems research, one will be able to truly make systems science the second dimension of knowledge. in terms of the future development of systems research, it is argued that calculus should be employed as a reference. and it should be emphasized that when students take courses of systems science, they need be taught, along all the fundamentals, with procedures that can be loosely followed to produce some more or less definite outcomes. that is actually why calculus has been used by generations after generations. in other words, since its inception over three hundred years ago, that is how calculus has provided livelihood for millions of people, and at the same time these people had helped to create the prosperity of calculus. this paper is organized as follows: in section 2, we look at the different origins of numbers and systems, and why additional dimension of knowledge is expected to bring about breakthroughs. then the systems yoyo model is introduced for the purpose of naturally leading the reader to the conclusion of the following section: the concepts of numbers and systems really complement each other; together they will make science more useful in addressing practical problems. in section 3, we analyse why from the root level up, the traditional science needs to be revisited from the viewpoint of systems. however, to satisfy such need of revisiting every corner of the traditional science requires more scientific advances in systems science and applications (2012) vol.12 no.4 307 manpower additional to what is currently available in the community of systems research. to resolve this problem, in section 4, we lay out a particular idea for developing educational programs to serve the greater audience of university students in order to eventually feedback to the systems research with additional manpower and scientific progress. section 5 concludes this paper. 2 some observations since 1924 when von bertalanffy pointed out that the fundamental character of living things is its organization, the customary investigation of individual parts and processes cannot provide a complete explanation of the phenomenon of life, this holistic view of nature and social events has spread over all corners of science and technology[2]. accompanying this realization of the holistic nature, in the past 80 some years, studies in systems science and systems thinking have brought forward brand new understandings and discoveries to some of the major unsettled problems in the conventional science[3-4]. due to these studies of wholes, parts, and their relationships, a forest of interdisciplinary explorations has appeared, revealing the overall development trend in modern science and technology of synthesizing all areas of knowledge into a few major blocks, and the boundaries of conventional disciplines have become blurred (“mathematical sciences,” 1985). underlying this trend, we can see the united effort of studying similar problems in different scientific fields on the basis of wholeness and parts, and of understanding the world in which we live by employing the point of view of interconnectedness. as tested in the past 80 plus years, the concept of systems has been widely accepted by the entire spectrum of science and technology[1,5]. similar to how numbers are theoretically abstracted, systems can be seen in each and every object, event, and process. for instance, behind collections of objects, say, apples, there is a set of numbers such as 0, 1, 2, 3, ; and with each organization there is an abstract, yet very realistic, system within which the relevant whole, component parts, and the related interconnectedness are emphasized. in other words, when internal structures are out of the concern, numbers can be very useful; otherwise the world dominantly consists of systems. historically speaking, traditional science has been developed on top of numbers and quantities; while along with systemhood comes the systems science. that gives rise of a 2-dimensional spectrum of knowledge, where the classical science, which is categorized by the thinghood of the objects it studies, constitutes the first dimension, and the systems science, which investigates structures and organization, forms the genuine second dimension[1]. that is, systems thinking focuses on those properties of systems and associated problems that emanate from the general notion of structures; the division of the classical science has been done largely on properties of particular objects. that explains why systems research naturally transcends all the disciplines of the classical science and becomes a force 308 yi lin:a development plan for the new millennium that unites all existing disciplines. the importance of this supplementary second dimension of knowledge cannot be in any way over-emphasized. for example, when studying dynamics in an n-dimensional space, there are difficulties that cannot be resolved within the given space without getting help from a higher-dimensional space. in particular, when a one-dimensional flow is stopped by a blockage located over a fixed interval, the movement of the flow has to cease. however, if the flow is located in a two-dimensional space, instead of being completely stopped, the 1-dimensional blockage would only create a local (minor) irregularity in the otherwise line movement of the flow (that is how nonlinearity appears[6]). additionally, if one desires to peek into the internal structure of the 1-dimensional blockage, he can simply take advantage of the second dimension by looking into the blockage from either above or below the blockage. that is, when an extra dimension is available, science will gain additional strength in terms of solving more problems that have been challenging the survival of the mankind. (a) (b) (c) fig.1 the eddy motion model of the general system even though systems research holds such a great promise, the systems movement has suffered a great deal in the past 80 some years of development due to the reason that this new science does not have its own speaking language and thinking logic. conclusions of systems research, produced in this period of time, are drawn either on ordinary language discussions or by utilizing the conventional mathematical methods, making many believe that systems-thinking is nothing but a clever way of rearranging conventional ideas. in other words, due to the lack of an adequate tool for reasoning and an appropriate language for speaking, systems research has been treated with less significance than they were thought initially since the 1970s when several publications criticized how systems enthusiasts derived their results without sufficient rigor[7-8], even though most of the criticized results turned out to be correct if seen through our 20-20 hindsight. considering the importance of the cartesian coordinate system in modern advances in systems science and applications (2012) vol.12 no.4 309 science[9-10] realizes that the concepts of (sizeless and volumeless) points and numbers are bridged beautifully together within the cartesian coordinate system. that is how this cartesian coordinate system plays the role of intuition and playground for modern science to evolve; and within this system, important concepts and results of modern mathematics and science are established. recognizing the lack of such an intuition and playground for systems science, on the basis of the blown-up theory[10], the yoyo model in fig.1 is formally introduced by[11] in order to establish the badly needed intuition and playground for systems science. in particular, on the basis of the blown-up theory, which shows how the common form of motion of systems is eddy motion, and the discussion on whether or not the world can be seen from the viewpoint of systems[12], the concepts of black holes, big bangs, and converging and diverging eddy motions are coined together in the model shown in fig.1. this model was established in[10] for each object and every system imaginable. in other words, each system or object considered in a study is a multi-dimensional entity that spins about its either visible or invisible axis. if we fathom such a spinning entity in our 3-dimensional space, we will have a structure as shown in fig.1(a). the side of black hole sucks in all things, such as materials, information, energy, etc. after funneling through the short narrow neck, all things are spit out in the form of a big bang. some of the materials, spit out from the end of big bang, never return to the other side while some will (fig.1(b)). for the sake of convenience of communication, such a structure, as shown in fig.1(a), is called a (chinese) yoyo due to its general shape. more specifically, what this model says is that each physical entity in the universe, be it a tangible or intangible object, a living being, an organization, a culture, a civilization, etc., can all be seen as a kind of realization of a certain multi-dimensional spinning yoyo with either an invisible or visible spin field around it. it stays in a constant spinning motion as depicted in fig.1(a). if it does stop its spinning, it will no longer exist as an identifiable system. what fig.1(c) shows is that due to the interaction between the eddy field, which spins perpendicularly to the axis of spin, of the model, and the meridian field, which rotates parallel to axis of spin, all the materials returning to the black-hole side travel along a spiral trajectory. note:this yoyo model for each and every system is theoretically established on newtons second law of motion, the concept of blown-ups existing widely in evolutions of systems, mathematical characteristics of nonlinear models, the principles of mathematical modeling, and the universal existence of eddy motions. empirically, this model is manifested by diracs large number hypothesis, the mystery of solar systems angular momentum, and the measurement analysis of the movement of earths atmosphere. for all the technical details, please consult with[13]. 310 yi lin:a development plan for the new millennium to show this yoyo model can indeed, as expected, play the role of intuition and playground for systems research[2,13], have successfully applied it to investigate newtonian physics of motion, the concept of energy, economics, finance, history, foundations of mathematics, small-probability disastrous weather forecasting, civilization, business organizations, the mind, among others. at this junction, let us look at how a workplace can be investigated theoretically as such a spinning structure. in fact, each social entity is an objectively existing system that is made up of objects, such as people and other physical elements, and some specific relations between the objects. it is these relations that make the objects emerge as an organic whole and a social system. for example, let us look at a university of higher education as a workplace. without the specific setup of the organizational whole (relationships), the people, the buildings, the equipment, etc., will not emerge as a university (system). now, what the yoyo model says is that each imaginable system, which is defined as the totality of some objects and some relationships between the objects[3], possesses the yoyo structure so that each chosen social system, as a specific system involving people, has its own specific multi-dimensional yoyo structure with a rotational field. to this end, there are many different ways for us to see why each social entity spins about an invisible axis. in particular, let us imagine an organization, say a business entity. as it is well known in management science, each firm has its own particular organizational culture. differences in organizational cultures lead to varied levels of productivity. now, the basic components of an organizational culture change over time. these changes constitute the evolution of the firm and are caused by inventing and importing ideas from other organizations and consequently modifying or eliminating some of the existing customs. the concept of spin beneath the systemic yoyo structure of the firm comes from what ideas to invent, which external ideas to import, and which existing customs to eliminate. if idea a will likely make the firm more prosperous with higher level of productivity, while idea b will likely make the firm stay as it has been, then these ideas will form a spin in the organizational culture. specifically, some members of the firm might like additional productivity so that their personal goals can be materialized in the process of creating the extra productivity, while some other members might like to keep things as they have been so that what they have occupied, such as income, prestige, social status, etc., will not be adversely affected. these two groups will fight against each other to push for their agendas so that theoretically, ideas a and b are actually spin around each other. for one moment, a is ahead; for the next moment b is leading. and at yet another moment no side is ahead when the power struggle might very well return to the initial state of the organizational affair. in this particular incidence, the abstract axis of spin is invisible, because no one is willing to openly admit his underlying purpose for pushing for a specific advances in systems science and applications (2012) vol.12 no.4 311 idea (either a or b or other ones). as for the concept of black hole in a social organization, it can be seen relatively clearly, because each social organization is an input-output system, no matter whether the organization is seen materially, holistically, or spiritually. the input mechanism will be naturally the “black hole”, while outputs of the organization the “big bang”. again, when the organization is seen from different angles, the meanings of “black hole” and “big bang” are different. but, collectively these different “black hole” and ‘big bang” make the organization alive. without the totality of “black hole” and that of “big bangs”, no organization can be physically standing. other than intuition, to this end the existing literature on civilizations, business entities, and individual humans readily does testify the conclusions used here. from this example, a careful reader might have sensed the fact that in this yoyo model, we look at each system, be it a human organization, a physical entity, or an abstract intellectual being, as a whole that is made up of a physical body, its internal structure, and its interactions with the environment. this whole, according to the systemic yoyo model, is a high dimensional spin field. considering the fact that the body is the carrier of all other (such as cultural, philosophical, spiritual, psychological, etc.) aspects of the system, in theory the body of the system is a pool of fluid realized through the researchers sensing organs in the three-dimensional space. the word fluid here is an abstract term totalling the flows of energy, information, materials, etc., circulating within the inside of, going into, and giving off from the body. and in all published references that we have searched these flows are studied widely in natural and social sciences using continuous functions, which in physics and mathematics mean flows of fluids and are widely known as flow functions. on the other hand, as it has been shown and concluded in[2,13] that the universe is a huge ocean of eddies, which changes and evolves constantly. that is, the totality of the physically existing world can be legitimately studied as fluids. to make this presentation complete for the reader, let us look briefly at the justification of this yoyo model of systems. in theory, the justification for such a model of general systems is the blown-up theory[10]. it can also be seen as a practical background for the law of conservation of informational infrastructures. more specifically, the blown-up theory establishes that in the evolution of a system, what are commonly observed are transitional changes, known as blown-ups, so that it supports the input-output structure of the yoyo model. and based on empirical data, the following law of conservation is proposed[14]: for each given system, there must be a positive number α such that at ×bs × cm ×de = α (1) where a, b, c, and d are some constants determined by the structure and at312 yi lin:a development plan for the new millennium tributes of the system of concern, and t stands for the time as measured within the system, s the space occupied by the system, m and e the total mass and energy contained in the system. because m (mass) and e (energy) can exchange into each other and the total of them is conserved, if the system is a closed one, equ. (1) implies that when time t evolves to a certain (large) value, space s has to be very small. that is, in a limited space, the density of mass and energy becomes extremely high. so, an explosion (a big bang) is expected. following the explosion, space s starts to expand, while time t starts to travel backward or to shrink. this end gives rise of the well-known model for the universe as derived from einsteins relativity theory[1516]. in terms of systems, what this law of conservation implies is: each system goes through such cycles as: . . .→ expanding→shrinking→expanding→shrinking →. . . now, the geometry of this model of universe established from einsteins relativity theory is given in fig.1. empirically, the multi-dimensional yoyo model in fig.1 is manifested in different areas of life. for example, each human being, as we now see it, is a 3-dimensional realization of such a spinning yoyo structure of a higher dimension. to this end, consider two simple and easy-to-repeat experiences. for the first one, imagine we go to a swim meet. as soon as we enter the pool area, we immediately fall into a boiling pot of screaming and jumping spectators, cheering for their favorite swimmers competing in the pool. now, let us pick a person standing or walking on the pool deck for whatever reason, either for her beauty or for his strange look or body posture. magically enough, in a brief moment, the person from quite a good distance will feel our stare and she/he will be able to locate us instantly out of the reasonably sized and boiling audience. the reason for the existence of such a miracle and silent communication is that each side is a high dimensional spinning yoyo. even though we are separated by space and possibly by informational noise, the stare of one side on the other has directed that side′s spin field of the yoyo structure into the spin field of the yoyo structure of the other side. that is the underlying mechanism for the silent communication to be established. as the second example, let us look at the situation of human relationship. two individuals are said to be in a congruent position when compared to each other, if none of them is in a more powerfully authoritative height over the other. for example, in a typical marriage relationship the husband and wide in general are in a congruent position, where no one has much authoritative control of the other. however, the relationship between a parent and a child is different, where the parent to a degree and in certain sense has an overarching control of the child. for individuals a and b in a congruent position, all clinic trials in family psychology have shown that when a has a good impression about b[17], advances in systems science and applications (2012) vol.12 no.4 313 magically, individual b also has a similar and almost identical impression about a. when a does not like b and describes b as a dishonest person with various undesirable traits, it has been clinically proven in family psychology that what a describes about b is exactly who a is himself. once again, the underlying mechanism for such a quiet and unspoken evaluation of each other is that each human being stands for a spinning yoyo and its rotational field. our feelings about other people are formed through the interactions of our invisible yoyo structures and their spin fields. on the other hand, when individual a possesses an authoritative control over individual b, it simply means that the yoyo field of a is more powerful than that of b. in this case, deviations in their mutual evaluations of each other appear, for more details, see[13]. historically, although the word system was never emphasized in science, we can still find many similar concepts. for example, nicholas of cusa, that profound thinker of the fifteenth century, linking medieval mysticism with the first beginning of modern science, introduced the notion of coincidentia oppositorum, the opposition or indeed fight among the parts within a whole which nevertheless forms a unity of higher order. leibnizs hierarchy of monads looks quite like that of modern systems; his mathesis universalis presages an expanded mathematics. that is not limited to either quantitative or numerical expressions and is able to formulate much conceptual thought. hegel and marx emphasized the dialectic structure of thought and of the universe it produces: the deep insight that no proposition can exhaust reality but only approaches its coincidence of opposites by the dialectic process of thesis, antithesis, and synthesis. gustav fechner, known as the author of the psychophysical law, elaborated, in the way of the natural philosophers of the nineteenth century, supraindividual organizations of higher order than the usual objects of observation c for example, communities of life and the entire earth, thus romantically anticipating the ecosystems of modern parlance, for a more comprehensive study of history along this line, please consult with[18]. in other words, various theories of systems had been considered throughout history. however, for one reason or another, each of these theories came and gone without leaving much trace and impact. in this section, after realizing the difficulty of not having its own particular speaking language and thinking logic for systems research, we introduced the badly needed systemic yoyo model. on the basis of this model, the next section shows vividly how such fundamental concepts of modern science as numbers are introduced cleverly with internal structures taken out of concern, and how the yoyo model can help to resolve this problem of the missing internal structures. that in turn establishes the need to develop systems science, the second dimension of knowledge, with the added responsibility of reshaping the traditional science from its bottom up. that is the underlying rationale for why we need to develop 314 yi lin:a development plan for the new millennium a comprehensive curriculum of systems science at all levels of formal education. 3 problems investigated using systems thinking and methodology because of the availability of the yoyo model, we now have an intuition and playground that is commonly available for systems theorists and practitioners to house their abstract reasoning and thinking, just as what people are accustomed to do with the cartesian coordinate system when they think about how to resolve a problem in the classical science. now, one of the most important problems of systems research is what kinds of problems systems science could and should attack and attempt to resolve. historically speaking, the importance of this question is that only when systems science can provide new and powerful means to resolve at least some of the age-old whys which have challenged the mankind since the dawn of the recorded history, systems science will have a chance to become firmly recognized as a legitimate branch and the second dimension of knowledge. the never-fading effort of the man invested in studying these forever important whys fundamentally signals the relevance of these endeavours to the very survival of the human race. here is a common belief currently existing in the community of systems researchers about what kinds of problems systems science could and should attempt to address[19]: systems science resolves problems that are related to systemhood instead of thinghood. in other words, systems science is good at addressing such a problem that when one looks at one aspect, he realizes that several other aspects of the issue should be addressed first. that is, the issue seems to be messy with neither any beginning nor an ending and is surely not a linear causality. many factors influence the outcome, while the outcome simultaneously affects the influencing factors. here is a common belief currently existing in the community of systems researchers about what kinds of problems systems science could and should attempt to address: systems science resolves problems that are related to systemhood instead of thinghood. in other words, systems science is good at addressing such a problem that when one looks at one aspect, he realizes that several other aspects of the issue should be addressed first. that is, the issue seems to be messy with neither any beginning nor an ending and is surely not a linear causality. many factors influence the outcome, while the outcome simultaneously affects the influencing factors. although such a belief is in line with how systems science has been perceived historically, it has also been quite misleading. in fact, based on this belief, it should be readily recognized that the limitation of modern science is that it takes all structure related aspects of issues out of concern. for instance, for the very basic arithmetic fact that 1 + 1 = 2 to hold true, modern science has purposefully give away many structural characteristics of the issue of concern. in particular, advances in systems science and applications (2012) vol.12 no.4 315 if 1 + 1 = 2 stands for the fact that when one object is placed together with another object, there will be two objects totally, then the internal structures of the objects have been ignored. it is because when the internal structures of the objects are concerned with, the togetherness of the objects can be zero, or one, or two, or any other possible natural number. to illustrate this fact, let us assume that 1 + 1 = 2 represent placing two systems together. (that is, the objects we put together now have their individual internal structures.) does that mean consequently we will have two systems totally? the answer is: not necessarily. in particular, each system now is a spin field, as what we have seen in the previous section. when we place two spin fields together, what do we have? do we really have two spinning fields? to answer this question, let us consider all possibilities when two spin fields n and m are placed alongside of each other, fig.2. considering their directions of spin and their divergence and convergence of the fields n and m, fig.2 (a) will produce the outcome of two systems. the spin fields n and m in fig.2 (b) will remain separate while creating a joint rotational field. due to their convergence, the fields n and m in fig.2 (c) will merge together to become one bigger field. if we look back at fig.2 (b), then they are simply the “big-bang” sides of the convergent fields in fig.2 (c). so, if the fields n and m do combine with each other, then there will be only one rotational field resulted. if these fields do not combine into a greater field, then while they stay separate from each other, they also create many smaller fields in the areas between them. similar to the situation in fig.2 (a), the fields n and m in fig.2 (d) will stay separate without creating any small field. similar analysis indicates that the fields n and m in fig.2 (e, i) will either destroy each other so that no more rotational field is resulted, or if they stay separate, they will create many smaller fields in the zones between n and m. for the fields n and m in fig.2 (f, h), they either destroy each other or simply stay separate. in short, 1 + 1 = 2 is a very special case among all the eight possibilities as depicted in fig.2. as a matter of fact, we can conclude the same fact that 1 + 1 = 2 is a very special case as follows: when one positron is placed together with an electron, the outcome is nothing; that is 1 + 1 = 0. when a woman and man combine into a family, the outcome can be (in theory) any number of people. in comparison, of course, the previous spin-fields analysis is more systemically and scientifically significant than this short version of modelling. now, let us look at how modern science resolves problems by analysing the following basic algebraic scenario: suppose that john and ed together can finish doing a job in 2 hours, john and paul together can do the same job in 3 hours, while ed and paul together can complete the job in 4 hours. the question that needs to be resolved is: if john, ed, and paul work together, how long do they 316 yi lin:a development plan for the new millennium need to finish the job? (a) (b) (c) (d) (e) (f) (h) (i) fig.2 the internal structure of a two-body system the standard method of solving this algebraic problem is first assume that john can do the job alone in x hours, ed alone in y hours, and paul alone in z hours. then, the following system is established to describe the relationship among the quantities x, y, and z. 1 x + 1 y = 1 2 1 x + 1 z = 1 3 1 y + 1 z = 1 4 next step is to solve this system mathematically, producing the answer that in 24/13 ≈ 1.85 hours john, ed, and paul can complete the job. advances in systems science and applications (2012) vol.12 no.4 317 to confirm the answer is correct, one is required to go back to the original problem to check the validity of his answer. here is one way to do just that: because john and ed can finish the job in 2 hours together, with paul added, the additional manpower should make complete the job quicker. so, 24/13 is the correct answer. now, if we look at the resolution of the previous problem from the angle of systems thinking, we can see at least two problems: one is that the job might have a structure so that extra hand may not help at all. the other is that if each of the three workers is seen righteously as a living system, then when they are put together, they will surely interact with each other so that instead of speeding up the work progress, the interaction may also very well slow down the progress. although a greatly simplified version of the first problem has been addressed in operations research, the second problem represents an extremely difficult issue for modern science, known as the three-body problem. for more detailed discussion about this end, please consult with[13]. what these two examples are intended for is that by simply holding onto the thinking of systemhood, almost all, if not all, basic concepts and methods of modern science can be either generalized or improved or both. for example, the following are some of the success stories among many others of thinking and doing just along this line: (1) by applying the yoyo model to the forecasting of near-zero probability disastrous weather conditions, the current prediction accuracy that is commercially available has been greatly improved[13,20]. (2) by employing the rotation structure of the yoyo model and the observational facts of the dishpan experiment, we are led to the discovery of the fourth crisis in the foundations of mathematics[13]. (3) by relying on the yoyo model as a road map, a sufficient and necessary condition is established for when beckers rotten kid theorem can hold true, where the theorem is widely used in the research of economics. note:whats listed above represents some of the most difficult, unsettled problems in modern science with which recent progress has been made only through applications of systems thinking. as for other examples of how systems science can address the important whys that have plagued the mankind for centuries, let us look at medicine. in particular, chinese traditional medicine is a welldeveloped systems theory about the human body. on the basis of four basic, not defined concepts c blood, qi, yin, and yang, the entire chinese traditional medical theory has been developed since nearly 5 thousand years ago along with a whole set of practical procedures of how various ailments can be treated, for details see[2] and references listed there. to this end, one specific example is that if a fetus is not positioned right inside its mother at the time of birth, 318 yi lin:a development plan for the new millennium the newborn will most likely suffer from physical disabilities due to birth related injury. a chinese medical claim is that by needling a specific acupuncture point on the mothers feet, the fetus will reposition itself to the desired position. this ancient mystery (claim) is only reconfirmed recently by researchers from oxford university, england, while why the needle works as it does is still not known in terms of modern medical theory[21]. in short, what the discussion in this section teaches us is that by bringing organization, structure, and systemhood into modern science as they should be naturally in the first place, the entire spectrum of science will be enriched and become more useful in terms of how well it could resolve real-life problems. on the other hand, by taking the responsibility of reshaping and enriching modern science with the previously unconcerned structures and systemhoods from its very bottom up as part of the future works of systems research, we will be able to truly make systems science the second dimension of knowledge, where the first dimension, the traditional science, considers less or no internal structures. to this end, a natural question arises: to accomplish this task of reshaping and enriching modern science plus those systems researchers are currently working on, the community of systems scientists does not seem to have the adequate manpower. 4 educational programs in systems science one proven effective way to solve the problem of manpower is to follow the disciplinary development of mathematics, which, as an educational program, has been offered to every level of schooling. that of course has provided livelihood and good living for millions of people from around the world. these millions of people surely provide an abundant supply of manpower, with which one can expect to have systems science developed at a rapid speed. for the community of systems scientists to achieve this goal, it will surely be a long and difficult journey. and no matter how long and how difficult this journey can be, we have to start with the first step, the initial planning. here are some of the details of our idea, which is empirically developed in reference to how programs of mathematics from around the world at different educational levels are offered. we first start at the undergraduate level by (1) offering service courses of systems thinking to all majors in a similar manner as how basic mathematics and science courses are offered currently; (2) providing major specific courses to those students who are currently required to learn such materials; if these imagined courses can be offered successfully, there will be thousands of college-teaching positions suddenly become available worldwide. after talking to many colleagues from around the world, this idea seems to have the potential of success. advances in systems science and applications (2012) vol.12 no.4 319 practically, to make this idea work, one needs to secure the financial backing for these positions. here is one way to get around this problem. if a full time professor teaching x students can provide enough revenue for the university, then the initial step to create the very first teaching position for systems science is to secure at least a steady stream of x students per semester. then gradually increase this number to 2x, 3x, ..., by adding additionally required systems science courses in as many majors as possible. based on the current landscape of systems science education, the first service courses could be readily offered to business majors and engineering majors, creating the initial momentum of our planned large-scale programs. in most universities, these two areas represent about 50% or more of the student body. the next in line will be social science and humanity majors, and then followed by the traditional science majors. as an example, let us imagine a small university of 8,000 undergraduate students, more than 1/8 of which are in the business school with about 300 freshmen each year. if these students are required to take courses in methods, which are most likely the case for many universities from around the world, then we can readily make one of the method courses to contain two parts: the first consists of an examination of the main concepts of quantitative thinking, and the second an introduction to systems science and methodology. if each class section contains a maximum of 30 students, then we are talking about 20 sections for both parts of the sequel. if the teaching load of each instructor is 3 sections per semester, then we have an independent systems science program of about 3.5 faculty members. a similar calculation also indicates an impressive course offering opportunity for engineering majors. so, counting on either the business school or the engineering school, the imagined program in systems science can be started without any trouble. if a systems science major is created accordingly, that of course also translate into additional faculty positions. this idea of building up educational programs of systems science by focusing on providing service courses is more practical than that of using external research funds. the latter method has been tried during the systems movement in the past decades. one problem with this approach is that the systems science programs established in such a way tends to be limited in size; and when external funds dwindle, the established educational programs also die out consequently, creating an undesirable aftermath for the next round of rapid development of systems science. at the same time, the previous idea of offering service courses can of course be combined with research funding opportunities, because the teaching faculty can surely engage in joint research on various applied topics. by doing so, the teaching load can be accordingly lessened, producing additional faculty positions. when enough teaching positions at the undergraduate level become available, 320 yi lin:a development plan for the new millennium corresponding graduate programs in systems science will consequently flourish, because graduate assistantships will be more widely attainable and there is a wellestablished job market for the graduates of the graduate programs. at the same time, to make the students majoring in systems science more marketable in the job place, we can again follow what programs in mathematics do for their students: each student has to have a special interest area in traditional disciplines, which generally help mathematics majors to locate their future jobs. in terms of the curriculum, there is a strong and desperate need to develop a whole set of courses of systems science to be taught to systems science major and non-major students. first of all, these courses collectively need to cover all the main blocks of the relevant knowledge. not only so, the curriculum should be designed in such a way that they form an organic whole and satisfy the following four criteria of a glorious and long-lasting scientific theory[13] in order to bring about historical success for systems science: a. the basic theories must be readable by as many people as possible; b. the theoretical conclusions must coincide with peoples intuition; c. each course of the curriculum must possess a certain kind of beauty, which can be easily felt; d. the theories must be capable of producing meaningful results and insights that excite the population. the first three criteria guarantee a wide-range acceptance of systems thinking and methodology by both the scientific and non-scientific population; and the last criterion provides the livelihood of the discipline of systems science, and the “milk and bread” for all those scholars who pursue a career in systems science. this last criterion also means that systems researchers have to, without any choice, put in their efforts to resolve at least some of the age-old challenges that have faced science and the mankind. as a side note, to see why what is stated in the previous paragraph is important, let us look at the following fact among many other similar ones: although systems science and engineering consider various theoretical and practical systems, these systems do not really have much in common even at the level of abstract thinking. this end constitutes a real challenge to the community of systems researchers. specifically, when a systems project or idea needs public support, such as locating reputable reviewers for a research grant application, the principal investigator or the funding agency in general has a hard time to crystalize the base of supporters other than a few possible personal contacts. due to this reason, it generally takes a long time, if it ever happens, for a new rising star in systems research to be recognized even within areas of systems science and engineering. in particular, most rising stars, based on my professional contacts, in systems research have experienced difficulties in getting their research projects funded and in obtaining advances in systems science and applications (2012) vol.12 no.4 321 their well-deserved promotions. as a consequence, most of these young scholars have turned away from systems research in order to acquire a professional safe harbor. without a steady supply of new blood, the effort of systems research will surely stay on the sideline and as secondary to the traditional sciences, as it is currently the case, instead of being complementary to the traditional sciences. comparing this state of systems science to that of, for instance, calculus as the core of the traditional science, one can see the clear contrast. in particular, calculus possesses the following characteristics: (1). it gives one the feeling of a holistic body of thoughts where each concept is tightly developed on the previous ones in a well-accepted playground, the cartesian coordinate system. (2). it possesses a high level of theoretical beauty. (3). it contains a large reservoir of procedures scientific practitioners can follow to obtain their desired consequences. that is, calculus provides solutions to practical problems of different disciplines. that is, by either further developing calculus and related theories or using these theories, thousands of people from around the world in the generations both before us and after us in the foreseeable future have made and will continue to make their satisfactory living. that in turn feeds back to the livelihood and prosperity of calculus. on the other hand, systems science does not have a tightly developed system of theory and methods, which new comers can firstly feel excited about and consequently identify themselves strongly with, and scientific practitioners can simply follow established procedures to produce their needed results. in other words, (1). there is a definite need for the community of systems researchers to develop a cohesive spectrum of systems theories that underlies all methods of systems technology; (2). to accomplish line 1 above, a speaking language and logic of reasoning particular to systems research have to be established so that currently seemingly unrelated publications could be unified on the same foundation; (3). each and every practically applicable methods of systems science has to be theoretically established on the cohesive theoretical spectrum so that conclusions of the methods are scientifically sound and more readily acceptable by the greater community of science; and (4). theoretical beauty and practical applicability of systems research have to be established in order to produce a steady follow of new blood for the community of systems scientists. the most urgent of these tasks of systems research is develop a specific speaking language and logic of reasoning for systems research so that new scientific 322 yi lin:a development plan for the new millennium progresses can be authentically and convincingly shown to be fruits of systems thinking. the current practice of identifying various new theories from various traditional disciplines as new progresses in systems research just wont make systems research a science that truly complements the traditional science as it promises. this end has been well witnessed by the recent and fast disappearance of systems programs from around the u.s.a. considering the historical successes of calculus for our purpose, the community of systems research should really use all the activities around the development of calculus as its reference to design its relevant courses of action. in particular, we need to develop a curriculum of systems science and methodology to be offered to all levels of formal and informal education, because as we have shown earlier in this paper, in any corner of knowledge that involves basic operations of the traditional scientific thinking, the corresponding systemic reasoning should also be involved. in the rest of this section, let us discuss more specific course contents to be used in the imagined curriculum. to this end, due to the fact that the field of systems science, the topics of systems research, and the interpretations of the basic terms, such as system, systems methodology, systems thinking, etc., are both broad and diverse, our discussion here will be naturally in principle only. all the specific details will be figured out and improved later when particular course contents are concerned with. what follows next in this section is based on[22] and is intended to be a starting point for future thinking and efforts. it does not aim to be definitive or authoritative. it represents only a partial account of the vast literature existing in systems science. (1) general systems theory c the foundation: it seeks to establish studies of systems as a discipline in its own right. it aspires to uncover from different scientific disciplines concepts, laws, and models that are applicable to systems of all types. as what is shown in[13], one needs to be reminded that some of the laws of nature are not expressible in the language of mathematics. considering the diverse audience of our imagined educational program, several versions of this theory should be made available for teaching to different majors of undergraduate students in order to be more readily accessible by these students. at the same time, at this foundation level, the basics of systems science and engineering should be laid out firmly so that on these basics all other courses of the curriculum will be established. (2) organizations as systems: studies in this direction seek to understand different levels of human organization through interactions of subsystems and relationships with the environment. courses along this line will mainly target students in management and social sciences. (3) hard systems approach: it investigates the general systems thinking by using the traditional natural scientific methods. all real-world systems of conadvances in systems science and applications (2012) vol.12 no.4 323 cern are modelled by using mathematical systems and then these systems are optimized in order to pursue some predefined goals. theories and methods established here should be well organized into an organic whole so that all students who pursue careers in science and technology could benefit from this course. (4) cybernetic thoughts: it stands for the science of communication and control among the elements of a system and is about effective organization. some of the laws established in this area are considered to be as important to managers as einsteins law of relativity is to physicists. similar to line 1) above, several versions of the textbook should be prepared for this area in order to be useful to students with different academic backgrounds. (5) systems dynamics: it concerns with continuous processes and the dynamic behaviours of feedback loops or processes, and covers model formulation, equation writing, approaches of validation, and policy analysis. computer simulation techniques are established. courses along this line should be designed for students who are more science and technology inclined. (6) soft systems: it seeks to apply systems ideas and laws to situations where objectives are not well-defined, and the systems of concern are too complex to be modelled by using mathematics. the complexity arises partially because human beings and ethical concerns are involved. courses on soft systems thinking should be made available to all students because methods and conclusions learned here can be useful in day-to-day lives. (7) emancipatory systems thinking: studies in this area attempt to provide help to groups in disadvantageous situations by applying systems approaches. so one course should be designed for this area along with the teaching of strategies developed and well tested throughout the history of various conflicts. (8) critical systems thinking: as the name suggests, studies in this area attempt to bring unity to the vast number of different systems approaches so that the issue at hand, which might be characterized with seemingly unmanageable large scale, unfathomable complexity, uncontrollable uncertainty, impermanence, and imperfection, can be resolved successfully. this subject matter can be arranged for graduating seniors to take. because calculus is surely one of the few scientific theories that has enjoyed a long-lasting glory, in order to make systems science as recognized and as widely accepted as it seems to promise, we should employ calculus as our reference when we compose the materials of the imagined courses in order to satisfy the four criteria of a glorious and scientific theory as listed above. another important insight we learned from calculus is that when students take courses of systems science, they need be taught, along all other fundamentals, with procedures that can be loosely followed to produce some more or less definite outcomes, which of course depends on which systems area is involved. this end actually explains 324 yi lin:a development plan for the new millennium why calculus has been used by generations after generations. in other words, that is how calculus has provided livelihood for millions of people since its inception over three hundred years ago. to conclude this section, let us look at how specifically one can get started with our imagined curriculum in systems science by looking at a typical university in the u.s.a. assume that a systems scientist teaches in an engineering school. other than what he does already in teaching and research, to succeed with his offering of a new course in systems methodology, he would normally go through the following steps: (1). chooses a book on systems methodology that is available in the marketplace, and that is good enough to be used as a textbook for all engineering majors. if no such book is available, then he can readily write one to fill the market need. (2). fill out some particularly designed forms for creating a new course. here what is presented earlier in this paper can be modified for the new course proposal for explaining why such a new course is needed. (3). get approvals of the departmental, college, and university curriculum committees; (4). advertise on campus to attract at least the minimum required number of students to take the course; and (5). actually teach the class. if the said systems scientist is in the department of philosophy instead of the engineering school as mentioned above, then he can more readily offer the course of systems science that is more philosophical in nature as the first class. if the systems scientist is in the department of mathematics, he can then conveniently offer a course like systems analysis that is more mathematically oriented. in short, as we discussed in this paper, systems research had popped up as a fad once in a while at different times in various languages throughout the history. however, when compared to the successes of mathematics and natural science, none of these fads lasted very long. on the contrary, the current wave of systems research has been quite long lasting, about one hundred years old, wide spread and accepted throughout the entire spectrum of learning. that is the scientific history has presented us a golden opportunity to make the past short-lived fads into a prominent dimension of knowledge. so, let us hold hands together and make our impacts in the history. 5 some final words as analyzed in this paper, the golden opportunity, which appears along with the breadth and diversity the vast amount of the literature of systems science and engineering represents, for the next stage development of the systems movement could be more effective if a little purposeful planning is first given. by referencing back to the development history of calculus, it is expected that as long as a advances in systems science and applications (2012) vol.12 no.4 325 worldwide undergraduate education program in systems science can be initiated, a 2-dimensional landscape of knowledge will be formally established, where many of the important whys the mankind has been asking for thousands of years will be ultimately answered. so, let us join hands to collectively make the world in which we live a better place for everyone. acknowledgements the first author likes to use this opportunity to thank professor markus schwaninger, institute of management, university of st. gallen, switzerland, for his suggestion of writing up this article due to its foreseeable impacts on the community of systems researchers. this work was supported by national natural science foundation of china (71171116). references [1] klir g. (2001), facets of systems science, new york: springer. [2] lin y, and forrest b. (2011), the systemic structure beyond human organizations: from civilizations to individuals, new york: crc press, an imprint of taylor and francis. [3] lin y. (1999), general systems theory: a mathematical approach, new york: plenum and kluwer academic publishers. [4] klir g. (1985), architecture of systems problem solving, new york: plenum press. [5] blauberg i.v, sadovsky v.n, and yudin e.g. (1977), systems theory, philosophy and methodological problems, moscow: progress publishers. [6] lin y. (2008), systemic yoyos: some impacts of the second dimension, new york: auerbach publications, am imprint of taylor and francis. [7] berlinski d. (1976), on systems analysis, cambridge, ma: mit press. [8] lilienfeld d. (1978), the rise of systems theory, new york: wiley. [9] kline m. (1972), mathematical thought from ancient to modern times, oxford: oxford university press. [10] wu y, and lin y. 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(1985), albert einstein: the great explorer, beijing: beijing people’s press. [17] hendrix h. (2001), getting the love you want: a guide for couples, new york: owl books. [18] von bertalanffy l. (1972), “the history and status of general systems theory”, in: g. klir (ed.), trends in general systems theory, new york, pp.2141. [19] armson r. (2011), growing wings on the way: systems thinking for messy situations, devon, uk: triarchy press. [20] lin y, and ouyang s.c. (2010), irregularities and prediction of major disasters, new york: crc press, an imprint of taylor and francis. [21] weiss r. (1995), “medicine’s latest miracle”, health, vol.9, pp.71-78. [22] lane d.c, and jackson m.c. (1995), “only connect! an annotated bibliography reflecting the breadth and diversity of systems thinking”, systems research, vol.12, no.3, pp.217-228. [23] von bertalanffy l. (1924. may), einfuhrung in spengler’s werk, literaturblatt kolnische zeitung. corresponding author yi lin can be contacted at: jeffrey.forrest@sru.edu advances in systems science and applications (2012) vol.12 no.4 399-405 the identification of outliers in armax models via genetic algorithm ping chen1 and ying chen2 1department of mathematics, southeast university, nanjing 210096, china 2department of forensic science, jiangsu police institute , nanjing, 210012, china abstract this paper proposes a procedure to identify additive and innovational outliers by genetic algorithm in autoregressive moving average with exogenous variable(armax) time series models. we use some methods to delete the influence of input process in armax model and then detect outliers in time series based on the previous work, which is an improvement and extension of the detection method on arma models. empirical and simulation studies show that the proposed procedure is effective. keywords dynamic systems, innovational outliers, armax model, genetic algorithm 1 introduction outliers in dynamic systems or engineering time series can have adverse effects on model identification and parameter estimation. several procedures are available in literature to handle outliers in a time series. however, the case of multiple additive outliers and innovational outliers is very difficult to study because of the great number of alternatives and of the masking and swamping effects. compared to other search algorithms, genetic algorithms allow many candidate solutions to be considered simultaneously at each step. baragona et al.[1] showed how to use genetic algorithms for outlier detection and classification in arma series. peña and sánchez[2] presented a new procedure for multifold predictive validation in armax models. also, chen et al.[3] and chen et al.[4] developed some methods for detecting outliers, change point and outlier patches in bilinear time series models. on the other hand, huang et al.[5] discussed the improved genetic algorithm for vehicle routing problem with time windows. in this paper, a genetic algorithm is proposed to identify additive and innovational outliers in armax series. we are using the standard genetic algorithm with complete replacement of the past population and elitist strategy. the relationship between inverse correlations and outliers is helpful to simplify the fitness function, which may be quickly computed by trench’s algorithm. for the case of large series, it is better to divide the series into several parts so that the two neighbor subseries share a length of same data, and then to detect the subseries respectively. thus, the problem of large population in genetic algorithms has been avoided, as well as 400 ping chen:the identification of outliers in armax models via genetic algorithm the loss of outliers around the cut point. at last, simulation studies are carried out, which show promising results. 2 genetic algorithm and outliers models in armax series the genetic algorithm(ga) is known to be able to provide us with a powerful optimization tool when the solution space happens to be both discrete and large, and the objective function does not fulfill the usual regularity requirements.the key feature of a ga is the manipulation of a population whose individuals are characterized by possessing a chromosome. this latter can be coded as a string of characters of given length. each string represents a feasible solution to the optimization problem. the link between the ga and the problem at hand is provided by the fitness function (ff). the ff establishes a mapping from the chromosomes to some set of real numbers. the greater the ff is, the better the adaptation of the individual. the procedure is iterative. it makes use of three evolutionary operators: reproduction, crossover and mutation. an arma model with input process is called armax model, which is defined as zt = d∑ i=1 υi(b)xi,t + nt, where υi(b) = (δ−1 i (b) · ωi(b))bki is the transfer function of ith input process, nt = (θ(b)/ϕ(b))εt is noise process. {zt} is called response process. and xi,t denotes the ith input process or the difference of ith input process at time t, ki presents the influence’s time delay of ith input process, εt is normal white nose process. θ(b) = 1 + θ1b + · · ·+ θqb q, ϕ(b) = 1− ϕ1b − · · · − ϕpb p where b is the backshift operator. when υi(b) = 0, i = 1, · · · , d, it is arma model, when some υi(b) is nonzero constant, i = 1, · · · , d, it is regression model with arma error. (1) additive outliers(ao) model: suppose that only the jth point zj be ao, whose influence magnitude is wtj , then we have zt = d∑ i=1 υi(b)xi,t + wtjδt,tj + θ(b) ϕ(b) εt where δt,tj is kronecker symbol: if t = tj, then δt,tj = 1, else δt,tj = 0. (2) innovational outliers(io) model suppose that only the jth point zj be io, whose influence magnitude is wtj , advances in systems science and applications (2012) vol.12 no.4 401 then we have zt = d∑ i=1 υi(b)xi,t + θ(b) ϕ(b) (εt + wtjδt,tj) = d∑ i=1 υi(b)xi,t + wtj θ(b) ϕ(b) δt,tj + θ(b) ϕ(b) εt. 3 identification of armax models suppose that the armax model of only one input process is as follows: zt = δ−1(b)ω(b)xt−b + nt = υ(b)xt + nt, (1) where δ(b) = 1− δ1(b)− · · · − δr1b r1, ω(b) = ω0 − ω1(b)− · · · − ωr2b r2 and υ(b) = δ−1(b)ω(b)bb suppose the input process xt is stationary and is able to be represented by some member of the general linear class of autoregressive-moving average models. given a set of data, similar to box et al.[6], then we can carry out our usual identification and estimation methods to obtain a model for the xt process ϕ(b)θ−1(b)xt = αt which, to a close approximation, transforms the correlated input series xt to the uncorrelated white nose series αt. at the same time, we can obtain an estimate s2α of σ2α from the sum of squares of the α̂′s. if we now apply this same transformation to zt to obtain βt = ϕ(b)θ−1(b)zt, then the model(1) may be written βt = υ(b)αt + εt, multiplying αt−k on both sides and taking expectations, we obtain γαβ(k) = υkσ 2 α, where γαβ(k) = e[αt−kβt]is the cross covariance at lag k between α and β. thus υk = [ραβ(k)σβ]/[σα], k = 0, 1, 2 · · · . hence, after “prewhitening” the input, the cross correlation function between the prewhitened input and correspondingly transformed output is directly proportional to the response function. in practice, we do not know the theoretical function ραβ(k), so we must substitute estimates in υk to give υ̂k = [rαβ(k)sβ]/[sα], k = 0, 1, 2... where rαβ(k) = cαβ(k)/[sα/sβ], cαβ(k) = 1 n n−k∑ n=1 (αt − ᾱ)(βt+k − β̄), sα = √ cαα(0), sβ = √ cββ(0), k = 0, 1, 2, ... 402 ping chen:the identification of outliers in armax models via genetic algorithm the preliminary estimates υ̂k can provide a rough basis for selecting suitable transfer function model. first, we may use the estimates υ̂k so obtained to make guesses of the order r1 and r2 of δ(b) and ω(b), and of the delay parameter b. second, we do not consider the noise nt now, substituting zt = υ̂(b)xt in the equation δ(b)zt = ω(b)bbxt, based on equating coefficients of b, to obtain initial estimates of the parameters δ(b) and ω(b). 4 the identification of outliers via genetic algorithm we let {yt, t = 0, 1, 2, · · · } be a zero mean and stationary time series: yt = ∞∑ j=0 ψjαt−j , where {αt} is gaussian zero mean white noise and v ar(αt) = σ2, {ψj , j = 0, 1, 2, · · ·} form a absolutely summable sequence. let γih denote the inverse autocovariance function of the process, for integer h. also let ρih = γih/γi0 denote the inverse autocorrelations. we have γik + ψ1γik−1 + ψ2γik−2 + · · · = 0, k > 0 (2) when outliers are present, {yt, t = 0, 1, 2, · · ·} is unobservable. instead the time series {zt, t = 0, 1, 2, · · ·} is observed which follows the model: zt = yt + dt, where dt is a deterministic perturbation. let ψj = 0 if j < 0, then we have dt = ψt−t0w0, if time is io; or dt = w0δt,t0 , if time t0 is ao , where w0 denotes the outlier’s magnitude at t = t0. in the present setting, a chromosome ξ is a string of characters of assigned length n that can be evaluated in terms of the ff, where n is the number of observations of the time series, as each locus ξj is corresponding to an observation zj where an outlier may occur. so, ξ = (ξ1, ξ2, · · · , ξn). then a gene ξj = 0, if the locus is an outlier-free time point, ξj = 1 if the observation at this time point is an additive outlier, and ξj = 2 if it is an innovational outlier. if k outliers are located at t1, t2, · · · , tk and denoting by z = (z1, z2, · · · , zn)′ the observed time series and by y = (y1, y2, · · · , yn)′ the unobserved realization of yt, then z = ψw + y , where w = (w1, w2, · · · , wk) ′ is the vector of the outliers’ magnitudes at t1, t2, · · · , tk and ψ is the matrix of the n × k elements ψjh defined as follows: if th is the timing of an io, then ψjh = ψj−th if j > th and 0, otherwise. if an ao is occurring in th, then ψjh = 1 if j = th and 0, otherwise. the idea is to seek for the matrix ψ which maximizes the likelihood function. when n is large, γ−1 may be replaced by the matrix of inverse autocovariances γi . then we have the likelihood function of the time series z l = p(z | ξ,w ) = 2π−n/2(detγi)1/2 exp{−1 2 (z −ψw )′γi(z −ψw )} (3) the joint maximum likelihood estimate w ∗ of w , given the pattern of the outlying observations and γi, is w ∗ = (ψ′γiψ)−1ψ′γiz and y ∗ = z −ψw ∗ (4) advances in systems science and applications (2012) vol.12 no.4 403 in practice, however, γi is seldom known, so we have to estimate it from the data. similar to baragona et al.[1], using the interpolator estimate of yt. we may obtain inverse autocorrelations interpolation estimates ρ̂ij , j = 1, · · · ,m and the estimate γ̂i0 of inverse variance and the estimates of inverse autocovariance γ̂ij = γ̂i0 · ρ̂ij , j = 1, · · · ,m. once the inverse autocovariances have been estimated, the ψj , j = 1, · · · , q may be computed using equation (2). nevertheless, these estimates are biased because of the presence of the outliers. so we have to resort to an iterative scheme for (4), which is repeated until convergences. if convergence is attained, and since the likelihood is maximized with respect to the inverse autocorrelations, the argument of the exponential in likelihood’s formula (3) is simply −n/2 , which is shown by the following proposition(the proof is omited). proposition 1 using the former representation and estimations, then we have −1 2 (z −ψw )′γ(z −ψw ) = −1 2 n therefore, we have logl = −n 2 log(2π) + 1 2 log(detγi)− 1 2 n thus, the likelihood function basically depends on detγi. because that the fitness function(ff) has to be positive, we let f (ξ) = a · blog(detγi)−ck, where b is a real constant such that b > 1, and the constant terms in the exponent were dropped. we use a = 10, b = 1.0001 and c = 14 in this paper. we adopted (1) population size s = n + 1; (2) probability of crossover pc = 0.8; (3) probability of mutation pm = 0.05; (4) the maximum number of outliers within a chromosome g = 10; (5) the number of iterations of the series’ adjustment/parameters’ estimation procedure was 3;(6) we perform as many iterations of the genetic algorithm as possible within a reasonable time period. 5 simulation studies example a in the following example, we consider the model (1− 0.8b + 0.3b2)xt = εt zt = (1− 0.5b)xt − 4δt,30 + 5δt,31 − 5δt,80 − 4δt,90+ 6× 1− 0.36b + 0.85b2 1− 0.6b δt,40 + 1− 0.36b + 0.85b2 1− 0.6b et where {εt} and {et} are all normal white noise, their means are zero and variance σ2 = 1. we create 101 observations x0, x1, · · · , x100 of xt and 100 observations z1,...,z100 of zt by simulation. obviously, it is ao at t = 30, 31, 80, 90 singly and io at 404 ping chen:the identification of outliers in armax models via genetic algorithm t = 40 , and outlier magnitudes are w30 = −4, w31 = 5, w80 = −5, w90 = −4 and w40 = 6, respectively. applying our method to the above data and prewhitening the input series. making {xt} follows an arma model: (1− 0.88088b + 0.32738b2)xt = εt. then we take the same manipulation to prewhiten {zt}. by analyzing filtered cross correlation coefficient of {zt} and {xt}, we obtain the transfer function 1.03416−0.73967b for {xt}. delete the influence of input process {xt} in response process {zt}, and let z∗t = zt − (1.03416 − 0.73967b)xt. we have that {z∗t } is an arma series include outliers. we detect the outliers in {z∗t } by applying the above method. let m = 2, q = 9, g = 10, s = 101. because the 30th point is very close to 31th point and they influence each other, it is difficult to identify the outliers. in this case, one needs larger number of iterations. we take 1000 iterations by standard genetic algorithm. and obtain the best individual at 612th iterations: t = 30(ao), w∗ 30 = −4.7172; t = 31(ao), w∗ 31 = 4.1010; t = 40(io), w∗ 40 = 4.0487; t = 80(ao), w∗ 80 = −5.6440; t = 90(ao), w∗ 90 = −4.6228 the outcome is consistent with our prearrangement. the outliers in {zt} process are detected successfully, and there is no misjudgement. 6 conclusions there are ao and io in our model, and also the outliers(ao) present consecutively. it is quite difficult to detect outliers, however, all of the outliers in the above model have been detected successfully, which shows our method is also efficient for the ao and io problem in armax model. some other case studies also show that our method is effective in detecting outliers’ location and type and in estimating their size for armax model. acknowledgements the research is supported by national natural science foundation of china (no.11171065) references [1] baragona r, battaglia f, calzini c. (2001), “genetic algorithms for the identification of additive and innovation outliers in time series”, computational statistics & data analysis, vol.37, pp.1-12. advances in systems science and applications (2012) vol.12 no.4 405 [2] peña d, sánchez i. (2005), “multifold predictive validation in armax time series models”, journal of the american statistical association, vol.100, pp.135-146. [3] chen p, li l, liu y and lin j.g. (2010), “detection of outliers and patches in bilinear time series models”, mathematical problems in engineering, vol.2010, pp.1-10. [4] chen p, yang j, li l.y. (2013), “synthetic detection of change point and outliers in bilinear time series models”, international journal of systems science, http://dx.doi.org/10.1080/00207721.2013.777983. (in press) [5] huang l, pang w, wang k.p, zhou c.g, xiao y. (2004), “improved genetic algorithm for vehicle routing problem with time windows”, advances in systems science and applications, vol.4, pp.118-124. [6] box g.e.p, jenkins g.m, reinsel g.c. (1994), time series analysis: forecasting and control, third edition, prentice-hall, englewood cliffs, nj. corresponding author author can be contacted at cp18@263.net.cn advances in systems science and applications (2014) vol.14 no.2 190-198 design & simulation of smitha: a structured and scalable architecture for 3d network on chip systems sanju v1, niranjan chiplunkar2, m khalid3 and p venkata krishna3 1nitte meenakshi institute of technology, yelahanka, bangalore, india 2nmamit, nitte p.o, karkala, india 3vit university,vellore, india abstract network on chip is a new paradigm in asic design, and new topologies are emerging from this domain of work, scalable modular interconnect for three dimensional high performance applications is a typical 3d architecture(smitha) exactly. in this paper, smitha will be analyzed, and be designed in verilog hdl. keywords smitha, network on chip, design and simulation 1 introduction early vlsi designs were implemented using common bus architecture. as time passed by the density of the ic’s rose dramatically.bus architecture failed to give the required amount of performance. this gave rise to a new architecture called network on chip[1-3]. the idea behind this is to use concepts of data networking on silicon. however, network on chip is becoming the backbone of all high performance chips, many topologies have been introduced in this novel field. mesh and torus[2,4] being the most commonly used topology. in this paper we like to introduce smitha, a new three dimensional topology for network on chip based systems. in this paper, a new three dimensional topology for network on chip based systems is introduced, and then, the internals of the topology and the implementation of the topology are explained. 2 a new 3d topology : smitha smitha: scalable modular interconnect for three dimensional high performance application (smitha) is a new three dimension topology for network on chip based systems. the construction of the topology can be explained as follows. the base topology is obtained by removing the root node and interconnecting the nodes of a complete binary tree along the same layer as shown in fig.1. the nodes are addressed based on the layer in which they belong and relative position within the layer. the three dimension topology is obtained by placing the above discussed base topology one over the other and by interconnecting the adjacent levels as follows. advances in systems science and applications (2014) vol.14 no.2 191 fig.1 smitha base topology with 3 layers fig.2 smitha base topology with three levels having three layers / level 192 sanju v: design & simulation of smitha: a structured and scalable architecture ... • the connections between odd level and an even are done by connecting the odd layers though the left and even layers through the right. for example interconnection between level one and two. • similarly the connections between even level and an odd are done by connecting the odd layers though the right and even layers through the left. for example interconnection between level two and three. this is depicted in fig.2. 3 design and sumulation of smitha the architecture discussed above was designed and implemented using verilog hdl. the code was tested and verified using modelsim and synthesised for cyclone ii fpga on altera de2 70 board. the design starts from the smallest unit in the topology which is the node which interconnect to form the topology. the node is designed with five interfaces which connect to the external world. fig.3 gives a representation of a node. fig.3 internals of a smitha node fig.4 internals of a smitha node interface advances in systems science and applications (2014) vol.14 no.2 193 each node has five interfaces which are named left interface (li), right interface(ri), top left interface (tli), top right interface (tri), bottom interface (bi). these connect to the neighbouring node to do their specified operation. fig.4 shows the internals of each of the interface. it consists of the connecting pins namely req, ack , data, clk which are request pin,acknowledgement pin, data pin and the clock pin. it consists of the routing algorithm in the routing logic and the whole control of the node in control logic. it also consists of temporary storage purpose namely the registers – send and receive and the buffers – send and receive. 3.1 packet format the packet has three components namely the destination address, source address and the data bits. fig.5 smitha packet format fig.5 depicts the same. the source and destination address consists of the level number to which the node belongs, the ring in the level and the position of the node in that ring. the level numbering starts from 0 to n 1 where n is the largest number of level the architecture holds. similar arguments holds good for ring number and node number which are within each level. similar argument hold good for source addressing also. data bits can be of any size depending on the problem. for example a node addressing (1, 0, 0, 2, 0, 0, 25) represents a data 25 from level two ring zero node zero to level one ring zero node zero. the total size of the packet is 2 ∗ ( log2 (n) + log2 (m) + ∑m k=1 2 k ) + t data bits. 3.2 communication between nodes fig. 6 shows how to nodes in smitha topology communicates. the nodes in smitha communicate with each other with first raising the req pin. the receiving node checks whether the receive buffer is full or not. if the receive is not full then busy bit and the ack bit is set. the code illustrates the same. pueduo code on recieve req // pseudo code when request signal is received { if(!full(recievebuffer) // checking receive buffer is full or not 194 sanju v: design & simulation of smitha: a structured and scalable architecture ... { busybit = 1; // if receive buffer is not full then make busy bit ack = 1; // and ack bit high } } fig.6 two nodes communicate in smitha once receiving the request signal, the data sending module will clock and send the data one bit at a time. after sending the data the busy bit of the sending module is kept low. on the receiving side, the data is received on with the help of receive register. after the reception of data, the busy pin is kept low. now the packet is checked by the routing algorithm to decide the next path. if the send buffer of the next interface is not full then the packet is send to the next interface else it is kept in the receive buffer of the current interface. the pseudo code below illustrates the same. pseudo code onack sendingside { for(i = 0 ; i< length(packet); i++) { clk = 1; data = sendregister[i]; clk = 0; } busybit = 0; advances in systems science and applications (2014) vol.14 no.2 195 } pseudo code onack recieveside { for(i = 0 ; i< length(packet); i++) { recieveregister[i] = data; } busybit = 0; nextinterface = route(recieveregister); if(!full(sendbuffer(nextinterface))) { store(nextinterfacebuffer); else store(recievebuffercurrentinterface); } } 3.3 control logic it is a integral part of the system. it basically checks and starts the operation of sending the packet. furthermore the packets which in the receive buffer are forwarded to their respective location by the logic. the pseudo code below depicts the same. pseduo code controllogic { if(!empty(sendbuffer)&&!busybit) { busybit = 1; req = 1 ; } if(!empty(receivebuffer)) { nextinterface = route alg(packet) ; if(!full(sendbuffer(nextinterface))) enqueue(packet) } } 3.4 output verification the above circuit was coded in verilog and tested using modelsim and was downloaded to de2 – 115 board to test the code. the code was tested for absolute 196 sanju v: design & simulation of smitha: a structured and scalable architecture ... fig.7 output waveform advances in systems science and applications (2014) vol.14 no.2 197 accuracy with different conditions. one of such output is as below (fig.7)it describes the packet going from (1,2,1) to (1,2,0 ) and then it moves to the second level to (2,2,0) and ends at ( 2,2,1). 4 conclusion the paper briefly draws a light into the new three dimensional architecture and discusses the fpga implementation using verilog hdl by means of altera’s quartus and modelsim for simulation purpose. the paper starts with the node implementation to the topology. the implementation discusses the internals of the node with every part of the node in detail. it clearly brings out the working of the module by the waveform shown. references [1] william j. dally , brian towles. (2001), “route packets not wires : on chip interconnection network”, dac 2001. [2] tobias bjerregaard , shankar mahadevan. (2006), “a survey of research and practices of network on chip”, acm computing survey. [3] ahmed hemani, axel jantsch, shashi kumar, adam postula, johnny berg, mikael millberg,dan lindqvist, “network on a chip: an architecture for billion transistor era”, dac. [4] ville rantala, teijo lehtonen, juha plosila. (2006), “network on chip routing algorithms”, tucs technical report, no.779. [5] radu marculescu, ieee, natalie enright jerger, yatin hoskote, umit y. ogras and li shiuan peh, “outstanding research problems in noc design: system, micro architecture and circuit perspectives”, ieee transactions on computer aided design of integrated circuits and systems,vol.28,no.1. [6] j. nurmi. (2005), “network on chip: a new paradigm for system on chip design”, proceedings 2005 international symposium on system on chip, no.17. [7] l.benini, g.de micheli, “networks on chips: a new soc paradigm”, ieee computer, vol.35. [8] s. kumar, a. jantsch, j. soininen, m. forsell, m.millberg, j. oberg, k. tiensyrja, a. hemani. (2002), “a network on chip architecture and design methodology”, proceedings international symposium vlsi (isvlsi), pp.117-124. 198 sanju v: design & simulation of smitha: a structured and scalable architecture ... [9] dan marconett, “a survey of architectural design and implementation tradeoffs in network on chip systems”, university of california, davis. [10] nilanjan banerjee, praveen vellanki, karam s.chatha. (2005), “power and performance model for network-on-chip architectures and test in europe conference and exhibition (date)”, transactions on computers, vol.54, no.8, pp.1025c1040. corresponding author sanju v can be contacted at:sanjuv21@gmail.com. advances in systems science and application (2015) vol.15 no.1 21-36 big data and big control d.a.novikov abstract this paper analyzes and structures modern challenges in the field of big data handling, arising before hardware developers, experts in applied mathematics and artificial intelligence, as well as researchers focused on different application domains. the author outlines further development of the theory and applications of big control, i.e., control based on big data. keywords big data, data analysis, big control, big data management 1 introduction i am a great believer in the simplicity of things and as you probably know i am inclined to hang on to broad & simple ideas like grim death until evidence is too strong for my tenacity. c sir ernest rutherford from letter to irving langmuir (june 10, 1919). the term “big data” means unstructured data whose volume exceeds the available handling capabilities in required time. this term appeared just a few years ago [1], but became very popular among it experts, scientists, business analysts and other specialists. for instance, a retrieval request to google returns millions of links on the subject. what are the capabilities and risks carried by big data? what challenges and problems do they formulate before scientists, experts in different application domains (particularly, specialists in systems science and control theory-big data usage in control problems and big data management), the education system as a whole? 2 big data. big analytics. big visualization in information technology, big data (perhaps, the term was first mentioned in the special issue of nature[1]) represents a direction of theoretical and practical investigations on the development and application of handling methods and means for the big volumes of unstructured data. big data handling comprises their1: acquisition; transmission; storage (including recording and extraction); processing (transformation, modeling, computations and analysis); 1in certain classifications, big data handling is associated with 4d (data discovery, discrimination, distillation and delivery/dissemination). 22 d.a.novikov: big data and big control usage (including visualization) in practical, scientific, educational and other types of human activity. in the narrow interpretation, the term “big data” sometimes covers only the technologies of their acquisition, transmission and storage. in this case, big data processing (including construction and analysis of corresponding models) is called big analytics (including big computations), whereas visualization of the corresponding results (depending on users cognitive capabilities) is called big visualization (see fig. 1). the universal cycle of big (generally, any) data handling is illustrated by fig. 2. here the key role belongs to an object and a subject (a “customer”); the latter requires knowledge on the state and dynamics of the former. however, sometimes there exists a chasm between the data acquired on an object and the knowledge necessary for a subject. primary data must be preprocessed, i.e., transformed into more or less structured information. subsequently, necessary knowledge is extracted from this information depending on a specific task solved by a subject. fig.1“the big triad”2: data, analytics, visualization particularly, a subject may adopt this knowledge for object control , viz., exerting purposeful impacts on an object to ensure its required behavior. control can be automatic in a special case (an inanimate subject). perhaps, the term “big control”3 will become common soon for indicating control based on big data, big 2we will not discuss another fashionable triad (big data, high-performance computations, cloud technologies 3for justice’ sake, note that in the recent fifteen years experts in control theory have tended to consider the problems of control, computations and communication jointly (the so-called c3 problem (control, computation, communication)). according to this viewpoint, control actions are synthesized in real time taking into account the existing delays in communication channels and information processing time (including computations). there is another generally accepted term (large-scale systems control), but big data can be generated by “small” systems.] advances in systems science and application (2015) vol.15 no.1 23 analytics and, possibly, big visualization.4 fig.2 the universal cycle of big data handling the qualitative analysis of numerous publications on big data leads to the author’s subjective expert appraisal for the current distribution of attention paid by researchers and developers (but not users!) to big data handling problems. this expert appraisal is demonstrated in fig.3. fig.3 the current distribution of attention paid by researchers and developers to big data handling problems 4an alternative interpretation of “big control” concerns control of big data handling processes. actually, this represents an independent and nontrivial problem. 24 d.a.novikov: big data and big control in other words, the overwhelming majority of big data investigations create the technologies of big data acquisition, transmission, storage and preprocessing, whereas big analytics and visualization receive by far less consideration. big control problems are almost not studied, despite that control objects become more and more complicated and multi-scale, while “networkism” represents a leading trend of modern control theory [2]! 3 civilization-scale problems however, is the current state of affairs (see the distribution in fig.3) reasonable? on the one hand, the answer is affirmative. indeed, technologies followed exactly this path of development; moreover, data analysis and visualization requires data acquisition and storage (no doubt, with the feasibility of rapid accessing and processing). on the other hand, the existing “disbalance” is the result of the following. nowadays, mankind realizes the potential utility of any data, but does not completely understand what should be done with the growing avalanche of data. this problem seems not novel, since a class of similar civilization-scale problems has appeared recently, formally called the problems of anticipatory technologies development. to elucidate this idea, consider the correlation among science, technologies and practice (see fig.4). during different periods of mankind development, science often initiated creation and adoption of certain technologies; sometimes, the chain was “inverse” (presently, we observe exactly this picture!). fig.4 science, technologies and practice really, appeal to history. starting from the middle of the 19th century, for over 100 years mankind had experienced the triumphal development of science and the technological boom (science anticipated and predetermined technologies, the latter were widely adopted in practice; generally speaking, before science was advances in systems science and application (2015) vol.15 no.1 25 stimulated by practice).in this context, we mention electricity, communications, nuclear power engineering, electronics, etc. most technologies were clear and available to average men.5 later on (approximately, since the 1970s) the situation gradually changedcthe accumulated fundamental scientific results were enough for technologies growth (i.e., some technologies even “anticipated” science). consequently, their “demand for science” was reduced (perhaps, living systems made an exception). nevertheless, technologies demonstrate further development, and the pace of their development even increases. for instance, in the recent decade advances in information and communication technologies (ict) have been even anticipating practice6, including human recognition of new technological capabilities, development prospects and associated threats.7 in other words, the threshold of the third millennium was remarkable for a turning point: mankind had earlier mastered science and technologies to satisfy its needs, whereas afterwards technologies rather dictated the directions, restrictions and conditions of scientific development, at the level of individuals and at the level of states and mankind as a whole.8 exactly this effect is called “anticipatory technologies development”. mankind will have to realize the corresponding civilization-scale problems and learn to respond to them. similar things happen with big data: human beings have mastered technologies for accumulating giant volumes of data, but are still unable to process and utilize them. the major problem concerns the comprehension of why should big data processed (rather than how should it be done). to have a clear view of the challenges faced by scientists and engineers, we 5amusingly, the heroes of fantasy books on technological breakthrough, owing to isolation (e.g., cyrus smith from j. verne’s the mysterious island) or a trip to the past (e.g., the hero from m. twain’s a connecticut yankee in king arthur’s court), are often engineers or average men but not scientists. 6as a positive feature, we acknowledge that advances in ict were stimulated by the development of applied mathematics, particularly “network mathematics” (random graphs, largedimensional graphs, network games and others, see the surveys [3-6]). 7scattered manifestations of this phenomenon took place earlier; e.g., a. noble and the participants of the manhattan project thought about the ethical responsibility of scientists. 8the problems of information security have become common. today is the right time for thinking not just about technological security (cybersecurity), but about the social, economic and other security of information technologies, i.e., the security of ict users, their groups and society as a whole against informational impacts (here a striking example concerns social media including online social networks [7]). recall that all important decisions (at the level of an enterprise and at the level of a government) are made on the basis of information coming from certain sources and processed by certain methods (far from always, a decision-maker knows these sources and methods). therefore, we have to acknowledge the relevance of the socioeconomic security of ict, i.e., the security of an individual, an economy, a society and a country against the consequences of decisions made by modern ics (including decision support systems applied in economics, military science, politics, etc.). 26 d.a.novikov: big data and big control discuss a series of questions. which data are actually big? where do big data arise? what are the modern applications of big data? how can one improve the efficiency of their usage in future? what is the role of science in big data handling? 4 sources and customers of big data they include large groups, namely: science (astronomy and astrophysics, meteorology, nuclear physics, highenergy physics, geoinformation systems and navigation systems, distant earth probing, geology and geophysics, aerodynamics and hydrodynamics, genetics, biochemistry and biology, etc.); internet (in the wide sense) and other telecommunication systems; business, commerce and finances, as well as marketing and advertising (including trading, targeting and adviser systems, crm-systems, rfidcradiofrequency identifiers used in sales, transportation, logistics and so on); monitoring (geo-, bio-, eco-; space, air, etc.); security (military systems, antiterrorist activity, etc.); power engineering (including nuclear power engineering), smartgrid; medicine; governmental services and public administration; production and transport (objects, units and assemblies, control systems, etc.). numerous applications9 of big data in these fields can be found in popular science literature (or even “glossy” journals) available at public internet sources. we will not describe these applications here to avoid embarrassing “zettabytes” and “yottabytes”. in almost all fields cited, the modern level of automation is such that big data have automatic generation. therefore, the following question gains growing importance. what is the volume of “lost” data flows (due to insufficient capabilities or time for their storage or processing)? this question seems correct for an engineer in ict, but not for a scientist or a user of big data processing results. rather, the former and the latter would ask “what are essential losses in this case?” and “what are the changes if we successfully acquired and processed all data?”, respectively. 9the principal idea of using big data is revealing “implicit regularities”, i.e., answering nontrivial questions: epidemic prediction based on information from social networks and sales in drugstores; medical and technical diagnostics; retention of clients by analyzing sellers’ behavior in stores (the spatial movements of rfid-tags of products); and others. advances in systems science and application (2015) vol.15 no.1 27 5 which data are big? scientific challenges traditionally, big data are unstructured data whose volume exceeds the available handling capabilities in required time. however, this definition appears somewhat“cunning”: data considered big today cease to be such tomorrow owing to the progress of data handling methods and means. data that looked big several hundreds or even thousands of years ago (in the absence of automatic treatment) are easily processed today by home computers. the competition between the (hypothetic) computational demands of mankind and corresponding technical capabilities has been known very long ago. of course, the capabilities have been always chasing the needs. and the gap between them represents a monumental stimulus for science development. researchers have to suggest simpler (yet, adequate) models, design more efficient algorithms, etc. sometimes, the definition of big data includes the so-called 5v properties (volume, velocity, variety, veracity, validity). alternatively, the difference between the big volume of conventional data and big data proper is that the latter form the big flow of unstructured10 data (in the sense of volume and velocity as the volume per unit time). in the wide comprehension, the unstructuredness of big data (text, video, audio, communications structures, etc.) is actually their characteristic feature and a challenge for applied mathematics, linguistics, cognitive sciences and artificial intelligence. creation of real-time processing technologies11, including the feasibility of implicit information revelation, for large flows of text, audio, video and other information forms the mainstream of applications of the above sciences12 to ict. therefore, we observe a direct (and explicit) query from technologies to science. the second explicit query concerns adaptation of traditional statistical analysis, optimization and other methods to big data analysis. furthermore, it is necessary to develop new methods with due consideration of big data specifics. a modern fashionable trend is boosting analytics tools (generally, business analytics) for big data. but their list almost coincides with the classical kit of statistical tools (or is even narrower, since some methods are inapplicable to big data). this is also the case for: machine learning methods (neural networks, bayesian networks, fuzzy inference and other logical inference, etc.); 10data unstructuredness can be the result of their omissions and/or different scales of studied phenomena and processes (in space and time, see the so-called multi-scale systems). 11in the first place, these technologies must perform data aggregation (e.g., detecting changes in technological data or storing aggregated indices). really, one does not need all data (especially, “homogeneous” data). 12mathematics rather easily operates structured data; and so, data structuring makes an important problem. 28 d.a.novikov: big data and big control high-dimensional optimization problems (in addition to traditional parallel computing, intensive research focuses on distributed optimization, l1-optimization; e.g., see [8-13]); discrete optimization methods (here an “alternative” lies in application of multiagent program systems [14]) and others. the common feature in the stated queries of technologies to science is adaptation or small modification of well-known tried-and-true methods. we have to be aware of the following. generally, automatic modeling (by traditional tools13) based on raw data represents just a fashionable delusion.14 we expect to suggest algorithms and apply them to bulky volumes of unstructured (often irrelevant) information, thereby improving the efficiency of decision-making. such delusions occurred in the history of science at the early development stages of cybernetics and artificial intelligence.15 they resulted in numerous disappointments and considerably impeded the development of these scientific directions. there exist no miracles in science: generally, new conclusions require new models and new paradigms (e.g., see the books on science methodology [15-16]). 6 general challenge the complexity of the surrounding world grows at a smaller rate than the capabilities of data detection (“measurement”) and storage. perhaps, these capabilities have exceeded the ability of mankind to realize the feasibility and reasonability of their usage. in other words, we “choke” with data, trying to find what to do with them. however, there exists an alternative viewpoint of this situation as follows. obtaining big data (having an arbitrary large volume) is possible and easy enough (obvious examples arise in combinatorial optimization, nonlinear dynamics or thermodynamics, see below). but we have to understand how to manage big data (and ask the nature correct questions). furthermore, it is possible to construct an arbitrary complex model using big data and then try to reach a higher accuracy within the model. but the associated dilemma is whether we obtain new results or not (in addition to very many new problems16). long ago math13an additional encumbrance is the accumulated experience of a researcher/developer and the traditions of its scientific school. successful solution of a certain problem leads to the conviction that same methods (only!) are applicable to the rest open problems. 14in some cases, additional information can be obtained by increasing the volume of data (under correct processing). 15the behavior demonstrated by a cybernetic system is always the result of embedded algorithms (stochastic, nondeterministic, etc.), despite the seeming generation of new knowledge or revelation of new (“unexpected”) behavior. this is especially the case for interaction of very many elements. 16we recognize the importance of model’s adequacy and stability of modeling results, but omit these problems. advances in systems science and application (2015) vol.15 no.1 29 ematicians and physics knew that increasing the dimensionality and complexity of a model (aspiration for considering more factors and relations among them) does not necessarily improves the quality of modeling results; sometimes, it even carries to the point of absurdity.17 let us study a series of examples. example 1. the book [17] by nobel prize winner h. simon considered the following example. imagine an ant walking along a beach. the ant may try to minimize the efforts required for moving from one point to another; and so, it escapes rocks, sometimes turns back, etc. if we observe just the horizontal projection of the ant’s path (without the knowledge of relief), explaining its behavior (a very winding, complex path) seems difficult. h. simon arrived to an important conclusion. the existing variety and complexity of human behavior are explained not by their complex principles of decision-making (actually, these principles appear simple), but by the diversity of related situations. one would hardly disagree with this opinion. really, nontrivial results can be provided by a complex model based on simple input data and by a simple model based on complex input data. ideally, nontrivial results should be generated by simple models owing to the correct choice of relevant and simple input data (mathematicians are used to say that“simplicity is the sign of truth”). example 2. suppose that, as if by magic, scientists of the 18th century receive a modern laptop and a tomographic scanner of laptops (with user manuals). the scientists make the tomographic image of the laptop and save it in the memory. by analyzing these bulky and very detailed data on the physical structure of the laptop, they would hardly understand the operation principle of the laptop. a correct paradigm, a correct conceptual model forms a necessary (yet, not sufficient) condition of a success. example 3. take any np-hard problem of combinatorial optimization [18], e.g., the travelling salesman problem. there are about n! variants to-be-analyzed for exact solution search; in the case of n∼100, the number of variants exceeds the computational capabilities of mankind. the network excitation control problem for a real social network possesses the dimensionality of n∼ 106[19]. this property of np-hard problems has been known for decades, motivating researchers to develop (generally, heuristic) methods of approximate solution search with estimated guaranteed accuracy at reasonable time. a similar example concerns nonlinear dynamics models: the“observations” of dynamic chaos demonstrated by a rather simple (medium-dimensional) nonlinear dynamic system may require the memory of all computers in the world, but generate no new knowledge. example 4. a classical example from the history of physics is the discovery of 17not to mention situations, when existing scientific paradigms make it impossible in principle to model system behavior on a large time horizon (e.g., accurate weather forecasting). 30 d.a.novikov: big data and big control the law of universal gravitation. for two decades, t. brahe (1546−1601) observed planetary motion in the solar system. for those days, his records represented big data. based on brahe’s records, i. kepler (1571−1630) formulated his empirical (!) laws of planetary motion: the orbit of a planet is an ellipse with the sun at one of the two foci; the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit. keplers laws of planetary motion aggregated brahes data, and the motion of any planet could be calculated by them (instead of brahe’s many-volumed recordings) with a high accuracy. in other words, brahe learned to describe18 planetary motion, whereas kepler learned to describe and predict it. however, kepler’s laws did not explain why planets move so. the answer was later given using the law of universal gravitation established by i.newton (1643-1727). perhaps, kepler’s laws would be derived by any modern computer (with adequate algorithms embedded and initial data entered). at the same time, modern computers would be unable to obtain the law of universal gravitation without the corresponding model of mass interaction. kepler’s laws are the “corollaries” of the law of universal gravitation (can be deduced from it), just like brahe’s results follow from kepler’s laws. therefore, the law of universal gravitation made useless (epistemologically superfluous) both kepler’s laws and brahe’s big data. today’s experience in big data handling testifies that, in most cases, we are at the level of brahe and apply titanic efforts to reach the level of kepler. but a qualitative jump occurs only under the appearance of generalizations (at the level of newton) that radically simplify the situation. once again, note that the whole secret is“putting correct questions to the nature.” example 5. the second classical example from the history of physics relates to the development of molecular-kinetic theory. it shows that generation of a big data flow appears easy. the whole point is what to do with this flow and what questions to answer. consider the following mental experiment (the problem of ideal gas behavior description). one cubic meter of air is in the normal conditions. it contains approximately 1025 molecules. the motion of molecules and their collisions are completely described by kinematics and dynamics (within the framework of the ideal gas model). in other words, there exist no fundamental obstacles in such description. during a second, each molecule suffers from 109 collisions with other molecules. the real-time description of such a system (the coordinates and velocities of all molecules) requires the minimum data flow of 1035 bytes per second. 18recall the basic functions of scientific cognition (particularly, modeling) [16]: description (the phenomenological function)explanationprediction (the prognostic function) control (the normative function). advances in systems science and application (2015) vol.15 no.1 31 this data flow exceeds even the modern processing capabilities of mankind (to say nothing of the processing capabilities in the 1850s when the theory appeared)! perhaps, physicians recognized the senselessness of such detailed analysis (even despite its possibility in principle) and passed to macrodescription in terms of aggregated characteristics (temperature, volume, pressure) and, subsequently, to the description of probabilistic distributions within the framework of statistical physics. but if physicians of that period were able to perform necessary calculations, science would never have statistical physics! today a similar effect arises, e.g., in informational control problems for online social networks. transition from microdescription using graphs with tens of millions of nodes and billions of connections (no doubt, this is a big control problem) to macrodescription in terms of probabilistic distributions [20] leads to realistic theoretical study, still preserving the key properties of an object. and finally, note the following. depending on their source, big data can be natural or artificial. in the former case, data are generated by some independent object and we (“investigators”) decide what should be “measured” (examples 1, 2 and 4). in the latter case, the source of data is a model (examples 3 and 5); complexity (data flow) is partially controlled and defined during simulation. “recipes.” there exist four large groups of subjects (see fig.5) operating (explicitly or implicitly) big data in their professional (scientific and/or practical) activity: manufacturers of big data handling tools (software/hardware developers, suppliers, consultants, integrators, etc.); designers of big data handling methods (experts in applied mathematics and computer science); specialists in application domains (scientists focused on real objects or their models) that represent big data sources; customers utilizing or planning to utilize the results of big data analysis in their activity. representatives of the mentioned groups interact with each other (see the dashed lines in fig.5). the normative (“ideal”) division of “responsibility areas” is illustrated by fig.6; here the thickness of arrows corresponds to the level of involvement. proceeding from sensus communis and not claiming to be constructive, we formulate the following general “recipes” for the listed groups of subjects. for manufacturers of big data handling tools: with the course of time, it will be difficult to sell big data solutions (including analytical ones) without suggesting new adequate mathematical methods and stipulating for the feasibility of close cooperation between customers, the developers of appropriate methods and specialists in application domains. for mathematicians (the author’s “brothers-in-arms”): a topical query con32 d.a.novikov: big data and big control fig.5 subjects operating big data fig.6 division of “responsibility areas” cerns adapting well-known methods and developing new processing methods (in the first place, with nonlinear complexity!) for the large flows of unstructured data representing a good testing area for new models, methods and algorithms (to the extent possible, at the expense of manufacturers and/or customers). for specialists in application domains: big data technologies lead to new capabilities for acquiring and storing the bulky arrays of “experimental” information, conducting the so-called computing experiments; the associated methods of applied mathematics enable systems generation and rapid verification of hypotheses (revelation of implicit regularities). for customers: the expensive technologies of big data acquisition and storage would hardly be economically sound without involving specialists in appropriate methods and subject areas (only if it is absolutely clear which questions a customer would like to answer using big data19). some threats. in addition to the emphasized necessity of searching for ade19though, it is possible to store data de bene esse (e.g., to verify a certain hypothesis in future based on them). advances in systems science and application (2015) vol.15 no.1 33 quate simple models and the alerting trend of anticipatory technologies development, we expect the future relevance of the following problems (the list below is unstructured and incomplete). • the informational security of big data. this requires adaptation of wellknown methods and tools, as well as development of fundamentally new ones. • the energy efficiency of big data. even today, data processing centers represent a considerable class of power consumers. the bigger are data to-beprocessed, the higher is energy needed. • the principle of complementarity was established in physics long ago; it declares that measurements modify the state of a system. however, does it apply to social systems whose elements (people) are active, i.e., possess their own interests and preferences, choose their actions independently, etc. [21]? a demonstration of this principle lies in the so-called information manipulation (strategic behavior). according to theory of choice [4,22], an active subject reports information by forecasting the results of its usage; generally speaking, an active subject does not adhere to truth-telling. another example concerns the so-called active forecasting: a system changes its behavior based on new knowledge about itself [23]. are these and similar problems (e.g., crowdsourcing [4,24], conformity behavior [25], etc.) eliminated or aggravated in the case of big data? • as far as we have mentioned the principle of complementarity, it is necessary to recall the principle of uncertainty in the following (epistemological) statement [16]: the current level of science development is characterized by certain mutual constraints imposed on results “validity” and results applicability. in the context of big data, this principle means the existence of a rational balance between the level of detail in the description of a studied system and the validity of results and conclusions to-be-made on the basis of this description. • a traditional assumption in design and operation of information systems (corporate systems, decision support systems of governmental services, inter-agency circulation of documents, etc.) is that all information in such systems must be complete, unified and publicly available (under existing access rights). but it is possible to show the “distorting-mirror” reality to each person, i.e., to create an individual informational picture20, thereby performing informational control [4, 7, 21, 23]. should we strive for or struggle against these effects in the field of big data. 20at the very least, a fragment of the “objective” picture (hushing up the whole truth); at the most, an arbitrary inconsistent system of beliefs about the reality. 34 d.a.novikov: big data and big control 7 instead of conclusion thus and so, data have always been big. new, more and more perfected tools of big data acquisition, storage and processing appear intensively. we wish we managed to perform these operations in real time. this calls for developing appropriate directions of applied mathematics and computer science (a topical query from technologies and practice to modern science). there also exists a pressing need for mass training of specialists in big data, big analytics and big visualization (with focus on concrete application domains). however, this is not enough: we have to accumulate knowledge (in appropriate branches of science) and create models for compact and adequate description of studied phenomena and processes (with due consideration of a solved problem). in other words, it is desired to advance from the level of brahe to the level of newton in each possible area of big data applications. otherwise, we are doomed to handle particulars, not seeing the wood for the trees (see e. rutherford’s citation above). furthermore, the anticipatory development of technologies has become the civilization-scale problem to-be-accounted by scientists and engineers (including the field of big data and big control), as well as by the consumers of appropriate methods and tools created by them. references [1] special issue. 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(2013), “models of mob control”,automation and remote control, vol.74, no.12, pp.2143-2154. corresponding author d.a.novikov can be contacted at:novikov@ipu.ru advances in systems science and application (2015) vol.15 no.1 60-71 tackling foundations of systemic complexity on reality and decision systems hermı́nio duarteramos emeritus professor, new university of lisbon, lisbon abstract what is reality? that’s an old question, a big question raised by so many thinkers, since very ancient times. many responses has been given, so many answers are purposed in these days. here we advance on quantum territories and we also approach the human decision theory to interpret the complex meaning of reality under new systemic complexity viewpoints. keywords reality, systemic complexity, decision system, qbism, bayesian probability. 1 introduction traditionally, thoughts on reality were put forward among philosophers and scientists, around force lines of idealism and materialism dipoles, as plato (427-347 bc) and lucretius (99-55 bc) traced on matter, or over generated currents between theism and deism bipoles, as wilhelm leibniz (1646-1716) and albert einstein (1879-1955) did conduct about the universe [1]. today we find some other ways of thinking, based on new scientific interpretations, using abstract fields of mathematics, as john von neumann (1903-1957) supposing a consciousness creation for object attributes, and quantum physics, as niels bohr (1885-1962) negating deep reality down phenomenal facts [2]. nowadays, what can we say about the meaning of reality? first of all, it is a general topic, which appears inevitably at the front panel of globalization’s science, thinking on worldwide vertiginous trends of scientific research without national frontiers. the reality meaning is a complex problem to understand systems we meet on nature, focusing our consciousness on things, because we aren’t aware of essentials on material structures and human-nature interactions.nevertheless, today we know much better what reality can be than some centuries ago, and even decades behind or from late years. here i join three basic ideas, integrating them with the general system theory to interpret what reality is, namely the wavefunction for the quantum formulation of the ultimate matter, the complexity of an observed system and the subjective probability on decision making.in such way, i briefly tackle how to set up general foundations of the so called systemic complexity, hopping to contribute with a new and powerful vision on system sciences. it is inevitable to approach physical and mathematical concepts, at least evocating some descriptive principles, because physics and mathematics are sciences advances in systems science and application (2015) vol.15 no.1 61 of the reality modelling, that humans use to understand the world around. 2 what is reality? outside my mind there are things on the objective (or concrete) world, which are only matter and signals, and inside the mind it runs a subjective (or abstract) world with forms of ideas and signs1. the human detects signals that are radiated from matter and he sees or hears the impressive reality, and then he judges on what he is seeing or hearing by decision making on some important cues, which are detected signals contributing to the awareness of reality. on one hand, the physics (the natural philosophy in past times) measures and relates features of the objective reality, since electromagnetic waves as signals of light until quantum waves as signals of the ultimate matter, i.e., the physics objectivity interprets the external reality by its fundamental properties, and so the objectivity in physics doesn’t describe the world “as it really is” [3], but it is an epistemic knowledge devoted to measurable features of the world, on which an open worldwide community of independent observers do agree. on the other hand, cognition sciences (encompassing the brain, the mind and the soul) study features of the subjective reality, since sensing signals and their perception until the ultimate decision on inner appearances, i.e., the cognitive subjectivity also represents interpretations on the outer reality. when plato’s prisoners emerged from the cave, they were only able to make out shadows and reflexions as reality, not just because the sunlight burned their eyes but also due to the subjective probability of decision making in the brain’s mind. within this frame, i wish to participate freely on the open discussion of some fundamental thoughts on complexity, to better understand systems both of objective and subjective realities, trying to discover an all-embracing model for a single scientific paradigm of reality systems. each one of us makes decisions taking systems from the natural matter and observing their behaviour, and then we interpret what it is going on actually in our mind as a real representation. how can we do that?basically, a system is one concrete structure of several interconnected parts within a limited space and interacting in concrete modes (doesn’t a matter what they are at the moment), emerging to us inherent black box outputs2, as reactions to eventual stimulating inputs or owing to its own 1we don’t discuss here the mind’s cognition nature of brain’s imagerial and soul’s imaginal information processes. 2the system science uses efficiently the concept of “black box” and its “outputs”, being equivalent to the whole functional behaviour and ignoring the real system structure. 62 hermı́nio duarteramos: tackling foundations of systemic complexity on reality... autonomous existence, besides the necessary vital energy feeding things to exist with an own intentionality. 3 systemtic complexity we call systemics the science on the function integration of whatever composite system in order to process its behaviour by a functional model.in open systems the outputs emerge from the internal operation to the outward environment, and otherwise inward outputs define closed systems. generally speaking, the functional organization of a given system is characterized by the following four operational systemic essentials: the acrony or composition, the axony or interactivity, the aquadry or boundary, the adaptacy or optimum workability.however, the most notable feature is that the real system always denotes an existential intentionality that must be truly expressed by their outputs into the environment, which may be detected by instruments or human observers. these external information signals establish a quintessential systemic principle, the telonomy or aim outputs representing the intrinsic intentionality of the system in action [4].that is to say, a system accomplishes its existential purpose if its bulk operation obeys to those five systemic principles, and we need to know them clearly in order to acknowledge its real behaviour. in complex systems (distinct from complicated ones) we don’t know quite well one or more systemic essentials, otherwise the correct knowledge of all these characteristics defines a simplex system (distinct from simple ones).in the case of a simplex system all systemic essentials are well known, and so we have a definite acrony and complete axony and certain aquadry and determinate adaptacy and an accurate telonomy. the four operational systemic essentials of a simplex system may be expressed mathematically onto a transfer function3 to calculate its right behaviour, denoting the systemic telonomy at each time, but complex systems incorporate additional barriers for obtaining reasonable outcomes, which are variable with the correctness of systemic essentials. the degree of complexity for the system’s observer is related to the imperfect knowledge he has on systemic essentials, for instance, the complexity of a simplex system is zero degree, and a complex system of 2nd degree has two systemic essentials that we don’t known entirely, either indefinite acrony or incomplete axony or uncertain aquadry or indeterminate adaptacy or inaccurate telonomy. 3the transfer function of a system requires initial conditions to be defined in the complex frequency domain s = σ + iω, where σ is a real frequency and iω is the imaginary angular frequency. advances in systems science and application (2015) vol.15 no.1 63 the concept of systemic complexity4 was introduced to approach complexities encountered in material or concrete (objective) systems, and the same functional methodology was extended to mind or abstract (subjective) systems, both trying to reduce complexity into simplexity. here they are some foundations of the systemic complexity of general and singular systems to interpret the observed natural world, fusing accessible conscious information on imagerial and imaginal mind processes.we use it to acknowledge both objective and subjective reality, for the near contact world till the ultimate mater in the void real space. the world scientific community in the twentieth century did interpret the reality using thoughts derived from the quantum theory, since the famous “copenhagen interpretation” by the physicist niels bohr supposing that reality is created by observation, and the wholeness of david bohm (1917-1992), until parallel universes purposed by hugh everett (1930-1982), and also the world consisting of potentials and actualities under the uncertainty principle of werner heisenberg (1901-1976) considering somewhat as probability waves in the middle between possibility and reality, after louis de broglie (1892-1987) having translated the reality into a “statistical interpretation” [2]. what is the quantum interpretation of the ultimate reality? it’s known that the wavefunction ψ(r, t) was formulated by the physicist erwin schrödinger (1887-1961) describing the behaviour of elementary particles through an equation5, which is interpreted as representing probability values of particle states on its existential trajectory (traced by the space-vector r at each time t). this mode of thinking do gives us a way to predict the electron behaviour on the move, calculating its own evolution in space, and pointed out a potential way to interpret the real face of things. the wavefunction corresponds to the frequency probability6 of the state occurrence, which is an objective probability, for instance, it’s value is 50 % in case 4the adjective systemic clearly qualifies the complexity we are talking about, distinguishing it from other complexity types as we find in mathematics (kolmogorov complexity) or in computer sciences (algorithm complexity) and also in physics (pattern complexity) [5]. 5the time dependent schrödinger equation for a single non-relativistic particle moving in an electric field is i~ ∂ ∂t ψ(r, t) = [ − ~2 2m ∇2 + v (r, t) ] ψ(r, t) where ~ is the constant’s planck, m is the particle’s mass, v is its potential energy, ∇ is the laplacian, and ψ (r, t) is the position-space wavefunction. 6frequency (or objective) probability po is defined by the ratio of the number of favourable cases (nfav) and the number of possible cases (npos), or mathematically po = nfav / npos. 64 hermı́nio duarteramos: tackling foundations of systemic complexity on reality... of an electron7, because this particle assumes at each time just one natural state between two possible (up/down) spin states. the logic consequence is that the quantum reality itself has a probabilistic character, while our common world reality seems to exclude a statistical temper. from that quantum narrative, the so called “schrödinger’s cat paradox” shows an intriguing quantum ambiguity of the cosmic nature, having been very much discussed among thinkers on the quantum world [6].it can be analysed considering a closed system formed by a cat inside a sealed room, where there is a glass flash full of poison, besides a radioactive source emitting particles into the inner space and also a particle decay detector which will trigger automatically the fall of a heavy hammer on the flash immediately after detecting a preset particle decay threshold, shattering it and pouring the poison out.thus, the cat will be killed, but in the meantime the cat was simultaneously alive and dead, according to the wavefunction of the particle represented by state probabilities.however our direct observation certificates correctly if the cat is alive or not, since it cannot be in both states at the same time. the paradox lies in the fact that the animal is alive and dead inasmuch we don’t observe what is going on inside the hermetic room (the cat doesn’t look at the emitted particles for sure), and when an observer looks at the particles he can say rigorously the cat is either alive or dead. this happens after the superposition of quantum states ends by observation, producing the collapse of the wavefunction, and then it doesn’t represent any more an objective probability, being the observed reality inevitably either one or another possible concrete real state. that was the reason why albert einstein felt serious doubts on quantum theory implications, and he did write to his friend max bohr (1882-1804) “i cannot believe that god plays dice with the universe” [7].during decades many scientists endeavoured to explain such logic contradiction of the strange quantum paradox.nevertheless, they struggled without success.let us now analyse the schrödinger’s cat system under the systemic complexity point of view. the acrony is definite because we are aware of all components inside the closed room; the axony is complete once interactions will follow exactly some serial causations of the automation mechanism breaking the glass with poison; the aquadry is certain due to rigid boundaries build by room walls; the adaptacy is determined supposing the structure will work perfectly as it was planned. so it seems to be a simplex system, but we can’t attribute a single state to the cat, the quantum ambiguity remains and we continue ignoring the true reality of 7the electron is an elementary particle of fixed mass and electric charge (corpuscular reality, discovered in 1897) and also found to be a wave (waving reality, discovered in 1927). advances in systems science and application (2015) vol.15 no.1 65 the system before its observation. why? because the system is closed, and the systemic theory can be effectively applied only to open systems, measuring their outputs or system telonomy, i.e., the observer belongs to the global system of reality acknowledgement, and he cannot be ignored.notice that the so called observer can be a human or an instrument, and for this the observation may correspond to a measurement whose result is a real measure. in the closed system we can say that the non-observed (or not measured) reality is represented by the state objective probability ascribed to the wavefunction, according to the quantum theory, and the observed reality on the consequent open system is described by a physical state convincing the observer about the reality character.before observation, in spite of the observer awareness on the inner operating systemic essentials, he didn’t get any external system telonomy and the reality was complex to him. when we observe the reality of the open system we verify the cat is alive or dead without any doubt, and before observation the reality of the closed system was ambiguous (due to the quantum wavefunction), being the cat alive and dead, although we were conscious that just one of two possibilities could arrive, saying that the cat is either alive or dead. that’s the real problem, a restless and/or question to the reality in those closed/open systems, revealing a fundamental difference between the quantum world, which is an ocean of waves, and the macroscopic world on the human scale, where we live.at the existential limit of the nature, the quantum theory makes out the reality with all possible objective probability states, and at the human existential nature, the reality to observers means something concrete far from objective probability of real states. each part of the reality on our newtonian world exists each time apparently with only one form, because multiple own states may not superpose at the same time in any physical system. what is such potential concreteness of our observation? maybe it is the old concept potentia of the aristotle (384-322 bc) philosophy, maybe it isn’t.recently that contradiction of the quantum theory has been overshoot by the so called qbism 8, applying the subjective probability of bayes to the wavefunction. 8the term qbism is derived from the quantum bayesian view, using the bayesian probability of elementary particle states in the wavefunction. 66 hermı́nio duarteramos: tackling foundations of systemic complexity on reality... 4 tackling foundations of systemic complexity in the eighteenth century, the english presbyterian and statistician thomas bayes (1701-1716) had a different idea on the probability notion, and in 1812 the french mathematician and astronomer pierre-simon, marquis de laplace (1749-1827), disseminated it among other scientists of the natural philosophy, gaining credibility last century in several statistical applications, as in economics and also engineering decision and in cognition sciences. in fact, the bayesian probability 9 measures the human conviction on estimated cues or contents of signals that we detect from the reality to make decisions.for instance, it happens when a physician regards his patient and evaluates the symptoms to diagnose a disease, and he would be self believed on the most credible hypothesis, guessing it by means of subjective probabilities, even he uses a practical method giving weights to the collected data.such subjective probability quantifies the human conviction level on the observed reality, being entirely different from the objective probability based on illness occurrences, which results from the disease frequency and it is not related to the human convincement about the reality itself. the modern decision theory has applied successfully the bayesian probability, conveying an important tool for engineers, economists, sociologists and other users, and even to solve trivial situations we have in our everyday life ourselves.undoubtedly, we are observers of the reality and making decisions always with subjective probabilities on all things we experience around us. now we pay attention to quantum reality supposing that wavefunction superposes subjective (or bayes) probability states instead objective (or frequency) probability states.reasoning like that we don’t see any paradox. indeed, a human can guess the schrödinger’s cat state in the closed system, assigning it a state either alive or dead, according to the subjective probability of his creed, and because the quantum reality to him is what the wavefunction assigns to be in reality. 9bayesian (or subjective) probability is defined by the bayes’ rule p (h|e) = p(e|h) p(e) .p (h) where | denotes a condition probability; h stands for competing hypothesis (competitors) from which one chooses the most probable; the evidence e (cue) corresponds to new data that were not used in computing the prior probability; the prior probability p(h ) is the probability of h before e is observed; the model evidence p(e) or the marginal likelihood (diagnosticy weight) is the same for all possible hypothesis being considered; the likelihood function p(e |h ) is the probability of observing e given h (impact weight); the posterior probability p(h |e) is the probability of h given e, i.e., after e is observed (credibility weight); the impact factor p(e |h ) / p(e) represents the impact of e on the probability of h. advances in systems science and application (2015) vol.15 no.1 67 such reasoning affirms that a quantum state is just a subjective probability assignment, and the reality of the quantum world tosses the observer’s beliefs to elect that one most evident.this is hard to accept for the traditional scientific thought, “it is really hard to believe something you don’t actually believe” [8], yet for a qbist it is a consistency question, considering the subjective probability a logic extension of the truth table.that extended logic encourages us to generalize the quantum reasoning to the macroscopic world, as we find it explicitly in decision system methodologies. we concern the collapsed wavefunction after observation (or measurement) which is characterized by subjective probabilities, and so the observers’ personal information gives us expectations or degrees of belief.when i look at the world nature my mind interprets detected cues by sensing and perception of the natural world as appearances in correspondence to reality. whether i don’t get information signals from the reality, this reality exists but i can’t say anything about it, and whether i detect some real signals i will build a creed on reality, assembling captured cues in my mind and weighting them to obtain the most evident subjective probability to make decisions.this is the apparent reality or the reality we think it is. consequently, the matter on macroscopic world relies on the subjective probability, as well as it must occur on the quantum level when it is observed (or measured).that’s a question of believing, but to believe grounding on a probability induced by the own reality.the behaviour of the observed quantum world is the same as that of our macroscopic world, taking in account the reality reveals the same physical principles. all this means that the natural interpretation of the reality is the reality itself, even when it exists inside a closed quantum system or whenever we are not observing it, and so the material substance is nothing more than a human judgement, reasoning on detected signals radiated from matter to process the subjective probability according to the intrinsic character of things.the reality remains “the thing-in-itself”, as immanuel kant (1724-1804) said, however to me it is the appearance i judge to be the reality in extremely rapid decision processes. we arrive to the conclusion that the reality exists as it is, but our consciousness decides its appearance using a very pragmatic heuristic, estimating some cues by perception in very short time, and seeking the accumulated evidence to adopt cue weights for diagnosticy and owing to their credibility proportions for each competitive hypothesis, integrating both weights in products as evidence factors, and 68 hermı́nio duarteramos: tackling foundations of systemic complexity on reality... choosing the evident hypothesis with the maximum sum of those factors 10.this optimization heuristic translates the bayesian probability into our everyday mental process, for instance buying a shirt or choosing a new car to buy it. whether the decision time is very short we feel acting under intuitive pulses, judging without any elaborate reasoning, and we can say the intuition feeling relies upon a very rapid decision making.this is an interesting by-product of the present search.the truth is that when we are observing our world we are always reasoning with subjective probability pulses. i see the green forest because i detect light cues from the trees using my eyes (and the integrated biological vision system), and they convince me that the tree colour is green, matching the wave frequency 11 of the light detected from trees with the learned language stored in my neural memory. the consciousness answers to the subjective probability of the real wavefunction whenever the human observes the reality, because it appears as an open system, whose telonomy is the appearance of the reality that affects him.therefore, the observer is not separated from the observed [9], they both interact intimately. it comes now the opportunity to ask if all systems are real or not.taking systems from the reality we observe them without any doubt, each concrete system belongs to the reality, doesn’t matter it is a macroscopic or a quantum system, and its systemic model is also real, because i design it in a representative abstract nature but its functional operation corresponds to a concrete system. a system model makes decisions exactly as the natural system itself, supposing systemic essentials are perfect in the case of simplex systems, or it approximates more or less right outputs in complex model systems with imperfect systemic es10the decision theory defines a heuristic based on the cue diagnostic [p(e)], that ranks detected cues with normalized weights according to their importance for the decision making, and also on the cue credibility [p(e |h )] of each hypothesis representing cue confidences by their impact weights, as we can see in the following table with three cues, and choosing the best hypothesis between two competitors (hypothesis a and b), defining evidence factors [p(e).p(e |h )] for each cue and each competitor by the product of diagnosticity weights (wdi) and credibility weights (wci), and electing the maximum sum of evidence factors: 11here frequency means the number of wave cycles per second (not the frequency of occurrences in an arbitrary gap of time, which is used to define objective probabilities). advances in systems science and application (2015) vol.15 no.1 69 sentials.all this is what an observer can infer observing both natural and model realities and using detected cues from outside of both black boxes (of natural and model systems), because results are exactly the same for human consciousness if system outputs are equal, or we approach them as much as they seem to be similar. for instance, a refrigerator’s user doesn’t need to know how the refrigerator system works, being only concerned on their cold outputs, but in the case of a refrigerator’s designer he must be aware of its five systemic essentials to produce good equipments. concluding, a concrete (natural or model) system means a black box to the observer, and we don’t need any especial knowledge about what is going on inside, supposing it operates as it was designed, because we reduce the reality to telonomy measurements and excluding the side-by-side information that doesn’t matter to the intentionality target. beyond that, decision systems are also real?as we know, in such situations we organize data sets which inform us on reality features, and we decide electing one of them through subjective probabilities, and the choice represents the most valuable or convincing hypothesis.only later real facts will confirm whether our choices are acceptable or unacceptable judgements. in any case, the methodology is universal, based on wavefunction collapses by observation and using the bayesian probability tool, i.e., a decision system is an abstract system but the systemic complexity governs their behaviour as well as in concrete natural systems.this important corollary is proved by systemic essentials of decision systems analysing the character of detected cues. the acrony is definite because in general we can fairly estimate all diagnosticity weights before comparing hypothesis; the axony is complete when we can clearly guess credibility weights of cues for competitive hypothesis; the aquadry is uncertain because it is possible missing some relevant cues, owing to the proper finiteness of cue boundaries; the adaptacy is determined in the case of impartial or not biased choices, otherwise it will come some errors by overconfidence bias and psychological anchoring heuristics, provoking undetermined optimum working points. following these main lines, normally a decision system is an open complex system, the telonomy being the best possible output from global weighted cues and their mutual relationships, although we cannot always assume the optimum decision. what we must do is trying to reduce the degree of complexity, recollecting better information on systemic essentials, endeavouring to attain simplexities or at least approximating as much as possible to simplex systems. 70 hermı́nio duarteramos: tackling foundations of systemic complexity on reality... here it is the recommended strategy. 5 conclusion in conclusion, this approach lengthens the copenhagen interpretation, which judges that reality is created by observation, defending the thesis it is much more than that by reasoning on the reality as a mind formulation founded on subjective probabilities. indeed, i can summarize the previous discussion in few basic items, raising important issues to be debated in general and in particular cases, as it follows: 1. the reality means my belief on decision making by perception of signals, which are radiated from things as system acrony (measurement problem). 2. i use a subjective probability scheme in my mind to decide what reality is, choosing the most convincing accumulated axony (subjective problem). 3. i can’t know the total meaning of the observed reality, because the biological structure of my body limits every systemic aquadry (boundary problem). 4. the correlation among collected data and psychological bias and heuristics may deviate from the best choice, and i cannot assure attaining the optimum system adaptacy (optimum control problem). 5. common decisions in our current life are intuitive, we make them in extremely short time and so the reality seems apparently to be evident from the system telonomy (consciousness problem). we are now seeing why i am convinced that kant’s appearance of reality is complex under the systemic standpoint, and so i hope we will be able to handle complex systems of worldwide realities developing the systemic complexity theory by scientific methodologies. actually, science research must get down to essential fundamentals, and science theory consists of human rational attempts to represent both appearance and reality of things and ideas. are reality and appearance a unique thing?the science and technology will develop detecting tools for supra-human sensing contacts with non-observable reality by humans, and we research on human-machine interfaces [10] to integrate simplexicity into complexity of nature systems inserting trends to observe the reality itself in a new systemic supra-appearance. that is my conviction, which emerges from decision making in my mind based on some detected signals and memory ones.in reality, that’s the reality. advances in systems science and application (2015) vol.15 no.1 71 references [1] a. north whitehead. (1933), adventures of ideas, cambridge at the university press, london. [2] nick herbert. (1985), quantum reality. beyond the new physics, archor books, new york. [3] john ziman. (2000), real science, what it is, and what it means, cambridge university press, cambridge. [4] hermı́nio duarteramos. (2006), “hard and soft systems intentionality”, 7th congress of ues, lisbon; also in res-systemica, no.7, afscet. [5] g. t. arecchi. (1997), “pattern generation and competition in nonequilibrium optical media: a case study for complexity”, the physics of complex systems, ios press, oxford. [6] david lindley. (1996), where does the weirdness go?, basic books, new york. [7] max born, albert einstein. (1971), the born-einstein letters, walter and company, new york. [8] christopher fuchs. (2012), “interview with a quantum bayesian”, maximilian schlosshauer, (2011). elegance and enigma: the quantum interviews, springer, frontiers collection. [9] david bohm. (1996), on dialogue, routledge, london. [10] r. antunes, f. v. coito, h. duarteramos. (2013), “skill evaluation in point-to-point human-machine operation”, applied mechanics and materials, vol.394, trans tech publications, zurich-durten. corresponding author hermı́nio duarteramos can be contacted at: hduarteramos@gmail.com. microsoft word 3.dan luo, jianhua jiang, buyun sheng--design and implementation of network based collaborative design system. 226-231 advances in systems science and applications (2011), vol.11, no.3-4 issn 1078-6236 international institute for general systems studies, inc design and implementation of network based collaborative design system dan luo, jianhua jiang and buyun sheng school of mechanical and electronic engineering, wuhan university of technology, wuhan 430070, china abstract in order to support collaborative design in network environment, according to the aim and requirements of network based collaborative design system (ncds), the functional architecture of ncds was designed and its key technology includes improving of collaborative application sharing, multimedia communication in real-time, management of collaborative interaction and network environment of collaborative work were researched. finally, a ncds was developed and its correctness and feasibility was validated by applied in practice. keywords collaborative design; ncds; product designing; audio & video communication; member management 1. introduction with the popularity of computer network and the implementation of parallel engineering, collaborative design technology[1] has become the research focus in advanced manufacturing domain. to construct an effective, convenient and practical ncds is the precondition of realizing internet based different design for users dispersed geographically. ncds supports product designing, information sharing and exchanging, cax accessing and invocation, product design projects discussing and modifying among product designing engineers in different locations by the communication of audio & video, word and images. in this way, the process of product designing can be carried out across time and space and its cycles be shortened, product development costs be reduced and personalized product development capacity be enhanced substantially. 2. system design 2.1 aims and requirements of ncds the purpose of ncds is to make each participant watch and participate in the real-time collaborative design process, and provide a management tool for design organizer to organize and control the whole process of produce designing. the aim and requirement of ncds is to: (1) provide a platform for information exchanging. it should support real-time and mutual communication in audio & video, text interactive and usb interface based camera images, more than 256 people online simultaneous, audio and video information exchanging with 4 users for each user at the same time. (2) tracking the entire process of remote designing. it achieves screen broadcasting of remote designing and other participants receiving in real-time smoothly. (3) support remote operation of collaborative design. organizer initiates a remote designing and designates the participants at the principle of one designer any time and the designer can only do the operation allowed by the organizer. (4) organize and manage the process of collaborative design. it manages the identity data, password and permissions of designers, and designates the jobs to be finished, allocates permissions to participants or expel a designated staff and manages the process of designing. (5) other requirements. the system should be scalable, interface attractively and easy to advances in systems science and applications (2011), vol.11, no.3-4 227 issn 1078-6236 international institute for general systems studies, inc use. 2.2 functional architecture of ncds according to the aims and requirements of the system, a ncds was designed, it includes audio & video meeting and remote screen broadcasting, remote operation control and console management four parts, as shown in figure 1. n c d s audio and video meeting remote screen broadcast remote operation control console management video & audio communication text charting network communication screen transmit screen receive control client controlled client user management role management figure 1 functional structure of ncds (1) audio and video meeting. it realizes information sharing by audio & video, text and network communication modules. audio communication process includes audio signal acquisition, compression, send and playback while video communication collects data from video card and send it to network after compressed, at same time it extract the received data from other members and playback it on the display, text charting provides real-time text messages interactive and supports text sending for a group, network communication transmits and receives data and choose transmission way according the type of message. (2) remote screen broadcast. it includes screen transmit and receive two modules. the screen transmit captures the screen of server, calculates images difference and sends it after compressed. while screen receive incepts the screen of server and extracts it in real-time, synthesizes images and displays it. (3) remote operation control. the control client simulates the images in local machine according the screen received from server client (controlled client), encodes the operation commands into characters and sent them back to server client. the server client receives control commands from control client and simulates the scene. (4) console management. the user management is responsible for constructing workgroup and managing user data, the role management defines and manages every role in designing process, and manager can change the designer and speaker according the need. 3 .key technology 3.1 improvements of collaborative application sharing collaborative application sharing refers to dispatch application view of server to other clients and collect all other input command to the server. it includes centralized sharing and replication sharing. in allusion to the inherent defects of the two types of sharing technologies, this paper improves the sharing method according to perceived degree of collaboration, collaborative function of collaboration tools and fast transmission of cad models and images. (1) improve the perceived degree of collaboration. for perceiving client information, perceptual information and perceptual result table was used to store online users’ status information both in server and clients as shown in figure 2. node 1 sends the perceived results received by the event sensing module from clients to the server and updates the perception table. in receiving the results, the server updates perceptual 228 luo: design and implementation of network based collaborative design system table and sends the updated information to other nodes, those other nodes then update their perceptual list according received information from the server. in this way, all the nodes can perceive the changing status of every node. figure 2 event sensing process of clients in addition, multi-cursor technology also supports the parallel perceiving of collaboration. all users’ cursors displayed in a screen at the same time and distinguished by different colors for different group or by showing designers’ name under the cursors[3]. (2) improve the function of collaboration tools in order to make common application software supports displaying of multi-cursors, revocation and redoing of operation, decomposition and analysis of input event stream, the interface of common software must be extended and it can be solved by the locking mechanism in application-level. the process of the improving is shown in figure 3 where solid line denotes information flow of collaborative operation and dotted line denotes information flow of collaborative tool. the collaborative tool can realize many control and cooperation mechanism such as certification and conflict coordinate among members, direct interaction, multi-input processing, locking in operation level. figure 3 improvement of collaborative tool to software (3) fast transmission of cad models and images for the problem of low transmission speed caused by large quantity of images data, incremental transfer technology and images compressing from the aspects of time-dependent, image compression and color transformation were used in this paper. time-dependent of screen images refers to changes in the adjacent images confined to a few places and the changes of image color limited in a few tint. so, only the data reflecting changed parts of the screen images should be compressed and transmitted. in this process, some technologies including the message mechanism used to capture the changes of screen images, comparing these changes in a relatively large time interval, inter-frame prediction and compensation were used to solve the inconsistencies between screen changes and the messages. advances in systems science and applications (2011), vol.11, no.3-4 229 issn 1078-6236 international institute for general systems studies, inc small piece based encoding was used to compress images of the screen and its encoded form of data packets as shown in figure 4. any changes in a rectangular area of screen images will be divided into a 16*16 sub-block named a piece, and then appropriate encoding form is used to each piece and a byte is adopted to as sign to show the encoding form of the piece. figure 4 data format of small piece based encoding color transformation refers to change screen data into the transition color (16-bit color) and divide the screen images into small pieces size of 16*16, use small piece comparative law to capture changes of the screen images, and adopt small piece based encoding algorithm to compress changes and re-assemble them into a packet. 3.2 multimedia based communication in real-time the technology of multimedia based collaborative design supported communication includes audio & video capture, its compression and special treatment. as to audio & video capture, video for window (vfw) sdk is used to capture, play and edit audio & video information into avi format, and mpeg4/divx is used to compress, store and transmit audio & video. special treatment of audio system includes voice consistency and delay, mute detection and interactive approach in full-duplex. data buffer technology is used to solve the voice consistency and delay[4]. by coding the sample value of voice and using a serial number to indicate the sample time, the missed number was discarded as to avoid sending mute information. limited input & output scope of audio is used to avoid the echo in full-duplex interactive process. special treatment of video ensures the video receiving quickly by throwing away some video frames. the video frame includes i, p and b three types. i frame is coded frame and it is the most important, while b frame is the least important. so, the principle of abandoning frame is: (1) abandon b frame firstly, then p frame and i frame lastly. (2) all the abandoned frames should distribute in the media flow as uniformly as possible. (3) adjust the rate of abandoning frames according to the network status. 3.3 management of collaborative interaction (1) member management member management in ncds refers to permission management and access control of members, permission altering, authorization and cancellation dynamically. role based access control can be used to achieve the management of permissions. roles including administrator, head designer, general member and spokesman. the administrator is responsible for assigning roles to members, designer has the operation right, general member and spokesman can send messages to other members. a member can be assigned multiple roles at the same time and there is only one head designer and one designer in a collaboration group any time. (2) speaking management in process of collaboration, each user can obtain the speaking right or give up it in preventing some speaker occupied the right too long or refused to give up speaking right. so, four types of speaking management mechanism is used[5]: 1) centralized control. an administrator is in charge of assigning speaking right to members. 230 luo: design and implementation of network based collaborative design system 2) grab initiatively. users sent application to the control machine, it then deprives the speaking right of current speaker and assigns the right to the user according to agreed strategy. 3) give up initiatively. the current speaker gives up its right initiatively. 4) speaking freely. everyone can speak and receive messages at freedom while obtain the operation right by competing freely. in the process of speaking, each participant can receive any other audio & video and only 8-way audio & video is allowed in order to ensure smooth communication. (3) token control. token is used to avoid inconsistency caused by multiple designers in operating on a same model at the same time. a head designer manages a token can be applied by any designer. during the designing process, only the token owner can operate the model and only his operation commands can be transmitted to the client of sharing the application, while members without token only can see the operation process or communicating each other by the multimedia environment provided by the ncds. 3.4 network environment for ncds for the ip is allocated by isp dynamically, it’s impossible to establish a permanent association between distributed members by ip address. moreover, there is a communication problem of how to penetrate nat. in these cases, a management server may be used to resolve them by storing user information and their current status of collaboration. in addition, all the data stored in the server transmitted to the recipient as to establish stable communication channels between recipient and server and solve the issue of nat. in order to ensure the security of enterprise information, port mapping technology is used to build a link between internal lan ip and the real ip. port mapping can also map a virtual ip of a host to a real ip or map a number of ports in a host with permanent ip address into different port of different machines in lan. 4.system implement and application example 4.1 system implement a ncds was developed based on tcp/ip protocol and can be run on windows platform, the hardware including camera, headphones, microphone and pc, the system developed on borland delphi and socket technology was used. the system including user management, main application interface, audio & video service, design service, screen broadcast and receive, text communication and port mapping. the user management is used to add, delete or change workgroup and its members, define roles and configuration network. main application provides an interface for users and audio & video service developed on vfw technology. design service uses vnc as kernel and called in main application interface. screen broadcast and receive adopts vncx.dll and port mapping realized by port mapping technology. 4.2 application example figure 5 running example of ncds advances in systems science and applications (2011), vol.11, no.3-4 231 issn 1078-6236 international institute for general systems studies, inc in the process of a collaborative design example as shown in figure 5, multiple members watching on the model and only one designer has the permission of operating on the model while other members can only watch or communicate each other by multimedia communication. in figure 5, the design service region was showing the shared screen and only one designer can operate on it at one time, the text communication region was showing the content of communication information between members, all the online users were listed in user region, there are four video images can be seen in the figure and the lower right corner is the audio & video service region. 5 conclusions collaborative design is an important technology of advanced manufacturing model while ncds is an effective tool for realizing it. from the viewpoint of developing a ncds, several key technologies were analyzed and a ncds was developed successfully in this paper. the current version is only provided an environment for collaboration design and our future work is to integrate it with pdm software. acknowledgements this paper is supported by national natural science fund project of china (contract no.50620130441), scientific and technological project of wuhan city (contact no. 200810321153) and youth science and technology chen guang project of wuhan city (contact no. 200750731289). references [1] gao shuming, he fazhi. survey of distributed and collaborative design. journal of computer-aided design & computer graphics, 16(2) (2004) 149~157. [2] tian ling, chen jizhong, zhao huishe. net-based collaborative design tools. journal of china mechanical engineering, 15(19) (2004) 1774~1777. [3] xu baomin,wanglixin. concept and implementation of improved application shared. journal of computer engineering, 29(11) (2003) 161~162. [4] ci jianwei，zhang huazhong. research on real-time video dealing methods of multimedia network. journal of computer engineering, 31(2) (2005) 193~194. [5] he fazhi. research on collaboration support technology and tool for cscw based cad system. wuhan: dissertation of wuhan university of technology,2000. advances in systems science and application (2015) vol.15 no.4 366-383 deduce user search progression with feedback session b.bazeer ahamed1 and t.ramkumar2 1faculty in computer science & engineering, sathyabama university, chennai,tamil nadu,india 2department of computer applications, avc college of engineering, mayiladuthurai,tamil nadu,india abstract the web is a medium for accessing a great variety of information stored in various locations. as data on the web grows rapidly it leads to several problems such as increased difficulty of finding relevant information. when a user submits a query to the search engine, it must be able to retrieve information according to the user’s intention. but search engine retrieves the list of pages ranked based on that similarity to the query. sometimes the results are not according to users interests, because many relevant terms may be absent from queries and words may be ambiguous. therefore, the results produced by the search engine are not satisfactory to fulfill the user query request. in order to solve this ambiguity, the proposed work is to discover the number of diverse user search goals for a query and represent each goal with some keywords automatically. keywords clustering; hits; restructuring search results; classified typical meticulousness; feedback session. 1 introduction with the fast growth of the web, a user can obtain abundant information easily by submitting a query to a search engine. many existing search engines use keyword matching as the search mechanism, which usually causes the situation that a large number of non-relevant documents containing query terms are founded out, and the user will make strenuous efforts to browse these non-relevant documents. thus, it is not simple to find out the real user goal from such short queries. data mining involves the use of sophisticated data analysis tools to discover previously unknown, valid patterns and relationships in large data sets. these tools can include statistical models, mathematical algorithms and machine learning methods (algorithms that improve their performance automatically through experience such as neural networks or decision trees). consequently, data mining consists of more than collecting and managing data, it also includes analysis and prediction. data mining can be performed on data represented in quantitative, textual or multimedia forms. data mining applications can use a variety of parameters to examine the data. they include association (patterns where one advances in systems science and application (2015) vol.15 no.4 367 event is connected to another event such as purchasing a pen and purchasing paper), sequence or path analysis (patterns where one event leads to another event such as birth of a child and purchasing dress), classification (identification of new patterns such as coincidences between duct tape purchases and plastic sheeting purchases), clustering (finding and visually documenting groups of previously unknown facts, such as geographic location and brand preferences) and forecasting (discovering patterns from which one can make reasonable predictions regarding future activities, such as prediction that people who join an athletic club may take exercise classes). information retrieval (ir) is essentially a matter of deciding which documents in a collection should be retrieved to satisfy a user’s need for information. the user’s information need is represented by a query or profile, and contains one or more search terms, plus some additional informations. hence, the retrieval decision is made by comparing the terms of the query with the index terms appearing in the document itself. the decision may be binary (retrieve/reject), or it may involve estimating the degree of relevance that the document has to the query. a stemming algorithm is a process of linguistic normalization, in which the variant forms of a word are reduced to a common form. it is important to appreciate that we use stemming with the intention of improving the performance of information retrieval systems [1]. unfortunately, the words that appear in documents and in queries often have many morphological variants. thus, pairs of terms such as “computing” and “computation” will not be recognized as equivalent without some form of natural language processing (nlp). improved hypertext-induced topic selection (hits) algorithm is a very popular and effective algorithm to rank documents based on the link information among a set of documents. the algorithm presumes that a good hub is a document that points to many others, and a good authority is a document that many documents point to. hubs and authorities exhibit a mutually reinforcing relationship: a better hub points at many good authorities, and a better authority is pointed to by many good hubs. to run the algorithm, we need to collect a base set, including a root set and its neighborhood, the inand out-links of a document in the root set [2]. because the hits algorithm ranks documents only depending on the in-degree and out-degree of links, it will cause problems in some cases. for example, a) mutually reinforcing relationships between hosts and b) topic drift. both problems can be solved or alleviated by adding weights to documents. the first problem can be solved by giving the documents from the same host much less weight, and the second problem can be alleviated by adding weights to edges based on text in the documents or their anchors. the simple modification to the hits algorithm for the first problem achieves a remarkable better precision, while further precision can be obtained by adding content analysis [3]. clustering is a 368 b. bazeer ahamed and t.ramkumar:deduce user search progression with ... common descriptive task where one seeks to identify a finite set of categories or clusters to describe the data. clustering is the process of identification of classes, also called clusters or groups, for a set of objects whose classes are unknown. the objects are so clustered that the intra class similarities are nearly maximized and the interclass similarities are minimized based on some criteria defined on the attributes of objects. once the clusters are decided, the objects are labeled with their corresponding clusters, and common features of the objects in a cluster are summarized to form the class description. agglomerative hierarchical clustering techniques is used to cluster the pseudo documents since agglomerative hierarchical clustering is a classical clustering algorithm, originating analogically like k-means from the statistics domain. the main advantage of ahc is to create descriptions of clusters, removes redundant descriptions and attaching cluster to another one whose description is a subset of its description [4]. improved hits algorithm is used to rank cluster results relevant for a particular topic. the web results are restructured. finally, we introduce user editable browser that allow the user to perform editing operations such as deletion and emphasis while browsing the search results. the rest of the paper is organized as follows. section 2 reviews various techniques for effective inferring user search goals and generalized feedback session techniques. deduce user search progression is presented in section 3. experiments measures in section 4.section 5 concludes the paper and shows our future directions on this topic. 2 literature review 2.1 identifying user goals from web search results yao-sheng chang et al propose a novel probabilistic inference model which effectively employs syntactic features to discover a variety of confined user goals by utilizing web search results.[5] on the basis of analyzing the user goals in the viewpoint of natural language processing (nlp) process. assume that the user goal should be expressed with the form of a hidden sentence in his/her mind. in general, a typical sentence includes a subject (s), a verb (v), and an object (o). also, assume that the subject of the hidden sentence in user mind is the user himself/herself and the combined pair of the verb and object is called vo pair. on the basis of vo pairs, potential user goal can be represented. for example when users submit a query “michael jackson”, predict that the hidden sentence in the user mind is “i want to download michael jacksons music,” and the potential user goal is the vo pairs “download music” (verb + object). the object can be regarded as the noun after the verb [5]. the user senses his own sentence according to his own mindset but awareness to the mechanism is much different. advances in systems science and application (2015) vol.15 no.4 369 2.2 an overview of personalization in web search the above mentioned technique has some drawbacks as follows: more challenges to adopt vo-pair classes to certain languages and time consuming to identify vo-pair. in order to overcome these drawbacks indu chawla introduces web search personalization algorithms to improve the web search experience by using an individuals data e.g. user’s domain of interest, preferences, query history, browser history etc . using these factors they extract the results that are the most relevant to that individual. personalization can be broadly categorized in two types: context oriented and individual oriented [6]. context oriented personalization include factors like the nature of information available, the information currently being examined, the applications in use, when, and so on. individual oriented personalization uses user interests, query history, browser history, pages visited etc [7]. even though the web search personalization algorithm improve the web search experience by using an individual’s data, but has some drawbacks as follows [8]: in context oriented personalization expecting searchers to provide context information explicitly as part of their search is not ideal. many users are simply unwilling to provide this type of additional information and even asking for it can lead to frustration certainly asking the user for anything close to personal information is liable to alienate many users because of privacy concerns [9]. and in the second option where the user can choose from among the categories provided, the problem is to know about the users’ interest for displaying the categories to the user. moreover, searchers often do not have enough knowledge available to them to explicitly express such context information even if they were inclined to do so [10]; in individual oriented personalization a single user profile or model can contain a too large variety of different topics so that new queries can be incorrectly biased and also users are becoming more concerned about threats to privacy in the online environment. 2.3 query recommendation using query logs in search engines baeza et al [11] present an algorithm to recommend related queries to a query submitted to a search engine. the related queries are based in previously issued queries, and can be issued by the user to the search engine to tune or redirect the search process. the method proposed is based on a query clustering process in which groups of semantically similar queries are identified. the clustering process uses the content of historical preferences of users registered in the query log of the search engine [12]. the method not only discovers the related queries, but also ranks them according to a relevance criterion. ranking the queries according to two criteria: first one by the similarity of the queries to the input query (query submitted to the search engine).trail by the next one by the measures, how much the answers of the query have attracted the attention of users.the combination 370 b. bazeer ahamed and t.ramkumar:deduce user search progression with ... of measures (a) and (b) defines the interest of a recommended query. 2.4 automatic identification of user goals in web search uichin lee et al [13] study whether and how to identify the user goal automatically without any explicit feedback from the user. two types of features for the goal identification task,one is past user-click behavior and another one is anchor-link distribution. first feature is based on the intuition that the user’s goal for a given query may be learned from how users in the past have interacted with the returned results for this query. if the goal of a query is navigational, then in the past users should have mostly clicked on a single website corresponding to the one they have in mind. on the other hand, if the goal is informational, in the past users should have clicked on many results related to the query. thus by observing how the results for a particular query have been clicked so far, and easily tell whether the current user who issues that query has a navigational or an informational goal. the anchor link distribution is used to find the destination of the links with the same anchor text as the query. for example, for a navigational query pub med, a single authoritative website exists (which is www.ncbi.nlm.nih.gov)[9]. as a result, if extract all the html links with the anchor text pub med, to find that a dominating portion of these links point to that single website. on the other hand, for an informational query hidden markov model, because of lack of a single authoritative site, can expect that the links with the anchor text hidden markov model point to a number of different destinations. 2.5 learn from web search logs to organize search results xuanhui wang et al propose a different strategy for partitioning search results, which addresses these two deficiencies through imposing a user-oriented partitioning of the search results.[10] learn “interesting aspects” of similar topics from search logs and organize search results based on these “interesting aspects”. finally generate more meaningful cluster labels using past query words entered by users. to know what the users are really interested in given this query, first retrieve its past similar queries in preprocessed history data collection. based on the similarity scores, we rank all the documents in history data set. the top ranked documents provide us a working set to learn the aspects that users are usually interested in. each document in history data set corresponds to a past query, and thus the top ranked documents correspond to q’s related past queries. 2.6 query-sets: using implicit feedback and query patterns to organize web documents barbara poblete et al present a new document representation model based on implicit user feedback obtained from search engine queries.[14] the main objective advances in systems science and application (2015) vol.15 no.4 371 of this model is to achieve better results in non-supervised tasks, such as clustering and labeling, through the incorporation of usage data obtained from search engine queries. this type of model allows user to discover the motivations of users when visiting a certain document. the query document model reduces the feature space dimensions considerably, because the number of terms in the query vocabulary is smaller than that of the entire website collection. this model is very similar to the vector model, with the only difference that instead of using a weighted set of keywords as vector features, we will use a weighted set of query terms. fig. 1 example of the query document representation. 2.7 user intent based searching haibo yu et al propose a user intent based searching mechanism (uibs )in order to enable precise discovery of a web site (for navigational searching) as well as aggregating of their information (for informational searching) and performing further activities (for transactional searching).[15] this mechanism includes three main components: a web site capability description interface for the explicit description of a web site’s capability, a search interface that enables explicitly describing user’s search intent and requirements and their relevant responses, and a query engine which provides relevant search results based on user’s intent and requirements. in a uibs enabled user, a uibs client should be installed which can issue uibs search requests based on user inputs or other sources of user intents, preferences and query [9]. the client should also process the received uibs results, and presents the processed result to the user or other applications. it explicitly describes the general information of the web site and the links to published content as well as web services that the web site can provide. the 372 b. bazeer ahamed and t.ramkumar:deduce user search progression with ... wscd works like a site map and presents all information that the web site wants to publish. 2.8 generalized feedback session techniques zheng lu et al extended the previously discussed algorithm in an effective manner. he focused on user search goals by organizing search results by aspect learned from user click through logs.[16, 17] in the feedback sessions related to the given query will be extracted from user clickthrough logs. the feedback session is defined as the series of both clicked and unclicked urls and ends with the last url that was clicked in a session from user click-through logs. the clicked urls tell what users require and the unclicked urls reflect what users do not care about [18]. then, map each feedback sessions to pseudo-documents using keywords which can efficiently reflect user information needs. by using kmeans clustering techniques to cluster the pseudo-documents to infer user search goals [19]. finally restructure the web search results inferring user search goals. fig.2 represents the data flow diagram for the existing system. but there is some confines that we are focused in obtainable system i.e. generate noisy and redundant search results problem and cluster labels generated are not informative enough to allow the user to identify the right search result. do not generate the search results if the query has been entered for the first time. fig. 2 dataflow diagram for the generalized feedback session techniques. 3 deduce user search progession with the fast growth of the web, a user can obtain abundant information easily by submitting a query to a search engine. many existing search engines use keyword matching as the search mechanism, which usually causes the situation that advances in systems science and application (2015) vol.15 no.4 373 a large number of non-relevant documents containing query terms are founded out, and the user will make strenuous efforts to browse these non-relevant documents. thus, it is not simple to find out the real user goal from such short queries. a variety of ranking algorithms have been proposed and used in many web search engines. people search for information using search engines. search engines, however, cannot always return good ranked search results which satisfy user’s search intentions adequately. hence, it is difficult to recognize users’ search intentions just by analyzing their input queries. the search results sometimes do not correspond to the user’s search intentions because of this diversity. in this case, the user must check the search results sequentially until he obtains sufficient information from the linked pages from the page containing the search results. the query contains parts of user general verbal communication and special characters which are not required for analysis as they do not truly reflect the relevance of a search result. if this query is used for analysis, it may give inconsistent and inaccurate results. therefore the user query will be pre-processed to identify the root words. the feedback sessions has been introduced to infer user search goals for a query. then, map each feedback sessions to pseudo-documents using keywords which can efficiently reflect user information needs and relate to the given query will be extracted from user clickthrough logs. the feedback session is defined as the series of both clicked and unclicked urls and ends with the last url that was clicked in a session from user click-through logs. the clicked urls tell what users require and the unclicked urls reflect what users do not care about. in order to apply the evaluation method to large-scale data, the single sessions in user click-through logs are used to minimize manual work. because from user click-through logs, we can get implicit relevance feedbacks, namely “clicked” means relevant and “unclicked” means irrelevant. a possible evaluation criterion is the typical meticulousness (tm) which evaluates according to user implicit feedbacks. tm is the average of precisions computed at the point of each relevant document in the ranked sequence however, tm is not suitable for evaluating the restructured or clustered searching results. therefore we introduce classified typical meticulousness (ctm) system has been initiated to evaluate the performance of the restructured web search results. where the ctm of the class including more clicks namely confer is calculated, ctm selects the tm of the class that user is interested in (i.e., with the most clicks/confer). then, map each feedback sessions to pseudo-documents using keywords which can efficiently reflect user information needs. k-mean clustering is too simple and it performs poorly for large set of data. and also it requires prior knowledge of number of clusters to be generated. therefore, instead of using k-mean clustering we had used agglomerative hierarchical clustering to cluster the pseudo documents. 374 b. bazeer ahamed and t.ramkumar:deduce user search progression with ... 4 experiments and measures the query submitted by the user contains parts of speech and special characters which are not required for analysis as they do not truly reflect the relevance of a search result. if this query is used for analysis, it may give inconsistent and inaccurate results. therefore, the user query will be pre-processed to identify the root words. fig. 3 data preprocessing fig. 4 general feedback session progression. feedback session consists of both clicked and unclicked urls and ends with the last url that was clicked in a single session. the clicked urls tell what users require and the unclicked urls reflect what users do not care about. table advances in systems science and application (2015) vol.15 no.4 375 1 shows a feedback session with lists of 10 search results of the given query “computer”, where “0” in the click sequence represents the “unclicked url” and remaining represents “clicked url”. the pseudo-document can be used to infer user search goals. the mapping of feedback sessions into a pseudo-document includes two steps. they are as follows: first enrich the urls with additional textual contents by extracting the titles and snippets of the returned urls appearing in the feedback session. in this way, each url in a feedback session is represented by a small text paragraph that consists of its title and snippet. then, some textual processes are implemented to those text paragraphs, such as transforming all the letters to lowercases, stemming and removing stop words. fig. 5 map feedback sessions to pseudo-documents. improved hits (hyperlink induced topic search) algorithm is applied to the clusters generated by agglomerative hierarchical clustering to rank cluster results relevant for a particular topic. ranking is done by assigning relevance weight to the cluster results. restructure the web search results based on result generated by improved hits algorithm. each webpage will be treated as a node, with hyperlinks treated as directed links from one node to another. each node i is assigned an authority score a(i) and hub score h(i). given a directed graph, the authority and hub score is defined as follows: a(i) = the sum of the hub scores of the nodes pointing to node i. h(i) = the sum of the authority scores of the nodes that node i is pointing to. the higher a node’s authority/hub score is, the better authority/hub is. this makes intuitive sense; a node is a good authority if good hubs are pointing to it, and a node is a good hub if it is pointing to good authorities. these authority and hub updates can be described through matrix notation. suppose that we define matrices a, u and v as follows:a = the adjacency matrix of the graph. that is a(ij) = 1 if node i has a directed link to node j, and 0 otherwise. u = a column matrix containing the hub score of all of the nodes. 376 b. bazeer ahamed and t.ramkumar:deduce user search progression with ... v = a column matrix containing the authority score of all of the nodes. with these definitions, it follows that for any iteration k: u(k) = a ∗ v (k)andv (k) = transpose(a) ∗ u(k) fig. 6 hubs are pages that link to authorities. search engines always return millions of search results, it is necessary to organize them to make it easier for users to find out what they want. restructuring web search results is an application of inferring user search goals. the inferred user search goals are represented by the vectors and the feature representation of each url in the search results can be computed. then, categorize each url into a cluster centered by the inferred search goals. perform categorization by choosing the smallest distance between the url vector and user-search-goal vectors. by this way, the search results can be restructured according to the inferred user search goals. the adjacency matrix of the graph a =  0 0 1 0 0 1 0 0 0 at =  0 0 0 0 0 0 1 1 0  assume the initial hub weight vector u =  1 1 0  advances in systems science and application (2015) vol.15 no.4 377 compute the authority weight vector by: v = at · u =  0 0 0 0 0 0 1 1 0  •  1 1 1  =  0 0 2  then, the updated hub weight v = a · u =  0 0 1 0 0 1 0 0 0  •  0 0 2  =  2 2 0  editable browser for selecting the results & re ranking the result based on user editing strategies. we enhanced an editing operation such as deletion and emphasis that can be employed while users are browsing web search results. this system enables users to edit any portion of the page of web search results at any time while searching. our system detects user’s search intentions from the editing operation. then our work propagates user’s search intentions based on their editing operation to all of the web search results. this system guesses the user’s search intentions, for example assuming, “this user does not want this kind of the result”, if the user deletes a part of the search results, or “this user wants this kind of the results more”, if the user emphasizes a part of the search results. after guessing the user’s search intentions, our system re-ranks the search results according to the intention and shows the re-ranked results to the user. in this way, the user can easily obtain optimized search results. fig. 7 editing browser processing. 378 b. bazeer ahamed and t.ramkumar:deduce user search progression with ... fig. 8 restructured outputs with editable options. fig. 9 restructured result after deletion operation. advances in systems science and application (2015) vol.15 no.4 379 fig. 10 feedback session for the query “the sun”. user editable browser has been introduced to perform editing operations such as deletion and emphasis while browsing the search results. when the user deletes a part of the search result, the system degrades search results which include the deleted term or sentence. when the user emphasizes a part of the search result, the system upgrades the search results which include the emphasized term or sentence. the system re-ranks the search results according to the user intention and shows the re-ranked results to the user. deletion operation: deletion is an operation that indicates what types of search results the user does not want to obtain from the system. consider the sample class shown in the fig.7 , if the user does not want the 36rd link the user uses the deletion operation to remove the link from the generated search result. performance evaluation based on restructured web search results each url in the click session is categorized into one class, we introduce the typical meticulousness(tm) process for calculating the user click through log. tm will always be the highest value namely 1 no matter whether users have so many search goals or not. therefore, there should be a risk to avoid classifying search results into too many classes by error. so, we can further extend tm by introducing the above risk and propose a new criterion called “classified tm” it is calculated by ctm = ( 1 y r=1∑ y rr r )× (1− dij/c 2 y ) x where y is the number of relevant (or clicked) documents in the retrieved ones, r is the rank, rr is the number of relevant retrieved documents of rank r and x is used to adjust the influence of risk on ctm which can be learned from training data. we select 10 queries and empirically decide the number of user search goals of these queries. then, we cluster the feedback sessions and restructure the search results with inferred user search goals. we tune the parameter x to make 380 b. bazeer ahamed and t.ramkumar:deduce user search progression with ... ctm the height. based on the above process, the optimal x is from 0.6 to 0.8 for the 10 queries the mean and the variances of the optimal x are 0.697 and 0.005, respectively. thus, we set x to be 0.7. table 1 user editable browser comparisons with other methods for 100 ambiguous queries method probabilistic inference model personalizati on algorithms query recommender algorithm identification of user goals generalized feedback session user editable browser i 0.7124 0.787 0.755 0.562 0.7173 0.8055 ii 0.7911 0.625 0.584 0.742 0.8031 0.8875 iii 0.801 0.749 0.611 0.632 0.9895 1 iv 0.5391 0.6245 0.512 0.654 0.6646 0.6787 v 0.632 0.745 0.741 0.524 0.7845 0.833 sample manual calculation for single class result: 1.ctm = 1/3(1/1 + 2/3 + 3/4) ∗ (1− 0)0.7 = 0.8055 2.ctm = 1/4(1/1 + 2/2 + 3/4 + 4/5) ∗ (1− 0)0.7 = 0.8875 3.ctm = 1/2(1/1 + 2/2) ∗ (1− 0)0.7 = 1 4.ctm = 1/5(1/1 + 2/3 + 3/5 + 4/7 + 5/9) ∗ (1− 0)0.7 = 0.6787 5.ctm = 1/2(1/1 + 2/3) ∗ (1− 0)0.7 = 0.833 in order to demonstrate that when inferring user search goals, clustering our proposed feedback sessions are more efficient than other clustering search results and clicked urls directly in order to further compare our method with the existing method, we test the 100 most ambiguous queries such as “apple”, “the sun”, “car”, “mobile”, “college”, “earth” and so on. ctm has the highest mean average which is significantly higher than existing method. fig. 11 ctm vs. mean average of other methods. 5 conclusion this paper aims to discover the number of diverse user search goals for a given query and predict with some keywords automatically. first, the given user query is pre-processed to find the root word. then, a feedback session has been introadvances in systems science and application (2015) vol.15 no.4 381 duced to infer user search goals for a query. the feedback session is defined as the series of both clicked and unclicked urls and ends with the last url that was clicked in a session from user click-through logs. each feedback sessions are mapped to pseudo-documents using keywords which can efficiently reflect user information needs. by using agglomerative hierarchical clustering techniques to cluster the pseudo documents. improved hits (hyperlink induced topic search) algorithm is used to rank cluster results relevant for a particular topic. we introduced classified typical meticulousness(tm) process for calculating the user click through log, the web results are restructured. finally, user editable browser has been introduced allow the user to perform editing operations such as deletion and emphasis while browsing the search results. this paper can be extended to support query recommendation there by suggesting queries that can helps the user to form queries more precisely. acknowledgement the authors would like to express their sincere thanks to the editorial board members and anonymous reviewers for providing their valuable comments. the first author would like to express their thanks to management and other authorities for providing an feasible environment to carry out the research work successfully. references [1] beeferman. d and berger. a (2000), “agglomerative clustering of a search engine query log”, acm sigkdd proc. sixth int’l conf. knowledge discovery and data mining(sigkdd ’00), pp.407-416. [2] joachims .t text mining, j. franke, g. nakhaeizadeh and renz, eds.(2003), “evaluating retrieval performance using click-through data”, physica/springer verlag, pp.79-96. [3] beitzel. s, jensen .e., chowdhury. a, and frieder. o (2007), “varying approaches to topical web query classification,” proc. 30th ann.int’l acm sigir conf. research and development (sigir ’07) , pp.783-784. [4] joachims .t,(2002), “optimizing search engines using click through data,”,proc. eighth acm sigkdd int’l conf. knowledge discovery and data mining (sigkdd ’02), pp.133-142. [5] yao-sheng chang, kuan-yu he, scott yu and wen-hsiang lu (2006), “identifying user goals from web search results”,in ieee. 382 b. bazeer ahamed and t.ramkumar:deduce user search progression with ... [6] indu chawla,(2010), “an overview of personalization in web search”, in ieee. [7] cao.h, jiang.d, pei.j, he.q, liao.z, chen.e and li.h (2008), “contextaware query suggestion by mining click-through”, proc.14th acm sigkdd int’l conf. knowledge discovery and data mining (sigkdd’08), pp.875-883. [8] chen .hand dumais. s, (2000), “bringing order to the web: automatically categorizing search results”, proc. sigchi conf. human factors in computing systems (sigchi ’00), pp.145-152. [9] zeng .h.-j, he. q.-c, chen .z, ma. w.-y and ma. j.(2004), “learning to cluster web search results”,proc. 27th ann. int’l acm sigir conf. research and development in information retrieval (sigir ’04), pp.210-217. [10] shital c. patil, prof. r.r. keole, (2013), “web usage mining and webcontent mining c a combine approach for enhancing search result delivery”, vol.3, no.10. [11] baeza-yates .r, hurtado. c and mendoza. m, (2004), “query recommendation using query logs in search engines”, the journal of supercomputing(edbt ’04), pp.588-596. [12] joachims .t,“optimizing search engines using click through data”. [13] lee.u, liu.z and cho. j, (2005),“automatic identification of user goals in web search”,proc. 14th int’l conf. world wide web (www ’05), pp.391400. [14] poblete.b and ricardo .b.-y (2008), “query-sets: using implicit feedback and query patterns to organize web documents,”,proc. 17th int’l conf.world wide web (www ’08), pp.41-50. [15] haibo yu , tsunenori mine and makoto amamiya(2011), “towards user intent based searching”, 2011 international joint conference of ieee trustcom-11/ieee icess-11/fcst-11. [16] wang .x and zhai. c.-x (2007), “learn from web search logs to organize search results,”,proc. 30th ann. int’l acm sigir conf. research and development in information retrieval (sigir ’07),pp.87-94. [17] zheng lu, hongyuan zha, xiaokang yang, weiyao lin and zhaohui zhengr, (2013), “a new algorithm for inferring user search goals with feedback sessions,”, ieee transactions on knowledge and data engineering,vol.25, no.3. advances in systems science and application (2015) vol.15 no.4 383 [18] takehiro yamamoto, satoshi nakmura and kutsumi tanaka, “an editable browser for reranking web search results”. [19] huang c.-k, chien l.-f, and oyang y.-j, (2003), “relevant term suggestion in interactive web search based on contextual information in query session logs,”,j. am. soc. or information science and technology, vol.54, no.7, pp. 638-649. corresponding author b.bazeer ahamed can be contacted at:bazeerahamed@gmail.com advances in systems science and application (2015) vol.15 no.4 392-399 the market laws declaration of economic development of the countries worldwide s.baizakov1, a. oinarov2, j. yi-lin forrest3 and n.baizakov4 1jsc, economic research institute, astana, kazakhstan; 2jsc, kazakh center of public and private partnership, astana, kazakhstan; 3pennsylvania state system of higher education, astana, kazakhstan; 4jsc stana-project, astana, kazakhstan abstract the mathematical formulation of a problem of market balance of levels of production, employment, the income and the prices is considered. the new market model of work which focuses economy on the end result is constructed. it becomes clear why development of market forces of work and the capital in economy branches, contrary to desires of businessmen on accumulation of competitiveness of the enterprises, happens according to laws of development of market economy. keywords equlibrium; balanced growth; table input-output; model; laws 1 introduction the analytical work to assess the costs and benefits of the regulatory impact on the development of the national economy was successfully carried out in europe and in other developed countries. appropriate tools in russian-speaking countries are known under the following names: regulatory impact analysis (ria) and regulatory impact assessment (ria). the ability to analyze the influence of the regulatory impacts on the economy, and to use them in the evaluation of scientific and technological changes in the economy, is very useful for developing countries. if we study the regulatory impact analysis tools on the development of the national economy, it is possible to distinguish two types of them. the first type of regulatory impact influences on macroeconomic indicators,and the second type of regulatory impact influences on the performance of enterprises at the microeconomic level. the first one mainly influences on the cost of capital, in the form of money, while the second one has an impact on the cost of capital, in the form of the product. it should be noted that changes in the cost of capital in the form of money, and the cost of capital in the form of the product are measured with units of national or global currency banknotes. the president of the republic of kazakhstan nursultannazarbayev in his article “the fifth way”, which has been written during the crisis of 2007-2009, marked the presence of the third type of regulatory impact influencing on the level of development of the national economy[1]. the “fifth way” demonstrates the necessity of an objective assessment of the social and economic subsequences advances in systems science and application (2015) vol.15 no.4 393 of those three types of regulatory impacts. the conclusion that appears from the logic of the “fifth way” is to ensure the mutual consistency of the indicators of all types of regulatory impact that have a crucial importance in determining the true value of money capital in macroeconomics and the true cost of the commodity-capital in microeconomics. this principle of mutual consistency of regulatory impacts defined in the “fifth way” has become the foundation of the concept of building cross-sectoral assessment models of levels of production market balance, employment, incomes and prices developed by the initiative group of kazakhstan in early 2010. thus, the initiative group of kazakhstan in 2010, sparing no effort and energy performed the implementation of the ideas proposed by the president of kazakhstan on the assessment of the influence of economic governance policies not only on the true cost of capital in the form of money, and on the true cost of capital in the form of the product. this initiative group reviewed in detail the basic principles of market economy management models associated with the name of keynes, monetarists and mundell-fleming[2]. as we know, those models are now extensively used in the analysis of the market economy development. firstly, it appeared that the main indicator of market forces analysis of capital in the form of money in all those three models is a nominal gdp, and the main indicator of market forces analysis of capital in the form of goods – a real gdp in the keynesian and monetarist models, gdp, and according to purchasing – power parity in the mundell-fleming model. secondly, the kazakhstani initiative group has proved that the principles of building models of the monetarists and mundell-fleming have a single macroeconomic root which was defined by a known keynesian theory of the equation of demand and supply for the goods and services. since the principles of all those models of economy analysis and management have been developed with the use of the system of the same macroeconomics indicators which had been created by keynes. thirdly, it was discovered that in the system of equations of market balance, created by keynes and his followers, there is no microeconomics indicator: all models of the economy analysis are based on macroeconomic theory and are not related to microeconomic theory and microeconomic indicators. fourthly, it became known that keynes is the author of separation of a unified indicators system of the national economy, which was defined in the classical economic science, in the part of macroand microeconomics. fifthly, it is determined that all of those models are based on the one assumption between the capital in the form of money and the capital in the form of the product. keynes has based his principle of construction of his model on the macroeconomic stability of the prices for goods and services, and the monetarists 394 s.baizakov, a. oinarov, j. yi-lin forrest and n.baizakov: the market laws declaration... – on macroeconomic stability of the money turnover velocity, but fleming and mundell – on the macroeconomic stability of the national currency exchange rate. consequently, representatives of the initiative group during the formation of the kazakh model of economic management had a reason to accept the hypothesis of a possible omission of keynes made in the course of a single economic system breakdown on macroeconomics and microeconomics, and to mark the uncritical attitude of monetarists as well as of mundell and fleming towards this omission. 2 the market laws declaration of economic development of the countries worldwide there are no words that all of those models were popular management tools of market economies of that time, this can be proved by the sustainable development of the world economy until 2007. however, keynes in due time determining the indicators of macroeconomics used the method to minus the material costs from sales of goods and services. and he created a new macroeconomic indicator called the gross domestic product. this arithmetic operation allowed the authors of macroeconomic theory to get rid from microeconomics indicators. as a result, in the study of the cost of the financial capital are mainly engaged macroeconomists, and with cost of commodity capital microeconomists. as it became known much later, after keynes, the determination of the solve equations of mutual dependence of the final result of the production from the gross sales of goods and services does not require arithmetic but deeper calculations using matrix algebra. the searches of the kazakh initiative group showed that the foundation of this dependence is determined by the system of equations describing the dependence between macroeconomic and microeconomic indicators as follows[3]: by economic sectors– t(i)x(i)− t (i)y (i) = ±θ(i), i = 1, 2, ..., n by national economy– c ≡ t t = y x , where t(i)= l(i) x(i)– time spent by the people themselves, in relation to one tenge, received from the sale of goods and services by economic activity, expressed in hours, days, months, years; t (i) = t(i)b– the total time spent by people on the job, per one tenge, received from the sale of goods and services by economic activity, expressed in hours, days, months, years; tx = l– national fund of working time, calculated by gross domestic product (x) is determined by the balance of the time model of inter-sectoral working advances in systems science and application (2015) vol.15 no.4 395 people, expressed in man-hours per year, man-days a year, man-year, people a year; ty = l– national fund of working time, calculated by the volume of the end product (y) is determined by the balance of the time model of inter-sectoral working people, expressed in man-hours per year, man-days a year, man-year, people a year; tiyi − tixi = θij– the difference between the cost of working time, defined in terms of the final product (y) and the working time defined in terms of gross product (x), expressed in man-hours per year, man-days a year, man-year, people a year; b = (e−a)(−1)– inverse matrix of the full costs of the kazakhstan economy, as determined by inter-branch model, where a the technological matrix of the economy of kazakhstan by types of economic activity, and e special algebraic matrix units. the final result of the analysis of the regulatory impact on the development of the national economy of kazakhstan is the difference between the growth rate of capital in the form of money in macroeconomics and of capital in the form of goods in microeconomics: ċ c = ẏ y = ẋ x , this difference arises from the opposite direction of capital flows in the form of money and of capital in the form of the product which have different flow rate. it can be called the coefficient of scientific and technological change, as it has appeared, as a manna from heaven, from the ceiling, with strict equality of the working time spent on the creation of the end product (y = c + g + i + nx) in macroeconomics and gross domestic product (x) in microeconomics. so that, this difference is a net contribution of scientific and technological progress determined by the difference between productivity of the end product and productivity of gdp.in other words this success comes from within the country’s economic system determined by its interdisciplinary development model. now we can explain any innovative initiative which is linked to scientific and technological changes in the national economy not only by an abstract definition of “scale” production or solow residue, but by the cost spent on the production of specific working hours, the specific productivity of labor and capital, the ratio of scientific and technological progress. in addition, every entrepreneur can perform the same calculation for each type of their economic activity. now, the true value of the national currency can be estimated by dividing the rate of scientific and technological changes on the gdp deflator, and the index of inflation will be determined by a formula which is different from the formula of the gdp deflator. the system of laws of the market economy determined due to the coefficient of scientific and technological change, includes the following 396 s.baizakov, a. oinarov, j. yi-lin forrest and n.baizakov: the market laws declaration... related equations: • equation of the law determining the overall impact of stimulating scientific and technological improvements – c(t): c(t) = y (t)/x(t) • equation of the law determining the purchasing power of money – pp: pp(t) = (c(t) ∗ i2(t))/i1(t); • equation of the law determining the prices of goods and services – pc(t)=1/pp(t): pc(t) = 1/pp(t) = i1(t)/(c(t) ∗ i2(t)); • basic law equation determining the real volume of the end product, measured by the real cost of capital in the form of money, as an indicator of real economic growth – i3(t): i3(t) = pp(t) ∗ i1(t); • basic law equation determining the real volume of the end product, measured by the real cost of capital in the form of goods as the benchmark of real economic growth – i3(t): i3(t) = c(t) ∗ i2(t); • equation of the law of general price deflation – b(t): b(t) = c(t)/pp(t) ≡ i1(t)/i2(t); • equation of the law determining the net benefits of stimulating scientific and technological improvements – dc(t)%: dc(t)% = ċ c = ẏ y = ẋ x . it should be noted that the index i1(t), is the rate of growth of gross domestic product for the price of the current year, while the index i2(t) the price of the previous year. progress of the us economy and china for 2002-2011 according to the inputoutput balance of these countries. as the table shows, in the period from 2002-2008, both countries were characterized by trends to reduce the cost of capital in the form of money (y), which falls per unit of capital in the form of goods (x) .this means that both countries, return on equity, in the form of money decreased, rates of stp were declining trend. this situation lasted until 2009 flesh. in spite of the substantial difference in the level of indicators of scientific and technological progress, the two countries after the crisis of 2007-2008 has a regulating effect on the level of the coefficient advances in systems science and application (2015) vol.15 no.4 397 of ntp so that stop further depreciation of the national currency. but they have different approaches. for example, in the united states since 2009 have been lowered growth rate of the cost of capital in the form of money and the cost of capital in the form of the product, so as to achieve better rate of scientific and technological progress. in china, by contrast, produced faster growth rate of growth capital in the form of money, compared to the growth rate of capital in the form of the product. table 1 comparative assessment of indicators of scientific and technological progress of the us economy and china for 2002-2011 according to their inputoutput balance (mm in usd). years usa china end product (y) gross product (x) ntp coefficient=y/x end product (y) gross product(x) ntp coefficient =y/x 2002 10416078.53 18873334.93 55.2 1324753 3794147 34.9 2003 10938122.88 19828498.07 55.2 1485275 4457621 33.3 2004 11736535.23 21264711.1 55.2 1760018 5372764 32.8 2005 12549008.41 23072266.04 54.4 2001206 6527490 30.7 2006 13280785.08 24479922.11 54.3 2354040 8160175 28.8 2007 13844412.73 25795266.08 53.7 3019764 10740915 28.1 2008 14214545.69 26565031.96 53.5 3951859 13913136 28.4 2009 13775598.08 24802899.21 55.5 4538041 15149965 30 2010 14230107.43 25810105.94 55.1 5401801 18070490 29.9 2011 14770666.86 26918120.33 54.9 6777332 22271025 30.4 source: developed by the author based on the table “input-output” of these countries. 3 discussion and conclusion now when we already know the trends of scientific and technological progress, which are determined by the correlation between the two main indicators of the market economy, namely, between the cost of the end product, which expresses the cost of capital in the form of money and the value of gross domestic product, which expresses the cost of capital in the form of the product, you can do strong statement: the law of scientific and technological progress and related laws of the market economy are a reflection of objective processes occurring in it, and they are independent of the will of the people. no one in the world can not change or cancel them. therefore, people can only identify, understand and learn these laws. do not consider their actions in the practice of economic management equivalent to curb the development of a market economy, its inhibition of further liberalization. this is to ensure that the laws of market economy, equivalent to the laws of natural science and nature. i can even say so that a natural disaster happens in one place, in one country, and the collapse in the developed economies as a 398 s.baizakov, a. oinarov, j. yi-lin forrest and n.baizakov: the market laws declaration... contagious disease spreads rapidly on the economies of developing countries. therefore, science and technology indicator changes justified by the kazakhstani initiative group, as the law of the market economy, is suitable for use in the analysis of the economies of all countries of the world. and it will be best if we call these economic laws system tools capable for economic analysis and regulatory impact assessment models such as the keynesian model, the monetarists and the model of mundell and fleming. this is due to the fact that any of these models is reduced to the particular case of the following generalized equation, kazakhstan proposed an initiative group to assess the actual final product anywhere in the world, including even such large countries like the us and china: pp ∗ngdp = c ∗rgdp (a) this indicator measuring the end product is significantly different from the nominal gross domestic product (gdp), real gdp and the gdp determined by purchasing-power parity, which are included in the current model of keynesianism deystvuyushie, monetarists, mundell and fleming. model (a) determines the power of the final product, used in the country and consists not only of the domestic product of the country, but also its product to come from external trade turnover. the new indicator measuring economic growth is the development of money equilibrium assignment (pp * ngdp) capital level of trade (with * rgdp) capital. it is the final product, and not the nominal (ngdp) and real gdp (rgdp), let alone the gdp determined by purchasing power parity, is a tool for sustainable development of the national economy and the real measure of wellbeing of its people. the measure defined by the equation of market equilibrium (a), is common for the economy and the economy of consumption of goods and services, as well as for the economy of the monetary and financial system. it is a new measure, adapted to the level of development of the productive forces of labor and capital in a globalized world economy. it can be used to assess the regulatory impact on the economic development of any country, any region of the world, since the value is the true purchasing power of the national currency, and the value is a measure of realindicator of scientific and technological change. this measure determining indicators of scientific and technological changes in the past never been used and is a new phenomenon in the development of world economics. and because the core of this declaration on market laws of economic development of the countries of the world is the progress in economic science, especially in the science of economic management, which are hidden multibillioneffects from the use of reliable tools of the regulatory impact. advances in systems science and application (2015) vol.15 no.4 399 references [1] the fifth way, (news), september 22, 2009. [2] y. hasanli, s. bayzakov, v. valiyev, g. sarsembaeva. (2011), “modeling of the multiplicative effects of opening of the work places on the bases of ‘intersectoral labor balance’ (on example of azerbaijan and kazakhstan)”, ecomodazkz full pape eng. [3] ba�zakov s. (2014), “model of the labour market, based on the final product”, science and innovation, no.12, pp.31-34. corresponding author sailau baizakov can be contacted at: baizakov37@mail.ru delay-dependent stability criteria of stochastic uncertain hopfield neural networks with unbounded distributed delays and impulses r. raja1, r.sakthivel 2 and s.marshal anthoni 3 1department of mathematics, periyar university, salem-636 011, india 2department of mathematics, sungkyunkwan university, suwon 440-746, south korea 3department of mathematics, anna university of technology, coimbatore-641 047, india email: antony.raja67@yahoo.com abstract this paper is concerned with the stability analysis problem for a class of delayed stochastic uncertain hopfield neural networks with unbounded distributed delays and impulses. a new lyapunov-krasovskii functional is constructed for the addressed system and several freeweighting matrices combined with the s-procedure are employed to derive the delay-dependent stability criterion. the criterion is derived and formulated in terms of linear matrix inequality (lmi). in addition to that, two illustrated examples with simulation results are given to show the effectiveness of the obtained theoretical results. keywords delay-dependent stochastic hopfield neural networks distributed delays lyapunov krasovskii functional linear matrix inequality impulses. 1. introduction during the past several years, the stability of a unique equilibrium point of hopfield neural networks [2] with delays have received especially considerable attention due to their extensive applications in solving optimization problem, traveling salesman problem and many other subjects in recent years [4, 5, 6, 14, 15, 16, 17, 18, 23, 24]. basically, the stability results of delayed hopfield neural networks can be classified into two categories: delay dependent stability and delay independent stability. delay-dependent stability results are generally less conservative than delay-independent stability when the delays are small. on the other hand, time-delays occurring in the interaction between neurons will affect the stability of a network by creating instability, oscillation and chaos phenomena. recently, a number of global stability criteria of hopfield networks with time-delays have been proposed (see [13, 15, 16, 17, 18]). the dynamical systems are often classified into two categories of either continuous-time or discrete-time systems. apart from this two systems, yet there is a somewhat new category of dynamical systems, which is neither continuous-time nor purely discrete-time, these are called dynamical systems with impulses. a basic theory of impulsive differential equations has been developed in [11]. the stability conditions in [19, 20, 22] were established by using the impulsive condition. in the real world, there are two common disturbances that affects the issn 1078-6236 international institute for general systems studies, inc. advances in systems science and applications (2011), vol. 11, no. 1-2 93-109 network process, one is stochastic perturbations and the other one is uncertain parameters. recently, there are some research papers about stochastic neural networks, has been investigated, see for example [3, 5, 6, 7, 9, 10, 16, 17, 18]. in [12] the stability problem for both discrete and distributed delays were discussed. yang [21] has been investigated the stability of neural networks with distributed delays. in practical, uncertainties often exist in most engineering and communication systems and may cause undesirable dynamic network behaviors such as oscillation, instability and chaos. more specifically, the connection weights of the neurons are inherent dependent on certain resistance and capacitance values that inevitably bring in uncertainties during the parameter identification process. in the literature, uncertainties can possibly be described by norm bounded, polytopic or linear fractional uncertainties characterizations and have been widely employed in the field of robust control and performance analysis. based on the above descriptions, this paper aims to develop the problem of asymptotic stability for delayed stochastic uncertain hopfield neural networks with unbounded distributed delays and impulses. by constructing an appropriate lyapunov-krasovskii functional, employing several free-weighting matrices and s-procedure, we obtain a delay-dependent stability criterion in terms of lmis. finally, two numerical examples with simulation results are provided to demonstrate the usefulness of the main results in this paper. 2. network model and preliminaries the delayed stochastic hopfield neural network model with unbounded distributed delays and impulses is defined by the following state equations: dxi(t) = [ − aixi(t) + n∑ j=1 bijfj(xj(t− τ)) + n∑ j=1 cij ∫ t −∞ kj(t− s)fj(xj(s))ds+ ji ] dt + n∑ j=1 σij(t, xj(t))dwj(t), t 6= tk (1) xi(tk) = ikx(t−k ), t = tk, k = 1, 2, ... where xi(t) is the state of the ith neuron at time t; ai > 0 denotes the passive decay rate; bij and cij are the synaptic connection strengths; fj denotes the neuron activation functions; ji is the constant input from outside the system; τ represents the continuous delay and the delay kernel kj is a real valued continuous function defined on [0,+∞] and satisfies, for each i, ∫∞ 0 kj(s)ds = 1. the stochastic disturbance w(t) = (w1(t), w2(t), ..., wm(t))t is an mdimensional brownian motion; σij(·, ·) is locally lipschitz continuous and satisfies the linear growth condition as well; xi(tk) = ikx(t−k ) is the impulse at moment tk, the fixed moment of time tk satisfy t1 < t2 <, ..., limk→+∞ tk = +∞ and x(t−) = lims→t− x(s); ik is a constant real matrix at the moments of time tk. let pc([−τ, 0],rn) denotes the set of piecewise right continuous functions φ : [−τ, 0]→ rn with the sup-norm |φ| = sup−τ≤s≤0‖φ(s)‖. for given t0, and φ ∈ (pc[−τ, 0],rn), the initial condition of system (1) is described as x(t0 + t) = φ(t), for t ∈ [−τ, 0], φ ∈ 94 raja: delay-dependent stability criteria of stochastic uncertain hopfield . . . . . . pc([−τ, 0],rn). the following assumptions is utilized throughout this paper: (c1) the activation function f(x) is boundless and satisfies 0 ≤ fi(ξ1)− fi(ξ2) ξ1 − ξ2 ≤ li, for any ξ1, ξ2 ∈ r, ξ1 6= ξ2, i = 1, 2, ..., n (c2) the function fi(xi(·)) = 0, i = 1, 2, ..., n satisfy 0 ≤ fi(xi(t)) xi(t) ≤ li, fi(0) = 0, ∀xi(t) 6= 0, i = 1, 2, ..., n where li, i = 1, 2, ..., n are positive constants. remark 2.1 the above conditions ensures that the nonlinear resulting neuron activation functions should be non-monotonic and be more general than the usual sigmoid function as well as the commonly used lipschitz condition. the equilibrium point y∗ = [y∗1, y ∗ 2, ..., y ∗ n] of system (1) will be shifted to the origin by the transformation y(·) = x(·)− x∗, transforms system (1) into the following form dy(t) = [−ay(t) +bg(y(t− τ)) + c ∫ t −∞ k(t− s)g(y(s))ds]dt+ σ(t, y(t))dw(t), t 6= tk (2) y(tk) = iky(t−k ), t = tk, k = 1, 2, ... y(t0 + t) = ψ(t), t ∈ [−τ, 0] where y = [y1, y2, ..., yn]t , a = diag[a1, a2, ..., an], b = [bij ], c = [cij ], k(t − s) = diag[k1(t − s), k2(t − s), ..., kn(t − s)], g(y) = [g1(y1), g2(y2), ..., gn(yn)] with gj(yj(t)) = fj(yj(t) + x∗j )− fj(x∗j ). note that since each function fj(·) satisfies the assumptions (c1) and (c2), hence each gj(·) satisfies g2j (ξj) ≤ l2 jξ 2, ξjgj(ξj) ≥ g2j (ξj) lj ∀ξj ∈ r, gj(0) = 0 (c3) there exist a constant matrix d0 such that trace[σt (t, y(t))σ(t, y(t))] ≤ yt (t) d0 y(t) lemma 2.2 [1] (s-procedure) let ti ∈ rn×n (i = 0, 1, ..., p) be symmetric matrices. the conditions on ti, (i = 0, 1, ..., p) αtt0α > 0, ∀α 6= 0 s.t. αttiα ≥ 0 (i = 0, 1, ..., p) hold, if there exist τi ≥ 0 (i = 0, 1, ..., p) such that t0 − p∑ i=1 τiti > 0 advances in systems science and applications (2011), vol. 11, no. 1-2 95 lemma 2.3 let u, v,w andm be real matrices of appropriate dimensions withm satisfying m = mt , then m + uvw +w tv tut < 0, ∀ v tv ≤ i if and only if there exist a scalar ε > 0 such that m + ε−1uut + εw tw < 0 definition 2.4 [25] the function v : [t0,∞)× rn → r+ belongs to class v0 if (1) the function v is continuous on each of the sets [tk−1, tk) × rn and for all t ≥ t0, v (0, t) ≡ 0; (2) v (x, t) is locally lipschitzian in x ∈ rn; (3) for each k = 1, 2, ..., there exist finite limits lim (q,t)→(x,t−k ) v (q, t) = v (x, t−k ) lim (q,t)→(x,t+k ) v (q, t) = v (x, t+k ) with v (x, t+k ) = v (x, tk) satisfied. in the following section, we will develop delay-dependent condition for the given system such that the origin of the delayed stochastic hopfield neural network (2) is asymptotically stable. 3. asymptotic stability criterion before discussing the stability analysis of the problem, we firstly introduce the ito’s formula for a general stochastic system. let v (y(t), t) : c([−τ, 0],rn × r+ → r+) be a positive function which is continuously twice differentiable in y and once differentiable in t. thus, an operator l acting on v (y(t), t), is defined by lv (y(t), t) = vt(y(t), t) + vy(y(t), t)[−ay(t) +bg(y(t− τ)) + c ∫ t −∞ k(t− s)g(y(s))ds] + 1 2 trace[σt (t, y(t))vyy(y(t), t)σ(t, y(t))] (3) where vt(y(t), t) = ∂v (y(t), t) ∂t , vy(y(t), t) = (∂v (y(t), t) ∂y1 , ∂v (y(t), t) ∂y2 , ..., ∂v (y(t), t) ∂yn ) vyy(y(t), t) = (∂2v (y(t), t) ∂yi∂yj ) n×n now, the following theorem gives a new stability criterion for system (2) without uncertain parameters. 96 raja: delay-dependent stability criteria of stochastic uncertain hopfield . . . . . . theorem 3.1 suppose that the assumption (c1)− (c3) is satisfied. if there exist matrices p = [ p11 p12 p t12 p22 ] ≥ 0 with p11 > 0, r = [ r11 r12 rt12 r22 ] ≥ 0 q1 > 0, q2 > 0, h1, h2, h3, h4, diagonal matrices e > 0, s > 0 and a positive scalar ρ > 0 such that the following inequalities hold: p < ρi (4) itk p11ik + 2itk p12ik + itk p22ik − p11 − 2p12 − p22 < 0 (5) ω =  ξ11 ξ12 ξ13 ξ14 −τatp t12 + τp t22 −τh1 −τatr22 ∗ ξ22 ξ23 −ht 4 −τp t22 −τh2 0 ∗ ∗ ξ33 0 τbtp12 −τh3 −τbtr22 ∗ ∗ ∗ −e τctp12 −τh4 −τctr22 ∗ ∗ ∗ ∗ −τr11 −τr12 0 ∗ ∗ ∗ ∗ ∗ −τr22 0 ∗ ∗ ∗ ∗ ∗ ∗ −τr22  < 0 (6) where ξ11 = −p11a−atp t11 + p12 + p t12 + ρd0 +q1 + lel+ τr11 +h1 +ht 1 − τr12a ξ12 = p12 −h1 +ht 2 , ξ13 = p11b +ht 3 + τr12b ξ14 = p11c +ht 4 + τr12c, ξ22 = −q1 −h2 −ht 2 , ξ23 = ls −ht 3 , ξ33 = −q2 − 2s, l = {l1, l2, ..., ln}. then the origin of system (2) is the unique equilibrium point and it is globally asymptotically stable. proof. define new state variables g1(t) = −ay(t) +bg(y(t− τ)) + c ∫ t −∞ k(t− s)g(y(s))ds (7) g2(t) = σ(t, y(t)) (8) to prove the asymptotic stability result, let us consider the following lyapunov functional candidate for system (2) as v1 = δt1 (t)pδ1(t), v2 = ∫ t t−τ yt (s)q1y(s)ds, v3 = ∫ t t−τ gt (y(s))q2g(y(s))ds v4 = ∫ 0 −τ ∫ t t−τ δt2 (s)rδ2(s)dsdσ, v5 = n∑ j=1 ej ∫ ∞ 0 kj(ξ) ∫ t t−ξ g2j (yj(γ))dγdξ (9) advances in systems science and applications (2011), vol. 11, no. 1-2 97 where p = [ p11 p12 p t12 p22 ] ≥ 0 with p11 > 0, r = [ r11 r12 rt12 r22 ] ≥ 0 and δ1(t) = [ y(t)∫ t t−τ y(s)ds ] , δ2(t) = [ y(t) g1(t) ] , by newton-leibnitz formula, the following equation is true for any matrices hi (i = 1, 2, 3, 4) with appropriate dimensions: 2 [ yt (t)h1 + yt (t− τ)h2 + gt (y(t− τ))h3 + (∫ t −∞ k(t− s)g(y(s))ds ) h4 ] × [ y(t)− y(t− τ)− ∫ t t−τ g1(s)ds ] = 0 (10) when t 6= tk the derivative of v can be calculated by using ito’s differential formula. then the trajectories of the system (2) is given as: lv1 = 2δt1 (t)p δ̇1(t) = 2  y(t)∫ t t−τ y(s)ds t p11 p12 p t12 p22 −ay(t) +bg(y(t− τ)) + c ∫ t −∞k(t− s)g(y(s))ds y(t)− y(t− τ)  = −2yt (t)p11ay(t) + 2yt (t)p11bg(y(t− τ)) + 2yt (t)p11c (∫ t −∞ k(t− s)g(y(s))ds ) + 2yt (t)p12y(t)− 2yt (t)p12y(t− τ)− 2 ∫ t t−τ yt (s)p t12ay(t)ds+ ∫ t t−τ yt (s)p t12b × g(y(t− τ))ds+ 2 ∫ t t−τ yt (s)p22y(t)ds− 2 ∫ t t−τ yt (s)p22y(t− τ)ds+ 2 (∫ t t−τ yt (s)ds ) × p t12c (∫ t −∞ k(t− s)g(y(s))ds ) + trace[σt (t, y(t))pσ(t, y(t))] (11) lv2 = yt (t)q1y(t)− yt (t− τ)q1y(t− τ) (12) lv3 = gt (y(t))q2g(y(t))− gt (y(t− τ))q2g(y(t− τ)) (13) lv4 = τδt2 (t)rδ2(t)− ∫ t t−τ δt2 (s)rδ2(s)ds (14) lv5 = n∑ j=1 ej ∫ ∞ 0 kj(ξ)g 2 j (yj(t))dξ − n∑ j=1 ej ∫ ∞ 0 kj(ξ)g 2 j (yj(t− ξ))dξ = gt (y(t))eg(y(t))− n∑ j=1 ej ∫ ∞ 0 kj(ξ)dξ ∫ ∞ 0 kj(ξ)g 2 j (yj(t− ξ))dξ ≤ yt (t)lely(t)− n∑ j=1 ej (∫ ∞ 0 kj(ξ)g 2 j (yj(t− ξ))dξ )2 = yt (t)lely(t)− (∫ t −∞ k(t− s)g(y(s))ds ) e (∫ t −∞ k(t− s)g(y(s))ds ) (15) 98 raja: delay-dependent stability criteria of stochastic uncertain hopfield . . . . . . it is noted from (c2) that, gi(yi(t− τ))[gi(yi(t− τ))− liyi(t− τ)] ≤ 0, i = 1, 2, ..., n (16) now, by applying the s-procedure, we find that system (2) is asymptotically stable, if there exist s = diag{s1, s2, ..., sn} such that lv = lv1 + lv2 + lv3 + lv4 + lv5 + 2 [ yt (t)h1 + yt (t− τ)h2 + gt (y(t− τ))h3 + (∫ t −∞ k(t− s)g(y(s))ds ) h4 ][ y(t)− y(t− τ)− ∫ t t−τ g1(s)ds ] ≤ lv1 + lv2 + lv3 + lv4 + lv5 + 2 [ yt (t)h1 + yt (t− τ)h2 + gt (y(t− τ))h3 + (∫ t −∞ k(t− s)g(y(s))ds ) h4 ][ y(t)− y(t− τ)− ∫ t t−τ g1(s)ds ] −2 n∑ i=1 sigi(yi(t− τ))(gi(yi(t− τ))− liyi(t− τ)) ≤ lv1 + lv2 + lv3 + lv4 + lv5 + 2 [ yt (t)h1 + yt (t− τ)h2 + gt (y(t− τ))h3 + (∫ t −∞ k(t− s)g(y(s))ds ) h4 ][ y(t)− y(t− τ)− ∫ t t−τ g1(s)ds ] −2gt (y(t− τ(t)))sg(y(t− τ(t))) + 2gt (y(t− τ(t)))sg(y(t− τ(t))) ≤ yt (t)[−p11a−atp t11 + p12 + p t12 + ρd0 +q1 + lel+ τr11 +h1 +ht 1 − τr12a] ×y(t) + yt (t)[p12 −h1 +ht 2 ]y(t− τ(t)) + yt (t)[p11b +ht 3 + τr12b]g(y(t− τ)) +yt (t)[p11c +ht 4 + τr12c] (∫ t −∞ k(t− s)g(y(s))ds ) + yt (t)[−τatp t12b + τp t22 × (∫ t t−τ y(s)ds ) + yt (t)[−τh1] (∫ t t−τ g1(s)ds ) + yt (t− τ)[−q1 −h2 −ht 2 ] ×y(t− τ) + yt (t− τ)[−ht 3 + ls]g(y(t− τ)) + yt (t− τ)[−τp t22] (∫ t t−τ y(s)ds ) +yt (t− τ)[−τh2] (∫ t t−τ g1(s)ds ) + gt (y(t− τ))[−q2 − 2s]g(y(t− τ)) + yt (t− τ)) ×[−ht 4 ] (∫ t −∞ k(t− s)g(y(s))ds ) + gt (y(t− τ)[τbtp12] (∫ t t−τ y(s)ds ) +gt (y(t− τ)[−τh3] (∫ t t−τ g1(s)ds ) + (∫ t −∞ k(t− s)g(y(s))ds )t [−e] advances in systems science and applications (2011), vol. 11, no. 1-2 99 × (∫ t −∞ k(t− s)g(y(s))ds ) + (∫ t −∞ k(t− s)g(y(s))ds )t [τctp12] (∫ t t−τ y(s)ds ) + (∫ t −∞ k(t− s)g(y(s))ds )t [−τh4] (∫ t t−τ g1(s)ds ) + (∫ t t−τ y(s)ds )t [−τr11] × (∫ t t−τ y(s)ds ) + (∫ t t−τ y(s)ds )t [−τr12] (∫ t t−τ g1(s)ds ) + (∫ t t−τ g1(s)ds )t [−τr22] × (∫ t t−τ g1(s)ds ) (17) it is easy to see y(t)− y(t− τ)− ∫ t t−τ g1(s)ds = ∫ t t−τ(t) g1(s)ds = 1 τ ∫ t t−τ(t) τ(τ−1τ(t)g1(s))ds (18) this together with (16), implies lv = 1 τ ∫ t t−τ(t) βt (t, s)ωβ(t, s)ds where βt (t, s) = [yt (t) yt (t− τ) gt (y(t− τ)) (∫ t −∞ k(t− s)g(y(s))ds )t yt (s) gt1 (s)] this implies that lv (y(t), t) < 0. when t = tk, we obtain the following result: v (y(tk), (tk))− v (y(tk), t − k ) = δt1 (tk) p11 p12 p t12 p22  δ1(tk)− δt1 (t−k ) p11 p12 p t12 p22  δ1(t−k ) = yt (t−k ) { itk p11 p12 p t12 p22  ik − p11 p12 p t12 p22 }y(t−k ) = yt (t−k )itk p11iky(t−k ) + yt (t−k )itk p12iky(t−k ) + yt (t−k )itk p t 12iky(t−k ) + yt (t−k )itk p22iky(t−k )− yt (t−k )p11y(t−k )− yt (t−k )p12y(t−k ) − yt (t−k )p t12y(t−k )− yt (t−k )p22y(t−k ) = yt (t−k ) [ itk p11ik + 2itk p12ik + itk p22ik − p11 − 2p12 − p22 ] y(t−k ) based on the lyapunov stability theorem, it follows that the delayed stochastic hopfield neural network (2) is globally asymptotically stable in the mean square. the proof of the theorem is completed. � 100 raja: delay-dependent stability criteria of stochastic uncertain hopfield . . . . . . 4. robust asymptotic stability criterion consider the system (2) with norm bounded parameter uncertainties that is dy(t) = [−(a+ ∆a(t))y(t) + (b + ∆b(t))g(y(t− τ)) + (c + ∆c(t)) ∫ t −∞ k(t− s) g(y(s))ds]dt+σ(t, y(t))dw(t), t 6= tk (19) y(tk) = iky(t−k ), t = tk, k = 1, 2, ... where a+ ∆a(t), b + ∆b(t) and c + ∆c(t) are of the following structure: [∆a(t) ∆b(t) ∆c(t)] = mf (t)[n1 n2 n3] where m,n1, n2, n3 are known constant matrices with appropriate dimensions and bounded which satisfies f t (t)f (t) ≤ i, t ≥ 0 theorem 3.2 suppose that the assumption (c1)− (c3) is satisfied. if there exist matrices p = [ p11 p12 p t12 p22 ] ≥ 0 with p11 > 0, r = [ r11 r12 rt12 r22 ] ≥ 0 q1 > 0, q2 > 0, h1, h2, h3, h4, diagonal matrices e > 0, s > 0 and four positive scalars ρ > 0, ε1 > 0, ε2 > 0, ε3 > 0 such that the following inequalities hold: p < ρi (20) itk p11ik + 2itk p12ik + itk p22ik − p11 − 2p12 − p22 < 0 (21) advances in systems science and applications (2011), vol. 11, no. 1-2 101 ω1 = θ11 θ12 θ13 θ14 −τatp t12 + τp t22 −τh1 p11m 0 −τatr22 0 ∗ ξ22 ξ23 −ht 4 −τp t22 −τh2 0 0 0 0 ∗ ∗ θ33 0 τbtp12 −τh3 0 0 −τbtr22 0 ∗ ∗ ∗ θ44 τctp12 −τh4 0 0 −τctr22 0 ∗ ∗ ∗ ∗ −τr11 −τr12 0 τr12m 0 0 ∗ ∗ ∗ ∗ ∗ −τr22 p22m 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ −ε1i 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ −ε2i 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −τr22 τr22m ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −ε3i  < 0 (22) where θ11 = −p11a−atp t11 + p12 + p t12 + ρd0 +q1 + lel+ τr11 +h1 +ht 1 − τr12a + ε1n t 1 n1 + τε2n t 1 n1 + τε3n t 1 n1 θ13 = p11b +ht 3 + τr12b − ε1nt 1 n2 − τε2nt 1 n2 − τε3nt 1 n2, θ14 = p11c +ht 4 + τr12c − ε1nt 1 n3 − τε2nt 1 n3 − τε3nt 1 n3, θ33 = −q2 − 2s + ε1n t 2 n2 + τε2n t 2 n2 + τε3n t 2 n2, θ44 = −e + ε1n t 3 n3 + τε2n t 3 n3 + τε3n t 3 n3 and ξ22,ξ23, l are stated as in theorem 3.1. then the origin of system (2) is the unique equilibrium point and it is globally asymptotically stable. proof. in order to prove the robust asymptotic stability, we use the same lyapunov-krasovskii functional as defined in (9). by replacing a,b and c in (2) with a+ ∆a(t), b + ∆b(t) and c + ∆c(t), respectively and by ito’s differential formula, we can calculate the trajectories of 102 raja: delay-dependent stability criteria of stochastic uncertain hopfield . . . . . . the system (19), then we have ω1 = ω + ε−11  p11m 0 0 0 0 p22m  × [ mtp11 0 0 0 0 mtp22 ] + ε1  −n1 0 n2 n3 0 0  × [ −nt 1 0 nt 2 nt 3 0 0 ] +τε−12  0 0 0 0 r12m 0  × [ 0 0 0 0 mtr12 0 ] + τε2  −n1 0 n2 n3 0 0  × [ −nt 1 0 nt 2 nt 3 0 0 ] +τε−13  0 0 0 0 0 r22m  × [ 0 0 0 0 0 mtr22 ] + τε3  −n1 0 n2 n3 0 0  × [ −nt 1 0 nt 2 nt 3 0 0 ] therefore, under condition (20)-(22), system (19) is robustly globally asymptotically stable with respect to the uncertain parameters ∆a(t), ∆b(t) and ∆c(t). this completes the proof of the theorem. � advances in systems science and applications (2011), vol. 11, no. 1-2 103 if we neglect the impulsive term and stochastic perturbations in (2), then it reduces to ẏ(t) = −ay(t) +bg(y(t− τ)) + c ∫ t −∞ k(t− s)g(y(s))ds (23) corollary 3.3 suppose that the assumption (c1)− (c3) is satisfied. if there exist matrices p = [ p11 p12 p t12 p22 ] ≥ 0 with l11 > 0, r = [ r11 r12 rt12 r22 ] ≥ 0 q1 > 0, q2 > 0, h1, h2, h3, h4 and diagonal matrices e > 0, s > 0 such that the following inequalities hold: ω =  ξ11 ξ12 ξ13 ξ14 −τatp t12 + τp t22 −τh1 −τatr22 ∗ ξ22 ξ23 −ht 4 −τp t22 −τh2 0 ∗ ∗ ξ33 0 τbtp12 −τh3 −τbtr22 ∗ ∗ ∗ −e τctp12 −τh4 −τctr22 ∗ ∗ ∗ ∗ −τr11 −τr12 0 ∗ ∗ ∗ ∗ ∗ −τr22 0 ∗ ∗ ∗ ∗ ∗ ∗ −τr22  < 0 (24) where ξ11 = −p11a−atp t11 + p12 + p t12 +q1 + lel+ τr11 +h1 +ht 1 − τr12a ξ12 = p12 −h1 +ht 2 , ξ13 = p11b +ht 3 + τr12b ξ14 = p11c +ht 4 + τr12c, ξ22 = −q1 −h2 −ht 2 , ξ23 = ls −ht 3 , ξ33 = −q2 − 2s, l = diag{l1, l2, ..., ln}. then the origin of system (2) is the unique equilibrium point and it is globally asymptotically stable. proof. by arguing similar to the proof of theorem 1, we can show that the equilibrium point of system (2) is globally asymptotically stable in the mean square. this completes the proof of the theorem. � remark 3.4 the authors chen and cao, [2] discussed the global asymptotic stability of delayed hopfield neural networks. wan et al. investigated the mean square exponential stability of stochastic delayed hopfield neural networks. in [18], wang et al. proposed the robust stability for stochastic hopfield neural networks with time delays and zhang et al., [24, 26] obtained the global stability results for delayed hopfield neural network. as a result, in all the above mentioned references, impulsive effect has not been taken into account. however, in our paper, we 104 raja: delay-dependent stability criteria of stochastic uncertain hopfield . . . . . . derived the delay-dependent stability results for stochastic hopfield neural networks with impulsive effects. therefore, our main result is new, quite effective and leads to less conservative results when compared with some existing works [2, 8, 19, 20]. remark 3.5 it is noteworthy that in our paper, we employed several free weighting matrices and s-procedure to derive the delay-dependent stability criterion. the derived criterion is obtained in lmi forms whose feasibility can be readily checked by using the matlab lmi toolbox. different from the conventional stability criteria that depend on the m-matrix computation, no tuning of parameters will be needed when employing our lmi-based stability criteria. moreover, two numerical examples with simulation results will show the effectiveness of the stability conditions in this paper. 5. illustrated examples in this section, we provide two numerical examples to demonstrate the effectiveness of the main results presented in this paper. example 4.1 consider a delayed stochastic hopfield neural network (2) with parameters as: a = [ 1.7679 0 0 1.8860 ] , b = [ −0.2376 −0.4769 −0.6707 −0.7654 ] , c = [ −0.1052 −0.5069 −0.0257 −0.2808 ] , l = [ 0.5219 0 0 1.8993 ] , d0 = [ 0.33 0 0 0.25 ] , ik = i = [ 0.2 0 0 0.2 ] it can be checked that system (2) satisfies the assumptions (c1) − (c3). for the delay bound τ = 0.957, we have obtained the following feasible solutions to the lmis (4) (6) in theorem 1 p11 = [ 25.7039 −1.6632 −1.6632 39.0949 ] , p12 = [ 4.3286 −0.9483 −0.9483 3.8977 ] , p22 = [ 8.8576 −1.6157 −1.6157 7.3501 ] q1 = [ 18.9787 −6.6658 −6.6658 16.4166 ] , q2 = [ 29.9330 20.6811 20.6811 51.6907 ] , r11 = [ 21.3444 −6.6192 −6.6192 19.6772 ] r12 = [ 6.4801 −1.9881 −1.9881 4.5859 ] , r22 = [ 7.2495 −2.6422 −2.6422 3.5791 ] , h1 = [ −1.5463 0.9452 −1.2486 1.3717 ] h2 = [ 1.6552 0.8485 −0.6090 1.9241 ] , h3 = [ 1.8527 1.2408 2.4549 1.7661 ] , h4 = [ −1.0116 2.4402 −1.5775 1.8661 ] advances in systems science and applications (2011), vol. 11, no. 1-2 105 s = [ 18.3098 0 0 8.9915 ] , e = [ 22.2513 0 0 16.6959 ] , ρ = 1.8663× 103 in order to show the significant improvement of our results, we summerize the comparisons between the previous works and the obtained result. for this example, the delay-dependent stability analysis in [3, 26, 27], cannot be satisfied for any τ > 0. table 1 shows the maximum upper bound of the previous works [8, 19, 20] as 0.4121, 1.7484 and 1.7644, respectively. however, by theorem 1, we have that the origin of delayed stochastic hopfield neural networks with impulsive effect is globally asymptotically stable for any constant allowable upper bound τ > 0. hence, it is clear that the proposed method shows the less conservativeness than the existing works [2, 8, 19, 20]. fig. 1 state trajectories of y1, y2 for example 1 example 4.2 consider a delayed stochastic uncertain hopfield neural network (2) with the following parameters: a = [ 0.7679 0 0 0.8860 ] , b = [ −0.1746 −0.8642 −0.2892 −0.7300 ] , c = [ −0.8252 −0.4912 −0.4732 −0.8858 ] , m = [ 0.07051 0 0 0.0342 ] , n1 = [ 0.3526 −0.1904 0.3322 −0.1564 ] , n2 = [ 0.2446 0.3674 −0.1753 0.2956 ] , 106 raja: delay-dependent stability criteria of stochastic uncertain hopfield . . . . . . table 1: maximum allowable bound of the delay method maximum upper bound of τ in ref [2, 26, 27] in ref [8] 0.4121 in ref [19] 1.7484 in ref [20] 1.7644 in this paper for any large finite τ > 0 n3 = [ 0.1981 −0.1313 0.1185 0.1645 ] , l = [ 0.07051 0 0 0.0342 ] , d0 = [ 0.85 0 0 0.85 ] , for τ = 0.957 and by solving the lmis (20)-(22) in theorem 3.2, we get the following feasible solution as follows: p11 = [ 133.1390 −39.2261 −39.2261 103.1396 ] , p12 = [ 5.2825 −5.7032 −5.7032 10.4098 ] , p22 = [ 18.1026 −10.6945 −10.6945 25.6053 ] q1 = [ 30.8319 −27.8887 −27.8887 47.6789 ] , q2 = [ 90.1163 28.7709 28.7709 147.2412 ] , r11 = [ 39.5332 −37.1370 −37.1370 66.5761 ] r12 = [ 21.2050 −14.1850 −14.1850 21.1791 ] , r22 = [ 31.5030 −17.8344 −17.8344 21.7477 ] , h1 = [ −13.2142 4.3068 −3.1365 −1.7307 ] h2 = [ 13.9993 −1.7353 −3.6838 8.4480 ] , h3 = [ −9.5150 11.0562 −15.8051 17.3187 ] , h4 = [ 12.1911 −1.3902 −7.3453 12.0990 ] s = [ 72.1101 0 0 263.5610 ] , e = [ 491.5915 0 0 339.6219 ] , ρ = 7.1793× 103 ε1 = 26.1631, ε2 = 15.5658, ε3 = 16.3759. thus, all the conditions in theorem 3.2 are satisfied. therefore, the model (19) with above given parameters is globally asymptotically stable in the mean square. 6. conclusion this paper has studied the problem of 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[27] q. zhang, x. wei and j. xu, delay-dependent exponential stability of cellular neural networks with time-varying delays, chaos, solitons and fractals, 23 (2005) 1363-1369. advances in systems science and applications (2011), vol. 11, no. 1-2 109 advances in systems science and applications (2017) vol.17 no.1 46 dynamic models of mob excitation control1 ivan n. barabanov a and dmitry a. novikov a,b a) v.a. trapeznikov institute of control sciences of russian academy of sciences, 117997, profsoyuznaya street, 65, moscow, russia; b) moscow institute of physics and technology, 141700, institutskiy per., 9, dolgoprudny, moscow region, russia e-mail: ivbar@ipu.ru, novikov@ipu.ru abstract. this paper formulates and solves the mob excitation control problem in the continuous-time setting by introducing an appropriate number of “provokers” at each moment of control. key words: collective behavior, granovetter’s model, stochastic models of mob control. 1. introduction in the case of conformity decision-making of agents that perform binary choice between “passivity” and “activity,” the associated control problems involve the models of collective behavior based on classical granovetter’s model [1] (see the surveys in [2, 3]). within this model, each agent is characterized by his threshold, a number from the interval [0; 1], in the following way. he decides to be active if the fraction of active agents in his neighborhood exceeds the threshold; otherwise, the agent prefers passivity. the dynamics of the fraction of active agents depends on its initial value and the distribution function of the agents’ thresholds. hence, a goal-directed (exogenous) change in the number of agents at the initial and/or subsequent moments affects the behavioral dynamics of the whole group. consider the collective behavior of agents forming a mob [4]. in this case, control consists in choosing a fraction of always active agents to-be-introduced to the mob (the so-called “provokers”). the problems belonging to this class can be classified by several bases such as discrete-time setting or continuous-time setting, single or multiple application of the control actions (constant controls and time-varying controls, respectively), and open-loop or feedback control. the discrete-time optimal choice problem for a single constant control applied by a control subject (principal) to a mob was formulated and solved in [4]. in [5], this problem was generalized to the case of multiple open-loop controls in the discrete-time setting. the present paper is dedicated to the continuous-time mob control models. further exposition follows the paper [6], in particular, the proofs of propositions 2-6 can be taken there. the mob model proper is imported from [7, 8], actually representing a generalization of granovetter’s model to the continuous-time case as follows. suppose that we know the initial fraction x0  [0; 1] of active agents at the zero moment. then the evolution of this fraction x(t) in the continuous time t ≥ 0 is governed by the equation ( )x f x x  , (1) where f(∙) is a known continuous function possessing the properties of a distribution function, f(0) = 0 and f(1) = 1. actually, this is the distribution function of the agents’ thresholds [2, 4]. similar to [4, 5], by applying a control action u(t)  [0; 1] (introducing provokers), we obtain the controlled dynamic system ( ) (1 ( )) ( )x u t u t f x x    . (2) this paper is organized in the following way. section 2 studies the reachability set and the monotonicity of the system trajectories in the control action. next, section 3 is focused on the case of constant controls according to the above classification. section 4 considers the models 1 this work was partially supported by rsf grant 16-19-10609. 47 i.n. barabanov and d.a. novikov: dynamic models of mob excitation control where control excites the whole mob. and finally, section 5 deals with the case of feedback control. 2. reachability set and monotonicity first, we formulate a lemma required for further analysis. consider functions 1( , )g x t and 2 ( , )g x t : 0[ , )r t r   that are continuously differentiable with respect to x and continuous in t . by assumption, the functions 1g and 2g are such that the solutions to the cauchy problems for the differential equations ( , )ix g x t , 1,2i  , with initial conditions 0 0 0( , ),t x x r , admit infinite extension in t . denote by 0 0( ,( , )), 1,2ix t t x i  , the solutions of the corresponding cauchy problems. lemma [6]. let 0 1 2, ( , ) ( , )x r t t g x t g x t      . then 0 1 0 0( ,( , ))t t x t t x   2 0 0( , ( , ))x t t x . note that, for validity of this lemma, one should not consider the inequality 1 2( , ) ( , )g x t g x t for all x r . it suffices to take the union of the reachability sets of the equations ( , )ix g x t , 1,2i  , with the chosen initial conditions 0 0( , )t x . denote by xt(u) the fraction of active agents at the moment t under the control action u(∙). the right-hand side of the expression (1) increases monotonically in u for each t and ∀ x ∈ [0; 1]: f(x) ≤ 1. hence, we have the following result. proposition 1. let the function f(x) be such that f(x) < 1 for x ∈ [0; 1). if 0 1 2( ) ( )t t u t u t    and x0(u1) = x0(u2) ( 0 1x  ), then 0t t  : xt(u1) > xt(u2). indeed, by the premises, for all t and x<1 we have the inequality 1 1 2 2( ) (1 ( )) ( ) ( ) (1 ( )) ( )u t u t f x x u t u t f x x       , as the convex combination of different numbers (1 and f(x)) is strictly monotonic. the point x=1 forms the equilibrium of the system (1) under any control actions u(t). and so, it is unreachable for any finite t. using the above lemma, we find that xt(u1) > xt(u2) under same initial conditions. suppose that the control actions are subjected to the constraint   0    ,     ,u t t t  (3) where δ ∈ [0; 1] means some constant. we believe that t0=0, x(t0)=x(0)=0, i.e., initially the mob is in the nonexcited state. if the efficiency criterion is defined as the fraction of active agents at a given moment t > 0, then the corresponding terminal control problem takes the form ( ) ( ) max, (2), (3). t u x u     (4) here is a series of results (propositions 2-4) representing the analogs of the corresponding propositions from [5]. proposition 2. the solution of the problem (4) is given by u(t) = δ, t ∈ [0; t]. denote by ˆ ˆ( , ) min { 0 | ( ) }tx u t x u x    the first moment when the fraction of active agents achieves a required value x̂ (if the set ˆ{ 0 | ( ) }tt x u x  is empty, just specify ˆ( , )x u = +∞). within the current model, one can pose the following time-optimal problem: ( ) ˆ( , ) min, (2), (3). u x u     (5) proposition 3. the solution of the problem (5) is given by u(t) = δ, t ∈ [0; τ]. by analogy with the discrete-time models [5], the problem (4) or (5) has the following practical interpretation. the principal benefits most from introducing the maximum admissible number of provokers in the mob at the initial moment, doing nothing after that (e.g., instead of first decreasing and then again increasing the number of introduced provokers). this structure of advances in systems science and applications (2017) vol.17 no.1 48 the optimal solution can be easily explained, as in the models (4) and (5) the principal incurs no costs to introduce and/or keep the provokers. what are the properties of the reachability set d = ( ) [0; ] ( )t u t x u   ? clearly, [0;1]d , since the right-hand side of the dynamic system (2) vanishes for x = 1. in the sense of potential applications, a major interest is attracted by the case of constant controls (u(t) = v, t ≥ 0). here the principal chooses the same fraction v  [0; ∆] of provokers at all moments. let xt(∆) = xt(u(t) ≡ δ), t ∈ [0; t], and denote by d0 = [0; ] ( ) [0;1]t v x v    the reachability set under constant control actions. according to proposition 1, ( )tx v represents a monotonic continuous mapping of [0; ∆] into [0; 1] such that (0) 0tx  . this leads to the following. proposition 4. d0 = [0; xt(∆)]. consider models taking into account the principal’s control costs. given a fixed “price” λ ≥ 0 of one provoker per unit time, the principal’s costs over a period τ ≥ 0 are defined by   0 (      .)u t dtc u     (6) suppose that we know a pair of monotonic functions characterizing the principal’s terminal payoff h(∙) from the fraction of active agents and his current payoff h(∙). then the problem (4) can be “generalized” to 0 ( ( )) ( ( )) ( ) max, (2), (3). t t t u h x u h x t dt c u         (7) under existing constraints on the principal’s “total” costs c, the problem (7) acquires the form 0 ( ( )) ( ( )) max, (2), ( ) . t t u t h x u h x t dt c u c        (8) a possible modification of the problems (4), (5), (7), (8) is the one where the principal achieves a required fraction x̂ of active agents by the moment t (the cost minimization problem): ( ) min, ˆ( ) , (2). t u t c u x u x      (9) the problems of the form (7)-(9) can be easily reduced to standard optimal control problems. example 1. consider the problem (9), where ( )f x x , x0 = 0 and the principal’s costs defined by (6) with 0 1  . this yields the following optimal open-loop control problem with fixed bounds: [0, ] 0 (1 ), ˆ(0) 0, ( ) , 0 , ( ) min . t u x u x x x t x u u t dt           (10) 49 i.n. barabanov and d.a. novikov: dynamic models of mob excitation control for the problem (10), construct the hamilton-pontryagin function  (1 )h u x u   . by the maximum principle, this function takes the maximum values in u . as h is linear in u , its maximum is achieved at an end of the interval [0, ] depending on the sign of the factor at u , i.e.,    sign 1 1 1 . 2 u x      (11) the fact that the hamilton-pontryagin function is linear in control actions actually follows from the same property of the right-hand side of the dynamic system (2) and the functional (6). in other words, we have the result below. proposition 5. if the constraints in the optimal control problems (7)-(9) are linear in control actions, then the optimal open-loop control possesses the structure described by (11). that is, at each moment the control action takes either the maximum or the minimum admissible value. the hamilton equations acquire the form (1 ), . h x u x h u x              the boundary conditions are imposed on the first equation only. for 0u  , its solution is a constant; for u   , the solution is    0 0( ) 1 1 ( ) . t t x t x t e      the last expression restricts the maximum number of provokers required for mob transfer from the zero state to x̂ : 1 1 log . ˆ1t x    and there exists the minimum time min 1 1 log , ˆ1 t x    during which control actions take the maximum value  , being 0 at the rest moment. particularly, a solution of the problem (10) has the form min min , , 0, t t u t t t       (12) when the principal introduces the maximum number of provokers from the very beginning, maintaining it during the time mint . the structure of the optimal solution to this problem (a piecewise constant function taking the value of 0 or  ) possibly requires minimizing the number of control switchovers (discontinuity points). such an additional constraint reflects situations when the principal incurs extra costs to introduce or withdraw provokers. if this constraint appears in the problem, the best control actions in the optimal control set are either (12) or min min , [ , ], 0, . t t t t u t t t        3. constant control in the class of the constant control actions, we obtain cτ(v) = λ v τ from formula (6). under given functions f(∙), i.e., a known relationship xt(v), the problems (7)-(9) are reduced to standard scalar optimization problems. example 2. choose f(x) = x, t = 1, x0 = 0, h(x) = x, and h(x) = γ x, where γ ≥ 0 is a known constant. it follows from (2) that   0   1   – ex ( .)p  t t u yx dyu         (13) for the constant control actions, xt(v) = 1 – vte . advances in systems science and applications (2017) vol.17 no.1 50 the problem (7) becomes the scalar optimization problem [0; ] 1 maxv v e v v v               . (14) next, the problem (8) becomes the scalar optimization problem [0; ] 1 maxv v e v v             . (15) and finally, the problem (9) acquires the form [0;1] min, ˆ1 . v v v e x       its solution is described by v = 1 log ˆ1 x       . 4. excitation of whole mob consider the “asymptote” of the problems as t → +∞. similarly to the corresponding model in [5], suppose that (a) the function f(∙) has a unique inflection point and f(0) = 0, (b) the equation f(x) = x has a unique solution q > 0 on the interval (0; 1) so that (0; ) ( ) , ( ;1) ( )x q f x x x q f x x      . several examples of the functions f(∙) satisfying these assumptions are provided in [5]. the principal seeks to excite all agents with the minimum costs. by the above assumptions on f(∙), if for some moment τ we have x(τ) > q, then the trajectory xt(u) is nonincreasing and lim ( ) 1t t x u   even under u(t) ≡ 0 t   . as mentioned in [5], this property of the mob admits the following interpretation. the domain of attraction of the zero equilibrium without control (without introduced provokers) is the half-interval [0; q). in other words, it takes the principal only to excite more than the fraction q of the agents; subsequently, the mob itself surely converges to the unit equilibrium even without control. denote by u τ the solution of the problem : ( ) [0; ], ( ) 0 ( ) min u u t x u q u t dt       (16) calculate qτ = 0 ( )u t dt    and find τ * = arg 0 min   qτ. the solution to the problem (16) exists under the condition [0, ] * ( ) max 1 ( )   x q x f x f x      (17) for practical interpretations, we refer to [5]. owing to the above assumptions on the properties of the distribution function, the optimal solution to the problem is characterized as follows. proposition 6. if the condition (17) holds, then u τ (t) ≡ 0 for t > τ. example 3. the paper [9] constructed the two-parameter function f(∙) describing in the best way the evolvement of active users in the russian-language segments of online social networks livejournal, facebook and twitter. the role of the parameters is player by a и b. this function has the form      , arctan( ( )) arctan( ) , arctan( (1 )) arctan( )   7;1  5 ,   0;1  .a b a x b ab f x a b a b ab        choose a = 13 that corresponds to facebook and b = 0.4. in this case, q ≈ 0.375 and * ≈ 0.169; the details can be found in [5]. 51 i.n. barabanov and d.a. novikov: dynamic models of mob excitation control 5. feedback control in the previous sections, we have considered the optimal open-loop control problem arising in mob excitation. an alternative approach is to use feedback control. consider two possible statements having transparent practical interpretations. within the first statement, the problem is to find a feedback control law ( ) :[0;1] [0;1]u x  ensuring maximum mob excitation (in the sense of (4) or (5)) under certain constraints imposed on the system trajectory and/or control actions. by analogy with the expression (3), suppose that the control actions are bounded:  ( ) , 0;1 ,xu x   (19) and there exists an additional constraint on the system trajectory in the form ( ) ,    0,tx t   (20) where δ > 0 is a known constant. the condition (20) means that, e.g., a very fast growth of the fraction of excited agents (increment per unit time) is detected by appropriate authorities banning further control. hence, trying to control mob excitation, the principal has to maximize the fraction of excited agents subject to the conditions (19) and (20). the corresponding problem possesses the simple solution * ( ) ( ) min ; max 0; 1 ( ) x f x u x f x            , (21) owing to the properties of the dynamic system (2), see the lemma. the fraction in (21) results from making the right-hand side of (1) equal to the constant δ. note that, under small values of δ, the nonnegative control action satisfying (20) may cease to exist. the second statement of feedback control relates to the so-called network immunization problem [4]; here the principal seeks to reduce the fraction of active agents by introducing an appropriate number (or fraction) of immunizers–agents that always prefer passivity. denote by w ∈ [0; 1] the fraction of immunizers. as shown in the paper [4], the fraction of active agents evolves according to the equation  (1 ) ( ) ;1  ., 0x w f x x x   (22) let ( ) :[0;1] [0;1]w x  be a feedback control. if the principal is interested in reducing the fraction of active agents, i.e., ( ) 0, 0,x t t  (23) then the control actions must satisfy the inequality ( ) 1 ( ) x w x f x   . (24) the quantity min [0;1] max 1 ( )x x f x         characterizes the bottom restrictions on the control actions at each moment when the system (22) is “controllable” in the sense of (23). 6. conclusion this paper has described the continuous-time problems of mob excitation control using introduction of provokers or immunizers. a promising line of future research is analysis of a differential game describing informational opposition of two control subjects (principals) that choose in continuous time the fractions (or numbers) of introduced provokers u and immunizers w, respectively. the corresponding static problem [7] can be a “reference model” here. the controlled object is defined by the dynamic system [4, 10] (1 ) (1 2 ) ( )x u w u w uw f x x       . another line of interesting investigations concerns the mob excitation problems with dynamic (open-loop and/or feedback) control, where the mob dynamics is modeled by the transfer advances in systems science and applications (2017) vol.17 no.1 52 equation [7] of the form       , (1 ) , 0p x t u u f x x p x t t x           . in this model, the mob state at each moment is described by a probability distribution function p(x, t), instead of the scalar fraction of active agents. references [1] granovetter, m., “threshold models of collective behavior”, ajs, vol. 83, no 6, pp. 1420-1443, 1978. [2] breer, v. v., “models of conformity behavior: a survey”, control sciences, vol. 1, pp. 2-13, 2014 [in russian]. [3] slovokhotov, yu. l., “physics and sociophysics. part 1”, control sciences, vol. 1, pp. 220, 2012 [in russian]. [4] breer, v. v., novikov, d. a., and rogatkin, a. d., “stochastic models of mob control”, automation and remote control, vol. 77, no. 5, pp. 895-913, 2016 [5] barabanov, i. n., and novikov, d. a., “dynamic models of mob excitation control in discrete time”, automation and remote control, vol. 77, no. 10, pp. 1792-1804, 2016. [6] barabanov, i. n., and novikov, d. a., “dynamic models of mob excitation control in continuous time”, upravlenie bol'simi sistemami (large scale systems control), vol. 63, pp. 71-68, 2016 [in russian]. [7] rogatkin, a. d., “continuous-time granovetter’s model”, upravlenie bol'simi sistemami (large scale systems control), vol. 60, pp. 139-160, 2016 [in russian]. [8] akhmetzhanov, a.r., worden, l., and dushoff, j., “effects of mixing in threshold models of social behavior”, phys. rev. e, vol. 88, no. 1, 2013, article number 012816. [9] batov, a. v., breer, v. v., novikov, d. a., and rogatkin, a. d., “microand macromodels of social networks. ii. identification and simulation experiments”, automation and remote control, vol. 77, no. 2, pp. 321-331, 2016. [10] novikov, d. a., “models of informational confrontation in mob control”, automation and remote control, vol.77, no.7, pp. 1259-1274, 2016. microsoft word 1 d.h. chen, k. ushijima--estimation of maximum compressive load for circular tubes under axial impact.doc advances in systems science and applications (2011), vol.11, no.3-4 203-213 issn 1078-6236 international institute for general systems studies, inc. estimation of maximum compressive load for circular tubes under axial impact d.h. chen1 and k. ushijima2 1department of mechanical engineering, faculty of engineering, tokyo university of science, tokyo, 1628601, japan 2department of mechanical engineering, faculty of engineering, kyushu sangyo university, fukuoka, 8138503,japan abstract the influence of impact velocity on the crushing behaviour of cylindrical shells subjected to an axial impact was investigated using a finite element analysis. the effects of the material properties, tube geometries and impact velocity v0 on the initial peak stressσ1 are explored. in this study, the applied material is assumed to be insensitive to the strain rate, and the effect of impact velocity is discussed as an inertia effect. it is shown that the initial peak stressσ1 during dynamic loading increases with increase of the impact velocity v0, which is due to the fact that the displacement in radial direction is delayed as the velocity v0 increases. also, based on our numerical simulations, the peak stressσ1 can be regarded as a function of the ratio of tube thickness to radius t/r, hardening modulus to young's modulus eh/e and impact velocity to elastic stress wave speed v0/c. moreover, an approximate equation to evaluate the peak stress is proposed and in good agreement with the fem results and other researcher's results under a relatively low impact velocity (v0<40m/s). keywords elastic-plastic cylindrical shells, axial impact, energy absorption, fem 1.introduction thin-walled structures such as circular and square tubes have been widely used in automotive and aerospace engineering as impact energy absorbing devices. a large number of studies concerning the static and dynamic response of thin-walled tubes subjected to axial load have been conducted by many researchers, and investigated some important parameters such as crushing distance, the peak load and the buckling shape in crashworthiness design [1-13]. figure 1 shows a schematic of compressive axial stressσx and displacement ux for a cylindrical tube subjected to axial loading. the purpose of this study is to explore the effect of impact velocity on the peak load for circular tubes, and to propose an empirical equation to estimate the peak load by numerical simulation. 2.method of analysis in this study, the dynamic numerical simulation of the impact crush test was carried out using the non-linear fe commercial code, msc.dytran. the geometry of the fe model and its boundary condition is shown in fig.2. the model is struck from the upper edge by a rigid mass m having an initial kinetic energy t0=mv0 2/2=99 kj. in the fe model, 4-node key-hoff shell elements(quad4, pshell) with three integration points are used to evaluate domain integrals, and the whole model is divided into 1440 elements. here, parameters l, t and r are the tube length, thickness and mean radius, respectively. also, the lower end of a tube is fixed to another rigid body, and a contact condition between the tube and the striker, and a self-contact condition at the inner and the outer surface of the tube are defined with the dynamic and static frictional coefficients of 0.2 and 0.3, respectively. 204 chen: prediction of maximum moment of rectangular tubes subjected to pure bending σ1 axial displacement , ux ax ia l s tre ss , σ x fig.1. schematic of axial compressive stress-displacement relationship for a tube subjected to axial impact fig.2. shell geometry and loading condition the analyzed fe model with a densityρ=2685 kg/m3 is assumed to be isotropic, and to obey the mises yield criterion with strain hardening, and a strain rate insensitive bilinear relationship between the uniaxial stress and strain as: ( ) ( ) ( )⎩ ⎨ ⎧ >−+ ≤ = eee ee yyhy y σεσεσ σεε σ (1) here, the young's modulus, e, yielding stress, σy and hardening coefficient, eh are assumed to be 72.4 gpa, 72.4 mpa and 3.62 gpa, respectively. all models in our calculation have the same tube length l=150 mm, mean radius r=25 mm and thickness t=1 mm, unless otherwise mentioned. 3.results and discussion 3.1 effect of impact velocity v0 on the initial peak stressσ1 0 0.01 0.02 0.03 0 50 100 :v0=5 (km/h) ax ia l f or ce (k n ) axial displacement (m) b a c d :v0=180 (km/h) :v0=360 (km/h) fig.3. comparison of axial force and displacement behaviour for a tube under different impact velocity v0 figure 3 shows comparisons of axial compressive force and displacement diagram for a tube under the impact velocity v0=5, 180 and 360 km/h. advances in systems science and applications (2011), vol.11, no.3-4 205 issn 1078-6236 international institute for general systems studies, inc. also, the deformed shapes at the initiation of the initial peak stress and soon after the stress for the case of v0=5 km/h and 360 km/h are shown in fig.4. it is evident from fig.4 that the initial peak stress is associated with the initiation of local buckling deformation which occurs near the upper and lower ends of a tube. from the relationship between the axial load and displacement for each v0 which is shown in fig.3, the initial peak stress can be calculated and summarized in fig.5. also in fig.5, quasi-static buckling stress for the tube obtained by implicit finite element code, msc.marc, is shown by a dashed line. it is found from this figure that the initial peak stressσ1 for a lower impact velocity is almost equal to the value of the quasi-static result, and the stress value becomes higher as the impact velocity v0 increases. fig.4. comparisons of deformed shape at points 'a', 'b', 'c' and 'd' in fig.3. 'a': buckling point for v0=5(km/h); 'b': point just after buckling for v0=5(km/h); 'c': buckling point for v0=360(km/h); 'd': point just after buckling for v0=360 (km/h) 0 100 200 300 400 0 250 500 in iti al p ea k s tre ss σ 1 (m p a) impact velocity v0 (km/h) quasi−static buckling stress fig.5. variation of peak stress σ1 with impact velocity v0 0 0.08 0.16 0 200 400 a xi al s tre ss σ x (m pa ) position x (m) impacted endv0=5 (km/h) t=6.2 (μsec) 4.16 3.16 2.16 1.16 0.16 0.08 0.05 0.03 0.02 0.01 :yield stress :quasi−static buckling stress (a) v0=5(km/h) figure 6 illustrates propagations of the stress wave in the axial direction for v0=5 km/h (fig.6(a)) and 360 km/h (fig.6(b)) until the initial peak stress occurs. also in figure 6, values of the yield stressσy and the quasi-static buckling stress are shown by dotted and dashed lines, respectively. it is evident from fig.6(a) that for the case of a lower impact velocity(v0=5 km/h), 206 chen: prediction of maximum moment of rectangular tubes subjected to pure bending the amplitude of the stress wave in axial direction at the initiation of impact is relatively small, and the stress wave travels and reflects many times along the tube until the local buckling deformation arises near the fixed end. in the end, the axial stress distribution developing over the tube becomes almost uniformly at t=0.16μs, and its amplitude increases stably to the quasi-static buckling stress. on the other hand, for the case of a higher impact velocity (fig.6(b)), the development of axial stress distribution apparently differs from that for a lower impact velocity(fig.6(a)). 0 0.08 0.16 0 400 800 a xi al s tre ss σ x (m p a) position x (m) impacted endv0=360 (km/h) t=0.14 (μsec) 0.12 0.1 0.03 0.02 0.01 :yield stress :quasi−static buckling stress (b) v0=360(km/h) fig.6. axial stress distribution until the initiation of the peak stressσ1 in fig.6(b), the stress concentration can be found at x=0.156 m. such a stress concentration occurs by the existence of the flange at the impacted end, and does not affect the overall buckling behaviour of a circular tube. it is evident from fig.6(b) that the amplitude of the stress wave bigger than the value of quasi-static buckling stress develops near the impacted end at the beginning. however, such the high stress field is relatively narrow, and no buckling behaviours seem to be observed even if the amplitude of stress wave is bigger than that of the quasi-static result. moreover, even though the axial stress developing over the tube becomes larger than the quasi-static result at t=0.12μs, no buckling can be observed. finally, the local buckling can be observed at t=0.14μs when the amplitude of the axial stress is almost twice larger than the quasi-static buckling stress. the mechanism of the buckling for circular tubes under higher impact velocities will be discussed as follows. fig.7. comparisons of deformed shape near the fixed end at the initiation of the peak stressσ1 based on the previous work concerning the quasi-static axial compressive behaviour for circular tube, it is evident that the initial peak stress is associated with the sufficient local bending deformation which is observed near the tube end. that is, during the axial compression, the tube seems to expand in radial direction, but near the tube end, such a movement is restricted by the existence of the fixed boundary condition. as a result, the local bending deformation can be observed near the tube end, and the local bending deformation is necessary for initiating the buckling behaviour. figure 7 shows comparisons of deformed shape near the fixed end when the initial peak stress is observed for some cases of impact velocity v0=5 km/h, 180 km/h and 360 advances in systems science and applications (2011), vol.11, no.3-4 207 issn 1078-6236 international institute for general systems studies, inc. km/h. it is evident that for the case of a higher impact velocity, the amount of the local bending deformation is smaller than that under a slower impact velocity. in order to initiate the buckling behaviour where the smaller amount of the local bending deformation occurs under higher impact velocity, that is, in order to increase the radial displacement, a large amount of axial stress should be needed. therefore, it could be understood that the reason why the initial peak stress increases as increase of the impact velocity v0 is due to the fact that the impact for a higher impact velocity causes a smaller radial displacement than that for a lower impact velocity. moreover, the reason why the bending deformation decreases as v0 increases can be explained by the radial inertia effect. that is, the more faster the impact velocity becomes, the more rapidly the axial stress increases, but the expansion in radial direction would be delayed by the radial inertia effect. such a mechanism can be observed in fig.8. 0 200 400 0 0.8 1.6 0 400 800 ra di al d is pl ac em en t u r ( m m ) impact velocity v0 (km/h) :ur :σx a xi al s tre ss σ x ( m p a) ur σx fig.8.variation of radial displacement ur and axial stress σx at the apex of wrinkle with impact velocity v0 0 1 2 0 200 400 600 a xi al s tre ss σ x ( m p a) radial displacement ur (mm) :peak stress :σx−ur curve v0=300 (km/h) v0=360 (km/h) v0=180 (km/h) v0=5 (km/h) fig.9. variation of axial stressσx with radial displacement ur at the apex of wrinkle for v0=5, 180, 300 and 360 (km/h) figure 8 presents the relationship between the radial displacement ur and the impact velocity v0 under almost the same amount of the axial stressσx. it is evident that the displacement ur decreases as increase of v0 even if the same amount ofσx occurs. moreover, the relationship between the axial stressσx and the radial displacement ur for four cases of v0= 5, 180, 300 and 360 km/h are chased and the initial peak stress for every v0 is summarized in fig.9 by marks ●. from the figure, the reason why the higher velocity causes the larger initial peak stress can be explained as follows. while the axial stress increases by propagating the stress waves over the tube, the radial displacement ur develops by the axial compression, but the rate of ur relatively decreases as increase of v0. as a result, the axial stress increases by the stress wave reflecting many times until the sufficient radial displacement can be reached for initiating the local buckling. in order to discuss the influence of v0 on the peak stress quantitatively, an effective parameter considering the effect of mechanical properties on the inertia effect is needed. figure 10 shows the relationship between the normalized axial stress σx/e and displacement ux/l for 208 chen: prediction of maximum moment of rectangular tubes subjected to pure bending three cases of a tube problem having different elastic modulus e, tube density ρ and the impact velocity v0. in the figure, parameter c represents the elastic wave speed for a one-dimensional rod, and can be shown as (e/ρ)1/2. here, all models have different values of e, ρ and v0, but keep the same ratio v0/c. it is evident that all models behave the same response of the axial stressσx/e and ux/l diagram. strictly speaking, the elastic and the plastic wave speed for a circular tube are distinct from the elastic wave speed c for a one-dimensional rod. however, based on the fact that the relationship between the normalized axial stressσx/e and displacement ux/l is the same under the same v0/c, the non-dimensional parameter v0/c can be used for evaluating the effect of impact velocity on the initial peak stressσ1. fig.10. normalized axial compressive stress and displacement behaviour for tubes having the same ratio of v0/c 3.2 effects of tube geometries and material properties on the initial peak stress as the other effective parameters for the dynamic initial peak stressσ1, tube geometries (such as mean radius, r and thickness, t) and material properties (such as young's modulus, e, hardening coefficient, eh and yield stress,σy) can be given, and the effects of these parameters onσ1 are discussed as follows. 0 0.1 0.2 0 100 200 300 σ x (m pa ) ux / l :t=0.5(mm) , r=12.5(mm) v0=40 (km/h) l=160(mm) :t=1.0(mm) , r=25.0(mm) :t=1.5(mm) , r=37.5(mm) fig.11. comparison of the compressive stress and displacement behaviour for tubes having the same ratio of tube thickness to radius t/r figure 11 summarizes effects of mean radius r and thickness t on the axial stress and displacement behaviour for tubes having different tube radius r and thickness t, but the same ratio of t/r. it is evident from fig.11 that if the ratio of t/r has the same value, the initial peak stressσ1 has the same value. advances in systems science and applications (2011), vol.11, no.3-4 209 issn 1078-6236 international institute for general systems studies, inc. 0 0.1 0.2 0 0.002 0.004 σ x / e ux / l e ρ v0=40 (km/h) σy=e/1000 e0 2e0 ρ0 2ρ0 c e0=72.4 (gpa) ρ0=2685 (kg/m3) eh e0/20 e0/10 e0 /ρ0 fig.12. normalized axial compressive stressσx/e and displacement ux/l behaviour for tubes having the same ratios of eh/e and v0/c figure 12 shows the comparison of the relationship between two tubes having the different e and eh but the same ratio of eh/e. in the figure, the stress value is normalized by e. it can be observed from fig.12 that the normalized initial peak stressσ1/e can be arranged as a function of the ratio, eh/e. moreover, the yield stressσy for a tube also affects the initial peak stress, but if the valueσy is quite smaller than the peak stressσ1, the effect seems to be negligible. from the above results and discussions, the normalized initial peak stressσ1/e for a tube obeying a bilinear stress and strain relationship can be expressed by a function composed of four parameters, v0/c, t/r, eh/e andσy /e as: ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = ee e r t c vf e yh σσ ,,,0 1 1 (2) figure 13 shows the relationship between the normalized impact velocity v0/c and the peak stressσ1/e for some combinations of ratios, t/r, eh/e andσ1/e. it is clear from the figure that the stressσ1/e increases linearly with the parameter (v0/c)2. also, the slope and intersect of the relationship depend on these ratios, eh/e and t/r. 0 20 40 0 0.004 0.008 0.012 σ 1 / e (v0 / c)2 10−5 t/r σy=e/1000 σy=3e/1000 eh/e 0.01 0.05 0.04 0.08 0.01 0.05 fig.13. normalized impact velocity v0/c and the peak stressσ1/e for some combinations of t/r, eh/e andσy/e based on these charateristics, the normalized initial peak stressσ1/e can be written by the following type of equation as: 2 2 0 1 1 c c vc e +⎟ ⎠ ⎞ ⎜ ⎝ ⎛= σ (3) where, 210 chen: prediction of maximum moment of rectangular tubes subjected to pure bending .,,,,,,, 3221 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ =⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = ee e r tfc ee e r tfc yhyh σσ equation (3) means that the initial stress can be expressed by the sum of the inertial term including the parameter v0/c and the quasi-static term for v0=0. here, the quasi-static term c2 has already been discussed in the previous study[11], and proposed an approximate equation as follows: ( ) ( ) ( ) ( ) ( ) . 1 2 13 0 17.0 02 1 2 ⎪ ⎪ ⎩ ⎪⎪ ⎨ ⎧ >−⎟ ⎠ ⎞ ⎜ ⎝ ⎛⋅+ ≤ − = = − xrte e e r t e xrt r t e c k ee hy static hσ ν σ (4) where, ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = − = 0 2 0 22, 13 x r t e eek e x y hy σ νσ figure 14 shows comparisons of the parameter c2 obtained by fem and its approximation obtained by eq.(4). in the figure, results forσy/e=1/1000 and 3/1000 correspond to solid and dotted lines, respectively. it is clear that both results coincide with each other, and the approximate equation which is shown as eq.(4) can be used to estimate the quasi-static peak stress with a good accuracy. 0 0.05 0.1 0 0.005 0.01 0.015 c 2 t / r σy/e=0.001 σy/e=0.003 eh/e 0.01 0.05 0.1 line :equation (4) :σy/e=0.001 :σy/e=0.003 fig.14. comparisons of quasi-static term c2 in eq.(3) for tubes having some combinations of t/r,σy/e and eh/e in figure 15, results of the parameter c1 for some ratios, eh/e,σy/e and t/r are summarized. it is found from the figure that the parameter c1 increases with increase of eh/e, and its value becomes larger as the ratio of tube thickness to radius t/r decreases. also, the value c1 depends on the yield stress σy. for example, c1 for t/r=0.02 and 0.08 under the same ratios, eh/e=0.1 andσy/e=0.001 are almost equal to 13.0 and 8.0, respectively, so that the difference between them is almost 5.0. however, by concerning the term (v0/c)2, for example, if the impact velocity v0 is 100km/h, the order of the parameter (v0/c)2 is about 10-5, and the scale of the difference is one digit smaller than the parameter c2. based on the fact, the parameter c1 can be written by the following equation as: 7.0 1 50 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛= e ec h (5) and shown in fig.15 as a solid line. advances in systems science and applications (2011), vol.11, no.3-4 211 issn 1078-6236 international institute for general systems studies, inc. 0 0.04 0.08 0.12 0 5 10 15 c 1 eh / e t/r 0.04 :equation (5) σy/e=0.001 0.08 σy/e=0.003 fig.15. comparisons of quasi-static term c1 in eq.(3) for tubes having some combinations of t/r,σy/e and eh/e consequently, an approximate equation for the non-dimensional initial peak stress for a cylindrical tube subjected to axial impact load is proposed in this paper as follows: static h ec v e e e 1 2 0 7.0 1 50 σσ +⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛= (6) 0 200 400 0 500 1000 σ 1 (m p a) v0 (km/h) :eh/e=0.01 :eh/e=0.05 :eh/e=0.1 t/r=0.04 :equation (6) (a) for the case of t/r=0.04 0 200 400 0 500 1000 1500 σ 1 (m pa ) v0 (km/h) :eh/e=0.01 :eh/e=0.05 :eh/e=0.1 t/r=0.08 :equation (6) (b) for the case of t/r=0.08 fig.16. estimation of the peak stress σ1 under some cases of impact velocity v0 figure 16 shows the relationship between the initial peak stress and the impact velocity for some cases of eh/e for the ratio of t/r=0.04 (fig.16(a)) and 0.08 (fig.16(b)). in these figures, solid lines correspond to the approximate results obtained by eq.(6), and the symbols show the numerical results obtained by fem. it is clear from these figures that the predictedσ1/e agrees well with the numerical results for a wide range of impact velocity v0. 4. validation of the proposed prediction for σ1 212 chen: prediction of maximum moment of rectangular tubes subjected to pure bending in order to check the validity of the proposed prediction forσ1 as shown in eq.(6), the same impact problem studied by karagiozova and jones[5] is examined and compared the prediction with their results in fig. 17. in the figure, the solid line shows the approximation by eq.(6), and dotted line and solid circles are the predictions and fem results which can be found in karagiozova and jones' paper[5]. here, the model is intended for a circular tube made of aluminium alloy, and the material and geometrical parameters of the model are shown in fig. 17. it is evident that for relatively low impact velocity (v0<40m/s), the proposed approximation gives a good agreement with numerical results obtained by karagiozova and jones[5], which means that the proposed approximation can be effectively used for estimating the initial peak stressσ1 under a relatively low impact velocity. 0 40 80 200 400 600 v0 (m/sec) σ 1 (m pa ) : fem : prediction karagiozova and jones(2001) : eq.(6) e =72.4(gpa), eh =542.6(mpa), σy=295(mpa) ρ =2685(kg/m3), t =1.65(mm), r =11.875(mm) fig.17. comparison of estimating the peak stressσ1 under some cases of impact velocity v0 5. conclusion in this paper, large displacement numerical simulation based on fem is undertaken to explore the relationship between the impact velocity v0 and the initial peak stressσ1 for circular tubes when subjected to an axial impact. based on our numerical results, the following points have been revealed. (1) the initial peak stressσ1 becomes higher with increases of the impact velocity v0. that is because the local displacement ur in radial direction decreases as v0 increases, under the same axial stressσx. (2) the initial peak stressσ1 can be expressed by the sum of a term including v0/c and a term obtained by quasi-static numerical simulation. (3) the proposed approximate equation forσ1 can be effectively used under low impact velocity (v0<40m/s). references [1] karagiozova d. and alves m., jones n., inertia effects in axisymmetrically deformed cylindrical shells under axial impact. int. j. impact engng. 24(10)(2000) 1083-1115. [2] karagiozova d. and alves m., transition from progressive buckling to global bending of circular shells under axial impact – part i: experimental and numerical observations. int. j. solids struct. 41(5-6)(2004) 1565-1580. [3] karagiozova d. and alves m., transition from progressive buckling to global bending of circular shells under axial impact – part ii: theoretical analysis. int. j. solids struct. 41(5-6)(2004) 1581-1604. [4] karagiozova d. and jones n., dynamic effects on buckling and energy absorption of cylindrical shells under axial impact. thin-walled struct. 39(7)(2001) 583-610. [5] karagiozova d. and jones, n., influence of stress waves on the dynamic progressive and dynamic plastic buckling of cylindrical shells. int. j. solids struct. 38(38-39)( 2001) 6723-6749. advances in systems science and applications (2011), vol.11, no.3-4 213 issn 1078-6236 international institute for general systems studies, inc. [6] karagiozova d. and jones, n., on dynamic buckling phenomena in axially loaded elastic-plastic cylindrical shells. int. j. of non-linear mech. 37(7)(2002) 1223-1238. [7] karagiozova d. and jones, n., on the mechanics of the global bending collapse of circular tubes under dynamic axial load dynamic buckling transition. int. j. impact engng. 35(5)(2008) 397-424. [8] karagiozova d., nurick g.n. and yuen s.c.k., energy absorption of aluminium alloy circular and square tubes under an axial explosive load. thin-walled struct. 43(6)(2005) 956-982. [9] lu g. and mao r., a study of the plastic buckling of axially compressed cylindrical shells with a thick-shell theory. int. j. mech. sci. 43(10)(2001) 2319-2330. [10] rusinek a., zaera r., forquin p., and klepazko, j.r., effect of plastic deformation and boundary conditions combined with elastic wave propagation on the collapse site of a crash box. thin-walled struct. 46(10)(2008) 1143-1163. [11] ushijima k., haruyama s. and chen d-h. evaluation of first peak stress in axial collapse of circular cylindrical shell. trans. jsme. series a 70(700)(2004) 1695-1702. [12] wei z.g., yu j.l. and batra r.c. dynamic buckling of thin cylindrical shells under axial impact. int. j. impact engng. 32(1-4)(2005) 575-592. zhao h. and abdennadher s. on the strength enhancement under impact loading of square tubes made from rate insensitive metals. int. j. solids struct. 41(24-25)(2004) 6677-669. advances in systems science and applications (2013) vol.13 no.3 249-275 agent-based models simulations for high frequency trading d. dezsi, e.scarlat and i. măries department of economic cybernetics, bucharest university of economics, romania abstract high frequency computer-based trading (hft) represents a challenging topic nowadays, mainly due to the controversy it creates among investors on the financial market. the hereto paper compares two types of agent-based models, one with zero-intelligence traders and the other with intelligent traders in order to simulate the tick-by-tick high frequency trades on the stock market for the selected u.s. stocks. the simulations of the agent-based models are done with the help of adaptive modeler software application which uses the interaction of 2,000 heterogeneous agents to create a virtual stock market for the selected stock with the scope of forecasting the price. within the intelligent agent-based model the population of agents is continuously adapting and evolving by using genetic programming in forming new agents by using the trading strategies of the best performing agents and replacing the worst performing agents in a process called breeding, while the zero-intelligence agent based model does not evolve, agents do not breed, and they trade in a random manner. after comparing the fitting of the two models with the real data, the results show that in almost all the cases the intelligent agent-based model performed better when compared to the zero-intelligence agent-based model, which could be interpreted as lower market efficiency, allowing for predictions of the stock market price, or even stock market manipulation. also, the zero-intelligent agent-based model generates more trades and lower wealth for the population, compared to the intelligent agentbased model. the high-frequency data turns out to be very hard to simulate and analyse due to its particularities which differentiate them from daily data, as price changes are discrete, being multiples of the minimum price increment, the price changes not being independent. keywords high frequency trading, agent-based modeling, zero-intelligence traders, double auction, financial market. 1 introduction the changes of the stock market structure due to the technological improvement which led to the high speed computer-based trading have switched the market from an investor-focused mechanism to a trader-focused mechanism, where the investors’ trust and concerns are ignored. according to a study conducted by beddington in 2012 on the impact of the computer trading on the financial markets based on which the report entitled the future of computer trading in d. dezsi, e.scarlat and i. măries :agent-based models simulations for high... 250 financial markets-an international perspective [1] was released, although the investors are worried regarding the possible abuse of the market generated by the high-frequency trading (hft) and claims on market manipulation using hft techniques are reported by institutional investors all over the world, it seems like so far economic research has provided no direct evidence that hft has increased market abuse, but the authors of the report also underlined that the research regarding the measurement of market abuse is at an early stage and incomplete, not being the main focus of the report. furthermore, brogaard (2010) [2] examines the impact of hft on the u.s. equity markets and the results show that hft activities are not detrimental to non-hft activities and that hft tends to improve market quality. despite the lack of research on this subject, and even though the abuse is present or not in the market, regulators and policy makers should take in consideration the perception of the investors which show worries in this regard, because this is what determines trading behavior, portfolio management and investment decisions. thus regulators should increase their ability to detect abuse and obtain significant empirical proof on either to confirm or deny the manipulation related to hft in order to restore market confidence. market abuse through hft is studied by the economists due to its implications on market liquidity, volume, pricing efficiency and even social welfare. according to the studies conducted by cumming et al. (2012) [3] on the possible risks of market abuse generated by hft, the authors find that hft lead to lower incidence of manipulation, as also another study conducted by aitken et al. (2012) [4] reports that hft improves market efficiency without harming the market integrity. the papers mentioned above analyze market abuse at the end of the trading day, while empirical studies for the hft market abuse during the continuous trading period have not been conducted yet. these results are in contrast with the surveys regarding the perception of the investors over the degree of manipulation in the market generated by hft, which were conducted on the stock markets all over the world, and which underline the considerable concerns showed by the large investors regarding the market abuse and the lack of action and detection of manipulation by the regulators. the statistical modeling facts, models and challenges for high frequency financial data have been studied by cont [5] who outlined the empirical characteristics of high frequency financial time series, providing an overview of stochastic models for the continuous-time dynamics of the limit order book described as queuing systems, pointing that the gap between microstructure models and stochastic models should be filled in by inputting theoretical issues from microstructure models to design new stochastic models with a better economic interpretation. furthermore, ponta et al. [6] propose a non-homogeneous normal compound 251 advances in systems science and applications (2013) vol.13 no.3 poisson process for describing non-stationary returns for high-frequency financial time series for the italian stock exchange, also testing if the model can reproduce some stylized facts of high-frequency financial time series. in order to model the high-frequency foreign exchange market, aloud et al. [7] constructed an agentbased model which is able to reproduce the stylized facts of the trading activity on the foreign exchange market, by using zero-intelligence directional-change event trading strategy. according to the results obtained by li and krause (2009) [8] by comparing the market structures with near-zero-intelligence traders with the use of agentbased model simulations, they have observed that the properties of returns arising in double auction markets are not very sensitive to the trading rules employed. also, sunder (2004) [9] underlines that artificial and computer intelligence are very important in understanding the distinction between individual behaviour and market outcomes, outlining that computer simulations helped to discover that allocative efficiency is largely independent of variations in individual behaviour under classical conditions, while herbert simon was convinced that “the possibility of building a mathematical theory of a system or of simulating that system does not depend on having an adequate microtheory of the natural laws that govern the system components. such a microtheory might indeed be simply irrelevant” [9]. the aim of our research is to identify the main distinctions between the highfrequency financial data simulation results of two agent-based models, one with intelligent agents and the other with zero-intelligence agents. in order to achieve our research aim, we use the adaptive modeler software to simulate the two types of agent-based market models-the zero-intelligence agent-based model and the intelligent agent-based model-for price forecasting of real world market-traded securities such as stocks. thus, heterogeneous agents trade a stock floated on the stock exchange market, placing orders depending on their budget constraints and trading rules, where the virtual market is simulated as a double auction market. for the intelligent agent-based model, the software uses evolutionary computing such as the strongly typed genetic programming [10] in order to create adaptive, evolving and self-learning market modeling and forecasting solutions, while for the zero-intelligence agent-based model the agents do not breed nor evolve, as they trade in a random manner. the results obtained from the simulations are compared revealing that the high frequency tick-by-tick data is best modeled by the intelligent agent-based model, although the high-frequency data turns out to be very hard to simulate and analyse due to its particularities which differentiate them from daily data, such as discrete price changes, which are multiples of the minimum price increment, thus the price changes are not independent. to the best of our knowledge, none of the works in the literature have attemptd. dezsi, e.scarlat and i. măries :agent-based models simulations for high... 252 ed to compare the results obtained from simulating zero-intelligence agent-based models with the results obtained from simulating intelligent agent-based models for high frequency financial data. the organization of this paper is as follows. section 2 provides an overview of the high frequency financial data and its statistical properties, section 3 describes the datasets used in this study, section 4 presents the agent-based models used in the simulations, while the results of the simulations that have been performed are presented in section 5, the paper ending with the conclusions and avenues for future work. 2 high frequency financial data and their statistical properties the high frequency financial data or high frequency trading (hft) represents a subset of algorithmic trading or a trading strategy where a large number of small size orders are sent into the market at high speed, therefore the securities are bought and sold by a computer algorithm and held for a very short period, usually seconds or milliseconds. the automated trading applies on the order-driven electronic platforms which aggregate all outstanding orders in an order book and market orders are executed in a mechanical manner, usually a double auction mechanism on the stock market. for a better comprehension, table 1 illustrates the hierarchy of time scales for financial data, as pointed out by cont in [5], while thousand of buying or selling orders are submitted in a 10-second interval. the hft is one of the most significant market structure developments in table 1 financial data series time scales regime time scale issues ultra-high frequency (uhf) 10−3 − 0.1s microstructure high frequency (hf) 1-100s trade execution daily 103 − 104s trading strategies recent years, as stated by the securities and exchange commission (sec), but is also controversial due to its divers impact on investors’ perception, which are worried about the degree of manipulation in the market generated by hft, according to surveys conducted on the stock markets. despite the investors worries described above, research studies on the impact of hft over the market show that hft tends to improve market quality [2], while vuorenmaa (2012) [11] reached the conclusion that the benefits of the hft have the most weight and dominate the negative aspects of hft. recently, eu parliament voted in favour of mifid ii, a draft law for strengthening the regulation of hft and other financial instruments. the new rules will apply starting with 2014 and aims to reduce speculation without harming 253 advances in systems science and applications (2013) vol.13 no.3 the real economy, by introducing a synchronised clock for trading shares, bonds, commodities and other instruments across the eu so regulators can spot abuses more easily in a market where many exchanges and platforms trade the same shares, and by extending the minimum time in which high-frequency orders have to remain in the market from 3 milliseconds currently to at least 500 milliseconds. that way, purely speculative business with high-frequency transactions will be discouraged. recall that high-frequency and algorithmic trading impact on the financial market has raised worries from behalf of the regulators and the investors since may 6, 2010, when about usd 862 billion was erased from stock values in 20 minutes before share prices recovered from the plunge, while on august 1, 2012 a trading malfunction at knight capital group which poured in the market unintended orders lead to high traded volume and high price volatility, which finally led to a usd 440 million trading loss for the company. according to a study conducted by beddington in 2012 [1] on the impact of the computer trading on the financial markets based on which the report entitled the future of computer trading in financial markets-an international perspective was released, although the investors are very worried regarding the possible abuse of the market generated by the hft, so far economic research has provided no direct evidence that hft has increased market abuse. furthermore, according to the studies conducted by cumming et al. (2012) [3] on the possible risks of market abuse generated by hft, the authors find that hft lead to lower incidence of manipulation, while also another study conducted by aitken et al. (2012) [4] reports that hft improves market efficiency without harming the market integrity. the papers mentioned above analyze market abuse at the end of the trading day, while empirical studies for the hft market abuse during the continuous trading period have not been conducted yet. on the other hand, the surveys conducted regarding the perception of the investors over the degree of manipulation in the market generated by hft underline the considerable concerns showed by the large investors regarding the market abuse and the lack of action and detection of manipulation by the regulators. also, hft allows the program trader to “see” ahead of others the major incoming orders and to act benefitting from its speed advantage, in order to gain profits, as hft has been shown to have a potential sharpe ratio, which is a measure of reward per unit of risk, thousands of times higher than the classical buy-and-hold strategies [12]. a major disadvantage is that it can generate trade crashes, such as the one on may 6, 2010, and the one on august 1, 2012, thus eroding the trust of investors in the market. as regards to statistical properties of the tick-by-tick high frequency financial data compared to daily frequency data, the main differences are the fact that d. dezsi, e.scarlat and i. măries :agent-based models simulations for high... 254 price changes are discrete, being multiples of the minimum price increment as it can be observed in the histograms from fig.1 below. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) fig.1 histogram of log returns for adbe(a), aa(b), aapl(c), bac(d), c(e), xom(f), ge(g), ibm(h), pfe(i), wfc(j) the autocorrelation function of high frequency price returns is significantly negative at the first lag and for higher lags it rapidly decreases to zero, meaning 255 advances in systems science and applications (2013) vol.13 no.3 that price changes are not independent [13]. another particularity is that trades occur at irregular intervals, they are random and endogenous, linked to the behaviour of the price and to previous trading history, and also seasonality effects is observed, as the trading activity is more intense at the market opening and closing [5]. 3 data the high frequency financial data in this paper was retrieved from the website http://www.finam.ru/analysis/profile041ca00007/default.asp and contains tickby-tick data from bats stock exchange. the tick-by-tick data was retrieved from this website, for the trades that occurred on the bats (better alternative trading system) stock exchange during regular trading hours on december 3rd, 4th, 5th and 6th of 2012 limited to 20,000 quotes for each stock, and includes the date, time and closing price information for each stock. the unique dataset used in this paper, namely the tick-by-tick high-frequency financial dataset contains real trading data for the following stocks floated on the bats, a computerized u.s. stock exchange market: addobe systems inc. (symbol: adbe), alcoa inc. (symbol: aa), apple inc. (symbol: aapl), bank of america (symbol: bac), citigroup inc. (symbol: c) exxon mobil (symbol: xom), general electric (symbol: ge), ibm (symbol: ibm), pfizer inc. (symbol: pfe), wells fargo (symbol: wfc). the bats stock exchange has been chosen for the data retrieval because it is one of the leading u.s. venues for hft, covering 12-13% of all u.s. equity trading on a daily basis according to the stock exchange’s website http://www.batstrading.com/about/. the distinction between the hft and non-hft was not defined in this paper, although taking in consideration the nature of the bats stock market, we consider the trades to be mainly hfts. table 2 symbols and numbers of observations for the 10 analysed stocks stock name symbol sector no. of obs. (bars) period of time addobe systems adbe technology 20,000 3rd − 6th of december 2012 alcoa inc. aa basic materials 12,612 3rd − 6th of december 2012 apple inc. aapl technology 19,999 3rd − 5th of december 2012 bank of america bac financial 20,000 3rd − 4th of december 2012 citigroup inc. c financial 20,000 3rd − 4th of december 2012 exxon mobil xom basic materials 20,000 3rd − 6th of december 2012 general electric ge industrial goods 20,000 3rd − 5th of december 2012 ibm ibm technology 6,144 3rd − 6th of december 2012 pfizer inc. pfe healthcare 20,000 3rd − 5th of december 2012 wells fargo wfc financial 20,0000 3rd − 5th of december 2012 the time period of trades is expressed in milliseconds, the number of observations and period of time for each stock is summarized in table 2, while the descriptive statistics for the tick-by-tick log returns are presented in table 3, in d. dezsi, e.scarlat and i. măries :agent-based models simulations for high... 256 which the excess kurtosis can be observed. also, fig.1 shows the histogram for the log returns of the analysed stocks, in order to show how the returns are distributed, being possible to see the discrete character of returns even after the logarithmic transformation. table 3 descriptive statistics for the tick-by-tick log returns stock name symbol mean×106 std.dev.×104 skewness kurtosis addobe systems adbe 0.486 1.46 -0.360791 43.17925 alcoa inc. aa 1.120 2.64 0.086025 19.32359 apple inc. aapl -3.700 1.90 -16.53734 979.9840 bank of america bac -5.060 1.37 -0.334374 23.19635 citigroup inc. c -0.909 1.10 -0.400060 32.14271 exxon mobil xom -0.443 1.10 -1.226893 41.91198 general electric ge 0.0472 1.24 0.436346 18.27284 ibm ibm -0.890 1.94 0.449297 32.84855 pfizer inc. pfe 1.230 1.05 0.267775 16.65013 wells fargo wfc -0.653 1.21 3.666686 156.9951 4 the agent-based models specifications an agent-based model is a computational model for simulating the actions and interactions of multiple agents in order to analyze the effects on a complex system as a whole, and represents a powerful tool in the understanding of markets and trading behavior. an agent-based model of a financial market consists of a population of agents (representing investors) and a price discovery and clearing mechanism (representing a virtual market). as regards to the financial markets, agent-based models can successfully replicate time series features like fat-tailed distributions and volatility clustering, on which standard financial models offer few explanations. conventionally, financial markets have been studied using analytical mathematics based on a generalization of market participants and other simplifications and idealizations. however, the behavior of financial markets as observed in reality can’t be fully described by such mathematical models. in reality, market prices are established by a large diversity of investors with different decision making methods and different investment goals. the complex dynamics of these heterogeneous investors and the resulting price formation process require a simulation model of multiple heterogeneous agents and a virtual market [14]. research has shown that complex behavior can emerge from simulations of agents with relatively simple decision rules. furthermore, commonly observed stylized facts of financial time series (price and order flow) have been reproduced by several agent-based market models [7,15-17]. although agent-based research has been introduced over 30 years ago in economics, it is still considered a niche field because of its detailed and complex 257 advances in systems science and applications (2013) vol.13 no.3 platforms. another impeachment in using computational agent-based modeling is represented by the fact that it uses source codes which usually are not disseminated along with the paper, thus being hard to be evaluated, tested and developed for further research by the rest of the community, as underlined by barr et all (2008) [12]. furthermore, agent-based modeling can generate important facts when used for institutional design. at a large scale they are already used for computer simulation by government agencies, an important example being the traffic simulations, while as a small to intermediate size model it can be used by the government agencies as an advisory and prediction instrument. fig.2 agent-based model cycle the agent-based models referred to in this paper are simulated in adaptive modeler software, which supports up to 2,000 agents and 20,000 observations for each simulation. to the best of our knowledge, none of the work in the literature has attempted to compare the simulation results generated from the simulations of zero-intelligence agent-based models with the results obtained from simulating intelligent agent-based models for high frequency financial data. each model consists of a population of agents and a virtual market on which the agents trade the envisaged security. the agents are autonomous entities representing the traders of the stock market, each having their own wealth (cash and shares) and their own trading strategy. the general cycle of the agent-based models used in this paper, which repeats at each of the 20,000 analyzed quotes, is illustrated in fig.2 and fig.3 in more details. it can be described as follows: 1. import real stock market data: real stock market data is imported for the simulated stock. 2. initialization of the model: the cycle starts after the model initialization and after the parameters are set. at the initialization, the agents receive an d. dezsi, e.scarlat and i. măries :agent-based models simulations for high... 258 initial wealth of 100,000 and no shares. each agent receives a trading rule which is called the genome, which is randomly created by taking in account the selected genes (which represent functions) using genetic programming described in [10]. fig.3 agent-based model flowchart there are no broker fees in these simulations, and no market maker. all the parameters for each of the two models and their values are described in table 4. 259 advances in systems science and applications (2013) vol.13 no.3 table 4 general settings of the models. market and agents’ parameters configuration in the simulations model parameter type parameter name parameter value no. of trading periods (bars) max. 20,000; see table 1. market no. of agents 2,000 parameters spread 20.005% variable broker fee 0% zi agentwealth distribution equal for all agents: 100,000 based position distribution equal for all agents% model : initial position 0% min. position unit 20% agent min. initial genome depth 2 parameters max. initial genome depth 5 genes rndpos, rndlimit, advice breeding cycle frequency 1,000,000 bars (so that the agents never breed, don’t evolve) no. of trading periods max. 20,000; see table 1. market no. of agents 2,000 parameters spread 20.005% variable broker fee 0% wealth distribution equal for all agents: 100,000 position distribution equal for all agents% : initial position 0 min. position unit 20% max. genome size 1000 max. genome depth 20 min. initial genome depth 2 max. initial genome depth 5 intelligent genes curpos, levunit, rmarket, agentvmarket, long, short, cash, bar, based invpos, rndpos, close, bid, ask, model agent average, min, max, volume, >, parameters change, +, dir, isupbar, upbars, pos, breeding cycle frequency 1 bar (so that the agents breed at each new bar, conditioned of the fact that they are of minimum breeding age) minimum breeding age 80 initial selection: 100% of agents of minimum randomly select breeding age or older parent selection 5% agents of initial selection will breed mutation probability 10% per offspring 3. receive new quote bar: a new quote bar from the imported real stock market data series is received by the virtual market. 4. agents evaluate trading rules and place orders: agents get access to historical prices and evaluate their trading rules according to the genomes allocated in the initialization process, resulting in a desired position as a percentage of wealth limited by the budget constraints, and a limit price. for the zero-intelligence agent-based model, a desired position and a limit price order are generated in a random manner. for the intelligent agent-based model, the position is also gend. dezsi, e.scarlat and i. măries :agent-based models simulations for high... 260 erated in a random manner, while the limit price is generated after a technical analysis has been performed, according to the genome structure which represent trading functions. 5. virtual market clearing and forecast generation: the virtual market determines the clearing price, executes all executable orders, and forecasts the price for the next bar, using a double auction mechanism. 6. breeding (only for the intelligent agent-based model): during the breeding process, new agents are created from best performing agents in order to replace the worst performing agents, creating new genomes by recombining the parent genomes through a crossover operation. the breeding process repeats at each bar, with the condition that the agents must have a minimum breeding age of 80 bars, in order to be able to assess the agents performance. 7. the model waits for a new quote: if the model receives new quotes, it will repeat the process described at points 4-6. if there are no more quotes to be processed, the simulation ends. in order to obtain random seed, the adaptive modeler software uses the mersenne twister algorithm [18] to generate pseudo random number sequences for the initial creation of trading rules or genomes and for the crossover and mutation operators of the breeding process. more information regarding the adaptive modeler software and how it works can be found at http://altreva.com/adaptive modeler users guide.htm. 4.1 zero-intelligence traders model the concept of zero intelligence (zi) traders has been introduced by gode and sunder (1993) [19] in order to study the lower limit of rationality required to participate in a double auction market, which implies that agents have no strategy and they behave in a random manner subject to a budget constraint, so that the market mechanism can be observed. in case the zi traders model generate good results, this means that the market mechanism used is incentive-compatible and most probably is not influenced by the trading strategies used by the investors. this represents a very important result for the market mechanism design, as a market mechanism which performs well despite the trader’s irrationality is preferred over the mechanism that performs well only with perfectly rational traders, as underlined in walia (2003) [20]. in ladley’s (2012) [21] point of view, any effects that are not observed in the zi model cannot solely be due to the market mechanism and requires the interaction of the investors strategies, therefore it can be possible to separate the effects of the market mechanism and trader strategy and to determine the driving forces within markets. in this context, a model that assumes stock market traders have zero intelligence has been found to mimic the behavior of the london stock exchange very closely, the research being conducted by farmer et al. (2004) [22] at 261 advances in systems science and applications (2013) vol.13 no.3 the santa fe institute, which suggest that the movement of the markets depend less on the strategic behavior of the traders and more on the design of the trading system. a double auction is a trading mechanism in which buyers and sellers can enter bid or ask limit orders and accept asks or bids entered by other traders [23]. this trading mechanism was chosen to be used for the virtual market simulation in the adaptive modeler models because most of the stock markets are organized as double auctions, which is considered to be a continuous-time game of incomplete information [24]. double auction stock markets converge to the equilibrium derived by assuming that the traders are profit-maximizing bayesian, being an example of a microeconomic system, as described in hurwicz (1986) [25] and smith (1982) [26]. in the double auction markets, agents introduce bid or ask orders, each order consisting of a price and quantity. the bids and asks orders received are put in the order book and an attempt is made to match them. the price of the trades arranged must lie in the bid-ask spread (interval between bid price and ask price). furthermore, the use of double auction trading mechanism generates stochastic waiting times between two trades, as stated by scalas [27]. an example of the order book resulted from buy and sell orders collected by the virtual market can be viewed in the fig.4, 5, 6 and 7, where blue bars represent buy orders, red bars represent sell orders and yellow bars represent buy and sell orders at the same price before the market clearing. fig.4 and 6 illustrate the order book before the clearing, while fig.5 and 7 illustrate the order book after clearing, so only the unexecuted buy (blue) and sell (red) orders remain visible in these two figures. the clearing price can be observed in the top of the figures. in the hereto paper, the zi traders model is designed within the context of the double auction mechanism, which sets the order placement, clearing price, the supply and demand. the model is implemented in the adaptive modeler software and will be used in this paper to simulate tick-by-tick high-frequency data. the adaptive modeler software in the zi agent-based model uses only three types of genes or trading strategies, as mentioned in table 4, namely rndpos, rndlim and advice. the rndpos gene returns a position value ranging from -100% to 100% which is randomly generated from a uniform distribution, the rndlim gene returns a random limit price that is generated as follows: the last closing price is taken from either the virtual market or the real market which is chosen randomly, then the price is multiplied with a normally distributed random value with µ = 1 and σ = 3.5 · σm where σm is the standard deviation of the log returns of the last 20 bars of the chosen market. the advice gene combines the position generated by rndpos function and the limit price value generated by the rndlim function into a buy or sell order advice. the buy or sell order is introduced in the market after comparing the desired position with the agent’s d. dezsi, e.scarlat and i. măries :agent-based models simulations for high... 262 fig.4 order book example from the zi-abm simulation for aapl stock. blue bars represent buy volume, red bars represent sell volume, yellow bars represent buy and sell volume at the same price before clearing; clearing price can be seen above the chart fig.5 order book example from the zi-abm simulation for aapl stock. blue bars represent buy volume, red bars represent sell volume after clearing; clearing price can be seen above the chart 263 advances in systems science and applications (2013) vol.13 no.3 fig.6 order book example from the zi-abm simulation for aapl stock. blue bars represent cumulative buy volume, red bars represent cumulative sell volume, yellow bars represent buy and sell volume at the same price before clearing; clearing price can be seen above the chart fig.7 order book example from the zi-abm simulation for aapl stock. blue bars represent cumulative buy volume, red bars represent cumulative sell volume after clearing; clearing price can be seen above the chart d. dezsi, e.scarlat and i. măries :agent-based models simulations for high... 264 current position and calculating the number of shares that need to be bought or sold, taking also in consideration the available cash. this order generation process is explained in detail in the paper agent-based simulation of a financial market by raberto (2001) [17], together with the ability of this model to exhibit the stylized facts of financial time series, such as fat tails and volatility clustering, using simple trading rules, budget restriction of the agents, order limit prices, and the creation and matching of demand and supply curves. the zero-intelligence agent-based model always skips the breeding step by setting the breeding cycle frequency parameter at a very high value (1,000,000) as mentioned in table 4, therefore agents are not replaced with better ones, the population does not evolve, nor adapt to the market conditions, agents trading in a random manner. 4.2 the intelligent agent-based model the intelligent agent-based model cycle starts by receiving a new quote bar, so that agents can place a new order or remain inactive according to their trading strategy. after all agents have evaluated their trading strategy, the virtual market determines the clearing price, executes all executable orders and releases the price forecast for the next bar. after that, breeding of new agents and replacement by evolutionary operations such as crossover and mutation can take place, a process which repeats itself for each bar. the trading rules of the model uses historical price data as input, either from the virtual market either from the real market, and return an advice consisting of a desired position, as a percentage of wealth, and an order limit price for buying or selling the security. the trading rules are implemented by genetic programming technology explained bellow. through evolution the trading rules are set to use the input data and functions (trading strategies) that have the most predictive value. the agents’ trading rules development is implemented in the software by using the strongly typed genetic programming (stgp) approach, and use the input data and functions that have the most predictive value in order for the agents with poor performance to be replaced by new agents whose trading rules are created by recombining and mutating the trading rules of the agents with good performance. the stgp was introduced by montana (2002) [10], with the scope of improving the genetic programming technique by introducing data types constraints for all the procedures, functions, variables and constants, thus decreasing the search time and improving the generalization performance of the solution found. therefore, the genomes (programs) represent the agents’ trading rules and they contain genes (functions), thus agents trade the security on the virtual market based on their analysis of historical quotes. in order to obtain random sequences for the initial creation of trading rules or genomes and for the crossover and mutation operators of the breeding process, the adaptive modeler software 265 advances in systems science and applications (2013) vol.13 no.3 uses the mersenne twister algorithm [18] to generate pseudo random number sequences. more information regarding the adaptive modeler software and how it works can be found at http://altreva.com/adaptive modeler users guide.htm. also, all the genes as functions are described in the guide, along with the types of the arguments used. during the breeding process which is specific only to the intelligent agent-based model, new offspring agents are created from some of the best performing agents to replace some of the worst performing agents. in order to achieve this, at every bar, agents with the highest breeding fitness return are selected as parents, and the genomes (trading rules) of pairs of these parents are then recombined through genetic crossover to create new genomes that are given to new offspring agents. these new agents replace agents with the lowest replacement fitness return. the fitness functions are a measurement of the agents investment return over a certain period, therefore the breeding fitness return is computed as a short term trailing return measure of the wealth over the last 80 analyzed quotes and represents the selection criterion for breeding (best agents), while the replacement fitness return is computed as the average return per bar and represents the selection criterion for replacement (worst agents). fig.8 oaddobe stock simulation-intelligent agent-based model 5 simulation results the results of the simulations are illustrated in fig.8-25, and were conducted over each of the ten analyzed stocks, for both types of agent-based models, with intelligent agents and zero intelligence agents. the results of the simulations mainly illustrate a better performance of the intelligent agent-based model in terms of forecast accuracy, except for the stocks addobe and citigroup. the results must be further tested for how stylized facts are reproduced with these models, compared to real stock quotes. d. dezsi, e.scarlat and i. măries :agent-based models simulations for high... 266 fig.9 addobe stock simulation-zero-intelligence agent-based model fig.10 apple stock simulation-intelligent agent-based model fig.11 apple stock simulation-zero-intelligence agent-based model 267 advances in systems science and applications (2013) vol.13 no.3 fig.12 bank of america stock simulation-intelligent agent-based model fig.13 bank of america stock simulation-zero-intelligence agent-based model fig.14 citigroup stock simulation-intelligent agent-based model d. dezsi, e.scarlat and i. măries :agent-based models simulations for high... 268 fig.15 citigroup stock simulation-zero-intelligence agent-based model fig.16 exxon mobil stock simulation-intelligent agent-based model fig.17 exxon mobil stock simulation-zero-intelligence agent-based model 269 advances in systems science and applications (2013) vol.13 no.3 fig.18 general electric stock simulation-intelligent agent-based model fig.19 general electric stock simulation-zero-intelligence agent-based model fig.20 ibm stock simulation-intelligent agent-based model d. dezsi, e.scarlat and i. măries :agent-based models simulations for high... 270 fig.21 general electric stock simulation-zero-intelligence agent-based model fig.22 pfizer stock simulation-intelligent agent-based model fig.23 pfizer stock simulation-zero-intelligence agent-based model 271 advances in systems science and applications (2013) vol.13 no.3 fig.24 wells fargo stock simulation-intelligent agent-based model fig.25 wells fargo stock simulation-zero-intelligence agent-based model it is worth mentioning that wealth is much better and uniformly distributed among agents in the case of zero-intelligence agent-based model, as it can be observed in the fig.8-25, which is of great importance for market design, while intelligent agent-based models generate discrepancies among agents’ wealth distribution. on the other hand, the wealth in the intelligent agent-based model simulations is much better preserved compared to the zero intelligence agentbased models, due to the fact that intelligent agent-based model is not a closed economy due to the replacement of the agents which bring new capital in the virtual market, while the zero-intelligence agent-based model uses the same traders during all the simulation period. this generates a divergence between the buy orders and sell orders due to lack of cash, thus buy orders. as illustrated in the average genome size window in the fig.9-25 for the zero-intelligence agent-based models the value remains constant at 3, as there are only 3 genes (functions) that generate the trades of the population, while for the intelligent agent-based model the value ranges between 30 and 80, showing how the agents strategies constantly adapt and evolve during the simulations. d. dezsi, e.scarlat and i. măries :agent-based models simulations for high... 272 in order to compare the fit of the models to the real data and to compare the efficiency of the models, the root mean squared error and mean absolute error have been computed and listed in table 5. the results show that the intelligent agent-based model performed better compared to the zero-intelligent agent-based model in terms of price forecast. table 5 root mean squared error and mean absolute error for the simulations stocks model simulated root mean squared error mean absolute error addobe (adbe) zi-abm 0.010 0.006 i-abm 0.032 0.003 alcoa (aa) zi-abm 0.004 0.003 i-abm 0.002 0.001 apple (aapl) zi-abm 0.279 0.124 i-abm 0.110 0.056 bank of america (bac) zi-abm 0.007 0.003 i-abm 0.002 0.000 citigroup inc. (c) zi-abm 0.027 0.011 i-abm 0.110 0.003 exxon mobile (xom) zi-abm 0.040 0.020 i-abm 0.010 0.005 general electric (ge) zi-abm 0.013 0.006 i-abm 0.003 0.001 ibm (ibm) zi-abm 0.062 0.041 i-abm 0.037 0.020 pfitzer inc. (pfe) zi-abm 0.035 0.011 i-abm 0.003 0.001 wells fargo (wfc) zi-abm 0.016 0.007 i-abm 0.009 0.002 6 conclusions taking into account the previous studies on the impact of high frequency trading on the financial markets, the results have shown that this algorithmic trading has led to higher liquidity, improved market efficiency without harming market integrity and lower incidence of market manipulation. although the results of the research in this field is rather optimist thus far, regarding the impact of hft over the financial market, the studies are few and do not comprise all the intra-day price impact, nor the impact of the unexecuted orders which are introduced into the market by these trading algorithms. thus, investors are worried regarding the possible manipulation of the market following these high frequency trades, and currently regulators focus more and more on a better policy for regaining the trust of the investors in the market by limiting the hft in the markets by imposing that the trading systems are configured such as not to cause market disturbances, a higher control of the market strategies is intended to be implemented, and a fee is also to be charged in the case of excessive use of the system and a limit introduced on the ratio of unexecuted orders. the results show that in almost all the cases the intelligent agent-based mod273 advances in systems science and applications (2013) vol.13 no.3 el performed better when compared to the zero-intelligence agent-based model, which could be interpreted as lower market efficiency, allowing for predictions of the stock market price, or even stock market manipulation. this has to be further studied and efficient market hypothesis should be tested. the hereto paper brings to light the importance and accuracy of the agent-based models when it comes to stock market price forecast and modeling, and more important it applies these types of models on high frequency trading data, obtaining high accuracy in forecasting. further research should focus on improving the fit of these models with the real stock market by adding brokerage fee and allowing zero-intelligence agents to breed (but not to adapt), thus renewing the population to obtain a zerointelligence agent-based model closer to reality. acknowledgements this work was co-financed from the european social fund through sectorial operational programme human resources development 2007-2013, project number posdru/107/1.5/s/77213 “ph.d. for a career in interdisciplinary economic research at the european standards”. references [1] j. beddington. 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(2007), “mixtures of compound poisson processes as models of tick-by-tick financial data”, chaos, solitons & fractals, vol.34, pp.33-40. corresponding author d. dezsi can be contracted at: dianadezsi@yahoo.com microsoft word 2. k. masuda and d. h. chen--prediction of maximum moment of rectangular tubes subjected to pure bending.doc 214-225 advances in systems science and applications (2011), vol.11, no.3-4 issn 1078-6236 international institute for general systems studies, inc. prediction of maximum moment of rectangular tubes subjected to pure bending k. masuda and d. h. chen department of mechanical engineering, tokyo university of science, 1-3 kagurazaka, shinjuku-ku, tokyo, 162-8601, japan abstract in this paper, the collapse behaviors of rectangular tube subjected to pure bending are studied by using the finite element method. such bending collapse has been studied for a long time, including the landmark study by kecman. according to these studies, there are two types of collapses. the first type is a collapse due to buckling at the compression flange. the second type is a collapse due to plastic yielding at the flanges. however, there may be another collapse. for a rectangular tube in which the web is wider than the flange, it is found that collapse due to buckling at the compression web may occur. further, an approximation prediction method is proposed for estimating the maximum bending moment of rectangular tubes in which the web buckling is also taken into account. its validity is verified by comparing with the numerical results by fem under various conditions. keywords fem, pure bending, rectangular tube, buckling, effective width 1．introduction evaluation of a car’s crush behavior when the car is subjected to an oblique load, such as in an offset crash, is becoming increasingly important for general car design. it is thus vital to understand the crushing characteristic of rectangular tubes that are used as general components in car. because the oblique load can be decomposed into axial load and pure bending, the pure bending of the rectangular tubes has been widely studied for a long time [1, 2, 3], including the landmark study by kecman [1]. according to these studies, there are two types of collapses. the first type is a collapse due to buckling at the compression flange. the second type is a collapse due to plastic yielding at the flanges. however, when the web is wider than the flange, it is considered that a collapse due to buckling at the compression web may occur because it was already reported in bending of open section beams [4]. in the present study, the effects of the material and geometrical properties of rectangular tubes on their bending collapse are studied by using the finite element method. further, based on the numerical results obtained, a method for estimating the maximum bending moment of rectangular tubes subjected to pure bending is proposed. in addition, a validity of fe analysis result under bending collapse has been already verified by comparing the experimental results by kyriakides [5] with the previous numerical results by authors [6] under pure bending with cylindrical tubes. 2．analytical method the commercial fem analysis package msc. marc[7] is used in this study to analyze large elastoplastic bending of the rectangular tubes shown in fig. 1. in the present calculation, one end of the rectangular tube was completely fixed to a rigid wall. pure bending was applied from the other end by modeling a lid rotating about the axis of z under rotary control. the effects of various geometric parameters, such as tube thickness t, tube flange width c1, and tube web width c2, on the bending collapse were investigated. the value of lid thickness tf was set to five times t as referred to guarracino [8] because the lid must be stiff enough to prevent distortion. advances in systems science and applications (2011), vol.11, no.3-4 215 issn 1078-6236 international institute for general systems studies, inc. the tube material used in the analysis was assumed to be homogeneous and isotropic elastic perfectly plastic material that conforms to von mises yield conditions. in this study, it was assumed that young’s modulus e = 72.4 gpa, and poisson’s ratio v = 0.3. the influence of the material properties on the bending collapse of the rectangular tube was investigated in terms of the yield stress yσ . in this study, the updated lagrange method was used to formulate the geometric nonlinear behavior, and the algorithm based on the newton-raphson method and the return-mapping method were used to solve the nonlinear equation. the rectangular tubes were modeled using four-node quadrilateral thickness shell elements (element type 75). the elements were divided the flange and wed width into 20 sub lengths, and divided the axial length in a way that the elements become almost square. in addition, the rectangular length used in the analysis was assumed to be long enough in order to exclude the influence of the boundary conditions. the ratio of the length and flange width l/c1 was set to 1/ 6l c > . l t t x y z2c 1c z y x m output zθ input tf =5t a aa-a fig.1.tube geometry and loading condition 3．results and discussion 3.1 comparison between proposal method by kecman and results of present numerical analyses first, we show kecman’s method for estimating the maximum bending moment of rectangular tubes subjected to pure bending. for a rectangular tube subjected to pure bending, the buckling stress bucσ of the compression flange was derived in the following equation. ( ) ( ) 22 2 5.23 0.16 1 12 1buc e a t b a πσ ν ⎛ ⎞⎛ ⎞= +⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠ 　　 　　 b a 2 eayσ 1y 1 1 y b y y σ − (a) b yσ (b) yσ 2 b a fig.2. schematic representation of axial stress distribution proposed by kecman: (a) buc yσ σ< ; (b) 2buc yσ σ≥ where e, v, a, b, and t are respectively, young’s modulus, poisson’s ratio, flange width, web width and tube thickness. in addition, a is 1c t+ and b is 2c t+ . kecman presented a proposal method in which was decided by relations of the buckling stress bucσ and yield stress yσ . (1) in the case of buc yσ σ< if the buckling stress bucσ is less than the yield stress yσ , the compression flange buckles 216 masuda: prediction of maximum moment of rectangular tubes subjected to pure bending and the edges stress come up to yield stress yσ . in order to consider this phenomenon, an effective width eσ is introduced in the following simplified equation. ( )0.7 0.3 2buc ea a σ σ ⎛ ⎞ = +⎜ ⎟⎜ ⎟ ⎝ ⎠y 　 　 　 as a result, stress distribution in the maximum moment is shown in fig. 2(a). in the figure, y1 in which the distances from a compression flange to the neutral axis is derived from the condition of zero axial loads. therefore, y1 is given by the following equation. ( )1 3 2e y a b b a a b + = + + 　 　 　 by summing moments through the cross-section, the maximum bending moment is derived in the following equation. ( ) ( )2 max 2 3 2 4 3 e aa b a bm t b a b σ ⎛ ⎞+ + ⋅ +⎜ ⎟ ⎝ ⎠⋅ ⋅ ⋅ +y= 　 (2) in the case of 2buc yσ σ≥ in this case, stress distribution in the maximum moment is shown in fig. 2(b). namely, it is assumed that the maximum moment is equal to a fully plastic moment mp. the maximum bending moment is derived in the following equation. ( ) ( ) ( )2 max 0.5 2 5pm m t a b t b tσ ⎡ ⎤⋅ − + −⎣ ⎦y= = 　 (3) in the case of 2y buc yσ σ σ≤ < first, if the buckling stress bucσ is equal to the yield stress yσ , it is assumed that the maximum moment is equal to a elastic moment me in which the stress of flanges are equal to the yield stress yσ . this elastic moment me is derived in the following equation. ( )6 3e bm t b aσ ⎛ ⎞⋅ ⋅ ⋅ +⎜ ⎟ ⎝ ⎠ y= 　 and in the case of 2y buc yσ σ σ≤ < , the maximum bending moment is derived from linear interpolation: ( ) ( )max 7buc y e p em m m m σ σ σ − − y = + 　 0 0.005 0.01 0.015 0.02 0.025 0 1 2 t / c1 m m ax / (σ y c 1 c 2 t ) σbuc<σy 2σy≤σbuc l = 300 mm c1 = 50 mm σy/ e = 1/1000 kecman [1] c2/c1=1 c2/c1=2 c2/c1=1 c2/c1=2 2σy≤σbuc σbuc<σy fig.3. comparison between kecman’s proposal and results of fem with relation of t/c1 and ( )max 1 2/ ym c c tσ ⋅ ⋅ ⋅ advances in systems science and applications (2011), vol.11, no.3-4 217 issn 1078-6236 international institute for general systems studies, inc. figure 3 compares these proposal methods by kecman and results of present numerical analyses for two levels of aspect ratio c2 /c1 with c1 =50mm, l=300mm, / 0.001y eσ = . as can be seen from this figure, in the case of high-aspect ratio in which the web is wider than the flange, the results of maximum moment under various t/c1 between the kecman’s proposal and results of fem have a margin of error. in particular, the error increases with decreasing t/c1. therefore, it is found that a region which does not apply to kecman’s proposal exists. in order to estimate the maximum moment, it is vital to reveal the bending collapse mechanism of rectangular tubes. 3.2 two types of collapse mechanism pointed out by kecman an investigation of two types of collapse mechanism pointed out by kecman was presented by using square tubes in which the aspect ratio c2 /c1 was set to 1. figure 4 shows the relation of a tube curvature / lκ θ= and moment m for a square tube with t=0.9mm, c1=50mm, c2=50mm, / 0.001y eσ = ( 1.52buc yσ σ= ). and figure 4 also shows the relations of the tube curvature / lκ θ= and axial stress /x yσ σ at point b and c (refer to schematic representation of cross-section in figure 4). as can be seen from the figure, the maximum moment is in good agreement with the value of kecman’s proposal equation (7). the axial compression stress /x yσ σ at point b in the middle of compression flange increases until the moment becomes maximum moment, and the value /x yσ σ comes up to 1. in addition, the axial compression stress /x yσ σ at point c in the quarter of web width increases until the moment becomes maximum moment. figure 5 shows the axial stress distribution of cross-section at phase ( )α and ( )β corresponding to 1/ 0.025l mθ −= and 10.065m− in fig. 4. as can be seen from the figure, the absolute value of the axial stress when the maximum moment occurs is greater than the value at phase ( )α in all cross-section positions. in addition, the axial stress distribution when the maximum moment occurs is in good agreement with kecman’s proposal. it is confirmed from the above investigation that in the case of 2 1/ 1c c = and y bucσ σ≤ , the collapse type is not due to buckling at the compression flange and web, but due to plastic yielding at the flanges. therefore, to estimate the maximum moment by kecman’s theory is possible in this case. 0 0.05 0.1 0 50 100 150 200 250 0 0.5 1 1.5 θ / l [m−1] m [ n m ] σ x /σ y moment σ x (point b) σ x (point c) eq.(7) l = 300 mm t = 0.9 mm c1 = 50 mm c2 = 50 mm extension side compression side b cc2/4 σy / e = 1/1000 σx /σy =1(α ) (β ) fig.4. relations of / lθ and , /x ym σ σ for square tube with t=0.9mm, c1=50mm, c2=50mm 218 masuda: prediction of maximum moment of rectangular tubes subjected to pure bending 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5 σ x / σ y s / (2c1+2c2 ) s extension side compression side a b a b l = 300 mm t = 0.9 mm c1 = 50 mm c2 = 50 mm σy / ε =1/1000 kecman (α ) (β ) fig.5. axial stress distribution of cross-section for the square tube shown in fig.4 0 0.05 0.1 0 40 80 0 0.5 1 θ / l [m−1] m [ n m ] σ x /σ y moment σ x (point b) σ x (point c) eq.(4) eq.(1) l = 300 mm t = 0.4 mm c1 = 50 mm c2 = 50 mm extension side compression side b cc2/4 σy /e= 1/1000 (α ) (β ) fig.6. relations of / lθ and , /x ym σ σ for square tube with t=0.4mm, c1=50mm, c2=50mm figure 6 shows the relation of a tube curvature / lκ θ= and moment m for a square tube with t=0.4mm, c1=50mm, c2=50mm, / 0.001y eσ = ( 0.31buc yσ σ= ). and figure 6 also shows the relations of the tube curvature / lκ θ= and axial stress /x yσ σ at point b and c (refer to schematic representation of cross-section in figure 6). as can be seen from the figure, the maximum moment is in good agreement with the value of kecman’s proposal equation (4). the axial compression stress /x yσ σ at point b in the middle of compression flange decreases before the moment becomes maximum moment, and the maximum value /x yσ σ is in good agreement with the equation (1) of elastic buckling stress. in addition, the axial compression stress /x yσ σ at point c in the quarter of web width increases until the moment becomes maximum moment. figure 7 shows the axial stress distribution of cross-section at phase ( )α and ( )β corresponding to 1/ 0.012l mθ −= and 10.038m− in fig. 6. as can be seen from the figure, although the axial compression stress in the middle of compression flange decreases due to buckling in the middle of compression flange, the axial compression stress in both edges advances in systems science and applications (2011), vol.11, no.3-4 219 issn 1078-6236 international institute for general systems studies, inc. of compression flange increase because buckling doesn’t occur in both edges. immediately after buckling, stress increment in the both edges is greater than stress decrement in the middle of compression flange. therefore, total force of compression side and moment increase. in addition, the stress distribution of the web changes linearly because buckling doesn’t occur in the web. therefore, the axial stress distribution when the maximum moment occurs is in good agreement with kecman’s proposal by using an effective width in the compression flange. it is confirmed from the above investigation that in the case of 2 1/ 1c c = and y bucσ σ> , the collapse type is due to buckling at the compression flange. therefore, to estimate the maximum moment by kecman’s theory is possible in this case. 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 kecman σ x / σ y s / (2c1+2c2 ) s extension side compression side a b a b l = 300 mm t = 0.4 mm c1 = 50 mm c2 = 50 mm (α ) (β ) fig.7. axial stress distribution of cross-section for the square tube shown in fig.6 3.3 collapse mechanism which is different from kecman’s indication in the case of high-aspect ratio in which the web is wider than the flange, it was confirmed that collapse due to buckling at the compression web occur. 0 0.005 0.01 0.015 0 100 200 300 0 1 2 θ / l [m−1] m [ n m ] σ x /σ y moment σ x (point b) σ x (point c) eq.(5) l = 300 mm t = 0.5 mm c1 = 20 mm c2 = 100 mm extension side compression side b cc2/4 σy / e = 1/1000 σx /σy =1 (α ) (β ) fig.8. relations of / lθ and , /x ym σ σ for rectangular tube with t=0.5mm, c1=20mm, c2=100mm 220 masuda: prediction of maximum moment of rectangular tubes subjected to pure bending figure 8 shows the relation of a tube curvature / lκ θ= and moment m for a rectangular tube with t=0.5mm, c1=20mm, c2=100mm, / 0.001y eσ = ( 2 12.83 , / 5buc y c cσ σ= = ). and figure 8 also shows the relations of the tube curvature / lκ θ= and axial stress /x yσ σ at point b and c (refer to schematic representation of cross-section in figure 8). as can be seen from the figure, the maximum moment is less than the value of kecman’s proposal equation (5). in addition, the axial compression stress /x yσ σ at point b in the middle of compression flange increases until the moment becomes maximum moment, and the value /x yσ σ comes up to 1. and the axial compression stress /x yσ σ at point c in the quarter of web width decreases before the moment becomes maximum moment. figure 9 shows the axial stress distribution of cross-section at phase ( )α and ( )β corresponding to 1/ 0.036l mθ −= and 10.048m− in fig. 8. as can be seen from the figure, the axial stress distribution in the compression flange is constant value and the absolute value is almost 1 when the maximum moment occurs. and the axial stress distribution in the compression web doesn’t increase linearly. therefore, the sum of axial stress when the maximum moment occurs is less than the kecman’s proposal as much as it is shown by arrows of figure 9. 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5 σ x / σ y s / (2c1+2c2 ) s extension side compression side a b a b l = 300 mm t = 0.5 mm c1 = 20 mm c2 = 100 mm σy / ε =1/1000 kecman (α ) (β ) fig.9. axial stress distribution of cross-section for the rectangular tube shown in fig.8 it is found from the above investigation that in the case of high-aspect ratio and y bucσ σ< , the collapse type is not due to buckling at the compression flange but due to buckling at the compression web. therefore, to estimate the maximum moment by kecman’s theory is impossible in this case. figure 10 shows the relation of a tube curvature / lκ θ= and moment m for a rectangular tube with t=0.4mm, c1=50mm, c2=100mm, / 0.001y eσ = ,( 0.30buc yσ σ= ), ( 2 1/ 2c c = ). and figure 10 also shows the relations of the tube curvature / lκ θ= and axial stress /x yσ σ at point b and c (refer to schematic representation of cross-section in figure 10). as can be seen from the figure, the maximum moment is less than the value of kecman’s proposal equation (4). the axial compression stress /x yσ σ at point b in the middle of compression flange decreases before the moment becomes maximum moment, and the maximum value /x yσ σ is in good agreement with the equation (1) of elastic buckling stress. in addition, the axial compression stress /x yσ σ at point c in the quarter of web width decreases before the moment becomes advances in systems science and applications (2011), vol.11, no.3-4 221 issn 1078-6236 international institute for general systems studies, inc. maximum moment. figure 11 shows the axial stress distribution of cross-section at phase ( )α and ( )β corresponding to 1/ 0.007l mθ −= and 10.016m− in fig. 10. as can be seen from the figure, the axial stress in the compression flange is concentrated in the edges when the maximum moment occurs. and the axial stress distribution in the compression web doesn’t increase linearly. therefore, the sum of axial stress in the maximum moment is less than the kecman’s proposal as much as it is shown by arrows of figure 11 because the equation (2) applies to the axial stress distribution of compression flange, and linearly approximation doesn’t apply to the axial stress distribution of compression web. it is found from the above investigation that in the case of high-aspect ratio and y bucσ σ≥ , the collapse type is not only due to buckling at the compression flange but also due to buckling at the compression web. therefore, to estimate the maximum moment by kecman’s theory is impossible in this case. 0 0.02 0.04 0.06 0 100 200 0 0.5 1 θ / l [m−1] m [ n m ] σ x /σ y moment σ x (point b) σ x (point c) eq.(4) eq.(1) l = 300 mm t = 0.4 mm c1 = 50 mm c2 = 100 mm c b extension side compression side c2/4 σx /σy = 1/1000 (α ) (β ) fig.10. relations of / lθ and , /x ym σ σ for rectangular tube with t=0.5mm, c1=20mm, c2=100mm 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 kecman σ x / σ y s / (2c1+2c2 ) s extension side compression side l = 300 mm c1 = 50 mm c2 = 100 mm t = 0.4 mm a b a b (α ) (β ) fig.11. axial stress distribution of cross-section for the rectangular tube shown in fig.10 222 masuda: prediction of maximum moment of rectangular tubes subjected to pure bending 3.4 proposal method of maximum moment considering the web buckling 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 fem kecman’s proposal σ x / σ y s / (2c1+2c2 ) s extension side compression side l = 300 mm c1 = 50 mm c2 = 30 mm t = 0.4 mm present proposal (b) 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 fem kecman’s proposal σ x / σ y s / (2c1+2c2 ) s extension side compression side l = 300 mm c1 = 50 mm c2 = 100 mm t = 0.4 mm present proposal (a) fig.12. axial stress distribution in the range of buc yσ σ< : (a) by kecman’s method; (b) by present method figure 12 shows a schematic representation of axial stress distribution in the maximum moment after buckling of the compression flange ( buc yσ σ< ). figure 12(a) shows kecman’s proposal in which doesn’t consider the web buckling, and figure 12(b) shows present proposal in which considers the web buckling. as can be seen in the figure(b), an effective width ae applies to the compression web as well as the compression flange. it is assumed that the effective width ae is independent of initial web width as referring to karman’s theory [9]. a coefficient α in which represents the axial tension stress is derived in the following equation. ( ) ( ) 1 2 8 2 ea t a b y t α − = + − − 　 　 　 figures 13(a) and (b) show comparison between results of fem and proposals with the axial stress distribution. as can be seen in the figures, in the case of high-aspect ratio 2 1/ 2c c = , present proposal in which considers the web buckling is in good agreement with the result of fem. and in the case of low-aspect ratio 2 1/ 0.6c c = , kecman’s proposal in which doesn’t consider the web buckling is in good agreement with the result of fem. b a 2 eayσ 1y 1 1 y b y y σ − (a) b a 2 ea yσ 1y yασ b a 2 ea yσ 1y yασ (b) fig.13. axial stress distribution by results of fem, kecman’s proposal and present proposal: with (a) 2 1/ 2c c = ; (b) 2 1/ 0.6c c = figure 14 shows a schematic representation of axial stress distribution in the maximum moment when the compression flange doesn’t buckle ( buc yσ σ≥ ). figure 14(a) shows kecman’s proposal in which doesn’t consider the web buckling, and figure 14(b) shows present proposal in which considers the web buckling. as can be seen in the figure, an effective width ae applies to only the compression web. a coefficient β in which represents the axial tension stress is derived in the following equation. advances in systems science and applications (2011), vol.11, no.3-4 223 issn 1078-6236 international institute for general systems studies, inc. ( ) 1 2 9 2 ea a t a b y t β + = + − − 　 　 　 figures 15(a) and (b) show comparison between results of fem and proposals with the axial stress distribution. as can be seen in the figures, in the case of high-aspect ratio 2 1/ 5c c = , present proposal in which considers the web buckling is in good agreement with the result of fem. and in the case of low-aspect ratio 2 1/ 2c c = , kecman’s proposal in which doesn’t consider the web buckling is in good agreement with the result of fem. we show present method for estimating the maximum bending moment of rectangular tubes subjected to pure bending. in the case of buc yσ σ< , a position of the center of gravity in the tension web g is derived in the following equation. b ayσ / 2b yσ (a) b / 2ea yσ 1y yβσ (b) a fig.14. axial stress distribution in the range of buc yσ σ≥ : (a) by kecman’s method; (b) by present method 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 fem kecman’s proposal σ x / σ y s / (2c1+2c2 ) s extension side compression side l = 300 mm c1 = 20 mm c2 = 100 mm t = 0.5 mm present proposal (a) 0 0.5 1 −2 −1 0 1 2 fem kecman’s proposal σ x / σ y s / (2c1+2c2 ) s extension side compression side l = 200 mm c1 = 20 mm c2 = 40 mm t = 0.5 mm present proposal (b) fig.15. axial stress distribution by results of fem, kecman’s proposal and present proposal: with (a) 2 1/ 5c c = ; (b) 2 1/ 2c c = ( )1 1 1 10 3 2 g b y⎛ ⎞= +⎜ ⎟ ⎝ ⎠ 　 　 therefore, in the case of buc yσ σ< , the maximum moment in which the compression web buckles is derived in the following equation. ( )( ) ( ){ ( )( )max 1 1 2 2 2 2 (11) 2 2 4 e y e e abm t a t b t b y g a t b t aσ α α ⎫⎛ ⎞= − − + − + − − + − ⎬⎜ ⎟ ⎝ ⎠⎭ 　　　　　　　 where 1,y α and g are respectively, the value of equation (3), (8) and (10). it is found from the above investigation that in the case of buc yσ σ< , the maximum moment is derived in the following equation. ( ) ( )( ) ( )max . 4 , . 11 12m min eq eq= 　 　 224 masuda: prediction of maximum moment of rectangular tubes subjected to pure bending { }1 ( 2 )( ) 2 ( ) ( 2 )( ) 2 ( ) (13)max 12 2 4 ab em t a t b t b y g a t b t ay eσ β β= − − + − + − − + − moreover, in the case of buc yσ σ≥ , the maximum moment in which the compression web buckles is derived in the following equation. where 1,y β and g are respectively, the value of equation (3), (9) and (10). it is found from the above investigation that in the case of buc yσ σ≥ , the maximum moment is derived in the following equation. ( ) ( ) ( )( ) ( )max . 5 , . 7 , 13 14m min eq eq eq= 　 figure 16 shows comparison between results of fem and kecman’s proposal and present proposal with the maximum moment for two levels of /y eσ . as can be seen in the figure, the lower values of kecman’s proposal and present proposal is in good agreement with the results of fem. 0 0.005 0.01 0.015 0.02 0.025 0 0.5 1 1.5 2 2.5 t / c1 m m ax / (σ y c 1 c 2 t ) l = 300 mm c1 = 50 mm kecman’s proposal 2σy≤σbuc σbuc<σy σbuc<σy σy/ e = 1/1000 σy/ e = 1/500 present proposal c2 = 100 mm σy/ e = 1/1000 σy/ e = 1/500 comparison between results of fem and kecman’s proposal and present proposal with relation of 1/t c and ( )max 1 2/ ym c c tσ ⋅ ⋅ ⋅ for two levels of /y eσ . 4. conclusion in this paper, the investigation of the bending collapse for rectangular tubes by using numerical analysis of the finite element method was presented. for a rectangular tube in which the web is wider than the flange, it is found that collapse due to buckling at the compression web may occur. it is possible to estimate the maximum moment under various the material and geometrical properties by using the present proposal in which applies an effective width to the web, and the kecman’s proposal. references [1] kecman d. bending collapse of rectangular and square section tubes, international journal of mechanical sciences, vol.25 (1983) 623-636. [2] kim t. h. and reid s. r. bending collapse of thin-walled rectangular section columns, computers and structures, vol.79 (2001) 1897-1911. [3] lu g. and yu t. x. energy absorption of structures and materials, (2003) section5 crc pr i llc. [4] parh m. s. and lee b. c. prediction of bending collapse behaviours of thin-walled open section beams, thin-walled structures, vol.25(3) (1996) 185-206. [5] kyriakides s. and ju g. t. bifurcation and localization instabilities in cylindrical shells advances in systems science and applications (2011), vol.11, no.3-4 225 issn 1078-6236 international institute for general systems studies, inc. under bending, international journal of solids and structures, vol.29 (1992) 1117-1171. [6] chen d. h. et al. study on elastoplastic pure bending collapse of cylindrical tubes, transactions of the japan society of mechanical engineers, series a, vol.74(740) (2008) 520-527. [7] msc. marc manual, 2003. [8] guarracino f. on the analysis of cylindrical tubes under flexure: theoretical formulations, experimental data and finite element analyses, thin-walled structures, vol.41 (2003) 127-147. [9] karman v. t. et al. strength of thin plates in compression, trans asme, vol.54 (1932) мягкая вероятность ошибки при передаче информации advances in systems science and application(2016) vol.16 no.4 71-80 specifics of functional planning and architectural organization of religious educational complexes as a new type of educational institutions olga i. zhovkva kyev national university of construction and architecture, ukraine annotation the article identifies the specifics of shaping the architecture of religious educational complexes as a new type of religious educational institutions. on the basis of the consolidation of international scientific and practical experience in the design and construction of religious educational institutions, the author substantiated the functional structure and functional relationships of religious educational complexes, provided the typology and characteristics of main functional groups of premises. the article specifies the methods of functional and planning organization of religious educational complexes, including modular, pavilion, combined and compact, wherein the latter is regarded as optimal in case of designing religious educational complexes in an urban area. the planning schemes and a composition structure of religious educational complexes are also discussed. alongside with the development of complete compositional solutions of religious educational complexes, it is proposed to create dynamic evolving compositions, making it possible to carry out further reconstruction or construction of new functional units. the article also discusses the types of volumetric and spatial composition, identifies the main approaches to creating architectural expression and harmonizing the architectural space of religious educational complexes. key words: architecture of religious educational institutions, sacral architecture, synamic architecture, functional and planning organization, volumetric and spatial composition, universal space, transformability, harmonization. 1 introduction the experience of the development of civilization shows that education is the foundation of spiritual, social and economic development of society. in recent years, religious life in ukraine has significantly intensified. according to statistics, a steady increase in the number of religious organizations has been observed in ukraine since 1997. new religious buildings and complexes are constructed, which determines the need for clergymen, superiors and counsellors – graduates of religious educational institutions. the relevance of the research issue for ukraine is also demonstrated by the amendments recently introduced to some laws of ukraine by the supreme council of ukraine, related to the possibility of the foundation of educational institutions by religious organizations. for instance, official religious organizations are allowed to found higher, vocational, general education, preschool and out-of-school institutions that will give children from religious families an opportunity to receive education in a religious environment and strengthen moral values of the younger generation. currently, although ukraine is a multi-religious society, the architecture of educational institutions of some christian (catholic, protestant) denominations, as well as muslim and 72 olga i. zhovkva: specifics of functional planning and architectural organization of religious educational complexes… jewish institutions, remains understudied, despite the presence of a fairly large number of representatives of these denominations in the territory of ukraine. the issue of designing religious educational complexes as a new type of religious educational institutions has not been studied in both domestic and foreign science. identification of the specifics of shaping the architecture of religious educational complexes will make it possible to improve design solutions, optimize and improve their functional and planning structure, harmonize the relationships between the architecture and architectural environment. 2 methodology the study is based on the approaches that regard religious educational complexes as a complex system phenomenon, which requires a comprehensive study by related sciences such as philosophy, art history, religious studies, history. the following methods were used in the course of the study: field survey of religious educational institutions of ukraine, cis countries and countries outside the former soviet union; questionnaire survey of representatives and management of educational institutions; comparative analysis of foreign and domestic experience of design; system approach; environmental approach; graphic-analytical method and experimental modeling method. the method of field survey was used to study the current state and the experience of design and construction of christian, jewish and muslim religious educational institutions in the territory of ukraine and abroad (in russia, belarus, uzbekistan, poland, italy, turkey). the review of the current state of the design and usage of religious educational institutions in ukraine provided an opportunity to identify both disadvantages and positive aspects, while comparative analysis made it possible to determine the ways of further improvement and development of these institutions. the disadvantages of religious educational institutions in ukraine are the following: facilities and resources of 50-60% of the surveyed objects don’t meet contemporary requirements to modern educational process. the majority of religious educational institutions do not have their own premises, designed for academic process, and are housed in adapted buildings not having all necessary functional groups of premises. the applied method made it possible to identify the need for the modernization of existing religious educational institutions, as well as the need for the introduction of a new type of building – a religious educational complex designed with regard to the basic modern educational requirements. the method of questionnaire survey involved interviews and a survey among teachers and management of religious educational institutions. respondents had to answer, what the spiritual education will be like in the near future, in which direction it is necessary to improve exiting institutions, what a religious educational institution of the 21st century should be like. clergymen were also asked to answer the questions about the desired optimal functional composition of modern religious educational institutions and their spatial organization. on the basis of the analysis of a large number of answers and interviews it was assumed that there is a need for the introduction of a new type of integrated religious educational institution with a sufficiently large amount of functions – a religious educational complex. the introduction of the concept of a religious educational complex to the architectural vocabulary made it possible to offer new trends of shaping the architecture of religious educational institutions, aimed at improving the conditions for obtaining spiritual education. the method of comparative analysis was used to study the typology of religious educational institutions of ukraine, cis countries and countries outside the former soviet union. this method made it possible to identify similarity and differences in the architecture of religious advances in systems science and application(2016) vol.16 no.4 73 educational institutions of different countries, different denominations and types, to propose a universal functional structure of institutions that can be used by christian, muslim and jewish denominations when designing religious educational complexes. 3 the main part the creation of religious educational complexes (campuses) is one of perspective lines of the development of religious educational institutions. for the development of the concept of the creation of religious educational complexes it is necessary to recall the experience of medieval universities. towns-universities, such as oxford or cambridge, are the examples of modern cooperated educational establishments, comprising several educational institutions [20]. the contemporary experience of design and construction of religious educational institutions of the catholic church should also be taken into account [21]. therefore, alongside with the modernization, reconstruction and expansion of the resource base of existing religious educational institutions, an attempt should be made to organize campuses – complexes comprising several religious educational institutions of different levels. however, in order to provide the conditions for educational process in an educational establishment, it is necessary to carry out a series of studies to substantiate the functional composition and zoning of the complex with regard to the technological process, power supply and the specifics of operation. an educational unit comprises: a group of classrooms for training specialists of primary, secondary (including vocational) and hither level, as well as teachers’ rooms and rooms of departments. these rooms can be housed either in one unit or in several separate educational units. functional longevity of an educational unit will provide the flexibility and dynamism of its planning organization and structure [4]. if territorial resources permit, it is expedient to use a separate unit of a religious educational complex for the organization of production and training workshops (for icon-painting, goldembroidery, carpentry -specialized in manufacturing kiots and iconostases), as well as classrooms for choral singing and teaching choirmasters. a sacred unit comprises sacred rooms and a group of auxiliary premises (baptistery, acolyte’s room, sacristy). in the general composition of a religious educational complex, the sacred unit is usually placed separately, but if the territorial resources are limited it can have a form of an extension to educational, refectory or housing units of the complex, or have the form of a house church. the architecture of a sacred unit brings uniqueness and specificity into the composition of religious educational complexes of different denominations [15]. the architectural and planning composition of this unit has a number of traits that are common for the studied five major religions [17, 18,]. for example, the sacred core of an orthodox educational institution (an orthodox church) is formed by the following three parts: the altar, the place of prayer and the narthex [19]. the altar in the orthodoxy is regarded as a sacred spot; the middle part is a place of prayer for the congregation; the narthex is a scope of earthly existence. an orthodox church is considered to be a symbol of the universe, and its entire architecture symbolically embodies the christian idea of the connection between earthly and heavenly [3]. the altar of an orthodox church is always oriented to the east and separated from the prayer hall by the iconostasis, which is regarded as a baffle separating the two worlds [12]. a catholic church (the sacred core of a catholic educational complex) has, in general, the same structure as an orthodox church. an essential difference of its internal structure is the absence of the iconostasis. a catholic church also consists of three parts: the central part of the church; altar, or presbytery, where the holy gifts are stored; and the narthex [11]. the planning structure of a catholic religious building, in a similar way to the configuration of an orthodox 74 olga i. zhovkva: specifics of functional planning and architectural organization of religious educational complexes… church, can include special places for chorus and separate premises for church ministers and for the storage of vestments [9]. the altar of a catholic church is also oriented to the east. protestant houses of prayer are free from exuberant decorations, icons and sculptures. such decoration is unnecessary for protestants, because in this faith there is no worship of icons in the form in which it can be observed in catholic and orthodox churches [1]. a church can be housed in any building, which is rented or purchased; there are no strict prescriptions regarding the orientation of certain groups of premises. in general, however, the protestant theology does not contradict the theological decisions of oecumenical councils. protestant houses of prayer usually also have a three-part structure: the entrance area (a counterpart of narthex), the prayer hall itself and the sacred area with a place for a sermon. in the prayer hall of a mosque, next to the mihrab, a minbar (a pulpit for a preacher) is constructed, which is a counterpart of the pulpit of protestant religious buildings, the ambon in orthodox and catholic churches, the bimah in synagogues. as in the orthodox christianity, the rectangular foundation of mosques symbolized the earth, while the spherical dome – the heaven [13]. the internal arrangement of a synagogue is also based on the construction of a temple, which in turn recreates the internal design of the tabernacle (framed rectangular space with the sanctuary in the middle). that is why a synagogue usually has a rectangular shape. in the part where in a christian church there is a sanctuary, in synagogue there is a receptacle called aron kodesh, containing torah scrolls. in the center of a synagogue there is an elevated platform from which the torah is read – the bimah [8]. synagogues must be placed in such a way that the wall, near which the aron kodesh is placed, is directed towards jerusalem. for jewish synagogues this means orientation to the east, as well as for christian and muslim religious buildings. in the planning structure of a settlement a synagogue is usually placed at the highest point, which is also characteristic for religious buildings of the above-mentioned denominations. for this very reason the sacred core of the general composition of a religious educational complex should play a dominant part in the general architectural and spatial composition. thus, the above-discussed sacred units of religious educational complexes of five major religions as an integral component of such complexes, while definitely having their distinctive features, have also much in common. an administrative unit of religious educational complexes comprises the following service rooms: an office of a superior (a rector), offices of deputies, accounting department, administration, etc.; a group of health care premises; a group of auxiliary and subsidiary premises (premises for the production of candles); a refectory group. multifunctional religious educational complexes may also comprise a missionary and charity unit, in which groups of premises for missionary-volunteer activities should be provided; a group of premises for psychological and physical rehabilitation of children and adults who find themselves in difficult circumstances; rooms for counselling with priests, psychologists, lawyers; rooms for temporary stay of people who need shelter; a medical aid station; a refectory; sacred rooms. the analysis of the structures of existing religious educational institutions showed that in order to ensure their functional longevity or the possibility of technical modernization with regard to modern requirements of constantly changing educational process it is reasonable to use universal units with flexible planning, on the basis of which dynamically developing systems can be organized [4]. due to the principle of universality, a unit can perform an educational, administrative or recreational function. such a unit is a single flowing space, divided in a way determined by functional reasons. only basic structural elements of this unit are fixed: vertical structures and utility systems, as well as main load-bearing structural elements [5]. advances in systems science and application(2016) vol.16 no.4 75 a general functional structure of major religious educational complexes (intended for 700 and more students) contains the maximum number of functional units, which are united into architectural and planning schemes by means of functional and compositional relationships. the units of of religious educational institutions form various volumetric and spatial compositions, among which the following types can be specified: closed, semiclosed and open (table 1). table 1. types of volumetric and spatial composition of religious educational complexes closed open now let us focus on the types and the analysis of the basic groups of premises, which form the above-discussed units of religious educational complexes, and propose a typology of premises (table 2). the entrance group of premises of various functional units usually consists of an entrance hall, a cloakroom and sanitary facilities. the primary purpose of the entrance hall is to rationally organize the flows of students, teachers and guests of the complex. the entrance hall of a religious educational complex can be used not only for serving visitors, but also for educational, missionary and club work; it can also serve as a reception room. such an entrance hall/reception room can be designed as open space, forming a part of back rooms or a corridor. the optimal solution is the one that makes it possible in case of necessity to separate the entrance hall as an independent space using sliding partitions. the main groups of premises in modern religious educational complexes are training and applied training groups of premises comprising classrooms, lecture rooms, music and rehearsal classrooms, choral singing classrooms, icon-painting, gold-embroidery, cabinetry and carpentry workshops. classrooms intended for 10-25 students are the main premises of the complex, because students spend there most of their time. the most optimal form of a school room may be a rectangular shape with a longitudinal external wall and the dimensions in the axes 3 * 6 m, 6 * 9 m. such proportions make it possible to arrange students’ desks in three rows, thus ensuring the optimum illumination of all work areas. using another constructive decision makes it possible to obtain a classroom of the third type, intended for 25-50 students. when designing religious educational complexes, more complex forms of classrooms can also be used: trapezoidal, pentagonal, hexagonal, etc. classrooms having a trapezoidal shape make it possible to conveniently group desktops with the possibility of allotting a separate area for a personal prayer. in the premises having a pentagonal or square shape it is easier to organize workplace illumination using natural light [6]. however, nowadays the requirements to the illumination of premises cannot be considered to be most important ones, because students often use a pc for capturing information received from a teacher; in this case the presence of natural lightning is not a mandatory requirement. nonstandard forms of classrooms open new possibilities for architectural and artistic solution of the interior of a religious educational complex, providing more freedom in arranging equipment and making it possible to organize a sacred area in classrooms. 76 olga i. zhovkva: specifics of functional planning and architectural organization of religious educational complexes… table 2. typology of premises multipurpose room classroom for 20-25 students, audience workshop reading room assembly hall exhibition room by virtue of the latest technologies, currently it is not necessary to organize equipment of a classroom in such a way that all the attention is focused on a teacher’s work area. the possibility of installing several monitors in a classroom to broadcast the lecture material eliminates the need to focus attention on one area, which also makes it possible to arrange workplaces not in a straightline scheme, but radially or in segments. the survey showed that the process of training in higher religious educational is effective in classrooms with a capacity of no more than 25-30 students for seminaries and 20-25 students for academies. advances in systems science and application(2016) vol.16 no.4 77 lecture rooms should be intended for 50-100 students. a progressive and cost-effective direction of the universalization of educational units and creating audiences of large capacity is the possibility of the transformation of medium sized lecture rooms into larger ones. handicraft rooms (icon-painting, gold-embroidery, cabinetry workshops) should be placed on the ground floor; it is also possible to place them in basements with mandatory natural lighting. the cutural and recreational group of premises comprises library premises, a publishing center, internet rooms; a group of premises for meetings and discussions; club and entertainment premises, exhibition rooms, halls, lounges. a library is the embodiment of an educational institution, its intellectual center, where the knowledge of previous generations is stored. it is formed on the basis of the premises of a reading room with an area for working with books and periodicals, reading newspapers and viewing microfilms, listening to audio records, the premises of circulation department, catalogue, storeroom, open-access fund. the size of a reading room depends on the number of students in the religious complex. the transition to the electronic storage of information eliminates the need for the organization of reading rooms in the classic sense with fixed workplaces. students can receive required materials, books or publications in an electronic form and process them in any place of the educational complex, which is most comfortable for the work. the most important features of a library are flexibility and multi-functionality – the possibility of using the working rooms also for classes, meetings, exhibitions. a group of entertainment premises is very important in the structure of a religious educational complex. the assembly hall of a religious educational complex is generally used to hold solemn meetings, deliver lectures, etc. when designing an assembly hall, it is necessary to provide for the possibility of its multiple use, for example, as a conference hall, a concert or an exhibition hall. one of the options is using the assembly hall as a large lecture room. besides, an assembly hall should have the capability to be transformed into a number of smaller audiences intended for 50-100 places, which will increase the efficiency of its use. modular planning should be used when designing medium-capacity religious educational complexes. efficiently connected functional units will ensure the optimal arrangement of the basic functions of religious educational complexes in individual units, provide a communication link between units through covered walkways or galleries. the advantage of this method of designing religious educational complexes is the possibility to isolate units and simultaneously provide a direct connection with the entrance area. pavilion planning is most widely used when designing medium-sized and large religious educational complexes. on the basis of this planning method it is possible to extend religious educational institutions through the construction of additional buildings and link units. this type of planning structure makes it possible to use plots with a complex terrain [10]. a disadvantage of this composition is the elongation of the functional links between units and groups of premises. combined planning is used when designing large-capacity religious educational complexes. it makes it possible to create expressive architectural compositions, optimize functional links, ensure optimal orientation of premises. according to the planning scheme, several types of compositions of religious educational complexes can be identified: t-shaped, h-shaped, e-shaped, o-shaped, perimetral, rectilinear, curvilinear, etc. all existing compositions are the result of the improvement of functional and technological schemes and the aspiration to improve the architectural space of religious educational complexes. according to the communication structure, the following compositional schemes of religious educational complexes can be identified: corridor-type, non-corridor, hall, enfilade and combined. 78 olga i. zhovkva: specifics of functional planning and architectural organization of religious educational complexes… corridor schemes are characteristic of linear buildings of religious educational complexes. a corridor system consists of cells that are linked by a common linear communication – a corridor. a non-corridor system integrates cells containing a part of the cycle of the general process in a religious educational complex. the link between the cells is represented by premises – a hall or a lounge, rather than a linear communication. such premises usually have their own particular purpose – a place of meetings, recreation, discussions and debates. the schemes described above can be used when designing religious educational complexes of different denominations (orthodox, catholic, protestant, muslim, jewish). when designing religious educational complexes, it is reasonable to combine the above-listed schemes, thus obtaining complex mixed and dynamic compositions. the prevalent compositional techniques of shaping and harmonizing the internal space of religious educational complexes are coloristics and ergodesign techniques that help create a truly comfortable artistic environment for effective learning and spiritual development. 4 discussion the study is based on the works by both domestic and foreign scientists in different areas of science: scientific works on religion by p. florensky; scientific works on the history of sacral architecture and the history of the church by n. budur, i. bondar'; scientific works, devoted to the study of architecture of religious educational institutions, by o. zhovkva, r. stots'ko and v. proskuryakov. the study is also based on the works, devoted to the general theoretical problems of designing educational institutions, by such scientists as v. ezhov and p. solobay. the study also relies on the works, devoted to the general theoretical and practical problems of architecture and the development of sacral architecture, by a. gutnov, a. gayduchenya, e. voznyak, sh. shukurov, o.boyko, e. kotlyar, v.kutsevich et al. these authors address the issues of shaping an architectural and artistic image, compositional solutions of architecture objects, including sacred ones. the architecture of orthodox and catholic religious educational institutions is also discussed in the monograph by r. stots'ko and v. proskuryakov “architecture of religious educational institutions of the ukrainian greek catholic church” and the monograph by o. zhovkva “architecture of orthodox religious educational institutions of ukraine", in which the authors conducted a study of the architecture of religious educational institutions of two influential christian denominations and analyzed the typology of these institutions. "selected works on art" by the priest p. florensky provide a deeper understanding of the symbolic essence of an orthodox temple and sacred art. in the work “iconostasis” florensky paid much attention to the canonical interpretation of the configuration of an orthodox temple and its altar. according to florensky, “... the altar means a human soul, and the temple – a body" [15]. this work made it possible to deeper understand the essence of the configuration of a temple as an integral part of an orthodox educational complex. the issues of the universality and transformability of space were discussed in the works by a number of scientists and architects, such as a. gutnov, a. gayduchenya. for example, a. gutnov in the book "world of architecture" discussed the issue of universal and free space of buildings and structures, relying on the idea of free planning declared by le corbusier [14]. using universal spaces and units in designing religious educational complexes makes it possible to improve the rationality of functional planning organization of such complexes. the above research and writings offer an opportunity to understand the state of knowledge on the issue of designing religious educational complexes and sacred buildings as their integral part. advances in systems science and application(2016) vol.16 no.4 79 these works cover the issue of temple construction and shaping the architecture of religious educational institutions. however, the issue of the complex shaping of the architecture of religious educational complexes of the major world religions remains understudied. that is why the issue of the development of architectural and planning solutions of religious educational complexes requires a comprehensive study and the preparation of relevant scientific guidelines and design recommendations. 5 conclusion a modern trend in designing religious educational complexes is the pursuit of universality and ensuring the possibility of the versatile use of premises and units, which would eliminate the problems of fast “moral” depreciation and obsolescence. the internal segmentation of such universal structure is carried out using facilities not associated with basic design structures: mobile partitions, walls, screens, etc. the basic requirement to the functional planning organization of premises of religious educational complexes is the possibility of flexible transformation, free planning. free planning will not only ensure the freedom in the search for optimal planning solutions and flexible transformation during the operation of a complex, but also brings into the internal space something new through the use of transformable partitions and bay window partitions that will make the interior more expressive. currently, a religious educational complex is understood as an environment that corresponds to the aesthetic and philosophical views of our time and provides favorable conditions for study, spiritual and physical development, self-improvement, teaching and scientific activity, accommodation and recreation. a religious educational complex should ensure the proper functioning of the educational process, provide an opportunity to hold meetings, exchange thoughts and have a rest. a religious educational complex is not only a temple of science, but also an environment designed for the education of an intelligent and humane personality. therefore, the architecture of a religious educational complex as a temple of science should keep up with the times [16]. the proposals, obtained as a result of the study, may be used in the process of designing religious educational institutions and complexes. the study determined the prerequisites for the development of a modern religious educational institution, as well as the need for the systematization of the scientific knowledge on architectural design and forecasting the development of religious educational complexes. on the basis of the consolidation of international scientific and practical experience in the study, design and construction of religious educational institutions the author substantiated the need for the development of a new type of a religious institution – a religious educational complex, its functional composition, structure and functional links. the study proposed a typology of the premises of religious educational complexes and a list of functional groups of premises of a modern religious educational complex including: educational, applied training (workshops), sacred, cultrural, recreational, sports, administrative, medical, refectory, utility and housing groups of premises. the paper specifies the methods of functional and planning organization of religious educational complexes – modular, pavilion, combined and compact, wherein the latter is regarded as optimal in case of designing religious educational complexes in an urban area. acknowledgment the study was conducted within the framework of the plans of research works of the kiev national university of building and architecture. 80 olga i. zhovkva: specifics of functional planning and architectural organization of religious educational complexes… the study was carried out with scientific and methodical support of the saint petersburg state university of architecture and civil engineering. some of the study results have been more than once reported at international scientific-practical conferences. the author expresses special gratitude to the staff of the design institute “mosproekt-2" named after m.v. posokhin for design and engineering support in the preparation of this study. references [1] bondar' and i.a.(2011). "lutheranism in the context of common european spiritual development: specifics of religious and cultural relations", thesis for the degree of candidate of philosophical sciences, kiev. p. 13 [2] boyko, o.g.(2010). "architecture of stone synagogues of right-bank ukraine of the 16th20th centuries", thesis for the degree of candidate of architecture, lviv. [3] budur, n.(2009). orthodox temple. moscow: rossa. p. 17 [4] gayduchenya, a.a.,(1983). dynamic architecture: main directions, development, methods. kiev: budivelnik. pp. 10-11 [5] gutnov, a.e. (1985). world of architecture: language of architecture. moscow: molodaya gvardiya. pp. 74-77 [6] yezhov, v.i.(1977). technology of designing school complexes. moscow: centre of scientific and technical information on civil engineering and architecture. p. 11,14. [7] zhovkva, o.i.(2009). architecture of orthodox religious educational institutions of ukraine. chernivtsi: llc druk art. [8] kotlyar, e.a.(2001). "synagogues of ukraine in the second half of the 16th century – early 20th century as a historical and cultural phenomenon", thesis for a degree of a candidate of art criticism, kharkiv. p. 9 [9] kutsevich, v.v. (ed.)(2002). religious buildings and structures of various denominations: design guidebook. kiev: kievzniiep. p. 70 [10] solobay, p.a.(2012). "typological foundations of shaping the architecture of higher education complexes", thesis for a degree of doctor of architecture, kiev. p. 15 [11] stots'ko, r.z. and v.i. proskuryakov(2009).architecture of religious educational institutions of the ukrainian greek catholic church: monograph. lviv: publishing house of the national university “lviv polytechnic”. p. 60 [12] florensky, p.a.(1996). selected works on art. moscow: mysl'. p. 97 [13] shukurov, sh.m.(2014). architecture of a modern mosque. origins. moscow: progress – tradition. pp. 46-47 [14] hasan uddin khan, p.j.(1998). international style: modernist architecture from 1925 to 1965. cologne: taschen. pp. 27-36 [15] jodidio, ph.(1997). new forms: architecture in the 1990s. cologne: taschen. pp. 113-114. [16] radoslav zuk’s seventh in a series of ukrainian catholic churches, built on geometric systems in holy trinity church, kerhonkson ny(1978). progressive architecture, no.10, pp.64. [17] religious architecture(1992-1993). techniques et architecture, no.405, pp. 34-66. [18] two churches(1986). progressive architecture, no.2, pp.81-82. [20] schöber, u.(2007). 1000 kirchen und kloster. cologne: naumann & göbel. pp. 49 [21] architektura. warszawa (1980). "ukrainian catholic clergy in diaspora (1751-1988): annotated list of priests who served outside of ukraine". rome. no.5, pp. 19-34 [23] contemporary materials recreate heritage(1988). building & construction, no.5, pp.47. [24] dom suits both sacred and secular events(1988). building & construction, no.5, pp.4445. corresponding author olga i. zhovkva can be contacted at: yal05@mail.ru. advances in systems science and applications (2012) vol.12 no.1 18-26 empirical analysis of credit risk measure using fuzzy adaptive neural-networks:for small and medium-sized enterprises zhichun xu and zongjun wang school of management, huazhong university of science technology,wuhan 430074 p.r.china abstract measuring credit risks of small and medium-sized enterprises should take a series of unique characteristics into consideration. firstly, non-financial conditions should be combined to produce more accurate evaluation results. secondly, to overcome the vague of non-financial and sharp boundaries of financial conditions, fuzzy approaches and a nonlinear classifier based on fuzzy adaptive neuralnetwork(hereinafter referred to as “fan”) were introduced. a sample family of 65 firms was collected to test the performance of the above classifier, among which 29 firms were default and 36 firms were good. the result was shown as roc curves. to compare the performance of the model, the empirical results of another 4 classifiers were shown at the same figures and tables. the results showed that fan has the best performance and can effectively distinguish the good and default firms. keywords fuzzy adaptive neural-network, small and medium-sized enterprises, credit risk, non-financial conditions 1 introduction it is well known that small and medium-sized enterprises (smes) have a series of unique characteristics differing from large enterprises. for instance, many small and medium-sized and businesses are lack of audited financial statements, and the reliability of their financial statements is doubtful to lenders. also, non-financial factors of the borrowers have deep impacts on the probability to meet their future obligations. these non-financial factors include the owners’ personal finances, as well as other attributes like their morality and beliefs. at last, macroeconomic environments and industry policies have important influence to operating of the firms. based on the above reasons, when we construct model to measure credit risk of smes, the unique characters should been taken serious consideration. in this paper, we will concentrate on these characters and put forward to a nonlinear classifier based on fuzzy adaptive neural-network to solve these problems.(einstein, 1954). the paper is structured as follows. the next section provides a survey of the technologies about credit risk modeling for smes and related work. section 3 describes our data set used in the paper. section 4 presents the feature extraction and the procedure of factors formation. section 5 discusses the empirical results advances in systems science and applications (2012) vol.12 no.1 19 and analyzes the results together with a comparison between the two methods. we conclude the paper in section 6. 2 technologies about credit risk measure for smes unlike many literatures emphasizing financial conditions, edmister (1988) claimed that numerical financial ratios and human credit analysis could be combined to produce more accurate evaluation results for credit risk measure. neglecting the information provided by these qualitative factors might result in undesirable consequences[1]. based on the judgments, many non-financial conditions were introduced to model the credit risk of smes. but the non-financial conditions always were linguistic and vague, they could not be easily treated like financial ratios in linear discriminant models such as mda and logistic regression model. even for financial information, the sharp boundaries also need seriously treated[2].to overcome this problem, fuzzy approaches were introduced. rommelfanger (1999) proposed the use of fuzzy logic to check the credit solvency of small business firms [3]. weber (1999) used fuzzy logic for credit worthiness evaluation[4]. syau et al. (2001) used fuzzy membership function to model the credit worthiness of an enterprise[5]. chen (1999) presented a fuzzy credit-rating approach to deal with the problem arisen from the credit rating table used in taiwan. the evaluation criteria were modeled as a hierarchical decision structure, and fuzzy integral was employed for aggregating credit information[6]. resent research has been done by jiao et al (2007). in their paper, fuzzy adaptive network was used to model the credit rating of small financial enterprises. the data of the credit rating problem was first represented by the use of fuzzy numbers. the network based on inference rules was constructed and was trained or learned by using the fuzzy number training data. because of the learning and the linguistic representation ability of the fuzzy adaptive network, they showed that it was ideally suited for the modeling of credit rating[2]. compared to jian’s work (2007), we consider the non-financial information when we classify smes by using fuzzy approaches together with neural-network. furthermore, we conduct factor analysis as a pretreatment process to reduce the complexity and detect structure in the relationships between variables.in addition, we apply analytic hierarchy process to get the weights of non-financial conditions. the input vectors have 5 dimensions, the output has only one dimension. the result will show on roc curves. 3 data acquisition statistics on smes are problematic almost by nature: the majority of them aren’t public companies and the financial information together with non-financial infor20 zhichun xu:empirical analysis of credit risk measure using fuzzy adaptive neural-networks. . . mation can’t be obtained publicly. in contrast to most other western countries, there is no long official default history that can be used. only by combining different sources, and expensive and labor-intensive process, can a usable database of default be constructed. as far as default companies are concerned, we adopt the definition in the basel accord ii. the basel ii (2004) suggest a conservative definition of default for a bank, that a default is considered to have occurred with regard to a particular obligor when the obligor is past due more than 90 days on any material credit obligation to the banking group, or the bank considers that the obligor is unlikely to pay its credit obligations to the banking group in full, without recourse by the bank to actions such as realizing security. the accounting data used in this study are collected by a state-owned bank’s credit management system. because the systems have been run since 2002, we get the data in the period between 2002 and 2008. when a firm is granted a credit rating by the bank, the bank’s loan managers are obliged to register their customers’ fixed information at first time together with accounting report each quarter of every year. during the period of 2002 to 2008, a total of 65 loans are selected. the data set comprises of 36 good loans (gl) and 29 bad (or default) loans (bl). all loans are under the normal loan scheme. all of the 65 loans correspond to 65 firms. the financial conditions are collected from the credit management system. the ratios are calculated based on the data in the balance sheets and income statements. the ratios of the default firms are calculated by the data of the years before default. the ratios of the good firms are computed by the newest data we can get. to obtain the objective evidence according to a company’s non-financial conditions, a reliable way is to collect the views of experienced bank loan officers who are asked to assess the degree of each information source using five linguistic terms, namely“very good”,“good”,“fair”,“bar”, and“very bad”. we collect the 65 firms’ basic information, and select 12 factors representing 4 kinds of abilities. the 12 non-financial factors are selected referring to the practices of loan risk classification of chinese commercial banks. we produce consulting tables and distribute them to 30 bank’s account managers and loan officers, and receive 24 answers. the answers are expressed by the above five linguistic. all the financial conditions and non-financial conditions are listed in table 1. 4 feature selection and factor formation described as above section, the financial conditions listed as table 1 have 10 items and non-financial conditions have 12 items. if all of them are used as inputs when we train fuzzy adaptive neural network, the nodes in hidden layer will become too large to properly work. in order to reduce the number of variables and to advances in systems science and applications (2012) vol.12 no.1 21 detect structure in the relationships between variables, an effective method to deal with financial conditions is the application of factor analytic techniques. we can combine the variables into several factors and reduce the complexity. table 1 variables used in the model financial conditions(f) non-financial conditions −−debt payment ability −−debt payment ability(nf1) debt-to-asset ratio(f1) period matching of liabilities and assets(nf11) quick ratio(f2) ability of financing from outside in time(nf12) current ratio(f3) collateral(nf3) −− earning ability(nf2) −− earning ability sustained profitability(nf21) return on assets(f4) core business(nf22) return on net sales(f5) quality of profit(nf23) −− management ability −− management ability(nf3) total assets turnover(f6) innovation ability(nf31) inventory turnover(f7) customer degree of satisfaction(nf32) equipment and technologies(nf33) −− growth ability −−characteristics of company and owner(nf4) sale growth ratio(f8) education degree of the owner(nf41) net asset growth ratio(f9) moral behavior of the owner(nf42) net profit growth ratio(f10) specifications of management(nf43) differ from those financial conditions can be expressed by numerical ratios, non-financial conditions are the problem we must seriously consider. as a result, we firstly put our focuses on the fuzzy processing of non-financial information. similar to the work of jiao(2007), our purpose is to get an aggregative score to express the information. in order to get the total score, we should calculate the objective evidence of each information source. to obtain the objective evidence according to a company’s non-financial conditions, we get five linguistic expressions by the above section’s methods. for the subsequent fuzzy operations, these linguistic terms should be translated into fuzzy numbers. we use five triangular fuzzy numbers to describe these linguistic terms according to a kind of conversion scale. the definitions of these fuzzy numbers are (80, 90, 90), (70, 80, 90), (60, 70, 80), (50, 60, 70), and (50, 50, 70), respectively, as shown in fig.1. by the fuzzy numbers, we can compute the average numbers of the non-financial conditions of each firm. it should be noted that another problem is to decide the weighting of each nonfinancial criterion. since this weighting depends on the particular problem at a particular time, the decision-maker usually determines it. an effective method to 22 zhichun xu:empirical analysis of credit risk measure using fuzzy adaptive neural-networks. . . fig.1 membership functions of the five rating levels for non-financial conditions build up the weighting structure is analytic hierarchy process. since the focus of this paper is to illustrate the application of fuzzy logic and neural network, the process of analytic hierarchy won’t be described in detail. we simplely give the result in table 2. table 2 normalized weights for the non-financial conditions category variables nf11 nf12 nf13 nf21 nf2 nf23 weights 0.23 0.45 0.32 0.25 0.35 0.30 variables nf31 nf32 nf33 nf41 nf42 nf43 weights 0.19 0.23 0.42 0.31 0.53 0.16 based on fuzzy arithmetic the aggregation of n fuzzy triangular numbers xi =( xi l, xi c , xi r ) ,, i=1, 2, 3, .n, with normalized weights w = (w1,w2, . . . ,wn) can be carried out using the following equations: ac = n∑ i=1 wixi c al = n∑ i=1 wixi l ar = n∑ i=1 wixi r (1) the final aggregated triangular fuzzy number is assumed as a = ( al, ac , ar ) . using the normalized weights listed in table 2, the triangular fuzzy numbers for non-financial conditions and eq. (1), the aggregated triangular fuzzy numbers for the four non-financial conditions categories of each sample can be obtained, respectively. in order to compare or rank the credit ratings of the fuzzy numbers obtained above, the overall existence ranking index (oeri) will be used. for triangular fuzzy numbers, the overall existence measure om for fuzzy set a can be written advances in systems science and applications (2012) vol.12 no.1 23 as: om (a) = 4al − ac + ar 4 (2) using eq.(2), we can obtain the oeri ratings of each sample company. we describe the process of feature selection as follows. step1. use factor analytic approach to treat the financial conditions. as a result, we obtain four main components. step2. build up the membership functions of the five rating levels for each non-financial criterion and the result criterion. step3. decide the weights of each criterion using analytic hierarchy process. step4. calculate the objective evidence of the four non-financial conditions categories by aggregating the triangular fuzzy numbers and rank the fuzzy numbers using the overall existence ranking index approach. finally, a ranking numerical value for each sample can be obtained. using factor analysis technique, we can combine 10 variables into several main components. by analyzing the data described above, we get four main components representing the four categories respectively. together with the aggregative score representing the non-financial conditions, we get five factors which represent the characters of a company and construct input vectors with five dimensions. 5 classification procedure and results based on the above discussions, every sample company is represented by a five feature vector. the classifier will be constructed by fuzzy adaptive neural network. in order to check the performance of the classifier, a generalized dynamic fuzzy neural network (hereinafter referred to as “gdfnn”) introduce by wu et al.(2007), a common bp neural network and logistical regression classifiers are used to process the data respectively. a method name round-robin will be used to extract training samples and checking samples. during every round, n-1 samples are used as training samples and the remaining one is used as checking sample. the process will be repeated until all samples are checked once. the performance of classification will be evaluated by roc methodology. the experiments were run on matlab7.0. the results are shown as fig.2,fig.3 and table 3. by using fan classifier, the roc curve line area(az) reached 0.959, and corresponding true default rate and false default rate were 89.6% and 11.2%,respectively, and the overall rate of accuracy was 89.2%. another classifier is constructed by gdfnn, a general dynamic fuzzy neural network. the network is constructed based on rbf network and had four layers. the first layer is input layer. the second layer is membership function layer. the third layer is if-then rule layer and the last layer is normalizing layer. the gdfnn is equivalent to tsk fuzzy logic system. the classifier is trained 24 zhichun xu:empirical analysis of credit risk measure using fuzzy adaptive neural-networks. . . like the above fan classifier. and the result is shown as fig.2 either. the roc curve line area(az) reach 0.893, and the overall rate of accuracy is 81.5%. the classifier bp haven’t carried out fuzzy computation, and az=0.714, the rate of accuracy is 64.5%. the result of logistic regression classifier is listed as fig.2 roc curve created by fan,gdfnn and bp classifier fig.3 roc curve created by fan table 3. one can find that the rate of accuracy was lower than fan and gdfnn, but higher than bp. the above four classifiers are tested by financial information and non-financial information. in order to test the effect of non-financial information, we build up a contrast experiment. the classifier is fan, and the input vectors are conditions advances in systems science and applications (2012) vol.12 no.1 25 including non-financial and not including non-financial condition. the result is shown as roc curves. the higher curve is for the input with non-financial conditions just as fig.1, and the lower curve line is for the input without non-financial conditions. the az is equal to 0.959 and 0.757 respectively. the results show that non-financial conditions can raise the rate of accuracy of default. table 3 classification table predicted classifier observed group percentage correct .00 1.00 group 0 28 8 77.7 logistic regression group 1 4 25 86.2 overall percentage 81.5 group 0 32 4 88.8 fan group 1 3 26 89.2 overall percentage 89.2 group 0 30 6 83.3 gdff group 1 6 23 79.3 overall percentage 81.5 group 0 25 11 69.4 bp group 1 12 17 58.6 overall percentage 64.5 6 conclusion small and medium-sized enterprises have uniquely financial features, models and methods to measure their credit risk need special treatments. in this paper, we have presented that non-financial conditions are helpful to produce more accurate evaluation results. in order to overcome the vague of non-financial and sharp boundaries of financial conditions, fuzzy approaches and a nonlinear classifier fuzzy adaptive neural network are conducted. our numerical results demonstrated that fuzzy adaptive neural network is ideally suited for the modeling of problems such as credit rating or other financial systems when the variables or even the problem are vaguely defined. in addition, we showed the learning ability can improve the initially represented approximate model as more data become available. references [1] r.o. edmister. (1988), “combining human credit analysis and numerical credit scoring for business failure prediction”, akron business economic re26 zhichun xu:empirical analysis of credit risk measure using fuzzy adaptive neural-networks. . . view, vol.19, no.3, pp.6-14. [2] y. jiao, y.r. syau, and s.e lee. (2007), “modeling credit rating by fuzzy adaptive network”, mathematical and computer modeling, vol.45, pp.717731. [3] h.j.rommelfanger. (1999), “fuzzy logic based systems fro checking credit solvency of small business firms”, soft computing in financial engineering, pp.371-387. [4] r.weber. (1999), “applications of fuzzy logic for credit worthiness evaluation”, soft computing in financial engineering, pp.388-401. [5] y.r. syau, h.t. hsieh, e.s. lee. (2001), “fuzzy numbers in the credit rating of enterprise financial conditions”, review of quatitative finance and accounting, vol.45, pp.351-360. [6] l.h. chen, t.w.chiou. (1999), “a fuzzy credit rating approach for commercial loans, a taiwan case, omega international”, journal of management science, vol.27, pp.407-419. corresponding author zhichun xu can be contacted at: spring0267@yeah.net advances in systems science and application(2015) vol.15 no.1 37-44 a subspace multi-relation model for the expression of system complexity dongyun yi1,2, yangyang liu2, xiaojie wang2, zheng xie2, xianggui qu3, chengli zhao2 and chengping hou2 1 national key laboratory of high performance computing, changsha, hunan, 410073, china; 2 department of mathematics and system science, national university of defense technology, changsha, hunan, 410073, china; 3 department of mathematics and statistics, oakland university,rochester, michigan, 48309, usa. abstract modeling expression of complexity is one of the frontier research fields in system science. classical graphical models characterize two-relations but lack of the ability of depicting multi-relations among objects in a system. in this paper, a multi-relation expression model is proposed based the subspaces of the attribute space of objects in a system. properties of the multi-relation model are analyzed and an algorithm is provided to find the multi-relation subspaces in a data set. the relationship between the multi-relation model and a hyper graphical model is also discussed. keywords complexity; relation; subspace; hyper-graph. 1 introduction there are two types of research in studying the complexity of a system [1-3], one considers how to characterize and compute the overall complexity of a system [4]. boltzmann entropy in statistical physics is such a representative which gives an overall measure for the system status. the other type of research describes the interactive relationships among various objects in a system. graphical models from graph theory or complex network are typical representatives [5]. a graphical model regards the objects in a system as vertices and relationships among objects as edges that can be undirected, directed or weighted. graphical models have a lot of applications in the analysis of e-mail data, communication data, web data, biological data, financial data, social networks, actor networks, etc. [6-8]. however, graphical models only depict two-relation between objects, lack of the description of multi-relation in a system in practice. recently, some research articles discuss the multi-relation using hyper-graph models [9-13], but no rigorous definitions and generating methods of hyper-graphs from data are given. in this paper, we propose a subspace multi-relation model to characterize the this research is supported by the natural science foundation of china(no.61473302), the basic research project of nudt and the open fund from hpcl. 38 dongyun yi : a subspace multi-relation model for the expression of system complexity different types of relationships in the attribute space of objects. in this model, each relationship exists in its own subspace, which is a part of the attribute space. the traditional graph model is a special case of our model, which can be called one-dimension relation model.based on the subspace clustering principle [14-16], we give the algorithm to find subspace multi-relation from the object attribute space, which is helpful to understand the complexity of data from multiple perspectives. real data are used to verify the validity of our model. we discuss the vector hyper-graph models at the end. 2 methods 2.1 definitions of subspace multi-relation in this section, we define the subspace multi-relation model. definition 2.1 if m objects have a common property of p , we call the m objects as m-relation. in the examples provided by tables 1 and 2, there are three events and three individuals. property p means that there are individuals appearing in the same event. in table 1, any two individuals appear simultaneously in one event, and therefore there exists a two-relation between any two individuals, which can be expressed by the fully connected network model. in table 2, we notice that three individuals simultaneously appear in the event a. individuals 1, 3 appear in the event b and individuals 2, 3 appear in the event c. thus there are one threerelation and two two-relations by definition 2.1. the difference between table 2 and table 1 shows that: firstly, three-relation cannot be obtained from tworelation; secondly, the definition of the multi-relation needs consider the subspace arising relation. thus we should make some modification to definition 2.1. table 1 any two individuals appear simultaneously in one event, and hence, two-relations between any two of them exist. event a event b event c individual 1 1 1 0 individual 2 1 0 1 individual 3 0 1 1 table 2 there are three individuals appearing in the event a at the same time, which indicates that a three-relation exists. event a event b event c individual 1 1 1 0 individual 2 1 0 1 individual 3 1 1 1 establishing relations among objects needs to consider the object attribute space. for example, a person can be described by an attribute space including gender, age, education, hobby, strength, event, and so on; an email attribute space includes sender, recipient, delivery time, subject, body, attachments, and people names that appear in the mail, and so on. hence each object can be expressed by a numerical vector in its attribute space. assuming there are m objects denoted by xi, i = 1, 2, · · · ,m, of which attribute space is n-dimensional. then each object can be represented as a n-dimensional vector xi = (xi1, · · · , xin)t ∈ rn, i = advances in systems science and application(2015) vol.15 no.1 39 1, 2, · · · ,m. in order to be able to obtain the multi-relations among objects from the attribute space, we give the definition of subspace multi-relation. definition 2.2 suppose that rn = x1 × x2 × · · · × xn,and if there exists a sub-sequence denoted by i1 < i2 < · · · < ik, aij ⊂ xij is a subset of xij , j = 1, 2, · · · , k,which makes the subset b = ai1 × ai2 × · · · × aik in the k-dimension subspace xi1 ×xi2 × · · · ×xik has a common property denoted by p . then we call set b as the k-dimension subspace m-relation, where m is number of element in set b. definition 2.2 shows that the relation not only means that the objects in set b have a common property, but also tells us that the relation occurs in which attribute space. as defined, the multi-relation in table 1 can be written as (x1, x2;a), (x1, x3;b), (x2, x3;c), and the multi-relation in table 2 can be written as (x1, x2, x3;a), (x1, x3;b), (x2, x3;c). therefore there are three one-dimension subspace two-relations in table 1, and two one-dimension subspace two-relations and one one-dimension subspace three-relation in table 2. we have defined the kdimension subspace multi-relation. how can we compute subspace multi-relations in real data? the following definition 2.3 provides a solution. definition 2.3 given ϵ > 0, let rn = x1 × x2 × · · · × xn, if there is a subsequence of 1, 2, · · · , n denoted by i1, i2 < · · · < ik, aij ⊂ xij is a point set of xij , which makes the subset b = ai1 × ai2 × · · · × aik in xi1 ×xi2 × · · · ×xik be covered by the hypercube u(ϵ), ϵ > 0 in rk, then the subset b is called the k-dimension subspace m-object relation, m is the number of element in subset b. note: in definition 2.3, u(ϵ) refers to the k-dimensional hypercube of side length 2ϵ in rk, where ϵ > 0 depends on the specific situation. for example, if we define u(ϵ) = (−ϵ, ϵ), ∀ϵ > 0 in the space of r1, then, there are one one-dimension subspace three-relation and two one-dimension subspace two-relations in table 2 by simple calculation. 2.2 properties of subspace multi-relation theorem 2.1 subspace multi-relation is of downward compatibility. that is to say, if b is a k-dimension subspace m-relation, then for any k1,m1, 0 < k1 < k, 0 < m1 < m, b is also k1-dimension subspace m1-relation. proof. let b = {xi|i = 1, 2, · · · ,m} has the k-dimension m-relation. given ϵ > 0, then there exists a positive integer k,rk ⊂ rn, which makes the projection xi|rk = (xi1 , xi2 , · · · , xik)t of xi covered by the hypercube of u(ϵ), ϵ > 0. thus for any subspace rk 1 with rk 1 ⊂ rk, and any subset {xi|i = i1, · · · , im1}, ∥ xi − xj ∥rk1≤∥ xi − xj ∥rk< ϵ. so b is k1-dimension subspace m1-relation. 40 dongyun yi : a subspace multi-relation model for the expression of system complexity theorem 2.2 subspace multi-relation is of upward aggregation. assuming that all one-dimension subspace multi-relation in the set of objects s denoted by si, i = 1, 2, · · · ,m . if si1i2···ik = si1 ∩si2 ∩· · ·∩sik ̸= ∅, 1 ≤ i1 < i2 < · · · < ik ≤ m , then si1i2···ik is k-dimension subspace mi1i2···ik-relation, where mi1i2···ik is the number of element in set (si1i2···ik . proof. for any xi, xj ∈ si1i2···ik , xi = (xi1 , · · · , xik), xj = (xj1 , · · · , xjk), to eliminate the influence of the different subspace calculation, we use the manhattan distance which is normalized on dimension as following: drk(xi, xj) = k∑ s=1 |xis − xjs |/k as xi, xj ∈ si1i2···ik ,then |xis − xjs | < ϵ,thus drk(xi, xj) = ∑k s=1 |xis − xjs |/k < ϵ note: here we assume that the ϵ > 0 is uniform on different subspace in the proof, but the ϵ > 0 can be different according to the actual problems. 3 algorithm we have given the definitions of subspace multi-relation and discussed its properties. in this section we will discuss the searching algorithm of subspace multirelation and give some practical examples. the subspace multi-relation is downward compatibility and upward aggregation, which allows us to adopt a bottom-up searching algorithm to obtain subspace multi-relation from low-dimension to high-dimension. firstly, multirelation is searched in one-dimension subspace. secondly any two of one-dimension subspace multi-relation are used to compute intersection, and which being not empty is to form two-dimension subspace multi-relation. thirdly, three-dimension subspace multi-relation could be found by computing the intersection of any two of two-dimension subspace relations. finally, based on this process all the existing subspace multi-relations in attribute space will be found. this method is capable of avoiding the effect of the high-dimension disaster problem which is encountered during the direct search in the high-dimension space, and hence can greatly improve the computational efficiency. assume that there are m objects xi, i = 1, 2, · · · ,m, which are expressed as xi = (xi1 , · · · , xin)t ∈ rn, i = 1, 2, · · · ,m in the attribute space. here we want to find the complex relations among the objects in attribute space based on subspace multi-relation model. the main steps of subspace multi-relation searching algorithm are given as follow: step 1: search one-dimension subspace multi-relation. let j = 1, according to some clustering methods, we cluster the projection {xij , i = 1, 2, · · · ,m} in advances in systems science and application(2015) vol.15 no.1 41 one-dimension subspace rj with each cluster covered by an interval of length 2ϵ. the result is denoted by cj(i), i = 1, 2, · · · ,mj , and let sj(i) = {x : xij ∈ cj(i)}, i = 1, 2, · · · ,mj , which is belong to rn. j = j + 1 until j = n. step 2: search two-dimension subspace multi-relation. if sj1j2(i1i2) = sj1i1 ∩ sj2i2 ̸= ∅ let mj1j2 (i1i2) = num(sj1i1 ∩ sj2i2 ) then there exists a two dimension mj1j2 (i1i2) -relation,1 ≤ j1 < j2 ≤ n, 1 ≤ i1 ≤ mj1 , 1 ≤ i2 ≤ mj2 , i1 < i2. step 3: similarly, we can search for higher dimension subspace multi-relations. k=3; if sj1j2...jki1i2...ik = sj1i1 ∩ sj2i2 ∩ ... ∩ sjkik ̸= ∅ 1 < i1 < i2 < · · · < ik ≤ m, 1 ≤ it ≤ mi, t = 1, 2 · ··, k then we say sj1j2···jk(i1i2···ik) = sj1i1 ∩ sj2i2 ∩ · · · ∩ sjkik has k-dimension mi1i2···ik -relation, where mi1i2···ik = num(sj1j2···jk(i1i2···ik)) k=k+1,if all of the intersection is empty, then stop searching. note 1: the clustering method of one-dimension space above can be suitably selected according to actual data such as k-means method or hierarchical clustering method; note 2: when dealing with big data, in order to control the number of subspace multi-relation, the number n of objects in which is asked to be greater than a given positive integer that is used to characterize how many objects are interesting, and parameter ϵ refers to the tightness of relation. 4 examples 4.1 experiment 1 in the department of mathematics of some college, the data set consists of 84 staff vertices, where each vertex has a total of 52 attributes including gender, education, job title, hobby, physical health, etc. the searching algorithm given above is used to find subspace multi-relation in data. the parameters ϵ is set to 0.5 and n (the number of staffs in subspace multi-relation) is 5. the total 125 subspace multi-relations are found. by use of traditional graph model we can only obtain fully connective graph which lost a lot of information 42 dongyun yi : a subspace multi-relation model for the expression of system complexity in data. that is because classical two-relation just indicate whether two objects have relation, but do pay attention to neither the simultaneously relation of how many objects nor in which subspace the relation happened. the more complex relation that exists in actual data can be expressed by the subspace multi-relation model, which can not only tell us how many objects have relation but also point out the attribute space which the relation exists in. in example given above, staffs link together differently because of schoolfellows, countrymen, or the same research field and so on. 4.2 experiment 2 enron was one of the most important companies in the u.s. energy industry. in 2001, the accounting fraud scandal of the enron was exposed. then the federal energy planning board took e-mail communications between enron employees posted online [17] for publicity and academic research. here, we want to discover some interesting groups based on these email. from the body content of those emails which happened before the fraud scandal we extract 2062 different people as vertices and set 144 staff’s mailbox as the attributes of vertices. let ϵ be 0.5 and n be 60, we finally find 275 subspace multi-relations by our searching algorithm. it is very interesting that the highest dimension of subspace in all subspace multi-relation is 8 with 86 mailboxes, referring to 8 people including chief operating officer, director of risk management, vice president and chief, online president of enron, enron ceo in north america, and other two important figures, who all are suspected of fraud, which tell us maybe the event come from people’s talking. 5 subspace multi-relation and vector hyper-graph as two-dimension relation corresponds to the graph model, subspace multirelation corresponds to the hyper-graph model. recently studies on hyper-graph model have mainly focused on how to expand the existed properties in graph theory into hyper-graph, such as hyper-path, hyper-chain, graph partitioning and so on [9-13]. furthermore, hyper-graph models do not indicate that the multi-relation exists in different subspaces. here we propose the concept of vector hyper-graph corresponding to subspace multi-relation. definition 5.1 let x = {x1, x2, · · · , xs}, xi ∈ rn be a finite set. a vector hypergraph of x is denoted by g = (e1,rk1), · · · , (ep,rkp) where ei is the finite collection of subset of x, such that: (1)ei ̸= θ, i = 1, 2 · ··, p advances in systems science and application(2015) vol.15 no.1 43 (2) p ∪ i=1 ei = x,rki < rn, 1 ≤ ki ≤ n, i = 1, 2 · · · p in the vector hyper-graph g, xi is called a vertex, (ei,rk i ) is called vector hyperedge. there exists a natural connection between the vector hyper-graph and subspace multi-relation. assume that a vector hyper-graph x has a vector hyperedge (ei,rki) = (xi1 , xi2 , · · · , ximi , rki), then the objects xi1 , xi2 , · · · , ximi has multi-relation on subspace rki , which means ki-dimension mi-relation, and vice versa. hence we can use the vector hyper-graph to express subspace multi-relation and study the complex relation in system. 6 conclusions we discuss the method to express the complexity of a system by a subspace multi-relation model. based on the attribute space of objects, we establish the rigorous definition of subspace multi-relation. the traditional graph model or complex network is a special case which is equivalent to the one-dimension tworelation. moreover, an algorithm is given to search for the subspace multi-relation and its effectiveness is verified by real data. finally, vector hyper-graphs related to multi-relation models are discussed. the equivalence of the subspace multirelation and the vector hyper-graph is pointed out, which provides a new method to study system complexity through vector hyper-graphs. references [1] nicholas w. watkins and mervyn p. freeman. 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[17] enron email dataset, http://www.cs.cmu.edu/ enron/ corresponding author dongyun yi: dongyunyi@sina.com advances in systems science and application (2016) vol.16 no.3 21-32 survey on secured password authentication for iot vanitha m school of information and technology, vit university, vellore 632014, tamilnadu,india abstract password is easy and mostly used method to provide security and authentication. now in iot environment many devices like rfid, smartcard, and wireless sensor devices use passwords for security. but passwords are susceptible to many attacks like loss of password, eavesdropping, forgetting passwords, etc., by saving passwords in a single server it is more prone to loss of password if the server is compromised. in order to avoid this it is better to divide and save passwords in multiple server and this helps us to overcome loss of password on servers. this paper has used light weight cryptographic algorithm to secure these passwords in the network because this type of algorithm occupies less space and the execution speed will be high. some of the recent light weight cryptographic algorithms are humming bird, present, tea, xxtea and humming bird which was surveyed by eisenbarth et al.[1]. these are specially designed for smart cards, rfid etc. this paper has implemented an algorithm that divides and saves password on two servers and use present algorithm on one server and use humming bird algorithm on another server for providing the efficient password security. so our model will be more secured than any other existing works. keywords rfid; smart card; iot; present; humming bird 1 introduction currently, internet of things (iot) is a communication paradigm that predicts a near future, in which the objects of normal life will be prepared with microcontrollers for digital communication that will make them able to communicate with one another and with the users, becoming an essential part of the internet discussed by l. atzori et al.[2]. the iot aims at making the internet even more pervasive by enabling easy access and providing good security. this model finds application in different domains, such as home automation, medical aids ,industrial automation, elderly assistance, mobile healthcare, traffic management, intelligent energy management and smart grids, automotive, and many others. in order to provide security for those devices we require some sort of light weight cryptographic algorithms for the two reasons such as 1. for the low resource-devices, e.g. battery-powered devices, the cryptographic operation with a limited amount of energy. 2. lower resource devices that are the footprint of the lightweight cryptographic primitives are smaller than the conventional cryptographic ones. 22 vanitha m: survey on secured password authentication for iot previously password based authentication model transmitted the hash value of the password in the public channel which is easily accessible by the attacker. burande et al surveyed the data and said that 55 percent of the password can be cracked in 8hours using the password recovery toolkit which is commercially available[3]. the first protocol for the password based authentication is pki(public key infrastructure) model. the purpose of a pki is to validate the information being transferred in a network and to confirm the identity of the clients using passwords. this model is used by gong et al[4], the client can send the password to the server by using public key encryption. bellovin et al have proved that the password is unaffected to offline dictionary attack[5]. halevi et al given the proofs for pki model with the formal definitions[6]. the second protocol for the password authentication is password-only model, bellare et al[7], boyko et al[8] and yang et al[9] have introduced a method called encrypted key exchange. the password is used as a secret key for key exchange. they were the first to give password only protocol which is more secure under cryptographic assumption. but consider the scenario that all the passwords are stored in the single server; if that server is compromised due to attacks the password will be disclosed. the overcome this issue , the third protocol called two server password only model were introduced ,where two servers cooperate , even if one server is compromised the attacker cannot act like a client with the information from the compromised server. katz et al protocol called koy protocol can run in parallel and generates secret session key between client and the two servers[10]. goldreich et al discussed about the session key generation[11]. due to denial of service attack if one of the servers shuts down another server can continue to provide services to the clients. all the above mentioned protocols two servers share the random password pw1 and pw2 = pw. yi et al proposed a new symmetric solution for the password only authenticated key exchange (pake)[12], one server s1 with an encryption of the password pk1 and another server s2 with the encryption of the password pk2, where pk1 and pk2 are the encryption keys of s1 and s2. they have concluded that their protocol is secure against passive and active attacks even if one of the servers is compromised. this paper has propose the new solution for the two servers pake. current works considered diffie-hellman key exchange protocol for the clients to establish a shared key over a communication channel. another protocol they have used is elgamal encryption which consists of encryption, decryption and key generation algorithms. we use light weight cryptographic algorithm which is specifically designed for smart cards, rfid tags, and wireless sensor devices. password is advances in systems science and application (2016) vol.16 no.3 23 divided in to two halves, one in server1 and second in server2. both the servers are not aware of each others encryption keys. security analysis has shown that our protocol is secure against the attacks like smart card loss attack, forgery attack and dos attack. moreover proposed work have improved the efficiency of the system by reducing the execution speed as well as the memory occupied by the algorithm is also reduced. the paper is organized as follows, section 2 introduces related work, and light weight cryptographic algorithm named humming bird algorithm in section 3 and section 4 discusses the proposed work, section 5 shows the security analysis and last section deliberates the conclusion. 2 related work password-authenticated key exchange is an authentication method where a client and a server who share a password and verify each other with that password and both will agree on a cryptographic key discussed by katz et al[13]. passwords which are required to verify the clients are stored on a particular server. if the server is cooperated, due to some horrible operations like hacking or fixing a trojan horse, passwords which are stored in the server gets discovered. to overcome this problem, ford et al proposed the threshold password authenticated key exchange with n servers[14]. di raimando et al proposed a protocol[15], which requires 1/3 of the servers to be compromised. brainard et al developed the first two server protocol with pki based setting and implemented using the public key techniques such as ssl[16]. katz et al proposed a koy protocol with the proof of security[10]. here two servers cooperate to authenticate a client and if one server is cooperated, the hacker still cannot act as a client with the evidence from the granted server. this protocol was based on katz-ostrovsky-yung protocol called koy protocol proposed by katz[17]. in this protocol the client randomly choose a password pwc, and two servers a and b are delivered with random password parts pwc1 and pwc2 where pwc1+pwc2=pwc. at higher level this protocol can be seen as implementation of two koy protocol, one between client and server a where server b helps for confirmation and one between client and server b where server a helps for confirmation. koy protocol is symmetric in the sense two servers correspondingly contribute to the authentication and key exchange. this protocol is secure under passive adversaries but everyone roughly performs twice work as koy protocol. in order to be secure from active attacks the user work remains same but the servers work increases by 2-4 times. the advantage of koy protocol is its structure and its disadvantage is ineffectiveness of practical use. subsequently in 2005, yang et al[18] system built on brainard et al.[16] work, yang et al suggested an asymmetric setting, where a service server, interacts with the client and a back-end server called control server. control 24 vanitha m: survey on secured password authentication for iot server helps service server with the authentication, and only service server and the client decide on a secret session key in the end. they suggested a pki based asymmetric two-server pake protocol in 2005 and several pake protocols has been proposed. recently in 2013, yi et al proposed a two server password-authenticated key exchange (pake) will share a password between the client and two servers for authentication[12]. so it is difficult to hack both the servers because if one server is compromised , the attacker wont pretend to be the client on other server. they proposed a solution for two-server pake, where the client found a different cryptographic keys with the two servers and it runs in parallel and is more efficient. in their protocol they have provedh(k1, 1) ⊕ b1 = h(k1, 1) ⊕ (h(k1, 0) ⊕ h1) = h1, the server1 accepts the message m6 and computes the secret session key sk1 = sk1 because k1 = k1. also they proved that their protocol is secured against the active and passive attacks in case that one of the two servers is compromised. 3 preliminaries 3.1 present algorithm present is a lightweight block cipher designed by bogdanov et al[19], the algorithm is notable for its compact size i.e., 2.5 times smaller than aes algorithm. light weight algorithms are suitable for constrained environments such as rfid tags, smart cards and sensor networks. following pseudocode shows that each of the 31 rounds consists of an xor operation to familiarize a round key ki for 1 ≤ i ≤ 32, where k32 is used for post-whitening, a linear bit wise permutation and substitution layer. it uses a single 4-bit s-box , which is applied 16 times in parallel in each round. genroundkeys() for i = 1 to 31 do arkey(state,ki) sblayer(state) player(state) end for arkey(state,k32) in applications that claim efficient use of space, the block cipher will be implemented as encryption-only. in this way it can be used for challenge-response authentication protocols, it could be used for both encryption and decryption by using the counter mode. taking such considerations into account so decided to make present a 64-bit block cipher. it supports both encryption and decryption is smaller than an encryption-only aes and produces an ultra-lightweight solution. the encryption sub keys can be calculated on-the-fly . advances in systems science and application (2016) vol.16 no.3 25 differential cryptanalysis.the case of differential cryptanalysis is captured by the following theorem discussed by biham[20]. theorem 1.characteristics that involve 10 sboxes over 5 rounds. the following two-rounds involves two s-boxes per round and holds with probability 2−25 over five rounds. ∆ = 0000000000000011 → 0000000000030003 → 0000000000000011 = ∆ linear cryptanalysis.the case of the linear cryptanalysis of present algorithm is handled by wang[21] in the following theorem. theorem 2.let ε4r be the maximal bias of a linear approximation of four rounds of present.then ε4r ≤ 1 27 . the theorem is formally proved in a. bogdanov1 et al[19] paper, and it is used directly to bound the maximal bias of a 28-round linear approximation by 26 × ε4r = 26 × (2−7)7 = 2−43 therefore a cryptanalyst need only approximate 28 rounds in present to mount a key retrieval attack, linear cryptanalysis of the cipher would require of the order of 284 known plaintext/cipher texts. such data requirements exceed the available text. 3.2 humming bird algorithm hummingbird, which is recently designed by smith et al[2] is an ultra-lightweight cryptographic basic for encryption and authentication in severely resource-constrained devices like passive rfid tags. it is an elegant combination of a block cipher and stream cipher with a 16-bit block size, 256-bit key size, and 80-bit internal state. the size of the key and the internal state of hummingbird provide a security level which is adequate for many rfid applications. moreover, hummingbird is resistant to the most common attacks to block ciphers and stream ciphers including differential and linear cryptanalysis, structure attacks, algebraic attacks, birthday attacks and cube attacks. hummingbird encryption process differential cryptanalysis: here ek(x) denote the encryption function of hummingbird with 256-bit key k. recall that ek(x), denotes the 16-bit block cipher encryption . then ek(x) is the composition of four ek(x). for a function f (x) from fm 2 to fm 2 , the differential between f (x) and f (x + a), where + is the bit-wise addition by df (a, b), is given by df (a, b) = |x|f (x) + f (x+ a) = b, xεfm 2 engels et al proved that, the differential of ek(x) has the same upper bound as ek(x), the block cipher component in ek[22]. they have tested the reduced version of hummingbird for more instances of different pairs. from those experimental results, the standard differential cryptanalysis method is not valid for 26 vanitha m: survey on secured password authentication for iot table 1 add caption algorithm : encryption process input: a 16-bit plaintext pti and four rotors ria(i = 1, 2, 3, 4) output: a 16-bit cipher text cti [block encryption] 1: v 12a = esr1(pti + nr1a) 2: v 23a = esr2(v 12a + nr2a) 3: v 34a = esr3(v 23a + nr3a) 4: cti = esr4(v 34a + nr4a) [internal state updating] 5: lfsra+1 ← lfsra 6: r1a+1 = r1a + nv 34a 7: r3a+1 = r3a + nv 23a + nlfsra+1 8: r4a+1 = r4a + nv 12t+ nr1a+1 9: r2a+1 = r2a + nv 12t+ nr4a+1 10:return cti hummingbird with time complexity. linear cryptanalysis: for the linear cryptanalysis of ek(x), consider |ek(a, b)|, the absolute value of the walsh transform of ek(x), where ek(a, b) = σx∈f 16 2 − 1<a,ek(x)><b,x>, a, b ∈ f 16 2 , a ̸= 0 where < x, y > is the inner product of two binary vectors x and y. the absolute value of the walsh transform of encryption function could be limited by the square root of 216. so, hummingbird is resistant to linear cryptanalysis attack in iot applications. 4 novel protocol for two-server password only authentication in this work , used two different light weight cryptographic algorithms for encrypting the passwords in two servers. algorithms such as present and humming bird help to encrypt the password and store it in the servers. two servers jointly work to authenticate the client and offer services proposed by yang et al[23]. client broadcast the message to both the servers , the servers cooperate to authenticate the password. our protocol uses two different algorithms so it will be very hard for the attacker to break because even if one of the servers is compromised its very difficult to hack the other one. our model used more efficient algorithm which is specially designed for the resource constraint devices when compared to the other. the protocol runs mainly in three stages initialization, registration and authentication. fig.1 explains the working process of password authentication. advances in systems science and application (2016) vol.16 no.3 27 fig. 1 working process of proposed work 1. server1 initialises runs present and sends its secret key of server1 to client. 2. server2 initialises runs humming bird and sends its secret key of server2 to client. 3. server1s secret key and sever2s secret key are received by client. 4. client registers to server1 and sends information after encrypting using secret key of respective server to respective server. the password is separated and sent to server1 and server2. 5. id and pwd1 is sent to server1. 6. id and pwd2 is sent to server2. 7. user logins to server1 with his registered id and password. 8. server1 and server2 communicates each other and server2 receives the full password and compares with user entered password. returns success if password matches else returns failure. 4.1 initialization the server then performs the following operations. 1. the server1 s1 generates the secret key according to present algorithm. 2. the server2 s2 generates secret key according to humming bird algorithm. 4.2 registration before authentication, each client c is essential to register to both s1 and s2 through different channels. first of all, the client c accesses the secret key of both servers s1 and s2. next, the client c picks a password pwc and encrypts the password using the secret key. after that, the client c recalls the password pwc. the two secure channels are essential for all two server pake protocols, where a password is encrypted by means of two different secret keys, which are safely broadcasted to the two servers, during registration. although, the idea of private key cryptosystem, the encryption key of one server should be unfamiliar 28 vanitha m: survey on secured password authentication for iot to another server and the client needs to memorize the secret code or password just behind registration. the two servers s1 and s2 have settled on the password confirmation information of the client c during registration. the id of each client has to be unique. 4.3 authentication/login 1. the client c broadcast a request message m1 to the s1 server. the message includes the authentication information of the client and a nonce. 2. the two servers exchange messages m2 and m3 based on the authentication information gathered during the registration phase. 3. now if the client is genuine (based on password check) then the client is allowed to have access. else the request for login is denied. advantage is to achieve better performance because parallelism is used and it prevents replay attack because of nonce. 4.4 encryption 1.here the plain text is the password which is divided as per the two-server password only authentication and according to novel architecture each half of the passwords are encrypted by client using present and humming bird. 2.but the change here is the password are not sent to same receiver but are sent to different servers and they use different algorithms namely present and humming bird. the detailed process of encryption is illustrated in fig.2. fig. 2 encryption process advances in systems science and application (2016) vol.16 no.3 29 4.5 decryption here the two parts of plain text are sent to different servers which use present and humming bird algorithms and these receive the respective plain text and decrypt them using present and humming bird. the two parts of plain text are merged only during login phase. the detailed process of decryption is illustrated in fig.3. fig. 3 decryption process 5 security analysis 5.1 smart card loss attack suppose if the customer lost his/ her smart card, by accessing the storage medium the attacker can try to access the data. then the attacker may try to use this information to login to the server. in out model its very hard to hack the password because passwords are divided and stored in two different servers. our protocol is against to smart card loss attack. 30 vanitha m: survey on secured password authentication for iot 5.2 dos attack suppose an attacker has stolen the smartcard, and the attacker try to update the information. in that case our model is not vulnerable to the attacker because two different algorithms have been used. even if one server is compromised the other server wont respond. smart card will be logged if it exceeds the number of login attempt limit. therefore out protocol can withstand dos attack. 5.3 forgery attack the attacker cannot create legal login request and cannot act as an authorized user, because without knowing the users password the attacker has no way to get the exact value. 6 conclusion this project presents a novel protocol for two server password only authentication. this protocol is secure against many attacks in case if one of the servers is compromised still the intruder will not be able to gain access to the password because only half of the password will be revealed. the performance is also efficient because of the parallel computation. therefore, our scheme is more suitable, particularly for applications which need enhanced security in the iot environment since making use of efficient light weight cryptographic algorithm. references [1] t. eisenbarth, s. kumar, c. paar, a. poschmann, a. and uhsadel, l. (2007), “a survey of lightweight-cryptography implementations”, ieee design & test of computers, (6), 522-533. [2] atzori, luigi, antonio iera and giacomo morabito. (2010),“the internet of things: a survey”, computer networks, 54.15, pp. 2787-2805. [3] burande, n. and gumaste, s. v. (2013), “survey on public key encryption for two server password only authenticated key exchange”, available at http://www.schneier.com/blog/archives/2006/12/ realworld passw.html. [4] gong, l., lomas, m., needham, r. m. and saltzer, j. h. (1993), “protecting poorly chosen secrets from guessing attacks”, selected areas in communications,ieee journal on, 11(5), 648-656. [5] bellovin, s. m. and merritt, m. (1992), “encrypted key exchange: passwordbased protocols secure against dictionary attacks”, in research in security and privacy, proceedings, 1992 ieee computer society symposium , pp. 72-84. advances in systems science and application (2016) vol.16 no.3 31 [6] halevi, shai and hugo krawczyk. (1999), “public-key cryptography and password protocols”, acm transactions on information and system security (tissec) 2.3, pp. 230-268. [7] m. bellare, d. pointcheval and p. rogaway. (2000), “authenticated key exchange secure against dictionary attacks”, proc. 19th intl conf.theory and application of cryptographic techniques (eurocrypt 00), pp. 139-155. [8] v. boyko, p. mackenzie and s. patel. (2000), “provably secure passwordauthenticated key exchange using diffie-hellman”, proc. 19thintl conf. theory and application of cryptographic techniques(eurocrypt 00), pp. 156-171. [9] yang, y. and bao, f. (2010), “enabling use of single password over multiple servers in two-server model”, ieee 10th international conference on in computer and information technology (cit), 2010, vol.34, pp. 846-850. [10] katz, p. mackenzie, g. taban and v. gligor,(2005), “two-server passwordonly authenticated key exchange”, roc. applied cryptography and network security (acns 05), pp. 1-16. [11] goldreich, o. and lindell, y. (2001), “ession-key generation using human passwords only”, in advances in cryptology-crypto 2001, pp. 408-432. [12] yi, xun, san ling and huaxiong wang. (2013), “efficient two-server password-only authenticated key exchange.”, ieee transactions on parallel and distributed systems, 24.9,pp. 1773-1782. [13] katz, j. and yung, m. (2007), “scalable protocols for authenticated group key exchange”, journal of cryptology, 20(1), vol.18, pp. 85-113. [14] ford, w. and kaliski jr, b. s. (2000), “ server-assisted generation of a strong secret from a password”, in enabling technologies: infrastructure for collaborative enterprises, 2000.(wet ice 2000). procedings, ieee 9th international workshops, pp. 176-180. [15] m. di raimondo and r. gennaro. (2003), “provably secure threshold password authenticated key exchange”, proc. 22nd intl conf.theory and applications of cryptographic techniques (eurocrypt 03), pp. 507-523. [16] brainard, j. g., juels, a., kaliski, b. and szydlo, m. (2003), “ultralightweight cryptography for low-cost rfid tags: hummingbird algorithm and protocol, a new two-server approach for authentication with short secrets”,in usenix security, vol.3, pp. 201-214. 32 vanitha m: survey on secured password authentication for iot [17] jonath katz, philip mackenzie, gelareh taban and virgil gligor. (2005), two-server password only authenticated key exchange, springer, pp. 1-16. [18] y. yang, f. bao and r.h. deng,(2005), “a new architecture for authentication and key exchange using password for federated enterprise”, proc. 20th ifip intl information security conf. (sec05), pp. 95-111. [19] a. bogdanov, l. r. knudsen, g. leander, c. paar, a. poschmann, m. j. b. robshaw, y. seurin and c. vikkelsoe (2007), “present an ultra lightweight block cipher”, proc. ches, springer, pp. 450-453. [20] biham, e. (1994),“new types of cryptanalytic attacks using related keys”, journal of cryptology, 7(4), 229-246. [21] m.wang(2007), “differential cryptanalysis of present”, proc .ches, pp. 1-4. [22] engels, x. fan, g. gong, h. hu and e. m. smith (2009), “ultra-lightweight cryptography for low-cost rfid tags: hummingbird algorithm and protocol”, centre for applied cryptographic research (cacr) technical reports cacr. [23] y. yang, r.h. deng and f. bao (2006), “a practical password-based twoserver authentication and key exchange system”, ieee trans. dependable and secure computing, vol. 3, no. 2, pp. 105-114. corresponding author vanitha m can be contacted at: mvanitha@vit.ac.in. investigation on fuzzy degree criterion of electric power equipment 180-187 advances in systems science and applications (2011), vol.11, no.1-2 issn 1078-6236 international institute for general systems studies, inc. research on energy management of hev lithium ion battery wu tiezhou 1,2 , chen xueguang 1 , zhang yongfei 2 and xiao qing 2 1 department of control science & engineering, huazhong university of science & technology，wuhan 430074 2 hubei university of technology school of electrical & electronic engineering，wuhan 430068 abstract battery system is a more complicate and fragile link in the hev vehicle. the performance of battery system directly affects the whole energy and the function and cost of hev system. therefore, energy management of battery system can optimize the working state of battery and make it match better with other system. the paper mainly discusses the charging management of lithium ion battery and energy feedback of braking. results of experiment demonstrate the proposed energy management strategy has an obvious effect of saving energy and can reduce the charging time and improve efficiency. keywords energy management hev lithium ion battery energy feedback of braking 1. introduction with the rapid development of automobile industry, energy and environmental issues become even more acute. in order to solve the energy consumption and pollution caused by these two problems, hybrid electric vehicle (hev) has been rapid development [1]. storage battery can store and provide clean power, and has been widely used in hybrid electric vehicles. compared to other batteries, lithium ion battery [2] has many advantages, such as the high voltage, energy density, no memory effect and the long lifetime. although battery technology has been great progress, traditional methods such as constant voltage battery charging method, constant current method, phase-wise constant current method and the constant current constant voltage method [3] have some problems, such as charging too long, and seriousness of polarization effects, reduced cycle life of the battery, taking noting of energy feedback during the period of braking into account, so it is necessary for new battery fast charging technology [4]and new method of energy feedback during the period of braking. according to maas laws [5], to charge the battery fast, the charge current should comply with the optimal acceptable current curve, which is a negative exponential curve. in practice, it is very difficult to meet the requirement. common methods such as fast charging pulse charge method [6], intermittent charging method [7], achieve fast-charging by eliminating the polarization effects [8]. pulse charging method is an effective method of fast charging. but, it is difficult to determine frequency of pulse [9]. in this paper, the pulse charging method is adapted, advances in systems science and applications (2011), vol. 11, no. 1-2 181 using variable frequency pulse charge method to reduce the charge time and improve charging efficiency. there is a transient and frequent braking in the process of hev vehicle’s driving. it will offer much motive to vehicle if the inertia energy be accumulated. the paper will research how to recycle the maximum possible feedback braking energy without destroying the battery. 2. principles of variable frequency pulse charge method and description of algorithm 2.1 principles of variable frequency pulse charge method in accordance with the battery emf equivalent model [10], the resistance of the battery has three parts: ohmic resistence r , electrochemical polarization resistance dz and concentration polarization resistance zk , the total resistance is as follows: kdb zzrz   (1) the less the zb is, the less chemical energy converted from the electrical energy is. so in order to reduce the energy loss during charging, it is necessary to determine the optimal frequency fop of charging to reduce the battery impedance zb. the battery resistance changes as the battery soc, so the charge frequency varies with the soc. to find the optimal pulse frequency conveniently, we need to charge the battery with different frequencies of pulses, acquire the corresponding average currents, and then select the pulse frequency with maximum average current as the best pulse frequency. 2.2 algorithm description the charging of lithium-ion battery continues with a constant period tsum . a cycle of tsum can be divided into three time sub-periods: tfull for fullness detection, tseek for determining of the optimal frequency and tch for charging. according to experimental results, tsum is set as 5min. the charging process is repeated until the charging completed. tch f should be much higher than tfull + tseek. work model of variable frequency pulse charging as shown in fig. 1. initialization, t = 0 start timer whether t≥tsum judge soc>95% or vb>vsup estimate soc , measure battery voltage vb search optimal frequency f beginning charging, the frequency is f y n n end y fig.1 work model of variable frequency pulse charging the optimal frequency is determined as follows: 1) calculate the average charging current with frequency fn. if the m-th sampling value of current is dib,n(m), and the sum of the sampled currents id dacc(n),and sample value, sample design and for the currents, setting the initial = 0: ;)()()( , mdndnd nibaccacc  m=m+1 (2) after calculating the average current of m sub-samples: m nd nd acc avg )( )(  (3) 2) calculating the average current of sampling value of current at varied frequencies, n=n+1. nnmd m nd m m nibavg ...3,2,1)( 1 )( 1 ,    (4) 3) determining optimal frequency according to (5):  nnndmaxff avgnop ,...2,1)),((|  (5) 182 wu: research on energy management of hev lithium ion battery182 wu: research on energy management of hev lithium ion battery182 wu: research on energy management of hev lithium ion battery182 wu: research on energy management of hev lithium ion battery 182 wu: research on energy management of hev lithium ion battery 182 wu: research on energy management of hev lithium ion battery 182 wu: research on energy management of hev lithium ion battery 182 wu: research on energy management of hev lithium ion battery 182 wu: research on energy management of hev lithium ion battery wu: research on energy management of hev lithium ion batterywu: research on energy management of hev lithium ion batterywu: research on energy management of hev lithium ion batterywu: research on energy management of hev lithium ion batterywu: research on energy management of hev lithium ion battery 182 wu: research on energy management of hev lithium ion battery 182 wu: research on energy management of hev lithium ion battery 182 wu: research on energy management of hev lithium ion battery ? ? advances in systems science and applications (2011), vol. 11, no. 1-2 183 3. battery group charge management with energy braking energy 3.1 determination feedback current and braking time according to mass’ law, in the charging process, the maximum acceptable charging current could be expressed as equation(6)： t tt eii  (6) charge the lithium-ion battery which had released capacity oc with maxci under the constant-current charging mode， t later, the battery’s voltage reaches the charging final voltage sv , then maxci is the maximum acceptable charging current, and sv could be found in the battery instruction manual of manufacturer. t is determined as following: there are two factors: one is whether it influences the battery’s cycle life or not when charging battery with a high current in t , the other is considering the durable time of peak power rate. we choose t =2min, considering the braking time of 510 bus in wuhan are mostly within 2minutes. 3.2 braking energy feedback charging strategy first of all, we should judge battery’s charge final voltage. if battery’s voltage does not reach it, then detection feedback current ri , and the maximum acceptable charging current was 0maxci in the brake moment. under the circumstance of 0maxcr ii  , the braking energy could be feedback. when clock the braking time rt , if ttr  and 0maxcr ii  , start energy feedback; with the continuous braking, energy feedback stop if ttr  or 0maxcr ii  . strategy implementation process shown in fig. 2 braking begin sb vv  read ri 0maxcr ii  flag=0 flag=1 0maxcr ii  ttr  output charge signal,then begin charge output uncharged signal braking over n y y y y n n n y n ，0 r t 0flag calculate 0maxc i beginning time output charge signal,then begin charge fig. 2 flowchart of braking feedback charging current 4. experiments and conclusions 4.1 lithium-ion batteries variable frequency pulse charging experiment batteries are selected: the gbs-lfmp90ah square lithium-ion batteries are selected for our variable frequency pulse charging experiments, which are produced by zhejiang jia-beth green energy co., ltd. experimental method: the number of sampling value of current at varied frequencies is selected and the tsum is determined for our experiments. 182 wu: research on energy management of hev lithium ion battery 182 wu: research on energy management of hev lithium ion battery 184 wu: research on energy management of hev lithium ion battery advances in systems science and applications (2011), vol. 11, no. 1-2 185 there are 40 frequencies for sampling, which start with f1=100hz,…fn=fn-1+500hz(n=2,…49). when the cycle tsum is too short or too long, the charging time will be extended. in our experiments, the tsum is 6min. the results are shown in fig. 3 and fig. 4. fig 3 shows a charging curve of the optimal frequency. 0 2000 4000 6000 8000 10000 12000 14000 12024 48 72 96 time（min） fr eq u en cy （h z ） fig.3 optimal charging frequency as shown in fig. 4, the optimal charge frequency is effected by factors such as soc(state of charge). we have charged the batteries using constant current constant voltage charging method and have charged the same batteries at 1khz, 100 khz pulse, and then we have charged the same batteries using the method given in this paper. the curves are as follows: fig.4 flowchart of different charging mode as can be seen from fig4, the voltage/time curve for variable frequency pulse charging method is steeper than those for other methods, implies that it requires less time for charging. 4.2 comparative experiment of energy feedback we have a experiment using dongfeng’s 1500kg hev, the friction coefficient: f = 0.02, frontal area: a = 2.872m * m, drag coefficient: cd = 0.4. the experiment data show as table 1: table 1 comparative table of energy feedback parameter some feedback all feedback no feedback initial soc 0.9 0.7 0.8 end soc 0.87 0.68 0.77 energy consumer(kwh) 4.52 4.68 4.49 feedback energy(kwh) 0.31 0.87 0 energy conservation (%) 6.85 18.58 0 according to table 1, when in average circulation, while battery’s soc under 0.7, hev could feedback all of the braking energe, and could saving many energy. 5. conclusions at present, the battery management system takes insufficiently the battery energy management into account, or lacks the energy management strategy, thus causes drop of battery's state of health, affects the using and the service life of the battery. in this paper, a fast charging technology for charging pulse is proposed, whose key idea is to accurately determine the pulse frequency and to charge with variable optimal frequency, and a kind of energy feedback strategy is proposed too. experimental results show that the fast charging technology for charging pulse can significantly shorten the charging time, and can be used in hev battery management system, the energy feedback strategy can reduce energy’s consumption, they can be used in hev. references [1] li jianping. the controlling & suggestion of vehicle’s emission [j]. research & application for car.1999,5:18-23. [ch] [2] becker h., schuh h., brunarie j.. high power lithium-ion battery: first field trial results of an advanced backup system for advanced cellular networks. intelec, twenty-seventh international, 2005, 257 – 262. 186 wu: research on energy management of hev lithium ion battery advances in systems science and applications (2011), vol. 11, no. 1-2 187 [3] r. c. cope and y. podrazhansky, “the art of battery charging,” in 14thannu. battery conf. applications and advances, 1999, pp. 233–235. [4] p. m. hunter and a. h. anbuky, “vrla battery rapid charging under stress management,” ieee trans. ind. electron., vol. 50, no. 6, pp.1229–1237, dec. 2003. [5] blake dickinson, jay gill. issues and benefits with fast charging industrial batteries[j]. battery conference on applications and advances, 2000, 223-229. [6] j. zhang, j. yu, c. cha, and h. yang, “the effects of pulse charging on inner pressure and cycling characteristics of sealed ni/mh batteries,”j. power sources, vol. 136, no. 1, pp. 180–185, sep. 2004. [7] lopez, j. ,gonzalez, m. ,viera, j.c. ,blanco, c. fast-charge in lithium-ion batteries for portable applications[j]. telecommunications energy conference, 2004,19-24. [8] p.-h. cheng and c.-l. chen. high efficiency and nondissipative fast charging strategy[j]. electric power applications, 2003,539-545. [9] tristan juergens, robert f. nelson, michael a. ruderman. a new high rate, fast charge, sealed, lead acid battery[c].wescon, 1994, 230-235. [10] martin coleman, chi kwan lee, chunbo zhu, william gerard hurley. state-of-charge determination from emf voltage estimation: using impedance, terminal voltage, and current for lead-acid and lithium-ion batteries. ieee transactions on industrial electronics, 2007,54(5):2550-2557. advances in systems science and applications (2012) vol.12 no.3 258-271 a geometric interpretation of gravity theory n.n. popov dorodnitsyn computer centre of russian academy of science abstract a system of postulates which give a geometric meaning to the fundamental notions of gravity theory is introduced. the fundamental equation of geometric gravity theory is derived. the consistency of the system of postulates is shown. a series of examples adequately describing the gravitational field are considered. mathematically rigorous definitions of a black hole and dark energy are given. keywords pseudo-riemannian space, scalar curvature, metric gravity equation, spherically symmetric spatial black hole, dark energy 1 introduction hilbert’s sixth problem of axiomatizing those branches of physics in which mathematics is prevalent, posed by hilbert in 1900 among other problems, has the status of being too vague. however, in the context of some particular area of physics, such as gravity theory, this problem can be stated rigorously. the foundation of physical gravity theory is two fundamental physical concepts, of the gravitational field and of the mass of a body. in general relativity theory (grt), which is essentially relativistic gravity theory, a substantial progress in the mathematical interpretation of one of the basic concepts of the theory, namely, that of the gravitational field, has been made. from the geometrical point of view, the gravitational field is interpreted as a metric pseudo-riemannian 4space. however, grt does not provide such a precise mathematical definition of mass, or, to be more precise, the distribution density of mass. thus, the grt fundamental equation contains both a purely mathematical left-hand side, which is generated by a metric space, and a purely physical right-hand side, which is the energy-momentum tensor of the physical system under consideration. note that grt imposes no constraints on the choice of the energy-momentum tensor; this leads to the possibility of constructing unrealistic models and, thereby, provides evidence for the insufficiency of the principles on which grt is founded. our purpose in this paper is to present a complete translation of gravity theory into the language of differential geometry, in which the physical notion of the mass distribution density has a purely geometric interpretation. 2 the first postulates of mathematical gravity theory the notion of a metric space is the basis on which mathematical gravity theory is constructed. to construct the theory, it suffices to choose a pseudo-riemannian 4-space of signature (−−−+), on which a twice covariant symmetric nondegenerate tensor field gij(x1,k, x4) is defined; we have advances in systems science and applications (2012) vol.12 no.3 259 det |gij | ̸= 0, gij = gji, i, j = 1, . . . , 4 we refer to the tensor as the metric of the pseudo-riemannian space. general requirements to a metric space are as follows: (1) smoothness, i.e., the continuity of all components of the metric on the entire space and the continuous differentiability of the components up to the second order with respect to all variables almost everywhere except, possibly, on singular sets; (2) the preservation of the metric signature at each point of the space. note that the metric smoothness condition is stated in a somewhat relaxed form in order to make it possible to consider pseudo-riemannian spaces with discontinuous scalar curvature. note that the space structure and dimension chosen above are not regarded to be final. they are only sufficient for constructing geometric gravity theory; thus, we introduce them as sufficient conditions for constructing an adequate theory rather than as fundamental postulates. we proceed to state two postulates of geometric gravity theory. as mentioned above, the main physical objects of gravity theory are the gravitational field and the distribution density of the matter mass. the objective of gravity theory is determining laws governing the interaction of the gravitational field with the distribution density of gravitational mass. the first postulate of geometric gravity theory can be stated as follows. postulate 1. the gravitational field is a metric of a pseudo-riemannian space. this assertion is the basis of grt and does not need any comments. before stating the second postulate, we introduce the following notation[1-2]: γk ij is the pseudo-riemannian connection, which is defined by γk ij = −1 2 gkl( ∂glj ∂xi + ∂gli ∂xj + ∂gij ∂xl ); (1) rij is the ricci tensor, that is, rij = γk ij ∂xk − γk ik ∂xj + γk pkγ p ij − γk pjγ p ik; (2) r is the scalar curvature, that is, r = gijrij ; (3) where the summation over repeated indices is implied; and is the contravariant metric tensor. postulate 2′ (weak statement). if the mass density of the matter is nonzero at some point of the space, then so is the scalar curvature of the space at this point, 260 n.n. popov:a geometric interpretation of gravity theory and vice versa. this postulate says nothing about the form of the dependence between these scalar functions. it only indicates the existence of a relation between them. to determine this relation, we use two principles; namely, the principle of minimal action and the correspondence principle. these principles make it possible to derive the fundamental equation of geometric gravity theory and refine the statement of postulate 2. 3 the third postulate, the fundamental equation of gravity theory, and the strengthened statement of postulate 2′ according to postulate 2′, there is a dependence between the scalar function of the matter mass density and the function of the space scalar curvature. to find this dependence, we introduce a composite scalar function χ (ρ) , which depends on the matter mass density ρ in an unknown way, and a scalar curvature function r(gij), which is a composite function depending on the metric gij . the general form of the gravitational field equations can be obtained by applying the principle of minimal action. the field equations are obtained as the eulerclagrange equations under the variation of the field action. for the field action functional we take the quantity sg = ∫ ω (r + χ) √ |g|d4x, (4) where dω = √ |g|d4x is the standard volume 4-form on the pseudo-riemannian space,g = det(gij), and the integral is the whole 3-space (x1, x2, x3) and over the interval x41 ≤ x4 ≤ x42 of the time coordinate x4. postulate 3. in geometric gravity theory, the following relation holds: δsg δgij = 0. postulate 3 means that the sought dependence between the composite functions r(g) and χ(ρ) minimizes the action functional (4) with respect to the contravariance metric of the pseudo-riemannian space gij . theorem 1. if postulate 3 holds for functional (4), then rij − 1 2 rgij = 1 2 χgij . (5) proof. relation (4) implies δsg δgij = δ ∫ ωr(g) √ |g|d4x δgij + δ ∫ ω χ(ρ) √ |g|d4x δgij . advances in systems science and applications (2012) vol.12 no.3 261 the first term is the hilbert variational derivative[3] δ ∫ ωr(g) √ |g|d4x δgij = rij − r 2 gij . the second term can be represented in the form δ ∫ ω χ(ρ) √ |g|d4x δgij = ∫ ω(δχ(ρ) √ |g|+ χ(ρ)δ √ |g|)d4x δgij . taking into account the relation δ √ |g| = −1 2 √ |g|gijδgij and the fact that the variation of the scalar. function χ(ρ), which does not depend on the metric gij , vanishes, we obtain δ ∫ ω χ(ρ) √ |g|d4x δgij = −χ(ρ) 2 gij . it follows that δsg δgij = rij − r 2 gij − χ(ρ) 2 gij = 0, which completes the proof of the theorem. equation (5) implies the presence of a linear dependence between the functions r and χ. indeed, multiplying the rightand left-hand sides of equation (5) by the contravariant metric tensor gij and convolving both sides over the indices i and j, we obtain the relation χ = −r 2 . (6) according to (6), the scalar curvature r turns out to depend on the matter mass density ρ. however, we do not know the particular form of this dependence so far. to determine it, we apply the correspondence principle, which can be stated as follows: if the metric gij weakly converges to the minkowski metric ηij and, moreover, g44 = 1+ aφ c2 + o( 1 c3 ) and gij = −δij + o( 1 c3 ) for i, j = 1, . . . , 4, i ̸= j , where c−1 is a small parameter and φ is a twice differentiable function, then equation (5) must degenerate into the poisson equation for newtonian gravity theory, which is ∆φ = 4πgρ, (7) where ∆ is the laplace operator, φ is the newtonian gravitational potential, g is the gravitational constant, and c is the speed of light. simple calculations show that equation (5) does degenerate into the poisson equation (7) under the condition ρ = c2 32πg r. (8) relation (8) can be regarded as a stronger statement of postulate 2’. postulate 2 (strong statement). the matter mass distribution density in space 262 n.n. popov:a geometric interpretation of gravity theory is directly proportional to the scalar curvature of the pseudo-riemannian space: ρ = ær , where æ = c2 32πg .it follows from postulate 2 and equation (5) , based on postulate 3, that the fundamental equation of gravity theory can be represented in the form rij − 1 2 rgij = −8πg c2 ρgij , i, j = 1, . . . , 4, (9) or in the form of the system of two relations rij = r 4 gij , i, j = 1, . . . , 4, (10) r = 32πg c2 ρ the first of which is a direct consequence of postulate 3 and the second, of postulate 2. thus, physical theory of gravitational fields can be translated into the language of differential geometry. according to postulates 1 and 2, the fundamental physical concepts of gravitational theory, such as gravitational field and matter mass density, are interpreted in the geometric as the metric and the scalar curvature (up to proportionality), respectively, of a pseudo-riemannian space. 4 consistency and physical adequacy of postulates 1-3 to verify the consistency of the system of postulates stated above and the physical adequacy of these postulates, consider a model of a spherically symmetric space with a ball of radius r1 at the center of symmetry. the ball is uniformly filled with a matter of mass ρ and constant density; outside the ball, there is no matter. from the geometric point of view, we have a spherically symmetric pseudo-riemannian space with constant scalar curvature r = 32πg c2 ρ inside a ball of radius r1 and vanishing scalar curvature outside the ball. the problem is to determine the metric of such a space. hereafter, we always assume that the system units used for measuring physical quantities is chosen so that g, c = 1 . the general form of a stationary spherically symmetric metric of a pseudoriemannian 4-space in spherical coordinates t, r, θ, φ is[4] ds2 = g44(r)dt 2 + g11(r)dr 2 + g22(r)(dθ 2 + sin2θdφ2), (11) where g11 and g22 are negative unknown functions and g44 is a positive function, which depend on the variable r, and r, t ∈ (0,∞), r ∈ (0,∞), θ ∈ [0, π], φ ∈ [0, 2π). the components of metric (11) must satisfy system (10) . using relations (1) advances in systems science and applications (2012) vol.12 no.3 263 and (2), we can reduce system (10) for metric (11) to the form ( g′22 g22 )′ + 1 2 ( g′44 g44 )′ − 1 2 ( g′11 g11 )( g′22 g22 + g′44 2g44 ) + 1 2 ( g′22 g22 ) 2 + 1 4 ( g′44 g44 ) 2 = r 4 g11, 1 2 ( g′22 g11 )′ + g′22 2g11 ( g′11 2g11 + g′22 g22 + g′44 2g44 )− 1 2g11g22 (g′22) 2 − 1 = r 4 g22, (12) −1 2 ( g′44 g11 )′ − g′44 2g11 ( g′11 2g11 + g′22 g22 + g′44 2g44 ) + 1 2g11g44 (g′44) 2 = r 4 g44, where g′ij = dgij dr , r > 0 ,at r ≤ r1, and r = 0 at r > r1 . for the sake of generality, we assume that the unknown scalar function r in system (12) depends on the r coordinate. the following theorem is valid. theorem 2. if g22(0) = 0 and g44(0) < ∞, then the scalar curvature r(r) does not depend on r in the domain where it is nonzero, and system (12) has a unique solution depending on one free parameter. if, in addition,g44(0) = 1, then the solution of system (12) is unique; moreover, at r ≤ r1, where r1 < √ 12 r , it coincides with the de sitter metric[5] ds2 = (1− r 12 r2)dt2 − dr2 1− r 12r 2 − r2(dθ2 + sin2θdφ2), (13) and at r > r1, it coincides with the schwarzschild metric[6] ds2 = (1− r 12 r31 r )dt2 − dr2 1− r 12 r31 r − r2(dθ2 + sin θ2dφ2). (14) proof. system of equations (12) can be simplified by passing to the generalized spherical coordinates x1, . . . , x4,where x1 = r3 3 ,x2 = − cos θ,x3 = φ,and x4 = t. let us introduce the following new notation for the components of the metric tensor: g11 = −(3x1) 4 3 f1(x1), g22 = −f2(x1), g44 = f4(x1). in this notation, metric (11) takes the following form in the generalized spherical coordinates: ds2 = f4dx4 2 − f1dx1 2 − f2( dx2 2 1− x22 + (1− x2)dx3 2). (15) using the arbitrariness of the scaling multiplier of the coordinate x1 , we can achieve f1f2 2f4 = 1. (16) 264 n.n. popov:a geometric interpretation of gravity theory system (10) for metric (15) can be represented as −1 2 ( f ′ 1 f1 )′ + 1 2 ( f ′ 2 f2 ) 2 + 1 4 ( f ′ 1 f1 ) 2 + 1 4 ( f ′ 4 f4 ) 2 = −r 4 f1, 1 2 ( f ′ 2 f1 )′ − 1 2f1f2 (f ′ 2) 2 − 1 = −r 4 f2, (17) −1 2 ( f ′ 4 f1 )′ + 1 2f1f4 (f ′ 4) 2 = r 4 f4. condition (16) gives the additional equation f ′ 1 f1 + 2f ′ 2 f2 + f ′ 4 f4 = 0, (18) where f ′ i = dfi dxi . system (17), (18) contains four unknown functions f1 ,f2 ,f4 , and r ; only two of these functions, say f2 and r , can be regarded to be independent. this follows from relations (3) and (16). let us express the unknown functions f1 and f4 in terms of f2 and r . for this purpose, note that the third equation in system (17) can be represented as 1 2f1f4 f ′ 1f ′ 4 − 1 2 ( f ′ 4 f4 )′ = r 4 f1. (19) adding the first equation in system (17) to equation (19) and using (18) ,we obtain the relation ( f ′ 2 f2 )′ + 3 2 ( f ′ 2 f2 ) 2 = 0. twice integrating it, we obtain f2 = λ(3x1 + α) 2 3 , where λ and α are arbitrary constants of integration. by the assumption of the theorem, we have f2(0) = 0 , which implies α = 0 and f2 = λ(3x1) 2 3 . (20) we seek f ′ 4 f4 in (19) in the form f ′ 4 f4 = c(x1)f1 . substituting the last relation into equation (19) , we obtain f ′ 4 = c − 1 2 ∫ rdx1 λ2(3x1) 4 3 , where c is a constant. integrating both sides of this relation, we arrive at the formula f4 = β − c λ2(3x1) 1 3 + 1 2λ2 ∫ rdx1 (3x1) 1 3 − 1 2λ2 ∫ r (3x1) 1 3 dx1, (21) advances in systems science and applications (2012) vol.12 no.3 265 where β is constant of integration. by the assumption of the theorem, the function f4(0) is bounded; therefore, considering a solution in some neighborhood of zero, we must set c = 0 , and the formula for f4 takes the form f4 = β + ∫ r(x1)dx1 2λ2(3x1) 1 3 − 1 2λ2 ∫ r(x1) (3x1) 1 3 dx1. (22) condition (16) implies f1 = 1 λ2(3x1) 4 3 1 f4 . (23) the functions f1 ,f2, and f4 specified by (20), (22), and (23) satisfy system (17) of differential equations only if βλ3 = 1,∫ r(x1) (3x1) 1 3 dx1 = r(x1) 2 (3x1) 2 3 . (24) it follows from relation (24) that the function r(x1) must be constant in the domain where it is nonzero. thus, inside the ball of radius r1 , the components of metric (25) have the form f2 = λ(3x1) 2 3 , f4 = 1 λ3 − r 12 (3x1) 2 3 λ2 , f1 = 1 λ2(3x1) 4 3 1 f4 . (25) outside the ball, the scalar curvature vanishes, and relation (21) implies that the components of metric (15) have the form f2 = λ(3x1) 2 3 , f4 = 1 λ3 − c λ2(3x1) 1 3 . (26) these components satisfy also system (17),(18). the constant c in (26) is found from the condition that the metric components must be continuous on the entire space, which implies c = r 12r1 3. system (17), (18) has the unique solution (25), (26), which contains one free parameter λ. taking into account the additional condition g44(0) = 1 in the theorem, which implies λ = 1, and passing to the usual spherical coordinates, we see that relations (13) and (14) hold. this completes the proof of the theorem. thus, in the framework of the proposed axiomatics, we have constructed a mathematically rigorous model of a spherically symmetric space adequately describing the spherically symmetric gravitational field generated by a spherical gravitational source with constant mass density. theorem 2 are easy to extend to a more general case. 266 n.n. popov:a geometric interpretation of gravity theory 5 stationary spherically symmetric spaces with scalar curvature having finitely or countably many discontinuities of the first kind and a mathematically rigorous definition of spherically symmetric black holes consider a stationary spherically symmetric space endowed with a metric of the general form (11). suppose that the scalar curvature of this space is a piecewise smooth function r(r) with at most countably many discontinuities of the first kind at points r1, r2, . . . , rn, . . . numbered in increasing order. for such a space, the following theorem is valid. theorem 3. if the components of metric (11) satisfy the conditions g22(0) = 0 and g44(0) = 1 , then the scalar curvature of the spherically symmetric space under consideration is a piecewise constant function taking the constant values r(r) = rk at rk−1 < r ≤ rk for k = 1, . . . , n, . . ., where r0 = 0. the metric of such a space is everywhere continuous, provided that 2m(r) r < 1 for r ∈ (0,∞), and has the form ds2 = (1− 2m(r) r )dt2 − dr2 1− 2m(r) r − r2(dθ2 + sin2θdφ), (27) where m(r) = r∫ 0 ∑ k>0 rk 4 [θ(x− rk−1)− θ(x− rk)]x 2dx. the components of metric (27) satisfy system (12) of differential equations everywhere except at a finite or countable set of points r1, . . . , rn, . . . . proof. this theorem is proved by the same method as theorem 2. as in theorem 2, we show that, under the assumptions of theorem 3, the components of metric (11) in the spherical coordinate system have the form g44 = 1− 1 4r r∫ 0 r(x)x2dx, g22 = −r2, g11 = −g44 −1, (28) where r is a piecewise smooth function with at most countably many discontinuities of the first kind at points r1, r2, . . . . the functions in (28) satisfy system (12) only under the condition dr dr = 0, (29) which is an analogue of condition (24) in the proof of theorem 2. relation (29) implies that r(r) must take constant values in its domains of continuity. suppose that r(r) takes a value rk at rk−1 < r ≤ rk ; then r(r) = ∑ k=1 rk(θ(r − rk−1)− θ(r − rk)), advances in systems science and applications (2012) vol.12 no.3 267 which implies the assertion theorem 3. the components of metric (27) are everywhere continuous if 2m(r) r < 1 for r ∈ (0,∞). if 2m(r) r = 1 at some r = rg. in this case, there are two possibilities: either the spherically symmetric space is bounded by a hypersphere with radial parameter rg (if 2m(r) r > 1 at r > rg ), or this space can be extended (if 2m(r) r < 1 at r > rg ). in the latter case, the space contains a spherically symmetric body of radius rg with generally nonuniform mass distribution density, and on the boundary of this body, the metric exhibits an irregular behavior. we refer to such bodies as spherically symmetric stationary black holes. below we give the definition of the simplest stationary black hole. definition 1. a globular body of radius rg with constant mass density satisfying the condition r = 12 rg2 is called a stationary spherical black hole. the sphere of radius rg being the surface of a black hole is called its horizon level, or the schwarzschild sphere. it follows from the definition that rg = √ 12 r . note that the signature of the space inside a black hole and outside it remains invariable, which agrees with condition (2) on the metric of a pseudo-riemannian space. all components of metric (27) are continuously differentiable on the entire space except on the surface of the black hole, on which the behavior of metric (27) is irregular, namely, g11(rg) = −∞. the mathematical properties of a stationary spherically symmetric black hole in the formalism suggested here substantially differ from the properties of black holes investigated in the framework of grt[7]. if the spherically symmetric space has the same scalar curvature r at all points, then the space is the de sitter closed elliptic space[5] determined by metric (27) of the form ds2 = (1− r 12 r2)dt2 − dr2 1− r 12r 2 − r2(dθ2 + sin2θdφ) with r < √ 12 r . this space is bounded by a sphere with radial parameter r < √ 12 r . such a model corresponds to a homogeneous closed space filled with a matter with constant mass density. consider yet another parameter of the spherically symmetric space generated by a ball of radius r1 with constant mass density ρ1 and a matter with constant mass density ρ2 filling the whole space outside the ball. the scalar curvature of the space inside the ball is calculated by r1 = 32πρ1 and outside the ball, by 268 n.n. popov:a geometric interpretation of gravity theory r2 = 32πρ2. the components of metric (27) for each space have the form g44 = 1− r1 12 r2 at r < r1, g44 = 1− r1 −r2 12 r1 3 r − r2 12 r2 at r ≥ r1, and g22 = −r2, g11 = −g44 −1(r) if r1 12 r1 2 < 1. the spherically symmetric space is bounded by a sphere of radius r2, which is the least positive root of the cubic equation r3 − 12 r2 r + r1 −r2 r2 r1 3 = 0. a criterion for the transformation of the ball into a black hole is r1 12 r1 2 = 1. 6 a model of a nonstationary spherically symmetric pseudo-riemannian space with constant scalar curvature and the definition of dark energy in the preceding section, we described a stationary spherically symmetric space with discontinuous scalar curvature. here, we consider a mathematical model of a spherically symmetric pseudo-riemannian space with nonstationary fridmantype metric[8] ds2 = dx4 2 − r2(x4)(dx1 2 + sin2x1(x2 2 + sin2x2dx3 2)), (30) where r is the curvature radius of the three-dimensional hypersphere, which depends on the parameter x4; x1 ∈ (0, 2π); x2 ∈ (0, π); x3 ∈ (0, 2π); and x4 ∈ (0,∞). the following theorem is valid. theorem 4. system (10) for metric (30) reduces to the single second-order nonlinear differential equation r d2r dx42 − ( dr dx4 ) 2 − 1 = 0 for the unknown function r(x4). this equation has a real solution r = r0 cosh x4 r0 ,where r0 is a some constant. the scalar curvature of the space is everywhere constant and has the form r = 12 r02 . proof. according to (30), the nonzero components of the covariant metric tensor gij have the form g11 = −r2, g22 = −r2sin2x1, g33 = −r2sin2x1sin 2x2, g44 = 1. (31) advances in systems science and applications (2012) vol.12 no.3 269 the components of the contravariant metric tensor gij have the form g11 = −r−2, g22 = − 1 r2sin2x1 , g33 = − 1 r2sin2x1sin 2x2 , g44 = 1. (32) substituting the metric components (31) and (32) into relation (1), we find all nonzero elements of the pseudo-riemannian connection; these are γ22 1 = − sinx1 cosx1,γ33 1 = − sinx1 cosx1sin 2x2,γ14 1 = 1 r dr dx4 , γ12 2 = cotx1,γ33 2 = − sinx2 cosx2,γ24 2 = 1 r dr dx4 , (33) γ13 3 = cotx1,γ23 3 = cotx2,γ34 3 = 1 r dr dx4 , γ11 4 = r dr dx4 ,γ22 4 = r dr dx4 sin2x1,γ33 4 = r dr dx4 sin2x1sin 2x2. using the components (33) of the connection and applying formula (2), we obtain all nonzero components of the ricci tensor rij : r11 = −2( dr dx4 ) 2 − r d2r dx42 − 2, r22 = sin2x1r11, (34) r33 = sin2x1sin 2x2r11, r44 = 3 r d2r dx42 . it follows from (10), (32), and (34) that the scalar curvature is r = 6 r d2r dx42 + 6 r2 ( dr dx4 ) 2 + 6 r2 . (35) by virtue of (31), (34), and (35), system (10) degenerates into a single equation of the form r d2r dx42 − ( dr dx4 ) 2 − 1 = 0. (36) we seek a solution of equation (36) in the form r = a1e β1x4 + a2e −β2x4 , where a1, a2, β1, and are some constants. then equation (36) transforms into the relation a1a2(β1 + β2) 2 = e(β2−β1)x4 , which implies β1 = β2 and a1a2 = 1 4β1 2 . since r > 0, it follows that a1 > 0 if β1 > 0, and we can set a1 = ea 2β1 and a2 = e−a 2β1 . if dr dx4 | x4=0 = 0, then a = 0. setting a1 = r0, we finally obtain r(x4) = r0 cosh x4 r0 . 270 n.n. popov:a geometric interpretation of gravity theory there exists yet another, complex, solution of equation (36), namely, r = ix4, but we are interested only in real positive solutions. substituting the solution r(x4) = r0 cosh x4 r0 into (35), we obtain the following expression r = 12 r02 for the scalar curvature, which proves the theorem. at first glance, the physical interpretation of the assertion of theorem 4 may seem rather contradictory. on the one hand, the hyperspherical space expands by an almost exponential law, i.e., the curvature radius increases as r = r0 cosh x4 r0 , while the scalar curvature r itself does not depend on the parameter x4 and remains everywhere constant. according to postulate 2, this means that the mass density of the matter uniformly filling the expanding space remains constant as well. in reality, these results involve no contradiction. at present, the existence of a new type of matter in our universe, which is known as dark energy, has been reliably established in cosmology; this matter fills uniformly whole space and is characterized by constant mass density not depending on the time parameter x4. the model suggested above can be regarded as an example confirming the existence of this type matter with such unusual physical properties. 7 conclusion the axiomatization of gravity theory proposed in this paper and the fundamental gravity equation obtained on the basis of these axioms make it possible to demonstrate the effectiveness of the suggested approach for a number of physical examples considered in the paper. it suffices to mention that the problem of constructing an everywhere continuous spherically symmetric stationary metric of a pseudo-riemannian space with discontinuous scalar curvature, which is solved in general form in section 4, still remains unsolved in the framework of grt, in which solving this problem involves fundamental difficulties. in the framework of the formalism suggested here, the concepts of a stationary black hole and dark energy are defined more rigorously from the mathematical point of view. importantly, dark energy arises in a natural way as one of the solutions of the system (10) of gravity equations, which, unlike in grt[9], does not require introduce any additional empirical constants into the main equation, such as the cosmological constant and its comparatively recent interpretation as the mass density of dark energy[10]. references [1] p. k. rashevskii. (1967), riemannian geometry and tensor analysis, (nauka, moscow) [in russian]. [2] b. a. dubrovin, s. p. novikov and a. t. fomenko. (1979), modern geometry, (nauka, moscow) [in russian]. advances in systems science and applications (2012) vol.12 no.3 271 [3] d. hilbert. (1915), “die grundlagen der physik (erste mitteilung), nachr. königl. gesellschaft d. wiss. göttingen”,math. phys. klasse, heft 3, pp.395407. [4] a. z. petrov. (1967), newest methods in general relativity theory, (nauka, moscow) [in russian]. [5] w. de sitter. (1917), “on einstein’s theory of gravitation and its astronomical consequences”, third paper, monthly notices roy. astron, soc. vol.78, no.3. [6] schwarzschild, k. (1916), “on the gravitational field of a point-mass, according to einstein’s theory sitzungsber”, preuss. akad. wiss., phys. math. kl. vol.189. [7] s. chandrasekhar. (1983), “the mathematical theory of black holes, research supported by nsf”, international series of monographs on physics, vol.69 (clarendon press-oxford university press, oxford-new york). [8] a. a. fridman. (1966), on space curvature, sselected works, (nauka, moscow) [in russian]. [9] a. einstein. (1917), kosmologische betrachtungen zur allgemeinen relativitätstheorie, ber. preuß. akad. wiss., berlin. [10] a.d. chernin. (2008), “dark energy and universal antigravitation”, uspekhi fiz. nauk, vol.178 no.3, pp.267-298 [physics-uspekhi, vol.51, no. 3, pp.253282]. corresponding author n.n. popov can be contacted at: nnpopov@mail.ru advances in systems science and applications (2012) vol.12 no.1 76-88 the performance study on two-stage vibration isolation system with active magnetic suspension control xiaojing liu and yefa hu mechanical and electronic engineering school, wuhan university of technology, wuhan 430070, china abstract two-stage vibration isolation system is normal equipment on naval vessel, which has good function of vibration isolation. active control on vibration isolation system is better than passive control. active magnetic suspension control has many virtues like alterable control methods and variable stiffness and damping. so that it is applicable to complex multi-interferes and multi-coupling floating raft. this paper set up mathematic models of two-stage vibration isolation with active magnetic suspension control system and solves the transfer functions. after simulating in matlab it analysis the effectiveness of kp ,τd,τi of pid controller to vibration isolation and the parameters which influence the vibration isolation performance including up and down layer stiffness, damping and mass ratio. the simulation results indicate that active magnetic suspension has good works to vibration isolation. at the same time, on the basis of these results optimize the construction. keywords two-stage vibration isolation system, active control of magnetic suspension 1 introduction two-stage vibration isolation systems is one kind of device of power equipment for damping vibration, which has better isolation vibration performance compared with one layer device. internal and external research focus on dynamic performance of two-stage vibration isolate system affected by design parameters[1], effects of nonlinear spring and damping on two stage vibration isolate system[2], effect of pedestal stiffness on two stage vibration isolate system[3] and study of flexible mass on two stage vibration isolate system[4], and so on. compared with passive control of isolation vibration system, active control can get better effectiveness by adjusting stiffness and damping. on this aspect, there are many researchs about methods[5], the fix position of active vibration isolator[6] and coupling vibration control[7]. active magnetic suspension control system has many virtues including stiffness and damping controllable and tunable. furthermore, it can apply classical or modern control theories according to different simulation force value and place to get the best effect, which make it become the best control system for multi-turbulent and multi-coupling floatin raft.this paper establish two-stage vibration isolation system with active magnetic suspenadvances in systems science and applications (2012), vol.12, no.1 77 sion control equipment and its mathematic models. then simulate in matlab and research how the three parapmeters of pid controller affect the peroformance of the system. furthermore, study the influence of parameters such as up and down layer stiffness, damping and mass ratio to vibration isolation. simulation results indicate that active magnetic suspension does good works to vibration isolation. in the end, on the basis of these results optimize the construction. 2 active magnetic suspension vibration isolation control system 2.1 theory of active magnetic suspension put active magnetic suspension vibration isolation system into the layer between middle mass m2and pedestal m3 .construction and theory is showed in fig.1 and fig.2.there are three masses —m1 is up layer mass,m2 is middle layer mass,m3is pedestal mass.k1and c1 is stiffness and daming of the first stage.k2 and c2 is stiffness and daming of the second stage.u is the system of active magnetic suspension.u is the system of active magnetic suspension.f = f0 sinϖt is outer simulate force act on up layer mass m1.x1 is displacement of m1.x2 is placement of m2.m3 is fixed to foundation firmly. so ignore its displacement. active fig.1model of active magnetic suspension in two-stage vibration isolation system magnetic suspension feed back system can provide damping force direct proportion to absolute speed of isolated object. when vibration happened and isolated vibration object elasticity mass increase or actual elasticity factor decrease, isolation vibration spring damping static deflection still keep constant. therefore active magnetic suspension feed back isolate vibration system is better than passive isolated vibration system, especially in low frequency area. middle mass m2 displacement is control variable quantity which is recorded by displacement sensor. after comparing with setting amount, pid controller calculate discrepant value and change electric current to produce magnetic force which help middle mass keep balance position to get the purpose about decrease the force from up layer to base m3. 78 xiaojing liu:the performance study on two-stage vibration isolation system... fig.2 theory of magnetic suspension system 2.2 transfer function establish system block plant (see fig.3).f(s) is outer simulation force,f1(s) is force from system to base. hk(s) is pid controller transfer function.h2(s) is power amplifier transfer function.h1(s) is the first stage transfer function.h3(s) is the second stage transfer function. fig.3 active magnetic suspension vibration isolation system block plant without magnetic force, system force transfer function is t (s) = f1(s) f(s) = h1(s)h3(s) when apply feed back control, system force transfer function is t1(s) = f1(s) f(s) = h1(s)h3(s) 1 +h2(s)hk(s) fe = k i2 x2 linearization force to fe = ki(i− i0)− kx(x− x0) advances in systems science and applications (2012), vol.12, no.1 79 hk(s) = kp + τi s + τds 1 + tfs h3(s) = k2 + c2s,h2(s) = ks substitute all parameters into system differential equation. ( m1 0 0 m2 ) .. x1 .. x2 + ( c1 −c1 −c1 c1 + c2 ) . x1 . x2  + ( k1 −k1 −k1 k1 + k2 )( x1(t) x2(t) ) = ( f fe ) use laplace transformation to equations and arrange to get h1(s) = x2(s) f (s) = s+ 440 (s+ 3.62)2(s+ 1.01)2 2.3 simulations in matlab simulink build block plant of system transfer function (see fig4.) system with and without active magnetic suspension control forcetime curves are list at fig.5. fig.4 block plant to simulate in simulink of matlab fig.5 showed that active magnetic suspension system indeed helps to improve the performance of two-stage isolate vibration. force transfer from up layer outer 80 xiaojing liu:the performance study on two-stage vibration isolation system... simulate to base become quit so small apparently and vibration curve become smooth. (a) nno feedback time-force transfer (b) with feedback time-force transfer fig.5 comparation with before and after using magnetic suspension feed back control 3 influence of pid controller parameters to isolate vibration 3.1 effective of proportion parameter kp without change other parameters, kp = 100, 500, 1000 result curve showed at fig.6 from curves we can see increase kp will decrease force to base tremedously. fig.6 different kp time-force curves 3.2 effective of differential constant τd without change other parameters, τd=100,500,1000. result curve showedat fig.7. form fig.7, τd has a little influnce to force max value and the speed of isolation vibriation. however, increase τd can make smooth of force vibraition and improve the performance of the system. advances in systems science and applications (2012), vol.12, no.1 81 fig.7 different τd time-force curves 3.3 effective of integral constant τi without change other parameters,τi=0.5,5,50. result curve showed at fig.8. fig.8 integral constant τd time-force curves curve tell that τi has no big influence to max force value but only effect to min force value slightly. 4 effectiveness of two-stage vibration isolation system ather parameters 4.1 natural frequency and force coefficient of transmission calculation without thinking of damping, system natural frequency will be calculated by listed formula(1) ωn 2 = k1 + k2 2m1 + k2 2m2 ± √ ( k1 + k2 2m1 + k2 2m2 )2 − k1k2 m1m2 substitute parameters into the formula: ξ2 = 0.06n/(m/s), ξ1 = 0.04n/(m/s) 82 xiaojing liu:the performance study on two-stage vibration isolation system... the natural frequencies are: ωn1 = 78hz,ωn2 = 2hz when force excited, the force coefficient of transmission is formula (2) tf = √ (α2 − 4ξ1ξ2αϖ2 1) 2 +ϖ2 1(2ξ1α 2 + 2ξ2α)2 a2 +b2 in the formula a = ϖ1 4 −ϖ1 2(α2 + 4ξ1ξ2α+ µ+ 1) + α2 b = ϖ1 3(2ξ2α+ 2ξ1µ+ 2ξ1)−ϖ1(2ξ1α 2 + 2ξ2α) µ = m1/m2, ϖ1 = ω/ω1, ϖ2 = ω2/ω1 ω1 2 = k1/m1, ω2 2 = k2/m2 ξ1 = c1/2 √ k1m1, ξ2 = c2/2 √ k2m2 when substitute the parameters into the formula, we get the curve of forcecoefficient transmission—simulation frequency (see fig.9). fig.9 curve of force-coefficient transmission-simulation frequency from the curve we can see the influence of simulation frequency to forcecoefficient transmission: • after 53hz, forcecoefficient of transmission is near zero that indicates the effectiveness of isolation vibration is good; • before 53hz, there are two peak values on resonance vibration that indicates this frequency region is the target we will research. to improve the effectiveness. advances in systems science and applications (2012), vol.12, no.1 83 4.2 influence of stiffness k2 to natural frequency according to formula (1), we can get the influence of stiffness to natural frequency (see fig.10). from the curve, we can see stiffness is almost linear direct proportion to natural frequency. so, the more stiffness is the more natural frequency is. on different working condition, we can change stiffness aim at natural frequency to avoid resonance vibration. fig.10 curve of stiffness to natural frequency fig.11 influence of stiffness to force-coefficient of 4.3 influence of stiffness k2 to force-coefficient transmission according to formula (2), choose three different value of stiffness k2500×102n/m, 500× 104n/m, 510×103n/m.we get three curves about k2 — force-coefficient of transmission (see fig.11). 84 xiaojing liu:the performance study on two-stage vibration isolation system... curves indicates that when stiffness is small the peak value move to left. if change up-layer stiffness, result is on the contrary that means when becomes larger the peak value move to left. so, increase up-layer or decrease down-layer stiffness all is good for isolation vibration. 4.4 influence of damping k1 ratio k1 to force-coefficient transmission without changing other parameters, damping ratio ξ1 = 0 ξ2 = 0 become ξ1 = 0.04 ξ2 = 0.06 we get the curve of simulation frequency—force e-coefficient of transmission (see fig.12).fig.12 indicates that damping ratio influence peak value directly but it doesn’t influence the position and tendency of peak value. the smaller damping ratio is the bigger peak value is. when ξ = 0.04, ξ2 = (0.01, 0.06, 0.2, 2) respectively, we get the curves (see fig.12 ξ1 = 0, ξ2 = 0; ξ1 = 0.04, ξ2 = 0.06 simulation frequency—force ecoefficient of transmission fig.13).when damping ratio is less than 1, it can’t influence peak value position and tendency, but influence peak value. the bigger damping ratio is the smaller peak value is, that is good for isolation vibration. while when damping ratio is more than 1, one hand it can decrease the peak value, on the other hand, peak value position move from left to right. advances in systems science and applications (2012), vol.12, no.1 85 fig.13 ξ1 = 0.04,ξ2 = (0.01, 0.06, 0.2, 2) simulation frequency—force e-coefficient of transmission fig.14 ξ2 = 0.06,ξ2 = (0.01, 0.04, 0.1, 2, 20) simulation frequency—force ecoefficient of transmission fig.15 change ξ2 and ξ1 simulation frequency—force e-coefficient of transmission 86 xiaojing liu:the performance study on two-stage vibration isolation system... when ξ2 = 0.06,ξ1 = (0.01, 0.04, 0.1, 2, 20) respectively, we get the curves (see fig.14).when damping ratio is less than 1, it can’t influence peak value position and tendency, but influence peak value. the bigger damping ratio is the smaller peak value is, that is good for isolation vibration. while when damping ration is more than 1, peak value position move from left to right and peak value become bigger when damping ratio is bigger. so, damping ratio than close to 1 is the best value to isolation vibration. 4.5 influence of mass ratio to force-coefficient transmission when m1 = 104kg, ξ1 = 0.04, ξ2 = 0.06, µ = 0.04042, 2, 5, 10 respectively, we get the curves showed in fig.16. isolation vibration system brings about two resonance points which position are determined by mass ratio. when µ ≥ 1, the first peak value position move to right but not very clearly and the second peak value position move to right quiet obvious. there is small influence to curve tendency in lower frequency region ,such as 40hz. but in low frequency region, the bigger mass ratio is the smaller peak value is. in middle frequency region, from 40hz to 180hz, curve change apparently, the second peak value decrease quickly. it is obviously that mass ratio influence middle frequency greatly. to make sure system work at frequency far away from resonance points we hope the distance between the first peak value and the second peak value is large, which means µ ≥ 1(mass m1 ≥ m1) is good for isolation vibration. fig.16different curve of simulation frequency—force e-coefficient of transmission 5 improve isolation vibration system according to above analysis, we take three proposals to improve the system performance. because mass has been chosen, then • without changing other parameters, decreasing under layer stiffness by readvances in systems science and applications (2012), vol.12, no.1 87 ducing spring from 6 to 4, then stiffness k2 = 430× 103n/m • without changing other parameters, raise up and under layer damping ξ1 = 0.4n/(m/s), ξ2 = 0.6n/(m/s) • decrease under layer stiffness and increase up layer stiffness at the same time, k2 = 430× 103n/m, ξ1 = 0.4n/(m/s), ξ2 = 0.6n/(m/s) results showed in fig.17, from which we can see that damping is the most important influence factor and only change stiffness is good for middle-high frequency isolation vibration. the best solution is decreasing under layer stiffness and increase up and under damping simultaneously. fig.17 change stiffness and dampin simulation frequency—force e-coefficient of transmission 6 conclusion this paper research on the performance of two-stage vibration isolation system and the effectiveness of system parameters including three parameters of pid controller,stiffness, damping and mass ratio. conlusion are • essence of active magnetic suspension control system act on two-stage vibration isolation system is change the second layer stiffness and damping to decrease vibration. • adjusting stiffness by active magnetic suspension to avoid system nature frequency is a good method to prevent resonance vibration. • decreasing the second stiffness by adjust active magnetic suspenion control has good effect. 88 xiaojing liu:the performance study on two-stage vibration isolation system... • adjusting damping by active magnetic suspension control to keep damping less than 1is good effect. • increasing proportion parameter kp is good for vibration isolation speed. • increasing differential constant τd ralatively is good for decreasing vibration. • integral constant τi has no apparent influence to prevent vibration. references [1] su ronghua, peng chenyu, ding wenwen. 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(2003), “research on dynamics of two-layer active vibration isolation system”, china mechanical engineering, vol.7, pp.1232-1235. [7] yang tie-jun, chen yuqiang, huang jinge. (2001), “simulation research on active control of coupling vibration of a two-stage isolation system for diesel”, ship engineering, vol.3, pp.24-27. microsoft word 10 li guangyu wei fengying wang ke--some results for self-stabilization of stochastic differentialequation.d 280-285 advances in systems science and applications (2011), vol.11, no.3-4 issn 1078-6236 international institute for general systems studies, inc. some results for self-stabilization of stochastic differential equation * li guangyu1, wei fengying 2 and wang ke 3 2,1 1dept of math, wenzhou univercity oujiang college, wenzhou 325027, zhejiang,pr china； 2college of mathematics and computer science, fuzhou university, fuzhou 350002, fujian, pr china； 3dept of math, harbin institute of technology, weihai 264209, shandong, pr china abstract a class of stochastic differential equation is studied about the self-stabilization in this paper. by constructing suitable hypothesis, sufficient criteria for the stochastic differential equation stabilized itself are established. keywords self-stabilization; stochastic neural networks; it’s formula; borel-cantelli lemma mr(2000) subject classification 92b20; 93e15 1. introduction the stability of solutions is very important in the theory of differential equations. many authors have done very excellent work [1-3] on the fields of stability. however, the stability for stochastic differential equations are somewhat difficult than that of deterministic differential equations, related literature can be found in recent work. for stochastic differential equations, [4] discussed a criterion that determines the region of mean square stability for second-order weak numerical schemes, [5] had given some sufficient conditions corcerning stability of solutions of stochastic differential evolution equations with general decay rate and [6] considered the one-step approximations of solutions, respectively. here we just mentioned the work for stability about mao xuerong, for example, he established stochastic versions of the well-known lasalle stability theorem in [7] and investigated exponential stability of paths for a class of hilbert space-valued non-linear stochastic evolutions in [8]. our work is motivated by mao [1], he had showed that the trivial solution of equation (1.1) in [1], namely, ( ) ( ( ), ) ( ( ), ) ( )dx t f x t t dt ug x t t db t= + (1) was almost surely exponentially stable for all sufficiently large u ,let 0u > be the noise intensity parameter and )(tb be an m -dimensional brownian motion. mao [1] had discussed if the intensity parameter u was replaced by 0 ( ) ( ) pt r s x s ds∫ , then the equation (1.1) becomes 0 ( ) ( ( ), ) ( ( ) ( ) ) ( ( ), ) ( ) pt dx t f x t t dt r s x s ds g x t t db t= + ∫ (2) where 0p > and ( )r s was a continuous $r^{n\times d}$-valued function defined on r+ satisfying ( ) tr t meγ≤ for all 0t ≥ ,which be called a convergence rate function. the standing hypothesis (h1) and (h2) are imposed in [1] as follows: *supported by nnsf of pr china(no.10701020) ; supported by nnsf of p.r.china (no.10726062). advances in systems science and applications (2011), vol.11, no.3-4 281 (h1.1) there exists a symmetric positive-definite d d× -matrixq and three positive constants , ,k α β with 2β α> , such that 2 2 2 ( , ) , ( ( , ) ( , )) , ( , ) , t t t t t x qf x t k x trace g x t qg x t x qx x qg x t x qx α β ≤ ≤ ≥ for all 0t ≥ and dx r∈ . now, we will prove that equation (4) stabilizes itself, that is a alternative theorem given by expressions (7)and (8). as far as the author's knowledge that there is no corresponding results. to make the statement more clear, mao [1] had stated the condition on the convergence rate function ( )r t as another hypothesis: (h1.2) there exists a pair of constants 0m > and 0γ ≥ such that ( ) tr t meγ≤ for all 0t ≥ 。 mao [1] had proved that if (h1.1) and (h1.2) hold, then for every 0 dx r∈ , the solution of equation (1.2), either 0 min 2( ) ( ) (2 ) ( ) p kr t x t dt qβ α λ ∞ ≤ −∫ (3) or ( )0 1limsup log ( ; ) 0 t x t x t→∞ < (4) holds for almost allω∈ω . if u is replaced by 0 sup ( ) ( ) s t r s x s ≤ ≤ , then equation (1.1) becomes 0 ( ) ( ( ), ) (sup ( ) ( ) ) ( ( ), ) ( ) s t dx t f x t t dt r s x s g x t t db t ≤ ≤ = + (5) for general 0 0t t≥ = and denote 0(0) dx x r= ∈ . mao [1] had also shown that if (h1) and (h2) hold, then for every 个 0 dx r∈ , the solution of equation (5) has the property 0 sup ( ) ( ) t r t x t ≤ <∞ < ∞ as. (6) furthermore, (i) if t →∞ as min ( ( ) ( ))r t x tλ →∞ , then lim ( ) 0 t x t →∞ = as. (7) (ii) if minliminf log[ ( ( ) ( ))] / 0t t r t r t tλ λ →∞ ≥ > ，then ( )1limsup log ( ) 2t x t t λ →∞ ≤ − as. (8) it is very interesting for us to analyze whether the system 0 ( ) ( ( ), ) (sup ( ) ( ) ) ( ( ), ) ( ) s t dx t f x t t dt r s x s g x t t db t ≤ ≤ = + (5) has the altermnative theorem like the results given by expressions (3) and (4). li: some results for self-stabilization of stochastic differential equation 282 to the author's knowledge, there is no corresponding results. now we will put our efforts on system (5), by use of exponential martingale formula, lyapunov function and some special inequalities in our paper, the trivial solution of equation (5) is self-stabilization will been shown in the next section. let us begin with our paper now. 2. main results lemma 2.1 let hypothesis (h1) hold. then the solution of equation (5) has the property that 0{ ( ; ) 0p x t x ≠ for all 0} 1t ≥ = provided 0 0x ≠ . we consider the following problem of stochastic self-stabilization in this section. suppose we are given a stochastic differential equation 0 ( ) ( ( ), ) (sup ( ) ( ) ) ( ( ), ) ( ) s t dx t f x t t dt r s x s g x t t db t ≤ ≤ = + (9) on 0 0t t≥ = with initial value 0(0) dx x r= ∈ (it is just for convenience to set 0 0t = and the theory clearly works for general 0 0t ≥ . we have the following result. theorem 2.1 let (h1) and (h2) hold. then for every 0 dx r∈ , the solution of equation (9), either 0 min 2sup ( ) ( ) (2 ) ( )s kr s x s qβ α λ≤ ≤∞ ≤ − (10) or ( )0 1limsup log ( ; ) 0 t x t x t→∞ < (11) proof. since hypothesis (1) guarantees ( ;0) 0x t ≡ , one only need to show the conclusions for all 0 0x ≠ . fix 0 0x ≠ arbitrarily, and write 0( ; ) ( )x t x x t= , by lemma 2.1 0t ≥ for all 0( ; ) 0x t x ≠ almost surely. suppose (10) is false, then there exists some 0 0x ≠ for which _ ( ) 0p ω > , where 0 min 2: sup ( ) ( ) . (2 ) ( )s kr s x s q ω β α λ − ≤ ≤∞ ⎧ ⎫⎪ ⎪ω = ∈ω >⎨ ⎬−⎪ ⎪⎩ ⎭ clearly, one only needs to show that (11) holds for almost allω∈ω , for each 1, 2, ,i = … define 1 1 2 0 min 2: sup ( ) ( ) (1 ) . (2 ) ( )i s kr s x s i i q ω β α λ − − − ≤ ≤∞ ⎧ ⎫⎪ ⎪ω = ∈ω > +⎨ ⎬−⎪ ⎪⎩ ⎭ now, 1 i i − −∞ = ω∈ ω∪ , and hence one only needs to show that for each 1i ≥ ,（10）holds for almost iω − ∈ω . fix any 1i ≥ let ( ( ), ) log( ( ) ( ))tv x t t x t qx t= , one then derives that advances in systems science and applications (2011), vol.11, no.3-4 283 2 2 ( ) 2 ( ) ( ) 2 ( ) 2 ( )( ( ), ) 0, ( ( ), ) , ( ( ), ) . ( ) ( ) ( ( ) ( )) t t t t t t x t q qx t qx t qx t x t qv x t t vx x t t vxx x t t x t qx t x t qx t − = = = i by it ô ’s formula, we obtain, we obtain 0 0 0 0 ( ( ), ) ( ( ), ) ( ( ), ) 1 [ ( ( ), )(sup ( ) ( ) ) ( ( ), )(sup ( ) ( ) ) ( ( ), )] 2 ( ( ), ) ( ( ), ) (sup ( ) ( ) ) ( ( ), ) ( ( ), ) ( ) 1 (sup ( ) ( ) ) 2 t x t xx s t s t x x s t s t dv x t t v x t t dt v x t t dx trace g x t t r s x s v x t t r s x s g x t t dt v x t t f x t t dt r s x s v x t t g x t t db t r s x s ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ = + + = + + 2 0 2 2 0 0 ( ( ( ), ) ( ( ), ) ( ( ), )) 2 ( ) ( ( ), ) ( ) ( ( ), )2(sup ( ) ( ) ) ( ) ( ) ( ) ( ) ( ) 1(sup ( ) ( ) ) ( ( ( ), ) ( ) ( ) ( ( ), )) ( ( ) ( )) 2(sup t xx t t t t s t t t t s t trace g x t t v x t t g x t t dt x t qf x t t x t qg x t tdt r s x s db t x t qx t x t qx t r s x s trace g x t t qx t qx t g x t t dt x t qx t ≤ ≤ ≤ ≤ ≤ = + + − 2 t 2 1( ) ( ) ) ( ( ( ), ) ( ) ( ) ( ( ), )) , ( ( ) ( )) t t s t r s x s trace g x t t qx t x t qg x t t dt x t qx t≤ this yields that 0 0 00 0 2 00 2 0 log( ( ) ( )) 2 ( ) ( ( ), ) ( ) ( ( ), )log( ) 2 (sup ( ) ( ) ) ( ) ( ) ( ) ( ) ( ) ( ( ( ), ) ( ( ), ))(sup ( ) ( ) ) ( ) ( ) ( ) 2 (sup ( ) ( ) ) t t tt t t t t s t t t t s t t s t x t qx t x s qf x s s x s qg x s sx qx ds r s x s db s x s qx s x s qx s trace g x s s qg x s sr s x s ds x s qx s x s r s x s ≤ ≤ ≤ ≤ ≤ ≤ = + + + − ∫ ∫ ∫ 2 2 0 ( ( ), ) . ( ( ) ( )) t t qg x s s ds x s qx s∫ by hypothesis (h1.1) and the fact that 2 2 min max( ) ( ) ( ) ( ) ( ) ( )tq x t x t qx t q x tλ λ≤ ≤ for q is a symmetric d d× matrix, one can show that for any 0t ≥ 2 0 0 0min 0 2 2 2 00 log( ( ) ( )) 2log( ) ( ) (sup ( ) ( ) ) ( ) ( ) ( ( ), ) 2 (sup ( ) ( ) ) ( ( ) ( )) t t t v s tt t v s x t qx t ktx qx m t r v x v ds q x s qg x s s r v x v ds x s qx s α λ ≤ ≤ ≤ ≤ ≤ + + + − ∫ ∫ (12) where 00 ( ) ( ( ), )( ) 2 (sup ( ) ( ) ) ( ) ( ) ( ) t t t v s x s qg x s sm t r v x v db s x s qx s≤ ≤ = ∫ is a continuous martingale vanishing at 0t = , let 1, 2, ,k = … then by the exponential martingale inequality li: some results for self-stabilization of stochastic differential equation 284 1 1 2 0 2 4 (1 ) log 1( : sup ( ) ( ), ( ) ) , 4 (1 ) 2t k i kp m t m t m t i k β α βω β β α − − ≤ ≤ ⎡ ⎤− + − > ≤⎢ ⎥+ −⎣ ⎦ where 2 2 2 00 ( ) ( ( ), ) ( ), ( ) 4 (sup ( ) ( ) ) . ( ( ) ( )) tt t v s x s qg x s s m t m t r v x v ds x s qx s≤ ≤ = ∫ wherehence the well-known borel-cantelli lemma yields that for almost all ω∈ω there exists a random integer 1( )k ω uch that for almost all 1k k≥ , 1 1 0 2 4 (1 ) logsup ( ) ( ), ( ) 4 (1 ) 2t k i km t m t m t i β α β β β α − − ≤ ≤ ⎡ ⎤− + − ≤⎢ ⎥+ −⎣ ⎦ that is，for 0 ,t k≤ ≤ 1 1 2 1 2 1 2 00 4 (1 ) log 2( ) ( ), ( ) 2 4 (1 ) ( ) ( ( ), )4 (1 ) log 2 (sup ( ) ( ) ) . 2 (1 ) ( ( ) ( )) tt t v s i km t m t m t i x s qg x s si k r v x v ds i x s qx s β β α β α β β β α β α β − − − − ≤ ≤ + − ≤ + − + + − = + − + ∫ (13) substituting (13）into (12) and then applying (h1) one obtains that for each ^ ,ω∈ω−ω with ^ ω a p-null set, there exists a random integer 2 ( ),k ω such that, for every 个 iω − ∈ω there exists a random number 3( )k ω such that 1 1 2 0 min 2sup ( ) ( ) (1 ) (2 ) ( )s t kr s x s i i qβ α λ − ≤ ≤ ≥ + − for almost all 3.t k≥ it then follows from that for almost all ^ iω − ∈ω −ω， if 2 31 , ( 1)k t k k k k− ≤ ≤ ≥ ∨ + 3 1 2 0 0 1 0min 1 1 0 0 3 min min 1 3 0 0 min log( ( ) ( )) 2 4 (1 ) log 2log( ) (sup ( ) ( ) ) ( ) 2 (1 ) 2 4 (1 ) log 2 (1 )log( ) ( 1 ) ( ) 2 ( ) 2 ( 1) 4 (1 ) llog( ) ( ) t t t v sk t t x t qx t kt i kx qx r v x v ds q i i kt i k k ix qx k k q i q k k ix qx q β β α λ β α β λ β α λ β λ − − ≤ ≤ − − − + − ≤ + + − − + + + ≤ + + − − − − + + ≤ + + ∫ 3 min og 2 ( 1 ). 2 ( ) k k k k i qβ α λ − − − − this implies that 1 3 0 0 3 min min 2 ( 1)1 1 4 (1 ) log 2log( ( ) ( )) log( ) ( 1 ) . 1 ( ) 2 ( ) t t k k i k kx t qx t x qx k k t k q i q β λ β α λ −⎡ ⎤+ + ≤ + + − − −⎢ ⎥− −⎣ ⎦ it then follows that min 1 2limsup log( ( ) ( )) ( ) t t kx t qx t t i qλ→∞ ≤ − for almost all ^ .iω − ∈ω −ω (14) advances in systems science and applications (2011), vol.11, no.3-4 285 thus, for almost all ^ ,iω − ∈ω −ω there exists a random number 4 ( )k ω and 0δ > arbitrarily such that min 1 2log( ( ) ( )) 2 ( ) t kx t qx t t i q δ λ ≤ − + for almost all 4 ,t k≥ from the expression 2 min ( ) ( ) ( ) ( )tq x t x t qx tλ ≤ one derives that min min exp{( ) } ( )( ) ( ) k t i qx t q δ λ λ − + ≤ for almost all 4.t k≥ consequently, 0 min 1 2limsup log( ( ; ) ) ( )t kx t x t i q δ λ→∞ ≤ − + for almost all ^ .iω − ∈ω −ω since 0δ > is arbitrary, we must have that 0 min 1 2limsup log( ( ; ) ) 0 ( )t kx t x t i qλ→∞ < − < for almost all ^ .iω − ∈ω −ω the proof is now complete. references [1] x. mao, stochastic differential equations and there applications, horwood publication, chichester, 1997:100-141 [2] j. k. hale, theory of functional differential equations, springer-verlag, new york inc, 1977:50-70 [3] geng liu, study on capacity expansion problems on directed networks, advances in systems science and applications 2007, 7(4): 679-684 [4] xianguo wu, lieyun ding, xintian cai, deformation predication of deep excavation pit support structure based on fuzzy neural network, advances in systems science and applications 2006, 6(2), 119-126 [5] xiaoya hu, desen zhu, bingwen wang, delay analysis of switched ethernet for networked control systems, advances in systems science and applications 2006, 6(2) : 201-206 [6] chen wanyi, globally exponential asymptotic stability of hopfield neural network with time-varying delays, act scientiarwn naturaliwn universitatis nankaiensis, 2005, 5 (3):70-75 [7] jiang minghui, shen yi, liao xiaoxin, stability of stochastic neural networks with multi-delay, mathemics applicata, 2006, 3 (19): 61-65 [8] x. mao, some contribution to stochastic asymptotic stability and boundedness via multiple lyapunov functions, j. math. anal. appl. 2001 260: 325-340. the article e.v. korolyuk, e.v. mezentseva “program-target approach based on the institutional environment of entrepreneurship in the krasnodar territory”, advances in systems science and application (2016), vol.16, no.4, p. 13-21 was withdrawn by the editorial board due to violation of assa publication ethics (please, see publication ethics statement). the reason is that the article’s content lacks originality due to substantial intersections with the following paper: e.v. korolyuk, e.v. mezentseva "establishing programme-target approachbased institutional entrepreneurship environment in krasnodar krai” in mediterranean journal of social sciences (2015), vol. 6, no. 6, pp. 583-589 editorial board of advances in systems science and application, 22.08.17 http://ijassa.ipu.ru/ojs/ijassa/ethics advances in systems science and application (2016) vol.16 no.2 54-69 adaptive background modeling for dynamics background n.a. zainuddin, y.m. mustafah, a. a. shafie, a.w.azman, m.a. rashidan, and n.n.a. aziz department of mechatronics, kulliyyah of engineering, international islamic university malaysia abstract an increasing number of cctv have been deployed in public and crime-prone areas as demand for automatic monitoring system is increasing to counterbalance the limitation of human monitoring. to have a good monitoring system in such places, a good background model is needed in order to reduce amount of the video processing needed for tracking, classification, counting and etc. this paper proposes an adaptive background modeling that is able to model a scene under review at real-time. the proposed modeling system is also expected to be able to handle dynamic backgrounds and common problems in detection methods. a novel patch-based background reconstruction based on highest frequency of occurrences assumption and past pixel observation is proposed. contrast adjusting method is used to reduce the problem of incorrectly classified foreground which is shadow problem. the proposed algorithm is focused to be tested and analytically compared with the dynamic background at the indoor and outdoor environment. the main challenges of background subtraction such as illumination changes, geometrical changes, stationary moving object problem and high speed object problem are taken care of and extensively discussed in this paper. the experimental results show that the algorithm is able to reconstruct a background model and produce accurate and precise foreground that can be used for other processing stages. keywordsadaptive background modeling; background subtraction; surveillance; foreground segmentation; dynamic background. 1 introduction there are two basic elements in video analysis that need to be identified, which are background and foreground region. the background which is also known as the reference frame usually consists of non-interested objects, either static or dynamic. foreground image on the other hand, is the objects of interest that need to be identified, detected or further analyzed. the shape of a foreground region unnecessarily be a rectangle. it can be arbitrary in shape and the background is in the complementary shape of the foreground region. the most common technique used by many researchers in handling this problem is by using background subtraction[1,2]. there is a lot of study in background subtraction, but the problem is still yet to be properly solved. the problem of ghost, stationary advances in systems science and application (2016) vol.16 no.2 55 moving object and repetitive movement of complex background scenes have been attempted to be solved, however, the result is still unsatisfactory. in the early days, many researches have focused on improving the general background subtraction process such as temporal differencing, double temporal differencing and many more. the recent trend shows that researchers tend to work on the specific stage in the background subtraction algorithm, which is background modeling and background reconstruction. the reconstruction of the background model is essential in order to correctly identify the foreground. a good background model will benefit the analysis of the foreground and produce accurate and effective background subtraction. the simplest method is setting the image without any moving object as the background[3]. however, this assumption is too idealistic, and not possible to solve many problems especially in outdoor environments, such as on a crowded street with vehicles and pedestrians or an outdoor environment like in a street will experience various levels of illumination at different times of the day[4]. plus, outdoor environment has a tendency of experiencing adverse weather condition like fog, rain or strong wind that could modify the reference image[5]. in these cases, the background must be adaptively refreshed and updated. therefore, in this paper, an adaptive background modeling technique is presented. there are five sections in this paper. in section two, previous works related to the research are described. meanwhile, in section three, the proposed algorithm is presented and discussed extensively. in section four, the experimental results at various scenes are presented with qualitative and quantitative results from various situations are illustrated. finally, the conclusion is drawn in section five. 2 related works recently, a lot of works are concentrating on adaptive background reconstruction[69]. several notable methods were introduced including temporal smoothing, pixel intensity classification, running gaussian average, gaussian mixture model, hidden markov model and kernel density estimation. gaussian mixture model (gmm) is a common method used by researchers in the foreground detection. the main idea of the gaussian mixture model is by having n number of gaussian distribution function, in order to reconstruct the background pixel. the implementation of gmm has been proposed as the adaptive background reconstruction technique, especially in surveillance system since late 1990s by stauffer et al.[10]. as it is still in the early stage, a lot of unwanted small blobs are scattered all over the frame and the main blobs aren’t completely detected as foreground. since then, many of the researchers have continued to improve the proposed algorithm[11-13]. mukherjee et al. have incorporated the horpreset colour model in order to make the algorithm to be able to detect 56 n.a. zainuddin etal: adaptive background modeling for dynamics background shadows[11]. mukherjee et al. work focuses more on the shadow handling, the experimental results show that the algorithm is tested for a non-complex indoor scene with one foreground handling. literature[12,13] managed to reconstruct the background with shadow consideration better with a more complex scene than mukherjee et al.’s work. they use the train station dataset in order to have multiple object handling. they have implemented the use of window-based decision rules for the shadow and the background model is reconstructed by gmm. to sum up, many implementations of gmm are focusing on the indoor scene for the surveillance system. elgammal et al. claimed that gmm is ideal for indoor scenes only[14]. thus, they introduced the use of kernel density estimation (kde) for background reconstruction. in kde method, the background and foreground pixel are model of probability density function (pdf) and the pdf is estimated by a kernel function or also known as window function. however, the major drawback of this method is the expensive computational cost. later gao et al. improved the method by introducing the marr wavelet equation in estimating the pdf[5]. marr wavelet is a second derivative of the gaussian smooth function. they combine the gaussian and kernel method together. however, from the experimental results, the background model is not reconstructed correctly as some of the background pixels are counted as a target; thus in some cases, the target is not detected correctly. lee et al. managed to reduce the need of storage[15]. they initialized the first frame and update it at every frame by setting the learning rate. for dynamic background cases, they used threshold method. the first frame was set as the background model and later, updated by learning method. this method is only ideal for the video sequence at has no target or foreground the beginning of the video sequence. for the crowd and complex scene, it is quite impossible to get such video sequence. the other method that commonly used by researchers in reconstructing the background is temporal smoothing[16]. the main idea of temporal smoothing is combining the stored image with the new image on pixel-based. pixels on the smoothed image will be replaced by a part from previous value combined with a new value from new image at the same position. ridder et al.[17] have improved the method by modeling each pixel with kalman filter. they managed to make the system more robust to cope with the illumination changes, however, the algorithm update the background slowly. later, in recent year, hung et al. improved the method by combining with the median filtering[6]. by doing so, they managed to reduce the computing frequency of median operations. however, they focus on the computational performances of the algorithm with no real life situation is considered in their experiment. motivated by [6], asif et al. use the temporal smoothing and median filtering for modeling the background[7]. as advances in systems science and application (2016) vol.16 no.2 57 their research focus more on the human gait, the background and foreground are non-complex. based on their experimental result, the temporal smoothing seems to be ideal for the non-complex scene. thus, the implementation of temporal smoothing algorithm is ideal for surveillance system that involves non-complex scene (indoor surveillance). in early of 2000s, hou et al. have introduced the pixel intensity classification (pic) method as the background reconstruction technique[18]. in this approach, at every frame, the difference of pixel intensity is calculated. then, the classification is made based on the calculated difference. the background model is assumed to be the highest frequency in the intensity value. the simulation results of hou et al. literature[18] used the outdoor dataset with parking lot environment. the works on the same method is continued by xiao et al. in 2006 and 2008[2,8]. xiao et al. managed to handle more adverse weather by adapting the raining situation, however, their experimental results shown that the adaption of the algorithm to the rain is still at the early stage. next, cao et al. have employed the improved version of pic in reconstructing the background model for light flow traffic movement video sequence[19]. they managed to compare their experimental results with gmm and time averaging algorithm. the results however are only ideal for slow moving to medium moving foreground detection. some of the foreground in high speed foreground is not detected and small blobs appeared as a result from the ghost problem also known as the blending of high speed object movement in the video sequence. after analyzing these methods, several assumptions were made. in non crowded scenes, the background pixel would be the maximum frequency in the image sequence. for crowded condition, most of the pixels in the image frame are expected to be modified. from the assumptions, we developed a background reconstruction algorithm based on pixel intensity classification and mode filtering. however, the difference in the inter-frame pixel intensity value is not calculated in the first step, the mode filtering is done first in order to model the background. 3 methodology in order to produce a good background model, several techniques are deployed through several stages with pre-determined assumptions. fig.1 shows the overall process of proposed background modeling algorithm. an elaborated discussions on the proposed algorithm is presented in this section. 3.1 pre-processing the video will be converted into uniform size of image sequences. next, the image sequence will be gray-scaled and undergone median filtering process in order to get rid of noises, especially from the camera pixel noise and impulse noise. 58 n.a. zainuddin etal: adaptive background modeling for dynamics background fig. 1 process of proposed background modeling algorithm 3.2 background model in order to model the background, an assumption of background-foreground pixels needs to be made. thus, based on the assumption that the background pixel is the most frequent appeared in the entire video sequences, the proposed algorithm is as follows: step 1 : calculate the mean of each patch let f1, f2, f3, · · · , fn represent the frames from the same video sequence. first, each of the frames must be segmented into m × n patches, where m and n can be the same number. assume n patches are obtained, where we let patches be marked as p1, p2, p3, · · · , pn . experiments are conducted to find the optimum size of the patches. the result of the experiments shows that the size of the advances in systems science and application (2016) vol.16 no.2 59 patches is not as sensitive as the sampling frame value. it does affect the result, but in small percentage error. then, at each frame,f1, f2, f3, · · · , fn the intensity value of pixel (x,y) in the area of specified patch, pi are marked as i1, i2, i3, . . . , in, where i = 1, 2, 3, . . . , n and total number of pixels in one patch can be written as: q = m× n (1) then, mean mp,f at each frame for the specified patch is calculated as in the formula below: mp,f = ∑q j=1 ij(x, y) q (2) step 2 : calculate the preliminary mode in order to calculate the mode, the calculated mean is grouped into matrices called mean matrixp,f according to the specified sampling time. assume that the sampling time is denoted as t . thus, the preliminary mode at that time interval for patch 1 is: nmodepre = max(m1,1,m1,2, · · · ,m1,t−1) (3) this formula is repetitively used for all other patches. step 3 : check the preliminary mode there are two possibilities of the mean matrices. first, there is repetitive mean value and the second one is there is no repetitive mean value. for the former case, there is no issue as it will follow equation 3 correctly. however, for the latter case, the result of the mode is not correctly calculated. the mode algorithm will set the minimum mean value as mode if there is all frequencies of the matrix elements are 1. to avoid this, the checking algorithm is set. assume that the result is the final mode and marked as,nmodefinal and a threshold value, ε is introduced. meanwhile, subp,f matrices are introduced to store the result of subtraction. if the number of zero in the subp,f matrix equal to or greater than a threshold number, ε, the final mode, nmodefinal is set to the calculated preliminary mode, nmodepre indicating there is repetitive mean value in the mean matrixp,f . likewise, if the number of zero in the subp,f matrix less than a threshold number, ε, the final mode, nmodefinal is set to the previous frame values, as there is repetitive mean value in the mean matrixp,f . subp,f = |mean matrixp,f − nmodeprep,f | (4) nmodefinalp,f = { nmodeprep,f n(sub) = 0 ≥ ε nmodeprep,f−1 n(sub) = 0 < ε (5) 60 n.a. zainuddin etal: adaptive background modeling for dynamics background step 4 : construction of background model and adaptive update to build the background model, the calculated final mode values are utilized. it is important to note that, in the earlier stage, we have built storage to store the patches in rgb image form matrices. thus, the final stage to build the background model is by pointing the right rgb image of patch to the right patch according to the final mode value. 3.3 background subtraction and binarization in binarization process, the images will be transformed into black and white images. the binarization algorithm used in this research is motivated by global-used binarization method, otsus method, in which threshold algorithm is deployed. this method is computationally inexpensive thus ideal for real time applications. in this method, we need to set for an initial threshold, ti. we estimate average of the minimum and maximum pixel value of the image, ti, where mathematically it is calculated as: ti = max(i) +min(i) 2 (6) then, the whole pixels are segmented based on the ti value. for the intensity values that greater than or equal to the ti value, it is grouped into g1. otherwise, the intensity values will be grouped to g2 and the average intensity values of each group are calculated and noted as ave1 and ave2 respectively, as shown in equation 7,8,9 and 10 respectively. group = { g1 ij ≥ ti g2 ij < ti (7) ave1 = ∑ ig1 n(ig1) (8) ave2 = ∑ ig2 n(ig2) (9) based on the ave1 and ave2 value, new threshold value is computed: tfinal = ave1 + ave2 2 (10) 3.4 shadow removal shadow is the incorrectly classified foreground pixels that need to be eliminated. foreground mask without shadow removal will adversely affect further processing by providing the false information such as inaccurate blob size, centroid and etc. thus, the efficiency of the method will be affected. there are many proposed algorithms for shadow removal, however, as we already have a good background advances in systems science and application (2016) vol.16 no.2 61 model based on the proposed algorithm, we just need a simple shadow removal. thus, the selection of algorithms of the shadow removing is focused on the computational aspects. one of the simplest methods for shadow removal is by adjusting the contrast. to simplify that we just adjust the luminance value of the pixel by multiplying the current luminance value, ycurrent with a predefined contrast factor, c. thus, the new luminance value is: ynew(x, y) = ycurrent(x, y)× c (11) 3.5 morphological in order to filter the small unwanted pixel in the binary images of the foreground mask, the operation of mathematical morphological is used. there four are basic morphological operations, which are erosion, dilation, opening and closing. in this research, we used the closing morphological operator to close the gap of highly-deformed of interested object shape. closing morphological operator mathematical definition is basically the morphological dilation followed by erosion operation. thus, the equations are: h • i = (h ⊕ i)⊖h (12) where i is the binary image of the foreground and h is the structuring element. holes in the foreground those are smaller than h will be filled. thus, it will reduce the deformation on interested shapes. 4 experimental results in this section, evaluation of the methodology is presented in order to compare the proposed algorithm with several existing methods. both qualitative and quantitative perspectives are discussed. in order to prove the robustness of the proposed algorithm, several video sequences with various scenes are tested in which including the complex dynamic background at the indoor and outdoor environment. 4.1 comparison methods several works on background modeling and object detection techniques are chosen to be compared in terms of performance with our proposed method. the methods are stochastic approximation[20], bayesian learning-based[21], adaptive gaussian mixture model[22], gaussian mixture model[23], and pfinder[24]. all the methods are tested using matlab software and performed on windows 7 on intel r coretm i3-2330m cpu 2.2ghz processor. 4.2 performance measurements to quantitatively evaluate the performance, well-known parameters in gold standard test measurements[25,26] are used. in this method, there are four important 62 n.a. zainuddin etal: adaptive background modeling for dynamics background parameters that need to be defined which are true positive (tp), true negative (tn), false positive (fp) and false negative (fn). in this research, the true positive is defined as the number of pixels that correctly identified as foreground, conversely false positive is number of background pixels that incorrectly identified as foreground. meanwhile, true negative is defined as the number of pixels that correctly identified as background. on the contrary, false negative refers to number of foreground pixels that incorrectly identified as background[27]. from these four parameters, the performance measures are defined as follows: recall(tpr) = tp tp + fn (13) precision(ppv ) = tp tp + fp (14) accuracy(acc) = tp + tn tp + tn + fp + fn (15) f1score = 2tp 2tp + fp + fn (16) specificity(tnr) = tp tp + fn (17) negativepredictivevalue(npv ) = tn tn + fn (18) fallout(fpr) = fp tn + fp (19) falsenegativerate(fnr) = 1− tpr (20) falsediscoveryrate = 1− ppv (21) 4.3 results fig.2 shows the results of the studied methods with our proposed methods in movingcurtain, wavebeach, campusroad and fountain video sequences1. each row shows the foreground mask generated by the corresponding methods for each video sequence respectively. row 2, row 4, row 6 and row 8 of fig.2 show the overlapping results of the foreground generated after background subtraction and ground truth. the overlapping results are used to quantitatively measure the performances of each method. the true positive (tp), true negative (tn), false positive (fp) and false negative (fn) are defined based on the overlapped pixel colours. true positive denoted by white pixels and black pixels are true negative. meanwhile, magenta pixels are for false positive and green pixels indicate false negative. on the other hand, the first column shows the selected frames from the respective video sequences and the second column shows the ground truth advances in systems science and application (2016) vol.16 no.2 63 information of that frame. the other columns are the foreground mask results from the studied methods and our proposed results. fig. 2 experimental results on complex scenes of dynamic background frame number for movingcurtain, wavebeach, campusroad and fountain are 2774,1499, 2348 and 1196 respectively from fig.2, we can deduce that in the high variability scenes like movingcurtain and video sequences, all algorithms handle the continuous moving curtain quite well, except for pfinder[24] as several of the moving curtain pixels are falsely detected as foreground and shadow suppression also failed to be handled by pfinder[24] and uniform-gaussian[20] that causing the additional white pixel formation at the bottom of human foreground for both foreground masks. in the wavebeach video sequences, the variability of the background scene increases as the wave at the beach are continuously moving at a higher frequency than the moving curtain, plus as it is in an outdoor environment, the illumination condition is tested with the addition of stationary moving object condition. this is because the foreground (human) in this sequence stays for quite a long time before moving, thus, the moving object (human) tends to be falsely classified as background. in this video sequence, we can see that most of the algorithms could 64 n.a. zainuddin etal: adaptive background modeling for dynamics background not handle the continuously moving wave as several pixels are falsely classified as foreground except for our proposed algorithm and uniform-gaussian[20] that able to give a clean foreground mask. for the stationary moving object cases, most of the algorithms lost the information of bottom part of the human (foreground) however, the upper part of the foreground are perfectly detected. in the campusroad video sequences, the problem of similar pixel colours and intense noise are evaluated. out of five methods, two methods are able to separate the noise from the negative effect of the waving trees which are our proposed algorithm and the bayesian[21] method. as the dark-coloured car is quite similar to the colour pixels of background, the shape of the car in the foreground mask is deformed except for our proposed algorithm and uniform-gaussian[20]. meanwhile, in the fountain video sequences where the dynamic background (running water from the fountain) is more than 70 percent of the frame size and the pedestrian keep moving in the entire video sequences requires the algorithm to be able to update the background model in a short time. the problem of ’ghost’ detection or slow update can be detected in bayesian approach foreground mask as the foreground mask tend to be bigger in size than the actual size. meanwhile, the repetitive movements from the fountain fail to adapt by adaptive gaussian mixture model (agmm), gmm and pfinder. to quantitatively measure the performance of the studied methods and proposed method,the performance measurements discussed in section 4.2 are used and tabulated in table 1, 2, 3 and 4 for each of the test sequences. meanwhile, table 5 summarizes the results by averaging the result from table 1, 2, 3, and 4. in each table, the best performance is highlighted in bold. to sum up, in the campusroad test sequences, we can see that our proposed algorithm outperforms all other methods by having high value in all measurement parameters. in movingcurtain test sequences, bayesian learning-based is the best in term of accuracy, however, the difference is only by 0.0028 compared with our proposed method. in addition, in this test sequence, our proposed method has the highest precision value and harmonic mean of precision-recall, f1 scores which make our proposed algorithm surpassed the bayesian method holistically. meanwhile, in wavebeach test sequences, uniform-gaussian[20], performs in most measurement parameters compared with the rest of the studied methods, however, our proposed algorithm still performs the best in term of precision and only have a small difference in term of accuracy. in fountain test sequences, our proposed algorithm scores five best values out of nine test parameters and the highest precision value that is very huge difference compared to other methods. based on table 5, we can summarize that our proposed method is the best in term of accuracy, precision and recall. the consistent high precision value of our proposed algorithm is outstanding and make the algorithm as a reliable detection advances in systems science and application (2016) vol.16 no.2 65 t a b le 1 q u an titative resu lt for m ov in gc u rta in test seq u en ces a lg o rith m a ccu ra cy p recisio n r eca ll f n r f d r n p v t n r f 1 s co re f a llo u t u n ifo rm + g a u ssia n [2 5 ] 0 .9 8 4 0 .8 1 3 1 0 .8 3 0 4 0 .1 6 9 6 0 .1 8 6 9 0 .9 9 2 1 0 .8 3 0 4 0 .8 2 1 7 0 .0 0 8 9 b a y esia n [2 6 ] 0 .9 9 0 1 0 .9 1 3 0 .8 0 1 6 0 .1 9 8 4 0 .0 8 7 0 .9 9 2 6 0 .8 0 1 6 0 .8 5 3 7 0 .0 0 2 9 a g m m [2 7 ] 0 .9 8 7 0 .9 3 3 2 0 .8 0 1 9 0 .1 9 8 1 0 .0 6 6 8 0 .9 8 9 5 0 .8 0 1 9 0 .8 6 2 5 0 .0 0 3 1 g m m [2 8 ] 0 .9 9 0 .9 4 2 8 0 .8 2 3 7 0 .1 7 6 3 0 .0 5 7 2 0 .9 9 1 9 0 .8 2 3 7 0 .8 7 9 2 0 .0 0 2 3 p fi n d er[2 9 ] 0 .9 8 3 3 0 .8 0 0 7 0 .9 6 0 5 0 .0 3 9 5 0 .1 9 9 3 0 .9 9 7 4 0 .9 6 0 5 0 .8 7 3 3 0 .0 1 5 3 p ro p o sed 0 .9 8 7 3 0 .9 6 2 4 0 .8 4 6 5 0 .1 5 3 5 0 .0 3 7 6 0 .9 8 8 9 0 .8 4 6 5 0 .9 0 0 7 0 .0 0 2 4 t a b le 2 q u an titative r esu lt for w aveb ea ch test seq u en ces a lg o rith m a ccu ra cy p recisio n r eca ll f n r f d r n p v t n r f 1 s co re f a llo u t u n ifo rm + g a u ssia n [2 5 ] 0 .9 9 0 2 0 .9 2 4 8 0 .8 8 5 2 0 .1 1 4 8 0 .0 7 5 2 0 .9 9 3 7 0 .8 8 5 4 0 .9 0 4 5 0 .0 0 4 b a y esia n [2 6 ] 0 .9 7 2 0 .8 1 6 9 0 .7 3 7 8 0 .2 6 2 2 0 .1 8 3 1 0 .9 8 1 7 0 .7 3 7 8 0 .7 7 5 3 0 .0 1 1 6 a g m m [2 7 ] 0 .9 8 6 0 .9 4 9 5 0 .7 9 5 2 0 .2 0 4 8 0 .0 5 0 5 0 .9 8 7 8 0 .7 9 5 2 0 .8 6 5 5 0 .0 0 2 5 g m m [2 8 ] 0 .9 8 6 5 0 .9 5 1 9 0 .7 9 9 8 0 .2 0 0 2 0 .0 4 8 3 0 .9 8 8 3 0 .7 9 9 8 0 .8 6 9 2 0 .0 0 2 4 p fi n d er[2 9 ] 0 .9 8 3 4 0 .9 4 9 8 0 .7 4 5 9 0 .2 5 4 1 0 .0 5 0 2 0 .9 8 4 9 0 .7 4 5 9 0 .8 3 5 6 0 .0 0 2 4 p ro p o sed 0 .9 8 8 8 0 .9 6 3 6 0 .8 0 6 7 0 .1 9 3 3 0 .0 3 6 4 0 .9 8 9 9 0 .8 0 6 7 0 .8 7 8 2 1 .6 0 e -0 3 t a b le 3 q u an titative r esu lt for c am p u sr o a d test seq u en ces a lg o rith m a ccu ra cy p recisio n r eca ll f n r f d r n p v t n r f 1 s co re f a llo u t u n ifo rm + g a u ssia n [2 5 ] 0 .9 8 5 4 0 .8 2 0 4 0 .9 1 1 6 0 .0 8 8 4 0 .1 7 9 6 0 .9 9 5 3 0 .9 1 1 6 0 .8 6 3 6 0 .0 1 0 6 b a y esia n [2 6 ] 0 .9 7 9 9 0 .6 8 6 8 0 .6 9 3 7 0 .3 0 6 3 0 .3 1 3 2 0 .9 8 9 8 0 .6 9 3 7 0 .6 9 0 2 0 .0 1 0 6 a g m m [2 7 ] 0 .9 4 0 8 0 .3 1 7 2 0 .2 8 5 0 .7 1 5 0 .6 8 2 8 0 .9 6 6 8 0 .2 8 5 0 .3 0 0 3 0 .0 2 8 6 g m m [2 8 ] 0 .9 7 0 2 0 .6 9 2 7 0 .5 4 3 5 0 .4 5 6 5 0 .3 0 7 3 0 .9 7 9 8 0 .5 4 3 5 0 .6 0 9 1 0 .0 1 0 8 p fi n d er[2 9 ] 0 .9 3 7 3 0 .3 6 8 8 0 .5 0 .5 0 .6 3 1 2 0 .9 7 5 3 0 .5 0 .4 2 4 5 0 .0 4 1 5 p ro p o sed 0 .9 9 4 1 0 .9 2 8 0 .9 7 2 6 0 .0 2 7 4 0 .0 7 2 0 .9 9 8 3 0 .9 7 2 6 0 .9 4 9 8 0 .0 0 4 6 66 n.a. zainuddin etal: adaptive background modeling for dynamics background t a b le 4 q u an titative resu lt for f ou n ta in test seq u en ces a lg o rith m a ccu ra cy p recisio n r eca ll f n r f d r n p v t n r f 1 s co re f a llo u t u n ifo rm + g a u ssia n [2 5 ] 0 .9 8 5 8 0 .7 0 0 3 0 .5 6 5 4 0 .4 3 4 6 0 .2 9 9 7 0 .9 9 0 7 0 .5 6 5 4 0 .6 2 5 7 0 .0 0 5 2 b a y esia n [2 6 ] 0 .9 8 1 4 0 .6 3 4 1 0 .8 8 3 2 0 .1 1 6 8 0 .3 6 5 9 0 .9 9 6 4 0 .8 8 3 2 0 .7 3 8 2 0 .0 1 5 6 a g m m [2 7 ] 0 .9 8 6 3 0 .6 6 8 6 0 .6 2 4 3 0 .3 7 5 7 0 .3 3 1 4 0 .9 9 2 3 0 .6 2 4 3 0 .6 4 5 7 0 .0 0 6 3 g m m [2 8 ] 0 .9 8 9 2 0 .7 1 2 7 0 .7 1 6 7 0 .2 8 3 3 0 .2 8 7 3 0 .9 9 4 5 0 .7 1 6 7 0 .7 1 4 7 0 .0 0 5 6 p fi n d er[2 9 ] 0 .9 8 6 2 0 .7 9 1 5 0 .5 2 0 9 0 .4 7 9 1 0 .2 0 8 5 0 .9 8 9 1 0 .5 2 0 9 0 .6 2 8 3 0 .0 0 3 2 p ro p o sed 0 .9 8 9 9 0 .9 0 4 1 0 .7 9 5 4 0 .2 0 4 6 0 .0 9 5 9 0 .9 9 2 6 0 .7 9 5 4 0 .8 4 6 3 tex tb f3 .1 0 e -0 3 t a b le 5 a v erage of gold stan d ard m easu rem en ts p a ra m eters fo r a ll test seq u en ces a lg o rith m a ccu ra cy p recisio n r eca ll f n r f d r n p v t n r f 1 s co re f a llo u t u n ifo rm + g a u ssia n [2 5 ] 0 .9 8 6 4 0 .8 1 4 7 0 .7 9 8 2 0 .2 0 1 9 0 .1 8 5 4 0 .9 9 3 0 .7 9 8 2 0 .8 0 3 9 0 .0 0 7 2 b a y esia n [2 6 ] 0 .9 8 0 9 0 .7 6 2 7 0 .7 7 9 1 0 .2 2 0 9 0 .2 3 7 3 0 .9 9 0 1 0 .7 7 9 1 0 .7 6 4 4 0 .0 1 0 2 a g m m [2 7 ] 0 .9 7 5 0 .7 1 7 1 0 .6 2 6 6 0 .3 7 3 4 0 .2 8 2 9 0 .9 8 4 1 0 .6 2 6 6 0 .6 6 8 5 0 .0 1 0 1 g m m [2 8 ] 0 .9 8 4 0 .8 2 5 0 .7 2 0 9 0 .2 7 9 1 0 .1 7 5 0 .9 8 8 6 0 .7 2 0 9 0 .7 6 8 1 0 .0 0 5 3 p fi n d er[2 9 ] 0 .9 7 2 6 0 .7 2 7 7 0 .6 8 1 8 0 .3 1 8 2 0 .2 7 2 3 0 .9 8 6 7 0 .6 8 1 8 0 .6 9 0 4 0 .0 1 5 6 p ro p o sed 0 .9 9 0 .9 3 9 5 0 .8 5 5 3 0 .1 4 4 7 0 .0 6 0 5 0 .9 9 2 4 0 .8 5 5 3 0 .8 9 3 8 0 .0 0 2 9 advances in systems science and application (2016) vol.16 no.2 67 method. 5 conclusion this research presented a novel real time based background modeling technique for the scene with varied dynamic background. the proposed algorithm are tested in four complex scenes and compared with five recent algorithm studied. experimental results show that our proposed algorithm is able to give a consistent, accurate and precise detection results. the common problems in detection method are also successfully handled. acknowledgment this research work is supported by fundamental research grant scheme (frgs13030-0271) under the ministry of higher education of malaysia (mohe). references [1] m. piccardi.(2004), “background subtraction techniques: a review”, in ieee international conference on systems, man and cybernetics, the hague, netherlands. 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[25] z. zivkovic and f. v. d. heijden. (2006), “efficient adaptive density estimation per image pixel for the task of background subtraction”,pattern recognition letters, vol. 27, no. 7, [26] c. stauffer and w. grimson.(2000), “learning patterns of activity using real-time tracking”, ieee transactions on pattern analysis and machine intelligence, vol. 22, no. 8, pp.747-757. [27] c. wren, a. azarbayejani, t. darrell and a. pentl. (1997), “pfinder: real-time tracking of the human body”, ieee transactions on pattern analysis and machine intelligence, vol. 19, no. 7, pp.780-785. corresponding author n.a. zainuddin can be contacted at: fiqahzainuddin@gmail.com adv syst sci appl 2017; 17(2); 14-28 published online http://ijassa.ipu.ru/ojs/ijassa/article/view/12 copyright ©2017 assa. adv. in systems science and appl. (2017) economic security under disturbances of foreign capital kurt schimmel 1) , sifeng liu 2,3) , jeananne nicholls 1) , nicholas a. nechval 4) , jeffrey yi-lin forrest 1) 1) school of business, slippery rock university, slippery rock, pa16057, u.s.a. e-mail: kurt.schimmel@sru.edu; jeananne.nicholls@sru.edu; jeffrey.forrest@sru.edu 2) institute for grey system studies, nanjing university of aeronautics and astronautics, nanjing 211106, pr china 3) centre for computational intelligence, de montfort university, leicester le1 9bh, uk; e-mail: sfliu@nuaa.edu.cn 4) department of mathematics, baltic international academy, lomonosov street 4, riga lv1019, latvia e-mail: nechval@junik.lv abstract: considering spillover effects of foreign capital, this paper establishes a method for the receiving economy to constantly monitor and forecast the movement of foreign capital within itself so that methods of regulation could be established to guarantee the health and stable development (or the security) of the domestic economy. methods of control theory are employed to model the movement of foreign capital, to estimate the initial state of foreign capital’s movement. on the method established and the systemic yoyo model, this paper introduces ways to prevent foreign capital from adversely impacting the receiving economy for establishing theoretical guidance for insuring the healthy development of the economy. this paper provides several practically useful strategies that could counter disturbances caused by foreign capital within the receiving economy and protect the economic security of the system in order to avoid the disastrous aftermath of the currency war that occurs along with large scale withdraw of foreign capital that was initially invested within the system in friendly terms. keywords: movement of money, economic stability, method of control, estimation, systemic yoyo model, strategies of regulation 1. introduction sequel to [7], this paper establishes a theoretical model on how to monitor the movement of foreign capital within a nation’s economy in order to keep the economic security of the economy intact. in particular, this paper is organized as follows: section 2 establishes a theoretical model for monitoring the dynamics of foreign capital within a nation’s economy. section 3 studies how to estimate the state of motion of foreign capital within the domestic economy. section 4 fills the need of estimating the initial state of foreign capital’s movement. section 5 explores some potential measures useful for countering the fluctuations of foreign capital within a national economy. section 6 concludes the presentation of this work. 15 k. schimmel1, s. liu, j. nicholls, n.a. nechval, j. yi-lin forrest copyright ©2017 assa. adv. in systems science and appl. (2017) 2. a model for monitoring dynamic foreign capital within an economic system the receiving economic system foreign investment is really a double-edged sword. it can not only provide sufficient capital, advanced technological equipment, and excellent management experience for the development of the economic system, but also seriously constrain the development of domestic enterprises, weaken the innovation capacity of the economic system, and create serious security risks for the economic system [7]. hence, practically speaking, there is an urgent need to appropriately formulate and implement strategic measures regarding foreign investments in order to accelerate the economic development by sufficiently taking advantage of foreign capital while lowering the security risk that accompanies the influx of foreign capital to a certain level or within the controllable range of the economic system. this section attempts to provide theoretically sound suggestions regarding the design of foreign capital policies and strategies for the receiving economic system by modeling the scale, regions, and areas of foreign investment, and by constructing a control-theory model for foreign investment. generally speaking, the investment within an economic system by foreign capital may be maintained at a certain stable speed without being affected by regional economic development cycles, financial shocks, and major changes in the environment of the economic system. for example, since 1992, china has been continuously attracting a large scale of foreign investment [1]. and only in some very particular situations, such as major financial shocks, drastic changes in the political environment of the economic system, etc., the invested foreign capital would show speculative, withdrawal and other behaviors. from what has been discussed in [7], it follows that foreign capital affects the economic security of the receiving economic system in three different ways: one is the magnitude of the invested foreign capital, two is the geographic distribution of the foreign investments, and three the difference between the actual distribution of invested foreign capital and the need distribution for foreign capital investment of the economic system. in terms of the magnitude of invested foreign capital, a rising magnitude increases the holding of foreign currencies of the economic system while it also increases the internal base money supply. that might affect adversely the effectiveness of the adopted monetary policies of the economic system. additionally, the rising amount of foreign investment also increases the uncertainty of how the economic system would be influenced by the external world. because of its drive for profit, foreign capital might leave the economic system in large scales within a short period of time when the economic system experiences shocks from the external environment. that exit could cause disastrous consequences for the healthy development of the economic system, and such effects had been sufficiently manifested in the southeast asian financial crises of the 1990s. in terms of the geographic distribution of foreign investments, when the imbalance in the geographic distribution of foreign investment goes up, the internal development imbalance within the economic system will expand, which might cause chaos in the regional development of the economic system so that the orderly domestic allocation of resources will be disrupted, such wasteful phenomena of resources as duplicated constructions, production overcapacity, etc., will likely to appear. in terms of the difference between the actual distribution of invested foreign capital and the need distribution for foreign capital investment of the economic system, it creates an imbalanced local distribution of regional industries within the economic system. that causes the internal industrial structure of the economic system to lose its balance so that when affected by external uncertainties, the development of the economic system will be badly economic security under disturbances of foreign capital 16 copyright ©2017 assa. adv. in systems science and appl. (2017) hindered by production imbalances and insufficient productions of some industries. for example, in some of the emerging economies foreign investment is relatively concentrated in a few industries. when such an economic system is affected by financial shocks, a quick withdraw of foreign capital generally leaves behind damaging effects for the development and stability of the economic system [3]. therefore, in the following, we use r(k) to stand for the magnitude of foreign investment at time tk within the economic system, (k) – the geographic distribution of foreign investment at time tk, and (k) for the difference between the actual industrial distribution of foreign investment and the actual distribution of areas that need foreign investment at time tk. here, we utilize the difference (k) to measure the effect of foreign capital on the economic security of the economic system at time tk . next, let us establish a control and monitor equation of these three variables in order to materialize the estimation and regulation of the state of foreign capital at any chosen time moment. additionally, in order to produce meaningful anticipation on the state of investment of foreign capital, let us use ( )r k , ( )k , and ( )k to respectively denote the rate of change of the investment magnitude r(k) of foreign capital, the rate of change of the geographic distribution of foreign investment (k), and the rate of change in the difference (k) between the actual industrial distribution of foreign investment and the actual distribution of areas that need foreign investment at time tk. if we let x(k) be the state vector of foreign capital within the economic system, where then we can use the following state equation to describe the movement of foreign capital within the economic system: ( ) ( 1)x k x k  (2.1) where  is a matrix defined below with t representing the time interval between consecutive observations, 1 0 1 1 0 1 1 0 1 t t t                     . however, due to the existence of various disturbances and errors in real life, the movement, when seen as a system, of foreign capital within the economy is constantly affected somehow. for example, no matter how we establish the dynamic equation, it will not be an accurate 17 k. schimmel1, s. liu, j. nicholls, n.a. nechval, j. yi-lin forrest copyright ©2017 assa. adv. in systems science and appl. (2017) expression of the realistic system; because of the interferences of the environment that is external to the economic system, exchange rate, politics, etc., the state of foreign capital movement is affected by countless many random factors. let us aggregate all these various known and unknown factors into one concept: process (or dynamic) noise. in order to more realistically reflect the true dynamics of the foreign capital movement, let us add a noise term ( )w k that describes all kinds of interferences into the previous equation. so, the state equation in eq. (2.1) can be rewritten as follows: ( ) ( 1) ( 1)x k x k tw k    (2.2) where 1 2 3( ) [ ( ), ( ), ( )]tw k w k w k w k is the random interference of the foreign capital movement system within the economic system with w1, w2, and w3 respectively being the random accelerations in the directions of r, , and , and matrix t is given below where t is the time interval between two consecutive observations 2 2 2 / 2 / 2 / 2 t t t t t t t                     . of particular interest is that the interference {w(k)} as introduced above is a stochastic sequence, which comprehensively (and approximately) reflects the interference as experienced by the foreign capital when it moves within the economic system. although there might be a large number of factors that interfere with the movement of foreign capital within the economic system, the effect of each factor might be quite small, which is likely unclear to the regulator of the system and cannot be readily measured. even so, the total effect of all the interfering factors on the system’s dynamics can be described by using a stochastic sequence (or a stochastic process if the movement system is modeled by a continuous model). however, it is still not enough for us to only employ eq. (2.2) to describe the movement of foreign capital within the economic system. for the general scenario, the economic system can acquire and observe some traces of the movement of the foreign capital by using various methods of statistics. however, what is acquired and observed tends to be only about some components of the state variable or the values of some linear transformations of the state variable. for example, when using econometric data to observe or measure the state of foreign capital, only the magnitude, locations of investment, and industries of investment of foreign capital can be observed and measured. however, the acceleration and other parameters of movement of foreign capital are difficult to measure directly. let us denote the data values that are actually observed as y(k), where economic security under disturbances of foreign capital 18 copyright ©2017 assa. adv. in systems science and appl. (2017) hence, the observation (or measurement) equation that describes the movement system of foreign capital within the economic system can be established as follows: ( ) ( )y k hx k (2.3) where 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 h           it can be expected that when econometric data are employed as observations of the movement of foreign capital, there is always error. such error could originate either from the inaccuracy of the observation equation or the method used during the statistical process and the random error that exists in the available data [6]. let us refer this sort of error as observation (or measurement) noise. to realistically reflect the relationship between the observed variables regarding the movement of foreign capital within the economic system, let us add a random variable v(k) into eq. (2.3) to describe such random interference. hence, the observation (or measurement) equation becomes: ( ) ( ) ( )y k hx k v k  (2.4) where v(k)=[v1(k), v2(k), v3(k)] t , and v1(k), v2(k), and v3(k) are respectively the observed values of r, , and  at time moment tk, and { v(k)} is a stochastic sequence. now, if we combine eq. (2.2), which is the state equation that describes the movement system of foreign capital within the economic system, and the observation equation (4), which describes how the movement system of foreign capital is measured, we then obtain the following state equation of the foreign capital that expresses how the capital moves within the economic system: ( ) ( 1) ( 1) ( ) ( ) ( ) x k x k tw k y k hx k v k         (2.5) because the movement system of foreign capital within the economic system, as described by equ. (5), contains random interferences, we will refer the system described by eq. (2.5) as a stochastic control system. for each of such systems, all obtained information (data) is “polluted” by noise. in order to obtain relatively more accurate information of the state, we will have to make our best estimation of the true state of the system from the available information (that is generally a sequence of observations) with interfering noise, which might even be incomplete. additionally, for stochastic control systems, the process of movement itself and the observation of the system’s output are all contaminated with noise [6]. so, to study the movement of the system quantitatively we must first describe the interfering noise quantitatively. from the knowledge of stochastic processes, it follows that the distribution of the given stochastic sequence determines all the statistical characteristics of the sequence [2]. however, generally it is difficult to fully know the distributional structure of a stochastic sequence. therefore, when solving practical problems, it will be mostly enough to simply know some of the main statistical characteristics of the given stochastic sequence. assume that {v(k)} is a stochastic sequence. if for any positive integer n and t, and n arbitrary time moments k1 < k2 <…< kn, v(k1),…, v(kn) and v(k1+t),…, v(kn+t) have the same joint distribution function, then the sequence {v(k)} is known as stationary. if the mean of the stationary stochastic sequence {v(k)} is 0, and its auto-covariance satisfies 19 k. schimmel1, s. liu, j. nicholls, n.a. nechval, j. yi-lin forrest copyright ©2017 assa. adv. in systems science and appl. (2017) 2( , ) ijr i j   , 0, 1, ij if i j if i j      (2.6) then {v(k)} is known as a white noise sequence. here, a white noise sequence can be fathomed as a pure stochastic sequence, satisfying that the values of any two consecutive terms are independent. if in eq. (2.5), which describes the movement of foreign capital within the economic system, we know the state equation and the statistical characteristics of the noise, then we can move ahead to estimate the state of the movement system of foreign capital. in order to establish a formula to estimate x(k) in the state equation in eq. (2.5), let us impose some conditions on the noises {w(k)} and{v(k)}. first, let us assume that the observation noise {v(k)} and the process dynamic noise {w(k)} are completely independent of each other; otherwise, they can no longer be known as noise. that is, we have [ ] 0t k je w v  , for any 0, 0k j  . (2.7) secondly, we assume that both {w(k)} and {v(k)} are zero mean white noise stochastic sequences. that is, these sequences satisfy the following: for {w(k)}, we have [ ( ) ( )] ( )t kje w k w j q k  , where 1 2 3 ( ) ( ) ( ) ( ) q k q k q k q k           , 1, 0, kj k j k j      which is known as the covariance matrix of the process noise, and 1( )q k , 2 ( )q k , and 3( )q k are respectively the stochastic acceleration variances of the objective along the directions r, , and  at time moment tk, which can be selected based on how freely foreign capital can move within the economic system and how accurate the estimate of the state needs to be. and for{v(k)}, we have [ ( ) ( )] ( )t kje v k v j r k  , where 1 2 3 ( ) ( ) ( ) ( ) r k r k r k r k           which is known as the covariance matrix of the measurement; and r1(k), r2(k), and r3(k) are respectively the observation noise variances of the objective along the directions of r, , and  at time moment tk. their values are related to the accuracy of the observation statistics and the state of foreign capital. as a matter of fact, if the noises are not of zero means or if {w(k)} and {v(k)} are not independent stochastic sequences, etc., then necessary mathematical treatments will be needed. of course, that will surely involve a lot of computational complexities. economic security under disturbances of foreign capital 20 copyright ©2017 assa. adv. in systems science and appl. (2017) other than the afore-mentioned basic assumptions about the random inference variables, we also need to assume that the initial state 0x is also a random vector satisfying 0 0[ ]e x x , 0 0 0 0 0[( )( ) ]te x x x x p   , and that 0x and kw , kv are independent. 3. estimate the state of motion of foreign capital at the initial time moment t=0, the ideal estimate of 0x is of course 0 0 0x̂ ex x  . if we assume that we have already obtained 0x , and let 0 0x̂ x , then the covariance matrix of the estimation error of the state is 0 0 0 0 0[( )( ) ]tp e x x x x   . by employing the thinking logic of mathematical induction, we can derive the needed recursive formula. assume that we have obtained the observations up until time moment k: y1,…,yk, and the optimal (unbiased) estimate ˆ kx of the state kx at time moment k. before the next new measurement data value yk+1 becomes available, to estimate the state xk+1 at time moment k+1 we have to start with ˆkx by making use of the evolutionary rule described in eq. (2.5). because wk is a zero-mean white noise, we cannot know what value it takes. so, the most reasonable choice is let wk=0, the mean value. on this basis, we obtain the predicted estimation as follows: 1| 1, ˆ ˆ k k k k kx x  . (3.1) because 0kew  , ˆ k kex ex , and 1| 1, 1, 1 ˆ ˆ k k k k k k k k kex ex t ew ex      , we conclude that 1| ˆ k kx  is an unbiased estimate of 1kx  . so, an optimal estimation is provided by eq. (3.1). and by using the measurement expression in eq. (2.4) and by letting 1 0kv   (for the same reason as above), we obtain the following predicted output value that corresponds to 1| ˆ k kx  : 1| 1 1| 1 1, ˆ ˆ ˆ k k k k k k k k ky h x h x      . (3.2) at the time moment t=k+1, we immediately obtain the actual output value yk+1 of the economic system. by comparing this actual value with the predicted value 1| ˆ k ky  , we can compute the error of the predicted output: (3.3) this expression includes not only the comprehensive information of the system’s noise kw , measurement noise vk, filtering error, and other random variables, but also some new information about the state xk+1 as the measurement value yk+1 at time moment k+1 becomes available. hence, let us refer to 1|k ky  as a vector of new information (innovation). it is now natural for us to think about employing the new information at moment k+1 to correct the predicted state 1| ˆ k kx  . that is, we can now derive a better estimate (filtering estimate) for the state at moment k+1. so, the estimation for the state at moment k+1 should consist of the following two parts: one is to derive the predicted estimate 1| ˆ k kx  of the state based on the observations up to time moment k, and the other is to correct the prediction 1| ˆ k kx  by employing the newly collected observation at moment k+1. when these two parts are put together, we establish our estimation for the state xk+1 at the time moment k+1. symbolically, we have 1 1| 1 1 1 1| ˆ ˆ ˆ[ ]k k k k k k k kx x k y h x        , (3.4) 21 k. schimmel1, s. liu, j. nicholls, n.a. nechval, j. yi-lin forrest copyright ©2017 assa. adv. in systems science and appl. (2017) where kk+1 is referred to as a gain matrix or a correction factor. it is a real matrix. the key for our optimal filtering is to select correctly the gain matrix kk+1 so that the estimation 1 ˆ kx  of the state obtained from eq. (3.4) has the minimum covariance matrix of error. that is, we have 1 1 1 1 1 ˆ ˆ[( )( ) ] mint k k k k kp e x x x x        . (3.5) next, we need to establish a formula from which we can produce the optimal gain matrix kk+1 that satisfies eq. (3.5) so that a complete recursive computational scheme can be developed for the purpose of estimating the state of movement of foreign capital within an economic system. let 1 1 1 ˆ k k kx x x    . then the covariance matrix pk+1 of the filtering error can be written as 1 1 1[ ]t k k kp e x x   . from 1 1 1 1 1| 1 1 1 1| 1| 1 1 1 1 1 1| 1| 1 1 1| 1 1 1 1| 1 1 ˆ ˆ ˆ[ ( )] ˆ[ ] [ ] [ ] k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k x x x x x k y h x x k h x v h x x k h x v i k h x k v                                           (3.6) where 1| 1 1| ˆ k k k k kx x x    , it follows that 1 1 1 1 1 1| 1 1 1 1 1| 1 1 1 1 1| 1| 1 1 1 1 1 1 1 1 1| 1 1 1 1 1| [ ] [(( ) )(( ) ) ] [ ] [ ][ ] [ ] [ ] [ ] [ t k k k t k k k k k k k k k k k k t t t t k k k k k k k k k k k k t t k k k k k k k k k p e x x e i k h x k v i k h x k v i k h e x x i k h k e v v k i k h e x v k k e v x                                             1 1][ ]t t k k ki k h  because 1| 1, ˆ ˆ k k k k kx x  only depends on 0x̂ and y1,y2,…,yk, from the knowledge of statistics, it follows that 1 1|[ ] 0t k k ke v x   and 1| 1[ ] 0t k k ke x v   . so, the formula for computing the covariance matrix pk+1 of the filtering error can be simplified as follows: 1 1 1 1| 1 1 1 1 1[ ] [ ]t t k k k k k k k k k kp i k h p i k h k r k            , (3.7) where 1| 1| 1|[ ]t k k k k k kp e x x   is referred to as the covariance matrix of the prediction error, and 1 1 1[ ]t k k kr e v v   . if we expand eq. (3.7) and regroup the terms, where, without causing confusion, we omit all the subscripts of 1kh  , 1kk  , and 1kr  , we have 1 1| 1| 1| 1| 1| 1| 1| 1|( ) t t t t t k k k k k k k k k t t t t k k k k k k k k p p khp h k khp p h k krk p k hp h r k khp p h k                    (3.8) our purpose here is to derive the gain matrix k such that pk+1 reaches its minimum. therefore, we rewrite eq. (3.8) as a complete square, such as the form of ( )( )tks a ks a  . because ( )( )t t t t t tks a ks a kss k aa ksa as k      , (3.9) by comparing eq. (3.9) and (3.8), we can see that because 1| t k khp h is a non-negative definite matrix and r is positive definite, 1| t k khp h r  is a positive definite matrix. so, from matrix theory [4], it follows that each positive definite matrix can be expressed as the square of a certain symmetric positive definite matrix. so, let us assume economic security under disturbances of foreign capital 22 copyright ©2017 assa. adv. in systems science and appl. (2017) 1| t t k khp h r ss   (3.10) where s is a symmetric positive definite matrix. next let sa t =hpk+1|k, where matrix a actually exists because the square matrix s is of full rank. by substituting eqs. (3.9) and (3.10) into eq. (3.8) we obtain the following: 1 1| 1| ( )( ) t t t t t k k k t t k k p p kss k ksa as k p ks a ks a aa             (3.11) from eq. (3.11), we see that only the second term on the right hand side has something to do with the gain matrix k, while this term is a non-negative definite matrix. hence, to make pk+1 reach its minimum, we only need to select such a matrix k so that the second term in eq. (3.11) is equal to zero. so, we have 1 1 1 1| 1| 1 1| 1| ( ) ( ) t t t t k k k k t t k k k k k as p h s s p h ss p h hp h r               (3.12) if we select k according to eq. (3.12), then the covariance matrix pk+1 of the filtering error is 1 1 1| 1| 1| 1| 1| 1|( ) t k k k k k k k k k k k k k p p aa p as hp p khp i kh p                 (3.13) next, let us derive a computational formula for the covariance matrix 1|k kp  of the prediction error. from the formula for pk+1|k, it follows that 1| 1| 1| 1| 1, 1| 1, 1, 1, 1, 1, 1, 1, 1, 1, [ ] {[ ][ ] } [ ] [ ] t t k k k k k k k k k k k k k k k k k k t t t t k k k k k k k k k k k k t t k k k k k k k k k k p e x x e x t w x t w e x x t e w w t p t q t                              (3.14) by combining what is obtained above, for the system in eq. (2.5), we have obtained all the recursive formulas necessary for estimating the state of movement of foreign capital within the economic system. however, to employ this theory to practically estimate the magnitude of foreign capital’s investment, regional distribution, and differences between industries within the economic system at any chosen time moment, we still need to estimate the initial state. 4. estimate the initial state of foreign capital’s movement from our discussions above, it follows that once we have the knowledge of the initial state of the movement of foreign capital within the economic system of our concern or a good estimation of that initial state, we then can estimate the state of foreign capital’s movement system step by step. to this end, the general method is to determine the initial state x(0) by using the prior (known) knowledge and then calculate the initial covariance matrix p(0) by employing the degree of accuracy of x(0). in this section, we assume that we have the initial observations at time moments t1 and t2. we then roughly estimate the state of the foreign capital’s movement system at moment t2 and its variance. these values will be used as the initial state of foreign capital’s movement within the economic system. in particular, assume that the initial two observations are respectively 23 k. schimmel1, s. liu, j. nicholls, n.a. nechval, j. yi-lin forrest copyright ©2017 assa. adv. in systems science and appl. (2017) 1 2 3(1) [ (1), (1), (1)]ty y y y and 1 2 3(2) [ (2), (2), (2)]ty y y y . then we calculate the following estimation of the initial state: 1 2 3 4 5 6 ˆ ˆ ˆ ˆ ˆ ˆ ˆ(0) [ (0), (0), (0), (0), (0), (0)]tx x x x x x x ， where 2 1 ˆ (0) (2)i ix y  , 2 ˆ (0) [ (2) (1)] /i i ix y y t  , 1,2,3i  , (4.1) and 1 2 2 2 3 (0) ˆ ˆ(0) [( (0) (0))( (0) (0)) ] (0) (0) t p p e x x x x p p              , where 2 (2) (2) / (0) (2) / [ (1) (2)] / i i i i i i r r t p r t r r t        , 1,2,3i  , (4.2) in the following, we use a sub-matrix p1(0) as an example to illustrate the detailed steps of computation. because 1 1 1 1 ˆ (0) (2) (2) (2)x y x v   , we have 2 1 1 1 1 1 1 1 1 1 1 ˆ (0) [ (2) (1)] / [ (2) (2) (1) (1)] / {[ (2) (1)] [ (2) (1)]}/ x y y t x v x v t x x v v t           because 2 1 1(2) [ (2) (1)] /x x x t  , we have 1 1 1 ˆ(2) (0) (2)x x v   , 2 2 1 1 ˆ(2) (0) [ (2) (1)] /x x v v t    1 1 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 2 1 1 1 ˆ ˆ(2) (0) (2) (0) (0) ˆ ˆ(2) (0) (2) (0) (2) (2) [ (2) (1)] / [ (2) (1)] / (2) (2) / (2) / [ (1) (2)] / t t x x x x p e x x x x v v e v v t v v t r r t r t r r t                                                  similarly, we can obtain the other sub-matrices of eq. (4.2). as of this point in our presentation, we have established a complete theory on how to estimate the state of movement of foreign capital within the economic system. and on the basis of the estimations, we can design appropriate strategies and measures to counter the adverse effects of foreign investments while insuring the economic security of the economic system. economic security under disturbances of foreign capital 24 copyright ©2017 assa. adv. in systems science and appl. (2017) 5. how disturbances of foreign capital affect economic security because of the entry of foreign capital, the money supply within the receiving economic system is increased, making the system have additional resources to produce so that the economic strength of the system consequently grows rapidly. however, with the accumulation of foreign capital to a large magnitude, when the foreign capital suddenly withdraws, a severe shock to the economic activities of the economic system will be created. for example, let us look at a fictitious economic system whose financial system is open to the outside world. assume that at a certain time moment, the nation’s total money supply is $50,000, including domestically invested foreign capital of $7,500. that is, the foreign investment occupies 15% of the entire national money supply. with the additional money supply, as created by the foreign capital, the circulation speed of money within the economic system accelerates, which indicates that this nation spends more on various economic activities and on the promotion of its economic level. if at a later time moment the capital invested by foreign merchants is suddenly withdrawn from the economic system, then it will inevitably, to a great extent, cause the liquidity of money within the economic system to drop suddenly, affecting adversely over 15% of all the economic activities within the economy. hence, the economic system has to strengthen the monitoring of foreign capital in order to prevent the disastrous aftermath created by the currency war into which the initial friendly investment was evolved by sudden large scale withdraw. the essential reason why an economic system introduces foreign capital is to stimulate its economic development. that is to say, there is a pre-defined purpose within the economic system for how the entering foreign capital will develop. this pre-defined purpose is referred to as the ideal trajectory of the movement of foreign capital. on the other hand, the fundamental reason for foreign capital to enter the economic system is to make profit, which includes taking advantage of the promised preferential conditions, low-priced resources available from within the economic system, developing the market resources of the new economy, etc. in the process of materializing the desired profit, the actual state of movement of foreign capital most likely deviates from the ideal trajectory that is pre-defined by the economic system. when such deviation expands, it might gradually erode the economic security of the economic system that is additional to the significant damage that might occur to the economic system when a collective withdraw of foreign capital happens at certain time moment. based on what has been discussed and analyzed earlier, the economic system could acquire “truthful estimations” of the state of domestic movement of foreign capital and possible trends of future development. so, for the policy makers of the economic system, they must design and introduce a series of policies to guide the “truthful estimations” to approach the ideal trajectory in order to materialize the desired control of the adverse effect of foreign capital on the economic security of the economic system. such dynamic adjustment on the part of policy makers could also potentially prevent a currency war created by large scale sudden withdraws of foreign capital although the capital initially entered the economic system in friendly terms. when applying a strategy to regulate the state of movement of foreign capital, it is inevitable that some of the limited resources of the economic system will have to be occupied and exhausted. so, a control strategy is considered ideal if it could not only reach its purpose of control within a limited time interval but also consume as little amount of resource as possible. to this end, as a form of expression for energy consumption, quadratic functions will be used due to their excellent properties. these functions can describe not only how resources are wasted by implementing excessive control, but also the distance from the target as caused by insufficient 25 k. schimmel1, s. liu, j. nicholls, n.a. nechval, j. yi-lin forrest copyright ©2017 assa. adv. in systems science and appl. (2017) control. therefore, in the following, we will choose a quadratic function as our evaluation standard to measure control strategies. let us define the objective function as follows: 1 0 ˆ ˆ[( ( 1) ( 1)) ( )( ( 1) ( 1)) ( ) ( ) ( )] n t t n k j x k x k q k x k x k u k r k u k           where n stands for the final time moment when the actual state ( )x n could approach the predefined ideal state ( )x n “perfectly”, ( )q k the effect on the economic system as caused by the difference between the “truthful estimation” and the “ideal trajectory” at moment k, r(k) the price that has to be paid for adopting control strategy u(k) at moment k. because this objective function contains the effect on the economic system of the difference between the “truthful estimation” and the “ideal trajectory” and the price paid to implement the control strategy u, the policy makers should design an optimal control strategy that makes the previous objective function reach its minimum. however, the design of any policy strategy is constrained by the resources available within the economic system, that is, u(k)ω, where ω stands for the space of constraints of all strategies. so, the purpose of the entire control problem is to find a control sequence u*(k)ω such that ( *(0), *(1), , *( 1)) min nj u u u n j    because the effect of policies can lead to changes in the state of foreign capital movement system, the “truthful estimation” ˆ( 1)x k  is a function of policy u(k). according to the necessary conditions for a function to reach its extreme values, we have 0k k j u    . that is, ˆ( 1) ˆ( )( ( 1) ( 1)) ( ) ( ) 0 ( ) t x k q k x k x k r k u k u k         so, we have 1 ˆ( 1) ˆ( ) ( ) ( )( ( 1) ( 1)) ( ) x k u k r k q k x k x k u k          theoretically speaking, the feasible region of the constraint variables is bounded, the objective function is nonnegative and convex. therefore, generally there is an optimal control sequence u*(k)ω such that jn reaches its minimum. however, because it is difficult to quantitatively or analytically describe the feasible region of the constraint variables, it is also difficult to provide a formal expression for an optimal strategy. even so, we will try to develop a general description of the necessary control strategy by using the state equation and possible evolution tendencies of foreign capital movement within the economic system combined with the systemic yoyo model. therefore, in the following we will provide several strategies based on logical analysis and reasoning strategies that could counter disturbances caused by foreign capital within the economic system and protect the economic security of the system in order to avoid the disastrous aftermath of the currency war that occurs along with large scale withdraw of foreign capital that was initially invested within the system in friendly terms. first, moderately monitor and supervise the activities of foreign investments, and strengthen the control of foreign capital if it attempts to enter the virtual economy, such as the financial market, stock market, etc. although the assets of foreign capital that enters the virtual economy economic security under disturbances of foreign capital 26 copyright ©2017 assa. adv. in systems science and appl. (2017) could provide investment capital for the economic system, such capital can often perturb the operation of the domestic financial system through unconventional ways, disrupt the smooth operation of the system, and withdraw from the system as soon as a high rate of return is acquired within a short period of time by making use of its scale and other advantages. such short-term rapid movement of capital can cause turbulence to the otherwise normal operation of the domestic financial market of the economic system, which in practice materializes damage to the domestic financial system. the southeast asian financial storm of the year of 1998 was an example of how foreign capital acquired short-term high rates of returns by making use of the imperfections of the regional financial systems. this financial storm ransacked the economic achievements of the past several decades of the relevant regions within a short time period, while leaving the development of these economies on the verge of collapse [3]. therefore, the receiving economic system must strengthen its supervision of those foreign capital investments that intend to enter the domestic financial market or the virtual economy, crackdown any illegal entry behaviors of foreign capital, and improve the control mechanism over the foreign capital that flows into any of the non-real economies. at the same time, receiving economic systems must constantly perfect emergency plans and formulate control measures for temporary, abnormal capital movements by considering the potentially most severe scenarios based on the logic of balancing the abnormal inflow and outflow of foreign investment capital. second, implement a flexible exchange rate mechanism in order to maintain investment grade foreign capitals while expelling those that are speculative in nature. among all the capital of foreign investments, the portion that enters into the real economy is the one needed by the domestic economic system for its economic development. this capital intends to acquire relatively long-term profits by valuing the development potential of the economic system and by actively investing in the economic system. on the other hand, this capital could be relied on for the construction and development of the real economy and for its relatively long-term stay within the economic system. however, the entry of the speculative portion of foreign capital into the domestic economic system is mainly for the purpose of obtaining speculative profits by making use of the imperfections that exist in the market place and the financial system of the receiving economy. hence, the economic system should employ its flexible exchange rate mechanism to discourage the influx of such foreign capital and to convert the influx of such foreign capital into investments in the real economy. for example, assume that merchant c is an international speculator and that he is upbeat about the stock market of the particular economic system and enters the stock market with an amount of us$1 billion. if the exchange rate between the u.s. dollar and the domestic currency is 1:5, then his entry into the stock market represents 5 billion units of the local currency. after obtaining a short-term speculative profit in the stock market, c wants to close out his positions. if at this time moment the monetary authority of the economic system has somehow changed the exchange rate to 1:6 by making use of their flexible exchange rate policy, then c’s original speculation amount of 5 billion units of the local currency can only be converted to us$5/6 billion = us$0.833 billion, where us$0.167 billion has evaporated. that is, the flexible exchange rate mechanism has discouraged c’s speculation in the stock market, while the short-term varying exchange rate has little adverse effect on the foreign capital that is invested in the real economy. hence, when a relatively large amount of foreign capital enters into the economic system, the receiving economy could employ a flexible exchange rate mechanism to discourage speculative capital while retaining the capital that is invested in the real economy. 27 k. schimmel1, s. liu, j. nicholls, n.a. nechval, j. yi-lin forrest copyright ©2017 assa. adv. in systems science and appl. (2017) third, implement a flexible foreign investment policy to guide foreign investment capital into the target industries and geographic regions by making use of the economic system’s market demand, cost advantage, and other resources. the profit-driven characteristics of foreign capital determine where and which industries foreign capital will enter; from how foreign capital flows into the economic system to obtain its desired profit, the investment types of foreign capital can generally be classified into the following five categories: cost seeking, policy seeking, resource seeking, market seeking, and technology seeking. with increasing economic strength of the economic system, the accumulation of investment capital, and the rising of labor cost, the foreign invested enterprises of policy-seeking and cost-seeking types, and some of the market-seeking types with relatively weak competitiveness will have to leave the economic system or remake themselves into different types, while those foreign capital assets of market-seeking and technology seeking types might very well occupy the high end positions of their individual markets and supply chains. hence, the receiving economic system of foreign capital should sufficiently utilize its consumer market and related fundamental advantages, such as opening its domestic market, already established supporting industries, existing resource conditions, etc., to guide the investment capital of market-seeking, technology seeking, and other types into the high technology areas that are needed by the development of the economic system. these investments will not only stay within the economic system for the long term, but also strengthen their collaboration with the economic system and improve the position of the economic system within the machining and manufacturing chain. additionally, by making use of the differences between investment policies internal to the economic system, such as the various preferential policies in terms of taxation, land use, regulations, etc., as introduced by different regions with varying degrees of capital scarceness, the receiving economic system could direct the foreign invested enterprises of policy-seeking and cost-seeking types, and some of the investment assets of market-seeking type with less competitiveness to migrate into regions with less foreign investment capital. these different investment policies could also be used to improve the uneven investment distribution within the economic system while decreasing the disparity between the development gradient forces of different regions. last, promote the capability of innovation and creativity of the enterprises within the economic system and the public of the entire society. according to the systemic yoyo model, the competition between foreign capital and domestic business entities makes some domestic firms been absorbed by foreign capital, while some others been pushed and pulled (or bullied), where some of these bullied entities grow with the intensified competition and eventually become able competitors while some others are eliminated out of the existence. those economic organizations that are either absorbed by foreign capital or eliminated are mainly those that are less competitive, and the competitiveness of an organization is mainly originated from the organization’s creativity. when an economic organization possesses a strong capability of innovation, the yoyo field of this organization has a strong force of attraction and a steady ability to stand and take shocks from the environment. that strong capability of innovation can help promote the magnitude of the organization’s yoyo field, while carrying the field to a higher level. as soon as the magnitude of the yoyo field of the organization rises to a new level, it would generally have acquired additional energy to better counteract the attraction and pressure of the yoyo field of foreign capital, while it might conversely absorb the yoyo field of foreign capital and make this external field a part of itself. in other words, improving the endogenous creativity is the fundamental way for the economic system to increase the magnitude of its yoyo field. to this end, the economic system can improve its internal innovation environment by fostering a economic security under disturbances of foreign capital 28 copyright ©2017 assa. adv. in systems science and appl. (2017) good atmosphere for innovation, increasing investment in innovation, encouraging innovative behaviors, and providing innovators with conditions to materialize their creativities. on top of all these suggested remedies the economic system would be able to reach a new level of creativity of the entire society. 6. concluding remarks because foreign capital can adversely affect the security of the receiving economy [3,7], this paper addresses the following challenging question: while taking advantage of foreign investments, how can a nation protect itself against the potential disastrous aftermath in case some of the foreign investments were actually part of an active currency war against the nation? to this end, this paper establishes a theoretical model for monitoring the movement and dynamics of foreign capital within a nation’s economy. then potential measures that are useful for countering the adverse effects of foreign investments on the national economy are developed. references [1] cheng, l.k. & kwan, y.k. (2000). what are the determinants of the location of foreign direct investment? the chinese experience. j. int. economics, 51, 379-400. [2] cinlar, e. (2013) introduction to stochastic processes (reprint edition), new york, ny: dover publications. [3] forrest, j. (2014). a systems perspective on financial systems, new york, ny: crc press, an imprint of taylor and francis. [4] horn, r.a. & johnson, c.b. (2012). matrix analysis (2nd edition), cambridge, uk: cambridge university press. [5] lin, y. (2008). systemic yoyo model: some impacts of the second dimension, new york, ny: auerbach publications, an imprint of taylor and francis. [6] liu, s.f. & lin, y. (2006). grey information: theory and practical applications, london, uk: springer. [7] schimmel, k., liu, s.f., nicholls, j. & forrest, j. yl. (2017). effects of foreign capital on economic security, advances in systems science and applications. advances in systems science and applications (2013) vol.13 no.2 100-115 set-theoretic design and analysis of l systems xiaojun duan1, bing ju1 and yi lin2 1department of mathematics and systems science, national university of defense technology, changsha 410073, pr china 2department of mathematics, slippery rock university, slippery rock, pa 16057 usa abstract from the perspective of set theory, this paper establishes a rigorous mathematical definition of l systems, and proves the common characteristics of simple l systems by providing a theoretical framework for a systematic research of l systems. additionally, explanations for the characteristics of systems’ emergence based on l systems are provided, and some design methods of l systems are developed and interesting cases of design are constructed. this work is the first of its kind that investigates the l system using a rigorous mathematical definition based on set theory. keywords set theory, l system, characteristics of systems 1 introduction although the figures produced out of fractals are generally complicated, the description of fractals can be quite straightforward. among the commonly utilized methods are the l-system and the ifs (iterated function system). in terms of the fractals they respectively describe, the l-system is simpler than the if system, where the former contains simple iterations of character strings, while the latter is much more complicated in this regard. aristid lindenmayer, a hungary biologist, introduced the lindenmayer system, or l system in short, as the mathematics theory for describing the growth of a plant[1]. it is a kind of subsequent string replacement system; its theory focuses on the topology of plants, and attempts to describe the adjacency relations between cells or between larger plant modules. while lindenmayer and others proposed the initial solution[1], prusinkiewicz used turtle graphics to implement a lot of fractal shapes and herb model based on turtle shapes[2]. his works made turtle the most commonly used form and explanatory schemes of l systems. in order to avoid the models constructed using l systems inflexible, eichhorst and others proposed the concept of stochastic l-systems to enhance the flexibility of the earlier l-models[3]. herman generalized the concept of l-systems to a context sensitive model in order to establish associations between different modules of the plant model[4]. then, lindenmayer introduced parameters to make l-systems even more forthright and efficient. the most typical applications of l-systems are done by scholars at calgary university, canada. they were involved in the parametricalization of l-systems, the establishment of differential l-system, open advances in systems science and applications (2013) vol.13 no.2 101 l-system, and other relevant theoretical researches[2]. additionally, they developed the plant simulation software l-studio[2,5]. because there has not been any rigorous mathematical definition established for l systems, all published studies on the subject have stayed only at the level of innovative designs without much theoretical support. on the other hand, as shown in (lin, 1999)[6], set theory is a good tool for the theoretical framework of systemics. in this paper, starting from the basics of set theory, we will develop a rigorous mathematical definition for simple l systems. on the basis of this definition, we establish some of the general properties simple l systems satisfy so that our constructed theoretical framework is expected to provide the needed fundamental ground for further investigation of l systems. 2 the definition of simple l systems the fact is that l-systems represent a formal language; it can be divided into three classes: 0l system, 1l system, and 2l system. a 0l system stands for an l system that is context free. that is, the behavior of each element is solely determined by the rewriting rule; the current state of the system has something to do only with the state of the immediate previous time moment and has nothing to do with any surrounding element. among l systems are 0l systems the simplest; that is why each 0l system is also referred to as a simple l system. each 1l system stands for a context sensitive l system that considers only one single-sided grammatical relationship. the current state of the system has something to do with not only the state of the immediate previous time moment but also the state of the elements on one side, either left associated or right associated. each 2l system is also a context sensitive l system that, different of 1l systems, it considers grammatical relationships from both sides. that is, the current state of the system is related to not only the state of the immediate previous time moment, but also the states of the surrounding elements. it represents a method that is most sensitive to the context. these three classes of l systems are further divided into deterministic and random l systems depending on whether or not the rewrite rules are deterministic. let s = {s1, s2, . . . , sn} be a finite set of characters of a language, say, english, and s∗ the set of all strings of characters from s. because s ⊂ s∗, s∗ is a non-empty set. ∀α, β ∈ s∗, define that α = β ⇔ both α and β are identical, meanings that they have the same length and order of the same characters. now, define the addition operation ⊕ on s∗ as follows: α ⊕ β = αβ = the string of characters of those in α followed by those in β, ∀α, β ∈ s∗. the scalar multiplication on s∗ is defined as follows: ∀α ∈ s∗, k ∈ z+ = the set of all whole 102 xiaojun duan:set-theoretic design and analysis of l systems numbers, kα = αα . . . α︸ ︷︷ ︸ k times . then, the following properties can be shown: (1)both addition and scalar multiplication defined on s∗ are closed; (2)(α⊕ β)⊕ γ = α⊕ (β ⊕ γ),∀α, β, γ ∈ s∗; (3)k(lα) = (kl)α,(α⊕ β)⊕ γ = α⊕ (β ⊕ γ),∀α, β, γ ∈ s∗;and (4)(k + l)α = kα⊕ lα,∀α ∈ s∗, k, l ∈ z+. before we develop a rigorous definition of simple l systems, let us first look at a specific mapping s∗ → s∗, known as l mapping. 2.1 the l mapping definition 2.1(production rules). for any si ∈ s, 1 ≤ i ≤ n, if an ordered pair (si, α) ∈ s × s∗ can be defined, then this pair defines a relation pi from si to α, known as a production rule from s to s∗. definition 2.2 (same class production rules). for a given si ∈ s, if there are r ≥ 1 production rules p (1) i , p (2) i , . . . , p (r) i defined for si such that ∀j ̸= k(1 ≤ j, k ≤ r), p (j) i (si) ̸= p (k) i (si), then p (1) i , p (2) i , . . . , p (r) i are referred to as r same class production rules of the character si, and pi = {p(1)i , p (2) i , . . . , p (r) i } the set of same class production rules of the character si. definition 2.3 (l mapping). suppose that m(≥ n) production rules p = n∪ i=1 pi from s to s∗ are given, where each pi is non-empty and stands for the same class production rules of the character si ∈ s, 1 ≤ i ≤ n. for any group of production rules p1, p2, . . . , pn ∈ p , satisfying pi ∈ pi, i = 1, 2, . . . , n, the set φ = {p1, p2, . . . , pn} defines a mapping s∗ → s∗, still denoted φ, by φ : sk1sk2 . . . skr → pk1(sk1)pk2(sk2) . . . pkr(skr) (1) ∀sk1sk2 . . . skr ∈ s∗, ki ∈ {1, 2, . . . , n}, 1 ≤ i ≤ r, and r stands for the length of the character string. then, this mapping φ : s∗ → s∗ is referred to as an l mapping. note: from the definition of l mappings, it follows that the set of m(≥ n) production rules from s to s∗: p = {p(1)1 , . . . , p (r1) 1 , p (1) 2 , . . . , p (r2) 2 , . . . , p(1)n , . . . , p(rn)n } (2) can define (r1r2 . . . rn) many l mappings from s to s∗, where n∑ i=1 ri = m. the set of all l mappings determined by the set p is denoted by φ = {φ1, φ2, . . . , φr1r2...rn}. proposition 2.1 (properties of l mappings). for any α, β ∈ s∗ and m,n ∈ z+, each l mapping φ satisfies the following properties: (i) φ(α) ∈ s∗ is uniquely defined; (ii) if φ|s is surjective s → s, then φ|s must be bijective; advances in systems science and applications (2013) vol.13 no.2 103 (iii) if α = sk1sk2 . . . skp ∈ s∗, then φ(α) = φ(sk1)φ(sk2) . . . φ(skp); (iv) φ(α⊕ β) = φ(α)⊕ φ(β); (v) φ(mα) = mφ(α); (vi) ∀φ, ϕ ∈ φ, φϕ(mα⊕ nβ) = mφϕ(α)⊕ nφϕ(β); proof. because both (i) and (ii) are evident, it suffices to show (iii) (vi). (iii) for any α = sk1sk2 . . . skp ∈ s∗, the definition of the l mappings implies that φ(sk1sk2 . . . skp) = pk1(sk1)pk2(sk2) . . . pkp(skp) (3) and, ∀ski ∈ s∗(1 ≤ i ≤ p), we have φ(ski) = p(ski) (4) hence, φ(α) = φ(sk1sk2 . . . skp) = pk1(sk1)pk2(sk2) . . . pkp(skp) = φ(sk1)φ(sk2) . . . φ(skp). (iv) for any α, β ∈ s∗, from the addition operation on s∗ and property (iii), it follows that φ(α⊕ β) = φ(αβ) = φ(α)φ(β) = φ(α)⊕ φ(β) (5) (v) for any α ∈ s∗ and m ∈ z+, from the definition of scalar multiplication on s∗ and property (ii), it following that φ(mα) = φ(αα...α︸ ︷︷ ︸ m times ) = φ(α)φ(α)...φ(α)︸ ︷︷ ︸ m times = mφ(α) (6) (vi) for any φ, ϕ ∈ φ, by employing properties (iv) and (v), we obtain: φϕ(mα⊕ nβ) = φ(mϕ(α)⊕ nϕ(β)) = mφϕ(α)⊕ nφϕ(β) (7) qed. 2.2 simple l-systems with the concept of l-mappings in place, let us now look at how to define simple l-systems. according to the classification of simple l systems, deterministic and random simple l systems, we now establish the relevant definitions by using the concept of l mappings. definition 2.4(deterministic simple l systems). let s = {s1, s2, ..., sn} be a finite set of characters and s∗ the set of all strings of characters from s. assume that φ : s∗ → s∗ is a given l mapping. for any given initial string ω ∈ s∗, the system that is made up of the nth iterations φn(ω), n ≥ 1, is referred to a deterministic simple l (d0l) system (of order n), denoted by the ordered triplet ⟨s, ω, φ⟩. definition 2.5(random simple l systems). let s and s∗ be the same as in 104 xiaojun duan:set-theoretic design and analysis of l systems definition 2.4, φ = {φ1, φ2, ..., φk} a set of k l mappings from s∗ to s∗, and ξ the l mapping randomly drawn from φ such that the probability p (ξ = φi) for ξ to be φi is πi, where k∑ i=1 πi = 1. for a given initial string of characters ω ∈ s∗ , the system that is made up of the nth generalized iterations ξnξn−1...ξ1(ω), the composite of the mappings ξ1, ..., ξn−1, ξn, n ≥ 1, is referred to as a random simple l (r0l) system (of order n), where ξ1, ξ2, ..., ξn are independent and identical distributions. this system is written as the following ordered quadruple ⟨s, ω,φ, π⟩. 3 properties of simple l systems from the definitions, it follows that each simple l system is produced by iterations or generalized iterations of l mappings. hence, the properties of simple l systems are determined by those of l mappings. theorem 3.1(property of linearity). each simple l system is a linear system. proof. assume that the deterministic nth order simple l system is given as l1 = ⟨s, ω, φ⟩, and the random nth order simple l system l2 = ⟨s, ω,φ, π⟩, where n ≥ 1. to show both l1 and l2 are linear systems, it suffices to show that both l1 and l2 systems respectively satisfy the superposition principle. (i)we use mathematical induction to prove that φn satisfies the superposition principle. when n = 1, ∀α, β ∈ s∗, k1, k2 ∈ z+, properties (iv) and (v) of l-mappings imply that φ(k1α⊕ k2β) = k1φ(α)⊕ k1φ(β) (7) so, φ satisfies the superposition principle. assume that when n = k, k ≥ 1,φk satisfies the superposition principle. that is, ∀α, β ∈ s∗, k1, k2 ∈ z+, the following holds true: φk(k1α⊕ k2β) = k1φ k(α)⊕ k2φ k(β) (8) then when n = k + 1 , we have φk+1(k1α⊕ k2β) = φ(φk(k1α⊕ k2β)) = φ(k1φ k(α)⊕ k2φ k(β)) = k1φ k+1(α)⊕ k2φ k+1(β) (9) hence, ∀n ≥ 1, φn satisfies the superposition principle. (ii)we show that ξnξn−1...ξ1 satisfies the superposition principle. from the definition of random simple l systems, it follows that ξi , 1 ≤ i ≤ n, stands for the l mapping φj , 1 ≤ j ≤ n , where n is the total number of elements in the set φ , randomly selected from φ at step i. because each φi ∈ φ , 1 ≤ i ≤ n , satisfies the superposition principle, from advances in systems science and applications (2013) vol.13 no.2 105 property (vi) of l mappings, it follows that ξnξn−1...ξ1 satisfies the superposition principle. qed. from theorem 3.1 it follows that in terms of a simple l system, when its order n is relatively large, to shorten the computational time, one can decompose relatively long strings of characters into sums of several shorter strings, which can be handled using parallel treatments to obtain the state of the next moment. theorem 3.2 (property of fixed points). the restriction of the l mapping φ in a d0l system on s is bijective from s to s, if and only if for any α ∈ s∗, there is a natural number k such that φk(α) = α. before proving this result, let us first look at the following lemma. lemma 3.1 if the restriction of the l mapping φ on s is a bijection from s to s , then ∀si ∈ s , ∃ki ∈ z+ such that φki(si) = si , 1 ≤ i ≤ n. proof. by contradiction, assume that ∃s ∈ s, ∀k ∈ z+ , φk(s) ̸= s. then let si1 = φ(s), si2 = φ2(s), ..., sin = φn(s) so we have s ̸= si1 , s ̸= si2 , ..., s ̸= sin (10) step 1: from the first (n − 1) inequalities in equ. (10) and that fact that φ|s : s → s is bijective, it follows that φ(s) ̸= φ(si1), φ(s) ̸= φ(si2), ..., φ(s) ̸= φ(sin−1). that is, si1 ̸= si2 , si1 ̸= si3 , ..., si1 ̸= sin (11) step 2: from the first (n − 2) inequalities in equ. (11) and that fact that φ|s : s → s is bijective, it follows that φ(si1) ̸= φ(si2), φ(si1) ̸= φ(si3), ..., φ(si1) ̸= φ(sin−1). that is, si2 ̸= si3 , si2 ̸= si4 , ..., si2 ̸= sin (12) by continuing this procedure, we obtain at step (n− 1) that sin−1 ̸= sin (13) therefore, s ̸= si1 ̸= si2 ̸= ... ̸= sin . it means that there are n + 1 different elements in the set s. however, s has only n elements. a contradiction. so, the assumption that ∃s ∈ s, ∀k ∈ z+, φk(s) ̸= s does not hold true. in other words, ∀si ∈ s , ∃ki ∈ z+ such that φki(si) = si , 1 ≤ i ≤ n. qed. in the following, let us prove theorem 3.2. (⇒)for α = sm1sm2 ...smp ∈ s∗, from lemma 3.1, it follows that ∀smi ∈ s, ∃ki ∈ z+, φki(smi) = smi ,1 ≤ i ≤ p . let k =< k1, k2, ..., kp >. so, φk(α) = φk(sm1sm2 ...smp) = φk(sm1)φ k(sm2)...φ k(smp) = φk1(sm1)φ k2(sm2)...φ kp(smp) = α 106 xiaojun duan:set-theoretic design and analysis of l systems so, there is a natural number k such that φk(α) = α . (⇐)if ∀α ∈ s∗, there is natural number k such that φk(α) = α, let us take α = s1, s2, . . . , sn. then ∃k1, k2, ..., kn such that φk1(s1) = s1, φ k2(s2) = s2, ..., φ kn(sn) = sn (14) according to property (ii) of l mappings, to show the restriction of φ on s is a bijection from s to s, it suffice to prove that φ|s : s → s is surjective. again, we prove this end by contradiction. assume that ∃s ∈ s , ∀si ∈ s, 1 ≤ i ≤ n ,φ(si) ̸= s . then we have that ∀k ≥ 1 , φk(s) ̸= s , which contradicts with equ. (14). so, φ|s : s → s is surjective. hence, φ|s : s → s is bijective. qed. from theorem 3.2, it follows that in terms of d0l systems, if the restriction φ|s : s → s of l-mapping φ is a bijection, then there are at most k =< k1, k2, ..., kp > different states. therefore, to produce complicated fractal figures using d0l systems, one should avoid using any such l mapping whose restriction on s is a bijection from s to s. theorem 3.3(property of fixed points). the restriction of the l mapping φ in a r0l system on s is a bijection from s to s, if and only if for any α ∈ s∗ , there is a natural number k such that φk(α) = α. proof. although as a r0l system, the probability of different mappings at each step is different, the maximum number n of total states of each step is fixed. since the length p of the mapping is finite, the number of overall states is k = np at most. so there exists a integer k = np to satisfy φk(α) = α. other parts of the proof are similar to those of theorem 3.2 and omitted. qed. remark:in terms of numbers, the finite states of rol are more than those of dol. however, from theorem 3.3, to follows that in order to produce complicated fractal figures using r0l systems, one should still avoid using any such l mapping that its restriction on s is a bijection from s to s. 4 an explanation of holistic emergence of systems using dol although the mechanism for a simple l system appears is quite straightforward, it clearly shows the attribute of holistic emergence of systems. so, simple l systems can be employed as an effective tool to illustrate the emergence of systems’ wholeness. in the following, we will design a simple binary system to explain the emergence of systems’ wholeness. in the cartesian coordinate system r2, assume that the initial location of particle a is (0,0) and its initial angle is 0. let f stand for moving forward one unit step 1, and + turning an angle of π/3 counterclockwise. now, let us construct the following d0l system l2 = ⟨s, ω, φ⟩, where s = {f,+},ω = f , and φ = { f → f ++f ++f + → + advances in systems science and applications (2013) vol.13 no.2 107 the first iteration φ(ω) produces: f++f++f; the second iteration φ2(ω) leads to: f++f++f ++ f++f++f ++ f++f++f; . . .; when the nth order, n ≥ 1 , l system l2 is applied on the particle a, one can obtain the output state as shown in fig.1. as shown in fig.1, such an nth order l system, a binary sysfig.1 the output state of the l2 system tem, is very simple. its function can be comprehended as follows: drive particle a from the starting point at (0,0) to make 3n−1 counterclockwise turns along the triangular trajectory. now, let us employ this simple binary system to illustrate the emergence of systems’ wholeness: (1) the whole is greater than the sum of its parts; and (2) the function of the whole is more than the sum of the functions of the parts. for our purpose, by the word “sum”, it means the collected pile of parts without any interactions between the parts. as for the word “function”, it is understood as follows: as long as a system is identified, its functionality is a physical existence; however, any function of the system has to be manifested through the system’s act on a specific object that is external to the system. so, to investigate the overall function of a system and the sum of the functions of its parts, one needs to consider the respective effects of the system as a whole and each of its parts on an external object. through analyzing their effects on this object, one can compare the overall effect of the system and the aggregated effect of the parts. before we illustrate the emergence of the system’s wholeness, let us first establish the following assumptions: (a) the whole can always be divided into several distinguishable parts in terms of components, attributes, or functionalities; (b) similar parts of the system (or parts with similar attributes) satisfy the 108 xiaojun duan:set-theoretic design and analysis of l systems additive property, while different parts (or parts with different attributes) do not comply with this property. here, the additivity stands for the algebraic additivity and is different of the meaning of the “sum” in “the sum of parts”; and (c) when studying the sum of parts’ functionalities, if an identified part does not have any function, then the functionality of this part is seen as “0”. illustration 1: the whole is greater than the sum of its parts let us symbolically denote the constructed nth order simple l system l2 = ⟨s, ω, φ⟩, n ≥ 1 , as φn(f ), and sum of parts as ∑ i si , where si , i = 1, 2, ..., stands for a division of the whole φn(f ), including all characters. in the following, we will discuss from two different angles according to the following different divisions of the whole. (1) the whole is greater than the sum of all the elements (the whole is divided using components). if we treat the system l2 as one containing only two components (operations) f and +, then in the nth order l2 system, there are 3n components f and 2(3n -1) components +. from the basic assumptions, it follows that f and + respectively satisfy the additive property. their algebraic sums are respectively written as s1 = 3n(f ) and s2 = 2(3n − 1)(+) . then, the sum of the elements of the l2 system can be expressed as 2∑ i=1 si = s1s2 or 2∑ i=1 si = s2s1. applying the output of the elements’ sum 2∑ i=1 si on the particle a produces a line segment of length 3n , while applying the output of the whole φn(f ) on the particle a creates a regular triangle with each side’s length 1. the whole constitutes a figure of the 2-dimensional space, possessing a special structure, while the sum of the elements represents a line segment of the 1-dimensional space without any qualitative mutation. that is to say, the whole has a spatial structural effect that is not shared by the sum of the parts, therefore, φn(f ) > 2∑ i=1 si . (2) the whole is greater than the sum of its parts, where the whole is divided using attributes. let us treat the l2 system as being composed of 3n unit vectors of the plane: i⃗1, i⃗2, ..., i⃗3n . that is, we see the vectors i⃗1, i⃗2, ..., i⃗3n as having the same attributes. then this l2 system is made up only of these 3n components of the same attributes. now, we desire to show φn(f ) > 3n∑ k=1 i⃗k . evidently, 3n∑ k=1 i⃗k = 0. so, when the output of the sum 3n∑ k=1 i⃗k of the parts is advances in systems science and applications (2013) vol.13 no.2 109 applied on the particle a, it is like that no operation is ever applied on a so that the particle a is fastened at the origin without moving. because the output of the whole φn(f ) is a string of characters, which is equivalent to a sequence of operations with a before and after order, when φn(f ) is applied on the particle a, this particle will repeatedly travel counterclockwise along the triangle and eventually return to the origin. that is to say, the whole possesses a structural effect of time that is not shared by the sum of the parts, therefore, we have φn(f ) > 3n∑ k=1 i⃗k . in fact, the essential difference between the whole and the sum of parts is that the whole has some structural effect in terms of space or time, while the sum of parts does not have. for instance, when n bricks are used to build a house, the whole stands for a building along with the spatial structure of rooms, etc. however, when the components of this building are divided, because there is only one kind of component, the bricks, the sum of the parts satisfies the algebraic additivity and is equal to the n bricks, which do not have the spatial structure of the house. illustration 2: the functionality of the whole is greater than the sum of parts’ functionalities. let us first introduce a new kind of set [s], where each element is allowed to appear more than once. to avoid creating any conflict with the conventional set theory, we will only apply the operation of drawing elements out of the set [s] with the following convention: if a non-empty set [s] contains element s at least twice, then drawing one s from [s] means that we take any of the elements s’s. for example, [s] = {a, a, b, a, b} . then, drawing an a from [s] means that we take any one of the elements a’s. for the nth order simple l2 system, n ≥ 1 , the individual characters (operations) f and + stand for the smallest units of functionalities. divide this system into (3n+1−2) parts, which include 3n functional units f and 2( 3n-1) functional units +. the set of these (3n+1-2 ) functional units is written as set [s] . we first consider the effect of the whole on the particle a. the function of this l2 system is to order the elements of [s] according to some specific rules. then, the effect of the system on a is force the particle a to counterclockwisely travel along the triangle 3n−1 times. now, let us look at the effect on a of each part. when the set [s] is given, the sum of the parts’ functions can be understood as follows: there are a total of m operations, each of which takes mi arbitrary elements from [s] without replacement to act on a, until all the elements in [s] are exhausted. evidently, as long as the order of the elements taking out of the set [s] is different from that of φn(f ) , the total effect of these elements that are individually taken out of [s] will not reach that of the l2 system. therefore, the whole possesses a functionality the sum of the parts does not share. that is, the function of the whole is greater 110 xiaojun duan:set-theoretic design and analysis of l systems than the sum of the parts’ functions. in fact, the essential difference between the function of the whole and the sum of parts’ functions is that the whole has an organizational effect, while the sum of the parts does not have. 5 the design of simple l systems 5.1 the basic graph generation principle of simple l systems in essence each l system is a system that rewrites strings of characters. its working principle is quite simple. if each character is seen as an operation and different characters are seen as distinct operations, then strings of characters can be employed to generate various fractal figures. that is, as long as strings of characters can be generated, one is able to produce figures. the character strings of l systems that are used to generate figures can be made up of any recognizable symbols. for example, in the design of programs, the symbols f, -, and + can be used respectively so that “f” means move one unit length forward from the current location and draw a segment, “-” stands for turning clockwise from the current direction a pre-determined angle, and “+” turning counterclockwise from the current direction another pre-determined angle. when generating character strings, start from an initial string and replace the characters of this string by substrings of characters according to the predetermined rules. that completes the first iteration. then, treat the resultant character string from the first iteration as the mother string and replace each character in this string by strings determined by the rules. by continuing this procedure, one can finish the required iterations of an l system, where the length of the resultant string is controlled by the number of iterations. 5.2 a fractal structure design based on d0l in terms of a tree, it stands for a fractal structure. in particular, each trunk carries a large amount of branches, and each branch has an end point, representing a figure with one staring point and many ending points. this fact implies that when one draws a branch to its end, he has to return his drawing pen to draw other structures. let us take the following conventions: “f” stands for moving forward a unit length 1, and “+” turning an angle of π/8 clockwisely, “-” turning an angle of π/8 counterclockwisely, and the characters within “[ ]” represent a branch; when the characters within a pair of [ ] are implemented, return to the position right before “[” and maintain the original direction, and then carry out the characters after “]”. assume that the starting point is at (0,0) on the complex plane and the initial direction at π/2 . now, we design a d0l system g1 = ⟨s1, ω1, φ1⟩ as follows: s1 = {f,+,−, [, ]}; ω1 = f ;and advances in systems science and applications (2013) vol.13 no.2 111 fig.2 the fractal structures: a tree dancing in breeze (of different iteration steps) fig.3 the fractal structures: a standing tree (n = 5, α = π/8, with design form f→ff+[+f[−ff−]−f]− [−f[−ff+]f]) 112 xiaojun duan:set-theoretic design and analysis of l systems φ1 =  f → ff + [+f− f− f]− [− f + f + f] + → + − → − [ → [ ] → ] then, we can produce the fractal structures as shown in fig.2. we can obtain other illustrations by just changing the design forms as fig.3-5. fig.4 the fractal structures: a tree towards sun (n = 5, α = π/8, with design form f→ff+[+f[−f−f]−f]− [−f[f−ff+f]f]) fig.5 the fractal structures: a floating grass ball (n = 4, α = π/8, with design form f →ff+[[+f−ff−f]− [−f+ff+f]] ) 5.3 a fractal structure design based on r0l systems in nature, the forms plants take are not invariant. even for the same kinds of plants, their shapes can vary from one plant to another. such varieties are caused advances in systems science and applications (2013) vol.13 no.2 113 by the effects of the environment. fig.6 the different fractal structures for a r0l system with each iteration unchanged (n = 4, α = π/16 ) in terms of the simulation effects of plants, the figures created by using d0l systems seem to be quite stiff. under the prerequisite of maintaining the main characteristics of plants, in order to generate varieties on the details, we can utilize the plants? structures produced out of r0l systems. the advantage of these figures is that these simulated plants are more real-life like and much closer to the true forms of natural plants. to this end, let us design a r0l system g2 = ⟨s1, ω1,φ, π⟩ as follows: s1 = {f,+,−, [, ]}; ω1 = f ; φ = {φ1, φ2, φ3}; where φ1 =  f → f[+f]f[− f]f + → + − → − [ → [ ] → ] ,φ2 =  f → f[+f]f[− f[+f]] + → + − → − [ → [ ] → ] , 114 xiaojun duan:set-theoretic design and analysis of l systems φ3 =  f → ff[− f + f + f] + [+f− f− f] + → + − → − [ → [ ] → ] and π = p (ξ = φi) = 1/3, i = 1, 2, 3 . then the fractal structures can be produced. for instance, fig.6 shows the different fractal structures of a r0l system with each iteration unchanged. fig.7 shows the different fractal structures of a r0l system where each iteration is different. fig.7 the different fractal structures of a r0l system with each iteration different (n = 5, α = π/16 ) 6 summary although the design principle underlying the l systems is quite straightforward, these systems can be employed to produce many complicated fractal patterns. after many years of research, l systems have evolved from the original rewrite systems of characters to such capable systems that can describe complex 3-dimensional systems. they have evolved from the simplest d0l systems to random l systems, and then to open l systems. the l systems have provided simulations of fractal structures that have become over time much closer to real-life like, and have been employed as an important tool for creating virtual plants. from the point of view of set theory, this paper first establishes a rigorous mathematical definition of simple l-system, and then proves the common characteristics of simple l-systems. by doing so, this paper developed the badly needed fundamental ground for further investigation of l systems. it is expected to be applicable to form the theoretical framework of other l systems. acknowledgements advances in systems science and applications (2013) vol.13 no.2 115 this work is supported by the natural science foundation of china (60974124), the program for new century excellent talents in university and the projectsponsored by srf for rocs, sem in china. references [1] lindenmayer, a. (1968), “mathematical models for cellular interaction in development”, journal of theoretical biology, vol.18, pp.280-289. [2] prusinkiewicz, p, lindenmayer, a. (1996), the algorithmic beauty of plants, london: springer. [3] eichhorst, p, savitch, w. j. (1980), “growth functions of stochastic lindenmayer systems”, information and control, vol.45, pp.217-228. [4] herman, g.t, rozenberg, g. (1975), developmental systems and languages, amsterdam: north-holland publishing company. [5] prusinkiewicz, p, hammel, m, mjolsness, e. (1993), “animation of plant developmen”, acm siggraph, pp.351-360. [6] lin, y. (1999), “general systems theory: a mathematical approach”, new york: kluwer academic and plenum publishers. corresponding author xiaojun duan can be contracted at: xjduan@nudt.edu.cn microsoft word 6-刘秋梅.doc 34-40 advances in systems science and applications (2010), vol.10, no.1 issn 1078-6236 international institute for general systems studies, inc. the research on the early warning system of soybean import dependence degree based on decision tree∗ qiumei liu1 and jun meng*1,2 1science college, northeast agricultural university, harbin heilongjiang, 150030, china 2national study center of soybean engineering, northeast agricultural university, harbin heilongjiang, 150030, china email: merd@mail.neau.edu.cn(*corresponding author) abstract the early warning, which is usually built on the base of forecast, is the premise of remove-warning. the paper presents a new warning way-decision tree warning which can warn all kinds of conditions directly. first, the paper sets up a soybean early warning index system by analyzing soybean market of china in nearly the recent ten years. then, some regulations are extracted according to the warning sign index’s effects on the warning alert index in the way of decision tree. at last, the early warning system of the soybean import dependence degree is set up. keywords decision tree soybean import dependence degree early warning system 1. introduction looking back the soybean export and import conditions, china is always the soybean net export country before 1995, but from then on, china has became a soybean net import country until to now. at present, soybean market of china is in turmoil, and seriously dependents on the foreign soybean. foreign soybean has formed a great shock to the soybean production of china. soybean import dependence degree is identified to the ratio of soybean net import to soybean demand amount. according to the constructed warning sign index system, the soybean import dependence degree is warned, and the degree is divided into non-warning area, gentle warning area, medium warning area and serious warning area. decision tree, by which extracted regulations are accurate convenient and simple, is the basic content of data mining. so the soybean import dependence degree is researched in the way of decision tree, so that the serious condition in the soybean import can be found ahead, and scientific bases can be provided for macro regulations and control departments to make decision. 2. the establishing of warning index system the warning of soybean market in china mainly warns the factors which effect the market. the paper combines the soybean demand, soybean price and so on, warns the abnormal condition, and then provides the corresponding precautions. warning is the basis process to ensure the safe of the soybean market, and only the serious conditions and degrees are timely warned, can we ensure the safety of soybean market in china. 2.1 the choice of warning indexes whether the choice of warning indexes is reasonable affects directly the effect and accuracy of warning, so in the process of determining the warning index system of the soybean market, we must consider all kinds of factors which should reflect every side of the soybean market and have representative characters, especially consider the effect of international factors on soybean * this research has been supported by education ministry emphases science and technology project in heilongjiang (project number: 1153lz10) and irtneau. advances in systems science and applications (2010), vol.10, no.1 35 market of china, so we can present the development condition of soybean market more accurately. the paper considers overall the soybean production, demand, export, price, country policy and so on. the corresponding data can be seen in table 1. table 1 the corresponding data of soybean market in china from 1995 to 2006 year import dependent degree（%） soybean production (ten thousand tons) soybean import (ten thousand tons) soybean export (ten thousand tons) price ratio agriculture basis constructing expenditure (a hundred million) soybean production in usa (ten thousand tons) soybean sowing area (k hectare) 1995 0.02 1350.00 29.8 37.6 0.75 110.0 68444 8127 1996 0.08 1322.00 111.4 19.3 0.75 141.5 59174 7471 1997 0.17 1473.00 288.6 18.8 0.71 159.7 64780 8346 1998 0.18 1515.20 320.1 17.2 0.63 460.7 73176 8500 1999 0.24 1425.00 432.0 20.7 0.59 357.0 74598 7962 2000 0.41 1541.00 1041.9 21.5 0.64 414.4 72224 9307 2001 0.48 1540.70 1394.0 26.2 0.62 480.8 75055 9481 2002 0.41 1651.00 1131.5 30.5 0.69 423.8 78672 8720 2003 0.58 1539.32 2074.1 29.5 0.70 527.3 75010 9313 2004 0.54 1740.15 2023.0 34.9 0.65 542.3 66778 9589 2005 0.63 1634.78 2659.1 41.3 0.58 512.6 85013 9590 2006 0.64 1596.70 2827.0 39.5 0.57 504.3 83368 9279 the data come from web of statistics of national bureau of china directly or have been calculated. warning alert index is the ratio of soybean net import to soybean demand amount, named as soybean import dependence degree, which reflects the dependence degree of soybean market in china on foreign soybean market, in which, the soybean demand amount is calculated as follows: soybean demand amount = soybean production + soybean import – soybean export warning sign indexes are soybean production, the ratio of soybean price in usa to that in china which is named as price ratio, soybean demand amount and agriculture basic construction expenditure, in which, price ratio = soybean price in usa × exchange rate of money ÷soybean price in china. 2.2 decide the foregoing years of warning sign indexes the dividing of foregoing, synchronous and lag indexes is very important in the process of soybean warning, which is the key part of the whole warning system of soybean market. usually, the approach of time difference relevant analytics is used. the foregoing warning sign indexes, which are the most important sign indexes of the warning system, have a role of predicting the future of soybean market in china. the paper decides the foregoing years by many ways and comprehensive analysis, the foregoing years can be seen in the table 2. table 2 the foregoing years of warning sign indexes warning sign indexes foregoing years warning sign indexes foregoing years soybean production in china 3 soybean demand amount 1 price ratio 2 agriculture basic construction expenditure 2 soybean production in usa 3 soybean sowing area 3 liu: the research on the early warning system of soybean import dependence degree based on decision tree 36 2.3 dividing alert limits and degrees of warning alert index and warning sign indexes dividing the alert limits and degrees is the key part in the design of warning system. alert degrees reflect the level and intensify of warning contained in the actual values of warning alert index. in order to decide the warning degree of each index, first we ensure the alert limits of each index, and then ensure the prosperity station of each index each year according to the values of alert limits. consulting ge huiling’s paper ‘the forecast and early warning research about soybean market in china’, the station of soybean market import dependant degree and alert limits are described as table 3. table 3 status description of soybean market import dependent degree condition feature described alert limit non-warni ng area soybean market is stable and boom, soybean mainly comes from china, there are little import, doesn’t depend on foreign soybean. [0 0.2] gentle warning area soybean market is stable and boom, soybean mainly comes from china, there are a little of import, doesn’t depend on foreign soybean. [0.2 0.35] medium warning area soybean market is relatively stable, soybean mainly comes from china, there are a little of import, depends on foreign soybean in some degree. [0.35 0.5] serious warning area soybean market is turbulent, soybean coming from china is a very small part, there were mass import soybean, largely depends on foreign soybean. [0.5 1] making use of feedback theory, the warning sign indexes’ limits are decided according to the warning alert index’s limits. to represent in the boundary of two warning areas of warning alert index )(ix , and the one of warning sign index )(iy . so the boundary is between the non-warning and the gentle warning area, or between the gentle warning and medium warning area, or between the medium warning and serious warning area. to represent in the maximal of x and y of xmax and ymax , the minimal of x and y of xmin and ymin , then according to the radial principle, the warning area of x and y are decided by the following formula： yy xx y x iy ix minmax minmax )(max )(max − − = − − by calculating, the warning areas of warning sign indexes can be seen in table 4. table 4 the warning areas of warning sign indexes warning sign index serious warning area medium warning area gentle warning area non-warning area soybean production [1643 +∞) [1537 1643] [1441 1537] (-∞ 1425] soybean demand amount [3677 +∞) [2944 3677] [2211 2944] (-∞ 2211] price ratio [0 0.62] [0.62 0.66] [0.66 0.70] [0.70 +1] agriculture basic construction expenditure [441 +∞) [337 441] [233 337] (-∞ 233] usa soybean production [8021 +∞) [7364 8021] [6706 7364] (-∞ 6706] soybean sowing area [9098 +∞) [8587 9098] [8076 8587] (-∞ 8076] thought out the data and results, we set up the training table of soybean market import dependence degree in table 5. advances in systems science and applications (2010), vol.10, no.1 37 table 5 training table of soybean market import dependence degree year import dependent degree soybean production soybean demand amount price ratio agriculture basis constructing expenditure usa soybean production soybean sowing area 1995 non-warning non-warning non-warning non-warning non-warning gentle warning gentle warning 1996 non-warning non-warning non-warning non-warning non-warning non-warning non-warning 1997 non-warning gentle warning non-warning gentle warning non-warning non-warning gentle warning 1998 non-warning gentle warning non-warning medium warning serious warning gentle warning gentle warning 1999 gentle warning gentle warning non-warning serious warning medium warning medium warning non-warning 2000 medium warning gentle warning gentle warning medium warning medium warning gentle warning serious warning 2001 medium warning gentle warning gentle warning serious warning serious warning medium warning serious warning 2002 medium warning serious warning gentle warning gentle warning medium warning medium warning medium warning 2003 serious warning medium warning medium warning gentle warning serious warning medium warning serious warning 2004 serious warning serious warning serious warning medium warning serious warning non-warning serious warning 2005 serious warning medium warning serious warning serious warning serious warning serious warning serious warning 2006 serious warning medium warning serious warning serious warning serious warning serious warning serious warning 3. decision tree warning 3.1 decision tree arithmetic decision tree that is a tree structure is similar to a flow chart. each node designates an attribute, the branch designates the value of the attribute and the leaf designates a category. the highest level node is root. the way of setting up a tree is recursion from up to down. 3.1.1 the creation of decision tree (1) if all the examples in the trained sample set belong to the same type according to the category attribute, then the set can be called leaf node. the content of the leaf node is the sign of the type. (2) otherwise, according to some tactic (information gain ratio of each attribute), a non-category attribute is chosen whose information gain ratio is biggest. by the value of the attribute, the sample set is divided into several subsets. (3) then, deals with each subset by the way of recursion till all samples in the same subset have the same value of category attribute. 3.1.2 choice of determinant attribute about the recursion way of decision tree, usually the attribute which have the most information gain is the test attribute of present node, so the demand information is smallest for sorting the remained test sample set. that is, dividing the sample set by the attribute will make the mixture degrees of the subsets be slowest. however, this calculating way based on information theory is partial to the attribute which has more values, but the attribute is not always the best. so, information gain is represented by gain ratio.definition of entropy ： ( ) ( )∑ = −= c i ii ppsentropy 1 2log s is the training set, c is the number of categories of category attribute and pi is the ratio of the number of samples of category i to that of s. definition of information gain ( ) ( ) ( ) ( ) ( ) ∑ ∈ −= avaluev vv sentropysssentropyasgain /, sv is the subset whose value of attribute a is v and entropy(s) is the entropy of the training set. the information gain of attribute is the reductive expected value after dividing. liu: the research on the early warning system of soybean import dependence degree based on decision tree 38 definition entropy of each non-category attribute. ( ) ( ) ( )∑ = = c i ii ssssassplitinfo 1 2 /log/, is is the subset which has the ith value of attribute a in s. for the more average of the attribute divided by attribute value, the bigger of splitinfo, information gain is represented by information gain ratio to avoid choosing that attribute. definition of information gain ratio ( ) ( ) ( )assplitinfo asgainasgainratio , ,, = gain ratio is information gain ratio，gain is information gain，splitinfo is the entropy of attribute. 3.2 actual application of decision tree putting the training data in table 5 into the decision tree arithmetic, the information gain ratio of non-category attribute is obtained in table 6. table 6 information gain ratio of each attribute soybean production in china soybean demand amount price ratio 0.5068 0.4856 0.6652 agriculture basis constructing expenditure soybean production in usa sowing area 0.5406 0.4192 0.6475 it can be seen from table 6, the information gain ratio of price ratio is biggest, so it is the root node, then deal with other attribute by the way of recursion. we obtain the following tree: figure 1 the decision tree obtained rules： if price ratio is non-warning, soybean import dependence degree is non-warning if price ratio is gentle warning and soybean demand is non-warning, soybean import dependence degree is gentle warning if price ratio is gentle warning and soybean demand is gentle or medium warning, soybean import dependence degree is serious warning if price ratio is medium warning and soybean production is gentle warning, soybean import price ratio soybean demand amount gentle warning serious warning agriculture basis constructing expenditure non-warning serious warning medium warning soybean production medium warning non warning medium warning gentle warning gentle warning non warning medium/ serious warning medium warning serious warning serious warning medium warning medium warning advances in systems science and applications (2010), vol.10, no.1 39 dependence degree is medium warning if price ratio is medium warning and soybean production is medium warning, soybean import dependence degree is serious warning if price ratio is serious warning and agriculture basis constructing expenditure is medium warning, soybean import dependence degree is medium warning if price ratio is serious warning and agriculture basis constructing expenditure is serious warning, soybean import dependence degree is serious warning. making use of this tree, the import dependence degree of the soybean market in china in 2007 is tested, and that of 2008 is warned ahead, corresponding warn degrees of foregoing warn sign indexes are in table 7. table 7 the training table of the import dependence degree of the soybean market in china in 2007 and 2008 year soybean production soybean demand price ratio agriculture basis constructing expenditure 2007 serious warn serious warn serious warn serious warn 2008 medium warn serious warn serious warn serious warn putting the training data in 2007 into decision tree, we obtain the result: the import dependence degree of soybean market in china in 2007 is serious warning. it has been known that the soybean production in china in 2007 is 1440 ten thousand tons, the soybean imports 3082 ten thousand tons and the soybean exports 39.2 ten thousand tons. by calculating, import dependence degree of soybean is 0.69 and in serious warning area. this accords with the result obtained by the way of decision tree. furthermore, the paper warns the import dependence degree of soybean market ahead in 2008, and the result is that it is in the serious warning. the tree comes true the warning by combining the soybean price, soybean production, soybean demand and country policy. usa supplies the most of soybean in the world, and china is one of the biggest import countries. the price ratio has a direct influence on china soybean market. the lower of the ratio, china is more partial to import foreign transgenic soybean in order to satisfy the demand, so resulting in warning affairs. at the same time, along with the rapid development of economy in china, people’s consumptions in albumen and oil are increasing gradually. because of the limit of the sowing area in china, the soybean production is limited. in recent years, the gap between inland soybean production and market demand amount is being enlarged gradually, following which the amount of soybean import is increasing, so resulting in warning affairs. besides, agricultural basis construction expenditure has an obvious effect on soybean market of china, in which storage and construction of traffic have a direct effect on soybean purchase price. if the costs of storage and transportation are reduced, the amount of the soybean import will be less. however, the ability of transportation in some areas of china is very low, which results in the rise of the cost of transportation and indirectly stimulates the rise of the inland soybean price, so the chinese soybean businessmen import the foreign soybean. therefore, the investment in agriculture basis construction expenditure is the direct factor influencing soybean import degree. the investment in agriculture basis construction expenditure of china hasn’t gone up for 4 years, which accords with the fact of serious warning of soybean import degree in recent years. in order to improve the present condition of china, first, the change of foreign and native soybean prices should be paid more attentions, and the inspecting degree of import soybean should be intensified at any moment. then, not only preventing dump of foreign country to soybean market in china, but also taking measures to retort the inferior position of domestic soybean price. at the same time, the policy is necessary and native soybean planting industry should also be stable. at last, the disadvantageous condition of depending foreign soybean should be changed overly by enhancing the completing ability. liu: the research on the early warning system of soybean import dependence degree based on decision tree 40 4. conclusion the basic functions of decision tree are prediction and classifying. both of the two functions are the basis of warning. first, the foregoing property of the warning sign indexes ensures the success of warning, and future import condition is estimated by the values of warning sign indexes, then the import condition is classified into 4 parts. the paper considers overall the factors affecting on the soybean import condition, so the non-category attributes are a little more, but there were only parts of the attributes which provides more information are used. therefore, the advantage of decision tree can be seen, that is the way can auto-filter factors and needn’t combining other ways. because of the limits of the foregoing years of warning sign indexes, the model can only warn the soybean import dependent degree of future in a year. besides, the number of data in this paper is small and taking use of every province’s data in china may be a good thing. the more of the data, the warning result is more accurate. references [1] jun meng. the analysis of soybean potential yield in jiansanjiang farm by logistic model. advances in systems sciences and applications, 2007, 7(1): 143-147. [2] jun meng. the research of agricultural producing industry structure’s evaluating and demonstration analysis. advances in systems sciences and applications, 2007, 7(2): 173-177. [3] fanliang kong, zhaowu ni. a martingale method of perpetual american options pricing with ddefault-risk. advances in systems sciences and applications, 2007, 7(1): 7-11. [4] minfen shen, bin li, qianhua zhan, p.j.beadle. estimation of visual erp signals with nonextensive entropy. advances in systems sciences and applications, 2007, 7(1): 26-31. [5] ming zhu. data mining[m]. he fei: china university of science and technology publishing house, 2002: 67-74. (in chinese) [6] jingmin chen. technology of data warehouse and data mining[m]. publishing house of electronics industry, 2007, 6: 203-206 (in chinese) [7] danyang cao, jinhong li, jinqiang wei, yanfang zhang. cet-4 grade performance analysis based on decision tree. journal of north china university of technology beijing china, 2007, 1: 38-41. (in chinese) [8] xue bai, fu duan. the study and application of decision tree on teaching access. the development and application of computer, 2007, 20(2): 24-26. (in chinese) advances in systems science and application(2015) vol.15 no.1 1-20 the paradoxes of the world’s progress(ii) sailau baizakov1 and jeffrey yi-lin forrest2 1jsc “iei” economic development and trade of kazakhstan, jsc “economic research institute”, str. temirkazyk 65, 0100000, astana, kazakhstan 2department of mathematics, slippery rock university, slippery rock, pa 16057, usa abstract an unbiased systematic view on the modern innovation allows you to see three key groups of innovations that are still very few people differentiate. the first is the technical and technological innovations that are the basis of development and change in technological ways of the world. secondly, it is monetary and financial innovation, the progress of which determines the change in monetary ways of life of the world. and third, it is the socio-political innovation, progress of which is in the basis of the change of socio-political models of the world. a clear distinction between these three “floors” of innovations is crucial for understanding the global crisis and the ways to quit it. since the essence of it lies in the tangle of clearly long overdue and contradictions between the rates of introduction of the world’s technical and technological, financial and socio-political innovations. as the global crisis clearly shows, technical and technological way today, is not decisive for the country’s prosperity and peace. exactly the countries with the highest level of technology development have become the main source and a key cause of the global crisis. keywords global crisis, sona analyzer, technical and technological innovation, monetary and financial innovation, socio-political innovation 3 the range of application of the analyzer sona 3.1 the main directions of analytics of macroeconomic policy area of practical applications for the sona analyzer in macroeconomics is very promising and productive. the basic formula that can be successfully used in the management of market economies are: 1.to estimate the rate of stp and modeling of balanced economic growth c(t) = gdp (t)/(qp (t) +gdp (t)) (1) 2. calculation of purchasing power of money and the price of world currencies pp(t) = (c(t) ∗ i2)/i1 (2) 3. to determine the index of market prices of goods and services 1/pp(t) = i1(c(t) ∗ i2) (3) 2 sailau baizakov and jeffrey yi-lin forrest:the paradoxes of the world’s progress(ii) 4. to determine the index of balanced economic growth with ppp i3 = c(t) ∗ i2 (c1) i3 = pp(t) ∗ i1 (c2) c(t) ∗ i2 = pp(t) ∗ i1 (c) (4) 5. to determine the nature of the gdp deflator as the ratio of the nominal indices (i1) and real (i2) of growth p(t) = c(t)/pp(t) (5) 6. calculation of net growth rate of stp and comparative analysis of the competitiveness of the countries dc(t) = c(t)− 100% (6) since purchasing power of the national currency is picked for each country, based on the principles of calculation of sdr, we have: pp(sdr, t) = i=n∑ i=1 ngdp (i)/[pp(1) ∗ngdp (1) + pp(2) ∗ngdp (2) +...+ pp(n) ∗ngdp (n)] (7) where n total number of imf member countries willing to be engaged in foreign trade. 3.2 debates about economy management regulators: questions and answers during the debates of s. baizakov conducted with famous russian professor, doctor of economic sciences, corresponding member of the ras k.valtuh, the following questions and answers that can actually reveal the effectiveness of anticrisis measures in the countries of the world and the capability of market economy management had appeared. question 1. how is stp coefficient defined? answer. stp coefficient is defined by formula: c(t) = µ(t) 1 + µ(t) (8) after conversion, it turns out that c(t) = gdp/x, where x is output, which is the sum of costs of intermediate qp consumption and gdp for production qp +gdp . at any anti-crisis event involving the purchase and sale, a coefficient c(t) is always known value, it can be determined with an accuracy of gdp (t) and x(t). question 2. what determines the rate of scientific and technological progress (stp ) if it does not reflect the performance? advances in systems science and application(2015) vol.15 no.1 3 answer : indeed, the indicator called the coefficient of scientific and technological progress (stp ), and defined by the formula: c(t) = µ(t) 1 + µ(t) (9) where the indicator µ(t) = ngdp (t) qp (t) (10) expresses the organic composition used in the production of gdp of material resources, which are not directly related to labor productivity. comment by valtuh k.k. concept of organic structure is occupied. it does not applied here. this itself is substantively different (see “capital” t i) answer : in this formula, the money in one case serve as money capital (ngdp). in another case acts as a commodity-capital (x). in this formula, commoditycapital is represented by its denominator. according to the theory of emerson, it is the equivalent of the capital expended plus a normal profit [emerson, twelve principles of productivity. c.62]. comment by valtuh k.k. macroeconomic indicator of labor productivity exists and is systematically used (see, for example, the annual reports of the president of the united states). in general, the opposition of macroeconomics and the real economy is unacceptable. as for economics (not to be confused with the widespread literature, claiming without evidence for non-marxist economic theory), it is precisely the subject of the real economy (and for this literature largely fictional). answer: i am here for the organic composition of social labor meant the ratio of money capital (ngdp) to the commodity-capital (x). my statement follows logically from marx’s remarks on the work shtibelinga in “capital”, volume iii, part i, p. 25. question 3. what does the coefficient of µ = gdp/qp expresses, which is included in the formula for determining the rate of scientific and technological progress? answer : output(x) = qp + gdp is the sum of the intermediate material resources (qp ) and gdp . and coefficient µ = gdp/qp expresses the performance of material resources qp . comment by valtuh k.k. output is not the amount of material resources and gdp , but the amount of current material costs and gdp . answer : agree. question 4. scientific and technological progress (stp ) is always associated with labor productivity. why do not you have that connection! answer : if the level of macroeconomics goes to the level of the real economy, contribution directly to the dynamics of stp performance will appear, and then 4 sailau baizakov and jeffrey yi-lin forrest:the paradoxes of the world’s progress(ii) its share of the output will be determined. advises by valtuh k.k.: representative calculations of this coefficient should be hold. it turns out that it varies in a narrow range. but the main quickly changing, usually growing the effect of technological progress does not change the value c(t) and the growth of productivity of human labor. this effect is not expressed in the formula. there are also a number of other ntp effects. answer: performance of living labor is reflected at the level of the coefficient of stp c(t). thus, the formula coefficient ntp c(t) = gdp (t)/x(t) can be replaced with an equivalent ratio of the productivity of living labor in gdp to performance of output x. but the growth of productivity of human labor will have a positive effect with domination of productivity growth for gdp over growth in productivity of the production. we are not interested in productivity growth, and its impact on the price of goods and services. as you know, the prices of goods and money change after the drop of productivity of labor and capital. it should be noted here that my opponent propose a coefficient of production to replace with the sources of real, not nominal growth. but, we are referring to the fact that the highest levels of productivity can be extinguished corresponding increase in average wages. the important point for a market economy it is not a productivity growth, the so-called living labor, but the importance of economic dynamics of labor productivity, which is defined as the ratio of labor productivity of living to the average annual salary. as has been justified above, the variation coefficient of stp in this case expresses the integrated expression of the effects of the anti-crisis measures and subjective actions of managers of production from a position of economic growth and stability in the country. table 1 shows the ntp coefficient calculated by formula c(t) = gdp (t)/x(t). as shown in the table 1, the range of the ratio of ntp does not vary within a narrow range, as k. valtuh thought, and its actual range of variation increased from 31% in cyprus and 54% in estonia to 148% in brazil and 162% in the czech republic. these range changes occurred in only 10 years, so we cannot speak about the narrowness of the range of variation of this indicator. do the obtained results allow us to provide answers to the question how to determine the ratio of scientific and technological progress (stp)? indeed, for a long time economists could not estimate a contribution of scientific technological progress, which remained exogenous, into the model of growth. mathematical models of economic growth, which appeared in the 80s, included the externality of economic growth.?only paul romer argued persuasively that the growth of capital by 10%increases the production of nearly 1%, rather than 0.25, according to calculations based on the solow’s model.?the costs of research and experimental development (niekr) provide the growing impact of scientific advances in systems science and application(2015) vol.15 no.1 5 table 1 coefficient of scientific and technological progress (stp), 2000 = 100% country/year 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 1 austria 98 97 97 96 94 91 92 88 90 88 2 belarus 104 114 117 115 117 116 115 115 108 105 3 belgium 99 85 71 65 74 87 87 89 95 94 4 bulgaria 100 100 99 94 93 90 87 89 93 98 5 brazil 98 96 95 94 94 103 111 116 113 148 6 united kingdom 100 101 101 101 98 101 100 103 77 96 7 hungry 101 106 107 108 106 103 105 99 109 106 8 germany 100 102 102 101 99 96 97 92 98 95 9 greece 102 102 103 104 105 106 105 120 105 107 10 denmark 98 99 100 100 98 97 94 100 95 96 11 india 101 100 99 97 96 96 96 88 89 114 12 ireland 76 69 69 68 68 66 65 70 64 64 13 spain 100 99 99 98 97 96 96 96 103 104 14 italy 100 100 100 100 99 98 98 95 103 101 15 kazakhstan 117 107 93 98 93 106 107 166 123 130 16 cyprus 95 87 70 60 55 51 26 31 31 31 17 china 101 101 100 100 100 99 102 100 121 131 18 latvia 100 101 98 94 91 93 96 100 77 65 19 lithuania 98 98 99 97 91 92 96 95 75 74 20 luxemburg 99 104 106 98 93 88 83 83 89 78 21 malta 107 107 104 103 103 96 109 108 106 107 22 netherlands 101 103 104 103 103 102 101 100 100 99 23 poland 100 100 99 96 97 97 99 97 101 110 24 portugal 100 102 102 101 101 100 97 96 91 114 25 russia 97 98 98 99 100 100 99 99 98 98 26 romania 98 98 99 97 100 100 101 94 75 74 27 slovakia 99 100 100 104 109 115 140 161 110 73 28 slovenia 101 101 102 101 104 102 100 100 105 103 29 united states 102 104 103 103 101 101 100 99 107 104 30 finland 100 102 102 101 101 100 87 85 98 97 31 france 100 98 99 98 97 96 97 96 99 98 32 czech republic 99 100 100 98 99 96 98 109 103 162 33 sweden 99 101 102 101 98 96 98 98 102 102 34 estonia 117 123 105 92 70 70 66 67 56 58 35 japan 82 89 101 102 92 88 91 100 102 109 36 turkey 100 86 100 100 100 100 101 101 101 101 37 iran, islamic republic 107 85 97 94 90 85 86 104 102 107 38 afghanistan 99 112 70 70 71 70 71 59 54 59 39 pakistan 101 102 103 104 105 106 107 108 109 111 40 uzbekistan 98 99 100 102 103 104 105 107 108 109 41 turkmenistan 100 100 100 101 101 101 101 101 101 102 42 tadzhikistan 86 86 87 92 94 97 101 102 93 93 43 azerbaijan 106 106 102 100 107 114 117 124 117 140 44 kyrgyzstan 102 102 102 101 101 101 100 85 92 94 45 south africa 98 105 108 105 105 107 111 109 116 127 6 sailau baizakov and jeffrey yi-lin forrest:the paradoxes of the world’s progress(ii) and technological progress [1]. reversibility principle has allowed us for the first time in world economics make a significant step forward in measuring the contribution of scientific and technical progress in the development of the real economy through the exact formula c(t) = gdp (t)/x(t). question 5. the purchasing power of money, how is it determined? answer : the purchasing power of money is relative measure. ntp coefficient is directly related to the index of purchasing power of money. the index formula of purchasing power of money, as indicated above, is defined by the ratio of the coefficient of stp to the gdp deflator index of official statistics. table 2 shows the changes in indices of purchasing power of money obtained by calculation according to global statistics. it defined the purchasing power of money is the national currencies of 45 countries for 2000 2010, which have over 80% of the gdp of the world economy. estimated formula: pp(t) = a(t)/p(t), where c(t)-factor of stp, p(t) the gdp deflator. as it can be seen from table 2, the range of changes in the purchasing power of money is wide as the range at the rate of stp. it is changed over the past decade in the range of 0.12 to 1.20 in cyprus in japan. the nature of change in the index of purchasing power of money in this case expresses the effectiveness of monetary and financial management system in the countries.?only one-quarter of these countries has saved the purchasing power of their currencies at a level higher than 50% of their value in the base year 2000.this group includes japan (1.20), the u.s. (0.81), china (0.66) and brazil (0.53).in the same group there were majority of oecd countries.on the contrary, outsiders with weak national currencies are the half of the european union countries, including the ten countries that have a power of their currencies less than one-third of their base power: bulgaria with a coefficient of 0.25, hungary 0.30 ireland 0.22, cyprus?0.12, latvia 0.20, lithuania 0.22, luxembourg 0.30, romania 0.18, slovakia 0.17 estonia 0.27. these figures increase the likelihood of permanent crisis in europe and create difficulties for?these countries to implement the pact for economic growth and stability, adopted in mid-2012. question 6. can you comment on the formula rgdp (t) = ngdp (t)/p(t)? answer. formula ngdp = p ∗rgdp refers to the work of official statistics. statistics are watching over the production and determine the physical volume index (pvi) of goods and displays the gdp deflator (p) by the formula: p = ngdp/rgdp (11) the ratio of the growth rate of ngdp to rgdp growth rate represents the rate of inflation or gdp deflator. emphasize: the gdp deflator is a product of pv i. the more growth of pv i index, the lower the gdp deflator. therefore, the more stable, seems to be evolving economy. and vice versa. the lower the advances in systems science and application(2015) vol.15 no.1 7 table 2 the dynamics of the purchasing power of national currencies of the world for the period 2000-2010 (2000 = 1) country/year 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 1 austria 1,00 0,92 0,74 0,65 0,60 0,58 0,52 0,46 0,49 0,39 2 belarus 0,92 0,89 0,81 0,68 0,58 0,51 0,45 0,38 0,43 0,39 3 belgium 1,01 0,96 0,79 0,68 0,63 0,59 0,53 0,49 0,55 0,44 4 bulgaria 0,99 0,82 0,62 0,49 0,42 0,38 0,31 0,27 0,29 0,25 5 brazil 1,19 1,31 1,20 0,95 0,71 0,61 0,55 0,50 0,50 0,53 6 united kingdom 1,03 0,95 0,83 0,70 0,66 0,64 0,63 0,60 0,53 0,51 7 hungry 0,98 0,82 0,64 0,53 0,47 0,46 0,38 0,34 0,41 0,30 8 germany 1,03 0,98 0,81 0,72 0,70 0,67 0,61 0,54 0,59 0,47 9 greece 1,00 0,89 0,69 0,59 0,55 0,54 0,48 0,48 0,47 0,45 10 denmark 1,01 0,93 0,76 0,67 0,62 0,60 0,53 0,50 0,50 0,44 11 india 0,97 0,92 0,78 0,67 0,56 0,50 0,40 0,34 0,34 0,35 12 ireland 0,73 0,57 0,43 0,36 0,33 0,29 0,25 0,25 0,26 0,22 13 spain 0,99 0,87 0,68 0,57 0,52 0,47 0,40 0,37 0,41 0,35 14 italy 1,01 0,93 0,75 0,65 0,63 0,59 0,52 0,48 0,52 0,42 15 kazakhstan 1,09 0,99 0,75 0,62 0,49 0,43 0,37 0,46 0,40 0,35 16 cyprus 0,95 0,77 0,51 0,36 0,31 0,26 0,11 0,12 0,12 0,12 17 china 0,91 0,90 0,87 0,81 0,75 0,70 0,66 0,57 0,65 0,66 18 latvia 0,95 0,88 0,71 0,57 0,51 0,39 0,31 0,26 0,22 0,20 19 lithuania 0,92 0,80 0,61 0,51 0,45 0,38 0,31 0,26 0,22 0,22 20 luxemburg 0,99 0,95 0,78 0,60 0,54 0,44 0,35 0,32 0,36 0,30 21 malta 1,11 1,04 0,86 0,75 0,72 0,64 0,56 0,50 0,51 0,52 22 netherlands 1,02 0,94 0,76 0,66 0,63 0,59 0,51 0,45 0,48 0,39 23 poland 0,92 0,89 0,79 0,69 0,57 0,49 0,40 0,31 0,41 0,42 24 portugal 1,01 0,94 0,77 0,66 0,63 0,61 0,50 0,45 0,44 0,44 25 russia 0,89 0,82 0,67 0,53 0,47 0,37 0,30 0,30 0,32 0,28 26 romania 1,24 0,98 0,76 0,55 0,42 0,34 0,24 0,19 0,18 0,18 27 slovakia 1,00 0,86 0,62 0,53 0,47 0,42 0,39 0,35 0,25 0,17 28 slovenia 1,07 0,94 0,74 0,62 0,57 0,51 0,42 0,36 0,39 0,35 29 united states 1,00 1,00 0,97 0,94 0,89 0,86 0,83 0,81 0,85 0,81 30 finland 1,00 0,92 0,74 0,65 0,62 0,60 0,45 0,40 0,43 0,35 31 france 1,02 0,91 0,74 0,64 0,62 0,58 0,51 0,47 0,50 0,40 32 czech republic 0,98 0,81 0,65 0,52 0,46 0,39 0,33 0,29 0,30 0,31 33 sweden 1,11 1,02 0,83 0,70 0,67 0,62 0,54 0,50 0,60 0,47 34 estonia 0,95 0,81 0,63 0,50 0,46 0,39 0,32 0,31 0,27 0,27 35 japan 0,95 1,07 1,14 1,10 0,98 0,97 1,04 1,02 1,10 1,20 36 turkey 1,44 1,11 1,04 0,88 0,78 0,76 0,65 0,58 0,72 0,67 37 iran, islamic republic 0,98 0,82 0,86 0,73 0,62 0,54 0,45 0,48 0,49 0,40 38 afghanistan 1,13 0,75 0,49 0,47 0,45 0,42 0,38 0,26 0,25 0,23 39 pakistan 1,06 1,09 1,01 0,93 0,91 0,84 0,79 0,70 0,79 0,74 40 uzbekistan 1,23 1,53 1,54 1,41 1,29 1,17 0,99 0,88 0,82 0,75 41 turkmenistan 0,87 0,81 0,72 0,68 0,64 0,57 0,52 0,86 0,89 0,91 42 tadzhikistan 0,89 0,86 0,75 0,66 0,64 0,59 0,50 0,39 0,38 0,36 43 azerbaijan 1,08 1,09 0,99 0,90 0,80 0,72 0,63 0,50 0,54 0,57 44 kyrgyzstan 0,97 0,91 0,82 0,76 0,68 0,60 0,48 0,38 0,39 0,39 45 south africa 1,18 1,32 0,92 0,76 0,73 0,75 0,75 0,83 0,80 0,64 8 sailau baizakov and jeffrey yi-lin forrest:the paradoxes of the world’s progress(ii) growth rates of pv i, the greater the rate of inflation, and inflation the scourge of the market economy. comment by valtuh k.k.: inflation index the concept is much broader than that of the gdp deflator. answer : we do not think so. we put the question differently: when earlier economists following a. marshall asked “what determines the price of the goods or services?” and what the rate of inflation, our initiative group asks “what determines the purchasing power of money and what is the level of balanced growth?” comment by valtuh k.k.: this question was discussed long before marshall. see also, in particular, my results in statistical test value theory. question 7. what is the relationship q of a particular product with a q in the fisher’s equation of exchange? do you want to aggregate the value of a specific product j and associate it with the indicators of macroeconomics? answer: in fact, the relationship q of a specific product j and q in the fisher’s equation does not exist. comment by valtuh k.k.: there is no direct connection. but it means in the “fisher’s equation” that the product of this equation is the sum of the prices of specific products. continued answer : as for the aggregation of indicators specific product j and link it to macroeconomic indicators, such problems in our economic management technology does not exist. it can be explained by the fact that we are willing to work with four indicators that are available in the statistical reports and businesses, and sectors of the economy, and in sna issue, the nominal gdp (gdp ) and real gdp (gdp ) deflator and v v p −ngdp/rgdp . aggregation of indicators is the work of statisticians, and the gdp deflator is defined by them alsongdp/rgdp . scientist has to work with analysis, and not the aggregation of indicators. conclusion by valtuh k.k.: i replied on your thoughts. further discussion between us is not necessary:in fact, the cause of our differences lies in the fact that we work in different ways, and not just have different understandings of a particular issue. conclusion by s. baizakov : for the conclusive answers to a system of all questions, i have reinterpreted presented the findings of the study. if to be precisely i introduced the concept of “balanced growth rate”(i3), which allowed me to remove many unclear answers to the questions in that discussion: i3 = pp (t) ∗ngdp (t) = c (t) ∗rgdp (t) (12) table 3 shows the comparative data rates of balanced growth in the european union and neighboring countries and regional organizations in developed countries. advances in systems science and application(2015) vol.15 no.1 9 table 3 the calculation of the rate of balanced economic growth in 2010, 2000 = 100 a coefficient of scientificreal growth balance nominal purchasing countries, technological rate growth rate growth rate power parity regions progress (stp) (i2 index) (i3 index) (i1 index) (ppp) 4=(2)*(3) 1 2 3 =(5)*(6) 5 6 u27 98,50 100,8 99,34 239,08 0,42 u15 97,13 100,1 97,21 233,39 0,42 u9 96,15 100,1 96,28 226,66 0,42 eec 99,83 165,0 164,74 566,25 0,29 bric 126,22 183,1 231,16 437,77 0,35 35 countries 105,46 121,8 128,43 200,97 0,64 eco 108,02 179,3 193,63 331,54 0,58 united states 103,70 114,5 118,74 146,22 0,81 japan 109,35 129,1 141,20 117,40 1,20 united kingdom 95,85 104,5 100,12 197,78 0,51 germany 94,51 109,3 103,28 219,53 0,47 france 97,55 101,2 98,75 245,13 0,40 as can be seen from the calculation formula of balanced growth rate and results of calculations based on this formula (table 3), the changes in coefficient in real gdp leads to changes of coefficient in nominal gdp . the source of any changes in rates of economic growth is scientific technological progress, in particular changing of its coefficient c(t). the overall conclusion of our discussions with valtuh k.k. valtuh k.k. believes that “in fact, the cause of our differences lies in the fact that we work in different ways, and not just have different understandings of this or that particular issue”. if so, then the k. valtuh’s desire to stop further discussion of the current issues in the world economy, i consider premature. my challenge is based on a k. valtuh’s study on value theory [2]. the results of this study are shown in section 4. 4 the mathematical substantiation of necessity of determining the rate of balanced growth 4.1 econometric models are meaningful if they relate to the economic laws of the market the modern theory of value is based, in addition to the works of marx and marshall on the major achievements of the twentieth century in the development of economic science. in this direction of economic theory a lot of work was done by novosibirsk school of economists led by corr. ras k. valtuh. k. valtuh said 10 sailau baizakov and jeffrey yi-lin forrest:the paradoxes of the world’s progress(ii) the current prejudice against labor theory of value extremely negative affects the state of economic science, including formulated of practical recommendations on behalf [2,3]. more specifically, it is recommended to give a general theory of the cost a place in economic science, which it in its content deserves. theory of value responds well known concept of the theory in general meets the criteria of external justification and internal perfection. first of all, it is justified by the classical law of value in its findings related to pricing, which in measuring the outcome of the economy has not yet taken its rightful place. as we know, in modern economic science the difference between cash and trade flows is generally measured by the gdp deflator. according to these scientific and theoretical considerations, the official statistics determines it by the ratio (index) of the amount of the final internal current prices of products manufactured of a certain the country to amount of the same output as measured in the prices of some base year. subsequently, the gdp deflator is identically defined as function (the product) of the three variables identified in the labor theory of value: (1) reciprocal relationship of payroll to gdp , measured in current prices, (2) average payment of unit of labor and (3) direct labor intensity of gdp , measured in base year prices. accordingly, the dynamics of the gdp deflator is identically determined by the dynamics of these three variables (for example, the growth rate of the deflator for a certain year the growth rate of these quantities for the same year). however, as a result of this expansion, we have only the identity [2]. so to express deflator in motion, as a model of the gdp deflator valtuh has taken advantage of econometric analysis methods. but, in our opinion, he is trapped in a simple modeling technique and missed the point of research of the economic content of the gdp deflator, before engaging its modeling. valtuh just wrote “in modern economics the difference between cash and trade flows is generally measured by the gdp deflator” [ibid.]. further, he was in the wrong chain of reasoning, taking at face value the econometric model of the gdp deflator. one of the examples of failed economic modeling of inflation and gdp deflator in kazakhstan we have given above. these econometric models do not take into account the contributions aspirations, successes and risks of the entrepreneurs to the economy. “the first most important of the innate properties of matter karl marx wrote a movement, not only as a mechanical and mathematical movement, but even more as a desire, a life spirit, stress, or, to use the expression of jacob boehme, flour of [gual] matter” [4]. as you know, the driving force, the spirit of the economy is every desire, which is called the scientific and technological improvement of production, in short, scientific and technological progress (stp) in the economy. the level of scientific and technological progress in each moment advances in systems science and application(2015) vol.15 no.1 11 of time locked by the one or more indicators. and the result of this desire can also be described as the contribution of the ntp activities undertaken. in our system of models the result of stp is its ratio c(t), and the level of scientific and technological progress gdp (t)/x(t) : c(t) = gdp (t)/x(t). and the purchasing power of money, which sets the cost of money, therefore, the value of the currency is defined by the formula: pp(t) = c(t)/p(t). here, the gdp deflator is given by official statistics on all types of goods and services, industries and economic activities. its clearance from the influence of scientific and technological progress is precisely defined by the formula (t) = (t)/(t). these indicators of stp contribution and the purchasing power of money successfully complement the gdp deflator (inflation index) for analytical work and predictive calculations of economic development. the advantage of these indicators is to analyze the three indices of economic growth which reveal the essence of the true value and balanced economic growth rate (fig.2). usa 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 ngdp 100 103 107 112 119 127 134 141 144 143 146 rgdp 100 101 103 106 110 112 115 118 118 115 118 sgdp 100,0 108,3 101,4 119,2 128,3 137,4 146,6 157,1 168,1 176,0 193,6 japan ngdp 100 88 84 91 99 98 94 94 105 108 117 rgdp 100 100 99 100 105 102 101 105 105 115 126 sgdp 100,0 83,4 89,7 104,1 108,7 96,0 91,1 97,5 106,7 119,3 141,2 kazakhstan 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 ngdp 100 121 135 169 236 312 443 573 729 630 809 rgdp 100 114 125 136 149 164 181 197 204 206 221 sgdp 100,0 132,5 133,3 126,9 146,2 152,6 192,2 210,2 338,1 252,7 286,9 bric ngdp 100 103 110 127 153 185 229 291 342 360 438 rgdp 100 101 106 113 119 126 137 150 162 169 183 sgdp 100,0 102,3 109,9 118,1 128,6 138,5 150,1 161,8 169,9 156,9 164,7 12 sailau baizakov and jeffrey yi-lin forrest:the paradoxes of the world’s progress(ii) united kingdom 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 ngdp 100 121 135 169 236 312 443 573 729 630 809 rgdp 100 114 125 136 149 164 181 197 204 206 221 sgdp 100,0 132,5 133,3 126,9 146,2 152,6 192,2 210,2 338,1 252,7 286,9 germany ngdp 100 103 110 127 153 185 229 291 342 360 438 rgdp 100 101 106 113 119 126 137 150 162 169 183 sgdp 100,0 102,3 109,9 118,1 128,6 138,5 150,1 161,8 169,9 156,9 164,7 fig.2 example of illustrations of the possible outcomes of the analysis of three levels of balanced economic growth on the fig.2 to illustrate the possible outcomes of the analysis there are the three levels of balanced economic growth. in the first row is the state of the economies of the u.s. and japan in 2000 and 2010. in these countries the rate of balanced growth is higher than the growth rate of nominal (second place) and real growth. on the second row there are the states of the bric and kazakhstan economies for the same years. their balanced growth rate was lower than the nominal, but higher than the actual growth rate. the third row shows the states of the economies of united kingdom and germany. as the chart above shows the development of the economies of these countries in the same years, both, the rate of balanced growth was lower not only advances in systems science and application(2015) vol.15 no.1 13 than the nominal, but the real rate of development. these analytical calculations confirm the findings of the president of kazakhstan nursultan nazarbayev made on the vi astana economic forum that definitely the developed countries caused the recession in some countries of the world[5]. in the practice of predictive calculations of economic development of countries stp contribution is still accepted, mainly exogenously. in the simplest case, non-decreasing function of time is introduced to the production function as a multiplier. these calculations in soviet times allowed us to determine the stp type (capital-intensive or labor-intensive) to take into account the structure of ntp, which is defined by technology used in the production, grouped according to their degree of progressivity [6,7]. but since the end of the twentieth century intensive work was started to integrate the stp function into models of economic analysis. thus, in the collective monograph sops at the academy of sciences of the russian federation, published under the imprint of the academic series “the problems of the soviet economy” in 1988 there were stress that the key “direction of account of stp progress is to use so-called stp functions (for example, the first equation of the model kaldor-mirrlees, the function proposed by baizakov s. in the institute of economic research). here, dependence of the relative indicators of the stp development is accepted as a function. according to kaldor-mirrlees, stp function is defined by dependence of growth of labor productivity on growth of capitallabor. in a model of s. baizakov it is determined by the dependence of growth in capital productivity per unit of labor embodied in the organic composition of production (capital-degree lead over costs to pay for it). in contrast to the first type of production functions, where ntp is autonomous, in these models, the latter a driving force of macroeconomic dynamics. similar analysis of unit of production functions as a method of accounting for ntp in the prediction was given in the works of s. baizakov, v. dadayan, n. kulbovskoy, etc.[8,9]”. indeed, as our russian colleagues point out, the implementation of stp functions in macroeconomic models would allow defining more precisely the most important synthetic indicators of the economy development (gross domestic product growth rate of labor productivity, capital productivity, etc.). if using a macroeconomic model the contribution of stp can be estimated, it would open the possibility for the treatment of price indices of goods and services from the influence of the gdp deflator. however, the analysis carried out by the institute for economic research in recent years has shown that the problem is not so simple. even if the amount of money corresponds to the number of produced commodity supply and circulation process is going well, the choice of indicators of measuring the final result pro14 sailau baizakov and jeffrey yi-lin forrest:the paradoxes of the world’s progress(ii) duction in terms of money is not a simple technical problem. in fact measuring the final product by the value of nominal gdp , real gdp and the gdp deflator is only a necessary condition, but the system of indicators is not sufficient for the detection of defective mechanisms bottlenecks in economic management. indeed, the gdp deflator is one of the most important indicators of economic management. because it is determined by the ratio of nominal gdp this year to real gdp , the gdp deflator itself needs meaningful economic expertise. our conclusion is that before engaging in mechanical treatment the gdp deflator by econometric methods, its economic content should be thoroughly understood. econometrics will give meaningful results when economic models, based on it, meet the aspirations of entrepreneurs. it’s not a secret that the most widely used econometric models and production functions do not fully reflect the contribution of scientific and technological progress (stp ) in the economy. 4.2 limitations of monetary theory of economic growth there is given a new interpretation of monetarist model, based on which the limitations of the gdp deflator (p(t)) and its inadequacy are mathematically proved as a key instrument of financial management: qtxpt = vtxmt (13) here, qt is the product of the base year, which equals the nominal gdp , pt is inflation index, defined by the velocity of circulation of money of the base period vt = v0. for example, if according to the national economy one of the countries had rates qt = $14 366 700 000, mt= $12 460 100 000, and vt =v0 = q0 /m0 = 9 817 000/7 173 800=1,36845, then pt = vt x mt /qt =1,36845 x 12 460 100 / 14 366 700 = 1,187 or 118,7%. as can be seen from these calculations, they do not take into account change in counter of technological excellence of production c = µ/(1 + µ), where µ = gdp/qp , qp is a product of intermediate consumption in the system of national accounts or the cost of current material expenses in the amount of salesc = qp +gdp . increasing dynamics of indicator µ(t) by m. porter is an increase in productivity of current material costs with the cost of qp , used in the production of gdp , and their sum is constant at any given point in time -x = qp+gdp = const. from the standpoint of environmental protection and green economy indicator µ(t) is the most important criterion for competitiveness, as well as the gdp/qp at x = const means a maximum of gdp production at current low cost of energy and use of raw materials and other natural resources. in turn, the monetarists formula qtxpt= vtxmt at vt = v0 is represented as ytxp2,t =v0xmt, where yt = qtxpt nominal gdp , and p2,t = v0/vt money inflation index. now let’s consider the ratio of the data of values of the current advances in systems science and application(2015) vol.15 no.1 15 year t to cost in the base period of time: qt × pt q0 × p0 = vt ×mt v0 ×m0 (14) or yt y0 = vt ×mt v0 ×m0 (15) in another way, it appears as a product of growth of indicators q and m, as well as the growth rate of the control indicators p and v: qt q0 × pt p0 = vt v0 × mt m0 (16) or yt y0 = vt v0 × mt m0 (17) so, the product of the growth rate of real and gdp deflator is equal to the product of the growth rate of money turnover (= index value of one unit of money in relation to nominal gdp ) and growth rate of the money supply. or, in other words, we can say: the growth rate of nominal gdp is equal to growth rate of the product of the circulation of money multiplied by the money growth rate. hence the price index is equal to one unit of money: vt v0 = qt/q0 mt/m0 × pt p0 (18) or vt v0 = yt/y0 mt/m0 (19) that is, the growth index of the price of one unit of money equals to the ratio of real gdp growth and money supply multiplied by the gdp deflator. or, growth index of value of one unit of money equals to the ratio of growth rates of nominal gdp and the money supply. if the prices of goods and services for final use did not change, the rate of growth of the price of one unit of money would be determined only by the ratio of real gdp growth in the money supply. since the price index for goods and services in real-life is greater than one, then multiplying this ratio to the gdp deflator leads to an increase in index of the price of money. the real index of the “price” of money expressed by physical benchmark is the ratio of the index of price of money received by the gdp deflator and in other words is the ratio of the rate of real gdp to growth rate of the money supply: vt v0 / pt p0 = qt/q0 mt/m0 (20) 16 sailau baizakov and jeffrey yi-lin forrest:the paradoxes of the world’s progress(ii) equation (4) shows how the physical contents of one unit of money changes when the real volumes of end-products and the money supply change. the reciprocal of the growth rate of money turnover represents an index of monetary inflation (relative to nominal gdp ): v0 vt = mt/m0 qt/q0 × p0 pt (21) or v0 vt = mt/m0 yt/y0 (22) if the prices of goods and services have not changed, then the monetary inflation would be determined by the ratio of the rate of growth of the money supply to growth rate of real gdp . and because the prices of goods and services change, the monetary inflation is determined by the ratio of the rate of growth of the money supply and nominal gdp . let us call the ratio of the rate of growth of the money supply to growth rate of real gdp in the base year price as index of real monetary inflation. it will be equal to the product of monetary inflation index for inflation in prices of goods and services for final use: mt/m0 qt/q0 = v0 vt × pt p0 (23) hence it is not difficult to get a formula to determine the nature of the gdp deflator, familiar to us from the official statistics: pt p0 = mt/m0 qt/q0 . vt v0 = yt/y0 qt/q0 = i1/i2 (24) wherei1 = yt/y0 is the rate of economic growth in the current year prices, andi2 = qt/q0 is the rate of economic growth in base year prices or the same as index of physical volume. as can be seen from (a1), the gdp deflator expresses the ratio of the rate of economic growth in the prices of the current year to the rate of growth in the prices of the base year. it does not take into account possible changes in the cost of the currency. as a result, the index of the gdp deflator is the ratio of a fixed state of the economy in two different time points, which does not react to changes in a rapidly changing reality, above all, to changes in the purchasing power of the national currency. 4.3 limitations of the duality theory of economic growth clarifying the roles of the gdp deflator is possible through the use of the principle of duality of kantorovich-koopmans, which is written as: qtxpt = ctxxt (25) advances in systems science and application(2015) vol.15 no.1 17 here, as noted above, indicator c represents the efficiency of the trade or expresses the contribution of scientific and technological progress in the real economy. at ct = c0 it is represented asytxp1,t = c0xxt, where yt = qtxpt is nominal gdp , and p1,t = c0/ct is the index of counter of technological excellence of production. now consider the formulas analogous to 1-6 with a contribution of scientific and technological progress in the real economy. first, we consider the ratio of the data values of the current year t to cost in the base period of time: qt × pt q0 × p0 = ct ×xt c0 ×x0 (26) or yt y0 = ct ×xt c0 ×x0 (27) in another way, it appears as a product of growth rates of indicators q and x, as well as the growth rate of the control indicators p and c: qt q0 × pt p0 = ct c0 × xt x0 (28) or yt y0 = ct c0 × xt x0 (29) that is the product of the growth rate of real gdp to the gdp deflator is equal to the product of the rate of counter of technological excellence of production of output of goods and services (including products for intermediate consumption). the term “technological improvements” for the first time in the economic cycle is entered, apparently, by alfred marshall. or, in other words, we can say: the rate of growth of nominal gdp is equal to the product of rate of counter of technological excellence of production to the rate of production of goods and services (including products for intermediate consumption). hence, index of counter of technological excellence of production (stp coefficient) equals: ct c0 = qt/q0 xt/x0 × pt p0 (30) or ct c0 = yt/y0 xt/x0 (31) that is, index of counter of technological excellence of production (stp coefficient) equals to the ratio of the rate of growth of real gdp and gross output multiplied by the gdp deflator. 18 sailau baizakov and jeffrey yi-lin forrest:the paradoxes of the world’s progress(ii) or, index of counter of technological excellence of production (stp coefficient) equals to the ratio of nominal gdp growth rate and gross output. if the prices of goods and services for final use did not change, then stp coefficient would be determined only by the ratio of real gdp growth to gross output. since the price index for goods and services in real-life is greater than one, then multiplying this ratio to the gdp deflator leads to an increase in the rate of counter of technological excellence of production. real growth rate of stp coefficient expressed by physical benchmark is the ratio of the resulting index of growth rate of stp coefficient for the gdp deflator and in other way is the ratio of real gdp rate to gross output: ct c0 / pt p0 = qt/q0 xt/x0 (32) equation (10) shows how the physical contents of one unit of output (one unit of money needed to trade x) changes when the real volume of the final product and gross output changes. the reciprocal of the stp coefficient represents the index of excess of the gross output relative to nominal gdp : c0 ct = xt/x0 qt/q0 × p0 pt (33) or c0 ct = xt/x0 yt/y0 (34) if the prices of goods and services have not changed, then the index of exceeding the gross output of the nominal gdp would be determined by the ratio of the rate of growth of gross output to growth rate of real gdp. and because the prices of goods and services change, the index of exceeding the gross output relative to the nominal gdp determined by the ratio of growth rate of gross output and nominal gdp. let us call the ratio of the rate of growth of gross output to growth rate of gdp as real index of exceeding the gross output relating to the nominal gdp. it will be equal to the product of the index of exceeding the gross output and the nominal gdp deflator: xt/x0 qt/q0 = c0 ct × pt p0 (35) hence it is not difficult to obtain a new specification of the formula of the gdp deflator, which reflects the changes in the real economy, and the changes in the monetary and financial system: pt p0 = ct/c0 ppt/pp0 (36) advances in systems science and application(2015) vol.15 no.1 19 wherept/pp0 = qt/q0 xt/x0 means the rate of change of purchasing power of money, andpt and ct have the same values. as can be seen from (a2), the gdp deflator saves its money status and its quantitative value determined by the model of monetarism. but its economic nature has now become meaningful and richer. according to its new formula, the nature of the gdp deflator revealed the unity of the two mutually independent indicators of economic management. one of them represents the ratio of scientific and technological risk of entrepreneurs’ work in the real sector, and the other expresses the purchasing power of money in the monetary and financial system. one of them expresses the performance of the real sector; the other expresses the quality of the financial sector of the economy. thus, the “crossing” of keynesian and monetarists’ theory based on the duality theory of kantorovich-koopmans has been successful. 4.4 balanced economic growth as the key to the management of the market economy due to the formula (a2), the rate of balanced economic growth (growth index i3(t)) is defined as the product of the purchasing power of money and the growth rate of nominal gdp (growth index i1(t)): i3(t) = pp(t) ∗ i1(t) (37) on the other hand, the same rate of balanced economic growth (growth index i3(t)) is determined by the product of the ratio of stp coefficient and the real growth rate of gdp (growth index i2(t)): i3(t) = c(t) ∗ i2(t) (38) the equality of the given rate of nominal gdp growth pp(t) ∗ i1(t) with the given rate of real gdp growth c(t) ∗ i2(t) means that any point on the path of balanced economic growth t = t0 is an equilibrium point in prices of goods and services to the purchasing power of the national currency: pp(t) ∗ i1(t) =c(t) ∗ i2(t) (39) thus, the identification of “explosive feature” of the gdp deflator has allowed the initiative group of kazakhstan to present it by two conjugated indicators of scientific and technological progress of the real sector of the economy and the purchasing power of the currency. indicator of scientific and technological progress is a barometer of management of the real sector of the economy. managing the dynamics of change is a function of private sector entrepreneurs, and government agencies that monitor the development of the natural monopolies. indicator of the purchasing power of money is the barometer of management 20 sailau baizakov and jeffrey yi-lin forrest:the paradoxes of the world’s progress(ii) of the financial sector of the economy. managing the dynamics of change is a function of the representatives of monetary and financial system, and of the national bank, which oversees the activities of banks. the initiative group of economists of kazakhstan believes that the speech of the president of kazakhstan nursultan nazarbayev at the vi astana economic forum has given a new impulse to the study of sustainable development issues and served as a key to unlocking the essence of the gdp deflator. thus it opened the way to the justification of the rate of balanced growth as the criterion of economic governance of the world economy. references [1] m. ratner and d. ratner. (2004), “nanotechnology: a simple explanation of the next brilliant idea, translated from english”, williams, moscow, pp.240. [2] v. l. makarov. (1985), “on the performance of scientific and technical progress”, economics and math. methods, vol xxi, 2nd ed. [3] (2005), “theory of value: statistical verification. informational generalization. actual conclusions” ,herald of the ras, no.9. [4] k. k. valtuh. (1965), “public utility of product and labor costs of its production”, m:thought. [5] n. nazarbayev. (2013), speech on th vi astana economic forum. [6] k. marx and f. engels, op. t. 2. pp.142. [7] v.s. dadayan. (1973), “modeling of national economic processes”, economics, moscow, pp.433-438. [8] v. l. makarov and a. p. torzhevsky. (1986), “effect of changes in the technological level of production on macro indicators of economic development”, cemi as ussr, moscow, pp.4-7. [9] b.m. shtulberg, e.g. chistyakov and v.v kotilko etc. (1988), “problems and methods of study of the territorial plans”, science, moscow, pp.192. corresponding author sailau baizakov can be contacted at: baizakov37@mail.ru advances in systems science and applications (2012) vol.12 no.2 141-152 rotor, bearing and dynamic equations in energy storage flywheels for vehicles jinguang zhang1 and yefa hu1,2 1school of mechanical and electronic engineering,wuhan university of technology, wuhan, 430070,china 2hubei digital manufacturing key laboratory wuhan,university of technology, wuhan, 430070,china abstract energy storage systems for vehicles present significant challenges for rotor and bearing design. this paper discusses rotor and bearing design technology in energy storage flywheels for vehicles, with particular emphasis on orientation of flywheel rotors, rotor geometry and magnetic bearings. material, rotational speed and geometry are mainly factors of flywheel rotor design. in order to achieve an attractive specific energy, the rotor speed should be as high as possible. the bearings must be capable of extremely high speed, have very low friction, and be stiff to adequately constrain the rotor, have long life, and have high load capacity. these requirements frequently lead to choosing magnetic bearings for flywheel energy storage system applications. a flywheel energy storage system prototype with active magnetic bearings was designed. the flywheel was suspended by the permanent magnetic bearings and stabilized by the active magnetic bearing. finally, we deduce differential equations of the magnetic suspended flywheel for the design of control system. keywords flywheel, energy storage, active magnetic bearing, vehicle, dynamic equations 1 introduction traditionally, the energy storage requirement for vehicles has been satisfied by chemical batteries. however, batteries have a number of disadvantages such as limited cycle life, maintenance, conditioning requirements, and modest power densities which have been improved upon by newer technologies such as energy storage flywheels. flywheels in particular offer very high reliability and cycle life without degradation, reduced ambient temperature concerns, and construction free of environmentally harmful materials. the energy storage flywheel system mainly consists of flywheel rotor, motor/generator, magnetic bearings, housing and power transformation electronic system. one of the major advantages of flywheels is the ability to handle high power levels. this is a desirable quality in e.g. a vehicle, where a large peak power is necessary during acceleration and, if electrical breaks are used, a large amount of power is generated for a short while when breaking, which implies a more efficient use of energy, resulting in lower fuel consumption. individual 142 jinguang zhang:rotor,bearing and dynamic equations in energy storage flywheels for vehicles flywheels are capable of storing up to 500mj and peak power ranges from kilowatts to gigawatts,with the higher powers aimed at pulsed power applications. the flywheel energy storage packed in vehicle can operate at a nearly constant, optimum speed, reducing fuel consumption, air and noise pollution, and engine maintenance requirements, and extending engine life. short bursts of power, for climbing hills and acceleration, are taken from flywheel energy storage, which is replenished directly by the engine or by regenerative braking when the vehicle is slowed down. unlike friction brakes, which turn kinetic energy into waste heat, regenerative braking changes it to speed up the flywheel for subsequent acceleration[1-4]. the university of texas, center for electromechanics, developed and tested a high speed composite flywheel for an advanced technology transit bus. this flywheel operated at 40,000 r/min and could deliver 844 wh at a power rating of 150 kw. road testing of the bus revealed acceleration time to 75km/h was reduced by a factor of two with a simultaneous reduction in engine power of 25%,the design of a high performance flywheel energy storage system for use in a vehicle poses many challenges. in order to achieve an attractive specific energy (kwh/kg), it is necessary to construct the rotor from materials with high specific strength (ultimate stress/density), leading to selection of composite materials employing graphite fibers over metals. this allows a higher specific energy, and increases rotor tip speed. the high tip speed, in turn, leads to an enormous increase in parasitic windage loss. to reduce windage losses to acceptable levels requires spinning the rotor in a very tight vacuum[5-6]. this paper describes issues associated with rotor and bearing design technology in energy storage flywheels for vehicles, with particular emphasis on orientation of flywheel rotors, rotor geometries , magnetic bearings and differential equations of the magnetic suspended flywheel. 2 orientation of flywheel rotors the interaction between vehicle and flywheel dynamics produces many sources of bearing loads not present in a stationary application. the primary contributors to bearing loads are shown to be vehicle shock, vibration, maneuvering, and gyrodynamics[6]. among these loads, gyroscopic loads occur when the rotor is precessed as the vehicle angular velocity changes, for instance cornering, driving over a hill or through a dip. there is no vehicle axis in which angular velocity is avoided so the bearings must withstand the gyroscopic loads. a spinning flywheel has a relatively large angular momentum so changing its spin axis requires significant torque, which must be produced by the bearings. the required torque is: m = jθ̂ × ω (1) advances in systems science and applications (2012) vol.12 no.2 143 where j is the polar moment of inertia of flywheel rotor, is the spin speed of flywheel rotor, and θ̂ is the turning rate of the flywheel spin axis. on the basis fig.1 coordinate system of the coordinate system defined in fig.1, (1) can be expressed as follow: mx = j(θ̇yωz − θ̇zωy) my = j(θ̇zωx − θ̇xωz) mz = j(θ̇xωy − θ̇yωx) (2) in vehicle operating conditions,yaw rate ωz is greater than roll rate ωx and pitch rate ωy .if the flywheel is oriented vertically, flywheel rotor and the yawing (turning) axis of the vehicle in the same direction, and θ̇x = θ̇y = 0 ,so mx = −jθ̇zωy my = jθ̇zωx mz = 0 (3) this means that the flywheel spin axis should be vertical. in the case of a vehicle, the gyroscopic torque is too small to influence the motion of the vehicle. a way to reduce the impact is to employ two similar flywheels, each contra-rotating at the same speed. however, the torque is large enough to be a major contributor to loads on the radial bearings. in practice some mechanism or device must support the flywheel, isolating the flywheel from the motions of the bus and decrease the loads on the bearings, and allowing it to pitch and roll as freely as possible relative to the vehicle. the usual method to rigidly mounting the flywheel housing to the vehicle is to mount it in a two-axis gimbal. without the use of a gimbal, bearing loads induced by gyroscopic reactions to pitch and roll movements of the bus can significantly reduce bearing life and make the design impractical for long term use. 144 jinguang zhang:rotor,bearing and dynamic equations in energy storage flywheels for vehicles 3 rotor design in rotor design, there are mainly three fully-coupled design factors that have significant effect in the overall performance of flywheels, material strength, rotational speed and rotor geometry. the kinetic energy stored in a flywheel is proportional to the mass and to the square of its angular velocity. it is given as ek = 1 2 iω2 (4) where i is the mass moment of inertia and ω is the angular velocity. the moment of inertia for any object is a function of its shape and mass. it is obtained by the mass and geometry of the flywheel and given as, i = ∫ x2dmx (5) where x is the distance from rotational axis to the differential mass dmx . the way to increase the energy density and minimise the volume of the system is to use a flywheel with both a high rotational speed and a large inertia. the speed limit is set by the tensile strength of the flywheel material. the stored energy density with respect to mass is given by: em = kσ/ρ (6) where em is kinetic energy per unit mass, k is the shape-factor which relates the relative energy stored in the solid disk to that of a constant stress disk of infinite radius, σ is maximum stress in the flywheel and ρ is mass density. in case of planar stress, if the height of the disk is small compared with the diameter, and a homogenous isotropic material with poisson ratio of 0.3, i.e. steel, is used, the k factors are given in table 1[7] . in a three-dimensional flywheel there will be three-dimensional interaction of material stresses. for the flywheel design is based on a hollow cylinder and the outside radius is assumed to be large compared to the flywheel thickness, the two stresses of primary concern are the radial stress and the tangential stress. table 2 presents characteristics for common rotor materials. there are two basic classes of flywheels based on the material in the rotor. the first class uses a rotor made up of an advanced composite material such as carbon-fiber or graphite. these materials have very high strength to weight ratios, which give flywheels the potential of having high specific energy. the second class of flywheel uses steel as the main structural material in the rotor. the highest tensile flywheels are not made of steel, but of fiber-reinforced composites. as well as rotating faster and storing more energy than steel flywheels, advances in systems science and applications (2012) vol.12 no.2 145 these composite flywheels are much safer if the maximum safe speed is exceeded, since they tend to delaminate and disintegrate gradually from the outer circumference rather than explode catastrophically. the material at the outside diameter of the rotor is most effective in storing energy with energy storage of that material being proportional to square of the radius. the peripheral speed of the rotor should be as high as possible for maximum energy storage but this is limited by the stress levels the designer is willing to accept. for a chosen tip speed and rotor outer diameter, the rotational speed of the rotor is fixed and this also fixes the shape. there are three flywheel geometries were developed to meet the energy storage and power requirement needs for vehicles, shown in fig.2. table 1 shape-factor k for different planar stress geometries table 2 2 data for different rotor materials 146 jinguang zhang:rotor,bearing and dynamic equations in energy storage flywheels for vehicles fig.2 flywheel geometry disc flywheel is based on more traditional flywheel designs. the mass of the steel hub provides the main source of inertial energy storage in the system. the profile of the steel hub is based on modified equations for a constant stress disc which include a rim section for an increased radius of gyration. as long as the rotor speed is within acceptable limits and there are no rotor dynamics issues, a long tubular flywheel rotor is the preferred option for a number of reasons as opposed to a disc shaped flywheel. the flywheel designs were denoted by the name “powerbeams” for practical commercial application[8-12]. an arbor incorporates an inside out topography for the motor-generator. the inside-out topography makes better use of the available space and increases the specific energy and power densities of the design. the arbor flywheel shows significant advantages with respect to system size and energy storage capability[5]. 4 bearing design , prototype and dynamic equations 4.1 bearing design the spinning flywheel rotor must be supported on bearings. initially, both mechanical bearings and magnetic bearings were considered. the important parameters in assessing the use of bearings are weight, loss, cost, lifecycle life, and low losses. they also can isolate rotor and stiffness. if the rotor speed is within acceptable limit, mechanical bearings are ideal in that they can operate with low losses and have high life for the average load yet can accept high loads on an intermittent basis several times the average load. due to the high friction and short life, mechanical bearings cannot be adapted to modern high-speed flywheels. mechanical bearings have benefited greatly from material advances such as ceramics and very hard steels. the main life issues are not material fatigue life, but rather lubricant life. lubricant life depends primarily on temperature. instead magnetic bearing system is utilized, including permanent bearings, active magnetic bearings (electromagnetic bearings) and high temperature superconducting (hts) bearings. magnetic bearings do not have any contact with the shaft, has no moving parts, experience little wear and require no lubrication. active magnetic bearings present major advantages in terms of lifetime and advances in systems science and applications (2012) vol.12 no.2 147 rotational speed, and also favorably integrate into high-speed flywheel systems. unlike active magnetic bearings, the hts magnetic bearing can situate the flywheel automatically without need of electricity or positioning control system. however, hts magnets require cryogenic cooling by liquid nitrogen. it is not suitable for vehicles. due to the higher magnetic flux density reached by nd-fe-b magnets and their low cost, applications with permanent magnetic bearings have become attractive, in spite of being very unstable. therefore, permanent magnetic bearings can be used as an auxiliary bearing to reduce the load weight of the rotor and the flywheel and to increase the stiffness of the whole bearing system. 4.2 prototype of flywheel a flywheel energy storage system prototype with an arbor flywheel and a hybrid bearing set is shown in fig.3. fig.3 flywheel geometry according to the above discussion, high speed is desirable since the energy stored is proportional to the square of the speed but only linearly proportional to the mass. magnetic bearings can accommodate very high spin speeds and have theoretically unlimited imbalance induced vibrations. magnetic bearings offer very low friction enabling low internal losses during long-term storage. so, our design of a flywheel system using the magnetic bearing consists of a vertical arbor flywheel rotor, permanent magnetic bearings, and an active magnetic bearing. the flywheel axial stability is actively controlled by the active magnetic bearing 148 jinguang zhang:rotor,bearing and dynamic equations in energy storage flywheels for vehicles while the motions in other directions are restricted by other two pairs of active magnetic bearings. a motor/generator is located in the center region of the arbor flywheel rotor. 4.3 dynamic equations of flywheel active magnetic bearing usually use differential excitation. it uses a pair of symmetrical power amplifier circuit to drive electromagnet in differential model and get a pair of magnetic force in opposite direction. magnetic force of rotor is the difference value between upper magnets and lower magnets. assuming disturbance and control current are very small, in accordance with the taylor series expansion the magnetic force in static working point can be expressed as: fi = kyyi + kiii (7) here, ky = µ0a0n 2i0 2 y30 , ki = µ0a0n 2i0 y20 (8) where fi = total magnetic force, its direction and the y positive direction are consistent ky = displacement rigidity coefficient of magnetic bearing ki = current rigidity coefficient yi = current rigidity coefficient ii = displacement according to the balance position of rotor, its positive direction is upward i0 = offset current y0 = air-gap in balance position µ0 = air magnetic permeability a0 = area of electromagnet pole n = number of coil winding turns equation (7) is the linear model of resultant force in small deviation range.with the increase of distance of balance point, the precision of (7) is decrease. in some limit state, such as rotor contact with stator, strong current (iron-core saturation) or weak current in winding, (7) is incongruity. we consider the base as stationary firstly. inertial reference frames and flywheel reference frames were set up. fig.4 depicts the coordinate system and forced diagram. fi = magnetic force of radial direction fz = magnetic force of radial direction the centroid of flywheel is (xc, yc, zc) .the rotation angles of rotor in yz and xz plane are θx and θy . advances in systems science and applications (2012) vol.12 no.2 149 differential equations of the magnetic suspended flywheel are: mẍc = f1 + f3 mÿc = f2 + f4 mz̈c = fz −mg jθ̈x = −jzωθ̇y + h1f4 − h2f2 jθ̈y = jzωθ̇x + h2f1 − h1f3 t0 = t (8) t0 is motor torque. the above differential equations of the 5-dof magnetic suspended flywheel can decouple into a single dof differential equation and a 4-dof differential equation. the two equations can be separated into axial and radial direction and solved respectively. in this way, control system in axial magnetic bearing can be considered as a single dof magnetic suspension control system. control system in radial magnetic bearing can be considered as a multidof magnetic suspension control system. because θx and θy are small, the displacements may be defined as follows: y1 = xc + h2θy y2 = yc − h2θx y3 = xc − h1θy y4 = yc + h1θx (9) we have q(t) = [xc, yc, θx, θy] ′ i(t) = [i1, i2, i3, i4] ′ carrying (7) and (9) into (8) , we obtain:{ q̈ = p1q̇ + p2q + p3i y = p4q (10) in which p1 =  0 0 0 0 0 0 0 0 0 0 0 −jz j 0 0 jz j 0  p2 =  2ky m 0 0 kyh2−kyh1 m 0 2ky m kyh1−kyh2 m 0 0 kyh1−kyh2 j kyh1 2+kyh2 2 j 0 kyh2−kyh1 j 0 0 kyh1 2+kyh2 2 j  150 jinguang zhang:rotor,bearing and dynamic equations in energy storage flywheels for vehicles p3 =  ki m 0 ki m 0 0 ki m 0 ki m 0 −kih2 j 0 kih1 j kih2 j 0 −kih1 j 0  p4 =  1 0 0 h2 0 1 −h2 0 1 0 0 −h1 0 1 h1 0  we can set z = [ q q̇ ] equation (10) then becames ż = [ q̇ q̈ ] = [ 0 i p2 p1 ] z + [ 0 p3 ] i y = [ p4 0 ] z + [0] i (11) if flywheels is set up on vehiclesthe dynamic equations are given by the expression m(ẍc + ẍcar) = f1 + f3 m(ÿc + ÿcar) = f2 + f4 m(z̈c + z̈car) = fz −mg j(θ̈x + θ̈xcar) = −jzωθ̇y + h1f4 − h2f2 j(θ̈y + θ̈ycar) = jzωθ̇x + h2f1 − h1f3 t0 = t (12) where ẍcar, ÿcarandz̈car are accelerations of different directions of vehicle. θ̈xcarandθ̈ycar are angular acceleration of x and y directions of vehicle. (m+mcar)ẍcar = fxcar mcar ≫ m ẍcar = fxcar mcar and equation (11) then results in mẍc = f1 + f3 − m mcar fxcar mÿc = f2 + f4 − m mcar fycar mz̈c = fz −mg − m mcar fzcar jθ̈x = −jzωθ̇y + h1f4 − h2f2 − j jcar mxcar jθ̈y = jzωθ̇x + h2f1 − h1f3 − j jcar mycar t0 = t (13) advances in systems science and applications (2012) vol.12 no.2 151 here, mcar = mass of vehicle. fxcar, fycar, fzcar = applied forces of of different directions of vehicle, fzcar is connected with road surface profile spectral excitations. the differential equations of the magnetic suspended flywheel can be used for the design of control system. 5 conclusion in this paper, we presented a prototype of flywheel energy storage system for a vehicle. the prototype is viable and can achieve perfect robustness. more improvements in material, magnetic bearings and power electronics make flywheels a competitive choice for vehicle energy storage applications. the use of composite materials, optimized rotor shape and magnetic bearings enable high rotational velocity with power density greater than that of chemical batteries. finally, we deduce differential equations of the magnetic suspended flywheel for the design of control system. acknowledgements this work is funded by the national natural science foundation of china under grant no. 50675163. references [1] paul p acamnley, barrie c mecrow, james s burdess, et al. (1996), “design principles for a flywheel energy store for road vehicles”, in ieee transactions on industry applications, vol.32, no.6, pp.1402-1408. [2] hebner. r, beno. j, and walls. a. (2002), “flywheel batteries come around again”, spectrum, ieee, vol.39, no.4, pp.46-51. [3] c.c.chan. (2002), “the state of the art of electric and hybrid vehicles”, proceedings of the ieee, vol.90, no.2, pp.247-275. [4] j. vanmierlo, p. van den bossche, g.maggetto. (2004), “models of energy sources for ev and hev: fuel cells, batteries, ultracapacitors, flywheels and enginegenerators”, journal of power sources, vol.128, no.1, pp.76-89. [5] c.s hearn, m.m.flynn, m.c.lewis, et al. (2007), “low cost flywheel energy storage for a fuel cell powered transit bus”, vehicle power and propulsion conference, vppc 2007. ieee 9-12, pp.829-836. [6] b.t. murphy, d. a. bresie, and j. h. beno. (1997), “bearing loads in a vehicular flywheel battery”, sae special publications, 1243, feb, 1997, electric 152 jinguang zhang:rotor,bearing and dynamic equations in energy storage flywheels for vehicles and hybrid vehicle design studies, proceedings of the 1997 international congress and exposition, detroit, michigan, usa. [7] björn bolund, hans bernhoff, mats leijon. (2007), “flywheel energy and power storage systems”, renewable and sustainable energy reviews, vol.11, no.2, pp.235-258. [8] pullen k.r, ellis c.w.h. (2006),“ kinetic energy storage for vehicles, hybrid vehicle conference”, iet the institution of engineering and technology, pp.91-108. [9] yefa hu, zude zhou, zhengfeng jiang. (2006), basic theory and application of active magnetic bearing, china machine press. [10] jiqiang wang, feng-xiang wang, ming zong. (2007), “critical speed calculation of magnetic bearing-rotor system for a high speed machine”, proceedings of the csee, vol.27, no.27, pp.94-98. [11] amati n, brusa e, gianolio g. (2000), “effects on the dynamic behaviour of rotors on abm of the electromechanical coupling in induction motors. proceedings of the seventh international symposium on magnetic bearings”, eth zurich, switzerland, zurich: international center of magnetic bearing,eth-zurich, pp.595 -600. [12] kent davey. (2004), “new electromagnetic lift control method for magnetic levitation systems and magnetic bearings”, ieee transactions on magnetics, vol.40, no.3, pp.1617-1624. microsoft word 19 hong xu--group link gear locus modeling based on motion relativity.doc 348-354 advances in systems science and applications (2011), vol.11, no.3-4 issn 1078-6236 international institute for general systems studies, inc group link gear locus modeling based on motion relativity hong xu mechanical engineering department, chengdu electromechanical college, chengdu, 610031, china abstract in order to solve the comprehensive analysis problem of the mechanism link locus and motion, this paper bring forward the parameter calculation of locus based on motion relativity and differential geometry theory. the problem of point-group calculation is thus solved, and the comprehensive analysis of motion and position is realized. moreover, the method is simplified and the calculation accuracy is also improved. in the paper, a moving frame of axes is set up by using the lever axes to describe the complex structure of linkage. motion relativity theory is adopted to study the overall motion rule of the structure. by using the calculation result of the linkage axes, the calculation model is set up to analyze its motion, which simplifies the point-group motion calculation. the point-groups form the same component have the same angle velocity, angle acceleration velocity, base point, and calculation model, which makes the calculation procedure modularized and the program shortened. owning to complete digital analysis, the characteristic parameter calculations of locus are very accurate, the data-base is easy to establish, the transfer and analysis of data become convenient. this also makes it easy to establish the shape chart of the linkage locus and to analyze their resemblance. it’s a useful method to analyze the locus and motion of the link gear comprehensively. keywords link gear motion motion analysing group locus multi-dimensional space 1.introduction the basic task of computer-aided analysis of link gear is to calculate the motion parameter. but it is difficulty because of the diversification of complex trajectory and motion parameter on every point of the link gear. especially the comprehensive analysis on motion and trajectory has been always important and difficult. so it’s very necessary to find a easy theoretical method to calculate and improve efficiency. francois isnard worked over the motion analysis of link gear, and has found out two arithmetic methods. john r m utilized graphics to analyze the characteristics of trajectory and motion with high speed. zahn c t, roskies r z put forward a method to fit linkage curve by fourier series. luokang proposed a easy method to process moving axes. by using two dimensional array to describe and store the coordinates of linkage curve, it has many merits in data processing and calculation, but too complicated. these methods ignored the physical characteristic linkage motion. even the study on the internal rule of motion didn’t utilize the integrated characteristic of the space configuration and the motion to simplify calculation. at present, there is no fundamental method to solve the unified problem on motion and geometry, also no method to establish and analyze linkage motion parameter accurately and swiftly. 2.our approach according to motion relativity and differential geometry theory, this paper proposes a motion trajectory analysis method based on moving axes of link gear. a frame of axes is fixed on link lever which produces a moving frame of axes by using lever axes as coordinate. it can be very convenient to express the complex structure of linkage, and also can take full advantage of the linkage axes calculation to simplify the motion calculation on point groups, thus realize the comprehensive study on linkage motion and geometry trajectory. it’s an efficient method to the project is funded by the sichuan educational commission in the area of nature science (contract no. 2004a163). advances in systems science and applications (2011), vol.11, no.3-4 349 analyze the motion characteristic of link gear... 3.new method for design of link mechanism 3.1.1 the basis problem for trajectory design generally, motion design on plane link gear can be arranged as three basis problems. first, component position, that is, the link gear can lead a certain component pass some preset positions accurately or approximately according to prescribed sequence. second, preset motion rule, that is the driving part and the follower meet some required relationship of positions, including definite quick-return characteristic. or, when the motion of driving part is defined, the follower can move accurately or approximately according to prescribed rule. third, prescribed motion trajectory, some point on the linkage component in plane motion can move accurately or approximately according to prescribed trajectory. the problem can be summarized as: present digital design needs to analyze the motion of some point or a group points on the component, which requires record the data of its trajectory characteristic on different point. the specific task of trajectory motion analysis is to analyze and calculate several adjacent points, define the most accordant articulated points which meet the preset trajectory motion requirements. on the other hand, the motion parameters on nonadjacent points are calculated, thus to establish the database on different trajectory motion characteristic, and prepare for the design. the description of the integrated motion rule of link gear it’s simple and flexible to utilize lever group method to analyze link gear, moreover, easy programming. paper [1] has worked out a general subprogram for second grade group of five basic type, and also several three and four grade lever group in common use. the kinematics analysis system of plane link gear based on fundamental lever group and single lever automatic identification is a general system for plane linkage motional performance analysis, which can analyze the motion of single and multifreedom system in a wide range. as to its mathematic model and solving process, it’s too complicated. normally the results are angle velocity and angle acceleration velocity. this paper works on the mathematic modeling of motional link gear. to simplify the process, the model is a four lever linkage. the links on the articulated points are the object to study the overall motion rule of linkage. fig. 1 shows a plane four lever link gear in xoy right-angle coordinate system. supposing the length of the framework is r1 and the angle θ1 （θ1=0）; the length of the crank is r2 and the angleθ2; the length of the lever is r3 and the angleθ3; the length of the pendulum lever is r4 and the angle θ4. by using vector loop method to establish vector equation of linkage: →→→→ +=+ 4132 rrrr (1) supposing the angle between every lever and x axes is positive anticlockwise, and negative clockwise. the projections of every vector on coordinate system: b c d fig. 1 plane four lever mechanical linkage 350 xu: group link gear locus modeling based on motion relativity 0coscoscoscos 44113322 =−−+ θθθθ rrrr (2) 0sinsinsinsin 44113322 =−−+ θθθθ rrrr (3) 01 =θ tωθ =2 33 ωθ =′ 44 ωθ =′ 33 εθ =′′ 44 εθ =′′ 22441 2244 3 coscos sinsinarctan θθ θθ θ rrr rr −+ − = (4) cb cbaa − −+± = 222 4 arctan2θ (5) 2sinθ−=a 221 cosθ−= rrb 42142 2 3 2 4 2 2 2 )cos(2 1 rrrr rrrr c θ− −++ = 2 433 422 3 )sin( )sin( θ θθ θθ θ −− −− = r r 2 344 322 4 )sin( )sin( θ θθ θθθ ′ −− −− =′ r r )sin( )cos()cos( 343 2 2134 2 23422 3 θθ θθθθθθθ −− ′−−′+−− =′′ r rrcr )sin( )cos()cos( 341 34 2 44 2 3332 2 22 4 θθ θθθθθθθθ − −′−′+−′ =′′ r rrcr for displacement solution of linkage, the key problems are to solve angle, angle velocity and angle acceleration velocity, on the base of which to solve those of one point. when crank angle changes between0~360°,the displacement of that point changes accordingly. if the length of every lever and crank angleθ2 are preset, θ3 and θ4 can be produced by the equation (2) (3). because θ2 is the function of t, so are θ3 and θ4,supposing： )(13 tf=θ )(24 tf=θ differentiate θ3 and θ4,we can get the angle velocity and angle acceleration velocity ω3,ε3,ω4,ε4 from lever and pendulum lever. because the differentiation process is very complicated, so if the result cannot be in full use, it will be a big waste. we use these parameter in point group calculation. 4.motion analysis of moving axes on linkage 4.1.1 trajectory and displacement solution of any point m on link gear when link gear with diversified shape moves in a plane, the description will be very complicated in fixed coordinates. coordinates show in fig. 2 are established on lever, so relative position of any point on lever can be expressed by one group of local coordinate, which won’t x r fig. 2 vector rm point m on plane four lever linkage b c d m r m r r y r x m 1 m 2 m 3 m 4 m 6 m 7 m 5 advances in systems science and applications (2011), vol.11, no.3-4 351 change according to the moving of linkage. no matter how complex of linkage, the size of linkage can be expressed fully and normatively. the absolute position of any point can be expressed by relative coordinate and the origin of local coordinate. according to fig. 2, position vector equation of any point m is: rmm rrr += 2 (6) supposing the angle is positive anticlockwise, the coordinate of point m is: 322 coscos θθ rmm xrx += (7) 322 sinsin θθ rmm yry += (8) among which: xrm,yrm are the coordinates of point m in local axes, xm,ym are the absolute coordinates of point m. when preset a crank angle θ2, the displacements in both x and y direction can be produced on point m with angleθ2 by putting angle θ3,θ4 from equation(4), (5)into them. if preset:θ2=0~360°, step 5°, and make 72 step loop computing, then we can get the trajectory point of point m in this range. these coordinates are more precise than those data produced by graphic chart. differentiate equation(7), (8), the velocity and acceleration velocity on point m can be produced. 4.1.2 trajectory solution of point group on mechanical linkage by trajectory graphic chart, the size of lever and trajectory points can be defined approximately. component size can be determined by using the range of trajectory size, and point position can be chosen by trajectory shape. even lever size has been determined, but calculation and analysis are still necessary for determination whether point trajectory and motion parameter meet requirements. generally speaking, tiny difference on point position can produce huge one on trajectory and motion characteristic. trajectories form different points on lever are shown in fig. 3. the shape of trajectory on different point of lever differs. we can suppose that in xrbyr coordinate system, there be at least one point can meet trajectory requirements. the ideal point may be near the one picked by graphic chart, so we need to choose one group of points nearby for analysis and calculation. those points are all on one component, and share the same angle velocity and angle acceleration velocity ω3,ε3. as before, we take origin b as basing-point, the moving point for calculation. the motion of the moving point can be separated to translational moving of basing-point and rotary moving of moving point. as to moving points group, which have identical angle velocity, angle acceleration velocity and basing-point, the calculation load are no more than those of one point, but the results multiply. l n fig. 3 trajectory from different point on lever xr yr x y 352 xu: group link gear locus modeling based on motion relativity 5.data processing of lever’s moting 5.1.1 the establishing of trajectory data group of point group on link gear the calculation of adjacent point group solved the trajectory character problem of comparison and selection. calculation on different point can solve the moving character problem of analysis and storage for many trajectories. nonadjacent points on lever also have the same angle velocity, angle acceleration velocity and basing-point, the calculation load of point group are no more than those of one point, but the analysis can be very accurate for many kinds of trajectory motion rule. as to the plane link gear shown in fig.1, we can set the number of the lever as 1, other moving levers are 2, 3, … randomly. similarly, all the moving point for calculation can be set as 1,2,3,…,n2 randomly. form a three-dimensional data group grid1(n,q,l), which has n+l line, q+1 column and 2 page. suppose the coordinates of point m on link gear to be (xnm,ynm ) , grid(n,m,0)= xnm and grid(n,m,1)= ynm, according to the simple rule above, we can put all the point coordinates into data group grid. by accessing data group grid, we can get length between p and q on any component l: length=[(xlp-xlq)2+(ylp-ylq)2]1/2 among which, the local coordinates of point p on lever l are xlp=grid (l,p,0),ylq =grid (l,p,1), those of point q are xlq=grid (l,q,0),ylq=grid (l,q,1). 5.1.2 trajectory motion analysis based on local coordinate system by accessing data group grid, we can also calculate motion parameter of point group. the absolute coordinate of any point l equals to its local coordinate plus the absolute coordinate of point b. the calculation shows in (7), (8). as shown in fig.3, the motion of point m is an overlap of two motions (translational moving of basing-point b and rotary moving of point m around b). the velocities of b and m are vb and vm, the rotary velocity of point m relative to b is vmb: ( ) ( )jyixjyixrv bbbbb 22222 ωωωω +=+=×= ( ) jyixjyixrv mbmbmbmbrmmb 3333 ωωωω +=+=×= jyyixxvvv mbbmbbbmbm )()( 3131 ωωωω +++=+= the acceleration velocity of point m: 33 2 2 2 2 εωω τ ×+×+×= ++= rmrm mbmb n bm rrr aaaa the equations above are combination of coordinates, angle velocity and angle acceleration velocity. after the size of linkage and the angle velocity of crank have been given, we can calculate the position, velocity and acceleration velocity of moving point. equally, we can define two-dimensional data group, record position, velocity and acceleration velocity curve on trajectory. 5.1.3 trajectory programming of point group trajectory programming is computer-aided computing. because of the unification and standardization of equation, the programs for one motion parameter are the same model. given crank’s position (angle) and lever’s length, you can calculate trajectory (position) , velocity and acceleration velocity curve by accessing program model. this method can analyze point motion on component precisely. on the other hand, it can process many points together, and store the calculation results as a database, so as to analyze the motion character on different point with digital method. with fourier series proposed by zahn c t, roskies r z to fit lever’s curve, we can fit discrete point into mathematics equation. or, we can pick out the shape character of graphs, and with computer-aided function to analyze trajectory character of plane linkage curve and its distribution rule. advances in systems science and applications (2011), vol.11, no.3-4 353 the calculation process is the same for the same type of plane linkage, just to input different structure parameter and original condition, and you can get the trajectory and displacement. for other types of plane linkage such as crank block, we can deduce its overall motion equation with local axes, that is, the motion rule of moving axes, then the calculation equation of position, velocity and acceleration velocity on the component, then program according to process. the structure is different, but the method is the same. 6.analysis example for linkage motion for four-lever linkage n, imaginary line shown in fig.2 (rectangle with round angle), when move in a plane, we use local axes xrbyr, then the coordinate of any point m can be expressed as （xrmi,yrmi). θ3, ωn ,εn are angle movement of local axes xrbyr. all the points’ local coordinates are in data group grid. basing-point is b, solve position and motion parameters on point m1,m2,……,mn at any moment t: rmi,vmix,vmiy,amix,amiy。 solution: position vector equation at any moment 322 coscos 11 θθ rmnm xrx += 322 sinsin 11 θθ rmnm yry += …… 322 coscos θθ ii rmnm xrx += 322 sinsin θθ ii rmnm yry += velocity vector equation at any moment t: jyyixxvvv bmbbmbbbmm )()( 3131 1111 ωωωω +++=+= …… jyyixxvvv bmbbmbbbmm iiii )()( 3131 ωωωω +++=+= we can see from above, the equation for any point mi is the same as for point m1, so is the program. similarly, the calculation of acceleration velocity also has common calculation equation and program. the basic process to establish linkage curve database is: 1)set linkage basic size r1,r2,r3,r4, if only proportion factor matters for curve character calculation, then suppose r2=1; 2)set position parameter rrmi of point m; 3)use plane linkage kinematics based on basic lever group and single automatic identification to analyze system, and calculate the overall motion rule of component; 4)calculate motion trajectory parameter of point m; 5)solve character parameter t for trajectory graph; 6)store basic parameter p and character parameter t according to certain data group grid1(n,q,l) format. repeat the step above, we can get character parameter database of linkage curve for different linkage and point. 7.conclusion the trajectory parameter calculation method is based on motion relativity and differential geometry theory, which calculates points group motion parameter. it takes full advantage of the simple math relationship between local coordinates and the basing-point and overall motion rule, which makes the equation’s deduction more simple, flexible and systematic, the programming and accessing of subprogram more handy, and accessing of data and analysis of graph more convenient. local coordinate makes it more simple to construct storage model for linkage size information. it makes the best of the middle results for calculation, which reduced repetition in point group trajectory parameter calculation. it’s a very effective resolution method. acknowledgements the project is funded by the sichuan educational commission in the area of nature science (contract no. 2004a163). 354 xu: group link gear locus modeling based on motion relativity references [1] zahn c t,roskies r z. fourier descriptors for plane closed curves. ieee transaction on pattern analysis and machine intelligence. c-21(3) (1992):269~281. [2] francois isnard , gordon dodds. dynamic positioning ofclosed chain robot mechanisms in virtual reality environments. international conference on intelligent robots and systems , victoria , b. c. , canada ,1998. [3] luokang,xiongnan the structure identification of link gear based on connected matrix aggregtion operation, sichuan industry technology college scientific journal, 19 (2) (2000) 35∶ ～37. [4] wang shuang. functional analysis and optimization theory [m]. beijing: beijing aerospace university press, 2004. [5] isnard f , dodds g ,claude vallée. efficient multi arm closed chain dynamics computation for visualisation . international conference on intelligent robots and sys2 tems , grenoble , france ,1997. microsoft word 12 wang huajun, li yamin, zhang yang,huang jing，tang xuan--forming load and metal flow of rotary forging proc 294-300 advances in systems science and applications (2011), vol.11, no.3-4 forming load and metal flow of rotary forging process for spiral bevel gear wang huajun1, li yamin1, zhang yang1,huang jing1,2 and tang xuan1 1.school of material science and engineering, wuhan university of technology, wuhan, 430070, china 2the first aviation academy of chinese air force，xinyang,464000, china abstract rotary forging can overcome the shortcoming of cutting machining and reduces the forming load of the precision forging process. in this paper, the rotary forging process of spiral bevel gear was studied using finite element method, deformation and l-t (load–time) curve at every stage was analyzed. thereby tooth filling phase during rotary forging of spiral bevel gear was obtained. at the same time, the formation of the inside, mid-point and outside at the addendum and dedendum of the convex and concave surface on the blank was also analyzed. as a result, the reason that the outside is difficult to fill was explained. via theoretical calculation and the lead specimen test, the rotary forging process of spiral bevel gear was verified which demonstrated the validity of the rotary forging process of spiral bevel gear. keywords spiral bevel gear, rotary forging, punch, finite element, experiment 1.introduction spiral bevel gear is a core component of the differential，which has the following characteristics: high transmission efficiency, high carrying capacity, driving smoothly and space-saving. nowadays, its manufacturing mainly relies on special gear compound cutting machine[1] due to the complexity shape of spiral bevel gear. comparing with machining, plastic forming can reduce material waste, improve gear strength and service life. however, precision forging of large-diameter spiral bevel gear is very difficult. rotary forging, which is successive partial plastic forming, can decrease the forming load of spiral bevel gear[2] [3]. finite element simulation can be used to verify the validity of the initial design and confirm the new ideas. in recent years, some scholars make use of finite element (fe) methods to study on the plastic forming process of gears. soo-young kim studied on cold forging and heat treatment process for manufacturing of precision helical gear[3]. in order to reduce the forming load, sung-yuen jung studied on the two-step extrusion of helical gear [5]. the rotary forging process of straight bevel gear was studied by cheng peiyuan [6]. j.j.sheu studied on the rotary forging process of spur gear and found the ways to improve die life [7][8]. han xinghui studied the deformation characteristics of cold rotary forging of the ring workpiece[9]. finite element simulation has become an important tool to develop and improve gear plastic forming. because the tooth curve and metal flow of spiral bevel gear are complex，the die often fails from fracture. die failure and difficulty in filling become the key technology of rotary forging process of spiral bevel gear[10][11]. in order to investigate some problems about metal flow, die stress and load change during rotary forging of spiral bevel gear, fe model of rotary forging process is established and rotary forging course of spiral bevel gear is simulated and analyzed in this paper. 2.fe model of rotary forging process for spiral bevel gear taking an automobile differential device spiral bevel gear as the example, the rotary forging process of spiral bevel gear was studied. forging shape of gear components is shown in fig. 1 and the blank shape shown in fig. 2 .the blank material was aisi-1045. advances in systems science and applications (2011), vol.11, no.3-4 295 fig.1.the rotary forging model of spiral bevel gear fig.2.the blank of spiral bevel gear fig. 3 shows the finite element model of the rotary forging process for spiral bevel gear. the blank was fixed on the centre of die and the deformation on inner wall was constraint, so the gear was formed by the punch. the constant friction model was adopted for the contact between punch and blank as well as die and blank, and the friction coefficient was 0.12. fig.3.the fe model of spiral bevel gear for rotary forging 3.analysis of rotary forging load 3.1load-time curve the load-time curve during rotary forging process of spiral bevel gear was shown in fig.4. the figurowed that spiral bevel gear forming process is divided into three phases: free upsetting stage, tooth filling stage, tooth corner filling stage. 1) free upsetting stage: from the beginning up to 1.5sec; because of the existence of the gap between the blank and die, the deformation at this stage is equivalent to free upsetting process. the forming load increases quickly and the maximum load was 1×106n. fig.4.the load –time curve of rotary forging 2) tooth filling stage: from 1.5sec to 4.5sec. some metal close to the die tooth moves into the tooth impression. other metal flows back axially or horizontally due to the obstruction of tooth impression. deformation at this stage is equivalent to forward extrusion with small 296 wang: forming load and metal flow of rotary forging process for spiral bevel gear cone angle; the tooth profile forms gradually. the load increase slowly and the maximum load is 3×106n. 3) tooth corner filling stage: from 4.5sec to 5.6sec; metal overcome friction resistance and move into the tooth corner. the load surge and the maximum load is 7.5×106n. this stage is a decisive stage during forming, for it determines the tonnage of forging equipment, prefabricated blank and precision die shape and so on. 3.2the deformation process of rotary forging fig.5 shows the deformation of blank during the rotary forging process, which is obtained from the simulation. as shown, the blank deformation begins from the outside section near the middle, followed by a steady teeth body filling with a uniform flash at the outer and inner edge as shown in fig.5c. finally, while metal flows into the tooth corner, the flash becomes thinner and larger because the gap between the punch and the die reduces. (a) 0sec (b) 1.5sec (c) 4.5sec (d) 5.6sec fig.5.the deformation process of rotary forging 3.3the distribution of effective strain the distribution of effective strain on the blank at the final stage is shown in fig.11. it can be seen that greatest effective concentrates on the inner and outer lateral flashes. the width of the outer flash is 9.5mm and the inner flash is 8.5mm by measure. effective strain can reflect the forming load approximately. thus, large lateral flash consumes large forming road at the final stage of rotary forging process, which causes the load increase to 7.5×106n sharply. so reduce the flash area at the final forming stage is necessary. fig.11. effective strain of work piece 4.the metal flow 4.1 the overall flow of metal in the forming process, the metal flows along the circumferential direction shown in fig.6. the circumferential size becomes large and the axial size becomes small. at the area which rolling part contacted with punch , both sides metal was squeezed like a wedge then flow circumferentially and gathered to the opposite of the wedge, as shown in fig.6b. one side of the wedge, metal flows towards the die; the other side metal flows towards the punch, as shown in fig.6c. metal being rolled flows complexly and the tooth shape was changed, which advances in systems science and applications (2011), vol.11, no.3-4 297 lead to a deviation between the direction of metal flow and feeding, as shown in fig.6d. (a) cross-section (b) top view (c) side view (d) wedge section fig.6. total velocity 4.2 metal deformation at the addendum (a) the selected points on the formed part (b) the tracking points on the initial blank (c) the tangential strain (d) the effective strain fig.7. deformation of the selected points at the addendum the convex and the concave shapes of a spiral bevel gear are dissimilar. in order to study the deformation of the convex and concave surface at the addendum, some points are selected: p1, p2, p3, the inside, mid-point, outside points at the convex surface, respectively; corresponding to them , p4, p5, p6, the inside, mid-point and outside at the concave surface, as shown in fig.7a. after tracking their position and plastic deformation, the selected points at the addendum of the convex and concave can be found on the cone surface of the blank, as shown in fig.7b, and the points at the outside of the concave and convex produce the largest tangential displacement relatively than others, followed by the mid points, and the inner points are the smallest shown in fig.7c. effective strain of the selected points is shown in fig.7d which consists with tangential strain trend. the outside portion is difficult to fill except increasing the load. 298 wang: forming load and metal flow of rotary forging process for spiral bevel gear 4.3 metal deformation at the dedendum the same method was used to analyze the deformation at the dedendum, some points were selected: p1, p2, p3, the inside, mid-point, outside points at the convex surface, respectively, corresponding to them, p4, p5, p6, the inside, mid-point, outside points at the concave surface, as shown in fig. 8a. fig.8b shows the points on the initial blank, and it can be found that they are all under the cone surface of the blank. as fig.8c shows, the points at the outside of the concave and convex surfaces produce the largest effective strain, relatively than others, followed by the mid points, and the inside points are the smallest. (a)the selected points on the formed part (b)the tracking points on the initial blank (c)the effective strain fig.8.the deformation of the selected points at the dedendum 5.theoretical calculation and forming test 5.1 theoretical calculation the width of the outer flash is 9.5mm by measure and the inner flash is 8.5mm. so the outer radius of the blank is 95mm, and the inner radius is 44mm. radius values are inputted into the formula of contact surface area ratio of circular pieces during rotary forging process [2], and obtained (formula 1）. 1527.0 ) 95 4431.001.1() 3952 5.114.0 3952 5.14.0( )31.001.1)(14.04.0( 1 1 = ×−× °×× ×+ °×× ×= −+= tgtg r rqqλ （1） inputting λ into the calculating load formulation [2], it was obtained: 2 2 6 0.1527 3.14 (95 44 ) 1.9 903 5.83 10 sp f k n λ σ= = × × − × × = × （2） whereλ is the contact surface ratio, f is the contact surface, q is the relative feeding, k is the coefficient of upsetting. load-time curve of rotary forging for spiral bevel gear (fig.4) shows that the stable forming force was about 6×106 n, the theoretical value is 5.83×106n according to formula (2). the forming force based on the fe simulation is close to the theoretical calculation. this fact shows that the fe simulation about the rotary forging process for spiral bevel gear is successful. 5.2 forming test in order to verify the validity of simulation, a test was done using lead as the test material. the blank shape is shown in fig.2, and its volume designed by equal volume method. the forming process during rotary forging was shown in fig.9. it can be found that the tooth filling process and the flash deformation obtained by simulation show good agreement with that advances in systems science and applications (2011), vol.11, no.3-4 299 obtained by the test. tooth is full-profile, which demonstrates that the rotary forging process of a spiral bevel gear is feasible. fig.9.the experiment of tooth filling process 6.conclusion spiral bevel gears have different convex and concave shape, long and spiral teeth, so the rotary forging process of spiral bevel gear is a complicated engineering. in this paper, the rotary forging process of spiral bevel gear was studied using deform-3d and test methods. through the study on the rotary forging process of spiral bevel gear, the deformation and load at every stage and the tooth filling characteristics are obtained. the main results of specific studies are as follows: (1) during the rotary forging process of spiral bevel gear, the course of metal-filled cavity is divided into three stages and at the final stage the flash is large and the load is excessive. (2) the largest strain at the outside points is the reason why it is difficult to fill totally at the outer portion. (3) via theoretical calculation and the lead specimen test, the feasibility and validity of the rotary forging process for spiral bevel gear was verified. acknowledgements the research is supported by the scientific foundation for hubei in china (2007aba214) and the fundamental research funds for the central universities (2010-zy-cl-046). references [1] zeng tao. spiral bevel gear design and processing harbin: harbin institute of technology press,1989 [2] pei xinghua, zhang meng , hu yamin. rotary forging．beijing: machinery industry press, 1991. [3] soo-young kim, satoshi kubota and masahito yamanaka. ‘application of cae in cold forging and heat treatment processes for manufacturing of precision helical gear part.’[j] journal of materials processing technology. 2008,201(1-3) : 25~31. [4] wang g.c, zhao g. q. ‘simulation and analysis of rotary forging a ring workpiece using finite element method’[j]． finite elem. anal. des．2002,38 (12)：1151~1164 [5] sung-yuen jung, myung-chang kang and chul kim. ‘a study on the extrusion by a two-step process for manufacturing helical gear’[j]. the international journal of advanced manufacturing technology. 2008,41: 684~693. [6] cheng peiyuan, hu rui and hua lin. ‘finite element analysis of rotary forging in straight tooth bevel gear’[j]. hot working technology. 2006,35(21) : 65~68. [7] j.j. sheu and c.h.yu. ‘the die failure prediction and prevention of the orbital forging process’[j]. journal of materials processing technology. 2008,201(1-3) : 9~13. [8] j.j.sheu and c.h.yu. the cold orbital ‘forging die and process design of a hollow ring gear part’[c]. proc. of the the 35th international matador conference. taipei (2007) 111~114. [9] han xinghui, hua lin.‘3d fe modeling of cold rotary forging of a ring workpiece’ journal of materials processing technology.2009,209(12-13):5353~5362. 300 wang: forming load and metal flow of rotary forging process for spiral bevel gear [10] m.h.sadeghi，t. a. dean. ‘precision forging straight and helical spur gear’ [j]. journal of materials processing technology,1995,45(1-4) : 25-30.. [11] li yamin，wang huajun，wang xueyang，zhu chundong. ‘die fracture of bevel gear in the rotary forging process’[j].china metal forming equipment & manufacturing technology.2009,89(4):89-92. advances in systems science and application (2015) vol.15 no.3 242-253 image retrieval based on spatial organization of patches yan yu1,2 1hubei province key laboratory of systems science in metallurgical process, wuhan university of science and technology, china 2school of science, wuhan university of science and technology, china abstract this paper presents a novel approach for image retrieval by utilizing the spatial organization of image patches. firstly, each image is presented as a set of patches and each patch is descripted as a bag-of-visual-words (bow) feature. secondly, the matrix of matching costs between all pairs of patches that come from the two images respectively is computed by utilizing both the appearance feature and spatial information. finally, the distance between two images is obtained by minimize the total cost of matching using hungarian algorithm. the resulting distance is applied to image retrieval. the experimental results indicate that the proposed method outperforms the traditional bow method in image retrieval scenario. keywords image retrieval; bag-of-visual-words; image patch; image distance measure; hungarian algorithm. 1 introduction the most popular approaches for content-based image retrieval nowadays are based on the model of bag-of-visual-words (bow). they represent image as an orderless collection of local features and demonstrate impressive performance because of their compactness and invariance. the traditional approach based on bow contains the following steps. firstly, local features are extracted from a training set of images by interesting point sampling or dense sampling. secondly, all the feature vectors are quantized to construct a dictionary of visual words by k-means clustering. each cluster tends to contain image patches with similar appearance and its centroid represents a visual word. thirdly, each image is descripted as a histogram indicating the probability distribution of visual words by hard assignment or soft assignment. lastly, the distances between histograms are used for image retrieval. the major limitation of such approaches is that their image representation ignores encoding the spatial information or geometrical relationships among visual words, which has been discovered very important for understanding image content [1]. for solving the problem, some existing approaches leave the spatial verification as a post-processing step [2-3], however, they are only suitable for partial-duplicate image retrieval, as for semantic retrieval they are limited. spatial pyramid matching [4] is successful for representing the spatial information advances in systems science and application (2015) vol.15 no.3 243 of visual words, which subsquently turn into a general framework combined with various feature encoding methods [5-6] for image classification. for analogous approaches like spm, we uniformly call them patch-based methods. in methods like these, image is divided into several non-overlapping rectangular patches, and the feature histogram of each patches is computed and combined as the image feature. consequently, the distance between images is the weighted sum of distances between corresponding patches locating in the same spatial positions. but we know, the foreground objects of similar images maybe lie in different spatial locations, poses and orientations, which make the distance measure is not accurate in many situations. in this work, we present a more flexible patch-based distance measure method integrating with bow, aiming at increasing the precision of the traditional bow for semantic retrieval. each patch of the first image is assigned to exactly one patch of the second one considering both the bow feature and the spatial location of patches in the image, and the distance measure problem of images is transformed into square assignment problem. our idea is similar to [7] which works in pixel level and only makes use of image color feature. however, our method works in patch level and utilize image local features which have stronger descriptive power. the rest of this paper is organized as follows. section 2 discusses the related work. section 3 introduces the image feature representation as a set of patches. section 4 details the proposed image distance measure. section 5 gives the experimental results on image retrieval application. section 6 concludes the paper. 2 related work there are several recent works in the content-based image retrieval (cbir) area which present interesting advances for encoding spatial information of visual words. in the early days, researchers present color correlogram [8] to encode the spatial arrangement of colors. now the study object is turned into local feature rather than pixel color. the most popular approach to encode the spatial information of visual words is spatial pyramid matching, which involves repeatedly subdividing the image and computing histograms of local features at increasingly fine resolutions. it is succesful for image classification, however, it is not robust for some image transformations such as roation. many of the existing approaches leave the spatial verification as a post-processing step. they find the matching points between images, then compute the spatial representation which is used to identify and remove the false matches, compute the similarity between images by using the number of matched features and a penalty term from spatial verification. ransac exploits full geometric verification [9-10], which can greatly improve retrieval performance, but it is com244 yan yu: image retrieval based on spatial organization of patches putationally expensive. another recent approach explicitly encodes the spatial relationship among each pair of matching points by utilizing binary spatial maps [2]. this method is efficient and effective, but its representation is very specific for partial-duplicate search and it is not suitable for the semantic-search application. another recent approah learns the postions of visual words occurrences in an image dataset, called lp-norm pooling [11]. the method first transforms all the images into the same resolution, and use a dense sampling which generates m points in each image. each visual word w has a md vector, in which the m-th value representis the m-th point activate the visual word w. however, the method is designed for image classification and it represent the absolute postions of visual words, which is not robust to translation or rotation of the object in the image. a recent image content description is proposed for describing the spatial layout of visual words, called ∆−tsr [12]. triangle is used to describe the relationship among arbitrary three interesting points, which is represented as a 7d signature composed of labels of correspoind viusal words, angles of vertices, relative orientations and scales of points. after the strategy of triangle selection is applied, the similarity between images is defined as the ratio of the sum of similairity between the most similar triangles and the cardinality of the set of triangles. this method explicitly encodes the spatial relationship among visual words,but it only works with hard assignment. more recently, graph-based approaches for encoding spatial information are proposed. the nested multi-layered local graphs [13] is proposed by building upon sets of surf points with delaunay triangulation. a bow framework is applied on these graphs, giving birth to a bag-of-graphwords representation, visual dictionary is built for each layer of graphs and l1 distance function is used for image retrieval. this method explicitly encodes the spatial relationship of image points, however it has higher computational cost and only work with hard assignment. in this work, the content representation and similarity measure method of images can work with both hard assignment and soft assignment, furthermoe, it is suitable for semantic-retrieval. 3 proposed method 3.1 image as a set of patches for a given image, the set of patches can be represented as {fi}i=1,...,n, fi ∈ rm (1) where fi is the feature vector in the i-th image patch, n is the total number of image patches and m is the length of the feature vector. the pixel sizes of images in our dataset are various. in order to realize the assignment relation between advances in systems science and application (2015) vol.15 no.3 245 fig.1 original images and their sets of patches. patches of two images, all the images are divided into patches of the same amount, rather than limited to the same size of each patch. for example, figure1 shows that two images are divided into 8 rows and 8 columns respectively. although they have different aspect ratio, they all have a set of 64 patches in total. in our method, we encode the appearance feature for each patch rather than the global image. the feature vector of each patch is extracted via bow model according to the following steps. firstly, the images were all preprocessed into gray scale and local features are extracted from images in the training set by dense sampling. in this work, sift descriptor [14] is adopted as local feature descriptor, because it has averagely the best performance among local feature descriptors [15]. dense sampling is realized by a sliding window in the size of 16∗16 pixels, which sliding from top to down and left to right in an image by a step of 8 pixels. these regions are overlapping for obtaining enough abundant local features. secondly, all the feature vectors are quantized to construct a visual dictionary by standard k-means clustering algorithm [16]. lastly, each patch of each image is descripted as a histogram indicating the probability distribution of visual words by hard assignment. many more effective dictionary construction method such as hierarchical clustering, and more advanced coding schemes, such as soft assignment [17], sparse coding [5], fisher coding [18] et.al., all can be utilized in our method. however, they are not key for evaluating our method, so we adopt the easiest clustering algorithm and coding method. consequently, the signature for representing the content of an image is composed of n patch features. the i-th patch feature is represented as fi = {ai, pi}, where ai codes the appearance feature in the patch, whose dimension is the same as the size of visual dictionary. pi = (xi, yi) is composed of the horizontal and vertical spatial positions of the patch in the 2d plane space. note that xi and yi both are normalized to [0,1] for each patch. 3.2 distance between images the method to measure the distance between images is critical for retrieval precision. in this work, a flexible patch-based distance measure method is proposed. given the i-th patch of the first image u and the j-th patch of the second image 246 yan yu: image retrieval based on spatial organization of patches v, the distance between patches is given by d(fu i , f v j ) = wa · d(au i , a v j ) δa + wp · d(p u i , p v j ) δp (2) where d(au i , a v j ) is the distance between appearance features which can be computed by chi-square distance as (3) d(au i , a v j ) = d∑ k=1 (au i (k)−av j (k)) 2 au i (k) +av j (k) (3) where d is the length of the appearance feature vector,au i (k) represent the k-th component value of the vector. and d(p u i , p v j ) is the distance between the spatial positions of the two patches, then we have d(p u i , p v j ) = √ (xui − xvj ) 2 + (yui − yvj ) (4) parameters wa and wp are weight coefficients, which balance appearance and spatial contributions respectively and the sum of them always is 1. the proportion of them has a great influence on the performance of the method, which will be discussed in the next section. parameters δa and δp are chosen according to the appearance and spatial dynamics. typical values used in this paper are δa = 2 and δp = √ 2. the distance d(fu i , f v j ) between two patches can be seen as matching cost, which is denoted as cij . given the set of costs between all pairs of patches in the first image and in the second image, the minimum of the total matching cost can be seen the distance between two images, subject to the constraint that the matching be one-to one. then the distance between image u and image v is given by d(u, v) = min π n∑ i=1 d(fu i , f v π(i)) (6) where π is a permutation of {1, ..., n} to guarantee the constraint. the input to the assignment problem is a square cost matrix with entries cij . the result is a permutation π such that total matching cost is minimized. this is an instance of the square assignment problem, which can be solved using the hungarian algorithm [19]. given a cost matrix to find an optimal assignment using the hungarian algorithm is presented as follow: 1) subtract the smallest entry in each row from all the entries of its row. 2) subtract the smallest entry in each column from all the entries of its column. 3) draw lines through appropriate rows and columns so that all the zero entries of the cost matrix are covered and the minimum number of such lines is used. advances in systems science and application (2015) vol.15 no.3 247 4) test for optimality: (i) if the minimum number of covering lines is n, an optimal assignment of zeroes is possible and we are finished. (ii) if the minimum number of covering lines is less than n, an optimal assignment of zeroes is not yet possible.in that case, proceed to 5). 5) determine the smallest entry not covered by any line. subtract this entry from each uncovered row, and then add it to each covered column. return to 3). compared with the traditional patch based distance measure methods, our method can deal with the position change of the foreground objects in the image automatically. we should note that the distance that we proposed is not a rigid mathematical distance. however, it dose not matter because the distance can be used to represent the divergence between images and it can be utilized in ranking retrieval results. 4 experiments in this section, we evaluate the performance of our proposed method in image retrieval scenario. 4.1 experimental setups experiments are performed on a personal laptop in windows 8.1, with quad intel core i7-4500u running at 1.8ghz. most of our programs are implemented in matlab except dense sift and hungarian algorithm for which we use open source code in c. our dataset is composed of fifteen scene categories, including bedroom, kitchen, livingroom, store, coast, forest, et.al.. each category contains 200 to 400 images, and average image size is 300×250 pixels. the dataset is popular in image classification and retrieval scenario. fig.2 show examples of the dataset. we use 100 images per class randomly selected for generating visual words dictionary and the rest for testing. we use dictionary size 200 and hard assignment for testing our method. in practical application, the soft assignment can be used in our framework for improving retrieval performance. the experimental results are presented in terms of mean average precision (map) and precision for the top n retrieved images (p@n). although map is a very popular measure to assess the effectiveness of cbir methods, it dose not reflect the ranking quality in the first positions. the practical web users usually pay more attentions to the good set of top 10 or 20 retrieved images, so we are more interested in good p@n values than map values. precision measures the retrieval accuracy. it is defined as the ratio between the number of relevant images retrieved and the total number of images retrieved. relevant images are in the same category with the query example. 248 yan yu: image retrieval based on spatial organization of patches (a)livingroom (b)kitchen (c)bedroom (d)mitcoast (e)industrial (f)mitmountain fig.2 sample images from the dataset, highlighting 6 categories (livingroom, kitchen, bedroom, mitcoast, industrial and mitmountain). table 1 comparison between two kinds of partition patterns based on 10 queries. partition pattern p@5(%) p@10(%) average retrieval time(ms) 4× 4 62.00 63.00 23.64 8× 8 78.00 69.00 185.77 table 2 comparison between two kinds of weighting ratios based on 100 queries. weighting ratio p@5(%) p@10(%) p@15(%) p@20(%) 1:1 67.00 59.10 55.47 52.50 2:1 67.40 59.90 55.80 53.65 4:1 68.60 60.00 57.00 55.15 8:1 66.40 59.80 56.33 54.60 4.2 influence about size of patches in this section, the impact of the patch size is illustrated. we use n to present the total number of patches in each image. we divide each images into several rows and several columns, including two partition patterns which are 4 × 4 and 8 × 8 in our experiments. to compare the two pattern patterns, 10 images are randomly selected from the dataset and each of them is iteratively selected as a advances in systems science and application (2015) vol.15 no.3 249 query example, so the total number of queries in each pattern is 10. it is obvious that the size of each patch is decreasing when the total number of the patches is increasing. the details can be seen in table 1. it can be seen that increasing the total number of the patches has a improvent on the retrieval results, however, the computation become more time-consuming. in consideration of the efficiency, in the rest of the paper patches of n=16 are always used. 4.3 appearance-spatial weighting the weighting parameters wa and wp balance appearance feature and spatial contributions respectively. in this section, the impact of ratio of the weighting parameters is illustrated. to compare the influence of weighting parameters, 100 images are randomly selected from the dataset and each of them is iteratively selected as a query example, so the total number of queries in each experiment is 100. if we wish each patch to be matched with the patch which is close to it in spatial position, wp must be set at a high value. otherwise, if we wish each patch in the first image can be matches with any patch in the second image, wp must be set at a low value. we test the weighting parameters including four kinds of ratio patterns which are 1:1, 2:1, 4:1 and 8:1. the details can be seen in table2. it can be seen that increasing the proportion of wa has a slight improvement on results. but if the ratio keep going up and reach 8:1, the precision begin to decline. in the rest of the paper the ratio 4:1 are always used. 4.4 comparison with the traditional method the traditional bow method is also implemented for comparison. to compare our proposed method with it fairly, we iteratively select each image in the dataset as query example, so the total number of queries is 2985 in each query experiment. in table 3, it can be observed that the precision decreases while the number of retrieved images increases. in our proposed method, the p@n is always better than the traditional bow method. therefore, our proposed method is more suitable for image retrieval. table 3 comparison with the traditional bow based on 2985 queries. methods p@5(%) p@10(%) p@15(%) p@20(%) map(%) traditional bow 57.45 50.67 47.92 46.08 29.28 our method 66.11 58.87 55.30 53.15 32.54 in the end, we give some retrieval examples of the query image random selected from the dataset via the two kinds of methods in fig. 3, fig. 4 and fig.5. the query images come from “calsuburb”, “mittallbuilding”and “mitmountain” 250 yan yu: image retrieval based on spatial organization of patches category respectively. in each fig., the query image lies on the top left corner. retrieved top 12 images are arranged from left to right and top to bottom according to the increasing distances between the retrieved images and the query example. it can be seen that the vast majority of the retrieved images are in the same category with the query image in our proposed method. (a)traditional bow (b)our method fig.3 retrieval example for a query image taken from the “calsuburb” category. (a)traditional bow (b)our method fig.4 retrieval example for a query image taken from the “mittallbuilding” category. 5 conclusions in this paper we present a flexible patch-based distance measure method for images, which utilizes the abundant local feature of images and the spatial information of them. the proposed method is tested in image retrieval application. experiments show better retrieval results than the traditional bow. we use hard assignment for testing our method. in the future, we will test the retrieval performance of integrating the soft assignment with our method. in addition, we only utilize the absolute position of visual words in our method. the relative relationship of visual words is more robust to image transformation, which will advances in systems science and application (2015) vol.15 no.3 251 be our next research work. (a)traditional bow (b)our method fig.5 retrieval example for a query image taken from the “mitmountain” category. acknowledgments this work was supported by hubei province key laboratory of systems science in metallurgical process (wuhan university of science and technology):y201406, and the natural science foundation of hubei province: 22013cfa131. references [1] j. philbin, o. chum, m. isard, et al. (2007), “object retrieval with large vocabularies and fast spatial matching”, ieee conference on computer vision and pattern recognition. [2] w. zhou, y. lu, h. li, et al.(2010), “spatial coding for large scale partialduplicate web image search”, 18th acm international conference on multimedia, pp.511-520. [3] h. jégou, m. douze and c. schmid.(2010), “improving bag-of-features for large scale image search” international journal of computer vision,vol.87,pp.316-336. [4] s. lazebnik, c. schmid and j. ponce. (2006), “beyond bags of features: spatial pyramid matching for recognizing natural scene categories”, ieee conference on computer vision and pattern recognition, pp.2169-2178. [5] j. yang, k. yu, y. gong et al. (2009), “linear spatial pyramid matching using sparse coding for image classification”, ieee conference on computer vision and pattern recognition, pp.1794-1801. 252 yan yu: image retrieval based on spatial organization of patches [6] c. c. yan, l. li, z. wang, et al. (2014), “fusing multi-cues description for partial-duplicate image retrieval” journal of visual communication and image representation, vol.25, no.6, pp. 1726-1731. [7] t. hurtut, y. gousseau and f. schmitt. (2008), “adaptive image retrieval based on the spatial organization of colors”, computer vision and image understanding, vol.112, pp.101-113. [8] j. huang, s. r. kumar, m. mitra, et al. (1997), “image indexing using color correlograms”, ieee conference on computer vision and pattern recognition, pp.762-768. [9] h. e. j. egou, m. douze and c. schmid. (2008), “hamming embedding and weak geometric consistency for large scale image search”, 10th european conference on computer vision, pp.304-317. [10] o. chum, m. perdoch and j. matas. (2009), “geometric min-hashing: finding a (thick) needle in a haystack”, ieee conference on computer vision and pattern recognition, pp.17-24. [11] j. feng, b. ni, q. tian, et al. (2011), “geometric lp-norm feature pooling for image classification”,ieee conference on computer vision and pattern recognition, pp.2609-2704. [12] n. v. hoàng, v. gouet-brunet, m. rukoz, et al. (2010), “ embedding spatial information into image content description for scene retrieval ”, pattern recognition, vol.43, pp.3013-3024. [13] s. karaman, j. benois-pineau, r. megret, et al. (2012), “multi-layer local graph words for object recognition ”, 18th international conference on multimedia modeling, pp.29-39. [14] d. g. lowe. (1999), “object recognition from local scale-invariant features”, 7th ieee international conference on computer vision, pp.1150-1157. [15] k. mikolajczyk and c. schmid. (2005), “a performance evaluation of local descriptors”, ieee transactions on pattern analysis and machine intelligence, vol.27, pp.1615-1630. [16] c. m. bishop. (2006), pattern recognition and machine learning, springer, pp.424-430. [17] j. c. van gemert, j. geusebroek, c. j. veenman, et al. (2008), “kernel codebooks for scene categorization ”, 10th european conference on computer vision, marseille, pp.696-709. advances in systems science and application (2015) vol.15 no.3 253 [18] f. perronnin, j. s a nchez and t. mensink. (2010), “improving the fisher kernel for large-scale image classification”, 11th european conference on computer vision, pp.143-156. [19] c. h. papadimitriou and k. stieglitz. (1982), “combinatorial optimization: algorithms and complexity”, prentice hall. corresponding author yan yu can be contacted at: yuyan wust@163.com advances in systems science and application (2016) vol.16 no.1 19-46 research on extension strategy generation of sustainable utilization of regional water resources qiaoxing li1,2 and naiang wang2 1school of management, guizhou university, guiyang, guizhou province, 550025, pr china; 2college of earth and environmental sciences, lanzhou university, lanzhou, gansu province, 730000, pr china. abstract sustainable utilization of water resources has been a global issue of common concern. under the prerequisites to meet the development need of the region’s population, resources, environment and economy, we should solve simultaneously both water crisis and ecological environment construction, which may be more important for the arid and semi-arid regions. although a lot of results in other aspects of sustainable utilization of water resources have been achieved, it has not been discussed how to generate the strategies to deal with the contradictory problem between water crisis and ecological environment construction. in this paper, extension analysis and extension transformation were utilized to generate a variety of innovative schemes for sustainable utilization of water resources. by using an appropriate evaluation methodology, we chosen one or more excellent schemes and changed them into the specific programs that can be implemented in practice. the strategy generation process can be achieved by computer software, which makes the decision-making be intelligent. the extension generation method provides a formalized and quantitative way for effectively solving water crisis and ecological environment construction, and clarifies the relevant departments ideas. it is also fit for basin water resources. keywords sustainable utilization of water resources; water crisis; ecological environment; utilization strategy; extensible methods 1 introduction as society and economy developed, the demand of water increased. however, the available water provided by the natural world is limit. at present, the water resources problem in the world especially the arid and semi-arid regions is becoming more and more serious, and it mainly shows that water scarcity is increasing and water conflicts are of comprehensive intensification. initially, the water management only focused on some aspects of water utilization and protection such as supply, irrigation, hydropower and others. the traditional management ways of water supply and decentralized sectors ignored the integrity of natural environment and the diversity of water use. therefore, we could not coordinate the various water relations and did not achieve the sustainable utilization of water 20 qiaoxing li and naiang wang babu:research on extension strategy generation ... resources. in order to avoid this deficiency, some scholars have proposed integrated management of water resources which unified water, land and related resources into a jointly development[1-6]. other scholars believed that the fundamental way to ease the water crisis and the eco-environment destruction is to implement sustainable management of water resources[7,8]. they considered the development and utilization of water resources together with the complex system of social economy water resources ecological environment, and inquired the concrete pathway of water sustainable development, which means that we should support the coordinated development among population, resources, environment and economy, and meet the water needs of intrageneration and intergeneration of human being under the promise of maintaining the social continuity and the ecosystem integrity. sustainability is the most rational utilization pattern to integrate the development, utilization, protection, control and management of water resources, and the essence of sustainable utilization is to coordinate water utilization with environmental protection, economic growth and social development. many results of sustainable utilization of water resources have been achieved, such as efficiency evaluation of water use[9-11], water allocation problem[1214], ecological environment impacts[5, 6, 15], analysis for present situation and countermeasures[16], and rules establishment and construction which included effective system of water property, open water market, reasonable emerged mechanism of water price and complete management system of water resources[9-10], and so on. although scholars have made abundant achievements in many aspects of the sustainable utilization of water resources, we are still hard to coordinate the development among population quality, natural resources, living environment and national economy, and can not accurately grasp the quantitative relationship between the water crisis and the construction of ecosystem. to study the strategy generation methods of sustainable utilization of water resources is helpful for solving the relationship between the water crisis and the eco-environment construction in practice, and improving the decision-making capacity of water resources management. it has more practical significance to support the sustainable development of nature, economy and society. the following sections of this paper are below. we introduced some basic knowledge of extenics in section 2, and proposed extension strategy generation step of sustainable utilization of regional water resources in section 3. the extension planning process of sustainable utilization is introduced in section 4, and we give a modified case in section 5. at the end of this paper, a conclusion is given. advances in systems science and application (2016) vol.16 no.1 21 2 preliminaries of extenics extenics is an original interdisciplinary which was put forward by a chinese scholar in 1983. it discusses the rules and methods of extension and innovation of objective things by using the formalized models, which are utilized to solve the contradictory problems. contradictory means that people’s goals can not be reached in current conditions[17]. at present, the contents of extenics include basic-element theory, extension set theory and extension logic. for the sake of simplicity and convenience, we only introduce the knowledge of basic-element theory that will be utilized in this paper. basic-element theory includes extensible analysis theory (eat), conjugate analysis theory (cat) and extension transformation theory (ett). eat includes divergence analysis theory, correlative analysis theory, implication analysis theory and opening-up analysis theory. cat includes nonmaterial and material conjugate analysis, soft and hard conjugate analysis, latent and apparent conjugate analysis as well as negative and positive conjugate analysis. ett includes basic extension transformations, conductive transformation and conjugate transformation, calculation of extension transformations and nature of extension transformations. all of knowledge above will be introduced below and can be seen in [17]. firstly, we introduce the basic concept of basic-element, which includes matterelement, affair-element and relation-element. basic-element is a logical cell of extenics. definition 2.1: an ordered triple m = (om, cm, vm), which is as the fundamental element for matter description composed of matter om, the characteristic cm of omand the value vm of om about cm, is called as one-dimensional matterelement. furthermore, the following array composed of matter om, n-names of characteristics of cm1, cm2,· · · , cmn and the corresponding value vmi of om about cmi (i = 1, 2, · · · , n) m =  om cm1 vm1 cm2 vm2 ... ... cmn vmn  = (om, cm, vm) or m = [ om cm1 · · · cmn vm1 · · · vmn ] , is n-dimensional matter-element, where cm = [cm1, cm2, · · · , cmn] t and vm = [vm1, vm2, · · · , vmn] t . definition 2.2: the ordered triple a = (oa, ca, va,) as the fundamental element for affair description composed of the action oa, the characteristic cm of oa and the value va of oa about ca, is called one-dimensional affair-element. furthermore, the following array composed of action oa, n characteristics of ca1, 22 qiaoxing li and naiang wang babu:research on extension strategy generation ... ca2,· · · , can and the obtained value vai of oa about cmi (i=1,2,· · · ,n) a =  oa ca1 va1 ca2 va2 ... ... can van  = (oa, ca, va) or a = [ oa ca1 · · · can va1 · · · van ] , is n-dimensional affair-element, where ca = [ca1, ca2, · · · , can]t and va = [va1, va2, · · · , van]t . definition 2.3: the n-dimensional array composed of relative or relation symbol (refer to as relation name) or, n characteristics cri of or and the corresponding value vri of or about cri (i=1,2,· · · ,n) which is as the following: r =  or cr1 vr1 cr2 vr2 ... ... crn vrn  = (or, cr, vr) or r = [ or cr1 · · · crn vr1 · · · vrn ] , where cr = [cr1, cr2, · · · , crn]t and vr = [vr1, vr2, · · · , vrn]t , and r describes the relation between vr1 and vr2, is n-dimensional relation-element. because the relation-element r often contains the same characteristics such as antecedent cr1, consequent cr2, degree cr3, maintaining mode cr4, contact channel cr5, contact method cr6, location cr7, and so on, we simply denote it as r (or, vr1, vr2, · · · ). definition 2.4: matter-element, affair-element and relation-element are collectively referred to as basic-element which is expressed as b =  o c1 v1 c2 v2 ... ... cn vn  = (o,c, v ) or b = [ o c1 · · · cn v1 · · · vn ] , where o (object) indicates a certain object (matter, action or relation), and ci is a characteristic of object o, and vi is the corresponding value of o about ci (i=1, 2, · · · , n), and c = [c1, c2, · · · , cn]t and v = [v1, v2, · · · , vn]t . secondly, we introduce the principles of extensible analysis which include divergent, correlative, implication and opening-up ones below: principle 2.1 (principle of divergent analysis): from one basic-element, multiple basicelements with the same object (or characteristic) can be extended, and the set of basic-elements with the same object (or characteristic) must be non-empty. advances in systems science and application (2016) vol.16 no.1 23 inference 2.1: from one basic-element, multiple basic-elements with the same object and value (or with the same characteristic and value, or with the same object and characteristic) can be extended. definition 2.5: given two sets of basic-elements {b1} and {b2}, for any b1 ∈ {b1}, if there is at least one b2 ∈ {b2} to let b1 corresponds to b2 , then {b1} and {b2} are correlative and we denote them as {b1} ∼ {b2}. in particular, as to the sets of basic-elements {b1} and {b2} with c0 as their evaluated characteristic, for any b1 ∈ {b1}, if there is at least one b2 ∈ {b2} to let c0(b2) = f(c0(b1)), then {b1} and {b2} are correlative about the evaluated characteristics c0 and it is denoted as {b1} ∼ c0 {b2}. principle 2.2 (principle of correlative analysis): for a given matterelement m = (om, cm, vm), there is at least one matter-element with the same characteristic mc = (o′ m, cm, cm(o′ m)) or matter-element with the same matter mo = (om, c′m, c′m(om)) or matter-element with different matters m ′ = (o′ m, c′m, c′m(o′ m)), to let m ∼ mc, or m ∼ mo or m ∼ m ′. definition 2.6: suppose b1 and b2 are two basic-elements, and b1 is realized inevitably with the realization of b2, then we call that the basic-element b1 implies the basic-element b2, and it is denoted as b1 ⇒ b2. especially, if b1 ⇒ b2 under the condition l, then b1 ⇒ (l)b2. definition 2.7: let b, b1 and b2 be basic-elements, and (1) if both b1 and b2 are realized inevitably with the realization of b, then the basic-elements b1 and b2 imply the basic-element b, and it is denoted as b1 ∧b2 ⇒ b. (2) if either b1 or b2 is realized inevitably with the realization of b, then the basic-element b1 or b2 implies the basic-element b, and it is denoted as b1 ∨b2 ⇒ b. (3) if b is realized inevitably with the realization of both b1 and b2, then the basic-element b implies the basic-elements b1 and b2, and it is denoted as b ⇒ b1 ∧b2. (4) if b is realized inevitably with the realization of either b1 or b2, then the basic-element b implies the basic-element b1 or b2, and it is denoted as b ⇒ b1 ∨b2. principle 2.3 (principle of implication analysis): if b1⇒b2 and b2⇒b3, then b1⇒b3. definition 2.8: the possibilities of composing, decomposing and expanding/contracting that matter, affair and relation own are called as composability, decomposability and expandability/contractability, correspondingly, and they are identified as openness of basic-element. according to composability of basic-element, one matter can combine with other matter to generate new matter. by decomposability, one matter can be 24 qiaoxing li and naiang wang babu:research on extension strategy generation ... decomposed into several new matters with certain characteristics that may be different from ones of original matter. similarly, one matter can be expanded or contracted to provide possibility for solving contradictory problems. principle 2.4 (principle of opening-up analysis): for any basic-element, the opening-up analysis includes the analysis of composability, decomposability and expandability/contractability: (1) composability analysis: given a basic-element b1 = (o1, c1, v1), there is at least one basicelement b2 = (o2, c2, v2) to allow b1 and b2 to be composed into b, and b2 is called as the composable basic-element of b1, where b = b1 ⊕ b2 and 1) when o1 = o2 and c1 ̸= c2, b = (o1, c1 ⊕ c2, v1 ⊕ v2) = ( o1 c1 v1 c2 v2 ) ; 2) when o1 ̸= o2 and c1 = c2, b = (o1 ⊕ o2, c1, v1 ⊕ c1(o2)) ; 3) when o1 ̸= o2 and c1 ̸= c2, b = ( o1 ⊕ o2 c1 v1 ⊕ c1(o2) c2 c2(o1) ⊕ v2 ) ; (2) decomposability analysis: any basic-element b=(o,c,v) can be decomposed into several basic-elements b1, b2, · · · , bm with the same characteristics under certain condition l, where bi = (oi, c, c(oi))(i = 1, 2, · · · ,m), and we denote b//(l) {b1, b2, · · · , bm}. (3) expandability/contractability analysis: for a real positive number α , any basic-element b=(o,c,v) can be expanded (α > l) or contracted (l > α > 0) as αb = (o,c, αv) under certain condition. thirdly, we introduce the principle of conjugate analysis below: extenics help us to completely understand the matter from its physical, systematic, dynamic and antithetic properties. in terms of physical property of matter, all matters are composed of a physical part and a non-physical part, then the physical part is called material part and the nonphysical part is nonmaterial part by extenics, respectively. when considering a matters structure from its systematic property, the matters components as a whole are called the hard part of matter and the relations between the matter and its components as well as between the matter and other matters are the soft part of the matter. furthermore, from the dynamic property, any matter is changing continuously. stagnation is ever-relative while motion is permanent. the parts that have not been appeared are called as the latent parts of the matter and the appeared parts are the apparent parts of the matter. the latent part may become apparent under certain conditions. from the antithetic property, all matters have two parts that are antithetic. in extenics, the part producing the positive value in the measure of matter about certain characteristic is the positive part of matter and the other taking negative value is the negative part of matter. so matter’s conjugate advances in systems science and application (2016) vol.16 no.1 25 analysis includes nonmaterial-material, soft-hard, negative-positive and latentapparent ones. we should analyze a matter according to the following property. principle 2.5 (principle of conjugate analysis): all matters have four pairs of conjugate parts, i.e., the nonmaterial-material, the soft-hard, the negativepositive and the latent-apparent ones. each conjugate part of any matter has numerous characteristics and it can be denoted by an n-dimensional basic-element. at the same time, in each pair of conjugate part of any matter, one certain conjugate part has at least one characteristic that is relevant to certain characteristic of its corresponding conjugate. fourthly, we introduce the extension transformation below: the tool for solving the contradictory problem is extension transformation. by using certain transformations, an unfeasible problem can be transformed a feasible one. we only introduce the general concept and the basic types of transformation. definition 2.9: supposing that the object γ0 is a matter-element, affairelement, relationelement, criteria or any element in the universe of discourse, and the transformation from γ0 to the object γ or multiple objects γ1,γ2, · · · ,γn in the same class is called as extension transformation of γ and it is denoted as tγ0 = γ or tγ0 = {γ1,γ2, · · · ,γn}. definition 2.10: the object γ have five types of basic transformations below: (1) substitution transformation, i.e., tγ = γ0; (2) increasing/decreasing transformation, i.e., the increasing transformation tγ = γ ⊕ γ0 and the decreasing transformation tγ = γ− γ0; (3) expansion/contraction transformation: tγ = αγ, where it is expansion when α > 1 and contraction when 1 > α > 0; (4) decomposition transformation: tγ = {γ1,γ2, · · · ,γn}, where γ0 = γ1 ⊕ γ2⊕ · · · ⊕ γn; (5) duplication transformation: tγ = {γ,γ1}. principle 2.6 (principle of extension transformation): for any object γ , there must exist a certain transformation t to let tγ = γ0, where γ ̸= γ0. on the other hand, if there is a certain transformation t to let tγ = γ0, there should be another transformation t1 to let t1γ = γ0. furthermore, it can be utilization several transformations to successfully solve a contradictory problem. 3 extension strategy generation step of sustainable utilization of regional water resources during the process of sustainable utilization of water resources, we need to simultaneously solve two goals of water crisis and ecological environment construction in the planning area, and should meet the needs of population promotion, resource exploitation, environmental protection and economic development in the 26 qiaoxing li and naiang wang babu:research on extension strategy generation ... planning period. in order to effectively solve this problem, we propose the extension strategy generation of sustainable utilization of water resources according to the literature [17] below: 1) define the contradictory problem. combining with the regional goals and development needs, we draw up the basic-elements of goals (water crisis solution and ecological environment construction) and the basic-elements of conditions (population promotion, resource development, environmental protection and economic development) of sustainable utilization of water resources in the planning period. under the basic-elements of conditions, the two goals can not be achieved at the same time, and they constitute a contradictory problem. 2) assume that the targets are not changed and do extension analysis for the four basicelements of conditions. suppose that the premise of targets without change is that the tasks of targets are feasible, then we should obtain the appropriate basic-elements of goals by using correct investigation methods. at that moment, we get the extension analysis matter-elements through expanding the condition basic-elements by using correlative analysis, divergent analysis, conjugate analysis and opening-up analysis. 3) generate a variety of utilization strategies of water resources by extension transformation to the extension analysis matter-elements. the purpose of doing extension transformation is to resolve the contradictory problem and make two objectives be achieved simultaneously. extension transformation to the extension analysis matter-elements is mainly for the values of characteristics of matter-elements, and the transformations include substitution, increasing, decreasing, expansion, contraction and decomposition, as well as the integration of these basic transformations. 4) evaluate and select the utilization strategies of water resources. we may get a group of extension transformation matter-elements after doing a complete extension transformation, and then form an integral strategy. however, it is not true that every strategy can effectively solve the problem between water crisis and ecological environment construction, or that the solution effect is the same as others. therefore, we should evaluate and select the strategies by using the following steps: (1) at first, we should determine some indexes a1, a2, · · · , al to measure the strategies which make up a measurement condition set. the measurement condition set is utilized to judge whether the strategies are feasible or not. the measurement criterions of sustainable utilization of water resources in general are technical feasibility and economic viability, and the technical feasibility is usually viewed as the prerequisite which must be satisfied. (2) secondly, evaluate and select the strategies. first step, we abandon the strategies which do not meet the prerequisites, i.e., if the strategies whose technoladvances in systems science and application (2016) vol.16 no.1 27 ogy is not feasible, then they are removed, and we utilize the rest of measurement criterions to justify other strategies. next step, by using the correct methods of investigation and analysis to get the values of remaining strategies for other criterions, we select an appropriate function such as the comprehensive evaluation method to compute the superior degrees of these strategies, and choose the strategies whose degree is the maximum as the optimal ones. 5) draw up the specific programs of action according to the optimal strategies. the purpose to form the action programs is to make the decision-makers and executors understand the optimal strategies, so that they can correctly implement them and accomplish the goals. 4 extension planning process of sustainable utilization of regional water resources 1) define the contradiction problem of water utilization the process of water utilization involves the natural, economic and social systems, and induces many contradictory problems between the ecological environment and the economic development goal as well as the social development mode. in the arid and semi-arid regions, we should solve the contradiction between water crisis and ecological environment construction under the premise of regional development of population, resources, environment and economy. in extenics, the formalized model and the extension analysis methods are utilized to study how to resolve conflict issues, which are also suitable for selecting feasible and effective utilization strategies of water resources and provide a formalized approach for relevant departments. the extension analysis process of water utilization should be made efforts to solve the central issues of water crisis and ecological environment construction on the basis of development of population, resources, environment and economics in this regional which includes administrative region and watershed. in the arid and semi-arid regions, water crisis and ecological environment construction can not be achieved at the same time, so this is a contradictory problem. sustainable utilization of water resources means the synthesis between ecological environment construction and water utilization. therefore, the goal of contradictory problem that should be resolved within the region is ecological environment construction and water crisis, and the problems conditions are development needs of population, resources, environment and economics. then the regional government may formulate the general objective basic-elements g1 and g2 below: g1 = ( construct a1 a2 a3 a t b1 b2 b3 a t ) and g2 = ( w a c2 c3 t a w2 w3 t ) , where for the basic-element g1, construct is a verb, and the characteristics are a1=dominating object, a2 = acting object, a3 = receiving object, a = location 28 qiaoxing li and naiang wang babu:research on extension strategy generation ... and t = time, and the values of these characteristics are b1 = ecological environment, b2 = {administration, beneficiaries, contractors, collaborators, etc.}, b3 = {vegetation, land, species}, a = a certain region which may be an administrative region or watershed, and t = a planning period, such as three-year plan, etc.. apparently, g1 is an affair-element. for another basic-element g2, w=water resources, and the characteristics are c2 = ownership quantity and c3 = demand quantity, and the unit of the values w2 and w3 is hundred million cubic meters. here, the equation w3 > w2 holds, which indicates that the water demand within the region a is larger than the water owner, and the water crisis exists in this region. the quantity w2 is an average statistical value of water resources for recent years, and the quantity w3 is a forecasting value of water demand of the region a in the planning period. also apparently, g2 is a matter-element. in addition, the premise to resolve the contradictory problem between ecological environment construction and water crisis is to meet the development needs of regional population, resources, environment and economics, thus we establish the following condition basic-elements: l1 = ( upgrade a1 a2 a3 a t r11 r12 r13 a t ) , l2 = ( exploit a1 a2 a3 a t r21 r22 r23 a t ) , l3 = ( improve a1 a2 a3 a t r31 r32 r33 a t ) and l4 = ( develop a1 a2 a3 a t r41 r42 r43 a t ) , where the values of characteristic a1 are r11=population, which includes rural and urban ones, r21= natural resources, r31=living environment and r41=economics, and the values of characteristic a2 are corresponding administrations, and the values of characteristic a3 are r13=population qualities, which include population quantity, educational background, etc., r23= {land, minerals, wild beast}, r33={green belt, waste treatment, transportation belt}, and r43=economics. waste treatment mainly refers to exhaust, waste water and garbage, and transportation belt includes the road clean, green, construction and maintenance, etc.. because we can not simultaneously achieve the goals g1 and g2, it is a contradictory problem, and thus we establish the extension model p = (g1 ∧g2) ↑ (l1 ∧ l2 ∧ l3 ∧ l4). the purpose to do extension analysis is to transfer the contradictory problem into the co-existence issue by doing the correlative analysis, the divergent analysis, the conjugate analysis and the opening-up analysis on the goal basic-elements and the condition basic-elements. in general, there are five approaches to resolve contradictory problem with multi-objectives and multi-conditions: firstly, all targets are not changed, but make some (or all) conditions be transformed; secondly, all conditions are unchanged but some (or all) targets are transformed; thirdly, part of the targets are changed and some (or all) conditions are altered; fourthly, part of conditions advances in systems science and application (2016) vol.16 no.1 29 are changed and some (or all) targets are transferred; fifthly, all of goals and conditions are changed. in this paper, we only discuss the most simple issue, i.e., all targets are unchanged and some (or all) conditions are transferred. by transferring the conditions, we solve contradictory problem between water crisis and ecological environment construction. for the other issues, we will analyze them in other papers. firstly, we define the specific target basic-elements and simplify the condition basic-elements on the basis of the original problem p. according to the reality and overall targets of the region a, we decompose the goal affair-element g1 on the basis of the investigation and analysis results, then obtain the specific target matter-elements of sustainable utilization of water resources below: g′ 1 =  g1 a11 b11 a12 b12 a13 b13 a14 b14 a15 b15  , g′′ 1 =  g2 a21 b21 a22 b22 a23 b23 a24 b24 a25 b25  and g′′′ 1 =  g3 a31 b31 a32 b32 a33 b33 a34 b34 a35 b35  , where for the matter-element g′ 1, the object g1 is vegetation, and the characteristics of g1 are a11= administrative, a12= original acreage, a13= planning acreage, a14= original amount of water demand and a15 = incremental of water demand; for the matter-element g′′ 1, the object g2 is land, and the characteristics of g2 are a21= administrative, a22= remediation acreage, a23= incremental of vegetation, a24= incremental of building and a25 = incremental of water demand, as well as the equation b22= b23+ b24; for the matter-element g′′′ 1 , the object g3 is species, and the characteristics of g3 are a31= administrative, a32= original amount, a33= incremental of species, a34= original amount of water demand and a35 = incremental of water demand. then, the target affair-element g1 is transformed into three specific target matter-elements. furthermore, the land g2 mainly refers to the unutilized land in the region a. meanwhile, in the case of non-confusion, we can simplify the condition basicelements by omitting the characteristics location and time. therefore, we transform the condition basicelements l1, l2, l3 and l4 into the simple forms below: l′1 = r1 a1 r11 a2 r12 a3 r13  , l′2 = r2 a1 r21 a2 r22 a3 r23  , l′3 = r1 a1 r31 a2 r32 a3 r33  and l′4 = r1 a1 r41 a2 r42 a3 r43  , 30 qiaoxing li and naiang wang babu:research on extension strategy generation ... where r1=upgrade, r2=exploit, r3=improve and r4=develop. through the transformations above, the original problem p = (g1 ∧ g2) ↑ (l1 ∧ l2 ∧ l3 ∧ l4) is converted to p ′ = (g′ 1 ∧ g′′ 1 ∧ g′′′ 1 ∧ g2) ↑ (l′1 ∧ l′2 ∧ l′3 ∧ l′4). because the problem p is the specific form of the original one p, it can be resolved when p is achieved. thus we only do extension analysis for the condition basic-elements of the problem p. 2) extension analysis process of condition basic-elements the correlative analysis on condition basic-elements: the correlation of basicelement discusses the correlations among the same or different objects (matters, actions or relationships) as well as among their characteristics or values. it can make people more clearly understand the interactions between each of condition basic-elements and the sustainable utilization of water resources by using the formalized way. on the basis of the correlation between water resources and the development of population, resources, environment and economics, we get the correlative matterelements w1, w2, w3 and w4 of condition basic-elements below: w1 = w k1 w11 k2 w12 k3 w13  , w2 = w k1 w21 k2 w22 k3 w23  , w3 = w k1 w31 k2 w32 k3 w33  and w4 = w k1 w41 k2 w42 k3 w43  , where the matter w=water, and the characteristics k1, k2 and k3 represent user, use volume and use mode, respectively, and the values of k1 are w11 = population, w21 = natural resources, w31 = living environment and w41 = economy, and the values w12, w22, w32 and w42 of k2 are the average statistical value of regional water utilization in recent years which are consumed by population, natural resources, environment and economics, respectively, and the values of characteristic k3 are w13={drink, wash, rinse, etc.}, w23={produce, drink, etc.}, w33={irrigate, rinse, etc.} and w43= {irrigate, produce, wash, drink, rinse, etc.}. the new matter-elements w1, w2, w3 and w4, which come from the results after doing the correlative analysis on the condition affair-elements, are called the correlative matter-elements of condition basic-elements. divergent analysis on the correlative matter-elements: the divergence of matterelement includes: different characteristic with same matter, different value with same characteristic, different matter with same value, same characteristic with same matter, same value with same characteristic, same value with same matter, similar value with same characteristic, etc. by using the divergent analysis, we can understand the multiple aspects of correlative matter-elements which involve in the process of water utilization. doing the divergent analysis on the values of advances in systems science and application (2016) vol.16 no.1 31 characteristic k1 of correlative matter-elements according to same characteristic with same matter, we grasp all water users in region a and get following matterelements ( 7→ means divergence): w1 7→ w 1 1 = ( w k1 k2 k3 w1 11 w1 12 w1 13 ) , w 2 1 = ( w k1 k2 k3 w2 11 w2 12 w2 13 ) ; w2 7→ w 1 2 = ( w k1 k2 k3 w1 21 w1 22 w1 23 ) , w 2 2 = ( w k1 k2 k3 w2 21 w2 22 w2 23 ) , w 3 2 = ( w k1 k2 k3 w3 21 w3 22 w3 23 ) ; w3 7→ w 1 3 = ( w k1 k2 k3 w1 31 w1 32 w1 33 ) , w 2 3 = ( w k1 k2 k3 w2 31 w2 32 w2 33 ) , w 3 3 = ( w k1 k2 k3 w3 31 w3 32 w3 33 ) ; w4 7→ w 1 4 = ( w k1 k2 k3 w1 41 w1 42 w1 43 ) , w 2 4 = ( w k1 k2 k3 w2 41 w2 42 w2 43 ) , · · · , wm 4 = ( w k1 k2 k3 wm 41 wm 42 wm 43 ) , where w1 11 = rural residents, w2 11 = urban residents ; w1 21 = land, w2 21 = minerals, w3 21 = biology; w1 31 = green belt, w2 31 = waste treatment, w3 31 = transportation belt; and the matters w1 41 , w2 41 , · · · , wm 41 are m industrial departments in region a such as agriculture, tourism, catering, etc.; and the values w1 12 , w2 12 , w1 22 , w2 22 , w3 22 , w1 32 , w2 32 , w3 32 , w1 42 , · · · , wm 42 of characteristic k2 are the average statistical values of water utilization which consumed by corresponding users of region a in recent years, and the values of characteristic k3 are w 1 13 ⊆ w13 = {drink,wash, rinse} , w 1 23,w 2 23,w 3 23 ⊆ w23 = {produce, drink}, w 1 33,w 2 33,w 3 33 ⊆ w23 = {irrigate, rinse}, as well as w 1 43, · · · ,wm 43 ⊆ w43 = {irrigate, produce, wash, drink, rinse}. the basic elements which come from the correlative matter-elements by using the divergent analysis are called divergent matter-elements. conjugate analysis on divergent matter-elements: the properties of system, material, dynamic and opposition that things own are collectively known as conjugation. we can understand things in a more comprehensive perspective and reveal the development and variation nature of things in a more profound way by using the conjugate analysis. extension theory describes the structure of things 32 qiaoxing li and naiang wang babu:research on extension strategy generation ... from four pairs of conjugate and opposite concepts which are material and nonmaterial, hard and soft, apparent and latent, as well as positive and negative. in generally, we utilized to consider the problems from the material, hard, apparent and positive aspects of things. however, the nonmaterial, soft, latent and negative angles of things can make us obtain unexpected results. because the divergent matter-elements come from the water users which are population, resources, environment and economics through divergence analysis, we will do conjugate analysis by combining the user k1 with water resources w. from the nonmaterial part of things, we firstly determine the requirements of user k1 and get a new characteristic named water quality k4 whose values for all divergent matter-elements are vi4 ∈ {i ∼ v grade}, where the standard of water quality refers to the book named china’s environmental quality standard of surface water (gb3838-2002), and the subscript i denotes the serial number of the divergent matter-elements from w 1 1 to wm 4 . secondly, we justify the economic status and consumption of user k1 in the region a, and then get a new characteristic of water named importance k5, and the values vi5 of k5 satisfy vi5 ∈ {necessary, priority, current situation, decrease, no}, where no means that the certain user does not consume water and we may remove the corresponding matterelement. form the soft aspect of things, we view the relationship among the orders of water users, and get two characteristics of water resources named as former user k6 and later user k7 whose values vi6 and vi7 are the elements from the set composed by the users in matter-elements above. from latent aspect of things, we analyze the potential utilization values of the consumption and discharge of water and get a characteristic denoted as recyclability k8 whose value vi8 ∈ { yes, no}. furthermore, according to recyclability, we should determine the characteristic named recovered amount k9, and denote its value as vi9. from the negative aspect of things, we mainly view that whether waste water has negative impact for environment or not, and get a characteristic named emission k10 with value vi10 ∈ { direct emission, treatable emission, non-emission}, where nonemission means that the water has been completely consumed by the corresponding user. then we investigate the technology to treat the wastewater and get another characteristic named treatable technology k11 with the value vi11 ∈ { high, normal, low, no }, where no means that the value vi10 is direct emission or non emission. then we obtain the conjugate matter-elements by using conjugate analysis on divergent matter-elements. the opening-up analysis on conjugate matter-elements: the possibility of matter-elements combination or decomposition is called the opening-up property of matter-element which includes addition, multiplication and decomposition of matters, characteristics and values. the opening-up property of matter-element provides another approach to solve the contradictory problems. at first, we disadvances in systems science and application (2016) vol.16 no.1 33 cuss the decomposition of water resources, and obtain the characteristic named as source k12 of desirable water in region a whose value vi12 ∈ {surface water, groundwater, external water, recycled water, all water}, where recycled water is reutilized water while wastewater is retrieved, and all water contains surface water, groundwater, external water and recycled water. secondly, from the additive or multiplicative property of characteristics, we examine user k1, water quality k4 and former user k6, and get the characteristic named freshness k13 with the value vi13 ∈ {fresh water, circled water}, where fresh water includes surface water, groundwater and external water. finally, according to the additive property of values, we study on utilization volume k2, recyclability k8, source k12 and freshness k13 and obtain the characteristic increment k14 with the value vi14. from the values vi12 of utilization volume k2 and vi14 of increment k14, we get another characteristic named recycle volume k15 with the value vi15. at last, by using the values vi2, vi14 and vi15, we get a characteristic named total volume k16 with the value vi16. the new matterelements from the conjugate matterelements by using opening-up analysis are called opening-up matter-elements. condition matter-elements are now transferred into a group of new matterelements with the object named water and 16 characteristics through extension analysis which includes correlative analysis, divergent analysis, conjugate analysis and opening-up analysis, and they are called as the extension-analysis matterelements of condition basic-elements. from the analysis showed above, the final results, i.e., the opening-up matter-elements, are the extension-analysis matterelements. 3) the generation and optimal selection of sustainable utilization strategy of regional water resources and the program implementation of optimal strategy by using extension analysis, the condition-basic-elements are transferred into extensionanalysis matter-elements of sustainable utilization of water resources. for the sake of convenience, we re-order the characteristics of extension-analysis matter-elements as user k1, use mode k2, water quality k3, importance k4, former user k5, later user k6, recyclability k7, emission k8, treatable technology k9, source k10, freshness k11, use volume k12, recovered amount k13, increment k14, recycle volume k15 and total volume k16, where the characteristics with nonquantities are in the front and the others with quantities are at the back, and the quantitative characteristics satisfies the following relationship: (1) the value vi12 of use volume k12 ≤ the absolute value vi13 of recovered amount k13, and let vi13 ≤ 0 which represents an increase of water resources; (2) use volume k12, increment k14 ∈ {surface water, groundwater, external water} = {initial water}, and their values vi12 + vi14 = the volume of initial water, where vi12 ≥ 0 and vi14 ≥ 0, and they means the consumption; (3) the value vi15 of recycle volume k15 ≥ 0, which means that the amount 34 qiaoxing li and naiang wang babu:research on extension strategy generation ... of recycle water was consumed by user i, and vi15 ≤ |vi13|; (4) the value vi16 of total volume k16 satisfies vi16 = vi12 + vi14 + vi15, which represent the total amount of water resources consumed by user i. in general, extension transformation of extension-analysis matter-elements, which includes substitution, decomposition, addition, decrease, etc., is only in connection with those quantitative characteristics of water resources, then we obtain a set of new condition matter-elements with new values and we call them as extension-transformation matter-element below: t (w 1 1 ) = ( w k1 · · · k15 k16 w1 1,1 · · · w1 1,15 w1 1,16 ) , t (w 2 1 ) = ( w k1 · · · k15 k16 w2 1,1 · · · w2 1,15 w2 1,16 ) ; t (w 1 2 ) = ( w k1 · · · k15 k16 w1 2,1 · · · w1 2,15 w1 2,16 ) , t (w 2 2 ) = ( w k1 · · · k15 k16 w2 2,1 · · · w2 2,15 w2 2,16 ) , t (w 3 2 ) = ( w k1 · · · k15 k16 w3 2,1 · · · w3 2,15 w3 2,16 ) ; t (w 1 3 ) = ( w k1 · · · k15 k16 w1 3,1 · · · w1 3,15 w1 3,16 ) , t (w 2 3 ) = ( w k1 · · · k15 k16 w2 3,1 · · · w2 3,15 w2 3,16 ) , t (w 3 3 ) = ( w k1 · · · k15 k16 w3 3,1 · · · w3 3,15 w3 3,16 ) ; t (w 1 4 ) = ( w k1 · · · k15 k16 w1 4,1 · · · w1 4,15 w1 4,16 ) , t (w 2 4 ) = ( w k1 · · · k15 k16 w2 4,1 · · · w2 4,15 w2 4,16 ) , · · · , t (wm 4 ) = ( w k1 · · · k15 k16 wm 4,1 · · · wm 4,15 wm 4,16 ) . by carrying out a transformation t, we get a set of extension-transformation matter-elements, which corresponds to a set of sustainable utilization strategy of water resources. different transformation t corresponds to a different set of extension-transformation matter-elements and then the strategy is different. repeatedly doing transformation, we obtain a variety of sustainable utilization strategies. to doing extension transformation to extension-analysis matterelements is known as the sustainable utilization strategy generation of water resources. different strategies have different implementation effects in the process of sustainable utilization of water resources. the purpose of extension transformation is to find the best solutions as possible as can, so we need to select the optimal one in a variety of strategies. firstly, we should determine the evaluation characteristics, such as technological possibility and economic feasibility. secondly, we select an appropriate evaluation method to calculate the optimal degree of these matter-elements. finally, choosing the maximum in the optimal degrees and then viewing the corresponding set of matter-elements as the optimal strategy, we formulate the specific action plans according to the optimal strategy. the advances in systems science and application (2016) vol.16 no.1 35 action plan can achieve the sustainable utilization of water resources during the planning period t in the region a under the required conditions to meet the development of population, resources, environment and economics, and effectively resolve the contradictory problem between water crisis and ecological environment constructions, and then the overall goals can be achieved. the formalized generation process of sustainable utilization strategies of water resources can be shown in fig. 1. fig. 1 the flow chart of extension strategy 5 modified case study in order to illustrate the generation and optimal selection of sustainable utilization strategies of water resources and the program implementation process, we will utilize a specific case to describe it in this section. 1) the overview of socio-economics and water resources in the studied region 36 qiaoxing li and naiang wang babu:research on extension strategy generation ... one region a located in the middle of heihe river basin and the central place of hexi corridor is the political, economic and cultural center of zhangye city, gansu province, pr china. the total acreage is about 4000 square kilometers, where the propositions of mountainous, desert area and plain are 14.4%, 34.5% and 51.1%, respectively. the coverage rate of vegetation is only about 3%. the total population in the region is approximately 520,000, where the agricultural population is about 350,000 whose proposition is 65%. in recent years, gdp is about 6.8 hundred million rmb with about 15,000 rmb per person. the water which is utilized to maintain the basic life of population and the production in this region comes from the rivers flowing through the region and the groundwater. the region is a typical agriculture oasis and large irrigated agriculture area. the total volume of available water in the region is about 1.3 hundred million m3, and the average volumes per person and per acre are about 1330m3 and 570m3, respectively. so it is a relatively serious water shortage region. from the structure of water utilization, the ratio of agriculture, industry, people living and ecology is 83.7:2.7:6.4:7.2 in 2005, where the proportion of agricultural water is still relatively large. the region a is an important commodity grain base and one of the five bases named west vegetable to east in northwest of china, and thus the agricultural production plays an important role in the area. region a has superior condition with 30 kinds of mineral resources, where the reserve of coal has been proven for 1.05 hundred million tons, and the content of both tungsten and molybdenum ranks at the first order in north of china. according to the survey, region a has about 10 species of wild animals, where the total number is only 2,000 surplus and some of them are on the verge of extinction. the averages of annual rainfall and annual evaporation are 120mm and 2000-2350mm, respectively, so it is a typical temperate continental arid climate. there are many problems of water resources, such as serious water shortage, obvious contradiction between supply and demand, irrational utilization structure, low efficiency and effectiveness, thus the water resources is the most important constraints of sustainable development of society and economy in the region. 2) define contradictory problem and set goal basic-elements and condition matter-elements the local government of regional a viewed the eco-economic development as the main task. so the government transforms the economic development mode and highlights three goals which are ecological construction, modern agriculture and passage economy. then sustainable utilization of local water resources may be achieved. under the requirements to enhance the population quality, exploit the natural mineral resources, improve the living environment and promote economic development, local government makes its efforts to solve the two central goals of water crisis and ecological environment construction. then the govadvances in systems science and application (2016) vol.16 no.1 37 ernment develops a “five-year plan” according to local condition and obtains following total goal basic-elements g1 and g2: g1 = ( construct a1 a2 a3 a t b1 b2 b3 a t ) and g2 = ( w a c2 c3 t a 13 20 t ) , where the unit of c2 and c3 is hundred million m3, t=five years, and the meaning of others is the same with corresponding formers. because the value of c3 is larger than one of c2, the water crisis exists. then we develop the specific goal basic-elements and simplify the condition basic-elements according to g1 and g2 respectively. on the basis of the actual situation and total goals of region a, we decompose g1 and get following specific matterelements: g′ 1 =  g1 a11 b11 a12 120 a13 360 a14 0.9 a15 2.3  , g′′ 1 =  g2 a21 b21 a22 300 a23 240 a24 60 a25 3.1  and g′′′ 1 =  g3 a31 b31 a32 2000 a33 2300 a34 0 a35 0  , where the characteristics a11, · · · , a15, a21, · · · , a25, a31, · · · , a35 are the same as the formers, and the value of a13 =the value of a12+ the value of a23. because the species g3 usually live in vegetation g2 such as forest and grassland, its water requirement has been included in the vegetation water demand, so its characteristics a34 and a35 are 0. at the same time, a part of treated land g2 will be planted vegetation as woodland and grassland, and another part will be as construction land such as residential housing and factory building. so the value of a25 contains the value of a25. therefore, the total water increment of three specific targets is 3.1 hundred million m3. if it is together with the original demand 0.90 hundred million m3, the total water demand of target basic-elements is 4.0 hundred million m3. simplify the condition basic-elements l1, l2, l3 and l4 as follows: l′1 = r1 a1 r11 a2 r12 a3 r13  , l′2 = r2 a1 r21 a2 r22 a3 r23  , l′3 = r3 a1 r31 a2 r32 a3 r33  and l′4 = r4 a1 r41 a2 r42 a3 r43  . then we get the problem p′ = (g′ 1 ∧ g′′ 1 ∧ g′′′ 1 ∧ g2) ↑ (l′1 ∧ l′2 ∧ l′3 ∧ l′4)) . if the problem has been solved, then the total goal basic-elements g1 and g2 can be achieved. 38 qiaoxing li and naiang wang babu:research on extension strategy generation ... 3) extension analysis and extension transformation according to the process of extension analysis introduced above, we obtain the extensionanalysis matter-elements of condition basic-elements. firstly, we determine that the values of user k1 are w1 11 = rural residents, w2 11 = urban residents; w1 21 = land, w2 21 = minerals, w3 21 = biology; w1 31= green belt, w2 31 = waste treatment, w3 31 = transport belt. we also assume that there are four economic sectors in region a which are w1 41 = agriculture, w2 41 =livestock, w3 41 =building industry and w4 41 = catering industry, respectively; the values of other characteristics, such as use mode k2, water quality k3, importance k4, former user k5, later user k6, recyclability k7, emission k8, treatable technology k9, source k10 and freshness k11, are referred to the process of extension analysis in section 3. these characteristics only have important role for analyzing water utilization and need not do extension transformation. therefore, we list them separately as the innovation characteristics of extension analysis in table 1. the other characteristics of extension-analysis matter-elements involve the quantitative of water resources, such as use volume k12, recovered amount k13, increment k14, recycle volume k15 and total volume k16. doing extension transformation such as substitution, decomposition, increase, decrease, etc., to the quantitative of the five characteristics of extension-analysis matterelements, we get extension-transformation matter-elements with new value. suppose that some types of extension transformation, which are named as t1, t2, t3 and t4, respectively, have been done to the values of five quantitative characteristics of extension-analysis matter-elements, we get four sets of extension-transformation matter-elements, i.e., four types of innovation strategies of sustainable utilization of water resources, which can be seen in table 2 and table 3. for the transformation t1 in table 2, the value of use volume k12 is the original statistics of water resources in region a, and the value of increment k14 is water increases to meet the development needs of population, resource, environment and economics according to the current consumption structure of water, and the increases is from the fresh water. the value of recovered amount k13 is the recycled quantities from use volume k12 and increment k14 which have been consumed by corresponding users according to the current consumption structure. recycle volume k15 is the re-utilized amount of recovered amount k13. from table 2, we know that the water which is only consumed by minerals, livestock and catering industry has been retrieved, and part of the retrieved water has been utilized by waste treatment, agriculture and livestock. in table 2 and table 3, the transformations t2, t3 and t4 come from t1 in turns, and then produce three additional utilization strategies of water resources. transformations t2 and t3 only adopt the water-saving measures, namely that the corresponding users utilize the retrieved water at first for use volume k12 and increment k14 as posadvances in systems science and application (2016) vol.16 no.1 39 table 1 analysis table of extension innovation basic elememt user k1 use mode k2 water quality k3 importance k4 former user k5 later user k6 w1 1 rural residents drink, wash, rinse grade i-iii necessary, priority rural residents green belt, waste treatment, transportation belt, agriculture, animal w2 1 urban residents drink, wash, rinse grade i-iii necessary, priority urban residents green belt, waste treatment, transportation belt, agriculture, animal w1 2 land none grade i-v necessary agriculture none w2 2 minerals produce grade i-iv priority none green belt, waste treatment, transportation belt w3 2 biology drink grade i-iii priority none none w1 3 green belt irrigate grade i-v priority residents none w2 3 waste treatment rinse grade i-v necessary residents, transportation belt, minerals none w3 3 transportation belt irrigate, rinse grade i-v priority residents waste treatment w1 4 agriculture irrigate grade i-v necessary, decrease residents, animal land w2 4 animal drink, rinse grade i-iii necessary, decrease residents, catering agriculture w3 4 building produce grade i-v current situation, decrease decrease none waste treatment w4 4 catering drink, wash, rinse grade i-iii necessary, decrease none animal tabel 1 analysis table of extension innovation (continued) basic elememt recyclability k7 emission k8 treatable technology k9 source k10 freshness k11 w1 1 yes direct emission no surface water, groundwater, external water fresh water, recycled water w2 1 yes direct emission no surface water, groundwater, external water fresh water, recycled water w1 2 no non-emission no all water fresh water, recycled water w2 2 yes treatable emission high surface water, groundwater, external water fresh water w3 2 no non-emission no surface water fresh water w1 3 no non-emission no all water fresh water, recycled water w2 3 yes treatable emission high all water fresh water, recycled water w3 3 yes direct emission no all water fresh water, recycled water w1 4 no direct emission no all water fresh water, recycled water w2 4 yes direct emission no surface water, groundwater, external water fresh water w3 4 yes treatable emission normal surface water, groundwater, external water fresh water w4 4 yes direct emission no surface water, groundwater, external water fresh water 40 qiaoxing li and naiang wang babu:research on extension strategy generation ... table 2 the first innovation t1 and the second one t2 of extension-exchange matter-element (unit: hundred million m3 ) t1 k12 k13 k14 k15 k16 t2 k12 k13 k14 k15 k16 t1(w1 1) 0.53 1.2 1.73 t2(w1 1) 0.53 -1.4 0.8 0.4 1.73 t1(w2 1) 0.27 0.6 0.87 t2(w2 1) 0.27 -0.65 0.23 0.37 0.87 t1(w1 2) 0 0.06 0.06 t2(w1 2) 0 0 0.06 0.06 t1(w2 2) 0.25 -0.3 0.7 0.95 t2(w2 2) 0.25 -0.85 0.7 0.95 t1(w3 2) 0.01 0.02 0.03 t2(w3 2) 0.01 0.02 0.03 t1(w1 3) 0.03 0.05 0.08 t2(w1 3) 0.03 0 0.05 0.08 t1(w2 3) 0 0.6 0.32 0.92 t2(w2 3) 0 0 0.92 0.92 t1(w3 3) 0.01 0.03 0.04 t2(w3 3) 0.01 -0.01 0 0.03 0.04 t1(w1 4) 10.85 -1.53 0.1 9.42 t2(w1 4) 10.85 -2.73 -1.53 0.1 9.42 t1(w2 4) 0.58 -0.15 -0.08 0.2 0.7 t2(w2 4) 0.58 -0.6 -0.08 0.2 0.7 t1(w3 4) 0.9 -0.1 0.8 t2(w3 4) 0.9 -0.55 -0.1 0.8 t1(w4 4) 0.5 -0.2 -0.1 0.4 t2(w4 4) 0.5 -0.27 -0.1 0.4 total 13.93 -0.65 1.45 0.62 16 total 13.93 -7.06 -0.06 2.13 16 table 3 the first innovation t3 and t4 of extension-exchange matter-element (unit: hundred million m3 ) t3 k12 k13 k14 k15 k16 t2 k12 k13 k14 k15 k16 t3(w1 1) 0.53 -1.4 0.8 0.4 1.73 t4(w1 1) 0.53 -1.4 0.8 0.4 1.73 t3(w2 1) 0.27 -0.65 0.23 0.37 0.87 t4(w2 1) 0.27 -0.65 0.23 0.37 0.87 t3(w1 2) 0 0 0.06 0.06 t4(w1 2) 0 0 0.06 0.06 t3(w2 2) 0.25 -0.85 0.7 0.95 t4(w2 2) 0.25 -0.85 0.7 0.95 t3(w3 2) 0.01 0.02 0.03 t4(w3 2) 0.01 0.02 0.03 t3(w1 3) 0 0 0.08 0.08 t4(w1 3) 0 0 0.08 0.08 t3(w2 3) 0 0 0.92 0.92 t4(w2 3) 0 0 0.92 0.92 t3(w3 3) 0 -0.01 0 0.04 0.04 t4(w3 3) 0 -0.01 0 0.04 0.04 t3(w1 4) 9.95 -2.73 -1.53 1 9.42 t4(w1 4) 9.83 -2.73 -1.53 1 9.3 t3(w2 4) 0.18 -0.6 -0.08 0.6 0.7 t4(w2 4) 0.18 -0.6 -0.08 0.6 0.7 t3(w3 4) 0.9 -0.55 -0.1 0.8 t4(w3 4) 0.9 -0.55 -0.1 0.8 t3(w4 4) 0.5 -0.27 -0.1 0.4 t4(w4 4) 0.5 -0.27 -0.1 0.4 total 12.59 -7.06 -0.06 3.47 16 total 12.47 -7.06 -0.06 3.47 15.88 sible as they can. transformation t4 has adopted the water-saving technologies to agriculture on the basis of transformation t3. transformation t2 is substitution on the basis of transformation t1. firstly, if the value of later user k6 is not ”none”, then the corresponding user should take some measures to retrieve part of the utilized water, and then we get that the sum of recovered amount k13 is 7.06 hundred million m3. secondly, if the value of former user k5 is ”none”, then the value of corresponding increment k14 should be prior from recovered amount k13 that can be utilized, and we obtain the sum of recycle volume k15 is 2.13 hundred million m3. transformation t3 is a synthesis of substitution and decomposition on the basis of transformation t2, i.e. if the value of former user k5 is not ”none”, then part of consumption water of the corresponding use volume k12 is from recovered amount k13. therefore, the water resources of recycle volume k15 has been conadvances in systems science and application (2016) vol.16 no.1 41 sumed 3.47 hundred million m3 and the remaining part is 3.59 hundred million m3. apparently, the total consumption water of transformation t2 is the same as the one of transformation t3 which is 16 hundred million m3. adding the water consumed by ecological environment construction of target basic-elements, which is 4 hundred million m3, the total water consumed in region a is 20 hundred million m3. transformation t4 is a decrease on the basis of transformation t3, where the highest consumer of water resources agriculture adopts a certain water-saving technology to reduce the consumption of initial water. then, agricultural expends initial water 9.83 hundred million m3. transformation t4 consumed the sum of initial water 12.41 hundred million m3 and recycle water 3.47 hundred million m3, i.e., the actual amount of consumed water is 15.88 hundred million m3. 4) assessment and optimal selection of innovation strategies in order to evaluate these four innovation strategies, we establish a set of to measurement criterions: a1: technological feasibility; a2: economical feasibility and a3: to meet basic needs of regional water resources, where a1 and a2 are prerequisites, and a3 contains two meanings: first, the initial water utilized by target basic-elements is no less than 0.4 hundred million m3; second, the initial water consumed by condition basic-elements should meet the basic needs. furthermore, the first meaning is also prerequisite, which means that the initial water consumption of condition basic-elements is not more than 1.26 hundred million m3. in addition, because the total water demands of the target basic-elements is 4 hundred million m3, its recycle volume is not more than 3.6 hundred million m3. apparently, the four innovative strategies satisfy the conditions both a1 and a2, but t1 and t2 do not meet the first meaning of a3, so they can be deleted firstly. following, we select the optimal one between t3 and t4. in general, the pros and cons of extension strategies can be evaluated by using priority degree evaluation method, comprehensive evaluation method, or others. however, in this case, speaking on the water consumption of the two innovative strategies t3 and t4, only agriculture is different from each other. therefore, the optimal strategy selection is only to compare the costs of between 0.12 hundred million m3 water resources that t3 is surplus to t4 and the amount of saving water by doing the technological reform of t4. since the study area is located in the arid and semi-arid regions, the sustainable utilization of water resources occupies the prime location in natural, economic and social systems. therefore, the technological reform to save water in this region is the main goal of future work. then we choose t4 as the optimal strategy. 5) draw up the action programs according to table 1 and the strategy t4 of table 3, we formulate the specific 42 qiaoxing li and naiang wang babu:research on extension strategy generation ... five-year action programs of water resources in the study area below: (1) residents are limited the total amount of 1.83 hundred million m3 of water utilization, where the quotas of rural residents and urban residents are 1.33 and 0.5 hundred million m3, respectively. under the situation of water ratio systems, we should retrieve the wastewater. the amount of recycled water from residents is 2.05 hundred million m3, of which 0.77 hundred million m3 is utilized as non-drinking, such as washing (courtyard, toilet), etc. on the other hand, 1.28 hundred million m3 of water resources will be allocated to green belt, waste treatment, transportation belt, agriculture and livestock. (2) land should strictly limit their consumption of initial water, and only utilizes about 0.06 hundred million m3 of recycle water from agriculture. the retrieved water will be utilized to conserve water, prevent land from desertification and semi-desertification. (3) rationally develop mineral resources. within the 30 plus kinds of mineral resources in the studied area, we preferentially choose coal, tungsten and molybdenum, which have great development potentiality and mature technology, to be exploited. all water to develop mineral resources is of 0.95 hundred million m3, and the enterprises should adopt retrieved measures to collect wastewater about 0.85 hundred million m3, which will be utilized for green belt, waste treatment and transport belt. (4) protect the wildlife habitat and properly feed wild animals. the program will be carried out simultaneously with the ecological and environmental protection. at the same time, an additional water of 0.03 hundred million m3 is supplied to supplement wildlife habitat. (5) green urban and rural regions. in the towns and villages surrounding residential areas, plenty of trees and flowers should be planted. all required water of about 0.08 hundred million m3 should be from recycled water provided by the residents, and initial water is strictly prohibited for irrigation. (6) purify the living environment and let it be a healthy region, and vigorously clear up waste water, waste gas and waste disposal. the water of 0.92 hundred million m3 utilized to improve the health of towns and villages is provided by residents, transportation belt and minerals, and initial water is prohibited to purify the living environment. (7) protect and reinforce roads. shelterbelts will be planted and garbage should be collected on both sides along the roads. the water of 0.04 hundred million m3 to plant trees and clean roads is entirely provided by recovered water from residents. at the same time, the water which was secondly recovered by using water-saving measures will be utilized to clear up waste water, waste gas and waste disposal. (8) supported by scientific and technological progress and oriented by market, advances in systems science and application (2016) vol.16 no.1 43 the structure of agricultural production will be adjusted greatly. firstly, on the basis of ensuring the basic supply of grains, vegetables, fruits as well as other agricultural products, we should vigorously adjust the agriculture structure and develop water-saving agriculture; secondly, we may adopt water-saving technologies and measures on agricultural production in order to implement watersaving and recycling irrigations. the total of agricultural water would be less than 9.30 hundred million m3, where recycle volume is 1 hundred million m3 and initial water is not more than 8.30 hundred million m3. initial water has been saved 2.55 hundred million m3 than before, and recovered amount is 2.73 hundred million m3, where a small part was directly discharged to land and the majority is utilized for planting new vegetations. (9) appropriately reduce the development scale of livestock in order to save water resources and protect vegetation. arrange livestock for initial water of 0.10 hundred million m3 and recycle volume of 0.60 hundred million m3, where recycle water comes from recovered amount of residents and catering industry, and recovered water of 0.60 hundred million m3 from livestock by using watersaving measures is provided for agriculture. (10) control the development scale of building industry and pay attention to retrieve its utilized water. since building industry must utilize the initial water, we should appropriately control its scale of development. the water consumption amount of building industry was reduced from the original 0.90 hundred million m3 to 0.80 hundred million m3, and about 0.55 hundred million m3 of retrieved water by adopting saving-water measures is utilized for waste treatment. (11) maintain and reduce water consumption amount of catering industry. catering water should consume initial water, so we may pay attention to water conservation and against waste. the quota of initial water was reduced to 0.4 hundred million m3 and 0.27 hundred million m3 of water is retrieved for livestock by improving water-saving measurements. in order to effectively perform the schemes above, the local government needs to formulate the following supportable measures: (1) promote the establishment of water-saving society and encourage broad participation of resident. since the programs need to use a lot of recycle volume, residents and other water users must actively cooperate to retrieve the utilized water. full of recovered amount is a necessary condition for sustainable utilization of water resources. (2) develop relevant laws and regulations, and constraint the user behavior. under the rigid constraints to limit amount of available water resources, the integrated management of water resources should be comprehensively improved. regulatory reforms need to be clear the ways and patterns of water utilization, and determine the quota, and clear water rights, and implement the total control 44 qiaoxing li and naiang wang babu:research on extension strategy generation ... and quota management. (3) strengthen water conservation measures and transform water-saving technology, and vigorously popularize the water-saving infrastructure projects. by laying recovery line, we can provide the necessary conditions for residents to retrieve and re-utilize water resources. at the same time, by adopting water-saving technologies, such as the drip and the timing spray irrigations, we can improve water efficiency and effectiveness. (4) optimize the agricultural structure and enhance the development level of agricultural economics. agriculture is the largest water user and its effectiveness is also the lowest. to adjust the farming structure and enhance the water-saving technology is a sufficient condition of sustainable utilization of water resources. 6 conclutions the paper provided a formal approach to generate strategies for the sustainable utilization of regional water resources in a planning period. under the condition to maintain the development of regional population, resources, environment and economy, it can help the relevant departments to clarify their ideas and effectively resolve the contradictory problem between water crisis and ecological environment construction. in real applications, the relevant departments can adjust the contents and values of characteristics of the condition basic-elements. the extension analysis result of water utilization is an open thinking way. it can make the decision-making of relevant departments be more standardized and scientific. the approach is also compiled into software that enables the decisionmaking process be intelligent. the extension generation strategy of sustainable utilization of regional water resources alters the only method of qualitative analysis in the past to the perfect combination qualitative analysis with quantitative calculation. this method is also suitable for the strategy generation of sustainable utilization of river basin. references [1] icwe. (1992), “international conference on water and the environment: development issues for 21 century”, the dublin statement and report of the conference, dublin. [2] wei-hua zeng, zhi-feng yang and gen-suo jia. (2006), “integrated management of water resources in river basins in china”, aquatic ecosystem health & management, vol.9, no.3 , pp. 327-332. [3] hong yang and alexander zehnder. (2007), “ ‘virtual water’: an unfolding concept in integrated water resources management”, water resources research, vol.43, no.w12301. advances in systems science and application (2016) vol.16 no.1 45 [4] jordi gallego-ayala and dinis juzo. (2011), “strategic implementation of integrated water resources management in mozambique: an awot analysis”, physics and chemistry of the earth, vol.36, pp. 1103-1111. [5] brian r. cook and christopher j. spray. 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(2009), “a multistage fuzzy-stochastic programming model for supporting sustainable waterresources allocation and management”, environmental modelling & software, vol.24, pp. 786-797. 46 qiaoxing li and naiang wang babu:research on extension strategy generation ... [14] han mei, duhuan, yangxiaoyan and liuyuan. (2010), “research advances on water resources optimal distribution”, procedia environmental sciences, vol.2, pp. 1912-1918. [15] bossel h. (2000), “the human actor in ecological-economic models: policy assessment and simulation of actor orientation for sustainable development”, ecological economics, vol.34, pp. 337-355. [16] zhai jin-liang, feng ren-guo and xia jun. (2011), “constraining factors to sustainable utilization of water resources and their countermeasures in china”, chinese geographical science, vol.13, no.4, pp.310-316. [17] yang chunyan and cai wen. (2013), extenics: theory, method and application, science press, beijing. [18] pan hulin. (2009), “the evaluation of iwrm performance and the analysis of its influential factors in the arid area: a case study on the iwrm of ganzhouqu district”, northwest normal university, lanzhou. corresponding author qiaoxing li can be contacted at: gxqxli@163.com and liqx@lzu.edu.cn characterization of lp c -solutions for the dilation equations on r2 chun-tai liu1 and guo-tai deng2 1department of mathematics and physics, wuhan polytechnic university, wuhan 430023, p. r. china 2college of mathematics and statistics, huazhong normal university, wuhan 430079, p.r. china email: lct984@163.com, hilltower@163.com abstract in this paper, the author discussed the existence of compactly supported lp-solutions for the dilation equations on the plane. furthermore, two examples are given to illustrate the general theory. keywords dilation equation compactly supportedlp-solutions iteration function system 1. introduction a α-scale dilation equation is a functional equation of the form f(x) = n∑ n=0 cnf(αx− βn) where f : r → r(or c), α > 1, β0 < β1 < · · · < βn are real constants, and cn are real (complex) constants. the equation is called a lattice k-scale dilation equation if f(x) = n∑ n=0 cnf(kx− n) for an integer k ≥ 2. a special case of the functional equation (k = 3, n = 4, and cn = 1, 2/3, 1/3, 1) was first studied by de rham[1] as an example of a continuous nowhere differentiable function. recently this equation has attracted a lot of attention, especially for the lattice case with k = 2. in wavelet theory, the study of multiresolution and the search of various orthogonal, compactly supported wavelets has lead to the investigation of the existence, uniqueness, and smoothness of such continuous integrable solutions[2]. the equation also plays an important role in the “subdivision schemes” and “interpolation schemes” of constructing continuous spline curves, surfaces and fractal objects [3, 4] . there are two major approaches to the equation: the fourier method(the frequency domain approaches) and the iteration method(the time-domain approaches). using fourier transformation, daubechies and lagarias[3] proved that the equation has a nonzero integrable solution. by using the fourier transform of f and the paley-wiener theorem, it was proved in [3] that f has compact support in [0, βn/(α− 1)]. the fourier method, however, does not give sharp criteria for the existence of l1 − solutions in terms of the coefficients{cn}. some partial results are given in [5, 6]. the iteration method is restricted to the lattice case. it applies particularly well in the case of compactly supported solutions. the basic idea is to identify a given function f supported by issn 1078-6236 international institute for general systems studies, inc. advances in systems science and applications (2011), vol. 11, no. 1-2 61-70 [0, n ] with the vector-valued function f(x) = [f(x), · · · , f(x+ (n − 1))]t , x ∈ [0, 1], and to use the right side of the dilation equation to construct two n × n matrices t0 and t1. a constant vector v is used as the initial condition, followed by iteration with the matrices t0 and t1. the limit, if the sequence converges, will be the solution of the dilation equation. such an approach was used by daubechies and lagarias[4], and independently by michelli and prautzsch[7]. it was also used by collela and heil[8] and [9] and ka-sing lau and jianrong wang[10]. similarly, on the plane the dilation equation is defined as the form f(x) = m∑ m=0 n∑ n=0 cmnf(ax−qmn) (1) where f : r2 → r or c, a is an expand matrix, qmn are vectors, cmn are real (or complex) constants. the equation is called a lattice dilation equation if f(x) = m∑ m=0 n∑ n=0 cmnf(ax− ( m n ) ) (2) where a is an integer expand matrix. in this paper we will study the existence of the compactly supported lp-solutionof the equation(2) on the plane with cmn ∈ r and a = ( 2 0 0 2 ) . as usually the basic assumption on the coefficients is m∑ m=0 n∑ n=0 cmn = | deta| = 4. let d1 = ( 0 0 ) , d2 = ( 0 1 ) , d3 = ( 1 0 ) , d4 = ( 1 1 ) , ϕk(x) = a−1(x + dk), k = 1, 2, 3, 4, there exists an attractor t = [0, 1]× [0, 1] satisfing t = 4⋃ k=1 ϕk(t ) , 4⋃ k=1 tk. at the same time there exist vectors {eis = ( i s ) , 0 ≤ i ≤m − 1, 0 ≤ s ≤ n − 1} such that suppf ⊂ m−1⋃ i=0 n−1⋃ s=0 (t + eis). let pi0 = (ci,2s−t)0≤s, t≤n−1, pi1 = (ci,2s−t+1)0≤s, t≤n−1, i = 0, 1, · · · ,m. for example, p00 = (c0,2s−t)0≤s, t≤n−1 =  c0,0 c0,2 c0,1 c0,0 c0,4 c0,3 c0,2 c0,1 c0,0 · · · · · · · · · · · · · · · · · · 0 0 0 0 0 · · · c0,n c0,n−1  . 62 liu:characterization of lp c -solutions for the dilation equations on r2 and let m1 = (p2i−j,0)0≤i,j≤m−1 =  p0,0 p2,0 p1,0 p0,0 p4,0 p3,0 p2,0 p1,0 p0,0 · · · · · · · · · · · · · · · · · · 0 0 0 0 0 · · · pm,0 pm−1,0  , m2 = (p2i−j,1)0≤i,j≤m−1 =  p0,1 p2,1 p1,1 p0,1 p4,1 p3,1 p2,1 p1,1 p0,1 · · · · · · · · · · · · · · · · · · 0 0 0 0 0 · · · pm,1 pm−1,1  , m3 = (p2i−j+1,0)0≤i,j≤m−1 =  p1,0 p0,0 p3,0 p2,0 p1,0 p0,0 p5,0 p4,0 p3,0 p2,0 p1,0 p0,0 · · · · · · · · · · · · · · · · · · · · · 0 0 0 0 0 0 · · · 0 pm,0  , m4 = (p2i−j+1,1)0≤i,j≤m−1 =  p1,1 p0,1 p3,1 p2,1 p1,1 p0,1 p5,1 p4,1 p3,1 p2,1 p1,1 p0,1 · · · · · · · · · · · · · · · · · · · · · 0 0 0 0 0 0 · · · 0 pm,1  we define a vector function: f (x) = (f(x+e00), f(x+e01), f(x+e02), · · · , f(x+e0,n−1), f(x+ e10), · · · , f(x + e1,n−1), · · · , f(x + em−1,n−1)) t for x ∈ t = [0, 1] × [0, 1], then equation (2) will satisfy f (x) =  m1f (ϕ−11 (x)) x ∈ t1 = [0, 1/2)× [0, 1/2); m2f (ϕ−12 (x)) x ∈ t2 = [0, 1/2)× [1/2, 1); m3f (ϕ−13 (x)) x ∈ t3 = [1/2, 1)× [0, 1/2); m4f (ϕ−14 (x)) x ∈ t4 = [1/2, 1)× [1/2, 1); 0 x ∈ others. (3) let v is 4-eigenvector of (m1 + m2 + m3 + m4),we have (m1 + m3 − 2i)v = −(m2 + m4 − 2i)v.and let ṽ = (m1 + m3 − 2i)v, h(ṽ) be the subspace in rm×n spanned by {mσṽ : σ ∈ σ∗}. then the basic theorem is as follows. theorem 1.1. for 1 ≤ p ≤ ∞, the following are equivalent: (1) equation (2) has a nonzero compactly supportedlp-solution; (2) there exists a 4-eigenvector v of (m1 +m2 +m3 +m4) satisfying lim l→∞ 1 4l ∑ |σ|=l ‖mσṽ‖p = 0 advances in systems science and applications (2011), vol. 11, no. 1-2 63 (3) there exists a 4-eigenvector v of (m1+m2+m3+m4) such that there exists an integer l ≥ 1 such that 1 4l ∑ |σ|=l ‖mσu‖p < 1 for all u ∈ h(ṽ), ‖u‖ ≤ 1 2. preliminaries lemma 2.1. if equation(2) exists compactly supportedlp-solutionf , then suppf ⊂ [0,m ]× [0, n ]. proof let suppf ⊂ d, take x ∈ d with f(x) 6= 0 thenax− ( m n ) ∈ d, i.e. x ∈ a−1(d+ ( m n ) ). let e = {0, 1, 2 · · ·m} × {0, 1, 2 · · ·n}, then d ⊂ a−1(d + e) = a−1d +a−1e ⊂ a−1(a−1d +a−1e) +a−1e = a−2d +a−2e +a−1e ⊂ · · · ⊂ a−td +a−te +a−(t−1)e + · · ·+a−1e let t→∞, then d ⊂ { ∞∑ t=1 a−ty : y ∈ e} ⊂ [0,m ]× [0, n ] for the closed set e. proposition 2.2. let f be supported by [0,m ]× [0, n ], and let f be defined as above, then f is an lpc−solution of (2) if and only if f ∈ lp and f = mf , i.e. f satisfies equation(3). proposition 2.3. if m∑ m=0 n∑ n=0 cmn = 4, then 4 is an eigenvalue of (m1 +m2 +m3 +m4) with left eigenvalue [1, 1, · · · , 1]. proof obviously, the sum of each column is equal to 4 in the matrix (m1 +m2 +m3 +m4). � it follows that the right 4-eigenvector of (m1 +m2 +m3 +m4) exists also; it will play a central role in the existence of the solution of equation(2). let f4 be the average of f over 4, i.e., f4 = 1 l(4) ∫ 4 f . proposition 2.4. let f be an compactly supported lp-solutionof equation(2), v = [ft+e00 , ft+e01 · · · ft+em−1,n−1 ]t be the vector defined by the average of f on the m ×n subintervals as indicated. then v is 4-eigenvector of (m1 +m2 +m3 +m4). proof according to proposition2.2, f = mf , i.e., f (x) =  m1f (ϕ−11 (x)) x ∈ t1 = [0, 1/2)× [0, 1/2) m2f (ϕ−12 (x)) x ∈ t2 = [0, 1/2)× [1/2, 1) m3f (ϕ−13 (x)) x ∈ t3 = [1/2, 1)× [0, 1/2) m4f (ϕ−14 (x)) x ∈ t4 = [1/2, 1)× [1/2, 1) (4) 64 liu:characterization of lp c -solutions for the dilation equations on r2 when we integrate the expression over t1, t2, t3 and t4 separately, we have [f[0, 1 2 ]×[0, 1 2 ], · · · , f[m−1,n− 1 2 ]×[m−1,n− 1 2 ]] t = m1v [f[0, 1 2 ]×[ 1 2 ,1], · · · , f[m−1,n− 1 2 ]×[m− 1 2 ,n ]] t = m2v [f[ 1 2 ,1]×[0, 1 2 ], · · · , f[m− 1 2 ,n ]×[m−1,n− 1 2 ]] t = m3v [f[ 1 2 ,1]×[ 1 2 ,1], · · · , f[m− 1 2 ,n ]×[m− 1 2 ,n ]] t = m4v on the other hand, note that on each interval [i, i+ 1]× [i, i+ 1] the average satisfies f[i,i+ 1 2 ]×[i,i+ 1 2 ] +f[i,i+ 1 2 ]×[i+ 1 2 ,i+1] +f[i+ 1 2 ,i+1]×[i,i+ 1 2 ] +f[i+ 1 2 ,i+1]×[i+ 1 2 ,i+1] = 4f[i,i+1]×[i,i+1], hence we conclude that (m1 +m2 +m3 +m4)v = 4v. let σ = {1, 2, 3, 4}, σn = {(i1, i2, · · · , in) : ij ∈ σ}, σ0 = ∅, σ∗ = ∞⋃ n=0 σn, σ∞ = {(i1, i2, · · · ) : ij ∈ σ}. for each σ = (i1, i2, · · · ) ∈ σ∞, define σ|n = (i1, i2, · · · , in). let σ = (i1, i2, · · · , in) ∈ σ∗, τ = (j1, j2, · · · , jm) ∈ σ∗, define (σ, τ) := (i1, i2, · · · , in, j1, j2, · · · , jm).tσ := 4⋃ i=1 t(σ, i), and mσ := mi1mi2 · · ·min . so for any σ, τ ∈ σ∗, we have t(σ, τ) ⊂ tσ. lemma 2.5. let f0(x) = v for x ∈ t , and fk+1 = mfk for k ≥ 0. then fk = mσv for each x ∈ tσ. moreover, if f is an lpc −solution of equation(2) and v is the average vector of f defined in proposition 2.4, then fk = mσv = [ftσ+e00 , ftσ+e01 · · · ftσ+em−1,n−1 ]t , where (tσ + j) = {x+ j : x ∈ tσ}. also, fk → f in lp(t,rm×n ). proof we will use induction to show that fk(x) = mσv for x ∈ tσ with |σ| = k. suppose that fk(x) = mσv for x ∈ tσ. let x ∈ t(1,σ) = ϕ1(tσ); then ϕ−11 (x) ∈ tσ and fk+1(x) = mfk(x) = m1fk(ϕ −1 1 (x)) = m1mσv = m(1,σ)v. similarly, if x ∈ t(i,σ), then fk+1(x) = m(i,σ)v, i = 2, 3, 4. moreover, f = mf and f (x) = mσf (ϕ−1σ (x)) for x ∈ tσ. integrating this over the interval tσ, we obtain [ftσ+e00 , ftσ+e01 , · · · , ftσ+em−1,n−1 ]t = mσv. lemma 2.6. let v be a 4-eigenvector of (m1 + m2 + m3 + m4), and let fk be defined as above;then for each k, ∫ t fk(x)dx = v. (5) advances in systems science and applications (2011), vol. 11, no. 1-2 65 proof this follows from the following induction argument:∫ t fk+1 dx = ∫ t1 m1fk(ϕ −1 1 (x)) dx+ ∫ t2 m2fk(ϕ −1 2 (x)) dx + ∫ t3 m3fk(ϕ −1 3 (x)) dx+ ∫ t4 m4fk(ϕ −1 4 (x)) dx = 1 4 ( m1 ∫ t fk(x) dx+m2 ∫ t fk(x) dx+m3 ∫ t fk(x) dx+m4 ∫ t fk(x) dx ) = 1 4 ( m1 +m2 +m3 +m4 )∫ t fk(x) dx = 1 4 (m1 +m2 +m3 +m4)v = v. theorem 2.7. for 1 ≤ p ≤ ∞, the following are equivalent: (1) equation (2) has a nonzero compactly supportedlp-solution; (2) there exists a 4-eigenvector v of (m1 +m2 +m3 +m4) satisfying lim l→∞ 1 4l ∑ |σ|=l ‖mσṽ‖p = 0 (3) there exists a 4-eigenvector v of (m1 +m2 +m3 +m4) such that there exists an integer l ≥ 1 such that 1 4l ∑ |σ|=l ‖mσu‖p < 1 for all u ∈ h(ṽ), ‖u‖ ≤ 1 (6) proof let f0 = v and fn+1 = mfn. by lemma (2.5), for x ∈ tσ and |σ| = n, fn(x) = mσv. let gn = fn+1 − fn; then fn+1 = f0 +g0 + · · ·+gn, where gn(x) = { m(σ,1)v +m(σ,3)v − 2mσv = mσṽ if x ∈ t(σ,1) ∪ t(σ,3), m(σ,2)v +m(σ,4)v − 2mσv = −mσṽ if x ∈ t(σ,2) ∪ t(σ,4), and ‖gn‖p = 1 4n ∑ |σ|=n ‖mσṽ‖p. since (1) implies that ‖gn‖ converges to zero, (2) follows immediately. to prove that (2) implies (3), we note that h(ṽ) is finite dimensional and has a finite basis of mτ ṽ’s. let u = mτ ṽ with |τ | = k; then 1 4n ∑ |σ|=n ‖mσu‖p = 1 4n ∑ |σ|=n ‖mσmτ ṽ‖p ≤ 4k 1 4n+k ∑ |σ|=n+k ‖mσṽ‖p → 0 as n → ∞, and the convergence is uniform for all ‖u‖ ≤ 1. hence (6) follows by taking l = n for n sufficiently large. 66 liu:characterization of lp c -solutions for the dilation equations on r2 now assume (3) holds. sinceh(ṽ) is finite dimensional, there is a constant 0 < c < 1 such that for any u ∈ h(ṽ), 1 4l ∑ |σ|=l ‖mσu‖p ≤ c‖u‖p. for any |τ | = n, let u = mτ ṽ ∈ h(ṽ); then 1 4l ∑ |σ|=l ‖mσmτ ṽ‖p ≤ c‖mτ ṽ‖p. summing over all |τ | = n, we have 1 4l+n ∑ |σ|=l+n ‖mσṽ‖p = 1 4l+n ∑ |σ|=l ∑ |τ |=n ‖mσmτ ṽ‖p < c 4n ∑ |τ |=n ‖mτ ṽ‖p. it follows from the expression of ‖gn‖ given above that ‖gn+l‖p ≤ c‖gn‖p. for each fixed n, {‖gn+kl‖}∞k=1 is dominated by a geometric series, hence fn+1 = f0 +g0 + · · ·+gn converges in lp. the limit f is nonzero by lemma (2.6), and so by proposition (5), (1) follows. remark 2.8. we can also consider the equation g(x) = m∑ m=0 n∑ n=0 dmng(bx− ( m n ) ) with b = ( 1 1 1 −1 ) and m∑ m=0 n∑ n=0 dmn = |detb| = 2, since iterating this equation again, we obtain the equation (2). corollary 2.9. under the same hypotheses of theorem (2.7), assume that the solution f exists; then v /∈ h(ṽ), and the dimension of h(ṽ) is at most mn − 1. proof by theorem (2.7)(2), 1 4n ∑ |σ|=n ‖mσu‖p → 0 for any u ∈ h(ṽ). it follows that if v ∈ h(ṽ), then ‖v‖p = 1 4np ‖(m1 +m2 +m3 +m4) nv‖p ≤ 1 4n ∑ |σ|=n ‖mσv‖p → 0 as n→∞. this contradicts v 6= 0. advances in systems science and applications (2011), vol. 11, no. 1-2 67 3. some examples example 1: we consider a dilation equation : f(x) = 1∑ m=0 2∑ n=0 cmnf(ax− ( m n ) ) (7) where a = ( 2 0 0 2 ) , 1∑ m=0 2∑ n=0 cmn = 4. theorem 3.1. for 1 ≤ p <∞, equation(7) has a (nonzero)lpc−solution if and only if either c01 + c11 = 2 and 1 4 (|c00 + c10|p + |2− (c00 + c10)|p) < 1 or c00 + c10 = 2 and c02 + c12 = 2. proof note that m1 = ( c00 0 c02 c01 ) ,m2 = ( c01 c00 0 c02 ) ,m3 = ( c10 0 c12 c11 ) ,m4 = ( c11 c10 0 c12 ) , and m1 +m2 +m3 +m4 = ( c00 + c10 + c01 + c11 c00 + c10 c02 + c12 c01 + c11 + c02 + c12 ) . if (c00 + c10, c02 + c12) = (0, 0), then m1 + m2 + m3 + m4 = 4i . any nonzero vector v = [x, y]t will be a 4-eigenvector. it is a direct calculation that v ∈ h(ṽ) and, by corollary(2.9), no nonzero lpc−solution exists. we assume that (c00 + c10, c02 + c12) 6= (0, 0), then 4-eigenvector of m1 +m2 +m3 +m4 is v = [c00 + c10, c02 + c12] t , so that ṽ = (m1 +m3 − 2i)v = ( (c00 + c10)(c00 + c10 − 2) (c02 + c12)(2− (c02 + c12)) ) . (8) for a lpc−solution exists, h(ṽ) can only be {0} or one-dimensional(corollary(2.9)). in the first case, ṽ = 0, condition(6) is automatically satisfied. the only possible cases are (c00 + c10, c02 + c12) = {(2, 2), (0, 2), (2, 0)}. in the second case, ṽ 6= 0. since h(ṽ) is invariant under m1 +m3 and m2 +m4, (m1 + m3)ṽ = cṽ for some c. expression (8) yields the following cases: (a) c00 + c10 = 0 or c02 + c12 = 0. in this case v ∈ h(ṽ) and corollary(2.9) implies that (7) has no lpc−solution. (b) c00 + c10 = 2 or c02 + c12 = 2. in this case a direct calculation shows that (m1 + m3)ṽ, (m2 + m4)ṽ are independent. since h(ṽ) is 2-dimensional and by corollary(2.9) no lpc−solution exists. 68 liu:characterization of lp c -solutions for the dilation equations on r2 (c) c00 +c10 6= 0, 2 and c02 +c12 6= 0, 2. let a = c00 +c10, b = c02 +c12. by equating (8) and (m1 +m3)ṽ = ( a2(a− 2) b[(2− b)(4− a− b) + (a− 2)a] ) (9) with (m1 +m3)ṽ = cṽ, we have c = a, so that by (8) and (9), b[(2− b)(4− a− b) + (a− 2)a] = ab(2− b); that is (a+ b− 4)(a+ b− 2) = 0. hence, either (i) or (ii) below holds. (i) a+ b = 4. in this case v = [a, 4− a]t and ṽ = (a− 2)v. once again v ∈ h(ṽ) and no lpc−solution exists. (ii) a+b = 2. in this case a direct calculation shows that (m1+m3)ṽ = aṽ, (m2+m4)ṽ = bṽ. by theorem(6), equation(7) has an lpc−solution if and only if there exists an integer l ≥ 1 such that 1 4l (|a|p + |b|p)l‖ṽ‖p = 1 4l ∑ |σ|=l ‖mσṽ‖p < ‖ṽ‖p. this is equivalent to 1 4 (|a|p + |2− a|p) < 1, i.e., 1 4 (|c00 + c10|p + |2− (c00 + c10)|p) < 1. the theorem follows by summarizing all the cases. it follows directly from the theorem that if a+ b = 2 and if (i) a ∈ (−1, 3), then an l1 c − solution exists; (ii) a ∈ (0, 2), then an l2 c − solution exists; (iii) a = 1, then an lpc − solution exists for all 1 6 p <∞. example 2: considering a dilation equation as follows: g(x) = 1∑ m=0 1∑ n=0 dmng(bx− ( m n ) ) (10) where b = ( 1 1 1 −1 ) and 1∑ m=0 1∑ n=0 dmn = 2.. iterating the equation once, we obtain the suppg ⊂ [−1, 2]× [0, 3]. let f(x) = g(x− ( 1 0 ) ), a = b2 = ( 2 0 0 2 ) , and let (cmn)0≤m,n≤3 =  0 d00d10 d01d10 0 d200 d00d01 + d210 d00d11 + d10d11 d01d11 d00d10 d00d11 + d00d01 d201 + d10d11 d211 0 d01d10 d01d11 0  advances in systems science and applications (2011), vol. 11, no. 1-2 69 then (10) is rewritten as the form f(x) = 3∑ m=0 3∑ n=0 cmnf(ax− ( m n ) ) (11) with suppf ⊂ [0, 3]× [0, 3], so the discussion of the equation (11) is similar with the example 1. references [1] rham, g.de.. sur un example de fonction continue sans dérivée. enseign. math., 3(1957) 71-72. 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[10] lau k.l. & wang,j.r. characterization of lp−solutions for the two-scale dilation equations, siam j.math.anal., 26(1995),no.4 1018-1046. 70 liu:characterization of lp c -solutions for the dilation equations on r2 advances in systems science and applications (2014) vol.14 no.2 144-157 disease universe: visualisation of population-wide disease-wide associations max moldovan1,2, ruslan enikeev3, shabbir syed-abdul4, phung anh nguyen4, yo-cheng chang4 and yu-chuan li4 1australian institute of health innovation, university of new south wales, level 1 agsm building, sydney nsw 2052, australia 2school of population health, south australian health & medical research institute (sahmri), university of south australia, north terrace, adelaide sa 5000, ausralia 3the apac sale group, 3 petain rd., #05-07, singapore 208108 4graduate institute of medical informatics, college of medical science and technology, taipei medical university, taipei, taiwan abstract over a lifespan, a human organism is affected by multiple disorders of different origin and severity. we apply a force-directed spring embedding graph layout approach to electronic health records in order to visualise population-wide associations among human disorders as presented in an individual biological organism. the introduced visualisation is implemented on the basis of the google maps platform and can be found at http://disease-map.net. we argue that the suggested method of visualisation can both validate already known specifics of associations among disorders and identify novel, never noticed association patterns. keywords systems biology; phonemics; electronic health records; visualisation; graph layout 1 introduction it is known that many human disorders are positively associated, accompanying each other due to various, often unknown, genetic, bio-pathological or common risk factors [?]. there is also evidence that some disorders tend to be associated negatively, playing a preventative role against each other, or due to other hypothesised but not properly understood reasons [2-3]. we use population-wide electronic health records data to visualise how human disorders are positioned against each other in a population with respect to an individual biological organism. by doing so, we attempt to execute a systems biology approach in order to reveal the presence of common functional mechanisms influencing pathogenesis behind groups of human disorders through biological, epidemiological or environmental factors. it is important to note that, due to the specifics of electronic health records [?], together with biological and environmental mechanisms, the method may reflect certain aspects of a healthcare system. for example, closely related but distinct diagnoses are often recorded against the same medical condition. this would induce a positive association between disorders due to healthcare administration rather than biological reasons. advances in systems science and applications (2014) vol.14 no.2 145 so far the attempts to characterise interactions between human disorders, as observed in a population, have been mainly implemented through network approaches [5-6]. while being no doubt informative, network approaches predominantly focus on positive association patterns. one of the alternative approaches to the disease-wide association analysis has been reported by rzhetsky et al. [?]. among other things, the authors objectively characterised probabilities of a person to be affected by a certain disorder, say a, given that the same person has been actually affected by an alternative disorder, say b. this approach allowed us to identify not only positively associated disorders, but also disorders associated negatively – the disorders “competing for the same nucleotide site in the human genome”, as hypothesised by the authors. the shortcoming of the study by rzhetsky et al. [?] is that the authors utilised patient records obtained from a single hospital, also covering a limited pre-selected number of diseases. in the following study, we use electronic health records covering the entire population and much wider range of human disorders. a central objective of the method and its implementation presented below is to visualise association patterns, both positive and negative, among human disorders as observed in an entire population. this would bring to the surface not only already known empirical facts, but also information not previously available. given that the method implementation can reflect empirical information already known, e.g., a strong positive association between hypertension and diabetes mellitus, as well as unexpected association patterns never noticed before, the presented visualisation can serve as a starting point for formulating novel testable hypotheses in the areas of healthcare and medicine. this can further lead to a better understanding of the complex unobserved dynamics of human disorders intersecting in a single biological body. such understanding can practically improve the delivery of healthcare and medical treatments. 2 material and methods 2.1 force-directed spring embedding graph layout algorithm imagine a single pair of nodes, a and b, positioned on a plane and connected by a spring of a certain natural length, δab. when the distance between a and b is exactly dab = δab, the spring is in a state of equilibrium, creating neither attraction nor repulsion forces between the nodes (fig.1(a)). moving a and b further apart from each other would create an attraction force (fig.1(b)), while moving a and b closer to each other would create a repulsion force between the nodes (fig.1(c)). given the values of initial required distances δij between multiple pairs of nodes, it is rarely possible to locate more than three nodes on a plane such that all required distances between them are satisfied exactly. in fact, it is not even always 146 max moldovan: disease universe: visualisation of population-wide disease-wide ... fig.1 a single pair of nodes in three possible states. (a) equilibrium: the nodes neither attract nor repulse; (b) attraction force: the nodes are shifted far away from equilibrium and attempt to attract; (c) repulsion force: the nodes are closer than if they were in equilibrium and attempt to repulse. possible to locate three nodes, keeping the pairwise distances intact, see fig.2. when distances between the nodes are not satisfied, springs connecting them are not in equilibrium, creating a certain force – either attraction or repulsion. fig.2 a hypothetical system of three nodes. the initial distances δij , ij ∈ {ab,ac,bc} between the nodes are given by the theoretical lines a′b′, b′c ′ and a′c ′. the joint length of a′b′ and a′c ′ is less than the length of b′c ′, i.e., δab+δac < δbc . as a result, for the nodes to connect, one or more of the initial distances between the pairs have to be distorted. the three possible states of springs are equilibrium (ab), attraction (ac) and repulsion (bc). aggregated forces created by out-of-equilibrium springs can be expressed by a specific function, drawn from the principle of physics (hooke’s law), leading advances in systems science and applications (2014) vol.14 no.2 147 to system’s potential energy u . while a force can be positive (attraction) or negative (repulsion), an energy level is always non-negative irrespective of the sign of the force. the potential energy of a system of m nodes connected by springs of varying stiffness can be expressed as follows: u = 1 2 ∑ ij∈k ( (dij − δ̂ij) 2 · κij ) , k = ( m 2 ) , i ̸= j (2.1) where dij = √ (xi −xj)2 + (yi − yj)2 is an euclidian distance between nodes i and j with coordinates (xi, yi) and (xj , yj), respectively, δ̂ij is a natural length of a spring between nodes i and j, κij ≥ 0 is an arbitrary parameter that defines the stiffness of a spring between i and j, and k is a number of all possible springs connecting m nodes, with (· · ) being a binomial coefficient. by varying pairwise euclidian distances dij , the force-directed spring embedding graph layout algorithm [7-9] performs a search for the configuration of node locations such that system’s potential energy u is minimised. by finding the minimum energy u , we attempt to obtain a shape of a system of nodes in which competing forces largely compensate each other. minimising function (??) is a complicated task due to the presence of multiple local minima, and it can rarely be guaranteed that a true global minimum is reached, see appendix for details. however, we observed that most nodes have nearly constant “designated” locations with respect to other nodes across alternative local minima achieved when minimising (??). 2.2 defining natural distances between human disorders observing a (sub)-population of size n , suppose that over period t there were ca individuals with at least one occurrence of disorder a, and cb individuals with at least one occurrence of disorder b. further, cab individuals presented with both disorders a and b, each disorder observed at least once over the same period. then the information can be summarised as shown by table 1. table 1 occurrence counts of disorders a and b in population of size n . disorder a disorder b a present a absent total b present cab · cb b absent · · − total ca − n table 1 is an example of a 2 × 2 table with fixed margins. assuming that individuals are affected independently of each other (which can be violated, e.g., 148 max moldovan: disease universe: visualisation of population-wide disease-wide ... for infectious diseases), x follows a non-central hypergeometric distribution x ∼ hyper(n,ca, cb) given by [10-11]: pr(x = cab) = ( cb cab )( n−cb ca−cab )( n ca ) eθabcab (2.2) where max(0, ca+cb−n) ≤ cab ≤ min(ca, cb), θ ∈ (−∞,+∞) is a log-odds ratio, and e = 2.718 . . . is the base of a natural logarithm. for algorithm implementation, conditional maximum likelihood estimates of θ were approximated by unconditional log-odds ratios: θ̂ab = ln ( cab(n − ca + cab − cb) (ca − cab)(−cab + cb) ) (2.3) where ln(·) is a natural logarithm. switching the risk factor from being b for a to being a for b does not effect log-odds estimates. natural (equilibrium) lengths of springs between nodes i and j were obtained through the following reversed expit transform [?, p.121]: δ̂ij = exp(−θ̂ij) 1 + exp(−θ̂ij) (2.4) where δ̂ij ∈ [0, 1] by construction. note that the sign on log-odds estimate θ̂ was changed to the opposite (i.e., reversed), making stronger positive associations correspond to the smaller values of δ̂ij . we do so in order to make δ̂ij resemble euclidian distances between the nodes. due to estimation, there is uncertainty about δ̂ values obtained from the data. such an uncertainty is usually handled by reporting confidence intervals corresponding to δ̂. in the present version of method implementation, we intentionally avoided using confidence intervals, p-values or other statistical tools normally involved in hypothesis testing. we did so in order to reflect the empirical information contained in the data without any subjective interpretation that could otherwise be introduced through, for example, the choice of a significance level. 2.3 potential alternative implementations it should be noted that the force-directed spring embedding graph layout method for visualising relationships among multiple objects, human disorders in our case, is not the only approach available. together with network algorithms already mentioned above, multidimensional scaling [?] and biplot [?] methods are two more approaches for two-dimensional visualisation of relationships between multiple objects. however, it is important to emphasise that both methods, at least advances in systems science and applications (2014) vol.14 no.2 149 in their traditional form, focus on similarities between objects, which would correspond to positive associations between disorders as per our current settings. while an application of alternative methods to our empirical data set would be interesting and potentially informative, multidimensional scaling and biplot methods are unlikely to address negative associations between disorders in a proper way. at the same time, euclidian distances between disorders, as specified by (??) in our study, could be easily reversed, i.e., positive and negative associations can be made corresponding to longer and shorter euclidian distances between the nodes in figure 1, respectively. such an alternative vision of the problem could reveal an entirely new set of empirical information, not reflected by the current method implementation. we retain this research direction for later investigation. 3 practical implementation 3.1 empirical information the presented visualisation has been motivated by the internet map implementation [?]. we use electronic health records obtained from the taiwanese national health insurance research database covering the entire population of taiwan over the period of three years (2000-2002). the same three-year observation window of the maximum available length has been used to record the counts corresponding to table 1. disorder records are based on icd9-cm (international classification of diseases, ninth revision, clinical modification), five-digit version. the dataset has been stratified into male and female groups. each of these two groups has been further stratified into ten age sub-groups, i.e., 0-9, 10-19, . . . , 90+. each subject within a sub-group was noted by his of her first insurance claim starting from 01 january 2000, assigned to a certain age-gender group and followed for the rest of the period ending on 31 december 2002. codes corresponding to e and v categories of icd9-cm (external causes of injury and supplemental classification) were excluded from consideration. 3.2 prevalence threshold and spring stiffness we compute δ̂ij given by (??) for each observed pair of disorders i and j. an empirical examination of δ̂ij revealed that log-odds estimates θ̂ij that underlie δ̂ij , exhibit anomalous behaviour for smaller counts ci and cj , i.e., they tend to be much larger than it would be expected under a random process. such an anomaly would bias the attention of an optimisation algorithm applied to (??) towards diseases of a smaller prevalence. we attributed this anomaly to exceptionally high positive associations between certain low prevalence pairs of disorders as observed in the context of the entire population and reflected by odds ratio estimates. in particular, the expected value of x in the hypergeometric distribution function 150 max moldovan: disease universe: visualisation of population-wide disease-wide ... fig.3 broad disease categories as per icd9-cm classification and the corresponding colour codes as displayed at http://disease-map.net. (??) when θ = 0, i.e., there is no association between disorders, is given by [?]: recij = cicj n (3.5) where rec stands for random expected co-occurrence. we interpret recij as a value reflecting “visibility” of co-occurrences between i and j, with higher visibility (i.e., greater values of recij) leading to more reliable empirical outcomes cij in table 1. keeping this interpretation in mind, we have executed the following ad hoc solution for dealing with the identified anomaly. firstly, we imposed the threshold c = √ 2n on disease occurrence counts. this guarantees that recij > 2 for all possible ci and cj . the meaning behind this restriction is to ensure that only theoretically “visible” θ̂ij estimates are used for visualisation. the cost is that we dismissed smaller prevalence disorders that never exceeded recij = 2 in any of the age-gender groups. secondly to the imposed lower limit on the observed occurrence counts, we set the stiffness parameter of a spring between pairs i and j to κij = ln(recij). this modification makes sure that less theoretically “visible” co-occurrences cij are given less importance when minimising the energy function (??). 3.3 visualisation the google maps platform (https://developers.google.com/maps/) was used to visualise the outcomes. the sizes of the nodes are set to be proportional to the advances in systems science and applications (2014) vol.14 no.2 151 observed disease prevalence in the corresponding age-gender stratified sub-groups. the colour codes of the nodes correspond to the broad disease categories as per icd9-cm classification, see fig.3. all maps are displayed in the same coordinate system with the same scale so that they can be compared against each other. 4 some examples of using the maps 4.1 an accidental proximity? when exploring the maps, some regions and mutual locations can attract attention due to certain, often subjective, reasons. for example, observing the map for females age 40-49 (f40+), in the central region towards south-east we find that peptic ulcer (icd9-cm 533) is located side by side with neurotic disorders (icd9-cm 300). have these disorders fallen close together by chance? a search through the medical literature has quickly identified that an abnormal association between peptic ulcer and neurotic disorders was noticed years ago [15-16]. looking at a wider category of digestive system disorders (icd9-cm 520 to 579), this mutual location pattern remains largely the same over multiple maps (e.g., see m30+, m40+ and f30+), leading to several testable hypotheses. one example of such a hypothesis would be: “there is no direct psychosomatic link between digestive system disorders and neurotic disorders”. viewed from a different angle, the literature coverage on pairs of closely located disorders appears not to be accidental. syed-abdul et al. [?] found that the proximity of disorder pairs is positively correlated with the degree of literature coverage, the latter being represented by a number of hits for a pair of disorders returned from an appropriate query to the pubmed search engine. 4.2 unlikely neighbours: a cancer-schizophrenia association puzzle the topic of observed evidence of associations between schizophrenia and various cancers has been widely debated [?]. evidence tends to point towards the presence of a negative association between schizophrenia and several cancers, even though there is no absolute consensus [20-21]. a shared genetic architecture has been proposed as a reason for observed associations [20-21]. alternatively, there is evidence that the chances of schizophrenia patients being timely diagnosed with certain types of cancer, on average, are lower than for general population nonschizophrenic patients [?]. exploring the maps, it can be found that schizophrenic disorders (icd9-cm 295) consistently fall on the southern border of the maps, sometimes being the most distant points from the imaginary centre of a “galaxy”, e.g., see m50+. interestingly, various types of cancers also regularly fall on the same southern border even though, consistently with the literature, association estimates for schizophrenia-cancer pairs regularly cross to the negative side, i.e., δ̂ij given by (??) exceeds the value of 0.5. 152 max moldovan: disease universe: visualisation of population-wide disease-wide ... one potential explanation for such an anomaly would be that schizophrenia and some cancers have closely related underlying causes revealed through similar relationships with other disorders. in this way, schizophrenia and cancers are placed to their southern border locations by the forces generated within the system. often being negatively associated, schizophrenia and cancers are like two sides of the same coin, “competing for the same nucleotide site in the human genome” as per rzhetsky et al. [?] vision, but potentially disassociated due to deeper, not properly recognised and understood reasons which are still to be identified and investigated. the visualisation we introduced is a tool for originating and directing such investigations. 5 conclusion electronic health records have become an integral part of national healthcare systems worldwide, and it is essential to comprehensively utilise the information contained in the growing number of databases. the method we introduced is one of the effective and informative tools for doing so. while the current realisation of the method has its obvious limitations, the presented maps are the first implementation of this kind and intended to set a reference benchmark for further developments in the same direction. a formal empirical validation of the introduced visualisation is beyond the scope of this paper, but based on the broad examination of the resulted maps, we argue that the presented implementation can both assist with validation of already known phenomena as well as with identification of novel, previously never noticed, association patterns related to functional aspects of medicine and healthcare. we suggest that the maps be used for generating testable hypotheses and invite the reader to explore the vast amount of information contained in them at http://disease-map.net. acknowledgements: we are grateful to chris lloyd and nicholas nechval who provided their constructive feedback that helped to improve the manuscript. we thank hanna kalkova and andrey stepanov for preparing figures 1 and 2. author contributions: m.m., r.e., s.s-a. and y-c.l. invented and developed the concept (ipn-13-000062, newsouth innovations). s.s-a., m.m., y-c.l. and y-c.c. performed selective biological validation. r.e. wrote software, implemented the method and developed the website. m.m. and r.e. wrote the manuscript. y-c.l. obtained the data. pa.n. organised and validated the data. funding: no formal targeted funding was allocated towards the project. competing financial interests: the authors declared no competing financial interests. advances in systems science and applications (2014) vol.14 no.2 153 references [1] ferrannini, e. and cushman, w.c. 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(2007), “risk of malignancy in patients with schizophrenia or bipolar disorder”, archives of general psychiatry, no.64, pp.1368-1376. [20] catts v.s., catts s.v., o‘toole b.i., and frost, a.d.j. (2008), “cancer incidence in patients with schizophrenia and their first-degree relatives – a meta-analysis”, acta psychiatrica scandinavica, no.117, pp.323-336. [21] gal, g., goral, a., murad, h., gross, r., pugachova, i., barchana, m., kohn, r., and levav, i. (2012), “cancer in parents of persons with schizophrenia: is there a genetic protection?”, schizophrenia research, no.139, pp.189-193. [22] crump, c., winkleby, m.a., sundquist, k., and sundquist, j. (2013), “comorbidities and mortality in persons with schizophrenia: a swedish national cohort study”, american journal of psychiatry, no.170, pp.324-333. corresponding authors maxmoldovan (max.moldovan@gmail.com) and yu-chuan li (jaak88@gmail.com) appendix: energy minimisation method finding a global minimum of (??) is a complicated task due to the presence of multiple local minima of this function. different approaches of global minimisation can be applied, but it can be rarely known when and if the global minimum advances in systems science and applications (2014) vol.14 no.2 155 is reached, unless a minimum energy level is known in advance. our current implementation of energy minimisation is to use multiple local searches with the conjugate gradient algorithm from random starting positions in order to obtain a master map – the map that includes diseases across the entire spectrum of age groups and both genders. in each of multiple attempts, the nodes are dropped on the map with random positions (x,y ), and the conjugate gradient algorithm runs searching for the closest local minimum of u in (??). if the new local minimum is less than the best (smallest) minimum recorded over previous attempts, it becomes the new best minimum. the procedure is repeated until the best minimum stops changing even after a reasonably large (4000, in our implementation) number of random allocation attempts, see algorithm ??. the computational complexity of the algorithm is o(n2). the obtained master map served as a collection of starting points for the agegender stratified maps, see algorithm ??. minimising (??) from a single set of starting points leads to a local minimum that can be further improved by applying the minimisation approach used for obtaining the master map. however, we still used minimisation from the single set of starting points in order to make the maps comparable across age groups and genders. table a1 reports the achieved minimum energy levels using “partial” minimisation as per algorithm ?? compared to the “full” minimisation implemented through algorithm ??. 156 max moldovan: disease universe: visualisation of population-wide disease-wide ... table a1 minimum achieved energy levels from partial and (attempted) full minimisation approaches. group subjects followed (n) disorder numbers partial full per cent improve f 0-9 1,677,365 565 7,807.96 7,700.69 1.39 f 10-19 1,595,057 743 9,470.09 9,166.34 3.31 f 20-29 1,780,095 1041 22,897.04 22,268.39 2.82 f 30-39 1,765,866 1136 25,914.10 25,387.60 2.07 f 40-49 1,631,968 1243 31,126.22 30,913.37 0.69 f 50-59 930,496 1251 33,451.35 33,334.80 0.35 f 60-69 711,096 1271 36,129.76 36,056.87 0.20 f 70-79 427,821 1177 29,935.15 29,857.36 0.26 f 80-89 141,225 783 10,802.72 10,773.33 0.27 f 90-99 8,532 176 318.26 315.74 0.80 m 0-9 1,827,447 630 10,068.44 9,910.12 1.60 m 10-19 1,678,415 721 9,451.03 9,346.16 1.12 m 20-29 1,767,163 859 12,532.53 12,345.32 1.52 m 30-39 1,737,715 948 14,263.07 14,099.18 1.16 m 40-49 1,577,320 1090 19,485.22 19,454.02 0.16 m 50-59 898,150 1065 20,296.22 20,247.20 0.24 m 60-69 692,061 1163 26,737.58 26,563.97 0.65 m 70-79 532,308 1225 30,740.90 30,622.10 0.39 m 80-89 133,480 781 10,636.67 10,599.66 0.35 m 90-99 4,769 151 240.25 238.70 0.65 master 21,518,574 2298 − 130,381.91 − advances in systems science and applications (2014) vol.14 no.2 157 algorithm 1 energy minimisation for the master map. require: γ ← 0.01 /* tolerance for the change in objective function (??) require: s← 1 /* initial step size require: τ ← 0.9 /* step decrease rate require: smin ← 0.000001 /* minimum step tolerance require: δ̂ij for k = ( m 2 ) pairs, i ̸= j /* pairwise natural lengths given by (??) require: ucurrent ← +inf /* current energy level to be reduced require: cc← 0 /* random positions attempts counter require: ccmax ← 4000 /* maximum number of attempts with no energy reduction while (cc < ccmax) do (x0, y0)← random() /* drop nodes at random positions u0 ← fe(x0, y0) /* value of objective function (??) g← {−∇ (fe(x0, y0))} /* define antigradients for the first step (∆x,∆y )← fg(g) /* step direction (x,y )← (x0, y0) + (∆x,∆y ) · s /* current coordinates of nodes u ← fe(x,y ) /* current value of objective function (??) ∆u ← (u0 − u) /* change in energy while (∆u > γ) & (s > smin) do gc ← {∇c (fe(x0, y0;x,y ))} /* evaluate conjugate gradients (∆x,∆y )← fcg(gc) /* step direction (xtemp, ytemp)← (x,y ) + (∆x,∆y ) · s /* trial coordinates of nodes u ← fe(xtemp, ytemp) /* current value of the objective function if u < u0 then ∆u ← (u0 − u) /* update change in energy u0 ← u /* update preceding energy value (x0, y0)← (x,y ) /* update preceding coordinates (x,y )← (xtemp, ytemp) /* assign the values of current coordinates else s← s · τ /* reduce step size end if end while if u0 < ucurrent then ucurrent ← u0 /* update minimum energy value (xcurrent, ycurrent)← (x,y ) /* update coordinates cc← 0 /* set attempts count to zero else cc← cc+ 1 /* next attempt end if end while (xmaster, ymaster)← (xcurrent, ycurrent) return (xmaster, ymaster) /* nodes’ coordinates under minimum energy achieved 158 max moldovan: disease universe: visualisation of population-wide disease-wide ... algorithm 2 energy minimisation for age and gender stratified maps. require: γ ← 1e-5 /* tolerance for the change in objective function (??) require: s← 1 /* initial step size require: τ ← 0.9 /* step decrease rate require: smin ← 0.000001 /* minimum step tolerance require: δ̂ij for k = ( m 2 ) pairs, i ̸= j /* pairwise natural lengths given by (??) (x0, y0)← (xmaster, ymaster) /* use coordinates from the master map as starting points u0 ← fe(x0, y0) /* the value of objective function (??) g← {−∇ (fe(x0, y0))} /* define antigradients for the first step (∆x,∆y )← fg(g) /* step direction (x,y )← (x0, y0) + (∆x,∆y ) · s /* current coordinates of nodes u ← fe(x,y ) /* current value of objective function (??) ∆u ← (u0 − u) /* change in energy while (∆u > γ) & (s > smin) do gc ← {∇c (fe(x0, y0;x,y ))} /* evaluate conjugate gradients (∆x,∆y )← fcg(gc) /* step direction (xtemp, ytemp)← (x,y ) + (∆x,∆y ) · s /* trial coordinates of nodes u ← fe(xtemp, ytemp) /* current value of the objective function if u < u0 then ∆u ← (u0 − u) /* update change in energy u0 ← u /* update preceding energy value (x0, y0)← (x,y ) /* update preceding coordinates (x,y )← (xtemp, ytemp) /* assign values of current coordinates else s← s · τ /* reduce step size end if end while (xstratif , ystratif )← (x,y ) return (xstratif , ystratif ) /* nodes’ coordinates under minimum energy achieved advances in systems science and application (2016) vol.16 no.2 39-53 principal causes of financial frises in latvia for last 20 years valery i. roldugin baltic international academy, latvia abstract in the article the author analyses activity of the latvian banks with the purpose of search of ways from a crisis situation. in the middle of 2000th rapid development of crediting resulted in negative tendencies. it was assisted by not only superfluous liberalization of crediting but also decline of quality of supervision. successful development of economy of latvia in a great deal depends on stability of commercial banks that accumulated money resources. besides the commission of market of finances and capital, successful development of the banking system in a great deal depends on activity of central bank and government. therefore an exit from a financial crisis depends on latvian commercial banks will be able to tide over financial difficulties and pick up thread crediting of enterprises. secondly, on the basis of the analysis of events development in the banking area the author discloses the true reasons which brought the latvian economy to the deepest financial crisis among the countries of the european union. the period from 1995 till 2014 is being investigated. the author uses a wide range of research methods, such as: grouping method, method of comparison of financial ratios, etc. keywords central bank; commercial banks; financial market; bank lending; real estate market. 1 introduction after latvia had regained its independence the central bank began to reform the credit system. we can note both positive and negative moments of its activity. undoubtedly, one of the reasons of banking crises in 1995 and 1998 were the shortcomings of activity of the bank of latvia[1]. the fast price liberalization, cancellation of subsidies, manipulation with funds of the enterprises at exchange of old money to new one also contributed to the creation of certain problems. at the same time the essential changes in relation to streamlining of requirements for credit institutions occurred in the field of regulation of all banking system in latvia. this has allowed the strengthening of the position of the whole credit system of the country, increasing confidence in the banks and improving of their financial performance. prior to 1995 the number of failures did not exceed two or three cases per year. in a result of crisis, the number of licenses revoked by the bank of latvia reached 15: tautas banka, latgales komercbanka, latintrdes banka, depozitu banka, 40 valery i. roldugin: principal causes of financial frises in latvia for fast 20 years centra banka, alejas komercbanka, polarzvaigzne, liepajas komercbanka, kredo banka, olimpija, oltibanka, talsu komercbanka, lettika, rigas starptautiska banka and banka baltija. in 1996, more seven banks have lost their licence: tukuma banka, daugavas banka, bauskas banka, ako banka, dinastija, jelgava, atmoda. despite the fact that the number of closed banks has been high, the total bank capital had resumed quickly. a considerable reduction of banks and the fact that the top ten banks were in this list, including the country’s largest bank banka baltija, significantly affected the main indicators of the industry, what gave a reason to speak about the banking crisis in latvia in 1995. 2 bank crisis in 1995 one of the main reasons of the banking crisis of 1995 was the fact that banks did not manage the risks in due manner. banks have been working in markets which had not been enough good explored. the recession, slow rate of reforms and the imperfection of legislation kept the already high degree of risk in bank activity. many of them enjoyed the demand for bank loans. they provided loans and attracted deposits at high rates, which had fallen harshly subsequently. a tough position of the bank of latvia in relation to the supervision over commercial banks had a considerable importance for stabilizing the banking system. only a year after the crisis, the bank of latvia issued a dozen of rules, guidelines and regulations. the most important of them included the toughening of regulatory requirements for commercial banks, providing the bank of latvia with information about shareholders, restrictions on attracting by banks of deposits from individuals, which had limitations in financial activities, mandatory annual audits by international audit companies, coordination with the bank of latvia of any changes in the capital, in management, etc. though all these measures have complicated the activity of commercial banks, streamlining of their activity was justifiable because of positive efficiency in development of banking system. some banks have had incompetent managers who failed to forecast the development of financial and currency markets. notwithstanding to the valid legislation the bank of latvia was approving people without economic education, or even without experience of work in the banking system, for the positions of chairmen of the board and their deputies. there was a shortage of qualified specialists in the field of lending, foreign exchange and bank marketing. as a result, the mistakes in lending and evaluating business plans were quite often. the banking regulatory authorities did not carry out a due control. advances in systems science and application (2016) vol.16 no.2 41 3 bank crisis in 1998 according to the law “on credit institutions”, which came into force in october, 1995, the bank of latvia got new supervising powers in relation to both licensing of banks and essential changes in structure of shareholders[2]. bank of latvia considerably strengthened this control, especially with acceptance of new rules of licensing in the beginning of 1996. the bank of latvia regularly checked banks records and events connected with the change of shareholders, and if the rules weren’t observed in full completeness, it applied the financial sanctions. the bank of latvia established a wide set of standards and regulations for credit institutions. developing them, it proceeded from principles and the standards valid in banking systems of the developed countries, considering eu regulations in the bank sphere and principles of basel committee of bank supervision. some of them were even stricter, for example, the requirement to the capital sufficiency (10%) to cover the defined risks. basel committee set a sufficiency of the capital at a level of 8%. it is necessary to notice that this standard had no big importance at that time since the capital sufficiency of bank industry at that time was more than 17%. at that time the bank of latvia has played an important role in the regulation of bank liquidity in order to limit the fluctuations in interest rates. as a matter of fact the value of interest rate is defined by the cash flows and the value of supply and demand in the financial market. the bank of latvia set a refinancing rate, which served as the initial point for the whole banking system. the short-term loans have been provided to the commercial banks under this rate. one of the most popular instruments of the bank of latvia was the credit refinancing which using began in 1993 by issuing the short-term loans to ensure the liquidity. in the late 1990-s there was a tendency in latvia of lowering of the refinancing rate. the interest rate of refinancing established in 1993 at level of 120% per annum, decreased to 4% by 1997. such policy made it possible to create more favorable conditions for banks lending, what made it possible to carry out a policy of credit expansion. the commercial banks lowered the interest rates, making their loans more available for economy development. the increased demand for loans thus caused the increasing of investments. initially, the loans were given to each bank within specific maximum limits which depended on the fulfillment of regulatory requirements stipulated by the bank of latvia. such order was necessary because the loans were given without any collateral. the amounts of the refinancing were relatively limited, as most of them have been used for foreign currency converting. though the loans were small, nevertheless they allowed banks keeping the liquidity in cases when those couldn’t get them at the interbank market which was poorly developed at that time. 42 valery i. roldugin: principal causes of financial frises in latvia for fast 20 years the development of securities market has reduced the necessity to cover the domestic debt by the loans of the bank of latvia. as a result, there appeared a possibility to limit the additional emission and to reduce inflation. the internal state bonds were the main source of the budget deficit financing. in 1996, the demand was much higher than supply. the amount of reserve requirement established by the bank of latvia on july 15, 1993 was 8% of the balances involved in all types and terms of money. this meant that each bank having a license was obliged to reserve the average daily funds appropriate to the amount mentioned in the requirements in the period from 16th day of previous month until the 15th day of the reporting month. therefore, the amount of funds in different days could differ considerably. at the same time, 50% of the required reserves could be the average cash balances, while the rest part should be reserved on the correspondent account with the bank of latvia. as of december 1998, the volume of bank reserves amounted to 129.9 million lats, including 34.3 million in cash balances. cash balances in foreign currency were not taken into account in meeting the reserve requirements. assessment of standards of execution of reserve requirements under average value gave the banks chance to handle their funds more freely and to provide liquidity in settlements. in order to consider the reserve requirements as monetary instrument to control the money supply, in april 1, 1997 the bank of latvia gave the banks the right not to include the obligations related to the transactions with foreign central banks, local and foreign credit institutions into the calculation of reserve requirements. as a result, banks got an opportunity to increase the money supply in the frameworks of this conservative monetary policy instrument. after a year since the introduction of these changes, the volume of liabilities to foreign central banks and credit institutions increased by more than three times. this gave a chance to attract additional loan resources for a longer period of time. at the end of april 1997 they amounted to 57.1 million and increased to 183.5 million to the end of april 1998. in 1997, the lowering of credit rates promoted the lowering of discount rates for liabilities. in the beginning of 1997, they were 8-10.5 %, and decreased up to 3.5-5.5 % by the end of the year. the total amount of the state securities in 1998 fell to 18.4 %, and amounted to 127.0 million in the end of the year. respectively, the share of all term liabilities available in circulation decreased in comparison with the previous year, i.e. 12 month liabilities-36.5 %, 6 month-8.0 %, 3 month 1.7 %, 1 month-0.2 %, and the share of 2 year bonds was 53.6 %[1]. however, the sustainability of financial market at that time was achieved mainly due to foreign currency interventions. it was possible to reduce loans interest advances in systems science and application (2016) vol.16 no.2 43 rates owing to the operations of buying and selling of foreign currency, what favorably affected the formation of the business environment. a tough position in establishment of the rate of lat allowed limiting the inflation, increasing the fiscal discipline, creating the conditions for sustainability of business and the confidence of foreign investors. it is worth regretting about poor coordination between the government and the bank of latvia. should the national currency be supported timely by reasonable tax policy stimulating, it could be possible to cause a domestic investment demand, which would enable a more efficient structural change in the economy. in process of its development the monetary market and securities market would have a stimulating effect on production development, creating the conditions for linkage in a trade turnover of considerable monetary funds. since july 1, 1997 the “terms and conditions of trade (currency swap transactions) on the purchase with subsequent sale and sale with subsequent lats repurchase” came into force. the bank of latvia has developed them with the aim to introduce a new financial instrument. the terms of such transactions does not exceed three months. since may 12, 1998 the bank of latvia began organizing auctions of currency reverse transactions quarterly. the volume of currency “swap” transactions in december 1998 amounted to 23.1 million lats, including 7.5 million lats with 7 day period (average interest rate 6.8%), 6.9 million lats with 28 day period (average interest rate 7.5%), 8.7 million lats with 91 day maturity (average interest rate 8.6%). regarding the involved deposits the demand deposits dominated in the banking structure. in april 1998, their share in total deposits was 73.4%[1]. lets note, that the currency market traditionally has been one of the most developed and liquid sector of financial market in latvia. its average turnover per day amounted to 279.9 million lats as of april 1998. in contrast, the average daily turnover of interbank market was almost four times less (76.3 million lats) during this period. prior to 1994 it was impossible to carry out the operations on open market, as government securities werent available. after their issuance the volume of state bonds grew relatively slowly, and their liquidity was not enough high, because the secondary market of liabilities was inactive and poorly developed. such operations gained their significance after 1997. the volume of state securities in the secondary market amounted to 704.5 million lats in 1998. the share of transactions with residents has increased from 46.4% (in 1997) to 78.0% (in 1998), and non-residents transactions decreased from 18.0% to 2.3%. by carrying out the operations in the open market the regulation of the reserves has been made providing the fixed level of interest rate, defined by a tactical task in spite of the eventual changes of interest rates specified by impact of market 44 valery i. roldugin: principal causes of financial frises in latvia for fast 20 years processes. the bank of latvia carried out the open-market purchases of securities as a stabilizing measure, thereby extending bank reserves and increasing the money supply to the extent that was sufficient to keep interest rates on the same level. this increase in money demand put pressure on the interest rate. as the interest rate exceeded the target level, the bank of latvia started buying the securities from commercial banks. the process of using the excessive loan resources increased the money supply in circulation. open-market purchases continued until the moment when sufficient amount of new money has been formed ensuring the compliance with the demand and supply at a fixed interest rate. thus, the latvian market has been achieving a certain balance. when the bank of latvia was selling the securities to the banks, sterilizing the money supply, the interest rates were increasing. the volume of transactions on purchase and sale of securities amounted to 109.5 million lats in 1998 (in previous year 89.7 million lats). the most volume of transactions was in august (purchase 30.2 million lats) and september (sale 11.5 million lats). the bank of latvia was buying mainly the government bonds, increasing its securities portfolio and replenishing the amount of active money by such operations. the securities portfolio of the bank of latvia in the middle of 1998 was 40.6 million lats (at par). since october, 1995 the bank of latvia began actively using “repo” auctions. the interest rates on “repo” auctions fixed by the bank of latvia were differentiated on terms and could be both above and below the level of a rate of refinancing. the bank of latvia carried out “repo” auctions daily, offering the credits for term of 7, 28 and 91 days. the interest rate at auctions exceeded a refinancing rate a bit. in 1997, the volume of the credits of the “repo” auctions with validity period of 7 days amounted to 55,7 million lats , for 28 days-33,1 million lats, for 91 days-5,7 million lats. in 1997, the bank of latvia credited the commercial banks in amount of 112.6 million lats that was more on 3.5 % than in 1996. their structure looked as follows: repo auctions-83.9 %, auto pawn-14.8 %, a pawn on demand-1.3 %[1]. the volumes of lending have been increasing every year. for example, in 1998, the loans to commercial banks reached 458.0 million lats, which was 4.1 times more than in 1997. from them: 58.8% was issued in repo auctions, 33.1% of pawn demand, 4.2% auto pawn and 3.9% loans in emergency situations. in 1998, repo loans for 194.4 million lats were issued with 7 day term, 40.6 million lats28 day term and 34.5 million lats with 91 day maturity. therefore, the commercial banks could solve the problem of short-term liquidity, and the bank of latvia knowing the maturity of loans, could make short-term forecasts of amendments in the liquidity of all banking sector. advances in systems science and application (2016) vol.16 no.2 45 while and today, one of the problems slowing down the development of a national economy of latvia[3, 4], was a deficit of balance of payments. foreign trade still remained a weakest point of national economy. in 1997, there was a growth of both in export and import. and the gap between them increased by high rates, creating a considerable deficit in the countrys trade balance. the volume of foreign trade in 1997 amounted to 2554.1 million lats that more on 23.2 % than in previous year. the deficit reached 610.7 million lats at export amounting to 971.7 million lats in1997. the payment of the international services already couldn’t compensate the missing foreign currency to pay the imported goods. the missing currency could be received generally by the means of loans. therefore, the trade surplus of the balance of payments at that time depended on capital flow more and more. it was clear, that since the producers of the goods don’t find the new markets and new buyers, the country will import on credit, live in the account of converting the foreign loans. the only one solution of successful development of economy was to search ways of increase the inflow of foreign investments into manufacture and infrastructure. the current account deficit of balance of payments could be financed either by foreign funds coming to the country or by using of foreign exchange reserves. the balance of payments summarizes the flows of goods, services, capital and related income interest payments, dividends of latvian residents and foreign countries. considering the limitation of the state budgetary funds and caution of commercial banks at crediting of the long-term projects, only the foreign investments could be the main source of long-term capital. however, the inefficient economic management, when the ministers of economic departments have been changed several times a year, volatile fiscal policy, the absence of free land market, the disorder in legislation and other factors prevented the attraction of foreign investments. since 1996, the government tried to limit the growth of external debt. the main reason lied in inefficient use of earlier received credits. in relation to state debt and the budget deficit, latvia focused on the implementation of the maastricht requirements. the external debt included loans from international financial organizations and the private sector. a number of loans was provided to latvia for a considerable period and under very favorable terms, i.e. the loan of the international bank for reconstruction and development-for 17 years (under 7.6% per annum), exim bank of japan for 16.5 years (4.7%), the european bank for reconstruction and development for 15 years, the credit of the american “commodity credit corporation” 30 years (terms of initial payments were extended to 5 years). the balance of state debt at the end of 1998 amounted to 231.5 million lats (6.1% of gdp) , and the government has issued guarantees in amount of 42.9 46 valery i. roldugin: principal causes of financial frises in latvia for fast 20 years million lats in order the enterprises could get the foreign loans. in 1998, the government has spent 37.0 million lats to administrate the external debt. the amount spent for servicing the external government debt equaled to 3,5% of annual export amount. the domestic debt included the government borrowings in the local market. it grew every year. for example, it increased by 5.4% and amounted to 8.3 million lats during 1997[1]. contrary to the previous crisis, the bank crisis of 1998 was caused not only by internal shortcomings of activity of latvian banks, but also by many external factors. the financial crisis in russia burst in august, 1998 was the main reason. respectively, the russian crisis negatively affected the development of the banking sector of latvia in the second half of 1998, as the majority of banks formed the considerable part of their assets by investments to cis countries. the main aspect was that considerable monetary resources left latvia in the form of investments into foreign securities because the latvian banks had actively operated on securities markets of other countries. its known that the commercial banks way out from a crisis situation in 1995 was reached for the account of increase of the operations with investments and converting of currencies in russia and cis countries. the bank of latvia didnt see it. in november, 1996, the investment into foreign securities reached 121.5 million lats and amounted to 171 million lats in august, 1998. such migration of capital was stipulated by high interest rates on the russian government securities and securities of the entities what created a possibility to receive profit by the latvian banks. the interest rates nearly three times lagged behind their counterparts in the cis at that time in latvia. unfortunately, the bank of latvia didn’t take the timely actions to protect a national banking system from possible financial crashes in the monetary markets of cis countries. the financial crisis of 1998 brought serious changes to the development of bank sphere of latvia. it negatively affected the activity of all banks. the total losses of credit institutions according to financial reports of auditors for 1998 amounted to 56.6 million lats (only 8 banks got profit in amount of 3.5 million lats). two banks got bankrupt (latvijas kapitalbanka and komercbanka viktorija), and rigas komercbanka was in a state of insolvency with losses amounting to 9 million lats in the beginning of 1999. parekss banka had the greatest profit-1.21 million lats (the total declared profit before audit 5.6 million lats). the losses of latvijas unibanka were 15.1 million lats[1]. in spite of the fact that the bank of latvia revoked licenses of two banks during 1998: latvijas zemes banka (in connection with the merger with hansabanklatvija) and doma banka (due to insufficiency of the equity), we can state that the whole banking system has successfully overcome the negative consequences of crisis. the positive changes in key indicators of the banking industry were the advances in systems science and application (2016) vol.16 no.2 47 evidence as well as the fact that 19 banks from 24 finished the year of 1999 with a profit. the process of enhancement of the legislation continued. in 1998, two new laws came into force: “on the prevention of laundering of proceeds derived from criminal activity ” and “on guarantees of deposits of individuals”. the adoption of the law ”on the financial and capital market commission” was an important event in the process of improving the legislation base[5]. the introduction of unified supervision on the market of finance and capital in latvia became a significant event. as a result, agency on supervision of credit institutions of the bank of latvia, state inspection for supervision of the insurance market and the commission of the securities market were united into the financial and capital market commission. the number of banks in latvia table 1 bank and branches in latvia from 2005 to 2014 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 total number of the banks 22 21 21 21 21 21 20 20 19 17 branches of foreign banks 1 3 4 6 8 10 9 9 9 10 number of branches 215 224 235 247 231 223 406 342 313 average number of employees 10,520 11,611 13,334 14,381 12,628 11,616 10,256 9,845 9,362 customers current accounts, thousands 2,912 3,316 4,341 4,545 4,462 4,525 4,539 4,568 4,581 3,078 customer accounts available via internet, thousands 2,538 2,859 3,015 3,163 3,230 3,444 3,643 2,618 payment cards, thousands 1,711 2,107 2,389 2,518 2,478 2,424 2,325 2,381 2,369 2,305 number of atms 878 957 1,143 1,274 1,320 1,359 1,207 1,270 1,154 1,068 pos terminals accepting payment cards 18,495 17,571 20,367 23,350 24,381 24,366 25,430 26,259 29,066 31,121 has considerably reduced in a period from 1992 to 2005. thus, if there were 61 commercial banks in 1993, in 2005 the number of commercial banks reduced to 22 banks (in 2014 -17 banks). the licenses of commercial banks have been issued by the bank of latvia which carried out the supervision over banks activities. since 2001, the bank of latvia handed over these responsibilities to the financial and capital market commission. 4 financial crisis of 2008 the volume of loans increased up to 1,086.7 million lats at the end of 2000, 3.8 times more than in 1996. the year of 2001 has been the most successful for 48 valery i. roldugin: principal causes of financial frises in latvia for fast 20 years the banking sector of latvia. the best indicator of the annual profit in amount of 46.7 million lats has been achieved this year. the indicators of all banking industry have been growing rapidly, i.e. the assets grew by 28.2%, deposits by 24.6% and shareholders’ equity 35.4%, and the volume of loans increased by 50.5%[1]. by this time, the main role in development to a bank system was played by three largest latvian banks. parekss banka was the leader of banking system in the beginning of 2000. it was the largest bank in latvia by size of total assets, equity and the attracted funds of the clients. from the beginning of 2000 to 2003 the assets of parekss banka increased by 74 % and reached 954 million lats. speaking about the dynamics of development among the largest banks, the leader was hansabanka which assets have been doubled for the similar period. latvijas unibanka, the second-top assets bank of latvia showed less aggressive dynamics at that period. its assets increased only by 57 %. at the same time, latvijas unibanka had the greatest amount of the loans among the largest banks of latvia 543 million lats. it was the largest lender among the largest latvian banks, the share of its credit portfolio in assets composed 74.5 %. this indicator clearly reflected the credit specialization of latvijas unibanka. it should be noted that credit-oriented latvijas unibanka had 135 ratio as regards of the issued loans and clients funds. it means that the bank allocated not only all client funds for loans, but also 35 % of other involved funds and own resources. accounting the supervision by the scandinavian bank group (seb), most likely the loans were financed from parent bank as well. hansabanka had the second rank by amount of the issued loans amounted to 497 million lats. hansabanka had essentially increased the credit portfolio for the last two years and bypassed of parekss banka under this indicator. hansabanka share of the issued loans in assets had been also high-70.5 %. parekss banka had relatively small credit portfolio among three largest banks of latvia (381 million lats), and its dynamics lagged behind the growth of credit portfolios of other banks in that triplet. in whole, the structure of assets of parekss was oriented on the investments into securities and interbank operations. after all, the credit portfolio even in structure of assets of parekss banka composed approximately 40 %. so, the triplet of system banks approached to today’s crisis with such indicators. regarding the prospects of development of the largest commercial banks of latvia it was possible to notice the following aspect: the affiliated structures of foreign banks (latvijas unibanka and hansabanka) still continued the dynamic development, using the resources of parent structures and favorable position on the market. these two banks were targeted to the increase of amounts of loans. parekss banka has been managed by major shareholders and constantly was advances in systems science and application (2016) vol.16 no.2 49 in process of search of the strategic investor. the parex financial group enlarged what fostered to attract the additional capital. with time, it was extended by parex bankas (lithuania); insurance companies parekss apdrosinasanas kompanija (latvia) and baltic polis (lithuania); leasing companies parekss lizings (latvia) and parex lizingas (lithuania). the activities of broker company parekss brokeru sistema and the pension fund parekss atklatais pensiju fonds effectively supported the activity of parex business group. the volumes of mortgage loans have formed a significant part of all loans issued in latvia. within 2007, the total amount of mortgage loans increased by 40.6% and their share in total loans was 52.9% that was 1.1% more than at the end of the previous year. in comparison with the end of 2006, the total loans to businesses for current assets increase, has grown by 24.4%, and its share in total loan portfolio of commercial banks amounted to 20.3%. the aggregate loan portfolio of commercial banks at the end of 2007 was divided between residents and non-residents of latvia as follows: residents-87.9 %, non-residents-12.1 %. similarly, the loan portfolio of borrowers-residents was divided as follows: private non-financial companies-49.8 %, households-41.2 %, financial institutions-6.7 %[1]. on the end of 2006, the latvian banks issued 916.5 thousand loans. in 2007, the number of the issued loans increased on 211.4 thousand (30 %). the number of the loans issued to private residents for acquisition, reconstruction or repair of housing has increased by 28.5 thousand (27.6 %) and reached 131.7 thousand loans; the consumer loans increased on 43.5 thousand (38.3 %) and reached 157 thousand loans. the number of the loans on individuals payments cards and accounts reached 583.9 thousand and increased on 131.9 thousand (29.2 %) in comparison with previous year[1]. the real estate in latvia was one of the basic instruments of savings. throughout more than ten last years it has been bringing high profits not only to businesses but also to individuals. a rare company did not invest their money into real estate transactions. these years were marked by prices explosion for all types of apartments throughout latvia. for example, there was 100% growth of prices in riga and its vicinity only in 2006. we can assume that the main reasons for such prices increases were the insufficient amount of new housing, the increase in solvent demand of people and growth of speculative transactions.the real estate in latvia was one of the basic instruments of savings. throughout more than ten last years it has been bringing high profits not only to businesses but also to individuals. a rare company did not invest their money into real estate transactions. these years were marked by prices explosion for all types of apartments throughout latvia. for example, there was 100% growth of prices in riga and its vicinity only in 2006. we can assume that the main reasons for 50 valery i. roldugin: principal causes of financial frises in latvia for fast 20 years such prices increases were the insufficient amount of new housing, the increase in solvent demand of people and growth of speculative transactions. in a result of housing crediting the number of real estate transactions in the mortgage market increased considerably. if in 2000 there were registered 22,689 transactions of real estate sale and purchase, by the beginning of 2007 their number was doubled, and 65 % of them were financed by the bank loans[6]. the basic reasons of crediting growth were: first, availability of bank loan upon housing purchase, secondly, a rise in prices for real estate. the commercial banks themselves played a considerable role in real estate prices rise while demanding a security at crediting. therefore, even the inexpensive apartments were accepted as a security by the prices which was seemed unrealistic to the appraising firms even a year before. as well, the decrease in interest rates for long-term loans played its role on the latvian real estate market. at the end of 2004 the price explosion was observed on the market, the price grew even several times on specific segments. the greatest growth was observed on a segment of apartment houses, i.e. in riga they increased more than 7 times in average in comparison with 2000. such growth had forced bank of latvia to undertake the urgent measures on restriction of amounts of crediting, including increase in interest rates for longterm loans. as a result, if in 2004 the average interest rate in latvia decreased by 2.5 percent points compared to 2001, for the next 4 years it increased again by 6.9 percent points, and made 14.6 % per annum in 2007[1]. it is necessary to notice that the consumer loans have been issued not only by banks, but also by other companies having no relation to bank activity and not supervised by the financial and capital market commission. according to the research of dnb nord banka, the latvian real estate market was saturated by 1.8 billion lats of free money along with the bank mortgage loans since the beginning of 2007, i.e. about 150 million lats per month (5). the calculations were made on the basis of data of the real estate companies stating that a half of transactions on housing acquisitions for the last year were financed by such loans. in april, 2007, the government developed the program to fight against inflation. the most efficient of this 30-points program for chilling the market were the following points: the developer must brought to bank his/her own funds in amount of 10 % of a project cost prior to the beginning of construction as well as must submit the reference about the amount of the received income. as a result, the performance indicators of latvian real estate market changed extremely negative. the volumes of mortgage lending began to fall quickly even before world financial crisis. the prices for new housing gradually decreased as well as on secondary housing market. thus, according to the real-estate compaadvances in systems science and application (2016) vol.16 no.2 51 nies, the prices for serial housing in riga have decreased on 8-10 % in a year. the real estate agents werent able to find buyers for three thousand apartments in newly built houses. the number of real estate transactions fell considerably, the income of realtors and developers respectively fell. increasingly, they have started offering discounts on real estate aiming to attract new customers. the construction boom went on recession. the bankruptcies of construction campaigns began. the developers and other builders could hardly to obtain mortgage loans in banks. there passed the first auctions on sale of real estate which was mortgaged earlier in banks. the developers and real estate companies were not in a hurry to reduce the prices hoping that the situation in the market would change and demand for real estate will go up again. the potential buyers, in their turn, didn’t hurry to conclude the real estate transaction, assuming the prices for property to become even lower. such situation has coincided with world financial crisis which much more aggravated a situation in the latvian market. the hopes of developers, realtors and buyers of real estate couldn’t come true, and the fast development of a crisis situation equalized all sides regarding the responsibility for the situation occurred in the market. the predicted trend has been implemented into the actual figures. despite the bank of latvia in 2005 has been repeatedly raising the reserve rate and brought it to 8%, the banks continued to increase the lending capacities. according to the information of the financial and capital market commission, the loan portfolio of latvian commercial banks amounted to 4,380.6 million lats in december 31, 2004, and at the end of 2006-10,872.9 million lats[2]. the most significant increase in the volume of loans amounting to 16,588.9 million lats was in 2008 (see table 2). as the table shows, the growth of credit portfolio of the commercial banks was especially high when the financial crisis had begun. its share in total assets was 71.4% in the end of 2008. in comparison with 2008 the assets of banks have decreased by 2,775.2 million lats or 11.9 % in 2013. thus, the share of credit portfolio in total assets of commercial banks in comparison with 2008 decreased by 13.7 percent points and made 57,7 %. equities in capital of enterprises increased (29.6 %) for the same period. the due from other financial institutions increased on 738.2 million lats (29.6 %). analyzing the changes in total liabilities of commercial banks for the similar period, we can conclude that the balance of deposit accounts had a reverse tendency compared to the amounts of credit debt. thus, in 2013 the deposits amounted to 11,071,5 million lats and increased on 1207 million lats or by 12.5 %, in comparison with 2008, and their share in total liabilities of commercial banks reached 52 valery i. roldugin: principal causes of financial frises in latvia for fast 20 years 51.4 %[1]. table 2 structure of asset of latvian commercial banks from 2008 to 2013(the end of the year, in million lats) 2008 2013 share of items in total assets changes in interest points changes in 2013 compared to 2008 2008 2013 mln. % cash and balances with the bank of latvia 1,325.20 2,008.60 5.7 9.8 1.2 683.4 51.6 due from other financial institutions 2,496.90 3,235.10 10.7 15.8 6.3 738.2 29.6 loans 16,588.90 11,820.30 71.4 57.7 -13.7 -4,768.60 -28.7 bonds and other securities with fixed income 2,015.70 2,104.30 8.7 10.3 1.3 -88.6 4.4 shares and other securities with non-fixed income 51.9 210.6 0.2 1 0.8 158.7 305.8 equities in capital of enterprises 130.7 324.5 0.6 1.6 1 193.8 148.3 fixed assets and intangible assets 234.8 115.8 1 0.6 -0.4 -119 -50.7 other assets 275.9 490.7 1.2 2.4 1.2 214.8 77.9 prepaid expenses and accrued income 123.5 158.2 0.5 0.8 0.3 34.7 28.1 total assets 23,243.30 20,468.10 100 100 -2,775.20 -11.9 5 conclusion an effective banking system is one of the most important conditions of economic development of latvia. it is assumed that latvian economy will be able to implement positively the experience of monetary and credit regulation accumulated in the worlds practice. for the past years the modern two-level banking system was established and developed in the country. gradually, the competitive credit and financial infrastructure was formed, basic elements of which were the commercial banks. some of them have already received a high international rating. the association of the latvian banks turned into the national bank association. at the end of 2014, 27 commercial banks, including ten branches of foreign credit institutions were registered in the republic of latvia (see table 1). jsc ge money bank wound up its business in 2013. moreover, based on the applications submitted by the particular commercial banks, as of 1 january 2014 commercial banks licences were cancelled for jsc unicredit bank which left the baltic market due to changes in the group strategy and sjsc latvijas hipotku un zemes banka which was transformed into sjsc latvijas attistibas finansu institucija altum[1]. the latvian banks slowed the lending down and tried to reduce the credit risks, what had a negatively effect on national economy. the banks tried to increase the liquidity by transferring the bad credits to the structures specially advances in systems science and application (2016) vol.16 no.2 53 established for this purpose. at the same time, there is the unpredictability of actions of the latvian administrative bodies. a main goal of the bank of latvia and the financial and capital market commission is the ensuring of the general sustainability in the monetary and credit markets. thus, exercising all rights defined by the law, the bank of latvia should provide stability of the prices, and the financial and capital market commission should treat everyone who forms the instability in finances and capital market in whole or in its separate sectors by their activity or non-activity. references [1] the bank of latvia: annual report 1994-2014. available at: https://www.bank.lv/en/statistics/monetary-statistics/mfi-balance-sheetand-monetary-statistics [accessed: oct. 12, 2015]. [2] banks statistics. (2015), available at: http://www.lka.org.lv/ru/statistika/. [3] roldugin valery. (2007), international business explanatory dictionary. riga, jumava. [4] roldugin valery. (2006), international economic dictionary of states, riga, jumava. [5] law on financial and capital market commission: lr law. latvijas vstnesis. (2000), nr.230/232, available at: http://www.likumi.lv/doc.php?id=8172 [accessed: sep. 19, 2015]. [6] information about latvian market of real estate. available at: http://www.realty.lv [accessed: apr.11, 2015] corresponding author roldugin valery can be contacted at: vroldugin@hotmail.com microsoft word 7-王永红.doc advances in systems science and applications (2010), vol.10, no.1 issn 1078-6236 international institute for general systems studies, inc. 41-47 multidimensional structural regression model for causal inference under strongly ignorable treatment assignment∗ yonghong wang school of mathematics and computer, harbin university, harbin 150086, china email:wangyonghong418@163.com abstract in the study of epidemiology aetiology, we usually cannot measure exposed effect relative to an individual, but under some assumptions, we approximately replace the exposed effect by estimator of population average causal effect. a multidimensional structural regression model for causal inference is established to estimate the population average treatment effect under strongly ignorable treatment assignment. under the normal distribution, the maximum likelihood estimator for population average treatment effect is proved to be consistent, unbiased and asymptotically normal. keywords strongly ignorable treatment assignment causal inference population average treatment effect multidimensional structural regression model maximum likelihood estimator 1. introduction the objective of epidemiology study is to search aetiology, and to measure its causal effect in the light of quantity, thereby prevent from the occurrence of disease[1]. the case -control study, which attempts to find a contrasted or exposed group which can be compared with the treated or exposed one, is the common method that we employ in the research of epidemiology aetiology. except the difference of being exposed and unexposed, the ideal treated group and the contrasted group are expected to share the rest of the other features, which serves as the guiding principle in the epidemiology research[2-3]. the dummy truth model laid a theoretical foundation for us to demonstrate this principle[4]. let u be the study population, and denote a generic individual in u by u u∈ . a variable e is defined on each u u∈ so that ( ) 1e u = if u is exposed to the casual agent of interest and ( ) 0e u = if u is not so exposed. )(1 uy and )(0 uy indicate the diseased status of u as in the exposed and non-exposed cases respectively. thus we can define the exposed causal effect of the individual u as )(1 uy — )(0 uy .as we know, in any real epdidemilogic study, one does not observe )(1 uy and )(0 uy simultaneously. therefore the casual effect of the exposure to the individual is not attainable. however, in some circumstances, we can get the population average causal effect, which can be defined as 1 0 e(y (u)-y (u)) , while )(⋅e denotes expection or population average (over u ). under the circumstances of random tests, for the fact that the independence of the exposed e and other variants, the result should be e （y1，y0），which indicates that the exposed e is independent of the random vector（y1，y0）. thus the population average causal effect is shown as )( 01 yye − = )()( 01 yeye − = )0()1( 01 =−= eyeeye .in other words, the causal effect can be estimated by the difference of the exposed population average diseased condition the non-exposed population average diseased condition[5]. the epidemiology, however, is an ∗this work has been supported by the science development foundation of harbin university(hxkq200701). wang: multidimensional structural regression model for causal inference under strongly ignorable treatment assignment 42 observatory science in nature, so the above formula is not to be established which necessitates the assumption of the state of being ignorable[5-6] . definition 1.1 given the covariant x , the exposed e is called being strongly ignorable. if (1)（y1，y0） xe , i.e. the observed covariant x is given, the response variant is independent of the exposed e. (1) (2) 1)pr(0 <=< xee , i.e. for each and every x, they may receive various treatment. when the e is strongly ignorable, by getting the matching-differencing-averaging between the exposed and non-exposed groups, we can get the unbiased estimate of population average causal effect[7]. but the above method is complicated in calculation and the additional sparse-data problems, excess covariant, though few, may arise. in addition, the method of ranking the individuals with the estimated propensity score is an advisable way to solve the problem, but the calculation is still very complicated[8]. [9] under strongly ignorable treatment assignment, a structural regression model for causal inference is established to estimates the population average treatment effect. they are the series of results achieved. with the response variant being one dimensional. the paper will discuss the inference problem with multi-dimensional response variants. 2. multidimensional structural regression model definition 2.1 in condition of strongly ignorable treatment assignment, denote multidimensional structural regression model as follows: ⎪⎩ ⎪ ⎨ ⎧ σ ++= tjtjttjxxtj tjtjtttj exvnenx exy oft independen is ),,0(~),,(~ μ βα (2) here t=0,1 means two treatment； tjy is the random vector of order 1×p ,indicated the response variant of the jth observation for the tth treatment; tjx is the covariant vector of order 1×k ; tje is the random error vector of order 1×p ; tα is the parameter vector of order 1×p ; tβ is the parameter matrix of order kp× ; xμ is the parameter vector of order 1×k ;the covariant matrix tv和xς of tjx and tje are both positive definite. under model （2） ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + =⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ xtt x tj tj y x e μβα μ ， ta ' ' tj x x t tj t x t x t t x cov y v β β β β ⎛ ⎞⎛ ⎞ σ σ = ⎜ ⎟⎜ ⎟ σ σ +⎝ ⎠ ⎝ ⎠ we can easily find txt va ⋅σ= ;because xς and tv are both positive definite, we can know 0≠σ x , 0≠tv ；so 0≠ta ;therefore 1− ta is valid, and we denote it as bt .again based on the principle of block matrix inversion, we can get bt= ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −+σ −− −−− 11 1'1'1 ttt tttttx vv vv β βββ so ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ t xtt x tj tj an y x ,~ μβα μ where: t=0，1 ； j=1，……，nt . we also get observation logarithm likelihood function of the sample ( )tjtj yx , with the advances in systems science and applications (2010), vol.10, no.1 43 capacity nt (t=0，1； j=1，……，nt ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −− − +=− ∑∑∑ = == xtttj xtj t t n j xtttj xtj t tt y x b y x anl t μβα μ μβα μ ' 1 0 1 1 0 lnln2 furthermore， 1 1 ' 1 ' 1 0 0 1 2 ln (ln ln ) ( 2 tn t x t tj x tj tj x x t t j l n v x x x μ− − = = = − = σ + + σ − σ∑ ∑∑ tjtttjtjttttjxxx yvxxvx 1''1''1' 2 −−− −+σ+ βββμμ （3） )22 1'1'1'1'' ttttttjtjttjttttj vvyyvyvx ααααβ −−−− +−++ 3. conclusion 3.1 two lemmas in order to get the likelihood estimates of all parameters in model (2) and the population average causal effect, we give the following two lemmas, of which lemma3.1 is shown at [10]. first, suppose x is the matrix of order nm× , )(xf is the real-valued function of matrix x , x xf ∂ ∂ )( indicate real-valued function )(xf settling partial derivation for each element of matrix x . thus the following lemmas are valid: lemma 3.1（1） '')( ba x axbtr = ∂ ∂ （2） '' ' )( xbaaxb x axbxtr += ∂ ∂ （3） '1 )( ln −= ∂ ∂ x x x of which a and b are two matrixes that can make matrix calculation with x , ( )tr x means the trace of matrix x . lemma 3.2 if model (2) is valid, when the covariant matrix and the random error matrix are independent of each other, then ∑ = = tn j tj t t x n x 1 1 and ∑ = −−= t tt n j ttjttj t xx xxxx n m 1 '))((1 ' ， ∑ = −−= t tt n j ttjttj t xe xxee n m 1 '))((1 ' are independent of each other, where ∑ = = tn j tj t t e n e 1 1 . proof: first to prove tx and ' tt xx m are independent. wang: multidimensional structural regression model for causal inference under strongly ignorable treatment assignment 44 let ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = ktntntn kttt kttt t ttt xxx xxx xxx x 21 22221 11211 set '1 tt x n x = 1 also '' 1(1 ' t t t t xx x n x n m tt −= 11 ' ）（ t t n x 1 − 11 ' tx ） = '1 t t x n (i t n nt 1 − 11 ' ) tx which 1 is the column vector of 1×tn . we can do the vector straight operations to matrix tx , then ⊗σ= xxveccov ))(( i tn , also, as for tx and each at the element ijm in ' tt xx m follow tx = tn 1 （ik⊗ 1 ' ） )( txvec ijm = ')( txvec [ ⊗+ )( 2 1 ' ijij ee (i t n nt 1 − 11 ' )] )( txvec of which⊗ indicates the kronecker product of matrix, ije indicates matrix of order kk × with （i,j）th is 1, and the rest is 0. now we can draw the following conclusion by making use of the characteristic of multidimensional normal random variant and the kronecker product of the matrix: tn 1 （ik⊗ 1 ' ）（ ⊗σ x i tn ）[ ⊗+ )( 2 1 ' ijij ee (i t n nt 1 − 11 ' )] = t ijijx n ee 1[)]( 2 1[ ' ⊗+σ 1 ' （i t n nt 1 − 11 ' ）]=0 thus we can prove ijm and tx are independent, and it’s valid to any i，j=1，……，k. so we have proved ∑ = = tn j tj t t x n x 1 1 and ∑ = −−= t tt n j ttjttj t xx xxxx n m 1 '))((1 ' are independent of each other. suppose ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = ptntntn pttt pttt t ttt eee eee eee e 21 22221 11211 ),( ttt exz = ，in the same way , and by mutual independence of tx and te , we can prove tx and ' tt xe m are also independent of each other. lemma3.2 is proved！ advances in systems science and applications (2010), vol.10, no.1 45 3.2 estimate of parameter theorem 3.1 suppose model（2）is valid, we can get the maximum likelihood estimates of all the parameter: txxxytt xmmy tttt 1)(ˆ '' −−=α ; 1)(ˆ '' −= tttt xxxyt mmβ ; ∑∑ = =+ = 1 0 110 1ˆ t n j tjx t x nn μ ; ' 1 0 110 )ˆ)(ˆ(1ˆ xtjx t n j tjx xx nn t μμ −− + =σ ∑∑ = = ; ∑ = −−−−= tn j tjtttjtjtttj t t xyxy n v 1 ')ˆˆ)(ˆˆ(1ˆ βαβα of which ∑ = = tn j tj t t y n y 1 1 ， ∑ = −−= t tt n j ttjttj t xy xxyy n m 1 '))((1 ' ； the form of tx and ' tt xx m is shown in lemma3.2. proof: first, we get the score function of tα : ∑ = −−− +− ∂ ∂ = ∂ −∂ tn j ttttttjttttj tt vvyvxl 1 1'1'1'' )22()ln2( ααααβ αα =∑ = −−− ∂ ∂ + ∂ ∂ − ∂ ∂tn j ttt t tttj t ttttj t vtrvytrvxtr 1 1'1'1'' )]()(2)(2[ αα α α α αβ α = )22( 11 1 tttjtt n j vxv t αβ −− = −∑ =0 we can get tttt xy βα −= . and the score function of tβ is as follows: 0)222()ln2( 1 '1'1'1 =+−= ∂ −∂ ∑ = −−− tn j tjtttjtjttjtjtt t xvxyvxxvl αβ β simplified as 0)( 1 ''' =+−∑ = tn j tjttjtjtjtjt xxyxx αβ fit tβ into above formula of tα , we set txxxytt xmmy tttt 1)(ˆ '' −−=α . in the same way, we can get the maximum likelihood estimates of the other three parameters. theorem3.1 is proved！ in fact, as for the multidimensional structural regression model with the response variants as shown in model (2), we can see from the above demonstration that the maximum likelihood estimates of parameter tα and tβ do not depend on the selection of the covariant matrix and random error matrix. moreover, we can also see that if we want to estimate the parameter that measure relationship between the covariant and the jth（j=1,……，nt）response variable, we should only note the jth response variable. i.e. the model(2) can be divided into p one dimensional models, which is evident in jinhua’s paper(2000). 3.3 population average causal effect definition3.1 provides the covariant x, if the assumption of strongly ignorable condition (1) is fulfilled, then treatment e=1 to e=0 average causal effect ate is ))()(()()( 0101 xxyeyeeyeyeate x =−=−= wang: multidimensional structural regression model for causal inference under strongly ignorable treatment assignment 46 xx x xxe xxeyexxeyee μββααβαβα )()()( )),0(),1(( 01010011 01 −+−=−−+= ==−=== theorem3.2 suppose model (2) is valid, and treatment assignment variant e is strongly ignorable, then (1) the maximum likelihood estimator of population average treatment effect ate is ate = )( ˆˆ 01 10 0110 01 xx nn nn yy − + + −− ββ (2) ate is consistent unbiased estimator of ate. (3) with the case of the large sample, eta ˆ is close to normal distribution n（ate, γ）. of which ' 0 1 0 1 01 0 1 0 1 ( ) ( )xv v n n n n β β β β− σ − γ + + + proof: (1) with model (2), if treatment assignment variant e is strongly ignorable, then the population average treatment effect ate= xμββαα )()( 0101 −+− based on theorem 3.1 and the invariance of maximum likelihood estimator, we can know ate xμββαα ˆ)ˆˆ()ˆˆ( 0101 −+−= )( ˆˆ 01 10 0110 01 xx nn nn yy − + + −−= ββ (2) consistency: we know based on the model (2) ttttt exy ++= βα , t=0,1 again based on theorem3.1, we get 1)(ˆ '' −= tttt xxxyt mmβ = 1))(( '' −+ tttt xxxet mmβ based on the law of large numbers and the nature of the random variant order converging by probability, we can know x p xx tt m σ⎯→⎯' , 0' ⎯→⎯p xe tt m then t p t ββ ⎯→⎯ˆ in the same way, x p tx μ⎯→⎯ , xtt p ty μβα +⎯→⎯ so far, again based on the nature of the random variant order converging by probability, we get pate ate⎯⎯→ , i.e. ate is consistent estimate of the population average treatment effect ate. unbiased：through the demonstration of the consistency, we know 1 1 ' )]())((1[ˆ ' − = ∑ −−+= tt t xx n j ttjttj t t mxxee n ββ whereupon })]())((1[{)ˆ( 1 1 ' ' − = ∑ −−+= tt t xx n j ttjttj t t mxxee n ee ββ = })]())((1[{ 1 1 ' ' ttxx n j ttjttj t xt xxmxxee n ee tt t =−−+ − = ∑β = tttxt xxe ββ ==+ )0( therefore xtttye μβα +=)( , so based on lemma3.2, we set advances in systems science and applications (2010), vol.10, no.1 47 0 1 1 0 1 0 1 0 0 1 ˆ ˆ ( ) ( ) ( ) ( _ )n ne ate e y y e e x x n n β β+ = − − + = atex =−+− μββαα )()( 0101 (3) asymptotically normality: based on above demonstration, we know ( )e ate ate= also because of 0 1 1 0 1 0 1 0 0 1 ( ) ( )n nate y y x x n n β β+ ≈ − − − + = ] )( [] )( [ 00 10 010 011 10 011 1 ex nn n ex nn n + + − +−+ + − + ββ α ββ α moreover x t t n xvar σ= 1)( ， t t t v n evar 1)( = therefore 1 1 ' 01012 10 1 11 10 011 1 1)()( )( ) )( ( v nnn n ex nn n var x +−σ− + =+ + − + ββββ ββ α 0 0 ' 01012 10 0 00 10 010 0 1)()( )( ) )( ( v nnn n ex nn n var x +−σ− + =+ + − + ββββ ββ α so ' 1 0 1 0 1 0 1 0 0 1 1 1 1( ) ( ) ( )xvar ate v v n n n n β β β β= + + − σ − γ + so based on the central limited theorem, we know ~ ( , )ate an ate γ . theorem 3.2 is proved！ references [1] rothman,k.j. modern epidemiology[m]. boston:litte, brown and company, 1986: 56. [2] guo,j.h. causal inference and confounding phenomenon. journal of northeast normal university, 2002, 34: 25-41. [3] habtemariam,t. and tameru,b. etc. epidemiologic modeling of hiv/aids: use of computational models to study the population dynamics of the disease to assess effective intervention strategies for decision-making. advances in systems science and applications, 2008, 8(1): 35-39. [4] tao,q.s. and li,l.m. aetiology effect model based on dummy truth theory. chinese journal of epidemiology, 2002, 23(1): 60-62. [5] rosenbaum,p. and rubin,d.b. the central role of the propensity score in observational studies for causal effects. biometrika, 1983, 70: 41-55. [6] holland,p.w. confounding in epidemiologic studies. biometrics, 1989, 45: 1310-1316. [7] kong,f.l. and wang,y.f. certain martingale methods in parameter estimation. advances in systems science and applications, 2004, 4(1): 1-6. [8] yu,j.w. and cheng,d.h. etc. factor analysis methods in the statistical literature of chinese medicine diabetes research applications. advances in systems science and applications, 2007, 7(2): 161-166. [9] jin,h. and fang,j.q. structural regression model for causal inference under strongly ignorable treatment assignment. journal of south china normal university, 2000, 4: 7-12. [10] wang,s.g. etc. theory of linear model[m]. beijing: science press, 2004: 44-47. microsoft word 8 lianhui jiang, guang yang--modeling and analysis of mechanical properties.doc 264-269 advances in systems science and applications (2011), vol.11, no.3-4 issn 1078-6236 international institute for general systems studies, inc modeling and analysis of mechanical properties for fdm parts* lianhui jiang and guang yang school of mechanical and electronic engineering, wuhan university of technology, wuhan 430070, china abstract fused deposition manufacturing has the potential to fabricate parts with locally controlled properties. modeling and analysis of the mechanical properties of the fdm parts are the foundation to produce this new class of products. a mechanics model is established for manufacturing fdm parts with required mechanical properties. a series of equations is proposed to determine the elastic constants that are used in the model. the results are evaluated by experiments. keywords solid freeform fabrication fused deposition manufacturing locally controlled properties 1.introduction solid freeform fabrication (sff), also referred to as rapid prototyping (rp) or layered manufacturing (lm), is a class of technologies that build objects in an additive fashion directly from a computer model. the ability of sff technologies to fabricate parts with locally controlled properties creates opportunities for manufacturing a whole new class of parts with functionally gradient structures [1,2,3]. the functionally graded structures could be useful in controlling the mechanical properties of parts and tools on a local scale. fused deposition manufacturing (fdm) has the potential to produce parts with locally controlled properties by changing deposition density and deposition orientation. in the fdm process, the deposit head that extrudes a semi-molten filament through a heated nozzle in a prescribed pattern follows the trajectory defined by the chosen deposition strategy of the layer. after the semi-molten material is deposited onto the worktable, it begins to cool and bond to the neighboring layer. the bonding between the individual roads of the same layer and of neighboring layers is driven by the thermal energy of semi-molten material. fdm processes use several types of materials, such as acrylonitrile-butadiene-styrene (abs), investment casting wax, ceramics and metals [5,6], to build conceptual or functional parts. most of the existing research efforts in fdm have been primarily directed to the development of new materials [2]. research efforts were also made to characterize fdm part quality. dutta and his associates investigated the relationship between deposition strategies and the resulting stiffness of fdm parts. matas characterized the meso-structural features of unidirectional extruded material as a function of fdm process variables. longmei li researched the relationship between parts’ properties and deposition parameters of fdm. [2,7] mechanical properties of fdm parts are governed by their meso-structures, which are determined by manufacturing parameters and bonding intension between filaments. manufacturing parameters include width of filaments, layer thickness, deposition orientation and gap sizes between filaments. by modulating manufacturing parameters, fdm processes can produce parts with desired properties. among the manufacturing parameters, deposition directions in layers and gap sizes between filaments are the most important parameters to control the mechanical properties. thus, it is essential to establish mechanical models of fdm parts in * this research has been supported by the national natural science foundation of china (60904073) advances in systems science and applications (2011), vol.11, no.3-4 265 relation to these two manufacturing parameters. the models can then be used to design parts with desired mechanical properties. the remainder of this article is organized as follows. section 2 is a study of the mechanical properties of fdm parts. experimental analysis is given in section 3. section 4 summarizes the conclusions of the paper. 2. modeling and analysis of mechanical properties of fdm parts 2.1 mechanical models. the fdm parts are composites of abs filaments, bonding between filaments, and voids. as composite materials, they can be viewed and analyzed at different levels and on different scales, depending on the particular characteristics and behavior under consideration. fdm parts are composed of layers, so lamination theory can be used to analyze the mechanical behaviors of fdm parts [8,9]. a unidirectional filament fdm part, fig. 1 definition of principal material axes and loading axes for one layer which can be regarded as a unidirectional fiber-reinforced composite, contains three orthogonal planes of material property symmetry, and is classified as an orthotropic material[8]. the coordinate system for analyzing mechanical behaviors of one layer of fdm parts was established as follows. two right-handed coordinate systems are defined, namely, 1-2-z and x-y-z, as shown in figure 1. both 1-2 and x-y axes are in the plane of the lamina, and the z-axis is normal to this plane. in the 1-2-z systems, axis 1, along the filament length direction, represents the longitudinal direction of the lamina. axis 2 is normal to the fiber length and represents the transverse direction of the lamina. in the x-y-z system, axes x and y represent the loading directions. the angle between positive axis x and axis 1 is called the fiber orientation angle, represented by θ . a laminate is constructed by stacking a number of laminas along the z direction. it should be pointed out that axes 1 and 2 are constant from one lamina to the next in a unidirectional laminate. for a general orthotropic ( 0≠θ or °90 ) lamina, the strain-stress relations can be expressed in matrix notation as (1): [ ] . 662616 262212 161211 ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ xy yy xx xy yy xx xy yy xx s sss sss sss τ σ σ τ σ σ γ ε ε (1) where [ ]s represents the compliance matrix for the lamina. the elements in [ ]s matrix can be expressed as (2): x yz 2 1 266 jiang: modeling and analysis of mechanical properties for fdm parts .cossin)22(cossin)22( .cossin)2(cossin)22( .coscossin)2(sin1 .cossin)()cos(sin .sincossin)2(cos1 3 661222 3 66121126 3 661222 3 66121116 4 22 22 6611 4 1122 22 662211 44 1212 4 22 22 6612 4 1111 θθθθ θθθθ θθθθ θθθθ ν θθθθ ssssssms ssssssms ssss e s ssss e s ssss e s y x yy yy xy xx −−−−−=−= −−−−−=−= +++== −+++== +++== (2) .111 ).cos(sincossin)422(21 12 66 22 22 22 21 11 12 2112 11 11 44 66 22 6612221166 g s e s ee ss e s sssss g s xy ==−=−=== ++−−+== νν θθθθ where 11e is longitudinal young’s modulus, 22e is transverse young’s modulus, 12ν is major poisson’s ratio and 12g is shear modulus. from (1) it appears that there are six elastic constants that govern the stress-strain behavior of a lamina. however, a close examination of these equations would indicate that these six elements are functions of the four independent elements. therefore, the four elastic constants can be used to determine the fully populated matrix of [ ]s . a laminate is constructed by stacking a number of laminas in the thickness direction. therefore, in macro mechanical approaches, the stiffness matrix of a laminate is composed of properties of every lamina according to the lamination theory [8], while the in-plane elastic behavior of a unidirectional lamina may be fully described in terms of four elastic lamina properties. for the design purpose, it is desirable to have reliable predictions of lamina properties as a function of constituent properties and geometrical characteristics. one of the objectives of composite analysis is to obtain such relationships. 2.2 modeling of elastic constants for fdm parts. when loaded in the longitudinal direction, a unidirectional fdm part is an aggregate of abs filaments. longitudinal young’s modulus 11e can be predicted by the rule of mixtures. therefore, longitudinal young’s modulus should be as follows: .)1( 111 ee ρ−= (3) where e is young’s modulus of abs filament and 1ρ is the area void density in the plane normal to filaments. the area void density can be calculated using the following equation: .1 2 11 1 dhv av −=ρ (4) where 1v is the velocity of the material supply system, 1a is the section area of the supply filament, 2v is the velocity of the manufacturing head, d is the distance between two filaments of the same layer and h is the thickness between two layers. advances in systems science and applications (2011), vol.11, no.3-4 267 in the case of transverse normal loading, the bonding among the filaments is the load carrier. because of imperfect bonding and the irregular interface between filaments, an accurate equation to calculate transverse modulus is impossible. an approximate equation is used to calculate transverse modulus: .22 ee ζ= (5) where 1~0=ζ is the coefficient that is decided by distance between filaments, bonding status, etc. to calculate ζ , the geometry of the cross section should be studied. fig. 2 shows the cross section image of road geometry [4]. fig. 3 shows the theoretical model of two adjacent roads. from the cross section images of road geometry, the shape of the hole between filaments can be assumed as lozenge. the horizontal cater-corner a of the lozenge can be calculated as follow: ).()32( dwra −−−= (6) where w is the deposition width as shown in fig. 3. the area of the lozenge is equal to the area of the cross section of the hole. so the perpendicular cater-corner of the lozenge b can be calculated as follow: ./2 1 adhb ρ= (7) the coefficient ζ can be calculated using the following equation: [ ] . 2)(/2 2/ 0 dxbxbhaahad d a ∫ +−+− =ζ (8) similarly, the prediction of the in-plane shear modulus takes the following form: .12 gg ζ= (9) the in-plane poisson’s ratio, 12ν , is the same as that of abs filaments, ν , as follows: fig. 2 image of cross section of road geometry fig. 3 theoretical model between two adjacent roads f w r table 1 fdm manufacturing parameters settings parameter value nozzle radius (r) 0.3[㎜] layer interval (h) 0.15[㎜] road width (w) 0.58[㎜] head temperature 220[℃] envelope temperature 60[℃] 268 jiang: modeling and analysis of mechanical properties for fdm parts .12 νν = (10) 2.3experimental analysis to evaluate the coefficient value in the equation, the experimental analysis was carried out. all the specimens were made using a fdm machine with the manufacturing parameters as shown in table 1. the designs of unidirectional specimens are shown in figure 4. the specimens with fiber orientation of °° 45,0 , and °90 , were tested to determine the elastic constants for different deposition orientation. the tests followed the standard procedures of gb 1040-79. in the 00 unidirectional specimen tests, the strain gages were mounted in the longitudinal and the transverse directions to determine the longitudinal young’s modulus 11e and poison’s ratio 12ν . transverse modulus 22e can be derived from the results of the °90 unidirectional specimen tests. with the resultant young’s modulus 45 xxe in the x direction, and with the other constants of ,, 1112 eν and 1222 ,ge can be determined as follows [8]: . 2114 1 11 12 2211 45 12 eeee g xx ν +−− = (11) the theoretic and experimental results are listed in table 2. table 2 theoretical results with experimental results 3.conclusion sff processes must become broadly used for manufacturing but not just prototyping. for fabrication of parts with locally controlled properties, a model of in-plane mechanical properties of fdm parts is established. a set of equations for calculating the elastic constants was found. the constants were used to determine the constitutive models of fdm parts. according to the model, different deposition densities, orientations and their combinations can be selected in producing constant, units experimental results theoretical estimation difference, % e11 [ mpa] 2238.7 2267.6 1.29% e22 [mpa] 2189.7 2156 1.54% 12ν 0.4211 n/a n/a g12 [mpa] 800 785.22 1.85% 1 20 0 11e 1 20 0 22e 1 20 0 45 xxe fig. 4 geometry of unidirectional i advances in systems science and applications (2011), vol.11, no.3-4 269 the parts with required stiffness properties. this research is useful for exploring fdm technology fabricating parts with locally controlled properties. 4.acknowledgement the authors would like to thank the national nature science foundation of china (grant no, 60904073) for its supporting for this research and to thank the people involved in the work. without their efforts and input, the completion of the research would not have been possible. references [1] cho.w, sachs,e.m., patrikalakis.n.m., cima.m.j., jackson.t.r, etc. “methods for distributed design and fabrication of parts with local composition control.” proceedings of the 2001 nsf design and manufacturing grantees conference. jan. 2001, tampa, florida. [2] longmei li, qiao sun, celine bellehumeur, peihua gu. “modeling and analysis for fabrication of fdm prototypes with locally controlled properties.” journal of manufacturing processes; 2002;vol4,no.2;pp129-141 [3] t.r. jackson, h. liua, n.m. patrikalakisa,u, e.m. sachsb, m.j. cimac “modeling and designing functionally graded material components for fabrication with local composition control” materials and design.2002, vol20, no.2-3,pp63-75 [4] wenbiao han, mohsen a. jafari, stephen c. danforth, ahmad safari, 2002, “tool path-based deposition planning in fused deposition processes”, transactions of the asme. journal of manufacturing science and engineering (v.124). pp462-472 [5] agarwala, m.k., van weeren, r., bandyopadhyay, a., whalen, p.j., safari, a., and danforth, s.c., 1996, ”fused deposition of ceramics and metals: an overview,” proceedings of solid freeform fabricaiton sypposium, austin, tx, pp.385-390. [6] jafari, m.a., han, w., mohammadi, f., safari, a., danforth, s.c., and langrana, n.a., 2000, “a novel system for fused deposition of advanced multiple ceramics.” rapid prototyping journal, 6, no.3, pp.161-174. [7] li, longmei (2002), “analysis and fabrication of fdm prototypes with locally controlled properties.” phd dissertation, calgary, alberta, canada: dept. of mecanical and mfg. engg, univ. of calgary. [8] mallick, p.k., 1988, fiber-reinforced composites: materials, manufacturing, and design, marcel dekker, inc, us. [9] p.gu, l.li, 2002, “fabrication of biomedical prototypes with locally controlled properties using fdm”, annals of the cirp, vol51(1), pp 181-184. advances in systems science and applications (2014) vol.14 no.1 50-65 investigation of the different type models for engine test data modelling m.h. wu school of technology, university of derby, marketon st., derby, de22 3aw, uk abstract the engine test bed has been used widely in automotive industry to test the protetypey developed engine in order to understand the engine operation process and find the optimum parameters for the engine operation. as a result, thousand and thousands data will be obtained from engine test bed.for this purpose. it is a time consuming task to get the data and it is hard to analyse those data directly from the huge number of the data. the best way to reduce the tested data from the test bed is to create a mathematics model from the limited engine tested data. the paper will list three different models used for the engine test data modelling, polynomial model, combined exponational model and neural network model. the principle and equations of each model has been introduced in the paper. the experiment results of each model have been listed as well. the results from last two developed models show that the models provide the achievement which can meet the customers requirement. keywords engine test data, engine performance, engine data modelling, neural network 1 introduction the engine test bed has been used widely in automotive industry to test the protetypey developed engine in order to understand the engine operation process and find the optimum parameters for the engine operation. as a result, thousand and thousands data will be obtained from engine test bed.for this purpose. it is a time consuming task to get the data and it is hard to analyse those data directly from the huge number of the data. the best way to reduce the tested data from the test bed is to create a mathematics model from the limited engine tested data. the model will be used to analyse the engine operation and to obtain the optimum operation data. figure 1 shows the structure of the engine test bed (etb). the newly developed or existed engines should be tested on it. fig.1 shows the format of the tested data from etb which are used to create the new model. the controled input variables and the responding out put are shows in fig.2. it includes: ignition time, ex valve timing, inlet valve timing , bmep and speed as the input variable and the bsfc as the output. there are three models developed for the engine data modelling. it is polyadvances in systems science and applications (2014) vol.14 no.1 51 fig.1 structure of engine test bed nomial model, combined exponational model and neural network model. fig.2 the form of the tested data from the etb 2 employing the polynomial model for engine test data modelling 2.1 the polynomial approximation model (1)shows the aplication of the polynomial approximation model in engine test data modelling. it called “quadratical model”(qm) which is denoted by voigt, lechner and hochschwarzer[1-3]. the qm is: y = xtqx+ atx+ b (1) here: y is an objective vector, x is an input variable vector, q is co-efficient matrix, a& b are constant vectors, 52 m.h. wu:investigation of the different type models for engine test data modelling the co-efficient of matrix q and the vectors a and b are calculated by leastsquares-method to minimize the deviation of measured values from the model values. 2.2 analysis of the application of the polynomial model this model has the advantage of simplicity, but is imperfect for real application. fig.3 shows the example which indicates the model’s imperfect. (a) draw by the original engine tested data (b) draw by the data calculated by qm fig.3 3d surface draws by the different data fig.3a shows the relationship between the bsfc and two input variables by the experiment data directly. fig.3 b shows the same relationship using the model qm. it is obviously that fig.3(a) is totally different to fig.3(b). in fig.3a it shows that the output will be varied as a irrgular wave when two inputs are varied. the fig.3b only provide a smooth raised convex area. in one word, the qm model can not trace the features in fig.3a. it is clearly that qm is not a suitable model to be used for engine parameters modelling and optimasation. the data from qm provides the large errors and losts nearly most of data features. another disadvantage of using quadratical equations is that the model only can be used to the case which has one convex (or concave) data set. 3 employing the combined exponational model for engine test data modelling 3.1 the combined exponational model (2) shows the aplication of the combined exponational model in engine test data modelling[4]. f(x1, x2, . . . xn) = a+ n∑ i=1 bixi + p∑ j=1 (hje(x1)e(x2) . . . e(xn)) (2) here: e(x) = exp(−8(x−m)2/w2); h : the heigth of the peak, it could be “+” or “-”; advances in systems science and applications (2014) vol.14 no.1 53 m : the position of the peak along the axis; w : the width of the peak; n = 5, (the model dealings the variables; f(x1, x2, ...xn) : break specific fuel consumption (bsfc); x1 : ignition time variable; x2 : inlet valve opening degree variable; x3 : exhaust valve opening degree variable; x4 : speed; x5 : toque (bmep); p : the number of peaks in nd space; it assumes p = 10; a : a constant; bi : a slope coefficient respect to input xi; fig.3a indicts that the output bsfc could be formed by constant element, linear element and multiconvex or concave peaks. the convex or concave peaks on the 3d surface can be expressed by the function of exponatioal. as a result, the (2) can be explained as follow: bsfc = constant + linear function + exponational function; 3.2 set up the coefficients of the combinated exponational model if the equation 2 has been set by 20 peaks (j=20) and 5 input variables (i=5), the total number of the coefficients of the equation, bi, hj , mi,j and wi,j , should be as following table 1. there is no suitable mathematics methods to set up the exactly values to 225 table 1 the number of the parameters of the combined exponational model with 5 inputs, 20 peaks coefficient bi hj mij wij total the numbers 5 20 100 100 225 coefficients through the existed tested data. as a result, genetics algorithm are used to define the appropriated values for each coefficient by the existed tested data. a vb based program is developed for this purpose. the software will be used to obtain the appropriate values of the coefficients in (2) by using the follow operations in ga: – random selection of the chromosome within certain range, – reproduction – crossover – evaluation – mutation in order to running the ga, the follow assumptions and definitions are re54 m.h. wu:investigation of the different type models for engine test data modelling quired: gene: the genes are used to form the chromosome in ga application. as a result, the number of the gene should be equal to the total numbers of the coefficients in the (2). for above example in table 1, the numbers of gene should be 225. population: the population is the numbers of the chromosome. it can be set from 1 to 100 in the software. the rule of setting up is that the more coefficients is , the larger population should be. in the case of table 1, the population is settled equal to 100. number of keep best members of the population: this is very important selection. if it is too small, too many poor chromosomes will be remained in the operator. on the other hand , it is easily tripped into the local best point. for the case in table 1, a range 10-20 is suggested. crossover method: crossover method can be selected from alternative methods, single point cross over, two points crossover, etc. type of mutation: random mutation hill climb and directional hill climb methods are available in the software. the first one is normally used for the fitness function is highly discontinuous. the 2nd one is for the continuous fitness function. our fitness function for ga is continuous so that the directional hill climb method is adopted. mutation probability of population: this percentage is employed to avoid trapped in local best. for engine test data modelling, the percentage is 10%. mutation probability of genes: if the percentage is too high, many “wild” mutations which have very poor fitness will be involved. if it is too low, some necessary genes will not be involved in crossover operator to improve the solution. it is normally 8-10%. fitness function in evaluation operation: fitness function is very important for chromosome parent selection in ga operation equation 3 is the fitness function used in the ga evaluation. min f(x1, x2, ...., xn) = ∑a+ n∑ i=1 bixi + p∑ j=1 hje(x1) · e(x2) · · · e(xn)− y0 2 (3) defined the ranges of variation of each variable: the ranges of variation of each variable in the engine test bed is settled as table 2, which will constraint the variables of x1, x2,...,xn; in (2) during the ga operation. 3.3 output results from the combined exponationa model as mentioned before, the parameters of the (2) are obtained from the existed test data from etb by ga system. as soon as the parametes is defined, the equation advances in systems science and applications (2014) vol.14 no.1 55 table 2 experimental data ranges variables range x1 input: speed 1237 to 4253 x2 input: bmep 0.9 to 6.8 x3 input: vanosin 80 to 132 x4 input: vanosex 70 to 135 x5 input: igtiming 4.5 to 51 y output: bfsc average: 346 2 can be used to calculate the engine output bsfc. fig. 4 shows the 3d surface drawings by the ordinary test data and the data calculated from (4) it is obviously that the major convex or concave peaks in (c) are repeated in (d). it means that the new model can generat the major features of the black box system (engine perfomance) and it is much more better than using the quadratical model in engine test data modelling. (a) draw by the original tested data (b) draw by the calculated data from new model fig.4 comparing the original tested data and the calculated data using the new model 4 employing the neural network model for engine test data modelling[56] 4.1 develop the radial basis function (rbf) networks for engine calibration modelling task[5] in 2003, dr. lin has published a paper. in the paper the neural networks used on the engine test data modelling was mentioned. on the application, radial basis function (rbf) networks is used for engine calibration modelling task. rbf networks are universal approximates, that is, given a network with enough hidden layer neurones, they can approximate any continuous function with any arbitrary accuracy [7-8]. an rbf is one whose output is symmetric around an associated centre µj . it is 56 m.h. wu:investigation of the different type models for engine test data modelling generally described as: ϕj(r) = ϕj(r)(∥x− µj∥); x ∈ rn; r ≥ 0 (4) where ϕj(r) is a continue function on(0,∞)and ∥ · ∥ denotes the euclidean norm. linear combination of rbfs represents a wide classes of functions: yj(x) = m∑ j=1 ωijϕj(∥x− µj∥) + ωi0 (5) the bias wi0 compensates for the difference between the average value over the fig.5 rbf network structure data set of the basis function activation and the corresponding average value of the targets. the rbf network is depicted as shown in fig.5. the bias wi0 can be incorporated into the summation by introducing an extra basis function ϕ0 and setting its activation to unity. thus the (2) can be written as yj(x) = m∑ j=1 ωijϕj(∥x− µj∥) (6) follows are the typical functions of the rbf: • the gaussian function: ϕ(r) = exp(−r2/2) (7) • the thin plate spline function: ϕ(r) = r2 × log r (8) advances in systems science and applications (2014) vol.14 no.1 57 • the multiquadric function: ϕ(r) = (r2 + 1)1/2 (9) • the inverse multiquadric function: ϕ(r) = 1 (r2 + 1)1/2 (10) • the pseudo cubic spline function ϕ(r) = r3 (11) • the logarithmic function: ϕ(r) = log(r2 + 1) (12) where r is a non-negative number and is the scaled distance from the input vector x to the rbf centre , which is defined as : rj = √√√√ n∑ l=1 (xl − µlj)2 σ2 j (13) in (13), n is the dimension of input vector and σj is scale factor or width of rbf. in engine calibration modelling task, all of six rbfs described above are applied. 4.2 develop neural network modelling tool for engine calibration modelling task[6] the fig.6 shows the structure of the neural network modeling tool used on the engine test bed. when the tested data is ready, they will be sending into the nn models tool box. in the nn tool box, there are three different types and totally ten nn structures. in it, the total tested data will be divided into three group. the first group data will take 50 % out of the total tested data. the second and third groups will take 25% out of the total data respectively. the first group data will be used to train the neural network and second group data will used to validate the nn model. final group data will be used to test the nn model. after nn tool box, the best fitted nn structural model is selected and sends to optimization section to defined the optimization data for engine management unit. in the optimization section, there are two options for optimization purposes according to the different requirement. soga is the model for single objective optimization and moga is one for multi objectives optimization. 58 m.h. wu:investigation of the different type models for engine test data modelling fig.6 the structure of the optimisation system of the engine performance as shown in figure 6 the nn model tool is employed to select a best suitable nn model out of ten nn models for the subsequence operation of optimization. fig.7 shows the common structure of the neural network in nn model tool and it is under the following assumptions: • it is a three layers nn model. • the numbers of the hidden layer note is defined by 2n + 1: here n is the number of inputs. the nn model tool includes three different type of neural networks, multi-layer perceptron (mlp); radial basis functions (rbf) and bar function (bar). a. mlp. the output of the mlp is denoted by yk(x) = m∑ j=0 f( n∑ i=0 xiωji)ωkj = m∑ j=1 f( n∑ i=1 xiωji + ωj0)ωkj + ωk0, k = 1, . . . , l where ωji and ωkj are the input-hidden weight and hidden-output weight, respectively. f is the activation function which has two types: (a) logistic function f(α) = 1 1 + e−α (15) advances in systems science and applications (2014) vol.14 no.1 59 fig.7 neural network structure (b) hyperbolic tangent function or tanh f(α) = eα − e−α eα + e−α (16) b. rbf. the output of the rbf is calculated using the following form in (5) and (6): yk(x) = m∑ j=1 ϕ(∥x− µj∥)ωkj + ωi0 = m∑ j=0 ϕ(∥x− µj∥)ωkj , k = 1, . . . , l where ωkj is the hidden-output weight and µj is the centre of j-th hidden unit. ϕ is the kernel function. to defining a distance from the input vector x to the rbf centre µj scaled by the scale factor or width σj , equation 13 is used. within the rbf networks, six kernel functions are widely used and defined by equations from (7) to (12). c. bar. the bar network has the same structure as the rbf network, but the kernel function is different. two kernel function for the bar network: (a) gaussian bar function ϕ(x) = n∑ i=1 exp [ − (xi − µji) 2 2σ2 ji ] (17) (b) sigmoidal bar function ϕ(x) = n∑ i=1 1/ { 1 + exp [ − (xi − µji) 2 2σ2 ji ]} (18) 60 m.h. wu:investigation of the different type models for engine test data modelling table 3 lists the summery of the neural network structure used in the nn model tool. table 3 nomenclature of neural network structures 4.3 experimental results 4.3.1 experiment results from the rbf nn model a set of engine test data with three inputs variables and one output variable was employed to evaluate the performance of the proposed approach. the input variables are “break mean efficient pressure”(bemp), “inlet valve opening degree”(vanosin), and “exhaust valve opening degree” (vanosex). the output is “break specific fuel consumption”(bsfc). the whole set of data are from 293 test points. the 196 data points out of 293 are used for training the ga-rbf network and another 97 data points are used for test the ga-rbf network. in the experiment, it run under the following conditions and definitions: • the number of rbfs is fixed at 20 • for comparison purposes, the six different rbf networks are executed 10 times • the normalised mean squares error (nmse) is used as a performance measure for the different rbfs: nmse = ( 1 k ∑k i=1(yi − ti) 2 )1/2 ( 1 k ∑k i=1(ti − t̄)2 )1/2 (19) where yi and ti are respectively the model output and the target value, and t̄ is the mean value of the target values on the training data set or test data set. this expression has the value 0 for a perfect match between model and target, and the value 1 if the model just outputs the target mean t̄. fig.8 shows the evolutionary process of different rbf. fig.8(a) is the average evolutionary process of 10 runs over the training data for six rbfs and fig.8(b) is the best process of 10 runs. the results review that advances in systems science and applications (2014) vol.14 no.1 61 the gaussian function provides the better results compare with other rbfs in terms of convergence speed and modelling accuracy. it is also shown that the local basis function has a faster convergence speed than non-local basis function. the table 4 summaries the final average nmse of 10 runs for different rbfs. the result indicates that the gaussian function has the smallest nmse while the pesudo cubic spline has the largest nmse. table 4 final average nmse over 10 runs for training data average nmse gaussian 0.1217 thin plate spline 0.1397 multiquadric 0.1304 inverse multiquadric 0.1368 pseudo cubic spline 0.1508 logarithmic 0.1274 table 5 compares the final best nmse of 10 runs for both training data and test data. it indicates clearly that the gaussian function performs better than other rbfs in both training data and test data. the figures from fig.5 to fig.10 plot the modelling performance of both table 5 final best nmse over training data and test data training data test data gaussian 0.1134 0.1119 thin plate spline 0.1231 0.1297 multiquadratic 0.1268 0.1245 inverse multiquadratic 0.1304 0.1281 pseudo cubic spline 0.1286 0.1419 logarithmic 0.1235 0.1240 training data and test data using the best final parameters of rbf networks by rcga run. these figures have shown that the rbf networks training by rcga are successfully applied in modelling of engine test data. fig.9 shows the results.of rbf network model for real engine test data analysis and modelling. the model provides a close fit to the ordinary tested data successfully. 4.3.2 experiment results from the nn-tool in this section, an experiment example for engine data modelling was carried out. the input data and output data are listed in fig.10 and fig.11. three nns, mlp, rbf and bar, with different activation functions, abbrevi62 m.h. wu:investigation of the different type models for engine test data modelling fig.8 evolutionary process of rbf networks over 10 runs fig.9 final results fig.10 input data set: speed (rpm), gn(%), ig(degree), vvt(degree) and egr(%) advances in systems science and applications (2014) vol.14 no.1 63 fig.11 output data set: bsfc (g/kwh) ated in table 3, were used throughout the example. the data was taken from engine test bed of the lander rover group, plc. in order to assess the goodness of modelling, the normalized mean squared error (nmse) over each output variable is used: nmse = ( 1 n ∑n i=1(yi − ti) 2 )1/2 ( 1 n ∑n i=1(ti − t̄)2 )1/2 (20) where n is the number of total data points. yi and ti are respectively the model output and the target value, and t̄ is the average value of the targets on the data set, defined as: t̄ = 1 n n∑ i=1 ti (21) nmse has the value 0 for a perfect match between model and target, and table 6 mean, standard (std. dev), minimum and maximum mse over 10 runs for each model 64 m.h. wu:investigation of the different type models for engine test data modelling fig.12 modelling performance of bsfc by sigmoidal bar model the value 1 if the model just outputs the target meant̄ .table 6 shows the result of the example. it is clearly that the model structure which has the minimum mse was the bar model with sigmoidal bar function. the modelling result by bar-sbar is plotted in fig.12. 5 conclusion in this paper, the models used by the engine test data modelling system is introduced. the details of rbf network and the neural network tool have been introduced and the experiment results show that the last two models are running successfully to catch the relationship between the input data and output data within the engine tested data. references [1] voigt k.u. (1993), model based on line optimization for modern engine management system, sae brazil. [2] lechner h. (1996), automated system for optimized calibration of engine management systems, rover group internal publication circa. [3] hochschwarzer h. (1992), fully automatic optimization of engine calibration, imece proceeding, pp.89-100. [4] w c lin & m h wu. (2003), optimization using genetic algorithm in engine calibration, icmr2003, glasgow, uk. [5] c lin, m h wu, s y duan. (2003), engine test data modelling by evolutionary radial basis function networks, imeche publication, journal of automobile eng, part d, vol.217, pp.489-497. advances in systems science and applications (2014) vol.14 no.1 65 [6] m h wu, wanchang lin, s y duan. (2008), develop a nn modelling tool for engine performance tested data modelling, icmr2008, brunel university, u.k. [7] park, j. and sandberg, i. w. (1991), universal approximation using radial basis function networks, neural computation, 3:pp.246-257. [8] poggio t. and girosi f. (1990), networks for approximation and learning, proceedings of ieee, vol.78, pp.1481-1497 corresponding author author can be contacted at: m.h.wu@derby.ac.uk advances in systems science and application (2015) vol.15 no.2 193-201 the physics of jet stream meandering walter e janach meggenhornstrasse 20, ch-6045 meggen, switzerland abstract large-amplitude jet stream meanders involve diabatic processes because conservation of angular momentum leads to vortex line stretching and shrinking. this requires latent heat release during poleward flow in contrast to cooling by sublimation and radiation during equatorward flow. moisture and its distribution, together with aerosols, play a key role. the increase of anthopogenic aerosols in the upper troposphere could be a cause for the more frequent large-amplitude jet stream meanders. keywords angular momentum; potential vorticity; latent heat; radiative cooling; sublimation cooling; rossby waves 1 introduction as the jet stream flows from west to east, rossby waves can form. large meridional excursions of the jet stream in form of meanders occur between alternating warm high-pressure ridges extending polewards and cold low-pressure troughs extending equatorwards. the balanced geostrophic flow of the jet stream is at right angle to the horizontal gradient of the pressure field, so that the resulting horizontal pressure force is balanced by the horizontal component of the coriolis force. at the same time the angular momentum of air parcels in the flow remains constant in the absolute system, which can be expressed by the conservation of potential vorticity. angular momentum remains constant also during horizontal convergence with vortex line expansion, visualized by a ballerina pulling her stretched arms close to her body in order to accelerate the spinning rate, or during horizontal divergence with vortex line shrinking. potential vorticity pv = −g(ζ + f)δθ/δp is the sum of relative vorticity ζ and planetary vorticity f (coriolis parameter) times the difference δθ of potential temperature between the bottom and the top isobaric surfaces of an air parcel divided by the corresponding pressure difference δp, with standard gravity g [1].the pressure difference δp between the bottom and top of an air parcel increases during horizontal convergence with vortex line stretching, while divergence lets the pressure difference δp decrease with vortex line shrinking. 2 conservation of angular momentum during meridional flow in synoptic-scale geostrophic flow of meandering jet stream, relative vorticity ζ is generally much smaller than the coriolis parameter f . as f changes with latitude, ζ of an air parcel cannot change sufficiently to conserve its angular momentum, 194 walter e janach: the physics of jet stream meandering so that a change of pressure difference δp between the bottom and top isobaric surfaces of the air parcel is needed in order to compensate the change of f . during poleward flow of an air parcel in the jet stream, the increase of coriolis parameter f (northern hemisphere) requires horizontal convergence to increase the pressure difference δp through a decrease of horizontal surface area of the air parcel (mass conservation). however the increase of the coriolis parameter f with latitude does not directly force the required increase of δp. what occurs is the forcing of a small ageostrophic deviation ζa of relative vorticity ζ in form of negative velocity shear (anticyclonic) parallel to the flow, superimposed on the velocity profile of the jet stream, as shown in fig.1. this causes a disturbance of the geostrophic balance in the jet stream, which is kept small by the existing pressure field. therefore this ageostrophic disturbance by ζa remains limited to a slightly faster flow on the left side of the jet stream and slower flow on the right side (looking in flow direction). fig.1 velocity profile of poleward jet stream and deviation caused by ageostrophic shear vorticity ζa. the result is a small ageostrophic imbalance between the horizontal component of the coriolis force and the horizontal pressure gradient across the jet stream. this leads to an up-gradient flow deviation on the left side of the jet stream and an opposite down-gradient deviation on the right side. the overall result is horizontal convergence from both sides so that mass continuity of the air parcel with mass δm leads to an increase of pressure difference δp between its bottom and top surfaces. and the increase of δp assures conservation of potential vorticity and angular momentum. this increase of δp gradually changes the pressure field. during poleward flow, the pressure difference δp of an air parcel must increase continuously in order to balance the increasing horizontal component of the coriolis force. by producing a small amount of superimposed velocity shear, which causes lateral convergence of the jet stream, the small ageostrophic deviation ζa of relative vorticity ζ acts as a forcing agent between the continuous increase of advances in systems science and application (2015) vol.15 no.2 195 f and the pressure difference δp. during equatorward flow, velocity shear with opposite sign causes lateral divergence, accompanied by a decrease of δp. in southern hemisphere, f is negative and ζa has opposite sign. the forcing between the change of coriolis parameter f and the pressure difference δp is stabilized by a negative feedback, which adapts the ageostrophic deviation ζa of shear vorticity ζ and the corresponding deviation of velocity in such a way that δp changes in proportion to f . fig.1 shows also the profile of this ageostrophic velocity deviation, whereby ζa is the derivative. the velocity deviation increases outwards from the jet stream center as long as the ageostrophic shear vorticity ζa is negative (anticyclonic). this lets the ageostrophic imbalance become larger with increasing distance from the center and therefore convergence with vortex line stretching also. as a result, starting from a certain distance from the center, the pressure difference δp becomes larger that required to compensate the increase of f . now conservation of potential vorticity forces a positive (cyclonic) correction of ζa in the outer region of the jet stream, accompanied by a reduction of the velocity deviation until it disapperars at the outer edge of the jet stream (fig.1). 3 heating in poleward flow and cooling in equatorward flow during meridional flow of the jet stream there is a continuous increase or decrease of pressure difference δp between the bottom and top isobaric surfaces of air parcels through vertical expansion or shrinking. because this affects an entire layer inside the jet stream, vertical expansion of the layer lifts the atmosphere above it and increases its geopotential height. this requires energy input to the upward expanding layer. in contrast, vertical shrinking of an entire layer reduces the geopotential above it and requires energy reduction. during poleward flow of the jet stream, the increase of δp from upward expansion requires latent heat release from condensation or deposition of water vapor in order to increase internal energy of the air parcel. in contrast during equatorward flow, cooling by radiation to space and by sublimation of ice crystals is needed to decrease internal energy. the upward expansion during poleward flow, which results from convergence and heating, increases the geopotential height of the air columns inside the jet stream as shown in fig.2. this results is westward propagation of the eastward pressure gradient and with it of the geostrophic flow and the jet stream, as a rossby wave. during equatorward flow, the downward shrinking caused by divergence and cooling lets the pressure gradient, which is westward now, propagate westwards also (fig.3). the jet stream meanders between warm high pressure ridges extending polewards and cold low pressure troughs extending equatorwards form a large-amplitude 196 walter e janach: the physics of jet stream meandering fig.2 vertical section of an isobar across poleward jet stream. fig.3 vertical section of an isobar across equatorward jet stream. rossby wave propagating westwards relative to the air mass. thereby air parcels enter into the jet stream from the west, are entrained by the meridional flow and slowly cross it until they exit to the east. this occurs in both poleward and equatorward jet stream. in poleward flow, the air parcels enter from the cold side with lower geopotential, are entrained by the flow and slowly traverse the jet stream towards the warm side with higher geopotential. in the equatorward jet stream, air parcels enter from the high geopotential side in the west and exit to the low geopotential side in the east. a stationary air parcel in the low geopotential region to the west of a poleward jet stream, displacing itself slowly sideways as a westward propagating rossby wave, is initially subjected to an increase of pressure difference δp (fig.2). conservation of potential vorticity at constant latitude and therefore constant f leads to an increase of relative vorticity in form of velocity shear, which accelerates the air parcel polewards. now the coriolis parameter f increases also, forcing a continuous increase of δp by means of a small negative ageostrophic shear vorticity ζa (fig. 1). this continues until the rossby wave and the jet stream have almost propagated past the air parcel and the pressure difference stops increasing. now δp becomes constant again as the air parcel comes to rest and reaches the high geopotential region in the east of the jet stream. heating during poleward flow by the release of latent heat becomes weaker with increasing height because the decrease of temperature strongly reduces the saturation water vapor concentration, a consequence of the nonlinearity of vapor pressure as a function of temperature. this lets the dry and moist adiabats almost coincide. towards the tropopause, long wave radiation to space by ice crystals in cirrus clouds increases strongly and its cooling offsets the heating. this prevents further upward expansion of the top cloud layer. in equatorward jet stream, ice crystals in cirrus clouds are instrumental for both radiative cooling to space and sublimation cooling. during downward shrinking, radiative cooling partly offsets compression warming with the effect that there is less sublimation and ice crystals can survive longer. as sublimation eventually lets ice crystals disappear, starting from the top cloud layer, lower advances in systems science and application (2015) vol.15 no.2 197 clouds get directly exposed to radiative cooling. heating and cooling in meridional flow of the jet stream are part of a dynamic system in which moisture and its distribution play a key role. heating and cooling however change the potential temperature, which violates conservation of potential vorticity. angular momentum can be conserved by replacing the difference of potential temperature between bottom and top isobaric surfaces of the air parcel by its mass m, so that now it is (ζ+ f)δm/δp of the air parcel that must be conserved. 4 changes of jet stream direction conservation of angular momentum of an air parcel requires that the ratio between absolute vorticity (ζ+f) and pressure difference δp must remain constant, δp being between the bottom and top isobaric surfaces of the air parcel with constant mass m. changes of coriolis parameter f or alternatively of pressure difference δp caused by convergence or divergence, lead to changes of relative vorticity ζ and subsequently to a change of the flow field. a change of ζ can be caused by either a change of f or a change of δp, depending upon the situation. diabatic processes play a key role in the evolution of the pressure and flow fields, including jet stream meandering. as an example, a poleward jet stream needs continuous release of latent heat to compensate the increasing f with latitude. an interruption of the moisture supply stops the convergence that increases δp. the result is that now ζ is forced to change, with opposite sign to the change of f . in northern hemisphere with positive f , the increase of f with latitude therefore produces a negative ζ (anticyclonic). this forces the poleward jet stream to veer to the right and eventually results in the cutting apart of the high pressure ridge to the right of the jet stream. thereby a blocking high can form to the north of the partition. in the southern hemisphere with negative f , all signs are opposite, including the sign of anticyclonic ζ, but the physics is the same. in a second example an oversupply of moisture and of latent heat release makes δp increase more than required to compensate the increase of f with latitude. the positive excess of δp causes a positive change of ζ and the formation of a cyclone to the left of the poleward jet stream. in an equatorward jet stream, moisture at higher level is in form of ice crystals that sublimate through compression heating during downward shrinking, which is needed to compensate the decrease of f . there is also radiative cooling to space, which is added to sublimation cooling. a slowdown or interruption of downward shrinking caused by a lack of ice crystals lets the decreasing f force an increase of ζ, which makes the equatorward jet stream veer to the left and east. at this point a cyclone can form at the cold front of the cold trough. now the warm conveyor belt of the cyclone can pick up moisture from the ocean and 198 walter e janach: the physics of jet stream meandering subsequently feed it into the jet stream, allowing it to flow polewards again in the east of the cold trough. 5 the role of aerosols aerosols form the nuclei of droplets and ice crystals by allowing condensation or deposition of water vapor. they play a key role in the formation of cirrus clouds in the upper troposphere [2,3] and in the jet stream. the size distribution and number concentration of aerosols and subsequently of ice crystals influence both the radiative properties and the evolution of cirrus clouds during upward expansion and downward shrinking in the jet stream. when aerosols are more numerous, smaller ice crystals with higher number concentration are formed. smaller ice crystals reduce the sedimentation rate, which influences their concentration at different levels. in addition smaller ice crystals increase the optical thickness for a given mass concentration with the effect that the top layer of the clouds which emits long wave radiation to space, becomes thinner. the higher optical thickness increases also the albedo of solar radiation by cirrus clouds, reducing the absorbed fraction and therefore solar heating of the affected top layer. the higher number density of the smaller ice crystals enhances the deposition of water vapor on ice nuclei because the average distance which a vapor molecule has to cover by diffusion decreases. the variety and subtlety of such processes can be observed in the formation and dissolving of contrails, the condensation trails of high flying airplanes. the dynamics of contrails varies strongly, depending on the properties of the air in which they form and to a smaller degree of the jet engines. 6 potential implications for climate jet stream meandering drives the advective meridional overturn of the troposphere between mid and high latitudes and influences the formation of extratropical cyclones. during meridional flow of the jet stream, aerosols play a key role in the formation of droplets and ice crystals, which in turn controls latent heat release, radiative cooling and sublimation cooling. aerosols have an influence upon the size distribution and number concentration of ice crystals. in the higher troposphere small size aerosols dominate because they have a longer lifetime. anthopogenic aerosols are generally smaller than natural aerosols. a technological reason for example are particle filters for the exhaust from coal fired power plants and diesel engines, which are less effective for small size particles. as a consequence the increase of anthropogenic aerosols at jet stream level is relatively higher than in the lower troposphere, where shortlived larger aerosols are more aboundant. the analysis of residual particles in cirrus crystals after sublimation of the ice by cziczo et al [3] shows that mineral advances in systems science and application (2015) vol.15 no.2 199 dust and metallic particles are the dominant ice nuclei. at present however the anthropogenic fraction of these ice forming aerosols is still uncertain. an increase of anthropogenic ice forming aerosols in the upper troposphere has several potential effects upon both heating and cooling during meridional flow of the jet stream: 1. the increase of the number concentration of small size aerosols leads to more numerous small ice crystals, which reduces their sedimentation speed. this increases their lifetime and in turn the mass concentration of the suspended ice crystals. 2. while this has no direct effect on latent heat release, the smaller size of ice crystals and their higher mass concentration increases the optical thickness, which concentrates the cooling by long wave radiation into a thinner top layer of cirrus clouds. this allows the decreasing amount of latent heat release with increasing height to remain effective up to a higher level and enhances the poleward flow of the jet stream. 3. once an air parcel exits from the poleward jet stream towards the high geopotential side on the east, upward expansion and latent heat release stop, allowing radiative cooling to become dominant. and the increased albedo of solar radiation resulting from the higher optical thickness of the more numerous small ice crystals reduces solar heating, which lets cooling by long wave radiation become more effective. 4. radiative cooling leads to subsidence, followed by compression warming, which lets ice crystals sublimate and generate additional cooling. continuous cooling by sublimation and radiation drives the process of downward shrinking during equatorward flow. and the increase of mass concentration of the smaller ice crystals, resulting from slower sedimentation, increases sublimation cooling during downward shrinking, which enhances the equatorward flow of the jet stream. because of these effects, the increase of anthropogenic aerosols has the potential to enhance both the upward expansion during poleward flow and the downward shrinking during equatorward flow of the jet stream. the overall result is an increase of jet stream meandering. evidence of the arctic jet stream becoming wavier, starting from the mid-1990s, has been presented by francis et al [4]. the regional effects of enhanced jet stream meandering are more frequent cold air outbreaks and the blocking of both low pressure troughs and high pressure ridges. francis [4] presents a linkage between arctic amplification and a wavier jet stream. the increase of anthropogenic aerosols through their role in jet stream meandering is a possible cause. starting from about the year 2000, a slowdown of global warming has appeared. more frequent cold air outbreaks, caused by the increase of jet stream 200 walter e janach: the physics of jet stream meandering meandering, could contribute to this in the following way. during the equatorward flow with downward shrinking of the cold air, high clouds sublimate and allow warmer low clouds and the warm ocean to radiate directly to space. this increases the heat loss by outgoing long wave radiation in a similar way as with the so called “iris effect” [5]. it is interesting to compare the slowdown of global warming during the past 15 years with a similar situation from the 40s to the 60s of the last century [6]. in both situations anthopogenic aerosols from coal burning, heavy industry and mining increased: in the first case during the second world war and the subsequent rebuilding of industry in europe and japan, while in the second case during the forced economic growth in china and india. finally it is possible that the enhanced cooling of a thinner top layer of cirrus clouds, resulting from the higher optical thickness of more numerous small ice crystals, creates a more effective barrier between the troposphere and the lower stratosphere. this could reduce the transfer of water vapor to the stratosphere, with the effect of lowering its concentration there. solomon et al. [7] found evidence of a 10 % decrease of water vapor in the lower stratosphere since the year 2000 and their simulations show a 25 % reduction of global warming. 7 summary and conclusions latent heat release is the main driver for convective overturn in the troposphere. the meridional advective overturn between mid and high latitudes depends also upon the presence of moisture and its distribution. we have shown that conservation of angular momentum and potential vorticity needs the release of latent heat during poleward flow and cooling by radiation and sublimation during equatorward flow of meandering jet stream. thereby aerosols form the nuclei for droplets and ice crystals and influence their size distribution. this in turn controls the rates of precipitation and sedimentation and influences the total mass of suspended ice crystals and their optical thickness, which in turn influences cooling by radiation and sublimation. we explain how the increase of anthropogenic aerosols can influence the meridional flow of the jet stream and suggest that this could increase both its meandering and the blocking of high pressure ridges and low pressure troughs between the meanders. characteristic for the physics of jet stream meandering are the multiple and complex interactions between rotational fluid mechanics, thermodynamics of moist air, aerosols and radiation. an example is the increase of stratification stability resulting from adiabatic convergence or divergence, which explains why latent heat release and cooling by radiation and sublimation are needed to reduce stability. often the distinction between cause and effect is difficult, a challenge for advances in systems science and application (2015) vol.15 no.2 201 numerical models. and a typical case is the evolution of the pressure field with synoptic-scale meridional flow, involving large changes of the coriolis parameter, which lead to the formation of rossby waves. acknoledgements the author would like to thank thomas stocker from university of bern and heini wernli from swiss federal institute of technology zurich for the critical and stimulating discussions. references [1] mcintyre m e. (2015), “potential vorticity”, encyclopedia of atmospheric science, second edition, pp.375-383. [2] lohmann u and feichter j. (2005), ”global indirect aerosol effects: a review atmos”, , atmos chem phys, vol.5, pp.715-737. [3] cziczo d j et al. (2013), “clarifying the dominant sources and mechanisms of cirrus cloud formation”, science, vol.340, pp.1320-1324. [4] francis a f and vavrus s j. (2015), “evidence for a wavier jet stream in response to rapid arctic warming”, environ. res. lett. vol.10. [5] mauritsen th and stevens b. (2015), “missing iris effect as a possible cause of muted hydrological change and high climate sensitivity in models”, tnature geoscience, vol.8, pp.346-351. [6] wilcox l highwood e j and dunstone n j. (2013), “the influence of anthropogenic aerosol on multidecadal variations of historical global climate”, environ. res. lett., vol.8.24-33. [7] solomon s. (2010),“ contributions of stratospheric water vapor to decadal changes in the rate of global warming”, science, vol.327, pp.1219-1223. corresponding author walter e janach can be contacted at: wjanach@gmx.ch the article y.m. tileubergenov, k.kh. shadiyev, b.m. koshpenbetov, y.a. buribayev, z.a. khamzina “fundamental causes of information systems vulnerability and their protection”, advances in systems science and application (2016), vol.16, no.4, p.53-61 was withdrawn by the editorial board due to violation of assa publication ethics (please, see publication ethics statement). the reason is that the article’s content lacks originality due to substantial intersections with the following paper: a.v. revnivykh, a.m. fedotov "main reasons of information systems vulnerability" in global journal of pure and applied mathematics (2016), vol. 12, no. 3, pp. 2133–2142 editorial board of advances in systems science and application, 18.08.17 http://ijassa.ipu.ru/ojs/ijassa/ethics advances in systems science and applications (2014) vol.14 no.1 76-83 exact conditional efficiency robust p-values from an arbitrary ranking of a sample space: an application to genome-wide association studied max moldovan1 and mette langaas2 1australian institute of health innovation, university of new south wales, level 1 agsm building, sydney nsw 2052, australia; 2department of mathematical sciences, norwegian university of science and technology, n-7491 trondheim, norway abstract we introduce a general method for computation of exact conditional efficiency robust enumeration p-values for detection of genotype-phenotype associations at a single bi-allelic genetic locus. our method can be based on any arbitrary ranking test statistics, such as efficiency robust test statistics or asymptotic p-values. the resulting p-values are exact conditional enumeration p-values and satisfy the basic statistical validity property pr(p ≤ α|h0) ≤ α for all parameters under the null hypothesis and all significance levels α. practically, the method allows performing statistically valid significance testing in genomic analyses with unknown modes of inheritance at individual bi-allelic genetic loci the situation typical in genome-wide association studies. we provide an open-source r code implementing the method. keywords: mode of genetic inheritance; efficiency robust statistics; exact conditional inference; enumeration; genome-wide association study. 1 introduction genome-wide association studies (gwas) consider hundreds of thousands of single nu-cleotide polymorphisms (snps) covering the entire human genome. each snp is nor-mally represented by a bi-allelic locus and assessed for association with a speci?c genetic trait, usually in the context of a case-control study. complex diseases such as asthma, diabetes and multiple sclerosis, among many others, are generally targeted by gwas in order to identify common genetic variations as potential disease risk factors. the number of genetic markers in a particular gwas can vary from several hundred thousands to several millions, depending on the platform used for genotyping and the type of genomes to be studied. for example, more snps are required for gwas that utilize african pop-ulations than for gwas involving european populations since the former are older, and thus being exposed to random gene recombinations for more genorations. see hirschhorn and daly[1] and manolio[2] for the interesting well illustrated introductions to gwas. at a given genetic locus, there is a pair of markers, called alleles, inherited from each of two parents. given a trait is passed through this locus, there are advances in systems science and applications (2014) vol.14 no.1 77 several modes of inheritance can be in effect. the dominant mode of inheritance requires the presence of a single “disease” allele from one of parents for a trait to be inherited. the recessive mode of inheritance requires “disease” alleles from both parents to be passed to the offspring for a trait to express. in case of the additive mode of inheritance, a trait is expressed only partly if a single disease allele is inherited, but express in full if both “disease” alleles are in place. there are a few more modes of inheritance can be specified depending on the degree to which a trait is expressed in an offspring, see visscher et al.[3] for the discussion of heritability concepts. when the mode of inheritance at a genetic locus is known, higher power of a test for genotype-phenotype association can be achieved through using the cochran-armitage trend test (catt) under the explicit assumption of the specific genetic model, see lettre et al.[4] and gonzlez et al.[5]. in practice, however, the mode of inheritance is usually unknown. under this typical scenario, the socalled efficiency robust tests (see podgor et al.[6]) can be used, the tests which remain sensitive to detection of genotype-phenotype associations even though the genetic model is either unknown or misspecified. there are several efficiency robust testing strategies. for example, the max test, first suggested by freidlin et al.[7], has been recommended by the several authors, see zheng and gastwirth [8] and gonzlez et al.[5]. this testing approach is implemented as aa sequential application of several statistical tests optimal for alternative genetic models with retaining the most significant result. the traditional version of the max test, normally referred to as max3, is based on the three catts with scores motivated by dominant, recessive and additive genetic models. alternatively, persons chi-square test (χ2) can be included within the same max testing strategy, leading to max4, see li et al.[9]. zheng et al.[10] demonstrated that χ2 test can be considered as a type of a trend test and also noted that this test is sensitive to detection of overdominant (underdominant) modes of inheritance. min2 is one more variant of the max test implemented as a combination of the additive catt and χ2 , see joo et al.[11]. a slightly different efficiency robust testing strategy is known as mert and is a weighted version of catt optimal for recessive and dominant models, see gastwirth[12] and freidlin et al.[7]. there are several more efficiency robust testing approaches can be specified and some authors even suggest applying a combination of different versions of efficiency robust tests within a single testing procedure, see joo et al.[11]. in finite sample settings, many of the currently known and used efficiency robust tests are not guaranteed to lead to statistically valid inference. this is because the under-lying computational procedures are based either on random sampling or on asymptotic distributions of efficiency robust statistics (gonzlez 78 max moldovan: exact conditional efficiency robust p-values from ... et al.[5], joo et al.[13], so anda sham [14]). for the methods that use random permutations (see sladek et al.[15]), the statistical inference will be valid, but a very high number of random permutations is needed to achieve the required precision for traditionally low gwas type significance levels (often in the order of 10−8 ). recently, loley et al.[16] attempted to unify the efficiency robust testing approaches by proposing a framework also leading to inference of unknown statistical validity. in the current paper, we introduce a computational procedure that takes as an input the ordering of a sample space imposed by any of test statistics or p-values, including the ones introduced above. the procedure outputs exact conditional enumeration p-values that satisfy the basic validity property pr(p ≤ α|h0) ≤ α, for all parameters under the null hypothesis and all significance levels α. 2 notation and the method let the information on a single snp be represented by the 23 contingency table given by table 1, where xi and yi are the counts of observed genotypes for n1 cases and n2 controls, respectively, with n = n1+n2 . we denote this empirically observed table by s ∗ (x1, x2|m1,m2, n1, n2), because all the other entries of the table can be calculated from these numbers. note that for given n1 , n2 , m1 and m2 , there is a finite number of possible contingency tables, called a reference set (verbeek[17]) and denoted here by (m1,m2, n1, n2). next let t be an arbitrary ranking statistic with the value t corresponding to the empirically observed table s ∗ (x1, x2|m1,m2, n1, n2). given a general hypothesis of ‘h0 : no association between genotypes and the case-control status of the subjects’ tested against ‘ha : there is association between genotypes and the case-control status of the subjects’, and larger values of t being more hostile to the null h0 , the set of tables ranked lower or equal than the observed table s ∗ (x1, x2|m1,m2, n1, n2) is given by the critical set: r(x1, x2|m1,m2, n1, n2) := {s(i, j|m1,m2, n1, n2) : t ≥ t} (1) by definition, a p-value is the probability of obtaining the outcome as extreme or table 1 genotype counts at a bi-allelic locus. aa ab bb total case x1 x2 x3 n1 control y1 y2 y3 n2 total m1 m2 m3 n worse than the empirically observed outcome s ∗ (·) under the null, which is just the probability of the critical set r(·). under the null and based on the assumed advances in systems science and applications (2014) vol.14 no.1 79 underlying hypergeometric sampling scheme (see lehmann[18] for descriptions of alternative sampling schemes), the probability of obtaining each individual table s(i, j|m1,m2, n1, n2) within the reference set can be computed as follows: f(i, j|m1,m2, n1, n2) = ( m1 i )( m2 j )( n−m1 −m2 n− i− j ) ( n n1 ) (2) see lloyd[19] for the generalization of the central multivariate hypergeometric probability function given by (2). the p-value ps∗;t corresponding to the empirically observed table s ∗ (x1, x2|m1,m2, n1, n2) is the probability of the critical set r(·) given by (1): ps∗, t (x1, x2|m1,m2, n1, n2) = ∑ s∈r f(i, j|m1,m2, n1, n2) (3) note that ps∗, t is a fisher-type conditional p-value by construction, inheriting positive(e.g. statistical validity and empirical relevance) as well as negative (e.g. conservatism and computational challenges) aspects of fisher’s p-values. 3 numerical illustration denote statistics obtained from catts optimal for dominant, recessive and additive models, respectively, by td, tr and ta (sasieni[20]): td = n(nx1 − n1m1) 2 n1m1(n− n1)(n−m1) tr = n(nx3 − n1m3) 2 n1(n− n1)(nm3 −m2 3) ta = n(n(x2 + 2x3)− n1(m2 + 2m3)) 2 n1(n− n1)(n(m2 + 4m3)− (m2 + 2m3)2) all three statistics asymptotically follow the chi-square distribution with one degree of freedom. the max3 test statistic is given by tmax3 = max(td, tr, ta) with the observed value tmax3 = max(td, tr, ta). for the empirically observed table s∗(0, 2|3, 4, 4, 5),the p-value ps∗, tmax3 can be computed as shown in table 2. specifically, there are 11 tables in the reference set s(3, 4, 4, 5) and only three tables in the critical set r(0, 2|3, 4, 4, 5) since only the tables with the values of tmax3 statistics equally or more extreme then observed are included in the critical set, i.e. tmax3 ≥ tmax3. the resulted exact conditional efficiency robust p-value ps∗, t (0, 2|3, 4, 4, 5) = 0 : 0952 and is the sum of f(·|m1,m2, n1, n2) given 80 max moldovan: exact conditional efficiency robust p-values from ... by (2) of the three tables in r(0, 2|3, 4, 4, 5). table 2 the illustrative example is based on (m1,m2, n1, n2) = (3, 4, 4, 5) with an observed value (x1, x2) = (0, 2). the critical region r is given by the lower part of the table under the horizontal line. table 2 genotype counts at a bi-allelic locus. x1 x2 td tr ta tmax3 f(x1, x2|m1,m2) 1 2 0.2250 0.0321 0.1636 0.2250 0.2857 2 1 0.9000 0.0321 0.2557 0.9000 0.1905 1 3 0.2250 2.0571 0.2557 2.0571 0.0952 2 2 0.9000 2.0571 2.0045 2.0571 0.1429 1 1 0.2250 3.2143 1.7284 3.2143 0.0952 2 0 0.9000 3.2143 0.1636 3.2143 0.0238 0 3 3.6000 0.0321 1.7284 3.6000 0.0635 0 4 3.6000 2.0571 0.1636 3.6000 0.0079 0 2 3.6000 3.2143 4.9500 4.9500 0.0476 3 0 5.6250 0.0321 2.0045 5.6250 0.0159 3 1 5.6250 2.0571 5.4102 5.6250 0.0317 ps∗, t = 0.0952 4 conclusion the method we suggested above is by no means new. the initial idea can be traced back to fisher [22] and ps;t given by (3) is based on the combinatorial results known for many decades, see freeman and halton [23]. our contribution to the original fisher’s methodology is the idea of ordering the sample space, given by the reference set s, based on any arbitrary chosen ranking statistics, the efficiency robust test statistics in our case. we have borrowed this approach from the unconditional exact testing literature, see barnard [24] and lloyd and moldovan [25] for the origination of the unconditional inference philosophy and one of the initial attempts to combine the conditional and unconditional types of exact inference, respectively. to conclude, it should be pointed out that only the basic form of the adjustment procedure has been given above. in practice, more special cases can arise, such as the presence of covariates (e.g. additional snps, environmental factors or baseline factors) or involvement of additional shifted parameters (e.g. in power studies). while this is clearly the limitation of the presented procedure, the basic general exact conditional method introduced above gives a solid basis for further investigations to these and possibly several more theoretical and applied research directions. we provide an open-source r code to encourage and facilitate such advances in systems science and applications (2014) vol.14 no.1 81 investigations. the r code is available upon request from the authors. references [1] hirschhorn, j.n., and daly, m.j. (2005), “genome-wide association studies for common diseases and complex traits”, nature reviews genetics, 6, 95108. [2] manolio, t.a. (2010), “genomewide association studies and assessment of the risk of disease”, new england journal of medicine, 363, 166-176. [3] visscher, p.m., hill, w.g., and wray, n.r. (2008), “heritability in the genomics era-concepts and misconceptions.”, nature reviews genetics, 9, 255266. [4] lettre, g., lange, c., and hirschhorn, j.n. (2007), “genetic model testing and statistical power in population-based association studies of quantitative traits” genetic epidemiology, 31, 358-362.. [5] gonzalez, j.r., carrasco, j.l., dudbridge, f., armengol, l., estivill, x., and moreno, v. (2008), “maximizing association statistics over genetic models” genetic epidemiology, 32, 246-254.. [6] podgor, m.j, gastwirth, j.l., and mehta c.r (1996), “efficiency robust tests of independence in contingency tables with ordered classi cations.” statistics in medicine, 15, 2095-2105. [7] freidlin, b., zheng, g., li, z. and gastwirth, j.l. (2002), “trend tests for casecontrol studies of genetic markers: power, sample size and robustness”, human heredity, 53, 146-152. [8] zheng, g., and gastwirth, j.l. (2006), “on estimation of the variance in cochranarmitage trend tests for genetic association using case-control studies.” statistics in medicine, 25, 3150-3159. [9] li, q., zheng, g., and yu, k. (2009), “robust tests for single-marker analysis in case-control genetic association studies.” annals of human genetics, 73, 245-252. [10] zheng, g., joo, j., and yang, y. (2009), “pearson’s test, trend test, and max are all trend tests with different types of scores.” annals of human genetics, 73, 133-140. [11] joo, j., kwak, m., ahn, k., and zheng, g. (2009), “a robust genome-wide scan statistic of the welcome trust case control consortium.” biometrics, 65, 1115-1122. 82 max moldovan: exact conditional efficiency robust p-values from ... [12] gastwirth, j.l. (1985), “the use of maximin efficiency robust tests in combining contingency tables and survival analysis”, journal of the american statistical association, 80, 380-384. [13] joo, j., kwak, m., and zheng, g. (2010), “improving power for testing genetic association in case-control studies by reducing the alternative space”, biometrics, 66, 266-276. [14] so, h.c., and sham, p.c. (2011), “robust association tests under different genetic models, allowing for binary or quantitative traits and covariates”, behavior genetics, 41, 768-775. [15] sladek, r., rocheleau, g., rung, j., dina, c., shen, l., serre, d., boutin, p., vincent, d., belisle, a., hadjadj, s., balkau, b., heude, b., charpentier, g., hudson, t.j., montpetit, a., pshezhetsky, a.v., prentki, m., posner, b.i., balding, d.j., meyre, d., polychronakos, c., and froguel, p. (2007), “a genome-wide association study identifies novel risk loci for type 2 diabetes”, nature, 445, 881-885. [16] loley, c.,, i.r., hothorn, l., and ziegler, a. (2013), “a unifying framework for robust association testing, estimation, and genetic model selection using the generalized linear model”, european journal of human genetics, to appear. [17] verbeek, a. (1985), “a survey of algorithms for exact distributions of test statistics in r × c contingency tables with fixed margins”, computational statistics and data analysis, 3, pp.159-185. [18] lehmann, e.l. (1986), testing statistical hypotheses, 2nd ed., new york: wiley. [19] lloyd, c.j. (1999), statistical analysis of categorical data, new york: wiley [20] sasieni, p.d. (1997), “from genotype to genes: doubling the sample size”, biometrics, 53, pp.1253-1261. [21] devlin, b., and roeder, k. (1999), “genomic control for association studies”, biometrics, 55, pp.997-1004. [22] fisher, r.a. (1935), “the logic of inductive inference (with discussion)”, journal of the royal statistical society, 98, pp.39-54. [23] freeman, g.h., and halton j.h. (1951), “note on an exact treatment of contingency, goodness of fit and other problems of significance”, biometrika, 38, pp.141-149. advances in systems science and applications (2014) vol.14 no.1 83 [24] barnard, g.a. (1947), “significance tests for 2 × 2 tables”, biometrika, 34, pp.123-138. [25] lloyd, c.j., and moldovan, m. (2007), “unconditional efficient one-sided confidence limits for the odds ratio based on conditional likelihood”, statistics in medicine, 26, pp.5136-5146. corresponding author authors can be contacted at: max.moldovan@gmail.com. advances in systems science and application(2015) vol.15 no.2 164-173 expert system for international crisis management -crisman rafael calduch cervera and nina wörmer nixdorf department of international law and international relations, madrid, spain abstract the prototype crisman is part of an ambitious project which claims to introduce computer simulation as a part of investigation labour fulfilled by spanish internationalists through the practicality of knowledge based systems and the establishment of multidisciplinary investigation groups. keywords expert system, international relations, rule-based, artificial intelligence, prediction, conflict. 1 computer simulation applied to international relations in the field of social sciences the discipline of international relations (ir) was included into degree studies in united kingdom and united states in 1919. nevertheless, its theoretical development and the use of quantitative techniques since the 1950s, allowed to incorporate mathematical models (game theory and probability theory) creating indicators which provided a more accurate description and explication of many phenomena of the international society: arms race, armed conflicts, “guerrillas” and terrorism, underdevelopment and conflicts of structural nature, mass media and public opinion, international bipolarity and multi-polarity or regional political integration, among others. progresses made in computational technology, artificial intelligence (ai) and communicational and informational media have accomplished to deal with international happenings through simulation[1]. in fact we can find important contributions of agent based simulations (abs) in social science[2]. however, the main obstacles for applying computer simulation in the field of ir are two: a) the vagueness of multiple theoretical concepts, used in this science, making quantitative data collection and its evaluation (using indicators or probability calculation) more difficult, and b) the existing ignorance among internationalists concerning the possibilities of new, both informational and computational, technologies. a fact that leads, quite often, to underestimate and undervalue the work carried out in the area of simulation. these two realities are more present among internationalists and investigators of the spanish speaking world due to the crucial influence of disciplines like legal or historical sciences during their training, being neither of those sciences really permeable concerning quantification and new technologies. crisman is the first step taken to introduce ai as an effective technique in the investigation labour fulfilled by spanish internationalists. we point out three consecutive phases of this project: 1st phase: creation of a prototype of a rule-based expert system (rbes) covering the analyse and foresight of international crises, with a direct application in different activities developed by investigators, intelligence analysts, political decision makers, high business executives, all of them guided in their international scope activity. advances in systems science and application(2015) vol.15 no.2 165 prepare a doctoral thesis focusing on the utility of expert systems (ess) teaching ir, including the development of a computer programme focusing on diagnostic, classification and training of international conflicts. 2nd.investigate about the applicability of fuzzy logic to ess created for international crisis management[3]. 3rd.develop a new ess concerning evaluation and international political risk management as well as training in international aid programmes through simulation. 2 prototype of expert system for international crisis management -crisman 2.1 problem framing the problem which is tried to be solved by using a rbex consists in defining the evolution and existing relations between two countries or group of countries a and b, taking as a starting point an initial situation (conflict or crisis) between those countries, in order to anticipate the more likely decisions which will be taken by political leaders and to provide explanations of those decisions. 2.2 technical specifications this es is constructed upon a theoretical model or simplified description of reality, in order to facilitate the learning of foreign policy analysis techniques. to achieve this goal, computer data and processing storage will be used with the aim of creating simulations of real-world cases through the use of rules or set of logical formulations which associate, in a conditional way, causal variables or attributes with consecutive variables or attributes. the shell or computer programme used for this expert system is clips, developed by nasa (http://www.ghg.net/clips) in its 6.23 version (2005). we also find an adapted version for java called jess ((http://www.herzberg.ca.sandia.gov/jess), as well as an application suitable for fuzzy logic called fuzzyclips, 6.10c version, developed by the national research council of canada. the knowledge base and the production rules have been carried out by professor rafael calduch cervera[4]. a forward chaining inference process allocated with certainty or trust coefficients will be applied to each of them. the case data, used as examples, and the results, obtained by executing the programme, will be stored in a data base using xml language. 2.3 theoretical model the theoretical model underpinning the knowledge base and the rules of the expert system are articulated on the existing relation between six basic attributes: 1) situations; 2) aims; 3) available means; 4) previous case experience; 5)future expectations; 166 rafael calduch cervera and nina wörmer nixdorf: expert system for international ... 6) relational strategy and 7)conduct. the user should define the variables choosing between the different options given by the es concerning the first four attributes, while the programme will deduce the categories 5, 6 and 7; which, in turn, become the initial situation for the new realisation cycle of the es. the biggest methodological problem that had to be solved to make the theoretical model applicable was to reduce the semantic options of each attribute category to its minimum, avoiding the exponential growth of number of rules. this has not been easy, mainly because of the huge amount of diversity of conditions, objectives, means and type of existing experiences concerning relations between two countries. graphic display of the theoretical model fig.1 graphic display of the theoretical model attributes 1.situations: advances in systems science and application(2015) vol.15 no.2 167 1.1).normalised 1.2).conflict 1.3).deepening conflict 1.4).crisis 1.5).armed conflict 2.aims: 2.1).compatible 2.2).perception of compatibility 2.3).incompatibility 2.4).perception of incompatibility 3.available means 3.1).equivalence 3.2).superiority 3.3).inferiority 4.previous case experience 4.1).certainty of trust 4.1.1).trust 4.1.2).mistrust 4.2).uncertainty 5.future expectations 5.1).continuation of the situation 5.2).escalating of disputes 5.3).deescalate of disputes 5.4).unpredictable 6.relational strategy 6.1).cooperative 6.2).negotiating 6.3).competing 6.4).deterrent 6.5).aggressive 7.conducts: 7.1).keep cooperation 7.2).intensify diplomacy 7.3).diplomacy with no military pressure measures 7.4).diplomacy with deterrent military measures 7.5).military deployment with limited use of force 7.6).widespread use of force 168 rafael calduch cervera and nina wörmer nixdorf: expert system for international ... 3 description of attributes 3.1 situations it is referred to the mutual relations between two countries in a limited period of time. we can choose between the following options: 1.1.normalised: the situation in which the relations between two countries (a and b) are mainly cooperative and respecting international legal standards. 1.2.conflict: the situation in which the relation between two countries (a and b) is of conflict, although there exists no explicit threat of violence or the use of force. 1.3.deepening conflict: the situation of conflict between two countries (a and b) in which one or both countries turn to military measures of pressure or with deterrent character, although there exist no direct nor express threat of violence or use of force. 1.4.crisis: the conflict situation in which, one or both countries (a and b), threaten with the use of force or make a limited use of it to condition the behaviour of the other country. 1.5.armed conflict: the situation in which the general use of force is the main tool concerning the relation of both countries (a and b). 3.2 aims we understand this topic as those interests or purposes which are being tried to achieve by each of the countries according to their capacities and available means in a particular situation. according to the relation between purposes and targets of the countries we can find the following possibilities: 2.1.compatible: it takes place when the purposes and targets of both countries (a and b) can be achieved simultaneously and, as well, this possibility is sensed as part of reality. 2.2.perception of compatibility: it takes place when the subjective perception of achieving interests or targets from one of the countries (a) can be achieved simultaneously with the satisfaction of others interests or targets of the other country (b), even though elements which prevent this compatibility exist. 2.3.incompatibility: it takes place when the accomplishment of interests or targets of one or both countries (a and b) cannot be achieved simultaneously and, also, this possibility is clearly sensed as part of reality. 2.4.perception of incompatibility: it takes place when the subjective perception makes believe that the achievement of interests or targets of one of the countries (a) cannot be achieved simultaneously with the satisfaction of the interests or targets of the other country (b), even though there do not exist any elements for this incompatibility in reality. 3.3 available means this topic includes all kind of capacities a country is willing to use in the relations with any other country to ensure the consecution of its targets. e can find the following possibilities: 3.1.equivalence: it exists when the means used by one country (a) in its relation with another country (b), whatever its nature, hold or at least are valued with the same level of effectiveness in the same manner, to ensure the achievement of their respective targets. advances in systems science and application(2015) vol.15 no.2 169 3.2.superiority: it exists when the means used by one country (a) in its relation with another country (b), are in fact, or at least are valued with a higher grade of effectiveness than those of the other country to ensure the attainment of their particular targets. 3.3.inferiority: it exists when the means used by one country (a) in relation with the other country (b) are in fact, or at least are valued with a lower grade of effectiveness than those of the other country to ensure the attainment of their particular targets. 3.4 previous case experience it is formed by the evolution of the political relations between two countries (a and b), during the period of time in which a generation exercises leadership in both countries (25 to 30 years) in the way it is perceived by their leaders. we have to work with the following options: 4.1.certainty of trust: when the leaders of one country (a), based on former identical or analogous experiences, are convinced that the leaders of the other country (b) will adopt the necessary decisions and actions to fulfil their compromises to execute their threats assuming the consequences associated to it. 4.1.1.trust: when the leaders of a country (a), based on former identical or analogous experiences, are convinced that the leaders of the other country (b) will fulfil the compromises reached with them, although they may imply losses of interests or additional winnings in case of infringement, and in case of escalation, they would formulate a clear and explicit threat. 4.1.1.mistrust: when the leaders of a country (a), based on former identical or analogous experiences, are convinced that the leaders of the other country (b) will not fulfil the compromises reached with them, because of putting the unilateral satisfaction of their targets or the achievement of additional winnings through infringement, and in case of escalation, they would not formulate a clear and explicit threat. 4.2.uncertainty: when the leaders of a country (a) have a lack of former identical or analogous experiences or the ones existing are contradictory and, in consequence, they have no deep-seated conviction concerning the degree of compliance or infringement of the other country (b) will make in relation with the achieved compromises or, in case of escalation, they do not know if a previous, clear and explicit threat will be formulated. 3.5 future expectations we consider future expectations as the evolution expected by political leaders of every country concerning the present situation with the other country. we have to choose between four possibilities: 5.1.-continuation of the situation: no significant changes between the two countries (a and b) are expected. 5.2.escalating of disputes: a new, more conflictive, situation between countries (a and b) is expected due to changes in their relation. 5.3.de-escalating of disputes: a new, less conflictive, situation between countries (a and b) is expected due to changes in their relation. 170 rafael calduch cervera and nina wörmer nixdorf: expert system for international ... 5.4.unpredictable: the political leaders of countries (a and b) lack the knowledge or enough former experiences not being able to make future expectations concerning the evolution of their relations. 3.6 relational strategy these kinds of strategies are formed by the planning and organization of the relations of one country with another through the exclusive, or clearly dominating, remit of specific types of behaviour in the relation. we can find the following categories: 6.1.cooperative: the strategy which resorts to collaborative relations between two countries (a and b) to achieve the established targets together. 6.2.-negotiating: the strategy which uses communication or political and diplomatic negotiation to achieve some kind of agreement or understanding between two countries (a and b) to satisfy their targets. 6.3.competing: it is the kind of strategy that combines communication or political and diplomatic negotiation with pressure builds or conflictive, but no military, behaviours, trying that country (a), the one making them, achieves unilateral advantages in relation with country (b) or achieves the exclusive satisfactions of its targets. 6.4.deterrent: this kind of strategy combines communication or political and diplomatic negotiation, deployment and/or increase of the military capacity of country (a), with the explicit intention of using this military capacity to defend its vital interests and targets or to avoid an armed aggression fulfilled by the other country (b). 6.5.aggressive: this strategy implies the unilateral and extensive use of force by one country (a) to achieve specific targets or exclusive advantages at the expense of the other country (b). 3.7 conducts conducts are the dominating actions which characterize the relations between two countries in a specific situation. there are the following possibilities: 7.1.keep cooperation: it means that the behaviour of one country (a) concerning the other country (b) keep the same so that the mutual interests or joint targets can be satisfied. 7.2.intensify diplomacy: it means that there is an increase of communication or political and diplomatic negotiation between one country (a) regarding the other country (b) in order to facilitate or achieve any type of understanding or agreement to achieve their interests or targets. 7.3.diplomacy with deterrent military measures: this conduct creates the use of a combination of communication or political and diplomatic negotiation with the use of behaviours, excluding the use of force, that are intended to provoke direct damage to the other country (b) or to hinder the accomplishment of its targets. 7.4.diplomacy with deterrent military measures: this conduct means the use of different communicational or political and diplomatic conducts with the use of demonstrations measures, display or increase of its military capacity by one country (a) but only with a defensive character concerning the other country (b). 7.5.military deployment with limited use of force: this kind of conduct means the use of a combination of measures from one country (a) that entails a direct threat for the other country (b) through the display of its military capacity or the use of these advances in systems science and application(2015) vol.15 no.2 171 military capacities in a limited period of time and space. 7.6.widespread use of force: this conduct involves developing of all kind of strategic, tactical and logistical conducts needed to warlike use of military capacities of one country (a) against another country (b) in order to achieve its defeat. 4 priority of conditioning of attributes to draw up rules the order of priority concerning the application of causal attributes to determine consistent attributes is the following: 1.initial situation 2.aims 3.available means/previous case experience the main conditioning resides in the initial situation of the relations between two countries because of two main reasons: firstly, because this situation will also be the final situation, product of former relations between those countries and this, of course, affects the previous case experience and also because the initial situation conditions the category of aims which are tried to be fulfilled by each country. the second most important conditioning is set-up by the aims every country tries to achieve through its relation with another country. concerning this topic we have to bear in mind the different strategies and conducts that can be developed in their relation. the third level of conditioning considers the effectiveness concerning the means every country uses compared to the ones used by another country, because this relation of effectiveness will condition possible strategies to be followed by every country as well as the effectiveness of conducts adopted by each of them concerning the other. nevertheless, this third conditioning can also match with previous case experience because this experience is decisive to determine future expectations made by the leaders of every country and this will, as well, condition the election of the judged to be more effective or probable strategies and conducts to achieve the targets set. the priority of conditionals concerning every single attribute will determine, in case of conflict, the effects that should prevail in the rule formulation, considering the following criteria: a).when the conditionals of different levels are complementary they cause a strengthening or intensification of the resultant effects. b).when there is a conflict or opposition between conditions of different levels, the effects of the hierarchical higher level will be chosen. c).if the conflict is between two conditionals of the same level their result concerning the effects will be neutralized, using those effects derived from higher hierarchical level. examples of rules used in crisman rules 1)normalized initial situation r 1).if initial situation = normalised and aims a) and b) = compatible 172 rafael calduch cervera and nina wörmer nixdorf: expert system for international ... and available means a) and b) = equivalence and previous case experience a) and b) = certainty of trust then future expectations a) and b) = continuation of the situation then relational strategy a) and b) = cooperative then conducts a) and b) = keep cooperation so final situation = normalised r 2).if initial situation = normalised and aims a) and b) = incompatibility and available means a) and b) = equivalence and previous case experience a) and b) = certainty of trust then future expectations a) and b) = escalating of disputes then relational strategy a) and b) = negotiating then conducts a) and b) = intensify diplomacy so final situation = conflict r 3). if initial situation = normalised and aims a) = perception of compatibility and aims b)= perception of incompatibility and available means a) and b) = equivalence and previous case experience a) and b) = certainty of trust then future expectations a) = continuation of the situation then future expectations b) = escalating of disputes then relational strategy a) = cooperative then relational strategy b) = negotiating then conducts a) = keep cooperation then conducts b) = intensify diplomacy so final situation = normalised 5 theoretical operating model of the es a).the initial situation between the countries a and b at the moment (t-0) is the starting point. b).the aims of each country (a and b) to identify the possibility/probability of simultaneous and joint or unilateral success of those aims by each of the countries are analyzed. c).the relation of existing effectiveness of each country (a and b) to achieve their targets is assigned. d).historical precedents that have existed in the relations between both countries (a and b) in identical or analogous situations are evaluated to define their influence concerning the perception of the leaders. taking the options chosen by the user for former attributes into account, the es will deduce: 1).expectations that the political leaders of each country (a and b) set on the future development of relations between the two countries. 2).the most likely strategies political leaders will develop in each country (a and b) in their relation with the other country. advances in systems science and application(2015) vol.15 no.2 173 3).the behaviour that will prevail in each country’s relations with the other based on the established relations strategy. 4).-the type of situation that will result at the end of reciprocal behaviour made by both countries. this final situation concludes the cycle and it is established as the initial situation of the next cycle in the operation of the expert system. 6 user profile crisman can be used in the teaching of graduate students in the universities and research centres, as well as by international analysts who may carry out an evaluation of political risk assessment. * carlos e. calduch cervera developed the prototype of the computer program to perform the expert system crisman. references [1] taber, ch. s. (1992), “poli: an expert system model of u.s. foreign policy belief systems”, american political science review, vol.86, no.4, pp.888-904. goldspink, ch. (2002), “methodological implications of complex systems approaches to sociality: simulation as a foundation for knowledge”, journal of artificial societies and social simulation, vol.5, no.1. see http://jasss.soc.surrey.ac.uk/5/1/3.html. [2] davisson, p. (2002), “agent based social simulation: a computer science view”, journal of artificial societies and social simulation, vol.5, no.1, see http://jasss.soc.surrey.ac.uk/5/1/7.html. [3] cioffi-revilla, c. a. (1981), “fuzzy sets and models of international relations”, american journal of political science, vol. 25, no.1, pp.129-159. [4] calduch, rafael. (1991),international relations , ediciones de ciencias sociales, madrid. [5] calduch, rafael. (1993),dynamics of international society, ediciones ceura, madrid. corresponding author nina wörmer nixdorf∗ can be contact at: nwormer@ccinf.ucm.es advances in systems science and application (2016) vol.16 no.1 95-101 vlsi implementation of lightweight cryptography algorithm hinpreet kaur and sakthivel r school of electronics engineering, vit university, vellore, tamil nadu, india. abstract lightweight algorithms for cryptography are popular for resource stringent devices. now days, radio frequency identification techniques are gaining popularity because of their small size and low cost applications. in this paper, hummingbird algorithm is used to provide security in such devices like rfid tags, smart cards, etc. it is a hybrid algorithm which provides security against most of the common attacks encountered such as linear attacks and differential attack. the hybrid combination of block and stream cipher makes this algorithm more secure using lesser number of clock cycles. the algorithm is implemented on both fpga and asic platform. the main aim is to reduce the number devices utilized and using lesser area on the chip. the algorithm is implemented in xilinx vertex-5 family. this design consumes the total standard cell area of 0.255 mm2. the design is placed and route using cadence soc encounter at tsmc90nm technology. keywords hummingbird algorithm; rfid tag; lightweight cryptography; asic; fpga. 1 introduction the motivation behind this work is the increase in demand of resource stringent devices such as smart cards and devices using the rfid (radio frequency identification) technology. rfid tags are now being used in libraries to keep the records of the book, in hospitals for medical equipment detection, etc. the small size and low cost of the tags makes it popular. rfid devices consist of mainly three elements, a tag, a reader and a database system. there exist a communication between the tag and the reader. this communication is affected by outside environment if a third person or attacker tries to leak the information. so it becomes essential to incorporate an algorithm in order to make the communication secure. this work proposes an area efficient architecture on both fpga and asic platform. 2 background hummingbird is proved to be very secure algorithm because it is a hybrid algorithm of stream cipher and block cipher. despite of presence of many cryptographic algorithms such as des (data encryption standards) and aes (advanced encryption standard)[? ], we need algorithm which is lightweight on both hardware 96 hinpreet kaur and sakthivel r: vlsi implementation of lightweight cryptography algorithm and software side, which is not fulfilled by the mentioned algorithms. since rfid devices are small in size, the cryptographic unit thus used should be hardware friendly and provides the same level of security as in other non-resource constrained devices. several work has been carried out till related to this algorithm. many other lightweight algorithms such as present, klein, hight, led, iceberg are implemented in fpga platform but hummingbird is proved to provide the highest degree of security and is resistant to many attacks such as birthday attacks, algebraic attacks, structural attack and cube attack. the work related to hummingbird are tabulated in table1.the first hummingbird algorithm was implemented in 4-bit microcontroller with low power consumption[? ]. many implementations are being carried out on fpga platform are shown in the table1. table 1 different implementations of hummingbird algorithm paper year of publication implementation platform key feature ref[? ] 2009 microcontroller 4-bit microcontroller , low power consumption ref[? ] 2010 fpga larger throughput ; smaller area ref[? ] 2011 fpga co-processor approach ref[? ] 2011 fpga throughput oriented and area oriented ref[? ] 2011 fpga gen2 protocols for mac ref[? ] 2014 fpga low power and high speed 3 algorithm steps the hummingbird algorithm consists of a 256-bit secret key shared between the tag and the reader. there are four internal registers of 16-bit and a 16-bit galois field lfsr. the registers are first initialized in the initialization process with some random initial vectors which are then being updated by the lfsr in the encryption process. the input to encryption is the plaintext and the output is the 16-bit cipher text. fig.1 shows the working of the algorithm. the 16-bit plain text is given as the input along with the 256-bit key, which is segmented into four 64-bit subkeys. there are three blocks which perform initialization, encryption and decryption. in the initialization process, the registers are updated for encryption. after each set of plaintext and cipher text, the internal status registers are updated by the 16-bit lfsr. the initialization block and the encryption block consist of the block substitution box or s-box and the linear permutation. the predefined s-box and the inverse s-box is shown in table 2. the message is encrypted by the tag using the encryption process and the resultant cipher text is sent to the reader. the reader decrypts the message using advances in systems science and application (2016) vol.16 no.1 97 the same key (as the key is being shared between tag and the reader). fig. 1 algorithm flow of hummingbird cryptography table 2 s-box and the inverse s-box used in the algorithm x 0 1 2 3 4 5 6 7 8 9 a b c d e f s(x) 2 e f 5 c 1 9 a b 4 6 8 0 7 3 d s(x) c 5 0 e 9 3 a d b 6 7 8 4 f 1 2 4 proposed architecture the proposed architecture is shown in fig.2. this architecture works on the encryption only and decryption only processor. the initialization block consists of four status registers and a ready pin. a 5-bit counter is used to count the number of clock cycles. when the data rdy pin goes high, the counter starts counting and the status registers are initialized with some random values. the registers are then updated by undergoing encryption round of block cipher in next 16 clock cycles. for the algorithm refer[? ]. the encryption block uses the initialized status registers to encrypt the plain 98 hinpreet kaur and sakthivel r: vlsi implementation of lightweight cryptography algorithm text into cipher text. encryption undergoes the modulo 216 addition of register rs1 and plain text and undergoes through the block encryption using the secret key. this is repeated four times and the resulting cipher text is given to the decryption module making the enc complete signal high. the decryption is just reverse of encryption. the input is cipher text and output is plain text. modulo 216 subtraction is used in decryption. thus when both dec complete and enc complete signals are high the output is given in one clock cycle. the registers are updated for the next process. table 3 device utilization summary sheet logic utilization used available utilization ref.[? ] this work ref.[? ] this work number of slice register 4242 74 12480 33% 1% number of slice luts 3504 2309 12480 28% 18% number of bonded iobs 278 117 172 161% 68% number of fully used lutff pair 1907 70 ref.[? ]5839 this work-2313 32% 3% number of bufg/ bufctrls 8 1 32 25% 3% fig. 2 proposed architecture of hummingbird encryption and decryption 5 simulation results the encryption block along with the initialization module of the hummingbird algorithm are designed and simulated using the s-box using the lut based apadvances in systems science and application (2016) vol.16 no.1 99 proach. lut based s-box uses less area and power. the simulation results are shown in fig.3 the 16-bit plain text is encrypted to 16-bit cipher text. when the reset is high, there will be no initialization process. after the reset signal changes to high, the initialization starts and the 16-bit plain text is converted to its cipher text using a 256-bit secret key. the first cipher text is obtained after 8-clock cycles and then at the subsequent clock cycles we will get the other cipher texts. the simulation carried out using modelsim 6.5b. in the simulation results shown in fig.3, a plaintext is encrypted using a 64-bit key and the respective cipher text and the decrypted data is obtained after 20-clock cycles. the result of the console window is given below: # hummingbird input data==0011000000111001 # hummingbird key==1110011011100111001000001101101110111000001101101011000111011001 # hummingbird encrypt data==0101100100001110 # hummingbird decrypt data==0011000000111001 fig. 3 simulation result of encryption process the entire algorithm is designed and implemented in xilinx 14.7 ise suite with vetex-5 xc5vlx20t in package ff-323 and speed grade -2. the results obtained were compared with the mentioned results in [? ]. the device utilization sheet is shown in table.3. the total number of slices occupied in our design is 917, which is very less as mentioned in [? ]. the design works at a frequency of 14mhz and the power consumption is 322.80 mw at 2.5v. the device utilization summary sheet mentioned in table.3 shows that our design utilizes less space and is hardware friendly. the lightweight algorithm such as hummingbird can be used in resource stringent devices and are now a days used in password identification, library management system, hospitals embedded in the rfid tag. the asic implementation of the hummingbird encryption and decryption core is done using cadence soc encounter using tsmc 90nm technology. the final chip layout is shown in fig.4. the design is successfully placed and route using cadence soc encounter tool using tsmc 90nm technology with no setup and hold violations. the cadence rc compiler results in the area, power, timing and the gate counts used in the design. the die area, power, area in gate counts and maximum frequency of 100 hinpreet kaur and sakthivel r: vlsi implementation of lightweight cryptography algorithm operation are tabulated in table 4. the design has worst case skew of 19psec and the best case skew of 16.1psec with no timing violations. after the place and route the design was verified for the geometry and there were no violations. this proposed design is thus area efficient at both fpga and asic platform. fig. 4 final chip layout table 4 specifications area .2556mm2 power 2.234mw gate counts 11363 max. frequency 2.129 ghz 6 conclusion and future scope the hummingbird encryption and decryption module is implemented in both fpga and asic platform. the design is coded using verilog hdl and implemented in xillinx 14.7 ise suite in vertex-5 family. the results show reduced device utilization in slices. the asic implementation is done using cadence soc encounter using tsmc90nm technology library. the design is successfully placed and route with no timing violations and the area power of the final chip is noted. thus this design is suitable for resource stringent devices and hardware friendly as compared to aes and other cryptographic algorithms. the algorithm used here s-boxes, which can be designed using boolean expression instead of lut based approach. so the area and power can be reduced by using bdd reduction unit. the variable reordering reduces the boolean functions according to the variables being selected. learning cad tools to perform bdd reduction module is further scope of study. references [1] p.chodowiec and k.gaj. (2003), “very compact fpga implementation of aes algorithm”, in cryptographic hardware and embedded systems cches 2003, c. walter, c. koy and c. paar(ed.), springer berlin heidelberg, vol. 2779, pp. 319-333. [2] f. xinxin, g. guang, k. lauffenburger and t. hicks. (2010), “fpga implementations of the hummingbird cryptographic algorithm”, in hardwareadvances in systems science and application (2016) vol.16 no.1 101 oriented security and trust (host), 2010 ieee international symposium on, pp. 48-51. [3] t. san and n. at.(2011). “compact hardware architecture for hummingbird cryptographic algorithm”, in international conference on field programmable logic and applications (fpl), pp. 376-381. [4] m. biao, r. c. c. cheung and h. yan. (2011). “fpga-based high throughput and area-efficient architectures of the hummingbird cryptography”, in iecon 2011 37th annual conference on ieee industrial electronics society, pp. 3998-4002. [5] x. mengqin, s. xiang, w. junyu and j. crop.(2011), “design of a uhf rfid tag baseband with the hummingbird cryptographic engine”, in 2011 ieee 9th international conference on asic (asicon) , pp. 800-803. [6] x. mengqin, s. xiang, w. junyu and j. crop.(2011), “design of a uhf rfid tag baseband with the hummingbird cryptographic engine”, in 2011 ieee 9th international conference on asic (asicon), pp. 800-803. [7] nikita arora and yogita gigras. (2014), “fpga implementation of low power and high speed hummingbird cryptographic algorithm”, international journal of computer applications , vol. 92, no. 16, pp. 0975-8887. [8] xinxin fan. (2010), efficient cryptographic algorithms and protocols for mobile ad hoc networks, canada: ontario. corresponding author hinpreet kaur can be contacted at: hinpreetkaur@gmail.com advances in systems science and applications (2013) vol.13 no.2 116-143 modified yoyo model and its applications in chinese history zelong wang1,yi lin2 and jubo zhu1 1school of science, national university of defense technology,changsha, 410073, china 2department of mathematics, slippery rock university, slippery rock, pa 16057, usa abstract this paper focuses on the modified yoyo model, a hot issue of the system research called the second dimension of science, and it supplies a new theory and methodology to address and solve the practical problems, which have been extremely difficult in modern science. moreover, the new explanations of the chinese history about the periods of spring and autumn and the warring states are introduced by the modified yoyo model. the systemic yoyo model is a new tool to describe the general systems, and it is effective to explain the existing phenomenon. the modified yoyo model, based on the systemic yoyo model, is constructed by the forms of the definitions and laws, which are demonstrated and explained by lots of the existing system theories and practical examples. the application of this model in chinese history is a new look at the historical events, compared with the traditional study of the chinese history. the modified yoyo model is proposed based on the systemic yoyo model and it emphasizes on explaining some theoretic problems of the system model, such as the general system structure model, system’s characteristics, systems’ interaction and so on. firstly, the yoyo model can not sufficiently depict the attributes of the general system, such as the behavior, the function, the size of the system and so on. all of these are the indispensability of a system and the system is half-baked without them. secondly, the basic characters of the yoyo model are not given enough to describe the system roundly. these matters include the complexity, diversity, stability of the system and how the system comes into being or perdition. thirdly, there have been many examples that refer to the interaction between yoyos, but the interaction process is not detailedly discussed and the final result can be ascertained by some laws. the modified yoyo model is applied in the chinese history about the spring and autumn and warring states periods. the military system of each country in the ancient time is modeled by the modified yoyo model and their interactions are analyzed by the characters of the modified yoyo, which lead to the same results with the history. the modified yoyo model is firstly applied in the chinese history, and the battle effectiveness of the military system is modeled as a yoyo. the wars between the countries are described as the interaction of the modified yoyos, and the rea117 advances in systems science and applications (2013) vol.13 no.2 sonable results of the wars are also explained. in fact, since the general systems are modeled by the modified yoyo model, this model can be applied in many practical systems, such as three-body movement system, economic system and so on. in theory, the modified yoyo model embodies and develops the systemic yoyo model, and it introduces some new concepts and characters of the yoyo as well as their interaction, which can be effectively used for the general system analysis. when referring to the application, the modified yoyo model can explain or predict lots of phenomenon, which may be the puzzles in modern science. keywords modified yoyo model, yoyo’s character, yoyos’ interaction, spring and autumn and warring states, military system 1 introduction as the developments of the system science and system thinking, new theories and methodologies have brought new understandings and discoveries to some of the major unsettled problems in modern science. moreover, the view of wholeness and interconnectedness in system thinking has greatly changed the tendency of modern science, such as synthesizing all areas of knowledge into a few major blocks, the appearance of the second dimension science and the cross-disciplinary studies, etc. system is not only an entity that can be touched and felt actually but also a thinking that is used to observe the world around us. then we would have a new idea to find the potential connection and disciplines that have been ignored for the divide of modern science. in fact, lots of so-called puzzles are caused by this parochialism. therefore, it is reasonable that these puzzles may be solved easily by system thinking, which causes the development of system science inversely. the yoyo model[1], cared and developed by yi lin, is a useful systemic tool to describe the real world and nearly all kinds of phenomenon, and it stands for the freshest achievement of system science. however, the renascence of yoyo model causes its imperfectness in theory and application. many entities and phenomenon have been modeled by the yoyo model soundly, but most of the basic properties of the system are not explained precisely, leading to an uncertain result. so this paper focuses on modifying the yoyo model and applying it in chinese history of the periods of spring and autumn and warring states. 1.1 a history review of system and the origin of yoyo model systems methodology is an important concept in system science, and it is understood in different ways by scholars at different periods and different science fields. however, the understandings are roughly the same and they are unified together gradually. quastler[2] said: “systems methodology is essentially the eszelong wang: modified yoyo model and its applications in chinese history 118 tablishment of a structural foundation for four kinds of theories of organization”. the opinion of zadeh[3] is that the main task of systems science is the study of general properties of systems without considering their physical specifics. even though the concept of systems has been a hot spot of discussion in almost all areas of modern science and technology, which was first introduced formally by von bertalanffy[4] in the second decade of the 20th century in biology, as all new concepts in science, the ideas and thinking logic of systems have a long history. although the concept of system is not emphasized in history, its idea was proposed thousands of years ago and “the whole is greater than the parts” is good evidence. during most time of the science history, the first dimension method has played a leading role in scientific research. as time going by, system attracted more and more attentions for the conflicts from the isolation of subjects and lots of new characters of system, such as the wholeness, interconnection, and occlusion, have been proposed in different fields. in last century, there have been many advances in technology: energies produced by various devices such as steam engines, motors, computers and automatic controllers, self-controlled equipment from domestic temperature controllers to self-directed missiles, and the information highway that has resulted in increased communication of new scientific results. all of these made the system a individual subject, and some scholars began to research the system from many points of view. the concept of the second dimension gradually came to the view, and system science came to a new level. at the turn of 21st century, with his profound insights, independent creativity, and courage, shoucheng ouyang proposed the blown-up theory of nonlinear evolution problems[5]. uneven structures are eddy sources, leading to eddy motions instead of waves, the mystery of nonlinearity, which has been bothering humankind for long time, is resolved at once both physically and mathematically. on the basis of the blown-up theory, the concepts of black holes, big bangs, and converging and diverging eddy motions are coined together in the model shown in fig.1[6]. each system or object considered in a study is a multidimensional entity that spins about its invisible axis. if such a spinning entity is fathomed in threedimensional space, a structure like that shown in fig.1 would be achieved. for the sake of convenience of communication, such a structure is called a yoyo due to its general shape. more specifically, what this model says is that each physical entity in the universe, be it a tangible or intangible object, a living being, an organization, a culture, a civilization, etc., can be seen as a kind of realization of a certain multidimensional spinning yoyo with an invisible spin field around it. it stays in a constant spinning motion. if it does stop spinning, it will no longer exist as an identifiable system. 119 advances in systems science and applications (2013) vol.13 no.2 fig.1 yoyo model (the side of a black hole sucks in all things, such as materials, information, energy, etc. after funneling through the short narrow neck, all things are spit out in the form of a big bang. some of the materials, spit out from the end of the big bang, never return to the other side, and some will.) 1.2 wide applications of yoyo model because spin is the fundamental evolutionary feature and characteristic of materials, the yoyo model can be used in figurative analysis method. it also gives us a chance to generalize all three laws of motion so that external forces are no longer required for these laws to work[7]. in terms of applications of the yoyo model in social science and humanity areas, it is shown that in a market of free competition, a concept as fundamental as demand and supply is about mutual reactions and mutual restrictions of different forces under equal quantitative effects[7]. hence, each economic entity can be naturally modeled and simulated as an economic yoyo or a flow of such yoyos. when looking at how people think, one can show the existence of the systemic yoyo structure in human thoughts[7]. so, the human way of thinking is proven to have the same structure as that of the material world. in a word, what can be thought as a system, what can be considered as a yoyo model. why does the yoyo model have so wide applications in our world? the most important is that it captures the essential character, i.e. nonlinearity and unevenness, of the general systems, and scientifically abstracts the main conflict between problems. the yoyo model describes the uneven resources, so it also describes the nonlinear motion since the motion comes from the uneven resources. the second reason lies in the materialism, which implies that the world is made up of materials. no one can talk anything without the materials, including the yoyo model. the yoyo model firstly pays attention on the materials, the basis of general system, and then has the ability to uncover the disciplines of much phenomenon about the materials. moreover, the yoyo model, proposed and developed by lots greats, has a unique structure that is fit to all kinds of real systems. all of these zelong wang: modified yoyo model and its applications in chinese history 120 make the yoyo model have a leading role in system research. 1.3 the imperfection of yoyo model as what’s known, the yoyo model has attracted more and more attentions since its appearance. this depends on its wide applications in our lives; however, due to the renascence of the yoyo model, it has some imperfections that need to be developed. there have been many examples that the yoyo model can explain partly, which show that the yoyo model is correct to describe our world; on the other hand, it can not give us a clear sense about the systems, their basic characters and their interactions. so we need to consummate the yoyo model to give a perfect explanation about the general system. fig.1 has gives us a direct vision about the yoyo model, which stands for any objects that can be considered as a real system. however, when a system is modeled as a yoyo, we could not know the basic information about the yoyo model, such as its exact shape, height, its spin direction, etc. in any case, we only know that the yoyo model exists with a system, and all the things, such as the materials, information and energy, come into the yoyo from the black hole and come out from the big bang. how shall we measure the stability of the yoyo? how does the yoyo spin? what is the yoyo’s size? nobody knows. if the yoyo model and its basic attributes are strictly defined, all of the puzzles above would be solved easily. moreover, the characters of the yoyo, such as the complexity and the diversity, should be explained by the yoyo model. firstly, the behavior of the yoyo is not paid enough attentions, because there are lots of cases that we can not capture the yoyo directly, but the behavior of the system is easy to learn, by which we can conclude the system indirectly, i.e. the yoyo and its behavior have some equivalences. secondly, the characters of the yoyo can tell us how a new yoyo comes into being, how an old yoyo is destroyed and how the yoyo keeps its steady structure. therefore, the characters of the yoyo should be studied carefully before its applications. another problem is the interaction between yoyos. as explained in examples, the interaction between yoyos is complex and the yoyos always release a ‘force’ to each others, so the phenomenon of the birth, the development and the perdition of yoyos occur during the interaction. for example, the competition between two yoyos, standing for two companies, will cause the bankruptcy of the weaker or their contemporary existence, but how shall we know which would happen? what we have finished only is that both of them can be explained by yoyo model and their interaction leads to two results. therefore, the interaction between yoyos should be analyzed more precisely and interaction disciplines should be uncovered. then based on the interaction laws, we can easily predict what will happen after the interaction between yoyos. 121 advances in systems science and applications (2013) vol.13 no.2 therefore, the yoyo model should be modified and some indispensable parts should be added to the yoyo model. since lots of systems have been successfully explained by the original yoyo model, the modified yoyo model would be more efficient in applications. in the following, the modified yoyo model is proposed and it would be proved by some simple examples. 1.4 organization of this paper this paper focuses on the modified yoyo model and its application in chinese history. in section 2, the modified yoyo model is introduced, including the background of the modified yoyo model, the characters of modified yoyo and their interaction. this section mainly consummates the yoyo model and makes the yoyo model have enough information to stand for the system exactly. the disciplines of the interaction between yoyos are helpful to predict the possible interaction result. section 3 gives us a new example about modified yoyo model, which has not been illuminated by the yoyo model. chinese history about spring and autumn and warring states is a wonderful period, and the transition from slave society to feudalism society causes the complexity and the diversity of the politics, militancy and ideology. these can be all explained by the systemic yoyo theory, but only the military system is modeled by the modified yoyo model. the last section is the conclusions about the whole paper and two open problems about the yoyo model are introduced for the continuing research. 2 modified yoyos and their interaction in order to give a more refined explanation about the system, the modified yoyo model and their interaction laws are proposed in this section. 2.1 background of modified yoyo model the key concept of system science is “system” and it has been a hot issue since its birth. although it is still not clear enough, the basic properties, such as the wholeness, interaction and so on, has been accepted by lots of scholars. now if we define a system, every one will understand what we mean. the yoyo model, a new tool as well as other system models, is proposed to describe the general system, and it has explained lots of properties of the system, such as the blown-ups and shrinking. it uncovers the essential characters of the general systems, i.e. the nonlinearity and unevenness cause the eddy motion. it is a great discovery that many unsettled problems in modern science have been solved by the yoyo model. however, the yoyo model is not perfect, because another important factor, i.e. the behavior of the system, is not included in the model. most of the time, we not only focus on the system itself, but also put emphasis on the interaction between systems. if a system can not have any influence on the environments, its existence will not make any sense. the system and its behavior should be zelong wang: modified yoyo model and its applications in chinese history 122 trussed together and modeled by yoyo model. except for the system itself, the behavior of the system is another aspect that can help us to understand the system’s properties. for example, even if we never hear of or visit some company, we can know its size easily by its raw material or production scope. the raw material and production are considered as the input and output of the company, which can also be seen as the behavior of the company. when we research the company, the input and output are also important factors except for the company itself. sometimes, we even can not capture the system; instead, we can only describe its behavior. taking the black hole as an example again, nothing can escape from the black hole, including the light, let alone the scientists. so nobody can look into it, but we can conclude its properties by its behavior. therefore, the behavior of the system supply us another way to describe the system. on one hand, the basic characters of the yoyo depend on its behavior and can also be depicted by the behavior. for example, the stability, complexity and diversity of the yoyo can be reflected by its behavior. on the other hand, the interaction between yoyos is also based on their behavior. when a system receives an action from others or it has an action on others, the acceptation and the impartment are both the behavior of the system. for example, there is gravitation among three-body problem, and the competition exists between two similar companies. when a system has no behavior, it would not have any connection with the environments and then it would not make any sense for the development of the human, which would not be the research context of human. as to the close system, it actually would not exist in nature, i.e. the systems in nature are open and the close ones are ideal. therefore, the yoyo model should include not only the system but also their behavior, from which the basic characters and the interaction laws of the yoyos. in the following, the modified yoyo model and its new properties are proposed. 2.2 modified yoyo model the basic concept of yoyo model has been introduced in detail, and every system can be considered as a yoyo model. in this subsection, the modified yoyo model is introduced, including its structure, intensity, polarities, size and its representation. all of them are reconsidered from a new point of view. 2.2.1 spin field the yoyo model has been shown in fig.1, where the eddy stands for the system, reflecting its nonlinearity and unevenness. what are the things that come into or out the yoyo from the black hole or the big bang? as to the company, they may be the input and output of the company; they can also be gravitation field for the celestial bodies; or they can be air current for the fan. they are actually the behavior ability of the system and can not be separated from the system, just as 123 advances in systems science and applications (2013) vol.13 no.2 fig.2. the behavior of the system is also important for the interaction between systems, and they either receive the behavior from others or give off behavior to others. if a system does not have any behavior, maybe it would not exist any more. therefore, the system and its behavior are trussed together and the yoyo model should also reflect both of them instead of the system alone. in the yoyo model, they are spin field through the yoyo, as along as the yoyo spins all the time. without the spin field, the yoyo would not spin and the yoyo would be destroyed. what’s more, the spin field is the medium that the yoyos connect fig.2 spin field of yoyo model with each other. it is just like that the systems have connection with others by their behavior, instead of the system themselves. for example, the company changes its raw material and production as along with the outside market, and the output and input are medium for connection. without these things, the company nearly has no behavior and no connection with the market. therefore, the spin field is a necessary for the yoyo model; the yoyo will be destroyed without the spin field. the spin field is not the part of the yoyo, i.e. the system, but it denotes all the behavior ability of the yoyo, including the action that the yoyo receives or gives off. since the function of the system is carried out by its behavior, every system should have the behavior ability to realize some function; otherwise, the existence of the system would not make any sense. therefore, as to the yoyo, there must be spin field through it, permeating the whole space. 2.2.2 spin line in order to make the spin field exact and visible, the spin line (sl) is assumed to describe the spin field of the yoyo model and it does not exist actually. as to a company, each spin line stands for a flow of input or output. generally speaking, a truss of spin lines of the yoyo reflects a kind of behavior ability by which the system has connection with other systems. all the spin lines of the yoyo form the whole behavior ability of the system, so the spin field is made up of all the spin lines of the yoyo, i.e. as to discretionarily given point in the spin field, there is a zelong wang: modified yoyo model and its applications in chinese history 124 spin line through it, shown in fig.3. since the spin lines denote the behavior fig.3 spin lines of yoyo model ability of the system, the behavior intensity, i.e. the spin field intensity (sfi), should also be defined by the spin lines. the sfi at some point in the spin field is defined as: sfi(p) = lim a→0 1 a ∫ s(p) sl(s)ds (1) where p is the observed point; sl(s) is the spin line that pass through the point s; the s(p), covering the observed point p, is the observed area and a denotes the acreage of the observed area, shown in fig.4. returning to the system archetype, sfi reflects the behavior intensity of the system, and where the sfi is larger, where the density of the spin lines is bigger, meaning that the system has a greater behavior. fig.4 spin field intensity there are two important properties of the spin lines, i.e. the partial occlusion and the non-intersection. generally speaking, the partial occlusion means that some of the spin lines that depart from some points would return to these points after period of time and their tracks are just like some close loops, but others not. as to a system, the close spin lines denote that the action that this system takes to the environments has influence on the system itself actually. however, not 125 advances in systems science and applications (2013) vol.13 no.2 all the action from the system has influence on itself, so there also are non-close spin lines except for the close ones. for example, if a company is considered as a system, i.e. a yoyo model, the spin lines would be the flow of the input and output of the company. some stuff of the input maybe just part of its production, which indicates some of the output returns to the company as part input of the company; and other output not. therefore, the spin lines of the yoyo are classified as the close and the non-close, i.e. the spin lines are partial occlusion. the second property of the spin lines is the non-intersection, i.e. each two spin lines in spin field of the yoyo do not have the same point. it means that the behavior of the system (a yoyo) would not have influence on its other behavior or each two categories of the behavior of a system are independent of each other. actually it may be impossible that two categories of the behavior of a system do not have any connection with each other, but the puny connection is always ignored and they are supposed to be independent for a convenient analysis. this method of assumption has been used in nearly all the modern science, and it is also so in system science. taking the three-body problem as an example, the popular method used in astrophysics is that the two heaviest bodies are firstly researched without the third body and then the third one is added to the two-body system. what to do next is just some modification of the system. the analysis results by the independent assumption are very close to the factual instance, although the modification is not discussed in this paper. 2.2.3 polarity of the yoyo we have known that all the things come into the yoyo by the black bole and out from the yoyo by the big bang, so the black hole and the big bang are two basic properties or polarities of the yoyo. however, the black hole and the big bang do not have universality, and they are replaced by the polarity of the yoyo. in fact, each yoyo, i.e. each system, has two polarities, such as the input and output of the system. generally speaking, each system has the ability of giving off influence on and receiving influence from the environments, and these different directions of behavior abilities of a system are considered as the system’s polarities. moreover, the two polarities of the yoyo can not be separated, such as the positive and negative polarities of a cell, the south and north poles of a magnet, etc. as to general system, if it only has influence on the environments and can not receive action, the system can not be controlled; if it only receives action from the environments and can not produce any action, the system would be useless and not exist any more. since two polarities of yoyo exist, they should be defined and their meanings should also be explained clearly. they are defined as the following. every yoyo has two polarities, the positive and the negative polarities; the positive polarity is located in the side where the spin lines of the yoyo come out from the yoyo, zelong wang: modified yoyo model and its applications in chinese history 126 and the negative one is located in the other side of the yoyo, shown in fig.6. the polarities can reflect the direction of the behavior of the yoyo, i.e. the yoyo takes action on others or receives action from others. for example, when a car is running by the control of the motorman, the motorman is located in the negative polarity and the running of the car is located in the other polarity. both of the two polarities are important for the car, the car would make no sense without anyone. fig.5 two polarities of the yoyo 2.2.4 size of the yoyo when we focus on a yoyo (a system), its size is an important content. actually it has great influence on the behavior intensity of the yoyo and even the behavior category, so it is discussed here. generally speaking, the size is a relative concept and it is hard to define this concept among the systems belonging to different layers. in fact, the size of the yoyo depends on the research intention and content, and the size of even the same system would be different if it is researched by different intention and content. if two yoyos are researched for different contents, it would make no sense to compare the size of a yoyo with the other. for example, when we experiment on the mass of the earth and the velocity of the light respectively, the mass and the velocity can be seen as a measure of their sizes. however, they do not have any comparableness for the different dimensions. as to the yoyos with the same research content, how shall we define the sizes of them by the yoyo model? when studying a yoyo, mostly we do not directly analyze the system itself; we can put an emphasis on its behavior instead. the definition of the size can also come from this thinking. if we study the mass of the earth, we can also measure it from the eart’s gravitation to another known system, i.e. the outside behavior of the earth. and sometimes it is the only method to measure the yoyo’s size, because the yoyo has not been mastered by the human in detail, such as the black hole, the universe, some particles and lots of social phenomenon. in sub-subsection 2.2.1 and 2.2.2, we have defined the spin field and the spin 127 advances in systems science and applications (2013) vol.13 no.2 lines of the yoyo, and both of them depict the behavior of the yoyo. so it is natural to define the size of the yoyo by its spin field or spin lines. as to a given research content , the behavior of a yoyo is only some of all the spin lines except for studying any aspects of the yoyo, shown in fig.5. in other words, the size of a yoyo is its behavior intensity at the research aspect. fig.6 size of the yoyo for a given research content therefore, the size of a yoyo is defined as: size(γ) = ∫ s(γ) sl(s)ds (2) where γ is the research content and s(γ) is the total spin lines that stands for the whole behavior at the aspect γ. if the spin lines denote the input or output flow of a company, then the total input or the output, defined as the size of the yoyo, can reflect the manufacture size of the company sufficiently. 2.2.5 denotation symbol of the yoyo in the book “systemic yoyos” written by professor lin, the denotation symbol of the yoyo model has been given in detail, just like figure 7. they are very visual, but there is some discommodiousness by these symbols. first of all, they can not contain enough information of the yoyo, such as the spin field, the size and the polarity of the yoyo, so they can not denote yoyos sufficiently. secondly, these symbols are inconvenient to paint on the papers, because there is a long line round some fixed point and it is hard to paint roundly. moreover, they take up a larger area of the paper. thirdly, when two yoyos have interaction with each other, their shapes change to elliptic type. if these symbols denoted the yoyos, does that mean the yoyos themselves have been changed? as we known, the interaction between yoyos need not change the yoyos, but we can not see the interaction clearly if the shape does not change. then a contradiction occurs. fourthly, as to a given yoyo model, we can see different spin directions from different sides, so how shall we make a decision about the spin direction when painting them? zelong wang: modified yoyo model and its applications in chinese history 128 fig.7 denotation symbol of the yoyo given by professor lin therefore, a simple and convenient symbol is given in fig.8. this symbol is a line with the same direction to the spin lines of the yoyo; its length is decided by the size of the yoyo; the line points from the negative polarity to the positive polarity and the jumping-off point of the line is located in the neck of the yoyo. moreover, when a yoyo line is given, the spin yoyo can be only ascertained by the representation laws. the representation law can be described as the following: the direction of the spin lines of the yoyo is accordant to that of the symbol, and spin direction of the yoyo has dextro rotation relation with that of the symbol, i.e. when the yoyo line is grasped by the right hand and the thumb points to the direction of the spin lines, the spin direction of the yoyo is along to the direction of other four fingers; the size of the yoyo is also ascertained by the length of the line. this representation symbol is not only simple but also contains nearly all the information of the yoyo. what’s more, they have interaction with each other by the spin field, i.e. their behavior, instead of the yoyos. fig.8 yoyo model and its denotation symbol 2.3 properties of modified yoyo model since the yoyo model has been modified and some dimensions have been added to the yoyo model, the new properties of the yoyo model are discovered naturally. these include the characters of the yoyo and the interaction between yoyos, which are described lively by the new dimensions. 129 advances in systems science and applications (2013) vol.13 no.2 2.3.1 basic characters of the modified yoyo as to the simplex yoyo, the most interesting things are how the yoyo comes to being and perdition, how the yoyo keeps steady and why the yoyo can endure the little change of the outside environments. all the answers come from the modified yoyo model and they can be explained in the following by the form of properties. property1 : the cause and effect relation leads to the continuity of the spin line through the yoyo, so the fluxes through two polarities are equivalent. the cause and effect relation widely exists in the world. for example, we are living for food; the output is produced for hard work; and the circle movement of celestial bodies keeps for their gravitation. so the cause and effect relation is general in the word, including the modified yoyo model. every effect of a system is begotten by some cause that has been known completely or incompletely. maybe there is more than one reason that leads to a selfsame result; to analyze these complex reasons conveniently, all of them are summed up to a spin line that links to the spin line of the result at the neck of the yoyo. therefore, the spin lines that have been painted in the figures continuously pass through the yoyo, which indicates the cause and effect relation of the system. since every cause spin line leads to an effect spin line and every effect spin line is begotten by a cause spin line, it is concluded that the quantities of spin lines through each polarity, i.e. the two fluxes of spin lines through two polarities correspondingly, are equivalent. when the “cause” spin lines change, the “effect” spin lines would change correspondingly. for example, if a factory is modeled as a yoyo, the change of the producers’ goods must leads to a change of the yield. the equivalence of the spin lines through two polarities can bring the convenience for analyzing the characters and interaction of the yoyos, so it is a basic and important property. for example, according to the research content and intuition, the size of the yoyo can be defined by each flux of spin lines through two polarities. another example is that the direction of the spin lines can be ascertained by each polarity of the yoyo for their equivalence. property2 : the behavior size of a yoyo has equipollence with its tsme size. the size of a yoyo is defined by its behavior ability, i.e. its spin field, and this definition is one of many possible definitions about the yoyo’s size. it can be used to describe the size of a yoyo if the yoyo’s structure and component are not learned clearly by us or we pay our attentions to the yoyo’s behavior ability instead of itself. when a yoyo has some invariability with regard to its structure or component, maybe it also can be used to describe the yoyo’s size. because for each given yoyo there must be a positive number such that: at ×bs × cm ×de = a (3) zelong wang: modified yoyo model and its applications in chinese history 130 where a,b,c,d and d are some constants determined by the structure and attributes of the system of our concern,t is the time as measured in the system,s is the space occupied by the system, and m and e are the total mass and energy contained in the system. if the yoyo keeps steady, the positive constant a, called tsme size in this paper, measures the essential size of the yoyo. moreover, the behavior size has some equivalence with the tsme size, i.e.: a ⇔ ∑ γ∈ω size(γ) (4) where γ is a sort of behavior ability, and ω is the set of all behavior abilities. property3 : the diversity and complexity of a yoyo are reflected by that of its behavior abilities, i.e. the spin field. now we have known that the world is wonderful for all kinds of species, so why the world is composed of so many objects, including the organics and the inorganics? the great natural historian darwin gives the answer by his darwinism, stating that all species of organisms arise and develop through the natural selection of small, inherited variations that increase the individuals ability to compete, survive, and reproduce. in other words, the national selection of behavior abilities causes the biology evolution as well as the inorganics. another visible example is that lots of artificial objects, such as the car, the computer, the airplane and so on, arise as the development of the human civilization, which increases the diversity and complexity of the world. their appearance lies in that their various behavior abilities can service for the human. we can not affirmatively conclude that the diversity and complexity of the system are brought to by their various behavior abilities, but it is affirmative that the diversity and complexity of the system can be reflected by that of its various behavior abilities. as to the yoyo model, its structure, component and attributes are not discussed in detail, so it is hard to talk about its diversity and complexity by the yoyo itself; however, its spin field, i.e. the various behavior abilities, can reflect its diversity and complexity. so the different behavior abilities and intensities stand for the difference between different yoyos. property4 : the stability of the yoyo is equivalent to that of its spin field. firstly, let us discuss the stability of the system. when a system is in the state of stability, its component, structure and function all do not change as time going by. of course its behavior is also steady. if the system’s component or structure changes, its behavior would change correspondingly, which even can not be discovered easily by us. moreover, if the behavior of a system keeps steady, the system must also keep steady. in other words, the behavior of a system can reflect the stability of the system. for example, when a cup is broken up, it can not be used for drinking, i.e. the 131 advances in systems science and applications (2013) vol.13 no.2 behavior of the cup has changed greatly. sometimes the destruction of a system can not bring a timely change of its behavior, but this change would come sooner or later. taking the assumptive vanishing of the sun as an example, all the planets in solar system maybe can not experience this change at once, i.e. they can still receive the gravitation and light for about 8 minutes[8], but their orbits must change for the disappearance of the sun finally. the change of the behavior of a system must lie in the change of the system; otherwise, we even can not control the simplest system to service for the human. in modified yoyo model, the system and its behavior are modeled as the yoyo and its spin field, so the stability of the yoyo is equivalent to that of its spin field. property5 : the change of the spin field leads to the corresponding size change of the yoyo. the change of the spin field means that the behavior of the yoyo has changed, so the size of the yoyo changes for its definition. when an additional “force” is operated on a system and its function is changed, the behavior of the yoyo must be changed correspondingly. for example, the input and output of a company, i.e. the spin field of a yoyo, are reduced for the furious market competition, and then the company has to reduce its size, such as reducing the employees, decreasing the salary and saving the cost. otherwise, the company would take the opposite action. property6 : the yoyo has inertia to keep the original state. the concept of inertia, explicitly proposed by great physical scientist newton, is used to describe the kinetic character of all the objects. the inertia counterworks the change of the movement of an object, but it can not stop this change. does a system (yoyo) have similar inertia to counterwork the change of its state or stability? yes, it not only can counterwork this change but also can stop this change within some range, i.e. when the yoyo is influenced by some outside factors, the yoyo, including its structure and its component, would not change if the influence is limited in some extent. the influence that begins to make the yoyo change is defined as the stability boundary. as to a relationship between two persons, denoted by a and b, when one of them, supposed as a, satirizes the other person for some reason, b would accept the advance and their relationship holds out in all probability if b is given enough prestige; however, if a speaks ill of the person b now and then instead of proposing his advance by courtesy, his action would go beyond the stability boundary of their relationship and the relationship would break up. therefore, the relationship system has the inertia to keep the original state. the inertia is important for the stability of the yoyo; otherwise, the yoyo would be changed as long as it is affected by the environment, no matter how little the effect is. the influence boundary describes the stability degree, and the yoyo is zelong wang: modified yoyo model and its applications in chinese history 132 steadier if its influence boundary is larger. the inertia helps the yoyo to have a steady spin field, i.e. the yoyo has a steady behavior, and then the yoyo has a steady function to servicing for the human. property7 : the yoyo comes into being or perdition both for the poignant change of the spin field of the environment, i.e. all the systems outside the observed system. the development of the yoyo is important for the system research and it also reflects the phenomenon of everything’s movement. the development of a yoyo means the processing of the yoyo from an old state to a new state, and these change including the variety of the components, structure or attributes of the yoyo. in other words, the development of a yoyo can also be considered as the perdition of the “old” yoyo and the being of a “new” yoyo. therefore, if the perdition and the being of a yoyo are researched clearly, the development of the yoyo is also known by us. how does the yoyo come into the being or perdition? property 7 gives us the answer. when the spin field that the observed yoyo is located in changes greatly, the yoyo would receive great “force” from the spin field and it would be destroyed if the “force” exceeds the stability boundary of the yoyo. when two or more yoyos have interactions with each others, their composed spin field would greatly change, and the new yoyo would come into being among their poignant interaction. for example, in the chinese history of the spring and autumn and warring states, the country jin was overthrown under the condition of unstable political, economic and cultural positions, which had exceeded the stability boundary of the country jin, and three new countries came into being from the “old” country jin. there are two basic processing by which the old yoyo is destroyed and the new yoyo comes into being. one of them is the composition, i.e. two or more yoyos form a new yoyo, such as the composition of two companies; the other is the decomposition of an old yoyo into small new yoyos, such as the dividing of a country. in a word, the yoyo’s being or perdition lies in the poignant change of the spin field. 2.3.2 interaction between modified yoyos the modified yoyo model and its some basic properties have been introduced, and then the interaction between modified yoyos is studied, because all the objects in the world have connections with each other generally and they lead to all the change instead of the isolated system. moreover, the birth of new yoyo and the perdition of the old yoyo both are created by the interaction between yoyos. the clear knowledge of the interaction between yoyos is helpful for us to grasp the properties of the yoyo. property8 : the interaction between the yoyos depends on the spin field. 133 advances in systems science and applications (2013) vol.13 no.2 as is argued above, the spin field of the modified yoyo is used to describe the behavior of the yoyo and sfi scales the behavior intensity. when yoyo a is in the spin field of yoyo b, a would receive the action from b. for example, the competition between two companies means that their productions scrabble for the market, i.e. their spin fields fight with each other instead of the companies. another example is about the sun and the earth, both of which are considered as two systems. their spin fields, i.e. their gravitation fields, are the medium that they attract each other. property9 : the locations and the polarity directions of spin field where a yoyo is located determine the influence that the yoyo receives. the interaction between the yoyos depends on their spin field, and this thesis tells how a yoyo receives the influence by the spin field of another yoyo. as to the same two yoyos, if they have little connection with each other, their behavior on each other would have small intensity. as we known, where the behavior intensity is smaller, where the density of spin lines is smaller and the distance between two yoyos is larger, vice versa. because the polarity of the spin lines stands for the direction of the behavior ability, the direction of a yoyo is determined by its behavior. if two yoyos have the same behavior abilities, they have the same directions, i.e. the polarities, vice versa. if some of their behavior abilities are the same, some are opposite and the other are different, their polarities would have a angle that ranges from zero to π. for example, the workers work in the factory and the farmers work on the farm, so their behavior abilities, i.e. their work skills, nearly are different from each other. they are assumed to be vertical to each other, shown in fig.9. in fact, there also are farkers in our country in recent years, who work on the farm when the harvest is coming and in the factory without harvest, so its yoyo is located between the worker yoyo and the farmer yoyo. since the size of farkers, i.e. the product of their quantity and their working load, is less than that of the workers and that of the farmers, the length of the farker yoyo is less correspondingly. fig.9 the polarities of different yoyos property10 : superposition law if there are only two yoyos, they would be in the spin field of each other, which zelong wang: modified yoyo model and its applications in chinese history 134 is simple to analyze. when a yoyo is in the spin field of two or more yoyos at the same time or two or more yoyos are considered as a whole, how shall we seek the spin field of the whole from the yoyos? superposition theorem would tell us the answer. superposition law tells that the spin field of the whole yoyos is equivalent to the superposition of the spin field of each yoyo. the law means that the spin field of yoyos is independent with that of each others. please note that this law does not mean that the spin field of the yoyos does not have interaction with each other; instead the whole spin field is composed of each independent spin field of the yoyos. for example, the earth is in the gravitation field of the sun and the moon, and the gravitation of them independently effects on the earth; the whole gravitation felt by the earth is composed of the gravitation of the sun and the moon together, and the composition method of the spin lines is introduced by the spin line composition law in the following. property11 : spin line composition law the whole spin field is achieved as long as its spin lines are gotten. suppose that there are two yoyos in the field space, such as yoyo a and yoyo b in fig.10. at a randomly given point p in the field space, the spin lines of yoyo a and yoyo b are a and b with regard to and respectively. by vector composition laws, the whole spin line c at point p is composed by vectors a and b, shown in fig.10. fig.10 composition of spin field the spin line composition law reflects the nonlinearity of the interaction of the yoyos, i.e. the systems. the whole behavior of two yoyos is not the simple linear sum of each, and the direction and the intensity of the whole behavior has parallelogram relation with the original spin field. now another problem that seems hard to solve is the phenomenon “1 + 1 > 2”, i.e. the whole behavior 135 advances in systems science and applications (2013) vol.13 no.2 seems stronger than the sum of each individuals. in fact, there has been another new yoyo produced by the poignant interaction of the yoyos. property12 : yoyo composition law in fact, the yoyo composition law can be concluded by the superposition theorem and spin field composition law. however, the yoyo composition law uncovers the discipline of the composition of yoyos from a high level and it let us see the processing of new yoyo’s birth clearly. the law can be used to explain the combination of the factories, the combined armies, etc. the yoyo composition law tells that the composition of yoyos satisfies the vector composition law, shown in fig.11. yoyo a and b from the new yoyo c, and their spin field are piled in the field space; the spin field of the new yoyo c is gotten by the spin field composition law, and then we get the new yoyo c. the yoyo c also has parallelogram relation with the original yoyos. because the composition of yoyos does not simply sum the yoyos linearly, this composition reflects the nonlinear relation between yoyos. therefore, this law can be fit to the complexity of the systems. fig.11 composition of yoyos 3 modified yoyos of chinese history spring and autumn and warring states periods, from 770 bc to 221 bc, is in the interim from ancient civilization to middle ages civilization and is also a transformation with sharp social movements, complex political dispute, endless warfare and wonderful academic culture. spring and autumn began with the east-transfer of the king ping, and then the western zhou dynasty became weaker and weaker. because the vassals contended for the slavery and the strong country oppressed the weak, many wars began to act on the historical arena. the partition of the country jin by three vassals indicated the end of spring and autumn and the beginning of warring states. during this period, except for the appearance of reformation, the wars between countries were more and more for zelong wang: modified yoyo model and its applications in chinese history 136 more ground and wealth. in 238 bc, the first emperor of qin began to prepare for overthrowing other six countries; finally he established the first centralized multinational country in chinese history with the perdition of qi in 221 bc. the chinese history came into a new era. during the period of spring and autumn and warring states periods, there was a homology between different countries, because they were all located at the era from slave society to feudalism society. one of them was the wars that happened between numerous countries. during the 242 years of spring and autumn, there had been more than 170 vassal countries recorded in history; 36 vassals were killed ferociously; 52 vassal countries were overthrown by more than 480 wars, and the vassal meetings were more than 450. the wars became more and more and the size of wars were much larger than that in spring and autumn as the history came into warring states. in the ancient slave society with the underdeveloped science and technology, the number of the soldiers is an important factor for the war, so the total belligerent soldiers of both countries maybe reached tens of thousands even hundreds of thousands. therefore, the history of spring and autumn and warring states periods also can be called a martial history. in this section, both sides of the war are considered and modeled by the modified yoyo model as well as their interaction correspondingly. since these matters occurred more than two thousands years ago, there were more word depictions instead of integrated and detailed record data. however, the chinese history is modeled by the modified yoyo model to the best of our abilities. 3.1 military yoyos every new overlord was established with bleedings and sacrifice by the wars. during the period of spring and autumn and warring states, there were so many wars that lots of famous generals and specific example of a battle, such as the battle of zhongyuan, the battle of handan and the battle of changping, had been studied by scholars. how shall we predict the result of a war by the war factors? the yoyo model supplies a useful tool to study the war and the interactions between the both sides of the war can help us to know the result. generally speaking, there are many factors that commonly decide the result of the war and these factors can be changed at different periods. economy of a country is a key factor that has influence on its munitions, such as the food supplies and armament, that service for the army; moreover, in the ancient time, the country population is also decided by the economy. the science and technology, deciding the power of the weapon, are universal underdeveloped, so the weapons of different countries during spring and autumn and warring states are almost the same. although the daily training had been formed at that time, the abilities of a single soldier of different countries are also nearly the same. the morale of the soldiers is important for the war, and it is decided by the prompting actions 137 advances in systems science and applications (2013) vol.13 no.2 of the generals. heroical historical view pays great attentions on the heroes or the generals of a war, so they are considered as a factor of the war. according to the records in history[9], the situations of the war were always turned by the wisdom of the generals. now the military organization of a country in spring and autumn and warring states is modeled by the yoyo model. according to the analysis above, the number of soldiers and the wisdom of generals are the most important factors that decide the result of the war; and other factors are ignored for their similarities between different countries. assume that the population of the country is n and the number of its soldiers is n1, where n1 ≤ n . therefore, the number of the people that are arranged to work for the economy is n2, where n1 +n2 = n (5) generally speaking, n2 affects the battle effectiveness indirectly by affecting the economy of the country, because the food, the clothes and the weapons of the soldiers come from the economy contributed to by its people. so the change rate of the battle effectiveness p can be expressed as: dp dn1 = an1n2 (6) where α is a constant, which means that the battle effectiveness will increase with the growth of the numbers of the soldiers and the workers. please note that equ.(6) is only fit for the ancient wars instead of modern wars, because the former is much simpler than the latter. combining equ.(5) and equ.(6), we have: dp dn1 = an1(n −n1) (7) as we known, the army is used to protect its country and overcome other countries. so an important rule of the war is that the battle effectiveness aspired after by the sovran of the country is to overcome other countries and protect the security of its country instead of maximization of the power. therefore, the number of the soldiers is also affected by its battle effectiveness, i.e. when the battle effectiveness is strong, the number of the soldiers would decrease, vice versa. so we have: n1 = γ(p0 − βp ) (8) where p0 > 0, which avoids the appearance of negative battle effectiveness; and γ > 0, β > 0, which means that the country would decrease its soldiers if its battle effectiveness has became strong enough. taking equ.(7) into equ.(8), we have: dp dn1 = ap 2 +bp + c (9) zelong wang: modified yoyo model and its applications in chinese history 138 where a = −αγ2β2 < 0, b = 2αγ2p0β − αnγβ and c = αγnp0 − αγ2p 2 0 . the wisdom of generals is always a key factor. if one side of the war does not have enough stratagems, it sometimes would be defeated by the other side with a relative weak power. the battle effectiveness should be affected by a factor k ∈ [0, 1] about the wisdom of the generals. when the generals have great providence, stratagems and brightness, such as lianpo, limu and sunwu, the factor k=1; if the general is an armchair strategist, such as zhaokuo, the factor k=0; otherwise, the factor k ∈ (0, 1]). therefore, equ.(9) can be rewritten as: dp dn1 = k(ap 2 +bp + c) (10) how does the battle effectiveness keep steady between all the countries? when the battle effectiveness is balanceable, the possibility of wars is small, vice versa. so the stability is the focus that most of the scholars pay attentions to. the discriminant of equ.(9) is: ∆ = b2 − 4ac (11) so there are three cases: ∆ = 0, ∆ > 0 and ∆ < 0. the evolution models are also different at different cases. in the following, let us take the first case as an example. if ∆ = 0, there is only one balance, i.e. p1 = −2a/b, that all the countries do not invade each others. so we have: p (n1) = k ( p1 − 1 p2 +an1 ) (12) where p2 is the initial constant determined by the country condition. if p2 < 0, so n0 = −p2/a < 0. the battle effectiveness decreases with the increasing of the soldier number. when p2 > 0, n0 = −p2/a > 0. if c, the larger the soldier number, the stronger battle effectiveness. if n1 = n0, the discontinuity occurs to the battle effectiveness. if n1 > n0, the battle effectiveness would decrease as the soldier number’s rising. this processing indicates that the battle effectiveness could not blindly increase as the soldier number. by the restriction of economy of the country, the army would keep a balance or fluctuate at some size. now let us analyze the representation of the yoyo model of the military organization. the spin field, depicted by the spin lines, naturally denotes the behavior ability, i.e. the battle effectiveness and the resistance. the spin field is important for the yoyo to have connection with others, because the behavior abilities of a yoyo is given off or received by its spin field, just like invading other countries. except for the soldiers and the generals, there should be other factors that affect the military organization, such as the weapons, logistics and geography. when modeling the yoyo, they are all ignored for the puny differences between them. 139 advances in systems science and applications (2013) vol.13 no.2 this disposal is consistent with the known word records, because most of the wars recorded in history only come down to the soldiers, generals and the stratagems[10]. what’s more, the stronger the behavior ability is, the larger its spin field is, which leads to a larger size of the yoyo. when interacting with other yoyos, the yoyo with larger size would be the leading role. for example, the victory of the country zhao in the war of fei[9], with the leader of general limu, showed that the country zhao had stronger behavior ability than the country qin for the wonderful stratagem. the polarities of the yoyo reflect the directions of the behavior ability, i.e. the military organization gives off its action or receives the force from others. if two countries had the same battle goal and defeated their common enemy, their yoyos would have the same polarities. according to the definition of the size of the yoyo, the size of a yoyo means the battle effectiveness; in fact, it also can be measured by its munitions, i.e. its food supplies and armament, although this measurement is inconvenient in this paper. therefore, the yoyo of a military organization can be expressed in fig.12. fig.12 the yoyo of military organization 3.2 explanations of the yoyo’s characters the explanations of the yoyo’s characters are considered by the qualitative analysis, such as the figurative analysis method, of the historical events in spring and autumn and warring states. that the soldiers and the generals determine the military effectiveness means that the spin lines at different polarities have a corresponding relation. taking the country wu and chu as examples, when the king helv of wu paid great attentions to the development of the army and selected the militarist sunwu as general, the military effectiveness of wu became larger and larger and defeated the chu finally[10]. so we can conclude that the investment and the harvest, i.e. the received behavior and the external behavior, are equivalent. moreover, the stability of the military effectiveness is equivalent to that of the zelong wang: modified yoyo model and its applications in chinese history 140 yoyo, i.e. only if the military organization has a steady military effectiveness, the organization would be also steady, and vice versa. when the military organization was not defeated by others, its military effectiveness was not changed, i.e. it was steady. it was sure that the organization was still steady by the protection of the steady military effectiveness. if the military effectiveness of a organization was decreased by other organizations, the military effectiveness would be unsteady and so was the military organization. the country qi was nearly conquered thoroughly in the war of yan and qi, because its military effectiveness was extremely unsteady, only 7000 persons left after that war[10]. the change of the effectiveness would lead to the change of the size of the yoyo. when we select the military organization as the yoyo model, its behavior ability will be the research context and intuition. therefore, the military effectiveness is used to describe the size of the yoyo exactly. it seems that the number of the soldiers and the generals is fit to measure the yoyo’s size, but it can not capture the essentials of military organization. as to a given military organization, it has inertia to keep steady and it would be steady if the outside force acted on the yoyo does not exceed the stability boundary. it is well explained by the phenomenon that not all the wars lead to perdition of the military organization. for example, during the war of yan and qi, the country qi did not be conquered finally although there were only 7000 persons left; this means that the violent military shock of yan still did not exceed the boundary of qi’s perdition. on the other hand, this boundary of a yoyo can reflect the inertia of the stability of the yoyo, returning the example above, the qi had great attributes to keep its country, i.e. the structure and component of the yoyo. the furious change of the spin field, exceeding the inertia boundary, would lead to the birth of a new yoyo and the perdition of an old one. for example, the war of wu and yue had been lasting for more than 30 years[10]; it showed that when the change of the spin field could not exceed the inertia boundary, the yoyo, i.e. the country, could not be changed. finally the perdition of the country wu shows that the force acted on it by the country yue had exceeded its inertia boundary and the yoyo came to perdition. at the same time, the country yue became a new yoyo with the addition of wu’s land and wealths. 3.3 the interaction of the yoyos what about the interactions between different countries? the combination of countries reached the climax at the end period of warring states, such as the combination of six countries for resisting qin[9]. let us analyze this phenomenon by the yoyo model. the combination of six countries means that the six yoyos with different sizes are united together; by the composition laws of the yoyos, the new yoyo of the combination should be stronger than that of the qin, shown in 141 advances in systems science and applications (2013) vol.13 no.2 fig.13, why would the qin be the winner finally? fig.13 the perfect combination of six countries and qin the problem lies in the polarities of the six countries, and the fig.13 shows that the polarities of the six countries are the same. however, in history the six countries had never been actually united together, and they acted by the motivation of their own benefits instead of defeating their enemy qin. therefore, the polarities of the six yoyos were not consentaneous, shown in fig.14, and their combination was still weaker than the qin. moreover, the country qin tried itself to destroy the combination of the six countries by alienating these countries one by one, so there even were some countries that attacked their allies surreptitiously, such as the country wei. when the number surviving countries was fewer and fewer, they could not defeat the strongest country qin even if they shared a bitter hatred of the enemy qin for their puniness finally. in 221 bc, the country qin defeated the six countries and established a centralized multinational country in chinese history, i.e. a new yoyo appeared by their complicated interactions. fig.14 the factual combination of six countries and qin 4 conclusions this paper focuses on modifying the yoyo model and proposes the modified yoyo model. firstly, the modified yoyo model enriches the yoyo model. the concept of the spin field is endowed with new meanings, i.e. the behavior abilities of the yoyo, and the spin lines are used to depict the spin field exactly, including the intensity and polarity of the spin field and the size of the yoyo. the symbol denotation of the modified yoyo is convenient to describe the yoyos and their zelong wang: modified yoyo model and its applications in chinese history 142 interaction. the modified yoyo model has more basic concepts about the system. secondly, the characters of the modified yoyo are researched in detail. the stability, diversity and complexity of the system can be reflected by the modified yoyo model; the equipollence of the stability between the yoyo and its spin field, the spin line between two different polarities and the yoyo’s size between the behavior and the tsme is also discussed; finally, the perdition and the birth of yoyo are explained by the inertia of the stability effectively. at last, the interaction between yoyos is studied and the composition laws of the spin lines and the yoyos are proposed. in order to prove the practical usefulness of the modified yoyo model, the chinese history of spring and autumn and warring states, mainly about the military affairs, is explained by this model. according to the analysis results, the military organization of each country can be considered as a yoyo, and the basic characters of the yoyos and their interaction can be reflected by the word records in history. therefore, the modified yoyo model has the rationality, to some degree, as the complementarities of the yoyo model. the modified yoyo model has referred to many basic properties of the system, but these are only restricted to the word description, i.e. qualitative analysis, or the figurative analysis method instead of the quantitative analysis, which should be an open problem for the development of the yoyo model. another open problem is that since almost any objects can be seen as systems, how these systems are modeled by the yoyo model exactly and effectively. references [1] yi lin. (2008), systemic yoyos, impacts of the second dimension, published by taylor and francis. [2] quastler h. (1965), “general principles of systems analysis”, in theoretical and mathematical biology, ed. t. h. waterman and h. j. morowitz. new york: blaisdell publishing. [3] zadeh l. (1962), “from circuit theory to systems theory”, proc. ire, vol.50, pp.856-65. [4] von bertalanffy l. (1924), einfuhrung in spengler’s werk, literaturblatt kolnische zeitung, may. [5] ouyang s. c, k. zhang, l. p. hao, and l. r. zhou. (2005), “structural transformation of irregular time-series information and refined analysis on evolutions”, eng. sci. china, vol.7, pp.36-41. [6] yi lin. (2008), systemic yoyos, impacts of the second dimension, published by taylor and francis, pp.9-10. 143 advances in systems science and applications (2013) vol.13 no.2 [7] yi lin. (2008), systemic yoyos, impacts of the second dimension, published by taylor and francis, pp.11-14. [8] http://www.thenakedscientists.com/forum/index.php?topic=22668. [9] shouqian wang. (1992), zhanguoce full translation, guizhou press (china). [10] qingxiang meng. (1986), zhanguoce translation, heilongjiang people’s press (china). corresponding author zelong wang can be contacted at: zelong wang@163.com. advances in systems science and applications (2014) vol.14 no.2 101-128 the theory of parametric control of macroeconomic systems and its applications(ii) a. ashimov, zh. adilov, r. alshanov, yu. borovskiy and b. sultanov kazakh national technical university named after k. satpayev, 22 satpaev street, almaty, 050013, kazakhstan abstract this work consists of three parts and presents the recent results of development of the theory of parametric control of macroeconomic systems and some its applications for solving a number of concrete problems. keywords mathematical model, structural stability, parametrical identification, parametric control part 2. mathematical foundations of the parametric control theory 2.1 sufficient conditions for the existence of solutions for the problems on synthesis and choice of optimal parametric control laws 2.1.1 conditions for the existence of solution for the variational calculus problem on synthesis of optimal parametric control law of continuous dynamical system consider continuous controllable system* ẋ(t) = f(x(t), µ(t), a(t)), t ∈ [0, t ] (1) x(0) = x0 (2) where t is time; x = x(t) = (x1(t), ..., xm(t)) is a vector function of system’s state; µ = µ(t) = (µ1(t), ..., µq(t)) is a vector function of control; a = a(t) = (a1(t), ..., as(t)) is a known vector function; s0 = (x10, ..., x m 0 ) is an initial condition of the system, a known vector; f-a known vector function of its arguments. the problem of synthesis of optimal economic tools values consists in finding the extremum of the following criterion k = ∫ t 0 f (t, x(t))dt (3) where f is a known function, at the phase constraints x(t) ∈ x(t), t ∈ [0, t ] (4) where x(t) is a given set and at explicit control constraints u(t) ∈ u(t), t ∈ [0, t ] (5) * all of the formulated in this part results for non-autonomous dynamical systems remain true for autonomous dynamical systems as well, when an exogenous vector function a(·) is taken as constant. 102 a. ashimov: the theory of parametric control of macroeconomic systems and ... where u(t) is a given set. define the variational calculus problem on synthesis of optimal parametric control laws for the continuous dynamical system. problem 2.1. at given function a(·) to find the control u(·), satisfying the condition (5), so that dynamical system (1), (2)solution meets the condition (4) and gives the maximum (the minimum) to functional (3). to prove the solubility of the problem 2.1 we first need the single-valued solubility of cauchy problem (9), (10). to obtain this result we use known result from the theory of ordinary differential equations. let be given: number t > 0, metrizable compact u and continuous function φ : [0, t ] × rm × u → rm such that for any ρ ≥ 0 exists such σ ≥ 0, that the following inequality is true |φ(t, y, µ)− φ(t, y′, µ)| ≤ σ|y − y′|∀t ∈ [0, t ], y, y′ ∈ ρb, µ ∈ u (6) where is a unit ball in rm and there is such a constant η ≥ 0 that the following inequality is true |yφ(t, y, µ)| ≤ η(1 + |y|2)∀t ∈ [0, t ], y ∈ rm, µ ∈ u (7) consider cauchy problem ẏ(t) = φ(t, y(t), µ(t))∀t ∈ [0, t ] (8) y(0) = y0 (9) where y0 ∈ rm. the following statement is true ([1], lemma 4.1): lemma 2.1. given above mentioned constraints (6), (7) for any measurable mapping µ : [0, t ] → u the problem (8), (9) has the only solution y : [0, t ] → rm, satisfying the estimate |y(t)| ≤ ( |y0|2 + 2ηt )1/2 eηt∀t ∈ [0, t ] (10) the following statement is true. theorem 2.2. let function a(·) be continuous on segment [0, t ], u is a compact in rq, function f is continuous, for any ρ ≥ 0 there is such σ ≥ 0, that the following inequality is true |f(x, µ, a(t))− f(x′, µ, a(t))| ≤ σ|x− x′|∀t ∈ [0, t ], x, x′ ∈ ρb, µ ∈ u (11) and exists such a constant η ≥ 0 that the following inequality is true |xf(x, µ, a(t))| ≤ η(1 + |x|2)∀t ∈ [0, t ], x ∈ rm, µ ∈ u (12) advances in systems science and applications (2014) vol.14 no.2 103 then for any measurable mapping µ : [0, t ] → u , cauchy problem (2.1), (2.2) has the only solution x : [0, t ] → rm satisfying the estimate |(t)| ≤ ( |x0|2 + 2ηt )1/2 eηt∀t ∈ [0, t ] (13) proof. let us introduce notation: φ(t, x, µ) = f (x, µ, a(t)) and f functions continuity results in continuity of function φ. (6), (7) inequalities truth follows from relations (11), (12). thus the theorem statements follow from lemma 2.1. the theorem is proved. for proof of solubility of the problem 2.1 we use known result from the optimal control theory for systems, described by differential equations. let us specify a closed subset e ⊂ [0, t ] × rm × u . consider the system, described by cauchy problem (8), (9) when lemma 2.1 conditions are satisfied, herewith by control we mean a measurable mapping µ : [0, t ] → u . acceptable pair for the system in question is such a pair “state-control”, which satisfies the relations (8), (9) and inclusion (t, x(t), u(t)) ∈ e (14) let us specify carathéodory function φ with non-negative values on the set [0, t]× (rm × u). let us specify a functional i = t∫ 0 φ (t, x(t), u(t)) dt. problem 2.2. to find the minimum for the functional i on the acceptable pairs set of the system (8), (9). for any point (t, x) ∈ [0, t]×rm define a section et,x = {u ∈ u : (t, x, u) ∈ e}. let us specify a set γt,x = {φ(t, x, u) : (t, x, u) ∈ et,x} and a function g : [0, t ]× rm×rm → r using an equality g(t, x, y) = min{φ(t, x, u)| (t, x, u) ∈ e,φ(t, x, u) = y}. it is known the following statement about solubility of optimization problem ([1], statement 4.2), where r̄ implies the set of real numbers, supplemented with the values −∞ and +∞: lemma 2.3. let, when lemma 2.1 condition is satisfied, the set γt,x be convex for all t ∈ [0, t ].x ∈ rm, and the function g(t, x, ·) : rm → r̄ be convex for all (t, x) ∈ [0, t ]×rm. then the problem 2.2has a solution. let be a closure of the sum ∪ t∈[0,t ] x(t). the following statement is true. lemma 2.4. let be a compact, function f be continuouson [0, t ]×x. then function φ, specified by equality φ(t, x, u) = f0 − f (t, x) where f0 is the maximum of function f on the set [0, t ]×x satisfies lemma 2.3 conditions. proof. existence of the maximum for f0 follows from the weierstrass theorem. non-negativity of the determined above function φ is obvious. it is carathéodory function because of the continuity of f. at last, the convexity of function g(t, x, ·) 104 a. ashimov: the theory of parametric control of macroeconomic systems and ... for all (t, x) ∈ [0, t ]×rm is implemented, as function f does not depend on control u. the lemma is proved. let u be a closure of the sum ∪ t∈[0,t ]. for reduction the constraints (4), (5) to the form (14) we specify the set e ⊂ [0, t ] × rm × u the way to hold the relation: et0 = {(t, x, u) ∈ e| t = t0} = x(t0)× u(t0), t0 ∈ [0, t ]. lemma 2.5. let the mappings t → x(t), t → u(t) be continuous at each point t ∈ [0, t ] in the following sense: if the inclusions xk ∈ x(tk), uk ∈ u(tk) are true, where tk ∈ [0, t ], k = 1, 2, ... and convergence of the sequences is met tk → t, xk → x, uk → u, then the inclusions x ∈ x(t), u ∈ u(t) are true.then the set is closed. proof. consider such an sequence {(tk, uk, xk)} of elements of the set , that there is a convergence (tk, uk, xk) → (t, u, x). from the inclusion (tk, uk, k) ∈ e, because of specifying the set , follow the inclusions tk ∈ [0, t ], xk ∈ x(tk), uk ∈ u(tk), from closure of segment [0, t ] follows the inclusion t ∈ [0, t ]. using lemma conditions, ascertain that x ∈ x(t), u ∈ u(t) therefore, the inclusion (t, u, x) ∈ e is true, that is the set is closed. the lemma is proved. let us make sure that lemma conditions are not excessively restrictive. consider, for instance, a typical situation for scalar case, when the set x(t) = {x| a(t) ≤ x ≤ b(t)} , t ∈ [0, t ] is given, where functions and b are continuous. let the inclusion xk ∈ x(tk) be met, where tk ∈ [0, t ], k = 1, 2, ... and there is the convergence tk → t, xk → x, ipso facto, the inequalities a(tk) ≤ xk ≤ b(tk), k = 1, 2, ... are true. by proceeding here to the limit allowing for continuity of functions and b, we get the result a(t) ≤ x ≤ b(t) what results in that x ∈ x(t) thereby, the mapping t ∈ x(t) is continuous on the segment [0, t ]. analogously ascertains the truth of lemma 2.5 conditions and in more general cases, when boundaries of sets of acceptable control and condition values are continuous functions of time. for reduction the equation (1) to the form (8) it is enough to specify φ(t, x, u) = f(x, u, a(t)) then the set γt,x, appearing in lemma 2.2 conditions, defines in the following way γt,x = {f (x,w, a(t))|w ∈ u(t)} (15) denote by v the set of acceptable pairs “state-control” of the system (1), (2) in question at given known function , that is such vector function pairs (x, u) which satisfy the relations (1), (2), (4), (5). from lemma 2.3 directly follows the next statement. theorem 2.6. let function (·) be continuous on the segment [0, t], u be a compact in rq, function f be continuous in x × u × a and for any ρ ≥ 0 there is such σ ≥ 0, that the following inequality is true∣∣f(x, u, a(t))− f(x′, u, a(t)) ∣∣ ≤ σ ∣∣x− x′ ∣∣ , x, x′ ∈ ρb, u ∈ u (16) advances in systems science and applications (2014) vol.14 no.2 105 and there is such a constant η ≥ 0, that the following inequality is true |xf(x, u, a(t))| ≤ η ( 1 + |x|2 ) ∀t ∈ [0, t ], x ∈ rm, u ∈ u (17) let x be a compact, function f be continuous on [0, t ]×x. let, moreover, the mappings t → x(t), t → u(t) be continuous for t ∈ [0, t ] in the following sense: if the inclusion xk ∈ x(tk), uk ∈ u(tk) are true, where tk ∈ [0, t ], k = 1, 2, ... and there is convergence of sequences tk → t, xk → x, uk → u, then the inclusions x ∈ x(t), u ∈ u(t) are true. then, in the case of non-emptiness of the set va and convexity of the set γt,x for all t ∈ [0, t ], x ∈ x(t) the problem 2.1 has a solution in the class of measurable functions. 2.1.2 conditions for the existence of solution for the variational calculus problem on choice (among given finite algorithms set) of optimal parametric control law of continuous dynamical system the continuous controllable system in question (1), (2). control u is chosen here among the set of given control laws: uj(t) = gj (v, x(t)) , t ∈ (0, t ), j = 1, ..., r (18) here gj is known vector function of its arguments; υ = (υ1, ..., υl) is vector of control parameters. on control parameters lay the constraints like v ∈ v (19) where v is some subset of the space rl. moreover, it is assumed that control parameters should be such that corresponding control law (18) satisfies the condition (5), that is the following inclusion is met gj (v, x(t)) ∈ u(t), t ∈ (0, t ) (20) where u(t) is given set. there are phase constraints on the system: x(t) ∈ x(t), t ∈ (0, t ) (21) where x(t) is given set. consider optimality criteria kj = kj(a, v) = ∫ t 0 f [t, xj(t)]dt (22) where xj = xj(t) = ( x1j (t), ..., x m j (t) ) is solution of cauchy problem (1), (2) at given function (·) and control u = uj(t) = ( u1j (t), ..., u q j(t) ) , that is for chosen 106 a. ashimov: the theory of parametric control of macroeconomic systems and ... j-th control law (26). consider the next subsidiary extremal problem: problem 2.3*. at given function a(·) for each of r control laws to find such a vector of control parameters v, that corresponding it solution x = xj of the problem (1), (2) with control law u = uj , determining by formula (18), satisfies the conditions (19)-(21) and gives the maximum for the functional (22). define the following variational calculus problem on choice (among given finite algorithms set) of optimal parametric control law for non-autonomous continuous system. problem 2.3. given known function a(·) among all optimal control laws in sense of the problem 2.3* to choose the one, which corresponds to the maximal optimality criterion value (22). substituting into the equation (1) control value from formula (19), we get ẋ(t) = f ( x(t), gj (v, x(t)) , a(t) ) , t ∈ (0, t ) (23) where for short we omit the index j in denoting condition function of the system, corresponding to given control law. denote by w j a a set of acceptable pairs “state-control parameter” of the system in question, that is such pairs (x, v), which satisfy both the equalities (23), (2), and the inclusions (19)-(21). thus, the problem 2.3* comes to the maximization of functional k = ∫ t 0 f [t, x(t)]dt on the set w j a . ascertain first a solubility of cauchy problem (23), (2). the following theorem directly results from lemma 2.1. theorem 2.7. let the function (·) is continuous on the segment [0, t ], sets u and v are compact, functions f and gj are continuous, for any ρ ≥ 0 there exist such σ ≥ 0, χ ≥ 0, that the following inequalities are true |f(x, u, a(t))−f(x′, u′, a(t))| ≤ σ(|x−x′|+|u−u′|)∀t ∈ [0, t ], x, x′ ∈ ρb, u, u′ ∈ u (24)∣∣gj (v, x)−gj ( v, x′ )∣∣ ≤ χ ∣∣x− x′ ∣∣∀x, x′ ∈ ρb, v ∈ v (25) and exists such a constant η ≥ 0 that the following inequalitiy is true |xf(x,gj(v, x), a(t))| ≤ η(1 + |y|2)∀t ∈ [0, t ], x ∈ rm, v ∈ v (26) then for any v ∈ v the problem (2), (2) has the only solution x : [0, t ] → rm,satisfying the estimate |(t)| ≤ ( |0|2 + 2ηt )1/2 eηt∀t ∈ [0, t ] (27) ascertain solubility of the problem 2.3*. the following theorem is true. theorem 2.8. assume that when the conditions of the theorem 2.7 are satisfied for given function (·) and given value of j ∈ {1, ..., r} the setw j a is not empty, advances in systems science and applications (2014) vol.14 no.2 107 the sets v, u, x are compact, the sets x(t), u(t) are compact for all t ∈ [0, t ], and function f is continuous. then the problem 2.3* has a solution. proof. from the condition (21) and continuity of function f follows that this function is bounded. then, because of non-emptiness of the set w j a , upper boundary sup k of function k exists on set w j a of acceptable pairs of the system in question. then there exists such an elements sequence {(xk, vk)} from the set w j a , that convergence takes place (here kk = ∫ t 0 f [t, xk(t)]dt) kk → supk (28) the inclusion (xk, vk) ∈w j a assumes the following relations to be true ẋk(t) = f ( xk(t), gj (vk, xk(t)) , a(t) ) , t ∈ (0, t ) (29) xk(0) = x0 (30) xk(t) ∈ x(t), t ∈ (0, t ) (31) vk ∈ v (32) gj (vk, xk(t)) ∈ u(t), t ∈ (0, t ) (33) from the theorem 2.7 results the estimate |xk(t)| ≤ (|x0|2 + 2ηt ) 1 2 eηt∀t ∈ [0, t ] (34) thus, using bolzanocweierstrass theorem allowing for boundedness of set v, after subsequence separation we get the convergences k(t) → (t)∀t ∈ [0, t ] (35) vk → v (36) limiting value v satisfies the inclusion (19) because of closure of set v and condition (32). using the conditions (35), (36) and continuity of function gj , we get the convergence f ( xk(t), gj (vk, xk(t)) , a(t) ) → f ( x(t), gj (v, x(t)) , a(t) ) , t ∈ (0, t ) (37) function xk satisfies the integral relation xk(t) = x0 + ∫ t 0 f ( xk(τ), gj (vk, xk(τ)) , a(τ) ) dτ, t ∈ (0, t ) (39) by proceeding here to the limit allowing for the conditions (35), (38), we get the equality x(t) = x0 + ∫ t 0 f ( x(τ), gj (v, x(τ)) , a(τ) ) dτ, t ∈ (0, t ) (40) 108 a. ashimov: the theory of parametric control of macroeconomic systems and ... by differentiating both parts of the equality (40) with respect to t, we ascertain that the equation (1) is true. consequently, after proceeding to the limit in equality (30) taking into account the condition (35), we ascertain that the initial condition (2) is true. thus, the limiting pair (x, v) is acceptable. continuity of function f from the condition (35) results in convergence f [t, xk(t)] → f [t, x(t)] , t ∈ [0, t ] (41) therefore ∫ t 0 f [t, xk(t)]dt→ ∫ t 0 f [t, x(t)]dt (42) from (42) and (28) follows the equality t∫ 0 f [t, x(t)]dt = supk. thus, we found the pair (x, u) ∈ w j a for which the upper boundary of functional k(on the set of all acceptable pairs of the system in question) is obtained. the theorem is proved. obviously, the problem 2.3 comes to the search of maximum of function φa(j) = max (x,v)∈w j a kj on finite set {1, ..., r} at given function (·). the maximal value of considered functional at given function (·) and j-th control law is placed on the right side of previous equality. the existence of this value is proved in the theorem 2.8. as the maximum of function on finite set is obtained always, we get the following statement. theorem 2.9. when theorem 2.8 conditions are satisfied, the problem 2.3 has a solution. 2.1.3 conditions for the existence of solution for the variational calculus problem on synthesis of optimal parametric control law of discrete dynamical system consider discrete non-autonomous controllable system x(t+ 1) = f(x(t), u(t), a(t)), t = 0, 1, ..., n− 1 (43) x(0) = x0 (44) where t is time. here x = x(t) = ( x1(t), ..., xm(t) ) is vector function (of discrete argument) of systems state; u = u(t) = ( u1(t), ..., uq(t) ) is control, vector function of discrete argument; a = a(t) = ( a1(t), ..., as(t) ) is known vector function of discrete argument; x0 = ( x10, ..., x m 0 ) is initial condition of the system, known vector; f is known vector function of its arguments. the problem of choosing optimal economic tools values consists in finding the extremum of the following criterion k = n∑ t=1 f [t, x(t)] → max (min) (45) advances in systems science and applications (2014) vol.14 no.2 109 where f is known function, at phase constraints on system (43)-(44) solution like x(t) ∈ x(t), t = 1, ..., n (46) where x(t) is given set, and following constraints on control: u(t) ∈ u(t), t = 0, 1, ..., n− 1 (47) where u(t) is given set. define the variational calculus problem on synthesis of optimal parametric control laws for discrete dynamical system. problem 2.4. given known function a(·), to find control u(·), satisfying the condition (47), so that dynamical system solution (43), (44), corresponding it, satisfies the condition (46) and gives the maximum (minimum) for functional (45). denote by va the set of acceptable pairs “state-control” of the system in question at given known function a(·), that is such pairs of vector functions (x, u), which satisfy the relations (43), (44), (46), (47). introduce notations: x = n∪ t=1 x(t), u = n−1∪ t=0 u(t). the following statement is true. theorem 2.10. let for given function a(·) the set va be not empty; sets x(t) and u(tc1) are closed and bounded for all t = 1, ..., n; mapping f is continuous by the first two arguments on set x × u , and function f is continuous by the second argument on set x. then the problem 2.4 has a solution. proof of this theorem is based on properties of functions which continuous on compact. 2.1.4 conditions for the existence of solution for the variational calculus problem on choice (among given finite algorithms set) of optimal parametric control law of discrete dynamical system consider discrete non-autonomous controllable system (43), (44). the following phase constraints are imposed on this system: x(t) ∈ x(t), t = 1, ..., n (48) where x(t) is given set. in equation (43) control is chosen among the set of given control laws: uj(t) = gj (v, x(t)) , t = 1, ..., n, j = 1, ..., r (49) where gj is known vector function of its arguments, v = (v1, ..., vl) is vector of control parameters. the following constraints are imposed on control parameters v ∈ v (50) 110 a. ashimov: the theory of parametric control of macroeconomic systems and ... where v is some subset of space rl. moreover, we will assume that control parameters should be such that corresponding control law (49) satisfies the condition gj (v, x(t)) ∈ u(t), t = 0, ..., n− 1 (51) where u(t) is given set. consider the following optimality criteria: kj = kj(a, v) = n∑ t=1 f [t, xj(t)] (52) where xj = xj(t) = ( x1j (t), ..., x m j (t) ) is problem (43), (44) solution at given function (·) and control u = uj(t) = ( u1j (t), ..., u q j(t) ) , that is for chosen j-th control law. define the next subsidiary extremal problem: problem 2.5*. given known function (·) for each of r control laws to find such a vector of control parameters v, that corresponding it solution x = xj of problem (43), (44) with control law u = uj , determining by formula (2.49), satisfies the conditions (48), (50), (51) and gives the maximum for functional (52). the following problem is called as variational calculus problem on choice among given finite set of algorithms of optimal parametric control laws for discrete non-autonomous system. problem 2.5. given known function (·) among all optimal control laws in sense of the problem 2.5* to find the one, which corresponds to the maximal optimality criterion value (52). ascertain first the solubility of the problem 2.5* for fixed control law. substituting into equation (43) the control value from formula (49), we get x(t+ 1) = f ( x(t), gj (v, x(t)) , a(t) ) , t = 0, 1, ..., n− 1 (53) where for short we omit the index j in denoting condition function of the system, corresponding to j-th control law. denote by w j a the set of acceptable pairs “state-control parameter” of the system in question, that is such pairs (x, v) which satisfy both the equalities (44), (53), and the inclusions (48), (49), (50). thus, the problem 2.5* comes to the maximization of functional kj = on the set w j a . let the sets and u be determined by relations x = n∪ t=1 x(t), u = n−1∪ t=0 u(t). the following theorem is true. theorem 2.11. let given known function (·) and j-th control law the set w j a is not empty, the sets v, x(t) and u(t − 1) are closed and bounded for all t = 1, ..., n, function f is continuous by the first two arguments on the set x × u advances in systems science and applications (2014) vol.14 no.2 111 function gj is continuous on the set v ×x and function f is continuous by the second argument on the set . then the problem 2.5* has a solution. proof of this theorem is based on property of continuous on compact functions to reach their maximal and minimal values. as on finite set the maximum of function is obtained always, we get the following statement. theorem 2.12. when the theorem 2.11 condition is satisfied, the problem 2.5 has a solution. 2.1.5 conditions for the existence of solution for the variational calculus problem on synthesis of optimal parametric control law of discrete stochastic dynamical system consider discrete stochastic controllable system like x(t+ 1) = f(x(t), u(t), a) + ξ(t), t = 0, 1, ..., n− 1 (54) x(0) = x0 (55) here x = x(t) ∈ rm is function of the (54), (55) systems state, random vector function of the discrete argument (vector random process); x0 is initial condition of the system, deterministic vector; u = u(t) ∈ rq is vector of controllable parameters, vector function of the discrete argument; a ∈ rs is vector of uncontrollable parameters, a ∈ a,a ⊂ rs is given set. ξ = ξ(t) = (ξ1(t), ..., ξm(t)) is known vector random process, expressing noises (as such a noise can be, for instance, gaussian noise); f is known vector function of its arguments. specify optimality criterion, which is subject to maximization for the present problem is of the form k = e { n∑ t=1 ft(x(t)) } (56) here ft are known functions, e is mathematical expectation. there are phase constraints on the system: e[x(t)] ∈ x(t), t = 1, ..., n (57) where x(t) is given set. determined earlier constraints on control hold in the problems considered further: u(t) ∈ u(t), t = 0, 1, ..., n− 1 (58) where u(t) ∈ rq is given set. define the following variational problem, called as the variational calculus problem on synthesis of optimal parametric control law for discrete stochastic dynamical system. problem 2.6. given known vector of uncontrollable parameters a ∈ a to find the parametric control law u, satisfying the condition (58), so that dynamical 112 a. ashimov: the theory of parametric control of macroeconomic systems and ... system (54), (55) solution corresponding it satisfies the condition (57) and gives the maximum for functional (56). determine the set of acceptable controls uad for system under study in terms of the set of such control laws u(t), satisfying the constraint (58), for which the mathematical expectation e[x(t)] of corresponding solution of stochastic system satisfies the inclusion (57). the following theorem is true. theorem 2.13. let in the problem 2.6 given a ∈ a for any t = 1, ..., n random variables ξ(t) are absolutely continuous and has zero mathematical expectations, the sets x(t), u(t) are closed and bounded for all t, function f satisfies the lipschitz condition, and functions ft are continuous by lipschitz. functions f (for u ∈ u and a ∈ a) and ft by absolute value does not exceed some linear relatively |x| functions. then, if the set of acceptable controls uad is not empty, the problem 2.6 will be soluble. proof. according to the weierstrass theorem, continuous function on nonempty bounded set reaches its maximum. thus, it is enough to show that the multivariable function k = k(u), determined by (56), is continuous, and the set uad is closed and bounded. its non-emptiness is one of the conditions of the theorem. we will show that mathematical expectations of the magnitudes, entering into phase constraint (57) are exist. indeed, according to equation (54), we have e[x(t+ 1)] = e[f(x(t), u(t), a)] + e[ξ(t)] the second summand of the right-hand-side of this equality has sense under the theorem conditions, and the first is calculated by formula e[f(x(t), u(t), λ)] = ∫ rn f(ω, u(t), a)px(t)(ω)dω if the last integral absolutely converges (here by px(t) is denoted probability density function of random variable x(t)). the latter fact is really takes place under constraints on increase of function f and existence of mathematical expectation of the magnitude x(t) for any t = 1, ..., n (this fact is tested using the mathematical induction method). existence of mathematical expectation on the right-hand-side of equality (56) results from constraints on increase of function ft and existence of mathematical expectation of the variable x(t). let convergence of vectors uk → u, uk ∈ uad takes place. from equation (54) follows the equality |xk(t+ 1)− x(t+ 1)| = |f(xk(t), uk(t), a)− f(x(t), u(t), a)| advances in systems science and applications (2014) vol.14 no.2 113 where uk and x are solutions of the problem (2.54), (2.55) at controls uk and u, correspondingly. then the following relation is true |xk(t+ 1)− x(t+ 1)| ≤ lf [|xk(t)− x(t)|+ |uk(t)− u(t)|] where lf is the lipschitz constant of function f . by repeating analogous reasoning and taking into account that under the condition (55), xk(0) = x(0) we will have |xk(t+ 1)− x(t+ 1)| ≤ (lf ) 2|xk(t− 1)− x(t− 1)|+ (lf ) 2|uk(t− 1)− u(t− 1)|+ lf |uk(t)− u(t)| ≤ t∑ s=0 (lf ) s+1|uk(t− s)− u(t− s)| ≤ εk where εk ≤ 0 under k → ∞. by denoting the maximal of the lipschitz constants of functions ft by lf for t = 1, ..., n we get the estimate |ft[xk(t)]− ft[x(t)]| ≤ lf εk after calculating mathematical expectations of both of parts of this inequality, we get inequality e{|ft[xk(t)]− ft[x(t)]|} ≤ lf εk this implies that e{|ft[xk(t)]−ft[x(t)]|} → 0 and convergence of the sequence in question e{|ft[xk(t)] − ft[x(t)]|}. hence, function k = k(u) is continuous because of (56). boundedness of the set uad results from boundedness of the set u(t). closure of the set uad results from continuity of the mapping uad → x, determined by defining of the set uad and compactness of the set x(theorem about closure of complete preimageof compact at continuous mapping). now, existence of solution for the problem under study follows from the weierstrass theorem. the theorem is proved. 2.1.6 conditions for the existence of solution for the variational calculus problem on choice (among given finite algorithms set) of optimal parametric control law of discrete stochastic dynamical system controllable discrete dynamical system with given additive noise, described by equations (54), (55) with phase constraints (57) is considered again in the following parametric control problem for discrete dynamical system. in this problem control is chosen among the set of given control laws: uj(t) = gj(v, x(t)), t = 0, ..., n− 1, j = 1, ..., r (59) 114 a. ashimov: the theory of parametric control of macroeconomic systems and ... where gj is known vector function of its arguments, v = (v1, ..., vl) is parameters vector of control law gj . on adjustable coefficients v impose the constraints v ∈ v (60) where v is compact in space rl. moreover, it is assumed that parameters of control law should be such that corresponding control law (59) satisfies the condition (58), that is the following inclusion would be met e [ gj(v, x v j (t)) ] ∈ u(t) (61) here xvj is solution of problem (54), (55) at chosen v coefficient values, uncontrollable parameter a and j-th parametric control law. consider optimality criteria kj = kj(v, a) = e { n∑ t=1 ft(x v j (t)) } (62) define the following variational problem, called as the variational calculus problem on choice of optimal parametric control law for discrete stochastic dynamical system. problem 2.7. given known vector of uncontrollable parameter a ∈ a to find each of r control laws to find such a vector of adjustable coefficients v, so that corresponding it solution x = xj of problem (54), (55) with control law u = uj , determined by formula (59), satisfies the conditions (60), (61) and gives the maximum for functional (62) with consequent choice of the best of found optimal control laws, i.e. the one, to which corresponds the maximal value of optimality criterion. now we get sufficient conditions for the existence of solution of the problem 2.7. denote by xvj solution of the system (54), (55) for chosen j-th parametric control law (59), its adjustable coefficient v and parameter α: xvj (t+ 1) = f ( xvj (t), gj ( v, xvj (t) ) , a ) + ξ(t), t = 0, 1, ..., n− 1 (63) xvj (0) = x(0) (64) for considered problem denote the set of acceptable values of adjustable coefficients as the set v j ad, consisting of such values v ∈ v satisfying the condition (60), for which corresponding solution of the problem (63), (64) satisfies the inclusions e [ gj(v, x v j (t)) ] ∈ u(t), t = 0, 1, ..., n− 1 (65) advances in systems science and applications (2014) vol.14 no.2 115 e [ xvj (t) ] ∈ x(t), t = 1, ..., n− 1 (66) we will call the problem 2.7 as nontrivial, if the set v j ad, corresponding it, is not empty and contains some open set for each j = 1, .., r. theorem2.14. let in the problem 2.7 a ∈ a, the sets u(t), x(t), v are compact, functions f , gj , ft satisfy the lipschitz condition. these functions satisfy constraints on increase as well: functions |f(x,gj(v, x), a)|, |ft(x)| do not exceed linear relatively |x| functions uniformly by v ∈ v . random variable ξ(t) is absolutely continuous and has zero mathematical expectation. then in the case of non-emptiness of the sets v j ad the problem 2.7 has a solution. proof. it is enough to ascertain that all of the functions (62) kj are continuous, and all of the sets v j ad are closed and bounded, where j = 1, ..., r. the existence of all used below mathematical expectations is proved in the same way as when proving the theorem 2.13. allowing for mathematical expectation additivity, we find the values of kj = kj(v) = n∑ t=1 e { ft[x v j (t)] } whence it follows inequality for v, w ∈ v j ad |kj(v)−kj(w)| = | n∑ t=1 e { ft[x v j (t)] } − n∑ t=1 e { ft[x w j (t)] } | ≤ n∑ t=1 |e { ft[x v j (t)] } − e { ft[x w j (t)] } | from relations (63), (64) follows that |xvj (t+ 1)− xwj (t+ 1)| = ∣∣f (xvj (t), gj ( v, xvj (t) ) , a ) − f ( xwj (t), gj ( w, xwj (t) ) , a )∣∣ ≤ lf [ |xvj (t)− xwj (t)|+ |gj ( v, xvj (t) ) −gj ( w, xwj (t) ) | ] where lf is the lipschitz constant of function f . after denoting by la the maximal one of the lipschitz constants of function gj , we will get inequality |xvj (t+1)−xwj (t+1)| ≤ lf (1+la)|xvj (t)−xwj (t)|+lflg|v−w|, t = 0, 1, ..., n−1 taking into account equality |xvj (0)− xwj (0)| = 0, we get the estimate |xvj (t+ 1)− xwj (t+ 1)| ≤ lflg t∑ l=0 [lf (1 + lg)] l |v − w| ≤ β|v − w|∀v, w ∈ v j ad 116 a. ashimov: the theory of parametric control of macroeconomic systems and ... where β = lflg t∑ l=0 [lf (1 + lg)] l after denoting by lf the maximal one of the lipschitz constants of function ft, we will have∣∣f [xvj (t)]− f [xwj (t)] ∣∣ ≤ lf ∣∣xvj (t)− xwj (t) ∣∣ ≤ lfβ|v − w|∀v, w ∈ v j ad thus, in the case of sufficient smallness of difference of adjustable coefficients v and w the values of xvj (t) and x w j (t) (and the same f [xvj (t)] and f [x w j (t)]) will be arbitrary close to each other. determine converged sequence w = vk → v. then, after finding mathematical expectations of lhs and rhs of the last inequality, we will get the following inequality: e [∣∣f [xvj (t)]− f [xwj (t)] ∣∣] ≤ lfβ|v − w| hence results the following convergence e { ft [ xvkj (t) ]} → e { ft [ xwk j (t) ]} from which follows continuity of function kv j . as v j ad ⊂ v , then all of the sets v j ad are bounded. closure of the sets v j ad results from closure of the sets u(t), x(t + 1), of proved above continuity of mappings v → e[xvj (t + 1)], v → e[gj(v, x v j (t))] and determining the set v j ad as complete preimageof pointed sets at continuous mappings. the theorem statement follows from the weierstrass theorem about reaching continuous on compact function of its upper boundary. 2.2 sufficient conditions for continuous dependence of optimal criteria values of the parametric control problems on uncontrollable functions in this section, within the framework of developing the 4th component of the parametric control theory, there are derived sufficient conditions of continuous dependence on uncontrollable functions a(·) of optimal criteria values for all of considered above parametric control problems for non-autonomous deterministic dynamical systems and sufficient conditions of continuous dependence on uncontrollable parameters of optimal criteria values for considered above parametric control problems for autonomous stochastic dynamical systems. all of the results defined for non-autonomous systems remain true for autonomous dynamical systems as well, where exogenous vector function a(·) is taken as constant. the following theorem determines sufficient conditions of continuous dependence of optimal criterion values for the problem 2.4. theorem 2.15. assume that when conditions of the theorem 2.10 are met, advances in systems science and applications (2014) vol.14 no.2 117 in neighborhood (in euclidean topology) of function a(·) function f is continuous by the third argument and satisfies the lipschitz condition by the first argument on uniformly by the second and third arguments.then optimal criterion value for the problem 2.4 continuously depends on uncontrollable function at the point a(·). proof. let the following convergence occurs ak → abrsn (67) according to the theorem 2.10 the problem 2.4 given the values of uncontrollable functions and ak has a solutions, which we denote, correspondingly, by u and uk. denote by x[b, w] solution of condition equation (43), (44), corresponding to uncontrollable function b and control w. ipso facto the function x[b, w] satisfies relations x [b, w] (t+ 1) = f (x [b, w] (t), w(t), b(t)) , t = 0, 1, ..., n− 1 (68) x [b, w] (0) = x0 (69) then the following inequalities are true 0 ≥ k (x [a, uk])−k (x [a, u]) ,k (x [ak, uk])−k (x [ak, u]) ≥ 0 where by k(y ) is denoted the value of criterion k at given value of system state . consequently, we get the following relations: 0 ≤ k (x [a, u])−k (x [a, uk]) ≤ k {(x [a, u])−k (x [ak, u])}+ {(x [ak, u])− (x [k, uk])}+ {k(x [ak, uk])−k (x [a, uk])} ≤ 2 sup w∈u |k(x [a,w])−k (x [ak, w])| (70) from the conditions (68) we derive inequalities |x [ak, w] (t+ 1)− x [a,w] (t+ 1)| = |f (x [ak, w] (t), w(t), ak(t))− f (x [a,w] (t), w(t), (t))| ≤ |f (x [ak, w] (t), w(t), ak(t))− f (x [ak, w] (t), w(t), (t))| + |f (x [ak, w] (t), w(t), a(t))− f (x [a,w] (t), w(t), a(t))| ≤ sup ∈x,φ∈u |f (y, φ, ak(t))− f (y, φ, a(t))|+ l |x [ak, w] (t)− x [a,w] (t)| , t = 0, 1, ..., n− 1, where l is the lipschitz constant of function f by the first argument, which does not depend on w. consequently, we get the following relation |ψk(t+ 1)| ≤ ηk + l |ψk(t)| , t = 0, 1, ..., n− 1 118 a. ashimov: the theory of parametric control of macroeconomic systems and ... where ψk(t) = x [ak, w] (t)− x [a,w] (t), t = 0, 1, ..., n− 1, ηk = max t=0,1,...,n−1 sup ∈x,φ∈u |f (y, φ, ak(t))− f (y, φ, a(t))| then we determine relations |ψk(t+ 1)| ≤ ηk + l|ψk(t)| ≤ ηk + l(ηk + l|ψk(t− 1)|) = (1 + l)ηk + l2|ψk(t− 1)| ≤ (1 + l)ηk + l2(ηk + l|ψk(t− 2)|) = (1 + l+ l2)ηk + l3|ψk(t− 2)| ≤ ... ≤ t∑ r=0 lrηk + lt+1|ψk(0)|, t = 0, 1, ..., n− 1 from initial condition (69) follows that |ψk(0)| = 0. then the following estimate is true |ψk(t+ 1)| ≤ ηk t∑ r=0 lr, t = 0, 1, ..., n− 1 from which follows that |x[ak, w](t+ 1)− x[a,w](t+ 1)| ≤ ηk t∑ r=0 lr, t = 0, 1, ..., n− 1 note that right-hand-side of this inequality does not depend on w. from the condition (67) follows the convergence ηk → 0 under continuity of function f by the third argument, as well as closure and boundedness of the sets x and u . then we derive from the last inequality that x [ak, w] (t) → x [a,w] (t) uniformly by w, t = 1, ..., n− 1. hence under continuity of function f follows that f [t, x [ak, w] (t)] → f [t, x [a,w] (t)] uniformly by w, t = 1, ..., n− 1.a therefore, k(x[ak, w]) → k(x[a,w]) uniformly by w, t = 1, ..., n− 1.a whence the convergence sup w∈u |k(x [a,w])−k (x [ak, w])| → 0 (71) follows. taking the limits in inequality (70) and taking into account the conditions (58), we will have k (x [a, uk]) → k (x [a, u]) (72) advances in systems science and applications (2014) vol.14 no.2 119 the following estimate is true. |k(x [ak, uk])− (kx [a, u])| ≤ |k(x [ak, uk])− (kx [a, uk])| + |(kx [a, uk])− (kx [a, u])| taking the limits and taking into account the conditions (71), (72), we will determine the convergence k (x [ak, uk]) → k (x [a, u]) hence under defining controls u and uk results that the maximal value of criterion k, corresponding to uncontrollable function ak, converges to its maximum, corresponding to the extreme value of uncontrollable function . the theorem is proved. determine now sufficient conditions of continuous dependence on uncontrollable functions of optimal criterion values for the variational calculus problem 2.5 on choice (among given finite algorithms set) of optimal parametric control laws based on discrete non-autonomous dynamical system. determine first corresponding result for the problem 2.5*. theorem 2.16. assume that when conditions of the theorem 2.11 are met in neighborhood of point function f is continuous by the third argument and satisfies the lipschitz condition by the first two arguments on x×u uniformly by the third argument, and function gj satisfies the lipschitz condition by the second argument on uniformly by the first argument.then optimal criterion value for the problem 2.5* continuously depends on uncontrollable function at point . proof. let the convergence occurs ak → abrsn (73) according to the theorem 2.11 the problem 2.5* at values of uncontrollable functions and ak has a solution, which we denote, correspondingly, by v and vk. denote by x[b, w] solution of state equations (53), (44), corresponding to uncontrollable function b(·) and controllable parameter w. ipso facto function x[b, w] meets relations x [b, w] (t+ 1) = f ( x [b, w] (t), gj (v, x [b, w] (t)) , b(t) ) , t = 0, 1, ..., n− 1 (74) x [b, w] (0) = x0 (75) analogously with inequality (70) the following relation is determined 0 ≤ k (x [a, v])−k (x [a, vk]) ≤ 2 sup w∈v |k(x [a,w])−k (x [ak, w])| (76) 120 a. ashimov: the theory of parametric control of macroeconomic systems and ... after denoting a = x [a,w] , ak = x [ak, w] from equalities (74) we derive inequalities |xk(t+ 1)− x(t+ 1)| = ∣∣f (xk(t), gj (w, xk(t)) , ak(t) ) − f ( x(t), gj (w, x(t)) , a(t) )∣∣ ≤ ∣∣f (xk(t), gj (w, xk(t)) , ak(t) ) − f ( xk(t), gj (w, xk(t)) , a(t) )∣∣ + ∣∣f (xk(t), gj (w, xk(t)) , a(t) ) − f ( x(t), gj (w, x(t)) , a(t) )∣∣ ≤ sup y∈x,φ∈u |f (y, φ, ak(t))− f (y, φ, a(t))| + l [ |xk(t)− x(t)|+ ∣∣gj (w, xk(t))−gj (w, x(t)) ∣∣] ≤ sup y∈x,φ∈u |f (y, φ, ak(t))− f (y, φ, a(t))| + l(1 +m) |xk(t)− x(t)| , t = 0, 1, ..., n− 1, where l is the lipschitz constant of function f by the first two arguments, and is the lipschitz constant of function gj by the second argument. consequently, we get inequality |ψk(t+ 1)| ≤ ηk +n |ψk(t)| , t = 0, 1, ..., n− 1 where n = l(1 +m) ψk(t) = xk(t)− x(t), t = 0, 1, ..., n− 1, ηk = max t=0,1,...,n−1 sup y∈x,φ∈u |f (y, φ, ak(t))− f (y, φ, a(t))| using technique, described above in proof of the theorem 2.15, we determine the estimate |ψk(t+ 1)| ≤ ηk n∑ r=0 n r, t = 0, 1, ..., n− 1 and the convergence ηk → 0. thereby, |ψk(t)| → 0, t = 1, ..., n therefore, x [ak, w] (t) → x [a,w] (t) uniformly by w, t = 1, ..., n− 1. hence under continuity of function f follows that f [t, x [ak, w] (t)] → f [t, x [a,w] (t)] uniformly by w, t = 1, ..., n− 1. therefore, k (x [ak, w]) → k (x [a,w]) uniformly by w. advances in systems science and applications (2014) vol.14 no.2 121 then under finite dimensionality of the space of controls and boundedness of the set u we get sup w∈v |k(x [a,w])−k (x [ak, w])| → 0 (77) taking the limits in inequality (76) and taking into account the condition (77), we will have k (x [a, vk]) → k (x [a, v]) (78) the following estimate is true |k(x [ak, vk])−k (x [a, v])| ≤ |k(x [ak, vk])−k (x [a, vk])| + |k(x [a, vk])−k (x [a, v])| taking the limits here and taking into account the conditions (77), (78), we determine the convergence k (x [ak, vk]) → k (x [a, v]) hence under definition of control parameters v and vk follows that the maximal value of criterion k, corresponding to uncontrollable function ak, converges to its maximum, corresponding to extreme value of uncontrollable function. the theorem is proved. as the maximum of two (therefore, any finite number as well) of continuous functions is continuous, then from the theorem 2.16 follows similar result for the problem 2.5. theorem 2.17. assume that when conditions of the theorem 2.11 are met in neighborhood of point , function f is continuous by the third argument and satisfies the lipschitz condition by the first two arguments on x × u uniformly by the third argument, and function gj satisfies the lipschitz condition by the second argument on uniformly by the first argument.then optimal criterion value for the problem 2.5 continuously depends on uncontrollable function at point a. the purpose of the next study is to determine sufficient conditions of continuous dependence on uncontrollable functions of optimal criterion values for the problem 2.1 based on continuous non-autonomous dynamical system. theorem 2.18. assume that when conditions of the theorem 2.6 are met in neighborhoods of point , function f is continuous by the second argument and satisfies the lipschitz condition by the first and third arguments on x × a uniformly by the second argument.then optimal criterion value for the problem 2.1 continuously depends on uncontrollable function at point . proof. let the convergence occurs ak → a (c[0, t ])s (79) 122 a. ashimov: the theory of parametric control of macroeconomic systems and ... according the theorem 2.6 the problem 2.1at values of uncontrollable functions and ak has solutions, which we denote, correspondingly, by u and uk. denote by x[b, w] solution of state equations (1), (2), corresponding to uncontrollable function b and control w. ipso facto function x[b, w] meets relations ẋ [b, w] (t) = f (x[b, w] , w(t), b(t)) , t ∈ (0, t ) (80) x [b, w] (0) = x0. (81) then, by repeating reasoning from proof of the theorem 2.13, analogously with relation (70) we determine inequality 0 ≤ k (x [a, u])−k (x [a, uk]) ≤ 2 sup w∈u |k(x [a,w])−k (x [ak, w])| (82) after denoting x = x [a,w] , xk = x [ak, w] from the conditions (80) we derive inequalities ẋk(t)− ẋ(t) = f (xk(t), w(t), ak(t))− f (x(t), w(t), a(t)) , t ∈ (0, t ) by integrating by t and taking into account equalities (81), we get |xk(t)− x(t)| ≤ ∫ t 0 |f (xk(τ), w(τ), ak(τ))− f (x(τ), w(τ), (τ))| dτ ≤ l ∫ t 0 |xk(τ)− x(τ)| dτ+l ∫ t 0 |ak(τ)− (τ)| dτ ≤ l ∫ t 0 |xk(τ)− x(τ)| dτ+lt∥ak−∥θ, t ∈ (0, t ) where l is the lipschitz constant of function f , not depending on w. using the gronwall lemma, we will have |xk(t)− x(t)| ≤ ∥ak − a∥θ, t ∈ (0, t ) where positive constant depends only on l and on . using the condition (79), we get that xk(t) → x(t), t ∈ (0, t ) and therefore, x [ak, w] (t) → x [a,w] (t) uniformly by w, t ∈ (0, t ) hence under continuity of function f follows the convergence k (x [ak, w]) → k (x [a,w]) uniformly by w advances in systems science and applications (2014) vol.14 no.2 123 consequently, we get sup w∈u |k(x [a,w])−k (x [ak, w])| → 0 (83) taking the limits in inequality (82) and taking into account the conditions (83), we will have k (x [a, uk]) → k (x [a, u]) (84) the following estimate is true |k(x [ak, uk])−k (x [a, u])| ≤ |k(x [ak, uk])−k (x [a, uk])|+ |k(x [a, uk])−k (x [a, u])| taking the limits here and taking into account the conditions (83), (84), we will get the convergence k (x [ak, uk]) → k (x [a, u]) hence under definition of controls u and uk it follows that the maximal value of criterion k, corresponding to uncontrollable function ak, converges to its maximum, corresponding to extreme value of uncontrollable function. the theorem is proved. determine sufficient conditions of continuous dependence on uncontrollable functions of optimal criterion values for the variational calculus problem on choice (among given finite algorithms set) of optimal parametric control laws based on continuous non-autonomous dynamical system. theorem 2.19. assume that when conditions of the theorem 2.8 are met in neighborhood of point , function f satisfies the lipschitz condition on the set x×u ×a, and function satisfies the lipschitz condition by the second argument on uniformly by the first argument.then optimal criterion value for the problem 2.3* continuously depends on uncontrollable function at point . proof. let the convergence occurs ak → ab(c[0, t ])s (85) according the theorem 2.8 the problem 2.3* at values of uncontrollable functions and ak has a solutions, which we denote, correspondingly, by v and vk. denote by x[b, w] solution of state equations (23), (2), corresponding to uncontrollable function b and control w. ipso facto function x[b, w] meets relations ẋ [b, w] (t) = f ( x [b, w] (t), gj (w, x(t)) , b(t) ) , t ∈ (0, t ) (86) x [b, w] (0) = x0 (87) 124 a. ashimov: the theory of parametric control of macroeconomic systems and ... then, by repeating reasoning from proof of the theorem 2.15, analogously with relation (70) we determine inequality 0 ≤ k (x [a, u])−k (x [a, uk]) ≤ 2 sup w∈v |k(x [a,w])−k (x [ak, w])| (88) after denoting x = x [a,w] , xk = x [ak, w] from the conditions (86) we derive inequalities ẋk(t)−ẋ(t) = f ( xk(t), gj (w, xk(t)) , ak(t) ) −f ( x(t), gj (w, x(t)) , a(t) ) , t ∈ (0, t ) by integrating by t and taking into account equalities (87), we get |xk(t)− x(t)| ≤ ∫ t 0 ∣∣∣f (xk(τ), gj (w, xk(τ)) , ak(τ) ) − f ( x(τ), gj (w, x(τ)) , a(τ) )∣∣∣ dτ ≤ l ∫ t 0 [ |xk(τ)− x(τ)|+ ∣∣∣gj (w, xk(τ))−gj (w, x(τ)) ∣∣∣+ |ak(τ)− a(τ)| ] dτ ≤ l(1+) ∫ t 0 |xk(τ)− x(τ)| dτ+lt∥ak − a∥θ, t ∈ (0, t ), where l is the lipschitz constant of function f , and is the lipschitz constant of function by the second argument. using the gronwall lemma, we will have |xk(t)− x(t)| ≤ ∥ak − a∥θ, t ∈ (0, t ) where positive constant depends only on l and on . using the condition (85), we get that x [ak, w] (t) → x [a,w] (t) uniformly by w, t ∈ (0, t ) hence under continuity of function f follows that f [t, x [ak, w] (t)] → f [t, x [a,w] (t)] uniformly by w therefore, k (x [ak, w]) → k (x [a,w]) uniformly by w consequently, we get sup w∈v |k(x [a,w])−k (x [ak, w])| → 0 (89) taking the limits in inequality (88) and taking into account the condition (89), we will have k (x [a, uk]) → k (x [a, u]) (90) advances in systems science and applications (2014) vol.14 no.2 125 the following estimate is true |k(x [ak, uk])−k (x [a, u])| ≤ |k(x [ak, uk])−k (x [a, uk])|+ |k(x [a, uk])−k (x [a, u])| taking the limits here and taking into account the conditions (89), (90) we will get the convergence k (x [ak, uk]) → k (x [a, u]) hence under definition of controls u and uk follows that the maximal value of criterion k, corresponding to uncontrollable function ak, converges to its maximum, corresponding to extreme value . the theorem is proved. using the theorem 2.19 analogously with the theorem 2.17 (in this case system continuity is not a matter of principle), we come to the following statement. theorem 2.20. when for all j = 1, ..., r conditions of the theorem 2.19 are met, optimal criterion value for the problem 2.5 continuously depends on uncontrollable function at point . present statements of the theorems about continuous dependence on uncontrollable parameter of optimal criterion values for the parametric control problems of stochastic dynamical systems. proves of these theorems are similar to those presented above. theorem 2.21. let given any a ∈ a conditions of the theorem 2.13 are met. then optimal criterion value for the problem 2.6 are continuous function of the parameter . now, study the conditions of continuous dependence of optimal criterion value for the variational calculus problems on choice of parametric control laws on uncontrollable parameters. theorem 2.22. let given any a ∈ a conditions of the theorem 2.14 are met. then for chosen number value of the law j kj optimal criterion values for the problem 2.7 are continuous function of the parameter . consequence 2.23. when conditions of the theorem 2.22 are met for all j = 1, ..., r, optimal criterion value k = maxj=1,...,rkj for the problem 2.7 are continuous functions of the parameter a ∈ a. obtained results will be used in the next section in proving the existence of corresponding bifurcation points of extremals of the variational problems. 2.3 sufficient conditions for the existence of extremals’ bifurcation points of the problems on choice of optimal parametric control laws let us introduce a notion of extremals’ bifurcation point of the variational calculus problems on choice (among given finite algorithms set) of optimal parametric control laws.existence of such a bifurcation point for some uncontrollable function a(·) means that in its neighborhood for the problem in question occurs transfer from one optimal parametric control law to another. 126 a. ashimov: the theory of parametric control of macroeconomic systems and ... consider abstract variational calculus problem on choice (among given finite algorithms set) of optimal parametric control laws, generalizing the problems 2.3, 2.5, and 2.7. given: the set of uncontrollable functions (or parameters), set of the acceptable controls sets (adjustable coefficients values) v j a , a ∈ a, j = 1, ..., r and set of functionals (optimality criteria) kj = kj(a, v) where v ∈ v j a , a ∈ a, j = 1, ..., r. give definition for extremals’ bifurcation point for set of the maximization problems for given functionals on corresponding sets of acceptable controls, i.e. abstract variational calculus problem on choice (among given finite algorithms set) of optimal parametric control laws. definition. the value a ∈ a call asextremals’ bifurcation point for the maximization problem for mappings u→ kk(a, v) on the sets v k a , k = 1, ..., r if there are exist two different numbers i, j ∈ 1, ..., r such that the following relation is true max v∈v l a ki(a, v) = max v∈v j a ki(a, v) = max k=1,...,r max v∈v k a ki(a, v) and in any neighborhood of point there is such a point b ∈ a for which the value max k=1,...,r max v∈v j b kj(b, v) reachs for the only value of k. theorem 2.24. assume that when conditions of the theorem 2.17(or 2.20, or 2.23) are met, is a connected set, and there are two different values a0, a1 ∈ a, such that the maximal values by j = 1, ..., r of function maxima v → kj(a, v) on the sets are reached for the different values j0, j1 that is the following inequalities are true: max j=1,...,r,j ̸=j0 max v∈v j a0 kj(a0, v) < max v∈v j0 a0 kj0(a0, v) max j=1,...,r,j ̸=j1 max v∈v j a1 kj(a1, v) < max v∈v j1 a1 kj1(a1, v) then there exists a bifurcation point of extremals of the problem 2.3 (or 2.5, or 2.7) on choice (among given finite algorithms set) of optimal parametric control law. proof. under connectivity of the set points a0, a1 can be connected by continuous line a = a(s), s ∈ [0, 1], lying in the set , and the following equalities are true a(0) = a0, a(1) = a1 denote kj(s) = max v∈v j a(s) kj1(a(s), v), s ∈ [0, 1] advances in systems science and applications (2014) vol.14 no.2 127 from the theorems 2.17 (or 2.20, or2.23) follows continuity of functions s → kj(s) in the segment [0, 1], and therefore, continuity in this segment of function as well, where determine the set ∆(j) = {s ∈ [0, 1]|kj(s) = k∗(s)} , j = 1, ..., r it is closed, being complete preimageof closed set, consisting of the only point (zero) for continuous function y = y(s) where y(s) = kj(s) − k ∗ (s) thereby, we present the segment [0, 1] in terms of the following sum [0, 1] = ∪ j=1,...,r ∆(j) consisting, according to the theorem conditions, as minimum, of two non-empty closed sets. from theorem conditions it follows the following relations as well: 0 ∈ ∆(j0), 1 /∈ ∆(j0) then the set of boundary points of the set ∆(j0) which are situated in the interval (0, 1), is not empty. consequently, there exists lower boundary s0 of such boundary points. this value is a boundary point of some another set ∆(j2) and is a part of it as well. thereby, at a = a(s0) the maximum by j = 1, ..., r the values is reached, as minimum, for two different numbers j0 and j2. at the same time, at 0 ≤ s ≤ s0 this maximum is reached for the only value j0. thus, a(s0) actually corresponds to desired bifurcation point. the theorem is proved. the following statement is direct consequence of the theorem 2.24. consequence 2.25. assume that when conditions of the theorem 2.17 (or 2.20, or 2.23) are met, is connected set, and at value control using the law provides solution of the problem 2.3 (or 2.5, or 2.7), and at , ( control using this law does not provide solution of the problem in question, that is the following inequalities are true max j=1,...,r,j ̸=j0 max v∈v j a0 kj(a0, v) < max v∈v j0 a0 kj0(a0, v) max j=1,...,r,j ̸=j1 max v∈v j a1 kj(a1, v) > max v∈v j0 a1 kj0(a1, v) then there is at least one bifurcation point of extremals of mentioned problem 2.3 (or 2.5, or 2.7). at the end we present description of the numerical algorithm for finding bifurcational value of function (or parameter) a of one of the problems 2.3 or 2.5 or 128 a. ashimov: the theory of parametric control of macroeconomic systems and ... 2.7 on choice (among given finite algorithms set) of parametric control laws and when conditions of the theorem 2.24 are met. connect the points and by smooth curve. divide this curve to equal parts with sufficiently small step. for obtained values (points) are determined the numbers of parametric control lawsbringing solution of the problem 2.3 or 2.5 or 2.7 given values. then we find the first value i, at which corresponding law number differs from. in this case bifurcational value lies on arc of the curve. for found part of the curve, the algorithm for determining bifurcation point with given accuracy consists in using the method of bisections. consequently we find the point, on the one hand from which on this arc within the range of deviation from the value the optimal law is, and on the other hand-within the range of deviation from the value this law is not optimal. from the consequence 2.25 follows that extremal’s bifurcation point for the problem under solution lies on mentioned arc, and as its estimate can be taken any point of the are. references [1] ekeland i., temam r. (1976), “convex analysis and variational problems”, amsterdam: nord-holland publishing company. corresponding author a. ashimov can be contacted at: ashimov37@mail.ru adv syst sci appl 2018; 1; 85-91 published online at http://ijassa.ipu.ru. copyright ©0000 assa. adv. in systems science and appl. (0000) management of the development of the region: the attractiveness of the region and human potential vadim kleparskiy 1 , victor sheinis 2 1) v.a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia e-mail: kleparvg@ipu.ru 2) v.a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia e-mail: scheynis@yandex.ru abstract: using a comparative analysis of the data of the "passive" experiment, it was shown that an increase in the attractiveness of the region can be achieved by long-term, purposeful increase in human potential and by attracting corresponding additional investments keywords: predictive control, “passive” experiment, attractiveness, human potential, investments. 1. introduction the top-level control of the region – a large-scale socio-economic system (ses) – is the complicated process of realizing adequate administrative measures aimed at maintaining commensurate flows of material, labor, information and other resources in a regime that ensures a sustainable increase in the level of well-being. such a goal can be achieved by a steady increase in the total productive potential (tpp) the main driving force for the evolution of the system. from the positions of the top level of the regional control hierarchy, such a multidimensional and multilevel process is reduced to the activity of forming the desired channel of "attracting" solutions (the attraction channel in terms of nonlinear dynamics). at the same time, the top level of regional governance should be taken into account that the total productive potential is determined not only by material assets (physical capital, investments), but also by intangible (nonmaterial) assets. and the main component of intangible assets is human potential. it should also be noted that for developed economies of the world, the share of human potential in the total volume of tpp is 50-60% (see, for example [1]). increasing the attractiveness of the region in physical (material) capital (for investment), as well as the desire to increase the human potential of the system is therefore the main objective of long-term predictive control of the region. in the competitive environment for resources, this task is quite difficult to fulfill. success is largely determined by the long-term control activity to create favorable conditions for ensuring the socialeconomic merits of the governed region using the cultural and historical heritage. in the proposed letter, using the modified poincare transversal surface method, using the "passive" experiment, the possibilities of improving the population welfare of the region by attracting qualified human resources and additional investments are considered. 2. investigation concepts given the relatively slow process of establishing a dynamic equilibrium in the flow of a dissipative administrative environment, the rate of change in the chosen "order parameter" c 86 v. kleparskiy, v. sheinis copyright ©2018 assa. adv. in systems science and appl. (2018) (the level of social-economic well-being) determined largely by the specific value of the regional gross product (of gdpreg per person) сan be determined with sufficient accuracy in the gradient approximation by the expression )( )( )( ),( t c pp t c rcw dt dc phnonmat         (1) here, w(c,r) is the total productive potential (tpp) of the system, r(t) is the vector of control actions, pph the physical potential of the ses (based assets, investments), pnonmat – is nonmaterial assets (human potential and its components). the second term – ξ(t) on the right side of the equation expresses the possibility of the influence of random perturbations on the trajectory of the ses development. dynamic equation (1) allows to study the process of long-term planned approximation of the region to the realities of the outside world from the results of observations of the course of socio-economic development using the "passive" experiment using the basic provisions of the qualitative theory of nonlinear dynamics and physics of society analyzing (1) it can be seen that with the increase in the share of nonmaterial assets in the tpp system the dynamics of socio-economic development of the region will increasingly be determined by a skilful increase in the human potential. and, simultaneously, increasing the effectiveness of its use at all levels of the hierarchy of management of the system. the strategy of long-term predictive control therefore requires an adequate assessment of the main components of the human potential of the system and the identification of possible ways to increase them. a qualitative assessment of the value of the human potential components of ses can be performed using the human development index (hdi) – until 2013 as an index of human development. the determination of the hdi value is performed taking into account three groups of basic indicators.. these groups of indicators are estimated: life expectancy – longevity (determined mainly by the level of development of medicine); the level of literacy of the population (determined by the availability of educational institutions); the standard of living, estimated through the value of gdpreg per capita at purchasing power parity.. it should be expected that it is the first two components of the hdi (the state of medicine and, accordingly, the level of literacy) that will determine the level of the third component of the hdi gdpreg/capita. creation of conditions ensuring the increase of human.potential in the predicted control can be ensured by the steady increase of its social and cultural components increasing in this way the attractiveness of the region for a growing population (and, accordingly, the saturation of the region by initiative, efficient skilled personnel) therefore seems to be the main task of the top level of the governance hierarchy in planning long-term development of the region. as the object of the "passive" experiment, the federal lands of germany were chosen. earlier it was noted that for germany there are two groups of federal lands: "new" (former gdrs) and "old", more developed in industrial relation (see, for example, [2]). the newest statistical data confirm the sharp heterogeneity of the level of regional well-being achieved by various regions (federal lands and individual cities in germany), estimated by the level of gdpreg per person employed. (see fig.1). 3. results and discussion analyzing the spstial distributions (gdpreg/person empl. data) presented in fig. 1, it can be seen that the largest values of gdpreg/person employed. were achieved in the "old" federal lands of germany: bavaria, baden-württemberg, north rhine-westphalia, hessen, hamburg). this level of the state of the objects of research makes it possible to compare the effectiveness of long-term planned control of the development of the region more fully based on the representative results of the "passive" experiment management of the development of the region 87 copyright ©2018 assa. adv. in systems science and appl. (2018) fig.1. spatial distribution of regional gdpreg/person employed. the brightest areas are the gdpreg/person empl.< 52.7 10 3 euro. the darkest areas are gdpreg/person empl. > 65.2 10 3 euro. fig. 1 obtained on the base of [3]. 3. results and discussion analyzing the spstial distributions (gdpreg/person empl. data) presented in fig. 1, it can be seen that the largest values of gdpreg/person employed. were achieved in the "old" federal lands of germany: bavaria, baden-württemberg, north rhine-westphalia, hessen, hamburg). this level of the state of the objects of research makes it possible to compare the effectiveness of long-term planned control of the development of the region more fully based on the representative results of the "passive" experiment as it follows from the definition of the human development index (hdi), the influence of nonmaterial assets (human potential and its components) on the dynamics of the region's long-term development is largely realized through skilled and highly motivated work at all levels of regional control activity (including also and the institutional stage of governance). it is the achieved level of social and cultural development of the region that determines the possibility of increasing the share of scientific and productive activities in the process of ses-control operation. the process of this increase is determined by the requirements of the modern (global) economy of high technology production. at the same time, regions with previously developed industrial infrastructure have the advantage of reinforcing the transition to a new "postindustrial" period – to the knowledge economy – with the corresponding additional investments in physical capital and human potential. such a 88 v. kleparskiy, v. sheinis copyright ©2018 assa. adv. in systems science and appl. (2018) complex relationship between necessity and obligation (the need for a transition to a knowledge economy, the necessety for additional investment and the need to increase the effectiveness of human potential use) largely determines the planned increase in well-being – and the value of the specific regional gross product gdpreg/capita. at the same time, the historical past of the region is of great importance. for example, the agglomeration of large cities (duisburg, dusseldorf, essen, cologne, bonn) in north rhine-westphalia (based on the ruhr industrial region) is characterized by a close relationship between the gdpreg per worker (parameter, which characterizes the labor.productivity achieved) – fig.1 – and the share of highly skilled workers engaged in high technology production – fig. 2. fig.2. proportion (in %) of workers employed in the high-technology industry. the brightest areas the share of workers employed in science-intensive production is less than 1%.the darkest areas – this share is more than 10%.fig. 2 obtained on the base of [3]. comparison of the spatial distributions of gdpreg per worker and the share of highly skilled workers (figure 1 and figure 2) makes it possible to visually compare and estimate the effectiveness of predicted control implemented in the course of evolution using the planned increase in the components of human potential for improving the welfare of the population of the region. comparative analysis presented in fig. 1 and fig. 2 spatial distributions of labor productivity (gdpreg per worker employed in production) and the share of workers. employed in science-intensive production allows you to notice numerous local deviations from the mean (by federal land) values. these local positive outbursts of gdp are characterized not only by certain regions, but also by individual cities, attractive both for investment in physical capital and for increasing human potential. this ability of the region to social and economic development is largely determined by the long-term activity of the top level of the control hierarchy of the region (city) in creating and maintaining an attractive image of a controlled ses. the hastily developing metropolitan cities (munich, stuttgart, düsseldorf, hamburg, bremen) and urban agglomerations (for example, the urban management of the development of the region 89 copyright ©2018 assa. adv. in systems science and appl. (2018) agglomerations of north rhine-westphalia and bavaria) are a good example of the implementation of such positive predictive control. one of the tools for creating an attractive image of the region is the predictive control activity associated with employment in scientific and technical research and innovation (stria) this control activity is impossible without the corresponding costs for highly qualified scientific and engineering specialists (stria-expenses). in their origin, these stria-expenses are due to the costs of training and saturating the region's economy by graduates of the higher school of science and technology. the values of stria-expenses largely depend on the region's economics possibilities – on region’s values of gdpreg/capita. there are peculiar feedbacks: the dependence of gdpreg/capita on the amount of investment (the specific value of investment) and the amount of expenditure on stria, and, at the same time, the dependence of striareg/person on gdpreg/capita. this dependence for two groups of german federal lands was detected using a modified poincare method of a transversal cutting surface (see, for example, [2]).. the striareg/person dependences on the gdpreg/capita for the "old" and the "new" federal lands of germany are shown in fig. 3. . it can be seen that the highly developed bavaria could have specific costs for stria/person  900 euro – within  2.4% of gdpreg/ person. such opportunities bavaria could have due to the implementation of a multi-year development program to create the conditions for a more effective exchange between scientific achievements and the economy (see, for example, [4])..at the same time, most of the funds for development programs were invested in education and training, research and development, in the development of information and communication technologies. as a result of the chosen strategy, bayern was able to reach the front lines in the aviation and space industries, biotechnology and medical technology, and other high-tech industries. this provided bavaria with an average gdpreg/capita of about 40,000 euros and, accordingly, a sufficiently high attractiveness. 0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 20 25 30 35 40 45 gdp reg/person, 10^3 euro s t r ia r e g /p e r. , 1 0 ^ 3 e u ro 5 4 7 3 2 5 4 6 1 3 1 2 fig. 3. influence of the value of the regional gross domestic product gdpreg/person on the specific value of expenditure for striareg/person. the upper line is the "old" lands of germany. . (1bavaria, 2badenwürttemberg, 3hessen, 4lower saxony, 5north rhine-westphalia, 6rhineland-pfalz, 7-schleswigholstein). lower (left) line new "lands" of germanylower (left) line new "lands" of germany. (1thuringia, 2saxony-anhalt, 3saxony, 4mecklenburg, 5brandenburg). for a relatively backward thuringia (as an example), it was possible to provide striareg/person only in the amount of 220 euros – within 1% of the achied level of the 90 v. kleparskiy, v. sheinis copyright ©2018 assa. adv. in systems science and appl. (2018) gdpreg/person. insufficiently large gdpreg/person. values in germany's “new” lands cause relatively small financial allocations for scientific and technological development. and innovations (striareg/person expenditures). under such circumstances, the region's achievement of attractiveness for additional investments in physical capital and human potential becomes difficult. accordingly, the subsequent achievement of a high standard of living for the “new” german lands (for lands as a whole) is also difficult. as a result, the achievement of a high standard of living in the “new” federal lands of germany was mainly realized only in the regional metropolises and individual centers of the previously achieved scientific and technological development. illustrative examples of such metropolitan cities are potsdam (brandenburg land), leipzig and dresden (saxony). in these cities, the leading scientific centers, knowledge-intensive industries are concentrated. these large cultural and historical centers have an anomalous attracting effect. these conglomerate cities have.the advantage of competing for skilled labor, which gives an additional impetus to their development. as a result, these cities were included in the top ten cities of germany with the largest population growth (potsdam – 6.3%, leipzig – 7%, dresden – 5.8%) for the federal land of thuringia (the average gdpreg per capita for the country = 23,000 euros), cities – islands of local development – are (as can be seen in figure 2) jena, erfurt, weimar. the city of jena – has a university and is the center of electronic-optical engineering – and has a regional gdpreg/person employed = 58,2 10 3 euro. the city of erfurt – the capital of the federal land and the university sity – has gdpreg/person employed = 54,3 10 3 euro. at the same time in erfurt there is a well-established system of social insurance of employees, which contributes to the inflow of labor.the city of weimar known for its university and its adjoining research institutions (precision instrumentation making)– has gdpreg/person employed = 54,3 10 3 euro. the listed cities have gdpreg per capita roughly twice the average for the federal state of thuringia [5]. but in the nearby city of ingolstadt (bavaria) – a flourishing center of the automobile industry – the value of the gdpreg per capita is 123 10 3 euro. such cities in bavaria as erlangen and regensburg have the value of gdpreg per capita = 83 10 3 euro. naturally, the "old" federal lands of germany continuously "suck" the qualified specialists from the "new" lands. especially noticeable is this effect in small provincial towns and rural areas [3, 6].. 3. results and discussion the study of the level of social and economic development of the "old" and "new" german lands (using a comparative analysis of the data of the "passive" experiment – figures 1, 2 and figure 3) shows that a balanced growth of regional welfare requires from the regional governance structures a long-term program for switching the economy to knowledge-based industries, aimed at creating a new value. the implementation of such a program requires, in turn, the expansion and modernization of the relevant educational institutions and the creation of an advanced training system at all levels of the regional management hierarchy. in the conditions of the existence of the federal land, as an open ses, the fulfillment of all these requirements for the “new” lands of germany is quite a difficult task.to prevent the outflow of human and financial resources into the "old" federal lands, it is therefore necessary to develop and implement appropriate preferences and benefits. references [1] meljanzev, v. o a. (2005) problemy i faktory formirovanija sovremennogo (intensivnogo) ekonomnitsheskogo rosta v stranah zapada, vostoka i v rossii [problems and factors of formation of modern (intensive) economic growth in the management of the development of the region 91 copyright ©2018 assa. adv. in systems science and appl. (2018) countries of the west, the east and in russia]. moscow, ussr. history and synergetics. methodology of the study m.: urss, 2005.[in russian] [2] kleparskiy, v. o g. & scheinis v. o e. (2016) upravleniye dolgosrotchnym razvitijem regiona: investicii, tchelovetcheskiy kapital, institucionalnye struktury. management of long-term development of the region: investments, human potential, institutional structures. proc. of the 6th int. conf. mlsd’2016. moscow, 2: 67-76. [in russian]. [3] albrech, j., fink, ph. & tiemann, h. (2015) ungleiches deutschland: sozioökonomischer disparitätenbericht 2015. [the disparate germany: the researchreport for social and economy disparate 2015]. 1-56. friedrich-ebert-stiftung [in german] www.fes-2017plus.de [4] grinberg, r. o s., knogler, m. & zedilin, l.i. (2008) ekonomitcheskaya i sozialnaya politika v bavarii: uroki dlya rossiyskih regionov [economic and industrial policy in bavaria: lessons for russian regions]. moscow, russia. econ-inform, [in russian] [5] http://www.statistik.thueringen.de/datenbank/tabauswahl [6] schwaldt, n. (2015, august 14) pulsirende metropolen, veroedete doerfer. [the trob capitals, the devastated country-sides]. die welt, [in german] http://www.fes-2017plus.de/ http://www.statistik.thueringen.de/datenbank/tabauswahl advances in systems science and applications (2012) vol.12 no.2 153-161 effect of laser shock processing residual stress on pellet traveling grate surface crack x.d.ren, y.k.zhang, d.w.jiang, a.x.feng and t.zhang jiangsu key laboratory of laser manufacture science and technology ministry, jiangsu university,zhenjiang,jiangsu 212013 abstract the effect of the laser shock processing on pellet traveling grate surface fatigue crack growth performance were investigated from the theory of the fracture mechanics. a stress formula was set up to indicate the function of the residual stress on fatigue life. the effect of the compressive stresses was deemed responsible for increasing the resistance to fatigue crack growth of the pellet traveling grate. it was observed that the dynamic stress intensity factor of the crack and the stress ratio were declined due to laser shock processing, which would make the spread doorsill of the fatigue crack increasing. the results indicate a significant reduction in fatigue crack growth rates using laser shock processning specimens. it is shown that the near-surface microstructures, which in pellet traveling grate consist of a layer of work hardened nanoscale grains, play a critical role in the enhancement of fatigue life by mechanical surface treatment. keywords laser shock, residual stress, pellet traveling grate, crack expand 1 introduction in recent years, the pellet has obtained favor and been taken seriously as the high quality raw material, and has the tendency of replacing the agglomerate gradually. the chain fire grate machine is a core equipment in the production of pelletizing mineral aggregate, and as a non-sign design large-scale equipment, which core part movement chain system operates in an alternation temperature environment from normal temperature to 1050◦c for a long time. the main bottleneck question is that the alternation high temperature environment forms the complex heat expansion and the alternation thermal load, which causes the heat-resisting service life to be shorter and the pelletizing chain fire grate machine requires extremely high performances of heat shock fatigue, strength at high temperatures and wear-resisting ability of spare part material[1]. therefore, enhancing the fatigue resistance ability of the crucial element and guaranteeing the reliability of the movement are the difficult problems of the entire equipment technology. laser shock processing is to use the high power density (gw/cm2 magnitude), short pulse (ns magnitude) to impact the energy conversion body in the metal surface. after laser energy absorption, the temperature is elevated rapidly, forming the shock-wave with a high peak-to-peak value, which means that the 154 x.d.ren:effect of laser shock processing residual stress on pellet traveling... energy of light is transformed into the shock-wave mechanical energy. the shockwave pressure which reaches as highly as counts gpa causes that the microscopic plastic deformation happens in the material surface layer, and forms the residual compressive stress layer, lsp improves the mechanical property of the metallic material effectively, which would enhance the fatigue life and the anti-stress corrosion performance of the material in a large scale specially. compared with the conventional method, this kind of high rate of strain strengthening technology has a unique superiority [2-3]. regarding to the stress destruction in the surface of essential spare part axis in chain fire grate machine, this article uses the technology of laser shock processing to enhance the fatigue life and the wear resistance of the axis, and the results indicate that using advanced surface treatment technology of the shock-wave mechanics effect induced by the strong laser can effectively improve the surface layer stress condition of the axis material, and specially can enhance the fatigue life performance of the material obviously. 2 residual stress analysis in the laser shock processing, the coating raises the laser energy coupling efficiency, and the coating is able to enhance the absorption of the laser energy, which causes the absorption coating gasification to form the plasma to explode, producing the high-pressured shock-wave separately disseminates to the metal target and the restraint level. the high-pressured shock-wave plays the strengthening and the distortion roles to the metal target, and the superficial residual compressive stress is enhanced greatly. researching the fatigue short fracture growth problem in laser shock processing residual compressive stress field has also solved the problems of the crack rate of expansion caused by residual stress and life prediction[4-6]. the model of metal surface’s influence layer depth and superficial residual compressive stress after laser shock processing proposed by ballard[7] is based on in this article, and the shock wave parameters of peak pressure is optimized in condition of ideal surface stress. a residual stress model which is used to calculate the longitudinal and plane shock-wave system’s ideal elastic-plasticity response for metallic material is proposed, and some basic suppositions are made as fellow: (1) distortion induced by the laser shock processing is single axle and plane; (2) distribution of pressure pulse induced by laser is even in space; (3) the material obeys von the mises yield criterion, and the elastic deformation and the strain strengthening effect of the material are neglected. the residual stress’s production has two stages, as shown in fig.1. (a) during the time of laser pulse, a pure uniaxial stress is generated along the shock-wave propagation direction because of rapid spraying by plasma, while a tensile stress is generated in the plane parallel to the material surface ; (b) after laser pulse advances in systems science and applications (2012) vol.12 no.2 155 vanishing, the plastic strain happens to the laser impact zone’s volume, and two axle compressive stress field is produced in the plane parallel to the impact surface due to that the plastic strain is limited by the metallic material around. fig.1 schematic of the residual stress induced by the lsp according to the hooke’s law, the metal surface plasticity strain εp can be written as[8], εp = −2hel 3λ+ 2µ ( p hel − 1 ) (1) wherehel is the hugoniot limit of elasticity, p is the laser shock-wave pressure, and λ and µ are the lame’s parameters. to determine the residual stress field in the materials impact area, regarding any impact condition assigned, the plastic influence depth was calculated as, lp = celcplτ cel − cpl (p −hel 2hel ) (2) cel and cpl represent the elasticity speed and the plastic speed separately, and τ is the duration of laser pulse. and ρ represents the target material’s density, while cel and cpl can be defined as, cel = √ λ+ 2µ ρ ,cpl = √ λ+ 2µ/3 ρ (3) therefore, according to the influence level depth, the compressive residual stress produced in the elastic-plasticity semi-infinite body can be shown as, σsurf = σ0 − [ µεp 1 + ν 1− ν + σ0 ][ 1− 4 √ 2 π (1 + ν) lp r0 √ 2 ] (4) where ρ0 represents the laser facular’s radius, and σ0 represents the initial superficial residual stress which may be zero in the ordinary circumstances. the formula (4) indicates that the metal surface residual stress’s peak-to-peak value increases with the laser peak pressure in the laser impact process, while the peak pressure is related to the incident laser power density. 156 x.d.ren:effect of laser shock processing residual stress on pellet traveling... 3 laser shock experiment the laser treatment was carried out at the jiangsu university strong laser laboratorys high power and nd:glass laser implement, with its laser pulse wave is 1.06µm,and syntory laser stick is made by ϕ6mm × 80mm yttrium aluminum garnet. one stair nd:glass laser magnify beforehand and four steps main amplifier by two ways, and big caliber quarter wave piece and high strength big caliber polarize film was adopt, q switch, the output laser film wave at half maximum is about 20ns, the pulse repletion rate is 0.5hz and the pulse energy was 28j to 36j corresponding to laser energy densities at the metals surface of 56j/cm2 and 72 j/cm2. the experiment material is 12crmov, whose thickness is 5mm. before the experiment, the test sample surface was polished with the ethyl alcohol. the axis material was made into the standard test specimen. fig.2 residual stress of the staff in different conditions the depths of the impact zone indicate that after laser shock processing, its superficial residual stress is transformed from tensile stress to compressive stress, as shown in fig.2. the surface residual stress is qualitatively changed from the 135.7mpa tensile stress to 230.6mpa compressive stresses, which scope is reached as highly as 3 times. and the hardness is obviously enhanced, which has an important value to enhance fatigue life of the work piece. 4 results and discussions 4.1 material performance influence after laser shock processing, the residual stress of the 12crmov is qualitatively changed from the 135.7mpa tensile stress to 230.6mpa compressive stress, correspondingly surface hardness is enhanced from hv252 to hv530. fig.3 shows the result of metallography observed, and the big arrow shows the crack that the naked eye sees, which assumes the bending shape and branches out many small cracks; the small arrow shows the small crack which branches out from two sides advances in systems science and applications (2012) vol.12 no.2 157 of the big crack. by magnifying part of the crack that naked eye sees in fig.3, it can be seen that the small cracks along with both sides of the crack are along crystal cracks. obviously these along crystal small cracks indicate that the crack that can be seen by naked eye is also to expand along the crystal. it is only because the latter is a little wider, and its characteristic of along crystal crack is not clearer than the small cracks.fig.4 shows the microstructure of the axis test specimen by chemical reagent corrosion. it is the axis high-temperature steel normal tissue. and fig.4 demonstrates that crack is to expand along the crystal. fig.3 metallographic grinding in the surface morphology of the cracks fig.4 microstructure of the small axis 4.2 influence to crack growth after laser shock processing, residual compressive stress remains in the surface, and the residual stress will weaken in the process of test specimen surface withstanding cyclic loading, and the residual stress on the test sample section is a 158 x.d.ren:effect of laser shock processing residual stress on pellet traveling... distribution rather than a definite value. in order to analyze that simply, the effect of the residual stress was estimated with mean stress’s viewpoint, and the influence of mean stress to fatigue limit is often described with the goodman relations [9], which is shown in fig.5. in fig.5, σm is the mean stress, and σb is the corresponding tensile strength, while σ0 p is fatigue limit whenσm = 0 . as average pressure exists, the fatigue limit may be represented as. σm p = σ0 p − (σ0 p/σb)/σm = σ0 p −mσm (5) fig.5 goodman sketch map of the relationship between stress and metal fatigue limit in the formula(4.1),m = σ0 p/σb denotes the slope of the line between σ0 p and σb in fig.5, which is called the mean stress sensitive coefficient. when the residual stress σr exists and also it is thought to be equivalent with mean stress, the formula (5) can be rewritten as. σr+m p = σ0 p −m(σm + σr) (6) compare formula (5) with formula (6), it is known that the change of materials fatigue limit induced by the residual stress is. ∆σr p = σr+m p − σm p = −mσr (7) it could be seen that the materials fatigue limit would decrease if the residual tensile stress remains in test specimens surface, while the materials fatigue limit would increase when the residual compressive stress remains in test specimens surface, and m may also be called as the residual stress function coefficient. therefore, the residual stresss influence may be estimated quantificationally if advances in systems science and applications (2012) vol.12 no.2 159 the materials residual stress function coefficien m and the test samples residual stress value are known. the problem of crack growth in residual compressive stress field, besides the influence of residual stress to fatigue limit, and also influence of the residual stress to the crack growth threshold should be considered. according to the revision relations proposed by ei haddad [10] and so on, the fatigue limit of the short crack in the metal surface can be calculate as ∆σ = ∆kth y[π(r+ a0)]0.5 (8) where ∆σ is the fatigue limited stress, and ∆kth is the cracks threshold stress intensity factor, and y is the scoop channel’s shape factor, and a0 is the materials critical crack length. ∆kth = ∆σy[π(r+ a0)] 0.5 (9) after laser shock processing, the existence of the residual compressive stress in the metallic material surface greatly enhances the fatigue limit in the formula (9), and its result will cause the threshold value to obtain the enhancement, and in this condition the crack is not easily generated; the cracks that have extended will stop extending in the high residual compressive stress region and become the non-extension cracks, and a more universal influence is that the fatigue crack growth rate is reduced obviously in the residual compressive stress field. this is because that the residual compressive stress reduces the mean stress in alternating load, and reduces the alternating tensile stress that the test sample surface actual withstands. meanwhile, laser shock processing enhances the plastic deformation resistance of the sample surface hardened layer, which enhances the fatigue strength of the material. when ratio of materials fatigue strength (σr) and value of test sample surface withstanding stress (σω)σr/σω > 1, he probability of the formation of fatigue crack is greatly reduced [11]. the influence of the residual stress to the fatigue crack growth rate can be described with forman formula, da dn = c(∆k)m/[(1−r)k −∆k)] (10) in the formula (10), ∆k is the cracks threshold stress intensity factor, and r is the stress ratio, while c and m are constants. when residual compressive stress exists after laser shock processing, it can be known that the stress ratio r is smaller than the stress ratio rw when residual compressive stress doesnt exist. obviously the residual compressive stress not only could reduce the expansion speed of the fatigue cracking, but also enhance the fatigue cracking expansion resisting force of the test sample. 160 x.d.ren:effect of laser shock processing residual stress on pellet traveling... 5 conclusions the compressive residual stress is formed in the impact metal surface during laser shock processing,. the residual compressive stress is equivalent to negative average residual stress, and it can enhance the anti-fatigue strength of the workpiece. the compressive residual stress increases the locking force of the crack and reduces the expansion speed of the fatigue cracking obviously, and also can close the metal plate crack effectively.after a large number of experiments, the empirical datum is summarized and theories are consummated unceasingly, and the high rate of strain strengthening theory and the laser shock processing database are established. acknowledgements support provided by the state key program of national natural science of china (grant no. 50735001) and project supported by the natural science foundation of the jiangsu higher education institutions of china (grant no. 07kjb460012 ) and the ph.d. programs foundation of jiangsu university(cx07b-02x). references [1] zhang xiliang, zhang jian, feng ai-xin, zhu shen-cai, wu ti-chang, wan xue-gong. (2007), “reliability design on key parts of chain grate”, sintering and pelletizing, vol.32, no.4, pp.5-9. [2] ren xudong, zhang yongkang, zhou jianzhong, feng aixin, kong dejun. (2006), “study of the effect of coatings on mechanical properties of tc4 titanium alloy during laser shock processing”, materials science forum, no.532-533, pp.73-76. [3] ren xu-dong, zhang yong-kang, zhou jian-zhong, feng ai-xin. (2006), “effect of laser shock processing on residual stress and fatigue behavior of 6061t651 aluminum alloy”, transactions of nonferrous metals society of china, vol.16, pp.1305-1308. [4] li fukai. (1998), “fatigue behavior of the metal materials in remanent stress field”, xi’an university of science & technology journal, vol.18, pp.154157. [5] cui zhenqi, xu kewei, hu naisai. (1994), “model and experiments on fatigue short crack growth in residual stress field”, acta metallrugica sinica, vol.30, no.2, pp.a65-a69. [6] li hangyue, hu naisai, he jiawen, zhhou huijiu. (1998), “an analytical model of compressive residual stress effect on closure”, acta metallrugica sinica, vol.34, no.8, pp.847-851. advances in systems science and applications (2012) vol.12 no.2 161 [7] dubrujeaud b, vardavoulias m, jeandia m. (1974), “dry sliding wear behavior of a p/m ferrous alloy superficially densified by laser shock proeessing”, surface and coatings technology, vol.67, pp.125-132. [8] g.banas et al. (1990), “laser shock induced mechanical and microstructural modification of welded marging steel”, j. appl. phys, vol.67, no.5, pp.23892401. [9] zhang ding quan. (2002), “the effects of residual stresses on the fatigue strength of metal”, physical testing and chemical analysis parta physical testing, vol.38 no.6, pp.231-235. [10] ei haddad m h, topper k j, pook l p. (1974), metal fatigue, london: oxford univ. press, pp.130-195. [11] wang shuqin, wang jingyi. (1997), “effect of laser hardening treatmeng on fatigue stength of 34crnimo steel”, ordnance material science and engineering, vol.20 no.4, pp.29-34. corresponding author corresponding author: renxd@ujs.edu.cn advances in systems science and application (2015) vol.15 no.4 338-350 simulation modelling for computer aided design of secondary aerodynamic wing surfaces aleksandr a. gorbunov, aleksej d. pripadchev, irina s. bykova and valerij v. elagin federal state educational government-financed institution of higher professional education, orenburg state university, orenburg region, russia abstract the simulation modelling technique of secondary aerodynamic wing surface of the long-haul aircraft using a high-precision mathematical simulation in the aerodynamics and hydrodynamics computer programme has been formulated in the present article. the method proposed is based on the modern methods in the field of computer-aided design of aircraft using catia three-dimensional simulation and simulation modelling in salome environment. the use of highprecision mathematical simulation in computer-aided design of secondary aerodynamic wing surfaces allows us to determine the aerodynamic characteristics of the secondary aerodynamic surface of the model developed and to verify the previous results of the project definition. keywords secondary aerodynamic surfaces; high-precision computer mathematical simulation; simulation model; a long-haul aircraft; synthesis; design automation; project procedures; three-dimensional modelling 1 introduction building the new long-haul aircraft (a/c) with improved performance characteristics and enhancing the currently used aircraft is effected in various ways. one of them is improving its aerodynamics depending largely on the a/c look. in our view, the most rational way to improve the aerodynamic characteristics of the a/c is to install the secondary aerodynamic surfaces (sas) at the wingtip. the use of sas reduces the induced drag of the aircraft, enhances the effective wing aspect-ratio and ascensional power at the wingtip, improves a/c longitudinal/transverse stability, reduces specific fuel consumption, cuts takeoff run and landing roll of the aircraft. currently, there are many wing sas designs installed at the long-haul a/c differing in geometrical and aerodynamic characteristics [1-3]. however, despite years of experience based on the empirical approach, there is no uniform method and engineering tools of computer-aided design of sas both individually and as a part of the wing in the available scientific matter. in this regard, development of the scientific foundations and the method of computer-aided designing of sas wing as a component of the simulation modelling technique is a priority task. sas designing involves numerous aerodynamic, energetic, structural and geadvances in systems science and application (2015) vol.15 no.4 339 ometrical, technological and operating characteristics. thus, modern computer technologies are to be used for the analysis, synthesis and appropriate design solutions. in this connection, the design and construction of sas is reasonable to do using a high-precision mathematical simulation in the computer programs of computational aeroand fluid dynamics (cfd). 2 methods the purpose of this research is to develop methods of simulation modelling of the designed sas of the wing of the long-haul aircraft using the latest methods of high-precision mathematical simulation and engineering analysis. the object of the research is simulation modelling of aircraft sas of the wing within the proposed method of simulation. the subject of the research is the methodology of high-precision mathematical simulation of aircraft sas wing. the methodological provision of the study is in using the latest methods of mathematical simulation in the computer programs of computational aeroand fluid dynamics. due to the fact that designing and construction of the sas wing is a complex and modifying with time process producing a large number of iterations, there is a necessity to describe rationally the design process with simulation and physical models. we shall consider the process of simulation modelling of the secondary aerodynamic wing surface of the long-haul aircraft. simulation modelling means a method of a high-precision mathematical simulation in the computational aeroand hydrodynamics programme [4-6]. currently, to solve cfd problems, several software products can be used, for example: cosmosfloworks; nafems efd. lab; efd.v5 forcatiav5; salome; ansys. we propose salome product to be used. it is intended for solving problems of computational aeroand hydrodynamics. this is an open integrated software platform for numerical computations and simulation modelling. the salome user interface of the computational aeroand hydrodynamics provides the following options: it sets the master data and displays the results immediately in the graphic design window; it has a user friendly interface; it takes minimal time to prepare the data and view the results of the experiment. cad used is a finite-element pre-post processor, being the computational environment kernel surrounded by the united multitude of cae solvers. with 340 simulation modelling for computer aided design of secondary aerodynamic wing surfaces... salome, one can also develop proprietory software solutions, which has been done by us, program for investigation and calculation of aircraft aerodynamic characteristics being the resulting product [7]. this programme based on cae solver open foam (liquids and gases flow analysis) helped us to carry out simulation of sas of the long-haul aircraft. the revised open foam package uses parasolid geometric simulation kernel which is a platform for the latest cad such as: ansys, icem-cfd, femap, solid works having full technical support. 2.1 discussion development of the wing sas 3d model was performed in catia high-level cad system. use of catia cad is conditioned by the fact that the system possesses a hybrid design function, combining both surface and solid elements in a single model. another important thing is the fact that the system has a function of free parameterization and constructs models with the help of the earlier created drawings, fig.1. fig.1 general view of sas, patent no.2481242 ni the number of working surfaces; χ1 leading edge sweep angle of the upper surface; χ2 leading edge sweep angle of the lower surface; bk tip chord advances in systems science and application (2015) vol.15 no.4 341 of the upper surface; b0 root chord of the upper surface; bk tip chord of the lower surface; b0 root chord of the lower surface; γ angle between the working surfaces; α installation angle end plate relative to the end rib axis; βleading edge sweep angle of the end plate. based on the previously developed sas drawing, its 3d model was created, fig.2. the 3d model developed was converted into step format to support to carry out simulation modelling in cad salome programme. step format makes it possible to describe the aerodynamic surfaces obtained using mathematical model of cad programme. step data format selection was done according to the requirements of the software used for simulation modelling. fig.2 3d sas model of the wing in catia system the figure shows that all the coupling parts of sas, namely the interface between the working surfaces have smooth fairings with the given radii. obtaining accurate aerodynamic shapes has become possible owing to the high-level catia cad system. in the dialogue box, a sas 3d model is mounted on the main wing of the long-haul aircraft on the left side in the flight direction [8]. the problem of computational aeroand hydrodynamics being solved leads to the analysis of the impact of air on the body. to make calculations it is necessary to set the initial data in the form of the initial and boundary conditions. the 342 simulation modelling for computer aided design of secondary aerodynamic wing surfaces... body is restrained by the surfaces which limit the range of the fluid. such conditions are called a wall. due to the fact that for sas the case of external flow is used, which is characterized by such parameters as speed, pressure, temperature, etc., this type is called ambient conditions (environmental conditions). in accordance with the specified conditions, the external type of task will be solved. as a result, the design area is limited on one side by the field of the set boundaries in the form of the walls forming a rectangular hollow parallelepiped and solid sas model, placed in a hollow parallelepiped. the size of the design area is set either automatically or manually by the user. in this case, the design grid had been done by the user, which allowed us to obtain a more accurate mathematical problem solution by means of a smaller calculation grid in the design area, fig.3. the design area is divided by the grid with a different step: a smaller one immediately in the design area for more accurate solutions, further with a larger step as the high accuracy of the calculation is not required for the remaining area. the design area construction method used results in the efficient use of the computation resources leading to the significant reduction of the computation time [9, 10]. fig.3 shows the view of a 3d sas model on the right located in the design area. fig.3 design area thus, the design area shown in fig.3 is a spatial cube, which contains a 3d sas model of the wing of the long-haul aircraft, in what connection, boundary conditions wall in the form of four faces on axes oy and ox are set around the model, and the initial conditions graphically displayed in the form of two faces on axis oz are set at the front and at the back of the 3d model. the boundary advances in systems science and application (2015) vol.15 no.4 343 and initial conditions are set in accordance with the conditions, which sas in natural scale will be found under. the initial condition in the form of the face, located in front of the sas in the program is referred to as (inlet), and a face located behind the sas is called (outlet) i.e. an output parameter. setting inlet and outlet conditions, we thereby define the mode of the fluid flow in the design area. to eliminate the influence of the boundary conditions wall on the model sas to neutralize vortex formation around wall faces, the movement of the flow rate in the direction of the fluid has been set. the parameters of the fluid are set by the following initial data given in table 1. in addition to the parameters listed in table 1 and initial conditions, we set the table 1 initial data for the software experiment implementation item no name unit of measurement value 1 name air 2 density kg/v3 1.205 3 dynamic viscosity pa·s 0.000019137 4 specific heat capacity l/kg· 1.006 5 kinematic viscosity m2/s 0.0000158813 6 laminar number 0.9 7 turbulent prandtl number 0.85 8 thermal conductivity w/m· 0.024 9 reference (absolute) pressure pa 101.325 10 thermal expansion coefficient k1 0.00333 11 reference temperature k 300 turbulence model which suits in the best way to the current design case and select the model of turbulence (k-e), where the calculation of two additional equations for kinetic energy turbulence transport and turbulence dissipation transport. other turbulence models implemented in the program can be used, but in this design case it is advisable to use the very (k-e) model [11]. all further calculations will be performed for the case when the body is at rest while the medium is moving. the selected design case is applicable for finding ascensional power, drag force, induced drag and fluid force acting on the body located therein. the problem of computational aeroand hydrodynamics for sas wing being solved is made to study the movement of air around the body. solving the problem using cad gives us the parameters describing the fluid flow, namely: speed, pressure, temperature, etc. simulation modelling of sas of the a/c wing in cfd in salome programme has been performed within the time equal to 200 calculation steps. the number of the calculated steps is set by the user and selected in the view of the following consideration: to obtain the sufficient experimental data under the steady flow process. 344 simulation modelling for computer aided design of secondary aerodynamic wing surfaces... eventually, repeatedly conducted purges in cfd programme allowed us to establish that the steady flow process starts with the 65th design step when the interference processes become steady, and the number of iterations of 200 design steps selected from the obtained information volume of 250 gb. a larger volume of information is not required, since the available data is sufficient for making design decisions. fig.4 fluid distribution rate change over sas fig.5 pressure distribution in sas section the results of simulation modelling presented further feature the same moment of time corresponding to design step no.180 in various design sections. fig.4 shows sas of the wing with the air distribution rate change in the form of relief advances in systems science and application (2015) vol.15 no.4 345 growths that change their colour with the distribution rate over sas. flow distribution value is found using the graduated scale changing its colour from red to blue and graph curves shown below. the field of low rates is marked with the red colour and the field of high rates is marked with the blue colour. fig.6 pressure distribution in the wingtip section. fig.7 pressure distribution over sas of the wing fig.5 shows sas section using which we can find the change in pressure over the sas. this section clearly shows pressure rise at the leading edge of sas and also a marked increase at the trailing edge, and further decrease due to the distance from the model owing to air circulation. fig.6 presents a section of the wingtip, which shows pressure distribution over 346 simulation modelling for computer aided design of secondary aerodynamic wing surfaces... the surface of the wing airfoil located close to sas. fig.7 is a 3d sas model showing the pressure distribution fields and the set cutting plane at the wingtip shown by the section in fig.6. the pressure distribution over sas makes it possible to trace the character of pressure change over time. fig.8 shows the enlarged section of the wingtip close to sas displaying air particles vector movement and rate distribution fields. fig.8 pressure distribution over sas of the wing a graphic curve in the form of the bar graph in fig.9 shows the change in the reynolds number (re) and kinematic speed over sas conforming to fig.4. fig.9 graphic curve of re number change and kinematic speed over sas advances in systems science and application (2015) vol.15 no.4 347 fig.10 a curve of a temperature change, kinematic speed and pressure change over sas (a) (b) (c) fig.11 secondary aerodynamic surface, patent no. 2481242 348 simulation modelling for computer aided design of secondary aerodynamic wing surfaces... fig.10 shows a curve describing air parameters, namely, a temperature change, kinematic speed and pressure change over sas conforming to fig.4. simulation modelling for sas wing made it possible to find the aerodynamic characteristics of the designed sas and to confirm flow processes around sas. let us consider the flow for sas wing, patent no.2481242. while the air is flowing the wing, the air overflows from the lower plane of the wing to the upper one, at the same time, at the endplate 1 supplied with the additional wind swept surface 2 of low aspect with a sharp front edge 3, mounted on the outer side of the endplate 1, there formed a vertical chamfering field transforming into a stable vortex flow with a conical vortex formation on the leading edge 3 of the secondary aerodynamic surface 2 mounted on the endplate 1, fig 11. at lower vertical aerodynamic surface 8, the vertical chamfering field due to the low aspect the lower surface, does not result in the early formation of the vortex at the leading edge 9 but it is transformed at the surface end into an end conical vortex. 3 results the sas proposed improves aerodynamic efficiency of the long-haul a/c and provides a maximum effect from air overflowing throughout the entire field of the effective values [12, 13]. in this way, it makes it possible to enlarge the effective wingspan reducing the induced drag generated by the vortex flying from the sweptwing tip and, consequently, increasing the ascensional power at the wingtip; to increase the effective aspect of the wing, without changing its span; to improve fuel efficiency of the a/c and the flying range. 4 conclusions 1. high-precision computer mathematical simulation and simulation modelling techniques in the analysis of the flow process around the designed sas model provides a means to find out the aerodynamic characteristics for given initial and final conditions affecting the induced drag of the aircraft and ascensional power generated by the system of lifting surface areas of the a/c. 2. using a simulation modelling technique can be used for complex multivariate iterative calculations of the new sas designs, high quality design solutions being secured due to the accumulation of output data followed by statistical analysis with the time from the previous to the next point in time set with the predefined step. 3. the behaviour of the flow processes around the sas model designed identified as a result of the high-precision mathematical simulation modeling enables us to confirm or deny (in the case of unsound designing) aerodynamic characteristics obtained at the preceding stages and to develop physical sas models. advances in systems science and application (2015) vol.15 no.4 349 acknowledgement the project has been performed under contract no.14.z56.15.5527-mk of february 16, 2015, grant of the president of the russian federation for the state support of young russian scientists to carry out research on the subject of “computer-aided design of secondary aerodynamic wing surfaces as an element of the aircraft life cycle”. the outcomes of the scientific research have been accepted for implementation in jsc. kapo tupolev (kazan). references [1] v.a. komarov et al (2013), conceptual design of the aircraft, house of samara state aerospace university, samara. pp.120. 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[8] a.a. gorbunov,a.d. pripadchev (rf).(2013), “aircraft wingtip”, patent ru no. 2481242, ecc 643/10.no. 2011148436. [9] jeppe johansen and niels n. sorensen, (2006), “aerodynamic investigation of winglets on wind turbine blades using cfd”, roskilde, denmark: riso r-1543. [10] hepperle, m.(2005),“aerodynamic optimisation of a flying wing transport aircraft”, new results in numerical and experimental fluid mechanics v 350 simulation modelling for computer aided design of secondary aerodynamic wing surfaces... (notes on numerical fluid mechanics and multidisciplinary design), vol.92, pp.69-76 [11] g. j. kennedy.(2012), “aerostructural analysis and design optimization of composite aircraft”, phd thesis, university of toronto. [12] schiktanz, daniel.(2011), “conceptual design of a medium range box wing aircraft”. department fahrzeug technik und flugzeugbau, master thesis, hamburg, haw hamburg. [13] joseph katz, allen plotkin.(1991), “from wing theory to panel methods”, low speed aerodynamics, singapore, pp.632. corresponding author tatiana can be contacted at:yal05@mail.ru adv sist sci appl 2017; 17(2); 52-62 published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/253 copyright ©2017 assa. adv. in systems science and appl. (2017) a system approach to the regional sustainable management gennady a. ougolnitsky southern federal university, rostov-on-don, russia e-mail: ougoln@mail.ru abstract. a system approach to the regional sustainable management based on game theoretic models and information technologies is considered. a notion of the regional active system is specified. the author's approach uses hierarchical dynamical game theoretic models formalizing and integrating many different aspects of the control in active systems. a concept of the regional sustainable development is added by the requirement of system compatibility. sustainable development means that a human activity should provide sufficiently high values of indicators of the social-economic development, and in the same time sustain an environmental equilibrium. formally, all indicators of the state of a regional socio-ecological-economic system must belong to a given range. in turn, system compatibility means that interests of the active agents should be considered by the center in a maximal degree for the achievement of the center's purposes. for a quantitative measurement of the system compatibility based on the proposed model an index of system compatibility is used. administrative and economic control mechanisms providing the system compatibility are described. on the regional level the problem of coordination of interests has the following specific features. first, a regional administration which should be a leader of sd, in a higher degree is interested in the approval from the federal center that estimates the regional activity by other criteria. second, the principal control levers, both legislative and economic, are also situated in the federal level. third, the municipal districts that should provide a practical implementation of the requirements of homeostasis, have not enough resources for it. a hierarchical structure of the model system in this domain is considered. a practical implementation of the methodology of sustainable management on the regional level requires a regional information-analytical system, the structure of which is proposed. key words: computer simulation, game theoretic modeling, homeostasis, regional development, sustainable management, system approach, system compatibility. 1. introduction system approach and its applications were extremely popular in 1960-1980s [25]. since that time up to the moment this approach has been developed not so actively though the system methodology is not completely exhausted [6]. it should be noticed that a system, complex approach was always used by russian scientists starting from michael lomonosov. vassily dokuchaev, kliment timiryazev, dmitry mendeleev, konstantin tsiolkovsky, alexander chizhevsky, alexander bogdanov and especially vladimir vernadsky belong to that glorious cohort. a great role in the system research based on mathematical modeling was played by works of andrey kolmogorov, leonid kantorovich, andrey lyapunov, nikita moisseev, alexander samarsky, sergey kurdyumov and many others. an essential contribution to the system methodology was made by the notion of active system proposed by vladimir burkov and his school in the institute of control sciences [5]. this notion formalizes relations in active systems of different nature including human beings (organizational, economic, social systems). an active system consists of active elements (agents) having their own objectives and interests and possibilities to achieve them. as a rule, one of the agents is separated as the center reflecting the objectives and interests of the whole system. the interests of the agents and the center do not coincide but are not also antagonistic that allows for some trade-off solutions acceptable for everybody. 53 g. a. ougolnitsky copyright ©2017 assa. adv. in systems science and appl. (2017) an informational interaction of the agents with the center and each other may include a strategic manipulation (a deliberate misrepresentation of the reported information in private purposes). the main problem of the theory of active systems is a concordance of the interests of the center (the whole system) and the active agents as elements of the system. for this purpose special administrative, economic, and informational control mechanisms are used [17]. close ideas are developed in the information theory of hierarchical systems [8], theory of incentives and mechanism design [1,12]. a widely spread since 1980s concept of sustainable development has also a system character. thus, a key approach of "three pillars" in the analysis and implementation of the sustainable development requires to consider social, economic, and environmental factors in their totality as a whole system. it should be also noticed an important idea that a global sustainable development is formed by the sum of local efforts in this direction [21]. an important class of active systems is generated by territorial formations. here the problem of coordination of private and public interests is also of the main importance and may be formulated in terms of state regions, regions municipal districts, municipal districts economic agents depending on the level of consideration. in russia this problem was especially essential in 1990s in the connection with the development of federalism, and it is still actual [15,16]. the author's approach to the analysis of regional active systems is based on the general complex theory of sustainable management in active systems [20]. the approach uses hierarchical dynamical game theoretic models formalizing and integrating all the named aspects. the complexity of these models and big data volumes make necessary an application of modern information technologies of the research support. the regional management is an example of the possible applications of the developed universal theory. the rest of the paper is organized as follows. in the section 2 a notion of a regional active system is considered. a specified concept of the regional sustainable development with consideration of the condition of system compatibility is proposed in the section 3. in the section 4 mechanisms of system compatibility and monitoring on the regional level are described. in the section 5 a hierarchical structure of the model system of the sustainable management is considered. section 6 concludes. 2. a regional active system a general structure of the regional active system is presented in fig.1. fig.1 a regional active system a system approach to the regional sustainable management 54 copyright ©2017 assa. adv. in systems science and appl. (2017) on the higher control level a regional administration (ra, center) is situated. the middle control level is presented by municipal districts (md). a more complicated variant is also possible when urban municipal formations and their districts are considered separately. the lower control level is formed by local active agents (la): enterprises, organizations, firms, individual entrepreneurs. the control object is the regional socio-ecological-economic system considered as a hierarchically controlled dynamic system [20]. in the majority of cases la are economic agents having certain financial, human, and other resources and tending to maximize their income (profit). however, non-commercial organizations and local control agencies can also be considered as la having non-economic objectives. economic la can have social and ecological objectives too. mds solve the problems of social-economic development of the respective territories with consideration of the environmental requirements using certain budget resources [15]. the objectives and possibilities of ra are structurally similar to the ones of md but differ from them by the volume and sources of financing [16]. ra can establish its own laws, and have additional means of influence to md and la. a control object is characterized by three groups of indicators describing the social sphere, economic potential, and environmental state respectively. for determination of the specific values of the indicators the data of official statistics are used which may be complemented by expert estimates. notice that only las exert a direct influence to the regional socio-ecologicaleconomic system. the higher control levels just create some administrative, legislative, and economic framing conditions of the influence. the laws of change of the state indicators due to natural factors and la's impact are described by dynamic balance relations. the left hand side contains a new value of the indicator, the right hand side contains an old one plus increase minus decrease generated by the considered processes. it is more appropriate to set the problems of sustainable management in an infinite period of time. however, a finite, long enough period may be more acceptable in practice. some discount rate should reflect a relation between incomes and expenses in different moments of time. most typical political technologies of regional elites include:  a concentration of financial resources;  a power pressure to the opponents;  an establishment of control and strategic partnership with the most powerful regional economic agents;  using of information flows from the federal center for the protection of the elites' interests;  negotiation process with the political leaders;  a direct or indirect control on the information flows and media, an active construction of a positive image of the regional administration by media;  a formation of the regional mythology;  a harder selection of the new candidates to the regional elites [13]. technically, the administrative controls of ra and md include legislative restrictions of the activity of subordinated control agents (norms, quotas, quality standards). the economic controls represent tax rates, parameters of privileges, subsidies, grants. the laws of state dynamics are determined by natural processes and social-economic activity. for their description specific models of the mathematical ecology and economics, and dynamic forecasting models of the regional level are used [18]. the respective three-level game theoretic model of regional control has the form     t t ttttttt xuqpsrgej 1 00 max),,,,,( (1) ;; ssrr tt  (2) 55 g. a. ougolnitsky copyright ©2017 assa. adv. in systems science and appl. (2017)     t t tt i t i t i t ii t i xuqprgej 1 max),,,,( (3) );();( t ii t i t ii t i sqqrpp  (4)     t t tt i t ij t iij t ij xuprgej 1 max),,,( (5) );,( t ij t iij t ij qsuu  ;,...,1;,...1 nikj i  (6) ;),,( 0 01 xxuxfxx tttt  (7) );,...,();,...,( 11 t n ttt n tt sssrrr  );,...,();,...,();,...,();,...,( 1111 t ik t i t i t ik t i t i t n ttt n tt ii qqqpppqqqppp  ),...,,...,,...,(),...,( 11111 1 t nk t n t k tt n tt n uuuuuuu  . here t is a period of consideration (in years);  is a discount rate; iji jjj ,,0 and iji ggg ,,0 summary and current payoff functions of ra, mds, and las respectively; tt sr , are economic and administrative controls of ra in the year t; sr, are the respective sets of feasible controls; t i t i qp , are economic and administrative controls of the i-th md in the year t; ii qp , are the respective sets of feasible controls; t iju is a control of the ij-th la in the year t; iju is the respective set of feasible controls; tx is a state vector of the regional socio-ecological-economic system in the year t; 0x a vector of initial values of the state indicators on a base year; f a set of models that describe the state dynamics; n a number of mds; ik a number of las in the ith md. the model (1)-(7) represents a hierarchical differential game, a discrete form of which is oriented to simulation analysis based on the scenario method [14]. solutions of the game (1)-(7) are understood in the sense of stackelberg [3]. as a rule, the ra chooses only ts while tr is fixed (administrative control, or compulsion) or vice versa (economic control, or impulsion). similarly, given ts or tr the mdi choose t iq while t ip are fixed or vice versa. at last, given t iq and t ip as well as ts and tr the laij choose their control parameters t iju . in fact, simulation modeling is almost unique possibility to solve the game (1)-(7) in the general setup. some simplified versions may be investigated by means of standard techniques like hamilton-jacobibellman equations or pontryagin' s maximum principle together with numerical methods. 3. sustainable development a serious attention to the problem of sustainable development (sd) was attracted by the report “our common future” (1987) prepared by the un world commission on environment and development, known as “brundtland commission”. conclusions of the brundtland report formed a base for the decisions of the un conference on environment and development (rio de janeiro, 1992). the brundtland report defines sustainable development as the development which satisfies the needs of present generation and does not undermine the possibility of future generations to satisfy their needs [21, p.43]. the declaration on environment and development accepted at the rio-92 conference includes 27 principles, such as principle 3 “a right to the development should be realized to provide a just satisfaction of the needs of the present and future generations in the domains of development and environment” and principle 4 “to provide the sustainable development an environmental protection should be an integral part of the development process and cannot be considered without it”. a system approach to the regional sustainable management 56 copyright ©2017 assa. adv. in systems science and appl. (2017) in spite of the active discussion of the concept of sustainable development and a number of accepted official documents, a unity in the definition and interpretation of the notion is still absent. already in the early work [23] more than 60 definitions of sustainable development given by different authors are cited. the general transformations of the environmental scene are shown in table 1 [26, p.84]. table 1 evolution of the views on sd in 1970 2000s environmental policies 1970-1980s 2010 iconic policy instrument command and control collaborative and market-based instruments key group relied upon government government and stakeholders dominant mode of action work with industry, mostly through technology work with industry and consumers, through technology, economy, finance knowledge about the environment superficial, or limited to specialists extensive, diffused in many realms of society discourse on green issues view of green as important (before 1973) and then as rather marginal acknowledgment of the seriousness of some problems: a trend of “going green” and resulting business opportunities social issues superficial or neglected concerns concerns for environmental justice and the influence of environmental degradation on poverty based on the initial notion of sd of the society it is possible to say about sd of any active system [20]. according to the existing approach, the main role in sd is played by the notion of homeostasis. it means that a human activity should provide sufficiently high values of indicators of the social-economic development, and in the same time sustain an environmental equilibrium. formally, all indicators of the state of a regional socio-ecological-economic system must belong to a given range that can be written as the condition *,...,1 xxtt t  . (8) a stronger formulation is also possible when the condition of homeostasis is treated as an asymptotic approximation of the values of indicators to their ideal values [20]. it seems that the main problem of the practical implementation of the concept of sd consists in the absence of interested and plenipotentiary agents of the implementation. the criteria used by acting politicians not always coincide with the conditions of homeostasis of the territories governed by them. that's why the concepts of transition to sd accepted by many countries and regions remain only as declarations in a great part. it should be noticed in the same time that from an objective point of view all politicians who wish to enlist the support of the population and secure the power for a long time, should be interested in the conditions of homeostasis. it concerns also the owners of companies thinking about conservation of their family business for many generations. on the level of a firm the considered problem is tightly connected with strategic delegation (see [24]), i.e. sharing of the powers between owners, topmanagers, and managers of lower levels. as for territories, the problem is in the distribution of resources and powers between federal, regional, and local agents of power and control (the problem of federalism). it is especially important that the values of sd be shared by active agents of the lower level whose activity has a direct impact to the state of a controlled system. that's why the concept of homeostasis, playing a key role in the traditional approach to sd, should necessarily be complemented by a requirement of system compatibility [2,7]. it means that interests of the 57 g. a. ougolnitsky copyright ©2017 assa. adv. in systems science and appl. (2017) active agents should be considered by the center in a maximal degree for the achievement of the center's purposes. for a quantitative measurement of the system compatibility based on the model (1) (7) it is expedient to use an index of system compatibility in the form * 0 max 0 jjsci  , (9) where );,,,,,(maxmaxmax 0 ),( )( )( max 0 xuqpsrjj qpuu sqq rpp ss rr       (10) ),,,,,(minminmax 0 ),(),(, * 0 xuqpsrjj qpneusrneqp ss rr     . (11) the substrahend in the formula (9) gives a guaranteed payoff of the center in the worst nash equilibrium in the game of agents regulated by her, and the minuend shows her payoff in the case of complete cooperation of the agents with the center (globally maximal value of the payoff in the team solution). the value 0sci indicates the complete (ideal) system compatibility. this approach generalizes an idea of the price of anarchy proposed for network games [22], and its development in other works [4]. an indicator of the ideal compatibility is proposed in [8] as well. given (9) a requirement of the system compatibility is expressed by the condition *scisci  , (12) where *sci is an expert threshold value of the system compatibility index. the condition (12) may be used additionally after the solution of the game (1)-(7) with phase constraints (8) to check whether the system compatibility holds. the relations (1)-(12) determine a holistic formal description of the problem of sustainable management in any active system [19]. on the regional level the problem of coordination of interests has the following specific features. first, a regional administration which should be a leader of sd, in a higher degree is interested in the approval from the federal center that estimates the regional activity by other criteria. second, the principal control levers, both legislative and economic, are also situated in the federal level. third, the municipal districts that should provide a practical implementation of the requirements of homeostasis, have not enough resources for it. thus, the problem of coordination of interests on the regional level has a special importance and deserves a detailed investigation by means of systems analysis, mathematical modeling, and information technologies. 4. control mechanisms in the case of a low system compatibility ( 0sci ) the center should use special control mechanisms to provide it. to develop a classification of the control mechanisms three attributes characterizing the center's strategy can be used: 1) absence/presence of a feedback of the center's strategy on the state of a controlled dynamic system (cds). this attribute has two basic values: open-loop strategies (ol) which depend only on the instant of time t , and closed-loop strategies (cl) which depend on the game position ))(,( txt [3]; 2) absence/presence of a feedback of the center's strategy on the agents' strategies. in the first case we deal with stackelberg games (st), and games of the second type we propose to call germeier games (ger), or incentive stackelberg games [9-11]; 3) methods of hierarchical control. here we differentiate compulsion, when the leader influences the followers' sets of feasible strategies, and impulsion, when she influences the followers' payoff functionals [20]. so, in the case of compulsion the leader chooses ts in the a system approach to the regional sustainable management 58 copyright ©2017 assa. adv. in systems science and appl. (2017) higher control level and tq in the middle control level while impulsion means the respective choice of tr and tp . a classification of the control mechanisms on example of the russian traffic rules (rtr) is given in table 2. here an illustrative active system is considered. in this system the center is the federal state that establishes rtr and the penalties for their violation exposed in the administrative codex (ac) and the criminal codex (cc); an active agent is a driver and his vehicle; a controlled object is a road and its environment. the respective clauses of the legislative documents are presented in brackets. administrative control mechanisms explicitly forbid some driver's actions, and economic ones charge penalties for their violation. from the mathematical point of view, a type of the center's control mechanism determines an information structure of the difference game (1) (7). for example, а/st/cl is a stackelberg game in closed-loop strategies in which the center restricts the agents' sets of feasible strategies. the solutions of respective differential games are defined in [20]. table 2 classification of the control mechanisms on example of the russian traffic rules st/ol st/cl ger/ol ger/cl administrative mechanisms it is forbidden to injure or pollute a road cover (1.5 rtr) in populated areas it is allowed to drive a vehicle with the speed not more than 60 km/h (10.2 rtr) it is forbidden to drive a vehicle with a working brake system disrepair (2.3.1 rtr) it is forbidden to drive a vehicle with not burning (absent) head-lights and back marker lights in the dark period (2.3.1 rtr) economic mechanisms absence of an insurance policy 800 roubles (12.37 p.2 ac) exceeding of the allowed speed: more than 20, but not more than 40 km/h 500 r. (12.9 p.2 ac); more than 40, but not more than 60 km/h 1000-1500 r. (12.9 p.3 ac) driving a vehicle by a driver in the state of intoxication 30000 r. (12.8 p.3 ac), repeatedly 200000 300000 r. (cc 264) non-compliance of the requirement to stop before the stop line: first time 800 r. (12.12. p.2 ac); repeatedly 5000 r. (12.12 p.3 ac) a very important role in the practical implementation of the control mechanisms is played by the monitoring system that provides to the control agents information about the state of the object and therefore ensures a feedback. a general form of the monitoring in a simplified hierarchically controlled dynamic system with one agent is shown in fig. 2 [20]. here slx is a center's information about the state of cds; fslx is a center's information about the impact of the agent on cds; flx is a center's information about the agent's state; sfx is the agent's information about the state of cds. in conformity with a regional active system the presented model can be used both on the level ra md, and on the level md la (see fig. 1). in the former case cds is treated as a socio-ecological-economic system of the respective md, and in the latter one as a part of that system controlled by the given la. a union of those systems forms a regional monitoring system, the visual presentation of which is huge. the information is collected by means of official statistics which can be complemented by special sample audits and expert estimates. 59 g. a. ougolnitsky copyright ©2017 assa. adv. in systems science and appl. (2017) fig.2 monitoring in a hierarchically controlled dynamic system (hcds) 5. hierarchical structure of the model system the regional level has an intermediate position in the structure of state control (fig. 3). here fc corresponds to ra in fig. 1, rai corresponds to mdi, and mdij corresponds to laij. the dynamics of a regional socio-ecological-economic system is described by a system of nonlinear equations (7), and the interests of agents and their possibilities by the relations (1)-(6). the respective system connections are presented in table 3. fig.3 hierarchical structure of the model system table 3 connections of the regional control level with federal and local levels inputs outputs fc federal laws and instructions informal directions of the federal power federal budget financing tax revenues to the federal budget information about the state of the socioecological-economic system of a region (indicators of the official statistics) md tax revenues to the regional budget information about the state of the socioecological-economic system of a region by separate md regional laws and instructions informal directions of the regional administration regional financing of the local budgets a system approach to the regional sustainable management 60 copyright ©2017 assa. adv. in systems science and appl. (2017) a practical implementation of the methodology of sustainable management on the regional level requires a regional information-analytical system, the structure of which is shown in fig.4. fig. 4 structure of the regional information-analytical sustainable management support system here the optimization subsystem includes mathematical models like the game (1)-(7) while the simulation subsystem allows for their solution. the expert subsystem additionally uses an expert knowledge. 6. conclusion the system approach plays an important methodological role in the solution of difficult complex interdisciplinary problems of the regional sustainable management. a specific character of the regional level of control consists in its intermediate, dual nature. from one side, a regional administration plays a role of the center relating to the municipal formations within the region, from the other side it is itself an active agent in the interrelation of the regions with their federal center. in spite of it, ra can and must be an agent of the sustainable management. it follows from the hierarchical nature of sustainable development, the achievement of which on the global level is provided by the totality of efforts of all agents on the lower control levels. this fact determines a necessity of the delegation of an essential volume of administrative and economic powers on the ra level. it should be noticed that the need of solution of the problems of sustainable management is not always realized by the agent, and the mission of science and public opinion formed by the science is of great importance here. a key role in the solution of the sustainable management problems is played by the coordination of interests of the center as the subject of sustainable development and the active agents who implement it directly. the requirements of homeostasis will not be satisfied by themselves unless specific agents be interested in it. in the great majority of cases the interests of active agents contain only maximization of their economic payoff in short periods of time. that's why it is necessary to design and implement administrative and economic control mechanisms that incorporate the conditions of homeostasis in the agents' payoff functions and feasible sets of strategies. later it is planned to analyze horizontal connections in a regional active system that leads to the cooperative games as conflict control models and investigation of the time consistency of their solutions. it is supposed particularly to apply the described methodology for decision support of the sustainable development of the rostov region (russian federation). 61 g. a. ougolnitsky copyright ©2017 assa. adv. in systems science and appl. 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(2012). over two decades in pursuit of sustainable development: influence, transformations, limits. environmental development, 1, 79-90. advances in systems science and applications (2017) vol.17 no.1 9 regional security: analysis of the emergency management effectiveness based on the scenario approach vladimir l. schultz a , vladimir v. kulba 1b, oleg a. zaikin c, alexei b. shelkov b, igor v. chernov b a) institute of socio-political research of the russian academy of sciences, 119333, fotievoi street, 6, building 1, moscow, russia. b) v.a.trapeznikov institute of control sciences of the russian academy of sciences, 117997, profsoyuznaya street, 65, moscow, russia. c) warsaw school of computer science, 00-169, lewartovskiego street, 19, warsaw, poland. abstract. the paper analyses the effectiveness of the scenario analysis and modeling methods for solving the planning and operational control problems of in the processes of prevention and elimination of the causes and effects of man-made disasters, industrial catastrophes and emergencies. the scenario approach belongs to a class of object-oriented methods used to represent information about the state of a plant and an environment, which is necessary to generate control actions for the rescue measures and activities, as well as disaster management. the main feature discussed in this paper is a class of industrial safety management tasks based on the scenario approach which have to be developed as well as the general normative framework that allows to fundamentally change the approach to building models of an emergency. the research shows that in this case the procedure of forming the base model using a comprehensive analysis of existing technical regulations, norms and standards were quite effective. the results of modeling and scenario analysis of the processes, elimination of technological accidents consequences on transport infrastructure of metropolises are presented in the paper. key words: management, scenario analysis, man-made disaster, emergency, simulation, signed graphs. 1. introduction in the last decades, the world has witnessed a stable trend of substantial growth of material losses as a result of industrial catastrophes, emergencies and natural disasters. one of the main reasons for this, apart from global climate changes accelerating in the coming century and the saliently manifested synergetic nature of many industrial catastrophes, is the lack of willingness of warning management systems to the prompt, adequate and effective response to such events. the main peculiarities of functioning of management processes in conditions of industrial disasters consist in the fact that an emergency (disaster) arises and develops suddenly and unexpectedly. since its appearance in the management system poses the problems which essentially aren’t inherent for the stationary mode of operation. thus, that is especially important that the measures to counter the emergency development and elimination of its consequences should be taken without delay and been as efficient as possible. at the same time the fundamentally new problems arise in management systems, which have been complicated by the powerful stream of information that is required to examine and analyze promptly. analysis of management processes in emergency situations (es) has allowed to identify a number of features compared to the daily activities mode. the most typical of them are given in the table 1 [1]. a management of prevention and dealing with consequences of technological disasters should cover the whole range of issues relating to emergencies, the most important of which are the stages of emergency forecasting, proactive planning and operational management of the liquidation of the causes and consequences of the emergency under high uncertainty. the research is focused on the effectiveness of different methods for scenario analysis and modeling in the planning and operational management measures to prevent and eliminate causes and consequences of man-made disasters and emergencies. 1 corresponding author. email: kulba@ipu.ru 10 v.l. schultz et.al.: regional security: analysis of the emergency management effectiveness based on the scenario approach table 1. comparative characteristics of management systems daily activity mode management in emergency situations permanent mode of operation different operating modes strict structure and clear allocation of functions for a long period the lack of a strict structure and a clear allocation of functions for a long period, flexibility, aggressiveness the narrow functional focus broad and partially unpredictable scope monostructure polystructure regulated information flows the dependence of the information flows on the current situation accurate information uncertain information redundant information insufficient information the low rate of change a high rate of change unpredictability of the situations unpredictability of the situations; orientation on past experience usually does not make sense the principle of unity of powers and responsibilities combination of the principles of unity of command and the distribution of powers and responsibilities functional potential organizational capacity the predominance of mostly socio-economic objectives and performance criteria the objectives – efficiency and effectiveness in liquidation the causes of emergencies and their consequences; criteria minimizing the time to achieve the objectives, a minimum loss (victims) in liquidation of emergencies 2. planning of prevention and liquidation the causes and consequences of manmade disasters and emergencies unlike traditional planning and control systems [2], which are intended to solve strategic tasks for a long time period, the control system in emergency should operate in real time. strategic tasks should be solved in a limited time interval promptly and continuously. in practice this also means periodic adjustment of the list of key strategic tasks and continuous monitoring of new emergency events. moreover, a control system in emergency situations must be ideologically focused on the need, the possibility and the ability to operate in extreme conditions and to adapt quickly to changing conditions. the most important task of the control system under these conditions is not only operational, but also long-term planning for prevention and elimination of consequences of emergency situations on the transport infrastructure of urban agglomerations. the principal features of the planning and management processes of eliminating the consequences of manmade disasters and emergencies are: (1) partial predictability of serious problems and possibilities of their solution; (2) partial predictability of places of appearance and development of emergencies; (3) low predictability of the scale and the time of a disaster; (4) unpredictability of adverse events and situations arising from the emergence and development of emergencies, i.e. availability of strategic surprises. these conditions dictate the need for pre-emptive use of planning and management methods, based on anticipation of problems, situations and events, making flexible emergency solutions, oriented to the external system environment (natural environment, living conditions of population and staff of enterprises and organizations, socio-political situation etc.). such methods can be effectively implemented in the unified modern concept of strategic and tactical planning and operational management. full planning and control cycle of risk of industrial catastrophes and emergencies, as well as the liquidation of their consequences include:  forecast risk and severity of the consequences of emergency situation by generating scenarios; advances in systems science and applications (2017) vol.17 no.1 11  forming objectives and risk management criteria;  strategic (long-term) planning of preventive measures;  tactical (current) planning of alternative response to the emerging threat of disaster;  strategic and operational management in emergency situations. the risk of occurrence and development of es of natural and industrial type is forecasted on the basis of constructed scenarios of these situations. the concept of the scenario approach in control theory is to some degree new, though now has been used widely, especially in the analysis of strategic decisions in organizational management [3]. during the study of management processes the scenario of prevention of es and liquidation the consequences of industrial catastrophes can be considered as a tool for formal analysis of alternative variants of es development on a control object for the given objectives and criteria under conditions of uncertainty, when it is impossible to directly form a specific and detailed plan of measures on liquidation of es consequences under the existing time constraints. at its core, the scenario approach belongs to a class of object oriented methods of presenting information on control object’s state and environment, which is necessary to generate control actions for rescue measures and activities, as well as disaster management. in other words, the main task to be solved in terms of the scenario approach is to form the necessary data for preparation and making effective strategic and operational decisions, as well as a comprehensive analysis of the consequences of these decisions’ implementation in a variety of conditions. thus, a scenario of a problem situation development is needed as an intermediary between the stage of goal setting and the stage of formation and implementation of specific management decisions aimed at achieving the objectives for the prevention and liquidation of es consequences. in its content, a scenario of a plant’s behaviour is a model of changes, which is related to appearance and development of an es and is determined in discrete time with a given step. the scenarios as a tool for management of complex systems belong to a class of so-called partial mathematical models, i.e. such models which include only the essential factors that can be formalized with an acceptable degree of accuracy. the main application of such models is a class of tasks that can be reduced to finding both optimistic and pessimistic assessments of key quantitative and qualitative parameters of the objects and the processes when a particular set of management decisions is implemented. in general, the scenario construction task may be stated as follows: a complex, dynamic, open, controlled, not fully observable system is given. describe the possible ways of change in several alternative directions thus to provide the most complete picture of the possible future states and paths of the studied system development. the process of scenario construction of industrial catastrophes and emergencies includes the following main stages [3]. (1) the possible moment of emergency and / or its consequences on the interval [0, t] is divided into discrete times ti with steps i. the period t describes the time during which the damage is liquidated and the emergency stops its further development, depending on the scale of the damage and the disaster. time instances correspond to control points of the es development when control actions aimed to improve the situation are implemented. the step in a scenario is determined on the basis of the condition of effective using of forces and means, as well as the preparation of the necessary resources for countermeasures to the development of emergency. (2) formation of the initial conditions (t = 0) and the conditions under which emergency occurs and develops and the evaluation of losses (damage). on this stage the development of es is simulated and the possible conditions of its course are described on the basis of initial and observed data. concretize targets of countering, assess their effectiveness, determine limitations on the course and consequences of emergency situations as well as resources support. define pessimistic preliminary assessment of 12 v.l. schultz et.al.: regional security: analysis of the emergency management effectiveness based on the scenario approach losses and damage of the disaster. develop measures to counter the emergency in different variants of its development. (3) organization and putting into operation forces and means necessary to eliminate causes and consequences of emergencies, rescue and other urgent works in industrial catastrophes, depending on scale, character and amount of damage. it describes the process of mobilization and deployment of forces and resources, providing materials and food, emergency medical care, including the preparation for the reception of additional operational units, vehicles of all types, hospitals deployment points for the rehabilitation of victims, etc. thus, at this stage briefly all the activities and resources are described as well as the forces and means necessary for an effective response to emergency situations. (4) formation of the current situation (at the time ti) and the conditions of emergency situations, as well as the adjustment of the expected damage assessment, i.e. at this stage the current range of activities is described (similar to p.2). the situation and the most important indicators obtained at the time ti-1 are taken as initial conditions. further the measures considering new conditions prevailing at a given moment are described. this process has to be described until the complete liquidation of the es. (5) verify the completeness and reality of the obtained scenario and its correction for maximum adequacy to the real development of the es. scenario is checked by experts and is included into education and training. (6) formation for a given scenario of the actions course and operational plans for countering an es. based on the developed scenario a course of actions is being defined (a graph of goals and tasks with logic ‘and’), preventive and operational action plans to reduce risk and damage in the form of many types of documents and descriptions (at the time ti) which are distributed across the entire organizational structure for the execution. thus, based on a scenario the general objective of the operation is formed, providing creation of the desired situation in the future. the formulated general objective is represented in the hierarchy of goals and called course of actions. course of actions is a graph of goals and tasks, formed as a result of the decision-making on a complete graph of alternatives. there are two ways to form general objectives with the appropriate hierarchy of goals and tasks, and therefore two ways to form a course of action: from all the alternative scenarios the most probable one is selected and the general goal and the course of actions are produced in accordance with it; for each alternative scenario its own general goal is built with its hierarchy of goals and tasks (the variant planning). however, in all cases for each scenario several alternative general goals and courses of actions are considered. a scenario formed this way allows to reflect the development of emergencies, develop a strategy of the organization and implementation of preventive and operational measures against emergencies, create strategic and tactical plans of actions to carry out a qualitative analysis of the impact, as well as to forecast data on the estimated losses and damage. 3. usage of scenario analysis in management of prevention and liquidation of emergencies the basic feature of use the scenario approach in the planning and management of the prevention and liquidation of consequences of man-made emergencies is the need to expand its capabilities to conduct advanced comprehensive analysis of the current situation. one of the most important features of man-made emergencies, as a research domain, is the sufficiently powerful normative base. its development and improvement are continuously expanding over a number of areas, such as industry, transport, fire, radiation, chemicals, energy, environmental, social, public safety, security of life activity, certification of potentially dangerous objects, medicine disaster etc. it allows to improve efficiency of the scenario analysis methodology in the management of the liquidation of consequences of emergency situations and to change traditional methodology and technology of research of the simulation models. advances in systems science and applications (2017) vol.17 no.1 13 at present the functional approach is widely used to study the problems of increasing efficiency of socio-economic systems. to study, for example, one or a few ‘close’ control functions is often not effective enough for solving the control tasks for the liquidation of consequences of man-made es. the main reason for this is the need for a broader, comprehensive analysis of the situation and taking into account the synergistic nature of adverse processes and phenomena, which significantly expands the boundaries of the study subject area and thus reduces the efficiency of the traditional approach. using the functional approach in these conditions creates objective difficulties for combining expert knowledge in various subject areas into a unified picture. this doesn't allow to carry out a comprehensive analysis of states and tendencies of the situation development. this is caused by the difference of professional languages (terminology) of experts, contradictory procedures and discrepancy of expert assessments of the results, as well the lack of common tools and mechanisms for pooling expertise. this leads to great time required to collect basic data and expert assessments under sufficiently rigid restrictions on the time of decision making. in addition, the traditional approach focuses on expert assessment of the current situation on a plant. this results in that a simulation model or at least its methodology basis is used. in this case, substantial expansion of domain area for a comprehensive analysis of the situation leads to difficulties with the assessment of the model’s adequacy, as well as with the validity of its borders and the level of detail. the presence of a general and developed normative base allows to fundamentally change the approach to formation of es patterns. it is more effective to form a basic model analyzing existing procedures and regulations, and further modify by the use of detailed information about the object’s specifics and the operational information about the development of the situation. this approach provides several quite apparent advantages: (7) the use of technical regulations and norms as an information base can significantly improve the adequacy of the developed multi-factor model, since it is based on reliable data about the object of research. the quality of such model is guaranteed by multiple checks in the development, coordination and approval of technical regulations and norms. (8) efficiency of diagnosis of sources of vulnerability in the control object is increased for different kinds of threats, as well as accuracy of risk assessment. the risk is characterized by a relatively low probability of es occurrence and by catastrophic consequences when it is realized. (9) the complexity of model development is substantially reduced, because it is done on the agreed and (at least partially) formalized documents that contain extensive, reliable and, most importantly, relevant information for the research scenario. (10) there are almost no serious problems of coordination and integration of expertise in the development of the base model, because, in fact, the most complex procedures of this type have been conducted in the development of regulations and their results can be directly used in developing of the model. (11) time spent on diagnosis and detailed analysis in the development of multi-factor model is reduced substantially, because the procedures of expert assessments, needed in the description of domain, carried out not on the full range of problems, and serve only to clarify the necessary details or analysis of emerging unexpected circumstances in the development of the situation. (12) using of normative base with a high level of detail gives possibility to conduct a study of a simulation model based on quantitative assessments and absolute scales. a simulation model allows to conduct experiments in real time and provides (1) increase of the validity of the generated scenarios of situation development, (2) precision of forecasts, formed on their basis, (3) as well as reliability of the evaluation of the effectiveness of management decisions. 14 v.l. schultz et.al.: regional security: analysis of the emergency management effectiveness based on the scenario approach (13) the most complex procedure of modification of a model is considerably simplified. it refers to an expansion and covering of adjacent domains and also allows to obtain complex strategic decisions, taking into account synergies in the development of es. (14) using a normative base allows to simplify solutions of many technological problems of scenario study for complex systems, associated with the laborious procedures of operative change of the models of a managed object in a limited time, taking into account the dynamics of change of key factors and parameters, as well as their relationships. (15) at last, perhaps the most important advantage of the widespread use of a normative base in the course of the scenario study is the possibility of an integrated approach to a solution of management tasks of prevention and liquidation of consequences of es, allowing at the same time to consider the related but different in nature phenomena and processes. this allows effectively integrate in a single model the factors and threats of terrorism, as well as fire, radiation, chemical, energy and environmental security. 4. scenario analysis of safety underground station in an emergency situation let us consider the task of scenario analysis of safety for potentially dangerous objects of urban transport infrastructure. the solution of this task is necessary in the process of planning and decision making on the prevention and liquidation of the causes and consequences of possible es. here we must briefly recall the main principles and approaches to the design and study of models of the situation development, which provide the implementation of management decisions [3-5]. the process of modeling and synthesis of alternative scenarios of the situation development is carried out using the apparatus of functional signed graphs. the mathematical model of these graphs is an extension of the classical models of oriented digraphs. in addition, the model of digraph g (x, e), where the x – a finite set of vertices, and e – a set of arcs of the graph includes additional components. so, each vertex of the graph has the parameters  xnivv i  , and each arc of the graph has a sign, weight or functional transformation f (v, e). as known, underground facilities of subways are the objects of high potential danger, that is related by a number of factors. the most important of them are:  a large number of passengers served by the subway;  high concentration of people inside trains, station premises and on inter-exchange transitions during peak hours;  the possibility of panic and need of evacuation of large numbers of passengers;  considerable depth of the tunnels and station facilities;  a limited number of inclined tunnels and vertical shafts, coming to the surface;  a large extent and a limited capacity of escape routes;  difficult maintenance of smooth and safe evacuation of passengers from a train stopped inside a tunnel;  presence of power grids under high voltage;  high-speed of smoke pollution of tunnels and space stations, the complexity of reconnaissance and control of fires;  the need for laying hose lines over long distances, taking into account the complexity of the layout and the availability of wagons;  the impact of ventilation on gas exchange of fire;  rapid increase in dangerous factors of fire values to a critical level;  threat of deformation and loss of structural elements carrying capacity station etc. in addition, metro stations and their environment are complex socio-technical systems, characterized by a significant amount of power, escalator, control-crossing, ventilation, climate, advances in systems science and applications (2017) vol.17 no.1 15 lighting and other kinds of process equipment; availability of premises of different categories for explosion and fire hazard, and fire resistance of load-bearing structures; a significant number of businesses for various purposes (kiosks, stalls, trade and promotional stands) to be placed in ground and underground hallways, in the transitions and on the platforms of underground stations, etc. consider the complex scenario models to predict the prevention planning and management, and liquidation of the causes and consequences of technological disasters and es in the underground facilities. as an example, we will use a hypothetical es initiated by a conventional explosion in a typical subway station of deep foundation, which moscow has more than 70. this type of station is most typical for the central administrative district (cad) of moscow, which will be considered in the following model example. when constructing model the risk zone is limited to only outside of the station. to ensure generality of the model we also exclude from consideration peak hours of the metro, which represent the greatest danger from the point of view of the consequences of es, as well as the busiest station and interchange nodes with the total daily load, considerably more than 50 million passengers (is known that the busiest station daily serves 100-150 thousand people). we assume that the number of people at the station at the time of a es corresponds to its average occupancy and is about 800 people. the designed simulation model consists of three parts: (16) the security model of the station and the people; (17) model of the work of the moscow city warning and es liquidation system; (18) model of the power and means involved for liquidation of es. the model can be further expanded to include information about the route to the nearest metro station firehouse (fh). in particular, one can consider a situation related to difficulties in extension of the given fh due to the congestion of highways or other reasons. also, data for the location of hospitals, in which the suffered people can come can additionally be used, as well as major socio-relevant and potentially dangerous objects, located near the considered metro station (if any). 4.1. the security model of station and people the model includes the following security issues: (19) the safety of people; (20) the safety of stay conditions; (21) mechanical safety; (22) fire safety. if necessary, the number of security issues may be increased. each of these aspects includes several related factors. 4.2. model of the work stages of the moscow city warning and liquidation of es system (mces). the main stages of the simulated operation of the district-level mces in the event of a es in the subway stations are as follows. (23) the first stage – the adoption of emergency measures:  alerting the population;  adoption of emergency measures to protect the public, victim assistance and localization of the accident;  arrival of the operational group (og) mces;  organization of a reconnaissance in the disaster zone;  notification about es and organization of an operational headquarter and the commission on prevention and liquidation of es and fire safety;  clarification of tasks for mces forces;  alerting forces of mces on the district level. (24) the second stage – operational planning of emergency. rescue and other urgent works:  collection of information for the decision-making mces; 16 v.l. schultz et.al.: regional security: analysis of the emergency management effectiveness based on the scenario approach  organization of work planning;  organization of interaction between the involved forces. (25) the third stage – liquidation of the es:  deployment of es staff, organization of communication with the management bodies operating in the area of the emergency and the parent body;  collection of executors and interacting of the management bodies deployed in the disaster area, clarifying the situation, the composition of forces, the action plan, the report for the elimination of es;  develop proposals of mces for operational decisions;  ensuring timely delivery of tasks to subordinates and interacting controls bodies;  organization of information exchange about the situation, measures taken to interacting, neighboring authorities. (26) the fourth stage – the implementation of measures on social protection and rehabilitation of areas affected by emergencies factors:  complete deactivation, decontamination and disinfection facilities;  repair of electrical, water, and gas utilities, communication lines, etc. in addition to these activities in the first stage rescue and other urgent works are carried out (asdnr), which include:  search of victims, the evacuation of the focus of infection;  conduct health suffered partial processing;  delivery of the victims at the place of loading on ambulances;  pre-hospital medical care, evacuation – transport sorting;  delivery of the victims to special medical institutions;  providing of skilled care. the structure of the model takes into account that the activities of the individual steps are directly related to each other. 4.3. model of forces and means involved for liquidation of emergency situations. the list of factors is based on the information about the presence of forces and means of liquidation of es, normative data about grouping mces forces in 10 minutes after the disaster, and includes:  technical services of subway;  squad of the subway protection;  operational groups of mces;  search and rescue team;  police units of internal affairs (uvd) cad;  units of road – patrol uvd cad;  public joint stock company for energy and electrification of moscow (mosenergo);  joint stock company ‘mosvodokanal’ (a company providing services in the sphere of water);  supply and sanitation in the city of moscow;  emergency gas service of moscow (mosgaz);  forces of emergency medical aid center;  forces of the specialized enterprises of fuel-energy sector of moscow operation communication;  collectors (moskollektor);  volunteer emergency rescue teams. additional factors in the model are the following peaks:  activation of forces and means;  arrival time of fire and rescue services; advances in systems science and applications (2017) vol.17 no.1 17  the number of people at the station. the main arcs in the model are circular, which means a continuous activity of forces and means. if necessary, the model can be developed, replacing the factors of new structures (for example, drive factor and hrc can be replaced with a model fire fighting, etc.). 4.4. joint model of the emergency liquidation. topology of unified situational model, which is the unification of the previously discussed models, is shown in figure 1. consider baseline scenarios obtained by simulation of an emergency at the facility underground. scenario 1: ‘the fire caused by an explosion and subsequent quenching’. at the first stage fire-fighting forces is done by underground workers (valid up to 20 steps). at first the human safety of the plant is growing. after 20 steps, gradually the all important characteristics become negative trend. then, starting with the 100 steps of action of fire brigades and other forces alter the situation in a favorable direction (figure 2). scenario 2: ‘the spread of panic in liquidation of emergencies’. we consider the situation in which the occurrence of emergency situations and fire causes panic among the passengers in the station. panic prevents the fire-rescue units and makes it difficult to search for survivors. this situation corresponds to the following modification of the original model:  increased weight feedback relationship ‘panic’ --> ‘access firefighters’ 10 times;  addition of a new arc ‘access firefighters’ --> (+1) --> ‘fire propagation’;  increased weight feedback relationship ‘panic’ --> ‘search of victims’ twice. dynamics of the major factors is presented in figure 3. fig.1. joint model of the emergency liquidation 18 v.l. schultz et.al.: regional security: analysis of the emergency management effectiveness based on the scenario approach fig.2. the fire caused by the explosion and subsequent quenching (scenario 1) fig.3. spread of panic in liquidation of emergencies (scenario 2) advances in systems science and applications (2017) vol.17 no.1 19 initially, there is a decrease in the safety of people and station structures. outbreaks of fire cannot be located and the fire spread process continues. significant interference to the undertaken measures to extinguish the fire are the presence of people in the station, and not to reduce panic. a further negative development of the situation can lead to the appearance of the victims of the man-made disaster. over time, due to held fire-rescue actions the fire safety factor first stabilized. it reduces the risk of destruction of bearing structures of plant facilities while reducing the spread of fire intensity. accelerate the arrival of additional fire-rescue services accelerates the growth of the security of stations (in terms of the developed model), however, that is especially important, given the operational situation is after the onset of the fire victims. scenario 3: ‘efficiency of the actions the fire-rescue service on liquidation of emergency’. under this scenario, the variant of the situation, in which the strong and well-coordinated actions of rescuers help reduce the panic, that is reflected in the following modification of the model:  added a negative relationship ‘the effectiveness of the fire and rescue services’ --> ‘panic’;  added a nonlinear relationship ‘panic’ --> ‘the efficiency of the fire and rescue services’ (link is positive if there is the panic growth and negative otherwise). the simulation results are presented in figure 4. as seen from represented in figure 4 graphics dependencies, shortly after the arrival of the additional fire and rescue services the panic reduced. this increases the efficiency of fire suppression and leads to subsequent growth of human safety. scenario 4: ‘modeling of losses during working escalators’. the aim of this scenario is an analysis of the casualties among the passengers of the station as a result of a disaster. the basis for the building a model are the characteristics of the underground station. carry out temporary normalization of modeling steps as follows: 100 steps = 10 minutes, that is 1 step = 6 sec. accordingly, the fire and rescue services come into place on the emergency regulations in 10 minutes, i.e. after 100 steps. fig.4. efficiency actions the fire rescue services on liquidation of emergencies (scenario 3) 20 v.l. schultz et.al.: regional security: analysis of the emergency management effectiveness based on the scenario approach the activity of metro services unit for the protection of underground rescue people is 20 steps = 2 min. in the period from 20 to 100 steps (4.8 min.) the assistance in rescuing on their own is limited. as before, it is assumed that metro stations are around 800 people. this corresponds to an average workload of the station. assume that the escalators on all saving time is not damaged, and the only way to save. the speed of the escalator is the motion on the rise – 3.0 thousand people / hour. thus, the rate of evacuation is 5 people / sec. it should also be noted that the speed of evacuation decreases with a decrease in the safety of people. moreover, the evacuation of injured people is slower than healthy people. the evacuation of everyone injured takes twice more time. every 6 sec. (step 1) add one injured. under these initial conditions the following modification of the model is conducted: (27) new factors are added:  ‘the number of people at the metro station’,  ‘evacuation speed’,  ‘the rate of destruction of people’,  ‘the number of evacuees’,  ‘the number of injured’,  ‘the number of rescued without injuries’. (28) new relationships are added:  ‘evacuation speed’ --> (-1) --> ‘number of people at the station’,  ‘evacuation speed’ --> (+1) --> ‘number of evacuees’,  ‘evacuation speed’ --> (+1) --> ‘number of rescued without injuries’,  ‘destruction speed’ --> (+1) --> ‘number of affected’,  ‘destruction speed’ --> (-1) --> ‘number of rescued without injuries’,  ‘the rate of evacuation’ --> (5, if the safety of the people is increasing 2 or otherwise) --> ‘evacuation speed’ (circular arc). (29) the conditions of factors activity:  ‘evacuation speed’ is active, if the number of people at the station is greater than zero,  ‘destruction speed’ is active, if the number of people at the station is greater than zero. (30) initial impulses:  --> ‘evacuation rate’ +1;  --> ‘destruction speed’ +1. the simulation results are presented in figure 5 (qualitative results) and figure 6 (quantitative results). the simulation results show that as a result of works on es liquidation 800 people are evacuated, 271 is injured (get damaged), and 529 people escaped without injury. scenario 5: ‘modeling of losses during non-working escalator’. the modification of the previous model: the speed of evacuation is 3 person / sec. the rate of evacuation of injured persons 2 / sec. evacuation speed cannot be negative. qualitative and quantitative simulation results are shown in figure 7 and figure 8, respectively. as a result of work on es liquidation on the results of the simulation of the 800 evacuated people were injured 447 people, 353 people escaped without injury. the results were almost twice as bad. furthermore, the evacuation time increased to 12 minutes. advances in systems science and applications (2017) vol.17 no.1 21 fig.5. modeling of losses during working escalators (scenario 4, the qualitative results) fig.6. modeling of losses during working escalators (scenario 4, the quantitative results) 22 v.l. schultz et.al.: regional security: analysis of the emergency management effectiveness based on the scenario approach fig.7. simulation of losses during nonworking escalators (scenario 5, the qualitative results) fig.8. modeling of losses during nonworking escalators (scenario 5, the quantitative results) advances in systems science and applications (2017) vol.17 no.1 23 5. conclusions the main feature discussed in this paper, a class of industrial safety management tasks based on the scenario approach is to have developed and the general normative framework that allows to fundamentally change the approach to building models of emergency. studies have shown that in this case the procedure of forming the base model using a comprehensive analysis of existing rules and norms were quite effective. in the second stage – modifications of the model taking into account the detailed information about the specifics of the object of study and the incoming operational information about the nature of the emergency. using the normative base in the process of forming the simulation model allowed in the generation of alternative scenarios of situation development to base not only on qualitative but also on quantitative indices. this improves the accuracy and reliability of the scenario analysis results, and the quality and efficiency of decisions made in the process of liquidation the es consequences. the developed methodology of advanced scenario analysis allows to solve a wide range of process management tasks of prevention and liquidation of consequences of technogenic catastrophes and emergencies. the proposed methods provide the complex multidimensional study of alternatives of an emergency at the given target criterion under conditions of uncertainty. the main advantage of this approach is the ability to predict the behavior of the simulated object through the formation of alternative development scenarios. this approach allows to make conclusions about the most likely and appropriate directions of dynamic processes development, their stability, and other significant characteristics on the basis of information about the structural features of the study domain. the practical application of scenario approach enables a comprehensive analysis of the current situation at a given time horizon, to form the short-term and long-term forecasts of its development and to counter the emerging threats, to assess the effectiveness and consistency of distributed in time and space strategic and tactical management decisions for the prevention and elimination of consequences of emergencies and man-made disasters and catastrophes. modeling and scenario analysis of a wide range of safety of potentially dangerous for transport and infrastructure, residential and industrial buildings, etc. it comprises:  diagnostic analysis and evaluation of the situation;  the development of the object model, the choice of performance criteria and an assessment of their relative importance;  generation of possible development scenarios;  assess the scenarios developed (primarily management processes of prevention and liquidation of consequences of es), and selecting the best of them for a given criterion of efficiency  continuous analysis of monitoring information on the situation and making the appropriate changes in the structure of the models on the basis of the data obtained;  assessment and selection of management actions on liquidation of emergency situations and improve safety;  dynamic analysis of the possible consequences of control actions;  collection of the results of the implementation of scenarios and assessment of data. the opportunity to develop the methodology of scenario analysis and modeling is certainly not limited to the examples given in this paper – they are much wider. currently, one of the most promising directions of development of the proposed methodology is unification of planning and scenario modeling. this area, which requires a separate study, involves the selection, analysis and certification of standard subclasses of potentially dangerous objects on the basis approved by the ministry of the russian federation for civil defense, emergencies and liquidation of consequences of natural disasters (emercom of russia) classification, which includes five basic classes, technological accidents which may are the source of federal (cross-border), regional, territorial and local emergencies respectively. unification (the reduction of all the variety of plans, control and monitoring actions to quite a limited set) should enable the development and use of standard tools and resources planning and 24 v.l. schultz et.al.: regional security: analysis of the emergency management effectiveness based on the scenario approach management processes of prevention and liquidation of consequences of es, as well as a large community of semantic and information content of the standard (basic) plans. on the one hand, it provides the ability to use quite limited set of information elements, and on the other hand – a significant invariance for the different levels of government and departmental affiliation, which should simplify the implementation of agreed plans of the system in various modes. development of theoretical and applied research in this field will provide an opportunity to solve a wide range of practical problems of planning and management processes of prevention and liquidation of consequences of emergency situations at the facility, municipal, regional and federal levels. references [1] arkhipova n., kulba v. emergency management. moscow, russian state humanitarian university,1998 (in russian). [2] arkhipova n., kulba v., kosyachenko s., chanhieva f., shelkov a. organizational management. moscow, russian state humanitarian university, 2007. (in russian). [3] schultz v., kulba v., kononov d., kosyachenko s., shelkov a., chernov i. models and methods of analysis and synthesis of scenarios of socio economic systems. moscow, nauka, 2012. (in russian). [4] schultz, v., kulba v., shelkov a., chernov i. methods of the planning and management of technogenic safety based on the scenario approach. national security / nota bene, 2013, vol.2, no. 25, p.198-216 (in russian). [5] kulba, v., zaikin o., shelkov a., chernov i. scenario analysis in the management of regional security and social stability. in “new frontiers in information and production systems”, eds. p. rozewski, d. novikov, n. bakhtadze and o. zaikin. intelligent systems reference library, switzerland: springer international publishing, 2016, vol. 98., p.249–268. adv syst sci appl 2018; 4:39–51 http://ijassa.ipu.ru/index.php/ijassa/article/view/572 estimation of stress-strength reliability for quasi lindley distribution m.m. mohie el-din1, a. sadek 1∗, shaimaa h. elmeghawry2 1 dep. of mathematics, faculty of science, al-azhar university, egypt 2 faculty of engineering, benha university, egypt received april 14, 2018; revised october 17, 2018; published december 31, 2018 abstract: this paper discussed the problem of estimating of the stress-strength reliability r = pr(y < x). it is assumed that the strength of a system x, and the environmental stress applied on it y, follow the quasi lindley distribution(qld). stress-strength reliability is studied using the maximum likelihood, and bayes estimations. asymptotic confidence interval for reliability is obtained. bayesian estimations were proposed using two different methods: importance sampling technique, and mcmc technique via metropolis-hastings algorithm, under symmetric loss function (squared error) and asymmetric loss functions (linex, general entropy). the behaviors of the maximum likelihood and bayes estimators of stress-strength reliability have been studied through the monte carlo simulation study. finally analysis of a real data set has also been presented. keywords: quasi lindley distribution; stress-strength reliability; maximum likelihood estimation; asymptotic confidence interval; bayesian estimation; importance sampling technique; mcmc technique via metropolis-hastings algorithm. 1. introduction the stress-strength models have been widely used for reliability design of systems. in these models the reliability is defined as the probability that the strength is larger than the stress r = pr(y < x). as the strength x is larger than the stress y, the system work efficiently, otherwise the the system fails. estimation of stress-strength reliability was studied by several authors, for example, stress-strength model and its generalizations has been discussed in [10]. the estimation of r when x and y are normally distributed was introduced by church and harris [7]. krishnamoorthy et al. [11] introduced an inference on reliability in two-parameter exponential stress-strength model. al-mutairi et al. [1, 2] presented the stress-strength reliability for lindley and weighted lindley distributions respectively. stress-strength reliability estimation for generalized lindley distribution has been introduced by singh et al. [18]. recentely khan and jan [9] studied the estimation of stress-strength reliability model using finite mixture of two parameter lindley distributions. this paper is focused upon studying the problem of the estimation of the stress-strength reliability for the quasi lindley distribution (qld) introduced by shanker et al. [17] of which the lindley distribution is a particular case. we estimated the parameter of the stressstrength reliability r using the maximum likelihood, and bayesian estimation methods. we ∗corresponding author: a sadek@azhar.edu.eg 40 m. mohie el-din, a. sadek, shaimaa elmeghawry construct the asymptotic confidence interval of r based on the asymptotic distribution of the mle of r. in bayesian estimation we introduced two sampling methods (importance sampling and mcmc). the qld has the following probability density function (pdf): f(x) = θ α + 1 (α + θx) e−θx , (1.1) and the following cumulative distribution function (cdf): f (x) = 1− [(1 + α + θx) α + 1 e−θx ] , (1.2) where; x > 0, θ > 0, α > −1. this paper is organized as follows. in section 2, stress-strength reliability issue is studied to obtain the reliability function of the parameters of qld distribution. maximum likelihood estimation for stress-strength reliability is discussed in section 3. asympototic confidence interval of reliability is proposed in section 4. in section 5, a general procedure for deriving the bayesian estimator of reliability is introduced, wherein we applied the importance sampling and mcmc techniques to compute the approximation of this estimator. section 6 presented simulation study to investigate and compare the performance of each method of estimation. section 7 presented analysis of a real data set for illustrative purposes. finally, conclusions appear in section 7. 2. stress-strength reliability assume x ∼ qld(θ1, α1) and y ∼ qld(θ2, α2) are independent random variables with pdf f(x) and g(y), respectively. then the stress strength reliability can be obtained as: r = pr(y < x). = ∫ ∞ 0 ∫ x 0 f(x)g(y) dydx. = ∫ ∞ 0 f(x)g(x) dx. = ∫ ∞ 0 θ1(α1 + θ1x) α1 + 1 e−θ1x [ 1− [ (1 + α2 + θ2x) α2 + 1 e−θ2x ]] dx. = ∫ ∞ 0 θ1(α1 + θ1x) α1 + 1 e−θ1x − ∫ ∞ 0 θ1(α1 + θ1x)(1 + α2 + θ2x) e−(θ1+θ2)x (α1 + 1)(α2 + 1) dx. = 1− θ1 ( 2θ1θ2 + (θ1 + θ2) (α2θ1 + α1θ2 + θ1) + α1 (α2 + 1) (θ1 + θ2) 2) (α1 + 1) (α2 + 1) (θ1 + θ2) 3 . (2.3) from eq.(2.3), we noticed that r is a function of parameters θ = (θ1, α1, θ2, α2). therefore, for maximum likelihood estimate (mle) of r we need to obtain the mles of these parameters. copyright c© 2018 assa. adv syst sci appl (2018) estimation of stress-strength reliability 41 3. maximum likelihood estimation for reliability suppose that x1, x2, · · · , xn is random sample from qld(θ1, α1), and y1, y2, · · · , ym is random sample from qld(θ2, α2), then the log likelihood function can be written as: logl(x, y; θ) = n log (θ1) +m log (θ2)− n log (α1 + 1)−m log (α2 + 1) −θ1 n∑ i=1 xi − θ2 m∑ j=1 yj + n∑ i=1 log (α1 + θ1xi) + m∑ j=1 log (α2 + θ2yj) . (3.4) the mle of θ = (θ1, α1, θ2, α2) can be obtained as a solution of the following equations: ∂l ∂θ1 = n θ1 − n∑ i=1 xi + n∑ i=1 xi α1 + θ1xi = 0, ∂l ∂θ2 = m θ2 − m∑ j=1 yj + m∑ j=1 yj α2 + θ2yj = 0, ∂l ∂α1 = − n α1 + 1 + n∑ i=1 1 α1 + θ1xi = 0, and ∂l ∂α2 = − m α2 + 1 + m∑ j=1 1 α2 + θ2yj = 0. solving these equations numerically using an iterative process as newton raphson to get θ̂1, α̂1, θ̂2, α̂2, then the mle of r can be obtained as following: r̂ = 1− θ̂1 ( 2θ̂1θ̂2 + ( θ̂1 + θ̂2 )( α̂2θ̂1 + α̂1θ̂2 + θ̂1 ) + α̂1 (α̂2 + 1) ( θ̂1 + θ̂2 ) 2 ) (α̂1 + 1) (α̂2 + 1) ( θ̂1 + θ̂2 )3 . (3.5) 4. asymptotic confidence interval of r the asymptotic variance-covariance matrix of all parameters can be approximated by the inverse of observed information matrix, and then derive the asymptotic distribution of r̂. based on the asymptotic distribution of r̂, we obtain the asymptotic confidence interval of r. the fisher information matrix of θ = (θ1, α1, θ2, α2) is given as: i(θ) = −  e( ∂ 2l ∂θ1 2 ) e( ∂2l ∂θ1∂α1 ) e( ∂2l ∂θ1∂θ2 ) e( ∂2l ∂θ1∂α2 ) e( ∂2l ∂α1∂θ1 ) e( ∂ 2l ∂α1 2 ) e( ∂2l ∂α1∂θ2 ) e( ∂2l ∂α1∂α2 ) e( ∂2l ∂θ2∂θ1 ) e( ∂2l ∂θ2∂α1 ) e( ∂ 2l ∂θ2 2 ) e( ∂2l ∂θ2∂α2 ) e( ∂2l ∂α2∂θ1 ) e( ∂2l ∂α2∂α1 ) e( ∂2l ∂α2∂θ2 ) e( ∂ 2l ∂α2 2 )  . = i11 i12 i13 i14 i21 i22 i23 i24 i31 i32 i33 i34 i41 i42 i43 i44  . copyright c© 2018 assa. adv syst sci appl (2018) 42 m. mohie el-din, a. sadek, shaimaa elmeghawry where: i13 = i31 = 0; i14 = i41 = 0, i23 = i32 = 0; i24 = i42 = 0, i11 = − n θ1 2 − n∑ i=1 xi 2 (α1 + θ1xi)2 , i22 = n∑ i=1 1 (α1 + θ1xi)2 − n (α1 + 1)2 , i33 = −m θ2 2 − m∑ j=1 yj 2 (α2 + θ2yj)2 , i44 = m∑ j=1 1 (α2 + θ2yj)2 − m (α2 + 1)2 , i12 = i21 = n∑ i=1 xi (α1 + θ1xi)2 , and i34 = i43 = − m∑ j=1 yj (α2 + θ2yj)2 . using the central limit theorem, we obtain the following theorem : theorem 1: as n→∞, m→∞; then ( √ n(θ̂1 − θ1), √ n(α̂1 − α1), √ m(θ̂2 − θ2), √ m(α̂2 − α2)) d→ n(0, i−1(θ)). where d→ means converge in distribution, and i−1(θ) is the inverse of the matrix i(θ). in order to establish the asymptotic normality of r, we first define: d(θ) = ( ∂r ∂θ1 , ∂r ∂α1 , ∂r ∂θ2 , ∂r ∂α2 )t = (d1, d2, d3, d4) t , where t is transpose operation, and d1 = −θ1(2α1(α2 + 1)(θ1 + θ2) + α1θ2 + (α2 + 1)(θ1 + θ2) + α2θ1 + θ1 + 2θ2) (α1 + 1)(α2 + 1)(θ1 + θ2)3 + 3θ1 (α1(α2 + 1)(θ1 + θ2) 2 + (θ1 + θ2)(α1θ2 + α2θ1 + θ1) + 2θ1θ2) (α1 + 1)(α2 + 1)(θ1 + θ2)4 −α1(α2 + 1)(θ1 + θ2) 2 + (θ1 + θ2)(α1θ2 + α2θ1 + θ1) + 2θ1θ2 (α1 + 1)(α2 + 1)(θ1 + θ2)3 , copyright c© 2018 assa. adv syst sci appl (2018) estimation of stress-strength reliability 43 d2 = θ1 (α1(α2 + 1)(θ1 + θ2) 2 + (θ1 + θ2)(α1θ2 + α2θ1 + θ1) + 2θ1θ2) (α1 + 1)2(α2 + 1)(θ1 + θ2)3 −θ1 ((α2 + 1)(θ1 + θ2) 2 + θ2(θ1 + θ2)) (α1 + 1)(α2 + 1)(θ1 + θ2)3 , d3 = 3θ1 (α1(α2 + 1)(θ1 + θ2) 2 + (θ1 + θ2)(α1θ2 + α2θ1 + θ1) + 2θ1θ2) (α1 + 1)(α2 + 1)(θ1 + θ2)4 −θ1(2α1(α2 + 1)(θ1 + θ2) + α1(θ1 + θ2) + α1θ2 + α2θ1 + 3θ1) (α1 + 1)(α2 + 1)(θ1 + θ2)3 , and d4 = θ1 (α1(α2 + 1)(θ1 + θ2) 2 + (θ1 + θ2)(α1θ2 + α2θ1 + θ1) + 2θ1θ2) (α1 + 1)(α2 + 1)2(θ1 + θ2)3 −θ1 (α1(θ1 + θ2) 2 + θ1(θ1 + θ2)) (α1 + 1)(α2 + 1)(θ1 + θ2)3 . hence; using theorem 1, the asymptotic distribution of r̂, the mle of r is defined as √ n+m(r̂−r) d→ n(0, b), where b = v ar(r̂) = dt (θ)i−1(θ)d(θ). (4.6) therefore, using eq.(4.6), an asymptotic 100(1− α)% confidence interval for r can be obtained as: r̂± zα 2 √ v ar(r̂), where zα 2 is the upper α 2 precentile of the standard normal distribution. 5. bayesian estimation in this section, we provide the bayes estimate of r where θ1, θ2, α1, α2 are unknown parameters and all of these parameters having independent gamma prior distributions as following: π(θ1) ∼ gamma(a1, b1), π(θ2) ∼ gamma(a2, b2), π(α1) ∼ gamma(a3, b3), and π(α2) ∼ gamma(a4, b4). copyright c© 2018 assa. adv syst sci appl (2018) 44 m. mohie el-din, a. sadek, shaimaa elmeghawry the joint posterior pdf is defined as g(θ1, θ2, α1, α2/data) = l(x, y/θ1, θ2, α1, α2)π(θ1)π(θ2)π(α1)π(α2)∫∞ 0 ∫∞ 0 ∫∞ 0 ∫∞ 0 l(x, y/θ1, θ2, α1, α2)π(θ1)π(θ2)π(α1)π(α2)dθ1dθ2dα1dα2 . then g(θ1, θ2, α1, α2/data) ∝ θ1 n (α1 + 1)n θ2 m (α2 + 1)m n∏ i=1 (α1 + θ1xi) m∏ j=1 (α2 + θ2yj) e −θ1 ∑n i=1 xi × e−θ2 ∑m j=1 yj θ1 a1−1e−b1θ1θ2 a2−1e−b2θ2α1 a3−1e−b3α1α2 a4−1e−b4α2 . (5.7) 5.1. bayes estimators under symmetric and asymmetric loss function: the bayes estimate of reliability formula depending on the choice of the loss function. two different loss functions are used, symmetric and asymmetric loss function. if the amount of loss assigned by a loss function to a positive error is equal to the negative error of the same magnitude, then the loss function is called a symmetric loss function. in most of the studies on estimation and prediction problems, authors prefer to use the squared error loss function which is symmetric in nature. however, the use of the squared error loss function is not appropriate particularly in these cases, where the losses are not symmetric. thus in order to make the statistical inferences more practical and applicable, we often needs to choose an asymmetric loss function. a number of asymmetric loss functions proposed for use, first is the linex loss function which suggested by varian [20], and studied by several others including basu and ebrahimi [3], zellner [21]. second is the general entropy loss function which introduced by calabria and pulcini [5]. these asymmetric loss functions are also studied by braess and dette [4], pandey and rao [14], parsian and kirmani [15], sanku dey [16] and soliman [19], who have used these loss function in different estimation and prediction problem. then the equation of the bayes estimate of the reliability is depend on the loss function, here is the general equation for each type: -the bayes estimate under the squared error loss function, which is the posterior mean of r, is given by: r̂se = ∫ ∞ 0 ∫ ∞ 0 ∫ ∞ 0 ∫ ∞ 0 r g(θ1, θ2, α1, α2/data)dθ1dθ2dα1dα2. -the bayes estimate under the linex loss function is given by: r̂lx = −1 c ln [ ∫ ∞ 0 ∫ ∞ 0 ∫ ∞ 0 ∫ ∞ 0 e−cr g(θ1, θ2, α1, α2/data)dθ1dθ2dα1dα2 ] , where c is constant, c > 0, see [21]. -the bayes estimate under the general entropy loss function is given by: r̂ge = [ ∫ ∞ 0 ∫ ∞ 0 ∫ ∞ 0 ∫ ∞ 0 r−q g(θ1, θ2, α1, α2/data)dθ1dθ2dα1dα2 ]−1/q , where q is constant, q > 0, see [5]. it is impossible to compute these integrals analytically. two approaches can be used to approximate these integrals, namely, importance sampling technique and mcmc technique. copyright c© 2018 assa. adv syst sci appl (2018) estimation of stress-strength reliability 45 5.2. importance sampling technique importance sampling technique has suggested by chen and shao [6]. in statistics, importance sampling is the name for the general technique of determining the properties of a distribution by drawing samples from another distribution. the focus of importance sampling here is to determine as easily and accurately as possible the properties of the posterior from a representative sample from the second distribution. using importance sampling technique, eq.(5.7) can be written as g(θ1, θ2, α1, α2/data) ∝ g1(θ1/data)g2(θ2/data)g3(α1/data)g4(α2/data)h(θ1, θ2, α1, α2/data), where: g1(θ1/data) ∝ gamma(n+ a1, b1 + n∑ i=1 xi), g2(θ2/data) ∝ gamma(m+ a2, b2 + m∑ j=1 yj), g3(α1/data) ∝ gamma(a3, b3), g4(α2/data) ∝ gamma(a4, b4), and h(θ1, θ2, α1, α2/data) = ∏n i=1 (α1 + θ1xi) ∏m j=1 (α2 + θ2yj) (α1 + 1)n (α2 + 1)m . as shown, all the above functions from g1(θ1/data) to g4(α2/data) follow gamma distributions with different parameters, so it is quite simple to generate qld parameters from them. assuming that a1, · · · , a4 and b1, · · · , b4 are known, and assuming initial values for θ1, θ2, α1, α2. we can use the following importance sampling algorithm: • step1: generate θ11 from g1(θ1/data). • step2: generate θ21 from g2(θ2/data). • step3: generate α11 from g3(α1/data). • step4: generate α21 from g4(α2/data). • step5: repeat steps from 1 to 4, n times to obtain the vector (θ11, θ21, α11, α21), · · · , (θ1n , θ2n , α1n , α2n). then -an approximate bayes estimate of r under squared error loss function can be obtained as r̃impse = ∑n i=1ri h(θ1i, θ2i, α1i, α2i/data)∑n i=1 h(θ1i, θ2i, α1i, α2i/data) , an approximate bayes estimate of r under linex loss function can be obtained as r̃implx = −1 c log [ ∑n i=1 e −cri h(θ1i, θ2i, α1i, α2i/data)∑n i=1 h(θ1i, θ2i, α1i, α2i/data) ], -an approximate bayes estimate of r under general entropy loss function can be obtained as: r̃impge = [ ∑n i=1ri −q h(θ1i, θ2i, α1i, α2i/data)∑n i=1 h(θ1i, θ2i, α1i, α2i/data) ]−1/q, where ri = r(θ1i, θ2i, α1i, α2i). as defined in eq.(2.3), for i = 1, · · · , n . copyright c© 2018 assa. adv syst sci appl (2018) 46 m. mohie el-din, a. sadek, shaimaa elmeghawry 5.3. mcmc technique the most general mcmc algorithm is the metropolis-hastings (mh) algorithm, which was originally introduced by metropolis et al. [13], and hastings [8]. the metropolis-hastings (mh) algorithm simulates samples from a probability distribution by making use of the full joint density function and (independent) proposal distributions for each of the variables of interest. the joint posterior density function of θ1, θ2, α1, and α2 is given in eq.(5.7). it is easily seen that the posterior density functions of θ1, θ2, α1, and α2 are, respectively: π1(θ1/data) ∝ gamma ( n+ a1, b1 + n∑ i=1 xi ) , (5.8) π2(θ2/data) ∝ gamma ( m+ a2, b2 + m∑ j=1 yj ) , (5.9) π3(α1/θ1, data) ∝ α1 a3−1e−b3α1 ∏n i=1 (α1 + θ1xi) (α1 + 1)n , (5.10) and π4(α2/θ2, data) ∝ α2 a4−1e−b4α2 ∏m j=1 (α2 + θ2yj) (α2 + 1)m . (5.11) therefore, easily samples of θ1 and θ2 can be generated by using gamma distribution as shown in eqs.(5.8), and (5.9) respectively. however, the posterior distribution of α1 , α2 cannot be generated from a well known distributions. the metropolis-hastings algorithm, can be used to solve this problem, as shown in the following algorithm. • step1: start with initial value of α1,α2 such that α1 (0) = α̂1, and α2 (0) = α̂2. • step2: set i = 1. • step3: generate θ1(i) from π1(θ1/data). • step4: generate θ2(i) from π2(θ2/data). • step5: generate α1 (i) from π3(α1/θ1, data) using the metropolis-hastings algorithm with the proposal distribution q1 as following: – generate α1 (∗) from the proposal distribution q1 = n(α1 (i−1), v ar(α1 (i−1))). – calculate the acceptance probability r1(α1 (i−1), α1 (∗)) = min[1, π3(α1 (∗)/θ1 (i),data) π3(α1 (i−1)θ1 (i),data) ]. – generate u from uniform(0, 1). – if u ≤ r1(α1 (i−1), α1 (∗)), accept the proposal distribution and set α1 (i) = α1 (∗) , otherwise set α1 (i) = α1 (i−1). • step6: generate α2 (i) from π4(α2/θ2, data) using the metropolis-hastings algorithm with the proposal distribution q2 as following: copyright c© 2018 assa. adv syst sci appl (2018) estimation of stress-strength reliability 47 – generate α2 (∗) from the proposal distribution q2 = n(α2 (i−1), v ar(α2 (i−1))). – calculate the acceptance probability r2(α2 (i−1), α2 (∗)) = min[1, π4(α2 (∗)/θ2 (i),data) π4(α2 (i−1)θ2 (i),data) ]. – generate u from uniform(0, 1). – if u ≤ r2(α2 (i−1), α2 (∗)), accept the proposal distribution and set α2 (i) = α2 (∗) , otherwise set α2 (i) = α2 (i−1). • step7: compute r(i) at (θ1 (i), θ2 (i), α1 (i), α2 (i)) using eq.(2.3). • step8: set i = i+ 1. • step9: repeat steps from (3− 8) n times. then; -an approximate bayes estimate of r under squared error loss function is given as: r̃mhse = 1 n −m n∑ i=m+1 r(i). -an approximate bayes estimate of r under linex loss function is given as: r̃mhlx = −1 c log [ 1 n −m n∑ i=m+1 e−cr (i) ] . -an approximate bayes estimate of r under general entropy loss function is given as: r̃mhge = [ 1 n −m n∑ i=m+1 ( r(i) )−q]−1/q . where m is the burn-in units, n is the mcmc samples. 6. simulation study in this section, we mainly present some simulation experiments to see the performance of the mentioned methods for different sample sizes, (n,m) = (10, 10), (20, 20), (30, 30), (50, 50), (70, 70), (100, 100). we simulated 1000 complete samples from quasi lindely distribution with the parameter values; θ1 = 0.2, θ2 = 1.5, α1 = 2, α2 = 0.8 with true reliability value is 0.87399. we also compute the 95% confidence intervals of r based on the observed fisher information matrix. we compared the performances of the mle and the bayes estimates in terms of mean squared errors (mse’s). also two different techniques of bayesian estimation (importance, mcmc) are compared for different loss error functions. bayesian estimation for different loss error functions was proposed with many values of c, q such that; c1 = −3(lx1), c2 = 5(lx2), q1 = −3 (ge1), q2 = 5 (ge2). bayesian estimation studied under the informative gamma priors. for choosing suitable hyper-parameters, the experimenters can incorporate their prior guess in terms of location and precision for the parameter of interest. the gamma distribution for the priors has mean = a/b, and variance = a/b2. we assume a small value of prior variance (0.01), and take the mean equal to the true value of the parameter of interest. for each parameter prior we solve the two equations of the mean and the variance, we obtain the following values of copyright c© 2018 assa. adv syst sci appl (2018) 48 m. mohie el-din, a. sadek, shaimaa elmeghawry hyper-parameters : a1 = 4, a2 = 225, a3 = 400, a4 = 64, and b1 = 20, b2 = 150, b3 = 200, b4 = 80. we also computed the bayes estimates based on 11000 samples and discard the first 1000 values as burn-in. the maximum likelihood estimator and asymptotic confidence intervals of r for different (n,m) are obtained in table 6.1. bayes estimates of r using different techniques under different loss error functions are obtained in table 6.2. therefore, from this study of the simulation results we observed that: • the performance of the bayes estimators is better than maximum likelihood for all different sample sizes. • mean squared error(mse’s) for all estimation methods decreased as sample size increased. • as sample size increased, the asymptotic confidence intervals for r are improving, and their lengths are decreasing. that means the estimated reliability becomes in the most accurate interval. • when the sample size increased, both bayesian and maximum likelihood results become close to each other. • for bayes estimators, importance sampling technique gives less mse’s values, so it is better than mcmc technique for the same priors values, and same number of generated samples. • general entropy, and linex loss error functions gave less mse’s at specified values of c, q. as shown lx2, ge2 acheived the best results for mcmc, but for importance sampling technique lx1, ge1 are the best methods . table 6.1. average estimate (mean squared error) for mle, and average confidence length of the simulated 95%confidence intervals of r. all mse values are multiplied by 10−3 estimator mle c.i.l c.i.u c.i. length (10,10) 0.866095 0.734952 0.997237 0.262 (3.2129) (20,20) 0.878225 0.787222 0.969227 0.182 (1.1262) (30,30) 0.874831 0.797586 0.952077 0.155 (0.7609) (50,50) 0.875794 0.816741 0.93706 0.120 (0.459333) (70,70) 0.875149 0.823703 0.926595 0.103 (0.3349) (100,100) 0.874276 0.834959 0.919893 0.084 (0.1554) 7. real data analysis in this section we present the analysis of real data, introduced by singh et al. [18]. the data represent the waiting times (in minutes) before customer service of two banks a and b, respectively. the use of lindley distribution for the waiting times (bank a) data has been originally discussed by lindley [12]. since then, many authors have suggested the data under different set-up for lindley distribution. we are interested in estimating the stress-strength copyright c© 2018 assa. adv syst sci appl (2018) estimation of stress-strength reliability 49 table 6.2. average estimates(mean squared error ) of r for different bayes estimators under different error loss functions. all mse values are multiplied by 10−3 est. importance sampling mcmc technique (m,n) se lx1 lx2 ge1 ge2 se lx1 lx2 ge1 ge2 (10,10) 0.86713 0.86862 0.86457 0.86829 0.86346 0.88787 0.88908 0.88577 0.8888 0.88494 (0.7516) (0.7058) (0.8421) (0.7134) (0.8934) (0.7611) (0.7748) (0.7459) (0.77) (0.7477) (20,20) 0.87063 0.87147 0.86919 0.87128 0.86861 0.89199 0.89268 0.89081 0.89251 0.89038 (0.4769) (0.4611) (0.5075) (0.4637) (0.5239) (0.6747) (0.692) (0.6479) (0.6870) (0.6403) (30,30) 0.87073 0.87135 0.86967 0.87121 0.86925 0.89157 0.89208 0.89071 0.89196 0.8904 (0.3591) (0.3491) (0.3784) (0.3508) (0.3883) (0.5684) (0.5822) (0.5464) (0.5785) (0.5396) (50,50) 0.87168 0.87206 0.87104 0.87198 0.87079 0.88997 0.89032 0.88938 0.89023 0.88916 (0.2536) (0.2486) (0.2626) (0.2496) (0.2670) (0.4366) (0.4458) (0.4216) (0.4433) (0.4167) (70,70) 0.873434 0.873697 0.872995 0.873636 0.872827 0.88874 0.889014 0.88828 0.888946 0.888114 (0.2189) (0.217) (0.2226) (0.2173) (0.2244) (0.3755) (0.3823) (0.3645) (0.3805) (0.3609) (100,100) 0.875249 0.875418 0.874965 0.875378 0.874857 0.885949 0.886165 0.885586 0.886112 0.885455 (0.1384) (0.1387) (0.138) (0.1386) (0.138) (0.2432) (0.2477) (0.2359) (0.2465) (0.2334) parameter r = p (y < x) where x and y denotes the customer service time in bank a and b (data set 1, 2) respectively. the data sets are presented below: data set 1: x (n=100) 0.8, 0.8, 1.3, 1.5, 1.8, 1.9, 1.9, 2.1, 2.6, 2.7, 2.9, 3.1, 3.2,3.3, 3.5, 3.6,4.0, 4.1, 4.2, 4.2, 4.3, 4.3, 4.4, 4.4, 4.6, 4.7, 4.7, 4.8, 4.9, 4.9, 5.0, 5.3, 5.5, 5.7, 5.7, 6.1, 6.2, 6.2, 6.2, 6.3, 6.7, 6.9, 7.1, 7.1, 7.1, 7.1, 7.4, 7.6, 7.7, 8.0, 8.2, 8.6, 8.6, 8.6, 8.8, 8.8, 8.9, 8.9, 9.5, 9.6, 9.7, 9.8, 10.7, 10.9, 11.0, 11.0, 11.1, 11.2, 11.2, 11.5, 11.9, 12.4, 12.5, 12.9, 13.0, 13.1, 13.3, 13.6, 13.7, 13.9, 14.1, 15.4, 15.4, 17.3, 17.3, 18.1, 18.2, 18.4, 18.9, 19.0, 19.9, 20.6, 21.3, 21.4, 21.9, 23.0, 27.0, 31.6, 33.1, 38.5. data set 2: y (m=60) 0.1, 0.2, 0.3, 0.7, 0.9, 1.1, 1.2, 1.8, 1.9, 2.0, 2.2, 2.3, 2.3, 2.3, 2.5, 2.6, 2.7, 2.7, 2.9, 3.1, 3.1, 3.2, 3.4, 3.4, 3.5, 3.9, 4.0, 4.2, 4.5, 4.7, 5.3, 5.6, 5.6, 6.2, 6.3, 6.6, 6.8, 7.3, 7.5, 7.7, 7.7, 8.0, 8.0, 8.5, 8.5, 8.7, 9.5, 10.7, 10.9, 11.0, 12.1, 12.3, 12.8, 12.9, 13.2, 13.7, 14.5, 16.0, 16.5, 28.0. first,we checked the suitability of quasi lindley distribution for the considered real data sets. we, therefore, have provided the kolmogorov-smirnov (k-s),anderson-darling(a-d) and cramér-von mises statistics to test the goodness-of-fit of above data sets to the quasi lindley distribution. the fitting summary has been presented in table 7.1, which indicates that the qld fits well to data set 1 and data set 2. table 7.3. p-value (statistic) of different goodness-of-fit tests for data set 1, 2. k-s a-d cramér-von data set 1. 0.0654 (0.1290) 0.0217 (3.2033) 0.0501 (0.4610) data set 2. 0.9287 (0.0677) 0.965 (0.2597) 0.9310 (0.0404) based on the mles θ̂1, α̂1, θ̂2, α̂2 the point estimate of r is 0.59 and the 95% confidence interval of r is (0.25, 0.93). for real data sets, the maximum likelihood and bayes estimates of the stress-strength parameters and reliability are summarized in table 7.2. copyright c© 2018 assa. adv syst sci appl (2018) 50 m. mohie el-din, a. sadek, shaimaa elmeghawry table 7.4. the mles and bayes estimates of stress-strength parameters and reliability r from real data sets est. θ̂1 θ̂1 α̂1 α̂2 r̂ mle 0.1 0.27 84.09 0.41 0.59 bayesimp 0.13 0.59 1.44 0.66 0.81 bayesmh 0.1 0.54 22.13 0.69 0.77 8. conclusion in this paper, maximum likelihood and bayesian estimation methods for stress-strength reliability r were discussed, when x and y both follow a quasi lindley distribution with different parameters. we obtained the 95% confidence intervals of r based on the observed fisher information matrix. we proposed the bayesian estimation based on independent gamma priors under different error loss functions(squared, linex, and general entropy). we suggested the importance sampling, and mcmc techniques to generate samples from the posterior distributions and then compute the bayes estimates. simulation study has been introduced to investigate the performance and compare among all mentioned methods. simulation results suggest that the performance of the bayes estimator is better than maximum likelihood for all different sample sizes. also, maximum likelihood method provides very satisfactory 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(2010). bayesian estimation of the shape parameter of the generalized exponential distribution under different loss functions. pak. j. statist. oper. res, 62, 163-174. 17. shanker, r., sharma, s. & shanker, r. (2013). a quasi lindley distribution. african journal of mathematics and computer science research, 6(4), 64-71. 18. singh, s. k., singh, u., & sharma, v. k. (2014). estimation on system reliability in generalized lindley stress-strength model.journal of statistics applications and probability, 3(1), 61-75. 19. soliman, a.a. (2000). comparison of linex and quadratic bayes estimators for the rayleigh distribution. commun.statis.theory meth, 29(1), 95-107. 20. varian, h.r.(1975). a bayesian approach to reliability real estate assessment. amsterdam, north holland, 195-208. 21. zellner, a. (1986). bayesian estimation and prediction using asymmetric loss functions. journal of american statistical associations, 81(394), 446-451. copyright c© 2018 assa. adv syst sci appl (2018) advances in systems science and applications (2012) vol.12 no.1 67-75 the research on effective video scene character extraction algorithm in natural scene images jinghua hu1 and huafeng kong2 1wuhan university of technology, wuhan 430070 2key laboratory of information network security, the ministry of public security, the third research institute of ministry of public security, shanghai 201204 abstract pcb technology is particularly important in electronic industry. however, the increasing of technology complexity makes it difficult to inspect the quality of pcb. aoi system greatly improves the pcb inspection efficiency, and it has become one of the most instructive subjects that how to identify the text information in the chip image rapidly and accurately. since the chip images have complex natural background, and are greatly affected by light, shadow, noise, font and size, texture, color, position and arrangement., it is often difficult to inspect, extract and identity the text. this paper presents an effective algorithm of scene character segmentation and recognition in natural scene images. the algorithm segments lines of character, then segments every line of character into individual words for further processing, such as feature extraction and character recognition according to the known features of character. after the segmentation of character, we use an advanced bp algorithm to recognize the character. it improves bp mainly through restructuring gradient in the sigmoid. the experiment shows a significantly improvement in pc: the recognition accuracy rate of this system is above 96% and the response time is 5ms/100 words. keywords scenes character segmentation, feature extraction, bp algorithm, and image processing 1 introduction as the optical character recognition (ocr) technology comes in vogue, many scholars begin to research on the character extraction from the document images. till the 1990s, during the rapid development of computer technology and multimedia technology, the content-based multimedia retrieval has become a research hotspot. then, the character extraction in the natural scene images has again gradually become one of the research hotspots. usually, the natural scene image character has substantial changes in font, size, color, alignment and arrangement; also with the complex character background, low resolution and high noise. furthermore, many systems also require the algorithm has higher processing speed in the application. all of these causing difficulties to effectively extract the characters from the natural scene images, especially for the natural 68 jinghua hu:the research on effective video scene character extraction algorithm... scene images based on the videos. many domestic and foreign scholars have made beneficial explorations and attempts in this field. y.zhong[1] firstly put forward the resolution that position the characters in the complex images. however, this resolution mainly aims at the character positioning in the image scanning for the colorful disc covers, and fails to be directly applied into the natural scene images. a.k.jai[2] and others proposed a kind of character positioning method, which applies to the newspaper, web pages and general images and video frames, but not ideal to identify the small character fonts. m.a.smith[3] and others developed a kind of method detecting the characters on the images. however, because of the limitation in size, this method can only detect the characters within the specific font scope. it can not use the feature that the same characters will appear in multi frame to further enhance the character detection performance. also, the word segmentation prepared for the ocr is not done. sato[4] and others developed a character segmentation and recognition system aiming at the static low resolution headlines. the system will take advantage of the method mentioned in literature[3] to recognize the character. then magnify as 4 times as the recognized characters, which will use the smallest time-based image pixed value to conformify the headline. such system has a good effect on news program. r.lienhart[5-6] and others have developed systems in character detecting, identifying, recognizing successively. the color-based algorithm system of inter infiltration segmentation adopted in literature[5], only deals with image segmentation and recognition in single frame, without consideration in successive images in multi frame. however, the system in literature [6] takes further consideration on texture feature in title character, and tracking and conforming in successive images in multi frame, which makes it a better recognition in ocr. character recognition has a close connection with character segmentation which is one of critical factors in character recognition[7]. since the threshold process in natural scene image tends to bring about noise and low quality problems, and probably results in character conglutination. such traditional character segmentation will not be satisfied with people. in order to identify characters in the natural scenes fast and accurately, localizing the chunks should be taken as first priority, and later extracting smaller chunks. therefore, the small chunk could be used respectively to sharpen the noise tremendously, without the complex impact from other parts in images.as a result, this paper presents a algorithm of scene character segmentation and recognition in natural scene images, based on feature feedback, which will segment chinese, english and adherent character. after the segmentation of character, an improved bp algorithm is afterwards used to recognize the character. advances in systems science and applications (2012), vol.12, no.1 69 2 feature feedback based algorithm of character segmentation & recognition between chinese, english and adherent character several traditional methods in character segmentation & recognition[8] 1⃝ imagebased analysis. search for rational segment point between characters, chiefly adopt static projection analysis; 2⃝ recognitionbased. select various kinds of current segmentations via identifying ability before actual segmentation. 3⃝ synthetical based on image analysis and recognition. reduce and filter vertical segmentation hypothesis by image analysis. 4⃝ unity recognition. use the whole word as recognizing object, according to the features of the whole word, to avoid segmentation harm on character. however, in the case of word conglutination, those methods can not segment adherent characters accurately. this paper deals with this kind of situation by an algorithm of feature feedbackbased character segmentation and recognition between chinese, english and adherent characters. first segment each line of character, and then search and extract the adherent character in each line, finally transfer small segmenting module to discover the most reliable segment point, so as to achieve the segmentation purpose.key points as follows: 1.identify the category of adherent characters after initial identification, the recognition and length of each image as follows: 1⃝ suppose the length of each character image is w, and the average length of each character image is w. when w is much wider than wv, and pick up a threshold via experiment, we could identify whether it is adherent character or not. 2⃝ after identifying adherent character image, according to english character image length is wider than the chinese character image length, and the separation distance is small between the adjoining characters, we could identify whether the adherent character image is chinese or english; according to average english adherent character length is shorter than the chinese one, we could identify those character images with wrong structure as adherent character image. 2.identify the category of adherent characters for the adherent chinese character, we can segment the chinese character by judging the potential character image length. we use identifying module to judge the border to segment the character. in this process, we confirm the most reliable segmenting point by right-to-left and left-to-right searching methods.algorithm as follows: step1, record the value of left border bl, right border br and line height hl. step2, cobfirm x0 value (x0 = hl)from the left border, choose one threshold value t0, the segmenting point should be picked up from ( x0-t0 , x0+t0), modulate it according to steplong. 70 jinghua hu:the research on effective video scene character extraction algorithm... step3, transfer identifying module, and judge whether the result is in the range or not. if not , do step2; if is , do step 4. step 4, save segmenting result br1, take br1 as the left border , and then do step2 and the following steps successively, till the right border br. to compare the validity of the final segmenting result, do all steps in turns from the right border, and save the best segmenting result. 3.segment adherent english number for the adherent english number, first use border searching algorithm to segment these adherent character with gaps in between, which can not be segmented by projection method. then suppose the potential height of each adherent character string is x, according to x, transfer the identifying module to segment the adherent character string so as to locate the segmenting point. border searching algorithm as follows: step1, take note of left rect urlli ,tiand right rect urlri ,bi, and pick up a threshold value t,find a p point whose gray-scale value is 1, and take it as initial point of the contour line. step2, start with the fund point and continue with the searching job. if the gray-scale value is 1, please carry on to the left; if the gray-scale value is 0, please carry on from the right turn. once meet the point whose gray-scale value is 1, these points would be contour point. step3, repeat step 2 till the contour line point coincides with the initial point; step4, confirm the right border via contour line searching. if br1 coincides with the right border, there is no adherent character; if not,for identified character we can pick up a suitable threshold via experiment, and change it to the left or eight. then use it as the segmenting position and save the result.if character width to height ratio is large, the character is adherent; if the ratio is relatively small, we could transfer the identifying module to check it is or not, after it being segmented. if it is adherent we can calculate its potential height value, according to whose height we adjust a threshold value and use it as its possible width. then we take advantage of identifying module to segment the adherent character and confirm the segmenting position via identified result. save the correct final segmenting result. the english character baseline feature as follows: for the adherent english number, if the line height is close to 5/6 of chinese fig.1 english character baseline feature line height, there are up-protruding letters (e.g. t) as well as down-protruding letters (e.g. p). we confirm the baseline approximately as much as 1/4 as adheradvances in systems science and applications (2012), vol.12, no.1 71 ent character height, and height x is approximately as much as 1/2 of adherent character height; if the line height is close to 2/3 of chinese line height, there are up-protruding letter or down-protruding letter. we confirm the baseline approximately as much as 1/3 as adherent character height, and height x is approximately as much as 2/3 of adherent character height; if the line height is close to 1/3 of chinese line height, there are neither upprotruding letter nor down-protruding letter. we confirm the baseline approximately as much as 1/2 as adherent character height, and height x is approximately as much as adherent character height. and we can adjust the potential segmenting position to meet height x via threshold-adjusting. and the same method could be used in english number adhesion and chinese adhesion. feature feedback based algorithm of character segmentation & recognition between chinese, english and adherent character identify result as fig.2 and fig.3. fig.2 english character segmenting result fig.3 chinese character segmenting result 3 character size normalization after being segmented,each character is in different size; therefore the segmented character region is different.in order to extract the character feature,we need to normalize the segmented character region. suppose height and width to represent respectively normalized character vertical and horizontal length. and h and w respectively represent height and width of the character rectangle. therefore, the ratio of vertical to horizontal hr and wr would be: hr = height h ,wr = width w 72 jinghua hu:the research on effective video scene character extraction algorithm... then we make coordinate mapping, and we suppose coordinate (i new,j new) is the mapped coordinate (i,j) after it has been normalized. similarly we map the following coordinates: i new = top+ (i− top)/hr, j new = left+ (j − left)/wr note: top is the vertical coordinate of the point a in the mornalized character region, and left is the horizontal coordinate of the point a in the mornalized character region. the character size normalized result is shown in fig.4. assign the gray-scale value of the character coordinate (i,j) before being normalized to the normalized character coordinate (i new,j new) to accomplish the character normalization. (a) before normalization (b) after normalization fig.4 contrasts two normalization results 4 improvement of bp algorithm base on the traditional bp algorithm, this paper presents an improved algorithm which improves bp mainly through restructuring gradient in the sigmoid. if the value is supposed too much, the input 0 or 1 of each layer would be scattered, and the study result is worse. otherwise, if too little, the system linearity is enhanced whereas nonlinearity is weakened. therefore, the best value is between the above both. the most important point is to amend the gradient of each allnodes in the sigmoid so as to find a best value. in the improved bp algorithm, sigmoid is turned into binary driving function on the connection of gradient and the potential value of the junction point: φ(α, υ) = 1 1 + e−∞ we can easily deduce that after all the amendment, adjusting method of the weight and threshold is the same with bp algorithm. suppose the actual output vector o = {o0o1, . . . , ol}, teacher vector t = {t0, t1, . . . , tl}, mean square error e: e = 1 2 ∑ k (tk −ok)2 advances in systems science and applications (2012), vol.12, no.1 73 gradient α is changed according to e ′ negative gradient change, where mean square error e reduces the least. hence the improved gradient algorithm would be: ∆α∞ = ∂e ∂α the node point of the output layer is: ok = 1 e−αk(µk−θk) ,then ∆αk = −η ∂e ∂αk = −η ∂e ok · ∂ok ∂αk = −η(tk −ok) ∂ ∂αk 1 1 + e−αk(µk−θk) = η(tk −ok)(µk − θk)ok(1−ok) η is studying step length,okis the actual output of output layer neuron k,tkis teacher signal,µkis the input signal linear combination of the output layer node, then µk = ∑ k hjwkj ,wkj is weight;θkis threshold;hj is the input signal of output layer nodepoint, that is the output of intermediate layer neuron. for the node point of ∆αj = −η ∂e ∂αj = −η ∂e ∂hj · ∂hj ∂αj hidden layer ∂hj ∂αj = ∂ αj · 1 1 + e−αj(µj−θj) = (µj − θj)hj(1−hj) ∂e ∂hj = ∑ k ∂e ∂µk · ∂µk hj = ∑ k ∂e ∂µk ·wkj ∂e ∂µk = ∂e ∂ok · ok µj = −(tk −ok)αkok(1−ok) substitute in turns would be: ∆αj = η(µj − θj)hj(1−hj) ∑ k (tk −ok)αkok(1−ok)wkj using driving function via the improved gradient, we can adjust the gradient of sigmoid to its best in the process. to study the photo sample, we memorize the character feature in the sample and find the weight and threshold, and then we establish and improve bp neural network. then we can input a photo, using bp neural network, to find a closest character to the input chinese character in the sample. the steps as follows: (1)network topologymost characters to be identified are chinese characters. because the chinese character structure is quadrel, we divide them into 26×26 unit. each unit corresponds to one input; hence the output nodes in the neural 74 jinghua hu:the research on effective video scene character extraction algorithm... network would be 26×26=676. the numbers of the output nodes in the neural network is the same with the characters to be identified. (2)output node confirmnesscalculation of node output is equivalent to calculation of the function sigmoid output. because of the functions nature, we would be aware of its value range [0,1]. the closer the value is to the limits, the less sensitive the function value changes to its independent variable. therefore, we can enhance the nodes precision by its output-to-input sensitivity. we suppose t is 0.9, and f is 0.1; for input, if with stroke, then input 0.9, if not, we input 0.1. and we deal with output the same way. (3)edit of training samplethe more samples the collection owns, the more corresponding sample each chinese character has. also the better identifying ability the neural network is after being trained, however, the more time the training takes. here we only take the sized 18 thick song ti character as sample. there are english characters, numbers, 3,775 chinese characters of one-level, and basic punctuation in the sample collection. (4)experiments and analysis the identifying result of bp-algorithm-based character identifying algorithm is shown in fig.5. this system is designed to deal with images including character, whose recognition rate is above 96%, and its response time is 5ms / 100 words. compared to the traditional recognition method it has been improved, whose recognition result is better. however, the worse the images quality will result in the worse the recognition result. in the image treatment phase, the wiping-out of useless information is partial, which has an effect on character recognition and the whole recognition result. fig.5 results of character identifying algorithm 5 conclusion according to the above discussion, the identifying result of bp-algorithm-based character identifying algorithm can improve the efficiency of ic identifying in aoi system. advances in systems science and applications (2012), vol.12, no.1 75 references [1] yu zhong, k.karu, a.k.jain. (1995), “locating text in complex color images”, in proceedings of the third international conference on document analysis and recognition, vol.1, no.14, pp.146-149. [2] a.k.jain, b.yu. (1998), “automatic text location in images and video frames”, pattern recognition, vol.13, no.12, pp.2005-2076. [3] m.a.smith, t.kanade. (1995), “video skimming for quick browsing based on audio and image characterization”, 3technology report cmu-cs-95-186, pp.13-15. [4] t.sato, t.kanade. (1995), “video ocr: indexing digital news libraries by recognition of superimposed caption”, multimedia systems, vo.l7, no.5, pp.385–395. [5] h.j. zhang, c.y. low, s.w. smoliar, and j.h. wu. (1995), “video parsing, retrieval and browsing: an integrated and content-based solution”, proc. acm multimedia , pp.15-24. [6] r.lienhart, w.effelsberg. (2000), “automatic text segmentation and text recognition for video indexing”, multimedia systems, vo.l8, no.2, pp.69-81. [7] huiping li, omid kia, david doermann. (2004), “text enhancement in digital videos”, proc. spie99-document recognition and retrieval, pp 3651:2-9 [8] casey r g, lecolinet e. (1996), “a survey of methods and strategies in character segmentation”, ieee transactions on pattern analysis and machine intelligence, vol.18, no.7, pp.690-706. advances in systems science and applications (2014) vol.14 no.4 302-324 the paradoxes of the world’s progress (i) sailau baizakov1 and jeffrey yi-lin forrest2 1jsc “iei” economic development and trade of kazakhstan, jsc “economic research institute”, str. temirkazyk 65, 0100000, astana, kazakhstan 2department of mathematics, slippery rock university, slippery rock, pa 16057, usa abstract an unbiased systematic view on the modern innovation allows you to see three key groups of innovations that are still very few people differentiate. the first is the technical and technological innovations that are the basis of development and change in technological ways of the world. secondly, it is monetary and financial innovation, the progress of which determines the change in monetary ways of life of the world. and third, it is the socio-political innovation, progress of which is in the basis of the change of socio-political models of the world. a clear distinction between these three “floors” of innovations is crucial for understanding the global crisis and the ways to quit it. since the essence of it lies in the tangle of clearly long overdue and contradictions between the rates of introduction of the world’s technical and technological, financial and socio-political innovations. as the global crisis clearly shows, technical and technological way today, is not decisive for the country’s prosperity and peace. exactly the countries with the highest level of technology development have become the main source and a key cause of the global crisis. keywords global crisis, sona analyzer, technical and technological innovation, monetary and financial innovation, socio-political innovation 1 analyzer sona the system weapon of independent analysts of kazakhstan 1.1 why the global crisis can not be regarded as finished ? the tone of the work on the vi astana economic forum was given by the president of kazakhstan. his conclusion is that the global crisis can not be regarded as finished. nazarbayev believes that the crisis now acquires explosive features, with local manifestations, such as in cyprus. what is the reason ? according to the president of our country, this is the artificial “inflation” of financial bubbles and desire for “easy money”. this is the lack of proper responsibility of national financial institutions, the ineffectiveness of anti-crisis measures because of the weakness of the mechanisms of global financial governance. anti-crisis measures taken at national level by the dictating of the imf and other un institutions themselves become the cause of recession. thus, the pact of growth and stability, adopted recently in europe, did not give real results. a year advances in systems science and applications (2014) vol.14 no.4 303 has not passed since the pact ; there is a new pact on the promotion of growth and competitiveness preparing. headache for leaders of european countries is unemployment, a large proportion of non-employed people among youth. to accept radical solutions to the global economy, sufficient will and responsibility have not yet been manifested. there are neither effective global anti-crisis mechanisms, nor reliable global reserve currency. the rules of international consensus were not worked out, meeting the mutual interests of global financial institutions and nation-states actors of the financial sector and the real economy of the world. any pact of economic growth and stability will be reduced to practice of the “patching holes up”, if the boundary of the regional union, as in the european union, expanding blindly, mechanically, without regard to the interests of each country separately. the conclusion of the president of the republic of kazakhstan is original : the world needs a new economic model of governance, which should be based on welldeveloped tools of their integration. thus, the financial sector can not and should not develop in isolation from the real production at all levels of management. and, it is important to study and solve the problem of monetary system off-shore in some countries. according to n. nazarbayev, the current global crisis is multidimensional. its speed and dynamics are determined not only by the economic but also the political, humanitarian and moral-value causes. in the end, the president of kazakhstan proposed to begin work on the pact of global regulation. it is said about the multidimensional innovative tool for construction a new global financial system and the creation of global regulator, which determines a uniform playing rules. the basis of its design is capacity of five simple and clear principles : i evolution and rejection of revolutionary change in policy ; i justice, equality, consensus ; i global tolerance and trust ; i global transparency ; i constructive multipolarity. 1.2 three types of innovation in the economy of the world : analysis and problem there is an urgent objective need for a number of analytical researches, including but not beyond one-sided judgments about the benefits of the existing models of management of the national economies of individual countries, such as anglosaxon. today’s rapidly changing world of market forces in the world economy makes innovation management to lift to new heights and to achieve consistency with the development of its real and financial sectors. the steady growth of the national economies of individual countries and the building of its respective management model, as pointed out by n. nazarbayev in his article “the fifth way”, are closely related to the harmonization of the rates 304 sailau baizakov, jeffrey yi-lin :the paradoxes of the world’s progress of development of the three types of innovations : type 1 innovations in the production of real goods ; type 2 innovations in the field of renovation of monetary system ; type 3 innovations in the field of social and political life of people, business, organization, region, etc. of course, the economic model of the future, that will enable implementing the principles of mutual coordination of rates of development of these types of innovations, defines a roadmap for sustainable post-crisis economic growth of not only individual countries but for all countries in the world[1-3]. market economy itself is based on the principles of competitiveness and innovation development in the production of real goods and services. due to the modern information and computer technology there is a scope for the smooth implementation of the first type of innovation in the real sector of the economy. there are no obstacles to the development of innovation type 1, except legal obstacles and restrictions on private sector development and further liberalization of its work. as for the second type of innovation, now in most countries of the world unlimited possibilities for the development of innovative technologies in the monetary and financial system are created. here, same as in production, there are no obstacles in the way of innovation type 2. the main obstacle, which is now holding back the potential of both of these types of innovative development, is the third type of innovation. innovations in the field of social and political life of people, business, organization, region, etc. in most countries are based on a mechanical copying the anglo-saxon model and implementation of the principles of the washington consensus. but these models and principles of its construction have already reached their limits and were inadequate for preventing the crisis manifestations and overcome the consequences of the crisis. this explains the incompleteness of the global crisis, the permanent duration of local manifestations and long term duration of its consequences. thus, the stagnation in the development of economic science and the lack of technical technological innovation in governance does not enable control on the gaps between turnover of money and goods flows between the rates of development of the real and financial sectors of the economy, not to mention the innovative rate of development of tools of economic management. the rate of the third type of innovation and the development level is the weakest link, which is an obstacle in the whole chain types of innovation development of economy and ensuring its financial stability. at the moment, the only obstacle in the way of innovation development of the world economy is exactly the type of innovation type 3, innovations in the field of socio-political life of people, business, organization, region, etc. an innovative tool type 3 analysis and management is needed not only at the advances in systems science and applications (2014) vol.14 no.4 305 level of macroeconomics : it is needed in each area of the economy from home economics, including the economics of the business sector and ending with the national economy. otherwise making the systematic and right decisions excluding the formation of imbalances in the implementation phase of projects and programs are not possible. to date, the initiative group of kazakh analysts developed a system tool with innovative components that meet the principles of “fifth way” realistic “estimate, measure, exchange, transfer the true cost of goods and services”[1]. this versatile tool of economic management is analyzer sona, which is fully belongs to the “new financial instruments of a new quality : the real measuring instruments of cost of goods and services”[ibid]. 1.3 three indexes of economic development and financial stability sona analyzer is able to determine the rate of balanced growth (growth index i3, which determines the level of innovation of the type 3), which will connect the index of the nominal growth rate of financial stability i1, which determines the level of innovation of type 1 with an index of the rate of real growth in the sphere of production i2, which determines the level of innovation type 2. thus, the establishment of a balanced growth rate measures the gaps between the development of the real sector and the financial system. and measurable economy is manageable economy. what does the index for balanced growth mean ? the answer to this question, just like the question of what the index of the real growth means is directly related to the concept of index of nominal growth. so first you should understand the index of real growth. if the index of the nominal growth is determined by the ratio of nominal gdp in the prices of the current year t to nominal gdp of base year 0 by the formula i1(t) = ngdp (t)/ngdp (0), then the index of real growth rate is determined by the index of the physical volume of a good or service by formula i2(t) = rgdp (t)/rgdp (0). the first index is the gdp growth rate in prices of current year. but the economic content of the index is determined not only by market prices of the current year, but the nominal value of the national currency of the country concerned. this is important, the fundamental advantage of a market economy compared with a directive command economy, which is focused on the principles of hard pricing. it should be noted that the index of growth of nominal gdp is expressed in nominal charge of the national currency of current year. thus, the nominal price of the national currency of kazakhstan in 2008, according to official statistics was 120.30 tenge per u.s. dollar, and three years later, in 2010 it was 147.35 tenge per u.s. dollar. its rate of growth for only three years was 122.5%. the second index, which is called the index of the real growth, is the growth 306 sailau baizakov, jeffrey yi-lin :the paradoxes of the world’s progress rate of gdp in 2010, but prices of the base year 2008. the official statistics, using special methods for calculating, transfers nominal gdp of current year to gdp of the same year, restated to prices of the base year. in our example, the nominal gdp in 2010 prices is transferred to gdp-2010 at 2008 prices. so far, to ascertain the nature of the gdp deflator, we have two gdp growth rates of 2010 relative to the same, letąŕs say, a base 2008. one is index of gdp growth in the nominal value of the national currency of the current year, the other is in the nominal value of the national currency of the base year. if we define the growth rate of nominal gdp relative to 2008, we have growth index i1(t) = ngdp (t)/ngdp (0), and if we define the growth rate of real gdp relative to 2008, we have index i2(t) = rgdp (t)/rgdp (0). basis for calculating both indexes of growth is 2008 and their performances in the base year equal to each other : rgdp(0) = ngdp(0). both of these figures represents a monetary phenomenon, and except tenge, kazakhstan national currency, does not contain any ounces of gold, or a watt of energy, not an ounce of food or a basket of rational budget or one unit of another substance in nature, and not even a single unit foreign currencies, including the u.s. dollar. one more step to determining the rate of balanced growth brings us the concept of the nature of the gdp deflator. denoting index of gdp deflator to official statistics p(t), we can express its meaning in classical method : p(t) = i1(t)/i2(t), which measures the gap between the nominal and real indexes of growth. analysis of the formula of the gdp deflator shows that at equal rates of growth of nominal gdp and real gdp (i1(t) = i2(t)), the gdp deflator is constant and equal to one. the rate of growth of the real sector and the financial system are in an equal distance from the bisector of the coordinate system (x, y). this equilibrium line of growth rates in both sectors of the economy represents an ideal balanced growth dynamics the index i3(t). of course, the nature of the change of each index of growth is influenced by its source of innovation development. thus, the source of market forces development of the growth rate of nominal gdp of financial stability i1(t) is the potential of innovative development of the monetary system. and source of market forces development of the growth rate of real gdp i2(t) is the potential of the innovative development of the production in the real sector of the economy. on the equilibrium trajectory of the development rates of the real and financial sectors of the economy equality of not two, but three indexes of growth is attained. this third index, as already indicated as i3(t), is the index of balanced growth. the function of this third type of innovation is to ensure the comparability of development the real and financial sectors of the economy not only in the indexes of prices of goods and services, but also on the purchasing power of the national advances in systems science and applications (2014) vol.14 no.4 307 currency, to provide scientific management by the first two types of innovation. however, in reality a balanced of growth rate (i3(t)) does not coincide with any index of real or nominal growth, and may be freely deflected to either side of the line of the ideal dynamic equilibrium. let his deviation from the growth rate of nominal gdp is α : α ∗ i1(t) = i3(t), and the growth rate of real gdp is β : β ∗ i2(t) = i3(t). the growth rate of the economy i3(t) on the equilibrium line αi1(t) = βi2(t) is said to be the dynamics of the rate of balanced growth. how accurately does a triple measurement tool for economic growth meet the criteria of democratic governance and the principles of liberalization of the market economy ? the section of the article“the fifth way” n. nazarbayev “the paradox of the global progress” will help to answer this question : “an unbiased systematic view on the modern innovation allows you to see three key groups of innovations that are still very few people differentiate. the first is the technical and technological innovations that are the basis of development and change in technological ways of the world. secondly, it is monetary and financial innovation, the progress of which determines the change in monetary ways of life of the world. and third, it is the socio-political innovation, progress of which is in the basis of the change of socio-political models of the world. a clear distinction between these three “floors" of innovations is crucial for understanding the global crisis and the ways to quit it. since the essence of it lies in the tangle of unsolvable contradictions between the rates and levels of development of technological, financial and monetary, social and political structure of each country and world as a whole. or, in other words, it is the tangle of clearly long overdue and contradictions between the rates of introduction of the worldąŕs technical and technological, financial and socio-political innovations. as the global crisis clearly shows, technical and technological way today, is not decisive for the country’s prosperity and peace. exactly the countries with the highest level of technology development have become the main source and a key cause of the global crisis ”[1]. 1.4 the transformation of the gdp deflator into indicators of balancing real growth rates of production areas with the rate of monetary and financial system remains to find out what forces withdraw the initial equity indices of growth in both sectors of the economy in 2010 from balance, when in the base 2008 they were equal : rgdp(2008) = ngdp(2008). no force will make us to consider nominal gdp balanced to real gdp in 2010, when at the point p (2010) = 113.4% economy has become an equilibrium with only two indices of growth i1(t) and i2(t) : 113.4 ∗ i2(t = 2010) = i1(t = 2010). at this point, equilibrium in the face 308 sailau baizakov, jeffrey yi-lin :the paradoxes of the world’s progress of the national currency of kazakhstan has not yet reached because in 2008 the price was 120.30 tenge per u.s. dollar, and three years later, in 2010 it was 147.35 tenge per u.s. dollar. what kind of balance can be discussed if the price of the national currency in only three years has deviated from its base by 22.5%, and the magnitude of the gap was even 8 percentage points more than the value of the gdp deflator (inflation rate) for the same years that’s why n. nazarbayev in his “paradox of the global progress” highlights the need to overcome “tangle of clearly and long overdue contradictions between the rate of introduction of the world’s technical and technological, financial and socio-political innovation” [1]. according to claims of training manual of sachs and laren on the economy, the gdp deflator used to calculate the inflation rate in 2010 is not the same as the index of growth of actual prices of goods and services of that year : “note j. sachs and f. laren support our conclusion that we calculate the price index indirectly. at first, we take the nominal gdp (ngdp) in current prices, then find a real rgdp in constant prices, ie q = rgdp. therefore price deflator calculated in this way is sometimes called implicit price deflator of gdp ”[13]. only the determination of the nature of the gdp deflator (inflation index) will help to understand the “tangle of unsolvable contradictions between the rates and levels of development of technological, financial and monetary, social and political structure of each country and world as a whole” which was mentioned by president [1]. to assess those market forces that unbalance the economy, we assume that the gdp deflator in 2010 represented the integrated expression of two opposite market forces. the first of these forces α = pp(t)represents purchasing power of national currency, and the second one β = c(t)represents market force of technical and technological structure of modern production sphere, expressed by the ratio of ntp in the real economy. as a result of this assumption, the rate of balanced growth is represented by two components of the gdp deflator, one of which serves as the weighting factor of equilibrium with the rate of nominal gdp growth (growth index i1(t)), and the other is with the rate of growth of real gdp (growth index i2(t) ). now easy to prove that the rate of growth of the real sector with a weighting factor ntp c(t) is balanced with the rate of growth of the financial sector of the economy on the line i3(t) : c(t) ∗ i2(t) = pp(t) ∗ i1(t). thus, the objective necessity of studying the nature of the gdp deflator does not derive only from the assessment of purchasing power of the national currency, but also from the assessing the contribution of scientific and technological developments in real growth rate of production of material goods. advances in systems science and applications (2014) vol.14 no.4 309 1.5 the function of scientific and technological progress however, the theory of economic growth does not allow us to calculate the size of the contribution of scientific and technical progress, the more the contribution of innovations to economic growth. thus one of the most important conclusions drawn from the theory of “romer and lucas according to russian sources for economic research is the fact that the economy, which manages large resources of human capital and the development of science in the long run is more likely to increase than economy, not having these benefits” [5]. it emphasizes only opportunities and points to the overall usefulness of technical and technological improvements. an attempt to solve the problem of estimating the contribution of stp, independently as the effect of investments in fixed assets at the time was made by n. kaldor and j. mirrlees [6]. james mirrlees is an active member of astana economic forum in recent years, the nobel prize in 1996. thus, the formulation of this problem in the model of kaldor mirrlees is stated as follows : let the balanced growth path is described by the interrelated functions of exponential following type [7-8] : gt(x) = g0(x)e z(x)t where g0(x) required initial level of indicator x to the balanced growth path ; gt(x) level of indicator x on the balanced growth path at time t ; z(x) required parameter value that determines the rate of increase in the index x on the same trajectory. kaldor mirrlees model forms a system of 11 equations with 11 unknowns. the peculiarity of this system is the presence within it of the equation that determines the function of scientific and technical progress. the function of scientific and technological progress is a performance increase depending on the growth of capital-labor ratio. in the simplest case, the function of scientific and technical progress is linear. in the latter case, the rate of balanced growth λ = z (x) is determined from the linear regression equation : φ t φt = σ1 + σ2 i′t it ;λ = it it = w′t wt = σ1 1− σ2 (1) σ1, σ2 > 0, σ2 < 1 where σ1, σ the coefficients of the regression equation. the economic content of the regression equation (1) reduces to the fact that with increasing intensity of labor productivity increases, but to a lesser extent, because it is assumed σ2 < 1. in fact, this limitation restricts the border of the 310 sailau baizakov, jeffrey yi-lin :the paradoxes of the world’s progress use of stp when the σ2 < 1 condition is not satisfied. models of this type do not take into account the explosive effects of individual scientific and technological activities. and because we believe that the contribution of scientific and technological progress allows us to understand more deeply the following two indicators. the first of them is the technical and economic level of production, which expresses the qualitative characteristics of scientific and technical activities at any given point in time. the second is effectiveness of the scientific and technical solutions that characterizes his return [13]. due to this mating pair of economic indicators a realistic model of the contribution of scientific and technical progress in the economy can be built. and the models as of the kaldor-mirrlees and other macroeconomic models are mostly theoretical and unlikely in the near future will find practical expression in economic management. however, the nbrk in its analytical work practices using the following econometric models, including : inflation equation, which is determined by the model : dlog(cpi)=0.3*diog(cpi(-1))+0.2*d(ulc)+0.02*gap+ +0.2*dlog(p_imp)-0.3*(log(cpi)-1.3-0.4*ulc-0.2*log(p_imp)), • cpi consumer price index,%, december1999=100 • ulc labor costs per unit of output • p_imp import price index, 2000q1=100 • gap deviation of gdp, equation of the gdp deflator by the other model : dlog(pgdp)=-1.3*(log(pgdp)+6.5-0.9*ulc-0.3* log(p_imp(-1)))+0.4*dlog(pgdp(-1))+0.3*dlog(pgdp(-4)) • pgdp gdp deflator,%, 2000 q=100 • ulc labor costs per unit of output • p_imp import price index, 2000q1=100 since the nbrk for determining an indicator of inflation applies one econometric regression equation, and to determine the gdp deflator applies other economic regression equation, it is easy to estimate the difference between them and it can be represented as a contribution to the scientific and technological activities. but the difference between two related indicators obtained in this way will be far from its real value. shown here a brief overview of the features of scientific and technological progress and econometric models of the gdp deflator (inflation index) allows us to understand the nature of the gdp deflator. so, in terms of content gdp deflator is different from the inflation index. inflation, as we know, is an immediate harm to the sustainable development of the market economy. however, the methodology for determining the inflation index and the gdp deflator is still single. in teaching aids and textbooks of such famous authors as sachs, dornbusch, advances in systems science and applications (2014) vol.14 no.4 311 mcconnell, menkyu these terms are used interchangeably and even are written together “gdp deflator (inflation rate).” however, the gdp deflator and the inflation rate needs to be systematically studied as indicators having different roles in determining the rate of balanced growth. thusin the construction of the analyzer sona, the contribution of stp is determined by a formula, the purchasing power of money to a different formula, and the gdp deflator by the third formula. these formulas of three mutually independent indicators allow to estimate the contribution of each of the real and financial sectors on the dynamics of qualitative and quantitative parameters of the development of the national economy. 1.6 analyzer sona is a versatile tool to support economic management analyzer sona is versatile tool that allows assessing, firstly, the contribution of scientific and technological developments in the real economy, and secondly, the purchasing power of the national currency, not only of kazakhstan but also the u.s. dollar, as well as the national currencies of other countries. the strength of analyzer sona is that the foundations of its construction are the laws of political economy. in the words of british economist john stuart mill, the laws of political economy act like the law of gravity, which is “without any compunction breaks neck to the best and dear man”, if he does not care about the consequences of the law of nature. unfortunately, in most countries of the world, the economy is dominated by legal laws that do not always conform to the objective economic laws. in some countries, political reform is not linked to the level of economic development, iron principle of scientific management of the market economy is disturbed, “economy first, then politics”. the main advantage of the analyzer sona is the use of an entire system of economic laws to ensure the sustainable development of a market economy. the analyzer shows that in a market economy is not market forces rule, but the system of economic laws by which equilibrium is established between the rate of nominal, real and balanced growth in the economy. the principle that has allowed them to balance is the principle of reversibility between the purchasing power of money and the market prices of goods and services. due to the principle reversibility indicators of the real and financial sectors are equally involved in the management of the market economy. both indicators conform to the methodological provisions of the statistics operating in the republic of kazakhstan. it is encouraging that in these methodological positions on statistics the difference between the deflators and price indices of goods and services is clearly indicated : “ comparing with the price index for goods and services, gdp deflator measures the change of wages, earnings (including mixed income) and consumption of fixed capital as a result of changes in prices and nominal net of 312 sailau baizakov, jeffrey yi-lin :the paradoxes of the world’s progress taxes” [9]. the formula for calculating the difference between the gdp deflator and the price indices of goods and services in kazakhstan is defined as clearly defined as ohm’s law in physics. the formula of this law is mathematically proved in section 4 of this paper, and it is economically justified that it is meaningful. the difference between them (its designation, c(t)) was equal to the production of gdp gdp (t), divided by gross output x (t), defined by amount of current material costs for the production of gdp qp (t) and of the gdp at current prices gdp (t) : c(t) = gdp (t)/x(t) = gdp (t)/(qp (t) +gdp (t)) thus, the impact of stp defined by its coefficient c (t), represents the effectiveness of the scientific and technical solutions. and the techno-economic level of production that defined the impact of stp, is gdp (t) / x (t). the peculiarity of the construction rate of stp is its expression ratio of two key indicators of national accounts : gdp and gross output. these indicators form the foundation of leontief table “input output”. it is considered that the total cost of production of specific types of products are gross output gross output (x). if you subtract from it a work in progress, it is determined by commodity products, expressed in money. if the cost of raw materials is deducted from commercial products, the newly created value or gdp will be determined. at one time, the index of gross output, called “gross social product” and the indicator “gross domestic product (gdp)” served as an apple of discord between the supporters of the private and public modes of production. the fact remains that the economists of the former soviet union, for the most part, considered gross output as the main indicator of economic development of the country, it is output x (the total social product x), giving parity to its naturalmaterial structure, the european union economists believe that gdp is the main indicator of the economy, giving the parity to its monetary cost structure. in the first case the goods were deficit in nature ; in the second case the money were deficit. the above formula for the contribution of the technical and technological progress shows that the dynamics of the growth rate of the indicator c(t) is expressed in a well-defined mathematical function defined by the ratio of gdp to be released. thus, it is strictly proved that the parity status should be fixed for the output and gdp, without mutual exclusion : either gdp or gross output. moreover, the gdp deflator is just one of the equivalent indicators of control and itself can serve as the function of the indicator ratio c (t) to the index of the purchasing power of money (pp(t)) : p(t) = c(t)/pp(t) advances in systems science and applications (2014) vol.14 no.4 313 in general, the originality of the principle of reversibility of commodity prices and the purchasing power of money is that the difference between the gdp deflator and price indices of goods and services determines the level of scientific and technological competitiveness (stc) of the real sector. and this difference between the gdp deflator and the price indices of goods and services can be called “contribution” of scientific and technological improvements aimed to increase the competitiveness of the real economy. the information necessary to calculate the difference between them is in the official statistics of the world, which is in line with international standards. however, the “contribution” of scientific and technological competitiveness (growth factors ntc) can be either positive or negative. negative growth of the stc rate in the real sector of the economy is formed when the current material costs of production qp (t) per unit of product will be greater than in the base year. conversely, the positive contribution of the stc in the real economy means lower production costs per unit of output of goods on the market conditions of development. in the case of the positive contribution of stp rate of economic growth is incremented by the value : ∆c(t) = c(t)− 100 > 0 and the index of the gdp deflator is reduced by the same amount. in the second case, on the contrary, when the contribution of stp is negative (∆c(t) < 0), the index of the gdp deflator is incremented, and it increases on this multiplicity −(∆c(t)). at the same time the rate of economic growth decreases by the same amount. thus, the objective the need to define index of market prices of goods and services is brewing other than the level of the gdp deflator. to save the gdp deflator as an independent regulator of the economy and deal with anti-crisis measures leads to a waste of money and time, as its value can be larger or smaller than the index of prices of goods and services by the amount of the contribution of stp. as evidenced by international experts, the losses of the world, gone with the wind of the global crisis of 2008-2009 exceed 10-12 trillion u.s. dollars. it is recognized that n. nazarbayev was right, pointing back in 2009 at a bottleneck in the chain of development of the three types of innovation as the paradoxes of world progress. despite the huge losses of the world economy, a tangle of “ unsolvable contradictions between the rates and levels of development of technological, financial and monetary, social and political structure of each country and world as a whole” remains unsolved to this day. the initiative group of economists kazakhstan believes that the gdp deflator is “explosive substance” containing in its composition “contribution” of scientific and technological improvements. the more real the positive effects of scientific 314 sailau baizakov, jeffrey yi-lin :the paradoxes of the world’s progress and technological progress is, the more powerful the explosive force of the gdp deflator. purification of the real price index of goods and services from this “explosive mixture” of the gdp deflator is impossible to carry out without the knowledge of the forces of economic laws and their practical application skills. its purification is performed only by those who know the power of these laws. otherwise, as shown by the economic crises of recent years it is impossible to pass the permanent shocks in different countries of the world. 2 the function of the gdp deflator in defining the principles of balanced growth 2.1 reciprocity principle is the core of the theory of balanced growth of the real economy and financial stability it is recognized that inflation is the scourge of the market economy. to control inflation, this section studied the nature of the gdp deflator, as a factor determining the trajectory of change in inflation. schematic diagram of the presentation of the gdp deflator by two mutually independent indicators and the definition of balanced economic growth is shown in fig.1. the designations : p(t) = i1/i2 − gdp deflator, where i1 = ngdp and ris. 1. ill�straci� vzaimosv�zi treh indeksov rosta. i2 = rgdp c(t)=ngdp(t)/x(t) risk management index (rmi), where x(t)= nqp + ngdp, where nqp(t) current material production costs ngdp(t). rmi is a synonym of stp coefficient or rate of scientific and technological competitiveness. pp(t)=c(t)/p(t)index of the purchasing power of money. as can be seen from figure 1, the gdp deflator (inflation index) is expanded by a factor of stp c (t) and the indicator of purchasing power of the national currency (pp (t)). by the coefficient of stp further integrated expression of risk management in manufacturing is understood, which is synonymous with the level of scientific and technological excellence of production. more precisely, the coefficient of stp is hiding all the integrated effect of all risk management decisions of the current period of analysis. main thing in theory of reversibility of indices of commodity prices and the advances in systems science and applications (2014) vol.14 no.4 315 purchasing power of money is that it opens the door to reveal the entire system of economic laws that govern the development of the market economy. thus, based on the above formula economic laws are defined, which are very useful for the stability of the market economy. so if the appropriate formulas of these laws are used in the analysis of the market economy not the market elements will dominate, but the system of economic laws by which the principle of market equilibrium is realization. and the indicators of of real and financial sectors in equal measure will participate in the analysis and management of the market economy. basic economic laws that meet the above logic, analysis, and outlined in the terms of the indices of growth of major management indicators, the following : i law of determining the overall impact of the adopted incentives for innovation and investment in the economy of scientific and technological improvement c(t) : c(t) = gdp (t)/(qp (t) +gdp (t)) i law of determining the purchasing power of money pp(t) : pp(t) = (c(t) ∗ i2(t))/i1(t) i law of determining prices of goods and services 1/pp(t) : 1/pp(t) = i1/(c(t) ∗ i2(t)) i leading law of determining the real growth of the economy i3(t) : i3(t) = pp(t) ∗ i1(t). i control law of definition of the real growth of the economy i3(t) : i3(t) = c(t) ∗ i2(t) i law of general price deflation p(t) : p(t) = c(t)/pp(t) = i1(t)/i2(t) i law of definition of net benefits from the stimulation of scientific and technological improvements -∆c(t) : ∆c(t) = c(t)− 100 it should be recalled that, in our notation, i1 (t) is the index of gdp growth in the prices of the current year, i2 (t) the index of growth of gdp at comparable prices. and qp (t) indicated the current material costs necessary for the production of gdp (gdp (t)). 316 sailau baizakov, jeffrey yi-lin :the paradoxes of the world’s progress if earlier political economy as the ideological basis of private or socialized economy disconnected the world into two camps, but now it is special science freed from ideology and will be used for the benefit of all, starting with the economy of the individual entrepreneur and completing with the individual economies of the world. all participants of production will have to study and use these laws in economic activity, just as natural scientists in their work using ohm’s law, or the law of gravity without regard to ideological awnings hanging over them. learning about and using them in their work is forced not the directive from above or below, and their own economic interests. they are equally beneficial and at the enterprise level and at the level of the economy. 2.2 investment in fixed assets as carriers of innovation the relationships between the key indicators of economic will change under the influence of capital investments, innovations in technology, science and other technical or organizational and technological measures. the main carrier of innovation is an investment that has a direct impact on the change in the structure of the market forces of the real economy, hence, on the purchasing power of the currency. of course, these investments will be effective if they are focused on the major discoveries in the natural sciences, which are expected in the areas of space technology and nano technology. space technology is only in the field of telecommunications has opened doors to new scientific and technological revolution, which is already being implemented in almost all human activities, ranging from educational processes, ending municipal services to the population. it is widely implemented in the management process and monetary institutions. it is expected that within the framework of e-government in the near future high technology breakthroughs of intelligent systems of state management of the economy and finance will be created. based on the current intensity of research and development work carried out at universities in the u.s. and western europe in the field of nanotechnology, breakthrough technology in the near future will be implemented in the real sector rapidly. as m. ratner and d. ratner write [11], nano technology will become the foundation of many advanced technologies. feature of this technology is that it can be developed in university laboratories and individual scientific research centers. and, in these institutions intellectual work comes to the forefront. even today, without noticing it, we use “smart materials” obtained through the use of nanotechnology. under the “smart materials” the latest developments in the field of materials consumer or industrial purposes in the nano world of electrons and neutrons are meant. the basic technology of their production is realized using the theory of quantum mechanics. from the point of view of science, here is the integration and sharing of existing knowledge of various branches of basic advances in systems science and applications (2014) vol.14 no.4 317 sciences chemical, biological, physical, mathematical and others. in developing such a “smart material” skilled professionals and financial capital are involved, the integration of which will give a new impetus to a high-performance production. sona analyzer is highly effective and innovative product that allows transferring responsibility for the managerial decisions, including, forecasting and planning at the appropriate levels of microeconomics. macroeconomic policy stops focusing on the development of planning and forecast indicators and, thus, increases the analytic function of public service staff of macroeconomic management. once management functions of microeconomics remains at a real sector, the main condition for profit maximization at the enterprise level is the dynamics of total factor productivity and attraction of additional labor. what is the reason for changes in the dynamics of total factor productivity and capital ? first of all, they are due to the intensity of capital investment in the real economy and its effectiveness. investments in fixed assets contribute to changes in the labor armament with basic production assets and return of capital. in the analyzer sona, armament of labor with fixed assets is measured by a special indicator that expresses the level of industrialization of the economy and the price of capital is measured by other special indicator that expresses the change in the level of innovation and contribution to innovations, carrier of which is the newly introduced in the production of new capacity and replacement of fixed assets. but the dynamics of changes in total factor productivity and fixed capital is associated by productivity of economic labor, which is represented by the ratio of the total factor productivity to the average annual wage per worker. these special indicators in the analysis of management decisions are of great practical meaning, since not all the “innovative activities and innovations”, in practice, increase the level of productivity of economic labor and often work in the opposite direction due to miscalculations in the stages of their design and implementation. thus, the two key indicators of interrelated capacities total factor productivity and productivity of economic labor determine the levels of innovation development of any part of the real, and after aggregation, also the national and global economy. 2.3 resource productivity of intermediate consumption as a measure of growth of production of the final product both labor productivities are arguments of the function of the third productivity the resource productivity of intermediate consumption for the production of the final product. this includes all products and services, including, first and foremost, energy, environmental, and other natural resources that are annually involved in the turnover of the current material costs of the real economy and are constantly in circulation. they are annually deducted from turnover. however, these resources of inter318 sailau baizakov, jeffrey yi-lin :the paradoxes of the world’s progress mediate destination reserve the mark elusive at first sight in the measurement of progress, and the money in circulation, and the mass of commodities in circulation. and the growth of the total number of employed people and other factors of increasing the working time can only affect the quantitative growth of economic indicators. productivity of intermediate product for the production of the final product (gdp) is a function, which reflects the dynamics of total factor productivity of labor and capital, hence, productivity of economic labor. as it is known, natural resources, primarily fuel and energy resources and mining and smelting clusters that are involved in the production, form the bulk of intermediate product values. therefore, this key indicator, defined by the relation between the final and intermediate products is the relationship between the natural environment and human society. and therefore the relevant performance indicator of intermediate resources of consumption is a crucial tool of economic management in a single technology of analyzer sona. what caused the changes in the productivity of intermediate product for the production of the final product (gdp) ? capital expenditures on scientific, technical, organizational, technological, and any other events, including, events aimed at resource conservation, improvement and updating of the commodity nomenclature lead to changes in productivity of the intermediate product for the production of the final product (gdp). these changes take place mainly under the influence of scientific and technological developments, including technical, technological and organizational measures that are used by certain people’s knowledge and monetary resources. but sona analyzer does not work out and does not offer a specific administrative decision and, therefore, does not control the quality of science and technology and other measures implemented in the real economy. it only establishes a de facto existing levels of three key performance indicators in the annual real time, and assesses changes in the level of armament of capital for labor (indicator of industrialization) and in the level of prices of fixed capital (the indicator of innovation) due to management decision-making and the definition of economic laws. that is, it does not replace the persons taking management solutions. subordinated system of indicators of sona analyzer is used only for audit and examination of the quality of the management decisions they made. a key indicator of the analyzer along with the number of employed people in the economy allows us to determine the quantitative parameters of the growth of its main indicators. it includes : • productivity of intermediate product for the production of the final product ; • total factor productivity of labor and capital ; • productivity of economic labor ; • the number of employed people in the economy. advances in systems science and applications (2014) vol.14 no.4 319 all other indicators needed to analyze the economic development of the country are derived from these four basic concepts and categories. official annual statistics of the national economy has background information on the definition of these key and additional indicators of analysis and management. thus, global practice of measurement of purchasing power of money is limited to assessment the physical volume index (pvi) of goods and services and the establishment of the gdp deflator in the real sector of the economy, as the ratio of the index of growth in prices of the current year to an index of growth in the prices of the base year. and the theory of balanced growth is still out of sight of management practices. until now, application tools of analysis of key sectors of the national economy are not synchronized, there are no direct and inverse relationships between indicators of the real and financial sectors, as part of a single economic system. as a result of the lack of mechanisms to control deviations of interrelated indicators of development of market forces, the gap “between those who are doing business and those who make money” is increasing. the proposed technology of work based on the analytical formulas of the system of economic laws is highly effective. and the appropriate navigation system for the analysis of planned, project and other management decisions gives concrete results in the form of a system of the control indicators based on data from official statistics. in particular, the indicator of growth of the price of a particular product or service in the real sector is clearly defined, which is the reciprocal of the purchasing power of money in relative terms. 2.4 the index of prices of goods and services as the inverse of purchasing power of money the scientific basis of the principle of reversibility as well as the principle of determining the sona analyzer is based on the use of the global economic thought and practice. it is based on a procedure that meets the basic rule of international consensus :the equality of the indices of growth of the sum of the sellerąŕs commodity prices and the reciprocal of the growth indexes of purchasing power the buyer’s sums of money. this means that all price indicators of economic analysis and management of the real economy are under the authority of the power unit of the national currency, determined, in this particular case, in accordance with the principles of international consensus : ⨿ᵀ ∗kᵀ/kπ = 1 (2) where ⨿ᵀ ∗kᵀ as the numerator of this equation expresses the value of commodity mass of the seller, and kπ is the money supply from the buyer at face value of the national currency. but this is only the rule ; “one” in the right-hand side of (1) is a relative quantity, necessary to compare purchasing power of the national currency of the 320 sailau baizakov, jeffrey yi-lin :the paradoxes of the world’s progress current time with its level in the base year. since the gross profit ⨿ᵀ ∗kᵀ is actually output in monetary terms -x= ⨿ᵀ ∗kᵀ, and money supply m = kπ, then velocity of money circulation at face value of the national currency will be vx : vx = x/m, or by the conditions of international consensus : x = vx ∗m. hence : x/(vx ∗m) = 1 (3) it is clear that the output x contains the repeated recording of individual components of products that are sold and resold, then nominal gdp is defined as : ngdp = x qp, where qp is intermediate consumption, and ngdp is gdp for the production, the product without double counting. since the equation of exchange of the monetarists : ngdp/vn ∗m = 1 (4) then replacing ngdp by x qp we have the equality that defines the difference between the three speeds : vx − vz = vn (5) which are clearly determined by the gross profit x= ⨿ᵀ ∗kᵀ. this rule is fully consistent with international and kazakhstani practice of analysis of nominal and real gdp and gdp deflator. introduction to the analyzer sona of three key performance indicators allows focusing the purchasing power of money at one point. and this point of consensus core of world currency can not be determined without consideration of gross profit x= ⨿ᵀ ∗kᵀ, on the basis of which three velocities of money are determined by the equation of the vn = vx − vz. the initiative group of kazakhstani analysts, after the analogical settlement of the economies of states of the customs union and the economy of the customs union as a whole, has come to the conclusion that there was an opportunity to focus their national currencies to a single point. in this case, the currency of the customs union will have its consensus core, other than the core of the national currencies of countries of the customs union. the point of this consensus itself, adopted as a basis for currency of cu, as well as world currency itself is in constant motion. thus, the price of the u.s. dollar in 2000 was equal to 0.75, in 2008 0.62, and in 2010 regained the relatively high level 0.81. similarly, as the u.s. dollar, all other national currencies will have their equilibrium prices different from world currency prices. price of world currency is needed for countries of the world to support the effectiveness of the exchange rate of its currency and the favorable choice for the country’s economic policy. in this case, the relative value of the power unit of the nominal value of world currency at the point defined by the application tool of the analyzer such as sona, will become advances in systems science and applications (2014) vol.14 no.4 321 a concentrated expression of the purchasing power of all national currencies of the states concerned un member states, including the united states. once for each country purchasing power of its currency is determined, then the basis for calculating the sdr can be taken the following formula, which does not contradict the general methodological principles of the imf’s definition of “price” of world currency : pp (sdr, t) = ∑i=n i=1 ngdp (i)/[pp(1) ∗ngdp (1) + pp(2) ∗ngdp (2) + ... +pp(n) ∗ngdp (n)] where n expresses the number of countries included to the zone of foreign trade. the cost of one million of kazakhstan tenge, as well as other national currencies in their official rate, which were included in the determination of the value of world currencies, is an equivalent product, and can be freely exchanged for a certain amount of other goods and services at their market prices. but its purchasing power, as well as other national currencies, undergoes a change with changes in the velocity of circulation not just one product, say, gold or oil. it varies with the velocity of circulation of all goods and services, including the prices of intermediate goods and services. 2.5 function of the velocity of circulation of goods and services in determining the rate of balanced growth all indicators of the analyzer sona are a function of main performance indicator of intermediate product for the production of the final product, which determines the economic content of the relative velocities of the key indicators of macro-and microeconomics. they include : • dynamics of intermediate product performance (qp) for gdp at face value (gdp(t))−gdp (t) qp (t) = µ(t), the reciprocal of material consumption of gdp ; • rate of stp, the value of which is equal to gdp(t)/x(t) in macroeconomics in this case is expressed by formula — c(t) = µ(t) 1+µ(t) . this indicator defines the contribution of innovation factors in the development of the economy and serves as a general indicator of technical, technological and other organizational measures implemented in the real economy as a whole ; • rate of turnover (x) on the face value of gdp, in this case, is defined by the formula — vx(t) = x(t) gdp (t) = 1+µ(t) µ(t) = 1 c(t) ; • turnover rate of the money supply m on face value of gdp is expressed by the formula — vn(t) = x(t) m(t) ∗ ( µ(t) 1+µ(t) ) = gdp (t) m(t) . as can be seen from this equation turnover rate indicator of the money supply does not take into account the contribution of all innovations in the real economy. • turnover rate of the money supply vn from the perspective of the circulation of goods and services is determined by a different formula : vn(t) = vx(t)− vz(t) 322 sailau baizakov, jeffrey yi-lin :the paradoxes of the world’s progress where vzthe turnover rate of the current material costs used in the production of each unit sold goods and services is determined by the formula — vz(t) = qp (t) m(t) . • the purchasing power of money is determined directly proportional to the ratio of ntp and inversely proportional to the gdp deflator : pp(t) = c(t)/p(t) • balanced economic growth, defined by multiplication of the purchasing power of money and nominal gdp : rngdp (t) = pp (t) ∗ngdp (t) • balanced economic growth, defined by multiplication of the coefficient ntc and real gdp : nrgdp (t) = c (t) ∗rgdp (t) • the principle of invertibility : nrgdp (t) = rngdp (t) or pp (t) ∗ngdp (t) = c (t) ∗ngdp (t) a key indicator in determining the efficiency of the financial sector of the economy is the value of money, expressed as purchasing power of the currency. the value of money, as the prices of goods and services, changes over time. but the value of money does not contain a single atom of natural ingredients : no gold, no coal, no oil, no labor, no capital, no bread, no meat and other goods. and therefore it makes no sense to look for another form of money, other than paper form. there are no prospects, for example, of the return to the gold standard, the more no prospects of energy (watt, joule) and other material substitutes for paper money. only deviation of index of prices of goods and services relative to the gdp deflator can accurately estimate the purchasing power of money, and set the real price and exchange rates of the national currencies of the world. its formula is defined from equation p(t) =c(t)/pp(t), where p(t) is gdp deflator, c(t) is the coefficient of stp and pp(t) is an indicator of purchasing power of money. analyzer sona, as the technology of management of unity development of the real, financial and monetary sectors of the economy is used to measure the purchasing power of money and the gap between the real and nominal gdp. as a result, the third dimension of economic growth is defined as the product of nominal gdp and purchasing power of money (pp(t)*ngdp) and a double of this measurement, defined by the product of real gdp of and the stc ratio advances in systems science and applications (2014) vol.14 no.4 323 (c(t)*rgdp). exactly this dual pair is the economic content of the principle of mutual convertibility, with the help of which the indices of growth of prices of goods and services (1/pp(t)) are defined as the reciprocal of the index of the purchasing power of money pp(t). the analytical model for determining the threshold levels of indicators of economic management based on official statistics of gdp deflators (p (t)) and on the human dimension of the purchasing power of money is composed of the following recursive system of equations : ngdp (t) = ngdpn (t) · ln (t) ·n(t). tw (t) = γ(t) · ln (t) ·n(t). tr(t) = ngdp (t)− tw (t). x(t) = ( 1 u(t) + 1) · q(t) · γ(t) · ln (t) ·n(t). c(t) = gdp (t) x(t) . p(t) = ngdp (t) ngdp (0) / rgdp (t) rgdp (0) . pp(t) = c(t) p(t) . pp(t) ∗ngdp (t) = c(t) ∗rgdp (t). (6) additionally accepted designations : n number of the country’s population,ln the proportion of people employed in the economy of the total population,tw -fund salaries,tr income on equity, ngdpn gdp per capita. equality between gdp by income, gdp by production and gdp by end-use remains in this analytical model. pp(t) ∗ngdp (t) = c(t) ∗rgdp (t). references [1] n. nazarbayev. (2009), “fifth way” , izvestia, sep. pp.22. [2] n. nazarbayev. (2009), “ keys of crisis”, russian newspaper, feb. pp.2. [3] n. nazarbayev. (2009), speech at a business forum in india, respublika.kz,no.3, jan. pp.27. [4] n. nazarbayev. (2013), speech on th vi astana economic forum, may. pp.23. [5] e-book http://www.monographies.ru/ 324 sailau baizakov, jeffrey yi-lin:the paradoxes of the world’s progress [6] n. kaldor, g. a. mirrlees. (1962), “a new model of economic growth”, review of economic studies, june, pp.174-192. [7] kaldor-mirrlees model. (1976), “in the book of a. pesenti”, essays on the political economy of capitalism, 2, translated from italian, moscow: progress publishers, pp.837-870. [8] s. baizakov. (1985), “scientific and technological progress a key lever of production efficiency (analysis of the theory of economic growth and efficiency)”, almaty, pp.89. [9] “scientific and technological progress a key lever of competitiveness and economic growth: 2nd ed”, ext, almaty: nc sti, 2007, pp.76. [10] “methodological guidelines on statistics. agency of the republic of kazakhstan on statistics”, astana, 2009, pp.198. [11] m. ratner, d. ratner. (2004), “nanotechnology: a simple explanation of the next brilliant idea, translated from english”, moscow: publishing house “williams”, pp.240 [12] e-book http://www.monographies.ru/ [13] j. d. sachs, f. b. laren. (1996), “macroeconomics” ,business, pp.52-54. [14] v. l. makarov. (1985), “on the performance of scientific and technical progress”, economics and math. methods, vol.xxi, no.2. [15] “theory of value: statistical verification. informational generalization. actual conclusions”, “herald of the ras” magazine, no.9, 2005. [16] k. k. valtuh. (1965), “public utility of product and labor costs of its production”, m: thought. [17] k. marx and f. engels. op. t. 2. pp.142. [18] editor v.s. dadayan. (1973), “modeling of national economic processes”, moscow: economics, pp.433-438. [19] v. l. makarov, a. p. torzhevsky. (1986), “effect of changes in the technological level of production on macro indicators of economic development”, moscow:cemi as ussr, pp.4-7. [20] b.m. shtulberg, e.g. chistyakov, v.v kotilko etc. (1988), “problems and methods of study of the territorial plans”, moscow:science, pp.192. corresponding author sailau baizakov can be contacted at: baizakov37@mail.ru. adv syst sci appl 2017; 4; 22-33 published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/255 hybrid papr reduction scheme for universal filter multicarrier modulation in next generation wireless systems pooja rani, silki baghla, dr. himanshu monga* j.c.d.m. college of engineering, sirsa, haryana, india e-mail: poojamehta0193@gmail.com, silky.er@gmail.com, himanshumonga@gmail.com abstract: universal filter multi carrier (ufmc) is one of the promising multi carrier modulation techniques for next generation wireless communication systems. ufmc seems to be most attractive because it provides better sub carrier separation like fbmc (filer bank multi carrier) and less complexity like ofdm (orthogonal frequency division multiplexing). but this technique suffers from limitation of higher peak to average power ratio (papr). in this paper a hybrid papr reduction technique scufmc have been proposed using slm (selective mapping) and clipping. the performance of proposed technique is evaluated for various design parameters including filter length, fft size and bits per sub carrier. the simulation results show that hybrid technique provides better papr reduction as compared with conventional slm and clipping techniques. keywords: ufmc, ofdm, fbmc, papr, slm. 1. introduction orthogonal frequency division multiplexing (ofdm) is the most popular multi-carrier modulation technique which is being used in 4th generation wireless communication [1]. but in the last few years, the number of users and the demand for higher data rates has increased exponentially so, next generation wireless communication systems must be able to deal with large numbers of users and provide a much higher data transmission rate using less complex systems. in order to serve all these requirements, various new multi carrier modulation techniques like filter bank multi carrier (fbmc), universal filter multi carrier (ufmc) and generalized frequency division multiplexing (gfdm) have been introduced [2,3]. in fbmc, each subcarrier is individually filtered and provides robustness against inter-carrier interference (ici) effects [4]. however, fbmc systems utilize filters, whose length is multiple times of samples per multi-carrier symbol resulting in increased complexity of the system. universal filtered multi-carrier (ufmc) is a novel multi-carrier modulation technique, which combines the features of fbmc and ofdm. ufmc filters groups of subcarriers instead of per subcarrier like fbmc or a complete signal in a single shot like ofdm. this allows reducing the filter length considerably as compared to fbmc. so, it is less complex like ofdm and provides better subcarrier separation like fbmc [5]. the main drawback of all these multicarrier modulation techniques is high peak to average power ratio [7,8]. various papr reduction techniques have been proposed in literature and are implemented on ofdm and fbmc. the conventional techniques for papr reduction include selective mapping (slm), companding, * corresponding author: himanshumonga@gmail.com http://ijassa.ipu.ru/ojs/ijassa/article/view/255 mailto:poojamehta0193@gmail.com mailto:silky.er@gmail.com mailto:himanshumonga@gmail.com mailto:himanshumonga@gmail.com 23 p. rani, s. baghla, h. monga copyright ©2017 assa. adv. in systems science and appl. (2017) tone reservation, clipping and filtering, partial transmit sequence (pts) and active constellation extension (ace) [9]. these techniques have been implemented with orthogonal frequency division multiplexing, but little efforts have been done for papr reduction in universal filtered multi-carrier [10]. clipping is the simplest and extensively used method for papr reduction. selective mapping is a suitable match for the subcarrier nature of ufmc. in this paper a novel hybrid papr reduction technique sc-ufmc for ufmc systems has been proposed. this technique is implemented by using slm and clipping papr reduction techniques. it is observed that this hybrid technique provides better results when compared with the individual performances of slm and clipping techniques. 2. ufmc waveform generation for next generation wireless communication systems, a new waveform is required which should achieve the asynchronous reception and transmission, non-orthogonal waveforms for better spectral efficiency and low latency. ufmc has been introduced as a new waveform design representing a generalization of this principle targeting to collect the advantages while avoiding the disadvantages of other modulation techniques [11,12]. ufmc is the method which combines the advantages of orthogonality of ofdm and concept of filter bank in fbmc. instead of filtering each carrier like in fbmc, blocks of carriers called sub-bands are filtered. each subband contains a number of carriers and the filter length will depend upon the width of the subband [13]. fig. 2.1 shows the process of transmission and reception in a ufmc system. here, the complex symbols generated from the modulator (qpsk or qam) are applied to serial to parallel converter resulting in a block of streams and fed as input to their respective ifft. the length of n points ifft output is converted back to serial per block and that output will be filtered with a pulse shaping filter of length l. the transmitted signal can be given as: 𝑥𝑘 = ∑ 𝐹𝑝,𝑘𝑉𝑝,𝑘 𝐵 𝑝=1 𝑋𝑝,𝑘 (2.1) as shown in fig. 2.1, the overall bandwidth and the total number of subcarriers are divided into number of sub bands. now, the input data stream of kth user, 𝑥𝑘 is divided into multiple sub streams denoted by 𝑋𝑝.𝑘 for pϵ{1,…, b}. here b is the total number of sub streams or sub bands. a sub band in ufmc may also correspond to physical resource block (prb) in lte. then, the signal of each sub band is applied to individual n point ifft represented by matrix v. the output of ifft is converted to the serial form and applied to the respective filter represented by matrix f. f is a toeplitz matrix, composed of the filter impulse response, performing the linear convolution in equation (2.1) [14]. for pth sub band, where p varies from 1 to b, 𝑋𝑝,𝑘 , 𝑉𝑝,𝑘 and 𝐹𝑝,𝑘 represents data blocks, ifft matrix and the filter respectively. for filter length l and ifft matrix of size n, the symbol xk of duration n+l-1 is generated at the output of transmitter section. hybrid papr reduction scheme for universal filter multi-carrier modulation 24 copyright ©2017 assa. adv. in systems science and appl. (2017) fig. 2.1. ufmc transceiver fig. 2.2 shows the spectrum for ufmc waveform with sub bands b=10 and each sub band is having 20 sub carriers. the complete process of data transmission and reception is shown in fig. 2.1. at the receiver, after passing through rf-link section the signal is applied to time domain pre-processing window to suppress interference. after windowing, the signal will be converted into 2n parallel streams; here n is the number of subcarriers. the demodulated signal is sent to the de-mapper, which is a demodulator to retrieve the data bits from the received symbols. the generated waveform is shown in fig. 2.2. the block-wise filtering provides flexibility to the system and may be used to avoid the main drawbacks of fbmc. ufmc supports short bursts data transmission, as well as operation in fragmented bands. the filter provides protection against inter-symbol interference (isi), as well as robustness for supporting multiple access users which are not perfectly time-aligned. due to the possibility to reduce guard bands, and to avoid need of cp, ufmc is spectrally more efficient than cp-ofdm [15]. serial to parallel converte r idft filter filter p/s p/s p/s idft idft filter base band to rf channel + noise rf to baseband time domain preprocessin g (e.g. windowing) + s/p fft 2n point symbol demapping complex symbols 25 p. rani, s. baghla, h. monga copyright ©2017 assa. adv. in systems science and appl. (2017) fig. 2.2. ufmc waveform the receiver processing can still be similar to cp-ofdm, single-tap per-subcarrier frequency domain equalizers can be used which equalize the joint impact of the radio channel and the respective sub band-filter. this leads to similar complexity order as cp-ofdm. so, it is thereby clear that ufmc provides advantages of both ofdm and fbmc system. 3. peak to average power ratio ufmc has numerous advantages over other modulation techniques, but it also suffers from high peak to average power ratio (papr).the papr is the relation between the maximum power of a sample in a given transmitted symbol divided by the average power of that symbol. papr occurs when in a multicarrier system the different sub-carriers are out of phase with each other. there are a large number of independently modulated subcarriers in a multicarrier system which are different with respect to each other at different phase values. when all the subcarriers achieve the maximum value simultaneously, this will cause the output envelope to suddenly increase which causes a 'peak' in the output, and when they are added up coherently for transmission purpose give a large peak value which is very large as compared to average value of the sample. the ratio of the peak to average power value is termed as peak-to-average power ratio [16]. the mathematical valuation of papr is defined in equation (3.1). papr = ( 𝑚𝑎𝑥{|𝑥[𝑛]|2} 𝐸{|𝑥[𝑛]|2} ) (3.1) where, |x[𝑛]| is the amplitude of x[n] and e denotes the expectation of the signal. this higher papr causes saturation in the power amplifier which produces inter modulation products among sub bands and also increases out of band radiation (oob). -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -120 -100 -80 -60 -40 -20 0 ufmc normalized frequency p s d (d b w /h z ) hybrid papr reduction scheme for universal filter multi-carrier modulation 26 copyright ©2017 assa. adv. in systems science and appl. (2017) 3.1.conventional papr reduction techniques there are various papr reduction techniques which can be used to improve performance of multicarrier modulation systems. some of them are: 3.1.1. selective mapping (slm) technique in slm, from the original data block several candidate data blocks are generated and all the data blocks are having the same information. after this a phase rotation is applied to each block and passed from its respective idft and a block with minimum papr is selected for transmission [17]. 3.1.2. companding companding is an easy and less complex method of papr reduction, the basic idea is to expand the small signal in the transmitter section and compression is carried out at the receiver side. in this technique, we enlarge the small signals while compressing the large signals to increase the immunity of small signals from noise. this compression is carried out at the transmitter end, after the output is taken from ifft block. there are two types of companders: μ−law and a-law companders [18]. 3.1.3. partial transmit sequence in pts original data block is partitioned into n disjoint sub blocks. the subcarriers in each sub block are rotated by the same phase factor such the papr of the combination can be minimized. pts scheme reduces papr with some additional complexity and it also affects spectral efficiency of the system because side information is also required to be transmitted. it does not produce any distortion in system [19]. 3.1.4. clipping and filtering this is one of the simplest techniques for papr reduction. the principle is to define a clipping level for data transmission above which the input signal is clipped off and peaks of signal are reduced [20]. let, there is a signal y[𝑛] which is to be transmitted and 𝑦𝑐[𝑛] is its clipped version which can be denoted as: 𝑦𝑐[𝑛] = { −a 𝑦[𝑛] ≤ −a 𝑦[𝑛] |𝑦[𝑛]| < a a 𝑦[𝑛] ≥ a (3.2) where a is the clipping level. after clipping out of band radiations are produced this can be reduced by using filtering after clipping. 3.1.5. tone reservation in this scheme, some subcarriers are reserved within the transmitted bandwidth and appropriate value is assigned to these reserved tones [21]. these reserved subcarriers don’t carry any data information, are only used for reducing papr. 3.1.6. active constellation extension (ace) in ace, at each block, some of the outer signal constellation points are extended towards outside of the constellation such that the papr of the resulting block is reduced. it is transparent to the receiver. there is no loss of data rate and no side information is required. however, all these papr reduction techniques were introduced for ofdm systems and due to different frame structure of ufmc signal, these techniques cannot be effectively utilized in ufmc systems. 27 p. rani, s. baghla, h. monga copyright ©2017 assa. adv. in systems science and appl. (2017) 4. proposed hybrid scheme various papr reduction techniques introduced for ofdm system are discussed in the previous section and we tried to apply all these techniques on ufmc systems. it has been observed that slm and clipping are suitable papr reduction techniques for ufmc systems and can reduce papr very effectively with a small increase in system complexity. on the other hand, schemes like pts, ace and tr makes the system very complex and it is very difficult to apply these techniques on the frame structure of ufmc systems. in this work, we have proposed a novel hybrid technique for papr reduction in ufmc by using slm and clipping. unlike existing slm and clipping schemes for ofdm systems, the proposed method exploits the nature of sub-block data transmission of ufmc. in ufmc data is generated in the form of sub blocks which are groups of subcarriers. the advantages of slm and clipping have been exploited in this hybrid technique. in clipping some distortion is produced but in slm there is no any distortion while on the other hand slm produces data rate loss but in clipping there is no any data rate loss. so, in this work we have combined these two schemes so that we can exploit the advantages of both schemes. both techniques have the advantage that the power of the system doesn’t increase. hence, we can decrease papr with the same power which is used by the system when no papr reduction technique is applied. 4.1. hybrid (sc-ufmc) papr reduction technique the basic block diagram of hybrid technique is shown in fig. 4.1. the waveform generated by the ufmc modulator is given to the serial to parallel converter where several candidate blocks are generated from original data block. this is done to find a block with minimum papr for transmission. after generating candidate blocks a phase rotation is applied to each block and applied to ifft of their respective. because of the varying assignment of data to the transmit signal, it is called selective mapping. the core is to choose a particular signal which is having desired properties out of n signals representing the same information. finally, we select a block with minimum papr, the signal generated by this selector is applied to clipper for removing the higher peaks. for this a perticular thershold value is defined above which all the signal is clipped of so that peak to average power ratio can be reduced. the signal generated by ufmc is given in equation 2.1 and after applying a phase rotation the signal become as given in equation (4.1). 𝑥 = ∑ 𝐹𝑃,𝑘𝑉𝑝,𝑘 𝐵 𝑝=1 𝑋𝑝,𝑘𝑝(𝑛) (4.1) fig. 4.1. block diagram of hybrid (sc-ufmc) technique clipping data from ufmc modulator serial to parallel conversion select sequence with minimum papr ifft p (n) ifft p (1) ifft p (2) hybrid papr reduction scheme for universal filter multi-carrier modulation 28 copyright ©2017 assa. adv. in systems science and appl. (2017) where, 𝑝(𝑛) denotes the phase rotation of the signal, after this signal after fft is applied to clipper. let the threshold for the signal is a. then the final signal will be: 𝑥𝑐 = { 𝑥, |𝑥| < 𝐴 𝐴, |𝑥| > 𝐴 (4.2) hence, by using this technique we selected a block with minimum papr and then a clipper circuit is used to clip of the peaks of that block so that we can reduce peak to average power ratio of signal to a great extent. 5. simulation setup and results to compare the proposed hybrid (sc-ufmc) technique with conventional slm (s-ufmc) and clipping (c-ufmc) techniques, matlab platform has been used. all the simulation work, including comparison and performance evaluation of proposed technique with variation in design parameters was done by using matlab platform. 5.1. simulation setup table 5.1 provides the simulation set up to evaluate the performance of proposed papr reduction technique. the proposed technique is also compared with three other techniques as original (ufmc), with selective mapping (s-ufmc), with clipping (c-ufmc) and hybrid (sc-ufmc) to analyze the effectiveness in papr reduction. the performance of the proposed method is evaluated and compared on the basis of variation in fft size, bits per sub carrier, filter length and modulation order. table 5.1. simulation setup parameter values fft size 2048, 1024, 512, 256 sub band size 20 number of sub bands 10 modulation order qam (4, 16, 64) bits per sub carrier 2, 4, 6 filter length 43, 63, 83 in fig. 5.1 peak to average power ratio of ufmc, slm scheme (s-ufmc), clipping scheme (c-ufmc) and hybrid scheme (sc-ufmc) with fft size 1024, filter length 43, bits per subcarrier 2 and modulation order 4 are shown. it can be concluded that the proposed hybrid schemes have improved papr reduction performance as compared with the two conventional schemes for papr reduction. 29 p. rani, s. baghla, h. monga copyright ©2017 assa. adv. in systems science and appl. (2017) fig. 5.1. papr reduction of different schemes 5.2. performance evaluation of sc-ufmc with variation in design parameters this section presents performance evaluation of sc-ufmc with variation in design parameters (fft size, bits per sub carrier and filter length). 5.2.1. variation in fft size fig. 5.2 shows the papr of ufmc signal and performance of three papr reduction techniques at fft size 2048. it can be observed that sc-ufmc is providing better papr reduction than other two techniques. fig. 5.2. performance of various papr reduction techniques with different fft sizes -60 -40 -20 0 20 40 60 80 0 0.2 0.4 0.6 0.8 1 papr in db ufmc s-ufmc c-ufmc sc-ufmc 0 5 10 15 20 25 ufmc s-ufmc c-ufmc sc-ufmc p a p r in d b papr reduction techniques papr with varition in fft size 2048 1024 512 256 hybrid papr reduction scheme for universal filter multi-carrier modulation 30 copyright ©2017 assa. adv. in systems science and appl. (2017) as shown in fig. 5.2, c-ufmc shows better performance than s-ufmc, so effectiveness of sc-ufmc is analyzed with different values of design parameters with respect to c-ufmc. table 5.2 is showing that sc-ufmc performs more effectively with larger fft size. table 5.2. effectiveness with variation in fft size 5.2.2 variation in bits per sub carrier performance of various papr reduction techniques with 2 bits per sub carrier is shown in fig. 5.3 from here it can be observed that proposed scheme is showing better results than conventional schemes. all the schemes are showing different values when we change any of the design parameters as shown in fig. 5.3, here we are changing bits per sub carriers. fig. 5.3. performance of various papr reduction techniques with different bits per sub carrier as shown in table 5.3 with increase in bits per sub carrier, sc-ufmc is being more effective and at 6 bits per sub carrier it is having maximum papr reduction. table 5.3. effectiveness with variation in bits per sub carrier 0 5 10 15 20 25 ufmc s-ufmc c-ufmc sc-ufmc p a p r in d b papr reduction techniques papr with variation in bits per sub carrier 2 4 6 fft size % effectiveness of sc-ufmc 256 4.23% 512 27% 1024 28.15% 2048 30.14% bits per sub carrier %effectiveness of sc-ufmc 2 30.14% 4 37.62% 6 58.60% 31 p. rani, s. baghla, h. monga copyright ©2017 assa. adv. in systems science and appl. (2017) 5.2.3 variation in filter length with changing filter length, papr of original ufmc signal as well as performance of papr reduction techniques also get affected as shown in fig. 5.4. fig. 5.4. performance of various papr reduction techniques with different filter lengths it can be observed from table 5.4 that at higher filter length proposed scheme performed less effectively and a better papr reduction can be obtained with lower filter length. table 5.4. effectiveness with variation in filter length filter length %effectiveness of sc-ufmc 43 36.79% 63 14.57% 83 8.40% 6. conclusion in this paper, a novel hybrid scheme sc-ufmc for papr reduction of ufmc signals is proposed. simulation results and analysis shows that the hybrid scheme is an efficient papr reduction method for ufmc systems, and it can provide better papr reduction performance than the conventional slm and clipping schemes. further, the effect of various design parameters on papr reduction in sc-ufmc has been analyzed and it is concluded that the proposed technique provides effective papr reduction than conventional schemes. references [1] 5gnow, (2013, november 21) d3.1: 5g waveform candidate selection. tech. rep. [online]. available: http://5gnow.eu/wp-content/uploads/2015/04/5gnow_d3.1_v1.1_1.pdf [2] rohde & schwarz (2016). 5g waveform candidates. application note. [online]. available: https://www.rohde-schwarz.com/us/applications/5g-waveform-candidatesapplication-note_56280-267585.html 0 5 10 15 20 25 ufmc s-ufmc c-ufmc sc-ufmc p a p r in d b papr reduction techniques papr with variation in filter length 43 63 83 hybrid papr reduction scheme for universal filter multi-carrier modulation 32 copyright ©2017 assa. adv. in systems science and appl. (2017) [3] schaich, f. & wild, t. (2014). waveform contenders for 5g: ofdm vs. fbmc vs. ufmc. 2014 6th international symposium on communications, control and signal processing (isccsp), ilissia athens, greece, 457–460. [4] farhang-boroujeny, b. (2011). ofdm versus filter bank multicarrier. ieee signal process. mag., 28, 92–112. [5] vakilian, v., wild, t., schaich, f., ten brink, s. & frigon, j.-f. 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(2017). performance evaluation of multi-carrier modulation techniques for next generation wireless systems. international journal of advanced research in computer science, 8(5), 508-511. [16] tellado, j., & cioffi, j. m. (1999). papr reduction in multicarrier transmission system. ph.d. thesis, stanford university. [17] bauml, r. w., fischer, r.f.h. & huber, j.b. (1996). reducing the peak to average power ratio of multi carrier modulation by selected mapping. ieee electronics letters, 32(22), 2056-2057 [18] wang, x., tjhung, t.t., ng, c.s. (1999). reduction of peak to average power ratio of ofdm system using a companding technique. ieee transactions on broadcasting, 45(3), 303-307. 33 p. rani, s. baghla, h. monga copyright ©2017 assa. adv. in systems science and appl. (2017) [19] muller, s.h. & huber, j.b. (1997) ofdm with reduced peak-to-average power ratio by optimum combination of partial transmit sequences. ieee electronics letters, 33(5), 368-369. [20] o'neill, r. & lopes, l.b. (1994). performance of amplitude limited multitone signals. proceedings of ieee 44th vehicular technology conference, stockholm, sweden, 1675-1679. [21] krongold, b.s. &. jones, d.l. (2002). a new tone reservation method for complex baseband par reduction in ofdm system. 2002 ieee international conference on acoustics, speech and signal processing (icassp), orlando, fl, usa, 2321-2324, doi: 10.1109/icassp.2002.5745110 microsoft word 18 xiushan wang, youzhou jiao, yongchang yu--thermal error modeling analysis and compensation based on the gre advances in systems science and applications (2011), vol.11, no.3-4 339-347 issn 1078-6236 international institute for general systems studies, inc thermal error modeling analysis and compensation based on the grey system theory on two turntable 5-axis nc machine tools xiushan wang, youzhou jiao and yongchang yu college of mechanical & electrical engineering, henan agricultural university, zhengzhou china abstract in this paper a novel concept of thermal error mode analysis is proposed in order to develop a better understanding of the thermal deformation on two turntable 5-axis nc machine tools. the correlation analysis theory is used as finding the correlation between temperature data and thermal errors. four key temperature points of two turntable 5-axis nc machine tools were obtained based on the temperature field analysis. a novel thermal error model on the basis of grey system theory was proposed. the experiments proved that there was averagely a 40% increase in machine tool precision. keywords thermal error, modeling, grey theory, compensation 1. introduction because inaccuracy of machine tools is a major source of workpiece errors, control of machine error sources is critically important. among the sources of machine error, thermally induced errors and geometric errors are known to be key contributors [1]. these errors could be reduced with structural improvement of the machine tool itself through design and manufacturing technology. however, there are, in many cases, physical limitations to accuracy improvement that cannot be overcome solely by production and design techniques. therefore, error compensation is a cost effective way to solve the problem [2]. accurate modeling of errors is a key part of error compensation. the thermal errors of a machine tool origin from the non-linear and time-varying thermal deformations caused by the non-uniform temperature variations in the machine structure. the temperature variations are related to the heat source location, heat source intensity, thermal resistance coefficient and the machine system configuration. therefore, the thermal error model is usually achieved by non-linear empirical modeling approaches which correlate machine thermal errors to temperature measurements of a machine. in this paper a novel concept of thermal error modeling analysis is proposed in order to develop a better understanding of the thermal deformation on two turntable 5-axis nc machine tools. the thermal error prediction model, which based on grey system theory, was built on the x, y, and z directions. the prediction precision of model is very high. the robustness of model is very strong. experiments proved that the model is more optimal model among available thermal error modeling of machine tools. 2. temperature field identification and thermal error measurement of machine tools 2.1 sensor location distribution on two turntable 5-axis nc machine tool two turntable 5-axis nc machine tools is a precise machine tool, with high geometric and position accuracy. so, the thermal error is the important factor affecting the machine tool accuracy. to detect the temperature field of two turntable 5-axis nc machine tools, a total of 24 temperature sensors were installed on the machine tools as shown in fig. 1. the locations of the temperature sensors can be divided into 8 groups [1], as shown in table 1. as shown in fig. 1, the spindle house of the machine tools is installed on the z direction slide, so the temperature field from the area is very complicated. in order to analyze the complex area, the more temperature 340 wang: thermal error modeling analysis and compensation based on the grey system theory on two turntable 5-axis nc machine tools sensors are applied. table 1 temperature sensor location ＆ function group number sensor serial installment place ＆ function 1 1,2,3,4,5,6,7,19,23 measuring the temperatures of the ball screw bearings and nuts of z axis 2 8,12,15,18 measuring the temperatures of the machine tool bed 3 21,22,24 measuring the temperatures of the spindle housing 4 10,11 measuring the temperatures of the ball screw bearings and nuts of x axis 5 13,14 measuring the temperatures of the ball screw bearings and nuts of y axis 6 9,17 measuring the temperatures of the ball screw bearings and nuts of a axis 7 16 measuring the temperatures of the ball screw bearings and nuts of c axis 8 20 measuring the ambient temperature fig.1. sensor locations of the thermal sensing system 2.2 temperature field under a cutting cycle in order to investigate the thermal behavior of the machine tool, simulation working conditions has been tested and the temperature field was recorded by the compensation control and temperature sensing systems. firstly, simultaneously moving the x, y, and z axes at low speed (800mm/min), with a spindle speed of 1500rpm for 2.5 hours. secondly, stopping and cooling down the machine for 1.5 hours. thirdly, moving the x, y, and z axes at high speed (1200mm/min) for another 2.5 hours, with a spindle speed of 3000rpm at the same time. in addition, during the testing moving a and c axes with 50rpm. advances in systems science and applications (2011), vol.11, no.3-4 341 fig. 3(a)–(d) show the temperature variations of the 8 groups of temperature sensors over time series respectively. fig.2. measurement of thermal errors for machine tool 2.3 thermal error measurement of machine tool the experimental set-up for measuring thermal errors is shown in fig. 2. three displacement sensors mounted on the turntable were used for measuring the thermal drift errors of the spindle in the x, y, and z directions. the inclination error was neglected because the workpiece is short. measurement results of thermal errors can be seen in fig. 4. 2.4 measurement result analysis of temperature and thermal error the conclusions are drawn from figs. 3figs.4: (1) temperature change is slow with machining process. the highest measuring temperature is about 45 0c . (2) the temperatures of the spindle are about 13 0c during the foregoing 2.5 hours and 21 0c the latter 2.5 hours. the temperatures of the nut and screw bearing of the x, y, a, and c-axis rise about 10 0c during the foregoing 2.5 hours and 18 0c the latter 2.5 hours. (3) the thermal error range is about 28 mμ , which is larger than the expected value. (4) the part error decreases with the machine temperature rise but having some time-lag. (5) when stopping cutting, the machine tool temperature does not drop immediately. thermal error does not decrease with stopping machine tool. 2 4 3 1 5 25 30 35 40 45 0 0 1 2 5 6 73 4 run2.5 hours stop 1.5 hours stop 6 t/sec 3106.3 ×× t/ ( ) run2.5 hours (a) 1-6 342 wang: thermal error modeling analysis and compensation based on the grey system theory on two turntable 5-axis nc machine tools 3106.3 ×× (b) 7-12 25 30 35 40 45 0 0 1 2 5 6 73 4 t/sec 3106.3 ×× t/ (º ) cutting 2.5 hours stop 1.5 hours cutting 2.5 hours stop (c) 13-18 3106.3 ×× t/ ( ) (d) 19-24 号 fig.3. temperature field and temperature collection systems 3. choice of the optimal temperature measuring point the amount of the temperature variable is a key factor to the accuracy of the thermal error model. too fewer amount induces that the accuracy of the thermal error model is very low, which can’t insure compensation precision. however, too much amount leads that the calculating time and additional cost are high. therefore, the temperature variable amount should be determined before the thermal error modeling. in this study, the temperature variables are determined based on correlation analysis between the thermal errors and temperature variables [3,4]. the optimal sensitive temperature measurement point should meet two factors (1) one is that the location should lie in the thermal sources (2) the other is that the correlation between the advances in systems science and applications (2011), vol.11, no.3-4 343 thermal errors and temperature variables is highest. correlation analysis is one of the most effective data processing methods on mathematic statistics, which describe correlative degree among multi-variables. the correlation among variables is often expressed by related coefficient ,x yρ : ( , ) ( ) ( )xy cov x y d x d x ρ = ⋅ (1) 33.6 10× × fig.4. output of displacement sensors from three directions table 2 the related coefficients among the temperature and error sensor coefficient error t21 t2 t24 t15 x 0.693909 0.985567 0.704328 0.969573 y 0.622667 0.710333 0.983336 0.958945 z 0.987637 0.692565 0.732559 0.985667 in this article, the related coefficients between temperature and thermal error variables are acquired through data processing program and matlab software. through calculating and analyzing the related coefficients between 24 temperature data series and x, y, and z direction thermal error series, the correlation coefficients 21, 2, 24, 15,, , ,x x x xρ ρ ρ ρ , as shown in table 2, are more sensitive than other points. what’s more, the thermal deformations of installment locations of four temperature measuring points seriously affect the machining precision of machine tool in the x, y, and z directions [1,3]. so the data series variable from 21, 2, 24, 15 temperature sensors are used for the thermal error modeling. 4. error modeling based on grey system theory 4.1 data prediction and process of grey system theory if (0){ ( )}( 1, 2, , )x i i n= is an original data series [5], new data series can be gotten by once accumulation calculation: (1) (0)( ) ( ) 1 k x k x j j = ∑ = 1, 2, ,k n= (2) then the differential equation relative to (1,1)gm model is: (1) 2 (1)dx ax u dt u v ∂ ω + = ∂ ∂ (3) 344 wang: thermal error modeling analysis and compensation based on the grey system theory on two turntable 5-axis nc machine tools here, a is development grey parameter; u is the endogenous control grey parameter. it is assumed that a is the estimating parameter, a a u ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ is acquired by the least square method. ( ) 1t t na b b b y − = (4) here ( ) ( ) (1) (1) 1 1 (1) (1) 1 (1) (2) 1 2 1 (2) (3) 1 2 1 ( 1) ( ) 1 2 x x x x b x n x n ⎡ ⎤⎡ ⎤− +⎢ ⎥⎣ ⎦ ⎢ ⎥ ⎢ ⎥⎡ ⎤− +⎣ ⎦⎢ ⎥= ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤− − +⎢ ⎥⎣ ⎦⎣ ⎦ (5) ( ) ( ) ( )( )0 0 0(2), (3), , ( ) t ny x x x n= (6) solving (3), grey model can be gotten: ( )0( 1) (1) aku ux k x e a a −⎡ ⎤+ = − +⎢ ⎥⎣ ⎦ 0,1, 2, ,k n= (7) 4.2 improvement of grey system theory during the thermal error modeling, the prediction model is acquired both by the collected all data and only a part of data. generally, the model from different data series is different. so the parameter a is not same. if (0) (0) (0) (0)( (1), (2), , ( ))x x x x n= , (0) ( )x n can be seen the origin of the time axis, and t i∈ , then t n< is considered as past, t n= as present, t n< as future. so (1) the grey model from (0) (0) (0) (0)( (1), (2), , ( ))x x x x n= is called as all data model (1,1)gm . (2) if 0 1k∀ > , the grey model from (0) (0) (0) (0) 0 0( ( ), ( 1), , ( ))x x k x k x n= + is called as part data model (1,1)gm . (3) if (0) ( 1)x n + is the new information, which is added to (0)x , the grey model from (0) (0) (0) (0) (0)( (1), (2), , ( ), ( 1))x x x x n x n= + is called as new model (1,1)gm . (4) adding new information (0) ( 1)x n + and eliminating the old information (0) (1)x , the grey model from (0) (0) (0) (0)( (2), , ( ), ( 1))x x x n x n= + is called as the metabolism (1,1)gm . 4.3 application of the grey system modeling on the basis of grey system modeling analysis and selected temperature variables above sections, the metabolism (1,1)gm of grey model is individually used as building thermal error model in the x, y, and z directions. detailed modeling process is neglected for the paper length limitation. this paper only offers built model based on experiment data, which include measurement values, forecast values, and residuals. experiment data graph, model predication graph, and residuals graph of three directions have been drawn as shown in fig.5. fig. 5 shows that experiment data graph and model predication graph are very coincident, the residuals of any data point is less 5 mμ . it can be seen that the prediction models is precise, which can be written in compensation control device. advances in systems science and applications (2011), vol.11, no.3-4 345 fig.5. the output results of real measurement and model 5. real time compensation of the thermal error in order to evaluate the performance of the metabolism grey model (1,1)gm , other experiments are carried out according to the standard iso.232. the experimental parameter setting is similar with the experimental setting described in section 2. four temperature sensors (no. 2, 15, 21 and 24) and the displacement sensors are only used in the experiments. two group comparison experiments have been done, one group is 60 parts without compensation, and the other group is also 60 parts with compensation. the material of experiments parts is 45# fe. the fig.6 is the trial result. the averagely thermal error of two turntable 5-axis nc machine tools is 20 mμ before compensation, and is 7 mμ after compensation. there is averagely a 60% 346 wang: thermal error modeling analysis and compensation based on the grey system theory on two turntable 5-axis nc machine tools increase on precision. compensation experiments proved the metabolism grey model (1,1)gm is with higher precision, and the modeling method is accurate. part num. e/ μ m 10 20 30 40 50 600 -5 0 5 10 15 20 25 before compensation after compensation (b) y direction fig. 6. x, y, z direction experiments before and after compensation 6. conclusions in this paper, the metabolism grey model (1,1)gm was proposed for on-line prediction of the dynamic and highly non-linear thermal errors on two turntable 5-axis nc machine tools. the proposed model not only enhances the prediction accuracy of the thermal error but also reduces the computation cost. relative analysis method is used as assuring the optimal temperature measurement point location of machine tool. the experimental results demonstrated that the machining accuracy was improved significantly after implementation of the compensation of the thermal errors. advances in systems science and applications (2011), vol.11, no.3-4 347 acknowledgments the authors are thankful to the education department of henan province of china for supporting this research under grant science research project 2010a460012. references [1] yang, j., yuan, j., ni, j.. thermal error mode analysis and robust modeling for error compensation on a cnc turning center , int. j. mach. tools manuf. vol.39, 1999, 1367–1381. [2] wu h., zhang, h., wang, x.s.. thermal error optimization modeling and real-time compensation on a cnc turning center, j. mater. process. technol. vol.207, 2008, 172–179. [3] wang xiushan, yang jianguo. synthesis error modeling and thermal error compensation of five-axis machining center. mater, sci. forum. vol.532-533, 2006, 49-52. [4] zhang zhiyong. master matlab 6.5 version . bhu publishing company, beijing, 2004. [5] li yongxiang, researchment and application of high-effective precise testing and reeor modeling for nc machine tools. shanghai: shanghai jiaotong university, 2007. advances in systems science and applications (2013) vol.13 no.1 68-79 optimization of statistical decision for personnel management problems in tourism konstantin n. nechval1, nicholas a. nechval2, gundars berzins2 and maris purgailis2 1applied mathematics department, transport and telecommunication institute,lomonosov street 1, lv-1019 riga, latvia 2statistics department, evf research institute, university of latvia,raina blvd 19, lv-1050 riga, latvia abstract a large number of problems in production planning and scheduling, location, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty. in the present paper, for improvement or optimization of statistical decisions under parametric uncertainty, a new technique of invariant embedding of sample statistics in a performance index is proposed. this technique represents a simple and computationally attractive statistical method based on the constructive use of the invariance principle in mathematical statistics. unlike the bayesian approach, an invariant embedding technique is independent of the choice of priors. it allows one to eliminate unknown parameters from the problem and to find the best invariant decision rule, which has smaller risk than any of the well-known decision rules. in order to illustrate the application of the proposed technique for constructing optimal statistical decisions under parametric uncertainty, we discuss the following personnel management problem in tourism. a certain company provides interpreter-guides for tourists. some of the interpreter-guides are permanent ones working on a monthly basis at a daily guaranteed salary. the problem is to determine how many permanent interpreter-guides should the company employ so that their overall costs will be minimal? we restrict attention to families of underlying distributions invariant under location and/or scale changes. a numerical example is given. keywords personnel management problem, invariant embedding technique, optimization 1 introduction most of the operations research and management science literature assumes that the true distributions are specified explicitly. however, in many practical situations, the true distributions are not known, and the only information available may be a time-series (or random sample) of the past data. analysis of decisionmaking problems with unknown distribution is not new. several important papers have appeared in the literature. when the true distribution is unknown, one may either use a parametric approach (where it is assumed that the true distribution belongs to a parametric family of distributions) or a non-parametric approach advances in systems science and applications (2013) vol.13 no.1 69 (where no assumption regarding the parametric form of the unknown distribution is made). under the parametric approach, one may choose to estimate the unknown parameters or choose a prior distribution for the unknown parameters and apply the bayesian approach to incorporating the past data available. parameter estimation is first considered in [1] and further development is reported in [2]. scarf [3] considers a bayesian framework for the unknown demand distribution. specifically, assuming that the demand distribution belongs to the family of exponential distributions, the demand process is characterized by the prior distribution on the unknown parameter. further extension of this approach is presented in [4]. within the non-parametric approach, either the empirical distribution [2] or the bootstrapping method (e.g. see [5]) can be applied with the available past data to obtain a statistical decision rule. a third alternative to dealing with the unknown distribution is when the random variable is partially characterized by its moments. when the unknown demand distribution is characterized by the first two moments, scarf [6] derives a robust min-max inventory control policy. further development and review of this model is given in [7]. in the present paper we consider the case, where it is known that the true distribution function belongs to a parametric family of distributions. it will be noted that, in this case, most stochastic models to solve the problems of control and optimization of system and processes are developed in the extensive literature under the assumptions that the parameter values of the underlying distributions are known with certainty. in actual practice, such is simply not the case. when these models are applied to solve real-world problems, the parameters are estimated and then treated as if they were the true values. the risk associated with using estimates rather than the true parameters is called estimation risk and is often ignored. when data are limited and (or) unreliable, estimation risk may be significant, and failure to incorporate it into the model design may lead to serious errors. its explicit consideration is important since decision rules that are optimal in the absence of uncertainty need not even be approximately optimal in the presence of such uncertainty. the problem of determining an optimal decision rule in the absence of complete information about the underlying distribution, i.e., when we specify only the functional form of the distribution and leave some or all of its parameters unspecified, is seen to be a standard problem of statistical estimation. unfortunately, the classical theory of statistical estimation has little to offer in general type of situation of loss function. the bulk of the classical theory has been developed about the assumption of a quadratic, or at least symmetric and analytically simple loss structure. in some cases this assumption is made explicit, although in most it is implicit in the search for estimating procedures that have the “nice” statistical properties of unbiasedness and minimum variance. such procedures are usually satisfactory if the estimators so generated are to be used 70 konstantin n. nechval: optimization of statistical decision for personnel management ... solely for the purpose of reporting information to another party for an unknown purpose, when the loss structure is not easily discernible, or when the number of observations is large enough to support normal approximations and asymptotic results. unfortunately, we seldom are fortunate enough to be in asymptotic situations. small sample sizes are generally the rule when estimation of system states and the small sample properties of estimators do not appear to have been thoroughly investigated. therefore, the above procedures of the statistical estimation have long been recognized as deficient, however, when the purpose of estimation is the making of a specific decision (or sequence of decisions) on the basis of a limited amount of information in a situation where the losses are clearly asymmetric-as they are here. in this paper, we propose a new technique to solve optimization problems of statistical decisions under parametric uncertainty. the technique is based on the constructive use of the invariance principle for improvement (or optimization) of statistical decisions. it allows one to yield an operational, optimal information-processing rule and may be employed for finding the effective statistical decisions for many problems of the operations research and management science. the illustrative application of the invariant embedding technique to personnel management problems in tourism is given below. 2 invariant embedding technique this paper is concerned with the implications of group theoretic structure for invariant performance indexes. we present an invariant embedding technique based on the constructive use of the invariance principle for decision-making. this technique allows one to solve many problems of the theory of statistical inferences in a simple way. the aim of the present paper is to show how the invariance principle may be employed in the particular case of improvement or optimization of statistical decisions. the technique used here is a special case of more general considerations applicable whenever the statistical problem is invariant under a group of transformations, which acts transitively on the parameter space [8-13]. 2.1 preliminaries our underlying structure consists of a class of probability models (x, a, p), a one-one mapping ψ taking p onto an index set θ, a measurable space of actions (u,b), and a real-valued function r defined on θ× u. we assume that a group g of one-one a-measurable transformations acts on x and that it leaves the class of models (x,a,p) invariant. we further assume that homomorphic images g and g̃ of g act on θ and u, respectively. (g may be induced on θ through ψ; g̃ may be induced on u through r). we shall say that r is invariant if for every (θ,u) ∈ θ×u r(gθ, g̃u) = r(θ,u), g ∈ g. (1) advances in systems science and applications (2013) vol.13 no.1 71 given the structure described above there are aesthetic and sometimes admissibility grounds for restricting attention to decision rules φ: x → u which are (g,g̃) equivariant in the sense that φ(gx) = g̃φ(x),x ∈ x, g ∈ g (2) if g is trivial and (1), (2) hold, we say φ is g-invariant, or simply invariant. 2.2 invariant functions we begin by noting that r is invariant in the sense of (1) if and only if r is a g•invariant function, where g• is defined on θ×u as follows: to each g ∈ g, with homomorphic images g, g̃ in g, g̃ respectively, let g•(θ,u) = (ḡθ, g̃u), (θ,u) ∈ (θ× u). it is assumed that g̃ is a homomorphic image of g. definition 1 (transitivity). a transformation group g acting on a set θ is called (uniquely) transitive if for every θ,ϑ ∈ θ there exists a (unique) g = g such that gθ = ϑ. when g is transitive on θ we may index g by θ: fix an arbitrary point θ ∈ θ and define gθ1 to be the unique g = g satisfying gθ = θ1. the identity of g clearly corresponds to θ. an immediate consequence is lemma 1. lemma 1 (transformation). let g be transitive on θ. fix θ ∈ θ and define gθ1 as above. then gqθ1 = q gθ1 for θ ∈ θ, q ∈ g. proof.the identity gqθ1θ=qθ1=q gθ1θ shows that gqθ1 and q gθ1 both take θ into qθ1, and the lemma follows by unique transitivity. theorem 1 (maximal invariant). let g be transitive on θ. fix a reference point θ0 ∈ θ and index g by θ. a maximal invariant m with respect to g• acting on θ× u is defined by m(θ,u) = g̃−1 θ u, (θ,u) ∈ θ× u (3) proof. for each (θ,u) ∈ (θ× u) and g ∈ g m(gθ, g̃u) = (g̃−1 gθ )g̃u = (g̃g̃θ) −1g̃u = g̃−1 θ g̃−1g̃u = g̃−1 θ u = m(θ,u) (4) by lemma 1 and the structure preserving properties of homomorphisms. thus m is g•− invariant. to see that m is maximal, let m(θ1,u1) = m(θ2,u2). then g̃−1 θ1 u1 = g̃−1 θ2 u2 or u1 = g̃u2, where g̃ = g̃θ1 g̃ −1 θ2 . since θ1 = gθ1θ0 = ḡθ1 ḡ −1 –θ2 θ2 = ḡθ2, (θ1,u1) = g•(θ2,u2) for some g• ∈ g•, and the proof is complete. corollary 1.1 (invariant embedding). an invariant function, r(θ,u), can be transformed as follows: r(θ,u) = r(g θ̂ −1θ, g θ̂ −1u) = r̈(v,η) (5) where v = v(θ, θ̂) is a function (it is called a pivotal quantity) such that the distribution of v does not depend on θ; η = η(u, θ̂) is an ancillary factor; θ̂ is 72 konstantin n. nechval: optimization of statistical decision for personnel management ... the maximum likelihood estimator of θ (or the sufficient statistic for θ). corollary 1.2 (best invariant decision rule). if r(θ,u) is an invariant loss function, the best invariant decision rule is given by φ∗(x) = u∗ = η−1(η∗, θ̂) (6) where η∗ = arg inf η eη {r̈(v, η)} . (7) corollary 1.3 (risk). a risk function (performance index) r(θ,φ(x)) = eθ{r(θ,φ(x))} = eη{r̈(v0,η0)} (8) is constant on orbits when an invariant decision rule φ(x) is used, where v0 = v0(θ,x) is a function whose distribution does not depend on θ; η0 = η0(u,x) is an ancillary factor. for instance, consider the problem of estimating the locationscale parameter of a distribution belonging to a family generated by a continuous cdf f : p = {pθ : f ((x− µ)/σ), x ∈ r,θ ∈ θ}, θ = {(µ, σ) : µ, σ ∈ r, σ > 0} = u . the group g of location and scale changes leaves the class of models invariant. since g induced on θ by pθ → θ is uniquely transitive, we may apply theorem 1 and obtain invariant loss functions of the form r(θ,φ(x)) = r[(φ1(x)− µ)/σ, φ2(x)/σ] (9) where θ = (µ, σ) and φ(x) = (φ1(x), φ2(x)) (10) let θ̂ = (µ̂, σ̂) and u = (u1, u2), then r(θ,u) = r̈(v,η) = r̈(ν1 + η1ν2, η2ν2) (11) where ν = (ν1, ν2), ν1 = (µ̂− µ)/σ, ν2 = σ̂/σ (12) η = (η1, η2), η1 = (u1 − µ̂)/σ̂, η2 = u2/σ̂ (13) 3 application to personnel management problem in tourism personnel management forms a significant proportion of overall costs in hotels, tourism companies and fast food restaurants. a reduction in this by even 1% represents considerable cost savings. demand for services is not generally known with certainty before hand and management often relies on a combination of intuition, software systems and local knowledge (particularly of marketing campaigns, events and attractions). staff scheduling is a key element of management planning in such circumstances. there have been a number of general survey advances in systems science and applications (2013) vol.13 no.1 73 papers in the area of personnel management; these include [14] and [15]. the latter survey concentrates on general labour scheduling models. a survey of crew scheduling is given in [16]. surveys of the literature in airline crew scheduling appear in [17-18]. a good survey of tools, models and methods for bus crew scheduling is [19]. a survey of the nurse scheduling literature is provided in [2021]. as can be seen from this review, a large amount of work has already been done in the area of personnel scheduling. nevertheless there is still significant room for improvements in this area. we see improvements occurring not only in the area of tools, models and methods for personnel management, but also in the wider applicability of these tools, models and methods. in this paper, we consider the following personnel management problem in tourism. a certain company provides interpreter-guides for tourists. the number of permanent interpreterguides employed by the company is such that u of them are permanently working on a monthly basis at a daily guaranteed salary c1 (in terms of money); when the demand for their services exceeds u, supplementary interpreter-guides or extras are taken on at a daily salary c2(> c1). sometimes the shortage of extras will necessitate canceling a tour, and when this happens, the loss is reckoned at c3(> c2). how many permanent interpreter-guides should the company employ so that overall costs will be minimal? following kaufmann and faure [22], we review the personnel management model and provide a broader interpretation to the structure of its solution. in development of the personnel management model, we will assume that the daily demand for tours x is a continuous nonnegative random variable with the probability density function fθ(x) and cumulative distribution function fθ(x). the notation, we use for the personnel management model, is given below. x random variable representing the daily demand for tours fθ(y) probability density function of a demand x fθ(y) cumulative distribution function of a demand x θ parameter (in general, vector) y random variable representing the daily supply of extras p(y) probability of a supply y, where y=0, 1, ...,∞ c1 daily guaranteed salary for the permanent interpreter-guide c2 daily salary for the supplementary (or extra) interpreter-guide c3 shortage cost per unit of x u variable representing the number of the permanent interpreter-guides u∗ optimal quantity of the number of the permanent interpreter-guides c(u) expected overall costs as a function of u 74 konstantin n. nechval: optimization of statistical decision for personnel management ... thus, the function of overall costs is given by c(u,x, y ) =  c1u, 0 ≤ x ≤ u c1u+ c2(x − u), u ≤ x ≤ u+ y c1u+ c2y + c3(x − u− y ), u+ y < x < ∞ (14) we write the expected overall costs as c(u) = e {eθ {c(u,x, y )}} = ∞∑ y=0 p(y) ∫ ∞ 0 c(u, x, y)fθ(x)dx = ∞∑ y=0 p(y)c(u, y), (15) where c(u, y) = ∫ ∞ 0 c(u, x, y)fθ(x)dx = c1u+ c2 ∫ u+y u (x− u)fθ(x)dx + ∫ ∞ u+y [c2y + c3(x− u− y)]fθ(x)dx. (16) the function c(u) can be shown to be convex in u, thus having a unique minimum. taking the first derivative of c(u) with respect to u and equating it to zero, we get ∞∑ y=0 p(y) ( c1 − c2 ∫ u+y u fθ(x)dx− c3 ∫ ∞ u+y fθ(x)dx ) = 0. (17) the value of u that minimizes (17) is the one that satisfies c2f̄θ(u ∗) + (c3 − c2) ∞∑ y=0 p(y)f̄θ(u ∗ + y) = c3 − c1, (18) where fθ(x) = 1− fθ(x) (19) if p(y = 0) = 1,then fθ(u ∗) = c3 − c1 c3 . (20) in this case, we should choose the u∗ such that the cumulative distribution function of u∗ equals the ratio of the difference of the underage and overage costs to the underage cost. a relatively high underage cost results in a higher number of the permanent interpreter-guides, whereas a relatively high overage cost leads to a lower number of the permanent interpreter-guides, as one would expect. if the advances in systems science and applications (2013) vol.13 no.1 75 daily demand for tours x follows the exponential distribution with the probability density function fσ(x) = σ−1exp(−x/σ), σ > 0 (21) and the cumulative distribution function fσ(x) = 1− exp(−x/σ) (22) where σ is the scale parameter (σ > 0), then c(u) = ∞∑ y=0 p(y)c(u, y) (23) where c(u, y) = σ[c1 u σ + c2exp(− u σ ) + (c3 − c2)exp(− u+ y σ )] (24) and the value of u that minimizes (23) is the one that satisfies c2exp(− u∗ σ ) + (c3 − c2) ∞∑ y=0 p(y)exp(−u∗ + y σ ) = c1 (25) if p(y = 0) = 1, then u∗ = σln( c3 c1 ) (26) and c(u∗) = c1[1 + ln( c3 c1 )]σ (27) parametric uncertainty. consider the case when the parameter σ is unknown. let x1 ≤ ... ≤ xn be the past observations (of the daily demand for tours) from the exponential distribution (21). then s = n∑ i=1 xi (28) is a sufficient statistic for σ; s is distributed with gσ(s) = [γ(n)σn]−1sn−1exp(−s/σ)(s > 0) (29) to find the best invariant decision rule ubi , we use the invariant embedding technique [8-14] to transform (24) to the form, which depends on the pivotal quantity υ = s/σ, the ancillary factor η = u/s and y/s, c(u, y) = σ[c1 u s s σ + c2exp(− u s s σ ) + (c3 − c2)exp(− u+ y s s σ )] = σ[c1ην + c2exp(−ην) + (c3 − c2)exp(−η + y s )ν] = c(η, y, ν|s) (30) 76 konstantin n. nechval: optimization of statistical decision for personnel management ... we find the expected overall costs for the statistical decision u = ηs as c(η|s) = ∞∑ y=0 p(y)c(η, y|s) (31) where c(η, y|s) = ∫ ∞ 0 c(η, y, ν|s)g(ν)dν = σ[c1ηn+ c2 1 (1 + η)n + (c3 − c2)(1 + η + y s )−n] (32) g(ν) = [γ(n)]−1νn−1exp(−ν)(ν > 0) (33) the value of η that minimizes (31) is the one that satisfies c2 1 (1 + η∗)n+1 + (c3 − c2) ∞∑ y=0 p(y)(1 + η∗ y s )−(n+1) = c1 (34) thus, ubi = η∗s (35) if p(y = 0) = 1, then η∗ = ( c3 c1 )1/(n+1) − 1 (36) and c(η∗|s) = σ[c1η ∗n+ c3 1 (1 + η∗)n ] = c1[( c3 c1 )1/(n+1) − n]σ (37) comparison of decision rules. for comparison, consider the maximum likelihood decision rule that can be obtained from (26) as uml = σ̂ln( c3 c1 ) = ηmls (38) where σ̂ = s/n is the maximum likelihood estimator of σ, ηml = ln( c3 c1 )1/n (39) since ubi and uml belong to the same class c = {u : u = ηs} (40) it follows from the above that uml is inadmissible in relation to ubi . if, say, c1 = 50, c3 = 3500 (in terms of money), and n=1, we have that rel.eff.c(η|s){uml, ubi , σ} = c(η∗|s)/c(ηml|s) = c1η ∗n+ c3 1 (1+η∗)n c1ηmln+ c3 1 (1+ηml)n = 0.9 (41) advances in systems science and applications (2013) vol.13 no.1 77 thus, in this case, the use of ubi leads to a reduction in the expected overall costs of about 10% as compared with uml. the absolute expected overall costs will be proportional to σ and may be considerable. predictive inference. it will be noted that the predictive probability density function of the daily demand for tours, x, which is compatible with (15), is given by f(x|s) = n+ 1 s (1 + x s )−(n+2)(x > 0) (42) using (42), the predictive overall costs are determined as c(p)(u|s) = ∞∑ y=0 p(y)c(p)(u, y|s) (43) where c(p)(u, y|s) = c1u+ c2 ∫ u+y u (x− u)f(x|s)dx+ ∫ ∞ u+y [c2y + c3(x− u− y)]f(x|s)dx = s n [c1 u s n+ c2(1 + u s )−n + (c3 − c2)(1 + u s + y s )−n] (44) which can be reduced to c(p)(η, y) = s n [c1ηn+ c2 1 (1 + η)n + (c3 − c2)(1 + η + y s )−n] (45) thus, it follows from (32) and (45) that ubi can be found immediately from (43) as ubi = argmin u c(p)(u|s). (46) 4 conclusions and directions for future research in this paper, we propose a new technique to improve or optimize statistical decisions under parametric uncertainty. the method used is that of the invariant embedding of sample statistics in a performance index in order to form pivotal quantities, which make it possible to eliminate unknown parameters (i.e., parametric uncertainty) from the problem. it is especially efficient when we deal with asymmetric performance indexes and small data samples. more work is needed, however, to obtain improved or optimal decision rules for the problems of unconstrained and constrained optimization under parameter uncertainty when: (i) the observations are from general continuous exponential families of distributions, (ii) the observations are from discrete exponential families of distributions, (iii) some of the observations are from continuous exponential families of distributions and some from discrete exponential families of distributions, (iv) the observations 78 konstantin n. nechval: optimization of statistical decision for personnel management ... are from multiparametric or multidimensional distributions, (v) the observations are from truncated distributions, (vi) the observations are censored, (vii) the censored observations are from truncated distributions. references [1] conrad s.a. 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(1968), introduction to operations research, new york: academic press. corresponding author author can be contacted at: nechval@junik.lv. adv syst sci appl 2018; 1; 20-40 published online at http://ijassa.ipu.ru. the contract theory: a 3-dimensional reflection on commodities & capital conversions sailau baizakov 1 , azamat ryskulovich. oinarov 2 , dana akylbekovna eshimova 3 , jeffrey yi-lin forrest 4 1) scientific supervisor, jsc "economic research institute", 65 temir-kazyk street, astana 010000, kazakhstan; e-mail: 37baizakov@gmail.com; 2) chairman of the board of the jsc "kazakhstan center for public-private partnership", 65 temir-kazyk street, astana 010000, kazakhstan; e-mail: azamat.oinarov@gmail.com; 3) deputy chairman of the board of the jsc "kazakhstan center for public-private partnership",65 temir-kazyk street, astana 010000, kazakhstan; e-mail: eda.07@mail.ru; 4) school of business, slippery rock university, slippery rock, pa 16057, usa; e-mail: jeffrey.forrest@sru.edu abstract: contract theory is a relatively young field of the economic science. the uniqueness of this branch of investigation is established on the basis that it is crucial to study the origin of microeconomic and macroeconomic indicators. in particular, because of this theory of contracts, it becomes possible to properly evaluate the performance of the real balanced growth. the study of the theoretical foundations of contracting paves the way for further development of particular tools for analyzing economic policies. keyword: trade flow, cash flow, reproduction, tio, circulation, input-output. 1. introduction the analytical techniques developed for evaluating the domestic product fail to ensure the assessment accuracy of the cost of consumed goods and services. therefore, measuring the real final product is only based on the inflation indicator that is entailed from the nominal gdp indicator. at the same time, (piketty, 2015, p.592) discovers that the indicators of inflation and real economy growth, as analyzed by using the currently available analytical techniques, may not always be accurate. the members of kazakhstan’s economists’ interest group are currently developing a program to enable the process of decomposing intra-industry input-output tables into smaller regional components. such regionalization of the intra-industry balance sheet is deemed to clearly define the deviations between the indicators of the real and financial sectors in the development of the market economy. specifically, the decomposition of the country’s intraindustries balance sheet, as reflected in the country’s input-output tables, into smaller regional components could better mirror economic activities of the subordinate administrative and territorial subdivisions within the common national management system, and would mailto:37baizakov@gmail.com mailto:azamat.oinarov@gmail.com mailto:eda.07@mail.ru mailto:jeffrey.forrest@sru.edu 21 s. baizakov, a.r. oinarov, d. eshimova, j. y.-l. forrest copyright ©2018 assa. adv. in systems science and appl. (2018) enable the setup of a new analyses model for qualitatively overseeing regulatory policies for sustainably developing the national economy. the concept that lies in the foundation of the program was initiated by the president of the republic of kazakhstan nursultan a. nazarbayev (2009). the core of the concept focuses on the right “choice of the innovation types, on which the ‘currency-and-financial system’ of the country and of the rest of the world would be based. such choice is viewed as the nucleus of the socio-political, technical, and technological milieu, in which every citizen would want to live. that is being desired by not only every citizen but also every family, every country, and the entire world. the development of the analytical tools and techniques in line with nazarbayev’s concept suggests that wreathing of benefits that item from technical and technological innovations in the real economy should be commensurate to socio-political innovations, especially in managerial decision-making. the latter has been set on a sound footing based on currency and financial innovations. measuring the true costs of goods and services, as reflected in (nazarbayev, 2009), is a distinctively new concept when compared to the existing approaches in the development analysis of a market economy. the unique content of the concept is linked to such a methodology that the key is the assessment of the performance quality of the institutions that implement the following three major innovational technologies within the cycle of reproducing capital in its monetary form, and capital in its commodity-form:  the real sector technical and technological innovations,  currency-and-financial innovations in managerial decision making, and  socio-political innovations in managerial decision making the specificity in measuring the true cost of goods and services under nazarbayev’s concept is given through the fact that production growth is closely aligned with conservation of resources. likewise, it is in line with the output increase of intermediate consumer goods. the latters are in essence directly involved in the production of output. the socio-economic effect of nazarbayev’s concept is in its negligence of the idea of generating profit at any cost. at the same time, according to (nazarbayev, 2009), profit may be obtained through efficient utilization of material, technical, and financial resources. such efficiency needs to be addressed in the production of each and every unit of the final product. 2. the purpose of this work the analysis of the macroeconomic dynamics using the three-component reproduction model, developed by russian academician alexander g. granberg (1985), has shown that economic growth indicators, specifically identified for sectors of an economy, and their growth rates have to be defined by decision makers. the key value of the granberg model has been attached to the reduction of a weighted average of material intensity in the gross domestic product. however, such contraction, according to the granberg model, “may be a cause of a more compounded impact on the dynamics of the three components” (granberg, 1985, p. 109). it is because the changes in the expenditure coefficient of one component necessitate changes in the other component. the solution as for how to satisfy the granberg-identified need for a reduction of material intensity in the gross product may be found in the comments by f. engels and were reflected in the supplement by f. engels to capital. on this matter, f. engels wrote the following: “the development of the productive power of labour reacts also on the original capital already engaged in the process of production. a part of the functioning constant capital consists of instruments of labour, such as machinery, which are not consumed, and therefore not reproduced, or replaced by new ones of the same kind, until after long periods of time. but the contract theory: a 2-dimensional reflection on commodities & capital 22 copyright ©2018 assa. adv. in systems science and appl. (2018) every year a part of these instruments of labour perishes or reaches the limit of its productive function. if the productiveness of labour has, during the using up of these instruments of labour, increased (and it develops continually with the uninterrupted advance of science and technology), more efficient and (considering their increased efficiency), cheaper machines, tools, apparatus, replace the old. the old capital is reproduced in a more productive form, apart from the constant detail improvements in the instruments of labour already in use. the other part of the constant capital, raw material and auxiliary substances, is constantly reproduced in less than a year. every introduction of improved methods, therefore, works almost simultaneously on the new capital and on that already in action. like the increased exploitation of natural wealth by the mere increase in the tension of labour-power, science and technology give capital a power of expansion independent of the given magnitude of the capital actually functioning. they react at the same time on that part of the original capital which has entered upon its stage of renewal. this, in passing into its new shape, incorporates gratis the social advance made while its old shape was being used up. of course, this development of productive power is accompanied by a partial depreciation of functioning capital. labour transmits to its product the value of the means of production consumed by it. on the other hand, the value and mass of the means of production set in motion by a given quantity of labour increase as the labour becomes more productive. though the same quantity of labour adds always to its products only the same sum of new value, still the old capital value, transmitted by the labour to the products, increases with the growing productivity of labour.” 1 by carefully reading through engels’ comments on k. marx’ capital, the accurate measurements of capital in its monetary form ( ), and capital in its commodity form ( ), one may find that these equations may well serve the basis for solving the granberg urge, targeting at the reduction in material intensity of the gross product. in other words, the three dimensional measurements of the indicators of the nominal gdp (ngdp = ) help to define the cost of the final product, i.e., by means of the indicators of the gross aggregate product ( ). that represents the sum of costs, including materials, in the form of the annual income ( ). without using the three dimensional method to measure the final product indicators, the solution of the granberg puzzle is deemed impossible or difficult. the research subject matter of this work, as first revealed by granberg, relates to the subjective need to track down the material cost and the efficiency of production resources. as such it echoes the following definition of this problem, as formulated by m. porter (2002, p. 496, 220): “any motion in a developed economy requires developing a sound local competitiveness. competition should be in line with the shift of the major focus from low wages to low costs. that would require improvements in the efficiency of production and of services”. 3. granberg’s contribution of unveiling the leontief paradox a. granberg not only discovered the function of the costs of production resources, but also led his followers to a search of the true cost of goods and services. moreover, he succeeded in explaining the leontief paradox. the approach developed by w. leontief consisted of numerical measurements of not only production, but also distribution of the common good. he constructed the reproduction schemes of the gross product. according to granberg, leontief discovered a new area in the economic science by blending the economic functions theory with mathematical modeling, systemic techniques, and processing of the economic information (suslov v.i., 2016). granberg named leontief as the most pragmatic economist-theoretician. leontief’s research methodology was built on practical observations and analyses of structural shifts in 1 marx, k., engels, f. collection. ed.2., v. 25, p. i, p. 286. 23 s. baizakov, a.r. oinarov, d. eshimova, j. y.-l. forrest copyright ©2018 assa. adv. in systems science and appl. (2018) economy and trade. unexpected conclusions were quite frequent outcomes of his research. the leontief paradox is now well studied academically. granberg acquiesced to leontief’s thought that the united states of america must interact with the rest of the world in such a way that its exports are capital intensive and imports labor intensive. such stance is stemmed from the stature of the u.s.a. as an excessively capitalized country with a relatively limited highly paid labor market. in this regard, leontief found that the u.s. did export labor and import capital. such a statement though had found no reflection in the well-known theories on foreign trade (suslov v.i., 2016, p. 163-164). for instance, the heckscher-olin theory explained it differently. the scientific essence of leontief’s approach lies in applying systemic thinking and logic to find economic solutions. then, what is the leontief paradox? the assessment of changes in the structure of the costs of materials, labor, and capital resources per a unit of production of the final product is the core of the leontief paradox. absence of juxtaposition between the methods of market-based planning and those of planned economic management, as shown in leontief’s approach, is important. leontief viewed an economy as a ship where private initiative is the wind, while planning is the steering wheel, which shows the direction. he pointed out the need for setting an equalizing balance between market and state (suslov v.i., 2016, p. 181). in our view, the leontief concept fully complies with the three dimensional measurement of capital in its monetary form, as a mechanism of pushing the market-based economy forward, and capital in its commodity form, as a mechanism making the management of state-run economy with limited resources effective. 4. the foundation of the contract theory the contract theory is based on the principle of gaining mutual benefits from a trade deal. in this theory, market relations are represented as a cycle of reproduction schemes of goods and services that mainly consist of the conversion of the capital in its monetary form, to the capital in its commodity form. the contractual relations between economic actors form the foundation for mutual conversions within the common cycle of capital and commodity. the basis of the contract theory is similar to the cyclical cost of capital in its monetary form, and the cost of capital in its commodity form. in formulaic terms, measuring the indicators of the cost of capital in its money form ( ), and capital in its commodity form ( ), has been confirmed by the fact that the cost of capital in its money form ( ) has originally been determined in monetary terms. however, the cost of production ( ) has been determined in monetary terms, and also in terms of time, which has been utilized for labor. let the average price of a unit of a national currency, as measured by purchasing power, in relative terms, be indicated as purchase power of the national currency (ppnc) following a balanced equation, which, in legal terms, has been notary-verified and signed as a contract. such a contract-based equation has close linkages between aggregate labor costs ( ) in the form of a reward of labor, and final outcomes of labor ( ) in the form of a chain of surplus value. as with the account of ‘ppnc’ and ‘wages’, such legally binding act of purchase and sale may be presented by the following equation of the contract theory: (1) since the contract has the power of a law, equ. (1), as a theoretic reflection of the noted specific law, has a legally binding force. this described pattern works in such a manner that is analogous to the power of either the ohm’s law in physics or the natural law of gravitation. however, equ. (1), upon having been converted to economic terms, is itself an economic law. the contract theory: a 2-dimensional reflection on commodities & capital 24 copyright ©2018 assa. adv. in systems science and appl. (2018) when this particular economic law, as represented in equ. (1), is extrapolated onto various sectors of the economy and various types of economic activities, the economic law, well known to experts in the area of analysis of intra-industries input-output tables, may easily be obtained as follows: (2) where is the direct labor intensity of the product, and the overall labor intensity of the product, as complied with the cost of final product ( ) and the aggregate sum of the costs of materials and financial resources that have been utilized in producing product ( ). let the coefficient ‘ ’ be the proportionality of in monetary terms. by measuring the variables in labor terms, we obtain the following equation: . both of these proportions are economically essential functions of time. in fact, they are equal to each another. here, the first proportion represents the cost of final product ( ), the sum of the costs of materials, labor, and capital resources, vs. the production ( ). it changes over time. each increase in the relevant coefficient ‘ ’ over time reflects an increase in the volume of the final product ( ) that items from the utilization of resources for production ( ). any reduction, of this coefficient over time indicates a decrease in the volume of the final product. the above-noted changes over time occur under the pressure of new, innovative technologies, etc. in a series of turnovers of capital and commodities, the growth rate of the technological potential of any economy may be determined. therefore, the indicator, referred to in this paper as ‘ ’, shall be named as the ‘coefficient of science-and-technology potential’. the second proportion also tends to change over time. however, its reverse measurement ( ), which represents the proportion of full (direct and indirect) labor costs vs. direct costs ( ) may be interpreted differently. in this regard, the leontief paradox is based on this particular economic law. any theory is enlivened only when it is positively tested in real time practices. for that matter, the contract theory is not an exception. and, the reciprocal technological cycle of the mutual conversion of physical measurements of commodity masses into monetary masses has clearly been observed in practice. for example, we may admit the hypotheses relating to the annual volumes of goods and services as ones that have been realized in-kind, denoted as ‘ ’, of the currency unit. those that constitute gross proceeds are marked as ‘ ’. then, the price of commodities ( ) is given by , and the proceeds from sales are . in that case, the velocity of money in the national currency unit will be equal to ‘ ’. the marxian reproduction schemes formula may be rewritten as follows: . (3) from the this, the following equation is derived: , (4) or . (5) the afore-described hypothesis, as accepted by this paper, fully complies with the clark concept where any realized commodity-based product is represented as a sum of the elementary utilities of the material wealth, which has been utilized in producing the materialized commodity-driven product (clark, 2000, p.220). 25 s. baizakov, a.r. oinarov, d. eshimova, j. y.-l. forrest copyright ©2018 assa. adv. in systems science and appl. (2018) in spite of the opportunity readily provided in evaluating the effectiveness of regulatory policies by the three-dimensional measurements of the cost of capital in its monetary form and capital in its commodity form, in compliance with the marxian reproduction scheme, additional analyses are currently undertaken by means of a more simplified technique following a. smith’s one dimensional income method (smith, 2007, p. 960). under the smithian income method, the nominal gdp is determined by simply deducting the costs of materials ( ) from the proceeds ( ): – . such calculations do not account for the contract theory, of which the main processes have been represented in equ. (1). in this regard, equ. (4) loses its original macroeconomic content and acquires quite a new setting. in the end, it converts to a macroeconomic equation incapable of solving the granberg puzzle: , (6) where ‘ ’ stands for the velocity of money. equ. (6) contains no physical dimension and is therefore represented by its monetary form, only. a. smith who defined labor as a sole method of measuring the annual income turns out to contradict himself. his followers are using a specific indicator, namely, the gdp deflator, to determine the physical volume of the final product. in reality, however, the gdp deflator, which has been derived from physical indices of goods and services, does serve as both a universal economic indicator, which is necessary to install the required dynamics of market prices on goods and services, and an indicator of the dynamics of the purchasing power of money in the financial system. since the latter indicator supports the balance in the nominal gdp as well as in real gdp ( ), equ. (6) is then helpful in determining . as with ceteris paribus, we have the following equation: (7) where the velocity of money ‘ ’ is itself a function in the velocity of money in the proceeds ‘ ’, and also dependent on the pace of money turnover in the intermediate product ‘ ’: and so, we have: . (8) therefore, equ. (8) relates the velocity of money ‘ ’ to the pace of the turnover of the nominal gdp and the pace of the turnover of intermediate commodities. the indicators of proceeds are therefore directly linked to the indicators of consumption of the intermediate products qp and the nominal gdp. based on equ. (8), emerges an opportunity for developing an operational technique, capable of transforming real economies into the engine of sustainable economic growth, meaning that the gdp price deflator (inflation indicator) and purchasing power of money may be transformed into the key indicators of managing innovational investments. the gdp price indicator (inflation indicator) is an integral factor consisting of such components that possess not only destructive, but also, constructive forces in developing the market economy. investigating its structure may help analyze the differences between the pace of implementation of technical and technological, ‘currency-related’ and financial, and socio-political innovations. the contract theory: a 2-dimensional reflection on commodities & capital 26 copyright ©2018 assa. adv. in systems science and appl. (2018) 5. contracts-based methods of analysis for economic policies the sraffa’s model: production of commodities by means of commodities. the main idea behind this model (sraffa, 1960) is found in the intra-industries nature of reproduction. the final product of one type of production serves the raw resource for the other type. in reality, the reproduction cycle forms a cycle of reproduction, where the final product is fully defined, and wages may contain the component of a surplus value, because the profit margin is defined by the interest rate. and that feature of sraffa’s model is distinct from the marginal utility theory, where the correlation between supply and demand may serve as the law on distribution. it also differs from the labor theory, where wages are defined by the means that provides the living to the worker and his/her family. similarly, the profit margin is defined by production technologies. the ronald coase conjecture (1960). the theory (kapelushnikov, 2017) stated that firms are created when transaction costs are lower inside the firm than outside it in the open market. it means that if the property rights of all parties are carefully defined and transaction costs nullified, then the final outcomes do not depend on the changes in the distribution of property rights. however, in the process of accounting the transaction costs, the desirable outcomes may not always be attainable. in this regard, high costs of obtaining the required information, conducting negotiations, and settling disputes may exceed potential benefits of the deal. besides, when accounting losses, there may appear essential differences in propensities of contractors on recording the losses. a reference to the potential impact of the income (that later became the core of the neo-institutional theory) has been introduced to capture all the afore-described differences. nevertheless, the coase theory fails to reflect the details of why some firms grow owing to the integration of sequential production stages, while others focus more on just one or a few production stages. the energy sector serves as an example of such integration where coal mines are paired with hydropower stations that work on coal. the williamson’s theory of transaction costs 2 . conceptually, this theory, established in 2009, is closely associated to the costs of contract settlements and regaining the right to property, or other services within the acts of interaction between two or more participants of a contract. according this theory, the hierarchical structure prevails in the market until it ensures an inexpensive and expeditious method of resolving conflicts. if the three agents cannot resolve their disputes relating to work load distribution and income, the manager steps in to resolve those disputes. not only from the economic stance has williamson’s concept been thoroughly studied, but also from the legal standpoint where a contract is regarded as an organizational component in harmonizing work processes. the theory ensured the unity of economies and legal acts, backed up by contracts and, thus, led to a more profound understanding of the aims and objectives of a collaborative work. the williamson’s firm has been depicted through the prism of institutional notions, not industrial. a firm and a market are being judged by their capacities to conduct various transactions, enabling resource minimization. the theory also pointed out that production facilities standing afar from one another are more likely to group together under the same owner. oliver hart and bengt holmstrom (kornelyuk, 2016): market economy is the economy of contracts. the theory is developed by hart and holmstrom to define the parameters of contracts. the theory has been valued for its advocacy of mutually beneficial decisions 2 the theories of transaction costs. chapters on economics. history of the economic science. retrieved from: (http://ecouniver.com/economik-rasdel/istekuz/215-teorii-transakcionnyx-izderzhek.html) 27 s. baizakov, a.r. oinarov, d. eshimova, j. y.-l. forrest copyright ©2018 assa. adv. in systems science and appl. (2018) relating to contracts’ parties. it is of practical value for any of the macroeconomic models geared to balance economic growth. in macroeconomics, the theory may serve as the basis for analyses of varied contracting models, such as rewards of top managers, franchises, surplus insurance payments, and privatization of public enterprises. it has been effective in decision-making on investments and incentivizing economic activities through a choice of optimal efforts. the sagadiev theory of surpluses’ exchange. among kazakh economists, the specific subject matter of a surplus exchange has been thoroughly studied by nurlan sagadiev. sagadiev (2004) concluded that neither special effects nor production costs, nor income, determine the final outcomes of market relations. the sagadiev approach resonated with the theories developed at different times by williamson and hart. sagadiev developed linkages in the formation of the surplus value and connected those to the sales profit. to show the appropriateness of his theory’s stance, sagadiev made a reference to the research by d. rosenberg, who provided his valuable comments relating to k. marx’ capital as follows: “it may seem that k. marx did not provide all the components of the whole chain that are necessary to transit from simple commodities turnover to the production of surplus value. it may also seem that he did not explain the details pertaining to the trade. those have been named by prof. n. sagadiev as a perfected cycle of commodities, which sets the grounds for capital. according to him, profit is a category, which is unknown for a casual turnover of commodities. it has first emerged in trade. in this regard, the then k. marx had proceeded to investigating the details of the surplus value, which was based on the mature capitalist method of production. the profit faced by sellers at different times, relating to the existence of trade, had various sources. those, sometimes, were mere robberies” (rosenberg, 1984). sagadiev (2004) concluded that “consumption, which delivers wealth, and labor, which creates wealth, in fact, are the characteristic features of wealth. accordingly, there exists the difference between the cost of the commodity and its consumer cost. and, that may be named as the surplus value in order to be abstracted from the postulates formulated by the marxian surplus theory. in other words, it is that component of the surplus value of wealth, which surpasses the cost”. additionally, sagadiev considered that condillac not marx was the first to discover the true source of creation of the surplus. it was condillac who questioned the surplus, the phenomenon, which came to be the subject of exchange. according to sagadiev (2004, p. 90), the latter should be regarded as an alternative to the paradigm of the cost, which has governed economic thinking since the times of aristotle. the results of sagadiev’s work and the logic of his methodological approach captured researchers’ attention, especially, when he presented his theory of contracts. in particular, he stated the following: “by purchasing a commodity, a buyer enters into the relationship with a producer of the commodity. the producer wants to be rewarded for his labor. therefore, his cost is based on the volume of those benefits that are necessary for maintaining his labor continually. after the sales of his commodity, the producer becomes a buyer and enters into the relationship of exchange, possibly, with the same seller of a consumer commodity. as a buyer, the producer is interested in the consumer cost of the commodity. by means of the consumer cost of the commodity, he should reward the costs of his own labor. no doubt, the demand of a producer of commodities is not for one specific commodity but for a set of commodities. the labor costs are rewarded by some set of consumer costs. at the same time, if we consider the aggregate of all commodities as one commodity, and also, all sellers of commodities as one seller, then we can consider the act of the exchange as one deal where the aggregate consumer cost of the aggregate commodity is greater than the labor cost of the commodity. the cost, which the producer receives, in exchange of his produced the contract theory: a 2-dimensional reflection on commodities & capital 28 copyright ©2018 assa. adv. in systems science and appl. (2018) commodity, in the form of money, becomes the means of the evaluation of the consumer cost of the commodities that he needs for his life. the metamorphosis of commodities for the producer concludes by the fact that the producer receives one good instead of the other: т-d-т. the producer materialized it in the capacity of the cost, i.e. as that volume of consumption, which is necessary to reward his expenses, in exchange of the consumer cost of another commodity, which satisfies his needs. eventually, for the producer, the three different commodities have similar costs of exchange because those have been exchanged for the similar quantities of money. however, in one instance of exchange, the subject of exchange has reflected the true cost of the commodity and, in another instance, the subject of exchange has turned out to be the consumer cost of the commodity. the final metamorphosis of commodities for the seller acquires the reflection of the following sequence: ‘d-т-d’. here, all the evolutions went upside down. one and the same commodity had different exchange value. in buying the commodity, the buyer should have paid its cost. however, the same buyer was selling the commodity at its consumer cost. the difference in money had been the consumer surplus cost. at first, such method of creating the surplus value may seem to depend on the will or whim of certain individuals. in the real world, things do happen in accordance with this particular method. however, things are not solely confined to such a method. a buyer and a seller are presented in the personified actors of economic activities. the role played by a seller has been in converting the production costs to consumer costs. in this regard, the producers have been in need of sellers. the sellers have been in need of producers. to perform his duties, a seller has to have money and commodities. to ensure such possessions, he has to accumulate excess commodity and excess money in the form of consumer surplus costs. as mass volumes of commodities and money are being exchanged through sellers, the aggregate seller in the eyes of an autonomous producer is associated with the world of monies and commodities. the wider the exchange of commodities and monies is, the wider the division of labor, and the more powerful the role of the seller in the life of the society. by carefully looking at the evolution of the capitalist production relationship, specifically, the example of england, one may observe that trade surplus has been the first ever to create the initial form of the accumulation of capital (sagadiev, 2004, p. 90). by substituting ‘economics of the buyer’ in sagadiev’s theory with ‘economics of the currency-financial sector’, we may see the full picture of the cycle of the reproduction processes of capital in its money form ( ), and capital in its commodity form, ( ). 6. a geopolitical model of developing countries: evaluating real sector growth as mentioned earlier in this paper, thomas piketty noted that the concepts of ‘inflation and growth’ have not always been accurately defined: the division of the nominal growth into ‘real’ and ‘inflation-derived’ components are arbitrary, and thus may be disputed. the reasons underneath the inaccurate definition of growth and inflation, in our view, are not related to the low quality of the measurement tools of the balanced economic growth. instead, they are linked to the necessity of balancing them against real time conditions of the globalization process of the world economy. what piketty reflected was about the models developed on the basis of one dimensional measurement of macroeconomic indicators of the sectors of developed countries, where money of the current year has been measured by money in the previous year. according to 29 s. baizakov, a.r. oinarov, d. eshimova, j. y.-l. forrest copyright ©2018 assa. adv. in systems science and appl. (2018) piketty, the traditional tools of economic analyses do not ensure full compliance of macroeconomic interests in the development of the financial sector with the real sector interests. that entails certain risks. the development the economies of the uk and france are examples that show that such incompliances occur in spite that these economies move along the trajectories of balanced economic growth. moreover, such risks tend to increase. so, any application of the traditional analysis techniques is subject to effects of risks. the need for qualitatively new analytical tools in measuring and evaluating indicators of balanced economic growth is there. the construction of growth models for developing countries is linked to the marker capacities of all types in managing their growth rates, or in reducing the material intensity of the gross product. under the condition of the globalizing world economy, the real economy mainly depends on real proceeds as well as on the efficient utilization of intermediate consumer goods and natural resources. as reflected earlier in this paper, macroeconomic models for defining the nominal gdp are incapable of solving the current challenges related to accounting the material intensity of the gross product in view that they are not defined as part of the whole after deductions of current costs of materials and costs of wear and tear of fixed assets. potential solutions to the outstanding challenges regarding the regulation of the costs stemming from material expenditures and the costs of utilizing natural resources lay beyond the domain of macroeconomic regulators. problems in this area cover the linkages of the macroeconomic indicators and are directly linked to the solution of environmental problems and the issue of green economy. that means, if we want to develop green economy and maintain a healthy environment, we should not only be directly involved in growth of current incomes and profits, but also responsibly engage in efficient utilization of materials and natural resources. stating the problem: the reasons behind the derivation of unbalanced indicators of growth in the above-noted sectors of economy are linked to the use of the outdated tools of assessment and evaluation of the indicators of the balanced economic growth that do not allow full compliance of interests of the real sector economy with interests in the financial sector. according to the monetarists, the keynesian model of balanced growth was immature because in it capital in its money form played a secondary role. a much improved model, built later by keynes, is presented as follows: . in the initial formula, ‘рр’ stood for purchasing power of the national currency unit. in the improved model, the equation has been upgraded as follows: where ‘b’ stood for the gdp deflator. both of these models suffer from one-sidedness in measuring the indicators of the real and financial sectors of an economy, which define the volume of the real final product. the methodology of constructing these models is based on the assumption that prices for goods and services remain unchanged as well as the velocity of money. that followed the one-sided principle of measuring the quality of balanced indicators of economic growth. and that principle paved the way to the introduction of arbitrary definitions. there also is a third model named after mandell and fleming. however, its close similarity to the keynesian models can easily be spotted. the only difference between them is in that the latter are derived from the unit of a world’s reserve currency instead of the units of individual national currencies. the grave one-sidedness of measurements in the afore-mentioned models has been formed by the three concurrent economic theories. the first is the marginal inutility theory, the contract theory: a 2-dimensional reflection on commodities & capital 30 copyright ©2018 assa. adv. in systems science and appl. (2018) which defines the price for goods and services by supply and demand. the second is the labor cost theory, which defines prices for goods and services by the costs required for producing the commodity following the marxian expenditure method. the first theory presents income based on annual product without accounting for materials and capital costs required for production ( ). the second presents annual product by balancing the costs of production ( ). in short, both approaches suffer from limitations of one-sidedness and one-dimensionality in measuring the indicators of balanced economic growth. one of them focuses on income in the annual production of capital in its monetary form, which is defined by using the macroeconomic approach based on . the other focuses on the capital in its commodity form, which is defined by the macroeconomic approach based on . the piketty model of balanced economic growth is developed based on economic laws of development. to resolve the problem of measurements inefficiencies of inflation and growth indicators, piketty attempted to use the cobbs-douglas production function by taking the aggregate elasticity factor within the limits between 1.3 and 1.6. however, these limits do not conform to the two economic laws given in his work (piketty, 2015, p. 225). the first of piketty’s laws is related to the dynamics of proportionality of the national capital and national income (piketty, 2015, p. 67): , where denotes the capital profitability, β the accumulated capital expressed in years of the national income α. the second law, named as the law of cumulative growth and of cumulative profitability, defines the size of the cumulative capital as the proportion of the form of the accumulation ‘ ’ to the pace of the economic growth ‘ ’: (piketty, 2015, p. 171). according to these laws and laws on population growth, any insignificant increase in the capital profitability (which prevails in the growth of the economy over the lengthy period of time) leads to significant increase in the growth of capital and, thus makes a destabilizing impact on the structure and dynamics of the social inequality (piketty, 2015, p. 90). however, these piketty’s laws do not account for linkages between macroeconomic and microeconomic indicators of growth, while the linkages are, in fact, products of reproduction of capital in its income form, and capital in its commodity form within cycles of the development of common good. the geopolitical model for analyzing the management efficiency of public-private partnership projects: our discussion above shows that the evolution of piketty’s economic laws, as the principle of defining the model structure of balanced economic growth based on input-output tables, enables the use of the three-dimensional method to measure the indicators of the market equilibrium. in the following macroeconomic analyses, with due reference to equ. (2), which represents part of the three-component equ. (1) of the macroeconomic contracts theory, we will justify the macroeconomic origin of the contents of equs. (3) (8). the macroeconomic contents of equ. (2) have been revealed by using the algorithm of scaling (akimov, 2014), where the algorithm of scaling vectors and matrices represents the measurements of the model of intra-industries balance reflected in input-output tables. let represents the labor intensity of the gross output, and the labor intensity of the gross product. then the concurrent measurements of aggregate labor intensity may be written respectively as and , which indicate respectively the labor intensity of the final and gross product to the i th extent of the sector ( ) of the economy. so , and can be written as follows (akimov, 2014, p. 176): and (9) 31 s. baizakov, a.r. oinarov, d. eshimova, j. y.-l. forrest copyright ©2018 assa. adv. in systems science and appl. (2018) then we obtain the set of equations for the full cost of physical labor that has been utilized for the production of the unit of the final product for each of industries (akimov, 2014, p. 176): (10) thus, the full labor cost of producing the final product is defined by scaling the product of direct labor intensity of the gross output of all industries by relevant columns of the matrix that reflect the full labor cost in the model of intra-industries balance, which is represented in monetary form. the true cost of money is obtained by solving the concurrent problems of the threedimensional measurement of the balanced economic growth, on the one hand, and the product’s labor intensity on the other, where the unit of money is where ‘ ’ stands for the coefficient of the science-and-technology potential of the country, of which the increment: ( , ) is defined by either the margin of changes in the price for goods and services or the measurement of rent stemming from the utilization of natural resources. in that case, the real growth rate of the science-and-technology potential of any country may be measured by the difference between the growth rate of the production cost of the final product and the cost of resources that have been utilized in the production. it is because both had been sourced from the common pool of time, spent for labor ( ): (11) since equ. (1) defines the difference between the three growth rates of different indicators of the national economy, given the equality of the time, spent for labor that had been utilized for production ‘l’, it may be represented as an indicator of acceleration of the economic growth . so, the whole of the science-and-technology potential may be defined by . 7. a model for the science-and-technology potential by using gdp deflator models of the science-and-technology potential may be built on the basis of converting the monetarists’ equation of exchange. let the gdp deflator be expressed by based on the monetarists’ formulation. by multiplying both sides with the purchasing power of the national currency ‘рр’, we obtain the following: which is a qualitatively new expression of the balanced economic growth and defines the real volume of the final product – : which represents the nominal gdp that defines the cost of the final product and its product with the true cost of money represents the real final product. if , then the purchasing power of money may be defined under the new economic law as follows: the contract theory: a 2-dimensional reflection on commodities & capital 32 copyright ©2018 assa. adv. in systems science and appl. (2018) in this case, the coefficient ‘с’ may be defined as the proportion of the direct labor intensity to the full labor intensity ( ), or as the proportion of the cost of the utilized final product to the aggregate cost of resources utilized in the production ( ). however, the assessment of the coefficient is subject to that measuring the real gdp is done by following the method of three approaches. the first is the income method of a. smith, according to which, capital has only one dimension, its money form. under the one-dimensional approach to measuring the indicators of the balanced economic growth, the coefficient of the science-and-technology potential ‘ ’ is defined as follows: under the smithian theory, the purchasing power of money is defined by: , and the gdp deflator by: . the second approach is represented by the marxian expenditure approach, according to which capital has three dimensions. in this case, capital in its money form has the money dimension and capital in its commodity form has the labor dimension. so under the threedimensional measurement of indicators of balanced economic growth, the coefficient of the science-and-technology potential ‘ ’ is defined by : where represents the price index for goods and services, and the indicator index ‘ ’, and the material cost of producing the final product . since the real final product ‘ ’ is equal to , all the previously discussed cases present an opportunity to re-assess the true cost of the quasi-real gdp by using the following formula: here, the main equation of assessment of the real final product ‘ ’ at the macroeconomic level is fully defined owing to the coefficient ‘ ’, as a result of analyzing the inputs and outputs at the microeconomic level: the result of analyzing the assessment and evaluation of the impacting efficiency of regulatory policies pertaining to the development of the national economy is presented in the equation, which defines the mutual convertibility of the cost of capital in its commodity form, back to its monetary form as follows: (12) 33 s. baizakov, a.r. oinarov, d. eshimova, j. y.-l. forrest copyright ©2018 assa. adv. in systems science and appl. (2018) 8. economically assessing the efficiency of innovative technologies in the monetary dimension the difference between the marginal growth rates of capital in its money form in macroeconomics, and capital in its commodity form, is defined by the following formula: (13) for simplicity, let (standing for acceleration), representing the difference between the three growth rates of the cost of the final product as an indicator of capital in its monetary form and the output of goods and services as an indicator of capital in its commodity form . it may be named as the marginal coefficient of the science-and-technology change. this indicator of the acceleration ‘a’ emerged as ‘deus ex machina’, given equality of labor and capital that have been utilized for producing the final product ( ) in macroeconomics, and output of goods and services ( ) in microeconomics. so, in formulaic terms the above stated may be presented as follows: (14) the final result evolves from inside of the socio-economic system, which is defined by juxtapositioning the outcomes of the intra-industries balance models of the development of a country, as reflected in labor and monetary dimensions. such an effect of the science-andtechnology potential of a country, as portrayed j. clark, relating to the macroeconomic dimension has been named as the enterprisers’ profit by (baizakov and oinarov, 2015), and the surplus profit in the reproduction schemes, as lenin formulated, with accounting for the scientific-technical progress (baizakov and oinarov, 2015, p. 172). 9. convert science and technology effect in its monetary form to its labor form equ. (14) may be rewritten by representing it as the difference between the growth rates of capital profitability in its form of the cost of the final product , and capital in its form of the resources utilized for production . such this difference represents the acceleration of the input of science-and technology potential in the development of a national economy ‘ ’, defined by equating the costs of labor time in hours, days, years ‘ ’. accordingly, the overall potential of science-andtechnology innovations in the aggregate expression ‘ ’ is defined by the product of the total labor time ‘ ’ and the indicator of the acceleration of economic growth such effect is defined by the difference between the three growth rates of the varying indicators of the development of the national economy. if the input of science-andtechnology potential of the country was previously defined in obscure terms, a scaling effect or a solow residual now enables an accurate assessment by employing the cost of the actually utilized labor time in production, labor productivity, capital profitability, and the coefficient of the science-and-technology potential. additionally, each entrepreneur may readily perform formulae-derived calculations corresponding to types of his/her economic activities. the contract theory: a 2-dimensional reflection on commodities & capital 34 copyright ©2018 assa. adv. in systems science and appl. (2018) 10. convert the science and technology effect in its labor form to its energy components the measurement ‘ ’, which represents the product of the number of people employed in an economy ‘ ’ and the indicator of the acceleration ‘ ’, may also be considered as the capacities in the energy units. according to the fao data, an average person needs a minimum 1,800 kcal (7,500 kilojoules) per day. each country has its own conversion coefficients on the efficiency of labor time in energy unit, which takes into account of climatic conditions of living in the country, the type of economic activities the citizens are engaged in, ages, genders, etc. example, in the uk, an average aged female would consume 2,200 kcal per day, a man 2,500 kcal. it adds up to 2,350 kcal per day, on average. these indicators in the usa would be: female 2,200 kcal, man 2,700 kcal, which adds up to 2,400-2,500 kcal per day on average. the energy formula used to define the science-and-technology potential becomes clear and comparable with formulae on measuring based on the thermodynamics theory and relativity theories. those measurements with the above-noted impacts in money, labor time, and energy units, turn out to be authentic with each other. they clearly define the level of the development of the production forces and that of capital. assessment of the energy impact, as well as that of money and labor, is defined by comparing the growth rates of key indicators of the macroeconomic dynamics, capital in its money form, and capital in its commodity form. the formula for the energy efficiency of innovations in the production and in the innovations technology is measured by , where ‘ ’ stands for the energy impact measured in kcal, kilojoules, кwт, and ‘ ’ the number of people employed in the economy, also measured in (kcal, kilojoules, кwт), and the indicator ‘ ’ the acceleration, defined as the difference between labor productivity and capital measured by the final product, and labor productivity as well as capital profitability measured by costs of the gross product. the unit of the indicator of acceleration is the relative measurement defined as the difference between the three productivities measured by their growth rates. the novice of the present research is in its ability to enable the measurement to be interpreted as the net input of the science-and-technology potential of the economy, in developing the formula where labor time utilized for the production of product ( ) and of goods and services ( ) enables the measurement to be considered as a net contribution of the science-and-technology potential in the economy, which in turn is defined as the difference between the marginal labor productivity in the final product and the marginal labor productivity in the aggregate costs of production. the afore-described formula may serve as a solution to the granberg puzzle. such solution has been obtained by science-driven measurment of inputs and outcomes. in measuring the macroeconomic dynamics, if the indicators of one of the threecomponent dimensions, either capital in its money form or capital in its commodity form, are missing, the resolving the granberg puzzle will become impossible. an application of the full, three-component matrix of the intra-industries input-output tables is the necessary prerequisite for solving the granberg puzzle. no doubt, not always, the science-and-technology potential of a country yields positive impact. it is because not every investment can ensure sustainable enterprisers’ profits evenly across all industries. in this regard, the negative surplus profit is often being registered at the national economy level. moreover, not always, the invested capital works at its maximum capacity, and often the expected impact proves to be nil. in order to avoid such unexpected 35 s. baizakov, a.r. oinarov, d. eshimova, j. y.-l. forrest copyright ©2018 assa. adv. in systems science and appl. (2018) outcomes, the assessment of the input of the science-and-technology potential into an economy at the country level is needed. the cycles theory versus the contract theory: the literature suggests the following three approaches that are applicable in explaining natural cycles. a) a cycle is a phenomenon that is external to the economic system; and its evolution is influenced by noneconomic factors. such factors may be solar radiation, political shake-ups, revolutions, demographic booms, inventions and innovations, influencing the economic and environmental milieu for production processes; b) a cycle is an internal phenomenon, driven by causes endogenous to the economic system, initiated by the self-reproduction of economic cycles, such as demand and benefits, consumption and investments; and c) a cycle is a multispectral phenomenon, driven by a synthesis of both internal and external factors that occasionally but sustainably impact the economic system. according to american researchers, 1,380 types of economic cycles have been identified. the duration of these cycles vary from 20 to 700 years (klinov, et al., 1989, p. 7). from the practical point of view, there exist different classifications of cycles. the following two criteria are given in the core of the above-referred classification: (1) the duration of cycles; and (2) the driving forces of cycles. according to the first criterion, we suggest the following classifications: 1) seasonal cycles with the duration of one week to several months; 2) business cycles with the duration of one to several years; 3) the kitchin cycles with the duration of 3 to 5 years; 4) the jugular cycles with the duration of 7 to 11 years; 5) the kuznets cycles with the duration of up to 20 years, and 6) the kondratiev cycles with the duration of 40 to 60 years. the driving forces of business cycles are primarily linked to investment activities. the mechanism underneath the market dynamics is vested on principles of acceleration and multiplier effects. the principle of acceleration suggests that the scale of investments depends on the increment and changes in the demand versus the final product. business cycles have well been studied in the western economic science, for details please consult with the works by p. samuelson, j. hicks, and others. short-term cycles with the duration of 3 to 5 years evolve out of the dynamics proportionate to the size of the reserves, which consist of materials and commodities stocked by firms. they have been named in honor of d. kitchin, an english researcher who first stidied how business cycles work. he particularly noted: “first, they emerge as outcomes of investing in materials, raw resources, and stock capital in an intention to make the best use of the market demand. gradually, the demand wanes off, and capital, invested in stocks, becomes excessive. stock investments sharply diminish in size, and the balance between stocks and demand slowly restores. thus, the cycle helps restore the market equilibrium of supply and demand" (menshikov, 1989). mid-term cycles, also known as industrial cycles, having the duration of 7 to 11 years, are related to renewal of fixed assets, i.e. inventory, equipment, facilities, cars. the life cycle of those assets depend on the extent of their wear and tear, while the latter defines the factual duration of the mid-term cycle. the important role in sustaining the cycles of this type is played by the scientific-technical progress. along similar lines, other types of mid-term cycles evolve as outcomes of mass revolutionary innovational technologies. those cycles are transmitted from one industry to another across the entire chain of industries in a domino effect, thereby completely changing the very foundation of the production (klinov, et al., 1989, p. 26). the contract theory: a 2-dimensional reflection on commodities & capital 36 copyright ©2018 assa. adv. in systems science and appl. (2018) s. kuznets developed the construction cycles theory for the cycles lasting 15 20 years. these cycles are linked to periodical mass renewals in housing and dwellings. y. rostow, h. biskhar, a. kleinknekht, among others consider the kuznets cycles as specific characteristics of the american economy only because they reflect huge migrants’ inflows and their construction-related activities. the super cycles theory or k-waves, established by kondratiev (1989), is one of the most promising directions in exploring long-term tendencies in the development of world economy. this area, by some reasons, has remained a less investigated domain compared to short-term and mid-term cycles. even so, the practical interest has gravitated towards the study of long-term changes in an economy since relatively recent past. of interest is also the account of managing the processes of production and sales of produce. such tendencies, often repetitive in nature, are clearly subjective. however, the lengthier is the tendency the slower is the process of accumulating statistical data and information required for uncovering and investigation of such tendencies. the economic policies that are currently adopted in developing countries are mainly designed on the bases on the monetarist’s concepts that are strictly aligned with the requirements of international monetary institutions and organizations. they, however, might lead to continual stagnation of developing economies and ineffective utilization of economic potential of these countries. based on the afore-discussed observations, equipped with the common aim of painlessly overcoming economic hardships imposed by a series of economic and financial crises in order to create prerequisites for the subsequent more sustainable economic growth of developing countries, a timely and effective change of the models of economic development for developing countries is much needed. in this regard, this paper specifically recommends to replace the monetarists-developed model for regulating economies by geopolitical model, which is primarily oriented at developing the function of the science-and-technology potential (stp) of a developing economy. if such model is accepted, the main indicators of the stp may objectively signal about the accumulation of negative consequences of any given economic policy in a developing country, undertaken at any given time interval, and identify the need to make remedial corrections in order to prevent an economic decline. the growth rate of labor productivity in terms of the cost of the final product and the growth rate of labor productivity increments in relation to the aggregate cost of productizing the final product may serve as the key indicators of the stp. the need for proceeding with active research in this direction is acute with the reasons listed below: the more is explored in the area of cyclical economic dynamics, the more accurate shall market economic forecasts be, and the more effective shall the impact of state be on the given economy. with appropriate knowledge in this research direction the governments of developing countries will be able to timely implement required economic policies in order to remove inefficiencies and work out action-based implementation plans, inclusive of investment, financial, credit, tax policies, as relevant. such sets of measures will help reduce negative and disastrous consequences of global economic and financial crises, and smooth out at least some of the cyclical nature of economic development. improving the pace of economic development towards its balanced and thus sustainable growth will ensure favorable conditions of the long-term prospective development and provide increasing rates of economic growth, which thereby improves the quality of life styles and wealth of people from around the world. as the main set of variations in assessing the short-, mid-, and long-term perspectives of economic growth, the growth model with kazakhstan’s regional specificities is suggested in this paper to be used as a pattern. the main criterion, which serves as the basis for replacing 37 s. baizakov, a.r. oinarov, d. eshimova, j. y.-l. forrest copyright ©2018 assa. adv. in systems science and appl. (2018) one model by another is the economy’s potential of ascending economic growth. it may be defined, according to equ. (14), as follows: . indeed, the development of market-driven forces of certain sectors of a developing economy does not underscore essential and insignificant deviations in the growth rate of another economy of either developed and developing countries. however, the contract theory, as a reflection of economic laws, enables one to timely react to such deviations not only at the macroeconomic level, but also at the level of individual firms, be they small or medium sized. the point is that the dynamics of proportions between the annual cost of the final product ( ) to that of the commodities of intermediate consumption (qp) that have been utilized in the process of producing the output represents the productivity of commodities of intermediate consumption in terms of the final product. if this indicator is denoted by ‘ ’, we then obtain the following function of time: . the system of national accounts of world countries reflects the costs of final product ( ) as well as of intermediate consumption ( ). in other words, both these indicators are known for calculations. that means that one can easily derive the dynamics of change in the productivity of commodities of intermediate consumption by using the final product. analyses of economies of both developed and developing countries reveal the fact that this particular function complies with the definition given by michael porter in terms of the productivity of the materials resources 3 . it defines the cyclical nature of the economic development of national economies of the world as well as the duration of the cycle, and also, the driving forces of the cycle, which are defined by the input of the stp in the overall economic growth rate. thus, in an ideal case, possible values of the productivity of intermediate consumption goods in terms of the final product, μ, are positioned within the range from 0.1 to 9.0. in a concrete case, one may define equations as functions of time, and identify the form of possible deviations. 11. growth rate of stp as a function of the productivity of intermediate consumption goods what is surprising is the fact that the growth rate of the stp, which is the coefficient ‘ ’, is a function of the productivity of the intermediate consumption goods ( : . which can be rewritten as follows: . if we let , , and , then the function of the stp takes the standard form of . in this particular case, the equation, involving the stp may be expressed as follows: . to chart of the function of the stp, which is equal to , we may involve the range of indicators of the independent variable ‘ ’ and then derive from the relevant values of the dependent variable ‘ ’. 3 it is michael porter who introduced the term ‘productivity of resources’, of which the reverse meaning defines the efficiency of production resources. special attention needs to be paid to the shift of ‘the focus from low wages to the overall low costs’ as the most important factor of the ‘development of strong local competition’ along the path of ‘moving towards the developed economy’. porter, m. e. (2002). on competition. in russian. m: williams publishing house. st. petersburg, moscow, kiev. (p.496), (p.220). the contract theory: a 2-dimensional reflection on commodities & capital 38 copyright ©2018 assa. adv. in systems science and appl. (2018) by using , , and , we can construct the following table by deriving the values of productivity of intermediate consumption goods from final product ‘ ’ within the range from 0.1 to 9.0: table 1. the stp function 0.45 1.45 -0.69 0.31 3.60 4.60 -0.22 0.78 0.50 1.50 -0.67 0.33 4.10 5.10 -0.20 0.80 0.55 1.55 -0.65 0.35 4.60 5.60 -0.18 0.82 0.60 1.60 -0.63 0.38 5.10 6.10 -0.16 0.84 0.65 1.65 -0.61 0.39 5.60 6.60 -0.15 0.85 0.10 1.10 -0.91 0.09 6.10 7.10 -0.14 0.86 0.60 1.60 -0.63 0.38 6.60 7.60 -0.13 0.87 1.10 2.10 -0.48 0.52 7.10 8.10 -0.12 0.88 1.60 2.60 -0.38 0.62 7.60 8.60 -0.12 0.88 2.10 3.10 -0.32 0.68 8.10 9.10 -0.11 0.89 2.60 3.60 -0.28 0.72 8.60 9.60 -0.10 0.90 3.10 4.10 -0.24 0.76 9.10 10.10 -0.10 0.90 as a result, the stp function has been derived from intermediate consumption goods , which, by the nature of changes, is fully dependent on the productivity of intermediate consumption goods. figure 1 showed the three instances of development of the stp. in the first instance, the stp function has been reflected at the background of the growth rate of productivity of intermediate consumption goods (function 1a), which acquires values from 0.10 to 9.0 and shows the ascending trend. this curve is shown in diamond symbols in figure 1. fig. 1. the function of the stp and intermediate consumption goods in the second instance, when the function of the stp , given that the growth rate of productivity of intermediate consumption goods ‘ ’ (curve 1б) acquires values 0,0000 0,1000 0,2000 0,3000 0,4000 0,5000 0,6000 0,7000 0,8000 0,9000 1,0000 0,00 2,00 4,00 6,00 8,00 10,00 с μ с=μ/(1+μ) 39 s. baizakov, a.r. oinarov, d. eshimova, j. y.-l. forrest copyright ©2018 assa. adv. in systems science and appl. (2018) from 2.0 to 9.0, we assume, has the descending trend, which at point 2.0, acquires the value of 0.89 and, at point = 9.0 equals 0.07. this trend has been shown in square symbols in function 1б. three more possible instances, as reflected in figure 1, have been shown by direct lines that are parallel to the axis of the productivity of intermediate consumption goods . the first of them characterizes the real time situation in the developing market given that the stp coefficient remains stable (curve 1с). thus, figure 1 reflects the individual case that for any . the second trend line parallel to the axis of the productivity of interdemediate consumption goods transcends the points of intersection of both curves (for details, see function 1d). above this trend line, a set of the indicators of balanced economic growth of different countries has been reflected. as per the trajectory of changes in the productivity of interdemiate consumption goods ( ), as derived from the cost of the final product, it may serve as a forecast indicator for the analysis of causes of distortions in any economy, since the stp growth rate is defined depending on the dynamics of the changes in the productivity of intermediate consumption goods. 12. some final remarks generally speaking, the results of recent economic discussions are mostly derived out of either keynesian type models or monetary-policy type models, developed respectively by the followers of these respective schools. although the thoughts of these schools are still used in the practical management of the world’s economies, many of their results are not consistent with the modern realities of the globalizing economy. as a matter of fact, both of these types of models are special cases of our generalized model of market equilibrium, developed on the basic ideological positions of the “fifth way”, proposed by nursultan nazarbayev, the president of kazakhstan. even so, this does not means that the theories of these schools should be sent to the archive of the theories on economic growth and equilibrium. in particular, keynesian type models are more in line with the economic interests of development of real economic sectors so that they can continue to provide service as the theoretical basis for relevant decision makings. and the models of monetary-policy type can continue to serve the economic interests of developing the financial sector. since our generalized model proposed in this paper is obtained by integrating three different market equilibrium indicators used in the keynesian type and monetary-policy type models, it is expected that this new model can help harmonize relevant economic interests and therefore acts as an instrument to ensure the implementation of regulatory policies. references [1] akimov, n. (2014). ot capitalisma k capitalismu [from capitalism to capitalis. volume i]. almaty, kazakhstan: raritet. [in russian] [2] baizakov, s., oinarov, a.r. (2015). systema modeley sbalansirovannogo razvitiya rynochnoi economiki [the system of models of the balanced development of market economy: theory of construction]. astana, kazakhstan: center of public-private partners. [in russian] [3] clark, j. (1908). the distribution of wealth: a theory of wages, interest and profits. new york, usa: macmillan. [4] granberg, a.g. (1985). dinamicheskiye modeli narodnogo khozyaystva [dynamic models of a national economy]. мoscow, ussr: ekonomika. [in russian]. the contract theory: a 2-dimensional reflection on commodities & capital 40 copyright ©2018 assa. adv. in systems science and appl. (2018) [5] coase, r. h. (1960) the problem of social cost. journal of law and economics, 3, 1–44. [6] klinov, v.g. manukovskii, a.b., hartukov, e.p., tsygichko, l.p. 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(2014). capital in the twenty-first century. cambridge, ma, usa: harvard university press. [12] porter, m. (2008). on competition. usa: harvard business review. [13] rosenberg, d. (1984). kommentarii k «kapitalu» k. marksa [the comments on the capital of k. marx]. moscow, ussr: ekonomika. [in russian] [14] sagadiev, n. (2004). ponyatie stoimosti v contexte probleme universaliy v nauke. perspektivy nereduktivnoi filosofii. [the notion of the cost in the context of the problem of the universality of science. perspectives of the non-reductive philosophy]. almaty, kazakhstan. [in russian]. [15] smith, a. (2007). issledovaniye o prirode i prichinakh bogatstva narodov. [an inquiry into the nature and causes of the wealth of nations]. moscow, russia: eksmo. [in russian] [16] sraffa, p. (1960). production of commodities by means of commodities: prelude to a critique of economic theory. cambridge, uk: cambridge university press. [17] granberg, a. g. (2016). vasily leontyev v mirovoy i otechestvennoy ekonomicheskoy nauke [wassily leontief in the world and national economic science]. in suslov, v.i. & suspitsyn, s. a. (eds.) k 80-letiyu so dnya rozhdeniya aleksandra grigoryevicha granberga: ucheny, uchitel, chelovek, (pp. 153-184). novosibirsk, russia: ieie so ras. [in russian] http://forbes.net.ua/nation/1422320-teoriya-kontraktov-v-chem-sut-otkrytij-nobelevskih-laureatov-2016 http://forbes.net.ua/nation/1422320-teoriya-kontraktov-v-chem-sut-otkrytij-nobelevskih-laureatov-2016 http://izvestia.ru/news/353298 adv syst sci appl 2018; 1; 132-141 published online at http://ijassa.ipu.ru. copyright ©2018 assa. adv. in systems science and appl. (2018) neuroprotection and timely troubleshooting of electric drive equipment viktor m. buyankin bauman moscow state technical university, moscow, russian federation e-mail: viktor12buyankin@yandex.ru abstract:. the article discusses the neurodiagnostics drives with the prediction of the future fault is a relevant task to avoid failure of electric motors and actuators. such systems neurodiagnostics increase the reliability and safety of operation, anticipating contingencies. an important aspect in the management of complex dynamic objects designed to operate in harsh operating conditions, such as aggressive, explosive and dusty environments, extreme temperatures, increased vibration, is the use of real sensors. the use of state observers in such facilities will improve the operational reliability of the electric drive, avoid shock currents, reduce weight and size characteristics, etc. in recent years, there are high requirements for modern control systems of electric drives, namely: accurate speed control, maintaining high torque at low control speeds, limiting the starting and shock currents, high dynamic characteristics, high signal processing accuracy, improved coordinate accuracy. keywords: neural network, extended kalman filter, fuzzy logic, genetic algorithm, induction motor, asynchronous electric drive. 1.introduction today there are various methods of identification of parameters and variables of the state of the asynchronous electric drive which include the extended kalman filter; the observer created on the basis of a neural network and using fuzzy logic; genetic algorithms [12-16]. most well-known observers do not provide for maintaining the necessary accuracy of identification of parameters and variables of the state in the entire range of speed control at different modes of operation of the electric drive and non-sinusoidal forms of stator currents [9-11]. the last remark is typical when driving an asynchronous motor from a thyristor voltage regulator and an autonomous voltage inverter [2-5]. the purpose of the study presented in this paper is a comparative analysis of the most common in practice, the observer of the state of the asynchronous electric drive on the simplicity of their implementation (in terms of mathematical description) and reliability. on the basis of this analysis, practical recommendations on their application in this type of electric drive in terms of their efficiency are developed [1]. today, the use of real sensors in complex dynamic systems, and in particular, drives operating under severe operating conditions, is undesirable for a number of reasons, including a significant increase in installation, increase in weight and size characteristics, decrease in operational reliability, etc. [17-20] methods of identification of parameters and state variables attracted both domestic and foreign scientists [6-8]. to determine the parameters and state variables used by the observers based on extended kalman filter, fuzzy logic, neural networks and genetic algorithms. http://ijassa.ipu.ru/ neuroprotection and timely troubleshooting of electric drive equipment 133 copyright ©2018 assa. adv. in systems science and appl. (2018) 2. literature review the ann used to identify the parameters and state variables of complex dynamic objects is composed of three main layers: input, hidden and output. more than one hidden layers are possible. to date, the most commonly used activation functions of the neuron include threshold, linear, sigmoidal, tangential, radial-basis activation functions. in practice, a linear function is used as the activation function of neurons in the output layer. the first layer of the neural network is a relay. the activation function of the neurons of the hidden layer is mainly non-linear. according to the work [15], the most suitable activation function for neurons of the hidden layer is the tangential activation function. to identify the parameters and state variables of complex dynamic objects it is necessary to use a dynamic neural network, which includes a dynamic neural network with delay at the input [16], a jordan network [17], an elman network [18], the combination of a dynamic neural network [19, 20]. the peculiarity of these neural networks is the presence of signal delays at the input, output and both input and output, which allows to provide the best learning and filter out strong pulse interference. before training a neural network, the developer must decide on the array of data needed for its training, and the choice of training algorithm. an excessively large array of data from each of the input signals can lead to a retrain effect, as a result of which, within the training sample, the neural network gives a minimum evaluation error, and when working with a test sample (different from the training sample), the evaluation error is very large. to date, there are a large number of training algorithms, the main of which are the gradient descent algorithm, gradient descent algorithm with perturbation, moller's learning algorithm and levenberg–markvardt learning algorithm. the presented first three methods of training require small computing power of the computer, but are not able to find a global minimum learning errors. the levenberg-markvardt learning algorithm requires significant computing abilities of the computer, but is able to get out of the local minimum and find a global one. 3. materials and methods rapidly expanding the range of functional requirements for automated systems the electromechanical energy conversion in industry, transport, special equipment, stringent requirements for the dynamic characteristics of such systems and their energy efficiency would require the construction of control algorithms, which for some features can be called "intelligent". it is generally recognized that the synthesis of such algorithms is currently the central problem of the modern theory of automatic control of electric drives. the conflict between obviously insufficient methodical basis of construction of "intellectual" algorithms of management of ep and growing opportunities of hardware becomes more and more acute as the modern condition of means of power and information electronics already allows to realize elements of "intelligence" (intelligent control) even in rather inexpensive serial sau. according to the author, the sign of "intelligence" of the controlled electric energy converter for electric drive systems is not the use of "exotic" methods (fuzzy logic, neural networks, genetic algorithms, etc.), as it is often presented, but functional completeness in solving the main problems of modern control theory in the appendix to such complex objects as general industrial electric drive. "intelligence" in this sense – is to provide a comfortable interface between a person and a microprocessor control system, a minimum of manually adjustable parameters of the ep, but in any case not the ability to independently set and solve new non-standard problems. particularly important tasks of identification and adaptive control become in the construction of common industrial frequency-controlled electric drives, one of the main requirements for which is the rejection of the use of external in relation to the controlled source of electrical energy (pm) sensors, including sensors directly controlled coordinates of mechanical motion. 134 v.m. buyankin copyright ©2018 assa. adv. in systems science and appl. (2018) 4. results and discussions electric drives are used in many branches of production such as machine-tool building, mechanical engineering, in the mining and oil-extracting industry. the motors work with a variety of loads in various modes: long lasting, intermittent, short-term modes. during operation, the drive equipment wears out over time, which leads to deterioration of static and dynamic characteristics, and sometimes to emergency situations. therefore, the prediction and timely detection of faults of electric equipment is an urgent task. however, predicting the malfunctions of the electric drive equipment is quite difficult and time-consuming task. a large number of diagnostic systems have been developed for fault detection, but it is not possible to determine the whole range of faults of the electric drive equipment by one hundred percent. to improve the quality of diagnostics, we propose systems with neural networks that have proven themselves quite well in pattern recognition and approximation of complex nonlinear dependencies [1]. fault diagnosis by many criteria coincides with pattern recognition and therefore, using neural networks, it is possible to achieve higher results of fault detection of electric drive equipment compared to other diagnostic systems. there is a wide variety of neural networks that differ from each other in their advantages and disadvantages. when designing a system neurodiagnostics stop your choice on neural networks like: newff and anfis [2]. that is, we will develop a combined neuroprotection system with an expert neural network approach. for initial identification of faulty components of the electric drive is advisable to use neuroprogenitor on the basis of neural networks like: newff and anfis, and for a more detailed examination of the health elements of the drive it is advisable to use the system neuroprogenitor with expert neural network approach fig. 1 fig. 1. system neuroprogenitor neuroprotection and timely troubleshooting of electric drive equipment 135 copyright ©2018 assa. adv. in systems science and appl. (2018) the purpose of building a neural network expert system is the initial prediction of faulty electric drive units, which consists of a neuroregulator, a power converter, an electric motor, a load mechanism fig. .2 fig. 2. block diagram of electric drive the operation of the actuator must comply with the rated static and dynamic characteristics. at rated load, the actuator must have a nominal armature voltage, rated current, nominal speed, which is displayed on the family of mechanical characteristics fig. 3 deviation from the nominal parameters leads to malfunctions, and sometimes to emergency situations of the electric drive equipment. we will consider as faulty the electric drive of deviation of parameters which exceed the maximum deviations. 136 v.m. buyankin copyright ©2018 assa. adv. in systems science and appl. (2018) fig. 3. mechanical characteristics of the electric drive will develop a system of neurodiagnostics in the beat of a forward current in the excitation winding iob(n+1) (and voltage in the armature of the motor (s) using a neural network newff [3]. suppose there is a database containing the current values of the iov table 1, fig. 4. table 1. data structure iов(n-9) iов(n-8) iов(n-7) iов(n-6) iов(n-5) iов(n-4) iов(n-3) iов(n-2) iов(n-1) iов(n) iов(n+1) 203ма 192ма 203ма 230ма 200ма 198ма 205ма 199ма 190ма 206ма ? fig. 4. value of current iов neuroprotection and timely troubleshooting of electric drive equipment 137 copyright ©2018 assa. adv. in systems science and appl. (2018) six lines of training samples for a neural network consists of four inputs i1, i2, i3, i4 data, and the desired output is the data of the subsequent clock i5 (table 2). the neural network training was carried out in the matlab environment. table 2. matrix for training the neural network 1i 2i 3i 4i 5i 203 192 203 230 200 192 203 230 200 198 203 230 200 198 205 230 200 198 205 199 200 198 205 199 190 198 205 199 190 200 the newff predictive neural network has four inputs and one output (fig. 5). fig. 5. newff predictive neural network system neurodiagnostics in tact ahead of the current in the excitation winding i5, iов(n+1) described by the following system of neuromante [4]. ,tan , ,tan ,tan ,tan ,tan , , , , ' 05 ' 1 ' 44 ' 33 ' 22 ' 11 ' 0 44 33 22 11 44444334224114 334433332213113 22442332222112 11141131121111 sigyi bwrwrwrwry siger siger siger siger bwiwiwiwie bwiwiwiwie bwiwiwiwie bwiwiwiwie = ++++= = = = = ++++= ++++= ++++= ++++= (1) 138 v.m. buyankin copyright ©2018 assa. adv. in systems science and appl. (2018) where i1, i2, i3, i4 the input signals of the neural network; i5 the output signal of the neural network; x0 the input signal of the neural network; y0 the output signal of the neural network; y1, y2, y3 the output signals of the neural network were detained in 1,2,3 tact; e1…e10 output signals of the first layer of neurons; w1…w215 weight of the first layer of neurons; b1…b10 displacements of the first layer of neurons; b1…b10 signals at the output of blocks of activation of the first layer of neurons; y0` signal at the output of the second layer of neurons; w1`…w10` weight of the second layer of neurons; b1` the offset of the second layer neurons, the signals output from the neural network were detained in 1,2,3 tact; displacements of the first layer of neurons; r1…r10 signals at the output of blocks of activation of the first layer of neurons; y0` signal at the output of the second layer of neurons; w1`…w10` weight of the second layer of neurons; b1` displacement of the second layer of neurons. after modeling and neural network learning algorithm fig. 6 we obtain the required weights and biases [5]. fig. 6. neural network modeling and learning algorithm weights of the first layer of neurons: w11= 0.0218, w12= 0.0203, w13= 0.0186, w14= 0.0213, w21=-0.0019, w22=0.0213, w23=0.0074, w24=0.0202, w31=-0.0097, w32=-0.0171, w33= -0.0092, w34=-0.0329, w41=0.0148, w42= -0.0029, w43=0.0252, w44=0.0145. weights of the second layer of neurons w’1=40.5136, w’2= 40.4765, w’3= -39.8220 w’4= 40.4293. displacements for the first layer of neurons b1= -5.9110, b2=-0.1183, b3=1.4517, b4=-0.9472 neuroprotection and timely troubleshooting of electric drive equipment 139 copyright ©2018 assa. adv. in systems science and appl. (2018) displacements for the second layer of neurons b’1=38.7585 for fig. 7 the dependence of learning errors depending on the number of epochs is given. fig. 7. the dependence of the training error depending on the number of epochs when testing and predicting neural network, we get the future value i5=200.0000 after predicting the future value iов(n+1) compare it with iовн. if the difference iовн и iов(n+1) greater than the maximum value allowed, the variable x1 assigns 1 otherwise 0. a value of 1 indicates an emergency condition in the motor excitation winding. 5. conclusion based on the data provided about the state observers, it can be concluded that it makes no sense to allocate a specific identifier. each of them has its advantages and disadvantages. everything will depend on the area in which the observer will be used and what performance criteria he or she should support. robustness have the majority of observers state. state observers, implemented on the basis of a mathematical model, require knowledge of the internal parameters of the object of identification. for creation of observers of a state at the same computing abilities of the computer the observers constructed on the basis of the genetic algorithm and the extended kalman filter are the most laborconsuming. the work of such observers is possible for most identifiers, except for the genetic algorithm. the scope of application of state observers is quite wide – from social sciences to technical sciences (up to the construction of new computer architectures). 140 v.m. buyankin copyright ©2018 assa. adv. in systems science and appl. 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(2017). encoding syntactic representations with a neural network for sentiment collocation extraction. sci china inf sci., 60(11), 110101. doi:10.1007/s11432-016-9229-y. [19] zhilin v v, filist sa, rakhim ka, shatalova o v. (2008). a method for creating fuzzy neural-network models using the matlab package for biomedical applications. biomed eng (ny), 42(2), 64-66. doi:10.1007/s10527-008-9019-y. [20] zhong x, wang b-z, wang h. (2001). artificial neural network model for the gap discontinuity in shielded coplanar waveguide. int j infrared millimeter waves, 22(8), 1267-1276. doi:10.1023/a:1015079619009. adv syst sci appl 2017; 4; 34-45 published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/257 copyright ©2017 assa. adv. in systems science and appl. (2017) development of the agent-based demography and migration model of eurasia and its supercomputer implementation valery. l. makarov, albert. r. bakhtizin, elena. d. sushko, gennady. b. sushko central economics and mathematics institute cemi ras, moscow, russia e-mail: albert.bakhtizin@gmail.com abstract: in this work we describe the development of a scalable agent-based modelling framework for simulation of eurasia population described in terms of demography, migration and transport flows. the simulated system will consist of agents representing individuals and sets of links to other agents, which represent the social interactions of individual. the individual agents in the model will participate in several independent processes, for which different sets of social links is important such as family and neighbors. as a base for our simulation system we have used a combination of a base native layer implemented using c++ language which uses mpi library, and microsoft .net platform as an environment for model code written in high-level c# programming language. to perform a load balancing of agents between processes the metis/parmetis algorithms were used. these algorithms allow to split the graph of agents and links into parts of similar size with the least possible number of links between them. a number of numerical experiments were carried out for test model to estimate the influence of the parameters of the model on its performance and parallel scalability. for each combination of parameters a number of simulations were performed to average the results. keywords: agent-based modelling, demography, numerical modelling, parallel computing. 2 1. introduction in this work we describe the development of a scalable agent-based modelling framework for simulation of population of eurasia described in terms of demography, migration and transport flows. the goal of the simulation is to describe the influence of large transport infrastructure projects on the development of population of the region. the simulated system consists of agents who represent individuals and a set of links to other agents, which represent the social interactions of individual. the individual agents in the model participate in several independent processes, for which different sets of social links is important such as family and neighbors. the agents of the system participate in two processes: 1) the process of reproduction of the population, and 2) the process of migration. in the first process they use messages exchange to search for the partner to form a family. in the second process the message exchange mechanism is used to obtain information about available jobs in different regions to determine the direction of migration. there are several tools for high performance computing for abm. microsoft axum is a domain-specific concurrent programming language, based on the actor model that was under active development by microsoft between 2009 and 2011. it is an object-oriented language based on the .net common language runtime using a c-like syntax which, being a domain-specific language, is intended for development of portions of a software application that is well-suited to concurrency. but it contains enough generalpurpose constructs that one need not switch to a general-purpose programming language (like c#) for the sequential parts of the concurrent components. http://ijassa.ipu.ru/ojs/ijassa/article/view/257 35 v.l. makarov, a.r. bakhtizin, e.d. sushko, g.b. sushko copyright ©assa. adv. in systems science and appl. (2017) the main idiom of programming in axum is an agent (or an actor), which is an isolated entity that is being executed in parallel with other agents. in axum parlance, this is referred to as the agents executing in separate isolation domains; objects instantiated within a domain cannot be directly accessed from another. agents are loosely coupled (i.e., the number of dependencies between agents is minimal) and do not share resources like memory (unlike the shared memory model of c# and similar languages); instead a message passing model is used. to coordinate agents or having an agent request the resources of another, an explicit message must be sent to the agent. axum provides channels to facilitate this. the axum project reached the state of a prototype with working microsoft visual studio integration. microsoft had made a ctp of axum available to the public, but it was removed later. although microsoft decided not to turn axum into a project, some of the ideas behind axum are used in tpl dataflow in .net 4.5 (more information at [5]). repast for high performance computing (repast hpc) 2.2.0, released on 30 september 2016, is a next generation agent-based modeling and simulation (abms) toolkit for high performance distributed computing platforms. repast hpc is based on the principles and concepts development in the repast simphony toolkit. repast hpc is written in c++ using mpi for parallel operations. it also makes extensive use of the boost [2] library. repast hpc is written in cross-platform c++. it can be used on workstations, clusters, and supercomputers running apple mac os x, linux, or unix. portable models can be written in either standard or logo-style c++. repast hpc is intended for users with: • basic c++ expertise. • access to high performance computers. • a simulation amenable to a parallel computation. simulations that consist of many local interactions are typically good candidates. models can be written in c++ or with a “logo-style” c++ [1]. cybergis toolkit is a suite of loosely coupled open-source geospatial software components that provide computationally scalable spatial analysis and modeling capabilities enabled by advanced cyberinfrastructure. cybergis toolkit represents a deep approach to cybergis software integration research and development and is one of the three key pillars of the cybergis software environment, along with cybergis gateway and gisolve middleware [3]. the integration approach to building cybergis toolkit is focused on developing and leveraging innovative computational strategies needed to solve computingand dataintensive geospatial problems by exploiting high-end cyberinfrastructure resources such as supercomputing resources provided by the extreme science and engineering discovery environment and high-throughput computing resources on the open science grid. a rigorous process of software engineering and computational intensity analysis is applied to integrate an identified software component into the toolkit, including software building, testing, packaging, scalability and performance analysis, and deployment. this process includes three major steps: 1. local build and test by software researchers and developers using continuous integration software or specified services; 2. continuous integration testing, portability testing, small-scale scalability testing on the national middleware initiative build and test facility; and 3. xsede-based evaluation and testing of software performance, scalability, and portability. by leveraging the high-performance computing expertise in the integration team of the nsf cybergis project, large-scale problem-solving tests are conducted on various supercomputing environments on xsede to identify potential computational bottlenecks and achieve maximum problem-solving capabilities of each software installation. the implementation of the scalable modelling framework 36 copyright ©2017 assa. adv. in systems science and appl. (2017) pandora is a novel open-source framework created by the social simulation research group of the barcelona supercomputing centre. this tool is designed to implement agentbased models and to execute them in high-performance computing environments. it has been explicitly programmed to allow the execution of large-scale agent-based simulations, and it is capable of dealing with thousands of agents developing complex actions. pandora has full geographical information system support, to cope with simulations in which spatial coordinates are relevant, both in terms of agent interactions and environment. the results of each simulation are stored in hierarchical data format (hdf), a popular format that can be loaded by most gis. this feature is particularly useful, as we will also use gis to analyze simulation results. pandora is complemented by cassandra, a program developed to analyze the results generated by a simulation created with the library. cassandra allows the user to visualize the complete execution of simulations using a combination of 2d and 3d graphics, as well as statistical figures (more information at [6]). swages [10], a distributed agent-based life simulation and experimentation environment that uses automatic dynamic parallelization and distribution of simulations in heterogeneous computing environments to minimize simulation times. swages allows for multi-language agent definitions, uses a general plug-in architecture for external physical and graphical engines to augment the integrated simworld simulation environment, and includes extensive data collection and analysis mechanisms, including filters and scripts for external statistics and visualization tools. moreover, it provides a very flexible experiment scheduler with a simple, web-based interface and automatic fault detection and error recovery mechanisms for running large-scale simulation experiments [10]. a hierarchical parallel simulation framework for spatially-explicit agent-based models (hpabm [11]) is developed to enable computationally intensive agent-based models for the investigation of large-scale geospatial problems. hpabm allows for the utilization of highperformance and parallel computing resources to address computational challenges in agentbased models. within hpabm, an agent-based model is decomposed into a set of sub-models that function as computational units for parallel computing. each sub-model is comprised of a subset of agents and their spatially-explicit environments. sub-models are aggregated into a group of super-models that represent computing tasks. hpabm based on the design of superand sub-models leads to the loose coupling of agent-based models and underlying parallel computing architectures. the utility of hpabm in enabling the development of parallel agent-based models was examined in a case study. results of computational experiments indicate that hpabm is scalable for developing large-scale agent-based models and, thus, demonstrates efficient support for enhancing the capability of agent-based modeling for large-scale geospatial simulation [11]. the growing interest in abm among the leading players in the it industry (microsoft, wolfram, esri, etc.) definitely shows the relevance of this instrument and its big future, while exponential growth of overall data volumes related to human functioning and the need for analytical systems to obtain new-generation data needed to forecast social processes, call for the use of supercomputer technologies. in march 2011, an abm was launched at the lomonosov supercomputer to simulate the development of russia’s socio-economic system for the next 50 years [8]. the implemented abm was based on the interaction of 100 mln agents who conditionally represented russia’s socio-economic milieu. the behavior of each agent was specified by a set of algorithms that described the agent’s actions and interaction with other agents in the real world. the adevs library for multiagent simulation, which the authors had already tested during the multisequencing of russia’s demographic model in 2011, showed itself quite well. in addition, the latest adevs versions support java to a certain extent, which is also a plus. however, the adevs developers have not yet implemented multisequencing on 37 v.l. makarov, a.r. bakhtizin, e.d. sushko, g.b. sushko copyright ©assa. adv. in systems science and appl. (2017) supercomputers except for the openmp technology for multiprocessors; therefore, our previous work required many updates for mpi support. when multisequencing the previous, quite simple, model, it was fully rewritten in c++, which was superfluous: the preand postprocessing of data, as well as the creation of the initial state of the multiagent environment, are not time-critical operations. usually, it is quite enough for supercomputers to multisequence only the algorithm’s computing core, i.e., the phase of converting the population’s state in this case. the stages and methods of the efficient reflection of the computing core of a multiagent system on the architecture of the state-of-the-art supercomputer using the supercomputer technology for agent-oriented simulation (stars) developed by the authors are analyzed in later article [9]. analysis of the latest software technologies has shown that embeddable tools are being actively developed lately to execute java programs that use the so-called ahead-of-time (aot) compilation. in this case, the result of the aot compiler is a typical self-contained executable module that contains a machine code for the target platform. it is interesting to note that this approach is used in the new versions of the android operating system, which, in our opinion, is not accidental: the efficiency of code execution is the main factor both for embeddable systems and for supercomputers. experiments with a similar product – the avian aot compiler – allowed us to conclude that, first, it helps obtain a selfcontained executable module as an mpi application for supercomputers; in addition, a random additional code, including initialization and binding to the mpi communication library, is easily implemented in c++; second, the operating speed of the obtained program module is close to the speed of the adevs operation. this made it possible to shift a large part of the work to the aot compiler and to implement only the most necessary in c++, fixing the function of supporting accelerated stages with complex interagent communication to adevs. 3 2. the model description in this work we describe the development of an agent-based modelling framework for simulation of large scale societies which consists of large number of agents. the described technology is to be applied for the implementation of the large-scale agent based model of countries of eurasia describing economy, migration and the results of implementation of large infrastructural projects. the main types of agents in these simulations are individuals, enterprises, regions and governments. the main processes described by the model are demographical evolution of the society, education, jobs and career of individuals and the migration of the workforce according to changing economic conditions. individual agents in the system are described in terms of their age, sex, education level, income and the set of social links to other agents. this set of parameters defines the formation of families, awareness about working conditions in different areas and the possibility of the labor migration. the regions in the model are characterized by the transport connectivity graph, the level of economic development and the labor market conditions. due to that structure of the model agents form the following graph: each agent is linked with a dozen of other individuals in the same or other regions. an individual agent is also linked to his place of work and a region. agents describing enterprises are linked with each other by trade contracts. the described structure of the model leads to formation of a largescale graph of agents of different types which can be partitioned into connected blocks linked to regions. the model uses a change in the transport connectivity graph as an input which leads to the change in economic activity in neighbor regions and the migration of the population. starting from the implementation of the infrastructural project the living conditions are the implementation of the scalable modelling framework 38 copyright ©2017 assa. adv. in systems science and appl. (2017) changing: new jobs are created in neighbor areas which leads to a change in incomes and migration of workforce. 2.1 the technology on the computational level the model should be scalable for system up to 109 agents. in order to perform efficient simulations of such systems the model should support running on modern supercomputers. to simplify the development of the model we use the high-level microsoft .net platform which has become available for running on supercomputers. the most usual architecture of modern supercomputers is a cluster of multicore computing nodes connected by high-performance low-latency network. to run efficiently on such cluster the program should be split into multiple separate processes exchanging messages through the network. the most common way of writing such programs is to use c++/fortran language and mpi library which are available on all supercomputers. as a base for our simulation system we have used a combination of a base native layer implemented using c++ language which uses mpi library, and microsoft .net platform as an environment for model code written in high-level c# programming language. as most of modern supercomputers run on linux operating system, we have decided to use microsoft .net core and mono [4] implementations of .net platform available for this os. the choice of these technologies was determined by the following criteria: 1) the system has to be scalable across multiple computational cluster nodes (i.e. use resource of multiple nodes for speedup) therefore the multithreading calculation model was not suitable as it is limited to single cluster node. 2) the model should be easy to develop and maintain and therefore the high-level programming language c# was used. 3) the system should be efficient and therefore the native mpi library was used instead of tcp/ip sockets or .net libraries like windows communication foundation as these technologies are not optimal for supercomputers and hpc applications. mpi libraries installed on each cluster computer are usually tuned for particular proprietary network system which is used on the cluster such as infiniband. on the level of individual agents the simulation of agent’s internal state evolution, the formation of constant and temporary links between agents, message exchange and formation and destruction of agents in the system are supported. to carry out these simulations efficiently the system has to implement a dynamic load balancing mechanism for agents taking into account their links to neighbors. the proper description and taking into account of links of agents in the process of decomposition is crucial for reduction of network exchange traffic which is necessary for scalability of the model up to hundreds and thousands of cpu cores. to perform a load balancing of agents between processes the metis/parmetis [7] algorithms were used, which are commonly used for decomposition of big graphs (up to 109), computational grids and matrices. these algorithms allow to split the graph of agents and links into parts of similar size with least possible number of links between them. the algorithm can be applied recursively in order to calculate hierarchical splitting of the system in efficient way. the use of the algorithm allows both initial decomposition of the system and refinement of the decomposition in the process of calculation which is necessary to maintain load balancing as new agents are added to the system and some of old agents are being removed. the dynamic decomposition and redistribution of agents should allow us to use efficiently up to 1000 cpu cores. 2.2 implementation of the model the implementation of the agent-based simulation platform requires the proper definition of classes for agents, messages, model, time-steps and utility classes for file input and calculation of characteristics. 39 v.l. makarov, a.r. bakhtizin, e.d. sushko, g.b. sushko copyright ©assa. adv. in systems science and appl. (2017) an efficient mechanism of message exchange between agents was implemented by means of message queue and native mpi collective operations. the message queue accumulates a buffer of messages to different processes and then uses mpi alltoall exchange operation to deliver contents of messages. this operation delivers the buffers of arbitrary size from each mpi process to all other processes in most efficient way by splitting the buffer into chunks of optimal size for network transfer and hiding the latency of network operations by performing simultaneous several send and receive operations. to use native operations with managed c# objects operations of binary serialization and deserialization of objects were implemented and c# wrappers for native functions were written. fig. 1. the agents of the model are implemented using c# programming language and run in .net virtual machine. the interaction between vm instances is implemented through the native library and mpi library. the implementation of native wrappers for c++ library was carried out using interopservices library and “dllimport” annotations in managed c# code. in fig. 1 the scheme of interaction of model agents through native library and mpi operation is shown. 2.3 the test model description in order to test the message delivery mechanism the test model was implemented and a set of simulations were carried out to estimate the scalability of the designed model. the model is characterized by the following parameters: the total number of agents n. the ratio of agents participating in message exchange s. message exchange intensity i. total simulation time t. the number of mpi processes u. as it is usual for all mpi programs the program starts as a set of separate processes running the same program. at the initial stage of the model the initial set of agents is created with the number of agents n. each agent has the only numerical parameter the number of messages it should send at each simulation time (a). the set of agents is distributed over all mpi processes, i.e. each mpi process creates only n/m agents according to its process identifier. the process with index 0 creates agents 0...n/m, the process with index 1 creates agents n/m+1...2n/m and so on. on each simulation step for each agent of the system the random number is generated which determines if the agent will participate in message sending process (the probability of the event is s/n). if the agent is sending messages on this simulation step, that the random number of messages a is generated with uniform distribution of probability between 1 and i. after that the implementation of the scalable modelling framework 40 copyright ©2017 assa. adv. in systems science and appl. (2017) the agent sends a messages to agents with random numbers and then receives replies. each simulation step consists of 5 stages: 1. the loop over all agents on the current process, execution of the performstep method of each agent which results in generation of random messages according to corresponding sending probabilities. all messages are put into outgoing message queue. 2. after the generation of all initial messages the method sendreceivemessages is called which initiates the exchange of the parts of message queue between all mpi processes using collective alltoall operation. each process is sending data buffers to all other processes and receives corresponding buffers from all other processes. at this stage the messages in the queue are serialized into binary arrays, these arrays are transmitted into the native library which uses mpi library to perform the exchange, after that new buffers are transmitted to c# part of the program and messages are deserialized. 3. after the exchange of buffers and deserialization of all messages the delivery of messages to corresponding agents on each mpi process is performed which results to the generation of new set of reply messages. 4. the exchange of the parts of message queue between all mpi processes using collective alltoall operation. each process is sending data buffers to all other processes and receives corresponding buffers from all other processes. 5. the delivery of messages to corresponding agents on each mpi process. the use of the delayed delivery of messages through the message queue allows us to optimize the message exchange which is now bound not to the latency of the network but to its bandwidth. after processing all agents in the population, the output characteristics are calculated and put into output file. the following output parameters of the model were written: 1. step number; 2. the average number of message recipients for agents participating in message exchange; 3. the total calculation wall time. 2.4 messages exchange procedure the message queue implemented in the program is a managed c# object which receives objects of abstract class message from agents, each mpi process contains one instance of the message queue object. the simulation model consists of agents of different types and messages can also have different types derived from the message class. for each message type the operations of reading and writing to binary stream are implemented which use binaryreader and binarywriter classes of c# standard library. these operations encode and decode the type of message object, the numbers of the sender and receiver objects and all additional message data fields into binary format. each message queue contains a set of binarywriter objects which are used as buffers for outgoing messages. for each outgoing message put into queue the number of destination process is determined, that the message is written to the corresponding binary stream. the message queue implements a sendreceivemessages method which performs the delivery of all messages to the corresponding agents. in order to deliver messages message queue objects of all mpi processes exchange the corresponding message buffers, for that all binarywriter objects are written to memorystream object which generates an outgoing array of bytes. in order to deliver the outgoing array of bytes first the mpi_alltoall function is used to exchange the sizes of all message buffers between all processes. after that the mpi_alltoallv function is used to deliver parts of the outgoing byte array to the corresponding processes. 41 v.l. makarov, a.r. bakhtizin, e.d. sushko, g.b. sushko copyright ©assa. adv. in systems science and appl. (2017) after the exchange of the message buffers each message queue has separate buffers with messages from all other processes, i.e. process 0 has k1 bytes from process 1, k2 bytes from process 2… for all of these buffers the binaryreader objects are created and the deserialization process is started. the program reads messages from all incoming buffers and generates an array of message objects. for each of these messages the receiver agent index is determined, and for the corresponding agent the notify method is called with the corresponding message object passed as argument. the process of the delivery of the message is illustrated in fig. 2. fig. 2. the message sending procedure. the message from agent 0 is transmitted first to message queue of process 0, then the message is serialized to outgoing buffer for sending to process 1, then the buffers are exchanged between processes and the outgoing buffer number 1 becomes the incoming buffer number 0 on mpi process number 1, then the message is deserialized and delivered to agent 1 by the message queue. 3. numerical experiments a number of numerical experiments was carried out to estimate the influence of the parameters of the model on its performance and parallel scalability. for each combination of parameters a number of simulations was performed to average the results. 3.1 test cluster configuration to test the performance and scalability of the model we have used the cluster consisting of 4 dual-processor nodes using amd opteron 6172 (12 cores) processors. the total number of cores in each node was 24 and the same number of mpi processes on each node was used. the high-performance network (qdr infiniband) was used to connect nodes of the cluster. 3.2 the results of the numerical experiments to study the parallel efficiency of the model the following test configuration was used: n = 10000000 (the number of agents) s = 200000 (the number of agents sending messages on the simulation step) i = 10 (the maximal number of messages for one agent on each step) t = 3000 (the number of simulation steps) the simulations were carried out for the number of mpi processes m = 1,24,48,96 and results of these simulations were compared in terms of parallel speedup (the ratio of total computation time for parallel and serial cases) and parallel efficiency (the speedup divided by number of cpu cores used). the implementation of the scalable modelling framework 42 copyright ©2017 assa. adv. in systems science and appl. (2017) fig. 3. the dependence of the parallel speedup of the test model on the number of cpu cores. in fig. 3 the dependence of the parallel speedup on the number of cpu cores is shown. the increase of the number of cpu cores leads to nearly linear speedup of the calculations. the use of 96 cores results to speedup of calculations by factor of 60 which means 65% efficiency of the cluster use. the dependence of the parallel efficiency on the number of cpu cores is shown in fig. 4. fig. 4. the dependence of the parallel efficiency of the simulation on the number of mpi processes. to estimate the influence of the parameters of the model on scalability the simulations were performed with parameter values n = 10000000, s = 200000, t = 3000 using m = 96 mpi processes with different values of message exchange intensity i = 30, 50, 100, 200, 1000. the increase of intensity of message exchange leads to linear increase of the network traffic on each simulation step, and also increase of computations (random number generation) on each simulation step. in fig. 5 the dependence of the speedup on message 43 v.l. makarov, a.r. bakhtizin, e.d. sushko, g.b. sushko copyright ©assa. adv. in systems science and appl. (2017) exchange intensity shows that these factors are balanced and increase of intensity doesn’t lead to degradation of parallel efficiency. fig. 5. the study of the influence of the message exchange intensity (i) on the scalability of the simulation. to study the influence of parameter s the parameter i was fixed (i = 10) and the calculations were carried out for the values s = 300, 500, 1000000, 10000000. the increase of the parameter s also leads to the linear increase of both network traffic and calculations of the random numbers, which shouldn’t affect much the scalability. in the case of lower values of the parameter the size of the data is rather small and the speedup is determined more by the latency of the exchange network. this effect leads to the decrease of efficient for low values of s and much better efficiency for larger values of the parameter. 3.3 the influence of the interprocess communications in the second variant of the test model the agents are divided into groups of size g, the exchange of the messages is done only between agents inside the group. for that each agent has additional parameter the number of group ng, which is defined at the beginning of the simulation to establish the uniform distribution of agents between groups. the use of such groups corresponds to the case of ideal decomposition of the graph of agents where most of the interprocess communications are removed. the implementation of the scalable modelling framework 44 copyright ©2017 assa. adv. in systems science and appl. (2017) fig. 6. the dependence of the parallel speedup on the number of agents participating in message exchange (s). the number of mpi processes m = 96. in order to study this effect a set of simulations was performed and the dependence of the parallel speedup on the number of participating agents s was plotted. the results of these simulations are plotted in fig. 6. the plot shows that the influence of the presence of blocks is much higher for high values of s when the number of agents participating in message exchange is high. 4. conclusion in this work a new framework for parallel calculations of agent-based models was presented and tested. the framework links the use of the high-level c# programming language and high-performance platform for messages exchange written using native c++ library and native mpi library of supercomputer. the provided results of the test simulations show good scalability of the program across multiple computational nodes. acknowledgements this work was supported by the russian science foundation (grant # 14-18-01968). references [1] collier n. (2013, august). repast hpc manual. [online] available: http://repast.sourceforge.net [2] boost c++ libraries (2017) [online]. available: http://boost.org [3] cybergis (2017) [online]. available: http://cybergis.cigi.uiuc.edu [4] mono (2017) [online]. available: http://www.mono-project.com [5] axum_(programming_language) (2017), [online]. available: https://en.wikipedia.org/wiki/axum_(programming_language) [6] pandora: an hpc agent-based modelling framework (2017), [online]. available: https://www.bsc.es/research-and-development/software-and-apps/softwarelist/pandora-hpc-agent-based-modelling-framework [7] karypis, g. & kumar, v. (1995). metis-unstructured graph partitioning and sparse matrix ordering system, version 2.0. technical report. http://boost.org/ http://cybergis.cigi.uiuc.edu/ http://www.mono-project.com/ https://en.wikipedia.org/wiki/axum_(programming_language) 45 v.l. makarov, a.r. bakhtizin, e.d. sushko, g.b. sushko copyright ©assa. adv. in systems science and appl. (2017) [8] makarov v.l., bakhtizin a.r., vasenin v.a., roganov, v.a. & trifonov i.a. (2011). capacities of a supercomputer system for work with agent-based models, programnaya injeneriya [software engineering], 3, 2-14 [in russian] [9] makarov, v.l., bakhtizin, a.r., sushko, e.d. et al. (2016) supercomputer technologies in social sciences: agent-oriented demographic models, her. russ. acad. sci, 86 (3), 248-257. doi:10.1134/s1019331616030047 [10] scheutz, m., connaughton, r., dingler, a., & schermerhorn, p. (2006). swages an extendable distributed experimentation system for large-scale agent-based alife simulations. in proc. of artificial life x, 412-419. [11] tang w. & wang s. (2009). hpabm: a hierarchical parallel simulation framework for spatially‐explicit agent‐based models, transactions in gis 13(3)315-333. advances in systems science and applications (2017) vol.17 no.1 25 a novel defense solution towards currency wars qionghao chen a) , yirong ying a) , jeffrey yi-lin forrest b) a) college of economics, shanghai university, shanghai, 200444, china; e-mail: qionghao_chen@sina.com, yrying@staff.shu.edu.cn; b) school of business, slippery rock university, slippery rock, pa16057, u.s.a.; e-mail: jeffrey.forrest@sru.edu. abstract. this paper investigates the following problem: how could a nation possibly design a measure to counter large-scale sudden flight of foreign investments in order to avoid the undesirable disastrous consequences? continuing [2], and based on how a currency war could be potentially raged against a nation, all results herein are established by making use of the results of feedback systems. based on theoretical reasoning and systemic analysis, this paper develops a self-defense mechanism that could conceivably protect the nation under siege. when a nation tries to accelerate its economic development, large amounts of foreign investments would generally be welcomed. and at the same time, a lot of such foreign investments would strategically rush into the nation in order to ride along with the forthcoming economic boom. however, recent financial events from around the world indicate that how to avoid the disastrous aftermath when a large-scale flight of foreign capital appears suddenly is still not well understood and well planned out. this fact vividly shows theoretical and practical value of this work. key words: purchasing power, feedback system, monetary policy 1. introduction 1.1. currency war since world war ii, the form of war has changed. the main battlefield of modern warfare has quietly shifted from direct military operations to the economic maneuvers. fundamentally, all modern forms of warfare are currency related. although there is no physical battleground, both the scales and benefits the conflicts eventually generate out of the competition for the financial highlands are no less any those of any war in history. each financial crisis can be seen as the signal of a currency war. our present world on the average experiences about 10 massive financial crises per years. the results are that the relevant countries lose their leadership, if there was any, and have to stay in the consequent economic shadows for years to come, or might be worse, they can no longer recover and return to their previous glory. for example, the british sterling crisis, japan's decade of recession after the plaza accord, southeastern asia's financial crisis, and so on. these related countries did not experience any military conflict, and did not have the time to use any advanced weaponry that had been prepared for use. compared to a military warfare, currency wars had made these countries pay a much greater economic cost. the reality is that most nations today have no need, and are unable to resolve conflicts by employing conventional wars, because by using the idea of currency wars one can achieve his desired goals. klein [11] showed in his empirical analysis that capital account liberalization has different effects on economic growth in different countries. henry [8], based on event study, discussed the effects of stock market liberalization on the emerging market countries’ economic reform and equity market prices. li and zhang [12], based on a dynamic model of aghion, analyzed the impact of capital account liberalization on economic and financial stability. they also used cross-sectional data model involving 57 countries, further studied the economic consequences of opening direct investment for different country samples. zhang [31] analyzed the conduction of a financial crisis as the breakthrough point based on the demonstration effect of the conduction, and tried to investigate financial crises from one side. li [13] established the currency substitution vec model and made a dynamic analysis of extent of china's currency substitution and the relationship between its influence factors. after reviewing previous theoretical analyses with recent cases of speculative attacks in the arena of international finance, we surely see the following predicament. when a nation tries to mailto:qionghao_chen@sina.com mailto:yrying@staff.shu.edu.cn mailto:jeffrey.forrest@sru.edu 26 q. chen, y. ying, j.y-i. forrest: a novel defense solution towards currency wars accelerate the economic development, due to its capital account, liberalization and loose monetary policies, large amounts of foreign investments would be welcomed; and at the same time, a lot of such foreign investments would strategically rush into the nation in order to ride along with the forthcoming economic boom. therefore, a serious question at this junction is: how could a nation possibly design a measure to counter such foreign investments to leave suddenly in order to avoid the undesirable disastrous consequences? in this work, based on the studies and discussions of how china’s capital account liberalization affects international capital flows and the economy and finance, we study the evolution mechanism of a currency war by using systems theory. 1.2. literature review 1.2.1. importance of capital account liberalization capital account liberalization is an important proposition of international finance, and has been a focus of attention of economists. as a result of the liberalization in industrialized countries, the issue of capital account liberalization is almost entirely concentrated on developing countries. in the economic globalization today, with the influence of liberalization of external and induced current accounts, implementation of capital account liberalization in developing countries in fact has become an inevitable choice. but since the 1990s, the world has been prone to financial crises, such as the mexican financial crisis in 1994, the asian financial crisis in 1997, and, a few years after that, the outbreak of the brazil's financial crisis, venezuela’s financial crisis, and argentina's financial crisis. the varying degrees of the crises reflect negative impacts on economies and finances as caused by international capital flows consequent to the liberalization of capital accounts [1]. 1.2.2. inevitability of currency wars with the development of civilization, economies, and societies, after world war ii, several of the world's major military powers have found that the use of force, including nuclear weapons, has become increasingly handicapped in resolving disputes. so, the economy has become a battleground, currency wars have already been referred to as a frequent agenda through news outlets. removing potential threats in the global monetary system has been the goal of the globalized economy of the modern world. previous reports on defense strategy towards currency wars can be categorized into active defense and passive defense. gagnon [5] indicated that 22 countries had boosted their economy and created employment opportunities by intervening in foreign-exchange markets. gagnon and bergsten [6] estimated that 91 economies increased their external deficits as some other countries manipulated their currencies so that they had to devalue their currencies as a hedge. the capitalist world is bound to the outbreak of economic crises, which represent the best time to repudiate debts. for example, in 1847, the then wealthy british even got out of its debts in china and the united states by using bankruptcy, creating a precedent example of how debts could be repudiated [18]. this technique can be surely employed today more readily than any time in history, creating more pains and uncertainties in the world. mathematically, an economic system can be written approximately in the following form of an n-dimensional constant coefficient linear system [29]: dx ax bu dt y cx       , where its state space is assumed to be x, and all relevant terms in these equations were defined in [2,3]. assume that xc is the subspace of x that consists of all controllable states and xno the subspace that consists of all non-observable states. then the state space x can be decomposed into the following four subspaces: 1 no cx x x  , advances in systems science and applications (2017) vol.17 no.1 27 2x such that 1 2cx x x  , 3x such that 1 3nox x x  , and 4x such that 1 2 3 4x x x x x    . if we assume the dimensionality of ix is in , then we have 1 2 3 4n n n n n    . by choosing a basis 11{ ' ' }ne e， , in 1x , 1 1 21{ ' ' }n n ne e ， , in 2x , 1 2 1 2 31{ ' ' }n n n n ne e   ， , in 3x , and 1 2 3 1{ ' ' }n n n ne e   ， , in 4x , we can introduce the following coordinate system: 1 1 1 2 1 2 1 2 3 1 2 31 1 1 1{ ' ' , ' ' , ' ' , ' ' }n n n n n n n n n n n n nco e e e e e e e e         ， , ， , ， , ， , . for any x x , let t be the transformation from the original coordinate system  to the new system co so that 'x tx , where the i-th column is the coordinates of 'ie in the original system  . so, t is obviously non-singular. now, in co we denote 1 2 3 4' [ , , , ]t t t t tx x x x x , where 11 1[ ' ' ]nx x x ， , , 1 1 22 1[ ' ' ]n n nx x x  ， , , 1 2 1 2 33 1[ ' ' ]n n n n nx x x    ， , , and 1 2 34 1[ ' ' ]n n n nx e e   ， , . with this coordinate transformation, the kalman canonical decomposition theorem of the control theory [10] implies the following: the previous constant coefficient linear system approximation of an economic system can be written in the coordinate system co into the following canonical form   11 12 13 14 1 22 24 2 33 34 44 2 4 0 0' ' 0 0 0 0 0 0 0 0 0 ' a a a a b a a bdx x u a adt a y c c x                              and the original system can be decomposed into the following four subsystems: 1 :s 1 11 1 1 1 1 10 0 dx a x b u f dt y x         , where 1 12 2 13 3 14 4f a x a x a x   2 :s 2 22 2 2 2 2 2 2 dx a x b u f dt y c x        , where 2 24 4f a x 3 :s 3 33 3 3 3 30 0 dx a x f dt y x        , where 3 34 4f a x 4 :s 4 44 4 4 4 4 dx a x dt y c x      , where the subsystem s1 does not have any observable output, the subsystem s2 can be completely observed and controlled, the subsystem s3 does not have any output and cannot be affected by any control variable, and the subsystem s4 is not influenced by any control variable. what this theoretical result implies is that in the most general circumstance, each economic system is vulnerable to external influences through first the subspace s1, and then the subspaces s3 and s4. and, domestic monetary and fiscal policies can only effectively affect the subsystem 28 q. chen, y. ying, j.y-i. forrest: a novel defense solution towards currency wars s2 and the consequent results on s2 can also be readily observed. the rest of the presentation of this paper turns to the topic of how to defend oneself against currency attacks without much symbolic connection with what is presented in this subsection. 1.2.3. currency superiority of the united states in order to maintain the central position of power in the world economy, the u.s. is ready to fight against the threat of any currency system. as of this writing the u.s. has deployed three strategic fronts centered on the dollar. (1) expansion of global influence in history, the first to the dollar's dominance as the center of the strategic front formed shortly after world war ii. on june 5, 1947, george marshall, the 50th secretary of state of the u.s.a., who had a long-term strategic mind, delivered a historic speech, declaring that the u.s. was ready to help with the european recovery that was to be funded by the u.s. with the most potentially far-reaching marshall plan. this program aimed to make europe american. all allies receiving the u.s. economic aids had to be centered on the dollar; and the recovery from the aftermath of world war ii made europe a mixture consisting largely of american interests, values and informal contact networks. and european countries also undertook political and economic commitments and obligations. that warranted that the united states would have a good hand in the following contest with the economy of the soviet union, and greatly curbed the forces of soviet communist in europe to advance further from the east to the west. however, the dollar, whose value tumbled in europe, was used to purchase additional u.s. goods to enter europe. the consequence was a substantial devaluation of the dollar, leading to a confidence crisis in the dollar. european countries then began to hedge the value of the dollars in their hands with gold. in october 1960, the first dollar crisis erupted. on august 15, 1971, nixon announced a "new economic policy". the u.s. provisionally stopped the convertibility of the dollar into gold, and refused to sell gold to foreign central banks, which not only made the gold-linked dollar in name only but also caused a complete collapse of the bretton woods system. however, as the sole oil pricing currency, the dollar gained a qualitative leap and drew for the first time the boundaries, beyond which the u.s. would employ public actions and deploy military forces. this strategic front of protecting the unique relationship between oil and the dollar, as the time goes on, has become an integral part of american global power. the establishment of cultural, economic and political networks around oil countries has become a global force for american expansion of global influence [26]. (2) expansion of its economic system and political will the second strategic front of consolidating the center position for the dollar was soon employed after the formation of the first one. because liberal economic reforms, as proposed by chicago school of economics, took place successfully in latin america, the u.s. was successful in promoting its capitalist form of economy in terms of geopolitics and in radiating the particular economic form to other socialist countries. the u.s. subversively tried to make chile, bolivia, mexico, and some other countries to abandon the influence of the soviet economic model. with the collapse of soviet union and economies in eastern europe, the united states achieved its important strategic goal to changing the political map of europe that was formed since world war ii, and become a dominant force in the global monetary system and acquired its political advantage over others [25]. it should be noted that since the start of the new century, with the initial regionalization of the rmb, sino and u.s. currencies have been in competition for global influence; and because of the marginalization of the dollar by china, the u.s. has determined to exclude china from the economic power center [26]. in 2011, the most important strategic goal of the united states for the next 10 years consisted of: the united states must expand to the west, dramatically increase diplomatic, economic, strategic and other investments in asia pacific to play a leading role in the entire 21 st century, because the pacific region will serve as the center of nation’s prosperity and global leadership [26]. (3) globally comprehensive containment capability advances in systems science and applications (2017) vol.17 no.1 29 this third strategic front was formed much later in time. the previous two strategic fronts made the united states maintain a multifaceted, cross-cutting, integrated containment capability: even if there were such a country that could surpass the united states in terms of military might, that country would still be hampered by its economic capacity, technological innovation, and social development so that it would no longer be a potentially threaten. more importantly, the advantage of technological innovations of the u.s. led to strategic advantages in many areas, such as the deployment of conventional forces around the world, competition in energy production, climate warming, space network, computer technology, and so on. military force becomes less critical in terms of defending national interests, while its importance only lies in helping the united states to establish a decisive advantage in the world. weak dollar monetary policy was used to hinder economic influence of developing countries. however, unexpectedly, such monetary policies that might shift the u.s. debt crisis also made the economic strengths of these developing countries both constrained and strengthened at the same time. in other words, america’s massive qe monetary policy was introduced at the expense of its national political influence, while its long-term strategic goals are immeasurable. the inconsistency between the monetary dominance of the dollar and the health of the world economy will continue to evolve. in order to reduce the risk of foreign exchanges, various countries will put aside the dollar, and settle their international trades with its own currency. a new currency pattern will be formed [25]. 1.2.4. increasing influence of chinese currency to reduce the pain created by large-scale quantitative easing in the u.s.a., and to maintain its currency sovereignty, china has adopted a strategy to marginalize the dollar while expanding the influence of its rmb [26]. (1) the center front of marginalizing the dollar strategy in the center front, to support and strengthen the rmb’s global influence, china in recent years relied mainly on signing rmb’s bilateral currency swap agreements, expanding the regionalization of trade settlement in rmb, and selectively deepening multilateral cooperation (e.g. free trade agreements). in order to ensure the safety of all its assets that are priced in the dollar, china exercises strategic relations with other developing countries in order to promote its rmb’s global ambition while restrict the effect of the dollar. due to the increasing global influence of rmb, china indeed poses a potential and visible threat to the u.s. through extensive global-scale deployments of its rmb while providing a new currency alternative to the world. china’s activities have resulted in such a directional trend in the international arena that many governments, including some in europe, tend to "marginalize the dollar" rather than oppose to the expanding global influence of rmb. in addition, because china's overall economic level has been below that of the united states, the trend has positively contributed to the implementation and expansion of china’s strategy of marginalizing the dollar [26]. (2) the international front of marginalizing the dollar strategy in the international front, china relies more heavily on political and diplomatic methods. some countries have recognized the fact that heavy dependency on the united states is not appropriate in terms of international trades and currencies [25,26]. this is the political foundation for china to implement the rmb trade settlements and bilateral currency swap agreements in the world. and, it represents a time to establish a broad, united front of currency union in politics, which would include mainly developing currencies and emerging countries in the currency markets, secondly all of the capitalist countries that are seriously, and adversely affected by the dollar hegemony. the union would establish a multilateral currency swap fund for its member countries in order for them to resolve all crises as caused by the instability of the current international currencies and the world financial system due to the instability of the dollar. (3) the outside front of marginalizing the dollar strategy in the outside front, the u.s. is clearly aware of the growing, worldwide movement of marginalizing the dollar. this will certainly encourage the u.s. to exploit any existing weaknesses or economic friction in order to slow down china's economic growth and its 30 q. chen, y. ying, j.y-i. forrest: a novel defense solution towards currency wars expanding global currency influence. as a currency policy tool, the exchange rate of the dollar has shown its dual effect. on one hand, depreciation of the dollar leads to a sharp rise in the trading price of the dollar-denominated commodities, such as oil and gold, forces countries from around the world to increase their demand for the dollar, triggers global inflation, and exploit people including the american people themselves. on the other hand, in order to avoid the dollar exchange rate to rise from the increased demand for the dollar, the u.s. government increased the supply of the dollar through massive qes, which not only reduce the fiscal deficit and valuation of foreign debts, but also enables the government to devalue the dollar to achieve the relative appreciation of other currencies so as to realize comparative advantage of the weak currency in exports. to counter such uncertainties, china has maintained the stability of its exchange rate with the dollar and appropriate trade scale [25,26]. 2. framework description 2.1 preliminaries lemma. for any positive definite symmetric matrix w and any constant 0y  , the following inequality holds true 0 ( ) ( ) ( ( ) ( )) ( ( ) ( ))t t t y y x t wx t d x t x t y w x t x t y            (1) proof: because w is a positive definite symmetric matrix, there is 0d  such that tw d d . let 1(g , ,g ) r , 1n ng g   , be a constant vector g of length one. then in the light of cauchy inequality, we have: 0 0 0 0 ( ) ( ) ( ( ) )( ( ) )t t t t y y y y dx t dx t d g gd g dx t d dx t gd                    (2) therefore, we have 0 0 01 ( ) ( ) ( ( ) )( ( ) )t t t y y y x t wx t d g dx t d dx t gd y                        ytxtxdggdytxtx y tttt  ()( 1 1 ( ( ) ( )) ( ( ) ( ))t tx t x t y w x t x t y y      .q.e.d. 2.2 the main result consider the following situation with a polynomial lag 1 1 1 1 ( ) ( ) ( ) ( ) 3(a) ( ) ( ) ( ) ( ) 3( ) ( ) ( ), ,0 n i i i n i i i x ax t a x t h bw t b u t z cx t c x t h dw t d u t b x t t t h                          (3) let 1 1 ( ) ( ) i i m mt t t t t i i i t h t h i i v x px x q xdt h t x hr xdt            (4) then we have the following theorem. theorem 1. if the decision matrix m = m1 + m2 + m3 of system (3) satisfies m1+m2+m3 < 0, then system (3) is stable. proof. a stability condition of system (3) is that the eigenvalues of m are negative numbers. this condition of equivalence means that the decision matrix m must be negative definite, that is, m = m1 + m2 + m3 < 0. in fact, we can do the following symbolic calculation: advances in systems science and applications (2017) vol.17 no.1 31 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) m m t t t t i i i i i i v t x t px t x t px t x t q x t x t k q x t k          2 1 1 ( )( ) ( ) ( )( ) ( ) i m m t t t i i i i t k i i x t k r x t x k q x d         (5) 1 1 2 ( ) ( ) 2 ( ) ( ) 2 ( ) ( ) ( ) m m t t t t t t t i i i i i x t a px t x t k a px t w b px t x t q x t         2 1 1 1 ( ) ( ) ( )( ) ( ) ( )( ) ( ) i m m m t t t t i i i i i i i t k i i i x t k q xt k x t k r x t x k q x d              . notice that (a) 2 1 ( )( ) ( ) m t i i i x t k r x t     2 1 2 1 1 1 ( ) ( ( ) ( ) ( ) ) ( ) ( ) m t t t t t t t t m i i m i m m a x a x t h x t x t k x t k w k r a a a w a x t h b w                                     1 1 1 ( ) ( ( ) ( ) ( ) ) ( ) t t t t m m x x t h x t x t k x t k w m x t h w                   where 2 2 2 2 1 1 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) m m m m t t t t i i i i i i m i i i i i i m m m m t t t t i i i i i i m i i i i i i m m m m t t t t i i i i i i m i i i i i i a k r a a k r a a k r a a k r w a k r a a k r a a k r a a k r w m b k r a b k r a b k r a b k r w                                                  (6) (b) ( )( ) ( ) ( ( ) ( )) ( )( ( ) ( )) i t t t i i i i i i t k x k r x d x t x t k k r x t x t k         1 1 2 1 ( ) ( ) ( )( ) ( ) ( ( ) ( ) ( ) ) ( ) i m t t t t t t i i m t k i m x t x t h x k r x d x t x t k x t k w m x t h w                          where 32 q. chen, y. ying, j.y-i. forrest: a novel defense solution towards currency wars 1 1 1 1 2 0 0 0 0 0 0 0 0 m m m m m m r r r r r r m r r r r                 (7) plugging eqs. (6) and (7) into eq. (5) gives 1 1 ( ) ( ) ( ) ( ( ) ( ) ( ) ) ( ) t t t t m m x t x t h v t x t x t k x t k w m x t h w                   because m = m1 + m2 + m3, 1 2 1 1 1 2 23 0 0 0 0 0 0 0 0 0 0 0 m t i m i t t t m m t a p pa q pa pa pa pb a p q a p qm a p q b p                            (8) now, the stability condition of the general system with polynomial lag is: m1 + m2 + m3<0. (9) q.e.d. 4. case study as in [2,3], we continue to use w to represent the vector  1 2 3 t w w w of categorized monetary policies 1 2, , nw w w , which are accordingly grouped into three categories as follows: 1w = the set of all those monetary policies that deal with the population meeting the minimum need to maintain the basic living standard; 2w = the set of all those monetary policies that deal with the population’s need for acquiring desired living conditions; and 3w = the set of all those monetary policies that deal with the population’s need for enjoying luxurious living conditions. similar to the concept of overall balance of international payments, we introduce an economic index vector  1 2 3 t z z z z such that iz measures the state of the economic sector i, i = 1, 2, 3. when the purchasing power increases, people will purchase more assets and products from foreign countries, the overall balance of international payments will decrease (foreign exchange expenditure increase); when the purchasing power decreases, people will sell more assets and products from the domestic country, and the overall balance of international payments will increase (foreign exchange revenue increase). we established the systemic model (eq. (3)) with polynomial lag. in this model, we use symbol z to represent the state of the national economy, w1, w2, and w3 the positive and negative effects of the monetary policies on the performance of the economy directly, or on the currency demand and supply to have an impact on the economy indirectly. here u(t) is a random vector with a nonzero mean. due to the fact that economic development can be seen as a continuous process, the current change in the money stock is determined by the current monetary policies, advances in systems science and applications (2017) vol.17 no.1 33 money stock, and the previous money stock. and the current performance of the economy is also determined by the current monetary policies, money stock, and the previous money stock. here x is the 3  1 matrix [d1 – s1 d2 – s2 d3 – s3] t of the categorized difference of demand and supply of money, referred to as the three parts of the state of the economic system. specifically, our systemic model divides the economy into three sectors e1, e2, and e3. compared to sector e2, which consists of such goods, services that are used by citizens to acquire desired living conditions, sector e1 consists of living necessities. sector e3 consists of such goods, services that are used by the citizens for their enjoyment of luxurious living. our systemic model of the national economy indicates that our separation of the economy into these three sectors can help properly manage the market reaction to the monetary policies. when the monetary policies have positive effect on the performance of the economy, people in every economic sector will purchase more assets and products with the increase of the purchasing power of their income, (foreign exchange expenditure increases); when monetary policies have negative effect on the performance of the economy, people in every economic sector will sell more assets and products with the decrease of the purchasing power of their income (foreign exchange revenue increases). we obtained the stability criterion m1 + m2 + m3 < 0 (eq. (9)) for the general time-delay system based on the systemic model structure with the first-order lag. let 1 1 1 1 2 2 2 2 3 3 3 3 0 0 0 0 0 0 0 0 0 0 , 0 0 , 0 0 , 0 0 0 0 0 0 0 0 0 0 i i i i a b c d a a b b c c d d a b c d                                            because airi = ki, biri = qi, and w are positive definite matrix, from eqs. (6), (7), and (8) we have 3 3 3 3 3 2 2 2 2 2 1 2 3 1 1 1 1 1 3 3 3 3 3 2 2 2 2 2 1 1 1 1 2 1 3 1 1 1 1 1 1 3 3 2 2 1 2 2 1 2 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( t t t t t i i i i i i i i i i i i i i i t t t t t i i i i i i i i i i i i i i i t t i i i i i i a k r a a k r a a k r a a k r a a k r w a k r a a k r a a k r a a k r a a k r w m a k r a a k r a a                          3 3 3 2 2 2 2 2 3 2 1 1 1 3 3 3 3 3 2 2 2 2 2 3 3 1 3 2 3 3 3 1 1 1 1 1 3 3 3 3 2 2 2 2 2 1 2 3 1 1 1 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( t t t i i i i i i i i i t t t t t i i i i i i i i i i i i i i i t t t t i i i i i i i i i i i i i i k r a a k r a a k r w a k r a a k r a a k r a a k r a a k r w b k r a b k r a b k r a b k r a b k r                         3 1 ) t i w                                 1 1 2 3 1 1 2 3 2 2 1 2 3 3 3 3 3 0 0 0 0 0 0 0 0 0 r r r r r r r r m r r r r r r r r                  3 1 2 3 1 1 1 3 2 2 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 t i i t t t t a p pa q pa pa pa pb a p q m a p q a p q b p                        34 q. chen, y. ying, j.y-i. forrest: a novel defense solution towards currency wars so we can obtain an expression for m1 + m2+ m3 explicitly. if the sum is negative, then the systemic model above is stable. to facilitate the detailed calculation, we only select a one-dimensional case for explanation so that the three sectors in [2] become one sector. by substituting the demand and supply of money, x is defined as an exchange rate. we will still use the same symbol w to represent the vector [w1 w2 w3] t of categorized monetary policies. eq. (3a) indicates that the current exchange rate is not only determined by the current monetary policies, but also by the previous monetary policies. in this paper, our model established by empirical studies shows that when the financial crisis occurred (from 2008 to 2010), the government made the exchange rate of the rmb against the u.s. dollar remaining at around 6.8 through the implementation of a series of effective policies and instruments. based on the systemic model structure for the second-order lag, the fitting degree of the model that contains parameters for policy implications increases 16.8% from that of the model without any parameters for policy implications, for details see figure 1. this result means the effectiveness of the policy parameters. fig. 1: delayed effects of monetary policies (fitting after the second order difference in 68 weeks from 2008 to 2010) when we define z as the overall balance of international payments, eq. (3b) indicates that the overall balance of international payments is determined by the current exchange rate and the previous exchange rates. the result also shows the fitting degree of model that contains parameters for policy implications is better than that of the model without any parameter for policy implications. the policy parameters are useful and necessary in the fitting process. we also know that z is determined by the current exchange rate and the previous exchange rates directly, and determined by the current monetary policies and the previous monetary policies indirectly. so the nature of the changing z is determined by quantitative continuous-deferred monetary policies. 5. implications of the established theory in the internationalization process of a currency, government policy is an extremely important factor. to this end, let us consider the policy implications process of some major currencies from around the world. gbp: britain was the first country in the history to build modern financial institutions that grew the fastest and developed with the most perfection. british national order passed a bill to establish a bank of england in 1694 so that britain became the first country in the world to have a central bank. from 1816 to 1819, british government issued a series of regulations and policies about mint and exchange, and implemented a true gold standard, which also made britain the world's first to implement such a standard. from the middle ages to the 19th century when britain became the "sun" empire, the british had dominated the world's finance. after world advances in systems science and applications (2017) vol.17 no.1 35 war i, in the "dollar bloc" and "franc bloc" supplant, pound was no longer used as an international currency. after world war ii, with the establishment of the bretton woods system, pound was degraded to a national currency [30]. usd: after a century of dormancy, with the establishment of the bretton woods system, the dollar became the world currency. however, as of 1971, a deficit, not seen since 1893, in the overall balance of payments of the united states emerged, and the gold reserve of the united states amounted to less than 1/5 of the foreign short-term liabilities. to prevent countries to exchange their holdings of the dollar into gold, president nixon announced the new economic policy on august 15th, 1971, and his administration issued policies and laws to save the crumbling bretton woods system; and the group of ten reached the smithsonian agreement in december, 1971. however, these efforts failed to curb the selling wave of the dollar and the buying of gold and other currencies of the world. the bretton woods system collapsed in 1973. since then, german mark, french franc, britain pound, and other currencies began to enter the international currency system. even since, the dollar began to embark on its long downward spiral [30]. jpy: after the meiji restoration, japan established the bank of japan, the central bank of the country, in october 1882. diverted to the gold standard in 1897, japan became the asian financial pacesetter. after world war i japan began its dominance of the far east and the pacific region. after world war ii japan was taken over by america's "allied command". since then, japan implemented a series of democratization reform measures. as a result of the war on the korean peninsula as well as the dodge plan in june 1950, japanese economy quickly recovered to the pre-war levels. in 1952, japan recovered its sovereignty and joined the international monetary fund and the world bank. however, the nationalization and free convertibility of the yen is not synchronized. since the early 1970s, the yen has become an international currency, but japan did not issue a decree to allow foreigners to issue bond in japan until the mid-1970s. in the meanwhile, japanese investment in foreign securities began to liberalize. during 2013-2014, the proportion of japanese yen traded in new york foreign exchange market was hovering around 23% [14]. mark: after world war i, germany became the country in the world that suffered the most from inflation. since the end of world war ii, germany has always stressed the independence of its central bank; and the german territories occupied by the west followed the united states and established a two-stage system to avoid the government from manipulating the center bank. on july 26, 1957, the federal german parliament enacted the bundesbank act, making bundesbank a unified central bank. after the 1973 collapse of the bretton woods system, the german decree introduced a floating exchange rate system. since then, german mark became the second-largest international currency in the world only after the dollar of the united states [30]. eur: the euro was controversial. based on the euro zone's gdp, import and export volume when comparable to that of the u.s., many european scholars held their widespread optimism that the euro will challenge the dollar with weight tilting to the euro. but scholars of the u.s. held bearish view on the euro (samuelson, 2000; frankel, 2015; soros, 2010). on january 4, 1999, the euro came into the world. since then, the history seems to suggest that although the european union has introduced a series of policies and regulations, the effectiveness of the policies has impacted the international community slightly [14]. china is now the second largest economy and the largest exporter in the world. with its growing economy and deepening financial reform, its rmb has the ambition to become an international currency. however, the internationalization will be a long process due to the following reasons. the chinese government deems free trade agreements (ftas) as a new platform to further open up the country to the outside world and speed up domestic reforms, an effective approach to integrate the country into the global economy and to strengthen economic cooperation with other economies, as well as particularly an important supplement to the multilateral trading system. currently, china has 20 ftas under construction or implementation. the rmb aimed at 36 q. chen, y. ying, j.y-i. forrest: a novel defense solution towards currency wars eventually becoming a regional financial settlement instrument and commodity with the ongoing development of the multilateral economic cooperation. it is seen as a natural and regional monetary integration which will eliminate the cost of currency trading and the risk of exchange rate fluctuations, and gradually expand the trading network, promote the economic cycle. secondly, the internationalization of rmb will provide favorable conditions for the development and implementation of china’s economic policies. this factor has a strong appeal to the chinese government. the internationalization of rmb will enhance the autonomy of monetary policy in china and help china to get rid of problems that exist in developed countries, such as the "mead dilemma” that to maintain external balance developing countries will have to sacrifice internal balance. this end will help china reduce the risk of economic instability and safeguard the national economic security, which is particularly important in the course of economic globalization. finally, the internationalization of rmb will help china to guard against financial crises [28]. within the imperfect dollar-dominated international monetary system, the internationalization of rmb will improve china’s capacity of finance. when financial assets are priced (at least partially) in rmb in the world markets, china would be able to decide prices and control risk to some extent. and in the process of internationalization of rmb, china must face two types of risks: (1) the liquidity risk. because of the integration of the global finance, when the assets china holds are priced in a foreign currency, the government will have no means of control to provide liquidity as lender of the last resort. this risk is difficult to manage once the market becomes volatile. the assets and liabilities held by either the government or the citizens in the current account will lead a basic process of money creation. we believe that a contraction of the money supply indicates a greater liquidity problem. even for those countries that have enough foreign exchange reserves, like china, trying to provide adequate liquidity within a brief moment of time will cause a huge impact on assets markets and create a vicious cycle. (2) the risk of price fluctuations under currency mismatch. under the floating exchange rate regime, one has to face the risk of price fluctuations. however, if assets are denominated in the local currency, the risk of price fluctuations will to certain extent be reduced. the fact that the assets held by china are denominated in foreign currencies means that china has surrendered part of its power to regulate its financial market to some foreign governments. so, whether it is for the purpose for china to maintain the stability of its economy and the security of its financial assets or for the purpose for it to balance the stability of its international trades, it is important for china to accelerate the internationalization of its currency, the rmb. the recent rise of emerging-market economies provides a special opportunity for the internationalization of rmb, because diversified international currency represents the interests of the majority of countries. because of these particular circumstances, china enjoys many favorable conditions to promote the internationalization of its rmb. first, china's comprehensive national strength has increased significantly. its fast growing economy in the past decades has laid a firm foundation for people from around the world to use rmb. due to the introduction of prudent monetary policies, china has enjoyed favorable social reputation. for example, china has maintained the value of its rmb by curbing its domestic inflation. rmb has played a notable role in combating the recent global financial crisis. the relatively stable exchange rate has also laid a solid, reliable foundation for promoting the internationalization of rmb. secondly, it is the relatively liberal regime environment in china and improvement of the convertibility of rmb. in china’s neighboring countries and regions, rmb has been employed in similar fashions as the dollar and other hard currencies that are relatively easy to acquire. in order to promote rmb in cross-border trade settlements, some border provinces have developed innovative institutions. asean free trade area for overseas circulation of rmb has provided a broad space for promoting rmb. under the premise of stable exchange rate with rmb, some countries and regions, which trade and invest frequently or with large amounts with china, have advances in systems science and applications (2017) vol.17 no.1 37 been willing to accept rmb as the denominated settlement currency. for example, in trades with vietnam, thailand, myanmar, and other countries, rmb has in fact become one of the monetary instruments for settlements [26]. thirdly, hong kong, a recognized center of international finance, represents a unique advantage for china. in the process of internationalizing rmb, hong kong can play and has played an important role. the city has a complete and mature legal system, perfect market infrastructure, and the hkma (hong kong monetary authority) operates independently from the government. so, hong kong can help with the implementation of rmb’s regionalization, provide useful experience for the chinese central government. before the chinese financial market could become mature enough to handle the excessive volatility of the international financial market, the offshore rmb market in hong kong will play the role of "firewall" for the mainland. and, the financial supervisory system in hong kong can help the mainland government and currency authority to grasp market trends and to detect potential threats. although many favorable conditions exist, the regionalization of rmb is still plagued with various internal and external difficulties. first, the degree of regional economic integration affects the scale of rmb’s regionalization. in areas where trades and personnel exchanges happen frequently, especially in border areas, rmb has seen greater liquidity, and even enjoyed more popularity than the relevant national currencies. however, if a neighboring country did not reach any institutional arrangement with china, then the existing economic integration measures along the border between the two countries would not in general be extended to the whole region, where the regionalization of rmb could only be confined to the border areas. in such areas, financial services are generally in short supply and there would be no unified basis for the exchange rate of the rmb. in some areas of southeastern asia that have liberalized trades and traded heavily with china, including cafta (china and asean free trade area), many countries suffer from their underdeveloped banking systems so that these countries have few financial institutions that handle the business of rmb exchange. in addition, many of these countries do not have their official exchange rates for direct conversion of their local currencies into rmb other than the black market. also, the current security issues and conflicts in northeastern and southeastern asia might very well place a damp on the internationalization of rmb and affect the confidence for rmb to be widely used in trade settlements and investments in the region. because of the constraints of these listed factors, the internationalization of rmb will be a gradual and long drawn-out process. that surely posts challenges for china’s authorities to formulate and to implement wise currency policies. with the increase in capital liquidity and overseas business of chinese commercial banks, the authorities better have an ability to predict, to manage, and to control risks. 6. conclusion each currency war represents a battle without involving gunpowder. during the fight, countries try to compete for acquiring biggest economic benefits mainly by adjusting the supply and value of their domestic money. today, more than 100 countries have been actively or passively involved in currency wars. although some countries can benefit from currency competitions in the short term, it is doubtful whether this benefit is sustainable. and one thing is for sure: most countries, especially those involuntarily involved in the wars, will suffer great economic and social unrest. by taking the impact of capital account liberalization of china on international capital flow and economic and financial system as an entry point, this paper establishes a dynamic systemic model with lag variables, and the stability condition of the dynamical system. then, a onedimensional case is developed to explain the significance of this work. through the model we can know the following. first, stability condition of the dynamical system shows monetary policy and its subsequent effects can play an important role in the economic stability of the country under the free flow of capital, and this condition has certain warning effect. second, how 38 q. chen, y. ying, j.y-i. forrest: a novel defense solution towards currency wars monetary policy regulates the economy in the system. third, the impact of monetary policy and its subsequent effects will play a decisive role in regulating economic equilibrium. based on what has been accomplished in this paper, we have following important suggestions on defense solutions towards currency wars. first, it is comparatively limited in theory to study simply the dynamic systems model between two countries. such study should be expanded to a much bigger dynamic system involving many mutually reciprocal feedback countries so that more convincing results with real policy effects can be established. second, the following is truly a quite complex problem: how can one improve the accuracy of assessing and quantifying the impact of different monetary policies on the economy? this problem and related issues need to be further investigated. acknowledgements. this research was supported by national natural science foundation of china (71301064; 71171128), and research fund of program foundation of education ministry of china (10yja790233). references [1] forrest, j. a systems perspective on 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[2] forrest, j. hopkins, z. and liu, s. f. “currency wars and a possible self-defense (i): how currency wars take place”, advances in systems science and application, vol. 13, no. 3, pp. 198-217, 2013. [3] forrest, j. hopkins, z. and liu, s. f. “currency wars and a possible self-defense (ii): a plan of self-protection”, advances in systems science and application, vol. 13, no. 4, pp. 298-315, 2013. [4] frankel, j., “the euro crisis: where to from here?”, journal of policy modeling, vol. 37, no. 3, pp. 428-444, 2015. [5] gagnon, j. e., “currency wars”, the milken institute review, vol. 15, no. 1, pp. 47-55, 2013. [6] gagnon, j. e., and bergsten, c. f., “currency manipulation, the us economy, and the global economic order", policy briefs, pp.1-25, 2012. [7] he, h. g., “rmb internationalization: mode selection and path arrangements”, finance & economics, vol. 2, pp. 10 – 15, 2007. 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[31] zhang, s.m., “an analysis on demonstrative effect of financial crisis and devaluation effect of competitiveness – the revelation of china’s transitional economy with the opening condition”, world economy study, vol. 3, pp. 36-40, 2003. http://fta.mofcom.gov.cn/article/shidianyj/ adv syst sci appl 2018; 1; 92-101 published online at http://ijassa.ipu.ru. copyright ©2018 assa. adv. in systems science and appl. (2018) a two-stage algorithm for generating a set of paretooptimal trajectories of an object elena l. kulida 1 1) institute of control sciences, russian academy of sciences, moscow, russia e-mail: lenak@ipu.ru abstract: a two-stage algorithm is proposed for generating trajectories and parameters of the object's motion based on searching for a set of pareto-optimal paths from the initial vertex to the terminal vertex. the features of graph construction in the first and second stages of the algorithm are described. at the first stage, the set of vertices of the graph uniformly cover the area of the object's motion, and the edges connect only the nearest neighboring vertices. at the second stage, the set of vertices of the graph includes only those vertices through which the paths constructed in the first stage pass. for this set of vertices, a complete graph is constructed: the edges connect each vertex to all the others. an algorithm for constructing a set of pareto-optimal characteristics of graph paths is described. the two-stage approach allows to significantly reduce the calculation time in comparison with the application of the described algorithm for a complete graph in one stage. two variants of the algorithm are considered. the first algorithm requires a longer calculation time, but allows to obtain more trajectories that are diverse. in particular, it is possible to obtain trajectories of different classes, for example, avoiding obstacles from different sides, etc. for the case when time for calculations is not enough, an abridged algorithm is proposed. the results of computational experiments are presented. keywords: approximation of trajectories by graph paths, a set of pareto-optimal characteristics, search for a path in a graph, algorithm for generation of trajectories. 1. introduction algorithms for generating trajectories that satisfy specified conditions are needed to effectively control the movement of an object moving in a conflict environment. it is necessary that flight management systems are able to comply with 4d trajectories, which needs to be done in computationally efficient ways due to the limited computational resources available [3, 5]. in [6], analytical and discrete approaches are considered to optimize the trajectory of an aircraft in a dangerous environment. the discrete approach allows solving the problem in case of the detection risk of an aircraft by several radars. however, the use of discrete optimization methods requires very large computational costs to generate optimal trajectories, especially if optimization by several criteria is required. in this paper, we propose a decomposition of the optimization process into two stages, which allows to reduce the time required for solving the problem. in [4], an algorithm for controlling the trajectory and parameters of the motion of an object in a conflict environment is given. the risk of detecting an object and the possibility of collision with the earth's surface are taken into account. optimization is carried out in accordance with specified quality criterion of optimizing the trajectory and traffic parameters with a restriction on the time of motion. for this purpose, the specified quality criterion and the criterion of the minimum time of completion of a route are considered. a graph is formed, and its paths approximate all possible trajectories of the object's motion in the given region. the graph must include vertices that cover the region of motion tightly enough to approximate the trajectories with sufficient accuracy. it is desirable that each vertex is a two-stage algorithm for generating a set of pareto-optimal trajectories of an object 93 copyright ©2018 assa. adv. in systems science and appl. (2018) connected by edges to all other vertices to move to an arbitrary direction if the motion along such an edge does not violate restrictions associated with the terrain [1]. two-component characteristics are set on the edge of the graph. the characteristic consist the values of two optimization criteria. sets of pareto-optimal characteristics of paths from a given initial vertex are constructed for the vertices of a graph. however the calculation of the set of pareto-optimal characteristics of the vertices of such a graph requires time-consuming unsatisfactory large for the practical use of the algorithm. the paper proposes an algorithm to solve this problem more effectively in two stages. 2. formulation of the problem the area of motion of the object m is set on the map. when the object moves in this area conflicts with the relief may occur. a known height matrix of the relief for the area is assumed. we consider the set z of all possible trajectories of the object motion from the initial point a m with coordinates and height to the endpoint with coordinates and height . where are the coordinates of the object, is the height of the object, is the speed of the object's motion at time t, and t is the trajectory z passing time. the trajectory should not have conflicts with the relief, at any point of the trajectory the height of the object above the relief must be greater than the specified value . the functional of the cost (the risk) of passing the point with the speed is the cost of the trajectory z is the task is to generate trajectories of the object from point a to point b of the paretooptimal by two criteria: the minimum cost and the minimum time of passing of the route. the problem cannot be solved analytically due to the complexity of the cost functional and the need to take into account the terrain. discrete optimization methods are used to solve the problem. a graph is constructed to approximate the trajectories from the set z, and the problem of finding the pareto-optimal trajectories of the graph by two specified optimality criteria is solved. 3. discrete optimization method for solving the problem 3.1. the construction of the graph in the first stage of optimization we proceed to a discrete solution to obtain an approximate solution from a continuous problem. the solution is sought in the form of a piecewise linear trajectory with piecewise constant speed and height of motion (speed and height is constant on segments of the trajectory and can be changed when moving from one segment to another). we consider a finite set of heights (depths) of the motion , l is the number of heights, and a finite set of velocities of the object , m – the number of considered speeds of movement of the object. 94 e.l. kulida copyright ©2018 assa. adv. in systems science and appl. (2018) the trajectories of the object's motion are approximated by the paths of the graph , n is the set of vertices of the graph, and e is the set of edges of the graph. the path is sought in the form of a sequence of edges where is the speed of motion along the edge . the edge either lies in the horizontal surface at height , then , or is intended to go to another horizontal surface, then . the given area of the object's motion is covered by a uniform grid of points at a distance d from each other to construct the set of vertices of a graph. a separate grid layer is created for each of the l different heights of the object. the vertices are projected onto all layers. if there is an obstacle to the relief: , the corresponding vertex is excluded from the set of vertices of the graph n. in addition the points a and b are included in the set n. it is assumed that the constant velocity is ascribed to the edge of the graph. this speed determines the cost and time of movement along the edge. m edges connect each vertex from the set n with adjacent (nearest) vertices in the horizontal layer. one of the m speeds corresponds to one of these m edges. in addition, each vertex is connected by edges to vertices in adjacent horizontal layers; these vertices are assigned vertical velocities (ascending or descending) of the object. the construction of the graph in the first stage of optimization is illustrated in fig. 3.1. in this figure in the horizontal layers the vertices form a square grid, each vertex is connected by edges with four neighboring vertices in the horizontal layer and with vertices under and above it in adjacent horizontal layers. the number of vertices of the obtained graph can be approximately estimated by the number , where s is the area of the region of motion, d is the distance between neighboring vertices in the layer, and l is the number of layers. the number of edges . these are estimates from above, because some vertices and edges are excluded from consideration due to the obstacles of the relief fig. 3.1. graph construction in the first stage of optimization 3.2. the algorithm for constructing the set of pareto-optimal characteristics each edge is assigned a characteristic in the resulting graph. definition 3.1: the characteristic of the edge at the speed of motion is the two-component vector , where is the cost of passing the edge with the speed , is the time of the passing with the speed , . a two-stage algorithm for generating a set of pareto-optimal trajectories of an object 95 copyright ©2018 assa. adv. in systems science and appl. (2018) definition 3.2: the characteristic of the trajectory z is the two-component vector , where is the cost of the trajectory z, is the trajectory z passing time, defined by the formulas: the problem is to construct in the graph the set of paths from vertex a to vertex b, pareto-optimal by two criteria: and . an algorithm is used to construct the sets of pareto-optimal characteristics of paths from the initial vertex to other vertices of the graph to solve this problem. then the paths corresponding to the pareto-optimal characteristics obtained are constructed for the final vertex. since there are two criteria, each vertex is associated with a set of incomparable twocomponent characteristics. consider two arbitrary characteristics and . definition 3.3: the characteristic and are equal: , if . definition 3.4: the characteristic is smaller than the characteristic : , if . definition 3.5: the characteristics and are incomparable if . definition 3.6: we call the set q of path characteristics from the vertex a to the vertex b pareto-optimal if all the characteristics of this set are pairwise incomparable and for the characteristic of an arbitrary path z from the vertex a to the vertex b . the algorithm for constructing the set of pareto-optimal characteristics notation: is an intermediate set of pareto-optimal characteristics of paths from vertex a to vertex n; is the final set of pareto-optimal characteristics of paths from vertex a to vertex n; q is the set of vertices n for which the set . step 1: the initial vertex a is considered. the characteristic (0,0) is stored in the set , the vertex a is stored in the set q. cycle: a cyclic process is performed until the set q is not empty. 96 e.l. kulida copyright ©2018 assa. adv. in systems science and appl. (2018) an iteration of the cycle consists of two stages. the first stage is to select the vertex for processing: among all vertices , we look for a vertex whose set contains the minimal characteristic . the characteristic is excluded from the set and is included in the set . if after this the set becomes empty, then the vertex n is excluded from the set q. the second stage is processing the selected vertex: all vertices connected by edges with vertex n are considered. for each vertex m, the characteristic , where is the minimal characteristic in the set , is the characteristic of the edge that connects the vertices n and m. if in the sets and there is no characteristic smaller than , then is added to the set . all the characteristics , are removed from – this ensures the pareto optimality of the set . if before the insertion of in this set was empty, then the vertex m is stored in q. after the completion of the algorithm, the set is the set pareto-optimal characteristics of paths from vertex a to vertex b. 3.3. the algorithm for generation of pareto-optimal trajectories the goal of the first stage of optimization is to construct a set n of vertices of the graph for the second stage of optimization. the first stage of optimization step 1: the construction of a graph in the form of a uniform grid. calculation of the characteristics of edges of a graph. step 2: construction of sets of pareto-optimal path characteristics from the vertex a to vertices of the graph. step 3: construction of the set of pareto-optimal paths from vertex a to vertex b based on the constructed set . we denote by n * all the vertices through which the paths corresponding to the set pass. examples of the pareto-optimal trajectories generated in the first stage are shown in figures 3.2 – 3.4. dangerous areas for the movement of the object are represented in the figures with filled rectangles. the color intensity characterizes the degree of danger of the zone. optimization according to two criteria allows obtaining different trajectories, including bypassing dangerous zones from different directions or passing through dangerous zones, as this degrades the quality of the trajectory by one criterion, but improves by the other – it allows reducing the trajectory transit time. a two-stage algorithm for generating a set of pareto-optimal trajectories of an object 97 copyright ©2018 assa. adv. in systems science and appl. (2018) fig. 3.2. trajectory bypasses dangerous areas fig. 3.3. this trajectory is more dangerous but the travel time is less 98 e.l. kulida copyright ©2018 assa. adv. in systems science and appl. (2018) fig. 3.4. the trajectory is passing through dangerous zones but the travel time is less the second stage of optimization step 4: construction of a graph for the second stage of optimization. n * is the set of vertices through which the pareto-optimal paths constructed in the first stage pass. each of the vertices of the set n * is projected onto all l layers of the grid to construct the set n of vertices of the graph for the second stage of optimization: a complete graph is constructed in each horizontal layer in the second stage of optimization; each of the vertices of the layer is joined by edges with all the other vertices of this layer to construct real trajectories of the object's motion. the vertices are connected by edges between the horizontal layers one under the other. a characteristic is calculated for each edge in the resulting graph. in the first step, the set of vertices of the graph includes vertices evenly covering the area of motion, and the edges of the graph connect only to the nearest neighboring vertices. at the second stage, the set of vertices of the graph contains only those vertices through which the paths built at the first stage pass. each of the vertices is connected by edges to all other vertices of the graph. a set of pareto-optimal paths is constructed in the complete graph. this two-step approach allows to significantly reduce the computation time in comparison with the use of the algorithm for constructing a set of pareto-optimal paths in the initial graph. step 5: construction of sets of pareto-optimal characteristics of paths from vertex a for the vertices of the graph. step 6: a two-stage algorithm for generating a set of pareto-optimal trajectories of an object 99 copyright ©2018 assa. adv. in systems science and appl. (2018) an analysis of the constructed set of pareto-optimal characteristics; the choice of characteristics satisfying the requirements and constraints [2]; the construction of appropriate paths. 4. simulation results a series of experiments was performed to evaluate the efficiency and accuracy of the proposed algorithm using the example of generating object trajectories moving in a conflict environment, where it is detected by means of detection of different nature. below are the results characterizing the use of the proposed optimization algorithm in one of such experiments. as can be seen from the table 4.1, a decrease in the distance d between vertices in the construction of the grid at the first stage of optimization predictably allows to build more diverse trajectories and substantially improve the accuracy of the solution, but the calculation time increases dramatically. the evaluation of the trajectory is the probability of not detecting an object during the trajectory, so the evaluation lies in the range [0,1]. the best trajectory has greater evaluation. table 4.1. results for different length of edge of graph length of edge of graph, km evaluation of the trajectory travel time, hours calculation time hour:min:sec number of trajectories 32 0,4078 28,29 00:01:50 254 16 0,5612 21,05 00:26:23 1140 8 0,6148 18,4 06:20:21 2639 when analyzing the set of generated trajectories, it turns out that the parameters of incomparable trajectory characteristics can differ by negligibly small values. in order to reduce the number of generated trajectories and to shorten the calculation time, the following method was used: during the calculation, the values of the parameters of the segments characteristics were rounded up to four decimal places. the results of the computational experiment are shown in the table 4.2. table 4.2. results for different length of edge of graph with/without rounding length of edge of graph, km evaluation of the trajectory travel time, hours calculation time hour:min:sec number of trajectories without rounding 32 0,4078 28,29 00:01:50 254 with rounding 32 0,4078 28,29 00:01:12 245 without rounding 16 0,5612 21,05 00:26:23 1140 with rounding 16 0,5613 21,05 00:19:44 1064 without rounding 8 0,6148 18,4 06:20:21 2639 with rounding 8 0,6149 18,4 04:04:00 2420 as follows from the table, the characteristics of the generated trajectories did not change to two decimal places, and the calculation time decreased. in order to reduce the calculation time significantly, one can use a shorter algorithm. the algorithm for generating pareto-optimal trajectories can be reduced as follows: step 2: instead of constructing sets of pareto-optimal path characteristics, calculate the length of the optimal path by the main criterion for each level of the graph, for example using the dijkstra algorithm. 100 e.l. kulida copyright ©2018 assa. adv. in systems science and appl. (2018) step 3: for each layer of the graph construct a path from vertex a to vertex b, which is optimal by the main criterion, i.e. with a minimal cost. we denote by n* all the vertices through which constructed paths pass. the other steps of the algorithm remain unchanged. the results of the comparison of the algorithm 1 (source) and the algorithm 2 (reduced) are presented in the table 4.3. table 4.3. the results of the comparison of the algorithm 1 and the algorithm 2 length of edge of graph, km evaluation of the trajectory travel time, hours сalculation time hour:min:sec тumber of trajectories algorithm 1 32 0,4078 28,29 00:01:12 245 algorithm 2 32 0,332 27,79 00:00:14 65 algorithm 1 16 0,5613 21,05 00:19:44 1064 algorithm 2 16 0,5291 20,67 00:01:33 262 algorithm 1 8 0,6149 18,4 04:04:00 2420 algorithm 2 8 0,5968 19,16 00:12:16 553 5.conclusion the article presents two algorithms for constructing pareto-optimal trajectories of object motion. optimization by the two criteria makes it possible to obtain trajectories that are optimal by the main criterion and satisfy a given time limit. the results of numerical experiments are presented. the following result is obtained by comparing the two proposed algorithms. in algorithm 2, the number of generated trajectories is reduced significantly, while the accuracy of the best solution decreases insignificantly, and the calculation time decreases very significantly. as can be seen from the table 3, in the case where the edge of the graph is 8 km, the calculation time has decreased by more than 20 times! references [1] bazhenov, s.g., egorov, n.a., kulida, e.l., & lebedev, v.g. (2016). control of aircraft trajectory and speed to avoid terrain and traffic conflicts during approach maneuvering, automation and remote control 77(10), 1827-1837. [2] bazhenov, s.g., korolyov, v.s., kulida, e.l., & lebedev, v.g. (2014). simulation of on-board model of airliner to evaluate capability of trajectories and flight safety. 29th congress of the international council of the aeronautical sciences, icas 2014. [3] diaz y., lee s., egerstedt m., & young s. (2013) optimal trajectory generation for next generation flight management systems. 32nd digital avionics systems conference. [4] dobrovidov, a.v., kulida, e.l., & rudko, i.m. (2015) path optimization for a moving object in an anisotropic environment using the probabilistic criterion in the passive sonar mode. automation and remote control, 76(7), 1271-1281. [5] hehn, m., & andrea r.d. (2011) quadrocopter trajectory generation and control. 18th ifac world congress, vol. 18, 1485–1491. a two-stage algorithm for generating a set of pareto-optimal trajectories of an object 101 copyright ©2018 assa. adv. in systems science and appl. (2018) [6] zabarankin, m., uryasev, s. and pardalos p. (2002). optimal risk path algorithms. cooperative control and optimization (r. murphey and p. pardalos ed.), kluwer academic publishers, dordrecht, vol. 66, 271–303. microsoft word 13 ge zhenghao, zhang kaikai, jiang meng , yang fulian--an exact reverse design approach for disk cam mechanis advances in systems science and applications (2011), vol.11, no.3-4 301-308 issn 1078-6236 international institute for general systems studies, inc. an exact reverse design approach for disk cam mechanisms ge zhenghao, zhang kaikai, jiang meng and yang fulian shaanxi university of science & technology, xi’an, shaanxi, 710021, china abstract in the view of the difficult problem of receiving the original design of the cam and its follower motion specification, an exact new reverse design method for disk cams is provided. the essential difference between the proposed method and other existing approaches is its ability to make the cam profile smooth while still exactly satisfying boundary condition of follower displacement，velocity and acceleration. it takes computer movement simulation technology as the main instrument. firstly，in order to reverse the follower motion specifications accurately, an new method is proposed in the paper, first of all， smooth disposal is processed to the cam profile which is formed by the equalized measurement data，and then the cam profile is processed into a series of discrete data. secondly，this paper does research on motion specification of disk cam follower through establishing mathematical modeling and analysis treatment，finally it works out the expression of follower motion specifications. in addition，the way of distinguish the real motion specification of cam mechanism is put forward. thirdly， the simulation modeling is established and the follower motion specifications which is the accurate gauge of the reversed disk cam is also achieved. an example shows that the proposed method can be a powerful tool of cam profile smoothing， which verifies the method can realize exact reverse design for disk cam mechanisms rapidly. the approach is not only suitable for own program but also for cad/cae software, and it can be used for the spatial cam reverse design as well. keywords reverse design motion simulation motion specification cam mechanism 1. introduction a cam is a mechanical device for transmitting mechanical work to another component （the follower） according to the transmission，direction and control function. a planar cam mechanism is the common type to transform rotary motion of the cam into translating or oscillating motion of the follower. since the motions of a follower depend on the cam profile and the follower type, the exact profile of the cams must be given to obtain the prescribed output of the follower after reasonable choice of a follower type and parameters for the cam follower systems. therefore, how to achieve process of the introduced products without drawing, and how to test for processed products, which not only play an important rule on import machinery accessories， but also can speed up the digestion and absorption of foreign advanced technology. so as to create and develop our own new products has very important practical significance. 2. relevant research many scholars make a great deal of reverse investigation on disk cam mechanisms. for example, the reverse design equation of follower motion specification is founded basing on the data which are measured by cmm and the meshing relation of cam and roller. using cubic spline function, the interpolation functions of actual cam contour is obtained, and then academic contour line equation is worked out, finally the motion specification of follower and the implementation are reversed. nowadays, there remain lots of difficulties on reverse design of disk cam mechanisms, such as realize measurement and evaluation reversed disk cam fleetly and precisely. as the fig. 1 shows，a new approach was introduced. first achieved the cam profile by the measured data and processed smooth disposal， second calculated the follower motion specification via motion 302 ge: an exact reverse design approach for disk cam mechanisms simulation，finally the new method of detection and evaluation for the cam profile fitting result using the motion specification of cam mechanism was put forward. this method of reserve design is not only suitable for own program conveniently, but also for existing cad/cae software. fig. 1 the flow of cam reverse design 3. curve fitting and discrete of disk cam contour the data which were measured by cmm can not be used directly in reverse design of disk cam motion specification. it should process smooth disposal. however, it is complicate to dispose the profile formed by the measured data smooth directly. thereby, in this paper, it took the method that firstly fitting profile via the measured data and then transforming the smooth curve into discrete points. nowadays, nurbs curve is widely used for various curves fitting .because it is convenient to adjust the curve slightly due to the localized performance of basis function of . its k-th power curve equation is: ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ]1,0 0 , , , 0 , 0 , 0 , ∈ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪⎪ ⎪ ⎪ ⎨ ⎧ = == ∑ ∑ ∑ ∑ = = = = u wub wub ur ur wub wub u n j jkj iki ki n i ikin i iki n i iiki ， v v p （1） iv –control vertex; iw weighted factor; ( )ub ki, k-th power basic function of b-spline. first the cam contour curve was processed into discrete points after fitted, and then the discrete point-group of cam contour curve can be obtained. ( )ii yx ′′′′, (i= 1, 2, 3……n). advances in systems science and applications (2011), vol.11, no.3-4 303 4. reverse design of motion specification as figure 2 shows, o xyz− was basic coordinate system, 1 1 1 1o x y z− was variation coordinate system which was consolidated with cam. 2 2 2 2o x y z− was variation coordinate system firmed with follower. roller and cam are meshed in points . basing on conjugate meshing principle, it can be concluded that the normal of cam which get across must get across roller-center point ifo , . fig.2 cam mechanism and its coordinate system 4.1 calculate the coordinate of roller center point when cam rotates counterclockwise, the angle between coordinate system and coordinate system can be described as: n i i ⋅ = πθ 2 (2) suppose the theoretic coordinate of cam contour curve is ( ), the roller center point can be concluded as follow: ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ′′′+′′′− ′′′+′′′ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ′′′ ′′′ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ −= ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ 1 cossinsin sincos 1100 0cossin 0sincos 1 , , iii iiii i i ii ii if if yx yx y x y x θθ θθ θθ θθ (3) the slope of cam counter curve normal via meshing point can be calculated by formula (4): 11 11 −+ +− ′′−′′ ′′−′′ =′′ ii ii i yy xx n (4) then the expressions of theoretic cam contour point can be worked out. ( ) ( )⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ +′′ ⋅′′ ±′′=′′′ +′′ ±′′=′′′ 1 1 2 2 i fi ii i f ii n rn yy n r xx (5) the coordinate of roller center can be deducted putting formula (5) into formula (3). 4.2 calculate the swing angle of rocker 304 ge: an exact reverse design approach for disk cam mechanisms in coordinate system o xyz− the original point of 2 2 2 2o x y z− coordinate system is known, the original angle between o xyz− and 2 2 2 2o x y z− coordinate system is ϕ , hypotheses the angle between rocker and x axis is iϕ′ . then the swing angle of rocker can be described as follow while the cam rotating. ii ϕϕπϕ ′−−= 2 (6) according to the triangle geometric relations: if if i xc y , ,tan − =′ϕ (7) then put formula (7) into formula (6)，obtained formula (8): ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − −−= − if if i xc y , ,1tan 2 ϕπϕ (8) the swing angle of rocker can be constructed by formula (3), (5) and (8). 4.3 calculate velocity and acceleration of follower using is 、 iv and ia represented discrete actuality displacement, velocity, and accelerate of follower separately, the expressions were constructed as follow: ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ − − = − − = = −+ −+ −+ −+ 11 11 11 11 ii ii i ii ii i ii vva v s θθ θθ ϕϕ ϕ (9) 4.4 calculate dimensionless motion specification of follower the actual motion specification curve of follower can be obtained by processing the follower actual motion specifications into dimensionless. because the follower actuality displacement, velocity, and accelerate are discrete, so they can be dimensionless processed directly. the time , displacement , velocity , and accelerate can be expressed as follow: ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ = = = = h ta a h tv v h s s t tt hi i hi i i i h 2 (10) in the expressions: iii avst ,,, ——dimensionless discrete time, displacement, velocity, and acceleration of follower; ht ——intervals of cam lifting or returning time; h ——position corresponding to of follower. advances in systems science and applications (2011), vol.11, no.3-4 305 5. example of cam mechanism reverse design a piece of cam which belongs to conjugating cam is shown in fig.3. for this disk cam mechanism, the distance between a and b is 228mm, the length of pendulum rod is bc=145 mm, and radius of roller circle is 65 mm. fig. 3 disk-cam mechanism with swinging follower the discrete data is measured by cmm， the diameter of probe-radius is 1mm. cam contour curve fitting shown as in figure 4 was obtained by equalized data. the curvature quality of contour curve directly fitted via compensation data is poor in fig. (a). however, the curvature of contour curve shown in fig. (b), which was fitted by points and processed smoothing, is more smooth and high-quality. (a) curvature of cam contour curve which is fitted directly(b) curvature of cam contour curve which is treated smoothly fig. 4 curvature of cam contour curve according to fitting cam contour curve, the paper built a motion model and simulate motion，shown in fig. 5. the follower actual motion specification curves were obtained. (fig.6) then compared the motion specification of the follower with the desired curve of design requirements(the expected tract of the actual follower is known )，if the difference is in the scope of permissible error of the track, the reverse cam quality is eligibility, otherwise it need to re-reverse design cam. fig.5 mechanism simulation modeling 306 ge: an exact reverse design approach for disk cam mechanisms (a) displacement curve (b) velocity curve (c) acceleration curve fig. 6 actual motion specification curve advances in systems science and applications (2011), vol.11, no.3-4 307 according to the actual motion specification curve, the cam rotates 90°during lifting and returning time separately. processed actual motion specification curve into dimensionless motion specification, the dimensionless motion specification of lifting and returning curve were constructed in fig.(7) and fig.(8). (a) lifting dimensionless displacement curve (b) lifting dimensionless velocity curve (c) lifting dimensionless acceleration curve fig. 7 lifting curve of dimensionless motion specification (a) returning dimensionless displacement curve (b) returning dimensionless velocity curve 308 ge: an exact reverse design approach for disk cam mechanisms (c) returning dimensionless acceleration curve fig. 8 returning curve of dimensionless motion specification basing on lifting motion specification curve, it can be concluded that the lifting motion specification is polynomial motion of 4-5-6-7. while it can be concluded that the returning motion specification is polynomial motion of 4-6-8-10， basing on returning motion specification curve. 6. conclusion an exact approach to profile generation of disk cams and evaluation of the reverse designed cam is presented for cam-follower systems. the way of identification the real motion specification of cam mechanism was put forward also. the approach is not only suit for own program but also for cad/cae software, and it can be adopt to the reverse design of spatial cam mechanisms as well. references [1] liu changqi， muye yang and cao xijing,. the design of cam mechanism.beijing：china machine press, (2005). [2] der-min tsay and meng-hung huanghui-chun ho. producing follower motions through their digitized cam coutours. journal of mechanical transmission, vol. 2, issue 2(2003) 98～105. [3] yang dongming. the studies of reverse design，error analysis and test for cam mechanism. machine design and research, (1999) 39-62. [4] wang yiming and zhao jibin. the reverse design for nozzle body of paper. machine design, 22(6) (2005) 51-53. [5] wang shuang, yen guo fu, luo zhong xian and wang ying – guo. development of planar cam mechanism cad system. coal mine machinery，27(6) (2006)999-1000. [6] yu xiaoyan and lan zhaohui. a method of determining the curvature radius of a planar cam profile by cubic spline differentiation. mechanical transmission，32(1) (2008)50-51. 188-200 advances in systems science and applications (2011), vol. 11, no. 1-2 issn 1078-6236 international institute for general systems studies, inc. analysis for gene networks of colon cancer based on logical relationships * 1 college of information science and engineering, shandong university of science and technology,qingdao, shandong 266510, china 2 college of applied science, beijing university of technology, beijing 100124, china ３ department of mathematics, southern illinois university carbondale, makanida 62958, usa email: wangshd2008@yahoo.com.cn abstract genome forms gene networks in terms of complicated interaction to realize its functions. the fundamental principle that “the structure of system determines its function” inspires researchers to contrast the differences between the structure of gene networks for experimental and control groups to get useful information of molecular mechanism of the formation of cancer. in this research, the higher-order logical networks are constructed through reverse network modeling based on the expression data of phylogenetic profiles of cancer-related genes in the normal and diseased stages. through contrasting of the network structure between experimental and control groups, such obvious differences in the structural components of networks are found as the number of non-isolated nodes and the logical relationships, the distribution of logic motifs of 2-order. furthermore, the dynamical behaviors of logical networks of the commander genes of experimental and control groups are simulated and analyzed respectively and significant differences in dynamical stability are discovered. when the above method is applied to the expression data of different stages of other cancers, revelatory differences of the variation of reaction modes for the formation and development of cancer can be given, which is useful for the study of the mechanism of the development of cancer. keywords systems biology gene network logical network dynamics 1. introduction the invasion and metastasis of colon cancer is an important reason that would influence the prognosis of the patients and lead to death. in recent years, through studying for the etiology and pathogenesis of colon cancer, it is generally accepted by the biomedical scientists that the formation and development of human‟s colon cancer is a complicated process which involves the alteration of many genes and stages. the process of canceration of the cells contains the stages of initiation, development and diffusion, and every stage involves the activation of oncogenes and the inactivation of the cancer suppressor genes. hence, finding the functional genome related with disease characteristics from many virulence genes, the reaction modes among genes and the dynamic of oncogenes etc. is of great significance to the diagnosis and cure of the cancer and drug design. this is also an important project in the research of *supported by the national natural science foundation of china (grant nos. 60874036, 60503002) and sdust research fund xiuzhu xu 1, shudong wang 1, kaikai li 1, dashun xu 3, dazhi meng1,2 mailto:wangshd2008@yahoo.com.cn app:ds:fundamental%20principle javascript:showjdsw('jd_t','j_') app:ds:contrast http://dict.cnki.net/dict_source.aspx?searchword=profiles app:ds:contrast app:ds:structural javascript:void(0) javascript:void(0) javascript:showjdsw('jd_t','j_') javascript:showjdsw('jd_t','j_') javascript:showjdsw('jd_t','j_') javascript:showjdsw('jd_t','j_') app:ds:process advances in systems science and applications (2011), vol. 11, no. 1-2 189 bioinformatics [1-2] . in recent years, the theoretical methods for studying the pathogenesis of cancers are mostly based on the expression data of phylogenetic profiles [3−11] , and the computational methods have just been developed to detect functional linkages between proteins or genes, such as pearson correlation coefficient, euclidean and hamming distances, mutual information, the hypergeometric distribution and shortest-path analysis [12−14] . however, traditional pairwise relationships cannot adequately illustrate the complexities that arise in cellular networks because of branching and alternative pathways. for instance, shikimate 5-dehydrogenase (cog0169) is present if and only if 3-dehydroquinate dehydratase (cog0757) or 3-dehydroquinate dehydratase (cog0710) is present [15] . in order to further reveal the ubiquitous logical relationships mentioned above in biology, a computational approach-logic analysis of phylogenetic profiles (abbreviated as lapp) was proposed for identifying detailed relationships among proteins on the basis of genomic data, which was applied into 4873 distinct orthologous protein families of 67 fully sequenced organisms, and identified 750, 000 triplets from analysis of the original, unshuffled biological protein profiles [15] . lapp can reveal the causal or restricted relationships in complex network, help biologists to understand the unknown logical relationships among genes or proteins, and further discover the new biological mechanisms through experiments. as an application of triplet logic analysis to 85 diffuse infiltrating gliomas quantified using oligonucleotide arrays, the meaningful sets of genes that matched clinical outcomes were explored [16] . a bayesian modeling framework for combining phylogenetic profile data via a likelihood with rosetta stone data via a prior probability was described [17] . a three-way gene interaction model was proposed that captured the dynamic of co-expression relationships between two genes [18] . a logical network with 16 active genes of shoot in different external stimuli was constructed, and the dynamics of the network was analyzed [19] . in this work, the logical networks of the normal and diseased stages are constructed through lapp method based on the expression data of phylogenetic profiles of cancer-related genes. through contrasting of the network structure between experimental and control groups, such obvious differences in the structural components of networks are found as the number of non-isolated nodes and the logical relationships, the distribution of logic motifs of 2-order. the differences inspire biomedical researchers: the reason that organism is normal in the normal stage might be that the biochemical reaction mode, the extent of gene relationships and most of the genes play normal roles. furthermore, the dynamical behaviors of logical networks of the commander genes of experimental and control groups are simulated and analyzed respectively and significant differences in dynamical stability are discovered that the number of attractors for control group is far less than the one of the experimental groups. this paper is organized as follows: in the first part, the background of our work and the present researching situation of logical network of genes. in the second part, the method of lapp is clarified briefly and the data sources and the processing methods of the data are given in the third part. the logical networks corresponding to experimental and control groups are constructed using lapp method respectively and the structural components of networks are contrasted and analyzed. the dynamical attractors of logical networks of the commander genes in the experimental and control groups are contrasted respectively in the fifth part. the above results are analyzed and discussed, and the problems to be solved further are put forward in the sixth part. 2. the method of lapp in order to understand and predict the functions of biological systems, we must first identify the structure of the system. for example, in order to show the regulatory relationships among the genes, we must identify all the components, the functions of every component, their interrelations and all the parameters related in the system. now, there are two main methods to http://dict.cnki.net/dict_source.aspx?searchword=profiles http://dict.cnki.net/dict_source.aspx?searchword=profiles app:ds:contrast app:ds:structural javascript:void(0) javascript:void(0) javascript:void(0) app:ds:structural app:ds:contrast app:ds:contrast build the model of biological system structure: forward and reverse network modelings. reverse logical network modeling is a method of reverse network modeling and can determine the logical relationships and types among elements of systems. in this work, we identify the logical structure of cancer-related genome using lapp method based on the data of gene expression profile. we entitle the sample data of gene obtained in many different experiments as its expression profile. for the convenience of writing and computing, we also denote the sample data by the genes. for example, the logic of genes a and b indicates the logical relationships of expression data of genes a and b. lapp method is determining the logical relationships of 1-, 2-, and 3-order by calculating the uncertainty coefficient (abbreviated as u ) of logical functions of every order among the genes based on the expression profiles of genes. the 1-order logical relationship between genes a and b is determined as follows:            )( , | 11 1 bh afbhafhbh afbu   , (1) where h refers to the entropy of the individual or joint distributions. )( xyu denotes the uncertainty coefficient of the influence of x on y . the size of u value denotes the statistical possibility of the uncertainty logic of 1-order of x to y . 1f is one of the proper functions of 1-order logic of a to b. the proper functions of 1-order logic are divided into two types (table 1): ab  , namely, the presence of a leads to the presence of b , which is called the synchronization (equivalent with ab  ); and ab  , namely, the presence of a leads to the absence of b , which is called the asynchronization (equivalent with ab  ). table 1 the list of the proper functions and logic types of 1-, 2-order. obviously, the proper functions of 1-order and logic types are the same. order the proper function and its serial number the logic type and its serial number 1-order 1．b a 1．b a 2．b a 2．b a 2-order 1． bac  1． bac  2． )( bac  2． )( bac  3． bac  3． bac  4． )( bac  4． )( bac  5． bac  6． bac  5． bac  bac  7． bac  8． bac  6． bac  bac  9． )( bac  7． )( bac  10． bac  8． bac  similarly, the expressions of 2-, 3-order logics are as follows:             ch bafchbafhch bafcu ,,, ,| 22 2   , (2)             dh cbafchcbafhdh cbafdu ,,,,, ,,| 33 3   , (3) where 32 , ff are one of the proper functions of 2-, 3-order logics respectively. the logics of 190 xu:analysis for gene networks of colon cancer based on logical relationships 190 xu:analysis for gene networks of colon cancer based on logical relationships 190 xu:analysis for gene networks of colon cancer based on logical relationships 190 xu:analysis for gene networks of colon cancer based on logical relationships 190 xu:analysis for gene networks of colon cancer based on logical relationships http://www.iciba.com/expression/ 191 2-order among a, b and c contain 10 proper functions and 8 logic types (table 1). the logics of 3-order among a, b, c and d contain 218 proper functions and 68 logic types. of course, the higher order logics can be calculated, but the complexities are higher. in this research, we only consider 1-, 2-, and 3-order boolean logics among genes. the detailed can be found in references [15]-[17] . 3. data source and processing 3.1 data source national center for biotechnology information (ncbi) and the european bioinformatics institute (embl-ebi) provide the database generally used in field of bioinformations. for description convenience, we refer to the sample data of stage a, b and c of colon cancer as experimental groups, normal stage as control group. the sample data is selected from gpl570 in ncbi. the sample numbers of stage a, b and c are 39, 103, and 92 respectively. they are all from gse2109 in gpl570. the 53 samples of normal stage consist of 10 samples of gds2609, 11 samples of gse10715 and 32 samples of gse8671. the data mentioned above include p_value and p-m-a, where p, a and m indicate presence, absence and margin respectively. we represent p with 1, a and m with 0. every sample database includes p_values and 0-1 expression profiles of 20827 genes of human (corresponding to 54676 probes). the four databases (normal, stage a, b and c) mentioned above form the original database used in this work. 3.2 selection of cancer-related genes in the original database mentioned above, every database includes the sample data of more than 20,000 genes of human genome. to construct and analyze the logical networks of more than 20,000 genes, the computational complexity is so high that it is beyond our computational ability. in order to simplify the computation, take the genes corresponding to 801 probes into account according to the known cancer-related genes [21] . if several probes correspond to the same gene, then we choose the data expressing the most in probes as the expression profile data of the gene. thus, 286 cancer-related genes in the original database are obtained. because we are concerned about the structure of the gene network, among these 286 cancer-related genes, the data almost 0 or 1 contribute very little to the difference of the network structure. hence, we delete these genes in whose expression profile the numbers of 1 are less than 15% or more than 90%. and then, the numbers of the remaining cancer-related genes of these four databases is normal: 91, stage a: 79, b: 70, c: 60 respectively. after the above processing, there are some genes with the same 0-1 expression profiles, which contribute equally the construction of network. hence, we preserve one of them, and delete other genes with the same expression profiles. thus, the number of genes in normal database is reduced to 79. after simplifying the samples treated as above, the obtained databases are the working databases. 4. numerical experiments, methods and results 4.1 determination of every order logics based on the above working databases, all the 1-order u values between genes are calculated respectively by the formula (1). obviously, there are 1-order u values of two directions between any two genes, denoted by  abu and  bau respectively. if the absolute value of the ratio of the difference between the two directions and the average value is less than advances in systems science and applications (2011), vol. 11, no. 1-2 http://www.iciba.com/contribute/ http://www.iciba.com/absolute%20value/ http://www.iciba.com/less/ certain threshold d , namely,         2 b a a b b a a b u u d u u    , then by the certainty of logical relationship, there is not 1-order logic between the two genes. only when         2 b a a b b a a b u u d u u    and the u value is greater than 1-order threshold, there is 1-order logic between genes a and b. if    baab uu  , then we consider that a regulates b, otherwise, b regulates a. by the formula (1) in the algorithm of u value,    a|ba|b uu , which means that the logic type between a and b might be ba  or ba . then we determine the logic type between a and b according to the support of ba  and ba , namely, the logic type with greater support is the one between a and b. here, the support refers the ratio of the number of the two states being 1 in the expression profile data of the genes and the total number of samples. for instance, suppose the expression profiles of the genes m and n are  0,0,0,1,1 and  1,0,0,1,1 respectively, then the supports of logic type m=n and m= n are 5 2 and 5 1 respectively. take the threshold of 1-order 3.01 u , then 143.0)m|n()m|n( uuu  , the support m=n m= n 2 1 5 5 s s    , thus we think the logic type of genes m and n is m=n . the proper functions of 2-, 3-order among genes in every database are determined by the formula (2), (3) and the support. that is to say, for any three genes a, b and c, calculating the u values corresponding to the 10 proper functions, then, the function 2f corresponding to the maximum u value is the proper function of a, b to c. similarly, for any four genes a, b, c and d, calculating the u values corresponding to the 218 proper functions, then, the function 3f corresponding to the maximum u value is the proper function of a, b , c and d. if the u values corresponding to several proper functions are the maximum value, then take the proper function with greater support as the one among genes (similarly to determining the logic type (or proper function) of 1-order). by the above numerical calculation, the complete directed logical networks corresponding to four databases of experimental and control groups, namely, each network contains all the proper functions of 1-, 2-, 3-order of its own database. 4.2 determination of threshold of every order in order to highlight the characteristics of the network and obtain the valuable biological information, we need to choose a threshold to carry out coarse graining to the complete directed logical networks mentioned above. the threshold is the expression of coarse-grained level and the structural difference between logical network structures should be contrasted in the same coarse-grained level. in order to make the logical network structures for experimental and control groups comparable, we normalize the u values of 1-, 2-, 3-order of the four networks, namely: logic u values of every order of each database are replaced with u max u , where max u is the maximum u value of each order in the own database. for simplicity, the normalized values are also denoted by the u values. let the thresholds of 1-, 2-, 3-order be 321 ,, uuu respectively. the 2-order logic of a , b to c is considered only when the 1-order logics of a to c and b to c do not exist, and claim that 192 xu:analysis for gene networks of colon cancer based on logical relationships http://www.iciba.com/consider/ 193 2u is greater than 1u :                12 11 11 22 )( )( )),(( uu ubfcu uafcu ubafcu ， (4) the 3-order logic of a , b , c to d is considered only when the 1-order logics of a to d , b to d , c to d , and the 2-order logics of any two of a , b ,c to d do not exist, and 3u is not less than 2u :                  23 111111 222222 33 )(&)(&)( )),((&)),((&)),(( )),,(( uu ucfduubfduuafdu ucbfduucafduubafdu ucbafdu ， (5) through the above numerical calculation, the four logical networks for experimental and control groups are obtained (fig. 1). the detailed situations can be seen in table 2. (a) normal (b) stage a (c) stage b (d) stage c fig. 1 taking 0.42, 0.75 and 0.85 as the threshold of 1-, 2-, 3-order respectively, the logical networks of 111 genes of the four databases are located, where (a), (b), (c) and (d) represent the logical networks corresponding to normal, stage a, b and c respectively. the nodes in the networks are divided into two types: the node labeling i denotes gene i, the node labeling “j(. . .)” (j = 2, 3) denotes the intermediate node of j-order logical relationships. for example, “2(5_1)” denotes logic type 5_1 of 2-order. the out-degree and in-degree of the intermediate nodes are fixed, i.e. when j =2, the in-degree and out-degree of the node is 2 and 1, respectively; when j = 3, the in-degree and out-degree of the node is 3 and 1, respectively. for instance, the node “2(3)” in fig. 1(b) denotes that gene cxcl and mmp2 regulate gene extl3 through logic type 3 of 2-order. table 2 the detailed list of the logical networks for control and three experimental groups stage no. of 1-order logics no. of 2-order logics no. of of 3-order logics no. of all the logics no. of genes no. of the non-isolated nodes normal 90 1614 39 1743 79 74 stage a 22 33 18 73 79 60 stage b 37 51 2 90 70 55 stage c 4 32 9 45 60 35 4.3 distribution of logic motifs of 2-order the subgraph with higher frequency in complex network is called the network motif [22] . the identification to the motifs of complex network helps to identify the typical local connections advances in systems science and applications (2011), vol. 11, no. 1-2 of network. previous studies have showed that: the motif in complex biological network has a direct biological significance. for example: the motif in the protein interaction networks of yeast highly evolves to protect the components, the transcription regulatory networks of different species have an evolutionary trend to the same motif [23] . in the logical networks, each of 8 logic types of 2-order is corresponding to a motif, called the logic motif. each type of logic motifs is corresponding to a typical mode of biochemical reactions. in order to explore the differences of reaction modes in the internal mechanism of normal and diseased organisms, we contrast and analyze the logic motifs of 2-order of the logical networks for experimental and control groups. fig. 2 and table 3 are the distribution histogram and the detailed list about logic motifs of 2-order of the four logical networks for experimental and control groups respectively. from fig. 2 and table 3, we can discover that the logic motifs of 2-order in control network cover six logic types 1, 2, 3, 4, 5, 6. however, in the networks for experimental groups, there only contain three logic types 1, 3, 6 (table 3) except that the network corresponding to stage a contains logic types 1，2，3，5，6，7 of 2-order. moreover, three logic motifs of types 1, 3 and 6 appear more in the network for control group and the corresponding proper functions are bac  , c a b  and bac  or c a b  , respectively; one logic motif of type 1 appears more in the network for stage a and the corresponding proper function is bac  ; two logic motifs of types 1 and 3 appear more in the networks for stage b and c and the corresponding proper functions are bac  , respectively. 1 2 3 4 5 6 7 8 0 100 200 300 400 500 600 logical type t h e n u m b e r o f l o g ic t y p e stage a stage b stage c normal 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 x 1 0 -3 logical type l o g ic a l ty p e f re q u e n c y stage a stage b stage c normal fig. 2 the distribution histogram of logic types of 2-order in the logical networks for control and three experimental groups table 3 the list of logic types of 2-order in the logical networks for control and three experimental groups logic type 1 2 3 4 5 6 7 8 normal 275 177 311 78 199 574 0 0 stage a 13 1 5 0 2 7 5 0 stage b 21 0 28 0 0 1 1 0 stage c 12 0 17 0 0 3 0 0 4.4 dynamical analysis of logical network although the study and analysis of topological structure of the above logical networks can obtain some biological principles and information, the structure of actual biological system is a changing network over time and the dynamical behaviors cannot be reflected by topological structure in most situations. therefore, we research the evolution of gene regulatory logical network from the dynamical point of view. because the number of the non-isolated nodes are too great in the logical networks corresponding to experimental and control groups to analyse the dynamical transfer laws, “the structural key genes” must be selected appropriately in every stages. the logical network is directed and every node in the network has out-degrees and in-degrees. if node i has an out-arc to j , then node i has a logical regulatory relationship to j . on the other hand, if node i has an in-arc, then some node has a logical regulatory relationship to i . the degree-difference of node is defined as the difference of out-degree and in-degree. the nodes 194 xu:analysis for gene networks of colon cancer based on logical relationships app:ds:on app:ds:the app:ds:other app:ds:hand 195 whose degrees and degree-differences are relatively great are called “the commander genes”, and the changes of states of these genes can influence most of genes of the network. in this work, the method of discovering the commander genes is as follows: (1) calculate the maximum degree and the maximum degree-difference of network; (2) normalize the the maximum degree and the maximum degree-difference of the non-isolated nodes respectively, namely: the degree and degree-difference of each non-isolated node divide the maximum degree and the maximum degree-difference of the network respectively. (3) set the parameters 1 0.05c  and 0.15c 2 . take these genes with degree 1c times greater than the maximum degree and degree-difference c2 times greater than the maximum degree-difference as the commander genes. when taking the parameters 1c and c2 , the number of commander genes must be appropriate, because some key genes might be lost if the commander genes are too few, and we cannot calculate if the commander genes are too many. the detailed situations of the commander genes of control and three experimental groups can be seen in table 4 and 5. table 4 taking the parameters 1 0.05c  and 0.15c 2 , the detailed list of the commander genes of control and three experimental groups stage the maximum degree the maximum degree-difference no. of the commander genes no. of the non-isolated nodes normal 855 109 12 74 stage a 30 15 12 60 stage b 42 6 12 55 stage c 32 5 13 35 table 5 the list of the commander genes of every stages normal stage a stage b stage c faslg araf araf ar axl bok bak1 bak1 etv3 esr2 esr1 elk1 gli3 ptk2b ptk2b erbb4 il1a gli2 fgf2 etv1 plag1 mafg hnf4a fgf2 rbl1 mdm2 ihh ihh tal1 pdgfb abcb1 mapk7 vav1 plag1 shh sele zap70 stk11 vav1 stk11 tcl1a tp73 arhgef5 tp53bp1 bag2 hpse hpse zap70 chek2 using the method of constructing the logical networks in (1) (2) of section 4, we can obtain the networks of the commander genes of control and three experimental groups (fig. 3) advances in systems science and applications (2011), vol. 11, no. 1-2 app:ds:relatively app:ds:influence app:ds:maximal app:ds:maximal app:ds:normalization app:ds:maximal app:ds:maximal app:ds:divide app:ds:maximal app:ds:maximal app:ds:maximal app:ds:maximal app:ds:maybe app:ds:maximal app:ds:maximal (a) normal (b) stage a (c) stage b (d) stage c fig. 3 the logical networks of the commander genes of control and three experimental groups. now we begin to analyse the dynamics of the above logical networks of the commander genes. assume the network have n genes. let the state of gene i at moment t be }1,0{)( txi , and the state-vector of gene i at moment 1t be ))1(),...,1(),1(()1( 21  txtxtxts n . without loss of generality, the triple set comprised of the u values, the corresponding proper functions of 1-order and the genes affecting gene i at moment t is 1 1 {( , , ) {1,2,..., }, }k i kz k u f k n k i   , which denotes that gene k regulates gene i by the proper function 1 kf and the uncertainty coefficient is iku  ; the triple set consisting of the u values, the corresponding proper functions of 2-order and the genes affecting gene i is 1 2 1 2 2 2 1 2 , , 1 2 1 2 1 2{(( , ), , ) , {1,2,..., }, , , }k k i k kz k k u f k k n k i k i k k     , which denotes that gene 21,kk regulate gene i by the proper function 1 2 2 ,k kf and the uncertainty coefficient is ikku 21 , ; the triple set consisting of the u values, the corresponding proper functions of 3-order and the genes affecting gene i is 1 2 3 1 2 3 3 3 1 2 3 , , , , 1 2 3 1 2 3 1 2 1 3 2 3{(( , , ), , ) , , {1,2,..., }, , , , , , }k k k i k k kz k k k u f k k k n k i k i k i k k k k k k        , which denotes that gene 321 ,, kkk regulate gene i by the proper function 1 2 3 3 , ,k k kf and the uncertainty coefficient is ikkku 321 ,, . in this research, we consider that the state of gene i at moment 1t is related to the states of itself and its adjacent nodes. at moment t , let genes 1 2, , , jk k k regulate gene i by proper function 1 2, , , j j k k kf  （  3,2,1j ）. if the truth-value of proper function 1 2, , , j j k k kf  with independent variable being the state of genes jkkk ,,, 21  at moment t is 1, then these j genes jkkk ,,, 21  activate the expression of gene i . let 1 2, , , sign ( ) 1 jk k k i t    . otherwise, these j genes jkkk ,,, 21  repress the expression of gene i . let 1 2, , , sign ( ) 1 jk k k i t     . at moment t , for gene i , let )(sign)(sign)(sign)()( 3213 321 212 21 1 ,, ,, , , tutututxtp ikkkz ikkk ikkz ikk ikz ikii         . (6) then, the transition rule is denoted by 196 xu:analysis for gene networks of colon cancer based on logical relationships app:ds:consist app:ds:of app:ds:consist app:ds:of app:ds:adjacent%20points app:ds:truth-value app:ds:independent%20variable 197       ,5.0)(,0 ,5.0)(,1 )1( tp tp tx i i i ni ,...,2,1 . (7) according to the above transition rules, the state transition configuration network for control and three experimental groups are obtained. the detailed situations are listed in table 6, fig. 4 and fig. 5. in the networks corresponding to control and three experimental groups, the numbers of nodes (genes) have certain differences. in order to eliminate the differences caused by the number of attractors, the stability of system is measured by the frequency of attractors instead of the number of attractors. table 6 the situations of attractors of control and three experimental groups stage no. of genes no. (frequency) of 1-periodic attractor no. (frequency) of other attractor no. (frequency) of all the attractors normal 12 262（6.40%） 22（0.54%） 284(6.94%) stage a 12 594（14.50%） 6（0.15%） 600(14.65%) stage b 12 407（9.94%） 0（0%） 407(9.94%) stage c 13 1812（22.12%） 116（1.42%） 1928(23.54%) fig. 4 2-periodic domain of attraction with the most nodes (46 nodes) of control group, where the node is labeled by the decimal number corresponding to the binary state vertor. (a) normal (b) stage a (c) stage b (d) stage c fig. 5 in the state transition configuration networks for control and experimental groups, the 2-periodic domain of attraction with the most nodes, where the transition networks for normal, stage a, b and c contain 111, 42, 40, 16 nodes respectively. 5. conclusions and discussions in this work, we construct the logical networks using lapp method based on the expression data of phylogenetic profiles of cancer-related genes in normal, stage a, b and c of colon cancer . through contrasting and analyzing of the structures of four networks corresponding to advances in systems science and applications (2011), vol. 11, no. 1-2 app:ds:eliminate app:ds:attractor app:ds:stability app:ds:measure app:ds:decimal%20number app:ds:binary http://dict.cnki.net/dict_source.aspx?searchword=profiles app:ds:colon%20cancer app:ds:colon%20cancer app:ds:colon%20cancer app:ds:contrast experimental and control groups, we find the significant differences in the structural components (fig. 1 and table 2). (1) the significant differences in the number of non-isolated nodes: in the network for control group, the number of non-isolated nodes is 74; in stage a, b and c of colon cancer, the numbers of non-isolated nodes are reduced obviously, moreover, the fewer the numbers of non-isolated nodes are, the more cancer deteriorates. (2) the significant differences in the the number of logical relationship: the number of logics of the network for control group is far more than the one of the three experimental groups. in the complex networks, the number of non-isolated nodes reflects the intensity of correlation relationship of nodes and the number of the relationships of nodes indicates the richness of the relationship of nodes of network. in the complex networks, most of genes in the normal stage play cancer suppression roles with other genes cooperating perfectly. for some (internal and external) reasons, the regulatory relationships of several genes with some other genes are lost completely or the regulatory pathways malfunction seriously. the normal tissue appears abnormal; the organism suffers from the colon cancer. with the intensifying of the failure of regulation and broadening of the failure propagation, the illness increase gradually. (3) the significant differences in the distribution of logic motifs of 2-order (see fig. 2 and table 3): in the network for control group, there are three logic motifs: type 1, 3 and 6; in the network corresponding to stage a, there are one logic motif: type 1; in the networks for stage b and c, there are two logic motifs: type 1 and 3. we find from the above results that the logic type 1 appears in the networks for control and experimental groups, yet the logic type 6 appears only in the network for control group. in this research, we consider that logic types (motifs) of 2-order are corresponding to typical modes of biochemical reactions in life actions. the above phenomena might show that logic type 1 ( bac  ) is the basic mode maintaining human existence and physiological metabolism reactions, yet logic type 6 (c a b  or c a b   ) is the reaction mode suppressing the normal organism not suffering from the colon cancer. the materials [24] have showed that logic type 1 does appear frequently in the complex biochemical reactions of the human body. for example: the coupling protein x and y plays a regulatory role in the expression of protein z, where x, y and z are the specific proteins. in conclusion, the logical networks for control and experimental groups are the models of structural mechanism of the normal organism suffering from the colon cancer gradually. the organism is normal in the normal stage because the biochemical reaction mode (type 6), the extent of genes relationship (the number of logical relationship) and most of the genes (the number of non-isolated nodes) play normal roles. with the change of the reaction mode of the organism and the decreasing of the number of acting genes and their relationships, many intrinsic biological functions are weaken or lost gradually, thus the organism suffers from the colon cancer. however, the significant differences between the networks for control and experimental groups mentioned above are the causes or the consequences of the organism suffering from the colon cancer, which will also need to be verified by the clinical experiments and annlysis of biomedical researchers. table 5 shows the commander genes of control and three experimental groups. we can find that: three genes plag1, vav1 and zap70 appearing in control group are also active in the experimental groups, and the three genes might regulate the basic survival activities of human. the material shows: gene vav1 as a kind of signal transduction molecule can happen quickly tyrosine phosphorylation with ifn-α, il-3, gm-csf, growth factor or antigen receptors after the activation and plays an important role in the process of cell differentiation, proliferation [21] . gene zap70 plays a very important role of transfer signal in the process of activation of t-cell mediated by cd3 and/or cd3 chain [21] . other genes faslg, axl, etv3, gli3, il1a, rbl1, tal1, tcl1a, bag2 of control group do not appear in the experimental groups and they might be the suppressor genes of colon cancer in human genome. furthermore, genes araf，bak1，ptk2b，fgf2，ihh，stk11，hpse appear in at least two stages of the 198 xu:analysis for gene networks of colon cancer based on logical relationships app:ds:significant app:ds:difference app:ds:significant app:ds:difference app:ds:significant app:ds:difference app:ds:reflect app:ds:indicate app:ds:tissue app:ds:organism app:ds:significant app:ds:difference app:ds:biological%20phenomena app:ds:suppressor app:ds:organism app:ds:material app:ds:specific app:ds:mechanism app:ds:organism app:ds:gradually app:ds:organism app:ds:extent app:ds:normal app:ds:change app:ds:organism app:ds:decrease app:ds:acting app:ds:biological app:ds:weaken app:ds:gradually app:ds:organism app:ds:significant app:ds:difference app:ds:mention app:ds:organism app:ds:clinical%20experiment app:ds:active app:ds:material app:ds:growth%20factor 199 experimental groups and not in control group, thus the seven genes might be the oncogenes of colon cancer. the facticity and reliability predicted in the above will also need to be verified by biological experiments. in the process of the dynamic analysis of logical network of the commander genes, we find from the numerical experiments that the number of attractors of control and three experimental groups has obvious difference (table 6). the former is far less than the latter. according to the principle and theory of dynamics: the less the number of attractors is, the more stable the system is. from the view of biology, the construction should be more stable in the normal organism than the stages of colon cancer. the views from two points are just coincident. thus, we conclude that the instability of the commander genes of organisms might be an important reason or result in the process of the colon cancer disease. we utilize lapp to search 1-, 2and 3-order logical relationships and logic types among the genes or proteins. using lapp to determine the logical 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[24] fuchu he, pengyuan yang, yunping zhu. systems biology in practice: concepts, implementation and application. fudan unversity press, shanghai, 2007.12. 200 xu:analysis for gene networks of colon cancer based on logical relationships http://www.cqvip.com/qk/94860x/200811/28883778.html adv syst sci appl 2019; 03; 140-162 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/676 scada based remote monitoring and control of pressure & flow in fluid transport system using imcpid controller e.b.priyanka1*, c.maheswari2, m.ponnibala3, s.thangavel4 1) senior research fellow, department of mechatronics engineering, kongu engineering college, perundurai, india. e-mail: priyankabhaskaran1993@gmail.com 2) faculty, department of mechatronics engineering, kongu engineering college, perundurai, india. e-mail:maheswari@kongu.ac.in 3) faculty, department of mechatronics engineering, kongu engineering college, perundurai, india. e-mail:ponnibala@kongu.ac.in 4) faculty, department of mechatronics engineering, kongu engineering college, perundurai, india. e-mail:thangavel.mts@kongu.ac.in received may 5, 2018; revised september 9, 2019; published october 1, 2019 abstract: ubiquitous sensing enabled for monitoring and control in the industrial sector are mostly wsn (wireless sensor network) based systems or scada systems. wsn based systems are homogenous or compatible systems and they support coordinated communication and transparency among regions and processes. hence in order to enhance the proprietary of remote monitoring and control technology, this paper emphasizes the development of architecture with local intelligent using imc based pid controller can be integrated with dcs/scada and this research work spectacle only the development and application of local intelligence architecture for remote monitoring and control of concerned field parameters. this proposed local intelligence tactic governs the input and output from a hardware platform positioned on a process plant, organize added processing power for meticulous analysis in the control center by the use of software integration, data acquisition and logging on a created hybrid database, and spectacle significant processed post-data statistics to operators via a standard dcs/scada hmi. the established local intelligence is experimentally validated in the lab scale experimental fluid transport system to monitor and control pressure and flow rate parameters remotely by incorporating with centum cs 3000.the simulation and experimental results of local intelligence using an imc-pid controller on a lab scale fluid transport system are conveyed with its numerical data. keywords: local intelligence, imc based pid controller, remote monitoring & control. 1. introduction now-a-days, supervisory control and data acquisition (scada) systems [1, 8] are not climbable due to derisory memory and processing time, offers high cost of hardware drivers and management, not malleable when a process plant hardware needs to be updated, inflexible when there is a need for protocol change and software updating, and provides data and result with long delay. hence, human interfaced scada [4, 5] and the engineering internet are reconnoitered towards the improvement of modules used to identify the manifestation of impairment. in this upgradation era, all renovated industries has been assured using the emerging methodology remote monitoring and automation [19] which permit a pervasive bonding between the process plant hardware (sensors, actuators, control panel, etc.), smart hybrid objects and networked implanted devices to accumulate applicable data that will be * corresponding author: priyankabhaskaran1993@gmail.com scada based remote monitoring and control of pressure & flow rate 141 copyright ©2019 assa. adv. in systems science and appl. (2019) communicated wireless manner to the remote information analytics center to be examined and modulated in order to construe the process and afford the appropriate decision in case of calamitous circumstances [7, 11]. this paper emphasizes a reliable monitoring and control with local intelligence using imcpid controller modular architecture design along with scada/dcs. this architecture with local intelligence using imc-pid controller facilitates minimum human intervention will provide better workplace safety, maintenance of assets and will perform predictive maintenance of various industrial assets by analyzing various parameters (sensed data) and detecting failure modes either before they are going to take place or when the equipment will likely to fail or need service. this paper enlightens the brief information on the development of local intelligence using an imc-pid controller which is present in the proposed iot architecture. to overcome the limitations of conventional remote monitoring and control systems [2, 3, 6], the proposed design of local intelligent offers a new utility monitoring and control system can amenably put up variations made to progress the functioning efficacy using an imc-pid controller. the imc pid tuning rubrics have the benefit of utilizing only a solitary tuning parameter to accomplish a vibrant trade-off between the closed-loop enactment and robustness. over the unified incorporation of self-governing practicalities in a integrated checking and control system, an complete process system can be supervised and functioned in real time from the dcs/scada hmi through this local intelligence incorporated with imc-pid controller which creates conceivable based automatic activation of tuning window block with trends, substantively enlightening engineering proficiency and working safety by exploiting centum cs 3000. 2. lab scale experimental setup of fluid transport system the lab scale experimental fluid transport system consists of the pumping unit, the transmitting pipes having a diameter of 1 inch about 15m long for the well-organized and extended transmission and manifold analog sensors at different distinctive locations with its corresponding control valves installed to compensate its drops/loss during the fluid transportation in the pipelines. the block diagram with local intelligence for the lab scale investigational setup envisaging the physical categorizations of apparatus positioned and also exhibits all the piping with directional tracks and equipment’s facts with their controls is shown in figure 1. the process plant consists of two sections consisting of an electric pump, differential pressure transmitter holding a range of 03-15psi (0.1-3kg/cm2), and pressure control valve at one section along with orifice flow meter having the limit of 0-1800lph, pump and flow control valve at another section. when the process plant is on track to run, initially reservoir tank fills up to 20% of its capacity, and then the electric pump is actuated to suck the fluid from the tank in order to transmit the fluid to each section. when the fluid starts to pass through the pipelines its corresponding pressure and flow transmitter send its present pressure and flow rate data to the i/o hub module station. during transportation to regulate the pressure and flow rate of the transmitting fluid, the opening and closing of the control valves are operated by the i/p converter in order to maintain its preferred effective range limits until the destination of the long run, till it drains from the process tank as shown in figure 2. the local intelligence will take the role to regulate and control the process plant parameters before it reaches the state of over limit/threshold. since this paper encompass on the development and application of local intelligence using imc-pid controller, the designed local intelligence can remotely monitor and control the pressure and flow rate as a separate loop through dcs/scada hmi which can be operated as stand-alone station without depending to a central server with mutual backup configuration by regulating the operation of the corresponding control valves to reach its desired operating points of pressure and flow rate. 142 e.b.priyanka, c.maheswari, m.ponnibala, s.thangavel copyright ©2019 assa adv. in systems science and appl. (2019) fig. 1.block diagram with local intelligence for the process plant. fig. 2.the lab scale experimental setup of the fluid transport system 3.functional deployment of local intelligence scada frame work the scada incorporating local intelligence using an imc-pid controller is developed for the experimental setup of the fluid transport system is shown in figure 3 to deal with remote surveillance and control of flow and pressure and it also encompasses of individual interface displays for flow rate and its respective pressure parameters of the fluid as shown in figure 3 input/ output hub module input ports pressure-ios flow-ios pressure transmitter pneumatic actuated control valve i/p converter flow transmitter pneumatic actuated control valve i/p converter pump 2 pump 1 output ports field control station (fcs) high level engineering interface with controller engineering station oil station 2 lab scale setup scada based remote monitoring and control of pressure & flow rate 143 copyright ©2019 assa. adv. in systems science and appl. (2019) and 4. the developed scada provides more functional utility options like system message banner, graphic view with graphics and control attributes, trend view, browser bar and tuning window. the system message indication window articulates the alarm existence eminence visually. the alarm existence eminence is presented by colors and blinking of action buttons, and the message presentation. the system message indication window is constantly shown at the header of the scada hmi, so will never be veiled overdue by other windows. the browser bar is utilized to call up process and manipulative windows to entact the scheduled tasks. it also spectacles a list of task oriented manipulating windows and process hierarchical configurations in a tree-like manner, enabling the whole process plant to be effortlessly functioned. the graphics attributes view tool parades plants conditions graphically and can be instinctively activated and monitored. the control attribute graphic window shows the function block eminences by elaborating instrument faceplates. fig. 3.scada view of the lab scale experimental fluid transport system 4. system modeling for pressure and flow control loop 4.1.open loop response analysis for modeling the fluid transport system, a transient response curve is chronicled by modifying the control valve opening in order to acquire the equivalent liquid pressure and flow rate changes on the pipeline in the open loop structure. this open loop experimentation reveals pressure of the liquid is at a maximum rate and the flow is at a minimum value when the 144 e.b.priyanka, c.maheswari, m.ponnibala, s.thangavel copyright ©2019 assa adv. in systems science and appl. (2019) opening of the control valve is around 10% of the overall opening and vice versa when the control valve opening reaches its full stretch of 100%. figure 4.individual scada view of pressure and flow control loop the open loop test run is carried out by linearly adjusting the percentage of control valve opening. the experimental analysis call ups with the initial flow rate of 179 lph and pressure of 2.2 kg/cm2 and unremitting readings were documented until the flow rate and pressure influences a balanced/fixed state. the result divulges that for 100% control valve opening the ultimate flow rate and pressure accomplished are 1792 lph and 0.22 kg/cm2. open loop readings were noted for the percentage of control valve opening versus flow rate and pressure through centum cs 3000 as displayed in figure 5 and obtained data are presented in table 1 by which the first order model considerations (process gain kp and process time constant p) are calculated. the evaluated model parameters for pressure and flow rate are given in table 2. . table 1.open loop response analyses of pressure and flow for a different level of control valve openings to enumerate first order strictures such as process gain kp, process time constant ʈp and time delay 𝜃, with 𝐾𝑝 = [initial value − final value of parameter]/ [maximum value of parameter] [initial value of opening − final value of opening]/ [maximum value of control valve opening] percentage of control valve opening (in %) flow rate (in lph) pressure (in kg/cm2) 10 179 2.20 20 230 1.92 30 373 1.83 40 468 1.71 50 585 1.48 60 757 1.21 70 919 0.91 80 1137 0.55 90 1429 0.36 100 1792 0.22 scada based remote monitoring and control of pressure & flow rate 145 copyright ©2019 assa. adv. in systems science and appl. (2019) ʈ𝑃 = 1.5 ∗ [t2 − t1] (4.1) fig. 5. pressure and flow rates of the liquid for the different level of control valve opening t1 at a1 = [initial value of parameter((initial value of parameter– final value of parameter)*0.632)] t2 at a2 = [initial value of parameter ((initial value of parameter– final value of parameter)*0.283)] θ = (𝑇2 − ʈ𝑃) (4.2) 146 e.b.priyanka, c.maheswari, m.ponnibala, s.thangavel copyright ©2019 assa adv. in systems science and appl. (2019) table 2.recognized model parameters at different % of control valve opening percentage of control valve opening (%) flow rate pressure kp ʈp (sec) 𝛉 (sec) kp ʈp (sec) 𝛉 (sec) 20 0.329 11.941 7.91 1.681 4.95 13.58 40 0.485 8.317 12.83 1.364 6.43 22.84 60 0.914 9.084 15.72 1.649 4.026 35.59 80 0.046 13.51 9.89 1.024 9.88 21.91 100 0.871 17.46 5.16 0.995 14.32 29.92 4.2.model identification for pressure and flow control loop from table 2, it shows that the behavior of a process is nonlinear and stable. hence, first order plus time delay (foptd) transfer function ( g(s) = kp ps+1 𝑒−s ) model is used to exemplify the pressure and flow rate maintenance of fluid transport system, where kp = process gain, p = time constant and  = process delay. then, the worst case model with the leading process gain and lowest time constant is designated to epitomize the process plant model [12, 13]. the identified foptd model for the flow control loop in the process plant is represented as, g(s) = 0.914 8.317s+1 𝑒−5.16s (4.3) similarly the foptd model for the pressure control loop is exemplified as, g(s) = 1.681 4.026s+1 𝑒−13.58s (4.4) the system model identification is arrived by formulated the real-time experimental data obtained from the fluid transport system in open loop performance analysis on process plant [16]. 5. robust controller design the pid (proportional-integral-derivative) controller serves as one of the popular and extensively applied controllers in the industrial sector due to its uncomplicatedness, robustness and affords wide applicability to near-optimal enactment. even though innovative control techniques can deliver substantial improvements, an elegant-characterized pid controller has ascertained to be suitable for an enormous quantity of engineering control loops. a pid type controller is used to optimize the performance of the control valve in edict to regulate and uphold pressure and flow-rate of the fluids being transported in the oil pipeline system. the proportional term ensures the trade of fast-acting modification which makes necessary alteration of output as rapidly as the error ascends. the integral part proceeds after a finite time but has the proficiency to create the steady state error as zero and the derivative term can advance proficiently the control loop performance. 5.1 ziegler-nicholas pid (zn-pid) controller the ziegler-nichols pid (zn-pid) controller is the utmost universally employed heuristic technique of fine-tuning a pid controller in all the engineering oriented feedback control application. the ability to predict the future errors in the process is possible in the pid scada based remote monitoring and control of pressure & flow rate 147 copyright ©2019 assa. adv. in systems science and appl. (2019) controller, meanwhile, it can eliminate oscillations and can decrease the rise time in the performance [9]. since the pressure and flow control process is the first order based system with time delay characteristics, hence the implementation of zn-pid controller is the benchmark of conventional techniques used for the comparative purpose. the parameters of zn-pid controller is exposed in table 3. table 3.zn-pid controller tuning parameters with values tuning rules tuning parameters for pressure tuning parameters for flow rate k c i (sec) d (sec) k i = (k c /i) k d = (k c *d) k c i (sec) d (sec) k i = (k c /i) k d = (k c *d) kc = ap θkp a∈ [1.2,2] i = 2θ and d = 0.5θ 1.84 3.77 1.42 0.490 2.629 1.28 5.14 0.699 0.25 0.9 where kc is controller gain, i, d indicate integral time and derivative time. 5.2 cohen-coon pid (cc-pid) controller cohen-coon tuning system is the more difficult and complex form of open loop ziegler nicholas method and its tuning parameter relations were established empirically to afford closed loop reaction for a feedback system which sustains ¼ decay ratio. the reformed cohencoon tuning procedures [9] is an outstanding scheme for attaining fast response on practically all control loops with self-adapting practices and to use on a non-interactive controller algorithm to provide a fast response on the process plant. the tuning parameters relations are given and their corresponding controller parameter of cc-pid is presented in table 4. 𝐾𝑐 = 1 𝐾𝑝 . ʈ𝐩 θ ( 4 3 + θ 4ʈ𝐩 ) (5.1) 𝑖 = 𝑑 32+6θ/ʈ𝐩 13+8θ/ʈ𝐩 (5.2) 𝑑 = 𝑑 4 11+2θ/ʈ𝐩 (5.3) where kc is controller gain,i and d are integral time and derivative time respectively. table 4.cc-pid controller parameters controller tuning parameters for pressure tuning parameters for flow rate k c i (sec) d (sec) k i = (k c /i) k d = (k c *d) k c i (sec) d (sec) k i = (k c /i) k d = (k c *d) cohencoon (ccpid) controller 1.84 3.77 1.42 0.490 2.629 1.28 5.14 0.699 0.25 0.9 148 e.b.priyanka, c.maheswari, m.ponnibala, s.thangavel copyright ©2019 assa adv. in systems science and appl. (2019) 5.3. internal mode pid (imc-pid) controller the efficiency of the internal model control (imc) strategy norm [9,11] has prepared it eye-catching in the process industrial sectors, where numerous endeavors have been done to accomplish the imc standard to develop pi/pid controllers in cooperation with stable and unstable process system. the eminent internal model control-pid (imc-pid) tuning rules have the improvement of solitary using of single tuning constraint to succeed a clear transaction between closed-loop presentation and robustness to exemplary inexactitudes and also serve as a newfangled online controller fine-tuning technique in closed-loop approach. the direct synthesis upon disturbance (ds-d) technique proposed by edghar and seborg [14] can be considered into the identical class of the imc-pid procedures, in that they achieve the pi/pid controller strictures by calculating the epitome feedback controller which contributes a predefined anticipated closed-loop retort. even though the ultimate feedback controller upholding both the imc and ds is habitually more problematical than the pid controller for time delayed practices, the controller arrangement can be condensed to that of either a pi or a pid based controller cascaded along with lower order filter by execution of applicable guesstimates of the dead time in process plant model. it is important to accentuate that the pi/pid controller premeditated according to the imc procedure affords first-rate setpoint tracking and consents the supervision of stable, unstable, and integrating system processes. this closed-loop fine-tuning method overwhelms the inadequacy of the renowned ziegler nichols unremitting cycling routine and contributes a reliably enhanced performance and forcefulness for a wide-ranging scheme of control process. 5.3.1 imc-pi/pid controller design for fluid transport process plant figure 6(a) and (b) show the block diagrams of the imc control [15-17] and corresponding traditional feedback control configurations, respectively, where 𝐺𝑝 points the process, 𝐺�̃� denotes the process model, q and 𝑓𝑟indicates the imc controller, the set-point filter, and 𝐺𝑐 corresponds to the comparable feedback controller. for the trifling case (i.e.,𝐺𝑝 =𝐺�̃�), the setpoint and disturbance reactions in the imc scheme can be streamlined as: 𝑌 = 𝐺𝑝𝑞𝑓𝑟 + (1 − 𝐺𝑝𝑞)𝐺𝑝𝑑 (5.4) conferring to the imc characterization (morari and zafiriou [10]), the process model 𝐺�̃� is sub-mani folded as factor of two parts: 𝐺�̃� = 𝑃𝑚𝑃𝐴 (5.5) where pm is the percentage of the model reversed by the controller, pa is the ration of the model not overturned by the controller and pa(0)= 1. the non-reversible part typically embraces the dead time and right side occupied half plane zeros and is elected to be all-pass. to acquire a enhanced reaction for unstable or stable processes with poles situated near zero, the imc controller q must gratify subsequent conditions: if the process gp possess unstable poles or poles positioned near zero at z1, z2… zm, then (i) q must ensure zeros at z1, z2, …, zm (ii) 1− gpq would also obligate zeros at z1, z2, …, zm scada based remote monitoring and control of pressure & flow rate 149 copyright ©2019 assa. adv. in systems science and appl. (2019) fig. 6.block diagram of the imc and corresponding conventional feedback control systems. for a fluid transport system process the methodology is encouraged by the enactment enhancement of the disturbance rejection predominantly. the overhead design principles are obligatory for the internal steadiness of the unstable method. the supplementary profit of utilized criteria is that they support in the performance progress of the control loop systems. meanwhile the imc controller q is intended as 𝑞 = 𝑃𝑚 −1𝑓, the paramount condition is fulfilled inevitably. the second condition can be contented by conniving the imc filter (f) as 𝑓 = ∑ 𝛼𝑖𝑠𝑖𝑚 𝑖=1 (𝜏𝑐𝑠+1)𝑟 (5.6) where τc is an modifiable parameter of the fluid pipeline system which reins the adjustment between the enactment and robustness; r is nominated to be great ample to sort the imc controller (semi-)accurate; αi are evaluated by equation (12) to abandon the poles 1 − 𝐺𝑃𝑞|𝑠=z1,z2,…,z𝑚 = |1 − 𝑃𝐴(∑ 𝛼𝑖𝑠𝑖𝑚 𝑖=1 ) (𝜏𝑐𝑠+1)𝑟 | 𝑠=z1,z2,…,z𝑚 = 0 (5.7) situated near zero around 𝐺𝑝. then, the imc controller emanates to be 𝑞 = 𝑝𝑚 −1(∑ 𝛼𝑖𝑠𝑖𝑚 𝑖=1 +1) (𝜏𝑐𝑠+1)𝑟 (5.8) as a result, the resultant set-point and disturbance retorts are attained as: y r = (gpq)fr = pa(∑ αisi+1m i=1 ) (τcs+1)r fr (5.9) y d = (1 − gpq)gp = (1 − pa(∑ αisim i=1 ) (τcs+1)r )gp (5.10) the numerator countenances ∑ 𝑎𝑖 𝑚 𝑖=1 𝑠𝑖 + 1 in equation (5.9) reason to be occasionally an unwarranted overshoot in the servo response that can be eradicated by presenting the set-point 150 e.b.priyanka, c.maheswari, m.ponnibala, s.thangavel copyright ©2019 assa adv. in systems science and appl. (2019) filter fr to recompense for the overshoot in the servo response. as of the beyond design practice, a stable, closed-loop reaction can be reached by exhausting the imc controller. the ultimate feedback controller that is correspondent to the imc controller [18] can be articulated for the fluid transport system in expressions of the internal model 𝐺�̃� and the imc controller q: gc = q 1−gpq (5.11) relieving equations. (10) and (13) into (16) contributes the ideal feedback controller: gc = pm −1∑ αisim i=1 (τcs+1)r 1− pa(∑ αisim i=1 ) (τcs+1)r (5.12) the resultant controller in equation (5.12) is substantially realizable for the fluid process plant, but it does not possess the customary pi/pid usage. the anticipated form of the controller can be attained by actuating the guesstimate of the dead time strategy in the process plant. meanwhile, the final derivation exploits both easiness and estimate error due to dead time mode deliberated carefully during the pi/pid controller strategy. first order plus dead time (fopdt) form is a descriptive model normally followed in the control process industries. on the source of the above detailed strategy principle, the fopdt process plant has been reflected as g(s) = kp ps+1 𝑒−s (5.13) where kp gives process gain, p and  denotes process stimulated time constant = process delay, the imc filter nominated is f = αs+1 (𝜏𝑐𝑠+1)2 (5.14) after exploiting the above examined principle the ultimate feedback controller is specified as 𝐺𝑐 = (𝜏𝑠+1)(𝛼𝑠+1) 𝐾[(𝜏𝑐𝑠+1)2−𝑒−𝜃𝑠(𝛼𝑠+1) (5.15) subsequently the ultimate feedback controller in equation (5.15) does not contain pi controller term; the residual assignment is to plan the pid controller that estimates the ideal feedback controller best meticulously. the ultimate feedback controller, gc, corresponding to the imc controller [20], can be attained later by the calculation of the dead time allocated for the poles by taylor series expansion, 𝑒−s =1−θs and outcomes in 𝐺𝑐 = (𝜏𝑠+1)(𝛼𝑠+1) 𝐾[(𝜏𝑐𝑠+1)2−(1−𝜃𝑠)(𝛼𝑠+1) (5.16) subsequently reorganizing of equation (5.16) gives 𝐺𝑐 = (𝛼𝑠+1) 𝐾(2𝜏𝑐−𝛼+𝜃)𝑠 (𝜏𝑠+1) [ (𝜏𝑐 2+𝜏𝑐𝜃) (2𝜏𝑐−𝛼+𝜃) 𝑠+1] (5.17) from equation (5.17), the consequential pid controller can be achieved after vulgarization as kc= 𝛼 𝐾𝑝(2i −α+) ; i=α ; d=2 (5.18) where kc points controller gain, ti and d are integral time and derivative time of the model designed process [21-24]. the significance of α (non-minimum phase element) is designated so that it terminates out the pole located at s=−1/τ and the rate of α is attained as 𝛼 = 𝜏{1 − (1 − 𝜏𝑐 𝜏 )2 𝑒− 𝜃 𝜏 ; c=2θ (5.19) by implementing this tuning procedure, it is conceivable to develop the enriched disturbance rejection action by adjusting the single fine-tuning of parameter in the controller. the important feature of this controller is that it compacts with the nonlinear and stable oriented plant system in a incorporated way. the parameters of the imc-pid controller are revealed in table 5. scada based remote monitoring and control of pressure & flow rate 151 copyright ©2019 assa. adv. in systems science and appl. (2019) table 5.imc-pid controller parameters controller tuning parameters for pressure tuning parameters for flow rate k c i (sec) d (sec) k i = (k c /i) k d = (k c *d) k c i (sec) d (sec) k i = (k c /i) k d = (k c *d) imc-pid controller 6.43 2.64 0.67 2.428 4.323 1.89 1.26 0.158 1.5 0.3 6. results and discussion the simulation and real-time investigational work have been conceded on the flow and pressure control loop outfit to examine the performance of the zn tuned pid, cc-pid and imc tuned pid controllers. to contrivance the closed loop control purpose, a pid block is established in the centum cs 3000 in which feedback regulating signal is programmed to pass through the control valve to form closed loop feedback structure in fluid transport process plant. 6.1.simulation results in the simulation work, the recitations of different controllers are equated by setting the peak of maximum uncertainty (ms) value for comparison. it is noteworthy to ensure iae, tv and ms (integral absolute error, total variation of the input, uncertainty margin value) to be lesser, but for a best characterized tuned controller causes the existence of trade-off, which indicates a declination in iae infers an escalation in tv and ms (and vice versa) [25, 26]. in the mean of the imc and direct synthesis implied tuning mode τc is an changeable parameter and hence it can be altered it for the anticipated robustness (ms) level. in the simulation work of the current paper, recitals of various controllers are associated by setting the similar ms value for a fair-minded comparison (i.e. ms=1.26). the controller structure of the pid block for the fluid transportation system to monitor pressure and flow rate is developed using simulink in matlab platform is presented in figure 7 and 8. fig. 7. matlab platform based created simulink model for pressure control loop 152 e.b.priyanka, c.maheswari, m.ponnibala, s.thangavel copyright ©2019 assa adv. in systems science and appl. (2019) fig. 8.matlab platform based created simulink model for flow control loop 6.1.1. performance analyses for pressure and flow control loop of process plant the performance of zn, cc and imc-pid controller are investigated at required setpoint ranges in order to normalize both pressure and flow rate of the fluid being transmitted. the closed-loop simulated transient responses for pressure and flow rates at the operational choice of 500 lph and 3kg/cm2 for the time interval of t = 0–50 s are unveiled on figures 9a and 9b. from the figures 9a and 9b, it is clear that the imc-pid controller is enforced to trail the fixed operating point of pressure and flow rate at short duration of time of about 17 seconds and 15.5 seconds and maintain the steady state with overshoot as compared to zn-pid and cc-pid control techniques which settle at about 34 seconds and 26.5 seconds for flow rate and 38.59 seconds and 47.987 seconds for pressure respectively. it confirms that imc-pid controller offers 16.72% of improved quality indices on comparing with zn tuned pid and cc-pid controllers. the presence of overshoot in the imc-pid controller performance ensures high rise time with quickly settling on its desired set point in order to improve the experimented operating condition of the fluid transport system. scada based remote monitoring and control of pressure & flow rate 153 copyright ©2019 assa. adv. in systems science and appl. (2019) fig. 9a .performance analyses of zn tuned pid, cc tuned pid, and imc tuned pid controllers for flow control loop at a set point of 500 lph. fig 9b. performance analyses of zn tuned pid, cc tuned pid, and imc tuned pid controllers for pressure control loop at a set point of 3kg/cm2. 6.1.2.setpoint tracking for pressure and flow control loop closed loop simulated transient responses attained at different operating points such as 0.7 and 1.6 kg/cm2 for pressure and 900 and 1400 lph for the flow rate to confirm the robustness of the imc-pid controller are shown in figure 10 and 11. the table 6 and table 7 reveal that the performance of imc-pid on comparing with zn tuned pid and cc tuned pid controllers affords superior performance with the similar settings for altered operating points. among the controller tuning rules, imc-pid sustains the trepidations in the model parameters and pay for the extreme reliable and robust response when the operating point diverges. figure 10 and 11 also ensures that imc-pid controller endows least possible error indices of 20.94% as compared with zn tuned pid and cc-pid controllers. this pragmatic imc-pid controller confirms the boosted performance meanwhile it has the talent to trajectory on faults online. 154 e.b.priyanka, c.maheswari, m.ponnibala, s.thangavel copyright ©2019 assa adv. in systems science and appl. (2019) fig. 10. performance analyses of zn tuned pid, cc tuned pid, and imc tuned pid controllers for pressure control loop at different setpoint tracking. table 6.pressure-performance measures of the controllers at different operating points operating points in kg/cm2 error and quality indices measures zn-pid cc-pid imc-pid 0.7 ise 0.00031 0.000435 0.001375 iae 0.0434 0.0405 0.037 itae 0.1077 0.0749 0.04332 tr (sec) 1.8 nil nil ts (sec) 39.9 47.9 15.5 %mp 36.214 nil nil 1.6 ise 0.00031 0.000441 0.001375 iae 0.04342 0.040529 0.037077 itae 0.10772 0.040529 0.037077 tr (sec) 1.76 nil nil ts (sec) 39.102 46.32 16.2 %mp 36.347 nil nil 3 ise 0.00034 0.000491 0.000253 iae 0.03590 0.024387 0.01444 itae 0.01447 0.013514 0.012359 tr (sec) 1.91 nil nil ts (sec) 38.59 47.987 15.37 %mp 37.102 nil nil scada based remote monitoring and control of pressure & flow rate 155 copyright ©2019 assa. adv. in systems science and appl. (2019) figure 11.performance analyses of zn tuned pid, cc tuned pid, and imc tuned pid controllers for flow control loop at different setpoint tracking. table 7.flow rate-performance measures of the controllers at different operating points operating points in lph error and quality indices measures imc-pid cc-pid zn-pid 500 ise 0.000417 0.00044 0.000435 iae 0.02471 0.027021 0.028951 itae 0.03691 0.039539 0.047324 tr (sec) 2.1 6.3 8.99 ts (sec) 17 26.5 34 %mp 15.36 3.075 22.98 900 ise 0.01667 0.0176 0.001741 iae 0.04943 0.054043 0.057901 itae 0.02967 0.030249 0.030684 tr (sec) 2.3 6.41 9.14 ts (sec) 17 27.01 35.102 %mp 14.89 3.152 21.39 1400 ise 0.01887 0.000356 0.018872 iae 0.02409 0.02658 0.025112 itae 0.03895 0.042130 0.043897 tr (sec) 2 5.98 8.93 ts (sec) 16.5 26.12 34.89 %mp 14.57 2.936 21.134 6.1.3. disturbance rejection test for pressure and flow control loop the disturbance rejection performance is investigated at the operating point of pressure at 2.1kg/cm2 and flow rate at 1300 lph. a step disturbance is introduced into the process by way of increasing the pressure to 3kg/cm2 and flow rate to 1600 lph after the time period of 100 and 150 seconds as shown in figure 12a and 12b. as of the analysis result endorses the merit by pointing only imc-pid controller damp the disturbance in a smaller time period of 13 seconds with less undershoot of 11.33% as compared to the cc-pid and zn-pid pid controllers on pressure control loop. the application of imc-pid controller on flow control 156 e.b.priyanka, c.maheswari, m.ponnibala, s.thangavel copyright ©2019 assa adv. in systems science and appl. (2019) loop divulges the sluggish disturbance rejection of zn tuned pid and cc-pid controllers where imc-pid tolerates the disruption with a short span of 21 seconds with enhanced quality indices. the error and quality indices of the output signal are applied to estimate the disturbance rejection presentation of the controllers for pressure and flow rates are given in table 8 and 9. fig. 12a.performance analyses of zn tuned pid, cc tuned pid, and imc tuned pid controllers for pressure control loop after a disturbance at a set point of 2.1kg/cm2. fig. 12b.performance analyses of zn tuned pid, cc tuned pid, and imc tuned pid controllers for flow control loop after a disturbance at a set point of 1300 lph. scada based remote monitoring and control of pressure & flow rate 157 copyright ©2019 assa. adv. in systems science and appl. (2019) table 8.performance measures after a disturbance at the operating point of 2.1 kg/cm2 performance measures zn-pid cc-pid imc-pid ise 0.000453 0.000418 0.000389 iae 0.014475 0.013514 0.012359 itae 0.007482 0.008391 0.006453 tr (sec) 1.792 nil nil ts (sec) 39.02 48.13 16.19 %mp 40.127 nil nil table 9. performance measures after a disturbance at the operating point of 1300 lph error and quality indices measures imc tuned pid cc tuned pid zn tuned pid ise 0.039257 0.043302 0.042816 iae 0.026801 0.028617 0.027843 itae 0.0141 0.018 0.0268 tr (sec) 5.34 8.921 13.973 ts (sec) 24.137 31.856 52.965 %mp 19.357 3.954 33.045 hence by simulation results, imc-pid controller contributes optimal smallest settling time with least error integral value and highest robustness on comparing with zn-pid and cc-pid controllers and hence affords 34.18% enhanced performance in monitoring and regulating the pressure and flow rate variations through the control valve opening and closing to attain the preferred operating choice at the destination in the fluid transport system process plant. 6.2.real-time experimentation of local intelligence on fluid transport system the proposed local intelligence using imc-pid controller design performance is experimentally substantiated in real-time on lab-scale experimental set up of the fluid transport system. the control drawing for the process plant architecture is developed by creating two separate control loop blocks of pid controller for pressure and flow rate as shown in figure 13. the resultant tuning values of pid controller using imc technique which is confirmed through simulation is put on to the established local intelligence tune window using centum cs 3000 r4 as shown in figure 14. through this control drawing builder option, hardware configuration gets synchronized with the flexibility of software customized application to the process plant in order to provide monitoring and control capabilities through online at a remote location from the process plant. the process plant is starting to run by enabling the auto mode initiated by the local intelligence, when the fixed operating point for flow rate and its pressure is given in the corresponding monitoring particular field parameters faceplate present in the scada front end panel of the fluid transport system. after fixing the required operating range, the pump will be on track to run which is enabled by an operator remotely. the real-time successive data of both pressure and flow rate parameters of the fluid being transported is displayed continuously in local intelligence trend view window in pic100.pv/ fic100.pv tag tab which is present below the trend graph and these data can be exported to excel by exhausting local utility data box option. 158 e.b.priyanka, c.maheswari, m.ponnibala, s.thangavel copyright ©2019 assa adv. in systems science and appl. (2019) fig. 13.control drawing for fluid transport system process plant fig. 14.developed local intelligence tune window of pid controller for the process plant. scada based remote monitoring and control of pressure & flow rate 159 copyright ©2019 assa. adv. in systems science and appl. (2019) fig. 15.real-time monitoring and control of pressure through local intelligence using an imc-pid controller. fig. 16.real-time monitoring and control of flow rate through local intelligence using an imc-pid controller. figure 15 and 16 reveals the performance of local intelligence using an imc-pid controller on remote surveillance and control of flow rate and its pressure on the implemented process plant. the setpoint of pressure and flow rate is given as 0.99kg/cm2 and 1200 lph respectively. based on the operating set point, the developed imc-pid controller running on the back end of the local intelligence scada adjusts the feedback signal going from the remote master control panel to the i/p converter incorporated with corresponding pressure control valve and flow control valve to regulate its opening and closing installed on the process plant control loops. the real-time experimentation discloses that when the percentage of control valve opening gets increased, the field parameters as like pressure and flow rate of the fluids passing through the pipelines get decreases and increases consistently. the developed local intelligence using 160 e.b.priyanka, c.maheswari, m.ponnibala, s.thangavel copyright ©2019 assa adv. in systems science and appl. (2019) imc-pid controllers authorizes the enriched performance meanwhile it has the capability to trajectory of errors occurrences through online by making real-time process plant information easily accessible through the online applications by its connectivity and interoperability. 7. conclusion and future scope the development and investigation of local intelligence using an imc-pid controller on the real-time fluid transport process plant are carried out in this paper. the real-time experimental results show that this local intelligence using imc-pid controller acts with provisioning smart decision making on the real-time data of the field parameters in order to optimize the process plant with minimum losses by means of its affordability and the reliability to conduct measurements remotely and in real-time of monitoring and control of field parameters. the experimental validation emphasizes has been engineered to condensed downtime, upgraded system accessibility, enriched control consistency, and unremitting system access. i/o remotehub module with control server, operator and engineer stations handling data organization, and gateway utilities are disseminated on an ethernet linkage to confirm system veracity and welltimed data broadcast communication. the evaluation of this local intelligence ensures that condition based maintenance with the proactive event-driven computing paradigm for fully exploiting its capabilities by enabling proactive maintenance decisions ahead of time on the process plant, so that the operator can react pre-emptive prior to sort out occurrences of any failure on the process plant. regarding the future work, an internet of things (iot) based reliable monitoring and control with local intelligence using imc-pid controller modular architecture design along with scada/dcs which is outlined in this paper will be implied and investigated on this lab-scale fluid transport system process plant. it is evident that the proposed iot architecture will offer a promising and a novel infrastructure to remote monitoring and control in any industrial sectors. conflict of interest all the contributors in this research work have no clashes of attention to announce and broadcasting this article. acknowledgment this research work is carried out under the senior research fellowship received from csir (council for scientific and industrial research) with grant no.678/08(0001)2k18 emr-i. references 1. galloway, b. & hancke, g.p (2013) introduction to industrial control networks, ieee communications surveys & tutorials, 15(2), 860. 2. davis, p., sullivan, e., marlow, d. & marney, d. 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(2002). imc-based control system design for unstable processes. industrial & engineering chemistry research, 41(17), 4288-4294. word template for assa manuscript adv syst sci appl 2019; 02; 1-7 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/758 technology of quality and reliability of complex technical systems characteristics stage-by-stage improvement on examples of rocket-space technology objects v.k. zawadzkii1* , v.p. ivanov1 , e.b. kablova1 , l.g. klenovaya1 1) v.a. trapeznikov institute of control sciences russian academy of sciences moscow, russia e-mail: vladguc@ipu.rssi.ru received june 20, 2019; revised july 1, 2019; published july 10, 2019 abstract: the principles of the evolutionary process of creating complex technical systems are formulated. the possibility of using the technology of stage-by-stage manufacturing of each specific sample of the system is considered. in the process of stage-by-stage manufacturing, the system improves its properties and characteristics, moving from relatively simple versions to more complex and effective versions. based on the results of the control, the system adapts to the real operating conditions. keywords: reliability, improvement of quality characteristics, rocket and space technology, complex technical system 1. introduction when improving the quality and reliability characteristics of complex technical systems (cts), which include modern objects of rocket and technical equipment (rte), problems have arisen that affect the conceptual basis of technology of their creation [1]. these systems are characterized by a multi-level hierarchical structure, a variety of "behaviors", a high dimension of the state space and a large number of elements, the ability to interact with the environment (adapt to and effect on the environment). for such objects, the incompleteness of physico-chemical processes knowledge underlying in the basis of their functioning, manufacturing technology and application conditions is characteristic. for the method of manufacture based on the decomposition of the product into the simplest elements, and production on the sequence of the simplest technological operations, these factors serve as the causes of design errors, production defects and violations in the operational procedures. in connection with the mentioned factors, the creation of complex systems is not a onestage act of manufacturing the finally chosen variants according to previously known technologies, but occurs through trial and error on simpler product prototypes [1, 2]. by experimental development of each new version, the missing information is filled and the correctness of design, technological and other innovations is confirmed. improvement of the technical characteristics of each new version of the system involves the use of new more effective principles of the system elements operation and a deeper understanding of the physical processes that underlie its functioning. as a result of a number * corresponding author: vladguc@ipu.rssi.ru mailto:vladguc@ipu.rssi.ru 2 v.k. zawadzkii , v.p. ivanov , e.b. kablova , l.g. klenovaya copyright ©2019 assa. adv. in systems science and appl. (2019) of iterations, the system approaches a certain limiting level of quality characteristics achievable within the product of this type. an example is the on-board control systems of launcher rockets, which in their development went from the forms of construction based on disintegration to the simplest subsystems and the use of the element base in the 1950s. to modern integrated control systems [3, 4], which include high-performance on-board computational facilities. let us stop on the fundamental principles of the stage-by-stage technology of creating cts: at all stages cts should retain the basic properties inherent in products of this type; transition to a new, more sophisticated product modification is associated with innovations; new cts maintains continuity with old, prior product samples; new product is created on the basis of a workable, reliable prototype; each product with innovations passes the stage of development test. the principles of stage-by-stage (additive) manufacturing are the basis of the additive manufacturing technology. this technology is associated with the use of three-dimensional printing and is considered by several authors as one of the important directions of the unfolding new production revolution. the most obvious possibility is the technology of stage-by-stage manufacturing in the production of dynamic systems and software (control algorithms) for these systems. elements of formalization of this technology within the life cycle of a dynamic system were considered in [5]. the improvement of the characteristics of the system was linked to the possibility of implementing a wider range of movements [6]. the existing technology of creating a system's software (control algorithm) resembles the assembly of a technical product from individual parts. with respect to the algorithm, the assembling elements are computational operations and the data of aprior and current information used in them. all the disadvantages of this method of manufacturing indicated above are also present in this case. a new, additive technology involves a stage-by-stage improvement of the properties and capabilities of the algorithm due to the sequential complication of the synthesis problem. the experiment in this case is simulation modeling. 2. formulation of a steady-state control problem let us consider the problem of complex multiply connected objects controlling, which include liquid-propellant launcher rockets and booster blocks. the control object is described by a differential equation of the form: )),(),(( ttwtxfx  , k extx ⊂∈)( ,  ew , r eutu ⊂∈)( , r , (2.1) ], 0 [∈ ttt , 0 ∈ 0 ) 0 ( xxtx  . there x(t) is the combined vector of the state coordinates of the object and the equations of the perturbation factors model; )( 0 tx extended vector of unknown initial conditions; f known vector function, differentiable with respect to the set of its arguments; )(tw control, which is a known vector function of time t; t the moment of achievement of the required final state (terminal time moment). for the final state of the object, boundary conditions are designated:  з)),((  ttx , l,1 , kl ≤ , lr  . (2.2) the value of the terminal time moment can be specified in advance. in general case, its magnitude is unknown in advance, and a certain degree of freedom in the assignment of the quality and reliability of complex technical systems 3 copyright ©2019 assa. adv. in systems science and appl. (2019) moment t is allowed. under perturbed conditions, the terminal time moment can be considered as a control parameter, chosen together with the parameters u from condition (2.2). for equation (2.1), we shall assume that the existence conditions and the uniqueness of the solution conditions are fulfilled. taking into account the possibility of control errors in solving problem (2.1)(2.2), we introduce the notion of boundary conditions discrepancies and reformulate the goal of control (2.2):  kllttxzz ≤ ,,1 ,з-)),((   , z , (2.3) where δ the set of admissible values of the vector of discrepancies, characterizing the accuracy of the solution of the problem (2.1)(2.2). in addition to the terminal conditions (2.3), we also consider indicators of integral type:  dwx t t ngni ),,,( 0 , nn ,...,2,1 , (2.4) where gn known function of fixed sign, the physical content of which can compose the cost of energy resource, time, loss due to control. to control the object, x(t) process measurements are made, defined by equation (2.1). the measuring devices are usually represented by static links. in the general case, the equation of the measuring part of the system for the control object (2.1) is written in the form: ))()(()( t,htx t =ty  , (2.5) where y(t) – dimension measurement vector p, h(t) measurement error random vector. equations (2.1) describe the processes of various physical nature occurring on a rocket carrier: the mechanics of rigid body motion, fluid hydrodynamics, heat exchange and thermodynamic processes, etc. the boundary conditions (2.1) and the discrepancies (2.3) are determined by the problems of launching the spacecraft and the conditions for the safe shutdown of the engine. criterion (2.4) is formed, based on the constant need to improve the energy characteristics of launching missile systems. the control problem formulated above should be solved taking into account the probabilistic nature of the perturbing factors, the relationship between the coordinates of the equations of various physical processes in the object, the diversity of the launching problems and the modes of operation of the control system. due to the complexity of the overall task, the object management is decomposed into specific tasks: navigation, guidance and stabilization, fuel consumption, tank pressurization, engine control. the mathematical support of each subsystem is formed from separate functional blocks and computational operations. at the same time, the understanding of the unity of the management of the object is lost, and the interconnections are not properly taken into account. when upgrading and improving the characteristics of the control system, developers constantly faced with the difficulties of making changes to the existing, proven technology of forming the software of the system. 3. decomposition of the synthesis problem by the degree of use of aprior and current information and definition of the main classes of terminal control it is advisable to consider the possibility of building a new technology based on the procedure for the stage-by-stage solution of partial problems of synthesis and continuity of these solutions. 4 v.k. zawadzkii , v.p. ivanov , e.b. kablova , l.g. klenovaya copyright ©2019 assa. adv. in systems science and appl. (2019) in this connection, let us consider possible ways of decomposition of the above terminal control problem into a sequence of simpler synthesis problems. as a result of solution of these problems, various classes of terminal control can be formed. by the nature and efficiency of aprior information and current information in the formation of physically realizable control, there are four main classes of terminal control: 1. program control (or open-loop control), where the formation of the control completely ignores the current information about the measured coordinates and realized controls. the program control is formed as a function of the time wпр(t) for the undisturbed motion of the object under given initial conditions and the terminal time t . under these conditions, the nominal (undisturbed) trajectory of the object xн(t) is realized. for the formation of software control, aprior information about the equations of the object and the specified boundary conditions for the nominal operating mode of the system is used. as a rule, optimal control is chosen as program control, which ensures the maximization of criterion (2.4), for example, on the basis of the maximum principle [7]. in accordance with this principle, a hamiltonian is constructed, which is the scalar multiplication of the vector of the right-hand sides of the equations of the object by the coordinate vector of the conjugate system. as a result of maximization, the control is defined as the vector function w(u,t) of the parameters u and the current time. parameters u are determined as a result of the solution of the main and conjugate system under given initial conditions. the initial conditions for the conjugate system must be linked to the terminal conditions of the terminal problem. as an example of a control program, it is possible to give an optimal law of variation of the pitch angle θ at given boundary conditions with respect to the position and velocity of the rocket's motion in a homogeneous plane-parallel gravitational field: tgθ = u1+ u2t. the law in the form w(u,t) turns out to be convenient for control with feedback. 2. control with feedback on the full vector of coordinates. when forming the control, the data of the current information arriving via the feedback loop is used. in the synthesis of control, a deterministic aprior description of the system is used. in this case, a considerably idealized formulation of the synthesis problem for an undisturbed object is considered. the deviations of the coordinates from the nominal trajectory and the discrepancy of the boundary conditions are caused by variable initial conditions, previously unknown before the time of measurement. the discrepancies are predicted for boundary conditions under a known vector of the current values of the object coordinates and given, for example, program control. the synthesis problem is to choose control with feedback in the form w(u,t). in the previous section, this vector-function was determined when solving the optimal control problem. the dynamics of the control process is characterized by a change in the vector of discrepancies of the boundary conditions. on that basis, it is advisable to choose a control from the lyapunov stability condition of the object's motion in the part of the vanishing of the predicted value of the residual vector at tt. stability of motion can be considered from the linear approximation of the object for deviations from the nominal trajectory: )()()()( twtbtxtax  , ], 0 [∈ ttt , (3.6) )()(),()( twtbtttxz  , з-)(),()( xtxtttxz  , where a(t), b(t) – matrix of order kk, k, (t,t)=x(t)x-1(t), x(t) – fundamental system of solutions. for the quadratic lyapunov function, the asymptotically stable motion and the condition zx(t)=0 are provided by the control of the following form: quality and reliability of complex technical systems 5 copyright ©2019 assa. adv. in systems science and appl. (2019) )(),( т )( т -)( tutttbtw  , )()( -1 )( txztctu  ,  dd t t tc )(∫)( , ),( т )( т )(),()( tttbtbtttd  . consider the case when the dimension of the control vector w(t) coincides with the dimension of the coordinate vector of the state x(t) (=k), and the matrix b(t) is such that the components of the control vector w(t) directly affect the derivatives of all coordinates of the state vector. as a rule, restrictions are imposed on the deviations of coordinates. in connection with this, for such deviations, it is possible to determine the values that are minimally needed for the solution of the terminal problem. the minimum required deviations are provided for a pulsed control action w() on a small time interval  in the t surroundings, for the remaining time interval w() =0: w()=w(t), tt+ , w() =0, >t+. here >0 is a small value in comparison with t. impulse control action on the object (3.6) from the condition 0) 1 (ˆ t x z for all t1: tt1>t+, is defined as follows:  /)(),( -1 -)( txztt b tw , )(),(),( tbtttt b  . such control is used in fuel consumption control systems for liquid-propellant launcher rockets. 3. deterministic control with feedback on the incomplete vector of object coordinates measurements. when measuring )()( thxty  , h – is a constant matrix of order kn , nk  , to recover x(t) let’s use the values of y()n a limited interval of prehistory ],-[ ttt  . as a criterion for the quality of recovery x(t) we take a definite-positive scalar function of the form:    t tt dyyqyyts ))(ˆ-)()(( т ))(ˆ-)((5,0)( , where q() diagonal matrix of positive coefficients. let )()-( 0 tttx x . at this )()-,()( 0 txttttx  . for )(ˆ 0 tx equation is specified: )()()()()( 000 ˆˆ tuttwttbttta xx  , ],[∈ 0 tt t , k etu ∈)(0 , (3.7) )( 0 ˆ∂ )(∂ )( 1 -)( 0 tx ts tktu  ,    dyyqhtt t tttx ts ))(ˆ-)()(( т )-,( т ∫ -)( 0 ˆ∂ )(∂ , where k1(t) diagonal matrix of algorithm parameters of order kk. algorithm (3.7) provides an asymptotically stable process of convergence )(ˆ 0 tx to )(0 tx . 4. stochastic control with feedback by a disturbed object. probability distribution laws and interference are assumed to be known. to solve the synthesis problem, the terminal quality control criterion is represented as a conditional mathematical expectation of the loss function, which depends on the discrepancies of the boundary conditions. for a wide class of problems, the synthesis of stochastic control is divided into independent estimation and control problems by estimating the extended coordinate vector of the object, including the parameters of disturbance models. to solve the estimation problem it is convenient to use the discrete analogue of the control object and the measurement channel in the form: i w i b i x i a i x  1, 6 v.k. zawadzkii , v.p. ivanov , e.b. kablova , l.g. klenovaya copyright ©2019 assa. adv. in systems science and appl. (2019) i=0,1,2,…,i+1, i hx i y  . as a result of solving a discrete equation, a linear function of the following form can be obtained: ) 1-,, ( jri w ri x jj x   , ) 1,..., ( 1-,j w ri w jri w   , irij ,...,1 , kr  . the estimation algorithm is presented in the following form: iri y i l iri x ri a iri x ,1-1-/ˆ 1-/ˆ      , (3.8) ),..., 1( ,1i y ri y iri y       , ) 1-,, 1-/ˆ(jri w iri x j h j y j y   , irij ,...,1 . the resulting estimation algorithm is a discrete analogue of the algorithm for reconstructing the full vector of state coordinates. in the case of an excessive number of measurements (r>k) with an appropriate choice of weighting matrix li in the algorithm (3.8) measurements error filtering is performed. conclusion in conclusion, we note the following. based on the analysis of a number of known, more complicated problems of synthesis, continuity of solutions has been revealed from program control to stochastic control with feedback. in the process of stage-by-stage improvement, approaching the real conditions of functioning, the possibilities of control are expanding, it acquires new useful properties. the carried out analysis can be used at creation of additive technology of control synthesis in complex technical systems. references 1. ivanov v.p. & portnov-sokolov iu. p. (1998) the issues of creating complex technical objects from the point of their safety (on the example of rocket and space technology objects), reliability and quality control, 1, 24–35. 2. martino d. (1977). technological forecasting. – m.: progress. 3. ivanov v.p., zavadskii v.k., guskov a.d., dishel v.d., vasyagina i.v., et.al. (2008). terminal control of launcher rocket and fuel consumption in the mode of its full development. international scientific and technical conference "systems and complexes of automatic control of aircrafts" dedicated to the 100th anniversary of the birth of academician of the russian academy of sciences nikolai alekseevich pilyugin. p. ii. proceedings of the plenary meeting (reports and announcements) mirea, april 23, 2008m.: “research and publishing center "engineer", 56-65. 4. zavadskii v.k., ivanov v.p., kablova e.b. & klenovaya l.g. (2013). integration of on-board control systems to improve the energy and reliability characteristics of quality and reliability of complex technical systems 7 copyright ©2019 assa. adv. in systems science and appl. (2019) launch vehicles, materials of iii all-russian scientific and technical conference "actual problems of rocket and space equipment": iii kozlov readings, september 16 – 20, 2013, samara) / samara scientific centre russian academy of sciences. – samara, 122–126. 5. ivanov v.p., kablova e.b. & klenovaya l.g. (2015). elements of formalization of the evolutionary process of improving the characteristics of complex dynamic systems, control problems, 4, 41-50. 6. krasnoshchekov p.s., fedorov v.v. & flerov iu.a (1997). elements of the mathematical theory of making design decisions, design automation, 1, 15–23. 7. pontriagin l.s., boltianskii v.g., garmkrelidze r.v. & mishchenko e.r. (1969) mathematical theory of optimal processes. – m.: nauka. advances in systems science and applications (2013) vol.13 no.4 317-333 latvia’s participation in the exchange rate mechanism of the european monetary system valery i. roldugin baltic international academy, latvia abstract the first objective is to investigate the on-going monetary policy of the bank of latvia, to analyse the basic principles of its operations and influence on national economic growth. the second objective of this article is to follow the main trends in the development in the latvian’s monetary policy and the european monetary system accession process, with a focus on the local currency stability problems. it discusses the process and strategies for choice of the strategy as well as the main issues that have arisen in the accession process. both objectives fully corresponding to the article’s research object, i.e. to a monetary policy of the bank of latvia. regarding the developed countries the general monetary policy objectives deals not only with maintenance of stability of the exchange rate and general price level, but also with stimulation of economic development, growth of employment and incomes of the citizens. the period from 2004 till 2011 is being investigated. the author uses a wide range of research methods, such as : grouping method, method of comparison of financial ratios, etc. keywords central bank, economic growth, monetary aggregate, monetary instruments, monetary policy. 1 introduction the monetary policy instruments of the bank of latvia are already in line with those used in the euro area. like the european central bank, the bank of latvia also uses the reserve requirements, market operations, as well as standing facilities of lending and deposit of funds. assets of bank of latvia including gold and exchange currency reserves, serve as maintenance of money issue in latvia. external reserves of bank of latvia which include gold reserves and foreign currency, and also currency from basket sdr in the end of 2012 has reached 7366,1 mill. lats (in 2003-830,5 mill. lats). the choice of the strategy depends on the size of country, level of openness, economic growth, relations between the key objectives and the intermediate target, features of financial and capital markets and other economic factors. the key objective of the central bank’s monetary policy is to facilitate the favourable macroeconomic environment for growth of the national economy in the long term. the course of global economic development assumes that monetary policy, employment and financial stability can foster the economic growth most of all by ensuring the low inflation rate. economic growth has stopped and even decreased in latvia, as well as baltic states and euro area in general thus restricting pos318 valery i. roldugin: latvia’s participation in the exchange rate mechanism... sibilities of profitable transactions carried out by latvian economic basic unitsenterprises in both domestic and export markets. rapid economic growth in recent years in latvia was mostly based on private consumption increase and large credit resource injections mainly in activities related to real estate market development. 2008 marked a turning point in latvia’s economic development, which began to decelerate after several years of buoyant growth. insufficient improvement in performance and efficiency of national economy, public administration and public services structure reduced the overall economic competitiveness, which was particularly influenced by recession in the export market. first signs of a slowdown of the economic growth became apparent in the second half of the year, when the implementation of the anti-inflation plan produced by the government, moderation of funding from parent bank and tightening of the banks’ lending policies resulted in a rather abrupt deceleration of the domestic lending growth. the reasons of failures of monetary policy include the methods of monetary regulation used by the bank of latvia. there is a danger of traps in the course of application of the monetary policy instruments. it is necessary that the bank of latvia should carefully chose the necessary financial tools of monetary policy. the author offers the ways of the solving of these problems on the basis of main principles using the achievements of world economic thought. in the conditions of a financial crisis the bank of latvia is able to achieve its targets while using the various financial instruments. 2 preparation for the participation in the european monetary union accession to the eu implies preparation for the participation in the economic and monetary union and introduction of the euro, for the new member states of the eu may not choose to stay outside the euro area. hence, after joining the eu, latvia will have to demonstrate its capability to meet the criteria for joining the economic and monetary union. the cabinet of ministers and responsible public administration institutions will take steps necessary to comply with maastricht convergence criteria to ensure euro changeover as soon as possible. in the analysis of any definition there is the first important question, the answer to which is connected to all systems definitions and classifications. this question will be worded as follows : “does a system really exist ?” not many researchers ask this question, but those who are wondering, as a rule, respond to it positively. it seems to us that the question canąŕt be answered at all. why ? let’s think constructively. economic convergence is a serious pre-requisite for successful operation of a monetary union ; therefore, prior to joining the euro area the eu member states have to ensure compliance with the maastricht convergence criteria : achieve price advances in systems science and applications (2013) vol.13 no.4 319 stability, sustainability of the government’s financial position, exchange rate stability against the euro and long-term interest rate convergence. maastricht convergence criteria : price stability is measured as the average consumer price increase of the last 12 months in three eu member states with the lowest inflation plus 1.5 percentage points ; sustainability of the government’s financial position means that the government deficit and the government debt do not exceed 3% and 60% of gdp, respectively ; long-term interest rates convergence criterion is interpreted as the average yield rate of 10-year government bonds within a 12 months period in three countries with the lowest inflation plus 2 percentage points ; exchange rate stability criterion provides for participation in the exchange rate mechanism ii (erm ii) without severe tensions, which, in turn, primarily means without currency devaluation against the euro. if required, other adequate indicators may be used to assess stability. in order to achieve the key objective as well as join the economic and monetary union successfully, the bank of latvia has chosen its exchange rate strategy for the implementation of monetary policy. now latvia almost complies with the majority of the maastricht convergence criteria necessary for joining the economic and monetary union (except for a parameter of interest rates on government long-term securities). table 1 the compliance with the maastricht criteria [1-4] criterion latvia’s criterion in latvia’s criterion latvia’s in performance in performance in performance 2005 in 2005 2005 in 2008 2011 in 2011 budget deficit -3.0 -1.0 -3.0 -4.0 -3.0 -4.1(% of gdp) government 60.0 14.7 60.0 19.5 60.0 42.6debt (% of gdp) average annu2.5 6.7 4.1 15.3 2.4 4.0al inflation rate (%) interest rates 5.4 3.96 6.24 6.43 7.80 6.00on government long-term securities(%) fixed fixed fixed fixed fixed fixed exchange exchange exchange exchange exchange exchange rate against rate against rate against rate against rate against rate against exchange the euro and the sdr the euro and the euro and the euro and the euro and rate regime participation since 1994 participation participation participation participation in erm ii in erm ii in erm ii in erm ii in erm ii for at least for at least for since for at least for since two years two years may 2005 two years may 2005 statistical data available for the last years indicate that the initial plan of latvia to adopt the euro in 2008 cannot be implemented due to the high inflation 320 valery i. roldugin: latvia’s participation in the exchange rate mechanism... rate. the schedule has not been revised as yet but according to the information released by the ministry of finance, in 2007 the government would discuss a new target for the changeover to the euro, tentatively in 2012-2014. the introduction of the euro in latvia will be an issue of the eu multilateral relations affecting common interests of all eu countries. therefore, the projected time frame for the introduction of the euro is merely tentative and will gain an official status only after the completion of all negotiations and other formal procedures. the high rate of lat stimulated a wide import of consumer goods displacing the latvian producers from domestic market that brought to the decrease of internal accumulation. in 1936, the gold content of lat was 0.29032254 grams of gold, and its parity to dollar equaled usd/lvl 3.03. the rate of lat at the riga exchange quoted on the same level. however, after adoption of law «on currency reform» on september 28, 1936 which provided the devaluation of lat on 40%, its rate at the riga exchange decreased to usd/lvl 5.04 [5]. the latvian exporters returned what they have lost due to the growth of lat rate before the currency reform, i.e. the same 40%. considering the indicators of national economy of latvia after lat devaluation in 1936, we see that it promoted the improving of economic situation. in 1937, in comparison with 1935 the amount of bank deposits increased on 67%, employment grew on 18%, a number of the unemployed people was reduced by 50%. the u.s. scientists carried out a study which stated that the fixed rate would bring latvian economy to such a decline, which’s incomparable with the u.s. recession during the great depression years of 1929-33 [6]. therefore, it is not casual that the rating agency moody’s and british analytical center capital economics point out the risk of devaluation of the lat. the news agency bloomberg referring to the experts of the u.s. bank brown brothers harriman & co. reported that the latvian currency may be devalued against the euro by 50% [7]. many western economists spoke about the necessity of devaluation of national latvian currency. kenneth rogoff, the professor of the harvard university and the former chief economist of the international monetary fund, considers that latvia should devaluate lat to avoid the toughening of economic crisis. it should be noted that the imf has no univalent position concerning devaluation of lat. according to prime minister valdis dombrovskis at discussions about financial aid to latvia, the imf wouldn’t object the devaluation of national currency but without insisting on it [8]. ilmars rimshevich, the president of the bank of latvia expressed opinion that lat devaluation would pull down the economy of latvia for one night. it is necessary to agree with him as it is necessary to treat devaluation of lat extremely carefully. the situation is similar to the tense spring. the devaluation must be based on exact calculations and shouldn’t be carried out without any damage to advances in systems science and applications (2013) vol.13 no.4 321 the population. there required the thorough economic calculations for search of an optimal rate which would correspond to the parity of purchasing power of lat to foreign currency more precisely than today. the price reform will demand many efforts because the standard of price expressed in a new rate of lat will change that should affect the size of salaries, pensions and other social payments, the state budget and its external balance of payments. the volume of export of the latvian industrial and agricultural output will increase. accordingly, the taxes and also the sums required for repayment of received earlier foreign loans will increase. all these points should be considered while starting such painful operation as devaluation of national currency. 3 problems of devaluation of lat the bank of latvia should be regarded as a state administrative body which carries out the main responsibility for development of events of the monetary sphere. its priority direction was always the external stabilization of the national currency which strengthening shall promote the internal sustainability as well. over the past 20 years, the latvian currency had a fixed exchange rate. since june 1993 lat had a floating rate, and in february 1994, the bank of latvia pegged the latąős exchange rate to the sdr basket (1 sdr=0.7997 lvl). on january 1, 2005 it was pegged to the euro (1 eur=0.702804 lvl). the fluctuations are admitted within +/-1% of the pegged rate. on may 1, 2004, when latvia has joined eu, the bank of latvia became a member of the european central banks system and began the preparations on euro adoption that was planned to take place in early 2008 according to the plans at those times. no big changes in the activities of the bank of latvia have happened. bank continued to perform the former economic functions, including monetary policy to ensure the operation of the payment system, the circulation of cash, preparation of financial statistics and balance of state payments and perform the macroeconomic analysis and research including the identification of trends in economic development in the scales of the european union. the administrative functions including licensing and control over the banks have been handed over to the financial and capital market commission in 2001. since may 1, 2004, the bank of latvia became the holder of the shares of the capital of the european central bank (ecb). the total share in the registered capital of the european central bank was established at 0.2978%, or 16,571.6 thousand euro. the share of the bank of latvia in the ecb’s capital was calculated on the basis of volume of the country’s economy and population. the bank of latvia brought 750 thousand lats and 1763 thousand lats into the fixed capitals of the european central bank and bank respectively for international payments. the bank of latvia uses the available foreign currency for the corre322 valery i. roldugin: latvia’s participation in the exchange rate mechanism... sponding installments. the bank of latvia aspired to direct the currency rate policy on achievement of low inflation and convergence of the prices for the goods and services and level of inflation similar the countries, which national currencies are included into sdr basket. latvia enjoys one of the most liberal currency regimes in the world. both the citizens and foreigners can open accounts in lats and in other currencies without restrictions and freely sell and buy lats by other currencies. the commercial banks credited in foreign currency not only the entities, but also the individuals. as a result, the real estate transactions have been carried out not in foreign currency rather than in national one. the same picture could be observed in the automobile market. the bank of latvia have been buying and selling the basket of sdr currencies on requests of commercial banks without restrictions. the primary long-term goal of approximately 65% of the world’s central banks is the prices stabilization ; the second most common goal is the sustainability of national currency. the small countries with open economies (like latvia) usually stick to the second long-term goal. it is impossible to achieve both goals at once because the central bank can control just one direction : either inflation or exchange rate. initially, the activity of the bank of latvia in the conditions of high inflation was directed on fight against inflation by the means of strict monetary policy. in 1992-1994, the monetary base ğő0 was the main indicator of bank of latvia because it was easier to supervise the monetary base, rather than other monetary aggregates in the conditions of high inflation (58.7% in 1992, 34.9% in 1993). the situation analysis after inflation decrease showed that the fiscal policy and monopolists’ price regulation incl. power industry, transport etc., became the determining factors. the internal assets of bank of latvia became the target parameter ; and the minimum limit of net external assets of the bank of latvia was set to maintain the stability of lat what was connected with an unofficial pegging of lat to sdr. the baltic states started their programs of sustainability of national currency in the middle of 1992. the economists had faced a problem when choosing the exchange rate mode : to accept the floating exchange rate or to operate the fixed rate. though the programs of sustainability of the baltic states differed from each other, however, all of them came nearly to the same result, i.e. estonia stuck the national currency to the fixed rate of dm, lithuania stuck to us dollar and latvia stuck to sdr. the main criteria were : to prevent the national economy from external shocks in a best possible way and to ensure the financial and economic sustainability of the restructuring of the economy. advances in systems science and applications (2013) vol.13 no.4 323 in 1992, estonia exchanged the soviet rubles at the market rate of 10 :1 and equated krone at a rate of dm/eek 8. lithuania replaced the intermediate currency in the form of coupon by lit in june, 1993, and pegged it to us dollar at a rate of usd/ltl 4 since april, 1994. it is interesting that in 1936 the rates of estonian krone, lithuanian lit and latvian lat against us dollar were following : usd 1/eem 6.95/ltl 4.38/lvl 5.04. in 1999, the rates of three baltic currencies to us dollar were respectively : usd 1/eek 11.2/ltl 4/lvl 0.59. there is a question how the current rate of lat against us dollar could be established almost ten times higher in comparison with 1936. on september 28, 1936 one british pound had been cost 25.11 lvl [5]. presently, the latvian lat is the main competitor for the pound. the latvian lat was heavier than pound, i.e. its rate was gbp/lvl 0.7910 as of march 1, 2010 [6]. so, the purchasing power of lat was approximately set on the same level as in 1936. let’s view the chronology of events related to the establishment of the new national currency rate in 1990-ies. it is known that latvia lost independence on july 17, 1940. the latvian lat was in circulation as a legal payment instrument till november 25, 1940 when the soviet ruble had been introduced in parallel at a rate of 1 lat=1 ruble. since march 25, 1941 all lat cash and deposits over 1000 lats (rubles) have been cancelled without any warning. such monetary confiscation has been carried out in three baltic republics. two currencies were functioned in latvia from 1941 to 1945. there were the german occupational money reichsmark (rm), and also the soviet rubles at a rate 10 rubles=1 reichshsmark. after the second world war till 1992 only soviet rubles were in circulation. only after latvia had regained its independence and the law “on the bank of latvia” had been adopted on may 12, 1992, there was an opportunity to carry out the monetary reforms targeted to establishing of a national currency circulation. to do this, the bank of latvia has specially established a committee for monetary reform (latvijas republikas augstākās padomes naudas reformas komiteja) that decided to introduce a temporary currency-the latvian ruble on may 2, 1992. the first stage of reform began on may 7, 1992, when the bank of latvia issued a temporary money the latvian ruble, which were in circulation along with the soviet rubles at a rate 1 latvian ruble = 1 ruble. since july 20, 1992 the parallel circulation was stopped, and the latvian ruble became the sole legal payment instrument. finally, the first five-banknotes came into circulation on march 5, 1993, which were used in parallel with the latvian rubles at a rate of 1 lat=200 latvian rubles. the latvian ruble was in parallel circulation with the lat till october 18, 1993. within 1992 there was a sharp shortage of cash in circulation in the state what 324 valery i. roldugin: latvia’s participation in the exchange rate mechanism... was threatening to cause a serious social crisis. so, if in february, 1992 the amount of cash exported from latvia exceeded the quantity of soviet rubles imported to latvia on 122 million banknotes (5.9%), in april of the same year this amount equaled to 686 million banknotes (29.2%) [10]. moreover, at the beginning of 1993 the latvian parliament discussed a case connected with unauthorized export of cash in amount of 665 million soviet rubles in railway car to russia. the commodity exchange carried out its operations in the building of the ministry of agriculture. commodity brokers have been changing the meat to paper, shoes to cereal, etc. the employees have been paid often by the enterprises products. in november 1993, there were 66 registered banks. the annual interest rate for deposits was as follows : banka baltija-90%, latvijas depozitu banka-45%, latvijas industriala banka-60%. now it is difficult to apprehend, but the liter of milk cost 14 centimes at the end of 1992, and the us dollar buying rate fixed by the bank of latvia was 0.8350 lvl and 0.5919 lvl already at the end of 1993. for january 1, 1995 the official rate of us dollar was fixed by the bank of latvia on a level 0.5480 lvl. as we can see, the bank of latvia took as a basis the rate of soviet ruble while establishing a rate of lat. therefore, on a moment national currency introduction, its rate initially was artificially overestimated. the high rate of lat allowed supporting one of the highest rates of economic growth among the countries of eastern and central europe (see tab. 2). the nominal gdp has been growing on more than 20% a year in pre-crisis years. for example, its growth was 23.3% in 2006, and 32.3% in 2007. 4 trends of latvia’s monetary policy the major quantity indicator of monetary circulation is the money supply representing a total purchasing volume and legal tenders, which serve for economic circulation of financial recourses of private persons, enterprises and state. the analysis of structure and dynamics of money supply has a great significance for development of reference points of the monetary policy of central bank. the monetary aggregates are applied to money supply definition. the cash assets are the basis of all monetary aggregates. the share of cash in money offer can vary depending on the circumstances. so, the more economy of the state is developed, the more banking system is stable and wide and the fewer shares of cash. the bank of latvia applies the following main monetary aggregates to the characteristic of monetary system and banking : (m0) monetary base monetary base is calculated on the basis of the bank of latvia’s methodology and comprises the lats banknotes and coins issued by the bank of latvia and demand deposits of resident mfis and financial institutions (overnight deposits) with the bank of latvia. the central bank directly creates advances in systems science and applications (2013) vol.13 no.4 325 this part of the monetary offer. by increasing the assets the central bank creates the money of high efficiency, by reducing it destroys them. the opportunities of central bank towards the creation of money of high efficiency are extremely great, as its liabilities (passives) are the money itself. m2x (wide money) broad money aggregates are calculated on the basis of the bank of latviaąŕs methodology and comprise the lats banknotes and coins issued by the bank of latvia (less vault cash of mfis) and overnight deposits and time deposits in lats and foreign currencies (including deposits redeemable at notice and repurchase agreements) held with mfis by resident non-financial corporations, financial institutions, households and non-profit institutions serving the households. m2x incorporates the deposits placed by local governments as a net position on the demand side. this parameter characterizes the total money in national economy. as foreign currency is of great significance for latvian economics, this parameter includes both the deposits of the enterprises and private persons in foreign currency [11]. let’s review the monetary parameters presented in table 2[1-4]. table 2 the basic indicators of monetary policy in latvia from 2004 till 2011(mill. lats) 2004 2005 2006 2007 2008 2009 2010 2011 gdp (current 7434.5 9059.1 11171.6 14779.8 16243.2 13070.6 12784.1 14275.2prices) growth gdp 116.3 121.0 123.3 132.3 109.9 80.5 97.8 111.7(%) m0 957.2 1350.7 2248.8 2471.2 2111.5 1645.8 1755.2 2168.9 m2x 2816.5 3905.8 5479.9 6171.3 5931.4 5796.2 6390.0 6486.1 growth m2x 127.0 138.7 140.3 112.6 96.1 97.7 110.2 101.5(%) m2x/m0 294.2 289.2 243.6 249.7 280.9 352.2 364.1 299.1(%) gdp/m2x 2.7 2.3 2.0 2.3 2.7 2.3 2.0 2.2 the velocity of money testifies the communication between monetary circulation and processes of economic development. aggravation of the macroeconomic risks and lower savings induced acceleration of the velocity of money, growing from 2.3 in 2007 to 2.7 in 2008. resident financial institution, non-financial corporation and household deposits with mfis decreased by 205.9 million lats or 3.9% in 2008 in comparison with an increase of 16.9% in 2007. in 2004, the velocity of money made 2,7 times per year. the economic situation stimulated the decrease of rate of the money turnover from 2.7 in 2008 till 2.2 in 2011. in relation to this parameter 326 valery i. roldugin: latvia’s participation in the exchange rate mechanism... latvia approaches to the developed countries where the speed of money turnover does not exceed 1.5 times. the decrease of rate of turnover of monetary volume (in 1.5 times) for the last 8 years cannot lower the negative influence of prompt monetary growth which increases more than by 40 % in 2006. the behaviour of the monetary aggregates in 2008 mirrored the sharp downturn of the economic development with both domestic and external demand shrinking, as well as the impact of the global financial crisis on the latvian banking system and money market. in 2009 m2x decreased by 2.3% (a growth of 10.2% in 2010) and amounted to 5796.2 million lats at the end of 2009 (see table 2). with the economic development coming to a halt in the second half of the 2008, banks cutting down on their lending business remarkably and confidence with regard to the financial sector deteriorating. the negative rate of the monetary expansion was primarily a result of the decelerating growth of mfi loans to the private sector, with the total loans outstanding shrinking in the last three year of the 2000s. the monetary situation in latvia is characterized also by other parameters : monetary multiplicator (m2x/m0) and velocity of money (nominal gdp/m2x). the great importance has monetary multiplicator a parameter describing the opportunities of economy as a whole and banking system in particular to increase a money stock in a turnover. its size pays off as the attitude of m2x to monetary base(m0). the monetary multiplicator is necessary to control over the monetary volume dynamics and the rate of inflation in latvia. in 2011, the monetary multiplicator has decreaseed essentially, and by the end of the year it consisted 299.1% (in 2010-364.1%). monetary base m0 decreased by 14.6% in 2008 and totalled 2111.5 million lats at the end of the year, whereas the cash component of the monetary base grew to 48.2% in comparison with 42.5% at the end of 2007. deposits from credit institutions and other financial institutions held by the bank of latvia declined by 328.3 million lats or 23.1% in 2008 as opposed to a 21.0% increase in 2007. for the second consecutive year, the demand for cash decreased, and currency in circulation shrank by 31.4 million lats or 3.0% (by 2.3% in 2007). the development trends of monetary aggregates were influenced by drying up capital inflows and foreign exchange interventions of the central bank reflected in the changes of net external assets. negative net foreign assets of mfis grew by 31.9% during the year, amounting to 5914.6 million lats, whereas the respective indicator of the bank of latvia decreased by 16.0% and totalled 2332.3 million lats at the end of 2008. thus the growth of the negative net foreign asset position decelerated considerably in comparison with 2007, when it expanded 1.7 times. nevertheless, latvia experience significant economic growth beginning in 2004 with eu accession. this followed the u.s. strategy to prevent an economic recession through asset inflation following the collapse of its stock markets in 2000. advances in systems science and applications (2013) vol.13 no.4 327 the usa and other developed countries flooded a world economy cheap credit resources. thus, developed countries created credit found its way into latvia through swedish banks. combine with eu structural funds, the latvian economy headily increased of gdp, until the inevitable global economy crisis led to latvia’s disastrous fall. but, latvia’s economy was not purely the victim of induced events. the bank of latvia and other state regulators are also responsible. the basic problem of latvia for the last 20 years was a balance of payments deficit of current transactions. balance of payments deficit arises as a result of excess of import over export. within last 8 years the volume of foreign trade of latvia has considerably increased in 3 times. the volume of foreign trade has been increasing year from a year, except 2009 when the consequences of the financial crisis have occurred (table 3) [1-4]. table 3 parameters of foreign trade of latvia from 2004 till 2011(mill. lats) 2004 2005 2006 2007 2008 2009 2010 2011 import 2989.2 4866.9 6378.5 7780.2 7484.4 4709.8 5911.9 7719.4 export 1650.6 2888.2 3293.2 4040.3 4406.0 3602.0 4695.0 5998.5 balance -1338.6 -1978.7 -3085.3 -3739.9 -3078.4 -1107.8 -1216.9 -1720.9 in 2008, the foreign trade dynamics was affected by the global financial crisis and weakening domestic and external demand. exports of goods expanded by 9.1% (a 15.1% rise in the first nine months of the year and 7.2% drop in the fourth quarter). imports of goods continued to shrink gradually, recording a 3.8% downslide on an annual basis. the excess of imports of goods over exports of goods decreased to 69.9% (92.6% in 2007) and the foreign trade deficit narrowed by 17.7%. the role of the eu in latvia’s foreign trade is strengthening : 76.1% of total export from latvia went to the eu countries in 2011 (73.1% in 2008). let’s consider the main financial instruments and principles of monetary policy in latvia. the choice of monetary instruments is wide enough. the basic tools of monetary regulation are : the standing facilities of lending and deposit of funds (refinancing) ; the reserve requirements ; the market operations. the choice and combination of instruments of monetary regulation depends, first of all, on issues, which are settled by central bank at the stage of economic development. the regulation of discount rate relates to market instruments of monetary regulation. the mechanism of regulation is simple enough ; therefore it is widely used in developed and developing countries. the official discount rate is a reference point for other market rates. the above level of official discount rate is the above cost of central bank refinancing. it means that the policy of change of discount rate represents a variant of regulation of qualitative parameter of money market the cost of bank credits. 328 valery i. roldugin: latvia’s participation in the exchange rate mechanism... the refinancing of commercial banks is carried out by holding of credit auctions, granting of lombard loans, etc. the bank of latvia has started using of refinancing as monetary policy instrument in 1993 only by granting the short-term credits to commercial banks for liquidity maintenance. originally the credits were granted to each bank within the limits depending on the bank’s performance in accordance with regulatory requirements established by the bank of latvia. such an order was necessary because the credits were granted without collateral. since november 1993, when the demand for credits exceeded the supply of credit resources, the bank of latvia has started carrying out of the credit auctions. as earlier, the credits were granted without collateral, therefore the quantity of the participants of auction was defined depending on the size of the equity capital of the bank, liquidity, and the bank performance in accordance with regulations of the bank of latvia. the bank of latvia has started granting the lombard loans in september 1995. it is the form of refinancing when the central bank grants the credit under pledge. the bank of latvia grants two types of lombard loans : the automatic lombard loans and lombard loans on demand. the commercial banks can exceed the balance of the correspondent account within the limit of the lombard loan during a payday for maintenance of efficiency of the interbank payment system. the bank of latvia grants the credits to commercial bank for one day in amount of the debit balance of corresponding account, in case the commercial bank could not involve the money resources till the end of a payday to liquidate the lack of resources on a corresponding account. the bank of latvia grants the lombard loans to commercial banks automatically in the end of a payday without special requirement. the basis for the lombard loan issuance is the special agreement between the bank of latvia and commercial bank. usually the lombard loan interest rate is higher than credits repo interest rate. it is a kind of “penalty" for the usage of resources of the central bank. the interest rate of lombard loans can vary depending on the terms of drawdown. as the lombard loans interest rate is higher than the interbank market interest rates and the interest rate of refinancing of the bank of latvia, the demand for these credits is usually small. the commercial banks use the lombard loans only in case of emergency. the bank of latvia supports the money volume in the set parameters and adjusts the level of liquidity of commercial banks by changing the minimal reserve requirements. it is assumed that credit institutions have to hold a certain share (currently 3.5%) of the attracted non-bank deposits with the bank of latvia. in event the reserve requirements are increased, these credit institutions will have to hold more funds with the central bank. it means that the amount of funds attracted by credit institutions, which is at their disposal and could be freely placed in the economics, thus increasing the level of credit and broad money, will advances in systems science and applications (2013) vol.13 no.4 329 decrease. the reserve requirements as a monetary policy instrument ensure the higher stability in the monetary base demand and facilitate the effectiveness of market operations, preventing the excessive interbank interest rate daily fluctuations. 5 three dogmatic “rule” there is a dogmatic “rule” of the monetary policy : the actual rate of refinancing should be positive (though it was negative in many developed countries in specific years, and in 2007 the refinancing rate was below the rates of inflation in the majority of these countries). in latvia, the monetary market rate is below the inflation rates. it is obvious there is an imported inflation in the conditions of prices rise which is underestimated by the central bank. accordingly, it overestimates the measurements of the base inflation depending only on the monetary factors. therefore, the refinancing rate should be minimal, i.e. at a level of correctly estimated base inflation. in the first half of 2007, the bank of latvia continued to pursue the tough monetary policy and in two occasions raised the refinancing rate by a total of 100 basis points (to 6.0%), thereby dampening the excessive domestic demand. later, when the signs of economic overheating abated, the bank of latvia left its interest rates unchanged, but in the first half of 2008 due to the slowdown in the growth of lending and the associated deterioration in the banks’ role in fuelling the domestic demand, it lowered the minimum reserve requirement for bank liabilities with agreed maturity of over 2 years by 2.0 percentage points (to 6.0%). the reasons of failures of monetary policy include the methods of monetary regulation used by the bank of latvia. they are reduced by application of several “rules” which are considered to be suitable for any condition in any country. the first “rule” : in order to decrease the rates of inflation it is necessary to limit the monetary offer or to apply the quantitative credit restrictions or to overestimate the refinancing rate. one more dogmatic “rule”-in order to decrease the inflation it is necessary to strengthen the rate of national currency ; on other hand, it reduces the price competitiveness of national commodity producers. it could be raised by depreciation of the credits, but it is forbidden by the first “rule”. the application of such “rule” of monetary policy by no means is rather offensive : the economics can get to a condition called by the “trap”. in such conditions the measures of state regulation do not bring positive results. these traps are known : “the debt”, “the liquidities” and “the negative effect rendered by strengthening of actual exchange rate of national currency on economic development”. the debt trap occurs at excessive debts of the state and private sector at small duration of debt. by involving the short-term and intermediate term loans under 330 valery i. roldugin: latvia’s participation in the exchange rate mechanism... low interests the borrowers count them as refinance by new loans. in case of steep increase of interest rates in the financial markets its long service sharply rises. if the borrower is not in position to extinguish it, the avalanche growth of debts begins, and it is impossible to extinguish it even after decrease of the interest rates. now, there is a growth of interest rates which were on rather low level for the long period. the latvian enterprises borrowed the financial resources abroad and can shortly face the refinancing problem and service of debts. the liquidity trap occurs at too low nominal interest rates when the central bank decreases refinancing rate, but it do not lead to expansion of the credits and stimulation of economic growth. the third trap is caused by the increase in balance of the international payments. it leads to the strengthening of national currency and shifting of employment to sphere of services. the bank of latvia can use the maintenance of the exchange rate by buying up the foreign currency, and it conducts to the growth of monetary offer and inflation strengthening. at high inflation the strengthening or stability of the nominal exchange rate leads to strengthening of the actual exchange rate. in other words, the efforts of the central bank do not achieve the object on maintenance of price competitiveness. thus, there is a necessity of sterilisation of superfluous liquidity. the escalating of the gold and exchange currency reserves by latvia basically is crediting of the usa and the european union countries. the inflation is mostly supported by excessive inflows of a variety of financing : the credit resources from foreign parent banks, the foreign direct investment, the eu funding, the workers’ remittances from abroad. to avoid the imported inflation the high validity of monetary policy is required. for example, the most effective remedy of the negative effect rendered by strengthening of actual exchange rate is crediting of the enterprises by the government and the central bank by replacing the foreign loans. certainly, such replacement of credits is better than the accumulation of superfluous currency provisions. however, from the macroeconomic point of view the replacement of foreign loans of the latvian enterprises by local credits is similar to repayment of external debt. now, the latvian commercial banks involve the foreign loans, and the central bank is compelled to buy up the foreign currency, generating the superfluous monetary offer which needs to be sterilised immediately. with the view of restriction of money growth the bank of latvia does not refinance sufficiently the credit organisations by establishing the refinancing rate at high level. it leads to the overestimate of credit resources cost for banks and enterprises. therefore, the latvian enterprises increase the foreign loans. the vicious circle turns out-the bank of latvia is compelled to get the additional volumes of the foreign currency arriving in the form of foreign credits to private sector advances in systems science and applications (2013) vol.13 no.4 331 of economy, increasing thereby the monetary base. the government is compelled to “freeze” huge budgetary funds on accounts of the bank of latvia in order not to admit an excessive monetary issue and inflation strengthening. the failure includes the preservation of high average rates of inflation which exceed 15.4% in 2008. the ways of their decrease in conformity with anti-inflationary program developed by the government are not enough clear. at low technological efficiency of the majority of latvian economic branches and growing world prices for energy sources it is impossible to stop inflation, including the attempts to limit the growth of monetary weight. however, the high inflation at stable or raising exchange rate of lat leads to the strengthening of the actual rate of exchange, and consequently-to corresponding decrease of competitiveness of the latvian enterprises. though the indicator of parity of purchasing capacity is considerably underestimated regarding the currencies of the developed countries, the actual exchange rate strengthening brakes the economic growth and promotes the advancing growth of import. at high inflation the investors of banks receive negative real percent on deposits, and the enterprises pay the overestimated income and added cost taxes. 6 conclusions for the past years the modern two-level banking system was established and developed in the country. gradually, there formed the competitive credit and financial infrastructure which basic elements were the commercial banks. some of them have already got a high international rating. the association of the latvian banks turned into national bank association. at the same time, there is an unpredictability of actions of the latvian administrative bodies. a main goal of the bank of latvia and the financial and capital market commission is the ensuring of the general sustainability in the monetary and credit markets. thus, exercising all rights defined by the law, the bank of latvia should provide a stability of the prices, and the financial and capital market commission should treat everyone who forms the instability in finances and capital market in whole or in its separate sectors by their activity or non-activity. the results of research testify the mistakes of the central bank made at the moment of establishment of national currency rate. the country had a possibility to provide unreasonably high rates of gdp growth due to the overestimated currency especially in the middle of the 2000th. in order to solve these problems it is necessary to boost the access of latvia into euro zone since the adaption of euro will help the bank of latvia not to carry out a number of aforementioned actions. an effective banking system is one of the most important conditions of economic development of latvia. it is assumed that latvian economy will be able to implement positively the experience of monetary 332 valery i. roldugin: latvia’s participation in the exchange rate mechanism... and credit regulation accumulated in world practice. in spite of the fact that devaluation of lat is matured, euro adoption is more of big modern importance. therefore, referring to a world experience, we should not abandon the possibility of joining the euro zone de facto, although the process of co-ordination with the european central bank may not be less complex than the joining of euro zone. in latvia, a market undergoes a process of euroization for a long time . euro has pressed lat long ago in the conclusion of credit and trading agreements. in order to find the correct options it is possible to make use of experience of the european and other countries which use euro as national currency, but didn’t enter the euro zone : montenegro, kosovo, andorra, monaco, san marino, mayotta, etc. references [1] the bank of latvia: annual report 2003-2010, available at: http://www.bank.lv/statistika/datu-telpa/apraksti/apraksti (accessed mar.28, 2012). [2] data of the central statistical bureau of latvia, available at: http://www.csb. gov.lv/en (accessed dec.12, 2011). [3] data of eurostat, available at: http://epp.eurostat.ec.europa.eu/portal/page/portal/eurostat/home/ (accessed may.11, 2012). [4] ārzemju valūtu tirgus latvijas republikā(2011), available at: http://www.bank.lv/lat/main/all/pubrun/lbgadaparsk/lb1993gadparsk/valstekon/arvvalutas/ (accessed nov.8, 2011). [5] ducmane k. (2011). latvijas nacionālā valūta-vēsture uns̆odiena, available at: http://www.bank.lv/nauda/latvijas-nacionala-valuta (accessed jan.16, 2012). [6] asv pētnieki: lata fiksētās piesaistes dēl. latvija nonāks dzil.as recesijas slazdā (2011). bns. 2010. gada 4. februāris, available at: http://www.diena.lv/lat/business/hotnews/asv-petnieki-lata-fiksetaspiesaistes-del-latvija-nonaks-dzilas-recesijas-slazda (accessed dec.23, 2011). [7] mvf ne trebuet ot latvii deval~vacii nacional~no@i vai�my, ria\novosti". kategorii: latvi� mvf. [8] glavny@i �kspert mvf: lamvi� dop�na deval~virovat~ lat, qtoby izbe�at~ u�estoqeni� �konomiqeskogo krizisa. [9] exchange rates of the bank of latvia, available at: http://www.bank.lv/lat /main/all/finfo/notvalkur/ (accessed nov.2, 2011). [10] the latvian ruble versus the russian ruble, available at: http://www. bank.lv/eng/main/all/pubrun/lbgadaparsk/lb1992gadparsk/v-alstekon1992 /rublelvvsrus/ advances in systems science and applications (2013) vol.13 no.4 333 [11] bel.kovskis k.(2008), “short-term forecasts of latvia’s real gross domestic product growth using monthly indicators”, riga., pp.45-54. [12] roldugin v.(2002), “nav izslēgta latvijas tirgus eirozācija”, latvijas ekonomists, no.4(88), pp.91-97. corresponding author author can be contacted at: vroldugin@hotmail.com. advances in systems science and applications (2014) vol.14 no.1 66-75 towards a new philosophical foundation for physics – speculations on the nature of space, gravity,inertia and mass philip j. tattersall1 and benjamin p. sidebottom2 1lenborough st, beauty point, tasmania, australia 7270 2apartment 5 block a, albion mill, pollard street, manchester. m4 7aj. united kingdom abstract recognizing that physics is now at an important turning point, the authors put forward ideas relating to the nature of space and its role in the emergence of gravity, inertia, mass and, ultimately, the ‘reality’ that derives from (scientific) observation and measurement. the essay cites relatively recent experiments and observations relating to phenomena such as the casimir effect,unruh effect, zitterbewegung and the results from the ‘moving mirror’ experiment. it argues that, when combined with older problems such as quantum entanglement,these phenomena provide new evidence that might inform a better understanding of the role of space in its interaction with particle/field entities. furthermore, it suggests that space may have a significant role in the creation of these entities. the authors suggest that fresh creative insight will be needed for physics to address the scale of the challenge implicit in this new, and exciting, territory. however, this is unlikely to emerge without revising the philosophical framework that underpins physics. this would need to reconcile quantum ontologies with non-quantum ontologies that may be scale dependent. in order to meet the many emerging challenges a more open, participatory and permissive physics is envisioned. keywords entropy, gravity, inclusionality, inertia, mass, quantum vacuum, reality, space. 1 introduction recent theoretical work and experimental findings each refute the idea that space is a benign nothingness, or void. our essay on the significance of space moots a philosophy that we see as a first step in enhancing current theoretical frames leading to a more comprehensive and compatible system of exploration and understanding. while all sciences represent their observations in domains that are, at least, implicitly 3-dimensional, the philosophical framework underpinning the more intensely theoretical sciences, such as physics, make it necessary to theorize in higher dimensions1 in order to ‘explain’, or to account for, the results of the new 1for example time as a 4th dimension in general relativity theory. advances in systems science and applications (2014) vol.14 no.1 67 experiments and observations. as experiments become more sophisticated and sensitive the resultant observations lead to ever more intense debate and speculation as to the ‘real’ nature of reality. while some see this as evidence of a perplexing universe, we are more inclined to see it as an epistemological overhead that stems from continuing to observe and to measure in only 3-dimensions when physics is increasingly multi-dimensional. this would explain why current approaches seem to offer only partial, or illusory, glimpses of what we are exploring. it accounts for the growing and, possibly, unparalleled sense of mystery and ‘weirdness’ that many scientists experience. in the past, science has usefully drawn upon a vital source of ideas and valuable insights from the ‘philosophical spring’ in order to make advancements. today, a similar process is no less important. this paper therefore advances a number of ideas as a way to initiate a conversation about how physics might refocus its philosophical base, in order to invite and encourage creative and informed debate, and research, on the above topics. we believe that the present interest in the nature of space and, in particular, quantum vacuum activity is an early indicator as to the future direction of physics. a successful continuation of this new trajectory in physics will require a (not unprecedented) process of reinvention, or possible revolution, that includes developing a new philosophical foundation. to that end we acknowledge the significant role natural inclusionality [1] has played in influencing our thinking during the development of this paper. according to natural inclusionality the classical description of space requires a reinterpretation: space is regarded merely as the distance over which mass, force and energy are stretched (or stretch themselves), such that they have variable density or frequency, and has no other influence beyond their limits. in this default condition, matter is inert and space passive. the very possibility of motion is therefore made ultimately dependent on some inscrutable external forceful agency or ‘unmoved mover’ to get it going. but if such agency can only be contained or applied locally, where is it? there is clearly something, or rather somewhere, missing from this classical description, which leads energy in the guise of mass and force paradoxically to be mentally confined within and excluded from the boundaries of discrete, completely quantifiable units-i.e. as atomic particles in material bodies, photons in electromagnetic radiation and phonons in heat. that missing somewhere, according to natural inclusionality, is everywhere, without limit-the intangible receptive presence of space. with the dynamic inclusion of this non-local omnipresence within, throughout and beyond local form, movement and change become understood in terms of processes of flow as a continuous energetic reconfiguration of space, not as the travel of independent particles or waves through space. by the same token, massy bodies and electromagnetic radiation are un68 philip j. tattersall: towards a new philosophical foundation for physics ... derstood as distinctive energetic configurations of space, neither solely ‘particles’ nor ‘waves’, but ‘flow-forms’ [1]. 2 inertia, mass and gravity the relatively recent discoveries of a physical manifestation in ‘empty space’ have shown space to be a sea of virtual2 activity3. the casimir and davies-unruh effects [2], the results of the ‘moving mirror’ experiments [3], fermion chirality and zitterbewegung [4-5] all suggest that space plays an active role in the manifestation of certain physical phenomena. we venture to suggest that the virtual sea of activity resident in space may well be the domain of so called ‘hidden variables’ that mediate and, perhaps, cause the emergence of ‘phenomena’ including non-locality, mass conference, inertia and, perhaps, plays a pivotal role in the emergence of ‘gravitational’ influence. rueda and haisch [5-6] have suggested that inertial and gravitational mass each arise from interactions of the electric charges and quarks of matter with the quantum vacuum. they suggest that matter distorts or polarizes the quantum vacuum, leading to an attraction of virtual particles with opposite charges and repulsion of those with like charges (cited by chown [5]). this idea resonates with the idea of matter interacting with ‘space’, as has been proposed elsewhere [1,7-8]. whilst offering a different interpretation of the nature of space itself, we agree that the quantum vacuum somehow interacts with mass and this is what causes the emergent properties of gravity and inertia; hence our proposed mechanism is somewhat similar to that proposed by rueda and haisch [6]. as matter is mostly ‘empty space’, and as space is considered to be a sea of quantum vacuum activity, it would seem that matter itself is permeated with quantum vacuum activity. it follows that the quantum vacuum interactions within matter may play a significant role in the emergence of mass and inertia. when energetic bodies are in uniform motion, the quantum vacuum permeates as laminar flow throughout them, whereas under acceleration we suggest there is a disturbance4 of the quantum vacuum activity, which is manifest as a davies-unruh effect [9-10]. as has been suggested [2,6], the result is an increase in inertia, which we term virtual mass. by way of analogy this could, perhaps,be understood as some kind of induction phenomena whereby quantum vacuum energy becomes stored within the body during acceleration5. radiation emission 2virtual in the sense that such activity is not easily measurable, is barely observable and does not have extension in 3 dimensional ‘space’. 3also termed quantum vacuum noise. 4the quantum vacuum when moving through mass under acceleration induces radiation effects. 5under acceleration there is a kind of ‘induction effect’ (with photon emission) in which the quantum vacuum interacts with fundamental matter fields [6] causing the affected body to advances in systems science and applications (2014) vol.14 no.1 69 during acceleration suggests virtual particle conversion[11], which in our view is not unlike the hawking black hole radiation effect6[2]. it is only at the large accelerations in the vicinity of black holes that we see effects that are barely detectable in our everyday experiences (e.g. at low accelerations, photon numbers are small and their wavelengths are very large [11]). davies sums up the present paradoxes in the following way: a further set of unsolved problems concerns the deeper significance of the relationship between acceleration and quantum vacuum noise. does the existence of “acceleration radiation” suggest a link between the quantum vacuum and inertia? haisch et al.41 claim that the very existence of inertia can be traced to the activity of vacuum noise on an accelerating particle. although this claim has not received widespread support, it is tempting to believe that the distinction between inertial and accelerated motion provided by acceleration radiation is telling us something fundamentally new about the principles of dynamics[2]. davies is suggesting that there is something new, maybe at a deeper level, that has been missed or perhaps misunderstood. in any case, researchers still find themselves having to make sense in the 3 dimensional domain of reality. verlinde [12], in making a conclusion on the origin of gravity, touches on the dynamic role of space in the creation of inertia and gravity. he posits that differences in entropy is the primary cause of gravity: other authors have proposed that gravity has an entropic or thermodynamic origin, see for instance [14]. but we have added an important element that is new. instead of only focussing on the equations that govern the gravitational field, we uncovered what is the origin of force and inertia in a context in which space is emerging. we identified a cause, a mechanism, for gravity. it is driven by differences in entropy, in whatever way defined, and a consequence of the statistical averaged random dynamics at the microscopic level. the reason why gravity has to keep track of energies as well as entropy differences is now clear. it has to, because this is what causes motion! the presented arguments have admittedly been rather heuristic. while we agree that entropic considerations are important and, indeed, relevant we view entropic effects as emergent and, therefore, only indicators of ‘causes’ resident at a deeper level, that is, beyond 3-dimensions. in what follows we attempt to explore these ideas further. become progressively ‘saturated’ with a form of quantum vacuum activity, thus progressively retarding the movement of quantum vacuum through it. this causes a progressive resistance to increasing acceleration. 6see also new scientist. “hawking radiation glimpsed in artificial black hole”. accessed december 21, 2012. http://www.newscientist.com/article/dn19508-hawking-radiation-glimpsedin-artificial-black-hole.html?full=true&print=true for parallels with the unruh effect. 70 philip j. tattersall: towards a new philosophical foundation for physics ... 3 the role of entropy in the quantum vacuum it has been suggested that gravity, rather than a ‘physical field’, emerges from quantum field theory (sakharov cited in [13]). entropy has been suggested to be an element of a similar ‘gravitational’ induction effect, emergent from the energy flux of unobservable degrees of freedom (jacobson cited in [13-14]). whilst entropy is not directly measureable [15] in the 3-dimensional domain it is nonetheless produced there. it is not a physical element of the thermodynamic equilibrium itself (as conceptualized in 3-dimensions); rather it is a virtual effect or flux produced during a reaction, which remains hidden. we propose that entropy, as understood by thermodynamic theory, and when emitted by bond resonances at equilibrium, is a disturbance or flux in quantum vacuum activity caused by the local presence of energetic flux and mass. entropy viewed in this way would be analogous to the casimir effect and is a part of the underlying mechanism for the induction of mass, gravity and inertia, as described later in this paper, and as explored in previous work [7]. we propose that the quantum vacuum of space, rather than being the source of gravitational influence via a polarisation mechanism in itself, also includes ‘hidden variables’, ‘dark’ forms of energetic flux. these ‘hidden variables’ would exist in states that, whilst not being accessible to direct detection or measurement in 3 dimensions when ‘entropic’, none-the-less play a role in the energetic interactions of mass when ‘gravitational’ or ‘inertial’. we suspect that what are currently theorized as dark energy, or dark matter,are varieties of these virtual manifestations permeating the quantum vacuum of space. it is possible that they will provide a route for the investigation of further dimensions and provide evidence of the so-called ‘hidden’ variables that would explain quantum phenomena, such as non-locality. in the case of non-locality, its apparent manifestation in 3 dimensions might suggest that, at a deeper level, ‘distance’ as such, may not exist; furthermore, that the notion of definitive ‘locality’ (as distinct from dynamic locality) is purely a manifestation of 3-dimensional ‘reality’. 3.1 some ideas on the phenomenon of gravity we contend that, in the vicinity of a body, there exists a disturbance in the quantum vacuum proportional to its mass7. in our view, in its interaction with matter, the quantum vacuum induces the emergence of gravity (see sakharov cited in [13]). this disturbance results from a casimir-like effect such that, in the vicinity of mass, there is an imbalance of quantum vacuum activity8. this in turn causes bodies in close proximity to move together, but not necessarily via 7mass is an extension in 3-d of interactions from within the quantum vacuum. mass ‘soaks’ up quantum vacuum activity. 8this is similar to the active gravitational mass idea of haisch and rueda (see [6]). advances in systems science and applications (2014) vol.14 no.1 71 the same mechanism as that proposed by le sage or brush cited in edwards [16]. when two bodies approach each other in ‘free space’, the strength of apparent ‘attraction’ is directly proportional to the magnitude of disturbance of the quantum vacuum activity in the intervening space, caused by the presence of the bodies. within this disturbance, the vacuum bodies induce a ‘suction-like’ reaction that causes them to move together. it is almost as though two bodies that appear to undergo gravitational attraction are pushed together by the higher vacuum activity in the surrounding space. although mass is mostly space there are still residual field, or energy centres observable at any particular scale (we loosely term these ‘horizons’ or ‘points of diminution’ or ‘diminished mass’). this explains why gravitational affect is related to size and mass. it also follows that the interaction of mass and the quantum vacuum can provide an explanation for the equivalence of inertial and gravitational mass. thus, two bodies of unequal mass will fall at the same acceleration in a ‘gravitational field’ because the casimir effect is proportionally influenced by an opposing davies-unruh effect. a comparatively large mass will experience a higher casimir effect, compared with a less massive body, but its ‘fall’ will also be proportionally retarded by an increase in its virtual mass (inertia) due to the davies-unruh effect. on the other hand while the less massive of the two bodies will experience a lower casimir ‘push’, it will also experience a proportionally lower inertial increase or retardation. 4 suggestions for physics – some tentative conclusions the observations and theorising that led to the idea of a quantum vacuum have now taken physics to an important new horizon and we are beginning to glimpse a ‘reality’ far stranger than the one we have become used to. this poses new challenges, because theorizing is tending to take the place of experiments and observations due to the constraints of working in 3 dimensions. on the other hand, certain empirical observations, such as those from the moving mirror experiment, casimir, unruh, the double slit experiment and from non-locality experiments in general are providing tantalizing glimpses of what appears to be a deeper world. however, while these sophisticated experiments are being used to explain the many new and exciting ideas, such as the many world theory, string theory, and so on, their conclusions are limited by being interpreted in only 3 dimensions. in order to explain their strangeness, physicists are forced into making their theorizing processes more elaborate, thus creating a new virtual reality as a surrogate. in the meantime the search continues for the ground-breaking observations and or experiments that might tell us how it all works and what it all means. the excitement over the work on the higgs boson is a recent example. the experiments and observations employed by physics are cosmic in scale, now 72 philip j. tattersall: towards a new philosophical foundation for physics ... that we can take excursions, via our telescopes or interstellar vehicles, into black holes, neutron stars and far-off galaxies. despite some wonderful observations and measurements made from these ‘natural’, massively powerful laboratories, we still try to explicate the data using only 3 dimensions. thus, the problem remains. while we are getting tantalizing glimpses of a mysterious, ‘hidden world’, how can we ‘explain’ them without having to devise increasingly complex experiments and ever more complex theories? in this essay, our missions not to offer a clear solution but, rather, to invite a philosophical conversation about the nature of space. this, we argue,is needed in order to invite the much-needed creative input and ideas that will be needed to define the new order. such a platform is already being suggested by many researchers and distinguished commentators including davies [2], rueda and haisch [6], johnson & walker[17], wuthrich [13] and david tong [18]. for instance, tong’s recent article in scientific american has already initiated an important debate that is at the heart of physics. moreover, its proposition, namely, that reality is a non-quantized continuum is germane to this essay. in our view, this is an exciting and much-needed first step, as we continue to move into a new era of physics. the paper by rafelski et al [19] demonstrates the depth of interest in the problem of the quantum vacuum and in our view straddles an important physicsphilosophy coupling. in their discussion of the ‘three riddles’ at the nexus of quantum theory, particle physics and cosmology the authors bring to the fore not only plausible arguments, but as important suggest new directions for inquiry into the nature of space. the authors argue: contemporary physics faces three great riddles that lie at the intersection of quantum theory, particle physics and cosmology, they are 1. the expansion of the universe is accelerating the extra factor of two appears in the size. 2. zero-point fluctuations do not gravitate a matter of 120 orders of magnitude. 3. the “true” quantum state does not gravitate. the latter two are explicitly problems related to the interpretation and physical role and relation of the quantum vacuum with and in general relativity. their resolution may require a major advance in our formulation and understanding of a common unified approach to quantum physics and gravity. to achieve this goal we must develop an experimental basis. so not only is more research needed, but moreover a new perspective from which to formulate ideas and problems is also needed. we suggest that new and perhaps alternative ontologies are required that encourage new, innovative and creative insights. advances in systems science and applications (2014) vol.14 no.1 73 the maturing philosophy of natural inclusion (ni) offers one such perspective. alan rayner [8] the founder, describes ni in these words: a term introduced by alan rayner and ted lumley, in conversation with others, intended to distinguish a form of reasoning that includes intangible presence and so is more comprehensive, comprehensible and realistic than abstract rationality. eventually it became necessary for alan rayner to distinguish his understanding of inclusionality as ‘natural inclusionality’, which takes account of local influence and identity, from ted lumley‘s understanding, which considers only nonlocal influence and regards locality as illusory. space in the context of ni is described as [8]: according to the logic of natural inclusionality, ... space cannot be pluralized into discrete particularities; it can only be distinguished into distinct, dynamically and permeably bounded regions. this is because a presence that has no resistance can neither be cut nor resisted by a tangible frame. it is inescapably present throughout and beyond the boundaries of tangible figures. a tangible frame is an inclusion of and is included in space but the frame is not the space. the tangible frame can move (or be moved) and be cut, but not the space. when the frame moves the space stays where it is: in relative terms by remaining still space permeates freely through the frame, the frame does not cut through the space. moreover, if the frame is to move without being forced to do so by a force situated somewhere outside of it, it must have the capacity for movement within itself, i.e. the frame is itself a manifestation of energy, not inert structure-it is a variably fluid ‘framing’, not a permanent, absolutely rigid ‘framework’. this tangible ‘framing’, or ‘dynamic interfacing’, has to be present for form to be distinguishable in a feature-full cosmos, but it can neither ‘occupy’ nor ‘exclude’ the space that it includes and is included in. ni accommodates continuousness and therefore perhaps at certain scales posits that reality is in fact non-quantum in nature, as suggested by david tong [18]. the introduction of such philosophical innovations would, in our view, invite new and perhaps very productive conversations leading to ideas and maybe new insights that might create conditions for thinking about old and current problems in new ways. after all that is what has happened many times before as ‘breakthroughs’ have arisen in the most unexpected of ways. in this essay metaphor and analogy have been employed as they help to invite new ideas and imaginings. our approach is therefore very much in keeping with the inquiry trajectory proposed by bohm [20] who suggested that future scientists would be less dependent on mathematics and modelling as they begin to draw upon new approaches that in the end would lead to a merging of art and science. acknowledgements 74 philip j. tattersall: towards a new philosophical foundation for physics ... the authors acknowledge the support of emeritus professor john wood who offered several very helpful suggestions and recommendations during the final stages of manuscript preparation. also our thanks to dr. alan rayner who offered useful suggestions during the early drafting of this paper. references [1] rayner a.d.m. (2011), “space cannot be cut: why self-identity naturally includes neighbourhood”, integrated psychological behaviour, vol.45, pp.161184. [2] davies p.c.w. (2001), “quantum vacuum noise in physics and cosmology”, chaos, vol.11, no.3, pp.539-547. [3] brumfiel g. (2011), “moving mirrors make light from nothing”, accessed november 15, 2012. http://www.nature.com/news/2011/110603/full /news.2011.346.html [4] huang, k. (1952), “on the zitterbewegung of the dirac electron”, american journal of physics, vol.20, no.8, pp.479-484. [5] chown,m. (nd), “mass medium why are loaded fridges difficult to budge? because empty space impedes them”, accessed december 15, 2012. http://www.calphysics.org/articles/chown2007.html [6] rueda a. and haisch b. (2005), “gravity and the quantum vacuum inertia hypothesis”, annals of physics (leipzig), vol.14, no.8, pp.479-498. [7] rayner a. d. m., b., peleshok d. and tattersall p. (2012), “place-time: the flow geometry of space”, accessed december 1, 2012, http://www.bestthinking.com/articles/science/math/place-time-theflow-geometry-of-space. [8] rayner a. d. m. (2012), “a natural inclusional glossary of terms”, accessed january 15, 2013. http://www.bestthinking.com/articles/society and humanities/languages/english language/a-natural-inclusional-glossary -of-terms. [9] davies p.c.w. (1975), “scalar particle production in schwarzschild and rindler metrics”, journal of physics a: mathematical and general, vol.8, no.4, pp.609-616. [10] unruh w.g. (1976), “notes on black-hole evaporation”, physical review d, vol.14, no.4, pp.870c892. advances in systems science and applications (2014) vol.14 no.1 75 [11] mcculloch m. e. (2012), “testing quantised inertia on galactic scales”, accessed january 2, 2013. http://arxiv.org/abs/1207.7007v1 [physics.genph] [12] verlinde e. p. (2010), “origin of gravity and the laws of newton”, accessed november 30, 2012. http://arxiv.org/abs/1001.0785 [hep-th] [13] wuthrich, c. (2005), “to quantize or not to quantize: fact and folklore in quantum gravity”, philosophy of science, vol.72, pp.777-788. [14] jacobson t. (1995), “thermodynamics of spacetime: the einstein equation of state”, physical review letters,vol.75, no.7, pp.1260-1263. [15] angrist s. w., and helper, l.(1973), laws of energy and entropy: order and chaos, ringwood, victoria: pelican books. [16] edwards m. r. (2007), “photon-graviton recycling as a cause of gravitation”, apeiron, vol.14, no.3, pp.214-233. [17] johnson g.w., and walker, m.(2005), “sir michael atiyah‘s einstein lecture: the nature of space”, notices of the american mathematical society, vol.53, no.6, pp.674-678. [18] tong d. (2012), “the unquantum quantum”, scientific american, vol.307, no.6, pp.32-35. [19] rafelski j., labun l.,hadad y., and chen p.(2009), “quantum vacuum structure and cosmology”, accessed december 23, 2012. http://arxiv.org/pdf/0909.2989v1.pdf [20] bohm‘s alternative to quantum mechanics, david albert scientific american may 94. last words of a quantum heretic, john horgan, new scientist 29 feb 93 accessed august 12, 2013. http://www.dhushara.com/ book/quantcos/bohm/bohm.htm. corresponding author philip j. tattersall can be contacted at: soiltechresearch@bigpond.com. adv syst sci appl 2020; 03; 24-35 published online at https://ijassa.ipu.ru. original russian text © m.a. gorelov, 2018, published in upravlenie bol’shimi sistemami / large-scale systems control, 2018, no. 72, pp. 6-26. the “value at risk” principle in hierarchical game mikhail gorelov1* 1) dorodnicyn computing centre, federal research center “computer science and control”, russian academy of sciences, moscow, russia e-mail: griefer@ccas.ru abstract: a hierarchical game of two persons with random factors is considered. it is assumed that the top-level player has the right of the first move. it is believed that the lower level player at the time of decision making knows exactly the realization of the random factor and the choice of partner. and the top-level player at the time of decision making knows only a probabilistic measure on the set of values of an uncertain factor. the principle of optimality is new: it is believed that a top-level player is ready to neglect some of the “unpleasant” events, the total probability of which is given, but otherwise he is careful. under these assumptions, the maximum guaranteed result of the top-level player is calculated. the structure of strategies providing such a result is clarified. two cases were investigated: a game with and without feedback. to solve the problem, an original definition of the maximum guaranteed result is proposed. it is equivalent to the classical definition, but is simpler. using this technique, solving of the problem reduces to identical transformations of the formulas for predicate calculus. as a result of the solution, the optimal strategy and the set of “unpleasant” cases which are excluded from consideration search task is reduced to calculating multiple maximins on finite-dimensional spaces. in this case, the operation of calculating the expected value with respect to given probabilistic measure is considered to be “elementary”. models of this type can have different interpretations. one can use them for methodological justification of the principle of maximum guaranteed result. one can use them when solving risk management tasks. one can consider them as models for managing the “customer base” of the service company. the proposed method allows to study such models at a qualitative level, and in some cases to obtain quantitative results. keywords: informational theory of hierarchical systems, hierarchical games, decision making under risk, maximal guaranteed payoff, risk management 1. introduction a study of hierarchical games with uncertain factors was started in the early seventies of the last century [8,9,15]. around the same time, similar models were investigated in the theory of active systems [4,5,18] and in the theory of contracts [2,3,16]. in this case, two methods of eliminating uncertainty were mainly studied. in one of them, it is assumed that the players know only sets of possible values of uncertain factors and they are careful i.e. reckons upon the worst option for themselves. another considered that the probability measure is defined on the set of uncertain factors, and the players are risk-neutral, that is they are ready to be guided by the expectations of their payoffs. a third, in a sense, intermediate way of eliminating uncertainty is also possible. in the financial engineering it received title principle “value at risk” [6,17]. the development of these ideas can be found in the monograph [1] and other works of the same author. at a meaningful level, its essence can be explained as follows. i do not think that making decisions in everyday life, people are very often appreciate some probability distributions, * corresponding author: griefer@ccas.ru the “value at risk” principle in hierarchical game 25 copyright ©2020 assa. adv. in systems science and appl. (2020) and even more so calculate mathematic expectations. and on a professional level, i have never had to deal with the customer who formulates the problem in probabilistic terms. but the tendency to the principle of guaranteed results, customers several times clearly formulated. but the principle of maximum guaranteed result has one not a very attractive property. if it is conducted quite consequentially, the possibility cannot be ruled out that the player “does not fall down a brick on his head” or something else unpleasant happens. and constantly focusing on such cases, it is hardly possible to make really effective decisions. in practice and in theory [7], from this situation, the next way out is used. a part of the completely “fatal” values of the uncertain factor is excluded from consideration, and with respect to the remaining part, the principle of maximal and minimal guaranteed result is used. on what basis are some possibilities excluded from consideration? probably, it happens because the operating party regards them as “unlikely”. of course, there is the temptation to postpone the work for exclusion of unfavorable factors from the level of the model building to the level of its research. for this purpose, it is necessary the operating party accept some probability distribution and said that with a given probability  it wants to get a payoff not less than the value of . of course, it is preferably to the value of  to be larger. thus, we come to the statement of the problem considered below. it is unlikely that all this makes sense in the analysis of the problems at the household level. but in the analysis of business decisions risk management problems has recently become very relevant, if not fashionable. moreover, the risk assessment method discussed above is one of the two most popular (as far as i know, game-theoretic models in which risk is estimated using dispersion also have not yet been investigated). but this method is usually used to investigate problems of centralized decision making under risk. game-theoretic models of this kind, apparently, have not yet been studied. further presentation is constructed as follows. the next section gives a formal statement of the problem. the rest of the article is devoted to its “solution”. by tradition the solution of the problem of calculating of maximal guaranteed result in hierarchical game with a “complicated” structure is regarded as it’s reduction to some “elementary” operations such as computation of maxima and minima on the “simple” sets. we will follow this tradition. in a problem under consideration “non-elementary” operation of selection of a set of uncertainties to be excluded from consideration arises already in games not endowed with additional structure. section 3 is devoted to their study. in the fourth paragraph, the more traditional problem of finding the maximal guaranteed result in a “feedback games” is solved. 2. games with random factors so, we begin to describe the simulated conflict. game with random factors will be hereinafter called a six-tuple  =  u , v , a , g , h ,  . here, u , v and a are the sets, g – function which maps the cartesian product u  v to the set of real numbers , : ,h u v a  → and  is probability measure on the set a. these constructions are interpreted as follows. it is assumed that two participants take part in the game, which we will call the first and second players. a set u is interpreted as the set of controls of the first player, a set v – the set of controls of his partner. we assume that the interests of the first and second players are described by the desire to maximize the functions g and h, respectively. the value of the indefinite factor   a is chosen by some third party – nature. this choice is made randomly in accordance with the distribution . we make the traditional technical assumptions, which significantly simplify the further narration. the sets u, v, and a are assumed to be endowed with topologies and compact ones. the functions g and h are assumed to be continuous. measure  will be considered to be borel. 26 m.a. gorelov copyright ©2020 assa. adv. in systems science and appl. (2020) unfortunately, not all results can be obtained in such a general assumptions. additional conditions on game under consideration, if required, will be indicated when corresponding results will be formulated. game  describes the possibilities and interests of the players. for the model completion one must also describe the dynamics of decision-making and the attitude of the players to the existing uncertainty. let’s start with the interpretation. we believe that all the game  options are exactly known to the first player. we assume that events take place as follows. at first, the first player chooses his control u  u. then, the specific value of the indefinite factor   a is realized (in accordance with the given probabilistic measure ). the values of u and  become known to the second player. thus, for the second player, no uncertainty remains. consequently, for him all the controls v  v will be divided into “reasonable”, in case of the choice of which he will receive a payoff greater than or equal to a certain number of , and “unreasonable” , the choice of which promises him payoff smaller then  (of course, number  depends on u and ). this principle of behavior is known to the first player. but he is careful and therefore he reckons upon the worst result that can happen when “reasonable” choice of partner will be made. but since the value  is not known to the first player, for him this result is a random variable. the attitude of the first player to this uncertainty is as follows. he agrees to exclude from consideration a certain number of “force majeure” events, the total probability of which does not exceed a given value 1 – . but other than, he focuses on the worst case for himself and wants to get the maximal guaranteed result. formally, the above is described as follows. definition 2.1: let a real number   [0,1] be given . number  is -guaranteed result of the first player in the game , if there exist a measurable set b  a, measure (b) of which is greater than or equal to , and such strategy u  u, that for every   b there exist a number , for which the following conditions are hold true: 1 . there exists w  v for which h(u,w,) ≥ ; 2 . for any v  v, either g(u,v) ≥  or h(u,v,) < . supremum of -guaranteed results of the first player is called its maximal -guaranteed result. remark 2.1: it would be possible to refuse the assumption of measurability of set b in this definition, replacing measure (b) with the corresponding outer measure. it will be seen from what follows that, under the assumptions that the measure  is borel, and the function h is measurable, such a modification of the definition do not essentially change anything. remark 2.2: the definition 2.1 is a modification of definition of maximal guaranteed result, proposed by the author in [10]. in more conventional terms the maximal -guaranteed result can be defined by the formula ( , ) supsupinf min ( , ) b v br ub u u g u v    wherein the outer supremum is taken over all measurable subsets b of the set a, for which (b) ≥ , and  ( , ) : ( , , ) max ( , , ) . w v br u v v h u v h u w    =  = the “value at risk” principle in hierarchical game 27 copyright ©2020 assa. adv. in systems science and appl. (2020) the proof of the equivalence of these definitions only insignificant differs from the reasoning in [10]. a significant part of this work will, in fact, be done in the next section. for more complex problems definition 2.1 is more convenient, therefore we will use it. remark 2.3: all probabilities in this paper can be regarded as subjective, namely, as an evaluation of the operating party (first player) the feasibility of certain events. in the future, no results such as the law of large numbers are used. therefore, there is no need to take care of any kind of statistical stability. indeed only readiness of operating party to eliminate uncertainty by the method described in the definition 2.1 is important. 3. game without feedback calculation of maximal -guaranteed result, for example, by the formula from the remark 2.2 involves computation of least upper bound on the class of subsets of the set a. standard methods for calculating of such supremum not exists even for the relatively simple case where the set a is a segment. in this section, we simplify the solution of the problem by replacing the operation of computing such an upper bound with the operation of calculating the expected value. traditionally such an operation is considered to be “elementary”. introduce the following notation ( , ) max ( , , ). v v m u h u v   = let  be -guaranteed result of the first player in the game . choose a set b  a and strategy u  u, the existence of which is provided for the definition 2.1. fix an arbitrary   b. for strategy w  v, the existence of which postulate the item 1 of definition the inequality h(u,w,) ≥  is satisfied, the more this inequality must be satisfied for the strategy w0 determining by the equality h(u,w0,) = m(u,). consequently the number , appearing in definition 2.1, must satisfy the condition   m(u,). therefore, if the item 2 of definition is satisfied for someone value , it moreover holds for  = m(u,). but with such a value of  item 1 also is obviously satisfied: it suffices to choose, for example, w = w0. thus, the number of  is -guaranteed results of the first player in the game , if there is a measurable set b  a , measure (b) of which is greater than or equal to , and such a strategy u  u , that for every   b and any v  v either g(u,v ) ≥  or h(u,v,) < m(u,). consider the set  ( , ) : ( , , ) max ( , , ) . w v br u v v h u v h u w    =  = if v  br(u,), then h(u,v,) = m(u,); therefore , by the second item of definition 2.1, the inequality g(u,v) ≥  must hold. conversely, if the last inequality holds for all v  br(u,), then the number  is -guaranteed result. indeed, for w  br(u,) and  = m(u,) the first item of definition is satisfied. for the same values of  and v  br(u,) the item 2 is met because of in this case h(u,v,) < m(u,), and for v  br(u,) it satisfies as by the assumption g(u,v) ≥ . thus, the number of  is -guaranteed result of the first player in the game , if there exist a measurable set b  a, measure (b) of which is greater than or equal to , and such a strategy u  u, that for every   b and any v  br(u,) the inequality g ( u , v ) ≥  is true. the same condition can be formulated equivalently: the number  is -guaranteed result of the first player in the game , if there exist a measurable set b  a, measure (b) 28 m.a. gorelov copyright ©2020 assa. adv. in systems science and appl. (2020) of which is greater than or equal to , and such a strategy u  u, that for any   b, the inequality ( , ) min ( , ) v br u g u v     (3.1) holds. consider the set   ( , ) ( ) : min ( , ) v br u c u a g u v     =   . since condition (3.1) holds for all   b, the set b must be contained in the set c(u), and hence , the condition (c(u)) ≥ (b) ≥  must holds. conversely, if the condition (c(u)) ≥  is valid for some strategy u, then condition (3.1) will be satisfied for all   b = c(u) , and hence  is -guaranteed result. let define the function (x) by the condition 1, if 0, ( ) 0, if 0. x x x   =   the measure of the set c(u) is equal to ( ) ( , ) min ( , ) , v br u g u v      − where the symbol  designate operator of calculating mathematical expectations with respect to measure . this immediately yields the following result. theorem 3.1: in order for the number  to be -the guaranteed result of the first player in the game , it is necessary and sufficient that either ( ) ( , ) max min ( , ) , v br uu u g u v       −  (3.2) or ( ) ( , ) sup min ( , ) , v br uu u g u v       −  if the upper bound in the last formula is not reached. remark 3.1: in the papers [11] and [12] in the same manner was formulated conditions characterizing maximum guaranteed result for games with undefined interval uncertainty and games with risk-neutral first player. in earlier papers [14] and [15], although with some additional assumptions explicit formulas were obtained for the maximum guaranteed results in these problems. in the problem considered in this paper, such an alternative is not visible even for simpler analogue problem of optimization corresponding the game , wherein a set v consists of one point. remark 3.2: formula (3.2) could be a starting point in an attempt to give a more traditional definition of maximal -guaranteed result. but such a definition in this case would require clarification. and apparently, this explanation inevitably would be similar to the definition 2.1. in addition, quite a “classical” form of this definition can’t be given, because the number  is part of the argument of function . thus, in this case it is not very convenient to the “value at risk” principle in hierarchical game 29 copyright ©2020 assa. adv. in systems science and appl. (2020) follow the traditions. in the next section, the advantage of the new definition will become even more obvious. in order not to be distracted by the technical details in the proof of theorem 3.1 was left a gap. to fill it, it is necessary to prove the following statement. lemma 3.1: for every u  u, the set c(u) is measurable. proof. fix u  u. it is sufficient to prove that the function ( , ) ( ) min ( , ) v br u g u v     = is measurable. to do this, it sufficient to prove that the function –() = 0 – () is measurable. therefore, it suffices to prove that for any  the set    0 ( , ) ( ) : ( ) : min ( , ) v br u c u a a g u v         =  −  − =   is measurable (for convenience, this part of the proof uses only the facts explicitly stated in [13]). since the measure  is assumed to be borel, it is sufficient to prove that the set c0(u) is open. suppose the contrary. then there exist   c0(u) and a sequence , ... such that lim k k   → = and  k  c0(u) for all k = 1,2, ... the set br(u, k) is defined by the condition of equality type, so it is closed due to the continuity of function h. since a set v is assumed to be compact the set br(u,k) also will be compact. hence, at some point vk  br(u,k) the minimum of the function g(u,v) on the set br(u,k) is achieved. since by assumption k  c0(u), the inequality g(u,vk)   holds. since the set v is compact it is possible without loss of generality to assume that the sequence v1, v2, ... converges to an element v  v. fix an arbitrary w  v. since vk  br(u,k), the inequality h(u,vk,k) ≥ h(u,w,k) holds. since the function h is continuous, going to the limit in this inequality, we obtain h(u,v,) ≥ h(u,w,). since w is arbitrary, it follows that v  br(u,). and going to the limit in the inequality g(u,vk)  , we get g(u,v)  , and even more so ( , ) min ( , ) v br u g u v     which contradicts the condition   c0(u). the obtained contradiction proves the lemma. □ thus, theorem 3.1 is completely proved. remark 3.3: the assumptions formulated in the previous section about the topological and metric structure of the game  can be changed, and, perhaps, simply weakened. the question of the weakest assumptions under which lemma 3.1 and further results of this type remain true, is of some interest, but beyond the scope of this article. the most interesting model examples certainly satisfy the conditions formulated above, so we omit further discussion. remark 3.4: the proof of lemma 3.1 is quite standard, but rather long. for this reason, further, the proof of similar results is omitted. a question of search optimal strategy of the first player in this game is quite meaningful. given the results obtained, the answer to it is not difficult to find. first of all, we note that the supremum in the definition of maximal -guaranteed result can’t be achieved. therefore, there may not exist a strategy allowing one to obtain such a result with probability . this effect is understandable because similar fact takes place already in the game with no uncertainty (i.e., with single-point set a). if the number  is a -guaranteed result, then any solution of the inequality 30 m.a. gorelov copyright ©2020 assa. adv. in systems science and appl. (2020) ( ) ( , ) min ( , ) , v br u g u v       −  if the upper bound in formula (3.2) is reached, or the inequality ( ) ( , ) min ( , ) , v br u g u v       −  otherwise, is the desired strategy. in both cases, the inequalities have solutions, since by assumption  is -guaranteed result. one of sets b “suitable” for this strategy can be defined by the condition b = c(u). of course, such selection of the set b is not the only possible. however, in the general case, the same applies to the choice of the optimal strategy u. 4. game with feedback consider another game * = u*, v*, a, g*, h*, , in a certain way related to the game . denote by (x,y) the class of all functions from the set x into the set y. let u* = (v  a,u), v* = v  a, and the functions g* and h* are determined by the conditions g*(u*,v*) = g(u*(v,),v), h*(u*,v*,) = h(u*(v,),v,), where v* = (v,). the set a and the measure  on it are the same as in the game . these constructions can be interpreted as follows. players choose their “physical” controls from the sets u and v. but by the time of the selection of its control u  u the first player receives reliable information about control v  v selected by his partner. in addition, the second player can transmit to the first player information on the realized value of the uncertain factor. however, this information does not have to be reliable, that is, the second player has the right to select some message   a, which he will transmit to the partner. his physical control u*(v,) the first player chooses on the basis of all the information received, and both players’ payoffs depend only on the physical choices made by them, and do not depend on information they exchanged. game  has the same structure as the game , so the question can be put of finding the maximal -guaranteed result in this game. we will deal with this task. when analyzing this problem, using the english language is already quite inconvenient. therefore, we turn to the language of predicate calculus. for the game  the definition of -guaranteed result  will look as follows:     * * * * ( , ) : ( ) & & : ( ( , ), , ) & & ( ( , ), ) ( ( , ), , ) . b u v a u b b w v a h u w w v v a g u v v h u v v                                   (4.1) this formula is not “elementary”, because it contains one existential quantifier refers to a class of all measurable subsets b of the set a, and another – to the class of all functions u from the set v  a to the set u. it can be simplified, but one has to make the following assumption. hypothesis 4.1: the function h is such that there exists such control up  u, that for any v  v and any   a the equality ( , , ) min ( , , )p u u h u v h u v   = holds. in fact, it assumes the existence of a universal (independent of ) strategy of punishing the second player by first. the “value at risk” principle in hierarchical game 31 copyright ©2020 assa. adv. in systems science and appl. (2020) now we can start converting the formula (4.1). as in the previous section let’s start with the specification of the value . put  ( ) ( , ) : ( , ) ,h u v u v g u v =    ( , ) ( ) ( , ) max ( , , ). u v h l h u v      = in meaningful terms, h() is the set of “acceptable” outcomes for the first player. the number l(,) characterizes the maximum payoff that the second player can get, provided that the first one somehow ensures that he gets an “acceptable” result (of course, this maximum payoff depends on the indefinite factor ). according to the condition (4.1), there exist a set b and a function * such that     * * * : ( ) & : ( ( , ), , ) & & ( ( , ), ) ( ( , ), , ) , b b w v a h w w v v a g v v h v v                                  (4.2) fix such a set and a function. then there exists a function u*, for which the condition is satisfied:     * * * ( ) & & : ( ( , ), , ) ( , ) & & ( ( , ), ) ( ( , ), , ) ( , ) . b b w v a h u w w l v v a g u v v h u v v l                               (4.3) let’s prove it. let the condition (4.2) be satisfied. then the set h() is not empty. indeed, let’s fix any   b. then by virtue of condition (4.2) we have h(*(w,),w,) ≥ . then, due to the same condition g(*(w,),w) ≥  and therefore (*(w,),w)  h(). for each   a let’s fix a pair (u,v)  h() such that h(u,v,) = l(,). put * * , if and , ( , ) ( , ) in other cases. u v v u v v         = = =   then for any   a a condition *: ( ( , ), , ) ( , )w v a h u w w l         is satisfied (one can take w = v and  = ). in addition, from the inequality h(*(w,),w,) ≥  it follows that   l(,), so condition (4.2) implies * *( ( , ), ) ( ( , ), , ) ( , ).g v v h v v l          (4.4) if *(v,)  u*(v,), then by construction inequality g(u*(v,),v) ≥  is true, hence the condition * *( ( , ), ) ( ( , ), , ) ( , )g u v v h u v v l        (4.5) is satisfied. otherwise, conditions (4.4) and (4.5) are equivalent. thus, it is proved that condition (4.2) follows condition (4.3). the reverse implication is obvious. therefore, conditions (4.2) and (4.3) are equivalent. exactly the same “modification” of the strategy of the first player proves that the condition     * * * * ( , ) ( ) & & : ( ( , ), , ) ( , ) & & ( ( , ), ) ( ( , ), , ) ( , ) b u v a u b b w v a h u w w l v v a g u v v h u v v l                                   is equivalent to a simpler condition 32 m.a. gorelov copyright ©2020 assa. adv. in systems science and appl. (2020)   * * * ( , ) ( ) & & ( ( , ), ) ( ( , ), , ) ( , ) . b u v a u b b v v a g u v v h u v v l                         the relevant reasoning is practically the same as the above, so we omit them. change the order of the generality quantifiers in the last formula:   * * * ( , ) ( ) & & ( ( , ), ) ( ( , ), , ) ( , ) . b u v a u v v a b g u v v b h u v v l                         now, one can use the structure of the set of strategies of the first player to change the order of quantifiers of existence and generality:  : ( ) & ( , ) ( , , ) ( , ) .b v v a u u b g u v b h u v l                    the variable  in square brackets has “disappeared”, so this formula can be further simplified:  : ( ) & ( , ) ( , , ) ( , ) .b v v u u b g u v b h u v l                 denote  ( ) : max ( , ) . u u e v v g u v   =   then the previous condition can be rewritten in an equivalent form: ( ) : ( ) & ( , , ) ( , ),b v e u u b b h u v l               or ( ) ( ) & : ( , , ) ( , ).b v e b u u b h u v l               now let us use hypothesis 4.1 to change the order of the generality and existence quantifiers: ( ) ( ) & : ( , , ) ( , ).b v e b b u u h u v l               once again changing the order of the generality quantifiers, we get ( ) & ( ) : ( , , ) ( , ).b b b v e u u h u v l               let 1, if 0, ( ) 0, if 0. x x x   =   replacing the quantifiers of generality and existence with the operators of maximum, minimum and expectation, how was it done in the previous section, we get the following result. theorem 4.1: let hypothesis 4.1 be fulfilled. then in order for the number  to be a -guaranteed result, it is necessary that ( )( ) ( ) inf max ( , ) ( , , ) , v e u u l h u v          −  and it is sufficient that ( )( ) ( ) inf max ( , ) ( , , ) . v e u u l h u v          −  (4.6) the “value at risk” principle in hierarchical game 33 copyright ©2020 assa. adv. in systems science and appl. (2020) remark 4.1: there are games  for which the least upper bound of the numbers  satisfying the necessary condition in the theorem differs from the least upper bound of the numbers  satisfying the sufficient condition. for such games, the results obtained do not give a definitive answer to the question, what is the maximum -guaranteed result? but it does not make sense to refine the obtained necessary and sufficient conditions in this case, since it is clear that for such games the problem of calculating the maximal -guaranteed result is not stable with respect to small changes in the parameters of the game . therefore, additional study this problem is required, which is beyond the scope of this article. however, such games are in a sense “exceptional”. remark 4.2: the acceptance of hypothesis 4.1 from a formal point of view seems rather restrictive, since it is essentially assumed that all functions from some parametric family have saddle points. but in many meaningful models, its use seems justified. for example, if the first player chooses a price, he can choose it as the minimum possible, if he allocates a resource to a partner, he can allocate it “at a minimum” under all conditions. a. f. kononenko generally believed that in economic models, hypothesis 4.1 is always fulfilled. i do not quite share this view, for the principle of proportion of the severity of the punishment to the severity of the offence must be considered. but in this case it is impossible to abandon this hypothesis. in this sense, the problem considered in this paper is more complicated than the problem with a risk-neutral player, where, as shown in [12], a similar hypothesis can be abandoned by introducing a “gauge” additive to the payoff function of the second player. analysis of the proof shows that hypothesis 4.1 can be replaced by the following assumption. hypothesis 4.2: there exist a control u  u such that the inequality max ( , , ) ( , ) v v h u v l     holds for all   a. since the inequality in hypothesis 4.2 is strict, it cannot be argued that it is weaker than hypothesis 4.1. however, it is quite possible to expect that there are quite a lot of meaningful models in which hypothesis 4.2 holds and hypothesis 4.1 does not. hypothesis 4.1 is accepted as the main one, since it is easier to interpret. if the sufficient condition (4.6) of theorem 4.1 is satisfied for some number , then the results obtained above allow us to construct one of the strategies that allow us to obtain the result  with probability . as the set b, which appears in the definition of the maximum guaranteed result, we can take the set ( )  ( ) : inf max ( , ) ( , , ) 0 . v e u u b a l h u v        =  −  for any  from the set b thus chosen let’s choose an arbitrary pair (u,v) from the set h() satisfying the condition h(u,v) = l(,). put * , if и , ( , ) in all other cases.p u b v v u v u      = =   it is directly verified that the so-defined strategy u* is the desired one. the interpretation of these constructions is standard. the second player is asked to select control v, if the value of the undefined factor   b has been realized, and to report the true information about this factor. in this case, the first player promises to use the “encouraging” control u. otherwise, the second player faces punishment. controls u and v are chosen so 34 m.a. gorelov copyright ©2020 assa. adv. in systems science and appl. (2020) that the message of reliable information is really beneficial to the second player. cases   b are excluded from consideration by the first player. therefore, in particular, for such values , the set h() can be empty. in these cases, the strategy of punishment is used for greater reliability. 5. conclusion in addition to the “methodological” interpretation described in the introduction, the studied model has another, perhaps more interesting one. the value 1 –  in this model can be naturally considered as a measure of risk. thus, the model explicitly describes both the “yield”, estimated by the value of the payoff g(u,v), and the risk. this seems important enough. it is quite natural to assume that the value  is the control of the operating party (the first player), along with u. here we can assume that the order of decision-making is as follows. first, the first player fixes the value  and the strategy u (or u*, respectively), then the value of the uncertain factor  is realized, then the second player chooses his control. in this case, it does not matter whether the second player receives information about the selected value , since his payoff does not depend on him. however, the model is not fully formed, because in this case it is natural to assume the presence of two criteria: the risk measure 1 –  and the corresponding -guaranteed result. it follows directly from the definition that, as  increases, the corresponding -guaranteed result does not increase. the choice of balance between the two criteria is left to the operating party. but if the researcher of the operation has an effective way of calculating the -guaranteed result, it will be a serious help in solving this problem. the described method of risk accounting, of course, is not the only one possible. but already on the basis of the studied model it is possible to construct other meaningful problem statements. for example, one can assume that the operating party selects a number of values  1, 2,…, n. for each strategy u  u it is possible to find the payoff of the first player  i(u) which with a probability of  i is provided with the choice of strategy u under the rational actions of a partner. and then a multi-criteria problem is solved with the criteria  1(u), (u)2,…, (u)n. to these criteria, one can add the expected value of a guaranteed payoff of the first player. thus, we get a fairly wide range of statements, each of which can be “tried on” for the simulated situation. apparently, the key step in the study of the corresponding problems is made in this article. references 1. agasandyan, g.a. (2011). primeneniye kontinual'nogo kriteriya var na finansovykh rynkakh [application of the var continuum criterion in financial markets]. moscow, russia: ccas, [in russian]. 2. bolton, p. & dewatripont, m. (2005). contract theory. mass.: mit press. 3. bremzen, a.s. & guriev, s.m. (2005). konspekty lektsiy po teorii kontraktov [lecture notes on contract theory]. moscow, russia: resh, [in russian]. 4. burkov, v.n. (1977). osnovy matematicheskoy teorii aktivnykh system [fundamentals of the mathematical theory of active systems]. moscow, ussr: nauka, [in russian]. 5. burkov, v.n. & novikov, d.a. (1999). teoriya aktivnykh sistem: sostoyaniye i perspektivy [the theory of active systems: state and prospects]. moscow, russia: sinteg, [in russian]. the “value at risk” principle in hierarchical game 35 copyright ©2020 assa. adv. in systems science and appl. (2020) 6. dempster, m.a.h. (ed.). (2002). risk management. value at risk and beyond. cambridge: cambridge university press. 7. germeĭer, yu.b. (1971). vvedeniye v teoriyu issledovaniya operatsiy [introduction to operations research theory]. moscow, ussr: nauka, [in russian]. 8. germeĭer, yu.b. (1986). nonantagonistic games. dordrecht, germany: d. reidel publishing co. 9. gorelik, v.a., gorelov, m.a. & kononenko, a.f. (1991). analiz konfliktnykh situatsiy v sistemakh upravleniya [analysis of conflict situations in control systems]. moscow, ussr: radio i svyaz', [in russian]. 10. gorelov, m.a. (2011). maximal guaranteed result for limited volume of transmitted information. automation and remote control, 72(3), 580–599, doi: 10.1134/s000511791103009x. 11. gorelov, m.a. (2016). iyerarkhicheskiye igry s neopredelennymi faktorami [hierarchical games under uncertainty]. upravleniye bol'shimi sistemami, 59, 6–22, [in russian]. 12. gorelov, m.a. (2016). iyerarkhicheskiye igry so sluchaynymi faktorami [hierarchical games under risk]. upravleniye bol'shimi sistemami, 63, 87–105, [in russian]. 13. kolmogorov, a.n. & fomin, s.v. (1961). elements of the theory of functions and functional analysis, volume 2. albany, ny: greylock press. 14. kononenko, a.f. (1973). the role of information on the opponent's target function in a two-person game with a fixed sequence of moves. ussr computational mathematics and mathematical physics, 13(2), 49–56, https://doi.org/10.1016/0041-5553(73)90130-4. 15. kononenko, a.f., khalezov, a.d. & chumakov, v.v. (1991). prinyatiye resheniy v usloviyakh neopredelennosti [decision making under uncertainty]. moscow, ussr: vc ran, [in russian]. 16. laffont, j.-j. & martimort d. (2002). the theory of incentives: the principal–agent model. princeton: princeton university press. 17. marshall, j.f. & bansal, v.k. (1992). financial engineering. ny: allyn & bacon. 18. voronin, a.a., gubko, m.v., mishin, s.p. & novikov, d.a. (2008). matematicheskiye modeli organizatsiy [mathematical models of organizations]. moscow, russia: lenand, [in russian]. https://www.sciencedirect.com/science/journal/00415553 https://www.sciencedirect.com/science/journal/00415553 https://doi.org/10.1016/0041-5553(73)90130-4 1. introduction 2. games with random factors 3. game without feedback 4. game with feedback 5. conclusion 14. kononenko, a.f. (1973). the role of information on the opponent's target function in a two-person game with a fixed sequence of moves. ussr computational mathematics and mathematical physics, 13(2), 49–56, https://doi.org/10.1016/0041-5553(73)9013... adv syst sci appl 2018; 03; 144-153 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/634 on classification and assessment of regional economic and social effects of the implementation of projects for mining of solid minerals deposits andrey m. valuev1, maria a. lozinskaya2 1) mechanical engineering research institute, russian academy of sciences, moscow, russia e-mail: valuev.online@gmail.com 2) the national university of science and technology misis, moscow, russia e-mail: mary-loz@mail.ru abstract: the main regional economic, ecological and social effects resulting from the implementation of design solutions in the field of mining of solid mineral deposits are systematized in general. two cases are distinguished, for which the new enterprise production is mainly consumed within the region and outside it, resp. typical impact on transport branch and social sphere is studied for both cases. the paper mainly concentrates on the case of russian federation, since differences in legal system (in particular, the fiscal system), culture of everyday life, territorial distribution and mobility of population, as it is shown in scientific literature, lead to different form of interrelation between these factors in involved countries and regions. their influence on incomes and expenses of local budgets depending on the economic conjuncture and the state of the manpower market is considered. direct and indirect effects are taken in to account, the latter resulting from additional amount of services due to increase of total wages in the local mining industry. a formula for budget increment because of the new production including the indirect effects is derived. particular attention is paid to the problem of selecting the level of completeness of the stock extraction in the project. economic and ecological consequences of the latter are considered in detail. the problem rests to reveal significance of impact of expected mineral industry activities on different sectors of the local economy, for which probably the statistical method by toda and yamamoto may be applied. keywords: solid minerals, mining projects, budget, tax revenue, economic effects, social effects. 1. introduction the problem of regional economic and social effects caused by development and expansion of mining production as well as by its reduction is important for all mining countries. many factors have universal significance and are to some extent characteristic for the development processes of other branches of heavy industry, but their combination has significant national and even regional specifics. in this regard, the vast majority of studies are focused on the problems of a specific country and a particular region, and generalizing works that could formulate patterns for the world as a whole are absent. e.g., in the paper [7], despite its title, only the case of canadian province british columbia is studied and its fiscal system is addressed; nevertheless, some way of quantitative assessment of dependency between local mining industry activities and regional economic development for more general use is proposed. the lack of universality is caused by significant differences in legal system, culture of everyday life, territorial distribution and mobility of population of involved countries and regions. despite that, conclusions of researches may be useful for analysis of situations lying far beyond the scope of the papers. 145 a.m.valuev, m.a.lozinskaya copyright ©2018 assa. adv. in systems science and appl. (2018) the impact of rise or decline of local mining industry to most economical sectors of regional economy of different countries (canada, poland, turkey, sweden) is investigated in [1, 3, 4, 7]. most effects are seen via changes of employment in main sectors of regional economy. economic and social effects of migration caused by mining industry development in peripheral areas “characterized by remoteness, a scattered population, and limited access to infrastructure” [4] are studied in [4, 9, 11]. it must be emphasized that such conditions are typical for russian mining industry as well to most other large northern countries. the paper [10] as well as many other ones concentrates on development of knowledgeintensive service activities directly serving australian mining sector. most principal features of impact of mining industry activities on public health that are typical not only to australia are studied in [8]. taking into account results of the above mentioned (and other) researches, we restrict our attention to the case of russian mining industry. russian mining industry, on the one hand, is a very important part of the economy, and on the other, is extremely unevenly distributed. enterprises that do not belong themselves to the leading enterprises of the national mining industry, nevertheless, can make a very significant contribution to the regional economy and social sphere. when selecting and approving mining projects for solid mineral deposits (smd), the interests of the state-proprietor of subsoil (subsoil owner) and subsoil user organization are taken into account [14, 18]. the significance of a particular mining enterprise being designed at the federal and regional levels is not the same, thus it is not possible to speak of a single state interest in connection with the mining project. in some cases, the national interest dominates, because of which local interests may not be sufficiently taken into account when selecting a project, for local enterprises it is vice versa. when considering the project variant, one should try to take into account all projected revenues to budgets, as well as budget expenditures (for social support measures, environmental protection). the indicators that determine the commercial efficiency of a mining project are basically the same as for other industries (adjusted for the more pronounced random nature of their estimates), but the national and regional economic effects are much more diverse and much less accountable. thus, in [14, p. 235], in addition to tax revenues, payments for the use of subsoil and customs duties received directly from the subsoil user, “additional tax revenue (or losses) from third parties, due to the impact of the project on their financial situation”. in this article, an attempt is made to reveal more details of the situation in question and to take into consideration, in addition to revenues, also budget expenditures, as well as indirect economic effects. 2. general characteristics of regional economic and social effects of progects of industrial enterprises creation emergence of the new production in itself creates several principal effects for the region in which it is located. if its products are consumed mainly within the region (which is typical, for example, for the production of steam coal that provides local power stations and population needs as well as for construction materials quarries), then its consumers pay to the new manufacturer instead of the former suppliers (presumably from other regions). in addition to increasing the profitability of production for consumers, which justifies shift to new suppliers, the effect of the latter is that the payment for the products they supply remains in the region. as a result, this leads to an increase in the profits of regional producers and the total wages of workers. for some transport enterprises, on the contrary, owners loose profits and employees their salary; however these losses are obviously much less than the increase for consumers and new manufacturers. regional economic and social effects of the implementation of projects for mining 146 copyright ©2018 assa. adv. in systems science and appl. (2018) if the produced products are mainly intended for consumers outside the region, then the income of the created enterprise is added to the income of the regional economy. in this case, additional income can also be received by transport companies in the region, if they are engaged in the delivery of these products. at the same time, a number of industries, including mining enterprises, can have a negative impact on the environment, mainly through pollution. penalties that companies pay for identified cases of pollution cannot always compensate this negative effect. the consequences of pollution, first of all, are the increase in the morbidity, which requires additional investments in the health care system. at the same time, only the maintenance of the health of the employees of polluting enterprises can be taken on the responsibility of the company and carried out at the expense of its revenues. 3. the main aspects of the implementation of mining projects taking into account the economic and social environment before considering the issue of comparing alternatives for a specific mining company’s project taking into account indirect effects, we note the following fundamental aspects of the impact of such enterprises activities on the region. 3.1. market for sales of the new enterprise production and its relations with project indicators production at the new enterprise can either compensate for the outgoing capacities of other mining enterprises, or create the opportunity to replace imported raw materials (but thereby reduce the burden on transport infrastructure and employment in transport), or serve as a means for creating new industries in the region (supply of raw materials or energy) or in other regions, or, at last, for exports increase. thus, for projects in question, it is necessary to take into account their effect on the functioning or development of industry and transport, and in the case of exports, on payment of the export duty. industrial production (or power generation) on a regional scale can depend heavily on the extraction of a particular raw material at a particular location due to high transport expenses, or, on the contrary, it may be economically acceptable to replace this production with imported raw materials or substitute raw materials or energy transfer from other regions. in the latter case, the possible volume of sales in the local market (and beyond it, indeed) is closely related to the offer price. project alternatives differ not only in terms of the estimated level of prime cost, and consequently, in the level of prices of sales of products, but also in terms of the achieved quality values of products. it would be more accurate in many cases to suppose the relationship between cost and the main (or cumulative) quality indicator. the latter refers to cases where the use of raw materials is permissible in a fairly wide range of quality, but the efficiency of the production usage depends on it. this is typical, in particular, for power stations generating energy by coal combustion. thus, project alternatives differ in economic indicators not only for the mining enterprise itself, but also for consumers of its products, including local consumers. with regard to the objectives of maintaining existing industries or energy companies, one should take into account the sensitivity of their profits and other elements of the tax base to the costs of raw materials and to assess the additional effect as the difference between budget payments for various project options, provided that the raw material in the corresponding project is purchased at the projected enterprise, and the missing volume — on the external market beyond the region. in the case when the supply of raw materials from other sources is unprofitable, one 147 a.m.valuev, m.a.lozinskaya copyright ©2018 assa. adv. in systems science and appl. (2018) must take into account the sensitivity of the profit of the consumer of raw materials to the volume of production. similarly, the contribution of a mining enterprise to the economy of the region (and even of the country) needs to be assessed when its creation is a prerequisite for the creation of new industries. 3.2. significance of transport and energy transfer infrastructure for most types of mineral products, significant physical volumes are typical, and as a result, its transportation creates a considerable, sometimes the main load of transport system of the region. handling this load may impose restrictions on the economic activities of the region. the existing road network and transport fleet provides a certain amount of transport work, and its increase requires investments, transport infrastructure development demanding sufficient time for its expansion. if the enterprise being created is sufficiently profitable, it can afford, in case of a shortage of transport resources, to pay a higher price for transport services. but in this way transport needs for other, less profitable activities will not be fully met. in this case, the costs of the regional transport system development for satisfaction of requirements for the transportation of the products of the company being established or expanded should be attributed to its account. on the contrary, in the case of a significant reserve of the transport infrastructure capacity, investments in its development are not required, but transport fleet increase should be provided at the expense of the created enterprise except for the situation of prevalence of transport services by interregional organizations. the most important processes of mining, namely excavation, conveyor transportation and processes of primary processing (crushing, sorting) are energy-intensive and carry a high load on the energy system of the region. the latter is especially important for energy-deficient regions and can have a negative effect on other economic activities. 3.3. ecological aspects the environmental effects of mining of smd are very diverse and may not always be expressed in monetary terms. among other things, the territory of a mining allotment is withdrawn from other types of activities, including agriculture and forestry. obtaining a land tax does not necessarily compensate for the loss from the shortfall in payments to the budget from other commercial activities that have not taken place on this territory. on the other hand, a significant part of mining enterprises is located where there are no conditions for efficient agriculture. unfavorable environmental effects can have varying degrees of significance. for example, in the pavlodar region of kazakhstan, where the largest ekibastuz coal deposit is located, there is also a fairly developed grain farming, but not in the vicinity of the field itself. therefore, the pollution of soils by coal and rock dust, which dissipates from the faces and dumps, does not have a significant negative economic effect. quite the reverse situation is in the zone of the kursk magnetic anomaly, abundant with fertile black soil. unfavorable ecological environment can affect not only other types of economic activity, but also the health status of the population. consequently, nature protection measures have different costs depending on the state of the local economy, especially agriculture and forestry, and the resettlement of residents. according to the authors of [20], the existing penalties do not fully cover the costs of compensating for the negative impact of industrial activities on the state of the environment and suggest a formula for correcting the grp (gross regional product) taking into account the following:  costs for wastewater treatment in accordance with the cost of cleaning; regional economic and social effects of the implementation of projects for mining 148 copyright ©2018 assa. adv. in systems science and appl. (2018)  the cost of greenhouse gas emissions (co2 equivalent) in accordance with prices in the carbon market;  costs for removal and disposal of production and consumption wastes in accordance with the fee for negative impacts. but the above mentioned factors do not cover the entire pollution cost. perhaps the most significant impact of them is on the health of the population. it should be noted that in some cases, employees of an enterprise exposed to harmful working conditions and the population not directly related to production, but experiencing its negative impact, are in an unequal position. the mining company is to a certain extent interested in maintaining the health and physical fitness of its employees and, at the expense of its revenues, can pay for recreational activities and other health measures, and, if necessary, also treatment. at the same time, it will hardly be engaged in charity with respect to the entire population of the territory. a typical example of this is the main subdivision of pjsc mmc norilsk nickel its polar division with a center in norilsk. by common belief, the environmental situation in norilsk is very unfavorable. meanwhile, in respect of the employees, there is a whole system of measures. employees are provided with various social guarantees, benefits and compensations aimed at health improvement, treatment and recreation on russian and foreign resorts [12]. as for the rest of the population, additional health care costs, along with the mandatory medical insurance fund, must be paid by the local budget. therefore, regional interests should be taken into account in the form of a requirement to implement a sufficiently high level of environment protection measures in the project. however, this level may be the result of a compromise between the requirements of maintaining a favorable ecological environment and the economic interests of the regional budget and the population in obtaining additional revenues as a result of the projected enterprise. 4. social importance of a mining enterprise an enterprise or a branch of industry that is a consumer of a particular type of raw material may have high social importance being the main (city-forming) enterprise in some territory. in this case, the raw material supply of such production becomes an important social and economic task. the region should seek the adoption of a project for a mining enterprise that maximizes the volume of break-even production in a supported enterprise and offers a compromise option for the selling price of this raw material. regional labor market, including occupations related to mining operations, may have a different ratio between supply and demand, as established it the level of wages. different versions of the project, the differences in volume and profitability differ, and what wages they can offer to employees. but it must be estimated from the actual level in the region — the idea put forward by f. w. taylor who was one of the founders of the management theory [16]. 5. the effect of completeness of deposit extraction as it was noted in [2], one of the key parameters distinguishing project variants proposed for selection and approval is the completeness of extraction of balance reserves, in other words, the percentage of their losses during extraction. in the same works, a fact is noted that seems paradoxical for authors, that in typical cases, the most profitable project variant for both the subsoil user and owner is the variant with the highest loss level. the authors of [2] link this fact to insufficient consideration of budget revenues not only directly from the project (i.e., paid by the subsoil user), but also “in connection with the project” (paid by consumers of mineral raw 149 a.m.valuev, m.a.lozinskaya copyright ©2018 assa. adv. in systems science and appl. (2018) materials). we accept this position, but we do not absolutize the importance of the completeness of the extraction of mineral reserves in the light of the side effects discussed in this paper. the main factors associated with the completeness of extraction of reserves: 1. the volume of production and sales. 2. the cost of extraction and primary processing. 3. the effect on cost efficiency among consumers of raw materials. 4. the need for workers and their level of payment. the increase in the completeness of the extraction of reserves either requires an increase in the degree of selectivity of their development, which leads to a complication of technology, poor performance of mining machinery (or the use of less productive and thus less economical machines), or leads to an increase in the dilution of mined minerals. an example of the first case is the project of using layer-by-layer cutters instead of singlebucket excavators when working out complex-structural coal deposits in eastern siberia. clogging of coal with rock and loss of coal in this case, indeed, decrease, but the cost of production also increases. it should be noted that the increase in the completeness of the extraction of reserves at the expense of their dilution has an ambiguous effect on the total profit across the entire production chain, and thus on budget payments. when burning high-ash coal, the output of electricity per unit of calorific value of fuel falls, so to ensure the power generation process it is necessary to add fuel oil to the furnace, which increases the cost of production, which generally depends on the technologies of fuel combustion [6, 15, 19]. the cost of ash removal influences the growth of the cost price. diluted ore not only increases the cost of its enrichment, but also cause losses in the process of enrichment, so the concentrate amount may be even smaller than when enriching a smaller amount of unsoiled ore. therefore, the loss of a certain portion of the reserves, accompanied by an increase in the amount of the final product, is beneficial both for producers and for the budget. in the case of a certain reduction in the volume of the final product, but a significant reduction in the cost of production along the entire production chain, the total effect depends on the correlation between economic indicators. 6. direct and indirect relationship between project indicators and budget revenues and expenditures alternatives associated, in particular, with the different completeness of the extraction of balance reserves (and, consequently, with the different quantity and quality of the produced raw materials) can be viewed from several viewpoints, and the role of the factors listed below depends on the type of mineral, the place of extraction, the state of the budget and the labor market at both the federal and the local scale. the mining enterprise, depending on the project variant, employs a different number of workers in extraction, primary processing (sorting, enrichment) and transportation to its consumers (including transportation within the region). in addition to the main payments to the budget from the subsoil user (the tax on the extraction of minerals and the tax on profit), their employees also pay income tax. the total wage non-linearly depends on the number of hired employees. with an excess of labor supply, an increase in the number of employees, together with a decrease in the enterprise’s income per employee, entails a decrease in wages, and in the case of a deficit, on the contrary, the employment of new workers becomes possible only with an increase in wages, even if the income per employee decreases. the change in the unemployment rate changes the costs of benefits, retraining, housing subsidies and other measures of social support. regional economic and social effects of the implementation of projects for mining 150 copyright ©2018 assa. adv. in systems science and appl. (2018) the social consequences of the adoption of a particular design decision take place, first of all, at the regional level. it was noted above that the variants differ in the different number (and composition) of the labor force involved in the development of the deposit and the accompanying and related business processes. the increase in the aggregate wage of these workers makes it possible to develop the services sector to “absorb” a part of this increase, which in turn increases payments to the budget from the services sector. such an indirect effect can be identified for any production, not just mining. the state statistics provide publicly available data on the structure of the nation-wide gross product. obviously, the same data can be obtained for the regions. among the highlighted classification headings, along with mining, processing industries, construction, as well as various categories of services. in turn, the main ways of using the gross domestic product (gdp) are assessed quantitatively. data from the source “on production and use of (gdp) for 2016” [13] that are of the greatest interest to us, are shown in table 6.1. table 6.1. data on the structure of the gnp of the russian federation in 2016 components of gdp in basic prices share (year), % share for quarters, % i ii iii iv agriculture, hunting and forestry 4.5 2.3 3.4 7.2 4.5 fishery, fish farming 0.3 0.4 0.2 0.3 0.2 mining 9.4 8.6 10.1 9.7 9.1 manufacturing industries 13.7 12.5 14.1 13.6 14.4 generation and distribution of electricity, gas and water 3.1 3.9 2.7 2.5 3.4 construction 6.2 4.5 5.5 6.4 7.8 wholesale and retail trade; repair of motor vehicles, motorcycles, household goods and personal items 15.9 16.9 15.9 15.3 16.0 hotels and restaurants 0.8 0.8 0.9 0.9 0.8 transport and communication 7.8 8.4 8.1 7.8 7.2 health and social services 3.8 3.9 3.8 3.7 3.7 other communal, social and personal services 1.7 1.8 1.8 1.6 1.7 other important data from [13] are: as to the structure of gdp use: final consumption expenditure as a whole it is 76.6% and 55.8% for consumption of households; as to the structure of gdp by source of income: compensation of employees (including labor remuneration and mixed incomes not observed by direct statistical methods) is 46.7%. as it can be seen, the share of final consumption varies significantly even within a year. analysis of the correlation between such indicators allows to predict the change in the volumes of various types of services, accompanying changes in the volume of industrial activity and total wages in this area. according to the budget code of rf [15], taxes received by the regional budget directly from the activities of a mining enterprise are: 151 a.m.valuev, m.a.lozinskaya copyright ©2018 assa. adv. in systems science and appl. (2018) 1) the tax on the extraction of minerals (for general-distributed solid minerals and natural diamonds — 100% of the amount of tax, for other solid minerals — 60%); 2) the tax on profit of organizations — 18% of its value; 3) the tax on income of individuals (company employees) at a standard of 85%; 4) the tax on property of organizations — 100%; 5) transport tax. an increase in the total wages of workers in both the created and consuming and servicing organizations will be spent in large part within the region. to do this, the total volume of services (in particular, trade, which, obviously, in the amount of grp, as well as in the amount of gnp is a very significant share). in turn, some of these revenues turn into wages of organizations and individual private entrepreneurs working in the service sector. let the increment of the total wages in the non-consumer sector be amounted to w1. on the basis of statistical processing of the data in the form of linear regression, we assume that the elasticity of local consumption with respect to wage k12w1. in turn, we will assume that for workers in the services sector (the consumer sector) the share of their wages spent on services is k22 and that a unit of income in the consumer sector yields the amount of profit kp2 and the amount of wages kw2. to define the unknown increase in the volume of services in monetary terms, caused by w1, we employ the balance of revenues in the services sector be expressed by the ratio kw2(k12 w1+k22w2)= w2 (6.1) thus, we obtain for the increment of the total wage the formula w2=kw2k12w1/(1-kw2k22) (6.2) in addition, the increment in profit in the services sector will be p2=kp2(w1+w2). (6.3) this means that the cumulative increase in the regional budget revenues from the profit tax of the listed organizations of the non-consumer sector and all consumer sector organizations, as well as the personal income tax of workers of these organizations and individual private entrepreneurs will be 0.18(p1+p2)+0.130.15(w1+w2). (6.4) this is the main budget effect of the new production, which will be different for different project variants. here, the effect of increased consumption by the owners of the organizations in question is not taken into account, which is more difficult to assess, in particular, because some of them do not live permanently in the region. but in any case, it can only increase the revenues of the regional budget. other effects, such as a change in the cost of social support due to a possible reduction in the level of unemployment or to eliminate the effects of environmental pollution, are less predictable and require special study of the current factors. this is the main budget effect of the new production, which will be different for different project variants. here, the effect of increased consumption by the owners of the organizations in question is not taken into account, which is more difficult to assess, in particular, because some of them do not live permanently in the region. but in any case, it can only increase the revenues of the regional budget. 7. the time factor in the assessment of project options regional economic and social effects of the implementation of projects for mining 152 copyright ©2018 assa. adv. in systems science and appl. (2018) all that was said above did not take into account the time factor. as a general rule, in order to determine the cumulative effect for a subsoil user, the company’s estimated profit is summed by years, taking into account discounting; the same is done with respect to budget revenue. the net discounted income expresses (approximately, due to fluctuations in the economic conjuncture and technological changes) the real economic interest of the entrepreneur and the investor. for a state or region that, in general, spends the budget not for the economic benefit and is not a creditor even in the case of a budget surplus (although potentially it can be a creditor and even an investor), the goal is rather the guaranteed receipt of budget revenue as permanently (or, better, in everincreasing volumes). on the other hand, not being a creditor, the region is more often a debtor who is interested, if possible, to pay off loans more quickly. this justifies the use of the traditional formula of net discounted income for the project's budget contribution, but with a lower discount rate than for the subsoil user, because the state or a region usually takes a loan at a lower interest rate than the business. 8. conclusion the proposed approach yields the preliminary estimation of main regional economic effects, direct or indirect, that may result from a certain project for the development of a smd. it does not, however, reveal significance of impact of expected mineral industry activities on different sectors of the local economy. with the use of more sophisticated statistical methods, such as the method by toda and yamamoto [17], it looks likely that the most significant effects may be found in advance on the base of the study of analogs. in that case it would be possible to explore these effects in more detail for more perfect decision making. acknowledgements the work was partially supported by the ministry of education and science of the russian federation within the framework of the basic part of the state task no. 2014/113 for nust “misis” (project no. 952). we thank the members of the organizational committee of mlsd’2017 for their kind invitation to contribute the paper presented at the conference to the journal as well as mikhail goubko, associate editor-in-chief, and andrey shevlyakov, the secretary of advances in systems science and applications, for technical assistance. references [1] aksoy, e. 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(2012). variable-speed technology used in power plants for better plant economics and grid stability. energy, 45(1), 588594. [20] zabelina, a. & klevakina, e.a. (2011). otsenka ekologicheskikh zatrat v proizvedennom valovom regional'nom produkte [estimation of environmental costs in the produced gross regional product]. region: ekonomika i sotsiologiya, 2, 223–232, [in russian]. adv syst sci appl 2017; 17(2); 1-13 published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/491 copyright ©2017 assa. adv. in systems science and appl. (2017) new analytics of international relations: system forecast of cold war’s outcomes victor a. svetlov, nikolay m. sidorov 1 1) emperor alexander i st. petersburg state transport university, 190031, 9 moskovsky ave., st. petersburg, russian federation e-mail: victsvetlov@yandex.ru abstract. systemic forecasts of international relations evolution for quite a long time were quite a rare phenomenon. the main reason for this is the lack of independent of authors' ideological and political predilections reliable analysts, and this fact determines relevance of the current study. the main goal of the article is to develop new analytics that allows prediction of long-term trends in the evolution of international relations world system. therefore, the algebra of relations and the corresponding section of predicate logic are used. the authors proved sixteen basic theorems on the properties of the world system. as the initial opposition, a pair of relations "dependence-independence" was chosen. the empirical conditions of the current state of affairs make it possible from the outset to exclude from the analysis the state of independence of states as a long-term factor of international politics. it was established that the world system of international relations can be strictly in one of three states – conflict, synergistic or antagonistic. the authors also carried out the forecast of the world states system dynamics after the end of the cold war. in regards with impossibility of achieving by the international relations world system in the next thirty years any of two possible attractors – states of synergism and antagonism, predicts – its stable oscillation between these points of stability until at least the middle of the nineteenth century is forecasted. in practice, this means, depending on the direction of the trend, the emergence of a variety of waves of instability, primarily in the field of international security keywords: relational algebra, conflict, synergism, antagonism, one-pole state, double-pole state. 1. introduction problem statement. in the late 20th century the warsaw treaty organization dissolved and then the soviet union collapsed. the cold war ended, and the world system’s bipolar division into two antagonistic military and political blocs, established after the world war ii, disappeared. the end of the age of the bipolar division of the world system makes the search for common regularities of international relations’ structural changes relevant more than ever. without theoretical solution of this problem, including its mathematical simulation, it’s impossible to comprehend, for example, what trend currently dominates and what configuration of relations between states is most probable in the nearest future. international relations play an important role in the implementation of the foreign policy of states, since they contribute to solving many economic, environmental issues, as well as issues related to the settlement and prevention of conflicts. regional issues which may be related to a military conflict in a particular territory and to affect the interests of many states are also resolved at the international level. international relations are also viewed as human activity where individuals from more than a single state interact individually or in groups. international relations can be also presented as interaction between two or more states, and foreign policy as an external action of a nation that proves the relevance of the current topic [1]. there is very little agreement among international relations specialists about estimation of the most probable structure of international relations after the cold war’s end. huntington s. 2 v.a. svetlov, n.m. sidorov copyright ©2017 assa. adv. in systems science and appl. (2017) distinguished the following possible configurations based on the analysis of most authoritative research paradigms [2]. • there will be one relatively harmonious world. – fukuyama’s forecast [3]. • despite inevitable economic interaction, the cultural opposition of west and east will never disappear. – northrop’s forecast [4]. • the world system will split into many autonomous nation states, each of them will strive for survival, entering alliances with other states or augmenting its potential independently. – waltz’s forecast [5]. • the world system will sink into an utter chaos, national states will dissolve, tribal and ethnic conflicts will escalate, terrorism will become a common phenomenon, international criminal groups will appear. – brzezinski’s forecast [4]. • a polycivilizational world divided into two global groups will appear – west civilization and a small array of non-western civilizations not related to each other, in total about 7-8. national states will remain leading players on the world stage, but conflicts between them will be caused by their belonging to two specified groups. – huntington’s forecast [2]. each of the forecasts distinguished by huntington, including his own one, predicts a certain result of rebuilding the system of states’ international relations started after the cold war’s end. there is no doubt any one of them has its more or less ample grounds. however, it’s crucial that none of them is based on the system nature of changing international relations. this circumstance explains analysts’ dispersed opinions on the forming historical trend and the world system’s future configuration. it is commonly known that the choice of forecast point has a substantial effect on its result. at this point we associate ourselves with american historian and theoretician lipschutz r. who clearly demonstrated how the end of the cold war led to the paralysis of an idea of national state and collapse of the previous system of international safety [6]. the study of issues related to international relations was a subject of high interest for scientists since ancient times: e.h. carr, g. morgenthau, l. waltz, g. clark, j.s. nye, r. cohen, h. bull, a. rapoport, j. burton, e. haas, a. walfers, k. wright, o. holsty. a lot of modern scientists also deeply study the issues of international relations: r. powell, s.a. lantsov, m.a. muntean, b. buzan, j.b. mannheim, r.k. rich, m.a. khrustalev, f. moreau-defarge, m. kaplan and others. researchers of the concepts of dependent development (a. cordova, o. zunkel, f.kh. cardoso, etc.) believe that the main reason for most countries of the world is the "covert" use of the poorly economically developed countries by more advanced ones [7]. the study of international relations by k. wright is based on the fact that they represent a body of knowledge with the help of which it is possible to assess and control the relationship between states. a. kaminsky considers this from the position that international relations are decisive factors, levers, mechanisms of mutual relations and finds regularities and randomness in these relationships. concept headings. taking into account a total absence of analogues of mathematical solution to the stated problem in literature, new analytics of system forecast of the world system’s dynamics of the international relations after the cold war’s end is justified below. to this end a special discipline of predicate logic, often named logic of (binary) relations, is used. necessary theorems are stated and proved. our main hypothesis is that the world system cannot have more than two points of stability represented by its synergetic and antagonistic states. the system dynamics is surprisingly simple: it either reaches one of the points and remains there for a long time or oscillates between them [8, 9]. strange as it may seem, the suggested system forecast of cold war’s outcomes doesn’t depend on the content of relations between states recorded empirically. it takes into account only structural and dynamic properties of the relations themselves [10]. that's its advantage and at the same time its drawback. the forecast’s – as any mathematical model’s – strength is a high new analytics of international relations: system forecast of cold war’s outcomes 3 copyright ©2017 assa. adv. in systems science and appl. (2017) degree of credibility. the weakness is that its general conclusions require particular historical, economic and political details while explaining and forecasting specific events. 2. materials and methods let ws = (a, b, c, …) designates the world system of states denoted with symbols a, b, c, … . first of allб we’re interested in possible relations between states. to this end the framework of relational algebra is used. assuming this comment, term ws can be interpreted as a world system of international relations [11]. the analysis’ long-run objective is statement of general laws the change of relations between states follow and determination of dominating trend after the cold war’s end. let x, y, z, … – individual variables, running the elements ws; (х), (eх) – generality and existential quantifiers, respectively. let signs , , , ,  denote complementary operations (to complete relation), entailment, equivalence, multiplication and relation addition, respectively. let’s denote probability measure determined on the set of all subsets of the ws system with рr. let’s set positive (р), negative (n), conflict (с), relevant (r) and irrelevant (ir) relations (impact, dependence) on cartesian product ws  ws according to the following definitions. definition 1: state a has a positive impact on state b if and only if probability of existence of b providing existence of a is more than 0.5: рr(b/a)  0.5. definition 2: state a has a negative impact on state b if and only if probability of existence of b providing existence of a is less than 0.5: рr(b/a)  0.5. definition 3: state a has no impact on state b (а is related to b irrelevantly) if and only if probability of existence of b providing existence of a is 0.5: irаb = рr(b/a) = 0.5. definition 4: state a has an impact on state b (а is related to b relevantly) if and only if state а has a positive or negative impact on state b: rаb = раb  nаb. definition 5: state a is in conflict with state b if and only if a relates to b both positively and negatively: саb = раb  nаb. definition 6: states a and b are in synergetic dependence if and only if they’re both related to each other only positively. definition 7: states a and b are in antagonistic dependence if and only if they’re both related to each other only negatively. the categories of conflict, synergism and antagonism have a special role to play in the building of qualitative forecasts of system transformations. conflict denotes the inner cause of system change, synergism and antagonism – stable, although opposite outcomes of conflict solution, its points of stability (attractors). the outbreak of a system conflict indicates the beginning of a system change, synergism and antagonism – achieving steady state of stability by the system. a conflict with the course of time either transfers the system to a higher or lower level of conflict or transforms it into a conflict-free – synergetic or antagonistic state. the essence of synergism as a steady conflict-free system state reveals the following significant dynamic properties: (1) states, which are elements of a synergetic system, all together either progress or regress. (2) all synergetic systems with the course of time and continuous generation of energy from outside only increase their synergism and strive to remain, therefore, conflict-free. (3) a synergetic system ceases to exist only when strengthening or weakening of all or several states becomes incompatible with ensuring general synergism of its components. antagonism as a steady conflict-free system state is characterized by the following distinctive dynamic properties: 4 v.a. svetlov, n.m. sidorov copyright ©2017 assa. adv. in systems science and appl. (2017) (1) the progress of some states within antagonistic systems always takes place at the expense of other states’ regress. this means that the one pole of antagonistic system prospers with the course of time, while the other one certainly comes down. (2) all antagonistic systems with the course of time and continuous generation of energy from outside only increase its antagonism and strive to remain, therefore, conflict-free. (3) an antagonistic system ceases to exist only when the degrading pole cannot stand its antagonism with the stronger pole anymore. let’s name positive, negative relations and their additions to complete relations initial. multiplicative unification of positive and negative relations with their additions can give rise to both relevant, i.e. positive or negative relations, and conflict and irrelevant relations as derivatives. all possible kinds of derivatives relations between states a and b based on their initial relations’ multiplication is summarized in table 1. table 1. all possible kinds of derivatives relations nab  nab pab сab pab pab nab irab let pl denotes a pole (alliance, bloc, coalition) of the world system of states composed by its elements based on the combinations of certain kind of relations with ws. definition 8: state a forms with state b a one-pole system if and only if a and b relate to each other only positively: plab = pab  nаb. the one-pole system’s extensive definition is as follows: plxy = (x)(y)[(x  y)  (pxy  nxy )]. it follows from def. 6 and def. 8 that one-pole system can be synergetic only. definition 9: state a forms with state b an antagonistic double-pole system if and only if a and b relate to each other only negatively: pla plb = nab  pаb. the double-pole system’s extensive definition is as follows: plx  ply = (x)(y)(z){[( x  y) & (x  z)]  [(nxy  pxy)  ((nxz  pxz pyz  nxz)  (nyz  pyz  pxz  nxz))]}. it follows from def. 7 and def. 9 that double-pole system can be antagonistic only, while each pole consists of synergetically interacting elements. relations introduced using def. 15 can be summed up and multiplied, forming more complex relations. for the purpose of the present paper it’s sufficient to determine the matrix of multiplication of the relations of various modalities (see table 2). table 2. relations of various modalities  p n c r ir p p n c r ir n n p c r ir c c c c c ir r r r c r ir ir ir ir ir ir ir for example, the following system of relations, despite even number of negative relations, is nonetheless conflict: new analytics of international relations: system forecast of cold war’s outcomes 5 copyright ©2017 assa. adv. in systems science and appl. (2017) pnnpppс = [pn]npppc = [nn]prpc = [pp]rpc = [pr]pc = [rp]c = rc = c it follows from table 2 that the relation of irrelevance ir is the most stable: being multiplied by any relation, it always remains. it can be considered a peculiar null (dominant) relation in the logic of studied relations between states. gaining complete independence as distinct from other types of relations offers a solution to all international problems: a state can have no difficulties in relations with neighbors, if only it exists independently of them. the second most stable is с conflict relation. it dominates all types of relations, except for irrelevance relation ir. conflict stability before all other dependence relations confirms a worldly wisdom: it’s easy to come into conflict, but difficult to come out of it [12]. the relation of relevant (positive or negative) relationship r is the next in the hierarchy of stability. this combined relation dominates only positive and only negative relations, giving up in stability only to irrelevance and conflict relations. the relationships of positive relation р and relevant relation r are reflexive, symmetrical and transitive, i.e. they’re a relation of equivalence. it is two types of relations (and only them) that create a basis for emergence of a stable system of international relations. strictly speaking, each of them is only necessary, but they can become adequate grounds for international stability if conditions are right. the relation of negative relation n is neither reflexive nor transitive, but it is symmetrical. the relations of irrelevant relation ir and conflict с are not reflexive, but symmetrical and transitive. these types of relations do not form equivalence classes and in no event can ensure uprising – let alone ensuring – of stable world order. it follows from the above that the problem of searching for stable architecture of the world system ws is mathematically comes to breakdown of all states into many non-crossing and jointly exhaustive classes (poles) offering equivalence. only two relations (from the considered above) have the equivalence property – the relations of relevance r and positive p relation. it means that each of them may become not only necessary but adequate grounds for dividing the world system ws into non-crossing and jointly exhaustive classes if conditions are right. this result is notable on its own, since it indicates relevance (positive or negative dependence) and positive dependence of states as two indispensable and, notably, only conditions of the international relations’ system stability [13]. the relations of positive (p) and negative (n) relevance represent peculiar atoms, various combinations of which give rise to all the rest types of relations (compare tables 1 and 3). table 3. the relations of positive (p) and negative (n) relevance u = all relations r = relevant ir = irrelevant (pn) conflict conflict-free c = (pn) c = ( pn ) ( p n ) let’s state the main theorems of the logic of international relations necessary for system forecast justification. however, it should be noted that in a nonformal sense the logic of relations between states is principally the same as the logic of interpersonal relations that are governed by four well-known rules of “the golden rule of morals” [1]: 1. the friend of my friend is my friend. 2. the friend of my enemy is my enemy. 3. the enemy of my friend is my enemy. 6 v.a. svetlov, n.m. sidorov copyright ©2017 assa. adv. in systems science and appl. (2017) 4. the enemy of my enemy is my friend. тheorem 1: in the world system ws states are dependent on each other if and only if it’s false that they’re independent on each other: (x)(y) (rxy  irxy). proof: 1. (x)(y) rxy (assumption of direct proof) 2. (x)(y)   rxy (1) 3. (х)(y)   [(pr (y/x)  0.5)  (pr (y/x)  0.5)] (2, def. 4) 4. (х)(y)  [pr (y/x) = 0.5] (3, def. 3) 5. (х)(y) irxy (4, def. 3) 6. the proof of converse implication is analogous. qed according to theorem 1, dependent states form an equivalent class, the common feature (relevance of relations) of which is not typical for none independent state (see table 3). from this perspective, being independent means having neither positive nor negative, nor conflict relations with any other state of the world system ws. тheorem 2: in the world system ws dependent states are positively related to each other if and only if it’s false that they’re negatively related to each other: (х)(y) (рху  nxy). proof: 1. (х)(y) pxy (assumption of direct proof) 2. (х)(y)   pxy (1) 3. (х)(y)   [pr (y/x)  0.5] (2, def. 1) 4. (х)(y)  irxу   [pr (y/x)  0.5] (3, assumption of dependency) 5. (х)(y)  [pr (y/x)  0.5] (4, t 1) 6. (х)(y) nxy (4, def. 2) 7. the proof of converse implication is analogous. qed according to theorem 2, positively related states among dependent states form their own equivalent class, the common feature of which (positive relevance) is not typical for none negatively dependent state. this theorem states that positive dependence is not the only type of dependence. there are also such types as conflict and antagonism, including not only positive but also negative relations of states. тheorem 3: in the world system ws pairs of dependent states are conflict-free if and only if they’re related in each pair either only positively or only negatively: (х)(y) {(rxy   сxy)  [(рxy   nxy)  (рxy  nxy)]}. proof: 1. (х)(y) (rxy   сxy) (assumption of direct proof) 2. (х)(y) [(рxy  nxy)   (рxy  nxy)] (1, def. 4 and 5) 3. (х)(y) [(рxy  nxy)  (рxy  nxy)] (2) 4. (х)(y) [(рxy  nxy)  (рxy  nxy)] (3) 5. the proof of converse implication is analogous. qed according to theorem 3, two interdependent states form an elementary dynamic cycle, which is conflict-free in two cases: either two ways of cycle are positive (synergism occurrence) or they’re both negative (antagonism occurrence). it is obvious that pairwise conflict-free nature of states does not generally guarantees the conflict-free nature of the whole world system ws. тheorem 4: in the world system ws, which is in conflict, each state is in a negative selfreference: new analytics of international relations: system forecast of cold war’s outcomes 7 copyright ©2017 assa. adv. in systems science and appl. (2017) (х)(y) сxy  nxч. proof: 1. (х)(y) сxy (assumption of direct proof) 2. (х)(y) (рxy  nxy) (1, def. 5) 3. (х)(y) (рxy  nyх) (2, symmetry of relation nxу) 4. (х)(y) [(рxy  nyх)  nxх]( theorem of logic of relations) 5. (х)nxх (3, 4) qed theorem 4 indicates a required feature of the world system’s conflict state: each its element is a relation of negative converse relation with itself. it means that whatever measures a nation in such a state may take, it will only aggravate its situation, thereby, escalating the system conflict. тheorem 5: in the world system ws, which is conflict-free, each state is in a positive selfreference: (х) рхх. proof: (cpr – calculus of probability) 1. (ех)  рхх (assumption of indirect proof) 2. (ех) (nхх  irxx) (1) 3. (ех) [(pr (x/x)  0.5)  (pr (x)  0)] (1, 2, cp, def. 2) 4. (ех) [pr (x/x)  0.5] (3) 5. (ех) [pr (x)  0] (4) 6. (х) [is pr (x)  0, pr (x/x) = 1] (theorem cpr) 7. (х) pr (x/x) = 1 (5, 6) 8. contradiction (4, 7) 9. (х) рхх (1, 8). qed theorem 5 indicates a required feature of the world system’s conflict-free state: each its element should be in a relation of positive converse relation with itself. тheorem 5 expresses a peculiar principle of state (self) preservation. in order to prosper each state must be able to support itself, advocate its interests, maintain consistent relations with its friends and enemies. тheorem 6. the world system ws is conflict-free if and only if it has none negative relation: (х)(y) ( nxy   cxy). proof: 1. (х)(y) nxу (condition) 2. (х)(y) nуx (1, symmetry of relation nxу) 3. (х)(y) (nxу  nуx) (1, 2) 4. (х)(y) [(nxy  nуx)  pxх] (theorem of logic of relations) 5. (х)(y) pxх (3, 4) 6. (х) nxх (5, t 2) 7. (ех)(еy) cxy (assumption of indirect proof) 8. (ех)(еy) (nxy  рху) (2, def. 5) 9. (ех)(еy) [(nxy  рух)  nxх] (theorem of logic of relations) 10. (ех) nxx (8, 9) 11. contradiction (6, 10) 12. (х)(y)  cxy (7, 11) 13. the proof of converse implication is analogous. qed the absence of negative relations in the world system ws is an indispensable and sufficient conditions for its conflict-free environment. this statement is fair for the admitted above assumptions on positive and negative relations as primary ones for systems of any type. however, a deeper analysis shows that a conflict is possible even when a system has only positive relations, which vary in its impact (see [1]). 8 v.a. svetlov, n.m. sidorov copyright ©2017 assa. adv. in systems science and appl. (2017) тheorem 7: the world system ws is conflict-free if and only if it has only positive relations: (х)(y) (рху   cxy). proof: follows from the combination of тheorem 2 and theorem 6. qed a system that has only positive relations is synergetic. it is known that synergism can have both positive and negative effects. in the former case the system with its elements progresses, in the latter – degrades. thus the system’s conflict-free state doesn’t guarantee its positive development trend. тheorem 8: the world system ws is conflict if and only if none of the states is independent on each other: (х)(y) (сху   irxy). proof: 1. (х)(y) сху (assumption of direct proof) 2. (х)(y) (nxy  рху) (1, def. 5) 3. (х)(y) [(nxy  рху)  (nxy  рху)] (theorem of logic of relations) 4. (х)(y) (nxy  рху) (2, 3) 5. (х)(y) rxy (4, def. 4) 6. (х)(y)  irxy (5, т1). qed according to theorem 8, a conflict is possible not only between dependent states. in other words, dependence is an indispensable, although not sufficient condition of conflict. dependence between states is potential of not only conflict-free but also conflict development. if several states become members of the same economic, political or war system, they not only benefit from mutual cooperation but also increase chances for conflicts between themselves. it explains why every union, stable as it may be conceived, dissolves sooner or later. тheorem 9: the world system ws is conflict free if it is consists of states dependent on each other: (х)(y) (irxy  сху). proof: theorem т9 is a contraposition to theorem т8 and, therefore, is equivalent to it. qed according to theorem 9, independence and conflict environment are incompatible system features. theorems 8 and 9 highlight state’s independence as the only guaranteed solution to any international conflict. however, this formula for establishing a stable world order can scarcely be put into practice worldwide. achieving complete independence by all states in the foreseeable future seems a utopian project because of the evident scarcity of resources necessary for progressive development. it means that strengthening of world economy’s globalization tendencies will certainly enhance the likelihood of international conflicts among its members. тheorem 10: if each state of the world system ws is in a positive self-reference, the system is conflict-free: (х)(y) (рхх   сху). proof: 1. (х)рхх (condition) 2. (ех)(еy) сху (assumption of indirect proof) 3. (ех)(еy) (nxy  рху) (3, def. 5) 4. (ех)(еy) (nxy  рух) (symmetry of relation рху) 5. (ех)(еy) [(nxy  рух)  nxх] (theorem of logic of relations) 6. (ех) nxх (4, 5) 7. (ех) рxх (6, т 2) 8. contradiction (1, 7) new analytics of international relations: system forecast of cold war’s outcomes 9 copyright ©2017 assa. adv. in systems science and appl. (2017) 9. (х)(y)  сху (2, 8). qed according to theorem 10, positive self-reference of each state is a sufficient feature of a conflict-free state of the world system ws. if we translate the “positive self-reference” term into the language of international relations, it means a positive effect of state’s self-regulation resulting from successful combination of system-wide and national interests. тheorem 11: the world system ws is conflict-free if and only if all states are dependent and each of them is in a positive self-reference: (х)(y) [(рхх  rху)   сху)]. proof: 1. (х)(y) (рхх  rху) (assumption) 2. (х)(y) [(рхх  rху)   сху] (т10) 3.  сху (1, 2) 4. the proof of converse implication is analogous. qed according to theorem 11, in case of dependence positive self-reference is not only a sufficient but at the same time necessary condition of conflict-free existence. collapses of coalitions always start when some of its members loose positive self-reference for one reason or another. for example, brexit took place when about a half of british population stopped to feel self-identification regarding membership in the eec. тheorem 12: if the world system of interdependent states ws has exactly one pole, it is conflict-free: (х)(y) (plxy   сху). proof: 1. (х)(y) plхy (assumption of direct proof) 2. (х)(y) (рху  nxy) (1, def. 6) 3. (х)(y) [(рху  nxy)   сху] (conclusion of theorem т6 and т7) 4. (х)(y) сху (2, 3). qed theorem 12 states that the absence of conflict in the community of dependent states is a necessary condition of one-pole and, therefore, synergetic systems. if the world economy achieved a complete and universally beneficial state of globalization for all ws states, and their political regimes and institutes corresponded to and supported it, the mentioned one-pole and synergetic word order would appear. in such a world order conflict would be impossible. тheorem 13: if the world system of interdependent states ws has exactly two poles, it is conflict-free: (х)(y) [(plx  ply)  сху]. proof: 1. (х)(y) (plх  ply) (assumption of direct proof) 2. (х)(y) (nху  pху) (1, def. 7) 3. (х)(y) (nxy  nxy) (2, т2) 4. (х)(y) (nxy  nxy)  pху (logic of relations theorem) 5. (х)(y) pxy (3, 4) 6. (х)(y) [pxy  (nxx  pxx  nyy  pyy)] (logic of relations theorem) 7. (х)(y) (nxx  pхx  nyy  pyy) (5, 6) 8. (х)(y) [(nxx  pxx  nyy  pyy)  сху] (conclusion of theorem т6 and т7) 9. (х)(y) сху (7, 8). qed theorem 13 is of important methodological value. it proves that antagonism, which everyone’s prone to identify with conflict, is actually one of its (in addition to synergism) 10 v.a. svetlov, n.m. sidorov copyright ©2017 assa. adv. in systems science and appl. (2017) oppositions. like synergism, antagonism features high resistance and ceases to exist until one of antagonists dies out or stops fighting because of eclipse of powers and resources depletion. the cold war between the soviet union and usa and its allies that lasted a little more than 40 years and ended only with the collapse of the soviet union, can be a case in point, demonstrating antagonism stability. тheorem 14: if the world system of interdependent states ws is polarized into one or two poles, it is conflict-free: (х)(y) [(plxy  (plx ply))  сху] proof: follows from the combination of theorems 12 and 13. qed the meaning of theorem 14 is that the absence of conflict in the system of international relations of interdependent states can be a consequence of only two causes – its one-pole (synergetic) or double-pole (antagonistic) organization. тheorem 15: the world system of interdependent states ws is conflict-free if only it is polarized into one or two poles: (х)(y) [сху  (plxy  (plх ply))]. proof: 1. (х)(y) сху (condition) 2. (ех)(еy)  [plхy  (plх ply)] (assumption of indirect proof) 3. (ех)(еy) [ plхy   (plх ply)] (2) 4. (ех)(еy) [ (pхy  nху)   (nху   pхy)] (3) 5. (ех)(еy) [( pхy  nху )  (nху  pхy)] (4) 6. (ех)(еy) [(nхy  nху )  (рху  pхy)] (5, т 2) 7. (ех)(еy) (nхy  pхy) (6) 8. (ех)(еy) схy (7, def. 5) 9. contradiction (1, 8) 10. (х)(y) [plхy  (plх ply)] (2, 9). qed theorem 15 is converse to theorem 14. it establishes that if the system of international relations is conflict-free, it means it is either one-pole (synergetic) or double-pole (antagonistic). тheorem 16: the world system of interdependent states ws is conflict-free if and only if it’s polarized into one or two poles: (х)(y) [(plхy  (plх ply))  сху] proof: follows from the combination of theorems 14 and 15. qed the important meaning of theorem 16, which can be called central one for our forecast justification, is that it determines necessary and sufficient conditions of conflict-free development of the system of international relations – synergism within one pole or antagonism between two synergetic poles. all other combinations of relations between states are knowingly conflict, i.e. induce the system to move to one of the stated points of stability. 3. results and discussion forecast and its justification. let’s divide all system forecasts into qualitative and quantitative. qualitative forecasts do not depend on historical, political, economic and cultural factors and are more significant to this effect in the degree of community of their conclusions. the credibility degree of a qualitative system forecast depends on three conditions: (1) the choice of forecast base point; (2) identification of dynamics or the main trend of the system’s new analytics of international relations: system forecast of cold war’s outcomes 11 copyright ©2017 assa. adv. in systems science and appl. (2017) change; (3) calculation of most probable outcomes of the system dynamics in a chosen period of time. we choose the cold war’s end as base point of suggested forecast. the reasons are the following. from the system perspective, the cold war represented antagonism of two superpowers (together with their allies) – the soviet union and usa. the world system ws was divided into two poles and according to theorem 13 was conflict-free and, therefore, a steady point of stability. each superpower strongly tried to maintain power balance in the military, first of all, nuclear field, which gave rise to the stability effect. however, the logic of antagonism development appeared stern and one of the antagonists – ussr eventually ceased to exist. antagonism and the resulting division of the states’ world system ws into two poles disappeared. however, the following 15-year domination of usa didn’t make the world system ws more stable and predictable for one reason only: usa didn’t become a powerful source of synergism for the whole world system neither in economic nor in political and military terms. on the contrary, the usa strategy focused on its isolated domination led and has been leading till now to strengthening and expansion of regional wars, emergence of new centers and waves of terrorism, and destruction of traditional states. according to theorem 16, it means that the world system ws reached not the “history end” but a new system conflict and consequently uprise of whirling motion to a new point of stability. chaotic and therefore poorly predictable movement of the world system ws from the burnt-out antagonism of two superpowers to a new point of stability – the main trend of the modern history after the cold war’s end. it follows from theorem 16 that the world economy can have only two points of stability– association of all (or the major part) states into one synergetic system or their dissociation no longer on the ideological basis into two antagonistic poles. before discussing the chances of each of these outcomes, it’s reasonable to assess the resiliency of an idea popular in political and diplomatic quarters – the idea of “multipolar world”, “polycentric architecture of international relations” that appeared as counterbalance to the idea of usa unconditional domination. let’ consider the simplest example of multipolarity. let’s assume the world system ws consists of three states – russia, china, and usa. pure algebra shows that in this case four conflict-free and four conflict states are possible. conflict-free states, in their turn, include one synergetic (all three states are friends) and three antagonistic (any two countries confront the third one) states. therefore, the world system multipolarity is possible only as purely quantitative expansion of synergism or antagonism of its members with binding preservation of its pole number (one or two). otherwise, when it has no poles or there are two poles, the world system becomes conflict, the sole destiny of which is to drift to new points of stability. thus, our theory states that there are no any other conflict-free states of the world system ws except oneor double-pole ones. the appearance of configurations with three and more poles in a conflict-free world system or the lack of any poles as long-term and stable state is impossible on the purely formal grounds. if we take into account that negative and positive relations can vary in their impact (lovefriendship-sympathy; hatred-enmity-antipathy), there are the following new opportunities [1]. firstly, any pole of synergetic or antagonistic system without conflict occurrence can be divided into several subsets (subpoles), the members of which are positively related to each other more strongly than all the other members. relations of incompatibility of the members of different poles can also vary within antagonistic systems, but in this case from strongly negative to weakly negative. secondly, any synergetic or antagonistic system can coexist conflict-free with any other number of poles symmetrically independent on them. one-pole system of international relations is a synergetic system (follows from def. 6 and def. 8). if synergism as a universal tendency has no any strong opposition, then what is now called “global community” would quickly emerge. however, the major obstacle to achieving this outcome by the world system is not so much significant differences in economic, political, ethnic, 12 v.a. svetlov, n.m. sidorov copyright ©2017 assa. adv. in systems science and appl. (2017) religious and cultural conditions of different nations’ existence, sometimes coming to their total incompatibility, as the state’s need to defend its national interests, which always fatally counteracts the unconditional “victory” of panhuman interests [13]. in view of the above stated, achieving the state of universal synergism by the humanity appears an extremely unlikely outcome of the transformation of the world system of relations after the cold war’s end in the foreseeable future. a double-pole system of international relations is an antagonistic system (follows from def. 7 and def. 9). so far, today there are no any prerequisites of military, economic or cultural nature, which indicate the imminence of arising two opposing world centers of forces. the economic competition of china and usa will never result in worldwide economic antagonism because of their economic interdependence, i.e. it will always be local and temporal. the similar conclusion can be made in relation to the military competition of russia and usa. any tendency of the world system division into two new poles will always be opposed by the tendencies of global labour and capital market formation, scarcity of resources, and consequently inevitable mutual dependence of all countries. thus, antagonism as a point of stability and inevitable division of the world system into two poles now and in the foreseen future can be considered not less problematic solution to its conflict of the world system of relations than universal synergism [14]. in view of the foregoing, our forecast is based on the following system regularity. if neither of the two possible points of stability is reachable in a definite period of time for a changing system, then the system most likely starts steadily oscillating (fluctuating) between them within its limits. forecast: after the cold war’s end the world system of states ws reached a stage of sustained oscillation between two conflict-free and equally unreachable states – unipolarity (synergism) and bipolarity (antagonism). taking into account present tendencies counteracting to the achievement of each of the stated points of stability, this stage will last at least till the middle of the 21 st century. empirical characteristics of the oscillation of the world system of states in a given time period will be determined by particular correlation of two counter forces, forming the basic conflict of any system, – a tendency to universal association of states into one global community and a tendency to protection of their national interests [10]. 4 conclusion ● there are two stable prejudices among analysts and practitioners of international relations. the both of them are related to misestimation of the conflict essence and functions in the evolution of the world system of states. ● according to the first prejudice, conflict is a solely deconstructive state, and all disputes, as a rule, unfold around searching for efficient methods of its prevention or resolution. ● according to the second prejudice, antagonism and such its important kind as war, are identified with conflict or are considered its type. ● the both prejudices can be overcome with a new theoretical approach, called new analytics. conflict is considered the only condition of transformation of the world system of international relations between the points of stability (attractors); synergism and antagonism – two (and only two) steady conflict-free states. references [1] svetlov, v. a. (2013). vvedenie v edinuyu teoriyu analiza i razresheniya konfliktov [introduction to a unified theory of conflict analysis and solutions]. moscow, russia: librokom. [in russian] [2] huntington, s. (1996). the clash of civilizations and the remaking of world order , new york, ny: simon & schuster. new analytics of international relations: system forecast of cold war’s outcomes 13 copyright ©2017 assa. adv. in systems science and appl. (2017) [3] solomon, t. & steele, b. j. (2017). micro-moves in international relations theory, european j. of int. relations, 23(2), 267-291. [4] northrop, f., the meeting of east and west: an inquiry concerning world understanding. new york, ny: macmillan, 1947. [5] abraham, k. j. & abramson, y. (2017). a pragmatist vocation for international relations: the (global) public and its problems. european j. int. relations, 23(1), 26-48. [6] lipschutz, r. d. (2000). after authority. war, peace, and global politics in 21 st century. albany, ny: state university of new york press. [7] lebow, r. n. (2014). constructing cause in international relations. cambridge, u.k.: cambridge university press. [8] matti, j. (2017). new national organization of europe: nationalism and minority rights after the end of the cold war. int. relations, 31(1), 21-41. [9] wolff, j. & spanger, h.-j. (2017). the interaction of interests and norms in international democracy promotion. j. int. relations and development, 20(1), 80-107. [10] svetlov, v.a. (2012) konflict i evolucia. ot geneticheskih konfliktov k konfliktu pokolenii [conflict and evolution. from genetic conflicts to generation gap]. moscow, russia: librokom. [in russian] [11] glencross, a. (2015) from ‘doing history’ to thinking historically: historical consciousness across history and international relations. int. relations, 29(4), 413-433. [12] maslikov, v.a. (2015) aktualnie factory neustoichivisti obshestvennogo razvitia [actual factors of the instability of social development]. materialy afanasievskih chtenii, 1(13), 210-214. [in russian] [13] zadihailo, d. (2013) economic power in the context of the legal regulation of economic relations. j. the national academy of legal sciences of ukraine, 4(75), 163-171. [14] buyarov, d., kireev, a. & druzyaka, a. (2017) russian-chinese relations: from the decision of interstate problems to the strategic partnership. j. advanced research in law and economics, 6(4), 746-752. https://doi.org/10.14505//jarle.v6.4(14).03. advances in systems science and applications (2013) vol.13 no.1 37-52 static models of corruption in hierarchical systems andrey v. antonenko and guennady a. ougolnitsky southern federal university, russia abstract the principles of modeling and control of corruption in the hierarchical systems are formulated. a system of the theoretical static models of administrative and economic corruption as well as models of corruption in real estate development is built. the dependence of corrupted behavior on model parameters is investigated, and the analytical conditions in which corruption is not profitable for the agent or can be controlled by the principal for ensuring sustainable development of the system are received. keywords corruption, hierarchical systems, game theory, optimization, sustainable development 1 introduction the pioneering work on the mathematical modeling of corruption is susan roseackerman’s paper [1] with the consequent development of the proposed approach in her monograph [2]. the monograph specifies the ideas by gary becker [3] concerning the modeling of arbitrary crimes and punishments. later the static models of corruption were developed in two principal directions. the first one includes mathematical models that investigate corruption inside an organization (internal corruption) and between organizations (external corruption). within the framework of this class of models the problems of interaction of competence and corruption, non-optimal resource distribution, the principal’s impact to the model equilibrium and some other ones are studied. the paper [4] may be taken as an example in which the problems of federal resources distribution by bureaucrats and the forms of impact of the competence between both the bureaucrats and the agents on the bribery scope are investigated. models of corruption in tax authorities also belong to this class, for example, [5-6]. the second research direction is connected with modeling of corruption in social and political life (particularly the corruption patterns presented in the electoral process). models of the principal agent type that analyze such problems as anti-corruption incentives development, building of decision making rules that facilitate the indication of bribery facts, providing the hierarchical structure and control mechanisms for the principal that compel or impel the agent to abstain from corruption. one of the principal papers in this research domain is [7] that describes the social costs from corruption in the bureaucratic chain of an arbitrary length with consideration of the competence between agents throughout their hierarchy. the paper by m.bac [8] continues the investigations started by f.kofman and j.lawarree [9] and targeted to the analysis of corruption in the 38 andrey v. antonenko: static models of corruption in hierarchical systems hierarchy principal controller agent. as an extension of the basic model m.bac has built a derivative model in which a hierarchy of homogeneous agents is considered instead of one agent. the papers [10-12] also refer to this class. this research direction includes also the works dealing with studies of the political-economic corruption, especially in the voting systems [13-15]. problems of endemic corruption are discussed in the papers [16-17]. in [18] a game theoretic model predicts that resource rents tend to an increase in corruption if the quality of the democratic institutions is relatively poor, but not otherwise. the paper [19] investigates the role of guilt aversion in public administration. the authors’ approach to the modeling of corruption is represented in [20-21]. its essence consists in the following principles. 2 principles of modeling twelve principles are used in building of the models of corruption. 1). the basic modeling pattern is a hierarchical structure “principal supervisor agent object” in different modifications and its investigation by means of the optimization theory and stackelberg games theory. in the dynamic models the state of the object is described explicitly while in the static models only the impact of the agent to the object is considered. the supervisor may be corrupted, while the principal is not corrupted and controls corruption. so the elements of the above structure are bribe-controller, bribe-taker, bribe-giver and object of impact respectively. 2). certain requirements of the sustainable development of the controlled system (object) are supposed to be known. in the dynamic models they are formulated in terms of the object’s state while in the static ones in terms of the agent’s impact to the object. if the requirements of sustainable development are satisfied then the principal’s control problem is assumed to be solved even if corruption exists. 3). both pairs “principal supervisor” and “supervisor agent” are in the relations “leader follower”. the leading player on any level (principal or supervisor respectively) uses methods of compulsion (administrative or legislative impacts) or impulsion (economic impacts) for achievement of his/her objectives. the mathematical formalization of compulsion means an impact of the leader to the set of admissible strategies of the follower (as a rule, without a feed back), and impulsion means the impact to the follower’s payoff function (as a rule, with a feed back) [20]. 4). the cases of administrative corruption when administrative requirements or constraints are weakened for a bribe and the economic corruption when the economic ones are weakened by corruption are differentiated. the model of administrative corruption describes compulsion of the agent by the supervisor with advances in systems science and applications (2013) vol.13 no.1 39 a feed back on bribe, and the model of economic corruption describes impulsion in that pair with an additional feed back on bribe. 5). corruption threatens to the object’s sustainable development because it is profitable for the bribe-taker to weaken the requirements of sustainable development in exchange for a bribe. from the other side, corruption is a specific form of a feedback in the hierarchical systems subject to which the control variables become the functions of bribe. 6). corruption exists in the forms of capture and extortion. in the case of capture a basic set of administrative or economic services is guaranteed while additional indulgences are provided for a bribe. in the case of extortion a bribe is required already for the basic set of services, otherwise the requirements are enhanced. 7). the bribe-taker’s behavior is characterized by tractability (a willingness to weaken the administrative or economic constraints in exchange for a bribe) and greed (a price of the weakening). a set of quantitative indicators of tractability and greed is developed. 8). the bribe may represent a part of the payoff received by the agent subject to the administrative or economic indulgences or an absolute sum. in both cases it is convenient to suppose the bribe variable to be a part of the corrupted payoff or the whole agent’s income respectively. 9). for studying corruption in hierarchical control systems with consideration of the requirements of sustainable development both descriptive and normative approaches are applicable. in the case of descriptive approach the functions of administrative and economic corruption are given, and the main problem is to identify their parameters on statistical data. in the case of normative approach the corruption (bribery) function is found as the solution of an optimization or game theoretic problem. 10). the investigation of corruption in the system “principal-supervisor-agentobject” is possible from three points of view. if the bribery function is known then from the point of view of the agent the corruption can be described by an optimization model. from the supervisor’s point of view a hierarchical parametrical game of the class γ2 arises which solution in the form of bribery function with a feedback on the value of bribe is known in a general form [22, see appendix]. from the point of view of the principal the problem of corruption control consists in seeking of such values of control variables that the found optimal strategy of the supervisor satisfies the requirements of sustainable development. 11). the identification problem in this domain is not at all trivial and requires special investigations and expertises. each data set determines a specific social, economic and political system exposed to corruption. 12). it makes sense to build “genetic” series of sequentially complicated mod40 andrey v. antonenko: static models of corruption in hierarchical systems els that more and more precisely describe the real phenomena of corruption in hierarchical control systems. the principal logical pattern of this sequential complication has a form “optimization models hierarchical two-person games hierarchical three-person games”. with consideration of the possible modifications of the models of each type the “series” become the “genetic networks”. it is the last principle that determines the rest of the paper. 3 system of theoretical models of corruption the principles formulated above are used to build a system of static models of corruption in hierarchical control systems. 3.1 static models of economic corruption the basic optimization model of economic corruption has the form g(b) = b+ r(b) → min (1) 0 ≤ b ≤ 1 (2) where b is a part of the bribe, r(b) ∈ [0, 1] is a given function of the economic corruption (for example, a real diminishing of the tax rate, i.e. absence of sanctions in case of non-payment for the bribe). thus, the function g(b) means the total costs for tax payments and bribe that are to be minimized by the agent. in case of the linear parameterization r(b) = r0 −ab the model (1) takes the form g(b) = r0 + (1−a)b → min, 0 ≤ b ≤ 1 (3) here r0 is an official tax rate (0 ≤ r0 ≤ 1), a is a model parameter. considering that the function of economic corruption r(b) = r0 −ab monotonically decreases when 0 ≤ b ≤ 1 then a > 0. from the other side, the total costs g(b) are non-negative, therefore a ≤ 1 + r0. thus 0 < a ≤ 1 + r0. the parameter determines the qualitative characteristics of the bribe-taker’s behavior. if a = 0 then corruption is completely absent. as the value of a increases, the bribetaker’s tractability also increases and his greed decreases. the threshold value is a = r0: in this case r(1) = 0, i.e. the maximal greed ensures the maximal tractability. if a < r0 then the greed is over-limited and the tractability does not reach the maximal value (i.e. a positive tax is paid for any bribe). when a > r0 the agent can avoid the tax payments completely in exchange for a moderate bribe (maximal tractability and small greed). return to the solution of the optimization problem (1). having that dg(b) db = 1−a the function monotonically increases when 0 < a < 1 and its minimal value is reached in the left end of the admissible range: gmin = g(0) = r0. respectively when 1 < a < 1+r0 the function g monotonically decreases and its minimal value advances in systems science and applications (2013) vol.13 no.1 41 is reached in the right end: gmin = g(1) = 1 + r0 − a < r0. in the degenerate case a = 1 we get g(b) ≡ r0(the bribe is useless and corruption is absent). so, the parameter a again plays the key role and determines two qualitatively different behavior strategies of the agent. if 0 < a < 1 then the total costs of the agent g(b) increase, it is rational to abandon from bribe and to pay taxes equal to the legislative rate r0. if 1 < a < 1 + r0 then the costs g(b) diminish and an economic incentive arises to give the bribe and to pay in total 1 + r0 −a < r0. now consider the function of economic corruption in a more general form g(b) = b+ r(b) = b+ r0 −abk(k > 0). it is true that dg db = 1− kabk−1 = 0 ⇒ b∗ = (ka) 1 1−k ; d2g db2 = k(1− k)abk−2; d2g(b∗) db2 = (1− k)(ka) 1 k−1 . therefore, d2g(b∗) db2 { > 0, 0 < k < 1 ⇒ b∗ is the point of min imun; < 0, k > 1 ⇒ b∗ is the point of max imun. if b∗ is a point of maximum then min b g(b) = { g(0) = r0, 0 < a < 1, g(1) = r0 + 1−a, 1 < a < 1 + r0. finally bmin  (ka) 1 1−k , 0 < k < 1; 0, k > 1 ∧ 0 < a < 1; 1, k > 1 ∧ 1 < a < 1 + r0; (the case k = 1 is studied separately). so, if 0 < k < 1 then it is always profitable for the agent to give a bribe; if k > 1 then the reason to pay the bribe depends on the parameter . now consider a hierarchical game supervisor agent in the form gs (r, b) = b+ pr → max, 0 ≤ r ≤ r0; (4) ga(r, b) = b+ r → min, 0 ≤ b ≤ 1. (5) here the parameter p designates a part of the collected tax payments transferred by the principal (considered in the model implicitly) to the supervisor as a reward. in this model the function r = r(b) is not given and is found as an optimal strategy 42 andrey v. antonenko: static models of corruption in hierarchical systems of the leader in the game (4)-(5). using germeyer’s theorem (see appendix), we find the ε-optimal strategy in the form r̃ε(b) = { 0, b = r0 − ε ∧ p < 1− ε r0 , r0, otherwise. so, it is almost always profitable for the agent to give the bribe b = r0 − ε and to receive an arbitrary small but positive tax economy ε. the condition of effectiveness of the economic control of corruption is given by the inequality p ≥ 1 − ε r0 . however, it is hardly possible because in this case almost all tax payments should be assigned to the supervisors reward. thus let’s consider a principal’s problem of the administrative (compulsive) control of corruption as a hierarchical three-person game in the form gp = c(q) +k(r0 − r) → min, 0 ≤ q ≤ r0 (6) gs = b+ pr → max, q ≤ r ≤ r0 (7) ga = b+ r → min, 0 ≤ b ≤ 1 (8) here q is a variable of the principal’s administrative control that constraints from below the ability of supervisor’s corrupted behavior; c(q) is an increasing convex principal’s control cost function, c(0) = 0, c(r0) = ∞ is a parameter of the penalty charged on the supervisor if the condition of sustainable development r = r0 is violated. now the solution of the hierarchical game (7)-(8) takes the form r̃ε(b) = { q, b = r0 − ε ∧ p < 1− ε r0 , r0, otherwise. the first-order condition for the problem (6) gives q̂ = (c ′)−1(k). the values of the objective function in this point and in the ends of the admissible segment are equal to gp(q̂) = c ′(k) + k(r0 − r), gp(0) = kr0, gp(r0) = c(r0). having that k ≫ 1 or even k → ∞, i.e. the condition of sustainable development is unalterable for the principal, we get that his objective function reaches its maximum when q = r0 therefore the supervisor’s optimal strategy is identically equal to r0, the condition of sustainable development is satisfied and corruption is absent. thus, in this model the administrative control of corruption is more effective than the economic one. 3.2 static models of administrative corruption the basic optimization model of administrative corruption has the initial form ga(u, b) = (1− b)f(u) → max (9) advances in systems science and applications (2013) vol.13 no.1 43 0 ≤ u ≤ s(b), 0 ≤ b ≤ 1 (10) where b is a part of the bribe, u is the agent’s action, f(u) is the agent (bribegiver)/s production function, s(b) is a quota that constraints the agent’s action from above and may be extended for the bribe. having that the production function increases its maximum is always reached in the right end of the admissible segment. therefore the model (9)-(10) can be represented as an optimization problem with one variable g(b) = (1− b)f(s(b)) → max, 0 ≤ b ≤ 1 (11) in case of the linear parameterization of the function of administrative corruption s(b) = s0 + ab, where s0 is the official value of quota (notice that the function monotonically increases when 0 ≤ b ≤ 1 because it describes the quota extension in exchange for the bribe) and the linear production function f(x) = x the model (11) takes the form g(b) = (1− b)(s0 +ab) → max, 0 ≤ b ≤ 1 (12) as in the case of economic corruption, the parameter determines the qualitative characteristics of the bribe-taker’s behavior. if = 0 then corruption is completely absent. as the value of increases, the bribe-taker’s tractability also increases and his greed decreases. the threshold value is a = 1 − s0: in this case s(1) = 1, i.e. the maximal greed ensures the maximal tractability. if a < 1 − s0 then the greed is over-limited and the tractability does not reach the maximal value (i.e. any bribe nevertheless requires to obey a quota strictly less than 1). when a > 1−s0, the agent can ignore the quota completely in exchange for a moderate bribe (maximal tractability and small greed). return to the solution of the problem (12). we have g(0) = s0, g(1) = 0, dg(b) db = a− s0 − 2ab, d2g(b) db2 = −2a < 0, therefore b∗ = a−s0 2a is the point of maximum, g(b∗) = (a+s0)2 4a ≥ g(0). notice that b∗ { > 0, a > s0, < 0, a < s0, , and gmax = { g(b∗), a > s0, g(0), a < s0. so, the parameter again plays the key role and determines two qualitatively different behavior strategies of the agent. if a < s0 then there is no reason to give a bribe because the agent’s payoff is maximal when b = 0 and is equal to s0. however, if a > s0 then the optimal part of bribe is equal to b∗ = a−s0 2a that leads to the agent’s payoff (a+s0)2 4a ≥ s0. 44 andrey v. antonenko: static models of corruption in hierarchical systems in more general case g(b) = (1−b)(s0+ab)k, k ≤ 1, the maximal agent’s payoff is equal to gmax = { g(0), a ≤ s0 k , g(b∗), a > s0 k , where b∗ = ka− s0 (1 + k)a . so, it is disadvantageously to give a bribe when s0 ≥ ka. adding of a supervisor leads to the two-person hierarchical game in the form gs(s, u, b) = bf(u) → max, s0 ≤ s ≤ 1; ga(s, u, b) = (1− b)f(u) → max, 0 ≤ u ≤ s, 0 ≤ b ≤ 1. taking the agent’s production function in the form f(u) = aum and considering that its maximum is reached in the right end of the admissible segment, we get the following game: gs(s, b) = absm → max, s0 ≤ s ≤ 1; (13) ga(s, b) = a(1− b)sm → max, 0 ≤ b ≤ 1; (14) using germeyers theorem (see appendix) and considering that interests of the players coincide in s, we get the solution of the game (13)-(14) in the form s̃ε(b) = { 1, b = 1− ε− s0 m, s0, otherwise. so, it is profitable for the agent to give the bribe and to avoid quota completely. to prevent corruption it is necessary to add to the game (13)-(14) a principal with the control problem gp = c(q) +k(q − s0) → min, s0 ≤ q ≤ 1; (15) that is similar to the problem (6) in the model of economic corruption; the supervisor’s problem (13) takes the form gs(s, b) = absm → max, s0 ≤ s ≤ q; (16) i.e. the supervisor’s ability of corrupted behavior is restricted from above by the variable of non-corrupted principal’s administrative control q. then the solution of the game (16), (14) is s̃ε(b) = { q, b = 1− ε− ( s0 q )m , s0, otherwise. advances in systems science and applications (2013) vol.13 no.1 45 and the solution of the problem (15) has the form q∗ = { s0, c(s0) < c(1) +k(1− s0) 1, otherwise. considering that as in the model of economic corruption k ≫ 1 or even k → ∞, we get q∗ = s0, i.e. the principal ensures the condition of sustainable development s = s0, and corruption is absent. 4 models of corruption in real estate development in a general form an optimization model of the economic corruption in real estate development may be represented as g(b) = (1− b)[r(b) + ξ(1− r(b))] → max, 0 ≤ b ≤ 1, (17) where g(b) is the agent (developer)’s payoff function having the sense of his income from a real estate development project considering a bribe cost; b is an economic bribe as a part of the agent’s income; r(b) is a part of the social real estate redeemed with guarantee by the state at a fixed price (the price may be augmented for a bribe); ξ is a part of the social real estate that can be sold by the agent himself. the function of economic corruption r(b) is supposed to be known according to the descriptive approach. according to the economic sense it increases monotonically in the segment [0,1] and in the considered case of capture (extortion is analyzed similarly) r(0) = r0, where r0 is the legislative value of r (it is supposed that r = r0 is the condition of sustainable development of the real estate project). it is natural to use the function of economic corruption in the form r(b) = r0 +abk (a > 0), 0 ≤ r(b) ≤ 1, (18) restrict ourselves by the linear parameterization (k = 1). considering (18) we get r(b) = min{r0 +ab, 1}, a > 0. the parameter a determines the qualitative characteristics of the bribe-taker’s behavior. if a = 0 then corruption is completely absent. as the value of increases, the bribe-taker’s tractability also increases and his greed decreases. the threshold value is a = 1− r0: in this case r(1) = 1, i.e. the maximal greed ensures the maximal tractability. if a < 1− r0 then the greed is over-limited and the tractability does not reach the maximal value (i.e. the agent should sell a part of the social real estate himself with any bribe). when a > 1− r0 the agent can sell the total amount of the social real estate to the state at a fixed price in exchange for a moderate bribe (maximal tractability and small greed). the optimization problem (17) takes the form g(b) = (1− b)[r0 +ab+ ξ(1− (r0 +ab))] → max, 0 ≤ b ≤ 1, 46 andrey v. antonenko: static models of corruption in hierarchical systems we have g(0) = r0+ξ(1−r0), g(1) = 0, dg(b) db = −2a(1−ξ)b+a−aξ−r0−ξ+ξr0. from the first-order condition dg(b) db = 0,we find b∗ = (a−r0)(1−ξ)−ξ 2a(1−ξ) that is the point of maximum because dg2(b∗) db2 = −2a(1− ξ) < 0. by the structure gmax = { g(0) = r0 + ξ(1− r0), a < r0(1−ξ)+ξ 1−ξ . g(b∗) > g(0), otherwise. (19) thus, the agent’s optimal strategy is determined by the parameters a, r0, ξ subject to (19): in the first case it is the zero bribe, in the second one the bribe b∗ > 0 giving the payoff g(b∗) > g(0) . now consider the models of administrative corruption in real estate development. in a general case the optimization problem has the form g(b, u) = (1− b)[ru+ ξ(1− r)u+ γη(γ)(1− u)] → max, s(b) ≤ u ≤ 1, 0 ≤ b ≤ 1, (20) where g(b, u) is the agent(developer)’s payoff function that means his income from selling all types of the real estate considering corruption; b is a part of the administrative bribe; u is a part of the social type in the total amount of real estate; r – a part of the social real estate redeemed by the state at a fixed price with a guarantee; ξ – an estimated part of the social real estate that can be sold by the agent himself at the same price; γ > 1 – an increasing factor of the price of more expensive types of the real estate; η(γ) – an estimated part of the agent’s own sale of more expensive types of the real estate; s(b) – an obligatory minimal quota of the part of social real estate in the total amount (can be diminished for a bribe). notice that in the optimization problem (20) the payoff function depends on two variables. investigate the problem: ∂g ∂b = −[(r + ξ(1− r)− γη)u+ γη]; ∂g ∂b = 0 ⇒ µ∗ = γη γη − (r + ξ(1− r)) > 0, r + ξ(1− r) < γη; ∂2g ∂b2 = 0, ∂2g ∂b∂u = γη − (r + ξ(1− r)); ∂g ∂u = (1− b)(r + ξ(1− r)− γη); ∂g ∂u = 0 ⇒ b∗ = 1; ∂2g ∂u2 = 0; ∂2g ∂b∂u = γη − (r + ξ(1− r)). advances in systems science and applications (2013) vol.13 no.1 47 the hesse matrix has the form h = ∣∣∣∣∣∣∣∣ 0 γη − (r + ξ(1− r)) γη − (r + ξ(1− r)) 0 ∣∣∣∣∣∣∣∣ , its determinant is equal to |h| = −[γη−(r+ξ(1−r))]2 < 0. therefore (b∗, u∗) is a saddle point and maximum of the function g(b, u) is reached in the boundary of the set of admissible values. we have g(1, u) = 0;g(0, 1) = r + ξ(1− r)− γη; g(0, s0) = r + ξ(1− r) + (1− s0)γη > g(0, 1);g(0, s0) > 0. thus in any case maximum of the function g(b, u) is reached when u = s(b). then one-variable optimization problem arises in the form g(b) = (1− b)[(r + ξ(1− r)− γη)s(b) + γη] → max, 0 ≤ b ≤ 1. denote c = γη > 0, d = γη− (r+ ξ(1− r)). notice that if d < 0 then it is more profitable to sell the social real estate than more expensive one, and to give a bribe is not rational. therefore the agents payoff function can be written in the form g(b) = (1− b)(c −ds(b)), d > 0. as earlier, represent a function of administrative corruption in the form s(b) = s0 −abk (a > 0), 0 ≤ s(b) ≤ 1, where s0 is the legislative value of quota, and restrict ourselves by the case of linear (k = 1) parameterization of the function. s(b) = s0 −ab (a > 0), g(b) = (1− b)(c −ds0 +adb) = c −ds0 + (d(a+ s0)− c)b−adb2; ∂g ∂b = ad(1−2b) = 0 ⇒ b∗ = 1/2; ∂2g ∂b2 = −2ad < 0 ⇒ b∗is the point of maximum; g(0) = c −ds0; g(1) = 0; g(b∗) = 2c +d(a− 2s0) 4 . thus, gmax = { g(b∗), d(a+ 2s0) > 2c, g(0), otherwise i.e. the advantageousness of giving a bribe depends on the relation between the model parameters. notice that in the case b = b∗ the agent’s payoff is positive if s0 < 2c+ad 2d , i.e. an obligatory quota of the social real estate is not very big. 48 andrey v. antonenko: static models of corruption in hierarchical systems now consider a game theoretic model of the administrative corruption in real estate development in the form gp(q, s) = p1(s− s0)− q s0 − q → max, 0 ≤ q ≤ s0. gs(s, b) = p2(s− s0) + bc[(r + ξ(c)(1− r))u+ γη(γ)(1− u)] → max, q ≤ s ≤ s0. ga(b, u) = c(1− b)[(r + ξ(c)(1− r))u+ γη(γ)(1− u)] → max, s ≤ u ≤ 1, 0 ≤ b ≤ 1. (21) here b is a part of the administrative bribe; u is a part of the social type in the total amount of real estate; c – a price of the social real estate; r – a part of the social real estate redeemed by the state at the fixed price c with a guarantee; ξ(c) – an estimated part of the social real estate that can be sold by the agent himself at the same price; γ > 1 – an increasing factor of the price of more expensive types of the real estate; η(γ) – an estimated part of the agent’s own sale of more expensive types of the real estate; s(b) – an obligatory minimal quota of the part of social real estate in the total amount (can be diminished for a bribe); q – a parameter of the principal’s administrative control; p1, p2 > 0 – penalty factors charged on the principal and the supervisor respectively if the quota (condition of sustainable development s ≥ s0) is violated. it is supposed that r = const, ∼ s = s(b). the problem (21) is solved using a heuristic two-stage algorithm. on the first stage the hierarchical game between the supervisor and the agent is considered. notice that the function ga achieves its maximum in u independently from b, therefore subject to the linearity of the function ga in u the solution of the agent’s problem has the form u∗ = { 1, r + ξ(c)(1− r) ≥ γη(γ), s, otherwise consider these two cases separately. 1). r+ ξ(c)(1− r) ≥ γη(γ), i.e. it is more profitable to sell the social real estate. in this case a game does not arise because when u = 1 then the function ga does not depend on s. an evident solution of the optimization problem is b = 0, i.e. corruption is absent. 2). r + ξ(c)(1 − r) < γη(γ), i.e. it is more profitable to sell the expensive real estate. then a standard hierarchical two-person game arises in the form gs(s, b) = p2(s− s0) + cb[r + ξ(c)(1− r)− γη(γ)] → max, q ≤ s ≤ s0; ga(s, b) = c(1− b)[(r + ξ(c)(1− r)− γη(γ))s+ γη(γ)] → max, 0 ≤ b ≤ 1. advances in systems science and applications (2013) vol.13 no.1 49 application of germeyer’s theorem (see appendix) gives sp (b) ≡ s0; s d(b) = { s0, p2 > bc(r + ξ(c)(1− r)− γη(γ)), q, otherwise; la = c[(r + ξ(c)(1− r)− γη(γ))s0 + γη(γ)]; ea = {b = 0};da = {(s, b) : ga(s, b) > la} = ∅, s.t.; (1− b)[(r+ ξ(c)(1− r)− γη(γ))s+ γη(γ)] ≤ (r+ ξ(c)(1− r)− γη(γ))s0 + γη(γ); when 0 ≤ b ≤ 1, q ≤ s ≤ s0. it is assumed in this case that k1 = −∞ < k2, thus ∼ s ∗ (b) = { sd(b), b = 0, s0, otherwise, = { q, p2 < bc(r + ξ(c)(1− r)− γη(γ)) ∧ b = 0, s0, otherwise. however the conditions p2 < bc(r + ξ(c)(1 − r) − γη(γ))and b = 0 are disjoint, therefore ∼ s ∗ (b) ≡ s0. so, the solution of the agent’s optimization problem has the form b∗ = 0 (in fact, corruption is absent), and u∗ = s0, therefore it is not required to involve the principal to ensure the condition of sustainable development. respectively, in the second stage it is sufficient for the principal to solve the control costs minimization problem q s0−q → min, 0 ≤ q ≤ s0,the trivial solution of which has the form q∗ = 0. at last, consider a model of the economic corruption in the three-level system of real estate development gp = h(c) +m(r0 − r) → min, 0 ≤ c ≤ 1; gs = f(1)(b+ cr) → max, 0 ≤ r ≤ r0; ga = f(1)(1− b− r) → max, 0 ≤ b ≤ 1; where h(c) – an increasing convex principal’s cost function; m – a penalty factor (the penalty is charged if the condition of sustainable development r = r0 is violated). the supervisor’s optimal guaranteeing strategy has the form ∼ r ∗ (b) = { 0, b = r0 − ε ∧ c < 1− ε r0 , r0, otherwise, if b ̸= r0 − ε then r ≡ r0 and the evident solution of the principal’s optimization problem is c = 0. if b = r0 − ε then the principal can ensure the condition of 50 andrey v. antonenko: static models of corruption in hierarchical systems sustainable development only by choosing c = 1. therefore the solution of his optimization problem has the form c∗ = { 1, h(1) < mr0, 0, otherwise, i.e. the principal is obliged to compare the penalty for the violation of sustainable development condition and the costs required for its control. 5 conclusion the principles of modeling and control of corruption in the hierarchical systems are formulated. particularly, a problem of corruption control is set as the problem of realization of certain requirements to the state of controlled system (conditions of sustainable development). if the conditions are satisfied then the corruption control problem is supposes to be solved even if corruption in the system exists. this setting differs from the proposed by g.becker especially economic approach based on commensurability of the damage caused by corruption and its control costs [3]. another important methodical principle is building of “genetic” series of sequentially complicated models that more and more precisely describe the real phenomena of corruption in hierarchical control systems. this principle is realized in the paper by building of series of the static theoretical models of administrative and economic corruption as well as models of corruption in real estate development. the dependence of corrupted behavior on model parameters is investigated, and the analytical conditions in which corruption is not profitable for the agent or can be controlled by the principal for ensuring sustainable development of the system are received. appendix germeyer’s theorem [22]. suppose that functionsm1(x1, x2),m2(x1, x2) are continuous on the compacts x1, x2. introduce the following functions: a punishment strategy xp1 (x2) by the rule m2(x p 1 , x2) = min x1∈x1 m2(x1, x2), and the dominance strategy xd1 (x2) that satisfies the condition m1(x d 1 , x2) = max x1∈x1 m(x1, x2). besides, the following quantities and sets are introduced: l2 = max x2∈x2 m2(x p 1 , x2);e2 = {x2 ∈ x2 : m2(x p 1 , x2) = l2}; d2 = {(x1, x2) ∈ x1 ×x2 : m2(x1, x2) > l2}; k1 = sup (x1,x2)∈d2 m1(x1, x2) ≤ m1(x ε 1, x ε 2) + ε (d2 = ∅ ⇒ k1 = −∞); advances in systems science and applications (2013) vol.13 no.1 51 k2 = min x2∈e2 max x1∈x1 m1(x1, x2). then the guaranteed payoff of the player 1 (leader) in the game γ2 is equal to ω1 = max(k1,k2), and the respective ε-optimal strategy has the form x̃ε1(x2) =  xε1, x2 = xε2, k1 > k2, xd1 (x2), x2 ∈ d2, k1 ≤ k2, xp1 (x2), otherwise. acknowledgements the work is supported by rfbr (project 12-01-00017). references [1] rose-ackerman s. 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(1992), “trivial bribes and the corruption ban: a coordination game among rational legislators”, public choice, vol.78, pp.305-327. [16] levin m, satarov g. (2000), “corruption and institutions in russia”, european journal of political economy, vol.16, pp.113-132. [17] kahana n, qijun l. (2010), “endemic corruption”, european journal of political economy, vol.26, pp.82-88. [18] bhattacharia s, hodler r. (2010), “natural resources, bureaucracy and corruption”, european economic review, vol.54, pp.608-621. [19] balafoutas l. (2011), “public beliefs and corruption in a repeated psychological game”, journal of economic behavior and organization, vol.78, pp.5159. [20] ougolnitsky g. (2011), sustainable management, n.y.: nova science publishers, pp.287. [21] antonenko a.v, ougolnitsky g.a, usov a.b. (2012), “static models of corruption in hierarchical control systems”, contributions to game theory and management. vol.v. collected papers presented on the fifth international conference game theory and management / ed. leon a. petrosyan, nikolay a. zenkevich. spb.: graduate school of management spbu, pp.20-32. [22] gorelick v.a, kononenko a.f. (1982), game theoretic models of decision making in ecological-economic systems, moscow, pp.145(in russian). corresponding author guennady a. ougolnitsky can be contacted at: ougoln@gmail.com. advances in systems science and applications (2013) vol.13 no.4 369-378 in-solid acoustic sources extraction using a multi-sensor architecture d.t. pham, m. yang and m. al-kutubi manufacturing engineering centre cardiff university, cardiff cf24 3aa, uk abstract this paper presents a novel signal feature extraction method based on location pattern matching (lpm) for improving the reliability of pattern matching of received acoustic signals by using multi-sensor architecture. the proposed solution is to solve the reliability problem of signal matching by extracting specific features from the signal that is only associated with the source location not the source information with multi-sensor layout. experiments showed that this method can increase the tolerance of lpm on various types of impacts, thus the matching accuracy. keywords signal feature extraction, pattern matching, multi-sensor architecture 1 introduction fig.1 lpm system layout. according to the well described acoustic time-reversal theory [1], it is possible to reconstruct an acoustic signal at its initial excitation location in a scatter370 d. t. pham:in-solid acoustic sources extraction using a multi-sensor architecture ing medium by recording the received signal and sending back the time reversed version of the signal through the medium. this implies that the received signal carries its source location signature as a result of scattering in the transmission medium and reflections from its complex boundaries. in the lpm approach, the uniqueness of features embedded in a received acoustic signal, as a representation of its source location, is employed to determine if it matches one of the predefined locations on a tangible object. this is achieved by creating an array of template of received signals generated from taps on preknown locations [2-5]. the source location can then be determined by a cross correlation analysis from the best similarity match with a signal in the templates. fig.1 shows a block diagram of an lpm system comprising a board, one sensor located at the bottom right of the board, signal conditioning hardware, data acquisition device and a host pc. a tapping on the hardboard is detected by the sensor, conditioned by the signal conditioning hardware, digitised by the data acquisition card and processed by the pc. the matching result can be achieved by a cross correlation analysis which is widely used to measure similarity between two signals or images. 2 pattern-based localisation the theory which location identification from the received signal pattern is based on is that an acoustic wave propagating from its source to a destination carries a specific signature in its pattern associated with the source location as a result of scattering. when a driving force is applied into a medium, a mechanical wave is generated transporting energy away from the source of disturbance. in a bounded medium the waves propagate until they reach the boundaries and are reflected or absorbed. the reflections cause reverberation on the received signal acquired by a transducer at certain location in the medium. to comprehend the effect of boundary on the magnitude and phase of a received signal, a plane wave is assumed propagating in the x direction, the acoustic wave equation is given by: ∂2p ∂x2 = 1 ν2p ∂2p ∂t2 (1) where p is the acoustic pressure as a function of time t and distance x.the solution of the differential equation (1) is the propagating wave equation given by [6]: p = aej(ωt−kx) (2) where a is the amplitude constant, νp phase speed and k is the wave number. with this wave equation, it can be shown how a transmission medium with one reflection affects the received signal. assume a simple model of two signal paths advances in systems science and applications (2013) vol.13 no.4 371 from the transmitter to the receiver. one direct path with unity gain and a delay td , and the other is reflected with attenuation α and a delay td+∆t resulting from the path length difference. the overall transfer function of such a transmission medium h(ω) can be expressed by: h(ω) = e−jωtd + αe−jω(td+∆t) = |h(ω)|︷ ︸︸ ︷√ 1 + α2 + 2α cosω∆t) θh(ω)︷ ︸︸ ︷ e −j(ωtd+tan−1 α sinω∆t 1+α cosω∆t ) therefore, multi-path reflections cause the distortion in the magnitude |h(ω)| and the phase θh(ω) characteristics of the transmission medium. since the time delay is the product of the path length difference by the wave-number, it is clear from (1) that the magnitude and phase of the received signal at a certain location will vary as the source location changes. in random medium the received signal is a combination of the direct wave and multiple delayed scattered waves went through reflections and refractions plus the effect of non isotropic material, and therefore the received signal in reality has much more complicated relation to its source location than that given in equation (1). with this phenomenon, signals received from different source locations will have a distinctive feature that can be used to localize an unknown source signal if there is some knowledge about this feature. practically, location features are obtained from the received signal in the training stage. 3 focusing in time reversal theory when a source is applied into a medium at certain location and a received signal is recorded by an array of transducers as shown in fig.1. the received signals are reversed in time and then re-emitted into the medium. the re-emitted acoustic energy wave propagates back through the same medium and goes through all the multiple scattering, reflections and refraction that they underwent in the forward direction and refocuses on the source location. if only an aperture of limited area, called time-reversal mirror, is adopted in a time reversal operation, a small part of the field radiated by the acoustic source is captured and time reversed, thus limiting focusing quality [7]. however, in a bounded medium, multiple reflections along the medium boundaries significantly increase the apparent aperture of the time reversal mirror which means that the transducers can equivalently replaced by reflecting boundaries that redirect part of the incident wave towards the aperture. thus spatial information is converted into one dimension representation of the time domain and the reversal quality depends crucially on the duration of the time-reversal window, i.e. the length of the recording to be reversed. the heterogeneity of the medium 372 d. t. pham:in-solid acoustic sources extraction using a multi-sensor architecture or the boundaries which produces multi-paths contributes to have an aperture that is much larger than its physical size. it has been shown experimentally that in a cavity with specific geometrical property focusing with time reversal can be obtained using one transducer [7]. the time-reversal approach is clearly connected to the inverse source problem. (a) impulse transmission and reception. (b) time reversed transmission and impulse localization fig.2 sketch of time reversal focusing in random medium they both deal with propagation of a time-reversed field, but the propagation is real in the time reversal experiment and simulated in the inverse problem. moreover, the most important distinction is that the time reversal approach doesnt need knowledge of the propagating medium while the inverse problem method does. as any linear and time-invariant process, wave propagation through a multiple scattering medium may be described as a linear system with a certain impulse response. if the source sends a dirac pulse δ(t) function,the jth transducer of the time-reversal-mirror will receive a signal hj(t),which is the propagation impulse response from the source to transducer j. moreover, due to spatial reciprocity,hj(t) is also the impulse response describing the propagation of a pulse from the hj(t) transducer to the source. thus, if the transducer is able to record and time reverse the whole impulse response, the signal generated at the source can be represented by the convolution hj(t) ∗ hj(−t).this convolution product, in terms of signal analysis, is a typical matched filter which is a linear filter whose output is optimal in some sense. for whatever the impulse response hj(t), the temporal result is the convolution between this response and its time advances in systems science and applications (2013) vol.13 no.4 373 fig.3 time-reversed wave field observed at different times around the central point on a square of 15 15mm2 ([fp01]). reverse version hj(t) ∗ hj(−t).which is maximal at time t = 0 . this maximum is always positive and equals ∫ h2j t dt , i.e. the energy conveyed by hj(t) . for an n-element array, the signal recreated on the source can be written as, s(t) = j=n∑ j=1 hj(t) ∗ hj(−t) (3) even if hj(t) are completely random and apparently uncorrelated signals, each term in this sum reaches its maximum at time t = 0. so all contributions add constructively around t = 0 , whereas at earlier or later times uncorrelated contributions tend to interfere destructively one with another. thus the recreation of a sharp peak after time reversing on n-element array can be viewed as an interference process between n outputs of n matched filters. 4 realization of source localization from time reversal focusing as described in the previous section, it is possible with time-reversal theory to reconstruct an acoustic signal in its original location in a scattering medium by recording the received signals and sending back the time reversed version of these signals through the medium. this implies that the received signal carries its source location signature as a result of scattering in the transmission medium and reflections from its complex boundaries. with the same assumption of dirac delta source excitation, the response term hj(t) of the temporal correlation as given by (1) can be interpreted in lpm as a template obtained in the learning stage. here hj(−t) is the applied test signal with negative sign turns the convolution into a 374 d. t. pham:in-solid acoustic sources extraction using a multi-sensor architecture cross correlation operation. therefore, cross-correlation is a focusing process in time reversal but a similarity measure in template matching localization. although source reconstruction in time reversal is a transmission process or active, it is comparable to passive source localization with template matching by the operation of cross correlation as illustrated in the previous section. however, time reversal can help to provide explanation of physical limitations for source localization problems. ideally, the source reconstruction is achieved with array of sensors surrounding the source origin with element spacing of at least half a wavelength. practically a limited aperture area is used on the cost of focusing resolution. the smaller the array, the larger the focal spot. as a result of wave diffraction, the waves will refocus to a spot not smaller than the shortest wavelength [5]. accordingly, the achievable localization resolution using template matching can be increased with more sensors but still limited to the smallest wavelength, and since wavelength is inversely related to frequency, higher accuracy can be obtained with interactions that generate higher frequency signals such as using nail clicks and metallic object than those who generate lower frequencies as finger tap and damping material. 5 location feature extraction for enhanced reliability ideally to comply with the time reversal theory, an impulsive source is needed for both learning and recognition stages. practically, it is found that cross correlation is sensitive to the template type which means that similar type of impacts should be used in both stages since the variation in the signal will appear as variation in the location signature. one option to make the system more reliable is to use multiple templates for different type of interactions, but this is impractical as the calibration work will be intensified. signal filtering improves the resolution but found to be not effective to improve reliability because it filters the frequency components not the location feature. therefore an attractive novel solution is proposed here to solve the reliability problem by extracting specific features from the signal that is associated with the source location not the source information. let an unknown source signal given by s(t) emitted from location i on the surface of a tangible object as shown in fig.4, and two sensors are receiving the signals g1(t) and g2(t). the propagation path from the given source to sensor-1 and sensor-2 can be expressed by a specific transfer function denoted by h1(t) and h2(t) respectively. the transfer function is characterized by the complex propagation path and independent on the source signal information. accordingly, the transfer function for a specific source to receiver path represents the actual source location signature. treating the transmission medium as a black box of a single input/multiple output time invariant system, the output signal received advances in systems science and applications (2013) vol.13 no.4 375 by the ith sensor can be expressed analytically the convolution integral given by: gi(t) = ∫ ∞ −∞ hi(t)s(t− τ)dτ (4) for instance considering the output from one sensor only, it is possible to measure the transfer function for a given location by applying an impulse δ(t) t that location and measure the output. in that case the received signal is the transfer function which in turn can be used as location signature in the template. if the test signal is also an impulse, then the matching process is just the comparison of location signatures and therefore high accuracy is anticipated. otherwise, if the signal used for test, or for the template, are not an impulse, the resulting received signal will include source signal information plus location information, which accordingly will result in estimation error. this is why lpm works better with impulsive type of impacts. the task now is to develop a technique to extract the only source information from any type of interaction by employing two sensors. let the input in the system shown in fig.5 is a stationary random signal. the fig.4 received signals from two different paths. fig.5 black box lpm model 376 d. t. pham:in-solid acoustic sources extraction using a multi-sensor architecture input/output relationship in the frequency domain is given by fourier transform of equation (5) is given by: gi(f) = s(f)hi(f) (5) the hypothesis of extracting the location signature involves utilizing a measurable quantity that doesnt require any knowledge about the input excitation or medium transfer function. thus from the output/output relationship that is given by the cross spectral density function between the two outputs as given by: pg1g2 = g1(f)g2(f) (6) then a hypothetical transfer function can be defined as: hg1g2(f) = pg1g2(f) pg1g1(f) (7) where pg1g2 is the autocorrelation function of g1(t). it can be seen from the above three equations hg1g2(f) is a function of the two transfer functions h1(t) and h2(t) which still represents an independent location signature. a related subject in literature is the binaural localization in human which is simulated by the head related transfer function cue and defined by the ratio of the two output spectrums [7]. by rewriting the complex equation (7) in the form of magnitude and phase as: hg1g2(f) = ag1g2(f)∠φg1g2 (8) either the magnitude or phase patterns can be used as a location signature information. to use both of amplitude and phase information, the pattern given by: ĥg1g2(t) = ∫ ∞ −∞ ψ(f)pg1g2(f)/pg1g1(f)e −j2πft df (9) can be used, where φ(f) is a weighting function introduced to compensate the bandwidth variation and depends on the interaction method. with φ(f) = ag1g2(f), only phase information are extracted. obviously, utilizing only the phase or the magnitude pattern is computationally faster than using (9) since it is not necessary to convert them into time domain. an experimental result was carried out by registering a template from impacts at defined locations generated by pen tip hits on a glass sheet. the test database consists of different interaction types as pen tip hits, nail clicks and finger tapping. then with the evaluation procedure it is found that highest percentage of correct estimations were obtained using (9), than using the phase information only and lastly when magnitude information were used only. advances in systems science and applications (2013) vol.13 no.4 377 mdf board with 4 piezo disk sensors 3-d interactive object conditioning circuit for piezo sensors piezoelectric microphone on mdf board fig.6 experimental setups and sensors for lpm localization 6 experimentation the experimental setup consists of the interactive object, sensors, signal amplifier, data acquisition card and a pc to process the signals. different object materials and shapes have been tested including metal, glass, plastic, fibre boards and 3-d objects. the suitable sensors were the piezoelectric discs, electret microphones and accelerometers. for data acquisition, four channel pci card is used for evaluation and two channel sound card used for demonstrations such as the portable usb sound card and the wireless audio transmitter. some of them are pictured in fig.6. the lpm system found to be working well on variety of materials. the piezo ceramic sounders and electret microphones are the cheapest but can only pick up low frequencies when firmly attached to the surface. the piezoelectric microphone is very sensitive with wide bandwidth response and the most expensive among others. the piezoelectric shock sensor from murata is the best sensor with sufficient frequency response and a very reasonable price. 378 d. t. pham:in-solid acoustic sources extraction using a multi-sensor architecture 7 conclusion in this paper, a new method for in-solid acoustic source localisation based on learning and template match is presented. with this location pattern matching technique, it is possible to convert virtually any solid objects into an interactive interface by simply attaching sensors on the objects surface to transfer the acoustic signals resulting from natural interactions through signal conditioning hardware to data acquisition device for sampling before delivering the digital data to a pc where the localization algorithm is running. the reliability problem caused by the variance of impact patterns to the matching process has been improved by the concept of extracting the location signature pattern from received signals using multi-sensor layout. acknowledgements this work was financed by the european fp6 ist project “tangible acoustic interfaces for computer-human interaction (tai-chi)”. the support of the european commission is gratefully acknowledged. references [1] h.jeong, y.jang. (2000), “wavelet analysis of plate wave propagation in composite laminates”, composite structure, vol.49, pp.443-450. [2] r.ing, n.quieffin, s.catheline, m.fink. (2004), “tangible interactive interface usingacoustic time reversal process”, applied physics letters. [3] g.lai, p aarabi. (2003), “active object localization using speaker arrays”, proceedings of the ieee sixth international conference on information fusion, vol.1, pp.70-73. [4] e dijk. (2004), “indoor ultrasonic position estimation using a single base station, phd thesis”, eindhoven university of technology, netherlands. [5] bousseljot r. and kreiseler d. (2000), “waveform recognition with 10,000 ecgs”, ieee computers in cardiology proceedings, pp.24-27, cambridge, ma, pp.331-334. [6] o’hagan r and zelinsky a. (1997), “finger track-a robust and real-time gesture interface”, advanced topics in artificial intelligence, tenth australian joint conference on artificial intelligence proceedings, pp.475-484. [7] m.fink. (1999), “time reversed acoustics”, scientific america, pp.91-95. corresponding author author can be contacted at: my@cf.ac.uk. advances in systems science and applications (2012) vol.12 no.2 194-206 prevalence of clustering for current smoking, current drinking and components of metabolic syndrome among adults of lanxi, heilongjiang, china yuling duan1, jingbo zhao2,yujun zhao3, shiying fu3, fuman wang2 and liting yang2 1harbin center for disease control and prevention, harbin, heilongjiang provence, 150056, china 2department of epidemiology, school of public health, harbin medical university, harbin 150081, china 3department of cardiology, first clinical medical college of harbin medical university, harbin 150086, china abstract a total of 2967 participants were surveyed in a poor county of heilongjiang by a stratified randomized cluster sampling. information on cvd risk factors was collected with standardized questionnaires and laboratory measurements during 2006 to 2007.overall, 93.65%, 73.71%, 46.29%, 21.55% and 6.99% of rural residents had ≥ 1,≥ 2,≥ 3,≥ 4 and ≥ 5 modifiable cvd risk factors (serum triglycerides(tg), high density lipoprotein cholesterol(hdl-c), hypertension, diabetes, current smoking, current drinking and overweight. both hypertension prevalence and diabetes prevalence increased with age, but hdl-c prevalence decreased with age among both men and women (each p for trend, < 0.001). tg, current drinking and current smoking prevalence decreased with age (each p for trend, < 0.05, except for overweight and ms, each p for trend, > 0.05) among men. while, tg, current drinking, current smoking and ms prevalence increased with age (each p for trend, < 0.001) among women. in a multivariate model including age and sex, the odds ratio (95% confidence interval [ci]) of having ≥ 1,≥ 2,≥ 3,≥ 4, and ≥ 5 cvd risk factors versus none of the studied risk factors were 3.15(95%ci, 1.92-5.16), 3.73(95%ci, 2.27-6.13), 4.31(95%ci, 2.59-7.17), 5.24(95% ci, 3.04-9.03), and 9.29(95% ci, 4.78-18.06), respectively, for chinese adults 65 to 74 years old versus 35 to 44 years old; 1.77(95%ci, 1.282.45), 1.98(95%ci, 1.42-2.74), 2.39(95%ci, 1.70-3.35), 2.40(95% ci, 1.66-3.74), and 2.79(95% ci, 1.77-4.40), respectively, for men compared with women. keywords smoking, drinking, metabolic syndrome, clustering, prevalence 1 introduction the term “metabolic syndrome”(ms) refers to the clustering of visceral adiposity, hyperglycemia, high blood pressure(bp) and dyslipidemia[1-2] ,associated with increased risk of cardiovascular disease(cvd). drinking and smoking are considadvances in systems science and applications (2012) vol.12 no.2 195 ered as risk factors of cvd. cvd accounts for half of noncommunicable diseases deaths worldwide, 16.7 million in 2002[3]. nowadays, cvd are not exclusive in developed countries. the majority of those deaths (≈ 9.1 million) occurred to economically developing countries, and much of the burden of cvd in developing countries occurs to china[4-5]. since the reform and opening-up in 1978, infectious diseases has declined, but changes in lifestyle and diet have led to an increase in life expectancy, a greatly increased frequency of cvd and other chronic diseases[6-7] in china. heilongjiang province located in the northeast of china. lanxi county is about 60 kilometers in the north of harbin city, heilongjiang. although at the beginning of opening-up and reform in china, lanxis economy was among poor counties of nation, and its economic is still lagging behind other developed regions, the income and lifestyle of residents have greatly changed during recent decade. although prevalence and risk factors of cvd in the developed country or developed population were paid much attention to, but little we known about prevalence and risk factors of cvd in the poor population in the rural. in order to provide important information and health education programs to policy makers in economic poverty county of heilongjiang province, it is important to quantify the proportion of the population at high risk for cvd. such data provide an understanding of the size of the population in need of targeted interventions to lower the population burden of illness due to cvd. the goal of this study was to quantify the proportion of adult residents in poor economic county who had 1 or more of the following major modifiable cvd risk factors: components of metabolic syndrome(which was defined by the international diabetes federation(idf), including overweight, hdl-c, serum triglycerides(tc), hypertension, diabetes),current smoking and current drinking, to determine the percentage of adults with a clustering of 2 or more, 3 or more, 4 or more and 5 or more of these risk factors. this was completed with the use of data collected from rural residents in lanxi county of heilongjiang province during 2006 to 2007. 2 methods 2.1 study population multi-stage stratified sampling method was used to select a representative sample of the rural residents of heilongjiang. in the first stage, lanxi county was selected as poor county, according to economic condition and residents income of lanxi during 1978 to 2006. secondly, each town was chosen from lanxi county if its economy was among middle, and then pingshan was selected as its economy was in the middle level of lanxi county. lastly, 11 villages were chosen by randomized method from 35 villages of pingshan. total residents, who aged 35 year and lived in the locate villages more than 5 years were considered as subjects, were 3480 persons. 3012 individuals completed the survey and examination, 196 yuling duan:prevalence of clustering for current smoking, current drinking and... and others were excluded because of absence or denying to answer questions, as well as exclusion criterion: type diabetes, fever, acute infection disease and other factors which influenced measurement of blood pressure, blood glucose or blood lipids. the response rate is 86.55%. at last, 2967 participants aged 35 to 74 (1324 males and 1643 females ) were used for analysis. 2.2 data collection data collection was conducted in examination centers at local health stations in the participants residential area. in a few instances when participants were unable to go to the examination center, the interview and examination were conducted in their homes. during clinic or home visits, trained research staff administered a standard questionnaire. information on demographic characteristics include age, gender, education, ethnicity, occupation, household income, cigarette smoking, drinking, a self-reported history of stroke, myocardial infarction and congestive heart failure, and the previous diagnosis and treatment of hypertension, high cholesterol, and diabetes. smoking prevalence was calculated using data obtained from the self report questionnaire. current smoker is defined as a person who has smoked cigarettes continuously or accumulatively for over 6 months during his or her life and smoked at least once during the last month before the investigation[8]. current drinker was defined as a person who has drunk beer or wine or distilled spirits at least once a week, apart from drinking in festivals and holidays[9]. 2.3 blood pressure measurement for each participant, after 15 min of sitting, 3 blood pressure measurements were obtained by a trained nurse using a standard sphygmomanometer, which procedures recommended by the american heart association[10]. before their blood pressure measurement for at least 30 minutes, all subjects were advised to avoid alcohol, cigarette smoking, coffee, tea, and excessive exercise. the mean value of three blood pressure measurements was used for this study. hypertension was defined as an average systolic blood pressure (sbp) ≥ 140mm hg and/or an average diastolic blood pressure (dbp)≥ 90mm hg and/or self-reported current treatment for hypertension with antihypertensive medication[11-12]. 2.4 measurements of weight, height and waist circumstance body weight, height and waist circumstance were measured by trained and certified observers according to a standard protocol. subjects wore light clothing and no shoes. weight was measured to the nearest 0.1kg by using calibrated balance scales placed on a solid horizontal surface. height was measured to the nearest 0.1cm with a frankfort plane positioned at a 90◦ angle against a wall-mounted metal tape by using calibrated stadiometer. advances in systems science and applications (2012) vol.12 no.2 197 bmi was calculated as weight divided by height squared (kg/m2). waist circumference was taken at the midway point between the inferior margin of the last rib and the crest of the ilium in a horizontal plane and measured to the nearest 0.1 cm. overweight was defined as having a waist circumference≥ 90 cm (man),≥ 80cm (woman)[13]. 2.5 laboratory measurements participants were asked to fast overnight before their study visit, and blood samples, to measure serum lipids and plasma glucose, were drawn by venipuncture. blood specimens were processed at the field center and then sent to a central clinical laboratory of first affiliated hospital of harbin medical university in harbin by bus, where specimens were stored at -70◦c until laboratory assays were performed. hdl-c and serum tc were analyzed enzymatically with commercially available reagents[14]. lipid measurements were standardized according to the criteria of the centers for disease control and prevention-national heart, lung, and blood institute lipid standardization program[15]. abnormal tc was defined as tc ≥ 1.7 mmol/l. abnormal hdl-c was defined as hdl-c < 0.91 mmol/l or self-reported current treatment with cholesterol-lowering medication. for the glucose measurement, whole blood was collected in evacuated tubes containing naf. plasma glucose was measured with a modified hexokinase enzymatic method. diabetes was defined as having a fasting plasma glucose level≥ 7.0 mmol/l and/or self-reported current treatment with antidiabetes medication (insulin or oral hypoglycemic agents). 2.6 the idf consensus worldwide definition of the metabolic syndrome according to the new idf definition[13], for a person defined as having the metabolic syndrome they must have central obesity (defined as waist circumference ≥ 90cm for chinese men or ≥ 80cm for chinese women) plus any two of the following four factors: raised tg level: ≥ 1.7 mmol/l, or specific treatment for this lipid abnormality, reduced hdl cholesterol: < 1.03 mmol/l in males and < 1.29 mmol/l in females, or specific treatment for this lipid abnormality, raised blood pressure: systolic bp ≥ 130 or diastolic bp ≥ 85 mm hg, or treatment of previously diagnosed hypertension, raised fasting plasma glucose (fpg) ≥ 5.6 mmol/l, or previously diagnosed type 2 diabetes, if above 5.6 mmol/l , ogtt is strongly recommended but is not necessary to define presence of the syndrome. in addition, ethics committees of harbin medical university in china approved the study. written, informed consent was obtained from each participant before data collection. participants with untreated conditions, identified during the study, were referred to their usual primary healthcare provider. 198 yuling duan:prevalence of clustering for current smoking, current drinking and... 2.7 statistical methods database was set up by epidata 3.0, and all analyses were conducted with sas 9.1.3. analyses were conducted in these survey participants without a history of myocardial infarction, stroke, or congestive heart failure. crude rate were standardized according to the age distribution for chinese adults in the year 2000. participants missing key variables measurements such as smoking, drinking, height, weight, waist circumstances, hdl-c, tg, glucose, and blood pressure were also excluded from the analyses, leaving a final study population of 2967. the prevalence of each cvd risk factor (hdl-c, tc, hypertension, diabetes, current smoking, current drinking and overweight, ms) was determined for men and women separately, by age group (35 to 44, 45 to 54, 55 to 64, and 65 to 74 years), all from rural residence. the prevalence of 0, 1, 2, 3, 4, and 5 cvd risk factors was determined for the overall study population as well as for men and women separately. then, the percentage of the population with ≥ 1,≥ 2,≥ 3,≥ 4, and ≥ 5 cvd risk factors was determined by age group and sex separately. the significance of the differences in the prevalence of ≥ 1,≥ 2,≥ 3,≥ 4, and ≥ 5 cvd risk factors across subgroups was compared with the wald-χ2 test. the adjusted odds ratios(or) and 95% confidence intervals (ci) of having ≥ 1,≥ 2,≥ 3,≥ 4, and ≥ 5 major cvd risk factors versus no cvd risk factor were determined from multivariable logistic-regression models that included age group and sex. 3 results 3.1 prevalence of cvd risk factors in rural residents in heilongjiang in the 11 villages of lanxi county, population aged 35 to 74 years, the age standardized prevalence of hdl-c, tg, hypertension, diabetes, current drinking, current smoking, overweight(wc≥ 90(man),≥ 80(woman)) and ms was 31.67%, 13.82 %, 39.26%, 6.49%, 24.65%, 41.81%, 32.86%, 19.84% respectively (table 1). prevalence ± se ∗hdl−c < 0.91mmol/l, serum tg ≥ 1.7 mmol/l, and/or current cholesterollowering medication use. †sbp ≥ 140mm hg and/or dbp ≥ 90mm hg and/or current antihypertensive medication use. ‡fasting plasma glucose ≥ 7.0mmol/l and/or current antidiabetes medication use. §overweight was defined as waist circumference (waist circumference ≥ 90(man), ≥ 80(woman)) se indicates standard error. total sr indicates total standard rate, standardized on the basis of the year 2000 age distribution of the chinese population. the age standardized prevalence of overweight, tg and ms was higher in women than in men (each p < 0.001). while hypertension, diabetes, current advances in systems science and applications (2012) vol.12 no.2 199 table 1 age-standardized prevalence of cvd risk factors among rural study participants in heilongjiang province population no tg∗ hdl∗ hypertension† diabetes‡ current current overweight§ ms groups drinking smoking (90(man)) (80(woman)) total sr 2967 31.67±0.85 13.82±0.63 39.26±0.90 6.49±0.45 24.65±0.79 41.81±0.91 32.86±0.86 19.84±0.73 total 2967 32.69±0.86 13.38±0.63 41.32±0.90 6.81±0.46 25.45±0.80 42.67±0.91 33.50±0.87 20.63±0.74 man overall 1324 31.80±1.28 14.34±0.96 43.96±1.36 6.90±0.70 53.22±1.37 53.59±1.37 21.06±1.12 14.17±0.96 sr 1324 31.34±1.27 13.52±0.94 48.04±1.37 7.48±0.72 53.32±1.37 53.10±1.37 20.77±1.11 14.12±0.96 overall age, y 35321 31.78±2.60 17.45±2.11 25.86±2.44 4.67±1.18 52.02±2.79 53.89±2.78 21.81±2.30 14.02±1.94 45401 38.90±2.43 14.21±1.74 47.38±2.49 6.23±1.21 59.35±2.45 56.61±2.47 23.19±2.11 15.96±1.83 55331 29.31±2.50 10.88±1.71 56.19±2.73 9.37±1.60 56.50±2.72 54.08±2.74 21.15±2.24 14.20±1.92 65-74 271 22.14±2.52 11.07±1.91 65.31±2.89 10.33±1.85 42.07±3.00 45.76±3.03 15.50±2.20 11.44±1.93 woman overall 1643 33.36±1.16 13.24±0.84 35.39±1.18 6.30±0.60 2.91±0.41 38.03±1.20 43.01±1.22 25.35±1.07 sr 1643 33.78±1.17 13.27±0.84 35.91±1.18 6.27±0.60 2.98±0.42 34.27±1.17 43.76±1.22 25.87±1.08 overall age, y 35524 20.80±1.77 17.18±1.65 19.47±1.73 2.86±0.73 1.15±0.47 24.05±1.87 34.16±2.07 15.27±1.57 45533 36.77±2.09 12.76±1.45 35.65±2.07 6.00±1.03 3.38±0.78 39.77±2.12 48.78±2.17 27.95±1.94 55376 40.96±2.54 12.23±1.69 46.81±2.57 7.71±1.38 3.46±0.94 36.44±2.48 48.67±2.58 33.51±2.41 65-74 210 45.71±3.44 6.67±1.72 58.10±3.40 12.86±2.31 5.71±1.60 41.90±3.40 46.19±3.44 35.24±3.30 drinking and current smoking prevalence were higher in men than in women (each p < 0.001). there were no significant difference on hdl-c prevalence between woman and man(p > 0.05). hypertension and diabetes prevalence increased with age among both men and women (each p for trend, < 0.001). among men, tg, hdl-c, current drinking and current smoking prevalence decreased with age (each p for trend, < 0.05). among women, hdl-c prevalence decreased with age (p < 0.001). while, tg, current drinking, current smoking and ms prevalence increases with age(each p for trend, < 0.001). 3.2 prevalence of ≥ 1,≥ 2, and ≥ 3 cvd risk factors in rural residents in heilongjiang overall, 4.32% of rural men and 7.98% of rural women, respectively, did not have any of the risk factors investigated(tg, hdl-c, hypertension, diabetes, current drinking, current smoking and, overweight; (figure 1 and table 2). cvd risk factors: current smoking, current drinking and component of metabolic syndrome (tg, hdl-c, hypertension, diabetes, overweight) in contrast, 17.07%, 25.95%, 27.47%, 15.86%, and 9.33% of men and 22.11%, 28.75%, 22.53%, 13.52%, and 5.12% of women had 1, 2, 3, 4 and ≥ 5 of these risk factors, respectively. overall, 6.35%, 19.86%, 27.50%, 24.73%, 14.56%, and 6.99% of rural adults had 0, 1, 2, 3, 4 and 5 of these risk factors, respectively. in total, 93.65% of the population had 1 or more cvd risk factors (table 2). the prevalence of ≥ 1,≥ 2,≥ 3,≥ 4, and ≥ 5 major modifiable cvd risk factors was mainly higher at older ages (table 3; each p < 0.001). also, the age standardized prevalence of ≥ 1,≥ 2,≥ 3,≥ 4, and ≥ 5 risk factors were higher among men compared with women respectively (all p < 0.05)(table 3 ). prevalence±se; se indicates standard error; total sr indicates total standard 200 yuling duan:prevalence of clustering for current smoking, current drinking and... fig.1 age-standardized prevalence of cvd risk factors among men and women of poor county in heilongjiang in china table 2 standardized prevalence of 0, 1 ,2 ,3 ,4 and ≥ 5 risk factors among total and gender rural participants in heilongjiang province 0risk factors 1risk factors 2risk factors 3risk factors 4risk factors ≥ 5risk factors n(%se) n(%se) n(%se) n(%se) n(%se) n(%se) age group 35102(12.07±1.12) 227(26.86±1.52) 225(26.63±1.52) 178(21.07±1.40) 85(10.06±1.03) 28(3.31±0.62) 4534(3.64±0.61) 172(18.44±1.27) 266(28.51±1.48) 234(25.08±1.42) 151(16.18±1.21) 76(8.15±0.90) 5532(4.55±0.78) 107(15.22±1.35) 192(27.31±1.68) 190(27.03±1.67) 130(18.49±1.46) 52(7.40±0.98) 65-74 20(4.18±0.91) 82(17.12±1.72) 131(27.35±2.04) 130(27.14±2.03) 65(13.57±1.56) 51(10.65±1.41) woman 137(7.98±0.67) 363(22.11±1.02) 472(28.75±1.11) 370(22.53±1.03) 222(13.52±0.84) 84(5.12±0.54) man 57(4.32±0.56) 225(17.07±1.03) 342(25.95±1.20) 362(27.47±1.23) 209(15.86±1.00) 123(9.33±0.80) total 188(6.35±0.45) 588(19.86±0.73) 814(27.50±0.82) 732(24.73±0.79) 431(14.56±0.65) 207(6.99±0.47) total sr 6.92±0.47 20.80±0.75 27.63±0.82 24.24±0.79 13.83±0.63 6.59±0.46 rate, standardized on the basis of the year 2000 age distribution of the chinese population. prevalence±se; n indicates number of participants with risk factors; % indicate prevalence; se indicates standard error the adjusted odds ratio of having ≥ 1,≥ 2,≥ 3,≥ 4, and ≥ 5 major modifiable cvd risk factors versus none decreased progressively with increasing age (table 4). the adjusted odds ratios (95% ci) of ≥ 1,≥ 2,≥ 3,≥ 4, and ≥ 5 cvd risk factors for persons 65 to 74 years of age compared with their counterparts 35 to 44 years of age were 2.43 (95% ci, 1.46 to 4.04), 2.96 (95% ci, 1.77 to 4.96), 3.78 advances in systems science and applications (2012) vol.12 no.2 201 table 3 prevalence of 0,≥ 1,≥ 2,≥ 3,≥ 4, and ≥ 5 risk factors among rural participants in heilongjiang province 0risk factors ≥ 1risk factors ≥ 2risk factors ≥ 3risk factors ≥ 4risk factors ≥ 5risk factors n(%se) n(%se) n(%se) n(%se) n(%se) n(%se) age-group 35102(12.07±1.12) 743(87.93±1.12) 516(61.07±1.68) 291(34.44±1.63) 113(13.37±1.17) 28(3.31±0.62) 4534(3.64±0.61 899(94.36±0.61) 727(77.92±1.360) 461(49.41±1.64) 227(24.38±1.41) 76(8.15±0.90) 5532(4.55±0.78) 671(95.45±0.78) 564(80.23±1.50) 372(52.92±1.88) 182(25.89±1.65) 52(7.40±0.98) 65-74 20(4.18±0.91 ) 459(95.82±0.91) 377(78.71±1.87) 246(51.36±2.28) 116(24.22±1.96) 51(10.65±1.41) woman 131(7.98±0.67) 1511(92.02±0.67) 1148(69.91±1.13) 676(41.17±1.21) 306(18.64±0.96) 84(5.12±0.54) man 57(4.32±0.56) 1261(95.68±0.56) 1036(78.60±1.13) 694(52.66±1.37) 332(25.19±3.27) 123(9.33±0.80) total 188(6.35±0.45) 2772(93.65±0.45) 2184(73.78±0.81) 1370(46.28±0.92) 638(21.55±0.75) 207(6.99±0.47) (95% ci, 2.24 to 6.40), 4.62(95% ci, 2.62 to 8.15) and 7.39(95% ci,3.68 to 14.84), respectively. in addition, after multivariable adjustment, men were more likely to have ≥ 1,≥ 2, and ≥ 3 cvd risk factors compared with women, respectively (all p < 0.001)(table 4). table 4 standardized prevalence of 0, 1 ,2 ,3 ,4 and ≥ 5 risk factors among total and gender rural participants in heilongjiang province ≥ 1risk factors ≥ 2risk factors ≥ 3risk factors ≥ 4risk factors ≥ 5risk factors age group 351.00(ref) 1.00(ref) 1.00(ref) 1.00(ref) 1.00(ref) 453.56(2.38-5.31) 4.19(2.80-6.29) 4.86(3.20-7.39) 5.85(3.71-79.23) 7.28(4.01-13.22) 552.76(1.83-4.17) 3.41(2.25-5.17) 4.07(2.65-6.25) 5.07(3.17-8.10) 5.10(2.73-9.52) 65-74 2.43(1.46-4.04) 2.96(1.77-4.96) 3.78(2.24-6.40) 4.62(2.62-8.15) 7.39(3.68-14.84) male-gender 1.77(1.28-2.45) 1.98(1.42-2.74) 2.39(1.70-3.35) 2.40(1.66-3.47) 2.79(1.77-4.40) ∗all variables (age group, gender) were included in the same models. for example, the odds ratios for age group were adjusted for gender, the odds ratios for gender were adjusted for age group. †risk factors include components of metabolic syndromeoverweight, wc≥90 cm(man) or ≥80 cm(woman), hdl-c ≤0.91 mmol/l, tg≥1.7 mmol/l, or on cholesterol-lowering medication, hypertension(sbp≥140 and/or dbp≥90 or on high blood pressure medication), diabetes (fasting plasma glucose ≥7.0 mmol/l or on antidiabetes medication), current smoking and current drinking. 4 discussion the metabolic syndrome is a common risk factor for cardiovascular diseases. individuals with this syndrome have an increased risk of developing cardiovascular disease(cvd)[16]. sandra costa fuchs et al[17] revealed that hypertension, diabetes mellitus, obesity, low fruit and vegetable intake, and lack of vigorous or moderate physical activity were clustered into a combination of risk factors, which were independently associated with self-reported cardiovascular disease. a populationcbased, cross-sectional national health examination survey(1998) of 8,816 subject aged 15-79 [18] showed that clustering of 3 or more cvd risk 202 yuling duan:prevalence of clustering for current smoking, current drinking and... factors was 22.7% in man and 21.7% in women. using < 21kg/m2 as a referent, subjects with bmi of 23kg/m2 and 27kg/m2 had an odds ratio of 3.5 and 10.2 in men, and 3.1 and 6.7 in women, respectively, for clustering of cvd risk factors. using <65 cm as a referent, subjects with wc of ≥90 cm in man and ≥80 cm in women had an odds ratio of 13.4 and 13.6, respectively, for clustering of cvd risk factors. a study[19] explored how to influence effect of risk factors on coronary artery disease(cad) when the age gap between men and women narrows, indicated that hypertension, diabetes, dyslipidemia and family history were independent risk factors for women with stable cad. clustering of traditional risk factors may explain the precocity of cad in women who are near in age to men. a study from souss, tunisia[20] for clustering of cardiovascular risk factors among obese urban school children showed that obese children were found to have higher blood pressure, higher tg levels and lower hdl-c than children of normal weight. this results indicated that clustering of cardiovascular risk factors among obese is no longer limited to industrialized countries. another study from japan[21] for multiple risk factors clustering and risk of hypertension, which 5275 japanese male office workers aged 23-59 years were involved, showed that after controlling for potential risk factors of hypertension, the odds ratio of hypertension compared with the absence of risk factors was 1.91, 2.65, 3.88, 6.54, and 8.18 for the presence of 1, 2, 3, 4, and 5 risk factors, respectively ( each p for trend < 0.001). the results indicated that the accumulation of risk factors is highly associated with the increased risk of hypertension in japanese men. multicentre collaborative study showed that high levels of cvd risk factors are common in many economically developing countries[22]. in a study of 7 economically developing countries, as many as 78%, 46%, 50%, and 20% of adults in 1 or more countries were current cigarette smokers, had a high cholesterol level, were overweight, and had hypertension, respectively[22]. these risk factors have emerged as important characteristics in predicting cvd morbidity and mortality in economically developing countries, including china[23]. result from international collaborative study of cardiovascular disease in asia[24] showed that 80.5%, 45.9%, and 17.2% of chinese adults had ≥ 1,≥ 2, and ≥ 3 modifiable cvd risk factors (dyslipidemia, hypertension, diabetes, cigarette smoking, and overweight), respectively. by comparison, 93.1%, 73.0%, and 35.9% of us adults had ≥ 1,≥ 2, and ≥ 3 of these risk factors, respectively. in a multivariate model including age, sex, and area of residence, the odds ratio (95% confidence interval [ci]) of having ≥ 1,≥ 2, and ≥ 3 cvd risk factors versus none of the studied risk factors were 2.61 (2.09-3.27), 3.55 (2.77-4.54) and 4.97 (3.67-6.74), respectively, for chinese adults 65 to 74 years old versus 35 to 44 years old; 3.65 (3.21-4.15), 4.67 (4.06-5.38), and 5.60 (4.70-6.67), respectively, for men compared with women; 1.18 (1.07-1.30), 1.34 (1.21-1.50), and 1.84 (1.60advances in systems science and applications (2012) vol.12 no.2 203 2.12), respectively, for urban compared with rural residents; and 1.98 (1.76-2.22), 2.75 (2.42-3.13), and 4.36 (3.68-5.18), for residents of northern compared with southern china, respectively. the present study indicates that 93.65% rural adults aged 35 to 74 years have at least 1 of the following cvd risk factors: current smoking, current drinking and component of syndrome (tg, hdl, hypertension, diabetes, and overweight). in addition, clustering of 2 or more, 3 or more, 4 or more or 5 or more of these risk factors was noted in 73.78%, 46.28%, 21.55% and 6.99 of rural adults, respectively. the significantly higher prevalence of ≥ 1,≥ 2,≥ 3,≥ 4 and ≥ 5 risk factors in men compared with women may be due to the fact that 43.96% of rural men versus 35.39% of women were hypertension, 6.90% of rural men versus 6.30% of women were diabetes, 53.22% of rural men versus 2.91% of women were current drinker and 53.59% of rural men versus 38.03% of women were current smokers. although the prevalence of current smoking and current drinking of men was higher than those of women, the prevalence of current smoking and current drinking decrease with age in man. in contrast, the prevalence of current smoking and current drinking increase with age in woman. the prevalence of ms in woman was higher than that in man. in this study, we selected components of metabolic syndrome which is defined by idf, current smoking and current drinking to estimate prevalence of cardiovascular disease. indexes we selected were different from international collaborative study in asia[24]. there were some reasons. firstly, components of ms are considered as risk factors of cardiovascular diseases, and prevalence of ms increase with age. secondly, prevalence of cvd, which involved components of metabolic syndrome, is limited. lastly, drinking should be considered as risk factors of cardiovascular diseases in our clustering analysis. this study contains the facts that its results are based on findings in 11villages, representative sample of the residents in the poor county, which allows for calculation of rural representative estimates. otherwise, standard protocols and instruments were used, 86.55% response rate was achieved, and the trained investigators were very careful to collect data. this is a cross-sectional study, which assesses the prevalence of several cvd risk factors. a limitation of the study was its reliance on estimates derived from a cross-sectional study. cross-sectional study has demonstrated the importance of these risk factors to the development of cvd, but cross-sectional study do not allow for quantification of the importance of risk factor clustering in the incidence of cvd. further design and statistics methods are needed[25-27]. in all, 93.65%, 73.78%, 46.28%, 21.55%, 6.99% have ≥ 1,≥ 2,≥ 3,≥ 4,≥ 5 of the cvd risk factors investigated in the current study, respectively. it was noted that hypertension and diabetes prevalence increased with age among both men 204 yuling duan:prevalence of clustering for current smoking, current drinking and... and women (each p for trend, <0.001). at the same time, among women, current drinking, current smoking and ms prevalence increases with age(each p for trend, <0.001). effective interventions for farmers such as improved diet, suitable 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(2004), “the effectiveness of physical activity interventions for the treatment of overweight and obesity and type 2 diabetes”, j. sci med sport, vol.7, pp.52-59. [29] thomson cc, rigotti na. (2003), “hospitaland clinic-based smoking cessation interventions for smokers with cardiovascular disease”, prog cardiovasc dis, vol.45, pp.459-479. corresponding author corresponding author:zhaojb168@sina.com,zyjddm@sohu.com advances in systems science and applications (2013) vol.13 no.1 21-36 systems defined on dynamic sets and analysis of their characteristics xiaojun duan1 and yi lin2 1department of mathematics and systems science, national university of defense technology,changsha 410073, pr china 2department of mathematics,slippery rock university slippery rock, pa 16057 usa abstract (lin, 1999) describes the static structures of general systems and investigates systems dynamics using the concept of time systems without clearing pointing out where systems attributes would come into play. to resolve this problem, in this paper, we introduce the concept of dynamic sets so that the evolutions and changing structures of systems can be adequately described by using object sets, relation sets, attribute sets, and environments of systems. on the basis of this background, we revisit some of the fundamental properties of systems, including systems emergence, stability, etc. keywords dynamic set, system description frame, attribute set, system property 1 introduction systems research represents such a science that it investigates the structure, evolution, and control of systems. its goal is to study the interactions of the parts or relevant matters or events that constitute a whole and the consequent emerging holistic properties and behaviors of the whole. this whole that is made up of the parts, relevant matters or events is known as a system. in order to investigate different behaviors of systems, as of this writing, there has been a good amount of published literature, constituting a well developed theory, see (lin, 1999)[1] and listed references there[2-4]. zadeh (1962)[5] believes that the main goal of systems science is to study the organization and structure of systems, that is, how parts of a system are organized together to form the whole. from the basis of set theory, (lin, 1999)[1] develops a multi-relation systems theory of rigor by establishing and analyzing the elementary concepts and fundamental characteristics of systems. that is, lin establishes the basic concepts and properties of systems science on the solid foundation and rigor of mathematics. the general systems theory maintains (lin, 1999)[1] that each system stands for such a whole that it consists of a set of objects and a set of interactions between the objects. symbolically, a system can be written as s = (m,r) (1.1) where the set m = {mi : i ∈ i} contains all the elements that make up of the system s, known as the object set, i is an index set, r the set of all relations 22 xiaojun duan: systems defined on dynamic sets and analysis of their characteristics between the objects in m that describe the system s. for any relation r ∈ r, there is an ordinal number n(r), which is a function of r, such that r is a subset of the cartesian product of n(r) copies of m : r ⊆ mn(r). both the object and relation sets m and r together completely describe the given system s, its component parts, and the interactions between the parts. on this basis, leveled systems, time systems, etc., are effectively described and studied in general systems theory. because the objects of systems are highly abstract, the only difference between sets of objects will be their index sets. that is why such general theory can be powerfully employed to investigate systems with parts of identical properties. that is, the general system, as given in equ.(1.1), grasps the commonality of objects by ignoring their individual specifics. however, in terms of a realistic system, the objects’ specifics might play important roles in the operations of the system. they influence not only the composite of the system, but also the structure and functionality of the system, producing the relevant dynamic behaviors of evolution of the system. for example, a system that is made up of pure oxygen gas or of pure hydrogen gas is fundamentally different of that consisting of both oxygen and hydrogen gases. in particular, the interactions of the objects of the former system are mainly the repulsive and attractive forces between the gas molecules, while in the latter system, other than the similar repulsive and attractive forces, there are also following chemical reactions when the environment provides the needed condition: 2h2 +o2 → 2h2o (1.2) the resultant system after the chemical reactions is different of the system that existed before the reactions in terms of the object set and also the relation set. what is more important is that the property of further chemical reactions in the system resulted from the first round of chemical reactions is essentially different from that of the system that existed before the first round of chemical reactions. if the system that existed before the chemical reactions is written as s0 = (m0, r0) and the system that existed after as s1 = (m1, r1), then it is obvious that m0 ̸= m1 and r0 ̸= r1. systems s0 and s1 are very different due to that the objects in s0 have undergone substantial change with 2h2o molecules added. additionally, the interactions between the objects of s1 are totally different of those in s0. other than repulsive and attractive reactions between the gas molecules, the interactions of the objects of s1 also include those between liquid molecules and gas molecules, and those between liquid molecules. from this example, it follows that for certain circumstance, it is just not enough to employ only object and relation sets to describe the systems and their behaviors, especially if changes and evolutionary behaviors of the systems are concerned with. it is because in the current general systems theory, both h2 and advances in systems science and applications (2013) vol.13 no.1 23 o2 are treated as abstract objects in s0, while ignoring their differences in other aspects. when such differences do not affect the evolution of the systems much, the description of systems in equ.(1.1) will be most likely sufficient. however, the undeniable fact is that there are many such systems that the interactions between the objects of specific attributes great affect the structures, functionalities, and evolutions of the systems, just as the differences described by the two gases in equ.(1.2). therefore, we need to develop such a systems theory to deal with this situation. we have to further deliberate the description of the object and relation sets of systems so that the consequent evolutions of systems can be adequately investigated. 2 attributes of systems because the usage of rigorous mathematical language in the discussion of properties of general systems is very advantageous, we will continue to employ this approach. in this paper, we will not consider the case when a set is empty; and all sets are assumed to be well defined. that is, we do not consider such paradoxical situations as the set containing all sets as its elements. to this end, those readers who are interested in set theory and the rigorous treatment of general systems theory are advised to consult with (lin, 1999)[1]. as discussed earlier, our main focus here is the difference between various parts of a system, that is, the differences between the system’s objects and between the relations of the objects. we will employ the concept of attributes to describe such differences. by attribute, it means a particular property of the objects and their interactions and the overall behavior and evolutionary characteristic of the system related to the property. if speaking in the abstract language of mathematics, the attributes of a system are a series of propositions regarding the system’s objects, relations between the objects, and the system itself. due to the differences widely existing between systems’ objects, between systems’ relations, and between systems themselves, these propositions can vary greatly. for example, a proposition might hold true for some objects, and become untrue for others. in this case, we say that these objects have the particular attribute, while the others do not. as another example, a proposition can be written as a mapping from the object set to the set of all real numbers so that different objects are mapped onto different real numbers. in this case, we say that the system’s objects contain differences. additionally, the discussion above also indicates that each system evolves with time so that its objects, relations, and attributes should all change with time. so, when we consider the general description of systems, we must include the time factor. by doing so, the structure of a system at each fixed time moment is 24 xiaojun duan: systems defined on dynamic sets and analysis of their characteristics embodied in the object set, relation set, and attribute set, while the evolution of the system is shown in the changes of these sets with time. summarizing what is analyzed above, we can define the system of our concern as follows: definition 2.1. assume that t is a connected subset of the interval [0,+∞), on which a system s exists. then, for t ∈ t , the system s is defined as the following order triplet: st = (mt, rt, qt) (2.1) where mt stands for set of all objects of the system s at the time moment t, rt the set of the relations between the objects in mt, and qt the set of all attributes of the system s. for the sake of convenience of communication, t is referred to as the life span and st the momentary system of the system s. what needs to be emphasized is that we study not only the evolution of systems with time, but also the evolution of the system along with continuous changes of some conditions, such as temperature, density, etc. in such cases, t will be understood as one of those external conditions. the object set of the system of our concern is made up of the system’s fundamental units. so, for each t ∈ t , the objects in mt are fixed. let us write m̂ = {mt,a : a ∈ it, t ∈ t} = ∪ t∈t mt (2.2) where mt = {mt,a : a ∈ it} is the object set of the momentary system st, and it the index set of mt as a function of time t. if for any t ∈ t , mt = m , for some set m , then we say that the object set of the system s is fixed. one of the most elementary relations between objects is binary, relating each pair of objects. such a relation can be written by using the 2-dimensional cartesian product of the system’s object set: r0 t,2 = {(mt,a,mt,b) ∈ mt ×mt : φ 0 t,2(mt,a,mt,b)} ⊆ m2 t (2.3) for each t ∈ t , where φ0 t,2(, ) is a proposition that defines r0 t,2. corresponding to different properties, the system st might contain different binary relations as subsets of the 2-dimensional cartesian product m2 t of the object set. the set of all the binary relations in st is written as follows: rt,2 = {rk t,2 : k ∈ k2} (2.4) where k2 is the index set of all binary relations of st. similarly, the system st might contain trinary relations: r0 t,3 = { (mt,a,mt,b,mt,c) ∈ m3 t : φ0 t,3 (mt,a,mt,b,mt,c) } ⊆ m3 t (2.5) advances in systems science and applications (2013) vol.13 no.1 25 for each t ∈ t , where φ0 t,3(, , ) is a proposition that defines r0 t,3 . the set of all trinary relations of st is written as follows: rt,3 = {rk t,3 : k ∈ k3} (2.6) where k3 is the index set of all trinary relations of st. higher order relations of st can be introduced similarly. for convenience, let us define unitary relations of st as follows: r0 t,1 = {(mt,a) ∈ mt : φ 0 t,1(mt,a)} ⊆ mt (2.7) for each t ∈ t , where φ0 t,1() is a proposition that defines r0 t,1. the set of all unitary relations of st is written as follows: rt,1 = {rk t,1 : k ∈ k1} (2.8) wherek1 is the index set of all unitary relations of st. now, the set rt of relations of st can be written as follows: rt = ∪ α∈ord ri,α (2.9) where ord stands for the set of all ordinal numbers. when ord is taken to be n = the set of all natural numbers, equ. (2.9) stands for the set of all finite relations the momentary system st contains. from the discussion above, it can be seen that all kinds of algebras and spaces studied in mathematics are special cases of equ. (2.9) with ord replaced by n . without loss of generality, let us assume that mt ∈ rt,1. with this convention, it can be seen that in the following when we talk about the attributes of a system st, we only need to mean the attributes of the relations of st, because the object set is also considered a relation, a unitary relation and the attributes of objects are now also those of relations. the attribute set qt of the momentary system st are a series of propositions about the relations. symbolically, we can write: qt = {qt(r1, r2, ..., rα, ...) : qt(...) is a proposition of rα ∈ rt, α = 1, 2, 3...} (2.10) without any doubt, with this notation in place, these propositions in qt embody all aspects of the momentary system st. for example, the concept of mass of object in physics is a mapping: µt : rt → r+ (2.11) which assigns each element in an unitary relation the mass of the element, and each element x⃗ = (x1, x2, ..., xα, ...) in an n-nary relation the sum of the masses 26 xiaojun duan: systems defined on dynamic sets and analysis of their characteristics of the objects contained in the element, if the sum exists. that is, for any r ∈ rt, and any x⃗ = (x1, x2, ..., xα, ...) ∈ r, µt(x⃗) = µt(x1, x2, ..., xα, ...) = ∑ α µt(xα) (2.12) assuming that the sum on the right hand side converges. let us look at the network model of systems as an example, where only unitary and binary relations are considered. in this model, all objects of the system of concern are treated as nodes; each binary relation is modeled as the set of edges of the network. by doing so, each binary relation of the system corresponds to a network or a graph; and different binary relations correspond to different sets of edges. such correspondence can be seen as an attribute of the edges. accordingly, different unitary relations of the system correspond to different attributes of nodes in the network, such as size of the nodes, flow intensities of the nodes, etc. if different types of edges are treated as identical, then the network can be expressed as g = (v,e,q), where v stands for the set of all nodes, e the set of edges, and q some attributes of either the nodes or the edges or both, such as weights of the edges. the ordered pair (v,e) completely describes the topological structure of the network, which is sufficient for some applications. however, if the system we investigate is quite specific, for example, it is a network of railroads, a network of human relationships, etc., we may very well need to model multiple relations. in this case, the attribute set q can be employed to describe the scales of the stations in the railroad network, the traffic conditions or transportation capabilities between stations, etc. if the system is a network of human relationships, then q can be utilized to represent the social status of each individual person, the intensity of interaction between two chosen persons, etc. as a matter of fact, the concept of systems, as defined in equ.(2.1), is a generalization of that as defined in equ.(1.1) (lin, 1999)[1]. in other words, we can rewrite equ.(1.1) in the format of equ.(2.1) as follows. let all object sets be static. so, for any t ∈ t , we have mt = m and rt = r. and what is interesting is how an attribute is introduced. to this end, we can introduce a proposition q0 on the cartesian product m̂ = ∑ α∈ordm α of the object set m so that for any r ∈ m̂ , q0(r) = { 1, if r ∈ r 0, otherwise (2.13) then, we take q = {q0}. that is, the attribute set is a singleton. now, each system written in the format of equ. (1.1) is rewritten in the format of equ. (2.1). here, the attribute q0 describes if an arbitrarily chosen relation in the cartesian product m̂ belongs to the system’s relation set r or not. in essence, it restates advances in systems science and applications (2013) vol.13 no.1 27 the membership relation to the relation set r from the angle of attributes. in particular, because the relation set r of the system s is a subset of m̂ ; now the membership in the relation set r is determined by a proposition q0, while such a description is an attribute of the system s. that is, the general systems theory developed on set theory (lin, 1999)[1] has already implicitly introduced the concept of attributes. what we do here is to make this fact explicit. and because the systems we are interested in can have multiple attributes, our contribution to the general systems theory is to make the concept of attributes more general as a set q of attributes, including more than just the particular attribute q0. 3 subsystems just like each set has its own subsets, every system has subsystems, which can be constructed from the object set, relation set, and the attribute set of the system. in short, a system s is a subsystem of the system s, provided that the object set, relation set, and attribute set of s are corresponding subsets of those of s so that the restrictions of the attributes of s on s agree with the attributes of s. symbolically, we have definition 3.1. let st = (mt, rt, qt) and st = (mt, rt, qt) are two systems, for any t ∈ t . if the following hold true: mt ⊆ mt, rt ⊆ rt|st , and qt ⊆ qt|st ,∀t ∈ t (3.1) then st is known as a subsystem of st, denoted st < st, ∀t ∈ t, or s < s (3.2) where rt|st and qt|st represent respectively the restrictions of the relations and attributes in rt and qt on the system st and are defined as follows: rt|st = {r|mt : r ∈ rt} and qt|st = {q|mt∪rt : q ∈ qt}. in this definition, the notation of less than of mathematics is employed for the relationship of subsystems, because the relation of subsystems can be seen as a partial ordering on the collection of all systems. similarly, when equ. (3.1) does not hold true, we say that s is not a subsystem of s, denoted st ≮ st, ∀t ∈ t , or s ≮ s. let the set of all subsystems of s be s , then we have s ≡ {s : s < s} = {st : st < st, ∀t ∈ t} (3.3) for any given system s = (m,r,q), where m = {mt : t ∈ t}, r = {rt : t ∈ t}, and q = {qt : t ∈ t}, as defined by equ. (2.1), take a subset set of its object set 28 xiaojun duan: systems defined on dynamic sets and analysis of their characteristics a = {at ⊆ mt : t ∈ t}. then by restricting the relation set r and the attribute set q on a, we obtain the following subsystem of s induced by a: s|a = {st = (at, rt|at , qt|(at,rt|at ) ) : t ∈ t} (3.4) where the restrictions rt|at and qt|at are assumed respectively to be rt|at = {r|at : each element in r|athas the same length, r ∈ rt} (3.4a) and qt|(at,rt|at ) = {q|(at,rt|at ) : qt|(at,rt|at ) is a well defined proposition on st, q ∈ qt} (3.4b) proposition 3.1. the induced subsystem s|a is the maximum subsystem induced by a. proof. let s = {(at, rt,a, qt,a) : t ∈ t} < s be an arbitrary subsystem with the entire a as its object set. according equ. (3.1) we have rt,a ⊆ rt|s, and qt,a = qt|s = qt|(at,rt,a), ∀t ∈ t so, from equ. (3.4a), it follows that s ≤ s|a only when rt,a = rt|s, the equal sign holds true. qed. proposition 3.2. assume that a and b are subsets of m satisfying that b ⊆ a ⊆ m , then s|b ≤ s|a. proof. according equ. (3.4), we have s|b = {st = (bt, rt|bt , qt|(bt,rt|bt ) ) : t ∈ t}.b ⊆ a implies that bt ⊆ at, ∀t ∈ t . so, it follows that rt|bt ⊆ rt|at and consequently rt|bt ⊆ (rt|at)|bt and qt|(bt,rt|bt ) = (qt|(at,rt|at ) )|(bt,rt|bt ) . therefore, s|b ≤ s|a. qed. proposition 3.3. let s be a system. then the collection of all subsystems of s forms a partially ordered set by the subsystem relation “<”. proof. this result is a straightforward consequence of proposition 3.2. qed. assume that s1, s2, and s are systems with the same time span such that s1 < s and s2 < s. let si, i = 1, 2, denote the set of all subsystems of si. then each element in the set s1 − s2 = {s : s < s1, s ≮ s2} is a subsystem of s1 but not a subsystem of s2. similarly, each element in the set s2 − s1 = {s : s < s2, s ≮ s1} is a subsystem of s2 but not a subsystem of s1. let s1∆s2 = {s : s < s1, s ≮ s2} ∪ {s : s < s2, s ≮ s1} be the union of the previous two sets of subsystems of either s1 or s2; and s1 ∩ s2 = {s : s < s1, s < s2} the set of all subsystems of both s1 and s2. proposition 3.4. assume that s1 < s and s2 < s. then s1 ∪ s2 = {s : s < s1} ∪ {s : s < s2} is a subset of s|m1∪m2 . proof. s|m1∪m2 is a maximal element in the partially ordered set (s|(m1∪m2),⊆) , satisfying ∀s ∈ s|m1∪m2 , s < s|m1∪m2 . on the contrary,∀s < s|m1∪m2 , we advances in systems science and applications (2013) vol.13 no.1 29 have s ∈ s|m1∪m2 . so, ∀s ∈ s1 ∪ s2, we have s ∈ s|m1∪m2 . qed. what this result indicates is that the union s1 ∪ s2 of the sets of subsystems of two subsystems s1 and s2 is a subset of the s|m1∪m2 of the subsystems of the induced system on the union m1 ∪m2. proposition 3.5. given two arbitrary systems s1 and s2, there is always a system s12 such that s1 < s12 and s2 < s12. proof. without loss of generality, assume that s1 = {(m1 t , r 1 t , q 1 t ) : t ∈ t 1} and s2 = {(m2 t , r 2 t , q 2 t ) : t ∈ t 2}. to construct the system s12 = {(m12 t , r12 t , q12 t ) : t ∈ t 12}, t 12 = t 1 ∪ t 2, we first assume that the object sets m1 t and m2 t are disjoint, that is, m1 t ∩ m2 t = ∅, for any t ∈ t 1 ∩ t 2. then, the desired system s12 is defined as follows: for t ∈ t 1 ∩ t 2,  m12 t = m1 t ∪m2 t r12 t = r1 t ∪r2 t q12 t = q1 t ∪q2 t (3.5) for t ∈ t 1−t 2, define m12 t = m1 t , r 12 t = r1 t , and q12 t = q1 t , and for t ∈ t 2−t 1, define m12 t = m2 t , r 12 t = r2 t , and q12 t = q2 t , and is denoted as s12 = s1 ⊕ s2. now, if m1 t ∩ m2 t ̸= ∅, for some t ∈ t 1 ∩ t 2, we simply take two systems ∗s1 = {(∗m1 t , ∗r1 t , ∗q1 t ) : t ∈ t 1} and ∗s2 = {(∗m2 t , ∗r2 t , ∗q2 t ) : t ∈ t 2} with m1 t ∩ m2 t = ∅, for any t ∈ t 1 ∩ t 2, such that ∗si is similar to si, i = 1, 2, where similar systems are defined in the same fashion as in [1]. then, we define s12 =∗ s1⊕∗s2. up to a similarity, the system s12 is uniquely defined. therefore, it can be seen as well constructed such that s1 < s12 and s2 < s12, where the time spans t 1 and t 2 are seen as the same as t 12 such that when t ∈ t 1 − t 2 (respectively, t ∈ t 2 −t 1), we treat s2 t (respectively,s1 t ) as a system with empty object set. qed. from this proposition, it follows that when the interactions of some given systems are considered, these systems can always be seen as subsystems of a larger system. on the other hand, this proposition also shows that there is always some kind of interaction between two given systems, which is embodied in the fact that they are subsystems of a certain system. 4 interactions between systems when there is an interaction between two objects of a system s = (m,r,q) (of time span t ), where m = {mt : t ∈ t}, r = {rt : t ∈ t}, and q = {qt : t ∈ t}, it can be described by using a binary relation of the system. we say that objects m1 and m2 ∈ m , which means either mi = (mt)t∈t such that mt ∈ mt, for each t ∈ t , or mi = mt ∈ mt, for a particular t ∈ t, i = 1, 2, interacts with respect to an attribute q ∈ q, it means that the proposition q holds true for a binary relation 30 xiaojun duan: systems defined on dynamic sets and analysis of their characteristics r ∈ r that contains either (m1,m2) or (m2,m1) or both. the idea of interactions between systems is a natural generalization of that between two objects. based on the discussion of the previous section, we will discuss interactions of systems in the framework of subsystems. assume that si < s, i = 1, 2. now, let us look at how these subsystems could interact with each other. definition 4.1. the system s1 is said to have a weak effect on the system s2 with respect to an attribute q ∈ q, provided that for any r2 ∈ r2 there is r1 ∈ r1 such that the proposition q holds true for the ordered pair (r1, r2). when no confusion is caused, we simply say that the system s1 affects the system s2 weakly without mentioning q. if the system s2 also exerts a weak affect of s1, then we say that these systems interact with each other weakly. definition 4.2. the system s1 is said to have a strong effect on the system s2 with respect to an attribute q ∈ q, provided that for any r2 ∈ r2 and any r1 ∈ r1, the proposition q holds true for the ordered pair (r1, r2). when no confusion is caused, we simply say that the system s1 affects the system s2 strongly without mentioning q. if the system s2 also exerts a strong affect of s1 , then we say that these systems interact with each other strongly. from these definitions, it follows that strong interaction requires interactions between every ordered pair of objects, which is a more rigorous requirement than that of weak interactions. also, to maintain the intuition behind the concepts of interactions, in definitions 4.1 and 3.2, we only look at two relations ri ∈ ri, i = 1, 2. in order to capture the general spirit, these individual relations should be replaced by subsets {ri ∈ ri : φi(ri)} , where φi() stands for the proposition that defines the set, for i = 1, 2. by doing so, what are discussed in definitions 4.1 and 4.2 become special cases. definition 4.3. given a subsystem s = (m, r, q) of a system s = (m,r,q), the totality of all objects in m −m, each of which interacts weakly with at least one object in m, is known as the environment of the subsystem s in s, denoted es . 5 systems properties based on dynamic set theory 5.1 basic properties for two given systems s1 = {s1 t = (m1 t , r 1 t , q 1 t ) : t ∈ t 1} and s2 = {s2 t = (m2 t , r 2 t , q 2 t ) : t ∈ t 2} , let us consider definition 5.1. these systems s1 and s2 are equal, provided that m1 t = m2 t , r 1 t = r2 t , q 1 t = q2 t , and t 1 = t 2, (5.1) for each t ∈ t 1 = t 2. the systems s1 and s2 are said to be identical on the time period t ⊆ t 1∩t 2, it means that m1 t = m2 t , r 1 t = r2 t , q 1 t = q2 t , for each t ∈ t. (5.2) advances in systems science and applications (2013) vol.13 no.1 31 definition 5.2. the system s1 is said to be homomorphically embeddable into the system s2, provided that there is a non-decreasing mapping f : t 1 → t 2 such that for any t1 ∈ t 1, if t2 = f(t1) ∈ t 2, then s1 t1 = s2 t2 , or equivalently, m1 t1 = m2 t2 , r 1 t1 = r2 t2 , and q1 t1 = q2 t2 (5.3) the mapping f is referred to as an embedding mapping from s1 into s2. if the system s1 can be homomorphically embeddable into s2 and s2 into s1, then the systems s1 and s2 are said to be homophorhically equivalent. evidently, equal systems are homomorphically equivalent with the identity mapping on the time set as the canonical embedding mapping. proposition 5.1. if the embedding mapping f : t 1 → t 2 from the system s1 into s2 is bijective, then the systems are homomorphically equivalent. proof. it suffers to show that the inverse mapping f−1 : t 2 → t 1 is an embedding mapping from s2 into s1. because f is bijective, it is strictly increasing from t 1 into t 2; and its inverse f−1 is also a strictly increasing mapping from t 2 into t 1 satisfying for any t2 ∈ t 2, if t1 = f−1(t2) ∈ t 1, then m2 t2 = m1 t1 , r 2 t2 = r1 t1 , and q2 t2 = q1 t1 . therefore, f−1 : t 2 → t 1 is an embedding mapping from s2 into s1. qed. definition 5.3. a system s = {st = (mt, rt, qt) : t ∈ t} is said to be cyclic or periodic, provided that there is time tc > 0 such that st+tc = st, for any t ∈ t . evidently, in this case, for any natural number n ∈ n , ntc is also a period of the system s. the minimum period is named the period of s, denoted tc. 5.2 systemic emergence by systemic emergence, it means the properties of the whole system that parts of the system do not have. such a property might suddenly appear at a particular time moment. in terms of the system, the holistic emergence is mainly created and excited by the system’s specific organization of its parts and how these parts interact, supplement, and constrain on each other. it is a kind of effect of relevance, the organizational effect, and the structural effect. to be specific, let p represent such a property. it is defined on the entire system s. because of the system’s dynamic characteristics, the interactions between the system’s objects, relations, attributes, and the environment change constantly. so, the value of p also varies accordingly. define p (st) = { 1, if s has this property 0, otherwise , ∀t ∈ t (5.4) it satisfies the following properties: p (st) = 1 → p (st) = 1, ∀s < s, ∀t ∈ t (5.5) 32 xiaojun duan: systems defined on dynamic sets and analysis of their characteristics that is, as long as a subsystem s has this property, the overall system s also has the property. that is another way to say that the whole is greater than the sum of parts. of course, there are also such properties that some subsystems have, while the overall system does not have. for instance, let p be a property the system s does not have. define p−1(1) = {s < s : s ̸= s} be the collection of all proper subsystems of s, where the superscript (−1) stands for the inverse operation. then, for any s ∈ p−1(1), s < s and p (s) = 1. however, p (s) = 0 does not satisfy equ. (5.5). when a property p is said to be system s’s holistic emergence, provided that for any subsystem s < s, p (st) = 0, ∀t ∈ t , and p (st0) = 1 for at least one t0 ∈ t . in the traditional static description of systems, there is another definition of systemic emergence. in particular, consider a monotonically increasing sequence of subsystems ∅ ̸= s1 < s2 < ... < sk < ... < s, there is a k0 such that p (sk) = 0, ∀k < k0, p (sk0) = 1 (5.6) in this case, the system s is said to have the emergent property p , while its parts sk, k < k0, do not share this property until they reach the whole sk0 of certain scale. from a detailed analysis, it follows readily that what is just presented is a special case of the systemic emergence of dynamic systems, where we can surely treat the collection of all the superscript k as a subset of the time index t so that the sequence {sk : k = 1, 2, 3, ...} of subsystems a subsystem of a dynamic system by letting st = st, for t = 1, 2, 3, ... from equ. (5.6), it follows that p (st) = 0, for t = 1, 2, 3, ... < k0, and p (sk0) = 1 that is, the property p emerges at the time moment k0. 5.3 stability of systems by stability, it means the maintenance or continuity of a certain measure of the dynamic system on a certain time scale. that is, under small disturbances, the measure does not undergo noticeable changes. let us look at the trajectory system of a single point, where the focus is how the point moves under the influence of an external force. if a disturbance is given to a portion of the trajectory that is not at a threshold point, then the disturbed trajectory will not differ from the original trajectory much. however, if the same disturbance is given to the trajectory at an extremely unstable extreme point, then a minor change in the disturbed value could cause major deviations in the following portion of the trajectory. thus, only the trajectory of motion at stable critical extrema is stable, while the trajectory systems with instable critical points, such as saddle point, are instable. advances in systems science and applications (2013) vol.13 no.1 33 in general, assume that an attribute q ∈ q of the system s satisfies that q is a real-valued function defined for each momentary system st, for any t ∈ t . let t0 ∈ t . if for any ε > 0, there is a δt0,ε > 0 such that |q(st0)− q(st)| < ε, ∀t ∈ (t0 − δt0,ε, t0 + δt0,ε) (5.7) then the attribute q of the system s is regionally stable over time at t0 ∈ t . if for any ε > 0, there is δ = δ(ε) = δε > 0 such that |q(st1)− q(st2)| < ε, ∀t1, t2 ∈ t such that |t1 − t2| < δε (5.8) then the attribute q is said to be holistically stable over time or uniformly stable over time. when no confusion can be caused, the previous concepts of stability of the attribute q are respectively referred to as that the system s is regionally stable at t0 ∈ t , or uniformly stable over t . in addition, the structural stability of systems can also be defined. in particular, let d ∈ q be such that d : st → r+ is a positively real-valued function defined for each subsystem of the momentary system st, t ∈ t , satisfying (1)d(∅) = 0,where ∅ stands for the the subsystems with the empty set as their object set; (2)∀s1, s2 < s, if s1 < s2, then d(s1) ≤ d(s2); (3)∀s1, s2 < s, if s1△s2 = ∅, then d(s1 ∪ s2) = d(s1) + d(s2). it can be readily seen that such an attribute d can be employed to measure the difference between subsystems of s; and for any chosen s1 < s, and for any s2 ∈ s, the greater the attribute value d(s1△s2) is, the more different the systems s1 and s2 are. by making use of such a d ∈ q, which satisfies the previous properties, for any chosen s ∈ s and any real number δ > 0, we can define the δ -neighborhood nbrdd(s, δ) of s as follows: nbrdd(s, δ) = {s′ ∈ s : d(s′△s) < δ} (5.9) the attribute q ∈ q of the system s is said to be structurally stable in the neighborhood of a subsystem s < s with respect to attribute d ∈ q, provided that for any ε > 0, there is δ = δε > 0 such that |q(s)− q(s′)| < ε, ∀s′ ∈ nbrdd(s, δε) (5.10) similarly, the concepts of regionally structural stability and uniformly structural stability of a system s at all of its subsystems can be defined. if a system s is both uniformly stable over time and uniformly structural stable, then the system is referred to as a uniformly stable system. 34 xiaojun duan: systems defined on dynamic sets and analysis of their characteristics 5.4 evolution of systems when systems’ evolution is concerned with, the focus is how the system develops over time. speaking rigorously, each system can be described by using the dynamic format in equ.(2.1). as for those systems, which do not seem to change with time, when seen from the angle of dynamics, they are either evolving with time extremely slowly or considered within a very short period of time. that is how they project a mistakenly incorrect sense of being static. the evolution of systems takes two main forms: one is the transition of the system from one structure or form to another structure or form, and the other is that a system appears from its earlier state of non-existence and grows from a immature state to a more mature state. both of these two forms of evolution can be described and illustrated by using the language of the dynamic systems in equ.(2.1). assume that the system of our concern is indeed described in the form of equ.(2.1). then for the first form of evolution, the relation set rt and the attribute set qt change with time, while for the second form of evolution, the elements in the object set mt develop over time together of course with changes of the relation and attributes sets rt and qt. they represent two specific cases of the evolution of dynamic systems. 5.5 boundary of systems by boundary, it stands for the separation between a system and its environment. the boundary is a part of the system and interacts with the environment more closely than the interior of the system. by reviewing the definition of environments in definition 4.3, we can define the boundary of a system as follows. assume that s < s is a subsystem of the system s over the time span t . let us denote the subsystem s and the system s respectively as st = (ms t , r s t , q s t ) and s = (m,r,q), for t ∈ t . for the sake of convenience of communication, we write s = (ms, rs, qs) for a fixed time moment. then, the (external) environment es of s within the system s is defined to be: es = {(m −ms, r|me , q|me s ,re ) : r ∈ r(r|ms /∈ rs), q ∈ q(q|ms /∈ qs)} = (me , re , qe) where me = m − m, re = r|me , and qe = q|me ,re , for r ∈ r(r|me /∈ rs) and q ∈ q(q|me /∈ qs). assume that there is an attribute µ ∈ qt of the system s such that it assigns each relation to a positive real number µ(r) : rt → r+. intuitively, this attribute µ is an index that measures the intensity of each relation of the objects of s. now, let us fix a threshold value µ0 for the relational intensity, then by combining with the concept of weak interactions between systems (definition 4.1), we can obtain the set of all relations in s of intensity at least µ0 that relate the subsystem s advances in systems science and applications (2013) vol.13 no.1 35 and its environment es as follows: rb = {r ∈ r : r|ms ∈ rs, supp(r) ∩ms ̸= ∅ ̸= supp(r) ∩me , µ(r) ≥ µ0} (5.11) where supp(r) stands for the support of the relation r, which is the set of all objects that appear in the relation r. the set of all objects of s that interact with the environment es of intensity of at least µ0 is given by the following: mb = ∪ r∈rb supp(r)−me (5.12) then the system ∂s = (mb, rb|mb , qb|mb,rb|mb ) (5.13) satisfies that ∂s < s, and is referred to as the boundary (system) of s within the system s. the intensity of its interaction with the environment is no less than µ0, while any other subsystem of s interacts with the environment es with strictly less intensity than µ0. if in the evolution of the system, ∂s has good stability, then we say that the boundary of the system is clear. otherwise, we say that the boundary of the system is fuzzy. symbolically, let q ∈ q be an attribute and t0 ∈ t chosen. if for any ε > 0, there is δ = δt0,ε > 0 such that |q(∂st0)− q(∂st)| < ε, ∀t ∈ (t0 − δt0,ε, t0 + δt0,ε) (5.14) then we say that the boundary of the system s at time t0 ∈ t is definite. otherwise, the boundary is said to be fuzzy at t0 ∈ t . it is not hard to see that the definiteness of a system’s boundary is defined by using the stability of the boundary system ∂s. therefore, the stability of the boundary system at one time moment corresponds to the definiteness of the system’s boundary. similarly, we can study the concepts of regional definiteness of boundaries over time and structural definiteness of boundaries. 6 some final words by using the traditional set theory, the static structures and relations of systems are described. then, the dynamic changes of systems are studied by using the concept of time systems. in this paper, on the basis of the concept of systems, we describe the dynamic structure and change of a general system from a more delicate angle by determining the objects, relations, and attributes. by considering the affects of environment, we establish the basic concepts of systems using the idea of dynamic sets that are related to the variable of time. then on top of these developed concepts, we revisit the description and analysis of some of the most fundamental properties of systems, while making the necessary comparisons 36 xiaojun duan: systems defined on dynamic sets and analysis of their characteristics with the studies of systems developed on the classical set theory. as for the analysis of systems properties under the new setting, additional rigorous deduction of various results of systems is badly needed in order to enrich the relevant systems theory. acknowledgements this work is supported by the national science foundation of china (60974124), the program for new century excellent in university, and the project-sponsored by srf for rocs, sem in china. references [1] lin. y. (1999), general systems theory: a mathematical approach, new york: kluwer academic and plenum publishers. [2] mickens. r. e. (1990), mathematics and science, singapore: world scientific. [3] quastler. h. (1965), general principles of systems analysis, in: t. h. waterman and h. [4] j. morrowits (eds.), theoretical and mathematical biology, new york: blaisdell publishing. [5] zadeh. l. (1962), “from circuit theory to systems theory”, proc. ire, vol.50, pp.856-865. corresponding author xiaojun duan can be contacted at: xjduan@nudt.edu.cn. adv syst sci appl 2017; 4:46–60 published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/510 threshold analysis of a stochastic epidemic model with delay and temporary immunity rui xue, fengying wei ∗ college of mathematics and computer science, fuzhou university, fuzhou 350116, p.r. china abstract: a stochastic susceptible-infected-recovered model is formulated and investigated when the temporary immunity is fixed for the population in this paper. the existence and uniqueness of the global positive solution has been checked with probability one for any initial value. and the sufficient conditions for the extinction and the persistence of the stochastic epidemic model with temporary immunity are derived by constructing lyapunov functions and the generalized ito’s formula, where the threshold of the persistence does not depend on the temporary immunity, while the densities of the infected and recovered are obviously dependent on the temporary immunity when given a perturbation. illustrative examples and simulations show that the perturbations make the properties of the stochastic epidemic model different from the deterministic one. keywords: stochastic epidemic model; delay; persistence; extinction; threshold 1. introduction the susceptible-infected-removed epidemic model is one of the most important models in epidemiological patterns and disease control. epidemic models with delay always make the description of the epidemiological patterns more realistic, more interesting and more complicated. some authors investigated the effects of disease latency or immunity, in which the models are described by systems of ordinary differential equations with delay, for instance [1–10], and systems without delay [11–15]. among these recent results we would like to mention the work by wen and yang [10], in which they considered a type of delayed susceptible-infected-recovered epidemic model with temporary immunity: ṡ(t) = λ− µ1s(t)− βs(t)i(t) + γi(t− τ)e−µ3τ , i̇(t) = βs(t)i(t)− (µ2 + γ)i(t), ṙ(t) = γi(t)− γi(t− τ)e−µ3τ − µ3r(t), (1.1) where s(t) is the number of the susceptible individuals to the disease, i(t) represents the number of the individuals who are infected and r(t) denotes the recovered individuals who have been removed from the possibility of infection, λ denotes a constant input of new members into the population, µ1, µ2, µ3 represent the death rates of the susceptible, the infected, the recovered, and they assume that µ1 ≤ min{µ2, µ3} from the biological point of view, β denotes the transmission coefficient between compartments s and i , γ denotes the recovery rate of the infected individuals, i.e., the rate at which the individuals move from the infected compartment to the recovered compartment, τ > 0 is the length of immunity ∗corresponding author: weifengying@fzu.edu.cn http://ijassa.ipu.ru/ojs/ijassa/article/view/510 threshold analysis of a stochastic epidemic model with delay and temporary 47 period. wen and yang [10] assume that λ, µi, β, γ are all positive constants in model (1.1), and they obtain the expression of the basic reproduction number in terms of parameters, that is, r0 = βλ µ1(µ2+γ) , which measures how fast the diseases spread in the deterministic model provided that the epidemics take place. some common phenomena in the real world, such as the weather fluctuations, temperature changes and perturbations caused by human beings, can not been ignored when it comes to the dynamics of the infectious diseases. from both biological and mathematical perspectives, the stochastic models have more reasonable patterns and have been investigated by different approaches presented in the recent literatures [16–20]. motivated by model (1.1) and the approaches mentioned in [16–20], we try to investigate several properties of model (1.2) when transmission coefficient β in (1.1) is perturbed by the white noise β + σξ(t): ds(t) = ( λ− µ1s(t)− βs(t)i(t) + γi(t− τ)e−µ3τ ) dt− σs(t)i(t)db(t), di(t) = ( βs(t)i(t)− (µ2 + γ)i(t) ) dt+ σs(t)i(t)db(t), dr(t) = ( γi(t)− γi(t− τ)e−µ3τ − µ3r(t) ) dt, (1.2) where ξ(t) = db(t) dt , andb(t) is the independent brownian motion, and σ is the intensity of the white noise when model (1.2) works on a complete probability space (ω,f , {ft}t≥0,p) with a filtration {ft}t≥0 satisfying the usual conditions (i.e., it is right continuous and increasing function while f0 contains all p-null sets). what we concern about in model (1.2) in this paper are listed below, of course, for any positive initial value herewith: • existence and uniqueness of a global nonnegative solution with probability one; • the sufficient conditions that guarantees the extinction of the diseases; • figuring out the threshold presented in terms of model parameters; • demonstrating illustrative examples and their realizations to support the validity of the main results. 2. existence and uniqueness of the positive solution for model (1.2), the sum of three equations gives d(s(t) + i(t) +r(t)) dt ≤ λ− µ1(s(t) + i(t) +r(t)), (2.3) then one can obtain that s(t) + i(t) +r(t) ≤  λ µ1 , s(0) + i(0) +r(0) ≤ λ µ1 , s(0) + i(0) +r(0), s(0) + i(0) +r(0) > λ µ1 . (2.4) we denote n = max { λ µ1 , s(0) + i(0) +r(0) } , (2.5) therefore, s(t) ≤ n, i(t) ≤ n,r(t) ≤ n . we notice that the first two equations in model (1.2) do not depend on the third equation, we omit them without loss of generality. thus, we copyright c© 2017 assa. adv syst sci appl (2017) 48 r. xue, f.y. wei only discuss the simplified model of having two equations:{ ds(t) = ( λ− µ1s(t)− βs(t)i(t) + γi(t− τ)e−µ3τ ) dt− σs(t)i(t)db(t), di(t) = ( βs(t)i(t)− (µ2 + γ)i(t) ) dt+ σs(t)i(t)db(t). (2.6) in order to investigate other properties, we firstly study the fundamental property of the solution of model (2.6). next we will present the existence and uniqueness of the solution of model (1.2). theorem 2.1: for any initial value s(0) > 0 and i(ζ) ≥ 0 for all ζ ∈ [−τ, 0) with i(0) > 0, model (2.6) admits a unique positive solution (s(t), i(t)) on t > 0 and the solution will remain in r2 + with probability one, that is to say, (s(t), i(t)) ∈ r2 + for all t > 0 almost surely. proof according to the approach mentioned in [13–15, 21]. the proof will go as follows. since the coefficients of model (2.6) satisfy the local lipschitz conditions, then for any initial value s(0) > 0 and i(ζ) ≥ 0 for all ζ ∈ [−τ, 0) with i(0) > 0, there is a unique local solution (s(t), i(t)) on the half-closed interval [−τ, τe), where τe represents the explosion time of the solution. to verify the solution is global, what we prepare to check that explosion time τe =∞ holds almost surely. to this end, let k0 ≥ 1 be sufficiently large such that each component of the initial value (s(0), i(ζ)) always lies within the interval [ 1 k0 , k0]. for each integer k ≥ k0, let us define the following stopping time τk = inf { t ∈ [0, τe) ∣∣∣∣s(t) /∈ ( 1 k , k ) or i(t) /∈ ( 1 k , k )} . (2.7) throughout this paper, we set inf ∅ =∞ (as usual ∅ represents the empty set). obviously, τk is an increasing function as k →∞. let τ∞ be the limit of τk as k →∞. it is obvious that τ∞ ≤ τe is valid almost surely. in order to complete the proof, we only need to show τ∞ =∞ a.s.. we start the proof by contradiction, if this assertion is false, then there exists a pair of constants t > 0 and ε ∈ (0, 1) such that p{τ∞ ≤ t} > ε. (2.8) thereby, there is an integer k1 ≥ k0 such that p{τk ≤ t} ≥ ε for all k ≥ k1. (2.9) let us define a c2-function v : r2 + → r+ as follows: v (s(t), i(t)) = s(t)− a− a ln s(t) a + i(t)− 1− ln i(t) + γe−µ3τ ∫ t t−τ i(r)dr, (2.10) where a is a positive constant to be determined later. the nonnegativity of µ− 1− log µ is obvious for all µ > 0. let k ≥ k0 and t > 0 be arbitrary positive constants. the application of itô’s formula gives dv (s(t), i(t)) = ( 1− a s(t) ) ds(t) + a 2s2(t) (ds(t))2 + ( 1− 1 i(t) ) di(t) + 1 2i2(t) (di(t))2 + γi(t)e−µ3τdt− γi(t− τ)e−µ3τdt := lv (s(t), i(t))dt+ σ(ai(t)− s(t))db(t), (2.11) copyright c© 2017 assa. adv syst sci appl (2017) threshold analysis of a stochastic epidemic model with delay and temporary 49 where lv (s(t), i(t)) is defined from r2 + to r and computed as lv (s(t), i(t)) = ( 1− a s(t) )( λ− µ1s(t)− βs(t)i(t) + γi(t− τ)e−µ3τ ) + aσ2 2 i2(t) + ( 1− 1 i(t) )( βs(t)i(t)− (µ2 + γ)i(t) ) + σ2 2 s2(t) + γi(t)e−µ3τ − γi(t− τ)e−µ3τ = λ− µ1s(t)− (µ2 + γ)i(t) + γi(t− τ)e−µ3τ − aλ s(t) + µ1a+ aβi(t)− aγe−µ3τ i(t− τ) s(t) − βs(t) + µ2 + γ +γi(t)e−µ3τ − γi(t− τ)e−µ3τ + σ2 2 (s2(t) + ai2(t)) ≤ λ + µ1a+ µ2 + γ + (aβ + γe−µ3τ − (µ2 + γ))i(t) + σ2 2 (1 + a)n2. (2.12) choosing a positive constant a = µ2 + γ(1− e−µ3τ ) β (2.13) such that aβ + γe−µ3τ = µ2 + γ, then we have lv (s(t), i(t)) ≤ λ + µ1a+ µ2 + γ + σ2 2 (1 + a)n2 := k, (2.14) where k > 0 is a constant. the remainder of the proof follows the similar approach given in [22]. remark 2.1: theorem 2.1 implies that the stochastic model (1.2) admits a unique solution (s(t), i(t), r(t)) for any initial value (s(0), i(0), r(0)) ∈ r3 +, i(ζ) ≥ 0 for all ζ ∈ [−τ, 0). each component of the solution (s(t), i(t), r(t)) is positive for all t ≥ 0 almost surely. according to the statement of theorem 2.1, if the sum of each component with the initial value is almost surely bounded, say s(0) + i(0) +r(0) ≤ λ µ1 , so the region γ∗ = { (s(t), i(t), r(t)) ∈ r3 + : s(t) + i(t) +r(t) ≤ λ µ1 , t ≥ 0 } (2.15) is the positively invariant set of model (1.2). 3. the sufficient conditions of the extinction of the diseases let r̃0 = βλ µ1(µ2 + γ) − σ2λ2 2µ2 1(µ2 + γ) , (3.16) which can be presented in terms of the basic reproduction number of the deterministic model (1.1): r̃0 = r0 − σ2λ2 2µ2 1(µ2 + γ) . (3.17) copyright c© 2017 assa. adv syst sci appl (2017) 50 r. xue, f.y. wei for the stochastic model (1.2), we will investigate the sufficient conditions of the extinction of the disease. in other words, the infected individuals will eventually recover from the attack of the disease, and the capacity of the total population will keep kind of constant in the long run. theorem 3.1: let (s(t), i(t), r(t)) be the solution of model (1.2) with initial value (s(0), i(0), r(0)) ∈ γ∗. if the intensity of the white noise satisfies σ2 > β2 2(µ2 + γ) (3.18) or r̃0 < 1, σ2 ≤ βµ1 λ , (3.19) then the density of the infected declines exponentially lim sup t→∞ ln i(t) t ≤ −(µ2 + γ) + β2 2σ2 < 0 a.s. (3.20) or lim sup t→∞ ln i(t) t ≤ (µ2 + γ)(r̃0 − 1) < 0 a.s. (3.21) respectively; and lim sup t→∞ s(t) = λ µ1 a.s.. (3.22) proof adding up those three equations of model (1.2) gives d ( s(t) + i(t) + γe−µ3τ ∫ t t−τ i(r)dr ) = [λ− µ1s(t)− (µ2 + γ(1− e−µ3τ ))i(t)]dt. (3.23) we integrate the above from 0 to t and divided by t, which implies λ− µ1〈s(t)〉 − (µ2 + γ(1− e−µ3τ ))〈i(t)〉 = 1 t ( s(t) + i(t) + γe−µ3τ ∫ t t−τ i(r)dr − s(0)− i(0)− γe−µ3τ ∫ 0 −τ i(r)dr ) , (3.24) and then derives that 〈s(t)〉 = λ µ1 − µ2 + γ(1− e−µ3τ ) µ1 〈i(t)〉+ ϕ(t), (3.25) where ϕ(t) = 1 µ1t ( s(t) + i(t) + γe−µ3τ ∫ t t−τ i(r)dr − s(0)− i(0)− γe−µ3τ ∫ 0 −τ i(r)dr ) . (3.26) the strong law of large numbers and the fact s(0) + i(0) ≤ λ µ1 show that lim t→∞ ϕ(t) = 0 a.s.. (3.27) copyright c© 2017 assa. adv syst sci appl (2017) threshold analysis of a stochastic epidemic model with delay and temporary 51 for the second equation of model (1.2), the generalized itô’s formula leads to d ln i(t) = ( βs(t)− (µ2 + γ)− σ2s2(t) 2 ) dt+ σs(t)db(t). (3.28) integrating (3.28) from 0 to t and divided by t gives ln i(t) t = β〈s(t)〉 − (µ2 + γ)− σ2 2 〈s2(t)〉+ σ t ∫ t 0 s(r)db(r) + ln i(0) t . (3.29) we denote m(t) := σ ∫ t 0 s(r)db(r), (3.30) where m(t) is a local continuous martingale with m(0) = 0. moreover, lim sup t→∞ 〈m(t),m(t)〉t t ≤ σ2λ2 µ2 1 <∞ a.s.. (3.31) the strong law of large numbers for martingales presented in [23] thus implies that lim sup t→∞ m(t) t = 0 a.s.. (3.32) if σ2 > β2 2(µ2+γ) , rewriting (3.29) gives ln i(t) t = −σ 2 2t ∫ t 0 ( s(r)− β σ2 )2 dr − (µ2 + γ) + β2 2σ2 + σ t ∫ t 0 s(r)db(r) + ln i(0) t ≤ −(µ2 + γ) + β2 2σ2 + σ t ∫ t 0 s(r)db(r) + ln i(0) t . (3.33) taking superior limit on both sides of (3.33), then we get that lim sup t→∞ ln i(t) t ≤ −(µ2 + γ) + β2 2σ2 < 0 a.s.. (3.34) copyright c© 2017 assa. adv syst sci appl (2017) 52 r. xue, f.y. wei if r̃0 < 1 and σ2 ≤ βµ1 λ hold, the schwarz inequality and substituting (3.25) into (3.29) yield that ln i(t) t ≤ β〈s(t)〉 − (µ2 + γ)− σ2 2 〈s(t)〉2 + σ t ∫ t 0 s(r)db(r) + ln i(0) t = βλ µ1 − (µ2 + γ)− β(µ2 + γ(1− e−µ3τ )) µ1 〈i(t)〉+ βϕ(t) + σ t ∫ t 0 s(r)db(r)− σ2 2 [ λ µ1 − µ2 + γ(1− e−µ3τ ) µ1 〈i(t)〉+ ϕ(t) ]2 = βλ µ1 − (µ2 + γ)− β(µ2 + γ(1− e−µ3τ )) µ1 〈i(t)〉+ βϕ(t) + σ t ∫ t 0 s(r)db(r)− σ2 2 [ λ2 µ2 1 + ( µ2 + γ(1− e−µ3τ ) µ1 〈i(t)〉 )2 +ϕ2(t)− 2λ(µ2 + γ(1− e−µ3τ )) µ2 1 〈i(t)〉 + 2λ− 2(µ2 + γ(1− e−µ3τ ))〈i(t)〉 µ1 ϕ(t) ]2 , (3.35) which gives that ln i(t) t ≤ βλ µ1 − σ2λ2 2µ2 1 − µ2 + γ(1− e−µ3τ ) µ1 ( β − σ2λ µ1 ) 〈i(t)〉 −(µ2 + γ) + ψ(t), (3.36) where ψ(t) = βϕ(t) + σ t ∫ t 0 s(r)db(r) −σ 2 2 [ ϕ2(t) + 2λ− 2(µ2 + γ(1− e−µ3τ ))〈i(t)〉 µ1 ϕ(t) ] . (3.37) it is easy to check that lim t→∞ ψ(t) = 0 a.s.. (3.38) we take superior limit on both sides of (3.35) and derive that lim sup t→∞ ln i(t) t ≤ (µ2 + γ)(r̃0 − 1) < 0 a.s.. (3.39) further, the density of the infected individuals declines to zero, and the density of the susceptible individuals keeps constant in a long time run, mathematically speaking, these two expressions lim t→∞ i(t) = 0, lim t→∞ s(t) = λ µ1 (3.40) hold almost surely when condition (3.18) or condition (3.19) is satisfied. remark 3.1: wen and yang [10] had shown that the disease-free equilibrium e0( λ µ1 , 0, 0) of the deterministic model (1.1) was globally asymptotically stable if the basic reproduction number r0 < 1 (i.e., the disease disappeared under the condition r0 < 1). while for the stochastic copyright c© 2017 assa. adv syst sci appl (2017) threshold analysis of a stochastic epidemic model with delay and temporary 53 model (1.2), we obtain that the disease disappears when conditions r̃0 < 1 and σ2 ≤ βµ1 λ are being satisfied in this chapter. this indicates that the condition of extinction to the stochastic model (1.2) is weaker than that of the corresponding deterministic model (1.1). 4. the threshold for the persistence of the diseases theorem 4.1: let (s(t), i(t), r(t)) be any solution of model (1.2), (s(0), i(0), r(0)) ∈ γ∗ be any initial value with i(ζ) ≥ 0 for all ζ ∈ [−τ, 0). we assume that r̃0 > 1, and also assume that the intensity of the white noise satisfies σ2 ≤ βµ1 λ , (4.41) then the densities of the infected and the recovered individuals have the following properties: ĩ∗ ≤ lim inf t→∞ 〈i(t)〉 ≤ lim sup t→∞ 〈i(t)〉 ≤ ĩ∗ a.s. (4.42) and r̃∗ ≤ lim inf t→∞ 〈r(t)〉 ≤ lim sup t→∞ 〈r(t)〉 ≤ r̃∗ a.s., (4.43) where ĩ∗ = µ1(µ2 + γ) β(µ2 + γ(1− e−µ3τ )) (r̃0 − 1), r̃∗ = γ(1− e−µ3τ ) µ3 ĩ∗, (4.44) and ĩ∗ = µ2 1(µ2 + γ) (βµ1 − σ2λ)(µ2 + γ(1− e−µ3τ )) (r̃0 − 1), r̃∗ = γ(1− e−µ3τ ) µ3 ĩ∗. (4.45) proof from (3.28), the second equation of model (1.2) implies that d ln i(t) ≥ ( βs(t)− µ2 − γ − σ2λ2 2µ2 ) dt+ σs(t)db(t), (4.46) integrating (4.46) on both sides and substituting (3.25) into the integration, which derive that ln i(t)− ln i(0) t ≥ β〈s(t)〉 − µ2 − γ − σ2λ2 2µ2 + σ t ∫ t 0 s(r)db(r) = βλ µ1 − ( µ2 + γ + σ2λ2 2µ2 ) − β(µ2 + γ(1− e−µ3τ )) µ1 〈i(t)〉 +βϕ(t) + σ t ∫ t 0 s(r)db(r). (4.47) inequality (4.47) can be rewritten as 〈i(t)〉 ≥ µ1 β(µ2 + γ(1− e−µ3τ )) ( βλ µ1 − ( µ2 + γ + σ2λ2 2µ2 ) − ln i(t)− ln i(0) t + βϕ(t) + σ t ∫ t 0 s(r)db(r) ) . (4.48) copyright c© 2017 assa. adv syst sci appl (2017) 54 r. xue, f.y. wei according to (2.15), we have −∞ < ln i(t) < ln λ µ1 , lim t→∞ σ t ∫ t 0 s(r)db(r) = 0, (4.49) and lim t→∞ ϕ(t) = 0. (4.50) taking the inferior limit on both sides of (4.48), we have lim inf t→∞ 〈i(t)〉 ≥ µ1(µ2 + γ) β(µ2 + γ(1− e−µ3τ )) (r̃0 − 1) := ĩ∗. (4.51) on the other hand, integrating the first equation of (4.46) from 0 to t on both sides yields ln i(t)− ln i(0) t = β〈s(t)〉 − µ2 − γ − σ2 2 〈s2(t)〉+ σ t ∫ t 0 s(r)db(r). (4.52) we rewrite (3.35), then we get that 〈i(t)〉 ≤ µ2 1 (µ2 + γ(1− e−µ3τ ))(βµ1 − σ2λ) × ( βλ µ1 − ( µ2 + γ + σ2λ2 2µ2 1 ) − ln i(t)− ln(0) t + ψ(t) ) , (4.53) taking the superior limit on both sides of above equation, then one can obtain that lim sup t→∞ 〈i(t)〉 ≤ µ2 1 (µ2 + γ(1− e−µ3τ ))(βµ1 − σ2λ) (r̃0 − 1) := ĩ∗. (4.54) the last equation of model (1.2) yields 〈r(t)〉 = γ µ3 〈i(t)〉 − γe−µ3τ µ3 〈i(t− τ)〉 − r(t)−r(0) µ3t , (4.55) then lim inf t→∞ 〈r(t)〉 = γ(1− e−µ3τ ) µ3 lim inf t→∞ 〈i(t)〉 ≥ γ(1− e−µ3τ ) µ3 ĩ∗ = r̃∗, (4.56) lim sup t→∞ 〈r(t)〉 = γ(1− e−µ3τ ) µ3 lim sup t→∞ 〈i(t)〉 ≤ γ(1− e−µ3τ ) µ3 ĩ∗ = r̃∗. (4.57) remark 4.1: both theorem 2 and theorem 3 have the common condition σ2 ≤ βµ1 λ therewith. while the opposite properties of these two theorems depend on the value of r̃0, that is, r̃0 < 1 indicates the extinction of the disease and r̃0 > 1 means the persistence of the disease. the expression r̃0 plays the role of threshold of model (1.2). copyright c© 2017 assa. adv syst sci appl (2017) threshold analysis of a stochastic epidemic model with delay and temporary 55 5. illustrative examples and their realizations example 1 let the parameters of model (1.2) be λ = 0.2, µ1 = µ2 = µ3 = 0.2, β = 0.8, γ = 0.5, σ = 0.6, and the initial value be (s(0), i(0), r(0)) = (0.1, 0.6, 0.3). we can verify that r̃0 = r0 − σ2λ2 2µ2 1(µ2 + γ) = 0.8857 < 1, σ2 = 0.36 ≤ βµ1 λ = 0.8. (5.58) condition (3.19) of theorem 1.2 is satisfied, the solution of model (1.2) has the properties as follows: lim sup t→∞ ln i(t) t ≤ (µ2 + γ)(r̃0 − 1) = −0.08 < 0 a.s. (5.59) and lim sup t→∞ s(t) = λ µ1 = 1 a.s. (5.60) figure 5.1 indicates that the infected vanishes exponentially, and the susceptible individuals reach their maximum when given a long time run. obviously, the basic reproduction number of the deterministic model (1.1) can be computed as r0 = 1.1429 > 1 in this case. and the endemic equilibrium e∗ of model (1.1) is globally asymptotically stable according to theorem 5.2 in [10]. we conclude that in the case of medium perturbation, say σ = 0.6, the density of the susceptible approaches one almost surely, and is much higher than that of the deterministic model. compared with the deterministic model, the density of the infected declines fast to zero with the exponential rate −0.08 at early time scale 1.5× 104 days. and the density of the recovered is somehow affected by the infected and ends up at zero at 2.5× 104 days instead of persistence for the deterministic model. example 2 we keep the initial value and other parameters same as shown in example 1 except for σ = 0.9. here σ2 ≥ β2 2(µ2 + γ) = 0.4571, (5.61) and condition (3.18) of theorem 1.2 is being satisfied, then the solution of model (1.2) admits the following property: lim sup t→∞ ln i(t) t ≤ −(µ2 + γ) + β2 2σ2 = −0.3349 < 0 a.s. (5.62) the corresponding simulations would be shown in figure 5.2 to support the main results of theorem 1.2 we got in the previous section. under large perturbation, say σ = 0.9 in this case, we find that the density of the susceptible behaves the similar dynamics. while the curve of the infected shows more sharper than that in example 1, and still decays exponentially with a larger rate−0.3349 at early time scale 1× 104 days. and the density of the recovered is close to zero at 2× 104 days compared with the deterministic model. example 3 let the intensity of the white noise be σ = 0.1, and the initial value and other parameters be the same as shown in example 1. here r̃0 = r0 − σ2λ2 2µ2 1(µ2 + γ) = 1.1357 > 1, σ2 = 0.01 ≤ βµ1 λ = 0.8. (5.63) copyright c© 2017 assa. adv syst sci appl (2017) 56 r. xue, f.y. wei 0 0.5 1 1.5 2 2.5 3 3.5 4 t,days ×104 0 0.2 0.4 0.6 0.8 1 1.2 the stochastic system s(t) i(t) r(t) 0 0.5 1 1.5 2 2.5 3 3.5 4 t,days ×104 0 0.2 0.4 0.6 0.8 1 1.2 the deterministic system s(t) i(t) r(t) fig. 5.1. realizations for model (1.2) with r̃0 < 1 and model (1.1) with r0 > 1 respectively. by theorem 4.1, the following property of model (1.2) holds: 0.0817 ≤ lim inf t→∞ 〈i(t)〉 ≤ lim sup t→∞ 〈i(t)〉 ≤ 0.0827 a.s. (5.64) if we keep the initial value and other parameters same as shown in example 1 except for µ2 = 0.3, γ = 0.4. we easily check that examples 1 and 2 still keep the same conclusions. while the persistence level of model (1.2) is lower than that in (5.64) by theorem 4.1. that is, 0.0638 ≤ lim inf t→∞ 〈i(t)〉 ≤ lim sup t→∞ 〈i(t)〉 ≤ 0.0646 a.s. (5.65) figure 5.3 reveals that the prevalence of the disease takes place under small perturbation of the white noise. the properties of the solutions for the stochastic model (1.2) and the deterministic model (1.1) demonstrate the similar behaviors when set a small perturbation. we would like to conclude that the density levels for the susceptible, the infected and the recovered are all alike when given a small perturbation environment. especially, the higher the recovery rate for the infected individuals, the more the infected, such as, the density copyright c© 2017 assa. adv syst sci appl (2017) threshold analysis of a stochastic epidemic model with delay and temporary 57 0 0.5 1 1.5 2 2.5 3 3.5 4 t,days ×104 0 0.2 0.4 0.6 0.8 1 1.2 the stochastic system s(t) i(t) r(t) 0 0.5 1 1.5 2 2.5 3 3.5 4 t,days ×104 0 0.2 0.4 0.6 0.8 1 1.2 the deterministic system s(t) i(t) r(t) fig. 5.2. realizations for model (1.2) with σ = 0.9 and model (1.1) with r0 > 1 respectively. of the infected varies from [0.0817, 0.0827] when γ = 0.5 (the green line in figure 5.4) to [0.0638, 0.0646] when γ = 0.4 (the blue line in figure 5.4). 6. conclusions in this paper, we work on the susceptible-infected-recovered model, where the individuals stayed in the recovered compartment finally lost temporary immunity returned to the susceptible compartment. the research results of this paper demonstrate that the existence and uniqueness of the global positive solution of model (1.2) has nothing to do with the temporary immunity due to the construction of lyapunov function (2.10) therewith. while no matter how big the temporary immunity period is, the sufficient condition of the extinction of the diseases merely depends on the parameters of model (1.2), say condition (3.18) or (3.19) is λ, µ1, µ2, β, γ, σdependent, and τ -dependent instead in theorem 2. copyright c© 2017 assa. adv syst sci appl (2017) 58 r. xue, f.y. wei 0 0.5 1 1.5 2 2.5 3 3.5 4 t,days ×104 0 0.2 0.4 0.6 0.8 1 1.2 the stochastic system s(t) i(t) r(t) 0 0.5 1 1.5 2 2.5 3 3.5 4 t,days ×104 0 0.2 0.4 0.6 0.8 1 1.2 the deterministic system s(t) i(t) r(t) fig. 5.3. realizations for model (1.2) with σ = 0.1 and model (1.1) with r0 > 1 respectively. we therefore conclude that, in theorem 3, the expressions of ĩ∗ and ĩ∗ are the inverse functions of factor 1− e−µ3τ , where r̃0 is a τ -independent expression; and r̃∗ and r̃∗ are respectively the saturated functions of 1− e−µ3τ . the illustrative examples have shown that the larger the perturbation, the sharper the infected individuals decline exponentially, the earlier the recovered individuals meet extinction. acknowledgements this work is supported by the national natural science foundation of china (grants no. 11201075 and 61773122), natural science foundation of fujian province of china (grant no. 2016j01015). copyright c© 2017 assa. adv syst sci appl (2017) threshold analysis of a stochastic epidemic model with delay and temporary 59 0 0.5 1 1.5 2 2.5 3 3.5 4 t,days ×104 0 0.2 0.4 0.6 0.8 1 1.2 the stochastic system i(t) i(t) fig. 5.4. realizations for model (1.2) with γ = 0.5 (the green) and γ = 0.4 (the blue) respectively. references 1. kyrychko y. & blyuss k. 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59: pp. 87-93. 19. yu j., jiang d., shi n. global stability of two-group sir model with random perturbation. j. math. anal. appl. 2009; 360: pp. 235-244. 20. xue r., wei f.y. persistence and extinction of a stochastic sis epidemic model with double epidemic hypothesis. ann. appl. math. 2017; 33(1): pp. 77-89. 21. dalal n., greenhalgh d., mao x. a stochastic model of aids and condom use. j. math. anal. appl. 2007; 325: pp. 36-53. 22. mao x., marion g., renshaw e. environmental brownian noise suppresses explosions in population dynamics. stoch. process appl. 2002; 97; pp. 95-110. 23. mao x. stochastic differential equations and applications (2nd ed.), horwood, chichester, uk, 2007. copyright c© 2017 assa. adv syst sci appl (2017) introduction existence and uniqueness of the positive solution the sufficient conditions of the extinction of the diseases the threshold for the persistence of the diseases illustrative examples and their realizations conclusions advances in systems science and applications (2014) vol.14 no.3 230-243 robust design and its challenge for manufacturing system han-xiong li and xinjiang lu department of manufacturing eng & eng management, city university of hong kong, hong kong, pr china abstract robust performance is one of the most important concerns in design of any system in manufacturing industry. this performance can be achieved by the robust design. the paper will provide a brief overview of the robust design. it includes discussions about how to account for design uncertainty, and how to measure and evaluate robustness for both static and dynamic systems. by reviewing the strengths and weaknesses of different design methods, the challenges in this area will be discussed. keywords robust design, sensitivity analysis, stability, uncertainty 1 introduction in order to design and manufacture high quality productions at a minimal cost, increasingly accurate systems are required in practical industry. one serious problem in these systems is the inconsistent performance due to uncontrollable variations existing in the real-world, including manufacturing operations, variations in material properties and the operating environment. if these variations are not considered, they will degrade the performance and may result in a failure in practice [1]. thus, robust performance is one of the most important concerns in design of any system. the concept of the robust design was introduced by taguchi. the fundamental principle in the robust design is to improve the quality of a product by minimizing the effects of variations without eliminating the causes [2]. by making a design more robust to variations, it is possible to improve number of the eligible parts or use less experiment [3]. in past decades, much effort has been dedicated to the robust design. in general, robust design may be classified two groups, static model based design and dynamic model based design. • static model based robust design is relatively simple since it is irrelevant with time and only considers the effect of the static variations on performance ; • dynamic model based robust design is complex because it must consider the stability, flexibility and robustness of the system in the whole operating process. the aim of this paper is to review and compare the applicable methods, so that their underlined philosophy can be clearly shown and easily understood. though the review, the challenges in this area can be discussed. advances in systems science and applications (2014) vol.14 no.3 231 2 uncertainty in a robust design problem, the system includes three kinds of variables : • design variables (or control variable) ĺc since their nominal values can be selected between the range of upper and lower bounds, they are controllable ; • uncertainties that can not be adjusted by the designer, so they are uncontrollable ; • performances that are the objective of design, they depend on the system model, design variables and uncertainties. these uncertainties, which should be first identified to obtain the robust performance, mainly include : • noise it is caused by changes of operating conditions, such as, environmental temperature, pressure, humidity and material change, etc. thus, it is of stochastic nature. in a robust design problem, the system includes three kinds of variables : • model parameters it is often caused by manufacture error. parameters of a product can only be realized to a certain degree of accuracy due to machinery limitation. it could be either stochastic or non-stochastic. • model uncertainty it is often caused by approximations in the modelling process. basically, these uncertainties can have following three different natures : • the deterministic type defines the domains in which the uncertainties can vary ; • the probabilistic type defines probability measure describing the likelihood by which a certain event occurs ; • the possibilistic type defines fuzzy measures describing the possibility or membership grade by which a certain event can be plausible or believable. these three different types of uncertainties are usually modelled by crisp sets, probability distributions and fuzzy sets, respectively. 3 robust design the robust design framework is shown in figure 1. the key issue in this framework is the robust design strategy, which optimizes the design variables to make the system less robust to uncertainties. all existing robust design methods depend on the characteristics of a system given in table 1 and table 2. if a system model is unknown and design parameters are non-probabilistic type, then no robust design method is applicable to it. the indirect method for this case is data-based modelling methods that transfer it into the case of the known model. 232 han-xiong li : robust design and its challenge for manufacturing system 3.1 static model based robust design this robust design includes three different approaches : deterministic robust design, probabilistic robust design, and fuzzy analysis. table 1 existing robust design methods for static system system static model model known unknown uncertainty non-probabilistic probabilistic non-probabilistic probabilistic type type type type •monte carlo simulation •taguchi robust •deterministic •first and second method design robust methods order moment n/a •monte methods •fuzzy analysis methods carlo •probabilistic simulation sensitivity analysis • performance index table 2 existing robust design methods for static system system static model model known unknown uncertainty non-probabilistic probabilistic non-probabilistic probabilistic type type type type •monte carlo simulation •taguchi robust •deterministic •first and second method design robust methods order moment n/a •monte methods •fuzzy analysis methods carlo •probabilistic simulation sensitivity analysis • performance index 3.1.1 deterministic robust design the design parameters are deterministic. (1) euclidean norm method and conditional number method the methods obtain the system robustness by analytically measuring the sensitivity of a design using the gradient information of parameters. the sensitivity analysis is based on the taylor-series expansion of the performance. the euclidean norm method [1, 3] is to minimize the largest singular value σmaxof the sensitivity matrix. the condition number method [1, 4] is to minimize the condition number σmax σmin of the sensitivity matrix. caro, et al. [1] has compared these two methods when designing the damper, and shown that the euclidean norm is more suitable advances in systems science and applications (2014) vol.14 no.3 233 as the robust index than the condition number. (2) the sensitivity region measures method gunawan & azarm [5] proposed the sensitivity region measures for the single objective robust design optimization. this method first projects the desirable objective function space into the design parameter space, where the sensitivity region is constructed. then the most sensitivity direction in the sensitivity region can be found. this most sensitivity direction is actually a measurement of a robust performance. li, azarm & boyars [6] proposed another sensitivity region measures for robust optimization. this method first projects the design parameter space into the desirable objective variations domain, where the objective sensitivity region is constructed. then the maximum performance variation in the sensitivity region is found. this variation is used as a measurement of a robust performance. moreover, the comparison with the gunawanąŕs method is carried out. 3.1.2 fuzzy analysis the possibility methods are proposed to apply in areas where it is not possible to obtain accurate statistical data dueto restriction of resource or conditions. its foundation is possibility theory. the extension principle calculates the possibility distribution of the fuzzy response from the possibility distribution of the fuzzy input variables. recently, this method was applied to the robust optimal design to deal with the epistemic uncertainty [7]. also, this method was applied for the modeling of tolerances and clearances in the mechanism analysis [8]. the comparison of probability and possibility for design against catastrophic failure under uncertainty was presented by [9]. the review of this application can be found in he & qu [10] and beyer & sendhoff [11]. 3.1.3 probabilistic robust design this probabilistic robust design will be less conservative than the deterministic robust design since it makes use of the probabilistic information of parameters. (1) monte carlo simulation monte carlo methods are a class of computational algorithms that rely on repeated random sampling from probabilistic density function of each parameter to compute their results. then, an experiment is executed under the generated samples and the process data is collected. finally, the statistical method is used to estimate its probability density function from the process data. (2) first and second order moment methods the most widely used non-statistical uncertainty analysis method is moment methods. a taylor series expansion is employed to estimate response variance 234 han-xiong li : robust design and its challenge for manufacturing system based on variance of model parameters x as σfirstordery = ( ∂y ∂x ) |µx2σ(x) (3.1) σsecondordery = ( ∂y ∂x ) |µx2σ(x) (3.2) (3) probabilistic sensitivity analysis this method evaluates the effect of design variables on performances using the sensitivity information. probabilistic sensitivity analysis methods have been developed to provide insight into the probabilistic behavior of a model, which can be used to identify those non-significant variables and reduce the dimension of random design space. a review about probabilistic sensitivity analysis was presented by [12]. the sensitivity analysis includes : variance-based methods [13], probabilistic sensitivity coefficients [14], kullback-leibler entropy based probabilistic sensitivity analysis (psa) method [15]. (4) performance index various performance indexes are constructed to measure the performance and capability of a system. • suhąŕs information content the information axiom is used to evaluate quality of designs so that an appropriate design can be chosen from available design alternatives. according to the information axiom proposed by suh [16], a design candidate that has minimum information content should be selected. thus, the information content is regarded as a robust index. • design capability indices the six standard deviations(±3σ)is commonly a measure of process capability, which compares the variation of a process to the customer specifications through the equation cdl = usl− lsl 6σ (3.3) where usl is the upper specification limit, lsl is the lower specification limit, is the standard deviation of the process. we hope that cp is greater than one so that the process variation is less than the specification limits and the performance can satisfy the requirement. chen, et al. [18] extended the concept of measuring process capability to measure the approximate degree between the mean of the process and the target value. they proposed design capability indices (dci) as metrics for system performance and robustness. the indices cdu, cdl and cdk means that smaller is better, larger advances in systems science and applications (2014) vol.14 no.3 235 is better and target is better respectively, and are defined as cp = usl− lsl 6σ ,cdu = url− µ 3σ ,cdk = min(cdl, cdu) (3.4) where url and lrl are upper and lower requirement limits. the index is expected to be greater than unity so that the design will meet the requirement satisfactorily. forcing the index larger to unity is achieved by reducing performance deviation and/or locating the mean of performance deviation farther from requirement limits. (5) taguchi method taguchi robust design, also known as parameter design, is an approach to identify design variable values that satisfy a set of performance requirements despite variation in noise factor [2]. since 1960, taguchi methods have obtained a great success in improving the quality of products and design robustness. this method is based on experimental data and includes three parts : experimental design, quality loss function and signal-to-noise ratio. taguchi robust design approach for the variable design starts from the experimental design, where the orthogonal array is used for design. control factor resides in an inner array and noise factors conditions in an outer array. the experimental results in all combinations of control factors and noise factors are recorded. then, taguchi proposed a signal-to-noise ratio for measuring sensitivity analysis of response to variation of noise factors. based on the signal-to-noise ratio, the robust design is obtained. although taguchiąŕs method has obtained great success, there are certain assumptions and limitations associated with his methods. use of the taguchi method will not yield an accurate solution for design problems that embody highly nonlinear behavior [19]. the taguchi method has been criticized by the statistical community [20]. many of taguchiąŕs statistical methods, e.g., orthogonal arrays, linear graphs and accumulation analysis, are not statistically efficient [20]. shoemaker et al., [21] presented a combined single array for both control and noise factors instead of orthogonal array. 3.2 dynamic model based robust design these uncertainties are, in general, also dynamic in nature and correspond to variations in either external variables or internal process parameters [22]. 3.2.1 stability based design the recent integration of the steady-state design and the dynamic stability is to explicitly consider dynamic elements in the process design by use of the eigenvalue theory. blanco and bandoni [23] proposed the multi-period program to solve the lyapunovąŕs stability matrix equality that can guarantee the system stability 236 han-xiong li : robust design and its challenge for manufacturing system under uncertainty and disturbance. however, the accuracy of this method will depend on the degree of discretization. since lyapuovąŕs criteria for asymptotic stability can not be easily implemented within a design optimization framework, mohideen, perkins & pistikopoulos [24] and kokossis & floudas [25] proposed an alternative robust stability criteria based on the matrix measures, which can avoid the tedious calculation of all eigenvalues. matrix measures can provide a single upper bound for all eigenvalues. however, this bound should not be used because it is typically not tight, therefore, may result in an overestimation of the stability boundary. monnigmann & marquardt [26] and grosch, monnigmann & marquardt [27] proposed the stability design in the steady-state process optimization. this design employed manifolds method to figure out the bound of the parameter variations that guaranteed all eigenvlues of the process smaller than zero when the parameter variations were limited in this bound. 3.2.2 flexibility analysis the flexibility, which defines the ability to maintain feasible operation over a range of uncertain conditions, is a vital important characteristic for the operation of these plants. as the state of dimitriadis & pistikopoulos [22], the flexibility analysis problem generally consists of two tasks which are complementary to each other. • the first task is to determine if a given design can feasibly operate over the range of uncertainty considered. this problem is known as the flexibility test problem. • the second task is to calculate a measure to quantify the ability of the design to operate in the presence of uncertainty. this is known as the flexibility index problem and is usually tackled by establishing the maximum parameter range over which the design can operate feasible. the flexibility measure is often used to select the suitable design by comparing different design alternatives with respect to their flexible operation [28]. swaney and grossmann [29] defined the flexibility index for measuring the flexibility of steady-state processes where the uncertain parameters are described by bounds of a specified range of operation. this approach was also extended to the analysis of dynamic systems under time-varying uncertainties [22]. the stochastic flexibility index is a metric for quantifying the ability of a process to maintain feasible operation in the face of stochastic uncertainties [30]. 3.2.3 robustness index various robustness indexes can be designed to measure the degree to which a system can meet its design objectives despite external disturbance and uncertainties in its design parameters. since adequate robust index is a necessary part of the optimal process design, it is desirable to consider process resiliency assessment when determining the process structure and establishing the operating range advances in systems science and applications (2014) vol.14 no.3 237 the resiliency indexmeasures the effect of the disturbance on the control input u. skogestad & morari [31] and lewin [32] considered the ratio of the control input u to disturbance d as the resiliency index based on the linear process model. cao, rossiter and owens [33] applied it to select the control inputs. solovyev & lewin [34] extended this resiliency index into the nonlinear system. the condition number is defined as the ratio between the maximum and minimum singular values. the condition number provides a direct measure of the directionality of the system. a large condition number indicates that the gain of the plant changes significantly with the input direction and that the system is sensitive to input uncertainty [35]. the disturbance condition number is a measure of the input magnitude which is needed to reject a disturbance in the given direction, relative to rejecting a disturbance with the same magnitude, but in the direction requiring the least control effort. a small disturbance condition number are most effective for disturbance rejection [36]. the relative gain matrix (rga)was originally proposed by bristol [37]. its objective is to provide a measure of interactions for multivariable square systems [31, 35]. if the plant has large rga elements within the frequency range where effective control is desired, then it is not possible to achieve good reference tracking with feedforward control because of strong sensitivity to diagonal input uncertainty. manousiouthakis et al. [38] generalized the concept of the rga to block relative gain which is capable of handling partially decentralized control systems. chen and yu [39] extended this method to non-square multivariable systems for selection of square subsystem from non-square system. a dynamic relative gain was proposed by avoy, et al. [40]. 3.2.4 operability index the operability measure can quantify the inherent ability of the process to move from one steady state to another and to reject any of the expected disturbances. the operability index is defined by vinson and georgakis [41] to effectively capture the inherent operability of continuous processes. vison and georgakis [41] applied this index to analyze the steady state of the linear system. the technique has also been proven to be effective for nonlinear processes [42]. it was also extended to dynamic systems by uztĺźrk and georgakis [43]. a brief survey paper about the operability index was presented by georgakis, et al. [44]. 4 challenge any method has its strength and weakness. the fundamental difficulties in robust design are related to model/parameter uncertainties and nonlinearity of the system. most of the existing methods can deal with linear system with the known model, and are not capable to handle model uncertainty or nonlinearity 238 han-xiong li : robust design and its challenge for manufacturing system that in turn generates extra uncertainties to the system. the weakness of the existing design methods poses the challenges to unsolved problems as shown in table 3. 4.1 robust design for static system 1) there is still no method that can consider the model uncertainties in the deterministic robust design. the accurate model is needed for euclidean norm method and conditional number method to obtain the gradient information. the sensitivity region measure methods are based on the projection between parameter space and performance space, which requires the system model. 2) the model uncertainties still can not be handled in the probabilistic robust design, where all the existing methods can be classified two categories : modelbased robust design and data-based robust design. the data-based robust design includes monte carlo method and taguchi method. • since monte carlo method is a simulation method, all system knowledge, which has to be known beforehand, need to be translated into computer code. thus, the accurate system model is critical. moreover, it costs huge computational time that will limit its application. • taguchi method obtains the system robustness based on the experiment data. however, it is only suitable for the stochastic environment and not suitable to minimize the effect of parameter variations. the model-based robust design includes firstand secondorder moment methods. since these methods are based on the taylor series expansion, which requires the accurate system model. thus, the model uncertainty will lead to the significant performance degradation. so far, there is still no solution. 3) the membership function in fuzzy analysis is defined according to human experience, which could be too subjective and causes uncertainties. it would be a challenge if the experimental data can be used to reduce subjective uncertainties of the membership. 4.2 robust design for dynamic system in difference to the static system, another problem for the dynamic system is that the design optimization becomes extremely difficult when the system has both continuous and discrete design variables. 1) the current stability design can work for the linear system, where its eigenvalues are critical to the system stability. the weakness of the existing methods have not considered two influences : one is model uncertainties on the eigenvalues and their variations, and the other is parameter perturbation on the eigenvaluesąŕ variations. 2) the stochastic flexibility index is only applicable to the linear dynamic system. so far, there is still no study on its application to the nonlinear dynamic system. advances in systems science and applications (2014) vol.14 no.3 239 4.3 potential solutions for the static system, a novel model-based robust design is proposed to design the system using the nominal model. the system-model mismatch, model uncertainties, can be properly considered in the design. for the dynamic system, a novel stability based robust design is proposed to guarantee the robust performance as well as the system stability, so that the method can also be applied to the weak nonlinear system. 5 conclusion this paper presents a brief overview about advances in robust design. different approaches in robust design are reviewed and compared, upon which challenges have been proposed to the unsolved problems. two novel approaches are proposed to the unsolved problems. table 3 summary of evaluation of existing methods design classification existing method challenges for unsolved problems deterministic system deterministic robust design these methods need an accurate model and can not handle the model uncertainty. probabilistic data-based methods all these methods can not handle model uncertainties properly, furthermore, static • monte carlo • monte carlo simulation costs huge computational time, and requires the accurate model ; robust system • taguchi method • taguchi method is not effective to parameter variations ; model-based methods • model-based methods require the accurate model ; design fuzzy system • fuzzy analysis membership function strongly depends on human experience that causes subjective uncertainties. dynamic robust design • stability based design all existing methods can not work for the system with model uncertainties, and • feasibility design hybrid system withboth continuous and discrete design variables, furthermore, • other design • stability design has not been applied to the nonlinear system. • feasibility design has not been applied to nonlinear dynamic system. 240 han-xiong li: robust design and its challenge for manufacturing system références [1] caro s., bennis f., and wenger p. 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[44] georgakis, c., uztĺźrk, d., subramanian, s., and vinson, d.r., (2003), “on the operability of continuous processes”, control engineering practice, vol.11, pp.859-869. microsoft word 8-门可佩.doc 48-54 advances in systems science and applications (2010), vol.10, no.1 issn 1078-6236 international institute for general systems studies, inc. research on comprehensive evaluation of harmonious society in east china* kepei men, liangyu jiang and jing liu nanjing university of information science and technology, nanjing 210044, china email: menkp@yahoo.com.cn abstract the harmonious society is the general goal of social development. we select 5 subsystems and 35 evaluation index to establish an system for evaluating harmonious society, with the method of principle component-cluster analysis, according to 2007 statistical yearbook of china and some other statistical data. meanwhile, we use topsis method as an assistant certification to analyze the harmonious society development of east china. the result shows that the six provinces in east china and shanghai city could be divided into 3 parts. the first part includes shanghai. the second part includes zhejiang, jiangsu and shandong. and the third part includes fujian, jiangxi and anhui. the result is identical to practice developing of east china. keywords east china harmonious society comprehensive evaluation principle component-cluster analysis (pcca) 1. introduction realizing social harmony and building a nice society is the social ideal that human being assiduously seeks. it is also the social ideal of all the marxist party including the communist party of china. to construct a socialist harmonious society is a great task that put forward in whole under the new situation of initiating the socialist road with the specific practice in china. mr hu jintao proposed to realize harmony zone on the 5th anniversary forum of the foundation of shanghai cooperation organization. east china is the most active economic growth zone. the six provinces and shanghai city of it account for less than 1/4 of the whole country, the population is 28.74% of the whole nation and the land area only accounts for 8.13%. but it has made great contribution for china, because of its wonderful economy. according to the statistical bulletins of nation and east china in 2007, we knew that the resident income of the six provinces and shanghai city was kept growing, and the living standard was improved further. the per capital dominating income of urban residents had exceeded 11 thousand yuan. among them, shanghai, jiangsu, zhejiang, fujian and shandong were over 14 thousand yuan, which were higher than the national average 13.786 thousand yuan. the gdp of this area in 2006 and 2007 were reached 8826.513 billion yuan and 10406.21 billion yuan respectively, which accounted for 41.86% and 42.20% of china’s gdp. therefore, study on the problem about the construction of the harmonious society in east china is strongly representative and has a profound significance. in this paper, we focus on the construction idea of socialist harmonious society, to put forward a harmonious society index system with great operative, for the purpose of providing reference for the construction of harmonious society. 2. the introduction to research methods 2.1 the method of principal component and clustering analysis[1-4] suppose there are n samples, every sample has p indices, we get a original data matrix ∗ this work is supported by the key projects of national statistics research program (2008lz022). advances in systems science and applications (2010), vol.10, no.1 49 ( )ij n px x ×= , ( 1, 2, , ; 1, 2, )i n j p= = and we make linear combination (comprehensive index) with p vectors 1 2, , , px x x , that is 1 1 2 2 , 1, 2, ,i i i pi pf a x a x a x i p= + + = then we limit the combination coefficients 1 2( , , , )i i i pia a a a= with 2 2 2 1 2 1, 1, 2, ,i i pia a a i p+ + + = = in which, ia is a unit vector and 1i ia a = . the comprehensive index is determined by the following rules: (1) if and jf ( , , 1, 2, ,i j i j p≠ = ) are uncorrelated, that is ( , ) 0i jcov f f = . (2) 1f is the biggest variance in all the linear combinations of 1 2, , , px x x , namely, ' 1 1 ( ) max p i i c c i var f var c x = = ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ ∑ , in which ' 1 2,( , , )pc c c c= . as to the rule (2), we will get the similar conclusions. 2f , uncorrelated with 1f , is the biggest variance in all the linear combinations of 1 2, , , px x x . pf , uncorrelated with 1 2 1, , pf f f − , is the biggest variance in all the linear combinations of 1 2, , , px x x , and so on. the comprehensive vectors 1 2, , , pf f f , satisfied above requirements, are the principal components. the information extracted from the information content of original indices decrease successively. we use variance to measure information extracted by every principal component, and the contribution of principal component variance is equal to corresponding eigenvalue iλ to correlation matrix of the original data. 1 2( , , , )i i i pia a a a= , the combination coefficient of every principal component, is the eigenvector it to corresponding eigenvalue iλ . the contribution rate of variance is 1 / p i i j j α λ λ = = ∑ . the more iα , the more information corresponding principal component explains. when the variance contribution rate of one principal component is very small, we think the information it provides is little, then we may delete it. in general case, if the cumulative variance contribution rate of first q principal components reaches 85%, we just consider the first q principal components, then explain properties of random vector x with them. other principal components are the random errors caused by incorrect observation. research on comprehensive evaluation has made process[5-7]. papers [1-3] point the method popular in recent year is to structure a comprehensive evaluation function of principal components for ranking, which is based on the contribution rate of variance iα , but it is wrong. when the contribution rate of variance of the first principal component 1f is quite high (over 85%), we may think this principal component can almost reflect the information provided by the original variables. then we may rank and evaluate according to the scores of the first principal component. to the ranking problems of multiple index system, when the contribution rate of variance of the first principal component is not over 85%, that is, the original data information expressed by the first principal component is not enough, it has one-sidedness, if we still rank and evaluate only by the scores of the first principal component. at this time, we combine principal men: research on comprehensive evaluation of harmonious society in east china 50 component analysis with cluster analysis, which is named “principal component-cluster analysis (pcca)”. table 1 the index system subsystem index unit directivity m at er ia l c iv ili za tio n 1 2 3 4 5 6 7 8 9 10 yuan/person yuan yuan % % square meter/person % part/10 thousand agriculture=1 % positive positive positive negative negative positive positive positive medium positive po lit ic al ci vi liz at io n 11 12 13 14 person/10 thousand female=100 % part/10 thousand positive medium negative negative sp iri tu al ci vi liz at io n 15 16 17 18 19 % person/100 thousand % % kind/10 thousand positive positive positive positive positive so ci al ci vi liz at io n 20 21 22 23 24 25 26 27 person/10 thousand person/100 thousand old % % part/10 thousand % % positive negative positive positive positive positive negative positive h ar m on io us so ci et y ec ol og ic al ci vi liz at io n 28 29 20 31 32 33 34 35 ton/10 thousand yuan % % % % % 10 thousand stere % negative positive positive positive positive positive positive positive note: 1 means per capital gdp; 2 means per capital disposable income of urban residents; 3 means per capital net income of rural residents; 4 means the engle's coefficient of urban residents; 5 means the engle's coefficient of rural residents; 6 means housing areas of rural residents; 7 means the proportion that the tertiary industry accounts for gdp; 8 means numbers of patents application; 9 means consumption level of urban and rural residents; 10 means the proportion that r&d accounts for gdp; 11 means lawyer’s numbers that per ten thousand people have; 12 means sex ratio of senior middle school graduates; 13 means registered unemployment rate of urban residents; 14 means numbers of criminal case; 15 means the proportion that education and culture entertainment account for total consumption expenditure; 16 means average students number of colleges and universities; 17 means excellent rate of product quality; 18 means the proportion that education operating expenses account for fiscal expenditure; 19 means book publishing kinds; 20 means doctor’s numbers that per ten thousand people have; 21 means death toll from traffic accidents; 22 means average life expectancy; 23 means mobile subscription; 24 means gross enrollment ratio of higher education; 25 means numbers of urban community service facilities; 26 means gross divorce rate; 27 means the proportion that numbers of participate in advances in systems science and applications (2010), vol.10, no.1 51 medical insurance account for total amount; 28 means energy consumption in every unit; 29 means forest coverage; 30 means industrial wastewater treatment level; 31 means per capita park greenery area; 32 means water subscription; 33 means gas subscription; 34 means daily treatment ability of domestic sewage; 35 means harmless treatment rate of domestic waste. [11] “principal component-cluster analysis” is such a method. first we do the principal component to get some principal components, with which we do cluster analysis to samples, then we rank and classify the samples by the scores of the first principal component. specific ways are the following: (1) select first r principal components by cumulative contribution rate and calculate the scores of the principal components 1 1 2 2 , ( 1,2, , )l l l pl pf a x a x a x l r= + + = (2) do the systematic cluster analysis to the selected new matrix ( 1 2, , , rf f f ); (3) calculate the scores’ average value of the first principal component to determine ranking of every class; (4) determine ranking of every sample in the class to get comprehensive evaluation result, according to every sample’s score. 2.2 the method of topsis main steps of topsis: (1) select p evaluation index to n evaluation units for comprehensive evaluation, then get a evaluation matrix ( )ij n px x ×= , in which ijx is observation data in unit i . (2) make original data being dimensionless, then get a normalized evaluation matrix ( )ij n pz z ×= , in which 2 1 / ( 1,2, , ; 1,2, , ) n ij ij kj k z x x i n j p = = = =∑ . (3) determine the positive ideal solution z + and negative ideal solution z − of matrix z , then get 1 2( , , , )qz z z z+ + + += , 1 2( , , , )qz z z z− − − −= , in which 1 2max{ , , , } ,j j j njz z z z+ = 1 2min{ , , , }, ( 1, 2, , )j j j njz z z z j q− = = . (4) calculate the distances between every unit and positive ideal solution, and the distances between every unit and negative ideal solution: 2 1 ( ) q i ij j j d z z+ + = = −∑ , 2 1 ( ) q i ij j j d z z− − = = −∑ ( 1, 2, , )i n= (5) calculate relative approach degrees between every evaluation unit and the optimal solution: 100%, ( 1, 2, , )i i i i dc i n d d − + −= × = + (6) rank according to the relative approach degree. the more it closes to 100, the more ic . it indicates that unit i is more close to the optimal level. otherwise, unit i is less close to the optimal level. 3. establishment of index system and empirical analysis to build a harmonious society is a great social system engineering. from the horizontal point of view, it is closely associated with society, politics, economy and culture. from the vertical point of view, it involves in macro-view, medium-view and micro-view. the concept of harmonious society is extensive, comprehensive and systematic. it includes every aspect of men: research on comprehensive evaluation of harmonious society in east china 52 economy, politics, society, culture, environment and people’s life. therefore, the index system of harmonious society should embody the comprehensive and systematic connections among index, and reflect progress condition of construction in many aspects. in this paper, we will divide it into 5 subsystems to make it have good operability, that is, material civilization system, political civilization system, spiritual civilization system, social civilization system and ecological civilization system, (details in table 1) [8-11] . 3.1 calculation with the method of principal component and cluster analysis first we do pretreatment to the original data. then we make principal component analysis to the harmonious index of six provinces and shanghai city with sas program. finally, we get the following eigenvalues of the correlation matrix. eigenvalues of the correlation matrix eigenvalue difference proportion cumulative 1 18.6926827 12.4745482 0.5341 0.5341 2 6.2181345 2.0407184 0.1777 0.7117 3 4.1774161 1.4626788 0.1194 0.8311 4 2.7147372 0.8307516 0.0776 0.9087 5 1.8839856 0.5709418 0.0538 0.9625 6 1.3130439 1.3130439 0.0375 1.0000 ………… in which, prin1, prin2, prin3, prin4 denote the first four principal components. the second list is eigenvalues of sample correlation matrix, the fourth list is contribution proportions of variance, and the fifth list is cumulative contribution proportions. as cumulative contribution proportion of the first four eigenvalues reaches 90.87%, which is higher than 85%, we only need to select the first four principal components to summarize the all data, then get the scores of the principle components (see table 2). table 2 the scores of the principle components analysis cities scores the first principal component the second principal component the third principal component the fourth principal component shanghai 8.52652 -2.49745 -0.61749 0.58853 jiangsu 0.68228 1.08935 1.92012 -1.20837 zhejiang 1.99413 4.24675 -0.908082 -0.56337 anhui -3.85481 -2.3237 -2.88072 -1.98804 fujian -1.67628 1.85291 -0.83024 1.03237 jiangxi -3.62903 -0.86869 -0.02778 2.92133 shandong -2.04282 -1.49017 3.34413 -0.78246 the contribution proportion of first principal component’s variance is 53.41%, which is the largest of all linear combinations, and the information of the first principal component is the biggest. we get samples’ ranking the first time by calculating the scores of the first principal component. if we rank only according to the scores of the first principal component, as contribution proportion is not over 85%, the information will not be big enough and the result will have one-sidedness. thus, we do the cluster analysis to the first four principal components’ score matrix with sas program, then we get the cluster graph (see figure 1). from figure 1, we know that the six provinces and shanghai city can be divided into three classes, that is, {shanghai}; {jiangsu, shandong, zhejiang}; {fujian, jiangxi, anhui}. then we rank according to the scores of first principal component. that is, {shanghai}; {jiangsu, shandong, zhejiang}; {fujian, jiangxi, anhui}. finally, we get the ranking by the scores of the first principal component in every class. that is, {shanghai, zhejiang, jiangsu, shandong, fujian, jiangxi, anhui}. from table 3, we get the rankings by principal component-cluster analysis and the scores of the first principal component. we find that they are almost the same, the only different is ranking between shandong and fujian. however, 20 index in shandong are better than fujian, shandong should be in front of fujian. therefore, it is more reliable to get the ranking by principal advances in systems science and applications (2010), vol.10, no.1 53 component-cluster analysis. figure 1 cluster graph table 3 the compositors of the principal component cluster and the first principal component shanghai jiangsu zhejiang anhui fujian jiangxi shandong ranking by principal component-cluster analysis 1 3 2 7 5 6 4 ranking by the scores of the first principal component 1 3 2 7 4 6 5 3.2 calculation with the method of topsis we calculate relative approach degree ic of every subsystem by matlab program. the more it closes to 100, the more ic . it indicates that unit i is more close to the optimal level. otherwise, unit i is less close to the optimal level. details in table 4. the construction of material civilization, political civilization, spiritual civilization, social civilization and ecological civilization are closely linked with each other in the process of building a socialist harmonious society and a well-off society. they are very important, so we should treat them equally. we get the final scores of the six provinces and shanghai city after doing the equal weight. then we rank by the final scores, which is listed in table 4. the result is the same as the ranking by principal component-cluster analysis, which verifies rationality of the method of principal component-cluster analysis. table 4 the development level of six provinces in east china and shanghai city material civilization political civilization spiritual civilization social civilization ecological civilization score ranking shanghai 86.6131 46.87 80.45 71.89 32.28 63.62062 1 jiangsu 38.4696 32.37 19.50 32.84 57.44 36.12392 3 zhejiang 53.6959 23.66 20.32 47.55 52.83 39.61118 2 anhui 12.3374 31.65 9.48 17.73 24.29 19.09748 7 fujian 21.9262 28.24 23.78 22.12 49.08 29.02924 5 jiangxi 8.7141 35.27 27.04 27.24 41.41 27.93482 6 shandong 24.8297 60.62 14.78 26.26 44.15 34.12794 4 4. conclusions and discussions 2006 was the beginning of eleventh five-year plan. under the leadership of provincial and municipal governments in east china, we had acquired remarkable achievement in the field of social economic development, by insisting in carrying out macro-control policy and guiding with scientific development view. according to the methods of principal component-cluster analysis men: research on comprehensive evaluation of harmonious society in east china 54 and topsis, we divide six provinces and shanghai city into 3 classes on the level of social harmonious development. (1) shanghai city is the first class. the score of shanghai is far higher than any other province. it shows that shanghai city is superior to other places in the construction of harmonious society. since reform and opening up to the outside world, shanghai city has got great achievement as the first metropolitan. the index of per capital gdp, per capital disposable income of urban residents and per capital net income of rural residents are in the first place. (2) zhejiang, jiangsu and shandong are in the second class, which are high in harmony. the scores of zhejiang province and jiangsu province are high in material civilization system, ecological civilization system and social civilization system, but low in spiritual civilization system. these places have strong economic strength, good security systems and harmonious living condition. the scores of shandong province are low in material civilization system, spiritual civilization system and social civilization system, but high in political civilization system. shandong province is one of the best provinces in public security. (3) fujian, jiangxi and anhui are in the third class. it has significant gap to others. the relatively backward social economy restricts social development, leading to the development of society unbalanced. from above conclusions, we know that shanghai, zhejiang, jiangsu and shandong are in the leading positions of the whole nation. in the future, we should not only focus on the quantity of economic operation, but also focus on the quality. to other relative undeveloped regions in east china, we should speed up the development to shorten the difference. six provinces and shanghai city in east china will obtain new development in the new century through cooperation and exchange and will create a more bright future. references [1] xueming wang. applied multivariate analysis (2rd edition). shanghai: shanghai university of finance and economics press, 2004: 1-358. (in chinese) [2] xueming wang. query to the applied method of principal component. statistics and decision, 2007, 8: 31-32. (in chinese) [3] jingya xu, wang yuanzheng. improvement to the applied method of principal component. mathematics in practice and theory, 2006, 6: 68-75. (in chinese) [4] daoyuan zhu, chengou wu, weiliang qin. applied multivariate analysis and sas software. nanjing: southeast university press, 1998: 1-408. (in chinese) [5] chuncheng wu, shengbao yao, chaoyuan yue. the model of the evaluation of the comprehensive progress for large project. advances in system science and applications, 2006, 6(3): 410-415. [6] pu gong, qiang si, jianling meng. the evaluation method of multi-stage compound real option on human capital valude. advances in systems science and applications, 2006, 6(1): 101-106. [7] wei wang, zhuangzhi liu. comprehensive evaluation of project risk based on fuzzy network analytical methods. advances in systems science and applications, 2008, 8(2): 270-275. [8] state statistic bureau studying team. research on the index system of harmonious society. statistical research, 2006, 5: 23-29. (in chinese) [9] mei song, qi xin. the construction of harmonious society index system. beijing social science, 2006, 1: 62-66. (in chinese) [10] jianguo ouyang. research on comprehensive evaluation index system of harmonious society. zhejiang social science, 2006, 2: 16-22. (in chinese) [11] prc national statistics bureau. china statistical yearbook-2007. beijing: china statistics press, 2007: 1-1028. (in chinese) adv syst sci appl 2017; 4:61–77 published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/518 price of anarchy for maximizing the minimum machine load yulia. v. chirkova institute of applied mathematical research, karelian research centre ras, petrozavodsk, russia abstract: the maximizing the minimum machine delay game (or cover game) with uniformly related machines is considered. players choose machines with different speeds to run their jobs trying to minimize job’s delay, i.e. the chosen machine’s completion time. the social payoff is the minimal delay over all machines. for the general case of n machines we found the lower bound for price of anarchy (poa), and for the case of 3 machines we found its exact value. we proved that the poa does not change or increases when an additional third machine is included into the system with two machines. also we propose a method of computation the poa value and illustrate it for 3 machines. keywords: nash equilibrium, maximizing the minimum load, cover game, price of anarchy 1. introduction load balancing represents a major problem in networks and distributed computing systems, since load optimization guarantees efficient resource utilization. modern systems such as telecommunication networks, cloud computing systems, grid, etc. consist of independent components, in many cases without their centralized control. particularly, users located at nodes and data transmission protocols do not interact with each other for maintaining a certain load level. furthermore, in practice they demonstrate egoistic behavior with respect to free resources. and global optimization methods often become inapplicable due to the infeasibility of realizing optimal resource utilization plans in such systems (server request schedules, capacity norms of data transmission channels and so on). the game-theoretic approach allows treating load balancing as a game, where players have egoistic behavior and can reach some equilibrium state such that none of them benefits from unilateral deviation from a chosen strategy. system efficiency is assessed by comparing the above equilibria with the global optimum. the present paper focuses on the maximizing the minimum machine delay game (or cover game) [1–3] also known as the scheduling problem [4] in the form of a game equivalent to the kp-model (see [5, 6]) with parallel different-capacity channels where system optimization is the maximizing the minimum machine delay [1–3] instead of the minimization the maximum machine delay (makespan). it is necessary to distribute several jobs of various volumes among machines of nonidentical speeds. the volume of a job is its completion time on a free unitspeed machine. machine load is the total volume of jobs executed by a given machine. the ratio of machine load and speed defines its delay, i.e., the job completion time at this machine. each player chooses a machine for its job striving to minimize job’s delay. players have egoistic behavior and reach a nash equilibrium, viz., a job distribution such that none of them benefits from unilateral change of a chosen machine. in the sequel, we study pure strategies nash equilibria only; as is well-known [7, 8], such an equilibrium always exists ∗corresponding author: julia@krc.karelia.ru http://ijassa.ipu.ru/ojs/ijassa/article/view/518 62 y.v. chirkova in the described class of games. the system payoff (also called the social payoff) is the minimum delay over all machines for an obtained job distribution. the price of anarchy [1] (poa) is defined as the maximum ratio of the optimal social payoff and the social payoff in the worst-case nash equilibrium. the problem where the system tries to maximize the minimum delay over all machines appears from the concept of a fair resource sharing and efficient routing of traffic. the paper [1] first in the equilibrium efficiency studying for such model gives motivations coming from issues of quality of service, fair resource allocation, and fair queuing. the base idea is that each system component must be loaded as much as possible and not to idle. consider an example where each player pays the system a value which equals his delay for his job processing. fair system should not have privileged players who pay rather less than others due to successful machine choose. also such system should not have machines providing small or zero payoff. according to the earlier publications, the poa in the maximizing the minimum machine delay games with pure strategies can be estimated by • for n ≥ 2 machines with speeds 1 ≤ · · · ≤ s [1] the price of anarchy is not limited if s ≥ 2; • the price of anarchy is closed to and no more than 1.7 for any number of homogeneous machines [1, 9]; • the price of anarchy equals{ 2+s (1+s)(2−s) for 1 ≤ s ≤ √ 2, 2 s(2−s) for √ 2 < s < 2 for two machines with speeds 1 ≤ s [2]; • the price of anarchy equals 2 + s 2(2− s) for 1 ≤ s < 2 for three machines with speeds 1 = 1 ≤ s [2]; • the price of anarchy equals 1+s s for 1 ≤ s ≤ s0, 2+s (1+s)(2−s) for s0 < s ≤ √ 2, 2 s(2−s) for √ 2 < s < 2 in the hierarchical model of two machines with speeds 1 ≤ s and two types of jobs where first machine can process both types jobs and second machine can process only second type jobs [3]. here s0 is the largest root of the equation 1+s s = 2+s (1+s)(2−s) . in what follows, we derive a lower estimate for the poa in the case of n ≥ 3 machines. also we present the exact value of the poa for 3 machines with speeds 1 ≤ r ≤ s < 2:{ 2+s (1+r)(2−s) for rs ≤ 2, 2 r(2−s) for rs > 2. moreover we show that the poa increases or does not change under new machine inclusion into the system of two machines. in the case of n machines, a computing algorithm of the exact poa value is proposed based on solving a series of linear programming problems. the algorithm is described for the case of 3 machines and is implemented numerically in the form of a program which draws the curves of the poa as a function of the fastest machine and compares them with the curves of the corresponding estimates. copyright c© 2017 assa. adv syst sci appl (2017) price of anarchy for maximizing the minimum machine load 63 2. the model consider a system s = s(n, v) composed of n machines operating with speeds v1 = 1 ≤ · · · ≤ vn = s. note that such choose of machine speeds does not contradict with generality since one always can normalize speeds dividing them by the speed of the slowest machine. the system is used by a set of players u = u(n,w): each of n players chooses an appropriate machine for its job execution. for player j, the volume of job equals wj , j = 1, . . . , n. denote by w = n∑ j=1 wj the total volume of all jobs. free machine i with speed vi executes a job of volume w during the time w/vi. study the following pure strategies game γ =< s(n, v), u(n,w), λ >. each player can choose any machine. the strategy of player j is machine lj selected by this player for its job execution. then the strategy profile in the game γ represents the vector l = (l1, . . . , ln). the load of machine i, i.e., the total volume of all jobs assigned to the machine is defined by δi(l) = ∑ j=1,...,n:lj=i wj . the delay of machine i takes the form λi(l) = ∑ j=1,...,n:lj=i wj/vi = δi(l) vi . actually, this quantity is the same for all players selecting a given machine. we suppose that the goal of the system is minimizing of the least busy machine idling, that is maximizing of its working time or delay on it. the social payoff is described by the minimum delay over all machines: sc(l) = min i=1,...,n λi(l). designate by opt = opt (s, u) = max l is a profile in γ(s,u,λ) sc(l) the optimal payoff (the social payoff in the optimal case) where maximization runs over all admissible strategy profiles in the game γ(s, u, λ). a strategy profile l such that none player benefits from unilateral deviation (change of the machine chosen in l for its job execution) is a pure strategies nash equilibrium. to provide a formal definition, let l(j → i) = (l1, . . . , lj−1, i, lj+1, . . . , ln) signify the profile obtained from a profile l if player j replaces machine lj chosen by it in the profile l for another machine i, whereas the rest players use the same strategies as before (remain invariable). definition 2.1: a strategy profile l is said to be a pure strategies nash equilibrium iff each player chooses a machine with the minimum delay, i.e., for each player j = 1, . . . , n we have the inequality λlj(l) ≤ λi(l(j → i)) for all machines i = 1, . . . , n . definition 2.2: the price of anarchy in the system s is the maximum ratio of the social payoff in the optimal case and the social payoff in the worst-case nash equilibrium: poa(s) = max u opt (s, u) min l is a nash equilibrium in γ(s,u,λ) sc(l) . copyright c© 2017 assa. adv syst sci appl (2017) 64 y.v. chirkova 3. the general case of n machines in this section we give the following assumptions and results which will be employed in further analysis. consider a system composed of n ≥ 2 machines operating with speeds v1 = 1 ≤ · · · ≤ vn = s. if the number of jobs n is less than the number of machines n then obviously the social payoff is zero in any profile. in this case we assume by definition that the ratio of an optimal payoff to an equilibrium payoff is 1. further we suppose that n ≥ n . if s ≥ 2 then the price of anarchy is infinite [1]. therefore, we assume 1 ¡ s ¡ 2 in the following. if the number of jobs n is more than or equals the number of machines n then obviously all machines are loaded in an optimal profile. moreover in this case all machines are loaded in any equilibrium. the optimal social payoff is not larger than the social payoff in the case when the whole volume of jobs is distributed among machines proportionally to their speeds so that all machines have an identical delay: opt ≤ w n∑ i=1 vi . (3.1) further we determine estimates for equilibrium delays and volumes for some jobs processed on machines. we also restate the proof for results taken from cited papers for the sake of completeness. lemma 3.1: [2] if the number of jobs n is not less than the number of machinesn then in any equilibrium the load of any machine is more than zero. proof consider an arbitrary equilibrium profile l. suppose that some machine i has zero load. then there is a machine k receiving not more than two jobs. since v1 = 1 ≤ · · · ≤ vn = s < 2 then vi > vk 2 . let wk be the minimal job volume on k. if it moves to an idle machine i then its load becomes equal wk vi < 2wk vk ≤ λk(l), that is less comparing with its load in the profile l. denote the number of jobs on some machine k in a profile l by nk. lemma 3.2: [2] suppose that l is a nash equilibrium profile and sc(l) = λi(l). if nk > vk vi then λk(l) ≤ nkvi nkvi−vk λi(l) for any machine k. proof let w be the job with the smallest volume on some machine k. then w ≤ vk nk λk(l). since l is an equilibrium then λk(l) ≤ λi(l) + w vi ≤ λi(l) + vk nkvi and thus λk(l) ≤ nkvi nkvi−vk λi(l). lemma 3.3: suppose that l is an equilibrium profile and sc(l) = λi(l) and consider an arbitrary machine k. if nk ≥ 2 and 1 ≤ vk vi < 2 then the volume wj of any job j on the machine k is at most vivk 2vi−vk λi(l). moreover the total volume of remaining jobs on k is also no more than vivk 2vi−vk λi(l). proof let the machine k receive two or more jobs and w be the minimal job volume on k. then copyright c© 2017 assa. adv syst sci appl (2017) price of anarchy for maximizing the minimum machine load 65 the total volume of remaining jobs on k equals vkλk(l)− w. since l is an equilibrium then λk(l) = vkλk(l) vk ≤ λi(l) + w vi and thus vkλk(l)− w ≤ vkλi(l) + ( vk vi − 1 ) w ≤ vkλi(l) + ( vk vi − 1 ) wj:lj=k ≤ vkλi(l) + ( vk vi − 1 ) (vkλk(l)− w). then w ≤ wj:lj=k ≤ vkλk(l)− w ≤ vivk 2vi−vk λi(l). the next theorem determines the lower estimate for the price of anarchy in the system of n ≥ 3 machines. the estimate is determined by speeds of 3 machines in the system: the first one and the second one which are the slowest, and the last one which is the fastest. theorem 3.1: for the system composed of n ≥ 3 machines having speeds v1 = 1 ≤ v2 = r ≤ v3 ≤ · · · ≤ vn = s < 2 the price of anarchy is at least est(r, s) = min{ 2 + s (1 + r)(2− s) , 2 r(2− s) }. (3.2) proof as far as we need to prove the lower estimate it suffices to present examples of systems providing ratios of the optimal payoff and the worst-case equilibrium payoff given in the theorem condition. suppose that in the system each machine i has a speed vi for each i = 1, . . . , n . 1. let first rs ≤ 2. then est(r, s) = 2+s (1+r)(2−s) . consider the set of jobs: w1 = w2 = (1 + r)s, wi3 = vi(2 + s), where i = 3 . . . , n , w4 = 2r − s, w5 = 2− rs. in the equilibrium lmachinen receives jobsw1 andw2, each machine i− 1 receives each jobwi3, i = 3 . . . , n , jobs w4 and w5 are assigned to machine 1. we need to show that l is really an equilibrium and find the system payoff. the loads of machines n and 1 equal 2s(1 + r) and (1 + r)(2− s) respectively. the load of each machine i = 2, . . . , n − 1 equals vi+1(2 + s). since λn(l) = 2(1 + r) > (1 + r)(2− s) = λ1(l) and λi(l) = vi+1(2+s) vi ≥ (1 + r)(2− s) = λ1(l), i = 2, . . . , n − 1, due to 2 + s > 1 + r, vi+1 ≥ vi and 2− s ≤ 1, then machine 1 has the smallest delay which equals its load. denote the delay of machine i as λji (l) = λi(l) + wj vi in the case where some job j deviates from the profile l and moves to machine i from another machine. no one of jobs w1 or w2 moves to machine i, i = 2, . . . , n − 1, since λn(l) = 2(1 + r) ≤ (2 + s) + (1 + r) ≤ vi+1(2+s)+s(1+r) vi = λ1 i (l) = λ2 i (l). also no one of them moves to machine 1, since λn(l) = 2(1 + r) = (1 + r)(2− s) + s(1 + r) = λ1 1(l) = λ2 1(l). each of jobs wi3, i = 3, . . . , n , has not reason to move to machine n due to λi−1(l) = vi(2+s) vi−1 ≤ 2(1 + r) + vi(2+s) s = λi3n(l), that is equivalent to the inequality (s− vi−1)vi(2 + s) ≤ 2svi−1(1 + r) which holds true since s− vi−1 < 1, 2 + s < 4 2 s vi vi−1(1 + r) ≥ 4. also no one of jobs wi3, i = 3, . . . , n , moves to machine j > i− 1 since λi−1(l) = vi(2+s) vi−1 < 2vi(2+s) vj ≤ (vi+vj)(2+s) vj = λi3j (l). moreover, job wi3 does not move to slower machine 1 or j < i− 1, and no one job from machine 1 moves to another machine because the delay on machine 1 is minimal. therefore, the given profile is an equilibrium with the social payoff (1 + r)(2− s). consider the profile where each job wi3, i = 3, . . . , n belongs to machine i, jobs w1 and w4 are assigned to machine 2 and machine 1 receives jobsw2 andw5. the social payoff equals 2 + s in this profile, so, opt ≥ 2 + s. 2. let now rs > 2. then est(r, s) = 2 r(2−s) . consider the set of jobs: w1 = w2 = rs, wi3 = 2vi, i = 3, . . . , n , w4 = r(2− s). in the equilibrium l jobs w1 and w2 belong to copyright c© 2017 assa. adv syst sci appl (2017) 66 y.v. chirkova machine n , each job wi3, i = 3, . . . , n is assigned to machine i− 1, machine 1 receives job w4. we show that it is an equilibrium indeed and find the system payoff. since λn(l) = 2r > r(2− s) = λ1(l) and λi(l) = 2vi+1 vi ≥ r(2− s) = λ1(l), i = 2, . . . , n − 1, due to 2 vi ≥ 1, vi+1 ≥ r and 2− s < 1, then machine 1 has the smallest delay which equals r(2− s). job w1 or w2 does not move to machine i, i = 2, . . . , n − 1, since λn(l) = 2r = r + r ≤ 2vi+1+rs vi = λ1 i (l) = λ2 i (l), and also to machine 1 due to λn(l) = 2r = r(2− s) + rs = λ1 1(l) = λ2 1(l). no one of jobs wi3, i = 3, . . . , n , moves to machine n since λi−1(l) = 2vi vi−1 ≤ 2r + 2vi s = λi3n(l) due to 2vi(s−vi−1) s ≤ 2rvi. also no one of jobs wi3, i = 3, . . . , n , moves to machine j > i− 1, since λi−1(l) = 2vi vi−1 ≤ 4vi vj ≤ 2(vi+vj) vj = λi3j (l). moreover, job wi3 does not move to slower machine 1 or j < i− 1. no one job on machine 1 moves to other machines with not smaller delay. hence, the given profile l is an equilibrium with the social payoff r(2− s). consider the profile where each job wi3, i = 3, . . . , n , is assigned to machine i, jobs w1 and w4 belong to machine 2, and job w2 to machine 1. the social payoff equals 2 for this profile, thus, opt ≥ 2. in both considered cases the ratio of the optimal payoff and the equilibrium payoff equal est(r, s), hence, the price of anarchy is not less than this estimate. from the obtained estimate (3.2) we see that when the speed of the fastest machine increases and comes closer to the value of 2, the lower estimate of the price of anarchy grows infinitely. hence, we obtain the following corollary from the theorem 3.1. corollary 3.1: for the system composed of n ≥ 3 machines having speeds v1 = 1 ≤ v2 = r ≤ v3 ≤ · · · ≤ vn = s < 2 the price of anarchy tends to infinity as s→ 2− 0. according to the following result, for poa evaluation it suffices to consider only games, where the optimal social payoff equals 1. theorem 3.2: for the system s, the price of anarchy constitutes poa(s) = max u1:opt (s,u1)=1 1 min l is a nash equilibrium in γ(s,u1,λ) sc(l) . proof we show that one can normalize job volumes in any game γ(s, u, λ) such that the optimal social payoff becomes equal 1 and the ratio of the optimal social payoff and the worst-case equilibrium payoff does not change its value. assume that l is the worst-case equilibrium in the game γ(s, u, λ) with an arbitrary set of players u(n,w). for each player j, the volume of its job equals wj , and the vector lopt gives the optimal strategy profile in this game. let sc and opt be the social payoff in the profile l and the optimal social payoff, respectively. the ratio of the optimal and worst-case equilibrium social payoff is defined by opt sc . so long as l represents an equilibrium, then for any player j we obtain that ∑ k=1,...,n:lk=lj wk vlj ≤ ∑ k=1,...,n:lk=i wk+wj vi for any machine i. now, explore the game with the same set of machines and players, where each player j has the job of volume wj opt . the social payoff in the profiles l and lopt constitutes sc opt and 1, respectively. by virtue of the linear homogeneity of machine delays in their loads, the profiles l and lopt form the worst-case equilibrium and optimal profiles, respectively, in the new game. particularly, the profile l is an equilibrium in the new game, since for any player copyright c© 2017 assa. adv syst sci appl (2017) price of anarchy for maximizing the minimum machine load 67 j the inequality ∑ k=1,...,n:lk=lj wk vljopt ≤ ∑ k=1,...,n:lk=i wk+wj viopt holds true for any machine i. imagine that l is any non-worst-case equilibrium in the new game. then the game admits an equilibrium l′ with social payoff sc′ opt such that the social payoff in the profile l′ is less than that in the profile l, i.e., sc′ opt < sc opt . however, in the initial game the profile l′ corresponds to the social payoff sc ′ < sc, and the equilibrium l′ is worse than its counterpart l. similarly, lopt gives the optimal profile in the new game. then the ratio of the optimal and the worstcase equilibrium social payoff in the new game also equals opt sc . consequently, any game γ(s, u, λ) corresponds to a game γ(s, u1, λ) with normalized job volumes such that opt (s, u1) = 1. moreover, the ratio of the optimal and the worstcase equilibrium social payoff is same in both games. hence, for poa evaluation it suffices to consider only games with unit optimal social payoff. 4. the case of 3 machines as a matter of fact, the exact poa value in the two-machine model was found in the paper [2]. consider the case of 3 machines in the system s. without loss of generality, throughout this section we believe that the machines have speeds v1 = 1 ≤ v2 = r ≤ v3 = s, i.e., machine 1 is the slowest one, machine 2 has medium speed and machine 3 is the fastest one. lemma 4.1: for the system s composed of 3 machines having speeds v1 = 1 ≤ v2 = r ≤ v3 = s the inequality opt ≤ w−wk 1+r holds true for any job k with the volume wk. proof suppose that there is such job of the volume wk assigned to machine i in the optimal profile l, that opt > w−wk 1+r . then all optimal delays on machines exceed w−wk 1+r . moreover, it is clear that λi(l) ≥ wk vi . hence, w = viλi(l) + vjλj(l) + vlλl(l) > wk + (vj + vl) w−wk 1+r ≥ wk + (1 + r)w−wk 1+r = w . lemma 4.2: for the system s composed of 3 machines having speeds v1 = 1 ≤ v2 = r ≤ v3 = s, if two jobs of volumes wk1 and wk2 are assigned to the same machine in the optimal profile, then opt ≤ w−wk1−wk2 1+r . proof assume for the sake of contradiction that opt > w−wk1−wk2 1+r and jobs of volumes wk1 and wk2 are assigned to machine i in the optimal profile. then all optimal delays on machines exceed w−wk1−wk2 1+r and λi(l) ≥ wk1+wk2 vi . then w = viλi(l) + vjλj(l) + vlλl(l) > wk1 + wk2 + (vj + vl) w−wk1−wk2 1+r ≥ wk1 + wk2 + (1 + r) w−wk1−wk2 1+r = w . theorem 4.1: for the system s composed of 3 machines having speeds v1 = 1 ≤ v2 = r ≤ v3 = s < 2 the price of anarchy does not exceed est(r, s) = min{ 2+s (1+r)(2−s) , 2 r(2−s)}. proof in the proof we consider cases with a certain number of jobs assigned to each of two the most loaded machines having the largest delay. for each case we show that the price of anarchy estimate presented in the theorem condition is true. suppose that l is an equilibrium profile copyright c© 2017 assa. adv syst sci appl (2017) 68 y.v. chirkova and sc(l) = λi(l), that is machine i has the smallest delay. explore different cases of an equilibrium l. 1. each of machines j and l receives one job. in the optimal profile these two jobs occupy at most two machines. thus, there is some machine k in the optimal profile, taking partially or wholly the equilibrium load of machine i and nothing else. that is opt ≤ viλi(l) vk ≤ sλi(l). by lemma 8.1 s ≤ est(r, s). 2. machine j receives nj ≥ 2 jobs, machine l receives nl = 1 job. by lemma 3.2 λj(l) ≤ 2vi 2vi−vjλi(l). by lemma 4.1 opt ≤ viλi(l)+ 2vivj 2vi−vj λi(l) 1+r = λi(l) 2v2i+vivj (1+r)(2vi−vj) . a) assume first that vi ≥ vj . then 2v2 i + vivj ≤ 3v2 i , since this expression increases by vj . also 2vi − vj ≥ vi, so long as this expression decreases by vj . then opt ≤ λi(l) 3vi 1+r ≤ λi(l) 3s 1+r ≤ λi(l)est(r, s) by lemma 8.2. b) suppose now that vi < vj . then by lemma 8.3 2v2i+vivj (1+r)(2vi−vj) < 2+s (1+r)(2−s) . explore now two cases. in the first case assume that vi = r and vj = s. here by lemma 8.4 2r2+rs (1+r)(2r−s) < 2 r(2−s) . in the second case vi = 1. by lemma 3.3 wk ≤ vk 2−vk λi(l) ≤ s 2−sλi(l) and vjλj(l)− wk ≤ s 2−sλi(l) for any job of volume wk assigned to machine j. if all jobs assigned to machine j in the profile l remain there in the optimal profile, two cases are possible. if a single job assigned to machine l in the equilibrium keeps its position in the optimal profile, then the load of machine i can only decrease with system’s transition from the equilibrium to the optimal profile. if this single job leaves machine l, then in the optimal profile machine l receives the load at most λi(l) coming from machine i. in both cases opt ≤ λi(l). if jobs move from machine j only to machine l with system’s transition from the equilibrium to the optimal profile, similar two cases are possible. if a single job assigned to machine l in the equilibrium remains there in the optimal profile, then the load of machine i can only decrease with transition to the optimal profile. then opt ≤ λi(l). if this single job leaves machine l, then in the optimal profile machine l can receive the load at most λi(l) + s 2−sλi(l) coming from machines i and j. then opt ≤ λi(l) 1+ s 2−s vl ≤ λi(l) 1+ s 2−s r = λi(l) 2 r(2−s) . if some jobs move from machine j to machine i with system’s transition from the equilibrium to the optimal profile, we obtain the same two cases. if a single job assigned to machine l in the equilibrium remains there in the optimal profile, then machine j receives the load at most λi(l) + s 2−sλi(l) consisting from the load remaining on j and possibly coming from i. then opt ≤ λi(l) 1+ s 2−s vj ≤ λi(l) 1+ s 2−s r = λi(l) 2 r(2−s) . if this single job leaves machine l, then in the optimal profile machine l can receive the load at most λi(l) + s 2−sλi(l) coming from machines i and j. 3. each of machines j and l receives exactly two jobs: nj = nl = 2. the total number of jobs assigned to machines j and l is four, the number of machines is three, so in the optimal profile at least two of these jobs (wk1 and wk2) become assigned to the same machine. then by lemma 4.2 opt ≤ w−wk1−wk2 1+r = viλi(l)+wk3+wk4 1+r , where wk3 wk4 remaining two jobs assigned to machines j and l. consider machine k ∈ {j, l}. if vi ≤ vk, then by lemma 3.3 the volume of any of jobs assigned to machine k does not exceed λi(l) vivk 2vi−vk . let now vi > vk. l is an equilibrium, therefore λk(l) ≤ λi(l) + w vi , where w is the smallest volume job assigned to machine k. thus, w ≥ viλk(l)− viλi(l). by lemma 3.2 copyright c© 2017 assa. adv syst sci appl (2017) price of anarchy for maximizing the minimum machine load 69 λk(l) ≤ λi(l) 2vi 2vi−vk ≤ 2λi(l), 2vi − vk ≥ vi, w ≤ λi(l). another job of the largest volume assigned to machine k has the volume vkλk(l)− w ≤ vkλk(l) = viλi(l)− (vi − vk)λk(l) ≤ viλi(l)− (vi − vk)λi(l) = vkλi(l) ≤ rλi(l) ≤ λi(l) rs 2r−s ≤ λi(l) s 2−s . a) let vi = s, then opt ≤ λi(l) s+2r 1+r ≤ λl(l) 3s 1+r ≤ λi(l)est(r, s) by lemma 8.2. b) let vi = r, then opt ≤ λi(l) r+2 rs 2r−s 1+r = λi(l) 2r2+rs (1+r)(2r−s) ≤ λi(l)est(r, s) by lemma 8.3. c) let vi = 1, then opt ≤ λi(l) r+2 s 2−s 1+r = λi(l) 2+s (1+r)(2−s) . from the other side so long as the number of machines equals three there are surely two machines α and β receiving at most one job from considered four jobs and, perhaps, some part of the load of machine i. then opt does not exceed the minimal delay over these machines: opt ≤ min α6=β {λi(l) 2 vα(2−s) , λi(l) 2 vβ(2−s)} ≤ min{λi(l) 2 1(2−s) , λi(l) 2 r(2−s)} = λi(l) 2 r(2−s) . 4. machines j and l receive the following job allocation: nj ≥ 2, nl ≥ 3. by lemma 3.2 λj(l) ≤ λi(l) 2vi 2vi−vj and λl(l) ≤ λi(l) 3vi 3vi−vl . thus in accordance with the estimate (3.1), opt ≤ λi(l) vi+ 2vivj 2vi−vj + 3vivl 3vi−vl 1+r+s ≤ λi(l)est(r, s) by lemma 8.5 and lemma 8.6. the next theorem is a special case of theorem 3.1 for the system of three machines. theorem 4.2: for the system s composed of 3 machines having speeds v1 = 1 ≤ v2 = r ≤ v3 = s < 2 is at least est(r, s) = min{ 2+s (1+r)(2−s) , 2 r(2−s)}. then we obtain from theorems 4.1 and 4.2 an exact value of the price of anarchy for the tree-machine system. theorem 4.3: for the system s composed of 3 machines having speeds v1 = 1 ≤ v2 = r ≤ v3 = s < 2 the price of anarchy exactly equals { 2+s (1+r)(2−s) for rs ≤ 2, 2 r(2−s) for rs > 2. the exact poa value allows establishing a possibility for poa increase under new machine inclusion into the system, i.e., in a situation resembling the braess paradox [10–13] when system’s power increasing leads to its performance characteristic degradation. the next statement illustrates that the price of anarchy increases or does not change under new machine inclusion into the system of two machines. theorem 4.4: for the system s composed of two machines having speeds 1 ≤ s the price of anarchy does not decrease with adding a new machine of speed 1 ≤ q < 2. proof 1. suppose that new machine has a speed of q ≤ s. if qs ≤ s2 < 2 then the price of anarchy does not decrease since 2+s (1+s)(2−s) ≤ 2+s (1+q)(2−s) . if s2 > 2 and qs ≤ 2 then that does not decrease due to 2 s(2−s) ≤ 2+s (1+s)(2−s) ≤ 2+s (1+q)(2−s) . consequently, we have the same if s2 > 2 and qs > 2 since 2 s(2−s) ≤ 2 q(2−s) . 2. suppose now that new machine is more powerful than existing in the system, s < q < 2. if qs ≤ 2, then s2 ≤ 2, and the price of anarchy does not decrease since 2+s (1+s)(2−s) ≤ 2+q (1+s)(2−q) . if qs > 2 s2 ≤ 2, then that does not decrease due to 2+s (1+s)(2−s) ≤ 2 s(2−s) ≤ 2 s(2−q) . also if qs > 2 and s2 > 2, we obtain the same because of 2 s(2−s) ≤ 2 s(2−q) . copyright c© 2017 assa. adv syst sci appl (2017) 70 y.v. chirkova 5. evaluating the price of anarchy in the previous section, we have derived an analytic expression for the price of anarchy in the three-machine model, where the fastest machine possesses a rather high speed. in what follows, we suggest a computing method for the price of anarchy on the example of the system of 3 machines which is similar to a corresponding method for the load balance game [14]. this method can be generalized to systems composed of more machines. but such generalization increases the number of linear programming problems to-be-solved and the number of associated variables and imposed constraints. particularly the n -machine model requires n ! linear programming problems each of which includes (2n − 1)n−1 subproblems with n2 variables. consider the following system of linear equations in the components of the vectors a = (a1, a2, a3), b = (b1, b2, b3), c = (c1, c2, c3). a1+a2+a3 vi ≤ b1+b2+b3+ min k=1,2,3:ak>0 ak vj a1+a2+a3 vi ≤ c1+c2+c3+ min k=1,2,3:ak>0 ak vl b1+b2+b3 vj ≤ c1+c2+c3+ min k=1,2,3:bk>0 bk vl a1+a2+a3 vi ≥ b1+b2+b3 vj ≥ c1+c2+c3 vl max k=1,2,3 ak > 0 max k=1,2,3 bk > 0 ak, bk, ck ≥ 0, k = 1, 2, 3. (5.3) this system describes a set of hyperplanes passing through the point (0, 0, 0, 0, 0, 0, 0, 0, 0) in the 9-dimensional space, and the solution set represents a domain in the space bounded by the hyperplanes. the above system is feasible, as far as, e.g., the triplet a1 = a2 = a3 = αsi, b1 = b2 = b3 = αsj and c1 = c2 = c3 = αsl makes its solution for all α > 0. furthermore, the solution set is unbounded, since α can be arbitrarily large. study the system s composed of 3 machines having speeds 1 ≤ r ≤ s and n players. let l indicate a nash equilibrium in the system s such that machine i is slowest in this profile having the greatest delay, machine j has a medium delay and machine l is fastest. suppose that in the equilibrium l machine i receives the total volume of jobs defined by∑ k=1,...,n:lk=i wk = a1 + a2 + a3 and the corresponding volumes for machines j and l equal∑ k=1,...,n:lk=j wk = b1 + b2 + b3 and ∑ k=1,...,n:lk=l wk = c1 + c2 + c3, respectively. the volume of jobs on each machine is somehow divided into three parts so that each component of the three-dimensional vectors a, b and c is either zero or positive and includes at least one job. lemma 5.1: let l be a nash equilibrium in the game involving three machines i, j and l and n players such that λi(l) ≥ λj(l) ≥ λl(l),∑ k=1,...,n:lk=i wk = a1 + a2 + a3,∑ k=1,...,n:lk=j wk = b1 + b2 + b3,∑ k=1,...,n:lk=l wk = c1 + c2 + c3. here for all k = 1, 2, 3 component ak equals zero or the volume of at least one job on machine i, component bk equals zero or the volume of at least one job on machine j, and component copyright c© 2017 assa. adv syst sci appl (2017) price of anarchy for maximizing the minimum machine load 71 ck equals zero or the volume of at least one job on machine l. then the set of the vectors a, b and c is the solution of the system (5.3). proof suppose that l represents a nash equilibrium and λi(l) ≥ λj(l) ≥ λl(l). the lemma 3.1 claims that λk(l) > 0, k = i, j, l. in this case, the following inequalities take place: ∑ k=1,...,n:lk=i wk vi ≤ ∑ k=1,...,n:lk=j wk+ min k=1,...,n:lk=i,wk>0 wk vj∑ k=1,...,n:lk=i wk vi ≤ ∑ k=1,...,n:lk=l wk+ min k=1,...,n:lk=i,wk>0 wk vl∑ k=1,...,n:lk=j wk vj ≤ ∑ k=1,...,n:lk=l wk+ min k=1,...,n:lk=j,wk>0 wk vl∑ k=1,...,n:lk=i wk vi ≥ ∑ k=1,...,n:lk=j wk vj ≥ ∑ k=1,...,n:lk=l wk vl . since each nonzero quantity ak (k = 1, 2, 3) equals the volume of at least one job on machine i, then we naturally have that min k:ak>0 ak ≥ min k:lk=i,wk>0 wk that provides satisfaction of the first and the second inequality of the system (5.3). similarly, min k:bk>0 bk ≥ min k:lk=j,wk>0 wk. this means satisfaction of the system (5.3). lemma 5.2: any solution of the system (5.3) defines a nash equilibrium l in the game involving the system s composed of 3 machines i, j and l and players whose jobs correspond to the nonzero components of the vectors a, b and c and the delays are sorted in the order λi(l) ≥ λj(l) ≥ λl(l). proof assume that the set of the vectors a, b and c gives the solution of the system (5.3). consider the game with 3 machines i, j and l. let each nonzero component of the vectors a, b and c specify the job volume of a regular player. consider a profile l such that the jobs of volumes ak > 0, bk > 0 and ck are assigned to machines i, j and l, respectively. so long as all inequalities (5.3) hold true, the profile l gives the desired nash equilibrium. the following result is immediate. theorem 5.1: any nash equilibrium l in the game involving the system s composed of 3 machines i, j and l and n players corresponds to a nash equilibrium l′ in the game involving the same system s and at most 9 players, where each machine receives no more than 3 jobs and the delays on all machines in l and l′ do coincide. proof consider a nash equilibrium l in the game with the system s of 3 machines and n players. number the machines so that λi(l) ≥ λj(l) ≥ λl(l). according to lemma 5.1, for any nash equilibrium in the game involving the system s and any number of players there exist a corresponding solution a, b, c of the system (5.3). by virtue of lemma 5.2, this solution determines a nash equilibrium l′ in the game with the system s such that the nonzero components of the vectors a, b and c specify the job volumes on machines i, j and l, respectively. by definition, the element sum of the vector a represents the load of machine i in a profile l. hence, delays on machine i coincide in both equilibria l and l′. similarly, for machines j and l the delays in the equilibrium l coincide with the corresponding delays in the equilibrium l′. copyright c© 2017 assa. adv syst sci appl (2017) 72 y.v. chirkova this theorem claims that it is sufficient to consider only games, where in an equilibrium each machine receives at most three jobs and the equilibrium solves the system (5.3). and the domain of the social payoff coincides with the value domain of games with an arbitrary number of players. imagine that the components of the vectors a, b and c are chosen as follows. in the optimal profile yielding the maximum social payoff, machines i, j and l receive the total volumes of jobs a1 + b1 + c1, a2 + b2 + c2 and a3 + b3 + c3, respectively, and the lowest delay can be on each of them. furthermore, by theorem 3.2, the volumes of jobs are assumed to be normalized so that in the optimal profile the maximum delay among all machines equals 1. in our case, this means that a1 + b1 + c1 ≥ vi, a2 + b2 + c2 ≥ vj, a3 + b3 + c3 ≥ vl, and at least one of these inequalities holds as an equality. lemma 5.3: solution of the linear programming problem lpp (vi, vj, vl) :  c1 + c2 + c3 → min (r1) a1+a2+a3 vi ≤ b1+b2+b3+ min k:ak>0 ak vj (r2) a1+a2+a3 vi ≤ c1+c2+c3+ min k:ak>0 ak vl (r3) b1+b2+b3 vj ≤ c1+c2+c3+ min k:bk>0 bk vl (r4) a1+a2+a3 vi ≥ b1+b2+b3 vj ≥ c1+c2+c3 vl (r5) max k=1,2,3 ak > 0 (r6) max k=1,2,3 bk > 0 (r7) ak, bk, ck ≥ 0, k = 1, 2, 3 (r8) a1 + b1 + c1 ≥ vi (r9) a2 + b2 + c2 ≥ vj (r10) a3 + b3 + c3 ≥ vl (5.4) with respect to the components of the vectors a, b and c provides the minimal social payoff in a nash equilibrium among all games, where in an equilibrium at most 3 jobs are assigned to each machine, i, j and l indicate the numbers of the machines in the descending order of their delays and the optimal social payoff makes up 1. proof due to lemma 5.2, any solution of inequalities (r1)− (r7) in the problem lpp (vi, vj, vl) defines an equilibrium in the game with 3 machines, where each machine receives at most 3 jobs and i, j, and l are the numbers of machines in the descending order of their delays the goal function in this game is bounded above only by the hyperplanes corresponding to inequalities (r8)− (r10). actually, inequalities (r1)− (r7) admit arbitrarily small nonnegative values of the goal function, including zero. therefore, the minimum is reached on one of the boundaries answering to the last three inequalities. this guarantees that one of them is satisfied as an equality, ergo the optimal payoff in the game corresponding to the solution of the problem lpp (vi, vj, vl) equals 1. consequently, exact poa evaluation for the system s composed of 3 machines calls for solving a series of linear programming methods lpp (vi, vj, vl) for all permutations (1, r, s). and the minimum solution among them yields the value of poa(s). in other words, it is possible to establish the following fact. copyright c© 2017 assa. adv syst sci appl (2017) price of anarchy for maximizing the minimum machine load 73 theorem 5.2: for the system s composed of 3 machines having speeds v1 = 1 ≤ v2 = r ≤ v3 = s < 2, the price of anarchy constitutes poa(s), which is the inverse value of 1 poa(s) = min (vi,vj ,vl) are permutations of (1,r,s){ c1+c2+c3 vl |a, b, c is a solution of lpp (vi, vj, vl) } , where lpp (vi, vj, vl) is the linear programming problem (5.4). proof according to lemma 5.3, the solution of the problem (5.4) gives the minimum social payoff in a nash equilibrium, where i, j and l are the numbers of the machines in the descending order of their delays, among all games such that in an equilibrium each machine receives at most 3 jobs and the optimal payoff equals 1. the minimum solution among the problems for all admissible permutations (1, r, s) as the values of (vi, vj, vl) provides the minimum social payoff in a nash equilibrium among all games, where in an equilibrium at most 3 jobs are assigned to each machine and the optimal payoff equal 1. by theorem 5.1, for any equilibrium in the game involving the system s of 3 machines and an arbitrary number of players, it is possible to construct a corresponding equilibrium in the game with the same machines and a set of at most 9 players, where each machine receives no more than 3 jobs and the social payoff coincides for both equilibria. thus, for poa evaluation it suffices to consider only games, where in an equilibrium each machine has at most 3 jobs. using theorem 3.2, we finally obtain that for poa evaluation it suffices to consider only games, where the social payoff in the optimal profile equal 1. 6. computing experiments to estimate the price of anarchy in the three-machine model, we have developed a program implementation of poa evaluation method presented in the previous section. this program allows to compare visually the theoretic poa value and its exact value constructed by solving a series of linear programming problems. moreover the program provides the possibility to see the poa dynamics for the machine number n > 3 where no any theoretic poa estimates are obtained. the parameters of the system s act as the options in the program; by assumption, the speed of machine 1 equals 1, whereas an exact value and a certain range are specified for the speeds of machines 2 and 3, respectively. in this case, users can study the poa dynamics under variations in the speed of one machine. the figures 6.1 and 6.2 present examples of poa estimates for different values of speeds of machine 2 and 3. at the fig. 6.1 the speed of machine 2 is r = 1.1, and the speed of machine 3 is s ∈ [r, 2). at the fig. 6.2 the speed of the fastest machine 3 is s = 1.7 and the speed of machine 2 is r ∈ [1, s]. here we can see that theoretical and computed values of poa coincide. the next example is more interesting. consider the system of four machines with speeds v1 = 1 ≤ v2 = q ≤ v3 = r ≤ v4 = s < 2. figures 6.3 and 6.4 present poa comparing with the lower poa estimate (3.2), which in fact is poa for the system composed of 3 machines with speeds 1 ≤ r ≤ s < 2. fig. 6.3 presents poa for the following cases. at the area a the value of q changes in the range [1, r], r = 1.3, s = 1.5. at the area b q = 1.3, the value of r changes in the range [q, s], s = 1.5. at the area c q = 1.3, r = 1.5, s value changes in the range [r, 2). in these cases poa value for four-machine system coincide with its lower estimate (3.2). fig. 6.4 presents the poa dynamics for those systems where machine speeds differ rather little, that is normalized speeds are closed to 1. in this case one can see that the poa value copyright c© 2017 assa. adv syst sci appl (2017) 74 y.v. chirkova 1 2 3 4 5 6 7 8 9 10 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 s fig. 6.1. poa for the system s, where r = 1.1, s ∈ [r, 2). 4 4.5 5 5.5 6 6.5 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 r fig. 6.2. poa for the system s, where s = 1.7, r ∈ [1, s]. 1 2 3 4 5 6 7 8 9 10 1 1.2 1.4 1.6 1.8 2 q r s a b c fig. 6.3. poa for the four-machine system s. presented by the thin curve exceeds its estimate (3.2) presented by the bold curve. both curves coincide under machine speeds increasing. at the area a the value of q changes in the range [1, r], r = 1.05, s = 1.1. at the area b q = 1.05, the value of r changes in the range [q, s], s = 1.1. at the area c q = 1.05, r = 1.1, the value of s changes in the range [r, 1.3). copyright c© 2017 assa. adv syst sci appl (2017) price of anarchy for maximizing the minimum machine load 75 1 1.5 2 2.5 3 1 1.05 1.1 1.15 1.2 1.25 1.3 q r s a b c fig. 6.4. poa for the four-machine system s with small speeds 7. conclusion this paper has explored the service system composed of n machines and n players and derived the lower estimate for the price of anarchy in the maximizing the minimum machine delay game (or cover game). the three-machine model has been analyzed in detail. here we have determined the exact value of the price of anarchy and showed that the poa increases or does not change under new machine inclusion into the system of two machines. also we have proposed a computing algorithm of the exact poa value. the algorithm can be generalized to systems with more machines, but this increases the number of linear programming problems to-be-solved and the number of associated variables and imposed constraints. and finally, we have implemented the algorithm as a program and conducted numerical experiments for comparing the obtained estimates of the poa with its exact value. the results of these experiments have demonstrated the correctness of the derived estimates. for the case of fourmachine system computing experiments demonstrate partial poa coinciding for three and four-machine systems, the analytic confirmation of this fact needs further investigations. acknowledgements this work was supported by the mathematical sciences branch of the russian academy of sciences and by the russian foundation for basic research (project no. 16-51-55006), and by the russian humanitarian scientific foundation (project no. 15-02-00352 a). 8. appendix this appendix contains supporting lemmas which are used in proofs in section 4. lemma 8.1: for any real r, s such that 1 ≤ r ≤ s < 2 we have s ≤ min{ 2+s (1+r)(2−s) , 2 r(2−s)}. proof 2 r(2−s) ≥ 2 s(2−s) ≥ s, since s3 − 2s2 + 2 = s(s− 1)2 + (2− s) > 0. 2+s (1+r)(2−s) ≥ 2+s (1+s)(2−s) ≥ s, since s3 − s2 − s+ 2 > s3 − 2s2 + 2 = s(s− 1)2 + (2− s) > 0. copyright c© 2017 assa. adv syst sci appl (2017) 76 y.v. chirkova lemma 8.2: for any real r, s such that 1 ≤ r ≤ s < 2 we have 3s 1+r ≤ min{ 2+s (1+r)(2−s) , 2 r(2−s)}. proof first, 3s ≤ 2+s 2−s , due to 3s2 − 5s+ 2 = (s− 1)(3s− 2) > 0. second, 3s 1+r ≤ 2 r(2−s) , since 6rs− 3rs2 − 2− 2r = r(6s− 3s2 − 2)− 2 = r(1− 3(s− 1)2)− 2 ≤ r − 2 < 0. lemma 8.3: for vi < vj , vi, vj ∈ {1, r, s}, where real r, s are such that 1 ≤ r ≤ s < 2, we have 2v2i+vivj 2vi−vj ≤ 2+s 2−s . proof if vi < vj , then 2v2i+vivj 2vi−vj decreases by vi and increases by vj , since 4v2 i − 4vivj − v2 j < 0 and vi(2vi − vj) + 2v2 i + vivj > 0. lemma 8.4: for any real r, s such that 1 ≤ r ≤ s < 2 we have 2r2+rs (1+r)(2r−s) < 2 r(2−s) . proof the inequality in the condition is equivalent to f(r, s) = −r2s2 − 2s(r3 − r2 − r − 1) + 4(r3 − r2 − r) < 0, chech if it holds true. show that f ′r(r, s) = −2rs2 + 2(2− s)(3r2 − 2r − 1) < 0, then f(r, s) ≤ f(1, s) = −s2 + 4s− 4 = −(2− s)2 < 0. for each fixed s the function f ′r(r, s) is a parabola with branches directed upwards. thus, its largest value is achieved on one of two ends of the interval r ∈ [1, s]. at the left end f ′r(1, s) = −2s2 < 0. at the right end f ′r(s, s) = −8s3 + 16s2 − 6s− 4 = −8s(s− 1)2 − 2(2− s) < 0. lemma 8.5: for vi 6= vj 6= vl, vi, vj, vl ∈ {1, r, s}, where real r, s are such that 1 ≤ r ≤ s < 2, we have f(vi, vj, vl) = vi + 2vivj 2vi−vj + 3vivl 3vi−vl ≤ 1 + 2s 2−s + 3s 3−s . proof the function f(vi, vj, vl) obviously increases by vj and vl, hence f(vi, vj, vl) ≤ vi + 2svi 2vi−s + 3svi 3vi−s = g(vi). show that g(vi) decreases by vi. the derivative g′vi(vi) = 1− 2s2 (2vi−s)2 − 3s2 (3vi−s)2 increases by vi and, therefore, does not exceed g′vi(s) = 1− 2− 3 4 < 0. then g(vi) ≤ g(1) = 1 + 2s 2−s + 3s 3−s . lemma 8.6: for any real r, s such that 1 ≤ r ≤ s < 2 we have 1+ 2s 2−s+ 3s 3−s 1+r+s ≤ min{ 2+s (1+r)(2−s) , 2 r(2−s)}. proof we show first that 1 + 2s 2−s + 3s 3−s ≤ (1+r+s)(2+s) (1+r)(2−s) . the right part of the inequality decreases by r, thus it suffices to show that 1 + 2s 2−s + 3s 3−s ≤ (1+2s)(2+s) (1+s)(2−s) . this is equivalent to s ≤ s2, that holds true under s ≥ 1. show now that 1 + 2s 2−s + 3s 3−s ≤ 2(1+r+s) r(2−s) . the right part of the inequality decreases by r, so it suffices to show that 1 + 2s 2−s + 3s 3−s ≤ 2(1+2s) s(2−s) . this inequality is equivalent to −4s3 + 11s2 − 4s− 6 = −s(2s− 3)2 − (2− s)(3− s) < 0. copyright c© 2017 assa. adv syst sci appl (2017) price of anarchy for maximizing the minimum machine load 77 references 1. epstein l., kleiman e., & van stee r. 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(1970) braess’s paradox of traffic flow. transportation research, 4, 391–394. 11. roughgarden t. & tardos é. (2002) how bad is selfish routing? journal of the acm, 49 (2), 236–259. 12. korilis y. a., lazar a. a. & orda a. (1999) avoiding the braess paradox in noncooperative networks. journal of apllied probability, 36 (1), 211–222. 13. mazalov, v. v. (2014) mathematical game theory and applications. new york, ny: wiley. 14. chirkova yu. v. (2015) price of anarchy in machine load balancing game. automation and remote control, 76 (10), 1849–1864. copyright c© 2017 assa. adv syst sci appl (2017) introduction the model the general case of n machines the case of 3 machines evaluating the price of anarchy computing experiments conclusion appendix advances in systems science and applications (2012) vol.12 no.3 220-237 creativity & biodiversity: towards a synergy-of-synergies john wood emeritus professor of design, goldsmiths, university of london, london se14 6nw abstract the paper argues that, in order to curb the current acceleration of species losses, a massive systemic change is needed. however, we are unlikely to achieve this with current methods. paradigm change includes the need to rethink the prevailing ‘realities’ and governances upon which the current paradigm depends. as designers are expressly trained to ‘think creatively’ this paper argues that their methods might usefully augment those used by scientists and politicians. however, as these design methodologies evolved as part of the old paradigm they, too, need to be re-designed. the paper explores some research into ‘metadesign’ a non-hierarchical, self-organizing framework for practice that helps designers to re-design their own mission and working vernacular, when required. in work that began in 2002, instead of seeing design as the process of creating products and services, we sought to cultivate ‘synergy’ at many levels. however, synergies tend to operate outside familiar boundaries of language and custom, which means that metadesigners must develop a metadiscourse, with which to explore each new predicament. this process encourages the radical ‘relanguaging’ of purposes, roles and self-identities. although, in the hierarchical sense, this makes these tasks ‘unmanageable’, politicians can facilitate local change by encouraging citizens to cultivate a ‘diversity-of-diversities’. this would act as an attractor to hitherto unforeseen orders, entities and configurations. it would offer an interoperable framework, within which communities could self-orchestrate a more complex, and emergent, ‘synergy-of-synergies’. keywords biomimicry, biodiversity, co-sustainment, creativity, ecomimicry, languaging, metadesign, paradigm, sympoiesis 1 the 6th great extinction ‘economies of scale’ is a term that emerged from many imperialistic enterprises over the last five or ten thousand years. while it probably started with the invention of agriculture, humans later applied it to the building of heavy structures and the management of large bureaucracies and manufacturing plants. but the discoveries that led to these achievements engendered a presumptuous mode of formalism that migrated to many other disciplines, including engineering, education, business and economics[1]. while it has facilitated rapid population growth and made many lifestyles more comfortable, it has also led to environmental pollution, habitat destruction and biodiversity contraction. hence, while politicians advances in systems science and applications (2012) vol.12 no.3 221 compete with one another to squeeze yet more ‘efficiency’ and ‘productivity’ out of a damaging and dysfunctional economic system, the rate of species extinctions is rising. some distinguished scientists, such as e.o. wilson, warn that, over the next few decades, this could rise to 10,000 times the ‘background’ rates. although an event of this magnitude would probably be fatal for human beings, it would not be without precedent. richard leakey’s term ‘6th extinction’(leakey & lewin, 1996) gives a 600 million year context to what is happening now. if these experts are right, our offspring could witness a halving of the number of living species by 2100, thus putting the survival of homo sapiens in grave doubt. this should not be surprising. from this perspective, systems of biodiversity seem less manageable, useful or attractive than monocultures. this approach has made a global force majeure more likely, probably in the form of violent climatic changes and agricultural failures. known remedies are likely to be too weak or inapplicable because they reflect, and uphold, the paradigm that caused the malady. in short, whereas nature is a system that operates on an ‘ecologies-of-scale’ basis, we have sought to manage it using an ‘economies-of-scale’ approach. 2 dysfunctional governance in order to address these problems, a more systemic and ecomimetic model is required. as systemic changes would need to be self-inclusively regulated, at virtually every level, this calls for a more joined-up, self-reflexive and innovative approach. in other words, we need to introduce a second-order cybernetic system[2] in place of the, largely first-order, system that is in place. rather than following existing approaches, our ‘metadesign’ methodology is intended to help us understand how to adopt an ‘ecology-of-scale’ approach[3-4]. most human societies are accustomed to a top-down system of government in which the management of major crises are seen, primarily, as the duty of politicians, scientists and civil servants. in the 1990’s, concerned by the apparent inability of ngos to influence radical changes in behaviour, donella meadows (1997) found that the most familiar measures were the least effective. this is probably because those most commonly used were bureaucratic (i.e. targets, subsidies, taxes, legislation) and, therefore, the least direct. again, the formalism implicit in these processes can be traced back to aristotelean and pythagorean logic[1]. while many regard these as axioms of truth, their dependability is largely epistemological. by contrast, living systems are less predictable because they are a great deal more complex and distributed. even the setting, and attaining, of targets can be a dispiriting process, especially when they are politically ambitious or controversial. if they are set lower than what is required, they may not easily be reached within the agreed timescale. moreover, when targets are missed (this is not uncommon), the morale of the parties involved may be reduced, thus creating a 222 john wood:creativity & biodiversity: towards a synergy-of-synergies positive feedback loop that can trigger, or amplify, a shared sense of futility in the mission. in seeking to think beyond this kind of sub-optimal system meadows looked for what she called the ‘levers of change’[5]. she noted that it is important to identify, and to reframe, the agreed purpose of a given paradigm in order to change it. this lesson has yet to be adequately acknowledged and applied by governments, who persist in applying a targets-based approach. however, in 2010 the nagoya world biodiversity summit sought to move beyond the customary agenda of deepening scientific knowledge and raising public awareness. it also agreed to assign large regions of land and the sea to act as regions of wilderness, in order to give endangered species a chance to recover. unfortunately, it soon became clear that the plan would fail[6] because the natural rate of species replenishment in the wild is too low to reverse the current rate of losses. 3 the role of science it seems unlikely that applying current, hierarchical, top-down remedies for biodiversity losses will be effective within the timescale we have. instead, we might start by differentiating between, then merging, the functions of scientists, politicians and designers, as their respective professions reflect complementary approaches. from the designer’s perspective, for example, governments tend to deal with major problems by listening to ‘scientific’ advice, then trading expediencies in a way that maintains the political status quo. whereas designers are trained to deliver desirable, pragmatic solutions within a short timescale, scientists are trained to regard evidence-based knowledge and reasoning as vital prerequisites to prudent practice. dr. brundtland, former director-general of the world health organization famously likened the problem of biodiversity to a major fire: “the library of life is burning and we do not even know the titles of the books”[7]. perhaps it is because i am more of a designer than a scientist i find this to be a peculiar analogy. if i am watching the library burn, my instinct is to put out the flames, not to lament my lack of knowledge. however, a scientist would probably argue that finding sensible remedies cannot work unless we can differentiate between species that look identical, yet are at very different levels of risk. more than three centuries after his birth, carl linnaeus (1707-1778) is still highly praised for simplifying the elaborate nomenclature that previous scholars used to differentiate between species. he is also celebrated for his clear taxonomy of classification. nonetheless, while his efforts may have made it easier to document all life on earth, the task has proved much too big to accomplish. recent studies estimate that 86% of all land species and 91% of all sea species are unclassified or undiscovered[8]. advances in systems science and applications (2012) vol.12 no.3 223 4 can scientists work more closely with designers? much of the problem lies not with science, but with the way that scientific knowledge tends to be managed. in the fishing industry, for example, just as politicians may be tempted to exaggerate figures when setting catch quotas, so fisheries are equally tempted to exceed these quotas. this led to the scandalous process of ‘discards’, trawled fish that exceed the official quotas[9]. with all due respect to scientists, we cannot wait for ‘robust data’, evidence-based reasoning, or so-called ‘rigorous’ analysis of the problem in order to address it. in other words, we cannot expect to name and classify all of the relevant species. therefore, while we know that a ban on bottom trawlers would probably save some species of marine life, it is difficult to know which ones. on the other hand, it is enough to know that some ‘fish fingers’ (or ‘fish sticks’) may contain a proportion of unnamed and/or unknown species. this is closer to design practices, in which designers often have to make formative decisions before there is clear, evidence-based knowledge. moreover, it would be politically difficult to do so, unless alternative technologies and business models can be offered. this is where design thinking is useful, especially if it can be brought into synergy with the gathering of scientific truths, and the making of political decisions. these hybrid systems of governance would need to embrace what i have called ‘auspicious reasoning’[1], which is contingent and outcome-focused, rather than knowledge-based, or truth-focused. a well-known example of ‘inauspicious reasoning’ is the ‘tragedy of the commons’[10], which reminds us that humans often choose actions with selfish and, or, short-term gratification, rather than settling for small, immediate reductions in well-being that will ensure greater long-term safety, or abundance, for all. 5 the importance of synergy since darwin, the life sciences have offered a worldview in which living systems (appear to) defy entropy and create emergent forms of abundance. unfortunately, the dominant paradigms of governance appear to reflect a less optimistic view that emerged from physics, inspired by the exploitative logic of mining. both traditions depict the world as a materials and energy system that is ‘running out’, or ‘running down’. this unhelpful mindset can also be found in school curricula, and in the non-ecological terms used by most environmentalists (e.g. in the notions of ‘resources’, ‘entropy’ and ‘sustainability’). there are very few ‘resources’ that are beneficial when consumed by themselves. indeed, peter corning claims that synergy is the cause behind the evolution of complexity in living systems[11]. buckminster fuller was probably the first person to show that, by designing for synergy, rather than for better products or services, designers would achieve more with less. he defines it as “...the behaviour of whole systems unpredicted by the separately observed behaviours of their parts taken separately”[12]. in a simple 224 john wood:creativity & biodiversity: towards a synergy-of-synergies example, combining nickel, iron and manganese to makes stainless steel, which is up to 35% stronger than any of its constituents. ‘synergy’ also became a key term within our metadesign research. this is partly because it applies equally to systems, whether they are seen as animate, or inanimate. but, despite its invaluable benefits, the task of seeking, harvesting and harnessing synergies may raise complex epistemological, technological, or managerial difficulties. this is probably because it appears in such a diverse and, often, elusive forms. for example, spontaneous humour, or a spiritually uplifting experience, are no less synergistic than the combinatorial benefits of stainless steel. in practice, many are hard to disentangle from the processes with which one thinks, experiences, shares and acts. fig.1 laufrad campa vento bicycle wheel fig.2 millstone 6 creating synergies-of-synergies the bicycle is said to be the most energy-efficient mode of transport on the planet. we can learn a great deal from how its vast array of subtle alignments (i.e. a ‘synergy-of-synergies’) is configured. as it is easier to analyse some of its simpler synergies, we will use these as a basis from which to reflect on more complex features. fig.1 shows a modern design whose metallurgical synergies combine with synergies of form. the use of stainless steel (see previous paragraph) for making wheel spokes ensures that they are considerably lighter than a more solid design. this is because stainless steel is far stronger when under tension, than in compression. these, and a totality of other synergies, make it capable of supporting a load that is up to seven hundred times its own weight. as this example may sound like the traditional ‘economies-of-scale’ logic, how might we advances in systems science and applications (2012) vol.12 no.3 225 retrieve ‘ecologies-of-scale’? here, we should not forget that, like any ecosystem (or ‘paradigm’) the bicycle is part of a much larger system that includes, in this case, a suitable cyclist, a transport culture and a network of flat roads. 7 creating synergies-of-synergies let us look at a more qualitative mode of synergy. for example, when brought together in the appropriate way, the poisons, chlorine and sodium, become nutritionally useful and palatable in the form of salt. unfortunately, simple, ‘freestanding’ synergies are less common than complex, messy ones, because most synergies have more than one property (e.g. most stainless steels last far longer than ordinary steels because they do not rust). hence, we probably fail to notice new synergies[13] even though they may be abundant and ubiquitous. in many cases they operate outside what can easily be described. this underlines the subjective nature of synergy, and its dependence on our creative ability to imagine conditions beyond the current affordances of ones language. this is similar to the challenge of supporting hitherto unnoticed, unnamed or untaxonomised species. however, it also suggests that we can adapt to the universe in a myriad of unforeseen ways. how might we set about creating spiritual states, or even miracles[14]? the first step is almost an act of faith, because observation is an emotional, as well as a rational faculty. this can begin with the logical assertion that ‘unthinkable’ things may be attainable, once we begin to notice them. after all, until quite recently, ships made of iron, flying machines, and quantum logic were all ‘unthinkable’ or ‘impossible’. 8 learning from synergistic complexity for example, by exploring synergies, a good cook can combine ordinary, local, inexpensive cooking ingredients and turn them into an extraordinary experience for her, or his, guests. this example includes very many types of synergy from the way heat melts cheese, or butter (i.e. simple physics) to the more complex nutritional benefits of combining, say, broccoli and sprouts[15]. the way that people co-create a unique, indefinable atmosphere together is even more complex and intangible. one thing we can learn from food synergies is the way that complementary flavours work. some molecular gastronomists predict that any two foods with a common (molecular) ingredient will taste good when combined. also, some apparently incompatible substances, such as oil and water can be made to mix when an emulsifier is added. this works if the emulsifier’s molecular chain has a water-compatible atom at one end and an oil-compatible atom at the other. these lessons from food technology can be applied, via metaphors, to other systems. for example, ‘immiscible’ ingredients, such as oil and water, can be combined when an emulsifier is added. the same principle could be applied 226 john wood:creativity & biodiversity: towards a synergy-of-synergies to teams containing antagonistic members. hence, differences might be resolved (‘emulsified’) via a third party who seems benign to each of the antagonists. ecosystems can be used as the model for a new order of innovation. for one reason, their complexity and interdependency that makes them resilient within a community context. we know, for example, that ‘keystone species’ are critical to the survival of other species, so we might set about designing ‘keystone synergies’ that would engender, and sustain, subordinate synergies within a given environment or context(see fig.2). another way to explain this process is by thinking of it as a singular act that behaves as a ‘manifold innovation’ that brings many benefits to many stakeholders. a good historical example is the millstone, which attracted many subsequent social, cultural, nutritional and technological synergies. 9 continuously re-languaging nature while we might assume, hypothetically speaking, that ecological diversity can be enhanced, it may be hard to see how this can be achieved. this is partly because the ‘language’ we use to enframe nature is always less than adequate to the situation. systems theory helped us to recognise the important interdependencies between meanings and actions in ecosystems. however, this relationship embodies a paradox. although language shapes our ‘realities’ and guides our actions[16-17], nature is ineffable and emergent, and will therefore defy clear and enduring definition[18]. hence, while new ideas may only become popularly understood when suitable terminology is found, this may have a limited era of usefulness. sometimes, seemingly simple keywords prove too difficult, or unpopular, to be accepted. for example, the rather narrow term ‘ecological footprint’[19] soon gave way to the (even narrower) idea of ‘carbon footprint’[19], thus giving the false impression that the complex phenomena that cause climate change can be curbed by simple economic transactions that ‘offset’ carbon emissions, or by geoengineering solutions that aim to ‘remove’ it from the environment. arguably, in living systems, the task of creating, modulating and switching meanings cannot be managed successfully in a top-down, external, or hierarchical way, especially when the hierarchy grows and becomes many-layered. this follows from ross ashby’s law of requisite variety, which warns of the dangers of arbitrarily reducing the number of variables within subsystems[20]. maturana and varela offer a useful ecomimetic account of living systems, by explaining that they balance the ‘meaning’ of their internal and their external identities[21]. 10 re-languaging sustainability the above description suggests that the survival of a given species is, at least, explicable within systemic terms that transcend ethics. no organism deserves advances in systems science and applications (2012) vol.12 no.3 227 longevity because of an a priori moral right or because it enjoys a privileged status. this is because there a seldom a simple, intuitive logic of cause-and-effect, or a fixed hierarchy of relations. here, it is the ambiguity of the popular term ‘sustainable’[22] that has rendered it unhelpful. the verb ‘to sustain’ denotes a continuation over time, but it may also carry the non-temporal meaning of something ‘holding together’. using the verb ‘to sustain’ transitively and temporally (e.g. ‘b’ sustains the continued existence of ‘a’) would clarify the direction of causation. however, merely saying that something is ‘sustainable’ does not make clear who ‘sustains’ what. in any case, reciprocal (2-way) relationships are prevalent in ecosystems, even though this may seem counterintuitive. the death of a predator species, for example, may seem to ‘cause’ the death of its prey. i therefore use the term ‘co-sustainment’ instead of ‘sustainability’, as it acknowledges the mutual dependency of all living systems[23]. evolution adjusts relationships all the time, which means that everything is always changing. in order for a species to endure i.e. to remain a viable part of the whole it must always be ready to adapt to its changing habitat. this means adjusting our actions and identities, rather than trying to re-design nature in accordance with our expectations. inviting designers to work at this level would also mean re-designing the design paradigm that is part of the problem. 11 changing the change in practical terms, we have found it helpful to change other key terms that were being applied in what we saw as a dubious, or unhelpful way. for example, we stopped using certain terms (e.g. ‘creativity’, ‘sustainability’ and ‘biomimicry’) and designed more useful alternatives (i.e. ‘sympoiesis’, ‘co-sustainment’ and ‘ecomimicry’). the idea of ‘languaging’ change is not only a theory about the seamlessness of interplay between action and thought. it is, also, a practical way for design teams to co-create their ‘survival’ as whole, living systems. however, this active, radical, consensual reframing of meaning means that, of necessity, the participant’s perceived reality will also change (we are aware that this also applies to us, as ‘metadesigners’). we have, therefore, found it useful to follow maturana and varela’s practice of using the word ‘language’ as a verb. as this change would be both shared and self-reflexive, our new, shared self-identity cannot be separated from our behavioural culture. this pluralisation of identity, led us to coin the term ‘sympoiesis’[24], based on the idea of ‘autopoiesis’[25], in which living systems ‘create themselves’. this is also reminiscent of freud’s early notion of ‘polymorphous perversity’[26], which describes how an infant steers its habits and identities by making situated choices based on its personal experience of gratification or displeasure. these habits may later be guided by the values and responses of the society, which are also mediated by the framework of language 228 john wood:creativity & biodiversity: towards a synergy-of-synergies within which the infant will, in theory, be free to co-create, as a member of that society. it may also be that there insufficient creative optimism, or opportunism, within the metadesign team. this also has useful implications for the future of democracy. indeed, the work of donella meadows (1997) implies that the collective imagination is more powerful than the counting of individual votes. 12 how can we do it better? unless we decide to see our fate as a largely self-inflicted (and well-deserved) inevitability, we need some decisive action that will have the requisite effect on species decline. i have been arguing for many years that design professionals might be a helpful complement to the range of experts currently consulted by the most influential government agencies and corporations. this would mean commissioning designers to be servants of humanity, rather than working for the profit of a few. unfortunately, most are trained to be specialist mercenaries who apply their ‘creative’ visual intelligence to service the insatiable demands of a consumer-led economy. while designers have shown themselves to be indispensible to the commercial system, they are less influential for their thinking. western thought has tended to value logic and truth so many of the major professions are educated within a learning culture that tends to see knowledge as a set of truthclaims, backed-up by evidence. designers are different because they are trained to immerse themselves in the pleasurable immediacy of forms and images. but they also think in terms of anticipatory affordances (i.e. the optimistically hypothetical, the creatively provisional and the contingently possible). these not only include skills cultivated in schools and universities, but they also reflect a set of capabilities that evolved over the last million years, or so[27]. at the end of the 1960s, the scientist, herbert simon and the designer victor papanek[28] both made the claim that everyone is a designer. in simon’s version, we ‘design’ whenever we create a particular course of action that is likely to change an existing situation into a preferred one[29]. more recently, the sociologist, zigmunt bauman pushed the idea further by arguing that the similarity between design and management has existed since the late 1880s, when governments invited designers and managers to bring science and technology into everyday life[30]. recently, tim brown[31] has popularised what herbert simon[29] called ‘design thinking’, and what nigel cross called ‘designerly ways of knowing’[32]. before this, many non-designers had not fully appreciated the distinctiveness of these approaches. 13 design for survival according to john thackara (2005) the decisions we take at the design stage are the ‘cause’ of 80% of the environmental impact of the products. yet, historically, although designers are encouraged to explore alternative futures in a creative advances in systems science and applications (2012) vol.12 no.3 229 way, their status, as a ‘minor’, or relatively junior, profession has given them few opportunities to work at a more strategic level. this may help to explain why we are stuck in a paradigm of bad habits that perpetuates confused thinking. to be able to tackle highly complex problems, such as biodiversity losses, designers would need to rethink their traditional role as catalysts of the consumer society. rather than working as specialists, they would need to cultivate a more selfreflexive and comprehensive approach that enables them to intervene at many simultaneous points within a whole system. they might need to learn how to create synergies for all, rather than new products and services. the pivotal role of designers within the fashion industry highlights this, in quite an ironic way. in europe, the habits of the fashion industry cause around 10% of all the waste produced, yet the role assigned to designers makes them seem more like the problem than the solution. this is a systemic problem that cannot be blamed on one particular group or method. as mathilda tham, the trends forecaster, put it, ‘fashion thrives on innovation but resists change’[33]. in other words, although fashion designers may be capable of re-thinking long-term business futures, they are trained to ignore these possibilities, and to focus onto next season’s stylefutures. but, if designers were trained to envision long-term business futures, they would be able to deliver waste-free rewards for corporations. the ‘cradleto-cradle’ movement is an excellent example of this approach. by creating an adaptive, circular economy, rather than one that looks for unlimited innovation and growth, it will be possible to design complex synergies of combination, rather than offering short-life products that cost additional money for their dispose. fig.3 some of the co-sustaining paradigms that maintain the status quo. 230 john wood:creativity & biodiversity: towards a synergy-of-synergies 14 re-designing paradigms what is stopping the implementation of systems that will produce longer-term outcomes? to a great extent, the obstacles to change are caused by the poor economic thinking behind technological development. if designers are not paid to look beyond their short-term cycle of design, production and re-design they cannot take full responsibility for what happens in the longer-term. one way to reconcile design thinking with ecological thinking is to invent a systemic discourse within which complex systems are seen as ‘paradigms’[34], because they resemble ecosystems. they also contain both animate, and inanimate agencies, similar to james lovelock’s gaia theory[35]. and these paradigms sustain themselves by attracting new elements that serve to support them. while designers tend to be taught how to (re)design existing products, we also need more radical solutions at the paradigmatic level. for example, although architects are trained to come up with a variety of iterations to high-rise offices, few are able to re-think the underlying paradigm of concrete, steel and glass, even though these materials have an inordinately large carbon footprint. the high-rise office is ubiquitous to london, new york or beijing. like all paradigms, it resists change because it is sustained by an interconnected array of subsidiary systems (also paradigms, or quasi-species), such as insurance protocols, commercial habits, technological habits, economic assumptions each of which appears to depend on it for their own survival[36]. 15 design for biodiversity in order to encourage designers to achieve better levels of ‘co-sustainment’, we would need to develop a more ‘ecomimetic’ approach[37]. ecomimicry is significantly different from what is known as ‘biomimicry’[38]. although ‘biomimicry’ was defined within an admirably broad philosophy, the way that designers tend to apply it seems disappointingly narrow. perhaps this reflects the prevailing business mindset, in which projects are seldom seen as part of a circular[39], or long-term economic vision. even today, it is hard to find examples of biomimetic innovation that do much more than behave as discrete technological ‘fixes’, gadgets or products. instead of seeing the designer’s role as the fixer of links in a chain of transactions, ‘ecomimetic designers’ would be expected to design the “conditions” that support interdependencies within whole systems. this is not a simple, or trivial shift. for example, the massive complexity and ineffability of these interdependencies mean that we can no longer expect to work predictively, as we might have done with, say, product design. many other aspects of design, as we know it, will also need to change, therefore we will refer to the new approach as metadesign. one of the radical changes we need to make is to re-purpose what we understand as ‘creativity’ and acquire a wiser understanding advances in systems science and applications (2012) vol.12 no.3 231 of relations, rather than separate ‘things’. 16 re-inventing creativity integrating hitherto disconnected methodologies is useful, but it is not enough. we also need to modify the assumptions that motivate designers, especially where ‘creativity’ is concerned. this means analysing and, where necessary, challenging the received assumptions behind this term. in the last decade or two, we have seen particular models of creativity as tools for regenerating urban communities [40], for stimulating economic growth, or for transforming businesses to make the economy more ‘efficient’[41]. here, ‘creatives’, such as designers, have come to be seen as part of a dependable toolbox that can bring success to prestigious real estate deals, national re-branding exercises, or other tourist attractions, such as the hosting of an olympic games events. how might we think of ‘creativity’ if the main aim is to generate wellbeing and prosperity by achieving optimum biodiversity? i believe we must develop creativity’s capacity to help us to ‘adapt’ better to our changing habitat. this may include applying it, less to the defeat of existing, or rival, plans but to look for new ‘synergies’ that will deliver additional benefits that emerge from combining existing things. 17 the problem of ‘creative genius’ one of the problems with the modern idea of ‘creativity’ is that it became part of the culture of competitive advantage, rather than a way to adapt to new situations. often, one finds ecological terminology applied, shamelessly, to an economic context whose autonomy is seldom framed within the bigger picture. and it is increasingly associated with destructive terms, such as ‘disruptive innovation’, or ‘disruptive technologies’. these tendencies remind one of the aggressive, pitiless language of francis bacon, whose crude scientific methodologies are perpetuated in some global corporations, whose products continue to threaten the survival of certain key species. the same, swaggering, solipsistic stance can also be noted from books on creativity and innovation, such as ‘ignore everybody’ or ‘relentless innovation’. redefining creativity is not a simple step. the destructive and egoistic models that we inherited from the romantics remain part of the popular culture. after the enlightenment it came to valorize individual power, originality, and to be blunt arrogance. david hume (1711-1776) and others, such as arthur schopenhauer (1788-1860), strongly admired the notion of ‘genius’ as someone who is so extraordinarily self-styled and unfathomable that he would struggle to adapt to the ‘normal’ world around him. this also echoes nietzsche’s idea of ‘der übermensch’, which developed in the 1880’s. this theory asserted that homo sapiens has the potential to create a new order, provided there is a sufficient will to power, and a readiness to reject societal ideals and moral codes. 232 john wood:creativity & biodiversity: towards a synergy-of-synergies in the 20th century, inspired by psychoanalysis and by concepts such as ‘positive self-regard’, ‘presentation of self’, and ‘self-actualisation’ creativity came to be associated with the right to personal self-expression. 18 unreasonable men in a sense, then, the modern sense of the word ‘creativity’ may mean a cynical, or arrogant refusal to adapt to anything. perhaps this is what the architect, frank gehry, meant when he said, in 2005, “i don’t do context”. today, the invitation to ‘be unreasonable’, or to ‘think different’ (presumably, a fusion of ‘be different’ and ‘think differently’) has become familiar to today’s consumers. indeed, the apple corporation have used it many times to advertise their computers. however, sustaining and enhancing biodiversity will require us to reflect more deeply on the shareable and distributed nature of creativity and how can help us adapt to our habitat. our stridently humanistic idea of creative may be traced back several thousand years to horace, who inspired kant’s famous phrase ‘dare to know’ (1784). and, while it had encouraged a long development of careful reason and inquiry, this was not what we would understand as ‘creativity’, as we understand it today. by the 17th century the idea of ‘daring to know’ had inspired john locke’s radical insight (1689) that “the mind can furnish the understanding with ideas”. while the notion of ‘creative thinking’ may now be commonplace for agnostics in the 21st century, it had remained virtually unthinkable to philosophers before hume and locke. the idea that an individual can choose to think what s/he wants to, later acquired a powerful framework of thinking, with coleridge’s term ‘self-consciousness’, as celebrated by a series of narcissistic ‘genius’ figures who dominated the post-romantic era in art, literature and music. 19 can we find creativity in nature? in 1877, ten years before the first sherlock holmes publications made it popular, charles peirce announced his idea of ‘abductive reasoning’. this turned deductive argumentation on its head, by enabling thinkers to begin the process with what might, hitherto, have seemed like a conclusion. while we may see this type of logic as a characteristically human, or even modern, mode of thought, gregory bateson has suggested that abductive reasoning is part the natural order: “all thought would be totally impossible in a universe in which abduction was not expectable......”[42]. he believed that evolution is responsible for the parallels between the way we think, and the way nature works. arthur koestler’s ‘bisociation’ method is interesting in this respect, as it is designed to elicit a new idea when two things are combined[43]. this resembles sexual reproduction in that different ‘parent’ ideas are brought together to create a new hybrid outcome. when the two ideas (or creatures) creatures, ‘a’ and ‘b’, are combined we may advances in systems science and applications (2012) vol.12 no.3 233 find that we also have a third idea ‘c’. abductive reasoning is a way to ‘reverse engineer’ evolutionary logic. instead of taking the 2 parents and seeing what type of child we will get, we start with the child (‘c’) and look for an ‘a’ and a ‘b’ that might have been its parents. one lesson we can learn from evolution is that creativity is only important when it works to produce new opportunities. and one reason why the romantic era created so much interest and excitement is because it was less concerned with the strict languages of scientific ‘truth’ and more to do with enriching the semantic discourse that may, subsequently, become applicable to science. however, unlike scientific ‘truths’, artistic propositions are usually hard to fathom, without reflecting upon their aesthetic status. 20 the role of aesthetics it will be important to harness aesthetic criteria when changing the paradigm, because aesthetics acts as an interface between thoughts and actions. aesthetic discourse can change consensus because we share and modify our perceptions when we discuss personal tastes. arguably, beauty tells us what is good, or, rather, what used to be good. aesthetics is not only a philosophical discourse, it is also a multi-dimensional field of awareness, so it cannot be quantified using a small number of dimensions. claims to beauty exist as combinations of elements, or as patterns of sensory awareness, that are deemed to work together. the fact that we have fads in food tastes, or fashions in clothing, is evidence that aesthetic judgements are seldom fixed by a rigid biological ‘need’, or aesthetic code. however, it seems likely that certain predilections and habits are strongly ‘wired’ into the human body, especially when they remind it of tastes, flavours, or colours that have proved beneficial to us over many hundreds, or thousands, of years. this is useful to designers (and to artists), because creating new tastes and aesthetic forms will change appetites & habits. a good example of this is the pleasure that we take when they see varieties of the rosaceae (rose) plant family. humans have enjoyed products of the rose family for many thousands of years. they have give us many varieties of nutritious fruits, such as apples, apricots, plums, cherries, peaches, pears, raspberries, and strawberries. during this time we have made them bigger and sweeter, thus creating a strong level of co-sustainment between people and the plants themselves. in my opinion it is not a meaningless coincidence that the largest and most successful brand in the world is an ‘apple’. 21 re-languaging new realities in previous papers i have argued that designers have, traditionally, overlooked the importance of language in ‘design thinking’. this is not meant to imply that designers should try to think in a more ‘critical’ way, but that they could be more 234 john wood:creativity & biodiversity: towards a synergy-of-synergies creative with the terms that we use to describe things, experiences, values and ideas. how many colours are there in a rainbow? even though science tells us that it contains countless wavelengths of light, most people answer with a small number (usually 7) they learned at school. but how does this answer affect our perceptions? our reality is lived out in metaphors, adjectives, images and categories. these, in turn, shape our beliefs, actions and assumptions. if one language has more words for flavours, and for colours, than another, it seems likely that the speaker of this language will have a bigger horizon of experience. the pioneer eco-semiotician, jakob von uexküll (1864-1944), used the term ‘umwelt’ to describe the phenomenological ‘reality’ of different creatures. this is an important idea, because no two creatures experience the world in quite the same way. even though symbiotic relationships may develop within the affordances of an existing language system (i.e. what maturana & varela call ‘structural coupling’) there may be a very tenuous overlap between the two sides of the ‘conversation’ for example, one may be able to see but not to hear, the other may have the opposite capabilities. this suggests that, by augmenting the naming system we have, we can expand our umwelt, perhaps in order to re-map vested interests, empathetically, in conjunction with other species. hypothetically speaking, redefining the agreed purpose of a paradigm should make it easy to change[5]. however, social inertial is co-sustained by the language and customs of the old paradigm, and these tend to mask opportunities, making them seem difficult, unthinkable or impossible. crudely speaking, the answer is to create new words that would afford new concepts and, perhaps, serve to facilitate new ecological paradigms that enable the homo sapiens to survive a little 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(1967), the ghost in the machine, penguin: london (reprint 1990). corresponding author john wood can be contacted at: maxripple@gmail.com advances in systems science and application (2015) vol.15 no.4 316-325 computer investigation of a game theoretic model of social partnership in the system of continuing education vladimir k. dyachenko, guennady a. ougolnitsky and larissa v. tarasenko southern federal university, russia. abstract a game-theoretic model of social partnership in the system of continuing education is proposed and investigated in the simulation mode. results of the model identification and investigation based on simulation modeling are considered. a comparative analysis of egoistic and cooperative approaches to the social partnership is conducted. keywords social partnership; continuing education; simulation modeling; identification; difference games. 1 introduction a system of higher education should be flexible enough to prepare qualified and competitive specialists capable to improve their knowledge and skills in the changing environment. one of the important mechanisms providing the solution of this problem is social partnership that allows for the cooperation of employers, universities, and students. social partnership relations are very important in the system of higher education [1,2]. an overview of the papers concerned with mathematical modeling of the problems of social partnership is proposed in some previous studies of the authors [3]. social partnership in continuing education is a specific system of joint activities of the education system agents characterized by trust, common objectives and values, and providing highly qualified, competitive, and mobile specialists for the labor market. the main research hypothesis is that social partnership permits to increase a level of professional competence of the specialists. it seems natural to use the formalism of differential games [4] for description of social partnership relationships. due to the high complexity of the differential game model the techniques of simulation modeling [5] are applied for its investigation in the difference form. the paper develops authors approach exposed in [3]. in the section 2 a general description of the model is given and the new moments in comparison with the previous studies [3] are shown. in the section 3 the model identification is described. in the section 4 planning and implementation of the computer simulation experiments are discussed. section 5 deals with processing and analysis of the modeling results. section 6 concludes. advances in systems science and application (2015) vol.15 no.4 317 2 a general description of the model in the proposed model social partnership relations among employers, students, and university are considered. the payoff functions are as follows: jp = t∑ t=0 gp (up (t), ub(t), uc(t), x(t) → max, up (t) ∈ up ; jb = t∑ t=0 gb(up (t), ub(t), uc(t), x(t) → max, ub(t) ∈ ub; jc = t∑ t=0 gc(up (t), ub(t), uc(t), x(t) → max, uc(t) ∈ uc ; where n = {p, b, c} is a set of players, namely: p – employer; b – university; c – student. the systems dynamics are given by the equation x(t+ 1) = x(t) + f(x(t), up (t), ub(t), uc(t)), x(0) = x0; here up (t), ub(t), uc(t) are strategies of the players describing their efforts directed to the development of social partnership relations; up , ub, uc – domains of feasible strategies; jp , jb, jc – payoff functionals of the players; gp , gb, gc – instantaneous payoff functions; p= {p1, ... ,pr} – a finite set of employers; b ={b1, ... ,bm} – a finite set of universities; c = {c1, ... ,cs} – a finite set of students; t = 4 (period = 5 years). the strategies determine a share of the annual budget assigned by a player to the needs of continuing professional education (cpe): upi(t) – a share of the annual budget assigned to cpe programs by an employer, upi = [0,rp ]; ucj (t) – a share of the annual budget assigned to cpe programs by a student, ucj = [0,rc ]; ub(t) – a share of the annual budget assigned to cpe programs by the university, ub = [0,rb]; rp , rc , rb – the annual budgets of the players. the main contribution of this paper in comparison with [3] is that the payoff functionals are taken in the form jp = t∑ t=0 e−ξt[gp (rp − up (t)) +gp (x(t))] → max, up (t) ∈ up ; jb = t∑ t=0 e−ξt[gb(rb − ub(t)) +gb(x(t))] → max, ub(t) ∈ ub; jc = t∑ t=0 e−ξt[gc(rc − uc(t)) +gc(x(t))] → max, uc(t) ∈ uc ; 318 vladimir k. dyachenko, guennady a. ougolnitsky and larissa v. tarasenko:computer ... this idea was firstly proposed in [6] and means that each player shares her resources between common (ui) and private (ri ui) interests, i=p, b, c. the first summand describes the private interests, and the second one describes the common interest. in this model it is supposed that the common interest consists in the development of the cpe system and each player gains her own gain gi of it. the private interests of the players are different and describe their payoffs from investments beyond the cpe system. discounting is also used, and the random variable ξ reflects a state of the social environment. in the current investigation ten university departments, five employers and ten students are considered. the strategies of generalized players in any moment of time are calculated as an arithmetic mean of the strategies of those agents, namely: strategy of p (employer): up (t) = 1 r r∑ i=1 upi(t);up (t) ∈ up , r = 5; strategy of c (student): uc(t) = 1 s s∑ j=1 ucj (t);uc(t) ∈ uc , s = 10; strategy of b (university): ub(t) = 1 z z∑ k=1 ubk (t);ub(t) ∈ ub, z = 10; the state variable of the model considered as a time function x(t) is also specified in comparison with [3] and now characterizes a level of professional training of the student; f – a function of the system dynamics depending on the players’ strategies. it is assumed that the function of system dynamics increases in respect of all arguments (the efforts of players positively influence to the results of social partnership). to give the system dynamics the modified verhulst-pearl model is used, i.e. f is taken in the form f(x(t), up (t), ub(t), uc(t)) = h(up (t), ub(t), uc(t))x(t)(1− x(t) k ); where k – maximal feasible value of the state variable in the given conditions; h – function of impact of the players’ strategies, namely: h(up (t), ub(t), uc(t)) = 3∑ i=1 aiui(t); ai ≥ 0; 3∑ i=1 ai = 1; i = p,b,c; advances in systems science and application (2015) vol.15 no.4 319 ai – relative weights of the strategies. for the estimation of the relative weights the following considerations are used. the sum of the weights is equal to 1, and all weights are positive. the most important influence is made by students as key agents of the cpe system. the two other weights are approximately equal to each other and don’t differ from the former weight (students) too significantly. so, the weights are chosen as follows (table 1). table 1 relative weights of the strategies (ai) relative weight i employer university student ai 0.3 0.3 0.4 3 identification of the model suppose that the state variable x(t) characterizes a level of professional training of the students. for the estimation of the initial training level a known approach proposed by donald kirkpatrick [7] is used. d. kirkpatrick has developed a model of evaluation of the training effectiveness which considers four levels: 1) reaction: to what degree participants react favorably to the training. for the estimation the results of sociological polls conducted among pce students of the southern federal university are used. the sample consisted of 2,110 respondents, including 827 female (39%) and 1283 male (61%). 2) learning: to what degree participants acquire the intended knowledge, skills, attitudes, confidence and commitment based on their participation in a training event. here the data about the students’ progress together with data characterizing the material base of education are considered. 3) behavior: to what degree participants apply what they learned during training in their professional practice. the results of sociological polls conducted among employers in the rostov region are used. the sample consisted of 2,580 respondents, including 785 female (30, 4%) and 1795 male (69, 6%). 4) results: to what degree targeted outcomes occur as a result of the training event and subsequent reinforcement. the data of polls among employers on the topic “a level of professional knowledge and skills of the graduates” are used. as a result, an initial value of the professional training level is evaluated as x0 in the scale [0, 10]. also, the maximal feasible value of the state variable in the given conditions is taken equal to k = 10. the identification of payoff functions was conducted as follows. let gi be the i’s player payoff from the investments which are not concerned with cpe. the most important type are bank deposits; suppose that vklad i is a 320 vladimir k. dyachenko, guennady a. ougolnitsky and larissa v. tarasenko:computer ... dividend function: v kladi = αi · (ri − ui(t)) · st; i = p,b,c; αi –a share of the investments; st – an average annual percent rate. use also a function ini, i=p,b,c , that determines some fixed payoffs or income of a university professor from other sources: gb(rb − ub(t)) = v kladb +rpt+ inb; rpt – a gain from additional training services: rpt = time · (stavka− β · (rb − ub(t))); time – a total amount of lectures; stavka – an average price of a lecture; β – a share of expenditures required for a lecture; gp (rp − up (t)) = v kladp + ent+ inp ; ent – a function of income business (for an employer): ent = ∑ j productj · (pricej − γj · (rp − up (t))); product j – an amount of the good j ; pricej – a price of the good j ; γj – expense for purchasing and storage of the good j ; gc(rc − uc(t)) = v kladc + earn+ inc ; earn – an income function for student: earn = ∑ k (servk · valuek)− ε · (rc − uc(t)); servk – a volume of services; valuek – an average price of a service; ε – a share of the expenditures. farther,gi – a payoff of each player from increase of the students level of professional training: gi(x(t)) =(x(t) − x(t − 1)) · ( ∑lengthmati k=1 matik · pricematik + ∑lengthmati j=1 nemi j · pricenemi j), i = p,b,c; advances in systems science and application (2015) vol.15 no.4 321 matk i – a set of evaluations of the material payoffs of the player i from cpe; nemj i – a set of evaluations of the non-material payoffs of the player i from cpe; the evaluations are made in the segment [0, 10]. pricematk i – a price of one point of the material payoff of the player i ; pricenemj i – a price of one point of the non-material payoff of the player i ; lengthmati – a number of evaluations of the material payoff of the player i ; lengthnemi – a number of evaluations of the non-material payoff of the player i ; material and non-material payoffs for different players are represented by different factors and determine the characteristics and types of the payoffs. a preliminary expert estimation of the points is made. a maximal possible value of gi for each player is found. a logically substantiated relation between material and non-material payoffs is deduced. 4 planning and implementation of the computer simulation experiments the model investigation was conducted by computer simulation on the base of scenario method [5]. the scenarios are formed according to plausible behavior patterns of the players. for simplicity the arithmetic mean strategies are described. it is supposed that for all scenarios for any moment of time the values of strategies are equal: up (t) = uc(t) = ub(t) = uo(t), t = 0, ... ,4. six scenarios are considered: 1) maximal (max) one corresponds to the maximal possible financing when the whole budget of a player is assigned for training: uo(t) = ri; 2) medium (med) one assigns a half of the budget for training: uo(t) = 0.5ri; 3) minimal (min) allows for training only a small part of the budget: uo(t) = 0,2ri; 4) absence of financing (abs) is clear: uo(t) = 0; 5) decreasing of financing (dec) means that initially an eminent part of the players budget is assigned for training but then the share decreases to a small value, namely: uo(t) = (0.8 0.15t)ri; 6) increasing of financing (inc) describes the opposite strategies: uo(t) = (0.2 + 0.15t)ri. thus, in the former four scenarios the strategies are constant in time, while in the fifth and sixth scenarios the shares of financing are time-dependent. 5 processing and analysis of the modeling results the processing of modeling data includes a comparative analysis of the graphs of state variable and payoff functionals for different scenarios. for example, in fig.1 the graphs of all payoff functions for the scenario of maximal financing are shown. in contrary, in fig.2, the graphs for students payoffs for all scenarios are 322 vladimir k. dyachenko, guennady a. ougolnitsky and larissa v. tarasenko:computer ... given. fig. 1 a comparison of the graphs of payoff functions of employer (jp(t)), student (jc(t)), and university (jb(t)) for the scenario of maximal financing fig. 2 a comparison of the graphs of the students payoff function for different scenarios advances in systems science and application (2015) vol.15 no.4 323 the graphs of payoff functions show a moderate growth considering discounting. instead of abstract utilities in [3], in this paper financial estimations of the payoffs were made. the comparative analysis has shown that the biggest payoff goes to employer, a less to student and the smallest to university. this result is new and was absent in [3]. preliminary analysis of compensation of the investments has shown that in the first period only 71 % of the expenses of student, 63% of the expenses of employer and 18% of the expenses of university are returned. considering the future investments, only the expenses of student and employer for two scenarios: minimal one and increasing of financing – are compensated at all. probably, a more precise identification of the model parameters is needed. the received values of payoffs are very sensitive to the amounts of annual budgets of the players. as test data were used for calculations, the results are still qualitative. nevertheless, the investigation is very important for debugging of the new model and preparation of the special sociological research. more precise values are expected to be received after this research. the results about the impact of strategies on the payoff functions are similar to the previous ones [3]: the best results are provided by the maximal financing scenario, and the worst ones – for the case of no financing. a comparison of the level of professional training for different scenario is made in fig.3 (for periods). fig. 3 comparison of the level of professional training for different scenarios this graph is qualitatively similar to its counterpart in [3]. accordingly to the new questionnaires, the players will be able to estimate the changes in the level of professional training themselves. 324 vladimir k. dyachenko, guennady a. ougolnitsky and larissa v. tarasenko:computer ... 6 conclusion a new version of the model of social partnership in continuing education system is considered. the payoff functions were specified as sums of private and common payoffs, where common payoff depends on the level of professional training of students. the transitive model has determined a set of questions to be answered by a special sociological research among students, professors, and employers. the next model version will be implemented after the research. the current version of the model shows that even a small investment to the development of continuing education system gives a certain gain to its participants. a detailed analysis should be made for the problem of compensation: not all proposed scenario are substantiated from the point of view of investments. for the university the compensation is problematic. in the model the players receive the maximal payoffs if the financing is maximal. it is evident that to increase the level of professional training is possible only if investments are made. in general, the model emphasizes the necessity of development of the continuing education system by all its participants. acknowledgements the work is supported by the russian foundation for humanities, project # 14-03-00236. references [1] keith n. (2011), engaging in social partnerships: a professional guide for successful collaboration in higher education, routledge, pp.288. [2] siegel d. (2010), organizing for social partnerships: higher education in cross-sector collaboration, routledge, pp.224. [3] dyachenko v.k., ougolnitsky g.a. and tarasenko l.v. (2014), “computer simulation of social partnership in the system of continuing professional education”, system science and application, vol.14, no.4, pp.27-37. [4] dockner e., jorgensen s., long n.v. and sorger g. (2000), differential games in economics and management science, cambridge university press,pp.382. [5] law a.m. and kelton w.d. (2000), simulation modeling and analysis, mcgraw-hill, pp.784. [6] germeyer yu.b. and vatel i.a. (1974), “games with hierarchical vector of interests”, j journal of computer systems international. advances in systems science and application (2015) vol.15 no.4 325 [7] kirkpatrick d.l. and kirkpatrick j.d. (2007), implementing the four levels: a practical guide for effective evolution of training programs, berrett koeler publishers, pp.153. corresponding author guennady a. ougolnitsky can be contacted at: ougoln@gmail.com advances in systems science and applications (2014) vol.14 no.4 396-403 stability of solution maps for a η-parameter weak vector variational inequality changchang bu, yuqiang feng and hui li school of science wuhan university of science and technology wuhan 430065 china abstract in this paper, we introduce a new class of η-parameter weak vector variational inequality (for short, η-pwvvi) in banach space, which extends the existing parameter weak vector variational inequality. we use the concepts of η(y, x) function, invex set, η-hemicontinuous and η-strongly c pseudomonotone mapping to study (η-pwvvi) and we obtain new η-generalized linearization lemma. the stability of solution maps for (η-pwvvi) is obtained by this lemma. finally, we present an example to illustrate our results. keywords: η-parameter weak vector variational inequality, η-generalized linearization lemma, invex set, η-hemicontinuous, η-strongly c pseudomonotone. 1 introduction as very powerful and important tools in the study of nonlinear sciences, variational inequalities and vector optimization have attracted so much attention. over the last decades, variational inequality and vector optimization techniques have been applied extensively in such diverse fields as biology, chemistry, economics, engineering, game theory, management science and physics. in 1980, f. giannessi [1] introduced a well-known inequality in finite dimensional space, which was called vector variational inequality (for short,vvi). thereafter, g. y. chen and g. m. chen [2] discussed this problem in infinite dimensional space. g.y. chen [3] presented a new class of vector variational inequality problem in real banach space. the new vector variational inequality problem not only extended the classical vector variational inequality problem, but also related to the existence of non-dominated solution for vector optimization problem. inspired by g. y. chen’s result, a. h. siddiqi et al. [4] and k. l. lin et al. [5] introduced and researched general vector variational inequality problem. l. n. wang et al. [6] proved the stability of solution maps for a weak vector variational inequality in 2013. it is worth pointing out that the convexity plays a significant role while studying the continuity of the solution for vector variational inequality. the concepts of invex set and generalized convexity have been given by s. r. mohan, s. k. neogy [7] and x. m. yang [8] respectively. motivated by the work reported in [1]-[8], the aim of this paper is to introduce a new class of η-parameter weak vector variational inequality (for short, η-pwvvi) in banach space, which extends the existing parameter weak vector variational advances in systems science and applications (2014) vol.14 no.4 397 inequality. our results unify, generalize and complement various known comparable results from the current literature. the rest of the paper is organized as follows. in sect.2, we recall some basic definitions and notations which will be used in the sequel. in sect.3, we use the concepts of η(y;x) function, invex set, η-hemicontinuous and η-strongly c pseudomonotone mappings to study (η-pwvvi) and we obtain new η-generalized linearization lemma. as a consequence, the stability of solution maps for (ηpwvvi) is obtained in theorem 3.3. finally, we present an example to illustrate our results in sect.4. 2 preliminaries let x, y and w (parameter space) are banach s-paces, c ⊆ y is a non-empty closed convex cone with int c ̸= ∅. l(x; y) denotes the space which consist of all the continuous linear operators, define the value of linear operator t ∈ l(x, y) at x ∈ x by ⟨t, x⟩. throughout this paper, assume that η(y, x) : x × x → x satisfies all the conditions as follows: (c1) η(x, x+ λη(y, x)) = −λη(y, x); (c2) η(x, y) + η(y, x) = θ; (c3) η(y, ·) is continuous. here, θ denotes the zero element of x. consider the following η-weak vector variational inequality problem (for short, η-wvvi) of finding x ∈ k such that ⟨t (x), η(y, x)⟩ /∈ −int c, ∀y ∈ k where k ⊆ x is non-empty, t : x → l(x,y ) is a vector value function. when the operator t perturbed by the parameter µ with µ ∈ λ ⊆ w and λ is non-empty, for fixed µ, we deal with the following η-parameter weak vector variational inequality problem (η-pwvvi) of finding x ∈ k such that ⟨t (x, µ), η(y, x)⟩ /∈ −int c, ∀y ∈ k where k ⊆ x is non-empty, t : x×λ → l(x,y ) is a vector value bifunction. for any µ ∈ λ, sη(µ) denotes the solution set of (η-pwvvi), that is, sη(µ) = {x ∈ k | ⟨t (x, µ), η(y, x)⟩ /∈ −int c, ∀y ∈ k} in this paper, we assume that for any µ ∈ λ, sη(µ) is non-empty. now, we give some basic definitions and some properties needed in the following 398 changchang bu: stability of solution maps for a -parameter weak vector ... sections. definition 2.1. (see [7]) a set k ⊆ x is said to be invex with respect to a given η(y, x) : x ×x → x if ∀x, y ∈ k,λ ∈ [0, 1] ⇒ x+ λη(y, x) ∈ k. definition 2.2. let k ⊆ x and k is invex with respect to η(y, x), the operator t : k → l(x,y ) is said to be η-hemicontinuous if and only if for any x, y ∈ k,λ ∈ [0, 1], the mapping λ → ⟨t (x + λη(y, x)), η(y, x)⟩ is continuous at 0+. definition 2.3. let k ⊆ x and k is invex with respect to η(y, x), the operator t : k → l(x,y ) is said to be η-weakly c pseudomonotone on k if for any x, y ∈ k, ⟨t (x), η(y, x)⟩ /∈ −intc implies ⟨t (y), η(y, x)⟩ /∈ −intc. definition 2.4. let k ⊆ x and k is invex with respect to η(y, x), the operator t : k → l(x,y ) is said to be η-strongly c pseudomonotone on k if there exists λ > 0 such that for any x, y ∈ k, ⟨t (x), η(y, x)⟩ /∈ −intc implies ⟨t (y), η(y, x)⟨+λ ∥ η(y, x) ∥2 by ∈ c, where by denotes the unit, closed ball in y . remark 2.1 if we take η(y, x) = y − x in definition 2.1, then invex set run into convex set. similarly, in definition 2.2, η-hemicontinuous reduce to νhemicontinuous (see [6]) with η(y, x) = y − x. remark 2.2 it is evident from definition 2.3 and definition 2.4 that if an operator is η-strongly c pseudomonotone, then it is η-weakly c pseudomonotone. 3 main results the following lemma 3.1 (η-generalized linearization lemma) extends the generalized linearization lemma (see [3]) through the concepts of η(y, x) function and invex set. lemma 3.1. (η-generalized linearization lemma) let k ⊆ x and k is invex with respect to η(y, x). moreover, assume the operator t : k → l(x,y ) is η-weakly c pseudomonotone and η-hemicontinuous, then the following two problems (i) and (ii) are equivalent: (i) there exists x ∈ k, such that for any y ∈ k, ⟨t (x), η(y, x)⟩ /∈ −intc; (ii) there exists x ∈ k, such that for any y ∈ k, ⟨t (y), η(y, x)⟩ /∈ −intc; proof. in view of definition 2.3, it is obvious that (i) implies (ii). now, assume that (ii) holds, then there exists x ∈ k, such that for any y0 ∈ k, ⟨t (y0), η(y0, x)⟩ /∈ −intc (1) note that k is invex with respect to η(y, x), thus for any x, y ∈ k,λ ∈ [0, 1], x+ λη(y, x) ∈ k (2) advances in systems science and applications (2014) vol.14 no.4 399 by (2) we can take y0 = x+ λη(y, x) ∈ k and combine the result with (1), we have ⟨t (x+ λη(y, x)), η(x+ λη(y, x), x)⟩ /∈ −intc (3) in view of the conditions (c1) and (c2) of η(y, x), we obtain that η(x+ λη(y, x), x) = −η(x, x+ λη(y, x)) = λη(y, x) thus, we claim that (3) is equivalent to ⟨t (x+ λη(y, x)), λη(y, x)⟩ /∈ −intc dividing by λ, we get ⟨t (x+ λη(y, x)), η(y, x)⟩ /∈ −intc let λ → 0+, take into account the η-hemicontinuity of t , we have ⟨t (x), η(y, x)⟩ /∈ −intc therefore, (i) holds. the proof is complete. lemma 3.2. let k ⊆ x and k is invex with respect to η(y, x). if for fixed µ ∈ λ, t (·, µ) is η-strongly c pseudomonotone, then the solution set of (η-pwvvi) is single valued, i.e., for fixed µ ∈ λ, sη(µ) is a singleton. proof. suppose, to the contrary, that there exist x1, x2 ∈ sη(µ), but x1 ̸= x2. by definition of sη(µ), ⟨t (x1, µ), η(y, x1)⟩ ∈ y \ − intc, ∀y ∈ k (4) ⟨t (x2, µ), η(y, x2)⟩ ∈ y \ − intc, ∀y ∈ k (5) in particular, take y = x2 in (4) and y = x1 in (5), respectively, we obtain ⟨t (x1, µ), η(x2, x1)⟩ ∈ y \ − intc (6) ⟨t (x2, µ), η(x1, x2)⟩ ∈ y \ − intc (7) since (6) holds and t is η-strongly c pseudomonotone, we claim that there exists λ > 0, such that ⟨t (x2, µ), η(x2, x1)⟩+ λ ∥ η(x2, x1) ∥2 by ∈ c 400 changchang bu: stability of solution maps for a -parameter weak vector ... consider that x1 ̸= x2, we have ⟨t (x2, µ), η(x2, x1)⟩ ∈ intc (8) in view of the condition (c2) of η(y, x), (8) is equivalent to ⟨t (x2, µ),−η(x1, x2)⟩ ∈ intc that is to say ⟨t (x2, µ), η(x1, x2)⟩ ∈ −intc this is a contradiction to (7). therefore, for fixed µ ∈ λ, sη(µ) is a singleton. theorem 3.3. let k is a non-empty, compact subset of x. assume k is invex with respect to η(y, x). if the following conditions hold: (i) for fixed µ ∈ λ, t (·, µ) is η-hemicontinuous on k; (ii) for fixed µ ∈ λ, t (·, µ) is η-strongly c pseudomonotone on k; (iii) for fixed x ∈ k,t (x, ·) is continuous on λ. then, sη(·) is continuous on λ. proof. first of all, we invoke lemma 3.2 to conclude that sη(·) is single valued. thus, we assume, without loss of generality, that sη(µ) = x(µ), ∀µ ∈ λ. take µ0 ∈ λ, in order to obtain that sη(·) is continuous at µ0, we only need to show that x(µ) → x(µ0) as µ → µ0. for any sequence {µn} ⊆ λ that satisfies µn → µ0, we can find a solution set sequence x(µn) ∈ k. further, note that k is compact, there exists a convergent subsequence {x(µnk )} such that x(µnk ) → ν. next we prove that ν = x(µ0). note that x(µn) is the solution of (η-pwvvi), we obtain ⟨t (x(µn), µn), η(y, x(µn))⟩ ∈ y \ − intc, ∀y ∈ k (9) in view of lemma 3.1, (9) is equivalent to ⟨t (y, µn), η(y, x(µn))⟩ ∈ y \ − intc, ∀y ∈ k (10) we claim, bear in mind that for fixed x ∈ k,t (x, ·) is continuous on λ and η(y, ·) is continuous, that ∥ ⟨t (y, µn), η(y, x(µn))⟩ − ⟨t (y, µ0), η(y, ν)⟩ ∥ ≤ ∥ ⟨t (y, µn), η(y, x(µn))⟩ − ⟨t (y, µ0), η(y, x(µn))⟩ ∥ + ∥ ⟨t (y, µ0), η(y, x(µn))⟩ − ⟨t (y, µ0), η(y, ν)⟩ ∥ ≤ ∥ t (y, µn)− t (y, µ0) ∥ · ∥ η(y, x(µn)) ∥ + ∥ t (y, µ0) ∥ · ∥ η(y, x(µn))− η(y, ν) ∥ advances in systems science and applications (2014) vol.14 no.4 401 thus, let n → ∞, we have ⟨t (y, µn), η(y, x(µn))⟩ → ⟨t (y, µ0), η(y, ν)⟩ (11) note that y \ − intc is closed and (11), we get ⟨t (y, µ0), η(y, ν)⟩ ∈ y \ − intc, ∀y ∈ k combine this with condition (ii) and use lemma 3.1 again, we find ⟨t (ν, µ0), η(y, ν)⟩ ∈ y \ − intc, ∀y ∈ k hence, v ∈ sη(µ0). recall that, by lemma 3.2, sη(·) is single valued. thus, ν = x(µ0). the proof is complete. remark 3.1 note that in theorem 3.3, t is required to be a η-hemicontinuous operator which extend ν-hemicontinuous in [6] and hence it weaken the continuous condition in [13]. 4 example in this section, we present an example to show that there exists η(y, x) function that satisfies the condition (c1)-(c3). let k = r, take the function η(y, x) = { y − x, x ≤ 0, y ≤ 0 and x ≥ 0, y ≥ 0, x− y, x ≤ 0, y ≥ 0 and x ≥ 0, y ≤ 0. it is evident that k is invex with respect to η(y, x). the function η(y, x) above satisfies condition (c2) and (c3) is obvious. next we verify that it also satisfies condition (c1). (i) for x ≤ 0, y ≤ 0 and any λ ∈ [0, 1], x+ λ(y − x) = (1− λ)x+ λy ≤ 0, η(x, x+ λη(y, x)) = η(x, x+ λ(y − x)) = η(x, (1− λ)x+ λy) = −λ(y − x) = −λη(y, x). (ii) for x ≥ 0, y ≥ 0 and any λ ∈ [0, 1], x+ λ(y − x) = (1− λ)x+ λy ≥ 0, η(x, x+ λη(y, x)) = η(x, x+ λ(y − x)) = η(x, (1− λ)x+ λy) = −λ(y − x) = −λη(y, x). 402 changchang bu: stability of solution maps for a -parameter weak vector ... (iii) for x ≤ 0, y ≥ 0 and any λ ∈ [0, 1], x+ λ(x− y) = (1 + λ)x− λy ≤ 0, η(x, x+ λη(y, x)) = η(x, x+ λ(x− y)) = −λ(x− y) = −λη(y, x). (iv) for x ≥ 0, y ≤ 0 and any λ ∈ [0, 1], x+ λ(x− y) = (1 + λ)x− λy ≥ 0, η(x, x+ λη(y, x)) = η(x, x+ λ(x− y)) = −λ(x− y) = −λη(y, x). the analysis above shows that the η(y, x) function satisfies the condition (c1)(c3) but η(y, x) ̸= y − x. that is to say η-parameter weak vector variational inequality(η-pwvvi) extends the existing parameter weak vector variational inequality and the results we obtained is reasonable. acknowledgements this research is supported by the doctoral fund of education ministry of china (20134219120003), the natural science foundation of hubei province (2013cfa131) and the nature science foundation of china (f030203). references [1] f. giannessi. (1980), “theorems of alternative, quadratic programs and complementarity problems”, variational inequalities and complementarity problems, edited by r. w. cottle, f. giannessi and j. l. lion, new york: john wiley and sons, pp.151-186. [2] y. chen, g. m. chen. (1987), “variational inequalities and vector optimization”, lecture notes in economics and mathematical systems, berlin:springer-verlag, 285, pp.408-416. [3] g. y. chen. (1992), “existence of solutions for a vector variational inequality: an extension of the hartman-stampacchia theorem”. j. optimiz. theory appl., 74, pp.445-456. [4] a. h. siddiqi, q. h. ansari, a. khaliq. (1995), “on vector variational inequalities”, j. optimiz. theory appl., 84, pp.171-180. [5] k. l. lin, d. p. yang, j. c. yao. (1997), “generalized vector variational inequalities”, j. optimiz. theory appl., 92, pp.117-125. [6] l. n. wang, z. m. fang, j. z. li. (2013), “stability of solution maps for aweak vector variational inequality”, journal of southwest university (natural science edition), vol.35, no.1, pp.95-98. (in chinese). advances in systems science and applications (2014) vol.14 no.4 403 [7] s. r. mohan, s. k. neogy. (1995), “on invex sets and preinvex functions”, j. math. anal. appl., 189, pp.901-908. [8] x. m. yang, x. q. yang, k. l. teo. (2003), “generalized invexity and generelized invariant monotonicity”, j. optimiz. theory appl., 117, pp.607625. [9] s. j. yu, j. c. yao. (1996), “on vector variational inequalities”, j. optimiz. theory appl., vol.89, no.3, 749-769 . [10] s. chang, b. s. lee, y. q. chen. (1995), “variational inequalities for monotone operators in nonreflexive banach space”, appl. math. lett., vol.8, no.6, pp.29-34. [11] n. j. huang, y. p. fang. (2005), “on vector variational inequalities in reflexive banach space”, j. global optim., vol.32, no.4, pp.495-505. [12] b. s. lee, g. m. lee. (1999), “variational inequalities for (η; θ)pseudomonotone operators in nonreflexive banach space”, appl. math. lett., vol.12, no.5, pp.13-17. [13] a. barbagallo, m. g. cojocaru. (2009), “continuity of solutions for parametric variational inequalities in banach space”, j. math. anal. appl., vol.351, no.2, pp.707-720. corresponding author yuqiang feng can be contacted at: yqfeng6@126.com. adv syst sci appl 2018; 02; 1-10 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/620 copyright ©2018 assa. adv. in systems science and appl. (2018) organization of traffic flows simulation aimed at establishment of integral characteristics of their dynamics anatoliy a. solovyev1,2, andrey m. valuev1 1) mechanical engineering research institute, russian academy of sciences, moscow, russia e-mail: aa.solovjev@yandex.ru 2) moscow institute of physics and technology (state university), dolgoprudny, moscow region, russia e-mail: valuev.online@gmail.com abstract: the paper presents an approach aimed at the study of various aspects of traffic passage through a multi-lane road that is based on computational experiments. microscopic simulation based on models of the type “leader-following models” is considered to be the most adequate and the most accurate means for performing them. the leader-following model of the entire traffic flows in a city road network acquires the more holistic form via the formalism of hybrid dynamic systems or, in other words, event-switched systems. the general formalization of this approach is presented and the corresponding representation of multi-lane traffic according to this approach is presented. the latter includes the detailed formal description of the traffic flow carrier with a certain traffic organization as well as the description of conditions for drivers’ choice of acceleration/braking and lane change according to incentive motives for it and positions and speeds of neighboring vehicles and the vehicle itself. organization of computational experiments allowing to establish the dependence of the average speed of traffic on the density of incoming flows, the distribution of various types of vehicles and their drivers, road organization and other factors in a multi-lane road is considered. it is demonstrated in what way they allow to evaluate quantitatively the dependence of the road throughput on the above factors. results of calculations are presented and analyzed. keywords: hybrid systems; traffic flows; leader-following model; city road network; multi-lane traffic; computational experiments. 1. introduction the city road network (crn) as a whole and its individual segments limit satisfaction of transport needs of the population, which is expressed in slow motion, time waste in traffic jams, rejection of some desirable trips or choosing an uncomfortable time for them. effective use of the throughput of the existing crn and its fragments, as well as substantiation of measures to increase it, requires knowledge of the characteristics of individual roads and crossroads with the existing traffic organization and alternative options. meanwhile, these characteristics are approximately known only for the most elementary cases. so, for singlelane traffic, estimates of road throughput have been established, ranging between 2000 and 2790 vehicles per hour [6, 8]. the variance of estimates is related to the difference in driving culture and the use of a different fleet of cars and trucks, and these differences can manifest even within one megacity. the use of road throughput is also affected by the distribution of trips between transport correspondences, almost always preventing the full load at the maximum level of all roads and their lanes. 2 a.a. solovyev, a.m. valuev copyright ©2018 assa. adv. in systems science and appl. (2018) the busiest highways carrying the bulk of the traffic flow (tf) are characterized by multi-lane movement. multi-lane traffic also results in complex organization of motion at crossroads, for which a number of types are distinguished [3]. integral characteristics of the flow through homogeneous segments of roads and crossroads depend on many circumstances and numerical characteristics of the tfs, including the proportions in which the flows are divided between directions in the nodes of the road network, as well as location of segments of allowed passage between the lanes and parking along the roads. so it quite impossible to establish integral characteristics of tf dynamics directly from treatment of observed data because of multidimensionality of affecting factors. however, systematical observations must be the source of needed data concerning elementary components of tfs (such as motion of a pair of successive cars) on a representative set of drivers and vehicles. physical-style theories, like widely known three-phase traffic theory by b.s. kerner [4] yield the qualitative picture of the traffic phenomena but hardly may take into account all the affecting details. another way to establish the necessary integral characteristics of tf is realized by mathematical modeling (analytical or, more often, by simulation). in this area both macroscopic models (e.g., the modern versions of hydrodynamic lwr models [2]) and microscopic ones exist. the latter, in turn, may be relatively rude cellular automata based on discretization of space and time [5] or much more realistic models in terms of dynamic systems that unite the movements of individual vehicles described with odes subject to additional servo-links. the conceptual basis for modeling tf as a dynamic system is the “leader-following model”, the latter being a formal description of the vehicle control choice by a driver depending on the movement of the previous one [6]. the paper [3] put forward the way of integration of these particular models together with conditions resulting from traffic light regulation in the in the holistic model. the next step of evolution of this approach is the expression of such models via the formalism of hybrid systems [7]. this way of tf modelling may be applied to multi-lane traffic as well, but in that case the “leader-following model” must be coupled with a “lane-changing model” [6]. the paper presents the construction of such a model and the way to organize computational experiments with it and treat their results to obtain integral characteristics of tf dynamics. 2. general approach to traffic modeling via the formalism of hybrid dynamical systems the analytical study of the tf dynamics within the framework of the “leader-following model” is limited to simple cases, mainly related to the so-called homogeneous congested traffic along one lane with a number of limiting assumptions, the main one being the identity of vehicles and their drivers (for example, [6]). the same models can be used via simulation. advantages of such models of private tf modes consist in the explicit formulation of all interrelations. undoubtedly, modeling by computer simulation has repeatedly been used to investigate more complex regimes, but only in the form of agent modeling, which treats relationships between the movements of neighboring vehicles as some forms of interaction between them. however, an integrated description of such an interaction may be absent; in any case, it is clearly not formulated in publications and therefore gives no possibility for critical analysis. with respect to verification of any agent modeling system, if it was fulfilled, one cannot be sure whether it is the result of a really correct reproduction of the nature of the transport processes or a successfully performed parametric identification (in other words, fitting) of the working model for specific conditions. in accordance with common sense traffic members are considered as “vehicle–driver units” (vdu) [6], i.e., specific vehicles that are controlled by specific drivers. the proposed approach does not postulate a specific model for a separate vdu and its choice of motion direction (in the case of a shift between lanes, a stop for parking, or a crossroad pass) in organization of traffic flows simulation 3 copyright ©2018 assa. adv. in systems science and appl. (2018) interaction with the movements of neighboring vehicles. such elementary relationships, obtained empirically, can be integrated into the resulting tf model for the road network segment, provided they are expressed in the required form. the model also includes a formal representation of the traffic organization on the segment of the road network being studied. in this sense, not a specific model of the tf is proposed, but a metamodel, which, however, is uniquely concretized to a specific tf model under specific conditions when using the required model elements characterizing the conditions being considered definitely or presumably. components of the model are: 1. formal description of the tf carrier furnished with the means of traffic regulation (traffic lights, barriers) and the conditions for the permissibility of trajectories along tf carriers; 2. formal description of the dependencies that determine the drivers’ choice of control when driving on a standard trajectory; 3. formal description of the conditions for drivers’ choice of the direction of movement; 4. description of the input streams (inflows) and their links with outflows that means distribution of vdus entering the road network in consideration in a certain on-ramp between out-ramps. 1. the tf carrier is a segment of the road network in a detailed description. permissible directions of motion on crossroads must be indicated; for a multi-lane road they must be determined for each lane. each crossroad is described by its own network, or a “route web”, uniting permissible trajectories that connect the all permitted pairs of entering lanes ends and beginnings of exit lanes. points of their intersection must be indicated as well. for regulated crossroads, it is indicated for each trajectory at which phase of the traffic light cycle movement along it is permitted. if parking is allowed on the road segment at the road edge or in special parking pockets, the totality of such parking spaces is also treated as a lane. in general, the carrier is represented by an oriented labelled graph g with some peculiarities. they consist in the fact that each arc (oriented edge) of g, representing a certain lane, is assigned to a certain road segment and it is determined which lanes (if any) of the same segment are adjacent to it on the left and on the right. an arc may allow (along a certain zone of its border) the shift of vdus moving along it to the adjacent lane. zones of permissible transitions must be determined; they are treated as labels of arcs. arcs of a crossroad “route web” connecting the same road segments are also grouped into a segment of the crossroad network. the transition between arcs that do not belong to the same road segment is possible only at the points of their joining, i.e. at the end of the entering arc, coinciding with the beginning of the outgoing arc. these points are the vertices (nodes) of the graph g. intersection points of admissible trajectories at crossroads are considered as vertices (nodes) of the graph g only if through such a point the vdus travelling along more than one admissible trajectory may pass within certain time intervals. the latter is characteristic only for unregulated crossroads. in view of the inability to simulate the tfs in their natural limits (then we would have to consider the road network from vladivostok to lisbon), a fragment is cut from the network into which the vdus enter from outside through the in-ramps and exit outside through the out-ramps. 2. each edge of g at each instant of time contains a chain of vehicles, treated as a moving, and sometimes a motionless queue. the concept of a hybrid system means mixed, discretecontinuous dynamics. discrete dynamics in models of this type include the change of the set of vdus in some queue, as well as the change of the traffic light phases and the mentioned below modes of motion of individual vdus. the dependencies defining the control laws (in the sense that this concept has in the control theory) for vdus are based on safety conditions. the most important and mandatory condition of safety is aimed to exclude the collision with the previous car and can be 4 a.a. solovyev, a.m. valuev copyright ©2018 assa. adv. in systems science and appl. (2018) expressed as a formula of safe distance depending on the speed of this and, possibly, the previous car. it is supplemented by the conditions of speed reduction on curved sections of the road, as well as by the conditions of braking on the red phase of the traffic light (for the vehicle at the head of the queue). in the transition between the lanes, parking on the roadside, the safety condition is modified. another condition for control choice is determined by the restriction on the speed and the desired (for public transport means — the normative) speed (they are distinguished if the latter is less than the maximum allowed speed). it is assumed that the forms of the required dependencies are the same for all vdus, but have parameters whose values are different for different types of vdus. various researchers developing “leader-following models” treat in the same way the general nature of these dependencies: the safe distance measured from the front bumper of the follower to the leader’s rear bumper depends on the speeds of both. the parameters of the dependence are the characteristics of the driver (first, its reaction time) and the vehicle of the follower. it is more convenient to consider the safety conditions with respect to the distance between the front bumpers of both (their positions relative to the current lane start are denoted by si-1(t), si(t) and serve as phase variables in the dynamic system of the tf). then we need to add the length of the vehicleleader to the “natural” distance. it is assumed, therefore, that the basic safety conditions si-1(t)-si(t)≥ssafe(vi-1(t),vi(t),pi,pi-1) (2.1) are satisfied permanently for all pairs “leader-follower”. together with (2.1) for curved sections of the road, restrictions on the speed are considered depending on the local curvature crv(s): �̇�𝑖 ≤ 𝑉safe crv(𝐶𝑅𝑉(𝑠𝑖(𝑡))). (2.2) in addition, there is a restriction similar to (2.2), on the maximum speed dictated by the traffic organization; they both can be combined in the form (2.2). the prohibition of entry to a crossroad when the traffic light is red means zero speed of the first vehicle in the lane at the moment of reaching the crossroad border and further until the green light turns on. obviously, this condition is one of those that make hybrid the dynamic system of a tf: the leader of the queue changes the mode of motion from braking (until reaching the border of the crossroad) to a stop and then to acceleration from the moment when the green light turns on. such qualitatively different situations for choosing control are various but few). they are: acceleration at free movement when a speed is less than the desired speed; maintaining the latter; in the case when the minimum safe distance to the consequent vdu is reached — maintaining this distance; braking before a traffic light, or when stopping for parking, or when approaching a turn; transition between the lanes. for each of these cases the definite law of control acts, in other words, a definite motion mode. for each vdu, the current mode is treated as a variable of the qualitative state; other such variables characterize the position on the graph g (arc number) and the sequence number in the chain on the arc. the variables that characterize the current and forthcoming directions of motion may also be treated as variables of the qualitative state as well. together with the regulator states, they form the vector of the qualitative state d of the simulated tf as a whole. the movement (or its expectation) of all vdu in constant modes for some time eventually leads to a change in the mode of one or several vdus or a change of the vdus set for some chains. the transition from one mode to another is caused by the achievement of a certain boundary (hypersurface) in the combined phase space of the vdu (and perhaps of the adjacent vdus) extended by the variable of the current time. for example, the condition for reaching a certain speed, approaching the minimum safe distance, starting the braking before the red traffic light signal from the stopping condition at the crossroad boundary under normal acceleration of braking (similarly when trying to occupy a free parking place) are all obviously conditions of that type. the same type of conditions determine the change in the of the vdus set for chains: reaching the end of the lane by the head of the chain; termination of the shift of a certain vdu between lanes at the moment of when the lateral organization of traffic flows simulation 5 copyright ©2018 assa. adv. in systems science and appl. (2018) coordinate reaches its final value; entering the network by a next vdu from the on-ramp at a certain time instant. all these changes are traditionally known for hybrid systems as switches. it should be emphasized that the conditions of each switch, in addition to the switching hypersurface, are also characterized by the set of values of some components of the vector d. so, braking before the crossroad at red light begins only with the shift to the corresponding phase of the traffic light; formally it means the change of the proper component of d. to start a shift between lanes or a parking process, the conditions are not limited to one component of d. all previously listed switches — changing the driving mode by a separate car, entering or leaving one tf in question, switching traffic lights at the end of the road — are considered as events of a stage change. within the stage the phases of all traffic light cycles, the number of vehicles on each arc, the mode of each vehicle remain unchanged. as moments of switches are not known in advance, the same holds for time limits [t(l-1), t(l)) for each (l-th) stage. the discrete state variables in the l-th stage are combined into the vector d(l), the phase variables characterizing the positions and speeds of the vehicles at the time moment t during the l–th stage — into the vector x(t,l), its components for the i-th vehicle form vector xi(t,l). dimensions of both x(t,l) and d(l) and even of certain xi(t,l) vary from stage to stage, since the latter includes the coordinate and the speed of movement in the transverse direction only during the transition between lanes. constant values characterizing individual vehicles and driving them, including the purpose of movement, are also treated as components of d(l). to describe the dynamics of an individual vehicle, the generic representation is used ))),(),((),,(),((/),( ltxldultxldfdtltdx iii  (2.3) in which one or more components of d(l) characterize the motion mode, and the formula expresses the control law for this mode. it seems most natural to use acceleration as a control variable. the conditions for determining the type j(l) and the moment t(l) of the switch that completes the l-th have the general form ,0))(),,(),(()( ltltxldg lj jj(d(l)), ))(),,(),(( ltltxldg j <0, jj(d(l))\{j(l)}, (2.4) and follow from restrictions of type (2.1), (2.2). the result of switching is the change of the set of qualitative state variables ))(()),(()1( )()( ldiilddld ljdlji  (2.5) together with, maybe, some phase variables )).(()),(),),((),(()1),(( )()( ldiiltlltxldxlltx ljxlji  (2.6) formula (2.6), in particular, refers to the longitudinal coordinates that change when passing from one arc to another ( 0)(( ltsi ), since they are now measured from the other point. the similar change takes place for the transverse coordinate at the time of completion of the transition to a new lane. however, a mode change may require certain values of several components of d(l). in this case, it occurs at the moment when the last one receives the required value. this moment is also determined by the condition (2.4) for a component of d(l), but only for the case when all other ones already have the required values (and do not change them at the last switching). however, it doesn’t not matter which component of d(l) is the last to get the required value. this does not require the modification or addition of the relations (2.4)–(2.6), but only their detailed specification. namely: a change in the regime of a specific vehicle can be made for different combinations of j(l) and d(l), but in all cases, the value of the corresponding component di(l+1) expressing the motion mode of this vehicle, in all cases, will be the same. 3. the current direction of movement of a specific vdu is characterized by the goal of stopping at the current section or continuing to move on reaching the end of the current segment for a certain new segment. hence, according to the current scheme of traffic 6 a.a. solovyev, a.m. valuev copyright ©2018 assa. adv. in systems science and appl. (2018) organization, this goal results in the task of transition to the lane from which it can proceed to this new segment. the overall goal of a particular vdu is to reach a specific target point (off-ramp) or to park in a certain area. with the exception of public transport, in most cases it is possible to choose a route. in practice, it is possible either to pre-select a route or to select it step by step when receiving information about the status of tfs, which, however, does not guarantee the choice of the most effective route. if we take as a basis any method of forecasting the duration of the route according to the current and historical data, then its recommendation will be reduced to the proposal to select the same next section for all vdus having the same target node and currently passing a certain node. each such recommendation is expressed by a variable that has the properties of a component of d and, apparently, can somehow be calculated within the same formalism, although it is unlikely that the method of calculating it will be relatively simple (for some forecasting methods, the computations that realize them are guessed, but confirm the assumption about their complexity and high amount of calculation). we retain such an option as possible in principle, but at the present stage of research we confine ourselves to a more realistic way of modeling the behavior of drivers, namely by assigning each of them a certain route (with detailing to the sequence of passing the sections of the road network). 3. modeling of traffic on a multi-lane road in accordance with the objectives of the study, a one-way multi-lane road from the crn is selected for study, it is bounded at both ends by crossroads (junctions) and not contains them within itself. the traffic organization can allow vehicles to leave the tf and enter the tf only from parking places along a part of the length of the road. parking spaces are collectively considered as the rightmost lane of a road, traffic on which is not allowed. regarding the transitions between the lanes, the segments of each lane on which passages through its left or right border are permitted (including for parking purposes) are defined. in addition, the destination of the end segment of each lane is defined as to carry out further movement in a certain direction or directions (right, left, right). limitations on traffic for each lane include: maximum speed and category of vehicles (passenger car, bus, freight with a certain weight limit) for which movement along it is allowed. the definite number of categories is introduced. for each individual participant (vdu) of the simulated tf, the following parameters are specified: 1) category; 2) the purpose of the movement, namely the direction of movement when leaving the lane or the intention to park along the road; 3) the length; 4) a set of indicators that determine the dynamics and driving, including perception of the minimum safe distance to the leader. it is assumed that to achieve its goal, each driver uses several driving modes. in the process of moving along one lane these modes are: 1) acceleration up to the maximum (desired) speed, 2) maintaining the desired speed; deceleration aimed at 3) stopping at a given place or (for a curved road) at 4) non-exceeding the safe speed; 5) maintaining the minimum safe distance to the leader. in the process of transition between the lanes, the variants of the same modes are implemented, but the conditions for the maintenance of the minimum safe distance are modified, since it is calculated to the nearest leader on the old and new lane. each mode is assumed to be expressed by the corresponding model law of control (acceleration/deceleration). the change of driving mode by a separate vehicle, entering or leaving the tf in question by a vehicle, switching traffic lights at the end of the road are considered as events of phase change; within the phase of the phase of the traffic light cycle, the number of vehicles, the mode of each remains unchanged. for the 1st mode, the “normal” (for a specific vdu) acceleration ( ii altxldu norm)),(),((  ) and for the third one “normal” deceleration organization of traffic flows simulation 7 copyright ©2018 assa. adv. in systems science and appl. (2018) ( ii bltxldu norm)),(),((  ) are accepted, for the 2nd one the uniform motion ( 0)),(),(( ltxldui ) takes place. for the fourth mode, the acceleration is determined from the motion condition at a safe speed ( dttsdvltxldu iii /))(()),(),(( safe ). in all these cases, there is no dependence of the control of the vdu on the phase coordinates of other vdus, which expresses the specifics of a tf as the interconnected motion of many of its participants. in the fifth mode, this specifics is present, but can be expressed in different ways. the movement of a vehicle is considered, first of all, along the axes of the road lanes. even in the passage to another lane, the trajectory is directed at a small angle to the axes of adjacent lanes and the longitudinal velocity is practically equal to the actual velocity; the change in velocity during the transition time can be neglected. therefore, with respect to the quantitative dynamics, we may reduce representation of the vdu dynamics to the equations of longitudinal motion with respect to the current position of the vehicle si measured along the axis of the lane, namely dsi(t, l)/dt=vi(t, l); dvi(t, l)/dt=ai(t, l), ai(t, l)= )),(),(( ltxldui . (3.1) for any representation and for any of the above modes, switches between them occur when the condition represented with one equation is satisfied, which parameter may be components of the qualitative state in the mode before switching. so, to complete the set of speed and transition to a uniform motion, this is just a condition vi(t)=vmax i, (3.2) for inclusion in the cluster and transition to the fifth mode — si-1(t)-si(t)=ssafe(vi-1(t),vi(t),pi), (3.3) similarly for switching between the fifth mode options. to start the transition, the transition conditions are somewhat more complicated. let us first formulate the conditions for the beginning of the transition at the content level. the condition for the possibility of a transition is: the admissibility of such a transition (the intersection of the dividing line in the direction of the new lane); presence of sufficient advance of the potential "follower" on the adjacent strip s(k); presence of motivation for transition under conditions of speed advantage, consisting in that the safe distance to the leader on the neighboring strip exceeds the safe distance to the leader on the current lane by at least; the presence of motivation for the transition to the conditions of the timely occupation of the lane, at which the vehicle under consideration is required to be located at the end of the present road segment. the moment of the onset of each of the conditions, as well as the moment when the previously fulfilled condition is violated, are determined, as above, from relations of the form (2.4), the parameters of which relate to the considered vdu, preceding on the strip and conditional leaders and followers for it on adjacent lanes. the moment of the shift beginning begins a stage that is characterized with additional coordinates and equations for lateral motion and additional relations between (security conditions are associated with a tuning vehicle with its "leader" and "follower" on both strips). with respect to the vdus the lateral coordinate, the condition for the termination of the transition is recorded, the result of which is the change in the order of the vdu on both lanes and the restoration of the old system of constraints with new parameters. 4. organization of computational experiments in accordance with the research objectives, in a separate computational experiment, the tf dynamics are calculated with definite and on average constant values: 1) the intensity of the input flow (and exit from parking places along the road, if it is allowed); 2) the proportions between the target directions in the input stream; 3) the composition of the input stream (in the simplest case the shares of the introduced categories of vehicles). in accordance with the selected values of the experimental constants listed above, the characteristics of the 8 a.a. solovyev, a.m. valuev copyright ©2018 assa. adv. in systems science and appl. (2018) individual vehicles of the input sequence and the time intervals between their appearances are randomly generated. a separate experiment begins with an empty road gradually filled with vehicles. flow characteristics are calculated in steady state when the road is completely filled. average characteristics and their dispersion for a sufficiently long period are determined. in the case of simulation of the application of traffic light regulation, the dynamics of the tf for a period after reaching the steady state, a multiple of the duration of the traffic light cycle is considered. the result of the calculation is the sequence of values of the components of the vectors x(t,l) and d(l). at the same time, these quantities can be stored in the database, but only integral indicators of flows along the lanes. initial data for the computational experiments are recorded in the database tables (see tables 4.1 and 4.2) and extracted from them for tf simulation. they characterize both the content conditions of the experiments and calculation parameters (see the maximal step tmax for integrating motion equations in table 4.1). table 4.1. general data for computational experiments expt. tmax, s tmax, s averaging interval, s qlane, vdus per hour t in tst011 0,5 300 10 1500 0,5 tst011a 0,5 600 10 1500 0,5 tst021 0,5 300 10 2000 0,5 tst021a 0,5 420 10 2000 0,5 tst021b 0,5 420 10 2000 0,5 table 4.2. data оn vdu types in computational experiments expt. vdu type anorm i; bnorm i length; width vopt, m/s treac, s sin type quote in onramp flow tst011 1 4;;7 3; 1,8 25 0,3 0,1 75 tst011 2 2; 4 10; 3 17,5 0,2 0,15 25 tst021 1 4; 7 3; 1,8 25 0,3 0,1 75 tst021 2 2; 4 10; 3 17,5 0,2 0,15 25 the character of the tf dynamics is ambiguous and depends on the characteristics of the atp (see tables 4.3, 4.4). at a moderate tf intensity (in the examples it is 1500 vdu/h per one lane), flow characteristics are stable, although deviations from the mean values take place permanently preserved. with the equivalence of the lanes, neither one gets an advantage. table 4.3. statistics on atp, a series of experiments “tst011” expt. tbeg tend mean vdus count mean vdu speed vdus in shift, % tst011 20 300 11,75 21,36 8,81 tst011a 20 300 12,18 20,78 9,94 tst011a 300 600 12,55 20,85 10,97 tst011b 20 300 11,92 21,08 9,67 tst011b 300 600 11,72 21,18 8,00 organization of traffic flows simulation 9 copyright ©2018 assa. adv. in systems science and appl. (2018) table 4.4. statistics on atp, a series of experiments “tst011” expt. tbeg tend mean vdus count mean vdu speed 1st lane 2nd lane 1st lane 2nd lane tst011 20 300 6,02 5,73 21,50 21,18 tst011a 20 300 5,95 6,23 20,71 20,81 tst011a 300 600 6,46 6,10 20,64 21,01 tst011b 20 300 6,26 5,66 21,47 20,51 tst011b 300 600 5,74 5,88 20,93 21,44 on the contrary, the flow approaching its maximum intensity is unstable (see tables 4.5, 4.6), queues at the entrance and supersaturation of the road section can be formed. in this case, the speed can drop sharply. these phenomena are well known, but modeling by simulation allows determining the conditions of their occurrence (depending not only on the intensity of the flows, but also on their composition with respect to vdu types). table 4.5. statistics on atp, a series of experiments “tst021” expt. tbeg tend mean vdus count mean vdu speed vdus in shift, % tst021 20 300 19,18 17,36 12,72 tst021b 20 300 16,81 19,53 11,13 tst021b 300 400 22,35 9,62 24,95 table 4.6. statistics on atp, a series of experiments “tst021” expt. tbeg tend mean vdus count mean vdu speed 1st lane 2nd lane 1st lane 2nd lane tst021 20 300 9,79 9,38 17,25 17,52 tst021b 20 300 8,07 8,74 19,31 19,68 tst021b 300 400 9,75 12,61 11,59 8,30 repetition of a series of experiments with different values makes it possible to obtain a series of data for the characteristics of the unknown dependencies that are interpolated to the range of interest. 5. conclusion the paper presents the general approach to establishment of integral characteristics of traffic flow dynamics, notably for multi-lane roads, and illustrates it in the case of two-lane traffic with a relatively simple road organization. with the development of this approach more sophisticated integral characteristics of multi-lane tfs must be obtained, including the influence of the structure and parameters of the traffic light cycle at the end of the road in question and the distribution of vdus between desired directions of subsequent passage of the road network. acknowledgements we thank the members of the organizational committee of mlsd’2017 for their kind invitation to contribute the paper presented at the conference to the journal and andrey 10 a.a. solovyev, a.m. valuev copyright ©2018 assa. adv. in systems science and appl. 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(2010, october). critical evaluation of roadway capacity of multilane high speed corridors under heterogeneous traffic conditions through traditional and microscopic simulation models. journal of indian roads congress, 71(3), 235– 264. adv syst sci appl 2020; 02:71–81 published online at https://ijassa.ipu.ru. comparison of two dynamic models of economic growth alexander p. chernyaev1* 1moscow institute of physics and technology (state university), dolgoprudnyi, russia abstract: two dynamic models of economic growth with the same balance equation are considered. first, we establish the solution of the harrod-domar model with time-dependent coefficient of the capital intensity of income growth. (previously, the only constant coefficients were considered.) second, we show that in the solow model with the cobb-douglas production function, the capital intensity of income growth depends on time. comparing these models, we demonstrate the effectiveness of the setting optimal control problems (maximization of the integral discounted utility function) in the extended the harrod-domar model. keywords: economic growth, production function, harrod-domar model, solow model 1. introduction models of economic growth became very popular due to their universality. they were applied to various objects in economic structures of many kinds. in mathematical economics, there are two widely acknowledged models of economic dynamics: the harrod-domar model and the solow model, which are presented in scientific and educational literature. see [1] – [12]. in both mentioned models, the total income is the sum of the total investment and the total consumption. following [13], we establish the exact solution of the cauchy problem for the differential equation in the harrod-domar model of macroeconomic dynamics with the time-dependent coefficient of the capital intensity of income growth (ciig). previously, the only constant coefficients of ciig were considered; see [12]. to confirm economic validity of the assumption that the coefficient of ciig depends on time, we investigate the exact solution of the solow model with the cobb-douglas production function. see, e.g., [3, 4] and [7] – [10]. calculation of the income growth capital intensity factor of this solution shows that this coefficient is a time-dependent function. in the present paper, we also show that the harrod-domar model is quite convenient for using the apparatus of the optimal control theory and the calculus of variations. using these methods, one can find the maximum of the integral discounted utility function of consumption. there are also formulated and investigated several optimal control problems with various constrains that follows from natural economic conditions. problems of this type are to find the maximum of a functional that expresses the integral discounted utility function in the presence of a differential relation. consumption and phase constraints are investigated, extremal problems in the pontryagin and dubovitsky-milyutin forms are considered. ∗corresponding author: chernyaev49@yandex.ru 72 a.p. chernyaev 2. the extended harrod-domar model in the harrod-domar model, the differential equation of the macroeconomic dynamics with exogenous dynamics of the consumption of arbitrary character [12, 13] has the form y (t) = c(t) +by ′(t). (2.1) in this model, time t is continuous. the income y (t) is equal to the sum of the consumptionc(t) and the investment i(t). usually, y (t) refers to the gross domestic product, which is identified with the national income. the economy is supposed to be closed, therefore, the net exports are zero and the government expenses are not considered in the model. the main factor of the growth – the speed of income growth – is proportional to the investment. see [6, 12, 13]. that is, i(t) = by ′(t), where b is the coefficient of the capital intensity of income growth (ciig) and 1/b is the limit product of capital at the macroeconomic level. previously, the coefficient of ciig was supposed to be a positive constant [12, 14]: b = const > 0. (2.2) in the case (2.2), the solution of differential equation (2.1) is given by the formula y (t) = y0e t−t0 b − 1 b ∫ t t0 c(τ)e t−τ b dτ. (2.3) we consider the cauchy problem: differential equation (2.1) with the initial condition y (t0) = y0 > 0. (2.4) the main assumption is that b = b(t). (2.5) the solution of this cauchy problem is given by the formula (see [13]): y (t) = y0e ∫ t t0 ds b(s) − e ∫ t t0 ds b(s) ∫ t t0 c(τ) b(τ) e − ∫ t t0 ds b(s)dτ. (2.6) obviously, in the case (2.2) formula (2.6) becomes (2.3). the economic validity of the assumption (2.5) follows from the comparative analysis of the harrod-domar model and the solow model. from the economic viewpoint, it reflects the rate of the technical progress and the rapidly changing of the economic conditions. 3. the solow model there are several different ways of the presentation of the solow macroeconomic model. see, e.g., the original works [3, 4] and the papers [6] – [11]. in the solow model, the average per capita capital k satisfies a first-order nonlinear differential equation, which follows from the balance equation for funds: dk dt = −λk + ρf(k), (3.7) with the initial condition k(0) = k0 > 0. (3.8) equation (3.7) with condition (3.8) yield cauchy problem we shall deal with. here k = k/l, where k is the capital or funds, l means human resources (or labour) resources, time t is measured in years, ρ is the norm of accumulation (the share of gross copyright © 2020 assa. adv syst sci appl (2020) dynamic models of economic growth 73 investment in the gross domestic product). following [7], we shall assume that ρ = const, 0 < ρ < 1. the function f(k) = f (k,l) l = f (k, 1). (3.9) here f (k,l) is the neoclassical production function; see [7, 8, 15]. the coefficient λ = µ+ ν, where µ is the share of the annual decrease of main production funds. similarly to [7], we assume that µ = const. the second term ν means the annual growth rate of the labour force, i.e. 1 l dl dt = ν. (3.10) here µ and ν satisfy the restrictions 0 < µ < 1, −1 < ν < 1. the balance equation for funds reads dk dt = ρy − µk. (3.11) then for the average per capita capital we have the following chain of the equalities dk dt = d dt ( k l ) = k ′tl−kl′t l2 = k ′t l −kl′t l2 = ρy − µk l − νk l . this yields equation (3.7). the transition mode [7, 8] was investigated under assumption that the production function f (k,l) is the cobb-douglas function [7, 8, 14]. indeed, in this case f(k) is the power function with a constant factor; see [7, 8]. then equation (3.7) is explicitly integrated, and its solution is represented via elementary functions. if ν is constant, equation (3.7) has the stationary solution k = k∗, where k∗ is the positive root of the equation ρf(k)− λk = 0. (3.12) here we assume that k0 and k∗ belong to the interval of average per capita capital under consideration. for k 6= k∗, integrating (3.7) with the initial condition (3.8), we obtain the integral equation ∫ k k0 ds ρf(s)− λs = t (3.13) which determines the function k = k(t). remark that formula (3.13) is correct if the function (ρf(s)− λs)−1 is integrable on the corresponding interval. sufficient conditions for this can be formulated in several ways. one can impose some conditions on f (as it is done in [11]) or impose conditions on the neoclassical production function f (k,l) and use (3.9). to satisfy the basic condition of the transient mode in the solow model [7, 8] k∞ = lim t→+∞ k(t) = k∗, (3.14) equation (3.12) needs to have one positive root k∗ on the interval under consideration. in addition, the improper integral ∫ k∗ k0 ds ρf(s)− λs (3.15) needs to diverge. this follows from the limit transition in (3.13) as t→ +∞ and (3.14). therefore, for the existence of a transition regime in the solow model one need to assume that function f(k) generated by the neoclassical production function f (k,l) satisfies to copyright © 2020 assa. adv syst sci appl (2020) 74 a.p. chernyaev conditions mentioned above: the existence of a unique positive root k∗ 6= k0 of equation (3.12) in the interval under consideration and the divergence of the integral (3.15). it is worth observing that if the solution to the cauchy problem (3.7), (3.8) is known, then all endogenous variables can be found from the equality y = f (k,l) = c + i. (3.16) here, the final product y is used for the non-productive consumption c and the investment i . now let us discuss the solow model with the cobb-douglas production function [7,8,14]: f (k,l) = akαl1−α, a = const > 0, 0 < α < 1. (3.17) substituting (3.17) in formula (3.9), we obtain f(k) = f (k,l) l = f (k, 1) = akα. (3.18) taking into account (3.18), one can bring equation (3.7) to the bernoulli form: dk dt = −λk + ρakα. (3.19) solving equation (3.19) with the initial condition (3.8) and the additional assumption λ = const, (3.20) we get k(t) = e−λt [ ρa λ eλ(1−α)t − ρa λ + k0 1−α ] 1 1−α . (3.21) let us find the capital intensity of the income growth for (3.21), that is, for the average per capita capital in the solow economic growth model with the cobb-douglas production function (3.17) and the additional condition (3.20). using the basic premise of the harrod-domar model, the definition of the solow model accumulation norm and the balance formula (3.16), we have i(t) = by ′(t) = ρy, (3.22) that is, y ′(t) = ρ b(t) y (t). (3.23) integrating (3.23), we get y (t) = e ρ ∫ t t0 ds b(s) , where the exponent is the indicator of the income growth. theorem 3.1: under the above assumptions, the capital intensity of the income growth is given by the following formula: b(t) = i(t) y ′(t) = ρf (k,l) d dt [f (k,l)] = ρk (ν − αν − αµ)k + αρakα . (3.24) copyright © 2020 assa. adv syst sci appl (2020) dynamic models of economic growth 75 proof from (3.22) and (3.16), we have b(t) = i(t) y ′(t) = ρy (t) y ′(t) = ρf (k,l) d dt [f (k,l)] . using formula (3.17) and multiplying the numerator and the denominator of the right hand side by a−1lαk1−α, one can write the latter equality in the form b(t) = ρkl αldk dt + (1− α)k dl dt . applying the formulas (3.10) and (3.11) to the right hand side of this equality, we obtain b(t) = ρkl αl(ρy − µk) + ν(1− α)kl . finally, taking into account (3.16), (3.17) and multiplying the numerator and denominator of the right hand side by l−2, we have b(t) = ρ(k/l) αρa(k/l)α + (ν − αµ− αν)(k/l) . to complete the proof, recall the definition of the average per capita capital: k = k/l. remark 3.1: formula (3.24) justifies the assumption that the capital intensity of the income growth depends on time. remark 3.2: equation (3.19) can be considered without assumption (3.20), i.e., λ = λ(t) is an arbitrary integrable function. this makes sense, because the annual growth rate of the employment (growth rate of the labour force (3.10)) is not constant. remark 3.3: in the case λ = λ(t), the solution of bernoulli equation (3.19) with the initial condition (3.8) is given by the formula k(t) = e− ∫ t 0 λ(τ)dτ [ ρa(1− α) ∫ t 0 e(1−α) ∫ s 0 λ(τ)dτds+ k0 1−α ] 1 1−α . 4. optimal consumption in the extended harrod-domar model in the extended harrod-domar model, formula (2.6) determines the income through the consumption. a natural question: how to find the consumption? from the mathematical viewpoint, this question can be formulated as an optimal control problem. we consider this problem is the most general form, as the problem of maximizing the integral discounted utility of the consumption [17, 18]:∫ t1 t0 u(c(t)) exp(−δt)dt⇒ max, (4.25) where δ > 0 is the discount factor. here we use the analogy with problems of optimal consumption management in the household economy, see [19] – [23]. following [19] – [21], copyright © 2020 assa. adv syst sci appl (2020) 76 a.p. chernyaev we suppose that the utility of consumption is represented by the function u(c), which reflects constant aversion to risk by arrow-pratt: a = −u ′′(c)c u′(c) ≥ 0. (4.26) the economic meaning of (4.26) is clear from the notation g(c) = u′(c). (4.27) then g(c) is the limit utility of consumption. further, taking into account (4.26) and (4.27), we have the equalities: g′(c) = u′′(c), ec(g) = g′(c) g(c)/c = u′′(c)c u′(c) = −a. here, ec(g) is the elasticity of g with respect to c. further, we assume that g(c) is a monotonically decreasing function, therefore, a ≥ 0. conversely, the assumption that g(c) is increasing, is associated with the risk. it is natural to call the condition that g(c) does not increase the disgust to risk. lemma 4.1: the derivative of the utility function u describing the constant aversion to risk by arrow-pratt (4.26) is given by the formula u′(c) = g(c) = γ ca = γc−a, γ = const > 0. (4.28) proof considering (4.26) as a differential equation with the unknown function u, we get a c = −u ′′(c) u′(c) . integrating this equation, we obtain a ∫ dc c = α ln |c| = − ∫ u′′(c) u′(c) dc = − ln |u′(c)|+ const = ln ∣∣∣∣ c0 u′(c) ∣∣∣∣ . since logarithm is a monotonic function, this yields |c|a = ∣∣∣∣ c0 u′(c) ∣∣∣∣ . since the consumption is positive, one can put |c0| = γ > 0. taking into account (4.27), the last equality implies (4.28). corollary 4.1: the utility function u satisfying the condition of lemma 4.1 has the form u(c) =  γc1−a 1− a + χ, a 6= 1; γ lnc + χ, a = 1; γ = const > 0, χ = const. (4.29) proof integrating equation (4.28), we obtain (4.29). copyright © 2020 assa. adv syst sci appl (2020) dynamic models of economic growth 77 theorem 4.1: the consumption function c(t) = (u′) −1 [ 1 b(t) c1 exp { δt− ∫ t t0 dτ b(τ) }] (4.30) gives the maximum in the variation problem (4.25) with fixed boundaries. proof substituting the expression of the consumption from (2.1) into (4.25), we obtain the functional j(y ) = ∫ t1 t0 u(y −b(t)y ′) exp(−δt)dt, (4.31) whose increment j(y + h)− j(y ) = ∆j(y, h) has the form ∆j(y, h) = ∫ t1 t0 [u(y + h−b(t)(y ′ + h′))− u(y −b(t)y ′)] exp(−δt)dt. (4.32) from the equality y −b(t)y ′ = c(t), it follows that u(y + h−b(t)(y ′ + h′))− u(y −b(t)y ′) = u(c(t) + h−b(t)h′)− u(c(t)), ∆j(y, h) = ∫ t1 t0 [u(c(t) + h−b(t)h′)− u(c(t))]e−δtdt. (4.33) substituting the taylor expansion u(c(t) + h−b(t)h′) = u(c(t)) + u′(c(t))[h−b(t)h′] + 1 2 u′′(c(t))[h−b(t)h′]2 +r(t), where r(t) = o[h−b(t)h′]2 as h−b(t)h′ → 0, into (4.33), we obtain ∆j(y, h) = ∫ t1 t0 u′(c(t))[h−b(t)h′]e−δtdt+ 1 2 ∫ t1 t0 u′′(c(t))[h−b(t)h′]2e−δtdt+ ∫ t1 t0 r(t)e−δtdt. (4.34) then, integrating by parts, we have − ∫ t1 t0 u′(c(t))b(t)h′e−δtdt = − ∫ t1 t0 u′(c(t))b(t)e−δtdh = − u′(c(t))b(t)e−δth ∣∣∣∣t1 t0 + ∫ t1 t0 h d dt [u′(c(t))b(t)e−δt]dt. (4.35) since we consider the variation problem (4.25) with fixed boundaries, in addition to the initial condition (2.4), i.e., the boundary condition on the left edge, we have the boundary condition copyright © 2020 assa. adv syst sci appl (2020) 78 a.p. chernyaev on the right edge: y (t1) = y1 > 0. (4.36) from (2.4) and (4.36), it follows that for the function h = h(t) the equalities h(t0) = h(t1) = 0. (4.37) hold true. from (4.37), it follows that the first term in the right hand part of (4.35) is zero. therefore, the equality (4.35) reads − ∫ t1 t0 u′(c(t))b(t)h′e−δtdt = ∫ t1 t0 h d dt [u′(c(t))b(t)e−δt]dt. (4.38) from (4.38) and (4.34) we conclude that ∆j(y, h) = ∫ t1 t0 h ( u′(c(t))e−δt + d dt [u′(c(t))b(t)e−δt] ) dt+ 1 2 ∫ t1 t0 u′′(c(t))[h−b(t)h′]2e−δtdt+ ∫ t1 t0 r(t)e−δtdt. (4.39) the sign of the left hand side of (4.39) coincides with the sign of the first term in the right hand side of (4.39). indeed, replacing h with βh, where β = const, we get ∆j(y, βh) = j(y + βh)− j(y ) = = β ∫ t1 t0 h ( u′(c(t))e−δt + d dt [u′(c(t))b(t)e−δt] ) dt+ 1 2 β2 ∫ t1 t0 u′′(c(t))[h−b(t)h′]2e−δtdt+ ∫ t1 t0 r̃(t)e−δtdt, (4.40) where r̃(t) = o(β2[h−b(t)h′]2) as β(h−b(t)h′)→ 0. passing to the limit β → 0, one can see that the first term in the right hand side of (4.40) tends to zero with the first order of smallness, while the second term has the second order of smallness and the third term has the order greater than two. this proves the statement. now let us prove that the first term in the right hand side of (4.39) is zero. suppose the contrary. then, replacing β with−β, we change the sign of the first term in the right hand side of (4.40), while the sign of the left hand side of (4.40) remains the same. this contradiction shows that ∫ t1 t0 h ( u′(c(t))e−δt + d dt [u′(c(t))b(t)e−δt] ) dt = 0. by the fundamental lemma of the calculus of variations, this yields the euler equation u′(c(t))e−δt + d dt [u′(c(t))b(t)e−δt] = 0. (4.41) taking into account (4.41), one can simplify (4.39) as follows: ∆j(y, h) = 1 2 ∫ t1 t0 u′′(c(t))[h−b(t)h′]2e−δtdt+ ∫ t1 t0 r(t)e−δtdt. (4.42) let us check that the solution of the euler equation (4.41) with the boundary conditions (2.4) and (4.36), and consequently, (4.37), give the maximum of the functional (4.25), or copyright © 2020 assa. adv syst sci appl (2020) dynamic models of economic growth 79 equivalently, (4.31). first, we remark that the sign of the left hand side of (4.42) coincides with the sign of the first term in the right hand side of (4.42). indeed, replacing in (4.42) h with βh, where β = const, we have ∆j(y, βh) = 1 2 β2 ∫ t1 t0 u′′(c(t))[h−b(t)h′]2e−δtdt+ ∫ t1 t0 r̃(t)e−δtdt. (4.43) here r̃(t) = o(β2[h−b(t)h′]2) as β(h−b(t)h′)→ 0. passing to the limit β → 0, one can see that the first term tends to zero with the second order of smallness, while the second term the order greater than two. therefore, the sign of the left side of (4.43) coincides with the sign of the first term in the right hand side of (4.43). it remains to use the inequality u′′(c(t)) ≤ 0, which follows from (4.26). taking into account (4.27), we can write equation (4.41) in the form g(c(t))e−δt + d dt [g(c(t))b(t)e−δt] = 0. (4.44) the change of variables w = g(c(t))b(t)e−δt, (4.45) i.e., g(c(t))e−δt = w/b(t), transforms equation (4.44) into w b(t) + dw dt = 0. integrating the latter equation, we obtain w = c1 exp { − ∫ t t0 dτ b(τ) } , c1 = const. substituting the obtained equality in (4.45), after obvious transformations we obtain g(c(t)) = 1 b(t) c1 exp { δt− ∫ t t0 dτ b(τ) } , c1 = const > 0. (4.46) finally, recall that g(c) monotonically decreases. therefore, it is invertible, and c(t) = g−1 [ 1 b(t) c1 exp { δt− ∫ t t0 dτ b(τ) }] , c1 = const > 0. (4.47) taking into account (4.27), from (4.47) it follows (4.30). the proof is complete. remark 4.1: the consumption function (4.30) can be also written in the form c(t) = [ γb(t) c1 ] 1 a exp { 1 a [∫ t t0 dτ b(τ) − δt ]} . (4.48) proof from (4.28) and (4.46) we have the equality [c(t)]a = γb(t) c1 exp {∫ t t0 dτ b(τ) − δt } , (4.49) which can be resolves by c and it gives (4.48). copyright © 2020 assa. adv syst sci appl (2020) 80 a.p. chernyaev remark 4.2: the consumption function (4.30) or, equivalently, (4.48) gives the maximum in the variation problem (4.25) with fixed boundary conditions (2.4) and (4.36). formula (4.49) expresses the consumption under the condition (4.26), which means that the utility function satisfies the constant risk aversion according to arrow-pratt. it is very convenient to formulate the above problems using the terminology from the control theory. the problem of maximization of the functional (4.25) under constraints (2.1), (2.4), (4.36), and the additional restriction 0 < c ≤ c(t) ≤ c < +∞ (4.50) is called the pontryagin problem. in the inequality (4.50), the lower bound c means the total subsistence minimum and the upper bound c is the total subsistence maximum. in turn, the pontryagin problem can be also formulated with an additional phase constraint, for example, y (t) ≥ const ≥ 0. such a problem is often called the dubovitskymilyutin problem. see [19] – [21]. 5. conclusion we presented the comparative analysis of two models of economic dynamics: the model of the economic growth by harrod-domar and the model by solow. there were several earlier models of the economic growth, but they are not widely acknowledged; see, e.g., [24]. at present, more advanced models are gaining popularity, however, they are based on the models discussed in the present paper; see [25] – [27]. the comparative analysis confirms the economic viability of the assumption that the ciig depends on time. comparing these models and using the analogy with household economies [19] – [23], we demonstrate the efficiency of the control theory approach in the extended harrod-domar model. the obtained results shaw that despite significant differences between these models, their comparative analysis is substantial. it is worth observing that using the approach [28] – [31] based on the theory of covering mappings, the both considered models can be generalized to the market of many goods with various production functions. references 1. harrod, r.f. 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(2018) necessary conditions for closedness of the technology set in dynamical leontief model, proc. 11th int. conf.: management large-scale system develop., moscow, russia: ieee. copyright © 2020 assa. adv syst sci appl (2020) introduction the extended harrod-domar model the solow model optimal consumption in the extended harrod-domar model conclusion microsoft word 1155 article text, copyedited.doc adv syst sci appl 2021; 04; 57-64 published online at https://ijassa.ipu.ru. the multidimensional network models method of developing discrete microfluidics andrey v. balabanov 1*, asim m. kasimov 2 v. a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia 1) e-mail: fca07@mail.ru 2) e-mail: kasimov@ipu.ru abstract: the performances of the microfluidic units may vary within a wide range depending on the design, even if the operational geometry remains the same. in order to obtain the most effective results of the designing, it is reasonable to use the formal procedures of systematizing possible variants of the design, analyzing these variants, and selecting the best design variant based on the preset criteria. in this context, the problem of creating the effective formal methods of designing the microfluidics is topical. the main purpose of this paper is to propose a decision of this problem. to solve the problem mentioned above, the authors devised a method of analyzing and synthesizing the designs based on the multidimensional network models (mnmmethod). the mnm-method provides the possibility of generating and in-computable-formanalyzing qualitatively different variants of the design on the principle of the unity of the geometrical, structural-hierarchical, and functional characteristics, as well as the possibility of selecting the best design variant on the basis of the preset criteria by means of analyzing both the structure and features of the multidimensional networks. keywords: multidimensional network model, design analysis and synthesis, microfluidics, reserve control system 1. introduction the experimentally validated high performances of microfluidic control elements, in combination with the small sizes, in large part, set the trend for the implementation of microfluidic devices in the form of integrated circuit units – monolithic (indivisible) designs, without using discrete components. the integrated form of the control microfluidics provides possibilities to avoid (or to significantly minimize) mounted interconnections, reduce the energy loss in transmitting the data, and provide the high performances like those of the discrete microfluidic elements. therefore, the main directions of developing the control microfluidics, at present, are improvement of the weight-size parameters as well as transition to the integrated technology. to effectively develop these trends it is necessary to bring the gap between, at the one hand, the conceptual, schematic and algorithmic decisions, and, at the other hand, the design implementation. as a solution, this paper represents the formal method of designing of the microfluidics, which is based on the multidimensional network models – the mnm-method. the mnm-method of designing the microfluidics is a new formal method of the design analysis and synthesis, with the effectiveness proved by practice. the method uses a new kind of the generalized models – the multidimensional network models – that are based on the ‘structure class’ entity. in order to design the microfluidics, the formal definitions of the geometrical, structural-hierarchical, and functional structure classes have been developed. * corresponding author: fca07@mail.ru 58 a.v. balabanov, a.m. kasimov copyright ©2021 assa. adv. in systems science and appl. (2021) in this paper, the key stages of applying the mnm-method are represented by the example of designing the microfluidic generator of the 100-µm feature size. this is very important from the practical viewpoint, because the microfluidics is increasingly used to create high-technology products. for instance, the microfluidics provides potential to create promising non-electric reserve control systems (rcs). the prospectivity of rcs is mainly determined by the resistance of the microfluidics to multiple destabilizing factors resulting in failures of electronics (for instance, radioactive and corpuscular emissions, electromagnetic emission, high temperatures, etc.). the microfluidics has its own approaches to generating, saving, transforming, and transmitting the data. therefore, the microfluidics allows new original cybernetic systems to be developed by means of the non-electronic element base. at the present time, the fields taking advantage of the microfluidics also are microanalytics, micromechanics, biotechnology, bioengineering and other complex scientific areas. this paper represents one of the possible applications of the mnm-method – designing the microfluidics. in general, the field of application is wider than the above one. however, not to disturb the monographic composition of the paper, no alternative was considered within the research undertaken. therefore, the authors do not declare the universality of the mnm-method, despite its large potential to be used as a development tool. 2. background there are a number of the researches devoted different aspects of creating the microfluidics. these are the schematic development, the investigation of microfluidics features, the modelling, and the manufacturing. in this context, what should be noted firstly are [2-11]. however, having performed the literature overview, the authors revealed no formal methods to aid the microfluidics designing. that the microfluidics has no formal tools to be designed is a circumstance that has been inhibiting and would have inhibited the development in this field – if this had been ignored. this paper is an attempt to make the microfluidics designing both more comprehensive and more scientific-proved, at least, in relation to the devices similar those represented bellow in order to give an instance of the mnm-designing. 3. mnm-method analyzing and synthesizing the microfluidic design, it is necessary to take into consideration the set of the possible variants of this design in order to select the best design structure based on the requirements for its characteristics. the mnm-method supposes that the main origin of generating the implementation set is the decomposition of the characteristics into the functional, geometrical, and structurehierarchical classes, with these classes systematized in the form of mnms. identifying the classes within the microfluidics (mf) is performed by means of the following formal procedures of the decomposition: (3.1) the procedure of the geometrical decomposition ( ) is represented by (3.1) and allows the set of the geometrical implementations of the design to be generated. the result of is the set of the structure classes, such that for any pare of the instances ( ) of any of these classes, the value areas of the geometrical functionals ( ) of the instances can differ by up to . the completion of the procedure results in the set of structure classes defining the geometrical structure of the microfluidic unit being designed: { } ( ) ( ) ( ) g ib g ia g i g ib g ia g m i i gg fefessssmfo d£-î"® = ,,|: 1 go go i gs ib g ia g ss , ib g ia g ff , gd go the multidimensional network models method of developing discrete… 59 copyright ©2021 assa. adv. in systems science and appl. (2021) (3.2) the formula (3.2) reflects the procedure of the structure-hierarchical decomposition. the result of is the set of the structure classes, such that for any pare of the instances ( ) of any of these classes, the definition domains of the geometrical functionals ( ) of the instances can differ by up to . thus, setting different requirements for , it is possible to obtain qualitatively different sets of the structure classes. for instance, the structure-hierarchical decomposition can be performed by the criteria of the standardization, the product kind (assembly, detail, kit, and complex), and so on. it is the requirement setting that is the main origin of formation of a variant set of the hierarchical structure of the microfluidics: (3.3) the procedure of the functional decomposition is described by (3.3). the result of is the set of structure classes, such that for any pare of the instances ( ) of any of these classes, the definition domains and the value areas of the geometrical functionals ( ) of the instances can differ by up to and , correspondingly. the functional decomposition provides formation of a set of the classes defining the functional structure of the microfluidics. mnms are built on the basis of the sets of the structure classes resulting from the decomposition procedures. to design a microfluidic unit with required parameters it is necessary to systematize the sets of the instances of the structure classes by means of mnms. these models have the i, j, and k dimensions. the subranges of i, j, and k depend on the cardinalities of the classes sets obtained as the result of decomposition. for example, let the two functional classes would be generated. then, the number of i-subranges is equal to two (i.e. one has and ). thus, the mnm dimensions correspond to the classes sets. given below is the case of mnm-designing involving the i, j, and k simple (single-member) subranges. in general, the number of the dimension subranges may be varied, and it is chosen depending on the complexity of both analyzing and synthesizing the design being created. 4. practice in this section, the discrete-microfluidics designing based on mnms is considered by the example of the microfluidic three-stage generator. at the first stage of the designing, it is necessary to carry out the analysis and the classification of the product requirements in terms of the preset features, with the number of these features depending on the task being solved. in relation to the generator, three groups of the features are specified: geometrical, structuralhierarchical, and functional. corresponding to the three groups of the features, the set of the structure classes (in this case, these are the geometrical, structural-hierarchical, and functional classes) is to be formed. for each of the structure classes, the fixed-cardinality set of the instances representing possible physical implementations of the class within the design to be created shall be generated and rated by the implementation costs. given in fig. 4.1 are the schematic diagram of the generator as well as the common operational geometry of its elements. the operational element of the generator under design is a microfluidic trigger. it is the trigger function that represents the main functional class. as the generator is a three-stage one, it consists of three triggers. these may be both identical and different in implementation. each original implementation of the trigger is an instance of the corresponding structure class. { } ( ) ( ) ( ) h ib h ia h i h ib h ia h n i i hh fdfdssssmfo d£-î"® = ,,|: 1 ho i hs ib h ia h ss , ib h ia h ff , hd ho { } ( ) ( ) ( ) ( ) ( )( )feib f ia ffd ib f ia f i f ib f ia f k i i ff fefefdfdssssmfo d£-ùd£-î"® = ,,|: 1 fo i fs ib f ia f ss , ib f ia f ff , fdd fed 1i 2i 60 a.v. balabanov, a.m. kasimov copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 4.1. schematic diagram (a) and operational geometry (b) of three-stage microfluidic generator fig. 4.2. example of possible implementations of microfluidic trigger to form the sets of the geometrical and structural-hierarchical classes, one considers the three available implementations of the microfluidic trigger. these are represented in fig. 4.2. they are different in the operational geometries, but have the same functions (memory cell) and the same design kinds (in the form of a sheet detail). it means that t1, t2, and t3 belong to same functional and structural-hierarchical classes, but to different geometrical classes. it should be noted that this example of dividing into the classes relates to the elements, but not to the generator as a whole. however, it is the classification by the feature groups specified that is the basis to form the classes sets. at the second stage of designing, the sets of the classes instances shall be systematized in the form of a multidimensional network model (mnm). building the mnm is to arrange the instances by placing the corresponding nodes along the i, j, k axes in order of increasing the implementation costs. this process is equivalent to generating a set of the possible variants of the design. each of these design variants is represented by a unique path from the input to the output of the mnm. fig. 4.3. mnm of microfluidic generator (a) and mnm-level example (b) the multidimensional network models method of developing discrete… 61 copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 4.3a demonstrates the three-dimensional mnm of the microfluidic generator, built on the basis of the previous procedures of designing. the number of the dimensions (three) of this mnm is determined by that, as an example, one class of each of the feature groups has been specified: 1) the functional class (i) is defined as the trigger function, 2) the geometrical class (j) is defined as the operational profile core, and 3) the structuralhierarchical class (k) is defined as the design kind. in general case, the number of the dimensions is determined by the sets of the structure classes, and can be varied. thus, the nodes of the mnm are the instances of the structure classes, i.e. the elements of the generator design. each of the ellipsoids includes the set of physical implementations of one of the functional-class instances. the loops of the mnm correspond to the operations of building the instances of the structure classes. in fig. 4.3a, the loops reflect the assembly operations involving the operational elements (triggers). other edges (excepting the input and output ones) of the mnm represent the transition operations between the pairs of the class instances (for example, making interconnections of the triggers). the end nodes of the transition edges shall be matched by the input and output characteristics (flow-rate, frequency, pressure, etc.). the input and output edges of the mnm reflect the complex assembly operations of mounting the terminal elements (base, covers, external interconnections) of the design. the mnm having been built, the edges shall be weighted. a required design variant within the mnm is selected by means of the procedures of analyzing the mnm features, evaluating the weights of the edges, and calculating the mnm shortest path in terms of the preset criteria. given that the generator design must provide the maximally possible frequency, the main criterion to be used in selecting would be reasonably assumed as being defined by the hydraulic resistance of the connection channels. consequently, the selecting is to evaluate the different combinations of the cross-section and the route, which are available in the mnm. as an example, fig. 4.3a explicitly shows the set of the edges, which renders the shortest path calculated by the criterion of the minimal coefficient of the hydraulic resistance of the connection channels. calculating of the resistance as depended on the design parameters is described in [1]. the determination the hydraulic resistance coefficient aids mnm-modelling the generator design in the part which is concerned with forming and estimating the physical implementations involving different sets of the mnm-levels. this part of the mnm-modelling is very important due to its large structure forming influence on the design. the nodes corresponding to the physical implementations are placed in the mnm-levels and are, in fact, reference decisions to design. as an example, given in fig. 4.3b is one of the levels of the mnm. the instances of the functional structure class (trigger) represented in fig. 4.3b are assigned as t1, t2, and t3. the figure shows that these instances are three different operating geometries (the i-dimension). each of these geometries is represented by four implementations (the j-dimension) differing in relative positions within the reference system of the generator. provided the structure-class instances are those represented in fig. 4.3b, possible variants of the generator design could be built with different sets of the mnm-levels (in this case, from one to three – the k-dimension) and, accordingly, would be different in interconnections (lengths, cross-sections, trajectories). the mnm given in fig. 4.3a encapsulates a set of the design variants of the generator, with each of the variants modelled by an original path from the input to the output of the network (a set of the nodes and the edges). the path to be selected should meet the criteria of the design quality most advantageously. the influence this path has on the performances is reflected by the weights of the mnm edges. for example, if two paths would represent two different variants of the allocation of the operational elements, the interconnections should be routed differently. the differences would influence such performances of the generator as the frequency, the branching factor, the hydrodynamic resistance, and others. therefore, the total weights of the two paths should differ in value. if some of the interconnections cannot be implemented because of design and technological factors, or the implementation results in 62 a.v. balabanov, a.m. kasimov copyright ©2021 assa. adv. in systems science and appl. (2021) inoperable state of the device, the weight of the corresponding edge is set equal to infinity (given that the best design variant is to provide the minimal total path weight). thus, the selection of the shortest path is defined as depended on the criteria of estimating the quality of the device being designed. the generator is given to be designed in order to evaluate the maximal frequency response of the operational elements. therefore, it is this response that is the main criterion of estimating the design. in general, the criteria of estimating the design quality by using the mnm-method may be both elementary and complex (taking into account technical, economic, ergonomic and other characteristics). the mnm shown in fig. 4.3a is a generalized design model of the generator. this model represents the generator in the form of the set of the structure class instances (nodes) and the operations (edges) on these instances. this set is ordered by the specified (meaningful) characteristics of the generator. the i and j dimensions define, correspondingly, functional and geometrical characteristics of the elements (triggers) at the mnm-levels of the design. the k dimension defines the number of these levels. as a result of analyzing the mnm, the following is revealed. allocation of the elements in the only planar level provides a possibility of manufacturing the generator as a detail, i.e. in the form of a monolithic design, without using assembly operations. however, the planar design does not allow the length of the interconnections between the generator stages to be reduced to minimum, which decreases the generator frequency. therefore, allocation of all the operational elements on the only plane does not provide a possibility of experimentally investigating the extreme frequency response. in addition, the use of the initial operational elements (see fig. 4.2) in the form of discrete units causes the need for assembly operations. the element designs given in fig. 4.2 provide the minimal length of the feed-back channels under the condition of the three-level generator design, with the structure class instances of this design outlined with the thick circle lines in fig. 4.3b. the figure shows that the orientations of the operational elements t1, t2 coincide. the element t3 is rotated by 180° in relation to the two others. therefore, the j-coordinates of t1 and t2 instances coincide, and the j-coordinate of t3 is different. it should be noted that fig. 4.3b demonstrates the differences in the i and j coordinates, but it does not determine allocation of the instances to the mnm-levels. as it is mentioned above, the design must be three-level. hence, the outlined instances are to be allocated to the three different levels, i.e. they are to have different k-coordinates. fig. 4.4. schematic and assembly diagrams of generator fig. 4.4 represents the schematic and assembly diagrams of the generator. the output signal of the first stage is transmitted to the input of the second stage directly, without using the multidimensional network models method of developing discrete… 63 copyright ©2021 assa. adv. in systems science and appl. (2021) mounted connections. the same is truth in relation to the output of the second stage and the input of the third stage. then, the signal is looped from the output of the third stage to the input of the first stage. the thicknesses of the matching boards are negligible. in this case, the lengths of the feed-back channels between the first and third stages are approximately equal to the thickness of the operational element (200 µm), which allows the design frequency to be considered as close to the maximum. thus, the design variant represented in fig. 4.4 is the best one from the view point of investigating the frequency response. the shortest path rendered in fig. 4.3a in the form of the set of the mnm edges and their end nodes corresponds to this design. fig. 4.5. experimental model of generator in accordance to the design variant selected by means of the mnm, the experimental model of the three-stage microfluidic generator has been created (see fig. 4.5). the mnm edge 1 in fig. 4.3a corresponds to the technological operation of mounting the top cover featuring the in/outlets in order to supply the power and transmit the data signals from the monitoring points of the first stage to a pneumoelectric converter. the further operations are concerned with mounting the three operational elements (edges 2, 4, 6) and their interconnections in the form of matching boards (edges 3, 5). the edge 7 corresponds to the technological operation of mounting the bottom cover, as well as the fixings and the output connectors. 5. conclusion the undertaken research has given rise to the development of the mnm-method regarding the discrete microfluidics. mnms are proved by the authors to be applicable to generate and compactly represent sets of the possible variants of microfluidics designs, as well as select the best design variant through calculating the shortest path within the network. using mnms, the designs may be synthesized, comprehensively analyzed, variously modified and repeatedly transformed, estimated, and ultimately properly structured. the number of the possible variants and the best design variant may be varied with requirements for the quality of the microfluidic device being created. these requirements may cover multiple aspects concerned with the creating, the exploitation, the repair, and the service. focused in this paper is the frequency response of the generator. more concretely, it is the maximal performance. this characteristic is provided by the corresponding design decision made by means of the mnm involving the structure and technology aspects. thus, the results reported in the paper prove both the reasonability and the feasibility of using the mnm-method to create devices of the discrete microfluidics. 64 a.v. balabanov, a.m. kasimov copyright ©2021 assa. adv. in systems science and appl. 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(2017). multilayer patterning technique for microand nanofluidic chip fabrication. microfluidics and nanofluidics, 21(12), 174,https://doi.org/10.1007/s10404-017-2013-5. advances in systems science and applications (2013) vol.13 no.2 167-181 e-market segmentation for internet-mediated fashion brands: a conceptual framework with mean-variance consideration tsan-ming choi, pui-sze chow and jin-hui zheng institute of textiles and clothing, the hong kong polytechnic university, hung hom, kowloon, hong kong. abstract e-market segmentation takes a crucial role in marketing for fashion brands. its applications for collaborative functions such as estimating the advertisement budget, retaining customers, carrying out direct tailored marketing, and implementing dynamic pricing are essentially important and critical for their e-business. unlike the bricks-and-mortar traditional store retailers, fashion brands which operate online can keep track of the details of the online customers easily and precisely. how to make use of these customers details in segmenting the e-market for each particular function becomes an important issue. as a result, we propose and discuss in this paper a conceptual model for carrying out e-market segmentation and we focus on the areas of dynamic pricing and advertisement budget estimation. through extensive discussions with mean-variance consideration, we believe that the model can be incorporated into other existing market segmentation analyses. managerial implications are discussed. keywords fashion branding, e-market segmentation, internet marketing, meanvariance 1 introduction for fashion brands which operate solely in the traditional bricks-and-mortar retailing mode, keeping track of the buying behaviours and preferences of each specific customer is a difficult, if not impossible, task. even if the fashion brands can collect some customers’ data by observation, the results are relatively sketch [1]. unlike the bricks-and-mortar stores, fashion brands which have its online channel (we call them efbs (e-fashion-brands)) can precisely keep track of the online customers’ buying behaviours easily. details such as the surfing habit, buying preference, price sensitivity, loyalty, etc can all be recorded and estimated. obviously, information of these details can help the efbs in making good business decisions. one of the most intuitive uses of these observed customers buying data is for marketing purpose (see [2] for the discussion of the issue of marketing on the internet). market segmentation∗ , as defined in chaffey [4], is the “identification of different groups within a target market in order to develop different offerings for ∗see [3] for a recent review and discussion on a framework for market segmentation and its applications. tsan ming choi : e-market segmentation for internet-mediated fashion brands... 168 the groups”. it involves identifying segments in the market, grouping customers into segments, and targeting, positioning and developing a differential advantage over competitors [5]. it also represents a rational and precise adjustment of the products and services for customers [6]. a good market segmentation scheme implies a good analysis of the target market and an identification of the true requirements of each specific group of customers [4, 7-9]. in modern business world, market segmentation has been realized as an essential part in nearly all marketing projects [4,10-11]. according to dibb and stern [8], the literature of market segmentation mainly goes to two distinct streams. the first stream treats market segmentation as a technique in determining and studying different segments in the market. the second stream views market segmentation as an approach for an efficient allocation of resource to different segments in the market. this paper basically follows the viewpoint of the second stream. in both the academic literature [10-12] and the industrial practice [13], in general, we need two types of information for market segmentation. the first type is called the “classification variables” which include four types of variables: demographic variables (e.g. age, gender, etc), geographic variables (e.g. city, country, etc), psychographic variables (e.g. risk attitude, lifestyle, etc) and behavioral variables (e.g. brand loyalty, usage level, etc). the second type is the descriptor variables which describe each particular segment and are used to distinguish one segment from another. many classification variables would function as the descriptor variables, too. market segmentation can be a complicated process in business. for example, suppose that after adopting the conventional segmentation steps with the classification according to the geographical and demographic variables, we have obtained a number of different segments. an important question to ask is: “should we continue to break down the segments into smaller segments?” no matter the answer is “yes” or “no”, we need to have a good reason for it. in other words, a good market segmentation scheme requires a good stopping condition. it is true that there are a few rules of thumbs for decision makers to decide the number of segments and when to stop the segmentation process. some factors [13] that decision makers will bear in mind include: (1). the size of the segments must be large enough. (2). the segments must be reachable by the company’s marketing strategies (e.g. promotion, pricing, etc). (3). the segments must be relevant to the company’s products and different segments should be clearly and sufficiently different. these guidelines are important and practical but they do not provide a precise decision model for decision makers for making a wise and optimal decision for a specific collaborative function with market segmentation. 169 advances in systems science and applications (2013) vol.13 no.2 with the advance of science and technology, market segmentation can now be done by a computerized decision support system. methods which rely on data mining and artificial intelligence [14-15], advanced database system [16], operations research and management science optimization techniques [6,10,17-18] , and others [11-12,14,19-20] have been widely tested and applied. however, none of these methods dominate the literature and each of them has its pros and cons. as many recent research projects revealed and discussed, market segmentation is very important for internet-enabled e-commerce [4,16,21-22]. this also gives rise to the term “e-market segmentation” which represents the segmentation of the electronic market. in fact, good e-market segmentation is generally believed to be not only important for maintaining the company’s competitive edge, but it is also essential and crucial, for efbs in the knowledge-based economy. when we look deeper into the potential collaborative applications of the e-market segmentation scheme for efbs, we can identify several major functionality areas for it. some of them include the following: 1. estimating the advertisement budget for acquiring different groups of customers: different customers have different values for the efbs. some of them only surfed around and did not buy a single item at all; some made one purchase and then did not come back; some made repeated purchases and showed a strong sense of trust on the efb (see moe and fader [23] and betts [24] for the identifications of the four types of online shopping visits). since it can be very expensive to acquire new customers on the internet [25], the advertising strategies should be more focused. the expenses spent on advertisement for different groups (i.e. segments) of customers should be made different, too. this can be achieved by a good e-market segmentation scheme. on the other hand, the information from the estimation of advertisement budget can be useful for e-market segmentation. 2. direct marketing by providing tailor-fit services and products to the interested customers: different customers have different preferences and requirements. if an efb can provide the right product to the right customer at the right price at the right time in the right place, then a transaction results. even though it is impossible for the efb to exactly predict all of these aspects, good market segmentation can help to narrow down the error in terms of the direct marketing offers of product type, selling price, etc. it can also help in turning browsers into buyers [23-25] and improve customer relationship management [26]. 3. dynamic pricing: different consumers have different sensitivities towards prices and other attributes for the products [1,27]. some consumers are highly price-sensitive while some care more about service and reliability. surveys also show that many customers purchasing online do not shop around and they just buy from the efb which they first visit. according to a survey as reported in baker et al. [27], the percentage of consumers who buy online from the first tsan ming choi : e-market segmentation for internet-mediated fashion brands... 170 website they visit ranges from 76% to 89% for various products including cds, books and electronics. in light of this, efbs should try to attract the customers at the very beginning by bringing the products in front of the consumers when they are in need (it is mentioned in the last paragraph). at the same time, how to set the selling prices dynamically for different groups of customers so that the profit for the efb is maximized becomes vitally important. failing to do so can lead to the collapse of the business (see the example of sun country in dutta et al.[28]). in order to exercise a profit-making dynamic pricing scheme for different groups of customers, we need to segment the market properly. 4. retaining customers: for all kinds of retail businesses, there are always some loyal customers. for every loyal customer, there is a sense of trust between the retailer and herself. however, the importance of customer’s loyalty varies among different retailers. for instance, a retail store whose customers are mainly the tourists from overseas countries care relatively less about the loyalty of the customers since it is unlikely for the overseas tourists to come back again in the near future while a cosmetics retailer always wants to maintain a loyal customer base (since the loyal customers are more likely to try some other products and have repeated purchase’). since most of the customers buying online have to contribute some private data to the efb (e.g. the credit card number) but they cannot touch the product and cannot visit the store to talk with the sales staff face-to-face, they simply won’t make a purchase without trusting the efb. as a consequence, compared with the bricks-and-mortar stores, the trust and loyalty of customers is especially important for efbs. moreover, having the loyal customers can help the efbs in at least two ways. first, a loyal customer is more likely to repeat her purchase. second, a loyal customer tends to refer new customers to the efb she is loyal to. it is why reichheld and schefter [1] have proposed that “price does not rule the web; trust does”. in order to gain the trust and keep the loyalty of the customers, we have to focus on their needs. without proper market segmentation, building and keeping the loyalty of the customers becomes more difficult. from the above description, we can see that a good e-market segmentation scheme is undoubtedly crucial for the success of efbs and it can be applied to different collaborative functionality areas. however, what is a good e-market segmentation scheme for each particular collaborative function? obviously, a good e-market segmentation scheme depends on its specific targeted functionality area. for example, a good e-market segmentation scheme for dynamic pricing may not be good for the estimation of advertisement budget. as a result, we propose in this paper a decision model for carrying out e-market segmentation for different specific functionality areas. we focus our attention on two important functionality areas: advertisement budget estimation, and dynamic pricing. 171 advances in systems science and applications (2013) vol.13 no.2 these are important functions for efbs and they can be collaborated with the conventional e-market segmentation scheme. through the incorporation of the performance measure (for the collaborative function) into the e-market segmentation decision model, we can provide more precise e-market segmentation results for the marketing managers to make an optimal decision (with each particular function). the idea behind this proposed decision framework is inspired by the classical markowitz’ mean-variance theory in financial portfolio management [29] with which we quantify and control the uncertainty associated with a decision by the variance of that measure (this point will be discussed in the next section). with the illustrative examples, we demonstrate the applicability of the proposed model. the organization of the rest of this paper is as follows. we first present the mean-variance decision framework which combines the conventional e-market segmentation and the collaborative function in section 2. the detailed e-market segmentation schemes for estimation of advertisement budget and dynamic pricing are proposed in sections 3 and 4. the e-market segmentation schemes for other functionality areas are discussed in section 5. we conclude with the discussion of managerial insights in section 6. 2 mean-variance decision models before we present each particular e-market segmentation model for the specific function area, we propose in this section the basic general decision model. in performing e-market segmentation, as we mentioned earlier, it is usual that the marketing managers would make use of the demographic variables, geographic variables, psychographic variables and behavioral variables. despite the intuitive physical meanings behind these variables, some of the classifications with these variables may not be very helpful for all specific collaborative functions. for example, when the objective of a particular market segmentation project is to decide the advertisement budget (and hence the advertisement strategy), the profit that can be generated by each customer becomes a key measure. as a result, we should incorporate a measure of the profit generated by the customers in the market segmentation scheme. however, different customers can carry different profit-values to the company, a precision control rule is hence essential for building a good market segmentation decision. in the advertisement budget estimation example we mentioned above, a precision control rule can be imposed on the degree of uncertainty of the profit. thus, a mean-variance consideration with which the average profit is used as the performance measure variable and the variance of profit is applied as a precision control variable. in this paper, we call the market segmentation scheme which includes the average objective performance measure and the variance of this performance measure for a particular tsan ming choi : e-market segmentation for internet-mediated fashion brands... 172 market segmentation project the “mean-variance performance measure market segmentation scheme (mvpm)”. with mvpm, we can carry out market segmentation following a mean-variance consideration with which the segmentation is performed with a measure by the “mean” and its precision is controlled by the “variance”. since the variance is an absolute measure, we make use of the relative measure of the coefficient of variation, defined as “the standard deviation divided by the mean” as the precision measure. the general idea behind mvpm is that: the e-market segmentation scheme follows the conventional type of segmentation policy and yields a number of emarket segments. this segmentation process is treated as an initial and basic e-market segmentation. after that, depending on different collaborative functions, the manager of each function would impose an additional measure on each of the segments. systematic and precise evaluation is carried out and further segmentation or re-segmentation may be required depending on the evaluation results. by doing so, tailor-fit e-market segmentation results are provided to each collaborative function and optimal decision can hence be made. in the following sections, we outline the use of the concept of mvpm for several important collaborative functionality areas with e-market segmentation. 3 segmentation for estimation of advertisement budget the expenses companies spent on online advertisement is expected to increase in the coming years. as reported in bhatnagar and papatla [21], forrester research has estimated that the spending on online advertisement in the united states will reach us$22 billion by 2004, which is more than 8% of the total spending for advertising in the united states. in fact, acquiring customers on the internet can be very expensive. as estimated and shown in hoffman and novak [25], many efbs have spent more than us$100 to acquire a new customer and some have even spent us$500! however, the “values” of most new customers, as measured by the expected lifetime spending on the efb, are less than these advertising expenses. in fact, some consumers shopping around the internet only surf and buy nothing; some of them may only make one purchase and never come back; some may have repeated purchases and are loyal customers. as we mentioned above, since it is expensive to acquire new customers on the internet, the advertising strategies should be more focused. recalling from the well-agreed pareto rule (or called the 80-20 rule), the majority of profit is actually generated by a relatively small amount of customers. it is thus a wise decision to focus the company’s resource on promoting to the customers which can bring higher profits. furthermore, the expenses spent on advertisement for different groups (i.e. segments) of customers should be made different. this can be achieved by a good e-market segmentation scheme. in the 173 advances in systems science and applications (2013) vol.13 no.2 literature, there are a number of methods proposed to help. they include the use of the customer’s searching behavior [21] and the use of personalized advertisement [9,14]. in this section, we apply the mvpm to build a decision model for the estimation of the advertisement budget for different market segments. under our proposed mean-variance model, mvpm, the efb should incorporate the performance measure and the precision control measure for “the profit that can be generated by the customer” into the decision model for market segmentation. thus, when efbs try to evaluate the “value” of the customers purchasing online, they should investigate the e-market segments with respect to the profit generated by the customers in each of these segments with a precision control. to be specific, we have the following market segmentation decision model: “the efb first segments the customers according to the conventional segmentation scheme by, for example, the geographic and demographic variables. then, the efb checks (and/or further segments) each group with respect to the average generated profit and the variance of profit from the members of that group. the objective is to ensure that the coefficient of variation of profit, defined as the standard deviation of profit divided by the average profit, for each segment is under the efb’s precision control and the segments size is large enough. after that, the final outcome from each market segment will have an average value for the customers in that segment and this average value can be a good representative measure since its variation is under the efb’s control.” to illustrate the above statement, let us consider a simple example. suppose that an efb has classified his customers according to the conventional segmentation scheme with respect to the customers’ variables of location, gender, age and education, and under a constraint on the size of each segment. after this segmentation scheme, he has obtained different market segments. when the efb looks deeply into each obtained market segment, he can identify the profit that has been generated by each customer inside each market segment. he can then obtain the average profit and the variance of profit generated by all the members in each market segment. for a particular segment, if the coefficient of variation of profit is larger than a certain threshold (decided by the efb), the level of uncertainty of the profit generated by the members inside this market segment is too large. further market segmentation should be carried out by adding another attribute (e.g. the purchear frequency of the customers). if the coefficient of variation of profit is less than a certain threshold (decided by the efb), the market segmentation that has been done is good enough and the efb can stick with it. however, if the coefficient of variation of profit is too large but the market segments size is also relatively small, further market segmentation should not be carried out. in this case, the efb needs to reconsider carrying out the market tsan ming choi : e-market segmentation for internet-mediated fashion brands... 174 segmentation in another way with consideration of other variables at the very beginning. we summarize this approach in the following proposition. proposition 1. the e-market segmentation scheme for estimating the advertisement budget can be stopped when the coefficient of variation of profit generated by the members of that segment is less than a threshold α. if the coefficient of variation of profit is larger than α, then further segmentation or re-segmentation is needed. to give a better picture of the proposed decision model in proposition 1, let us have an illustrative numerical example below. example 1. an efb has performed a market segmentation scheme by using the conventional variables of city, gender and age and identified 18 e-market segments. suppose that this efb looks into two distinct segments: segment 1 and segment 2, where segment 1 refers to the group of customers who live in city 1, male, and aged between 22-25, and segment 2 refers to the group of customers who live in city 2, female, and aged between 18-21. the desirable minimum size of each segment is 500. for segment 1, there are 1600 customers and for segment 2, there are 850 customers. the profit generated by each one of the customers can be found from the efb’s database. when the efb calculates the average profit per head (ap), variance of profit (vp), standard deviation of profit (sdp), and coefficient of variation of profit (cvp) generated by the members of each segment, he has the following results, table 1.1 example 1 segment 1 segment 2 average profit (in $) 50 180 variance of profit (in $2) 352 802 standard deviation of profit (in $) 35 80 coefficient of variation of profit 0.70 0.44 obviously, even though the vp of the members in segment 1 is smaller than the members in segment 2, the cvp is much larger. in fact, the profit uncertainty for segment 1 is too large to be ignored. thus, if the precision threshold of the efb (α) is 0.5, then the e-market segmentation for segment 2 is good enough because its cvp is less than 0.5. however, the e-market segmentation for segment 1 is not good enough because segment 1’s cvp is larger than α. since the size of this segment is 1600 and the minimum segment size is 500, the efb can consider carrying out further segmentation on segment 1 by using another classification variable. after the e-market segmentation scheme with the precision control over the profit generated by the members of each segment, the efb can make use of the estimated average profit for each segment as an indicator to decide the 175 advances in systems science and applications (2013) vol.13 no.2 amount of budget for advertisement on each segment. in this example, the value for each customer in segment 2 is $180 and a reasonable budget (say $100 per head) for advertising towards customers in this market segment can hence be estimated. as summarized in proposition 1 and illustrated by example 1, we can see that a decision model which provided a tailor-made stopping condition for the e-market segmentation process has been proposed. as we mentioned earlier, advertisement for e-commerce can be very expensive. nowadays, on-site banner, pup-up screen, emailing, affiliate program, tv and radio commercials, etc are all popular means of advertisement. however, obviously, different means of advertisement carry different costs. as a result, it is a wise decision to decide the advertisement budget for each specific group of customers before considering the specific means of advertisement. by having the market segmentation scheme as described in proposition 1 where the members of each segment are grouped together with the consideration of the average profit generated under precision control, we can identify precisely the “value” of each member of that particular market segment. as a consequence, the efb can allocate the optimal advertisement resource to focus on the most profitable customers, and decide the most appropriate advertisement scheme for them. 4 segmentation for dynamic pricing the online consumers have different sensitivities towards price and other nonprice factors. some consumers are highly price-sensitive and they like to use the shop bots for finding the efbs which offer the lowest prices while some care more about service, reliability and trust. as a result, how to set the right selling prices dynamically for different groups of customers so that the efb’s profit is maximized becomes very important and, in fact, crucial (the failure stories due to the lack of good pricing policy can be found in [28]). one of the effective ways for dynamic pricing is to carry out dynamic price testing. the idea of the dynamic price testing [27,30] is that: from changing the listed price showing on an efb’s website and keeping track of the customers’ purchasing rates at that price, the efb can know the expected profitability of each listed price. for example, when the efb sets the product’s listed price as $10, it is observed that 2 out of 10 visitors will buy the product; when the efb changes and tests the price at $9.5, it is found that 3 out of 10 visitors will buy the product. the purchasing rates are 20% and 30% for $10 and $9.5, respectively. the efb can then decide whether $9.5 is better than $10 or not based on the corresponding observed purchasing rates and profit margins. however, in order to exercise an effective dynamic price testing scheme (and hence a good dynamic pricing), the efb needs to segment the e-market properly. from the sales record of the customers for that product tsan ming choi : e-market segmentation for internet-mediated fashion brands... 176 (or a closely related product if the data for that product is not sufficient or available) in the past, the efb can keep track of the exact purchasing price of each customer. as a result, the efb can first segment the e-market for that product according to the conventional approach by the customers’ age, gender, etc. after that, the efb can check the average purchasing price and the variation of the purchasing price per each segment following the concept from mvpm. similar to the proposed method for the segmentation for advertisement budget estimation, we have the following model: “from the database of the customers, the efb first segments the customers according to the conventional segmentation scheme by, for example, the geographic and demographic variables. then, the efb further segments each group with respect to the average purchasing price and the variance of purchasing price from the customers in that group. the objective is to ensure that the coefficient of variation of the purchasing price, defined as the standard deviation of purchasing price divided by the average purchasing price, for each segment is under the efb’s precision control and the segments size is large enough.” with the above method, the efb can effectively identify the group of customers with a specific average purchasing price. after that, a tailor-made dynamic price testing scheme can be arranged for that particular market segment. it is thus more effective than performing the dynamic price test blindly. we summarize this proposed method in proposition 2 below. proposition 2. the e-market segmentation scheme for dynamic price testing (and hence dynamic pricing) can be stopped when the coefficient of variation of the purchasing price of the members in that segment is less than a threshold θ. if the coefficient of variation of profit is larger than θ, then further segmentation or re-segmentation is needed. example 2 below gives an illustrative numerical example for the proposed emarket segmentation model presented in proposition 2. example 2. an efb has performed conventional market segmentation (by using the variables such as age, city, gender, etc) for the customers of a specific product and has identified 20 e-market segments. suppose that this efb looks into two distinct segments: segment a and segment b. the desirable minimum size of each segment is 500. for segment a, there are 1300 customers and for segment b, there are 1500 customers. when the efb calculates the average purchasing price, variance of purchasing price, standard deviation of purchasing price, and coefficient of variation of purchasing price generated by the members of each segment, he has an khown in table 1, from table 1, the coefficient of variation of the purchasing price for customers in segment b is much larger than the customers in segment a. suppose that the efb has set the precision threshold θ(for the coefficient of variation of the pur177 advances in systems science and applications (2013) vol.13 no.2 table 1.2 example 2 segment a segment b average purchasing price (in $) 100 90 variance of purchasing price (in $2) 252 502 standard deviation of purchasing price (in $) 25 50 coefficient of variation of purchasing price 0.25 0.56 chasing price) to be 0.5. by proposition 2, the e-market segmentation result for segment a is good enough because the coefficient of variation of the purchasing price is less than θ. thus, for consumers in segment a, the efb can carry out the dynamic price test with a reference price of $100 and the dynamic testing prices can be set within a reasonable range. on the other hand, the e-market segmentation for segment b is not good enough because segment b’s coefficient of variation of the purchasing price is larger than θ. since the size of this segment is 1500 and the minimum segment size is 500, the efb can consider further segmenting segment b by using another classification variable. 5 segmentation for retaining customers, direct marketing and others in sections 3 and 4, we have discussed the e-market segmentation schemes, following the concept of mvpm, for two important collaborative functions for efbs. we will discuss more functionality areas where mvpm can be applied for emarket segmentation in this section. as we mentioned earlier, trust and customers’ loyalty are two important issues for efbs doing business on the internet. without proper e-market segmentation, building and keeping the loyalty of the online customers becomes more difficult. as a result, when the efb performs the e-market segmentation, it is important for him to bear in mind that he has to be able to identify precisely the need of the customers inside that market segment and be focused [1]. this objective follows exactly the mvpm where the need of the customers inside each e-market segment refers to the average measure of that need and the precision for the understanding of this need in the corresponding e-market segment is reflected by the coefficient of variation of that need measure. this is what the mvpm captures. as an example, suppose the efb would like to retain the customers by providing a membership system. in order to attract the customers, the efb would like to provide bonus points for the customers who join the membership and make some purchases. obviously, in order to attract more customers to be members, the amount of the bonus points and the amount of required purchases should be offered differently to customers in different segments. the concept of mvpm can hence be applied for providing a measure for this purpose. tsan ming choi : e-market segmentation for internet-mediated fashion brands... 178 similarly, for an efb who wants to provide the right product and service to the right customer for the right price at the right time in the right place with direct marketing strategy [31-32] , the mvpm also works where it provides a mechanism for e-market segmentation which helps to narrow down the error in terms of the product type, selling price, etc. for instance, the efb can make use of the data on the products bought by the customers in the past to help in segmenting the customers and measuring the customers likelihood of buying the product that will be direct marketed. 6 conclusion and managerial implications we have proposed in this paper a conceptual decision model, called the “meanvariance performance measure market segmentation scheme (mvpm)”, for emarket segmentation for different specific collaborative functions. as we all know, the traditional market segmentation relies on the classification variables like the demographic, geographic, psychographic and behavioral variables. however, in spite of the intuition behind the classification by using these variables, researchers have questioned the reliability of the available market segmentation techniques (e.g. [8]). furthermore, the classifications with these conventional variables may not be very helpful for a specific collaborative function. as a result, we propose to incorporate the collaborative function’s performance measure and its precision into the e-market segmentation scheme for effective decision making by the managers of the corresponding functions. a mean-variance consideration with which the average or expected performance measure (e.g. the average profit) is used as the performance measure variable and the degree of variation of the performance measure (e.g. the coefficient of variation of profit) is applied as a precision control variable. this market segmentation scheme is thus called the mean-variance performance measure market segmentation scheme (mvpm). with mvpm, efbs can carry out e-market segmentation for each particular collaborative functionality area following a mean-variance consideration. we have discussed the use of mvpm for efbs with the estimation of advertisement budget, dynamic pricing, retaining customers and direct marketing. since brand managers are involved with the resource allocation and decision making for the efbs, effective and precise market segmentation schemes can provide them with good assistance in making a sound and optimal decision. under mvpm, the e-market segmentation decision is under a control on the precision of the specific performance measure. as a consequence, better resource allocation decisions for different collaborative operations can be made by the fashion brand managers optimally and precisely. we can thus view mvpm as an uncertainty control model which targets at reducing and constraining the degree of uncertain179 advances in systems science and applications (2013) vol.13 no.2 ty associated with the performance measure. we believe that mvpm is especially important for e-commerce (and mobile commerce) because the e-market is highly volatile with a large amount of uncertainty sources and customers details can 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(2001), “e-service and the consumer”, international journal of electronic commerce, vol.5, no.3, pp.85-101. corresponding author tsan-ming choi can be contacted at jason.choi@polyu.edu.hk. advances in systems science and application(2015) vol.15 no.2 186-192 systemic complexity of empires and the globalization hermínio duarteramos new university of lisbon, lisbon, portugal abstract generally, very wide systems are complex, and they afford particular methodology implementations to obtain reasonable operation outputs. here we are looking for a strategy to rational interpretations on empire data collected from different civilizations in the universal history, in order to understand complex globalization trends as it occurs in our days. keywords systemic complexity; empire system; globalisation; worldwide integrated system 1 introduction the concept of complexity is linked to real systems through the integration of multiple interactive components.from such unitary structure emerges a systemic output called telonomy, which is significant for the system intentionality and for the human interpretation of the observed reality [1]. the system operation response may be difficult to be interpreted, and so it will be impossible to warranty a sure comprehensive operation prediction for the future.this reasoning conducted us to define complexity on the basis of difficulties we experiment to point out the essential features of the system functionalities. these systemic essentials are the following: acrony or structural components, axony or interactions among components, aquadry or real and virtual boundaries of the functional set, and adaptacy or the evolution system to the optimum working point in order to get the best telonomy, according to the system intentionality. whenever we don’t know the right conditions of one or more systemic essentials, we can say that the system is complex (even if we recognise it is simple), and otherwise it will be simplex (even we say it is complicated).in current human lives, we can handle trivial and sophisticated cases of simplexity, using the science and art or the societal common knowledge, but instances of complexity do offer some unknown process data. 2 worldwide societal systems a system in any society includes natural and technological components, which are inserted in all normal social activities. each worldwide system integrates human actions and natural objects or signals and artificial manufactured products. human and technology and universe phenomenology are three classes of general sets operating partial subsystems within societal systems. we know that very well in our current life, for instance it happens in remote communication systems advances in systems science and application(2015) vol.15 no.2 187 using the trivial internet. we can see it in everyday activity, and also regarding the historical evolution in all geographic regions until the contemporary globalization perspective of the world. in fact, today we have the possibility to interact in very short time at long distances as if we were remotely present. nevertheless, such remote action is very ancient. primitive people did migrate from africa to asia and europe and to america, they moved away with some adventure purposes and building several social structures to better profit from natural goods, organizing different human communities to live according to their own collective wishes. some of them tried to exert forced influences imposing empire charges impelled by ambitious and audacious adventurers, using newer power materials (as weapons or animal powers) to submit neighbour populations to their rules. the empires did grow up, and fell down after terrible and declining experiences. however nations hope to be free, living according to their cultures and developing freely own traditions. after the 15th century, maritime navigations over the atlantic ocean did carry europeans to the indic ocean and later to the pacific, and the entire world became known for all peoples. the time did pass along, and the 20th century developed sufficient scientific knowledge and new technologies to amplify some relations between countries over the world to a very high extent. the basic idea states that all organized societies will change global interactions by wide networks of relationships. is that the real trend for the future? what are the necessary global conditions to live happy in peace? how to defeat any possible threat by an eventual empire outburst? these questions inspire us to analyse the complexity of global interlinked systems over the world in comparison to local forced interconnected countries composing empires. between empires and the globalization we have colonization systems, with features from both basic social system types. for that we endeavour to understand evident systemic essentials associated to empires and the globalization. 3 what is an empire? typically any empire is an extensive territory governed by a single supreme authority, subjugating several peoples and different cultures to a dominant power. analysing the ancient history, since four millenniums behind, mostly at east and far east territories, we note the birth of several empires, as qin and han in china, cyrus in persian, alexander the great, roman empire, maurya and gupta in india, genghis khan, ming, and inca and aztec in america, the ottoman empire and others [2-4]. from such information we can extract some common features to identify the general empire concept under a systemic standpoint, neglecting political or so188 hermínio duarteramos: systemic complexity of empires and the globalization cial references. an empire sets down in forced blocs of populations with different cultures, using dominant strategies and a serial of tactic adaptations in order to harmonise suitable rules over the space and the time. imposing authority to subjugated people the empires use powerful technologies (transports, weapons, new inventions) and human capabilities (brave warriors, smart advisers, loyal governors). the empire evolution always strengthens higher hegemony levels, forcing social value uniformities, including a vehicular language, and destroying culture diversities. nevertheless, the empire implementation requires a convenient intellectual background warranting somewhat social stability. 4 what is the globalization? much more than world relations, the globalization do fall upon the free integration of several organizations inside many countries. each organization implements proper structures and pursues its own objectives. it is a singular societal organization within a general independent sovereignty endeavouring social aims without external impositions, but it grows by mutual acceptable requirements. today we observe a tremendous increase on a commercial globalization, practicing business everywhere through international corporations (are they only a powerful finance globalization?) operating as national enterprises in many independent countries and working in special industrial sectors or as services providers [5]. so, we detect a few trends on the globalization reality, extending actions over all human activities (science, art, literature) [6] and social environments (university, finance, energy) [7].in such global processes, the globalization will induce value uniformities when the time elapses, but without programmed forcibleness, preserving culture diversities by local behaviour adaptations to general imported influences. 5 systemic essentials of an empire history data gathered from asian empires, which has been created and died several thousands of centuries ago, can give us important information about their main features as systems. we observe similar properties in european empires and also in primitive american civilizations. the following description summarises empire systemic essentials as we did notice it. acrony: the composition structure of an empire depends on the occupied planetary space and on living people inside their boundaries, showing an authority expansion to contiguous spaces by subjugation of resident populations to external dictated rules. an empire has always certain heterogeneity among subsystems, and each one advances in systems science and application(2015) vol.15 no.2 189 exhibits a particular culture following their old traditions. that is to say the empire acrony can be definite, and it is not very much homogeneous. axony: distinct people do live integrated and in peace only if the empire system reveals an interactivity to aim at getting a minimum harmony, pursuing same accepted ways of live. but the empire domination compels behaviours against people desires, otherwise the dominator will be immediately defeated. this means the empire axony is not always well known being incomplete. aquadry: each partial country belonging to an empire tries to maintain its community frontier according tradition although it will be not clear. however the empire external boundaries are perfectly defended, separating the empire territory to neighbour spaces, and may be extended if the empire decides to struggle for greater dimensions. for that reason we can assert the empire aquadry may be certain in some history periods. adaptacy: the law code of an empire must consolidate a dominant culture, forcing it to be followed everywhere by the integrated people. guide lines to control all subsystems claim to attenuate culture differences, imposing a common paradigm to general behaviours even against people reactions. therefore empire adaptacy is mostly indeterminate. telonomy: some outputs from the empire process will go to the external world, but the majority results emerging from the organic activity are self oriented to maintain the auto-authority. the empire system must assure the sovereignty and independence, spending a lot of resources to continue alive, and the external actions try to create an image of internal paradise, pretending to influence neighbours to adhere to the expanded project, even under several threats. but the empire telonomy is almost isolated from other wide systems in the world, and outputs are not very much accurate due to some secret and not transparent actions. 6 systemic essentials of the globalization basic globalization features are very different from empire ones, because the global aim is quite distinct from the empire intentionality. as a corollary we note the consensual existence of multiple interconnected systems and so they can design an intricate societal network over the world. acrony: the organization structure of a global system is homogeneous in each functional subsystem. partial components are distributed over separated regions in the 190 hermínio duarteramos: systemic complexity of empires and the globalization world, instead of adjacent territories. following we can tell without any doubt that the global acrony is well definite. axony: all parts of a global system interact in different spaces, and they are perfectly controlled converging to the same global intentionality. this network paradigm means a fundamental feature of global systems. as a consequence the global axony is considered complete in general cases. aquadry: a global system has necessarily a virtual frontier, because concrete boundaries of all partial subsystems are distant one from others, and the global system boundaries are not physical, being distributed over many countries. this means that a global boundary may be not rigid, depending very much from exogenous variables, following political or culture environments in each country of implementation. we affirm that the global aquadry may be not exactly known and thereafter will be uncertain. adaptacy: operating subsystems optimise global work functionalities in an easy way, owing to the structure homogeneity and to the interaction accuracy. in fact, the network operation by agent interconnections will make easier to optimize working points. consequently global adaptacy appears to be perfectly determinate. telonomy: multiple outputs from a global system may be active in several subsystems to various geographical regions, giving all capabilities to interact in the internal network at remote locations. the result is a reasonable efficient control over the expected system intentionality. finally, we may say the global telonomy interacts accurately with many environments. 7 comparing systemic essentials although brief, the foregoing analysis reveals how far the systemic complexity of empires and globalization is to be considered. to better see the problem we summarize the most important features of the systemic essentials in the comparison table 1. the table teaches us that the empire system is a complex system (and also very complicated) denoting a very difficult scientific approach, giving the incompleteness of the axony and the indetermination of the adaptacy, although the acrony definiteness and the aquadry certainty can ease necessary specifications for a right system description. this means that in general empire systems have a 2nd or 3rd degree of complexity. on counterpoint, a concrete globalization represents a complex system (even advances in systems science and application(2015) vol.15 no.2 191 it could be very complicated) with a 1st or 2nd degree of complexity, seeing that we can get higher quality for scientific approaches, giving the definiteness of the acrony, the completeness of the axony and the determination of the adaptacy, although somewhat aquadry uncertainty may difficult system specifications. table 1 systemic essential comparison systemic empire system global system essential acrony definite contiguous components definite separated components axony incomplete forced interaction complete free programmed interaction aquadry certain rigid external boundary uncertain soft internal boundaries adapatacy indeterminate the operation depends on many unknown factors determinate the operation depends on many known factors telonomy accurate multiple internal on many unknown few external outputs inaccurate multiple external outputs and few internal outputs 8 conclusions a worldwide process requires induced perspectives from very high observation positions, and paradoxically we notice much better a global system far away from their details. in fact, we see better the earth from the sky, because the distance allows us to have a clearer vision on the part under global observation. really, we get a finer understanding about ourselves if we go outside from us and look at our profound inner states. any global system must be regarded from outside interpreting external outputs of its telonomy, but we must know inner components and interactions to understand the best operation modes inside their organization. an example can be the global finance system and its consequences for the natural world, requiring a suitable global control to eliminate perversion and inequity, and avoiding dangerous austerities [6]. all this is very complex to face it. but all this must be rationally workable, processing it by science and ethics in order to get a superior knowledge on reality to better survive and being happy. references [1] hermínio duarteramos. (2008), “hard and soft systems intentionality”, 7 th congress of the ues, res-systemica, afscet, lisbon, vol.7. [2] edward gibbon. (1998), the history of the decline and fall of the roman empire,wordsworth classics of world literature , london. [3] laurent testot (ed.). (2013), sigma: a new open economy model for policy analysis,les grands dossiers des sciences humaines, paris. 192 hermínio duarteramos: systemic complexity of empires and the globalization [4] dietmar pieper (ed.). (2014),another world: the inca, maya, aztecs, the mysterious kingdoms, der spiegel geschichte, nr.2, hamburg. [5] jacques adda. (1966), economic globalization, la découverte, paris. [6] zygmunt bauman. (1998), globalization: the human consequences, blackwell publishers, cambridge/oxford. [7] world energy council. (2014), 2014 world energy issues monitor, world energy council, london. [8] jean-pierre warnier. (1999),cultural globalization, éditions la découverte et syros, paris. corresponding author hermínio duarteramos can be contacted at: hduarteramos@gmail.com microsoft word 11-wu jianhua.doc advances in systems science and applications (2010), vol.10, no.1 67-72 issn 1078-6236 international institute for general systems studies, inc. characteristics of item replacement in weibull distribution jianhua wu, dexin tao and hongxiang li wuhan university of technology, wuhan, china abstract the aim of this paper is to study preventive replacement in order to increase system’s mtbf by replacing item following the weibull distribution. here, we discuss the periodic preventive replacement and random preventive replacement as preventive replacement. according to item preventive replacement following weibull distribution, based on the mtbf evaluation of item to study the characteristics of item replacement. keywords weibull distribution preventive replacement mtbf 1. introduction outline and background of the preventive replacement theory notation used in this paper. f(t): the failure distribution function of replacement item. dt/)t(df)t(f);t(f1:)t(f =− g(t): the distribution function of the preventive replacement dt/)t(dg)t(g);t(g1:)t(g =− <t>: first moment or mtbf under the preventive replacement <tn>: n moment μ: the replacement factor of preventive replacement distribution function g(t)=1-exp(-μt) t : time interval of preventive replacement г(·): gamma function; г(·,·): in-complete gamma function; δ(·): derta function β: shape parameter of weibull distribution η: scale parameter of weibull distribution cv: coefficient of variation τ: reliability improvement rate now let’s gather up the result of fundamental and general theory about preventive replacement (1) first moment or mtbf under preventive replacement mtbf or the first moment<t> is given as: 0 0 ( ) ( ) 1 f(t)g(t)dt f t g t dt t ∞ ∞< >= − ∫ ∫ (1) (2) second moment and variance under preventive replacement the second moment<t2> is given as: 68 wu: characteristics of item replacement in weibull distribution 2 0 0 2 0 0 0 2 0 2{1 f( ) ( ) }{ tg( ) ( ) } (1 ( ) ( ) ) 2{ g( ) ( ) }{ tf( ) ( ) } (1 f( ) ( ) ) t g t dt t f t dt t f t g t dt t f t dt t g t dt t g t dt ∞ ∞ ∞ ∞ ∞ ∞ − < >= + − − ∫ ∫ ∫ ∫ ∫ ∫ (2) thus the varianceδ2 is easily got fromδ 2 =<t 2 >-<t> 2 2. periodic preventive replacement and random preventive replacement the periodic preventive replacement is the most general way in the preventive replacement, so that we study on this case, and then take up the random preventive replacement as extreme example 2.1 the average <t>p and the 2nd moment <t2>p of the periodic preventive replacement if theb periodic preventive replacement is done at time t, nonpreventive replacement )t(g is as figure 1, then 1 0 ( ) 0 t t g t theother ≤ ≤⎧ = ⎨ ⎩ (3) g(t)=δ(t-t) (4) figure 1 consequently, the molecule and the denominator of (eq.1) is described as following ∫∫ ∫∫ ∞∞ ∞ =−δ= = 00 t 00 )t(fdt)tt()t(fdt)t(g)t(f dt)t(fdt)t(f)t(g and )t(f)t(f1dt)t(g)t(f1 0 =−=− ∫ ∞ therefore the 1st moment <t>p 0 ( ) ( ) t p f t dt t f t < > = ∫ (5) on the other hand, the 2nd moment <t2>p can be also obtained in the same way. 2 0 0 2 2 ( ) 2 ( ) ( ) ( ) ( ) t t p tf t dt t f t f t dt t f t f t < > = +∫ ∫ (6) so the variance of preventive replacement can be obtained through formula (eq.5) and (eq.6). advances in systems science and applications (2010), vol.10, no.1 69 2.2 the 1st moment and the 2nd moment of the random replacement in the case of the random preventive replacement, as maintenance function we put g(t)=1-exp(-μt) into formula (eq.1) and (eq.2), then 1st moment <t>r and 2nd moment <t2>r are obtained by 0 0 ( ) exp( ) 1 ( )exp( ) r f t t dt t f t t dt μ μ μ ∞ ∞ − < > = − − ∫ ∫ (7) 2 0 0 0 0 2 0 2 ( )exp( ) 1 ( ) exp( ) 2( ( )exp( ) )( ( ) exp( ) ) (1 ( ) exp( ) ) r tf t t dt t f t t dt f t t dt tf t t dt f t t dt μ μ μ μ μ μ μ ∞ ∞ ∞ ∞ ∞ − < > = + − − − − − − ∫ ∫ ∫ ∫ ∫ (8) 3. applying to weibull distribution when applying these theoretical, we consider preventive replacement characteristic from average value increase of the failure interval. we suppose the weibull distribution as ( ) exp{ ( / ) }f t t βη= − (9) 3.1 the periodic preventive replacement we insert it into (eq.5), the average value is as follow: 0 exp{ ( / ) } 1 exp{ ( / ) } t p t dt t t β β η η − < > = − − ∫ (10) moreover, it can be rearrange as [(1/ ), ( / ) ] [1, ( / ) ]p tt t β β η β η β η γ < > = γ (11) on the other hand, the 2nd moment can be got by (eq.6) 2 2 2 2 [(2 / ), ( / ) ] [1, ( / ) ] 2 exp{ ( / ) } [1/ , ( / ) ] [1, ( / ) ] p tt t t t t t β β β β β η β η β η η η β η β η γ < > = + γ − γ γ (12) 3.2 the random preventive replacement we apply the weibull distribution into (eq.7) and (eq.8) yield the following: average value i.e. 0 0 exp( ) 1 exp( ) r t dt t t dt λ μ λ ∞ ∞ − < > = − − ∫ ∫ (13) 70 wu: characteristics of item replacement in weibull distribution 2 0 0 0 0 2 0 2 exp( ) 1 exp( ) 2 ( exp( ) )( exp( ) ) (1 exp( ) ) r t t dt t t dt t dt t t dt t dt λ μ λ μ λ λ μ λ ∞ ∞ ∞ ∞ ∞ ⋅ − < > = + − − − ⋅ − − − ∫ ∫ ∫ ∫ ∫ (14) simplify, here (t/η)β +μt =λt is used 4. consideration 4.1 condition for calculation first, for convenience and simple expression, we suppose average of the weibull distribution e(t)=ηг(1/β+1)=1 and normalize the real time. if the periodic normalized time is t=0.1, the real time is 0.1ηг(1/β+1). more once we define the replacement rate μof the preventive replacement distribution function g(t)=1-exp(μt) in the random replacement because reverse of the μ is average replacement intervals. suppose t is 0.1, μ is 10, it show random maintenance had 0.1 intervals in average, the real time is 0.1ηг(1/β+1).moreover in order to compare the preventive replacement characteristics, we use the evolution reliability improvement rate in mtbf and coefficient of variation in dispersion. the reliability improvement rate is defined as following (1/ 1) 1 (1/ 1) t tη βτ η β < > γ + = =< > − γ + (15) 1 t t t tt ff f − >< = −>< =τ and the coefficient of variation cv is defined as 2 2 2 2 2 var 1iance t t tcv average t t < > − < > < > = = = − < > < > (16) 4.2 reliability improvement rate in the case of the periodic replacement, according to formula (eq.15) andηг(1/β+1)=1 improvement rate of periodic replacement is [(1/ ), ( / ) ] 1 1 exp( ( / ) )p t t β β η β ητ η γ < > = − − − (17) about random replacement use the same way. improvement rate of random replacement is 0 0 exp( ) 1 1 exp( ) r t dt t dt λ τ μ λ ∞ ∞ − < > = − − − ∫ ∫ (18) 4.3 coefficient of variation the cv value of the periodic replacement is advances in systems science and applications (2010), vol.10, no.1 71 2 2 { } 1 [2 / , ( ( )) ](1 exp( ( / )) ) [1/ , ( ( )) ] ( ) exp( ( ( )) ) [1/ , ( ( )) ] pcv a b t ta t t tb t β β β β β β β η β β = + − γ γ ⋅ − − = γ γ ⋅ γ ⋅ − γ ⋅ = γ γ ⋅ (19) here, г(·) is г(1/β+1),η=1/г(1/β+1) on the other hand, the cv value of the random replacement is 0 2 0 2 exp( ) 1 ( exp( ) ) r t t dt cv t dt λ λ ∞ ∞ ⋅ − < > = − − ∫ ∫ (20) 4.4 numerical results table 1 shows an example of calculated results in the periodic preventive replacement β= 2.5 β= 3.0 β = 3.5 β= 4.0 t τp cvp τp cvp τp cvp τp cvp 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 479 84 30 14 7 4 2 1 1 0 0.664 1.305 1.688 1.986 2.250 2.510 2.791 3.126 3.557 4.144 1571 195 57 23 11 6 3 2 1 0 0.973 1.605 2.000 2.305 2.575 2.844 3.147 3.526 4.052 4.837 5085 448 107 38 17 8 4 2 1 0 1.234 1.882 2.287 2.595 2.865 3.136 3.451 3.866 4.481 5.486 16344 1020 200 62 25 11 5 2 1 0 1.469 2.141 2.554 2.862 3.128 3.397 3.716 4.158 4.857 6.099 comparing to above table, we can get two picture as following: figure 2 improvement rate of periodic replacement from the figure 2 of above, we can see that: the interval t of exchange is smaller, reliability improvement rate τp is bigger. turn over, the interval t of exchange is bigger, reliability improvement rate τp is smaller. this kind of trend has nothing to do with the size of β. the interval of exchange around one, the reliability improvement rate will become very small, the effect of preventive replacement will disappear. in a word, it is very important to exchange 72 wu: characteristics of item replacement in weibull distribution with the interval which is smaller to the average of item. the shape parameter β is bigger, the reliability improvement rate is bigger, the interval between breakdown and breakdown will become bigger. figure 3 cv value of periodic replacement from the figure 3 of above, we can see that: the interval t of exchange is bigger, the coefficient of variation cvp is bigger. turn over, the interval t of exchange is smaller, the coefficient of variation cvp is smaller. this kind of trend has nothing to do with the size of β. the shape parameter β is bigger, the coefficient of variation is bigger. if we want to get the optimum value, we don’t only consider the reliability improvement rate but also consider the coefficient of variation cvp synthesize: it will obtain the good effect when short the interval time of exchange. this is in accordance with our experience. 5. discussion and conclusion in this report, we obtain the results of theory about periodic preventive replacement and random preventive replacement and then introduced clearly the formula of average and variance in the both preventive replacement, more applied these result to the weibull distribution, and showed the example of replacement characteristics about preventive replacement. references [1] barlow. r, proschan. f, hunter. l. “mathematical theory of reliability”, john wiley&sons inc., pp.61-63, 1965. [2] weiss. g. h. on the theory of replacement of machinery with a random failure time. naval research logistics quartely, 1956, vol.3, no.4, pp.279-293. [3] jinming su, shenyong ruan. matlab6.1 practical guide, publishing house of electronics industry, 2002, pp.212-213. [4] jianjun zhang, xiaoping wu. the c-k class of estimators of the coefficients. advances in systems science and applications, 2004, vol.4, no.3, pp.393-399. [5] jun fei, mianyun chen, chuanfeng zheng. audit inherent risk assessment based on fuzzy ahp. advances in systems science and applications, 2004, vol.4, no.4, pp.557-561. [6] jun meng. the analysis of soybean potential yield in jiansanjiang farm by logistic model. advances in systems science and applications, 2007, vol.7, no.1, pp.143-147. [7] songfa xie , jiaxiong peng, nanzhong he. automatic recognition of elevation values in scanned topographical maps. advances in systems science and applications, 2006, vol.6, no.3, pp.361-368. advances in systems science and applications (2014) vol.14 no.4 388-395 optimal location model and algorithm of the emergency exits hong-jie wang and yan gao school of management, university of shanghai for science and technology, 334 jun gong road shanghai 200093, china abstract in evacuation, the emergency exits are important. the exit’s location is a key which decides the success of evacuation. so how to build the emergency exits is worth to researching. in this paper, the effect of the emergency exit and main door are considered. the emergency exit’s ordinate is defined as the independent variable. then a model of exits is proposed and the golden section algorithm is used to find the most suitable location. at last, it is verified that the method is effective and feasible. keywords evacuation emergency exits golden section algorithm simulation. 1 introduction emergency evacuation is a mass movement of people from areas affected by disasters, such as earthquake, hurricane, and fire, to safer in a timely manner to mitigate disastrous consequences. it involves the choice of routes. if people can quickly find the exit and through the door, then hope of escape is big in the danger. so the location of emergency exit is important. and the right location of the exit can increase the rate of evacuation. in recently, the researchers have found many factors which affect the evacuation [1-5] and provide many evacuation models [6-9]. although some researches show that the building’s structure can affect the evacuation [10-11], the most of the research presupposes the building’s layout. they didn’t consider the effect of the exit’s position. it means the emergency exit, main door and access are designed and don’t change in the researches. obviously, the effect of layout is ignored. if we don’t consider the main door in evacuation process, obviously the middle sides is the most suitable exit’s location in some symmetrical architect. but if main door can be used well in the evacuation, the new evacuation route is added. so the exits and main door is in some suitable locations, maybe it can shorten the evacuation time. it’s useful to evacuation. so it’s important to finding the suitable locations. in this paper, we try to find the suitable location of exit’s location. so we only consider the effect of exit’s location. and in the research, we also consider the main door in the evacuation. in order to finding the suitable location of exit, we use the golden section algorithm. at last, we discuss and conclude the numerical results by numerical examples. advances in systems science and applications (2014) vol.14 no.4 389 2 description the authors consider a population of n individuals and assumed that each individual nk(k = 1, 2, · · · , n) moving by the speed vkin the emergency condition and randomly select the evacuation route. generally, the evacuation time is that how many times is used between the beginning of escape and all people arriving at the safe place. because we hope that we can save more people in shortest time, so the evacuation problem can be described by the object function as follows: min f = max{ti(ni) | i = 1, 2, · · · , n } (1) where ti(ni) is the evacuation time of the ith individual. in the articles, we consider the situation as follow. all individuals were randomly distributed in space according to the population density, and their initial velocities equal to zero. everyone has two choose, main door or emergency exits, to escape. we assume that each individual is a particle without quality and each individuals speed is same. we set up a coordinate system as fig.1 and use the array (xi, yi) to show the location of the ith particle. (a) the rectilinearly shaped floor (b) the vertical shaped floor fig.1 the initial distribution of the people on the floor we assume the number of main door is l,and the one of exits is w.then the kth particle’s evacuation time by the each door can be as:(k = 1, 2, · · · , n) ti(nk) = ski vk i = 1, 2 · · · , l (2) tj(nk) = skj vk j = l + 1, l + 2, · · · , l + w (3) which the ski is the shortest distance between theith main door and the kth particle,the skj is the shortest distance between the jth exit and the kth particle.so the kth particle′s evacuation time tk is satisfied the equation: tk(nk) ≤ max{ti(nk) |i = 1, 2 · · · l, l + 1, l + 2, · · · , l + w} (4) 390 hong-jie wang: optimal location model and algorithm of the emergency exits summary, the evacuation problem can be as: min f = max{tk(nk) | k = 1, 2, · · · , n } ≤min max 1≤k≤n {ti(nk) |i = 1, 2, · · · l, l + 1, l + 2, · · · , l + w} (5) obviously, when the each people choose the shortest path, the equation is established. the people’s location is fixed. perhaps, the each evacuation time maybe change, if the emergency exit’s location is changed. because the distances between the exits and the kth particle are changed.so the location of emergency exit effects the evacuation. as follow, we consider the exit’s location. we assume(x, y) is the coordinate of emergency exit and f(x) is the max evacuation time. in articles, we only consider the situation that there are two emergency exits which are distributed on the right and left sides of building respectively. in the situation, the coordinate y is 0 and the x is the independent variable. so we only consider the coordinate of emergency exit x.absolutely, the f(x) is decided by x. because we hope to find the most suitable location of emergency exit, which the evacuation time is the shortest, so we can setup the function as follows: min f(x) = max{tk(nk, x) | k = 1, 2, · · · , n } s.t. a ≤ x ≤ b (6) where a is the index of wall-width,b is the superscript of wall-width, and x is the coordinate of emergency exit’s center point. 3 algorithm according to the shortest path theory, when everyone selects the shortest path, the each individual’s evacuation time and total time are the shortest. we select the shortest path as the each individual’s escaping path. let each individuals distance function as: (k = 1, 2, · · · , n) sk(x) = min{ski(nk, x) |i = 1, 2, · · · , l, l + 1, l + 2, · · · , l + w} (7) absolutely, the function sk(x) is the linear and the f(x) = max{tk(nk, x) |k = 1, 2, · · · , n} is the linear, too. we can solve the problem by the optimization algorithm. we hope that we can exclude the unsuitable position in the process of searching. so we select the golden section algorithm to solving the problem. the golden section algorithm is a kind of optimization algorithms for solving one-dimensional problem as follow: min f(x) s.t. a ≤ x ≤ b (8) advances in systems science and applications (2014) vol.14 no.4 391 where f(x) is a uni-modal descent function. the method is a iteration algorithm and find the best solution by shorting the independent variables interval in each iteration, which reduced probability is the 0.618. the algorithms step is as follows: step1:define the a,b,ε and (xi, yi), i = 1, 2, · · · , n ; step2:define: x2 = a+ 0.618 ∗ (b− a) (9) sk(x2) = min{ski(nk, x2) |i = 1, 2, · · · , l, l + 1, l+2, · · · , l + w} (10) f2 = max{tk(x2) = sk(x2) vk | k = 1, 2, · · · , n } (11) go to step3; step3:define x1 = a+ 0.382 ∗ (b− a) (12) sk(x1) = min{ski(nk, x1) |i = 1, 2, · · · , l, l + 1, l + 2, · · · , l + w} (13) f1 = max{tk(x1) = sk(x1) vk | k = 1, 2, · · · , n } (14) go to step4; step4: if | b− a | ≤ ε,define x∗ = a+ b 2 (15) and stop; else go to step5; step5: if f1 < f2, define b = x2, x2 = x1, f2 = f1 then go to step3; if f1 = f2, define b = x2, x2 = x1,f2 = f1, then go to step3; if f1 > f2, define a = x1, x1 = x2, f1 = f2,then go to step6; step6: define x2 = a+ 0.618 ∗ (b− a) (16) sk(x2) = min{ski(nk, x2) |i = 1, 2, · · · , l, l + 1, l + 2, · · · , l + w} (17) f2 = max{tk(x2) = sk(x2) vk | k = 1, 2, · · · , n } (18) then go to step4. 392 hong-jie wang: optimal location model and algorithm of the emergency exits 4 simulation results 4.1 architectural attributes to design the suitable position of exit, the authors have designed the building layout to be two shaped. one is the rectilinearly shaped building, which each floor is a 10×30 units orthogonal area. the another is the vertical shaped building, which major semi-axis of floor is 14units, and the minor semi-axis is the 7units. in the two layouts, we consider the one kind of scenario that main entrance is placed on the bottom middle-south side of the floor. because the exits are usually placed on the right and left sides in most buildings, we consider the exits on the right and left sides, and they are opposite. the space occupancy levels vary in square meter per person ranging within the limits of space and occupancy density standards. in the two scenarios, the authors used occupancy densities of 0.15, 0.25, 0.5, 0.7, and 1.2 person/units 4.2 simulation and numerical tests to verify the feasible and effective of the way, it’s run each scenario for six hundred times, take the average of the evacuation time, and analyze the all results. in the all tables, where x, is the center point’s coordinate of emergency exit,p is occupancy densities, and p is the percentage of people selecting the emergency exit. in order to clearly verify our conclusion, we compare the number results to the evacuation time of building with emergency exits in center sides. simulation numerical results reveal the following: table 1.1 the results of rectilinearly shaped with one main entrance situation of emergency exit p number of number of number of number of number of number of( person/m2 ) 0 < x < 1 1 ≤ x < 2 2 ≤ x < 3 3 ≤ x < 4 4 ≤ x < 4.9 4.9 ≤ x < 5.1 0.15 1 7 4 176 44 3 0.25 0 1 5 98 55 14 0.5 0 0 0 128 39 8 0.7 0 0 0 100 49 12 1.2 0 0 0 87 34 12 table 1.2 the results of rectilinearly shaped with one main entrance. situation of emergency exit p number of number of number of number of number of( person/m2 ) 5.1 ≤ x < 6 6 ≤ x < 7 7 ≤ x < 8 8 ≤ x < 9 9 ≤ x < 10 0.15 40 171 50 81 53 0.25 0.45 191 63 70 32 0.5 82 208 50 50 35 0.7 103 201 53 44 38 1.2 128 200 66 40 33 advances in systems science and applications (2014) vol.14 no.4 393 table 2 the results of vertical shaped with one main entrance. occupancy building with emergency exits densities in center sides results of simulations the percentage of the percentage of p average-time people selecting the average-time people selecting the( person/m2 ) (unit) emergency exit p (unit) emergency exit p 0.15 6.51 0.51966 6.43 0.51049 0.25 6.66 0.51896 6.56 0.51319 0.5 6.78 0.52090 6.67 0.51368 0.7 6.82 0.51860 6.71 0.51167 1.2 6.89 0.51890 6.77 0.51450 table 3.1 the results of vertical shaped with one main entrance. situation of emergency exit p number of number of number of number of number of number of number of( person/m2 ) 0 < x ≤ 1 1 < x ≤ 2 2 < x ≤ 3 3 < x ≤ 4 4 < x ≤ 5 5 < x ≤ 6 6 < x ≤ 6.9 0.15 13 9 20 8 10 15 2 0.25 11 5 10 13 4 7 1 0.5 2 0 7 6 2 3 0 0.7 0 2 8 3 9 4 1 1.2 0 0 1 2 0 2 0 table 3.2 the results of vertical shaped with one main entrance. situation of emergency exit p number of number of number of number of number of number of number of( person/m2 ) 6.9 < x ≤ 7.1 7.1 ≤ x < 9 9 ≤ x < 10 10 ≤ x < 11 11 ≤ x < 12 12 ≤ x < 13 13 ≤ x < 14 0.15 7 19 22 115 99 209 52 0.25 6 11 11 103 115 256 47 0.5 1 2 8 101 65 341 62 0.7 0 2 3 87 52 378 60 1.2 0 2 4 79 22 411 77 table 4 the compare of vertical shaped with one main entrance. occupancy building with emergency exits densities in center sides results of simulations the percentage of the percentage of p average-time people selecting the average-time people selecting the( person/m2 ) (unit) emergency exit p (unit) emergency exit p 0.15 7.78 0.64214 5.62 0.68798 0.25 8.22 0.64292 5.94 0.67642 0.5 8.62 0.65034 6.18 0.68156 0.7 8.81 0.64968 6.26 0.68131 1.2 9.01 0.64573 6.37 0.68303 394 hong-jie wang: optimal location model and algorithm of the emergency exits 5 discussions and conclusion from the table1.1, table 1.2, table 3.1, and table 3.2, it can be known that the x arent distributed randomly, but most in some areas. the occupancy density is more, the number of x’s appearance is larger in this areas. in the rectilinearly shaped building, there are about 50% x in the area from 5.1 to 7. in the vertical shaped building, the x is the most in the areas from 12 to 13. and both of numbers are rising with the occupancy density’s raising. in the centre area, the percentage of x’s appearance is less than 2.5% in each simulation, both of two shaped. obviously, the more the occupancy density is, the smaller the individual’s distributive space is. so the regularity is clearer as the density raising. these show that the effect of emergency exit is existent and the center point is not the suitable place for buildings of this size and shape. because that if the effect isn’t existent, the distribution of x is random and no regular. from the table 2 and table 4, the evacuation times of simulations are shorter than the one of situations which emergency exit is in the center of right and left sides, although the p are close in each comparison. it shows the location of emergency exits is effect the evacuation, and the center point is not necessarily the most suitable position. the suitable location of emergency exit can shorten the evacuation time. so the authors conclude the conclusions as follow: 1. the location of emergency exit effects the evacuation in some way. 2. the suitable place of emergency exit is related to the shape of building, occupancy density, and the layout and people distribution. the center point of left and right sides is not necessarily the most suitable position in different situation. 3.golden section algorithm about searching the emergency exit is effective. because the distribution of people is random and irregular in each simulation, so the results are not clustering some point. but when a building is designed and built, the building’s shape, the buildings layout and the distribution of people can be estimated. so for buildings, we can use the golden section algorithm to search the emergency exit in base of some constraint, such as the shape of building, the width of exit, the pass-rate per unit of exit and the estimable distribution of person. references [1] y. lim, and s. rhee. (2010), “an efficient dissimilar path searching method for evacuation routing”, ksce journal of civil engineering, vol.14b, pp.61-67. [2] r. stamatina, s. constantinos. (2010), “escape dynamics in office buildings: using molecular dynamics to quantify the impact of certain aspects of human behavior advances in systems science and applications (2014) vol.14 no.4 395 during emergency evacuation”, environmental modeling assessment, no.15, pp.411418. [3] g. y. jeon, j.y. kim, w.h. honget al. (2011), “evacuation performance of individuals in different visibility conditions”, building an environment, vol.46,1094-1103. [4] h. frantzich. (2001), “occupant behavior and response time cresults from evacuation experiments”, in: proceeding of 2nd international symposium on human behavior in fire,pp.159-165. [5] r. a. kady. (2012), “the development of a movement-density relationship for people going on four in evacuation”, safety science , no.50, pp.253-258. [6] z.x. fang, q.q. li, q.p. li, et al. (2011), “a proposed pedestrian waiting-time model for improving space-time use efficiency in stadium evacuation scenariosa”, building an environment, no.46, pp.1774-1784. [7] p. a . thompson, e. w. marchant. (1995), “a computer model for the evacuation of large building populations”, fire safety journal, no.24, pp.131-148. [8] t.s. shen. esm. (2010), “a building evacuation simulation model”, regional studies, vol.45, no.6, pp.733-754. [9] buch, c.m. (2005), “why do banks go abroad? evidence from german data”, building and environment, no.40, pp.671-680. [10] w j. lei, a g. li, r. gao et al. (2012), “influences of exit and stair conditions on human evacuation in a dormitory”, physica a no.391, pp.6279-6286. [11] z.m. fang, w.g. song, j. zhang, et al. (2010), “experiment and modeling of exitselecting behaviors during a building evacuation”, physica a, no.389, pp. 815-824. [12] m. kobes, i. helslootb. de vries, et al. (2010), “way finding during fire evacuation: an analysis of unannounced fire drills in a hotel at night”, building an environment, no.45, pp.537-548. [13] y. yuan, d.w. wang, z.z. jiang, et al. (2008), “mult-i objective path selectionmodel for emergency evacuation taking into account the path complexity”, operations researchand management science,, no.17, pp.73-79. [14] jf. yang, y. gao, lh. ling. (2010), “emergency evacuation model and algorithm in the building with severalexits”, systems engineering-theory & practice, no.31, pp.147-153. corresponding author author can be contacted at: gaoyan@usst.edu.cn microsoft word 1151 article text, copyedited.docx adv syst sci appl 2021; 04; 45-56 published online at https://ijassa.ipu.ru. choosing directions for investments in the development of companies under uncertainty valery akinfiev* v.a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia e-mail: akinf@ipu.ru abstract: the problem of choosing investment decisions of companies in the oligopoly market in conditions of uncertainty in demand for products is considered. investment decisions include the choice of the size and ratio of funds for the implementation of projects of two types: projects to expand production and, accordingly, to increase the supply of products on the market; as well as projects aimed at reducing production costs, which only affect the profitability of production and the free cash flow of companies. we propose an approach based on the joint use of the dcf model and algorithms that describe the investment behavior of companies in the market. the model takes into account the relationship between the choice of investment decisions by companies and the dynamics of the market price. the solution of the problem is reduced to the analysis of a matrix game in which the payoff matrix is formed as a result of numerical simulation. an illustrative example of using the proposed approach is given. keywords: investment decisions, oligopoly, demand uncertainty, company behavior model 1. introduction we consider the problem of analyzing and choosing investment strategies for companies under conditions of uncertainty in market demand. we are looking at oligopolistic markets, where two or more companies compete with each other for market share and therefore profit share. companies can choose different investment strategies knowing that the future dynamics of market demand is uncertain. different investment strategies, depending on the implementation of the demand dynamics scenario, can lead to both profits and losses for the company. in addition, when setting the problem, an important factor of mutual influence of the investment strategies of companies and the dynamics of prices for products will be taken into account [24]. the investment strategy determines the amount of funds directed by the company to invest in expanding production and (or) reducing production costs. the cost of products (below the industry average or above the industry average) is of great importance in the competitive struggle. consequently, an investment strategy aimed at reducing costs is often preferable to a production expansion strategy. the modern theory of investments in conditions of uncertainty has been developing over the past twenty to twenty-five years and is associated with the development of analysis methods using the ideology of evaluating the value of real options of various types in continuous time. the fundamentals of the methodology of this approach are outlined in a number of publications [6, 10, 11], which had a great influence on the development of this area of research. currently, it is characterized by a wide arsenal of methods and many problems statements [5-11, 13]. * corresponding author: akinf@ipu.ru 46 v. akinfiev copyright ©2021 assa. adv. in systems science and appl. (2021) for most variants of these problems, qualitative or quantitative results were obtained under the following fairly strong assumptions: 1. it is assumed that investments occur instantly and immediately after that there is an infinite cash flow, which is the main factor of uncertainty. in a number of works, the uncertainty of the cash flow is determined by the stochastic demand and supply of products on the market. 2. it is also assumed that cash flow or demand behaves as a random process of a certain kind. the overwhelming majority of works use geometric brownian motion (gbm) to model this uncertainty, which was originally used in the black-scholes-merton option pricing model in financial markets, and consists of the following: a random process st is a geometric brownian motion (gbm) if it satisfies the following stochastic differential equation [12]: (1) in this model, the parameter µ is a non-random trend, σ is the degree of volatility, and wt is the standard wiener process (brownian motion). note that this approach assumes that the parameters µ and σ in the stochastic equation are constant. this approach, according to its supporters, describes well the volatility of asset prices in financial markets and can be applied with great caution to the valuation of real assets, in particular, to the assessment of investment projects and investment strategies. there is some pessimism regarding the practical value of this approach [12]. in addition, these models provide only a first approximation to the complex processes taking place in modern financial and especially commodity markets. it should be emphasized that in equation (1) the parameter µ (non-random trend) is considered constant and does not depend on time. this assumption significantly reduces the realism of modeling the dynamics of demand in the market, the rate of change of which can also vary significantly over time if we consider time intervals of several years, which is a characteristic feature of investment projects in real sectors of the economy. and, of course, the function µ (t) is also unknown in advance. for large investment projects, the forecasting horizon is usually 10-15 years. demand shocks can occur during this period, when demand can change dramatically. for example, “demand shocks” in the markets occurred in 2008-2009. (world economic crisis) and in 2014-2015. (falling oil prices). for example, the demand for metal on the russian market fell by 20-25% in 2009 and began to partially recover only by 2010-2011. in 2015, domestic demand for metal also decreased by 10-15% due to a significant decrease in production and sales in the automotive industry (-30%) and a decrease in the construction industry. another important note. for the purposes of evaluating the investment decisions of companies, the parameter µ (t) is more important than the second stochastic term. the influence of the second term in formula (1) decreases if we take into account that time-averaged integral characteristics are used to assess the efficiency of investment decisions, such as, for example, the npv indicator. regarding the first assumption, it is well known that the investment process of a company is continuous. the investment effect (cash flow) occurs with a lag, which depends on the duration of the investment phase of the project. in addition, cash flow can change significantly over different periods of time, including under the influence of random demand, the results of the choice of investment strategies of companies and product prices. as shown, there is a significant gap in the degree of realistic modeling of investment processes between theoretical works based on the methodology of real options and traditional methods of modeling investment projects based on the dcf methodology. dcf models more realistically allow simulating cash flows at different stages of the implementation of investment projects, which explains their wide application in practice. however, these methods do not work well in the face of uncertainty in the model input data, including price and demand volatility associated with company competition in the market. this t t t tds s dt s dwµ s= + choosing directions for investments in the development of companies... 47 copyright ©2021 assa. adv. in systems science and appl. (2021) is one of the main arguments against them on the part of the proponents of the real options approach [6]. the article attempts to mitigate the disadvantages of both approaches. for this, it is proposed to use jointly methods of scenario modeling of market uncertainty and aggregated dcf models, which are supplemented by the behavioral models of companies making investment decisions based on changing market information. 2. the model consider a market with n companies. each company can make investment decisions in accordance with the chosen strategy and based on information from the market. the criterion for the success of the company's investment strategy is the npv indicator (total discounted free cash flow of the company for the forecast period). as noted earlier, the choice of investment decisions is significantly influenced by the forecast of demand and prices for products. we will consider a situation of high demand volatility, including when market demand can change the trend several times during the forecast period. this situation is the most difficult to assess and analyze the effectiveness of investment decisions. the difficulty lies in the fact that companies cannot predict changes in demand for the entire forecast period and are limited only to assessing the trend that is observed during the period in which the company's investment budget is formed. the question arises what should be the investment strategy of companies in the face of uncertainty in demand and uncertainty in the behavior of competitors? the situation becomes even more complicated if we take into account the influence of companies' investment activity on the dynamics of market parameters. excessive investment activity of companies, as a rule, leads to the emergence of "extra" production capacity and, during periods of declining demand, to a significant decrease in the price of products [2, 3]. we consider a situation where campaigns at the beginning of each period form investment budgets based on some rational rules, using the results of analyzing trends in the dynamics of demand and product prices. in this paper, the term "choice of an investment strategy" includes: the choice of the company's financial resources allocated for investment, and the choice of the investment direction, which determines the ratio of funds allocated for the implementation of projects of two types: § projects to expand production and, accordingly, increase the supply of products on the market; § projects aimed at reducing production costs, which only affect the profitability of production and the free cash flow of companies. consider a time period (forecast period) equal to t periods, . free cash flow of company i in period t, equal to the company's net profit for this period minus investments, and is calculated by the formula: (2) where: the market price of products in the period t. in each period of time, the price is formed based on the ratio of demand for products and the total supply of manufacturers . is determined in each period t as , where is the production capacity of company i. then , where the parameter is the price elasticity on the value of the excess of demand over supply. here is the market price at the beginning of the forecast period (initial conditions). if there is a shortage of supply in the tt ,1= ( )tncfi ),1( ni = )()1()())()(()( tiptbtctptncf iiii --××-= )(tp ( )td ( )ts ( )ts ( ) å = = n i i tsts 1 )( )(tsi ( ) ( ) ( ) ( ) ( ) 0 (1 ( ) d t s t p t p d t g = × + × g ( )0p ( ) ( ) 0³tstd 48 v. akinfiev copyright ©2021 assa. adv. in systems science and appl. (2021) market, and the price rises, otherwise, there is an excess of supply and, accordingly, the price falls. it should be noted that may also depend on price dynamics. an increase can lead to a decrease , which is taken into account by introducing negative feedback into the model. the degree of price influence on demand is specified through the parameter of demand elasticity in relation to changes in the market price. further, markets with inelastic demand are considered, which, in particular, include the metallurgical industry. let, further, be the total sales in the market in period t, which is calculated in the model using the following formula: . suppose that the capacity utilization of all companies is the same, then the sales revenue of company i is calculated as follows: is the cost of production of company i in period t, p is the income tax rate. note that all the quantities included in the calculation formula depend on random demand and the choice of investment strategies of companies that determine the dynamics of their production capacity and production costs . next, we will consider a model that allows us to evaluate the effectiveness of investment strategies of companies, taking into account the market uncertainty of demand. the general structure of the developed model for the duopoly market is shown in fig. 1. demand dynamics . exogenous variable of the model, the graph, which sets various external macroeconomic scenarios in relation to the model. fig. 1. model structure ( )td ( )tp ( )td ( )tb ( ) ( ) ( ){ }tstdtb ;min= )( )( )()( ts ts tbtb i i ×= )(tci ( )tncfi ( )td )(tsi )(tci ( )td ( )td choosing directions for investments in the development of companies... 49 copyright ©2021 assa. adv. in systems science and appl. (2021) 2.1 company behavior model we assume that companies make investment decisions in the face of market volatility and uncertainty in the dynamics of demand for companies' products. companies can observe in each period t only the change in their financial indicators (net profit, product price, sales volume) and (or) predict their change for the next several periods. the model allows varying the depth of a “reliable” forecast of market dynamics available to market participants. this makes it possible to take into account the factor of "foresight" of companies in the analysis. obviously, in case of low market volatility, the depth of the “reliable” forecast can be increased. at the beginning of each period t, the company makes an investment decision based on this available information in accordance with some predetermined algorithm, which will be described below. let further, if or (which signals the company about an upward trend in the market), then part of the company's net cash flow accumulated over period t in a share equal to this can be directed to investments in its development . the value determines the investment activity of the company i. the higher the value (share), the higher its investment activity. thus . as noted earlier, a company can direct total investments in projects of two types: projects aimed at increasing production capacity (projects of the first type) and projects aimed at reducing costs (projects of the second type) in a certain ratio and , ( ). the quantities , and are the parameters of the model that companies can choose based on their forecasts of market dynamics. thus, the company chooses investments based on the analysis of actual data and a possible forecast of market dynamics. cannot exceed , which is determined, in turn, through a variable parameter of investment activity. , where: and investments in projects of the first and second types, respectively, which are determined in accordance with some algorithms described below. 2.2 first type investment company i invests in projects of the first type according to the following algorithm: if during the period t there is an upward trend in the market (supply-demand> 0), then the company invests in projects of the first type as follows: , where is the maximum allowable level of investments in projects of the first type for the period. if there is a downward trend in the market during period t, then this signals the company about the appearance of excess production capacity and, in accordance with this, the company does not invest in projects of the first type, i.e., . suppose there is a lag between the investment period and the period of increasing production capacity . let also the value characterize the increase in production capacity per unit of investment. then , production capacity is calculated using a recurring formula . at t = 1, the initial production capacity of the company i is set. 0)1()( >-tptp 0)1()( >-tbtb ii ia ( )ti i * ia ( ) å = ×= 1 1 * )( t t iii tncfti a 1 ia 2 ia 12 1 ii aa -= ia 1 ia 2 ia ( )ti i ( )ti i ( )ti i * )()()( 21 tititi iii += )(1 ti i )(2 ti i }),(min{)( 1*11 прiii ititi ×= a прi 1 0)(1 =ti i 1t ( )tvi 1e )()( 1 1 1 t-×= tietv ii ( ) ( ) ( )tvtsts iii +-= 1 ( )0is 50 v. akinfiev copyright ©2021 assa. adv. in systems science and appl. (2021) 2.3 second type investment. let us consider the influence of the company's investment decisions on the dynamics of changes in the cost of production. the cost of production in period t is calculated using the following formula: (3) where: the specific efficiency of investments in projects to reduce production costs, the lag between the investment period and the period of the corresponding change in the cost. for t = 1, set is the production cost at the beginning of the forecast period. the value characterizes the reduction in the cost of production per unit of investment in projects of the second type. the calculations are based on the “decline in investment efficiency” model. in accordance with this model, it decreases as the cost of production changes and approaches a certain threshold value . assessment of the maximum possible reduction in production costs, upon reaching which the investment efficiency becomes equal to zero . (4) the company invests in projects of the second type according to the following algorithm: , where is the maximum allowable level of investments in projects of the second type for the period. in accordance with the described algorithm, if starting from period t becomes equal to zero, then the amount of investment will also become equal to zero. in accordance with this algorithm, the volume of investments of company i in period t depends on the accumulated cash flow for this period, the selected parameters and , as well as on the ratio of the current investment efficiency indicator to the investment efficiency at the beginning of the period. forecast period. 3. the problem of choosing an investment strategy the presented model makes it possible to calculate the free cash flow of companies based on the choice of parameters of investment strategies and scenarios of market demand dynamics . let be the free cash flow of company i , which depends on the choice of investment strategies by all companies and the implementation of the scenario of market demand dynamics. then the indicator of the effectiveness of the chosen investment strategy is calculated as the difference between the free cash flow of the company implementing the investment strategy with parameters and the free cash flow of the company in the absence of investments, i.e., . (5) where: d is the discount rate. companies make decisions to invest in expanding production or reducing production costs in accordance with the rules described in the previous sections. for each scenario of market )()()1()( 2 2 2 t-×--= titetctc iii )(2 te 2t )0(ic )(2 te )(2 te )(tci прc прc 0)(2 =te ) )()0( 1()0()( 22 пр ii c tcc ete -×= }, )0( )()1()(min{)( 2 2 21*2 прiii i e tetiti ×-×= a прi 2 )(2 te )(2 ti i ia 1 ia ),1,,( 1 niii =aa y )(tncfi ),1( ni = 1, ii aa 0,0 1 == ii aa t tt t iiiii d tncftncfnpv )1( 1))0,0,(),,(( 1 1 + ×-=å = = aa choosing directions for investments in the development of companies... 51 copyright ©2021 assa. adv. in systems science and appl. (2021) demand dynamics , companies choose investment strategies that maximize (5), taking into account the possible choice of investment strategies of competitors. in the above formulation of the problem, each company solves its own maximization problem (5). the sought-for task variables are parameters and . the difficulty in solving this problem is that the dependence cannot be specified in the form of an explicit analytical expression on and . the value of the optimality criterion (5) is calculated each time for given and for all agents using the simulation model described in the previous section. this problem belongs to a class of optimization problems that is quite difficult to solve, in which the values of the criterion and constraints are set using a simulation model. standard mathematical programming methods are not applicable here. a grid search algorithm for finding a solution to the problem is proposed, which consists in discretizing the set of solutions. for each point, a simulation experiment is carried out, and on the basis of the data obtained, matrices of criteria values are constructed for each agent. the general solution of the problem for n agents must satisfy the nash equilibrium conditions (the saddle point of function (5) in the space of the sought variables). the final stage of the search for a solution is reduced to the analysis and search for a solution to the game, which is described below. the problem can be reduced to the study of a model of a one-step continuous non-zero sum game, in which the payoff functions of players (companies) are specified by a simulation model. therefore, numerical modeling methods will be used as the main method for studying this problem. note that the set of possible investment strategies for each company coincides with the set of points in the unit square on the plane. without loss of generality, one can consider a finite set of strategies using, for example, grid search methods. let, further, for greater clarity and simplicity of presentation, companies use several basic investment strategies. consider, for example, the following set of strategies. strategy 1 ( ). invest in production expansion (intensive development option). this strategy allows you to increase production volumes and, in a growing market, leads to an increase in sales and, accordingly, cash flow. in a falling market, this strategy leads to a decrease in the utilization of production equipment and an increase in production costs and, accordingly, a decrease in cash flow. strategy 2. ( ). invest in lower production costs (extensive development option). this strategy allows you to reduce production costs without increasing production. in a growing market, this leads to maintaining sales and, accordingly, increasing cash flow by increasing profitability. in a falling market, this strategy allows you to maintain the amount of cash flow due to the lower "break-even point". often, companies use more cautious strategies in which the estimated volume of investment is evenly distributed between investments of the first and second types ( ). at the same time, only investment activity changes, which depends on the predicted dynamics of the market situation. as a rule, with the expected improvement in market conditions, companies increase their investment activity ( ). strategy 3. ( ). high investment activity strategy 4 ( ). moderate investment activity strategy 5 ( ). low investment activity. y ),1,,( 1 niii =aa ia 1 ia 1( , , )i i incf t a a ia 1 ia ia 1 ia 1,1 1 == ii aa 0,1 1 == ii aa 5,01 =ia ia 5,0,1 1 == ii aa 5,0,5,0 1 == ii aa 5,0,0 1 == ii aa 52 v. akinfiev copyright ©2021 assa. adv. in systems science and appl. (2021) next, we'll look at the duopoly market. let the set of investment strategies of company 1 ( ) and company 2 ( ). based on the results of a series of calculations for each scenario, it is possible to construct companies' payoff matrices in this formulation, the solution of the problem is reduced to the analysis of a bimatrix game with payoff matrices . this problem has been well researched. as is known, the condition for the existence of at least one nash equilibrium point in pure strategies (k0, j0) is the fulfillment of the following inequalities: (6) (7) if such a point exists, then this is considered a solution to this problem. the possibility of obtaining a solution (nash equilibrium point) of a bimatrix game in pure strategies, as a rule, is not guaranteed and depends on the properties of the matrix . the method for solving this problem includes carrying out a series of numerical calculations on a simulation model, constructing a payment matrix, analyzing it, and finding a solution. as an example of using the proposed approach, the following are the results of calculations and analysis of solutions to the problem for the case of three investment strategies and two scenarios of market demand dynamics. it will be shown that in many cases there are nash equilibrium points in pure strategies; the analysis of game solutions allows us to draw a number of conclusions that are interesting for practice. the result of this stage of the analysis of the investment strategies of companies is the answer to the following question: if the scenario is implemented, then what should be the companies' strategies that are optimal in the sense of criterion (5). the solution to the problem assumes that both companies consider the most likely implementation of the same scenario of market demand dynamics. 4. modeling and analyzing results this section provides an illustrative example of the application of the proposed approach. consider the duopoly market in the steel industry. a simplified situation is considered: there are two companies on the market that produce one type of metal products. the following model parameters were used in the calculations: it is assumed that in the period t = 0 supply and demand in the market are balanced, i.e., b (0) = d (0) = s (0), and equal to 10 million tons per year. p (0) is the market price of the product, equal to usd 500 per tonne; c (0) the cost of producing one ton of products is usd 400. the parameter of price elasticity g is equal to 0.5. this means that if the value of the imbalance deviates in the period t from the equilibrium value (in the period t = 0) by b%, respectively, the price of the product will change in relation to its equilibrium value by 0.5b% (with the same sign). e1(t)=0,03, i.e., with an investment of us $ 100 million, the company's production capacity is increased by 3.0 thousand tons over the period. is equal to 2 years, which corresponds to the average duration of the implementation of investment projects in metallurgy aimed at increasing production capacity. the maximum allowable volume of investments in projects of the first type for the period was adopted at the level of usd 100 million per year. e2 (0) = 0.015, i.e., with an investment of usd 100 million, the production cost of 1 ton of products is reduced by usd 1.5. the value of in the calculations is taken to be 1 year. these parameters of the model were obtained on the basis of an analysis of data on the implementation of investment projects in a large metallurgical company (nlmk group) and, naturally, reflect the assessment of the averaged values of these parameters. this information kk ,1= jj ,1= 2 , 1 , , jkjk npvnpv 2 , 1 , , jkjk npvnpv kknpvnpv kjjk ,1,11 000 =³ jjnpvnpv jkjk ,1,22 000 =³ 2 , 1 , , jkjk npvnpv y 1t прi 1 2t choosing directions for investments in the development of companies... 53 copyright ©2021 assa. adv. in systems science and appl. (2021) is presented in detail in [1]. for other sectors of the economy, these parameters of the model should be revised. the forecast period from 2020 to 2035 is considered. let both companies have the same initial capacity (5 million tons of products per year). scenario 1 (steady growth in demand). in accordance with this scenario, market demand for products throughout the forecast period grows at a steady 4% per year (fig. 2). the payoff matrix is presented in table 1. table 1. payoff matrix (scenario 1) j=1 j=2 j=3 k=1 7120\7120 8097\6414 9221\4262 k=2 6414\8097 7481\7485 9056\4801 k=3 4262\9221 4801\9056 6965\6965 the first element of the matrix corresponds to the winning of the first player, and the second element corresponds to the winning of the second player. in the table, the maximum elements of the columns of the matrix of the first player and the maximum elements of the rows of the matrix of the second player are highlighted in bold. analysis of the resulting matrix shows the presence of a single nash equilibrium point, which corresponds to the choice of investment strategy 1 by both companies (k = 1 and j = 1). at this stage, the elements of the matrix of the first and second players are highlighted in bold. figure 2 shows the dynamics of the expected market demand for products for scenario 1 and the dynamics of the growth of production capacity and, accordingly, the supply from companies 1 and 2. fig. 2. demand production capacity (scenario 1) it is important to note that a solution to a bimatrix game can have multiple nash points. this situation is illustrated by an example (table 1). at the first nash point, the winnings of both players are the same and equal to usd 7120 million. however, if companies agree to choose a different point that corresponds to investment strategy 2 (k = 2 and j = 2), then the players' gains will amount to usd 7481 million, which exceeds their winnings at the first nash point. the way out in such situations is the cooperation of the players. in real economic situations, competitors can interact with each other, entering into negotiations and concluding agreements, 54 v. akinfiev copyright ©2021 assa. adv. in systems science and appl. (2021) often informal. if the payoff matrix of the game has a set of points that are pareto optimal (i.e., others do not dominate), then such a set is called negotiated. players can agree to jointly select a point from the negotiating set in a variety of ways, including various arbitration schemes. the analysis of the game matrix (table 1) illustrates the usefulness of such cooperation for both companies. if cooperation between players is impossible, it is more profitable for them to adhere to equilibrium strategies. scenario 2 (2020-2024 is the period of growth in demand, then, 2025-2026 is a sharp drop in demand, and, further, 2027-2035 is a slow steady recovery of demand to the level of 2024) (fig. 3). the payoff matrix for scenario 2 is shown in table 2. in the table, the maximum elements of the columns of the matrix of the first player and the maximum elements of the rows of the matrix of the second player are highlighted in bold. analysis of the resulting matrix shows the presence of a single nash point, which corresponds to the choice of investment strategy 1 by both companies (k = 1 and j = 1). the elements of the matrix of the first and second players are highlighted in bold at this point. table 2. payoff matrix (scenario 2) j=1 j=2 j=3 k=1 4750\4750 4285\3507 5764\2980 k=2 3507\4285 4087\4087 5941\3715 k=3 2980\5764 3715\5941 5098\5098 note that the nash point for both scenarios is the same. this suggests that in conditions of competition and lack of cooperation, regardless of the scenario, the most profitable strategy for both companies are the strategy of maximum investment activity. the negotiation set consists of a single point that corresponds to the choice of investment strategy 3 by both companies (k = 3 and j = 3), which corresponds to the strategy of minimum investment activity. this set of player strategies provides them with the maximum payoff in case of their cooperation. figure 3 shows the dynamics of the expected market demand for products for scenario 2 and the dynamics of the growth of production capacities of company 1 and 2. fig. 3. demand production capacity (scenario 2) choosing directions for investments in the development of companies... 55 copyright ©2021 assa. adv. in systems science and appl. (2021) 5. conclusion the problem of analysis and selection of investment strategies of a company in the duopoly market, aimed at increasing their competitive advantages (increasing production capacities and reducing production costs), has been investigated. the problem is reduced to the analysis of a bimatrix game, in which the payoff matrix is formed as a result of numerical simulation. the method for solving this problem includes carrying out numerical calculations on a simulation model, constructing a payoff matrix and analyzing it. it is shown that in many cases there is a solution of a given game (nash equilibrium point) in pure strategies. the analysis of decisions taking into account possible coalitions of players and various types of agreements between them made it possible to draw a number of qualitative conclusions that are interesting for practice. it should be noted that the proposed approach to the study of the problem of choosing the investment strategies of companies opens up ample opportunities to study various modifications of this problem. for example, by varying the parameters of the model, one can consider various types of asymmetry in the market: § one of the companies is the industry leader in terms of production volumes and market share (d1 (0)> d2 (0)); § one of the companies is the technology leader in the industry. has lower production costs (s1 (0)> s2 (0)); § one of the companies has the best management, which translates into more efficient use of funds allocated for investments. e1 (0)> e2 (0). the methodology presented in the article was used to study the investment strategies of companies in the rolled metal market, as well as in the oil market [3, 4]. references 1. akinfiev, v. (2010) management of the development of integrated industrial companies: theory and practice (on the example of ferrous metallurgy). moscow: lenand, 224 pages. [in russian] 2. akinfiev, v. (2017) a modeling and choice of strategic investment decisions in oligopoly markets / proceedings of the 10th international conference "management of large-scale system development" (mlsd). moscow: ieee, [online]. available: https://ieeexplore.ieee.org/document/8109587/ 3. akinfiev, v. (2017) a model of competition between oil companies with conventional and unconventional oil production. large-scale systems control. issue 67. moscow: ics ras. 52-80, [in russian] 4. akinfiev, v. (2019) modeling and estimating the impact of the opec agreement on oil production in russia. advances in systems science and applications, 19(3), 131-139. doi.org/10.25728/assa.2019.19.3.718 5. bolton, p., yang, n. & wang, n. (2014) investment, liquidity, and financing under uncertainty // working papers/ columbia university. [online]. available: https://www0.gsb.columbia.edu/faculty/pbolton/papers/ssrn-id2364067 6. dixit, a. k. & pindyck, r. s. (1994) investment under uncertainty // princeton university press, princeton, р. 488. 7. grenadier, s. r. & wang, n. (2005) investment timing, agency, and information // journal of financial economics 75, 493–533. 8. grenadier, s.r. (2002) option exercise games: an application to the equilibrium investment strategies of firms // review of financial studies, vol. 15, no. 3, 691–721. 56 v. akinfiev copyright ©2021 assa. adv. in systems science and appl. (2021) 9. mason, r. & weeds, h. (2010) investment, uncertainty and pre-emption // international journal of industrial organization, 28, 278–287. 10. schwartz, e. s. & trigeorgis l. (2001) real options and investment under uncertainty: classical readings and recent contributions // the mit press, cambridge, usa, р. 261 11. smit, h. t. & trigeorgis, l., (2004) strategic investment: real options and games // princeton university press, princeton, nj, usa. 12. subbotin, a. (2009) modeling of volatility: from conditional heteroscedasticity to cascades on multiple horizons // applied econometrics, no. 3 (15), 94 138 [in russian]. 13. xiumei, l., shiqin, x. & xiaoling, t. (2014) investment timing and capacity choice under uncertainty // hindawi publishing corporation. abstract and applied analysis. volume 2014, article id 801862. adv syst sci appl 2018; 02; 53-58 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/582 use of neural network models in market risk management mariya radosteva1, vladimir soloviev1*, vera ivanyuk1,2, anatoliy tsvirkun2 1) financial university under the government of the russian federation, moscow, russia e-mail: vsoloviev@fa.ru 2) institute of control sciences ras, moscow, russia e-mail: tsvirkun@ipu.ru abstract: this topic is of high relevance due to the fact that many market risk assessment mathematical models currently available contain many limitations for their effective use. however, these limitations are often not feasible, which leads to a decrease in forecast accuracy. to avoid this, more accurate models are necessary. neural network-based models can show a more accurate result due to their basic property – nonlinearity. the goal of this paper is to build a model that can enable us to assess a market risk for a company. keywords: risk, forecasting, neural networks 1. introduction fig. 1. the primary goal of this paper is to determine a lower bound of the yield to be forecast by the neural network model with a certain level of significance. current actual yields will be fed to the neural network output, and some factors will be fed to the neural network input. starbucks stock prices for the period from 2011 to 2016 were taken as the factors fed to the neural network input. * corresponding author: vsoloviev@fa.ru 54 m. radosteva, v.soloviev, v.ivanyuk, a.tsvirkun copyright ©2018 assa. adv. in systems science and appl. (2018) fig. 2 however, there is no point in using this data as is, so the values of stock yield were fed to the neural network input. it was decided to take weekly yields with a lag of one day as the yield basic indices. in other words, the yield values were calculated as follows: 𝑟𝑡 = 𝑃𝑡 − 𝑃𝑡−5 𝑃𝑡−5 , where the t is a time interval assumed as a day (the t assumes the values of 6,7,8,... from the beginning of the counting period taking into account that the initial value is indexed by the one); the 𝑃𝑡 is the closing price on the t day without taking into account dividends (this choice is due to the fact that we cannot forecast dividends upon stocks and we do not need this because we assess the market risk of a company); the 𝑟𝑡 is the earned weekly yield which has been used for modeling. fig. 3 to improve the model quality volatility was also used as input and output data, so the next step was to calculate daily yields and weekly volatilities (standard deviations) on their basis with a lag of one day similarly with the yields. the volatilities finding procedure can be described as follows: 𝑟𝑘 = 𝑃𝑘 − 𝑃𝑘−1 𝑃𝑘−1 𝜎 = √∑(𝑟𝑘 − 𝑟�̅�)2 4 𝑘=1 another input parameter was trade volume change values. use of neural network models in market risk management 55 copyright ©2018 assa. adv. in systems science and appl. (2018) 𝑣𝑜𝑙𝑢𝑚𝑒𝑡 = 𝑉𝑜𝑙𝑢𝑚𝑒𝑡 − 𝑉𝑜𝑙𝑢𝑚𝑒𝑡−5 𝑉𝑜𝑙𝑢𝑚𝑒𝑡−5 , where 𝑉𝑜𝑙𝑢𝑚𝑒𝑡 is an absolute trade volume value on the t day; 𝑣𝑜𝑙𝑢𝑚𝑒𝑡 is a trade volume relative change on the t day. 2. distribution of the neural network errors when forecasting the yield the model is based on the architecture of the artificial neural network of a multilayer perceptron or a direct propagation network with error back propagation learning method. the model is aimed at forecasting yield lower bound for the time being, which can be crossed by the actual yields graph not more than α in 100% of cases. to solve this problem, it was decided that the neural network should learn based on historical data. it was also assumed that the current yield and volatility depended on lagged yield and lagged volatility values. an empirical distribution function was constructed and a quantile of 0.95 level was taken. the error distribution density is shown in fig. 4. fig. 4 the resulting value was calculated as a correction level to the initially specified var curve. that is the final formula for the var based on the neural network is as follows: 𝑉𝑎𝑅𝑛𝑒𝑢𝑟 = 𝑟𝑛𝑒𝑢𝑟 − 𝜎𝑛𝑒𝑢𝑟 − 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛 this curve was constructed in two ways. the first way of constructing is based on the fact that the neural network learns once and then a forecast is made using it. that is essentially a model with static weights in that the weights inside the network are calculated once and no longer change. this approach to the var curve constructing gives an advantage in terms of program running time, since the learning occurs once. for the sake of convenience this model will be hereinafter referred to as a neural network model with static weights. the second way is based on the assumption that the market structure is changeable (which is actually the case) and to make a forecast more accurate the neural network learns each time when calculating the var. that is, to forecast the yield for the time being the neural network learns using n size sample, which includes the previous n values of this index. this is a model with dynamic weights. at each calculation of the var value the neural network learns again. of course, such method of the var constructing is inferior to the first one, but it should be more accurate. for the sake of convenience this model will be hereinafter referred to as a neural network model with dynamic weights. 56 m. radosteva, v.soloviev, v.ivanyuk, a.tsvirkun copyright ©2018 assa. adv. in systems science and appl. (2018) kupic test consists in checking the following statistical hypothesis: 𝐻0: 𝛼0 = 𝛼 against alternative 𝐻𝑎𝑙𝑡: 𝛼0 ≠ 𝛼, where the 𝛼 = 𝐾 𝑁 , 𝐾 is the number of the var line breaks, the 𝑁 is the quantity of forecast data, and the 1 − 𝛼0 is a specified level of significance. checking is performed using statistics 𝑆𝐶𝑢𝑝 = −2 ln((1 − 𝛼) 𝑁−𝐾𝛼𝐾) + −2 ln((1 − 𝛼0) 𝑁−𝐾𝛼0 𝐾), which has the 𝜒2(1) distribution if the null hypothesis is true. another quality index that helps to choice between the models that passed the kupic test is the loss function value, which measures the average value of the var level excess by actual losses. the smaller the loss function value, the more adequately risk is assessed by the considered model. the loss functions most commonly used are the lopez and blanco-ihle’s ones: 𝐿𝐿𝑜 = 1 𝐾 ∑(𝑦𝑡 − 𝑉𝑎𝑅𝑡) 2𝐼(𝑦𝑡 < 𝑉𝑎𝑅𝑡) 𝑇 𝑡=1 , 𝐿𝐵𝐼 = 1 𝐾 ( 𝑦𝑡 − 𝑉𝑎𝑅𝑡 𝑉𝑎𝑅𝑡 ) 𝐼(𝑦𝑡 < 𝑉𝑎𝑅𝑡) the lopez’s loss function is different in that it gives a greater weight to significant deviations. this is justified from a substantive point of view, since single large excesses are usually more dangerous than a few small ones. when constructing the var curve, the programming language r with the neuralnet library was used, which allowed to construct direct propagation neural networks with the error back propagation methods in different versions. in particular, the rprop learning method was used. during the research, experiments to change the number of layers in the multilayer perceptron, as well as the number of neurons in each layer in order to improve forecast accuracy were carried out. the learning sample-based forecast error formula built into the neural network was used as a guide. such error formula was calculated as follows: 𝐸 = 1 2 (𝑦𝑓𝑎𝑐𝑡 − �̃�) 2 and was output by the program itself. since the neural network showed different results at each learning session, it was decided to take the arithmetic mean of this error in order to generalize this parameter for the neural network with such layers and quantity. 𝐸ср = 1 𝑛 ∑𝐸𝑖 𝑛 𝑖=1 , where the n was taken equal to the order of 30. the result of the research is the error! reference source not found.. table 1. number of neurons in layers yield volatility with a relative change in trade volume without a relative change in trade volume without a relative change in trade volume (6, 3) 0.14399 0.163759 0.00813 use of neural network models in market risk management 57 copyright ©2018 assa. adv. in systems science and appl. (2018) (8, 3) 0.135139 0.157891 0.00813 (11, 6) 0.12661 0.14632 0.00813 (12, 3) 0.12789 0.139695 0.00813 (12, 6) 0.13096 0.132457 0.00813 (22, 6) 0.1176 0.14441 0.00801 (22, 11) 0.10761 0.15441 0.00771 (24, 3) 0.107 0.14177 0.00713 (24, 6) 0.11826 0.17130 0.00712 (22, 11, 3) 0.12405 0.1789 0.01519 (22, 11, 6) 0.10574 0.17111 0.02907 (24, 12, 6) 0.11581 0.1753 0.01813 from the implementation point of view the back propagation method converges for a long time, therefore its various modifications are often used. in particular, during the neural network learning the rprop method has been used which is as follows. the algorithm is based on the partial derivative sign. new weights are updated according to: 𝑤𝑖𝑗 (𝑡+1) = 𝑤𝑖𝑗 (𝑡) + ∆𝑤𝑖𝑗 (𝑡) { ∆𝑤𝑖𝑗 (𝑡) = −𝑠𝑖𝑔𝑛 ( 𝜕𝐸 𝜕𝑤𝑖𝑗 (𝑡) )∆𝑖𝑗 (𝑡), if 𝜕𝐸 𝜕𝑤𝑖𝑗 (𝑡−1) 𝜕𝐸 𝜕𝑤𝑖𝑗 (𝑡) ≥ 0 ∆𝑤𝑖𝑗 (𝑡) = −∆𝑤𝑖𝑗 (𝑡−1) и ( 𝜕𝐸 𝜕𝑤𝑖𝑗 (𝑡) ) = 0, if 𝜕𝐸 𝜕𝑤𝑖𝑗 (𝑡−1) 𝜕𝐸 𝜕𝑤𝑖𝑗 (𝑡) < 0 . where the 𝑠𝑖𝑔𝑛() is a function that assumes the value of +1 or -1 depending on which sign has a value in parentheses; ∆𝑖𝑗 (𝑡)= { min(𝜂+∆𝑖𝑗 (𝑡−1) , δmax ) , if 𝜕𝐸 𝜕𝑤𝑖𝑗 (𝑡−1) 𝜕𝐸 𝜕𝑤𝑖𝑗 (𝑡) > 0 max(𝜂−∆𝑖𝑗 (𝑡−1), δmin ) , if 𝜕𝐸 𝜕𝑤𝑖𝑗 (𝑡−1) 𝜕𝐸 𝜕𝑤𝑖𝑗 (𝑡) < 0 ∆𝑖𝑗 (𝑡−1), if 𝜕𝐸 𝜕𝑤𝑖𝑗 (𝑡−1) 𝜕𝐸 𝜕𝑤𝑖𝑗 (𝑡) = 0 where the zero values ∆𝑖𝑗 (0) and 0 < 𝜂− < 1 < 𝜂+ are chosen arbitrarily. such an algorithm converges faster than a usual back propagation method. the result of this study is the var curve constructed on the basis of the neural network model. one of the ways to implement the neural network model for the var level forecasting is shown in figure 5. 58 m. radosteva, v.soloviev, v.ivanyuk, a.tsvirkun copyright ©2018 assa. adv. in systems science and appl. (2018) fig. 5 the var curve constructed on the basis of the neural network model with static weights and a significance level of 0.95 (the black line is the yield curve, the red line is the var level). in this case, the actual percentage of break is at the level of 4.65%. the kupic test has shown a p-value= 0.6809912418, which is a good result. the lopez and blanco-ihle’s tests for such a situation assumed the following values: 𝐿𝐿𝑜 = 0.0001344051456, 𝐿𝐵𝐼 = 0.1513434227, which generally evidences a good quality of the model, since the break depth is relatively small. in particular, the lopez’s index indicates that existing breaks are light, which is a good sign. the result of the paper is the market risk assessment neural network model, which has shown good results according to the tests performed. references [1] ben, k. & van der smagt, p. (1996). an introduction to neural networks. university of amsterdam. [2] haykin, s. (1999) neural networks: a comprehensive foundation, prentice hall [3] ivanyuk, v. & tsvirkun, a. (2013). intelligent system for financial time series prediction and identification of periods of speculative growth on the financial market. ifac proceedings volumes, 46(9), 1128-1133. [4] ivanyuk, v. & pashchenko, f. (2015) methods and models for the forecasting and management of time series. // itise 2015, granada, 283-292 [5] simonov, b., ivanyuk, v., & simonova, i. (2016). existence of best approximation elements in the spaces l. journal of mathematical sciences, 217(5), 1-21. [6] koroteev, m., terelyanskii,, p. & ivanyuk v. (2016) approximation of series of expert preferences by dynamical fuzzy numbers. journal of mathematical sciences, 216(5), 692-695. [7] rumelhart, d. e., hinton, g. e., & williams, r. j. (1985). learning internal representations by error propagation (no. ics-8506). california univ san diego la jolla inst for cognitive science. adv syst sci appl 2017; 3:22–33 published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/497 conformal kernel expected similarity for anomaly detection in time-series data aleksandr safin1,2, evgeny burnaev2,3* 1 national research university higher school of economics, moscow, russia 2 skolkovo institute of science and technology, skolkovo, moscow region, russia 3 institute for information transmission problems, moscow, russia abstract: the problem of anomaly detection arises in many practical applications. currently it is highly important to be able to detect outliers in data streams, as recent years have seen a rapid growth in the amount of such data. only a few techniques are applicable to real-time data and even fewer could provide an interpretable anomaly score. probabilistic interpretation of the anomaly score could allow an analyst to choose the anomaly threshold based on the desired false alarm rate, which is highly important in a number of real-life applications. we propose a modification of the expose algorithm for anomaly detection in time series data, which produces a probabilistic score of abnormality. the proposed algorithm is developed within the framework of conformal anomaly detection and utilizes the expected similarity as a measure of non-conformity. keywords: anomaly detection, conformal prediction, time series, kernel methods, expected similarity 1. introduction there are many cases in which it is highly important to determine whether a new observation comes from the same distribution or not. this problem is referred to as outlier or anomaly detection. e.g. when a fitted model is applied to new data, it should be checked whether a test data set belongs to the same population as the training data set. to address the issue of novelty detection, anomaly detection techniques can be used. anomaly detection has proven to be helpful for certain medical purposes, fraud detection and machine diagnostics, to name but a few. for instance, in [1] failure prediction for aircrafts is considered. the definition of an anomaly varies between algorithms and applications. in general, an anomaly “is an element whose properties differ from the majority of the other elements under consideration which is called as normal data” [14]. in [15] anomaly detection is described as follows: “anomaly detection refers to the problem of finding patterns in data that do not conform to expected behavior”. summing up, the problem of anomaly detection can be formulated as follows: the task is to determine for every object in a test set whether it is a normal or abnormal instance in comparison with observations from a training set. anomaly detection approaches can be divided in the following three groups [15]: • unsupervised approaches use only the assumption that most observations are normal. such assumption favours incremental and autonomous learning in data streams. ∗corresponding author: e.burnaev@skoltech.ru http://ijassa.ipu.ru/ojs/ijassa/article/view/497 conformal kernel expected similarity for anomaly detection in time-series data 23 • supervised approaches require availability of a labelled training set containing instances of both normal and abnormal objects. • semi-supervised approaches require a small amount of labeled data with a large amount of unlabeled data. in many practical cases a number of outliers is significantly smaller than a number of target observations, and thus usual classification methods may yield unsatisfactory results as classes in a dataset are very imbalanced. the significant dominance of target instances over outliers is a natural property of real-life data: e.g. in case of air traffic safety problems accidents happen very rarely. another reason for that is the impossibility or very high costs of reproducing faulty conditions when we consider a machine diagnostic task. in the light of known and outlined difficulties, classical methods are not applicable to solving these problems, thus a variety of outlier detection methods have been developed. such problems justify the need for specialized approaches to anomaly model selection [2], learning with privileged information [6], construction of ensembles of non-parametric anomaly detectors in data streams [3,7,8], usage of specific time-series models [9–13], and explicit rebalancing of normal and abnormal classes [4, 5], among others. unsupervised anomaly detection does not require the training dataset to be labelled, thus it is applicable to various problems, as in general it is not feasible to collect labels. therefore a number of applications adopts unsupervised approaches, e.g. based on density estimation or clustering [16]. according to the surveys [15] and [17] unsupervised anomaly detection techniques could be generally categorized as probabilistic (distributionand density-based), prediction-based, distance-based, classification-based, clustering-based and information-theoretic approaches. distribution-based methods estimate parameters of a target data distribution and determine whether a test object comes from the same distribution that generated samples from the training set. the main drawback of these methods is the necessity to select some parametric class of data distributions. one of the tricks is to model the target distribution as a mixture of gaussians, however the number of gaussians still have to be determined. to mitigate this problem, other non-parametric techniques could be utilized, for instance histogram-based or kernel density estimator (kde). however, the number of bins should be firstly specified, and the performance is highly sensitive to this hyper-parameter. for multivariate problems a basic approach is to estimate a histogram per each input feature. however some features could be correlated, in that case the information about such dependency will be lost. prediction-based techniques predict future observations based on previous items and then compare predicted and real data to identify anomalies. another type of approaches is based on the distance to the k-th nearest neighbour (knn). one of such techniques is the knn-based outlier detector [18]. all objects are sorted w.r.t. the average distance to k nearest neighbours and top n of them with the highest average distance are claimed to be anomalies. lof method, proposed in [19], exploits density based approach: it uses the distance to the k-th nearest neighbour as an inverse estimate of a local density value. however data could contain clusters with different densities leading to significantly increased false anomaly detections. main drawback of distances-based anomaly detection methods is poor interpretability of their output. to address this issue conformal anomaly detector (cad) was proposed by laxhammar [16]. having a probabilistic interpretation of the degree of anomalousness allows choosing a threshold with a false alarm rate guarantee. zhao and saligrama stated in [20] that “while [modern anomaly detection] approaches provide impressive computationally efficient solutions on real data, it is generally difficult to precisely relate tuning parameter choices to desired false alarm probability”. at the same time according to burnaev and nazarov, conformal prediction could be used for constructing non-parametric confidence intervals with a specified confidence probability [21, 22]. copyright © 2017 assa. adv syst sci appl (2017) 24 aleksandr safin, evgeny burnaev some techniques adopt approaches used for classification tasks. tax and duin proposed support vector data description [23] and later it was refined in [24]. the task of data description is formulated as follows: given the unlabelled training data, construct a closed boundary (or a set of them) that contains predominantly the target data, and outliers are outside this boundary. in a simple case, boundary is supposed to be spherical, but in general it is possible to determine an arbitrary-shaped flexible boundary by using kernel functions. moreover, svdd is robust against the training data containing outliers and also is capable of improving the accuracy by incorporating additional information about negative examples, in case when the training dataset is labelled. according to the results of their study, svdd is shown to yield mostly comparable or even better results for sparse and complex multidimensional datasets. an extension of svm to the case of unlabelled data is outlined in [25]. this approach which is referred to as one-class svm has been adapted by ma and perkins [26] for time series. the problem of anomaly detection has many dimensions. in particular, data could be not fixed but represented as a stream. a variety of techniques could be used for anomaly detection. however, only a few can be used for data streams. classical algorithms for anomaly detection are not applicable due to their computational complexity and memory consumption, since the number of elements in the set is growing and therefore it is not possible to store all previously observed data. it is worth emphasizing that frequently the concept of anomaly could change in the course of time. this phenomenon is called as concept drift. in general, a streaming version of an anomaly detection algorithm should have an ability of adaptation to a concept drift; therefore, such algorithms are usually based on one of the following strategies of accumulating information about current changes in data: namely windowing and exponential smoothing, which is also referred to as decay. the above-mentioned issues are considered in the paper [14]. the proposed expose algorithm is developed within the framework of reproducing kernel hilbert space (rkhs) and exploits the concept of kernel mean embedding. in a nutshell, the estimator used in the algorithm could be described as a dot product of a kernel mean map and a feature map of the observed data point. however, the produced anomaly score has a lack of intuitive interpretability and therefore the false alarm rate could not be guaranteed when anomaly threshold is chosen. in order to eliminate this drawback we utilize lazy drifting conformal detector procedure proposed in [32] to construct the algorithm with probabilistic interpretation of the anomaly score while based on the idea of kernel mean embedding proposed in [14]. the structure of the paper is the following. sections 2 and 3 shed light on kernels and conformal anomaly detection respectively. the proposed algorithm is described in section 4. in section 5, the results of empirical evaluation (using numenta anomaly benchmark) of the proposed algorithm are outlined. finally, section 6 describes achieved results and the work still to be done. 2. overview of kernel-based methods for anomaly detection in machine learning kernels are broadly used for handling data of diverse nature. therefore it is not surprising that a number of anomaly detection methods are based on the kernel framework. in this section we give some necessary definitions and provide an overview of such anomaly detection methods which uses kernels and therefore are applicable to data of various types. 2.1. introduction to kernels reproducing kernel hilbert space (rkhs) is a hilbert space (h, 〈·, ·〉) of functions f: x → r if the evaluational functional δ̄x : f → f(x ) is continuous. copyright © 2017 assa. adv syst sci appl (2017) conformal kernel expected similarity for anomaly detection in time-series data 25 reproducing kernel of h is a function k: x × x → r which satisfies the reproducing property: 〈f,k(x , ·)〉 = f(x ), 〈k(x , ·), k(y , ·)〉 = k(x ,y). the map φ: x → h with the property that k(x ,y) = 〈φ(x ), φ(y)〉 is referred to as a feature map. definition 2.1 (expected similarity estimation): the expected similarity [14] of z ∈ x given the probability distribution p(x) is defined as: η(z) = ex [φ(z)] = ∫ x k(z, x)dp(x). definition 2.2 (kernel embedding): kernel embedding of the distribution p has the form µ[p] = ∫ x k(x, ·)dp(x). expectation of any f ∈ h ex [f ] = 〈f, µ[p]〉h. thus, η(z) = 〈φ(z), µ[p]〉h. given the empirical distribution pn(x) by observing n realizations {x1, . . . xn} independently sampled from p, one could approximate µ[p] as follows: µ[p] ≈ µ[pn] = 1 n n∑ i=1 φ(xi). this approach is referred to as empirical kernel embedding [29] and given that ‖φ(x)‖ ≤ c,c > 0 the following guarantee has been proved by schneider [30] for all ε > 0: p (‖µ[p]− µ[pn]‖ ≥ ε) ≤ 2e− nε2 8c2 . considering above mentioned, having observed {x1, . . . xn}, the expected similarity estimation for z ∈ x is η(z) = 〈φ(z), µ[p]〉h ≈ 〈φ(z), µ[pn]〉h = 1 n n∑ i=1 k(z, xi). 2.2. expose expected similarity estimation (expose) that was proposed in [14] is the method for anomaly detection which could handle data streams. for every new observation z the algorithm computes an anomaly score η(z) based on computed empirical kernel mean map wt of previously observed items η(z) = 〈φ(z), wt〉 ‖wt‖2 . copyright © 2017 assa. adv syst sci appl (2017) 26 aleksandr safin, evgeny burnaev kernel mean map could be evaluated using one of the following strategies. the first strategy is to use a sliding window of length l: wt = 1 l t∑ i=t−l+1 φ(xi). more flexible approach is to apply exponential smoothing to all previous observations: wt = γφ(xt) + (1− γ)wt−1, t > 1. the parameter γ reflects the influence of a new data item. however, as already was indicated, the anomaly score provided by this algorithm could not be interpreted in a probabilistic manner, therefore we propose an approach to transform the anomaly score produced by expose. to that end, the idea of conformal anomaly detection is adopted to build an anomaly detector for online data. 3. conformal anomaly detection laxhammar [16] proposed a conformal anomaly detection (cad) which is a distributionfree procedure for probability-like confidence measure estimation based on non-conformity measure (ncm) provided by some detector. the ncm a(x, y) reflects how different the investigated object y is from other observations x. ncm could be for instance the average distance to k neighbours, the distance to the k-th neighbour, residual in a regression model, to name just a few. let us consider a time series xt, then compute scores ats = a(x−s:t , xs), s = 1, . . . , t, where a(x, y) is an ncm used by the algorithm. then the empirical p-value is defined as: p(xt, x:(t−1), a) = 1 t |{s = 1, . . . , t : ats ≥ att}|. the lower it is, the lower the probability of falsely rejecting the null hypothesis (xt is anomaly) is, thus the more likely xt is an anomaly instance. shafer and vovk proved [31] the fact that cad could provide the following guarantee when xt is i.i.d: px∼d(p(xt, x −t, a) < ε) ≤ ε,x = (xs) t s=1. it is clear that cad could be computationally heavy as it requires computations of a(x−s:t , xs) for s from 1 to t. to mitigate this problem, an inductive conformal anomaly detection (icad) was proposed by laxhammar and falkman in [27]. this approach relies on scores computed on training set x̄ for every instance of the calibration set. for further simplicity, let us consider relabelled sequence xt that starts from−n+ 1. then, icad has the following setup for every t ≥ 1: x−n+1, . . . ,x0︸ ︷︷ ︸ x̄ training , calibration︷ ︸︸ ︷ x1,x2, . . . ,xt−1,xt. in that setup, the conformal p-value of a test object xt is computed on the basis of modified scores: {ats = a(x̄,xs), s = 1, . . . , t}, x̄ = (x−n+1, . . . ,x0). however, by relaxing deterministic guarantee to probably approximately correct guarantee, it is achievable to adapt icad to use only fixed size calibration set. offline icad copyright © 2017 assa. adv syst sci appl (2017) conformal kernel expected similarity for anomaly detection in time-series data 27 was developed to use a calibration set only with fixed size m, sliding along the time series, as illustrated: x−n+1, . . . ,x0︸ ︷︷ ︸ x̄ training , . . . , calibration︷ ︸︸ ︷ xt−m,xt−m+1, . . . ,xt−1,xt. it should be highlighted that the conformal p-value in this case uses a subsample of the icad non-conformity scores: p(xt, x:(t−1), a) = 1 m+ 1 |{s = 0, . . . ,m : att−s ≥ att}|. vovk proved [28] the following guarantee for the offline icad: px∼d(p(x, x, ā) < ε) ≤ ε+ √ log 1 δ 2m . 4. proposed approach for anomaly detection in time series data in this section we outline the proposed algorithm for anomaly detection in time series. it is worth emphasising that the developed approach does not require any assumption about the data distribution and it outputs a probabilistic measure of anomality based on nonconformity scores. such measure of anomality is calculated using an adaptation of icad to the case of potentially non-stationary and quasi-periodic time series. cad and online icad are computationally complex, therefore offline icad seems much suitable for the task. nevertheless, as offline icad uses a fixed training set, it should be noticed that one could face problems in case of non-stationary time series. in the light of the discussed details and difficulties, lazy drifting conformal detector (ldcd) has been proposed in our paper [32]. for simplicity we consider a univariate time-series x = (xt)t≥1 ∈ r, although our approach is valid for multivariate data as well since we are going to use kernel-based non-conformity measure. to begin with, the time series x should be embedded into l-dimensional space. to that end, we further consider the sequence of xt = (xt−l+1, . . . , xt) ∈ rl constructed by moving window of the width l on the time series x: . . . , xt−l−1, xt−1 xt−l, xt−l+1, . . . , xt−1, xt xt , xt+1, . . . . it should be noticed that such approach obviously produce t− l+ 1 embeddings from the sequence of the length t. in other words, to produce the first such embedding, we need to observe l instances initially. as ncm we are using the expected similarity: a(tt,xt) = 1 n n∑ i=1 k(xt,xt−m−i), where tt = {xs : s = t−m− n, . . . , t−m− 1}, k(x, y) is a kernel function. data: . . . , tt training︷ ︸︸ ︷ xt−m−n, . . . ,xt−m−1, xt−m, . . . ,xt−1, test xt, . . . scores: . . . , at−m−n, . . . , at−m−1, at−m, . . . , at−1︸ ︷︷ ︸ at calibration , at test , . . . copyright © 2017 assa. adv syst sci appl (2017) 28 aleksandr safin, evgeny burnaev the proposed approach could be described as follows: 1. construct time series embedding in a sliding window, 2. compute an+s = a(tn+s,xn+s), s = 1, . . . ,m+ 1, 3. evaluate the empirical p-value of its non-conformity score: p(xt, tt, a) = 1 m+ 1 |{i = 0, . . . ,m : at−i ≥ at}|. the algorithm is depicted in the figure 4.1. fig. 4.1. flowchart of the algorithm it is worth emphasising that the proposed approach could be implemented with time complexity equal to o(n) +o(logm) in the case of using red-black tree for the calibration set. 5. results on numenta anomaly benchmark the results of the expose ldcd comparison with several other algorithms is presented in this section along with the testing methodology. the numenta anomaly benchmark (nab) is utilized to test the proposed algorithm. 5.1. datasets the nab corpus consists of 58 both real-world and artificial time series datasets. real-world data are obtained from such sources as aws server metrics, twitter volume, advertisement clicking metrics, traffic data, to name just a few. we also conduct experiments using numenta anomaly benchmark on yahoo! s5 dataset [34] which has been created to gauge the anomaly detectors performance on different types of anomalies. this corpus is divided into 4 groups: first one contains real production metrics from different yahoo! properties, and the rest are synthetic time series. copyright © 2017 assa. adv syst sci appl (2017) conformal kernel expected similarity for anomaly detection in time-series data 29 5.2. scoring algorithm commonly used metrics for performance evaluation such as accuracy, precision and recall do not suit well for anomaly detection, since they do not consider time. the nab proposes such an approach for scoring which rewards only early true detection, meanwhile penalizes late detections and punishes false alarms very hard. to capture early detections, nab considers the area which is centred around the anomaly point which is referred to as anomaly window. the window length is defined as 10% of the length of the time series. all detections within this window are true positives, but only the earliest one contributes in the total score, the others will be ignored. the detections outside the anomaly window are false positives, missed anomalies are false negatives. true negatives are not considered in the scoring mechanism. bellow we described scoring scheme used in nab [33]. an example of time series is provided in figure 5.2. the first 15% of the time series is considered as probationary period and during this period an algorithm learns patterns from the data and is not required to do any detections. then the algorithm is evaluated on the remaining part of time series. the weights for accuracy calculation is evaluated using the smooth sigmoid function as depicted in figure 5.3. fig. 5.2. the purple shaded area is the probationary period. anomalies are depicted as red points and red shaded regions represent anomaly windows [33]. as the costs of true positive (tp), false positive (fp) and false negative (fn) vary among distinct applications, in nab this is captured by an application profile which reflects the contribution of weights for tp, fp an fn detections. the “standard” application profile reflects scenarios in which misdetections have identical costs. the “reward low fp” and “reward low fn” profiles penalize harder for fp and fn respectively. profile atp afn afp atn standard 1.0 -1.0 -0.11 1.0 reward low fp 1.0 -1.0 -0.22 1.0 reward low fn 1.0 -2.0 -0.11 1.0 table 5.1. the detection rewards on nab application profiles the reward for the detection depends on the relative position t of the alarm (about possible anomaly) to the left side of the anomaly window: σa(t) = (atp − afp ) ( 1 1 + e5t ) − 1. the raw performance score on the dataset x with respect to application profile a is the sum of the scores over all detections plus the impact of missed anomalies (false negatives) copyright © 2017 assa. adv syst sci appl (2017) 30 aleksandr safin, evgeny burnaev captured by the number of anomaly windows with no detections fdet: sadet(x) = ∑ y∈ydet σa(y) + afnfdet. the overall performance of the algorithm is the sum of raw performance scores over the all datasets d: sadet = ∑ x∈d s a det(x). the final normalized performance score is determined by: sanab = 100 sadet − sanull saperfect − sanull . fig. 5.3. nab weighted scores: detections outside the anomaly window are false positives and punished; only earliest detection inside window is true positive and it will be counted, other will be ignored [33]. 5.3. results since proposed expose ldcd is conservative and demonstrates high level of false alarms, we have applied the following simple pruning strategy to reduce the false alarm rate: we output 1− p as anomaly score for the observation xt and if p is greater than 99.65%, then output of the detector is fixed at 0.5 for the next n 5 observations (n is the length of probationary period). the proposed approach has been validated on both the numenta anomaly benchmark corpus and the yahoo! s5 dataset. tables 5.2 and 5.3 reflect the results of the algorithms comparison. 5.4. automated kernel bandwidth tuning kernel-based methods are sensitive to the choice of bandwidth, therefore we modify the algorithm to choose the bandwidth of the kernel based on the best value of the bandwidth for kernel density estimator obtained by 3-fold cross-validation. the proposed modification demonstrates significantly better results on nab dataset and is able to increase the score on both low fn and low fp profiles, meanwhile it results in slight score decrease on standard profile. 6. conclusion in this paper we propose an algorithm for anomaly detection in time series data, utilizing the concept of expected similarity and applying framework of conformal anomaly detection. copyright © 2017 assa. adv syst sci appl (2017) conformal kernel expected similarity for anomaly detection in time-series data 31 table 5.2. results on numenta anomaly benchmark detector profile standard reward low fp reward low fn numenta htm 70.1 63.1 74.3 expose ldcd +tuning 45.53 25.77 54.78 windowed gaussian 39.6 20.9 47.4 expose ldcd 37.93 20.14 45.11 etsy skyline 35.7 27.1 44.5 bayesian changepoint 17.7 3.2 32.2 expose 16.4 3.2 26.9 table 5.3. results on yahoo! s5 dataset detector profile standard reward low fp reward low fn expose ldcd 51.88 38.76 58.95 expose ldcd +tuning 49.79 43.73 61.45 numenta htm 41.0 37.5 44.4 bayesian changepoint 35.7 17.6 43.6 expose 32.09 7.00 45.45 windowed gaussian 31.1 25.8 40.7 etsy skyline 23.6 18.0 28.9 table 5.4. average running time performance on nab dataset detector performance items per second ms per item windowed gaussian 1984.862 0.504 expose ldcd 1500.224 0.667 bayesian changepoint 428.639 2.333 expose 398.496 2.51 numenta htm 98.012 10.202 etsy skyline 4.582 218.229 table 5.5. average running time performance on yahoo! s5 dataset detector performance items per second ms per item expose ldcd 2548.293 0.392 windowed gaussian 2348.041 0.426 bayesian changepoint 1217.888 0.821 expose 383.711 2.606 numenta htm 103.777 9.636 etsy skyline 4.656 214.773 this approach has been rigorously validated on nab corpus and yahoo! s5 dataset using numenta anomaly benchmark. on both datasets the proposed approach excel the expose, which produces expected similarity as anomaly score and expected similarity is used as nonconformity measure in the ldcd procedure. moreover, the developed algorithm shows great running time performance, which is important for online detectors and achieves high results on standard profile on yahoo dataset. also, the implementation of the algorithm could be enhanced, as it has not been thoroughly optimised and it could be one of the directions for future research. we also propose a tuning procedure for the kernel bandwidth parameter, however there is still a significant room for improvements. acknowledgements the work was supported by the ministry of education and science of russian federation, grant no. 14.606.21.0004, grant code: rfmefi60617x0004. copyright © 2017 assa. adv syst sci appl (2017) 32 aleksandr safin, evgeny burnaev references 1. alestra s., bordry c., brand c., burnaev e., erofeev p., papanov a. & silveirafreixo c. 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(2017) conformal k-nn anomaly detector for univariate data streams. arxiv e-prints, [online]. available: https://arxiv.org/abs/1706.03412 33. lavin a. & ahmad s. (2015) evaluating real-time anomaly detection algorithms the numenta anomaly benchmark. 14th international conference on machine learning and applications (ieee icmla), 38-44, https://arxiv.org/abs/1510.03336 34. yahoo! webscope (2017, december 26) s5 a labeled anomaly detection dataset, version 1.0. [online]. available https://webscope.sandbox.yahoo.com/catalog.php?datatype=s&did=70. copyright © 2017 assa. adv syst sci appl (2017) https://doi.org/10.1109/icmla.2016.0017 http://ai2-s2-pdfs.s3.amazonaws.com/a244/422ba339713d0c9eaa153b378e9f9fc08263.pdf https://doi.org/10.1007/978-3-540-75225-7_5 https://arxiv.org/abs/1706.03412 https://arxiv.org/abs/1510.03336 https://webscope.sandbox.yahoo.com/catalog.php?datatype=s&did=70 introduction overview of kernel-based methods for anomaly detection introduction to kernels expose conformal anomaly detection proposed approach for anomaly detection in time series data results on numenta anomaly benchmark datasets scoring algorithm results automated kernel bandwidth tuning conclusion adv syst sci appl 2018; 02; 63-83 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/576 copyright ©2018 assa adv. in systems science and appl. (2018) wgw: a hybrid approach based on whale and grey wolf optimization algorithms for requirements prioritization amjad hudaib1, raja masadeh2, abdullah alzaqebah2 1computer information systems department, the university of jordan, amman, jordan e-mail: ahudaib@ju.edu.jo 2computer science department, the world islamic sciences and education university, amman, jordan e-mail: raja.masadeh@wise.edu.jo , abdullah.zaqebah@wise.edu.jo abstract: requirement engineering is the base phase of any software project, since this phase is concerned about requirements identification, processing and manipulation. the main source of these requirements is the project stakeholders with considering the project constraints and limitation. number of requirement is varying for each project, so the requirements prioritization term comes for prioritizing the order of execution for software requirements according to the stakeholder's opinions and decisions. various proposed optimization algorithms are employed to solve optimization problems; recently whale optimization (wo) algorithm is proposed in 2016 by mirjalili which mimics the main characteristic of humpback whales which is the foraging method that is called bubble-net technique. on the other hand grey wolf optimization (gwo) algorithm was proposed in 2014 in order to solve optimization problems by imitating the grey wolves hunting behavior. in this paper, a hybrid approach based on whale and grey wolf optimization algorithms (wgw) is proposed by combining the advantages of each algorithm in order to prioritize the software requirements. moreover, the data set that used in this paper is ralic which a real software project’s requirements is in order to evaluate the proposed method. thus, the proposed method shows 91% accuracy of requirements prioritization comparing with ralic data sat. keywords: requirement prioritizations (rp), whale optimization algorithm (woa), grey wolf optimization (gwo), replacement access, library and id card project (ralic), rp-woa. i. introduction requirement engineering (re) in one of the most significant branch in the domain of software engineering. in addition, it is considered as the most important phase in software development life cycle [1]. this phase contains identification and elicitation of requirements, analysis and requirements validation and documentation. in almost software projects, there are restrictions on development process like budget and time to market production, this leads to deliver the software projects in consecutive releases, hence large projects have more than one of stakeholders that makes a difficulty in decision making about what release should be developed firstly. this difficulty contributes the software engineers to prioritize the requirements in efficient way to make the right decision about project's delivery and development. [1, 2, 3] thus, requirement prioritization (rp) is the most important section of re that comes under the requirement analysis stage. rp is considered as one of the most significant activities in the process to construct software project and deliver good system as the customer need. in case the project has strict execution plan, insufficient resources and the expectations of the consumers are so high, then it must publish the most important characteristics as early as possible. thus, this reason leads the requirements to be prioritized. 64 a. hudaib, r. masadeh, a. alzaqebah copyright ©2018 assa adv. in systems science and appl. (2018) requirement prioritization process increases the stakeholder's involvement by including them in deciding what requirements should be included in the software with its importance and effects on development process, and this involvement aids the stakeholders to really understand the restrictions and limitations on project's resources, and negotiate the conflicts between viewpoints that really effect in software development, and these conflicts come from different objectives and roles of stakeholders [43]. thus, in projects that have large number of requirements, the prioritization become a base of software success or failure according to the project's constraints and limitations, these participation and involvement aims to prioritize the requirement according to their importance in efficient manner and useful order of execution [1, 2, 4]. according to that, the clustering technique could be used in order to classify the requirements based on their importance. data clustering is one of the most important functions in data mining, which has attracted many authors, researchers and experts recently. clustering is an unsupervised ranking of types in groups [5]. the major purpose of data clustering depends on the notion and the idea of grouping the objects into groups that based on the similarity and dissimilarity between these objects [6, 7]. in general, clustering is categorized into two main classifications; the hard clustering and the soft clustering. in hard clustering, the data objects only belong to one cluster, also in the soft clustering; all the individual data objects belong to the individual classes to some range [6, 8]. the major goal of data clustering is which it optimally sorting all n data into k clusters in the opinion that the whole unseen types in the data are visible [9, 10]. thus, each cluster contains the similarity data objects, as well as, the clusters are various from others. one of the most significant methods that are employed in the data exploration process, neural computing, image segmentation and other engineering is cluster analysis [11]. for the time being, many clustering techniques are proposed by researchers which categorized into model-based clustering algorithm, hierarchical clustering algorithm, partition-based clustering algorithm, grid-based clustering algorithm and density-based clustering algorithm [9, 13]. the data that are divided into k clusters utilizing the euclidean distance as a measure in the partition-based clustering algorithms, as well the tree of groups that are created in the hierarchical clustering algorithms. recently, many researchers have proposed much of meta-heuristics and heuristic algorithms in order to solve the issues which happen as a result of complicated datasets. however, the most techniques that proposed in order to resolve the issue of the optimization relies on the meta-heuristic algorithms [11, 40-42]. the meta-heuristic algorithms main goal is to define the optimal solutions for fulfilling data cluster and reduces the issue of the local minima [14, 15]. the latest meta-heuristic optimizations algorithms are grey wolf optimization (gwo) algorithm and whale optimization (wo) algorithm. grey wolf optimization (gwo) algorithm is proposed by mirjalili et al in 2014 [12]. this mimics the hunting behavior of grey wolves in nature. grey wolves are one of the well-known predators in the nature. usually, they live in pack within group size is 5 to 12 on average. these wolves have robust rules in social dominant hierarchy. according to [12] grey wolves include alpha, beta, omega and delta wolves. alpha wolf represents the leader of the pack and it is responsible for making decision about hunting and other activities. while the beta wolf helps the higher level to make decision. omega wolves are responsible to submit any information to the highest levels. all other wolves in the pack are called delta. whale optimization (wo) algorithm is recently suggested by mirjalili and lewis in 2016 [13]. where its main objective is to define the global optimal solution for any given optimization problem. the major distinction between this algorithm and other algorithms is the principle that develops the candidate solutions in each iteration of optimization. in addition to that, bubblenet feeding process represents the hunting process in humpback whales in order to find and attack the prey. a hybrid approach based on whale and grey wolf optimization algorithms 65 copyright ©2018 assa adv. in systems science and appl. (2018) the reset of this paper is organized as follows: section ii contains backgrounds of requirement prioritization, gwo algorithm and wo algorithm in details. while section iii describes the proposed wgw and how it works. section iv outlines the suggested algorithm “rp-wgw”. the experimental results are presented in section v. finally, section vi draws the conclusion and future work. ii.background a.requirements prioritization the meaning of requirements prioritization is seen from several angles. summerville defined the requirements prioritization as one of the most significant task for decision makers [16]. while, firesmith defines it as the major process in software engineering as it gives perfect implementation order of the requirements in order to planning software versions and supplying desirable functionality as early as possible or the process to define the priority of the requirements to stakeholders based on the requirements importance [17]. therefore, we can conclude that the requirements prioritization denotes the prioritization by importance or by implementation. implementing the most significant operations that leads to get incremental feedback from the customer, set schedule, solve mistakes and resolve any misunderstand between the customer and the corporation in premature phases that lead up to customer contentment. moreover, it is valuable by eliminating needless requirements which may be inefficiently costly and choosing the most suitable requirements for each version; which leads to assist in future planning, reduce the risk of cancellation, evaluate the benefits, prioritize the investments and determine the financial effect with regards to the implementation of each requirement [18]. requirement prioritization is the most significant and critical portion of requirements analysis due to the restrictions in project resources. in other words, it is so hard to implement the whole requirements simultaneously due to the restrictions in resources whence of schedule, staff and budget. moreover, to improve some projects may require many months or often several years, wherefore it is important to determine the requirement that should be implemented at the beginning. furthermore, budget plays an important factor, especially when transaction with requirement prioritization process because budget is considered as small activity with regards to requirement engineering compared to other activities in software engineering. lastly, as mentioned before concludes that requirements have various levels with regards to their importance and it is complicate to determine which one is the most important. as mentioned above, the project stakeholders are the base of the prioritization process with respect to business and regulations factors because they have various viewpoints and each one must determine the highest priority for requirements in order to impose stakeholders to clearly gather all the relative importance requirements that guides to raise the communication between stakeholders, supplies a reasonable base for requirement negotiation and enables engineering to schedule the development activities in reasonably. b.grey wolf optimization grey wolf optimization algorithm (gwo) is one of the latest bio-inspired optimization techniques that proposed by mirjalili et al in 2014 [12]. which imitate the hunting behavior of grey wolves in nature. whereas the major purpose of gwo algorithm is determining the optimal for a given problem by using a population of search agents. these wolves usually live in groups; their members between five and twelve. the major difference between gwo algorithm and the other optimization algorithms is the social dominant hierarchy which develops the candidate solution in each iteration of optimization. in facts, the gwo simulates the foraging behavior of the wolves in finding and attacking the victims [12, 19]. the social hierarchy of the wolves pack is shown in fig.1 [12]. 66 a. hudaib, r. masadeh, a. alzaqebah copyright ©2018 assa adv. in systems science and appl. (2018) fig.1. hierarchy of grey wolf. alpha displays the leader which is the best candidate solution. in addition to that, alpha are dominant wolves which followed by the other wolves. while beta represents the second candidate solution which helping alpha in decision making and represent bridges between the leader and the rest of the pack that plays the lowest levels. delta displays the third candidate solution that responsible to submit information to the two higher levels (alpha and beta). while omega's displays the rest of solutions. moreover, its responsible to submit information to the three higher levels. in fact, hunting mechanism contains three steps: tracking, encircling and attacking the prey. thus, gwo represents the hunting technique of the grey wolves mathematical that is used to resolve complicated optimization problem. thus, the optimal solution of the given problem is considered as victims. the three top levels movement simulates the victim encirclement by grey wolves, which is the following formula that is suggested in this regard [12]. �⃗⃗� = |�⃗⃗� . �⃗⃗� 𝒑(𝒕) − �⃗⃗� (𝒕)| (1) where t displays the current iteration, xp indicates to the prey position vector, x represents the grey wolf position and c is a coefficient vector. thus, the result of vector d is used to move the particular element toward or away from the area that the best solution is located which represents the prey by using the following equation [12]: �⃗⃗� (𝒕 + 𝟏) = |�⃗⃗� 𝒑(𝒕) − �⃗⃗� . �⃗⃗� | , with �⃗⃗� = 𝟐�⃗⃗� . �⃗� 𝟏 − �⃗⃗� (2) where r1 is selected randomly in [0, 1] and a is minimized from 2 to 0 through predetermined number of iterations. in case |a| < 1, this matches to the exploitation behavior and simulates the behavior of attacking the prey. otherwise, if |a|> 1, this matches exploration behavior and imitates the wolf spacing from the victim. the suggested values for a are in [-2, 2]. thus, three higher levels α, β and δ will be computed using the following mathematical expressions [12]. a hybrid approach based on whale and grey wolf optimization algorithms 67 copyright ©2018 assa adv. in systems science and appl. (2018) �⃗⃗� 𝜶 = |�⃗⃗� 𝟏. �⃗⃗� 𝜶 − �⃗⃗� | with �⃗⃗� 𝟏 = �⃗⃗� 𝜶 − �⃗⃗� 𝜶. (�⃗⃗� 𝜶) (3) �⃗⃗� 𝜷 = |�⃗⃗� 𝟐. �⃗⃗� 𝜷 − �⃗⃗� | with �⃗⃗� 𝟐 = �⃗⃗� 𝜷 − �⃗⃗� 𝜷. (�⃗⃗� 𝜷) (4) �⃗⃗� 𝜹 = |�⃗⃗� 𝟑. �⃗⃗� 𝜹 − �⃗⃗� | with �⃗⃗� 𝟑 = �⃗⃗� 𝜹 − �⃗⃗� 𝜹. (�⃗⃗� 𝜹) (5) in order to mathematically imitative the hunting process of grey wolf, assume that α, β and δ have enough knowledge about the possible position of the victim. moreover, the first three best solutions that gained are saved and force the other agents to update their locations according to the best agents α, β and δ. this behavior is represented mathematical by the following expression [12], and the pseudocode of the gwo is shown in algorithm 1 [12]. �⃗⃗� (𝒕 + 𝟏) = �⃗⃗� 𝟏+ �⃗⃗� 𝟐+�⃗⃗� 𝟑 𝟑 (6) algorithm 1: pseudocodes of gwo [12] c.whale optimization whale optimization algorithm (woa) is a recently suggested randomly optimization algorithm by mirjalili and lewis in 2016 [13]. this algorithm purposes to determine the global optimum for the problem by using a population of search agents (whales). the search process is the first step begins with generating a collection of candidate solutions that are selected randomly for the given problem. then, it ameliorates this collection during many numbers of iterations until the satisfaction of an end condition. in fact, whales imitative private hunting technique that was called bubble-net feeding method as shown in fig. 2 [13]. 68 a. hudaib, r. masadeh, a. alzaqebah copyright ©2018 assa adv. in systems science and appl. (2018) fig. 2. bubble-net hunting behavior [13] it is obvious from fig. 2 that a humpback whale generates ambush with shifting in a spiral route around the victims, then generates bubbles all the way ahead. thus, this search process is the major inspiration of the whale optimization algorithm. in addition to that, the encircling method is another simulated technique in woa. whereas the humpback whales surrounding around the victims in order to begin hunting them using the foraging mechanism which is called the bubblenet technique. this behavior is represented mathematical by the following equations [13]: 𝑋 (𝑡 + 1) = { 𝑋 ∗ (𝑡) − 𝐴 . �⃗⃗� 𝑖𝑓 𝑝 < 0.5 𝐷’. 𝑒𝑏𝑙 . 𝐶𝑜𝑠 (2 𝜋 𝑙) + 𝑋 ∗ (𝑡) 𝑖𝑓 𝑝 ≥ 0.5 (7) where p is a random number in [0, 1], b is a constant for determining the shape of the logarithmic spiral, and l indicates to a random number in [−1, 1], t presents the current iteration, and d’ = |x∗ (t) – x (t)| which mentions the distance between the ith whale and the victim. in other words, the first phase is presented in this equation is the foraging mechanism that mimics the encircling technique, whereas the second phase simulates the bubble-net mechanism. the variable p exchanges between these two phases with similar probability. the potential cases those using these equations are shown in fig.3 [13]. fig.3.mathematical models for prey encircling and bubble-net hunting [13] a hybrid approach based on whale and grey wolf optimization algorithms 69 copyright ©2018 assa adv. in systems science and appl. (2018) the main two phases of the optimization algorithm by using population based algorithms are exploration and exploitation phases. whereas both of them ensured in woa by adaptively setting a and c in the major equation. in case, a problem is given, the woa begins optimizing this problem by generating collection of random solutions. in each iteration, the search agents update their location depends on the randomly chosen search agent or the best search agents that will gain so far. to assure the exploration phase, the other agents update their positions based on the best solution that represents the pivot point when |a|>1. in other case, when |a|<1 the best solution plays another role with the pivot point. the pseudocode of the woa is shown in algorithm 2 [13]. algorithm 2: pseudocodes of woa [13] iii.related work as mentioned before, the requirements prioritization phase is an operation that presenting the priority of a requirement over other requirements. moreover, it is the most motivating field in the requirement engineering domain [21]. many techniques are proposed in order to give the precedence to a software requirement over other requirements at the same project. fig.4 shows the classification of requirements prioritization techniques according to [22]. 70 a. hudaib, r. masadeh, a. alzaqebah copyright ©2018 assa adv. in systems science and appl. (2018) fig.4. classification of requirement prioritization techniques [22]. as shown in fig.4 the requirement prioritization techniques are classified as ordinal scale, ordinal scale and ratio scale in terms of powerful prioritization. the most powerful prioritization is ratio scale that is giving how much more significant a requirement than the others requirements [42, 44]. while the least powerful prioritization is the ordinal scale which is responsible to give ranked list of requirements and giving which requirement is more important than others requirements but without giving how much more significant. the analytic hierarchy process (ahp) is the most popular and traditional technique that is joined ratio scale class. thus, it is mentioned by large number of studies [23-28]. furthermore, it is considered as a systematic decision making technique which has been performed for precedence of requirements for a particular project [29, 30]. it compares all potential pairs of requirements to order them and define which has higher priority. ahp is not suitable for a project that has large number of requirements [31, 32]. thus, many researchers have attempted to minimize the number of comparisons such as [33, 34] and different techniques are presented in order to minimize the number of comparisons by approximately 75% such as [35]. the most precise level software requirements are placed at the base of the hierarchy while the most complicated requirements are placed on the top of the hierarchy [23]. hierarchy ahp is another type of requirements prioritization technique that prefers the requirements that are presented at the same level. cumulative voting is called the 100-dollar test which is a straightforward technique; where 100 imaginary units are given to the stakeholders in order to divide among the given software requirements [25]. the finding of the prioritization is given on a ratio scale. numerical assignment is considered as one of the most popular mechanism that is proposed in [24, 27, 47]. this technique depends on clustering requirements into various priority classes. however, the number of classes are vary and there are three classes are very popular [25, 28]. thus, in this technique it is significant that each class displays something which the stakeholder can communicate to such as optional, critical, etc. while moscow (museum of a hybrid approach based on whale and grey wolf optimization algorithms 71 copyright ©2018 assa adv. in systems science and appl. (2018) soviet calculators on the web) is considered as a type of numerical assignment mechanism that depicted in [36, 37]. moreover, this technique has four priority classes; must have, could have, should have and wont have. must have means the requirements that joined this class must be implemented at the first then goes to version, while could have means the requirement that joined this class exist then it will be great for the software product. should have means the requirements that present in this class are implemented then will be great for the software product and finally wont have means that requirements join this cluster cannot be implemented in the present iteration as other requirements that are of low precedence. bubble sort mechanism is used to rank the element such as requirements. in this technique, two requirements are taken to compare with each other. in case the requirement is not in series then exchange between them and then compare it with another requirement and continue until get ranked list of requirements descending (from higher to lower) as used in these studies [23, 37]. binary search tree technique is another type that is used for ranking which suggested by [37]. furthermore, this technique was presented at the first time by [23] for requirement prioritization. each node in this technique indicates to a software requirement where all requirements that located in left side of the tree are of lower priority comparing with other nodes while the other requirements that are located at the right side of the tree are of higher priority. however, at the first one requirement is selected to be the root node then will compare with unsorted requirement. in case this requirement is lower priority than the root, it searches the left side of the tree. otherwise, it searches the right side of the tree. the operation is repeated until get sorted tree. in top ten requirements technique, the stakeholders select top ten requirements in terms of their importance for them without giving inner rank among these requirements. this leads the technique to be suitable for many stakeholders of equal importance [38]. grey wolf optimization (gwo) is applied by [40] which is one of the most recently metaheuristic algorithms; in order to prioritize the software project’s requirements. in addition; it is evaluated and compared with analytical hierarchy process (ahp) mechanism; where the proposed work performed better than the traditional technique (ahp) by approximately (30%). while, [41] is applied whale optimization algorithm (woa) which is recently used in optimization problems since it imitates the humpback whale hunting behavior by employing bubble net hunting technique. it is likewise evaluated with ahp; where the results shown the proposed work outperform the ahp mechanism by approximately (40%). iv.the proposed algorithm (wgw) as shown in fig.5 the approach that suggested in this study is mainly designed by combining the woa and the gwo algorithm in order to design a hybrid approach that called "wgw". the combination process is done by combining the advantages of each algorithm that are utilized in this study. the main advantage of gwo that is from each cluster the top three highest values will be considered as α, β and δ and these values represents the first three best solutions from that cluster, with keep into account the group that represents the cluster is limited 5-12 nodes on average. on the other hand the woa has no limitation in group members and finally the woa did not have a top three highest solution instead of one. from theses points the wgw is proposed to make a combination between these two algorithms by employing the woa algorithm to skip the limitation on the group team members and make it unlimited and take the top three highest solutions as gwo works. [12, 13] as any meta-heuristic optimization algorithms, the proposed algorithm is composed of two stages. the first is the exploration stage while the second is the exploitation stage. 72 a. hudaib, r. masadeh, a. alzaqebah copyright ©2018 assa adv. in systems science and appl. (2018) in this study, the proposed wgw algorithm is used to search in the overall search space in order to find all possible xrands, these xrands play an important roles in exploration phase. on the other hand, any search agent sees prey, it is considered to be xrand, then this foraging process is done by generating bubbles along the path that forms spiral shape and it called the bubble-net hunting technique. thus, when any search agent sees the bubbles that mean it belongs to this spiral. in case, there are many spirals at the same time from different xrand, each one is considered as cluster in the search space and each one of them has first three best solutions xα, xβ and xδ. whereas this advantage is driven from gwo algorithm, whereas, the previous scenario represents the exploitation phase.[12, 13] in each iteration, after determining the best solutions xα, xβ and xδ, the other search agents update their positions in the cluster. taken in account the value of absolute a, if it is greater than one means the search agent goes beyond the cluster (spiral) and it must search to another xrand to belong to new cluster. otherwise, the absolute value of a is less than one means the search agent stills in the cluster and closes to the prey. fig.5. flowchart of the proposed wgw algorithm. a hybrid approach based on whale and grey wolf optimization algorithms 73 copyright ©2018 assa adv. in systems science and appl. (2018) v.algorithm “rp-wgw” in this work, an improvement approach is suggested by combining the recently bio-inspired optimization techniques that proposed by mirjalili et al which are gwo and wo algorithms which imitate the hunting techniques in nature. then, the wgw algorithm is used for requirements prioritization. to get the prioritization of the requirements, the wgw algorithm is applied. fig.6 presents the suggested pseudo code for “rp-wgw” algorithm. “rp-wgw” algorithm begin 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. initialize the agents’ population xi (i = 1, 2, 3…..… n) initialize c, r and a select xrand randomly calculate the distance between each whale (i) and all xrand by eq. (1) if the agent (i) is not assigned assign agent (i) to its closet xrand calculate the fitness for each agent (i) invoke cluster function invoke requirement prioritization function return the best solution end fig. 6. pseudo – code for “rp-wgw” algorithm a.initialization stage at the beginning, initialize the agents’ population as shown in fig.6. number of agents is selected randomly to generate the bubble that performs as cone (spiral) shape. then, measure the distance between all agents and all xrands to assign to the closet one and join that group to be a member of it. b.fitness function based on eq. (8), xrand generates bubble-net when it sight the victim, all agents that see the bubble-net will associate to the cluster (spiral). thus, the other agents who joined the cluster must update their locations towards the location of xrand as illustrated in fig. 7. in other hand, these agents will update their locations depending on the value of absolute a. in case, |a| is less than one, this means the agent is still entering the cluster and updates its location by the following eq. (8). otherwise, |a| is greater than or equal one, which means the agent is not belong to the cluster, so it search for another xrand to associate by using eq. (9). this behavior is represented mathematical by eq. (8, 9) [13] and eq. (10). 𝑋 (𝑡 + 1) = �⃗⃗� . 𝑒𝑏𝑙 . 𝐶𝑜𝑠 (2𝜋𝑙) + 𝑋 (𝑡) (8) 𝑋 (𝑡 + 1) = 𝑋𝑟𝑎𝑛𝑑⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ − 𝐴 ⃗⃗ ⃗. 𝐷 ⃗⃗ ⃗ (9) 𝑉 = ℎ ∗ 𝑟 ∗ 𝜋 ( 1 3 ) (10) where r is the radius of the bubble-net which is constant value (r = 15) [20], h is the height of bubble-net, which is selected randomly between 6 and 12 [12, 20], and π is constant value that equal 3.14. 74 a. hudaib, r. masadeh, a. alzaqebah copyright ©2018 assa adv. in systems science and appl. (2018) fitness function begin 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. xrand creates bubble-net by eq. (8) for each search agent (i) update a, a, c and l calculate the distance between each agent (i) and xrand by eq. (1) if (a<1) update the position of the current agent (i) by the eq. (8) else select new xrand randomly update the position of the current agent (i) by the eq. (9) end if end for end fig. 7. fitness function c.clustering as shown in fig.8, each cluster k has xrand that selected randomly. the fitness function will be calculated for each agent in order to check if this agent sights the bubble that generated by xrand. in other words, this agent is still joining this spiral. in case the agent didn’t see the bubble, it searches for another xrand to belong. thus, each cluster k will get xα, xβ and xδ that represent the first three highest priority of all requirements in this cluster. cluster function begin 1. 2. 3. 4. 5. 6. 7. 8. 9. for each cluster k select xrand randomly select xα, xβ and xδ while (t< = max_iteration) invoke the fitness function invoke the rp function end while return xα, xβ and xδ end for end fig.8. cluster function d.requirements prioritization function as discussed in clustering step, each agent in the search space represents a requirement with its importance; this importance represents its importance in the development process of the target project. this importance calculated according to the stakeholders ranking for each requirement with respect the importance of that stakeholder and their effect on the project. since each stakeholder belongs to a role in the target environment that the system built for, the role's rank effect directly to the stakeholder importance so when calculating the stakeholder importance the role's rank plays an important role in it. according to the project environment, first the importance of the role based on the stakeholders ranking on it will be calculated as equation (11) [39]. 𝐼𝑛𝑓𝑙𝑢𝑛𝑐𝑒 𝑟𝑜𝑙𝑒(𝑖) = 𝑅𝑅𝑀𝑎𝑥 + 1 − 𝑅𝑎𝑛𝑘(𝑅𝑜𝑙𝑒(𝑖)) ∑ 𝑅𝑅𝑀𝑎𝑥 + 1 − 𝑅𝑎𝑛𝑘(𝑅𝑜𝑙𝑒(𝐽))𝑛 𝐽=1 (11) a hybrid approach based on whale and grey wolf optimization algorithms 75 copyright ©2018 assa adv. in systems science and appl. (2018) where rrmax is the maximum rank of roles list, rank (role (i)) is the rank of the i's role and n is the total number of roles, rrmax+1 is used to invert the value of rank since the lowest rank is highly effect. after that the influence of each stakeholder in each role will be calculated to find the effect of that stakeholder on the project, equation (12) [39] shows the stakeholder's importance calculation. 𝐼𝑛𝑓𝑙𝑢𝑛𝑐𝑒 (𝑖) = 𝑅𝑆𝑀𝑎𝑥 + 1 − 𝑅𝑎𝑛𝑘(𝑖) ∑ 𝑅𝑆𝑀𝑎𝑥 + 1 − 𝑅𝑎𝑛𝑘(𝐾)𝑛 𝐾=1 (12) where i represents a specific stakeholder, rsmax is the maximum rank of stakeholder in that role, rank(i) is the rank of the i'th stakeholder in the role and n is the total number of stakeholders in the same role, rsmax+1 is used to invert the value since the lowest rank is the highest effect. then the influence of that stakeholder on the project at all will be calculated by multiplying the role influence and the stakeholder influence as equation (13) [39]. 𝑃𝑟𝑜𝑗𝑒𝑐𝑡 𝐼𝑛𝑓𝑙𝑢𝑛𝑐𝑒 (𝑖) = 𝐼𝑛𝑓𝑙𝑢𝑛𝑐𝑒 𝑟𝑜𝑙𝑒(𝑖) × 𝑖𝑛𝑓𝑙𝑢𝑛𝑐𝑒 (𝑖) (13) since each stakeholder ranked the requirement list, the importance of that requirement is calculated by summation of all stakeholders' influence on the project multiplied by its rating on that requirement, equation (14) shows the requirement importance calculation )[39]. 𝐼𝑚𝑝𝑜𝑟𝑡𝑎𝑛𝑐𝑒 𝑅 = ∑𝑃𝑟𝑜𝑗𝑒𝑐𝑡 𝑖𝑛𝑓𝑙𝑢𝑛𝑐𝑒(𝑖) × 𝑟(𝑖) 𝑛 𝑖=1 (14) where r (i) represents the i'th stakeholder's rank on that requirement and n is the total number of stakeholders that rating the requirement r (i). as shown in fig. 9, the first three top values will be chosen from each cluster. thus, this process will be executed for each cluster iteratively, as well as a result from this process the set of best solutions from each cluster will be gained in a temporary list, so this list contains the α’s, β’s and δ’s from the clusters. since this list contains the best solution, the top three solutions will selected as first result of prioritization process. the selected results will be moved to the final ranked list as best three solutions, since they are already taken and prioritized; these requirements will be removed from original clusters. this process will be repeated until all clusters have no requirements inside, as mention above the selected best solutions will be removed from the original clusters, so the number of requirements inside the cluster will be decreased while iteration keep running. finally the proposed approach will return the final ranked set of requirements according to the importance of each requirement. 76 a. hudaib, r. masadeh, a. alzaqebah copyright ©2018 assa adv. in systems science and appl. (2018) requirements prioritization function begin 1. 2. 3. 4. 5. 6. 7. 8. 9. while each cluster is not empty (1,2,3,…k) for each cluster k sort the requirements based on eq. (14) select xα, xβ and xδ and move it to temp_list end for select xα, xβ and xδ from the sorted temp_list and move them to final ranked list remove xα, xβ and xδ from original clusters end while return the final results set end fig.9. requirements prioritization function vi.experimental results in order to evaluate the proposed method for requirements prioritization in term of accuracy, the ralic project was selected from a soo ling lim phd thesis as a case study in this paper; ralic stands for the replacement access, library and id card project. it was a software project to enhance the existing access control system at university college london (ucl). located in the central part of london, ucl is concerned with its security [39]. many ucl buildings require authorized access, such as the libraries, academic departments, and computer clusters. in 2005, ucl had various methods of identification and access control, such as swipe cards, contactless cards, photo id cards, library barcode, digital security code, and metal door keys. ucl staff and students had to use different mechanisms to access different buildings, which meant they had to carry various cards with them [39]. furthermore, some of the security systems were already obsolete; others would cease to be operable in a few years’ time. ralic’s aim was to replace the obsolete access control systems, consolidate the various existing access control mechanisms, and at the minimal, combine the photo id card, access card, and library card [39]. ralic was a combination of development and customization of an off-the-shelf system [39]. the project started in 2005 and its duration was 2.5 years. the system has been deployed in ucl. the project scope is summarized in table 1 [39]. a hybrid approach based on whale and grey wolf optimization algorithms 77 copyright ©2018 assa adv. in systems science and appl. (2018) table 1. ralic project scope [39] ralic is a large-scale software project. it had more than 60 stakeholder groups and approximately 30,000 users. some of the stakeholder groups included students, academic staff, short course and academic visitors, administrators from academic departments, security staff, developers, managers, and front line staff from supporting divisions such as the estates and facilities division that manages ucl’s physical estate, human resource division that manages staff information, registry that manages student information, information services division, and library services. ralic has approximately 30,000 students, staff, and visitors, who use the system to enter ucl buildings, borrow library resources, use the fitness center, and gain it access. ralic had a complex stakeholder base, with different stakeholders having conflicting requirements. for example, members of the ucl development & corporate communications office preferred the id card to have ucl branding, but the security guards preferred otherwise for security reasons in case the cards were lost. some administrators were worried that the new system, which promised to reduce manual labor, would threaten their job. the project involved many divisions in ucl, some of which had low stake in the project but were critical to its success. in particular, the project team found it difficult to engage with the divisions that manage the interfacing systems that supply data to ralic, such as the student registry that provide student data, and human resources that provide staff data, because they had little stake in the project. some representatives from these departments were often absent in project meetings. the author presents a ground truth in order to evaluate her proposed method; this ground truth contains a real list that prioritized in the real project after deploying it, the data set contains 10 project objectives, 49 requirements and 80 specific requirements after cleaning and processing the data which described in [39]. in this research; the work of [39] was implemented specifically on requirements with keep into account the rank of project objectives and discard her work on specific requirements since the proposed method concerns about requirements prioritization. the tested data set contains 57 requirements with each importance of them, the result of [39] work is shown in table 2 which shows a prioritized list of requirements. 78 a. hudaib, r. masadeh, a. alzaqebah copyright ©2018 assa adv. in systems science and appl. (2018) table 2. prioritized list of requirements [39] id requirement description priority a.3 all in 1 card 1 a.1 easier to use 2 a.2 use the same access control for library entrance 3 c.3 enable visual checking 4 c.4 access control to include movement tracking/logs 5 c.5 increase access control to buildings 6 c.2 control access to ucl buildings 7 c.1 ensure appropriate access for each individual 8 d.5 granting access rights 9 d.1 faster issue of cards 10 d.2 reduce queuing time 11 d.3 id card status: ability to check if a user has collected an id card 12 d.6 able to create access reports 13 d.4 easy to replace lost cards 14 d.7 procedures for dealing with fraud 15 b.3 card should be secure 16 b.1 card to include user details 17 b.4 card to include ucl branding 18 b.5 easy identification/card is clear looking 19 b.6 card should be sturdy/robust 20 b.2 card to include barcode 21 b.7 card should be attractive 22 g.1 centralized management of access and identification information 23 g.2 export data to other systems 24 g.3 import data from other systems 25 g.4 data access: able to view, update, delete remotely and securely 26 g.5 clear policies on use of access data 27 e.1 save money on cards 28 e.2 save processing time 29 e.3 reduce paper trials 30 f.5 compatible with current network infrastructure 31 f.6 impact to other systems 32 f.4 compatible with upi 33 f.2 compatible with library systems 34 f.1 compatible with bloomsbury system (gladstone mrm) 35 f.3 compatible with hr system 36 h.3 used for computer logon 37 h.2 include payment mechanism 38 h.4 upgradable (software revisions) 39 h.1 include digital certificate 40 h.5 increase security 41 j.2 conform to standards and legislations 42 j.6 available 43 j.5 reliable 44 a hybrid approach based on whale and grey wolf optimization algorithms 79 copyright ©2018 assa adv. in systems science and appl. (2018) j.3 technology 45 j.1 fail safe 46 j.7 network infrastructure 47 j.4 lifecycle 48 j.9 the chosen manufacturer must have a proven tracker record within institutions with access control, id pass production, odbc, smart card technologies. 49 j.10 photo id pass software must be an embedded feature of the access control software and only require software/license upgrades. 50 j.11 the system manufacturer must be a microsoft™ certified partner. 51 j.12 the system must utilize microsoft™ windows 2000 and/or xp operating system. 52 j.13 the database platform must support microsoft™ sql server and/or oracle server. 53 j.8 be capable of having direct printing to both sides of the card, which will include the library barcode 54 i.3 project management activities 55 i.2 technical documents 56 i.1 supplier support 57 in order to evaluate the proposed technique, the theoretical approach was implemented and tested on the same requirements data set with same parameters, the proposed technique prioritize the set of requirements with 52 matching prioritization values and 5 mismatching values. the error rate is calculated using equation (15) and the accuracy is calculated using equation (16) 𝑬𝒓𝒓𝒐𝒓 𝑹𝒂𝒕𝒆 = 𝒎𝒊𝒔𝒎𝒂𝒕𝒄𝒉𝒊𝒏𝒈/𝒔𝒆𝒕 𝒔𝒊𝒛𝒆 ∗ 𝟏𝟎𝟎 (15) so the error rate of proposed method is approximately 9% so the approximate accuracy is 91% while comparing the proposed method's result with the used requirement list. table 3 shows the ordering comparison between the proposed method and [39] work, and fig.10 shows the error rate and accuracy between the two methods. table 3. comparison of requirements ordering between wgw-rp and work of [39] wgw ordering (req id) original ordering (req id) matching order wgw ordering (req id) original ordering (req id) matching order a.3 a.3 yes e.3 e.3 yes a.1 a.1 yes f.5 f.5 yes a.2 a.2 yes f.6 f.6 yes c.3 c.3 yes f.4 f.4 yes c.4 c.4 yes f.2 f.2 yes c.5 c.5 yes f.1 f.1 yes c.2 c.2 yes f.3 f.3 yes c.1 c.1 yes h.3 h.3 yes 𝑨𝒄𝒄𝒖𝒓𝒂𝒄𝒚 = 𝟏𝟎𝟎% − 𝑬𝒓𝒓𝒐𝒓 𝑹𝒂𝒕𝒆 (16) 80 a. hudaib, r. masadeh, a. alzaqebah copyright ©2018 assa adv. in systems science and appl. (2018) d.5 d.5 yes h.2 h.2 yes d.1 d.1 yes h.4 h.4 yes d.2 d.2 yes h.1 h.1 yes d.3 d.3 yes h.5 h.5 yes d.6 d.6 yes j.2 j.2 yes d.4 d.4 yes j.6 j.6 yes d.7 d.7 yes j.5 j.5 yes b.3 b.3 yes j.3 j.3 yes b.1 b.1 yes j.1 j.1 yes b.4 b.4 yes j.7 j.7 yes b.5 b.5 yes j.4 j.4 yes b.6 b.6 yes j.9 j.9 yes b.2 b.2 yes j.11 j.10 no b.7 b.7 yes j.8 j.11 no g.1 g.1 yes j.10 j.12 no g.2 g.2 yes j.12 j.13 no g.3 g.3 yes j.13 j.8 no g.4 g.4 yes i.3 i.3 yes g.5 g.5 yes i.2 i.2 yes e.1 e.1 yes i.1 i.1 yes e.2 e.2 yes fig.10. error rate and accuracy for wgw-rp vii. conclusion requirement engineering is the most important phase in software development since it dealing with stakeholders and other activities. since the number of requirements is varying for each project the requirements prioritization is an important process in order to deliver a good ordering of project's phases with satisfying stakeholders and end users. in this paper a hybrid approach was suggested by combining the advantages of gwo and woa algorithms as metaheuristic approach in order to prioritize the software project requirements. 0% 20% 40% 60% 80% 100% error rate accuracy error rate accuracy a hybrid approach based on whale and grey wolf optimization algorithms 81 copyright ©2018 assa adv. in systems science and appl. 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(2010). a comparison of nine basic techniques for requirements prioritization. helsinki university of technology. advances in systems science and applications (2012) vol.12 no.4 388-398 optimal designs in random intercept model with heteroscedastic errors jing cheng1 and rongxian yue2 1department of mathematics, chaohu college, anhui 238000, china 2department of mathematics, shanghai normal university, shanghai 200234, china scientific computing key laboratory of shanghai universities, and division of scientific computation of e-institute of shanghai universities abstract this paper considers optimal designs based on the d-, g-, a-, iand ds-optimality criteria for a random intercept model with heteroscedastic errors. it is shown that the search of optimal approximate designs can be confined at extreme settings of the design region if heteroscedastic structure satisfies specified conditions. closed expressions for the optimal proportions are given. keywords optimal design, random intercept model, heteroscedastic errors, identical design 1 introduction random coefficient models have been widely used for the researching in the area of biosciences, psychology and population pharmacokinetics, where repeated measurements are available from different individuals. these models have been introduced by longford[1], for recent researching we refer to pena and yohai[2] and yu[3]. in recent years, the problem of optimal designs for random coefficient models has attracted growing interest. schmelter[4-5] showed that optimal designs in the linear mixed models could be restricted to the class of group-wise identical designs, and optimal designs in the class of single-group designs were also optimal designs in the larger class of more group designs when the design criteria satisfied some assumptions. schwabe and schmelter[6], schmelter et al[7] and luoma et al[8] . investigated optimal designs in random intercept model, random slope model and random coefficient cubic regression model, respectively. entholzner et al[9] obtained optimal and efficient designs in mixed models. debusho and haines[10] provided v-optimal and d-optimal designs with longitudinal data in linear regression models with a random intercept. there are many other results of optimal designs are obtained, such as wang et al[11], yu[12] and wen et al[13]. in this article, we investigate the problem of optimal designs based on some common optimality criteria for a random intercept model with heteroscedastic errors. in section 2, we introduce the model with necessary notations. section 3 provides a lemma which makes it sure that we can confine the search of optimal designs at extreme settings of the design region if the optimality criteria satisfy an assumption. simple expressions of these optimal advances in systems science and applications (2012) vol.12 no.4 389 designs are given in this section. section 4 introduces some examples. proof of lemma 1 is given in appendix. 2 the random intercept model with heteroscedastic errors we investigate a linear regression model on the unit interval with a random intercept and heteroscedastic errors. it is assumed that there are individuals with observations each, and the jth observation of ith individual is described by yij = µi + xijβ + e(xij), i = 1, ..., n; j = 1, ...,mi. (1) where, xij ∈ [0, 1] is the experimental setting; µi denotes the ith individual effect with unknown mean µ and known variance σ2 µ; β is the unknown slope parameter; observational errors e(xij) are assumed to be heteroscedastic with zero mean and variance σ2/λ(xij), here σ2 is known and λ(xij) is a positive real-valued continuous function defined on [0,1]. we assume that cov (µi, µi′) = 0, i ̸= i′ cov (µi, e(xi′j)) = 0, ∀i, i′; cov (e(xij), e(xi′j′)) = 0, (i, j) ̸= (i′, j′). for the ith individual, denote yi =  yi1 ... yimi  , xi =  xi1 ... ximi  , e(xi) =  e(xi1) ... e(ximi)  , fi = (1mi , xi) here 1mi is a vector of length mi with all entries equal one. then the model (1) can be expressed by yi = fi ( µ β ) + 1mi(µi − µ) + e(xi) , fiθ + 1mi(µi − µ) + e(xi), i = 1, . . . , n. by the assumptions we have (µi − µ) ∼ (0, σ2 µ) and vi , cov (yi) = σ2diag{1/λ(xi1), . . . 1/λ(ximi)}+σ2 µ1mi1 t mi , σ2(di+d1mi1 t mi ). here di = diag{1/λ(xi1), . . . 1/λ(ximi)} and d = σ2 µ/σ 2. for all n individuals, the vector of all observations can be expressed by y =  y1 ... yn  =  f1 ... fn  θ +  1m1 0 . . . 0 1mn   µ1 − µ ... µn − µ +  e(x1) ... e(xn)  (2) 390 jing cheng:optimal designs in random intercept model with heteroscedastic errors the design matrix for random intercepts is block diagonal, e.g., 1m1 0 . . . 0 1mn  . consequently, the covariance matrix of y is cov (y ) = diag{v1, . . . , vn}. the best linear unbiased estimate of θ is given by θ̂ = ( n∑ i=1 f t i v −1 i fi )−1 n∑ i=1 f t i v −1 i yi. (3) and we can get cov (θ̂) = ( n∑ i=1 f t i v −1 i fi )−1 . 3 optimal designs in this section, we investigate the optimal designs based on d-, g-, a-, iand dsoptimality criteria for the models described in previous section. the d-optimal design minimizes the generalized variance of parameter estimates, the g-optimal design minimizes the maximum variance of the predicted value of the response over the design region, the a-optimal design minimizes the total variance of the parameter estimates, the i-optimal design minimizes the integrated mean squared error and the interest of ds-optimal design is in estimating the slope. in some practical situations like human or animal pharmaceutics studies or medical diagnostics there are often restrictions, e.g., technical implementations, which force the experiment to be performed with identical regimes for all individuals. this means that for each individual the number mi of repeated measurements equals m and experimental settings xij = xj are identical across all the individuals. so we only consider identical designs in the following, i.e., mi = m, xi = x1 and hence, fi = f1, vi = v for all i. then the best linear unbiased estimate of θ can be written as θ̂ = ( nf t 1 v −1 1 f1 )−1 f t 1 v −1 1 n∑ i=1 yi. here v −1 1 = 1 σ2 ( d1 + d1m1tm )−1 = 1 σ2 ( d−1 1 − dd−1 1 1m1tmd−1 1 1 + d1tmd−1 1 1m ) = 1 σ2 ( diag{λ(xj)} − dd−1 1 1m1tmd−1 1 1 + d ∑n j=1 λ(xj) ) . advances in systems science and applications (2012) vol.12 no.4 391 without loss of generality, we assume σ2 = 1 in the followings. furthermore, we will consider approximate designs. for any approximate design ξ of the following form ξ = ( x1, . . . , xp ω1, . . . , ωp ) , 2 < p < m, p∑ j=1 ωj = 1. (4) denote νk = ∫ 1 0 xkλ(x)dξ(x) = p∑ j=1 ωjx k jλ(xj), k = 0, 1, 2. then the information matrix corresponding to the design ξ of the form (4) can be expressed by m(ξ) = mn 1 + γν0 ( ν0 ν1 ν1 ν2 + γ(ν0ν2 − ν21) ) . (5) here we note γ = md. for regression model without any random effects, optimal designs are obtained at extreme settings of the design region and schwabe et al[6] discussed optimal designs of random intercept models. we can’t use the conclusions in schwabe et al[6] directly in the random intercept model with heteroscedastic errors, but we have the following lemma. lemma 1 in the model (2), assume that mi = m, (i = 1, . . . , n) and λ(x) satisfies the following condition 1 λ(x) ≥ 1− x λ0 + x λ(1) , x ∈ [0, 1]. (6) where λ0 = λ(0) and λ1 = λ(1). then for any approximate design ξ of the form (4), there exists an approximate design of the form ξ∗ = ( 0, 1 1− ω, ω ) , 0 < ω < 1. such that m(ξ∗) ≥ m(ξ). the proof of lemma 1 can be found in the appendix. the criteria, φ(·), considered in this paper are functions of the information matrices which are required to satisfy the following assumptions: a1 φ(·), is a real-valued function defined on the whole set m of 2×2 symmetric non-negative definite matrices, φ : m → (−∞,∞]; a2 φ(·) is monotone (the loewner order (e.g., pukelsheim[14], p.101)) on m in the sense that m1,m2 ∈ m,m∞ ≥ m∈ ⇒ ⊕(m∞) ≤ ⊕(m∈). 392 jing cheng:optimal designs in random intercept model with heteroscedastic errors these assumptions are satisfied for most of the common criteria including the d-, g-, a-, iand ds-optimality. so, by majorization we can confine the search of optimal designs at extreme settings x = 0 and x = 1 if λ(x) satisfies the condition (6) in lemma 1. therefore, in what follows we only consider approximate designs ξ of the form ξ = ( 0 1 1− ω ω ) . (7) for approximate designs of the form (7), we have ν0 = λ1ω + λ0(1− ω), ν1 = ν2 = λ1ω. first, we consider the d-optimality φ(m(ξ)) = ∣∣m−1(ξ) ∣∣. note that |m(ξ)| , |m(ω)| = (mn)2λ0λ1ω(1− ω) 1 + γ[ωλ1 + (1− ω)λ0] . it is easy to verify that |m(ω)| is maximized at ω = √ 1 + γλ0/( √ 1 + γλ0 + √ 1 + γλ1) and hence ∣∣m−1(ω) ∣∣ is minimized. therefore we have theorem 1 for the model (2) with mi = m (i = 1, . . . , n) and λ(x) satisfying (6) the d-optimal design is ξ∗d = ( 0, 1 1− ωd, ωd ) , ωd = √ 1 + γλ0√ 1 + γλ0 + √ 1 + γλ1 for the ds-optimality, note that the covariance matrix of θ̂ can be calculated by m−1(ω) = 1 mn(ν0 − ν2) ( 1 + γ(ν0 − ν1) −1 −1 ν0 ν1 ) = 1 mn ( 1 λ0(1−ω) + γ − 1 λ0(1−ω) − 1 λ0(1−ω) 1 λ0(1−ω) + 1 λ1(ω) ) the variance of the estimate for β is given by cov (β̂) = [m−1(ω)]22 = 1 mn [ 1 λ0(1− ω) + 1 λ1ω ]. the variance of β̂ is minimized at ω = √ λ0/( √ λ0 + √ λ1). therefore we have theorem 2 for the model (2) with mi = m (i = 1, . . . , n), and λ(x) satisfying (6) the ds-optimal design is ξ∗ds = ( 0, 1 1− ωds , ωds ) , ωds = √ λ0√ λ0 + √ λ1 . advances in systems science and applications (2012) vol.12 no.4 393 consider the g-optimality, then φ(m(ω)) = max x∈[0,1] d(ω, x). here d(ω, x) is the variance of the predicted value of the response, which is given by d(x, ω) = ( 1 x ) m−1(ω) ( 1 x ) = 1 mn {[ 1 λ0(1− ω) + 1 λ1ω ] x2 − 2x λ0(1− ω) + 1 λ0(1− ω) + γ } . as d(ω, x) is a polynomial of degree 2 with positive leading term, its maximum is attained either x = 0 or x = 1 or both, i.e.,max d(ω, x) = max x∈[0,1] {d(0, ω), d(1, ω)} . note that d(0, ω) = 1 mn { 1 λ0(1− ω) + γ } is strictly increasing in ω, d(1, ω) = 1 mn { 1 λ1(ω) + γ } is strictly decreasing in ω. thus min ω∈[0,1] max x∈[0,1] d(x, ω) is attained when d(0, ω) = d(1, ω), i.e., 1 λ0(1− ω) = 1 λ1(ω) . so we have theorem 3 for the model (2) with mi = m (i = 1, . . . , n), and λ(x) satisfying (6) the g-optimal design is ξg ∗ = ( 0, 1 1− ωg, ωg ) , ωg = λ0 λ0 + λ1 . for the i-optimality, φ(m(ω)) = ∫ 1 0 d(x, ω)dx = 1 mn [ γ + 1 3λ0(1− ω) + 1 3λ1ω ] . which is minimized at ω = √ λ0/( √ λ0 + √ λ1) = ωds . so we have theorem 4 for the model (2) with mi = m (i = 1, . . . , n), and λ(x) satisfying (6) the i-optimal design is ξ∗i = ( 0, 1 1− ωi , ωi ) , ωi = √ λ0√ λ0 + √ λ1 . 394 jing cheng:optimal designs in random intercept model with heteroscedastic errors for the a-optimality φ(m(ω)) = tr ( m−1(ω) ) = 1 mn [ 1 λ1ω + 2 λ0(1− ω) + γ ] . it is easy to verify that tr ( m−1(ω) ) is minimized at ω = √ λ0√ λ0+ √ 2λ1 .therefore we have theorem 5 for the model (2) with mi = m (i = 1, . . . , n), and λ(x) satisfying (6) the a-optimal design is ξa ∗ = ( 0, 1 1− ωa, ωa ) , ωi = √ λ0√ λ0 + √ 2λ1 . from above discussion, we observe that the g-, ds-, iand a-optimal designs only depend on the variances at extreme settings; the d-optimal design depends repeated times m and variance proportion d and error variances at the extreme settings. note that the particular shape of λ(x) is immaterial for the results, but only its values at 0 and 1, as long as condition (6) is satisfied. specially, when heteroscedastic structure satisfies λ0 = λ1 = max x∈[0,1] λ(x) , the optimal designs discussed above are independent of the variance ratio d. these optimal designs are the same as the corresponding optimal designs in the linear regression model without any random effects, i.e., ωd = ωg = ωi = ωds = 1 2 , ωa = √ 2− 1. 4 examples in this section, we consider three random intercept models with the following heteroscedastic errors λ(x) = x2 + 1, λ(x) = 1 1 + x , λ(x) = x4 + 1 x2 + 1 . it is easy to verify that these three λ(x) satisfy the condition (6). these heteroscedastic structures are also considered in chang[15] for d-optimal designs in weighted polynomial regression models. we will give the optimal designs for the three models in terms for the results given in section 3. we also compare the dand g-optimal designs for the three models with the equireplicated design ω0 = 0.5 which is simultaneously dand g-optimal for the fixed effects only model (d = 0) in terms of the dand g-efficiency which are defined as following effd(ω0) = ( |m(ω0)| |m(ω)| ) 1 2 , effd(ω0) = max x∈[0,1] d(x, ωg) max x∈[0,1] d(x, ω0) (8) advances in systems science and applications (2012) vol.12 no.4 395 example 1 for the model (2) with mi = m(i = 1, . . . , n), and λ(x) = x2 + 1 , from the theorems in section 3, we obtain the d-, g-, a-, iand ds-optimal proportions as follows: ωd = √ 1 + γ√ 1 + γ + √ 1 + 2γ , ωg = ωa = 1 3 , ωi = ωds = √ 2− 1. the dand g-efficiencies defined by (8) of the equireplicated design ω0 = 0.5 are as follows: effd(ω0) = √ 1 + 2γ + √ 1 + γ√ 4 + 6γ , effd(ω0) = γ + 1.5 γ + 2 . it is clear that effd(ω0) decreases strictly in γ and ultimately tends to ( √ 2 + 1)/ √ 6 , and effd(ω0) increases strictly in γ and ultimately tends to one. fig.1 shows the plots of these two efficiencies. fig.1 the efficiencies of effd(ω0) and effg(ω0) with different λ example 2 for the model (2) with mi = m(i = 1, . . . , n),and λ(x) = 1 1+x , the d-, g-, a-, iand ds-optimal proportions as follows: ωd = √ 2 + 2γ√ +γ + √ 2 + 2γ , ωg = 2 3 , ωa = 1 2 , ωi = ωds = 2− √ 2. the dand g-efficiencies defined by (8) of the equireplicated design ω0 = 0.5 are as follows: effd(ω0) = √ 2 + 2γ + √ 2 + γ√ 8 + 6γ , effd(ω0) = γ + 3 γ + 4 . it is clear that effd(ω0) decreases strictly in γ and ultimately tends to ( √ 2 + 1)/ √ 6 = 0.9856, and effd(ω0) increases strictly in γ and ultimately tends to one. fig.2 shows the plots of these two efficiencies. 396 jing cheng:optimal designs in random intercept model with heteroscedastic errors fig.2 the efficiencies of effd(ω0) and effg(ω0) with different λ example 3 for the model (2) with mi = m (i = 1, . . . , n), and λ(x) = x4+1 x2+1 , the d-, g-, a-, iand ds-optimal proportions as follows: ωd = ωg = ωi = ωds = 1 2 , ωa = √ 2− 1. that is, the d-, g-, iand ds-optimal designs are all the equireplicated designs. appendix proof of lemma 1 from liski et al[16], we get m−1(ξ) = m−1 0 (ξ) + ( d n 0 0 0 ) here m0(ξ) is the corresponding generalized information matrix when there are no individual intercepts, i.e., m0(ξ) = mn ( ν0 ν1 ν1 ν2 ) . let the proportion ω in ξ∗ be of the form ω = ν1/λ1. it follows that m0(ξ ∗) = mn ( ν∗0 ν∗1 ν∗1 ν∗2 ) , and m0(ξ ∗)−m0(ξ) = mn ( ν∗0 − ν0 0 0 ν∗2 − ν2 ) . here ν∗0 = λ1ω + λ0(1− ω) and ν∗2 = ν∗1 = λ1ω . since 1 λ(x) ≥ 1− x λ0 + x λ1 ≥ x λ1 , advances in systems science and applications (2012) vol.12 no.4 397 so λ1 ≥ xλ(x). it implies 0 < ω < 1. [ m0(ξ ∗)−m0(ξ) ] 11 = mn [ λ1ω + λ0(1− ω)− p∑ j=1 ωjλ(xj) ] = mn p∑ j=1 ωj [ λ(xj)xj(λ1 − λ0)− λ1λ(xj) + λ1λ0 ] condition (6) implies λ(x)x(λ1 − λ0)− λ1λ(x) + λ1λ0 ≥ 0. so we obtain [ m0(ξ ∗)−m0(ξ) ] 11 ≥ 0 by ν∗2 = ν∗1 = ν1 ≥ ν2 , we have[ m0(ξ ∗)−m0(ξ) ] 22 ≥ 0 so we get m0(ξ ∗) ≥ m0(ξ) and hence m(ξ∗) ≥ m0(ξ). acknowledgements this work was partially supported by a nsfc grant (11071168), special funds for doctoral authorities of education ministry (20103127110002), e-institutes of shanghai municipal education commission (e03004), shanghai leading academic discipline project (s30405), the innovation program of shanghai municipal education commission (11zz116), and the scientific research foundation of chaohu college. references [1] n.t. longford (1993), random coefficient regression models, clarendon press, oxford. [2] d. pena, v. yohai (2006), “a dirichlet random coefficient regression model for quality indicators”, journal of statistical planning and inference, vol.136, pp.942-961. [3] f. yu (2007), “quadratic design criterion for nonlinear models”, advances in systems science and applications, vol.7, no.2, pp.155-160. [4] t. schmelter (2007), “the optimality of single-group designs for certain mixed models”, metrika, vol.65, pp.183-193. 398 jing cheng:optimal designs in random intercept model with heteroscedastic errors [5] t. schmelter (2007), “consideration on group-wise identical designs for linear mixed models”, journal of statistical planning and inference, vol.137, pp.4003-4010. [6] r. schwabe, t. schmelter. (2008), “on optimal designs in random intercept models”, tatar mountains mathematical publications, vol.39, pp.189-195. [7] t. schmelter, et al. (2007), “some curiosities in optimal designs for random slopes”, in: contributions to statistics. moda 8 advances in modeloriented design and analysis,physica-verlag heidelberg, pp.189-195. [8] a. luoma, et al. (2007), “optimal designs in random coefficient cubic regression models”, journal of statistical planning and inference, vol.137, pp.3611-3617. [9] m. entholzner, et al. (2005), “a note on designs for estimating population parameter”, listy biometryczne-biometrical letters, vol.42, pp.25-41. [10] l.k. debusho, l.m. haines (2007), “vand d-optimal population designs for the simple linear regression model with a random intercept term”, journal of statistical planning and inference, vol.138, pp.1116-1130. [11] g. z. wang, j. zhao, j. b. chen (2006), “the uniform design modeling and analysis”, advances in systems science and applications, vol.6, no.4, pp.699-703. [12] s.h. yu (2007), “the linear minimax estimator of stochastic regression coefficients and parameters under quadratic loss function”, statistics and probability letters, vol.77, pp.54-62. [13] g.y. wen, et al. (2008), “orthogonal experimental design in matrix form and its application to the introduction selection of thymus genus”, advances in systems science and applications, vol.8, no.3, pp.437-446. [14] f. pukelsheim (1993), optimal design of experiment, johnwiley, new york. [15] f.c. chang (2005), “d-optimal designs for weighted polynomial regression– a functional approach”, annals of the institute of statistical mathematics, vol.57, no.4, pp.833-844. [16] e.p. liski, n.k. mandal, k.r. shah, b.k. sinha (2002), topics in optimal designs, springer, new york. corresponding author nicholas nechval can be contacted at:yue2@shnu.edu.cn advances in systems science and applications (2013) vol.13 no.4 334-354 agent based modeling of integration of organizational cultures in mergers and acquisitions albert r. bakhtizin1 and svetlana v. denisova2 1central economics and mathematics institute of russian academy of sciences, moscow, russia 2moscow aviation institute (national research university), moscow, russia abstract in the article a new approach to the analysis of compatibility of organizational cultures during merges and acquisition using the agent based simulation model (abm) is considering. the choice of this method is substantiated. using abm makes the considerable economy of time possible. usually companies spend a lot of time on estimation of possible results of merge. also abm allows promptly change the initial parameters of the model. results of the research experiments are presented in the article. they showed that the agent based simulation model can be successfully used as a tool of prompt analyzing and prediction of the result of organizational culturesąŕ integration. in addition to this it can take into account features of each culture separately. keywords agent based model, modeling organizational culture, merger and acquisition, culture integration, corporate mutual relations 1 introduction the number of mergers and acquisitions is growing up every year. russian companies also are using potential of such transactions for development of their successful businesses. mergers and acquisitions open opportunities to enter new markets, gain access to promising modern technologies, and transition to the new, more promising economic sectors. regular changes of economic environment, together with increase of information flows with fast access to data and acquisition of new knowledge, create preconditions for development of internal sources of economic growth, which allow the company to operate progressively in rapidly changing environment. organizational culture of company provides resources designed to support flexible, adaptive and efficient business systems. organizational culture determines how and in what way created and controlled business processes are. organizational culture creates foundation for joint activities of company team and formation of organizational culture, corresponding to the current external environment, and it also provides effective organizational development of the company. on the other hand, study of organizational culture itself makes it possible to obtain objective assessment of many processes occurring in the company, which becomes especially important in the process of implementing mergers and acquisiadvances in systems science and applications (2013) vol.13 no.4 335 tions, when merging of companies with already established organizational cultures greatly increase resistance of their personnel to conducted organizational changes. influence of organizational culture upon results of merger or acquisition deal is often underestimated. organizational culture, as a cohesive core of company, can produce significant impact on many aspects of company activities, promoting or delaying its development. it depends primarily on characteristics of the culture itself and its compliance with current situation and goals, in the context of which it manifests. creating of organizational culture of such type, which will be more consistent with structure and goals of the company formed as the result of merger or acquisition, is one of challenges that must be met at an early stage of planning of the deal. large-scale changes in mergers and acquisitions initiate resistance of the personnel of merging companies to these changes, which may result in substantially lower to expected effectiveness of merger or acquisition deal. solution to this problem can become establishment of informational support of the coming changes, when employees are regularly informed about all events inside the company, which creates a favorable working atmosphere and greatly impedes spread of rumors and thus mitigates negative reaction of team. as far as possible, company employees should be involved in reorganization process, which significantly increase the role of communications. but if resistance of personnel to changes in the period of integration is a special case of reaction of employees to any organizational changes, and the ways to solution of this problem are indicated, than solution to wide-range problems still is not fully determined. in this way, the potential conflict of organizational cultures should be analyzed at the early stage of merger or company takeover deal. but in practice, to conduct such analysis is very problematic case. the main difficulty is that it is impossible to predict the future culture of a new company after merger deal and to evaluate its effectiveness. options may vary depending on many factors, like : organizational cultures type prior to merging, their strength and level, similarity and ability to changes. according to consultants any change of organizational culture requires at least three years. in other words, it takes a long time for assessment of the results of integration of organizational cultures. while if in the process of cultural integration were made mistakes, than negative results show up only with time, and to make adjustments and get the desired results would take more time and additional capital inputs. the purpose of all the mergers and acquisitions transactions, without any exception, is integration carried-out successfully. integration is understood as element joining-up resulted in forming a single whole. as a rule, the process of merging and acquiring companies, integration, results in creation of a new organizational structure capable to dispose available resources in a way more efficient 336 albert r. bakhtizin : agent based modeling of integration of organizational cultures in ... and rational, to optimize material, financial, labor and informational flows of the integrated companies. statistical data show that 70% cases of prospectively advantageous transactions fail because of the low-quality preparation and carrying out integration. so, selection of an organizational form for companies integration in compliance with the purpose and objectives stated is an important step that can facilitate to unify resources and flows efficiently. integration is carried out in various directions. though, whereas corporate strategy integration, product series management systems, product distribution and delivery management systems are predictable and subject to numerical computation on the base of data available, organizational culture integration is often unpredictable. integration of united enterprises organizational cultures is a crucial component in forming the corporate interaction efficiently. resistance exhibited by the personnel to the process in the period of companies mergers and acquisitions is conditioned, first of all, by the difference among various (in the majority of cases) corporate values and organizational cultures. the mergers of ąőequalsąŕ finishes frequently with the matter that a stronger group imposes their organizational culture on a unilateral basis. there is no doubt, mergers and acquisitions propose a lot of advantages for business development. nevertheless, some erroneous illusion might be created that such transactions are relatively easy, inexpensive and represent the only way to increase business considerably. however, most investigations on mergers and acquisitions efficiency evidence that 60 to 80% companies even armed with potentially advantageous strategy do not accomplish the objectives. it is often concerned with mistakes committed in the course of enterprise integration as well as incorrect organization of the transaction itself. mistakes and inadvertences can appear in every phase of mergers and acquisitions. thus, in addition to the incorrectly chosen object for merger and acquisition, low-quality preparation for transactions, and erroneous financial estimation, there could be chosen erroneous ways to implement integration. particularly, serious mistakes can be made in the course of changing and forming the enterprise organizational culture resulted from mergers or acquisitions. in this phase, the mistakes committed can be conditioned by the lack of detailed integration plan and the lack of appropriate approach. the main cause for most failed mergers is inconsistence and incompatibility of organizational cultures. the cause for most failed mergers and acquisitions consists in the joining companies inability to overcome organizational culture contradictions. cultural problems are to be solved even for most successful mergers. in other words, cultural and organizational problem solving acquires the utmost importance for each integration, both successful and failed one. although this area has not been investigated advances in systems science and applications (2013) vol.13 no.4 337 properly yet, experts working in this area stated that enterprises which completed integration successfully had paid a lot of attention to the following aspects : management team forming : how to target the management ranks to the tasks issued by the director general and the board ; organizational structure : how to create a structure that would mostly correspond to the new enterprise strategy ; highly-efficient culture : how to work-out and develop a culture that would facilitate for efficiency increase and would help the new enterprise to realize their long-term objectives ; expert employee administration : how to reveal the most various employees in both enterprises and what actions to undertake for involving them into the process of new enterprise creation. in order to avoid mistakes, it is necessary to elaborate a plan for mergers and acquisition procedure, and, as it has already been noted and never before stated, the question on organizational culture and emerging companies personnel is to be formulated in the phase of choosing the object for mergers and acquisitions. in case the merged enterprises remain existing independently from each other, as a rule, there are no big problems with the personnel. however, if the enterprises start functioning as a single whole, the question on organizational cultures integration becomes exceptionally acute. the new organizational culture could not be acquired by simply joining two old cultures. in theory, if promoting people, introducing new values, orientations, introducing new behavioral models, remunerating them, forming models to emulate, these behavioral models will be repeated and will be fixed in the personnel minds. this way, a new culture is being created. though, in practice, organizational culture integration is rather complicated, as far as it is a multiphase complex process that needs comprehensive planning and accurate implementation. detailed calculations along with competent and thoughtful actions of the managing personnel will guarantee the successfully implemented integration. nevertheless, the potential of organizational culture confrontation is rarely analyzed in this phase, preceding the mergers or acquisition transaction. as a result, the culture confrontation creates a serious problem for merger procedure. from this we can conclude that during the period of mergers and acquisitions the most important is issue of compatibility of organizational cultures of the merging organizations. and at planning stage of the deal, for future development of effective merging strategy, it is very important to be aware of possible results of integration of cultures. at the moment there is no tool of diagnostics of organizational cultures compatibility, which can during short period of time to play off a large number of possible scenarios for merging organizational cultures and get expected results of 338 albert r. bakhtizin : agent based modeling of integration of organizational cultures in ... integration, in order to subsequently select among all possible variants the most optimal one. in this regard, the main objective of study is building of tool that allows during short period of time to study large number of possible scenarios of merging of organizational cultures targeted on getting of expected integrated culture, and then choose from all options the most suitable one. in this way, one of the objectives of study is task of taking into account individual characteristics of micro level individual agents in order to obtain more realistic assessment of impact of organizational changes on production indicators. as the above mentioned tool, we decided to develop agent-oriented computer simulation model, related to the class of models based on individual behavior of agents. due to its advantages, namely : 1) ability to simulate close to reality system, 2) emergency, 3) flexibility, 4) possibility of specification of model parameters without knowledge of global dependencies within the frameworks of simulating of relevant subject area, we assume that the agent based model will allow us to achieve objectives of study. 2 literature review it should be noted that in russia an agent-oriented models have been developed relatively recently and their development is mainly concentrated in the central economics and mathematics institute (cemi ras) (under leadership of academician v.l. makarov), as well as the company’s “xj technologies” (st. petersburg). for more details about theoretical aspects of this model class see articles of v.l. makarov and a.r. bakhtizin [1]. in the world practice there is some experience in developing agent-oriented models of organizational culture and corporate relations. examples of the most advanced works are shown below. a multi-agent simulation platform for modeling perfectly rational and boundedrational agents in organizations [2]. this paper presents an agent-based simulation framework for the analysis of the equilibria that emerge in a complex structure such as an organization ; we can think of some of these equilibria as corporate culture. authors concentrate on modeling the effort exerted by heterogeneous agents in an organization, and how the interaction between them may lead to a common level of effort (corporate culture). the simple model authors propose is a system in which agents interact in a dynamic, adaptive and evolving way. the model shows how different compositions of the population may lead the system to different common behaviours ; the implications of result findings are both descriptive and normative, and shed advances in systems science and applications (2013) vol.13 no.4 339 light on some core problems of the economics of organization design. how groups can foster consensus : the case of local cultures [3]. this work is based on an idea that a local culture denotes a set of rules on business behaviour among firms in a cluster. similar to social norms or conventions, it is an emergent feature of interaction in an economic network. to model its emergence, authors consider a distributed agent population, representing cluster firms. the model introduces a feedback mechanism of agent behaviour and in-group structure. studying its consequences by means of agent-based computer simulations, authors find that for narrow-minded agents the feedback mechanism helps find consensus more often, whereas for open-minded agents this does not necessarily hold. overall, the dynamics of agent interaction in clusters as modelled here, are conducive to consensus among all or a majority of agents. computer mediated communication and organizational culture : an agentbased simulation model [4]. this paper examines the mutual relationship between the organizational use of computer mediated communication and organizational culture. computer mediated communication supplements communication among members of an organization to maintain the culture, especially when those persons cannot communicate by other means. on the other hand, a strong organizational culture allows a more effective use of computer mediated communication by providing members with some of the necessary common ground to better understand the information exchanged. these relationships are investigated using an agent-based model. this agent-based model incorporates many partial theories into a coherent and fully defined model, which helps formalize and integrate those theories. in this paper, authors present some of the results of the agent-based model that show that organizational culture can influence the effectiveness of computer mediated communication and that computer mediated communication can help maintain and stabilize a culture. social construction of organizational culture : an agent-based model [5]. this model, called “orgnorms”, assumes that culture is important to organizations, and companies in particular, on two levels. first, the homogeneity of an organization’s culture affects communication and efficiency. second, the ‘cultural fitness’ of an organization to local society affects its competitive advantage. orgnorms is an agent-based model designed to simulate the development of and changes in organizational culture in a culturally changing society, tracking the organization’s internal homogeneity and its external fitness to its societal environment. a central assumption is that homogeneity demands that new members adopt the organizational norms, while fitness demands that the organization adopts the views of the new members. that connection breeds similarity, that knowledge is local, and that agents take after those who are similar and the local 340 albert r. bakhtizin : agent based modeling of integration of organizational cultures in ... majority, are other key assumptions. these examples of agent-oriented simulation models related to organizational culture show that aom can be successfully applied in studies and forecasting of various processes of transformation of organizational culture. 3 agent-based modeling of organizational mergers with regard to experience of our colleagues from russia and other countries, in 2010 we developed an agent-based model of organizational mergers, as is described below. 3.1 characteristics of agent . table 1 the characteristic parameters of agent n/n parameter description parameter value 1 age from 18 to 60 years 1.1 first age group 20-25 years 1.2 second age group 25-40 years 1.3 third age group 40-50 years 1.4 fourth age group 50-60 years 2 marital status 0 agent has no family, 1 agent has a family 3 professionalism of agent 3.1 education 0 secondary, 1 higher 3.2 experience from 1 to 40 years 3.3 work experience in particular organization before merging from 1 to 10 years 4 loyalty from 1 to 10 points, where minimum value of parameter means intolerance to the values of company, and 10 points means his strong commitment 5 ability to adaptation from 1 to 10 points 6 satisfaction with working conditions after integration 1 to 10 points 7 labor market demand of agent in times of integration from 1 to 10 points 8 ability to work from 1 to 10 points (1). age. for these parameters, a distribution among four age groups : 20-25 years ; 25-40 years ; 40-50 years ; 50-60 years was specified. (2). marital status. it includes : has no family, has a family. advances in systems science and applications (2013) vol.13 no.4 341 (3). professionalism of agent, which consists of concepts such as : education(c1), experience(c2), work experience in particular organization before merging(c3). these variables during initializing of models have random values with standard deviations enclosed in brackets. in order to create the model, 8 main parameters were chosen : age, marital status, professionalism of agent, loyalty, ability to adaptation, satisfaction with working conditions after integration, labor market demand of agent in times of integration, ability to work. input parameter data are given in table 1. professionalism is determined by value within intervals from 0 to 100 points, as function of three components (c1, c2, c3) in the following as p = 33.3 · c1 + 33.3 · c2 40 + 33.3 · c3 10 (1) i.e. in case of maximum values of all components the level of professionalism of agent is also maximum close to 100 points. (4). loyalty. in this case, a loyal employee should share the core beliefs and values of the company (from 1 to 10 points, where minimum value of parameter means intolerance to the values of company, and 10 points means his strong commitment). (5). ability to adaptation. it also can be divided from 1 to 10 points. under adaptation we understand mutual adjustment of employee and company, which is based on gradual involvement of worker into labor activities in new professional, psychophysiological, psychosocial, organizational, administrative, and economic conditions. (6). satisfaction with working conditions after integration. it also can be divided from 1 to 10 points. (7). labor market demand of agent in times of integration. it also can be divided from 1 to 10 points. (8). ability to work. it also can be divided from 1 to 10 points. 3.2 characteristics of environment for functioning of agents and organizational cultures we come to the description of environment of agent model. the agent functioning medium comprises four organizational culture types clan culture, adhocratic culture, hierarchical culture and market culture. for determination of environment we used a simplified version of methodology for assessing of organizational culture, and below is brief description of four types of organizational cultures, used in probability function, which graph is shown in fig.5. type 1. clan culture. it means very friendly working place where people have much in common. companies are like big families. leaders or chiefs of companies 342 albert r. bakhtizin : agent based modeling of integration of organizational cultures in ... are perceived as educators, and perhaps even as parents. the company is cemented with loyalty and traditions. there is high responsibility of the company. it is focused on a long-term benefits for improving of individual, pays attention to a high degree of team unity and morale. success is measured in terms of good relations to consumer and care about people. the company encourages teamwork, participation of people in business and mutual consent. type 2. adhocratic culture. it means a dynamic, entrepreneurial and creative working place. people are willing to offer their own support and stand the risks. the leaders are considered innovators and persons who are ready to stand the risks. cohesive spirit of this company is devotion to experiments and innovation. necessity of work on the business forefront is emphasizes as need. the general long term policy of such company is its growth and acquiring of new resources. success means production and performance of unique and new products and / or services. it is important to be a leader on the products and services markets. such organization encourages individual initiative and freedom. type 3. hierarchical culture (bureaucratic). it is very formalized and structured place of work. procedures dominate activities of employees. leaders are proud of the fact that they are rationally minded coordinators and organizers. it is critical to maintain smooth running of the company. company is united with formal rules and official policies. long-term concern of organization is to provide stability and smooth running performance of cost-effective operations. success is measured in terms of supply, smooth schedules and lower costs. management is concerned about employment status and long-term predictability of employees. type 4. market culture. this kind of company is focused on results, the main concern of which is performance of task. people are ambitious and compete with each other. chiefs are hard leaders and tough competitors. they are unshaken and demanding. this company is united together with emphasis on the desire to win. reputation and success are things of common concern. focus of strategy is targeted to a specific action, achievement of tasks and measurable goals. success is measured in terms of markets penetration and increase of market share. important is competitive pricing and leadership on the markets. the working style of this company is hard line targeted on competition. 3.3 agent behavior behavior of agents is specified by diagram of state (state chart), transitions inside of which depend on the values of probability functions listed below. since within the frameworks of model occurs absorption of one company by another, the agent has two options : adapt to the new conditions or leave (for simplicity it is assumed that after reorganization the absorbing company does not change its type, and, on the other hand, the absorbed company takes leading style of absorbing organization). this chart shows process in the following way : agents advances in systems science and applications (2013) vol.13 no.4 343 of absorbed organization (fig.1 “organization 2”) through conversion pass to the absorbing organization (“organization 1”) or have to leave (this transition is shown along the arrow directed towards the ring with dot in center). in the process of work this model simulates the process of absorption, and some time after reorganization, when agent may resign (i.e. it is another transition along the arrow directed towards the ring). fig.1 state chart of model agent for simplicity, we do not consider optimization of personnel, i.e. possible redundancy of employees by company. next, we go to the more detailed description of agent state chart. first of all in the state chart of transition may work out transition 1, depending on agent’s loyalty towards values of company (in this case loyalty parameter of all agents is relevant only to absorbing company). if company’s values are alien to the agent, than he can adapt to them, depending on the values of corresponding parameter (i.e. may work out transition 2). the behavior of agent may be adjusted depending on other parameters. for example, if qualification of agent is in high demand on labor market, than probability of his resignation is high (transition 3). otherwise “thing, which can change up his mind,” may be his age (transition 4), as well as having a family (transition 5). after the process of absorption the agent may stay unsatisfied with new con344 albert r. bakhtizin : agent based modeling of integration of organizational cultures in ... fig.2 probability (y-axis) of transition of agent into absorbing organization based on his level of loyalty (x-axis) fig.3 probability (y-axis) of transition of agent into absorbing organization based on his adaptation ability (x-axis) fig.4 probability (y-axis) of transition of agent into absorbing organization based on the level of labor market demand in times of integration (x-axis) advances in systems science and applications (2013) vol.13 no.4 345 fig.5(a) probability (y-axis) of transition of agent (working in company with organizational culture of first type) into absorbing company, depending on type of organizational culture (x-axis) fig.5(b) probability (y-axis) of transition of agent (working in company with organizational culture of second type) into absorbing organization, depending on type of organizational culture (x-axis) fig.5(c) probability (y-axis) of transition of agent (working in company with organizational culture of third type) into absorbing company, depending on type of organizational culture (x-axis) 346 albert r. bakhtizin : agent based modeling of integration of organizational cultures in ... fig.5(d) probability (y-axis) of transition of agent (working in company with organizational culture of fourth type) into absorbing company, depending on type of organizational culture (x-axis) fig.6 probability (y-axis) of transition of agent into absorbing company based on age (x-axis) fig.7 probability (y-axis) of transition of agent into absorbing company based on satisfaction with working conditions after integration (x-axis) ditions, and in this case he may initiate transition 6 (this applies to employees of advances in systems science and applications (2013) vol.13 no.4 347 absorbed organization, only). transitions in state chart can work out depending on values of probability functions, defined by experts. fig.2 shows a graph of probability function, where the argument (x-axis) is level of loyalty and value of function (y-axis) probability of transition 1. fig.3 and 6 show graphs with probabilities of working of transitions 2 and 4, depending on capacity for adaptation and age of agent, respectively. for determination of probability of transition 3 we must calculate average probability on basis of functions, whose graphs are shown in fig.4 and 5(a)-5(d). 3.4 adequacy of model for testing of model adequacy we have conducted three experiments on merging of different types of companies, whose parameters are provided in tables 2-4. table 2 parameters of absorbing and absorbed companies for experiment no.1 parameter absorbing company absorbed company type of organizational culture bureaucratic market number of employees 20 000 20 000 age of employees 20-25 years 25% 20-25 years 60% 25-40 years 40% 25-40 years 20% 40-50 years 20% 40-50 years 10% 50-60 years 5% 50-60 years 10% table 3 parameters of absorbing and absorbed companies for experiment no.2 parameter absorbing company absorbed company type of organizational culture bureaucratic market number of employees 300 30 age of employees 20-25 years 25% 20-25 years 15% 25-40 years 50% 25-40 years 50% 40-50 years 20% 40-50 years 30% 50-60 years 5% 50-60 years 5% for experiments, were collected data on transactions of mergers and acquisitions during the period from 2004 to 2009. companies involved in this experiment represent large retail sector and it-sphere. in the first experiment, two organizations having manpower strength of about 20 000 persons each, were represented ; each company had associated companies. the experiment took into account the total number of employees for each enterprise including associate companies and representative offices. 348 albert r. bakhtizin : agent based modeling of integration of organizational cultures in ... table 4 parameters of absorbing and absorbed companies for experiment no.3 parameter absorbing company absorbed company type of organizational culture bureaucratic market number of employees 50 30 age of employees 20-25 years 20% 20-25 years 18% 25-40 years 60% 25-40 years 70% 40-50 years 16% 40-50 years 10% 50-60 years 4% 50-60 years 2% in the second experiment, the absorbing company before reorganization had 300 people, the merged company had 30 people. in the third experiment, small enterprises took part ; the absorbing company personnel counted 50 persons for when transaction started and the merged company personnel counted 30 persons. bellow in listing 1 adduces a code generated by anylogic program which is responsible for agent initialization and model initial state initialization. all the agent populations (people.size()) are distributed by groups (age groups, as well as agent groups pertaining to the first or the second organization) depending on the values preset (initial number of employees working for the first (agentfirm1) and the second organization and, also, four age frames. (age1, age2, age3, age 4)). for (int i = 0 ; i<people.size() ; i++) { if (randomtrue(agentfirm1) == true) { people.get(i).statechart.fireevent(“firm1”) ; people.get(i).firm = 1 ; people.get(i).oldfirm = 1 ; // for the first age group, age values of 20 to 25 years are assigned if (randomtrue(age1) == true) { if (age == false) { people.get(i).age = uniform (20, 25) ; age = true ; } } // for the second age group, age values of 25 to 40 years are assigned advances in systems science and applications (2013) vol.13 no.4 349 if (randomtrue(age2) == true) { if (age == false) { people.get(i).age = uniform (25, 40) ; age = true ; } } // for the third age group, age values of 40 to 50 years are assigned if (randomtrue(age3) == true) { if (age == false) { people.get(i).age = uniform (40, 50) ; age = true ; } } // for the fourth age group, age values of 50 to 60 years are assigned if (randomtrue(age4) == true) { if (age == false) { people.get(i).age = uniform (50, 60) ; age = true ; } } age = false ; } else { people.get(i).statechart.fireevent(“firm2”) ; people.get(i).firm = 2 ; people.get(i).oldfirm = 2 ; // for the first age group, age values of 20 to 25 years are assigned if (randomtrue(age1) == true) { if (age == false) { people.get(i).age = uniform (20, 25) ; age = true ; 350 albert r. bakhtizin : agent based modeling of integration of organizational cultures in ... } } // for the fourth age group, age values of 25 to 40 years are assigned if (randomtrue(age2) == true) { if (age == false) { people.get(i).age = uniform (25, 40) ; age = true ; } } // for the fourth age group, age values of 40 to 50 years are assigned if (randomtrue(age3) == true) { if (age == false) { people.get(i).age = uniform (40, 50) ; age = true ; } } // for the fourth age group, age values of 50 to 60 years are assigned if (randomtrue(age4) == true) { if (age == false) { people.get(i).age = uniform (50, 60) ; age = true ; } age = false ; } } } listing 1 piece of code responsible for initial model initialization it should be noted that the agents in the base model version are defined rather “rigidly”, as referred to the age frames ; nevertheless, in the same way, we can define other agent parameters (loyalty, ability to adaptation, labor market demand of agent, etc). for example, by means of the model, we can simulate a situation when the majority of employees working for a company possess a high-degree loyadvances in systems science and applications (2013) vol.13 no.4 351 alty or labor market demand. in the following we describe pieces of codes that determine numbered transitions in fig.1. here, two points are important : 1) the functions in use are anylogic package functions that operate with java language. 2) all the program structures adduced define the general class of agent-employee who takes part in the process of two organizations merging. when creating the direct instance of a class, for every agent a unique set of parameters is assigned, each of which is specified by normal distribution using uniform() function. transition 1 operates on the following condition : randomtrue (fnloyality(loyalty)/100) == true , where loyalty variable is specified by normal distribution uniform (1, 10), and fnloyality function is given as a graph in fig.2. transition 2 operates on condition : randomtrue (fnadoptation(adoptation)/100) == true, where adaptation variable is specified by normal distribution uniform (1, 10), and fnadoptation function is given as a graph in fig.3. transition 3 operates on condition : randomtrue ((fnfirm4(get_main().firm1ty -pe)/100)) == true, where varied functions fnfirm1, fnfirm2, fnfirm3, fnfirm4 are given as graphs in fig.5(a)-5(d), and independent variable firm1type, that determines the type of organization being absorbed (1, 2, 3 or 4) is specified on carrying out experiments. transition 4 operates on condition : randomtrue (fnage(age)/100) == true, age variable is specified on initializing the model, and fnage function is given as a graph in fig.6. transition 5 operates on condition : randomtrue (fnfamily(family)) == true, variable family is randomly specified randomtrue(0.6), and fnfamily function is specified in an expert way as follows : if (family == true) { return 0.2 ; } else { return 0.2 ; } transition 6 operates on condition : randomtrue(fnintegration(afterintegration) /100) == true, afterintegration variable is specified by normal distribution uniform (9, 10), and fnintegration function is given as a graph in fig.7. table 5 shows final results. as you can see, the difference between resulting values is not too large, and 352 albert r. bakhtizin : agent based modeling of integration of organizational cultures in ... table 5 number of employees, retired after merging, percentage of total number. actual data calculated data experiment no.1 20,0% 16,1% experiment no.2 2,12% 3,5% experiment no.3 12,5% 10,24% therefore we go to other calculations. 4 results of computer experiments after that using the developed model we evaluated effects of hypothetical merging of companies with different types of organizational culture in order to identify the best and worst variants suitable to both organizational cultures. for experiments we used companies with the same number of employees, as well as with similar values of other parameters. output data are shown in table. 6. these data represent relative values of changes in number of employees as percentage from the first scenario, which perform merging of two companies of first type. as you can see, the biggest number of employees resigned as a result of merging of companies of second type (absorbing company with adhocratic culture) and fourth type (absorbed company with market culture). table 6 relative changes in the number of retired employees of the merged companies of various types, in percentage from the first scenario type 1 type 2 type 3 type 4 type 1 0 -0.6 -0.9 -0.9 type 4 -0.7 -1 -1 -1.4 type 2 -0.9 -0.9 -0.3 -0.4 type 3 -0.6 -0.8 -0.8 -0.4 we propose a new method for assessing of compatibility of organizational cultures in mergers and acquisitions with help of agent-based model, which has several advantages. first, it allows during relatively short period of time to get preliminary assessment of the results of mergers with different types of organizational cultures, because aom can consider more variants of possible scenarios. second, aom allows you to specify unlimited number of basic parameters, such as number of employees, which can vary from several dozens to several thousands employees. during development of this aom with method of expert judgments, we selected eight parameters to describe agent. but it is important to note that in each case the merger or acquisition model can be changed depending on initial conditions of transaction, and there can be defined bigger or smaller number of parameters. advances in systems science and applications (2013) vol.13 no.4 353 possibilities of fast changes of operating parameters of the model, and its easy adaptation to specific conditions allow considering agent-based model as universal tool for preliminary assessment of merger or acquisition of companies with different types of organizational cultures. the other authors work has shown that the agent-oriented models could be successfully used for studying and solving organizational culture problems. the authors of the present article, by-turn, have worked-out such an agent-oriented model that makes it possible to solve several important problems concerning the organizational cultures integration when merging and acquiring companies, that is : on pre-setting selected parameters values for each company, determining the degree of integration success and the number of resigned and remained employees. in case the number of resigned employees does not satisfy the management, they will be able even on the preparatory phase of integration, to work out measures to retain employees and implement these measures in the course of merging. the number of employees maintained after merging is the direct indicator of transaction success. as far as if after merging, the majority of employees resigned, then in the phase of preparing and carrying out mergers, some outrage mistakes had been committed ; the company had not made the maximal efforts to retain them, had not taken into account employeesąŕ interests. in this case, the efficiency of a transaction carried out depends on the number of employees who decided to link their future to the new company in which every employee feels comfortable and finds it beneficial to work. that is why the results acquired by the using of the agent-oriented model are so significant and representative for predicting the results of organizational culture integration. the authors of the present work did not have a purpose to investigate all the existing integration types. the scope of their scientific interests covers only one of the integration types, organizational culture integration. the agent-oriented model was developed to study the selected integration type. in particular, aom presented in this article shows that, most of all, who will be not be satisfied with new conditions and quit the company as the result of merger are employees of two adhocratic and market type organizational cultures. in practice, to obtain such results takes several years, during which companies with these types of organizational cultures would suffer significant loss to resolve inevitable conflicts. but with aom results can be obtained at the planning stage of potential deal and it allows management to take appropriate measures to prevent massive layoffs of employees or even cancel the deal, which action could help company to save unnecessary expenditure. 354 albert r. bakhtizin: agent based modeling of integration of organizational cultures in ... references [1] canessa e. and riolo r. l. (2006), “computer mediated communication and organizational culture: an agent-based simulation model”, journal of information technology, vol.21, pp.272-283. [2] dal forno a. and merlone u. (2002), “a multi-agent simu-lation platform for modeling perfectly rational and bounded-rational agents in organizations”, journal of artificial societies and social simulation, vol.5 no.2, pp.3, http://jasss. soc.surrey.ac.uk/5/2/3.html. [3] groeber p., schweitzer f. and press k. (2002), “how groups can foster consensus: the case of local culture”, journal of artificial societies and social simulation vol.12, no.2, http://jasss.soc.surrey.ac.uk/12/2/4.html. [4] makarov v. l. and bakhtizin a.r. (2009), “new tool in social science agent based simulation model: the general description and concrete examples”, economics and managemen, december, pp.13-25. [5] raaterova m. (2005), “social construction of organizational culture: an agentbased model north american association for computational social and organizational science”, http://www.casos.cs.cmu.edu/events/conferences/2005/ 2005_proceedings/raaterova.pdf. corresponding author albert r. bakhtizin can be contacted at: albert.bakhtizin@gmail.com advances in systems science and applications (2014) vol.14 no.1 36-49 an agent-based distributed process planning and scheduling system y.f. wang, y.f. zhang and j.y.h. fuh department of mechanical engineering, national university of singapore, abstract this paper reports an agent-based approach to develop a distributed process planning and scheduling system to allow geographically dispersed entities to work cooperatively towards the global goals. the system comprises several intelligent functional agents, each performing a specific task while accomplishing the global tasks through the corporation with other agents. for a schedule with an unsatisfactory performance measure, a facilitator agent is able to automatically generate instructions, which is fed back to the corresponding process planning agent (ppa) for updating the solution space. the ppa then employs an optimization algorithm to find a new optimal plan. this new plan, together with the unchanged plans, will come out a new schedule in the scheduling agent. the coordination process continues until a satisfactory process plans/schedule solution is derived. in this way, the schedule is improved while the process plans still possess high quality in terms of the process planning objective. to ensure effective coordination and communication for the involved agents, the system implementation employs the architecture based on a multi-tier application model and an agent development framework. keywords process planning, scheduling, agent. 1 introduction effective process planning and scheduling is increasingly becoming a core component of the manufacturing system. process planning is the task of transforming a product design into a set of instructions in sequence while scheduling is the task of allocating resources over time to perform a collection of activities. traditionally, they are treated as two separate tasks, between which the generated gap will cause several problems. firstly, since the process planning is usually done before the process plan is executed, the process plan may not be applicable when the job is dispatched in an overall schedule due to change of resource availability. secondly, in process planning, it is often not possible to use the knowledge of the actual situation in the shop floor condition, thus resulting in the lower overall resource utilization and poor on-time delivery performance. moreover, the dynamic shop floor is full of different kinds of disruptions, which will easily disturb the process plans and schedule being executed infeasible. thus, in order to maintain the optimum and feasibility of the process plans and schedule, the collaboration between the process planning and scheduling becomes more and more necessary. on the advances in systems science and applications (2014) vol.14 no.1 37 other hand, industrial growth is increasingly dependent on situations where large businesses are distributed. to account for these requirements, it is possible to be realized by the agent technology. agent technology derived from distributed artificial intelligence is increasingly being recognized as a promising paradigm for collaborative manufacturing systems. the term “agen” has been defined as“a computer system situated in some environment and capable of autonomous action in this environment, in order to meet its design objectives”[1],with the properties of autonomy, social ability, reactivity, and pro-activeness [2]. a group of such agents form a multi-agent system (mas), where the participated agents can communicate, cooperate, and interact with each other to achieve their individual objectives as well as global goals of the system [2]. other advantages of the agent-based system include scalability, modularity and re-configurability. in this paper, we present an agent-based distributed process planning and scheduling system (adpps), which consists of several functional agents. each agent is able to conquer its individual local problems while accomplishing the global tasks through the corporation with other agents. the overall aim is to improve the performance of the schedule while minimum process planning cost is also desirable. with the help of an optimization algorithm, the process planning agent is able to generate a high quality of process plan for the designed part according to the predefined objective functions. the scheduling agent is capable of generating a schedule based on the involved process plans and the available resources in the shop floor. the facilitator agent plays a coordination role among the different agents to iteratively achieve the global objectives. such a system is capable of balancing the machine utilization and reducing the tardy jobs for the production of prismatic parts [3,4]. in addition, it also has the capability to deal with machine breakdown scenario that commonly occurs in the production [5]. 2 related work since the early 1990s, agent technology has been applied to develop the intelligent manufacturing system by negotiation, coordination, and cooperation. due to the page limitation, only some typical examples in distributed collaborative systems are investigated. shade [6] proposed the information sharing and decision coordination approach for integrated collaborative manufacturing. ciimplex [7] presented an agent-based system for intelligent enterprise integration for manufacturing planning and execution. metamorph ii [8] developed a hybrid agentbased mediator-centric architecture to integrate process planning and scheduling through the higher level coordination within a supply chain network. jia et al. [9] demonstrated an agent based system for coordinated product development and manufacture. wong et al. [10] implemented a hybrid-based multi-agent system 38 y.f. wang: an agent-based distributed process planning and scheduling system for integrating process planning with scheduling in job shops, where each of the jobs and machines is modeled as a local agent. one limitation for this approach is that with the increasing number of resources in reality, the amount of agents will also increase, leading to the overhead communication. the functional decomposition approach adopted in our work can overcome this problem, where each manufacturing activity is encapsulated as an intelligent and autonomous agent. such an agent-based encapsulation approach can greatly improve the integration of heterogeneous software components, which also facilitates the coordination and cooperation among geographically dispersed departments. 3 a proposed agent-based distributed process planning and scheduling system 3.1 system overview fig.1 lpm system layout. fig.1 presents the framework for the agent-based distributed process planning and scheduling system, which consists of six kinds of agents: resource agent (ra), production monitor agent (pma), process identification agent (pia), process planning agent (ppa), scheduling agent (sa), and facilitator agent (fa). these functional agents can be configured in different computers located in geographically different places. all the agents cooperate with each other to achieve a common goal through the internet. in recent years, extensible markup language advances in systems science and applications (2014) vol.14 no.1 39 (xml) has become an increasingly accepted language for internet-based applications, which provides the benefit of neutral data representation. in compliance with this trend, xml technology is used to encapsulate, extract, and transfer the knowledge among the participated agents. the functionality of each agent is described in the following sections. 3.1.1 resource agent (ra) the ra plays a fundamental role in the distributed manufacturing system, which is able to retrieve the resource data for the other agents according to their demands. it is also responsible for managing the resource information in the shop floor, including the specification of machines and tools such as maximum size, achievable accuracy, and surface finish, etc., as well as their manufacturing capabilities. in order to ensure the data consistency, the resource information is dynamically updated. when a machine breaks down, the ra updates its status to be unavailable. similarly, when it gets recovered, its status turns to be available again. by connecting to each resource entity in the production, ra sends the working status of the current resources to the production monitor agent so as to react the unexpected disruptions. 3.1.2 process identification agent (pia) the pia is used to identify the process information for an input part and the stock used. it follows two steps. firstly, a feature extraction algorithm [11] is used to partition the delta volume (difference between the stock and part) into a set of volumetric features (vfs), each of which can be removed by a set of operations (e.g. a cylinder can be removed by central drilling + drilling) along one or several tool approach directions (tads). secondly, after vf extraction, the process information for all vfs is identified based on the available machining resources managed by the ra. for each vf, all the operation types (opts) are determined based on the feature type and the technological requirements (tolerance and surface finish). for instance, to remove a cylinder with certain accuracy requirement, either “central drilling + drilling + boring” or “drilling + milling” can be selected. therefore, each vf could lead to several sets of opts and each may have different number of opts. to make every opt set for a vf having the same number of opts, the concept of dummy opt is introduced, which incurs no manufacturing cost and imposes no precedence relationships with other opts. up to this point, each vf corresponds to a number of opt sets, each having the same number of opts. for each opt, all the feasible operation methods (opms) are formed by the combination of specific machine (m), cutter (t), and tool approach direction (tad). furthermore, the precedence relationship between different opts is also generated based on fixture constraints, datum dependence, and knowledge of good manufacturing practice. fig.2 presents the hierarchical representation of the generated process information. in order to transfer this pro40 y.f. wang: an agent-based distributed process planning and scheduling system cess information using xml file, we define the following xml format according to the presented hierarchical representation: <?xml version= ‘ ‘1 .0 ’ ’ ?> <part pa r t i d = ‘ ‘ part 32 ’ ’> <vo l ume t r i c f e a tu r e f e a t u r e i d = ‘ ‘vf 7 ’ ’> <ope ra t i on type op t id = ‘ ‘opt 7 ’ ’> <machines>m1,m2,m3,m4,m5</machines> <t o o l s>t14</ t o o l s> <tads>+z ,−z</ tads> <pr ede c e s s o r s></ p r ede c e s s o r s> </ ope ra t i on type> <ope ra t i on type op t id=”opt 8”> <machines>m1,m2,m3,m4,m5</machines> <t o o l s>t11</ t o o l s> <tads>+z ,−z</ tads> <pr ede c e s s o r s>opt 7</ p r ede c e s s o r s> </ ope ra t i on type> </ vo l ume t r i c f e a tu r e> <vo l ume t r i c f e a tu r e> </ vo l ume t r i c f e a tu r e> </ part> fig.2 the hierarchical representation of process information based on an input part and stock used. advances in systems science and applications (2014) vol.14 no.1 41 3.1.3 production monitor agent (pma) due to the dynamic and stochastic characteristic in nature, a job shop production always faces different kinds of uncertainties (e.g. machine breakdown and rush order), leading to a predictive schedule not applicable. thus, in order to minimize the impact of the disruption on the production, we need to quickly react to these disturbances and revise the schedule in a cost-effective manner. the task of pma is to monitor any disruption occurred in the production. once a disruption is detected, a pre-processing procedure is triggered to pre-process the disruption so that the rescheduling procedure can further handle it. generally, with respect to the different disruptions, the pre-processing procedure will perform in different ways. for instance, when a rush order comes, it will inform the ppa to generate a feasible process plan for the input part based on the current available machines. when a machine breaks down, it will identify the affected jobs, which are to be processed on this breakdown machine, and then inform the ppa to generate new process plans for them. note that the stability of the schedule will be maintained and measured in terms of the sequence deviation when revising the disturbed schedule. 3.1.4 process planning agent (ppa) the ppa is used to obtain a high quality of process plan for a part, with its input process information from the pia. the objective of ppa is to select a set of opms for each vf and place them into an ordered sequence such that the sequence satisfies the precedence constraints and process plan achieves the predefined objective function. obviously, process planning problem is a combinatorial optimization problem. typically, the criteria for process plan evaluation include minimum number of setups, shortest processing time, minimum manufacturing cost, etc. from the economic point of view, the minimum manufacturing cost is taken as the objective function in this work, which can be considered from the following five cost aspects. (1) machine cost (mc) mc = n∑ i=1 mcii (1) where n is the total number of opts and mcii is the cost index for the machine used to perform opti, which is a constant for a particular machine. (2) tool cost (tc) tc = n∑ i=1 tcii (2) where tcii is the tool cost index for the tool used to perform opti, which is a constant for a particular tool. 42 y.f. wang: an agent-based distributed process planning and scheduling system (3) machine change cost (mcc): a machine change cost is required when two adjacent opts are performed on different machines. mcc = mcci n−1∑ i=1 (1− ω(mi+1,mi)) (3) ω(x, y) = { 1 if x = y 0 otherwise (4) where mcci is the machine change cost index and mi is the identity of the machine used to perform opti. (4) set-up change cost (scc): a setup change cost is required when two adjacent opts performed on the same machine have different tads. scc = scci n−1∑ i=1 ω(mi+1,mi)(1− ω(tadi+1, tadi)) (5) where scci is the setup change cost index. (5) tool change cost (tcc): a tool change cost is required when two adjacent opts performed on the same machine use different tools tcc = tcci n−1∑ i=1 ω(mi+1,mi)(1− ω(ti+1, ti)) (6) where tcci is the tool change cost index. the above five cost items can be taken either individually or collectively as a cost compound based on the actual requirement and data availability of the job shop. in this paper, all these five items are considered in the objective function, i.e., the total manufacturing cost (tmc): tcm = mc + tc +mcc + scc + tcc (7) two steps are followed to find an optimal process plan. the first step is to generate the process plan solution space of the input part, formed by all feasible opms that can be used for the fabrication of the part subject to the precedence relationship. the second step is to find the best solution based on a given optimization objective, i.e., minimum manufacture cost, as formulated in eq.(7). due to its vast solution space, it is difficult to find the optimal solution for this combination problem in a reasonable amount of time. two optimization methods based on genetic algorithm (ga) and simulated annealing algorithm (saa), respectively, have been developed to resolve this intractable problem and promising results were reported [12-13]. recently, we have developed an alternative algorithm advances in systems science and applications (2014) vol.14 no.1 43 based on particle swarm optimization [14]. particle swarm optimization (pso), being one of evolutionary computation approaches, was firstly proposed by kennedy and eberhart [15]. it is a class of population-based optimization algorithm that imitates the social swarm behaviours. members in the population interact with one another by learning from their own experience and gradually individuals move into better regions of the problem space. the attractive features of pso include inexpensive computation, individual improvement, and the ability of effective exploration and exploitation search. due to the characteristic of discrete process planning solution space and the continuous nature of the original pso, a novel solution representation scheme is introduced for the application of pso in solving the process planning problem. moreover, a local search algorithm is incorporated and interweaved with pso evolution to improve the best solution in each generation. the numerical experiments and analysis have demonstrated that the proposed algorithm (pso-ls) is capable of gaining a good quality of solution in an efficient way. one example optimal process plan in terms of xml format is shown below: <?xml version= ‘ ‘1 .0 ’ ’ ?> <part pa r t i d = ‘ ‘ part 32 ’ ’> <ope ra t i on type op t id=”opt 18”> <sequence>1</ sequence> <opm>m1,t1,+x</opm> </ ope ra t i on type> <ope ra t i on type op t id=”opt 1”> <sequence>2</ sequence> <opm>m1,t1,+x</opm> </ ope ra t i on type> <ope ra t i on type> </ ope ra t i on type> </ part> 3.1.5 scheduling agent (sa) the sa is used to generate a high quality of schedule based on the involved process plans from the different ppas, each specified with the information of weight (1-10), due date, and batch size. a set of dispatching rules [16], including earliest due date (edd), shortest processing time (spt), are developed for the generation of schedule. since the scheduling problem is a typical non-polynomial (np) hard problem, the quality of solution obtained by dispatching rules may not be good. we therefore developed an approximation algorithm by incorporating 44 y.f. wang: an agent-based distributed process planning and scheduling system a tabu search in the pso for minimizing the tardiness in a flexible job shop scheduling. by this integration, pso provides a diversity of initial solutions for the tabu search while tabu search performs an exploitation search so as to affect the particles swarm search behavior. this hybrid procedure, taking the advantage of the difference between these two algorithms, proves to be an effective algorithm. the proposed algorithm (pso-ts) has been tested on various experiments and the results have demonstrated its robustness, efficiency and efficacy in all sets of experiments. 3.1.6 facilitator agent (fa) the objective of the fa is to improve the performance measure of the schedule through the coordination with the other agents. the fa evaluates the schedule obtained from sa, followed by invoking a heuristic rule, which is able to automatically issue a modification suggestion by identifying a particular operation of one job from the scheduled jobs and the resource to be modified. although the rule can choose several jobs for process plan modification, the strategy employed here is for fine-tuning, thus only one job is chosen in each round of iteration. the generated suggestion will be fed back to the corresponding ppa that the job belongs to and impose the constraints on its solution space. based on the update solution space, the affected ppa will then re-generate a process plan using an optimization algorithm. this generated optimal process plan, together with unchanged process plans, forms a new schedule in the sa. the coordination process continues until a satisfactory result is achieved. it is noted that the heuristic rules are in association with the optimization scenario and thus performs in different ways. currently, the system is capable of balancing the machine utilization and minimizing the number of tardy jobs and their tardiness, as well as accommodating the disruption of machine breakdown. the carried out simulation results have proven its effectiveness to achieve a satisfactory plans/schedule solution in a reasonable amount of time [3-5]. 3.2 agent structure in a multi-agent system, each agent is essentially an autonomous cognitive entity, which is able to communicate, reason, and react to the events from the external environment. to start, the agent receives a message and stores it in the receiving message queue. by decoding a message from the message list, an agent resolves the problem using its domain knowledge. once the problem is resolved, the agent will encode the solutions in a well-defined message and return it to the sender agent. the internal structure of an agent typically includes the following components: (1) network interface: it couples the agent to the network. (2) communication interface: it enables the agent to exchange and understand the messages. advances in systems science and applications (2014) vol.14 no.1 45 (3) interaction interface: it allows the agent to have conversation with other agents. (4) domain knowledge: it provides enterprise expertise knowledge required to perform the specific task. (5) problem solving module: owning the reasoning and decision making capability, it enables the agent to resolve the problem according to the received message and domain knowledge. 3.3 system architecture fig.3 presents the system architecture for the agent-based distributed process planning and scheduling system, which is based on the multi-tier application model, consisting of presentation tier, business tier, and eis (enterprise information system) tier. moreover, in order to ensure the effective coordination and communication among the involved agents, each agent sits on a platform based on jade (java agent development framework), which is a software framework to develop distributed agent-based applications (http://jade.tilab.com). in the presentation tier, a user can operate one or more agents to accomplish the required tasks in the local machine through the graphic user interfaces (gui). depending on the role of the user, one may have different privileges to access different kinds of agents. as such, these agents configured at geographically dispersed locations forms a peer-to-peer network, where the loosely coupled agents can communicate and cooperate with each other to fulfill the task. being located at the jade platform, each agent has a lifecycle. some agents are shut down when its behavior is completed while some are always active to repetitively execute the specific task. to manage these agents, a directory facilitator agent (dfa) is used to search and modify the description of registered agents. when an agent starts up (shuts down), it will be registered (deregistered) to the dfa. the business-tier located on the server side provides a set of functionality for business logic processing. it comprises two parts. the first part is to create an instance of runtime container for all active agents based on jade platform. the agent in the container is taken as an independent and autonomous process with a unique identity. in order to fulfil the task, it requires communicating with other agents. such a communication is realized through asynchronous message passing with an agent communication language (acl) in terms of a well-defined semantics. to implement it, some kernel packages are utilized, including jade core, jade acl, jade content, jade domain, jade mtp (message transport protocol), and jade protocol. however, the modeled agents cannot perform the specific task, since they are not endowed with capabilities except those of communication and interaction. we therefore develop different kinds of algorithms placed in the business tier, which are eligible for each agent to invoke in order to accomplish the assigned task. note that the feature extraction algorithm is written in c++ 46 y.f. wang: an agent-based distributed process planning and scheduling system fig.3 system architecture language. since the framework is developed based on the java virtual machines (jvm) and remote method invocation (rmi), agents cannot directly invoke this algorithm. we therefore utilize the java native interface (jni) technology, which allows a java application running in the jvm to operate with other applications or libraries written in other languages. the eis-tier plays a critical role in providing information infrastructure to the business process of an enterprise, which may include data integration, the existing application system integration, and legacy system integration. in this work, we only concern about the data integration, focusing on integrating the existing data with the developed system. in this way, the business process and data can be easily shared. the connection to the data stored in a relational database is realized by the jdbc technology. based on this architecture, the agents in the client tier communicate and exchange the data with the eis-tier through business tier while the business-tier functions manipulate the data from eis-tier and client-tier. by logically separating the presentation tier, the business tier, and the eis tier, the presented architecture brings the following benefits: (1) it only needs to define the business logic once within the business tier, which can be shared by any agent in the presentation tier. (2) it is possible to change the contents of any tier without having to make corresponding changes in the others. (3) it enables parallel development of the different tiers for a complex application. advances in systems science and applications (2014) vol.14 no.1 47 4 system implementation a prototype system has been developed to realize the integration of distributed process planning and scheduling based on the proposed architecture, using java language. mysql database is used to manage the production entities so as to ensure timely information sharing and maintain the data consistency among different functionalities. moreover, database connection pool is designed to improve the performance of the data retrieval and manipulation. in this way, users in geographically dispersed departments are able to cooperate with each other in a distributed, transactional, and portable environment. fig.4 illustrates the guis of the ppa, sa, and fa. fig.4 system interfaces of the process planning agent, scheduling agent, and facilitator agent. 5 conclusion in this paper, an interoperable intelligent multi-agent system is implemented to realize the integration of process planning and scheduling in a distributed manner. since both process planning and scheduling are known to be np-hard in nature, the combined solution space therefore makes it more difficult to find a 48 y.f. wang: an agent-based distributed process planning and scheduling system good solution. we therefore developed an intuitive approach based on the agent technology. through the iterative coordination among the modelled functional agents, the user delivery requirement will be finally satisfied while the costs of process plans are maintained as low as possible. to develop the prototype system, multi-tier system architecture and the jade platform, taking the advantage of its flexibility, scalability, reusability, and interoperability, were employed. although the prototype system based on the agent technology has gained certain success to find a satisfactory solution, more efforts still need to realize its applicability to the actual manufacturing environment. firstly, the functionalities in a manufacturing system usually interact with each other to give a better product design and achieve a higher production efficiency. likewise, the design process and shop floor control, which are respectively performed before the process planning and after the scheduling, have a significant impact on the quality of the obtained process plans and schedule. thus, to achieve a more robust solution, it would be better to incorporate the design agent and the shop floor agent. secondly, due to various unexpected events in the actual manufacturing, future work is expected to extend its capability to handle more disruptions. moreover, to meet different users requirement, more performance measures in the scheduling agent should be included. references [1] [1] jennings, n. r., and wooldridge, m. 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(1974), “introduction to sequencing and scheduling”, new york: wiley publications. corresponding author author can be contacted at: mpefuhyh@nus.edu.sg advances in systems science and applications (2013) vol.13 no.2 182-197 finding unbiased simultaneous prediction limits for order statistics of future samples with applications nicholas a. nechval1, konstantin n. nechval2, gundars berzins1, juris krasts1, maris purgailis1, uldis rozevskis1 and vladimir f. strelchonok3 1 statistics department, evf research institute, university of latvia, raina blvd 19, lv-1050 riga, latvia 2 applied mathematics department, transport and telecommunication institute, lomonosov street 1, lv-1019 riga, latvia 3 informatics department, baltic international academy, lomonosov street 4, lv-1019 riga, latvia abstract this paper provides procedures for finding unbiased simultaneous prediction limits on the observations or functions of observations of all of k future samples using the results of a previous sample from the same underlying distribution belonging to invariant family. the results have direct application in reliability theory, where the time until the first failure in a group of several items in service provides a measure of assurance regarding the operation of the items. the simultaneous prediction limits are required as specifications on future life for components, as warranty limits for the future performance of a specified number of systems with standby units, and in various other applications. prediction limit is an important statistical tool in the area of quality control. the lower simultaneous prediction limits are often used as warranty criteria by manufacturers. the initial sample and k future samples are available, and the manufacturer wants to have a high assurance that all of the k future orders will be accepted. it is assumed throughout that k + 1 samples are obtained by taking random samples from the same population. in other words, the manufacturing process remains constant. the results in this paper are generalizations of the usual prediction limits on observations or functions of observations of only one future sample. in the paper, attention is restricted to invariant families of distributions. the technique used here emphasizes pivotal quantities relevant for obtaining ancillary statistics and is applicable whenever the statistical problem is invariant under a group of transformations that acts transitively on the parameter space. applications of the proposed procedures are given for the two-parameter exponential and weibull distributions. the exact prediction limits are found and illustrated with a numerical example. keywords future samples, order statistics, simultaneous prediction limits 1 introduction statistical intervals used by engineers and others include confidence intervals on a population parameter, such as the mean, and tolerance intervals. confidence intervals give information about parameter of the population or a function of 183 advances in systems science and applications (2013) vol.13 no.2 population parameters such as a percentile; tolerance intervals give information about a region which contains a specified proportion of a population. often one desires to construct from the results of a previous sample an interval which will have a high probability of containing the values of all of k future observations. for example, such an interval would be required in establishing limits on the values of some performance variable for a small shipment of equipment when the satisfactory performance of all units is to be guaranteed, or in setting acceptance limits on a specific lot of material, when acceptance requires the values of all items in a future sample to fall within the limits. an interval which contains the values of a specified number of future observations with a specified probability is known as a prediction interval. such an interval need be distinguished both from a confidence interval on an unknown distribution parameter, and from a tolerance interval to contain the values of a specified proportion of the population. research works on prediction intervals related to a single future statistic are abundant (see hahn and meeker [1], patel [2], and references therein). in many situations of interest, it is desirable to construct lower simultaneous prediction limits that are exceeded with probability γ by observations or functions of observations of all of k future samples, each consisting of m units. the prediction limits depend upon a previously available complete or type ii censored sample from the same distribution. for instance, two situations where such limits are required are: 1. a customer has placed an order for a product which has an underlying timeto-failure distribution. the terms of his purchase call for k monthly shipments. from each shipment the customer will select a random sample of m units and accept the shipment only if the smallest time to failure for this sample exceeds a specified lower limit. the manufacturer wishes to use the results of a previous sample of n units to calculate this limit so that the probability is γ that all k shipments will be accepted. it is assumed that the n past units and the km future units are random samples from the same population. 2. a system consists of n identical components whose times to failure follow an underlying distribution. initially one component is operating and the remaining n-1 components are in a standby mode; a new component goes into operation as soon as the preceding component has failed. the system is said to fail when all n components have failed. thus, the system time to failure is the total of the failure times for the n components. a simultaneous lower prediction limit to be exceeded with probability γ by the system time to failure of all of k future systems is desired. this limit is to be calculated from the times to failure of n previously tested components. similar problems also arise in various product maintenance and servicing problems. nicholas a. nechval : finding unbiased simultaneous prediction limits for order ... 184 prediction limits can be of several forms. hahn [3] dealt with simultaneous prediction limits on the standard deviations of all of the k future samples from a normal population. hahn [4] considered the problem of obtaining simultaneous prediction limits on the means of all of k future samples from an exponential distribution. in addition, hahn and nelson [5] discussed such limits and their applications. mann, schafer, and singpurwalla [6] gave an interval that contains, with probability γ, all m observations of a single future sample from the same population. fertig and mann [7] constructed prediction intervals to contain at least m − k + 1 out of m future observations from a normal distribution with probability 1− β. they considered life-test data, and the performance variate of interest is the failure time of an item. their lower prediction limit constitutes a “warranty period”. in this paper we give an expression for obtaining unbiased simultaneous prediction limits on order statistics of all of k future samples. in order to obtain the unbiased simultaneous prediction limits, attention is restricted to invariant families of distributions. in particular, the case is considered where a previously available complete or type ii censored sample is from a continuous distribution with cumulative distribution function (cdf) f ((x−µ)/σ) and probability density function (pdf) 1/σf((x − µ)/σ), where f (.) is known but both the location (µ) and scale (σ) parameters are unknown. for such family of distributions the decision problem remains invariant under a group of transformations (a subgroup of the full affine group) which takes µ (the location parameter) and σ (the scale) into cµ+ b and cσ, respectively, where b lies in the range of µ, c > 0. this group acts transitively on the parameter space and, consequently, the risk of any equivariant estimator is a constant. among the class of such estimators there is therefore a “best” one. the effect of imposing the principle of invariance, in this case, is to reduce the class of all possible estimators to one. in the present paper we investigate this question for the problem of constructing the unbiased simultaneous prediction limits on order statistics in future samples. the technique used here emphasizes pivotal quantities relevant for obtaining ancillary statistics. it is a special case of the method of invariant embedding of sample statistics into a performance index [8-11] applicable whenever the statistical problem is invariant under a group of transformations which acts transitively on the parameter space (i.e., in problems where there is a unique best invariant procedure). the exact unbiased simultaneous prediction limits on order statistics of all of k future samples are obtained via the technique of invariant embedding and illustrated with numerical example. 185 advances in systems science and applications (2013) vol.13 no.2 2 mathematical preliminaries the main theorem, which shows how to construct lower (upper) simultaneous prediction limit for the order statistics in all of k future samples when prediction limit for a single future sample is available, is given below. theorem 1 (lower (upper) simultaneous prediction limit under complete information). let (y1j , ..., ymj) be the jth random sample of mj “future” observations from the cdf fθ(.), where θ is the parameter (in general, vector), j ∈ 1, ..., k, and let y(rj ,mj) denote the rjth order statistic in the jth sample of size mj . assume that all of k samples from the same cdf are independent. then a lower simultaneous (1− α) prediction limit h on the rjth order statistics y(rj ,mj), j = 1, , k, of all of k future samples may be obtained from pθ{y(r1,m1) ≥ h, ..., y(rj ,mj) ≥ h, ..., y(rk,mk) ≥ h} = r1−1∑ i1=0 ... rj−1∑ ij=0 ... rk−1∑ ik=0 ( m1 i1 ) ... ( mj ij ) ... ( mk ik ) × pθ{y(iς+1,mς) ≥ h} − pθ{y(iς,mς) ≥ h}( mς iς ) = 1− α (1) where iς = k∑ j=1 ij ,mς = k∑ j=1 mj (2) (observe that an upper simultaneous α prediction limit h may be obtained from a lower simultaneous prediction limit by replacing 1− α by α.) proof. we have: pθ{y(r1,m1) ≥ h, ..., y(rj ,mj) ≥ h, ..., y(rk,mk) ≥ h} = k∏ j=1 pθ{y(rj ,mj) ≥ h} = k∏ j=1 rj−1∑ ij=0 ( mj ij ) [fθ(h)] ij [1− fθ(h)] mj−ij = r1−1∑ i1=0 ... rj−1∑ ij=0 ... rk−1∑ ik=0 ( m1 i1 ) ... ( mj ij ) ... ( mk ik ) [fθ(h)] iς [1− fθ(h)] mς−iς (3) since nicholas a. nechval : finding unbiased simultaneous prediction limits for order ... 186 [fθ(h)] iς [1− fθ(h)] mς−iς = ( mς iς )−1 [ iς∑ i=0 ( mς i ) [fθ(h)] i[1− fθ(h)] m∑−i− iς−1∑ i=0 ( mς i ) [fθ(h)] i[1− fθ(h)] mς−i ] = pθ{y(iς+1,mς) ≥ h} − pθ{y(iς,mς) ≥ h}( mς iς ) (4) the joint probability can be written as pθ{y(r1,m1) ≥ h, ..., y(rj ,mj) ≥ h, ..., y(rk,mk) ≥ h} = r1−1∑ i1=0 ... rj−1∑ ij=0 ... rk−1∑ ik=0 ( m1 i1 ) ... ( mj ij ) ... ( mk ik ) × pθ{y(iς+1,mς) ≥ h} − pθ{y(iς,mς) ≥ h}( mς iς ) (5) this ends the proof. corollary 1.1 if rj = 1, ∀j = 1(1)k, then pθ{y(1,m1) ≥ h, ..., y(1,mj) ≥ h, ..., y(1,mk) ≥ h} = pθ{y(1,mς) ≥ h} = 1− α (6) theorem 2 (lower (upper) unbiased simultaneous prediction limit under parametric uncertainty). let (x1 ≤ ... ≤ xr) be the r smallest observations in a random sample of size n from the cdf fθ(.), where the θ is the parameter (in general, vector), and let (y1j , ..., ymj ) be the jth random sample of mj “future” observations from the same cdf, j ∈ {1, ..., k}. assume that (k + 1) samples are independent and the parameter θ is unknown. let h = h(x1, ..., xr) be any statistic based on the preliminary sample and let y(rj ,mj) denote the rjth order statistic in the jth sample of size mj . then an unbiased lower simultaneous (1− α) prediction limit h on the rjth order statistics y(rj ,mj), j = 1, , k, of all of k future samples may be obtained from 187 advances in systems science and applications (2013) vol.13 no.2 eθ { pθ{y(r1,m1) ≥ h, ..., y(rj ,mj) ≥ h, ..., y(rk,mk) ≥ h} } = r1−1∑ i1=0 ... rj−1∑ ij=0 ... rk−1∑ ik=0 ( m1 i1 ) ... ( mj ij ) ... ( mk ik ) · eθ { pθ{y(iς+1,mς) ≥ h} } − eθ { pθ{y(iς,mς) ≥ h} }( mς iς ) (7) proof. for the proof we refer to theorem 1. corollary 2.1. if rj = 1, ∀j = 1(1)k, then eθ { pθ{y(1,m1) ≥ h, ..., y(1,mj) ≥ h, ..., y(1,mk) ≥ h} } = eθ { pθ{y(1,mς) ≥ h} } = 1− α (8) remark. in this paper, in order to find the unbiased lower simultaneous (1−α) prediction limith on the rjth order statistics y(rj ,mj), j = 1, ..., k, of all of k future samples, the technique of invariant embedding [8-11] is used. 2.1 weibull distribution in this paper, the two-parameter weibull distribution with the pdf fθ(x) = δ β ( x β )δ−1exp [ −( x β )δ ] , x > 0, β > 0, δ > 0 (9) indexed by scale and shape parameters β and δ is used as the underlying distribution of a random variable x in a sample of the lifetime data, where θ = (β, δ). we consider both parameters β, δ to be unknown. let (x1, ..., xn) be a random sample from the two-parameter weibull distribution (9), and let β̂, δ̂ be maximum likelihood estimates of β, δ computed on the basis of (x1, ..., xn). in terms of the weibull variates, we have that v1 = ( β̂ β )δ, v2 = δ δ̂ , v3 = ( β̂ β )δ̂ (10) are pivotal quantities. further more, let zi = (xi/β̂) δ̂, i = 1, ..., n (11) it is readily verified that any n-2 of the zi’s, say zi, ..., zn−2 form a set of n-2 functionally independent ancillary statistics. the appropriate conditional approach, first suggested by fisher [12], is to consider the distributions of v1, v2, v3 nicholas a. nechval : finding unbiased simultaneous prediction limits for order ... 188 conditional on the observed value of z(n) = (zi, ..., zn). (for purposes of symmetry of notation we include all of (zi, ..., zn) in expressions stated here; it can be shown that zn, zn−1, can be determined as functions of zi, ..., zn−2 only.) theorem 3. (joint pdf of the pivotal quantities v1, v2 from the two-parameter weibull distribution) let (x1 ≤ ... ≤ xr) be the first r ordered observations from a sample of size n from the two-parameter weibull distribution (9). then the joint pdf of the pivotal quantities v1 = ( β̂ β )δ, v2 = δ δ̂ (12) conditional on fixed z(r) = (zi, ..., zr) (13) where zi = ( xi β̂ )δ̂, i = 1, ..., r (14) are ancillary statistics, any r-2 of which form a functionally independent set,β̂ and δ̂ are the maximum likelihood estimates for β and δ based on the first r ordered observations (x1 ≤ ... ≤ xr) from a sample of size n from the two-parameter weibull distribution (9), which can be found from solution of β̂ = ([ r∑ i=1 xδ̂i + (n− r)xδ̂r ] /r )1/δ̂ (15) and δ̂ = ( r∑ i=1 xδ̂ i lnxi + (n− r)xδ̂ rlnxr )( r∑ i=1 xδ̂ i + (n− r)xδ̂ r )−1 − 1 r r∑ i=1 lnxi −1 (16) is given by f(v1, v2|z(r)) = ϑ•(z(r))vr−2 2 r∏ i=1 zv2i vr−1 1 exp ( −v1 [ r∑ i=1 zv2i + (n− r)zv2r ]) = f(v2|z(r))f(v1|v2, z(r)), v1 ∈ (0,∞), v2 ∈ (0,∞) (17) where ϑ•(z(r)) = [∫ ∞ 0 γ(r)vr−2 2 r∏ i=1 zv2i ( r∑ i=1 zv2i + (n− r)zv2r )−r dv2 ]−1 (18) 189 advances in systems science and applications (2013) vol.13 no.2 is the normalizing constant, f(v2|z(r)) = ϑ(z(r))vr−2 2 r∏ i=1 zv2i ( r∑ i=1 zv2i + (n− r)zv2r )−r , v2 ∈ (0,∞) (19) ϑ(z(r)) = [∫ ∞ 0 vr−2 2 r∏ i=1 zv2i ( r∑ i=1 zv2i + (n− r)zv2r )−r dv2 ]−1 (20) f(v1|v2, z(r)) = [ ∑r i=1 z v2 i + (n− r)zv2r ]r γ(r) vr−1 1 exp ( −v1 [ r∑ i=1 zv2i + (n− r)zv2r ]) = 1 γ(r) ( v1 [ r∑ i=1 zv2i + (n− r)zv2r ])r−1 exp ( −v1 [ r∑ i=1 zv2i + (n− r)zv2r ]) ×[ r∑ i=1 zv2i + (n− r)zv2r ] , v1 ∈ (0,∞) (21) proof. the joint density of x1 ≤ ... ≤ xr is given by fθ(x1, ..., xr) = n! (n− r)! r∏ i=1 δ β ( xi β )δ−1exp(−( xi β )δ)exp(−(n− r)( xr β )δ) (22) using the invariant embedding technique [8-11], we transform (22) to fθ(x1, ..., xr)dβ̂dδ̂ = n! (n− r)! r∏ i=1 x−1 i δr r∏ i=1 ( xi β )δexp ( − r∑ i=1 ( xi β )δ − (n− r)( xr β )δ ) dβ̂dδ̂ = n! (n− r)! β̂δ̂r r∏ i=1 x−1 i ( δ δ̂ )r−2 r∏ i=1 ( xi β̂ )δ̂( δ δ̂ )( β̂ β )δ(r−1)× exp ( −( β̂ β )δ [ r∑ i=1 ( xi β̂ )δ̂( δ δ̂ ) + (n− r)( xr β̂ )δ̂( δ δ̂ ) ])( δ β ( β̂ β )δ−1dβ̂ ) (− δ δ̂2 dδ̂) = n! (n− r)! β̂δ̂r r∏ i=1 x−1 i vr−2 2 r∏ i=1 zv2i vr−1 1 exp ( −v1 [ r∑ i=1 zv2i + (n− r)zv2 r ]) dv1dv2 (23) normalizing (23), we obtain (17). this ends the proof. theorem 4. (lower (upper) unbiased prediction limit h for the lth order statistic yl in a new (future) sample of m observations from the two-parameter weibull distribution on the basis of the preliminary data sample) let x1 ≤ ... ≤ xr be the first r ordered observations from the preliminary sample of size n from the nicholas a. nechval : finding unbiased simultaneous prediction limits for order ... 190 two-parameter weibull distribution (9). then a lower unbiased (1−α) prediction limit h on the lth order statistic yl from a set of m future ordered observations y1 ≤ ... ≤ ym also from the distribution (9) is given by h = arg[eθ{pθ{y1 ≥ h}|z(r)} = 1− α] = z 1/δ̂ h β̂ (24) where eθ{pθ{yl ≥ h}|z(r)} = ∞∫ 0 vr−2 2 r∏ i=1 zv2 i l−1∑ k=0 ( m k ) k∑ j=0 ( k j ) (−1) j ( (m− k + j)zv2h + r∑ i=1 zv2 i + (n− r)zv2r )−r dv2 ∞∫ 0 vr−2 2 r∏ i=1 zv2i ( r∑ i=1 zv2i + (n− r)zv2 r )−r dv2 (25) zh = ( h β̂ )δ̂ (26) zi = (xi/β̂) δ̂, i = 1, ..., r; β̂ and δ̂ are the maximum likelihood estimates for β and β based on the first r ordered observations (x1 ≤ ... ≤ xr) from a sample of size n from the two-parameter weibull distribution (9). (observe that an upper unbiased α prediction limit h on the lth order statistic yl from a set of m future ordered observations y1 ≤ ... ≤ ym may be obtained from a lower unbiased (1− α) prediction limit by replacing 1− α by α.) proof. if there is a random sample of m ordered observations y1 ≤ ... ≤ ym from the two-parameter weibull distribution (9) with the pdf fθ(y) and cdf fθ(y), then for the lth order statistic yl we have pθ{yl ≥ h} = l−1∑ k=0 ( m k ) [fθ(h)]k[1− fθ(h)]m−k = l−1∑ k=0 ( m k )[ 1− exp ( − ( h β )δ )]k[ exp ( − ( h β )δ )]m−k (27) writing (27) as 191 advances in systems science and applications (2013) vol.13 no.2 pθ{yl ≥ h} = l−1∑ k=0 ( m k )[ 1− exp ( − ( h β )δ )]k exp ( −(m− k) ( h β )δ ) = l−1∑ k=0 ( m k )1− exp − ( h ⌢ β )⌢ δ ( δ ⌢ δ )(⌢ β β )δ   k exp −(m− k) ( h ⌢ β )⌢ δ ( δ ⌢ δ )(⌢ β β )δ  = l−1∑ k=0 ( m k ) [1− exp(−zv2 h v1)] k exp(−(m− k)zv2 h v1) = l−1∑ k=0 ( m k ) k∑ j=0 ( k j ) (−1)j exp[−v1(m− k + j)zv2h ] = p{zl > zh |v1, v2} (28) where zl = ( yl ⌢ β )⌢ δ (29) we have from (17) and (28) that eθ{pθ{yl ≥ h}|z(r)} = e{p{zl ≥ zh |v1, v2}|z(r)} = ∞∫ 0 ∞∫ 0 p{zl ≥ zh |v1, v2}f(v1, v2|z(r))dv1dv2 (30) now v1 can be integrated out of (30) in a straightforward way to give (25). this completes the proof. corollary 4.1. if l = 1, then h = arg  ∞∫ 0 vr−2 2 r∏ i=1 zv2 i m (h ⌢ β )⌢ δ v2 + r∑ i=1 zv2 i + (n− r)zv2r −r dv2 ∞∫ 0 vr−2 2 r∏ i=1 zv2 i ( r∑ i=1 zv2 i + (n− r)zv2 r )−r dv2 = 1− α  (31) theorem 5 ((lower (upper) unbiased prediction limit h for the lth order statistic yl in a new (future) sample of m observations from the left-truncated weibull distribution on the basis of the preliminary data sample) let x1 ≤ ... ≤ xr be the first r ordered observations from the preliminary sample of size n from the left-truncated weibull distribution with the pdf fθ(x) = δ σx δ−1 exp[−(xδ − µ)/σ], (xδ ≥ µ, σ, δ > 0) (32) nicholas a. nechval : finding unbiased simultaneous prediction limits for order ... 192 where θ = (µ, σ, δ), δ is termed the shape parameter,δ is the scale parameter, and µ is the truncation parameter. it is assumed that the parameter δ is known. then a lower unbiased (1−α) prediction limit h on the lth order statistic yl from a set of m future ordered observations y1 ≤ ... ≤ ym also from the distribution (32) is given by h = ( xδ 1 + whs )1/δ (33) where wh =  arg ( 1− m!(m+n−l)! (m−l)!(m+n)! (1− nwh) −(r−1) = 1− α ) , if α < m!(n+m−l)! (m−l)!(n+m)! , arg nl ( m l ) l−1∑ i=0  l − 1 i (−1)i[1+wh(m−l+i+1)]−(r−1) (n+m−l+i+1)(m−l+i+1) = 1− α  , if α ≥ m!(n+m−l)! (m−l)!(n+m)! , (34) s = r∑ i=1 (xδ i −xδ 1) + (n− r)(xδ r −xδ 1) (35) (observe that an upper unbiased α prediction limit h on the lth order statistic yl may be obtained from a lower unbiased (1 − α) prediction limit by replacing 1− α by α.) proof. it can be justified by using the factorization theorem that (xδ 1 , s) is a sufficient statistic for (µ, δ). we wish, on the basis of the sufficient statistic (xδ 1 , s) for (µ, δ), to construct the predictive density function of the lth order statistic yl from a set of m future ordered observations y1 ≤ ... ≤ ym. by using the technique of invariant embedding [8-11] of (xδ 1 , s), if x1 ≤ yl, or (y δ l , s), if x1 ≥ yl, into a pivotal quantity (y δ l − µ)/σ or (xδ 1 − µ)/σ, respectively, we obtain an ancillary statistic wl = ( y δ l −xδ 1 )/ s (36) it can be shown that the pdf of wl is given by f(wl) =  n(r − 1)l ( m l ) l−1∑ i=0 ( l − 1 i ) (−1) i [1 + wl(m− l + i+ 1)]−r n+m− l + i+ 1 , if wl ≥ 0, n(r − 1) m!(n+m−l)! (m−l)!(n+m)! (1− nwl) −r , if wl < 0. (37) 193 advances in systems science and applications (2013) vol.13 no.2 it follows from (37) that p (wl > wh) =  nl ( m l ) l−1∑ i=0 ( l − 1 i ) (−1) i [1 + wh(m− l + i+ 1)] −(r−1) (n+m− l + i+ 1)(m− l + i+ 1) , if wh ≥ 0, 1− m!(m+ n− l)! (m− l)!(m+ n)! (1− nwh) −(r−1) , if wh < 0. (38) where wh = ( hδ −xδ 1 )/ s (39) this ends the proof. corollary 5.1. if l = 1, then a lower (1−α) prediction limit h on the minimum y1 of a set of m future ordered observations y1 ≤ ... ≤ ym is given by h =  ( xδ 1 + s m [( n (1−α)(n+m) ) 1 r−1 − 1 ])1/δ , if α ≥ m n+m , ( xδ 1 − s n [( m α(n+m) ) 1 r−1 − 1 ])1/δ , if α < m n+m . (40) 2.2 two-parameter exponential distribution theorem 6 ((lower (upper) unbiased prediction limit h for the lth order statistic yl in a new (future) sample of m observations from the two-parameter exponential distribution on the basis of the preliminary data sample) let x1 ≤ ... ≤ xr be the first r ordered observations from the preliminary sample of size n from the two-parameter exponential distribution with the pdf fθ(x) = 1 σ exp[−(x− µ)/σ],(xδ ≥ µ, σ > 0) (41) where θ = (µ, σ), σ is the scale parameter, and µ is the shift parameter. it is assumed that these parameters are unknown. then a lower unbiased (1 − α) prediction limit h on the lth order statistic yl from a set of m future ordered observations y1 ≤ ... ≤ ym also from the distribution (41) is given by h = x1 + whs (42) where wh =  arg ( 1− m!(m+n−l)! (m−l)!(m+n)! (1− nwh) −(r−1) = 1− α ) , if α < m!(n+m−l)! (m−l)!(n+m)! , arg nl ( m l ) l−1∑ i=0  l − 1 i (−1)i[1+wh(m−l+i+1)]−(r−1) (n+m−l+i+1)(m−l+i+1) = 1− α  , if α ≥ m!(n+m−l)! (m−l)!(n+m)! (43) nicholas a. nechval : finding unbiased simultaneous prediction limits for order ... 194 s = r∑ i=1 (xi −x1) + (n− r)(xr −x1) (44) (observe that an upper unbiased α prediction limit h on the lth order statistic yl may be obtained from a lower unbiased (1 − α) prediction limit by replacing 1− α by α.) proof. for the proof we refer to theorem 5. corollary 6.1. if l = 1, then a lower (1−α) prediction limit h on the minimum y1 of a set of m future ordered observations y1 ≤ ... ≤ ym is given by h =  ( x1 + s m [( n (1−α)(n+m) ) 1 r−1 − 1 ]) , if α ≥ m n+m , ( x1 − s n [( m α(n+m) ) 1 r−1 − 1 ]) , if α < m n+m . (45) remark 2. let us assume that the parent distributions are the two-parameter exponential fθ(x) = 1− exp ( −x− θ2 θ1 ) , x ≥ θ2, θ1 > 0 (46) where θ = (θ1, θ2) and the pareto distribution fθ(x) = 1− (θ2/x) 1/θ1 , x ≥ θ2 > 0, θ1 > 0 (47) let x be a random variable with the pareto distribution (47), and define y = lnx. then y becomes a random variable with the exponential distribution (46), where θ2 is replaced by lnθ2. therefore it is enough to consider only the exponential distribution, because the results for the pareto distribution are easily obtained from those for the exponential distribution. 3 numerical example an industrial firm has the policy to replace a certain device, used at several locations in its plant, at the end of 24-month intervals. it doesn’t want too many of these items to fail before being replaced. shipments of a lot of devices are made to each of three firms. each firm selects a random sample of 5 items and accepts his shipment if no failures occur before a specified lifetime has accumulated. the manufacturer wishes to take a random sample and to calculate the lower prediction limit so that all shipments will be accepted with a probability of 0.95. the resulting lifetimes (rounded off to the nearest month) of an initial sample of size 15 from a population of such devices are given in table 1. goodness-of-fit 195 advances in systems science and applications (2013) vol.13 no.2 table 1 the resulting lifetimes statistical results x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 8 9 10 12 14 17 20 25 29 30 35 40 47 54 62 lifetime (in number of month intervals) testing. it is assumed that xi ∼ fθ(x) = δ σx δ−1 exp[−(xδ − µ)/σ], (x ≥ µ, σ, δ > 0), i = 1(1)15 (48) where the parameters µ and δ are unknown; (δ = 0.87). thus, for this example, r = n = 15, k = 3,m = 5, 1−α = 0.95, xδ 1 = 6.1,and s = 170.8. it can be shown that the uj = 1−  j+1∑ i=2 (n− i+ 1)(xδ i −xδ i−1) j+2∑ i=2 (n− i+ 1)(xδ i −xδ i−1)  j , j = 1(1)n− 2 (49) are i.i.d. u(0, 1) rv’s (nechval et al. [13]). we assess the statistical significance of departures from the left-truncated weibull model by performing the kolmogorovsmirnov goodness-of-fit test. we use the k statistic (muller et al. [14]). the rejection region for the α level of significance is k ≥ kn;α. the percentage points for kn;α were given by muller et al. [14]. for this example, k = 0.220 < kn=13,α=0.05 = 0.361 (50) thus, there is not evidence to rule out the left-truncated weibull model. it follows from (8) and (40), for α = 0.05 < km n+ km = 0.5 (51) that h = ( xδ 1 − s n [( km α(n+km) ) 1 n−1 − 1 ]) 1 δ = ( 6.1− 170.8 15 [( 15 0.05(15+15) ) 1 14 − 1 ]) 1 0.87 = 5 (52) thus, the manufacturer has 95% assurance that no failures will occur in each shipment before h = 5 month intervals. nicholas a. nechval : finding unbiased simultaneous prediction limits for order ... 196 4 conclusion and future work in this paper we propose the technique of constructing unbiased simultaneous prediction limits on observations or functions of observations in all of k future samples under parametric uncertainty of the underlying distribution. these unbiased simultaneous prediction limits are based on a previously available complete or type ii censored sample from the same distribution. we present an equation for this type of unbiased simultaneous prediction limits which holds for any distribution and any statistic from the previous sample when a prediction limit for a single future sample is available. the exact prediction limits are found and illustrated with a numerical example. the methodology described here can be extended in several different directions to handle various problems that arise in practice. we have illustrated the proposed methodology for the two-parameter exponential and weibull distributions. application to other distributions could follow directly. acknowledgements this research was supported in part by grant no.09.1544 from the latvian council of science and the national institute of mathematics and informatics of latvia. references [1] hahn g.j and meeker w.q. (1991), statistical intervals – a guide for practitioners, john wiley, new york. [2] patel j.k. (1989), “prediction intervals – a review”, communications in statistics theory and methods, vol.18, pp.2393-2465. [3] hahn g.j. (1972), “simultaneous prediction intervals to contain the standard deviations or ranges of future samples from a normal distribution”, journal of the american statistical association, vol.67, pp.938-942. [4] hahn g.j. (1975), “a prediction interval on the means of future samples from an exponential distribution”, technometrics, vol.17, pp.341-345. [5] hahn g.j, nelson w. (1973), “a survey of prediction intervals and their applications”, journal of quality technology, vol.5, pp.178-188. [6] mann n.r, schafer r.e, singpurwalla j.d. (1974), methods for statistical analysis of reliability and life data, john wiley, new york [7] fertig k.w, mann n.r. (1977), “one-sided prediction intervals for at least p out of m future observations from a normal population”, technometrics, vol.19, pp.167-177. 197 advances in systems science and applications (2013) vol.13 no.2 [8] nechval n.a, berzins g, purgailis m, nechval k.n. (2008), “improved estimation of state of stochastic systems via invariant embedding technique”, wseas transactions on mathematics, vol.7, pp.141-159. [9] nechval n.a, purgailis m, berzins g, cikste k, krasts j, nechval k.n. (2010), “invariant embedding technique and its applications for improvement or optimization of statistical decisions”, in: al-begain. k., fiems. d., knottenbelt. w. (eds.), analytical and stochastic modeling techniques and applications, lncs, springer, heidelberg, vol.6148, pp.306-320. [10] nechval n.a, purgailis m, cikste k, berzins g, nechval k.n. (2010), “optimization of statistical decisions via an invariant embedding technique”, in: lecture notes in engineering and computer science: proceedings of the world congress on engineering 2010, wce 2010, london, 30 june 2 july 2010, pp.1776-1782. [11] nechval n.a, purgailis m, nechval k.n, strelchonok v.f. (2012), “optimal predictive inferences for future order statistics via a specific loss function”, iaeng international journal of applied mathematics, vol.42, pp.40-51. [12] fisher r.a. (1934), “two new properties of mathematical likelihood”, proceedings of the royal society a, vol.144, pp.285-307. [13] nechval n.a, nechval k.n. (1998), “characterization theorems for selecting the type of underlying distribution”, in: proceedings of the 7th vilnius conference on probability theory and 22nd european meeting of statisticians, tev, vilnius, pp.352-353. [14] muller p.h, neumann p, storm r. (1979), tables of mathematical statistics, veb fachbuchverlag, leipzig. corresponding author nicholas a. nechval can be contacted at: nechval@junik.lv мягкая вероятность ошибки при передаче информации adv syst sci appl 2018; 04; 136-150 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/584 systems methodology and model tools for territorial sustainable management t.yu. anopchenko1, o.i. gorbaneva1, e.i. lazareva1, a.d. murzin1, g.a. ougolnitsky*1 1)southern federal university, rostov-on-don, russia received may 7, 2018; revised november 14, 2018; published december 31, 2018 abstract: no doubt, this research is topical in the context of new challenges arising for different russian territories due to intensive globalization processes and jumping trends of world economy. these challenges are stressing the need for an innovation-oriented improvement of territorial management systems in accordance with the realities of cyclicity and the conditions of fundamental resources to stimulate positive social and economic dynamics. the goal of the present paper is to justify a methodology and real ways of territorial sustainable development management system design based on efficient model tools and information technologies. for achieving this goal, several problems have been solved such as the dynamic modeling of a territorial social-ecological-economic system and coordination of public (social) and private interests, the elaboration of reasonable management approaches to territorial development risks, as well as the design of an appropriate methodology to integrate the suggested models and approaches within an information-analytical sustainable management support system. the novelty of an original scientific outlook introduced by the authors consists in the elaboration of a well-grounded systems methodology and model tools for territorial sustainable management and also in the justification and verification of qualitative and quantitative approaches to the analysis of development risks and their integration into a unified territorial sustainable management system.the methodology and model tools presented below can be used for enhancing the scientific and practical components of a territorial sustainable management system under dynamic and conflicting external conditions. keywords: homeostasis, simulation, econometric methods, territorial sustainable development, system compatibility, welfare capital reproduction, risk management, sustainable development management. 1. introduction this paper extends the results of the earlier publication [22], in which a systems approach to regional sustainable management was described. the methodology suggested below integrates territorial social-ecological-economic system simulation, econometric assessment of its innovative sustainable development potential and coordination of social and private interests into a unified territorial sustainable management system. a conceptual platform of the original modeling approach is the evolutionary-cyclic paradigm of subject-object relations in the “welfare capital reproduction –– territorial innovative sustainable development” system. in accordance with this paradigm, the network reality makes it necessary to elaborate a collectively compatible management strategy relying on the coordination and balancing of personalized interests for different subjects of innovative sustainable development, mutual benefit, trust, and public-private partnership (fig. 1.1). as shown in [12], an alternative way to implement the coordination principle is to identify an ideal hierarchical chain of interests of economic subjects, associating it with an adequate incentive policy. * corresponding author: ougoln@mail.ru model tools for sustainable management 137 copyright ©2018 assa adv. in systems science and appl. (2018) the models that produce analysis tools for analyzing the object in strategic management systems – the trajectories of territorial sustainable development must have the following major features [14]:  nonlinearity of the model and orientation for a long time interval;  taking into account the impact of economic activity on natural processes and the state of the social sphere;  •inclusion of feedback flows between the ecological, social and economic subsystems of the territorial system;  not only material or monetary valued services of social and natural systems should be included, but also the rest, "intangible", such as favorable conditions for production and residence;  taking into account the concern for future generations, which can be expressed in restrictions on natural capital, social and economic capital to ensure an even distribution of social, natural and economic potential between successive generations;  the possibility of describing qualitative structural changes;  reflection of the limited availability of resources, including the assimilation potential of the environment. there are two methods to add the sustainable development conditions of dynamic trajectories in the model as follows: – establishing temporal constraints for welfare level (at any time, welfare level must exceed a given threshold or welfare dynamics must have a uniform nondecreasing character); – establishing physical constraints for resources (stocks and flows). an important class of dynamic models of territorial economic development is formed by the so-called computable general equilibrium (cge) models, see [6, 23]. these models have good microeconomic grounds and also provide a complete description for the sectoral structure of economy and mutual effects of different economic sectors. however, the cge models suffer from a major drawback: in fact, they are mostly economic-mathematical models with a superficial treatment of ecological and social aspects and all the phenomena connected with dynamics and uncertainty. besides, their identification is also a complex procedure. the modeling framework of regional social-ecological-economic development using optimal control methods was suggested by gurman et al. [19]. regional development problems are studied by analytical methods in combination with simulation modeling. in the monograph [8], the well-known solow growth model [17] was modified taking into account the spatial aspect and environmental pollution. a detailed survey of the models and decision support systems for sustainable management was given in [21]; various economic growth models were described in [5]. the concept of homeostasis plays a key role for sustainable management. it can be formalized using aubin’s viability theory [4]. in particular, the so-called capture-viability kernels are of assistance here. the capture-viability kernel of a closed set k with closed target x under a set-valued dynamic f is the set of all initial states from which there exists at least one solution, remaining in k and reaching x at a given finite time horizon [7]. in the earlier works, the authors of this paper introduced an original complex approach to model the processes of accumulation and productive use of tangible and intangible assets – resources for territorial sustainable development as well as to model the coordination processes of social and private interests for resource allocation in hierarchical control systems. an implementation of this approach allowed to establish the system compatibility conditions for different control problem setups; to justify a control strategy design methodology for balancing the social and economic interests of national, regional (local) and global economic agents in the reproduction and utilization of welfare resources for the sustainable development of a regional system; to propose system coordination mechanisms and study their properties; to initiate application of the coordination models of social and private interests to real regional 138 t.yu. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2018 assa adv. in systems science and appl. (2018) management problems, more specifically, to the design of administrative and economic coordination mechanisms (engines) for the interests of territorial subjects [3, 9, 10, 12]. fig. 1.1 the conceptual model of subject-object relations [12] model tools for sustainable management 139 copyright ©2018 assa adv. in systems science and appl. (2018) the problems of rationalizing the mutual influence of the subjects of environmental and economic relations within certain territorial boundaries have been repeatedly considered by the authors [1, 2, 16, 20]. in accordance with the concept of sustainable development, which emphasizes the vital importance of current needs satisfaction taking into account the interests of future generations, risk is a significant quantitative indicator of balanced decisions. for these purposes, a control algorithm for the ecological and economic risks of urbanized territories development was suggested in [20]. for managing the social and economic risks of territorial development, earlier the authors introduced the economic and mathematical models based on specification of the latent mutual influences of welfare capital accumulation and the pace of innovative sustainable development of a territory [15] and also on the use of an econometrically identified relationship between population health and the variations of environmental parameters as a primary functional characteristic of social and economic damage [2]. an implementation of the sustainable management methodology at the regional level requires a regional information-analytical support system [22] as a basic technological tool of management. this system has a hierarchical structure, and the subsystems at the lower levels of management can be used independently. the remainder of this paper is organized as follows. in section 2, the structure of the dynamic model of a regional social-ecological-economic system, its main variables and processes as well as operation of this model are presented. section 3 is dedicated to the coordination mechanisms (engines) of interests in the public-private partnership and their use in the model. next, in section 4, risk management methods at territorial level are characterized. section 5 is to synthesize the above-mentioned methods and models for territorial sustainable management system design. finally, in section 6, some concluding remarks are given. 2. dynamic model of regional social-ecological-economic system the model of a regional social-ecological-economic system has the form )())(()()( 1 tlrtktaty ii iiiii    ; (2.1) )()()( tytsti iii  ; (2.2) )()](1[)( tytstc iii  ; (2.3) )()1()1( trtr iii  ; (2.4)    n j jjiiii tittktk 0 )()()()1()1(  ; (2.5) )()1()1( tlmbtl iiii  ; (2.6) )]()()][(1[)( tlbtkbtvctp i a lii a ki a i a i a i  ; (2.7) )]()()][(1[)( tlbtkbtvctp i w lii w ki w i w i w i  ; (2.8) ;)0(;)0(;)0( 000 iiiiii rrllkk  (2.9) 140 t.yu. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2018 assa adv. in systems science and appl. (2018) ;0)(;0)(;0;1)(0;1)()()( 0   tvtvtstvtvt w i a iiji w i a i n j ij  (2.10) ,...2,1,0;,...,1,0,  tnji as a rule, index i denotes a municipal unit within a region (e.g., within a federal subject). it can be associated with an enterprise but, in this case, productive and economic activity should be described by the firm model. in the sequel, it will be referred to as the agent’s index. the model has discrete times t = 0, 1, 2, ... with a step of 1 year. the model also includes other variables and parameters as follows. )(tyi as the final output of agent i in year t (in financial terms); )(tk i as the agent’s basic production assets (capital) in year t; )(tli as the agent’s labor resources in year t; ( )ir t as the efficiency of the agent’s labor resources in year t; )(tai as the influence function of the agent’s innovative activity on the final output in year t; i as the parameter of the agent’s cobb-douglas production function; )(ti i as the agent’s production investments in year t; )(tci as the agent’s nonproduction consumption in year t; )(tsi as the share of the agent’s production investments in its final output in year t; i as the efficiency growth parameter of the agent’s labor resources; i as the depreciation factor of the agent’s basic assets; )(tij as the share of the investments of agent i in the activity of agent j (the cooperation coefficient of these agents); here index j=0 describes an external agent for the whole system; ib and im as the reproduction and retirement coefficients of the agent’s labor resources, respectively; ( )a ip t and ( )w ip t as the agent’s pollutant emissions into air and water in year t, respectively; ( )a iv t and ( )w iv t as the agent’s allocations to prevent air and water pollution in year t, respectively; a ic and w ic as the efficiency coefficients of these allocations; a kib and w kib as the specific rates of industrial pollution into air and water, respectively; a lib and w lib as the specific rates of human pollution (labor resources) into air and water, respectively; finally, 0 0, ,i ik l and 0 ir as given initial values of the model variables. therefore, the agent’s state vector is ))(),(),(),(),(),(),(),(()( tptptrtltktctitytx w i a iiiiiiii  ; the control vector is ))}({),(),(),(()( 0 n jij w i a iii ttvtvtstu   ; and the parameter vector is ),,,,,,,,,,( w li a li w ki a ki w i a iiiiiii bbbbccmbz  . the whole system can be written as ),...,()),(),...,(()()),(),...,(()( 111 nnn zzztutututxtxtx  . (2.11) the innovative activity function )(tai is considered separately [17]. with these notations, model (2.1)-(2.10) takes the form )),(),(()()1( ztutxftxtx iii  ; (2.12) model tools for sustainable management 141 copyright ©2018 assa adv. in systems science and appl. (2018) ii tu )( ; (2.13) ,...2,1,0,,...,1,)0( 0  tnixx ii (2.14) the structure of model (2.1)-(2.10) has three blocks, namely, economy (2.1)-(2.5), demography (2.6), and ecology (2.7)-(2.8). the transboundary interaction of municipal units is described using the control variables )(tij . also note that innovative activity can be modeled by through model parameters instead of the function )(tai . more specifically, the parameter i characterizes labor productivity; the parameter i resource-saving technologies; the parameters , , ,a w a i i kic c b and w kib nature protection technologies; the parameter ib demographic policy; the parameter im innovations in public health; the parameters a lib and w lib the ecological consciousness of population. to identify the parameters of model (2.1)-(2.10), it is necessary to define the numerical values of all elements of the vector z. in a rough approximation, consider just two values of each parameter },{ h i l ii zzz  , where the second value corresponds to higher technological level. for further refinement, the parameter values can be taken from a discrete set },...,{ 1 ik iii zzz  . the sustainable development conditions (homeostasis) of a regional social-ecologicaleconomic system in model (2.1)-(2.10) can be defined by ,...2,1;,...,1;)(;)(;)( maxmaxmin  tniptpptpyty w i w i a i a iii (2.15) the first condition in (2.15) states the requirements to the agent’s economic growth; the other, the maximum permissible emissions of pollutants into the environment. these conditions can be interpreted as the goal of control. the agent’s objective function in model (2.1)-(2.10) takes the natural form max)( 0     dttcej i t i  , (2.16) where )(/)()( tltctc iii  is the current specific consumption of agent i;  denotes the discount factor. since the model has discrete time and will be studied using simulation, the objective function (2.16) should be replaced by its discrete analog    t t i t i tcej 0 )( max . (2.17) simulation scenarios for model (2.1)-(2.10) include some trajectories of control variables of vector (2.13). here it seems reasonable to perform optimization by solving a certain control problem. a classification of such problems is given in table 2.1. table 2.1. control problem setups within the model decentralized centralized optimal control optimization performed by each agent independently (optimal control problem) global optimization performed by an external agent (optimal control problem) conflict control conflict resolution with interaction of agents (differential normal form game) conflict resolution with the optimal response of agents (hierarchical differential normal form game) 142 t.yu. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2018 assa adv. in systems science and appl. (2018) the decentralized control problem setups treat all agents equally. if the situation is considered from the viewpoint of a single agent, then the optimal control problem with the objective function (2.16) or (2.17) subject to constraints (2.1)-(2.10) arises accordingly. attempts to study the interaction of several or all agents lead to differential normal form games, with nash equilibrium as a standard solution concept. on the other hand, the centralized setups interpret one of the agents as a principal responsible for the system goals. in practice, the principal’s role can be played by federal government, regional administration or even a special coordination body for the activity of several agents or the whole set of agents established on voluntary basis (e.g., coordinating committee, project directorate, etc.). as before, the principal may perform global optimization or take into account the agents’ response to its actions. in the latter case, hierarchical differential games are immediate, with stackelberg equilibrium or the germeier principle of guaranteed result as possible solution concepts, see [22]. 3. coordination models for social and private interests in [9] and other publications, the authors constructed and analyzed coordination models for social and private interests (spice-models). the suggested approach consists in the following. 1. consider n agents, each distributing his/her/its resource between private activity and the production of some social good. 2. in turn, the produced social good is allocated among the agents in given or controlled shares. this defines the distinction between social good and pure public good. 3. the payoff function of each agent includes two terms, the first reflecting his/her/its income from private activity while the second his/her/its share in social good. 4. the concept of system compatibility is defined to characterize in quantitative terms the degree of concordance of social and private interests in a corresponding organizational and technical system. perfect system compatibility is achieved if the aggregate of the individually optimal strategies of all agents (dominant strategies or nash equilibria) maximizes the social welfare function of the system. 5. in real economic organizations, perfect system compatibility is a rare thing due to the individualism of agents. therefore, a special agent (principal) should be assigned to represent the social interests (social welfare maximization) and ensure system compatibility. 6. the principal controls the agents in two ways. first, he/she/it may restrict the share of resource allocated by the agents to their private activity (administrative mechanisms, compulsion). second, the principal may determine the shares of all agents in social good depending on their actions (economic mechanisms, impulsion). speaking mathematically, these control mechanisms are formalized as the germeier games г1 and г2 (the stackelberg and inverse stackelberg games, respectively). an implementation of this approach yielded the following. – system compatibility conditions were obtained for different control problem setups. the major result is that, without external control, perfect system compatibility can be achieved only by partitioning the set of all agents into pure individualists (distributing all their resources to private activity) and pure collectivists (distributing all their resources to social good production). in addition, the spice-models were compared in the cases of independent equal agents (nash equilibrium), hierarchically organized agents (stackelberg equilibrium), and full cooperation of agents (the pareto maximal value of the total payoff function). it was demonstrated that equality is preferable to hierarchy in the sense of social welfare maximization; – system coordination mechanisms were suggested and their properties were studied. a mechanism is system compatible if it ensures perfect system compatibility. here empirical and theoretical approaches are possible as follows. in accordance with the former, it is necessary to analyze the mechanisms widely used in practice (e.g., proportional allocation). for instance, it was shown that the proportional allocation mechanism is system compatible only under the model tools for sustainable management 143 copyright ©2018 assa adv. in systems science and appl. (2018) linear social income function. the theoretical approach suggests to construct the control mechanism as the ε-optimal solution of the germeier game г2 (the inverse stackelberg game). the economic mechanism based on the germeier game г2 is system compatible if social and private incomes represent power functions with an exponential less than 1; – an application of the spice-models to real regional management problems was also initiated; more specifically, the administrative and economic coordination mechanisms for the interests of regional subjects were explored. a control problem was studied in which two or more neighbor subjects distribute their funds between the development of their own and common (transboundary) territory or between private activity and a joint project. a special control authority (principal) was introduced for activity coordination. the economic mechanism was considered in two modifications (financial participation in the development of a transboundary territory via income share control, and resource allocation). a detailed analysis of these mechanisms was carried out and their organizational and economic interpretation for specific regional management problems was given (public-private partnership, euroregions), see [3,10]. the static spice-model has the form max),...,()(),...,( 11  niiiini uucsurpuug (3.1) .,...,1, ,0,0 ,0:,1 ,0,0,0 1 ni si si ssrru n j i i jiiii          (3.2) the notations are as follows: },...,1{ nn  as a finite set of active agents; ],0[ ii ru  as the set of admissible strategies of agent i ; ir as an amount of resource available to agent i ; ),...,( 1 ni uug as the payoff function of agent i ; ;...,: 1 ni uuurug  )( iii urp  as the private interest function of agent i ; ),...,( 1 nuuc as a social income function; is as the share of social income allocated to agent i ; ),...,( 1 ni uucs as the social component of the payoff function of agent i . the following assumptions are applied to this model: c is a monotonically increasing function in all iu such that ;0)0,...,0( c ip are monotonically increasing functions in )( ii ur  and monotonically decreasing in iu such that 0)0( ip (for ii ru  ); 0is  if 0iu . construct the social welfare function    n j njjj n j njn uucurpuuguug 1 1 1 11 ).,...,()(),...,(),...,( (3.3) denote by },...,{ )()1( ne k ne uune  the set of nash equilibria in game (3.1)–(3.2). also let )}(),...,(min{ )()1(min ne k nene ugugg  and ).()(max max max ugugg uu   then the price of anarchy in model (3.1)–(3.2) (an indicator of system compatibility) is max min g g pa ne  . (3.4) obviously, 1pa . if pa is close to 1, then the equilibria have high efficiency and there is little need for system coordination in model (3.1)–(3.2) (for 1pa , even no need at all). the need for system coordination grows as pa is decreased. as a matter of fact, in itself the condition of system compatibility ( 1pa ) holds merely in some cases. to ensure this condition, it seems reasonable to use control mechanisms [18]. 144 t.yu. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2018 assa adv. in systems science and appl. (2018) suppose that maximization of the social welfare function (3.3) is the goal of a certain subject (principal, leader, or mechanism designer) that may influence the sets of admissible controls and/or payoff functions of agents to achieve it. denote by )( iii quu  the first possibility and by ),( iiii usgg  the second. then the following control mechanisms are immediate, see table 3.1. the principal may influence the sets of admissible controls of agents (administrative mechanism) or their payoff functions (economic mechanism). both types of influence are based on the germeier games г1 and г2 (the stackelberg and inverse stackelberg games, respectively). therefore, there are four types of control mechanisms illustrated in table 3.1. table 3.1. control mechanisms principal influences: based on germeier games г1 (stackelberg games) based on germeier games г2 (inverse stackelberg games) the sets of admissible controls of agents compulsion, or administrative mechanism, without feedback constqi  compulsion, or administrative mechanism, with feedback )(uqq ii  the payoff functions of agents impulsion, or economic mechanism, without feedback constsi  impulsion, or economic mechanism, with feedback )(uss ii  the economic control mechanisms in the spice-model (3.1)-(3.2) are implemented as the result of choosing values is by the principal. for the administrative mechanisms, an additional assumption is that the principal may restrict the admissible controls of agents, i.e. niquq iii  ,~ . (3.5) in the continuous-time setup with finite horizon, the agent’s objective function (2.16) takes the form                  t n i ij iijii t i dttifttcej 0 1 max])()()([  ; (3.6) in the discrete-time setup with the same horizon,                   t t n i ij iijii t i tifttcej 0 1 max])()()([  . (3.7) in both cases, f denotes the social welfare function while ρ is the discount factor. note that, by choosing a control trajectory, the agent defines the logical chain iiiiii ccyk  in model (2.1)–(2.10), which corresponds to the choice of allocations to private activity in the static spice-model. therefore, the first term in the integrand (summand) describes the agent’s model tools for sustainable management 145 copyright ©2018 assa adv. in systems science and appl. (2018) income from private activity; the second, his/her/its consumption of social good depending on the variable )(ti . the administrative and economic mechanisms can be described for the dynamic spicemodel with a principal. using administrative mechanisms, the principal explicitly restricts the “egoism” of agents by imposing the conditions )()( max tt iiii   . (3.8) note that conditions (3.8) can be the result of a voluntary agreement of the agents. if they are established by the principal, then his/her/its objective function also includes administrative control cost. economic mechanisms may have a share-based motivation of the agents, i.e. ))(()( tt iiii   , (3.9) or a resource allocation among them performed by the principal. 4. approaches to risk factors management in territorial development sustainable development is impossible without proper consideration of risks, which have to be identified first. a detailed identification of territorial risks facilitates the objective formalization and modeling of territorial development, yielding adequate assessments for possible consequences of decisions using the integral risk indicator of a territory. territorial risk assessment methods are mostly reduced to score calculation in some rating scales for comparing the development levels of different regions. speaking formally, the existing approaches to territorial risk analysis can be divided into several groups, namely, qualitative analysis, quantitative analysis, combined analysis, and structural analysis. qualitative analysis considers the weights of different factors affecting risks. note that the list and significance of such factors are defined via expertise. major disadvantages of this approach consist in the subjectivism and qualification of experts. therefore, qualitative analysis makes sense only in the case of well-defined goals of study and available highly qualified experts, who have rich experience and sufficient familiarity with regional situation. quantitative analysis is often performed for most important risk indicators: consideration of all risk factors would dramatically increase the complexity of calculations and further examination. this approach allows constructing a multifactor function for explicit quantitative assessments. there are two standard methods to define regional risk functions as follows. in accordance with the first method, it is necessary to consider only those factors that yield an objective quantitative characterization for the current state of economical, ecological, and social spheres. then the territorial risk function takes the form r = f (x1, x2, …, xn) = r (xi), i = 1,...,n. (4.1) the second approach involves a set of numerical qualitative assessments, i.e. r = f (r1, r2, …, rm) = r (ri), i = 1,...,m. (4.2) econometric modeling is intended for statistical data processing and prediction based on quantitative data arrays of key indicators. however, in real conditions, some significant factors 146 t.yu. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2018 assa adv. in systems science and appl. (2018) and criteria (e.g., social and ecological situation in a region, or the synergetic effect of several environmental conditions) cannot be assessed in quantitative terms. in this context, a reasonable solution is to integrate quantitative and qualitative assessment methods, which makes the essence of combined approaches. in this case, the integral risk indicator includes heterogeneous parameters of different risks in numerical form obtained by objective calculations and also by subjective qualitative assessments, which are considered in some scales or criteria. the identified factors are used for assessing the risks of comparable level. the integral risk indicator r (4.1), (4.2) can be expressed as a linear relationship of the corresponding indicators calculated for the risk of smaller level, i.e. r = f (x1, x2, x3, x4, x5) = a0 + a1x1 + a2x2 + a3x3 + a4x4 + a5x5. (4.3) here the notations are as follows: х1 = y (b1,…,bn) as the function of environmental and ecological factors; х2 = y(c1,…,cm) as the function of social and demographic factors; х3 = y (t1,…, tk) as the function of anthropogenic and production factors; х4 = y (s1,…,sj) as the function of financial and economic factors; finally, х5 = y (h1,…,hg) as the function of organizational and managerial factors. these variables characterize the specific levels of corresponding risks; their weights а1, …, а5 can be defined using qualitative methods in order to consider a basic level of risk formed by the associated factors. the described approach to territorial risk assessment is rather flexible and convenient for practical implementation. moreover, it takes into account any changes in structural parameters through the coefficients of the corresponding variables, which gives an objective description for the dynamics of external and internal territorial environment. the sustainable development programs of different territories can be elaborated using the scenario approach. the number of scenarios under study (alternatives) may vary depending on regional capabilities and resources. as a rule, developmental prospects are considered in view of three alternatives. the first regional development scenario is the preservation of current social and economic dynamics at the same level. this scenario implies that territorial authorities have no significant interference into social and economic development. their major concern and funding are focused on maintaining the current volumes of regional gross domestic product (gdp) and the existing sectors of regional industry; economy is developing with orientation towards export of resources and raw materials. the second territorial development scenario is based on the implementation of continuous investment projects and programs, not only in the sphere of raw materials and semi-products but also in financial and industrial groups, resource processing and resource supply. for this scenario, of crucial importance are the industries and productions oriented towards import substitution, high technologies, finished products and services. the third territorial development scenario is a logical continuation of the second; it can be implemented in several fields simultaneously through initiation of long-term projects (clusters and growth drivers in industries, high technologies, science and education). in contrast to the productions oriented towards export of resources and raw materials, these projects can be implemented in form of medium and even small enterprises, hence with smaller capital investments and higher rates of return. in the long run, this scenario yields the multiplicative effect owing to the creation of new jobs, the development of adjacent, auxiliary and supporting industries, and natural clustering. the structural method of territorial risk assessment is based on the expertise of given quantitative parameters––the probability and amount of losses. for the identified risks, this approach involves probabilistic weighting of each development scenario to distribute the final result. the territories with similar level and initial conditions of development can be compared using the integral risk indicator that includes several structural components, namely, (a) the ratio model tools for sustainable management 147 copyright ©2018 assa adv. in systems science and appl. (2018) of regional gpd and average gdp for a corresponding group of regions; (b) the ratio of the deviation of regional gdp from average gdp for a corresponding group of regions; (c) the ratio of regional gpd increase rate and average gdp increase rate for a corresponding group of regions. also this approach is a snap analysis tool for territorial risks in a given group of regions. note that the identification of possible risks and the assessment of average territorial risk must agree with the goals of development. for strategic development programming, such procedures allow to find territorial fields and spheres in which risk management is applicable and also reasonable. consequently, the resulting assessments can be used to refine territorial development programs taking into account risk management methods and tools, territorial risk management system optimization and the prediction of possible consequences of different risk events. in this regard, there exists an obvious need for integrating the risk management subsystem directly into the territorial sustainable management system: otherwise, the whole complex of well-known approaches to risks identification, assessment and management would be fruitless, not gaining an expected effect. integration must cover all levels of the management system, relying on available administrative resources and the support of all structural elements. this is the only way towards the required efficiency of risks monitoring and optimization with territorial sustainable development. the territorial risk management subsystem is deployed in the following way. the functions and authorities are divided by management levels, industries, and spheres of development. necessary management information is exchanged by the structural units of the system in the online mode. all necessary resources of each management level and any sphere of activity are immediately mobilized. the goals of risk management must agree with the goals of territorial development for achieving strategic stability based on the prediction of possible threats and negative factors in the long run. modern science has produced numerous methods to affect risks. almost any territorial risk can be managed properly. one may control the probability of risks and also the amount of incurred losses by preventive and protective measures. evidence suggests that purely preventive methods do not completely eliminate the negative factors of risk in form of damage and loss conditions. therefore, it is crucial to plan complex measures towards the abolition and/or absorption of territorial risks at the level of territorial administration, particularly with the engagement of large territorial taxpayers. the external and global risks that are uncontrollable at the territorial level must be transferred to other management levels or secured using efficient financial mechanisms and tools (e.g., insurance). in this case, possible damage is compensated at necessary level (in terms of time and amount), which makes the territory independent of environmental disasters and anthropogenic accidents. so the strategic goals of territorial development can be achieved. 5. territorial sustainable development management system design based on mathematical models and information technologies the territorial sustainable management system covers organizational and political as well as technical aspects. the former concerns the existence of well-defined sustainable development regulations and procedures for territorial (legislative and executive) authorities. in particular, of crucial importance is territorial monitoring that implements feedback in the management system through the acquisition of necessary information about the dynamics of the territorial socialecological-economic system. the technical part of the system is represented by the regional information-analytical sustainable development management support system, see fig. 5.1. 148 t.yu. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2018 assa adv. in systems science and appl. (2018) fig. 5.1 structure of regional information-analytical sustainable development management support system [22] this information-analytical system has the client-server architecture. the client part includes the interfaces of models and data; the other blocks belong to the server part. operative databases contain information about separate municipalities and large enterprises, which is provided by the monitoring system. for regional management, it is necessary to integrate these (often heterogeneous) data. this problem is solved by importing all available data to the regional data warehouse. the latter also implements other functions such as data refinement, aggregation and security, processing of large data arrays, creation of multilevel metadata directories, execution of user requests, and generation of different reports. the analytical block consists of three subsystems––simulation, optimization and expert ones. the simulation subsystem implements model (2.1)–(2.10) using data from the regional data warehouse. the optimization subsystem solves the control problems described in table 3.1 using the dynamic optimality criteria (3.7) subject to constraints (2.1)–(2.10). the expert subsystem is optional: this subsystem allows to consider the control rules used in practice. the internal system interface is responsible for the interaction of models and data as well as for the implementation of different calculation schemes. a major role is played by the external user interface, which provides a user-friendly environment for common users of the regional information-analytical sustainable management support system. 6. conclusions the intrinsic character of a territorial system as a multilayer polystructural complex of heterogeneous subsystems (economic, social, and ecological) suggests the idea to consider such systems as the subjects of sustainable economic growth and to review the methodological approaches to territorial management. these approaches must be focused on the territorial reproduction of qualitative resources (first of all, human capital) based on proper coordination of interests of different subjects and public-private partnership [11]. overcoming the shortcomings of modern territorial management and emerging risks in its system is inextricably linked with the enhancement of the effectiveness of the mechanism for managing sustainable development of the territory on the basis of the current model tools and information technologies model tools for sustainable management 149 copyright ©2018 assa adv. in systems science and appl. (2018) no doubt, the methodology and model tools suggested in this paper will accelerate the positive processes of the innovation-oriented transformation of territorial sustainable development management systems. for the new strategic policy and its efficiency, a significant methodological aspect of dynamic systems analysis is the gradual transition to the management processes of functional-spatial territorial development, with highest priority assigned to the formation of innovative clusters and behavioral rules (economic, social, ecological, etc.) for the elements of social environment, see [13]. this work was supported by the russian foundation for basic research, project no. 18-010-00594. references [1] anopchenko, t.yu. & murzin, a.d. (2012). upravlenie riskami razvitiya urbanizirovannykh territorii [risk management in development of urbanized territories]. rostov-on-don: rostov state university of civil engineering [in russian]. [2] anopchenko, t.yu. & murzin, a.d. (2014). economic-mathematical modeling of social and environmental risks management of projects of urbanized territories development, asian social science, 10(15), 249–254. [3] anopchenko, t.yu., murzin, a.d. & ougolnitsky, g.a. (2017). modelirovanie soglasovaniya interesov v zadachakh upravleniya ustoichivym razvitiem territorii [modeling of interests coordination in regional sustainable management problems], ekonomika prirodopol'zovaniya, 6, 35-47. [4] aubin, j.-p. (1991). viability theory. birkhauser: springer-verlag. [5] barro, r.j. & sala-i-martin, x. (2004). economic growth. massachusetts: mit press. [6] bohringer, c. & loschel, a. (2006). computable general equilibrium models sustainability impact assessment: status quo and prospects, ecological economics, 60 (1), 49-64. [7] bonneuil, n. & boucekkine, r. (2017). viable nash equilibria in the problem of common pollution, pure and applied functional analysis, 2(3), 427-440. [8] druzhinin, a.g. & ougolnitsky, g.a. (2013). ustoichivoe razvitie territorial'nykh sotsial'no-ekonomicheskikh sistem: teoriya i praktika modelirovaniya [sustainable development of regional social and economic systems: theory and practice of modeling]. moscow, russia: vuzovskaya kniga [in russian]. [9] gorbaneva, o.i. & ougolnitsky, g.a. (2015). system compatibility: price of anarchy and control mechanisms in the models of concordance of private and public interests, advances in systems science and applications, 15(1), 45-59. [10] gorbaneva, o.i., murzin, a.d. & ougolnitsky, g.a. (2018). mekhanizmy soglasovaniya interesov pri upravlenii proektami razvitiya territorii [mechanisms of interests’ coordination in project management of regional development], upravlenie bol'shimi sistemami, 71, 61-97. [11] lazareva, e. & karaycheva, o. (2017). human oriented reframing of the territories of innovative sustainable development system management model, sgem 2017 proceedings, book 4, vol. 2, 672-670. [12] lazareva, e.i. (2013). features of national welfare innovative potential parametric indication information-analytical tools system in the globalization trends' context, ceur workshop proceedings 9, integration, harmonization and knowledge transfer, 339-351. [13] lazareva, e., anopchenko, t. & lozovitskaya, d. (2016). identification of the city welfare economics strategic management innovative model in the global challenges conditions, sgem 2016 proceedings, book 2, vol. 4, 3–11. https://elibrary.ru/item.asp?id=23979708 https://elibrary.ru/item.asp?id=23979708 https://elibrary.ru/item.asp?id=23979700 https://elibrary.ru/item.asp?id=23979700 150 t.yu. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2018 assa adv. in systems science and appl. (2018) [14] lazareva, e.i. (2011). metody modelirovaniya innovatsionno-orientirovannykh ekonomicheskikh strategii ekologoustoychivogo razvitiya [modeling methods of innovationoriented economic strategies of ecological sustainable development]. rostov-on-don: southern federal university [in russian]. [15] lazareva, e.i. (2013). osobennosti modelirovaniya traektorii prirashcheniya kapitala natsional'nogo blagosostoyaniya v perspektive ustoichivogo innovatsionno-orientirovannogo razvitiya [modeling specifics for capital increment trajectories of national welfare capital subject to sustainable innovation-oriented development], partnerstvo tsivilizatsii, 4, 234-244. [16] lazareva, e.i. et al. (2012). modeli ekologicheskoi orientatsii gosudarstvenno-chastnykh strategii investirovaniya chelovecheskogo kapitala v innovatsionnoi ekonomike [models of ecological orientation of public-private strategies of human capital investment in innovative economy]. rostov-on-don: southern federal university [in russian]. [17] lotov, a.v. (1984). vvedenie v ekonomiko-matematicheskoe modelirovanie [introduction to economic-mathematical modeling]. moscow, russia: nauka [in russian]. [18] mechanism design and management: mathematical methods for smart organizations, ed. by prof. d. novikov. new york: nova science publishers, 2013. [19] modelirovanie sotsio-ekologo-ekonomicheskoi sistemy regiona [modeling of socialecological-economic system of region], ed. by gurman, v.i. & ryumina, e.v. moscow, russia: nauka (2001). [20] murzin, a.d. (2015). algorithmization of ecologo-economic risk-management in urban areas, asian social science, 11(9), 312-319. [21] ougolnitsky, g.a. (2015). sustainable management as a key to sustainable development, in sustainable development: processes, challenges and prospects, ed. by d. reyes. new york: nova science publishers, 87-128. [22] ougolnitsky, g.a. (2017). a system approach to the regional sustainable management, advances in systems science and applications, 17(2), 52-62. [23] partridge, m.d. & rickman, d.s. (2010). cge modeling for regional economic development analysis, regional studies, 44(10), 1311-1328.  2017 г adv syst sci appl 2018; 1; 59-84 published online at http://ijassa.ipu.ru. analytical study of the antitumor viral vaccine introduction regimens based on mathematical modeling nina a. babushkina1, ekaterina a. kuzina1 1) v.a. trapeznikov institute of control sciences of russian academy of sciences, 117997, profsoyuznaya street, 65, moscow, russia e-mail: babushkina_na@mail.ru, kate_k93@mail.ru abstract. the paper presents the algorithm for calculating the maximal effective antitumor viral vaccine introduction regimens, using a computing experiment method (in silico) based on the software implementation of two mathematical models in the matlab-simulink system. the first model of antitumor vaccine therapy describes a two-stage mechanism of the tumor cells’ death as a result of the immune response. the effectiveness of immune response is measured in the number of antibodies formed by the immune system against virus-infected tumor cells. the second model of antitumor therapy with discontinuous trajectories of tumor growth is designed to evaluate the rate of the tumor cells’ death after the introduction of the viral vaccine. the effectiveness of the therapy is measured in the number of dying tumor cells after the introduction of a viral vaccine. keywords: mathematical model, experimental oncology, tumor cells, kinetic curves of tumor growth, tumor growth delay, virus, vaccine therapy, immune response, number of antibodies, vaccine efficacy, time-frame for recurrent vaccine introductions, in silico. 1. introduction vaccine therapy is one of the methods of immune therapy of the oncological diseases. it implies vaccination against the malignant cells formed in the body. as with any vaccination, the introduction of viral vaccines stimulates the body's immune system and causes the formation of antibodies specific to the tumor type. experiments in tumor immunotherapy can be traced back to ehrlich [1], and since the middle of the twentieth century the studies aiming to find the viruses that are able to affect malignant tumors without causing any harm to the humans commenced in the soviet union and were further continued in russia [2-6]. one of approaches to developing effective antitumor vaccines implies using the viruses that are able to identify malignant cells and do not cause any damage to the normal tissues. the viruses selected for therapy should be clinically harmless and epidemiologically safe. therefore, oncotropic viruses must have a genetically fixed absence of pathogenic properties and be unable to cause an acute infectious disease in a human body. among the viruses that meet these criteria are the venezuelan horse encephalomyelitis virus (vee) [5-7] and rat parvovirus h-1 [15-23]. both are not harmful to humans because they do not affect healthy tissues since they cannot replicate in the non-dividing cells [16-18]. thus these viruses are able to identify the tumor cells based on their high proliferation rates, and specifically target them without attacking the normal tissues [18-19]. experimental studies of rat parvovirus h-1 indicate that there are two possible mechanisms of the tumor cells’ death as a result of vaccine introduction. firstly, the virus blocks the cycle of the tumor cells [16-18]. secondly, it produces specific protein formations on the tumor cell surface that stimulate the immune system to develop antibodies against these cells, resulting in their death [13-14, 19-21]. this corresponds to the results of the experimental studies of the impact of the vee virus on tumor growth [4-6]. however, since such experiments require mailto:babushkina_na@mail.ru mailto:kate_k93@mail.ru 60 n.a. babushkina, e.a. kuzina extensive funding, these studies did not determine the maximal and minimal effective dosages and the optimal moment of vaccine introduction. thus, the present study presents an algorithm that allows to determine the optimal dosage of the vee virus-derived vaccine and the moment of its introduction that will ensure the maximal rate of tumor cells’ death, building on the mathematical models of anti-tumor therapy and vaccine therapy developed in [7-9]. 2. the mathematical model of the vee virus vaccine therapy the mathematical model of vaccine therapy describes the mechanism of two-stage death of tumor cells caused by the virus itself and as a result of the immune system stimulation against the infected tumor cells. based on the experimental data on the growth of erlich adenocarcinoma in mice after a single introduction of the vee virus vaccine, one day after the tumor was transplanted to animals (figure 1), the parameters of the model were estimated and its adequacy was tested using experimental dosages of the vee virus vaccine [8-11]. the values of the model parameters are shown in appendix 1. fig. 1. experimental curves of the ehrlich adenocarcinoma growth without vaccine administration (control) and after single vaccine administration the mathematical model of the two-stage death of tumor cells after the viral vaccine introduction is described by the following system of differential equations (1-8) [7-10]: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) v v v v v av v v v v n n an n n n dn t t n t t t t t k v t t k a t t n t t t t dt t t k a t t n t t       =   − + − − − − −  − − +   + − − −  −  (1) where ( )v t is the number of viruses, ( )n t is the number of tumor cells without vaccine introduction, described by the following differential equation: ( ) ( ) ( ) dn t t n t dt =  , at 0 0( )n t n= (2) where  (t) is the rate of tumor cells’ proliferation, analytical study of antitumor viral vaccine 61 ( )vn t is the number of infected tumor cells after the vaccine introduction, 1v cvt z= + is the moment of the start of immune response against viruses, 1n cnt z= + is the moment of the start of immune response against tumor cells, 1 is the moment of vaccine introduction, cvz and cnz are the time lag of immune response against the virus and infected tumor cells, ( )v va t t− is the number of antibodies against the virus, ( )n na t t− is the number of antibodies against infected tumor cells, ( )vt t − and ( )nt t − are the heaviside step function, avk , ank , and vk are the dimension factors. the first component of equation (1) describes the dynamics of the tumor cells’ growth before the introduction of the viral vaccine. the second component of equation (1) describes the death of infected tumor cells caused by the virus. the first stage of the immune system stimulation is production of antibodies against the virus (figure 2 – ( )vc t , ( )va t ). at this stage the death of tumor cells is caused by the penetration of the virus inside the tumor cells (figure 3). the third component of the equation describes the death of tumor cells caused by antibodies produced against the infected tumor cells (figure 2 – ( )nc t , ( )na t ). reproducing in tumor cells, the viruses induce the development of new protein formations on the surface of the tumor cell membranes. such infected tumor cells are perceived by the immune system as foreign to the body. as a result, immune system produces antibodies specific to the tumor cells, and they destroy the infected tumor cells (figure 3). the patterns of immune response to the foreign cells are based on marchuk's mathematical model of infectious diseases [24-27]. this model treats the immune response (production of t and b cells) as a combined action. therefore in the present study the immune response means the formation of antibodies and plasma cells specific to the vee virus and infected tumor cells. using marchuk’s model, the dynamics of the number of viruses is described by the following equation: ( ) ( ) ( ) ( ),v v v dv t v t a t v t dt  = − (3) where 0 1( )v v = is the initial dose of the viral vaccine, 1 is the moment of the first administration of the viral vaccine, v is the rate of viruses’ reproduction within the cell, v is the death rate of viruses when they interact with antibodies ( )va t . in the vaccine therapy model, the initial condition 0 1( )v v = denotes the dosage of the vaccine. the immune response of the body to the virus introduction results in production of antibodies and plasma cells. their numbers are calculated using the following four differential equations [24-25] for each of the two stages of the immune response – against the virus and against the infected tumor cells. the dynamics of the number of antibodies to the virus ( )va t : 1 ( ) ( ) ( ) ( ) ( ),v a v v av v v v v v da t c t t a t t v t t a t dt    = − − − − − − (4) 62 n.a. babushkina, e.a. kuzina where a is the rate of formation of antibodies from a single plasma cell, av is the death rate of antibodies due to interaction with viruses ( )va t , v is the rate of decrease in the number of antibodies due to natural destruction. the dynamics of formation of plasma cells cv(t): 1 1 ( ) ( ) ( ) ( )v vc v cv v vn dc t v t a t t c t c dt         = − − − − − , at 1 ,( )v vnc c = (5) where с is the rate of formation of plasma cells, cv is the dimension factor, cvz is the time lag of the immune response to the formation of a plasma cell clone. the second term of this equation defines the deviation of the actual number of plasma cells from the norm cvn. the equation for the dynamics of the number of antibodies ( )na t acting against infected tumor cells is as follows: ( ) ( ) ( ) ( ) ( ),n an n n an n n v n nn n n da t c t t a t t n t t a t t dt   = − − − − − − (6) where an is the rate of formation of antibodies from a single plasma cell, an is the death rate of antibodies ( )na t due to interaction with infected tumor cells ( )vn t , nn is rate of decrease in the number of antibodies due to natural destruction. the dynamics of formation of plasma cells ( )nc t : , ( ) ( ) ( ) ( )n cn v n n cn n n nn dc t n t a t t c t t c dt        = − − − − (7) at 1( ) ,an nnc c = where cn is rate of formation of plasma cells, cn is the dimension factor, zcn is the time lag of the immune response to the formation of a plasma cell clone against the infected tumor cells. dynamics of antibodies ( )va t against the virus and against infected tumor cells ( )na t are shown in figure 2. fig. 2. population dynamics of antibodies ( )va t against the virus and against infected tumor cells an(t) for experimental dosage 0 0,015v = on 1 = day 1 (the first and second stages of the immune response) analytical study of antitumor viral vaccine 63 in the equations that describe the dynamics of the formation of antibodies (6) and plasma cells (7), the number of infected tumor cells was calculated as ( ) ( ) ( ),v n nn t n t p t=  where ( )np t accounts for the proportion of rapidly proliferating cell fraction as the size of the tumor increases [14]: 2 2 21 ( ) 1 ( ) , 1 p p p p t p t arctg k t       = −   −    (8) where p and kp are constant parameters, t is the current time of the tumor cell population growth, * 1 ,p t  = where *t is the moment when the numbers of the rapidly and slowly proliferating cells are equal. the fraction of rapidly proliferating tumor cells is located near the blood vessels (oxygenated fraction), while the fraction of slowly proliferating cells is pushed to the tumor periphery (hypoxic fraction). the process of the change in the proportion of the rapidly and slowly proliferating cells with the increase in the tumor size is described in the skipper model [31]. as the number of the fast-proliferating tumor cells decreases, the virus ceases to affect them. this causes the decrease of the tumor sensitivity to the viral vaccines and of the number of infected tumor cells ( )vn t that can induce the immune response. as a result, the number of the antibodies developed against tumor cells is reduced, and the effectiveness of the viral vaccines declines. thus, the mathematical model of vaccine therapy describes the mechanism of two-stage death of tumor cells caused by the virus itself and as a result of the immune system stimulation against the infected tumor cells. the calculated curves that describe the dynamics of the two-stage death of infected tumor cells ( )vn t and the production of antibodies and plasma cells at the each of the two stages of the immune system stimulation are shown in figure 3 [8-10]. fig. 3. calculated growth curve of the total number of tumor cells n(t) (full line) that approximates experimental data (+) based on the mathematical model of antitumor therapy (dosage 0 0,015v = on 1 = day 1); the dotted line indicates the dynamics of the number of infected tumor cells ( )vn t . 64 n.a. babushkina, e.a. kuzina calculations show that the curve that describes the dynamics of the number of infected tumor cells is located below the experimental curve of the tumor growth. this indicates that when the viral vaccine is administered, it does not infect the entire tumor cell population ( ).n t some cells remain uninfected ( )rn t and continue to multiply, causing a repeated growth of the tumor, as observed in the experiment (figure 1). some cells survive ( 0 vrn – after the 1st stage of immune response, 0 nrn – after the 2nd stage), causing a repeated growth of the tumor as observed in the experiment (figure 1). 3. the mathematical model of antitumor therapy with discontinous trajectories for assessing the tumor growth delay after the two-stage death of tumor cells the mathematical model of antitumor therapy with discontinuous trajectories was developed to control and optimize the regimes of applying different antitumor therapy methods based on kinetic curves of tumor growth [8-10]. figure 3 shows the calculated growth curve of the total number of tumor cells n(t) (full line), which was obtained based on the mathematical model of antitumor therapy by approximating the experimental data. the duration of tumor growth delay was estimated based on the following assumptions adopted in the mathematical model of antitumor therapy [8-10]: 1. the cell death occurs instantly at each of the two stages of the immune system stimulation. the abrupt decrease in the tumor size occurs at the moment when the maximal number of antibodies to the virus and infected tumor cells is produced. 2. reduction in the tumor size occurs only due to the death of the virus-infected tumor cells. 3. the surviving tumor cells continue to grow at the same reproduction rate. the kinetic curves of their growth are described by the gompertz function, keeping the parameters of the tumor growth without treatment (control), but accounting for the time shift caused by tumor growth delay 0( )v v and 0( )n v after each stage of their death (figure 4). according to the basic antitumor therapy model [8-10], the kinetic curves of tumor growth without treatment are described by a simple differential equation ( ) exp( ) ( )n n n dn t t n t dt   = − , at 0(0)n n= , (9) where 0n is the initial number of tumor cells at the time of tumor transplantation to animals at t = 0, n(t) is the number of cells in the tumor at time t, n and n are the parameters of the gompertz function, which approximates the experimental kinetic curve of ehrlich adenocarcinoma growth without treatment (figure 1) [5-7, 10-14]. based on the assumptions listed above, tumor growth after the introduction of viral vaccines follows a differential equation describing the two-stage death of tumor cells: ,0 1 0 , 1 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( )) ( ) ( ) ( ) ( ) ( ) ( ( )), v v v v v v n n v n n n n dn t t n t t t s v n t t t t t n t v t t dt s n n t t t t t n t v            = − − − + − − − − − − + − − (10) where ( )n t is the total number of tumor cells at time t, ( )t is the growth rate of a given type of a tumor, ( )vt t − and ( )nt t − is the heaviside step function, 1v cvt z= + and 1n cnt z= + are the moments of the tumor cells' death at each of the two stages of the immune system stimulation, 1 is the moment of the viral vaccine introduction, analytical study of antitumor viral vaccine 65 cvz and cnz denote the time lag of the immune response against the virus and infected tumor cells, ( )vt t − and ( )nt t − are the dirac delta function describing instant death of tumor cells at each of two stages of the immune system stimulation, ,0( )v vs v t and ,0( )v ns v t are the relative decrease in the tumor size at the time the tumor cells’ death at each of the two stages, ,0( )v vv t and ,0( )n nv t are the tumor growth delay after cells’ death at each of the two stages of the immune system stimulation (figure 3). in the vaccine therapy model, it was assumed that the relative decrease in tumor volume occurs only due to the death of the virus-infected tumor cells nv(t). thus in this model the number of dying cells was calculated as the difference between the maximum and minimum number of infected cells: 1 2( ) ( ) ( )v v v vv vn t n t n t = − and 1 2( ) ( ) ( )n n v vv nn t n t n t = − , (11) where 1 vt and 1 nt are the starting moments for the immune response against the virus and against the infected tumor cells, 2 vt and 2 nt are the ending moments of the immune response. according to the model of antitumor therapy, the relative decrease in tumor size at the time of the tumor cells’ death at each of the two stages was determined from the following equations: 0 ( ) ( , ) ( ) v v v v v n t s v t n t  = и 0 ( ) ( , ) , ( ) v n n n n n t s v t n t  = (12) where ( )vn t and ( )nn t denote the total number of cells in the tumor at the time points vt and nt . the death of tumor cells causes tumor growth delay, and its duration serves as a quantitative measure of the cells’ death depending on the dosage of the viral vaccine. the tumor growth delay ,0( )v vv t and ,0( )n nv t equals the time interval between the cells’ death and their subsequent recovery to the original size at the moment of vaccine introduction (figure 3): 0( ) ( ( , ))v v v vn t n t v t= + и 0( ) ( ( , )).n n n nn t n t v t= + (13) then the tumor growth after each stage of the infected cells' death is described by gompertz equations with a time shift that accounts for the duration of the tumor growth delay as follows: 0 0( ) exp( (1 exp( ( ( ))))),v v n n v vn t t rn t v  − = − − − (14) 0 0( ) exp( (1 exp( ( ( )))))n n n n n nn t t rn t v  − = − − − (15) where 0 vrn and 0 nrn is the number of surviving cells after the 1st and the 2nd stages of immune response. figure 3 shows the calculated curve that describes the total number of tumor cells n(t) (upper curve) derived from the mathematical model of antitumor therapy with discontinuous trajectories of tumor growth by approximating the experimental data. this curve describes the tumor growth n(t) following three gompertz equations (one for a control group) with a time shift that accounts for the tumor growth delay after each stage of the cells’ death [8-10]. appendix 1 shows the values of the parameters for the vaccine therapy model and the antitumor therapy model, estimated based on the experimental data provided in [10-14]. 66 n.a. babushkina, e.a. kuzina 4. algorithm for calculating effective administration regimens for viral vaccines mathematical model of vaccine therapy and mathematical model of antitumor therapy with discontinuous trajectories of tumor growth were developed in matlab-simulink to conduct a computing experiment (in silico). the experiment aimed to identify the efficient vaccine therapy regimes for a wide range of vee virus dosages and for different moments of vaccine introduction. the efficiency of the vaccine therapy was assessed after each of the two stages of immune response based on two parameters: 1 – number of produced antibodies ( )va t and ( )na t , 2 – number of dying tumor cells ( )v vn t and ( )v nn t . the efficiency of vaccine therapy method was estimated by two criteria. the first was the number of antibodies formed by the immune system at each of the two stages of its stimulation. we needed to determine the dosage that results in the maximum number of antibodies against infected tumor cells [10-13]. when this condition is met, the tumor-bearing organism is able to restrain the growth of the surviving tumor cells. the second criterion was the tumor growth delay. it depends on the number of dying cells on each of the two stages of immune response. therefore, the algorithm for calculating optimal regimen of the viral vaccine is based on defining a maximal effective dosage when the immune system produces the biggest number of antibodies. 4.1. calculating the viral vaccine dosage using the criterion of the production of antibodies against infected tumor cells based on the model of vaccine therapy with a single introduction of the varying dosages of the viral vaccines at the varying moments (from the 1st to 35th day), the graphs that show the changes in the number of antibodies against the virus ( )va t (figure 4) and antibodies against infected tumor cells ( )na t (figure 5) were developed [14]. these graphs reflect two fundamentally different types of the production of antibodies against the virus and against the tumor cells, depending on the vaccine dosage. the number of antibodies against the virus increases in proportion to the dosage increase, i.e. there is a linear dependence between these variables (figure 4). however, the increase in the number of antibodies against infected tumor cells 0 1( , )na v  is non-linear, and depends on the tumor size at the moment of vaccine introduction (figure 5). this non-linear dependence allows determining the virus dose max 0v = 0,024 that results in the maximal production of antibodies against the infected tumor cells. analytical study of antitumor viral vaccine 67 fig. 4. maximum number of antibodies against the virus max 0( )va v depending on the dose 0v administered on day 1 (the first stage of the immune system stimulation) fig. 5. maximum number of antibodies against infected tumor cells max 0( )na v depending on the dose of 0v administered at different moments (the second stage of the immune system stimulation); 1 = 1, 5, 15, 25, 35 days the non-linear dependence of production of antibodies on the dosage increase 0 1( , )na v  may be due to the changes in the number of infected tumor cells on the 1st and 2nd stages of immune response. when the dosage is small, the number of the dying cells is low, but many cells get infected by the virus. with the further increase of the dosage exceeding max 0v = 0,024, the number of cells dying at the 1st stage of immune response increases. this leads to reduction of the number of infected cells, and the consequent decrease in the number of antibodies against these cells on the 2nd stage of immune response. thus, with the administration of significant doses of the virus, almost the entire population of tumor cells is killed already at the 1st stage of immune response. this leads to a rapid and effective destruction of the tumor, but doesn’t stimulate the 2nd stage of immune response; therefore the antibodies against the infected tumor cells are not produced. 68 n.a. babushkina, e.a. kuzina the effectiveness of production of the antibodies against the infected tumor cells depends on the tumor size. as indicated by the calculations [14], after 35 days the number of fastproliferating tumor cells starts to decrease, and this leads to the decline in the number of infected tumor cells and antibodies against them at the later stages of vaccine therapy. thus, the calculations using the mathematical model of vaccine therapy allowed to define the maximal effective dosage of the vee virus-derived vaccine and to identify the moment of its administration that results in the maximal number of infected tumor cells maxt [14]. based on the model of vaccine therapy the graphs that show the number of infected tumor cells that die at the 1st and 2nd stages of immune response were developed based on equation (11) (figures 6 and 7). fig. 6. calculated curves describing the dynamics of the tumor cells’ death 0( )v vn v depending on the dosage on 1 = day 1, 15, 25 (1st stage of immune response) fig. 7. calculated curves describing the dynamics of the infected cells’ death 0( )n vn v depending on the dosage on 1 = day 1, 15, 25 (2nd stage of immune response) analytical study of antitumor viral vaccine 69 there are fundamental differences in the dynamics of the tumor cells’ death on each stage of the immune response. on the 1st stage the number of dying cells increases with the increase in the dosage, and depends on the tumor size at the moment of vaccine introduction (figure 6). on the 2nd stage the number of dying cells declines due to the decreasing number of produced antibodies, as shown by the 0 1( , )na v  graphs (figures 5 and 7). 4.2. kinetic curves of tumor cells’ growth after 1st and 2nd stages of immune response a model of antitumor therapy with discontinuous trajectories (equations (9)-(16)) was used to forecast the dynamics of the tumor growth after the introduction of the viral vaccines in different regimes. calculated curves of tumor growth are shown in appendix 2. the model includes a threshold value 0 cut offn − that indicates the minimal number of the tumor cells that can trigger the renewed experimental tumor growth. after each of the two stages of tumor cells’ death the tumor can resume its growth if the number of surviving cells 0 vrn and 0 nrn exceeds the threshold 0 cut offn − . threshold value depends on the normal level of plasma cells vnc and nnc (equations (5) and (7)). threshold value was defined based on proportion 0 0 0,76 0,4 2 2 cut off n n − = = = , where 0n is the initial number of tumor cells (equation (2)). if the number of surviving cells is below the threshold, then the tumor will not be able to resume its growth. in the model this is taken as complete elimination of tumor cells. in this case the lifetime of the vaccine-treated animals will be equal to the average lifetime of the experimental animals without treatment. the resulting kinetic curves that describe tumor growth after vaccine introduction on 1 = 1 day (figures 8-15) allow to identify two dosage ranges that determine two possible regimens for administering antitumor vaccines. the first regimen (with dosages exceeding exp 0 03 0,045v v=  = ) makes it possible to ensure that at the 1st stage of immune response the number of tumor cells drops below the threshold 0 cut offn − (figures 8-15), which is equivalent to elimination of tumor cells. the second regime (with dosages between exp 0 0,015v = and exp 03 0,045v = ) does not produce the decline in the number of tumor cells below the threshold level 0 cut offn − . however, these dosages cause the death of the tumor cells at both stages of immune response, and this initiates the production of antibodies against infected cells. when these antibodies exist, the process of tumor growth can be controlled through recurrent introductions of the viral vaccine. this regimen will stabilize the initial tumor size. if the initial tumor size is small, this regimen makes it possible to avoid the surgical intervention, with vaccine therapy acting as a regular vaccination against autologous tumor cells. the calculations show that if the vaccine is introduced on 1 =25 day (appendix 2) with dosages between exp 0 0,015v = to 0 0,1v = , the decrease of the number of tumor cells below the threshold level is not possible. therefore if the tumor size is big, it needs to be surgically removed. however, if the surgical intervention is contraindicated, the tumor size can be stabilized through recurrent vaccinations. in any case, this decision rests with the physician. 5. time frame for recurrent introduction of the viral vaccines a model of antitumor therapy with discontinuous trajectories of tumor growth (equations (9)(16)) was used to identify the time frame for recurrent viral vaccine introductions. the time length between the recurrent vaccine introductions t is equal to the interval between the moment of initial introduction 1 and 2 – the moment when the tumor size n( 1 ) = n( 2 ). 70 n.a. babushkina, e.a. kuzina kinetic curves of tumor growth after the viral vaccine is introduced on 1 =1 day, with dosages between exp 0 0,015v = to exp 0 02,67 0,04v v=  = , indicate that the time frame for subsequent introduction lies in the range of t = 14,5 days to t = 18,9 days depending on the dosage. therefore, if the tumor size is small, it can be stabilized through recurrent introduction of these dosages within 14,5 – 18,9 days. when the tumor is already big, its size can be stabilized while it is dominated by the fraction of the fast-proliferating cells ( * 35t = days). if the vaccine is initially introduced on 1 =25 day, stabilizing regimen can be implemented through subsequent administration of dosages that produce two-stage immune response. introduction of dosages in the range of exp 0 0,015v = to exp 0 03 0,045v v=  = within the time frame of t =30,5 can stabilize the tumor size on its pretreatment level as of day 25. this allows to avoid surgical removal of the tumor. 6. prediction of the frequency of repeated viral vaccine administration for the clinical setting the model used in this study allows to calculate the time frame for recurrent viral vaccine introductions for experimental animals. in humans, these time intervals will be different. however, it is possible to calculate their approximate duration by applying allometric relationship (16) that describes the dependence of the metabolic processes in mammals on their body weight [32, 33]: bm a =  (16) where m is the mammal's body weight,  is the speed of the mammal's metabolic processes, a and b are constant parameters. research indicates that body weight and lifespan also follow an allometric relationship for both humans and laboratory mice [32-34]. if we find the logarithm of equation (16), we obtain a linear equation that connects the average body weight of various mammals and humans with their average life span tl (figure 8): ln ln ln lnm b tl a b tl c=  + =  + (17) regression analysis allowed to define the value of regression coefficient b = 2,4981, which characterizes the slope of the regression line. figure 8. dependence of the life span on body weight for ○ mice, □ rats, + guinea pigs, х rabbits, ◊ dogs, and * humans analytical study of antitumor viral vaccine 71 allometric relationships that define the tumor growth delay for mice and humans are as follows: 0( )human human bm a t v=  and 0( )mouse mouse bm a t v=  (18) where 0( )humant v is the duration of the tumor growth remission in a human after the viral vaccine administration in clinical oncology, 0( )mouset v is duration of tumor growth delay in a mouse in the experiment. the duration of tumor growth delay for a mouse and a human can be calculated using the following proportion 1 0 0 ( ) ( ) human human b mouse mouse t v m t v m   =      this proportion makes it possible to calculate the duration of tumor remission in humans based on the estimated tumor growth delay in mice (experimental group), average body weight of mice (experimental group), and human body weight: 1 0 0( ) ( ) human b human mouse mouse m t v t v m    =      (19) for the vaccine dosage max 0 0 0,024v v= = that produces the maximal number of antibodies against infected tumor cells, the periods between recurrent introductions for a mouse and a human will be as follows: for small tumors, when the vaccine is initially administered on 1 =1 day: 0( )mouset v = 16,7 days, 0( )humant v =1,3 years; for bigger tumors, when the vaccine is initially administered on 1 =25 day: 0( )mouset v = 30,5 days, 0( )humant v =2,1 years. the period for the repeated vaccine introduction can be treated as a time when a patient needs to make a return visit to the therapist for examination. examination will indicate whether the repeated administration of the vaccine is needed. 7. conclusion the software developed in matlab-simulink based on two mathematical models describing tumor growth after the introduction of antitumoral viral vaccines makes it possible to study the effectiveness of vaccine therapy for a wide range of dosages and times of introduction. the proposed algorithm for defining the optimal regimens of viral vaccine administration can be used for various types of experimental tumors and types of antitumor viral vaccines. the results of the computing experiment presented in this paper make it possible to assess the effectiveness of different regimes of vaccine therapy. they suggest two strategies for the viral vaccine treatment. the first strategy allows a complete elimination of the tumor cells. this strategy implies a single-shot administration of a high dosage that produces the death of tumor cells at the 1st stage of immune response. the second strategy does not lead to the complete elimination of the tumor, but makes it possible to stabilize its size through recurrent introduction of the viral vaccine. if the initial tumor size is small, this stabilization gives the possibility to avoid the surgical intervention, with vaccine therapy acting as a regular vaccination against autologous tumor cells. the present study also proposes the method of using experimental results to predict the duration of remission for the humans, based on allometric relationships. this method makes it 72 n.a. babushkina, e.a. kuzina possible to define the time when a patient needs to make a return visit to the therapist for repeated examination and decision on whether further antitumor vaccine administration is required. the development of high-tech methods for the treatment of oncological diseases is directly related to the cost of experimental 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(2016) programmnaja realizacija metoda prognozirovanija jeffektivnosti vakcinoterapii ot jeksperimenta v kliniku [software implementation of the method for predicting the efficacy of vaccine therapy from experiment to the clinic]. proceedings of the xii int. conf. “physics and radioelectronics in medicine and ecology” phreme-2016. vladimir, russia, 1, 125-130, [in russian] analytical study of antitumor viral vaccine 75 appendix 1. values of the models' parameters after a single introduction of the viral vaccine on the first day of the tumor growth equation parameter description ( )dn t dt – (9) n = 3.3613, n = 0.0332 n= 23 0n = 0.75 parameters of the gompertz function approximating the experimental growth curves of the tumor cell population without the vaccine administration (control sample) ( )vdn t dt – (1) 1 = 1 moment of the first introduction of the viral vaccine cvz = 4.5 time lag of the immune response against the virus cnz = 10.5 time lag of the immune response against the infected tumor cells 1v cvt z= + starting moment of the immune response against the virus 1n cnt z= + starting moment of the immune response against infected tumor cells ( )dv t dt – (3) v = 0.1 rate of virus replication v = 15 death rate of viruses due to their interaction with antibodies 0v = 0.015 dosage of the viral vaccine – the initial condition of equation (2) ( )vda t dt – (4) a = 100 rate of antibodies’ formation from a single plasma cell av = 70 rate of decrease in the number of antibodies due to interaction with viruses a = 5 rate of decrease in the number of antibodies due to natural destruction max va = 1.05 maximum calculated number of antibodies ( )vdc t dt – (5) c = 100 rate of antibodies’ formation from a single plasma cell cv = 4.5 dimension factor vnc = 0.001 initial number of plasma cells ( )nda t dt – (6) an = 30 rate of antibodies’ formation from a single plasma cell an = 6.2 rate of decrease in the number of antibodies due to interaction with tumor cells nn = 6.3 rate of decrease in the number of antibodies due to 76 n.a. babushkina, e.a. kuzina natural destruction max na = 4.6445 maximum calculated number of antibodies ( )ndc t dt – (7) cn = 76.677 rate of formation of plasma cells cn = 38 dimension factor nnc = 0.0001 initial number of plasma cells ( )vdn t dt – (1) vk = 0.25 avk = 0.8 ank = 0.8 constant coefficients of the dynamic equation for infected cells after a single introduction of the vaccine ( )p t – (8) p = 0.3 pk = 0.95 p = 1/t* parameters of function p(t), describing the dynamic equation for reduction of the fast-proliferating fraction of tumor cells t*= 35 days moment when the number of fractions of rapidly proliferating cells is equal to the number of fractions of slowly proliferating cells ( )dn t dt –(10 ) v = 4.3 days length of the tumor cells’ growth delay after their death at 1st stage of immune system stimulation cn = 9.8 days length of the tumor cells’ growth delay after their death at 2nd stage of immune system stimulation sum = 14.1 days total length of the tumor cells’ growth delay (1st and 2nd stage of immune system stimulation) 1 vt = 5.82 1 nt = 12.32 starting moments of the 1st and the 2nd stage of immune response 2 vt =7.13 2 nt =14.288 ending moments of the 1st and the 2nd stages of immune response 1( )v v vn t =1.3125 1( )n n vn t =1.1458 maximal numbers of infected tumor cells before the start of the 1st and the 2nd stages of immune response 2( )v v vn t = 0.84 2( )n n vn t =0.1031 minimal numbers of infected tumor cells before the end of the 1st and the 2nd stages of immune response 1( )v v vn t = 0.472, 1( )n n vn t = 1.043 number of dead infected cells at the 1st and the 2nd stages of immune response analytical study of antitumor viral vaccine 77 appendix 2. kinetic curves of tumor growth kinetic curves are calculated for varying dosages on two moments of introduction: 1 = day 1 (figures (a) – (b)) and 1 = day 25 (figures (c) – (d)). for each dosage, figure (a) shows the dynamics of tumor growth ( )n t after viral vaccine administration (dotted line denotes the dynamics of the number of infected tumor cells ( )vn t ), figure (b) shows the dynamics of the number of antibodies ( )va t against the virus (red line), and ( )na t against infected tumor cells (black line) at two stages of the immune response. dosage 0v = 0,015 а) c) b) d) 78 n.a. babushkina, e.a. kuzina dosage 0v = 0,024 а) c) b) d) analytical study of antitumor viral vaccine 79 dosage 0v = 0,03 а) c) b) d) 80 n.a. babushkina, e.a. kuzina dosage 0v = 0,04 а) c) b) d) analytical study of antitumor viral vaccine 81 dosage 0v = 0,045 а) c) b) d) 82 n.a. babushkina, e.a. kuzina dosage 0v = 0,06 а) c) b) d) analytical study of antitumor viral vaccine 83 dosage 0v = 0,075 а) c) b) d) 84 n.a. babushkina, e.a. kuzina dosage 0v = 0,1 а) c) b) d) advances in systems science and applications (2013) vol.13 no.1 80-99 an extension of a logistic model for microbial kinetics anne talkington1, floyd inman iii2, leonard d. holmes2 and guo wei2 1duke university, durham, nc 27708, usa 2university of north carolina at pembroke, pembroke, nc 28372, usa abstract in contrast to the traditional logistic model, a series of asymmetrical models have been proposed for modeling bacterial growth. these models are similar to the logistic model for the lag-phase and exponential phase of the population growth, but quite different in the stationary phase − the growth becomes remarkably slower after the inflection point. at this point, limiting factors within the population such as competition or environmental stress inhibit further exponential growth. moreover, models as variations of the traditional exponential model are also proposed. the models presented demonstrate more general patterns and representative properties, from which relevant algorithms can be developed for calculating the population specific growth rate occurring in the exponential phase. keywords maclaurin series, logistic growth, exponential growth, point of inflection, carrying capacity, specific growth rate 1 introduction the study of microbial kinetics is of particular significance to research in the fields of microbiology and biotechnology [1]. although existing models and methods have been developed, an issue arises over the precision of potentially subjective, traditional methods of culture analysis. calculating population growth rate is essential to microbial kinetic studies of substrate-culture interactions, and the introduction of a new model for population growth increases the precision of the method. pinpointing this microbial specific growth rate, represented as µ or r, with accuracy and precision is at the heart of consistency in understanding laboratory results. furthermore, extending calculations to model the data uses growth patterns to provide insight into characteristics of the microbe. an exploration of four models incorporates a review and extension of the traditional exponential and logistic growth function, and introduces the potential for exponential-like and logistic-like functions using a modified form of the exponential maclaurin series. several models have been fit for microbial growth curves, modifications of the idealized exponential (malthusian) and logistic (limited) growth to fit the reality of microbial patterns. among growth models, competition models, and nutrient uptake models are included theta logistic functions, trans-theta logistic functions, time-delay logistic functions, general logistic functions. michaelis-menten kinetics, gompertz, von bertalanffy and general von bertalanffy, verhlust, and advances in systems science and applications (2013) vol.13 no.1 81 lotka-volterra are specific cases of these models [2-11]. exponential growth, or unlimited and ever-increasing growth, is represented by the general differential form dp dt = rp , and its solution p (t) = p0e rt[8]. logistic growth is sigmoidal and limited. it is represented by the general differential form dp dt = rp (1 − p m ), and its solution p (t) = mp0ert m+p0(ert−1) [8]. previous extensions to the logistic model involve the general logistic model and the time-sensitive carrying capacity logistic model [8-11]. the general logistic model takes the differential form dp dt = rpα[1 − ( p m )ν ]γ , which is simplified to p (t) = m [1+[(γ−1)βrm(α−1)t+[( m p0 )β−1](1−γ)]1/(1−γ)]1/β or p (t) = a + m−a (1+qe−r(t−t1))1/ν , (t1 is defined as time of maximum growth, a is the value of the lower asymptote, q is an initial condition constant, ν is a constant determining the point of inflection, and α and γ represent constant inflection value parameters) [8]. this introduces new parameters so that the curve can idealize scenarios previously considered “less ideal” by existing models. the time-varying model introduces the upper bound itself as a function of time, and it therefore cannot be definitely determined as an analytical solution. this model further complicates the function with the introduction of a variable parameter. it is represented as dp dt = rp (1− p m(t))[11]. the introduction of a series incorporates the concept of a traditional model modified through extension. this new function is explored as a differential, and characterized by its non-integrable form. the exponential and logistic models represent the first two terms of a maclaurin series. previously, the remaining terms were not considered significant, and the use of a series was disregarded [8]. however, discovering and incorporating the terms as an expanded series removes inconsistencies between methods of calculation for parameters, such as rate of increase. the convergence of the series represents an ideal model, that can be characterized by the properties of several functions, and to which these functions converge. the first terms reflect its most basic exponential and logistic properties. as it continues to expand and the series converges, truncation produces a series of polynomial functions with exponential and logistic-like (sigmoidal) growth patterns. the final closed model itself appears to grow infinitely, like an exponential function, but at a slower rate, like a hyperbolic pattern. indeed, elements of the series resemble the taylor series expansions for the hyperbolic sine (sinh) and cosine (cosh) functions, and the geometric series hyperbolic expansion, in addition to the inherently exponential elements. the potentially hyperbolic properties connect this model to the monod and michaelis-menten models [2], while they provide a basis for parametric data analysis through the function. however, it is not a distinct hyperbola because the values of dp dt approach infinity as p approaches infinity, without any distinct limit or asymptote. this new series relies on the parameters of growth limit, rate of increase, and 82 anne talkington: an extension of a logistic model for microbial kinetics population size related through time. it is unique in that it does not introduce new factors or parameters with each additional term, but builds upon known parameters. manipulation of the series model over its interval of convergence opens applications in both symmetrical and asymmetrical (semi-logistic) growth patterns. analysis of the curve to the upper bound of its convergence (below and including the inflection point) produces a precise estimate for the initial takeoff of the microbe; the rotation of the existing curve and intersection of another curve are two approaches for determining the upper bound. evaluating the series as a whole, or as an individual polynomial function at any point of truncation, allows for the ability to fit both limited and unlimited growth patterns. the models, likewise, are consistent with numerical (nonparametric) methods for determining microbial specific growth rate [12-13]. numerical methods analyze the data without assumption, whereas the models provide a framework for the data. each introduces a degree of precision and understanding of the data. 2 procedure to determine the value of the specific growth rate numerically, an algorithm was developed to analyze the data through elimination. it removes all subjectivity and assumptions found in previous methods. the new algorithm focuses on eliminating non-significant data points. initially, all points past the point of inflection are eliminated. this is defined as the point at which the concavity of the data changes from up to down, or the second derivative changes from positive to negative. thus, the deceleration phase is disregarded in analyzing growth. the next step is to isolate the growth phase from the lag and acceleration phases. further elimination is accomplished through regression. the natural logarithm of the function is taken, and the slope of the linearized function is determined from the line of best fit. points from the left that lower the slope are eliminated, and another regression line is fitted to the remaining points. the process is repeated as the slope increases and iterations continue until either the change in slope (indicative of growth rate) is not statistically significant, or until the number of remaining points becomes too small. the slope of the regression line for the final remaining points is the specific growth rate. 3 cases of the series discussion there are four notable models that serve appropriate data set fit. fitting an appropriate model to the data set is critical for the most accurate approximations of data parameters and descriptions of the populations. the most basic is the exponential model [8, 14-15]. also known as the malthusian model, it depicts unlimited growth. biologically, there is no carrying capacity of the population, and it therefore multiplies infinitely. mathematically, there is advances in systems science and applications (2013) vol.13 no.1 83 no asymptote to describe limiting factors. these may include competition, lack of nutrients, or density-dependent inhibition in the microbial population [16]. the model is represented as a direct proportion between the rate of population change and population size (equation 1, fig.1). dp dt = rp (1) fig.1 graphical representation of the exponential model for microbial population growth of course, the simplest model given above does not reflect most biological growth systems. more appropriate models were thus explored by researcher, as stated below. microbial growth may be logistic [17]. this model is a special case of limited growth in which rate of population decrease mirrors rate of increase. the population grows exponentially but encounters one or more obstacles that prohibit it from increasing indefinitely. as the definite limit m approaches infinity, growth patterns approach the exponential model. logistic growth, often used as a standard in population studies, is represented by the differential equation 2 (see also fig.2). dp dt = rp (1− p m ) (2) the logistic growth model introduces the first two terms of a maclaurin series that can be used to model microbial growth. limiting the series to two terms introduces truncation error [14-15]. this error can be reduced with the introduction of additional terms. successive terms alternate in sign, and each is smaller in value than the term it follows [14-15]. extending the series another step produces a differential function to the fourth power of p . this function is similar to logistic growth in the lower portion of the curve, up to the point of 84 anne talkington: an extension of a logistic model for microbial kinetics fig.2 graphical representation of the logistic growth model inflection. however, the point of inflection is no longer the midpoint between the upper and lower limits of growth. beyond the point of inflection, growth reduces rapidly and tapers until it reaches its actual limit. the population may have been subjected to extreme pressure over a short period of time, so realized growth does not reach the theoretical, ideal estimate of m . this reduced upper bound is characteristic of higher even degrees of p in the extension of the formula due to factorial growth in the denominator. the end behavior of odd degrees of p mimics exponential-like growth. the fourth-degree formula is represented as equation 3, and its graphical interpretation as fig.3. dp dt = rp − 2rp 2 m(2!) + 8rp 3 m2(3!) − 48rp 4 m3(4!) (3) fig.3 graphical representation of the fourth-degree model the most idealized model is the use of the full infinite series, represented in advances in systems science and applications (2013) vol.13 no.1 85 its closed form for precision. it states that the rate of change of the population and a function of the population size are directly proportional. the function of population size is the maclaurin series. the function, or the series, incorporates the exponential model when m (the upper limit) approaches infinity. it incorporates the logistic model as the ideal symmetrical case in which the point of inflection is the midpoint between the upper and lower limits of growth. if the lower limit is 0, p represents population size at the point of inflection, and m represents the upper growth limit, then this is represented as 2p = m . all of the previous models converge, as a series of overestimates and underestimates, to the model presented by the complete series. it is an exponential-like function, but its predictions for population (p (t)) lie between the exponential and logistic predictions. the modified alternating maclaurin series is convergent for all points up to and including the point of inflection. it satisfies the conditions of the ratio test and leibniz’s theorem for this domain and range [14-15]. past the point of inflection, divergence becomes more extreme with additional terms. as a model, therefore, it accurately produces the lower portion of the curve. if the growth pattern is symmetrical, a 180o rotation about the point of inflection projects the upper portion. if the growth is semi-logistic, it appears as portions of intersecting logistic curves, and the point of intersection is the point of inflection. in this scenario, the bottom portion of the curve is accurately described by the model (equation 4, fig.4). fig.4 graphical depiction of the modified alternating maclaurin series model dp dt = ∞∑ n=0 (−1)n(2nn!)rpn+1 mn(n+ 1)! (4) its expanded form appears as equation 5. 86 anne talkington: an extension of a logistic model for microbial kinetics dp dt =rp − 2rp 2 m(2!) + 8rp 3 m2(3!) − 48rp 4 m3(4!) + 384rp 5 m4(5!) − 3840rp 6 m5(6!) + ...+ (−1)n(2nn!)rpn+1 mn(n+ 1)! (5) its simplified form appears as equation 6. dp dt = rp ∞∑ n=0 (−1)n (n+ 1) × ( 2p m )n (6) its closed form appears as equation 7. dp dt = rln(1 + p m 2 ) m 2 (7) the similarities of the function to the exponential function are evident in the simplified form of the model. the factor rp represents exponential growth. in addition, the behavior of the series is similar to that of exponential growth for sufficiently small values of p (p << 1). dp dt = rln(1 + p m 2 ) m 2 dp dt = r × m 2 × ln(1 + 2p m ) ln(1 + 2p m ) ≈ 2p m dp dt = r × m 2 × 2p m = rp this property does not apply when 2p m is greater than 1, because ln(1 + x) = x− 1 2x 2 + 1 3x 3 − 1 4x 4 + ... holds only for −1 < x ≤ 1. also like an exponential function, the graph of the population p over time is concave up, as justified by the always positive second derivative: d2p dt2 = r m 2 1 1 + p m 2 (1 + p m 2 ) ′ t = r m 2 1 1 + p m 2 1 m 2 dp dt = r2 m 2 1 1 + p m 2 ln(1 + p m 2 ) > 0 however, the rate of growth as p approaches infinity is much slower than that of an exponential function since by limit comparison: lim p→∞ rp r × m 2 × ln(1 + p m ) = lim p→∞ 2p m × ln(1 + p m ) = ∞ advances in systems science and applications (2013) vol.13 no.1 87 it is known that the numerator becomes infinitely large at a faster rate than the denominator. l′hôpital′s rule confirms this assertion by removing the infinite term from the function, to arrive at the definite conclusion: lim p→∞ rp r × m 2 × ln(1 + p m ) = lim p→∞ (rp )′ lim p→∞ (r × m 2 × ln(1 + p m ))′ = r lim p→∞ (r × m 2 × 1 m 1+ p m ) = ∞ this property is the result of the alternating series succession of positive and negative terms. the positive and negative terms indicate the properties of the series at any point of truncation. if truncated up to an even degree of p (say up to pn+1 where n is odd), then the solution p has an upper bound, and an inflection point (see theorem 2). the inflection point will always occur when p = m 2 as justified by the second derivative of p (t) as follows. dp dt = n∑ n=0 (−1)n2nrpn+1 mn(n+ 1) (8) d2p dt2 = n∑ n=0 (−1)n2nr(n+ 1)pn mn(n+ 1) ( dp dt ) = r( dp dt ) n∑ n=0 (−1)n( 2p m )n (9) in the case 2p = m , we have d2p dt2 = r( dp dt ) n∑ n=0 (−1)n hence, when n is odd, it holds that d2p dt2 |p=m 2 = 0 it follows from the chain rule of derivatives [14-15]: d dt ( dp dt ) = d dp ( dp dt )× dp dt that the derivative of dp dt with respect to t is equal to the derivative of dp dt with respect to p . the value for dp dt decreases at it nears its upper bound but never becomes 0. it is always positive (theorem 3). continued simplification of the modified alternating maclaurin series and its truncations, as shown below, reveals its properties and its relationship to the form of a transcritical bifurcation. 88 anne talkington: an extension of a logistic model for microbial kinetics 4 simplified series dp dt = rp ∞∑ n=0 (−1)n n+ 1 ( 2p m )n = rp [1− 1 2 ( 2p m ) + 1 3 ( 2p m )2 − 1 4 ( 2p m )3 + 1 5 ( 2p m )4 − 1 6 ( 2p m )5 + ...] (10) theorem 1 for equation (10), (i) the closed form of the series on the right-hand side is dp dt = rln(1 + p m 2 ) m 2 ; (ii) it holds that dp dt > 0 at any time t. proof (i) equation (10) is simplified to dp dt = r ∑∞ n=0 (−1)n2npn+1 mn(n+1) (by cancelling n!). then we have dp dt = r ∞∑ n=0 (−1)n2npn+1 mn+1(n+ 1) = rp ∞∑ n=0 (−1)n (2pm )n n+ 1 = rp ∞∑ n=0 (−2p m )n n+ 1 = rp [1 + x 2 + x2 3 + x3 4 + ...+ xn n+ 1 + ...](for x = −2p m ) = rp x [x+ x2 2 + x3 3 + x4 4 + ...+ xn+1 n+ 1 + ...] = rp x (−1)ln(1− x) = r m 2 ln(1 + 2p m ) by the taylor expansion of natural logarithm: ln(1 − x) = ∑∞ n=0 xn n for −1 ≤ x < 1. hence, we have r = dp dt ln(1+ p m 2 ) m 2 , and this holds whenever −1 ≤ −2p m < 1, i.e., p ≤ m 2 . thus, the closed form of the series is given by dp dt = rln(1 + p m 2 ) m 2 . (11) (ii) this is implied by the closed form given in (i). theorem 2: for the following tail-truncated equation dp dt = rp n∑ n=0 (−1)n n+ 1 ( 2p m )n (12) advances in systems science and applications (2013) vol.13 no.1 89 (i) if n is odd (so the highest power of p is even), then d2p dt2 is positive when p < m 2 , is zero when p = m 2 , and is negative when p > m 2 . this is an asymmetrical model. (ii) if n is even (so the highest power of p is odd), then d2p dt2 is always positive. since the second derivative is always positive, the first derivative is always increasing. therefore, if the first derivative is positive initially, the first derivative is never zero or negative. (the least upper bound of p is finite and determined as described in theorem 3.) proof it has been established above that in the case 2p = m d2p dt2 = r( dp dt ) ∞∑ n=0 (−1)n = r( dp dt )[1− 1 + 1− 1 + ...] for an even number of terms, or odd n : case 2p = m . the truncation of the above equation results in an even pairing of terms that cancel out, and the derivative is equal to 0. case 2p < m . d2p dt2 = r( dp dt ) ∞∑ n=0 (−1)n = r( dp dt )[1−( 2p m )+( 2p m )2−( 2p m )3+( 2p m )4−( 2p m )5+ ...] a grouping of the first 6 terms of the second derivative results in d2p dt2 = r( dp dt ) [ [1− ( 2p m )] + [( 2p m )2 − ( 2p m )3] + [( 2p m )4 − ( 2p m )5] ] > 0. case 2p > m . d2p dt2 = r( dp dt ) ∞∑ n=0 (−1)n = r( dp dt ) [ 1−( 2p m )+( 2p m )2−( 2p m )3+( 2p m )4−( 2p m )5+... ] a grouping of the first 6 terms results in[ [1− ( 2p m )]+ [( 2p m )2− ( 2p m )3]+ [( 2p m )4− ( 2p m )5] ] < 0. (each bracket is negative.) for an odd number of terms, or even n : the truncation of d2p dt2 = r( dp dt ) ∞∑ n=0 (−1)n = r( dp dt )[1− 1 + 1− 1 + 1− 1 + ...] 90 anne talkington: an extension of a logistic model for microbial kinetics results in a pairing with an additional positive term that is not cancelled. so this second derivative is positive. case 2p < m . d2p dt2 = r( dp dt ) ∞∑ n=0 (−1)n = r( dp dt ) [ 1−( 2p m )+( 2p m )2−( 2p m )3+( 2p m )4−( 2p m )5+... ] a grouping of the first 5 terms of the second derivative results in[ [1− ( 2p m )] + [( 2p m )2 − ( 2p m )3] + ( 2p m )4] > 0. case 2p > m . d2p dt2 = r( dp dt ) ∞∑ n=0 (−1)n = r( dp dt ) [ 1−( 2p m )+( 2p m )2−( 2p m )3+( 2p m )4−( 2p m )5+... ] a grouping of the first 5 terms results in[ 1 + [−( 2p m ) + ( 2p m )2] + [−( 2p m )3 + ( 2p m )4] ] > 0. theorem 3: for the above tail-truncated equation (12), (i) dp dt is always positive (i.e., the right-hand side is always positive). (ii) when n is odd (i.e., including up to an even power of p in the partial sum), the equation rp ∑n n=0 (−1)n n+1 (2pm )n = 0 has a unique positive solution and the supremum value of p is determined by this positive zero of the equation. moreover, it holds that rp ∑n n=0 (−1)n n+1 (2pm )n = rm 2 ∫ 2p m 0 1−xn+1 1+x dx. these models are all asymmetrical. proof (i) if n is even, this is clearly true. let gn (x) = n∑ n=0 (−1)n n+ 1 xn. then dp dt = rp n∑ n=0 (−1)n n+ 1 ( 2p m )n = rpgn ( 2p m ). since xgn (x) = ∑n n=0 (−1)n n+1 xn+1, we have (xgn (x)) ′ = ∑n n=0(−x)n = 1−(−x)n+1 1+x , and hence it holds that[ xgn (x) ] | 2p m 0 = ∫ 2p m 0 ( xgn (x) )′ dx = ∫ 2p m 0 1− (−x)n+1 1 + x dx, advances in systems science and applications (2013) vol.13 no.1 91 implying 2p m gn (2p m ) − 0 =  ∫ 2p m 0 1−xn+1 1+x dx, if n + 1 even,∫ 2p m 0 1+xn+1 1+x dx, if n + 1 odd. therefore, dp dt = rpgn (2p m ) = rp×m 2p ×2p m gn (2p m ) = rm 2  ∫ 2p m 0 1−xn+1 1+x dx, if n + 1 even,∫ 2p m 0 1+xn+1 1+x dx, if n + 1 odd. it is obvious that dp dt = rm 2 ∫ 2p m 0 1+xn+1 1+x dx > 0 when n is even (n + 1 is odd). next we consider the case that n is odd (or n + 1 is even). it follows from the above proof that dp dt = rm 2 ∫ 2p m 0 1−xn+1 1+x dx. when p ≤ m 2 (2pm ≤ 1), dp dt is positive because 1 − xn+1 is always positive except at the single point x = 2p m , at which 1 − xn+1 is zero. we claim that dp dt never becomes 0. in fact, if dp dt becomes 0 at some infimum time t0, it must strictly be after the time at which the inflection point occurs. (at this point, dp dt changes from positive with upward concavity and begins to decrease. the value of the integral, likewise, begins to decrease after the shift in dp dt at this point, corresponding to the concavity, but the value still must be positive. it follows that dp dt cannot be 0 at the inflection point.) since d2p dt2 is still negative at t0, p (t) by definition would have attained its local maximum p (t0) (at a specific p (t0) > m 2 ). now, let t1 be a nearby point of t0, on the right of t0 such that m 2 < p (t1) < p (t0). (since dp dt |t=t0 = 0 and d2p dt2 < 0 after t = t0, dp dt is negative after t = t0). then the result would become 0 = dp dt |t=t0 = rm 2 ∫ 2p (t0) m 0 1−xn+1 1+x dx < rm 2 ∫ 2p (t1) m 0 1−xn+1 1+x dx = dp dt |t=t1 , a contradiction. because the value of p (t1) is less than the value of p (t0), the inequality is inconsistent with the fact that dp dt at t1 must be negative. hence, dp dt never reaches 0, and it follows from the continuity of dp dt that dp dt can never be negative. therefore, dp dt is always positive. proof of (ii) method for determining the supremum of p : let us illustrate the method using two (logistic), four, or six terms. if n = 1, n+1 = 2 (the traditional logistic model), then dp dt = ... = rp (1− p m ). hence, dp dt > 0 if and only if p < m . observe that if p could exceed m , then dp dt would become negative and consequently p would be immediately lowered making dp dt positive. therefore, dp dt is always positive. again, continuity indicates that dp dt must reach and cross through a point of 0 slope to become negative, which is impossible. 92 anne talkington: an extension of a logistic model for microbial kinetics if n = 3, n + 1 = 4 (four terms), then dp dt = rm 2 ∫ 2p m 0 1−x4 1+x dx = rp (1− 1 2 2p m + 1 3( 2p m )2− 1 4( 2p m )3). the equation 1− 1 2x+ 1 3x 2− 1 4x 3 = 0 has a real solution 1.621. for example, this gives the maximum of p as 81.05, assuming m = 100. if n = 5, n + 1 = 6 (six terms), then dp dt = rm 2 ∫ 2p m 0 1−x6 1+x dx = rp (1− 1 2 2p m + 1 3( 2p m )2− 1 4( 2p m )3+ 1 5( 2p m )4− 1 6( 2p m )5). the equation 1− 1 2x+ 1 3x 2− 1 4x 3+ 1 5x 4− 1 6x 5 = 0 has a real solution 1.462. this gives the maximum of p as 73.1, assuming m = 100. generally, for any odd n , dp dt is always positive, and the supremum value of p is determined by a positive real solution (that exists uniquely) of the equation dp dt = rm 2 ∫ 2p m 0 1−xn+1 1+x dx = rp ∑n n=0 (−1)n n+1 (2pm )n. the supremum values decrease as n is increased. rp ∑n n=0 (−1)n n+1 (2pm )n = rm 2 ∫ 2p m 0 1−xn+1 1+x dx cannot be zero when p ≤ m 2 ; the equation ∑n n=0 (−1)n n+1 xn = 0 does not have a solution less than 1 (hence divergence as the partial series extends to infinite form). ∫ α 0 1−xn+1 1+x dx > 0 if α ≤ 1 and ∫ β 0 1−xn+1 1+x dx < 0 if β is (sufficiently) large. there exists γ between α and β such that ∫ γ 0 1−xn+1 1+x dx = 0; ∫ α 0 1−xn+1 1+x dx > ∫ β 0 1−xn+1 1+x dx for α ≤ 1 < β. more precisely, an arbitrary α beyond the point of inflection presents a scenario that is consistent with the definition of a maximum point and downward concavity in this domain. hence, the equation n∑ n=0 (−1)n n+ 1 xn = 0 (13) has a unique positive solution x0. the formula for calculating the supremum of p is psup = x0p = x0 m 2 . in summary, each term is smaller than the term preceding it until p is equal to the least upper bound. at p = m 2 , dp dt has reached its maximum value and will decrease but will still be positive, so the negative terms cannot be larger than the positive terms before them. the slopes remain positive and decrease as dp dt approaches 0 at the least upper bound without ever reaching it. therefore, dp dt will always be positive despite negative concavity (reduction in value) past the point of inflection, and the limit of dp dt as p approaches a set of arbitrary values reveals an approximation of the least upper bound of the function. if the limit reveals that dp dt is less than 0 at a particular value of p , then the least upper bound is below that value; the value can never be reached. at the upper limit, a horizontal asymptote, dp dt is 0. therefore, dp dt approaches this value at the transition from a small positive value to a small negative value. when limit comparison reveals that one value results in a positive dp dt and the next value for p results in a negative dp dt , the bound must lie between these two points. in the advances in systems science and applications (2013) vol.13 no.1 93 logistic case, this limit is m . by contrast, a function with an odd degree of p resembles the properties of an exponential function. it is concave up as well as always increasing; the sum of the alternating terms is positive, plus the addition of a positive term. it has neither a point of inflection nor an upper bound. in terms of the second derivative defined above, the incomplete pairings ensure positive concavity (theorem 2). the series and its truncations can be analyzed as a transcritical bifurcation [18]. the concept of “harvesting”, initially applied to logistic growth, remains. for even degrees of p , the model exhibits logistic-like properties and therefore the potential for two positive zeros. in logistic growth (power of p is 2), zeros are found as the solutions to a simplified, basic quadratic form of the equation. in the general situation (power of p is even and larger than or equal to 4), by the integral representation given in theorem 3 (ii), dp dt = 0 has two positive zeros, one positive zero, or no positive (real) zeros. such zeros in the forms of the series can be determined by utilizing numerical methods. as the degree increases, the amount of symmetry as found in logistic models decreases. as the number of terms in the polynomial increases, and approaches the closed form of the series, the equation becomes increasingly complex to solve. in this parametric approach, fixed points are determined as the variable values that set the equation at equilibrium. a harvesting term introduces a parameter that affects the location of such points, and their position for approaching or separating from one another as the differential increases or decreases. the equation for transcritical bifurcation is as follows (equation 14): dp dt = rp ∞∑ n=0 (−1)n n+ 1 × ( 2p m )n −h (14) because the model is idealized, it can be used as a method to solve for rate of growth. such calculations are consistent with the numerical estimates. like theta logistic models, this model introduces precision through specific terms added to the traditional logistic model. it is unique in that it achieves this precision without the introduction of new parameters and accounts for lack of symmetry (semi-logistic scenarios) through the model itself rather than a term in the formula. this form of the series is appropriate to solve for growth rate of any indicator (equation 15): µ = r = dp dt∑∞ n=0 (−1)n(n!2n)pn+1 mn(n+1)! (15) 94 anne talkington: an extension of a logistic model for microbial kinetics closed form of the series (equation 16): r = dp dt m 2 ln(1 + 2p m ) . (16) consider that: ln(1 + 2p m ) = ln(2) when m = 2p . the modified alternating maclaurin series accurately models the data as it converges, over the domain of initial data collection to the point of inflection p . the model converges with the condition that p ≤ m 2 . the applicability of the maclaurin series as a method for determining growth rate is categorized as one of three cases, based on the relationship between p and m . the following procedure applies for the first case of the modified alternating maclaurin series: if data is symmetrical (ideally logistic) and p = m 2 . 1) the y-coordinate of the point of inflection (value substituted for p ) is the midpoint between the upper and lower limits of growth (asymptotes). it is calculated as lower limit + (1/2) (upper limit lower limit). 2) dp dt represents change in population (growth) over time. it should reach its maximum value at the point of inflection. dp dt should therefore reflect the change at the point of inflection as a tangent line. it is determined more precisely as the mean value of the rates of change surrounding the point of inflection. for even greater precision, arbitrary ranges chosen to include the point of inflection can be averaged and compared; the greatest value among these is taken as dp dt . 3) m is the upper limit of growth. it is identified as a horizontal asymptote. 4) values for p , dp dt and m are substituted into the series to solve for r (µ). example of the first case : fig.5 graphical traditional method: µ=1.0374 advances in systems science and applications (2013) vol.13 no.1 95 graphical traditional method: µ=1.0374 maclaurin series method: p = 3; dpdt = 2.2;m = 6 µ = 1.0580 error (disparity between the methods): 1.99 % the following procedure applies for the second case of the modified alternating maclaurin series: if the point of inflection is less than half of the carrying capacity, or p < m 2 . 1) the y-coordinate of the point of inflection (value substituted for p ) is less than the midpoint between the upper and lower limits of growth (asymptotes). it is calculated as the point in the center of the range of maximum change in population over time (largest values of dp dt ). when averages are taken for dp dt , p should be at or near the center of the range. 2) dp dt represents change in population (growth) over time. it should reach its maximum value at the point of inflection. dp dt should therefore reflect the change at the point of inflection as a tangent line. it is determined more precisely as the mean value of the rates of change surrounding the point of inflection. for even greater precision, arbitrary ranges chosen to include the point of inflection can be averaged and compared; the greatest value among these is taken as dp dt . 3) m is the upper limit of growth. it is identified as a horizontal asymptote. 4) values for p , dp dt , and m are substituted into the series to solve for r (µ). 5) because p is low with respect to m , convergence of the series occurs more rapidly than in the symmetrical first case. the result is more precise at a smaller number of terms. example of the second case : fig.6 graphical traditional method: µ=0.2817 graphical traditional method: µ=0.2817 96 anne talkington: an extension of a logistic model for microbial kinetics maclaurin series method: p = 0.15; dpdt = 0.04;m = 1.3 µ = 0.2964 error (disparity between the methods): 5.22 % the following procedure applies for the third case of the modified alternating maclaurin series: if the point of inflection is greater than half of the carrying capacity, or p > m 2 . 1) the y-coordinate of the point of inflection (value substituted for p ) is greater than the midpoint between the upper and lower limits of growth (asymptotes). it is calculated as the point in the center of the range of maximum change in population over time (largest values of dp dt ). when averages are taken for dp dt , p should be at or near the center of the range. 2) dp dt represents change in population (growth) over time. it should reach its maximum value at the point of inflection. dp dt should therefore reflect the change at the point of inflection as a tangent line. it is determined more precisely as the mean value of the rates of change surrounding the point of inflection. for even greater precision, arbitrary ranges chosen to include the point of inflection can be averaged and compared; the greatest value among these is taken as dp dt . furthermore, the use of means contributes to the robustness and applicability of the symmetrical case. 3) m is the upper limit of growth. it is identified as a horizontal asymptote. in this case, use of the horizontal asymptote results in divergence of the series. therefore, ratios are implemented to calculate r or µ. let the horizontal asymptote be denoted as m0. set m1, an alternate limit, as equal to 2p . 4) values for p , dp dt , and m1 are substituted into the series to solve for r (µ). the series converges as in the symmetrical first case. 5) the use of m1 results in an inflated value for r (µ). to account for the substitution of m1, the result of the series is divided by the value of m1 m0 . example of the third case : graphical traditional method: µ=0.1704 maclaurin series method: p = 0.38; dpdt = 0.05;m0 = 0.61;m1 = 0.76 µ = 0.1523 error (disparity between the methods): 10.62 % 5 limitations irregular data is the greatest limitation to any method of determining microbial kinetics. environmental factors such as ph inconsistency, culture contamination, advances in systems science and applications (2013) vol.13 no.1 97 fig.7 graphical traditional method: µ=0.1704 or other laboratory error introduce complications to the data [16]. furthermore, microbial cultures are inherently not ideal. the use of an ideal model provides a basis of comparison by which inconsistencies may be determined. a numerical method strictly relies upon the data. however, irregular data will produce an irregular result. neither method will produce an accurate or precise estimate. the expansion of the maclaurin series is currently under investigation to extend the model beyond the point of inflection. successful extension would eliminate divergence as a limitation to the microbial growth model and its applications. 6 conclusion microbial growth kinetics is the numerical analysis of a complex living system. a data-based approach is significant as it determines microbial growth rate through a procedure designed to fit the data exactly. the use of models is essential to understanding the population structure through the relationship of population, population growth rate, carrying capacity, and time. population, and the population growth rate, change as a function of time. the models relate this through direct variation and a more complex function. the explored models are built upon exponential and logistic growth, and related through the transcritical bifurcation. each model is unique in its ability to predict and determine the growth of a particular microbe. culturing technique is also a factor in that it imposes or prevents natural limiting factors. batch cultures, and secondary metabolite fed batch, for example, display a pattern that may fall into the logistic, the symmetrical series, or the fourth-degree model. primary metabolite and continuous cultures tend towards exponential-like growth patterns, as modeled by exponential growth or the closed form of the modified alternating maclaurin series. for calculations using these models, the point of manipulation of culture conditions 98 anne talkington: an extension of a logistic model for microbial kinetics represents the point of environmental change, indicated graphically as the point of inflection. in full expansion, the discussed models converge to the curve of the modified alternating maclaurin series, up to the point of inflection. the partial curve is precise for this domain and range, and presents a scenario of semi-logistic analysis viewing the growth curve as the intersection of separate curves. in all cases, the models can be solved for the microbial specific growth rate, and yield precise results consistent with traditional methods of analysis. acknowledgements the authors would like to thank farm bureau robeson for their support. references [1] brock td. 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(2011), “trans-theta logistics: a new family of population growth sigmoid functions”, acta biotheor, vol.59, no.3-4, pp.273289. doi 10.1007/s10441-011-9131-3. [8] tsoularis a. (2001), “analysis of logistic growth models”, res. lett. inf. math. sci., vol.2, pp.23-46. [9] birch c.p.d. (1999), “a new generalized logistic sigmoid growth equation compared with the richards growth equation”, annals of botany, vol.83, no.6, pp.713-723 advances in systems science and applications (2013) vol.13 no.1 99 [10] yin x, goudriaan j, lantinga e.a, spiertz h.j. (2003), “a flexible sigmoid function of determinate growth”, annals of botany, vol.91, no.3, pp.361371. [11] yukalov v.i, yukalova e.p, sornette d. (2009), “punctuated evolution due to delayed carrying capacity”, physica d, vol.238, no.17, pp.1752-1767. [12] panikov ns. (1995), microbial growth kinetics, chapman & hall, london. [13] perni s, andrew pw, shama g. (2005), “estimating the maximum specific growth rate from microbial growth curves: definition is everything”, food microbiol , vol.22, no.6, pp.491-495. doi: 10.1016/j.fm.2004.11.014 [14] stewart, j. (2008), calculus, early transcendentals(6th ed.), belmont, california: thomson learning. [15] demana f. d, finney r. l, kennedy d, & waits b. k. (2003), calculus: graphical, numerical, algebraic, upper saddle river, new jersey: prentice hall. [16] gause g. f. (1934), struggle for existence, hafner, new york. [17] gause g. f. (1932), “experimental studies on the struggle for existence. i. mixed populations of two species of yeast”, j. exp. biol., vol.9, no.4, pp.389-402. [18] guckenheimer j. and holmes p. (1997), nonlinear oscillations, dynamical systems, and bifurcations of vector fields, 3rd ed. new york: springerverlag. corresponding author guo wei can be contacted at: guo.wei@uncp.edu. adv syst sci appl 2018; 04; 74-91 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/617 modeling and simulation of frequency response masking fir filter bank using approximate multiplier for hearing aid application raghavachari ramya1, sridharan moorthi*2 1) vlsi systems research laboratory, department of electrical and electronics engineering, national institute of technology, tiruchirapalli, india e-mail:407114003@nitt.edu 2) vlsi systems research laboratory, department of electrical and electronics engineering, national institute of technology, tiruchirapalli, india e-mail: srimoorthi@nitt.edu received july 20,2018; revised december 15,2018; published december 31,2018 abstract: the tremendous increase in the use of portable electronic devices is due to the development in the fields of signal processing and electronic technology. these battery operated devices needs reduction in power consumption with increased performance and long battery life. since cmos technology scaling fast approaches its physical limit of minimum supply voltage and smaller feature size, the hardware designer has to opt for new multiplier architectures for achieving low power and high speed performance. this paper proposes an area and power efficient approximate multiplier architecture. the error metrics and circuit characteristics are estimated to verify its performance advantage over other approximate multipliers. using frequency response masking approach, a 6-band non-uniform digital fir filter bank is developed using approximate multiplier for hearing aid application. audiogram matching is done with audiograms of two different types of hearing losses and the matching error is computed. simulation results show that the audiogram matching error falls within +/4 db range. keywords: approximate multiplier, audiogram, filter bank, hearing aid, wordlength 1. introduction audio signal processing to devise low power and low cost hearing aids become one of the major application areas in digital signal processing. it should also be noted that a large population affected by hearing defects of different levels requires medical assistance in the form of hearing aids. the hearing aid selectively amplifies the audio sounds such that the processed sound matches with that of one’s audiogram. the signal processing block involved in the design process is the digital filters. digital filter banks are networks of digital filters. the filter bank separates an input signal into many sub-band signals. these sub-band signals can be independently processed according to the requirements and can also be combined together to generate the desired output signal. non-uniform filter banks are more desirable because they can better match the physiological properties or the perceptual properties of the human ear. the major arithmetic block in digital filters is the multiply and accumulate (mac) unit. they are the largest power consuming units in digital filters. since the speeding up and energy reduction in vlsi technology fast approaching its physical limits [1-2], new approaches in digital design of arithmetic circuits are mandated for high speed, low power vlsi systems. thus, to achieve improved performance in terms of speed and power consumption it is essential to adopt power efficient * corresponding author: srimoorthi@nitt.edu mailto:srimoorthi@nitt.edu modeling and simulation of frequency response masking fir filter bank 75 copyright ©2018 assa. adv. in systems science and appl. (2018) hardware design of multiplier and adder to enhance the overall system performance. the adder and multiplier unit decide the speed, size and power dissipation of the mac unit and in turn of the filter bank module. approximate computing represents a paradigm shift in low-power vlsi design for error resilient signal processing applications. for multiplication involving large numbers, truncation and allowance of a small magnitude of error in the generation of partial products will improve speed and reduce power consumption in the multiplier to a large extent. digital signal processing for speech processing, multimedia, graphics, and computer vision can accommodate approximation methods to generate the outputs of multiplication and accumulation process. this is possible because of the inherent error resilience of above mentioned application areas. this paper proposes an area and power efficient approximate integer multiplier architecture. both unsigned and signed multiplier architectures are developed and their performances were studied. a 6-band non-uniform finite impulse response (fir) filter bank is developed using the frequency response masking (frm) technique and used to demonstrate the efficacy of the approximate multiplier in applications like hearing aids. the computationally intensive mac operation of fir filtering is performed using the proposed approximate multiplier. the approximate multiplier architecture is modeled using verilog hdl. the circuit characterization is done by evaluation of the area, power, and delay performance of the circuit. the error characterization of the proposed design is performed using standard error characterization techniques and compared with other similar approximate designs. the paper is organized as follows. the related research in approximate multiplier design and filter bank design for hearing aid are reviewed in section 2. the description of the proposed multiplier architecture and its error characteristics are presented in section 3. section 4 discusses the hardware implementation of the proposed approximate multiplier and its circuit characteristics are discussed. section 5 deals with the implementation of frm based fir filter bank for error tolerant hearing aid application using the proposed multiplier unit and audiogram matching for two different hearing losses are analyzed. finally, the conclusions are presented in section 6. 2. related works in this section, we describe some prior research that focuses on approximate multiplier design and filter bank design for hearing aid applications. the difficult design problem in fir filter design is to design filters with sharp transition band with less hardware complexity. this can be overcome by using frm design method [34]. frequency response masking technique is the most efficient method for the realization of digital fir filters with sharp transition bandwidth. the advantage of frm technique is that the filter has a very sparse coefficient vector and hence the hardware complexity is low. also, it has guaranteed stability and linear phase response. ying wei and yong lian [5] proposed an 8-band non-uniform digital fir filter bank for hearing aid applications using frm approach. two half-band filters are used as prototype filters and the filter bank provides a minimum stop band attenuation of 80 db. the filter bank is tested with audiograms for different hearing losses and matching errors are within +/5 db. nisha haridas and elizabeth elias proposed a set of variable bandwidth filters for hearing aid design using farrow structure [6]. a fixed number of bands are generated from the variable bandwidth filters by spectral shifting of the required bandwidth response. each filter of the bank is tuned to match different categories of audiograms. the reported matching errors are within 3 db for a 6-band filter bank and better matching is achieved with higher order filter banks. deng [7] proposed a three-channel variable filter bank for digital hearing aid applications. a normalized analog chebyshev type-i low pass filter is used as a prototype filter. using analog frequency 76 r. ramya, s. moorthi copyright ©2018 assa. adv. in systems science and appl. (2018) transformations and a modified bilinear transformation approach, variable low pass filter, variable band pass filter, and variable high pass filter are obtained from the prototype filter. these filters have adjustable gains and band edge frequencies which can be independently tuned for matching various hearing loss patterns. the variable filter bank was designed using infinite impulse response (iir) digital filters, which introduces non–linear phase to the system. the audiogram matching experiments shows that the maximum matching error is below 3.44 db. a 16-band low power non-uniform fir filter bank designed using multiplier less approach for a digital hearing instrument was proposed by setiawan, lesmana, and gwee [8]. the power reduction is mainly due to the replacement of multipliers with multiplexers and adder arrays and restricting the wordlength of filter coefficient and data word. minimum attenuation of 60 db is provided in the stop band for high programmability of magnitude gain. ying wei and debao liu [9] proposed a reconfigurable fir filter bank with adjustable sidebands. the frm technique used reduces the computational complexity and is able to show better results for the matching error. interpolation and decimation techniques are used to make the filter bank reconfigurable. a quasi-ansi s1.11 1/3 octave filter bank with 18 bands is proposed in [10] for hearing aid design with significant reduction in group delay and number of multiplications per sample. interpolated fir technique is adopted for reducing the computational complexity. ying wei and yinfeng wang [11] proposed an adjustable filter bank for personalized hearing aids. fractional interpolation technique along with symmetric filters and complementary filters are utilized to reduce the complexity. the filter bank can meet different types of hearing loss with acceptable delay. moving away from the exact arithmetic towards the approximate arithmetic for filter bank design gives enough opportunity to develop power and area efficient circuits and systems for error tolerant audio signal processing applications like portable hearing aids. discussions of the recent developments in the field of approximate multipliers are elaborated. several error tolerant approximate adders and multipliers have been proposed in the literature [12-15]. most of these multiplier designs use truncated multiplication method. truncation reduces the complexity of the multiplier unit by computing the most-significant bits of the product only. in order to reduce the error introduced by the truncation process, a correction factor may be added [16]. as more columns are eliminated, the error introduced by truncation also increases. kyaw et al. [17] developed an error tolerant multiplier (etm) in which the operands are split into a multiplication part with higher order bits and a nonmultiplication with lower order bits. every bit position from left to right is checked for the lower order bits and if either or both the operands are ‘1’, all the bit positions are set to ‘1’ from that bit onwards. for the higher order bits, normal multiplication operation is performed. the major drawback of the etm is the relatively large magnitude of the relative error. parag kulkarni et al [18] proposed a 2 × 2 underdesigned approximate multiplier block and used this block to construct large inaccurate multipliers. the inaccurate multipliers have shown to achieve an appreciable power savings over an accurate multiplier. the performance of the underdesigned multiplier decreases in terms of area and power with increasing wordlength. reza zendegani et al [19] presented three hardware implementations of approximate multipliers based on rounding of the inputs in the form of 2n was proposed. one unsigned and two signed rounding-based approximate multiplier (roba) architectures are implemented. in [20], two variants of signed 16-bit approximate radix-8 booth multipliers are developed using approximate recoding adder logic with and without truncation of a number of less significant bits in the partial products. venkatachalam & ko [21] proposed two variants of approximate multipliers. the partial products of the multiplier are altered using generate and propagate signals and the accumulation of generate signals is done column wise. approximate 4:2 compressors and adders are used to accumulate the remaining partial products. in the first variant, approximation is applied to all the columns of the partial product, and in variant 2, approximation is not applied to most significant column of partial products. even though multiplier design2 has lower relative error compared to multiplier design1, it offers less area modeling and simulation of frequency response masking fir filter bank 77 copyright ©2018 assa. adv. in systems science and appl. (2018) and power savings as compared to multiplier design1. narayanamoorthy et al [22] proposed dynamic segment method (dsm), static segment method (ssm), enhanced static segment method (essm) based approximation multipliers for various dsp and classification applications. a scalable dynamic range unbiased multiplier (drum) for approximate applications such as image filtering, jpeg compression, perceptron classifier is proposed in [23]. both [22] and [23] follows an approach for approximating the input operands based on the output of a complex leading one detection (lod) circuit. even though drum offers better accuracy than [22], the power and area gets increased due to the complex steering logic employed. the steering logic circuit extracts a predefined number of consecutive bits starting from the leading ‘1’ bit to be retained for further processing. as the size of the input grows, the complexity of the steering logic also increases. we propose an approximate multiplier circuit that has the same accuracy as that of dsm and better circuit characteristic than both dsm and drum. in this paper we target power and area reduction by reducing the wordlength of the input operands. the wordlength reduction is performed by right shifting and the amount of right shift required to truncate the operands is derived from the most-significant n/2-bits of the inputs. the approach retains a block of most significant information carrying n/2-bits beginning with the leading one for multiplication. this ensures much accurate result compared to a truncation process. the reduced circuit complexity and error characteristics thus achieved in the design of the approximate multiplier makes it very attractive for asic implementation. 3. proposed approximate multiplier the data wordlength affects the design parameters like speed, area, and power [24] in vlsi circuits. increased speed and reduced power consumption in dsp circuits can be achieved using data wordlength reduction in various arithmetic operations. this reduces the switching activity of cmos circuits and in turn the power consumption. the multiplier unit decides these performance parameters for dsp circuits. the data wordlength reduction can be applied to one or both the inputs of the multiplier and this greatly reduces the area and power in the multiplier unit. the main idea in the design of proposed multiplier lies in the fact that simple right shifting operation is exploited to reduce the wordlength of the input operands for multiplication with good accuracy along with reduced area and power compared to similar designs. the approximate multiplier architecture can be represented by three sub-blocks as shown in fig 3.1. fig.3.1 general block diagram of approximate multiplier the first sub-block is the wordlength reduction logic which reduces the input operand wordlength. second sub-block is an arithmetic unit which is an exact multiplier block of n/2bit wordlength and the last sub-block is a correction logic block to compensate for the reduction of input operand wordlength. in the proposed approximate multiplier, the 78 r. ramya, s. moorthi copyright ©2018 assa. adv. in systems science and appl. (2018) wordlength of the n-bit input operands are reduced to n/2-bits and multiplication is done with a single n/2-bit multiplier instead of n-bit multiplier. the block diagram of unsigned approximate multiplier is shown in fig.3.2. the wordlength reduction logic is composed of following blocks: n/2-bit priority encoder, a binary to excess one converter, 2:1 mux, and n-bit right barrel shifter. the priority encoder receives only the most-significant n/2-bit as the input. the location of one bit position present in the mostsignificant n/2-bit of the input i.e. a[n-1:n/2] and x [n-1:n/2] are encoded by the n/2-bit priority encoder. in order to obtain the amount of shift needed to obtain the truncated input operands, a binary-to-excess one converter and a 2:1 multiplexer (mux) are used. the select lines needed for the 2:1 mux are obtained by performing the bit-wise or operation of the most significant n/2-bit of the input operand. in case of input operands with wordlength less than n/2-bits, the select value will be set as ‘0’ and the 2:1 mux select the input as ‘0’ and the right barrel shifter will not provide any shift and input operands are not truncated. for operand size greater than n/2-bits, then select value will be set as ‘1’ and the 2:1 mux will select the required shift amount from the binary-to-excess one converter. the input operands are truncated to n/2-bit wordlength using right barrel shifters and multiplication is performed using n/2-bit multiplier. the correction logic is made up of 2n-bit left-barrel shifter which expands the truncated product to a 2n-bit number by left-shifting. the barrel shifter left shifts the product by an amount equal to the number of bit positions which is the sum of right shifts applied to both the input operands. the design can be extended for signed numbers by inserting a two’s complement block at the input of each branch. at the output the product value may be negated if necessary . for further simplifying the circuit complexity, the maximum negative input of magnitude -2n is left out from the computation and the resulting simplified architecture of proposed signed approximate multiplier is shown in fig.3.3. the signed multiplier comprises of a two’s complement block at the input and output, n/2-bit sign-extension encoder, inverter based control logic, right and left barrel shifters, unsigned exact multiplier block of n/2-bit wordlength. the multiplier receives the inputs in signed two's complement format. the sign of the input operand is determined and if the sign bit is set, the two’s complement block determines the absolute value of the negative operand. a modified priority encoder is used as a sign-extension encoder to calculate the effective size of the positive operands. the effective size of the positive operands is computed by finding the number of sign-extension bits (zeros) immediately to the right of the most-significant sign bit. for a n-bit multiplier, n/2 to log2(n/2) priority encoder is required since the multiplier block is of n/2-bit wordlength. the count of the sign-extension bits is passed to the control logic block which is used to calculate the amount of shift needed to truncate the inputs. if the size of the input operand is below n/2bits, then shift operation need not be performed and operands are not truncated. in case of operand wordlength above n/2-bits, the right barrel shifter right shifts the input operands to the required number of places as decided by the control logic to obtain the n/2-bit input operand. now the effective size of the input operands is limited to wordlength of size n/2-bit instead of n-bits and the multiplication is performed as an n/2×n/2-bit multiplication. unsigned exact wallace tree multiplier is adopted for the fundamental multiplier block. in order to compensate for the n/2-bit truncated multiplication, the correction logic block is used. the obtained truncated product is left shifted appropriately to compensate for the truncation performed. depending on the sign of the input operands, the unsigned product is negated to obtain the final signed product as the output. modeling and simulation of frequency response masking fir filter bank 79 copyright ©2018 assa. adv. in systems science and appl. (2018) fig. 3.2 block diagram of unsigned approximate multiplier fig. 3.3 block diagram of signed approximate multiplier 3.1. illustration of approximate multiplication of two 16-bit signed numbers as an example, a multiplication of two signed 16-bit numbers with multiplicand a= -259 and multiplier x = 517 is shown in fig. 3.4. the msb of the inputs are used to determine the sign of the operands. since the sign bit of a is ‘1’ the operand is negative and hence the absolute value of a is calculated and used for computation. hence the value of a becomes 259 and x =517. the control logic will decide the amount of shift needed for both the inputs based on the wordlength of the input. the right barrel shifter provides 1-bit right shift for operand a and 2-bit right shift for operand x. after right shifting, the operands are truncated 80 r. ramya, s. moorthi copyright ©2018 assa. adv. in systems science and appl. (2018) to 8-bit wordlength i.e., a′ =129 and x′=129 and process of multiplication is illustrated in fig 3.4. truncated product p′ 0100 0001 0000 0001 ap 1111 1111 1111 1101 1111 0111 1111 1000 ep 1111 1111 1111 1101 1111 0100 1111 0001 fig. 3.4 illustration of multiplication of two 16-bit signed numbers a= -257 and x=513; ap: approximate signed product , ep: exact product the truncated product p′ is expanded to 32-bits by left-shifting three bits to obtain the unsigned approximate product. since the multiplicand is negative, two’s complement of the unsigned product is taken and approximate signed product , ap = 1111 1111 1111 1101 1111 0111 1111 1000 = -133128 is obtained in place of the exact product value, ep = 1111 1111 1111 1101 1111 0100 1111 0001 = -133903. 3. 2. error characteristics of the proposed system the error metrics such as error rate, normalized mean error distance (nmed), absolute value of mean relative error distance (mred) and percentage mean accuracy are used to evaluate the performance of approximate multipliers [15] [25]. the definitions of the various error metrics are given as follows: 1. error distance (ed): for adders and multipliers, ed is the absolute difference between the accurate output (m) and the approximated output (m’). ed = | m – m’ | 2. mean error distance (med): it is computed by taking the average value of all possible eds . med = 1 n ∑ edi n i=0 where n is the total number of samples, and edi is the error distance in the ith value. 3. normalized mean error distance (nmed): it is the normalization of the mean error distance by the maximum output of the accurate multiplier. nmed = med/ mmax, where mmax is the maximum accurate product 4. mean relative error distance (mred): mred is computed as the average value of all possible relative error distances and is defined as: mred = 1 n ∑ edi mi ⁄n i=0 where edi and mi are the error distances and the accurate output of the ith input a 1111 1110 1111 1101 |a| 0000 0001 0000 0011 x 0000 0010 0000 0101 truncated input a′ 1000 0001 truncated input x′ 1000 0001 modeling and simulation of frequency response masking fir filter bank 81 copyright ©2018 assa. adv. in systems science and appl. (2018) 5. mean accuracy: it is defined as 100-mred. 6. error rate (er): it is defined as the probability of producing incorrect outputs for different combination of inputs. error rate = number of incorrect outputs/ total number of outputs in order to evaluate the error performance of the proposed multipliers two sets of hundred thousand random numbers are generated with uniform probability and the multiplication is performed. the error metrics for different 16-bit approximate multipliers were evaluated and summarized in table 3.1. table 3.1 arithmetic accuracy comparison of proposed 16-bit multiplier with state-of art designs unsigned designs error rate (%) nmed (%) mred (%) mean accuracy (%) proposed unsigned approximate multiplier 99.95 0.13 0.53 99.47 underdesigned multiplier udm[18] 80.85 1.37 3.33 96.67 unsigned roba [19] 99.96 0.69 2.93 97.07 venkatachalam and ko multiplier design1 [21] 99.80 1.78 7.63 92.37 dsm8×8 [22] 99.95 0.13 0.53 99.47 drum8 [23] 99.98 0.09 0.36 99.64 signed designs error rate (%) nmed (%) mred (%) mean accuracy (%) proposed signed approximate multiplier 99.87 0.032 0.52 99.48 signed roba [19] 99.90 0.172 2.88 97.12 approximate signed roba [19] 99.94 0.173 2.89 97.11 the results tabulated in table 3.1 shows, the current design of the approximate multiplier provides highest accuracy in terms of various error metrics. the proposed unsigned multiplier achieves the same error performance as that of the dsm8. this is due to the same functionality of the wordlength reduction logic even though the hardware implementation differs in them. due to the unbiased nature of error distribution of the drum8 design, it shows a better error 82 r. ramya, s. moorthi copyright ©2018 assa. adv. in systems science and appl. (2018) performance than the proposed multiplier. comparing with the unsigned roba design, the error performance of dsm8 is shown to be better [19]. the proposed multiplier also shows better error performance comparing with roba designs. smaller the values of nmed and mred shows that the proposed approximate multiplier gives higher accuracy over underdesigned multiplier, roba, multiplier design1 architectures. the relative error is also calculated for inputs of various wordlength and is tabulated in table 3.2. table 3. 2 variation of relative error (re) of outputs in percentage for various wordlength for the proposed approximate multiplier multiplier type input wordlength re <0.5% re <1% re <2% re <5% re <10% re <20% u n si g n ed 8-bit 3.79 % 5.4% 9.6% 31.7% 79.8% 100% 16-bit 47% 97 % 100% 32-bit 100% s ig n ed 8-bit 9.3% 10.72% 16.1% 41.7% 85.6% 100% 16-bit 48.6% 97.23% 100% 32-bit 100 % table 3.2 gives the percentage of outputs with the relative error less than the specified percentage for various wordlength such as 8-bit, 16-bit, and 32-bit proposed approximate designs. it must also be noted that the relative error reduces as wordlength of the operand increases. for a 16-bit design, the relative error is less than 2% and that for a 32-bit multiplier, it is less than 0.5%. the difference between the exact and approximate multiplier almost vanishes for 32-bit signal processing applications and it is negligible for 16-bit applications. 4. hardware implementation of the proposed approximate multiplier the proposed 16-bit approximate multiplier architectures are modeled using verilog hdl and synthesized in cadence rtl compiler using generic pdk (gpdk) 90-nm cmos technology with typical library settings. the functionality of the proposed multipliers is verified using cadence ncsim and all the designs are synthesized in rtl compiler with proper timing constraints. the post-synthesis circuit performance characteristics such as power, area and critical-path delay are tabulated in table 4.1. the power-delay-product (pdp) is a measure of energy and is defined as the product of average power and the corresponding delay of the circuit. since leakage currents are also contributing to the total power, better parameters for characterizing circuit performance are energy or power-delay product (pdp), and area-delay product (adp) [15] [26]. hence the compound metrics such as pdp, and adp were also computed and tabulated in table 4.1 for performance comparison. table 4.1 gives the post-synthesis circuit characteristics of various 16-bit multipliers. the circuit characteristics of the proposed design are compared against exact multiplier architectures like wallace tree (exact unsigned), and baugh-wooley multiplier (exact signed) architectures. for the comparison, a few of the state-of-art approximate multiplier architectures are considered and implemented using gpdk 90-nm cmos technology with the same timing constraints. the synthesized gate level netlist is used to extract the layout of the proposed approximate multiplier using cadence soc encounter and the physical layout of the proposed 16-bit multiplier is shown in fig.4.1. modeling and simulation of frequency response masking fir filter bank 83 copyright ©2018 assa. adv. in systems science and appl. (2018) table 4.1. post synthesis performance characteristics of various 16-bit multipliers unsigned designs power (µw) delay (ns) pdp (pj) area (µm2) adp (µm2. ns) proposed unsigned multiplier 303.47 3.97 1.21 3533 14026 underdesigned multiplier [18] 760.49 4.05 3.07 6241 25276 unsigned roba [19] 235.97 4.84 1.14 4522 21886 venkatachalam & ko multipier 1[21] 425.75 4.38 1.86 4527 19828 dsm 8 ×8 [22] 402.71 4.44 1.78 3548 15753 drum8 segment [23] 424.83 4.64 1.96 3806 17659 wallace tree (unsigned exact) 871.72 4.08 3.56 7012 28609 signed designs power (µw) delay (ns) pdp (pj) area (µm2) adp (µm2. ns) proposed signed multiplier 414.02 5.34 2.21 4012 21424 signed roba [19] 541.17 5.24 2.83 5640 29553 approximate signed roba[19] 537.40 5.15 2.76 5210 26831 baugh-wooley (signed exact) 769.07 4.56 3.51 6679 30456 the tabulated result in table 4.1. shows that the proposed unsigned and signed multiplier architectures have smaller values of area and power consumption (except that unsigned roba has lower power consumption) than other existing state-of-art multipliers, due to hardware efficient wordlength reduction logic employed, thereby their pdps and adps are also small. the area requirement of the proposed approximate unsigned (signed) multiplier is 50% (40%) lower than that of exact wallace (baugh-wooley) multiplier. the proposed 16-bit unsigned (signed) multiplier consumes 65% (45%) less than the total power consumed by exact wallace (baugh-wooley) multiplier. the above results show the efficiency of the multiplier in terms of area and power compared to standard designs. the energy or power-delay product (pdp), and adp of the proposed unsigned (signed) approximate multiplier are about 66% (37%), 51% (30%), lower than that of exact wallace (baugh-wooley) multiplier. table 4.2 illustrates the ranking of approximate multipliers in terms of both circuit metrics and error metrics such as pdp, adp, nmed and mred. the results reveal that the proposed unsigned multiplier gives minimum area, power, and adp; while drum8 gives lowest nmed and mred values and unsigned roba design gives lowest pdp value in comparison to all multipliers. but the nmed and mred error metrics of unsigned roba is higher than the proposed, dsm8, and drum8 designs. the other two multipliers have comparatively less performance in terms of design and circuit metrics. 84 r. ramya, s. moorthi copyright ©2018 assa. adv. in systems science and appl. (2018) table 4.2 ranking of approximate multipliers in terms of design metrics and error metrics unsigned designs pdp adp nmed (%) mred (%) proposed unsigned multiplier 2 1 1 2 underdesigned multiplier 6 6 4 4 unsigned roba 1 5 3 3 venkatachalam & ko multiplier design 1 4 4 5 5 dsm 8×8 3 2 2 2 drum8 segment 5 3 1 1 signed designs pdp adp nmed (%) mred (%) proposed signed multiplier 1 1 1 1 signed roba 3 3 2 2 approximate signed roba 2 2 3 3 the proposed signed multiplier gives lowest pdp, adp, nmed, and mred values in comparison with signed-roba and approximate signed roba architectures. the lowest value of pdp and adp is due to the reduced complexity wordlength reduction logic employed in the multiplier design. fig 4.1 layout of proposed 16-bit approximate multiplier 5. applicationapproximate fir filter bank the design method followed in the design of filter banks is the frequency response masking approach introduced in [3]. the method finds wide acceptance in the design of fir filters with sharp transition band. the filter bank proposed covers the frequency ranges from 0 to 8 khz with 6 non-uniform bands. the sampling frequency selected as 16 khz. the prototype filter is h(z) and the interpolated filters h(z4) and h(z2) are used for the filter design. the modeling and simulation of frequency response masking fir filter bank 85 copyright ©2018 assa. adv. in systems science and appl. (2018) higher order filters appropriately cascaded to generate the desired frequency response when masked with h(z). the transfer functions of different sub-bands in the 6-band filter bank are listed in the table 5.1. table 5.1 transfer functions for each sub-bands the prototype filter h (z) is designed in matlab using the least square method. the dependency of stop band attenuation on transition bandwidth was verified and is shown in fig.5.1. for a 40 tap filter. the simulation study to identify the dependence of transition bandwidth on stop band attenuation shows that a minimum transition bandwidth of 0.2 radians should be maintained to achieve the 50 db stop band attenuation. fig. 5.1 the variation of minimum stop-band attenuation with increasing transition bandwidth the normalized transition bandwidth of the prototype h(z) is fixed as 0.2 and the order of the filter is selected as 40 to ensure a minimum stop band attenuation of 50 db which is sufficient for hearing aids to compensate for the hearing loss satisfying the dynamic range requirements of the hearing impaired person. the structure of the 6-band filter bank is shown in fig.5.2. the lower bands are complementary bands of the upper bands formed by replacing h(z) with its complement hc(z) as the masking filter. the multipliers used in the circuit are integer multipliers. hence the integer filter coefficients are generated by scaling the real filter coefficients by a scaling factor 2n and rounded off to the nearest integer. the filtering process is the convolution of impulse response of the filter with the input audio samples. sub-band transfer function b1 h(z4)h(z2)h(z) b2 h(z2)h(z)h(z4)h(z2)h(z) b3 h(z)h(z2)h(z) b4 hc(z)-h(z2)hc(z) b5 h(z2)hc(z)-h(z4)h(z2)hc(z) b6 h(z4)h(z2)hc(z) 86 r. ramya, s. moorthi copyright ©2018 assa. adv. in systems science and appl. (2018) fig. 5.2 structure of 6-band filter bank approximate multipliers are used to compute the mac operation of the convolution sum which in effect can be interpreted as the convolution of approximate input samples with an approximate impulse response. the approximate filter thus results in a frequency response with less attenuation in the stop band. the frequency response of the 6-band filter bank designed in matlab is shown in fig.5.3. fig. 5.3 the frequency response of the 6-band approximate filter bank 5.1. audiogram matching the 6-band approximate fir filter bank is simulated in matlab and the gains of the different bands are adjusted to match the audiogram of hearing impaired. fig. 5.4a shows the audiogram matching result of the experiment. the audiogram of a patient with noise induced hearing loss (nihl) is selected for the experiment. the results show that the proposed 6-band approximate filter is a suitable one for the implementation of the hearing aids. the matching error is plotted in fig. 5.4b and is within +/3db. modeling and simulation of frequency response masking fir filter bank 87 copyright ©2018 assa. adv. in systems science and appl. (2018) (a) (b) fig.5.4 matching example 1 (a) nihl audiogram matching for the proposed approximate filter bank (b) plot of matching error another matching experiment is performed using old age related presbycusis hearing loss. fig 5.5a shows the plot of audiogram matching and the corresponding matching error plot are shown in fig. 5.5b. results show that matching error can be adjusted within +/4db for the particular case considered. 88 r. ramya, s. moorthi copyright ©2018 assa. adv. in systems science and appl. (2018) (a) (b) fig. 5.5 matching example 2 (a) presbycusis audiogram matching for the proposed approximate filter bank (b) plot of matching error 5.2. area and power savings the area and power savings achieved for the approximate filter bank are evaluated when using proposed approximate multiplier for the digital hearing aid application. table 5.2 gives the area and power savings achieved in the 6-band digital filter bank designed using with and without the utilization of proposed approximate multiplier. the filter-bank is synthesized in rtl compiler and post-layout design parameters are taken from cadence soc encounter. modeling and simulation of frequency response masking fir filter bank 89 copyright ©2018 assa. adv. in systems science and appl. (2018) table 5.2 area and power savings achieved for the proposed approximate filter bank exact design proposed approximate design savings achieved area (µm2) power (mw) area (µm2) power (mw) area (%) power (%) 487233 7.170 374268 6.17 23 14 the power and area results show the advantage of the proposed approximate design in comparison with exact design. the approximate filter bank consumes 14% less power and 23 % less area than the filter bank designed using exact multiplier. 6. conclusion in this paper, we proposed an area and power efficient approximate multiplier based on simple shifting operation to approximate a number for multiplication. the proposed multiplier consumes less area, power, and in energy compared to exact and other recently published approximate multipliers. by adopting frequency response masking approach, a 6-band nonuniform digital fir filter bank is developed using approximate multiplier for hearing aid application. audiogram matching is done with audiograms of two different types of hearing losses and the matching error is computed. it is found that the design offer comparable matching error performance with that of hearing aids implemented with exact arithmetic filter banks reported elsewhere. the large attenuation in the stop band ensures high programmability of the magnitude response of the filter bank, which may be utilized to attain arbitrary reduction of the matching error. the customary low clock rates of digital audio devices makes allowance for time sharing of the resources which may further improve the hardware efficiency. references 1. itoh, k., yamaoka, m., & oshima, t. 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(2007). generalized power-delay metrics in deep submicron cmos designs. ieee transactions on computer-aided design of integrated circuits and systems,26(1), 183-189, https://doi.org/ 10.1109/tcad.2006.883926 https://doi.org/10.1109/tvlsi.2016.2643639 https://doi.org/10.1109/tvlsi.2014.2333366 advances in systems science and application (2015) vol.15 no.2 176-187 making a causal contextualization with the four causes of aristotle j.-f. vautier french society of systems science (afscet), paris, france abstract this article presents an empirical study which proposes a chained contextualization based on the four causes of aristotle: material, formal, moving and final ones. this contextualization is a causal one, dedicated to help an analyst looking for root causes of a problem or an unwanted event. it consists of four different chains of causes beginning by the problem or the event and ending by root causes. a “why” question is asked at each step of questioning. two examples are studied: “why is this part brittle?” and “why is this valve opened?”. an aristotelian causal contextualization aims at helping an analyst to deepen the chains of causes and to extend the range of causes which can be identified with a traditional method. then, the first main interest of an aristotelian causal contextualization is to help an analyst to see, during an interview, the field of responses of the interviewee and to detect responses which stay on the same “plateau” of explanation (reformulations of symptoms or causes, or description of different steps of an activity). moreover, since the aristotelian causal contextualization permits to see the kind of field of causality which is used by an interviewee, it is possible to guide him more easily in other fields of causality by using appropriate questions related to other fields of responses. it is the second main interest of this contextualization. keywords causality, aristotle, contextualization, root cause 1 introduction a communication related to an approach of contextualization, based on a musical metaphor (the concept of systemic score), was presented at the 7th congress of eus in 2008 [1]. the four causes of aristotle [2-5] were used and grouped in a disc-shaped which was named a note. in this article, the four causes of aristotle, still grouped as a note, are also used, but to present a chained contextualization aimed at helping an analyst looking for root causes of a problem or an unwanted event. this contextualization consists of different chains of causes beginning by a problem or an event, occurring in a working situation of an industrial company, and ending by root causes. the root causes are defined as causes that condition the occurrence of more direct causes of a problem or an unwanted event. they are more general than direct causes. it means that the work on a root cause may lead to a larger variety of effects than the work on a direct cause. the general purpose of this article is to discuss the interest of this causal conadvances in systems science and application (2015) vol.15 no.2 177 textualization. then, the first part will present how to take place an aristotelian causal contextualization. secondly, two examples of this contextualization will be proposed and finally a discussion will be presented. 2 how does a causal contextualization using the four causes of aristotle take place ? 2.1 an aristotelian note and the four causes in the concept of systemic score, a note is represented as a disc-shaped set of the four causes of aristotle [1]. fig.1 representation of the four causes as an “aristotelian note” the description of the different blocks is as below. the material causes represent the components of a system. the formal causes represent the principles of functioning of the system (which explain also its shape). the moving causes represent the actors dedicated to the process of designing and managing the system. according to aristotle, the notion of movement is more general than the notion that is considered nowadays (which is only related to kinematic). in this way, the aging of a person is for example a movement too. the final causes represent the purposes which are aimed by the system. then, if the system is for example a car: the material causes represent the components of the car: the body, the engine, the wheels, the tyres...; the formal causes represent the principles of functioning: the propulsion of the car resulting from the movement of the four wheels on the ground...; 178 j.-f. vautier: making a causal contextualization with the four causes of aristotle the moving causes represent the build of the car by the manufacturer and the way of driving and maintaining the car by its driver(s) (indeed that strongly explains also the state of the car...); the final causes represent for what uses the car was designed and also the current uses of the car by the driver(s): to transport people, goods... in a sportive way or not, just in town or not? 2.2 principles and pratical aspects of contextualization in the present contextualization, the process for identifying the causes is based on “why” questions. the four causes are used to answer, in four different ways, questions which begin by “why”. for example, with a question like: “why is this car red ?”, the four responses may be: because a red paint is on the body of the car (material cause); because a red color is emitted by the painting of the body of the car (formal cause); because somebody painted the car in red (moving cause); because the manufacturer wanted to enjoy the buyer (final cause). in this paper, the two initial “why” questions of the two examples (cf. §3), which will permit to start the contextualization, are “why is this part brittle ?” and “why is this valve opened ?”. then, it means that the system which is firstly considered, to start the contextualization, is a local one: a working situation in a company for example. afterwards, for moving and final causes, a more general system is progressively considered to point out more general causes. in practice, an aristotelian causal contextualization is supported by a process of iterative question-answer which starts with an aristotelian note. new notes are built from each of the four causes, hence the previous denomination. the contextualization starts with a question which begins by “why”. the first answer is represented as an aristotelian note, whose the four parts are filled in, i.e. a set of four responses which correspond to the four parts of the note. afterwards, a new question (which still begins by “why”) built from each previous response is asked. the four new answers are also represented as aristotelian notes but just one part of these notes is filled in. the part dedicated to one cause is related to the same part of the previous note which is considered. this general process is repeated several times. to find the responses, a specific “connector” is used for each kind of previous responses. thus, this contextualization works with keys of questioning to go advances in systems science and application (2015) vol.15 no.2 179 back step by step to root causes. a connector is a set of few words which begins by “because”, introduces a response and connects each kind of response to the “why” question. then, to contextualize, it is used iteratively the connector: • “because it consists of ...” for the “material causes” (an analytical approach); • “because there is/are ... (the principle of functioning of this phenomenon)” for the “formal causes”; • “because there was... (a project manager of it)” for the “moving causes”; • “because the project manager wanted to...” for the “final causes”. the connector “because the project manager wanted to...” was preferred rather than “in order to” for the final causes since the focus is here on the process inside a company and the decisions which were taken. then, the notion of project manager was introduced since this function is very common in a lot of companies (and besides it is often a part of the job of a manager). for the moving causes, the focus is also on the role of the project manager of the system since the purpose is to present the root causes on which it can be acted on. indeed, the designer of the current system is generally not still present in the plant. in this way, as aristotle suggested, there are a lot of relationships between moving cause and final cause (hence the curved dotted lines in the two examples below). 3 examples of contextualization the first question which is considered is: why is this part brittle? the second one is: why is this valve opened? these two questions are “why” questions, respectively related to a problem or an unwanted event. in these examples, the problem or the unwanted event are related to the occurrence of decision failures. it means that, for instance, an operator did not use the right procedure... considering the part, which is brittle, there was a dysfunction in the process of making it, since the part ought to be resistant to breaking. furthermore, there was no sabotage (otherwise the final chain of causes would be different). considering the activity of closing the valve in a room of a plant, there is an unwanted event since the valve ought to be closed. 3.1 why is this part brittle ? the responses of this question is in the middle of the figure 2 and the other answers are built from these previous ones (look at the arrows). for example, to find the first response of “why is this part brittle?” in the “material causes” axis, an interviewee may use the connector “because it consists 180 j.-f. vautier: making a causal contextualization with the four causes of aristotle of” to find the response “brittle matters”. afterwards, during an interview, an analyst has to ask “why” and to propose that the interviewee uses again the same connector for the same axis or another one for the other axis. fig.2 example of an aristotelian causal contextualization 3.2 why is this valve opened ? the responses of this question are in the middle of the figure 3 and the other answers are built from these previous ones (look at the arrows). the valve is a butterfly valve which is manually opened and closed. it consists of a core and a disc which rotates around an axis and there is generally a fluid in the pipe when the valve is opened. 3.3 some remarks about the contextualization 1) representation of an aristotelian causal contextualization in the two examples, only one chain of aristotelian notes was presented in each field of causality. other chains of notes could be carried out with the same answer-formulation. for example, it could be possible to go back to root material causes from each components and afterwards sub-components of the system. nevertheless, our objective is not to propose another method of determination advances in systems science and application (2015) vol.15 no.2 181 of root causes. it is to discuss the interests of this kind of contextualization to help the analyst when he carries out a fact gathering and uses methods, like, for example, the 5 whys [6-7] or the causal tree analysis [8]. fig.3 example of an aristotelian causal contextualization 2) expression of the moving and final causes the focus was on two kinds of examples which are related to two domains: quality and safety. in the first one, the notion of quality, as a result [9], may be evaluated in terms of scope (of the product), cost and time/duration (to obtain the product) [10]. it is why the final causes are indicated with one objective and one or two characteristics of it (in terms of scope, cost or time/duration) which may mainly contribute to explain the occurrence of the problem (of quality). concerning safety, it is not provided, for final causes, characterizations of an objective in terms of scope, cost or duration. indeed, “to be opened” is not, for the valve which is considered, a defect by itself (sometimes the application of the procedures leads to open it). on the contrary, “to be brittle” is, for the part of 182 j.-f. vautier: making a causal contextualization with the four causes of aristotle the first example, a defect by itself (it is never a quality for it). it is why it is necessary to add some characteristic in terms of scope, cost or time/duration. note that for other things like for example some kind of cakes, “to be brittle” is a quality... in our context of looking for the root causes, it was considered that “to be brittle” or “to be opened” are respectively a problem or an unwanted event. for the part, it could have been related to some problems concerning raw materials used to make it. but, in this case, the final causes would have been different. it means that, in the two examples, operators made mistakes. more precisely, these mistakes are rule-based or knowledge-based errors [11]: rule-based error: for example, an operator opened the valve instead of keeping it closed. he did not analyze correctly the situation and then did not use the right procedure; knowledge-based error: in this case, the operator did not know he had to close the valve to achieve this operation. a skill-based error (for example an operator wanted to close the valve but he kept the valve opened: it was a slip) may be here considered only for the part since in our example the operator wanted to open the valve. the explanation of the occurrence of the human failures comes from the examination of the moving causes. in this axis, the explanation of the problem, the unwanted event or a direct moving causes results from the lack of adaptation between four sets of factors: local work organization, team and competences, technical devices and work environment (like premises, noise) which can be identified in each root moving causes. these groups of factors appear in the classical methods using “why” questions or in human and organizational factors methods. the aristotelian causal contextualization proposes here a framework to locate these factors in the working situation, the unit or the plant. the factors are very similar to those which can be found in a mto approach (man, technology, organization) [12]. then, in a few words, if there is a mistake in the operation which is considered, the explanation (the lack of interaction between the previous factors) is found in the moving causes which are identified. 4 discussion: interests of this aristotelian causal contextualization to help an analyst even if more examples would be necessary to confirm these main findings, here are some first elements. two points are proposed: deepen the chains of causes, and extend the range of causes. advances in systems science and application (2015) vol.15 no.2 183 4.1 deepen the chains of causes after the occurrence of a problem, an unwanted event the first step of an analysis is generally to carry out a fact gathering. it is often the step before an iterative question-answer session using a more or less complicated “why” question to find root causes and then to identify chains of causes that explain the occurrence of the event, the problem· · · afterwards, the objective is to propose solutions to remove the root causes. nevertheless, sometimes analysts may not really go back to root causes, even if there is a presentation of different causal chains in the analysis. in fact, there may be a treadmill effect at one or several levels of explanation of the event or the problem. there is a “plateau” in which, for example, it may be found reformulations or descriptions of different steps of a human activity or a mistake. it could partly explain that analysts may just consider the symptoms or very direct causes. it means also that the number of question-answer is not a sufficient criterion to provide an in-depth analysis and to find the real root causes. this problem may be observed especially when the focus is on the material or formal fields of causality. two phenomena may be mainly described: concerning the material causality, the evocation of a succession of steps of the activity of an operator instead of detailing one step into sub-steps... (for example a detailing process would be: make a mistake / press the wrong button / push the button a with one finger / push of 0.5 cm on the button a) concerning the formal causality, the reformulation of the cause instead of going back to a more general principle of explanation (for a mistake for example, a more general principle indicates the type of mistake and afterwards the type of characterization of the mistake...: make a mistake / realize an operation in a different way that is assigned / consider the results of the activity to evoke the mistake) to go back really to root causes, in the material causality and formal causality fields (cf. the two examples of §3), it is necessary to identify, at each step of questioning (with a “why” question), respectively, the basic components and the general principle of functioning (which underlies the previous formal cause which is considered). 1) material causality: succession of elements as some steps of the activity of a person here is an example which illustrates this phenomenon of treadmill effect: “why did he press the wrong button? because he confused the two buttons why did he confuse the two buttons? because he had a wrong mental representation why did he have a wrong representation? because he did not analyze correctly the working situation 184 j.-f. vautier: making a causal contextualization with the four causes of aristotle why did he not analyze correctly the working situation? because he did not detect the pertinent information” in this succession of question-answer, the focus is on a mistake. first of all, the movement of a finger of the operator is evoked, next some dysfunctions of the mental process are explored and, in the end, the problem of perception is considered. in other words, the model of j. rasmussen [13] is followed in the reverse direction (from execution of action to the perception). thus, it is taken into account several elements which are just different steps of the mental information processing and execution of action! 2) formal causality: some kinds of reformulation • example 1: going from action (or human failure) to emotion and vice versa “why did he move these objects nervously? because he was angry” or (in the opposite way) “why was he happy? because he was watching the sun set over the sea” • example 2: going from an expression of a mistake to another expression of the same mistake “why did he make a careless mistake ? because he did not pay attention” “why did he not pay attention ? because he worked too quickly” “why did he work too quickly ? because he used a mode of thought “system 1” (fast thinking) [14]” then, in cases of reformulations or successions of different steps of an activity, the analysts do not go back to root causes. they stand still! this phenomenon seems to occur particularly with material and formal causes. thus, the first interest of the aristotelian causal contextualization is both to help the analyst (during an interview) to see the field of responses of an interviewee and to avoid some current treadmill effect. when the analyst-interviewer detects responses which stay on the same “plateau” of explanation, for example reformulations of symptoms, or direct causes, or the description of different steps of an activity, it means that the interviewee stands still at one level of causality without going back really to root causes. 4.2 extend the range of causes a little variety of causes is sometimes taken into account in the analyses [6-7]. it may mean that analysts cannot go beyond their current knowledge about the working situations. the interests of an aristotelian causal contextualization is to propose four ways of questioning a problem and force the analysts to multiply the points of view. it leads analysts not to focus on a single set of root causes. the example of §4.1 about material causality, with a 4 why’s iterative questioning, shows that it is easy to stay in the “material causes” field of a mistake without going to another causality field like moving causes. for example, considering the question “why did he press the wrong button?” the response could advances in systems science and application (2015) vol.15 no.2 185 also have been: “because the buttons of displays were too small” or “because he did not know that the procedure had changed”. thus, since the aristotelian causal contextualization permits to see the kind of field of causality which is used by an interviewee, it is possible to guide him more easily in other fields by using appropriate questions related to the wanted field of responses. for example, if an analyst wants to identify some causal factors related to moving causes, he can turn to questions like “what technical devices or competences are necessary to make this product?” or “what are the objectives of the project manager of the unit?”. in other words, the range of fields proposed by an aristotelian causal contextualization is very large since, as r. caratini (2012) [15] indicated, the material and formal causalities are immanent (they depend only on the object) and the moving and final causalities are external. it is an important interest of this kind of formalization to permit to enlarge the set of points of view! 5 conclusion an aristotelian causal contextualization is an implementation of the notion of contextualization [16]. it is a support to identify and categorize the causes of a problem or an event [17-18]. then, it may be a way to help an analyst looking for root causes. to end this article, here are some perspectives of works for the future: the possibility to distinguish different ways of causality and the existence of specific connectors for each field of causality are perhaps a way to find, in a chain, a “good distance” from one cause to another. indeed, with a lot of approaches, the different causes may be more or less “distant” one to another. it means that, with these latest approaches, another analyst may often identify an intermediate cause between two successive causes of a chain; the use of an aristotelian causal contextualization aims at completing a common investigation about the root causes. it is perhaps a way to make results more repeatable from one analyst to another (each of them using their own method completed by this aristotelian contextualization). do we see the same evolution than the airplane traffic? in this latest domain, the air corridors were created so that all flights can be repeatable... may these air corridors be compared to the 4 causality fields of an aristotelian causal contextualization? then, going back in the past to the basics of aristotle will be perhaps a way to take a better jump forward! references [1] j.-f. vautier. (2008), “a systemic approach to question complexity: the systemic scores”, 7th congress of the european union for systemics (eus186 j.-f. vautier: making a causal contextualization with the four causes of aristotle ues), lisbon, portugal. [2] aristote. (2002), physics, book ii, chapters 3 and 7, translation of pierre pellegrin, editions garnier flammarion, paris. [3] aristote. (2008), metaphysics (the original is in french: métaphysique), book ∆, chapter 2, translation of marie-paule duminil et annick jaulin, editions garnier flammarion, paris. [4] aristotle. (1930), physics, book ii, chapters 3 and 7, translation of r.p. hardie and r.k. gaye , oxford press. [5] p. pellegrin. (2007), dictionary aristotle (the original is in french: dictionnaire aristote), ellipses. [6] fs. anderson. (2009), “root cause analysis: addressing some limitations of the 5 whys”, see http://www.qualitydigest.com/inside/fda-compliancenews/root-cause-analysis-addressing-some-limitations-5-whys.html. [7] t. minoura. (2007), “talks about problems with 5-whys”, see http://www.taproot.com/archives/710. [8] t. meyer and g. reniers. (2013), engineering risk management, walter de gruyter. [9] r. atkinson. (1999), “project management: cost, time and quality, two best guesses and a phenomenon, its time to accept other success criteria”, (international journal of project management, vol.17, no.6, pp.337-342, elsevier science ltd and ipma. [10] k. schwalbe. (2009), an introduction to project management, course technology cengage learning. [11] j. reason. (1990), human error, cambridge university press. [12] o. andersson and c. rollenhagen. (2002), “the mto concept and organisational learning at forsmark npp“, sweden (iaea international conference on safety culture in nuclear installations, rio de janeiro, brazil. [13] j. rasmussen, a. m. pejtersen and l. p. goodstein. (1994), cognitive systems engineering, wiley. [14] d. kahneman. (2011), thinking, fast and slow, macmillan. [15] r. caratini. (2012), initiation to philosophy (the original is in french: initiation à la philosophie), editions archipoche. advances in systems science and application (2015) vol.15 no.2 187 [16] e. morin. (2006), for a reform of the thinking (the original is in french: pour une réforme de la pensée), entretiens nathan, editions nathan. [17] j.-f. vautier. (2014), “the birth of systems according to a new approach: the birth triangle”, 9th congress of the european union for systemics (eusues), valencia, spain. [18] j.-f. vautier. (2011), “the interests of the lists of categories to deal with diversity and systemics” (the original is in french:“de lintérêt des listes de catégories pour appréhender la diversité et aborder la systémique) the interests of the lists of categories to deal with diversity and systemics”, acta europeana systemica, vol.1, pp.1-8. corresponding author j.-f. vautier. can be contacted at: jean-francois.vautier@cegetel.net advances in systems science and applications (2014) vol.14 no.4 378-387 computer simulation of social partnership in the system of continuing professional education vladimir k. dyachenko, guennady a. ougolnitsky and larissa v. tarasenko southern federal university, russia abstract a game-theoretic model of social partnership in the system of continuing professional education is proposed. some results of the model identification and investigation based on simulation modeling are considered. a comparative analysis of egoistic and cooperative approaches to the social partner-ship is conducted. keywords: social partnership, continuing professional education, simulation modeling, identification, dynamic games. 1 introduction a system of professional education should be flexible enough to prepare qualified and competitive specialists capable to improve their knowledge and skills in the changing environment. one of the important mechanisms providing the solution of this problem is social partnership that allows for the cooperation of employers, universities, and students. traditionally, social partnership is considered in conformity with relations between workers, employers, and trade unions[1]. a wider view is presented in[2]. social partnership relations also exist in the system of higher education. in order to be successful in these partnerships, agents must look carefully at how they are influenced by factors such as race, class, gender, age, culture, histories and other differences[3]. unequal power relations prevailing in such partnerships affect the ability to share information, solve problems and form honest relationships. the most complex social challenges such as post-secondary access and success for under-represented students, diversification of the workforce, poverty, environmental degradation, and global health exceed the problem-solving capacity of single organizations or societal sectors. social partnership provides colleges and universities, corporations, government agencies, non-profits, and other organizations with a model for how to effectively address these and other pressing social issues through strong, effective collaboration[4]. there are only a few papers that contribute to the mathematical modeling of the problems of social partnership. d. talman and z. yang[5] propose a general model of partnership formation. let a number of agents want to conduct some activities. they may act alone or seek a partner for cooperation and need in the latter case to consider with whom to cooperate and how to share the profit in a cooperative or competitive environment. necessary and sufficient conditions under which an equilibrium exists are given. in[6] the cobb-douglas production function is used to measure synergy effects of a public social private partneradvances in systems science and applications (2014) vol.14 no.4 379 ship project. the microeconomic approach opens up a possibility for allocating the synergy effects to partners according to a cooperative principle of optimality. analysis of a social partnership among a complex network of stakeholder organizations is fulfilled in[7]. m. diaconu and a. pandelica examine the partnership relationship between economic academic and business environment and propose a series of measures regarding how this relationship can shape the modern university[8]. the authors of[9] propose a synergy model of a smart tri-partite partnership among polytechnics industry students in the industrial training program in malaysia. social partnership in continuing professional education is a specific system of joint activities of the education system agents characterized by trust, common objectives and values, and providing highly qualified, competitive, and mobile specialists for the labor market. the main research hypothesis is that social partnership permits to increase a level of professional competence of the students. it seems natural to use the formalism of differential games for description of social partnership relationships[10]. due to the high complexity of the differential game model the techniques of simulation modeling are applied for its investigation[11]. some previous results of the authors’ approach in social systems modeling are presented in[12, 13]. 2 a general description of the model in the proposed model social partnership relations among employers, students, and university are considered. the payoff functions are as follows: jp = t∑ t=0 gp (up (t), ub(t), uc(t), x(t)) → max, up (t) ∈ up jb = t∑ t=0 gb(up (t), ub(t), uc(t), x(t)) → max, ub(t) ∈ ub jc = t∑ t=0 gc(up (t), ub(t), uc(t), x(t)) → max, uc(t) ∈ uc (1) where n = p,b,c is a set of players, namely: employer; university; student. the systems dynamics are given by the equation x(t+ 1) = x(t) + f(x(t), up (t), ub(t), uc(t)), x(0) = x0 (2) here up(t), ub(t), uc(t) are strategies of the players describing their efforts directed to the development of social partnership relations; up, ub, uc domains 380 vladimir k. dyachenko:computer simulation of social partnership in the system of ... of feasible strategies; jp , jb, jc payoff functionals of the players; gp , gb, gc instantaneous payoff functions; p = {p1, . . . , pr} a finite set of employers; university (only one is considered); c = {c1, . . . , cs} a finite set of students; = 4 (period = 5 years). the strategies determine a share of the annual budget assigned by a player to the needs of continuing professional education (cpe): upi (t) a share of the annual budget assigned to cpe programs by an employer, upi = [0, 1]; ucj(t) a share of the annual budget assigned to cpe programs by a student, ucj = [0, 1]; ub(t) a share of the annual budget assigned to cpe programs by the university, ub = [0, 1]. in the current investigation five employers and ten students are considered. the strategies of generalized players in any moment of time are calculated as an arithmetic mean of the strategies of those agents, namely: strategy of p (employer): up (t) = 1 r r∑ i=1 upi(t);up (t) ∈ up , r = 5 (3) strategy of (student) uc(t) = 1 s s∑ j=1 ucj (t);uc(t) ∈ uc , s = 10 (4) strategy of (university): ub(t);ub(t) ∈ ub (5) the state variable of the model considered as a time function x(t) characterizes a quantitative factor which determines relations in the social partnership system; f − a function of the system dynamics depending on the players strategies. it is assumed that the function of system dynamics increases in respect of all arguments (the efforts of players positively influence to the results of social partnership). to give the system dynamics the modified verhulst-pearl model is used, i.e. f is taken in the form f(x(t), up (t), ub(t), uc(t)) = h(up (t), ub(t), uc(t))x(t)(1− x(t) k ) (6) where k maximal feasible value of the state variable in the given conditions; h function of impact of the players’ strategies, namely: h(up (t), ub(t), uc(t)) = 3∑ i=1 aiui(t);ai ≥ 0; 3∑ i=1 ai =1; i = p,b,c; (7) advances in systems science and applications (2014) vol.14 no.4 381 ai relative weights of the strategies. for the estimation of the relative weights the following considerations are used. the sum of the weights is equal to 1, and all weights are positive. the most important influence is made by students as key agents of the cpe system. the two other weights are approximately equal to each other and dont differ from the former weight (students) too significantly. so, the weights are chosen as follows (table 1): table 1 relative weights of the strategies (ai) relative weight i employer university student ai 0.3 0.3 0.4 accordingly to the research hypothesis it is rational to consider two variants of parameterization of the payoff functions. 1) egoistic approach. speaking about the real situation in the cpe system it is natural to suppose that gi decreases on ui and increases on other arguments (“ free-rider principle ”). this variant determines an egoistic approach when all players save their personal efforts. thus, a problem of coordination of private (efforts saving) and common (social partnership development) interests in the cpe system arises. in this case the payoff functions are parameterized as follows: gi(up (t), ub(t), uc(t), x(t)) = bjuj(t) + bkuk(t) + bxx(t) 1 + biui(t) ; i, j, k = p,b,c (8) bi relative weights of the strategies. 2) cooperative approach. this parameterization describes a desirable (ideal) state of the relationships of social partnership in the cpe system when its elements voluntarily and consciously contribute to the development of social partnership relationships. this variant defines a cooperative approach when payoff functions become increasing on all arguments: gi(up (t), ub(t), uc(t), x(t)) = bipup (t) + bibub(t) + bicuc(t) + bixx(t),i = p,b,c (9) bj i a relative value of the factor for the player i (i=p , b, c; j=p , b, c, x). 3 identification of the model suppose that the state variable x(t) characterizes a level of professional training of the students. for the estimation of the initial training level a known approach 382 vladimir k. dyachenko:computer simulation of social partnership in the system of ... proposed by donald kirkpatrick[14] is used. d.kirkpatrick has developed a model of evaluation of the training effectiveness which considers four levels: 1) reaction: to what degree participants react favorably to the training. for the estimation the results of sociological polls conducted among students of the southern federal university are used. 2) learning: to what degree participants acquire the intended knowledge, skills, attitudes, confidence and commitment based on their participation in a training event. here the data about the students progress together with data characterizing the material base of education are considered. 3) behavior: to what degree participants apply what they learned during training when they are back on the job. the results of polls conducted among employers in the rostov region are used. 4) results: to what degree targeted outcomes occur as a result of the training event and subsequent reinforcement. the data of polls among employers on the topic “a level of professional knowledge and skills of the graduates” are used. therefore, an initial value of the professional training level is evaluated by an evident formula x0 = 1 4 4∑ i=1 xi (10) to determine the values of xi the results of polls are interpreted as follows. the answers “satisfied”, “rather satisfied”, “high” and “very high” are treated as satisfaction and their values for a question are added. the answer “i dont know” means that the respondent may have either positive or negative opinion. supposing their ratio as 1:1, the value is taken equal to 1/2 in this case. the summing up gives an array of the values of xi as follows: x = [0.911; 0.823; 0.559; 0.617]. also, the maximal feasible value of the state variable in the given conditions is taken equal to k = 1. the relative weights are identified accordingly to the following reasoning. 1) egoistic approach. this approach supposes an economy of the personal efforts (“free-rider problem”). the principal role belongs to the student whose efforts and professional training are of crucial importance. nevertheless, other players are important too. the next is university as the training base and then employer who determines the requirements on a labor market. if this approach is chosen then the players are more interested in resource economy than in the improvement of professional training. the respective relative weights of influence factors bi are presented in table 2. 2) cooperative approach. student. the aim of partnership for the student is knowledge acquisition and improved professional training. respectively, the factor of professional level is the principal one for the student and it has the biggest value. the factor of personal advances in systems science and applications (2014) vol.14 no.4 383 table 2 relative weights of influence factors (bi) relative weight employer university student professional level bi 0.25 0.3 0.4 0.1 table 3 relative values of the factor j for the player i, ( bij , i = p,b,c; j = p,b,c, x) relative values bj i employer university student professional level employer 2/8 1/8 2/8 3/8 university 1/8 3/8 2/8 2/8 student 1/7 1/7 2/7 3/7 effort is evaluated a bit smaller. the two other factors (significance of university and employer) are equal and have minimal values. university. the university gives a maximal value to the significance of its efforts because they provide a base for the professional training. the significance of student and professional level are evaluated by the university with equal and a bit smaller values. at last, the minimal value goes to the employer who is as a rule not very active participant of the training process. employer. this player is interested above all in the professional level of the potential employees, and efforts of the student and his own receive equal values. the last place belongs to the university because its training programs rarely satisfy the employers needs completely. the relative values of the factors are presented in table 3. 4 planning and implementation of the computer simulation experiments the model investigation was conducted by computer simulation on the base of scenario method [11]. the scenarios are formed according to plausible behavior patterns of the players. for simplicity the arithmetic mean strategies are described. it is supposed that for all scenarios for any moment of time the values of strategies are equal: up (t) = uc(t) = ub(t) = uo(t), t = 0, ..., 4. six scenarios are considered: 1) maximal (max) one corresponds to the maximal possible financing when the whole budget of a player is assigned for training: uo(t) = 1; 2) medium (med) one assigns a half of the budget for training: uo(t) = 0.5; 3) minimal (min) allows for training only a small part of the budget: uo(t) = 0.2; 4) absence of financing (abs) is clear: uo(t) = 0; 5) decreasing of financing (dec) means that initially an eminent part of the 384 vladimir k. dyachenko:computer simulation of social partnership in the system of ... players budget is assigned for training but then the share decreases to a small value, namely: uo(t) = 0.8− 0.15t; 6) increasing of financing (inc) describes the opposite strategies: uo(t) = 0.2 + 0.15t. thus, in the former four scenarios the strategies are constant in time, while in the fifth and sixth scenarios the shares of financing are time-dependent. 5 processing and analysis of the modeling results the processing of modeling data includes a comparative analysis of the graphs of state variable and payoff functionals for different scenarios. the graphs of the employer’s payoff functional for different scenarios are shown in fig.1-fig.2. the functional j1p(t) corresponds to the egoistic approach, and j2p(t) to the cooperative one. as it could be expected, the best results are achieved for the maximal scenario fig.1 comparison of graphs of the employers payoff functional for the scenarios “dec” and “inc” , and the worst results for the minimal one. the values of payoff functionals and state variable decrease in the following order of scenarios: maximal, decreasing, medium, increasing, minimal, absence. in fig.3 it is shown the comparative dynamics of the state variable (professional level) respec-tively to the six scenarios. the dynamics are described by the verhulst-pearl function. let‘s notice that for the decreasing scenario the values of payoff functionals are initially big but then they decrease in time, and for the increasing scenario the picture is opposite. it means that if the players make an eminent contribution from the very beginning then they can create a good base which allows for a certain decreasing in the future, and the level of development of social partnership advances in systems science and applications (2014) vol.14 no.4 385 fig.2 comparison of graphs of the employers payoff functional for the scenarios “max” and “abs” fig.3 comparison of graphs of the state variable for different scenarios will be still satisfactory. the comparative analysis shows that all players win from the social partnership. however, for any specific scenario the payoff functional of a player can achieve both maximal and minimal value in respect to the payoffs of other players depending on the players strategy. so, the strategic choice is critical for a player. for two variants of payoff functionals (egoistic and cooperative approach) a comparison of the payoffs is fulfilled. a difference between the approaches is clearly observable in time. the difference grows when financing increases, and the comparison demonstrates advantages of a higher level of the social integration. in the case of cooperation payoffs are greater than in the case of egoism: for the employer in 3.75 times (absence of financing) and 1.64 times (maximal scenario); for the student in 4.29 and in 2.04 times for the same scenarios respec386 vladimir k. dyachenko:computer simulation of social partnership in the system of ... tively; for the university in 2.5 times (absence of financing) and in 1.76 times (medium scenario). 6 conclusions in the paper a mathematical model of the social partnership in the cpe system is built using differential games techniques and investigated by computer simulation. the model identification is made on the base of sociological polls conducted in the southern federal university and rostov region (russia). one of the main results is a confirmation of the thesis concerning necessity to join efforts of the subjects of social partnership. in this case all considered scenarios lead to better results than in the egoistic approach. it is clear that more financing allows for more convincible results. acknowledgments the work is supported by russian foundation for humanities, project #14-0300236 references [1] frege c.m. (1999), “social partnership at work: workplace relations in post-unification germany”, routledge, pp.272. [2] eds. m. m. seitanidi, a. crane. (2013), “social partnership and responsible business: a research handbook”, routledge, pp.432. [3] keith n. (2011), “engaging in social partnerships: a professional guide for successful collaboration in higher education”, routledge, pp.288. [4] siegel d. (2010), “organizing for social partnerships: higher education in cross-sector collaboration”, routledge, pp.224. [5] talman d., yang z. (2011), “a model of partnership formation”, journal of mathematical economics, no.47, pp.206-212. [6] fandel g., giese a., mohn b. (2012), “measuring synergy effects of a public social private partner-ship (pspp) project ”, int. j. production economics, no.140, pp.815-824. [7] wilson e.j., bunn m.d., (2010), “savage g.t. anatomy of a social partnership: a stakeholder pers-pective ”, industrial marketing management, no.39, pp.76-90 advances in systems science and applications (2014) vol.14 no.4 387 [8] diaconu m., pandelica a. (2012), “the partnership relationship between economic academic and business environment, component of modern university marketing orientation ”, procedia social and behavioral sciences, no.62, pp.722-727. [9] zaharatul a.a.z., abd shukor h., ir. ghazari a.a. (2012), “smart tripartite partnership: polytechnic industry student ”, procedia social and behavioral sciences, no.31), pp.517-521. [10] dockner e., jorgensen s., long n.v., sorger g. (2000), “differential games in economics and man-agement science”, cambridge university press, pp.382. [11] law a.m., kelton w.d. (2000), “simulation modeling and analysis”, mcgraw-hill, pp.784. [12] ougolnitsky g. (2011), “sustainable management”, nova science publishers, pp.287. [13] antonenko a.v., ougolnitsky g.a. (2013), “static models of corruption in hierarchical systems”, advances in systems science and applications. vol.13, no.1, pp.32-46. [14] kirkpatrick d.l., kirkpatrick j.d. (2007), “implementing the four levels: a practical guide for ef-fective evolution of training programs”, berrett koeler publishers, pp.153. corresponding author authors can be contacted at: ougoln@gmail.com. adv syst sci appl 2017; 17(2); 29-42 published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/483 copyright ©2017 assa. adv. in systems science and appl. (2017) attitude to entrepreneurship in russia: three-dimensional institutional approach irina a. petrovskaya, sergey m. zaverskiy and elena s. kiseleva lomonosov moscow state university business school, 1-44 leninskie gory, moscow 119234, russia. e-mail: petrovskaya@mgubs.ru abstract. this paper aims to analyze the impediments to the development of entrepreneurship in russia from the institutional perspective. to describe the institutional environment we use a concept of a three-dimensional institutional profile which classifies the institutions into three types: regulatory, cognitive and normative. these three dimensions imply three bases of legitimacy: entrepreneurship can be legitimized if it conforms to legal requirements (regulatory dimension), if it is seen as legitimate through a common frame of reference (cognitive dimension) and if it conforms to the existent moral base (normative dimension). we argue that one of the impediments to entrepreneurship development in russia is that it is not seen as legitimate enough by the society at large. we explore the foundations for this through the regulatory dimension (the dynamic of the legal legitimation of entrepreneurial activity from the soviet epoch to the present times), in the cognitive dimension (the stereotype of entrepreneur and its origins), and in the normative dimension (basic assumptions which relate to the fundamental moral dimensions of entrepreneurial activity: assumptions about money, wealth, and work). key words: russia; entrepreneurship; institutional environment 1. introduction at the end of xx century russia faced a new reality which brought the new concepts, new words and new meanings. “market economy”, “free trade”, “competition”, “entrepreneurship” and “entrepreneur” – these new terms were understood by many very vaguely, and the perception of their meaning was largely shaped in the preceding historical period. russia was not the only country that faced this situation, and the variations across the success levels of economic transition in the ex-communist european countries show that the transition to market economy is highly dependent on the historical and social context. in this paper, we try to look at the context where one of the key actors of the market system – the entrepreneur – exists in russia. the market economy in general is grounded in the set of fundamental beliefs about the benefits of free market and competition, but the perceived meanings of these words may be different and the market actors may behave according to their own understanding. thus different elements of market economy need to be considered not only from the economic perspective, but also through the lens of deep-lying beliefs about what is good and what is bad. in this respect, the entrepreneurial activity should be also placed in this context if we attempt to explore its drivers and impediments. in russia entrepreneurial activity is lower as compared to countries with the same level of economic development. the global entrepreneurship monitor shows that in russia in 2014 30 i. a. petrovskaya, s. m. zaverskiy & e. s. kiseleva copyright ©2017 assa. adv. in systems science and appl. (2017) (latest available data) entrepreneurial activity measured as a share of population at ages 18-64 who are early-stage or established entrepreneurs was 8,6%, while for instance in argentina it was 25,5%, in brazil – 34,3%, in slovakia – 18,7%, in poland – 16,5% [1]. since entrepreneurial activity is one of the contributors to economic performance, the issue of what aids or impedes entrepreneurship has received much attention. we approach this issue from the institutional perspective [2] which implies that the level of entrepreneurial activity in a country is influenced by its institutional environment. north [2] defines institutions as formal and informal rules of the game in a society; they act as constraints and guidelines for human behavior, and their main role is to reduce uncertainty in human interaction. north’s definition of institutions is a very broad one, and consequently the research on entrepreneurship and institutions adopts multiple and varying views of what institutions are. one stream of research focuses on formal institutions and economic environment, such as secure property rights [3]. another stream of research focuses on culture [4-6] and is largely based on the models for cross-cultural comparison such as developed by hofstede [7] and inglehart [8]. however, the complexity of the issue calls for the broader framework for its analysis. therefore in this paper we will employ a concept of a threedimensional institutional profile [9], which follows scott’s concept of institutional pillars [10] and allows for analyzing country institutional environment in the context of entrepreneurship [11]. the concept of a three-dimensional institutional profile classifies the institutions into three types: regulatory (laws), cognitive (cognitive categories) and normative (values, beliefs and other culture-related notions), and therefore makes it possible to encompass a variety of institutions, including both formal and informal. the existent institutional research on entrepreneurship mostly focuses on how institutions influence entrepreneurial activity [12]. however, the focus of the present study is not on how institutions influence entrepreneurial activity per se, but the perception of entrepreneur and entrepreneurship in the society. following etzioni’s concept of legitimation of entrepreneurship [13], we argue that one of the impediments to entrepreneurship development in russia is that it is not seen as legitimate enough by the society at large. the notion that negative attitude to entrepreneurship affects its development in russia is voiced largely by business experts [14]. however, this issue didn’t receive sufficient coverage in the academic research. this paper aims to address this gap. 2. three-dimensional institutional profile as the legitimation framework etzioni [13] argues that the legitimacy of entrepreneurship in a society is a key factor that determines the level of entrepreneurship: “the extent to which entrepreneurship is legitimate, the demand for it is higher; the supply of entrepreneurship is higher; and more resources are allocated to the entrepreneurial function”. legitimacy is defined as a “generalized perception or assumption that the actions of an entity are desirable, proper, or appropriate within some socially constructed system of norms, values, beliefs, and definitions” [15]. accordingly, legitimation occurs through placing a phenomena within a framework through which it is viewed as right and proper [16]. the three dimensions of the institutional profile imply three bases of legitimacy [10,17]. entrepreneurship can be legitimized if it conforms to legal requirements (regulatory dimension), if it is seen as legitimate through a common frame of reference (cognitive dimension) and if it conforms to the existent moral base (normative dimension). in order to explore these sources of attitude to entrepreneurship in russia: three-dimensional institutional approach 31 copyright ©2017 assa. adv. in systems science and appl. (2017) legitimation, it is necessary to specify the contents of each dimension of the institutional profile. since the set of regulatory, cognitive and normative institutions within the institutional profile is issue-based [9], this implies that the contents of each dimension are defined by researchers according to the specifics of their study, and such is indeed the case [11,18-21]. adapting this framework to the aims of our research, we will specify the contents of each dimension as follows. the regulatory component includes the “existing laws and rules in a particular national environment that promote certain types of behaviors and restrict others” [9]. as the detailed revision of the russian legislation is not the aim of this paper, within this dimension we will focus on the two specific points: the dynamics of the legal legitimation of entrepreneurial activity from the soviet epoch to the present times, and the policy statements made by the authorities regarding the current attitude to business legitimation. the cognitive component in the context of entrepreneurship was defined as “the widely shared social knowledge and cognitive categories (for instance, schemata, stereotypes) used by the people in a given country that influence the way a particular phenomenon is categorized and interpreted” [9]. following this broad definition, we will focus specifically on the stereotypes. stereotypes are psychological representations of the characteristics of people that belong to particular groups which serve as aids to cognition by simplifying the person perception process [22]. consequently, in our study of this component of the institutional environment we will focus on the stereotype of entrepreneur defined as a set of widely-shared beliefs about his or her personal attributes and qualities. the normative dimension of the institutional environment reflects the values, beliefs, norms, and assumptions about human nature and human behavior held by the individuals in a given country [9]. the research on connection between values and various dimensions of entrepreneurial phenomena is extensive [4-6] and includes russian culture as well [23,24]. as mentioned above, this stream of research is mostly based on the models for cross-cultural comparison such as developed by hofstede [7], inglehart [8] and schwartz [25]. the application of these frameworks to the russian culture was researched by many authors [26-30], so we do not see the need to further explore this perspective. instead, we will build our discussion of the normative dimension around the concept of basic assumptions. trompenaars and hampdenturner [31] and schein [32] place the basic assumptions at the deepest layer of culture. schein [32] argues that when value leads to a certain behavior which solves the problem, it is transformed into “underlying assumption about how things really are”. these are assumptions about relationship to environment, nature of reality, time and space, nature of human nature, nature of human activity, and nature of human relationships – a classification very similar to the value orientations of kluckhohn and strodtbeck [33]. trompenaars and hampden-turner [31] describe basic assumption as “absolute presupposition about life”, “what is taken for granted, unquestioned reality”. there is no clear boundary in the existing literature between values and basic assumptions. however, since the domain of values rests mostly with cross-cultural research on entrepreneurship (etic approach), for the study of the meanings implicit to the russian culture we will employ the emic approach [34] and use the term “basic assumption” to avoid conceptual confusion. in view of our focus on entrepreneurship, we will discuss the assumptions which relate to the fundamental moral dimensions of entrepreneurial activity: assumptions about money and wealth, and about work as a means to acquire it. the rationale behind this set of assumptions lies in the notion of entrepreneurship and business. in essence, entrepreneurial behavior is aimed at profit generation [35] – the same can be said about business behavior in general. therefore the concept of money becomes central to the discussion of legitimation of entrepreneurship: if money-seeking 32 i. a. petrovskaya, s. m. zaverskiy & e. s. kiseleva copyright ©2017 assa. adv. in systems science and appl. (2017) is encouraged in the specific culture, we may expect that the activity aimed at acquisition of money will be legitimized. the important point here is that money is not just the medium of exchange – it is a symbol imbued with meaning [36]; money and wealth signify deepseated and complex values [37]. the idea that fundamental assumptions about the morality of striving for profit and wealth accumulation, as well as work ethic are connected to the entrepreneurial activity can be traced back to weber [4]. therefore the assumptions about money, wealth and work will be the focal point of our discussion of the normative dimension of the legitimation of entrepreneurship. finally, we need to add a caveat that the three institutional dimensions of legitimation are in interplay. the interaction of normative and regulatory components was demonstrated by edelman [38], of cognitive and normative dimensions – by aldrich & fiol [39]. cognitive dimension may interact with the regulatory one – e.g. the way of treating entrepreneurs as criminals can lead to the creation of the cognitive scheme that implies that entrepreneurs are indeed criminals. although for the purposes of the present study we will draw boundaries between the different components of the institutional profile, it is nevertheless important to keep in mind that they do not exist separately from each other. 3. attitude to entrepreneurship in the society attitudes to entrepreneurship in a society influence the level of entrepreneurial activity, determine peoples’ intentions to become entrepreneurs and influence the business climate. in this section we will explore the specifics of the general attitude to entrepreneurship in russia. as a method of research we use comparative analysis of sociological data revealing respondents’ attitudes to entrepreneurship activity and entrepreneurs themselves, respondents’ opinions about main characteristics of entrepreneurs and their role in economic and social development. we draw on the data from the surveys conducted by the leading russian opinion research companies such as levada-center and fom. samples of these surveys are representative for the russian adult population as a whole (at age of 18 and older). we also use data of international surveys on entrepreneurship, such as global entrepreneurship monitor and eurobarometer. in the soviet era entrepreneurial activity was a criminal offence – private property and private business became legitimate only in the late 1980s, and due to the rapid transition from command to market economy the attitude of russians towards entrepreneurs and entrepreneurial activity is now contradictory. fom surveys conducted in december, 2001 (10 years after the dissolution of ussr) and in march 2013, show that the majority of respondents reported having positive attitude towards entrepreneurs (58% in 2001 and 68% in 2013), and the recent data indicates that 42% of population agrees that entrepreneurship is beneficial for the country [40]. eurobarometer surveys [41] show that a very similar share of respondents in russia, european union and the usa agree that entrepreneurs bring economic benefits through creating jobs (87% of respondents in eu27, 88% in the usa and 89% in russia). if the public surveys show that the attitude to entrepreneurship is quite positive – much less positive than in the usa, but still comparable to eu27 – why the concerns about the negative attitude to entrepreneurship are voiced by the business itself [14]? to answer this question we need to look deeper into the subject and consider not only on the general attitude to entrepreneurship and to its outcomes, but the entrepreneurial behavior. in the fom 2001 survey, when asked an open question about “what do entrepreneurs do? what is their job about?”, the significant part of respondents characterized entrepreneurship attitude to entrepreneurship in russia: three-dimensional institutional approach 33 copyright ©2017 assa. adv. in systems science and appl. (2017) largely as a selfish profit-seeking activity. among the respondents 26% think that entrepreneurs make money: they “grab money”, “coin money”, “enrich themselves”. another 23% of the respondents point out that entrepreneurs speculate, repurchase goods: “buy cheaper, sell at higher price”, “in soviet times it was called “speculation”, “rip-off the consumers”. at the same time positive evaluation of entrepreneurial activity is less common. only 6% of respondents point out that entrepreneurs “launch manufacturing”, 5% – that entrepreneurs “develop the economy”, 2% – “work hard” and 2% – “create jobs” [42]. the eurobarometer survey results [41] also indicate that when it comes to entrepreneurial behavior, the inclination to ascribe negative properties to entrepreneurs in russia becomes more apparent. the share of respondents agreeing that “entrepreneurs take advantage of other people’s work” in russia exceeds the eu27 and the usa levels and amounts to 76%, and the same applies to the statement “entrepreneurs think only about their own pockets”. levada-center has conducted several surveys in 1991-2013 when the respondents were asked to describe the qualities of the russian and western entrepreneurs [43], and these indeed support the concerns about the negative image of an entrepreneur: according to the survey results, the respondents see the primary quality of the russian entrepreneur as the “itch for gains”, or profitseeking (with negative connotations), followed by “inclination for deception and fraud”, “cornercutting” and “unwillingness to work honestly”. moreover, when compared to the qualities attributed to the western entrepreneur, the negative image of the russian entrepreneur becomes even more pronounced: while a western entrepreneur is characterized by the business acumen (51%), rationality (44%) and competence (39%), the russian entrepreneur looks for gains mostly of the illicit quality (53%) [43]. thus, even though the public surveys show the general attitude to entrepreneurship in the russian society being rather positive than negative, the image of entrepreneurs, the motives of their behavior and their personal qualities are characterized in a primarily negative way, which supports the claim that entrepreneurship is not seen as a legitimate activity. in the following section we will discuss the sources of legitimizing entrepreneurship through the perspective of a three-dimensional institutional profile. 4. three-dimensional approach to legitimization of entrepreneurship in russia as discussed above, the three dimensions, or pillars, of the institutional profile imply three bases of legitimacy. entrepreneurship is legitimized through laws and regulations (regulatory dimension), through a common frame of reference (cognitive dimension) and through the existent moral base (normative dimension). we will start with the regulatory dimension and focus on the development of the legal legitimation of entrepreneurial activity from the soviet epoch to the present times, and on the policy statements made by the authorities regarding the current attitude to business legitimation. 4.1. the regulatory dimension in the soviet times entrepreneurship was officially qualified as a criminal offence. the criminal code of russian soviet federal socialist republic of 1960 imposed the punishment for “private entrepreneurial activity” of up to 5 years of imprisonment (article 153). profiteering (article 154) was punished with imprisonment for up to 7 years. currency transactions implied 34 i. a. petrovskaya, s. m. zaverskiy & e. s. kiseleva copyright ©2017 assa. adv. in systems science and appl. (2017) punishment ranging from imprisonment for up to 15 years to death penalty (by the special law) – actually, that was an equivalent of punishment for aggravated homicide (article 102). the legal emergence of entrepreneurs became possible only after 1986, when selfemployment and creation of joint ventures with foreign capital were allowed. after that it became possible to establish cooperative organizations (“cooperatives”) – prototypes of private companies able to rent their premises from state enterprises. at the same time, the prosecution of “underground entrepreneurs” was strengthened. however, those entrepreneurs who became rich overnight were distrusted by the society due to the dishonest nature of these earnings. partly, this was also facilitated by government policies [44]. the sources of immediate enrichment were often connected with certain regulations, providing exceptional opportunities to a small number of admitted individuals. the examples are provision of export licenses, import subsidies, preferential industrial loans and restrictions on domestic prices in some sectors. however, despite the formal legitimation of entrepreneurship in the legal sense, an ambiguous attitude of public authorities to business and, in particular, to entrepreneurs, was still publicly shown. even in 1990s, at the time of large-scale liberalization and promotion of entrepreneurship (at least formal), the state acted as a “grabbing hand” [45]. an entrepreneur in russia in 1990s de-facto was treated by government officials as an object for contrived arraignments and extortions, and not as an object for care and support. as a result, many entrepreneurs could not make their business in a law-abiding way, and had to “go into the shadows” [46]. since 2000 a number of initiatives were implemented to improve the image of an entrepreneur in russia and enhance the government support of business (at least in terms of lowering the administrative barriers). however, despite these initiatives, the laws that regulate business activity currently appear to inhibit it to a certain extent, primarily because of their inconsistency: “even when laws and regulations do not obstruct firms’ entry and exit, application and enforcement of rules often remain inconsistent” [47]. thus the analysis of the regulatory dimension of the institutional profile shows that even though the entrepreneurship is officially legalized, the issue with the regulatory legitimation is an environment which makes it difficult for entrepreneurs to follow the existent laws, thus often forcing them to become illegitimate by default. 4.2. the cognitive dimension in this study the cognitive dimension of the institutional profile deals with the stereotype of an entrepreneur – a set of widely-shared beliefs about his or her personal attributes and qualities. the beliefs about the qualities of an entrepreneur and the negative stereotype of an entrepreneur have already been discussed above. in the following section we will look at how this stereotype developed and how it was transferred to the population. specifically, two aspects of stereotype development should be considered: the soviet ideological attack at the entrepreneurs, and the stereotypes which developed in the transition period. 4.2.1. entrepreneurship and the soviet ideology stereotypes are formed largely through education. in the soviet school education the ideological tasks dominated over the educational ones [48]. a special role in the spread of ideology in the soviet time was played by primary and secondary schools. elements of ideology were present in many academic disciplines, but the main role in this process was given to history and literature. the history textbooks in the soviet union were the mirror of public policy [49]. the idea that history textbooks should play the central role in the ideological construction was formulated in attitude to entrepreneurship in russia: three-dimensional institutional approach 35 copyright ©2017 assa. adv. in systems science and appl. (2017) the ussr in 1930. in march 1934, the current state of history teaching in schools has been the subject of two sessions of the politburo, and stalin was personally involved in the editing of general history textbook [50]. one of the reviews on a history textbook, written in 1948, was rather typical and stated that “the idea of irreconcilable class struggle of the proletariat against all its enemies should permeate all textbook exposition. as a result of modern history studies, soviet schoolchildren should feel the hatred of capitalism and its’ political leaders, should feel contempt and disgust for the social-democratic lackeys of capitalism” [51]. later, in the post-stalin times, the situation has changed only slightly – in the resolution of the central committee of the cpsu and the ussr council of ministers of 1959 it was stated that the course of history in the secondary school should form the belief in the inevitable collapse of capitalism and the victory of communism. in line with this, one of the central places of the soviet history textbooks was devoted to a crisis of capitalism, and futility of this way of development was strongly substantiated. a capitalist was portrayed almost exclusively as an immoral oppressor of the working people. in line with this, it was stated that only centralized, authoritarian state is necessary for ussr. noteworthy, the key findings on these topics in new history textbooks of the post-soviet era were largely preserved, although the grounds have changed [49]. teaching of literature also occupied a significant place in soviet schools. this was largely due to the fact that russian literature was a synthetic phenomenon, exercising at one time the functions of philosophy, humanities, the social and political platform [52]. accordingly, the image of an entrepreneur as shown in the literature, especially in the classical literature of the xix century, has produced a number of stereotypes that are deeply embedded in the consciousness of many generations. moreover, the educational curricula focused only those works from the variety of classical literature that supported existence of such stereotypes. as a result, the idea of entrepreneurship as a free creativity was undermined. description of business was focused not on the business itself, but only on money acquisition, while business activities were associated with a number of negative characteristics: moral decay, deceit, exploitation of other people, even crime [53]. the origins of entrepreneurial success, vitality and energy of businessmen were seen in the illegibility in achieving the goals – people moved by a passionate desire to create value often found themselves unable to resist the temptation to “accelerate” it through fraud, corruption, forgery [54]. referring to the surveys quoted above [43] this set of characteristics remarkably coincides with the personal qualities attributed to entrepreneurs in the post-soviet times. 4.2.2. the image of entrepreneur in transition period even though in soviet times entrepreneurship was a criminal offence, some individuals were still involved in these activities, and some of them were convicted and sentenced to prison. thus in the post-soviet era it turned out that business experience and spirit of entrepreneurship can be often attributed to ex-prisoners. while being in prison, they had time to develop connections in the criminal world, and thus, later on, the newborn entrepreneurs were surrounded by former criminals. in a situation when property protection and law enforcement were extremely poor, the private businesses had to resolve problems with the help of the so-called "roof" (“krysha”) that provided protection of their lives and interests [55]. as a result, the public perception of business, even after it was legitimized by the authorities, was still associated with illegal and criminally bound activities. the other category of new-born businessmen which emerged in the transition period was former communist party officials, managers of public enterprises and government officials. their 36 i. a. petrovskaya, s. m. zaverskiy & e. s. kiseleva copyright ©2017 assa. adv. in systems science and appl. (2017) credibility among ordinary people was rather low at all times, but after legalization of business it turned out that what had previously been considered as public property suddenly became someone's undeserved, but legal private property, which created a strong feeling of unfairness in the general public. this was also stipulated by numerous cases of administration officials legally obtaining a privileged position during the privatization of enterprises. the large-scale fund withdrawals from these enterprises (for the benefit of privatizers) were held directly in front of workers who were paid with significant delay [56]. the privatization process largely affected the public attitude to entrepreneurship. after privatization the majority of russian citizens did not get any income-generating property, in contrast to propaganda of privatization. moreover, they felt themselves robbed. this became especially evident after the “shares for loans” auctions used for privatization of major state enterprises that invited questions among the public – why did somebody become the owner of resources and enterprises that were built by thousands of people and worked for decades almost for free [57]. and even if the new owners paid something for the assets, there still was the question about origins of the money. the media reports about dollar billionaires further led to negative and even hateful attitude to them. “itching for gains” as an attribute of an entrepreneur may be largely traced back to the privatization as well as to the soviet era. thus the complicated history and heritage of the transition period still influences the perceptions of entrepreneurship, and combined with the issues in the regulatory dimension contributes to the persistence of the negative general stereotype of the entrepreneur in the society. 4.3. the normative dimension as mentioned above, the normative dimension of the institutional environment deals with basic assumptions relate to the fundamental moral dimensions of entrepreneurial activity: assumptions about money and wealth, and about work as a means to acquire it. in essence, business behavior is aimed at revenue generation through seeking opportunities and rational allocation and utilization of resources. this contrasts sharply with the superiority of spiritual over material inherent in the russian culture. this opposition is well illustrated in chekhov’s “the cherry orchard”, where lopakhin, whose character doesn’t display any negative traits, nevertheless invites antipathy by his triumph after the acquisition of the cherry garden. his behavior, which makes perfect sense from the business perspective, doesn’t invite any compassion as opposed to the behavior and attitudes of the impractical and passive landowners living in the past. similar opposition can be found in dostoevsky’s protagonist’s speculation on “which is the worst of the two russian ineptitude or the german method of growing rich through honest toil”, followed by stating his preference for the former: “i would rather live a wandering life in tents, … than bow the knee to a german idol” [58]. why does “ineptitude” emerge as a more worthy option than “honest toil”? we argue that the roots of this can be found in the specific attitude to money, wealth and work shaped partly by religion and work ethic. 4.3.1. assumptions about money and wealth the concept of money and wealth is one of the focal points in the discussion of legitimation of entrepreneurship: if money-seeking is encouraged in the specific culture, we may expect that the activity aimed at acquisition of money will be legitimized. in russia, the specific attitude to money and wealth has developed under the influence of the religion and the general course of economic development. these will be considered further on. attitude to entrepreneurship in russia: three-dimensional institutional approach 37 copyright ©2017 assa. adv. in systems science and appl. (2017) the orthodox religion promotes the ideals of simplification and humility. the “mundane way of escape and pilgrimage”, “high estimate of begging and poverty” was preached already in ancient religious poetry [52,59]. accordingly, the entrepreneurs saw their activities not only and not so much directed towards accumulation of wealth, but rather as a kind of a mission entrusted by god or fate. a distinctive feature of orthodoxy is that the owner is not the master of his estates, but is the manager of the god’s belongings that he was given for temporary use during his life [60]. it was typically said that the wealth was granted by god for use, and god will require a report on it, which contributed to the development of philanthropy that was regarded as the fulfillment of a duty [61,60]: by 1900, in moscow there were more donations produced than in paris, berlin and vienna combined [60]. thus money and wealth were not considered as fundamental values, and were treated as perishable and incidental, subject to a high risk of loss. this may have continued into the modern period and manifest itself in the desire to immediately make a handsome fortune [62]. insecurity, lack of guarantees of the irreversibility of reforms, instability in the economy lead to the spread of “a one-time” business psychology, that has nothing to do with the care for reputation or business ethics. 4.3.2. assumptions about work assumptions about work – its meaning and its purpose – are as central to the discussion of entrepreneurship as are the assumptions about money. the soviet period undermined the meaning of work and the incentives for productive work. higher salaries were not necessarily aligned with productivity growth, the rates were normalized, and the employees knew that they get paid primarily for the time spent for work, and not for the result. moreover, motivation was restricted by the limited availability of career opportunities. as noted in the popular saying, “they pretend that they pay us and we pretend that we work” [44]. however, even in the presoviet period work had specific connotations that shaped the assumptions about work and its meaning. the etymology of the word “work” (“trud”) in russian implies that the work is seen as a necessary burden or necessary evil rather than the source of joy or fulfillment: work is ”everything that requires effort, diligence and care, any tension of bodily or mental powers, all that makes tired” [63]. religion and the work ethic it implies also have the fundamental influence on the assumptions about the meaning of work and its purpose. in pre-revolutionary period the work ethic that was dominant among peasants and industrial workers was minimalist or traditional [64], focused on meeting the modest, almost minimal needs for food, clothing, shelter [65]. minimalist consumption standards allow individuals not to worry about the accumulation of wealth [66]. this type of work ethic existed in europe as well, but with the start of industrialization it began to transform – the process which weber associated with the emergence of protestantism [67]. while the protestant ethic, according to weber, suggests that the criteria chosen for salvation is the welfare, and the path to welfare is multiplication of wealth through work [67], the orthodox church suggests that economic activity has nothing to do with the salvation of souls. in orthodoxy, not any work is useful, but only work that contributes to the improvement of the soul. orthodox religion doesn’t denounce work per se – on the contrary, work is seen as a natural mode of life [68]. however, the ultimate goal of any activity, including economic activity, is spiritual improvement, and material well-being is not connected with the prospects of salvation. moreover, the extent of this material well-being is set at the level which provides for meeting the basic needs, and anything exceeding this level is not seen as moral. the situation has partly changed in the soviet era. marxist ideology stated that work “transformed apes into men”, and the love for work was a measure of moral maturity. working 38 i. a. petrovskaya, s. m. zaverskiy & e. s. kiseleva copyright ©2017 assa. adv. in systems science and appl. (2017) was a primary activity, and those who were not engaged in public production without good reason were considered as “parasites” and were subject to criminal prosecution [66]. due to this emphasis on working soviet model is sometimes compared with protestantism [69] – however, the important difference was that in terms of the soviet ideology the purpose of work was not to obtain individual benefits, but contribute to meeting the goals of the society, as stipulated in 1977 constitution of the ussr. the assumptions about work and its meaning which constitute the normative dimension of the institutional profile allow an observation that the soviet experiment didn’t develop in a vacuum, but to some extent built on the existent normative environment. it can be seen that some aspects of these assumptions are deep-lying and even based on etymology. recent research suggests that work values have been changing since the start of the transition period, and the changes happen very quickly [70] – specifically, money comes to be seen as a primary reason for work, which is typical for the emerging economies [71]. however, research also suggests that the general attitude to entrepreneurship is defined primarily by the morality of aspiration for wealth [72], which supports the concept of moral legitimation as essential to legitimation of entrepreneurship [13]. 5. conclusion the issue of legitimacy of entrepreneurship in russia dates back many centuries and cannot be attributed only to the soviet period. however, the soviet times delivered probably the strongest blow to the legitimation of entrepreneurship. entrepreneurship was a criminal offense, and the system of education transferred a negative image of entrepreneur to the population. thus currently one of the problems that have to be dealt with is the generational gap in assessing entrepreneurial activity and its role in the economy. while the positive attitude towards entrepreneurs is typical for the young and educated population groups, the older generation sees it mainly through the negative lens [42]. however, the younger generation is placed in the complicated institutional environment which often provides conflicting messages, especially its regulative component. what is needed now is the change regarding all the three dimensions of the institutional profile. clear messages about the legitimacy of entrepreneurship should be sent by the state and the government, and the rules of the game should be set and observed (regulatory dimension). the positive stereotypes of entrepreneurs should be developed and transmitted (cognitive dimension). the normative dimension, which deals with the basic assumptions and with the deepest layers of culture, is the most difficult to change, however it should also be addressed. the globalizing world and the contact with different cultures that it brings may gradually influence the change of these basic assumptions. in any case, what is required for the further development of entrepreneurship is the consistent policy and the consistent effort that recognizes the interconnection between the different dimensions of the institutional environment and their specifics. entrepreneurship is one of the key drivers of the economic growth, and the current level of entrepreneurial activity in russia undermines its potential for diversifying the economy, thus making it susceptible to the fluctuations in oil and gas prices that can be subject to the market manipulations. development of the institutional environment with an aim to support and promote entrepreneurship is thus becoming a key to opening the unlimited dimensions of economic growth [73]. attitude to entrepreneurship in russia: three-dimensional institutional approach 39 copyright ©2017 assa. adv. in systems science and appl. 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(2016). the making of an economic superpower: unlocking china’s secret of rapid industrialization. singapore: world scientific. http://bd.fom.ru/report/cat/pow_pec/dd050322 http://slovardalja.net/ adv syst sci appl. 2019; 03;11–22 published online at https://ijassa.ipu.ru/index.php/ijassa/article/view/751 hybrid particle swarm optimization and pegasos algorithm for spam email detection lamiaa m. el bakrawy∗ faculty of science, al-azhar university, cairo, egypt e-mail: dr lamiaa el bakrawy@azhar.edu.eg received may 25, 2019; revised september 9, 2019; published october 1, 2019 abstract: email is one of the most popular communication tools for most internet users nowadays. it has become fast and an effective method to share and exchange information all over the world. despite the great advantages of emails, its usage is facing problem which is spam emails. spam emails are the huge presence of bulk and unsolicited emails which are expensive for the companies, consume a huge amount of mail servers, network bandwidth and waste of time. isolating and detecting these emails is known as spam detection. many spam detection methods have been proposed but there is still need to detect the email spam effectively with high accuracy. in this paper, hybrid particle swarm optimization and pegasos algorithm, which is called (psopegasos) is proposed for spam email detection. particle swarm optimization is employed as a search strategy to determine the optimal parameters for pegasos algorithm in order to achieve higher performance. the proposed algorithm has been applied on spambase dataset downloaded from uci machine learning repository. experimental results demonstrate that the proposed algorithm outperforms the performance of all the earlier proposed algorithms, considering the accuracy, recall, precision and f-measure on the same dataset. keywords: particle swarm optimization, pegasos algorithm, email spam, spam detection, accuracy. 1. introduction nowadays, email is one of the most popular communication tools for most internet users because of its free availability and efficiency [1, 2]. email is a method of receiving and sending information over electronic networks such as the internet. however, the major problem is the presence of bulk and unsolicited email which is known as spam. spammer is the person who sends mass quantity of spam emails and collects email addresses from chatrooms, viruses, customer lists and websites. spam email consumes a huge amount of mail servers, network bandwidth and wastes users’ time to remove all spam emails which causes lower productivity. thus, how to isolate and detect spam email in efficient way with high accuracy becomes an important study. spam email detection can be considered as classification problem which is used to detect the spam emails one by one to classify email as spam or non-spam [3]. in recent years, most of the spam detection algorithms based on machine learning techniques is used, but still the reported accuracy requires more work to accomplish better accuracy. sabri et al. in [4] presented continuous learning approach based on artificial neural network (cla ann) for spam email detection. they made core modifications in the input layer of artificial neural network to substitute the useless layers with new favorable layers and to be varied with time. ∗corresponding author: dr lamiaa el bakrawy@azhar.edu.eg 12 l.m. el bakrawy the results showed that applying cla ann using 300 input layers succeeded in achieving 3.668 % false negative and 0.534 % false positive. zhang et al. in [5] presented naive bayes model for spam email detection by applying cost-sensitive multi-objective genetic programming for feature extraction and achieved an accuracy of 79.3%. renuka et al. in [6] proposed spam classification algorithm using hybrid ant colony optimization and naive bayes classifier and applied it on spambase dataset. the accuracy obtained was 84% which indicated that the hybrid algorithm outperformed hybrid genetic algorithm and naive bayes tested on the same dataset. özgür et al. in [7] used artificial neural network and bayesian filter for spam email detection. they considered two artificial neural network structures, multi layer perceptron and single layer and the inputs are specified based on probabilistic and binary models. experimental results for 750 e-mails (410 spams and 340 non-spam), achieved 90% accuracy. temitayo et al. in [8] used genetic algorithm to optimize the support vector machines (svm) classification parameters. the hybrid algorithm achieved 90% accuracy for the testing set. liu et al. in [9] proposed a new learning method (pso-lm ) for process propagation neural networks (pnns) based on particle swarm optimization (pso) and gaussian mixture functions. experiments results showed that applying (pso-lm) on spambase dataset achieved 90.5% accuracy for the testing set which is better than back propagation neural networks (bpnns) and basis function expansion based learning method (bfe-lm). moreover, idris and selamat in [10] presented a hybrid model of negative selection algorithm (nsa) and particle swarm optimization (pso). they worked on spambase dataset and achieved 91.22% accuracy for the testing set. awad and foqaha in [1] proposed a hybrid algorithm of rbf neural network and particle swarm optimization (hc-rbfpso) for spam email classification. they used particle swarm optimization algorithm to optimize the parameters of radial basis function neural networks (rbfnn) based on the evolutionary heuristic search of pso. they divided spambase dataset into 70% training set and 30% testing set. experiments are measured by using a different number of hidden layer starting from 10 to 50. the accuracy obtained was 91.4% for the testing set which was concluded that the hybrid approach had good performance compared to other algorithms tested on the same dataset. olatunji in [11] proposed support vector machines-based model for spam detection. he used a systematic parameter search in order to achieve better spam detection accuracy. the accuracy obtained was 94.06% for the testing set. experimental results show that the proposed scheme outperformed other published algorithms tested on spambase dataset used in this work. considering the performance accuracy achieved till now, there is still need to try to achieve better results on the same dataset. the main aim of this paper is to propose an alternative algorithm that can accomplish a performance higher than previous algorithms. in this paper, hybrid particle swarm optimization and pegasos algorithm (pso-pegasos) is proposed to achieve better accuracy of spam email detection. pegasos algorithm is applied to solve the optimization problem cast by support vector machines (svm) while particle swarm optimization is used as a search strategy to select the optimal parameters (the weights) for pegasos algorithm, which means in each iteration of pso, the weights (w-parameters) are changed based on the fitness function (mean squared error). after running pso algorithm a number of iterations, it will obtain the best optimal w-parameter for pegasos algorithm. in this paper, comparison of performance measures of pegasos and hybrid algorithm (pso-pegasos) for training and testing sets is presented for spam email detection. the rest of this paper is structured as follows: the fundamentals of particle swarm optimization and the principles of the original pegasos algorithm are introduced in section copyright c© 2019 assa. adv. in systems science and appl.(2019) hybrid particle swarm optimization and pegasos algorithm for spam email detection13 2. section 3 describes the details of the proposed algorithm. experimental results and discussions are demonstrated in section 4. finally, section 5 concludes the paper. 2. preliminaries 2.1. particle swarm optimization particle swarm optimization (pso) was invented by kennedy and eberhart in 1995 [12]. pso is a widely used population-based stochastic optimization technique since it has strong global search capability, high convergence speed, high robustness and is conceptually very simple [13, 14, 15]. pso is still attracted the attention of a lot of researchers over nearly a quarter century. particle swarm optimization simulates the social behavior among species such as fish schools, bird flocks. the set of particles represent a population of the possible solutions. in canonical pso algorithm, particles are initialized with a population to get a random solution. then, the particles fly iteratively around in d-dimension search space to search the optimal solution, where the proper fitness function can be calculated according to the problem. each particle is indicated by a row vector ~xi, where i is the index of the particle, and a velocity indicated by ~vi. the best position of the particle (pbest) is indicated by vector ~x#i , and its j-th dimensional value is x#ij , while the best position among the swarm (gbest) is indicated by a vector ~x∗, and its j-th dimensional value is x∗j . in each iteration t, the velocity updating formula of particle is calculated by eq. (2.1) and the position updating formula of particle is determined by the sum of the previous position and the new velocity by eq. (2.2). vij(t+ 1) = { wvij(t) + c1r1(x # ij(t)− xij(t)) +c2r2(x ∗ j(t)− xij(t)) (2.1) xij(t+ 1) = xij(t) + vij(t+ 1). (2.2) where c1 and c2 are nonnegative constants called as learning factors, r1 and r2 are random numbers uniformly distributed in u(0,1) for the j-th dimension of the i-th particle. w is the inertia weight, which can increase the algorithm search capability and control the process of algorithms searching. eq. (2.1) makes each particle tends to move across the design space, considering its own experience, which is the memory of its best fitness function value achieved by the particle in the past, and the experience of its most successful particle in the swarm. in pso algorithm, the particles tend to search the solutions in the problem space with a range [−s, s] to prevent the particle from flying away out of the search space. if the range [−s, s] is not symmetrical, it will be changed to the corresponding symmetrical range and the maximum velocity during one iteration must be limited on the interval [−vmax, vmax] given in eq.(2.3) vij = sign(vij)min(|vij| , vmax). (2.3) where the value of vmax is p× s, with p ∈ [0.1, 1] but vmax is usually selected to be s, i.e. p = 1. the termination criterion for iterations will be determined according to whether the maximum number of iterations or minimum fitness function error is reached. 2.2. pegasos: primal estimated sub-gradient solver for svm pegasos was described and analyzed by shalev-shwartz et al. in [16] for solving the optimization problem cast by support vector machine (svm). it performed a stochastic subgradient descent based on the primal objective by chosen step size carefully to improve copyright c© 2019 assa. adv. in systems science and appl.(2019) 14 l.m. el bakrawy convergence [17, 18, 19]. pegasos has attracted research interest because it has better convergence bounds and robustly convex optimization objective. it uses theory of strongly convex optimization problems and hinge loss instead of the original linear constraints which makes the objective of svm unconstrained. given a binary classification problem with training set s = (xi, yi), (i = 1, . . . , n), where xi is a d-dimensional feature vector and yi = ±1 is the class label. the goal of linear support vector machines is to find a classifier in the following form h(x) = sign(wtx), (2.4) where w is the weight vector which can be learnt from training set to solve the following optimization problem after number of iterations t . min w = λ 2 ‖w‖2 + 1 n ∑ (x,y)∈s l(w, (x, y)), (2.5) where l(w, (x, y)) = max(0, 1− y(w, x)), and λ ≥ 0 is the regularization parameter. in each iteration t , pegasos algorithm aims to update w by choosing a random training set at ⊆ s with size k, where k is the number of training examples used for calculating sub-gradient through the following approximate objective function f(w,at) = λ 2 ‖w‖2 + 1 k ∑ (x,y)∈at l(w, (x, y)). (2.6) the sub-gradient of the approximate objective function f(w,at) at wt is calculated by ∇t = λwt − 1 |at| ∑ (x,y)∈a+ t yx, (2.7) where a+ t is the set of examples when w suffers a non-zero loss. finally, the sub-gradient is used to update the weight by using a step size of ηt = 1 |λt| as wt+1 = wt − ηt∇t, (2.8) where ηt is the learning rate. the last vector wt+1 is the output of pegasos algorithm after number of iterations t . 3. the proposed algorithm in this section, we describe the proposed pso-pegasos algorithm to determine the optimal values of pegasos parameters as shown in figure 3.1. the detailed description is as follows: 3.1. data preprocessing in this paper, the popular and often used corpus benchmark spambase dataset is utilized to classify email as spam or non-spam. the dataset is available in numeric form and the features are frequencies of various characters and words in emails. the main tasks in preprocessing are transformation, reduction, cleaning, integration and normalization. normalization is an important pahse to fast the algorithm, convergence and decrease the influence of imbalance in data. in spambase dataset, normalization is done before running pso-pegasos algorithm. each feature of spambase dataset is normalized in the range [0, 1] through the following function copyright c© 2019 assa. adv. in systems science and appl.(2019) hybrid particle swarm optimization and pegasos algorithm for spam email detection15 figure 3.1: flowchart of the proposed algorithm. a = a−min max−min (3.9) where a is the scaled value, a is the original value, max and min are the maximum and minimum bounds of the feature value. 3.2. pso-pegasos algorithm in this research, hybrid particle swarm optimization and pegasos algorithm (pso-pegasos) is proposed for spam email detection. particle swarm optimization has been utilized to optimize the parameters of pegasos algorithm. in pso each solution is called a particle. fitness function ( mean squared error) is used to evaluate the particles for the optimal solution. particle swarm optimization is used as a search strategy to determine the optimal parameters (weights ) for pegasos algorithm, which means in each iteration of pso, the weights (w-parameters) are updated depending the fitness function. no assumptions are needed about the w-parameter in pegasos algorithm since pso algorithm can help us to identify automatically the best optimal w-parameter (bw) that utilized to obtain the highest classification accuracy for pegasos algorithm. the major steps of the hybrid particle swarm optimization and pegasos algorithm (pso-pegasos) are shown as follows: copyright c© 2019 assa. adv. in systems science and appl.(2019) 16 l.m. el bakrawy 1. initialize the population for w-parameter individuals (particles) in a random manner from spambase dataset. suppose that, each particle swarm position is xi = {ai,j, j = 1, 2, ..., k}, where ai,j is j th w-parameter for the i th individual, k is the number of features (attributes) of spambase dataset and the value of the w-parameter for each individual is vector of k random numbers in range from -10 to 10. 2. initialize velocity of particle swarm optimization randomly in range from -100 to 100 3. calculate the fitness function for each particle which is acquired by pegasos algorithm to classify non-spam and spam emails correctly by fitness =mse = 1 n n∑ i=1 (xi − yi)2. (3.10) where mse is the mean squared error , x is a vector of n predictions, and y is the vector of true values. 4. if the fitness function is better than the best fitness function of the particle (pbest) then the current position will be (pbest) 5. select the best position among all particles (gbest) in current iteration 6. update the velocity of each particle depending on eq. (2.1). 7. update the position of each particle (w-parameter) depending on eq. (2.2). 8. search the the pbest of particle as (w-parameter) of pegasos algorithm in same iteration. 9. repeat steps 3 to 8 until obtaining the best optimal w-parameter (bw) which leads to get the highest accuracy for spam email detection with more exploration in the search space. 4. experimental results and discussions in this paper, the experiments were performed on a system with a 2.40 ghz intel(r) core(tm)i7 processor and 16 gb memory using written codes in matlab 15. 4.1. dataset description in this work, the dataset utilized is spambase dataset which is used to evaluate the proposed algorithm. hopkins et al. [20] presented spambase dataset in their colleagues. it has been collected from uci machine learning repository site. in the spambase dataset, the total email instances is 4601. 1813 from these email instances are characterized as spam (39.4%) and the remaining are non-spam. spambase dataset consists of 57 features and 1 classification attribute, which is the label of class indicating the status of each email instance whether it is spam (1) or non-spam (0). most of the features (1-54) show particular characters or words were repeatedly occurring in an email or not. the features from 55 to 57 present the measurement for length of consecutive capital letters. the definitions of the features can be shown as follows • features from 1 to 48 are real continuous features which are equal to the percentage of words in the e-mail that match word. • features from 49 to 54 are real continuous features which are equal to the percentage of characters in the e-mail that match char. • feature 55 is real continuous feature which is equal to the average length of continuous sequences of capital letters. • feature 56 is an integer continuous feature which is equal to the length of longest continuous sequence of capital letters. • feature 57 is an integer continuous feature which is equal to the total number of capital letters in the e-mail. copyright c© 2019 assa. adv. in systems science and appl.(2019) hybrid particle swarm optimization and pegasos algorithm for spam email detection17 4.2. evaluation measures in this research, the evaluation of the proposed algorithm is carried out based on popular and commonly performance measures such as accuracy, recall, precision, f-measure [21, 22]. the information about these measures is done depending on the confusion matrix presented in table 4.1. table 4.1: confusion matrix actual class spam non-spam predicted class spam tp fp non-spam fn tn brief overview of each performance measure is shown below. • accuracy is defined as the fraction of all emails (non-spam and spam emails) that are classified correctly by the algorithm. it can be represented by the following equation: accuracy = tp+tn fp+fn+tp+tn (4.11) where tp and tn are the number of spam emails and non-spam emails correctly classified, respectively. fp and fn are the number of spam emails and non-spam emails incorrectly classified, respectively. • recall stands for the proportion of spam emails being recognized and can be represented as follows: recall = tp fn+tp (4.12) • precision stands for the fraction of spam emails that are correctly classified as spam. precision = tp fp+tp (4.13) • f-measure (f-score), denotes the harmonic average of precision and recall and can be written as follows: f −measure = 2∗precision∗recall precision+recall (4.14) 4.3. results and discussion the experimental method applied here followed carefully the computational intelligence technique. spambase dataset was first divided into two phases, training set and testing set in the ratio 7:3, respectively. the data was chosen randomly for training and testing sets in order to exclude any particular behavior of the dataset. then, the training set ( 70% of data) was first entered to the algorithm for training and validation and the rest of dataset ( 30% of data) was used to test the algorithm to ensure the performance accuracy of the proposed algorithm. to evaluate the proposed algorithm, the parameters settings for original pegasos algorithm are regularization parameter (λ) with different values from 0.0001 to 0.1 and number of iterations t =1000. the original pegasos algorithm is applied on spambase dataset with different values of λ and the four performance measures are recorded for training and testing sets as shown in table 4.2. it can be observed in table 4.2 that using a small value of λ (0.0001) in a large dataset increases the accuracy, recall, precision and f-measure, respectively for spam detection by pegasos algorithm. according to this result, we fixed the copyright c© 2019 assa. adv. in systems science and appl.(2019) 18 l.m. el bakrawy table 4.2: performance measures for pegasos algorithm with different values of λ for spambase dataset (λ) accuracy recall precision f-measure 0.0001 training set 93.62 % 0.9360 0.9370 0.9360 testing set 92.71 % 0.9270 0.9280 0.9270 0.001 training set 93.04 % 0.9300 0.9310 0.9310 testing set 92.51 % 0.9250 0.9250 0.9250 0.01 training set 92.60 % 0.9260 0.926 0.926 testing set 91.67 % 0.9170 0.9180 0.9160 0.1 training set 90.43 % 0.9040 0.9060 0.903 testing set 89.19 % 0.8920 0.8970 0.8900 value of λ as 0.0001 in proposed algorithm (pso-pegasos). in pso-pegasos, the parameters of particle swarm optimization were set as learning factors c1 = c2 = 1.4, vmax = 4 and inertia weight (w) was linearly decreased from 0.9 to 0.4. the population size was fixed to 20 particles to reduce the computational cost and fast the convergence process of the algorithm. tuning the parameters for particle swarm optimization is important in designing the algorithm. figure 4.2 shows the effect of the number of iterations on the accuracy of the proposed algorithm pso-pegasos using different number of iterations from 5 to 30. as shown in fig. 4.2, we can observe that when the number of iterations was increased, the accuracy was increased until it accomplished an extent (number of iterations =20) at which increasing the number of iterations did not affect the accuracy of the proposed algorithm. figure 4.2: effect of the number of iterations on the accuracy of pso-pegasos algorithm for training and testing sets. according to parameter analysis and paper results, we put number of iterations in pso = 20 to run the proposed algorithm, therefore, the computational cost is small. figures 4.3 and 4.4 show the performance measures accuracy, recall, precision and f-measure of pegasos and pso-pegasos algorithms for training and testing sets, respectively. experimental results in figures 4.3 and 4.4 show that the accuracy of the proposed algorithm (pso-pegasos) for training and testing sets are higher than the accuracy of pegasos algorithm by about 3.39 and 3.48 respectively. it also shows that the proposed algorithm outperforms pegasos copyright c© 2019 assa. adv. in systems science and appl.(2019) hybrid particle swarm optimization and pegasos algorithm for spam email detection19 algorithm in terms of recall, precision and f-measure for training and testing sets due to the existence of particle swarm optimization, which has strong global search capability and high convergence speed to optimal solution. figure 4.3: comparison of performance measures of pegasos and pso-pegasos for training set. figure 4.4: comparison of performance measures of pegasos and pso-pegasos for testing set. finally, in order to indicate that the improvement obtained by the proposed algorithm (pso-pegasos) clearer, its accuracy compared with earlier used algorithms implemented on the same dataset is presented below. experimental results in table 4.3 show that the results of the proposed algorithm outperforms the results of other published classifier called svm-based spam detector [11]. the proposed pso-pegasos presented improvement of 2.13% over svm-based spam detector model, which is the best among the other earlier published classifiers for spam email copyright c© 2019 assa. adv. in systems science and appl.(2019) 20 l.m. el bakrawy table 4.3: comparison of accuracy of the proposed algorithm and other published classifiers on spambase dataset classifiers classification accuracy ga-naive bayes [6] 77% aco-naive baye [6] 84 % pso-lm [9] 90.5 % nsa [10] 68.86 % pso [10] 81.32 % nsa-pso [10] 91.22 % hc-rbfpso [1] 91.4 % svm-based spam detector [11] 94.06% pso-pegasos (proposed) 96.19 % detection. it also presented an accuracy improvement of 4.79 % over a hybrid approach (hcrbfpso), that combines radial basis function neural network (rbfnn) and particle swarm optimization (pso) algorithm [1]. the proposed algorithm also presented improvement of 4.97% over a hybridized negative selection algorithm and particle swarm optimization (nsapso) [10], yet the proposed algorithm in this paper outperformed all the three algorithms nsa-pso, nsa and pso including the hybrid schemes. it also presented an accuracy improvement of 5.69 % over learning method for process neural networks based on particle swarm optimization (pso-lm) [9] and an accuracy improvement of 12.19 % over hybrid ant colony optimization and naive bayes (aconaive bayes), while presenting an accuracy improvement of 19.19% over hybrid genetic algorithm and naive bayes (ganaive bayes) [6]. 5. conclusion primal estimated sub-gradient solver for svm (pegasos) algorithm was utilized to solve the optimization problem cast by support vector machines (svm). it is characterized by better convergence bounds and robustly convex optimization objective. in this paper, hybrid particle swarm optimization (pso) and pegasos algorithm, called (pso-pegasos) is proposed for spam email detection. pso is used to identify automatically the best optimal w-parameter for original pegasos algorithm. the proposed algorithm has been trained and tested using popular and often used spambase dataset, which consists of collection of spam and non-spam emails with 57 features and 1 classification attribute. excremental results indicated that the proposed pso-pegasos algorithm outperformed original pegasos algorithm and other recently published algorithms tested on the same popular dataset used in this paper. the need for more accurate spam email detection method cannot be overemphasized, the proposed pso-pegasos algorithm provides improvement of 2.13% over svm-based spam detector model, which is the best among the previous reported schemes for spam email detection. the results show that pso-pegasos improves the convergence accuracy and it is an effective algorithm, which is a powerful alternative for spam email detection. we can conclude that the aim of this paper has been achieved through training and testing proposed pso-pegasos algorithm on spambase dataset. this algorithm has enabled build an improved spam email detection system based on hybridization of particle swarm optimization and pegasos algorithm. references copyright c© 2019 assa. adv. in systems science and appl.(2019) hybrid particle swarm optimization and pegasos algorithm for spam email detection21 1. awad m, foqaha m (2016) email spam classification using hybrid approach of rbf neural network and particle swarm optimization, international journal of network security and its applications (ijnsa) vol.8, no.4, pp. 17-28. 2. saad o, hassanien a, darwish a, faraj r (2013) a survey of machine learning techniques for spam filtering, ijcsns international journal of computer science and network security, vol.13 no.1, pp. 103-110. 3. zhiwei m, singh m, zaaba z (2017) email spam detection: a method of metaclassifiers stacking, proceedings of the 6th international conference on computing and informatics, icoci, pp. 750-757.757. 4. sabri a, mohammads a, al-shargabi b, hamdeh m (2010) developing new continuous learning approach for spam detection using artificial neural network (cla ann), european journal of scientific research, 42(3), pp. 525-535. 5. zhang y, li h, niranjan m, rockett p (2008) applying costsensitive multiobjective genetic programming to feature extraction for spam e-mail filtering. springer, berlin, pp. 325-336. doi:10.1007/978-3-540-78671-9 28 6. renuka d, visalakshi p, sankar t, improving e-mail spam classification using ant colony optimization algorithm, international journal of computer applications (0975 8887)international conference on innovations in computing techniques (icict 2015) 7. özgür l, güngör t, gürgen f (2004) spam mail detection using artificial neural network and bayesian filter. pp. 505-510. doi:10.1007/978-3-540-28651-6 74. 8. temitayo f, stephen o, abimbola a (2012) hybrid ga-svm for efficient feature 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li h, zhu p (2019) diversity enhanced particle swarm optimization algorithm and its application in vehicle lightweight design, journal of mechanical science and technology, 33 (2), pp. 695-709. 15. cheng s, lu h, lei x, hi y (2018) a quarter century of particle swarm optimization, complex and intelligent systems, january 2018, accepted, 22 march 2018 16. shalev-shwartz s, singer y, srebro n, (2007) pegasos: primal estimated sub-gradient solver for svm, in proceedings of the 24th international conference on machine learning, pp. 807 -814. 17. shalev-shwartz s, singer y, srebro n, cotter a (2011) extended version: pegasos: primal estimated sub-gradient solver for svm, mathematical programming, series b, 127(1), pp. 3-30, springer and mathematical optimization society. 18. lu s, jin z (2017) improved stochastic gradient descent algorithm for svm, international journal of recent engineering science (ijres), issn 2349-7157, vol 4, pp. 39-42. 19. v. jumutc, x. huang, j. a. k. suykens,(2013) fixed-size pegasos for hinge and pinball loss svm, in proceedings of the 2013 international joint conference on neural networks (ijcnn), pp. 1122-1128, 2013. copyright c© 2019 assa. adv. in systems science and appl.(2019) 22 l.m. el bakrawy 20. hopkins m, reeber e, forman g, suermondt j (1999) spambase dataset. hewlett-packard labs, 1501 page mill rd., palo alto, ca 94304. https://archive.ics.uci.edu/ml/datasets/spambase. 21. el bakrawy l (2017) grey wolf optimization and naive bayes classifier incorporation for heart disease diagnosis, australian journal of basic and applied sciences, 11(7) m, pp. 64-70. 22. liu p, moh t (2016) content based spam e-mail filtering, in proceedings of the international conference on collaboration technologies and systems, 978-1-5090-23004/16 31.00, ieee, pp. 2018-2024. copyright c© 2019 assa. adv. in systems science and appl.(2019) introduction preliminaries particle swarm optimization pegasos: primal estimated sub-gradient solver for svm the proposed algorithm data preprocessing pso-pegasos algorithm experimental results and discussions dataset description evaluation measures results and discussion conclusion adv syst sci appl 2021; 02:8–19 published online at https://ijassa.ipu.ru. uncertain controllability and observability of an optimal control model tolulope latunde1*, adam a. ishaq2, adedayo f. adedotun3, joel o. ajinuhi4, olumuyiwa j. peter5 1department of mathematics, federal university oye-ekiti, oye-ekiti, nigeria 2department of physical sciences, al-hikmah university, ilorin, nigeria 3department of mathematical sciences, olabisi onabanjo university, ago-iwoye, nigeria 4department of mathematics, federal university of technology, minna, nigeria 5 department of mathematics, university of ilorin, ilorin, nigeria abstract: the interest of this paper is to examine the controllability and observability of a control system in the configuration state-space of an uncertain optimal control system. the control system is designed based on the realization of capital asset values where a special case of asset management is modelled and optimized. thus some necessary and sufficient conditions of the controllability and observability of the deterministic systems and the corresponding uncertain systems for the case of the uncertain optimal control system with application in capital asset management are considered. keywords: controllability, observability, uncertain systems, optimal control, capital asset management 1. introduction controllability and observability are important properties in control systems. they represent the ability to move a system around its entire configuration space using certain manipulations. the controllability and observability of a system are mathematical duals that play important roles in control problems such as optimal control. these have played important roles in control theories such as in [1–5]. recently, researchers such as [6–8] and a host of others have been considering controllability and observability problems in dynamic systems. however, most works done in this area have concentrated on the deterministic and stochastic controllability and observability problems. in this work, uncertain controllability and observability of dynamic systems are carried out by formulating a capital asset management control problem for the uncertain dynamic system such that the uncertain dynamic system is limited to systems involving uncertain processes. the choice of uncertainty theory over the conventional probability theory exists when the sample size is small to estimate a probability distribution and degree beliefs are ascertained from experts to work in place of frequency since human beings always over-weigh unlikely events. here, the general controllability and observability for the uncertain system in uncertainty theory are presented based on klamka and mahmudov works. ∗corresponding author: tolulope.latunde@fuoye.edu.ng uncertain controllability and observability of an optimal control model 9 2. preliminaries uncertainty theory is a branch of mathematics for modelling belief degrees. the theory is based on some concepts which may be referred to [9]. for easy interpretation, some of the concepts are given. let γ be a nonempty set and l be a σalgebra over γ such that (γ, l) is a measurable space. each element λ ∈ l is called an event. definition 2.1: a set function m defined on the σ-algebra over l is called an uncertain measure if it satisfies the following axioms: axiom 1. (normality axiom): m{λ} = 1 for the universal set γ. axiom 2. (duality axiom): m{λ} + m{λc} = 1 for any event λ. axiom 3. (subadditivity axiom): for every countable sequence of events, λ1,λ2, . . . , we have m { ∞⋃ i=1 λi } ≤ ∞∑ i=1 m{λi}. axiom 4. (product axiom): let (γk, lk,mk) be uncertainty spaces for k = 1, 2, . . . the product uncertain measure m is an uncertain measure satisfying m { ∞∏ k=1 λk } = min 1≤k≤∞ mk{λk}, where λk are arbitrarily chosen events from lk for k = 1, 2, · · · , respectively; see [9]. definition 2.2: let (γ, l,m) be an uncertainty space and let t be a totally ordered set. an uncertain process is a function xt(γ) from t × (γ, l,m) to the set of real numbers such that {xt ∈ b} is an event for any borel set b of real numbers at each time t; see [10]. definition 2.3: an uncertain process cσ is said to be a liu process if (i) c0 = 0 and almost all sample paths are lipschitz continuous, (ii) cσ has stationary and independent increments, (iii) every increment cs+σ − cs is a normal uncertain variable with expected value 0 and variance σ2. the uncertainty distribution of cσ is φσ(x) = [ 1 + exp ( −πx√ 3σ )]−1 , x ∈ r, (2.1) and the inverse distribution is φ−1 σ (y) = σ √ 3 π ln y 1− y , y ∈ r; (2.2) see [11]. definition 2.4: let ξ be an uncertain variable. then the expected value of ξ is defined by e[ξ] = ∫ +∞ 0 m{ξ ≥ x}dx− ∫ 0 −∞ m{ξ ≤ x}dx provided that at least one of the two integrals is finite; see [9]. copyright © 2021 assa. adv syst sci appl (2021) 10 t. latunde, a.a. ishaq, a.f. adedotun, j.o. ajinuhi, o.j. peter definition 2.5: an uncertain process xt is said to have independent increments if xt1 −xt0 , xt2 −xt1 , · · · , xtk −xtk−1 are independent uncertain variables, where t0 < t1 < · · · < tk. that is, an independent increment process means that its increments are independent uncertain variables whenever the time intervals do not overlap. it is noted that the increments are also independent of the initial state; see [10]. definition 2.6: suppose ct is a canonical liu process, and f and g are two functions. then dxt = f(t,xt)dt+ g(t,xt)dct is called an uncertain differential equation. a solution is a liu process xt that satisfies (2.1) and (2.2) identically in t; see [10]. definition 2.7: let xt be an uncertain process. then for each γ ∈ γ, the function xt(γ)is called a sample path of xt; see [10]. definition 2.8: an uncertain process xt is said to be sample-continuous if almost all sample paths are continuous functions with respect to time t; see [9]. definition 2.9: uncertainty distribution of solution. let α be a number with 0 < α < 1. an uncertain differential equation dx(t) = f(t,x(t))dt+ g(t,x(t))dc(t) is said to have an α-path x(t)α if it solves the corresponding ordinary differential equation dx(t)α = f(t,x(t)α)dt+ |g(t,x(t))|φ−1(α)dt, where φ−1(α) is the inverse uncertainty distribution of a standard normal uncertain variable, that is, φ−1(α) = √ 3 π ln α 1− α , α ∈ r; see [11]. 3. the asset management model asset management problem is mainly based on decision making and the understanding of probable asset degradation and trading-off capital investments, maintenance costs, risks and other uncertainties to optimize decisions made by investors. however, it is assumed that an individual invests his/her wealth in a capital asset, a(t), of a large business for a time, t, ranging from t0 to tn. supposing he/she starts with a known initial net worthx0(t). at the time t, what fraction of his/her net worth, ψ, must he/she choose to utilize on the capital asset and what fraction of his net worth, τ , must he/she choose to be incurred on the liability of the business such that the expected present value of the utility of asset, j(x), is maximized. table 3.1 represents the definition of the formulated model’s parameters. a dynamic optimization model of the expected present value of assets over a given life cycle based on uncertainty theory is herein presented following the study of portfolio copyright © 2021 assa. adv syst sci appl (2021) uncertain controllability and observability of an optimal control model 11 table 3.1. definition of parameters to the model. parameter description x(t) net worth at time t (state variable) τ(t) liability ratio (control) at time t, τ ∈ r σr(t) diffusion volatility of liability (with variance σ2 r per unit time) ψ(t) capital asset ratio at time t (control) ψ ∈ r σb(t) diffusion volatility of asset (with variance σ2 b per unit time) κ(t) capital gain on asset due to inflation at time t σp(t) diffusion volatility on asset price (with variance σ2 p per unit time) β(t) mean rate of return on the asset at time t ω(t) mean interest rate of liability at time t c(t) liu canonical process at time t µ(t) consumption level at time t j(t) tax ratio at time t g(t) depreciation ratio at time t h(t) asset supplies ratio at time t η subjective discount rate, e.g., a η+1 = present value λ degree of relative risk, where (1− λ) is the risk aversion u utility function selection by [12]. it is assumed that the goal of the asset management is to choose the optimal utilization and asset allocation policies for maximizing a value function that discounts exponentially future uncertain values of hyperbolic absolute risk aversion (hara) utility function over a given time horizon with the net worth of tangible assets as the state variable. the risky asset is assumed to earn an uncertain return and an uncertain gain with the mean rate of return and capital gain. furthermore, we express the change in liability as the sum of liability service with an assumption of uncertainty, consumptions, investment and net foreign supply, less taxation, depreciation and revenue over a period of time; see [13]. thus, we have j(x) = max ψ ec  tn∫ t0 1 λ e−ηt(ψx(t))λdt  subject to dx(t) = [(κ+ β)ψ − (ω(ψ − 1) + µ+ h− j − g)]x(t)dt +[ψσp + ψσb − σr(ψ − 1)]x(t)dc(t) . this model has been solved, characterised, analysed and applied to some real-life situations of sustainable finance; see [14–17]. 3.1. optimality of the solution it is important to derive the optimal solutions of the capital asset management problem as it would help in selecting the best available values. the following are utilised in deriving the optimality of the proposed model. definition 3.1: (principle of optimality) [19]: for any (t, x) ∈ [0, t )× r and ∆t > 0 with t+ ∆t < t , we have j(t, x) = sup d e [∫ t+∆t t f(xs, d, s)ds+ j(t+ ∆t, x+ ∆xt) ] , copyright © 2021 assa. adv syst sci appl (2021) 12 t. latunde, a.a. ishaq, a.f. adedotun, j.o. ajinuhi, o.j. peter where x+ ∆xt = xt+∆t. theorem 3.1: (equation of optimality, [18]) let j(t, x) be twice differentiable on [0, t )× r, then we have −jt(t, x) = sup d [f(x,d, t)) + jx(t, x)v (x,d)] , where jt(t, x) and jx(t, x) are the partial derivatives of the function j(t, x) in t and x respectively. proof see ( [19], pp. 15-16) 3.2. optimal control of the model the above equation of optimality is applied to the uncertain optimal control problem to evaluate the optimal controls analytically. applying equation (3.1), we obtain −jt = max ψ { 1 λ e−ηt(ψx)λ − ψ(κ+ β)xjx + (µ+ j + g + h− (ψ − 1)ω)xjx } = max ψ h where h stands for terms in the braces (condition the optimal ψ satisfies), ∂h ∂ψ = 0, ∂h ∂ψ = e−ηt(ψx)λ−1x − (κ+ β − ω)xjx = 0, ψ = 1 x [ (ω − κ− β)jxe ηt ] 1 λ−1 . hence, by solving the above equations, we obtained the optimal ratio of the net worth in capital assets as ψ∗ = (µ+ j + g + ω − h)λ− η (1− λ)(κ+ β − ω) . however, the optimal liability ratio, τ ∗ can also be obtained as a control to the system. since τ = ψ − 1 τ ∗ = [ (µ+ j + g + ω − h)λ− η (1− λ)(κ+ β − ω) ] − 1 or τ ∗ = (µ+ j + g − h)λ− (1− λ)(κ+ β) + ω − η (1− λ)(κ+ β − ω) . 3.3. solution to the model here, the analytical and numerical solutions are derived. for the analytic solution, the required problem under consideration is j(ψ) = min ψ ec  tn∫ t0 1 λ e−ηt(ψx(t))λdt  copyright © 2021 assa. adv syst sci appl (2021) uncertain controllability and observability of an optimal control model 13 subject to dx(t) = [(κ+ β)ψ − (ω(ψ − 1) + µ+ h− j − g)]x(t)dt +[ψσp + ψσb − σr(ψ − 1)]x(t)dc(t) with α-path equation dx(t)α = [(κ+ β)ψ − (ω(ψ − 1) + µ+ h− j − g)]x(t)αdt +|[ψσp + ψσb − σr(ψ − 1)]x(t)α|φ−1(α)dt. the analytical solution to the constraint is x(t) = x0 exp ([(κ+ β)ψ − (ω(ψ − 1) + µ+ h− j − g)]t +[ψσp + ψσb − σr(ψ − 1)]c(t)) and its inverse uncertainty distribution is ψ(t)−1(α) = x0 exp ( [(κ+ β)ψ − (ω(ψ − 1) + µ+ h− j − g)]t + [ψσp + ψσb − σr(ψ − 1)]t √ 3 π ln α 1− α ) . hence, ψ(t)−1(α) = e(x(t)α). 3.4. multifactor model the multifactor model can be expressed in the following form: j(x) = max ψ ec  tf∫ t0 1 λ e−ηt(uλ)tx1−λdt  (3.3) subject to dx = fxdt+ upxdt+ uqxdc(t), (3.4) where x =  x1t x2t ... xnt  , u =  ψ1 0 · · · 0 0 ψ2 · · · 0 ... . . . ... . . . 0 0 · · · ψn  , f =  µ1 + h1 − j1 − g1 − ω1 0 · · · 0 0 µ2 + h2 − j2 − g2 − ω2 · · · 0 ... . . . . . . ... 0 0 · · · µn + hn − jn − gn − ωm  , p =  κ1 + β1 − ω1 0 · · · 0 0 κ2 + β2 − ω2 · · · 0 ... . . . . . . ... 0 0 · · · κn + βn − ωn  , copyright © 2021 assa. adv syst sci appl (2021) 14 t. latunde, a.a. ishaq, a.f. adedotun, j.o. ajinuhi, o.j. peter q =  σ1p + σ1b − σ1r + 1 0 · · · 0 0 σ2p + σ2b − σ2r + 1 · · · 0 ... . . . . . . ... 0 0 · · · σnp + σnb − σnr + 1  . see, for example, [18]. equations (3.3) and (3.4), which are the model of risky capital assets, is an uncertain optimal control system whereby x(t) is the state, u is the control, f is n× n dimensional constant matrix while p and q are n×m dimensional constant matrices, c(t) is the liu process and j(x) is the objective functional. 4. controllability and observability of the uncertain system let (γ, l,m) be a complete uncertainty space with uncertain measurem on γ and a filtration {l(t)|t ∈ [0, t ]} generated by n-dimensional uncertain process {c(t) : 0 ≤ t ≤ t} defined on the uncertainty space (γ, l,m). let l2(γ, l(t),rn) represent the hilbert space of all l(t)-measurable square integrable uncertain variables with all values in rn. also, let ll2 ([0, t ],rn) represent the hilbert space of all square integrable and l(t)-measurable process with the values in rn. let x(t) = x(t+ s) for s ∈ [0, t ] represent the segment of the trajectory, that is, x(t) ∈ ll2 ([0, t ], l2(γ, l(t),rn)). letr be a linear operator on the hilbert space l2(γ, l(t),rn) with domaind1(r).r ≥ 0 if 〈rc, c〉 ≥ 0 for all c ∈ d1(r), r > 0 if 〈rc, c〉 > 0 for all nonzero c ∈ d1(r) and r is coercive (r− γi ≥ 0) if there exists a γ > 0 such that 〈rc, c〉 ≥ γ ‖c‖2for all c ∈ d1(r). now, using the proposed model with the multidimensional constraint of uncertain differential equation dx(t) = fx(t)dt+ upx(t)dt+ uqx(t)dc(t) (4.5) for t ∈ [0, t ] with the function initial condition x0 ∈ ll2 ([0, t ], l2(γ, l(t),rn)), where the state x(t) ∈ l2(γ, l(t),rn) and control ψ(t) ∈ rm = u . u and f are n× n dimensional constant matrix while p and q are n×m dimensional constant matrices. thus, suppose the admissible controls u = ll2 ([0, t ],rm), then for any given initial condition x0 ∈ ll2 ([0, t ], l2(γ, l(t),rn)) and any admissible control ψ ∈ u for t ∈ [0, t ], there exists a unique solution x(t;x0, ψ) ∈ l2(γ, l(t),rn) of the constraint uncertain differential state equation (4.5); see [20]. 4.1. controllability definition 4.1: the uncertain dynamic system (4.5) is said to be relatively exactly controllable on [0, t ] if r(t)(u) = l2(γ, l(t),rn). this implies that if all the points in l2(γ, l(t),rn) can exactly be reached at time t from any arbitrary initial condition x0 ∈ ll2 ([0, t ], l2(γ, l(t),rn)). definition 4.2: the uncertain dynamic system (4.5) is said to be relatively approximately controllable on [0, t ] if r(t)(u) = l2(γ, l(t),rn). copyright © 2021 assa. adv syst sci appl (2021) uncertain controllability and observability of an optimal control model 15 this implies that if all the points in l2(γ, l(t),rn) can approximately be reached at time t from any arbitrary initial condition x0 ∈ ll2 ([0, t ], l2(γ, l(t),rn)). the relationship between the controllability concepts for the uncertain dynamic system (4.5) and the controllability of the related deterministic dynamic system below dx(t) = fx(t)dt+ upx(t)dt for t ∈ [0, t ], (4.6) where the admissible control ψ ∈ l2([0, t ],rm). firstly, the deterministic system (4.6) is defined according to [1]. let qj(t) = fqj−1(t) for j = 1, 2, 3, . . . and t > 0, with the initial condition q(t) = q0(0) = j, t = 0, q(t) = q0(t) = 0, t 6= 0. for instance, the sequence of the n× n dimensional matrices qj(t) deduced from the determining equation gives: q0(0) = p, q1(0) = fp, q2(0) = f 2p. these can be written in a general notation as qj(t;t ) = {q0(t), q1(t), q2(t), . . . , qj−1(t) fort ∈ [0, t ]}. the following lemma is given with respect to [1], relating to the controllability of the deterministic system (4.6) in the time interval [0, t ]: lemma 4.1: the following conditions are equivalent: 1. the deterministic system (4.6) is relatively controllable on [0, t ], 2. the relative controllability matrix b(t) is non-singular, 3. rank qj(t;t ) = k. hence, the following lemma is formed with respect to [4, 20, 21] which will be useful in the proof of the uncertain controllability and observability. lemma 4.2: for every c ∈ l2(γ, l(t),rn), ∃ a process q ∈ ll2 ,rn×n such that the controllability operator is expressed as: g(t)c = b(t)ec+ ∫ t 0 b(t)(s)q(s)dc(s). lemma 4.3: the uncertain system (4.5) is relatively controllable on [0, t ] if and only if one of the following conditions holds true: 1. e 〈g(t)c, c〉 ≥ γe ‖c‖2 for some γ > 0 and all c ∈ l2(γ, l(t),rn), 2. r(λ1, g(t)) converges as λ1 → 0+ in the uniform operator topology, copyright © 2021 assa. adv syst sci appl (2021) 16 t. latunde, a.a. ishaq, a.f. adedotun, j.o. ajinuhi, o.j. peter 3. λ1r(λ1, g(t)) converges to the zero operator as λ1 → 0+ in the uniform operator topology, 4. ker(l(t))∗ = {0} and im(l(t))∗. lemma 4.4: the uncertain system (4.5) is approximately controllable on [0, t ] if and only if one of the following conditions holds true: 1. g(t) > 0, 2. λ1r(λ,g(t)) converges as λ1 → 0+ in the strong operator topology, 3. λ1r(λ,g(t)) converges to the zero operator as λ1 → 0+ in the weak operator topology, 4. ker(l(t))∗ = {0}. theorem 4.1: the following conditions are equivalent: 1. the deterministic system (4.6) is relatively controllable on [0, t ], 2. the uncertain system (4.5) is relatively exactly controllable on [0, t ], 3. the uncertain system (4.5) is relatively approximately controllable on [0, t ]. proof condition (i) implies condition (ii). suppose the deterministic system (4.6) is relatively controllable on [0, t ], then the relative controllability matrix b(t)(s) is invertible and strictly positive definite for all s ∈ [0, t ]. hence, for some γ > 0, 〈b(t)(s)x,x〉 ≥ γ ‖x‖2 for all s ∈ [0, t ] and x ∈ rn. in order to prove the relative exact controllability of the uncertain system (4.5) on [0, t ], the relationship between the controllability operator g(t) and the controllability matrix b(t) given in lemma 4.2 as e 〈g(t)c, c〉 to be expressed in terms of 〈g(t)ec,ec〉. firstly, e 〈g(t)c, c〉 = e 〈 b(t)ec+ ∫ t 0 b(t)(s)q(s)dc(s),ec+ ∫ t 0 q(s)dc(s) 〉 = 〈b(t), ec, ec〉+ e ∫ t 0 〈b(t)(s)q(s), q(s)〉 ds ≥ γ ( ‖ec‖2 + e ∫ t 0 ‖q(s)‖2 ds ) = γe ‖c‖2 . thus, in the view of the controllability operator, g(t) ≥ γi, which implies that the relative controllability operator g(t) is strictly positive definite and, the inverse operatorg(t)−1 is bounded, [1]. hence, the uncertain relative exact controllability of uncertain dynamic system (4.5) on [0, t ] is proved from the relative controllability of deterministic system (4.6) on [0, t ] . condition (ii) implies condition (iii). since the state space for the uncertain dynamic system (4.5) is finite-dimensional, which implies that the exact and approximate controllability coincide; see [1]. hence, it is easy to conclude that the uncertain system (4.5) is relatively approximately controllable on [0, t ]. condition (iii) implies condition (i). suppose the uncertain dynamic system (4.5) is uncertainly relatively approximately copyright © 2021 assa. adv syst sci appl (2021) uncertain controllability and observability of an optimal control model 17 controllable on [0, t ] and its controllability operator is positive definite, that is, g(t) > 0, then applying the resolvent operator λ1r(λ1, g(t)), where λ > 0, [20], gives e ‖λ1r(λ1, g(t))c‖2 → 0. this implies e ‖λ1r(λ1, g(t))c‖2 = ‖λ1r(λ1, b(t))ec‖2 + e ∫ t 0 ‖λ1r(λ1, b(t)(s))q(s)‖2 ds→ 0. therefore, e ∫ t 0 ‖λ1r(λ1, b(t)(s))q(s)‖2 ds→ 0 for all q(s) ∈ ll2 [0, t ],rn×n, and consequently there exists a subsequence λn such that for every c ∈ l2(γ, l(t),rn) we have ‖λnr(λk, b(t)(s))c‖ → 0 almost everywhere on [0, t ]. thus, the property holds for all 0 ≤ s < t because of the continuity of r(λ,b(t)(s)). this implies that the deterministic system (4.6) is relatively approximately controllable on [0, t ]. however, considering the state space for the deterministic system (4.6) is finitedimensional, that is, exact and approximate controllabilities coincides, [1]. therefore, it is concluded that the deterministic system (4.6) is relatively controllable on [0, t ]. 4.2. observability here, the dual concepts of observability for the uncertain dynamic system (4.5) are treated. suppose f and j generate c0-semigroup s1 and s2 on the hilbert space l2(γ, l(t),rn), that is, f and j generate a continuous representation of semigroup s. then the dual of the uncertain system (4.5) is: dx(t) = [f ∗ + j∗ +w ∗ 1 +w ∗ 2 ]x(t)dt+ uqx(t)dc(t). thus, the following concepts are described: • the observability map of the uncertain system (4.5) on [0, t ] is expressed as the linear operator o(t) : l2(γ, l(t),rn)→ ll2 ([0, t ]) which is defined by o(t)c = (w1 +w2)(s1 + s2)(t − s)e{c|l}. • the observability gramian of the uncertain system (4.5) on [0, t ] is defined by: θ = (o(t))∗o(t). definition 4.3: the uncertain system (4.5) is said to be relatively observable on [0, t ] if the operator o(t) is injective and its inverse bounded on the range o(t). the means that the initial state can be uniquely and continuously constructed from outputs in ll2 (γ, l(t),rn). definition 4.4: the uncertain system (4.5) is said to be approximately observable on [0, t ] if o(t) = {0}, that is, the initial state uniquely depends on the knowledge of the result in ll2 (γ, l(t),rn). copyright © 2021 assa. adv syst sci appl (2021) 18 t. latunde, a.a. ishaq, a.f. adedotun, j.o. ajinuhi, o.j. peter theorem 4.2: for the uncertain dynamic system (4.6), the following duality results hold true: 1. the uncertain system (4.5) is relatively observable on [0, t ] if and only if the dual system (4.2) is relatively controllable on [0, t ]. 2. the uncertain system (4.5) is approximately controllable on [0, t ] if and only if the dual system is approximately controllable on [0, t ]. proof since f and j generate a co-semigroup s1(t) and s2(t) respectively on ll2 (γ, l(t),rn), then f ∗ and j∗ generate the co-semigroup s∗1(t) and s∗2(t) respectively. also, since o(t) ∈ ll2 ([0, t ], l2(γ, l(t),rn)), (o(t))∗q = ∫ t 0 (s∗1 + s∗2)(t − s)(w ∗ 1 +w ∗ 2 )q(s)ds. thus, the range of o(t) implies that of the controllability operator of the dual system (4.2). let the controllability of the dual system (4.2) be represented by n(t), then (o(t))∗ = n(t), (n(t))∗ = o(t). 1. suppose the deterministic system (4.5) is relatively observable, there exists an inverse (o(t))−1 on the range of o(t). then,∥∥(o(t))−1q ∥∥ ≤ l ‖q‖ for all q ∈ ll2 ([0, t ],rn×n) and l > 0. hence, ‖c‖ = ∥∥(o(t))−1o(t)c ∥∥ ≤ l ∥∥ot c ∥∥ = l ‖(n(t))∗c‖ . therefore, the relative controllability of the uncertain system (4.5) follows from lemma 4.3 as thus. suppose that the uncertain system is relatively controllable, then (n(t))∗ is injective and has a closed range. this implies that from (n0)∗ = o(t), o(t) is injective and has a closed range. thus, by the closed graph theorem, the inverse of o(t) is bounded on the range of o(t). 2. however, by definition, the deterministic system (4.6) is approximately observable if and only if kero(t) = ker(n(t))∗ = {0}. therefore by lemma 4.4, ker(n(t))∗ = {0} if and only if the uncertain system (4.5) is approximately controllable. hence the proof of the equivalence. 5. conclusion the necessary and sufficient conditions for the uncertain controllability and observability of a finite-dimensional uncertain dynamic control system have been established and proved. thus, the effectiveness of each input and output in the general operation of the control system can be determined. the application of this work is to measure the controllability and observability degree of the system by using certain admissible input factors to determine how the system can move around its configuration space. subsequently, we can observe the analytic and numerical solutions to the multifactor system. copyright © 2021 assa. adv syst sci appl (2021) uncertain controllability and observability of an optimal control model 19 references 1. klamka, j. (1991) controllability of dynamical systems. dordrecht: kluwer academic, 1991. 2. klamka, j. (1993) controllability of dynamical systems a survey. archives of control sciences, 2(3/4), 281–307. 3. klamka, j. (2007) stochastic controllability of linear systems with state delays. international journal of applied mathematics and computer science, 17(1), 5–13. 4. mahmudov, n.i.& denker a. (2000) on controllability of linear stochastic systems. international journal of control, 73(2), 144–151. 5. mahmudov, n.i. (2003) controllability and observability of linear stochastic systems in hilbert spaces. progress in probability, 53, 151–167. 6. guo, t.l. (2012) controllability and observability of impulsive fractional linear timeinvariant system. computer and mathematics with applications, 64 3171–3182. 7. wang, l & chen, y. (2017) physical controllability of complex networks. scientific reports, 7, 1–14. 8. leitold, d., vathy-fogarassy, a. & abonyi, j. 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(2010) uncertain optimal control with application to a portfolio selection model. cybernetics and systems: an international journal, 41, 535–547. 20. mahmudov, n.i. (2001) controllability of linear stochastic systems in hilbert spaces. journal of mathematical analysis and application, 251, 64–82. 21. mahmudov, n.i. & zorlu, s. (2003) controllability of nonlinear stochastic systems. international journal of control, 76(2), 95–104. copyright © 2021 assa. adv syst sci appl (2021) introduction preliminaries the asset management model optimality of the solution optimal control of the model solution to the model multifactor model controllability and observability of the uncertain system controllability observability conclusion adv syst sci appl 2018; 04; 92-120 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/536 design of robust stabilizing pi/pid controller for time delay interval process plants using particle swarm optimization d.srinivasa rao1, m.siva kumar*2, m. ramalinga raju3 1,2) department of electrical and electronics engineering, gudlavalleru engineering college, gudlavalleru, ap, india. e-mail: dsrinivasarao1993@gmail.com, profsivakumar@gmail.com. 3) department of electrical and electronics engineering, university college of engineering, jntu university kakinada, ap, india. e-mail: rajumanyala@yahoo.com received december 16,2017; revised november 11,2018; published december 31,2018 abstract: generally, most of the real plants operate in a wide range of unknown operating conditions, but bounded parameter uncertainties in the control system. these uncertainties in the control system cause degradation of system performance and destabilization. in general, it is not easy to design a controller for interval time –delay process plant, because of interval dead time therefore, robust control of these uncertainties is a vital to operate the plant under stabilized condition. with a view to conquering the uncertainty, in this paper a new stability conditions are developed for determining the stability of interval process plants based on rouche’s theorem and then a robust pi/pid controller is designed for the interval process plant with and without time delay based on these newly developed stability conditions for stability of interval polynomial by using particle swarm optimization algorithm. a set of inequalities for a closed loop characteristic polynomial of an interval process plant in terms of controller parameters are derived from these newly developed stability conditions. these inequalities are solved to obtain controller parameters with the help of pso algorithm. the pi/pid controller designed in this proposed method stabilizes the given interval plant with and without time delay at all operating conditions. the proposed method has the advantage of having less computational complexity and easy to implement on a digital computer. the viability of the proposed methodology is illustrated through numerical examples of its successful implementation. the efficacy of the proposed methodology is also evaluated against the available approaches presented in the literature and the results were successfully implemented. keywords: kharitonov’s theorem, parametric uncertainty, robust controller, interval polynomial, particle swarm optimization. 1. introduction generally, many of the real plants operate in a wide range of unknown operating conditions bounded under parametric uncertainties called interval plants, in control systems. the large uncertainty present in the control system causes degradation of system performance and destabilization. therefore, robust control of these uncertainties is vital to operate the plant under stabilized condition. this necessitates a robust controller design which could stabilize the plant for all the operating conditions. hence designing a robust controller for the parametric uncertain plants having unknown, but bounded parameter uncertainties has become the problem of research nowadays. with a view to minimizing the stated * corresponding author: profsivakumar@gmail.com mailto:dsrinivasarao1993@gmail.com mailto:rajumanyala@yahoo.com design of robust stabilizing pi/pid using particle swarm optimization 93 copyright ©2018 assa. adv. in systems science and appl. (2018) uncertainties, many solutions are proposed in the literature for the simulation, design and tuning of controllers [1-3]. recently, affordable results have been reported on computation of all stabilizing p, pi and pid controllers which are mentioned here. the problem in [4] of stabilizing a linear time-invariant plant using a fixed order compensator was considered by using the hermite-biehler theorem. a feasible robust pid controllers have been developed in [5] using the minimax search, a co-evolutionary algorithm based on particle swarm optimization. the problem of designing robust and optimal pid controllers for a given linear time-invariant plant was proposed in[6-7]. design of a robust pid controller for a first-order lag with pure delay (folpd) model in [8] using pso enabled automated quantitative feedback theory (qft) and compared with manual graphical techniques. a design method was proposed for calculating the optimum values pid controller for interval plants using the pso algorithm. most of the practical systems operate based on approximate polynomial models; the parameters of these models would lie within an interval but not have specific values and are unknown. therefore, the stability analysis of polynomials subjected to parameter uncertainty has received considerable attention after the celebrated theorem of kharitonov [10], which assesses robust stability under the condition that four specially constructed extreme polynomials, called kharitionov polynomials are hurwitz. robust stability of interval polynomial is also discussed here by many researchers. among these discussions, some important methods have been presented here from the literature. a robust controller has been designed [11] for interval plants based on kharitonov’theorem and the results nie of [12] for fixed polynomials. a systematic optimization approach was proposed [13] to design a robust controller using the well-known kharitonov and hermite-biehler stability theorems for single-input/single-output process systems in the presence of unknown but bounded parameter uncertainties. the problem of robust stabilization [15] of a linear timeinvariant system was considered subject to variations of a real parameter vector used to design a robust controller. the design of a robust course controller was proposed in [16] for a cargo ship interacting with an uncertain environment using pso enabled automated quantitative feedback theory. a fractional-order proportional-integral controller was proposed and designed in [17] for a class of nonlinear integer-order systems to guarantee the desired control performance and the robustness of the designed controllers to the loop gain variations. a robust controller was designed in [18-19] for interval plants based on the result of kharitonov’theorem. with a view to reducing the test of hurwitz stability of the entire family, several investigations have been presented in the literature. among these, a few imperative investigations are discussed here, including; an algorithm has been presented in the design of a robust pi and pid controller [20]. this method is based on approximating the fuzzy coefficients by the nearest interval system and then a robust controller is designed using the necessary and sufficient conditions for stability of the interval systems. the inverse bilinear transformation (ibt) is proposed in [21] to design a robust controller using the necessary and sufficient conditions for a discrete-time interval plants. they designed a robust controller using the necessary and sufficient conditions for a chemical process plant with delay subjected to unknown, but bounded parameter uncertainties referred to an interval process plant with interval time delay. srinivasa rao et.al [22] presented a new algorithm for the design of the robust pi controller for a process control interval plant using routhe’s theorem and kharitonov’s theorem. a robust pi controller design approach was discussed in [23] by finding the controllers using pole placement method for active suspension system with parametric uncertainty. a pure gain compensator c(s) = k stabilizes the entire interval plant family such that a distinguished set of eight of the extreme plants are stable [24]. the first order controller is made by the experimental setup in [25] which was developed by ghosh. they prove that to robustly stabilize the extreme plants which are obtained by taking 94 d.srinivasa rao, m. siva kumar and m. ramalinga raju copyright ©2018 assa. adv. in systems science and appl. (2018) all possible combinations of extreme values of the plant numerator has degree m and the plant denominator is monotonic with degree n, the number of extreme plants can be high as next =2m+n+1 in [26]. an explicit equation of control parameters defining the stability boundary in parametric space was derived based on the plant model in time domain and by using the extraordinary feature results from the kronecker sum operation [27]. the stabilizing values of the parameters of a pi controller were computed based on plotting the stability boundary locus method in [28]. a complete survey of these extreme points is given in [29]. the necessary and sufficient conditions in [18] and [30] for interval polynomials are proposed using the results of [12] for fixed polynomials. in process industries due to the presence of transportation lag, recycle loops, and dead time corresponding to composition analysis, time delays frequently occur. the mathematical model of uncertain processes has described by the interval time-delay model in the presence of time delay. unfortunately, as compared with the successful development of controller design for the rational interval model, much less effort has been devoted to the processes described by an interval time-delay model; this is because the process delay is a source of instability and can render many established techniques inadequate. a new approach [32] was proposed to determine the entire set of stabilizing pi/pid parameters for time delay process with bounded uncertainties using the combination of the generalized kharitonov theorem and the hermit biehler theorem. the design of a smith predictor for the operation of processes under the variation of process gain, time constant and dead time based on the concept of an inferential control framework was presented in [32]. by considering the interval time-delay process, an alternative smith predictor design in [33] was proposed for the purpose of ensuring robust performance. the use of the structured singular value to design robust controllers was presented in [34] for interval time-delay processes. the designers of the robust stabilizing controller and construction of pre-filter with interval time delay have been considered in [30] to guarantee both robust stability and performance. particle swarm optimization (pso), first introduced by kennedy and eberhart [35] is one of the modern heuristic algorithms. it was developed through simulation of a simplified social system, and has been found to be robust in solving continuous nonlinear optimization problems [9] and [36-37]. the pso technique can generate a high-quality solution within shorter calculation time and stable convergence characteristic than other stochastic methods. in this note, a pi/pid controller is designed for an interval process plant with and without time delay based on the newly developed necessary and sufficient stability conditions. these conditions are used to derive a set of inequalities in terms of controller parameters. the inequality constraints from the characteristic polynomial are solved consequently to obtain the controller parameters with the help of pso algorithm. the efficacy of the proposed method is demonstrated by implementing with typical numerical examples available in the literature. in comparison with the method available in the literature [5], [8], [18], [30] and [31] the proposed method in this paper is simple and involves less computational complexity. the paper is organized as follows: section 2 describes development of stability conditions for robust stability of interval polynomial. section 3 gives the design of robust stabilizing pi/pid controller with and without time delay process plant. section 4 proposes pso algorithm to find the controller parameters. in section 5, the proposed method is applied to design a robust pi/pid controller for an interval process plant. 2. development of stability conditions for interval polynomial according to anderson et.al [38] the necessary and sufficient condition for robust stability of interval polynomials of order 3n  is positive lower bounds on the coefficients of an interval polynomial. design of robust stabilizing pi/pid using particle swarm optimization 95 copyright ©2018 assa. adv. in systems science and appl. (2018) therefore, consider an interval polynomial of order n=1 ].b,a[pwhere,sp)s(p iii i1 0i i   ].b,a[s]b,a[psp)s(p 001101  therefore, as per anderson [38], the robust stability condition is 0aand0a 01  i.e. 0a i for i=0,1. similarly for order n=2 ].b,a[s]b,a[s]b,a[pspspsp)s(p 0011 2 2201 2 2 i2 0i i   therefore, the robust stability condition is 0a,0a 12  and 0a0  i.e. 0a i  for i =0,1,2. lemma 2.1 consider a real hurwitz polynomial q(s) of the form 01 i i 1n 1n n n qsq....sq...sqsqq(s)    (2.1) n,.......,2,1,0i  where iq is real and positive, 0q0  . if any complex number z such that ,)z(f)z(f,0re  moreover, ,)z(f)z(f conzconz  where c is a closed contour, then, according to routhe’s theorem [39] the following two polynomials can be formulated. x2s0 ])s(q)s(q[ 2 1 q   (2.2) x2s1 ])s(q)s(q[ s2 1 q   (2.3) theorem 2.1: for stability of )s(q the two polynomials 0q and 1q formed by the alternate coefficients of a hurwitz polynomial in accordance with equations (2.2) and (2.3) must have negative real zeros. the proof of this is given in [39]. 2.1 necessary conditions for stability of an interval polynomials consider an interval polynomial of order n > 3 of the form ,psp...sp...spsp)s(p 01 i i 1n 1n n n    where ]b,a[p iii  for .n,.....,3,2,1,0i  the necessary conditions for an interval polynomial to be stable is given as 0ab ii  for n,......,3,2,1,0i  (2.4) 2.2 sufficient conditions for stability of interval polynomial for n > 3 2.2.1. for fourth-order interval polynomial (n= 4) consider the fourth-order interval polynomial as 01 2 2 3 3 4 4 pspspspsp)s(p  (2.5) where ]b,a[pand]b,a[p],b,a[p],b,a[p],b,a[p 444333222111000  using lemma 2.1, the p(s) can be represented into two polynomials p0 and p1 as given below. ]b,a[x]b,a[x]b,a[)]s(p)s(p[ 2 1 p 0022 2 44x2s0   (2.6) 96 d.srinivasa rao, m. siva kumar and m. ramalinga raju copyright ©2018 assa. adv. in systems science and appl. (2018) ]b,a[x]b,a[)]s(p)s(p[ s2 1 p 1133x2s1   (2.7) according to the theorem 2.1, for robust stability of interval polynomial p(s) the above polynomials p0 and p1 must have negative real zeros i.e. ]b,a][b,a[4]b,a[ 4400 2 22  (2.8) .0 ]b,a[ ]b,a[ 33 11   (2.9) apply interval arithmetic to the above equations (2.8) and (2.9), the stability conditions for the interval polynomial are 40 2 2 bb4a  (2.10) 40 2 2 aa4b  (2.11) 0 b a 3 1   (2.12) 0 a b 3 1   (2.13) from the above four equations, the sufficient conditions for the robust stability of fourth order interval polynomial p(s) are 40 2 2 bb4a  (2.14) 0 b a 3 1   (2.15) 2.2.2. for fifth-order interval polynomial (n = 5) consider the fifth-order interval polynomial as 01 2 2 3 3 4 4 5 5 pspspspspsp)s(p  (2.16) where ]b,a[p and]b,a[p],b,a[p],b,a[p],b,a[p],b,a[p 555 444333222111000   using lemma 2.1, the p(s) can be represented into two polynomials p0 and p1 as given below. ]b,a[x]b,a[x]b,a[)]s(p)s(p[ 2 1 p 0022 2 44x2s0   (2.17) ]b,a[x]b,a[x]b,a[)]s(p)s(p[ s2 1 p 1133 2 55x2s1   (2.18) according to the theorem 2.1, for robust stability of interval polynomial p(s) the above polynomials p0 and p1 must have negative real zeros i.e. design of robust stabilizing pi/pid using particle swarm optimization 97 copyright ©2018 assa. adv. in systems science and appl. (2018) ]b,a][b,a[4]b,a[ 4400 2 22  (2.19) ]b,a][b,a[4]b,a[ 5511 2 33  (2.20) apply interval arithmetic to the above equations (2.19) and (2.20), the stability conditions for the interval polynomial are 40 2 2 bb4a  (2.21) 40 2 2 aa4b  (2.22) 51 2 3 bb4a  (2.23) 51 2 2 aa4b  (2.24) from the above four equations, the sufficient conditions for the robust stability of fifth order interval polynomial p(s) are 40 2 2 bb4a  (2.25) 51 2 3 bb4a  (2.26) in a similar manner, the robust stability conditions for interval polynomial of degree 4n  can be determined. the robust stability conditions for higher-order interval polynomials are represented in a tabular form in table 2.1. . table2.1. robust stability conditions for various higher order interval polynomials order of the polynomial robust stability conditions necessary conditions sufficient conditions 3n  0ai  where i = 0,1,2,3. 20 2 1 3 bba  4n  0ai  where i = 0, 1,2,3,4. 40 2 2 4 bba  and .0 3 1   b a 5n  0ai  where i = 0, 1,..,4, 5. 40 2 2 4 bba  and 51 2 3 4 bba  . 6n  0ai  where i = 0, 1,...5, 6. 40 2 2 3 bba  and .4 51 2 3 bba  7n  0ai  where i = 0, 1,.... 6, 7. 40 2 2 3 bba  and 51 2 3 3 bba  . 8n  0ai  where i = 0, 1,2,..,7,8. 51 2 3 3 bba  , 80 2 4 4 aab  and 0 6 2   b a . 9n  0ai  where i = 0, 1,2,..,8,9.. 80 2 4 4 aab  91 2 5 4 aab  , 0 6 2   b a and 0 7 3   b a 80 2 4 4 aab  , 91 2 5 4 aab  , 98 d.srinivasa rao, m. siva kumar and m. ramalinga raju copyright ©2018 assa. adv. in systems science and appl. (2018) using these developed stability conditions, the stability of interval polynomials can be determined easily without formulating the four kharitonov’s polynomials, unlike kharitonov’s theorem. a robust pi/pid controller (which can stabilize the given plant under large uncertainty) can be designed easily using these stability conditions. the design procedure is given in following section. 3. design of robust stabilizing pi/pid controller 3.1 interval plant without time delay consider a plant with parametric uncertainty without time delay represented by its transfer function as n n 1n 1n10 m m 1m 1m10 sdsd...sdd scsc...scc )d,s(d )c,s(n )d,c,s(g        (3.1) where ]c,c[cc ii  for m,....,2,1,0i  ]d,d[dd ii  for n,....,,,i 210 and the bounds  iii d,c,c and  id are specified a priori and mn  . let the stabilizing pi/pid controller transfer function of the form given below )s(d )s(n s k k)s(c c ci ppi  for pi controller (3.2) )s(d )s(n sk s k k)s(c c c d i ppid  for pid controller (3.3) where pk = proportional gain, ik =integral gain and dk = derivative gain 3.2. interval plant with time delay consider a plant with parametric uncertainty with time delay represented by its transfer function as sdte )d,s(d )c,s(n )d,c,s(g   (3.4) where the numerator and denominator polynomials are of the form m m 1m 1m10 scsc...scc)c,s(n    n n 1n 1n10 sdsd...sdd)d,s(d    with the parameters being specified by their lower and upper bounds as follows: ]c,c[cc ii  for .m,....,2,1,0i  ]d,d[dd ii  for .n,....,2,1,0i    ddd ttt . 0td  it is not easy work to design the stabilizing controller for interval time-delay process plants, because of interval dead time. in order to extend the technique, the firstorder rational function of the approximation of the interval delay plant is obtained using the procedure given in corollary 3.1 to solve the design of the robust stabilizing controller problem of an interval time-delay plant. 10n 0ai  where i = 0, 1,2,..,10. 102 2 6 4 aab  and 0 7 3   b a design of robust stabilizing pi/pid using particle swarm optimization 99 copyright ©2018 assa. adv. in systems science and appl. (2018) 3.2.1. an interval approximate time delay model the firstorder rational function, )t,s( d of the approximation of the interval delay part is given by .)t,s(e d sdt   for   ddd ttt (3.5) the simplex form for )t,s( d is given by the following corollary which is expressed in[13]. corollary 3.1 : the interval function sdte  , where   ddd ttt can be approximated by a first-order interval rational function ),,s(  which is given by s 2 t t, 2 t 1 s 2 t t, 2 t 1 )t,s( d d d d d d d                      (3.6) with the following properties: 1 j 2 t 1 jt 2 t 1 )t,j( max d d d maxd                       (3.7) 1 jt 2 t 1 j 2 t 1 )t,j( min d d d mind                       (3.8) .e)t,j( jdt maxd    (3.9) .e)t,j( jdt mind    (3.10) where <  and  is the limited frequency that the inequalities in equations (3.7) to (3.10) hold. the arbitrarily extended approximation of a desired degree is the first-order approximation, since it covers the properties of phase frequency of original function for a wider range of frequencies, whereas the approximate original function can be in the other forms of approximations for control in the loss of some system information within a very limited frequency range. the rational interval function, then the approximate system model is given by ).t,s( )d,s(d )c,s(n )d,c,s(g d (3.11) after simplification the (3.11) will become 100 d.srinivasa rao, m. siva kumar and m. ramalinga raju copyright ©2018 assa. adv. in systems science and appl. (2018) )d,s(d )c,s(n )d,c,s(g  (3.12) where m m 1m 1m10 scsc...scc)c,s(n    n n 1n 1n10 sdsd...sdd)d,s(d    ]c,c[cc ii  for .m,....,2,1,0i  ]d,d[dd ii  for .n,....,2,1,0i    ddd ttt . and the bounds ddiiii tandt,d,d,c,c   are specified a priori and mn  . now the system with robust stabilizing controller for parametric uncertainty is as shown in fig.3.1. fig.3.1. block diagram of interval plant with a robust controller let the stabilizing pi/pid controller transfer function of the form given below )s(d )s(n ) s 1 1(k)s(c c c i cpi   for pi controller (3.13) )s(d )s(n ) s 1 1(k)s(c c c d i cpid    for pid controller (3.14) where kc = proportional gain, i =integral gain and d = derivative gain. then the closed loop transfer function with a pi / pid controller can be defined as )d,s(d)s(d)c,s(n)s(n )c,s(n)s(n )d,c,s(g)s(c1 )d,c,s(g)s(c )s(t cc c pi pi     for pi controller (3.15) )d,s(d)s(d)c,s(n)s(n )c,s(n)s(n )d,c,s(g)s(c1 )d,c,s(g)s(c )s(t cc c pid pid     for pid controller (3.16) the characteristic equation of this system with a pi / pid controller is given as )d,s(d)s(d)c,s(n)s(n)d,c,s(g)s(c1 ccpi  for pi controller (3.17) )d,s(d)s(d)c,s(n)s(n)d,c,s(g)s(c1 ccpid  for pid controller (3.18) where )c,s(n and )d,s(d are the numerator and denominator polynomials of the plant considered respectively, and )s(nc and )s(dc are the numerator and denominator polynomials of pi/pid controller transfer function respectively. this pi/pid controller robustly stabilizes the interval plants family, if for all cc and dd  , then the characteristic polynomial of a closed loop transfer function given in equations (3.17) and (3.18) has all zeros have negative real values. now apply the necessary and sufficient design of robust stabilizing pi/pid using particle swarm optimization 101 copyright ©2018 assa. adv. in systems science and appl. (2018) conditions of robust stability conditions given in table.1 to the closed-loop polynomial )d,s(d)s(d)c,s(n)s(n cc  which leads to a set of inequalities in terms of controller parameters. then these inequalities can be solved by using pso with matlab optimization [40] programming so as to minimize the objective function to obtain controller parameters. then after obtaining the controller parameters, form four sets of kharitonov’s polynomials to check the stability and the closed-loop step response to verify the results. the pso algorithm for the proposed method is given in section 4. fig.3.2. flowchart for the proposed algorithm. 4. application of particle swarm optimization algorithm kennedy and eberhart [35] first introduced the pso method. it is one of the optimization algorithms and a kind of evolutionary computation algorithm. the method has been found to be robust in solving problems featuring nonlinearity and no differentiability, multiple optima, and high dimensionality through adaptation, which is derived from the social-psychological theory. pso is inspired by social and cooperative behaviour displayed by various species to fill their needs in the search space. the algorithm is guided by personal experience (p best), the overall experience (g best) and the present movement of the particles to decide their positions in the next space. further the experiences are accelerated by two factors c1 and c2, and two random numbers generated between [0, 1]. whereas the present movement is multiplied by an inertia factor w varying between [wmin,wmax]. 102 d.srinivasa rao, m. siva kumar and m. ramalinga raju copyright ©2018 assa. adv. in systems science and appl. (2018) application of pso algorithm for determining the controller parameters is as follows: step 1: initialization: pso parameters are chosen as population size (p) = 100 number of iterations (n) =1000 cognitive coefficients c1 =2 and c2 =2 inertia weight n/)ww(xiterww minmaxmax  where wmax =0.9, wmin =0.4 step 2: initial search space limits of control variables in pi/pid controller are selected as for pi controller: 10k0 p  and 10k0 i  .(without time delay) 1k0 c  and 150 i  (with time delay) for pid controller: ,10k0 p  10k0 i  and 10k0 d  (without time delay) ,1k0 c  150 i  and 100 d  (with time delay) step 3: initial search space populations of xi are generated from specified intervals using the given below equation ),xx).((randxx min,imax,imin,i 0 i  (4.1) where i =1, 2......, n, and rand () represents a uniformly distributed random number within the range of [0,1] step 4: initialize the iteration index n=1 during the initialization, parameters of a pi / pid controller are randomly generated within allowable limits. step 5: evaluate the fitness function to get effective performance, in this paper, the fitness function j is defined as minimize j for pi controller: the fitness function j is chosen as integral square error (ise)   0 2 dt)t(ej (without time delay) (4.2) where output1)t(e  2 0 i 0 ii 2 0 c 0 cc k kk j       (with time delay) (4.3) for pid controller: the fitness function j is chosen as integral square error (ise)   0 2 dt)t(ej (without time delay) (4.4) where output1)t(e  2 0 d 0 dd 2 0 i 0 ii 2 0 c 0 cc k kk j           (with time delay) (4.5) j is determined when the controller parameters are subjected to ;kkk maxppminp  ;kkk aximiinim  and ;kkk maxddmind  ;maxmin cpc kkk  ;maxmin iii   and ;maxddmind   step 6: update velocity. for each particle, the velocity can be updated by design of robust stabilizing pi/pid using particle swarm optimization 103 copyright ©2018 assa. adv. in systems science and appl. (2018) )xgbest()(randc)xpbest()(randcvwv n i n i2 n i n i1 n i 1n i  (4.6) step 7: update position. each particle changes its position by adding the updated velocity to the previous position and it is represented as. 1n i 0 i vx)j,i(x  (4.7) step 8: repeat steps 5 to 7 until maximum generations are completed pso algorithm is run for each particle to evaluate fitness function several times and better results are saved and applied to the proposed pi/pid controller. 5. design of robust stabilizing controller in this section, a design procedure for a robust pi/pid controller of a plant with parametric uncertainty is illustrated. example 1 consider a wing aircraft [18] whose transfer function with parametric uncertainty is given by ]1.0,1.0[s]9.33,1.30[s]8.80,4.50[s]6.4,8.2[s ]166,90[s]74,54[ )d,c,s(g 234    (5.1) as the necessary conditions 0ab ii  (for i=0, 1, 2, 3, 4) are not satisfied for the above characteristic polynomial, hence the given interval plant is unstable. thereby, it is required to design a robust controller, which stabilizes the given plant. design of pi controller: the transfer function of the pi controller is given by )s(d )s(n s k k)s(c c ci ppi  then the closed loop transfer function with a pi controller becomes ]k166,k90[s]1.0k74k166,1.0k54k90[ s]9.33k74,1.30k54[s]8.80,4.50[s]6.4,8.2[s]1,1[ ]k166,k90[s]k74k166,k54k90[s]k74,k54[ )s(t iiipip 2 pp 345 iiipip 2 pp     (5.2) from the above equation, the characteristic equation of the closed loop interval system with pi controller can be taken as 0]k166,k90[s]1.0k74k166,1.0k54k90[ s]9.33k74,1.30k54[s]8.80,4.50[s]6.4,8.2[s]1,1[ iiipip 2 pp 345   (5.3) the step response pi controller ( pk = 1.3172 and ik = 1.9378) using ziegler-nichols settings [41] are shown in figure 5.1. from this figure 5.1, it has been observed that the designed pi controller from ziegler-nichols tuning method cannot stabilize the given interval plant at all operating conditions. hence it is necessary to redesign the pi controller to stabilize given interval plant. by applying the necessary and sufficient conditions from table 1 to the above 5th order polynomial (5.3), the following set of inequality constraints are obtained. in order to make this set of constraints into the feasible closed set, a small positive number ‘ε’ is introduced into the constraints. hence the optimization problem can be stated as to find pk and ik such that the objective function dt)t(ej 0 2   is minimized, subjected to the following constraints. inequality constraints for proposed method: necessary conditions: 104 d.srinivasa rao, m. siva kumar and m. ramalinga raju copyright ©2018 assa. adv. in systems science and appl. (2018) 0k90 i   01.05490  ip kk 01.3054  pk sufficient conditions: 04.305401.9068.32502916 2  ipp kkk 04.029666416.2540  ip kk fig.5.1. closed loop step response with pi controller for all extreme plants using the ziegler-nichols the linear programming problem consists of two decision variables and five constraints. the controller parameters pk and ik are restricted to small values by choosing the objective function j properly. the purpose of using a small positive number ε is to formulate a feasible set closed. in this work, the pso algorithm proposed in section 4 is used to minimize objective function. it attempts to explain the problems of minimization, subjected to linear as well as nonlinear constraints. by applying the proposed algorithm, then the values of controller parameters are obtained as pk = 0.5766 and ik = 0.01. the closed loop step response of the system with a pi controller for both proposed methods ( pk = 0.5766 and ik = 0.001) and the method given in [18] ( pk = 0.5 and ik = 0.1) are shown in figures 5.2 and 5.3 for ε = 1 respectively. the nominal pi controller parameters 655.11and6.0k 0 i 0 c   and nominal pid controller parameters 993.668.0k 0 i 0 c   and 74825.10 d are designed by the ziegler-nichols settings. the time domain specifications of figures 5.2 and 5.3 are shown in table 5.1 and which describes the efficacy of the proposed method for the design of the robust pi controller. the step response comparison of four extreme plants with a pi controller obtained by the proposed method and the method given in [18] is shown in figure design of robust stabilizing pi/pid using particle swarm optimization 105 copyright ©2018 assa. adv. in systems science and appl. (2018) 5.4. fig. 5.2. closed loop step response to the pi controller for all extreme plants using the proposed method. in order determine the robustness of proposed method, the given interval plant g(s,c,d) from equation (5.1) can be written as s32s64s7.3s 128s64 )s(g 234    (5.4) here g(s) is obtained by averaging the upper and lower bound of s coefficients of g(s,c,d). the closed transfer function of wing aircraft with pi controller is given by iip 2 p 345 iip 2 p1 k128s)k64k128(s)32k64(s6.65s7.3s k128s)k64k128(sk64 )s(t    (5.5) the closed loop step response of the system with pi controller ( pk = 0.5766 and ik = 0.001) from proposed methods is shown in figure 5.5. the pi controller designed from proposed method can stabilize the given plant if any changes occur in the uncertain parameters within the bounds . 106 d.srinivasa rao, m. siva kumar and m. ramalinga raju copyright ©2018 assa. adv. in systems science and appl. (2018) fig.5.3. closed loop step response to the pi controller for all extreme plants using the method given in [18]. fig.5.4. closed loop step response to the pi controller for all extreme plants for proposed method and the method given in[18]. design of robust stabilizing pi/pid using particle swarm optimization 107 copyright ©2018 assa. adv. in systems science and appl. (2018) fig.5.5. closed loop step response to the pi controller for changes in the uncertain parameter to determine the robustness for proposed method table 5.1. time domain specifications for proposed method and existing method in [18] it has been observed from figures 5.2 and 5.3 that the designed pi controller, which uses the proposed stability conditions, robustly stabilizes the plant very quickly when compared to the method given in [18]. from table 5.1, the designed pi controller stabilizes the plant with lesser time domain parameters than the existing method. it has been observed from figure 5.5, that the designed pi controller from the proposed method is used to determine the robustness in any changes in the uncertain parameters of the given plant. in our proposed method, the controller parameters are obtained based on the minimization of the objective function (ise). this integral square error obtained from our proposed method is less compared to other methods available in the literature. this shows the efficacy of the proposed method in terms of time domain specifications and the ise. the proposed method involves five sets of equations for nlp to solve. whereas the method in [18] requires eight set of equations. thus, the proposed method requires less computational complexity than the method given in [18]. it is also observed that the computation time required for solving the name of the kharitonov polynomial proposed method existing method in [18] % peak overshoot mp peak time tp(sec) rise time tr(sec) settling time ts(sec) ise 10-4 % peak overshoo t mp peak time tp(sec) rise time tr(sec) settling time ts(sec) ise 10-3 first 17.492 3.514 1.501 7.0534 4.22 32.4147 4.008 1.508 10.808 1.101 second 20.826 1.707 0.878 5.0156 0.093 24.3994 2.488 0.880 8.2771 0.158 third 29.702 3.683 1.349 14.229 1.53 57.6729 3.820 1.378 26.828 16.00 fourth 31.419 1.988 0.718 7.7175 0.631 42.659 2.166 0.741 8.326 0.022 108 d.srinivasa rao, m. siva kumar and m. ramalinga raju copyright ©2018 assa. adv. in systems science and appl. (2018) fig.5.6. closed loop step response with pid controller for all extreme plants using the ziegler-nichols nlp problem with minimum number of equations using proposed stability conditions and pso algorithm is much less than the method given in [18]. thus, the developed pi controller using necessary and sufficient conditions of interval polynomial robustly stabilizes the wing aircraft. these stability conditions can be implemented easily for determining the stability of higher order interval plants. design of pid controller the transfer function of the pid controller is given by )s(d )s(n sk s k k)s(c c c d i ppid  then the closed loop transfer function with pid controller becomes ]k166,k90[s]1.0k74k166,1.0k54k90[ s]9.33k74k166,1.30k54k90[ s]8.80k74,4.50k54[s]6.4,8.2[s]1,1[ ]k166,k90[s]k74k166,k54k90[ s]k74k166,k54k90[s]k74,k54[ )s(t iiipip 2 pdpd 3 dd 45 iiipip 2 pdpd 3 dd       (5.6) from the above equation, the characteristic equation of the closed loop interval system with pid controller can be taken as 0]k166,k90[s]1.0k74k166,1.0k54k90[ s]9.33k74k166,1.30k54k90[ s]8.80k74,4.50k54[s]6.4,8.2[s]1,1[ iiipip 2 pdpd 3 dd 45    (5.7) the step response pid controller ( pk = 1.3172, ik = 1.9378 and kd=0.1791) using design of robust stabilizing pi/pid using particle swarm optimization 109 copyright ©2018 assa. adv. in systems science and appl. (2018) ziegler-nichols settings [41] are shown in figure 5.6. from this figure 5.6, it has been observed that the designed pi controller from ziegler-nichols tuning method cannot stabilize the given interval plant at all operating conditions. hence it is necessary to redesign the pid controller to stabilize given interval plant. by applying the necessary and sufficient conditions for the above 5th order polynomial (5.7), six sets of inequality constraints are obtained. the controller parameters dip kandk,k are obtained by solving these inequality constraints using pso such that the objective function dt)t(ej 0 2   is minimized. then the controller parameters pk = 0.5933, ik = 0.001 and dk =0.252 are obtained. the closed loop step response of system with pid controller using the proposed method for all extreme plants is shown in figure 5.7 for ε = 1 respectively. the time domain specifications of figure 5.7 are tabulated in table 5.2 which describes the efficacy of the proposed method. from the equation (5.4) the closed transfer function of wing aircraft with pid controller is given by ]k128s)k64k128(s)32k64k128(s)6.65k64(s7.3s,1[ k128s)k64k128(s)k64k128(sk64 )s(t iip 2 pd 3 d 45 iip 2 pd 3 d1    (5.8) the closed loop step response of the system with pid controller ( pk = 0.5933, ik = 0.001 and dk =0.252) from proposed methods is shown in figure 5.8.the pid controller designed from proposed method can stabilize can stabilize the given plant if any changes occur in the uncertain parameters within the bounds. table 5.2. time domain specifications for proposed method name of the kharitonov polynomial % peak overshoot mp peak time tp(sec) rise time tr(sec) settling time ts(sec) ise *10-6 first 4.934 4.179 1.4975 6.3771 10.381 second 15.79 2.096 0.9614 5.0017 0.1072 third 12.41 4.015 1.8559 6.9367 56.76 fourth 15.26 2.189 0.9886 5.4216 0.4.348 it has been observed from the simulation results of figure 5.7 that the designed pid controller robustly stabilizes the plant very quickly. from table 5.2, it is evident that the designed pid controller stabilizes the plant with lesser time domain parameters. the proposed method involves six sets of equations for nlp to solve. thus, the proposed method requires less computational complexity than the methods in the literature it has been observed from figure 5.8, that the designed pid controller from the proposed method is used to determine the robustness in any changes in the uncertain parameter of the given plant this shows the efficacy of the proposed method in terms of time domain specifications. it is also observed in the computation time required for solving the nlp problem using stability conditions and pso algorithm is much less. thus, the developed pid controller using necessary and sufficient conditions of interval polynomial robustly stabilizes the wing aircraft. 110 d.srinivasa rao, m. siva kumar and m. ramalinga raju copyright ©2018 assa. adv. in systems science and appl. (2018) fig.5.7. closed loop step response with pid controller for all extreme plants using the proposed method. fig.5.8. closed loop step response to the pid controller for changes in the uncertain parameter to determine the robustness of the proposed method example 2 consider the dynamics of a vast range of chemical processes which is described by the interval first-order plus dead time (fopdt) model [30] is given by sdte 1s k )s(g     (5.9) where ,]9,3[k ]6,1[t,]18,10[ d  in this, the nominal value variations in the process gain k, time constant and dead time td are taken as ±50%, ±28.57%, and ±71.43%. the nominal design execution is continued based on the nominal values of .5.3,14,60  dtk  the nominal pi controller parameters 655.11and6.0k 0 i 0 c   and nominal pid controller parameters design of robust stabilizing pi/pid using particle swarm optimization 111 copyright ©2018 assa. adv. in systems science and appl. (2018) 993.668.0k 0 i 0 c   and 74825.10 d are designed by the ziegler-nichols settings [41]. the step response pi/pid controller using ziegler-nichols is shown in figure 9 and 10. from this figure 9 and 10,it has been observed that the designed pi/pid controller from zieglernichols cannot stabilize the given interval plant at all operating conditions. hence it is necessary to redesign the pi/pid controller to stabilize given interval time delay plant. from the corollary 3.1, se ]61[ can be written as . s]5.65.0[1 s]5.65.0[1   then the following delay-free model for original interval fopdt is given as 1s]5.24,5.10[s]117,5[ ]9,3[s]5.58,5.1[ )d,c,s(g 2    (5.10) fig. 5.9. : closed loop step response to the pi controller for all extreme plants using ziegler-nichlos method with time delay. fig. 5.10: closed loop step response to the pid controller for all extreme plants using ziegler-nichlos method with time delay. design of pi controller: 112 d.srinivasa rao, m. siva kumar and m. ramalinga raju copyright ©2018 assa. adv. in systems science and appl. (2018) consider the pi controller of the form )s(d )s(n ) s 1 1(k)s(c c c i cpi   then the closed loop transfer function with a pi controller becomes ]k9,k3[s]k5.1k9,k5.58k3[ s]k5.15.24,k5.585.10[s]117,5[ ]k9,k3[ s]k5.1k9,k5.58k3[s]k5.58k5.1[ )s(t cccicicici 2 iciici 3 ii cc ciccic 2 ic,ic         (5.11) from the above equation, the characteristic equation of the closed loop interval system with pi controller can be taken as 0]k9,k3[s]k5.1k9,k5.58k3[ s]k5.15.24,k5.585.10[s]117,5[ cccicicici 2 iciici 3 ii     (5.12) by applying the necessary and sufficient conditions for the above 3rd order polynomial (5.12), the following set of inequality constraints is obtained. in order to make this set of constraints into the feasible closed set, a small positive number ‘ε’ is introduced into the constraints. hence the optimization problem can be stated as to find ck and i such that the objective function 2 0 i 0 ii 2 0 c 0 cc k kk j       is minimized, subjected to the following constraints. necessary conditions: 0k3 c   0k5.58k3 icic   0k5.585.10 ici   05 i   sufficient condition: 0)k5.124(k27)k5.58k3( icic 2 cici   the linear programming problem consists of two decision variables and five constraints. the controller parameters kc and i are restricted to small values by choosing the objective function j purposely. in this work, the pso algorithm proposed in section 4 is used to minimize the objective function .j by applying the proposed algorithm, then the values of controller parameters are obtained. as shown in table 5.3, the values of the controller parameters ck and i increase as ‘ε’ are increased which shows the sensitivity of the controller parameters with respect to the nlp parameter ‘ε’. the closed loop step response of the system with the pi controller by the proposed method ( kc = 4.0432 and i = 0.00926) and the method given in [30] ( kc = 0.0684 and i = 12.5102) are shown in figures 5.11 and 5 .12 for ε = 0.05 respectively. the time domain specifications of figures 5.11 and 5.12 are shown in table 5.4 which describes the efficacy of the proposed method. in order determine the robustness of proposed method, the given interval plant g(s,c,d) from equation (5.10) can be written as 1s5.17s61 6s30 )s(g 2    (5.13) here g(s) is obtained by averaging the upper and lower bound of s coefficients of g(s,c,d). the closed transfer function of wing aircraft with pi controller is given by design of robust stabilizing pi/pid using particle swarm optimization 113 copyright ©2018 assa. adv. in systems science and appl. (2018) ccici 2 ici 3 i ccic 2 ic k6s)k30k6(s)k305.17(s61 k6s)k30k6(sk30 )s(t      (5.14) the closed loop step response of the system with pi controller ( kc = 4.0432 and i = 0.00926) from proposed method is shown in figure 5.13. the pi controller designed from proposed method can stabilize the given plant if any changes occur in the uncertain parameters within the bounds . table 5.3. variation of ck and i for different values of ε controller set  proposed method existing method ck i ck i 1 0.05 0.00926 4.0432 0.0684 12.5102 2 1 0.01389 8.0934 0.0683 12.5110 3 1.5 0.01803 10.4032 0.0683 12.5116 4 2 0.02059 11.9023 0.0683 12.5121 table 5.4. time domain specifications for proposed method it has been observed from the figure 5.11 and 5.12 that the designed pi controller, which uses the proposed stability conditions robustly stabilizes the plant when compared to the method given in [30]. from figure 5.12, it is observed that the closed loop interval time delay system with a controller is unstable by the method in [30]. whereas closed loop interval time delay system with proposed method is stable. it has been observed from figure 5.13, that the designed pi controller from the proposed method is used to determine the robustness in any changes in the uncertain parameters of the given plant. this shows that proposed method robustly stabilizes the interval time delay process plant. the proposed method has five inequality constraints which are less compared with the method (which has seven inequality constraints) in [30]. hence the proposed method is simple and requires less computation as it uses a lesser number of inequality constraints than the method given in [30]. it is also observed that the computation time required for solving the nlp problem using proposed stability conditions and pso algorithm is 3.872603 seconds, which is much less than the method given in [30]. thus, the developed pi controller using necessary and sufficient conditions of interval polynomial robustly stabilizes the interval process time delay system. these stability conditions can be implemented easily for determining the stability of higher order interval plants. name of the kharitonov polynomial rise time tr (sec) settling time ts (sec) first 222.2482 422.5395 second 294.5659 526.1731 third 43.3195 139.2935 fourth 92.4901 166.2970 114 d.srinivasa rao, m. siva kumar and m. ramalinga raju copyright ©2018 assa. adv. in systems science and appl. (2018) fig.5.11.closed loop step response with a pi controller for all extreme plants using proposed method with time delay. fig.5.12. closed loop step response to the pi controller for all extreme plants using the existing method [30]. design of robust stabilizing pi/pid using particle swarm optimization 115 copyright ©2018 assa. adv. in systems science and appl. (2018) fig.5.13. closed loop step response to the pi controller for changes in the uncertain parameter to determine the robustness for proposed method design of pid controller: consider the pid controller of the form )s(d )s(n ) s 1 1(k)s(c c c d i cpid    then the closed loop transfer function with pid controller becomes ]9,3[]5.19,5.583[]5.195.24 ,5.5835.10[]5.1117,5.585[ ]9,3[]5.19,5.583[ ]5.19,5.583[]5.58,5.1[ )( 2 3 23 ccciciciciicdici icdicidicidici ccciccic icdicicdicdicdic kkskkkkskk kkskk kkskkkk skkkkskk st          (5.15) from the above equation, the characteristic equation of the closed loop interval system with pid controller can be taken as 0]k9,k3[s]k5.1k9,k5.58k3[ s]k5.1k95.24,k5.58k35.10[s]k5.1117,k5.585[ cccicicici 2 icdiciicdici 3 dicidici     (5.16) by applying the necessary and sufficient conditions of equations for the above 3rd order polynomial (5.16), set of inequality constraints are obtained. hence the optimization problem can be stated as to find ,kc i and d such that the objective function 2 0 d 0 dd 2 0 i 0 ii 2 0 c 0 cc k kk j           is minimized. by using the proposed method given in section 4 to the above nlp problem, the controller parameters ,kc i and d are obtained. as shown in table 5.5, the values of the controller parameters ,kc i and d increase as ‘ε’ are increased which shows the sensitivity of the controller parameters with respect to the nlp parameter ‘ε’. they are given by 1520.3,00914.0k ic   and d = 1.2040. the closed loop step response of system with pid controller for proposed method is shown in figure 5.14 for ε = 1. the time domain 116 d.srinivasa rao, m. siva kumar and m. ramalinga raju copyright ©2018 assa. adv. in systems science and appl. (2018) specifications of figure 5.14 are shown in table 5.6 which describes the efficacy of the proposed method in terms of the time domain specifications. table 5.5. variation of ,kc i and d for different values of ε table 5.6. time domain specifications for proposed method from equation (5.10) the closed loop transfer function of wing aircraft with pid controller is given by ccici 2 icdici 3 dici ccic 2 icdic 3 dic k6s)k30k6[s)k30k95.17(s)k3061( k6s)k30k6(s)k30k6(sk30 )s(t      (5.17) the closed loop step response of the system with pid controller ( 1520.3,00914.0k ic   and d = 1.2040.) from proposed methods is shown in figure 5.15. the pid controller designed from proposed method can stabilize the given plant if any changes occur in the uncertain parameters within the bounds. controller set  ck i d 1 0.05 0.00914 3.1520 1.2040 2 1 0.00925 6.8461 1.7481 2 1.5 0.00930 7.1864 1.8998 4 2 0.00977 7.4979 2.1075 name of the kharitonov polynomial rise time tr (sec) settling time ts(sec) first 152.3465 283.6998 second 230.6303 381.7915 third 29.0473 131.9949 fourth 72.5310 119.3718 design of robust stabilizing pi/pid using particle swarm optimization 117 copyright ©2018 assa. adv. in systems science and appl. (2018) fig.5.14. closed loop step response with pid controller for all extreme plants using the proposed method with time delay. fig.5.15. closed loop step response to the pid controller for changes in the uncertain parameter to determine the robustness with time delay for the proposed method it has been observed from the simulation results of figure 5.14 that the designed pid controller robustly stabilizes the plant very quickly. from table 5.6, it is evident that the designed pid controller stabilizes the plant with lesser time domain parameters. this shows that proposed method robustly stabilizes the interval time delay process plant. it has been observed from figure 5.15, that the designed pi controller from the proposed method is used to determine the robustness in any changes in the uncertain parameters of the given plant. the proposed method has five inequality constraints which are less compared with the method (which has seven inequality constraints) in [30]. hence the proposed method is simple and requires less computation as it uses a lesser number of inequality constraints than 118 d.srinivasa rao, m. siva kumar and m. ramalinga raju copyright ©2018 assa. adv. in systems science and appl. (2018) the method given in [30]. this shows the efficacy of the proposed method in terms of time domain specifications. it is also observed that the computation time (4.63 seconds) required for solving the nlp problem using the proposed stability conditions and pso algorithm is much less. thus, the developed pid controller using necessary and sufficient conditions of interval polynomial robustly stabilizes the interval process time delay system. 6. conclusions a robust stabilizing pi/pid controller is designed for interval process plant with and without time delay based on the newly developed necessary and sufficient conditions using pso algorithm. a set of inequality constraints in terms of controller parameters are derived from interval polynomial based on the new necessary and sufficient conditions. consequently, these inequalities are solved using pso to obtain controller parameters. the proposed pi/pid controller procedure is also applied and demonstrated through typical numerical examples. it is observed that the designed pi/pid controller robustly stabilizes the plant with and without time delay with lesser time domain parameters than the existing methods. the pi/pid controller designed from proposed method can stabilize the given plant if any changes occur in the uncertain parameters within the bounds the simulation results are evidence for its robustness in stabilizing the interval process time delay plant using a pi/pid controller. also, the proposed method is simple and involves less computational complexity in comparison with the methods available in the literature. references 1. astrom, k. j., hagglund, t., hang, c. c. & ho, w.k. 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(1942). optimum settings for automatic controllers, trans. asme, 64,759-768. advances in systems science and applications (2013) vol. 13 no. 1 1-20 mechanisms of organizational behavior control: a survey v.n. burkov, m.v. goubko, n.a. korgin and d.a. novikov institute of control sciences, russian academy of sciences, moscow, russia abstract we survey basics of a version of the mechanism design theory tailored to solve management problems, and introduce the core concept of a mechanism of organizational behavior control. we discuss methodological grounds of the theory, give classification of mechanisms, and characterize the mechanism implementation process. finally, we sketch basic mechanisms which help solving important management problems on all stages of management cycle and provide an example of the mechanism of incentive-compatible planning. keywords mechanism design, management theory, organizational behavior 1 introduction in xxi century economics is transforming into the economics of knowledge. contemporary trends in management science consider workforce not as just a “yet another production factor” (in line with land and capital), but promote it at least to the status of intangible asset, which possesses rational economic behavior with the abilities of self-regulation and self-development. the reason is that employees’ decisions, based on their skills and highly influenced by their motivation, now play the crucial role in a value added chain. business process management (bpm) [1] and business process reengineering (bpr) [2] techniques did not cover motivational and decision-making aspects of business processes and, thus, they alone were insufficient to drive corporate reengineering efficiently. we claim that it is lack of attention paid to employees’ behavioral response to changes was the main reason of bpr projects failure1. the major differences between a complicated technical system and a human (or a collective) as a control object are condensed in the concept of activeness. the first aspect of activity consists in human ability of independent goalsetting. a technical system has no interests beyond interests of its designer. but employees in an organization do pursue their own objectives, which can be inconsistent with objectives of the organization. the second aspect lies in human ability to choose actions independently; in particular, an employee can deliberately manipulate information (when he or she finds it profitable) and/or not fulfill the assigned plans and orders (again, if this promises some benefit). 1the citation attributed to m. hammer “i was reflecting my engineering background and was insufficient appreciative of the human dimension. i’ve learned that’s critical” is in line with the above proposition. 2 v.n. burkov: mechanisms of organizational behavior control: a survey the third important aspect of activity is the ability of reflexion regarding his or her personal activity and activity of the other subjects (including the ability of forecasting their behavior). in this paper we survey the approach, which allows accounting systematically for the phenomenon of activity of employees in organizations. the approach is based on the methodology of systems and control sciences, and widely employs results of mechanism design – a branch of game theory, which deals with conflict situations involving the principal and a set of active agents (usually, in the presence of asymmetric information). mechanism design theory delivers a solution to management problems in the form of a control mechanism, i.e., a formalized routine of decision-making. for a control mechanism to keep efficiency in the presence of active agents, it must be robust to information manipulation, plan non-fulfillment and other aspects of activeness [3]. the main challenge of mechanism design in management is search of the so-called correct mechanisms, which assure both interests coordination for all organization employees including stakeholders, management, and workers, as well as motivate them to report true information, which forms the basis of any efficient decision. to adopt mechanism design to practical problems of business administration we developed a tailored and, in some aspects, more straightforward theory. the main goal of this paper is to provide the reader with basic methodological and technical grounds of control mechanisms in organizations. in section 1 we briefly explain a formal scheme of accounting for behavioral response by the principal (a manager in an organization). the art of management is positioned as ensuring the desired behavior of subordinates. in section 2 models of organizational behavior are considered in the context of the production and management activity. in section 3 we introduce different types of mechanisms in an organization classified by the element of the organizational system model being affected by the mechanism. in section 4 we describe an applied technology of mechanism analysis, synthesis, and implementation in an organization. in section 5 the toolkit of mechanisms developed during last decades is briefly sketched, while in section 6 an example of a mechanism to control employees’ behavior is described in more detail. 2 principal-agent model consider a formal scheme of making a management decision by the principal (a manager in an organization). the problem of the principal is to analyze the available information and to choose the decision relevant to the current situation; so, the management decision appears to be the action of a principal. the principal compares possible options (feasible decisions) and their consequences and advances in systems science and applications (2013) vol.13 no.1 3 then chooses the best one. if the decision touches interests of the other persons (employees or some external parties), to predict properly the consequences of his or her decisions the principal has to forecast the response of such counterparties, with subordinates being the most important ones, a certain decision. a subordinate (an individual or a team) possesses exactly the same abilities and comprehension as a principal. indeed, they have personal preferences and interests; they make decisions and perform actions. the only difference is that the situation for a subordinate includes a decision made by the principal. the action of the subordinate, being it predicted or real, is accounted by the principal when making decisions. so, we obtain the simplified scheme of interaction between the principal and the subordinate (see fig. 1). fig. 1 the scheme of interaction between the principal and the subordinate in terms of control theory [4], we have a control subject (a manager) and a controlled system, i.e., a control object (a subordinate). state of the control object depends on external disturbances, actions of the control subject and, probably, on actions of the object, see fig. 1 (if the control object appears active, especially in organizational systems). a problem of the control subject consists in performing control actions (or, management decision, see fig. 1) to ensure a required state of the control object. this is done on the basis of information on external disturbances (or external situation). generally speaking, the art of management lies in ensuring the desired behavior of subordinates. consequently, the principal wants to select a mechanism such that subordinates choose desired actions. the problem is decomposed into, first, the analysis problem (given a behavioral model of a subordinate, find actions he or she would select under a certain control mechanism), and, second, the design problem (find a mechanism assuring the desired subordinates’ actions according to a given behavioral model). solving the design problem requires the ability to solve the analysis problem. hence, the principal should forecast behavior of subordinates as their response 4 v.n. burkov: mechanisms of organizational behavior control: a survey to his or her management decision. interaction demonstrated in fig. 1 represents a “closed loop”. let us replace the principal with a control mechanism, viz., a management decision procedure used by the principal. the inputs of this procedure are given by external environment and an action of a subordinate, while the output is a specific management decision. thus, we have the structure presented by fig. 2; here the control mechanism determines management decisions made by the principal. fig. 2 the scheme of interaction between the principal and the subordinate 3 models of employees’ behavior control theory considers a control object as passive one, possessing no individual preferences and information. the constraints imposed on his or her activity are unique and defined by the planned state, while the action (actual state) coincides with the plan. control of a passive object consists in assigning “plans” or in specifying the requirements to actual state, i.e., to the results of activity. in fact, there is no need in control subject’s pondering whether a passive control object performs the plan or not! the model of a passive object seems natural for technical systems. but, more surprisingly, many management theories imply that subordinates surely strive to implement the orders (in other words, it is assumed that the actual state equals to the planned one). planned and actual states may not coincide due to the following reasons: • presence of uncertainty (uncontrolled external factors that influence the result of the employee’s activity, making it differ from his or her action); • employee’s activity. both aspects should be considered when solving problems of management in organizations. the following fact has been emphasized earlier. the ability of employees to set goals, to choose actions independently, is reflected by the concept of activity of the control objects (agents) in organizational systems. it implies that the agents have certain opportunities and scope for independent goal-setting and making advances in systems science and applications (2013) vol.13 no.1 5 decisions regarding their actions. this is done, first, within the framework of the full-fledged structure of activity. next, we list the effects of activity: data manipulation, choosing a state varying from the plan, a negligent behavior, etc. these effects should be considered when designing and applying control mechanisms. efficient control makes it necessary to model agents’ behavior, i.e., to predict their reaction to specific control actions. thus is done using a concept of utility maximization, known in economics as the model of “economic man behavior” (we will use the term “agent” as a synonym). an agent acts (reports information, chooses actions, and so on) to maximize his or her utility. the stated concept has turned out fruitful and gained a dominating role in mathematical economics, decision theory and other scientific directions focused on the models of human behavior. as a rule, two types of economic agents (economics subjects) are identified, notably, • an economic subject of the market (the examples are an organization, a holding company, a firm, a corporation); • employee(s) of an economic subject. for both types of agents, the utility is typically represented by “economic profit”, that is, the difference between “revenue” and “costs”. yet, the content and methods used to represent the income and costs depend on the agent’s type. suppose that a department or a separate organization is studied; its revenue and costs are calculated in a management accounting system. the utility could be derived from the operational income according to the results of business activity (under the adopted policy of management accounting). if an agent is an individual employee, the management accounting system stores income, but not efforts exerted by the agent to perform the operational activity and corresponding costs. moreover, the information system does not weight costs against the income. thus, it is necessary to compare efforts and costs with rewards received by the agent. this sort of relationship is set by virtue of different supervision techniques, timing, as well as by the search for analogs. a specific situation being analyzed, one should choose an approach enabling the simplest and the most accurate definition of the agent’s effort function and utility function. the principal is also an “economic man”. he or she also has a utility function, but has different possibilities to implement his or her activity (in particular, management authorities). the principal also has internal constraints, utility, awareness and actions. the principal’s action consists in choosing a control mechanism, i.e., the relationship between control actions and actions of the agent. the major difference between the principal and agent lies in that the former possesses management authority; notably, he or she has the right “to make the 6 v.n. burkov: mechanisms of organizational behavior control: a survey first move” by establishing activity actions (a control mechanism) for the agent. the principal’s utility generally depends on the actions of the agent, i.e., control efficiency is determined by the utility of the principal (representing the interests of the whole organizational system) gained as the result of agent’s activity. therefore, the mechanism in the described environment specifies a “legislative, regulating and mandatory base”, i.e., the rules and procedures involved to perform actions of the principal and the agent during implementation of processes and projects. moreover, control rises to the level of meta-control, to formation of the “rules of the game”, conditions of the agent’s functioning and motivation. in a certain sense, the principal appears an architect of social and economic rules and a meta-player. according to the general model [4], an economic agent is described by four basic parameters aggregating the procedural components of his or her activity: 1) constraints and norms of activity; 2) an utility function; 3) awareness; 4) action of the agent (it is chosen on the basis of his or her awareness under existing constraints and utility). the action indicates of the agent’s state and, to a considerable degree, determines the result of his or her activity. under the existing constraints, economic agent has two types of possible actions. they are: • reporting information on uncertain parameters to the principal and/or to other agents; • choosing an action (production output, time consumed, etc). in fact, what is the incentive-compatible control of an economic agent? the matter concerns, e.g., the fact that the principal never exactly knows the agent’s subjective estimate function of his or her actions. in other words, the principal does not know how much the agent is willing to do for $1 reward. low level of motivation results in no productivity growth. starting from a certain moment (exactly when the reward exceeds the agent’s subjective estimate of his or her efforts), productivity increases. next, a moment comes when the principal no more benefits from raising the reward (costs to motivate additional efforts are greater than gains of productivity growth). roughly speaking, three intervals of rewards appear; the principal’s task lies in balancing the interests to make the situation convenient for him or her and for the agent (this is done over the admissible interval with respect to the costs and productivity). evaluating such balance of interests requires joint actions of economically rational persons (the principal and the agent). their common aim is making a decision that would be more efficient than in the case of independent actions, based on their individual interests. thus, a mechanism of control in an organization is a sort of a “visible advances in systems science and applications (2013) vol.13 no.1 7 hand”, which drives self-interested actors toward socially desirable outcomes. consider the case of a single principal and several agents. now the agents immediately choose their actions given the mechanism set by the principal. as soon as interests of agents differ, they appear to participate is a certain conflict. the model of such conflict situations is given by the game theory [5], which uses different concepts of equilibrium (e.g. nash equilibrium) to predict the outcome of conflict. so, the set of equilibrium outcomes of a game is considered as a behavioral response of a controlled system to a management decision in the case when a system consists of several agents. the assumption that the agents’ action coincides with the output of his or her activity makes a sort of simplification. actually, the result may vary from the action due to uncontrolled factors, namely, actions of the rest agents, the state of external environment, etc. in other words, the result of the agent’s activity generally depends on his or her action, actions of the other agents and impact exerted by the external environment. then, to predict agent’s behavior in the case of a single agent the principal employs the hypothesis of expected utility maximization, which implies that an agent chooses his or her action to maximize the expected utility function obtained by averaging agent’s utility over realization of random states of external environment. in the case of several agents the concept of bayesian nash equilibrium [5] or informational equilibrium [6] is employed to account for unknown external parameters. the case of asymmetric information in principal-agent models is also studied in detail by the contract theory [7]. so, game-theoretical and optimization models of human rational behavior allow analyzing the response of a control object to certain control action and form the basis of the synthesis phase of control mechanism design. 4 methods of control in organization above we defined control as purposeful impact on control object. yet, organization represents an intricate control object. hence, one should clarify the internal structure of an organization as a control object; in other words, it is necessary to find out what entities inside an organization can be influenced by the control actions (so as to change state, more specifically, the behavior of the organization). those components of organizational system being modified during the process (and as the result) of control are known as objects of control. expressing any complicated system as a complex of interacting elements makes a certain model. therefore, below we introduce the model of organizational system [4]. organizational system (os) is described via specifying: • staff (employees, their groups and collectives, its members); • structure (a set of informational, control, technological and other relations among the os members); 8 v.n. burkov: mechanisms of organizational behavior control: a survey • constraints and norms of activity imposed on os members; they are institutional, planned, technological and other of constraints and norms of individual and joint activity; • goals and preferences of os members; • information, i.e., data regarding essential parameters being available to os members at the moment of decision-making (choosing the strategies); • sequence of operations (of data acquisition and choice of actions by os members). the staff determines “who” is included into the system; the structure describes “who interacts with whom, who is subordinate”, etc. finally, feasible sets define “who can do what”, goal functions represent “who wants what”, and information states “who knows what”. control in os, being interpreted as an impact on the controlled system to ensure its required behavior, may affect each of the listed parameters. hence, taking the focus of control (the parameter of os, which is modified during the process of control and as a result of control) as the basis of classification of control in os, we obtain the following methods (types) of control [4]; staff control deals with the following issues: who is included into the organization or department, who should be dismissed or recruited? generally, staff control either includes the problems of personnel training and development. for example, the well-known model of signaling [8], one of the seminal models in contract theory, when considered from the principal’s point of view, can be treated as a mechanism of staff control, which allows a principal to implement a rational hiring policy in the presence of hidden information. structure control is as a rule performed in parallel to staff control. it provides answers to several questions, viz, what functions should be performed by whom, what participants are subordinate to whom, who should control and be controlled, what information should be transferred and acquired, etc. mechanisms to control structure are not well studied formally at this moment, although a number of competing models exists of a management hierarchy in a firm (see the surveys in [9-10]). institutional control appears to be the most stringent; it consists in that the principal restricts the sets of feasible actions and results of activity of the subordinates (in a purposeful way). such restriction may be implemented via explicit or implicit influence (legal acts, directives, orders, etc) or mental and ethical norms, corporate culture and so on. numerous variations of resource allotment mechanisms (with auctions being a very special case) commonly met in public institutions give us a good example of institutional control mechanisms [4, 11]. incentive control is in a certain sense “softer” than institutional one, and consists in purposeful modification of the preferences of control object (the subordiadvances in systems science and applications (2013) vol.13 no.1 9 nates). the described modification could be performed through introduction of a certain scheme of penalties and/or rewards for choosing specific actions and/or attaining definite results of activity. job contracts for individual employees and teams are a good example of incentive schemes, which lie in the very core of any organizational activity [4, 7]. against institutional and motivational counterparts, informational control appears the “softest” (indirect) type. it lies in formation of control object’s awareness such that the decisions made on its basis are the most beneficial to control subject. this sort of mechanisms is used in organizations from ancient times, but in recent years they started a new life with development of social networks in internet [12]. 5 technology of control mechanism design: from models to policies the version of mechanism design we suggest for organizational management technically is close to the mechanism design, which forms the grounds of modern social choice theory and agency theory. the only difference is a systematically promoted pragmatic normative approach – in contrast to a standard descriptive approach of economic theory any conflict is a priori considered in the view of a single part, i.e., a client. not-withstanding formal resemblance of mathematical models, motivation of the research often varies, mostly limited to optimal mechanism design (i.e., construction of the best mechanisms in the view of a principal), whereas efficiency (in the sense of welfare economics) is overshadowed. under a given control mechanism, one may separate out three major stages of designing and implementing management decisions, notably, 1) data acquisition for decision-making; 2) decision-making process; 3) implementation of the decisions made. let us characterize each stage in greater detail. 1) data acquisition for making decisions. information required for making decisions may be acquired by the principal from the agents. the principal should be on the watch for untrue information (the agents may either “color up the truth” or “get dramatized”). no doubt, the principal would desire to have a certain control mechanism when economic agents (as rational “economic men” with individual preferences striving to maximize their utility function) benefit from being fair. the stated mechanisms are known as fair play mechanisms (strategyproof mechanisms). however, when the agents benefit from truth-telling? the only situation is when information reported by no way harms them. this is the underlying property of fair play mechanisms. 2) making decisions. at the second stage, the principal should make an effi10 v.n. burkov: mechanisms of organizational behavior control: a survey cient decision. if the principal is unable to succeed, he should improve himself (otherwise, he will be definitely replaced by another principal). thus, a key component here consists in management capabilities of the principal. 3) implementing the decisions made. finally, the third stage is intended for implementation of the decisions made by the principal. the principal is then exposed to the danger of non-fulfillment (or incomplete fulfillment) of the decisions. mechanisms where agents benefit (again, according to their individual preferences) from implementing the decisions of the principal are said to be incentive-compatible. an incentive-compatible strategy-proof mechanism is a correct mechanism. surely, an ideal situation is when a correct mechanism appears optimal (i.e., the most efficient). the mechanisms studied are adapted to managerial practice. for instance, despite their formal reducibility to a social choice problem (in a certain statement), resource allocation problems and auctions are traditionally considered separately, as the ones arising at different stages of organizational control cycle. now turn from the theory to the practice. who should solve the above-stated problems of mechanism analysis and synthesis? the first alternative is for the principal to solve both problems involving the templates and personal experience. the second alternative is to design a control mechanism with the aid of some sort of external consulting. almost any control problem in an organization can be stated formally in the following way. find feasible control actions ensuring maximum efficiency (such control is called optimal). to succeed, one should solve an optimization problem, notably, choose optimal control (optimal control actions. this is the most general setting. consider a general technology of solving the problems of this sort, which covers all the stages (from organizational system modeling to implementation of the model in concrete regulations); see fig. 3 where inverse links between the stages are omitted for better clarity. • the first stage (model construction) consists in description of an organizational system (first and foremost, control object) and in its modeling, i.e., specification of staff, structure and functions of the modeled system. • the second stage (model analysis) lies in studying the behavior of control object under different control actions. • the analysis stage being completed, one may apply control theory methods to solve the formulated control problem. first, solving the direct control problem, i.e., synthesis problem for optimal control actions (find a feasible control ensuring the maximal efficiency). second, solving the inverse control problem (find a set of feasible controls rendering the os to the desired state). it should be emphasized that generally this stage causes major theoretical difficulties and seems the most advances in systems science and applications (2013) vol.13 no.1 11 time-consuming for the researcher. • having obtained the set of solutions to the control problem, one should move to the fourth stage, notably, study their stability. stability analysis implies solving (at the very least) two problems. the first problem is to study the dependence of optimal solutions on parameters of the model; in other words, this is the analysis problem for solution stability. the second problem turns out specific for mathematical modeling. it consists in theoretical study of model adequacy with respect to the real system; such study means efficiency evaluation for those solutions derived as optimal within the model when they are applied to real os (due to modeling errors the solutions may vary from the model). • thus, the above-mentioned four stages constitute general theoretical study of os model. using the results of theoretical study to control real os requires adjustment of the model (i.e., identifying the modeled system and carrying out a series of simulation experiments; these are the fifth and sixth stages, respectively). in many cases the stage of simulation appears necessary due to several reasons. first, far from always one is able to obtain analytical solution to optimal control synthesis problem and to study its dependence on parameters of the model. note that simulation may represent a certain tool to derive and assess the solution. second, simulation allows for verifying the validity of hypotheses adopted to construct and analyze the model. in other words, simulation gives additional information on the adequacy of the model without conducting natural experiment. finally (and this is the third reason), employing management games and simulation models for training aims lets the managing staff master and test the suggested control mechanisms. • the closing is, in fact, the seventh stage, or the stage of implementation; it includes training of the managing staff, implementation of control mechanisms (designed and analyzed at the previous stages) in real os, with subsequent efficiency assessment of their application, correction of the model, and so on. 6 toolkit of control mechanisms management activity is traditionally divided into stages, which form the cycle of management. one of most popular divisions belongs to classic of regular management henry fayol, who detached five stages: planning, organizing, commanding, coordinating, and controlling. in the course of management theory development the role of proper employees’ motivation was also emphasized. every stage gives rise to a number of complex management problems. most of them involve principal-agents interactions of some sort, and, thus, require adequate support in the form of organizational behavior control mechanisms. the following list of mechanisms summarizes briefly the authors’ long experience of both academic studies and consulting projects. the mechanisms are 12 v.n. burkov: mechanisms of organizational behavior control: a survey separated into four stages of management cycle (fayol’s stages of commanding and coordinating are replaced to motivating, which is more important for employees’ behavior management) – see fig. 3 – and address the major management problems, which arise on the appropriate stage. this brief description is intended to give the reader general expression about the set of problems covered by mechanisms of employees’ behavior control and some flavor of mechanism design approach to address these problems (see the detailed description of mechanisms in [4]). the list presented below is not complete, but, of course, basic mechanisms do not cover all the variety of management problems. at the same time, these basic mechanisms can serve as bricks in designing numerous complex mechanisms of organizational management, which are based on combinations of a comparably small set of ideas. fig. 3 the complex of control mechanisms within a fayol’s management cycle 6.1 mechanisms of planning the resource allocation mechanism allows distributing scared resource (typically, budget funds, but also production plans, water, etc) among agents to maximize total efficiency of resource usage in the presence of lack of information about agents’ abilities to use resource efficiently (depending on situation this may be advances in systems science and applications (2013) vol.13 no.1 13 real demand for resource, local production capacities, etc). the properly designed mechanism ensures credibility of agents’ reports. we suggest using variations of a sequential resource allocation scheme [4, 11] to guarantee maximum efficiency while keeping truth-telling property. the mechanism of active expertise supports managers in many common situations when a decision is made on the basis of opinions of some experts. it is always a big problem to find a competent expert in the field, but very often no external experts can be involved at all, while independency and equity of internal experts is open to questions. the mechanism of active expertize minimizes the consequences of possible opinions distortion by motivating experts’ truth-telling. a class of median schemes [4, 13-14] is shown to solve the problem of truth-telling, in contrast to typically employed variations of an averaging scheme. transfer prices (also known as internal or corporate prices) help to distribute profit between production units in a corporation, but also provide a tool for high performance compensation and incentive compatible planning. units report their production plans, while the principal balances external demand with local production capabilities adjusting internal prices. the mechanism of transfer pricing establishes relation between the total manufacturing plan and the internal price per unit of manufactured good. when the number of agents is large enough the properly designed mechanism of transfer prices ensures, first, reporting real production plans (which units are interested to fulfill) and, second, efficient plan allocation between production units. an idea of a contest is used in a rank-order tournament mechanism to select most efficient project portfolios. the simplest tournament reduces to the following procedure. first, for every candidate project its total discounted cost is calculated and the effect of the project is estimated. second, projects are ordered by effect to cost ration in the descending order, and included in a portfolio until the budget runs out. this tournament procedure is also called the cost-benefit analysis. tournaments differ in the procedure of winners’ selection. for example, in a more complex mechanism a knapsack problem [15] is solved for the set of project cost-effect pairs. 6.2 mechanisms of organization the mechanism of joint financing helps to share federal funds with that of local authorities to implement complex regional programs. mechanisms of this sort are used in public-private partnership to finance socially oriented projects. the same idea works well in corporations, when, for instance, some local projects of performance improvement are jointly financed both by corporate center (by the principal) and at the local level (by the agents). project costs are reported by the agents, and the whole amount of a central fund is allocated proportionally to the reports, while the remainder of the reported cost must be covered by the 14 v.n. burkov: mechanisms of organizational behavior control: a survey agent. cost-saving mechanisms are intended to motivate an agent to improve efficiency of his or her activity as much as possible. the mechanism stimulates an agent to keep high quality of the output with minimum costs. cost-saving mechanisms are based on the following general concept. assume that agent’s profit depends on variables of two types, viz., parameters selected by an agent (e.g., costs) and parameters specified by the principal (e.g., a flexible norm of profitability, pricing coefficients, taxation coefficients, and so on). the mechanism design problem is to choose the values of principal-driven parameters for agent’s profit to grow with the costs decrease; this should be either accompanied by price reduction. in other words, net profit of an agent should rise when costs fall; in contrast, the product price should decrease. 6.3 incentive mechanisms an individual incentive scheme serves, firstly, for motivating agents to choose actions being beneficial to a principal and, secondly, for increasing the employees intensity of labor and motivating them to achieve better production results. the idea is that the employee’s wage includes a tariff (a fixed part) and a bonus (a variable part). the latter depends either on (measurable) intensity of his or her labor, or on the results of his or her activity (the intensity being immeasurable). adjusting the mechanism allows for attaining the required performance of the employees with minimum payments. when an agent is paid merely for fulfillment (or overfulfillment) of the plan assigned by a principal, the agent is not interested in having a high (“tight”) plan. the reason is performing it would require additional efforts (costs) of the agent. for instance, an agent may inform the principal of his or her preferences, eo ipso reporting his or her estimate of the plan (referred to as a “counter plan”). within the framework of incentive scheme for counter plans, an agent is given rewards for reporting counter plans that better meet the principals interests (yet, are “tighter” for the agent). a collective incentive scheme is designed for situations when a principal turns out unable to separately observe the action of every agent (the principal merely knows a certain aggregated rate, e.g., the result of collective activity). imagine that the principal can evaluate the minimum costs to-be-incurred by the agents to achieve the required result of collective activity. in this case, the efficient incentive scheme takes the following form. the minimum costs of each agent are compensated (provided that the result of collective activity agrees with the requirements of the principal). moreover, sometimes the principal has no costs related to observing individual actions of every agent; thus, the principal’s workload to acquire and process information is substantially reduced. the unified incentives mechanism is employed in situations when a principal advances in systems science and applications (2013) vol.13 no.1 15 has to motivate large groups of agents, to involve “democratic” management methods and to decrease the amount of processed data. under unified incentives, the relationship between the reward and labor intensity of the agents (alternatively, the results attained by them) is identical for all agents. in several cases, the described unification leads to no loss of efficiency, while the wages fund is spent in an optimal way. yet, unified control may be inefficient, when non-consideration of individual features of the agents results in inefficient spending of financial resources. a special case of unified incentive scheme is represented by competition incentives mechanism. the mechanism of informational control in incentive problems is involved when a principal possesses complete information (in contrast to agents). since the latter choose their actions based on individual awareness, the principal may impact on their actions by manipulating information; in other words, the principal may modify awareness of the agents, e.g., by means of informing every agent of the intensity of labor and/or plans of the rest agents. the team incentive mechanism serves to motivate groups (e.g., a production area, a department, a shop floor, a team, etc) with collective organization of work (taking into consideration individual contribution of every employee). incentive procedures for team members are based on bonus fund distribution according to their labor participation factor. 6.4 controlling mechanisms in integrated rating mechanisms one passes from a detailed description of a complex object (involving numerous indicators and parameters) to an aggregated description based on few generalized characteristics of the object. these mechanisms allow for regular monitoring and timely estimating (taking into account priorities of a principal) the results of the object’s activity and changes within the object. note that such changes take place during operation of the object or depending on the impact exerted by an external environment. the mechanism of consent is intended for coordinated distribution of financial resources among several possible directions of investment. for this, expert commissions are created; each commission generates a coordinated decision concerning the ratio of financial funds to-be-spent on a given direction to that of a fixed (basic) direction. we emphasize that the number of commissions should be by unity less than the number of directions. using information acquired from all expert commissions, a distribution of financial resources is defined among all directions; the idea is making the ratio of financial funds (to-be-spent on a specific direction) to that of the basic direction equal to the estimate of the corresponding expert commission. the mechanism ensures truth-telling of the commissions. the primary concept of a two-channel mechanism lies in simultaneous involving two channels of decision-making. motivating actions are based on comparative 16 v.n. burkov: mechanisms of organizational behavior control: a survey analysis of the efficiency levels of the decisions suggested by different channels. notably, rewards are evaluated for every channel. the first channel makes a decision. certain alternatives exist for the second channel; it could be either an active channel (decisions are worked out by human beings) or an advising (a computer-type, a “normative”) channel. in the latter case, suggestions are used to establish the norms of control efficiency (to compare the actual efficiency of the decisions made by the first channel). mechanism of predictive self-control is intended for well-timed informing a principal of possible deviations (from a plan) in the agents’ activity. the earlier the principal gets aware of possible deviations from the plan (e.g., in due dates, financial investments, etc), the more efficient and well-timed would be his or her decision (e.g., additional measures to eliminate deviations and reduce losses, or plan correction); note the agents report deviations. the matter is that the penalties of the agents (in the case of plan correction) depend on the moment the agents report of the correction (they are smaller if the report is early); moreover, these penalties are less than in the case of plan non-fulfillment. 7 an example of the mechanism: incentive-compatible planning due to limited space we leave the detailed description of the whole toolkit of organizational control mechanisms for future papers, and just give references to the papers with detailed description of the models and techniques used to solve various problems of control mechanism analysis and synthesis. below we give just an example of a single incentive mechanism used to boost planning accuracy due to accounting for employees’ incentives. as it has been mentioned, an incentive-compatible mechanism is remarkable for that the agent benefits from performing the plan. the following questions arise then: 1) does the principal need exact fulfillment of the plan by the agent? 2) if yes, how could it be ensured? the first question appears not as easy as it could seem. for instance, consider the utility function of the agent defined by the difference between his or her income 0.5y (the agent gains 50% income) and the penalty for plan nonfulfillment: 1 4(x−y)2; here y denotes production output of the agent expressed in money (the sales volume), while x means the plan. this function is shown in fig. 4. simple computations indicate that the agent benefits from choosing the action (x+1), i.e., he or she would strive for overfulfilling the plan by the unity. hence, if the principal requires y = 5 units of products (exactly this quantity, since the residual amount exceeds demand), he or she should assign the plan x = 4. we underline that the plan differs from the action which is desired by the advances in systems science and applications (2013) vol.13 no.1 17 principal! such situation is common in control of active agents. the principal has to predict their behavior and assign the plan based on the forecast. the described mechanism is not incentive-compatible (the agent does not perform the plan). nevertheless, sensus communis suggests that it would be good to have incentive-compatible mechanisms. fortunately, there exists a wide range of the incentive/penalty functions such that optimal plan is proven to be incentive-compatible. an example is provided by fig. 5. fig. 4 the utility function of the agent in an incentive-compatible planning mechanism fig. 5 an example of the penalty function such functions have been shown to possess optimal (in the view of the principal) incentive-compatible plan. thus, the ideology of optimal planning is being replaced with the ideology of optimal incentive-compatible planning. the plan should be optimal exactly over the set of incentive-compatible plans (beneficial to the agents). fig. 6 demonstrates an example of the set of incentive-compatible plans, where the agent is equally penalized for the plan non-fulfillment or overfulfillment. 18 v.n. burkov: mechanisms of organizational behavior control: a survey fig. 6 an example of the set of incentive-compatible plans one easily observes that any plan belonging to the set of incentive-compatible plans is beneficial to the agent. if the principal is interested in maximal production output, the optimal plan corresponds to the point a in fig. 6. for further increasing the output, the principal has to either enlarge the penalty or pay a reward for the plan fulfillment. in the latter case optimal incentive-compatible plan is defined by the point b in fig. 6. increasing the penalties or rewards corresponds to growing centralization of control, since the set of incentive-compatible plans is enlarged and the principal possesses better opportunities for assigning the plans. therefore, growing centralization of control may improve the efficiency of functioning in an organizational system. however, an infinite growth of centralization is clearly impossible; first, it requires infinite resources of control and, second, it may contradict the existing social norms, as well as the democratic and autonomous norms of control. 8 conclusion thus, the theory of behavior control mechanisms reviewed in this paper is positioned as a branch of control science (more specifically, cybernetics) which deals with control in the so-called active systems. the elements of active systems are people possessing individual interests, being able to choose actions independently and to manipulate information. in fact, the subject of the theory is systematic accounting for activity phenomenon in control problems based on systems approach and using the methods and results of operations research and game theory. the primary task is designing efficient mechanisms of organizational control. game-theoretic modeling appears one of the basic research methods used by the theory. as compared to modern economic theory, application of game theory in the theory of control mechanisms is different. pragmatic normative approach (a game conflict is a priori considered in the view of a single part, i.e., a client) is typical, in contrast to standard descriptive approach in economics (when a conflict is viewed indirectly). advances in systems science and applications (2013) vol.13 no.1 19 therefore, the basic idea consists in combining maximum usability in formulation of organizational control problems with wide adoption of formal models (including game-theoretic ones). still, to solve a complex applied problem of improving the efficiency of project management, we supplement the mechanisms of employees’ behavior control with optimal project planning or supply network optimization techniques. the subject of the theory coincides with that of management theory; nevertheless, the corresponding methods have dramatic differences. management theory seems much flexible in the description of psychological aspects, while mechanism design brings all psychological factors to the concept of rational behavior based on utility theory. many implications of management theory and organizational theory per se supplement formal analysis (performed by mechanism design theory) with empirical components that could not be embedded in modern formal models, yet are relevant during implementation of theoretical results. for instance, incentive mechanisms with monetary rewards are successfully supplemented with motivation theories, such as f. herzberg’s motivator-hygiene theory [16] or w.g. ouchi’s theory z [17] and others. integrated rating mechanisms provide a certain tool to design control systems for company efficiency based on different concepts, such as management by objectives by p. drucker [18] or balanced scorecards by norton d.p. and kaplan r.s. [19]. thus, we are sure that blending mechanism design with management theory may serve as a bridge of the formal theoretical results towards managerial practice. references [1] thiault, d. (2012), managing performance through business processes, createspace independent publishing platform. [2] hammer, m. and champy, j.a. (1993), reengineering the corporation: a manifesto for business revolution, new york, harper business books. [3] novikov, d.a. and rusjaeva, e. (2012), “foundations of control methodology”, advances in systems science and application, vol.12, no.3, pp. 33-52. [4] novikov, d.a. (2013), theory of control in organizations, new york, nova science publishers inc. [5] myerson, r.b. (1991), game theory: analysis of conflict, london, harvard university press. 20 v.n. burkov: mechanisms of organizational behavior control: a survey [6] novikov, d.a. and chkhartishvili, a.g. (2003), “information equilibrium: punctual structures of information distribution”, automation and remote control, vol.64, no.10, pp. 1609-1619. [7] bolton, p. and dewatripont, m. (2005), contract theory, mit press. [8] spence, m. (1973), “job market signaling”, the quarterly journal of economics, vol.87, no.3, pp. 355-374. [9] gubko, m.v. (2008), “mathematical models of formation of rational organizational hierarchies”, automation and remote control, vol.69, no.9, pp. 1552-1575. [10] forrest, j. and novikov, d. (2012), “modern trends in control theory: networks, hierarchies and interdisciplinarity”, advances in systems science and application, vol.12, no.3, pp. 1-13. [11] shoham, y. and leyton-brown, k. (2008), multiagent systems: algorithmic, game-theoretic, and logical foundations, cambridge university press. [12] barabanov, i. n., korgin, n.a., novikov, d.a., and chkhartishvili, a.g. (2010) “dynamic models of informational control in social networks”, automation and remote control, vol.71, no.11, pp. 2417-2426. [13] arrow, k., sen, a., suzumura, k. (2002), handbook of social choice and welfare, north holland. volume. 1. [14] arrow, k., sen, a., suzumura, k. (2011), handbook of social choice and welfare, north holland. volume. 2. [15] kellerer, h., pferschy, u., and pisinger, d. (2004), knapsack problems, springer. [16] herzberg, f. and mausner, b. (1993), the motivation to work, new brunswick, transaction publishers. [17] ouchi, w.g. (1981), theory z. how american business can meet the japanese challenge, massachusetts, addison-wesley. [18] drucker, p. (2006), the effective executive: the definitive guide to getting the right things done, collins business. [19] kaplan, r. s. and norton, d. p. (1996), balanced scorecard: translating strategy into action, harvard, harvard business school press. corresponding author d.a. novikov can be contacted at: novikov@ipu.ru. microsoft word 825 network redundant dcs configuration management adv syst sci appl 2020; 01; 83-90 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/825 network redundant dcs configuration management n.a. zakharov1, v.i. klepikov1, d.s. podkhvatilin1 1) “dozor” subdivision jsc “kemz” moscow, russia e-mail: nazakharov@npp-dozor.ru abstract: an approach to the configuration management of a distributed control system with redundant components is proposed. the approach is based on a formalized vector-matrix representation of sets of system components, possible system configurations and functional relationships between them. keywords: distributed control system, configuration, configuration management, configuration supervisor, redundancy, fault detection 1. introduction the evolution of distributed control systems (dcs) at the present stage leads to the creation of models of complex automation of processes and production with automatic reconfiguration of their structure without the participation of personnel in order to continue normal operation in the event of failure in their components. dcs with digital communication channels can be represented as a set of jointly and purposefully functioning distributed dynamic objects and, in accordance with the modern theory of systems, is classified as a complex dynamic system. currently, a promising concept for the construction of dcs based is on maintenance-free modular electronics [1, 2]. such a dcs consists of redundant equipment and redundant communication network. this concept implies that the dcs possesses comprehensive system of collection and generalization information about equipment functioning. it uses highperformance algorithms for detection and isolation observable and unobservable failures. the key feature of the dcs under consideration is its capability of deep system reconfiguration based on component redundancy of functional systems and communication network. 2. redundancy the most promising way to parry failures in dcs is to use the redundancy of system resources. in general, there are the following types of redundancy:  system redundancy of individual components and subsystems;  structural (architectural) – adaptive reconfiguration of the system structure;  information addition to the main signal information on which it is possible to check its reliability (the checksum in the communication channel, estimates on model in dynamic control systems);  communication use of duplicated and different types of communication channels;  algorithmic using different algorithms to complete the same tasks;  software – use of different software tools;  technological use of a variety of software, information and other technologies; 84 n.a. zakharov, v.i. klepikov, d.s. podkhvatilin copyright ©2020 assa adv. in systems science and appl. (2020)  time re-execution of operations with subsequent processing of results;  semantic use of redundant semantic structures, digital images of controlled parameters;  organizational use of additional systems or subsystems that duplicate functions or information flows of the main system. 3. schedule-triggered protocol a promising solution to build a communication network is the use of the scheduletriggered protocol (stp) [3, 4]. the network distributed architecture of the control system based on the stp provides a high level of hardware, software, time, communication and information redundancy. this redundancy can be used not only to improve the reliability and fault tolerance of the control system, but also to improve the accuracy and quality of regulation and control. the large bandwidth of the stp channel allows to organize a single information space for all nodes of the distributed system. it allows several nodes to simultaneously perform calculations of control algorithms and transmit the results to the schemes of majority voting or data aggregation. 4. configuration supervisor method to control dcs redundancy a modified method for configuration supervisor (cs) is offered. in the initial method [5], cs refers to software and hardware modules used to monitor the operability of its configuration, participate in inter-service arbitration to activate its configuration either in case of winning the arbitration, or for parallel work together with other configurations on a common actuator. in the proposed modified cs method, the determination of the integrity of the configuration based on the information about the operability of its components is supplemented with estimates of the integrity of the configurations, formed on the basis of the analysis of the configurations output data. the obtained estimates of the integrity of the configurations are used as feedback to obtain estimates of the operability of the components forming the configuration. let’s consider dcs based on stp. the dcs has an excessive number of hardware, software, communication and software components (resources). resources in the general are:  sensors and input devices;  actuators and signal output devices;  computing nodes (controllers);  communication lines between the nodes and interconnecting systems;  built-in mathematical models. in this system, n variants of the sets of the available m components can be organized to perform the specified system functions. each such variant of the set will be called a configuration. different configurations can implement either different or the same system functions, configurations can have completely disjoint component sets, or they can share resources. the designed approach to management of redundancy of the distributed system is providing:  formalized representation in vector-matrix form of all components, used configurations and sets of components involved in each configuration;  formalized method for working configurations vector y direct calculation based on the original vector x component of operability;  calculation of the vector ŷ configurations integrities estimates based on configurations operation results;  formalized the procedure of reverse computations of the vector of estimated components operability based on vector serviceable configurations estimates ŷ; network redundant dcs configuration management 85 copyright ©2020 assa. adv. in systems science and appl. (2020)  replacement of existing faulty configurations by new or replacement in existing configuration of the failed components by the same serviceable. 5. gas turbine unit distributed control system let’s consider operation of the proposed method on the example of configuration management of a gas turbine unit drive distributed control system (gtu dcs). the block diagram of the gtu dcs is shown in fig. 1. the system contains k=46 components ci, i=1...46, which have the following functionality:  c1...c5 – u1...u5 – gtu dcs controllers;  c6, c12 – l1, l2 – redundant stp-bus;  c7...c11 – l11...l15 – controllers u1...u5 stp-bus l1 ports;  c13...c17 – l21...l25 – controllers u1...u5 stp-bus l2 ports;  c18 – l3 –controller u5 with the upper level control system communication channel;  c19, c32 – n11, n12 – main and backup speed sensors of the low-pressure compressor (lpc);  c20, c33 – n21, n22 – main and backup speed sensors of the high-pressure compressor (hpc);  c21, c34 – n31, n32 – main and backup speed sensors of the free power turbine (fpt);  c22, c35 – t11, t12 – main and backup air temperature sensors;  c23, c36 – p11, p12 – main and backup air pressure gauges;  c24, c37 – t41, t42 – main and backup temperature sensors of the gas before fpt;  c25, c38 – p21, p22 – main and backup combustion chamber pressure sensors;  c26, c39 – q11, q12 – fuel control valve position main and backup sensors;  c27, c40 – q21, q22 – lpc guide vanes position main and backup sensors;  c28, c41 – q31, q32 – hpc guide vanes main and backup position sensors;  c29, c42 – z11, z12 – main and backup control signals to the fuel control valve;  c30, c43 – z21, z22 – main and backup control signals to the lpc guide vanes;  c31, c44 – z31, z32 – main and backup control signals to the hpc guide vanes;  c45 – t13, p13, ns3 – information from the upper level control system: air temperature and pressure, set point to the free power turbine speed controller;  c46 – n14, n24, n34, t44, p24 – built-in engine model that produces real-time estimates of speed, gas temperature and pressure in the combustion chamber. the gtu dcs must provide:  depending on the current free turbine speed setpoint (ns) and the actual values of inlet temperature (t1) and air pressure (p1), by regulating the fuel consumption (q1) by the control signal (z1), ensure that the required value of the free turbine speed (n3) is maintained, while preventing the output of the combustion chamber pressure (p2) and gas temperature (t4) parameters for permissible values, which are also functions of temperature (t1) and air pressure (p1).  depending on the actual values of the engine rotor speeds (n1, n2) and the pressure in the combustion chamber (p2) adjust by means of control signals (z2, z3), the guide vanes position (q2, q3), which provides the required margin of gas-dynamic stability of the engine. the control system operation can be described by three functions: f1 – fuel consumption control, f2 – lpc guide vanes position control and f3 – hpc guide vanes position control: 86 n.a. zakharov, v.i. klepikov, d.s. podkhvatilin copyright ©2020 assa adv. in systems science and appl. (2020) z1 = f1(ns, n1, n2, n3, t1, p1, p2, t4, q1), z2 = f2(n1, p2, q2), (1) z3 = f3(n2, p2, q3). analysis of the gtu dcs block diagram (fig. 1) indicates that the values of the arguments of the functions f1, f2 and f3 can be obtained from various sources (redundant sensors, model, information links with the upper level system). calculation of functions values can be performed on one or several processors of various controllers, in-system information exchange can be carried out via one of two or simultaneously via both buses of the stp duplicated channel. issue of control signals both to the fuel control valve and to the guide vanes drives can be made via any of two or simultaneously on both control channels. during gtu dcs operation failures of individual components may occur, there may be failures in the processes of measurement, calculation, data transmission. due to the hardware, computing, information and time redundancy of the dcs, the gtu can continue operation using remaining serviceable components. fig. 1. dcs block diagram 6. configurations let's call configuration a set of components of the system c = {c1, c2, ..., cm}, providing execution of a certain function. the configuration can provide both the execution of the object control function, i.e. end with the calculation of the output value of the system, and perform intermediate calculations that determine the values of the parameters necessary for other configurations, for example, to calculate the most reliable values of the input parameters of the dcs based on the readings of several sensors. the same function in the control system can be implemented in different configurations depending on which components are currently in good condition. in complex objects distributed control systems can consist of hundreds of components, which, depending on their current state can be combined into dozens of different configurations to perform certain functional tasks. the formation of such configurations must be performed either in advance at the design stage of the system, or generated automatically in real time, depending on the current situation. without the use of formal design methods, both approaches are very time-consuming, require a lot of "manual" work at the stages of design and testing of the system. for this reason there is a need to develop analytical methods for representing the sets of configurations of available resources and managing these configurations in real time. network redundant dcs configuration management 87 copyright ©2020 assa. adv. in systems science and appl. (2020) the value of the fi function formed in some j-th configuration of cij will be denoted as zij. in the given system, for the functions f1, f2 and f3 can be formed, for example, three different variants of the calculation of the output values: z1 1 = f1(ns3, n11, n21, n31, t11, p11, p21, t41, q11), z1 2 = f1(ns3, n12, n22, n32, t12, p12, p22, t42, q12), z1 3 = f1(ns3, n14, n21, n32, t13, p13, p24, t44, q11), z2 4 = f2(n11, p21, q21), z2 5 = f2(n12, p22, q22), (2) z2 6 = f2(n14, p24, q21), z3 7 = f3(n21, p21, q31), z3 8 = f3(n24, p22, q32), z3 9 = f3(n22, p24, q31). in terms of the functional designation of the component, these variants can be implemented in the following configurations: c1 1 = {ns3, n11, n21, n31, t11, p11, p21, t41, q11, u1, u2, l1, l11, l12, l3, z11}, c1 2 = {ns3, n12, n22, n32, t12, p12, p22, t42, q12, u3, u4, l2, l23, l24, l3, z12}, c1 3 = {ns3, n14, n22, n32, t13, p13, p24, t44, q11, u1, u2, u5, l1, l15, l12, l2, l25, l22, l3, z11}, c2 4 = {n11, p21, q21, u1, u2, l1, l11, l12, z21}, c2 5 = {n12, p22, q22, u3, u4, l13, l14, l2, l23, l24, z22}, (3) c2 6 = {n14, p24, q21, u2, u5, l1, l15, l12, l25, l22, z21}, c3 7 = {n21, p21, q31, u1, u2, l1, l11, l12, z31}, c3 8 = {n24, p22, q32, u3, u4, u5, l2, l23, l24, z32}, c3 9 = {n22, p24, q31, u2, u3, u5, l1, l12, l13, l15, l2, l25, l22, z31}. the same configurations in terms of component numbers can be written as: c1 1 = {c45, c19, c20, c21, c22, c23, c25, c24, c26, c1, c2, c6, c7, c8, c18, c31}, c1 2 = {c45, c32, c33, c34, c35, c36, c38, c37, c39, c3, c4, c12, c15, c16, c18, c42}, c1 3 = {c45, c46, c33, c34, c26, c1, c2, c5, c7, c11, c8, c12, c17, c14, c18, c31}, c2 4 = {c19, c25, c27, c1, c2, c6, c7, c8, c30}, c2 5 = {c32, c38, c40, c3, c4, c9, c10, c12, c15, c16, c43}, (4) c2 6 = {c46, c27, c2, c5, c6, c11, c8, c17, c14, c30}, c3 7 = {c20, c25, c28, c1, c2, c6, c7, c8, c31}, c3 8 = {c46, c38, c41, c3, c4, c5, c12, c15, c16, c44}, c3 9 = {c33, c46, c28, c2, c3, c5, c6, c8, c9, c11, c12, c17, c14, c31}. components in formulas (3, 4) can be written in arbitrary order because these formulas are intermediate. they are used to clarify formal representation of configurations. expressions (4) for sets of ck components present in ci j configurations can be written as a matrix of k dimension (n×m), where n is the number of all configurations considered in the system, m is the number of all components of the system involved in the formation of configurations. the kij element of the matrix k is 1 if the i-th configuration uses the cj dcs component (see fig. 1). 7. state and availability vectors we define a vector x that characterizes the current state of all components of the dcs. if all components of the dcs are operable, the components of the vector x are represented as: 88 n.a. zakharov, v.i. klepikov, d.s. podkhvatilin copyright ©2020 assa adv. in systems science and appl. (2020) xs = 1, s = 1…m. (5) note that all components involved in the configuration are equally involved in the implementation of the system function, regardless of their physical nature and indicators of their own availability. for the example under consideration, m = 46. if there are faulty components in the system, the corresponding components of the vector x are zero. the value of this vector is formed in real time according to the results of the functioning of dcs software and hardware self-diagnostics. next, we define the availability vector of configurations y, which characterizes the readiness for operation of all considered configurations. yi = 1 if the i-th configuration is healthy, otherwise yi = 0, i = 1...n. configurations availability is a function of the components involved in configurations and their operability. y = f1(k, x), (6) where f1 is defined by the expression:   m s ssj xkf 1 ,1   , (7) the symbol  denotes the logical "and" function, the symbol  denotes the implication function. the meaning of the expression (7) is that each j-th element of the vector y will be equal to 1, i.e. the j-th configuration will work if only serviceable components are used in this configuration. in the stp-based dcs, several or all configurations can be executed in parallel or sequentially, which allows you to parry the failures of its components. the number of executable configurations is limited by the computing power of the dcs. if at execution of several configurations different values of the same output parameter are received, it is required to solve two problems: 1) reject incorrect results and mark the configurations which have formed them as faulty; 2) on the basis of the results obtained from the recognized working configurations to form the final agreed output value. 8. dcs components diagnostics an effective way to diagnose the dcs components operability is the inclusion in its composition of the built-in real time model of the object under control [6]. this allows you to get a number of additional features to improve the quality and reliability of control, improve the performance of the system: noise and fault filtering; the recovery of unmeasured parameters for the diagnosis and management; detection of abnormal states of the object and the control system; diagnostics of the state and parametric degradation of the object. some methods of rejection of incorrect values are considered in [7, 8]. next, consider the analysis of configurations based on tolerance control. let's write down the vector of output values obtained from the results of all n configurations operation. z = [z1, …zj, …zn] t (8) (upper indexes are omitted for clarity) and the initial configuration availability vector y = [y1, …yj, …yn] t. (9) network redundant dcs configuration management 89 copyright ©2020 assa. adv. in systems science and appl. (2020) using the vectors zmin and zmax, we set the minimum and maximum output values for each configuration, respectively. let’s define a tolerance control function as f2(y, z, zmin, zmax) = 1, if (yj=1) & (zmin j ≤ zj ≤ zmax j); (10) else 0. the evaluation of the configurations availability vector ŷ is defined by the function f2: ŷ = f2(y, z, zmin, zmax), (11) the zero value of any component of the configuration availability vector indicates that there is one or more failed components of the dcs in the corresponding configuration. next, we define the function f3 as   n j isj ykf 1 ,3 ˆ   , (12) the symbol  denotes a logical or function; the symbol  denotes an implication function. for the vector that characterizes the serviceability evaluation of the dcs components, we can write:  ykfx ˆ,ˆ 3 , (13) the meaning of the expression (13) is that for each s-th element of the state vector, the operability of all j = 1...n configurations, in which it is involved, is checked. if at least one configuration in which this component is involved, i.e. the condition cj,s ≤ ŷj is satisfied for at least one j = 1...n, then the s-th component is identified as serviceable. if all configurations in which this component is involved are found to be faulty, the component is identified as failed. 9. configuration management algorithm configuration management algorithm is the following. the algorithm is executed cyclically. the first step of the algorithm is the formation of the initial vector x of the dcs components serviceability. the vector ν of dimension m formed by the built-in dcs components self-diagnostics means and the vector x̂ of components serviceability estimates calculated on the previous cycle are used. at the initial cycle, all components of the vector x̂ are assumed to be equal 1. in the second step, based on the original serviceability vector x and the configuration matrix k the vector of configurations availability y is formed. next, according to the configurations availability verification method (in this example – tolerance control) configurations availability evaluations vector ŷ is formed. in the fourth step the dcs components serviceability evaluations vector x̂ is calculated. in case of faulty configurations detection, a new configurations matrix k is formed by replacing the used faulty configurations with new ones or replacing the faulty components with serviceable ones in the existing dcs configurations. 90 n.a. zakharov, v.i. klepikov, d.s. podkhvatilin copyright ©2020 assa adv. in systems science and appl. (2020) 10. conclusion in conclusion it should be noted that both of configurations set and their component composition can be optimized according to various criteria, such as the dcs computing and communication resources load, results interpretation unambiguity. in particular, it is obvious that if each of the system components will be involved in only one specific configuration and absent in all the others, its fault is clearly detected when recognizing this configuration failed. if two or more components are present in only one configuration (or a group of configurations) and are not present in all the others, it is not possible to distinguish their failures due to the failure of this configuration. optimization of the number and component composition of dcs configurations should be performed taking into account the depth and reliability of the built-in software and hardware self-diagnostics of dcs components. references 1. jin, h. lee s., han s., jo, h. kim d. (2012). wip abstract: challenges and strategies for exploiting integrated modular avionics on unmanned aerial vehicles. ieee/acm third international conference on cyber-physical systems, beijing, china, 211-211. https://doi.org/10.1109/iccps.2012.34. 2. wang, l. sun y., guo p., zhang y. (2015). an enhanced reconfiguration method for the second generation integrated modular avionics, 11th international conference on computational intelligence and security (cis), shenzhen, china, 433-436. https://doi.org/10.1109/cis.2015.110. 3. kopetz, h. (2011) real-time systems. design principles for distributed embedded applications. heidelberg, germany: springer. https://doi.org/10.1007/978-1-44198237-7. 4. zakharov, n.a., klepikov, v.i., podkhvatilin, d.s. (2013) sinkhronno-vremennoj protokol dlja raspredelennykh sistem upravlenia [schedule triggered protocol for distributed control systems], avtomatizatsiya v promyshlennosti, 2, 37-39. [in russian]. 5. ageev, a.m., bronnikov, a.m., bukov, v.n., gamayunov, i.f. (2017). supervisory control method for redundant technical systems, journal of computer and systems sciences international, 56 (3), 410-419. https://doi.org/10.1134/s1064230717030029. 6. klepikov, v.i., kalin, s.v., zakharov, n.a., podkhvatilin, d.s. (2008). algoritmicheskoe obespechenie otkazoustojchivosti raspredelennykh system upravlenia [algorithmic provision of distributed control systems fault tolerance], radioelektronni i komp'uterni sistemi, 34 (7), 43–48. [in russian]. 7. klepikov, v.i., podkhvatilin, d.s., dudorov, y.n. sharapov, g.v., zakharov, n.a. (2011). information-measuring diagnostics complex for technical maintenance. autom remote control, 72 (5), 1089-1094. https://doi.org/10.1134/s0005117911050171. 8. klepikov, v.i. (2014) otkazoustoychivost' raspredelennykh sistem upravleniya [fault tolerance of distributed control systems]. moscow, russia: zolotoe sechenie [in russian]. advances in systems science and application(2016) vol.16 no.3 52-75 load-aware congestion adaptive multipath multicasting h. santhi and n. jaisankar school of computing science and engineering (scope), vit university, vellore, india abstract in recent years, all communication system becomes wireless due to vast development and advancements in wireless technology. such network offers a platform to deploy a wide range of multimedia and other services with different data types. in mobile ad hoc networks one of the most prominent factors which affect the overall performance of the network is congestion. congestion is one of the most misinterpreted concepts in the context of the wireless network. in general, excessive data traffic in the network referred to as congestion. however, in the wireless network, the congestion occurs due to unavailability of resources such as insufficient bandwidth, low battery power, and so on which leads to high packet losses, bandwidth reduction, and wastage of energy and time in recovering congestion. in addition to this, another factor which degrades the performance is improper load balancing due to certain routing metrics limitation. from the literature, it is observed that many of the existing solutions handle the above problems separately in a not-adaptive manner. however, addressing these issues together in an adaptive manner provides an effective solution to balance the load in the network as well as congestion. the proposed work addresses congestion and load balancing problems parallel with the aim to improve and enhance the overall network performance and lifetime. the proposed load aware congestion adaptive multipath multicast (lacamm) routing approach adapt to current changes in the load and congestion level to find a suitable path even in the case of congestion scenario and node resource constraints. the proposed scheme measures the node resources such as residual bandwidth and the residual battery to predict the node stability. redirects the data transmission under congested scenario from the congested node through the non-congested alternate path thereby improves the overall network performance. this work performs well under burst traffic with the harsh environment. keywords multicasting, congestion, multipath, adaptive routing 1 introduction a mobile ad hoc wireless network consists of a set of wireless mobile nodes shaped dynamically with none central administration or existing network infrastructure[1]. every node in this network will act as a host and additionally as a router and contains a capability to move freely and at random in a direction at any speed. owing to limitation of mobile nodes transmission varies the packets advances in systems science and application(2016) vol.16 no.3 53 are forwarded to the destination during a multi-hop fashion with the assistance of intermediate nodes. one among the key problems in multihop mobile ad hoc network is congestion. the crucial factors that influence congestion throughout multi-hop relay are shared restricted wireless information measure, low device power, dynamically ever-changing configuration, and so on[2,3]. congestion will cause packet loss, degradation of information measure, waste of resources on congestion recovery. it’s troublesome to beat congestion problem however it’s potential that the congestion is often avoided by adapting bound appropriate mechanism and rules for the flow. normally routing algorithms in manets are generally classified into proactive and reactive routing. in proactive routing, the routes are established as like wired network approach and are updated either sporadically or on a progressive update fashion. this approach isn’t appropriate once the network is just too massive and, therefore, the nodes are extremely mobile. in reactive routing, the routes are created as and once required and is a lot of economical than the proactive routing approach[4-7]. however, the matter is that the reactive routing protocol creates high management overhead just in case of frequent path breaks. this drawback is self-addressed by use of multipath routing approach. the routing protocols in manets are often classified in our own way as congestion-aware routing and congestion adaptive routing. several existing solutions belong to congestion aware approach solely only a few are congestion adaptive. in congestion aware approach the congestion is taken into thought solely throughout route discovery and maintains a similar standing till the trail breaks. however in congestion adaptive routing the routes are adaptive to the present congestion standing of the network. congestion non-adaptiveness can cause the following[8-9]: long delay: congestion-aware routing takes long-standing to observe congestion. upon congestion, it’s quite essential to use a brand new route. but, the matter with congestion aware on demand routing protocol is that it takes longstanding to search out a much better non-engorged route this ends up in high delay. high overhead: upon congestion invoking re-route discovery involves with flooding of control packets to search out a brand new route to forward the information. the flooding of control packets creates high control overhead that degrades the performance of the network. many packet losses: as mentioned congestion could happen to any node at any time as a result of lack of resources that ends up in several packet losses. a typical congestion control try and scale back the traffic load, either by decreasing the rate at the sender or dropping packets at the intermediate nodes or doing each. these problems become additional visible throughout transmission of enor54 h. santhi, n. jaisankar: load-aware congestion adaptive multipath multicasting mous multimedia system applications during a large mobile accidental network and supply a negative impact on the network performance still as within the quality of service. 1.1 congestion congestion may be a drawback that happens on shared networks, once multiple users access to constant resources (bandwidth, buffers, and queues). once numbers of packets are present in a network is larger than the capacity of the network then this case is termed as congestion[10]. congestion in a network could occur once the load on the network i.e. the amount of packets sent to the network is bigger than the capacity of network[6,11]. 1.1.1 congestion control congestion control mechanism is performed once the network faces congestion. congestion control mechanism sometimes enhances network overall performance based on the load condition of the network. the congestion control mechanism is completed through controlling the sending rate of information streams of every source and conjointly results in high utilization of the offered bandwidth. the most objective of congestion control is to attenuate the delay and buffer overflow caused by network congestion and, therefore, alter the network to perform higher. as congestion is directly associated with the problem of dropping the packet, it’s needed that some technique is applied on the network so the drop of the packet can decrease. however to regulate on the quantity of dropping rate is tougher in manets as compared to the wired network due to the following characteristics [7,12-13]. a. dynamic topology as in manet, there’s no central point or base station, to regulate the entire network. each device will move freely in manet, therefore, the topology of the network isn’t mounted. thus, it can’t be expected whether or not a node that participates throughout some transmission can collaborate in the whole transmission or not[14]. a node will move any time instance thus a path detected by the source node to transfer its information will be a break at any time. if no path is found by the intermediate node to forward the information it’ll begin to drop the packet once a while. b. multi-hop routing each node in manet will receive and forward the data towards the destination nodes. however node forwarding capacity is restricted to its transmission range; it suggests that it will deliver the data packets to solely that node that come beneath its transmission range. if any 2 nodes that not come back beneath the transmission range of each other than the forwarding node depends on intermediadvances in systems science and application(2016) vol.16 no.3 55 ate nodes to relay data in a multi-hop fashion[10, 12]. a route has been detected by a routing protocol then sender begins to transfer the data to a node that comes beneath its transmission range this node referred to as an intermediate node, every intermediate node further transmitted data to its neighbor node and this process is repeated till information reach to the destination. arrival rate of packets at this specific node are often larger than its forwarding capacity so this node begins to drop the packet. c. heterogeneous environment in manet, any device will participate if it’s able to forward the data. these participating devices are totally different of various kind having a different storage capability and different resource. the transmission rate of every device could stay completely different. in manet addition of recent device is extremely simple if it comes beneath the transmission range of different node it becomes the part of that network. thus, it should be possible that a brand new device comes back and begin to transmit its own data on the route that is already detected by a different node. all devices taking part in communication are of various kinds and will become unavailable at any time that makes the period of communication not so long[5,15]. in such kind of condition, packets are dropped by the precursor node. sometimes a particular node becomes the intermediate node between several nodes. a scenario will arise at this node that several of its neighbor nodes forward the data to that the same time, thus there’ll be an excessive quantity of packets inward at these intermediate node. if the arrival rate of data on the nodes is bigger from its transmission rate node can begin to drop the packets. d. density of node the number of neighbor nodes of every node might also the reason in manet, as a result of if a node cannot deliver the data on to the receiver node then use another intermediate node to forward the data packet. in manet for every node, a lot of neighbor nodes mean a lot of link connections between the nodes and their neighbors. itll become the rationale of a lot of arrival rate of packets at a specific node, therefore, a lot of neighbor node of any intermediate node become the rationale of coming significant load as compared to the node reception capacity. in such kind of condition, a node can begin to drop the packet[13,16]. e. presence of malicious node reliability of the node is going to decrease as a result of the presence of malicious packet dropper node. in manet participating devices have restricted resource sometimes routing protocol select the path within which packet dropper node work as an intermediate node. a packet dropped node is self-seeking nodes that really not forward the data packets to next node however in place of this it 56 h. santhi, n. jaisankar: load-aware congestion adaptive multipath multicasting simply drops the packet to save lots of the resources[3,17]. the presence of packet dropper node could be a severe downside in manet and that they don’t seem to be the sole reason for the massive delay, however additionally become the reason of heavy traffic load on the network because the sender might become involved in causing packets again and again if no acknowledgment is received from the receiver. f. absence of physical protection in manet it’s impossible to guard a node type numerous kinds of threats because the node position isn’t fixed, a node will move in any direction within the network[18]. the nodes will be attacked from any direction wherever fixed physical protection like firewall and gateways can’t be applied. it means that for securing itself a node should be equipped to fulfill an offender directly or indirectly. however because of the absence of physical protection like in hard wired network, there’s a lot of likelihood for a node to become unreliable, and begin to drop the packet. 1.1.2 congestion prevention it is the mechanism to handle the network from congestion that involves play before network faces congestion. for this purpose nodes got to monitor their status and that they negotiate with the neighbor node within the network so no a lot of traffic than the required amount, the node will handle, are allowed to return to the network so no congestion can occur. congestion affects the performance of the network. therefore, some necessary congestion control technique is needed to stop the network from the congestion. prevention from congestion in manets is far difficult as compared to wired networks because of its specific characteristics. the subsequent are a number of the most qos provisioning and maintenance issues in manets[19]. a. stable route to prevent the network from the congestion it’s better to decide on a reliable path. for this purpose route are going to be analyzed so a perfect error free totally coverage path with high transmission delivery ratio is select. it needs data of the nodes which can be remain offered all the time, however because of the dynamic environment of manet choice of such node isn’t possible. b. reservation of bandwidth bandwidth reservation is a technique to stop the network from congestion, during which nodes reserve bandwidth for future communication through negotiation between the neighbors nodes which come back among 2 to 3 hops. it needs communication, and exchanges of a message between them because the channel is shared between the nodes. in manet environment, a node will moves from the reservation space of the node at any time even communication goes on. thus, advances in systems science and application(2016) vol.16 no.3 57 reservation of bandwidth means that additional overhead for communication and releasing messages. therefore, bandwidth reservation isn’t attainable in manet. c. service level agreement (sla) in manet, every participating node works as a host and as a router. any node isn’t responsible for performing some specific task. since all the nodes within the network work to produce services, there’s no clear definition of a service level agreement (sla). whereas in, an infrastructure network the services to the users within the network are provisioned by one or additional service providers. thus, estimation of the node behavior isn’t possible that is needed for prevention from the congestion. d. channel reliability since the wireless bandwidth and capacity in manets are suffering from interference, noise and multi-path attenuation, the channel isn’t reliable. moreover, the offered bandwidth at a node can’t be estimated precisely as a result of it involves large variations based on the quality of the node and other wireless device transmission within the neighborhood etc. e. routing difficulty routing is troublesome in manet because link breakage occurs overtimes. once any link of a path breaks, it got to find the other offered link or replaced with a new found path. this rerouting operation costs the scarce radio resource and battery power whereas rerouting additionally increase delay that additionally have an effect on quality of service of applications and degrade the network performance. thus, the routing operation has got to take care of such variety of challenge that is tough to handle. 2 previous work there are many congestion algorithms are proposed for mobile ad hoc networks some of them are explained below. in [19] developed a method for detecting congestion well in advance in order to prevent the network from the congestion. their work is based upon the calculation of approximate queue length in advance. for this purpose, they calculate the average queue length at the node level. network characteristics like congestion and route failure need to be monitored and resolved with a reliable mechanism. to solve the congestion problem, a novel dynamic congestion estimation technique has proposed that could analyze the traffic fluctuation. by the assessment of average queue length, a node is able to find that there is some probability of congestion so it sends a warning message to its neighbors. upon receiving the warning message they try to search some alternative congestion free path to the destination and resumes communication through an alternate path. so this dy58 h. santhi, n. jaisankar: load-aware congestion adaptive multipath multicasting namic congestion estimation procedure tries to provide a reliable communication within the manets by controlling upon the congestion in ad hoc networks. in [20] a new technique to detect the packet dropper nodes in the network by using a reliability factor. in manet each node has limited resources like limited battery power, a packet dropper node is that node in the network which may not cooperate properly in network operations as they not forward the coming data packets to the next node but instead of this they drop the data packet to save their resources. such nodes are called selfish or misbehaving nodes and these nodes are also the reason of congestion. the dropping of data packet not only affects the network connectivity but also can widely waste the network resources. to handle this situation a scheme based on mac-layer acknowledgments is used to detect the packet dropper nodes. to eliminate such nodes from the network its reliability is evaluated during the packet transformation. in this work the field of reliability factor is increased on the basis of acknowledgment received from the receiver, and all senders making the decision to send a packet to a node having higher reliability factor. the reliability factor identifies the packet dropper nodes based on the acknowledgment. hence, on the basis of node reliability factor, a packet dropper node can be detected and also can be isolated from the network. in [21], proposed congestion-aware routing (carm) to adapt to the congestion. the high throughput non-congested routes to any node in the network are selected based on the weighted channel delay (wcd) value. second, the proposed work adapts mismatched data-rate routes using effective link data-rate categories (eldc). in general, the protocol tackles congestion by switching between the above-said approaches to compact congestion in the network and efficiently increases the overall network performance. a method for reliability analysis for manet is presented by sreedhar and damodaram[22]. they proposed that the node performance is also influenced by the number of neighbor nodes of that node. in their work effect of node mobility and reliability in a real manet platform is proposed and analyzed. they proved that the wireless network has limited capacity, and the throughput of the wireless network granted to each user can be decreased to zero if the number of users increased. as the transmission capacity of the wireless network affect the throughput and it will affect the terminal reliability of manet. congestion means the arrival of an excessive amount of packets at a network which leads to many packet drops. a node can communicate with many nodes which are its neighbor nodes. as they come under the communication range of that node then there will always the chance that at the same time many neighbor nodes send their data packets to the same node, so there will be an excessive amount of packets arriving at these nodes become the reason of packets drop. hence, congestion is related to the density of the node in some area, and it will influence advances in systems science and application(2016) vol.16 no.3 59 the terminal reliability by reducing the intermediate node reliability. this work focuses on upon identifying the relationship between the number of link connections and the node reliability to reduce the congestion problem. type of service aware routing protocol (tsa) proposed in [23] is an improvement to aodv. this approach uses only a hop count as a metric for route selection. tsa is a cross-layer congestion-avoidance routing protocol in which the routes are used for extended periods of delay sensitive traffic. avoiding busy nodes alleviates congestion, leads to fewer packets drop and in a short end-to-end delay. in addition, tsa distributes the load on a large area, so by increasing the spatial reuse. a simulation study reveals that tsa significantly improves the throughput and reduce packet delay during high congestion state. to handle the network dynamics an optimized reliable ad-hoc on-demand distance vector (oraodv) scheme proposed in [24]. the proposed protocol (oraodv) is meant for best route discovery and reliability of packet delivery. a new idea of blocking expanding ring search (blocking-ers) is employed in it to avoid network wide broadcasting. the blocking-ers doesn’t begin its route search procedure from the source node whenever a broadcast is needed. the broadcast is initialized by any acceptable intermediate nodes on behalf of the source node that acts as a relay or an agent node. 3 proposed lacamm approach this work is adaptive to the current load and the tries to prevent congestion well in advance by warning its upstream and downstream forwarding group nodes. each node on the primary path generates a warn message when it is prone to be congested. upon receiving warn message the upstream node uses an alternate non-congested path along the primary path for avoiding the potential congestion area. traffic is distributed eventually over the available routes, thus, efficiently decrease the chance of congestion. lacamr is on-demand multipath multicast routing protocol which comprises the following components: 1.resource monitoring 2.congestion monitoring 3.construction of resource full node list 4.congestion free route primary route discovery 5.congestion adaptively and traffic redistribution and 6.route failure recovery these components are explained in detail in the forth-coming subsections. fig.1 depicts the proposed load aware congestion adaptive multipath multicasting approach. 60 h. santhi, n. jaisankar: load-aware congestion adaptive multipath multicasting fig. 1 proposed ladamr approach 3.1 resource monitoring 3.1.1 link stability in mobile ad hoc networks, the mobility induced by nodes as well as the propagation effects cause a packet to suffer fading effect. the link stability can be measured using signal-to-noise ratio (snr) and can be determined with the help of hardware component. if the estimated snr value is less than the threshold limits the packets will contain excessive errors due to high noise. this result in retransmission their by increases the overall delay thus degrades the performance significantly. the link stability can be estimated as follow using an equation (1). ber = 0.5 ∗ func( √ rsp ∗ cb np ∗br (1) where, ber = bit error rate, rsp = received signal power, cb = channel bandwidth, np = noise power and br = bit rate and func = error function the signal to noise ratio (snr) for multiple packet transmission can be estimated using the equation (2) as: snr = 10log rsp np + ∑n i=1rspi (2) where, ∑n i=1rspiis the signal strength of packets at the receiver.n is the number of packets received instantaneously. when a node is sending the data packet it appends its signal strength i.e., a transmitted power then the receiving nodes advances in systems science and application(2016) vol.16 no.3 61 estimates the received signal strength using the free-space propagation model using the wavelength of the medium,the distance between the communicating nodes and unity gain of sending and receiving antennas as shown in equation (3). rss = tss ( λ 4πd )2 gsgr (3) 3.1.2 available bandwidth the bandwidth availability is one of the very important factors which determine the connectivity of the network. in general, packet forwarding between the source and the destination follows a multi-hop communication. hence, it is very much essential to ensure that whether the intermediate forwarding nodes have sufficient bandwidth to forward the data or not. the nodes in the wireless networks rely on the shared wireless links and the links are severely affected by fading, inference, and path loss[23]. it is estimated by measuring the idle periods of the wireless channel. each node in the network listens to the channel and obtains the status to estimate the channel idle period using the channel observed time interval (coti). then the channel idle time (cii) can be estimated by increasing the count from the previous busy time to the start of the next busy time. let us consider the total channel idle time consists of several channel idle slots, say n. total channel idle time (citi) is the sum of all n idle times. thus, available bandwidth at a node is estimated using equation (4): ab = ∑n i=1cii coti ∗bwtotal (4) 3.1.3 estimation of residual battery battery lifetime (bli) of a node i is estimated using the residual energy (rei) and new and old drain rate (dr) of a node i and is estimated as shown in equations (5) and (6): bli = rei ∝ ∗drold(j, k) + (1− α)drnew(j, k) (5) drnew(j, k) = ej,k (1− perror)n (6) where, drold and drnew represents old and current calculated drain values and represents a constant value between 0 and 1. the number of neighbors of node says n is xk where n knows its entire neighbors total current load, tcl(i) then the probability of data forwarding can be computed as shown in equation (7): p (n) = 1− ( tcl(n)i macqi(i) ) (7) 62 h. santhi, n. jaisankar: load-aware congestion adaptive multipath multicasting 3.2 congestion monitoring detecting congestion in a reactive manner will produce longer delay, high packet loss, and high control overhead. unlike wired and high-speed networks congestion in manet becomes more viable during transmission of large-scale multimedia data. hence, eliminating congestion in such dynamic networks produce excessive overhead, delay, and a waste of resources. many solutions have adapted active queue management strategies to eliminate the congestion problems. this work aims to propose an approach which is adaptive to the incoming traffic and anticipates congestion by redistributing the traffic along the available congestion free path. each node in the network estimates the incoming traffic and updates the same in the neighbor table periodically. this helps to find out the neighbor node current load status. the current data traffic(cdt) is estimated to find out the congestion level of the node. each node ni samples the queue length in the mac layer periodically.suppose qj(k) is the kth sample value, and x is the overall sampling time period, then the current data traffic of node ni can be estimated using the equation (8). cdt (i) = ∑x k=1qj(k) x (8) the total length of the queue of node ni is the maximum capacity of the queue in the mac layer is macqi(i); then the total current load is defined as follows using equation (9). tcl(i) = cdt (i) macqi(i) (9) to monitor congestion well in advance, the average queue size is estimated by setting the static maximum and minimum threshold value for the queue length as qminth = 0.25 ∗ size of buffer and qmaxth = 0.75 ∗ size of buffer the current queue size can be estimated using equation (10) as: currentavgqsize(i) = (1− wq) ∗avgqold+ tcl(i) ∗ wq (10) where wq is the queue weight, is a constant (wq = 0.002) from red queue results in floyd, (1997). the current congestion status is computed as shown in equation (11). currentcs(i) = tcl(i)− currentavgqsize(i) (11) if the currentcs(i) is less than qminth ,then the incoming data traffic is below the buffer size and hence, a node can handle the traffic. if currentcs(i) ≤ advances in systems science and application(2016) vol.16 no.3 63 qminth and ≥ qmaxth , then the buffer overflow likely to take place perform packet drop probability to avid the packet loss. finally, if currentcs(i) ≥ qmaxth , then the node is congested, invoke redistribute the route through the available non-congested alternate path. table 1 list of symbolizations used in this work symbol description rsp received signal power cb channel bandwidth np bit rate br error function func interference ranges of the nodes snr signal to noise ratio rss received signal strength gs sending antennas gain gr receiver antennas gain propagation wavelength of the medium tss transmitted signal strength d distance between any two communicating nodes coti channel utilized time interval cii channel idle time tcii total channel idles time ab available bandwidth bli battery lifetime of node i dri drain rate of node i rei residual energy of node i xk set of neighbor nodes of k tcl total current load p(n) the probability of data forwarding cdt the current data traffic x overall sampling time macqi(i) current load on mac layer of node i qminth the minimum queues threshold limit qmaxth the maximum queue threshold limit wq the queue weight currentcs the current congestion status of node currentavgqsize the current average queue size rfn the set of resource full node 64 h. santhi, n. jaisankar: load-aware congestion adaptive multipath multicasting 3.3 construction of resource full node (rfn) list fig.2 shows sample network scenario. each node periodically updates its one-hop neighbor resource information. the periodic interval is set to 1sec. it helps each mobile node in the network to know about its one-hop neighbor resource information for constructing one-hop and two-hop congestion free resource-full node (rfn) list. after that, this set of congestion free (rfn) nodes will be used as a subset of a forwarding node to forward the datagram from the corresponding source to a destination. fig. 2 sample network scenario each mobile node updates its one-hop and two-hop neighbor list in its routing table and the same is used during the route discovery process to build congestion free primary path. the routing table contains the following fields information for each route entry: multicast rt { src addr is the source mobile node address, dst addr is the destination node address, grp addr is the multicast group address, hop cnt is the number of intermediate hops i.e. hop count, rfn node addr is the resource-full node address, rfn set is the list of resource-full node set, and advances in systems science and application(2016) vol.16 no.3 65 con status is the neighbors congestion status } table 2 resource full node list node id one-hop resource full node id two-hop resource full node id s 1, 2 4, 7 1 s, 4 5, 7 2 s, 7 4, r3 4 1, 5 7, r3 5 4, r1 7, 9, r3 7 2, r3 5,9 9 r2, r3 r1 r1 5 9 r2 9 5 r3 7,9 4,5 3.4 congestion free primary route discovery ldamm is an on-demand protocol initiates route discovery when a source mobile node has data to send. it first checks from its resource full node list whether the multicast receiver is in two-hop resource full node list or not. if the multicast receiver is in the two-hop resource full list, then it forwards the jrreq using the existing path in its routing table. if not, then the source node initiates a route discovery process by just forwarding the jrreq packet through its one-hop and two-hop resource full node set rather than flooding the jrreq packet into the network. this procedure helps in minimizing the control overhead to a certain extent. upon receiving this packet, the receiver node checks its two-hop resource full node list. if the multicast receiver found, then it forwards the jrreq packet directly to it. the multicast receiver then responds to the first received jrreq packet and sends back jrrep packet. if not found, it updates the received information and forwards jrreq packet to its one-hop resource-full node. this process repeats until the multicast receiver node found. the first jrrep path considered as a primary path between the source and the multicast receiver. finally, the source finds a resource full non-congested primary path to the destination. a primary route found in this case, are s→1→4→5→r1,s→2→7→r3,and s→2→7→r3→r2. these routes are used by the source to transmit a datagram towards the multicast receivers. thus, the proposed work finds a resource full congested free primary path from source to destination with the help of resource full node list and controls the overhead by avoiding unnecessary flooding of packets. table 3 describes the overall procedure involved in congestion free primary route discovery process after resource full node list selection. 66 h. santhi, n. jaisankar: load-aware congestion adaptive multipath multicasting table 3 procedure for congestion free primary route discovery process input: g= (v, e) output: the congestion free multicast tree begin 1) the source mobile node s checks its one-hop rfn list (1, 2) and a two-hop rfn list (4, 7) to find whether a multicast receiver is available or not. 2) if the multicast receiver nodes r1, r2, and r3 is not in one-hop and two-hop rfn list, then the source mobile node forwards jrreq packet to its one-hop resource full nodes 1 and 2. 3) now upon receiving jrreq node 1 and node 2 will check its one and two hop rfn list to find whether r1, r2, and r3 is available or not. if not 1 and 2 forwards a jrreq packet to its one-hop rfn nodes i.e.,(4, 7) 4) this process is repeated until jrreq reaches a multicast receiver. in this case, node 2 and node 4 finds the multicast receivers is its two-hop resource full node list and forwards the jrreq through intermediate nodes 5 and 7. 5) the multicast receiver node now sends a jrrep along the reverse path of the jrreq to reach the source. 6) the source mobile node now fixes the first jrrep as a congestion free primary path and starts the transmission along this path. end 3.5 congestion adaptive alternate route discovery each node in the primary path periodically estimates and finds its congestion status. if it is likely to be congested then warns its upstream and downstream node by sending congestion warning packet (cwp). upon receiving this packet, the upstream node checks it updated resource full node (rfn) list to find whether a multicast receiver is in it or not. if exists, exchange the new rfn list with its neighbors and resume the transmission along the newly available congestion free path. this new alternate route gets updated in its routing table. if not forwards the cwp to its previous node. if no rfn list found on the congestion free primary path, then the cwp, send to the source node. the source now assigns another alternate path if it is available otherwise initiates a new route discovery process to find a congestion free primary path. this alternate path finding process does not incur any significant overhead, because of the availability of one-hop and two-hop resource full node list in each node. for example, if the resource-full node 9 detects the congestion, sends a cwp to its neighboring nodes on the primary path in this case r2 and r3 and updates the rfn list in the routing table. in response, the upstream node r3 checks its routing table new rfn list along the primary path. if exits, traffic will be resumed through the available rfn nodes otherwise it forwards the cwp to advances in systems science and application(2016) vol.16 no.3 67 its previous node. in this case, there is no such alternate congestion free alternate path exists for node r2 it forwards towards its source node. the source node then initiates a new route discovery process. table 4 presents the overall procedure involved in finding the congestion free alternate routes. table 4 procedure for congestion free alternate route input: g=(v, e),multicast sessions. output: the congestion free of alternate route v ∈ rms begin 1) initialize the current queue buffer size, average queue size new and old as 0. 2) set minimum queue threshold limit to 0.25 * current queue buffer size and maximum queue threshold limit to 0.75 * current queue buffer size and queue utilization. also, set queue weight as 0.002 3) check if current avg que size is half of queue size. 4) for each arriving packet in queue increment the instantaneous queue size. 5) if it is a non-empty queue size, then apply the formula and if queue average new is less than queue minimum and queue average new together which is less then congestion warning limit, then set queue status as safe. 6) else if queue average new is greater than queue minimum and queue average new put together which is less than queue maximum then set queue status as likely to be congested 7) if the instantaneous queue size is greater than queue maximum and alternate path be false together then update queue maximum. 8) else queue status is congested 9) update queue average old as queue average new and queue weight. end 4 results and discussion a comparison of ladamm performance with that of maodv is done for various network scenarios using the network simulator (ns2.34) . the observations are presented below 4.1 simulation configuration and performance metrics the network consists of 100 nodes in a 1700 * 1700 m terrain size. the radio range is set to 250 m with bandwidth 2 mbps. to detect the link breaks using feedback mechanism ieee802.11 dcf is used. the channel propagation model used is two-way ground propagation model[24]. a queue size at each node is set to hold only 50 data packets and a routing buffer size is set to 64 data packets. the queue and buffer value is initially set to zero until the route discovery process. 68 h. santhi, n. jaisankar: load-aware congestion adaptive multipath multicasting the routing protocols used for performance analysis is maodv. the data flow used constant bit rate (cbr), which varies from 1 packet/sec to 50 packets/sec. the mobility speed of the node is varied from 5m/s to 30 m/s and each scenario is simulated for 600 s. table 5 describes a list of simulation configuration parameters used for performance analysis for various network scenarios. table 5 list of simulation parameters used simulation parameters simulation parameters node placement scheme random propagation model two-way ground propagation environment size 1500mx1500 m number of nodes 100 transmitter range 250m bandwidth 1mbps simulation time 600s traffic type constant bit rate (cbr) packet size 512bytes number of packets transmitted by sources 100 mobility model random way point model packet rate 5-50packets/s 4.1.1 end-to-end delay the average end-to-end delay is a measure of time consumed to deliver a packet from the source to the destination due to buffering of packets, transmission, retransmission and propagation delays. 4.1.2 packet delivery ratio percentage of data packets received at the receivers out of the number of data packets generated by the cbr traffic sources. pdr(%) = total number of packets received total number of packets transmitted ∗ 100 4.1.3 routing control overhead is the ratio of a total number of control packets received to the total number of control packets generated during the simulation time. 4.2 overall performance evaluation the simulated results discuss the different network scenarios. various performance metrics such as end-to-end delay, packet delivery ratio, and control overadvances in systems science and application(2016) vol.16 no.3 69 head are evaluated to facilitate the performance of the proposed lacamm protocol. 4.2.1 impact of lacamm with maodv the end-to-end delay, packet delivery ratio and control overhead results with respect to varying cbr packet rates are shown below from fig.3 to 5. these figures clearly show that the proposed lacamm yields better results when compared to maodv. fig.3 represents the end-to-end delay for lacamm and aodv with respect to varying the cbr packets rates from 5 packets/s to 55 packets/s. this figure shows that the end-to-end delay for proposed lacamm is much smaller that of maodv for all values of packet rates. the delay variation is lacamm was less than that of madov enables the proposed work more suitable for real-time multimedia applications. fig.4 shows the obtained packet delivery ratio with respect to varying the cbr packet rate is much higher than that of maodv. this is because the proposed lacamm has an ability to adapt to the load and congestion. but when the cbr packet rate increase maodv fails to handle congestion and hence leads to poor packet delivery ratio. fig.5 shows the routing control overhead for maodv and lacamm with respect to varying cbr packet rates. the figure reveals that proposed lacamm has less routing control overhead that of maodv. this is because lacamm uses suppressed flooding concept during route discovery process with the help of rfn list and redistributes the traffic in case of congestion or route failure using an alternate congestion free path along the primary path. the re-route discovery takes place only if no alternate congestion free paths exists along the primary path and thus make the lacamm superior to the modav. fig. 3 attained end-to-end delay with respect to various cbr packet rates 70 h. santhi, n. jaisankar: load-aware congestion adaptive multipath multicasting fig. 4 attained packet delivery ratio with respect to various cbr packet rates fig. 5 attained routing control overhead with respect to various cbr packet rates 4.2.2 impact of lacamm with edaodv fig.6 to fig.7 shows the results obtained for the end-to-end delay, packet delivery ratio and routing control overhead of the proposed lacamm with edaodv. fig.6 shows the attained end-to-end delay for lacamm and edaodv with respect to varying the cbr packet rates. both protocols attain somewhat same end-to-end delay when the data rate is between 5 packets/s to 15 packets/s but at higher data rates form 25 packets/s to 55 packets/s the proposed lacamm attain lower delay that of edaodv. this is because the availability of alternate congestion-free routes at each node along the primary path. fig.7 show the attain advances in systems science and application(2016) vol.16 no.3 71 packet delivery ratio for lacamm and edaodv with regard to the packet rates. the result reveals that for lower data rates 5 packets/s to 15 packets/s both protocols attain somewhat same packet delivery ratio but for higher data rates from 25 packets/s to 55 packets/s lacamm improves the packet delivery ratio. fig.8 shows the attained routing control overhead with respect to varying packet rates. from the figure, it is clearly understood that the proposed lacamm has lower routing control overhead that of edaovd for all packet rates. thus, these results reveal that the proposed lacamm achieves better result when compared with other two protocols madov and edaodv improving network performance. fig. 6 attained end-to-end delay with respect to various cbr packet rates fig. 7 attained packet delivery ratio with respect to various cbr packet rates 72 h. santhi, n. jaisankar: load-aware congestion adaptive multipath multicasting fig. 8 attained routing control overhead with respect to various cbr packet rates 5 conclusion congestion control techniques have been specifically made for multimedia applications in manets. a suitable mechanism needs to be implemented such that network characteristics like congestion and route failure need to be found out and an apt solution needs to be supplied. the proposed approach could analyze the fluctuation in traffic and categorize the congestion status accurately to solve the congestion problem which is robust and dynamic in nature to estimate congestion. the lacamm controls the congestion by 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(2006), “congestion adaptive routing in mobile ad hoc networks”, ieee transactions on parallel and distributed systems, vol. 17, no. 11, pp. 1294-1305. advances in systems science and application(2016) vol.16 no.3 75 corresponding author n. jaisankar can be contacted at: hsanthi@vit.ac.in adv syst sci appl 2019; 2:80–89 http://ijassa.ipu.ru/index.php/ijassa/article/view/707 aodv-based key management in vanet chaima bensaid1∗, sofiane boukli-hacene 1 1eedis laboratory , computer science department, djillali liabes university at sidi bel abbes , sidi bel abbes , algeria e-mail: chaimaa184@hotmail.fr , boukli@gmail.com received february 23, 2019; revised june 21, 2019; published july 10, 2019 abstract: vehicular ad hoc network (vanet) is a self-organized multi hop system comprised by multiple vehicles. this kind of network offers different kind of communication such as vehicle to vehicle (v2v) and vehicle to infrastructure (v2i) communication. the v2v communication a folk of mobile ad hoc network(manet). it is characterized by a high mobility, dynamic topology, lack of centralized control infrastructure and a non secured shared media. similar to manet, v2v is vulnerable to a many types of attacks such as blackhole, sybil and denial of service attack. security in such cases is primordial. many studies have addressed this issue and many solutions have been proposed. one of the most promising studies used distribution certification authority (dca) to secure communication within vanets networks. in this paper, we propose a new method of certification based on the study of the behavior of each node based on aodv routing protocol. the operational performance is evaluated with rigorous analysis and extensive simulation study. our approach reduces certificate with 6.01 to 69.42% less certificates. keywords: vanet,aodv, clustering , trust based mechanism. 1. introduction vanet is a network in which a mobile node is a smart vehicle equipped with a sensors. it offers three types of communication: vehicles to vehicles (v2v) and vehicles to road side unit infrastructure (v2i) and hybrid communication. many researches state that v2v vanet communication are vulnerable to a lot of attacks because the dynamic topology and the use of wireless links. to secure these communications a security mechanism is essential. a public key infrastructure assists that the users obtain the necessary public keys, these public keys are used to perform cryptographic operations. the ca (certificate authority) issues the digital certificates and provides the means operations to verify the validity of the certificates. however, this solution is costly and not suitable for the different requirements of the vanets. lightweight solutions have been also proposed such as dca and mobile certificate authority (moca). cluster based ca have been proposed for cluster based routing protocols (cluster based routing protocol). these solution suffer from cluster head (ch) overload because these nodes issue new certificates when a mobile cluster number moves between clusters. in our paper, we propose a new approach to adopt the pki in vanet based on the well known routing protocol aodv with a trusted system when an extension is used to predict node migration to adjacent clusters. the trusted system proposed is lightwight and it is based only on neighboring node behavior and the prediction process predict the destination cluster ∗corresponding author: chaimaa184@hotmail.fr aodv-based key management in vanet 81 and anticipate certificate issuing which minimize the number of issued certificate and allow certification renewal with in all clusters. our improvement gives a satisfactory result, where the number of issued certificates has declined by about 69%. 2. aodv routing protocol the aodv protocol is a reactive routing protocol [1, 2]. it operates using three types of messages: request messages (rreq) and route reply messages (rrep) and route error messages (rerr). when a node wants to send a data packet to a destination node, it looks in its routing table if a fresh route exists to the destination node. if there exist already a valid path it is use to route packets, else, it launches a route discovery process by broadcasting a rreq. each intermediate nodes check if it is the destination node or it has a fresh route to the destination node. if so, it sends a rrep to the source node. upon receiving the rrep, the source node uses the discovered route to exchange packets with the destination [3]. these route are not secured, for that many studies have focused on security in vanet. in the next section, we’re going to discuss security mechanism in vanet. fig. 2.1. rreq and rrep packets in aodv 3. key distribution in this section, we present key distribution mechanism present in literature. zhou and hass [4] proposed a partially distributed ca where the key management service (k, n) consists of n servers. each node has a public/private key, the private key is divided into n shares, one share for each server,and the public key is known to all nodes in the network. each ca generates a portion of the certificate using its private key and sends it to the combiner. with k correct partial signatures the combiner can construct the complete certificate for member node. it is always possible for a combining node to be compromised by an adversary or be unavailable . authors have not paid too much attention to the certificate revocation; they proposed a simple approach which is a certification revocation list (crl). yi and kravet [5] proposed a distributed certificate authority based on threshold cryptography where the signature scheme does not require a combiner c. the distribution certification authority is called mobile certificate authority (moca). all nodes are equipped with mp (moca certification protocol), the combination of different parts of the signatures is done by each node. this approach is unsuited for vanet because all certificates must be known by the dca servers certificates must be known by the dca servers before providing any access to certification . a fully distributed certification service based on clustering have proposed by kong and al [6] . the establishment of certificates in the network is provided by the nodes themselves, copyright c© 2019 assa. adv syst sci appl (2019) 82 c. bensaid, s. boukli-hacene which will be stored by a particular node called cmn (certificate management node) for each cluster. all nodes must request the cmn of the same cluster to collect the certificate strings for each authentication. this system converges to the centralized models where the certificates are stored by a set of special nodes, which puts into question the availability of the certification service especially when cmn became unreachable. in [7], a based cluster certification authority architecture is proposed. each cluster-head (ch) has a ca information table, which contains a list of ca nodes. when a member node requires a certificate, it sends a request to its cluster-head ch to get information about the ca servers. the ch collects information, and forwards to the member node. in this time the node sends the certification request to them. this approach generates lower certificate maintenance overhead by resolving certificate transaction problems. chaining is done only with trusted nodes that are cluster head and gateways. mukherjee et al in [8] proposed an extention of aodv protocol by adding a new trust mechanism and considering the successful cooperation frequency and average encounter rate as factors to compute for direct trust. modified dempster-shafer theory is used to build the recommended trust based on multiple pieces of trust evidence. the trust model is composed of two phases : route discovery and trusted route selection which selects the most reliable next hop for routing discard nodes with high mobility and high drop packets. ahmed et al. [9] proposed a modified route discovery algorithm to efficiently and securely route data to its destination. this flooding algorithm is used to define the link failure probability of misbehaving nodes and normal nodes and an enhanced multi-swarm optimization is used to optimize the discovered route. 4. proposed approach we propose a cluster based trust model (cbtm) to secure data exchanges in vanets networks. to degrade network efficiency in a vanet network, a malicious node sends a great number of rrep to intercept the data packets of its neighbors or to overload the network. for that it uses a very high sequence number in the rrep packet to attract the data packets of its neighbors to go through him to edit or delete them later. in our model, each node in the network is equipped by a cache memory, in which it saves the number of data packets, route request packets (rreq) and the number of route reply packets (rreps) received from this node and the sequence number. each node will update its cache memory with the following functions : • recv rreq () cache [0] [@src] ++; • recv data () cache [1] [@src] ++; • recv reply () cache [2] [@src] ++; copyright c© 2019 assa. adv syst sci appl (2019) aodv-based key management in vanet 83 n d: the number of data packets received from a node; n request: the number of rreq packets received from a node; n reply: the number of rrep packets received from a node; if n d +n request == 0 and n seq dest >>>>> n seq src then trusted value=0; else trusted value=1; end if n d +n request ! = 0 and n rply < n d +n request then trusted value=1; end if n d +n request == 0 and n reply ! = 0 then trusted value=0; end algorithm 1: computing trust value if the received packet is a rrep, it looks in its cache memory to check the stored values. it will use the following algorithm to decide whether the node is a trusted node or not. 4.1. clustering to divide the networks into clusters each node uses its neighbors table. in our implementation, the size of each cluster is empirically fixed to 5, and within each cluster there is a single ch which is the node that has the highest trust value and the higher sequence number in the cluster. in addition, only a trust node can be a cluster member. 4.2. certification hahn and al [10] proposed a model for manet, where the cluster-head acts as a ca. the certificate chain allows the exchange of session keys and the encryption / decryption of data. however, any node can be selected as a ca. due to mobility which is a important feature of the v2v communication, a member node request a new certificate each time it passes from one cluster to another which will overload the number the certificate generating. our approach is a trust model based on the study of the total behavior of the member node and if the member node passes of cluster to another, the certificate is broadcast by the gateways. in this paper, we develop this proposal with detailed simulations study by the well known network simulator ns2.35 [11]. when a node enters in the preemptive region, three signal values are collected and we used the lagrange interpolation to predict node mobility. the formula of the interpolation is: y = n∑ i=0 [ ∏n j=0,j 6=i (x− xj)∏n j=0,j 6=i (xi − xj) × yi ] (4.1) we store three signals values of received data packets and their corresponding receiving time. when two consecutive measurements give the same signal, we store the time of the second occurrence. the expected signal strength p of the packets received from the ch node is computed as follows [12, 13]: p = ( (t− t1)× (t− t2) (t0 − t1)× (t0 − t2) × p0 ) + ( (t− t0)× (t− t2) (t1 − t0)× (t1 − t2) × p1 ) + ( (t− t0)× (t− t1) (t2 − t0)× (t2 − t1) × p2 ) (4.2) where p0 , p1 , p2 are the measured power strengths at the times t0 , t1 , and t2 respectively. the time t is the sum of time required to send the certificate to cluster adjacent copyright c© 2019 assa. adv syst sci appl (2019) 84 c. bensaid, s. boukli-hacene (inonde period) and the difference between t2 and the average value of the measurement; this value has been determined empirically [14]. t = 2× t2 − ( t0 + t1 + t2 3 ) + inonde period (4.3) when p is less than the minimum acceptable power (81dbm) a warning message is sent to the ch. the ch sends the certificate to adjacent cluster [12]. when the join the adjacent cluster, the cluster-head compares the nodes address with the received one and saves its certificate. fig. 4.2. certficate generation 5. performance evaluation to evaluate the performance we used ns2 simulator . in our approach we used two mobility scenarios, the first is of the city of malaga and the second is generated by the simulator vanetmobisim . 5.1. malaga city scenarios fig.5.3 present a geographic card of urban vanet scenarios from the downtown of malaga, spain [15, 16] . it is composed of three areas u1 , u2 and u3. detailed parameters of the simulation area is presented in the table 5.1. copyright c© 2019 assa. adv syst sci appl (2019) aodv-based key management in vanet 85 table 5.1. vanet scenarios details scenario area size number of vehicules number of connections u1 120000m2 60 10 15 u2 240000m2 60 20 u3 360000m2 60 30 40 fig. 5.3. malaga urban areas in this scenario 60 vehicles are simulated. each of them use aodv routing protocol. a cbr application over udp is used to simulate data transmission between pairs of communicating nodes where packet of 1 kb are sent using a rate of 100 kpbs during 180s. table 5.2 summarize all used parameters. table 5.2. simulation paramaters parameters value propagation model nakagami phy layer ieee 802.11p mac layer ieee 802.11p routing layer aodv transport layer udp cbr packet size 1024 bytes cbr packet rate 100kbps simulation time 180 s 5.2. vanetmobisim scenarios the vehicular ad hoc networks mobility simulator (vanetmobisim) [17] is a set of extensions to user mobility modeling framework canumobisim, used by the canu (communication in adhoc networks for ubiquitous computing) research group, university of stuttgart. it includes a visualization module, mobility models, as well as various formats parsers for geographic data sources. this framework is easily extendable and it is based on the concept of pluggable modules. the set of extensions provided by vanetmobisim consists mainly on a vehicular spatial model using gdf-compliant data structures, and a set of vehicular-oriented mobility models. copyright c© 2019 assa. adv syst sci appl (2019) 86 c. bensaid, s. boukli-hacene table 5.3 details all parameters used in this scenario where protocol is under attack. table 5.3. simulation paramaters parameters value propagation model nakagami phy layer ieee 802.11p mac layer ieee 802.11p area size 1000*1000 m vehicule speed from 8.33 to 13.89 m/s cbr packet size 1024 bytes cbr packet rate 100kbps simulation time 900 s number of malicious nodes 3 the well-known metrics to evaluate routing protocol is used in our study : • packet delivery ratio (pdr): represents the percentage of packets delivered to their destinations . • the average latency of data packets (delay): this is the average time required to deliver data packets from the source to the destination successfully. • additive costs (overhead): this criterion illustrates the amount of additives cost required for each received data packet. • dropped packet: number of dropped packets due to either link failure or by the malicious node. • certificate overhead: the number of certificates sent in network. simulation scenarios are : • aodv vanetmobisim denotes aodv under attack. • cbtm vanetmobisim represent our approach under attack used vatenmobisim scenario. • aodv malaga denotes aodv under attack. • cbtm malaga represent our approach under attack used malaga scenario. fig. 5.4. certificate overhead fig.5.4 shows the certificate overhead. we observe that the overhead increase high because many nodes request a new certificate from ca and from adjacent ca when a node moves. however, we depict that, our approach outperforms the original with 6.01% to 44.88% less certificates by malaga mobility model and 40.40% to 69.42% less certificates by the model generated by vanetmobisim. copyright c© 2019 assa. adv syst sci appl (2019) aodv-based key management in vanet 87 fig. 5.5. packet delivery fraction the fig. 5.5 shows the decreasing evolution of pdf in protocol aodv under attack against to our approach with two mobility models. when the number of vehicles is high, the pdf of our approach register a small degradation. this is due to frequent network topology changes with a high number of connections. we also observe that if the number of vehicles increases the pdf increases for our solution. in the aodv protocol under attack the pdf reach 19.98% . fig. 5.6. dropped packet the fig. 5.6 presents the number of dropped packets depending on the number of vehicles. in our proposal, with 10 sources of connection, the number of lost packets is 69 and almost no significant against to the aodv under attack 9971 packets in malaga mobility model. whereas the number of connections increases, it means that there is a lot of data packet sent. in this case, the malicious node intercepts a large quantity of packet so the rate of dropped packet, but in our proposal is a little minimal. fig. 5.7. normalized routing load copyright c© 2019 assa. adv syst sci appl (2019) 88 c. bensaid, s. boukli-hacene the simulation results in fig. 5.7 show that our proposal is a small routing head relative to aodv under attack. this is due to the fact that when the packet is not received in the right destination, the source node attempts to repair the paths with rrer packets. fig. 5.8. communication delay the fig. 5.8 shows the evolution of the average end to end delay, depending on number of vehicles in our approach and the aodv under attacks with two mobility models. it is found that the time required by our proposal is higher than the aodv under attack. this can be justified by using an extra process in our solution to create the certificates and update him. 6. conclusion in this paper, we presented the different approaches proposed based on pki in vanet. we also presented a new proposal based on trusted model. in our proposal, we interested to v2v communication system where the cluster-head node acts as a virtual ca the main idea is to compute a confidence level for each node by traking its behavior and avoiding the issue of new certificate request in case a node moves. our approach reduces certificate with 6.01% to 44.88% less certificates by malaga mobility model and 40.40% to 69.42% less certificates by the model generated by vanetmobisim. references 1. gupta, p., goel, p., varshney, p., & tyagi, n. (2019). reliability factor based aodv protocol: prevention of black hole attack in manet. in smart innovations in communication and computational sciences (pp. 271-279). springer, singapore. 2. kandali, k., & bennis, h. (2018, july). performance assessment of aodv, dsr and dsdv in an urban vanet scenario. in international conference on advanced intelligent systems for sustainable development (pp. 98-109). springer, cham. 3. perkins, c. e., belding-royer, e. m., & das, s. r. (2002). mobile ad hoc networking working group-internet draft. 4. zhou, l., & haas, z. j. (1999). securing ad hoc networks. ieee network, 13(6), 24-30. 5. yi, s., & kravets, r. (2004). moca: mobile certificate authority for wireless ad hoc networks. 6. kong, j., zerfos, p., luo, h., lu, s., & zhang, l. (2001, november). providing robust and ubiquitous security support for mobile ad hoc networks. in icnp (vol. 1, pp. 251-260). copyright c© 2019 assa. adv syst sci appl (2019) aodv-based key management in vanet 89 7. dong, y., sui, a. f., yiu, s. m., li, v. o., & hui, l. c. (2007). providing distributed certificate authority service in cluster-based mobile ad hoc networks. computer communications, 30(11-12), 2442-2452. 8. mukherjee, s., chattopadhyay, m., chattopadhyay, s., & kar, p. (2018). eaer-aodv: enhanced trust model based on average encounter rate for secure routing in manet. in advanced computing and systems for security (pp. 135-151). springer, singapore. 9. ahmed, m. n., abdullah, a. h., chizari, h., & kaiwartya, o. (2017). f3tm: flooding factor based trust management framework for secure data transmission in manets. journal of king saud university-computer and information sciences, 29(3), 269-280. 10. g. hahn, t. kwon, s. kim, & j. song, ”cluster-based certificate chain for mobile ad hoc networks,” in international conference on computational science and applications (iccsa) , pp. 769-778, 2006. 11. issariyakul, t., & hossain, e. (2011). introduction to network simulator ns2. berlin: springer 12. boukli-hacene, s. , lehireche, a., & meddahi, a. (2006). predictive preemptive ad hoc on-demand distance vector routing. malaysian journal of computer science, 19(2), 189-195. 13. laroui, m., sellami, a., nour, b., moungla, h., afifi, h., & boukli hacene, s. (2018, december). driving path stability in vanets. in 2018 ieee global communications conference (globecom) (pp. 1-6). ieee. 14. boukli-hacene, s., ouali, a., & bassou, a. (2014). predictive preemptive certificate transfer in cluster-based certificate chain. international journal of communication networks and information security, 6(1), 44. 15. toutouh, j., & alba, e. (2011, july). an efficient routing protocol for green communications in vehicular ad-hoc networks. in proceedings of the 13th annual conference companion on genetic and evolutionary computation (pp. 719-726). acm. 16. malaga city downtown scenario . http://neo.lcc.uma.es/staff/jamal/vanet/?q=node/11 17. vanetmobisim manuel, (2006) institut eurcom/politecnico di torino copyright c© 2019 assa. adv syst sci appl (2019) adv syst sci appl 2019; 04; 45-57 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/748 evaluating comparative total factor productivity in the russian industry svetlana v. orekhova1*, evgeniy v. kislitsyn2 1) business economics dept., ural state university of economics, yekaterinburg, russia e-mail: bentarask@list.ru 2) information technology and statistics dept., ural state university of economics, yekaterinburg, russia e-mail: kev@usue.ru received may 14, 2019; revised november 28, 2019; published december 31, 2019 abstract: searching for domestic reserves of economic growth has lately become one of the central problems in russia. the paper examines the role of small industrial enterprises in stimulating economic growth. in contrast to the stereotype that small business serves as a driving force for the economic development and encourages innovation, the authors hypothesizes that in russia this statement is false. the conceptual and methodological framework of the study rests upon neoclassical models of economic growth. the authors investigate the existing approaches to the analysis of factors influencing economic growth of the state and choose the tools of total factor productivity analysis. total factor productivity is calculated using the translog production function, which allows determining the effect of the technological level on value added of the object under study. the choice of the type’s function is due to the low elasticity between the factors of production, as well as the imperfect competition in the industrial markets under review. the information base of the research includes the data of small, medium-sized and large enterprises of 10 industrial macro-sectors of the russian economy for 2013–2017. we use spark-interfax database to assess production functions. the results of the analysis prove that small enterprises operating in the russian industry demonstrate much lower values of average and weighted average total factor productivity than medium-sized and large enterprises. the general trend for such businesses is a decline in total factor productivity. only single leading companies produce a gain in value added in small entrepreneurship. thus, the economic situation in russia rejects the hypothesis about a higher entrepreneurial potential of small businesses, business models and technological innovations emerging on its basis. our further studies are assessing institutional traps and general context in the development of small business. keywords: total factor productivity, small enterprises, russian industry, economic development. 1. introduction harrod [20] and domar [12], who described a one-factor model for determining growth rates, provided fundamentals of the theory of economic growth. since the paradigmatic works by solow [30; 31] and swan [33], who proposed endogenous growth model, contemporary economic thought has been developing towards substantiating sources and refining growth models. the basic exogenous growth factors include scientific and technological progress and labor [27]. arrow [4] и uzawa [35] examine the role of knowledge and human capital in economic growth. the mechanisms of innovation growth are clarified in the schumpeterian growth model [1; 18] for russian economic development strategy, the key issue is the effective economic structure (and also institutional environment). this aspect “solves the question of the number of types of activities, markets, the size of the general welfare and its distribution (income * corresponding author: bentarask@list.ru mailto:kev@usue.ru 46 s.v. orekhova, e.v. kislitsyn copyright ©2019 assa adv. in systems science and appl. (2019) inequality)” [32, p.26]. equally important is the question of the contribution of different types of enterprises to economic growth. the history and practices of small business have been among the most controversial issues in economic development in different countries. science postulate that small business is an important object of economic growth [15; 17; 38; 39]. job creation, production of innovations and technologies, and finally, profits are the outcome of small businesses activity due to their entrepreneurial potential, flexibility and maneuverability of managerial decisions, optimization and red tape reduction of business processes. small businesses make a huge contribution to national prosperity. on the other hand, empirical studies demonstrate that the size of enterprises can influence their growth rate differently (the positive correlation is highlighted in [6; 13; 37] and the negative one is discussed in [3; 10; 15]). the role of small business in an economy has frequently been undermined and even misinterpreted. in the past, small businesses were believed to impede economic growth by attracting scarce resources from their larger counterparts [5]. these inconsistencies between theory and practice encourage more detailed studies to be conducted on the role of small enterprises in promoting economic development of the russian industry, and later – on the factors determining this role. the purpose of the research is to analyze total factor productivity (tfp) of small enterprises of russia’s industrial sector in comparison with medium-sized and large companies. such a research agenda is set for the first time and implies a consistent solution of a number of tasks. the first task is to perform a critical analysis of the existing theoretical approaches and russian empirical studies on measuring economic growth rates taking into account various factors. the second task is to substantiate the choice of a method for calculating tfp and its empirical testing. it is reasonable to carry out a comparative analysis of data on small and other enterprises in the context of individual industries, which allows taking into account the organizational and production specifics of enterprises. the authors state that industrial segments, serving as the basis for the rf government’s economic growth policy, are of the greatest academic interest. at the same time, regional, technologic and economic disproportions of industrial markets are very serios [2]. the third task of the research is to interpret the obtained results of calculating the total factor productivity from the standpoint of small industrial enterprises’ development in the russian economy. 2. theoretical approaches and empirical studies on factor productivity neoclassical economics theory showed that the level of technological development was the primary indicator of the degree to which society masters the forces of nature. the combination of productive forces and the type of production relationships constitutes a unique mode of production. neo-institutional economics largely associates the rate of technological growth with institutional environment for business development. [11; 28] total factor productivity is the most integrated indicator of technological progress and growing economic efficiency. it is calculated as the ratio of output to the volume of production factors used [23]. the logic of tfp calculation is as follows: an enterprise uses a certain set of factors of production (labour and capital) to produce the final product. then the growth of the factors of production used or a change in the combination of their use (technology) causes an increase in output of the final product. the dynamics of factor productivity may indicate the degree to which the growth of an enterprise, industry or economic sector is sustainable. measuring tfp means discovering the correlation between output (production quantity) and factors of production, such as resource evaluating comparative total factor productivity in the russian industry 47 copyright ©2019 assa. adv. in systems science and appl. (2019) costs and the level of technology. such a quantitative dependence was dubbed “production function”. there are numerous types of production functions and the most famous of them are the models of cobb–douglas, leontiev, ces, etc. the choice of these models is determined by a combination of factors that embrace data access, market specificity, a dynamic or static aspect of research. methods for assessing tfp are widely used throughout the world, including as a way of comparing the economic state of countries [14; 16; 21; 25] and the reasons for their differentiation [19; 26; 29]. in russia, the evaluation of the effectiveness of economic subjects are conducted on a regular basis using certain types of production functions. this fact is due to a number of reasons. first, the production function is an adequate tool for measuring the influence of various factors on economic growth. second, this tool allows analyzing intra-industry, crossindustry and inter-regional differentiation. third, in the context of uncertainty and volatility of incomes and expenditures of enterprises, such an assessment allows performing a real-time monitoring of the reasons behind the rise or decline in economic development and determining its trend. for example, bessonov [7] show that industries with relatively safe output dynamics and lacking sufficient incentives to increase productivity demonstrated the worst tfp dynamics in russia in 1989–2002. the tornqvist index was also calculated by orekhova [24] to analyze the growth sustainability in metallurgy. tornqvist index formula allows characterizing the change in the efficiency of resource use by factors. the tornqvist index deals with the change in the volume of two resources – labor (data on the average number of employees) and capital (data on the nominal value of fixed assets). the results of the analysis illustrate the overall technological and technical underdevelopment of the industry. timmer and voskoboynikov [34] prove that the growing role of capital expenditures in the growth of value added in russia resulted in the use of an extensive model of economic growth, which is typical of rental economies. bessonova [9] demonstrates that there is a significant gap in the total factor productivity of enterprises within the industry. such a gap may arise due to both organizational-efficiency-based reasons and resource or institutional constraints of the russian market. all evaluations of the russian industry’s effectiveness illustrate spasmodic and chaotic shifts, but also display a declining total factor productivity. at the same time, such assessments are more integrated and not explaining the behavior of individual firms, their contribution to the total efficiency of the industry or the entire economy. on the basis of the recent research [10; 39], we assume that small entrepreneurship, a priori not so rich in resources and power in comparison with large business, will be less efficient. 3. tfp research method when it comes to the method for calculating total factor productivity, our study rests upon a series of works by bessonova [8; 9]. total factor productivity is value added of the final product minus changes in labour and capital costs. empirically tfp growth can be calculated as an unexplained residue of the final product’s growth. this residue encompasses the effects from technological or organizational innovations that determine technological progress in the industry and influence the shift of the production function [9, p. 99]. however, such a calculation method is possible to be applied only on the basis of correct data on the share of costs of production factors by industry. the previous approach to the evaluation of tfp within the framework of the solow growth model [30] makes an assumption about competitive factors of production. but nowadays we knows that industrial markets are characterized by a hybrid form of organization, which obliges a researcher to first evaluate their production functions, and only after that calculate indicators of production factors and growth of tfp. 48 s.v. orekhova, e.v. kislitsyn copyright ©2019 assa adv. in systems science and appl. (2019) this fact rationalizes the choice of the translog production function for calculating tfp. the function allows one not to make prerequisites about absolute elasticity of substitution between factors of production and perfect competition in the markets of these factors [22]: ln 𝑌𝑖𝑡 = 𝛼0 + 𝛼𝐿 ln 𝐿𝑖𝑡 + 𝛼𝐾 ln 𝐾𝑖𝑡 + 𝛼𝑡𝑡 + 𝛼𝐿𝐿(ln 𝐿)2 + 𝛼𝐾𝐾(ln 𝐾)2 + 𝛼𝑡𝑡𝑡2 + 𝛼𝐿𝐾 ln 𝐿𝑖𝑡 ln 𝐾𝑖𝑡 + 𝛼𝐿𝑡 ln 𝐿𝑖𝑡𝑡 +𝛼𝐾𝑡 ln 𝐾𝑖𝑡𝑡 + 𝜀𝑖𝑡 (3.1) where yit denotes value added at the enterprise i for the period t; kit is fixed assets of the enterprise i for the period t; lit is wage at the enterprise i for the period t; t denotes a time factor ranging from 1 to n (where n is the number of observed periods). based on the proposed production function, the growth of tfp is calculated by formula: 𝐴𝑖,𝑡 = ln ( 𝑌𝑖,𝑡 𝑌𝑖,𝑡−1 ) − �̅�𝐿 ln ( 𝐿𝑖,𝑡 𝐿𝑖,𝑡−1 ) − �̅�𝐾 ln ( 𝐾𝑖,𝑡 𝐾𝑖,𝑡−1 ) (3.2) where a denotes a total factor productivity; is average elasticity of value added of labour; is average elasticity of value added of capital. average elasticity is calculated as average value of the elasticities of the added value of labour and capital for the periods (t – 1) and t, which in turn are measured as a partial derivative of the corresponding factor: �̅�𝐿,𝑡 = 𝜕 ln 𝑌𝑖𝑡 𝜕 ln 𝐿𝑖𝑡 = �̂�𝐿 + 2�̂�𝐿𝐿 ln 𝐿𝑖𝑡 + �̂�𝐿𝐾 ln 𝐾𝑖𝑡 + �̂�𝐿𝑡𝑡, (3.3) �̅�𝐾,𝑡 = 𝜕 ln 𝑌𝑖𝑡 𝜕 ln 𝐾𝑖𝑡 = �̂�𝐾 + 2�̂�𝐾𝐾 ln 𝐾𝑖𝑡 + �̂�𝐿𝐾 ln 𝐿𝑖𝑡 + �̂�𝐾𝑡𝑡, (3.4) to estimate value added and factors of labour and capital, it is required to use the following indicators: fixed assets, the revenue volume, total costs and wage. value added was calculated using formula: yit = volit-(tcit-wageit) (3.5) where volit is the revenue volume of the enterprise i for the period t; tcit is total costs of the enterprise i for the period t; wageit is labour cost borne by the enterprise i for the period t. the amount of capital was estimated as the average annual value of fixed assets, and labour – as the enterprise’s costs incurred in labour remuneration. 4. evaluating tfp of industrial enterprises in russia: small vs. large enterprises the study aims at primarily identifying the contribution of small enterprises to the economic growth of the russian industry. in accordance with the federal law no. 209-fz “on the development of small and medium-sized entrepreneurship in the russian federation” of july 24, 2007, a small enterprise is that which employs no more than 100 people and its revenue is below 800 million rubles. the empirical testing of the proposed method for calculating tfp was divided into several stages (fig. 4.1). evaluating comparative total factor productivity in the russian industry 49 copyright ©2019 assa. adv. in systems science and appl. (2019) stage 1. description of the research object stage 2. constructing production functions stage 3. evaluating total factor productivity stage 4. analysis and interpretation of results choosing the industries for investigation descriptive statistics data upload calculating value added forming panel data estimating production function s coefficients checking coefficients for reliability, unbiasedness and consistency calculating elasticities of value added of labour and capital for each period calculating total factor productivity for each period calculating average tfp of industry for each period calculating tfp of small enterprises for each period calculating average tfp of small enterprises in the industry for each period fig. 1. algorithm for comparative evaluation of tfp of small and large businesses hence, the authors use spark-interfax database to assess production functions; the period under study was from 2013 to 2017. the research object was industrial sectors of the russian federation. the entire set of industries is combined into 10 industrial sectors (tab. 4.1). this grouping is based on similar industry characteristics within groups. the group “small enterprises” embraces all the companies meeting the aforementioned criteria (micro-enterprises included). since tfp is an average of the indicators’ values for two years, tfp was calculated for 4 periods. table 4.1. number of enterprises in the study sample industry 2014 2015 2016 2017 enterprises large and medium -sized smal l and micr o large and medium -sized smal l and micr o large and medium -sized smal l and micr o large and medium -sized small and micro 50 s.v. orekhova, e.v. kislitsyn copyright ©2019 assa adv. in systems science and appl. (2019) mining 501 432 537 518 502 608 510 666 food industry 886 995 933 1223 1012 1489 1064 1817 light industry 77 317 83 373 87 428 94 497 woodworking , pulp and paper 240 464 242 574 260 664 264 817 petrochemical industry 525 570 569 742 591 913 697 1033 mineral industry 266 618 288 736 308 840 306 998 metallurgy 365 475 395 633 384 798 391 993 mechanical engineering 693 912 725 1099 749 1304 790 1622 industrial service 79 321 102 435 100 537 99 662 energy industry 631 2167 360 1927 850 1696 1296 1502 total 4263 7271 4234 8260 4843 9277 5511 1060 7 at stage 2, production functions were constructed for each enterprise belonging to the 10 industrial sectors. the data were retrieved from spark-interfax database and included the following details: company size, fixed assets, revenue, total costs and wages for 2013–2017. calculation of value added for each enterprise was carried out according to formula (3.5). to construct the production function, the authors used panel data, which contain statistical information about the same number of enterprises for several consecutive periods of time (2013–2017). the use of panel data allowed us to enhance the size of the sample under review, which provided greater efficiency in estimating the regression model’s parameters. to construct the production function, we applied the method of least squares. the obtained results were checked for reliability, unbiasedness and consistency through calculating the expected value, the durbin–watson statistic, constructing the correlation matrix and holding the white test. we obtained a total of 10 production functions for various groups of industries. it is worth noting that for different groups of industries the indicators’ coefficients vary significantly. nevertheless, we can notice that in all production functions, coefficients with cross-section indicators have a negative value, and in more than half of them – the coefficient with a logarithm of the labor factor. all the rest coefficients have a positive value. for example, for mechanical engineering enterprises, the production function is as follows: ln(𝑌) = 9.65 − 0,10 ln(𝐿) + 0,17 ln(𝐾) + 0.04(ln(𝐿))2 + 0.01(ln(𝐾))2 + 0.01𝑡2 − 0.03 ln(𝐿) ln(𝐾) − 0.01 ln(𝐿) 𝑡 + 0.01 ln(𝐾) 𝑡 (4.1) once the production functions are constructed, we proceed to stage 3: calculating total factor productivity. based on the coefficients of production functions, we calculate the elasticity of value added of labour and capital for each period through identifying partial derivatives (formulas (3.2) – (3.3)). the result is presented in table 4.2. evaluating comparative total factor productivity in the russian industry 51 copyright ©2019 assa. adv. in systems science and appl. (2019) table 4.2. elasticity of value added of labour and capital for industrial sectors for 2013–2017 year / industry m in in g f o o d in d u st ry l ig h t in d u st ry w o o d w o rk i n g , p u lp an d p ap er p et ro ch em i ca l in d u st ry m in er al in d u st ry m et al lu rg y m ec h an ic al en g in ee ri n g in d u st ri al se rv ic e e n er g y in d u st ry 2013 0,605 0,916 0,848 0,846 0,912 0,889 0,854 0,884 0,844 1,04 0,222 0,134 0,132 0,143 0,1 0,112 0,099 0,036 0,03 0,008 2014 0,51 0,902 0,84 0,83 0,899 0,863 0,829 0,869 0,819 1,015 0,31 0,135 0,128 0,151 0,099 0,111 0,1 0,041 0,041 0,019 2015 0,519 0,906 0,84 0,838 0,892 0,854 0,809 0,855 0,808 1,003 0,297 0,131 0,12 0,147 0,097 0,104 0,101 0,045 0,047 0,025 2016 0,522 0,904 0,852 0,839 0,89 0,836 0,798 0,845 0,794 0,998 0,294 0,13 0,108 0,148 0,095 0,099 0,101 0,048 0,053 0,028 2017 0,523 0,886 0,794 0,798 0,852 0,787 0,742 0,792 0,702 0,993 0,29 0,118 0,13 0,153 0,092 0,078 0,103 0,063 0,093 0,027 over the entire period, the greatest elasticity of value added of capital was typical of the industrial sectors, such as energy, food and petrochemical industries. for the mining industry, on the contrary, this indicator varies between 51–60%. most industries exhibit the same level of the elasticity of labor value added – 9–15%. however, in 2013, this indicator for the mechanical engineering and industrial service was quite low; but by 2017, there was a positive trend, i.e. an increase of 6% and 9% respectively. the lowest level of labor elasticity was observed in the energy sector. taking into account the elasticity values, we used formula (3.1) to calculate a rise in tfp for the industrial sector as a whole, as well as for large/medium-sized enterprises and small enterprises taken separately. tfp growth rate was analyzed using the indicator of simple average tfp growth rate. the method developed by bessonova [9] also implies the calculation of another indicator, i.e. weighted average tfp growth rate by the volume of value added. however, within the scope of the present research, the calculation of this indicator does not make sense. one of the study’s tasks is to compare tfp growth rates for large and small industrial enterprises. at the same time, the volume of value added of large and medium-sized enterprises will be greater, which means a strong preponderance of such businesses and, as a result, a distortion of the tfp indicator. for this reason, we deliberately exclude the calculation of this indicator from our research. dynamics of average tfp growth rates for 2013–2017 illustrates that their values for large and medium-sized enterprises significantly exceeds similar values for small enterprises (fig. 4.2). 52 s.v. orekhova, e.v. kislitsyn copyright ©2019 assa adv. in systems science and appl. (2019) note: based on the authors’ calculations. fig. 4.2. average tfp growth rates by size of enterprises in comparison with 2013, large and medium-sized enterprises’ tfp growth rates in 2014 was 4.4%, whereas small companies demonstrated a 1.4% drop in growth. in the post-crisis period (after 2014), small industrial enterprises experienced an increase of up to 1.2%, but it was still significantly lower than that of other enterprises. let us look at the dynamics of changes in tfp for the 10 industries. figure 4.3 shows the values of tfp average growth in 2014. note: based on the authors’ calculations. fig. 4.3. tfp average growth in 2014 the period under consideration is characterized by both the decline in the global economic situation and the worsening national institutional environment in russia. this results in the fact that more than 50% of the industries experience a negative tfp growth rate, regardless of the size of enterprises. however, for the mining, food, woodworking, pulp and paper industries and metallurgy, tfp growth of large and medium-sized industrial enterprises significantly exceeds that of small businesses. it is also noteworthy that tfp decline rate of large enterprises 0,044 -0,014 0,069 -0,023 -0,030 -0,020 -0,010 0,000 0,010 0,020 0,030 0,040 0,050 0,060 0,070 2017 2016 2015 2014 -8,0% -6,0% -4,0% -2,0% 0,0% 2,0% 4,0% 6,0% 8,0% large and medium-sized enterprises small enterprises mining food industry light industry woodworking, pulp and paper petrochemical industry mineral industry metallurgy mechanical engineering industrial service energy industry evaluating comparative total factor productivity in the russian industry 53 copyright ©2019 assa. adv. in systems science and appl. (2019) operating in industrial service and petrochemical industry are much lower than that of small ones. in general, in 2014, small enterprises of two industries only had a positive dynamic of tfp growth. note: based on the authors’ calculations. fig. 4.4. tfp average growth in 2015 figure 4.4 demonstrates that, compared with the previous period, tfp growth rate in 2015 changed dramatically. for all the industries, with the exception of energy production, tfp growth rate of large and medium-sized enterprises significantly exceeded that of small enterprises. the biggest gap – from 6 to 14% – was characteristic of mining, food, light and petrochemical industries. in 2014–2015, the strongest tfp growth was achieved by small enterprises engaged in woodworking, pulp and paper industry – 3.6%, petrochemical industry – 3.5% and light industry – 3.2%. the same indicators for large and medium-sized businesses equaled 6.1%, 9.1% and 8.6% respectively. note: based on the authors’ calculations. fig. 4.5. tfp average growth in 2016 -8,0% -6,0% -4,0% -2,0% 0,0% 2,0% 4,0% 6,0% 8,0% large and medium-sized enterprises small enterprises mining food industry light industry woodworking, pulp and paper petrochemical industry mineral industry metallurgy mechanical engineering industrial service energy industry -8,0% -6,0% -4,0% -2,0% 0,0% 2,0% 4,0% 6,0% 8,0% large and medium-sized enterprises small enterprises mining food industry light industry woodworking, pulp and paper petrochemical industry mineral industry metallurgy mechanical engineering industrial service energy industry 54 s.v. orekhova, e.v. kislitsyn copyright ©2019 assa adv. in systems science and appl. (2019) in 2016, the absolute growth of average factor productivity of large and medium-sized enterprises was typical of only 6 out of 10 industries under study (fig. 4.5): mining, woodworking, pulp and paper, mineral industry, mechanical engineering, industrial service and energy industry. moreover, in all sectors, excluding industrial service, the growth exceeded that of small and micro-enterprises. other industries experienced a decline in tfp, but only in light industry this decline was greater for small enterprises. the greatest tfp growth among small enterprises was observed in mining (2.6%), energy industry (2.4%) and industrial service (1.1%). note: based on the authors’ calculations. fig. 4.6. tfp average growth in 2017 in 2017, only 50% industries encountered an increase in tfp of large and medium-sized enterprises, which dominated over the same indicator for small businesses. only small enterprises operating in metallurgy demonstrated a positive dynamic of tfp growth (0.9%), in contrast to total factor productivity of large businesses (–2.2%). the research results indicate that small industrial enterprises are characterized by low tfp growth rates. the general trend for such businesses is a decline in total factor productivity. in 2017, the decrease in tfp for small enterprises in the petrochemical industry was 11.3%; food industry – 4.7%; woodworking, pulp and paper industry – 2.2. however, in 2016–2017, small enterprises of some industrial sectors exhibited a slight gain in tfp (mining – 2.7%; engineering – 1.7%; industrial services – 1.3%; metallurgy – 0.9%). most industrial sectors experienced an increase in total factor productivity of large and medium-sized enterprises. a sharp fall in tfp growth rates in 2017 was recorded only for large enterprises engaged in food and woodworking industries. having compared the data, we found that small enterprises’ tfp growth exceeded that of the rest of the enterprises only in 2016–2017 in metallurgy (small businesses’ tfp growth was 0.4 and 0.9% respectively). in all other cases, even if tfp growth rate of small businesses in absolute value is greater than that of large and medium-sized enterprises, the value of this indicator is negative, that is, only if there is no tfp growth per se. 5. conclusion the period under consideration is not long enough to project further tfp dynamics. however, it is possible to draw a number of important conclusions. -8,0% -6,0% -4,0% -2,0% 0,0% 2,0% 4,0% 6,0% 8,0% large and medium-sized enterprises small enterprises mining food industry light industry woodworking, pulp and paper petrochemical industry mineral industry metallurgy mechanical engineering industrial service energy industry evaluating comparative total factor productivity in the russian industry 55 copyright ©2019 assa. adv. in systems science and appl. (2019) firstly, the hypothesis that large enterprises develop faster than small ones was confirmed. there are several possible explanations for this phenomenon and each of them has to be checked and verified in further studies. on the one hand, there are objective reasons behind poor efficiency of small businesses. one of them is a vague possibility to get increasing returns due to economies of scale. since we only examine the industrial sector, this factor can be of considerable importance. in addition, industry in russia is vertically or horizontally integrated structures, where auxiliary production is outsourced. thus, small industrial enterprises in russia are poorly performing divisions of large businesses. on the other hand, there are subjective factors associated with small industrial business in russia lagging behind in terms of technology and innovation. it means that the technological factor of tfp growth is more attributed to the development of technical equipment, rather than technological innovation. large businesses possess more advanced and productive types of fixed assets, which supports the conclusions about the rental, extensive growth of the russian industry. secondly, the industrial sectors under review are characterized by a significant companies differentiation, which leads to a slowdown in the economic growth of the entire industrial production, and consequently, the processes of re-innovation of the russian economy. the results obtained are confirmed by other studies on total factor productivity of the russian industry. however, the previous research made conclusions “about a large group of inefficient enterprises that continue functioning in the market ...” [8, p. 31], but failed to reveal what kind of enterprises they were and what features they had. we suppose that the current paper lays the foundation for the study of such enterprises’ behavior. according to our study, small enterprises are among those companies forming the inefficient segment of the russian economy. yudin and cherkasov [39] state that “in russia, small business [...] does not fulfill the functions of diversifying production and introducing effective innovation processes. small enterprises develop primarily in the sphere of rapid capital turnover and are not involved in research and development”. the reason behind this is that flexibility and entrepreneurship skills are insignificant resources unable to provide competitive advantages in the russian institutional space. assessing institutional traps in the development of small business is the primary avenue for the authors’ further studies. references 1. aghion, p. & howitt, p. 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(2016) analiz specifiki razvitiya malogo predprinimatel'stva v rossii v ramkah razrabotki metodologii ocenki social'noehkonomicheskoj ehffektivnosti malyh predpriyatij [analysis of the specifics of small business development in russia in the framework of the methodology for assessing the socioeconomic efficiency of small enterprises], social'no-ehkonomicheskie yavleniya i process, 3, 117-123, [in russian]. advances in systems science and application (2016) vol.16 no.3 76-93 optical mems sensor for measurement of low stress using ptolemy ii i. mala serene, rajasekhara babu m, zachariah.c.alex school of computing science and engineering, vit university, school of electronics engineering, vit university, vellore-632 014, tamil nadu, india abstract modeling and simulation plays vital role in the micro electro mechanical systems (mems) field. optical mems comprise of three domains namely optical, electrical and mechanical. the existing mems software for modeling is very expensive. this cost of modeling software increases the design and development of optical mems sensors. this paper proposes the design and development of a novel optical read out mechanism. this mechanism is used to measure the maximum stress applied on the cantilever and its corresponding deflection of the cantilever. the experiments have been carried out using ptolemy ii software for design and simulation of mems optical sensors. laser actor, a photo detector and force actor have been created using ptolemy ii. comsol software has been used to model cantilever. a comparative study has been done for cantilever with three modes of eigen frequencies using comsol. the experimental result shows that the parylene optical mems force sensor can sense less range of stress 0.0003 n/m to 0.272 n/m when compared to the polyimide optical mems sensor. keywords microcantilever, optical mems, ptolemy, sensor, laserdiode, comsol 1 introduction optical mems can be defined as micro devices with three functionalities like electrical, mechanical and optical at the same time and can be fabricated using batch processing techniques developed from microelectronic fabrication [1]. for integrated micro-systems composed of electrical, optical and mechanical components, the need to model large numbers of linear and non-linear components with sufficient accuracy to analyze cross-talk, noise and tolerance in an interactive environment leads to the requirement of an efficient yet accurate mixed-technology simulation technique[2]. stevan p. levitan et al reported a computer aided design tool for free-space optoelectronic systems and achieved system-level modeling[3]. the advantages of optical mems sensors over electrical sensors are high adaptability in harsh environments high temperature, chemical corrosion, strong electromagnetic interference and high-energy radiation exposure[4]. currently, no single cad tool completely models the complexity of these mixed tools to model, simulate, and analyze each stage of the design[4] . hence we have chosen advances in systems science and application (2016) vol.16 no.3 77 ptolemy as our framework for developing optical mems based sensors. ptolemy ii is a system level design environment that supports heterogeneous modeling and design of concurrent systems. for simulating optical mems devices it is essential to integrate tools with different models of computation to simulate the whole system [5]. the ptolemy ii software provides an infrastructure that allows designers explore and integrate the different models of computation [6]. it is system levels tool it. it does not provide the functionality for implementation-level simulation. but external tools based on different model of computation can be integrated into each domain and ptolemy ii can serve as semantic glue. in this present work, the simple component of mems, a microcantilever is used to sense the stress. it can be operated in two modes: static and dynamic mode. in static mode, the bending of microcantilever depends upon the force or stress on the cantilever. in dynamic mode, the resonant frequency of microcantilever changes when the mass added to it. the different read out mechanisms of the microcantilever are optical readout, piezoelectric and piezoresistive [7]. many researchers reported that the microcantilever is made of materials like silicon, silicon nitride and polysilicon [8-9]. but the fabrication cost of the silicon based cantilevers is expensive. so silicon can be replaced by a polymer which offers a shining future for the development of chemical and biological sensors. the merits of the polymer microcantilever over silicon microcantilever are low cost, more flexibility, transparency to visible uv, easily mouldable capability, improved bio-compatibility[10]. in this paper, polyimide and parylene are identified as suitable polymers for microcantilevers given their low youngs modulus, high planarity, chemical resistance and biocompatibility [11]. 2 expermental and simulation laser source emits the light of wavelength (λ=850nanaometers).this laser beam is then passing through the two optical fibers separated apart axially. the cantilever structure is fixed at one end and free at other end. a slit is connected at the free end of the cantilever moves between the two optical fibers when force is applied. the deflection of the beam will be in y direction and by virtue of this deflection the output power detected at one of the fiber ends is varied continuously from maximum to minimum though the slit arrangement as shown below. this output power variation can be calibrated according to change in minute force variation over the cantilever which in turn will constitute an accurate optical mems sensor. the light coming out of the second optical fiber is detected by the photo detector. 78 i. mala serene, rajasekhara babu m and zachariah.c.alex:optical mems sensor for ... fig. 1 flowchart of individual particle update 3 details of software development in the present investigation, we have developed software codes for various actors that make an optical mems sensor in a software platform called ptolemy. the various actors are laser, a photo detector and a force actor. the individual figures of the various actors like laser actor, force actor and photodiode actor are given in the fig 2 (a)-(c) below: fig. 2 (a) laser actor (b) force actor and (c) photodiode actor 3.1 laser actor the abbreviation of laser is light amplification by stimulated emission of radiation. laser operates on the principle called stimulated emission. it was postulated by albert einstein before 1920. this is a semiconductor laser diode (gaas) which emits light when we apply a forward biased across the p-n junction. the laser diode actor is modelled using the mathematical equations which include internal power of the laser, external power of the laser and reverse leakage current of the diode. the external power of the laser diode is given by po = pint n(n+ 1)2 (1) where n, pint, po is refractive index of the gaas, internal power and external power of the laser. advances in systems science and application (2016) vol.16 no.3 79 3.2 force actor the force actor made of a cantilever beam and two optical fibers. the deflection of the cantilever is modelled using stoneys equation, spring constant and the three modes of the resonant frequency. the optical fiber actor is created using the power output detected at the second fiber and the loss of light due to the force applied on the cantilever. 3.2.1 cantilever beam micro cantilever is a widely used component in micro electro mechanical system devices [12]. cantilever is a type of beam fixed at one end and suspended freely at the other end and the beam is originally straight. the equation (2) is the stoneys formula [13], which relates cantilever end deflection δ to applied stress σ: δ = 3σ(1− ν) e ( l t )2 (2) where δ,σ,l,t,e,ν are deflection, stress, length of the cantilever beam, youngs modulus, poissons ratio. the spring constant (k) of the cantilever beam is given by k = ewt3 4l3 (3) where e, w, t and l are the youngs modulus, width , thickness and length of the cantilever beam. the frequency at which a cantilever tends to oscillate in the absence of any force is the eigen frequency .the eigen frequency of a cantilever beam [14] can be find out from the optimized cantilever geometry for the l and t and density , for the two sensors is given by f = αn t l2 √ e ρ (4) αn = 1 4π √ ε λ2 n (5) where λn=1.8751, 4.6941, 7.8547 ...... 3.2.2 optical fibre we have designed two fibres with core diameter 2a= 175m coupled longitudinally such that the free end of the cantilever will move the slit vertically down between the fiber ends as force is applied on it. as a result the light coupled from fiber1 to fiber2 decreases gradually as the amount of force increases. there are two formulas used for calculating loss and power detected at the second at the second fiber is given below. pout = pin [ 1− ( w 2a )] (6) 80 i. mala serene, rajasekhara babu m and zachariah.c.alex:optical mems sensor for ... loss = 20log10a (7) where a stands for ratio of pout/pin. loss is a function f(w), where w is the cantilever deflection, which is numerically equal to w/2a, where 2a is the fiber diameter. 3.3 detector actor a photodiode is a semiconductor device, with a p-n junction and an intrinsic layer between p and n layers. the photo detector used is a reverse biased photodiode (pd) which converts the input optical power into the photo current (ip). the following formulas are applied to create a detector actor: the photocurrent is given by i = rpout (8) the responsivity measure the electrical output per optical input of the photodiode is given by r = ηqλ hc (9) where η, q, h, c, λ are internal quantum efficiency, charge of electron, plancks constant, velocity of light in vacuum, wavelength of light. using the above actors, the optical mems sensor model are created in the ptolemy framework as shown in the fig 3 and fig 4. in the present work, two force actors were created using the same geometrical parameters but the cantilever beam is made of different polymer materials like polyimide and parylene. the maximum stress sensed by the cantilever is measured for two different materials of the cantilever beam. the material properties of the cantilever beam include the youngs modulus (e), poisson ratio (ν) and density (ρ) is given in the table 1 : table 1 material properties of the cantilever beam material properties polyimide parylene youngs modulus (gpa) 3.2 2.8 poison ratio 0.42 0.4 density (kg/m3) 1300 1289 advances in systems science and application (2016) vol.16 no.3 81 fig. 3 model of the optical mems sensor(polyimide material) using ptolemy ii fig. 4 model of the optical mems sensor (parylene material) using ptolemy ii 3.4 model the mems cantilever beam using comsol comsol multiphysics version 5.0, a commercial fem tool for mems was used to develop a finite element model [15] of the polymer cantilevers. in the present work, the cantilever beam modelled using cost effective open source ptolemy software and its eigen frequency of the first three modes are compared with the rectangular beam of two different materials polyimide and parylene using the using comsol software. the free tetrahedral meshing is applied. 82 i. mala serene, rajasekhara babu m and zachariah.c.alex:optical mems sensor for ... fig. 5 5(a) model of the rectangular cantilever beam using comsol, fig 5(b) mesh model of the cantilever beam 4 results and discussion 4.1 optical mems sensor using ptolemy in the optical mems sensor model, the laser diode is modelled the fig.6 (a) and fig 7(a) represents the output power of the laser and output current of laser diode. the output power increases linearly with the applied current, when the applied current is larger than the threshold current. when the force applied on the cantilever, the cantilever bends and light passing from the optical fiber 1 to optical 2 is blocked based on the amount of force applied. the range of the force applied and the deflection of the cantilever is recorded for the two optical sensors are tabulated in table 2. in fig.6 (b)-(d) and fig. 7(b)-(d), the sample of the force applied in the cantilever and corresponding deflection of the cantilever is recorded, then the deflected laser power is converted into current by the photodiode and plotted in the graph. 4.2 comsol cantilever beam result the results of the first three modes of the cantilever beam of two materials modeled using comsol software are shown in the fig 8(a)-(f). the analytical values of the eigen frequencies are compared with eigen frequencies of the two cantilevers modelled using comsol are tabulated in the table 3 and the same is represented using bar chart is shown in fig 9(a)-(b). 5 optimization of the geometrical parameters the different lengths (200 µm, 300 µm, 400 µm, 450 µm, 500 µm) of the two different materials of the cantilever are kept constant and the thickness of the cantilever is varied from 0.5 µm to 3.0 µm. for each length and the maximum stress/force is recorded for each simulation is shown in table 4 and table 6 and the results are plotted is shown in figure 10. (a)-(f). for different thickness (t=0.5 µm, 1.0 µm,1.5 µm,2.0 µm,2.5 µm and 3.0 µm), the length is varied from 200 µm to 500 µm for each thickness and the maximum stress/force is recorded for each advances in systems science and application (2016) vol.16 no.3 83 fig. 6 simulation result of the polyimide optical mems force sensor (a) output power of laser diode (b), (c) and (d) deflected laser power vs output current at the photodetector fig. 7 simulation result of the parylene optical mems sensor1 (a) output power of laser diode (b), (c) and (d) deflected laser power vs output current at the photo detector 84 i. mala serene, rajasekhara babu m and zachariah.c.alex:optical mems sensor for ... table 2 stress/force applied vs. cantilever deflection of the two optical mems sensor using ptolemy polyimide optical mems sensor parylene optical mems sensor stress/force applied cantilever stress/force applied cantilever (n/m) deflection (n/m) deflection 0.0005 2.72e-07 0.0003 1.93e-07 0.05 2.72e-05 0.05 3.21e-05 0.2 1.09e-04 0.1 6.43e-05 0.321 1.75e-04 0.272 1.75e-04 0.322 1.75e-04 0.273 1.76e-04 fig. 8 eigen frequency of the result of the rectangular cantilever beam1 (a)-(c) and (d)-(f)cantilever beam2 using comsol table 3 comparison of eigen frequency values usingptolemy andcomsol software modes of polyimide cantilever parylene cantilever eigen frequency ptolemy(khz) comsol(khz) ptolemy(khz) comsol(khz) 1 0.5071 0.5074 0.4764 0.4678 2 3.178 3.2392 2.986 3.037 3 8.899 9.1789 8.3597 8.602 advances in systems science and application (2016) vol.16 no.3 85 fig. 9 (a-b): comparison of the modes vs. eigen frequency of the polyimide and parylene cantilever beam using ptolemy and comsol table 4 different length of the cantilever beam vs. maximum stress/force applied at constant thickness of the polyimide optical mems force sensor polyimide optical mems sensor cantilever max. stress/force applied (n/m) thickness (µm) l=200 µm l=300 µm l=400 µm l=450 µm l=500 µm 0.5 2.01 0.88 0.502 0.397 0.321 1 8.045 3.57 2.011 1.588 1.287 1.5 18.1 8.04 4.52 3.575 2.89 2 32.1 14.29 8.04 6.35 5.14 2.5 50 22.3 12.56 9.93 8.04 3 72.3 32.17 18.08 14.3 11.58 table 5 different thickness of the cantilever beam vs. maximum stress/force applied at constant length of the polyimide optical mems sensor polyimide optical mems sensor cantilever max. stress/force applied (n/m) length (µm) t=0.5µm t=1.0µm t=1.5µm t=2.0µm t=2.5µm t=3.0µm 200µm 2.01 8.045 18.1 32.1 50 72.3 300µm 0.88 3.57 8.04 14.29 22.3 32.17 400µm 0.502 2.011 4.52 8.04 12.56 18.08 450µm 0.397 1.588 3.575 6.35 9.93 8.04 500µm 0.321 1.287 2.89 5.14 8.04 11.58 86 i. mala serene, rajasekhara babu m and zachariah.c.alex:optical mems sensor for ... fig. 10 (a)-(f) different length of the cantilever beam vs maximum stress/force applied of the polyimide optical mems sensor at constant thickness advances in systems science and application (2016) vol.16 no.3 87 fig. 11 (a)-(f) different length of the cantilever beam vs maximum stress/force applied of the polyimide optical mems sensor at constant thickness 88 i. mala serene, rajasekhara babu m and zachariah.c.alex:optical mems sensor for ... fig. 12 (a)-(f) different thickness of the cantilever beam vs. maximum stress/force applied of the polyimide optical mems sensor at constant length advances in systems science and application (2016) vol.16 no.3 89 fig. 13 (a)-(f) different length of the cantilever beam vs. maximum stress/force applied at constant thickness of the parylene optical mems sensor 90 i. mala serene, rajasekhara babu m and zachariah.c.alex:optical mems sensor for ... fig. 14 (a)-(e) different thickness of the cantilever beam vs. maximum stress/force applied at constant length of the parylene optical mems force sensor advances in systems science and application (2016) vol.16 no.3 91 table 6 different length of the cantilever beam vs. maximum stress / force applied at constant thickness of the parylene optical mems sensor parylene optical mems sensor cantilever max. stress/force applied (n/m) thickness (µm) l=200µm l=300µm l=400µm l=450µm l=500µm 0.5 1.7 0.755 0.425 0.336 0.272 1 6.78 3.02 1.7 1.342 1.088 1.5 15.3 6.8 3.824 3.02 2.448 2 27.2 12.07 6.8 5.37 4.35 2.5 42.5 18.84 10.63 8.39 6.8 3 61.1 27.16 15.3 12.09 9.79 table 7 different thickness of the cantilever beam vs. maximum stress/force applied at constant length of the parylene optical mems sensor parylene optical mems sensor cantilever max. stress/force applied (n/m) length (µm) t=0.5µm t=1.0µm t=1.5µm t=2.0µm t=2.5µm t=3.0µm 200 µm 1.7 6.78 15.3 27.2 42.5 61.1 300 µm 0.755 3.02 6.8 12.07 18.84 27.16 400 µm 0.425 1.7 3.824 6.8 10.63 15.3 450 µm 0.336 1.342 3.02 5.37 8.39 12.09 500 µm 0.272 1.088 2.448 4.35 6.8 9.79 simulation is shown in table 5 and table 7 and the results are plotted is shown in figure 11. (a)-(f). from the recorded values, low stress/force is achieved at the length 500 µm and the thickness is 0.5m for both the sensors. 6 conclusion different actors like laser actor, force actor and photodetector have been developed and added in ptolemy framework. the physical functioning of each component of the optical mems force sensor device has been simulated using these actors. the results have been presented. the two optical mems force sensor are simulated in ptolemy ii. the parylene optical mems sensor can sense the low stress/force in the range of 0.0003 n/m to 0.272 n/m with the present fiber optic setup. the eigen frequency of the two cantilever beam is modelled using open source ptolemy ii and the three modes of eigen frequencies are compared with 92 i. mala serene, rajasekhara babu m and zachariah.c.alex:optical mems sensor for ... the results of the comsol. the low stress/force is measured for the optimized length (500 µm) and the thickness (0.5 µm) is found by varying the thickness and length of the cantilever using ptolemy. 7 acknowledgement we are thankful for support by the npmass mems design centre, sense, vit university for doing this project. references [1] e. ollier(2002), “optical mems devices based on moving waveguides”, ieee j.sel. topics quantum elect, vol. 8, no. 1, pp. 155c62. 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[15] g. louarn and s. cuenot(2008), “ finite element modelling of microcantilevers used as chemical sensors”, recent advances in modelling and simulation,pp.207-222. corresponding author rajasekhara babu m can be contacted at: rajababu.m1@gmail.com adv syst sci appl 2019; 04; 14-24 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/754 distributed data gathering system to analyze natural gas composition ivan brokarev1*, sergei vaskovskii2 1) national university of oil and gas «gubkin university», moscow, russia e-mail: brokarev.i@gubkin.ru 2) v. a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia e-mail: v63v@yandex.ru received june 07, 2019; revised september 30, 2019; published december 31, 2019 abstract: problems of development of gas analysis system based on computing fabrics have been studied. the structure of the data gathering system that will be used for the natural gas composition analysis is proposed. the most significant features and the main advantages of the proposed system are described. the most appropriate statistical models that can be used in solving the task of natural gas composition analysis are presented. the consecutive stages for statistical model development and correlation analysis results are shown. algorithms have been developed for effective solution for a number of gas analysis tasks for distributed systems. the mathematical basis of the learning algorithm and the architecture of the proposed neural network model are described. the different training completion cases for the developed neural network model are shown. recommendations are made for effective application of neural network tools as the main models for solving the task of the natural gas composition analysis. opportunities for further development of the proposed system are considered. keywords: distributed systems, data gathering system, composition analysis, neural network analysis. 1. introduction currently a variety of different gas analysis systems are under development [2]. distributed small size computerized systems are widely used to develop automatic process control systems including gas analysis systems. real time systems are the most effective systems to analyze gas. the modcon systems wireless gas analysis system is one of them; the system monitors the technological process online [7]. however, in spite of the past gains in this field, it is time-consuming to develop distributed real-time control and gas analysis systems. existing systems have a number of disadvantages: it is necessary to use expensive specialized equipment, the system is oriented mainly to monitor the technological process, the system detects limited amount of gas components. so, it is worthwhile to develop distributed gas analysis systems to overcome these shortcomings. it is one of the most complicated task to develop a distributed hardware and software for distributed microcomputerized systems. the main task solved by a distributed data gathering system is to provide access to a large amount of data in real time. this paper describes difficulties the engineers face while developing a distributed data gathering system to analyze natural gas composition; we take into account the specific features of such systems and how to apply neural network tools to investigate the features. * corresponding author: ivan brokarev (brokarev.i@gubkin.ru) mailto:v63v@yandex.ru distributed data gathering system to analyze natural gas composition 15 copyright ©2019 assa. adv. in systems science and appl. (2019) 2. development of the distributed data gathering system for natural gas composition let us see how to develop a distributed data gathering system to analyze natural gas composition. we want the system to be automated and based on the proposed data gathering system. the automated control system can be both information-logical system to monitor the technological process of gas transport and information-computing system to calculate a number of qualitative natural gas parameters, for example, the gas calorific value. we suggest to use a multiplex data gathering system for the task. in particular, multiplex systems, that utilize individual signal processing techniques at each measuring channel, are currently the most widespread systems. the most important features of distributed systems are the following: • transparency of the system – ability to hide physical distribution of resources, errors concerning access to resources and operation itself, duplication of resources, difficulties concerning concurrent operation of several users with a single resource; • scalability of the system – dependence of system characteristics on the number of its users and involved resources; • system security – data integrity protection, proper fall-over protection and error recovery capability; • openness of the system – availability of full description of interfaces and services of system operation; • system real-time operation – ability to respond to unpredictable flow of external events within predictable time. it should be noted that such features are inherent for the majority of distributed systems that can be applied to analyze gas. the main differences of the proposed system from the existing systems are commercially available and relatively inexpensive sensors as hardware components. in addition, our system is able to compute in real-time a number of natural gas energy characteristics, including its calorific value. finally, our system can detect main natural gas components (for example, eight components should be detected for russian natural gas [1]). all these can be detected and computed in real-time. when developing the data gathering system the process is considered to be carried out in several stages. the main stage is dedicated to independent development and debugging of interconnected components of applied software and hardware. the software of the designed data gathering system consists of the following main components: • programming of the low level of data gathering (sensor level); • interaction of the low and high levels (sensor level and data analysis level); • data analysis after monitoring, data documenting and data archiving. the structure of the proposed distributed system is shown in fig. 2.1. two measurement chambers with connected sensors are located at the first level of our system. two chambers are necessary to verify the collected measurement data. an important subsystem of our data gathering system is the sensor system. we suggest to use commercially available and relatively inexpensive sensors to measure physical parameters of gas mixtures – speed of sound [4], thermal conductivity [11], and molar fraction of carbon dioxide [3]. our data collecting system is located at the second level of the proposed system. the main functions of that subsystem are to scan for the information and transit the information to the higher level. the system processes the gathered data at the third level. to process the data is to eliminate errors from the measuring results, to average the collected measurement data, to compute the accessory parameters and visualize the collected data. the algorithms to analyze and evaluate the collected data are implemented at the fourth level of the system. these algorithms will be described later. at that level we compute how accurate the designed neural network is. this accuracy output shows if our model is adequate. 16 i. brokarev, s. vaskovskii copyright ©2019 assa adv. in systems science and appl. (2019) fig. 2.1. structure of the proposed distributed system 3. neural network tools for natural gas composition analysis currently, artificial intelligence solves a variety of applied problems. one of the important areas of artificial intelligence are statistical models. among a large variety of these models, we distinguish artificial neural networks (ann) mathematical models built on the principles of how biological neural networks are organized and function. ann are trained to detect complex dependencies between input and output. this ability is one of the main advantages of neural networks over traditional algorithms. one of the promising tasks, where anns are applied, is to analyze component composition of natural gas. if gases from different fields and sources are mixed during transportation and storage, the natural gas composition changes from that on the operation sites. this makes it harder to account the natural gas. since gas composition determines gas quality indicators for the equipment to operate properly and optimally, we need to measure or calculate certain properties of natural gas to transport the natural gas via a pipeline. this makes it very important to determine component composition of gas in real time; to solve this problem, correlation methods are currently being developed to analyze gas quality [2, 5]. statistical models, in particular ann, are often used to determine the desired properties and composition of natural gas by measuring the physical parameters of the gas. the fourth level of the shown distributed system carries out algorithms for determination of the target composition by measuring input physical parameters of natural gas. a statistical method is proposed as the main method for determination of the target composition. simplicity of mathematical calculations is the main advantage of the statistical method over distributed data gathering system to analyze natural gas composition 17 copyright ©2019 assa. adv. in systems science and appl. (2019) traditional algorithms. the main disadvantage the need for a large amount of initial data is eliminated by conducting the measurements via two channels. the following models can be used as statistical models for solving the task under discussion: • multi-parameter linear regression can serve as a reference model. its results can be used to compare accuracy of other regression models. for this model, it is necessary to calculate each component of the gas mixture separately. this is inefficient in comparison to other models; • model based on the support vector machine (svm). this is a set of similar supervised machine learning algorithms used to classify and regress analysis tasks. the main idea of the method is to translate source vectors into a higher dimension space and to search separating hyperplanes with a maximum gap in this space. we don't choose this model because of its significant shortcomings, in particular, complex interpretation of model parameters; also the model is applicable only to solve problems with two classes; • gaussian process regression (gpr); • ridge regression; • neural network model. on the basis of the previous research [5], it can be concluded that neural network is most effective as the main statistical model in the discussed task because of its learning capability and scalability in comparison with other models. to develop a model to analyze gas mixture composition, we implement these consecutive stages: • select data to train the model; • choose model architecture; • select a model training method; • assess accuracy of the model. the first stage consists in selecting input and output data for the model. we select a physical parameter as input, if there is a relationship between this gas parameter and the gas composition, as well as if the parameter can be measured with commercially available and relatively inexpensive measurement devices [3,4,11]. to determine if the parameters and the composition of the natural gas are related, a correlation analysis is carried out to verify inter dependencies within the sets of the input and output parameters and if the parameters are connected with each other. all data is cross-validated before it is used to train the model. cross-validation is a method to evaluate an analytical model and its behavior on independent data. this procedure is as follows: available data is divided into k parts, then the model is trained on the k 1 parts of the data, and the rest of the data is used to test the model. the procedure is repeated k times; each of the k pieces of the data is used to test the model. as a result, we assess effectiveness of the model if the available data is used uniformly. for this task, the sample of n = 700000 elements was divided into k = 10 parts. moreover, the training sample was normalized before training to improve results of prediction of the designed neural network model. the ranges of gas mixture components for the training samples are shown in table 3.1. table 3.1. ranges of components molar fractions for training sample component molar fraction, % methane 70 – 100 ethane 0 – 10 propane 0 – 5 carbon dioxide 0 – 10 nitrogen 0 – 10 18 i. brokarev, s. vaskovskii copyright ©2019 assa adv. in systems science and appl. (2019) the aim is to eliminate possible multicollinearity of the parameters linear interrelation of two or several variables. multicollinearity can lead to undesirable consequences, since the parameter estimates become unreliable. this means that the standard error increases, and it becomes impossible to isolate how the factors influence the effective indicators. we use correlation analysis (see table 3.2) with carbon dioxide molar fraction, sound velocity, and thermal conductivity coefficient as the input parameters. the output parameters are molar fractions of the gas mixture components. table 3.2. the correlation analysis results s p ee d o f so u n d , m /s t h er m al c o n d u ct iv it y , m w /( m * k ) m et h an e m o la r fr ac ti o n , % n it ro g en m o la r fr ac ti o n , % c ar b o n d io x id e m o la r fr ac ti o n , % e th an e m o la r fr ac ti o n , % p ro p an e m o la r fr ac ti o n , % speed of sound, m/s 1 0,99 0,93 -0,24 -0,61 -0,33 -0,67 thermal conductivity, mw/(m*k) 0,99 1 0,91 -0,15 -0,53 -0,45 -0,71 methane molar fraction, % 0,93 0,91 1 -0,52 -0,52 -0,48 -0,49 nitrogen molar fraction, % -0,24 -0,15 -0,52 1 0 0 0 carbon dioxide molar fraction, % -0,61 -0,53 -0,52 0 1 0 0 ethane molar fraction, % -0,33 -0,45 -0,48 0 0 1 0 propane molar fraction, % -0,68 -0,75 -0,49 0 0 0 1 the results of correlation analysis are shown on fig. 3.2 for parameters that have not been chosen as input parameters because of low correlation (speed of sound is shown as reference parameter for comparison). the results of correlation analysis are shown on fig. 3.3 for parameters that have been chosen as input parameters because of high correlation (methane concentration is shown as reference parameter for comparison). distributed data gathering system to analyze natural gas composition 19 copyright ©2019 assa. adv. in systems science and appl. (2019) fig. 3.2. the correlation analysis results for unchosen as input parameters fig. 3.3. the correlation analysis results for chosen as input parameters this neural network model is proposed as the main statistical model in the task [9]. to solve the problem, a three-layer network (multilayer perceptron) was chosen, which includes two neurons in the input layer, where two is the number of components of the input vector, eleven neurons in the second layer and four neurons in the output layer, where four is the number of components of the output vector. the input layer is a vector x = [x0 x1 x2]. the neurons of each next layer are connected to the neurons of the previous layer, and each input signal has a certain weight. the weight is the same for all input neurons since all input variables are equally important. each neuron has an activation function; the function argument is the input signal of the neuron. for the neurons of the hidden layer, the activation function is sigmoidal in the form of a hyperbolic tangent. for the neurons of the output layer, the activation function is linear. for training the network the levenberg-marquardt algorithm was chosen. this algorithm optimizes parameters of nonlinear regression models. note that in this algorithm, the optimization criterion is the root-mean-square error of the model on a training sample. the basic idea of the algorithm is as follows: to minimize the error locally, the initial values of the parameters are approximated. let a regression sample be a set of pairs of an independent variable x and a dependent variable y, and the regression model is a continuously differentiable function f. it is 20 i. brokarev, s. vaskovskii copyright ©2019 assa adv. in systems science and appl. (2019) necessary to find the value of the parameter vector w, where the error function fε reaches its local minimum: 1 ( ( , ) n i i i f = y f x w   (3.1) on the first iteration of the algorithm, the initial vector of parameters w0 is specified. on each following iteration, the vector is replaced by the vector w0 + δw. to estimate the increment δw, the following approximation of f is used: 0 0( , ) ( , ) *f w w x f w x j w     (3.2) where j is the jacobian of f. the increment δw at the minimum of fε is zero. so, to find the subsequent value of the increment δw, it is necessary to set the vector of partial derivatives of fε over w to zero. then we differentiate this expression over w and set the partial derivative to zero. after all transformations, we get the expression for δw: 1( * ) * *( ( ))t tw j j j y f w   (3.3) for this algorithm, the condition number of the matrix is important; the number shows how close the matrix is to the partial rank matrix (for square matrices it shows how close the matrix is to degeneracy). since the condition number of the matrix jt*j is equal to the squared condition number of the matrix j, the matrix jt*j may turn out to be degenerate. for this reason, in this algorithm we introduced a regularization parameter λ; the parameter is greater than or equal to zero. this parameter is selected on each iteration of the algorithm. given the regularization parameter, expression for δw takes the following form: 1( * * ) * *( ( ))t tw j j e j y f w     (3.4) where e is a unity matrix. there is a modification of this method, where the regularization parameter is multiplied by the matrix d, a diagonal matrix with the elements that coincide with the diagonal elements of the matrix jt*j. this approach is used to reduce the effect of the regularization parameter on the δw value. it is necessary, however, to note that this algorithm converges slower if the step is constant, which is a disadvantage. the problem is solved by introducing a coefficient k; the coefficient determines the step length and makes the method converge faster. given the two amendments described, the expression for δw takes the following form: 1*( * * ) * *( ( ))t tw k j j d j y f w     (3.5) the value of the vector w at the last iteration of the algorithm is the target. it is reached either if the calculated increment δw is less than the specified value, or if the error function fε is less than the specified value for the vector w. the architecture of the neural network model is shown in fig. 3.4. the number of neurons in the input layer is n = 3 for the case when the vector of input parameters contains carbon dioxide molar fraction, sound velocity, and thermal conductivity. the number of neurons in the hidden layer of our model is 11; to get this number, many different models were analyzed. the number of neurons in the output layer is m = 4 for the case of a five-component gas mixture. distributed data gathering system to analyze natural gas composition 21 copyright ©2019 assa. adv. in systems science and appl. (2019) fig. 3.4. the architecture of the proposed neural network model the next step is to train the network on the previously obtained data using the levenbergmarquardt algorithm. before that it is necessary to set the stopping criteria; for example, the criteria can be the maximum deviation value, when training is considered complete. in our case, to estimate the value, the sum of the squared deviations of the network outputs from the true values is calculated. also, when training the model, the maximum number of training cycles may be selected as the stopping criteria. however, as can be seen from fig. 3.5, this choice makes the network longer to train. fig. 3.5. cases of training completion for the developed neural network model after successful training of the network, the network is tested on the data, that were not in the training set. then, to assess the model accuracy, the following accuracy parameters are calculated: 22 i. brokarev, s. vaskovskii copyright ©2019 assa adv. in systems science and appl. (2019) • maximum absolute error; rgmax[| |]out ta ety y y   (3.6) where yout are the values of the neural network output, ytarget are target parameter values. • mean absolute error; rg[| |]out ta ety avg y y   (3.7) • maximum relative error (in %); rg rg | | max[ ] out ta et y ta et y y y    (3.8) • average relative error (in %); rg rg | | [ ] out ta et y ta et y y avg y    (3.9) • standard deviation; 2 rg 1 ( ) n out ta et i y y mse n     (3.10) • determination coefficient is a parameter that shows how the investigated model corresponds to the data. if the determination coefficient equals 1, the data is exactly described by the model. 2 rg 2 1 21 1 ( ) 1 ( ) n out ta et i n outn i out i y y r y y n           (3.11) according to the listed parameters, we conclude that our model can be used to solve the problem. if the model has low accuracy, it is trained again, or a different model architecture is used. 4. application of the neural network model according to the results obtained in this study, the neural network model can be used to calculate the component composition of natural gas, where the composition is measured by physical parameters. to check how accurate the method is, it was tested on a data sample. this sample included data that was not used for training the neural network model. the test sample includes 2218 gas mixtures, the ranges of its components are shown in table 4.3. to calculate the physical chemical parameters of the simulated sample, we used nist refprop software [8]. to calculate the speed of sound, the gerg-2008 gas state equation was used at standard temperature and pressure conditions. to calculate the thermal conductivity of the gas mixture, models for individual components and extended models for the corresponding states were used; the models are implemented in nist refprop. table 4.3. ranges of components molar fractions for test sample component molar fraction, distributed data gathering system to analyze natural gas composition 23 copyright ©2019 assa. adv. in systems science and appl. (2019) % methane 80,5 – 99,5 ethane 0 – 4,5 propane 0 – 2 carbon dioxide 0 – 9,5 nitrogen 0 – 9,5 table 4.4 and fig. 4.6 show how accurately the model predicts the natural gas composition of the test sample. carbon dioxide deviation was set to zero, because the content of this component in the gas sample is known since the non-dispersive absorption of infrared radiation (ndir) makes it easy to measure carbon dioxide molar fraction. calculation of the parameters, as well as constructing, training, and testing the neural network model was implemented by means of the matlab 2018a package with the deep learning toolbox [6]. table 4.4. accuracy of the test sample composition prediction using neural network model component maximum absolute error, % mean absolute error, % standard deviation determination coefficient methane 0,77 0,33 0,35 0,9999 ethane 0,38 0,16 0,16 0,9999 propane 0,27 0,12 0,12 0,9999 nitrogen 0,23 0,09 0,11 0,9999 carbon dioxide 0 0 0 1 fig. 4.6. neural network prediction deviation (in absolute scale) for the test sample 5. conclusion the proposed conception of distributed data gathering system for natural gas composition analysis was used to determine gas composition by values of the gas measurable parameters. however, expansion of such systems is limited by huge complexity of distributed software debugging [10]. the main tasks of further research are to develop a neural network model with a more complicated architecture, to optimize the number and set of input measurable parameters to make this model more accurate. 24 i. brokarev, s. vaskovskii copyright ©2019 assa adv. in systems science and appl. (2019) acknowledgements the authors are grateful to brisk ii ta and erasmus + 2017-1-se01-ka107-034292 staff mobility for providing the opportunity to conduct this study. references 1. altfeld k., schley p. (2012) development of natural gas qualities in europe, heat processing, 3, 77-83. 2. dorr h., koturbash t., kutcherov v. (2019) review of impacts of gas qualities with regard to quality determination and energy metering of natural gas, measurement science and technology, 30(2), 1-20. 3. dynament infrared gas sensors website [online]. available: https://www.dynament.com. 4. koturbash t., bicz a., bicz w. (2016) new instrument for measuring velocity of sound and quantitative characterization of binary gas mixtures composition, measurement automation monitoring, 3, 254-258. 5. koturbash t.t., brokarev i.a. (2018) metod opredeleniya svoystv i sostava prirodnogo gaza po izmereniyam ego fizicheskikh parametrov [the method of determination of the properties and the composition of natural gas by measuring of its physical parameters]. sensors & systems, 6(1), 43-50, [in russian]. 6. matlab 2018b software [online]. available: https://www.mathworks.com. 7. modcon systems ltd website [online]. available: https://www.modconsystems.com. 8. refprop software [online]. available: https://www.nist.gov/srd/refprop. 9. stepin u.p., trahtengerts e.a. (2007) kompyuternaya podderzhka upravleniya neftegazovymi tekhnologicheskimi protsessami i proizvodstvami.kniga 1. metody i algoritmy formirovaniya upravlencheskikh resheniy [computer-aided management of oil and gas technological processes and industries. book i. methods and algorithms of managerial decisions]. 221-223, [in russian]. 10. vaskovskii s.v. (2009) sredstva otladki raspredelennykh mikroprotsessornykh kompleksov realnogo vremeni [ways of debugging distributed microcomputerized real time systems]. sensors & systems, 1, 44-45, [in russian]. 11. xensor integration website [online]. available: http://www.xensor.nl advances in systems science and applications (2012) vol.12 no.2 186-193 a novel grey-fuzzy predictive control algorithm based on physical mode for roller kiln liang tang and mingzhong yang school of mechanical and electronic engineering, wuhan university of technology, wuhan 430070, china abstract this paper proposes a novel grey-fuzzy predictive control (gfpc) strategy using an on-line dynamic switching mechanism for controlling the sintering process of ceramic roller kiln. the grey predictor is applied to extract key information such as temperature parameters and gas pressure, and send the prediction information to the controller of actuator. the mathematical model which based on physical and heat conduction theory is derived and the real-time fuzzy controller is designed to make the output temperature in the sintering area follow the temperature-curve accurately. to achieve excellent transient performance and steady-state response, an on-line switching mechanism is adopted to regulate the forecasting step size of the grey predictor appropriately according to the error between reference value and real temperature of the kiln. additionally, this kind of method can control the temperature of kiln without accurate mathematical models for the kiln effectively. to verify the gfpc strategy, the control module based on matlab is presented. when the sintering operating condition is stable, it is a kind of effective control strategy for temperature of roller kiln. keywords roller kiln, grey-fuzzy predictive control, firing process, physical mode, heat conduction 1 introduction chinese construction industrial is booming, the ceramic tile manufacturing industry is one important industry related to this field. ceramic tile manufacturing line is a very complicated process which contained pressing machine, vertical dryer, enameling, firing kiln, sorting line, and packaging. roller kiln have important applications in ceramic industry and shows superiority over other intermittent kilns in process flexibility, production efficiency and energy consumption. roller kiln is nearly the last station and its performance is key process which control final tile quality. therefore, the finished ceramic tile has to satisfy the engineering parameters such as shape, size, durability and deformation as much as possible. if temperature values in roller kiln do not follow the reference curve accurately, then many serious defects, such as cracks and fractures, black heart, insufficient firing and over firing, tonality variations of the glaze will occur in the finished ceramic tiles [1]. however, if there are any flaws at the firing process all efforts of the preceding stations will become the waste. in view of these factors, in order to avoid the defects and obtain better quality, good controller is required to improve advances in systems science and applications (2012), vol.12, no.2 187 the control accuracy of temperature process. an appropriate control system designed to make temperature process control in roller kiln as accurate as possible. so far, researches on roller kiln have concentrated most on the kiln construction, linging material, loading and unloading machinery, and other installations. only a few studies considered the how to make the output temperature of roller kiln track the set-point values closely is a key factor affecting the firing process and the products quality. huang and fang [2] design the distributed intelligent control system for roller kiln, which using auto-tuning method based on relay feedback theory in order to decide the pid’s control parameter, and using the fuzzy control theory on this foundation. chen et al. [3] present logic control algorithm based on panboolean algebra, the control scheme relies on the correlation between e and ec of temperature. however, to be able to control this temperature process, in this study such as pid controller and fuzzy logic controller, for the best response characteristic of the system in time domain, you need choose the precision parameters. therefore, fuzzy logic controller improves the intelligent behavior, robustness, response speed, and accuracy of control system. due to the complexity of physical behaviors of temperature process in roller kiln, one of the ways to improve the system is by collecting the input-output data of plant and these details used to train nn or neuro-fuzzy model. thus, nguyen et al. present a comparative study using anns and co-active neuro-fuzzy inference system (canfis) in modeling a real, complicated multi input multi output (mimo) nonlinear temperature process of roller kiln used in ceramic tile manufacturing line[1]. in consideration of previous studies, to improve the systems performance, we proposed a novel control algorithm based on grey-fuzzy theory. the reason for using grey prediction in a fuzzy control system is that the grey predicted output from an unknown plant could always provide us some useful information for better control of the system before the system behavior runs into bad situations. the rest of this paper is organized as follows. in section 2 and 3, the structure and the mathematical model for roller kiln based on grey-fuzzy predictive control is presented. in section 4, matlab program with the proposed control scheme are obtained. finally, conclusion remarks and future work are presented in section 4. 2 grey prediction model it has been more than twenty years since the grey system theory proposed by a chinese professor julong deng in the 1980s. the grey predictive method has been successfully used to model the dynamic systems in different fields such as agriculture, ecology, economy, statistics, meteorology, industry, environment, and so on [4]. grey system theory has classified its research objects into three kinds of ‘black’, ‘white’ and ‘grey’, according to some cognitive hierarchy. 188 liang tang:a novel grey-fuzzy predictive control algorithm based... the general form of a grey differential model is gm (i, j), where i is the order of the ordinary differential equation and j the number of grey variables. the grey model gm (1, 1) with a single parameter has been most widely applied, therefore the model can be derived by the following basic steps in the following [5-6]. step 1 : rc and ago (accumulated generating operation) considering a temperature sequence as the original data, we have η s(0) = {t (0)(1), t (0)(2), . . . , t (0)(m)} (1) where m=2,. . . ,η,η ≥ 4 wherein (0) represents original data of grey prediction. the data sequence proceeded by ago once is s(1) = ago ◦ s(0) = {t (1)(1), t (1)(2), . . . , t (1)(m)},m = 1, 2, . . . , η, η ≥ 4 (2) which t (1)(1) = t (0)(1) t (1)(m) = t (1)(m− 1) + t (0)(m),m = 2, 3, . . . , η, η ≥ 4 (3) step 2: build grey model gm (1,1) the gm (1,1) model can be constructed by establishing a first order differential equation as follows: dt (1)(x) dx + at (1)(x) = b (4) where a and b are estimation parameters.z(1)(m) is affected by t (1) z(1)(m) = αmt (0)(m) + t (1)(m− 1) (5) m = 2, 3, . . . , η, η ≥ 4 however,the selection of αm value will influence the accuracy of predicted value. the solution for eq.(3) with discretization is t̂ (1)(m+ 1) = (t (0)(1)− b a )e−am + b a (6) m ≥ 0 wherein̂represents grey forecast and t̂ (1)(m+1) is the grey predicted value of the s(1) . advances in systems science and applications (2012), vol.12, no.2 189 here, we set b =  −z(1)(2) 1 −z(1)(3) 1 ... ... −z(1)(m) 1  , t =  t (0)(2) t (0)(3) ... t (0)(m)  (7) m = [ a b ] ,m = 2, 3, . . . , η, η ≥ 4 so, eq.(4) can be substituted as tn = bm (8) then the optimal parameter m can be obtained by using the minimum leastsquare estimation algorithm [ a b ] = (btb)−1btt (9) fig.1 measured points and predicted points for the temperature in roller kiln 3 modeling of grey-fuzzy predictive control 3.1 the derivation of physical model for roller kiln as shown in fig.1, in the ceramic roller kiln, dozens of nozzles distributed upper locations and lower locations alone roller kiln which controlled the temperature 190 liang tang:a novel grey-fuzzy predictive control algorithm based... at considered point within a limited range. the kiln shows non-linearity, due to the different behavior in the heating and in the cooling phases. however, it is very difficult, even impossible to build an accurate and perfect mathematical model for temperature process in ceramic roller kiln. so, with reference to fig. 1, our research assumes that the kiln has been wrapped up by the insulated materials and there is no any heat source besides nozzle inside such target. the governing equation for the one dimensional transient heat conduction is given by ∂t ∂t = ∂ ∂x (k ∂t ∂x ) (10) namely, ∂t ∂t = k ∂2t ∂x2 + ∂k ∂x ∂t ∂x (11) t means the temperature, t indicates the time, x indicates the positional parameter, and k indicates the thermal conductivity. wherein, eq.(11) can be discretizated into n-1 units as ti j+1 − ti j ∆t = ki j ti j+1 − 2ti j + ti−1 j (∆x)2 + ki+1 j − ki−1 j 2∆x ti+1 j − 2ti−1 j 2∆x (12) from manuscript [7], we can deduce at = dc (13) and, t = a−1dc = ec,e = a−1d (14) c = (ete)−1ettgrey (15) wherein tgrey indicates the temperature acquired by the grey prediction. according to eq.(13)-(15),it is dispensable to measure the temperature at each dispersed point. 3.2 grey-fuzzy predictive controller of roller kiln thus, in our control system, the control procedures are summarized as follows: (1) measure the temperature signal from thermocouples (sensors). (2) calculate output by using the control algorithm and switch the responses instantaneously. the temperature control system of the ceramic roller kiln consists of thermocouples (sensors), valves, actuators, electrically-driven conveyors and other advances in systems science and applications (2012), vol.12, no.2 191 components to form realtime control. schematic diagram of the proposed gfpc (greycfuzzy predictive control) with an on-line dynamic switching mechanism is shown in fig. 2. r(t) is the reference value, t(t) indicates the measurement signal, t ∗(t) = tgrey is the grey prediction value, u(t) is the output of the fuzzy controller, and e∗(t) is the offset: e∗(t) = r(t)− t ∗(t). fig.2 the structure of grey-fuzzy predictive controller in this paper the whole sintering will be dispersed into 8 units. according to eq. (6), thus t̂ (1)(5) = (t (0)(1)− b a )e−4a + b a (16) since the forecasting step size decides the predictive value and finally affects the control performance, an on-line switching mechanism is adopted to regulate the appropriate forecasting step size of the grey predictor [7]. this paper, we have also presented the comparison on selecting the measuring points for temperatures in order to reduce the synergistic effect on the grey prediction with errors of measurement. therefore, the grey predictor has the ability of predicting the ”previous” behavior of the system based on different states of feedback responses by either increase or decrease of control signal applied to actuator. we have the following switching mechanism: p =  p1 < 0 if e(t) > e1 p2 > 0 if eξ < e(t) < e1 p3 < 0 if e(t) < eξ (17) in kiln temperature control system, e and ec (e means error between reference value and real temperature of the kiln, and ec means the differential coefficient of e) are typical variables. any control rules of this temperature control system 192 liang tang:a novel grey-fuzzy predictive control algorithm based... should be based on these variables [3]. then, in our research, the input variables of the fuzzy controller are e∗(t) = r(t)− t ∗(t) ∆e∗(t) = [e∗(t)− e∗(t− t )]/t (18) where e∗(t) and ∆e∗(t) are the output error and the error change, respectively. 4 results and discussion due to the characteristics of long time-delay, nonlinearity, coupling and incompletion of parameter information in sintering process, the grey predictive controller is established by using the matlab. the algorithm is shown in fig.3. fig.3 algorithm of grey predictive controller 5 conclusions in this paper, a control algorithm for temperature in the roller kiln has been proposed. from the study we concluded that this temperature process is too complicated to build a certain mathematical model for identification purpose. this work focused on analyzes the heat conductive feature. basically, from the viewpoint of developing an effective control system, several key technologies were analyzed and the structure of controller was developed successfully in this paper. besides, we created matlab project to implement the predictive controller. however the limitation so far that all the control system are obtained by theory and simulation and in future we are planning to test the system for a pilot ceramic plant. advances in systems science and applications (2012), vol.12, no.2 193 references [1] nguyen quoc dinh, nitin v. afzulpurkar. (2007), “neuro-fuzzy mimo nonlinear control for ceramic roller kiln”, industrialsystems engineering, asian institute of technology, (ait), p.o. box 4, klong luang, pathumthani 12120. [2] huang yi-xin, fang yi-bin. (2002), “reaearch and application of distributed intelligent control system for temperature in rolling kiln”, proceedings of the csee, vol.22, no.5, pp.148-151. [3] chen jing, xu guocheng, xiao chun, yuan youxin, xiang kui, lang jianxun. (2008), “logic control algorithm based on panboolean algebra and its application for temperature control of ceramic roller kiln”, 2008 ieee pacificasia workshop on computational intelligence and industrial application. [4] g.r. yu, c.w. chuang, r.c. hwang. (2001), “fuzzy control of brushless dc motors by grey prediction”, ninth ifsa world congress and 20th nafips international conference, vol.5, pp.2819-2824. [5] lisheng wei, minrui fei, huosheng hu. (2008), “modeling and stability analysis of grey-fuzzy predictive control”, neurocomputing 72, pp.197-202. [6] f. cupertino, v. giordano, d. naso, l. delfine. (2006), “fuzzy control of a mobile robot”, ieee rob. autom. mag, no.13, pp.74-81. [7] jaw-yeong chiang, cha’o-kuang chen. (2008), “application of grey prediction to inverse nonlinear heat conduction problem”, international journal of heat and mass transfer, vol.51, pp.576-585. advances in systems science and applications (2014) vol.14 no.4 325-345 dynamic stackelberg games with requirements to the controlled system as a model of sustainable environmental management sergey a. kornienko and guennady a. ougolnitsky southern federal university, russia abstract a definition of sustainable environmental management based on the hierarchical game theoretic formalization is proposed. the definition includes requirements both to the state of the considered environmental system and to the control impact on it. a classification of leaders in the model context is given, a model example is described, and a qualitative assessment of the proposed game theoretic models is fulfilled. keywords dynamic stackelberg games, sustainable environmental management 1 introduction the term “sustainable development” was introduced by the international commission for environment and development in 1987. it determines a development that “meets the needs of the present without compromising the ability of future generations to meet their own needs” [1]. an apparently simple intergenerational rule is that development is sustainable “if it does not decrease the capacity to provide non-declining/capita utility for infinity” [2]. the weak sustainability rule requires that total net capital investment, or the rate of change of total net capital wealth, not allowed to be persistently negative. this definition entails the assumption that natural capital is similar to produced capital and can be substituted for it. proponents of the strong sustainability concept argue that natural capital is to a greater or lesser extent non-substitutable [3]. the concept of sustainable development (sustainability) is very vague and fuzzy. pezzey listed 60 published definitions of sustainable development[4]. a detailed analysis of the modern state of the art is made by zaccai [5]. glavic and lukman have proposed a hierarchical classification of concepts and terms concerned with sustainability[6]. some researchers argue for a special science about sustainable development (sustainability science)[7-9]. necessity of sustainability economics as a complement to ecological economics is discussed by baumgartner and quaas [10]. there are many principles of sustainability assessment and measurement [11]. an example of the assessment is given in literature[12]. nooteboom puts the problem of environmental assessment procedures in the context of complexity theory[13]. some authors tried to give the concept of sustainability a formal nature, either symbolic [14], discursive [15], or reflexive [16]. the axiomatic foundation of sustainable development based on the concept of sustainable preferences has 326 a. ashimov: the theory of parametric control of macroeconomic systems and ... been launched by chichilnisky [17]. asheim and mitra have introduced sustainable discounted utilitarianism, allowing to resolve intergenerational conflicts while satisfying the two main chichilnisky axioms[18]. d’albis and ambec addressed the question of fair intergenerational sharing of scarce natural resources[19]. a very interesting approach to the mathematical formalization of ecosystem sustainability based on optimal control theory is described in literature[20-28]. in these papers, fisher information as a sustainability measure for dynamic systems is proposed, and the sustainability hypotheses with particular focus on the natural ecosystems are formulated. it was also marked that strong efforts on different levels are necessary to provide sustainability. the sustainable development of an environmental system could be defined as: 1) an integral and balanced development of all aspects of the system functioning; 2) concordance of interests of all agents associated with the system; 3) a trade-off between short-term and long-term criteria of the system efficiency. it is very important to notice that to ensure sustainability it is necessary to provide some requirements not only to the state of the environmental system, but also to the control impact on it. as far any environmental system is impacted by many associated agents, the environmental management is a conflict process that should be described by game theoretic models. in many practical situations a traditional model of controlled dynamical system is not sufficient. for example, let the control subject be an industrial enterprise situated on the bank of a river which is the control object. the enterprise objective is to maximize its profit without consideration of the river water pollution by industrial sewage. so, in many cases the actions of control subject determined by his/her private interests and objectives are able to bring the control object in a state which is not acceptable from the point of view of sustainability. this suggests that an additional (higher) control level should be introduced to provide sustainability requirements to the state of control object. this new control subject is able to have his/her own interests too. to achieve his/her objective the new control subject can exert an impact to the initial control subject. for example, an environmental protection agency can control water quality by establishing sewage limits and charging penalties if an enterprise exceeds them. in this case a new subject of the hierarchical control arises which is a complexly structured system with its internal links and relations. dynamic stackelberg games [29] is a relevant model in this case. from the other side, sustainable development requires strong collaborative efforts of states, corporations, social organizations and individuals. from this point of view, a cooperative game theoretic formalization is required and the main problem is time consistency of the trade-off solutions. literature[30-35] presented classes of transferable-payoff cooperative games with solutions which advances in systems science and applications (2014) vol.14 no.4 327 satisfy group optimality and individual rationality. in literature[36-43] are presented solutions satisfying group optimality and individual rationality at the initial time in cooperative games with nontransferable payoffs. in literature[38-39] threats are used to ensure that no players will deviate from the initial cooperative strategies throughout the game horizon. the problem of time consistency in differential games has been intensively explored in the past decades [44]. haurie [42] raised the problem of instability when the nash bargaining solution is extended to differential games. petrosyan [45] formalized the notion of time consistency in differential games. kidland and prescott [46] introduced the notion of time consistency related to economic problems. petrosyan [47-48]and petrosyan and zenkevich [49-50] presented a detailed analysis of time consistency in cooperative differential games, in which a method of regularization was used to construct time consistent solutions. yeung and petrosyan [51] designed time consistent solutions in differential games and characterized the conditions that the allocation-distribution procedure must satisfy. petrosyan and zenkevich [52] proposed the conditions of sustainable cooperation. environmental game theoretic applications are considered in literature [3235,39,44,53-63] and others. in the author’s publications [64-67] an approach to the mathematical formalization of sustainable development and sustainable management is developed. our contribution in this paper is to describe a model of sustainable environmental management that includes requirements both to the state of the considered environmental system and to the hierarchical control impact on it. in the section 2 of the paper a mathematical formalization of sustainable environmental management is proposed. in the section 3 dynamic compulsion and impulsion stackelberg games with requirements to the controlled system are introduced. in the section 4 a classification of leaders in the games is given. section 5 is dedicated to an illustrative model example. in section 6 the principles of assessment of decision making models of sustainability [68] are used for the proposed model. section 7 concludes. 2 sustainable environmental management: mathematical formalization thus, to define sustainable environmental management it is necessary to formulate certain requirements both to the state of the environmental system and to the control impact on it. to some extent of conventionality the first group of requirements is called “homeostasis”, and the second group “compromise” [65]. requirements of homeostasis. highly organized systems of the real world should resist to the external impacts or accommodate to them providing a conservation of the conditions of their existence and goal-oriented development. as a french physiologist claude bernard has said: “the constancy of the internal 328 a. ashimov: the theory of parametric control of macroeconomic systems and ... milieu is a condition of the free life of organism” (1878). the term “homeostasis” was introduced in 1932 by walter cannon who treated it as a relative dynamic constancy of the whole organism [69]. there is no escape from the conclusion about a similarity between the concepts of homeostasis as a dynamic constancy of the organism and sustainable development as a combination of the economic development (dynamics) and the environmental stability (constancy) on the biosphere level. we think that the concept of homeostasis could be considered not only on the level of a separate organism but also on the levels of populations, ecosystems, environmental-economic systems, and arbitrary dynamic systems including human beings. let’s name the homeostasis of a system the domain of values of the essential parameters of the system in which its normal existence and development are possible. functioning of any dynamic system is characterized by a set of parameters the values of which change in the time. the condition of homeostasis means that all parameters of the system functioning during a considered period of time (long enough or even infinite) take their values from a given range, and in the specific case take given point values. for example, the point requirements to the physiological parameters of a human organism are well known: temperature 36.6 degrees centigrade, blood pressure 120 on 80 millimeters of mercury column, and so on. in the same time certain deviations from the standard values are allowable: they form the admissible ranges of functioning of a healthy organism. the point and interval requirements of homeostasis of any dynamic system including human beings can be given similarly. in the mathematical modeling a set of the essential parameters of functioning of a dynamic system is called its vector of state (phase vector) and is denoted x(t) = (x1(t), ..., xn(t)). its components (state variables) xi(t)(i = 1, ..., n) are the values of parameters which characterize the system state in the moment of time t from the point of view and with the degree of detail which are determined by the objectives and resources of the research. for example, a population can be characterized by one parameter (its number or biomass) or, more precisely, by dozens of parameters considering its sex, age, genetic, and other structuring. then a requirement (condition) of homeostasis can be formulated in two forms: weak and strong. in the weak form the condition can be written as (lagrange stability) ∀t ∈ [0, t ] : x(t) ∈ x∗ (1) where x∗ is the domain of homeostasis, t the length of the considered time period. for example, ∀t ∈ [0, t ] : x(t) > 0 (population does not extinct), or ∀t ∈ [0, t ] : x(t) > xcr (number of population is not less than a dangerous threshold), or ∀t ∈ [0, t ] : x(t) ≤ x̄ (concentration of a pollutant does not exceed the maximal allowable one). advances in systems science and applications (2014) vol.14 no.4 329 often it is possible to define the domain of homeostasis as x∗ = [x∗−ε, x∗+ε], where x∗i is an “ideal” value of the i-th state variable, ±εi is an allowable deviation from the value (i = 1, ..., n). in this case the condition (1) takes the form ∀t ∈ [0, t ] : x(t) ∈ [x∗ − ε, x∗ + ε] (2) and becomes in fact the condition of (neutral) lyapunov stability of a stationary point x∗ of the controlled environmental system. the strong form of requirement of homeostasis means satisfaction of (2) together with the additional condition lim t→∞ x(t) = x∗ (3) i.e. asymptotic lyapunov stability of the stationary point x∗ of the controlled environmental system [70]. let’s notice that it is possible to require not only stability of a stationary point, but also of another trajectory of a controlled dynamic system (periodical oscillations, linear or exponential growth, and so on). the choice of the specific form of the requirement of homeostasis (lagrange stability, neutral or asymptotic lyapunov stability of a stationary point or another trajectory) is determined by a specific environmental sustainable management problem under consideration. requirements of compromise. it is extremely important to notice again that the requirements of homeostasis are necessary but not sufficient to provide sustainable environmental management. many agents are associated with any environmental system. from one side, their objectives and interests are determined to an extent by the state and dynamics of the system. from the other side, the agents impact the system and exert some effect on its functioning. thus the sustainable development is possible only if a condition of the consideration and coordination of interests of the associated agents is satisfied. as the objectives and interests of agents do not coincide in the majority of cases then their interaction is conflict. but in the same time the objectives and interests are not antagonistic, therefore a compromise is possible. in the light of the requirements of sustainable development the compromise should consider the condition of homeostasis. it is this compromise between all agents associated with the system that forms another condition of its sustainable development. in the mathematical formalization of a conflict interaction by game theoretic models compromises are described by optimality principles for different classes of games. thus from the mathematical point of view the condition of compromise means an existence of solution of the game describing a conflict interaction of the agents associated with the system. the condition of homeostasis reflects the requirement to the state of the system meanwhile the condition of compromise formalizes the demands to the impact on 330 a. ashimov: the theory of parametric control of macroeconomic systems and ... it. if a compromise is not achieved then the system will be permanently threatened by not appropriate impacts violating the homeostasis. a simple question should be answered: who will ensure the condition of homeostasis and why does (s)he need it? note that any solution of a game (optimality principle) has some properties that provide a stability of the compromise. so, nash equilibrium is stable towards individual deviations, i.e. no player has incentives to violate an initial agreement. in stackelberg equilibrium the follower chooses an optimal answer to the leaders strategy which in turn is chosen such as to provide the leader the maximal payoff on the set of the optimal answers. in the case of a cooperative solution the key property is pareto optimality due to which a players payoff can be increased only at the cost of other players. another key problem of the sustainable management is a possible nonconformity of the short-term and long-term criteria of optimality. the condition of homeostasis is a long-term one because it should be satisfied along the whole period of existence and functioning of the environmental system. in the same time the agents associated with the system are often guided by short-term criteria with much smaller character times. as a result the compromise is under the threat of violation by those participants of the initial agreement for which it could be more advantageous in the current moment of time to take another strategy corresponding to their short-term interests. in this connection a realization of the requirements of sustainable management is possible only for those compromises which keep their optimality for all associated agents along the whole period of existence of the environmental system. in the game theoretic formalization this principle was called a time consistency [51-56, 70]. the property of time consistency of the solution of a game means that a truncation of the solution is still optimal in all subgames arising along the optimal trajectory of the system development. this property provides a practical realization of the compromise solution from which it is not advantageous for any agent to deviate all along the time of system functioning. thus, the weak form of the requirement of compromise is existence of a solution of a game theoretic model of conflict interaction of the agents associated with the controlled environmental system. in the case of a hierarchically controlled environmental system that is under consideration in this paper the stackelberg solution is used. the strong form of the requirement of compromise means additionally the time consistency of the solution. it could be stated that separately the characterized requirements of homeostasis and compromise are necessary, and in their totality also sufficient conditions of the sustainable development of any environmental system. the condition of homeostasis expresses basic requirements to all aspects of the system functioning, advances in systems science and applications (2014) vol.14 no.4 331 the condition of compromise provides the adequacy of impacts by all agents associated with the system with acceptable consideration of their interests, including a coordination of short-term and long-term optimality criteria of the agents and the consequent non-advantageousness for them to deviate from the initially agreed compromise solution all along the time. if the homeostasis is provided and a dynamic consistent compromise between all the associated agents is achieved then a sustainable environmental management of the system takes place. the sustainable environmental management ensures both conditions of sustainable development and means of their realization. the conditions of sustainable environmental management are characterized in table 1. table 1 conditions of sustainable environmental management. homeostasis compromise weak form lagrange (1) or neutral lyapunov stability (2) of a phase trajectory (for example, stationary point) of the controlled environmental system existence of a solution of the game theoretic model which has some strategic stability (for example, nash of stackelberg solution) strong form asymptotic lyapunov stability (2)-(3) of a phase trajectory (for example, stationary point) of the controlled environmental system time consistency of the solution 3 dynamic stackelberg games with requirements to the controlled system. in many practically important cases the stakeholders of an environmental system are organized hierarchically. then three methods of hierarchical control [65] can be used (table 2). their game theoretic formalization is proposed in [64]. consider the example from introduction in more details. the higher level control subject (leader) is an environmental protection agency, the lower level control subject (follower) is an industrial enterprise, and the control object is river ecosystem. it is natural to treat the desirable strategy for leader as such a strategy in which the industrial sewage doesnt exceed the maximum allowable concentration of pollutants in the river. in the case of compulsion the objective is reached by establishing some sewage limits and license recall from the enterprise if the limits are violated. in the case of impulsion if the enterprise exceeds the maximum allowable concentration of pollutants in the river then a penalty is charged. at last, in the case of conviction the enterprise administration is environmentally 332 a. ashimov: the theory of parametric control of macroeconomic systems and ... table 2 characteristic of the methods of hierarchical control. compulsion impulsion conviction general description of the method leader provides choosing the desirable strategy of follower by force the strategy desirable for leader is more profitable for follower than undesirable ones follower chooses the strategy desirable for leader voluntarily and consciously nature of impact administrative or legislative economic socialpsychological type of relationships subject-to-object subject-to-object with partial consideration of follower’s interests subject-to-subject mathematical formalization leader’s impact on the follower’s set of admissible strategies leader’s impact on the follower’s payoff function transition of leader and follower to cooperation and maximization of the summarized coalitional payoff function conscious and it provides the required sewage refinement voluntarily. to find the strategies of sustainable environmental management it is necessary to build game theoretic models with requirements to the state of controlled system. we will consider dynamic stackelberg games which formalize hierarchical relations between the stakeholders of an environmental system based on compulsion or impulsion. a possible transition to conviction (cooperation) is also considered. dynamic compulsion stackelberg game with requirements to the controlled system. the game can be written as follows: jl = ∫ t 0 e−αt[gli(u(t), x(t))− glc(q(t))−mρ(x(t), x∗)]dt → max (4) q(t) ∈ q (5) jf = ∫ t 0 e−αtgf (u(t), x(t))dt → max (6) u(t) ∈ u(q(t)) (7) ẋ = f(x(t), u(t)), x(0) = x0 (8) advances in systems science and applications (2014) vol.14 no.4 333 where jl, jf – payoff functionals of the leader and the follower respectively; t – period of consideration; α – discount factor; x(t) – state vector of the hierarchically controlled environmental system; u(t) – vector of controls of the follower (impact on the controlled environmental system); q(t) – vector of compulsive controls of the leader; gli – function of the leader’s personal interest (for example, income); glc – function of control cost of the leader (glc(0) = 0); m – penalty constant; ρ(x(t), x∗) = { 0, x(t) ∈ x∗, 1, otherwise; – indicator function; q,u – sets of admissible controls of the leader and the follower respectively; gf – instant payoff function of the follower; f – known function of controlled dynamics of the environmental system; x0 – given initial condition. compulsion in the game (4) – (8) consists in that the leader by controls q(t) exerts an impact to the set of the follower’s admissible controls u(q(t)) (without control dependence). requirements to the state of the controlled environmental system (without loss of generality in the form of lagrange stability) are described by means of an indicator function ρ and a penalty constant m in the leader’s payoff functional. therefore, if the requirements to the controlled system (treated as its homeostasis conditions) are violated then the leader is charged a penalty m . a stackelberg equilibrium [29] is accepted as solution of the dynamic game (4) – (8). its specific feature is presence of the indicator function ρ in the leader’s payoff functional subject to which the functional can become discontinuous when m → ∞. therefore a key role belongs here to the solution of a parametrical inverse optimal control problem, i.e. the problem of building of the set of “homeostatic” controls of the follower u∗ = {u(t) ∈ u(q(t)) : x(t) ∈ x∗} (9) if ∃q(t) : u∗ ̸= ∅ then solution of the game (4) – (8) is reduced to the solution of an ordinary dynamic stackelberg game without requirements to the controlled system, otherwise the game (4) – (8) has no solution. time consistency of a solution of the game (4) – (8) depends on its information structure. it is known [29] that in the class of open-loop strategies a stackelberg solution is not time consistent, i.e. in this case only the weak form of the requirement of compromise can be discussed. to provide time consistency (the strong form of compromise) closed-loop strategies should be considered. dynamic impulsion stackelberg game with requirements to the controlled system. the game can be written as follows: jl = ∫ t 0 e−αt[gli(p(t), u(t), x(t))− glc(p(t))−mρ(x(t), x∗)]dt → max (10) p(t) ∈ pu (11) 334 a. ashimov: the theory of parametric control of macroeconomic systems and ... jf = ∫ t 0 e−αtgf (p(t), u(t), x(t))dt → max (12) u(t) ∈ u (13) subject to (8), where in comparison with (4) – (7) p(t) – vector of impulsive controls of the leader. impulsion in the game (8), (10) – (13) consists in that the leader by controls p(t) exerts an impact to the follower’s payoff function. a control dependence (feedback) also takes place, namely p̃(t) = p(u(t)). therefore the set of the follower’s admissible strategies is a set of maps pu = {p : u → p}. considerations about the stackelberg solution are the same as in the case of compulsion. transition to conviction. both in the case of compulsion and in the case of impulsion a transition to conviction is possible. this transition means cooperation between the leader and the follower for joint solution of the problem of sustainable environmental management. from the mathematical point of view, conviction means a coalition of the leader and the follower and their joint maximization of the summarized payoff functional, or solution of the optimal control problem jl+f = ∫ t 0 e−αt[gli(u(t), x(t))− glc(q(t))−mρ(x(t), x∗) + gf (u(t), x(t))]dt → max (14) q(t) ∈ q; u(t) ∈ u(q(t)) subject to (8). suppose that u∗ ̸= ∅ (otherwise the sustainable environmental management problem has no solution). then it is evident that the follower chooses u(t) ∈ u∗. thus q(t) ≡ 0 (compulsion is not required), respectively glc(0) = ρ(x(t), x∗) = 0, and the problem (14) takes the form jl+f = ∫ t 0 e−αt[gli(u(t), x(t)) + gf (u(t), x(t))]dt → max, u(t) ∈ u (15) subject to (8), i.e. only the follower maximizes the coalitional payoff functional. after solution of the optimal control problem (15) the received payoff is shared between the leader and the follower accordingly to a cooperative optimality principle [36-37]. in the case of impulsion a transition to conviction is similar. 4 classification of leaders in the proposed conceptual framework the leader is treated as a regulator who is responsible for the requirements to the controlled environmental system, or a subject of the sustainable environmental management. the following classification attributes can be used accordingly to the form of the payoff functional (4) or (10). advances in systems science and applications (2014) vol.14 no.4 335 importance of the requirements to the controlled environmental system. from the point of view of the leader, the requirements can be: mandatory (m → ∞): in this case the leader can’t solve her control problem without ensuring the requirements because she is charged an arbitrarily big penalty otherwise; desirable (m ≈ gli , glc): in this case the leader may compare what is more profitable: to ensure the requirements, to reduce cost or to increase her income; neglectable (m ≈ 0): in this case the leader is in fact not interested to ensure the requirements. the first variant can be considered as an “ideal” setting of the problem of sustainable environmental management, the second one as a more practical model of the real life. the third variant is a degenerated one because the requirements of homeostasis are neglectable (in fact, absent). notice that the specific problem of sustainable environmental management arises only if the requirements are mandatory. presence of the personal interest. the leader can be disinterested (gli = 0) or interested (gli > 0). in the former case the only leaders objective is to ensure the requirements of homeostasis (considering her control cost) while in the latter case the leader has also her personal interest (for example, to get a share from the collected taxes or fines). control efficiency. a process of control generates cost for the leader. so, the efficiency of control can be high (glc ≈ 0) or low (glc >> 0). in the former case the costs are very small, and the leader can concentrate on the requirements of homeostasis (probably considering her personal interest). in the latter case the control costs become an essential factor. 5 a model example for the sake of parsimony, let’s consider as an illustrative example the most simplistic malthus model of a controlled environmental system ẋ = (a− u(t))x(t), x(0) = x0 (16) where x(t) – number (biomass) of a population; u(t) ∈ [0, 1] – share of its exploitation (harvesting). lets require that ∀t ∈ [0, t ] x0 − ε ≤ x(t) ≤ x0 + ε (17) be the weak condition of homeostasis (given that the initial population value is optimal for the habitat). then the strong form of homeostasis is (17) plus lim t→∞ x(t) = x0 (18) for each u(t) the solution of cauchy problem (16) takes the form x(t) = x0e (a−u(t))t (19) 336 a. ashimov: the theory of parametric control of macroeconomic systems and ... it is evident that both conditions (17) and (18) are satisfied for (19) when u(t) ≡ a (20) (the solution of the inverse optimal control problem). now lets investigate whether the leader can ensure (20). compulsion stackelberg game has the form jl = ∫ t 0 [kpu(t)x(t)− cq2(t)−mρ(x(t), x∗)]dt → max (21) 0 ≤ q(t) ≤ 1 (22) jf = ∫ t 0 (1− p)u(t)x(t)dt → max (23) 0 ≤ u(t) ≤ 1− q(t) (24) subject to (16). here discounting on the finite time interval [0,t] is omitted for simplicity, p – constant tax rate, k – share of the collected taxes in the leader’s income, c – factor of the leader’s control efficiency, m → ∞ – penalty constant. the leader can ensure the condition (20) by choosing q(t) ≡ 1− a (25) in this case is the stackelberg solution for the game (16), (21) c (24), and the players’ payoffs are jcomp l = (kpax0 − c)t, jcomp f = (1− p)ax0t (26) notice that jcomp l { > 0, kpax0 > c (the leader′s control is efficient), < 0, otherwise. . in the case of transition to conviction the coalitional control problem is jl+f = ∫ t 0 [(1− (1− k)p)u(t)x(t)− cq2(t)−mρ(x(t), x∗)]dt → max 0 ≤ q(t) ≤ 1, 0 ≤ u(t) ≤ 1− q(t) the team solution has the form ; in this case the condition (20) is satisfied and jconv l+f = (1− (1− k)p)ax0t > ((1− (1− k)p)ax0 − c)t = jcomp l + jcomp f impulsion stackelberg game has the form jl = ∫ t 0 [kp(t)u(t)x(t)− cp2(t)−mρ(x(t), x∗)]dt → max (27) advances in systems science and applications (2014) vol.14 no.4 337 0 ≤ p(t) ≤ 1 (28) jf = ∫ t 0 (1− p(t))u(t)x(t)dt → max (29) 0 ≤ u(t) ≤ 1 (30) subject to (16). here the leader can ensure the condition (20) by choosing the impulsive strategy p̃(t) = { 1− δ, u(t) = a, 1, otherwise, (31) and the follower’s optimal reaction is also u(t) ≡ a. thus, (1 − δ, a) is the stackelberg solution in the game (16), (27)–(30), and the players’ payoffs are j imp l = (kax0 − (1− δ)c)t, j imp f = δax0t (32) as in the case of compulsion, jcomp l { > 0, kax0 > (1− δ)c (the leader′s control is efficient), < 0, otherwise. let’s notice also that jcomp l < j imp l , but in the same time jcomp f >> j imp f . transition to conviction entails the following optimal control problem: jl+f = ∫ t 0 [(1− (1− k)p(t))u(t)x(t)− cp2(t)−mρ(x(t), x∗)]dt → max 0 ≤ p(t) ≤ 1, 0 ≤ u(t) ≤ 1. the team solution is also ; in this case the condition (20) is satisfied and jconv l+f = ax0t > ((k + δ)ax0 − (1− δ)c)t = jcomp l + jcomp f i.e. cooperation is more advantageous for both players. 6 qualitative assessment of the model in the paper [68] five methodological criteria for sustainability models are proposed: an interdisciplinary approach, uncertainty management, a long-range or intergenerational point of view, “glocality” and participation. let’s use these criteria for the qualitative assessment of our model of sustainable environmental management. interdisciplinary approach. this approach is followed in two aspects. first, state variables of the modeled controlled environmental system characterize its ecological, economic, social and other components. second, in the game theoretic models an equation of dynamics can describe environment, while payoff functions 338 a. ashimov: the theory of parametric control of macroeconomic systems and ... and sets of admissible strategies of the players determine socio-economic interests and possibility of their implementation. uncertainty. this requirement is naturally ensured by building and investigation of stochastic control (game theoretic) models. note that random variables can enter both in the dynamics equation and in the payoff functions. long-term perspective. this criterion is considered by the very setting of a dynamic game theoretic model. the game can be defined both on finite (long enough for the character scale of functioning) and on infinite period of time. an additional comparison of the temporal effects is provided by discounting of the payoffs. global-local perspective. this requirement means a consideration of the hierarchical nature of modeled processes that is ensured by using stackelberg games. in more complicated model settings the systems of equations describing the modeled system dynamics could include small parameters that reflect different scale of the modeled processes. participation. as in the case of “glocality”, this criterion is considered by the very game theoretic setting that means a compromise concordance of different interests of the stakeholders. thus, the qualitative analysis demonstrates that the proposed sustainable environmental management models satisfy the methodological criteria introduced by boulanger and brechet [68]. 7 conclusion in spite of the long-term discussion, the concept of sustainability remains vague and fuzzy. in this paper a mathematical formalization of the concept based on requirements of neutral or asymptotic lyapunov stability of an ideal trajectory in the state space of an environmental system is used. this approach can be considered as a formalization of the requirement of strong sustainability. however, this is not sufficient. while speaking about sustainable environmental management, it is necessary to add some requirements concerning the control process. it is provided by the description of interests of all stakeholders in a game theoretic model. as far in many practical important situations relations between the stakeholders are asymmetric, dynamic stackelberg games are used as the model. we consider compulsion stackelberg games in which the leader exerts an administrative impact to the set of the followers admissible controls without control dependence (feedback), and impulsion stackelberg games in which the leader exerts an economic impact to the followers payoff function and control dependence (feedback) takes place. in both cases a transition to conviction (cooperation) is possible that means a coalition of the leader and the follower and their team maximization of the joint payoff function. conviction is the most advances in systems science and applications (2014) vol.14 no.4 339 perspective and advantageous for both players if they are able to overcome different obstacles on the way to cooperation. all proposed game theoretic models include requirements to the controlled environmental system that are treated as conditions of its homeostasis. the proposed models are successfully tested by criteria described in [68]. it seems worthwhile to develop the proposed methodology for different classes of 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(1965), “stability and asymptotic behavior in differential equations”, heath mathematical monographs, boston. corresponding author author can be contacted at: ougoln@gmail.com adv syst sci appl 2020; 04; 45-59 published online at https://ijassa.ipu.ru. computational models of trajectory investigation of marine geophysical fields and its implementation for solving problems of map-aided navigation l.v. kiselev 1*, v.b. kostousov 2, a.v.medvedev1, a.e. tarkhanov2, k.v. dunaevskaya2 1) imtp feb ras, vladivostok, russia e-mail: imtp@marine.febras.ru 2) imm ub ras, ekaterinburg, russia e-mail: dir-info@imm.uran.ru abstract: comprehensive investigations of geophysical fields (gpf) of the ocean are among the top priority problems in underwater robotics. problems reside in developing automated systems capable of real-time operation for gathering, accumulating, and processing diverse geophysical information. the data’s total volume is applied for long-term or real-time monitoring of marine areas, environment or objects surveillance, mapping certain regions, or anomalies in geographical coordinates. one of the main elements of such systems is the computational models sensitive to properties of geophysical fields, features of search routes of underwater vehicles, and particular aspects of missions performed during trajectory measurements and features of navigational support tasks. the work considers computational models of comprehensive interpretation of the results of trajectory measurements of geophysical fields using an autonomous underwater vehicle (auv), and estimation of accuracy of map-aided navigation on the reconstructed map of geophysical fields. algorithms and software consider distinctive features of representation of bathymetry, magnetometry, and gravimetry data visualized in 2d and 3d. algorithm of mapaided navigation by the reconstructed fields is discussed. the final assessments of the considered models’ accuracy consider averaged errors of measurements, mapping, and inertial navigation. obtained assessments are based on theoretical investigations, results of model experiments, and experimental trials of auv’s systems under actual operating conditions. keywords: autonomous underwater robot (autonomous unmanned underwater vehicle), motion control, navigation, marine geophysical fields, trajectory measurements, mapping, computational models, bathymetry, magnetometry, gravimetry. 1. introduction nowadays, geophysical measurements using aerospace, ground, and marine means are widely used for precise navigation of moving objects and investigation of the earth’s geological structure. particular importance is attached to comprehensive research of anomalous fields in different regions of the earth’s surface, including aquatic areas promising in terms of its geological structure, exploration, long-term and real-time monitoring, and performing of various underwater missions. herewith, tasks measuring the parameters of fields with precise navigational referencing, 3d visualization of surveying results, bathymetry, magnetometry, and gravimetry of different marine areas are relevant for practical applications. prospects of using auvs for geophysical measurements at sea repeatedly served as subjects of research and development over the number of years. originally this topic was * corresponding author: imtp@marine.febras.ru 46 l.v. kiselev, v.b. kostousov, a.v. medvedev, a.e. tarkhanov, k.v. dunaevskaya copyright ©2020 assa. adv. in systems science and appl. (2020) raised in the monograph [1] and concerned the estimation of advantages, which can be introduced using auvs. later, similar problems were discussed in the works of russian [217] and non-russian specialists [18-22]. bathymetry measurements always accompany echo-ranging surveys of the sea bottom terrain, usually performed using modern side-scan sonars, interferometric and multi-beam sonars, and synthetic aperture sonars. studying marine geomagnetic fields and its anomalies and solving magnetized object search and detection tasks are carried out using highly sensitive field-level meters and gradiometers. some issues of solving search tasks utilizing auvs equipped with magnetic anomalies meters were considered in works [1-6, 10]. the use of auvs seems promising for trajectory measurements and mapping of local gravity fields or gravity anomalies (ga) with the prospect of using the results for navigation purposes. an important tool, along with the creation of equipment and conducting field tests, is the development and research of information and computational models [23-24] of specific processes for measuring geophysical fields, their recording and mapping, including models for the use of auv and the use of mission results. gravimetric computational models and their capabilities in solving navigational adjustments tasks were studied in research [12-17]. problems of mapping geophysical fields using auv are essential by itself, and besides that are directly related to the solving tasks of navigation by distinctive gpf elements. these tasks may include the follows: • search for the position of a fragment of the field map, recorded during the auv movement, on the field reference map, which is related to the task of matching the fragment to the reference field; • the best approximation of the field map in terms of accuracy of navigation for economical storage of the reference field onboard the auv; • estimating the informativeness of the field and developing the most meaningful route, i.e., the best route in terms of the accuracy of navigation by the gfp while moving along the route [25, 26]. let’s take a closer look at possible solutions to this kind of task. 2. search tasks during mapping and reconstructing the map of anomalous geophysical field based on trajectory measurements from auv let be the function describing the cross-section of the field in a coordinate plane , which can be denoted by the isoline map the field meter, while moving alongside the trajectory, records a representation of the field with some random error (2.1) suppose that coordinates and projections of the speed vector of the auv (field meter) are given by an internal navigation system (ins) with errors then, coordinates related to the measurement of differs from true ones (2.2) let’s denote the field’s measure of the variability alongside the trajectory by the value of field gradient or an increment , where is the moving velocity, is the time interval. ( , ) f x y ( , ),x y ( , ) .f x y const= ( ) f t ( ) : tx ( ) ( ( ), ( )) ( ). f t f x t y t tx= + ( , )x y ( , ) x yv v ( , ), ( , ).а а x yx y v vd d d d ( ( ), ( ))x t y t ( ) f t ( ) ( ) .( ) , ( )) a x a yx x t x v t y t y v tt y t= + d +d = +d +d! ! fñ f f v td = ñ d v td computational models of trajectory investigation of marine geophysical fields 47 copyright ©2020 assa. adv. in systems science and appl. (2020) the following interrelated tasks are of practical interest: task 1 consists of recovery of the field map in the form of an intensity matrix, defined on the regular orthogonal grid, or as a set of isolines. the recovery of the field map must be performed by covering the given area by the net of trajectories and measurements (2.1) of the field with referencing to points (2.2.) obtained by inaccurate navigational data. as usual, the control of an underwater robot consists of an assignment and correction of a navigational program in characteristic points of the trajectory. the points can be derived from conditions of passing the field local extremums or condition of maximal field gradient: (2.3) task 2 consists of the search of field anomalies, which appear in the function f as areas of local extremums. for that, it is necessary to arrange a search motion trajectory for an underwater robot for detection and contouring the anomaly by the fixed level and variability of the field meter signal under the masking influence of external noises. the search of coordinates with specific (extremal in particular) values the field parameters is vital for the determination of purposed motion cue. the search procedure can be performed based on orthogonal or gradient descent with an orientation towards a chosen target, such as an anomaly source. accumulation of the data about the extremal values of the field allows outlining the borders of anomaly or boundaries of area (background) with no anomalies. task 3 is connected with tracking the given isoline delimiting the boundaries of the anomaly by the given (extremal) field value. the motion is then organized by the orientation of the speed vector by the field behavior proportional to the value . generally, the search mission scenario includes the following processes: • approaching the area with the given field level using orthogonal tacks and course correction according to field gradient sign; • covering the fount area by the net of squarewave like rectilinear trajectories (tacks) with continuous measurements of the field parameters and mapping them to navigational data for further reconstruction of the field map; • performing search movements by orthogonal descent for determination of the probable position of the extremum; • tracking the given isoline delimiting the boundaries of the anomaly by pointing the speed vector according to the “curvature” of the field alteration or moving by calculated coordinates, which corresponds to the given field level. the primary scenario process consists is a restoration of the field map be the net of discrete measurements obtained by covering the target area by rectilinear trajectories (tacks). measured in nodes of the irregular net values are then interpolated with step depending on the value of the field gradient value in nodes. based on obtained data, isolines with given values of the field can be created. motion control in the field of anomaly is a unique but yet practically important case. while developing the particular algorithms of auv motion control during field anomaly investigation, it is necessary to consider features of the spatial structure of the field, including the presence of the natural background field, time and spatial distortions, and significant nonlinearities in their formal description. examples of modeling the search motions and restoration of the map of investigating areas of the field are presented in [7-9, 11]. ( , )f x y ( , )k kx y { } .( , ) ( , )loc loc loc max min maxk k k k f f f ff x y f x yº º ñ º ñ! ! ( )tf ( )tfd ( , ) ,f x y const= /x yf fñ ñ 48 l.v. kiselev, v.b. kostousov, a.v. medvedev, a.e. tarkhanov, k.v. dunaevskaya copyright ©2020 assa. adv. in systems science and appl. (2020) 3. features of the visual representation of geophysical information and results of its comprehensive processing as was mentioned before, the idea of using auv for solving tasks of marine geodesy and navigation appeared at the very beginning of their development and then continuously evolved in theoretical and applied studies. let’s consider some of the results of research and computational experiments in the field of marine gravimetry. first of all, it is worth mentioning that auv is a well-stabilized platform abled of compensating external forces’ actions, such as vertical accelerations of the carrier [2-4, 7-9]. examples of gravimeter signal records obtained by auv “tiflonus” when working on the depth of 70 m are shown in the work [14]. the procedure of recording and restoration of the map of the gravity anomalies (ga) field based on trajectory measurements while covering the given area by a planned grid of trajectories with predetermined measurement accuracy are illustrated in fig. 3.1. here, particular estimations use a geographical map of residual gravity anomalies in the the peter the great gulf, obtained in the laboratory of gravimetry of the v.i. il'ichev pacific oceanological institute (poi) feb ras using shipboard precision gravimetry chekan-am made by csri elektropribor [10]. the map contains a selected fragment, which has a ceased volcano with a volatile ga gradient in the lower part of it (fig. 3.1,a). a 50x40 km part of the field in this area was investigated using squarewave-like trajectories (fig. 3.1,b). the computational experiment assumed that the processing time of the small tack is 20 min, which equals to the distance of 1.2 km on the speed of 1 m/s. correspondingly, the processing time of the big tack is 14 hours, and the distance is 50 km. trajectories were created considering auv dynamics in modes of depth and heading control. obtained digital data then were used for the restoration of the field map using scripts of matrix transformation and functions of interpolation for 2d and 3d images (fig. 3.1,b,c). (c) fig. 3.1. the fragment of the ga geographical map with plotted squarewave-like trajectories (a); the 3d image of the recovered field fragment (b); a horizontal slice of the ga map recovered fragment (c) a subject of undoubted interest is extensive use of all the available geophysical information, including bathymetry, magnetometry, and gravimetry. it allows firstly to find a possible correlation in measurements of heterogeneous fields, and secondly to obtain more certain and precise data for navigational correction by the cumulative results of mapping. let’s have a closer look at the possibility of such an approach on the example of combining the data of mapping of sea depth field, geomagnetic field, ga field for the predetermined aquatic area. we will use an integrated mapping data for the studied area of the strait of tartary provided by the laboratories of gravimetry and magnetometry of the feb ras. these initial maps originate from the expedition using high-precision shipboard computational models of trajectory investigation of marine geophysical fields 49 copyright ©2020 assa. adv. in systems science and appl. (2020) measurement devices. there are no similar available maps obtained with the help of auv. therefore, real maps are used, on which it can work out all the procedures simulating the trajectory measurements results of the field from the board of auv. recovered maps are virtual maps obtained using computational procedures based on the achievable results of trajectory measurements from the auv, taking into account methodological errors (model errors) and measurement errors. these maps are actually analogs of those maps that can be obtained by using of auv in real work. further these maps are used as a reference ones to evaluate the accuracy of navigation by the gfp of certain moving objects, in the structure of which the map data is embedded. in the accepted computational model these maps are used to work out the evaluation algorithm described below. the initial 2d maps of these three fields are demonstrated in fig. 3.2,a,b,c, and the reconstructed maps on fig. 3.2,d,e,f. two-dimensional images of three fields shown in fig. 3.2 can be transformed into 3d images in the form of corresponding reliefs of the spatial structure (an example of it can be fount in fig. 3.1,b). additional capabilities of the visual representation in the form of threedimensional images can be obtained by a combination of the ga field, anomalous geomagnetic field (agmf), and the sea depth field (sdf). here, the sea depths axis is a vertical coordinate, and the other two form a projections of their 3d images on the upper (zero) level of the sea terrain field. such representation can be got in the form of images obtained from any angle and any step of the image rotation angle. note that according to the bathymetry matrix, the average depth of the selected area in the tatar strait is 2737 m. fig. 3.3 demonstrates examples of this visual representation with the covering the entire zone with squarewave-like trajectories when auv is in motion on the depth of 1000 m. during movement along a given trajectory, simultaneous and spatially combined measurements of three fields are made. a) b) c) d) e) f) fig. 3.2. fragments of the initial maps of the sea depths (а), ga field (b), agmf (c), map of the recovered fragmets of the sea depths field (d), ga field (e), agmf (f) 50 l.v. kiselev, v.b. kostousov, a.v. medvedev, a.e. tarkhanov, k.v. dunaevskaya copyright ©2020 assa. adv. in systems science and appl. (2020) a) b) fig. 3.3. examples of the angle pictures combining the sea bottom’s terrain field with ga field (a) and with the field of magnetic anomalies (b). the redline shows the coverage trajectory. the images are obtained using matlab graphic toolkit (functions surf(), mesh(), plot3()), where one space unifies data about physical fields of two types and trajectories with different color schemes (colormap()) 4. estimation of the navigational informativeness of an anomalous geophysical field works [10-16, 27-29] consider issues of the informativeness of the anomalous ga field and estimating the accuracy of navigational correction by recovered field map. here the estimations of navigational informativeness of the fields (sdf, ga fields, and agmf) recorded in the tatar strait are considered. thereunder, the method of estimation of errors of map-aided navigation [32-35] proposed in [30, 31] is studied by the example of the joint using of three geophysical fields. the task of the method of map-aided navigation is correction of errors of required navigational parameters (for example, coordinates (latitude and longitude) of the carrier) by the matching the measured field fragment to the reference map [33, 34, 36, 37]. the matching of the fragment and the reference map can be performed using the search of the global extremum of some matching functional of measured fragment and reference fragments obtained from the reference map. the navigational informativeness of the geophysical field is measured by the level of navigation errors when using this method. as a rule, the main source of errors is misclosures of mapmaking and errors in field measurements while the object is moving. however, the sensibility of the navigation results to these errors is mainly determined by the gradient characteristics of the field itself. here, the task is considered in the probabilistic statement, as in works [33-35]. computational models of trajectory investigation of marine geophysical fields 51 copyright ©2020 assa. adv. in systems science and appl. (2020) 4.1. theoretical estimation of errors of coordinates correction by the maps of geophysical fields since the map-aided method of navigation is used for the correction of errors of inertial navigation system, the task the navigation by geophysical field is often called the correction task of navigation system. here, the task of coordinates error correction in a horizontal plane is considered. let’s denote the reference map of the geophysical field defined in a certain rectangular area of the plane by . formally, the field map is a smooth function of the two-dimensional vector with values on the axis of real numbers. practically, the function is represented by samples in nodes of a regular 2d grid with a given step size. if necessary, the values of the function between the nodes can be obtained using a proper interpolation method, for example, bilinear interpolation [38, pp. 123-128]. let’s consider that measurements of the field are performed in points of a rectilinear interval, which begins in the unknown point in a given fixed direction defined by a unit vector . the model of the field measurement onboard the moving object can be expressed as: (4.1) where is the value of the geophysical field measured in the discrete moment . then, let’s denote the set of values (m-vector) by a measured fragment of the field or just a fragment. the expression (4.1) consists of the following notions: • is an area of a priori location of the initial point of the fragment in the area of the given reference map ; • is the actual location of the fragment; • is a given directing unit vector of the measurement route; • is a vector of the fragment distortion. equality (4.1) can be rewritten in the vector form as: (4.2) where m-vector is a set of samples recorded from the field map in time points : (4.3) at its simplest, the vector is a centered gaussian vector with a zero mean and covariance matrix (where is an identity matrix, is a given dispersion). in reality, the distortion of the fragment along with the field meter random noises include the inaccuracies of mapping and errors of the relative positions of measurement points, which are caused by navigational errors of the carrier. in the correction task, it is required to obtain the best estimation of the unknown vector x using given initial data and evaluate the accuracy of this estimation, for example, using the covariance matrix . in the framework of the bayesian method, vectors x and are assumed to be random vectors with known distribution , . here, the main research subject is a correlation-extremal search method of the navigational error correction [24-26], which is added with the new approach for the correction error estimation based on the analysis of matching functional. w 2 1 2( , ).r x x= x ( )g x ( )g x x ( )g x ( )g × x p ( ) ( ) , , 1,... ,k k k kt g t x q k mj j x= + î =! x x + p kj kt 1( ,... )tmj j=φ g q ìw qîx φ w g x p 1( ,... )mx x=ξ ( ) ,= +φ s x ξ s(x) kt 1( ) ( ( ),... ( )) .tmg t g t= + +s x x p x p ξ 2 xs i i 2 xs ( )!x φ ( )p φ 1( ,... )mx x=ξ ( )xf x ( )fx ξ 52 l.v. kiselev, v.b. kostousov, a.v. medvedev, a.e. tarkhanov, k.v. dunaevskaya copyright ©2020 assa. adv. in systems science and appl. (2020) let’s consider the following quadratic functional of matching measurements and field map as an example: (4.4) in this case, the estimation can be found as a result of the search of minimum: : (4.5) the calculation of the correction error estimation relies on the following inequality: , (4.6) where is a point of the global minimum, is a point of the second-biggest local minimum spaced apart from the first minimum on a significant distance, is a threshold value. in the case of a single extremum, it is possible as take the minimum value of the functional at the boundary of the uncertainty domain. inequality (4.6) shows how strongly the global minimum in (4.5) is pronounced against other local minimums, as the smaller the relation in (4.6), the more reliably the correction problem is solved in errors presence. in this context, this inequality describes the informativeness of the field in the . the threshold is derived from a statistical experiment using the neumannpearson criterion and corresponds to the minimal level of the errors of the second kind at a fixed level of the errors of the first kind. here the errors of the first kind are understood to be taking a wrong algorithm result as an accurate correction, and the errors of the second kind are understood to be a false rejection of the correction while the algorithm result is correct. in a statistical experiment, the correctness of algorithm output was defined by exceeding a predetermined threshold of the distance between the actual location of the fragment and the solution of the task (4.5). at a predetermined threshold , it is possible to calculate the threshold and then to find a diameter of the next set: (4.7) in this case, the diameter of the set is defined as the maximum distance between the points of the set. thus, the result of the correction algorithm is taken as an estimate (4.5), and the value (4.7) serves as an estimate error of the search algorithm, which can be calculated during the process work of the algorithm in real time without knowing the actual position x. in this case, the value of is an estimate of the maximum possible radial error of the correction. upon the availability of several spatially connected maps of different geophysical fields and a possibility of simultaneous measurement of these fields during motion, it is possible to co-use them to improve the accuracy of navigational error correction using a search algorithm (4.5). to do so, we will consider the matching functional of the following form: (4.8) where k is a number of used geophysical fields, is a map of the i-th field, is the kth sample of the measured fragment of the ith field, is a weight influence coefficient of the measurements of the i-th field on the matching functional. [ ]2 1 ( ) ( ) m k k k g t = f = + -åx x p j ( )!x j ( )f x ( ) argmin ( ) qî = f! x x xj min_1 min_ 2 ( ) ( ) trp f < f x x min_1x min_ 2x min_1x trp min_ 2( )f x q trp trp min_1( ) tr tr x p f f = max { : ( ) }.trd diam x x= f £f ( )!x j maxd / 2max maxdr = computational models of trajectory investigation of marine geophysical fields 53 copyright ©2020 assa. adv. in systems science and appl. (2020) further coefficients 𝛼�coefficients � are accepted to be equal to inverse values of dispersions of summary errors of the measurement and the reference map: (4.9) 4.2. results of computational experiments the most accurate and practically usable method of assessing the informativeness of geophysical field maps is a method of statistical imitational modeling of the process of navigational error correction, including repeated forming the sequence of measurements and their referencing to the reference field maps. recovered maps of three fields recorded in the tatar strait in coordinates e137º…e138º, n43,8º...n44,3º (fig. 3.2,d-f) were used as reference maps. maps were developed as matrices of samples assigned in nodes of the regular net with a step of 0.001° on both axes. herewith, experiments involved forming of model measurements of the field by the initial maps of corresponding fields. experimental conditions in correction zones for every three fields (their maps can be found in fig. 3.2), we perform the forming of the model measurements of fields with the step of 100 m alongside rectilinear routes of 40 km directed along an x-axis. the correction was made by the recovered maps, shown in fig. 3.2,d,e,f. confidence rectangle q (i.e., an area of the a priori location of the route beginning point, where a search of the functional minimum was performed (4.5)) had a size of 48x55 km. the size of the researched zones was approximately 85×55 km. the grid of the experiment containing the nodes, where the measurement paths began, covered the confidence rectangle with a step of 0.01 degrees (about 1 km). there were nэ = 2880 total experiments of correlation-extremal correction of navigation errors performed for each map. in each experiment, a set of measurements was modeled on the initial map, and the functional was formed (4.4) or (4.8) based on the reconstructed maps, the search for the functional minimum and the calculation of the radial -estimate were performed. the full generation model of the measurements of the sdf, ga field, and agmf included errors of autonomous navigation, random errors of the meters: echosounder, gravimeter, magnetometer correspondingly. the approach for error generation of the ga field measurements and inertial navigation system is provided in work [14]. fig. 4.1 comparatively demonstrates the results of statistical experiments of modeling the correction and obtaining the estimation of correction errors . the results in fig. 4.1 have the form of matrices with a size of 55×85 according to the grid in increments of one kilometer covering the research area. herewith, ga field experiments specified the rms deviation of the fluctuating part of the measurement error = 3 mgal, the computational threshold of the estimation was chosen to be =0.8. for the sdf, these parameters were taken equal to = 60 m, =0.6. and for the magnetic field, they were equal to = 7 nt, = 0.9. sdf: actual error sdf: error of dmax estimation (a) (d) maxd maxd xs maxd trp xs trp xs trp 54 l.v. kiselev, v.b. kostousov, a.v. medvedev, a.e. tarkhanov, k.v. dunaevskaya copyright ©2020 assa. adv. in systems science and appl. (2020) ga field: actual error ga field: error of dmax estimation (b) (e) agmf: actual error agmf: error of dmax estimation (c) (f) fig. 4.1. a comparison between distributions of actual errors along the correction zone (a,b,c) and estimations using dmax (d,e,f). the predetermined green-red color scale (the scale is shown on the left in each figure) shows the values of the actual radial errors of coordinate correction and their -estimations related to the initial points of the measurement routes located in the nodes of the kilometer grid. one of the measurement routes is shown in the center of every figure. all distributions are shown against the background of the corresponding halftone images, which represent the average radial modulus of the gradient of the study area in the gray scale (on the right). all the matrices shown in the figures correspond to the 85×55 km zones of the experiments. in the presented results, the estimation error vector for convenience is brought to the scalar value – radial error , where , are the squares of the vector coordinates. fig. 4.2 uses the same color map and shows distributions of the actual error of the comprehensive correction along with all three fields according to the algorithm (4.7) and distribution of the corresponding -estimation with the threshold =0.8. (a) combining fields: actual error (b) conbining ields: dmax-estimations fig. 4.2. a comparison between the distribution of the actual errors along the correction zones (a) and dmaxestimation (b) when correcting by all three fields. the given colormap shows the values of coordinate correction errors and their dmax-estimates related to the initial points of the measurement routes located in the nodes of the kilometer grid. one of the measurement routes is shown in the center of every figure. the distributions of the errors are shown against the matrix of the integral informativeness, obtained by the weighted sum of the matrices of component fields. the statistical mean values and statistical rms of estimation (4.7) of the correction errors of the considered geophysical fields and their combination are shown in table 4.1 in comparison with the actual errors for different locations of the fragment (nэ = 2880) and for different rms deviations of measurement errors . maxd ( )= !δ x x φ 1 2 2 2( ) x xr d d= +d 1 2 xd 2 2 xd δ maxd trp maxd qîx ξ computational models of trajectory investigation of marine geophysical fields 55 copyright ©2020 assa. adv. in systems science and appl. (2020) table 4.1. results of a statistical experiment for the separate fields and their combination correction by the ga field: = 3 mgal, ptr = 0.8 correction by the sea depths field: = 60 m, ptr = 0.6 correction by the magnetic anomalies field: = 7nt, ptr = 0.9 comprehensive correction by three fields: ptr = 0.8 actual error: / , m 980 / 811 761 / 752 1223 / 1015 671 / 737 radial estimation: / , m 1300 / 837 1175 / 710 966 / 842 737 / 528 in the table, we accepted the following notations: average actual radial error of the search algorithm, where – solution of task (4.5) in the i-th experiment, – the actual position of the estimated parameter in the i-th experiment, the expression described above; – statistical standard deviation of the actual radial error from its mean value; the average value of the error estimate of the search algorithm according to the method (4.7), где – where is the radial estimation in the i-th experiment; – statistical standard deviation of the radial – estimation from its mean value. matching the results shown in table 4.1 and distributions of the errors shown in fig. 4.1, and fig. 4.2 allow concluding that -estimation is close to the actual estimation of the correction accuracy, which proves the applicability of the method for solving the tasks of this kind. regarding the navigational informativeness of considered geophysical fields, it is mentioning that the most informative (or the most accurate in terms of correction) field is an sdf, and the least informative is an agmf. this actually shows the property of corresponding measuring systems and methods of measurement processing. it can be summarized that the complexing the measurements in three fields significantly increases the average accuracy of map-aided navigation: for the sea depths field of at least 10%, for the ga field of at least 30% and for anomalous geomagnetic field of at least 45%. 5. conclusion 1. the preceding experience of works connected with the use of auv for real-time and long-term monitoring of the geophysical fields of the ocean proves its viability for solving tasks of this kind. but on the other side, it demonstrated a number of theoretical and practical issues regarding the development of the proper computational models. these xs xs xs sd srms maxd maxr max rmsr 1 1( ( ) ) ( ( ) ) эn s s s i i iэ r r x x n d j = = = -åx φ x! ! ( )s ix j! ix ( )r × srms ( ( ) )s ir x xj -! max max, 1 1 эn i iэn r r = = å max,ir maxd max rmsr maxd maxd 56 l.v. kiselev, v.b. kostousov, a.v. medvedev, a.e. tarkhanov, k.v. dunaevskaya copyright ©2020 assa. adv. in systems science and appl. (2020) models are based on methods of trajectory investigation of the areas and objects with respect to the features of the motion control tasks, information properties of the fields itself, and precision properties of the means of navigational correction by the recovered field map. nowadays, gravimetry tasks are of particular importance thanks to the opportunity to perform it onboard the auv, which is the precision stabilized platform with a minimum of the vertical accelerations. that establishes close relationships with tasks of inertial navigation. the results of natural and computational experiments show that this is a way for providing high-precision gravimetric measurements with the cumulative error of the ga field mapping of units of mgal, which corresponds to the geographical referencing error of units of meters. 2. critical issues of the described research are: • development of the models of trajectory investigation of the fields, anomalies search, and reconstruction of the map using the results of trajectory measurements; • visual representation of the recovered fragments of the map in 2d and 3d formats and estimate the map reconstruction’s accuracy regarding the dynamic errors, measurement, and navigation errors; • development of computational models for estimating the information properties of reconstructed fragments of the field map for solving the task of navigational correction using a map-aided navigation approach. 3. accuracy evaluation of correlation-extreme correction by the maps of geophysical fields is considered using specially developed a new method, which does not impose excessive requirements for onboard computer. 4. one of the main advantages of auv equipped with the complex of meters of the geophysical fields is the joint processing of all the available information for the mapping and following use of it for navigational correction. using software processing and 2d and 3d visual representation of digital formats, it is possible to find a correlation in the fields’ scalar-vector properties, which can be an additional means for developing more precise computational models of map-aided navigation. references 1. ageev m.d., kasatkin b.a., kiselev l.v., molokov yu.g., nikiforov v.v., rylov n.i. 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(1992) numerical recipes in c: the art of scientific computing (2nd ed.). new york, ny, usa: cambridge university press. advances in systems science and applications (2014) vol.14 no.4 346-360 research and development of process knowledge management system in okp company w. l. chenli1 and s. q. (shane) xie2 1department of mechanical engineering,university of auckland, auckland,new zealand, f. f. zeng 2school of mechanical science and engineering, huazhong university of science and technology, wuhan, china abstract the coming of knowledge-based economic era has greatly changed manufacturing environment. knowledge has become the most important and valuable asset for manufacturing companies. for most of one-of-a-kind production (okp) companies, innovation based on knowledge is their opportunity and the sharpest competitive weapon to compete and thrive in current global market-driven environment. process knowledge is a special type of knowledge that controls how products are manufactured. the knowledge may help okp companies manufacture high value-added products with best quality and shorter times at competitive cost. managing process knowledge is very important, and its benefits are enormous. this paper proposes a framework and definition of process knowledge management system (pkms) after analysis of the characteristics of process knowledge and requirements in okp companies. a hybrid approach aiming at representation of three main types of process knowledge is discussed. one of the ways for representing dynamic process flow knowledge is introduced in detail, which is solved using an approach based on parameter flow charts (pfc). the proposed approach supports experts to create knowledge repository themselves. a prototype system is developed to validate the proposed framework and approach. keywords knowledge management, knowledge representation, one-of-a-kind production, knowledge-based engineering, process planning 1 introduction we are now in a knowledge-based economic era[1], in which manufacturing situation has changed greatly and can be characterized by the following buzzwords: customer-oriented, globalization, and time-driven competition. this is not only a challenge, but also an opportunity for manufacturing companies, especially for one-of-a-kind production (okp) companies to compete and thrive in this global market environment. these okp companies might be small and middle-sized, but their survival and success depend on producing newer, better, quicker and more innovative products and services, not simply rely on size and strength as before. innovation plays an important role in their strategies and is becoming advances in systems science and applications (2014) vol.14 no.4 347 one of the sharpest competitive weapons for them to manufacture high quality products at lower costs with shorter development times. the foundation and source of innovation is knowledge. knowledge and intellectual capital have become the most important and valuable asset for organizations. this has attracted much research interest in the areas of knowledge based engineering (kbe) and knowledge management (km) over the past years [2-7]. process knowledge in manufacturing industry is a special type of knowledge. most of them are the kernel technical secret that may help companies manufacture high value-added product with best quality and shorter times at competitive cost. the process knowledge may exist in the mind of experts or skilled employees. survey showed that to train an expert or a skilled employee of process planning might take at least 5 years or more. for most okp companies, the shortage of knowledge workers is one of their biggest obstacles to grow and the problem has to be solved. hence, exploiting and managing process knowledge is very important for okp companies and its benefits are enormous. process knowledge management system (pkms) can help okp companies develop, build, and deploy process knowledge assets effectively and profitably. considerable research effort has been placed in this area [8-13]. however, most of existing work are based on fixed process knowledge representation schema and fixed system architecture. they were applied to a particular kind of manufacturing environments and are very specific, and are difficult to adapt to continuous technology developments and changing manufacturing environment. to resolve these issues, a tiered framework of pkms is proposed in this paper. a hybrid and open approach for process knowledge representation is discussed in detail. the prototype version of system has been developed and demonstrated with case studies. 2 definition and framework of pkms 2.1 analysis of process knowledge in okp companies before discussing the framework of pkms, the characteristics of process knowledge in okp companies are analyzed firstly. for most okp companies, their market strategies are either make-to-stock, or assembly-to-order, or make-toorder. they are facing special critical problems including high customization, ‘once’ successful approach, loose or fatter production, and complicated product data and information flow, etc[14]. this determines their manufacturing environment is customer-oriented and has been undergoing changes often. hence, process knowledge in these okp companies also has special different characteristics that can be summed up as followings: (a) dynamic and changing. process knowledge should be updated along with the emergence of new techniques and new methods. 348 w. l. chenli: research and development of process knowledge management system ... (b) form diversity. there is a variety of process knowledge format, including description sentence, tables, pictures, drawings, formulas, and numerous other representations. (c) universality. process knowledge comprises of not only internal knowledge in a company, but also external knowledge of suppliers and customers. internal knowledge includes all the knowledge from market survey to product maintenance after sale. (d) uncertainty. process knowledge comes from experiences and skills of experts, and it is implicit and hard to represent by computer. 2.2 definition and framework of pkms the definition of pkms here refers to a system based on information technology for managing process knowledge in okp companies, supporting creation, capture, representation, storage, dissemination and share of information. pkms aims to enhance the innovation ability of okp companies by accumulating and sharing of personal, organizational, and industrial process knowledge. based on above analysis, the most important requirements and characteristics of pkms are: (a) supporting management of all kinds of process knowledge. not only the knowledge based on rules, but also the knowledge embedded in drawings, documents, formulas and other representations needs to be managed. (b) open and extensible. when manufacturing situation and technology change, the knowledge database can be updated and reconstructed. (c) people oriented. the source of knowledge is people. the knowledge database in pkms should be maintained and used by these skilled employees, not just by other it employees or system developers. companys own experts should be able to create, capture and storage their process knowledge in pkms. (d) combination of technological advancement and practicability. as process knowledge is tacit and implicit, its difficult to capture and transform this kind of knowledge clearly and automatically, new methods need to be developed. based on the requirements and characteristics mentioned above, a proposed framework of pkms has been constructed and is shown in fig.1. the framework includes five layers. each layer is devoted to a specific task and is designed to link the interface with the framework from different levels of abstractions and functionalities. (1) support layer: it provides the necessary computer net environment that supports the system running. support layer includes a computer network, a database management system (dbms), operation systems (os), desk pcs, and servers. (2) database layer: it is the core knowledge repository of pkms that stores all kinds of process knowledge of a okp company. the process knowledge is classified into four main types. process resource knowledge refers to information of advances in systems science and applications (2014) vol.14 no.4 349 fig.1 framework of pkms in okp companies machine tools, fixtures, cutters, materials, and quantity standard used in cutting, etc. product process knowledge is typical process knowledge used regularly in companies.feature process knowledge includes recommended process routes of all sorts of machining features. other process knowledge refers to other often used process planning information, such as basic concepts in process planning, formulas of working hour calculation, etc. (3) core layer: it is comprised of knowledge representation, knowledge discovery, and knowledge reasoning and explaining. knowledge representation is to provide environment and methods for representing various types of process knowledge. knowledge discovery is used to search for appropriate process knowledge in repository. knowledge reasoning and explaining deal with the search result and give helpful output process knowledge. (4) function layer: it is the functional implement layer for the purpose of process knowledge management. function modules in this layer include the creation, presentation, modification, searching, and browsing of all the process knowledge stored in the database. for the purpose of managing diverse forms of process knowledge, a process resource management module, a process flow knowledge definition module, and a formula management module are developed. (5) application layer: this layer provides industrial solutions for a given com350 w. l. chenli: research and development of process knowledge management system ... fig.2 design support strategy for pkms pany or industry, such as equipment, machine tools, mould, boat and ship, etc. the knowledge repository is deployed flexibly according to the requirements of a particular company or industry. 2.3 design support strategy for pkms according to the proposed system framework as shown in fig.1, the design support strategy is proposed to implement pkms as shown in fig.2. the following summarizes the key issues of the design support strategy. (1) requirements investigation. to know the current situation and requirements of process knowledge management in okp companies, collect process knowledge of personal, company and industry. the investigation results are used as input and foundation for pkms development and implementation (2) technical solutions planning. based on the result of investigation, the technical route, function models, and task schedule for pkms are planned and determined. (3) knowledge establishment. classify all kinds of collected process knowledge, create, modify, update knowledge database using function modules of system. (4) application framework deployment and releasing. configure pkmss running environment according the situation of a particular company and release it. (5) system deployment and implementation. establish implementation team, install and deploy the system, train companys employees to use the system. (6) system assessment and continuous optimization. after running for a period of time, collect feedback, assess, improve and maintain the system. to develop this proposed pkms, one of the key issues is the representation and management of a variety of process knowledge. as mentioned above, the diversity and dynamic characteristics of process knowledge makes it hard to be represented effectively through only one approach. a hybrid and open approach for process knowledge representation is required. advances in systems science and applications (2014) vol.14 no.4 351 3 hybrid approach for representation of process knowledge 3.1 classification of process knowledge process knowledge is classified into three main types (see fig.3) according to the form of process knowledge described as following: (1) knowledge of process flow is rule-based knowledge with process routes or working operations. it comprises knowledge of feature process/product process/typical process. feature is the definition of components basic geometry entity for manufacturing. popular form features include cylinder, hole, plane, etc, which have typical recommended process scheme for each of them. knowledge of product process refers to process route information of product family or similar products, which may change according to the input manufacturing data. knowledge of typical process is the mature process route information which has been validated by practice and is used frequently (2) knowledge of resource refers to static manufacturing resource information, which includes all kinds of process resource, such as machine tools, fixtures, cutters, machining data, materials, etc. (3) knowledge of calculation refers to information obtained through calculation. in process planning, the selection of working hour and material quota is a regular activity. for these three types of knowledge, process flow knowledge is most difficult to capture and represent, because they are rule-based, dynamic, determined by many factors. most of them come from experience of experts or skilled employees. in the proposed pkms, a visual expressing method based on parameter flow charts (pfc) is used to represent them. this method combines the parameter technology, flow chart technology, and visual technology. for resource knowledge, an approach of classification tree is used to represent them. a formula management tool is used to represent calculation knowledge. the focus of this paper is placed on the approach of pfc. 3.2 representation approach based on pfc 3.2.1 definition of pfc definition 1: parameter (p) p is an element that has effects on the description or reasoning of the knowledge, e.g., when machining a hole in a part, the diameter, machining precision, and surface roughness will influence the machining route, therefore the elements can be extracted as: p = f(pg, pa, pr, ccons) f()is the definition function. is the geometric information of p. is the attributes of p, such as name, id, data type. is the action range of p, such as global parameter, local parameter. defines the constraints. p has two types, one is 352 w. l. chenli: research and development of process knowledge management system ... fig.3 process knowledge composition and classification model individual, and the other is dependent. the value of the individual p is directly from designers input or is derived from the superior process; the value of the dependent p is from the calculation result among other individual p according to the constraints. definition 2: parameter table (pt) pt is a set of information that arrays the parameters of the knowledge using a table form. it can efficiently support parameter storage, classification, discovery, and comparison. pt = f(pta, p, ui,m) ptais the attributes of pt, p is the parameter set in pt; ui is the user interface set of the pt; m is the manipulation set of pt, e.g, add, delete and edit. for process planning, the evaluations of the parameters often have mixed number and string operations, including the illegibility operation. definition 3: chart unit (cu) cu is the basic node that makes up of the pfc, which presents an action of the knowledge expression simulating the human beings. in different application domains, the actions are different, and the tasks and the aims of the actions are different too, so cus are different. according to the requirements of ke in different domains, the types and the attributes of cus can be extended. generally, advances in systems science and applications (2014) vol.14 no.4 353 a cu consists of seven types. cu = f(bu,eu, v u,ru,ou, iu, spfc) bu refers to the beginning of the pfc(see fig.4,bu1 ). refers to the end of pfc (see fig.4, bu1 ). sets value for parameters (see fig.4,vu1,vu2 ). ru depicts the reasoning logics according to the parameters. ru has single branch (see fig.4, ru1 ) or multiple branches (see fig.4, ru2 ).the ru with single branch denotes the judgment of yes or no logic. the with multiple branches defines several decisions according with several conditions. oudefines the output (see fig.4,ou1, ou2, ou3 ). is the attention graphic unit (see fig.4,iu1 ). iu1 assists in judging and validating for ke, which can help the designers find the incorrectness and remind designers to revise it. spfc represents a sub-process and nesting is available in the (see fig.4,spfc1 ). by establishing the database of and calling the spfc in the main process, the whole knowledge is available. fig.4 is a simple example of representation form for process flow knowledge based on pfc. it’s a visual and direction knowledge chart, starts from chart unit bu1 and ends at chart unit eu1. vu1 and vu2 are used to set values for parameters. ru1 and ru2 are responsible for judging the input data and determining the next flow step. spfc1 is a sub-pfc that is defined in another place ou1,ou2,ou3 , and are chart units of output, such as work operations. through drawing such flow chart with predefined chart units, a rule-based process knowledge can then be represented. fig.4 an example of representation form based on pfc 354 w. l. chenli: research and development of process knowledge management system ... definition 4: parameter flow chart (pfc) pfc is a direction chart for knowledge description and reasoning that consists of parameter tables, chart units and the logical routes among the chart units (see fig.4). pfc presents the contents and the structure of the knowledge. pfc =< a,pt,cu,cr > a, pt, cu and cr refers to the set of attributes, parameter tables, chart units and the relations among the chart units of pfc respectively. cr is classified as joined cr, unjoined cr, paratactic cr and branchless cr. for representing complicated knowledge, it can be represented by several sub-pfc. 3.2.2 model of the software system based on pfc fig.5 model of the pfc software system based on the analysis of the process flow knowledge and the definitions of pfc above, the model of the software system based on pfc is shown in fig.5. the parameters are defined via interactive interfaces, including extraction and semantic description of the parameters. the process flow chart is defined via microsoft visio, a popular flow design software system based on the component object. visio is customized and integrated with the proposed software system. hence, the capabilities of extendibility, visualization and readability of system are achieved. when explaining and reasoning using the pfc, the system will firstly read the related information from the pfc file (vsd file), then call the main module to load corresponding file via automation interface, finally explain and reason the flow chart. the system provides two operations, reasoning in step and reasoning advances in systems science and applications (2014) vol.14 no.4 355 continuously, to meet the requirements of debugging and output of process planning knowledge. during the process of explaining and reasoning, if errors occur, the system will notify the user and ask for solutions. after the process flow chart has been explained and reasoned successfully, the results will be formalized with a table form via neutral file. 3.3 other representation approach resource knowledge comprises machine tools, fixtures, machining data, etc. they are static and have diverse forms. classification tree is used to represent them. the node of the classification tree indicates the type of resource knowledge, and its self-defined attributes show the description and information of resource knowledge. such hierarchical structure of tree is very similar to practical process resource management. the structure of node is defined as following: class cproceeresnodetree { public: int nnodekey; //node id int nparentnodekey; //parent node id int nsibling; //sequence number in sibling int nchildren; //whether has child node cstring snodename; //node name int nnodetablekey; //database tale id of node int nnodegraphicskey; //document id of node cstring snodeeditaccess; //permission of edit cstring snodeviewaccess; //permission of browse } for calculation knowledge, the key issues include creating user-defined formula, definition of variable value source, and definition of suitable condition for the formula. the data structure of calculation knowledge is abstracted as following: class cformulaknowledge { public: int nformulakey; // formula id cstring sformulatype; // type of formula cstring sformula; // formula expression cstring sdescription; // formula description int nconditiontablekey; // table id of condition int nvariabletablekey; // table id of variable value source } to represent the calculation knowledge, a formula management tool based on the above data structure is developed and is shown in following case study. 356 w. l. chenli: research and development of process knowledge management system ... 4 case study fig.6 manage machine tools in pkms fig.7 manage cutting tools in pkms a prototype system of pkms has been developed to validate and demonstrate the feasibility and the compatibility of the proposed pkms. microsoft visual c++ is employed to develop the framework and functional modules. microsoft sql server 2005 is utilized to construct the systems repositories. microsoft visio advances in systems science and applications (2014) vol.14 no.4 357 2003 is customized and used to draw process flow charts. a variety of standard industrial process knowledge has been established using functional modules of the prototype system. fig.6 and fig.7 are screenshots of process resource management module using classification tree method. the left side of fig. 6 shows the resource classification tree that comprises of process resource nodes. the data structure of node is defined as mentioned above, including its name, parent node, database table and related document, etc. the nodes of tree can be created, deleted, searched according companys situation given authorization. the right side of fig. 6 shows the information of current selected resource node, a list of horizontal lathes with their technical parameters, which may guide to select appropriate lathe when carry on process planning. fig.7 is an interface of managing standard cutting tools. the drawings of current selected end mill are also captured and displayed. fig.8 manage calculation knowledge in pkms fig.8 is the screenshots of formula management module, which takes charge of managing calculation knowledge. two types of formula used often in process planning are built in case study, one is for material calculation, and the other is for working hour calculation. a whole formula may have its expression, variables, applied conditions, and categorization, etc. its data structure is defined as mentioned above. fig.9 is the screenshots of representation approach based on pfc. the example 358 w. l. chenli: research and development of process knowledge management system ... fig.9 manage process flow knowledge in pkms flow chart is drawn in microsoft visio 2003 using predefined chart units. mentioned above. there are two, four and one in fig.9, which are pointed by arrows, is used to judge the next information flow direction according to the input conditions. stands for working operation of process. is a sub-pfc defined elsewhere. the directed flow chart represents knowledge of a parts process route. 5 conclusion in this paper, the characteristics of process knowledge in okp companies have been analyzed; a definition and framework of pkms in okp companies were introduced as well. to overcome the difficulty of representing diverse type of process knowledge, a hybrid approach aiming at three main types of process knowledge was proposed. a prototype system has been developed to validate the proposed ideas. from this research work, the following conclusions are drawn: (1) the proposed pkms is open and extensible, the process knowledge database can be established and updated through developed interfaces. 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[14] xie, s.q. and y.l. tu.(2006), ‘rapid one-of-a-kind product development”, international journal of advanced manufacturing technology. vol.27, no.56, pp.421-430. corresponding author author can be contacted at: s.xie@auckland.ac.nz advances in systems science and application(2015) vol.15 no.2 119-133 obsolescence of organizations: a modeling approach in system dynamics didier cumenal1,2 1 dynamic systems, group afscet the french association of systems sciences 2 system dynamics society abstract economic crisis as well as human crisis is well alive. bankruptcy filings, the collapse of financial/capital markets, strategy failures, the death of some organizations accuse the turbulent economic environment. but what about their potential activities, their internal capacities that change over time. it is unfortunate that some literature areas dealing with the decline of organizations focus on their disease at a specific time by giving importance to factors related to age, size and performance. this is wrong to underestimate the evolutionary aspect of tangible and above all intangible resources. the purpose of this communication is to highlight that before the decline and death of an organization, there is a preliminary stage, often misperceived: the obsolescence.obsolescence defines a condition that prevents the organization to perform correctly its vital functions and that without her being fully conscious. our model of system dynamics has the primarily objective of defining the state of obsolescence and secondly to highlight the importance of two strongly linked subsystems meaning: organizational capacity and managerial ability. these two dimensions, coupled together, are often misidentified or misunderstood by the top management, whereas they may be able to characterize and anticipate an emergency situation. keywords managerial capacity; organizational capacity; dynamics; obsolescence; model; system 1 why, what field and what subject for this research? 1.1 why this study our past experience has led us to lead a group of “sme”, to straighten firms in distress as a senior consultant and, many years, to conceptualize these achievements with a thesis phd at the sorbonne in the field of organizations dynamics (evolution and change of state). it was during this career and after in becoming a full faculty member and researcher that we often asked ourselves some questions about the future of organizations. the sole ambition of this study is to identify and teach the concept of obsolescence as a general phenomenon often unknowed for executives and managers of the company. yet it is a step often announcing the decline or disappearance of the organization. our research on the phenomenon of obsolescence is trying to reach an understanding which is not limited to specific cases, but that is applicable 120 didier cumenal: obsolescenceof organizations:amodeling approachinsystem dynamics across time and space and to different types of organization. 1.2 the research question is there a persistent step which would herald the decline and death of organizations? what would this step? is it possible to identify the precursor weak signals of the organization collapse? can we model this prior step announcing disaster? while there are many researches on organizations mortality, yet few qualitative studies have been conducted about symptoms often considered marginal, but harbingers of economic and social tragedy. because the model is based on concepts such as the organizational capacity or the managerial capacity (cognitive factors) it belongs to qualitative areas. we then discard ourselves from quantitative parameters such as the age of the organization, of the staff, the size, the strength of the company that abound in the studies. 1.3 the problem the above questions can be synthetized by the finding and the question that follows, giving rise to thought and discussion: some organizations becoming mature lock themselves increasingly into their adaptations and are pushed inexorably such ”lava”, unable to change course, heralding a likely senescence index. therefore, how to spot early enough the gradual slowing of vital functions of an organization? what are the warning signs? to address this problem, we will define semantically the concepts of complex organization, organizational capacity and obsolescence.then we will depict the literature on our research field. finally we will present our model, the methodology underlying it and results of simulations will be debated during the ”discussion” which will close this presentation. 2 definitions and foundations of the model on the obsolescence 2.1 organization this is an intangible concept. for the organization is a myth[1], “it exists only events linked by causal loops” that creates meaning (’sense-making’). it is the interactions between the components of the organization that create a coherent and stable set. it is also a system of collective problem solving. in this spirit, the organization is the product of the choices of leaders and of their social representation. this collaborative system attempts to combine, to arrange the components of the work to be done to achieve the best organizational capacity, concept we will define below. organizations move, change over time1. the uncontrolled change creates asym1this topic has previously been the subject of dynamic simulations. cumenal d[3,4]. advances in systems science and application(2015) vol.15 no.2 121 metry, disrupts the harmony and curbs the progression of the organization. this one changes shape both structurally and in terms of the functional organization due to its history, its environment, internal resources and its management. thus, we enter in a field named “dynamics of organizations”. we think we can characterize this one by different stages: a conservative state where the weight of history, customs of the past pushes the organization to inertia despite active economic context; an adaptive state due to the pressure of the environment causing the competitive intelligence, to imitation; a cooperative state characterized by the maintenance of cohesion, the desire to avoid conflicts, the development of inter-organizational networks; finally, a state called progressing state which is defined by the desire to innovate and operate the organizational inimitable assets. of course, these states may be overlapped or juxtaposed. to move from one state to another, there is always energy dissipation, a cost of change. 2.2 the organizational capacity the end of 20th century saw the development of a school of thought based on organizational resources and on organizational skills named resource based view. organizational capacity is defined as a collective ability. we explain that the performance of the company benefits from a human capital (knowledge, skills) but also from a good organizational architecture (optimized process, formal and informal network of relations and communications, etc.) and from technologies. thus, the results can be explained by the presence of intangible resources shared by the players in the organization, but also difficult to imitate by competitors. according to laurent and gilles renard e st-amant “the concept of organizational capacity defines the ability or the skill of the organization to deliver activities with efficiency and effectively in deploying, combining and coordinating its resources and its expertise across various processes creating value, depending on the objectives defined previously [2].” 2.3 the definition of obsolescence from “obsolere”, “to fall into disuse”, in latin, the obsolescence reflects a behavior, an processing that is no more adapted to needs, to current and desired methods and to changing desired by the management . it turns out, over time, by slowing the functions and vital activities of the company (the unsuitable routines and processes business and support, etc.). the behaviors, the actions of an organization are no longer adapted to environmental change because the processes, routines, professional reflexes that have developed during the learning have become rigid or unable to agree to new situations. symptoms of obsolescence are often veiled, because they grow moderately over 122 didier cumenal: obsolescenceof organizations:amodeling approachinsystem dynamics time. for example, a product loses slowly market share; a business process is becoming more expensive and less efficient than a new process developed by competitors; staff engagement in the business of the company gradually becomes weak; the indifference of the staff for customers is becoming more marked; the leaders do not move in order to avoid achanges; information flows less and less. in general the reactions are increasingly inadequate and unsuitable for external events and circumstances. 3 literature review existing approaches for understanding the decline and death of organizations are fertile, although empirical studies on the spot are leaner. however, works on the obsolescence are not common and do not define properly the concept. seventy five percent of the literature on the decline of the firm were written during the last decade of the 20th century. we can mention books and papers by david a. whetten and kim s. cameron, authoritative on the subject of decay and mortality of the organizations[5][6]. most recently, yitzhak samuel describes in his book, models for predicting bankruptcy[7]. it also addresses the age, size and niche factors, performance and even corruption. stewart thornhill and raphael amit in an article named ”bankruptcy, firm age, and the resource-based view conducted a literature review on organizational mortality[8]. through the number of publications, it is clear for many researchers, that the age of the organization is a critical factor in mortality. however, stewart thornhill and amit raphael, learning , establish that the firms failure is also caused by a lack of managerial knowledge and in general by the lack of tangible and intangible resources (such as skills and organizational capacity)[9]. these firms are often paralyzed by a change in their environment. barry m. staw, lance e. sandelands, jane e. dutton show how organizational rigidity evolves faced with a challenge[10]. faced with a threatening environment, the organization tends to respond based on past experience deemed relevant, but unfortunately inappropriate given the new circumstances. moreover, in an urgent and tense situation, policymakers can hide certain information by discarding solutions they consider abnormal with regard to the opinions and beliefs of the group they belong to. somehow, filtered information must be in resonance with the unique thought of the leaders! erik larsen and alessandro lami operate a systems dynamics model by demonstrating how the causal relationships and feedbacks between variables, simulate over time, evolution of organizations inertia[11]. these authors show the opposition between resistance to change and the ability of organizations to develop new capacities to generate performance. the model, rather schematic is, according to advances in systems science and application(2015) vol.15 no.2 123 these authors, based on ecological theories and evolution of organizations. inertia is mentioned and briefly explained by the following factors: size, age, cumulative experience. figure 1 shows their model. it will be seen that size is correlated with age, which is an exogenous (external) variable to the system. fig.1 the organnization evolution model 4 model 4.1 methodology the model we have developed is not intended to generate assumptions to be tested. it seeks to show the stages of evolution of an organization to explain the causes of obsolescence thereof. it is therefore not limited to analyze each step separately considered. our model generates situations, rather than describing them. it strives to provide an pedagogical insight into the obsolescence onset. the model aims to study the evolution of organizational behavior and to identify early signs of wear and the weakening of main functions and activities of the organization. our approach is keyed on systems dynamics that studies how things, objects evolve and change state over time. john d sterman , in his book“business dynamics ”, sets principles2 of this approach[12] the approach is structured around the concept of causal network that establishes a cognitive diagram highlighting structural feedback loops. these are the relationships between the variables that 2for a criticism of the system dynamics, we carefully read the book by david berlinski[12]. 124 didier cumenal: obsolescenceof organizations:amodeling approachinsystem dynamics explain the system behavior over time, and not the system parameters considered separately. system dynamics highlights the counter-intuitive effects in the medium and long term, related to decisions taken in the short term, taking into account the effects of random events that can occur within, or in the environment of, the system. the model breaks down the system in state variables (reservoir) and flow variables (transit rules from one state to another). delayed effects and adjustment time are integrated into the model structure. the model uses numerical algorithms to produce simulated results. we can present the results as time series plots and measure the sensitivity of a variable to evolution of other one. system dynamics can be modeled using a set of equations with that standard form: x(t+∇t) = x(t) + x(t) • ∇t (1) in which is the model integration time step. however, the choice of value greatly influences the results of the numerical simulation as shown with the three graphs below (figure 2 : the population development; time on x-axis and quantities on y-axis). clearly, there are analytical solutions, but as soon as the model becomes complex, it is difficult to solve the equations in a conventional way. fig.2 the population development we will simulate our organization over several years to observe the evolution of the properties that characterize obsolescence. but for that, and before its use, we must expose the assumptions that characterize the structure of the model. advances in systems science and application(2015) vol.15 no.2 125 4.2 the causal structure of the model our model is structured into two subsystems or two causal diagrams: 1) the organizational capacity (c.0), 2) the cognitive ability of management (ccm). the following properties are not included in our model, voluntarily: age of the organization, age and size of staff. these attributes are often considered by the authors as exogenous variables. we believe that there is not necessarily a correlation between size and age of the firm, for example. likewise, these variables influence very little the inertia of the organization as we think sometimes. rather, we focus on capabilities of the organization, i.e. the potential3 and cognitive factors of the “top management” that ensure organizational transformation. it is the hypothesis that we assume. 4.3 the causal diagram of organizational capacity we defined above the organizational capacity. we modeled it (see figure 3) using the following properties: fig.3 organizational capacity causal diagram (o.c.) -organizational learning4 i.e. collective acquisition of skills by recurrent actions, -the skills produced by both organizational learning and training croutines5 built and distributed by internal collaboration within the orga3tangible and intangible resources provide the firm a competitive benefit. 4for a study on organizational learning, reference may be made to argyris, c[14]. 5routines represent individual and collective, repetitive work habits which may be driven by operational capitalized modes. 126 didier cumenal: obsolescenceof organizations:amodeling approachinsystem dynamics nization the flow of information and communication that contribute to this synergy between actors. this causal diagram shows many feedback loops that explain the results that we will comment during the presentation of results and discussion. for example, the more the widely disseminated (and unfiltered) information in the organization, the more it contributes through exchanges, to enhance coordination between employees or vice versa (direct relationship). on the other hand, the more the turnover is increasing, the more organizational skills6 are diminishing, due to departures and overall impact the organizational capacity. 4.4 the causal diagram of the managerial capacity collective rationalization is a cognitive process of sorting and selection of information, but this process also products resonance, answers already learned, as consistent with prevailing views in the firm. this mental state generates a collective logic that justify the decisions. necessarily wanting to be in agreement with the “others”, and especially with the “top management”, can be a powerful engine of collective errors. this rationalization taken to the extreme, leads to a denial, an occultation of the environment, filtering of information, but also a refusal to take a look on the past and on current state of the organization. the strategic blindness with unrealistic goals, can lead the entire organization to a thought, a single depiction. we have, unfortunately, observed this fact many times when a department persists in its judgment error. therefore excess of hierarchical pressure (the authority) was included as a variable in our model. beyond a pressure threshold, it acts to discipline, to structure responses of the organization. there is a “mahatma” effect by which governance behaves as a spiritual leader! its possible that problem could be identifyed, but the illusion that what we’ve always done is the best thing, trumps reality, generating actions that may be the wrong answers. our model in figure 4, below, reflects this divergence of representation with a gap that symbolizes a “cognitive dissonance”7 . with the latter, the information, leading indicators of entrepreneurial tragedy are ignored because they are too discordant with the successes until recently. the intensity of this difference is as important to the overall performance8 of the organization. moreover, poor re6organizational skills are generally depicted as coordination of resources and capacity[15]. they are linked to an accumulated experience (organizational learning). 7pioneered the concept of cognitive dissonance is leon festinger.according to this theory, when events lead a person or organization to act at odds with his beliefs, these acts produce a state of discomfort andtensioncalleddissonance.to reduce this condition, the person or organization is allowed to exculpate himself by constructing an argument near a deception! and management can persist in his error justifying its decisions and actions despite the good sense of “field”[16]. 8we define the concept of performance as a business development index value created by taking into account the quality of the responses of the“top management” facing a problem. advances in systems science and application(2015) vol.15 no.2 127 sults reinforce hierarchical control by systematically operating (routines) modes, but also impacting organizational agents behavior of the organization, themselves sensitive to performance. fig.4 managerial capacity causal diagram (m.c.) 4.5 results and behavior of the model it is interesting to study properties of all variables only on a qualitatively basis. we focused on the behavior, i.e. on shape of the curves rather than looking numerical values hard to estimate when the concepts of organizational and managerial capacity is mentioned. the range of variation of each variable extends for most of them from 0 (minimum) to 1 (maximum). it corresponds to the assumptions, and common sense “on the ground”. for instance, increased internal information filtering reduces synergy, knowledge sharing or informal9 kills formal knowledge and consequently performance. the simulation performed using a system dynamics software (stella, from isee systems, v.10.0.6) produces a typical organization behavior over a period of 10 years. you can of course change the duration of the simulation. watching graphical patterns on charts : 5, 6 and 7, we note a performance in sharp decrease. the diagnosis of the curves shows a slowdown in performance from grade 8 (curve 4 in chart 1 below) and then decreased. it can be seen also from the sixth year a slowdown in the level of organizational skills (curve 3), that of organizational routines10 (curve 5) and finally a breathlessness of organizational learning (curve 1). these are state variables which gradually accumulate flows acting as drivers 9it is, for example, short circuits in the official channels of communication in the organization. 10as a reminder, the routine reflects the regularity of collective behavior induced by procedures. 128 didier cumenal: obsolescenceof organizations:amodeling approachinsystem dynamics of change over time.chart 2 below shows a bad fit between dominant and learned responses that are an expression of “top management” thought and timely responses (curve 2) produced by the community in light of the problems identified. the gap persists for many several years, peaking in the third year of the firm. then, the two curves are similar from grade 611 to deviate from each other the 7th grade. we believe this difference is a revealing leading indicator of future performance degradation. finally, it takes place in companies when some people say that top management persists in his error by maintaining an axis of inappropriate development. chart1 however, the entire management team adopts this guided behavior and looking for one or several arguments to justify their position according to the cognitive dissonance theory, which we mentioned earlier. the chart 3 bellow, analyzes several behaviors:the impact of information filtering (curve 1), the degree of formalization and rationalization of the weighted procedures organization (curve 2) and intra-organizational synergy (curve 3). the latter also depends on the information that flows and the level of technical and managerial skills move through the organization. these are also indicators revealing symptoms which anticipate possible resources shortage and organization’s capabilities. actually, we hear, for example, that information does not flow, or the procedures are too heavy, reducing more and more autonomy in the work of each. 11is this rapprochement leads to the reason? advances in systems science and application(2015) vol.15 no.2 129 chart 2 chart 3 5 discussion and conclusion our model consists of five stock variables12(levels) corresponding to five integrations or state equations described as follows: e(tk) = e(tj) + ∫ tk tj [input(t)− output(t)]dt (2) e(tk)= thestateatktime. e (tj) = thestateatpreviousjtime. (j = k − 1)∑ tj tk = algebraic sum of inputs and outputs between t time and t time + increment time step dt. inputs and outputs over time depends on “auxiliary 12skills-learning-behavior-routine sand performance variables that have been previously defined. 130 didier cumenal: obsolescenceof organizations:amodeling approachinsystem dynamics variables” that changes according to causality with other variables (stocks, flows or auxiliaries). auxiliary variables are expressed with mathematical functions often based on non-linear correlations. our systemic model is built on feedback processes with, often, a delay between cause and effect. our model can help us to understand the emergent properties of the obsolescence and to analyze the results of an a concept not easily accessible through statistical interpretations of reality. we support the following proposal: analysis of organizational capacity and of managerial ability, i.e. symbolic representations and the answers given by the leaders facing a problem, can anticipate a resources and vital functions slowdown of the organization. we did not cover theories that emphasize the organizations ecology (size, age, etc.) to help us focus on intangible properties. of these ,we may recall the important role of learning and organizational skills, filtering and retention of information, the coordination and collaboration process(organizational synergy), collective rationalization of responses from the leaders giving the illusion that what they do is the best thing (further reinforced by the weight routines or past habits). the simulations show that weakly filtered information combined with a normal hierarchical pressure (more or less) can achieve a good level of performance. in this case, the staff has the opportunity to provide possible answers to the context. obviously, these responses are determined by the skills level and by the received or perceived information quality. censorship or a bad collection of information weakens staff responses and strengthens, conversely, the traditional answers provided by the “top management”. these are related to the rationalization of produced responses and to the organizational routines that capitalize collective behaviors generated by the procedures and by the taken decisions. in the latter situation, the operational process integrates finally the dominant responses produced by the “top management”, in finding good reasons to accept (see the concept of “cognitive dissonance”). our simulation explain the importance of the gap between the collective responses learned by staff and dominant responses supplied by the “top management”. this gap is an important performance gear. 5.1 model limitations we may recall the important role of learning and organizational skills, filtering and retention of information, the coordination an collaboration process (organizational synergy), collective rationalization of responses from the leaders giving the illusion that what they do is the best thing (the rest reinforced by the routines weight or the habits of past). the dynamics systems’ mathematics is widely used in physics. however, social sciences resist using them. indeed, formalize mathematically organizational behavior is very optimistic! advances in systems science and application(2015) vol.15 no.2 131 the model quickly becomes a black box for an external observer who wants understanding the results produced (non-linearity is one of the reasons). another limit is the validation of the model. we don’t have neither historical nor quantitative data about concepts presented. as we have previously stated: our model is more generative than descriptive of existing situations. it suggests events and behaviors. it allows us to anticipate, and to have a better understanding. in conclusion, we have been always convinced that an organization gets its best performance when the representations of a problem, the supplied answers, aims of employees and those of top management are similar. fig.5 bayesian model outline when societies and civilizations are becoming complex and prosperous, they have difficulty adapting to their environment. their potential assets appear to be inadequate to move to their next evolution stage. 5.2 the potential model evolution we are currently developing a bayesian network based model (figure 5) to anticipate the phenomenon of obsolescence. we intend to test this tool on several professional situations. the result will be the topic of a future communication. references [1] weick, karl e. (1995), “making sense of the organization”. blackwell publishing, pp.179-224. 132 didier cumenal: obsolescenceof organizations:amodeling approachinsystem dynamics [2] st-amant gilles et renard laurent. (2003), “capacity, organizational capacity and dynamic capacity: a proposal for definitions”. research paper, uqam. [3] cuménal, d. (2005), “simulating organizational change: moving and shaking”. in: 23rd international conferenceof the system dynamics society, july, pp.17-21. [4] cuménal, d. (2010),“can we model and simulate the changes of state of the organization over time?” . management & sciences sociales,vol.5, pp.91122. [5] whetten david a. (1979), “the organizational life cycle, chapitre 10”. josseybass publishers, pp.343-373. [6] cameron kim s.,. kim myung u,. whetten david a.(1987), “organizational effects of decline and turbulence”. administrative science quarterly, vol.32, pp.222-240. [7] yitzhak samuel. (2010), organizational pathology,transactions publishers. [8] thornhill stewart, amit raphael. (2003), “bankruptcy, firm age, and the resource-based view”. organization science volume,vol.14. [9] thornhill stewart, amit raphael. (2003), “learning about failure: bankruptcy, firm age, and the resource-based view”. organization science,vol.14. [10] staw barry m., sandelands lance e., dutton jane e. (1981), “threat rigidity effects in organizational behavior: a multilevel analysis. ”. administrative science quarterly,vol.26, pp.501-524. [11] larsen erik., lami alessandro. (2002), “representing change : a system model of organizational inertia and capabilities as dynamic accumulation processes ”. elsevier simulation modelling pratice and theory,vol.10, pp.271-296. [12] sterman john d. (2000), business dynamic,mcgraw-hill. [13] berlinski david.(1978), “on system analysis: an essay concerning the limitations of some mathematical methods in the social”. political, and biological sciences, mit press. [14] argyris, c, and schön d. a. (1996), “apprentissage organisationnel”. théorie, méthode, pratique, de boeck université. advances in systems science and application(2015) vol.15 no.2 133 [15] grant r.m. (1991), “ the resource-based theory of competitive advantage : implications for strategy formulation ”. california management review,vol.33, pp.114-135. [16] festinger léon. (1957), a theory of cognitive dissonance,standford university press. [17] amit r., schoemaker p.j. (1993). “strategic assets and organizational rent”. strategic management journal, vol.14, pp.33-46. [18] bossuet jacques bénigne. (1722),organizational effects of decline and turbulence, hachette réédition. corresponding author dr. didier cumenal can be contacted at: cumenald@wanadoo.fr sv-lncs adv syst sci appl 2018; 02; 107-120 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/539 copyright ©2018 assa. adv. in systems science and appl. (2018) an approach for developing context-aware adaptive information systems mahmoud hussein1 1) faculty of computers and information, menoufia university, egypt e-mail: mahmoud.hussein@ci.menofia.edu.eg abstract. context-awareness and adaptability are highly desirable features for modern information systems that are operating in dynamic environments. unfortunately, such information systems are still difficult to build. issues like (a) lack of an approach to compose a system from a set of functional services and context providers and (b) lack of a mechanism to enable runtime adaptation of the system in response to changes in its operating context need to be solved for easy and effective development of such information systems. in this paper, we introduce a novel approach to developing context-aware adaptive information systems. in composing the system, our approach explicitly separates but relates the system model and the context model, so that they and their relationships and changes can be clearly captured and managed. we also support runtime changes to the system, the context model, and their adaptation logic in response to the context changes. we do so by maintaining runtime representations of their models, and then we use these representations to realize the required changes. we have developed a tool to enable modelling the system and generate its implementations from their models. to demonstrate the viability of our approach, we used it to develop a context-aware adaptive travel guide system. we also assessed the performance of our approach through measuring the overhead caused by performing the system adaptation at runtime. the results demonstrate that our approach is capable of performing runtime adaptation with a small overhead. keywords: information systems engineering, context-awareness, system adaptation, context and system modelling, model driven development. 1. introduction there is an increasing demand for information systems that are aware of their contexts and dynamically adapt themselves at runtime in response to changes in these contexts [19]. consider, for example, a context-aware travel guide system that a tourist can use to plan his trip to visit some attractions. this system needs to be composed of a set of functional services (e.g. route planner and attraction finder) that takes the context information (e.g. weather and traffic information) into account to operate effectively (e.g. give the tourist better suggestions for routes and attractions). existing approaches focus on composing the system’s functionality (e.g. [4, 6-8, 10, 14, 25-26, 29]), but they leave the task of capturing the context and relating it with the system to the system developers. as such, the system implementation stage become more complex and requires a developer with experience in building such type of systems. in practice this expertise is usually lacking, and it will be a tedious task for the developers. while the travel guide system is in operation, it needs to adapt itself to keep achieving the tourist needs. for example, when the tourist wants to include a route planning service that was not provided to him free initially, the system should adapt itself by adding such service (i.e. changing system’s functionality), the context providers needed by that service to operate effectively such as a traffic information provider (i.e. adapting the context model), and some adaptation logic to enable the switching between different route planning services based on 108 m. hussein copyright ©2018 assa. adv. in systems science and appl. (2018) their availability and provided qualities (i.e. changing the adaptation logic itself). most of existing approaches supports system’s functionality changes. however, existing approaches intertwined the context with the system as discussed above. as such, the addition of a context provider and relating it to the system becomes a difficult and error prone task. in addition, they use goal-policy [14, 29], utility-policy [7-8, 26], or action-policy [4, 6, 10] to capture the system’s adaptation logic. most of these approaches are not able to change the adaptation logic at runtime except the approach proposed by andrade et al. [2] that has the ability to change the adaptation logic manually at runtime by the system developer. however, the communications between the system developer and the running system are usually infrequent and sometimes impossible [18]. as such, the process of changing the system adaptation logic needs to be done automatically without the developer involvement. to address the above drawbacks, in this paper, we propose a model-driven approach to developing information systems that are able to dynamically adapt themselves in response to changes in their contexts, which we call context-aware adaptive systems. our approach has the following novel features. firstly, in composing the system, we explicitly separate but relate the system model and the context model, so that their relationships, changes, and changes impact across the system and its contexts can be clearly captured and managed. secondly, our approach enables the runtime changes to the system, its contexts, and their adaptation logic. we do so by maintaining runtime representations of their models and having two sets of rules: adaptation rules and adaptation meta-rules and. in response to context changes, the adaptation rules are used to decide the required changes to system and its contexts while the adaptation meta-rules specify changes that need to be applied into the adaptation rules themselves. in addition, our runtime environment enables applying the required adaptation actions. finally, our tool can be used to model a context-aware adaptive system and generate its implementations from their models. the remainder of the paper is organized as follows. we start by introducing a motivating scenario in section two. our approach to modelling context-aware adaptive information systems is described in section three. in section four, we discuss our tool support for modelling and realizing context-aware adaptive systems. in this section, we also measured the runtime overhead of adding the context-awareness and adaptability features to an information system to assure our approach’s applicability. section five analyzes existing work with respect to our approach. finally, we conclude the paper in section six. 2. motivating scenario and requirements analysis the travel guide information system helps a tourist to find attractions, plan his trip by providing suitable routes, and locate a restaurant. below are a few scenarios the tourist experiences in using this application during his visit to melbourne one day. scene 1: in the morning, the tourist starts to plan his trip. based on his preferences (e.g. outdoor attractions) and the weather forecast for that day (e.g. sunny), the travel guide suggests to him a number of attractions such as melbourne aquarium, royal botanic gardens, melbourne zoo, etc. he selected some of these attractions to visit using his rented car, and then a set of routes are displayed to him. these routes are calculated based on his current location, his attractions list, his driving preferences (e.g. shortest route), and current traffic information (e.g. blocked and congested roads). he selected a suitable route and started to explore the attractions. scene 2: at lunch time, the application suggested to him a number of nearby restaurants that matches his food preferences while taking into account the locations of the remaining attractions he is planning to visit. when he selects a restaurant, the trip route is re-planned automatically to take into account the restaurant location. an approach for developing context-aware adaptive information systems 109 copyright ©2018 assa. adv. in systems science and appl. (2018) to develop the travel guide system that meets the tourist needs, a set of general requirements need to be considered: compose a context-aware system (req. 1): the travel guide application need to be consisted of a set of functional services (i.e. attractions finder, restaurant locator, and route planner) that interact with each other to meet the tourist needs, while considering some quality requirements (e.g. fast route planner). in addition, these services should take into account the context information with a certain quality (if required) to give the tourist better suggestions. for example, the route planner needs the traffic information updated to the last minute to provide accurate estimations for the possible routes travel times and to display the routes that are less congested first. as such, there is a need for an approach that can be used to compose a set of functional services and context providers to provide the required context-aware functionalities while considering their quality requirements. adapt the system while it is in operation (req. 2): the travel guide provider may want to provide the attractions finder service free, while the tourist should pay to use the other services. as such, while the travel guide system is in operation, the tourist may wants to include the route planner service that is not provided free to him initially. to include such service, several changes need to be applied into the running system. firstly, the system needs to be adapted by incorporating the route planner service (i.e. adding a functional service). secondly, to find a suitable route for the tourist, there is also a need to acquire the tourist driving preference and current traffic information and use them in calculating and suggesting the routes (i.e. changing the context model by including context providers and their relationships with the system functionality). finally, the traffic information may become unavailable for a period of time (due to communication failure with road side units, for example), and then the travel guide provider need to have two route planners. one of them considers the traffic information in calculating the routes while the other does not take it into account. to switch between these two route planners while the system is running, the system adaptation logic needs to be changed, so that a suitable route planner can be selected based on the traffic information availability. 3. the approach to support the development of context-aware adaptive systems, we propose a model-driven approach. model-driven development is the notion of constructing a model of the system that can be transformed to a real system [9]. in this section, we introduce an organisational approach and associated notation that can be used to model such systems and in the next section we discuss how to transform a system model created in this notation to a real system. 3.1. composing context-aware adaptive systems: an organizational approach to develop a context-aware adaptive information system such as the travel guide system, it need to be composed of a set of functional services and context providers that interact with (related to) each other to meet the user needs (req. 1). in addition, while the system is in operation it needs to adapt itself in response to context changes to preserve the achievement of the user goals (req. 2). a system as an organisation is “a set of dynamic relationships between its roles to maintain the system viability in a changing environment” [21]. the relationships can be seen as defining the system, where they are used to specify the system roles position descriptions. these descriptions specify what tasks the system roles should do, while there are players who actually perform the tasks by playing these roles. in addition, in response to environment changes, the system manager may change the system roles, their relationships, and their players’ bindings to maintain the system viability. for example, a business organization is a 110 m. hussein copyright ©2018 assa. adv. in systems science and appl. (2018) collection of roles (e.g. public officer, secretary, etc.) that are related to each other through contracts (relationships) [12]. these contracts define the permissible interactions between the organization roles and their mutual obligations (i.e. what tasks a role can request from others). in addition, the organization roles are played by employees or outsourced to external organizations. furthermore, to maintain the business viability in response to business market changes, the business manager may add roles, hire employees, etc [24]. similarly, we can see the travel guide system as an organization. in this organization, there are a set of functional and contextual roles that interact with each other to provide the system required functionally while taking the context information into account. in addition, these functional and contextual roles are played by functional services and context providers respectively (i.e. req. 1). furthermore, there is an organizer role that is able to adapt this organization in response to context changes to keep achieving the user needs. for example, add the route planning service and the other elements related to it when the tourist needs such service (i.e. req 2). because of the above correspondences, we followed the organizational approach in modelling context-aware adaptive information systems. following the organizational approach, a meta-model for a context-aware adaptive system is shown in figure 1. the system composition is consisted of a set of roles that are related to each other through contracts and each role can be played by one or more players. we have three types of roles: functional, context, or organizer as shown in figure 1. the functional roles represent the systems functionality while context roles capture the context model. the organizer role bound to its player is used to manage the system by adapting it in response to context changes. in addition, to capture the system elements’ different relationships, we have two types of contracts: functional and contextual. the functional contract is formed between two functional roles (i.e. role a and role b) to capture their functional interactions and mutual obligations. on the other hand, the contextual contract is formed between a context source and a context consumer to capture the contextual requirements and their required quality. furthermore, for each type of role, we have a corresponding player (i.e. context provider, organizer player, and functional player). in the following, we describe these concepts in details with examples from the travel guide system. fig. 1. a meta-model for context-aware adaptive software systems 3.2 modelling system’s functionality and context model using the above meta-model concepts, we introduced a modelling language that can be used by the software engineer to model a context-aware adaptive system. the basic elements of this language are shown in figure 2 and listings 1 and 2. we introduced these specific graphical notations and textual representation to ease the system modelling task compared to using general purpose languages (e.g. uml) and to enable the system code generation from its model [17, 23]. a uml profile was an option for defining our domain-specific language [11], however the contracts in our model are more complex (see listings 1 and 2) and it cannot an approach for developing context-aware adaptive information systems 111 copyright ©2018 assa. adv. in systems science and appl. (2018) be captured easily using the uml profile concepts. in the following, we describe our modelling language elements and use them to model the travel guide system. the functional system model: the system’s functionality is modelled as a set of functional roles that interact with each other though functional contracts. in addition, each role can be played by one or more functional players. functional contracts: the functional contracts are used to capture the relationships between the system functional elements and they have the following items. first, each contract has an identifier and it is formed between two functional roles. for example, the contract “fc4” is formed between the user and route planner roles as shown in figure 2. second, it has a set of permissible interactions between the contracted roles. each interaction as shown in listing 1 has (a) an identifier (e.g. i2) and a name (e.g. request routes), (b) zero or more input parameters (e.g. destination and current location), (c) a direction to specify who is responsible for providing the operation included in that interaction (e.g. “atob” which mean the route planer role is responsible for providing route calculation operation), (d) zero or more context parameters (e.g. blocked and congested roads), and (e) a return type (e.g. routes). third, the contract defines a set of conversion clauses that specifies the acceptable sequences of interactions between the functional roles. we used interaction rule specification (irs) language to specify these temporal constraints as shown in listing 1 [15]. for example, “c1” specifies that “request routes” operation must be invoked before invoking “select a route” operation. forth, to define the system non-functional requirements (e.g. response time, availability, and reliability), we specify a set of obligations on the functional interactions and we followed the web service level agreement (wsla) language in defining these obligations [20]. for example, the obligation “o1” in listing 1 specifies that the request route operation “i1” should not take more than 5 seconds in calculating the routes. finally, the interactions between the functional roles can be permitted or blocked based on the context information. for example, the interaction “i2” cannot be performed when the traffic information is not available and then it needs to be blocked. to do so, each contract has a set of regulation rules to regulate its interactions. we used event-condition-action rules to decide permitting or blocking an interaction [22]. an example of such rules is the rule shown in listing 1. this rule is triggered when the interaction “i2” is requested (i.e. i2.activated). when the traffic information is not available, the interaction “i2” is blocked (r2) while it is permitted otherwise. fig. 2. the context-aware travel guide system model 112 m. hussein copyright ©2018 assa. adv. in systems science and appl. (2018) interaction clauses: i1: {requestroutes (destination, currentlocation), atob, routes}; i2: {requestroutes1 (destination, currentlocation), contextparameter (blockedroads, congestedroads), atob, routes}; i3: {selectroutes (routeid), atob}; conversion clauses (temporal constraints): c1: {i1 precedes i3 globally} c2: {i2 precedes i3 globally} obligations: o1: {i1, responsetime, lessthan , 5 seconds} o2: {i2, responsetime, lessthan , 7 seconds} regulation rules: r1: {block the interaction i2: when i2.activated; if trafficinformationavailability == false; do i2.block( )}; functional roles and their players: the functional roles represent the system’s functionality where each role position description is derived from its relationships (contracts) with other functional roles (i.e. an abstract definition of the tasks that need to be performed by a player who will play that role). the functional role also has one or more functional players to provide its actual functionality at runtime. for example, there are two route planning algorithms that can play the route planner role as shown in figure 2. routeplanner1 considers the traffic information in calculating the routes while routeplanner2 do not take the traffic information into account. the context model: the context model is represented through a set of contextual contracts that are formed between context sources and context consumers to capture the system contextual requirements. in addition, there are a set of context providers to make the context information available to the system. contextual contracts: the contextual contract defines what context information required by a system element (i.e. the context consumer in figure 1 such as functional contract, organizer role, or functional role) and the quality of this required context (e.g. accuracy, freshness, etc.). for example, in listing 2, a contextual contract “cc6” is used to specify that the functional contract four needs to know (a) the congested roads with freshness up to the last minute and accuracy greater than 80%, and (b) the blocked roads with accuracy greater than 95% so that the route planner can calculate the routes effectively. in addition, it needs to know the traffic information availability to decide permitting or blocking the interaction “i2”. listing. 2. the contextual contract between the traffic information role and the contract fc4 contextual contract id cc6: trafficinformation_fc4 parties: context source:trafficinformation; context consumer: fc4; context attributes: a1: congestedroads; a2: blockedroads; a3: trafficinformationavailability; context attributes quality: q1: {a1, freshness, lessthan , 1 minute} q2: {a1, accuracy, greaterthan, 80%} q3: {a2, accuracy, greaterthan, 95%} context roles and context providers: the system contextual requirements are captured through a set of contextual contracts as discussed above. these contracts are then used to drive the context roles position description, where each context role bound with a context provider (each context role can have one or more context providers) is responsible for providing some context information. for example, a weather role bound to its provider (i.e. the weather service) listing. 1. the functional contract between the user and the route planner functional roles functional contract id fc4: user_routepalnner parties: role a: user; role b: routeplanner; an approach for developing context-aware adaptive information systems 113 copyright ©2018 assa. adv. in systems science and appl. (2018) is responsible for providing current temperature and rain level to the attraction finder service, so that a correct suggestion is given to the user based on current weather conditions. 3.3 engineering system’s adaptability through organizer role and its player to make the system able to adapt itself in response to context changes, there is a need for a mechanism to first decide when and what to adapt and then apply the decided adaptation actions. we do so through the system organizer role and its player. we modelled the organizer player as a set of event-condition-action rules that can be used to decide when and what to adapt [22]. the event and condition part of a rule specifies when to adapt, and the action part of the rule defines what to adapt. the events that activate the adaptation rules are usually context changes where the system needs to adapt itself in response to these changes. the rule condition is used to specify the context situation that needs a system reaction(s). the rule action is a set of adaptation actions to cope with the context changes. in general, the adaptation actions are to add, remove, or modify a system element. for example, to change the system functional and context roles, we have three adaptation actions: add role, remove role, and change role-player binding. in the same manner, we have actions to add, remove, and change a functional contract and a contextual contract. to apply the adaptation actions, the organizer role is engineered with a set of standard methods that are corresponds to the adaptation actions so that the organizer player can invoke these methods to adapt the system while it is in operation. an example of an adaptation rule is shown blow. this rule is activated when the tourist wants to include the route planning service in his application (i.e. event). in response to the changes in the tourist needs (i.e. he wants the route planning service, condition), the application adapts itself (i.e. actions) by adding a set of roles (e.g. route planner), adding a set of contracts (e.g. fc3 and fc4), and bind players with the added roles (e.g. bind route planner role with route planner one). rule “adaptationrule1”: { when valuechanges (routeplannerselected); if routeplannerselected == true; do addrole(“routeplanner”), addcontract (“fc3”), addcontract (“fc4”), addrole (“trafficinformation”), addcontract (“cc6”), addcontract (“cc8”), bind(“routeplanner”, “routeplanner1”), bind(“trafficinformation”, “roadsideunit”)}; when the running system is changed, its adaptation rules may need to be adapted too. for example, after performing the adaptation actions in the above rule, we can see that the route planner role has two players and the traffic information role has two context providers as shown in figure 2. as such, there is a need for some rules to decide the switching between the added roles-players. to do so, we specified another set of rules (i.e. adaptation meta-rules) that adapt the adaptation rules themselves. these rules have the same structure of the adaptation rules describe above, but they have different adaptation actions (i.e. add rule and remove rule). an example of such meta-rules is show blow. in this rule, four adaptation rules are added to enable a correct selection of a route planner algorithm and a traffic information provider when the route planning service is included into the running system. rule “adaptationmetarule1”: { when valuechanges (routeplannerselected); if routeplannerselected == true; do addrule(“selectrouteplanner1”), addrule(“selectrouteplanner2”)}; addrule(“selectroadsideunit”), addrule(“selecttrafficinfoprovider”)}; 4. implementation and approach evaluation in this section, we describe the tool that has been developed to support the modelling and realization of context-aware adaptive information systems, and how to use this tool to develop 114 m. hussein copyright ©2018 assa. adv. in systems science and appl. (2018) the travel guide system described in section two. we also measure the performance overhead of adding the adaptability feature to a software system using our approach to assure its applicability. 4.1 tool support for modelling context-aware adaptive system we have developed a tool to enable the modelling of a context-aware adaptive system. it enables the software engineer to specify the system roles, players, contracts, and adaptation rules. screenshots from our tool during the system modelling are shown in figure 3. fig. 3. screenshots from our tool during the modelling of the travel guide application to simplify the process of specifying the adaptation rules, we provide a gui that helps the engineer in codifying these rules. this gui is used to specify the rule conditions and adaptation actions. the rule events are directly inferred from the rule conditions, where they usually are the changes in the context attributes that are used in the rule conditions. an example is shown in figure 4, where the engineer can specify the current situation i.e. the user wants to include the route planning service and a set of adaptation actions need to be performed in this situation such as add route planner role (figure 4-a), bind the route planner one player to route planner role (figure 4-b), and add functional contract “fc4” (figure 4-c). fig. 4. specifying the travel guide system adaptation rules using our tool 4.2 realizing context-aware adaptive systems to realize context-aware adaptive systems, we used road framework1 where it follows the organizational approach as our approach does. this framework is an extension to the apache axis22 to realize adaptive software systems [16]. to use this framework, we used our tool to transform the model described in the previous section to a model that is compatible with the road framework. in the following we describe the major transformations we did while the others are one-to-one mapping. 1 http://www.swinburne.edu.au/ict/research/cs3/road/ 2 http://axis.apache.org/axis2/java/core/ http://www.swinburne.edu.au/ict/research/cs3/road/ http://axis.apache.org/axis2/java/core/ an approach for developing context-aware adaptive information systems 115 copyright ©2018 assa. adv. in systems science and appl. (2018) first, the use of context information as extra parameter in a functional interaction is not considered in road model. but, during the system execution, the functional contracts are used to mediate the interactions between the system roles. when an interaction reaches a contract a set of rules are executed to decide permitting or blocking it (in road these rules are codified as drools rules3). as such, we added a rule to these rules that is activated when a contextualized interaction is received. this rule is responsible for updating the context information required by this interaction before sending it to the destination role. an example of such rule is shown blow. this rule updates the context information (i.e. congestedroads and blockedroads) of the request route interaction when it is received at the functional contract “fc4”. second, in road model, the context information is maintained as a set of facts. each fact contains one or more context attributes. these facts can be provided or consumed by the system roles or its functional contracts. in our model, the contextual contracts are used to drive context roles descriptions, and then each context role can be seen as a collection of context attributes. this makes a correspondence between a fact in road terms and context role derived from contextual contracts in our model. as such, we use the contextual contracts to drive context roles descriptions, and then we transform each role position description to a fact in road model. third, to enable the execution of the adaptation rules and meta-rules described above, we transform them to drools rules, so that the drools rule engine can be used for their execution to decide the required adaptation actions while the system is in operation. the result of transforming the “adaptatiorule1” described above is shown blow. in transforming the rules, we used the rule “when” part to specify the rule event (e.g. the user need of the route planning service is changed). in addition, the rule “then” part is used for capturing both the rule condition and action. to evaluate the rule condition part, we created a class called “conditionevaluator”. this class has a method called “evaluate” that takes a condition as an input, and then it replace the condition variables with current context values and returns true or false based on the condition evaluation. when the condition is evaluated to true (i.e. the user wants the route planning service for example), a set of adaptation actions are added to the adaptation script (e.g. actions.addaction(“addrole_routeplanner”). by the same manner, the adaptation meta-rule “adaptatiometarule1” can be generated. the adaptation rules are used to generate an adaptation script. this script is then used by the organizer player to adapt the running system by invoking the organizer role standard adaptation methods that are corresponding to the required actions. these standard methods require some details to execute. for example, to add a role there is a need for the role name, 3 http://www.jboss.org/drools http://www.jboss.org/drools 116 m. hussein copyright ©2018 assa. adv. in systems science and appl. (2018) identifier and description. here comes the role the maintained runtime representation of the system models where they have these required details. as such, the organizer player parses these representations to get the required details and generate the executable actions. for example, adding the route planner role in the above script is transformed to organizer.addnewrole(“fr2”, “routeplanner”, “role represents route planning service”). the organizer variable is a reference to the running system organizer role. a similar mechanism is used for changing the adaptation rules, where we have a reference of the loaded adaptation rules (i.e. instance of knowledgebase 4 class) and the methods addknowledgepackages and removerule are used to add and remove adaptation rules respectively. the above transformation process is automated in our tool. when the software engineer completes the system modelling, he can press a button that generates the files required by the road framework to deploy an instance of the system. this instance contains the system roles, their contracts and the generated organizer player. to have a fully running system, we have developed a set of functional players and context providers. for example, we used google maps5 services to develop he route planners, attractions finder, and restaurant locator players. we also have developed a gui to enable the user interactions with the provided services as shown in figure 5. in figure 5-a, the application only includes the attraction finder service which is provided free initially. this service suggested to the tourist a set of attractions based on his preference, his current location, and the weather forecast. he can select some of them to be included in his attractions list. to plan a route to see these attractions, the tourist requests the route planning service to be included in his application. as such, the application is adapted to include such service by performing the adaptations described above. after the internal adaptations are performed, the application gui is changed also by including the route planning service. when this service becomes available, it acquires tourist location, his attractions list, and traffic information to suggest a suitable route (see figure 5-b). fig.5. the context-aware adaptive travel guide application in action 4.3 performance evaluation6 the overhead in our approach is the extra time needed to adapt the system while it is in operation. this can be calculated by the time required to monitor the context, decide the required adaptation actions, and act these actions. monitor the context. in our approach, there is a need to keep track of some context variables that cause system adaptation. when any of the variables that the adaptation rules are interest in changes, this change is notified to the organizer player to decide the needed 4 http://docs.jboss.org/jbpm/v5.1/javadocs/org/drools/knowledgebase.html 5 http://code.google.com/apis/maps/documentation/webservices/ 6 a pc with intel core 2 due 3 ghz cpu and 3 gb ram is used as the test-bed and drools-5.1is used as the rule engine. http://docs.jboss.org/jbpm/v5.1/javadocs/org/drools/knowledgebase.html http://code.google.com/apis/maps/documentation/webservices/ an approach for developing context-aware adaptive information systems 117 copyright ©2018 assa. adv. in systems science and appl. (2018) adaptations. the time required to notify the system organizer with a context variable change equals to 14.59 milliseconds in average. decide required adaptation actions. when the context is changed, the adaptation rules need to be evaluated to decide the required adaptation actions. the overhead in the decision making process is laid in rules loading time at the beginning and their execution time in response to context changes. to measure that overhead, we used sets of rules with sizes 10, 20, 30, 40, and 50. figure 6-a shows that the rules loading time varies from 1.81 to 2.05 seconds based on the adaptation rules size. this is not much overhead where it is usually performed once at the system start-up. in addition, the time to execute the rules is between 14.8 to 29.9 milliseconds (see figure 6-b) which cannot be considered as an overhead also. fig.6. the adaptation rules loading and execution times apply the adaptation actions. we have different adaptation actions that can be performed to adapt the system in response to context changes. table 1 summarises the average time needed to apply some of the required adaptation actions in milliseconds. in table 1, we only show the actions for adding some elements to the system, where they are of interest from the user point of view (i.e. they are added where the user wants them). the adaptation actions for removing parts of the system have a small overhead and are performed while it is running. as such, they do not affect the user interactions with the system. due to space limitation, we do not present them here. table 1. the time required to apply the required adaptation actions in milliseconds adaptation action required time adaptation action required time add functional/context role 198.953 change role-player binding 0.0348 add functional contract 0.204 add adaptation rule 0.0096 add contextual contract 1.565 a delay that the tourist can experience in using the travel guide system is happened when he wants to include the route planning service. to include this service, it takes around 422 milliseconds which cannot be considered as a delay to the tourist. 5. related work and discussions a number of approaches have been proposed to support the process of developing contextaware adaptive systems. some of them propose a way to model and realize such systems [4, 8, 14, 25-27, 29] while the others provide a framework or a middleware to help the software engineer in realizing the context acquisition and interpretation [5, 13, 28] or the system runtime management [1-3, 7, 10]. in this section, we analyse existing and our approaches in relation to the requirements we have identified in section two. compose a context-aware system: existing approaches to developing context-aware adaptive systems can be classified into two categories. on the one hand, a set of approaches consider the system functionality as a single service and the context information is used to 118 m. hussein copyright ©2018 assa. adv. in systems science and appl. (2018) adapt this service operational parameters [27, 29]. on the other hand, a set of approaches has been proposed to compose a system from a set of components that are changeable at runtime to cope with context changes [4, 8, 14, 25-26]. however, they capture the relationships between the system and its context implicitly during the system development except few who only consider these relationships explicitly at design time [4, 27, 29]. as such, the system implementation complexity is increased and the system-context relationships changes become difficult and error prone. in addition, in composing system’s functionality, existing approaches connect the composed components directly (i.e. linking the components required and provided interfaces directly) [4, 8, 14, 25-26]. as such, in large scale systems, the composed elements complex interactions and their mutual obligations become difficult to capture and manage. in our approach, we explicitly represent the relationships between the system and its context and between the system functional elements themselves. as such, we can clearly compose and realize a context-aware adaptive information system. similar to our approach sheng et al. [27] propose approach who only consider these relationships explicitly only at design time and they also consider the system as a single service runtime adaptation of the context model: at runtime the context model may change by including a context attribute or remove an existing one as shown in our motivating scenario. however, most of existing approaches have only a design time context model and usually it disappears during the system implementation where it is intertwined with system’s functionality and/or management. few approaches keep the context model explicit at runtime [26, 28], so that they can switch between different context providers or include new providers while system is in operation. however, they do not have the ability to change the context model itself by adding, removing, or modify the system required context attributes. our approach has an adaptable runtime representation of the context model and its management enables its runtime changes. adaptation logic runtime changes: existing approaches use goal-policy [14, 29], utilitypolicy [8, 26], or actionpolicy [1-2, 13] to capture the system adaptation logic. most of these approaches do not support the runtime changes to the system adaptation logic except andrade et al. [2] who enable the system developer to change this logic manually. however, the adaptation logic of the travel guide system need to be changed automatically without the developer involvement as the system may include or exclude parts of its adaptation rules while it is running. to do so, our approach maintains a runtime representation of the adaptation rules and we have a set of adaptation meta-rules that decides the required changes to the adaptation rules in response to context changes. in this paper, we adopted the action-policy approach in capturing the adaptation logic because of its expressiveness and availability of tools that supports runtime changes of the rules. 6. conclusion in this paper, we have proposed a model-driven approach to developing context-aware adaptive information systems. we have considered the system model, the context model, and their relationships explicitly from modelling to realization and to runtime execution and adaptation. in addition, we have developed a prototype tool for modelling the system and generating its implementations from their models. furthermore, we have demonstrated our approach through the development of the context-aware adaptive travel guide system. we also measured the overhead of adding the runtime adaptability feature to a software system using our approach. compared to existing approaches, our approach has the following key contributions. firstly, we explicitly separate but relate the system model and the context model, so that their relationships and changes can be clearly captured and managed. secondly, the relationships an approach for developing context-aware adaptive information systems 119 copyright ©2018 assa. adv. in systems science and appl. (2018) between the system functional elements are represented explicitly, so that the functional elements interactions, mutual obligations, and changes can be clearly captured and managed. thirdly, our approach supports the runtime adaptation of the system, the context model, and their adaptation logic in response to context changes. finally, our tool supports the system modelling and the generation of its implementations from their models. as future work, our approach can be enhanced in several directions. first, software systems are usually deployed in environments which are not totally anticipated at the system design time. while we have a runtime representation of the system aspects (including rule-based management) to be able to cope with unanticipated changes, runtime system management strategies and decision-making techniques are required to fully realize this capability. second, we have applied our approach to the tourist travel guide case study, and the results were promising. we will perform more validations to assess the applicability and practicality of our approach. references [1] adamczyk, j., chojnacki, r., jarząb, m., & zieliński, k. (2008). rule engine based lightweight framework for adaptive and autonomic computing international conference on computational science (vol. 5101, pp. 355-364). [2] andrade, s. s., & de araujo macedo, r. j. (2009, 18-19 may 2009). a non-intrusive component-based approach for deploying unanticipated self-management behaviour. paper presented at the software engineering for adaptive and self-managing systems, 2009. seams '09. icse workshop on. [3] asadollahi, r., salehie, m., & tahvildari, l. (2009, 18-19 may 2009). starmx: a framework for developing self-managing java-based systems. paper presented at the software engineering for adaptive and self-managing systems, 2009. seams '09. icse workshop on. [4] ayed, d., delanote, d., & berbers, y. (2007). mdd approach for the development of context-aware applications. paper presented at the proceedings of the 6th international and interdisciplinary conference on modeling and using context, roskilde, denmark. 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(2007). model-driven development of complex software: a research roadmap. paper presented at the 2007 future of software engineering. [10] garlan, d., cheng, s. w., huang, a. c., schmerl, b., & steenkiste, p. (2004). rainbow: architecture-based self-adaptation with reusable infrastructure. computer, 37(10), 4654. [11] giachetti, g., marín, b., & pastor, o. (2009). using uml as a domain-specific modeling language: a proposal for automatic generation of uml profiles advanced information systems engineering (vol. lncs 5565, pp. 110-124). http://dx.doi.org/10.1109/ms.2006.61 120 m. hussein copyright ©2018 assa. adv. in systems science and appl. (2018) [12] governatori, g. (2005). representing business contracts in ruleml. international journal of cooperative information systems, 14(2), 181-216. [13] gu, t., pung, h. k., & zhang, d. q. 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(2005). when and how to develop domainspecific languages. acm computing surveys (csur), 37(4), 316-344. [24] mintzberg, h. (1994). rounding out the manager's job. sloan management review, 36, 11-26. [25] morin, b., barais, o., nain, g., & jezequel, j.-m. (2009). taming dynamically adaptive systems using models and aspects. paper presented at the proceedings of the 31st international conference on software engineering. [26] rouvoy, r., barone, p., ding, y., eliassen, f., hallsteinsen, s., lorenzo, j., . . . scholz, u. (2009). music: middleware support for self-adaptation in ubiquitous and serviceoriented environments. in b. cheng, r. de lemos, h. giese, p. inverardi & j. magee (eds.), software engineering for self-adaptive systems (vol. 5525, pp. 164-182): springer berlin / heidelberg. [27] sheng, q. z., jian yu, segev, a., & liao, k. (2010). techniques on developing contextaware web services. international journal of web information systems, 6(3). [28] taconet, c., kazi-aoul, z., zaier, m., & conan, d. (2009). ca3m: a runtime model and a middleware for dynamic context management. paper presented at the proceedings of the confederated international conferences, coopis, doa, is, vilamoura, portugal. [29] zhang, j., & cheng, b. h. c. (2006). model-based development of dynamically adaptive software. paper presented at the proceedings of the 28th international conference on software engineering, shanghai, china. http://dx.doi.org/10.1016/j.jnca.2004.06.002 advances in systems science and application (2015) vol.15 no.4 351-365 high throughput area efficient architecture for light weight cryptography vanitha m and subha school of information and technology, vit university, vellore 632014, tamilnadu, india abstract hummingbird algorithm is one of the recently proposed light weight cryptographic algorithms targeted for resource constrained devices like rfid (radio frequency identification), smart cards and majority of wireless sensor nodes. the main advantage of this algorithm is that it provides adequate security with smaller block size. as per the previous works on this algorithm, area and performance are two main design tradeoff of this algorithm. performance is increased by loop unrolling and area is optimized by looping. so optimization in both area and performance is a big challenge. this work, proposes efficient hardware architecture for the hummingbird algorithm using partial loop unrolling which concerns both the area and performance of hardware implementation. the overall architecture was modeled using verilog hdl and synthesized using cadence rtl compiler with 45nm technology from tsmc. proposed design also implemented in low cost spartan 3 fpga board and the results are compared with the existing implementations. results show that there is an area reduction of around 6% and throughput almost get doubles. keywords hummingbird; lightweight cryptography; rfid. 1 introduction the importance of low cost devices like rfid (radio frequency identification), smart cards and various wireless sensor devices is increasing in our present day life. so the security of such devices is very important. since these devices are extremely resource constrained in terms of computing power, battery power supply and memory, the standardized cryptographic algorithms such as aes(advanced encryption standard) , des(data encryption standard), which are well focused on software implementation rather than hardware, cannot be used as it is in these devices. so a certain class of cryptographic algorithms known as light weight cryptography is evolved. the design metrics of light weight cryptography are security, cost, and performance. practically it is difficult to optimize all the three design goals. 1.1 related works many lightweight cryptographic algorithms are presented until now, a light weight algorithm named hight is proposed with 3048 gate equivalents (ge) which is much faster than aes[1]. scalable encryption algorithm (sea), with a block 352 vanitha m and subha s:high throughput area efficient architecture for light ... size and key size of 96 bits, and word size of 8 bits and 93 rounds operation can encrypt one data block within 1428 clock cycles and 3758 ge[2].slight modifications is done on classical block cipher lightweight des variant called desl (des lightweight)[3]. asic implementation of the same requires 1848 ge and it can encrypt one 64-bit data block in 144 clock cycles. the implementation of desl has approximately 20% smaller chip size than des. key whitening technique can be useful to improve the security of cipher in desxl algorithm. for encrypting the plaintext, it requires 2, 170 ges and 144 clock cycles. present algorithm is a lightweight substitution, permutation based block cipher, which operates on 32 rounds, key size of 80 or 128 bits and 64 bit block size. present serial version can be implemented with 1000ges[4]. the hummingbird algorithm is the one of the recently presented ultra-light weight cryptographic algorithm[5]. the size of the key and the internal state of hummingbird provides adequate security level for many embedded applications. the present researches are going on the development and different implementation of the same because of the simplicity in the architecture. until now the hummingbird has been implemented on different target platform, software as well as in hardware and shows good efficiency in both. xinxin fan implemented the algorithm on 16 bit as well as 4-bit microcontroller.[6] daniel engels implemented the architecture on 8-bit microcontroller.[7] xinxin fan implemented both area oriented and throughput oriented design of hummingbird on low cost fpga.[6,8] biao min proposed another efficient hardware implementation of the same fpga.[9] ismail san proposed yet another implementation on fpga using coprocessor approach.[10] all these implementation shows that this algorithm works well on different target platforms. as per the previous works, hummingbird can be implemented in different platforms but each of which is focused on either optimizing area or optimizing the speed. here we are presenting an efficient architecture for the hummingbird algorithm focusing to optimize the area as well as the speed. comparison is made with the existing implementations. the synthesized result shows the increase in throughput by 111% and reduction in area by 6.6%. the remaining portion of this paper is organized as follows. section 2 presents the standard hummingbird algorithm, with its initialization, encryption and decryption steps. next, section 3 discuss about the proposed architecture for increasing the throughput and reducing the area. section 4 presents the simulation results, synthesis outcome in fpga and asic platform and their comparison with the existing architecture section 5 concludes this work. advances in systems science and application (2015) vol.15 no.4 353 2 standard hummingbird algorithm hummingbird is the recently proposed ultra-light weight cryptographic algorithm, which is the combination of block cipher and a stream cipher. the design of hummingbird consists of 16-bit block size, 256-bit key size, and 80-bit internal state. the main advantage of this algorithm is, it is having smaller block size compared to other algorithms and provide sufficient security even though the block size is small. the overall structure of the hummingbird cryptographic algorithm includes four 16-bit block ciphers eki(i=1,2,3,4), four 16-bit internal state registers rsi (i= 1,2,3,4), and a 16-bit lfsr (linear feedback shift register). the 256-bit key with 4 internal registers and lfsr provides adequate security level. for each block eki., 256-bit secret key is divided into four 64-bit sub keys ki(i=1,2,3,4). fig. 1 initialization notations used rs1rs4 internal state registers to hold nonce value ek1ek4 state registers to hold the modulo addition values of 16 bit register rs with ek + modulo addition tv register to store the final output after 4 rounds of initialization 2.1 initialization the initialization process shown in fig.1 initialize the four internal states registers ek1 to ek4 and get the lfsr initial value before encryption starts. the four internal state registers are first loaded with four 16-bit random nonce values. taking rs1 + rs3 as input data, four block ciphers are consecutively executed four times and the states are updated accordingly as shown in fig.1. the final output after four iteration is shown in the register tv, which is used to get the initial value of lfsr and used to update the state rs3 in encryption process. 354 vanitha m and subha s:high throughput area efficient architecture for light ... the lfsr is used not only for rs3 updating but also to ensure that period of the internal states are at least 216. fig. 2 encryption notations used rs1rs4 internal state registers ek1ek4 16 bit register to hold the block cipher value after modulo addition + modulo addition pt plain text ct cipher text fig. 3 decryption notations used rs1rs4 internal state registers dk1dk4 16 bit register to hold the value after modulo subtraction modulo subtraction + modulo addition pti plain text cti cipher text lfsr linear feedback shift register advances in systems science and application (2015) vol.15 no.4 355 fig. 4 block cipher 2.2 encryption/decryption after the initialization process, encryption starts by taking the input as the plain text(pt) followed by the modulo 216 addition (shown in fig.2 as + ) of plain text with internal state register rs1 and the result is applied to the block cipher ek1. the whole process of encryption is shown in fig.2. 356 vanitha m and subha s:high throughput area efficient architecture for light ... this process is repeated for four times and produces the cipher text. when all the four block ciphers are completed, the rsi state register is updated accordingly. decryption process is just reverse operation of the encryption as shown in fig.3. 2.3 block cipher block cipher used in encryption process as shown in fig.4. it consists of four rounds of operation and a final round. one regular round operation consists of a key mixing step, a substitution step and a linear transformation step. the sub-key of 64-bit is separated into four 16-bit round keys which will be used in the next corresponding rounds respectively. in the key mixing process, plaintext block uses an exclusive-or with the round-key. the s-box produces the results step by step. the substitution round uses 4 serpent-type s-boxes with the input and output of 4-bits. table 1 shows the substitution value of 4bits used for substitution process. here each 4-bit input is substituted with another 4-bit to make confusion in output. for the final step it uses one substitute step. there is no linear transformation step in the final round. the linear transformation step is defined in equation (1) where an xor operation is performed between din and 6 times right shifted value of din and 10 times right shifted value of din. dout = din∧(din << 6)∧(din << 10) (1) table 1 substitution s box values 0 1 2 3 4 5 6 7 8 9 a b c d e f s1 8 6 5 f 1 c a 9 e b 2 4 7 0 d 3 s2 0 7 e 1 5 b 8 2 3 a d 6 f 6 2 9 s3 2 e f 5 c 1 9 a b 4 6 8 0 7 3 d s4 0 7 3 4 c 1 a f d e 6 b 2 8 9 5 3 proposed architecture until now, hummingbird is implemented across different target platform. each of that implementation is either focusing area optimization by looping technique or throughput optimizing by loop unrolling of block cipher. the proposed block cipher is similar to the previous models with a modification that cipher loop is partially unrolled. this architecture is a compromise between earlier looped and loop unrolled architecture so that it tries to optimize both area and throughput. it requires 6 xors, 12 s-boxes, a linear transform and 2 multiplexers. here a plain text can be encrypted in 8 clock cycles. so the proposed architecture is an optimization of area and speed. fig.5 shows the proposed block cipher architecture. the four rounds of the encryption are unrolled and the three operation in a round like substitution, permutation and linear transformation is executed advances in systems science and application (2015) vol.15 no.4 357 in sequence. since the four rounds are executed in parallel it requires 4 times duplication of hardware but at the same time the number of clock cycle required is only 1 and for the last round it require one more extra. this encrypted block of 16 bit data is passed for next 5 rounds in a looped pipelined manner with pipelining register r1 and r2. the final encrypted data is taken in a 64 bit register at every clock cycle so it require 5 more clock cycle so in total it require 7 clock cycle for getting the encrypted data. the encryption block (ee) is represented from ee1-ee4 which operates in parallel. instead of using the four sbox as such in the design, a hardware friendly sbox is selected from the sbox given in the table 1 and it is repeated four times to improve the area and performance. from the table 3, it is found that minimum hardware required for s3 implementation. so it is selected as the hardware friendly sbox’s and used in the further implementation. the overall encryption core architecture is shown in fig.6. system is made reset before the first encryption starts. upon reset, the control signals data sel, rss el, key sel,init encr and the counters reset to zero. data sel is used to select corresponding input data to the block cipher, rs sel is used to select required internal register during operation, key sel is used to select corresponding sub keys and init encr is used to differentiate between initialization and encryption. at first the internal registers are loaded with the 16-bit random nonce values and core starts encrypting the rs1+rs3 for four iterations. each iteration takes 7 clock cycles as well as internal state updating and same block cipher architecture is reused each time as shown in the fig.5 which ensures maximum architecture reuse and hence better area and performance. after four iteration, the initialization is completed and the control signal init encr set to 1. once the initialization is completed, i th plain text is taken as the input and after 7 clock cycles, we get i th cipher text. during the above procedure, the corresponding sub keys, internal register and input data are selected according to the control signals key sel, rs sel and data sel respectively. the control signals are generated based on two counters. control signals rs sel and data sel should be updated after each block cipher operation so it is controlled by a block counter. the init encr signal is updated after initialization process so it is controlled by a round counter. after each iteration, the internal states are updated accordingly. the lfsr is initialized after initialization process and updated after each encryption. the proposed architecture is not altering the algorithm; it is an optimized hardware implementation of the same. 358 vanitha m and subha s:high throughput area efficient architecture for light ... fig. 5 proposed block cipher advances in systems science and application (2015) vol.15 no.4 359 table 2 cipher comparison looped loop unrolled proposed (partial loop unrolled) area 5xor 8sbox 1linear transform 2mux 8xor 20sbox 4linear transform no mux 6xor 12sbox 2linear transform mux clock cycles (encryption) 16 4 7 fig. 6 overall architecture 360 vanitha m and subha s:high throughput area efficient architecture for light ... table 3 area requirement of different sbox implemented on spartan-3 xcs200 fpga platform sbox #lut #flipflop total occupied slices s1 187 16 101 s2 184 16 99 s3 184 16 98 s4 186 16 99 fig. 7 proposed architecture against scan attack 3.1 cryptanalysis of proposed hummingbird architecture cryptographic processors are subjected to side channel attack. if the processor uses the intermediate registers, the probability of easy hacking is more. the scan chain based flip-flop (ff) used in the design synthesis is more vulnerable to scan attack. the hackers can easily run the processor for 2 to 3 rounds by operating in normal mode and then switch over to test mode which can retrieve all the data stored in the intermediate register made of scan based ff. the hackers can hack the data by running the processor for few clock cycles by applying all possible inputs and outputs vectors in a reversal fashion. based on the hamming distance between the input pairs the plain text can be hacked. usage of scan based ff cannot be avoided because it facilitates the easy testability of a ics and this opens an easy way for the hackers to steal the data. a response compactor register is used instead of using normal ff to store the intermediate results. the response compactor is a multiple input signature advances in systems science and application (2015) vol.15 no.4 361 register (misr) which compresses the scan ff intermediate data in a compact register by xor operation as shown in the fig.7. this will reduce the area and simultaneously make the architecture resistance to scan attack. we could see this architecture provide two hamming distance (1 and 8) which are at extreme so that we get 40 ≈ 25.32 possible values for each key byte. therefore, on average, the final key hypotheses is (25.32)16 = 285.15, which cannot be brute-forced easily. basically this architecture is resistance to attacks like birthday attack, differential attack and linear attack. 3.2 validation of the proposed architecture the encryption and decryption block for 16 bit is modeled with verilog hdl and simulated using modelsim. the functional verification is shown in fig.8 and 9. the asic implementation is done by synthesizing the design with rtl compiler with 45nm technology file from tsmc. the backend of the design is done using soc encounter. the complete chip layout is shown in fig.10 with design specification. the cryptanalysis on the proposed architecture is performed, since we have used the iterative architecture for compressing the scan chain ff response in a response compactor it shows that it is very difficult for brute-forced attach on our architecture. 4 results and discussion the architecture for encryption core as well as decryption core is modeled using verilog hdl and simulated in modelsim. the architecture is synthesized and implemented using cadence rtl complier and encounter. for comparison with the previous implementations, the same architecture is also implemented using xilinx ise 13.2 by taking low cost fpga sparten-3 xc3s200 in package ft-256 with a speed grade of 5. fig. 8 simulation result of encryption fig. 9 simulation result of decryption 362 vanitha m and subha s:high throughput area efficient architecture for light ... table 4 cadence synthesis report in 45nm item area power(uw) frequency 4900(mhz) cell cell area encr cipher 535 1126 794.09 218.91 decr cipher 463 1025 735.92 244.61 encryption 1393 3996 845.86 164.17 decryption 1956 5494 1623.29 216.07 table 5 performance comparison of fpga implementations design algorithm key size block size fpga area (slices) memory blocks frequency (mhz) through put (mbps) efficiency (mbps/ #slice) x.fan et al.[6,8] hummingbird (throughput opt) 256 16 spartan3 xc3s2005 273 0 40.1 160.4 0.59 hummingbird (area opt) 253 0 66.1 66.1 0.26 ismailsan et al.[10] hummingbird 256 16 spartan3 xc3s2005 40 2 260.8 55.64 1.38 biaomin[9] hummingbird (throughput opt) 256 16 spartan3 xc3s2005 230 0 39.8 157.6 0.68 hummingbird (area opt) 183 0 72.9 72.9 0.39 poschmann et al.[4] present 80 64 spartan3 xc3s4005 176 0 258 516 2.93 128 64 202 0 254 508 2.51 kaps et al.[11] stea 128 64 spartan3 xc3s505 254 0 62.6 36 0.14 yalla et al.[12] hight 128 64 spartan3 xc3s505 91 0 163.7 65.48 0.72 mace et al.[13] sea 126 126 virtex-ii xc2v4000 424 0 145 156 0.368 this work hummingbird 256 16 spartan3 xc3s2005 255 0 69.96 140 0.54 table 4 shows the 45nm synthesis report in cadence. no comparison is made with this report because it is done for the asic implementation of the algorithm. comparison is done with fpga implementation of the same with existing implementations. table 5 shows the comparison of our implementation with other existing implementations as well as other few cryptographic algorithm implementations. x.fan proposed throughput optimized implementation of hummingbird with 273 slices with a throughput of 160.4mbps,[8] x.fan implemented area opadvances in systems science and application (2015) vol.15 no.4 363 timized hummingbird with 253 slices and 66.1mbps throughput.[6] comparing with [8] our implementation has better area performance (-6.6%) but throughput is slightly lower. comparing with [6] our implementation has a good throughput performance (+111%) but area is slightly higher. strictly speaking, our implementation is a compromise between area optimized design and throughput optimized design. fig. 10 simulation result of decryption this clearly shows that the proposed design is optimized in terms of both area and throughput. ismail san has very good area performance but it utilizes the embedded memory in the fpga and it utilizes instruction stored in the embedded memory.[10] so this implementation cannot be assumed as a complete hardware implementation. biao min throughput oriented design has better performance compared to our design.[9]but the proposed design have better throughput compared to area oriented design of the same [9]. the algorithm present [4] has better area and throughput performance than hummingbird implementation but it required comparatively larger fpga xc3s400 which intern increases the cost. xtea[11] and hight[12] have very less throughput compared to hummingbird implementation and for sea[13] requirement of area is very high. the loop unrolling can increase the throughput but simultaneously increase the area so an optimum of around 50% rounds is unrolled to have a compromising solution for area and performance. 5 conclusions this paper presented an efficient vlsi architecture for ultra-light weight cryptographic algorithm named hummingbird. previous implementation of this algorithm was focusing on optimizing either the throughput or area. but this paper presented a design which is optimized in terms of both in area as well as throughput. the proposed design is a compromise between both area optimized and 364 vanitha m and subha s:high throughput area efficient architecture for light ... throughput optimized schemes. the hummingbird algorithm has good efficiency compared to all other algorithms and also it has smallest block size compared to all. the experimental results shows that the hummingbird algorithm is very much useful for resource constrained embedded devices. the initialization of internal registers is done by loading some random nonce values which is highly influences the security of the algorithm. the generation of these random values on the hardware is done using an lfsr. references [1] d. hong, j. sung, s. hong, j. lim, s. lee, b. s koo, c. lee, d. chang, j. lee, k. jeong, h. kim, j. kim and s. chee. (2006), “hight: a new block cipher suitable for low-resource device”, proceedings of ches 2006, volume 4249 of lncs, pages 46-9, springer. [2] f. x standaert, g. piret, n. gershenfeld and j.-j. quisquater. (2006), “sea: a scalable encryption algorithm for small embedded application”, smart card research and applications, proceedings of cardis 2006, volume 3928 of lncs, pages 222-236, springer-verlag. [3] g.leander, c.paar, a. poschmann and k. schramm. (2007), “new lightweightdes variants”, fast software encryption. [4] a.poschmann, a.bogdanov and l.r knudsen. (2007), present: an ultralightweight block cipher, springer. [5] d. engels, x. fan, g. gong, h. hu and e. m. smith. (2010), “hummingbird:ultra-lightweight cryptography for resourceconstrained devices”, toappear in the proceedings of the 14th international conference on financial cryptography and data security fc 2010, berlin, germany:springer-verlag. [6] x.fan, g. gong, k.lauffenburger and t.hicks. 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(2007), “fpgaimplementation(s) of a scalable encryption algorithm”, ieeetrans, very large scale integ, (vlsi) syst.vol.16, no.2, pp.212-216. corresponding author vanitha m can be contacted at: mvanitha@vit.ac.in adv syst sci appl 2019; 03; 1-10 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/753 test bench algorithms for catamaran roll simulation ivan lipko* sevastopol state university, sevastopol, russia e-mail: ivanlipko@yandex.ru received may 28, 2019; revised august 21, 2019; published october 1, 2019 abstract: creation of catamaran seaworthiness system requires two stages: computer simulation of models and prototyping of software and hardware. to implement second stage we made test bench to perform catamaran roll simulation, computation and collecting experiment data. possibility of making computations like on real onboard vessel deck is required. thus test bench consists of a stewart platform with microcontrollers and computer connected to them. first performs catamaran deck roll and second performs calculations. 3-dof sensor mounted on movable plate of platform gives us roll and pitch measurements. using transfer function approximation of pierson-moskowitz spectrum and runge-kutta 4-order method with constant step size we developed algorithms for sea-wave simulation, roll generation, data filtering and common calculations. we made a set of command-line software based on these algorithms. utilizing all possibilities of test bench allows us to collect statistics and compare recorded data. keywords: stewart platform, catamaran roll, test bench, simulation, motion platform. 1. introduction this article is devoted to developing of algorithms for test bench. test benches are complex devices which consist of several parts and designed to test any electronic devices or software in natural environment. in comparison with numerical simulation researcher has conditions when his theoretical models are not limited by math representation of real world. the industrial scope application of the test bench is the testing of electronic devices for catamarans. simulation is a one of important steps in developing any electronic devices and systems. at first one performs a numerical simulation. the goal of this step is research of the system, synthesis of controls and algorithms. at second one performs a hardware testing on conditions close to real. the goal of this step is proof of theoretical results. then engineers carry out onboard experiments. and final is exploitation. the test bench we discuss aims at the simulation of the catamaran roll and to test and to debug onboard measurement systems like on real environmental conditions and collect data to verify theoretical results. it is known that in vessel researching and simulation procedures researches use wave generation pool [6]. in this pool a small prototype of real vessel is swinging in the water. special wave generator creates waves similar to a real sea. researcher watches on vessel position, roll angles, etc. this method is very good to research vessel seaworthiness but has several restrictions: difficult to repeat conditions of experiment, require special laboratory with expensive equipment. in our case, we need to reproduce vessel roll as many times as possible in a limited time. mathematical descriptions of the real sea disturbances and catamaran are considered sufficient. that is why we develop test bench instead of using the special pool. usually the necessary algorithms for the bench include methods to prepare, to conduct experiments and to collect the results. preparation of experiment includes a generation of theoretical path of events which must be registered. conduction and collection include a control of all actions executed by test bench, registration and saving of a data. in our case there *corresponding author: ivanlipko@yandex.ru mailto:ivanlipko@yandex.ru 2 i. lipko copyright ©2019 assa adv. in systems science and appl. (2019) are respectively: trajectory generation of catamaran roll, filtering the data, collecting data to files. the test bench consists of the stewart platform with bottom fixed part and top mobile part which corresponds to vessel deck and pc-based operator workspace and microcontrollers. there are known several examples of stewart platform [15] usage to simulate roll of vessels and sea-offshore construction under sea disturbances [13]. some of them use two merged platforms to simulate sea-waves (bottom platform) and vessel deck (top platform) [9]. the same situation is in the other fields: automobile [1, 8, 14], aircraft and helicopters [16], where simulators are used to train machine operators or to test devices. to move mobile part of the platform most of them use linear actuators such as hydraulic, electro-mechanic or pneumatic. these actuators have many advantages, but very expensive. in contrast to them we use rotated step motors which less expensive. we have several conceptual requirements for test bench and operator workspace: 1. simple intuitive inputs and outputs. there are many situations when one requires interpreting results according to reality: real vessel has horizontal velocity, vessel hull is influenced by wind and sea waves which have an angle to vessel direction, wind velocity, etc. therefore appropriate mathematical models of catamaran and external disturbances are required. 2. repeat experiment as many times as one needs. ideal actions sequence is a loop: set input, run experiment, collect, show and explore data, export data to other applications. e.g. we developed several predictive algorithms [2, 3, 12] which must be tested on the real applications. repeating of experiments allows us to verify the quality of the algorithms. 3. test bench must support mounting of any available microelectronic or other sensors. due to these requirements the key development points are: 1. compute trajectory of catamaran roll on a microcontroller or a pc. so we use linear models when it possible. 2. reproduce catamaran roll trajectory by stewart platform. 3. collect angle and velocity data from sensors mounted on mobile part of platform. using these measurements estimate full state vector and produce special-task computations (e.g. stability, overpitch probability, seaworthiness, etc.). the paper is organized as follow. test bench based on the stewart platform is briefly described in section 2, where block diagram of its construction is discussed. the mathematical models of simulated catamaran, sea-wave disturbances and state observation are presented in section 3. results of simulation and experimental study are given in section 4. concluding remarks and the future work intentions are presented in the last section. 2. construction this section will describe a bench construction and its function diagram. test bench (fig. 2.1) is designed for debugging of predictive algorithms of catamaran roll based on large deviation theory. physically it consists of stewart platform and pc connected to microcontroller which drives motors (fig. 2.2). at the same time there are two logical parts which allow to reproduce catamaran roll and to collect and to process data from the sensors. the first part is the stewart platform that reproduces the movement of the ship deck disturbed by sea wave. the bottom part of platform is fixed and top part is movable. this version of stewart platform uses rotated step motors and legs connecting bottom and top parts of the platform (details in [4]). ball joints in the sole of the legs are a weakness of this construction because it leads to small rattling in motion. noise is not visible by eyes but noticeable by sensor. to reduce this noise we use ballast weight on top surface. motivation of use it is cheap cost and availability, simple control. the pitching trajectories are generated on the computer and converted into data for the microcontroller which controls platform motors. test bench algorithms for catamaran roll simulation 3 copyright ©2019 assa adv. in systems science and appl. (2019) fig. 2.1.test bench the second part (blocks “sensor”, “filter and estimate”, “general computations”) is a prototype of device for reading the roll data and calculating probability estimates. roll data (roll angle and velocity, pitch angle and velocity) are taken by a sensor mounted on the movable surface of platform and transmitted to the computer where calculations are made and displayed on the screen: current state of the vessel, estimates of the probability of large deviations. the blocks described above are modules of developed programs and perform certain tasks. all programs (unless otherwise written) are written in c++ in the qt creator and executed under the console environment. console usage is convenient for creating a test bench user interface application. block “roll generation”. the pitching trajectory generator is program designed to create trajectories of the catamaran that the platform will reproduce. the program started using command line parameters or a special configuration file. the parameters include initial state of the catamaran, seed-number for the random number generator, service flags. as a result the program creates a csv-file containing in each line the time countdown and the current catamaran state: angle and velocity of roll, level of heave, angle and velocity of pitch. to generate trajectories we use 4-order runge-kutta integration method with a discrete step. the system of differential equations describing the catamaran under the influence of wind-wave disturbances discussed in section 3.2. block “motor calculation”. this block solves inverse kinematics problem and translates roll trajectory motion to the motors control signals of the stewart platform. csv file with trajectory from previous block is an input of the program (written in c#). output of this block is csv file with lines which contain the angles of platform motors. a special program implements the execution of platform movements [4]. block “filter and estimate”. this block performs data reading and restoring the state of catamaran. 3dof-sensor (mpu6050) and microcontroller nucleo-f401re are used for reading angles and their velocities. the values of the observable state vector are passed to the input of the filter and recovery module. filtration and estimation of the full state vector is performed in accordance with the section 3.3. the restored full state vector is saved to a file and passed into next block. 4 i. lipko copyright ©2019 assa adv. in systems science and appl. (2019) fig. 2.2. block diagram block “general computations”. this block implements the predictive algorithms which necessary for other research. input of the block is full state estimates the resulting vector is a task-special data displayed and saved to a file. 3. algorithms this section describes the mathematical model of the simulated catamaran and its control system aimed to reduce the amplitude of the roll and the angle of the roll; the mathematical description of the wind-wave disturbances acting on the catamaran, used in the generation of the trajectory of the catamaran and the estimation of state vector. 3.1. catamaran model the catamaran is a vessel with two identical symmetrical hulls connected by a deck [11]. to reduce pitching of the boat it have t-foil and thruster flaps, which create a forces (𝐹𝑇 and 𝐹𝐹 respectively) and moments (𝑀𝑇 and 𝑀𝐹, respectively). the main dimensions of the catamaran are presented in table 3.1, block diagram on fig 3.1. table 3.1 the catamaran dimensions name value length, m 90 width, m 25.96 draught, m 2.6 displacement, t 734.54 in general there is a relationship between all types of ship pitching in the motion model [7]. in this case we are interested in roll, pitch angles and heave of the catamaran. the space-state equations for the catamaran will be as follows: { �̇� = 𝐴𝑥 + 𝐵(𝑢 + 𝑤) 𝑦 = 𝐶𝑥 , (3) stewart platform roll generation motors sea state path computer sensor motor calculation motor angles filter and estimate general computations roll, pitch state vector results test bench algorithms for catamaran roll simulation 5 copyright ©2019 assa adv. in systems science and appl. (2019) fig. 3.1.catamaran controller and signals where 𝑥 = [η̇, ζ̇, η, ζ, θ̇, θ] is state vector including heave velocity, pitch velocity, heave, pitch, roll velocity, roll; 𝑢 = [ 𝐹𝑇 + 𝐹𝐹 𝑀𝑇 + 𝑀𝐹 ] is control vector including forces and moments of actuators; 𝑤 = [ 𝐹𝑤 𝑀𝑤 ] is wind-wave disturbances including forces and moments acting on vessel hull; state and input matrices are: a = [ −0.9073 −25.1097 −14.1503 −17.4945 0.001 0 0.0514 −0.503 0.2442 −12.4 0.001 0 1 0 0 0 0 0 0 1 0 0 0 0 0.001 0.001 −0.01 −0.01 −5 −15 0 0 0 0 1 0 ] , 𝐵 = [ 0.0082 0.0000083 −0.00016 0.000017 0 0 0 0 0.002 0.0000083 0 0 ] , and output matrix is 𝐶 = [ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ]. for simulation and path generation matrix 𝐶 is identity matrix, but in real situation measuring a heave is very difficult task. we have actuators that will be controlled by feedback controller. in this article we use lqregulator which will reduce pitching amplitudes. so matrices 𝐴, 𝐵, 𝐶 are suitable to make controller with feedback matrix 𝐾 = [ 59.3026 26.6427 18336.7 2006.9 −1807.6 −2950.1 −1.2133 2.5576 14.2237 −2.8566 2.1435 93.5241 ] and control 𝑢 = −𝐾𝑥. as a result we have a closed-loop system { �̇� = (𝐴 − 𝐵𝐾)𝑥 + 𝐵𝑤 𝑦 = 𝐶𝑥 . (4) equation (4) is integrated via 4-order runge-kutta method with constant step size 0.01. wind-wave disturbances 𝑤 is input of the system. sea-wind forces and moments controller catamaran actuators t-foil signal flaps signal x moments forces forces moments 6 i. lipko copyright ©2019 assa adv. in systems science and appl. (2019) 3.2. wave-wind disturbances to describe sea-wind disturbances we use pierson-moskowitz spectrum in common form 𝑆(ω) = 8.1 ∙ 10−39.82 ∙ ω−5 ∙ exp (−0.74 ∙ 0.92 ∙ ω−4). above numbers and parameters corresponds to sea state with 2.14 m significant wave height and 10 m/s wind velocity [5]. pitching of the catamaran arises as a result of the external forces and moments actions created by sea waves. the simulation of these forces and moments is obtained by using two transfer functions (see fig. 3.2). transfer functions are the linear approximations of the corresponding nonlinear relations (e.g. spectrum in fig. 3.3). first, white noise is applied to the input of the pierson-moskowitz spectrum linear approximation block and then to the block of linear approximation of forces and moments. pierson-moskowitz spectrum linearized by 2-order transfer function [5] has form 𝐻𝑃𝑀(𝑠) = 𝐾ω𝑠 𝑠2+2λω0𝑠+ω0 2, where𝐾𝜔 = 2λω0σ is amplifier constant, σ2 = max 0<ω<∞ 𝑆(ω) is wave disturbance intensity constant, σ is damping factor, ω0 is dominating wave frequency. damping factor 𝜎 is calculated using nonlinear least square method. fig. 3.2.sea-wind forces and moments generation fig. 3.3. pierson-moskowitz (pm) spectrum (dashed) and linear approximation (solid) block “linear approximation of forces and moments” approximates “wave to force” function by constant 𝐾𝑓. it is tuned so that the amplitude of roll catamaran satisfies sea state statistics [5]. resulting signal is an input 𝑤 to system (4). 3.3. state estimation in this section we describe in short about filtering the data from sensor. to do this we use kalman filter which performs data filtering and estimation of full state vector. remind that from sensor we receive only roll, pitch and their velocities and we also need to know heave and heave velocity. linear approximation of spectrum white noise linear approximation of forces and moments sea state, spectrum wave amplitude forces, moments p o w er , ra d /s frequency, rad/s test bench algorithms for catamaran roll simulation 7 copyright ©2019 assa adv. in systems science and appl. (2019) to programming filter we need to discretize object system (4) and made discrete version of kalman filter so called “delayed estimator” [10]. delayed estimator is very simple to programming, but it is enough to get sufficient quality: �̂�𝑛+1,𝑛 = (𝐴𝑒 − 𝐿𝑒𝐶𝑒)�̂�𝑛,𝑛+1 + [𝐵𝑒 − 𝐿𝑒𝐷𝑒 𝐿𝑒] ∙ [𝑂4,1 𝑦𝑟𝑎𝑤]𝑇, [ �̂�𝑛,𝑛−1 �̂�𝑛,𝑛−1 ] = [ 𝐶𝑒 𝐼6 ] �̂�𝑛,𝑛−1 + [ 𝐷𝑒 𝑂10,4 𝑂6,4 𝑂6,4 ] ∙ [𝑂4,1 𝑦𝑟𝑎𝑤]𝑇, where�̂�𝑛,𝑛−1 is catamaran state estimate, �̂�𝑛,𝑛−1 is catamaran output estimate, 𝑦𝑟𝑎𝑤 is raw sensor data vector. matrices 𝐴𝑒, 𝐵𝑒, 𝐿𝑒, 𝐷𝑒 are the following: 𝐴𝑒 = [ 0.8770 −0.0091 −0.0762 −0.0119 0.0053 0.0244 −1.0854 −0.0195 −0.7805 −0.1685 −0.0077 0.4085 0.0067 0.0975 0.5590 0.0360 −0.0055 0.0040 −0.1307 −0.7889 −5.9193 −0.0658 −0.0661 −0.0626 0.0004 −0.0007 0.0075 0.0013 0.9390 0.0930 0.0013 0.0231 0.0866 0.0200 −1.1861 0.8465] , 𝐵𝑒 = [ 0.0597 0.0857 0.0001 0.0003 −0.0674 0.5716 0.0026 0.0359 −0.0077 −0.0980 −0.0004 −0.0067 0.1209 0.7826 0.0026 0.0231 −0.0004 0.0007 0.0001 0.0003 −0.0013 −0.0229 −0.0002 0.0004] , 𝐶𝑒 = [ [ 𝐼2 02,4 02,4 𝐼2 ] 𝐼6 ], 𝐷𝑒 = 𝑂10,4, 𝐿𝑒 = [ 0.0597 0.0857 0.0001 0.0003 −0.0674 0.5716 0.0026 0.0359 −0.0077 −0.0980 −0.0004 −0.0067 0.1209 0.7826 0.0026 0.0231 −0.0004 0.0007 0.0001 0.0003 −0.0013 −0.0229 −0.0002 0.0004] . matrices above include 𝐼𝑛 that is identity matrix with size n x n; 𝑂𝑛,𝑚 is zero matrix with n rows and m columns; and computed via kalman filter synthesis procedure with parameters 𝑄 = 𝐸(𝑤𝑤𝑇) = 𝐼2 ∙ 107, 𝑅 = 𝐸(ζζ𝑇) = 𝐼4 ∙ 10−1, 𝑁 = 𝐸(𝑤ζ𝑇) = 𝑂2,4, where ζ is measurement noise. 4. results, experiments in this section we show several examples of test bench experiments. to make sure in quality of results we perform experiments in range of input parameters. here some of them. all test bench experiments are configured for better compare and interpretation via natural parameters: wind velocity, relative bearing, initial state vector of catamaran and seed-number. fig. 4.1 and fig. 4.2 show examples where test bench filter outputs are compared with numeric simulation results. raw data from sensors were scaled and filtered. the test bench reproduces catamaran roll with satisfactory quality for our applications. the initial state vector of catamaran is equal to zero in next examples. 8 i. lipko copyright ©2019 assa adv. in systems science and appl. (2019) fig. 4.1.errors in catamaran roll (top) and pitch (bottom) between simulated and experimental angles corresponding to a) wind 13.45 m/s, seed-number 11622; b) wind 13.23 m/s, seed-number 5; c) wind 12.96 m/s, seed-number 25 a) b) c) fig. 4.2.test bench and runge-kutta outputs a) wind 13.45 m/s, seed-number 11622; b) wind 13.23 m/s, seed-number 5; c) wind 12.96 m/s, seed-number 25 test bench algorithms for catamaran roll simulation 9 copyright ©2019 assa adv. in systems science and appl. (2019) experiment data in fig. 4.1 and 4.2 have some rattling and noise influenced by ball joints in the sole of the legs connecting the bottom and the top parts of platform. paths from these figures can be associated via indices “a”, “b”, “c”. 5. conclusion the algorithms for reproducing a catamaran roll on test bench are developed. for better interpretation of results inputs of algorithms have nature-like parameters: wind velocity, direction, catamaran initial state and other. mathematical models of algorithms include the catamaran and sea-wave disturbances. full state of catamaran is estimated by kalman filter. to improve estimating one should use another filter, e.g. extended kalman filter the test bench for reproducing a catamaran roll includes the stewart platform, sensors, microcontrollers and pc. the top part of the platform moves like a catamaran deck and results of experiments are collected by sensors. thus, it turns out to reproduce the same oscillations of the catamaran and to debug device prototypes that will be installed on the real vessel. software for test bench has been developed based on these algorithms. it helps to create the trajectories of the catamaran deck roll, to filter data obtained from the sensors, and to estimate the full state vector of the catamaran. these developments can also be used in any other similar area where it is necessary to accurately reproduce the specified motion of the object, for example, any other vessels, submarines or aircraft. acknowledgements studies were supported by the ministry of science and higher education of russian federation (project rfmefi57817x0259) mechanic and electro-mechanic components were assembled by osadchenko a.e. and lazarev v.b. we would also like to show our gratitude to the reviewer for his useful insights and for comments on an earlier version of the manuscript. references 1. andrievsky, b. et al. (2014). control of pneumatically actuated 6-dof stewart platform for driving simulator, 2014 19th international conference on methods and models in automation and robotics (mmar), miedzyzdroje, 663-668, doi: 10.1109/mmar.2014.6957433. 2. dubovik, s. a. & kabanov, a. a. (2017). quasipotentials in synthesis of control systems based on knowledge. proceedings of international conference on industrial engineering, applications and manufacturing (icieam), st. petersburg, 1-4. doi: 10.1109/icieam.2017.8076123. 3. dubovik, s. a. & kabanov, a. a. (2019). funkcional'no ustojchivye sistemy upravleniya: asimptoticheskie metody sinteza [functionally stable control systems: asymptotic methods of synthesis].moscow, infra-m, [in russian]. doi: 10.12737/monography_5b446a985cf9a5.11626044. 4. dubovik, s. a., kabanov, a. a. & lipko, i.u. 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(2009). motion visualization and control of a driving simulator motion platform, 2009 6th international symposium on mechatronics and its applications, sharjah, 1-5. doi: 10.1109/isma.2009.5164839 15. stewart, d. (1965-66).a platform with six degrees of freedom, proceedings of the institution of mechanical engineers, vol. 180, 371-378. 16. villacís, c. et al. (2017). real-time flight simulator construction with a network for training pilots using mechatronics and cyber-physical system approaches, 2017 ieee international conference on power, control, signals and instrumentation engineering (icpcsi), chennai, 238-247, doi: 10.1109/icpcsi.2017.8392169. adv syst sci appl 2017; 4:1–13 published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/481 predictive feedback control method for stabilization of continuous time systems lina a. shalby1,2 1moscow institute of physics and technology, dolgoprudny, moscow region, russia 2faculty of women, ain shams university, cairo, egypt abstract: recent researchers have studied the chaotic phenomenon and introduced methods to control chaos. one of these methods is the predictive feedback control (pfc). pfc method has been used to stabilize only the discrete chaotic maps. in this paper, we generalize the pfc method by extending it to stabilize continuous time systems. the numerical simulations are carried out to solve the system using euler method for its simplicity, and then the pfc method is applied to the discretized system. the controlled trajectory converges to an unstable equilibrium point via small control action. the stability analysis is shown compared to that of the continuous system in details. also the choice of the controlling input of the pfc method and its properties are discussed. lorenz system and rössler system are the well known chosen chaotic examples. the pfc method is applied to each of them, showing stability at each of their equilibrium point. with a very small control, the behaviour of the system is completely changed from chaos to stable. keywords: continuous time systems, chaos control, euler method, predictive feedback control, unstable equilibrium point 1. introduction chaos theory is one of the recent fields of study that attracts the interest of scientists. two most important properties of chaos are the high sensitivity of trajectories to initial conditions and to parameter values. in the last decades, many researchers concerned with how to control chaotic phenomena depending on these two properties. they aimed to bring a trajectory to a small neighborhood of a desired location in the chaotic attractor, using a small perturbation. pioneering work ott-grebogi-yorke [12] proposed ogy method for stabilizing chaos. it could only be applied to discrete data, so continuous time systems should first be made discrete time by using the poincare map. then it was followed by many other publications adjusting and extending the ogy method as in [6] and [7], see also survey [1]. pyragas [14,15] proposed the delayed feedback control (dfc) to stabilize the unstable periodic orbits (upo). the control input was given as the multiplication of a gain with the difference between current system state and a state with the time delay as it is discussed in [4]. dfc method is firstly constructed for continuous time systems and extended to the discrete-time case in [18]. many applications, such as in [3, 10], were controlled by using dfc method. also, many developments for dfc method were performed, see [8, 11]. one of the recent surveys on dfc is presented in [16]. ushio [19] introduced the idea of predicting control in his paper and it was extended in [2] to stabilize continuous time systems using predictive-based control method. predictive ∗corresponding author: lina.khamis@gmail.com, lina.khamis@women.asu.edu.eg http://ijassa.ipu.ru/ojs/ijassa/article/view/481 2 lina a. shalby feedback control (pfc) [13] is an easily implemented method to stabilize unknown upos in discrete time dynamical systems. in pfc, the control input was given as the multiplication of a gain with the difference between two predicted future system states. pfc depends on applying small controls that completely change the nature of system’s behavior and devotes its attention to the most important problem of stabilizing or controlling chaos. detailed survey on existing approaches and methods for chaos control can be found in the handbook of chaos control [17]. in the next section of the paper, we extend pfc to stabilize chaotic continuous time dynamical systems solved numerically by euler method and the pfc method is analyzed. in section 3, the numerical simulations results are given for the most popular examples of chaos: lorenz system and rössler system. 2. predictive feedback control method we start with introducing the pfc method [13] for discrete-time system which depends on the system state and a control action u(x) as follows xn+1 = f (xn)− u(xn), (2.1) where f, u : rn → rn and u(x) = e(f p+2(x)− f p+1(x)), for a chosen control gain matrix e and any arbitrary forward predicted state p (p > 0 ), by denoting f p+1(x) = f (f p(x)), f 1(x) = f (x) — prediction of the map f for p+ 1 iterations forward. the idea is to choose e to stabilize unstable orbits or unstable equilibria points via small control action u(x). indeed state x and control u are of the same dimension n , and this strongly limits the application of the proposed technique to real-life problems. however there are various situations where such controls are applicable. for instance if one deals with classical analytical examples (e.g. rössler and lorenz systems) it is of interest to deal with their fixed points and orbits which are unstable. then pfc allows to stabilize such points and orbits. and the main peculiarity of the approach is the arbitrary small control action required. suppose n = 1, x∗ is an unstable equilibria point point of f (x), that is x∗ = f (x∗), µ = |f ′(x∗)| > 1. then for e = 1 µp+1(µ−1) , iterations of (1) converge to x∗. the most important fact is obtaining control u(x) small enough. indeed, if we deal with chaos control, iterations f p(x) remain bounded, while e is small for p large, because µ > 1. it is effective to choose p large enough (because larger is p, smaller is the control action). however, large predictive horizons imply computational error. thus the choice of p is some trade-of. in [13] the result is extended for n > 1 and for unstable orbits (i.e. for x∗ such that x∗ = f k(x∗) with k > 1). similar results are available for µ known approximately. the goal of the present paper is the application of the above results for continuous-time systems. let us consider a system of nonlinear autonomous differential equations ẋ = f(x), (2.2) where x ∈ rn , and f is assumed to be differentiable. an equilibrium point x∗ necessarily satisfies f(x∗) = 0. the following result on asymptotic stability is standard (see e.g. [5]). the equilibrium point x = x∗ of ẋ = f(x) is asymptotically stable if all eigenvalues of the jacobian matrix a = f ′(x∗) satisfy re(λi) < 0. from now on, we reduce a continuous time system to a discrete time map. this map is created from sampling the flow at discrete times tn = t0 + nh, n = 0, 1, 2, . . . , where the sampling interval h can be chosen on the basis of convenience. thus, a continuous time trajectory x(t) yields a discrete time trajectory xn = x(tn). by using euler method, the simplest numerical method to solve a differential equation, the system will be discretized easily to take the form xn+1 = xn + hf(xn) = f (xn), (2.3) copyright c© 2017 assa. adv syst sci appl (2017) pfcm for stabilization of continuous time systems 3 where f (x) = x+ hf(x). equation (2.3) is the simplest discrete form corresponding to the continuous time system. lemma 2.1 highlights the relation between the discretized function f and the continuous system function f . let µi denote the eigenvalues of the jacobian matrix m = f ′(x∗) = i + hf ′(x∗), f(x∗) = 0, while λi are eigenvalues of a = f ′(x∗). lemma 2.1: f (x) satisfies the following: (i) f(0) = 0, and f (x∗) = x∗ + hf(x∗) = x∗. (ii) let re(λi) = ui and im(λi) = νi, if ui < 0 and 0 < h < 2|ui| u2i+ν 2 i , then |µi| = |1 + hui| < 1, and if ui ≥ 0, then |µi| ≥ 1. lemma 2.1 shows that the discrete form for the continuous system has the same equilibrium point, and unstable equilibrium points of the continuous system imply instability of the discrete system. thus for the discretized system, the predictive feedback control method in [13] can be used to stabilize the system as illustrated for the vector case. thus we arrive to the following stabilization scheme for continuous system (2), where the matrix e could be considered to be the diagonal with entries εi. the choice of εi depends on the value of the eigenvalues µi of the jacobian matrix m , which could be real or complex. firstly, for | µi |< 1, we take εi = 0. secondly, if | µi |≥ 1 and µi ∈ r , then εi = 1 µp+1 i (µi−1) . thirdly, if | µi |≥ 1 and µi ∈ c, such that µi = ai + bij, j = √ −1, then the control input εi will be a 2× 2 matrix: εi = d−p−1(d − i2)−1 , where d = [ ai bi −bi ai ] . we restrict our analysis with fixed points, because orbits are fixed points of the iterated maps. the next theorem and its proof illustrate local stability of the system for these choice of the control gain. theorem 2.1: let f ∈ c1, x ∈ rn and assume x∗ is an unstable equilibrium point of continuous system (2.2) and its jacobian matrix m is not hurwitz. then x∗ is stable fixed point of the predictive feedback control equation (2.1) after discretizing system (2.2) numerically by using euler method (2.3) provided h is small enough. proof at x = x∗, matrix a has eigenvalues λi, with ui = re(λi) > 0 for some i. suppose that f (x) is h-step sized euler approximation of equation (2.2), given by equation (2.3), then the jacobian matrix m has the eigenvalues µi = 1 + hλi. by the chain rule, the jacobian matrix of f p(x) is associated with the eigenvalues µpi . let j is the jacobian matrix of the right hand side in equation (2.1) at x = x∗, which will have the form, j =m − e {mp+2 −mp+1} = m − emp+1 {m − i}, and j has eigenvalues ηi , i = 1, ..., n . then | ηi |=| µi − δ {µi − 1} |< 1∀i, for the following cases of µi : for all | µi |< 1, εi = 0 and ηi = µi. if | µi |≥ 1 and µi ∈ r, for εi = δh|λi| µp+1 i (µi−1) = δ µp+1 i , 1 < δ < 1 h|λi| , then | ηi |=| µi − δ(µi − 1) |=| 1 + hλi − δhλi |=| 1− (δ − 1)hλi |< 1. while if | µi |≥ 1 and µi ∈ c, for εi = δh | λi | (d−p−1(d − i2)−1), 1 < δ < 1 h|λi| where d = [ ai bi −bi ai ] has the same eigenvalues µi and dp has the eigenvalues µpi . then | ηi |=| µi − δh|λi| µp+1 i (µi−1) µp+1 i (µi − 1) |=| 1 + hλi − δh | λi ||< 1. hence the system is stable since the eigenvalues of j are less than unity. copyright c© 2017 assa. adv syst sci appl (2017) 4 lina a. shalby the next section illustrates the examples of chaotic systems and their stabilization achievement after adding the control term. the examples are clearly showing easy implementation and the efficiency of the pfc method. 3. examples in this section, we provide two examples of the pfc method applied to classical chaotic system. lorenz model is 3d system example, and rössler system is 4d example. 3.1. lorenz model lorenz [9] was the first who introduced the strange attractor notion and coined the term “butterfly effect”. lorenz system is described by differential equations ẋ = −σ(x− y) ẏ = −xz + rx− y ż = xy − kz (3.4) equation (3.4) has two nonlinear terms, and exhibits both periodic and chaotic motion depending upon the values of the control parameters σ, r and k. by using euler method the system will be discretized to the form xn+1 = xn − hσ(xn − yn) yn+1 = yn + h(−xnzn + rxn − yn) zn+1 = zn + h(xnyn − kzn) (3.5) where h is the step size for discretization of time t. the stability analysis of equation (2.1) can be performed by studying the discrete 3d function corresponding to it, f (x, y, z) = x− hσ(x− y)y + h(−xz + rx− y) z + h(xy − kz). (3.6) the system (3.4) has three fixed points s1 = (−b √ r − 1,−b √ r − 1, r − 1), s2 = (0, 0, 0) and s3 = (b √ r − 1, b √ r − 1, r − 1), which are the same as the equilibrium points of the continuous system (we denote 3d vector (x, y, z) as s). set σ = 10 and k = 2.67, and make r the adjustable control parameter. varying the values of r reveals a critical value at rc = 24.74. below rc the system decays to steady, non-oscillating, state. once r increases beyond rc, the continuous oscillatory behavior occurs and the system shows aperiodic behavior which lorenz called deterministic non-periodic flow which refer to chaos. for r = 28, the chaotic case, the equilibrium points are s1 = (−8.4906,−8.4906, 27), s2 = (0, 0, 0) and s3 = (8.4906, 8.4906, 27) and the eigenvalues corresponding to them are λs1 = λs3 = {−13.8569, 0.0934± 10.2001j} and λs2 = {−22.8277, 11.8277,−2.67}. let us use the step size h to be a small value equal to 0.001 which satisfies the condition due to the complex value of λs1 (i.e h < 2×0.0934 (0.0934)2+(10.2001)2 = 0.00179), and then the solution of the system is chaotic as shown in fig. 3.1. for the discrete form, the eigenvalues are µs1 = µs3 = {0.9861, 1.00009± 0.0102j} and µs2 = {0.99733, 1.0118, 0.9771}. the complex eigenvalues have the absolute value 1.00014. let us choose p = 5, the initial condition to be (5, 5, 5), and step size h = 0.001. starting with stabilizing the system at the origin, at which ε1 = ε1 = 0 for | µ(1,3) s2 |< 1, and otherwise we copyright c© 2017 assa. adv syst sci appl (2017) pfcm for stabilization of continuous time systems 5 fig. 3.1. phase space plot for lorenz system. choose δ = 2, ε2 = 2 (µ (2) s2 )p+1 . the matrix e is taken to be in the form e = [ 0 0 0 0 1.8638 0 0 0 0 ] and the first iterate of the decaying additive control term is −0.2040. fig. 3.2. the distance between s and 0 before and after stabilization. fig. 3.3. phase space plot of the controlled trajectory. copyright c© 2017 assa. adv syst sci appl (2017) 6 lina a. shalby fig. 3.2 illustrates the difference between the equilibrium (0, 0, 0) and the values of xtrajectory, y-trajectory and z-trajectory respectively, before stabilizing the system in blue color compared to those after applying the controlling term to the system that is coloured by red. fig. 3.3 is the phase space plot of the stabilized system. both figures show that the system is stable and all variables time series are converging to the origin starting from initial point (5, 5, 5). in case of the complex eigenvalues, µ (1) s1 = µ (1) s2 = 0.9861 and µ (2) s1 = 1.00009± 0.0102j, ε1 = 0, and d = [ 1.00009 0.01020 −0.01020 1.00009 ] ,ε2,3 = δh | λ2,3 | d−p−1(d − i2)−1 =[ 0.01030 −0.1978 0.1978 0.01030 ] , and e = [ 0 0 0 0 0.0103 −0.1978 0 0.1978 0.0103 ] . fig. 3.4. comparison between the controlled and chaotic trajectories for the equilibrium point (-8.4906, -8.4906, 27). fig. 3.5. phase space plot for the controlled trajectory for the equilibrium point (-8.4906, -8.4906, 27). in fig. 3.4 and fig. 3.5, the equilibrium point (−8.4906,−8.4906, 27) is stable. to stabilize the system at (8.4906, 8.4906, 27), the negative sign of the imaginary part of the complex eigenvalue is reversed, so that the matrix d will be [ 1.00009 −0.01020 0.01020 1.00009 ] . both fig. 3.6 and fig. 3.7 show the slow convergence of the trajectory after stabilization due to the point (8.4906, 8.4906, 27). it is important and interesting to test the effect of different initial conditions. for 50 random initial conditions, the convergence has been tested. at t = 10, the trajectories succeeded to reach the fixed points (0, 0, 0) (see fig. 3.8) copyright c© 2017 assa. adv syst sci appl (2017) pfcm for stabilization of continuous time systems 7 fig. 3.6. comparison between the controlled and chaotic trajectories with respect to each dimension with the equilibrium point (8.4906, 8.4906, 27). fig. 3.7. phase space plot for the controlled trajectory due to the equilibrium point (8.4906, 8.4906, 27). and (8.4906, 8.4906, 27) (as shown in fig. 3.9) for all initial points x0, y0 ∈ [−10, 10] and z0 ∈ [0, 30]. all the statements rely on local analysis. but in some cases, there was an effect similar to global stabilization, as it is seen in fig. 3.8 and fig. 3.9. this fact is a motivation for research of global behavior in future works. 3.2. rössler model rössler model of 4d phase space is one of the most famous hyperchaos models. for x ∈ r4, its defining equations are ẋ = −y − z ẏ = x+ αy + w ż = β + xz ẇ = −γz + τw (3.7) the chaotic behavior of the system (3.7) is observed for the parameter values α = 0.25, β = 3, γ = 0.5 and τ = 0.05. in this case, the equilibrium points are eq1 = (5.40833, 0.55470, −0.55470,−5.54700) and eq2 = (−5.40833,−0.55470, 0.55470, 5.54700), which are associated with the eigenvalues λeq1 = {5.504039, 0.103929, 0.0501791± 0.971052j} (the step size h must be less than 0.1061) and λeq2 = {0.101890, 0.0493731± 0.9986873j,−5.308963} (the step size h must be less than 0.0987) respectively. by solving the system with euler method, copyright c© 2017 assa. adv syst sci appl (2017) 8 lina a. shalby fig. 3.8. the stability is reachable at (x(10), y(10), z(10)) = (0, 0, 0) for 50 random initial conditions. fig. 3.9. the stability is reachable at (x(10), y(10), z(10)) = (8.4906, 8.4906, 27) for 50 random initial conditions. adjusting the step size h = 0.001(suitable for stability conditions stated in lemma 2.1), the continuous system translated to the discrete form with eigenvalues µeq1 = {1.005504, 1.000103929, 1.00005017± 0.00097105j} and µeq2 = {1.000101890, 1.000493731± 0.0009986873j, 0.9946910}. let us use the initial condition (−10,−6, 0, 10), then the system (3.7) is hyperchaos as shown in fig. 3.10 and fig. 3.11 for the phase space xyw and x− y plot, respectively. by setting p = 3, δ1,2 = 1.5 and δ3,4 = 5 for ε1 = δ1 (µ (1) eq1) 4 = 1.46743, ε2 = δ2 (µ (2) eq2) 4 = 1.49937, and ε3,4 = δ3,4 × h | λ(3)eq1 | ×d−4(d − i2)−1 = [ 0.01243 0.26011 −0.26011 0.01243 ] , then eeq1 =  0.01243 0.26011 0 0 −0.26011 0.01243 0 0 0 0 1.46743 0 0 0 0 1.49937 . while δ1 = 1.5 and δ2,3 = 40 for ε1 = δ1 (µ (1) eq2) 4 = 1.49939 , ε2,3 = δ2,3 × h | λ(2)eq2 | ×d−4(d − i2)−1 = [ 0.09053 1.99260 −1.99260 0.09053 ] , then eeq2 =  0.09053 1.99260 0 0 −1.99260 0.09053 0 0 0 0 0 0 0 0 0 1.49939 . the first iterate of the control term values are -0.00031, -0.00158, 0.00419, and 0.00074 to stabilize the system at eq1. and its first iterate values to stabilize the system at eq2 are -0.00237, -0.01207, 0, and 0.00074. the control term values are small negligible in the real life applications but they are enough to change the behavior of the system from ergodicity to be stable. fig. 3.12 and fig. 3.13 are showing the controlled phase space xyw due to eq1 and eq2, respectively. and fig. 3.14-3.17 are the trajectories convergence of x, y, z and w to the equilibrium points compared to the uncontrolled trajectories, respectively. copyright c© 2017 assa. adv syst sci appl (2017) pfcm for stabilization of continuous time systems 9 fig. 3.10. the strange attractors of rössler hyperchaos system xyw phase space plot. fig. 3.11. x− y plot for rössler hyperchaos system. for each example, the pfc method is successfully stabilized all the equilibrium points. it is also important to discuss the relation between the continuous system and its discretized form numerically, as they are difficult to be solved analytically. at this point, a question may arise. what is the continuous system that has stable attractor while its numerical solution is chaotic? copyright c© 2017 assa. adv syst sci appl (2017) 10 lina a. shalby fig. 3.12. the xyw phase space after control at eq1. fig. 3.13. the xyw phase space after control at eq2. fig. 3.14. the x trajectory behavior before control blue colored trajectory and after control red colored trajectory compared to eq1 and eq2 in green respectively. copyright c© 2017 assa. adv syst sci appl (2017) pfcm for stabilization of continuous time systems 11 fig. 3.15. the y trajectory behavior before control blue colored trajectory and after control red colored trajectory compared to eq1 and eq2 in green respectively. fig. 3.16. the z trajectory behavior before control blue colored trajectory and after control red colored trajectory compared to eq1 and eq2 in green respectively. fig. 3.17. the w trajectory behavior before control blue colored trajectory and after control red colored trajectory compared to eq1 and eq2 in green respectively. copyright c© 2017 assa. adv syst sci appl (2017) 12 lina a. shalby 4. conclusion in this paper, we have extended pfc method to stabilize chaotic continuous time systems by discretizing the system numerically using euler method, and then applying the pfc method. the pfc method is used by adding small control term to the iterative solution of the system, this term is the multiplication of a controlling matrix by the difference between two consecutive predicted iterates. we have also discussed the stability analysis of the continuous system compared to its numerical discrete form, and the choice of the control matrix due to the eigenvalues of the discretized system type either real or complex, and we have proved that the method is stable. the method has been applied to the most popular systems, lorenz 3d system and rössler hyperchaos 4d system. their trajectories have been controlled to each equilibrium point after a few iterations. the extended pfc method is easy implemented and highly efficient method for all chaotic systems to be stabilized, as we have generalized pfc method for all chaotic systems either discrete or continuous time system. acknowledgements it is my pleasure to have the supervision of prof. b. t. polyak. and i would like to express my gratitude to him, for the patient guidance, support and advice he has provided me in this work. also we would like to express our thankful to the editor and anonymous reviewers of the journal, for their great efforts and valuable revision. references 1. andrievskii, b. r. & fradkov, a. l. 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(2016) pyragas stabilization of discrete systems via delayed feedback with periodic control gain. ifac-papersonline, 49 (14), 56–61. 9. lorenz, e.(1963) deterministic non-periodic flow. journal of the atmospheric sciences, 20 (2), 130–141. 10. mitsubori, k. & aihara, k. (2002) delayed–feedback control of chaotic roll motion of a flooded ship in waves. in proceedings of the royal society of london a: mathematical, physical and engineering sciences, 458 (2027), 2801-2813. 11. morgul, o. (2003) on the stability of delayed feedback controllers. phys. lett. a, 341(4), 278–285. 12. ott, e., grebogi, c. & yorke, j. a.(1990) controlling chaos. phys. rev. lett., 64(11), 1196–1199. 13. polyak, b. t. (2005) stabilizing chaos with predictive control. automation and remote control, 66(11), 1791–1804. copyright c© 2017 assa. adv syst sci appl (2017) pfcm for stabilization of continuous time systems 13 14. pyragas, k. (1992) continuous control of chaos by self-controlling feedback. phys. lett. a, 170 (6), 421-428. 15. pyragas, k. (1995) control of chaos via extended delay feedback. phys. lett. a, 206(5-6), 323-330. 16. pyragas, k. (2006) delayed feedback control of chaos. philosophical transactions of the royal society of london a: mathematical, physical and engineering sciences, 364(1846), 2309–2334. 17. schöll, e. & schuster, h. g.(2008) handbook of chaos control, john wiley & sons. 18. ushio, t. (1996) limitation of delayed feedback control in nonlinear discrete-time systems. ieee trans. circ. syst., 43(9), 815–816. 19. ushio, t., and yamamoto, s. (1999) prediction-based control of chaos. phys. lett. a, 264(1), 30–35. copyright c© 2017 assa. adv syst sci appl (2017) introduction predictive feedback control method examples lorenz model rössler model conclusion adv syst sci appl 2018; 04; 121-135 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/584 chaotic analysis and improved finite-time adaptive stabilization of a novel 4-d hyperchaotic system edwin a. umoh*1, ogechukwu n. iloanusi1 1) department of electronic engineering, university of nigeria, nsukka, nigeria e-mail: edwin.umoh.pg76768@unn.edu.ng, ogechukwu.iloanusi@unn.edu.ng received july 27, 2018; revised november 11, 2018; published december 31, 2018 abstract: chaotic systems are evidently very sensitive to slight perturbations in their algebraic structures and initial conditions, which can result in unpredictability of their future states. this characteristics has rendered them very useful in modelling and design of engineering and nonengineering systems. using the burke-shaw chaotic system as a reference template, a special case of a novel 4-d hyperchaotic system is proposed. the system consists of 10 terms and 9 bounded parameters. in this paper, after the realization of a mathematical model of the novel system, we designed an autonomous electronic circuit equivalent of the model and subsequently proposed an improved adaptive finite-time stabilizing controller which incorporates some augmented strength coefficients in the derived controller structures. these augmented coefficients greatly constrained transient overshoots and resulted in a faster convergence time for the controlled trajectories of the novel system. this novel system is suitable for application in the modelling and design of information security systems such as image encryption and multimedia security systems, due to its good bifurcation property. keywords: chaotic analysis, adaptive finite-time stabilization, hyperchaos, lyapunov stability. 1. introduction chaos is a unique phenomenon that often occur in a class of dynamic systems that are sensitive to perturbation in their initial conditions or mathematical models, consequently resulting in unpredictability of their future states [1]. chaos has been found to exists in natural and man-made systems, and its dynamics have been used in the modelling and studies of practical and hypothetical systems in communications engineering [2], medicines [3], robotics [4], thermodynamics [5], waste water treatment plant [6], electric power system [7], amongst others. however, for chaos to be of practical use, it must be controllable. consequently, during the past decades since the seminal works of ott, grebogi and yorke [8], extensive research on chaos control has resulted in the proposition of different methods in the literature and their applications by researchers in controlling and stabilizing chaotic dynamics such as active control [9], sliding mode control [10], fuzzy control [11] and adaptive hybrid control [12], amongst others. classical stability concepts such as the lyapunov stability, bibo stability and asymptotic stability concepts [13], [14] do not take into consideration the time intervals of stabilization of systems. rather, the main objective is the stabilization which can continue into infinite time. in recent years however, the concept of finite time stability and stabilization has gained wide attention due to its usefulness in time-critical system designs including secure communication systems, image and data encryption systems. a dynamic system is said to be finite-time stable (or synchronized), if given a bound on the initial conditions, its states does not exceed a certain threshold during a specified interval. finite-time stability is essential to ensure that the trajectories of a targeted system does not overshoot a specified bound in order to avoid undesirable consequences * corresponding author: edwin.umoh.pg76768@unn.edu.ng mailto:ogechukwu.iloanusi@unn.edu.ng 122 e. a. umoh, o. n. iloanusi copyright ©2018 assa. adv. in systems science and appl. (2018) during applications. there is a distinction between finite time stability and lyapunov stability. a system which is finite-time stable may be lyapunov asymptotically stable, whereas a system that is lyapunov asymptotically stable may not be finite-time stable, if during transient it uncontrolled dynamics exceeds a prescribed threshold of time [15]. many works on finite time control have appeared in recent year [16]–[18]. in some of the reference works listed under section four, however, the settling times are comparably large, even when the controllers featured the use of sign function. in the proposed adaptive controller in this paper, we proposes a control structure that offers a faster rate of convergence of the controlled trajectories of a novel special case of a hyperchaotic system [19], which was derived from the burke – shaw atmospheric model [20]. certain characteristics distinguished the novel system from several others in the literature. these include its unique bifurcation properties where the plots of control parameters depict bifurcation diagrams with fewer periodic windows, thus making the system a good candidate for application in chaos-based cryptosystem modelling and design. the elegance of the algebraic structure makes it possible to further reduce the algebraic structure to consists of fewer parameters, yet would still exhibit hyperchaotic features with some degrees of novelty in its qualitative features. the novel system is also highly sensitive to slightest change in its initial conditions and system parameters. thus, several attractors may be evolved through key sensitivity. 1.1 concepts and preliminaries throughout this paper, the following existing definitions and lemmas are used in the analysis and synthesis of the improved finite-time controller design for stabilization. consider a class of nonlinear systems given by 0 0( , ), ( ) (0)x f x t x t x x   (1.1) where 0( , ), ( ) nx f x t x t  is the system state and : n nx f    is a nonlinear function. assume that the origin o is an equilibrium point, 0(0,0,0,0) of (1). if there exist a constant 0t  where t may be influenced by the initial condition such that lim ( ) 0, t t x t   lim ( ) 0, t t x t   if t t then the system (1) is finite-time stable. definition 1.1 [15], [21] the origin of (1.1) is a finite-time stable equilibrium, if the origin is lyapunov stable and there exist a function : nt  called the settling time function, such that for every 0 nx  , the solution 0( , )x t x of (1.1) is defined on 0 0 0[0, ( ), ( , ) [0, ( )]]nt x x t x t t x   and 0 0 ( ) lim ( , ) 0 t t x x t x   . lemma 1.1 [22], [23] if there exist a continuous positive definite function ( ) : n nv t  , such that ( )v t is radially bounded (i.e. ( ( ))v x t  as ( )x t  ), and satisfies the following differential inequality: 0 0( ) ( ( )) , 0, , ( ) 0,qv t v t t t t v t      (1.2) where 0  and 0 1q  are two positive numbers. it follows that for any 0t , ( )v t satisfies the inequality: 1 1 0 0 0( ) ( ) (1 )( ),q qv t v t q t t t t t       (1.3) chaotic analysis of a novel 4-d hyperchaotic system 123 copyright ©2018 assa. adv. in systems science and appl. (2018) and ( ) 0,v t t t   . we conclude that the origin of (1.1) is globally stable in finite time t and the settling time t is given by the relationship: 1 0 0 1 ( ) (1 ) qt t v t q    (1.4) assumption 1 [24], [25] assume 1 2, ,..., n na a a  and 0 1  are all real numbers. then the following inequality holds: 1 2 1 2( ... ) ... rr rr n na a a a a a       (1.5) lemma 1.2 [25] for the dynamic system (1.1), if there exists a continuous differentiable function :[0, ) ,v d   class k function (.)a and (.)b , a function :[0, )k   , such that ( ) 0k t  for almost all [0, ),t  a real number (0,1)  and an open neighbourhood m d of the origin, such that: ( ,0) 0, [0, ) ( ) ( , ) ( ), [0, ), , ( , ) ( )( ( , )) , [0, ), , v t t a x v t x b x t x m v t x k t v t x t x m              (1.6) holds, then the equilibrium point ( ) 0x t  of the system (1.1) is uniformly finite-time stable with settling time function 0 0( , )t t x satisfying 1 1 0 0 0 0 ( , ) ( , ) [ ],( ), 1 v t x t t x k k        where 0 ( ) ( ) t t k t k s ds  . if nm d  , then the equilibrium ( ) 0x t  of the system (1.1) is globally uniformly finite-time stable. 2. main results it is well known in practical system applications that system parameters are not always known in advance due to the uncertainties that inevitable arises during operations. as a result, practical controllers are designed with uncertainties in focus. in this section, the aim is to design a finite-time adaptive stabilizing control laws that will stabilizing the unstable dynamics of the novel system and update the unknown parameters in a uniform finite time. 2.1 mathematical model of the novel system consider a general structure of a novel 4d hyperchaotic system which is inspired by the burke-shaw atmospheric model template [19], 1 1 1 2 2 3 2 3 1 3 4 2 5 4 3 6 1 2 7 4 8 1 9 2 10 3 11 4 ( )x x x x x x x x x x x x x x x x x                              (2.1) where 1 11, , 0i i      are bounded parameter vectors and 1 2 4[ , ,..., ]x x x x are the state variables. system (2.1) may be considered as a generic structure of a family of the novel hyperchaotic systems which can produce special cases that still exhibit hyperchaos, yet possessing qualitative and quantitative properties which are conditioned by nullifying one or 124 e. a. umoh, o. n. iloanusi copyright ©2018 assa. adv. in systems science and appl. (2018) more system parameters, in addition to appropriate selection of system parameter values. in this paper, we nullified two terms 2 11 0   , resulting in a new 10 terms and 9 parameter system given by 1 1 1 2 2 3 1 3 4 2 5 4 3 6 1 2 7 4 8 1 9 2 10 3 ( )x x x x x x x x x x x x x x x                          (2.2) when 1 37, 3.1,   4 53.5, 0.95,   6 71, 13,   8 9 100.01, 3, 0.01     , system (2.2) exhibit hyperchaotic behaviour depicted in fig. 1 (a) – (f) after 200s. fig. 2.1. 2-d phase portraits of the novel hyperchaotic system -15 -10 -5 0 5 10 15 -20 -15 -10 -5 0 5 10 x1 x 3 (a) -15 -10 -5 0 5 10 15 -20 -15 -10 -5 0 5 10 15 20 x1 x 4 (d) -25 -20 -15 -10 -5 0 5 10 15 20 25 -20 -15 -10 -5 0 5 10 x2 x 3 (b) -25 -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 x2 x 4 (e) -15 -10 -5 0 5 10 15 -30 -20 -10 0 10 20 30 x1 x 2 (c) -15 -10 -5 0 5 10 -20 -15 -10 -5 0 5 10 15 20 x3 x 4 (f) chaotic analysis of a novel 4-d hyperchaotic system 125 copyright ©2018 assa. adv. in systems science and appl. (2018) 2.2 qualitative and quantitative properties of the hyperchaotic system 2.2.1 dissipativity we can examine whether the system is dissipative or otherwise by applying the liouville divergence theorem [26], [27]. let the vector notation of the 4-d hyperchaotic system (2.2) be denoted by 1 1 1 21 2 3 1 3 4 2 5 42 3 3 6 1 2 7 4 4 8 1 9 2 10 3 ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( ) i i f x a x xf x f x a x x a x a xf xdx f x f x f x a x xdt f x f x a x a x a x                             (2.3) suppose  is a region in the phase space 4 with a smooth boundary and ( ) tt  where t is the flow of the system (2.3). if v is a hypervolume of the phase space at time 0t  , then by liouville’s divergence theorem [28], 1 2 3 4 ( ) . t dv t fdx dx dx dx dt    (2.4) where 4 1 ( ) . i i i df x f dx   is the divergence of the vector field f of the system (2.3). using the theorem, the rate of volume contraction is given by the lie derivative [33] 1 1,2,3...i i i dv i v dt     (2.5) where 1 1 2 2 3 3 4 4, , ,x x x x        are the state variables of the system (2.3). the divergence of the vector field f on 4 can be obtained by using the relationship 31 2 4 1 2 3 4 ( )( ) ( ) ( ) . df xdf x df x df x f dx dx dx dx      (2.6) by using (2.3), (2.4) and (2.6), the divergence is 1 4 1 4( )f           (2.7) for 1 47, 3.5   , 3.5f   . since the divergence is negative, it implies that the hyperchaotic system is dissipative. 2.2.2 equilibria and local stability hyperchaotic systems are essentially nonlinear models, and are therefore linearized in order to study their local stability for different parameters. the stability is determined by the sign of the real part of the eigenvalues of the jacobian matrix. the jacobian matrix is the matrix of the partial derivatives of the right-hand side with respect to state variables where all derivatives are evaluated at the equilibrium point ex x and is expressed by 126 e. a. umoh, o. n. iloanusi copyright ©2018 assa. adv. in systems science and appl. (2018) 1 1 1 1 1 2 3 4 2 2 2 2 1 2 3 4 3 3 3 3 1 2 3 4 4 4 4 4 1 2 3 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) f x f x f x f x x x x x f x f x f x f x x x x x j x f x f x f x f x x x x x f x f x f x f x x x x x                                                   (2.8) when 0x  , the equilibrium is at the origin, i.e. 1 2 3 4( 0, 0, 0, 0)e x x x x    . the equilibrium point of (2.3) at any point 4x r (other than the origin), that is, * * * * 1 2 3 4( 0, 0, 0, 0)e x x x x    is calculated by using the matrix (2.8). 1 1 * * 3 3 4 3 1 5 * * 6 2 6 1 8 9 10 0 0 ( ) 0 0 0 x x j x x x                              (2.9) analyzing (2.9), the equilibrium points are located at 10( , , , )e b b ab cb da   10( , , , )e b b ab cb da     (2.10) where 8 9 7 6 4 5 3 7 5 6 10 a b c d                 (2.11) to test for the type of stability associated with each equilibrium point, eq. (2.10) were computed as follows: 3.61 3.61 3.61 3.61 , 1078 1078 12670.28 12696.88 e e                             (2.12) using e and e respectively in the matrix (2.9) gives an indication of the nature of stability of these equilibrium points i.e. 7 7 0 0 3341.8 3.5 11.191 0.95 ( ) 3.61 3.61 0 0 0.01 3 0.01 0 j e                  (2.13) the eigenvalues of (2.13) are 1 2 3 40.000, 154.911, 150.886, 0.0251        , which implies that it is a saddle and unstable. next, we have 7 7 0 0 3341.8 3.5 11.191 0.95 ( ) 3.61 3.61 0 0 0.01 3 0.01 0 j e                (2.14) chaotic analysis of a novel 4-d hyperchaotic system 127 copyright ©2018 assa. adv. in systems science and appl. (2018) the eigenvalues of (2.14) are 1 2 3 40.000, 155.032, 151.000, 0.025        which implies it is a saddle and unstable. 2.2.3 lyapunov exponents and kaplan-yorke dimension the lyapunov exponent measures qualitatively the rate of exponential divergence or convergence of nearby trajectories of the system in state space, while the kaplan-yorke dimension (lyapunov dimension) gives a quantitative measure of this divergence. the lyapunov exponents was calculated based on the wolf algorithm [29], in conjunction with the procedure which implements the gram-schmidt ortho-normalization in the matlab environment. the numerically computed values are 1 2.216,le  2 1.280,le  3 0.000,le  and 4 6.997le   . the system has two positive lyapunov exponents, a null and a negative lyapunov exponent (+, +, 0, -), which confirmed its hyperchaoticity. moreover, 1 2 3 4le le le le   and 1 2 3 4 0le le le le    , hence the novel system is dissipative. the kaplan-yorke dimension [30] is given by 11 1 , 1,2,3 d ky j jd d d le j le     (2.15) where d is the topological dimension of the attractor, and must satisfy 1 0 d j j le   . for regular chaos with three dimensions, the topological dimension is 2, while for hyperchaos, it is 3. by using the numerically generated values of lyapunov exponents, the kaplan-yorke dimension is calculated as 1 2 3 4 3 3.4996ky le le le d le      2.2.4. bifurcation diagrams bifurcation diagram shows how the dynamics of a system changes with variation of control parameters. in this paper, we explored the parameter space to discover which parameters influence the dynamic characteristics of the system. two parameters 1 and 3 influences the route to chaos. fig. 2.2 depicts the bifurcation diagrams plotted by varying the two control parameters for the range 16.3 6.5  and 13.5 4.15  . fig. 2.2. bifurcation diagrams of control parameters 1 , 3 3. finite-time adaptive stabilization the objective of finite-time adaptive stabilization is to design an adaptive control law and parameter update laws such that the state and parameter update trajectories of the controlled system (2.16) converged in uniform finite time for any initial condition. let the controlled form of system (2.3) be given as follows: 128 e. a. umoh, o. n. iloanusi copyright ©2018 assa. adv. in systems science and appl. (2018) 1 1 1 2 1 2 3 1 3 4 2 5 4 2 3 6 1 2 7 3 4 8 1 9 2 10 3 4 ˆ ( ) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ x x x u x x x x x u x x x u x x x x u                              (3.1) where 1 3 10 ˆ ˆ ˆ,   are the unknown parameters to be estimated, , ( 1,2..4) i ii x xu d a i   is the finite-time adaptive control functions associated with each coupled equation and comprises of ixd , the model equation-based derivative of the control functions and sgn( ) ,( 1,2..4) i ix i u i ia x x x i      , the augmented control functions and iu are the augmented controller strength coefficients. lemma 3.1 the unknown parameters of system (3.1) can be estimated if there exists parametric error functions ,( 1,3,...10)i i  , where : 1 1 1 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 ˆ ˆ ˆ, , , ˆ ˆ ˆ, , , ˆ ˆ ˆ, ,                                              (3.2) 1 3 10 ˆ ˆ ˆ,   are the unknown parameters to be estimated. taking the derivatives of (3.2) yields the following time-varying functions: 1 1 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 ˆ ˆ ˆ ˆ ˆ, , , , ˆ ˆ ˆ ˆ, , ,                                     (3.3) remark 3.1 several works have featured the use of sign function in finite-time controller design due to its good tracking properties [18], [31] etc. in these references, the designed controllers exhibit good noise rejection and good convergence time than controllers that do not feature the signum variables. however, their convergence time are still appreciably large. in this present work therefore, we introduced coefficient terms called augmented controller strength coefficients and augmented parameter update strength coefficients respectively, in conjunction with signum variables to produces controller and parameter update structures that drastically cut down on the uniform convergence time of the dynamics of the controlled system. 3.1 the proposed controller and parameter update structures the controlled hyperchaotic system (3.1) can be finite-time adaptive stabilized and the unknown parameters can be accurately estimated in uniform finite time, if the following adaptive stabilizing control laws and parameter update laws are applied: chaotic analysis of a novel 4-d hyperchaotic system 129 copyright ©2018 assa. adv. in systems science and appl. (2018) 1 2 3 4 1 3 4 1 1 1 2 1 1 1 2 3 1 3 4 2 5 4 2 2 2 3 6 1 2 7 3 3 3 4 8 1 9 2 10 3 4 4 4 1 1 1 2 1 3 1 2 3 3 2 4 2 4 5 ˆ ( ) sgn( ) ˆ ˆ ˆ sgn( ) ˆ ˆ sgn( ) ˆ ˆ ˆ sgn( ) ˆ ( ) ˆ ˆ ˆ u u u u u x x x x x u x x x x x x x u x x x x x x u x x x x x x x x x x x x x                                                          5 6 7 8 9 10 2 4 5 6 1 2 3 6 7 3 7 8 1 4 8 9 2 4 9 10 3 4 10 ˆ ˆ ˆ ˆ ˆ x x x x x x x x x x x x                                                              (3.4) where 1  are the augmented parameter update strength coefficients ( i  has the same definition with iu ) and 0 1  is a non-negative index. proof. firstly, by using (3.2) and (3.4) in (3.1), the controlled system (3.1) becomes 1 2 3 4 1 1 1 2 1 1 1 2 3 1 3 4 2 5 4 2 2 2 3 6 1 2 7 3 3 3 4 8 1 9 2 10 3 4 4 4 ( ) sgn( ) sgn( ) sgn( ) sgn( ) u u u u x x x x x x x x x x x x x x x x x x x x x x x x x x x x                                            (3.5) secondly, based on lemma 1.2, a lyapunov function candidate is given by 4 10 2 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 3 4 1 3 4 5 6 7 8 9 10 1 1 ( ) 2 2 1 1 ( ) ( ) 2 2 x p i i i i v t v v x x x x x                               (3.6) the partial derivative of (3.6) produces 1 1 2 2 3 3 4 4 1 1 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 ( ) ... ... v t x x x x x x x x                                 (3.7) by making use of (3.3), (3.4) and (3.5) in (3.7), the derivative becomes 1 2 3 4 1 1 1 2 1 1 1 2 3 1 3 4 2 5 4 2 2 2 3 6 1 2 7 3 3 3 4 8 1 9 2 10 3 4 4 4 1 1 3 3 4 4 5 5 ( ) ( ( ) sgn( ) ) ( ... ... sgn( ) ) ( sgn( ) ) ... ˆ ˆ... ( sgn( ) ) ... ˆ ˆ... u u u u v t x x x x x x x x x x x x x x x x x x x x x x x x x x x x                                                       6 6 7 7 8 8 9 9 10 10 ˆ ˆ ˆ ˆ ˆ              (3.8) 130 e. a. umoh, o. n. iloanusi copyright ©2018 assa. adv. in systems science and appl. (2018) using the following convention 1 sgn( ) ; sgn( ) sgn( ) i i i i i i i i i i i i i x x x x x x x x x x x x x        (3.9) eq. (3.8) is then transformed to 1 2 3 4 1 1 1 1 1 2 3 4 1 2 3 4 2 1 1 1 2 3 1 2 3 4 2 5 2 4 6 1 2 3 7 3 8 1 4 9 2 4 10 3 4 1 1 3 3 4 4 5 5 6 6 7 7 8 8 9 9 ( ) ... ... ( ) ... ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ... u u u uv t x x x x x x x x x x x x x x x x x x x x x x x x x x x x                                                                     10 10̂ (3.10) and by using (3.9) in (3.10), the derivative reduces to 1 2 3 4 1 3 4 5 6 7 8 9 10 1 1 1 1 1 2 3 4 1 2 3 4 1 3 4 5 6 7 8 9 10 ... ( ) 0 ... u u u ux x x x x x x x v t                                                                  (3.11) eq. (3.11) is negative definite in 17 , thus lemma 1.2 is satisfied, and the equilibrium point ( ) 0x t  is uniformly finite-time stable about the origin because ( ) 0v t  and also implies that (0) 0v  and lim 0 t t x   . suppose 1 2 3 4 cu u u u u        (where cu is known as the controller strength coefficient) and 1 3 4 10 ... p             (where pu is known as the parameter update coefficient), then by setting c pu u g    , ( g is known as the global strength coefficient), (3.11) reduces to: 4 4 9 11 1 1 1 ( ) 0i g i g i i i i v t x x                (3.12) using assumption 1, it can be deducted that 14 4 1 12 2 1 1 ( )g i i i i x x x             (3.13) substituting (3.3) in (3.12) yields 4 11 11 1 1 ( ) i g i i i v t x x            (3.14) by virtue of assumption 1, it is easy to see that 1 1 2 2 21 1 1 1 2 2 2 2 ( ) ( ) 2 ( ) 2 2 v t x x x v                     (3.15) setting 1 2 1 2 , 2 p        , reduces (3.15) to chaotic analysis of a novel 4-d hyperchaotic system 131 copyright ©2018 assa. adv. in systems science and appl. (2018) ( ) pv t v  (3.16) also, by substituting  and p into (1.4), we have   1 12 0 20 0 01 2 ( ) 1 2 ( ) 11 2 2 v t t t t v t                  (3.17) where 22 2 2 0( ) (0) (0) (0) (0)x p i iv t v v x     remark 3.2 eq (3.17) implies that the finite-time stabilization of the controlled system (3.1) depends on the initial condition  and rational number p , while the uniform convergence time is either increasing or decreasing as the parameter p is varied as can be observed in the works of [18], [32] . in our case however, due to the complexity introduced by the global strength coefficient g , it was observed that the bounded value of g , min max [ , ]g g g   has a strong constraining effect on trajectory overshoot and also has domino effects on the uniform convergence time. accordingly, the following cases were observed when min max    and min maxg g g    . a. as ming g  and min  , the convergence rate was relatively slower and the uniform convergence time is an increasing function b. as maxg g  and max  , the convergence rate was relatively faster and the uniform convergence time is a decreasing function. 4. numerical simulation results the controlled hyperchaotic system is simulated using matlab for the following parameters 1 37, 3.1   4 5 6 73.5, 0.95, 1, 13       , 8 9 100.01, 3, 0.01     , with initial conditions set as 1 2 3 4[ (0), (0), (0), (0)] [4,2, 4, 2]x x x x    . the following results were obtained for two different cases. case 1: minmin 0.8, 1000g   0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 -4 -2 0 2 4 t(s) s ta b il iz e d t r a je c to r ie s x1 x2 x3 x4 fig.4.1. stabilized trajectories of the controlled system 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 -3000 -2000 -1000 0 1000 2000 3000 4000 t(s) c o n tr o ll e r d y n a m ic s u1 u2 u3 u4 fig.4.2. uniformly converged dynamics of the adaptive controller 132 e. a. umoh, o. n. iloanusi copyright ©2018 assa. adv. in systems science and appl. (2018) case 2: maxmax 0.99, 8000g   remark 4.1 the rate of asymptotic convergence of the system dynamics in case 1 is slower than those in case 2 due to the effects of the global strength coefficients. this is in consonance with the remark 3.2 on the effects of the global strength coefficient. it can be observed that the larger g , the faster the rate of uniform convergence of the system’s trajectories. when compared to controllers that featured in related works listed in table i, it is obvious that the designed adaptive controller provides a faster convergence time than those in these works. 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 -5 0 5 10 15 t(s) e s ti m a te d p a r a m e te r s a7 a8 a9 a10 fig.4.3 (b). estimated parameters of the controlled system 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 -10 -5 0 5 10 t(s) e s ti m a te d p a r a m e te r s a1 a3 a4 a5 a6 fig.4.3 (a). estimated parameters of the controlled system 0 0.5 1 1.5 x 10 -3 -4 -2 0 2 4 t(s) s ta b il iz e d t r a je c to r ie s x1 x2 x3 x4 fig.4.4. stabilized trajectories of the controlled system 0 0.5 1 1.5 x 10 -3 -4 -2 0 2 4 x 10 4 t(s) c o n tr o ll e r d y n a m ic s u1 u2 u3 u4 fig.4.5. uniformly converged dynamics of the adaptive controller 0 0.5 1 1.5 x 10 -3 -10 -5 0 5 10 t(s) e s ti m a te d p a r a m e te r s a1 a3 a4 a5 a6 fig.4.6 (a). estimated parameters of the controlled system 0 0.5 1 1.5 x 10 -3 -5 0 5 10 15 t(s) e s ti m a te d p a r a m e te r s a7 a8 a9 a10 fig.4.6 (b). estimated parameters of the controlled system chaotic analysis of a novel 4-d hyperchaotic system 133 copyright ©2018 assa. adv. in systems science and appl. (2018) 4.1. comparison with related works the rate of uniform convergence of system dynamics using the proposed finite-time adaptive controller was compared with related works in the literature. the different rates reported in the papers are given in table i. table i. comparison with related works related works approximated convergence time of error dynamic systems proposed work 0.005s ref [33] 0.2s ref [34] 0.3s ref [35] 0.5s ref [36] 1.0s ref [37] >1.0s ref [38] >1.0s ref [31] 2.0s ref [39] >1.0s ref [40] >2.0s ref [41] 8.0s 5. conclusion in this paper, we proposed an improved finite-time adaptive controller with a fast rate of uniform convergence of system trajectories of hyperchaotic systems. we examined the performance of the controller on a novel special case of a hyperchaotic system which was originally evolved by adding feedback control to the classical burke-shaw chaotic system that models some atmospheric phenomena. numerically obtained results shows that the proposed control structure offered better rate of convergence of the controlled system dynamics when compared to related works in the literature. overall, these systems can contribute to improve the design characteristics of chaos-based cryptosystems. references [1] e. n. lorenz, “deterministic nonperiodic flow,” j. atmos. sci., vol. 20, no. 2, pp. 130–141, 1963. 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[41] l. liu, x. cao, z. fu, and s. song, “guaranteed cost finite-time control of fractional-order positive switched systems,” j. control sci. eng., vol. 2017, p. 10pp, 2017. adv syst sci appl 2019; 01; 44-60 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/647 protecting a low voltage direct current system using solid-state switching devices for dc grid applications ibrahim almutairy1*, johnson asumadu2, zhang xingzhe3 1) majmaah university, western michigan university, michigan, usa e-mail: ibrahim.almutairy@wmich.edu 2,3) electrical and computer engineering, western michigan university, michigan, usa e-mail: xingzhe.zhang@wmich.edu received september 20, 2018; revised february 24, 2019; published april 15, 2019 abstract: low voltage dc (lvdc) distribution system has less distributional losses than ac grid, integrating renewable sources, greatly increasing the mix of clean energy sources in the standard grid in the next decade and this has become a trend that is likely to continue. as an additional benefit, dc electrical power is oftentimes seen as beneficial to applications that use cleanly generated, renewable power. considering lvdc power system design, and focusing on fault protection and the systems dc circuit breaker, are system priorities. only solid-state circuit breakers (sscb) should be considered to obtain advanced system topography. a new type of solid state circuit breaker is being developed as a new power device for lvdc power networks to replace the emcb. the only suitable candidate for this task is the insulated gate bipolar transistor (igbt). development of dc grid protection allows for the development and improvement of new power electronic devices, focusing in on the application for medium to low voltage dc grids, a rapidly acting switching function as well as fault current limiting features. to avoid nuisance tripping, fault current limiting function can be satisfactorily accomplished by extending the elapsed time using the same control circuit. this paper presents a novel lvdc circuit breaker model, which uses a coupled inductors circuit breaker, a designed model of igbt showed and discussed and developed an lvdc system for grid applications. keywords: dc protection, insulated-gate bipolar transistors (igbts), lvdc power system, solid-state circuit breakers (sscb). 1. introduction lvdc electrical energy using renewable environmentally friendly power is frequently seen in many applications as helpfully modern. considering the design of dc power systems, it is important to focus on fault protection and maintain the power system's dccb. only solidstate circuit breakers should be considered for installation and applied to these types of systems because these are the only types breakers that respond quick enough to faults to prevent damages even though they tend to have a greater rate of power loss than other breaker types. sscbs are pure semiconductor devices that monitor all the behaviors of the igbt which are tremendously important to switching transitions. solid state lvdccbs make use of a completely manageable power semiconductor device from the primary branch to interrupt electrical current and doesn’t require a mechanical switch. they interrupt dc current very effectively the idea being introduced in this system is a new type of dc solid state circuit breaker with transformer coupling that uses coupled inductors to detect and isolate faults. these types of circuit breakers gain advantages over other types of circuit breakers because they * corresponding author: ibrahim.almutairy@wmich.edu mailto:xingzhe.zhang@wmich.edu protecting a low voltage direct current system 45 copyright ©2019 assa. adv. in systems science and appl. (2019) don’t have as many parts and has an adjustable current threshold that can be set to operate when the level of current rises past the set threshold causing the circuit breaker to trip [1-6]. normally, the circuit breaker is in the system between the source and the load. sscbs are used in high voltage systems because they have a swift switching speed and interrupt high voltage dc current very efficiently. during fault interruption, the rapidly dropping current will expose the sscb to serious strain due to overvoltage. if this is left uncorrected, this overvoltage can easily cause the sscb to become unresponsive and might possibly damage the sensitive devices inside the sscb [7], [8]. to address this issue, it is necessary to install a snubber circuit to protect the system by suppressing overvoltage. having an installed snubber circuit becomes essential to the safe turn off of the system. there are some requirements that should be able to be met by the snubber circuit. it should be highly reliable for medium capacity, low voltage applications. overvoltage should cause a minimal level of stress to the sscb at turn off. the presence of a snubber circuit that has been designed and specialized for precise application in sscbs is of extreme importance. generally, methods of snubber design anticipate that snubbers be must be suited for converter switches and thus, they cannot be applied directly to the sscb snubber. there are many reasons for this. reduction of snubber loss as well as rapid suppression features do not get lots of attention in sscb snubber design because of inconsecutiveness as well as infrequency of sscb switching. suppression of overvoltage as well as ability to withstand fault current are the features which receive the most attention in sscb snubber design due to the magnitude of fault current that is met and, as a consequence, sscbs become exposed to high overvoltage [7-10]. this paper is organized as follows: section two covers related work, section three present the background of the lvdc system and protection using sscb for lvdc grid applications. in section four, analytical analysis of proposed dc protection circuit is presented. in section five, examination of igbts for implementation in lvdc sscb is covered. section six presents modeling the lvdc system under study and section seven conclusion. 2. related work certain advancements in the field of power electronics have been ongoing and lvdc systems are looking more advantageous than they ever and have also become increasingly dependable since the application of grid distribution, advanced circuit breaker technology, power converters and other lvdc protective devices. new microgrids will be perceived in the dc power format and many power systems or, power conversion components are readily available but, as it applies to dc circuit breakers, many different designs are still in the experimental stages. circuit breakers have certain functions and duties which must be performed. a circuit breaker must be capable of performing three basic duties. it must be capable of opening the faulted circuit and breaking the current. it must be able to be closed to a fault. lastly it must be able to carry fault current for a short period of time as part of a hierarchal protection coordination scheme. in addition, it must be completely controllable. a circuit breaker must have a rapid switching speed as well as low conduction losses, no arcing, and must ensure definitive tripping of the whole breaker system. detecting short circuits as quickly as possible is the key to minimize power dissipation in system components during fault events so that overheating and damages to a switching device, such as an igbt, can be prevented. various works address the need for a new and fast circuit breaker model for lvdc system applications. one recent research proposed a bidirectional magnetically coupled t-source inverter (tsi) for lvdc systems [11]. the paper investigated the performance of a magnetically coupled t-source inverter for lvdc systems. considering that building an ideal transformer is not possible, the coupling coefficient of the transformer, in respect to the leakage inductance, 46 i. almutairy , j. asumadu, z. xingzhe copyright ©2019 assa adv. in systems science and appl. (2019) determines its performance, settling upon practical lvdc application of the tsi and its limitations were exposed in the experimental results. analytical circuit analysis and derivation of mathematical equations were performed under ideal conditions. another study highlights protection of lvdc systems using a bidirectional z-source circuit breaker [12]. modified topologies of a z-source lvdc with bidirectional power flow capability are introduced in the study. conventional z-source breaker topologies make it possible to protect unidirectional power flow. proposed topologies in the study are a bidirectional zsb and bi directional zsb with a coupled inductor and can clear faults in any direction that is needed. both topologies make use of a scr as a switch for commutation. coupled inductors help reduce the size of inductors, the cost and also to eliminate any need for an additional z-source breaker capacitor. other recent research focuses on the design of solid-state circuit breaker-based protection for dc shipboard power systems [13]. protection is an important aspect of dc shipboard distribution system design. it should be thought about from the very beginning design stage for reliable, low cost and efficient system operation quick action is an essential requirement for dc shipboard distribution protection. sscb based dc protection offers very rapid protection speed and reduces the requirements on the fault withstanding limitations and fire hazards but has a high cost and losses of power because of semiconductor switching devices. dc protection methods are evaluated based upon speed and other criteria. any optimized sscb-based dc protection design is a compromise of the right levels of reliability, fast speed, reasonably low cost and good overall system efficiency. further work addresses dc microgrid protection using the coupled inductor sscb [14]. large scale implementation of dc microgrids depends upon the development of reliable dc protection options. the proposed dc breaker design uses an scr to interrupt the fault current instantly. a bidirectional version of the breaker is also provided so multiple dc breakers can be installed in a notional dc microgrid. the breaker has a control switch in the design so the breaker can be opened manually without creating a large disturbance. traditionally, a central control was proposed to supervise operations of multiple breakers in a microgrid. two other control strategies are studied. one involves independent, local control of breakers and the other is to control breakers in pairs. the merits of all three strategies are studied. it can be concluded that within certain limitations, the paired method of control provides the desired results either a much simpler communication architecture. the next research introduced a new dc sscb concept using a small inductor like that used in electro-mechanical circuit breakers [15]. the new sscb scheme uses the voltage inductor to activate short circuit protection. it is in this way that the circuit starts to open when the fault condition to prevent overload current is present [16]. some sscb technologies for dc are reported and can be classified into three main types, mechanicals, solid state and hybrids. each of them have their advantages and disadvantages in respect to the others however, there are some common characteristics that are extremely important in this device, such as, time of response, electric arc suppression, high efficiency, and low cost among others. the development of new semiconductor devices with better features is entirely feasible and may improve sscb technologies in many aspects. the new sscb is based on two power mosfets, having one latch circuit, one inductor and one high-pass filter within a voltage divider. 3. the idea of low voltage dc electrical distribution low voltage dc grids are not very different from their ac counterparts. even though some components are different in nature, characteristics, or function, the basic concept is always the same. the objective is to transport electricity through cables and components at low voltage to the end-user. the main difference between dc and ac systems are the way in which they are configured. the microgrid itself, is a distributed power system on a small protecting a low voltage direct current system 47 copyright ©2019 assa. adv. in systems science and appl. (2019) scale, that is made of distributed energy sources and associated loads. microgrids eliminate the need for connection to the main utility grid [12-15]. there are numerous advantages of dc microgrids over ac microgrids such as the absence of a skin effect, lower on-state losses as well as the ability to provide 1.41 times more power than ac, no problems with reactive power control and no proximity effect. dc also has unique issues of its own like arc interruption, fewer protection devices and is more difficult to control than an ac system. lvdc microgrids are a new concept in distribution systems. it is well suited for installation in offices that have sensitive computer loads, isolated or rural power systems. the grid that we have today in the united states of america is mostly ac and most of loads we consume using various household electronic devices which consume the energy in dc, requiring power converters to connect to the grid. if these loads are fed through the dc bus, then they will require fewer power conversion stages. since there are fewer power conversion stages, losses resulting from conversion are also lower. most of the resistive loads can be connected to either the ac or the dc bus. fig. 2.1. conceptualization of the lvdc bus microgrid system. the above diagram shows the concept of a lvdc bus microgrid system. the active power consumed by dc loads is the same as ac loads but, there is no reactive power function in dc. a dc microgrid has a wind turbine, dc/dc converter for its battery and the solar pv. generally, wind turbine and micro turbine uses both dc/ac and ac/dc converters but, only an ac/dc converter is used in the methodology of the latest technology. ac power grid systems possess various advantages that are easily applicable but, certain issues like synchronization, control of reactive power and stability of the ac bus are persistent problems in the ac bus microgrid. as a lvdc bus microgrid system is small, it can neglect the loss of transmissions within the system and provides a solution to these problems. the dc distribution protection system is of considerable benefit to a lvdc bus microgrid as the fault is detected and located within the system autonomously by the protection system of the lvdc bus microgrid. besides the benefits, dc bus microgrids also face challenges like lack of standard practices and guidelines, as well as breaking of dc arcs and dc protective devices. a lvdc microgrid is dynamic because operation of supply and demand equipment so, suitable power-flow control, voltage regulation, protection, coordination and adequate filtering are necessary to guarantee an acceptable quality of supply. over the last few decades, power electronics have been one of the main contributing factors behind the paradigm shift that started in the 1960s and have become increasingly power-dense while decreasing in physical volume. such a trend was made possible, mainly, by increasing switching frequency. these days, power electronics that are very small, can be found in all types of electrical appliances that have an efficiency ranging from 50% up to 99%. the dialog regarding voltage levels seems unending because voltage levels will always be the greatest difference between the two types of systems. proposals about voltage level were specifically limited to voltages below 120vdc [16] [17]. some recent pilot projects put 48 i. almutairy , j. asumadu, z. xingzhe copyright ©2019 assa adv. in systems science and appl. (2019) voltages from 24vdc to 48vdc demonstrating that this may be something worthy of investigation in future work. an approach such as this one is consistent with the definition of iec 60038 for extra low voltage to assure the protection of the user, it should be possible to have a lvdc system that runs that level of voltage even if the system had plug-in electric vehicles as well as the household’s kitchen loads. 3.1. lvdc short circuit currents characteristics a lvdc microgrid that has a new, complex topology of mixed ac and dc currents become dangerous as the system becomes more complex and new types of faults are introduced and different responses from the system are to be expected when faults occur on a lvdc system. the energy stored in the smoothing capacitors and storage devices begin to feed a large discharge current in a very short amount of transient time. this causes dc fault currents to undergo a transient discharge current that has high frequency oscillations between the smoothing capacitors and system inductances as well as short circuit current in the steady state. these transient fault currents cause a very high level of stress to the operation of protection and performance of lvdc systems [18] [19]. characterizing lvdc short-circuits is therefore, of extreme importance. ratings of appropriate equipment and correct protection settings, as well as selectivity, need accurate short-circuit characterization. the effectiveness of using standards that are already in place such as iec61660 for characterizing lvdc faults will be investigated. when an external dc fault in a lvdc microgrid begins, the charged smoothing capacitor acts instantly as a considerable source of dc fault current and begins to feed the fault. when the discharge peak current is reached, the capacitor is totally discharged and the dc terminal voltage become very small. in some cases, it is almost zero. when the antiparallel diodes of the converter become forward biased, the vsc loses control and the igbt switches become blocked for the purpose of self-protection. this results in the anti-parallel diodes functioning as a bridge rectifier and continuing to feed the fault during the transient. after fault occurrence, transient fault current ends, steady-state dc fault current is fed by the grid and local generation. 3.2. dc fault types in general, the most common causes of short-circuiting are component failure, lighting surges, high magnitude ground fault or line to line fault, external environmental stressors such as fires or insulation failure caused by operating at excessive temperatures that result from overload stress. in lvdc microgrids, converters like vscs igbt-based converters are sturdier to ac faults and more sensitive to dc faults. two types of faults may occur on the dc side. the first is an internal dc fault inside of the main converter and the other is an external dc fault, either on the converter terminals or at isolated places on the downstream dc feeders. internal faults aren’t as frequently occurring when compared to external faults [20], [21]. the most extreme external dc fault happens at the converter terminals. remote faults on dc feeders also have a considerable impact on converter performance. dc faults are classified as follows: line-to-ground fault: this type of fault occurs when a positive or negative pole is shorted to the ground. in this case, the voltage of the faulted dc line drops down depending on the fault impedance. line-to-line fault: this type of fault happens if the positive and negative dc poles are shorted. this could be the result of the dc cable insulation breaking down or a direct short circuit between the lines. table 3.1 shows the differences between dc fault types. protecting a low voltage direct current system 49 copyright ©2019 assa. adv. in systems science and appl. (2019) table 3.1 differences between line-to-line and line-to-ground fault line-to-line fault line-to-ground fault the positive and negative lines are normally shortcircuited whenever line-to-line faults happen. line will be shortcircuited to the ground when line-to ground faults happen. low fault impedance is the typical characteristic of line-to-line faults. both low and high impedance are the typical characteristics of line-toground faults. although overvoltage protection is vital during line to ground faults, when faults happen, the capacitor swiftly discharges to the ground [22]. current flows through the ground and back to the non-faulted line then finally goes back to the source, causing voltage on the healthy pole to rise. overvoltage also is concerning during the loss of an inverter. the loss of an inverter causes voltage to rapidly spike because of extreme unnecessarily high power, rapidly charging the dc link capacitors. a rectifier loss, only becomes concerning when the loss is temporary. when the rectifier abruptly returns, it causes overvoltages like an inverter loss does. 3.3. problems associated with lvdc protection system typically, power distribution systems have not been considered important when they are viewed from the perspective of protection complexity because of the high cost as well as the radial characteristic and nature of distribution networks, that have fault current directions that are always known. the principles of operating most protection systems are based on non-unit overcurrent protection schemes in which operating time decreases as the fault current increases. protecting lvdc distribution networks does not involve installing hardware of any kind. additionally, interrupting dc fault current is more difficult than interrupting ac fault current. dc fault current and voltage waveforms do not have a natural zero crossing ability, so they must be forced to zero meaning that the risk of fires or arcing are expected more often with dc systems than ac because of problems that develop from forcing dc currents to zero. because of this, standard circuit breakers and fuses have difficulty in protecting dc systems. if emcb are going to be utilized in a dc system, a larger size and heavier weight as well as slow performance can be expected. there is a solution. interruption of dc fault current is possible if current-limiting fuses or current limiting circuit breakers, such as sscbs, are used and their ratings are adjustable for application in lvdc systems [23]-[25]. these types of devices don’t require zero-crossing to extinguish dc fault arcs. fault current will be interrupted much sooner than the time needed for fault current to reach its peak. dc fault arcs are more difficult than with ac because they must reduce the voltage across the arc by increasing its length which is accomplished by increasing the length between the two contactors and using an arc splitter to split the arc. in addition, dc short circuit profile includes two main forms. one is a high transient discharge short circuit current path and the second is a steady state short circuit current path. transient discharge fault current can be high when compared with steady state fault current as well as causes high stress levels to network components and negatively affects protection system performance [26], [27]. two big problems are expected to happen because of high discharging dc fault current. the first is an increase in the risk of physical damages to the sensitive equipment because of high magnitude and the second is the negative impact upon selectivity of non-unit protection. during transient current discharge, dc voltages near the fault become very small, almost zero, and sometimes the dc voltages can even become negative because of oscillation between cable inductances and filter capacitors. 50 i. almutairy , j. asumadu, z. xingzhe copyright ©2019 assa adv. in systems science and appl. (2019) 4. analytical analysis of proposed lvdc protection circuit observe the suggested lvdc circuit breaker design shown in fig. 4.1 during normal steadystate operation, current flows from the source to the load passing through the igbt and coupled inductors. an inductor is a passive two-terminal electrical component that holds electrical power in a magnetic field when electrical current is flowing through it. in this circuit, the inductor is considered as an inductive value in the distribution line. fig. 4.1. proposed lvdc circuit breaker mode. insulated-gate bipolar transistor (igbt) is a three-terminal power semiconductor device primarily used as an electronic switch. when compared with a mechanical switch, igbt needs several microseconds to switch off the circuit. consider the dc model of igbt [28], shown in fig. 4.2. fig. 4.2. lvdc model of igbt. in the model above, the voltage transfer function is expressed as: (4.1) (4.2) in the above equations, the parameters are: vcet : collector emitter voltage k : boltz mann constant m : modulation factor to : temperature of the junction of diode protecting a low voltage direct current system 51 copyright ©2019 assa. adv. in systems science and appl. (2019) vg : gate emitter voltage vd : dc bus voltage (collector emitter voltage) a coupled inductor transfers electrical energy between two circuits during current or voltage alteration. during the igbt shutdown, the voltage rapidly spikes because the circuit’s inductance (v=ldi/dt). the transformer transfers voltage to the snubber circuit which absorbs and dissipates energy and is used to suppress the voltage spikes that are caused by the circuit's inductance [29]. the capacitor absorbs the voltage spikes and the resistor releases any energy stored in the capacitor. the equivalent circuit of the proposed lvdc breaker is shown in fig. 4.3. fig. 4.3. proposed lvdc circuit breaker mode. vd is the voltage for igbt, it is acknowledged as a negligible value; so it can be ignored, in order for the transfer function of the equivalent circuit of the suggested lvdc circuit breaker is expressed as: (4.3) (4.4) kcl at node v1 gives: (4.5) (4.6) vo=v1-vg (4.7) (4.8) in this model, the voltage to current relation is expressed as: (4.9) the suggested lvdc circuit model (without snubber) shows the circuit with only the igbt, without a snubber circuit. from the simulation with no snubber, it shows that there is obviously a voltage spike happening at the load during igbt. parameter calculations of custom designed snubber circuit resistor and capacitor value. current across igbt: = (4.10) 52 i. almutairy , j. asumadu, z. xingzhe copyright ©2019 assa adv. in systems science and appl. (2019) (4.11) then choose the value for (4.12) = (4.13) by calculation, when c = 1mf is used. the power dissipates in the resistor (4.14) the lvdc circuit (with snubber) is the circuit model with both igbt and snubber circuit installed. from the simulation with a snubber, fig. 4.4. illustrates how the voltage spikes are reduced to zero, because the spike is being absorbed and dissipated by the snubber. the proposed lvdc circuit breaker has a rapid switching time of 30µs. as displayed in fig. 4.5. fig. 4.4. proposed lvdc cb model overvoltage with snubber. fig. 4.5. proposed lvdc cb model overvoltage with snubber. 5. examination of igbts for application in lvdc sscb connected systems the igbt is the most frequently used switching device for the sscb. meaning that the reliability and durability of igbts are ceaselessly being confirmed by several converters that protecting a low voltage direct current system 53 copyright ©2019 assa. adv. in systems science and appl. (2019) are installed in real-world power systems. most sscbs are based upon implementation of three semiconductor devices in some type of way. semiconductor devices are used in sscbs because they have extensive commercial accessibility, low power requirements for operation of gate drives as well as, high current ratings [30-33]. igbts have many benefits in contrast to its meagre number of drawbacks. the benefits are low switching losses, minor snubber circuitry requirements and high input impedance. because of these things, igbts are reliable and durable enough to be used for the circuit breaker role. there are many topologies for the dc circuit breaker using igbts. based on fig. 5.1, the igbt model under study comprises of an nchannel mosfet and two bjts, bjts one and two. the first one is a p+n-p bjt and the second one is an n-pn+ bjt. r1 is the resistance that is given by the drift region. r2 is the resistance given by the body region. it has been noted that the collector of bjt1 is the same as the base of the collector in bjt2 and vice versa. an equivalent circuit model of igbt is shown below. fig. 5.1. p igbt model equivalent circuit. the simulation result displays the igbt voltage waveform in fig. 5.2. the switching time for igbt model shown in fig. 5.3, is 0.593 ms. to make the switching time of the igbt better for protecting the dc system, capacitors are introduced to the model in parallel with the resistor. fig. 5.4, shows the improved igbt model equivalent circuit with the added capacitors. fig. 5.2. voltage result of igbt model. 54 i. almutairy , j. asumadu, z. xingzhe copyright ©2019 assa adv. in systems science and appl. (2019) fig. 5.3. switching time of igbt model. fig. 5.4. improved igbt model equivalent circuit. fig. 5.4, shows the improved equivalent circuit of igbt model. the capacitor increases the speed of the switch’s response. the simulation result from fig. 5.5, shows the voltage of the improved igbt during the on-state of the igbt. the switching time for the improved igbt model is 0.553ms as shown in fig. 5.6 because of the shorter response time than the original model, the improved igbt model has been chosen for implementation in sscb for protecting lvdc microgrids. fig. 5.5. switching time of igbt model. protecting a low voltage direct current system 55 copyright ©2019 assa. adv. in systems science and appl. (2019) fig. 5.6. switching time of the improved igbt model. 6. modeling the lvdc system under study a circuit schematic has been developed for a post-regulated, isolated dc-dc converter that has two dc loads. a dc-dc converter, specifically, a unidirectional post-regulated isolated dc-dc converter is assembled from a solid-state transformer that has a controlled halfbridge on its primary side and a diode-based half-bridge on the secondary side, followed by a buck converter as displayed in fig 6.1. another important system component to have is a voltage follower. a voltage follower delivers high impedance transformation from one circuit to another, with the purpose of preventing the source from being affected by currents or voltages that the load produces. the corresponding parameters and their respective values are displayed in table 6.1. both loads have an equal rating, protected by sscbs and the load-side has freewheeling diodes that provide current continuity because of load side inductances when the load sscb opens. with the improved igbt model, the capacitor increases the speed of the switch’s response with the improved equivalent circuit of igbt. table 6.1 parameters used in the proposed model and their values parameters specifications c2,c3 c4,c5 v z2, z3, z5 l1 c11 l11 r10 d1-d8 r13, r15, r14, r16 c9, c12, c13, c16 r8, r9 l9, l10 l3, l4, l12, l13 c1, c14 r3, r17 10uf 100uf 60v igbts(ixgh40n60) 0.1h 1u 0.1h 0.003ω diode(d1n4002) 0.5ω 0.01f 0.0015ω 30.25uh 1h 5nf 1kω 56 i. almutairy , j. asumadu, z. xingzhe copyright ©2019 assa adv. in systems science and appl. (2019) fig. 6.1. modified circuit schematic based on existing lvdc sscb without snubber circuit. fig. 6.2. shows high levels of positively charged overvoltage during switching on times, and negatively charged overvoltage during switching off times which negatively effects equipment, causing overheating which damages electronic devices. fig. 6.2. simulation of modified circuit based on existing lvdc sscb model overvoltage without snubber circuit. the equations are based on capacitor voltage and inductor currents in the distribution system as shown below and, the complete system parameters can be obtained thru consideration of the following equations. (6.1) (6.2) (6.3) (6.4) the laplace transform protecting a low voltage direct current system 57 copyright ©2019 assa. adv. in systems science and appl. (2019) (6.5) (6.6) where: the following schematic of proposed lvdc sscb include improved igbt model and custom design snubber circuit in each load. the parameters of snubber circuits were previously discussed. fig. 6.3. circuit schematic of proposed lvdc sscb with snubber circuit. fig. 6.4. protected sscb model with snubber circuit. 58 i. almutairy , j. asumadu, z. xingzhe copyright ©2019 assa adv. in systems science and appl. (2019) fig. 6.4. shows a simulation of a circuit schematic of proposed lvdc sscb with a snubber circuit. when compared with fig. 6.2, which is without a snubber circuit, an extremely small overvoltage during switching on times, and negative overvoltage is reduced to zero during switching off times. the custom designed snubber circuit suppresses the voltage spikes caused by the circuit's inductance when operation modes switch. fig. 6.5. displays the switching time of the igbt, the time duration is preformed faster than the existing igbt model, as discussed above. fig. 6.5. proposed sscb model switching time. 7. conclusion this paper offers an illustration of the workings of sscbs in lvdc applications. the motives for the selection of sscbs are discussed along with the design of the suggested lvdc sscbs. advancement in semiconductor technologies have experienced substantial improvement these last few decades and many devices have been designed. in the future, these technologies are only going to get even better. sscbs are swift to respond to faults experienced in the lvdc grid. this makes the dc power system of the future a more reliably sturdy and advanced system that is much better protected than the systems already in place today. further studies on the use of different semiconductor devices such as igcts and gtos enables greater understanding of the difference between the systems of the past and the modern systems of today as well as future systems as new technologies become available. this will enable protection of many different configurations of lvdc distribution systems and microgrid applications. references [1] d. salomonsson, l. soder & a. sannino (2009). protection of low-voltage dc microgrids, ieee transactions on power delivery, 24(3), 1045 – 1053. 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(2016). dc microgrid protection: using the coupledinductor solid-state circuit breaker, ieee electrification magazine, 4(2), 58–64. [15] capilla, l. a., rodríguez, e. j. j., león, f. j., gordillo-tapia, c. & martínez, j. j. (2017). dc ultra-fast solid state circuit breaker using the voltage inductor to activate short circuit protection, ieee second international conference on dc microgrids (icdcm), nuremburg, 2017, 24–25. [16] gu, c., effah, f., castellazzi, a., waston. a. j. & wheeler, p. (2016) novel currentlimiting strategy for solid-state circuit breakers (sscb) without additional impedance, 2016 18th european conference on power electronics and applications (epe'16 ecce europe), karlsruhe, 1–10. [17] salonen, p., nuutinen, p., peltoniemi, p. & partanen, j. (2009). lvdc distribution system protection — solutions, implementation and measurements, 13th european conference on power electronics and applications, barcelona, 1–10. [18] cuzner, r. m., & venkataramanan, g. (2008) the status of dc micro-grid protection, 2008 ieee industry applications society annual meeting, edmonton, alta., 1–8. [19] park j. d., & candelaria, j. (2013) fault detection and isolation in low-voltage dc-bus microgrid system, ieee transactions on power delivery, (28)2, 779–787. 60 i. almutairy , j. asumadu, z. xingzhe copyright ©2019 assa adv. in systems science and appl. (2019) [20] saeedifard, m., graovac, m., dias, r. f. & iravani, r. (2010) dc power systems: challenges and opportunities, ieee pes general meeting, minneapolis, mn, 1–7. [21] li, w., mou, x., zhou, y., & marnay, c. (2012) on voltage standards for dc home microgrids energized by distributed sources, proceedings of the 7th international power electronics and motion control conference. [22] williamson, b. j. et al., (2011) project edison: smart-dc, 2011 2nd ieee pes international conference and exhibition on innovative smart grid technologies, manchester, 1–10. [23] salonen, p., nuutinen, p., peltoniemi, p., & partanen, j. (2009) protection scheme for an lvdc distribution system, iet conference publications. [24] salomonsson, d., soder, l. & sannino, a. (2009), protection of low-voltage dc microgrids, ieee transactions on power delivery, 24(3), 10451053. [25] yang, j., fletcher, j. e. & o'reilly, j. (2010) multiterminal dc wind farm collection grid internal fault analysis and protection design, ieee transactions on power delivery, 25(4), 2308–2318. [26] almutairy, i. & alluhaidan, m. (2017). protecting a low voltage dc microgrid during short-circuit using solid-state switching devices, in ieee green energy and smart systems conference (igessc), 1–6. [27] candelaria, j. & park, j. d. (2011). vsc-hvdc system protection: a review of current methods, 2011 ieee/pes power systems conference and exposition, phoenix, az, 1–7. [28] almutairy, i., & asumadu, j. & alluhaidan, m. (2017), investigation of igbt behavior in dc systems with advanced circuit breaker technology, 2017 ieee 8th annual ubiquitous computing, electronics and mobile communication conference (uemcon), new york city, ny, 504–508 [29] almutairy, i. & asumadu, j. (2018). a novel dc circuit breaker design using a magnetically coupled-inductor for dc applications, 2018 ieee international conference on consumer electronics (icce), las vegas, nv, usa, , 1–5. [30] magnusson, j., saers, r., liljestrand, l. & engdahl, g. (2014). separation of the energy absorption and overvoltage protection in solid-state breakers by the use of parallel varistors, ieee transactions on power electronics, 29(6), 2715–2722. [32] almutairy, i. & asumadu, j. (2018). examination of breaker-based protection systems for implementation in lvdc sscb applications, 2018 ieee 8th annual computing and communication workshop and conference (ccwc), las vegas, nv, usa, 509–514. [33] yang, j., fletcher, j. e. & o'reilly, j. (2010) multiterminal dc wind farm collection grid internal fault analysis and protection design, ieee transactions on power delivery, 25(4), 2308–2318. adv syst sci appl 2021; 01; 76-85 published online at https://ijassa.ipu.ru. on uniform convergence property of solutions for periodic differential inclusions with asymptotically stable sets mikhail morozov v.a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia e-mail: granmiguel@mail.ru abstract: the paper considers a periodic differential inclusion with an asymptotically stable set. the uniform character of convergence of solutions to an asymptotically stable set is established. an exponential estimate is obtained for solutions of a periodic differential inclusion homogeneous in state vector. examples of control systems leading to consideration of periodic differential inclusions are given. these results can find applications in the stability analysis of control systems with periodic parameters, in particular, servomechanisms whose elements operate on ac, control systems with pulse amplitude modulation, and systems used to solve problems related to investigating vibrations of milling machines. keywords: time-invariant differential inclusion, periodic differential inclusion, homogeneous differential inclusion, asymptotically stable set, control system 1. introduction studying control systems has led to the use of differential inclusions theory. consider a control system ),,,( uxtfx = (1.1) where nrx , dtdxx /= is the velocity vector, t is time, and )(tuu = is the control on which the next constraint is imposed ,)( utu  (1.2) u is an arbitrary set, nru  . under fairly general assumptions control system (1.1) with constraint (1.2) is equivalent to the differential inclusion ),,( xtfx (1.3) where ),( xtf denotes a multivalued mapping, i.e., a function that assigns a set nrxtf  ),( to each time t and each point nrx in the state space. differential inclusion (1.3) can be adopted not only for a control system with given control constraints but also for other objects. for example, such objects include systems of differential inequalities, implicit differential equations, control systems with state-space constraints, systems with variable structure and with sliding modes, and differential equations with discontinuous right-hand sides. theory of differential inclusions is well developed and presented in systematic form in [1,4,17]. the use of differential inclusions in control systems theory is covered in the monograph [6]. the examples leading to differential inclusions are given below in section 4. in some cases, e.g., such as the problem of absolute stability, the study of linear nonstationary systems, the matrix of the right-hand side of which satisfies interval on uniform convergence property of solutions for periodic differential… 77 copyright ©2021 assa. adv. in systems science and appl. (2021) constraints, and the stability analysis of control systems that contain elements with incomplete information linear-selectionable differential inclusions can be used. a linearselectionable inclusion is an inclusion of the form ),,( xtfx  ,)()( ,)(:),( ttaxtayyxtf == (1.4) where nryx  , and )(t is a set in the space of nn matrices. an inclusion of the form (1.4) is called a linear-selectionable inclusion, because the multivalued mapping ),( xtf in (1.4) is the union of linear single-valued mappings (selectors) ,)( xta )()( tta  . in the case of a time-invariant linear-selectionable inclusion, the right-hand side f of this inclusion − the matrix a and the set  − are time-invariant. a linear-selectionable inclusion is said to be asymptotically stable if its trivial solution 0x is asymptotically stable. time-invariant linear-selectionable inclusions are studied in a number of publications. for time-invariant linear-selectionable inclusions for which the set  is a compact or convex polyhedron, necessary and sufficient conditions of the zero solution asymptotic stability were obtained in [12,13] on the base of lyapunov functions method. the papers [7,12,13] give various algebraic criteria for the asymptotic stability of time-invariant linear-selectionable inclusions. the publications on time-periodic (in short, periodic) differential inclusions (e.g., [2,8,9]) were mostly devoted to the existence of periodic solutions. few investigations were focused on the analysis of solutions of periodic differential inclusions and their properties. for example, the weak asymptotical and weak exponential stability of an equilibrium of a periodic differential inclusion were studied in [5,18]. in accordance with the definitions introduced therein, an equilibrium of a given periodic differential inclusion is weakly asymptotically (weakly exponentially) stable if there exists at least a single solution satisfying the standard definitions conditions of the asymptotical (exponential) stability for a differential inclusion equilibrium. the method consists in the design of a firstapproximation inclusion and further analysis of the properties of its solutions. in addition, the theorems of the weak asymptotical (weak exponential) stability of an equilibrium of an original inclusion using the corresponding properties of an equilibrium of its first approximation-inclusion were established. it was demonstrated that the proposed method can be used to study the weak asymptotical (weak exponential) stability of the inclusions equivalent to control systems. the problems of absolute and robust stability of control systems with periodic variable parameters were solved in [10,11,14,15]. in particular, control systems with periodic parameters under consideration were proved to be equivalent to a periodic differential inclusion in the sense of the coincidence of the sets of absolutely continuous solutions. as was demonstrated in [16], in some cases solutions of periodic differential inclusions with the asymptotically stable trivial solution have the same properties as solutions of autonomous differential inclusions. this paper continues the research of [16]. it considers periodic differential inclusions with an asymptotically stable set. the remainder of this paper is structured as follows. in section 1 we consider periodic differential inclusion of general form and give preliminary remarks. the definition of an asymptotically stable set is also given. in section 2 the uniform character of convergence of solutions to an asymptotically stable set is established. for solutions of periodic differential inclusion that is homogeneous in state vector we derive an exponential estimate. in section 4 examples of control systems leading to the periodic differential inclusions are given. in the final section we offer concluding remarks. 2. statement of the problem consider the dynamic systems described by periodic differential inclusion 78 m.v. morozov copyright ©2021 assa. adv. in systems science and appl. (2021) 0.t const,t , 0, t),,(),( ),,( =+ nrxxttfxtfxtfx (2.1) everywhere below we assume that in some domain }},:{ , ,0{ 00 rxxggxttg rr == the multivalued function ),( xtf satisfies the main conditions [4, p. 60]; i.e., for all gxt ),( the set nrxtf ),( is nonempty, bounded, closed, and convex, and the function ),( xtf is upper semicontinuous [4, p. 52] with respect to ).,( xt a solution of inclusion (2.1) is understood as an absolutely continuous vector function )(tx defined on an open or closed interval i and satisfying (2.1) almost everywhere on i . by virtue of periodicity of the multivalued function ),( xtf in t , when studying the properties of solutions ),,( 00 xttx of inclusion (2.1), we can assume without loss of generality that ],0[0 tt  . by the definition of solutions and the periodicity of the right-hand side of (2.1) in t the solutions of this inclusion have two properties as follows. if a function )(tx is a solution of inclusion (2.1) (for   t ), then 1) the function ),( kttx + where kttkt −−  and k denotes any integer, is also a solution of inclusion (2.1); moreover, the solutions )(tx and )( kttx + have the same trajectory; 2) for any tttt and , ],,0[ 10  such that ,10 ttt  the equality =))(,,( 11 txttx ),,( 00 xttx holds, where ),,()( 0011 xttxtx = . let nra , nrb be points (vectors) with coordinates ia and ib respectively, ni ,1= , and also let nrb  be a set. the distance  between two points or between a point and a set is interpreted as the nonnegative values ,)(),( 2/1 2 1       −=−=  = n i ii bababa ),(inf),( baba bb = . as is well-known, the function ),( bx is uniformly continuous and for any points nrx and nry )y,(),(y-),(x xbb   . a closed  -neighborhood m of a set m is a set of such points x that  m),(x . let gm 0  , .00  definition 2.1: a set m is asymptotically stable for inclusion (2.1) if for any 0 there exists a value 0)(  such that, for each 0x satisfying the inequality )(),( 0  mx , there exists a solution with the initial condition 00 )( xtx = and all solutions with the abovementioned property are extendable on the interval  tt0 and also satisfy the conditions  )),(( mtx for  tt0 and 0)),(( →mtx as →t . the problem is to study solutions of inclusion (2.1) with an asymptotically stable set .m 3. results theorem 3.1: if inclusion (2.1) has an asymptotically stable set ,m then there exists a value 00  such that all solutions ),,( 00 xttx of inclusion (2.1) satisfy the condition on uniform convergence property of solutions for periodic differential… 79 copyright ©2021 assa. adv. in systems science and appl. (2021) 0)),(( →mtx as →t (3.1) uniformly with respect to ),( 00 xt for any ],0[0 tt  and . 0 0  mx  proof. first, let us show that there exists a value 00  such that all solutions of inclusion (2.1) with initial condition =)( 0tx ,0x , 0 0  mx  satisfy condition (3.1) uniformly with respect to 0 x for any given ].,0[0 tt  suppose the contrary. then for any ,0 there exists ,0)(  ],0[0 tt  , a sequence of solutions )(txk of inclusion (2.1), a numerical sequence ktttk + 0 , and a sequence of vectors ,...,2,1 ,0 =kxk such that mxk 0 and . tas ,...,2,1 ,0)()),,,(( k00 →= kmxttx k kk  since the set m is asymptotically stable, it follows that for any 0 we can choose a value  small enough to ensure that all solutions with mtx )( 0 satisfy the relations  ))),(,,(( 00 mtxttx ),( 0  tt 0)),(( →mtx ).(t → (3.2) consequently, there exists a number 0 such that all solutions of inclusion (2.1) with mtx )( 0 satisfy the inequality )())),(,,(( 00  mtxttx ).( 0  tt (3.3) let us show that all ),,( 00 k k xttx satisfy the inequalities , 0 mxk  1,2,...k ,k1,m ,)),x,,(( k 000 ==+  mtmttxk (3.4) indeed, otherwise there would exist k ~ and ) ~ (km ( kkm ~ ) ~ (1  ) such that ,)),x,,) ~ ((( k ~ 000~  + mttkmtx k . ~ 0 mxk  then the solution =+++=+= )),,) ~ ((,) ~ (,) ~ (()),,) ~ ((,,()( ~ 000~0~ ~ 000~0 k kk k k xttkmtxtkmttkmtxxttkmtxttztz ),,) ~ (( ~ 00~ k k xttkmtx += )( 0  tt of inclusion (2.1) would satisfy the relations =)),(( 0 mtz ,)),x,,) ~ ((( k ~ 000~  + mttkmtx k ),()),,,(())),,,) ~ ((,,)tk ~ ((( ~ 00~~ ~ 000~0~  =+− mxttxmxttkmtxtmtz k kk k kk which contradict (3.3). the sequence of segments of solutions )(txk contains a subsequence uniformly convergent for 10 ttt  ; in turn, the subsequence contains a new subsequence uniformly convergent for 20 ttt  , and so on. since ),( xtf satisfies the basic assumptions, it follows that the limit function )(tx is a solution of inclusion (2.1) [4, p. 60], which, together with (3.4), implies that mtx )( 0 and  + ))),(,,(( 000 mtxtkttx , 1,2,...,k = and we have arrived at a contradiction with (3.2). therefore, the assertion started at the beginning of the proof of the theorem is valid. 80 m.v. morozov copyright ©2021 assa. adv. in systems science and appl. (2021) let us now prove the assertion of theorem 2.1. suppose that is fails. then for any ,0 there exist a number ,0)(  a sequence )(txk of solutions of inclusion (2.1), a sequence of numbers kt0 ( ],0[0 tt k  and kttt k k + 0 ), and a sequence of vectors ,...,2,1 ,0 =kxk such that mxk 0 , . t,...;2,1 ,0)()),,,(( k00 →= kmxttx kk kk  (3.5) without loss of generality, we can assume that ,0  where 0 is a number for which the above-proved assertion holds. since , 0 mxk  ],0[0 tt k  , ,...,2,1 =k it follows that the sequence ,...2,1 =k contains a subsequence ),( }{ → kk such that the limits 00lim xxk k =  → ( m0x ) and 00lim tt k k =  → ( ],0[0 tt  ) simultaneously exist. to simplify the notation, we assume that kk tt 00 =  , kk xx 00 =  ,...,2,1 k == k and ,lim 00 xx k k = → .lim 00 tt k k = → (3.6) let .0m),( 00 =−  x by (3.6), there exist a 1k such that 2/)x,( 00  kx (3.7) for all 1kk  since all solutions of inclusion (2.1) are equicontinuous [2, p. 61] in any closed bounded domain lying in ,g it follows that they are equicontinuous in the domain , { 0mx ]},0[ tt as well. therefore, for the point 0t , there exist a neighbourhood ]},,0[ ,2/:{)( 00 tttttts −=  ],[)( 0 bats = ( ta 0= if 00 =t and 0tb = if tt =0 ), such that .2/))t ~ x(,)) ~ (, ~ ,(( 000  txtbx (3.8) for all solutions )) ~ (, ~ ,( 00 txttx of inclusion (2.1) with ],[ ~ 0 bat  and .) ~ ( 0 mtx  then, taking into account relations (3.7), (3.8) and uniform continuity of the function ),()( mxx  = we obtain two relations 2/)x,(),(x-),(x),(),( 000 k 000  − kk xmmmxmx , 2/)x),,,((),(x-)),x,t(b,(x),()),,,(( k 000 k 0 k 0 k 0k000  − kk k kkk k xtbxmmmxmxtbx for all solutions )(txk with .1kk  adding the last two inequalities, we obtain 0000 ),()),,,((  =+ mxmxtbx kk k , .1kk  this, together with (3.5) and (3.6), means that there exist a 2k such that the relations ,0 tbt k  000 )),,,((  mxtbx kk k , )()),,,(( 00  mxttx kk kk (3.9) are simultaneously valid for all .2kk  taking into account relations (3.9), from the sequence of solutions )(txk of inclusion (2.1) satisfying inequalities (3.5) for 2kk  , we pass to the sequence of solutions )(tyk ),,( 00 kk k xttx= defined by the relations ),()( 0 bxyby k k k == ,),( 00  myk ),()),y,,(( k 0  mbty kk .k →t (3.10) on uniform convergence property of solutions for periodic differential… 81 copyright ©2021 assa. adv. in systems science and appl. (2021) the existence of a sequence ),(tyk ,2kk = ,12 +k ,...22 +k ,(  tb ),0 tb  satisfying conditions (3.10) contradicts the already known fact that relation (3.1) is valid uniformly with respect to 0x (for 0 0  mx  and ],0[0 tt  ) for all solutions of inclusion (2.1). the proof of theorem 3.1 is complete.  in what follows, we consider differential inclusions homogeneous with respect to .nrx if b is a set in nr and c is a number, then cb stands for a set of points of the form cx for all .bx definition 3.2: a multivalued function ),( xtf is homogeneous (of degree one) in x if ),( cxtf ),( xtcf for all 0c . definition 3.3: a differential inclusion )0 ),,(),(( ),(  cxtcfcxtfxtfx (3.11) is homogeneous in .x homogeneous differential inclusion (3.11) is preserved under the substitution 1cxx = with arbitrary .0c it means that if a function )(tx = is a solution of inclusion (3.11), then the function )(tcx = with arbitrary 0c is also a solution. let us consider the differential inclusion ), 0,( ),,( nrxtxtfx  0),t const,(t ),,(),( =+ xttfxtf ),0( ),(),(  cxtcfcxtf (3.12) periodic in t and homogeneous in .x since inclusion (3.12) is homogeneous in ,x we have the following assertion. corollary 3.1: if a bounded set m is asymptotically stable for inclusion (3.12), then all solutions ))(,,( 00 txttx of inclusion (3.12) with the initial conditions ],0[0 tt  and rgx 0 (where r is an arbitrary positive number) satisfy condition (3.1) uniformly with respect to ).,( 00 xt for solutions of periodic homogeneous differential inclusion (3.12) with an asymptotically stable set m an exponential estimate is valid. theorem 3.2: if a bounded set m is asymptotically stable for inclusion (3.12), then there exist numbers 0 ,0 10  cc such that any solution ),,( 00 xttx of inclusion (3.12) satisfies the estimate ).(t )exp()),x,,(( 010000 − ttcxcmttx (3.13) for any 0t and .0tt  proof. by virtue of corollary 3.1, for some ,0 there exists a 0 (independent of ),( 00 xt ) such that tk ~ = ( k ~ is some positive integer) and ,2/)),x,,(( 00  mttx + t0t , for all solutions ),,( 00 xttx with the initial condition .0 x by virtue of theorem 3 in [4, p. 62] the set of solutions of inclusion (3.12) is compact on the closed interval ]t,[ 00 +t in the metric of ]t,[ 00 +tc , whence it follows that  200 )),x,,(( cmttx  )0( 2 c for .t00 + tt if ),,( 00 xttx is a solution with an arbitrary 00 x and ,/ 0xc = then the function ),,(),,( 0000 xttcxytty = is also a solution and ,0 =y whence it follows that 82 m.v. morozov copyright ©2021 assa. adv. in systems science and appl. (2021) 2/)),y,,(( 00  mtty )t( 0 + t and  200 )),y,,(( cmtty  ).t( 00 + tt returning from ),,( 00 ytty to ),,( 00 xttx , we obtain 0200 )),x,,(( xcmttx  ),t( 00 + tt 2/)),x,,(( 000 xmttx  ).t( 0 + t (3.14) for any 0t and .0x since after replacement of t by ktt + ( k is an arbitrary integer), a solution of inclusion (3.12) remains a solution, it follows from (3.14) that a solution ),,( 00 xttx with an arbitrary 0x satisfies the relations 000 2)),x,,(( xmttx i− ),t( i  t (3.15) where ,...2,1 ,t 0i =+= iit  for any ,0tt  we choose an i such that .t 1-i itt  then itt + 0 and ./)( 0 tti − therefore, the inequality )/2lnexp()/2lnexp()/)(2lnexp()2lnexp(2 00  ttttii −=−−−=− is fulfilled. the last inequality, together with (3.15), implies (3.13) with )/2lnexp( 00 tc = and ./2ln1 =c the proof of theorem 3.2 is complete.  4. exapmles example 4.1: in the paper [13] the nonlinear control system is considered ,),( 1  = += m j jj j tbaxx   = == n i i j i j j xcxc 1 ,, ,,1,0),,0( mjtj = (4.1) where nn ii rxxx = = ,)( 1 is n -dimensional vector, characterizing the deviation of the system from the mode prescribed by control objective (the zero solution otx )( of system (4.1) corresponds to this mode), a is a constant square matrix of order ,n and jb and ),1( mjc j = are constant n -dimensional vectors. the brackets .,. denote the scalar product. it is also assumed that nonlinear functions ,,1),,( mjtjj = that define the characteristics of nonlinear elements, satisfy conditions for the existence of an absolutely continuous solution of system (4.1) for any initial conditions and inequalities ( )mjkkktk jjjjjjjjj ,1, ),( 21 2 2 2 1 =+−  for all j and .0t system (4.1) is equivalent, see the paper [13], to time-invariant linear-selectionable inclusion (1.4), where  is a compact set in the 2n -dimensional matrix space. here and in the following examples, equivalence is understood in the sense of solutions sets identity for the system under consideration and the corresponding inclusion under the same initial conditions. example 4.2: in the paper [10] the linear nonstationary control systems ,)()( 1 xtatx k m k k = =  0)( tk , 1)( 1 = = t m k k (4.2) on uniform convergence property of solutions for periodic differential… 83 copyright ©2021 assa. adv. in systems science and appl. (2021) are considered, where nт ii rxx = =1)( is n -dimensional state vector of the system, =)(tak (t))(a n 1ji, k ij = are given continuous periodic matrices of the period ,0t =+ )( ttak )(tak , mk ,1= , and )(tk ( mk ,1= ) are bounded measurable functions. system (4.2) is equivalent to the periodic differential inclusion ),,(),( ),,( xttfxtfxtfx +  == === m kk m k k tttatyyxtf 1k k 1 }.1)( ,0)( ),()(:{),(  (4.3) example 4.3: in the paper [14] the linear nonstationary control system nrxxtax = ,)( (4.4) is considered, where n jiij tata 1,))(()( == is an arbitrary matrix ( )(taij , nji ,1, = , are measurable functions, generally speaking, not periodic). system (4.4) almost everywhere satisfies the inequalities ))(()( ,))(((t) (t),(t) (t) 1,1j, n jiij n nij tatataaaaa == == (4.5) on any finite interval of the semi-axis ).[0, matrix inequalities (4.5) are understood elementwise, that is ,,1, (t),)((t) ijij njiataa ij = , where ,)(a tij , ,1, (t), (t), ijij njiaa = are arbitrary measurable functions. it is assumed that the given bounded "extreme" matrices (t) a and (t)a are periodic with a period 0t  , i.e. conditions are valid (t) )( atta + , ).()( tatta + thus, by virtue of (4.5), not one fixed system (4.4) is considered, but a set of linear nonstationary systems (4.4) with periodic interval constraints (4.5). a series of examples leading to systems (4.2) and (4.4) with constraints (4.5) were considered in [19]. among such systems, note tracking systems with ac elements, control systems with pulse amplitude modulation and also the systems arising in the vibration analysis of milling machines. if the condition )()( tatta =+ is additionally satisfied, the set of linear nonstationary systems (4.4) with periodic interval constraints (4.5) is equivalent to the linear-selectionable periodic inclusion ),,( xtfx  )()( ,)(:),( ttaxtayyxtf == ,  )()()()()( :)()( 21 tattattatat  +== , )()( ttt =+ , where arbitrary bounded and measurable functions 2,1),( =ktk satisfy the conditions ,1)( ,0)( 2 1 =  =k kk tt  ),()( ttt kk  =+ .0tt  5. conclusion the paper considers solutions of periodic differential inclusions with asymptotically stable sets. theorem 3.1 establishes the uniform character of convergence to an asymptotically stable set. homogeneous in state vector periodic differential inclusions are also considered. corollary 3.1 is obtained for such inclusions. theorem 3.2 gives the exponential estimate for solutions. theorem 3.1 is a generalization of lemma 2, proved for autonomous differential inclusions [3]. theorem 3.2 generalizes the estimate well known for solutions of an autonomous homogeneous (first degree) differential inclusion [4]. the examples of control systems leading to consideration of periodic differential inclusions are given. the results obtained can find applications in the stability analysis of control systems with periodic parameters, in particular, servomechanisms whose elements operate on ac, control 84 m.v. morozov copyright ©2021 assa. adv. in systems science and appl. (2021) systems with pulse amplitude modulation, and systems used to solve problems related to investigating vibrations of milling machines. further research into the inclusions considered in the present paper can be related to producing weak asymptotic and weak exponential stability conditions. in addition, it seems interesting to distinguish the classes of lyapunov functions establishing necessary and sufficient conditions for the asymptotic stability of periodic differential and difference inclusions. references 1. aubin, j.-p., & cellina, a. (1983). differential inclusions. berlin heidelberg: springerverlag. 2. filippakis, m., michael e., & papageorgiou, nikolaos s. (2006). periodic solutions for differential inclusions in ,nr archivum mathematicum., 42(2), 115−123. 3. filippov, a. f. 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(2016). kriterii robastnoy ustoychivosti nestatsionarnih system interval’nimi ogranicheniyami [robust stability criteria for nonstationary systems with interval constraints]. proceedings of the institute for systems analysis, 66, 4, 4-9 [in russian]. 15. morozov, m.v. (2014). kriterii robastnoy absolutnoy ustoychivosti diskretnyh system upravleniya s periodicheskimi ogranicheniyami [criteria of robust absolute stability for discrete control systems with periodic constraints]. proceedings of institute for systems analysis, 64, 2, 13-18. [in russian]. 16. morozov, m. v. (2000). svoystva resheniy periodicheskih differentsial’nyh vklucheniy [properies of solutions of periodic differential inclusions]. differets. uravn., 36(5), 677682. [in russian], https://doi.org/10.1007/bf02754225 17. smirnov, g. v. (2002). introduction to the theory of differential inclusions. providence, rhode island, usa: amer. math. soc. graduate studies in mathematics, 41. 18. smirnov, g. v. (1995). weak asymptotic stability at first approximation for periodic differential inclusions, nonlinear differential equations and applications., 2(4), 445 461, https://doi.org/10.1007/bf01210619 19. shil'man, s.v. (1978). metod proizvodiatshikh funktsiy v teorii dinamitcheskikh sistem. [the method of generating functions in the theory of dynamic systems]. moscow, ussr: nauka, [in russian]. 1. introduction microsoft word 10-zhang yong.doc advances in systems science and applications (2010), vol.10, no.1 61-66 issn 1078-6236 international institute for general systems studies, inc. research on dynamic pricing model based on e-supply chain management* yong zhang1,2, yingjin lu1 and xianglan jiang1 1school of management, university of electronic science & technology of china, chengdu 610054, china 2 college of information science and technology, chendu university, chengdu 610054, china email: luyingjin@uestc.edu.cn, yingjin.lu@changhong.com abstract with the influence of the uncertain factor in business trade, the research on dynamic pricing considering the random fluctuation of price has become an important project in managerial economics. in this paper, we introduce the uncertain factor, which produced by the random errors, into the pricing model of the dominant manufacturers. by introducing the expectation of retail price, variance and transfer price, we change the game of incomplete information into the game of complete information. then we resolve the problem of optimal pricing about transfer price with extremum of function according to the optimal production and storage model. keywords random fluctuation dynamic pricing incomplete information game extremum of function 1. introduction based on the network and electronic commerce (ec), the internet can arrive any place of the world. it results in the substantial increase of demand or supply under non-prediction. at the same time, lower menu cost supports traders to change the price more frequently according to the condition of market. so it is less and less efficient to confirm the optimal price according to establish the static demand price model under the random fluctuation of demand and price. many scholars researched the dynamic pricing model. the article[4] discussed the dynamic pricing model of the dominant manufacturer after the time factor was added to the demand function. the article[5] suggested introducing uncertain factor into economic model and establishing the price model of rational expectation. the article[6] displayed the optimization of dynamic game and nonlinear pricing and the dominant firm price leadership model completely. based on the above articles, this paper discuss the pricing model of the dominant manufacturer further with uncertain factor, which produced by the random errors in ec. by introducing uncertain expectation and variance, it changes the game of incomplete information into the game of complete information. then it solves the problem of optimal pricing about transfer price with the extremum function method according to the optimal production and storage model. 2. analyzing and hypothesizing of the model assumed that a new kind of production is in monopoly position, and there is only one manufacturer and one seller. because of the new production, the manufacturer is in dominance and can predict the demand of market. the manufacturer confirms the transfer prices of its production according to the maximum profit. then, the seller confirms the order quantity according to the maximum profit. this process can be considered as the incomplete information game with the decentralized decision. this paper considers uncertain factor as an endogenous variable and introduce it into the model. then it assumes the demand in consumption market is * this work was supported by fund item for the doctoral program of high education (no. 20030614011), national scientific fund item for excellent youth (no. 79725002), chinese science fund item for post doctor (no. 79725002). zhang: research on dynamic pricing model based on e-supply chain management 62 related not only to the price, but also to the price fluctuation which is the uncertain factor of price, so it expresses price fluctuation with price variance. 3. supply chain model 3.1 optimal order strategy of sellers after manufacturer price the production 1p with maximum profit according to some information, sellers choose the optimal ordering quantity. supposed the lead time t k− is fixed, and the period of order t is constant. when sellers estimate the ordering quantity according to the profit maximization, they order t k− days in advance. then, the products will arrive after k days. after sellers introduce uncertain factor into economic model as inner variable, they can price according to the demand of the market in internet. the demand function of consumers in this period is: 1 1( ) ( ( ), , ) ( )u u s s s s sq t d p t p u a bp t p uλ= = − + + , (1) where ( )sp t is the sell price at point t, and ( )sq t is the demand amount at point t. both of them are random process. u sp is the variance of price (fixed), which reflects the fluctuation of price. if λ is negative, consumers hate the price undulation. if λ is positive, consumers like the price undulation. 1u is a random disturbance. and the parameters , 0a b > . (and the parameters a and b are greater than zero.) because of the predomination of manufacturer, transferring price 1p is assured. seller assures sell price ( )sp t , and market gives the amount of demand ( )sq t . assumed all the demands are satisfied under ( )sp t , the order quantity in this period is the accumulation of instantaneous demand ( )sq t , that is, 0 ( ) t sq q t dt= ∫ ， (2) the storage at point x is 0 0 0 ( ) ( ) ( ) ( ) x t x t s s s s sx q q q t dt q t dt q t dt q t dt= − = − =∫ ∫ ∫ ∫ ， (3) if the storage cost of unit production is 2c at unit time, the storage cost is 2 2 20 0 ( ) ( ( ) ) ( ) t t t s s s sx d c c q x dx c q t dt dx c q t dtdx= ⋅ = =∫ ∫ ∫ ∫∫ ， 2 20 0 0 ( ( ) ) ( ) t t t s sc q t dx dt c t q t dt= = ⋅∫ ∫ ∫ , (4) where d is a triangle, surrounded by 0, , ,x x t t x t t= = = = . the purchase cost is 1 1 1 10 ( ) t b sc p q c p q t dt c= + = +∫ , (5) where 1c is fixed purchase cost, so total cost is s b stc c c= + , (6) the revenue of the seller is si 1 0 0 1( ) ( ) [ ( ) ] ( ) t t u s s s s s uap t q t dt q t p q t dt b b b b λ = = − + +∫ ∫ advances in systems science and applications (2010), vol.10, no.1 63 2 1 0 1[ ( ) ( ) ( ) ( )] t u s s s s s ua q t q t p q t q t dt b b b b λ = − + +∫ , (7) and the profit of the seller in order period is s s s b sr i tc i c c= − = − − 2 1 1 2 10 0 0 1[ ( ) ( ) ( ) ( )] ( ) ( ) t t tu s s s s s s s ua q t q t p q t q t dt p q t dt c t q t dt c b b b b λ = − + + − − × −∫ ∫ ∫ 2 1 1 2 10 1[ ( ) ( ) ( ) ( ) ( ) ( )] t u s s s s s s s ua q t q t p q t q t p q t c t q t dt c b b b b λ = − + + − − × −∫ . (8) assumed supply quantity equal demand quantity in every period, the seller predicts and chooses supply quantity ( )sq t and its sell price ( )sp t to get maximize the profit sr . if 2 1 1 2 1( ) ( ) ( ) ( ) ( ) ( )u s s s s s s s uaf q t q t p q t q t p q t c t q t b b b b λ = − + + − − × . we will get maximum profit sr with euler equation. then ( ) 0 [ ] '[ ]s s f d f q t dt q t ∂ ∂ − = ∂ ∂ (9) solving this differential equation, we can get the sell function at point t when the profit is maximum: 1 2 1 1( ) ( ) 2 u s sq t a bp p bc t uλ= − + − + (10) *q is the optimal order quantity for sellers after the wholesale price is determined, that is, 2 2 1 2 1 1 10 2 1 1 0.51( ) ( ) 4* 0.5 0 . u t u s s s u s a p bc t u q t dt bc t a bp p u t p bq a p bc t u p b λ λ λ ⎧ + − + = − + − + + ≤⎪⎪= ⎨ + − +⎪ >⎪⎩ ∫ (11) 3.2 rational expectations of the best product plan and transferring price that manufacturer determined assumed the demand function of the market and the order interval of the seller are open. then, the order quantity of seller can be predicted, and the transferring price 1p can be determined under the maximum profit. the manufacturer signs a contract for goods with the seller. according the contract, the seller purchase *q productions with 1p and the manufacturer deliver at point k . for the manufacturers, they must consider sales revenue, production cost and storage cost. sales revenue is multiplication of price and order quantity. production cost depends on production rate, which is the quantity in unit time. and the production rate is higher, the production cost is more. storage cost is determined by the finished productions and expiration time. ( )x t is the cumulative production of product plan until point t without consideration of assets’ time value. because the product rate at point t (marginal product) is '( )x t , marginal product cost is ( '( ))f x t , marginal storage cost is ( ( ))g x t and marginal revenue is ( '( ))x tϕ . then, the total cost ( ( ))c x t is zhang: research on dynamic pricing model based on e-supply chain management 64 0 ( ( )) [ ( '( )) ( ( ))] k c x t f x t g x t dt= +∫ . (12) in order to confirm the actual expression of this function, we assume the following items: (1) the cost of increasing one production in uniform time is in proportion to the product rate, where the coefficient is 12k . so, we get 1 ( '( )) 2 '( ) '( ) df x t k x t dx t = , or, 2 1( '( )) [ '( )]f x t k x t= . (13) (2) the storage cost is in proportion to storage amount in unit time, where the coefficient is 2k , that is, 2( ( )) ( )g x t k x t= (14) the total revenue ( ( ))i x t from point 0 to point k is: 10 0 ( ( )) ( '( )) '( ) k k i x t x t dt p x t dtϕ= =∫ ∫ (15) the profit r from point 0 to point k is: 0 0 ( ( )) ( ( )) ( ( )) ( ( )) [ ( '( )) ( ( ))] k k r r x t i x t c x t x t dt f x t g x t dtϕ= = − = − +∫ ∫ 2 1 1 20 [ '( ) ' ( ) ( )] k p x t k x t k x t dt= − −∫ (16) (0) 0, ( ) *x x k q= = (17) the problem may be attributed to find the maximum of functional ( ( ))r x t , which can be solved by variation method. if 2 1 1 2( , , ') ' 'mr t x x p x k x k x= − − , we will get the maximum profit r according to euler equation, we have to let: ( ) 0 ' mr d mr x dt x ∂ ∂ − = ∂ ∂ then we can get the differential equation of second order: 2 12 ''( ) 0k k x t− + = under the constraint condition (17), its solution is: * 2 22 1 2 1 1 4 ( ) 4 4 k k q k k x t t t k k k − = + . (18) it’s the product plan of maximum profit. obviously, with ( ) 0,0x t t k≥ ≤ ≤ , we can get 2 * 2 14 k k q k ≥ . (19) that is, when the order quantity of the sellers satisfies condition (19), the product plan confirmed by condition (18) is optimal. if 2 * 2 14 k k q k < , we can delay the start time which make the time difference k satisfying equation 2 * 2 14 k k q k = from t1 to k, so the product plan is better. after we substitute ( )x t in the equation (16), the maximum profit is solved according to 1p and ( )x t : advances in systems science and applications (2010), vol.10, no.1 65 3 2 * * 2 *2 2 1 1 1 ( ) 48 2m k k kk q k q r p q k k = − + − (20) take non-zero of *q in equation (11) into equation (20), we get 3 2 22 2 1 2 1 1 1 1 ( ) 48 2 4 u m s k k kk r p bc t a bp p u t k λ⎛ ⎞ ⎡ ⎤= + − − + − + +⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠ 2 21 2 1 1 1 ( ) 4 u s k bc t a bp p u t k λ⎡ ⎤− − + − + +⎢ ⎥⎣ ⎦ (21) obviously, r is a quadratic and convex function of price 1p . if we get maximum r with appropriate 1p , that is 1 0dr dp = , then 2 2 2 2 1 1 1 2 2 1 1 1 1 2 4 ( 2 ) 4 ( ) 2 8 ( ) 8 ( ) u u s sb c k t a k bk t k p u bk k bc kt bk t p u p b k bk t λ λ− + + + + + − + + = + (22) 1 ep is the expectation of the best price 1p , 1 1( )ep e p= 2 2 2 1 2 1 2 2 1 1 1 4 ( 2 ) 2 2 2 ( ) 8 ( ) 2 ( ) u s a k bk t b c k t bk k bc kt k bk t e p b k bk t b k bk t λ + − + − + = + + + . assumed variance u sp of transferring price sp is a constant. then 2 2 2 1 2 1 2 2 1 1 1 1 4 ( 2 ) 2 2 2 8 ( ) 2 ( ) e u s a k bk t b c k t bk k bc kt k bk t p p b k bk t b k bk t λ + − + − + = + + + (23) so, 1 ep is made up of two terms. one is determined by the structure of model, the other involves retail price undulation. when 0λ > , 1 ep is in proportion to u sp . u sp is higher, 1 ep is higher. it is coincident with the theory, which is the expectation revenue of asset increased with the risk. in a word, under the incomplete information, the expectation of economical people is scaled to uncertainty rather than related to it only. 4. calculation examples 1( ) 970 3 ( )s sq t p t u= − + assumed 1000, 3, 2a b λ= = = − and 15u sp = in equation (1), the demand function of the consumer in order cycle time is: 1( ) 970 3 ( )s sq t p t u= − + if 7t = , we can confirm the lead time is two days according to some conditions, such as transportation, so 2 5k t= − = . the coefficient between rate of change and production rate 1 2 k is 1 when the product rate increase unit product cost, and the coefficient between storage cost and storage amount 2k is 0.05. and the purchase cost 1c is 100 yuan, holding cost per unit production in unit time 2c is 1 yuan. above data can be adjusted and modified according zhang: research on dynamic pricing model based on e-supply chain management 66 to the actual statistic and forecasting. because zero is the average of 1u , we substitute mathematical expectation ( )sq t for ( )sq t when we calculated. according to equation (23), we can get 1 314.78 0.32 305.31e u sp pλ≈ + = . and with 1 321.58ep ≤ from equation (11), the optimal sale *q is 360 when the manufacturer determine price. according to equation (21) and (8), the maximum profit of the manufacturers rm is 58028.6, and the maximum profit of the wholesalers sr is 1295.17. from the data simulation, we can know that the manufacturers have the active power and control the profit mostly because of the open and dominant price under the monopoly and competition. 5. conclusions comparing to definite situation，the hypothesis containing fluctuate price is more close to the fact. with the development of the electronic commerce, the forecast and decision should be more reliable and it is more important to study the random situations. we have made some effort on the related aspects in this paper and got the optimal strategy of pricing under the situation of the fluctuating price in random. (see formula (23)) the application of this strategy is important to resolve the questions in commodity exchange with uncertainty factors references [1] luo zhanghua, liu lu. research on dynamic pricing mechanism in transactions[j]. beijing university of aeronautics and astronautics journal (social science edition), 2005,18 (1). [2] kannan p k, praveen k kopalle. dynamic pricing on the internet: importance and implications for consumer behaviour[j]. international journal of electronic commerce, 2001, 5 (3): 63283. [3] gallego g, van ryzin g j. a multiple product dynamic pricing problem with applications to network yield management [j]. operations research, 1997, (45): 24241. [4] liu jian, lai ming-yong, zhang han-jiang. the optimization of supply chain ordering and pricing decisions based on the trend of demand [j]. system engineering, 2003, 21 (5). [5] cao caixia, zhang yan. rational expectations model with uncertainties [j]. journal of the beijing union university (natural science) 2004, 18 (1). [6] tang xiaowo, zeng yong, li shiming, et al.. management economic analysis -theory and application [m]. chengdu: university of electronic science and technology of china press, 2000. [7] jiang qiyuan, xie jinxing, ye jun. mathematical model [m]. beijing: higher education press, 2003. adv syst sci appl 2021; 03:40–62 published online at https://ijassa.ipu.ru. a controllability problem for causal functional inclusions with an infinite delay and impulse conditions maria afanasova1, valeri obukhovskii1, garik petrosyan1,2* 1faculty of physics and mathematics, voronezh state pedagogical university, voronezh, russia 2research center, voronezh state university of engineering technologies, voronezh, russia abstract: in this paper we study the controllability problem in a banach space for various classes of functional inclusions with causal operators with an infinite delay, and impulse effects. basing on the topological degree theory for condensing multimaps, we prove a global theorem on the existence of trajectories for systems governed by functional inclusions. as an application, we obtain generalizations of existence theorems for the controllability problem for a semilinear first order functional differential inclusions of this type and a semilinear functional differential inclusions of a fractional order 0 < q < 1. keywords: causal operator, functional inclusion, controllability problem, functional differential inclusion, fractional derivative, measure of non-compactness, fixed point, topological degree, condensing multioperator 1. introduction it is well known that the contemporary approach in the theory of control systems and in mathematical physics leads to models which are convenient to be described by using differential equations and inclusions. recently, the attention of many researchers (see [1], [2], [3] and references therein) has been attracted to generalizations of differential equations and inclusions, namely to the class of functional equations and inclusions with causal operators. the term of a causal or volterra operator in the sense of a.n. tikhonov (see [4]), was used in mathematical physics to solve problems of differential equations, integro-differential equations, functional-differential equations with a finite or infinite delay, integral equations of volterra type, functional equations of a neutral type, etc. (see, for example, [5]). the papers [6], [7], [8], [9] among others are devoted to the study of equations and inclusions with causal operators of various types, theorems on the existence of solutions, description of qualitative properties of solutions and various applications. at the same time, in recent decades the interest to the theory of fractional-order differential equations has significantly increased, thanks to applications in various branches of applied mathematics, physics, engineering, biology, economics, etc. (see, for example, monographs [10], [11] papers [12], [13], [14], [15], [16], etc.). the boundary value problems of various types for fractional differential equations and inclusions were considered in the works [17], [18], [19], [20], [21], [22], [23]. in this paper we develop and generalize the results of papers [2] and [3], and we study the controllability problem in banach spaces for various classes of functional inclusions with causal operators with an infinite delay and impulse effects. basing on the topological degree ∗corresponding author: garikpetrosyan@yandex.ru a controllability problem for causal functional inclusions with an infinite delay41 theory for condensing multimaps we prove a global theorem on the existence of trajectories for systems governed by functional inclusions. as an application, we obtain generalizations of existence theorems for solutions of the controllability problem for a first order semilinear functional differential inclusions and a semilinear functional differential inclusions of a fractional order 0 < q < 1. 2. preliminaries 2.1. multivalued maps and measures of noncompactness let x be a metric space and y a normed space. introduce the following notation: p (y ) denotes the collection of all non-empty subsets of y ; pb(y ) denotes the collection of all non-empty and bounded subsets of y ; c(y ) denotes the collection of all non-empty and closed subsets of y ; cv(y ) denotes the collection of all non-empty, closed and convex subsets of y ; k(y ) denotes the collection of all non-empty and compact subsets of y ; kv(y ) denotes the collection of all non-empty, compact and convex subsets of y. let us recall some notations (see, for example, [24], [25]). definition 2.1: a multivalued map (multimap) f : x → p (y ) is said to be upper semicontinuous (u.s.c.) at a point x ∈ x, if for every open set v ⊂ y such that f(x) ⊂ v, there exists a neighborhood u(x) of x such that f(u(x)) ⊂ v. definition 2.2: a multivalued map (multimap) f : x → p (y ) is called closed if its graph gf = {(x, y) : x ∈ x, y ∈ f(x)} is a closed subset of x × y. definition 2.3: a multivalued map (multimap) f : x → p (y ) is called quasicompact if its restriction to each compact subset a ⊂ x is compact. lemma 2.1: ( [24], theorem 1.1.12). if f : x → k (y ) a closed quasicompact multimap, then f is u.s.c. definition 2.4: for a given p ≥ 1, a multifunction g : [0, t ]→ k(y ) is called: • lp–integrable if it admits an lp–bochner integrable selection, i.e., there exists a function g ∈ lp ([0, t ];y ) such that g(t) ∈ g(t) for a.e. t ∈ [0, t ]; • lp–integrably bounded if there exists a function ξ ∈ lp([0, t ]) such that ‖g(t)‖ ≤ ξ(t) for a.e. t ∈ [0, t ]. let e be a banach space lemma 2.2: (see [24], theorem 4.2.1.) let a sequence of functions {ξn} ⊂ l1([0, t ];e) be l1–integrably bounded. suppose that χ({ξn} (t)) ≤ α(t) a.e. t ∈ [0, t ] for all n = 1, 2, ..., where α ∈ l1 +([0, t ]). then for every δ > 0 there exist a compact set kδ ⊂ e, a set mδ ⊂ [0, t ] of a lebesgue measure mδ < δ, and a set of functions gδ ⊂ l1([0, t ];e) with values in kδ, such that for every n ≥ 1 there exists a function bn ∈ gδ for copyright © 2021 assa. adv syst sci appl (2021) 42 m. afanasova, v. obukhovskii, g. petrosyan which ‖ξn(t)− bn(t)‖e ≤ 2α(t) + δ, t ∈ [0, t ] \mδ. moreover, the sequence {bn}may be chosen so that bn ≡ 0 onmδ and this sequence is weakly compact. definition 2.5: let (a,≥) be a partially ordered set. a function β : pb(e)→ a is called the measure of noncompactness (mnc) in e if for each ω ∈ pb(e) we have: β(co ω) = β(ω), where co ω denotes the closure of the convex hull of ω. a measure of noncompactness β is called: 1) monotone if for each ω0,ω1 ∈ pb(e), from ω0 ⊆ ω1 follows β(ω0) ≤ β(ω1). 2) nonsingular, if for each a ∈ e and each ω ∈ pb(e) we have β({a} ∪ ω) = β(ω). if a is a cone in a banach space, the mnc β is called: 3) regular, if β(ω) = 0 is equivalent to the relative compactness of ω ∈ pb(e); 4) real, if a is the set of all real numbers r with the natural ordering. as the example of a real mnc obeying all above properties, we can consider the hausdorff mnc χ(ω): χ(ω) = inf{ε > 0, for which ω has a finite ε-net in e }. as other examples, consider the measures of noncompactness defined in the space of continuous functions c([a, b];e) with values in the banach space e: (1) the modulus of fiber noncompactness: ϕ(ω) = sup t∈[a,b] χe(ω(t)), where χe is the hausdorff mnc in e and ω(t) = {y(t) : y ∈ ω}; (2) the fading modulus of fiber noncompactness: γ(ω) = sup t∈[a,b] e−ltχe(ω(t)), where l > 0 is a given number; (3) the modulus of equicontinuity: modc (ω) = lim δ→0 sup y∈ω max |t1−t2|≤δ ‖y (t1)− y (t2)‖ . these measures of noncompactness satisfy all the above properties, except for the regularity. definition 2.6: a multimap f : x ⊆ e → k(e) is called condensing with respect to a mnc β (or β– condensing) if for each bounded set ω ⊆ x which is not relatively compact, we have: β(f (ω)) 6≥ β(ω). let d ⊂ e a non-empty closed convex subset, v a non-empty bounded open subset of d, β a monotone nonsingular mnc in e and f : v → kv (d) be a u.s.c. β-condensing map copyright © 2021 assa. adv syst sci appl (2021) a controllability problem for causal functional inclusions with an infinite delay43 such that x /∈ f (x) for all x ∈ ∂v , where v and ∂v denote the closure and the boundary of the set v in the relative topology of d. in such a setting, the (relative) topological degree degd ( i−f , v ) of the corresponding vector field i−f , satisfying the standard properties is defined (see, for example, [24], [25]). in particular, the condition degd ( i−f , v ) 6= 0 implies that the fixed points set fixf = {x : x ∈ f(x)} is a nonempty subset of v. application of topological degree theory leads to the following fixed point principles, which will be used in the what follows. theorem 2.1: ( [24], corollary 3.3.1). let m be a convex closed bounded subset of e and f :m→ kv(m) a β–condensing multimap, where β is a monotone nonsingular mnc in e . then the fixed point set fixf is non-empty. theorem 2.2: ( [24], theorem 3.3.4). let v ⊂ d be a bounded open neighborhood of a point a ∈ v and f : v → kv(d) a u.s.c. β-condensing multimap, where β is a monotone nonsingular mnc in e , satisfying the boundary condition x− a /∈ λ(f(x)− a) for all x ∈ ∂v and 0 < λ ≤ 1. then fixf 6= ∅ is a non-empty compact set. 2.2. phase space we will use the axiomatic definition of the phase space b, introduced by j.k. hale and j. kato (see. [26], [27]).the space b we will be considered as a linear topological space of functions defined on (−∞, 0] with values in a banach space e endowed with the seminorm ‖ · ‖b. for all function x : (−∞, t ]→ e, where t > 0, and every t ∈ (−∞, t ], xt is a function from (−∞, 0] to e, defined as xt(θ) = x(t+ θ), θ ∈ (−∞, 0]. we will be assume that b satisfies the following axioms: (b1) if the function x : (−∞;t ]→ e is continuous on [0;t ] and x0 ∈ b, then for each t ∈ [0;t ] : (i) xt ∈ b; (ii) the function t 7→ xt is continuous; (iii) ‖xt‖b ≤ k(t) sup 0≤τ≤t ‖x(τ)‖+h(t)‖x0‖b, where the functions k,h : [0;∞)→ [0;∞) are independent of x, k is strictly positive and continuous and h is locally bounded. (b0) there exists l > 0 such that ‖ψ(0)‖e ≤ l‖ψ‖b, for all ψ ∈ b. notice that under these conditions the space c00 of all continuous functions from (−∞, 0] to e with compact support into phase space b( [27], proposition 1.2.1). in addition, we will assume that the following condition is satisfied: (bc1) if a uniformly bounded sequence {ψn}+∞ n=1 ⊂ c00 converges to a function ψ compactly (i.e. uniformly on each compact subset (−∞, 0]), then ψ ∈ b and lim n→+∞ ‖ψn − ψ‖b = 0. copyright © 2021 assa. adv syst sci appl (2021) 44 m. afanasova, v. obukhovskii, g. petrosyan the condition (bc1) implies that the banach space of bounded continuous functions bc = bc((−∞, 0];e) is continuously embedded into b. more precisely, the following assertion is true. theorem 2.3: ( [27], proposition 7.1.1). (i) bc ⊂ c00, where c00 denote the closure of c00 in b; (ii) if a uniformly bounded sequence {ψn} in bc converges to a function ψ compactly on (−∞, 0], then ψ ∈ b and lim n→+∞ ‖ψn − ψ‖b = 0; (iii) there exists l > 0 such that ‖ψ‖b ≤ l‖ψ‖bc for all ψ ∈ bc. finally, we will assume that the following condition is satisfied: (bc2) if ψ ∈ bc and ‖ψ‖bc 6= 0, then ‖ψ‖b 6= 0. this assumption implies that the space bc, endowed with ‖ · ‖b is a normed space. we will denote it as bc. we may consider the following examples of phase spaces satisfying all the above properties: (1) for γ > 0 let b = cγ be the space of continuous functions ϕ : (−∞; 0]→ e, having a limit lim θ→−∞ eγθϕ(θ) with ‖ϕ‖b = sup −∞<θ≤0 eγθ‖ϕ(θ)‖. (2) (spaces of ”fading memory”) let b = cρ be the space of functions ϕ : (−∞; 0]→ e such that (a) ϕ is continuous on [−r; 0], r > 0; (b) ϕ is lebesgue measurable on (−∞; r) and there exists a nonnegative lebesgue integrable function ρ : (−∞;−r)→ r+ such that ρϕ lebesgue integrable on (−∞; r); moreover, there exists a locally bounded function p : (−∞; 0]→ r+ such that, for all ξ ≤ 0, ρ(ξ + θ) ≤ p (ξ)ρ(θ) a.e. θ ∈ (−∞;−r). then, ‖ϕ‖b = sup −r≤θ≤0 ‖ϕ(θ)‖+ −r∫ −∞ ρ(θ)‖ϕ(θ)‖dθ. a simple example of such a space is given by ρ(θ) = eµθ, µ ∈ r. 2.3. causal multioperators with infinite delay let e be a separable banach space. by lp ([0, t ];e) , 1 ≤ p ≤ ∞, we denote the banach space of all bochner summable functions f : [0, t ]→ e with the usual norm. for each subset n ⊂ lp ([0, t ];e) and τ ∈ (0, t ) we define restriction n on [0, τ ] as n |[0,τ ]= {f |[0,τ ]: f ∈ n}. we split the segment [0, t ] by points 0 < t1 < ... < tm < t,m ≥ 1. for a function c : [0, t ]→ e we will denote c(t+k ) = lim ξ→0+ c(tk + ξ), c(t−k ) = lim ξ→0− c(tk + ξ), copyright © 2021 assa. adv syst sci appl (2021) a controllability problem for causal functional inclusions with an infinite delay45 for 1 ≤ k ≤ m. for a function g : (−∞, t ]→ e, we will assume that its restriction to [0, t ] belongs to the space pc([0, t ];e) of functions z : [0, t ]→ e, continuous on [0, t ] \ {t1, · · · , tm} and such that the left and right limits z(t+k ) and z(t−k ), 1 ≤ k ≤ m, exist and z(t−k ) = z(tk). it is easy to see that the space pc([0, t ];e), endowed with the norm ‖z‖pc = sup t∈[0,t ] ‖g(t)‖e, is a banach space. the classical space of continuous function c([0, t ], e) is its closed subspace. we denote by c((−∞;t ];e) the normed space of piecewise continuous functions x : (−∞;t ]→ e, endowed with the norm ‖x‖c = ‖x0‖bc + ‖x |[0;t ] ‖pc. definition 2.7: a multivalued mapq : c ((−∞, t ];e) ( lp ([0, t ];e) is said to be a causal multioperator, if for each τ ∈ (0, t ) and for every u(·), v(·) ∈ c ((−∞, t ];e) the condition u |(−∞,τ ]= v |(−∞,τ ] implies that q(u) |[0,τ ]= q(v) |[0,τ ] . let us give examples of causal multioperators. example 2.1: we assume that the multimap f : [0, t ]× bc × e → kv (e) satisfies the following conditions: (f1) for each (ψ, φ) ∈ bc × e the multifunction f (·, ψ, φ) : [0, t ]→ kv (e) admits a measurable selection; (f2) for a.e. t ∈ [0, t ] the multifunction f (t, ·, ·) : bc × e → kv (e) is u.s.c.; (f3) there exists a function α ∈ lp+[0, t ], 1 ≤ p ≤ ∞, such that ‖f (t, ψ, φ)‖e := sup{‖z‖e : z ∈ f (t, ψ, φ)} ≤ α(t)(1 + ‖ψ‖bc + ‖φ‖e) for a.e. t ∈ [0, t ] and (ψ, φ) ∈ bc × e. from above conditions (f1)− (f3) and (b1) it follows that the multimap pf : c((−∞;t ];e)→ p (lp([0, t ];e)), given in the following way pf (x) = {f ∈ lp([0, t ];e) : f(t) ∈ f (t, xt, x(t)) a.e. t ∈ [0, t ]} is well defined (see, for example, [24], [25]). it is clear that the multioperator pf is causal. example 2.2: let f : [0, t ]× bc → kv(e) be a multimap satisfying conditions (f1)− (f3) from example 2.1. suppose that {k(t, s) : 0 ≤ s ≤ t ≤ t} is a continuous (with respect to the corresponding norm) family of bounded linear operators in e and m ∈ l1([0, t ];e) is a given function. consider the volterra integral multioperator v : c ((−∞, t ];e) ( l1 ([0, t ];e) defined as v(u)(t) = m(t) + ∫ t 0 k(t, s)f (s, us)ds, i.e., v(u) = {y ∈ l1 ([0, t ];e) : y(t) = m(t) + ∫ t 0 k(t, s)f(s)ds : f ∈ pf (u)}. it is also clear that the multioperator v is causal. copyright © 2021 assa. adv syst sci appl (2021) 46 m. afanasova, v. obukhovskii, g. petrosyan 3. the controllability problem for functional inclusions with the causal operators we will assume that the causal operator q : c ((−∞, t ];e)→ c (lp ([0, t ];e)) satisfies the following conditions: (q1) q is weakly closed in the following sense: conditions {un}∞n=1 ⊂ c ((−∞, t ];e) , {fn}∞n=1 ⊂ lp ([0, t ];e) , 1 ≤ p ≤ ∞, fn ∈ q(un), n ≥ 1, un → u0, fn l1 ⇀ f0 implies f0 ∈ q(u0); (q2) there exists a function α ∈ l∞+ ([0, t ]) such that ‖q (u) (t) ‖e ≤ α (t) (1 + ‖u‖c) , for a.e. t ∈ [0, t ], for all u ∈ c((−∞, t ];e); (q3) there exists a function ω : [0, t ]× r+ → r+ such that (ω1) for all x ∈ r+ : ω(·, x) ∈ lp+([0, t ]), 1 ≤ p ≤ ∞, ; (ω2) for a.e. t ∈ [0, t ] a function ω(t, ·) : r+ → r+ is continuous, nondecreasing and quasihomogeneous in the sense that ω(t, λx) ≤ λω(t, x) for all x ∈ r+ and λ ≥ 0; (ω3) for each bounded set ∆ ⊂ c ((−∞, t ];e) we have χ (q (∆) (t)) ≤ ω ( t, sup s∈[0,t] ϕ (∆s) ) for a.e. t ∈ [0, t ], where the set ∆s = {ys : y ∈ ∆} ⊂ bc and ϕ is the modulus of fiber noncompactness in bc. note that the condition (ω2) means that ω(t, 0) = 0 for a.e.t ∈ [0, t ] and as an example of such a function we can consider ω(t, x) = k(t) · x, where k(·) ∈ lp+([0, t ]). consider a linear operator s : lp([0, t ];e)→ c([0, t ];e), which is causal in the sense that for every τ ∈ (0, t ] and f, g ∈ lp([0, t ];e) condition f(t) = g(t) for a.e. t ∈ [0, τ ] implies (sf) (t) = (sg) (t) for all t ∈ [0, τ ]. following [24], we impose the next conditions on operator s : (s1) for 1 ≤ p <∞ there exist d ≥ 0 such that ‖sf(t)− sg(t)‖pe ≤ d ∫ t 0 ‖f(s)− g(s)‖peds for all f, g ∈ lp([0, t ];e) and 0 ≤ t ≤ t ; if p =∞ there exist d1 ≥ 0 such that ‖sf(t)− sg(t)‖e ≤ d1 ∫ t 0 ‖f(s)− g(s)‖eds for all f, g ∈ l∞([0, t ];e) and 0 ≤ t ≤ t. (s2) for an arbitrary compact set k ⊂ e and a sequence {fn}∞n=1 ⊂ lp ([0, t ];e) , 1 ≤ p ≤ ∞, such that {fn(t)}∞n=1 ⊂ k for all t ∈ [0, t ] the weak convergence fn l1 ⇀ f0 implies sfn → sf0 in c([0, t ];e). also we suppose that s satisfies the relation: (s3) (sf) (0) = 0 for each function f ∈ lp([0, t ];e). copyright © 2021 assa. adv syst sci appl (2021) a controllability problem for causal functional inclusions with an infinite delay47 notice, that condition (s1) implies that the operator s satisfies the lipschitz condition: (s1′) ‖sf − sg‖c ≤ d‖f − g‖l1 . consider following important examples. (i) let a closed linear operator a : d (a) ⊂ e → e be the infinitesimal generata of a c0semigroup {eat}t≥0. the operator l : l1([0, t ];e)→ c([0, t ];e) defined as lf(t) = ∫ t 0 ea(t−s)f(s)ds is a special case of the operator s. note that taking a = 0 we obtain, as a special case, the usual integral operatorli : l1([0, t ];e)→ c([0, t ];e), lif(t) = ∫ t 0 f(s)ds; (ii) let a : d(a)→ e be a closed linear operator e generating a c0semigroup {u(t)}t≥0 . the operator g : lp([0, t ];e)→ c([0, t ];e), p > 1/q, defined as gf(t) = ∫ t 0 (t− s)q−1t (t− s)f(s)ds, 0 < q < 1, where t (t) = q ∫ ∞ 0 θξq(θ)u(tqθ)dθ, ξq(θ) = 1 q θ−1− 1 qψq(θ −1/q), ψq(θ) = 1 π ∞∑ n=1 (−1)n−1θ−qn−1 γ(nq + 1) n! sin(nπq), θ ∈ r+, is a special case of the operator s. lemma 3.1: ( [24], lemma 4.2.1, [12], lemma 3.4). the operators l and g satisfy conditions (s1)− (s3). consider a control system governed by a functional inclusion with causal operatorsq and s, of the following form: y(t) ∈ g(t) ( ψ(0) + σtk<tik(y(tk)) ) + s ◦ q(y)(t) + s ◦bu(t), t ∈ [0, t ], (3.1) y(t) = ψ(t), t ∈ (−∞, 0], (3.2) iky(tk) = y(t+k )− y(tk), k = 1, ...,m, (3.3) where ψ ∈ bc is a given function, ik : e → e are the impulse functions, b : u → e is a linear bounded operator, u is a banach space of the controls, g(t) = ∫ ∞ 0 ξq(θ)u(tqθ)dθ, copyright © 2021 assa. adv syst sci appl (2021) 48 m. afanasova, v. obukhovskii, g. petrosyan and a control function u ∈ l∞([0, t ];u). we will assume that the corresponding linear problem is solvable, that is, there exists an operator w−1 which is a right inverse operator for w : l∞([0, t ];u)→ e of the following form wu = ∫ t 0 (t − s)q−1t (t − s)bu(s)ds let the operator w−1 satisfy the following regularity condition: (w ) there exists a function σ ∈ l∞+ ([0, t ]) such that for every bounded set ω ⊂ e we have: χu ( w−1(ω)(t) ) ≤ σ(t)χe(ω) for a.e. t ∈ [0, t ], where χu is the hausdorff mnc in u. we impose the following conditions on the impulse functions ik: (i1) the functions ik : e → e, 1 ≤ k ≤ m, are completely continuous. (i2) the functions ik : e → e, 1 ≤ k ≤ m, are globally bounded, i.e. there exists n > 0 such that ‖ikx‖ ≤ n for all x ∈ e. lemma 3.2: (see [12]) the operator functions g and t possess the following properties: 1) for each t ∈ [0, t ], g(t) and t (t) are linear bounded operators, more precisely, for each x ∈ e we have ‖g(t)x‖e ≤m ‖x‖e , ‖t (t)x‖e ≤ qm γ(1 + q) ‖x‖e , where m = supt≥0 ‖u(t)‖ 2) the operator functions g(·) and t (·) are strongly continuous, i.e., functions t ∈ [0, t ]→ g(t)x and t ∈ [0, t ]→ t (t)x are continuous for each x ∈ e. suppose ψ ∈ bc is a given function. for a function y ∈ pc([0, t ];e) such that y(0) = ψ(0) we define the function y[ψ] ∈ c((−∞, t ];e) as y[ψ](t) = { ψ(t), −∞ ≤ t < 0, y(t), 0 ≤ t ≤ t . we denote by d the closed convex subset of pc([0, t ];e), consisting of all functions y satisfying the condition y(0) = ψ(0). definition 3.1: a function y ∈ c((−∞, t ];e) is called a mild solution of problem (3.1)-(3.3), if it satisfies conditions: (1) the function y|[0,t ] ∈ d and satisfies inclusion (3.1); (2) y(t) = ψ(t), for t ∈ (−∞, 0]; (3) iky(tk) = y(t+k )− y(tk), k = 1, ...,m. now, the controllability problem which we solve in this paper may be formulated in the following way: for a given initial function ψ ∈ bc and a given x1 ∈ e we consider the existence of a mild solution y of problem (3.1)-(3.3) and a control u such that y(t) = ψ(t), t ∈ (−∞, 0], and y(t ) = x1. (3.4) copyright © 2021 assa. adv syst sci appl (2021) a controllability problem for causal functional inclusions with an infinite delay49 the pair (y, u) consisting of an integral solution y of problem (3.1)-(3.3) and a control u ∈ l∞([0, t ];u) will be called the solution of controllability problem (3.1)-(3.4). consider the multioperator γ : d( d defined as γ(y) = {x ∈ d : x(t) = g(t) ( ψ(0) + σtk<tik(y(tk)) ) + s ◦ q(y[ψ]) + s ◦bw−1 ( x1 − g(t )ψ(0)− ζ ◦ s ◦ q(y[ψ]) ) − s ◦bw−1(σm k=1g(t )ik(y(tk)))}, where f ∈ q(y[ψ]) and the linear operator ζ : c([0;t ];e)→ e is defined as ζx = x(t ). it is clear that if the function y is a fixed point of the multioperator γ, then the pair (y[ψ], u) is a solution to controllability problem (3.1) (3.4), therefore, our goal is to prove the existence of fixed point of the multioperator γ. definition 3.2: a sequence of functions {ξn} ⊂ lp([0, t ];e) is called lp–semicompact if it is lp–integrably bounded, i.e., ‖ξn(t)‖e ≤ v(t) for a.e. t ∈ [0, t ] and for all n = 1, 2, ..., where v ∈ lp([0, t ]), and the set {ξn(t)} is relatively compact in e for a.e. t ∈ [0, t ]. lemma 3.3: (see. [24], proposition 4.2.1.) every lp–semicompact sequence is weakly compact in l1([0, t ];e). we need the following properties of the multioperator s ◦ q. since for every 1 < p ≤ ∞ : lp([0, t ];e) ⊂ l1([0, t ];e), we can formulate a modification of theorem 5.1.1 from [24] in the following form. lemma 3.4: let s : lp([0, t ];e)→ c([0, t ];e) be an operator satisfying conditions (s1) and (s2). then for every lp-semicompact sequence {fn}∞n=1 ⊂ lp([0, t ];e), the sequence {sfn}∞n=1 is relatively compact in c([0, t ];e) and, moreover, the weak convergence fn l1 ⇀ f0 implies that sfn → sf0 in c([0, t ];e). theorem 3.1: let the multioperator q satisfy conditions (q1)–(q3) and the operator s satisfy (s1), (s2). then the composition s ◦ q : c((−∞, t ];e) ( c([0, t ];e) is a u.s.c. multimap with compact values. proof let us show that the multioperator s ◦ q is closed. let {xn}∞n=1 ⊂ c((−∞, t ];e), {yn}∞n=1 ⊂ c([0, t ];e), xn → x0, yn ∈ s ◦ q(xn), n ≥ 1, and yn → y0. take an arbitrary sequence {fn}∞n=1 ⊂ lp([0, t ];e) such that fn ∈ q(xn), yn = s(fn), n ≥ 1. from condition (q2) it follows that the sequence {fn}∞n=1 is lp-integrally bounded. the condition (q3) means that χ ({fn (t)}∞n=1) ≤ ω ( t, sup s∈[0,t] ϕ ({(xn)s}∞n=1) ) = ω(t, 0) = 0 for a.e. t ∈ [0, t ], and therefore the sequence {fn}∞n=1 is lp-semicompact. from lemma 3.3 it follows that the sequence {fn}∞n=1 is weakly compact, so we can assume without loss of generality that fn l1 ⇀ f0. by lemma 3.4 we have yn = sfn → sf0 = copyright © 2021 assa. adv syst sci appl (2021) 50 m. afanasova, v. obukhovskii, g. petrosyan y0. on the other hand, applying the condition (q1) we obtain f0 ∈ q(x0) and moreover y0 ∈ s ◦ q(x0), i.e., the multioperator s ◦ q is closed. for each x ∈ c((−∞, t ];e), conditions (q2) and (q3) mean that the sequence {fn}∞n=1 ⊂ q(x) is lp-semicompact and, by lemma 3.4, the sequence {sfn}∞n=1 ⊂ c([0, t ];e) is relatively compact. the compactness of the set s ◦ q(u) follows from its closedness. finally, if we consider the converging sequence {xn}∞n=1 ⊂ c((−∞, t ];e) and the arbitrary sequence {fn}∞n=1 ⊂ lp([0, t ];e, ) such that fn ∈ q(xn), then the sequence {sfn}∞n=1 ⊂ c([0, t ];e) is relatively compact, which means that the multimap s ◦ q is quasicompact and applying lemma 2.1 we obtain that it is u.s.c. let us proceed to finding conditions under which the multioperator s ◦ q will be condensing with respect to a corresponding mnc. for this we need the following statements. lemma 3.5: let a sequence of functions {fn}∞n=1 ⊂ lp([0, t ];e) be lp-integrally bounded and there exist a function υ ∈ lp+([0, t ]) such that χ ({fn (t)}∞n=1) ≤ υ(t) for a.e. t ∈ [0, t ]. if an operator s satisfies conditions (s1) and (s2), then for 1 ≤ p <∞ we have χ ({sfn (t)}∞n=1) ≤ ( 4pd ∫ t 0 υp(s)ds )1/p , and for p =∞ χ ({sfn (t)}∞n=1) ≤ 2d1 ∫ t 0 υ(s)ds, where d,d1 are the constants from condition (s1). proof for ε > 0 we take δ ∈ (0, ε), such that for every m ⊂ [0, t ], with the measure meas(m) < δ, we have: ∫ m |υ(s)|p < ε, for 1 ≤ p <∞, and respectively for p =∞ : ∫ m |υ(s)| < ε. taking mδ and bn corresponding to {fn} from lemma 2.2, we, by using property (s1), obtain that the sequence {s(bn)} is relatively compact in c([0, t ];e). let 1 ≤ p <∞, then the following estimates hold: ‖s(fn)(t)− s(bn)(t)‖pe ≤ d ∫ t 0 ‖fn(s)− bn(s)‖pe ds ≤ d ∫ [0,t]\mδ ‖fn(s)− bn(s)‖pe ds+d ∫ [0,t]∩mδ ‖fn(s)‖pe ds ≤ d ∫ [0,t]\mδ [2υ(s)− δ]p ds+d ∫ mδ |υ(s)|p ds ≤ d ∫ t 0 |2υ(s) + ε|p ds+ εd ≤ d ∫ t 0 2p |2υ(s)|p ds+d ∫ t 0 2pεpds+ εd ≤ 4pd ∫ t 0 υp(s)ds+ 2pεptd + εd. copyright © 2021 assa. adv syst sci appl (2021) a controllability problem for causal functional inclusions with an infinite delay51 therefore, the relatively compact set sgδ(t) is a ( 4pd ∫ t 0 υp(s)ds+ 2pεptd + εd )1/p net for the set {s(fn)(t)} . since ε > 0 is arbitrary, we obtain the conclusion of the lemma for the case 1 ≤ p <∞. if p =∞, then the following estimates hold: ‖s(fn)(t)− s(bn)(t)‖e ≤ d1 ∫ t 0 ‖fn(s)− bn(s)‖e ds ≤ d1 ∫ [0,t]\mδ ‖fn(s)− bn(s)‖e ds+d1 ∫ [0,t]∩mδ ‖fn(s)‖e ds ≤ d1 ∫ [0,t]\mδ |2υ(s)− δ|ds+d1 ∫ mδ |υ(s)| ds ≤ 2d1 ∫ t 0 υ(s)ds+ εtd1 + εd1. thus, the relatively compact set sgδ(t) is a 2d1 ∫ t 0 υ(s)ds+ εd1(t + 1) net for the set {s(fn)(t)} . since ε > 0 is arbitrary, we obtain the conclusion of the lemma also for the case p =∞. let m1, m2 be positive constants, such that ‖b‖ ≤m1, ∥∥w−1 ∥∥ ≤m2. (3.5) consider the measure of noncompactness ν in the space pc([0, t ];e) with values in the cone r2 +. on a bounded subset of ω ⊂ pc([0, t ];e) we define the values of ν as follows: ν(ω) = (γ (ω) ,modc (ω)) , where modc is the modulus of equicontinuity, γ is the fading modulus of fiber noncompactness γ(ω) = sup t∈[0,t ] e−ltχ(ω(t)). the constant l > 0 is chosen so that max{q1, q2} < 1, where q1 = sup t∈[0,t ] ( 4d1/p ( 1 + 4m1σd 1/pt 1/p ) ∫ t 0 e−lp(t−s)ωp (s, 1) ds )1/p , q2 = sup t∈[0,t ] ( 2d1 (1 + 2m1σd1t ) ∫ t 0 e−l(t−s)ω (s, 1) ds ) , where the constants d,d1 are from condition (s1), ω is a function from condition (q3). it is easy to see that the mnc ν is monotone, nonsingular, and algebraically semi-additive. it follows from the arzela–ascoli theorem that it is also regular. theorem 3.2: let a causal multioperatorq : c((−∞, t ];e) ( lp ([0, t ];e) satisfy conditions (q2) and (q3) and for a causal operator s : lp ([0, t ];e)→ c ([0, t ];e) the conditions (s1)–(s3) be satisfied. then, under conditions (i1), (i2), (w ) the multioperator γ is ν-condensing. copyright © 2021 assa. adv syst sci appl (2021) 52 m. afanasova, v. obukhovskii, g. petrosyan proof by lemma 3.2 and conditions (i1), (i2), it is suffices to prove the assertion of the theorem for the multioperator s ◦ q+ s ◦bw−1 ( x1 − g(t )ψ(0)− ζ ◦ s ◦ q ) . let ω ⊂ d be a bounded set such that ν ( s ◦ q (ω[ψ]) + s ◦bw−1 ( x1 − g(t )ψ(0)− ζ ◦ s ◦ q(ω[ψ]) )) ≥ ν (ω) . (3.6) let us show that the set ω is relatively compact. inequality (3.6) means that γ({s ◦ q (ω[ψ]) + s ◦bw−1 ( x1 − g(t )ψ(0)− ζ ◦ s ◦ q(ω[ψ]) ) }) ≥ γ(ω). (3.7) applying the condition (q3) and by using the properties of the function ω, we obtain for a.e. t ∈ [0, t ] χ ({f(t) : f ∈ q (ω[ψ])}) ≤ ω ( t, sup s∈[0,t] ϕ ({y[ψ]s : y ∈ ω}) ) = ω ( t, ϕ ( {y|[0,t] : y ∈ ω} )) = ω ( t, elte−ltϕ ( {y|[0,t] : y ∈ ω} )) ≤ ω ( t, eltγ ( {y|[0,t] : y ∈ ω} )) ≤ ω ( t, eltγ (ω) ) ≤ ω ( t, elt ) · γ (ω) . at first, we consider the case 1 ≤ p <∞. by lemma 3.5 we have for each t ∈ [0, t ] : χ ({sf(t) : f ∈ q (ω[ψ])}) ≤ ( 4pd ∫ t 0 ωp ( s, els ) ds · γp (ω) )1/p ≤ 4d1/p (∫ t 0 eplsωp (s, 1) ds )1/p · γ (ω) . further, χ ( {bw−1 ( x1 − g(t )ψ(0)− ζ ◦ sf(t) : f ∈ q (ω[ψ])} ) ≤ m1σχ ({ζ ◦ sf(t) : f ∈ q (ω[ψ])}) = m1σχ ({sf(t ) : f ∈ q (ω[ψ])}) ≤m1σ ( 4pd ∫ t 0 eplsωp (s, 1) ds · γp (ω) )1/p = 4m1σd 1/p (∫ t 0 eplsωp (s, 1) ds )1/p · γ (ω) , where σ = supt∈[0,t ] σ(t). using lemma 3.5 again, we have for each t ∈ [0, t ] : χ ( s ◦bw−1 ( x1 − g(t )ψ(0)− ζ ◦ sf(t) : f ∈ q (ω[ψ]) ) ≤( 4pd ∫ t 0 4pmp 1σ pd (∫ t 0 eplsωp (s, 1) ds ) dτ · γp (ω) )1/p ≤ copyright © 2021 assa. adv syst sci appl (2021) a controllability problem for causal functional inclusions with an infinite delay53 16d2/pm1σt 1/p (∫ t 0 eplsωp (s, 1) ds )1/p · γ (ω) . inequality (3.7) and the last inequality imply the following γ(ω) ≤ sup t∈[0,t ] ( 4d1/p ( 1 + 4m1σd 1/pt 1/p ) ∫ t 0 e−lp(t−s)ωp (s, 1) ds )1/p γ (ω) = q1 · γ (ω) , therefore γ (ω) = 0, thus ϕ (ω[ψ]t) = 0 for all t ∈ [0, t ]. let us turn to the case p =∞. by lemma 3.5 we have for each t ∈ [0, t ] : χ ({sf(t) : f ∈ q (ω[ψ])}) ≤ 2d1 ∫ t 0 ω ( s, els ) ds · γ (ω) ≤ 2d1 ∫ t 0 elsω (s, 1) ds · γ (ω) ; χ ( {bw−1 ( x1 − g(t )ψ(0)− ζ ◦ sf(t) : f ∈ q (ω[ψ])} ) ≤ m1σχ ({ζ ◦ sf(t) : f ∈ q (ω[ψ])}) = m1σχ ({sf(t ) : f ∈ q (ω[ψ])}) ≤m1σ2d1 ∫ t 0 elsω (s, 1) ds · γ (ω) = 2m1σd1 ∫ t 0 elsω (s, 1) ds · γ (ω) , where σ = supt∈[0,t ] σ(t). using lemma 3.5, we have for each t ∈ [0, t ] : χ ( s ◦bw−1 ( x1 − g(t )ψ(0)− ζ ◦ sf(t) : f ∈ q (ω[ψ]) ) ≤ 2d1 ∫ t 0 2m1σd1 (∫ t 0 elsω (s, 1) ds ) dτ · γ (ω) ≤ 4d1m1σt ∫ t 0 elsω (s, 1) ds · γ (ω) . inequality (3.7) and the last inequality imply the following γ(ω) ≤ sup t∈[0,t ] ( 2d1 (1 + 2m1σd1t ) ∫ t 0 e−l(t−s)ω (s, 1) ds ) γ (ω) = q2 · γ (ω) , therefore γ (ω) = 0, thus ϕ (ω[ψ]t) = 0 for each t ∈ [0, t ]. now we will show that the set ω is equicontinuous. we take sequences {yn}∞n=1 ⊂ ω, n ≥ 1 and {fn}∞n=1, fn ∈ q(yn[ψ]). from conditions (q2) and (q3) it follows that the sequence {fn}∞n=1 is lp-semicompact, and therefore by lemma 3.4 the sequence {sfn}∞n=1 copyright © 2021 assa. adv syst sci appl (2021) 54 m. afanasova, v. obukhovskii, g. petrosyan is relatively compact. hence modc({sfn}∞n=1) = 0. from the conditions that the operatorsb,w−1, ζ are bounded and linear, we can conclude that modc ( s ◦bw−1(x1 − g(t )ψ(0)− ζ{sfn}∞n=1) ) = 0. thus ν ( {s ◦ q (ω[ψ]) + s ◦bw−1 ( x1 − g(t )ψ(0)− ζ ◦ s ◦ q(ω[ψ]) ) } ) = (0, 0), but then it follows from the inequality (3.6) that ν(ω) = (0, 0), and the last expression yields that the set ω is relatively compact. to prove the main theorem of the paper, we need the following statements, known as the gronwall bellman lemma and the generalized gronwall bellman lemma. lemma 3.6: let v(t) and f(t) be nonnegative continuous functions on the segment [a, b], moreover v(t) ≤ c+ ∫ t a f(s)v(s)ds, t ∈ [a, b], where c is a positive constant. then for each t ∈ [a, b] the inequality v(t) ≤ ce ∫ t a f(s)ds, holds. lemma 3.7: let h(t), u(t) and v(t) be nonnegative functions integrable on [a, b] satisfying the inequality: v(t) ≤ u(t) + ∫ t a h(s)v(s)ds, t ∈ [a, b]. then the following inequality holds: v(t) ≤ u(t) + ∫ t a e ∫ t a h(θ)dθh(s)u(s)ds, t ∈ [a, b]. theorem 3.3: let a causal multioperator q : c((−∞, t ];e)→ cv(lp([0, t ];e)), 1 ≤ p ≤ ∞, satisfy conditions (q1)–(q3) and a linear causal operator s : lp([0, t ];e)→ c([0, t ];e) satisfy conditions (s1)–(s3). then, under conditions (i1), (i2), (w ) the set σψ of all solutions to problem (3.1)-(3.4) is a non-empty compact set. proof let us show that the set of all solutions y ∈ d of a one-parameter inclusion y ∈ λγ(y), λ ∈ [0, 1], (3.8) is a priori bounded. we divide the proof into three cases: p = 1, 1 < p <∞, p =∞. copyright © 2021 assa. adv syst sci appl (2021) a controllability problem for causal functional inclusions with an infinite delay55 let p = 1, if y ∈ d satisfies condition (3.8), then for each t ∈ [0, t ], using assumptions (b0), (s1), (i1), (i2) and (3.5), we have the following estimates: ‖y(t)‖e ≤ ‖g(t) ( ψ(0) + σtk<tik(y(tk)) ) ‖e +d ∫ t 0 ‖f(s)‖eds+ dm1m2 ∫ t 0 ‖x1 − g(t )ψ(0)− ζ ◦ sf(s)‖eds+dm1m2 ∫ t 0 ‖σm k=1g(t )ik(y(tk))‖eds ≤ m(l‖ψ‖bc +mn) +d ∫ t 0 ‖f(s)‖eds+dm1m2 ∫ t 0 ‖x1 − g(t )ψ(0)‖eds+ dm1m2 ∫ t 0 ‖ζ ◦ sf(s)‖eds+dmm1m2mnt ≤ m(l‖ψ‖bc +mn) +d ∫ t 0 ‖f(s)‖eds+dm1m2 (‖x1‖e +ml‖ψ‖bc)t+ dm1m2 ∫ t 0 ( d ∫ t 0 ‖f(s)‖eds ) dτ +dmm1m2mnt ≤ m(l‖ψ‖bc +mn) +d ∫ t 0 ‖f(s)‖eds+dm1m2 (‖x1‖e +ml‖ψ‖bc)t+ d2m1m2t ∫ t 0 ‖f(s)‖eds+dmm1m2mnt ≤ m(l‖ψ‖bc +mn) +dm1m2t (‖x1‖e +ml‖ψ‖bc +mmn) +(1 +dm1m2t )d ∫ t 0 ‖f(s)‖eds, where f ∈ q(y[ψ]) and, therefore, by condition (q2) : ‖f(s)‖e ≤ α(s)(1 + ‖y[ψ]‖c). then ‖y(t)‖e ≤m(l‖ψ‖bc +mn) +dm1m2t (‖x1‖e +ml‖ψ‖bc +mmn) + (1 +dm1m2t )d ∫ t 0 α(s)(1 + ‖y[ψ]‖c)ds ≤ m(l‖ψ‖bc +mn) +dm1m2t (‖x1‖e +ml‖ψ‖bc +mmn) + (1 +dm1m2t )d‖α‖l∞ ∫ t 0 ( 1 + ‖y[ψ]s‖bc + sup s∈[0,t ] ‖y(s)‖e ) ds. from property b1 (iii) it follows that ‖y[ψ]s‖bc + sup s∈[0,t ] ‖y(s)‖e ≤ h‖ψ‖bc + (k + 1)‖y‖pc , (3.9) where h(t) ≤ h,k(t) ≤ k, t ∈ [0, t ]. then, we obtain the following estimates copyright © 2021 assa. adv syst sci appl (2021) 56 m. afanasova, v. obukhovskii, g. petrosyan ‖y(t)‖e ≤m(l‖ψ‖bc +mn) +dm1m2t (‖x1‖e +ml‖ψ‖bc +mmn) + (1 +dm1m2t )d‖α‖l∞ ∫ t 0 (1 +h‖ψ‖bc + (k + 1)‖y‖pc) ds ≤ m(l‖ψ‖bc +mn) +dm1m2t (‖x1‖e +ml‖ψ‖bc +mmn) + (1 +dm1m2t )d‖α‖l∞ (1 +h‖ψ‖bc)t+ (1 +dm1m2t )d‖α‖l∞ ∫ t 0 (k + 1)‖y‖pcds. the last expression is a non-decreasing function of t, so we get ‖y‖pc ≤m(l‖ψ‖bc +mn) +dm1m2t (‖x1‖e +ml‖ψ‖bc +mmn) + (1 +dm1m2t )d‖α‖l∞ (1 +h‖ψ‖bc)t+∫ t 0 (1 +dm1m2t )d‖α‖l∞(k + 1)‖y‖pcds. this means that the function v(t) = ‖y‖pc satisfies the estimate v(t) ≤m(l‖ψ‖bc +mn) +dm1m2t (‖x1‖e +ml‖ψ‖bc +mmn) + (1 +dm1m2t )d‖α‖l∞ (1 +h‖ψ‖bc)t + ∫ t 0 (1 +dm1m2t )d‖α‖l∞(k + 1)v(s)ds. applying lemma 3.6, we obtain the required a priori boundedness: v(t) = ‖y‖pc ≤ ue(1+dm1m2t )d‖α‖l∞ (k+1) = γ1, where u = m(l‖ψ‖bc +mn) +dm1m2t (‖x1‖e +ml‖ψ‖bc +mmn) + (1 +dm1m2t )d‖α‖l∞ (1 +h‖ψ‖bc)t. for the case 1 < p <∞, by using the same properties (b0), (s1), (i1), (i2) and (3.5), we have ‖y(t)‖e ≤m(l‖ψ‖bc +mn) + ( d ∫ t 0 ‖f(s)‖peds )1/p + d1/pm1m2t 1/p(‖x1‖e +ml‖ψ‖bc) +d1/pm1m2 (∫ t 0 ‖ζ ◦ (sf)(t)‖peds )1/p + d1/pmm1m2mnt 1/p ≤m(l‖ψ‖bc +mn)+ d1/pm1m2t 1/p(‖x1‖e +ml‖ψ‖bc +mmn) +d2/pm1m2t 1/p (∫ t 0 ‖f(s)‖peds )1/p + ( d ∫ t 0 ‖f(s)‖peds )1/p ≤m (l‖ψ‖bc +mn) + d1/pm1m2t 1/p (‖x1‖e +ml‖ψ‖bc +mmn) + copyright © 2021 assa. adv syst sci appl (2021) a controllability problem for causal functional inclusions with an infinite delay57 d2/pm1m2t 1/p (∫ t 0 αp(s)(1 + ‖y[ψ]‖c)pds )1/p + ( d ∫ t 0 αp(s)(1 + ‖y[ψ]‖c)pds )1/p ≤ m (l‖ψ‖bc +mn) +d1/pm1m2t 1/p (‖x1‖e +ml‖ψ‖bc +mmn) + d2/pm1m2t 1/p (∫ t 0 αp(s)(1 + ‖y[ψ]s‖bc + sup s∈[0,t ] ‖y(s)‖e)pds )1/p + ( d ∫ t 0 αp(s)(1 + ‖y[ψ]s‖bc + sup s∈[0,t ] ‖y(s)‖e)pds )1/p . using inequality(3.9), we obtain the following estimate ‖y(t)‖e ≤m (l‖ψ‖bc +mn) +d1/pm1m2t 1/p (‖x1‖e +ml‖ψ‖bc +mmn) + d2/pm1m2t 1/p (∫ t 0 αp(s)(1 +h‖ψ‖bc)pds+ ∫ t 0 αp(s)(k + 1)p‖y‖ppcds )1/p + ( d ∫ t 0 αp(s)(1 +h‖ψ‖bc)pds+d ∫ t 0 αp(s)(k + 1)p‖y‖ppcds )1/p ≤ m (l‖ψ‖bc +mn) +d1/pm1m2t 1/p (‖x1‖e +ml‖ψ‖bc +mmn) + d2/pm1m2t 1/p21/p (∫ t 0 αp(s)(1 +h‖ψ‖bc)pds )1/p + d2/pm1m2t 1/p21/p (∫ t 0 αp(s)(k + 1)p‖y‖ppcds )1/p + (2d)1/p (∫ t 0 αp(s)(1 +h‖ψ‖bc)pds )1/p + (2d)1/p (∫ t 0 αp(s)(k + 1)p‖y‖ppcds )1/p ≤ m (l‖ψ‖bc +mn) +d1/pm1m2t 1/p (‖x1‖e +ml‖ψ‖bc +mmn) + d2/pm1m2t 1/p21/p(1 +h‖ψ‖bc) (∫ t 0 αp(s)ds )1/p + d2/pm1m2t 1/p21/p(k + 1) (∫ t 0 αp(s)‖y‖ppcds )1/p + (2d)1/p(1 +h‖ψ‖bc) (∫ t 0 αp(s)ds )1/p + (2d)1/p(k + 1) (∫ t 0 αp(s)‖y‖ppcds )1/p . let us introduce the following notation: c0 = m (l‖ψ‖bc +mn) +d1/pm1m2t 1/p (‖x1‖e +ml‖ψ‖bc +mmn) + (d2/pm1m2t 1/p21/p + (2d)1/p)(1 +h‖ψ‖bc)‖α‖lp , h(s) = ( d2/pm1m2t 1/p21/p(k + 1) + (2d)1/p(k + 1) )1/p α(s). copyright © 2021 assa. adv syst sci appl (2021) 58 m. afanasova, v. obukhovskii, g. petrosyan then we get: ‖y‖pc ≤ c0 + (∫ t 0 hp(s)‖y‖ppcds )1/p . let v(t) = ‖y‖ppc , then from the last inequality we obtain the estimate: v(t) ≤ 2pcp0 + 2p ∫ t 0 hp(s)v(s)ds. now applying lemma 3.7 to the last inequality, we get v(t) = ‖y‖ppc ≤ 2pcp0 ( 1 + ∫ t 0 e2p ∫ t 0 hp(θ)dθhp(s)ds ) . then we have the final estimate for 1 < p <∞ : ‖y‖pc ≤ 2q0 p √ 1 + ∫ t 0 e2p ∫ t 0 hp(θ)dθhp(s)ds = γ2. for the case p =∞, in the same way as in the case p = 1, the following estimate holds: ‖y‖pc ≤ u1e (1+d1m1m2t )d1‖α‖l∞ (k+1) = γ3, where u1 = m(l‖ψ‖bc +mn) +d1m1m2t (‖x1‖e +ml‖ψ‖bc +mmn) + (1 +d1m1m2t )d1‖α‖l∞ (1 +h‖ψ‖bc)t. now, if we take r ≥ max{γ1, γ2, γ3}, then we can guarantee that the set v ⊂ d, given as v = {y ∈ d : ‖y‖pc < r}, contains all solutions of inclusion (3.8). thus, the multioperator γ satisfies on ∂v the condition of theorem 2.2 with a = 0, hence the set of its fixed points is non-empty and compact. 4. controllability problems for semilinear differential inclusions with a delay and impulse effects 4.1. controllability problems for a first order semilinear functional differential inclusions consider the following system governed by a differential inclusion in a separable banach space e : y′ (t) ∈ ay (t) + f (t, yt) +bu(t), t ∈ [0, t ] \ {t1, t2, · · · , tm}, (4.10) y (θ) = ψ(θ), θ ∈ (−∞, 0], (4.11) iky(tk) = y(t+k )− y(tk), k = 1, ...,m, (4.12) where f : [0, t ]× bc → kv(e) is a multivalued map, ψ ∈ bc is a given function. suppose that copyright © 2021 assa. adv syst sci appl (2021) a controllability problem for causal functional inclusions with an infinite delay59 (a) a : d (a) ⊂ e → e is a linear closed operator, generating a c0-semigroup eat, t ≥ 0; a multimap f : [0, t ]× bc → kv(e) is such that: (f1) for each ψ ∈ bc the multifunction f (·, ψ) : [0, t ]→ kv (e) admits a measurable selection; (f2) for a.e. t ∈ [0, t ] the multimap f (t, ·) : bc → kv (e) is u.s.c.; (f3) there exists a function α ∈ l∞+ [0, t ] such that ‖f (t, ψ)‖e := sup{‖z‖e : z ∈ f (t, ψ)} ≤ α(t)(1 + ‖ψ‖bc) for a.e. t ∈ [0, t ] and for all ψ ∈ bc; (f4) there exists a function ωf : [0, t ]× r+ → r+ satisfying the conditions (ω1)-(ω3) such that for each bounded set ω ⊂ bc we have χ (f (t,ω)) ≤ ωf (t, ϕ (ω)) for a.e. t ∈ [0, t ]. we impose the following conditions on the impulse functions ik: (i1) the functions ik : e → e, 1 ≤ k ≤ m, are completely continuous. (i2) the functions ik : e → e, 1 ≤ k ≤ m, are globally bounded, i.e., there exists n > 0 such that ‖ikx‖ ≤ n for all x ∈ e. notice that when q = 1 : g(t) = eat, t (t) = eat, therefore, in accordance with [24], a function y ∈ c((−∞, t ];e), is a mild solution of problem (4.10)-(4.12), if it can be represented in the form: y(t) =  eat ( ψ(0) + σtk<tik(y(tk)) + ∫ t 0 ea(t−s)f(s)ds+ ∫ t 0 ea(t−s)bu(s)ds, t ∈ [0, t ], ψ(t), t ∈ (−∞, 0], where f ∈ pf (y[ψ]), pf is a superposition multioperator (see example 2.1). the fact that the superposition multioperator pf : c((−∞, t ];e) ( l1([0, t ];e) satisfies condition (q1) can be verified by lemma 5.1.1 from [24]. conditions (q2) and (q3) for pf follow from (f3) and (f4), respectively. taking into account lemma 3.1, we can consider relation (4.10) as a special case of functional inclusion (3.1) with q = pf , and s = l is the cauchy operator. as a direct consequence of theorem 3.3, we obtain the following result. theorem 4.1: suppose that conditions (a), (f1)–(f4), (i1), (i2), (w ) hold. then the set of solutions to problem (4.10)-(4.12), (3.4) is a non-empty compact subset of the space c((−∞, t ];e). 4.2. controllability problems for fractional semilinear functional differential inclusions let us recall the notion of the caputo fractional derivative. definition 4.1: the caputo fractional derivative of an order q ∈ (0, 1) of a function g ∈ c1([0, t ];e) is the function cdq 0g of the following form: cdq 0g(t) = 1 γ(1− q) ∫ t 0 (t− s)−qg(s) ds, where γ is the euler’s gamma-function γ(q) = ∫ ∞ 0 xq−1e−xdx. copyright © 2021 assa. adv syst sci appl (2021) 60 m. afanasova, v. obukhovskii, g. petrosyan consider the following control system governed by a functional differential inclusion in a separable banach space e : cdq 0y (t) ∈ ay (t) + f (t, yt) +bu(t), t ∈ [0, t ] \ {t1, t2, · · · , tm}, (4.13) y (θ) = ψ(θ), θ ∈ (−∞, 0], (4.14) iky(tk) = y(t+k )− y(tk), k = 1, ...,m, (4.15) where ψ ∈ bc is a given function. suppose that (a) a : d (a) ⊂ e → e is a linear closed operator, generating a c0-semigroup {u(t)}t≥0. assume also that a multimap f : [0, t ]× bc → kv(e) satisfies conditions (f1)–(f4) from example 2.1, impulse functions obey the conditions (i1), (i2). a function y ∈ c((−∞, t ];e), is a mild solution to problem (4.13)–(4.14), if it can be presented in the form (see [12]): y(t) =  g(t) ( ψ(0) + σtk<tik(y(tk)) + ∫ t 0 (t− s)q−1t (t− s)f(s)ds+∫ t 0 (t− s)q−1t (t− s)bu(s)ds, t ∈ [0, t ], ψ(t), t ∈ (−∞, 0], where f ∈ pf (y[ψ]). the fact that the superposition multioperator pf : c((−∞, t ];e) ( lp([0, t ];e), p > 1/q satisfies condition (q1) can be verified as in the paper [28]. conditions (q2) and (q3) for pf follow from (f3) and (f4), respectively. taking into account lemma 3.1, we can consider the relation (4.13) as a special case of functional inclusion (3.1) with q = pf , and s = g is the cauchy type operator. as a direct consequence of theorem 3.3, we obtain the following result. theorem 4.2: suppose that conditions (a), (f1)–(f4), (i1), (i2), (w ) hold. then the set of solutions to problem (4.13)-(4.15), (3.4) is a non-empty compact subset of the space c((−∞, t ];e). acknowledgements the work of the first author was supported by the state contract of the russian ministry of education as part of the state task (contract fzgf-2020-0009). the work of the second author was supported by rfbr (project number 20-51-15003). the work of the third author was supported by rfbr (project number 19-31-60011). references 1. lupulescu, v. (2008). causal functional differential equations in banach spaces, nonlinear anal., 69(12), 4787-4795. 2. obukhovskii, v., zecca, p. (2011). on certain classes of functional inclusions with causal operators in banach spaces, nonlinear anal., 74(8), 2765-2777. 3. kulmanakova, m.m., ulyanova, e.l. (2019). o razreshimosti kauzalnih funkcionalnih vkluchenii s beskonechnim zapazdivaniem [on the solvability of causal functional inclusions with infinite delay]. tambov university reports. series: natural and technical sciences, 24(127), 293-315, [in russian]. copyright © 2021 assa. adv syst sci appl (2021) a controllability problem for causal functional inclusions with an infinite delay61 4. tikhonov, a.n. (1938). o funkcionalnih uravneniyah tipa volterra i ih prilozheniya v nekotorih zadachah matematicheskoi fiziki [on functional equations of volterra type and their applications to some problems of mathematical physics]. bulletin of moscow university, 1(8), 1-25, [in russian]. 5. corduneanu, c. (2002). functional equations with causal operators. stability and control: theory, methods and applications. london, taylor and francis. 6. bulgakov, a.i., maximov, v.p. (1981). funkcionalnie i funkcionalno-differencialnie vkluchenia s operatorami volterra [functional and functional differential inclusions with volterra operators]. differential equations, 17(8), 1362-1374, [in russian]. 7. drici, z., mcrae, f.a., vasundhara devi, j. (2005). differential equations with causal operators in a banach space, nonlinear anal., 62(2), 301-313. 8. drici, z., mcrae, f.a., vasundhara devi, j. 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(2018). fractional derivatives of convex lyapunov functions and control problems in fractional order systems. fract. calc. appl. anal., 21, 1238-1261. 15. kamenskii, m., obukhoskii, v., petrosyan, g., yao, j.-c. (2019). existence and approximation of solutions to nonlocal boundary value problems for fractional differential inclusions, fixed point theory and applications, 30(2). 16. mainardi, f., rionero, s., ruggeri, t. (1994). on the initial value problem for the fractional diffusion-wave equation. in waves and stability in continuous media (pp. 246-251). singapore, world scientific. 17. agarwal, r.p., ahmad, b. (2011). existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions, comput. math. appl., 62, 1200-1214. 18. gomoyunov, m.i. (2019). approximation of fractional order conflict-controlled systems, prog. fract. differ. appl., 5, 143-155. 19. kamenskii, m., obukhoskii, v., petrosyan, g., yao, j.-c. (2019). on a periodic boundary value problem for a fractional order semilinear functional differential inclusions in a banach space, mathematics, 7(12), 5-19. 20. kamenskii, m., obukhoskii, v., petrosyan, g., yao, j.-c. (2021). on the existence of a unique solution for a class of fractional differential inclusions in a hilbert space, mathematics, 9(2), 136-154. 21. kamenskii, m.i., petrosyan, g.g., wen, c.-f. (2021). an existence result for a periodic boundary value problem of fractional semilinear differential equations in a banach space, journal of nonlinear and variational analysis, 5(1), 155-177. 22. petrosyan, g. (2021). antiperiodic boundary value problem for a semilinear differential equation of fractional order, the bulletin of irkutsk state university. series: mathematics, 34, 51-66. 23. petrosyan, g. (2020). on antiperiodic boundary value problem for a semilinear differential inclusion of fractional order with a deviating argument in a banach space, copyright © 2021 assa. adv syst sci appl (2021) 62 m. afanasova, v. obukhovskii, g. petrosyan ufa mathematical journal, 12(3), 69-80. 24. kamenskii, m., obukhovskii, v., zecca, p. (2001). condensing multivalued maps and semilinear differential inclusions in banach spaces. berlin-new york, walter de gruyter. 25. obukhovskii, v., gelman, b. (2020). multivalued maps and differential inclusions. elements of theory and applications. singapore, world scientific. 26. hale, j.k., kato, j. (1978). phase space for retarded equations with infinite delay, funkc. ekvac., 21, 11-41. 27. hito, y., murakami, s., naito, t. (1991). functional differential equations with infinite delay. lecture notes in mathematics. berlin-heidelberg-new york, springer-verlag. 28. petrosyan, g.g. (2015). teorema o slaboi zamknutosti superpozicionnogo multioperatora [a theorem on the weak closure of superposition multioperator], tambov university reports. series: natural and technical sciences, 20(5), 1355-1358, [in russian]. copyright © 2021 assa. adv syst sci appl (2021) introduction preliminaries multivalued maps and measures of noncompactness phase space causal multioperators with infinite delay the controllability problem for functional inclusions with the causal operators controllability problems for semilinear differential inclusions with a delay and impulse effects controllability problems for a first order semilinear functional differential inclusions controllability problems for fractional semilinear functional differential inclusions microsoft word 20 donglin li--preliminary design on automatic production line of high frequency induction welding light gauge advances in systems science and applications (2011), vol.11, no.3-4 355-361 issn 1078-6236 international institute for general systems studies, inc preliminary design on automatic production line of high-frequency induction welding light gauge h-beam donglin li school of mechanical engineering, hubei univ. of technology, wuhan 430068, china abstract：the automatic production line of high-frequency induction welding h-beam is preliminarily designed, which is applied to produce the h-beam of the web plate with the height less than 100mm. in the first place, the technological process for the whole production line is framed based on the principle of the hf induction welding. subsequently, the scheme design is conducted by dividing the production line into pre-welding processing, hf induction welding and post-welding processing. finally, the software solidworks is applied to design the virtual prototype which consists of the feature modeling of the parts and the virtual assembly of the machine, and carry out the motion simulation in order to check the interference and movement coordination of mechanisms. process provides an effective basis and good guidance for the succeeding design of the production line. keywords: high-frequency (hf) induction welding h-beam quality control web plate wing plate virtual prototype 1.introduction with the rapid development of steel structure, the demand for h-beam is growing, especially for the hf induction welding h-beam of the web plate with the height less than 100mm which is now being widely applied because of its economic and rational cross-section and its highly accurate size in the construction, bridges, machinery, shipbuilding, light industry and other industries. the small size of the cross-section of this type of hf welding h-beam, however, usually leads to series of problems such as difficulties in adjusting the welding contact and controlling the welding quality as well as the large welding deformation[1,2]. so the automated production line technology for this specification has always remained the focus of the research and development at home and abroad. this paper will probe into the preliminary design of automatic production line of high-frequency induction welding h-beam of the web plate with the height less than 100mm. in the first place, the technological process for the whole production line is framed based on the principle of the hf induction welding. and then, the scheme design is conducted with the view of improving the welding quality. finally, the software solidworks is applied to design the virtual prototype of the whole production line and carry out the motion simulation, calculate out the smallest traction of the automatic production line. this process provides important data reference for the practical design. 2.the method of hf induction welding h-beam hf induction welding is the welding achieved by rapidly heating the steel through the use of skin and proximity effects of the hf current. when welding the h-beam, first of all, the upper and lower wings will form a v-angle[3,4]. and then, supply power to the wings and the web through electrode to form the back-and-forth loops and liquid beam at the joining point. with the forward movements of the work piece, the beam will continuously extrude the liquid metal and oxide from the edge under the compression of the rolling. the remaining metal will keep in close contact with each other, producing plastic deformation and re-crystallization and resulting 356 li: preliminary design on automatic production line of high-frequency induction welding light gauge h-beam in the solid welding line. hf induction welding method of the h-beam diagram is shown in figure 1. fig.1 the schematic diagram of hf induction welding method 3.the technological process the technological process of the entire automatic production lines is shown in figure 2. production line machines include the flattening machine, angle-controlled machine, hf induction welding machine, deburring machine, correcting machine, traction engine and others, totaling to 28 sets of unit constructions. by the application of these unit constructions, many functions such as sheet flattening, upsetting web plate, hf welding, deburring, post-welding cooling, profiles correcting are expected to be achieved. 4.the scheme design of the production line in order to effectively control the product quality, the design of the entire production line divides into pre-welding processing, hf induction welding, post-welding processing. 4.1 pre-welding processing 4.1.1 loading material, decoiling and clipping the raw materials of hf induction welding light gauge h-beam are hot-rolled strip steel coil[5], so the equipment on the production line in the first place is the loading and decoiling machine of strip steel. the decoiled strip steel will then enter into the process of clipping during which the strip steel will undergo cutting according to the required size for wings and web plate. 4.1.2 loop storage, cutting and butt-welding for uninterrupted production, loops and the equipment of cutting and butt-welding are required on the production line. loops are used to store strip steel, the equipment of cutting and butt-welding to cut the ends of the strip steel. and then the two volumes of steel will be welded together smoothly, in order to ensure the non-stop production. 4.1.3 sheet flattening the decoiled strip steel usually has certain deficiencies such as buckling, bending arc, wave shape and so on. therefore, the sheet flattening is very necessary. flattening is being carried out on the roll flattening machine equipped with the interlaced top and bottom working rollers. during the process of flattening, the clearance between the top and bottom working rollers should be adjusted a little less than the thickness of the steel plate being flattened. in addition, the intake side must be ensured to undergo complete plastic deformation, and the delivery side must undergo complete elastic deformation. the steel will undergo a couple of times of forward and backward alternating bending and finally become flat and smooth. 4.1.4 upsetting of the web plate relying on the self-melting welding, the deposition rate of the hf induction welding h-beam can only reach 85 %～90 %, and the width of the weld seam can only be the 85% of the web plate. the strength of the weld seam cannot exceed the strength of the base metal. in order advances in systems science and applications (2011), vol.11, no.3-4 357 to expand the width of the weld seam equal to the width of the web plate, it is necessary to upset both sides of the web plate by the use of rolling mill before welding in order to expand the area of the ends by 30%. by the use of the u-shaped groove within the rollers, the rolling mill upsets the web plate through the extruding force produced from between the top and bottom rollers. fig.2 the technological process 4.2. hf induction welding 4.2.1 control of the vee angle in order to make an effective use of the skin and proximity effects of the hf current, it is required to set an angle control machine in the front of a hf induction welding machine to form a vee angle by the upper and lower wings and web plate. the most crucial point in the design of the angle control machine is to control the size of the angle because the size has a large impact on the stability of the hf flashing process and the welding quality as well as the welding efficiency. in many cases, the vee angle is required to be less 10°. the size of the vee angle needs the succeeding study. 4.2.2 the best location of the welding contactor during the process of welding, the two contactors respectively belonging to the upper and lower hf induction welding machine heat the web plate and wings separately, as shown in figure 1. the position of the contactors determines the length of the heating period, and loading material decoiling and clipping cutting and butt-welding loop storage sheet flattening web plate upsetting controlling the vee angle hf induction welding deburring post-welding cooling correcting the welding deformation sizing and shearing stacking upper and lower wing plate web plate loading material decoiling and clipping cutting and butt-welding loop storage sheet flattening 358 li: preliminary design on automatic production line of high-frequency induction welding light gauge h-beam produces a large impact on the temperature field and stress field of the h-beam. in order to raise the efficiency of welding, contactors should be located close to the squeeze roller as much as possible. but as for the h-beam of the web plate with the height less than 100mm, the design and location of welding contactor have always remained a difficulty due to the limitation of spacing between the web plates. 4.2.3 extrusion between the welded joints since welding is completed under the extrusion pressure when the sheet material is heated up to the fusing temperature, it is required to set up a squeeze roller above the wings where the sheet material is welded. extrusion force has a strong impact on the quality of the weld seam. in production, the extrusion force is replaced by the extrusion quantity of the plates for welding which is adjusted and measured by alternating the gap of the squeeze rollers. on the premise of the welding quality to be guaranteed, the extrusion quantity should be reduced as much as possible in order to reduce the metal consumption. 4.3 post-welding processing 4.3.1 deburring it is inevitable to produce the unevenly scars in the four weld seams. deburring is a requirement because the weld scars not merely influence the appearance of the weldment, but also leads to the uneven scatter of the stress in the weld seam. two sets of deburring machine are arranged in symmetry on the production line, with a milling cutter and planning tool loaded respectively on each of the deburring machine. the first step is to mill the two weld seams in order to get rid of some larger scars. and then, some minor welding scars and burr after milling will be removed by the use of planning tool. in this way, the four weld seams will become even and smooth, which improves the quality of the h-beam’s appearance to a large extent. 4.3.2 post-welding cooling post-welding cooling is not only for decreasing temperature before correcting profiles, but also for the consideration of the changes in the structure property of the weld seams and the production of the residual stress. the post-welding processing of the hf induction welding h-beam generally adopts the normal water-cooling, with the control of the cooling rate and cooling temperature at the ordinary level. in order to produce the high quality h-beam, our design adopts the air-cooling craft. during the process of water-cooling, we increase the spray equipment and lengthen the period of air-cooling properly. 4.3.3 correcting profiles due to the uneven scatter of the temperature field, the steel in the post-welding will endure large deformation, especially regarding the light gauge h-beam, the condition is even worsening. failure in the control and correction of the deformation will have a direct impact on the erection and connection, operational life span as well as its bearing capacity. the welding deformation of the h-beam lies mainly in the deformation of the angle of the wing plate. this condition requires the symmetric arrangement of two sets of correction machine on the production line. each correction machine presses both sides of the wings through two sets of upper and lower rollers with conicity in order to produce the plastic deformation of the reversed direction during the process of continuous feed-in, completing the continuous correction of the h-beam steel wing plates. 4.4 design of the traction machine the traction machine is the main driving force on the production line because it mainly depends on the traction force of the traction machine to facilitate the whole continuous and stable process from flattening, upsetting, vee angle controlling, to hf induction welding, deburring, correction and so on. in the production line are set up many direct-current and adjustable speed traction machine, which, using the principle of loops control, through the differential speed and the swing of loops, conducts the micro tension control over the steel advances in systems science and applications (2011), vol.11, no.3-4 359 material and then facilitate the smooth going of the whole process, providing guarantee to the welding quality, size and shape tolerance of the h-beam. 5.design of the virtual prototype after determining the technological process and the design scheme of the production line, we take advantage of the software solidworks’ strong three-dimensional modeling function to carry out the virtual prototype of the production line, and conduct the motion simulation to test and verity the coordination of the movements of all the separate parts[6,7], providing the important design parameters for the concrete design of the framework. 5.1 modeling of the parts’ parametric features the design of the parts by the solidworks mainly depends on the basic operations such as drawing, rotation, scanning and so on to set up the three-dimensional solid model. as for the complicated parts, it is essential to choose correct way to generate because the incorrect choices are not only inefficient, but also cannot generate solid model at all in certain cases. take the modeling of the deburring machine as an example, due to its symmetric structure, half of the model can be built at first, and then another half can be completed through mirror. when constructing the half modeling, first, choose the sketch plane to enter the sketch mode and draw out the sketch of the framework. then, generate the three-dimensional solid entities using the drawing method. the completed framework model of the deburring machine is shown in figure 3. fig.3 the framework model of the deburring machine 5.2 virtual assembly after modeling the three-dimensional solid entities of the separate part, the virtual assembly is required for each of these parts. in the assembly module of the solidworks, it is workable to assemble these separate three-dimensional solid entities into a working machine, and check whether or not there is interference between different parts and whether or not the assembling entities move according to the requirements of design[8,9]. take the deburring machine as an example, first, insert the machine framework which the system defaults as a built-in fitment. then, insert the model of other parts and choose the location of each separate part. finally, select appropriate assembly constraint types to complete the location of the parts model. the deburring machine model after virtual assembly is shown in figure 4 (in order to achieve the true-to-life visual effect, h-beam is fitted together into the deburring machine). fig.4 the assembling model of the deburring machine 5.3 the motion simulation of the machine this paper chooses the cosmos motion in solidworks to carry out motion simulation. before simulation, it is necessary to enter the assembly environment to input some necessary 360 li: preliminary design on automatic production line of high-frequency induction welding light gauge h-beam parameters beforehand in order to confirm the moving parts, the fixing parts, the kinematics pair, and install the motion environment and simulation options. the motion simulation of the deburring machine is shown in figure 5. the highlighted parts shown in the picture are the kinematics pair between different machines. according to the requirements of the h-beam production line, with the given horizontal velocity, certain parameters such as the mean momentum and the average torque and others will be attained after the automated simulation calculation. fig.5 the motion simulation of the deburring machine 5.4 the motion simulation of the production line in order to get a clearer and more direct-viewing glimpse of the whole production motion process, we carry out the motion simulation to the entire production line. considering the fact that the entire production line is too bulky for the software to conduct a one-off calculation, we divide the production line into two stages: the pre-welded process of the hf induction welding which consists of the sheet material flattening machine, web plate upsetting machine and the angle control machine; the post-welding process of the hf induction welding which consists of the deburring machine and the correction machine. fig.6 the simulation model of the pre-welding processing. fig.7 the simulation model of the post-welding processing. the simulation steps for the entire production line are almost the same as that of the single machine. the simulation model of the pre-welding processing is shown in figure 6. the simulation model of the post-welding processing is shown in figure 7. after the process of simulation, the momentum and the torque of each moving parts can be accessible from the data-process module. for example, if the given horizontal velocity of the h-beam is 30m/min, the average traction torque of the h-beam will be 450nm based on the calculated mean momentum and average torque. 6.conclusion the automatic production line of the hf induction welding h-beam has the advantage of high productive efficiency, stable quality and the convenience to alternate the varieties and advances in systems science and applications (2011), vol.11, no.3-4 361 specifications and so on. this paper has delved into the preliminary design of automatic production line of hf induction welding h-beam with the web plate’s height less than 100mm. in the first place, the technological process of the production system is framed based on the principle of hf induction welding. subsequently, the scheme design is conducted with the view of improving the welding quality. finally, the software solidworks is applied to design the virtual prototype of the whole production line and carry out the motion simulation. all the preliminary design will provide a significant guidance for the succeeding engineering design in practice. how to promote the welding quality and control deformation, however, still remains the largest problem in the automatic production line of hf induction welding light gauge h-beam. in this paper, such measures as steel material flattening, web plate upsetting, profiles correcting are adopted to guarantee the quality; nevertheless, the best location of the welding contactor and the size of the welding vee angle still remain the focus of the succeeding research and design. 7.acknowledgements the emphasis scientific and technological key project of wu han city fund the project. (contract no. 030877). references [1] scott p. f. and smith w. “a study of the key parameters of high frequency welding”. tube china ’95. (1995) 161-181. [2] scott p. f. “the effects of frequency in high frequency welding”. transactions of tube 2000 toronto, ita conference. (2000) 74-80. [3] asperheim j. i., markegaard l., induksjon e. and lombard p. “temperature distribution in the weld vee cross section”. tube international. 17(70) (1998) 563-567. [4] asperheim j. i. and grande b. “temperature evaluation of weld vee geometry and performance”. tube international. 19(110) (2000) 497-502. [5] wesling v., schram a. and rekersdrees t. “high-frequency welding of martensitic hot-rolled strip”. welding research abroad. (1998) 145-162. [6] mao-peng chen and wei-fang. “automatic modeling for modular fixture components based on solidworks”. journal of south china university of technology (natural science). (2005) 56-59. [7] c.k.chua, s.h.teh and r.k.l.gay. “rapid prototyping versus virtual prototyping in product design and manufacturing”. the international journal of advanced manufacturing technology. 7(1999) 597-603. [8] yang fuqin, chang degong and wang xianlun. “dynamic simulation of the tripod sliding universal joint based on virtual prototype”. 2008 asia simulation conference 7th international conference on system simulation and scientific computing, icsc 2008. (2008) 711-714. [9] li feng, li shuguang, zheng lei and lou li. “dynamic virtual prototype of soldier system”. proceedings of the 27th chinese control conference. (2008) 329-333. adv syst sci appl 2019; 01; 61-74 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/632 identification of nonlinear systems having hard function adil brouri1*, tarik rabyi1, abdelmalek ouannou1 1) ensam, imsm, l2mc lab, moulay ismail university sciences, meknes, morocco e-mail: a.brouri@ensam-umi.ac.ma received august 14, 2018; revised march 18, 2019; published april 15, 2019 abstract: in this work, an identification approach of nonlinear systems is studied. presently, the nonlinear system can be described by wiener-hammerstein model. this latter is composed of a nonlinearity surrounded by two linear blocks. the linear blocks can be nonparametric. the nonlinear element is allowed to be discontinuous, specifically it can be of hard element (e.g. preload, coulomb friction, dead-zone). roughly, it is very difficult to model these types of nonlinearities by orthogonal decompositions (e.g. polynomial). keywords: nonlinear systems, linear system, static and dynamic systems, control system, system identification, discontinuous nonlinearity, hard element. 1. introduction nonlinear systems exist widely in industry and science applications [2,5-7,13], among which the wiener-hammerstein model (fig. 1.1) is one of the most typical cases [1,3,9,15-16]. the wiener-hammerstein models consist of a nonlinear block surrounded by two linear elements (fig.1.1). the identification problem have been paid considerable attention due to their benefits such as control [7,11-12]. the wiener-hammerstein like systems are used in a wide range of applications such as identification of skeletal muscle [1]. note that, wiener and hammerstein nonlinear systems can be modeled by wiener-hammerstein systems. the solution of this problem identification can be dealt using several method. the available methods have been developed following three main approaches i.e. iterative nonlinear optimization procedures [19]; stochastic methods e.g. [3,21]; in [27], an approach based on the standard svm (support vector machines) for regression was presented. the quite poor results obtained in that work highlighted some of the limitations of the method. in particular, only a nfir (nonlinear finite impulse response) model structure was taken into account, which did not perform well since the considered system has a long impulse response. another problem was given by the high computational time and memory usage, which made it difficult to work with a large amount of data. several svm-like approaches [18], based on the least squares svm (ls-svm), are characterized by a very high number of parameters. many approaches use the bla, or a similar correlation analysis, as a starting point for the algorithm (e.g. [24-26]). then, the user does not have to take order decisions needed to parametrize the bla (or the qbla). in [23], a nonparametric approach to separate the front and back dynamics starting from the best linear approximation (bla) is proposed. in [16], a recursive identification method of wiener-hammerstein system with internal noises is developed. in [17], a recursive leastsquares identification method of wiener-hammerstein system is proposed. this approach is treated in the case where the system nonlinearity is dead-zone. then, a wide multiplicity of approaches is currently under study. these contain parametric and non-parametric methods. there exist different class of non-parametric * corresponding author: a. brouri, a.brouri@ensam-umi.ac.ma mailto:a.brouri@ensam-umi.ac.ma 62 a. brouri, t. rabyi, a. ouannou copyright ©2019 assa adv. in systems science and appl. (2019) solutions. among these concepts the frequency modeling, for which control and output are relying using periodic signals [4,8,10,14,20,22]. depending on the nature of the control (the input excitation) to the system, several estimators have been developed. in this paper, a solution based on analytical geometry (frequency study) is proposed. presently, the linear elements are allowed to be nonparametric. the nonlinear block can be discontinuous or of hard nature. examples of popular hard functions are shown by figs. 1.2ab. for convenience, fig. 1.2a illustrates coulomb friction nonlinearity with dead-zone and fig. 1.2b shows example of viscous friction function. a hard function is often known as being a nonlinearity having several discontinuities and it is an affine nonlinearity elsewhere [9,14]. on the other hand, the modeling of hard function using orthogonal polynomials decomposition often remains a challenge. it is very complicated to approximate these types of nonlinearities using any orthogonal basis approximation (e.g. polynomial decomposition). for convenience, fig. 1.3 illustrates example of orthogonal polynomial decomposition of hard function. this example shows that remarkable modeling errors are thus induced especially around discontinuity points even though the troncature degree m is too big. presently, the nonlinear element is allowed to be discontinuous and not necessarily affine function between two consecutive discontinuities. except in a small interval where it is supposed to be parametric function (e.g. polynomial nonlinearity). then, recall that the identification method is based only on control signal u(t) and observed output signal y(t) (i.e. all inner signals are not accessible). in this work, the linear blocks are not necessarily parametric. accordingly, because of these last difficulties, it is not surprising to notice that they are very rare papers dealing the identification of wienerhammerstein models having hard function. fig. 1.1. wiener-hammerstein model the paper is organized as follows: the identification problem is formulated in section 2, which also introduces the problem of multiplicity of identification solutions; section 3 is devoted to the determination of nonlinear system parameters (linear block and nonlinear element); the performances of the identification method are illustrated by simulation in section 4. -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 example of hard nonlinearity fig. 1.2a. coulomb friction nonlinearity with dead-zone identification of nonlinear systems having hard function 63 copyright ©2019 assa. adv. in systems science and appl. (2019) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 nonlinearity with preload fig. 1.2b. viscous friction function -2 -1.5 -1 -0.5 0 0.5 1 1.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 true n.l m=5 m=10 m=20 fig. 1.3. decomposition of hard nl with series of polynomial basis 2. problem formulation presently, the problem of determination of nonlinear system parameters is discussed. the considered nonlinear system is structured by wiener-hammerstein model (fig. 1.1). let gi(s) and go(s) denote the transfer function of linear blocks. the associated impulse responses (or the inverse laplace transforms) are respectively denoted  1 )( ()i ig l g st  and  1 )( ()o og l g st  . the control signal ( )u t and the inner signal v(t) are thus related by the following relation: ( ) ( )* ( ) i v t g t u t (2.1) where the notation “ * ” designates the convolution product. then, the inner signals v(t) and w(t) undergo the following equation:    ( ) ( ) ( )* ( )iw t f v t f g t u t  (2.2) as far as that goes, the inner signal w(t) and the undisturbed output x(t) are related as follows: ( ) ( )* ( ) o x t g t w t (2.3) finally, it follows from (2.3) and fig. 1.1 that, the system output can be expressed as:   ( ) ( )* ( ) ( ) ( )* ( ) ( ) o o y t g t w t t g t f v t t       (2.4) where the extra input ( )t accounts for measurement noise and other modelling effects. on the other hand, note that this identification problem does not have a unique solution [6,12]. indeed, if the triplet  ( ), ( ), ( )i og s f v g s is solution of this wiener-hammerstein 64 a. brouri, t. rabyi, a. ouannou copyright ©2019 assa adv. in systems science and appl. (2019) identification problem, then any set of form    ( ) , ( / ) / , ( ) ; 0 and 0i i i o o o i ok g s f v k k k g s for k k  is also solution of this problem (by distributing nonzero constants between the blocks of system). in this respect, the question that arises is how to choose a solution to this problem? the answer to this will be dealt with in the next section. the considered wiener-hammerstein nonlinear system is completed by the following assumptions: assumptions 2.1:  the nonlinear element is hard function having multiple discontinuities. except in a small interval where it is supposed to be parametric or polynomial function of degree increased by a finite integer n.  the linear elements have a nonzero static gains, i.e. (0) 0ig  and (0) 0og  .  the extra input ( )t is supposed to be zero-mean ergodic and uncorrelated with the control input u(t). □ except of these assumptions, the system is arbitrary. in particular, the nonlinear element is allowed to be discontinuous and it is not necessarily an affine function between two consecutive discontinuities. then, the linear elements are nonparametric. 3. system parameters estimation 3.1. nonlinear element estimation the goal presently is to develop a solution allowing to give the estimate of a set of points belonging to nonlinear function f(.). the question: from the plurality problem discussed in section 2, what is the system to be estimate? a key idea is to get benefit from this model plurality to make the identification problem more tractable. in this respect, the following selection of the free scalars ( , )i ok k will prove to be judicious: 1 (0)i i k g  and 1 (0)o o k g  (3.1) without loss of generality, it is readily follows from (3.1) and model plurality that, the linear elements of system to be determined check the following property: (0) (0) 1i og g  (3.2) on the other hand, let excite the system by a constant value: 1( )u t u ,  0t t (3.3) where t is more superior to the system rise time rt . then, it follows from (2.1), (3.2) and (3.3) that, the inner signal v(t) boils down (in steady state) to: 1( )v t u (3.4) one immediately gets from (2.2) and (3.4) that (after transient regime):  1( )w t f u (3.5) accordingly, in view of (2.3), (3.2) and (3.5), it follows that x(t) is written as follows (after transient regime): identification of nonlinear systems having hard function 65 copyright ©2019 assa. adv. in systems science and appl. (2019)  11( )x t x f u  (3.6) then, it is readily seen from (2.4), (3.2) and (3.6) that the output y(t) undergoes (after transient regime) the following expressions:  1( ) ( )y t f u t  (3.7) at this point, it is worth emphasizing that, the signal y(t) is constant up to noise. furthermore, it readily seen (in steady state) that:      1 1 1 1, , , ( )u x u x u f u  (3.8) is a point belonging to the nonlinearity f(.). one difficulty with the considered identification problem is that, the output of the system y(t) is infected by the disturbance ( )t whose stochastic law is not known. then, it follows from the assumption on the noise ( )t , just as suggested in [6,12] the following estimator for 1( )f u is proposed: 1 1 1ˆ ˆ( ) ( ) r r t t t f u x y t t t      (3.9) indeed, one has from (3.7) and (3.9) that:  11 1ˆ ( ) ( ) r r t t t uf u f t t t       (3.10) bearing in mind that, the noise ( )t is zero-mean ergodic stochastic sequence, the last term in (3.10) converges (with probability 1) to zero. this implies that:  11 ˆ ( ) t uf u f   w.p.1 (3.11) this result shows that, an accurate estimate of a point belonging to the function f(.) can be obtained. accordingly, the estimate of set of points belonging to f(.) can be achieved using the same procedure, i.e. applying the control (input) sequence: ( ) ku t u , ( 1) t k t kt     for 1k n (3.12) then, ˆ( )kf u ( 1k n ) can be obtained using the estimator (3.9): ( 1) 1ˆ ( ) ( ) k r r kt t k t t f u y t t t       for 1k n (3.13) remark 3.1: consider the nonlinear system (wiener-hammerstein) described by (2.1)-(2.4). then, the system nonlinearity estimator, given by (3.9), enjoys the consistency property for any value .ku specifically, one has the following result:  ˆ ( ) kk t uf u f   w.p.1 (3.14) the proof of this can be found by combining (3.9)-(3.10) and the property of ( )t (zeromean ergodic stochastic). □ 66 a. brouri, t. rabyi, a. ouannou copyright ©2019 assa adv. in systems science and appl. (2019) remark 3.2:  the identification method can be easily applied in the case where f(.) is parametric (e.g. polynomial function of degree n). then, a number of points 1n n  (arbitrarily chosen by the user) is largely sufficient to determine the nonlinearity over the entire working range.  in the case where f(.) is hard element, it is thus not necessarily a parametric function and can be discontinuous. furthermore, in a small interval (e.g. between two consecutive discontinuities), f(.) is supposed to be parametric or polynomial function of degree increased by an integer n. then, using the set of estimated points   ˆ, ( ) 1; k k u f u k n , choose a candidate interval and verify if the nonlinear element can be decomposed with orthogonal polynomials approximation. to this end, we can excite the system by other inputs in this interval. then, find out if we can approximate the set of points  ˆ, ( ) k k u f u with a polynomial function of degree n.  the last statement can be checked using a sine control in the chosen interval. accordingly, by observing the spectrum of the output signal if it does not contain harmonics of higher rank than n. for further information, please see the following subsection « the linear blocks determination ». in this interval, the output of nonlinear block can thus be expressed as: 0 ( ) ( )( ) k k n k w t c vf v t     (3.15) where  0 ... t nc c c is the parameters vector corresponding to the nonlinear element.□ 3.2. the linear blocks determination presently, the aim is to present an identification approach permitting to provide the estimate of linear blocks parameters. for convenience, the nonlinear system (2.1)-(2.4) is excited by the following control signal:  ( ) cos( )u t u t   (3.16) where the parameter  is adjusted such that u(t) belongs to the chosen interval using remark 3.2 for any small amplitude u. this result can be practically depicted by observing the spectrum of the output signal. let ( )i  and ( )o  designate the phases (argument) of linear elements ( ) i g j and ( ) o g j , for any frequency  , respectively. then, it readily follows from (2.1), (3.2) and (3.16) that, the inner signal v(t) (in the steady state) is given by:  ( ) ( ) cos( ( )) i i v t u g j t       (3.17) one has then using (3.15) and (3.17) that:   0 ( ) ( ) cos( ( )) i i kk k n k w t c u g j t         (3.18) bearing in mind that:     0 ( ) cos( ( )) ( ) cos( ( )) i i i i k k lk l l k l g j t c g j t                (3.19) where: identification of nonlinear systems having hard function 67 copyright ©2019 assa. adv. in systems science and appl. (2019)   ! ! ! k l k c l k l   (3.20) then, it immediately follows from (3.18)-(3.20) that:   0 0 ( ) ( ) cos( ( )) i i k lk k l k l n k k l w t c u c g j t           (3.21) for convenience, the formulas of power identities of types 2cos m  and 2 1cos m  can also be given analytically as:  2 2 2 2 2 1 1 0 1 1 cos cos 2( ) 2 2 m m m m pm m m p c c m p        (3.22a)  2 1 2 1 0 1 cos cos (2 1 2 ) 4 m m pm m p c m p       (3.22a) in view of (3.22a-b) and grouping the components having the same harmonics, (3.21) becomes (for any limited integer n):       0 1 ( ) ( ) ( ) cos ( ) i i ik k n k w t a g j a g j k t          (3.23) where the unknown variables in the amplitude  ( ) ika g j ( 0 )k n and the phases  ( ) ik   ( 1 )k n are the parameters of linear element ( ) i g j (i.e. the modulus gain ( ) i g j and the phase ( ) i   ). therefore, one immediately gets using (2.3), (3.2) and (3.23):        0 1 ( ) ( ) ( ) ( ) cos ( ) ( ) i i o i ok k n k x t a g j a g j g jk k t k               (3.24) it readily seen that, the unknown parameters in the expression of undisturbed output signal are  ( ) , ( ) i i g j   and   ( ) , ( ) ; 1 o o g jk k k n    . in this respect, note that the inner signal x(t) is equivalent to sum of n sinusoidal signals and dc component. specifically, having the spectrum of x(t), the equation (3.24) leads to a system of (2 1)n equations and (2 2)n unknowns, i.e. more unknowns than equations. this problem can be overcome by repeating the same experiment using the input signal (3.19) having the frequency 2 , 3 , …. the difficulty that arises is that, the signal x(t) is not accessible to measurement. fortunately, an estimate of x(t) can be established using the fact that this latter is periodic of the same period 2 /t   of control signal u(t). this property suggests the following estimator: 1 1 ˆ( ) ( ) l p x t y t pt l    for [0, )t t (3.25a) ˆ ˆ( ) ( )x t kt x t  for any integer k (3.25b) where l is any integer preferably large. indeed, one immediately gets from (2.4) and (3.25ab) that: 68 a. brouri, t. rabyi, a. ouannou copyright ©2019 assa adv. in systems science and appl. (2019) 1 1 1 1 1 ˆ( ) ( ) ( ) 1 ( ) ( ) l l p p l p x t x t pt t pt l l x t t pt l                (3.26) the periodic stationarity of ( )t means that  ( ) ( ( ))e t pt e t   , for any p and t. then, the zero-mean ergodicity of ( )t means that: 1 1 ( ) 0 l l p t pt l      (3.27) this implies that (using (3.26)-(3.27)): ˆ( ) ( ) l x t x t   (3.28) 4. simulation presently, the system (2.1)-(2.4) is characterized by the linear elements of transfer functions: 0.1 ( ) (0.4 )(0.1 ) i g s s s    (4.1a) 1 ( ) (0.2 )(0.5 ) o g s s s    (4.1b) the curve of nonlinear block (.)f , used in simulation, is shown by fig. 4.1. the noise signal ( )t is a sequence of random numbers, with zero-mean and standard deviation 0.5  . in the first stage, we apply to the input of system the sequence plotted in fig. 4.2. the collected system output is illustrated in fig. 4.3a. then, using the estimator (3.9) or (3.13), the estimate values ˆ( )kf u ( 1k n ) are also given by fig. 4.3a. for convenience, a zoom of these results is given by fig. 4.3b. -2 -1 0 1 2 3 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 the n.l considered in simulation fig. 4.1. shape of the function f(.) considered in simulation identification of nonlinear systems having hard function 69 copyright ©2019 assa. adv. in systems science and appl. (2019) 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 time (s) u the input sequence u(t) fig. 4.2. the applied control sequence then, to compare the estimated and the true nonlinearities, fig. 4.4 shows the set of points   ˆ, ( ) 1 16; k k u f u k  and the true function (.)f (of rescaled nonlinear system using (3.1)-(3.2)). these results show that the estimated points   ˆ, ( ) 1 16; k k u f u k  are very close to their true values. it follows that the function does not seem to be of hard type in the interval  1 3 . in this respect, the system can be excited with other inputs within this interval and comparing the interpolation of these points with a polynomial of degree n. 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 0 50 100 150 200 250 300 time (s) y(t) and f(u ) the system output y(t) the estimates of f(u ) k ^ k fig. 4.3a. the system output signal and the estimate of x(t) 600 700 800 900 1000 1100 1200 1300 1400 1500 -10 -8 -6 -4 -2 0 2 4 6 8 time (s) y(t) and f(u ) fig. 4.3b. zoom of the signal y(t) and the estimate of x(t) in the second stage, we apply to the system input the sine signal (3.16) where  is chosen such that u(t) belongs to the interval  1 3 . taking e.g. 2  , the resulting system output signal y(t) for an amplitude 1u  and a frequency 0.02 ( / )rd s  is illustrated by fig. 4.5. 70 a. brouri, t. rabyi, a. ouannou copyright ©2019 assa adv. in systems science and appl. (2019) the measured signal y(t) is collected on a sufficiently large interval. then, the collected sample is used to generate the undisturbed output ( )ˆ tx using (3.25a-b). the obtained estimate is plotted in fig. 4.6 over one period of time. -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 0 50 100 150 200 250 300 the true nonlinearity f(.) the set of estimated points fig. 4.4. comparison between the true and estimated nonlinearities accordingly, it readily follows from the results of first stage and using an orthogonal polynomial decomposition of degree 3n  that, the obtained expression of ˆ(.)f within the interval  1 3 is given as: 3 2ˆ( ) 0.003 +31.24 0.02 24.8f v v v v    (4.2a) while the expression of the true nonlinearity is as follows: 2( ) 31.25 25f v v  (4.2b) let us consider: 3 ˆ 0.003c  ; 2 ˆ 31.24c  ; 1̂ 0.02c  ; 0 ˆ 24.8c   ; (4.3) denote the coefficients of the estimated nonlinearity. on the other hand, it readily follows from (3.18)-(3.23) and (4.2a) ( 1)u  that, the inner signal w(t) can be rewritten as:       0 3 1 ( ) ( ) ( ) cos ( ) i i ik k k w t a g j a g j k t          (4.4a) where:               2 2 0 0 1 2 3 2 3 2 2 1 1 2 3 2 2 2 3 3 3 3 ( ) ˆ ˆ ˆ ˆ ˆ ˆ( ) 3 ; 2 ( ) ˆ ˆ ˆ( ) 2 3 ( ) ; 4 1 ˆ ˆ( ) 3 ( ) ; 2 1 ˆ( ) ( ) ; 4 i i i i i i i i i g j a g j c c c c c c g j a g j c c c g j a g j c c g j a g j c g j                                          (4.4b) and:      1 2 3( ) ( ); ( ) 2 ( ); ( ) 3 ( ); i i i i i i                  (4.4c) then, one immediately gets using (3.2), (3.24) and (4.4a-c): identification of nonlinear systems having hard function 71 copyright ©2019 assa. adv. in systems science and appl. (2019)               0 1 2 3 ( ) ( ) ( ) ( ) cos ( ) ( ) ( ) ( 2 ) cos 2 2 ( ) (2 ) ( ) ( 3 ) cos 3 3 ( ) (3 ) i i o i o i o i o i o i o x t a g j a g j g j t a g j g j t a g j g j t                                 (4.5) where des parameters  ( ) ika g j ( 0 ... 3k  ) are given by (4.4b). further, note that the inner signal ( )tx is periodic of the same period of ( )tu (i.e. 2 /t   ). accordingly, ( )tx can be expanded in series fourier, where its parameters (the amplitudes and arguments) can be easily generated using the estimate ( )ˆ tx . then, it readily follows from (4.5) and ( )ˆ tx that, 7 equations are provided. specifically, using the estimate of dc component 0a , the value of harmonic amplitudes ka ( 1... 3k  ), and the estimate of harmonic phases. accordingly, it readily seen from (4.4b) and (4.5) that, the modulus gain ( ) i g j can be determined using the dc component of ˆ( )x t . then, the modulus gains ( ) o g jk , for 1... 3k  , can be immediately estimated using the amplitude of the first 3 harmonics (three unknowns and three equations). furthermore, using the argument of the fundamental component and that of the first two harmonics, one has 4 unknowns (i.e. ( ) i   , ( ) o   , (2 ) o   , and (3 ) o   ) and 3 equations (see (4.5)). we have more unknowns than equations. the nonlinear system is thus excited with the sine input (3.16) with the frequencies 2 and 3 . it readily follows that, these experiments generate 9 equations (from the phases) involving 9 unknowns ( ( ), i   (2 ) i   , (3 ) i   , ( ) o   , (2 ) o   , (3 ) o   , (4 ) o   , (6 ) o   and (9 ) o   ). finally, an estimate of ( ) i l  and ( ) o kl  , for 1... 3l  and 1... 3k  , can be obtained. 0 100 200 300 400 500 600 700 0 50 100 150 200 250 time (s) y true system output fig. 4.5. the system output y(t) then, it follows from these three experiments that, for any frequency  , the estimate of modulus gains ( ) i g jl and ( ) o g jkl ( 1... 3l  and 1... 3k  ) can also be given. on the other hand, in the case where the linear elements are parametric, the coefficients of transfer functions (numerators and denominators) can easily be determined using these estimates [6,12]. among the advantages of the proposed method, the estimate of the gains ( ) i g jl ( ( ) i g jl and ( ) i l  ) and ( ) o g jkl ( ( ) o g jkl and ( ) o kl  ), for 1... 3l  and 1... 3k  , can be obtained using only 3 experiments. then, repeating this steps for 0.04 ( / )rd s  and 0.06 ( / )rd s  , i.e.: excite the system by the control (3.16) with these frequencies and estimate the corresponding ( )ˆ tx . this latter allows us to get the fourier parameters. finally, the gains ( ) i g jl and ( ) o g jkl ( 1... 3l  and 1... 3k  ) can easily be determined using (4.4b) 72 a. brouri, t. rabyi, a. ouannou copyright ©2019 assa adv. in systems science and appl. (2019) and (4.5). table 4.1 gives the estimated numerical values. the obtained results show that the estimated parameter values are very close to their true values. 0 50 100 150 200 250 300 50 100 150 200 250 time (s) the filtered output fig. 4.6. the stimated signal ˆ( )x t over one period of time table 2.1. estimate of linear elements and their true values ( / )rd s 0.02 0.04 0.06  ( ) , ( )i ig j   (0.97, -0.25) (0.92, -0.48) (0.85, -0.69)  ( ) , ( )o og j   (0.99, -0.14) (0.98, -0.27) (0.95, -0.41)  ( 2 ) , (2 )o og j    (0.98, -0.27) (0.91, -0.54) (0.83, -0.77)  ( 3 ) , (3 )o og j    (0.95, -0.41) (0.83, -0.77) (0.7, -1.08)  ˆ ˆ( ) , ( )i ig j   (0.98, -0.24) (0.9, -0.46) (0.87, -0.67)  ˆ ˆ( ) , ( )o og j   (1.01, -0.15) (0.96, -0.25) (0.98, -0.39)  ˆ ˆ( 2 ) , (2 )o og j    (0.96, -0.25) (0.93, -0.57) (0.80, -0.81)  ˆ ˆ( 3 ) , (3 )o og j    (0.98, -0.39) (0.80, -0.81) (0.74, -1.04) 5. conclusion in this paper, an identification method of nonlinear systems is proposed. presently, the nonlinear system can be described by wiener-hammerstein model. the nonlinear element is allowed to be discontinuous or of hard shape. it is interesting to point that, it is very complicate to decompose or approximate these types of nonlinearities using polynomial decomposition. the estimation of system parameters is done using two stage. firstly, the nonlinear block is determined using a simple sequence of constant controls. in the second stage, the linear elements parameters are estimated using sine signal. the identification method also features the fact that the linear elements identification is made decoupled from the nonlinear element identification. to the author's knowledge very few previous studies have been dealt with discontinuous or hard nonlinearity and nonparametric linear elements. identification of nonlinear systems having hard function 73 copyright ©2019 assa. adv. in systems science and appl. 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(2009). identification of wiener-hammerstein systems using ls-svms. 15th ifac symposium on system identification, saint-malo, https://doi.org/10.3182/20090706-3-fr-2004.00136. https://doi.org/10.1063/1.4981971 https://doi.org/10.3182/20140824-6-za-1003.00303 https://controls.papercept.net/conferences/conferences/cifa12/program/cifa12_contentlistweb_1.html https://controls.papercept.net/conferences/conferences/cifa12/program/cifa12_contentlistweb_1.html https://doi.org/10.3182/20140824-6-za-1003.00678 https://www.scopus.com/record/display.uri?eid=2-s2.0-85050194294&origin=resultslist&sort=plf-f&src=s&sid=76dfaa66776891fe92b39fedca761532&sot=autdocs&sdt=autdocs&sl=18&s=au-id%2823567866000%29&relpos=3&citecnt=0&searchterm= https://www.scopus.com/record/display.uri?eid=2-s2.0-85050194294&origin=resultslist&sort=plf-f&src=s&sid=76dfaa66776891fe92b39fedca761532&sot=autdocs&sdt=autdocs&sl=18&s=au-id%2823567866000%29&relpos=3&citecnt=0&searchterm= https://doi.org/10.3182/20090706-3-fr-2004.00136 74 a. brouri, t. rabyi, a. ouannou copyright ©2019 assa adv. in systems science and appl. 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(2009). estimation of generalised wienerhammerstein systems. 15th ifac symposium on system identification, saintmalo, 1104-1109, https://doi.org/10.3182/20090706-3-fr-2004.00183. https://doi.org/10.3182/20090706-3-fr-2004.00135 https://doi.org/10.3182/20090706-3-fr-2004.00135 https://doi.org/10.3182/20090706-3-fr-2004.00183 adv syst sci appl 2018; 02:84–92 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/542. segregation model for dynamic frequency allocation alexander kuznetsov1∗, 1voronezh state university, voronezh, russia abstract: we apply the schelling type ii segregation model to the dynamic frequency allocation. an algorithm is introduced for agents segregation over initially unknown radio channels. we relate the number of algorithm iterations until complete agents’ segregation to the number of agents, networks, and the other parameters via numerical experiment. also, there are some parallels with the continuous-time model of segregation used in sociology. keywords: segregation model, dynamic frequency allocation, cellular automaton 1. introduction nowadays, the number of robotic systems consisting of a large number of autonomous mobile agents (terrestrial robots, uavs, etc.) with wireless radio communication network constantly increases. agents interact with each other to perform a common task, for example, for emergency response, intrusion detection to a protected area, reconnaissance or networking in areas where it is difficult to organize a communication network in the usual ways [4] and so on. human-mediated deployment or restoration of the communication system of such agents can be extremely difficult – for example, in an emergency situation it can be too dangerous for human operators. for this reason, it is necessary to provide the agents themselves with all the logic necessary to maintain the communication network, initially setting only the basic principles of agents’ interaction. this problem is investigated in detail in [5], and below we introduce a simple model of such self-organization. briefly, agent periodically scans radio channels to discover other agents or stops at one of the channels and transmits a beacons sequence in order to be discovered by other agents. the problem we consider is one of cognitive radio ad hoc networks problems. the article [8] introduces dynamic spectrum access (dsa) which uses spectrum policy reasoning to determine allowed frequencies, requesting sensing periods on those frequencies, classifying the results from sensing events, then providing the list of allowed frequencies for use in frequency assignment. the papers [3, 10] describe different physical and informational aspects of the dynamic spectrum access. according to these works, two main approaches to dsa exists: dynamic spectrum allocation and opportunistic spectrum access. dynamic spectrum allocation exploits temporal and spatial traffic statistics and aims at improving spectrum efficiency through timeand space-dependent spectrum sharing among coexisting radio services. different from dynamic spectrum allocation, which uses the statistics of spectrum occupancy, opportunistic spectrum access uses the instantaneous spectrum availability by opening the licensed spectrum to secondary users. the idea is to allow ∗corresponding author: avkuz@bk.ru segregation model for dynamic frequency allocation 85 secondary users to identify available spectrum resources and communicate opportunistically in a manner that limits the level of interference perceived by primary users. this article is motivated by the question, whether it is possible to create a completely autonomous cognitive radio network, without any base station or initial spectrum allocations, or coordination from a base station is necessary for agents to distribute over radio channels in a finite number of iterations? the purpose of the present article is the attempt to disconnect network self-organization from its radio-physics ground and to study it as a purely mathematical segregation problem. the dynamical network organization is, in its essence, a segregation model related to the schelling segregation model [9]. the schelling cellular automaton (fig. 1.1) is a grid where each cell corresponds to a person. a person can have one of several “colours” that symbolizes her belonging to a particular social group. colours are distributed randomly on the grid, and there is also some amount of unoccupied cells. every person wants a certain part of people around him to be like him. if the number of adjacent cells of the same colour is below the specified threshold, then the person goes to the next free cell, otherwise, it remains in place. however, unlike the two-dimensional classical schelling model, the proposed automaton is practically one-dimensional: we need the coordinate n of an agent only to simplify the description of the dynamics of the automaton. fig. 1.1. the schelling segregation model 2. the cellular automaton model denote as id ⊂ z, 0 ∈ id the set of possible agent’s identifiers, asq ⊂ n the set of channel qualities. definition 2.1: let us define the set of channels as the simply connected domain c ⊆ z2 where each (n, f) ∈ c associated with the cell (qnf , idnf ) ∈ q× id and qkf = qlf for every (k, f), (l, f) ∈ c. we assume that if idnf = 0 there is no agent in the cell (n, f). therefore, the f th column of cells {(i, f)} corresponds to the f th channel. the number nf of cells in the f th column corresponds to the maximum possible number of agents using the f th channel. definition 2.2: let us call the agent the following vector ag = (idag, colag, fself , fold, nself ; ftarget, rag,qag; sensed; t), copyright c© 2018 assa. adv syst sci appl (2018) 86 a. kuznetsov where idag ∈ id is the unique agent’s identifier, fself is the current channel number, nself is the position of the agent on the current channel fself , ftarget is the number of a target channel, fold is the auxiliary variable. further, colorag ∈ col is the agent’s colour, rag : col × id → {0, 1} is the grouping predicate, qag : q → [0, 1] is the agent preferences function, t is an agent’s overall functioning discrete time. the set of all agents we denote as ag. the sensed is the agent’s knowledge of environment (see below). define the predicate r : ag ×ag → {0, 1} as following r(ag1, ag2) = rag1(colag2 , idag2), ag1, ag2 ∈ ag. if agents ag1, ag2 ∈ ag need to occupy the same channel, then r(ag1, ag2) = 1, otherwise r(ag1, ag2) = 0. it is clear, rag1 should be defined that rag1(ag2) = rag2(ag1) for any ag1, ag2 ∈ ag. note that r determines the communication graph for ag. definition 2.3: let p, x are functions. we shall write p ∼ x, if it exists such monotonically non-negative increasing function ϕ that p(x) = ϕ(x). for simplification, assume that c = {(f, n)|f = 1, fmax, n = 1, nmax}. also, we denote qnf = qf , agf ⊆ ag is the set of agents on the channel f (i.e. with fself = f ), agag = {ãg ∈ ag|r(ag, ãg) = 1} is the set of agents from the same network as the agent ag, and agag f = {ãg ∈ agf |r(ag, ãg) = 1} is the set of agents on the channel f from the same network as the agent ag. discovering of agents can be unsuccessful. by this reason, introduce the function d : 2ag → 2ag . the agent ag ∈ agf falls into the d(agf ) with probability p1 ∼ qf . our automaton will function in a discrete time. an agent’s tact behaviour can be described by the following algorithm. 0. initialization. the agent ag scans all channels in c and randomly selects (n, f) with probability p0(n, f) ∼ qag(qnf ) and such that the cell (n, f) is not already occupied. 1. sensing. the agent ag senses the channel f . the agent sets fself := f, nself := n, t := t+ 1, sensed[f ] := {qf ,d(agf )}. 2. if fself = fmax, then the agent chooses a channel f̃ from sensed with probability p0(n, f). for example, p0 can be defined so that the agent selects the channel with the best quality. the agent sets ftarget := f̃ . 3. decision. if |d(agag f )| = |ag ag|, then go to the step 6, else if |d(agag f )| > |agag| α0 ∧ |d(agag f )| > |d(agf )| − |d(ag ag f )|, α0 > 1, then go to 5, else go to 4. 4. channel change. the agent ag memorizes the current frequency fold := fself copyright c© 2018 assa. adv syst sci appl (2018) segregation model for dynamic frequency allocation 87 and chooses a new frequency fself := fself + dir, 1 ≤ fself + dir ≤ fmax, fmax, fself + dir = 0, 1, fself + dir > fmax. if an arbitrary coordinate n such that the cell (fself , n) is unoccupied exists, the agent chooses the cell (fself , n) and occupies it. if there is no such cell, then the agent does not occupy any cell on this turn and remains on its old cell. in this case, the agent sets fself := fold, go to 1. if fself = ftarget, then go to 5 else go to 1. 5. waiting for agents. the agent waits twait = α1fmax + α2τ(|d(agag f )|, |ag ag|, |d(agf )|), α1 ≥ 0, α2 ≥ 0, (2.1) τ(x, y, z) ∼ min { x y , x z } , (2.2) turns and selects dir randomly from the set {−1, 1}. go to 1. 6. wait. the main problem is to find model’s parameters αi, i = 1, 3 and a function τ such that the segregation process would complete in a reasonable time. we should note that an unlucky choice of parameters will result in the algorithm not converging at all. physically, agents get information about other agents and about a state of channels, scanning the spectrum and exchanging beacons, as described in [5]. one tact of the cellular automaton corresponds to the full cycle of the pilot and beacons exchange. a detailed technical solution corresponding to the proposed model is described in the patent “telecommunication network data transmission means and telecommunication network” # ru 2 549 120. fig. 2.1. final configuration of the automaton the example of the automaton’s final state is shown on fig. 2.1. the lighter tone of cells in the picture corresponds to the better quality of corresponding channels. we can see the network net1 = {agi|i = 1, 4} formed on the channel f1, and net2 = {agi|i = 5, 6} formed on the channel f2. copyright c© 2018 assa. adv syst sci appl (2018) 88 a. kuznetsov 3. numerical experiment and discussion we performed simulation of the proposed algorithm with the “psychohod” simulator [7]. for these purposes, we introduced the new operating mode of the program (see fig. 3.1). fig. 3.1. “psychohod” program in the process of agents segregation we defined p1 = 1, the grouping predicate as the following r(ag1, ag2) = { 1, colag1 = colag2 , 0, colag1 6= colag2 . fig. 3.2 contains simulation results for 5 networks, 50 agents in each network, and for α1 = fmax + 1, α2 = (fmax + 1)β, τ(x, y, z) = min { x y , x z } , where α1, α2, τ are parameters from (2.1), (2.2). ordinates correspond to the average discrete time t , abscissas correspond to the α0. channels qualities q = 1, 9 were distributed uniformly over fmax = 100 channels, 100 experiments for each point were performed. fig. 3.2. average time to complete network segregation (50 agents per net, 5 nets) copyright c© 2018 assa. adv syst sci appl (2018) segregation model for dynamic frequency allocation 89 the simulation stopped when the segregation process was completed or when t > 30000. the points encircled correspond a case when for all 100 experiments the segregation was completed in less than 30000 turns. for other points, the segregation process was completed in 79–99% of experiments. fig. 3.3 represents histograms of segregation’s completion times for different values of α0 and β. we used function t (α0) = a+ b α0 − c , c > 0, b > 0 for approximations of data points. fig. 3.3. histograms of segregation’s times next, we fixed α0 = 2.2, β = 30 and performed experiments with different numbers of nets and agents in each net as described in table 3.1. note that segregation considered table 3.1. unsuccessful segregation percentage agents in net 10 20 30 40 50 60 70 80 90 100 unsuccesses number, five nets 37 15 3 4 1 0 0 5 9 30 unsuccesses number, eight nets 97 75 59 33 22 2 1 18 29 100 unsuccessful if it takes more than 30000 turns. therefore, we should change α0 and β proportional to the number of agents. for example, if we have 25 agents in each of five nets, we need to reduce β approximately in four times and slightly increase α0 (see fig. 3.4) to minimize the average segregation time. results of the numerical experiment immediately suggest an improvement of the previously proposed algorithm. we should add the following step: 2.5. the elapsed time check. if t > tr, then change αi, i = 0, 2 on arbitrary small values. also, we can memorize all changes of algorithm’s parameters and gradually choose the optimal ones. the further improvement can be dividing of the set of channels into subsets so that agents from one net would tend to search channels in the channel subset associated with their net number. copyright c© 2018 assa. adv syst sci appl (2018) 90 a. kuznetsov fig. 3.4. average time to complete network segregation (25 agents per net, 5 nets) the proposed cellular automaton is close to the so-called schelling type ii model. works [1, 2], which introduce continuous-time schelling dynamical system, characterize schelling ii model as the following. let the population is partitioned into disjoint sets, akin to the different districts in a city. the population is divided into “types”. let x and y denote the two types, inhabiting a space which is partitioned into m areas, denoted ai, i = 1, 2, ...,m. also, let xi and yi denote the total x-type population and y -type population in ai, |x| and |y | the total of each type in the population, and n = |x|+ |y | the total population. tolerances are allocated to a given type in a given area via a “tolerance schedule”. this function describes the maximum x-type population that would tolerate up to r(x)xi members of type y in the same area, where r(x) is necessarily monotone decreasing. it is also assumed that there is no lower bound on tolerance, i.e. no population insists on the presence of the opposing type. the simplest, single-area continuous-time schelling dynamical system has the following form: dx dt = [xrx(x)− y]x, (3.1) dy dt = [yry (y)− x]y, (3.2) where x, y are densities of populations x , y , |x| = k|y |, p > 0, and rx , ry are monotone decreasing tolerance schedules’ functions. here, for example rx(x) = a(1− x)p, (3.3) ry (y) = b(1− ky)p, (3.4) a > 0, b > 0, p > 0, k = |x|/|y |. we have found, that our automaton’s behaviour in some cases can be described by a model like (3.1), (3.2). denote as x(t) = xf (t)/|x(t)| the density of agents of the net 1 on the network channel f . let’s also denote y(t) = yf (t)/|y (t)| the density of agents of all other networks on the network channel f . as we have 50 agents per net and 5 nets, k = |x|/|y | = 1/4, x(18) = 1/50, y(18) = 1/200. at first, select p = 1.7, a = 1, b = 1/2. copyright c© 2018 assa. adv syst sci appl (2018) segregation model for dynamic frequency allocation 91 we can modify (3.1), (3.2) in the way described in [1], using the exponential schedule rx(x) = e−6x − e−6 1− e−6 , (3.5) ry (y) = e−6y/k − e−6/k 2(1− e−6/k) , (3.6) to obtain better fitting. we can see the comparison of experimental points with the solutions of schelling dynamical system on fig. 3.5, left. circular and diamond-shaped markers correspond to experimental values of x and y. the black dotted line corresponds to the linear schedule (3.3), (3.4), solid lines correspond to the exponential schedule (3.5), (3.6). interestingly enough, that such initial condition exists which can provide oscillatory behaviour for xf . this is the case of approximately equal initial numbers of agents of different nets at the channel f (see fig 3.5, right). unfortunately, the model (3.1), (3.2) can not explain this case at whole, and we should use more complex multi-area segregation model, but it is possible to model oscillations by multiplying (3.5), (3.6) with a periodic function. the aforesaid situation entails the potential impossibility of the network’s self-organization in a reasonable time. fig. 3.5. graphs of numerical experiment’s results in comparison with the solutions of schelling dynamical system 4. connection with the agents’ motion model previously, the author had developed the model of agents’ motion and conflict and related simulator “psychohod”. groups of agents move through a rough terrain and can destroy other groups of agents. this simulator can export data with coordinates of communicating agents, the list of destroyed agents, and landscape obstacles to communication. the data exported used by the communication simulator which contains described above automaton as well as by other telecommunication models (fig. 4.1). optionally, the communication simulator can export the message exchange data to a network simulator like ns-3 or omnet++. 5. conclusions we proposed the simulation of the automatic initial channels distribution in a wireless network. also, we studied the dependence of the channels distribution time on various parameters like the nets’ number, the agents’ number, the channels’ scanning speed. at last, we compared the simulation results with the solution of schelling dynamical system. we found that the proposed network self-organization is similar to social segregation schelling type ii models. copyright c© 2018 assa. adv syst sci appl (2018) 92 a. kuznetsov fig. 4.1. conjugation of the agents’ motion simulator, the communication simulator, and (optional) a network simulator in the future, we are planning to design an adaptive algorithm providing minimization of the channels distribution time, a model with relay agents based on the “psychohod” simulator, and to obtain numerical characteristics of aforesaid models. it is also planned to provide agents with additional characteristics, such as a memory of channels visited, and link the proposed model of self-organization of the communication system with the model of the movement of agents [6], previously developed by the author. it seems promising to compare the proposed cellular automaton with more complex continuous-time segregation models to explain the occurrence of the chaotic or oscillatory behaviour of the system. this work was supported by rfbr grant 18-07-01240 a. references 1. haw, d. (2016). measuring and understanding segregation (doctoral dissertation, bristol centre for complexity sciences). 2. haw, d. j., & hogan, j. (2018). a dynamical systems model of unorganized segregation. the journal of mathematical sociology, 42(3), 1-15. doi: 10.1080/0022250x.2018.1427091 3. horne, w. d. (2003). adaptive spectrum access: using the full spectrum space. in proc. of the international symposium on advanced radio technologies. 4. jones, m., & dudenhoeffer, d. (2000). a formation behavior for largescale micro-robot force deployment. in winter simulation conference (vol. 01, p. 972-982). los alamitos, ca, usa: ieee computer society. doi: doi.ieeecomputersociety.org/10.1109/wsc.2000.899900 5. kuznetsov, a. (2017). allocation of limited resources in a system with a stable hierarchy (on the example of prospective military communications system). large-scale systems control, 66, 68–93. 6. kuznetsov, a. v. (2017). a simplified combat model based on a cellular automaton. journal of computer and systems sciences international, 56(3), 397–409. doi: 10.1134/s106423071703011x 7. kuznetsov, a. v. (2018). model of the motion of agents with memory based on the cellular automaton. international journal of parallel, emergent and distributed systems, 33(3), 290-306. doi: 10.1080/17445760.2017.1410819 8. redi, j., & ramanathan, r. (2011, nov). the darpa wnan network architecture. in 2011 milcom 2011 military communications conference (p. 2258-2263). doi: 10.1109/milcom.2011.6127657 9. schelling, t. c. (1971). dynamic models of segregation. the journal of mathematical sociology, 1(2), 143-186. doi: 10.1080/0022250x.1971.9989794 10. zhao, q., tong, l., & swami, a. (2005). decentralized cognitive mac for dynamic spectrum access. in first ieee international symposium on new frontiers in dynamic spectrum access networks, 2005. dyspan 2005. (p. 224-232). doi:10.1109/dyspan.2005.1542638 copyright c© 2018 assa. adv syst sci appl (2018) advances in systems science and application (2015) vol.15 no.4 326-337 biodiversity of island ecosystems of the northern and middle caspian and a new outlook at the islands age and the caspian sea level regime g. m.abdurakhmanov, a.a.teymurov and a.a. gadzhiyev dagestan state university, institute of applied ecology, makhachkala, russia. abstract species diversity of some biota taxonomic groups of the larger islands (chechen, tyuleniy, nordovyy, kulaly) of the middle and northern has been established by the research expeditions of 2009-2013. endemic taxa of species and subspecies rank have been found in the flora and fauna of this island group. the results of the gis simulation (of the map), showing the water surface configuration under the sea level lowering have been presented and different approaches to explain the fluctuations in the caspian sea level and the mechanisms of islands and evolution and depositional bench lands of the northern and middle caspian have been briefly described in the article. the supposition on the presence of island firm land throughout the holocoen (even during the highest sea level) has been made on the basis of analysis of the island biota composition and the causal explanation of the endemic taxa presence in the water area of the middle and northern caspian. keywords the caspian sea; biodiversity; coastal and island ecosystems; level fluctuation,autochthonous speciation. 1 introduction level regime of the caspian sea is an issue to be much discussed in the scientific printed matter in the theoretical and applied perspective. there were a variety of views on the causes of the fluctuation in this sea-lake [1–4] and forecast expectations of its level regime [5-17]. in these and other papers different hypotheses on the causes of the caspian sea level regime instability are grouped together in the geological, hydrogeological, climatic and technological conceptions. we do not set a goal to give a detailed analysis of the causes and forecasts of the caspian sea level fluctuations. it should be noted that the climate hypotheses, in our opinion, are of greater importance. our interest to the caspian sea level fluctuations is connected with the necessity to give a causal explanation of the identified taxonomic characteristics of the caspian sea island biota. 2 materials and methods during the 2009-2013 large-scale comprehensive studies of flora and fauna of the coastal and island ecosystems of the northern and middle caspian were made jointly by ecology and geography department of dagestan state university and advances in systems science and application (2015) vol.15 no.4 327 institute of applied ecology of the republic of dagestan. the research covered the western and eastern coasts, as well as the large islands of this part of the sea. the main focus of the expedition was to study the specific diversity of flora and fauna. faunal material processing was performed in the zoological institute of the russian academy of sciences (saint petersburg). table 1 summary table of species diversity of the island middle and northern caspian no. taxa number of genera number of species new for the science new for russia 1 darkling beetles (coleoptera, tenebrionidae) 127 341 1 species 2 owletmoths (lepidoptera, noctuidae) 279 902 1subfamily 1genus 3 carabid beetles (coleoptera, carabidae) 98 608 1 genus 1species 4 spiders (aranei) 131 290 2 species 5 click beetles (coleoptera, elateridae) 6 12 6 oribatid mites (acariformes, oribatida) 39 49 2 species 12 species 8 snout beetles (coleoptera, curculionidae) 127 318 9 orthopterous insects (orthoptera) 24 30 10 dung beetles (coleoptera, scarabaeidae) 133 363 1 species 1 subspecies 11 higher plants (cormophyta) 186 269 1 species 2 species total: 1150 3182 7 20 3 results as a result of this processing work of floristic (higher plants) and fauna (some groups of invertebrates) field data, the general picture of species diversity of large islands (chechen, tyuleniy, nordovyy, kulaly) of the north and middle caspian was determined. firstly, it should be emphasized that the biota of the islands, in general, consists of taxa which are widely spread in the eastern and, to a lesser extent, in the western coast of the sea. there are also species which extend far beyond the caspian sea region. however, the fauna and flora data given in table 1 show that new species currently unknown for the science in 328 g. m.abdurakhmanov, a.a.teymurov and a.a. gadzhiyev:biodiversity of island... the continental part of the caspian are proven to have appeared in the modern biota of the caspian sea islands. these are taxa of species or subspecies rank. considering the above said, the modern floral or faunal status of the given species group is to be recognized as endemic. formation and evolution stabilisation of these taxa requires time sufficient for the diagnostically important features to emerge. undoubtedly, the present-day configuration of island biota taxa natural habitat of the middle caspian depends on the scale and time duration of transgressive and regressive sea cycles and on the causes predetermining those events. in line with the nature of these cycles changes in the species composition and structural organisation of the island biocenoses were taking place. the need for a causal interpretation of the present-day configuration of the natural habitat of the biota species composition of the islands investigated and the endemic taxa presence made us develop a hydrodynamic gis model of the caspian. to solve this problem, a three-dimensional model of the caspian sea has been made in map 2011 gis. matrices keeping the hydrodynamic gis models functioning are based on approximately forty thousand depth soundings. the base level of the caspian sea in this simulation is -28 m point. below (fig.1-10) is the configuration of the sea water surface at different points of sea level lowering. the analysis of these materials allows us to make some important points: 1. the catastrophic reduction of the sea water plane square takes place with a gradual lowering of its level up to -38 m (fig.6). 2. at the level of -30 m by the islands of chechen, nordovyy, kulaly overland connection with coastal land appears (fig.2). island tyuleniy has the same connection between -31 and -32 m (fig.3). 3. between -32 and -33 m within the ural furrow an independent pool is differentiated (fig.4). 4. from the level of -34 m the baring of underwater bench land chain from the area of chechen island in the direction of kulaly island begin to show (fig.4,5). 5. at the level of -38 m a separate pool of ural furrow disappears (fig.6). 6. at the level of -40 m the northern caspian is completely drained (fig.7). 7. a further drop in the level up to -45 and -50 m does not result in a large reduction of the water plane (fig.8). 8. a further drop in the level up to -100 m or to -150 m, does not result in a critical reduction of the water plane (fig.9,10). advances in systems science and application (2015) vol.15 no.4 329 4 discussion as it was previously shown [18], a causal interpretation of autochthonous profiles of the water biota of the caspian sea and, as a consequence, high levels of endemism of its taxa does not present a problem. a different situation arises with understanding and explaining the endemism pattern of the coastal and especially island taxa. the modern faunal or floral status of the new species for science is to be considered as endemic. the species and subspecies rank taxa were identified in this group of species. for those taxa to be formed and evolutionally stabilized sufficient time is required for diagnostically important features to appear. it is obvious that the modern natural habitat configuration of the island biota taxa of the middle caspian depends on the scale and the duration of transgressiveregressive sea cycles and the causes that predetermine the course of the events. according to the nature of these cycles the change in the species composition and structural organisation of the coastal island cenoses was taking place. the shelf area of the north caspian has a variety of numerous accumulative bench lands, shoals and islands. yet our attention is drawn to another feature in their distribution: they are found and grouped exclusively in the field of sea zone extensions of prikumskaya uplift zone and tersko-caspian trough. the same is observed in the eastern part of the northern caspian. according to o.k. leontyev [19], the concentration of alluvia is associated with the local uplifts. n.a. kasyanova [20] believes that concentration of alluvia is confined to the tectonically active currently local uplifts, proving this by the fact that along the sea extension of anticlinal zones of karpinski range there also was found a series of local uplifts and that accumulative bench lands here are not present (or their forms are few). this can be explained taking into account the features of modern geodynamics of those areas: prikumskaya uplift zone has a higher current geodynamic activity compared with that of karpinski range [20]. however, the high latest tectonic activity of the local uplifts, located within the sea extension of karpinski range proven by the power reduction of the upper levels in almost all uplifts should be noted [21]. while not rejecting the genetic connection between the accumulative bench lands and islands ye.n. badyukova et al. propose a different interpretation of the previous data and, accordingly, a new version of those accumulative forms origin. [22] she believes that the zonal arrangement of bench lands and islands is conditioned by the location of the ancient coastlines of the various stages of the new caspian transgression. some scholars proposed to reconstruct the pleistocene caspian basin based on summarizing the results of the malacofaunal analysis and the data of the comprehensive study of the caspian region sediments.t.a. yanina proposes another 330 g. m.abdurakhmanov, a.a.teymurov and a.a. gadzhiyev:biodiversity of island... sequence of the pleistocene events on the basis of the factual data collected over the years of field, and laboratory investigations of the key sections of the pleistocene sediments and the caspian malacofauna localities. [23]the main attention is paid to the leading for the caspian sea and endemic for the ponto-caspian brackish-water shell-fish of didacna eichw genus, which features rapid evolutionary development at the species and subspecies level played the essential role of the genus for stratification of the sea pleistocene of the caspian sea and paleogeographic reconstructions of its basins. to monitor the results of the malacofauna analysis they used a conjugated method of the latest sediments investigation and events reconstruction. the possibility of islands surface absolute altitude increase following the banked up water level [19] suggests that even during the periods of high water levels in the north caspian sea there were island lands where the process of new species formation took place. during the holocene (the last 10 thousand years), the level of the caspian sea did not rise above -20 m [23]. if it was so tyuleniy, chechen and possibly kulaly islands and the adjacent islands existed all this time since the highest points of the island rise above the present sea level by 5-8 metres and the emergence of the continental links of the western and eastern shores secured a rich representation of turanian species (up to 30-60%) in northern dagestan and the variety of reliquiae of sarykum sand dune with its typical central asian flora and fauna (to the level of subendemics). fig. 1 sea surface configuration at the level of 28,6 m. advances in systems science and application (2015) vol.15 no.4 331 fig. 2 sea surface configuration at the level of 30 m. fig. 3 sea surface configuration at the level of 32 m. 332 g. m.abdurakhmanov, a.a.teymurov and a.a. gadzhiyev:biodiversity of island... fig. 4 sea surface configuration at the level of 34 m. fig. 5 sea surface configuration at the level of 36 m. advances in systems science and application (2015) vol.15 no.4 333 fig. 6 sea surface configuration at the level of 38 m. fig. 7 sea surface configuration at the level of 40 m. 334 g. m.abdurakhmanov, a.a.teymurov and a.a. gadzhiyev:biodiversity of island... fig. 8 sea surface configuration at the level of 50 m. fig. 9 sea surface configuration at the level of 100 m. advances in systems science and application (2015) vol.15 no.4 335 fig. 10 sea surface configuration at the level of 150 m. acknowledgements the study was carried out with support of the ministry of education and science of the russian federation, agreement no. 14.574.21.0109 (a unique identifier for applied scientific researches (project) rfmefi57414x0032). references [1] berg l.s. (1934),“the level of the caspian sea in the historical time”, issues of physical geography, no.1, pp.11-64. [2] lilienberg d.a.(1994),“new approaches to the modern endodynamics evaluation of the caspian region and the issues of its monitoring ”, news of the russian academy of sciences. series geographer, no.2, pp.16-36 [3] rychagov g.i. (1997), the pleistocene history of the caspian sea , publishing house of mgu, pp.267. [4] varushenko s.i. varushenko a.n. and klige r.k.(1987),“ change in the caspian sea and stagnant basins in palaeotime”, nauka, pp.240. [5] apollov b.a. (1935),“water balance of the caspian sea and its possible changes” tr. tsiegm , issue vol.2, no.44, pp.11-18. 336 g. m.abdurakhmanov, a.a.teymurov and a.a. gadzhiyev:biodiversity of island... [6] bregman g.r. mikhalevsky a.i. (1935,“water balance of the caspian sea in connection with the great volga ”, news of the academy of sciences of azssr. physics-chemistry series.,vol.19, pp.7-138. [7] kritskiy s.n. korenistov d.v. and ratkovich d.ya, (1975),fluctuations of the caspian sea level, nauka, pp.160. [8] shlyamin b.a., the caspian sea, geografgiz, pp.128. [9] kislov a.v., panin a., toropov p.(2014),“current status and palaeostages of the caspian sea as potential evaluation tool for climate model simulations ”, quaternary international, pp.1-8 [10] bolgov m.v. filimonova m.k. (2005),“on the sources of uncertainty in forecasting the caspian sea level and evaluation of flooding risk of the coastal areas” , water resources, no.6, pp.664-669. [11] budyko m.i., yefimova i.a., v.v. lobanov (1988),“the future level of the caspian sea”, meteorology and hydrology , no.5, pp.86-94. [12] kroonenberg s.b., abdurakhmanov g.m., badyukova e.n., van der borg k., kalashnikov a., kasimov n.s., rychagov g.i., svitoch a.a., vonhof h.b.and wesselingh f.p., (2007), “solar-forced 2600 bp and little ice age highstands of the caspian sea ”, quaternary international , vol.173, pp.137143 [13] kroonenberg s.b., kasimov n.s. and lychagin, m.yu, (2008),“ the caspian sea: a natural laboratory for sea-level change”, geography, environment, sustainabilility ,vol.1, pp.22-37 [14] mikhailov v.n. “mysteries of the caspian sea”. http://journal.issep.rssi.ru/articles/pdf/0004s 063.pdf soros educational journal. [15] ratkovich d.ya., bolgov m.v. (1994),“ study the probable regularities of the long-term fluctuations in the caspian sea level”, water resources, no.6, pp.389-404. [16] shlyamin b.a.(1962),“super long-term forecast the caspian sea level”, news of vgo ,vol.94, no.1 ,pp.26. [17] abdurahmanov g.m., teymurov g.a., shokhin i.v., nabozhenko m.v., alieva s.v., eskendarova s.n. and elderkhanova z.m., (2010), “the caspian region: environmental consequences of the climate change”, moscow: faculty of geography, pp.60-64. advances in systems science and application (2015) vol.15 no.4 337 [18] abdurakhmanov g.m., karpyuk m.i., morozov b.n. and puzachenko yu.g. (2002),“current status and the factors influencing the biological and landscape diversity of the volga-caspian region of russia”, nauka,pp.416. [19] leontyev d.c.(1957),“ on the origin of some islands of the northern part of the caspian sea”, tr. oceanographer, commission of the ussr academy of sciences,vol. 2, pp.147-158 [20] kasyanova n.a.(1998), “new data on the structure and oil and gas content of the north-west caspian sea”, oil and gas geology,no.4,pp.10-16. [21] ulitsky yu.a. and turaev i.a. et al.(1967),“the main features of the structure of the upper-pliocene quaternary sediments of the north-west caspian region in connection with the finding the mesozoic structural plan features ”, structural-geomorphological investigations when studying oil and gas basins. l. pp.180-186. [22] badyukova ye.n., varushchenko a.n. solovyova g.d. (1996), “on genesis of the bottom topography of the northern caspian”, bulletin of moip. geology department. ,vol.71, no.5, pp.80-88. [23] yanina t.a., “paleogeography of the caspian sea in the pleistocene ”, geology of the oceans and seas: proceedings of the 18th international scientific conference (school) on sea geology, m: geos, vol.3, pp.355-360. corresponding author tatiana can be contacted at: yal05@mail.ru adv syst sci appl 2020; 04:125–131 published online at https://ijassa.ipu.ru. on some global properties of multivalued simple waves dmitry v. tunitsky1 1v.a. trapeznikov institute of control sciences of ras, profsoyuznaya 65, 117342 moscow, russia abstract: the cauchy problem for one-dimensional quasilinear wave equation is considered. in the case that its solutions are multivalued simple waves, we derive explicit expressions in quadratures by introducing global characteristic coordinates. the main result of the paper is a theorem on some global properties of multivalued simple waves. keywords: hyperbolic quasilinear wave equation; multivalued solution; cauchy problem; simple wave introduction a number of models in continuum mechanics concern the wave equation zxx − g2(zy)zyy = 0, (0.1) where g = g(q) is a given positive coefficient, g(q) > 0, (0.2) cf. [1], [2], and [3], and the monge notations p = zx(x, y), q = zy(x, y) (0.3) are used. the variable x designates time, y is the spatial coordinate, and z is the displacement of continuum. in particular, vibration of a string, cf. [3], [1]; wave propagation in a bar of elastic-plastic material, cf. [4]; and isentropic flows of a compressible gas with plane symmetry, cf. [4], [5], are described by equations of the type (0.1). equation (0.1) is a quasilinear second-order partial differential equation with two independent variables x and y and unknown function z = z(x, y). (0.4) due to condition (0.2) it is hyperbolic. it is common knowledge what is a classical solution of equation (0.1). classical solutions have a rather serious drawback: in finite time singularities, a so called gradient catastrophes, can develop in them, cf. [2] and [1]. the latter means that there is a point (x, y) such that in some its vicinity a classical solution z (0.4) itself and its first derivatives p and q (0.3) are bounded but at least one of second derivatives is unbounded. this is a motivation and reason to generalize a notion of classical solution and define the notion of a multivalued solution, which is a relatively well-known one, cf. [6], [7], [8], and [9]. ∗corresponding author: dtunitsky@yahoo.com 126 d.v. tunitsky 1. multivalued solutions the differential 2-form ω2 = dp ∧ dy − g(q)dx ∧ dq, (1.5) is in an obvious way associated with the left hand side of the equation (0.1), cf. [8], and the linear differential form and its exterior derivative ω0 = dz − pdx− qdy, ω1 = dω0 = dx ∧ dp+ dy ∧ dq (1.6) are associated with equalities (0.3). an immersion σ : s −→ r5 (1.7) of a two-dimensional hausdorff paracompact manifold s is a multivalued solution of equation (0.1) if it satisfies the following system of exterior differential equations σ∗ω0 = 0, σ∗ω1 = 0, σ∗ω2 = 0, (1.8) cf. [7] and [8]. obviously, the graphic of a classical solution is a multivalued solution but not vice versa. and it can be proven that any classical solution is a part of maximal multivalued solution, cf. [10]. if a gradient catastrophe takes place at a point s ∈ s, then (dσ∗x ∧ dσ∗y) (s) = 0. multivalued solutions have a serious advantage over classical solutions because gradient catastrophes do not happen to them; cf. [10]. the linear differential forms ωj,1 = dp− (−1)jg(q)dq, ωj,2 = dy − (−1)jg(q)dx (1.9) will be called characteristic. it is not hard to see that the equalities ω2 − g(p)ω1 = ω1,1 ∧ ω1,2, ω2 + g(p)ω1 = ω2,1 ∧ ω2,2, ω2 = 1 2 (ω1,1 ∧ ω1,2 + ω2,1 ∧ ω2,2), ω1 = 1 2g(p) (ω1,1 ∧ ω1,2 − ω2,1 ∧ ω2,2) are through for characteristic forms ωj,1, ωj,2 (1.9) and 2-forms ω2 (1.5) and ω1 (1.6). thence, an immersion σ (1.7) is a multivalued solution of the equation (0.1) iff σ∗ω0 = 0, σ∗(ω1,1 ∧ ω1,2) = 0, σ∗(ω2,1 ∧ ω2,2) = 0. (1.10) a curve γ : γ −→ r5, where γ is a one-dimensional connected manifold, will be called a characteristic curve of the equation (0.1) that belongs to the j-th family, j = 1, 2, if γ∗ω0 = 0, γ∗ωj,1 = 0, γ∗ωj,2 = 0. (1.11) suppose an immersion σ (1.7) is a multivalued solution of equation (0.1). then equations (1.10) are through for this immersion. hence the pull-backs σ∗ωj,1 and σ∗ωj,2 of the copyright c© 2020 assa. adv syst sci appl (2020) multivalued simple waves 127 characteristic forms ωj,1 and ωj,1 (1.9) are linearly dependent and the linear algebraic equations σ∗ωj,1 = 0, σ∗ωj,2 = 0 (1.12) uniquely define the one-dimensional subbundle of the tangent bundle ts for j = 1, 2. therefore, by the frobenius theorem [11], for any point s ∈ s and number j = 1, 2 there exists the maximal integral manifold γj,s : γj,s −→ s, (1.13) of the system γ∗j,sωj,1 = 0, γ∗j,sωj,2 = 0, (1.14) containing the point s, where γj,s is a connected one-dimensional manifold. it follows from the equations (1.11), (1.12), and (1.14) that the composition σ ◦ γj,s of the maps (1.7) and (1.13) is a characteristic curve of the equation (0.1), belonging to the j-th family. of cause, for classical solutions conventional characteristic curves are obtained this way. 2. cauchy problem an initial curve or initial value for equation (0.1) is a smooth curve l : (−∞,+∞) −→ r5 (2.15) that satisfies the conditions ω0(l̇(τ)) = 0, ω2 1,1(l̇(τ)) + ω2 1,2(l̇(τ)) 6= 0, ω2 2,1(l̇(τ)) + ω2 2,2(l̇(τ)) 6= 0 (2.16) for −∞ < τ < +∞. the last two conditions mean that the initial curve l (2.15) is not characteristic, i.e. it is free, see (1.11). a classical-type initial value problem at x = 0 for equation (0.1) is defined by conditions z(0, y) = z0(y), zx(0, y) = q0(y), (2.17) where z0 : (−∞,+∞) −→ r, q0 : (−∞,+∞) −→ r (2.18) are given functions. this initial values determine initial curve l (2.15) with the coordinates l∗x(τ) = 0, l∗y(τ) = τ, l∗p(τ) = p0(τ), l∗q(τ) = q0(τ), l∗z(τ) = z0(τ). (2.19) obviously, this curve is an immersion and meets conditions (2.16) since ω2 1,1(l̇(τ)) = ω2 2,2(l̇(τ)) = 1 copyright c© 2020 assa. adv syst sci appl (2020) 128 d.v. tunitsky according to expressions (1.9). if for a multivalued solution (1.7) of equation (0.1) there exists an imbedding l : (−∞,+∞) −→ s (2.20) such that l = σ ◦ l, (2.21) then σ is called a solution of cauchy problem (0.1), (2.15) and l – an initial embedding. thus a solution of the cauchy problem (0.1), (2.15) is a pair (σ, l) of an immersion σ (1.7), satisfying the system (1.10), and imbedding l (2.20), satisfying (2.21). a multivalued solution (σ, l) of the initial value problem (0.1), (2.15) is said to be definite if for any point s ∈ s and number j = 1, 2 the intersection γj,s(γj,s) ∩ l(−∞,+∞) of the image of initial imbedding l (2.20) and characteristic curve γj,s (1.13) consists of exactly one point. theorem 2.1: (characteristic uniformization, [10]) let (σ, l) be a definite solution (1.7), (2.20) of the cauchy problem (0.1), (2.15)–(2.16). then there exists a unique diffeomorphism φ : s −→ φ(s) (2.22) such that the following three properties hold. (a) the image φ(s) is a subset of r2 and contains its diagonal δ = {(τ, τ) ∈ r2| −∞ < τ < +∞}. (b) for −∞ < τ < +∞ the initial imbedding l (2.20) satisfies the equality φ ◦ l(τ) = (τ, τ). (c) images φ ◦ γj,s(τ) of characteristic curves γj,s (1.13), where s ∈ s, lie in coordinate straight lines u = const if j = 1 and v = const if j = 2 for τ ∈ γj,s. a diffeomorphism φ (2.22), which is uniquely defined by theorem 2.1, is called a characteristic uniformization of the definite solution (σ, l). the coordinate plane of the parameters u, v and the same coordinates in it are called characteristic as well. the image φ(s) of the uniformization φ (2.22) is a uniformized domain and the composition ψ = σ ◦ φ−1 (2.23) is a uniformized solution. 3. simple waves a solution σ (1.7) is said to be a simple wave, if the equality dσ∗p ∧ dσ∗q = 0 (3.24) holds for it, cf. [4]. it is possible to show that a definite solution (σ, l) of the cauchy problem (0.1), (2.17) is a simple wave iff either p0(y) = −g(z′0(y)), (3.25) or p0(y) = g(z′0(y)), (3.26) where g = g(q) is a primitive function of g = g(q). copyright c© 2020 assa. adv syst sci appl (2020) multivalued simple waves 129 by definition (3.24), in case of simple waves linearization of equation (0.1) by means of hodograph transformation is not applicable. but it is possible to apply theorem 2.1 and use characteristic uniformization φ (2.22) in this case instead. indeed, by theorem 2.1, definition (1.13)–(1.14) of characteristic curve, and initial values (3.25) and (3.26) we get for coordinates x, y, p, q, and z of uniformized solution (2.23) the following equations ∂ ∂v y ψ∗ω1,1 = pv + g(p)qv = 0, ∂ ∂v y ψ∗ω1,2 = yv + g(p)xv = 0, ∂ ∂v y ψ∗ω0 = zv − pxv − qyv, ∂ ∂u y ψ∗ω2,1 = pu − g(q)qu = 0, ∂ ∂u y ψ∗ω2,2 = yu − g(p)xu = 0, ∂ ∂u y ψ∗ω0 = zu − pxu − qyu (3.27) where the symbol y denotes the interior multiplication, see [11] (section 2.11), and the initial values z0(τ, τ) = z0(τ), q0(τ, τ) = z′0(τ), p0(τ, τ) = −g(z′0(τ)) (3.28) in the case of (3.25) and z0(τ, τ) = z0(τ)), q0(τ, τ) = z′0(τ)), p0(τ, τ) = g(z′0(τ)) (3.29) in the case of (3.26). integration of cauchy problems (3.27), (3.28) and (3.27), (3.29) allows to represent characteristic uniformization (2.23) of a simple wave in quadratures. proposition 3.1: in characteristic coordinates u and v the coordinates x, y, p, q, and z of uniformization (2.23) of any definite multivalued solution (σ, l) of the cauchy problem (0.1), (2.17) that is a maximal simple wave are representated in the following way: x(u, v) = h(u, v) 2 √ g0(v) , y(u, v) = v + √ g0(v) 2 h(u, v), z(u, v) = z0(v) + p0(v) + g0(v)z′0(v) 2 √ g0(v) h(u, v), p(u, v) = p0(v), q(u, v) = z′0(v) (3.30) copyright c© 2020 assa. adv syst sci appl (2020) 130 d.v. tunitsky in case (3.25) and x(u, v) = h(u, v) 2 √ g0(u) , y(u, v) = u− √ g0(u) 2 h(u, v), z(u, v) = z0(u) + p0(u) + g0(u)z′0(u) 2 √ g0(u) h(u, v), p(u, v) = p0(u), q(u, v) = z′0(u) (3.31) in case (3.26). here (u, v) ∈ r2, g0(τ) = g(z′0(τ)), h(u, v) = u∫ v dτ√ g0(τ) . put π : r5 3 (x, y, p, q, z) 7−→ (x, y) ∈ r2. the following statement is an immediate corollary of representations (3.30) and (3.31) from proposition 3.1. it gives sufficient conditions for projection π ◦ σ of a maximal simple wave (σ, l) to be a proper map of degree 1. theorem 3.1: if lim τ→∞ g0(τ) τ = 0, lim τ→∞ |h(τ, 0)| = +∞, then the projection π ◦ σ of the maximal simple wave (σ, l) is a proper map of degree 1 and, therefore, π ◦ σ(r2) = r2. acknowledgements the reported study was funded by rfbr and jsps (project number 1951-50005) and rfbr (project number 20-01-00610). references 1. n.j. zabusky, ”exact solution for the vibrations of a nonlinear continuous model string”, j. math. phys. vol. 3, no. 5, pp.1028–1039, 1962. 2. p.d. lax, ”development of singularities of solutions of nonlinear hyperbolic partial differential equations”, j. math. phys. vol. 5, no. 5, pp.611–613, 1964. 3. m. taylor, partial differential equations, vol. 1, basic theory, springer-verlag, new york, 1996. 4. r. courant and k.o. friedrichs, supersonic flow and shock waves, interscience publishers, inc., new york, 1948. 5. b. l. rozhdestvenskii and n. n. janenko, systems of quasilinear equations and their applications to gas dynamics. transl. math. monogr., vol. 55, amer. math. soc., providence, ri, 1983. copyright c© 2020 assa. adv syst sci appl (2020) multivalued simple waves 131 6. é. cartan, les systèmes différentiels extérieurs et leur applications géométriques, hermann, paris, 1945. 7. a. m. vinogradov, ”multivalued solutions and a principle of classification of nonlinear differential equations”, soviet math. dokl., vol. 14, pp.661–665, 1973. 8. a. kushner, v. lychagin, and v. rubtsov, contact geometry and nonlinear differential equations, encyclopedia math. appl., 101, cambridge univ. press, cambridge, 2007. 9. r.l. bryant, s.s. chern, r.b. gardner, h.l. goldschmidt, and p.a. griffiths, exterior differential systems, springer-verlag, new york–berlin–heidelberg–london–paris– tokyo–hong kong–barcelona, 1991. 10. d.v. tunitsky, ”on hyperbolic systems of monge-ampère equations”, sbornik: mathematics, vol. 200, no. 11, 2009, pp. 1681–1714. 11. f. w. warner, foundations of differentiable manifolds and lie groups, springer-verlag, new york–berlin–heidelberg–tokyo,1983. copyright c© 2020 assa. adv syst sci appl (2020) multivalued solutions cauchy problem simple waves microsoft word article final.doc adv syst sci appl 2025; 2; 1-10 published online at https://ijassa.ipu.ru. computer simulation of a magnetic separator for iron ore dressing under conditions of uncontrolled disturbances nina osipova* university of science and technology misis, moscow, russia financial university under the government of the russian federation, moscow, russia abstract: the article presents a theoretical material about the principle of operation, design and advantages of the magnetic iron ore separator. a review of researches is given, where special attention is paid to the study of factors that affect the quality of concentrate and the loss of a valuable component in the tailings. the specification of the pdm-sc-120/300 magnetic separator is given. regression models are obtained that reflect the relationship between the content of the class -0.074 mm and the solid phase flow rate in the separator feed with a high coefficient of determination. the statistical significance of this coefficient was verified using the fisher criterion. a mathematical description of the object elements such as governing valve, motor, and magnetic separator is given. these models are simplified by dropping small time constants and linearization in the vicinity of nominal modes using taylor series expansion. the main disturbances that cause the deviation of the dressing indicators from the optimal values are highlighted. a computer model of the magnetic separator was constructed using the simulink matlab application. the simulation results showed that with the specified control actions, an increase in the content of the class -0.074 mm in the feed to 94-95 % brings the mass fraction of iron in the concentrate to the acceptable limits but reduces the productivity of the separator. keywords: magnetic separator, concentrate, tails, matlab, ms excel, regression model, fisher criterion, correlation, coefficient of determination, taylor series. 1. introduction over the past century, the constant growth of human needs for iron has led to the development and improvement of new technologies for ore dressing. the most widespread process is the magnetic separation. its main purpose is to separate ore material particles into two phases: concentrate with a high iron content and tailings with a small fraction of the valuable component. by design the magnetic separator is made in the form of a drum that rotates at the given speed and a stationary magnetic system located in its inner part. separation of the ore stream is based on the magnetic susceptibility of its constituent particles. if this parameter has a high value, then the particles stick to the drum and enter the concentrate compartment, and the rest, without attracting, are immediately washed off into the tailings compartment. the advantage of the magnetic dressing method is the ability to create a high force of attraction to the drum, which is hundreds of times higher than the particles weight, as well as safety during maintenance of the separator and harmlessness to the environment [5]. there are dry and wet magnetic dressing. in the first case, the ore is fed to the separator after preliminary crushing and screening. in the second case, after grinding in the mill and classification by size, the ore enters the magnetic separator in a mixture with a liquid (pulp). the above processes are multi-stage. * corresponding author: nvo86@mail.ru 2 n. v. osipova copyright ©2025 assa adv. in systems science and appl. (2025) the iron ore raw material coming to the metallurgical plant in the form of agglomerate or pellets must be of the specified quality with permissible deviations from the norm established by technical conditions and regulations. otherwise, you have to adjust the modes of melting units or spend more additional materials, which increases the cost of steel production. it is also necessary to ensure that iron losses in the tailings that do not exceed the permissible value. 2. review of the research the main purpose of the research in the study of mineral processing is to identify the main factors that affect the iron content in the products of the magnetic separator, including controlling and disturbing influences. control actions can be changed by a human operator or by an automatic system. disturbing influences are random changes in the physical and mechanical properties of the ore or pulp. several researches have been devoted to the study of the influence of controlling and disturbing influences on the magnetic separation process. in [1], multiple regression equations for a specific type of ore are obtained. one of them relates the iron content in the concentrate and the control variables: the filling level of the mill at the second stage of grinding, the density of the hydrocyclone discharge. another is the association of loss of iron in the tails of the first stage of dressing and fill level of the mill, the water flow into the mill the first stage of grinding, drain density classifier. the coursebook [10] describes the principle of creating a regression model based on experimental planning, which can predict the mass fraction of iron in the concentrate for a predetermined step forward in time. this model is based on the calculated average iron content for a certain period, the deviation of the current value of the iron content from the average, and changes in the load on the ore section. the paper [4] presents a regression model, in which the dependent variable is the iron mass fraction in the enrichment products, and the factors are the load on the industrial product of dry magnetic separation, water flow into the classifying apparatus and magnetic separators at the 1st, 2nd, 4th stages of dressing. the disadvantage of these methods is a large delay between obtaining of input and output variables, which makes it difficult to quickly update the coefficients of the regression model and calculate the control variables. the above papers do not specify what specific disturbances can cause deviations of the dressing indicators from the set ones. in [6], static characteristics of magnetic separators are given, reflecting the dependence of the iron content in the concentrate and its losses in the tailings on the drum rotation speed and pulp density, which can be controlled by supplying additional water to the separator bath. however, the graphs do not indicate the numerical values on the axes and the equations that they were based on. it is only known that in a wide range of parameters, the dependencies are nonlinear, close to the second-order polynomial. therefore, it is necessary to solve the following problems: to select the control and disturbing effects that affect the content of the useful component in the concentrate and in the tailings, so that there is no lag between the input and output of the object; to develop a computer model of the magnetic separator, which allows to study its operation of the separator under nominal control effects under conditions of uncontrolled disturbances. 3. object of research the object of research is the drum semi-countercurrent magnetic separator pdm-sc-120/300, which is used in many iron ore mining and processing plants to separate particles of less than 1 mm in size at the final stages of dressing. it’s required for simulation specification is given in table 3.1 [2]. computer simulation of a magnetic separator for iron ore enrichment 3 copyright ©2025 assa. adv. in systems science and appl. (2025) table 3.1. specification of the pdm-sc-120/300 magnetic separator feed option drum rotation speed, min-1 permissible productivity for solid phase, tons/h content in the feed separator, % class -0.074 mm solid phase of the pulp 1 60-70 30 140-180 2 20 80-120 3 75-85 30 100-140 4 20 70-100 5 94-96 30 60-80 6 20 40-60 as you can see from the table, there are six separator feed options. in this case, the disturbing effects are the content in the feed of the class -0.074 mm and the of solid phase flow rate (productivity) at two limit values of the solid phase content in the pulp. before finding the degree of influence of factors on the dressing indicators, it is necessary to check their correlation. for this purpose, ms excel generated samples of random variables with a normal distribution law that characterize the content of the class -0.074 mm and the corresponding solid phase flow rate for two values of the solid phase content in the separator feed (table 3.2). table 3.2. characteristics of random values of disturbances for simulation in ms excel solid phase content in pulp, % mathematical expectation maximum deviation (3σ) standard deviation (σ) performance, tons/h the content of the class -0.074 mm, % performance, tons/h the content of the class -0.074 mm, % performance, tons/h the content of the class -0.074 mm, % 20 100 65 20 5 20/3 5/3 30 160 20 85 80 15 15/3 30 120 20 20/3 20 50 95 10 1 10/3 1/3 30 70 for each of the above six options, the sample consists of 10 values, i.e. 30 positions for the solid phase content in the feed, equal to 20 % and 30 %. the diagram with the trend line based on the linear dependence is presented and the coefficient of determination is calculated (fig. 3.1). fig. 3.1. correlation diagram of solid phase flow rate and -0.074 mm class content in the separator feed the high determination coefficients r1 2 = 0,9134 и r2 2 = 0,9767 indicate a strong dependence of the solid phase flow rate on the content of the class -0.074 mm in the 4 n. v. osipova copyright ©2025 assa adv. in systems science and appl. (2025) separator feed. let’s evaluate the statistical significance of the coefficients using the fischer criterion. in the first case, we find the observed f-value for two dependencies: 2 1 1 2 1 2 2 2 2 2 0.9134 28 38.55, 1 1 0.9134 1 0.9767 28 1173.72, 1 1 0.9767 1 r f f r m r f f r m           (3.1) where n = 30 is the number of experimental points; m = 1 is the number of factors; f = n-m-1 is the number of degrees of freedom; α = 0.05 is the level of significance. it turns out that the found value is greater than the critical value indicated in the table fcr = 4.2 (f1 > 4.2; f2 > 4.2), which confirms the statistical significance of r2 with a probability p =1 α = 0.95. therefore, if there is a correlation in the mathematical model, it is sufficient to use just one of the disturbances: the solid flow rate or the content of the class -0.074 mm in the feed of the separator. 4. computer model of a magnetic separator 4.1 nonlinear dynamic model of a magnetic separator as control variables, we will use the water flow rate into the separator bath and the rotation speed of its drum. this choice is justified by the absence of a lag between these impacts and the output indicators, as well as the fact that there is no need to change the separator design. the model block diagram is shown in figure 4.1. when controlling the degree of the valve opening and the motor shaft rotation speed by setting different values εs and ωs, the water flow rate into the bath wb and the rotation speed of the drum ω, respectively, are regulated. it causes a change in the content of magnetite iron in the concentrate β and tailings ν monitored by analyzers. the parameter ξ is a disturbance that characterizes the instability of the physical and mechanical properties of the pulp, which also causes fluctuations in the dressing indicators β and ν. fig. 4.1. block diagram of the magnetic separator model let’s look at the individual components of the model in detail. the model of the governing valve is described by the following differential equation: v v v s ( ) ( ) , d t t t dt      (4.1) computer simulation of a magnetic separator for iron ore enrichment 5 copyright ©2025 assa. adv. in systems science and appl. (2025) εs is the set value for the degree of opening of the valve, %; εv is the current value of the valve opening degree, %; tv is the time constant of the controlled valve, sec. for the most valves, tv is approximately 0.3 seс [7]. the water flow rate in the separator bath is related to the value εv and the ratio [9]: b v( ) 11 ( ).w t t  (4.2) the model of an asynchronous drive motor has a very complex mathematical description in the form of a system of nonlinear equations. however, in the case of operation in a small neighborhood of the point that corresponds to the rated mode, it is allowed to consider a linearized system equivalent to the dc motor model: m m m s ( ) ( ) , d t t t dt    (4.3) ωs is the set value of the motor drive shaft rotation speed, min-1; ωm is the current value of the motor drive shaft rotation speed, min-1; tm is the time constant of the drive motor, s. the model of the gearbox is determined based on the ratio of the nominal rotation speed of the separator drum ωs.nom = 19 min–1 and the motor ωm.nom = 1000 min –1: s.nom m.nom ω 19 0.02. ω 1000g k    (4.4) the rotation speed of the separator drum ωg using a reducer is equal to: mgω ( ) 0.0 .2ω ( )t t (4.5) according to the simulation of asynchronous motors the value of tm can be assumed to be equal to 0.03 seс [9]. the models must have "dead zones" δv and δm for the valve and drive, respectively. this means that if the input signal is |εv|≤ δv or |ωg|≤ δm, then the output signal is ε = 0, ω = 0, otherwise ε = εv δv or ω = ωg – δm. let’s take δv = 3 %, δm = 0,1 %. the dynamics of the content of magnetite iron in the concentrate and tailings is governed by the following system of differential equations: β β β β β ν ν ξ ξ v v ν β 1 1 1 β ,ω ( ) (ξ) ( ) ( , ν 1 1 1 ν ,ω ξ), d f w f dt t t t d f w f dt t t t      (4.6) where w = wl + wb is the sum of the liquid phase consumption in the pulp and the water consumption in the separator bath, fβ(w, ω) и fν(w, ω) describe the relationship of the dressing indices β, ν and the control variables w, ω and the control variables w, ω in steady state when βd dt = 0 и ν d dt = 0: 2 2 β 2 2 ν , ω – 0.0000042 0.0082 – 0.023ω 1.21ω 46.5, , ω – 0.0000025 0.0087 0.035ω – 1.3ω 10.2, ( ) ( ) f w w w f w w w         (4.7) where tβ, tv are the time constants of magnetic separator (sec), tβ = 1…10 sec, tv = 1...10 seс [7]; fβξ(ξ), fνξ(ξ) are the functions on uncontrolled disturbances ξ. to simplify the simulation, we put tβ = tv = 1 sec. the system (4.7) is obtained in [9] based on the reference data of the average concentration of iron ore mining and processing plants, provided that fβξ(ξ) = const, fνξ(ξ) = const. the units of measurement for w and ω are m3/h and min-1 respectively. 6 n. v. osipova copyright ©2025 assa adv. in systems science and appl. (2025) 4.2 linearization of the magnetic separator model for the convenience of research, simulation and automation of the magnetic separator, we linearize equations (4.6). the most of automatic systems are designed and configured under the assumption that the object under control is linear. using the rule for calculating the function increment δy = y'δx, we perform the system linearization by expanding the taylor series of equations (4.6) in the vicinity of the nominal modes (wnom, ωnom) with respect to δβ and δν with the rejection of nonlinear terms. given that tβ = tv = 1 seс and that fβ(w, ω), fν(w, ω) are found by equation (4.7), we obtain: nom nom nom nom nom nom nom nom β β – 2 0.0000042 0.0082 – 2 0.023ω ω 1.21 ω= = β – 0.0000084 0.0082 – 0.046ω ω 1.21 ω, ν ν – 2 0.0000025 0.0087 2 0.035ω ω-1.3 ω ν – 0.000005 0.0087 0.07ω ω – w w w w w w w w w w w w                                        1.3 ω. (4.8) for simulation, we assume that the required content of magnetite iron in the concentrate βs = 63.8 %, which is the average for many iron ore mining and processing plants [2]. substituting ωnom = 19 min-1 in the first equation in (4.7) for fβ(wnom, ωnom) = βs, we get wnom = 400 m3/h. then the loss of magnetite iron in the tailings at wnom, ωnom is fν(wnom, ωnom) = νs = 1.22 %. assuming the time constants of the valve and motor are negligible in compare with the time constants of the magnetic separator, the water flow rate into the separator bath is associated with a given degree of opening of the valve as w = 11εs, and the rotation speed of the drum is determined by the equality ω = ωs. we introduce new notation for the variables δβ = x1, δν = x2, δw = u1, δω = u2. we assume that the change δw is due to the control of the valve. is due to the control of the valve. taking into account the above, substituting wnom, ωnom in equation (4.8), we get: 1 1 1 2 2 2 1 2 0.00484 0.336 , 0.0067 0.03 x u u x u x x u          (4.9) or in matrix form: ( ) ,( ) ( )t x tx ta bu  (4.10) where: 1 0 0.0048 0.336 , . 0 1 0.0067 0.03 a b             (4.11) the linear representation of the system (4.7) in the interval (wnom, ωnom) has the form:         β nom nom βlin β nom nom nom β nom nom nom ν nom nom νlin ν nom nom nom ν nom nom nom , ω ) , ω , ω ) , ω ) ω ω 55.48 0.00484 0.336ω, ω , ω ) , ω , ω ) , ω ) ω ω 2.035 0.0067 0. ( ( ) ( ( ( ( ) ( 03ω. ( ω f w f w f w w w w f w w f w f w f w w w w f w w                            (4.12) as a disturbance, we will consider the change in the percentage q of solid and the solid phase flow rate in the feed q. as can be seen from table 4.1, the separator maximum performance is q = 180 tons/h with a percentage of solid q = 30 %, and the minimum q = 40 tons/h with a solid q = 20 %. let’s calculate the minimum and maximum values of the liquid phase flow rate in the pulp, taking into account that the water density ρw = 1 ton/m3: computer simulation of a magnetic separator for iron ore enrichment 7 copyright ©2025 assa. adv. in systems science and appl. (2025)       l w 3 l.min 3 3 l.max 3 100 % , 100 % 30 % 40 tons/h 93.3 m /h, 1 ton/m 30 % 100 % 20 % 180 tons/h 720 m /h. 1 ton/m 20 % q w q q w w                (4.13) substituting w = wb + wl in (4.12) and replacing the liquid phase flow rate of the with the expression from equation (4.13), we get:     ξ ξ βlin b β νlin b ν β l ν ξ w ξ w l ( ) ( , ) ( ) ( , ω 55.48 0.00484 0.336ω+ , , ω 2.035 0.0067 0.03ω+ 0.00484 , ), 100 ( , ) , 100 ( , ) 0.00484 0.0067 0.0067 . f w w f f w w f f q q q q q q q q q q q q q q w f w                    (4.14) according to the specifications, the permissible deviation for the content of magnetite in the concentrate x1,2max ≤ 1 %. then βmin = 63.8 % 1 % = 62.8 %, βmax = 63.8 % + 1 % = 64.8 %. obviously, the minimum losses in the tailings are 0. their limit is νlim = 1.22 % + 1 % = 2.22 %. then the deviation x2max = 1 %. consider the procedure for determining the upper limits on control actions. the main requirement for the linear representation of the model (4.7) is that at the boundary values of ±u1,2max, the absolute value of the difference in the dressing indicators calculated from the nonlinear and linearized models should be equal to the error of their measurement. according to russian standard, it is 0.9 % for magnetite iron in concentrate and 0.3 % for losses in tailings during laboratory research by chemical analysis [3]. let the error of the analyzers also be equal to these values, then restrictions can be found from the solution of the system: β nom 1max nom 2max βlin nom 1max nom 2max ν nom 1max nom 2max νlin nom 1max nom 2max ( ) ( ) ( , ω + , ω + 0.9, , ω + , ω + 0) ) 3.( . f w u u f w u u f w u u f w u u          (4.15) the expressions fβlin(wnom + u1max, ωnom + u2max), fνlin (wnom + u1max, ωnom + u2max) are found by substitution in (4.12) wnom + u1max, ωnom + u2max instead of w, ω: β nom nom β nom nom βlin nom 1max nom 2max β nom nom 1max 2max ν nom nom ν nom nom νlin nom 1max nom 2max nom nom 1max 2max , ω ) , ω ) , ω + , ω ) , ω , ω ) , ω ) , ω + ( ( ( ) ( ( ( , ω ) . ω ( ) ( f w f w f w u u f w u u w f w f w f w u u f w u u w                 (4.16) the system (4.16) is easily solved using the matlab computing package of the optimization toolbox application. in this case, among the real roots, u1max = ± 346.83 m3/h, u2max = ± 4.14 min-1. so wnmin = 400 m3/h -346.83 m3/h = 53.2 m3/h, wnmax = 400 m3/h + 346.83 m3/h = 746.8 m3/h, ωnmin = 19 min-1 4.14 min-1 = 14.86 min-1, ωnmax = 19 min-1 + 4.14 min-1 = 23.14 min-1, the following restrictions must be met: wnmin ≤ w ≤ wnmax and ωnmin ≤ ω ≤ ωnmax. the nominal value of degree of opening of the valve εnom = 36.36 %, which corresponds to w = 400 m3/h. we will assume that the data from the analyzers is transmitted with a very small delay, the voltage at the output of their microchip is related to the physical value through a coefficient equal to one. therefore, we will not take them into account in the model. 8 n. v. osipova copyright ©2025 assa adv. in systems science and appl. (2025) the diagram of the computer model of the magnetic separator is shown in fig. 4.2. it was also built using the matlab environment of the simulink application based on the block diagram from fig. 4.1. fig. 4.2. diagram of a computer model of a magnetic separator in matlab simulink 4.3 simulation result the simulation was performed as follows. disturbances were periodically fed to the input of the object at an interval equal to the end time of transients of the magnetic separator tend = 5 seс [8]: the change in the percentage of solid in the feed and the solid phase flow rate, which was set using a random number generator in the range according to table 4.1, i.e., six variants of pulp properties were modeled. let’s take the first option as an example. when the solid content in the feed is 30 %, the solid phase flow rate increases to 120 tons/h. the model calculates the water consumption in the pulp using the equation (4.13), which becomes equal to wl = 280 m3/h, which is within the range of wlmin ≤ wl ≤ wlmin. the change in this signal is fed to the magnetic separator model (4.14) and used in the functions fβξ(q, q), fνξ(q, q) (fig. 4.3). fig. 4.3. changing the liquid and solid phases flow rate in the pulp computer simulation of a magnetic separator for iron ore enrichment 9 copyright ©2025 assa. adv. in systems science and appl. (2025) fig. 4.4. the content of magnetite iron in the concentrate and tailings when the properties of the pulp are unstable in the absence of automatic control of the separator, constant nominal values of water flow rate wb = 400 m3/h and rotation speed ω =19 min-1. the following conditions are met: wnmin ≤ w ≤ wnmax. the content of magnetite iron in the concentrate in the time interval of 520 seс is greater than the value of βmax. the same can be said for tailings that are superior to the νlim. the largest of the bursts is β = 65.34 % и ν = 3.35 % (fig. 4.4). this corresponds to the content of the class -0.074 mm in the feed of 60-70 %. only after t = 20 sec, the iron content falls below βmax, when the pulp is 94-95 % of the class -0.074 mm, but there is a sharp decrease in the productivity of the separator, which becomes below 80 tons/h. 5. conclusion as control actions were selected the rotation speed of the magnetic separator drum and the water flow rate into its bath. the nominal operating modes corresponding to the specified dressing parameters in the absence of disturbances are determined by the content and the solid phase flow rate in the feed, which correlates with the content of the class -0.074 mm in the feed, are taken as the parameters. a computer linear dynamic model of the separator in the matlab simulink package is constructed. based on the results of the simulation, the following conclusions can be drawn. setting control actions with constant nominal values is not enough to stabilize the quality of iron ore concentrate. when changing the content of the class -0.074 mm in the feed in the range from 60 % to 85 %, the separator operates with a sufficiently high performance, but the iron content in the concentrate is higher than normal. the same can be said about tailings. increasing the content of the class -0.074 mm in the feed to 94-95 % brings the mass fraction of iron in the concentrate to the acceptable limits but reduces the productivity of the separator. therefore, controlling the grinding and classification processes before magnetic separation in order to stabilize the granulometric composition is not promising. 10 n. v. osipova copyright ©2025 assa adv. in systems science and appl. (2025) a more rational approach is to create an optimal control system. it should be based on the formulated criterion for minimizing deviations of the iron content in the concentrate and tailings relative to the set values with restrictions on the control variables selected based on the permissible linearization error at the boundaries of their change intervals. in the future, the model of optimal control of the magnetic separator will allow evaluating and predicting the possibility of maintaining the quality of iron ore concentrate with acceptable deviations. this will reduce the number of experiments in laboratory and industrial installations and thus increase their service life and reduce the number of consumables. references 1. anikin, a. i. (1984). research and development of an optimal control system for a section of a self-grinding magneto-concentrating plant as a subsystem of the automated control system of the lebedinsky mpp kma. ph.d. thesis, leningrad order of lenin, order of the october revolution and order of the red banner of labor plekhanov mining institute 2. bogdanov, o. s. (1984) spravochnik po obogashcheniyu rud. obogatitel'nye fabriki (tom 4) [handbook of ore dressing. processing plants (volume 4)]. moscow, ussr: nedra, [in russian]. 3. gost 16589-86 (1987) rudy zheleznye tipa zhelezistyh kvarcitov. metod opredeleniya zheleza magnetita [iron ores of the ferruginous quartzite type. method for determining iron magnetite]. moscow, ussr [in russian]. 4. kazakov y. m. (1994) control of the wet magnetic dressing process line. ph.d. thesis, ural state mining and geological academy 5. karmazin, v. v., (2013). problemy i perspektivy magnitnogo obogashcheniya [problems and prospects of magnetic dressing]. gornyj informacionno-analiticheskij byulleten', s1, 560–575, [in russian] 6. maryuta a. n., kachan yu. g., bun'ko v. a. (1983) avtomaticheskoe upravlenie tekhnologicheskimi processami obogatitel'nyh fabrik [automatic management of technological processes in concentrating plants] moscow, ussr: nedra, [in russian]. 7. osipova n. v. (2018). ispol'zovanie fil'tra kalmana pri avtomaticheskom kontrole pokazatelej magnitnogo obogashcheniya zheleznyh rud [the use of kalman filter in automatic control of indicators of iron ores magnetic concentration], izvestiya visshikh uchebnykh zavedenii. chernaya metallurgiya = izvestiya. ferrous metallurgy, vol. 61, iss. 5, 372-377. doi: https://doi.org/10.17073/0368-0797-2018-5-372-377 8. osipova n. v. (2018). sintez asimptoticheskogo nablyudatelya dlya sistemy upravleniya magnitnym separatorom pri obogashchenii zheleznoj rudy [synthesis of asymptotic observer for magnetic separator control in iron ore processing] gornyj informacionnoanaliticheskij byulleten', no 6, 153-160, [in russian] doi: 10.25018/0236-1493-2018-60-153-160 9. osipova n. v. (2018). model of stabilization of the quality of iron-ore concentrate in the process of magnetic separation with the use of extreme regulation, metallurgist, vol. 62, nos. 3-4, 303-309. doi 10.1007/s11015-018-0660-8 10. vinogradov, s. v. (1984) avtomatizaciya tekhnologicheskih processov gornogo proizvodstva [automation of technological processes of mining production]. moscow, ussr: nedra, [in russian]. microsoft word 9-linhai zhao.doc advances in systems science and applications (2010), vol.10, no.1 55-60  issn 1078-6236 international institute for general systems studies, inc. study on determinants of the abnormal returns of a-shares on the first listed day linhai zhao and ridong hu academy of quantitative economics, huaqiao university, quanzhou 362021, china email: longmen168@126.com abstract on the background of the reform of split share structure, applying weighted least squares method of multiple linear regression model, 206 a-shares issued and listed from january 1st,2004 to december 31st,2007 is empirically studied, and it is found that turnover rate on the first listed day, size of money raised, and market index，have significant impact on the abnormal returns of a-shares on the first listed day, whereas, the influence of issuing price, issuing p/e ratio, net asset per share before issuance, real negotiable share ratio after issuance, method of issuance and the reform of split share structure, is not significant. keywords abnormal returns ipo underpricing reform of split share structure 1. introduction in the stock markets throughout the world there is an obvious phenomenon of the ipo underpricing, that is, shares after an initial public offering will have high abnormal rate of returns on the first listed day[1]. in spite of the complexity of china's capital market[2], the regulators have carried out drastic reform and the market-oriented objective is approaching step by step. the method of issuance and the mode of pricing are changing, meanwhile the split share structure reform are putting forward and the regime of new stock issues is about to change. on the background of the great institutional innovation, the study of the determinants of the abnormal returns of a-shares on the first listed day will help understand the determinants of stock prices and the mechanism of changes in the prices, promote china's reform of the regime of the issue and the pricing of the new stocks, and is of practical significance to china’s stock market to play the reasonable and adequate role. 2. research method 2.1 research assumptions in view of the big going-up margin in the prices of china's stocks during their initial public offering, the paper made the following assumptions: first, the subscription costs of new shares are ignored. the subscription costs include the opportunity costs of the funds used for subscription and the subscription expenses incurred. due to the inaccuracy of the measurement of the opportunity costs, and the very small ratio of the subscription expenses to the funds used for subscription, they are all ignored here. second, transaction costs are not included. the transaction cost of stock mainly comprises the commission charged by stock brokers and the stamp duty. though the rate of stamp duty was increased to 3 ‰ on may 30th, 2007, the transaction costs accounted for the amount of the investment funds only about one percent. third, the won rate of the issuance is not considered. the won rate is often closely linked with the method of issuance. because of the seriously low won rates under each of the methods of issuance, they are not considered hereby. 2.2 sample selected this paper examined the a-shares listed during the period from january 1st, 2004 to zhao: study on determinants of the abnormal returns of a-shares on the first listed day 56 december 31st, 2007 in shenzhen stock exchange and shanghai stock exchange. all the data came from the "ccer capital market research database". the shares with incomplete data or not enough observations are excluded. in the sample period, there are 300 stocks newly listed in shenzhen and shanghai. 206 stocks in the 300 stocks meet the above requirement, of which 59 are in shanghai and 147 are in shenzhen. this study used a relatively large sample carefully selected, experienced the important process of the market-oriented evolution of china's securities market, and can more comprehensively and accurately reflect actual level of the ipo abnormal returns of the china's a-share market and the changes in the process. excel and spss 11.5 are applied to conduct the statistical analysis [3,4]. 2.3 the measurement of the ipo abnormal returns to measure it, the following formula is used: 1 0 1 0(ln ln ) (ln ln )j j j j jr p p m m= − − − (1) where, jr stands for the ipo abnormal return of the j-th stock adjusted by the market return, 1 jp stands for the closing price of the j-th stock on the first listed day, 0 jp stands for the issuing price of the j-th stock, and 1 jm and 0 jm are the closing market indices on the first listed day of the j-th stock and on the issuing day of this stock respectively. 2.4 the hypotheses to be examined through analysis, there are 8 hypotheses to be examined: hypothesis 1: the methods of issuance influence the abnormal returns on the first listed day. the methods of issuance of the sampled stocks are categorized as following: issuance through the trading network with fixed price, issuance off the trading network by inquiry, the secondary market placement, corporate placement (including the strategic placement, placement off the trading network and private placement). hereby, issuance through the trading network with fixed price and issuance off the trading network by inquiry are classified by the methods of pricing; the secondary market placement and corporate placement are classified by the types of investors. these methods of issuance are often combined. different method of issuance may affect the ipo abnormal returns differently. hypothesis 2: the issuing price is negatively related to the abnormal returns on the first listed day. for the stocks with lower issuing price, the rising intervals of their prices are larger and the magnitudes of the rise are larger. hypothesis 3: the issuing p/e ratio is related to the abnormal returns on the first listed day. hypothesis 4: the negotiable share ratio positively correlates with the abnormal returns on the first listed day[5,6]. for firms with lower negotiable share ratio, their risks of corporate governance tend to be larger, and the probability of mergers and acquisitions is bigger after the issuance. this may lead to the smaller magnitude of the rise of the price of the stock on the first listed day. hypothesis 5: the turnover rate on the first listed day and net asset per share before issuance positively correlates with the abnormal returns on the first listed day. the larger the turnover rate and the more the net asset per share of the newly issued stock, the more investors are in the secondary market for this stock. that too many investors rush for the stock will push the price upward. hypothesis 6: the size of money raised negatively correlates with the abnormal returns on the first listed day. the smaller size of money raised, the more speculators buy the stock and the higher of the abnormal returns on the first listed day. hypothesis 7: the market index positively correlates with the abnormal returns on the first listed day. the higher market index shows that the market is in better condition and investors are more confident. this will heighten the trading price and lead to higher abnormal returns on the first listed day. hypothesis 8: listed before or after the reform of split share structure is related to the advances in systems science and applications (2010), vol.10, no.1  57 abnormal returns on the first listed day. the reform of split share structure will change the expectations of the investors and cause the fluctuations of stock prices. so it is thought that listed before or after the reform of split share structure is related to the abnormal returns on the first listed day. 3. empirical studies 3.1 multivariate linear regression model ⅰ 3.1.1 variables according to the above-mentioned hypotheses, this model includes following variables, r (abnormal returns)、t (turnover rates)、p (issuing price)、n (net asset per share before issuance)、e (p/e ratio)、l (logarithm of the size of money raised)、i (market index, the sum of the comprehensive index of shanghai stock exchange and the composite index of shenzhen stock exchange is adopted hereby)、c (change of policy, a dummy variable, its value is 1 for the stock issued before the implementation of the reform of split share structure, otherwise 0)、r (negotiable share ratio)、m (methods of issuance, dummy variables. three kinds of them, m1、 m2 and m3 are involved hereby, where m1 is the combination of issuance through the trading network with fixed price and issuance off the trading network by inquiry; m2 is the combination of issuance off the trading network by inquiry and the secondary market placement; m3 is the secondary market placement). 3.1.2 model ⅰ 3 1 2 3 4 5 6 7 8 9 1 j j j r p t n e r l i c mα β β β β β β β β β ε = = + + + + + + + + + +∑ (2) where, r is dependent variable, p 、t 、 n 、 e 、 r 、 l 、 i 、 jm （ 1, 2,3j = ）、 c are independent variables, iβ ( 1, 2, ,8i = )and 9 jβ ( 1, 2,3j = ) are coefficients, α is constant, ε is stochastic error. 3.1.3 empirical analysis process first, exclude multicollinearity. this can be done in two ways. one way is to examine tolerance and variance inflating factor. when the tolerance is not bigger than or near 0.1 and variance inflating factor is bigger than 10, multicollinearity may exist. the variable correlated more with other variables should be deleted. the other way is to examine the condition index and variance ratio. when the condition index of some row in the collinearity diagnose table is bigger than 15 and at least two variables have variance ratios bigger than 0.90, multicollinearity may exist. the variable correlated more with other variables should be deleted. second, exclude abnormal points. abnormal points are points with extremely large or extremely small values. these points should be removed in order to keep robust. third, examine the goodness-of-fit. if the regression equation passes the f-test, the equation established. the more the determinable coefficient r2is, the better the goodness-of-fit is. the determinable coefficient used hereby is the adjusted determinable coefficient, adjusted r2. fourth, carry out t-test of the regression coefficients. *** stands for having passed t-test at the 1% significant level, ** stands for having passed t-test at the 5% significant level, and * stands for having passed t-test at the 10% significant level. fifth, select appropriate regression method. weighted least squares method with total shares is adopted. 3.1.4 analysis of the regression results the regression results are listed in table 1 and table 2. m1 should be removed due to its tolerance less than 0.0001. α 、t、r、l、i are significant at the 1% level, but n、p、e、m2、 m3、c are not significant at the 10% level. the condition index in the 11th row in table 2 is 17.020, bigger than 15, and the variance ratios of c and m3 are both larger than 0.90. this shows zhao: study on determinants of the abnormal returns of a-shares on the first listed day 58 that multicollinearity may exist and the model should be modified. table 1 regression result of model i coefficients non-normalized coefficients normalized coefficients statistics of collinearity model b std. error beta t significance tolerance variance inflation factor α 4.001 0.404 9.907 0.000*** t 0.270 0.115 0.123 2.339 0.020*** 0.615 1.625 n 0.005 0.026 0.010 0.187 0.852 0.644 1.552 p -0.007 0.005 -0.077 -1.401 0.163 0.559 1.789 e 0.002 0.002 0.058 1.031 0.304 0.533 1.877 r -0.740 0.186 -0.483 -3.975 0.000*** 0.116 8.650 l -0.164 0.015 -0.924 -11.184 0.000*** 0.250 3.999 i 0.000 0.000 0.888 9.531 0.000*** 0.196 5.091 m2 -0.179 0.152 -0.066 -1.175 0.241 0.548 1.824 m3 0.060 0.243 0.069 0.245 0.806 0.022 45.915 i c 0.094 0.229 0.110 0.411 0.682 0.024 41.786 r2 adjusted r2 f prob>f d.w. 0.667 0.650 39.122 0.000 1.785 variables removed statistics of collinearity model beta in t significance partial correlation tolerance variance inflation factor minimum tolerance i m1 . . . . 0.000 . 0.000 table 2 diagnosis of collinearity of model i variance ratio mod el numbers of dimensi on eigenval ue condit ion index α t n p e r l i m2 m3 c 1 3.343 1.000 0.00 0.01 0.01 0.00 0.01 0.01 0.01 0.00 0.00 0.00 0.00 2 2.071 1.270 0.00 0.00 0.00 0.08 0.06 0.01 0.00 0.03 0.00 0.00 0.00 3 1.342 1.578 0.00 0.15 0.10 0.00 0.00 0.00 0.02 0.01 0.03 0.00 0.00 4 1.102 1.742 0.00 0.06 0.28 0.05 0.06 0.00 0.00 0.00 0.09 0.00 0.00 5 1.000 1.828 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 6 0.963 1.864 0.00 0.12 0.01 0.02 0.02 0.00 0.00 0.00 0.40 0.00 0.00 7 0.452 2.720 0.00 0.03 0.02 0.19 0.20 0.03 0.13 0.06 0.01 0.00 0.00 8 0.357 3.058 0.00 0.16 0.38 0.57 0.45 0.00 0.00 0.00 0.00 0.00 0.00 9 0.293 3.376 0.00 0.43 0.09 0.09 0.14 0.01 0.27 0.16 0.03 0.00 0.00 10 0.064 7.213 0.00 0.03 0.10 0.01 0.06 0.90 0.56 0.71 0.00 0.01 0.03 i 11 0.012 17.020 0.00 0.01 0.02 0.00 0.00 0.03 0.01 0.03 0.43 0.98 0.96 3.1.5 the modification of model ⅰ to avoid multicollinearity, correlation coefficients are examined. if the absolute value of correlation coefficient of two variables is bigger than 0.5, they can be thought of as having strong correlation. this may cause serious multicollinearity. so stepwise method is adopted, variables not significant are removed and variables significant are left. (1) i and e：because the correlation coefficient between i and e is almost 0.6 and the p-value of e in the t-test is 0.304, e is deleted. (2) r：r is strongly related with l、c、m1 and m3. (3) m：m2 and m3 are not significant at 10% level. (4) c：c is not significant at 10% level. by the means of stepwise, advances in systems science and applications (2010), vol.10, no.1  59 model ⅰ can be modified as model ⅱ. 3.2 model ⅱ 1 2 3r i t lγ εγ γ γ= + + + + (3) 3.2.1 analysis of the regression coefficients from table 3, it can be found that γ、t、l、i are all significant at 1% level. multicollinearity does not exist in model ⅱ. 3.2.2 analysis of variance as it is shown in table 3, this model has passed the f-test and is significant. 3.2.3 autocorrelation test through the durbin-watson test, autocorrelation is examined. as it is shown in table 3, there does not exist autocorrelation in model ⅱ. table 3 regression result of model ii coefficients non-normalized coefficients normalized coefficients statistics of collinearity model b std. error beta t significa nce tolerance variance inflation factor γ 2.212 0.226 9.768 0.000** * i 2.729e-0 5 0.000 0.661 14.215 0.000** * 0.898 1.113 t 0.337 0.112 0.154 3.005 0.003** * 0.743 1.345 ii l -0.101 0.009 -0.566 -11.111 0.000** * 0.748 1.336 r2 adjusted r2 f prob>f d.w. 0.607 0.602 104.166 0.000 1.717 3.2.4 the economic explanation of the model based on the regression equation, it can be seen that r positively correlates with i and t , and negatively correlates with l .the regression coefficients γ 、 1γ 、 2γ 、 3γ are 2.212、 2.729e-5、0.337、-0.101 respectively, and their t-test values are 9.768、14.215、3.005 and -11.111. so they passed statistical tests. the concrete economic meanings can be explained as following: (1) t: the positive correlation between r and t shows that almost all the winners of the subscription wish to sell stocks subscribed in order to gain higher yield. meanwhile the losers of the subscription want to buy these stocks eagerly. because the quantity of the losers is more than that of the winners, supply cannot meet demand in the market. this pushes stock prices upward continually and the closing price on the first listed day is much higher than the issuing price. (2) i: the partial regression coefficient of the market index is 2.729e-05. this result presents that investors are more confident when the market is in good condition. this will pull the trading price further on the first listed day. (3) l: the logarithm of the size of the money raised negatively correlates with the abnormal returns of newly issued stocks. this shows that the stocks with smaller size of the money raised are favored by the short-term speculators and their abnormal returns tend to be higher. but the stocks issued by some large-sized companies supported by the state are not favored by the short-term speculators. from another view, this tells that asymmetric information theory of underpricing sponsored by some western economists[7-9] is not suitable for china’s stock market. zhao: study on determinants of the abnormal returns of a-shares on the first listed day 60 4. conclusion on the background of the reform of split share structure, through the empirical analysis of 206 a-shares issued and listed from january 1st,2004 to december 31st,2007, it is found that turnover rate on the first listed day, size of money raised, and market index，have significant impact on the abnormal returns of a-shares on the first listed day, whereas, the influence of issuing price, issuing p/e ratio, net asset per share before issuance, real negotiable share ratio after issuance, method of issuance and the reform of split share structure, is not significant. references [1] yunhui jin, wen yang. an empirical analysis of the phenomenon of underpricing of newly issued stocks. statistical research, 2003, 3: 49-53. [2] bowen sun, xiaoguang luo, baiyu sun and tianli liu. the study on the topology structure of china stock market and the self-organized criticality. advances in systems science and applications, 2007, 7(1): 127-131. [3] wei xue. data analysis based on spss. beijing: renmin university of china press, 2006. [4] shancun liu. application of excel in financial modeling analysis. beijing: people’s postal and telecommunication press, 2004. [5] huayang yin, xinping xia and pengcheng zhang. factors resulting in illiquidity premium of illiquid share in china. advances in systems science and applications, 2006, 6(1): 161-170. [6] bo qu. an empirical analysis on liquidity premium value of non-floating shares and influencing factors based on artificial neural network. advances in systems science and applications, 2007, 7(2): 306-312. [7] allen, f., faulhaber, g. r. signaling by underpricing in the ipo market. journal of financial economics, 1989, 23(2): 303-324. [8] grinblatt, m., hwang, c. y. signaling and the pricing of unseasoned new issue. journal of finance, 1989, 44(2): 393-420. [9] welch, i. seasoned offerings, imitation costs, and the underpricing of initial public offerings. journal of finance, 1989, 44(2): 421-450. advances in systems science and applications (2014) vol.14 no.1 84-100 optimized resource-to-task scheduler for cloud service r. a. karthika and a. shajin nargunam computer science & engineering department, noorul islam university, tamilnadu, india-629180 abstract a key concern in cloud computing setting is to capitalize on turnover by accommodating all arriving needs and to diminish bad consequences for cloud providers. attaining these purposes extremely depends on optimal usage of accessible resources in datacenters. to diminish the power and time consumption in cloud computing environment, the previous work evaluate the process of identifying level of each server peer consuming power and task execution time to perform web requests from client peers. this work explains inadequacies caused by the absence of resource organization method and propose a geometric representation using scheduler-based optimal resource allocation (s-ora). the proposed s-ora scheme investigates the suitability of commercial cloud service using amazon ec2 to hierarchical data exchange between multiple cloud instances with minimal resource utilization. the experimental evaluation shows that proposed performance of cloud computing services including amazon ec2, with suitable management of resources result in an increment of profit by reducing rejected requests of the cloud. keywords cloud instance, distributed networks, resource utilization, rejected requests. 1 introduction with the fast development of computing, storage space and system tools, dispersed calculating paradigms have undergone intense amends in the precedent decade. such a mutiny allows request service providers (asps) to organize treacle or even pet scale requests for manufacture reason. for instance, employing cluster or grid calculating amenities, scientists are capable to sprint large-scale climate anticipate representations with a huge quantity of data engendered from the most superior scientific tools, and regularly issue the estimate outcome to the universal public. nevertheless, owing to the vast computational intricacy and elevated quantity of data, such requests need demanding resource practice and place a serious financial weight on the association who systems, upholds and functions these possessions. in the kingdom of large-scale dispersed computing, the rising cloud computing notion, with its almost infifinite possessions and suppleness, is being accepted with the assurance to release asps from the upfront system and preservation cost of the communications. as an efficient means of giving subtracting resources in a shape of usefulness, advances in systems science and applications (2014) vol.14 no.1 85 cloud computing has lately concerned a considerable quantity of concentration for both manufacturing and academic world. the example move to cloud computing is determined by strapping command, particularly from endeavors, to progress the general efficiency by means of and running computing possessions. cloud repair providers employ datacenters to stipulation a common pool of calculation, storage space and bandwidth possessions, to be exercised by requests when the requirement arises. as resources at datacenters are common by using virtualization, requests are permissible to statistically complex such possessions in the structure of practical machines. completion of cloud arrangement has been probable with release source and profitable software letters. in the middle of them opennebula, eucalyptus, and openqrm are finest option for making iaas cloud communications as a release basis service. bringing services in iaas stage has been probable by means of datacenters as possessions with the assist of virtualization knowledge. virtualization is equipment that intangibles absent the particulars of corporeal hardware and gives virtualized resources for high-level requests. virtualization is a leading expertise in the countenance of resource operation sorting in huge kind of resources from mainframe, recall to storage and system. leveraging virtualization creates it probable to combine, disconnect and evolutes resource operation. spanning and dimension of the cloud possessions demand arrangements for resource administration and transmitting requests. this subject is an inspiration to suggest algorithms and machines for allocation, harmonizing and organization of resources in cloud surroundings. current years there have been planned different algorithms for organization and qos provisioning in the cloud resources. the objective of such a cloud is to supply competent good method questioning on the back-end data at a short cost with clever mode while being reasonably practical, and in addition, best advantageous and also receiving decrease of preparation cost on command changes. a worth over the commission charge for every structure can make sure proceeds for the cloud. and also inside cloud is occasionally inform his caches in turn as necessary on diverse cloud and other command will almost runs a diverse enthusiastic server at an instance on cloud. 2 related work cloud computing is a hopeful commercial framework that promises to do away with the obligation for supporting pricey computing amenities. nonetheless, the current cost-effective clouds created to continue web and minute database workloads tremendously vary from characteristic computing workloads. while services related to current cloud computing are insufficient for scientific computing, many-task computing [1] employing loosely coupled applications which did not provide with resources, required instantly. our work differs in 86 r. a. karthika: optimized resource-to-task scheduler for cloud service the problem that the address is towards the minimization of resource utilization obtained from the resource pool instantly. subsequently, vm migration algorithm based on nash equilibrium [2] solves resource utilization maximization problem in virtualized data centers. cloud computing is a deep uprising method in its calculation ability. the major purpose currently is to decrease the charge of organizing a service in the cloud and containing correct coordinative in among the models. public, private, and mixture cloud [3] surroundings all facade the presentation confines intrinsic in todays requests and networks. in turn for endeavors to exploit the plasticity and cost investments of the public, private, and cross cloud they have to conquer the similar latency and bandwidth restraints that confront dispersed it communications environments [4]. the optimization trouble of reducing resource cost in cloud for meeting service necessities is analyzed in literature[5]. social clouds provide the possibility to share resources among clients. the benefit provided to clients within a social network community using different economic patterns and evolved technical metrics, through simulation [6]. a novel framework called, network flow based resource allocation nfra, for reducing the energy consumption and increasing the profit was introduced in [7]. the ocrp [8] gave provisioning for computing resources to be used in multiple provisioning stages as well as a long-term plan in which cloud consumer minimize cost of provisioning in cloud environments. in literature[9], a novel highly decentralized information accountability framework to keep track of the behavior of the users in the cloud was maintained to provide the accountability. the problem for assigning with a set of clients for certain demands towards a set of servers with capacities and degree constraints are presented in literature[10]. the goal of heterogeneous resource allocation was to find an allocation, in such a way that the number of clients allocated with a server is smaller than the degree of assigning to the server. at the same time, their overall demand of the client should be smaller than the server’s capacity, while maximizing the overall throughput. however, achieving security in cloud was another major concern. the author presented fade [11], a mechanism to achieve full security goals upon a set of cryptographic key functions supported by a quorum of key managers. in literature[12], the task of assigning a third party auditor was the main focus that works on behalf of the cloud and client to verify the integrity of data stored in client. consequently, the choosing nodes for implementing a job in the cloud computing must be measured to develop the efficiency of the resources [13]. the author provided a method called, nephele, a first data processing framework that uses dynamic resource allocation [14] for both the tasks, scheduling and execution. with cpu and memory as constraints, a protocol called gossip advances in systems science and applications (2014) vol.14 no.1 87 [15] was designed that minimizes the cost for adapting an allocation that executes on dynamic input and does not require global synchronization. scheduling algorithms for parallel jobs [16] make efficient use of the two tier vms to improve the responsiveness which significantly outperforms commonly used algorithms such as extensible argonne scheduling system in a data center setting. a method for the efficient mapping of resource requests with a heuristic methodology [17] was addressed. routing and scheduling algorithm [18] for cloud architecture target minimal total energy consumption by switching off unused network and/or information technology (it) resources. the author provided maximized resource utilization with optimal execution efficiency [19] using the proportional share model. various workflow scheduling algorithms are listed and compared their characteristics and applicability for cloud scheduling and concluded that hcoc schedulers superior performance may partially be due to its multicore awareness, which is clearly a characteristic requiring consideration in hybrid cloud computing [20]. a mechanism for scheduling single tasks considering two objectives: monetary cost and completion time and dynamic scheduling of scientific workflows [21] was proposed. a cloud scheduler [22] considers both user requirements and infrastructure properties assured users that their virtual resources are hosted using physical resources that match their requirements without getting users about the details of the cloud infrastructure. all the works mentioned above provide mechanisms for resource management in cloud environment. certain other problems related to resource allocation for cloud in the existing p2p infrastructure has to be addressed. so, the problem specification is provided in depth, followed by the method adopted to solve the issue. 3 problem specification normally, cloud consists of set of resources. to manage the set of resources in the cloud, it is necessary to provide a resource manager (rm) to control and set up. the request sent by the clients (c) for the requisition of resources is processed based on dispatcher algorithm. once the resources are allocated and released, resource monitoring unit (rmu) set up a list of resources which are free from the data centers on it. the resource monitoring unit also provides information about the unavailability of the resources in the data centers. the sequence diagram described in fig.1 uses web service as an interface to enhance the communication between the clients and clouds. the workflow of managing the resources proceeds with the clients’ requests. the foremost step is that the client sends a resource request to the cloud, through the web interface (wi). the wi then passes onto the authorization unit 88 r. a. karthika: optimized resource-to-task scheduler for cloud service fig.1 resource allocation (without scheduler) (au). the au check for authorization and proceeds with the next process upon successful completion of the authorization. it will pass the clients requests to the resource manager (rm). upon analyzing the set of resources required by the client, the rm passes the requests to the resource dispatcher unit (rdu). before dispatching the set of resources to the client, the rdu checks the clients authorization information from the authorization unit (au). then the au allocates the resources to the clients who send requests. by this way, the resources are allocated to the clients from the clouds. 4 scheduler-based optimal resource allocation in cloud environment the previous section discussed about the process of managing the resources in the clouds in conventional manner. if allocation of resources is done improperly, then the resources which are free are simply assigned to the task accomplished. this results in the wastage of resources in the resource pool. to enhance and to further optimize the resource allocation in the cloud, in this section we describe a mathematical model using scheduler-based model with a set of constraints. as a solution for resource utilization in hierarchical distributed peer networks, inward requests to cloud primarily are sent to an essential component. this vital component called the scheduler, judge quantity of present resources in cloud instances and employs an improvised dispatcher algorithm to choose which cloud instance swarm this request and propels essential instructions to its virtual machine formation. at first we design the architecture, further explain the scheduler-based optimal resource allocation using improvised dispatcher algorithm. advances in systems science and applications (2014) vol.14 no.1 89 fig.2 architecture of s-ora 4.1 system architecture of scheduler-based optimal resource allocation the scheduler-based optimal resource allocation is designed based on sensible clouds like amazon ec2. the architecture of scheduler-based resource allocation is shown in fig.2. the process of allocating the resources is described individually based on the four components namely, the client, the allocator, the scheduler and finally the optimizer. the process starts with the (i) client (c), (ii) allocator comprises of web interface (wi) and authorization unit (au) (iii) scheduler schedules the availability of resources according to the availability and finally (iv) optimizer comprises of resource manager (rm), resource monitoring unit (rmu) and resource dispatcher unit (rdu). the notations will be used throughout the work. 4.2 process of scheduler-based optimal resource allocation to balance the set of utilized resources and free resources in the cloud, in this section, the resources are distributed among the clients to increase the availability with the help of scheduler. the scheduling of resources in the clouds are focused oriented towards the management of memory and cpu usage of every clients required. a mathematical model is presented here to introduce a parametric environment for better allocation of resources in cloud computing environment. before assigning the resources to the clients, the current stature of the client is to be noted. the stature includes the requirement of resources, memory and cpu usage. for this purpose, a parametric setting is initiated in a method to think about proximity and resource consumption on cloud instances as illustrated below 90 r. a. karthika: optimized resource-to-task scheduler for cloud service using the following constraints. let us consider the scheduler-based optimal resource allocation problem for processing and memory usage denoted by the equations (1) and (2) respectively. α = n∑ i=1 (procc[avail]− proci[cons] total procc (1) β = n∑ i=1 (memc[avail]−memi[cons] total memc (2) where i = 1, 2, ..., n requests placed by the clients to n cloud instances denoted by c = 1, 2, ...,m. the scheduler-based optimal resource allocation using processing and memory are further denoted by objective function as given below: x = α ∈ procnsymbolizes x =  x1 x2 ... xn  , xi ∈ proc (3) similarly, y = β ∈ memnsymbolizes y =  y1 y2 ... yn  , yi ∈ mem (4) where procn denotes set of n representation or n requests placed by the clients. procc[avail] and memc[avail] represent the current processing and memory availability in cloud instances denoted by ‘c’ for a particular request with total processing and memory available in cloud ‘c’ represented by total procc and total memc respectively. proci[cons] and memi[cons] represents the processing and memory consumed for the corresponding ith request. this design creates the model selfsufficient of quantity of resources in each cloud instance and creates a reliable representation for all cloud instances. the two parameters, proci and memi show the ratio of resource utilization for every machine. the two parameters are evaluated for every request and for every clients present in the cloud. the process of scheduler-based optimal resource allocation in cloud computing environment is performed using (a) cloud interface generation and (b) allocating optimal set of resources which will be discussed in the forthcoming section. 4.3 cloud interface generation the cloud interface generation (cig) forms the foremost step in schedulerbased optimal resource allocation. the cig generate, retrieve and modify the advances in systems science and applications (2014) vol.14 no.1 91 fig.3 model for scheduler-based cloud interface generation data items from the cloud instances. with the interface generated, the clients in the clouds derive the capability of the cloud to manage, link and data placed in it. the cloud interface send and process the data items through the set of cloud instances. with the cloud instances, the client sent the request for resources. most of the existing cloud process through these interfaces. a sample model for the cloud interface generation using scheduler-based model is shown below in fig.3. for the purpose of data storage operations, the client requires to know only about the link objects and data objects. as illustrated in fig.3, the client performs a put to the link url and creates a new link with the specified name. once the link for the client is created, the scheduler does the job of scheduling the processing and memory of the respective client and finally performs a put to form a new data object url. the subsequent get then fetches the actual data object and its corresponding value section. 4.4 allocating optimal set of resources once the cloud interfaces has been generated, the clients select the cloud instances from the set of cloud instances available in the network. next, the optimal resource allocation is taken place for each client who sends requests to the scheduler as illustrated in fig.4. the scheduler allocates the resources based on the resource availability. the optimal resource allocation is done based on the process of data exchange between the available cloud instances and assigning/releasing the resources based on the users tasks. the processes is based on two steps, one is task 92 r. a. karthika: optimized resource-to-task scheduler for cloud service selection and node selection. for the optimal set of allocating resources, the scheduler maintains resource pool and task pool. the resource pool consists of set of freed resources and the task pool consists of set of tasks which are not assigned to any resources. at first, the scheduler checks these two pools before assignment of resources. the scheduler identifies the rate of resource utilization and further identifies the task which is to be performed. once the resource from the resource pool is selected, the particular resource is locked by the scheduler and it cannot be assigned to any other tasks requested by the clients. once the resource is locked, the corresponding node which holds its task is ready to send the task to finish. once the task is received by the node, the resource starts its process to accomplish the task. fig.4 scheduler-based optimal resource utilization using improvised dispatcher algorithm for distributed hierarchical peer networks if any new task enters into the network, the scheduler checks the status of the task. if the task is unassigned, the scheduler determines the resource requirements of the corresponding task and further assigns it. based on the utilization of resources and task, the nodes will adjust its corresponding resource allocation advances in systems science and applications (2014) vol.14 no.1 93 processes. the performance of the scheduler-based optimal resource allocation is analyzed in the forthcoming section. 5 experimental evaluation the scheduler-based optimal resource allocation (s-ora) for hp2p networks with efficient data exchange among cloud instances are implemented in java using amazon ec2 a web service that provides resizable computing capacity in cloud. the s-ora uses the amazon ec2’s simple to use web service interface to obtain and configure that reduces the time taken to obtain and boot new server instances to minute. the performance evaluation tests aimed at comparing the direct invocation for cloud computing services using hierarchical distribution process with challenging interactions through traditional scientific computing. the cloud computing services at first recognizes the source and destination node for data transfer from one cloud instance to the other. the source nodes effectively chose the destination node based on hierarchical distribution process. so, the data transfer is successfully performed in cloud computing environment with amazon relational database service. then the fitness of commercial cloud service to hierarchical data exchange between multiple cloud instances is measured. presenting an efficient commercial cloud structure for service provider with optimal resource utilization and fast and accurate data exchange between the cloud service users. the performance of scheduler-based optimal resource allocation is evaluated by, number of rejected requests, time consumption to exchange data between cloud instances and resource utilization by clients from different cloud instances. in this section the experimental setup for designing s-ora that is used in our experiments is explained. the experiments were conducted on amazon’s ec2 infrastructure due to the popularity, feature rich, and stable commercial cloud available. it offers distinct resource configurations for virtual machine instances. amazon ec2’s interface minimizes the time required for various instances according to the changes observed in computing requirements. s-ora experiments with c1.medium, a compute optimized instance type, a 32-bit processor, 1.7 gb ram and 350 gb local disk storage. 6 results and discussion in s-ora, hp2p networks are designed for cloud computing service composition to capture the minimal resource allocation for service performance to other systems written in mainstream languages such as java using amazon ec2 web service. independent tests are run with growing number of applications, and constant number of service requests sent by each user. the performance graph and table describes the evaluation of the performance of s-ora. 94 r. a. karthika: optimized resource-to-task scheduler for cloud service 6.1 measure of number of rejected requests the number of rejected requests [clientrejreq] measures the number of request rejected as made by the client to the cloud instance through scheduler using amazon ec2 with windows server. the rejected request for a client is evaluated based on the difference between the requests made by the client and rejected request made by the cloud for the specific client at a particular time period. a rejected request for a client is given as: clientrejreq = clientreq − cloudrejreq (5) table 1 described the performance of s-ora and compared the results with existing works like bargaining towards maximized resource utilization in video streaming datacenters [1], ccs using scientific computing [2] based on requests to be rejected. table 1 number of requests vs. rejected requests number of requests (task/minute) number of rejected requests (task/minute) proposed s-ora existing nash bargaining solutions existing ccs using scientific computing 5 1 3 4 10 3 5 7 15 4 8 13 20 7 12 16 25 10 14 17 30 12 15 22 fig.5 shows the ratio of requests made by the client to cloud instances, to the number of rejected requests from the cloud instances according to the availability of resources. sending the client requests to cloud, either result in accept of the particular request, if the cloud has resource to be allocated from pool of resources or reject the clients request if the resource is not available. with the help of improved dispatcher algorithm, the number of rejected request is measured. the algorithms effectiveness is measured using the rejection rate. higher the number of rejection rate, lower the performance of algorithm. since the resource is allocated with the help of scheduler in the proposed model, the performance of s-ora is improved when compared to other algorithms. the number of rejected request using s-ora is 40%, 50% using nash bargaining solutions and 70% with ccs using scientific computing. when compared with the three algorithms, the rejected rate using s-ora is comparatively less as the scheduler schedules the advances in systems science and applications (2014) vol.14 no.1 95 fig.5 number of requests vs. rejected requests resource using the resource pool whereas the rejected rate using nash bargaining solutions is 50% which is less when compared to ccs as it uses the nash bargaining solution. 6.2 measure of time consumption for data exchange the time consumption [timecon] in s-ora measures the time consumed for data exchange between the cloud instances. the time consumption is evaluated on the basis of distance for particular data exchange by the cloud instances divided by the speed with which the operation is performed. the formula to evaluate time consumption is given below: timecon = distancedataexc/speed (6) the time consumed to exchange the data by the cloud instances are illustrated in table 2 for the s-ora with the existing works like bargaining towards maximized resource utilization in video streaming datacenters, ccs using scientific computing. fig.6 describes the consumption of time taken to exchange the data among the cloud instances in the distributed peer networks. in the proposed s-ora, the deployment of commercial cloud computing services is efficiently achieved using the scheduler-based cloud interface generation. the is due to clear separation between resource manager (rm) and resource dispatcher unit (rdu), where the scheduler maintains the set of resources and at the same time checks through the pool of resources and allocates accordingly. this in turn results in minimum 96 r. a. karthika: optimized resource-to-task scheduler for cloud service table 2 data size vs. time consumption data size (kb) time consumption for data exchange (seconds) proposed s-ora existing nash bargaining solutions existing ccs using scientific computing 10 10 15 20 20 18 24 26 30 22 33 34 40 30 42 45 50 38 56 58 60 43 68 70 fig.6 data size vs. time consumption time consumption to the hierarchical peer users. rather than existing works, the s-ora consume less time to exchange the data among the set of cloud instances. the variance in time consumption is 5-10% low in the s-ora whereas using nash bargaining solutions the time consumption is comparatively higher due to the use of vm migration algorithm and finally higher in ccs with many task computing. 6.3 measure of resource utilization by clients the resource utilization [resutil] in s-ora measures the utilization of resources by the clients from different cloud instances through scheduler. resource utilization is the summation of resource consumption by the total resource availability. advances in systems science and applications (2014) vol.14 no.1 97 table 3 algorithms vs. resource utilization. algorithms resource utilization (%) proposed s-ora 50 existing nash bargaining towards maximized resource utilization in video streaming datacenters 70 existing ccs using scientific computing 80 fig.7 algorithms vs. resource utilization resource utilization is evaluated using the formula given below: resutil = actualresutil/totalresutil = proci +memi/total proci + total memi (7) the comparison of utilization of resources for the purpose of exchange of data using three different algorithms is depicted in table 3. fig.7 describes the utilization of resources for the efficient data exchange between the cloud instances in the hierarchical distributed peer networks. when a network resource, for instance a cpu or a meticulous disk, is engaged by an operation or query, it is occupied for handing out further requests. awaiting requests have to stay for the resources to turn out until the resources have been freed by other clients. existing woks such as bargaining towards maximized resource utilization in video streaming datacenters and ccs using scientific computing has higher the percentage of time that the resource is occupied, the longer each operation must wait for its turn. so, rather than using existing schemes, the s-ora utilizes minimal resources for transfer of data which is handled efficiently 98 r. a. karthika: optimized resource-to-task scheduler for cloud service by the scheduler with the set of cloud instances. further, from the table 3 it is evident that s-ora uses minimum resources (50%) whereas the nash bargaining solutions uses (70%) that is minimum when compared to the ccs (80%) as the nash bargaining solutions equips the pivot point which is determined by the user, but comparatively higher to the s-ora as it uses the scheduler-based cig resulting in minimum resource utilization. finally, it is being observed that the scheduler-based optimal resource allocation technique efficiently manage the resources and tasks of the nodes in the cloud instances. by generating the scheduler-based cloud interface, the allocation of resources is done in an optimal manner. 7 conclusion this paper efficiently analyzes and resolves the issue of resource management in cloud environment by adapting scheduler-based optimal resource allocation technique. to manage the resources in the clouds, the cloud interface is generated primarily with the help of scheduler. with the cloud interface, the resources and the unassigned tasks are balanced efficiently using the scheduler. then the improved dispatcher algorithm is presented to manage the resources and tasks efficiently in the cloud instances. the scheduler-based optimal resource allocation using improved dispatcher algorithm is implemented on amazon ec2 as a practical cloud. the obtained results are compared with the existing works like bargaining towards maximized resource utilization in video streaming datacenters, 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(2013), “towards trustworthy resource scheduling in clouds”, ieee transactions on information forensics & security, vol.8, pp.973-984. corresponding author r. a. karthika can be contacted at: karthika78@gmail.com adv syst sci appl 2017; 3:34–41 published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/498 dynamic fracture tests data analysis based on the randomized approach marina v. volkova1∗, oleg n. granichin1,2, yuri v. petrov1,2, grigory a. volkov1,2 1saint petersburg state university, st. petersburg, russia 2institute of problems of mechanical engineering russian academy of sciences, st. petersburg, russia abstract: methods of measurement and prediction of materials dynamic strength are complicated and not standardized. usually dynamic tests are labour-consuming and each assay demands a lot of time in contrast to static experiments. the way of experimental data treatment based on incubation time criterion and randomized approach of sign-perturbed sums is considered. it is shown that a few experimental points are enough to determine strength parameter with accuracy proper for engineering. the lack of real experimental data is compensated in spsalgorithm by carrying out a large number of virtual random tests in which the bernulli signs are randomly generated in this method. keywords: sign-perturbed sums, incubation time, randomized approach, data analysis 1. introduction one of the most important problems in mechanical engineering is measuring the strength and rheological parameters of materials. determination of the static strength or young modulus is a common procedure, because these values result from direct observation. there are standard measurement techniques to obtain the average value of a required parameter with a certain degree of accuracy, for example σc ∈ [σc−;σc +]. also direct observations stipulate that the physical sense of the measured parameter is established and well-known, and all researchers have a unified conception about the final result. the more complicated situation is observed in dynamics, where material strength cannot be characterized by one parameter of the critical stress. experimental test results demonstrate that under intensive impacts, specimens can resist to stress level significantly higher than their static strength σc and stress value at fracture moment σ∗ depends on the rate of loading and the shape of breaking pulse [1–7]. these strain-rate dependencies are interpreted as passport specifications of materials in certain theories. the main difficulty of these approaches is that the variety of strain-rate curves is infinite due to strong influence of impact conditions to σd value. also it is impossible to describe material strength in the same way when fracture is initiated by threshold impacts. in this case the stress level at the breaking moment can be less even than the static strength, this phenomenon is called the fracture delay effect. the structure-temporal approach based on the incubation time criterion of fracture [8,9] is considered at the present work. the main idea of this criterion is that fracture does not occur instantly and there is some characteristic time for every transient process. implementation of only one additional strength parameter as the incubation time τ permits to predict fracture ∗corresponding author: m.volkova@spbu.ru http://ijassa.ipu.ru/ojs/ijassa/article/view/498 dynamic fracture tests data analysis based on the randomized approach 35 stress value and calculate strain-rate dependencies for all types of impacts. this structuretemporal approach was successfully applied for many different problems of dynamic strength estimation for different materials and condensed matters e.g. dynamic fracture of rocks and concretes, dynamic yielding of metals, acoustic ultrasonic cavitation of liquids, etc. [9–12]. the tests measuring incubation time directly are not realized, and now its value could be obtained only implicitly. the simplest way is to select an appropriate value of τ ensuring good correspondence between a model curve and a scatter of the experimental data. any method could be applied for fitting the model curve, e.g. least mean squares (lms) method, but it gives only one value of the incubation time and there is no any estimation of inaccuracy when we have an absence of any adequate variability in observations. standard estimation algorithms use typically the condition of persistent excitation in data. however, this condition is difficult to ensure in the considered problem since dynamic tests are very complicated and labour-consuming, so usually there are no many experimental points to treat. this often results in a degenerate observation data and complicated identification problems. the wellknown set-membership approach to identification uses no statistical properties of the noise, but assumes some known upper bounds on uncertain system components instead. the purpose of the approach is typically to compute some upper or lower set estimates on the set of dataconsistent parameters and no convergence of this set estimates to the true unknown parameter vector can be achieved without any significant additional assumptions (see, e.g. [13, 14]). in the context of the considered problem the robust estimates of min & max methods give a very wide interval for τ with high inaccuracy and this result is very conservative. randomness of the experimental values stipulates to use stochastic methods for treatment algorithms [15,16]. new sps (sign-perturbed sums) procedure proposed in [17] provides rigorously guaranteed non-asymptotic confidence regions for the unknown parameters τ of a linear dynamical control plant in the small-sample setting. in this paper we adopt this approach to the problem of the incubation time evaluation. this method is applicable to the problem because it can provide a confidence interval of τ with the rate of accuracy admissible for engineering. approximately ten data points are enough to determine an average value of incubation time with ε = 20− 35%. the applicability of the new algorithm to the incubation time approach will be illustrated on few dynamic fracture tests of different types of rocks [18]. 2. incubation time approach the main features of the structure-temporal approach are the following. the general form of the incubation time criterion is 1 τ ∫ t t−τ ( σ(t′) σc )α dt′ ≤ 1 (2.1) where σ(t′) is the loading stress function, τ is the incubation time of the fracture, α is a dimensionless parameter, for most brittle materials α = 1. according to (2.1) the fracture does not occur while left part of the criterion is less than one and the fracture moment t∗ corresponds to equality. experimental dynamic tests are often carried out in overthreshold regimes hence stresses grow linearly until the fracture. then the shape function of the loading impacts can be determined by strain-rate of load ε̇ and elastic modulus k: σ(t) = h(t)kε̇t, (2.2) where h(t) is heaviside step function. substitution function (2.2) to criterion (2.1) leads to the following equation for fracture time t∗ h(t∗) ( t∗ τ )α+1 − h(t∗ − τ) ( t∗ τ − 1 )α+1 = s (2.3) copyright c© 2017 assa. adv syst sci appl (2017) 36 marina v. volkova, oleg n. granichin, yuri.v. petrov, grigory.a. volkov fig. 1. dynamic fracture of gabbro-diabase [18]. blue points are experimental data. lines are theoretical curves plotted by the incubation time fracture criterion: solid line least mean squares method; dashed lines min & max robust method where s = (α + 1) ( σc/(kε̇τ) )α is a dimensionless parameter, which value depends on strainrate of load impacts. fracture time t∗ could not be negative, hence h(t∗) = 1 and there are two cases: when t∗ > τ or t∗ ≤ τ . from expression (2.3) it follows that s = 1 for t∗ = τ , hence the final equation set determining fracture moment t∗ is{ (t∗/τ) α+1 = s, s < 1, (t∗/τ) α+1 − (t∗/τ − 1)α+1 = s, s ≥ 1. (2.4) thus, incubation time approach is able to predict strain-rate dependence for the dynamic threshold of fracture σ∗(ε̇) = kε̇t∗. in most cases parameter α equals to 1 during calculation of dynamic strength of brittle materials such as rocks. then roots of equation (2.4) could be expressed in an explicit form and the critical fracture stress is the following: σ∗(ε̇) = ϕ(τ, ε̇) = { σc + τ 2 kε̇, ε̇ ≤ 2σc/kτ,√ 2σcτkε̇, ε̇ > 2σc/kτ. (2.5) an application of the incubation time criterion is demonstrated on the example of an impact fracture test on gabbro-diabase and marble. the theoretical curve in comparison with experimental [18] points is shown in fig. 1. the value of τ = 0.61 µs (solid line) is calculated by lms method, as it is mentioned above this result does not provide any information about inaccuracy of this value. another min & max method gives very wide interval for incubation time value τ ∈ [1.31; 2.21] µs, and there is no information about properties, that new experimental points would lie in this interval. hence, it is necessary to apply some method of data analysis, which gives, at first, a more accurate interval for possible values of τ and, at second, the degree of confidence in it. 3. problem description in dynamic test experiments we can choose acting factor ε̇ and to get the correspondent observation σ∗. dynamic test data are fitted using the following model of n noisy copyright c© 2017 assa. adv syst sci appl (2017) dynamic fracture tests data analysis based on the randomized approach 37 observations: σ∗i = ϕ(τ, ε̇i) + vi, i = 1, 2, . . . , n (3.6) where vi is a random noise (an inaccuracy) with symmetrical distribution. if the strain-rate dependence satisfies the principles of structure-temporal approach, then function ϕ(τ, ε̇i) is ϕ(τ, ε̇i) = kε̇t∗(τ), (3.7) where t∗ is the fracture time predicted by the incubation time criterion by solving equation (2.4). equation (3.7) allows to calculate fracture stress for different τ , then the lms method gives the best fitted value of the incubation time, for that follow sum has a minimum value n∑ i=1 ( ϕ(τ, ε̇i)− σ∗i )2 → min τ . however, we are not able to get a sufficiently good confidence interval for unknown τ without significant restrictions for the noise vi when n is small. the objective is to construct confidence regions for unknown τ that have guaranteed userchosen confidence probabilities for finite, and possibly small number of data points. it must be defined by the observations of outputs {σ∗i}ni=1 and known acting factors {ε̇i}ni=1 which may be chosen. the constructed regions are almost distribution-free, as the only assumption is the noise to have a property of symmetry. this is important since in practice the knowledge about the noise distribution is limited. additionally, the confidence regions should contain the least-squares point estimate. 4. sps procedure for constructing of confidence regions for a finite number of observations we can use the following procedure which is similar to sps procedure from [17]: the lms estimate is obtained as the solution of the equation h0(τ) = n∑ i=1 (σ∗i − ϕ(τ, ε̇i)) dϕ(τ, ε̇i) dτ = 0, where dϕ(τ, ε̇) dτ = { 1 2 kε̇, ε̇ ≤ 2σc/kτ, 1√ 2τ √ σckε̇, ε̇ > 2σc/kτ. we will try to exploit the information in the data as much as possible while assuming minimal prior statistical knowledge about the noise. our core assumption is the symmetry of the noise. for some m > 0 we generate n(m − 1) bernoulli random values βij = ±1 with probability of 1/2, and introduce m − 1 sign-perturbed sums hj(τ) = n∑ i=1 βij(σ∗i − ϕ(τ, ε̇i)) dϕ(τ, ε̇i) dτ , j = 1, 2, . . . ,m − 1. if τ ? is a nominal value of τ then h0(τ ?) and hj(τ ?) have the same distribution since {vi} are symmetric. therefore, there is no reason why a particular |hj(τ ?)| should be bigger or smaller than another |hj′(τ ?)| and the probability that a particular |hj(τ ?)| is the m-th copyright c© 2017 assa. adv syst sci appl (2017) 38 marina v. volkova, oleg n. granichin, yuri.v. petrov, grigory.a. volkov largest one in the ordering of {|hj(τ ?)|}m−1j=0 will be the same for all j, including j = 0 (the case where there are no sign-perturbations). as it can take different values, this probability is exactly 1/m . algorithm: 1. given a (rational) confidence probability p ∈ (0, 1), set integers m > q > 0 such that p = 1− q/m . 2. generate n(m − 1) i.i.d. random signs {βij} with prob{βij = 1} = = prob{βij = −1} = 1/2 for i = 1, 2, . . . , n and j = 1, 2, . . . ,m − 1. 3. set t := {τ : sps indicator(τ) = 1}. procedure: sps indicator(τ) 1. for the given τ compute the prediction error for i = 1, 2, . . . , n δi(τ) = σ∗i − ϕ(τ, ε̇i). 2. evaluate h0(τ) = n∑ i=1 δi(τ) dϕ(τ, ε̇i) dτ , hj(τ) = n∑ i=1 βijδi(τ) dϕ(τ, ε̇i) dτ , for j = 1, 2, . . . ,m − 1. 3. order scalars |hj(τ)| from smallest to biggest. 4. compute the rank r(τ) of |h0(τ)| in the ordering, where r(τ) = 1 if |h0(τ)| is the smallest in the ordering,r(τ) = 2 if |h0(τ)| is the second smallest, and so on. 5. return 1 ifr(τ) ≤m − q, otherwise return 0. note that the lms estimate τ̂ has by definition the property that h0(τ̂) = 0. therefore, the lms estimate τ̂ is included in the sps confidence region. the probability that τ ? belongs to t is given in the following theorem. theorem 1: if the observation noise in (3.6) is independent and it has a property of symmetry then for nominal value τ ? of the incubation time we have prob{τ ? ∈ t } = 1− q/m, where m , q, t from steps 1 3 of the algorithm described above. the proof of theorem 1 is similar to the correspondent one in [17] and it is provided at the expanded version of this paper [19]. the main difference is the nonlinearity of function ϕ(·, ·) from (2.5), but it is a convex function with monotonic property. 5. experiments results of sps procedure applied to experimental data of dynamic fracture tests for different types of rocks are shown in fig. 2-4. there are different amount of data points for each material, but it does not influence the applicability of sps algorithm. it provides values of the incubation time with the proper degree of accuracy approximately 20− 40%. copyright c© 2017 assa. adv syst sci appl (2017) dynamic fracture tests data analysis based on the randomized approach 39 fig. 2. dynamic fracture of gabbro-diabase [18]. red line is plotted by lms method; dotted lines are plotted by sps procedure for τ0.9 ∈ [0.39; 0.89] µs; dashed lines τ0.75 ∈ [0.48; 0.78] µs in the case of gabbro-diabase (see fig. 2) two values of confidence probabilities 90% and 75% were achieved by the following values of sps procedure parameters m = 200, q = 20 and correspondingly m = 200, q = 50. this gives us two intervals of the incubation fracture time τ0.9 ∈ [0.39; 0.89] µs and τ0.75 ∈ [0.48; 0.78] µs, while lms method provides τ = 0.61 µs. thus, sps procedure allows us to calculate the incubation time value with relatively small inaccuracy 25% and with sufficiently high degree of confidence 75%. fig. 3. dynamic fracture of coelga marble [18]. red line is plotted by lms method; dotted lines are plotted by sps procedure for τ0.9 ∈ [0.39; 0.89] µs; dashed lines τ0.75 ∈ [0.48; 0.78] µs in the case of coelga marble (see fig. 3) we have similar situation for following values of the sps procedure parameters: m = 100, q = 10 for 90% and m = 100, q = 25 for 75%. the inaccuracy does not exceed acceptable for engineering 30% for both confident intervals of the incubation time τ0.9 ∈ [0.72; 1.26] µs and τ0.75 ∈ [0.78; 1.1] µs, since lms method gives us τ = 0.91 µs. sps algorithm also provides good results for pervouralsky marble (see fig. 4) despite there are only five data points and lms method provides τ = 1.81µs. copyright c© 2017 assa. adv syst sci appl (2017) 40 marina v. volkova, oleg n. granichin, yuri.v. petrov, grigory.a. volkov fig. 4. dynamic fracture of pervouralsky marble [18]. red line is plotted by lms method; dotted lines are plotted by sps procedure for τ0.9 ∈ [0.39; 0.89] µs; dashed lines τ0.75 ∈ [0.48; 0.78] µs this number of points leads to relatively small number of iterations m = 20, so proper confidences are achieved by follows q = 2 and q = 5. these parameters give us incubation time intervals τ0.9 ∈ [1.35; 2.05] µs and τ0.75 ∈ [1.66; 1.91] µs with inaccuracy less than 20%. it should be noted, that 90%-confident and min & max intervals for pervouralsky marble are approximately the same. the advantage of the sps procedure in comparison with robust estimation is that it gives additional information about inaccuracy of the determined interval. 6. conclusion the incubation time approach proved itself as a good instrument to estimate and predict the limit parameters of loading impacts. the main problem is that there is no standard procedure to determine the incubation time value τ with estimation of its inaccuracy ε. the main difficulty is that the incubation time could be measured only implicitly and standard methods of data analysis do not work. the proposed application of sps procedure demonstrates how to calculate the incubation time with certain inaccuracy ε under limited number of experimental points. the advantage of this method is that it allows us to predetermine the confidence of the target intervals for parameter τ . the incubation time approach complemented by sps procedure for data analysis can become a universal instrument of engineering to determine material strength in dynamics. the further researches are aimed to improve developed method in case of arbitrary noises like in [20, 21] acknowledgements this work was supported by russian science foundation (project 16-19-00057). references 1. campbell j.d. & ferguson w.g. (1970) the temperature and strain-rate dependence of shear strength of mild steel. the philosophical magazine, 21 (1), 63–82. copyright c© 2017 assa. adv syst sci appl (2017) dynamic fracture tests data analysis based on the randomized approach 41 2. rosakis a.j., duffy j. & freund l.b. (1984) the determination of dynamic fracture toughness of alsi 4340 steel by the shadow spot method. j. mech. phys. solids, 32 (4), 443–460. 3. shockey d.a., seaman l. & curran d.r. (1983) advanced mathematical tools for automatic control engineers: deterministic techniques, new york, us: springer. 4. nikiforovsky v.s. & shemyakin e.i. (1979) dynamic fracture of solids [dinamicheskoe razrushenie tverdyh tel], novosibirsk, ussr: nauka [in russian]. 5. ravi-chandar k. (2001) experimental challenges in the investigation of dynamic fracture of brittle materials. physical aspects of fracture. nato science series, ser. ii : mathematics, physics and chemistry, 32, 323–342. 6. kalthoff j.f. & wincler s. 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(2010) rock dynamic fracture toughness tested with holed-cracked flattened brazilian discs diametrically impacted by sphb and its size effect. experimental mechanics, 50, 877–885. 13. bai e.-w., nagpal k.m. & tempo r. (1996) bounded-error parameter estimation: noise models and recursive algorithms. automatica, 32 (7), 985–999. 14. blanchini f. & sznaier m. (2012) a convex optimization approach to synthesizing bounded complexity hinf filters. ieee transactions on automatic control, 57 (1), 219– 224. 15. kolbin v.v. (2014) generalized mathematical programming as a decision model. applied mathematical sciences, 8 (69-72), 3469-3476. 16. volkova m.v. (2017) one method to solve a problem of nonlinear two-stage perspective stochastic planning. international journal of pure and applied mathematics, 112 (4), 817-826. 17. csaji b., campi m.c. & weyer e. (2015) sign-perturbed sums: a new system identificaiton approach for contructing exact non-asymptotic confidence regions in linear regression models. ieee trans. on signal processing, 63 (1), 169–181. 18. bragov a.m., konstantinov a.y., petrov y.v. & evstifeev a. (2015) structuraltemporal approach for dynamic strength characterization of rock. materials physics and mechanics, 23, 61–65. 19. volkova m., granichin o., petrov y. & volkov g. (2017) sign-perturbed sums approach for data treatment of dynamic fracture tests. proc.of the 56th ieee conference on decision and control(cdc), melbourne, australia, 1652-1656. 20. senov a., amelin k., amelina n. & granichin o. (2014) exact confidence regions for linear regression parameter under external arbitrary noise. proc.of the 2014 american control conference (acc), portland, usa, 5097–5102. 21. senov a.a. & granichin o.n. (2014) identifikacija parametrov linejnoj regressii pri proizvol’nyh vneshnih pomehah v nabljudenijah. xii vserossijskoe soveshhanie po problemam upravlenija (vspu-2014), moscow, russia, 2708–2719 [in russian]. copyright c© 2017 assa. adv syst sci appl (2017) introduction incubation time approach problem description sps procedure for constructing of confidence regions experiments conclusion advances in systems science and application (2016) vol.16 no.3 11-20 regions of russia economic alignment: trends and tasks for the policy of regional development albert r. bakhtizin1 ,evgeniy m. bukhvald2 and anastasiya kolchugina2 1 central economics and mathematics institute of russian academy of sciences, moscow; 2 institute of economics of russian academy of sciences, moscow. abstract the article provides an analysis of the various programs and concepts, anyway focused at the economic alignment of the regions of russia. the paper discusses the reasons for the stability of interregional economic disparities, its prospects in connection with the priority focus on the innovative modernization of russian economy. the authors provide suggestions concerning the tasks of positive regions economic alignment in russia within the emerging system of strategic planning and its documents, concentrating basic orienteer of spatial development of the russian economy and the federal regional policy. keywords strategic planning; subjects of federation; economic differentiation; policy of regional development 1 introduction during recent years, was clearly delineated the need for qualitatively new approaches to the system of goals and instruments of the federal policy of regional development in the russian federation, including the “classic” or traditional for this policy task of equalizing the levels of socio-economic development of regions of the country. necessary legal and institutional framework for the realization of task by the moment has been formed by the recently adopted federal law on strategic planning in the russian federation no.172[1]. as one of the main priorities of such strategizing we see the need to maintain a high of integrity or integration of the economic space of the country. however, the most significant obstacle to providing such integrated economic space in the country at the present is concerned primarily with continuing deep rupture in the level of social-economic development of regions in russia. these gaps are due both to the structural deformations of the national economy and as well to unsuccessful options of economic reforms in 1990-s and later to low effectiveness of regional policy of the federal center. long-term maintenance in the country regions with qualitatively different levels of socio-economic development has a powerful disintegration effect over the national economy. recognizing the importance of this problem, russian government repeatedly attempted to ensure positive, (i.e. oriented at pulling up lagging regions, and not at the deterrence of the leading regions) economic alignment of subjects of federation as a priority in 12 albert r. bakhtizin et al:regions of russia economic alignment: trends and tasks for ... the regional policy of the federal state. however, serious practical results in this direction havent been achieved. so, in the early 2000-s this range of problems in the sphere of regional development was address to the federal target program[2], which initially was planned to be completed by 2015. but since 2006 this program has been terminated. however, this program worth to be briefly considered since now we are just at the “time point”, when this program was designed to end its action. the program quite objectively estimated the situation with the spatial parameters of the russian economy development. the government document noted: “at the present time the differences in expanding regions of the russian federation in the basic socio-economic indicators has reached a critical level. sharp interregional differentiation is the inevitable consequence of the increase in the number of lagging regions, the weakening of mechanisms of interregional economic interaction and augmentation of interregional discrepancies, which greatly complicate the conduction of national policy of socio-economic transformation”. in this context, the program set very ambitious objectives, namely to reduce the differences in socio-economic development of regions of the russian federation due to the reduction of the gap in main indicators between the developed and lagging regions by 2010 to 1,5 times and by 2015 to 2 times. the reason, why this program was stopped and, accordingly, failed to meet its goals was not only in the insufficient volumes of its financing, but in the apparent isolation of the program from other key directions and priorities of the state economic policy. another reason institutional and instrumental “poverty” of the program, which in fact operated with only one target institution of the “economic alignment policy”, namely with so called regional development fund (doesnt exist now). today we are fully entrenched in the understanding that the economic alignment of the regions of russia can not be a result of one or more funds or other specialized financial institutions, although their role can be very substantial. this alignment in significant parameters is reachable only as a result of the deep changes in the driving factors of the russian economy development, via reducing its dependence on natural resources exploitation in favor of the more rapid growth of high-tech industries. however, as concerned the functions of the above regional development fund, the program actually contained some interesting ideas, which can be used today. according to the program, the fund was intended to act not as an independent financial institution with its “own” budget, but as a “regulating and control center” for various (by purpose and by government affiliation) financial flows, anyway affecting economic and social development of russian regions. in this program it was noted that the fund should be a supervisor for the combined set of relevant parts of federal and regional programs as well as programs and advances in systems science and application (2016) vol.16 no.3 13 projects of industry funding. of course, such multilateral functionality doesnt fully fit with the traditional concept of “fund”, however, the idea of the state institution, aimed to provide unified management, coordination and control of all resources of the regional policy of the federal center seems to us relevant and still in demand at modern time. later adopted conceptual documents of the rf government (for example, the famous “concept-2020”, strategy of innovative development of the russian federation for the period till 2020, etc.) contained certain goals regarding the spatial aspects of the socio-economic development of the country. however, these documents didn’t record either clear-cut priorities of positive economic alignment of the regions or, respectively, special institutions and interventions for achieving this goal. finally we have to admit: for more than 20 years of sovereign russia, several attempts have been made in order to prepare a long-term concept (strategy) of regional (spatial) development of the national economy; to define its priorities, the objectives of this policy and the tools, needed to achieve them. attempts to prepare such a document were numerously undertaken by the former ministry of regional development of russia, by profile committees of the state duma and council of federation of the federal assembly (parliament) of the russian federation, by various expert institutions, etc. however, none of such documents has been brought to the stage of legal approval and contained cost-based objectives, aimed to overcome the extreme parameters of interregional economic disparities in the country. noteworthy that all these papers on regional development policy issues were very similar. all of them operated with approximately same set of instruments, namely: intergovernmental fiscal relations (including inter-budgetary distribution of tax revenues); state programs of territorial development, localization of federal investment projects, etc. besides, the developers of these documents in determining the strategic direction of the federal regional development policy invariably “rotated” around one important contradiction. one position in this dispute insisted on the need to continue to maintain economic alignment as the leading priority for the federal regional development policy. the other position seemed it possible to overcome the absolutzation the idea of economic alignment of the regions towards optimally balancing it with the primary focus of the policy of spatial development on the group of regions-leaders, which can play the role of locomotives for the russian economy as a whole. none of these positions revealed obvious evidence of its benefits and up to now has taken the place of a clearly expressed priority in the economic policy of the federal center. 14 albert r. bakhtizin et al:regions of russia economic alignment: trends and tasks for ... 2 can economic alignment remain a priority of the federal regional development policy? as regards above mentioned dispute, today economic science in russia meets several important issues. first, what is the current degree of economic differentiation of regions of russia; secondly, what is the actual trend of this differentiation at present and what we can expect in the future; thirdly, what is the extent to which economic alignment of the regions of russia its necessary or simply possible to keep among the priorities of economic policy of the federal state. if positive socio-economic alignment of territories still remains among the most important goals of the federal regional development policy, this task should be considered as unsolved. according to our estimates, based on the latest published data on gross regional product grp (2013), the gap between economically extreme regions of russia (tyumen region and the republic of ingushetia) in per capita grp amounted to 15.7 times in comparison with 17.7 times in 1995. however, its hardly an irrefutable argument in favor of an apparent reduction of interregional economic disparities in the country; at best, we can resume only about the stabilization of the situation. this is due to the fact that these year-by-year calculations, based on the two “extreme” subjects of the federation have a high degree of conditionality, particularly in the context of the ranking of the economically less developed regions. the implementation in one of these regions a large, co-financed by the federal government investment project for some time can remove the region from the last positions in the ranking. then the situation changes in favor of some other region, etc. in this regard, more reliable estimates of economic differentiation of regions should be based not on individual points of two regions, but as the relation of upper and lower decile groups of russian regions (see table 1). table 1. the decile coefficients of interregional economic differentiation for the whole set of regions of the russian federation . 1995 r. 1996 r. 1997 r. 1998 r. 1999 r. 2000 r. 2001 r. 3.46 3.13 3.36 3.41 4.07 4.15 3.68 2002 r. 2003 r. 2004 r. 2005 r. 2006 r. 2007 r. 2008 r. 3.18 3.21 3.86 .91 3.78 3.55 3.27 2009 r. 2010 r. 2011 r. 2012 r. 2013 r. 3.33 3.61 3.52 3.86 3.73 source: russian statistical agency data. according to the data, presented in table 1 and based on the so-called ”decile advances in systems science and application (2016) vol.16 no.3 15 ratio of differentiation” (in this case calculated as the ratio for the relevant years between the lowest value of grp per capita among the 10% of regions with highest per capita grp to the maximum rate for the 10% of regions with the smallest per capita grp) within the period 1995-2013 there was no substantial increase in inter-regional economic differentiation in russia. this figure showed rather a wavy trend. the highest ratio took place in 1999-2000 (4.1 4.2) and at present (2012-2013) these figures are located at the level of 3.7 3.8. in economically developed countries similar differentiation coefficients are somewhat lower: they are within 1.5 times (france, usa) or about 2.0 times (germany, italy), reflecting more integrated spatial economic development of these countries and, consequently, a higher level of integration of the regional segments of their economies. this level of differentiation is typical as well for developed countries with a high share of extractive industries in the economy, such as canada, where the decile ratio also does not exceed twice level. at the same time, more pronounced inter-territorial differences can be found in developing countries, for example in china, where the degree of differentiation of regions by the decile ratio varies from 2.8 to 3.5 times[3]. similar is the situation in the investment sphere. in recent years up to half or more of all fixed capital investment in the russian economy was accumulated in 10 leading regions (in 2014 51.8%). another evidence of a high degree of economic differentiation of subjects of the russian federation is also a limited number of regions-donors, the number of which in the current system of intergovernmental fiscal relations still amounts to 14 (2015), although the substantial increase in the number of such regions (up to half of the total number of subjects of federation) numerously has been declared in policy documents on regional and budgetary policy in the russian federation. most likely, the current trend towards relative stabilization of the economic differentiation of russian regions will continue. first of all, it is connected with absence at the moment and in the near perspective evident regions-locomotives of the rapid growth of the economy as a whole. to day the potency of maintaining rapid growth for the former leading regions (regions rich in raw materials and regions, performing the role of leading transactional centers in the russian economy), has largely exhausted. as concerns regions potential “dots” of innovative development, they are still too weak to take on entirely the role of “locomotives of economic growth”. thus, at present its too early to remove economic alignment of the regions from the position of one of the main priorities of the state economic policy. sometimes, the hypothesize is proved that its not necessary to focus this policy specifically on the task of “alignment”; and its only necessary to create for all regions of russia equal favorable conditions for socio-economic development and the prob16 albert r. bakhtizin et al:regions of russia economic alignment: trends and tasks for ... lem of regional and municipal disparities will be resolved “automatically”. but both our and foreign experience of regional development doesn’t confirm such an opportunity. this is true as well as the fact that any isolated measures aimed to align regions will obviously be unproductive, if these measures are carried out in isolation from other positive changes in the russian economy. for positive alignment of the regions the country needs, first of all, to change key drivers of economic growth. it largely co-operates the priorities of regional development policy, including the positive alignment of the regions, with the implementation of structural reforms in the national economy. in this sense, the policy of innovative modernization represents an opportunity not only to achieve structural changes in the economy and, accordingly, radical increase of its competitiveness, but also to create the basis for a “breakthrough” in solving the problems of regional development, including the task of positive alignment of subjects of the russian federation. however, it is difficult to hope for the automaticity of market regulators of comparative regional development, especially in view of the fact that according to their innovative potential, regions of russia differ even in greater degree than in such indicators as grp per capita[4, pp. 35-52]. 3 differentiation of regions and the rates of economic growth in russia is it possible to prove that changes in the degree of economic differentiation of russia’s regions are subject to certain regularities? basing on the analysis of the available data its principally possible to state a hypothesis concerning the fact that in periods of faster economic growth (for example, at the beginning of the 2000s) the rate of differentiation of regions of russia slightly increased, and during the periods of slow growth and stagnation of the economy differentiation is reducing. this hypothesis can be illustrated with the data given in table 2. the data of table 2 illustrate the regularity, proving that a higher economic growth in russia rate usually corresponds to a higher economic differentiation of regions and, conversely, during the period of slowdown of the growth rates, ratio of differentiation is comparatively lower. of course, this is only a preliminary statement, which can not be a reason to think that increasing trend of this differentiation is a obligatory condition or prerequisite of high economic growth rates. accordingly, any slowdown in growth rates is not the very trend of the alignment regions, which could satisfy the country from the point of view of achieving greater integration of its economic space. here we meet one of the most difficult questions for economic theory and practice of regional development policy: is it possible to combine high rates of economic development of the country with a consistent reduction of the gaps which still exist between the regions and, if so, via what means of economic policy this result can be achieved? to our mind, there is a real chance to combine economic alignment of regions advances in systems science and application (2016) vol.16 no.3 17 table 2. decile inter-regional differentiation ratio and general growth rates of the russian federation economy the period of unstable development and crisis decile differentiation ratio for rf regions 1995 1996 1997 1998 3.46 3.13 3.36 3.41 mean rate for 1995-1998 3,34 gdp growth rates ( %) +3.3 -3.6 +1.4 5.3 mean rate for 1995-1998 1.05 the period of relative economic growth decile differentiation ratio for rf regions 1999 2000 2001 2002 2003 2004 2005 2006 2007 4.07 4.15 3.68 3.18 3.12 3.86 3.91 3.78 3.55 mean rate for 1999-2007 3.71 gdp growth rates ( %) 6.4 10.0 5.1 4.7 7.3 7.2 6.4 8.2 8.5 mean rate for 1999-2007 7.09 the period of declining of growth rates and entering stagnation decile differentiation ratio for rf regions 2008 2009 2010 2011 2012 2013 3.27 3.33 3.61 3.52 3.86 3.73 mean rate for 2008-2013 3.55 gdp growth rates ( %) 5.2 -7.8 4.5 4.3 3.4 1.3 mean rate for 1999-2007 7.09 source: russian statistical agency data; authors calculations. with sufficiently high and sustainable rates of the russian economy growth. first, this chance demand for changes in the structure of the economy, which should be clearly associated with the identification and implementation of reasonable measure to reduce economic differentiation of regions and macro-regions of russia. the national development strategy should not just declare this task in general, but should contain economically relevant indicators, quantitatively showing the necessary progress in the reduction of such a differentiation. but this progress is possible only via targeted government economic policy, based on the alignment as a strong priority of national economy spatial development. there is another important condition, which could contribute to the reduction 18 albert r. bakhtizin et al:regions of russia economic alignment: trends and tasks for ... of regional economic disparities simultaneously with the sustainable growth of the national economy. that is setting the entire consistency of “spatial activities of all institutions and instruments of the federal economic policy. by the moment the total picture of ”spatial activities of all these institutions and policy instruments isn’t anyway regulated by the rf government. the main tool of the well-known “social alignment” (i.e. alignment of provision of public goods of a social nature for the russian regions population) is the system of so-called “fiscal equalization” transfers from the federal budget to regional budgets. however, as has shown already the practice of several decades, the current model of financial equalization, partially offsetting the social effects of spatial disparities in the russian economy, actually leads to the conservation of economic background of inter-regional disparities and doesn’t ensure trend to growing financial self-sufficiency of subjects of federation. the grant component of these transfers has almost nothing to do with the intensification of investment processes and growth rates in the regions and for the country in total [5]; doesn’t stimulate sufficient efforts of the regional and local authorities, aimed to support entrepreneurial activity and attraction of investors, including on the basis of public-private partnership. in other words, the financial equalization mechanism acts, but real economic alignment, as shown above, doesn’t occur. practical implementation of the all-national vertical of strategic planning needs for a new model of fiscal federalism for russia with active stimulating mechanisms for the stabilization of sub-federal (regional and local) budgets [6]. however, in the current situation the sphere of inter-budget relations already isn’t the only or even the predominant instrument of the federal impact over socio-economic development of the regions. this impact at present is multilink, and, unfortunately, unsystematic. this situation puts the task of strategic coordination of all the channels (instruments, links) of federal funding for socioeconomic development of regions, in particular, resources going through state targeted territorial development programs, through the operation of institutions for territorial development, as well as in the framework of the “standard” system of inter-budgetary interactions. this practically means a binding of spatial strategic planning and monitoring the final picture of the spatial distribution of all expenditures of the federal budget. here, in fact, it would be possible to return to the idea, proposed in the early 2000s, in the framework of the target state program. that is the idea of setting special “regional development fund”. but now this idea should be implemented at a qualitatively higher level of institutional arrangement, i.e. not simply in the form of a financial institution, but as a supervising institution with the functions (authority) for coordination of the spatial distribution of all types (channels) of federal resources, directed for socio-economic development of regions of the country. advances in systems science and application (2016) vol.16 no.3 19 respectively, should be offered a system of clear and transparent criteria for the selection of regions (macro-regions), to which federal funds are implemented through state target territorial development programs or are established special territorial development funds and corporations, etc. according to the experience of some countries, may be approved a special status the regions, which on the basis of statutory criteria can be declared “economic disaster areas”. these regions will receive in such a case additional forms of government assistance while taking measures to conserve and rehabilitate the regional finances. for example, in germany such territories for a long time are the states (lands) of bremen and saar [7]. similarly, should be formalized principles of spatial localization and concrete functions of federal “development institutions” (“science-towns, special zones and other territories of “advanced” development, etc.), as well as conditions and forms of federal support to various regional development institutions (regional economic zones, industrial parks, etc.). it is important that each institution has a transparent, targeted adjusted spatial pattern (program) of its activities, i.e. among other functions, operated as a policy instrument for regional development. these institutions should be objective-oriented on the spatial distribution of “pulses” of industrial-innovative development and, thus, contribute to the integration of the economic space of the country on the basis of “economy of innovations. 4 conclusion to summarize the research, the following results can be formulated: 1) most likely, the current trend towards relative stabilization of the economic differentiation of russian regions will continue, but letter cannot serve as satisfactory solution of the problem of positive alignment of the total economic space of the country. 2) at present its too early to remove economic alignment of the regions from the position of one of the main priorities of the state economic policy. 3) national development strategy should not just declare this task in general, but should contain economically relevant indicators, quantitatively showing the necessary progress in the reduction of such a differentiation. but this progress is possible only via targeted government economic policy, based on the alignment as a strong priority of national economy spatial development. 4) the russian federation government must ensure the entire consistency of “spatial activities of all institutions and instruments of the federal economic policy and establish a special state institution, aimed to provide unified management, coordination and control of all resources of the regional policy of the federal center. 5) for the state regional policy should be approved a system of transparent criteria for the selection of regions (macro-regions), to which federal funds are 20 albert r. bakhtizin et al:regions of russia economic alignment: trends and tasks for ... implemented through state target territorial development programs; besides following the experience of some countries, may be approved special status the regions, which on the basis of statutory criteria can be declared as “economic disaster areas”. references [1] law principle. (2014),“on the strategic planning in the russian federation” federal law of june, the 28th, russian . [2] law principle. (2001),“on the federal target program :reduction of distinctions in socially-economic development of regions of the russian federation (2002-2010 and till 2015)”, the decree of the russian federation government of october, the 11s, no. 717, russian. [3] nikolaev i. and tochilkina o. (2011),“economic differentiation of regions: estimations, dynamics, comparisons.” analytical report: fbk. [4] valentey s.d. and bakhtizin a.r., bukhvald, e.m. and kolchugina a.v. (2014),“trends of development of russian regions”, economy of the region. ekaterinburg, pp. 9-22. [5] cosgrove michael and marsh daniel. (2013),“ why the slow u.s. economic growth”, journal of academy of business and economics , vol. 13,no. 4. [6] dahlberg matiz. (2001), “fiscal federalism and state-local finance”, regional science and urban economics, vol.31, no.1. [7] bukhvald e.m. (2003), special modes of intergovernmental relations: the realities of russia and the german experience., finance and statistics, moscow. corresponding author albert r. bakhtizin can be contacted at: albert.bakhtizin@gmail.com advances in systems science and applications (2012) vol.12 no.2 162-173 parametric modeling and simulation of orthogonal milling process based finite element method jing sheng1 and liping yao2 1dpt. of mechanical engineering, hubei automotive industries institute, hubei shiyan 442002, china 2dpt. of foreign languages, hubei automotive industries institute, hubei shiyan 442002, china abstract the key techniques of 2d modeling with msc.marc software and the whole modeling procedure of metal oblique cutting process was presented. the rule based on the modeling process was investigated. the finite element simulation of metal machining is a complex process. it is essential to exploit a system to construct a model of simulation so as to obtain simulation data more conveniently and rapidly. this research is significant for the development of parametric modeling. the system’s interface, designed using c++ builder, can access data which includes the geometrical angles and dimensions of tool, the sizes of work, the relative position between tool and work, properties of tool and work, cutting conditions, etc.. the procedure file that is able to model in the msc.marc environment automatically is generated by the program. the parametrical modeling of simulation is completed by the system which calls the procedure file. finally, case studies were performed to study simulation model, interface and simulation. so the parametric modeling is a kind of effective avenue for metal machining process simulation. keywords orthogonal milling machining, parametric modeling, fea, interface design 1 introduction the experimentation, analytic and numerical method are frequently applied on the research on the metal cutting process. the disadvantages of experimentation include the high cost and labor-intensive process, and the analytic methods are very difficult to understand and analyze the machining process in detail. nowadays numerical approaches have been growing acceptance in industries and academia as a method for characterizing machining process [1-10]. it is known that modeling is one of some key techniques, in which geometrical angle and structural parameters of cutting tool are important factors. the modeling procedure, however, is relatively complex and professional. very little research on parametric modeling in metal machining has been reported. numerical simulation will be taken as a tool by consumer, and the parametric modeling is the first problem to be solved because of the practicability. the advances in systems science and applications (2012) vol.12 no.2 163 paper described the general modeling procedure of metal cutting simulation, and presented some key techniques, especially parametric modeling. 2 the parametric modeling of milling process 2.1 geometrical modeling of milling system geometric modeling is very important for the simulation of milling. it directly affects the operation and results of the simulation system. the geometrical model of cutting system is presented in fig.1. for confirming the milling tool’s rotating center conveniently, the origin of the coordinate is placed on the milling tool’s rotating center. in fig.1, ∆x represents the transverse distance between the outer circle of the milling tool and the lateral surface of the workpiece, and ∆y is the height between the outer circle of the tool and the machining surface of the workpiece. fig.1 the mode of two-dimension milling 2.2 geometric modeling of milling tool fig.2 shows the polygon line of a milling tool during upmilling. created the shape of single tooth, the 2-d model of milling tool was derived from duplicating the shape of single tooth along origin o. computing formulas of the coordinates of the points on a tooth are seen in table 1. where θ1 is the angle between face and back of tooth,θ2 is the angle of chip flute; while γ, α01, α02 represent the rake angle, the first relief angle and the second relief angle respectively. the tooth-spacing angle, the width, the height of teeth and the diameter of cutting tool are denoted by ε, ba1, h and d0. the computing formulas of the coordinates of down milling can be obtained by changing the abscissa in the formula into negative number, while keeping the value of ordinate the same. 164 jing sheng:parametric modeling and simulation of orthogonal milling process... fig.2 the polygon line of milling tool table 1 computing formulas of the coordinates of the points on a tooth point coordinates of the points p1 x1 = −d0 2 + h; y1 = h× tgγ0 p2 x2 = −d0 2 ; y2 = 0 p3 x3 = −d0 2 + ba1 × tgα01; y3 = ba1 p4 x4 = y5−y3+ctgα02×x3−tg(θ2−ε+γ0)×x5 ctgα02−tg(θ2−ε+γ0) y4 = y5ctgα02−y3tg(θ2−ε+γ0)+tg(θ2−ε+γ0)(x3−x5)ctgα02 ctgα02−tg(θ2−ε+γ0) p5 x5 = − √( −d0 2 + h )2 + (h× tgγ0) 2 × cos ( ε+ arctg h×tgγ0 d0 2 −h ) y5 = − √( −d0 2 + h )2 + (h× tgγ0) 2 × sin ( ε+ arctg h×tgγ0 d0 2 −h ) p6 x6 = −d0 2 × cos ε; y6 = d0 2 × sin ε 2.3 geometrical modeling of workpiece the computing formulas of the coordinates of the points on workpiece was derived from fig.1. the formulas are shown in table 2. advances in systems science and applications (2012) vol.12 no.2 165 table 2 computing formulas of the coordinates of the points on workpiece point x y w1 d0/2 + ∆x d0/2−∆y w2 d0/2 + ∆x d0/2−∆y + b w3 a+ d0/2 + ∆x d0/2−∆y + b w4 a+ d0/2 + ∆x d0/2−∆y where a, b are length and height of workpiece respectively. 2.4 geometrical modeling of rigid walls two lines,r1r2 and r3r4, were used to represent two rigid walls.the two rigid walls are used to restrict the movement of workpiece on x-direction and ydirection. the coordinates of the points on two rigid walls are shown in table 3. table 3 the coordinates of the points on two rigid walls point x y r1 a+ d0/2 + ∆x d0/2−∆y + 1 r2 a+ d0/2 + ∆x d0/2−∆y + b− 1 r3 d0/2 + ∆x− 1 d0/2−∆y − b r4 a+ d0/2 + ∆x+ 1 d0/2−∆y − b 2.5 material modeling it is necessary to configure properties of material. the workpiece material during machining generates elastic-plastic deform under high temperature, large deformation and large deformation rate. considering the effect of strain hardening, strain-rate hardening, and thermal softening on the stresses, johnson-cook’s empirical model (see (1)) was adopted. the jcmaterial law parameters are obtained by shpb equipment (see table 4). σ̄ = [a+b(ε̄)n] [ 1 + c ln ( ˙̄ε ε̇0 )][ 1− ( t − troom tmelt − troom )m] (1) where ε̄ is the equivalent plastic strain, ˙̄ε the equivalent plastic strain rate, t the temperature; while a, b, n, c, m and ε̇0 is the parameters determined by a material itself; tmelt and troom represent the melting temperature and the room temperature respectively. 166 jing sheng:parametric modeling and simulation of orthogonal milling process... table 4 jc material law parameters parameter a/mpa b/mpa n c m value 626 3614 0.82 0.0268 1 meshing, configures of contact, boundary conditions, remeshing, analysis conditions, etc. are no longer mentioned here. 2.6 friction modeling between tool and chip there are two explicit areas on the rake surface: slip region and glue region. on the basis of research, constant coefficient friction is applied in slip region and constant friction stress is used in glue one. the friction stress is written as [2]. f = { µσn σf = µσn k σf = k (2) where σn is normal stress. µ is friction coefficient and k is shear stress. 2.7 the criterion of chip separation during the simulation, there are criterions that make the chip separate from workpiece and rake face. they are divided into geometric criterion and physical criterion. the geometric criterion decides the separation through the changes of geometric dimension of deformable body. the physical one is used to identify whether magnitude of physical quantity causes critical value or not. in fact, chips are separated by setting a minimum force or stress of the nodes as threshold. 2.8 equation of heat conduction because the system consists of workpiece, chip and tool generates heat continuously, the first and the second deformation zone of the workpiece go through plastic and elastic deformation. besides, the rake surface of the tool has severe friction [4-5]. equation of the heat conduction in unsteady-state temperature field (take variable thermal conductivity into account) is defined as follows: ρc ∂t ∂t = k ( ∂2t ∂x2 + ∂2t ∂y2 ) + dk dt [( ∂t ∂x )2 + ( ∂t ∂y )2 ] −ρc ( wx ∂t ∂x + wy ∂t ∂y ) +q∗ (3) where k represents the thermo conductivity coefficient and t is temperature.ρ is the material density and c is thermal capacity. x and y are cartesian coordinate advances in systems science and applications (2012) vol.12 no.2 167 system. wx and wy represent velocity component of kinetic heat-source in x and y axis respectively. q∗ is heat generation rate per unit volume. q∗ = whσ ˙̄ε/j (4) where wh is the ratio that plastic deformation workpiece turn into heat energy. σ̄ is equivalence stress. ˙̄ε is equivalence strain ratio. j is coefficient of thermal equivalent of workpiece. because the amount of radiant heat is little, it is ignored. 3 key techniques of parametric modeling of simulation of milling process 3.1 interface design 3.1.1 the interface design between c++ builder and database exploiting database, database module and database engine provided by c++ builder or ado(active data object)were employed to access database. the tables whose type is dbf and frequent bde engine were used, while parameters about bde were set, such as path, type and language drive. 3.1.2 the interface file of parametric modeling in msc.marc governed msc.marc software characters, the system’s knowledge base rules were established. thus, procedure files were written using c++ builder code according to the rule. based on the model of the tool and the workpiece, the topology and geometric information of the tool, such as points, line and surface, the procedure of their modeling as well as meshing was written in procedure file line-by-line. then the modeling of the rigid walls was done, too. while the relative position between tool and workpiece, material model, friction model between tool and chip, properties of tool and workpiece, cutting conditions, the configures about finite element simulation and so on, were written into the file in same way. therefore the created procedure file can be operated according to specified manner. so the modeling process becomes easy and rapid. fig.3 shows the block diagram of modeling process. the structure of procedure file is seen in fig.4. 3.2 creating the parametrical modeling file parametric modeling file (procedure file) can finish the scheduled task in finite element software msc.marc to model and simulate the process of milling. so through explanation facility of a process file, the geometric information of cutting system’s points and lines can be performed. 168 jing sheng:parametric modeling and simulation of orthogonal milling process... fig.3 the block diagram of modeling process fig.4 structure of procedure file 3.3 parametric setting of workpiece and property of milling tool. before the simulating of machining process, geometric properties and material property of elements, contact relationship of bodies, mechanics and thermal conductivity between milling tool and workpiece need to be defined and evaluated. because the dimensions of workpiece, structural sizes and geometrical angles of tool have influence on the number of elements, dynamically meshing workpiece and tool are important. here element sets were employed to store the workpiece and tool elements respectively. the system implemented the method that setting units invisible or visible instead of calculating the number of elements developed before. by programming, all of configures of the model parameters were carried out automatically. advances in systems science and applications (2012) vol.12 no.2 169 4 execution of example the user interface of modeling system is presented in fig.5. the interface has five functions that include file management, adding modeling database, browsing modeling data, parameters of tool and work and contact parameters between tool and work. in the example, the diameter of tool is 30.2 mm, the height of teeth is 4 mm, and other parameters are given. the model of down milling is shown in fig.1a and the model of up-milling is shown in fig.1b.when the amount of feed is 46 mm·min-1,the rotating rate of milling is 275 rpm, and the width of cutting 3 mm, the situations of down milling and up milling are shown in fig.6 and fig.7. fig.8 and fig.9 show the predicted forces of down milling and upmilling. fig.5 the user interface of modeling system fig.6 the simulation of down milling 170 jing sheng:parametric modeling and simulation of orthogonal milling process... fig.7 the simulation of up-milling fig.8 the milling force of down milling fig.9 the milling force of up-milling 5 experimental verification for the purpose of verifying the simulation result, a milling experiment was conducted. the milling equipment is xh719 manufactured by qingdao first machine advances in systems science and applications (2012) vol.12 no.2 171 tools factory, and its power is 28 kw. the material of a milling tool is ys2t. the diameter of milling tool with four teeth is d=30.2 mm. the measure equipment is kistler9237a. down milling was used. cutting conditions were the same as the value used in simulation. the history of cutting force is shown in fig.10. it is shown that the experimental result agree with the simulation data. then orthogonal table was designed to measure cutting temperature(see table fig.10 the simulation of cutting force of down milling table 5 orthogonal table axial radial amount cutting cutting cutting of feed speed width /mm width /mm /mm·min−1 /m·min−1 2.00 5.0 37.5 23.6 2.00 7.1 47.5 29.5 2.00 10.0 60.0 37.3 2.45 5.0 47.5 37.3 2.45 7.1 60.0 23.6 2.45 10.0 37.5 29.5 3.00 5.0 60.0 29.5 3.00 7.1 37.5 37.3 3.00 10.0 47.5 23.6 5). fig.11 shows experiment value and simulation date. 6 conclusions following the expatiations of the whole process of parametric modeling in msc.marc, the paper discussed the key techniques. it has been proven that the ways and means are effective. it is helpful to simulate under different cutting parameters, various dimensions and geometric angles of a tool. therefore parametric modeling will provide good foundation for creating further cutting databases 172 jing sheng:parametric modeling and simulation of orthogonal milling process... fig.11 the comparison of cutting temperature and designing tool. acknowledgements the project is funded by the grants from the doctor fundation of hubei automotive industries institute. references [1] domenico u. (2008), “finite element simulation of conventional and high speed machining of ti6al4v alloy”, journal of materials processing technology, vol.196, pp.79-87. [2] pantalé, bacaria j l, dalverny o, rakotomalala r, s caperaa. (2004), “2d and 3d numerical models of metal cutting with damage effects”, comput. methods appl. mech. engrg, vol.193, pp.4383-4399. [3] vernaza-pena k m, mason j j, li m. (2002), “experimental study of the temperature field generated during orthogonal machining of an aluminum alloy”, experimental mechanics, vol.42, pp.221-229. [4] calamaz m, coupard d, girot f. (2008), “a new materialmodel for 2d numerical simulation of serrated chip formation when machining titanium alloy ti6al4v ”, international journal of machine tools & manufacture, vol.48, pp.275-288. [5] raviraj shetty, laxmikant k r. (2008), “pai finite element modeling of stress distribution in the cutting path in machining of discontinuously reinadvances in systems science and applications (2012) vol.12 no.2 173 forced aluminium composites”, journal of engineering and applied sciences, vol.3, pp.25-31. [6] limido j, espinosa c, salaun, m. (2007), “sph method applied to high speed cutting modeling”, international journal of mechanical sciences, vol.49, pp.898-908. [7] mamalis a g, horváth m, branis a s, manolakos d e. (2001), “finite element simulation of chip formation in orthogonal metal cutting”, journal of materials processing technology, vol.110, pp.19-27. [8] wen q, guo y b, todd b a. (2006), “an adaptive fea method to predict surface quality in hard machining”, journal of materials processing technology, vol.173, pp.21-28. [9] simoneau a, elbestawi m a. (2005), “surface defects during microcutting”, international journal of machine tools & manufacture, vol.10, pp.1-10. [10] xie l j, schmidt j, schmidt c, biesinger f. (2005), “2d fem estimate of tool wear in turning operation”, wear, vol.258, no.10, pp.1479-1490. adv syst sci appl 2017; 3:58–66 published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/499 regression tree control of multidimensional static object ekaterina mangalova1∗ 1system analysis and control department, siberian state aerospace university, krasnoyarsk, russia abstract: in this paper the control problem of a static system under incomplete information is discussed. a nonparametric control algorithm based on classification and regression tree is proposed and evaluated for different task formulations: controlled inputs – one-dimensional output, controlled and observed uncontrolled inputs – one-dimensional output, controlled and observed uncontrolled inputs – multi-dimensional outputs. keywords: control, regression tree, machine learning, optimization 1. introduction the modeling of technical systems and processes is the essentially important tool used in engineering to design, improve, optimize and control systems. using simulation modeling is generally cheaper, safer and faster than conducting real-world experiments. this allows to use modeling for different alternatives analyses. there are two different ways to create simulation models: • manually built models. a specialist builds the simulation model manually based on his theoretical knowledge and technical experience. this method is the most useful and imminent for systems that do not yet exist. • statistical modeling. statistical modeling is based on generating models by observations. this method can be useful for complicated systems when input-output relationships are not evident or are unknown, internals are not essential for the technical problem. statistical models are used to detect a deviation of the normal behavior of a system and to control. model predictive control is a family of controllers in which there is a direct use of an explicit and separately identifiable model. control algorithms based on model predictive control concept have found wide acceptance in practical applications and have been studied by researchers. in [1] one of the first successful applications of model predictive heuristic control is described. there are two commonly used and well studied approaches to model predictive control: based on artificial neural networks and based on fuzzy logic. the use of artificial neural networks in model based control, both as process models and as controllers, is investigated by d. c. psichogios and l. h. ungar [2]. a. draeger, s. engell, h. ranke [3] implemented a feed-forward neural network as the nonlinear prediction model in an extended dmc-algorithm to control. h. sarimveis and g. bafas [4] introduceed the method based on a dynamic fuzzy model of the process to be controlled, which is used for predicting the future behavior of the output variables. in the paper [5] j. m da costa sousa and u. kaymak investigated the use of fuzzy decision making in model predictive control. ∗corresponding author: e.s.mangalova@hotmail.com http://ijassa.ipu.ru/ojs/ijassa/article/view/499 regression tree control of multidimensional static object 59 in contrast to neural network and fuzzy logic approaches, in [6] a control algorithm based on the nadaraya-watson estimator [7] (as a predictive model) and their sequence is proposed. however, there are major problems with this approach in the case of a multidimensional control task. on the one hand, it is connected with observations distribution in a highdimensional feature space (especially in the case of small number of observations). on the other hand, the nadaraya-watson estimator has the high computational complexity. the bigger feature space dimension, the harder to optimize the vector of bandwidths. decision trees are used for solving such regression tasks [8]. to avoid these problems it is suggested to use a decision tree instead of the nadaraya-watson estimator. in this paper an approach based on classification and regression tree (cart) is proposed to solve control tasks. the paper is organized as follows: in the second section, statements of the modeling and control problems are introduced; in the third section, the nonparametric modeling and control algorithm based on the nadaraya-watson estimator is presented; the fourth section is devoted to classification and regression tree (cart) and the control algorithm based on cart with its variants for different control task statements. 2. modeling and control tasks the block scheme of the control process is shown in figure 1. the following designations are taken: x̄ is the vector of output variables, x̄∗ is the vector of the desired output x̄ values, ū is the vector of controlled inputs, v̄ is the vector of observed uncontrolled inputs, ξ is the unobserved input (noise). fig.1 the block scheme of the control process it is evident from figure 1 that the output variables x̄ depend on the inputs ū, v̄, ξ. the control task is to build a control unit which generates ū such that e (x̄, x̄∗) is minimized, where e is some error measurement. the modeling (regression) and control tasks are adjacent. if a system reaction to an input (prediction) is known, it is possible to get the desired outputs by identifying the controlled inputs [9]. the forward model (regression model) x̂ (ū, v̄) is fit using the training set (ūi, v̄i, x̄i, i = 1, ..., n), where n is the observations number (figure 2). using the forward model x̂ (ū, v̄) the reactions to a given input can be predicted. for control task solving, it is necessary to invert the model x̂ (ū, v̄). it means that we should find controlled inputs ū∗ that could result in the desired outputs x̄∗ in the case of estimated copyright c© 2017 assa. adv syst sci appl (2017) 60 e. mangalova fig.2 forward model scheme (regression task) uncontrolled inputs v̄predicted (figure 3). in contrast to the forward model, the backward model û ( x̄∗, v̄predicted ) is fit using the training set (ūi, v̄i, x̄i, i = 1, ..., n) such that the outputs are given, uncontrolled inputs are predicted and we need to determine an appropriate controlled input. fig.3 backward model scheme (control task) 3. nadaraya-watson estimator approach the nadaraya-watson estimator approach was proposed to solve control tasks in such forward-backward models statement [6]. 3.1. modeling the nadaraya-watson estimator has been widely applied for the nonparametric regression, using the weighted average observations output in the neighborhood around (ū, v̄): x̂k (ū, v̄) = ∑n i=1 xi ∏mu j=1k ( uj, uji , c j u )∏mv j=1k ( vj, vji , c j v )∑n i=1 ∏mu j=1k ( uj, uji , c j u )∏mv j=1k ( vj, vji , c j v ) , k = 1, ...,mx, (3.1) where the kernel function k is a non-negative function that integrates to one and has zero mean, c̄u, c̄v are the vectors of bandwidths, mx is the number of output variables. to identify the forward models x̂k (ū, v̄), the bandwidths c̄u, c̄v should be optimized according to the selected accuracy measurement. 3.2. control backward models of nadaraya-watson estimators have the following form: copyright c© 2017 assa. adv syst sci appl (2017) regression tree control of multidimensional static object 61 ûm ( u1, ..., um−1, x̄, v̄ ) =∑n i=1 u m i ∏m−1 j=1 k ( uj, uji , c j u )∏mv j=1k ( vj, vji , c j v )∏mx j=1k ( xj∗, xji , c j x )∑n i=1 ∏m−1 j=1 k ( uj, uji , c j u )∏mv j=1k ( vj, vji , c j v )∏mx j=1k ( xj∗, xji , c j x ) , m = 1, ...,mu, (3.2) the bandwidths c̄u, c̄v in the equations (3.1) and (3.2) are not identical. there are some important limitations on the nadaraya-watson algorithm implementation: • ranking input features. the controlled input variables ū should be ranked by their importance. the process (3.2) starts from the most important controlled input feature and continues with less and less important ones. • bijective relationship. the forward models (the true relationships between input features and output features) should be bijective functions. suppose there are two significantly different controlled inputs ū∗1 and ū∗2 that produce the desired output. the result of the algorithm is a controlled input values between ū∗1 and ū∗2 (this control does not lead to the desired output) instead of choosing one of them. • curse of dimensionality. it is connected with observations distribution in a high dimensional feature space (especially in the case of a small number of observations). suppose, there are n = 1000 points uniformly distributed over the ten dimensional unit cube [0, 1]10. an average over the neighborhood of diameter 0.25 (in each coordinate) results in the volume of 0.2510 ≈ 0.00000095 for the corresponding ten-dimensional cube. hence, the expected number of observations in this cube will be 0.00095 and any averaging can not be expected. if we fix the count k = 1 of observations over which to average, the diameter of the typical neighborhood will be larger than 0.5. it means that the average is calculated over at least one-half of the range along each coordinate [10]. 4. decision tree approach decision trees are commonly used for solving regression tasks in multidimensional cases and allow preventing the nadaraya-watson algorithm limitations described in section 3. 4.1. modeling decision tree is a piecewise constant nonparametric model. the most popular decision tree model is classification and regression tree (cart) [8]. cart is a binary tree where each root node represents an input variable ujr and a split point br. let us aggregate controlled and observed uncontrolled inputs to make the description easier: ū = {u1, ..., umu , v1, ..., vmv}. each leaf node contains values of the output variable x which is used to make a prediction. cart fitting involves input variables and split points selection until a suitable tree is constructed. input variables and split points are chosen using a greedy algorithm to minimize the: min j min b ∑ i:uj i<b l ( xi, x̂ − (ūi, j, b) ) + ∑ i:uj i>b l ( xi, x̂ + (ūi, j, b) ) (4.3) where x̂− (ūi, j, b) or x̂+ (ūi, j, b) is the prediction in the point ūi based on the subspaces after splitting: copyright c© 2017 assa. adv syst sci appl (2017) 62 e. mangalova x̂− (ūi, j, b) = ∑ i:uj i<b xi∑ i:uj i<b 1 , x̂+ (ūi, j, b) = ∑ i:uj i>b xi∑ i:uj i>b 1 . (4.4) the tree construction ends using a predefined stopping criterion, such as the minimum number of observations assigned to each leaf node of the tree or the maximum depth of the tree. each node in the tree corresponds to a rectangular region of the predictor space sk, a subset of the observations lying in the region sk, a constant x̂k which is the average response of the observations in k-th rectangular region. thus, the binary tree model can be formalized as follows: x̂ (ū) = { x̂k : ū ∈ sk, k = 1, 2, ..., k } . (4.5) 4.2. control control tasks can be solved using the regression trees in different formulations depending on the number of input and output values, presence of observed uncontrolled input variables. consider the basic formulation of the problem. 4.2.1. controlled inputs one-dimensional output. assume there are only controlled variables ū and one output variable x. decision tree is fit using the training dataset containing simultaneous observations of ū and x. the control algorithm: 1. set the control target x∗. 2. search for such leaf node that k∗ = arg min k d1 (x̂k, x ∗) , (4.6) where d1 is a one-dimensional distance measurement. call k∗-th node ”target node”. 3. find control ū∗ contained in the region sk∗ . there are different ways to choose this control: • the center of the rectangle guaranties the most accurate forward model prediction (decision tree prediction). • points on the rectangle boundary allow to explore object and collect more information. • also the distance between the previous control and the current control can be minimized to reduce interference in the system. the experimental result. suppose the object is represented by the equation: x = sin (2πu1) + √ u2 − u3 + ξ . this equation is used in the computational experiment to simulate a real process. input variables u1, u2, u3, u4 are bounded by [0, 1]. the initial dataset contains 10 points distributed uniformly. control is selected as the center of the target node. figure 4 illustrates the desired outputs and the reactions to a generated control. figure 5 shows the initial dataset and the observations obtained during the simulation process. 4.2.2. controlled and observed uncontrolled inputs one-dimensional output. assume there are controlled and observed uncontrolled variables ū and one output variable x. the decision tree is fit using the training dataset containing simultaneous observations of ū and x. the control algorithm: 1. set the control target x∗. copyright c© 2017 assa. adv syst sci appl (2017) regression tree control of multidimensional static object 63 fig.4 simulation of a process. desired outputs and reactions to a generated control fig.5 simulation of a process. initial dataset and generated dataset 2. predict the uncontrolled input variables uj,predicted, j = mu + 1, ...,mu +mv. 3. cut nodes which can not be achieved with the predicted uncontrolled input variables. 4. search for a leaf node such that k∗ = arg min k,ū′∈sk d1 (x̂k, x ∗) , (4.7) where ū′ is the input values such that u′j ,j = 1, ...,mu can take any value and u′j = uj,predicted,j = mu + 1, ...,mu +mv. 5. find a control ū∗ contained in the region sk∗ . the experimental result. suppose now that u2 is uncontrolled and changed as follows u2 i = sin (πi/25). the previous value of u2 i−1 is used as a prediction for the next step u2,prediction. figure 6 illustrates the desired outputs and the reactions to a generated control. figure 7 shows the initial dataset and the observations obtained during the simulation process. 4.3. multi-dimensional output, controlled and observed uncontrolled inputs assume there are controlled and observed uncontrolled variables ū and the vector of output variables x̄. decision trees are fit using the training dataset containing simultaneous observations of ū and x̄. the control algorithm: 1. set the control targets x̄∗. 2. predict the uncontrolled input variables uj,predicted, j = mu + 1, ...,mu +mv. 3. find intersections of nodes which contain the predicted uncontrolled variables sk 1,...,mu , k = 1, ..., k ′ and the output values associated with these intersections x̂1 k,..., x̂mx k . copyright c© 2017 assa. adv syst sci appl (2017) 64 e. mangalova fig.6 simulation of a process with uncontrolled input. the desired outputs and the reactions to the generated control fig.7 simulation of a process with uncontrolled input. the initial dataset and the generated dataset 4. search for a leaf node such that k∗ = arg min k dmx ( x̂1 k, ...x̂ mx k , x̄∗ ) , (4.8) where dmx is a multi-dimensional distance measurement. 5. find a control ū∗ contained in the region sk∗ 1,...,mu . the experimental result. suppose now there are two outputs x1 = sin (2πu1) + √ u2 − u3 + ξ and x2 = √ u2 − u3 + ξ. figure 8 illustrates the desired outputs and the reactions to a generated control. the desired outputs for x1 and x2 are the same. figure 9 shows the initial dataset and the observations obtained during the simulation process. fig.8 simulation of a multidimensional process. the desired outputs and the reactions to a generated control copyright c© 2017 assa. adv syst sci appl (2017) regression tree control of multidimensional static object 65 fig.9 simulation of a multidimensional process. the initial dataset and the generated dataset) conclusion in this paper a control task of a multidimensional static system is discussed. the paper presents the advantages and disadvantages of the nonparametric algorithm based on the nadaraya-watson estimator and their sequence. in order to avoid the disadvantages of this nonparametric algorithm, the modification with piecewise constant approximation of the nonparametric estimator is put forward. in the proposed algorithm classification and regression trees are used as a predictive model instead of the nadaraya-watson estimator. the control algorithm based on cart is tested for different task formulations: controlled inputs and one-dimensional output, controlled and observed uncontrolled inputs and onedimensional output, controlled and observed uncontrolled inputs and multi-dimensional outputs. references 1. richalet j., rault a., testud j.l. & papon j. (1978). model predictive heuristic control: applications to industrial processes, automatica, 14 (5), 413–428. 2. psichogios d. c. & ungar l. h. (1991). direct and indirect model based control using artificial neural networks, industrial & engineering chemistry research, 30 (12), 2564– 2573. 3. draeger a., engell s. & ranke h. (1995). model predictive control using neural networks, ieee control systems magazine, 15 (5), 61–66. 4. sarimveis h. & bafas g. (2003). fuzzy model predictive control of non-linear processes using genetic algorithms, fuzzy sets and systems, 139 (1), 59–80. 5. da costa sousa j.m. & kaymak u. (2001). model predictive control using fuzzy decision functions, ieee transactions on systems, man, and cybernetics, part b (cybernetics), 31 (1), 54–65. 6. medvedev a.v., & raskina a.v. (2017). on the nonparametric identification and dual adaptive control of dynamic processes, journal of siberian federal university. mathematics & physics, 10 (1), 96–107. copyright c© 2017 assa. adv syst sci appl (2017) 66 e. mangalova 7. nadaraya e. a. (1964). on estimating regression, theory of probability and its applications, 9, 141–142. 8. breiman l., friedman j., stone c. j. & olshen, r. a. (1984). classification and regression trees. crc press. 9. kunh s. & guhmann c. (2008) modeling and control with local linearizing nadaraya watson regression, arxiv:0809.3690. 10. härdle w. (1990) applied nonparametric regression. cambridge university press. copyright c© 2017 assa. adv syst sci appl (2017) https://arxiv.org/abs/0809.3690 introduction modeling and control tasks nadaraya-watson estimator approach modeling control decision tree approach modeling control controlled inputs one-dimensional output. controlled and observed uncontrolled inputs one-dimensional output. multi-dimensional output, controlled and observed uncontrolled inputs bibliography paper title (use style: paper title) adv syst sci appl 2019; 04; 79-86 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/829 an optimal itinerary generation in a configuration space of large intellectual agent groups with linear logic dmitry maximov* institute of control science russian academy of science, moscow, russia e-mail: dmmax@inbox.ru received may 16, 2019; revised december 12, 2019; published december 31, 2019 abstract: a group of intelligent agents which fulfill a set of tasks in parallel is represented by the tensor multiplication of corresponding processes in a linear logic game category. an optimal itinerary in the configuration space of the group states is defined as a play with maximal total reward in the category. the reward is represented as a degree of certainty (visibility) of an agent goal, and the system goals are chosen by the greatest value corresponding to these processes in the system goal lattice. keywords: intellectual agent; itinerary choice; goal lattice; game semantics. 1. introduction the artificial intelligence is represented in the artificial general intelligence (agi) approach as an information processor which consumes and gives out information. investigations in the field are focused on systems which act rationally. a formal description of the most intelligent agent (aixi) behavior, in the sense of some intelligence measure, is suggested in agi framework [1]. the model is based on probabilistic modeling of the environment, and on the next system move determination based on previous experience, and on a numerical estimation of the system position reward and on the maximization of the expected reward along the trajectory. however, the method to obtain this numerical estimation is absent. also, there are no models to describe such agent groups’ behavior. it has been demonstrated in [2-4] that the structure existence (a lattice structure or else a monoid structure, i.e. the linear logic structure) in the system task [3, 4] or goal [2] set is sufficient for the system to behave quite reasonable. the behavior looks even like ants’ behavior in something [4]. but it is not supposed here the environment modeling unlike [1]. in this paper, the topic develops based on the idea that it is possible to represent different aim parallel achievement processes fulfilling by various agents in some environment as a tensor multiplication in linear logic. the logic is modeled in some game category [5]. thus, it is possible to describe goals achievement process by the intelligent agent system in some environment as a game. position rewards in the game are represented by sets which describe the information about goals or their distinctness degree. thus, the rewards are provided by the environment, and they are lattice elements but not numbers as in [1]. these lattice and linear logic structures are provided by the environment. but similar structures are also given by our ideas about the system and agent purposes. these are used to determine the priority of different parallel processes in such game category. the project was partly supported by rfbr grant 16-08-00832a * corresponding author dmmax@inbox.ru mailto:dmmax@inbox.ru 80 d. maximov copyright ©2019 assa adv. in systems science and appl. (2019) 2. mathematical backgrounds 2.1. lattices [6] definition 2.1: a partially-ordered set p is the set with such a binary relation x ≤ y for elements in it, that for all 𝑥, 𝑦, 𝑧 ∈ 𝑃 the following relationships are performed:  x ≤ x (reflexivity);  if x ≤ y and y ≤ x, then x = y (anti-symmetry);  if x ≤ y and y ≤ z, then x ≤ z (transitivity). the definition means that in the partially-ordered set not all elements are compared with each other. this property distinguishes these sets from linear-ordered ones, i.e., from numeric sets which are ordered by a norm. thus, the elements of the partially-ordered set are the objects of more general nature than numbers. in the partially-ordered set diagram, the greater the element (i.e., vertex, node) is the higher it lies, and the elements are compared with each other lie in the same path from the minor element to the greater one. two examples of a partially-ordered set diagram are represented in fig. 2.1which are also lattice diagrams. definition 2.2: the upper bound of a subset x in a partially-ordered set p is the element 𝑎 ∈ 𝑃, containing all 𝑥 ∈ 𝑋.the supremum or join is the smallest subset x upper bound.the infimum or meet defines dually as the greatest element𝑎 ∈ 𝑃 containing in all 𝑥 ∈ 𝑋. definition 2.3: a lattice is a partially-ordered set, in which every two elements have their meet, denoting as 𝑥 ⋀𝑦, andjoin, denoting as 𝑥 ⋁ 𝑦. in the lattice diagram the elements join is the nearest upper element to both of them, and the meet is the nearest lower one to both. the elements generating by joins and meets all other elements are called generators. they refer to the lattice as complete lattice if its arbitrary subset has the join and the meet. thus, any complete lattice has the greatest element “1”, and the smallest one “0” and every finite lattice is complete. fig. 2.1. examples of a goal lattice (a) and of an agent desire lattice (b) an optimal itinerary generation with linear logic 81 copyright ©2019 assa adv. in systems science and appl. (2019) 2.2. linear logic[7] if a multiplication operation is additionally defined at the lattice elements, then the operations of linear logic also exist at the lattice. we use the phase semantic of linear logic from [7]. definition 2.4: a phase space is a pare (𝑀, ⊥), where mis a multiplicative monoid (i.e., a triple (𝑀0,⋅, 𝑒) with 𝑀0 isa set and ⋅ is a multiplication with the unit e), which is also a lattice and the element false of the lattice ⊥ ⊂ 𝑀 is anarbitrary subset of the monoid. in linear logic, the element false differs from 0 (the minimal lattice element) in general in contrast to classical logic or intuitionistic one. the multiplication 𝑋 ⋅ 𝑌 = {𝑥 ⋅ 𝑦|𝑥 ∈ 𝑋; 𝑦 ∈ 𝑌 } is defined for arbitrary monoid subsets (i.e., the lattice elements) 𝑋, 𝑌 ⊂ 𝑀. the linear implication 𝑋 ⇒ 𝑌 = {𝑧|𝑥 ⋅ 𝑧 ∈ 𝑌, ∀𝑥 ∈ 𝑋} is also defined. for𝑋 ⊂ 𝑀 its dual is defined as 𝑋⊥ ⇒⊥. the dual element isa generalization of the negation in the case of linear logic. definition 2.5: facts are such subsets 𝑋 ⊂ 𝑀 that 𝑋⊥⊥ = 𝑋or equivalently 𝑋 = 𝑌⊥ for some 𝑌 ⊂ 𝑀. thus, facts are lattice elements coinciding with their double negations. e.g. ⊥⊥ = 𝐼 = {𝑒}⊥⊥; 1 = 𝑀 = ∅⊥; 0 = 1⊥ = 𝑀⊥ = ∅⊥⊥. here 1 is the maximal element of the lattice m, 0is its minimal element, e is the monoidal unit, and i is the neutral element of the multiplicative conjunction (see after this). it is easy to get the next properties: 𝑋⊥𝑋 ⊂ ⊥; 𝑋 ⊂ 𝑋⊥⊥; 𝑋⊥⊥⊥ = 𝑋; 𝑋 ⇒ 𝑌 ⊥ = (𝑋 ⋅ 𝑌 )⊥; (𝑋 ∨ 𝑌 )⊥ = 𝑋⊥ ∧ 𝑌 ⊥. from here we get only facts may be the values and the consequents of the implication. at facts the lattice operations of the additive conjunction& and the additive disjunction + are defined in the following way: 𝑋 & 𝑌 = 𝑋 ∧ 𝑌 = (𝑋⊥ ∨ 𝑌 ⊥)⊥; 𝑋 + 𝑌 = (𝑋⊥&𝑌 ⊥)⊥ = (𝑋⊥ ∧ 𝑌 ⊥)⊥ = (𝑋 ∨ 𝑌)⊥⊥ . the duality of the operationsunderstands here as in the set theory: ∨ ⊥ =∧ ∧⊥ = ∨ inwhich the duality means the negation. at facts, multiplicative operations are also defined. these are the multiplicative conjunction ⊗ and the multiplicativedisjunction ð: 𝑋 ⊗ 𝑌 = (𝑋 ∙ 𝑌 )⊥⊥ = (𝑋 ⇒ 𝑌 ⊥)⊥ = (𝑋⊥ ð 𝑌 ⊥)⊥; 𝑋 ð􀀀 𝑌 = (𝑋⊥ ∙ 𝑌 ⊥)⊥ = 𝑋⊥ ⇒ 𝑌 . theneutral element of the operation & is 1, the dual to it (neutralelement of the operation +) is 0. the neutral element of theoperation ð is ⊥, the dual to it, the neutral element of the operation ⊗ is i. the set of facts is divided into two classes dual to each other: the class of open facts op and the class of closed facts cl. the set op is closed by operations + and ⊗. its maximalelement by inclusion is i, and the minimal one is 0. the set cl is closed correspondingly by operations &and ð, and its maximal element is 1, and the minimal one is ⊥. in fig.2.1a, the class of open facts is encircled as op with 𝐼 = 𝑚1𝑒, and the class of closed facts is encircled as cl with ⊥= 𝑋3 2.3.game semantics in a linear logic category [5] definition 2.6: a conway game is defined as a rooted graph with vertices v as the game positions and edges 𝐸 ⊂ 𝑉 × 𝑉 as the game moves. each edge has a polarity ±1 which depends on whether it is the proponent or the opponent move. definition 2.7: 82 d. maximov copyright ©2019 assa adv. in systems science and appl. (2019) a trajectory or a play is some path from the graph root ∗. the path is alternated if the adjacent edges are of different polarities. definition 2.8: a strategy is defined as a non-empty set of alternated paths of even length, which are started from the opponent move, closed up to the prefix of even length and determined. determinism means that two paths with the common prefix, which differ in two moves, should coincide. definition 2.9: a dual play 𝑋⊥is obtained from the play x by reversing the polarity of moves. definition 2.10: the tensor product 𝑋⨂yof two conway games x and y is defined in such a way: positions 𝑥⨂y are 𝑉𝑋⨂𝑌 = 𝑉𝑋 × 𝑉𝑌with the root ∗𝑋⨂𝑌=∗𝑋×∗𝑌, moves are 𝑥⨂y ↦ { 𝑧⨂y, x ↦ z 𝑖𝑛 𝑋 𝑥⨂z, y ↦ z 𝑖𝑛 𝑌, and the polarity of a move in 𝑋⨂yis inheritedfrom the polarity of the underlying move in x or y. a generalized linear logic is modeled in a category of such games. the category objects are conway games and morphisms 𝑋 → 𝑌 are strategies in 𝑋⊥ðy.these multiplications are linear implications𝑋⊥ðy = 𝑋 ⇒ 𝑌. it should be mentioned that on game graphs the operations ð and ⨂ are the same so in [5] they are not even distinguished. a conway game with a payoff is the play with an additional weight 1, 1/2 or0 in each vertex. the weight depends on whether the position is winning (i.e., of the weight 1 or 1/2) or not. in the tensor product, these weights obey rules of boolean conjunction and implication. a strategy is winning if it terminates in the winning position. in the category of conway games with a payoff, morphisms are winning strategies now. it is possible to prove [8] that the categorical construction is conserved if the weights’ numbers are replaced with some sets form a lattice, and the boolean operations are replaced with the lattice operations. the greater set is connected with a position, the more advantageous it is, and it is winning if its weight is not 0. we suppose the existence of a universal set containing all the others. thus, all such estimation sets form a complete brouwer lattice. 3. behavior determination of an intellectual agent system an example of the goal lattice ms matches a system of l agents (fig. 2.1,a). in the lattice vertices, xi and e are the generators and denote the system goals. vertices ji are the generators joins and denote combined goals achieving. these combined goals, as well as generators, may be associated with the correspondent task fulfilling thus the vertices may have meets, i.e., some subtask included in different tasks (or correspondent goals). the higher goal lies (hence the more tasks it contains), the more important this behavior variant is. it supposed that the preferable behavior of the system is to achieve all its goals. this variant corresponds to the top lattice element 1. and the bottom element 0 corresponds to complete inactivity and to the least essential behavior variant. all the estimations may be considered as partially true truth values. thus, we can say that the more important the behavior is, the truer it is. the agent lattice mi (fig. 2.1, b) has another meaning. the same generators and their joins mean the agents’ desires. one of the vertices is marked as an intention for every agent. in the same manner, the more desires are included in the intention, the more essential the intention is. a variant of the open (op) and the closed (cl) classes definition is indicated in the system lattice (fig. 2.1, a). this definition (as well as the multiplication definition) defines the structure of linear logic in the lattice. the element multiplication is obtained (usually ambiguously, [4]) from the demand to implement the linear logic operations properties. in this case, it is possible to consider an optimal itinerary generation with linear logic 83 copyright ©2019 assa adv. in systems science and appl. (2019) parallel processes of combined goals achieving as the tensor product of corresponding lattice elements in the logic. and the priority of different processes is obtained from the demand of the greatest correspondent tensor product estimation in the lattice (the product is an element of the lattice; hence the higher it lies, the more important it is, i.e., the greater its truth-value is). it is supposed some initial agent goal distribution. for such a system, we consider the process of the interaction of the system with the environment as a conway game. in the game, the environment is the proponent which provides the system (the opponent) with the information about the environment objects. the opponent moves from one position in the environment to the other by the use of the information to achieve his goals. for the system of l agents, we, in fact, have l parallel processes and, therefore, the resulting game is their tensor product. thus, agents are placed initially in the configuration space (environment) in the root ∗=∗1 ⨂ … ⨂ ∗𝑙 of the system game 𝐴 = 𝐴1 ⨂ … ⨂𝐴𝑙 with the system goal lattice ms and agent intention lattices mi, 𝑖 = 1. . 𝑙. the game aj represents possible agent moves in the environment. but the real trajectory or the play is chosen from the demand of the maximal total position reward along the projected path. the agent move in the environment is estimated corresponding to an optimality criterion with the reward 𝑘(𝑝𝑖, 𝑏𝑗) in the position 𝑝𝑖of the goal 𝑏𝑗achieving process. it may be that the system does not see any goal initially and moves according to a criterion of an optimal move in goals absence. thus, the system has additionally l goals (tasks) 𝑎𝑗of movement in the environment, which are included as in the generator set of the lattice ms, as in the generator sets of lattices mi. the optimality criterion may represent the highest degree of the correspondence to the demanding system configuration, or the highest freedom in future moves or the most excellent visibility from a position or so on in the case of the free movement in the goals absence. and in the case of the system goals achieving we suppose the better a goal object is visible, the higher the reward is. it is supposed that agents see not the whole environment, but up to some horizon which may change in each direction fig. 3.1. moreover, the agent sees the environment as in the fog – the nearer the object, the better it is perceived†. therefore, we can predict a play only up to some finite step number with the increasing uncertainty along the path. let us n goals b1…bn are discovered in the environment by the system with information about them (or some other reward) 𝑘(𝑝𝑖, 𝑏𝑗) in positions 𝑝𝑖of the play 𝐴 = 𝐴1 ⨂ … ⨂𝐴𝑙of l agents. the game a corresponds to l parallel processes of achieving l movement goals 𝑎𝑗. then, a winning strategy of the game𝐴′ = 𝐴1 ⨂ … ⨂𝐴𝑙 ⇒ 𝐵1 ⨂ … ⨂𝐵𝑘defines a transition (morphism) to this new game 𝐴′. this game corresponds to l parallel processes of moving and achieving those k goals from discovered n ones, which may be better achieved in the next sense (that means that these k goals are the system intentions). the reward 𝑘(𝑝𝑖 , 𝑏𝑗) is some set. in the case of, e.g., unmanned vehicles, it may be an image of the object. the better the object is visible, the better the coincidence of the image with the original and the greater the reward. † it is possible to consider different horizons and different uncertainty degree for different goals. we do not do it here. 84 d. maximov copyright ©2019 assa adv. in systems science and appl. (2019) fig. 3.1.the visibility horizon it is reasonable to choose the trajectory (play) from the demand to maximize the reward along the path within the visibility horizon: 𝑘𝑝𝑙𝑎𝑦 (𝐴1 ⨂…⨂𝐴𝑙)⊥ð𝐵1 ⨂…⨂𝐵𝑘 = (3.1) = 𝑚𝑎𝑥𝑝𝑙𝑎𝑦𝑠 [ ⋃ 𝑘(𝐴1 ⨂…⨂𝐴𝑙)⊥ 𝑝𝑙𝑎𝑦 & ⋃ 𝑘𝐵1 𝑝𝑙𝑎𝑦 & … & ⋃ 𝑘𝐵𝑘 𝑝𝑙𝑎𝑦 here the reward 𝑘𝑝𝑙𝑎𝑦 (𝐴1 ⨂…⨂𝐴𝑙)⊥ð𝐵1 ⨂…⨂𝐵𝑘 is maximized in the game 𝐴′ and corresponds to that process of k goals achieving that has the highest priority (𝑎1 ⨂ … ⨂𝑎𝑙) ⊥ð𝑏1 ⨂ … ⨂𝑏𝑘 in the system goal lattice. the priority is maximal among all possible parallel processes of achieving n discovered goals. the maximum is taken among all possible plays and it joins the rewards along these plays (i.e., trajectories) in (𝐴1 ⨂ … ⨂𝐴𝑙)⊥, and in 𝐵1, and so on and in 𝐵𝑘. thus, the sign & means conjunction. if there are parallel processes which are not compared by priority, i.e., if there are several incompatible priorities(𝑎1 ⨂ … ⨂𝑎𝑙)⊥ð𝑏1 ⨂ … ⨂𝑏𝑘, it is possible to reorder the goal lattice in the manner that some lattice vertex has used as an additional priority [3]. in the reordered lattice, initially incompatible elements may become compatible, so we can choose the processes’ priority. such a lexicographical rule follows from the teleological system assignment, i.e., our vision of the system purpose. it may be an occasion that the lattice is so symmetric that does not allow make such a choice. in this case, the ambiguity of the linear logic structure allows the necessary variant choosing [8]. fig. 3.2. an example of a game (depicted only the opponent moves) an optimal itinerary generation with linear logic 85 copyright ©2019 assa adv. in systems science and appl. (2019) also, if we are not able to determine the trajectory in (3.1) uniquely, it is possible to use the goal lattices of different agents to determine the agent (and the correspondent trajectory) priority as in [2]. in this case, we assign a definite agent to achieve a particular goal from agents’ intention estimations. specifically, let us there are three agents’ plays a1, a2 and a3 in which the agents in positions p1 have discovered two goals b1 and b2 from possible three ones (fig. 3.2). let us also the rewards in the positions p1 satisfy the next relations: 𝑘𝐴1 (𝑝1, 𝑏1) > 𝑘𝐴2 (𝑝1, 𝑏1)< >𝑘𝐴3 (𝑝1, 𝑏1) (3.2) 𝑘𝐴3 (𝑝1, 𝑏2) > 𝑘𝐴2 (𝑝1, 𝑏2)< >𝑘𝐴1 (𝑝1, 𝑏2) (3.3) where the sign < > means the rewards’ incompatibility. this means that though the visibility degrees in positions p1 of a2 and a3 does not differ, the foreshortenings are different thus the images are different and, therefore, incompatible. therefore, we cannot define the system trajectory uniquely. it is clear that agent-1 should achieve the goal b1because it sees it better than the others. for the others let us, in this case, the desires’ lattices for these two agents have the form of fig. 3.3. thus, the agent-2 may reach goals b1 and b2, and the agent-3 has an additional desire b3. then, we may evaluate the lattice vertices’ weights by the formula [2]: vertex weight = joined in the vertex desires /total desires’ number (3.4) the formula evaluates vertices’ weights by two parameters: the number of desires joined in the intention and the vicinity of the vertex to the top element, i.e., to the most desirable behavior variant. the play of the two parameters allows weights’ comparing in different lattices. thus, e.g. the u12 weight is 𝑉𝑈12 = 2/3. since 𝑉𝑏2 = 1/2 in the agent-2 lattice and 𝑉𝑏2 = 1/3 in the agent3 lattice, we should choose the bigger value and correspondent agent-2 to achieve the goal b2.this is reasonable because the agent-3 is left free and can look for the goal b3. finally, the obtained play is depicted in fig. 3.3 by bold lines. fig. 3.3. the agent-2 lattice (a) and the agent-3 lattice (b) the additional subtle point is that the graphs of the games a and b do not always coincide, i.e., a goal may be seen in b, but the way to the goal may not exist in a (i.e., in the environment). thus, we have a deal with the game a of l parallel processes of agent moves, on the positions of which the rewards of k games b are possible but are not obligatory. it should be pointed out that the structures of tensorial multiplications ⨂ and ð in (𝑎1 ⨂ … ⨂𝑎𝑙)⊥ð𝑏1 ⨂ … ⨂𝑏𝑘 and in (𝐴1 ⨂ … ⨂𝐴𝑙)⊥ð𝐵1 ⨂ … ⨂𝐵𝑘 are different. in (𝑎1 ⨂ … ⨂𝑎𝑙)⊥ð𝑏1 ⨂ … ⨂𝑏𝑘the tensor ⨂ is a monoid in the goal lattice. these structures of ðand ⨂ are pre-existent and does not depend on the environment. it may be chosen from very general considerations [4] and determines the system purpose. but the tensor ⨂ and the co-tensor ðin(𝐴1 ⨂ … ⨂𝐴𝑙)⊥ð𝐵1 ⨂ … ⨂𝐵𝑘are determined by the environment, by the goals and obstacles 86 d. maximov copyright ©2019 assa adv. in systems science and appl. (2019) distributions, by visibility and so on. thus, there are two different linear logic structures in the approach. 4. conclusion in the paper, a movement of an agent group with its goal lattice was considered in some environment. each agent move was represented as a game in which the environment corresponds to the proponent which provides the opponent (the agent) with some information to achieve the agent goal. different agent moves are considered as parallel processes which are represented as the tensor product of correspondent games and form a comprehensive game. the game has rewards on its positions, which estimates the quality of the information provided by the environment. the reward may be very different: it may represent the highest degree of the correspondence to the demanding system configuration, or the highest freedom in future moves, or the most excellent visibility from a position, or the degree of the coincidence of the goal object image with the original or some else. we demand the greatest total reward along all agents’ plays to choose the group trajectory. and we choose those goal achieving processes (i.e., agent plays) from all possible, which have the highest estimation in the agent group goal lattice. it is so because every goal and the corresponding process of its achieving has a definite correspondent truth value in the lattice. the higher the value lies in the lattice diagram; the higher priority of the process is. thus, we consider two types of estimations: the goal lattice value defines the choice of the goal achieving processes from all possible ones, and the position rewards of the game determine the optimal trajectory of these chosen processes in the environment. when it is not possible to determine the most significant estimations in both these cases, some additional methods may be used to select the optimal variant. references 1. hutter m. (2012). one decade of universal artificial intelligence, in theoretical foundations of artificial general intelligence (pp. 66–88), atlantis press. 2. legovich y. s. & maximov d. y. (2017). selecting executor in a group of intellectual agents, automation and remote control, 78 (7), 1341–1349. 3. maximov d. y. (2016). reconfiguring system hierarchies with multi-valued logic, automation and remote control, 77(3), 462–472. 4. maximov d. y., legovich y. s. & ryvkin s. e. (2017). how the structure of system problems influences system behavior, automation and remote control, 78(4), 689–699. 5. mellies p. -a. & tabareau n. (2010). resource modalities in tensor logic, ann. pure. appl. logic, 161(5), 632–653. 6. birkhoff g. (1967). lattice theory. rhode island, providence. 7. girar j. -y. (1987 ). linear logic, theoretical computer science, 50, 1–102. 8. maximov d. y. (2019). formirovanie optimal’nogo marshruta bol’shih grupp intellectual’nyh agentov [an optimal itinerary generation of large intelligent agent groups], large-scale systems control, 78, 46–70. adv syst sci appl 2018; 02; 93-106 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/531 risk analysis in seawater desalination sector: a case study of beni saf water company “bwc” mohammed bouamri1, hassiba bouabdesselam1 1) lte research laboratory, national polytechnic school of oran, algeria e-mail:bouamrimohammed25@gmail.com, has.bouab@yahoo.fr abstract: in this present paper, a risk analysis approach is applied to an algerian reverse osmosis seawater desalination plant using the mads mosar method. mads mosar method is a stepwise risk analysis approach containing many phases. our work begins with analyzing a review of past accidents triggered by the ben in saf water company (bwc) seawater desalination plant locating in the algerian coast in ain temouchent region and analyzing their similar seawater desalination plants (or plants that using similar and potential equipment). then, the mads mosar method will apply essentially for the macroscopic vision (module a). the macroscopic vision corresponds to a main risk analysis. in the current case study, we were able to identify eight subsystems where sources and scenarios of hazards are identified, accident scenarios are assessed, recognized and ranked by "severity×probability" grid. we found twenty-six scenarios whose we were able to assess them in function of their probability and severity using "probability x severity" grid criteria. at the end of the analysis, we were able to define and suggest the most appropriate prevention and protection barriers for the potential elements in the studied seawater desalination plant, including pipelines, transformers, compressors, high pressure pumps, pressure vessels and energy recovery devices. keywords: risk analysis; beni saf water company (bwc); seawater desalination plant; mads mosar. 1.introduction a hazard is a situation or substance that has the potential to cause harm while risk is the likelihood or probability of a certain undesired event to occur within a certain period of time or under specified circumstances [1]. although the bwc seawater desalination plant has become a priority for the algerian freshwater production (provides 200.000 m3/d of water to meet the needs of ain temouchent and oran regions), there are also potential downsides. apart hazards inherent of any seawater desalination plant (chemicals occur in the desalination process from numerous origins. these include the source water and chemicals that are used in the treatment process to aid its efficient functioning, to ensure microbiological safety, to stabilize the water before it enters the distribution system, and to control corrosion from contact surfaces during storage and distribution to consumers [2] and the presence of electrical equipments), the plant studied using the reverse osmosis ro membrane type that operates with high pressures than what is normally found in conventional plants assured by high pressure equipments. membrane desalination processes use semi-permeable membranes with pumping pressure as the driving force-to separate a saline feedwater into two streams: a low-salinity product (permeate) and a high-salinity stream (concentrate or reject) [2]. according to [3] 40 % of total cost of water turns to seawater ro system (percentage could vary depending on project-specific factors). reliability, service time and safety of these 94 m. bouamri, h. bouabdesselam copyright ©2018 assa. adv. in systems science and appl. (2018) equipments must be considered. as regards death and injuries experienced in such desalination unit, one could refer to two deaths caused primarily by not observing the exiting safety and hse instructions [4]. safety precautions should be taken in the plant and shutdown-startup procedures were developed [5] and process monitoring devices are provided to prevent the damage of components due to process upsets [6]. also, employers should be aware of these hazards and even workplaces must be appropriately designed to ensure proper operations and better health and safety conditions (workers, the public, property and the environment safety). in chemical industries, risk assessment is an approach for better and more efficient management of the process safety, whereby risk is estimated and considering various factors involved, decisions are made on the appropriate tolerability required [6, 7]. 1.1 case study: beni saf water copany seawater desalination plant the first step of our risk analysis study is to describe study area and desalination process used. based on a-7-years history (during 2010) of the bwc plant operation, maintenance records and accident reports, the second step is getting past internal/external accident review (the return of experience ) for enhanced risk prevention. then, the mads mosar method is applied and recommended preventives and protective barriers are proposed. 2.study area description before hazard identification can begin, it is necessary to understand the system operations so that the risk analysis covers all activities of interest [7]. some though be given to grasping how the system works and how the hardware, software, people, and environment all interact [8]. clearly, we need to define the system and theirs functions to bind our analysis. this case study concerns beni saf water company bwc seawater desalination plant located on the mediterranean in the algerian coast in ain temouchent region, which has an area of 65,700m2, with a 200 000 m3/d production capacity. bwc has worked since march 2010. the location of the bwc plant is presented in the following map (fig. 1). fig.1. location of the bwc seawater desalination plant (source: author) risk analysis in seawater desalination sector 95 copyright ©2018 assa. adv. in systems science and appl. (2018) 3. process description bwc uses the reverse osmosis’s method for the seawater desalination. in general, the main treatment unit includes the following stages: seawater intake and pumping, pretreatment, membrane ro separation unit and post-treatment. seawater intake system allows seawater to flow to a desalination unit where minerals are then removed from the saline water through a desalination process. at present, practically all ro desalination plants incorporate two key treatment steps designed to sequentially remove suspended and dissolved solids from the source water. the purpose of the first step-source water pretreatment-is to remove the suspended solids and prevent some of the naturally occurring soluble solids from turning into solid form and precipitating on the ro membranes during the salt separation process [9]. pressures applied in reverse osmosis applications vary between 15 bar (brackish water desalination) and 60 to 80 bar in seawater desalination [10]. which is 65 bar in the bwc case. the ro system consists of a number (usually 2-18) of individual ro trains, each of which is capable of independently producing desalinated water from pretreated source water [11]. once the desalination process is complete, the freshwater produced by the ro system is further treated for corrosion and health protection and disinfected prior to distribution for final use. this third step of the desalination plant treatment process is referred to as post-treatment [9]. fig. 2 presents a general schematic process of the bwc plant. fig.2. schematic of bwc plant (source: author) 4. implementation of the mads mosar medthod: macroscopic vision risk management: a continuous management process with the objective to identify, analyze, and assess potential hazards in a system or related to an activity, and to identify and introduce risk control measures to eliminate or reduce potential harms to people, the environment, or other assets [12]. two main purposes of the risk management are to ensure that adequate measures are taken to protect people, the environment and assets from undesirable consequences of the activities being undertaken, and to balance different concerns, for 96 m. bouamri, h. bouabdesselam copyright ©2018 assa. adv. in systems science and appl. (2018) example safety and costs [13]. the risk analysis discussed here is one that anticipates and prevents undesired events during the implementation of the combustion pilot, using mads– mosar. mads refers to “the analysis method of dysfunctional systems”, and mosar refers to “the organized and systemically method of risk analysis”. mads proposes a general model of hazard, mosar builds a global methodology for the risk analysis [14,15]. the retained approach is the "mads-mosar" methodology was developed by a team of specialists from the cea (french atomic energy authority) [16]. it makes it possible to model our industrial field by a "man-installation-environment" system [17, 18]. it is an integrated approach that allows progressive analysis of risks an industrial site [19]. mosar method consists of two modules (module a and b) which can be used more or less independently. module a corresponds to a macroscopic analysis of the risks of an industrial site and requires a preliminary risk analysis. module b is used for a more detailed analysis of the scenarios identified in module a, realized with specific instruments of safe operation. the two modules are almost the same structure [20]. mads mosar structure process is shown in fig.3. fig.3. mads mosar process steps (adapted from [21]). as shown in the figure above, the first step of the mosar method consists of modeling the bwc plant by means of functional division into subsystems. so that for each subsystem (ssi) the type of hazard source is identified and the hazardous processes are defined (short and long scenarios). hazard identification is an iterative process. to identify each hazard, we need to decompose the system into many subsystems. 4.1. systems identification and modeling the main system constituting the study context is the industrial installation on which the risk analysis is carried out (the bwc desalination plant in our case). there are several possible divisions, the aim being to make the complex entity to be studied in sub-units (subsystems) simpler. from the previous description of the process, we can distinguish eight subsystems: ss1: seawater intake and pumping section; ss2: seawater pretreatment section; risk analysis in seawater desalination sector 97 copyright ©2018 assa. adv. in systems science and appl. (2018) ss3: reverse osmosis section; ss4: post-treatment section; ss5: electrical substation; ss6: administrative building; ss7: human; ss8: environment. our system can be broken down into following subsystems (fig.4). fig.4. system decomposition into subsystems. 4.2. identify the sources of hazards the source of hazard should not be ignored. the first step is to identify the sources of hazard for each subsystem or to identify how each subsystem can be a source of hazard. by making this identification for all subsystems, a list of the hazards of the installation is obtained. table 1 illustrates the areas, subsystems decomposition and hazard sources associate to each subsystem. table 1. sources of hazards presented the bwc plant area subsystems (ssi) hazard sources seawater intake and pumping area ss1: seawater intake and pumping -seawater intake pipelines -intake screens -feed pumps (11 pumps) -seawater storage reservoir -electric substation -sodium hypochlorite tank (120 m3) -2 compressors ss2 : seawater pretreatment -ferric chloride tanks (34 m3) -sodium metabisulfite tanks (14 m3) -antiscalant storage tank (7 m3) -sulfuric acid storage tanks (2x 100 m3) -sand filters (48 reservoirs) -antracite filters (28 reservoirs) -cartridge filter (20 filters) ss3 :ro unit -10 high pressure pumps (65 bar) -ro membrane trains racks (10 98 m. bouamri, h. bouabdesselam copyright ©2018 assa. adv. in systems science and appl. (2018) production area racks) :  1792 membranes -energy recovery devices (22 units)  pressure exchanger  booster pumps and valves -256 pressure vessels -caustic soda tank (1 m3) ss4 : post-treatment  -chemical dosing system: neutralisation or chlorination  -sodium hypochlorite tank  -produced water tank  -brine disposal  -10 pumps  -chemical cleaning tanks  -sand and anthracite filters cleaning  -03 membranes cleaning pumps.  -produced water tanks : 11 pumps & pipelines ss5 : electrical substation -transformer power station ss6 : administrative building -offices -laboratory -control room ss 7 : human -operators trainees ss 8 : environment climatic and natural conditions material environment to identify hazards, our work begins with analyzing a review of past accidents triggered by the bwc plant and their similar facilities. based on the analysis of the accident balance sheets, an example of the repartition per month by accident triggered by the bwc plant during 2010 is shown in the following diagram (fig.5). fig.5. accident distribution per month during 2010 (source: bwc) 0 0,5 1 1,5 2 2,5 3 3,5 number of accident number of accident risk analysis in seawater desalination sector 99 copyright ©2018 assa. adv. in systems science and appl. (2018) then, and according to accident reports and accident balance sheets, we analyzed accidents triggered by major desalination plants in algeria. as is apparent from the table 2, 37.5% of accidents are related to the transformer accidents while 25% of the accidents caused pipeline leak. also, 25% of the accidents present an environmental pollution while 12,5% of them are due to an electrical cause (short-circuit generated fire in the storage area). table 2. accident triggered by the algeria's desalination plants (source: author) desalination plant years accidents arzew (oran) 17-10-2005 transformer fire arzew (oran) 12-02-2006 hydrocarbon pollution skikda 10-02-2008 fire in the storage area ain temouchent 14-04-2010 explosion of water hp pipeline mers el hadjadj (oran) 28-07-2011 transformer fire kahrama (oran) 19-05-2012 hydrocarbon pollution honaine (tlemcen) 25-08-2012 transformer explosion honaine (tlemcen) 26-09-2016 water leak results transformer accidents 37,5 % pipeline leaks 25 % environmental pollution 25 % electrical causes 12.5 % 4.3. identify the scenarios of hazards mosar method insist on hazard chaining flow between component systems of industrial plants and is specially adapted for studying the effects of simultaneous accidents or "domino“ effects [19]. each subsystem in table a (from module a) is characterized by inputs (initiating events) and outputs (principal events). we can also make the relation to find the danger flux to compete the models. the table a is the best tool to establish that. as a part of our risk analysis study, table 3 presents initiating events, initial events and main events (flux of danger) for the ss5 for the element transformer. table 3. identify the source of hazard for the transformer source of danger phase of process initiating events initial events main events (flow of danger) ss5 electrical substation external internal container contonent transformer exp -transformer overheating -thermal radiation -no compliance with the safety instruction -maintenance operations -as a consequence of the lightning -hazardous discharge of static electricity -humidity -effect of defective joint -low impedance faults -electrical malfunction -corroded cables -internal heat -locking of the disc -electric arc -short-circuit -over-voltage -degradation of insulation like decay of the transformer oil due to moisture or ageing or decomposition -loss of containment -corrosion of the tank -crack in the wall of the oil conservator corrosion and rupture of the tank due to aging or magnetic shock insufficient maintenance or too infrequent inspection. -oil overheating -oil leak -accidental oil spill inflammation of the transformer oil contained inside the metal envelope the presence of flammable gas -explosion -fire -pollution human damages -production shutdown -atmospheric pollution due to smokes -loss of cooling oil in transformer -materials loss or damage (premises destruction, plant damage) 100 m. bouamri, h. bouabdesselam copyright ©2018 assa. adv. in systems science and appl. (2018) using the black-box concept by subsystems such as presented the fig.6, we can make the reduction of the variety. short scenarios of undesired events can be structured from the links between them. from table a and short scenarios, we can easily define long undesired scenarios (as an example, fig.7 presents long scenarios for transformer explosion). 4.4. assess the scenarios of risks probability assessment can be made, at the option of the analyst, qualitative, semiquantitative or quantitative, using classical instruments (trees of failures, trees of events). mosar method explicitly provides for identify and assess security measures and distinguish between technical measures and usage measures (called in other methods, human protective measures”) [22]. in this study, 26 accident scenarios were identified as those most relevant to our system and analyzed. the figure bellow (fig.8) shows the combination of risk severity and probability for each scenario. it allows scenarios to be classified and decided the relative priority. fig.8. severity x probability combination. as mentioned in the figure above, scenarios 5, 10, 11, 19 and 20 have a high “severity x probability” scores whereas scenarios as 16, 18 and 26 present the low “severity x probability” scores. the rest scenarios have medium “severity x probability” scores. sc 1 sc 2 sc 3 sc 4 sc 5 sc 6 sc 7 sc 8 sc 9 sc 10 sc 11 sc 12 sc 13 sc 14 sc 15 sc 16 sc 17 sc 18 sc 19 sc 20 sc 21 sc 22 sc 23 sc 24 sc 25 sc 26 probability 4 4 2 2 3 2 2 2 2 3 1 3 2 3 2 1 1 1 3 3 2 3 3 3 3 1 severity 2 2 2 1 4 2 2 2 2 4 3 4 2 2 2 1 3 1 4 4 2 2 1 2 2 1 0 1 2 3 4 5 6 7 8 severity x probability evaluation 101 ss1 ss3 ss5 ss8 ss7 -earthquake -mechanical shock -bad welding -breaking fiberglass reinforced polyster seawater collector -external/internal corrosion -design, operation or manufacture defects -loss of seal overpressure -pressure controller failure -compressor components failure -sodium hypochlorite injection pump failures -defective valve -flood -total/partial shutdown -explosion -disrupt operation or damage the apparatus -fragment projections -materials damage -pollution -insufficient maintenance or too infrequent inspection -stress and fatigue -lack of adequate training -human error -non-compliance with the safety instructions -non-compliance with service parameters -overpressure -suction pressure too high -suction and/or discharge valves closed or clogged -excessive air entrapped in liquid -pump is run dry -pump run off design point air leak in suction line -cavitation -inadequate lubrication inadequate lubricant cooling -internal/external corrosion -overheating lack of seal flush at seal faces chemicals in liquid other than specified -mechanical failure -bursting of simple pressure vessels -total/partial shut-down -explosion -fire -noises -fragment projections -materials damage -internal heat -locking of the disc due to wear or sand -advanced pollution -imprudence of a smoker -thermal radiation -transformer overheating -insufficient maintenance -mechanical/electrical malfunction -corrosion and rupture of the tank due to aging or magnetic shock -oil leak -hazardous discharge of static electricity -short-circuit -total/partial shut-down -explosion -fire -fragment projections -materials damage -dust storm -earthquake -mechanical degradation -pollution -atmospheric pollution -flood -mechanical or thermal shock lightning humidity -advanced electrical substation pollution -materials damage -overheating -overpressure -control instruments failures -explosion -fire -short-circuit -injuries fig.6. examples of short and long accident scenarios 102 copyright ©2018 assa. adv. in systems science and appl. (2018) dielectric oil lack neighbor equipment bringing of a thermal radiation nearby fire corroded cables maintenance operations crack in the wall of the oil conservator corrosion of the tank short-circuit imprudence of a smoker locking of the disc due to wear or sand faulty seal causing an oil leak lightning consequences internal heat accumulation of dust and grease deposits explosion of the transformer electric arc overheating transformer oil leak and gate or gate no smoking leak detector periodic inspection preventive maintenance leak detector lightning protection electrical grounding disc checking insulating oil retention pond periodic inspection corrosion protection emergency plan fig.7. long scenarios for ss5: transformer explosion risk analysis in seawater desalination sector 103 copyright ©2018 assa. adv. in systems science and appl. (2018) risk management plan is a step in the overall risk management procedure following the risk assessment step. after all risks are identified in the risk assessment step, risks that are not acceptable must be selected. the main task in the risk management plan step is to treat each selected unacceptable risk. to perform risk quantification, a risk matrix must be developed for each accident initiator from the corresponding accident matrix [7]. once scenarios have been identified, they must be assessed and grouped in a grid by their severity and probability as shown in the figure 9. fig.9. “severity x probability” grid the “severity x probability” grid (fig.9) shows that only 42.3% of the scenarios are located in the acceptable zone (green zone). however, 57.7% of the scenarios are located in the unacceptable zone (red zone). 4.5. define the means of prevention and qualify the barriers the last step of the mads mosar approach is to suggest/define and to quantify barriers. in order to improve safety approach, preventive and protective appropriate barriers are cited in table 6. the table below presents some proposed preventive and other protective barriers that should be available for potential elements (elements that may present a significant hazard) in the bwc seawater desalination plant. these elements are respectively: pipelines, transformers, high pressure pump, compressors, pressure vessels and energy recovery devices. 104 copyright ©2018 assa. adv. in systems science and appl. (2018) table 6. recommendations component recommendations pipelines 1the implementations of an accidental spill prevention plan for preventing and controlling accidental spills or discharges (especially brine discharges). 2damaged or leaking containers will be isolated, when possible, in a containment area or repackaged to prevent loss, exposure or hazards. 3leak detection of water pipeline. 4periodic inspection of the water pipeline systems (routine maintenance of piping system). 5the replacement of old pipes with new pipelines. 6prevention of corrosion. 7preventive and corrective maintenance of the fiberglass reinforced polyster seawater collector (estimate the lifetime of collectors and change them periodically). transformer 1transformer status check/control. 2preventive maintenance of the various safety and protection devices of the transformer (relays, fuses, circuit breakers, powder and carbon dioxide extinguishers). 3dielectric oil transformer quality control (transformer oil aging) and replacement. 4compliance with safety instructions. 5design to the appropriate electrical standards. 6periodically temperature oil checks. 7gas emission control and periodic oil analysis. high pressure pumps 1preventive and corrective maintenances are recommended. 2install flow indicators. 3increase pressure indication and alarms. 4preventive and corrective maintenance can reduce the physical hazard of noise. 5recommend anti-vibration pump core. compressors 1installing a pressure sensor. 2carbon dioxide extinguishers. 3emergency response for projections accidents. 4fire fighting equipment, procedures and alarms for emergency response. 5personal protective equipment (ppe) uses. pressure vessels 1periodic pressure vessels inspection. 2operating service pressure respect. 3respect of the safety distance (explosion prevention). 4pressure control instrument. erd (energy recovery devices) 1periodic verification is recommended. 2leak and seal detectors. 3ensure the erd support. 4never exceed permitted flow limits. 5parameters service respect (flow and pressure: erd recommends au minimum 1.0 bar). 6erd cavitation prevention (most recognized problem for erd). 5. conclusion in this study, we carried out the bwc plant risk analysis throughout its life cycle, including seawater intake and pumping, pretreatment, reverse osmosis desalination, posttreatment, electrical substation, environment and human. to control or mitigate risks those appear to be the most critical, we were applied the mads mosar method. the most important problem in the bwc plant or in the similar plants lies in ro membranes in several pressure vessels, which are placed in a certain order. this process requires that a higher pressure (more than 65 bars) be exerted on the high concentration side of the membrane which would render the unit being out of service and pipes being burst. https://www.travelers.com/boilerre/boiler-reinsurance-services/risk-control/risk-control-guides.aspx risk analysis in seawater desalination sector 105 copyright ©2018 assa. adv. in systems science and appl. (2018) many hazards can be related to the bwc plant equipments such as the transformers, compressors, high pressure pumps and the energy recovery devices. they can involve fires, explosions, high-voltage electrical arcs, fragments projections, oil ignition and dispersions, and potential injuries or may up to death. they make noises and they also give off heat during operation. there is also a wide range of chemicals in the workplace (for the seawater desalination process or laboratory analysis). hazardous chemicals can be a health, physicochemical or environmental hazards if not handled or stored correctly. the safe clean up of a chemical spill requires knowledge of the properties and hazards posed by the chemical to also protect the environment. this study offered valuable results in terms of safety as indeed the recommendations and improvements yielded are expected to lead directly/indirectly to help the bwc plant in its safety approach. the risk management plan is a continuous risk controlling and monitoring process. risk assessment is only one step in the overall approach to risk management. it belongs to a dynamic approach. in addition, it is recommended that a new risk assessment be carried out on a regular basis to determine whether the risks have been eliminated definitively or whether other risks have arisen since the last assessment. the actions resulting from the evaluations must be performed, formalized and documented. references [1] pollard, s.j.t. (2007). risk management for water and wastewater utilities. water and wastewater technologies, united kingdom: iwa publishing. [2] cotruvo, j., voutchkov, n., fawell, j., payment, p., curnliffe, d et al. (2010). desalination technology health and environmental impacts. chemical aspects of desalinated water. boca raton, florida: crc press. [3] voutchkov, n. (2018, august 21). seawater reverse osmosis design and optimization. [online]. available https://web.stanford.edu/group/ees/rows/presentations/voutchkov.pdf [4] rezaei, k & aziznezhad, v & chenani, a & tabibian, m & mozdianfard.m.r. (2016).risk analysis of the sea desalination plant at the 5th refinery of south pars gas company using hazop procedures. journal of fundamental and applied sciences. 8(2s) 602-613, doi: http://dx.doi.org/10.4314/jfas.8vi2s.75. [5] abdel-fatah, m. a & el-gendi, a & ashour. f. (2016). performance evaluation and design of ro desalination plant: case study. journal of geoscience and environment protection. 4, 53-63. [6] mannan, s. (2005). lee’s loss prevention in the process industries, 3rd edition, vol. 1, oxford, uk: elsevier butterworth heinemann,. [7] bley, d.c., droppo j.g., eremenko v.a., lundgren r. e. (2003). risk methodologies for technological legacies: proceedings of the nato advanced study institute, bourgas, bulgaria. [8] bahr, n.j. (2015). system safety engineering and risk assessment: a practical approach, 2nd edition, new york: taylor & francis. [9] voutchkov, n. (2013). desalination engineering: planning and design. mcgraw hill professional. [10] fritzmann, c., lowenberg, j., wintgens, t., melin t. (2007). state-of-the art of reverse osmosis desalination, desalination, 216, 1-76. https://web.stanford.edu/group/ees/rows/presentations/voutchkov.pdf http://dx.doi.org/10.4314/jfas.8vi2s.75 106 copyright ©2018 assa. adv. in systems science and appl. (2018) [11] voutchkov, n. (2014). desalination engineering: operation and maintenance. mcgrawhill professional publishing. [12] rausand, m. (2010). risk assessment: theory, methods, and applications, hoboken, n.j.: j. wiley & sons. [13] aven, t. (2011). quantitative risk assessment: the scientific platform, cambridge university press, cambridge. [14] périlhon, p. (2000). analyse des risques, élément méthodiques, phoebus, la revue de la sureté de fonctionnement, l’analyse de risques, 12, 31–49. [15] bultel, y., aurousseau, m., ozil, p., & perrin, l. (2007). risk analysis on a fuel cell in electric vehicle using the mads/mosar methodology, process safety and environmental protection, 85(3), 241-250. [16] ayrault, n. (2005). evaluation des dispositifs de prévention et de protection utilisés pour réduire les risques d’accidents majeurs (dra-039), rapport oméga 10 – evaluation des barrières techniques de sécurité, ineris, france. [17] marref, a.s., bahmed, b.l. & benoudjit, a.c. (2014).multi-field analysis of the systems sociotechnic in the industrial developing countries: case of the industry of cotton in algeria. international journal of innovation, management and technology. 5(4), 287-293. [18] périlhon, p. (1998). risk analysis risk: development of a method mosar organized and systemic risk analysis method, school of mines saint-etienne. [19] felegeanu, d –c., nedeff, v. & panainte, m. (2013).analysis of technological risk assessment methods in order to identify definitory elements for a new combined/complete risk assessment method, journal of engineering studies and research, 19 (3), 32-43. [20] perilhon, p. (2000). mosar, techniques de l'ingénieur, traité sécurité et gestion des risques, se, 4(060), 1-16. [21] laurent, a. (2013). sécurité des procédés chimiques connaissances de base et méthodes d’analyse des risques, lavoisier tec & doc. [22] perrin, l & felipe, m.g & dufaud, o & laurent, a. (2012). normative barriers improvement through the mads/ mosar methodology, safety science. 50(7), 1502 1512. http://eu.wiley.com/wileycda/section/id-302479.html?query=marvin+rausand adv syst sci appl 2017; 4; 78-92 published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/521 copyright ©2017 assa. adv. in systems science and appl. (2017) developing a strategy of environmental management for electric generating companies using dea-methodology svetlana ratner, pavel ratner v.a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia e-mail: lanaratner@gmail.com abstract: this paper investigates the possibility of utilization of the data envelopment analysis (dea), which is a non-parametric method of optimization, to solving problems of environmental management in electric generating companies. an advantage of dea is the possibility to work with dmus without any knowledge of the actual functional relation between inputs and outputs. we analyze methods of incorporating the negative ecologic effects into a model and propose an algorithm for applying the basic dea ccr input-oriented model twice in succession for the purpose of developing an optimal (ecologically and economically) strategy for environmental management in electricity energy generating companies. the developed method consists of sequentially solving several dea models: the first-stage model determines the effectiveness of dmus from an ecologic perspective and calculates target values for decreasing negative ecologic effects of non-effective dmus. the second stage requires solving one input-oriented ccr model for each non-effective object, using economic and social characteristics of projects meant to reduce negative environmental influence, and using the target values calculated in the first stage as outputs. besides the problem of evaluation the comparative efficiency of dmus, ecologically oriented studies also often needs to evaluate the changes in a dmu’s efficiency dynamically. for this, the malmquist productivity index (mpi) is used. mpi is a non-parametric method for analyzing time series that allows to track changes in dmu efficiency over time by means of dea models. we test this algorithm on the statistical data provided by russian electric companies for the period 2009-2011, and discuss methods for its practical application. the statistical data used in our calculations is averaged, and the results do not reflect the entire picture and should not be used to judge the quality of ecologic management in these companies. nevertheless, the calculations can be used to evaluate the progress of completion and the practicality of investment projects of companies, from an environmental viewpoint. they also may be used to help develop state programs for support of modernization in electric energy industry, ecologic standards or energy-saving programs. keywords: data envelopment analysis, non-parametric optimization, ecologic effects, environmental management, ecology management, electric companies. 1. introduction currently, electric generating objects that operate on hydrocarbon fuels are one of the largest emitters of greenhouse gases and other atmospheric pollutants (16% of total emissions from stationary sources in russia), as well as consumers of fresh water (35% of total water use in russia), pollutants of soil, underground and surface waters. increasing the ecologic efficiency of electric generating companies is one of the primary conditions for sustainable development of both this industry and the country itself. one of the most important problems for ecologic optimization of the development of electric energy generation industry is decreasing negative environmental influence as much as possible, via a variety of environmental protection measures (both technologic and organizational), while maintaining the existing volumes of electricity production [20]. investment priorities for energy companies directed towards decreasing negative ecologic effects are determined primarily by conventional system of ecologic penalties for overhttp://ijassa.ipu.ru/ojs/ijassa/article/view/521 79 s. ratner, p. ratner copyright ©2017 assa. adv. in systems science and appl. (2017) standard pollution and rarely correspond to the actual ecologic situation of the region. many sources (e.g. [5, 19]) consider the existing russian economic mechanisms and incentives for minimizing negative environmental influence ineffective. at the same time, there are currently no examples of successful transition to ecologically safe energy generation technologies at regional or national scales anywhere in the world. the well-known brazil energy crisis of 2001 that happened due to the droughts and lack of water for hydro-power plants [9], the recent chinese ecologic crisis due to the increase of coal use for power generation, the non-proportionally intensive growth of solar power plants in czech republic [10] show the complexity of the problem of optimizing the energy system configuration. a lack of complete understanding of the way certain energy-generating processes influence ecosystems as well as a lack of attention to ecologic aspects of energy systems on their planning stage can lead to unexpected and undesired results, where the decrease of negative influence from one parameter (or group) is completely overshadowed by the increase in another parameter (or group). for instance, attempting to capture со2 to decrease greenhouse gas emissions leads to a significant increase in water use by power plants [11]. increasing the amount of optimization criteria for incorporation while planning the structure of a region energy system (decrease of negative environmental influence, decrease of energy production price, maximization of useful social and economic effects) served as another reason for using non-parametric methods of data envelopment analysis (dea) for solving this class of problems. among the multitude of approaches to modelling energetic and ecologic problems in foreign literature, dea has attained a leading position. one of the main reasons is the possibility to model comparatively the efficiency of various energy sectors in a variety of countries, which has become especially relevant with the liberalization of energy markets [24]. currently, dea is a well-developed methodology for comparing the efficiency of various homogenous economic agents, operating in production or elsewhere, via a variety of mathematical programming models. the agents the efficiency of which is evaluated by dea are usually known as “decision-making units” (dmus). all dmus perform the same production function that transforms a set of inputs into a set of outputs. an advantage of dea is the possibility to work with dmus without any knowledge of the actual functional relation between inputs and outputs. russian economists conventionally employ dea to analyze the efficiency of the budgetary system, regional authorities or banking structures, etc. [17, 18]; however, during the recent years, studies that use dea to analyze ecologic aspects of economic activity, including electric energy [12], started appearing. the classic dea model known as ccr (named after its developers: charnes a., cooper w.w., and rhodes e. [4]) involves solving a fractional linear programming problem that maximizes the ratio of the linear combination of weighted outputs to the linear combination of weighted inputs:    k kjk i iji vu xv yu h , max (1.1) this ratio is known as the efficiency coefficient, and its value lies between zero and one. any dmus with their efficiency coefficient equal to one are considered efficient, and all the others are, inversely, deemed ineffective. if the efficiency coefficient is defined in the form of (1.1), the dea problem itself is considered to be “input-oriented”. the other form for defining the efficiency coefficient (a ratio of the linear combination of weighted inputs to the linear combination of weighted outputs) is known as “output-oriented”. in our case, we’re dealing with an output-oriented problem. developing a strategy of environmental management for electric generating 80 copyright ©2017 assa. adv. in systems science and appl. (2017) calculating the projections to the efficiency frontier for inefficient dmus in the input/output space allows us to determine the estimated targets for decreasing inputs or increasing outputs. achieving these projected values will allow the dmu to become efficient. a peculiarity of using dea for optimizing energy systems is the presence of so-called undesirable outputs, that is to say, the negative ecologic effects. for solving ecological problems, a special class of dea models was developed: these are known as “environmental dea (edea)”. the goal of this paper is to review methods and approaches of accounting for undesirable outputs in environmental dea models and developing an algorithm for applying the basic input-oriented dea model for performing a comparative analysis of the ecologic efficiency of large electric generating companies of russia. we tested the capabilities of the developed two-stage algorithm on 24 dmus on its first stage and on detailed data of 11 power plants (that are part of ogk-2) on its second stage. 2. methogology overview: dea models used for optimizing energy systems based on ecologic criteria in its coefficient form, the classic input-oriented ccr dea model is as follows: 0 1 , max m m m m vu yu  (2.1) s.t. ;,2,1,2,10, ,1 ,,2,10 0 1 11 nnmmvu xv kkxvyu nm n n n n nk n n n m m mkm          where 0 – index of the dmu being optimized, x – input vector of size n, y – output vector of size m, к – amount of dmus. or, in dual form:   min (2.2) s.t: kk mmyy nnxx k mok m m mk nok n n nk    ,2,1,0 ,2,1, ,2,1, 1 1           this model searches for the possibility of proportionally decreasing inputs without a decrease in outputs. the ccr production set is the following set of vectors (x, y):              n j n j jjjjj njyyxxyxт 1 1 ,1,0,,),(  (2.3) 81 s. ratner, p. ratner copyright ©2017 assa. adv. in systems science and appl. (2017) classic dea models, including ccr, assume their inputs and outputs are strictly monotonous, in other words, the production set follows this rule: if tyx );( and xx  or yy  , then tyx  );( or tyx );( (2.4) however, this property does not always describe the real production situation. for instance, energy generation via hydrocarbon fuel will always be linked to the production of sulfur dioxide, and decreasing this output without a decrease in the corresponding input is technologically impossible. thus, using a production set that follows (2.4) will lead to incorrect modelling results. a literature review allows one to conclude, that a significant number of attempts to consider undesirable outputs in dea models have been made. these fall into two main categories: i) recalculating (modifying) original data and using a traditional dea model [22]; ii) using original data with models based on the concept of weak disposability [6-8]. when using the first approach, the overall efficiency of a company may be divided into technical/economic efficiency, defined as a ratio of the weighted sum of wanted outputs to the weighted sum of inputs, and ecologic efficiency, which is defined as the ratio of weighted sums of wanted and unwanted outputs. let the first k of m outputs of the model (2.1) be desirable, and the others undesirable. then, the economic efficiency of dmu with the index of zero can be represented as:     m i ii k r rr economy xv y h 1 0 1 0 (2.5) and the ecologic efficiency as:     p ks ss k r rr yeco y y h 1 0 1 0 log   (2.6) to incorporate both efficiency measures in the basic ccr model, we have to somehow combine them in a way that corresponds to the general logic of the problem: maximization of desirable outputs and minimization of undesirable outputs and inputs. the following option (a) fits these conditions rather well:       m i ii p ks ss k r rr a xv yy h 1 0 1 01 0  besides that, undesirable outputs can be treated as inputs of the model (option b), which transforms the efficiency measure as follows:       p ks ss m i ii k r rr b yxv y h 1 01 0 1 0   . in this case, the decrease in undesirable outputs happens simultaneously with the decrease of inputs. paper [17] provides the proof that basic ccr models that use option a are analogous to those which employ option b. the production set corresponding to the property of weak disposability is defined thusly: developing a strategy of environmental management for electric generating 82 copyright ©2017 assa. adv. in systems science and appl. (2017)                                  ;,2,1,0 ,,2,1 ,,2,1 ,,2,1|),,( 1 1 1 kkz jjuuz mmyyz nnxxzuyx t k k k jjkk k k mmkk k k nnkk e     where u is a vector of undesirable outputs, e is an index with the meaning of “environmental”, since these types of production sets are used in environmental dea. besides that, et also fulfills the following condition: if tuyx ),,( and u=0, then y=0 (2.7) this indicates that a complete elimination of undesirable outputs is only possible via a complete halt of the production process. this approach can be used for undesirable inputs as well: models with weakly disposability of inputs are introduced in [7, 16]. besides disposability, operational characteristics of inputs and outputs (their unit measures) may also serve as a distinct characteristic of environmental dea. two models [12] often mentioned in the literature use categories for input and output variables. these models describe production processes during implementation of environmental laws rather well, and allow to consider the influence from other external factors that dmus themselves have no control of. return to scale is also an important characteristic of the dea production set. the basic ccr model uses constant return-to-scale. if we add the following condition to the set t:    k i i 1 1 , we’ll end up with the bcc model with variable returns to scale (while production grows, the scale effect changes from increasing to decreasing). another popular approach is the use of non-increasing return to scale, defined as:    k i i 1 1 . besides the orientation of the efficiency measure (input, output or undesirable output), an important property of environmental dea problems is the method for reducing inputs and increasing outputs, that is to say, the direction of movement towards the efficiency frontier. radial efficiency measures are the ones most frequently encountered in any dea models. in this case, the inputs decreasing proportionally by the same value of oc co  (radial movement from source point to efficiency frontier) (see fig. 1). 83 s. ratner, p. ratner copyright ©2017 assa. adv. in systems science and appl. (2017) fig. 2.1 graphical illustration of the radial efficiency measure in a basic ccr model (2d case) combining radial efficiency measures with different production sets allows us to obtain a variety of dea models, including the basics: ccr and bcc. for example, the papers [7, 22, 23] use the production set et with a radial efficiency measure: etuyx ),,(:min 000  non-radial efficiency measures allow to decrease inputs and increase outputs nonproportionally, and usually have a better discriminative power than radial measures. a particularly well-known example is the non-radial russell measure:         n n n tyx n 1 00 ),(: 1 min θ , where θ is a diagonal matrix containing n ,1 . if n ,1 have different weights, a weighted non-radial efficiency measure may be used. such measures reflect the preferences of specific dmus [25]. a hyperbolic efficiency measure, also known as the graphical measure, decreases inputs and increases outputs by the same value at the same time (moving towards the efficiency frontier hyperbolically):       t y x ),(: 0 0 θ θθ this efficiency measure works best when both desirable and undesirable outputs are present. the directional distance function (ddf) is an efficiency measure that allows one to simultaneously increase desirable outputs and decrease inputs using a directed vector. this is a generalized form of the traditional radial efficiency measure [3]. 3. results 3.1. analyzing the comparative efficiency of energy generating companies in russia with ecologic indicators let’s consider the problem of evaluating the efficiency of the russian energy generating companies based on a set of ecologic parameters. to calculate the ecologic efficiency with the basic input-oriented ccr model (2.1) with a radial efficiency measure, we’ll use freely available statistical information [15] on ecologic aspects of the activity of the main players on the electric energy market: five wholesale generating companies (ogk) that unify the largest heat power-plants, and some territorial generating companies (tgk) that unify the developing a strategy of environmental management for electric generating 84 copyright ©2017 assa. adv. in systems science and appl. (2017) power plants of several neighboring regions that did not become parts of ogk and work as isolated energy systems (24 dmus total). we view atmospheric emissions (in thousands of tons), solid waste (in thousands of tons) and freshwater consumption (in millions of cubic meters) as our unwanted outputs. we consider the generated electricity as our sole wanted output. the results of our calculations, done in the maxdea software, using a radial and a nonradial efficiency measure, represented in table 3.1. table 3.1. scores of ecologic efficiency of generating companies in 2011 company name radial efficiency non-radial efficiency ogk-1 0.374 0.190 ogk-2 0.168 0.115 ogk-3 0.142 0.101 ogk-4 ojsc «e.on rossiya» 0.611 0.543 ogk-5 «enel ogk-5» 0.132 0.093 tgk-1 0.300 0.265 tgk-2 0.112 0.111 ojsc «mosenergo» (tgk-3) 1.000 1.000 tgk-4 ojsc «kvadra» 0.558 0.477 tgk-5 0.428 0.402 tgk-6 0.591 0.412 ojsc «volzhskaya tgk» (tgk-7) 1.000 1.000 tgk-9 0.122 0.096 ojsc «fortum» (tgk-10) 0.366 0.299 tgk-11 0.532 0.206 ojsc «kuzbassenergo» (tgk-12) 0.115 0.081 ojsc «eniseyskaya tgk» (tgk-13) 0.089 0.075 tgk-14 0.717 0.475 generiruyushchiye kompanii «lukoyl» 1.000 1.000 ojsc «dal'nevostochnaya gk» 0.177 0.091 ojsc «irkutskenergo» 0.528 0.284 ojsc «tatenergo» 1.000 1.000 ojsc «bashkirenergo» 0.678 0.616 ojsc «sibeko» 0.136 0.101 the dmu efficiency score herein is to be interpreted as a ratio of the minimal possible negative ecologic effects to the real ones. that is to say, the effective dmus are those who use the best available technologies and the cleanest fuel (from the ecologic point of view). the efficiency coefficient for the effective dmus is equal to 1, and highlighted in bold. it is easy to note that the efficiency coefficient of non-effective dmus is greater when calculated with the radial measure than with the non-radial one. the target indicators (for 2011) that have to be reached by non-efficient companies (calculated under a radial efficiency measure) to become efficient are presented in table 3.2. table 3.2. values of target indicators for inputs that need to be reached in 2011 company name target indicators emissions waste water consumption ogk-1 34.256 84.108 332.035 ogk-2 63.444 155.770 614.933 ogk-3 26.513 65.097 256.985 85 s. ratner, p. ratner copyright ©2017 assa. adv. in systems science and appl. (2017) ogk-4 ojsc «e.on rossiya» 55.864 147.106 412.725 ogk-5 «enel ogk-5» 43.741 121.476 241.682 tgk-1 16.385 29.442 158.045 tgk-2 9.305 26.654 40.894 tgk-4 ojsc «kvadra» 11.685 19.870 112.633 tgk-5 10.929 31.453 46.131 tgk-6 15.532 18.012 270.476 tgk-9 14.927 42.799 65.070 ojsc «fortum» (tgk-10) 18.787 51.954 106.641 tgk-11 9.231 26.567 38.964 ojsc «kuzbassenergo» 19.991 52.513 149.342 ojsc «eniseyskaya tgk» 11.266 31.170 63.775 tgk-14 26.226 75.477 110.700 ojsc «dal'nevostochnaya gk» 23.341 67.172 98.520 ojsc «irkutskenergo» 63.063 181.488 266.183 ojsc «bashkirenergo» 20.362 53.769 148.491 ojsc «sibeko» 11.243 31.417 59.607 when the indicators given in table 3.2 are reached, each of these companies can match its reference point on the efficiency frontier in the multi-dimensional input/output space. the process of calculation of target indicators under non-radial efficiency measure has shown that their values change only in case then dmu has several benchmarks (see table 3.3.) table 3.3. changes in target indicators depending from the type of efficiency measure company name reference points δemissions δwaste δwater ogk-4 «mosenergo»; «tatenergo» 5.93 24.50 -71.27 ogk-5 «mosenergo»; «tatenergo» -3.51 -14.52 42.22 tgk-1 «mosenergo»; «volzhskaya tgk» 3.39 -2.45 32.13 tgk-2 «mosenergo»; «tatenergo» -0.10 -0.41 1.20 tgk-4 «mosenergo»; «volzhskaya tgk» 2.77 -2.01 26.27 tgk-6 «mosenergo»; 6.25 -4.77 180.53 tgk-9 gk lukoyl -0.13 -0.52 1.53 tgk-10 «mosenergo»; «tatenergo» -1.68 -6.96 20.24 tgk-12 «mosenergo»; «tatenergo» -4.00 -16.53 48.08 tgk-13 «mosenergo»; «tatenergo» -0.99 -4.13 12.01 ojsc «bashkirenergo» «mosenergo»; «tatenergo» 2.25 9.30 -27.05 ojsc «sibeko» «mosenergo»; «tatenergo» -0.75 -3.09 8.99 moving towards the efficiency frontier non-radially leads to some target parameters being greater than in the radial efficiency measure (such differences are indicated as negative developing a strategy of environmental management for electric generating 86 copyright ©2017 assa. adv. in systems science and appl. (2017) numbers), and in some cases, they will be smaller (which is indicated by a positive number). which of these two efficiency measures is the best depends on the expenditure of the specific company trying to reach these target indicators. 3.2. dynamic of ecology efficiency of generating companies in this section we investigate how ecologic efficiency of generating companies changes through time. table 3.4 shows efficiency scores of generating companies throughout 20092011, calculated according an input-oriented ccr model with a radial efficiency measure. table 3.4. ecology efficiency of generating companies in 2009-2011 company name 2009 2010 2011 ogk-1 0.374 0.387 0.374 ogk-2 0.173 0.165 0.168 ogk-3 0.171 0.158 0.142 ogk-4 ojsc «e.on rossiya» 0.624 0.577 0.611 ogk-5 «enel ogk-5» 0.147 0.126 0.132 tgk-1 0.534 0.378 0.300 tgk-2 0.130 0.115 0.112 ojsc «mosenergo» (tgk-3) 1.000 0.993 1.000 tgk-4 ojsc «kvadra» 0.588 0.400 0.558 tgk-5 0.505 0.464 0.428 tgk-6 0.338 0.366 0.591 ojsc «volzhskaya tgk» (tgk-7) 0.731 0.627 1.000 tgk-9 0.136 0.118 0.122 ojsc «fortum» (tgk-10) 0.438 0.308 0.366 tgk-11 0.667 0.714 0.532 ojsc «kuzbassenergo» (tgk-12) 0.123 0.107 0.115 ojsc «eniseyskaya tgk» (tgk-13) 0.097 0.078 0.089 tgk-14 0.099 0.079 0.717 generiruyushchiye kompanii lukoyl (before 2010: «yugk tgk-8») 1.000 1.000 1.000 ojsc «dal'nevostochnaya gk» 0.208 0.189 0.177 ojsc «irkutskenergo» 0.719 0.597 0.528 ojsc «tatenergo» 1.000 1.000 1.000 ojsc «bashkirenergo» 0.527 0.510 0.678 ojsc «sibeko» 0.142 0.126 0.136 in 2009 and 2010, out of all analyzed dmus, only 3 companies: ojsc “mosenergo” and the generating companies of the “lukoil” and “tatneft” groups, were ecologically effective. another company joined this list in 2011: ojsc “volzhskaya ttk”. however, it is not correct to judge changes in actual ecologic efficiency merely by these coefficients, since the efficiency frontier itself may shift from year to year. to evaluate the dynamic changes of dmu efficiency in dea problems, the malmquist productivity index (mpi) is used. this is a non-parametric method for analyzing time series [13]. in its general from, the mpi can be defined with a distance function as its base, however, it can also be presented as a ratio of efficiency measures. 87 s. ratner, p. ratner copyright ©2017 assa. adv. in systems science and appl. (2017) let ),( 00 ttt yx and ),( 00 1 ttt yx be the input-oriented efficiency measures for dmu0, calculated using inputs and outputs from the solution of (2.2) during the moment t and the production set t, in the moments t and t+1. let ),( 1 0 1 0  ttt yx и ),( 1 0 1 0 1  ttt yx be the inputoriented efficiency measures for dmu0, calculated using inputs and outputs from the solution of (2.2) during the moment t+1 and the production set t, in the moments t and t+1. then, the input-oriented mpi can be defined as: 2 1 00 1 1 0 1 0 1 00 1 0 1 0 0 ),( ),( ),( ),(          ttt ttt ttt ttt yx yx yx yx mpi     (3.1) values of mpi0<1, mpi0=1 and mpi0>1 signify, respectively, a decrease, an increase or constancy in the efficiency of dmu0 throughout the research period [13]. various sources often employ the following form for the malmquist index: ),( ),( ),( ),( ),( ),( 00 1 0 1 0 12 1 1 0 1 0 1 1 0 1 0 00 1 00 0 ttt ttt ttt ttt ttt ttt yx yx yx yx yx yx mpi                  (3.2) the form (3.2) defines changes in efficiency in a decomposed format, with the first part representing the frontier shift effect, and the second part – the catch-up effect. the calculated malmquist indices are shown in table 3.5. table 3.5. malmquist index values (output-oriented) throughout 2009-2010 and 2010-2011 company name mpi 20092010 mpi 20102011 ogk-1 0.9522 0.9426 ogk-2 0.8574 0.9784 ogk-3 0.9522 0.9426 ogk-4 ojsc «e.on rossiya» 0.6700 1.0242 ogk-5 «enel ogk-5» 0.8805 1.0170 tgk-1 0.5365 0.6175 tgk-2 0.8805 0.9484 ojsc «mosenergo» (tgk-3) 0.9723 0.9865 tgk-4 ojsc «kvadra» 0.6705 1.3079 tgk-5 0.8805 0.9200 tgk-6 0.8469 1.2539 ojsc «volzhskaya tgk» (tgk-7) 0.7708 1.2641 tgk-9 0.8805 0.9520 ojsc «fortum» (tgk-10) 0.8891 1.1160 tgk-11 0.8805 0.9200 ojsc «kuzbassenergo» (tgk12) 0.8556 1.0659 ojsc «eniseyskaya tgk» (tgk-13) 0.8157 1.1212 tgk-14 0.8805 1.2143 generiruyushchiye kompanii lukoyl 1.0171 1.0569 ojsc «dal'nevostochnaya gk» 0.8805 0.9200 ojsc «irkutskenergo» 0.8805 0.9200 ojsc «tatenergo» 0.9383 1.0000 ojsc «bashkirenergo» 0.7630 1.3424 developing a strategy of environmental management for electric generating 88 copyright ©2017 assa. adv. in systems science and appl. (2017) ojsc «sibeko» 0.8550 1.0862 thus, a real increase in environmental efficiency has only been observed within the “lukoyl” generating companies in 2009-2010, and it has decreased for the other companies. during 2010-2011, already 12 companies have exhibited an increase in ecology efficiency. malmquist index values signifying an increase in efficiency are highlighted in bold in the above table. 4. policy application: elaboration of an optimal environmental management strategy for generating companies it is necessary to mention that all of the calculations done so far only considered ecologic efficiency of dmus, without taking into account any economic parameters of power plant operating. in fact, incorporation of economic parameters in analysis is possible with the use of methods described in section 2. however, we believe that elaboration of proactive environmental investment strategy for a generating company can be done with a far simpler approach, which involves solving the two basic dea problem consistenly. on the first step, we solve the problem of evaluating ecologic efficiency of the company’s activity using primary ecologic indicators, and calculate target values if the company ends up ineffective. on the second stage, we build another dea model for each ineffective dmu, which uses economic parameters (cost, implementation time, etc.) of environmental projects (meant to reduce negative ecologic effects down to the calculated target value, or a value sufficiently close to that) as inputs. solving these models allows us to choose projects that are most effective at achieving positive ecologic results, from an investor’s point of view. sequentially processing data for each of the power plants that is part of an ogk or a tgk, we obtain a problem for optimizing the development of the entire energy system. let us consider the problem of evaluating the efficiency of specific power plants that are part of ogk-2 (which is recognized as ineffective in 2011). table 4.1 shows the results of a solving ccr dea model with a radial efficiency measure, calculated in maxdea, using data from the company’s official 2013 reports (http://www.ogk2.ru). table 4.1. ecologic efficiency of ogk-2 power plants (ccr model) power plant efficiency coefficient input target indicators emissions, thousands of tons waste, thousands of tons water, millions of cubic meters adlerskaya 1 743.25 3.09 642.33 kirishskaya 1 2415.57 3.40 574645.2 krasnoyarskaya-2 0.066 1572.01 1.26 34978.45 novocherkasskaya 0.057 3262.63 2.57 52722.81 pskovskaya 0.876 488.38 0.69 116182.1 ryazanskaya 0.422 3507.38 14.60 3031.13 serovskaya 0.039 645.04 0.49 3281.98 stavropol'skaya 0.744 2489.71 3.51 592281.4 surgutskaya -1 1 7432.52 5.57 21622.2 troitskaya 0.237 1818.42 7.57 1571.507 cherepovetskaya 0.041 932.33 0.72 11876.88 89 s. ratner, p. ratner copyright ©2017 assa. adv. in systems science and appl. (2017) it can be seen that three power plants that belong to ogk-2 (adlerskaya, kirishskaya and surgutskaya-1) are effective from an ecologic point of view. all three of these power plants use gas as their primary source of fuel; however, the pskovskaya and stavropolskaya power plants use gas also and are, nevertheless, ineffective. this means that the fuel type is not the only factor of ecologic efficiency. the “closest” statistical indicator (meaning-wise) to our calculated efficiency coefficient, that is used in the practice of energy management in generating companies is the ratio of fuel consumption to generated energy, which characterizes the efficiency of different generating equipment (the linear pearson correlation coefficient, calculated with ogk-2 power plant data, equals -0.81). improving ecologic indicators for generating companies is possible thanks to the implementation of the following projects, unrelated to changing fuel or the technology used for energy generation [12, 15, 19-20]: 1) reduction of atmospheric emissions by implementation of low-toxicity burners for highly-concentrated dust, thereby decreasing nitrogen oxides, installation (or repairing) ventilation technologies, energy filters and ash collectors; 2) reduction of water consumption by cleaning and discarding mineralized waste waters, installation (or repairing) treatment facilities for industrial wastewater, implementation of oil collectors; 3) reduction of waste generation by recycling ashes, and/or developing technologies to increase the reliability of their storage; 4) increasing overall environmental efficiency by implementation iso 14001 compliant environmental management systems (currently implemented on the stavropolskaya, pskovskaya, surgutskaya, serovskaya and troitskaya power plants), decrease fuel expenditure for energy generation. besides that, one also needs to consider the advanced technological capabilities for replacing existing fuel with cleaner alternatives or switching to completely different energy generation technologies (options of innovative development). the least expensive fuel option is currently to use natural gas for energy generation, and the least expensive options for new energy technologies, according to russian experts, are photovoltaics and wind energy [15]. atom energy and “clean coal” technologies that minimize atmospheric pollution are midrange in terms of expense. cost and time indicators of the aforementioned projects (as well as some indicators of their social efficiency) can be used as input parameters for a next-level ccr dea model. same model can be used to optimize economic and social parameters of investment project meant to decrease negative environmental influence of the non-efficient power plants. as outputs, we can use the target indicators of ecologic effects calculated during the last stage. the overall algorithm with which the model can be built is shown on figure 4.1. we cannot provide an example calculation for this case due to a lack of available statistical data on economic values of investment projects. developing a strategy of environmental management for electric generating 90 copyright ©2017 assa. adv. in systems science and appl. (2017) fig. 4.1. algorithm of second-level dea model creation 6. conclusion the main result of this study is the proof of possibility for using a basic input-oriented ccr dea model for solving the problem of choosing an optimal environmental management strategy for a generating company. it is important to point out that each of the large generating companies analyzed in this research consists of several separate dmus – power plants that work using different equipment and different generating technologies. therefore, the statistical data used in our calculations is averaged, and the results do not reflect the entire picture and should not be used to judge the quality of ecologic management in these companies. nevertheless, the calculations can be used to evaluate the progress of completion and the practicality of investment projects of companies, from an environmental viewpoint. they also may be used to help develop state programs for support of modernization in electric energy industry, ecologic standards or energy-saving programs. the developed method consists of sequentially solving several dea models: the firststage model determines the effectiveness of dmus from an ecologic perspective and calculates target values for decreasing negative ecologic effects of non-effective dmus. the second stage requires solving one input-oriented ccr model for each non-effective object, using economic and social characteristics of projects meant to reduce negative environmental influence, and using the target values calculated in the first stage as outputs. the choice of an efficiency measure (radial or non-radial) depends on the cost of the projects, and can only be determined while solving the second-stage dea problems (determining economic efficiency). besides the cost, the choice of an efficiency measure can be made by judging the importance of a specific ecologic effect for the specific region or territory. besides the problem of evaluation of the comparative efficiency of dmus, ecologically oriented studies also often need to evaluate the changes in a dmu’s efficiency dynamically. for this, the malmquist productivity index (mpi) is used. mpi is a non-parametric method for analyzing time series that allows to track changes in dmu efficiency over time by means of dea models. 91 s. ratner, p. ratner copyright ©2017 assa. adv. in systems science and appl. (2017) acknowledgements this study is partly supported financially by the russian foundation for basic research (rfbr), project no. 16-06-00147 “development of data envelopment analysis models for optimizing development of regional economic systems based on ecologic parameters”. references [1] banker, r.d. & morey, r.c. (1986) efficiency analysis for exogenously fixed inputs and outputs, operations research, 34, 513–521 [2] banker, r.d. & morey, r.c. (1986) the use of categorical variables in data envelopment analysis, management science, 32, 1613–1627 [3] chung, y.h., fare, r. & grosskopf, s. (1997) nproductivity and undesirable outputs: a directional distance function approach, journal of environmental management, 51, 229–240 [4] cooper, w.w., seiford, l.m. & tone, t. (2006) introduction to data envelopment analysis and its uses: with dea-solver software and references. springer, new york. [5] fadeyeva, a.v. 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(1996) data envelopment analysis with preference structure, journal of the operational research society, 47, 136–150. java based distributed learning platform adv syst sci appl 2019; 01; 116-140 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/591 models of continuous dynamics on the 2-simplex and applications in economics denis stijepic*1 1) university of hagen, department of macroeconomics, hagen, germany e-mail: denis@stijepic.com received may 16, 2018; revised january 21, 2019; published april 15, 2019 abstract: in this paper, we discuss the models of continuous dynamics on the 2-simplex that arise when different qualitative restrictions are imposed on the (continuous) functions that generate the dynamics on the 2-simplex. we consider three types of qualitative restrictions: inequality (or set-theoretical) conditions, monotonicity/curvature (or differential-geometrical) conditions, and topological conditions (referring to (transversal) non-(self-)intersection of trajectories). we discuss the implications of these restrictions for transitional and limit dynamics on the 2-simplex and the wide range of potential and existing applications of the resulting system-theoretical models in economics and, in particular, in economic growth and development theory. keywords: dynamics, trajectory, 2-simplex, continuous, monotonous, intersection, self-intersection, poincaré-bendixson theory, economics 1. introduction in this paper, we discuss the models of continuous dynamics on the 2-simplex that arise when different qualitative restrictions are imposed on the (continuous) vector function x(t) ≡ (x1(t), x2(t), x3(t)) that generates the dynamics on the 2-simplex (where t represents time). in particular, there are three major types of qualitative conditions that can be imposed on this function: (1.) inequality conditions of the type ∀t ∈ a ∀i ∈ b xi(t) ≶ ai = const., which can be treated by using set-theoretical concepts (referring to the points or segments of the corresponding trajectory and the partitions of the 2-simplex); (2.) (strict) monotonicity conditions referring to all or some of the functions xi(t), which can be treated by (differential) geometrical concepts of tangential vector angles and curvature; and * corresponding author: denis@stijepic.com mailto:denis@stijepic.com models of continuous dynamics on the 2-simplex and applications in economics 117 copyright ©2019 assa. adv. in systems science and appl. (2019) (3.) conditions regarding (transversal) trajectory non-(self-)intersections, which can be treated by using topological concepts (e.g., homeomorphisms). we discuss the implications of these restrictions for the transitional and limit dynamics on the 2-simplex (among others, fixed points, waves, or (limit) cycles may arise). the models that result from this discussion are relatively simple from the mathematical point of view, yet they seem widely applicable in economic growth and development theory and, thus, may be regarded as powerful system-theoretical constructs. as discussed in section 6, many core topics of long-run economic theory (e.g., savings rate dynamics, functional income distribution dynamics, and sector dynamics) can be modeled by continuous trajectories on simplexes, to which our system-theoretical results apply. in general, the system-theoretical analysis of economic dynamics seems a highly valuable complement to the standard approaches of economic modeling, which rely on quantitative theoretical micro-foundations: system-theoretical models can be relatively crude and do not necessarily require detailed/quantitative assumptions about the nature of economic phenomena. this can be an advantage, since, often, detailed/quantitative assumptions about economic agents and economic environments cannot be supported by empirical evidence and, thus, may open the door for speculation, ideology, and prediction errors. for these reasons, system-theoretical results are not only transferable across many economic topics, but can also be relatively robust in comparison to the results of quantitative micro-founded economic models, and this fact is particularly useful in prediction of long-run economic dynamics (see [20] and section 6 for a detailed discussion). moreover, the dynamics on the 2-simplex merit a detailed consideration from the (application oriented) mathematical point of view: although the 2-simplex can be regarded as a bounded subset of a plane (in ℝ3), the description of the dynamics on the 2-simplex requires a greater variety of analytical concepts in comparison to the description of the dynamics in ℝ2 (see, e.g., the discussion of the monotonicity concepts in sections 2.3 and 3). the rest of the paper is organized as follows. in section 2, we discuss the characterization of trajectory families on the 2-simplex via geometrical and topological concepts. sections 3-5 discuss the implications of these concepts for transitional and limit dynamics on the 2-simplex. this discussion yields system-theoretical models. the potential and existing applications of these models in economics and, in particular, in growth and development theory are discussed in section 6. concluding remarks are provided in section 7. 2. characterization of the trajectories on the 2-simplex in section 2, we summarize the concepts that can be used to characterize continuous dynamics on the 2-simplex as applied by [13-15] in structural change modeling. while there are different mathematical notational conventions, we choose the following notation for reasons of simplicity: small letters (e.g., x), bold small letters (e.g., x), capital letters (e.g., x), and greek letters (e.g., α) denote scalars, vectors/points, sets, and vector angles, respectively. cl(a) denotes the closure of the set a. if i denotes an open interval (e.g., (a, b)), then [i], [i), and (i] denote the corresponding closed (e.g., [a, b]), left-closed (e.g., [a, b)), and right-closed (e.g., (a, b]) interval, respectively. 118 d.stijepic copyright ©2019 assa. adv. in systems science and appl. (2019) 2.1 trajectories on the 2-simplex the (standard) 2-simplex (s), which is defined by (1), is a triangle in ℝ3, as depicted by fig. 1. the cartesian coordinates of the simplex vertices v1, v2, and v3 are stated by (2). s := {(x1, x2, x3) ∈ ℝ3: x1 + x2 + x3 = 1 ∧ ∀i ∈ {1, 2, 3} 0 ≤ xi ≤ 1} (1) v1 := (1, 0, 0) (2a) v2 := (0, 1, 0) (2b) v3 := (0, 0, 1) (2c) we define the vector function x(t, j) as follows: x(t, j) ≡ (x1(t, j), x2(t, j), x3(t, j)): t × j → s (3a) 0 ∈ t ⊆ ℝ (3b) j ⊆ s (3c) the trajectory x(t, j) and the trajectory segment x(t.+, j) are defined by (4). ∀j ∈ j x(t, j) := {x(t, j) ∈ s: t ∈ t} (4a) ∀j ∈ j x(t +, j) := {t ∈ t: t ≥ 0} (4b) x3 v3 v2 v1 x2x1 fig. 1. the standard simplex s in ℝ3. in fact, (4a) defines a trajectory family indexed by the set j, where each trajectory x(t, j) describes a path on s that is traversed over the period t. x(t +, j) is the segment of this path that is traversed over t ≥ 0. 2.2 set-theoretical trajectory classification equations (5) introduce a partitioning of s, which can be used for describing the location of relevant trajectory points or segments (e.g., initial segment/state, empirically observed segment, or segment representing the future dynamics), as we will see later. ∀i ∈ {1, 2, 3} svi := {(x1, x2, x3) ∈ s: xi > 1/2} (5a) sv0 := s \ (sv1 ∪ sv2 ∪ sv3) (5b) models of continuous dynamics on the 2-simplex and applications in economics 119 copyright ©2019 assa. adv. in systems science and appl. (2019) (5a) and (1) imply that the partition svi contains all the points of s that are dominated by xi; i.e., if a point (x1, x2, x3) is located in partition svi, then ∀j ∈ {1, 2, 3}\i xi > xj. the geometrical interpretation of the partitioning (5) is depicted in fig. 2. as we can see, for i ∈ {1, 2, 3}, the partition svi contains all the points of s that are closer to the vertex vi than to the other vertices (vj, j ≠ i). the following (set-theoretical) definitions allow us to assess the prediction range of monotonous models, as we will see later. let a(k) denote the area function, which assigns to a set k ⊆ s the (real number indicating the) area of k. b(t, f) := ⋃j∈f x(t, j) is the image of the family f of trajectories x(t, j), j ∈ f ⊆ j (cf. (4)). among all the path-connected and closed subsets of s that cover b(t, f), let m(t, f) denote one of the sets that cover the smallest area of s. a*(t, f) := a(m(t, f)) is the family image size of the family f. 23113 12332 31221 vwvv vwvv vwvv    sv1 sv2 sv3 sv0 v3 v2v1 w31 w23 w12 fig. 2. the partitioning of s. 2.3 differential-geometrical trajectory classification while the previous discussion can be used for a set-theoretical characterization of trajectories, we focus now on a differential-geometrical characterization of trajectories referring to the angles of the tangential vectors and expressing the monotonicity characteristics and the curvature of a trajectory. we say that the trajectory x(t, j) is continuous if for the given j, x(t, j) is continuous in t on the time interval t (cf. (4a)). moreover, a trajectory family is continuous if all the trajectories belonging to this family are continuous. let (a) d(t, j) be the directional (or tangential) vector associated with the point x(t, j), (b) ℓ be a line through the point x(t, j) that is parallel to the simplex edge v1-v2, and (c) δ(t, j) := ∡(d(t, j), ℓ) ∈ [0°, 360°] be the angle between the directional vector d(t, j) and the line ℓ (cf. fig. 3). 120 d.stijepic copyright ©2019 assa. adv. in systems science and appl. (2019) x(t, j) ℓ|| x(t, j) d(t, j) δ(t, j) v2v1 v3 fig. 3. the vector angle δ(t, j). moreover, we define the angles αi and the angle intervals ii by relying on the ‘saw tooth’ function ϕ: ℝ → ℝ as follows (cf. fig. 4): ϕ(z) = [(z – 1)/6 – floor((z – 1)/6)]360° (6a) ∀i ∈ ℕ ii ≡ (αi, αi+1) := (ϕ(i), ϕ(i + 1)) (6b) ∀i ∈ ℕ ∀j ∈ {n ∈ ℕ: i < n} [ii~j] := ⋃ j k = i [ik] ∧ [ii~j) := ⋃ j k = i [ik] \ αj+1 ∧ (ii~j]:= ⋃ j k = i [ik] \ αi ∧ ii~j := ⋃ j k = i [ik] \ αi \ αj+1 (6c) by using our definition of the vector angle δ(t, j) and the vector angles and intervals (6), we can formulate the properties 1-3 reflecting the relation between the tangential vector angles and the dynamics of x(t, j) in the case that ẋ(t, j) ≠ 0 (cf. fig. 1, 3, and 4). property 1: if ẋ(t, j) ≠ 0, then (a) δ(t, j) ∈ i3~5 ⟺ ẋ1(t, j) > 0, (b) δ(t, j) ∈ i0~2 ⟺ ẋ1(t, j) < 0, and (c) δ(t, j) ∈ {α3, α6} ⟺ ẋ1(t, j) = 0. property 2: if ẋ(t, j) ≠ 0, then (a) δ(t, j) ∈ i5~7 ⟺ ẋ2(t, j) > 0, (b) δ(t, j) ∈ i2~4 ⟺ ẋ2(t, j) < 0, and (c) δ(t, j) ∈ {α2, α5} ⟺ ẋ2(t, j) = 0. property 3: if ẋ(t, j) ≠ 0, then (a) δ(t, j) ∈ i1~3 ⟺ ẋ3(t, j) > 0 (b) δ(t, j) ∈ i4~6 ⟺ ẋ3(t, j) < 0, and (c) δ(t, j) ∈ {α1, α4} ⟺ ẋ3(t, j) = 0. we rely on the following definitions of monotonicity. first, xi(t, j) is monotonous (in t) if (∀t ∈ t ẋi(t, j) ≥ 0) or (∀t ∈ t ẋi(t, j) ≤ 0). second, xi(t, j) is strictly monotonous (in t) if either (∀t ∈ t ẋi(t, j) > 0) or (∀t ∈ t ẋi(t, j) < 0) but not both. third, the trajectory x(t, j) (associated with the function x(t, j)) is (strictly) monotonous in one dimension if (a) there exists an i ∈ {1, 2, 3} such that xi(t, j) is (strictly) monotonous and (b) for all k ∈ {1, 2, 3}\i xk(t, j) is not (strictly) monotonous. fourth, the trajectory x(t, j) is (strictly) monotonous in two models of continuous dynamics on the 2-simplex and applications in economics 121 copyright ©2019 assa. adv. in systems science and appl. (2019) dimensions if (a) there exist an i ∈ {1, 2, 3} and a k ∈ {1, 2, 3}\i such that xi(t, j) and xk(t, j) are (strictly) monotonous and (b) xl(t, j) is not (strictly) monotonous with l ∈ {1, 2, 3}\{i, k}. fifth, the trajectory x(t, j) is (strictly) monotonous (in three dimensions) if ∀i ∈ {1, 2, 3} xi(t, j) is (strictly) monotonous. ϕ(z)/100 z α1 = 0 α2 = 60 = α8 α3 = 120 α4 = 180 α5 = 240 α6 = 300 = α0 i6 = (300 , 360 ) = i0 i1 = (0 , 60 ) = i7 i2 = (60 , 120 ) i3 = (120 , 180 ) i4 = (180 , 240 ) i5 = (240 , 360 ) . s5(x0) s6(x0) = s0(x0) s4(x0) s3(x0) s2(x0) s1(x0) = s7(x0) v1 v2 v3 x0 ≡ (x01, x02, x03) fe d c b l1(x0) = l2(x0) = l3(x0) = l4(x0) = l5(x0) = l6(x0) = a fig. 4. the function ϕ(z), the angle intervals ii, the line segments li(x0), and the sets si(x0). 122 d.stijepic copyright ©2019 assa. adv. in systems science and appl. (2019) instead of using the curvature definition that is widespread in differential geometry (and which is difficult to apply in the proofs of our theorems), we use the following definition of curvature relying on vector angles: let β(q, r, j) denote the angle between the two tangential vectors d(q, j) and d(r, j) associated with the (monotonous) trajectory x(t, j) on s (cf. (4)), where q, r ∈ t. among all the tangential vector pairs (d(t, j), d(s, j)) associated with the (monotonous) trajectory x(t, j), where t, s ∈ t, let d(t*, j) and d(s*, j) be among the ones that are characterized by the largest angle β, i.e., β(t*, s*, j) =: κ(t, j) is the maximal tangential vector angle β associated with the trajectory x(t, j). the greater κ(t, j) is, the greater the curvature of the trajectory x(t, j) is. obviously, a linear trajectory has a curvature of 0. we say that a strictly monotonous trajectory x(t, j) (or a trajectory segment) that describes a clockwise (counterclockwise) movement on s has a positive (negative) signed curvature and write κ(t, j) > 0 (κ(t, j) < 0). let f be a family of trajectories x(t, j), j ∈ f ⊆ j (cf. (4)). then, κ*(t, f) := max(cl({κ(t, j): j ∈ f})) is the maximum curvature of the family f on the time interval t. 2.4 topological trajectory classification here, the topological characterization refers to the question whether a trajectory family is non-(self-)intersecting, which is a characteristic that can be expressed by homeomorphisms. moreover, it is deciding for the applicability of the poincaré-bendixson theory (cf. section 5) that the 2-simplex is homeomorphic to a bounded (and closed) subset of a plane. two trajectories x(t, j) and x(u, k) are non-intersecting if x(t, j) ∩ x(u, k) = ∅, where u ⊆ ℝ. otherwise they are intersecting. a trajectory x(t, j) is self-intersecting if ∃(r, s, t) ∈ t 3 r < s < t ∧ x(r, j) = x(t, j) ≠ x(s, j). otherwise, the trajectory is non-self-intersecting. a trajectory x(t, j) is transversally self-intersecting if ∃(t, s) ∈ t 2 t ≠ s ∧ x(t, j) = x(s, j) ∧ δ(t, j) ≠ δ(s, j). otherwise, the trajectory is not transversally self-intersecting. according to these definitions, a closed trajectory corresponding to a jordan curve is self-intersecting but not transversally self-intersecting. 3. implications of monotonicity in contrast to monotonous and bounded trajectories in ℝ2, monotonous trajectories on the 2-simplex can have (a) a wide range of different shapes and (b) omega limit sets consisting of more than only one (fixed) point. in this section, we discuss the geometrical aspects of the transitional and limit dynamics associated with continuous trajectories that are monotonous in one, two, or three dimensions. as we will see, these geometrical properties have interesting applications in economic dynamics modeling. 3.1 general properties of monotonous trajectories on the 2-simplex in this section, we show that continuous trajectories that are monotonous in three dimensions (two dimensions) are characterized by relatively low curvatures, allow for relatively weak waves, and are placed in relatively small subsets of the 2-simplex in comparison to the ‘related’ trajectories that are monotonous in two dimensions (one dimension). propositions 1-3 and corollary 1 summarize these results formally. the models of continuous dynamics on the 2-simplex and applications in economics 123 copyright ©2019 assa. adv. in systems science and appl. (2019) readers who are less interested in this formal discussion can also go directly to the discussion of figure 4 (see the paragraphs below proposition 3), which elaborates on the intuitive/graphical interpretation of these geometrical properties. given a point x0 ≡ (x01, x02, x03) ∈ s, (7) defines different subsets of s. as we will see (in proposition 3), each of the subsets s1-s6 defined by (7) corresponds to the closure of one of the vector angle intervals i1-i6 defined by (6), and each of the line segments l1-l6 defined by (7) corresponds to one of the angles α1-α6 defined by (6) (cf. fig. 4). l1(x0) := {(x1, x2, x3) ∈ s: x1 ≤ x01 ∧ x3 = x03} (7a) s1(x0) := {(x1, x2, x3) ∈ s: x2 ≥ x02 ∧ x3 ≥ x03} (7b) l2(x0) := {(x1, x2, x3) ∈ s: x1 ≤ x01 ∧ x2 = x02} (7c) s2(x0) := {(x1, x2, x3) ∈ s: x1 ≤ x01 ∧ x2 ≤ x02} (7d) l3(x0) := {(x1, x2, x3) ∈ s: x1 = x01 ∧ x2 ≤ x02} (7e) s3(x0) := {(x1, x2, x3) ∈ s: x1 ≥ x01 ∧ x3 ≥ x03} (7f) l4(x0) := {(x1, x2, x3) ∈ s: x1 ≥ x01 ∧ x3 = x03} (7g) s4(x0) := {(x1, x2, x3) ∈ s: x2 ≤ x02 ∧ x3 ≤ x03} (7h) l5(x0) := {(x1, x2, x3) ∈ s: x1 ≥ x01 ∧ x2 = x02} (7i) s5(x0) := {(x1, x2, x3) ∈ s: x1 ≥ x01 ∧ x2 ≥ x02} (7j) l6(x0) := {(x1, x2, x3) ∈ s: x1 = x01 ∧ x2 ≥ x02} (7k) s6(x0) := {(x1, x2, x3) ∈ s: x1 ≤ x01 ∧ x3 ≤ x03} (7l) s0(x0) := s6(x0) s7(x0) := s1(x0) (7m) ∀i ∈ ℕ ∀j ∈ {n ∈ ℕ: i < n} si~j(x0) := ⋃j k = i sk(x0) (7n) in propositions 1-3, we derive the characteristics of monotonous trajectories and, in particular, the ranges of tangential vector angles δ(t, j) and the sets in which the trajectory segments x(t +, j) are located. later, we will discuss the geometrical/graphical interpretation of these characteristics and their applications in economic dynamics analysis. proposition 1 focuses on the trajectories that are continuous and monotonous in one dimension on s and shows that there are only nine trajectory types belonging to this monotonicity class, where each type is identified by one of the condition-sets p11-p19. proposition 1: assume that (a) the trajectory x(t, j) defined by (4a) is continuous and monotonous in one dimension on s, (b) x(0, j) = x0 ≡ (x01, x02, x03) ∈ s, and (c) ∀t ∈ t ẋ(t, j) ≠ 0. then, x(t, j) and x(t +, j) (cf. (4b)) satisfy one and only one of the condition sets p11-p19, which are defined as follows (cf. (6) and (7)): a) for i ∈{1, 2, …, 6}, condition set p1i is: (∀t ∈ t δ(t, j) ∈ [i(i–1)~(i+1)]) ∧ (∃t ∈ t δ(t, j) ∈ i(i–1)~(i+1)) ∧ (∃(r, s) ∈ t 2 δ(r, j) ∈ [ii–1) ∧ δ(s, j) ∈ (ii+1]) ∧ x(t +, j) ⊂ s(i–1)~(i+1)(x0); b) for i ∈ {7, 8, 9}, condition set p1i is: (∀t ∈ t δ(t, j) ∈ {αi–6, αi–3}) ∧ (∃(p, q) ∈ t 2 δ(p, j) ∈ {αi–6} ∧ δ(q, j) ∈ {αi–6}) ∧ x(t +, j) ⊆ li–6(x0) ∪ li–3(x0). proof. as defined in section 2, x(t, j) is monotonous in one dimension if for one and only one i ∈ {1, 2, 3}, the function xi(t, j) is monotonous while for all other i, xi(t, j) is non-monotonous. thus, for proving proposition 1, we have to consider only three alternative scenarios of monotonicity in one dimension: (a) x1(t, j) is monotonous, (b) x2(t, j) is 124 d.stijepic copyright ©2019 assa. adv. in systems science and appl. (2019) monotonous, and (c) x3(t, j) is monotonous. moreover, since a monotonous function can be monotonously increasing or monotonously decreasing (or both), we have three alternative sub-scenarios for each of the three scenarios (a)-(c): (a) monotonously increasing, (b) monotonously decreasing, and (c) both, monotonously increasing and monotonously decreasing (which means constant). thus, overall, we have nine sub-scenarios: (aa)-(ac), (ba)-(bc), and (ca)-(cc). according to properties 1-3, each of the condition sets p11-p19, to which proposition 1 refers, represents one of the nine sub-scenarios (aa)-(cc). for example, condition sets p12, p15, and p17 represent the sub-scenarios (ca), (cb), and (cc), respectively. consider first the sub-scenario (ca), i.e., assume that x3(t, j) increases monotonously. property 3 and (6b) imply that (8) is valid in sub-scenario (ca). (∀t ∈ t δ(t, j) ∈ [i1~3]) ∧ (∃t ∈ t δ(t, j) ∈ i1~3) (8) moreover, according to the definition of ‘monotonicity in one dimension’, to which proposition 1 refers, (9) is valid in sub-scenario (ca). x1(t, j) and x2(t, j) are non-monotonous. (9) the interval [i1~3], to which (8) refers, can be partitioned into three subintervals [i1), [i2], and (i3]. properties 1 and 2 and (6b) imply (10). ∀t ∈ t δ(t, j) ∈ (i3] ⇒ x1(t, j) is monotonous. (10a) ∀t ∈ t δ(t, j) ∈ [i1) ⇒ x2(t, j) is monotonous. (10b) ∀t ∈ t δ(t, j) ∈ [i2] ⇒ x1(t, j) and x2(t, j) are monotonous. (10c) the statements (10) imply statement (11). x1(t, j) or x2(t, j) is monotonous if for all t ∈ t, δ(t, j) is within one and only one of the sub-intervals [i1), [i2], and (i3]. (11) (9) and (11) imply that over the period t, the tangential vectors δ(t, j) cannot stay within one and the same subinterval, i.e., at least one subinterval switch must occur over the period t. given the three subintervals [i1), [i2], and (i3], the set of all possible subinterval switches is: (i) switch from [i1) to [i2], (ii) switch from [i1) to (i3], (iii) switch from [i2] to [i1), (iv) switch from [i2] to (i3], (v) switch from (i3] to [i1), and (vi) switch from (i3] to [i2]. we analyze now these interval switches. in case (i), i.e., if (1.) initially, the tangential vector angles are within the interval [i1) and (2.) at some later time point, the tangential vector angles switch to the interval [i2], x1(t, j) is monotonous (cf. property 1). this contradicts (9). analogously, it can be shown that cases (iii), (iv), and (vi) contradict (9), since: in case (iii), x1(t, j) is monotonous; in case (iv), x2(t, j) is monotonous; in case (vi), x2(t, j) is monotonous. only, in cases (ii) and (v), both, x2(t, j) and x1(t, j), are non-monotonous, which is consistent with (9). in each of the cases (ii) and (v), (12) is true. ∃(r, s) ∈ t 2 δ(r, j) ∈ [i1) ∧ δ(s, j) ∈ (i3] (12) the fact that x3(t, j) increases monotonously in sub-scenario (ca) implies that ∀t ≥ 0 x3(t, j) ≥ x3(0, j), where x3(0, j) = x03 according to the assumptions made in proposition 1. in other words, in sub-scenario (ca), x(t +, j) ⊂ {(x1, x2, x3) ∈ s: x3 ≥ x03} =: sca(x0) (cf. proposition 1). if x(t +, j) ⊂ sca(x0) ⇒ x(t +, j) ⊂ s1~3(x0), then (13) is valid in sub-scenario (ca). x(t +, j) ⊂ s1~3(x0) (13) models of continuous dynamics on the 2-simplex and applications in economics 125 copyright ©2019 assa. adv. in systems science and appl. (2019) we prove now that x(t +, j) ⊂ sca(x0) ⇒ x(t +, j) ⊂ s1~3(x0). given the point x0 ≡ (x01, x02, x03) ∈ s (cf. proposition 1), the definition of sca(x0) (and (1)) implies that (14)-(16) are true if x(t, j) ∈ sca(x0). either x3(t, j) > x03 or x3(t, j) = x03 but not both. (14) x2(t, j) < x02 or x2(t, j) > x02 (or x2(t, j) = x02 ). (15) x1(t, j) < x01 or x1(t, j) > x01 (or x1(t, j) = x01). (16) the statement (16) can be divided into the two (disjunctive) cases (17a) and (17b). either x1(t, j) > x01 or x1(t, j) = x01 but not both. (17a) x1(t, j) < x01 (17b) if (14), (15), and (17a) are true and x(t, j) ∈ s, then x(t, j) ∈ {(x1, x2, x3) ∈ s: x1 ≥ x01 ∧ x3 ≥ x03} and, thus, x(t, j) ∈ s3(x0) (cf. (7f)). we consider now the cases in which (14), (15), and (17b) are true. these cases are: x1(t, j) < x01 ∧ x3(t, j) > x03 ∧ x2(t, j) > x02 (18a) x1(t, j) < x01 ∧ x3(t, j) > x03 ∧ x2(t, j) < x02 (18b) x1(t, j) < x01 ∧ x3(t, j) > x03 ∧ x2(t, j) = x02 (18c) x1(t, j) < x01 ∧ x3(t, j) = x03 ∧ x2(t, j) > x02 (18d) x1(t, j) < x01 ∧ x3(t, j) = x03 ∧ x2(t, j) < x02 (18e) x1(t, j) < x01 ∧ x3(t, j) = x03 ∧ x2(t, j) = x02 (18f) obviously, the cases (18e) and (18f) violate (1). thus, if (18e) or (18f) is true, then x(t, j) ∉ s. if (18b) or (18c) is true and x(t, j) ∈ s, then x(t, j) ∈ {(x1, x2, x3) ∈ s: x1 < x01 ∧ x2 ≤ x02 ∧ x3 > x03} =: sbc(x0). if x(t, j) ∈ s2(x0), then x3(t, j) ≥ x03, since, otherwise, (1) is violated (cf. (7d)). in other words, s2(x0) = {(x1, x2, x3) ∈ s: x1 ≤ x01 ∧ x2 ≤ x02 ∧ x3 ≥ x03}. obviously, sbc(x0) ⊂ s2(x0). thus, if (18b) or (18c) is true and x(t, j) ∈ s, then x(t, j) ∈ s2(x0). analogously, if (18a), (18c), or (18d) is true and x(t, j) ∈ s, then x(t, j) ∈ {(x1, x2, x3) ∈ s: x1 < x01 ∧ x2 ≥ x02 ∧ x3 ≥ x03} =: sacd(x0). moreover, if x(t, j) ∈ s1(x0), then x1(t, j) ≤ x01, since, otherwise, (1) is violated (cf. (7b)). in other words, s1(x0) = {(x1, x2, x3) ∈ s: x1 ≤ x01 ∧ x2 ≥ x02 ∧ x3 ≥ x03}. obviously, sacd(x0) ⊂ s1(x0). thus, if (18a), (18c), or (18d) is true and x(t, j) ∈ s, then x(t, j) ∈ s1(x0). overall, we have shown that if x(t, j) ∈ sca(x0) ⊆ s, then the statements (14)-(17) are valid, which imply several feasible cases. in each of these cases, x(t, j) is in one of the sets s1(x0), s2(x0), and s3(x0), i.e., x(t, j) ∈ sca(x0) ⇒ x(t, j) ∈ s1(x0) ∪ s2(x0) ∪ s3(x0). this implies that x(t +, j) ⊂ sca(x0) ⇒ x(t +, j) ⊂ s1(x0) ∪ s2(x0) ∪ s3(x0), since x(t +, j) is the union of the points x(t, j) ∈ s for which the statements (14)-(17) (and (1)) hold (cf. proposition 1). according to (7n), s1~3(x0) = s1(x0) ∪ s2(x0) ∪ s3(x0). this completes the proof that x(t +, j) ⊂ sca(x0) ⇒ x(t +, j) ⊂ s1~3(x0). by now, we have shown that in the sub-scenario (ca), the statements (8), (12), and (13) must be true. these three statements reduce to condition set p12. it can be shown in the same way that (1.) the sub-scenarios (cb) and (cc) correspond to condition sets p15 and p17, respectively, and (2.) each of the sub-scenarios (ba)-(cc) corresponds to one and only one of the condition sets p11, p13, p14, p16, p18, and p19. 126 d.stijepic copyright ©2019 assa. adv. in systems science and appl. (2019) this completes the proof that each of the alternative (sub-)scenarios of monotonicity in one dimension (i.e., each of the sub-scenarios (aa)-(cc)) corresponds to one and only one of the condition sets p11-p19. □ now, we turn to proposition 2, which is similar to proposition 1 except for the fact that it focuses on the trajectories that are continuous and monotonous in two dimensions on s. in particular, proposition 2 derives the corresponding ranges of tangential vector angles δ(t, j) and the sets in which the trajectory segments x(t +, j) are located and shows that there are only six types of trajectories that are continuous and monotonous in two dimensions on s, where each type is identified by one of the condition-sets p21-p26. proposition 2: assume that (a) the trajectory x(t, j) defined by (4a) is continuous and monotonous in two dimensions on s, (b) x(0, j) = x0 ≡ (x01, x02, x03) ∈ s, and (c) ∀t ∈ t ẋ(t, j) ≠ 0. then, x(t, j) and x(t +, j) (cf. (4b)) satisfy one and only one of the condition sets p21-p26, where for i ∈ {1, 2, …, 6}, condition set p2i is: (∀t ∈ t δ(t, j) ∈ [ii~(i+1)]) ∧ (∃(r, s) ∈ t 2 δ(r, j) ∈ [ii) ∧ δ(s, j) ∈ (i(i+1)]) ∧ x(t +, j) ⊂ si~(i+1)(x0) (cf. (6)/(7)). proof. according to the definition of monotonicity in two dimensions, two of the functions x1(t, j), x2(t, j), and x3(t, j) must be monotonous, while the remaining one must be non-monotonous. thus, we have to consider only three cases: (a) x1(t, j) and x2(t, j) are monotonous (while x3(t, j) is non-monotonous), (b) x1(t, j) and x3(t, j) are monotonous (while x2(t, j) is non-monotonous), and (c) x2(t, j) and x3(t, j) are monotonous (while x1(t, j) is non-monotonous). for each of these cases, we must distinguish between four subcases. for example, in case (a), we can distinguish between the following subcases: (a) x1(t, j) and x2(t, j) are monotonously increasing, (b) x1(t, j) is monotonously increasing, while x2(t, j) is monotonously decreasing, (c) x1(t, j) and x2(t, j) are monotonously decreasing, and (d) x1(t, j) is monotonously decreasing, while x2(t, j) is monotonously increasing. subcases (a) and (c) are infeasible, since they violate (1): for example, if x1(t, j) and x2(t, j) are monotonously increasing, then x3(t, j) must be monotonously decreasing (instead of being non-monotonous), since x1(t, j) + x2(t, j) + x3(t, j) must be equal to 1 for all t. thus, we must consider only the subcases (b) and (d) of case (a). properties 1 and 2 imply that in subcase (b) of case (a), the statement (19) is valid (cf. (6)). ∀t ∈ t δ(t, j) ∈ [i3~4] (19) moreover, since case (a) requires that x3(t, j) is non-monotonous, property 3 implies that (20) is valid in case (a). ∃(r, s) ∈ t 2 ẋ3(r, j) > 0 ∧ ẋ3(s, j) < 0 (20) according to (6), the interval [i3~4], to which (19) refers, can be partitioned into the following partitions: [i3), α4, and (i4]. property 3 and (6) imply (21). ∀t ∈ t δ(t, j) ∈ [i3) ∨ δ(t, j) ∈ [i3) ∪ α4 ⇒ ∀t ∈ t ẋ3(t, j) ≥ 0 (21a) ∀t ∈ t δ(t, j) ∈ (i4] ∨ δ(t, j) ∈ α4 ∪ (i4] ⇒ ∀t ∈ t ẋ3(t, j) ≤ 0 (21b) ∀t ∈ t δ(t, j) ∈ α4 ⇒ ∀t ∈ t ẋ3(t, j) = 0 (21c) models of continuous dynamics on the 2-simplex and applications in economics 127 copyright ©2019 assa. adv. in systems science and appl. (2019) the statements (21) imply that if (19) and (20) are true, the tangential vectors δ(t, j) cannot stay within one and only one of the subintervals [i3), [i3) ∪ α4, α4, α4 ∪ (i4], and (i4] for all t ∈ t, and there must occur a switch from subinterval [i3) to subinterval (i4] or from subinterval (i4] to subinterval [i3) over the period t. thus, (22) is valid. ∃(r, s) ∈ t 2 δ(r, j) ∈ [i3) ∧ δ(s, j) ∈ (i4] (22) since in subcase (b) of case (a), x1(t, j) increases monotonously and x2(t, j) decreases monotonously, the assumptions made in proposition 2 and (4b) imply that x(t +, j) ⊂ {(x1, x2, x3) ∈ s: x1 ≥ x01 ∧ x2 ≤ x02} =: sab(x0). sab(x0) can be partitioned as follows: sab(x0) = sab1(x0) ∪ sab2(x0), where sab1(x0) ∩ sab2(x0) = ∅ and sab1(x0) := {(x1, x2, x3) ∈ s: x1 ≥ x01 ∧ x2 ≤ x02 ∧ x3 ≥ x03} and sab2(x0) := {(x1, x2, x3) ∈ s: x1 ≥ x01 ∧ x2 ≤ x02 ∧ x3 < x03}. we can see immediately that sab1(x0) ⊂ s3(x0) (cf. (7f)) and sab2(x0) ⊂ s4(x0) (cf. (7h)). thus, sab(x0) ⊂ s3(x0) ∪ s4(x0). this result, (7n), and the previously shown fact that x(t +, j) ⊂ sab(x0) imply (23). x(t +, j) ⊂ s3~4(x0) (23) overall, we have shown that in the subcase (b) of case (a), condition set p23 must be true (cf. (19), (22), and (23)). analogously, it can be shown that in all the feasible subcases of cases (a)-(c), one and only one of the statements p21, p22, p24, p25, and p26 is true, which proves proposition 2. □ finally, we focus on the trajectories that are continuous and monotonous in three dimensions on s. in particular, proposition 3 derives the corresponding ranges of tangential vector angles δ(t, j) and the sets in which the trajectory segments x(t +, j) are located and shows that there are twelve types of trajectories that are continuous and monotonous in three dimensions on s, where each type is identified by one of the condition-sets p31-p312. proposition 3: assume that (a) the trajectory x(t, j) defined by (4a) is continuous and monotonous (in three dimensions) on s, (b) x(0, j) = x0 ≡ (x01, x02, x03) ∈ s, and (c) ∀t ∈ t ẋ(t, j) ≠ 0. then, x(t, j) and x(t +, j) (cf. (4b)) satisfy one and only one of the condition sets p31-p312, where (cf. (6) and (7)): a) for i ∈ {1, 2, …, 6}, condition set p3i is: (∀t ∈ t δ(t, j) ∈ [ii]) ∧ (∃s ∈ t δ(s, j) ∈ ii) ∧ x(t +, j) ⊂ si(x0); b) for i ∈ {7, 8, …, 12}, condition set p3i is: ∀t ∈ t δ(t, j) ∈ {αi–6} ∧ x(t +, j) ⊆ li–6(x0). proof. according to our definition of monotonicity (in three dimensions), x1(t, j), x2(t, j), and x3(t, j) must be monotonous if x(t, j) is monotonous (in three dimensions) on s. since a monotonous function can be (a) monotonously increasing, (b) monotonously decreasing, or (c) both (monotonously increasing and monotonously decreasing and, thus, constant), we have per function xi(t, j) three cases ((a)-(c)). moreover, we have three functions x1(t, j), x2(t, j), and x3(t, j). thus, overall, there are 33 possible combinations. this set of 27 combinations contains the combination (a) ∀i ẋi(t, j) ≤ 0, the combination (b) ∀i ẋi(t, j) ≥ 0, three times the combination (c) ẋi(t, j) ≥ 0 ∧ ẋk(t, j) ≥ 0 ∧ ẋl(t, j) = 0 ∧ i ≠ k ≠ l, three times the combination (d) ẋi(t, j) ≤ 0 ∧ ẋk(t, j) ≤ 0 ∧ ẋl(t, j) = 0 ∧ i ≠ k ≠ l, six times the combination (e) ẋi(t, j) = ẋk(t, 128 d.stijepic copyright ©2019 assa. adv. in systems science and appl. (2019) j) = 0 ∧ ẋl(t, j) ≠ 0 ∧ i ≠ k ≠ l, and the combination (f) ∀i ẋi(t, j) = 0. the combinations (a)-(e) are infeasible, since they violate (1) unless they reduce to combination (f). the combination (f) represents a fixed point (ẋ(t, j) = 0) and is excluded by the assumptions made in proposition 3. in the rest of the proof, we have to consider the remaining 12 combinations.1 each of these 12 combinations is covered by one of the conditions sets p31-p312. we leave it to the reader to prove the validity of proposition 3 in all these 12 cases; we prove the validity in only two representative cases. consider the case ∀t ∈ t ẋ1(t, j) ≤ 0 ∧ ẋ2(t, j) ≤ 0 ∧ ẋ3(t, j) ≥ 0, where ∃(r, s, p) ∈ t 3 ẋ1(r, j) < 0 ∧ ẋ2(s, j) < 0 ∧ ẋ3(p, j) > 0. then, (a) properties 1-3 imply almost directly that the tangential vector angles δ(t, j) satisfy the condition set p32, and (b) the assumptions made in proposition 3 and (4b) imply that x(t +, j) ⊂ {(x1, x2, x3) ∈ s: x1 ≤ x01 ∧ x2 ≤ x02 ∧ x3 ≥ x03} =: sz(x0), and, thus, (7d) implies that sz(x0) ⊂ s2(x0); thus, x(t +, j) ⊂ s2(x0) as stated by the condition set p32. alternatively, consider the case ∀t ∈ t ẋ1(t, j) ≤ 0 ∧ ẋ2(t, j) ≥ 0 ∧ ẋ3(t, j) = 0, where ∃r ∈ t ẋ1(r, j) < 0 ∧ ẋ2(r, j) > 0.2 properties 1-3 imply almost directly that in this case, the tangential vector angles δ(t, j) satisfy the condition set p37. moreover, the assumptions made in proposition 3, (4b), and (7a) imply that x(t +, j) ⊆ {(x1, x2, x3) ∈ s: x1 ≤ x01 ∧ x2 ≥ x02 ∧ x3 = x03} ⊆ l1(x0). thus, x(t +, j) ⊆ l1(x0) as stated by condition set p37. □ we discuss now the geometrical interpretation of propositions 1, 2, and, 3, as depicted by fig. 4. to construct fig. 4, we choose an arbitrary point (x0) in the interior of s. then, we draw three line segments going through x0 and each being parallel to one of the simplex edges v1-v2, v2-v3, and v3-v1. the intersections of the line segments with the simplex edges are denoted by the points a-f. we can see that the line segments that connect x0 with one of the points a-f are the line segments l1(x0)-l6(x0), which are defined by (7) and which localize the six (closed) subsets s1(x0)-s6(x0) defined by (7). the angles between the line segments l1(x0)-l6(x0) and the simplex edge v1-v2 (according to the definition of tangential vector angles and intervals (6)) are depicted in the middle panel of fig. 4. (7n) and fig. 4 imply almost directly that (1.) each of the six sets si~(i+1)(x0), to which proposition 2 refers, is simply the union of two neighboring sets sj(x0) and sk(x0), (2.) each of the six sets s(i–1)~(i+1)(x0), to which proposition 1 refers, is simply the union of three neighboring sets sj(x0), sk(x0), and sm(x0). in particular, propositions 1-3 can be interpreted easily by using fig. 4: 1.) proposition 3 implies three geometrical properties of a trajectory segment x(t +, j) that is monotonous in three dimensions. first, x(t +, j) is located in one of the line segments l1(x0)-l6(x0) or in one of the sets s1(x0)-s6(x0). second, if x(t +, j) is in li(x0), then for all t ≥ 1 these feasible combinations are: (1.) ẋ1 ≤ 0 ∧ ẋ2 ≤ 0 ∧ ẋ3 ≥ 0, (2.) ẋ1 ≤ 0 ∧ ẋ2 ≥ 0 ∧ ẋ3 ≤ 0, (3.) ẋ1 ≤ 0 ∧ ẋ2 ≥ 0 ∧ ẋ3 ≥ 0, (4.) ẋ1 ≤ 0 ∧ ẋ2 ≥ 0 ∧ ẋ3 = 0, (5.) ẋ1 ≤ 0 ∧ ẋ2 = 0 ∧ ẋ3 ≥ 0, (6.) ẋ1 ≥ 0 ∧ ẋ2 ≤ 0 ∧ ẋ3 ≤ 0, (7.) ẋ1 ≥ 0 ∧ ẋ2 ≤ 0 ∧ ẋ3 ≥ 0, (8.) ẋ1 ≥ 0 ∧ ẋ2 ≤ 0 ∧ ẋ3 = 0, (9.) ẋ1 ≥ 0 ∧ ẋ2 ≥ 0 ∧ ẋ3 ≤ 0, (10.) ẋ1 ≥ 0 ∧ ẋ2 = 0 ∧ ẋ3 ≤ 0, (11.) ẋ1 = 0 ∧ ẋ2 ≤ 0 ∧ ẋ3 ≥ 0, and (12.) ẋ1 = 0 ∧ ẋ2 ≥ 0 ∧ ẋ3 ≤ 0, where ∃t ∈ t ẋi(t) < 0 if it is stated that ẋi ≤ 0, and, analogously, ∃t ∈ t ẋi(t) > 0 if it is stated that ẋi ≥ 0. 2 note that the cases ẋ1(r, j) < 0 ∧ ẋ2(r, j) = ẋ3(r, j) = 0 and ẋ2(r, j) > 0 ∧ ẋ1(r, j) = ẋ3(r, j) = 0 are infeasible (see the discussion of combinations (a)-(e)). models of continuous dynamics on the 2-simplex and applications in economics 129 copyright ©2019 assa. adv. in systems science and appl. (2019) 0, the tangential vector angles δ(t, j) associated with x(t +, j) are equal to the angle that is associated to the line segment li(x0) in fig. 4 +/–180°. for example, if x(t +, j) is in l3(x0), then δ(t, j) ∈ {120°, 300°} for t ≥ 0. third, if x(t +, j) is located in one of the sets si(x0), then for t ≥ 0, the tangential vector angles δ(t, j) associated with x(t +, j) are within the angle range indicated by the angles associated to the line segments li(x0) and li+1(x0) that bound the set si(x0) in fig. 4. for example, if the trajectory segment x(t +, j) that is monotonous in three dimensions is in s3(x0), then δ(t, j) is within the angle range [120°, 180°] for t ≥ 0 (cf. proposition 3 and condition set p33). 2.) the geometrical interpretation of proposition 2 is analogous. in particular, the trajectory segment x(t +, j) that is monotonous in two dimensions is located in two neighboring sets sj(x0) and sk(x0), and for all t ≥ 0, the tangential vector angles δ(t, j) of x(t+, j) are within the angle range indicated by the angles associated to the two line segments lj(x0) and lk+1(x0) that bound the union of the sets sj(x0) and sk(x0) in fig. 4. for example, if the trajectory segment x(t +, j) that is monotonous in two dimensions is in s3~4(x0) = s3(x0) ∪ s4(x0), then δ(t, j) is within the angle range [120°, 240°] for t ≥ 0 (cf. proposition 2 and condition set p23). 3.) analogously, proposition 1 implies that the trajectory segment x(t +, j) that is monotonous in one dimension is located in three neighboring sets sj(x0), sk(x0), and sm(x0). moreover, for all t ≥ 0, the tangential vector angles δ(t, j) associated with this trajectory segment are within the angle range indicated by the angles associated to the two line segments lj(x0) and lm+1(x0) that bound the union of the sets sj(x0), sk(x0), and sm(x0) in fig. 4. for example, if the trajectory segment x(t +, j) that is monotonous in one dimension is in s3~5(x0) = s3(x0) ∪ s4(x0) ∪ s4(x0), then δ(t, j) is within the angle range [120°, 300°] for t ≥ 0 (cf. proposition 1 and condition set p14). this graphical interpretation highlights important implications of propositions 1-3: first, a trajectory that is monotonous (in three dimensions) is captured in a smaller subset of s than a related trajectory that is monotonous in two dimensions. second, a trajectory that is monotonous in two dimensions is captured in a smaller subset of s than a related trajectory that is monotonous in one dimension. moreover, the maximum curvature κ* of trajectories that are monotonous in three dimensions (two dimensions) is greater than the maximum curvature of related trajectories that are monotonous in two dimensions (one dimension). this intuitive discussion does not explicitly define the meaning of the term ‘related’. thus, we define the meaning of this term and then formulate corollary 1 (which is implied by propositions 1-3) on the basis of this definition such that the discussion becomes more precise. let f(x0) ⊆ j be a family of continuous trajectory segments x(t +, j) ⊂ s, j ∈ f(x0), satisfying ∀j ∈ f(x0) x(0, j) = x0 ∈ s and ∀t ∈ t + ∀j ∈ f(x0) ẋ(t, j) ≠ 0 (cf. (4)). moreover, let p11(x0), p12(x0), …, p16(x0), p21(x0), p22(x0), …, p26(x0), p31(x0), p32(x0), …, and p36(x0) denote the subfamilies of f(x0) satisfying the conditions sets p11, p12, …, p16, p21, p22, …, p26, p31, p32, …, and p36, respectively. that is, j ∈ pcd(x0) ⊂ f(x0) implies that x(t +, j) satisfies the condition set pcd, where c ∈ {1, 2, 3} and d ∈ {1, 2, …, 6}. for (h, k) ∈ {1, 2, …, 6}2, we say that the families p1h(x0) and p2k(x0) are related if ∃i ∈ {1, 2, 3} ∀j ∈ p1h(x0) ∪ p2k(x0) (∀t ∈ t + ẋi(t, j) ≥ 0) ∨ (∀t ∈ t + ẋi(t, j) ≤ 0) ∧ (∃tj ∈ t + ẋi(tj, j) ≠ 0). that is, a family defined by proposition 1 is related to a family defined by proposition 2 if there exists an i for 130 d.stijepic copyright ©2019 assa. adv. in systems science and appl. (2019) which the monotonicity characteristics of xi(t, j) are identical in both families. for example, the families p12(x0) and p21(x0) are characterized by a monotonously increasing x3(t, j), i.e., ∀j ∈ p12(x0) ∪ p21(x0) (∀t ∈ t + ẋ3(t, j) ≥ 0) ∧ (∃tj ∈ t + ẋ3(tj, j) > 0); thus, p12(x0) and p21(x0) are related. we define the relations between the families defined by propositions 2 and 3 analogously: for (p, q) ∈ {1, 2, …, 6}2, we say that the families p2p(x0) and p3q(x0) are related if ∃(v, w) ∈ {1, 2, 3}2 ∀j ∈ p2p(x0) ∪ p3q(x0) (∀t ∈ t + ẋv(t, j) ≥ 0) ∨ (∀t ∈ t + ẋv(t, j) ≤ 0) ∧ (∀t ∈ t + ẋw(t, j) ≥ 0) ∨ (∀t ∈ t + ẋw(t, j) ≤ 0) ∧ (∃tj ∈ t + ẋv(tj, j) ≠ 0) ∧ (∃sj ∈ t + ẋi(sj, j) ≠ 0) ∧ v ≠ w. that is, a family defined by proposition 2 is related to a family defined by proposition 3 if (a) there exists a v for which the monotonicity characteristics of xv(t, j) are identical in both families and (b) there exists a w ≠ v for which the monotonicity characteristics of xw(t, j) are identical in both families. for example, as implied by (6), properties 1 and 3, and propositions 2 and 3, the families p21(x0) and p31(x0) are characterized by (a) a monotonously decreasing x1(t, j), i.e., ∀j ∈ p21(x0) ∪ p31(x0) (∀t ∈ t + ẋ1(t, j) ≤ 0) ∧ (∃tj ∈ t + ẋ1(tj, j) < 0), and (b) a monotonously increasing x3(t, j), i.e., ∀j ∈ p21(x0) ∪ p31(x0) (∀t ∈ t + ẋ3(t, j) ≥ 0) ∧ (∃sj ∈ t + ẋ3(sj, j) > 0). thus, p21(x0) and p31(x0) are related. corollary 1: a) consider the trajectory family p1h(x0), where h ∈ {1, 2, …, 6} and x0 ≡ (x01, x02, x03) ∈ int(s). there exist two trajectory families p2k(x0) and p2m(x0), (k, m) ∈ {1, 2, …, 6}2, k ≠ m, that are related to p1h(x0) and satisfy the following condition: ∀n ∈ {k, m} a*(t +, p1h(x0)) > a*(t +, p2n(x0)) ∧ κ*(t +, p1h(x0)) > κ*(t +, p2n(x0)) (cf. sections 2.2. and 2.3). b) consider the trajectory family p2p(x0), where p ∈ {1, 2, …, 6} and x0 ≡ (x01, x02, x03) ∈ int(s). there exist two trajectory families p3q(x0) and p3r(x0), (q, r) ∈ {1, 2, …, 6}2, q ≠ r, that are related to p2p(x0) and satisfy the following condition: ∀u ∈ {q, r} a*(t +, p2p(x0)) > a*(t +, p3u(x0)) ∧ κ*(t +, p2p(x0)) > κ*(t +, p3u(x0)). proof. we only sketch here the proof. starting with corollary 1a, assume that h = 2, i.e., consider the family p12(x0). according to our definition of relatedness, p12(x0) is related to p21(x0) and p22(x0), since (6), property 3, and propositions 1 and 2 imply that p12(x0), p21(x0), and p22(x0) are characterized by a monotonously increasing x3(t, j). according to proposition 1, (7b), (7d), (7f), (7n), and the definitions of a and m given in section 2.2, the following statements are true: m(t +, p12(x0)) ⊆ s1~3(x0) = s1(x0) ∪ s2(x0) ∪ s3(x0) = {(x1, x2, x3) ∈ s: x3 ≥ x03} (24) m(t +, p21(x0)) ⊆ s1~2(x0) = s1(x0) ∪ s2(x0) ⊂ s1~3(x0) (25) m(t +, p22(x0)) ⊆ s2~3(x0) = s2(x0) ∪ s3(x0) ⊂ s1~3(x0) (26) as implied by (24), all the trajectories belonging to the family p12(x0) are located in s1~3(x0), where the latter is a triangle obtained by constructing a line on s going through x0 and being parallel to the simplex edge v1-v2 (cf. property 3a, (6), and fig. 3 and 5). according to proposition 1 and (6), all the trajectories belonging to the family p12(x0) satisfy the following vector angle condition: ∀j ∈ p12(x0) (∀t ∈ t + δ(t, j) ∈ [0°, 180°]) ∧ (∃(r, s) ∈ t + × t + δ(r, j) ∈ [0°, 60°) ∧ δ(s, j) ∈ (120°, 180°]) (27) models of continuous dynamics on the 2-simplex and applications in economics 131 copyright ©2019 assa. adv. in systems science and appl. (2019) if we allow for non-smooth trajectories and, in particular, trajectories that are unions of line segments, it is easy to show geometrically by referring to fig. 5 that such trajectories can be constructed to any point on s1~3(x0) while satisfying the condition (27).3 thus, m(t +, p12(x0)) = s1~3(x0) and, thus, a*(t +, p12(x0)) = a(s1~3(x0)) (cf. section 2.2). moreover, (25) and (26) imply that a*(t +, p21(x0)) ≤ a(s1~2(x0)) < a(s1~3(x0)) and a*(t +, p22(x0))) ≤ a(s2~3(x0)) < a(s1~3(x0)). thus, (28) is true. a*(t +, p12(x0)) > a*(t +, p21(x0)) ∧ a*(t +, p12(x0)) > a*(t +, p22(x0)) (28) if we require that the trajectories belonging to the family p12(x0) are smooth (i.e., ∀j ∈ p12(x0) ∀t ∈ t + x(t, j) is differentiable with respect to t), then it is not possible to construct a trajectory that obeys (27) and goes through the points/vertices (x01, 0, x03) ∈ s1~3(x0) and (0, x02, x03) ∈ s1~3(x0), which can be easily proven by referring to fig. 4 and 5. that is, the smooth trajectories belonging to the family p12(x0) cannot cover two infinitesimally small areas of s1~3(x0). however, even in this case, it is still ensured that a*(t +, p12(x0)) > a*(t +, p21(x0)), since (a) s1~3(x0) = s1~2(x0) ∪ s3(x0) (cf. (24) and (25)), (b) s3(x0) (cf. (24)) is not infinitesimally small (in generic cases), and (c) a*(t +, p21(x0)) ≤ a(s1~2(x0)). the definition of κ* (cf. section 2.3) and (27) imply that κ*(t +, p12(x0)) = 180°. analogously, proposition 2, the definition of κ*, and (6) imply that κ*(t +, p21(x0)) = 120°. thus, κ*(t +, p12(x0)) > κ*(t +, p21(x0)). overall, by now, we have (heuristically) proven corollary 1a for h = 2. the proof is analogous for h ∈ {1, 3, 4, 5, 6}. the proof of corollary 1b is very similar to the proof of corollary 1a. thus, we omit it here. □ overall, among related trajectories and trajectory families the following is true: the higher the dimension of monotonicity, the smaller is (a) the family image (and, thus, the set of predicted states) and (b) the maximal curvature (and, thus, the potential strength of waves). if two families are unrelated, then a higher degree of monotonicity does not necessarily imply a smaller family image and a smaller maximum curvature. we can see that corollary 1 does not categorize all the condition sets postulated by propositions 1 and 3 and, in particular, not the condition sets p17-p9 and p37-p312. these condition sets imply that one of the xi(t, j) is constant for all t ∈ t + and, thus, the trajectory segments x(t +, j) are located on line segments. obviously, the constancy requirement is much stronger than a monotonicity requirement; thus, in the cases represented by the condition sets p17-p9 and p37-p312, the size of the family image m(.) is relatively small and the curvature is zero. 3 exactly speaking, (a) each of the line segments constituting such a trajectory is characterized by an angle to the v1-v2-edge of s in the range of [0°, 180°], (b) each trajectory contains a line segment that has an angle in the range of [0°, 60°), and (c) each trajectory contains a line segment that has an angle in the range (120°, 180°] (cf. (27)). 132 d.stijepic copyright ©2019 assa. adv. in systems science and appl. (2019) . s1~3(x0) v1 v2 v3 x0 ≡ (x01, x02, x03). .(x01, 0, x03) (0, x02, x03) fig. 5. the set s1~3(x0). 3.2 implications for prediction of transitional dynamics if we regard t = 0 as now and t > 0 as the future (and, thus, t + as the predicted trajectory segment), corollary 1 implies that the trajectories that are monotonous in one dimension (two dimensions) are harder to predict than the trajectories that are monotonous in two dimensions (three dimensions), ceteris paribus, since (a) the set of all possible future states is relatively great and (b) relatively stronger curvatures/waves may arise in the former case (in comparison to the latter case). however, besides the monotonicity characteristics, the location of the initial state x0 is decisive for the predictability of the future dynamics. in particular, the family image m and its size a* depend on x0 (cf. corollary 1 and its proof). in general, monotonicity implies that the system moves from x0 along the trajectory segment t + towards a vertex or an edge of the 2-simplex. thus, if the initial state x0 is relatively close to this vertex/edge, t + is captured in a relatively small set, i.e., the set of potential future states of the system is relatively small. this is almost a direct implication of the boundedness of the 2-simplex. as implied by propositions 1-3, (6), and section 2.3 (cf. proof of corollary 1), the maximum curvature κ* of trajectories that are monotonous in three dimensions, two dimensions, and one dimension is 60°, 120°, and 180°, respectively. thus, the cyclical behaviors corresponding to a transversally self-intersecting trajectory or a closed trajectory (jordan curve) are prohibited in all cases of monotonicity, since these types of cyclical behavior require a curvature greater than 180°. yet, monotonous trajectories allow for a cyclical behavior corresponding to waves on the simplex (see fig. 6). the angle range of 180° associated with monotonicity in one dimension allows for waves of high amplitude and short wavelength on the 2-simplex. in contrast, monotonicity in three dimensions allows only for relatively low-amplitude/long-wavelength waves (cf. fig. 6). models of continuous dynamics on the 2-simplex and applications in economics 133 copyright ©2019 assa. adv. in systems science and appl. (2019) . s3(x0) s2(x0) s1(x0) v1 v2 v3 x0 d c b l1(x0) = || l2(x0) = || l3(x0) = || l4(x0) = || a x(t +, j) monotonous in one dimension (condition set p12) . s2(x0) s1(x0) v1 v2 v3 x0 c b a x(t +, j) monotonous in two dimensions (condition set p21) . s1(x0) v1 v2 v3 x0 b monotonous in three dimensions (condition set p31) a x(t +, j) fig. 6. examples of monotonous waves. obviously, all three types of monotonicity allow for curved trajectories on the 2-simplex. however, only monotonicity in three dimensions allows for unidirectional linear trajectories, while monotonicity in one dimension allows for linear (non-smooth) trajectories yet requires at least one direction change. 134 d.stijepic copyright ©2019 assa. adv. in systems science and appl. (2019) 3.3 limit dynamics in all three cases of monotonicity (monotonicity in one, two, and three dimensions), the following two facts are true for continuous dynamics. first, the cyclical limit dynamics where the omega limit set is a jordan curve are excluded, since such cycles require that x1(t), x2(t), and x3(t) are non-monotonous in the limit. second, the system may converge to a fixed point or reach the fixed point in finite time (and stay there). the proof of these facts is obvious. in the case of monotonicity in three dimensions, only the fixed point outcome is possible, as implied by the monotone convergence theorem: since each of the functions x1(t), x2(t), and x3(t) is monotonous and restricted by an upper/lower limit of 0 and 1, each of the x1(t), x2(t), and x3(t) converges to its fixed point (x1 *, x2 *, and x3 *, respectively) or reaches it in finite time (and stays there). thus, x(t) converges to a fixed point x* ≡ (x1 *, x2 *, x3 *) ∈ s or reaches it in finite time. additionally, in the case of monotonicity in one dimension, the system may converge to a line segment if the trajectory is a wave. in this case, the wavelength decreases and the vector-angle range converges to the range of 180° as the system converges to the line segment (see the first part of fig. 6). note that in the case of monotonicity in two dimensions, the convergence to a line segment is not possible, as explained in the following. if the trajectory converges to a line segment, the tangential vector angle range must increase to a range of 180° which is prohibited by the definition of monotonicity in two dimensions, which allows only for a vector angle range of 120°. the former fact follows from the definition of the omega limit set, where for each of the points on the line segment (constituting the omega limit set), a sequence of points on the wave must be found that converges to it. 4. implications of non-self-intersection for transitional dynamics while non-intersecting trajectories and limit dynamics are treated in section 5, we focus, now, on the implications of (transversal) non-self-intersection for transitional dynamics. the class of transversally non-self-intersecting continuous trajectories on the 2-simplex is a subclass of the class of continuous trajectories on the 2-simplex. moreover, the class of non-self-intersecting continuous trajectories on the 2-simplex is a subclass of the class of transversally non-self-intersecting trajectories on the 2-simplex, since the former does not allow for jordan-curves in contrast to the latter. thus, by imposing the condition of (transversal) non-self-intersection, we can reduce the set of feasible trajectories on the 2-simplex, which can be exploited in prediction of dynamics, as explained in the following. obviously, the non-self-intersection is an important constraint in systems of continuous trajectories on two-dimensional domains. in the case of discontinuous trajectories, non-self-intersection still may reduce the class of feasible trajectories significantly depending on the type of discontinuity and the physical/social system being analyzed. however, in extreme cases and, in particular, in the case of point sequences on the models of continuous dynamics on the 2-simplex and applications in economics 135 copyright ©2019 assa. adv. in systems science and appl. (2019) 2-simplex (e.g., discrete-time paths), non-self-intersection becomes obsolete as a restraint (in natural and social sciences where the exact position of a system on the simplex is not measurable). for the same reasons, non-self-intersection is an obsolete restraint in threeor higher-dimensional dynamical system domains (see, e.g., [13]). 4.1 qualitative simulation the (transversal) non-self-intersection constraint on continuous trajectories on the 2-simplex can be understood as a dynamic constraint: at each point of time t ∈ t +, we have a restriction on system dynamics x(t, j) preventing certain type of dynamics (namely the dynamics that correspond to a self-intersection of the trajectory). the constraint is dynamic in the sense that it changes over time. in particular, it depends on the current position of the system on the 2-simplex and the form of the trajectory segment x(t –, j) representing the dynamics over the past time period t – (e.g., the longer the latter segment, the stronger is the constraint on the current dynamics), i.e., the constraint is updated continuously. this fact can be used in qualitative simulation, as discussed in detail by [7-8]. 4.2 non-self-intersection in combination with a determined trajectory-segment the concept of (transversal) non-self-intersection can be very useful even if we do not assume the dynamic constraint view discussed in section 4.1. in particular, assume that the trajectory segment (x(t –, j)) representing past dynamics is given by empirical data on past dynamics or by empirical laws. then, in general, x(t –, j) can be used as a basis for a partitioning of the 2-simplex. for example, since the lines that are parallel to the 2-simplex edges have a clear intuitive interpretation, x(t –, j) and such lines can constitute an intuitively meaningful partitioning of the simplex (see [13]). then, paths on the 2-simplex can be understood as sequences of partition switches, and the non-self-intersection constraint as an exclusion of certain switches, as demonstrated in fig. 7, where (immediate) switches between the partitions a and c are prohibited by the non-self-intersection constraint. if we interpret t = 0 as present, t – as past, and t + as future, this prohibition corresponds to infeasible future scenarios, i.e., non-self-intersection can be used in prediction of future dynamics (cf. [13]). obviously, depending on the (natural/social sciences) topic analyzed by these concepts, a certain length and positioning of x(t –, j) may be necessary to derive significant predictions. in particular, if x(t –, j) is relatively short or located in a relatively small or peripheral subset of the 2-simplex, it may not be possible to establish a relevant partitioning inducing a prohibition of paths that is of significant relevance for the topic/theory being analyzed (cf. [13]). these requirements are well known in statistics, where the length of the past time series and avoidance of outliers is important for the (statistical) significance of the predictions based on empirical (time-series) data. 136 d.stijepic copyright ©2019 assa. adv. in systems science and appl. (2019) x(t –, j) x(0, j) x(t +, j) v2v1 v3 d c a b fig. 7. paths as partition switches. 5. poincaré-bendixson theory the poincaré-bendixson theory, which is one of the fundaments of the dynamical systems theory, can be used to predict the qualitative properties of the limit dynamics of a smooth dynamical system in the plane. it applies to continuous systems, yet requires additional restrictions on the system (to ensure a sufficient degree of smoothness). we discuss here these requirements from a rather topological point of view applying the concepts discussed in section 2.4. for a general, discussion of the requirements and predictions/statements of the poincaré-bendixson theory, see, e.g., [1, pp.362f.], [2], [3, p.45], [4, p.55], and [21, chapter 7.3]. assume that the dynamics on the 2-simplex are representable by a (relatively) smooth autonomous differential equation system in terms of the coordinates (y1, y2) of a two-dimensional coordinate system that is parallel to the 2-simplex (see fig. 8). then, the poincaré-bendixson theory states that the limit dynamics of this system are either cyclical or transitory. in particular, the omega limit set of a trajectory generated by such a system consists of a fixed point, a jordan curve, or a homo-/heteroclinic union (of curves and fixed points). the geometrical interpretation of the requirement of the representability by a smooth differential equation system in y1-y2-coordinates is that the trajectories of the dynamical system on 2-simplex constitute a simple covering (of a connected subset) of the 2-simplex. in particular, such a simple covering consists of non-intersecting and transversally non-self-intersecting trajectories, where the union of these trajectories is a connected subset of the 2-simplex (see [16,20] for a detailed discussion and literature references). 6. applications in economics the mathematical theories of continuous dynamics on the standard 2-simplex developed in the previous sections have almost direct applications in the analysis of economic dynamics. in particular, they can be used in prediction of economic structural change, discussion of models of continuous dynamics on the 2-simplex and applications in economics 137 copyright ©2019 assa. adv. in systems science and appl. (2019) sectoral production functions, assessment of structural change costs, and design of cost-minimal structural change policies, as discussed in the following sections. v3 v2v1 y2 y1 ẏ(t) = φ(y(t)) y(t) ≡ (y1(t), y2(t)) φ: u → u' u, u' ⊂ ℝ2 fig. 8. representation of a dynamical system on s by a two-dimensional differential equation system. 6.1 economic topics covered by the models of continuous dynamics on the 2-simplex a major pillar of economics is the study of long-run economic dynamics, where short-run fluctuations are neglected and the dynamic patterns that persist over long periods of time (e.g., 100 years) are studied. in this context, the concept of structural change is essential, where not only aggregate economic indices (e.g., gross domestic product, trade volume, and economy-wide employment) are studied but also their structure. in particular, the aggregate indices are subdivided into components and the significance of these components for the aggregate index is indicated by the components’ shares in the aggregate index. many of these ‘shares’, such as savings rate, investment rate, and sectoral employment shares, are well known even in public debates. in general, structural change refers to the dynamics of these ‘shares’, where the shares satisfy the conditions stated by (1). in other words, economic structural change, i.e., the long-run dynamics of the ‘shares’ can be depicted by trajectories on standard simplexes. moreover, the assumption of continuous-time frameworks and continuous functional forms is a general convention in long-run economic dynamics modeling (although there are exceptions from this convention), which, in general, yields continuous dynamics of the shares on standard simplexes. for an overview of the topics that are covered by the system-theoretical models of continuous trajectories on standard simplexes and for corresponding references from the economics literature, see [15]. to provide some details and references on the economic applications of the system-theoretical models derived in the previous sections, we focus on a specific sort of economic structural change, namely long-run labor allocation dynamics in the three-sector framework, in section 6.2. 138 d.stijepic copyright ©2019 assa. adv. in systems science and appl. (2019) 6.2 example: long-run labor allocation dynamics the three-sector framework is one of the major concepts for studying economic structural change (for an overview of the literature, see, e.g., [5-6,9-12]. it is based on the assumption that economic activities can be divided into three categories or ‘sectors’: agriculture, manufacturing, and services. one of the major indices for studying the importance of these sectors are the shares of each of the sectors in aggregate employment (abbr. ‘employment shares’). these shares and their dynamics can be depicted by continuous trajectories on the 2-simplex (cf. [13,14]). thus, we can use the concepts discussed in sections 2-5 in the analysis of labor allocation dynamics as discussed in the following. while it is difficult to derive any consensus statements on the quantitative characteristics of labor allocation dynamics, the literature implies that there seem to be some empirically observable qualitative laws of labor allocation dynamics, which can be expressed by using the geometrical concepts discussed in our paper. in particular, [19] shows that the typical long-run labor allocation dynamics of a nowadays highly developed country over the last two centuries can be described by a trajectory that has the following characteristics: (1.) it is monotonous in two dimensions (cf. sections 2.3 and 3) and (thus) non-self-intersecting (cf. sections 2.4 and 4); (2.) it has a negative signed curvature κ (cf. section 2.3); (3.) its initial segment is located in the simplex partition sv1 (cf. (5) and fig. 2); and (4.) its final segment is located in partition sv3 (cf. (5) and fig. 2). as discussed by [14], these empirical observations can be interpreted as “natural” laws of structural change (since, among others, they are supported by the theoretical literature consensus) and, thus, can be exploited for predictions of structural change. for example, proposition 2 and corollary 1 (cf. [14]) and the approach discussed in section 4.2 (cf. [13]) can be used to predict the future (transitional) dynamics of labor allocation in developing and developed economies. moreover, [16,20] uses the empirical findings of [19] and the topological approach discussed in section 5 for a discussion of the applicability of the poincaré-bendixson theory in the prediction of limit dynamics of labor allocation. beside these applications, which focus on prediction of structural change, the models of continuous dynamics discussed in our paper have further applications in in structural change modeling: [17] shows that labor allocation trajectories that are monotonous in three dimensions minimize the structural change costs (e.g., unemployment, geographical relocation costs, and environmental pollution) and uses this result to elaborate a development policy minimizing the labor reallocation costs in a developing economy. [18] uses the model of monotonous and continuous trajectories and the concept of curvature (cf. section 2.3) to discuss a widespread assumption in theoretical structural change modeling (namely the assumption of cobb-douglas production functions) by applying an axiomatic-geometrical approach. finally, [15] discusses how the topological concepts discussed in section 2.4 can be interpreted and applied in the context of labor allocation dynamics modeling. models of continuous dynamics on the 2-simplex and applications in economics 139 copyright ©2019 assa. adv. in systems science and appl. (2019) 7. conclusions in this paper, we have studied models of continuous dynamics on the 2-simplex that arise when set-theoretical, differential-geometrical, or topological restrictions are imposed on the trajectories of the model. we focused on the qualitative properties of transitional and limit dynamics of these models and discussed their applications in long-run economic dynamics modeling. many of our results (in general, the results of section 3 and their applications in economics) can be extended to discrete or discontinuous systems or higher-dimensional simplexes. yet, the rather topological concepts discussed in sections 4 and 5 (e.g., the poincaré-bendixson theory) are, in general, not applicable or not useful in discrete or higher-dimensional systems and their applications (cf. [13,15-16,20]). in the latter systems, the concept of chaos as well as existence theorems on fixed points are of interest. thus, further research could focus on them and, in particular, their system-theoretical significance for long-run economic dynamics. empirical evidence implies that there are fluctuations of the labor allocation shares that correspond to the waves on the 2-simplex discussed in sections 3.2 and 3.3. further research could focus on a detailed discussion of waves on the 2-simplex and the application of the resulting system-theoretical models in the explanation of the empirically observed waves arising in labor-allocation dynamics. acknowledgement no funding has been received for this research. references [1] andronov, a.a, vitt, a.a. & khaikin, s.e. (1987). theory of oscillators. dover publications, mineola, new york. [2] ciesielski, k. (2012). the poincaré–bendixson theorem: from poincaré to the xxist century, central european journal of mathematics, 10, 2110–2128. [3] guckenheimer, j. & holmes, p. (1990). nonlinear oscillations, dynamical systems, and bifurcations of vector fields. springer, new york. [4] hale, j.h. (2009). ordinary differential equations. dover publications, mineola, new york. [5] herrendorf, b., rogerson, r. & valentinyi, á. (2014). growth and structural transformation, in p. aghion & s.n. durlauf (eds.), handbook of economic growth, volume 2b. elsevier b.v., amsterdam. [6] krüger, j.j. (2008). productivity and structural change: a review of the literature, journal of economic surveys, 22, 330–363. [7] kuipers, b. (1986). qualitative simulation, artificial intelligence, 29, 289–338. 140 d.stijepic copyright ©2019 assa. adv. in systems science and appl. (2019) [8] lee, w.w. & kuipers, b.j. (1988). non-intersection of trajectories in qualitative phase space: a global constraint for qualitative simulation, proceedings of the seventh national conference on artificial intelligence (aaai-88), morgan kaufmann, los altos, ca. [9] van neuss, l. (2018). the drivers of structural change, forthcoming in journal of economic surveys, 32. [10] schettkat, r. & yocarini, l. (2006). the shift to services employment: a review of the literature, structural change and economic dynamics, 17, 127–147. [11] silva, e.g. & teixeira, a.a.c. (2008). surveying structural change: seminal contributions and a bibliometric account, structural change and economic dynamics, 19, 273–300. [12] stijepic, d. (2011). structural change and economic growth: analysis within the partially balanced growth-framework. südwestdeutscher verlag für hochschulschriften, saarbrücken. an older version is online: http://deposit.fernuni-hagen.de/2763/ [13] stijepic, d. (2015). a geometrical approach to structural change modelling, structural change and economic dynamics, 33, 71–85. [14] stijepic, d. (2017a). positivistic models of structural change, journal of economic structures, 6, 1–30. [15] stijepic, d. (2017b). empirical evidence on the topological properties of structural paths and some notes on its theoretical explanation, mpra working paper no. 82473. [online]. available https://mpra.ub.uni-muenchen.de/82473/1/mpra_paper_82473.pdf [16] stijepic, d. (2017c). on the predictability of economic structural change by the poincaré-bendixson theory. [online]. available https://ssrn.com/abstract=3015889. [17] stijepic, d. (2017d). on development paths minimizing the structural change costs in the three-sector framework and an application to structural policy. [online]. available https://ssrn.com/abstract=2919806 [18] stijepic, d. (2017e). an argument against cobb-douglas production functions (in multi-sector growth modeling), economics bulletin, 37, 1143–1150. [19] stijepic, d. (2018). empirical evidence on the geometrical properties of structural change trajectories, research journal of economics, 1, 1–10. [20] stijepic, d. (2019). on the predictability of economic structural change by the poincaré-bendixson theory, forthcoming in foresight. [21] teschl, g. (2012). ordinary differential equations and dynamical systems. american mathematical society, providence, rhode island. http://deposit.fernuni-hagen.de/2763/ https://mpra.ub.uni-muenchen.de/82473/1/mpra_paper_82473.pdf https://ssrn.com/abstract=3015889 https://ssrn.com/abstract=2919806 advances in systems science and applications (2013) vol.13 no.1 53-67 soft probability of large deviations d.a. molodtsov dorodnitsyn computing center, russian academy of sciences, moscow, russia abstract a concept of soft probability is presented. an analogue of chebyshev’s inequality for soft probability is proved. soft large deviation probabilities for a nonnegative random variable under a mean hypothesis is calculated. keywords soft probability, hypothesis replacement, soft chebyshev inequality, large deviations. 1 what is soft probability? in modern textbooks of probability theory, the exposition usually begins with a discussion of the subject matter of this science. all phenomena are divided into three types. phenomena of the first type are those characterized by deterministic regularity; this means that a given set of circumstances always leads to the same outcome. phenomena of the second and the third type are not deterministic regular. the second type consists of statistically regular phenomena, and the third type, of the remaining phenomena. by statistical regularity the statistical stability of outcome frequencies is usually understood. as a rule, this notion is associated with the example of coin tossing. some authors outline more constructive ways of interpretation [1], but a final formalization of statistical stability has never been proposed, and its verification is always left to the reader’s judgment and intuition. thus, the most important question of the applicability of the theory to real phenomena remains essentially unanswered. soft probability [2-7] is merely the logical completion of the construction of statistical regularity, whose approximate description is usually contained in probability theory textbooks. our approach is based on examine the conclusions to which the logical development of the notion of statistical regularity leads. first, we mention at once that the verification of statistical regularity suggested here requires a certain set of trial outcomes, that is, a statistical database. if there are no trial results, then there is no object of examination. 2 statistical database first, we introduce the base outcome space ω. each trial is associated with an element of the set ω, namely, the outcome of this trial. a statistical database is merely a finite sequence of outcomes: base = {ω1, ..., ωn}, ωi ∈ ω. 54 d.a molodtsov: soft probability of large deviations by an event a we mean a subset of ω : a ⊆ ω. we say that an event occurs under a certain trial if the outcome of this trial belongs to the given event. we have to define the statistical regularity of the occurrence of an event a. in effect, statistical regularity means the closeness of frequencies for a given set of samples. thus, in formalizing this concept, we must begin with specifying a set of samples for a statistical database. we identify a sample i with the positions in the sequence base = {ω1, ..., ωn} occupied by its elements; in other words, a sample is a subset i ⊆ ind(base) of the set ind(base) = {1, ..., n}. in specifying a set of samples, it is natural to first constrain their size; thus, we introduce a parameter determining the size of a sample. as admissible samples we consider any samples of size m which consists of consecutive elements of the set ind(base) and are not too “old”. we denote the set of such samples by s(base,m, τ): s(base,m, τ) = {(i, i+ 1, ..., i+m− 1), i = τ, ..., n −m+ 1}. apparently, the set s(base,m, τ) is a minimal set of samples which are natural to consider in defining the notion of statistical regularity. of course, other definitions of admissible samples are also possible, which lead to different definitions of statistical regularity; it is important that the set of admissible samples be precisely specified. we define the frequency of occurrence of an event a in a sample i as µ(base, χ(a, ·), i) = 1 |i| ∑ i∈i χ(a,ωi). here |i| denotes the cardinality of the set i and χ(a,ω) = {1,ω∈a0,ω /∈a. 3 statistical regularity now, it is natural to understand the statistical regularity of the occurrence of an event a as the closeness of the frequencies of occurrence of a in any admissible samples from the set s(base,m, τ). below we give a more formal definition of this notion. definition 1. an event a is said to be statistically (m, τ, δ) − regular on a database base if |µ(base, χ(a, ·), i)− µ(base, χ(a, ·), j)| ≤ δ. for any samples i, j ∈ s(base,m, τ). we see that the definition of statistical regularity involves several parameters. apparently, it is for this reason that this notion has not been used, because the advances in systems science and applications (2013) vol.13 no.1 55 dependence of statistical regularity on parameters inevitably makes probability dependent on parameters as well, while probability is traditionally thought of as a single real number in the unit interval. however, the parameterization of probability only means the detailing of the description of the situation under consideration. in different situations, probabilities corresponding to different parameters are important. the description using the same number as probability in all situations is coarser than that using parameterized probability. parameterized families are used very extensively in theory and practice. thus, different methods of measuring physical quantities yield different results, which naturally leads to parameterized families. a striking example of such a family is the description of mine deposits in geology. in computational mathematics, it often happens that only approximate solutions can be found numerically; such solutions form parameterized families too [2,8]. for dealing with such objects, the notion of a soft set was introduced and the theory of soft sets was developed, which has found numerous applications in various areas of mathematics [2-7, 9-24]. for this reason, the alternative to classical probability considered in this paper is called soft probability. let us contemplate definition 1. the statistical database consists of the outcomes of events which have already occurred, while the main purpose of the theory is to produce informative statements concerning future events. thus, the only interesting aspect of the notion of statistical regularity is its use as a hypothesis on the future behavior of trial outcomes. accordingly, the application of soft probability is divided into two processes: • verifying hypotheses at each step. • given a set of accepted hypotheses, constructing estimates, predictions, etc. note that, in classical probability theory, the former process is virtually absent; the statistical regularity hypothesis is accepted only once, before launching the apparatus of probability theory, after which this hypothesis is neither controlled nor verified anew. moreover, in classical probability theory, the statistical regularity hypothesis is accepted for all possible events together. it remains unclear how to handle situations in which some of the events are statistically regular and the other events are not. the definition of statistical regularity is easy to generalize to a random function. by a random function f we mean any real-valued function defined on the set ω, that is, any function f : ω → e, where e is the set of real numbers. an example of such a function is the characteristic function χ(a, ·) of a set. 56 d.a molodtsov: soft probability of large deviations the mean value of a random function f on a sample i is defined as µ(base, f, i) = 1 |i| ∑ i∈i f(ωi). definition 2. a random function f is said to be statistically (m, τ, δ) − regular on a database base if |µ(base, f, i)− µ(base, f, j)| ≤ δ. for any samples i, j ∈ s(base,m, τ). thus, the statistical regularity of an event means simply that the characteristic function of this event is statistically regular. the condition that a random function is statistically regular can be written in a different equivalent form as follows. we set 2a = max i∈s(base,m,τ) µ(base, f, i) + min i∈s(base,m,τ) µ(base, f, i), and 2b = max i∈s(base,m,τ) µ(base, f, i)− min i∈s(base,m,τ) µ(base, f, i). it is easy to see that the statistical regularity of f is equivalent to the inequality 2b ≤ δ. for any i ∈ s(base,m, τ), we have |µ(base, f, i)−a| ≤ b. this readily implies the equivalence of the statistical regularity of f to the fulfillment of the inequality |µ(base, f, i) − a| ≤ δ/2, or the inclusion µ(base, f, i) ∈ [a − δ/2, a + δ/2], for any i ∈ s(base,m, τ). the inclusion is also equivalent to [ min i∈s(base,m,τ) µ(base, f, i), max i∈s(base,m,τ) µ(base, f, i)] ⊆ [a− δ/2, a+ δ/2]. this suggests the following natural definition. definition 3. the interval λ(f,base,m, τ) = [ min i∈s(base,m,τ) µ(base, f, i), max i∈s(base,m,τ) µ(base, f, i)] is called the (m, τ)− approximate mean value of the random function f on the database base. we denote the left and right endpoints of this interval by an underscore and an overscore, respectively: λ(f,base,m, τ) = [λ(f,base,m, τ), λ(f,base,m, τ)]. advances in systems science and applications (2013) vol.13 no.1 57 4 hypotheses on the behavior of a random function on the basis of the notion of the statistical regularity of a random function, we can formulate hypotheses of two types on the future behavior of the values of a random function. in fact, these are hypotheses on the future values of the statistical database. definition 4. a databasebase is said to be statistically (m, δ)− regular with respect to a random function f if |µ(base, f, i)− µ(base, f, j)| ≤ δ. for any samples i, j ∈ s(base,m, 1). definition 5. a databasebase is said to be statistically significantly (m, δ, a)− regular with respect to a random function f if |µ(base, f, i)− a| ≤ δ. for any samples i ∈ s(base,m, 1). the difference between definitions 4 and 5 is in that definition 4 supposes only the closeness of mean values on any admissible samples, while definition 5 specifies the number to which these mean values must be close with a given accuracy. in other words, the mean values must belong to the interval [a− δ.a+ δ], or the approximate mean values of the random function under consideration must belong to [a− δ.a+ δ]. these definitions can also be regarded as hypotheses on certain properties of the approximate means of a random function. definition 4 specifies only the length of an interval containing the approximate mean, and definition 5 specifies the boundaries of the entire range of this mean. an analysis shows that these hypotheses are often insufficient for obtaining instructive results. in addition to hypotheses on approximate mathematical expectation, hypotheses describing the deviation of the random function under consideration from its mathematical expectation (that is, hypotheses similar to that of the existence of variance) are very useful. such hypotheses can be formulated in various forms. we give only one version. together with a random function f , consider the random function equal to the absolute value of the deviation of f from the interval [a− δ.a+ δ], that is, defined by g(ω) = max{|f(ω)− a| − δ, 0}. definition 6. we say that a database base satisfies the (m, δ, a,△) − variance hypothesis with respect to a random function f if base is statistically significantly (m,△, 0) − regular with respect to the random function g, i.e. µ(base, g, i) ≤ △. 58 d.a molodtsov: soft probability of large deviations for any samples i ∈ s(base,m, 1). the approximate mathematical expectation and approximate variance hypotheses describe a variable oscillating about a certain interval. it is also natural to consider other hypotheses, which describe the growth, decline, periodicity, and other properties of random variables [6-7]. after classes of hypotheses are chosen, dealing with hypotheses is a dynamical step process. at each step, new realization from the base space (trial outcomes) appear. thus, at each step, it is required to form a list of accepted hypotheses and perform calculations on the basis of this list. 5 properties of the approximate mean value of a random function properties of an approximate mean are similar to those of mathematical expectation. inequalities involving intervals are assumed to hold componentwise, that is, at each endpoint of the interval. 1). λ(c,base,m, τ) = [c, c], c ∈ e. 2). if f(ω) ≥ g(ω) for any ω ∈ ω, then λ(f,base,m, τ) ≥ λ(g,base,m, τ). 3). λ(cf,base,m, τ) = cλ(f,base,m, τ), c ∈ e, c ≥ 0. 4). λ(f + c,base,m, τ) = λ(f,base,m, τ) + c, c ∈ e. 5). λ(−f,base,m, τ) = −λ(f,base,m, τ) = [−λ(f,base,m, τ),−λ(f,base,m, τ)]. 6). λ(f + g,base,m, τ) ⊆ λ(f,base,m, τ) + λ(g,base,m, τ). 7). λ(f,base,m, τ) ⊆ λ(f,base,m, t), τ ≥ t. 6 an approximate variance of a random variable in classical probability theory, the variance of a random variable is defined as the mathematical expectation of the squared difference between this variable and its expectation. the mathematical expectation of a random function is merely a number. in the case under consideration, the approximate mean is a parametric family of intervals which characterizes the random variable on a certain database. hypotheses involving the notion of statistically significant regularity impose constraints on approximate means. the intervals describing the constraints are not required to equal the corresponding approximate means. thus, it is convenient to define approximate variance as the measure of deviation of a random variable from a certain interval rather than from the corresponding approximate mean. definition 7. the approximate (m, τ, a, δ) − variance of a random function f on a database base is the (m, τ)− approximate mean of the random function max{|f(ω)− a| − δ, 0}, that is, the interval d(f,base,m, τ, a, δ) = λ(max{|f(·)− a| − δ, 0}, base,m, τ). advances in systems science and applications (2013) vol.13 no.1 59 the simplest properties of approximate variance are as follows. 1). d(f,base,m, τ, a, δ) ≥ 0. 2). d(cf,base,m, τ, ca, cδ) = cd(f,base,m, τ, a, δ), c ∈ e, c ≥ 0. 3). d(−f,base,m, τ,−a, δ) = d(f,base,m, τ, a, δ). 4). d(f+g,base,m, τ, a+b, δ+γ) ≤ d(f,base,m, τ, a, δ)+d(g,base,m, τ, b, γ). 5). d(f+g,base,m, τ, a+b, δ+γ) ≤ d(f,base,m, τ, a, δ)+d(g,base,m, τ, b, γ). 6). d(f,base,m, τ, a, δ) ≥ d(f,base,m, τ, a, γ), δ ≤ γ. 7). d(f,base,m, τ, a, δ) ⊇ d(f,base,m, γ, a, δ), τ ≤ γ. 7 chebyshev’s inequality for soft probability statement 1 (chebyshev’s inequality). if a random function f is nonnegative everywhere on ω and ε > 0, then λ(χ({ω′ ∈ ω|f(ω′) ≥ ε}, ·), base,m, τ) ≤ 1 ε λ(f,base,m, τ). proof. if a random function f is nonnegative everywhere on ω, then, as is easy to see, we have f(ω) ≥ εχ({ω′ ∈ ω|f(ω′) ≥ ε}, ω) for any ε > 0 and any ω ∈ ω. property 2 of approximate mean implies λ(f,base,m, τ) ≥ ελ(χ({ω′ ∈ ω|f(ω′) ≥ ε}, ω), base,m, τ). the approximate means of the characteristic function of a set equals the soft probability of this set; therefore, the soft probability of the set {ω′ ∈ ω|f(ω′) ≥ ε} is estimated as λ(χ({ω′ ∈ ω|f(ω′) ≥ ε}, ·), base,m, τ) ≤ 1 ε λ(f,base,m, τ) (recall that the inequality is interval). this completes the proof of the statement. now, let f be an arbitrary random function. for the function |f |, we have λ(χ({ω′ ∈ ω| |f(ω′)| ≥ ε}, ·), base,m, τ) ≤ 1 ε λ(|f |, base,m, τ). since the inequality |f(ω′)| ≥ ε is equivalent to |f s(ω′)| ≥ εs, where s > 0, it follows that λ(χ({ω′ ∈ ω| |f(ω′)| ≥ ε}, ·), base,m, τ) ≤ 1 εs λ(|f |s, base,m, τ). therefore, λ(χ({ω′ ∈ ω| |f(ω′)| ≥ ε}, ·), base,m, τ) ≤ inf s>0 1 εs λ(|f |s, base,m, τ). 60 d.a molodtsov: soft probability of large deviations now, consider the nonnegative random function g(ω) = max{|f(ω)−a|−δ, 0}, where f is an arbitrary random function. applying chebyshev’s inequality to g, we obtain λ(χ({ω′ ∈ ω|max{|f(ω′)−a|−δ, 0} ≥ ε}, ·), base,m, τ) ≤ 1 ε d(f,base,m, τ, a, δ). elementary transformations yield λ(χ({ω′ ∈ ω||f(ω′)− a| ≥ δ + ε}, ·), base,m, τ) ≤ 1 ε d(f,base,m, τ, a, δ). this allows us to write the inequality in the equivalent form λ(χ({ω′ ∈ ω||f(ω′)− a| ≥ δ}, ·), base,m, τ) ≤ inf 0<ε<δ 1 ε d(f,base,m, τ, a, δ − ε). thus, chebyshev’s inequality gives an estimate of the soft probability of “large” deviations in terms of approximate mean or approximate variance. if the database is known, then this information is of little value, because it is easy to directly calculate the exact values of any probabilistic characteristics, including the probabilities of large deviations. apparently, it is of more interest to apply this inequality to estimating the probability of a random function on a future database. naturally, this requires hypotheses on the approximate mean or the approximate variance of the function under consideration. however, in the presence of hypotheses, of interest are sharp bounds for the probability of large deviations. 8 soft probability of large deviations for a nonnegative random function under an approximate mean hypothesis suppose that a database base is statistically significantly (m, δ, a) − regular with respect to a nonnegative random function f , i.e., given any sample i ∈ s(base,m, 1), we have |µ(base, f, i)− a| ≤ δ we assume that a ≥ 0. we define a large deviation as the event a(f, ε) = {ω ∈ ω|f(ω) ≥ ε}, where ε > 0. we are interested in the range of values of the soft probability of this event under the above hypothesis. clearly, the solution essentially depends on the range of the function f . consider the case where f(ω) = e+ = {x ∈ e|x ≥ 0}. let f(ωi) = xi ∈ e+. then the constraints on the database can be written as constraints on the vector x = {x1, ...xn} ∈ x(n,m, a, δ), where x(n,m, a, δ) = {x ∈ en +| | j+m−1∑ i=j xi − am| ≤ δm, j = 1, ..., n−m+ 1}. advances in systems science and applications (2013) vol.13 no.1 61 the boundaries of the range of the soft probability are u∗(n,m, a, k, τ, δ, ε) = sup x∈x(n,m,a,δ) max τ≤j≤n−k+1 1 k j+k−1∑ i=j χ({y|y ≥ ε}, xi)}. and u∗(n,m, a, k, τ, δ, ε) = inf x∈x(n,m,a,δ) min τ≤j≤n−k+1 1 k j+k−1∑ i=j χ({y|y ≥ ε}, xi)}. note that, at n = m, the set x(n,m, a, δ) takes the form x(m,m, a, δ) = {x ∈ em + | | m∑ i=1 xi − am| ≤ δm}. in the case k ≤ m, the evaluation of u∗ and u∗ is based on the following assertion. statement 2. if x ∈ em + and (xj , xj+1, ..., xj+m−1) ∈ x(m,m, a, δ) , then there exists a vector y ∈ x(n,m, a, δ) such that yi = xi for i = j, j+1, ..., j+m−1. proof. let yi = xj+(i−j)modm for i = 1, ..., n. then ∑l+m−1 i=l yi = ∑j+m−1 i=j xi for any l = 1, ..., n−m+1. therefore, y ∈ x(n,m, a, δ). this proves the required assertion. the u∗ and u∗ problems can be formulated as u∗(n,m, a, k, τ, δ, ε) = max τ≤j≤n−k+1 sup x∈x(n,m,a,δ) 1 k j+k−1∑ i=j χ({y|y ≥ ε}, xi)}. and u∗(n,m, a, k, τ, δ, ε) = min τ≤j≤n−k+1 inf x∈x(n,m,a,δ) 1 k j+k−1∑ i=j χ({y|y ≥ ε}, xi)}. at k ≤ m, in the problems sup x∈x(n,m,a,δ) 1 k j+k−1∑ i=j χ({y|y ≥ ε}, xi) and inf x∈x(n,m,a,δ) 1 k j+k−1∑ i=j χ({y|y ≥ ε}, xi) the function to be optimized depends only on the variables xj = (xj , xj+1, xj+m−1); hence, we can perform optimization over the projection of the set x(n,m, a, δ) 62 d.a molodtsov: soft probability of large deviations on the corresponding coordinates rather over the entire set. it follows from statement 2 that this projection coincides with the set x(m,m, a, δ); thus, at k ≤ m, the u∗ and u∗ problems take the forms u∗(n,m, a, k, τ, δ, ε) = sup x∈x(m,m,a,δ) 1 k k∑ i=1 χ({y|y ≥ ε}, xi)}. and u∗(n,m, a, k, τ, δ, ε) = inf x∈x(m,m,a,δ) 1 k k∑ i=1 χ({y|y ≥ ε}, xi)}. for x ∈ em + , we set π(x,m, ε) = |{i ∈ {1, ...,m}|xi ≥ ε}|; this is the number of components greater than or equal to ε. we have u∗(n,m, a, k, τ, δ, ε) = sup x∈x(m,m,a,δ) min{π(x,m, ε), k} k = 1 k min{ sup x∈x(m,m,a,δ) π(x,m, ε), k}. and u∗(n,m, a, k, τ, δ, ε) = 1 k max{k + inf x∈x(m,m,a,δ) π(x,m, ε)−m, 0}. thus, it is required to find the maximum and the minimum value of the function π(x,m, ε) on x(m,m, a, δ). let us introduce the set π(m, ε, p) = {x ∈ em + |π(x,m, ε) = p}. it is easy to see that the image of the function ∑m i=1 xi on the set π(m, ε, p) equals { [εp,+∞), p > 0 [0,mε), p = 0, therefore, the function π(x,m, ε) takes the value p on the set x(m,m, a, δ) if and only if π(m, ε, p) ∩x(m,m, a, δ) ̸= ∅. that is, 1 ≤ p ≤ ma+mδ ε or a− δ < ε at p = 0. let [x] denote the largest integer not exceeding x. then the condition on those positive values p which the function π(x,m, ε) can take on x(m,m, a, δ) can be written in the form 1 ≤ p ≤ min{m, [ m(a+ δ) ε ]}. advances in systems science and applications (2013) vol.13 no.1 63 the condition that π(x,m, ε) vanishes on x(m,m, a, δ) has the form a < δ + ε. thus, we have proved the following assertion. statement 3. let k ≤ m. 1). if 0 ≤ a < δ + ε, then u∗(n,m, a, k, τ, δ, ε) = 0. 2). if a > δ + ε, then u∗(n,m, a, k, τ, δ, ε) = max{k+1−m,0} k = {1/m,k=m 0,k<m . 3). if m(a+ δ) ≥ ε, then u∗(n,m, a, k, τ, δ, ε) = min{[m(a+δ) ε ],k} k . 4). if m(a+ δ) < ε, then u∗(n,m, a, k, τ, δ, ε) = 0. note that the solution has the useful feature that, for each of the u∗ and u∗ problems, there exists the same vector, no matter at what parameter the solution is attained. now, consider the case k > m. the function ∑j+k−1 i=j χ({y|y ≥ ε}, xi) depends only on those components of the vector x whose numbers belong to {j, j+1, ..., j+ k − 1}. thus, we introduce the set x(k,m, a, δ) = {x ∈ ek +| | j+m−1∑ i=j xi − am| ≤ δm, j = 1, ..., k −m+ 1}. for this set, an assertion similar to statement 2 is valid. statement 4. if x ∈ en + and (xj , xj+1, ..., xj+k−1) ∈ x(k,m, a, δ), then there exists a vector y ∈ x(n,m, a, δ) such that yi = xi for i = j, j + 1, ..., j + k − 1. proof. we set • yi = xj+(i−j)mod m for i = 1, ..., j − 1, • yi = xi for i = j, j + 1, ..., j + k − 1, • yi = xj+k−m+(i−j−k+m)mod m for i = j + k, ..., n. we have • ∑l+m−1 i=l yi = ∑j+m−1 i=j xi for l = 1, ..., j − 1, • ∑l+m−1 i=l yi = ∑l+m−1 i=l xi for l = j, j + 1, ..., j + k −m, • ∑l+m−1 i=l yi = ∑j+k−1 i=j+k−m xi for l = j + k −m, ..., n−m+ 1. therefore, y ∈ x(n,m, a, δ). this completes the proof of the statement. now, the problems for u∗ and u∗ with k > m take the forms u∗(n,m, a, k, τ, δ, ε) = 1 k sup x∈x(k,m,a,δ) π(x, k, ε). and u∗(n,m, a, k, τ, δ, ε) = 1 k inf x∈x(k,m,a,δ) π(x, k, ε). let us divide k by m with a remainder, that is, write k = mq + r,m > r ≥ 0. take an arbitrary vector x ∈ x(k,m, a, δ) and consider its decomposition into the parts x0 = (x1, ..., xm) ∈ em and xj = (xr+(j−1)m+1, xr+(j−1)m+m) ∈ em, j = 1, ..., q. 64 d.a molodtsov: soft probability of large deviations it is assumed that r > 0; if r = 0, then the vector x0 is absent. obviously, π(x, k, ε) = π(x0, r, ε) + q∑ j=1 π(xj ,m, ε) ≤ min{π(x0,m, ε), r}+ q∑ j=1 π(xj ,m, ε). note that the condition x ∈ x(k,m, a, δ) implies that xj ∈ x(m,m, a, δ) for any j = 0, ..., q. hence, we have sup x∈x(k,m,a,δ) π(x, k, ε) ≤ min{ max y∈x(m,m,a,δ) π(y,m, ε), r}+ q max y∈x(m,m,a,δ) π(y,m, ε). let us show that this inequality is, in fact, an equality. we take a vector y∗ ∈ x(m,m, a, δ) at which max y∈x(m,m,a,δ) π(y,m, ε) is attained and place all components of this vector greater than or equal to ε at the first positions. let x∗ ∈ ek + be the vector formed by as many copies of y∗ written one after another as needed to achieve the required dimension. it is easy to see that x∗ ∈ x(k,m, a, δ) and sup x∈x(k,m,a,δ) π(x, k, ε) ≥ π(x∗, k, ε) = min{ max y∈x(m,m,a,δ) π(y,m, ε), r} + q max y∈x(m,m,a,δ) π(y,m, ε). thus, we have proved the following assertion. statement 5. suppose that n ≥ k > m and k = mq + r, m > r ≥ 0 . 1). ifm(a+δ) ≥ ε, then u∗(n,m, a, k, τ, δ, ε) = min{[m(a+δ) ε ],r}+q min{[m(a+δ) ε ],m} k . 2). if m(a+ δ) ≤ ε, then u∗(n,m, a, k, τ, δ, ε) = 0. it is easy to see that statement 5 is also valid for n ≥ m ≥ k > 0. now, consider the u∗ problem. for the arbitrary vector x ∈ x(k,m, a, δ) under consideration and its partition constructed above, we have π(x, k, ε) = π(x0, r, ε)+ q∑ j=1 π(xj ,m, ε) ≥ max{k−m+π(x0,m, ε), 0}+ q∑ j=1 π(xj ,m, ε). since xj ∈ x(m,m, a, δ) for any j = 0, ..., q, it follows that inf x∈x(k,m,a,δ) π(x, k, ε) ≥ max{r−m+ inf y∈x(m,m,a,δ) π(y,m, ε), 0}+q inf y∈x(m,m,a,δ) π(y,m, ε). as above, this inequality is, in fact, an equality. to show this, we take a vector y∗ ∈ x(m,m, a, δ) at which min y∈x(m,m,a,δ) π(y,m, ε) is attained and place all components of this vector which are greater than or equal to ε at the last positions. advances in systems science and applications (2013) vol.13 no.1 65 consider the vector x∗ ∈ em + consisting of copies of y∗ written one after another. it is easy to see that x∗ ∈ x(k,m, a, δ) and inf x∈x(k,m,a,δ) π(x, k, ε) ≤ π(x∗, k, ε) = max{r −m+ min y∈x(m,m,a,δ) π(y,m, ε), 0}+ q min y∈x(m,m,a,δ) π(y,m, ε). thus, we have proved the following assertion. statement 6. suppose that n ≥ k > m and k = mq + r,m > r ≥ 0 . 1). if 0 ≤ a < δ + ε, then u∗(n,m, a, k, τ, δ, ε) = 0. 2). if a ≥ δ + ε, then u∗(n,m, a, k, τ, δ, ε) = q k . 9 conclusion the exact boundaries of the range of the soft probability of large deviations under a single mean hypothesis, which were found in this paper, show (although, for a very simple example) that it is quite possible to deal with soft probabilities, in spite of the presence of parameters and the interval form of soft probability. the next goal is to solve more complicated problems on evaluating various probabilities and other characteristics in the presence of several hypotheses, preferably of different types. of special interest is the application of the ideas and results presented in this paper to a real statistical problem, which would make it possible to verify the effectiveness of the approach for real data. all readers interested in such a practical experiment are kindly requested to send their suggestions to the 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(2008), “data analysis approaches of soft sets under incomplete information”, knowl.-based syst, vol.21, pp.941-945. corresponding author d.a molodtsov can be contacted at: dmitri molodtsov@mail.ru advances in systems science and applications (2013) vol.13 no.4 299-316 currency wars and a possible self-defense (ii) : a plan of self protection jeffrey forrest1, zack hopkins2 and sifeng liu3 1department of mathematics, slippery rock university, slippery rock, pa 16127, usa 211934 route 89, wattsburg, pa 16442, usa 3school of economics and management, nanjing university of aeronautics and astronautics, nanjing 210016, pr china abstract continuing what is presented in part one (forrest,hopkinsand liu, 2013)[1], and based on how a currency war could be potentially raged against a nation, this sequel makes use of the results of feedback systems to develop a self-defense mechanism that could conceivably protect the nation under siege. keywords purchasing power, supply and demand of money, feedback system, economic sector, monetary policy 1 introduction when combining the previous theoretical analysis in (forrest and hopkins, 2013)[1] with the recent cases of speculative attacks in the arena of international finance, we surely see the following predicament. when a nation tries to develop economically, due to its loosening economic and monetary policies, large amounts of foreign investments would be welcomed ; and at the same time, a lot of such foreign investments would strategically rush into the nation in order to ride along with the forthcoming economic boom. now what we have shown earlier is that if a large amount of foreign investments leaves suddenly, then the nation would most likely suffer from a burst of the economic bubble with a large percentage of economic activities interrupted either temporarily or indefinitely. so, a natural question at this junction is : how could we possibly design a measure to counter such sudden leaves of foreign investments in order to avoid the undesirable disastrous consequences ? we will address this problem in the rest of this paper. 2 a model for categorized purchasing power let us look at the following model that relates the purchasing power of money with the demand and supply of the money of a national economy : dp dt = k(d − s) (1) where d stands for the demand for money,sthe money supply, p the purchasing power of money, k > 0 is a constant,and t represents time. 300 jeffrey forrest :currency wars and a possible self-defense (ii) : a plan of self protection what this model says is that the rate of change in purchasing power is directly proportional to the difference between the demand and supply of money. in particular, the model says that with all other variables staying constant, if the money supply s increases by a large amount and satisfies s > d, then the purchasing power of the money p decreases so that more money is needed to buy essentials of living and inflation will increase due to the increase in the money supply s. now, let us divide the overall national economy into three sectors e1, e2 and e3 as follows : e1 stands for the goods, services, and relevant production of these goods and services that are needed for maintaining the basic living standard, e2 he goods, services, and relevant productions that are used to acquire desired living conditions, and e2 the goods, services, and relevant productions that are used for the enjoyment of luxurious living conditions. the reason why we divide the economy in such a way is that according to (allen and goldsmith, 1972)[2],the following are four base requirements that allow for a stable society to be achieved and maintained : (1). minimum disruption of ecological processes ; (2). maximum conservation of materials and energy, or an economy of stock rather than flow ; (3). a population in which recruitment equals loss ; (4). a social system in which individuals can enjoy rather than feel restricted by the first three conditions. so, in terms of economics, to maintain a stable society, a relative stability of the economic sector e1 has to be achieved first. next, let us accordingly divide the overall demand d of money into three corresponding categories as follows : d1 = the demand of money for meeting the minimum requirement to maintain the basic living standard ; d2 = the demand of money for acquiring desired living conditions ; d3 = he demand of money for enjoying luxurious living conditions. assume that in a stable economy, we have the following allocation of the money demand : d = d1 +d2 +d3 = α1d + α2d + α3d (2) where the weights αi, i=1,2,3, stands for the average allocation of the citizens of the economy over the three categories as described above, satisfying α1+α2+α3 = 1, and di = αid, i=1,2,3. for instance, in the stable economy, an average family allocates half of its monthly income on necessities of living, such as food, utilities, etc., 4 tenth of the income on acquiring the desired quality of life, and one tenth of the income on luxurious items, then α1 = 0.5, α2 = 0.4 and α3 = 0.1. if the money supply s increases drastically, along with the decreasing purchasing power of money, all goods will cost more. if somehow the goods in the category advances in systems science and applications (2013) vol.13 no.4 301 of living necessities rise more rapidly, then a re-allocation of household income will appear. for instance, due to rumors about potential interruptions in the supply of food and clean water accompanying a substantial increase in the money supply, the average family has to reallocate its income as follows :α1 = 0.625, α2 = 0.3 and α3 = 0.075. when such a reallocation of income of the average family is forced to take place, the stability of the economy would very likely be in trouble. so, to stabilize the economy, the purchasing power of money in category d1 should stay relatively constant, while in d2 increases somehow slightly, and in d3 it should be allowed to increase in order to attract and trap the additional money supply away from category d1. so, let us assume p1 = the purchasing power of money in category d1 ; p2 = he purchasing power of money in category d2 ; p3 = the purchasing power of money in category d3. similarly, let us define s1 = he money supply that goes into category d1 ; s2 = the money supply that goes into category d2 ; s3 = he money supply that goes into category d3. so, equ.(1) would look as follows : dp1 dt = k11(d1 − s1) + k12(d2 − s2) + k13(d3 − s3) + ∑n j=1 q1jxj dp2 dt = k21(d1 − s1) + k22(d2 − s2) + k23(d3 − s3) + ∑n j=1 q2jxj dp3 dt = k31(d1 − s1) + k32(d2 − s2) + k33(d3 − s3) + ∑n j=1 q3jxj (3) where (k)(ij), and (q)(ij) are constants, and (x)(j) stands for monetary policies, i=1,2,3, and j=1,2, . . . ,n,by using matrix notations, equ.(3) can be rewritten as follows : ṗ = kz +qx (4) where p = [p1 p2 p3] t , ṗ is newton’s original notation for derivatives such that ṗ = [dp1dt dp2 dt dp3 dt ] t ,k = [kij ]3×3 the coefficient matrix of the variables (di − si), i=1,2,3, q= [qij ]3×n the coefficient matrix of the variables xj , j=1,2, . . . ,n. in terms of systems research, the mode in equ.(2) can be seen as a feedback system as depicted in fig.1, where s represents the initial state of the economy. after the monetary policies x1, x2, . . . , xn are introduced, the participants of the economy introduce either consciously or unconsciously a feedback component system sf so that the overall system with the added feedback produces the desired output p1,p2 and p3. what is shown in fig.1 is the fact that each and every market economy, where the participants are allowed to design their own methods (without violating the 302 jeffrey forrest :currency wars and a possible self-defense (ii) : a plan of self protection fig.1 the feedback loop between monetary policies and categorized purchasing powers established laws) to achieve their individually defined financial successes, then the economy constitutes a “rotational” field. here, the word of rotation means that as soon as a monetary policy is introduced, the market participants will find ways to take advantage of the policy so that the policy and the individually designed methods jointly produce the individually desired outputs. 3 the functional relationship between p and (d − s) in this section, we investigate the relationship between the purchasing power vector [p1 p2 p3] t of money and the difference vector [d1 − s1 d2 − s2 d3 − s3] t of demand and supply of money. in particular, we discuss why the trends found in purchasing power of money could righteously be described by using linear models, although the purchasing power is clearly not linear in terms of the difference of the demand and the supply of money. to this end, let usintroduce the following three models, the first oneis linear, the second one quadratic, and the third one cubic, to demonstrate the effects of supply and demand of money on purchasing power of money in each case : p (t) = a(d(t)− s(t)) + ε (5) p (t) = a(d(t)− s(t))2 + b(d(t)− s(t)) + ε (6) and p (t) = a(d(t)− s(t))3 + b(d(t)− s(t))2 + c(d(t)− s(t)) + ε (7) where p (t) stands for the purchasing power of money, d(t) the demand of money, s(t) the supply of money,ε a random variable with mean c ̸= 0,and a, b and c are constant. more specifically, the random variable is the error term in the sense that it compensates for any unpredicted event or factor that could impact the purchasing power and that is not taken into account in the model. when the nonlinearity in the trend of purchasing power is considered, such as advances in systems science and applications (2013) vol.13 no.4 303 in the case of japanese yen (fig.2), the quadratic and cubic models in equs.(6) and (7) seem to model adequate. in particular, the non-linear graph of the purchasing power of japanese yen has two distinguishable patterns : one is parabolic and the other cubic. these two types of general patterns are respectively depicted by the nonlinear models in equs.(6) and (7). fig.3 depicts both the purchasing power and the amount of money in circufig.2 urchasing power and currency in circulation, from www.dollardaze.com as accessed on april 4, 2012 lation of the u.s. dollars over time. here a clear inverse relationship between the currency in circulation, supply of money, and the purchasing power of money can be seen. that is to say, as the amount of money in circulation increases the purchasing power of the us dollars decreases. the graph of purchasing power is fairly linear, especially if broken up into two segments : from 1971 to approximately 1981 and from 1981 until the present day. this trend in the purchasing power of u.s. dollars suggests that the linear model in equ.(5) seems to adequately reflect what happened to the purchasing power over time. in particular, if we employ the linear model in equ.(5) to describe the relationship between the purchasing power of the u.s. dollars and the difference between the demand and supply of the money with respect to time, then this model well explains how the u.s. dollar declines in purchasing power. for instance, the initial high purchasing power was due to the fact that the currency in circulation was fairly low and the value of the u.s. dollar was fixed at $35 an ounce of the gold. however, starting in may 1971 the u.s. dollar suffered from a major crisis, and consequently began to depreciate against the gold from the initial $ 35 an ounce to $38 an ounce on august 15, 1971, then to $42.22 an ounce at the start of 1973, and then to as high as $96 an ounce in march of the same year (wang and hu, 304 jeffrey forrest :currency wars and a possible self-defense (ii) : a plan of self protection 2005, p.12)[3]. in 1976 a new international agreement was reached in the capital city kingston of jamaica ; with several rounds of modifications the international financial system entered the jamaica system in 1978. within this new system, gold is no longer considered as a form of money. that is, when the demand for the u.s. dollars is brought into the equation, it can justify why there are more severe or gradual drops in purchasing power as the amount of money in circulation increases. for example, if the change in the supply of money is equal to the increase in demand of money, then the purchasing power of the money should remain constant. to cause the severe drop in purchasing power as seen from 1971 until approximately 1981, there was an increase in the money supply accompanied by a decrease in the demand for money. this decrease in the demand for money was caused by the transition from the u.s. dollar being backed by gold to that not having gold backing (dollardaze.org). this also makes sense intuitively. if we already know that a rise in the money supply decreases the purchasing power of money, then people wanting the money less would further exacerbate that decrease in purchasing power. from that point on, the demand of money must have increased but still not greater than the supply of money because the decline in the purchasing power flattens out instead of becoming more severely negative. as the graph of the stock prices of the dow jones industrial average indicates, fig.4, the stock market began to rise in the early 1980s and had a sharp incline until the start of the 21st century. this rise is an indicator that supports the claim that the demand for money increased during this time fig.3 the feedback loop between monetary policies and categorized purchasing powers period and, in conjunction with the supply of money, influenced the purchasing power to decrease less severely than during the time period from 1971 to 1981. advances in systems science and applications (2013) vol.13 no.4 305 fig.4 historical data of djia (from yahoo finance, accessed on april 17, 2012) looking at the same graph created for the japanese economy, fig.2, we see a similar but different story. by breaking the graph up into more intervals, fig.5 we fig.5 the fluctuation in the purchasing power of japanese ye are able to form various linear patterns. from 1971 until around 1986 there was a general decline in the purchasing power of japanese yen. then once it came to a peak again in 1996 there again was an overall decline in the purchasing power. these are clearly not linear graphs ; but looking at the general motion of the purchasing power in the graph they are fairly linear in two segments, like the situation with the u.s. dollar. therefore, it is also reasonable for us to use the linear model in equ.(5) to explain the evolutionary trends in the purchasing power of japanese yen. under the assumption that the supply and purchasing power of money are inversely related when all other factors remain constant, we can see that the purchasing power is a function of the demand and supply with respect to time. by using the linear model in equ.(5) and by breaking up the graph in fig.5 into two 306 jeffrey forrest :currency wars and a possible self-defense (ii) : a plan of self protection fig.6 the money supply and the nikkei 225 stock market (okina, shirakawa and shiratsuka, 2001)[4] intervals, we see a similar relationship that was discussed with the supply and demand of money in the u.s. dollar. if both the supply and demand increase at the same rate, then the purchasing power will remain constant. if the supply increases with the demand increasing at a faster rate than before but still less than the increase in he supply, then a less severe decline in the purchasing power would be observed. however, if the demand increases at a rate greater than that of the supply of money, then this would lead to an increase in the purchasing power. furthermore, an increase in the supply matched with a decreasing rate in demand would result in a more severe decrease in the purchasing power. these explanations help explain varying linear slopes that are reflected in the purchasing power of money in the japanese economy. for example, notice that for japan 1986 was the first year of the asset pricing bubble. in the graph of the money supply and the nikkei 225 stock market,fig.6, for details,see (okina, shirakawaand shiratsuka, 2001)[4], during this time there was a small increase in the money supply while the nikkei 225 increased drastically. with the stock market increasing during this time, the demand for money also increased because people wanted to take advantage of the market rise. as we mentioned earlier, a relatively low increase in the money supply accompanied by a large increase in the demand for money would result in an increase in the purchasing power. this explanation corresponds with the depiction of the purchasing power of japanese yen until 1990 when it burst reflecting the sharp decline in the purchasing power as reflected in the graph and by our model. advances in systems science and applications (2013) vol.13 no.4 307 the sharp rise in the purchasing power that appeared at around the year of 1973, fig.3, could have been due to the oil crisis which caused a shift in japanese economy toward huge investments in the electronic industries. from (hutchison, 1986)[5], we can see that japan’s money supply was fairly constant in the year of 1973 but the demand of money from the oil crisis increased drastically. that made the purchasing power of the money to increase drastically during the early 1970s ; this conclusion is further validated by the discussions of(okina, shirakawa and shiratsuka, 2001)[4]. fig.7 is a graph that reflects the japanese inflation rate over the time period from april 1971 until april 2012, where we can notice the continuing patterns and similarities with those of the purchasing power. fig.7 japanese inflation rate from april 1971 to april 2012. the original source was accessed on april 4, 2012 as a matter of fact, the similarities between the graphic patterns of the purchasing power of money and the inflation rate, as described above, are also seen in the current movements of oil prices. on the ed show of the msnbcat 11 :00 p.m., barny frank referenced certain people who bought crude oil to drive up the prices only to sell it at a later time for a windfall of profits. the market system unconsciously allows the oil prices to riseand to consequently manipulate the gasoline prices, fig.8. that situation works in conjunction with the demand and supply of money. summarizing what is discussed above we conclude that it is theoretically reasonable for us to analyze the relationship between the purchasing power of money and the difference of the demand and supply of money by using the linear model in equ.(5), where the random variable accounts for all the unexpected factors that are not included in the model. by combining what is obtained in section3 with the linear model in equ.(5), we have the following relationship between categorized purchasing power [p1 p2 p3] t of money and categorized difference [d1 − s1 d2 − s2 d3 − s3] t of demand and 308 jeffrey forrest :currency wars and a possible self-defense (ii) : a plan of self protection supply of money : p1 p2 p3  = r3×3  d1(t)− s1(t) d2(t)− s2(t) d3(t)− s3(t)  +  ε1 ε2 ε3  (8) where r3 × 3 is a constant square matrix with real entries, and [ε1 ε2 ε3] t a random vector with a none zero mean. fig.8 the prices of crude and gasoline move in concert, the original source was accessed on april 17, 2012 4 separating economic categories using feedback component systems if we consider the mathematical expectations of the variables in equ.(8), we have p1 p2 p3  = a3×3  d1(t)− s1(t) d2(t)− s2(t) d3(t)− s3(t)  +  c1 c2 c3  (9) whereeεi = ci ̸= 0, i = 1, 2, 3. by substituting equ.(9) into equ.(4), we have r3×3ż = kz +qx (10) without loss of generality, we assume that r3×3 is invertible. that is, we assume in general the categorized purchasing power of money is completely determined by the categorized differences of demand and supply of money. then, equ.(10) can be rewritten as follows : ż = az +bx (11) advances in systems science and applications (2013) vol.13 no.4 309 wherea = r−1k and b = r−1q. to make the model in equ.(11) technically manageable, we assume without loss of generality that b is a 3×3 matrix, meaning that the monetary policies x1, x2, . . . , xn are accordingly categorized into three groups : x1=the set of all those monetary policies that deal with the population meeting the minimum need to maintain the basic living standard ; x2=the set of all those monetary policies that deal with the populationąŕs need for acquiring desired living conditions ; x3=the set of all those monetary policies that deal with the populationąŕs need for enjoying luxurious living conditions. without loss of generality, we will still use the same symbol x to represent the vector [x1 x2 x3] t of categorized monetary policies. similar to the concept of consumer price index (cpi), let us introduce an economic index vector y = [y1 y2 y3] t such that yi measures the state of the economic sector i, i = 1, 2, 3. then from equ.(11) we can establish the following model for the national economy of our concern : ż = az +bx y = cz +dx z(0) = 0 (12) where z is the 3 × 1 matrix [d1 − s1 d2 − s2 d3 − s3] t of the categorized difference of demand and supply of money, referred to as the state of the economic system, a,b,c, and d are respectively constant 3 × 3 matrices, such that d is non-singular (meaning that each introduction of monetary policies does have direct, either positive or negative, effect on the performance of the economy), and the input space x and output space y are following : x = y = {r : [0,+∞) → r3 : r is a piecewise continuous function} (13) where r stands for the set of all real numbers and r3 the nth dimensional euclidean space. what is described by equ.(12) is that the state of the national economy is representable through the use of the state variable z that helps the economy to absorb the positive and negative effects of the monetary policies x1, x2 and x3.then, both the internal mechanism z of the economy and the monetary policies x jointly have a direct effect on the overall performance y of the economy. the condition, as imposed on the input space x and the output space y means that monetary policies, which form the input space x, are introduced based on the effects of the previously implemented policies, while the overall performance, indexes of which constitutes the output space y , of the economy evolves from previous 310 jeffrey forrest :currency wars and a possible self-defense (ii) : a plan of self protection states mostly continuous. such assumptions historically speaking are not always true, while failures occur rarely in terms of frequencies. for instance, in modern china, the so-called cultural revolution occurred abruptly (macfarquhar and schoenhals, 2008)[6] ; and several times in the history of russia, the tsars had introduced rushed social reforms creating a torn country (huntington, 1996)[7]. geometrically, the systemic model in equ(12) of the national economy is depicted in fig.9. the monetary policy input x splits into two portions f1f2f3f4 and f5f6f7f8.the portion labeled f1f2f3f4 directly affects the performance y of the economy. the other portion, labeled f5f6f7f8,is fed into the initial state of the economy s, leading to the introduction of the feedback component systemsf ,which stands for the market reactions to the introduced monetary policies, and the formation of a feedback loop within the economy. this feedback loop in fact constitutes the main body of the economy, while the overall performance vector y of indexes is merely an artificially designed measure. what needs to be noted is that within each of our three economic sectors e1, e2 and e3,the market reactions, which constitute parts of the feedback component system sf ,to monetary policies in general are unique and economic sector specific. that is, the market reactions in one economic sector are different from those of another sector. the inner most loop f1f2f3f4 is the second stage of the economy after the introduction of new monetary policies. they by the joint effect of the marfig.9 the geometry of the systemic model of the national economy ket reactions the introduction of new monetary policies. they by the joint effect of the market reactions (sf ) and the non-reactionary aspects (f5f6f7f8) of the monetary policies, the final numerical readings y of the economy are produced. to see how monetary policy input x could have aspects, one is reactionary and the other non-reactionary, let us assume that x stands for such a monetary policy that allows the inflation to inch higher. therefore, the price of crude oil is expected to rise accordingly. now, the reactionary aspects of the policy x make advances in systems science and applications (2013) vol.13 no.4 311 the gradual and calculated increase in the price of crude oil more or less random. on the other hand, the expected outcome of the policy is the theoretical, non-reactionary aspect of the policy. when the actual outcome deviates from the expectation, the difference is caused by the market reactions to the policy. according to (lin, 1994)[8], the 3-dimensional system in equ.(12), meaning that both the input x and the output y are elements from r3, can be decoupled into three independent systems of the same kind with one-dimensional input and out. specifically, if we let s = {(x, y) ∈ x × y : ∃z ∈ zsuch that x, y, z satisfy equ.(12)} = the system of all the ordered pairs (x,y) satisfying equ.(12) where the state space z = {r : [0,+∞) → r3 : r is a piecewise continuous functi -on}, and for each i = 1, 2, 3, define a system si as follows : ż = az +bixi y = cizi +dixi z(0) = 0 (14) where bi is the ith column of b,ci the ith raw of c,di a non-zero constant, and the input space xiand the output space yi are given as follows : xi = yi = {r : [0,+∞) → r3 : r is a piecewise continuous function} in particular, the system s can be decoupled through feedback into the factor systems si, i = 1, 2, 3, as follows. let α = [ a 0 0 0 a 0 0 0 a ] β = [ b1 0 0 0 b2 0 0 0 b3 ] γ = [ c1 0 0 0 c2 0 0 0 c3 ] δ = [ d1 0 0 0 d2 0 0 0 d3 ] then, the cartesian product system sd = s1s2s3 is represented by the set of all ordered pairs (x, y) satisfying  ż = αz + βx y = γz + δx z(0) = 0 (15) because δ is non-singular, the inverse system sd −1 is obtained as follows : sd −1 = {(y, x) : (y, x) satisfies equ.(16)} 312 jeffrey forrest :currency wars and a possible self-defense (ii) : a plan of self protection where equ.(16) is given as follows : ż = (α− βδ−1γ)z + βδ−1y x = δ−1γz + δ−1y z(0) = 0 (16) the particular feedback component system sf : y → x used in this decoupling is given as follows : [ ż ż′ ] = [ a−bd−1c 0 0 α− βδ−1γ ] [ z z ′ ] + [ bd−1 βδ−1 ] y x = [−d−1c δ−1γ] [ z z ′ ] + (d−1 − δ−1)y[ z z ′ ] (0) = 0 (17) in terms of economics, what the concept of decoupling the 3-dimensional system s into component systems si,i = 1, 2, 3,as discussed above, implies is that when monetary policies are established individually and respectively for each of the three economic sectors e1, e2, and e3, although these economic-sector specific policies will most definitely have joint effects on the economy, there is at least one way to design a feedback component system sf so that the overall feedback system f (s, sf ), which represents the whole system as depicted in fig.1, can be controlled through adjusting individually each of the economic sectors e1, e2and e3. 5 a strategy for national defense the significance of the previous discussion is that we can now propose based on the sound analytical reasoning presented above a strategy for national defense against currency warfare in case that initial‘friendly’ foreign investments turn out to be an aggressive act of war by suddenly withdrawing all or significant amount of the investments. in particular, let us assume that at time period t, the total money supply of the specific nation of our concern is $1,000, $100 of which is from foreign investments. that is, the foreign investments amount to 10% of the total domestic money supply. with the additional money supply (due to the foreign investments) the speed of money circulation increases, signaled by increased spending and elevated levels of economic activities. if these foreign investments leave the nation suddenly, then it is reasonable to expect that more than 10% of the economic activities from around the nation will be more or less affected adversely due to the sudden exhaustion of the money flow, because accompanying the foreign investments there advances in systems science and applications (2013) vol.13 no.4 313 tends to be domestic investments attached too, making the total investments on the related economic activities more than 10% of the national economy. now, if before the foreign investments depart while leaving behind disastrous aftermath, the national government has been keeping the exchange rate (to this end not all nations from around the world are currently able to do this successfully) the same while gradually and strategically increased its money supply, say, to the level of $10,000, then in the entire monetary circulation around the nation the proportion of the foreign investments shrinks to about 1% from the original 10%. and if the money supply had been increased to the level of $1,000,000, then the proportion of the foreign investments would have shrunk from the original 10% to about 0.01%, which is nearly zero. so, if at this moment of time the foreign investments are suddenly withdrawn as an aggressive act of war, only around 1% or 0.01% of the consumption and economic activities of the nation will be affected adversely and the overall economic health of the nation will be relatively stable. on the other hand, along with the drastically increased money supply, all prices in the nation will most certainly go through the roof, placing a large portion of the national population in financial crises due to the run-away inflation. for instance, a certain commodity a, which is piece of living necessity, used to cost $1.00 a unit ; now it requires $10.00 or even $1,000 to purchase. such dramatic increase in prices will surely cause hardships for a good number of citizens of the nation. to this end, the national government needs to work on how to redistribute the additional money supply. in fact, as long as the distribution of the increased money supply does not cause social upheaval, then to this nation, nothing disastrous will really happen and the potentially damaging impacts of sudden departure of the foreign investments will be under control, too. however in reality, the distribution of the extra money supply could easily lead to major societal instabilities to the nation due to the increased unevenness in the economic scene : the rich become richer while the poor become poorer. the situation here is similar to that of the normal inflation, which has been employed to make the economic structure more uneven than before so that the yoyo structure of the economy spins with more strength. in other words, with additional money supply injected into the economy, the redistribution of the wealth that is represented by the increased money supply is surely uneven, where some people receive more than their share, some simply keep pace with the decrease of the purchasing power of their income, while others fall behind or further behind with their financial status. so, if our suggested measure is adopted to counter the damaging effects of sudden departure of foreign investments, the nation needs to develop a practical plan to distribute the extra money supply in order to : (1). keep societal peace and national stability so that a normal and operational economy can be maintained. 314 jeffrey forrest :currency wars and a possible self-defense (ii) : a plan of self protection (2). increase the nation’s economic prosperity by taking advantage of the foreign investments even though they leave sooner or later either slowly or suddenly. specifically, as suggested by the systemic model in the previous section, to help protect the innocent citizens of the nation from suffering from the potential economic turmoil, the national government could purposely divide the economy into three sectors e1, e2, and e3 as described earlier to meet the following goals. in sector e1, which consists of living necessities, the sector performance, such as the sector specific cpi, evolves as normally as possible ; in sector e2,which consists of such goods, services, and relevant productions that are used by citizens to acquire desired living conditions, the sector performance index, say the particular cpi of e2, could outpace that of sector e1 by a large amount ; and critically, the national government needs to manage to trap most of the additional money supply in the economic sector e3, which consists of such goods, services, and relevant productions that are used by the citizens for their enjoyment of luxurious living. the previous discussion on the systemic model of the national economy indicates that by managing the market reactions correctly, that is the design of the feedback component system sf , these three economic sectors can be well separated from each other. when the economic sector e1 evolves normally as expected based on the history of the nation, the citizens would not need to worry about their basic living and survival. that naturally leads to the desired societal stability and peace. although the prices in the economic sector e2 increase drastically when compared to those in e1, that in general should not affect the mood or the happiness of the population much, because desired living conditions change from time to time and vary from one family to another. in other words, isolated, temporary personally desires,which are not yet satisfied and which are generally inconsistent with each other and even contradictory to each other, could not amalgamate into a political force to cause turmoil. now, the key is how to keep most of the additional money supply in the economic sector e3.to address this problem, let us pay attention to our earlier assumption that a large amount of foreign investments entered the nation. that means commercial products and services that are new to the people in the specific nation are expected to and will become available soon. because these products and services are brand new to the region, high prices can of course be charged for these commercial goods and services. the situation is similar to the scenarios with developed nations, where the focus has been placed on innovations, design and production of unprecedented products and services. it is these never-seen products and services that generate the most profits. at this junction, let us address the problem of how to classify a product or a service into one of the three economic sectors e1, e2, and e3. first of all, goods and services that should be classified into e1 are clear.they represent such commodiadvances in systems science and applications (2013) vol.13 no.4 315 ties that directly relate to the survival of any human being, such as foods, utilities, and shelters of the basic quality. now, if a family lives in a rented two-bedroom apartment and wants to own a house, then houses to that particular family will be in the economic sector e2.f the people in a community go to work, shopping, and entertainment mostly on foot, then individually owned transportation tools, such as bicycles, motorcycles, cars, private jets, etc., will belong to the economic sector e3.that is, as time evolves forward, what belongs to the economic sector e3 gradually lowers itself into the economic sector e2, and what used to be in e2 also gradually moves to e1.that is, the classification of one particular product or service is a function of time and space. just by looking around shopping centers, one can see vividly that such classification of commercial goods and services have been successfully done by merchants throughout the world without much trouble. now, let us consider the scenario that when the nation purposefully and strategically increases its money supply in order to prevent disastrous aftermaths that could be potentially left behind by sudden withdrawals of foreign investments, additional foreign investments could continue to pour in. in this case, if the nation could not pick up its corresponding speed of economic development, then it will be taken over by the run-way high rate of inflation. in this case, if the foreign investments depart strategically, then the nation will be in real major political, societal, and economic crises. to prevent such disastrous consequences, the continued inflow of foreign money has to been managed so that they cannot depart quickly. another practical scenario that is different of what has been discussed above is that at a high speed develops the nation itself into a super economic power such that there is no longer such a single amount of foreign investments that can amount to an influencing percentage in the nation’s domestic consumption and/or economic activities. in this case the nation will be safe in the face of any potentially sudden withdrawal of foreign investments. 6 a few final words similar to how many different methods are there to invest money in life, there should be that many different ways to launch a currency war. what is presented in this work of two parts simply considers only one possible way to launch a currency war and one possible way to protect oneself against the disastrous consequences of such an attack. in other words, more scientific efforts should be devoted to the studies of financial attacks (or wars) along the line as outlined in this work in the spirit of how conventional warfare is conducted, analyzed, and planned strategically. the purpose of doing so is not to launch financial attacks on the innocent people who simply work hard to make a living ; instead, one should always be prepared for the worst that might be imposed upon him by those who always look for 316 jeffrey forrest :currency wars and a possible self-defense (ii) : a plan of self protection ways to take advantage of others. in this regard, we hope that our presentation in this work of two parts will play the role of a brick that has been thrown out there to attract beautiful and practically meaningful gemstones. acknowledgements we like to use this opportunity to express our appreciation to all the participants of our weekly meetings. our specific thanks go to professor yirong ying of shanghai university and professor xiangdong li of jiangsu normal university of technology. their insightful comments and input helped to enrich this work greatly. références [1] forrest j., hopkins z. and liu s. f.(2013), “currency wars and apossible selfdefense (i) : how currency wars take place”, advances in systems science and application, vol.13, no.3, pp.197-216. [2] allen r., and goldsmith e. (1972), “towards the stable society : strategy for change.the ecologist archive”, the ecologist, located at http ://www. theecolo-gist.info/page32.html, accessed on may 22, 2012. [3] wang r. x., and hu g. h. (2005), international finance, wuhan, hubei : press of wuhan university of science and technology. [4] okina k., shirakawa m., and shiratsuka s. (2001), “the asset price bubble and monetary policy : japan’s experience in the late 1980s and the lessons”, monetary and economic studies (special edition), february, pp.395-450. [5] hutchison m. m. (1986), “japan’s ‘money focused’ monetary policy”, economic review(federal reserve bank of san francisco), summer, no.3, pp.3346. [6] macfarquhar r., and schoenhals m. (2008), mao’s last revolution. cambridge, ma : harvard university press. [7] huntington s. p. (1996), the clash of civilizations and the remaking of world order, new york : simon & schuster. [8] lin y. (1994), “feedback transformation and its application”, journal of systems engineering, vol.1, pp.32-38. corresponding author jeffrey forrest can be contacted at : e-mail : jeffrey.forrest@sru.edu. advances in systems science and application (2015) vol.15 no.4 299-315 variable precision rough set model based on set pair homeopathic similarity relationship yong liu1 and yi lin2 1school of business, jiangnan university, wuxi 214122, china. 2school of business,slippery rock university, slippery rock, u.s.a. abstract due to the complexity and uncertainty of the objective world, as well as the limitation of human understanding, it has been difficult for the classical rough set method to deal with incomplete information system with noise data, ambiguity and other informational uncertainties. in view of this, the thought and method of the set pair theory and variable precision rough set are used in this paper to construct a novel variable precision rough set model. after scrutinizing the advantages and limitations of the models previously established based on the set pair similarity relationship, we define the set pair homeopathic similarity relationship based on threshold α connection degree. then we employ this new concept to substitute the equivalence relationship of the variable precision rough set so that a novel variable precision rough set model based on the set pair homeopathic similarity relationship is established. upon developing the properties of this model, we construct an example to show the feasibility and effectiveness of our novel model. keywords incomplete information system; set pair analysis; variable precision rough set; equivalence relationship. 1 introduction as an useful mathematical tool to deal with information that involves inaccuracy, uncertainty and fuzziness, rough set theory was proposed by pawlak [1]. this theory has been widely applied in such areas as the knowledge discovery, data mining, decision analysis, pattern recognition, and other fields [2–4]. the classical rough set theory is developed to deal with complete information systems with discrete attribute, where all the attribute values are known, thus the upper and lower approximation sets of the object sets based on equivalence relations, which satisfy the reflexive, symmetric and transitive properties, are defined. however, due to the complexity of the real world, uncertainty and limitation of human knowledge, not all attribute values acquired are known. so it is difficult for the traditional rough set theory to deal with such information systems and to materialize the desired knowledge discovery and extraction of relevant rules in order to make rough set theory effectively deal with systems of incomplete information, many scholars have been involved in the relevant research, leading to some important results. in summary, there exist two main classes of relat300 yong liu and yi lin:variable precision rough set model based on set pair homeopathic ... ed works. on one hand, each incomplete information system is pre-processed in order to make up the missing data; then the given system is investigated as a system with complete information. on the other hand, the theoretical equivalence relation among the objects of the system of incomplete information is directly weakened so that some new kind of relation, such as tolerance relationship [5, 6], similarity relationship [7], dominance relationship [8], etc., is introduced. for the former class of research, due to the information loss caused by pre-processing of the information system, the related works will not be discussed in this paper. for the latter class of works, when all the unknown attribute values of the system of incomplete information are of the omission type, kryszkiewicz builds the tolerance relationship, which satisfies the properties of reflexivity and symmetry.[9] when the unknown attributes values of the incomplete information are of the lost type, stefanowski and tsoukias establish the similarity relationship, which satisfies the properties of reflexivity and transitivity.[10] in view of the characteristics of incomplete information systems, wang puts forward the limited tolerance relationship, which satisfies the properties of reflexivity and symmetry.[11] corresponding to the systems of incomplete information that contain preference information,greco, matarazzo and slowiski[12, 13],shao and zhang [14] respectively put forward rough set model based on tolerance dominance relationship, which satisfies the property of reflexivity. for systems with incomplete order information, yang, yang, chen, et al [15],xie, song, chen, et al [16] construct a rough set model based on the similar dominance relationship, which satisfies the property of reflexivity. they then analyze the shortcomings of the tolerance dominance relationship and the similar dominance relationship, leading to the establishment of a rough set model based on the limited tolerance dominance relationship, which satisfies the property of reflexivity. however, when these rough set models are used to classify the domain with many condition attributes, the definition of rough sets still seems too loose. in order to deal with this problem, the set pair analysis method is introduced into the study of incomplete information systems [17], and additional rough set models, which are based on set pair analysis, are constructed [18–26]. by carefully analyzing these models, it can be seen that most of them are based on a local aspect and consider the classification effect of some specific objects without an overall classification performance of all objects by taking into account of unified approach, relevant uncertainty, oppositions, etc. it has been difficult for each of these established models to deal with incomplete information systems with preference information. in view of the shortcomings of the existing literature on incomplete information systems and the fact that there exist noise data, fuzzy and grey information in the incomplete information of systems, in this paper we will employ the set pair analysis method, combined with the previous studies, to construct a novel advances in systems science and application (2015) vol.15 no.4 301 variable precision rough set model based on the set pair homeopathic similarity relationship. the organization of this paper is as follows. section 2 compares the advantages and disadvantages of the currently available set pair similarity relationships. section 3 introduces the concept of our set pair homeopathic similarity relationship. section 4 establishes the variable precision rough set model based on set pair homeopathic similarity relationship. a case study is given in section 5. then this presentation is concluded in section 6. 2 the set pair analysis as a useful tool of system analysis to characterize and to study a variety of certainty and uncertainty and their laws of transformation in systems, the methodology of set pair analysis is proposed by chinese scholar keqin zhao in 1989. its core idea is to holistically analyze and deal with the certainty relationship and the uncertainty relationship of the objective matter of concern. this methodology has been widely applied to the study of such areas as mathematics, physics, systems science, management science, decision-making science, theory of predictions, computer science, social science, artificial intelligence, and many others. it has shown theoretical significance and potential for success for applications[17]. definition 1[17].assume that two given setsa andb from a set pairh(a,b),and that under the particular background (w ) of the problem of concern, h have n attributes, where s attributes are shared by both of a and b ,p attributes are about the opposition of a and b,and f attributes are neither shared by a and b nor about the opposition of a and b.then the ratio s/n is called the identity degree of a and b under the background (w ) ; f/n the discrepancy degree of a and b under the background (w ) ;and p/n the contrary degree of a and b under the background (w ) ; and the connection degree of and under the background is defined as follows: µw (a,b) = s n + f n i+ p n j (1) the connection degree reflects the relationship between a and b under the background (w ),denoted as = a + bi + cj , 0 ≤ a, b, c ≤ 1 and a + b + c = 1 , where the parameters a, b, c stand for some relationship between the sets a and b under the background (w ) . definition 2[17].for the given connection degree = a + bi + cj ,the rate of the identity degree a and the contrary degree c under some specific background (w ) is referred to as the set pair situation, denoted as follows: s(h) = a c (2) 302 yong liu and yi lin:variable precision rough set model based on set pair homeopathic ... if c = 0 ,then s(h) → ∞ ,which is referred to as the infinite situation with respect to the specific background (w ) . 3 set pair homeopathic similarity relationship 3.1 incomplete information system definition 3. suppose that s = (u,a, v, f) is an information system, where u = {u1, u2, ..., un} is a finite and nonempty set, known as the universe, a = c ∪ d a finite and nonempty attribute set, c = {a1, a2, ..., am} the condition attribute set, d = {d1, d2, ...., dp} the decision attribute set, and v = ∪ vij the value range of u on a, where vij is the attribute value of ui on aj ∈ a, f : u × a → v is an information function. for ∀aj ∈ a,∀xi, xk ∈ u, f(xi, aj) ∈ vij ,∀b ∈ b,b ⊆ a ,the equivalence relationship of xi, xk with respect to the attritube b is defined as follows: r = {(xi, xk) ∈ u × u : f(xi, b) = f(xk, b)} (3) definition 4. suppose that s = (u,a, v, f) is an information system. if there is at least one object ui and one attribute aj ∈ a such that vij takes the null value, denoted by *, then information system s = (u,a,a, f) is called incomplete information system, denoted as s∗ = (u,a, v, f); otherwise, it is called a complete information system. 3.2 comparison of different kinds of set pair similarity relationships due to the incompleteness of information of an incomplete information system, it is difficult for the traditional method of rough sets based on equivalence relationship to solve problems involving such systems. in view of this fact, some scholars suggest to use the methodology of the set pair analysis to define the similarity relationship to investigate incomplete information systems. in short, there exist the following four categories of similarity relationships [18–26]. definition 5.in the incomplete information system s∗ = (u,a, v, f), the general similarity relationship is defined as follows: sim(x, y) = {(x, y) ∈ u × u | ∀a ∈ a, a(x) ≥ a(y)or a(x) = ∗or a(y) = ∗} (4) for ∀a ∈ a,∀x, y ∈ u,b ⊆ a, |b| = n, teh connection degree based on the incomplete information system is denoted as µb(x, y) = s n + f n i+ p n j where s = |{a ∈ b | a(x) = a(y) and a(x) ̸= ∗ and a(y) ̸= ∗}| stands for the number of the attributes that are clear and equal for x and y with respect to the attribute set b; while p = |{a ∈ b | a(x) ̸= a(y) and a(x) ̸= ∗ and a(y) ̸= ∗}| stands for the number of the attributes that are clear but not equal for x and y advances in systems science and application (2015) vol.15 no.4 303 with respect to the attributes set b; f = |{a ∈ b | a(x) ̸= ∗ or a(y) ̸= ∗}| stands for the number of the attributes that are not clear and equal for x and y with respect to the attributes set b. write µb(x, y) = s n + f n i+ p nj = a+ bi+ cj where a = s n , b = f n , and c = p n (1)set pair similarity relationship and similarity classes based on the connection degree definition 6.[18]. for the incomplete information system s∗ = (u,a, v, f), ∀a ∈ a,∀x, y ∈ u,b ⊆ a,the set pair similarity relationship of x and y is defined as follows: smh(b) = {(x, y) ∈ u × u | ub(x, y) = a+ bi+ cj} (5) definition 7.[18].for the incomplete information system s∗ = (u,a, v, f), ∀a ∈ a,∀x, y ∈ u,b ⊆ a,the set pair similarity classes of x and y with respect to the attribute set b are defined as follows: smh(b) = {y ∈ u | ub(x, y) = a+ bi+ cj} (6) obviously, this set pair similarity relationship satisfies the property of symmetry, but does not satisfy the properties of reflexivity and transitivity. in particular, in practical applications, it can be found that when the information system of concern contains a large amount of missing values, the result of model analysis tends to be not satisfactory, and the operational properties of the lower and upper approximations of rough sets are imperfect. (2) set pair similarity relationship and similarity classes based on a threshold value α definition 8. [19].for the incomplete information system s∗ = (u,a, v, f), ∀a ∈ a,∀x, y ∈ u,b ⊆ a, 0 ≤ α ≤ 1,the set pair similarity relationship of x and y based on the threshold value α is defined as follows: smhα(b) = {(x, y) ∈ u × u | ub(x, y) = a+ bi+ cj, a+ b ≥ α} (7) definition 9. in the incomplete information system s∗ = (u,a, v, f),for ∀a ∈ a,∀x, y ∈ u,b ⊆ a, 0 ≤ α ≤ 1,the set pair similarity classes of x and y with respect to the attribuye set b based on the threshold value α are defined as follows: smhα b(b) = {y ∈ u | ub(x, y) = a+ bi+ cj, a+ b ≥ α} (8) evidently, this set pair similarity relationship satisfies the property of symmetry, but not that of transitivity. the rough set model based on the set pair similarity relationship only considers and limits the known identity degree and the proportion of the uncertain 304 yong liu and yi lin:variable precision rough set model based on set pair homeopathic ... attributes among the objects. it ignores the influence of the discrepancy degree on the similarity, which will affect the performance of classification. in addition, identifying the influence of the identity degree and that of the discrepancy degree on the similarity will also to a certain extent increase the classification error. for the rough set based on the set pair similarity relationship, there exist two types of errors. on one hand, it ignores the difference between uncertain attributes and plays too much emphasis on their similarity. on the other hand, it ignores the interfering effect of the discrepancy degree on the decision-making system in which requirements of relatively higher accuracy are imposed, leading to major differences in the decision-making rules. (3) set pair similarity relationship and similarity classes based on identity and contrary degrees (α, λ) definition 10. [22].in the incomplete information system s∗ = (u,a, v, f), ∀a ∈ a,∀x, y ∈ u,b ⊆ a, 0.5 ≤ α ≤ 1, 0 ≤ λ ≤ 0.5,the set pair similarity relationship of x and y based on identity degree α and contrary degree λ is defined as follows: smhα(b) = {(x, y) ∈ u × u | ub(x, y) = a+ bi+ cj, a ≥ α, c ≤ λ} (9) obviously, this set pair similarity relationship satisfies the properties of reflexivity and symmetry, but not that of transitivity. definition 11.in the incomplete information system s∗ = (u,a, v, f), ∀a ∈ a,∀x, y ∈ u,b ⊆ a, 0.5 ≤ α ≤ 1, 0 ≤ λ ≤ 0.5,the set pair similarity relationship of x and y with respect to the attribute set b based on identity degree α and contrary degree λ are defined as follows: smhα,λ b = {y ∈ u | ub(x, y) = a+bi+cj, a ≥ α, c ≤ λ, 0.5 ≤ α ≤ 1, 0 ≤ λ ≤ 0.5} (10) (4) set pair similarity relationship and similarity classes based on threshold values (α, λ) definition 12[23].in the incomplete information system s∗ = (u,a, v, f), ∀a ∈ a,∀x, y ∈ u,b ⊆ a, 0 ≤ α ≤ 1, 0 ≤ λ ≤ 1,the set pair similarity relationship of x and y based on the threshold values (α, λ) is defined as follows: smhα(b) = {(x, y) ∈ u × u | ub(x, y) = a+ bi+ cj, a+ λb− c ≥ α} (11) definition 13[23].in the incomplete information system s∗ = (u,a, v, f), for∀a ∈ a,∀x, y ∈ u,b ⊆ a, 0 ≤ α ≤ 1, 0 ≤ λ ≤ 1,the set pair similarity classes of x and y with respect to the attribute set b based on the threshold values (α, λ) are defined as follows: smhα b(x) = {y ∈ u | ub(x, y) = a+ bi+ cj, a+ bλ− c ≥ α} (12) advances in systems science and application (2015) vol.15 no.4 305 the concept of set pair similarity classes reflects the object set that the similarity degree of x with respect to the attribute set b is greater or equal to(α, λ) . in view of the existing two kinds of errors the above set pair similarity relationship suffers from, tao, dai and zhang put forward the rough set model based on the threshold value (α, λ) .[23] here the threshold value α is introduced into the model by separating the contributions of both uncertain and certainty attributes to the relationship of similarity and by paying much more attention to the disturbance of the contrary degree. the relationship of similarities between things is considered and characterized in three aspects: the identity degree, discrepancy degree and contrary degree. due to the fact that the discrepancy degree possesses both positive and negative incentives on the connection degree, while the threshold value satisfies 0 ≤ λ ≤ 1 , the model only considers the positive incentives of the uncertain attributes on the similarity while neglecting their punitive effects. at the same time, because of the introduction of the threshold value λ , the subjectivity of the model is increased. additionally, the model does not satisfy the property of reflexive, which affects the quality of the operation of the model. 3.3 set pair homeopathic similarity relationship and similarity classes in view of the existing problems in connection degrees, we will use the homeopathic value method to make improvement on the effect of connection degrees. the idea of the homeopathic value method is that along with the identity degree, the value is determined while keeping the same potential class status. for example, for the connection degree µ = ai+ bj+ ck , the homeopathic value method is used to acquire the novel connection degree µ = a(1+ b)i+ b2j+ c(1+ b)k. from comparing the two connection degrees, we can see that by using the homeopathic value method the uncertainty of the original connection degree µ = ai+bj+ck is divided into three parts in the novel connection degree ab. one part is classified into the identity degree b2 ; one part into the discrepancy degree cb; and the third part into the contrary degree . at the same time, the set pair potentials of these two connection degrees are the same as a c = a(1+b) c(1+b) . for the connection degree a+ λb− c based on the fourth set pair similarity relationship, the homeopathic value method can be used to deal with the threshold λ so that a novel connection degree a + (a − c)b − c + b2λ is obtained. because the impact from the part of the uncertainty that is classified into the discrepancy degree b2 on the relationship of similarity is very little, b2 can be ignored. because of this reason, we define the set pair homeopathic similarity relationship as follows. definition 14.suppose that s∗ = (u,a, v, f)is an incomplete information system.for ∀a ∈ a,∀x ∈ u,b ⊆ a, 0 ≤ α ≤ 1, the set pair homeopathic similarity relationship of x and y based on the threshold value α is defined as follows: 306 yong liu and yi lin:variable precision rough set model based on set pair homeopathic ... smhα(b) = {(x, y) ∈ u ×u | ub(x, y) = a+ bi+ cj, a+(a− c)b− c ≥ α} (13) the set pair homeopathic similarity relationship satisfies the properties of reflexivity and symmetry, but not that of transitivity. definition 15.in the incomplete information system s∗ = (u,a, v, f).for ∀a ∈ a,∀x, y ∈ u,b ⊆ a, 0 ≤ α ≤ 1, the set pair homeopathic similarity classes of x and y with respect to the attribute set b based on the threshold value α are defined as follows: smhα b(x) = {y ∈ u | ub(x, y) = a+ bi+ cj, a+ (a− c)b− c ≥ α} (14) the set pair homeopathic similarity classes reflect such object set that the similarity degree of with respect to the attribute set b is greater or equal to α. 4 variable precision rough set based on set pair homeopathic similar relationship 4.1 construction of the model in any real-life information system, there always exist a variety of noise data, which is difficult for the traditional rough set method to deal with. in order to deal with such problem, ziarko [27–29] put forward the variable precision rough set model by introducing the threshold value β and approximation space to reflect this kind of restrictions. based on the thought and methodology of variable precision rough set model, we introduce the set pair homeopathic similarity relationship to substitute the equivalence relationship of rough set so that a novel variable precision rough set model can be constructed. accordingly, the lower and upper approximations of the variable precision rough set model based on the set pair homeopathic similarity relationship can be defined as follows. definition 16.suppose that s∗ = (u,a, v, f) is an incomplete information sysytem.for x ⊆ u,b ⊆ c, 0 ≤ α ≤ 10.5 ≤ β ≤ 1, the β-lower and β-upper approximations of xbased on the set pair homeopathic similarity relationship are defined respectively by: aprα,βb (x) = ∪{x ∈ u | |smhα(x) ∩x| |smhα(x)| ≥ β} (15) aprα,β b (x) = ∪{x ∈ u | |smhα(x) ∩x| |smhα(x)| ≥ 1− β} (16) the β-lower approximation aprα,β b (x) of the set x based on the set pair homeopathic similarity relationship can also be called as the positive region of the variable precision rough set model, denoted as posα,βb (x).in other words, for given confidence threshold values α and β,aprα,β b (x) is the set in which the universe advances in systems science and application (2015) vol.15 no.4 307 u can be classified definitely into all the element sets of the set x based on the set pair homeopathic similarity relationship. the β-upper approximation aprα,βb (x) of the set x reflects that for a given threshold value β ,the universe u can probably be classified into all the element sets of x definition 17.suppose that s∗ = (u,a, v, f) is an incomplete information sysytem.for 0 ≤ α ≤ 1, 0.5 ≤ β ≤ 1,the β-negative region and β-boundary of a subset x based on the set pair homeopathic similarity relationship are defined as follows: negα,βb (x) = ∪{x ∈ u : |smhα(x) ∩x| |smhα(x)| ≤ 1− β} (17) bndα,βb (x) = ∪{x ∈ u : 1− β < |smhα(x) ∩x| |smhα(x)| < β} (18) the β-negative region negα,βb (x) of x based on the set pair homeopathic similarity relationship reflects that for a given confidence threshold β , the universe certainly cannot be classified into all the elements of the collection set x. the boundary bndα,βb (x) of x based on the set pair homeopathic similarity relationship reflects that for the given confidence threshold β , the universe u certainly cannot be classified into all the element sets of either the set x or the set -x . definition 18.suppose that s∗ = (u,a, v, f) is an incomplete information sysytem.for 0 ≤ α ≤ 1, 0.5 ≤ β ≤ 1,the β classification quality of x based on the set pair homeopathic similarity relationship is defined as follows: rα,β(b,d) = ∣∣∣∪{x ∈ u : |smhα(x)∩r(x)| |smhα(x)| ≥ β} ∣∣∣ |u | (19) the quantity γα,β(b,d) measures the proportion that the possible correct classification knowledge is in the existing knowledge for the given values α, β in the universe. 4.2 properties of the model theorem 1.for the given incomplete information system s∗ = (u,a, v, f), let x ⊆ u, y ⊆ u,∀b ⊆ c, 0 ≤ α ≤ 1, 0.5 < β ≤ 1, the following hold true: (1)aprα,βb (x ∪ y ) ⊇ aprα,βb (x) ∪ aprα,βb (y ); (2)aprα,β b (x ∩ y ) ⊆ aprα,β b (x) ∩ aprα,β b (y ); (3)aprα,β b (x ∪ y ) ⊇ aprα,β b (x) ∪ aprα,β b (y );and (4)aprα,βb (x ∩ y ) ⊆ aprα,βb (x) ∩ aprα,βb (y ); proof.(1)for x ⊆ u, y ⊆ u, 0 ≤ α ≤ 1, 0.5 < β ≤ 1, we have |smhα(x) ∩ (x ∪ y )| |smhα(x)| ≥ |smhα(x) ∩x| |smhα(x)| 308 yong liu and yi lin:variable precision rough set model based on set pair homeopathic ... and |smhα(x) ∩ (x ∪ y )| |smhα(x)| ≥ |smhα(x) ∩ y | |smhα(x)| therefore, we obtain aprα,βb (x ∪ y ) ⊇ aprα,βb (x) ∪ aprα,βb (y ); (2)for x ⊆ u, y ⊆ u, 0 ≤ α ≤ 1, 0.5 < β ≤ 1, we have |smhα(x) ∩ (x ∩ y )| |smhα(x)| ≤ |smhα(x) ∩x| |smhα(x)| and |smhα(x) ∩ (x ∩ y )| |smhα(x)| ≤ |smhα(x) ∩ y | |smhα(x)| therefore, we obtain aprα,β b (x ∩ y ) ⊆ aprα,β b (x) ∩ aprα,β b (y ); similar to the argument of statement (1), statement (3) can be proved; and statement (4) can be proved in the same way as that of statement (2). qed theorem 2.for the given incomplete information system s∗ = (u,a, v, f), let x ⊆ y ⊆ u,∀b ⊆ c, 0 ≤ α ≤ 1, 0.5 < β ≤ 1, the following hold true: (1)aprα,β b (x) ⊆ aprα,β b (y );and (2)aprα,βb (x) ⊆ aprα,βb (y ). proof.(1)due to x ⊆ y ,it follows that x ∩y = x and x ∪y = y . so we obtain aprα,β b (x) = aprα,β b (x ∩ y ) = aprα,β b (x) ∩ aprα,β b (y ) therefore,we have aprα,β b (x) ⊆ aprα,β b (y ). (2)from x ⊆ y ,it follows that x ∪ y = y so that we have aprα,βb (y ) = aprα,βb (x ∪ y ) = aprα,βb (x) ∪ aprα,βb (y ) therefore,we obtain aprα,βb (x) ⊆ aprα,βb (y ).qed theorem 3.for the given incomplete information system s∗ = (u,a, v, f), let x ⊆ u,b1 ⊆ b2 ⊆ c, 0 ≤ α ≤ 1, 0.5 < β ≤ 1, the following hold true: (1)aprα,β b1 (x) ⊆ aprα,β b2 (x);and (2)aprα,βb1 (x) ⊇ aprα,βb2 (x). proof.(1)for ∀x ∈ aprα,β b1 (x),from b1 ⊆ b2 ⊆ a it follows that [x]α,βb1 ⊆ x such that [x]α,βb2 ⊆ [x]α,βb1 .so we obtain [x]α,βb1 ⊆ x such that x ∈ aprα,β b2 (x) therefore,we have aprα,β b1 (x) ⊆ aprα,β b2 (x) (2)for ∀y ∈ aprα,βb2 (x),we have [y]α,βb2 ∩x ̸= ∅,satisfying that [y]α,βb1 ∩x ̸= ∅.so, we get y ∈ aprα,βb1 (x).therefore,aprα,βb1 ⊇ aprα,βb2 (x) follows.qed advances in systems science and application (2015) vol.15 no.4 309 theorem 4.for the given incomplete information system s∗ = (u,a, v, f), let x ⊆ u,b ⊆ c, 0 ≤ α1 ≤ α2 ≤ 1, and 0.5 < β1 ≤ β2 ≤ 1. then following hold true: (1)aprα,β1 b (x) ⊆ aprα,β2 b (x); (2)aprα,β1 b (x) ⊇ aprα,β2 b (x). (3)aprα1,β b (x) ⊆ aprα2,β b (x); and (4)aprα1,β b (x) ⊇ aprα2,β b (x). proof.(1)for ∀x ∈ aprα,β1 b (x),from 0.5 < β1 ≤ β2 ≤ 2 it follows that [x]α,β1 b ⊆ x. so, we have [x]α,β2 b ⊆ [x]α,β1 b from which we get x ∈ aprα,β2 b (x). and consequently we obtain aprα,β1 b (x) ⊆ aprα,β2 b (x). (2)for ∀y ∈ aprα,β2 b (x), [y]α,β2 p ∩x ̸= ∅ holds true so that [y]α,β1 b ∩x ̸= ∅ follows. hence, we get y ∈ aprα,β1 b (x), and consequently aprα,β1 b (x) ⊇ aprα,β2 b (x) similar to the proof of statement (1), statement (3) follows; and statement (4) can be shown in the same way as that of statement (2). qed the results in theorem 4 indicate that β value is negatively related to the classification quality. in particular, when β value increases, the classification quality decreases and the positive and negative regions of the set x based on the variable precision rough set model, as proposed in this paper, will become narrower, while the boundary region of the set x will become wider. it means that only a small number of objects are classified. on the other hand, as the value of β decreases, the classification precision increases, and the positive and negative regions of the set x based on the variable precision rough set will widen, while the boundary region narrows. that means that most of the objects are classified, but possibly misclassified. 4.3 attribute reduction algorithm according to the similarity degree of the positive region, as constructed based on the proposed variable precision rough set model, the following attribute reduction algorithm is established for solving the reduction problem. the special steps are given as follows. input: the incomplete information system s∗ = (u,a, v, f) , the set pair homeopathic threshold value α and confidence threshold value β . output: a reductionb of the incomplete information system s∗ = (u,a, v, f). step 1: suppose b = c; step 2: compute the system classification quality γα,β(c,d); step 3: for any condition attribute a ∈ b, compute γα,β(b − a,d) and the dependence degree sigα,β(a) of the attribute a; step 4: if γα,β(b − a,d) ≥ γα,β(c,d) and sigγ(a) is the least, then b = b − {a} . if for some a ∈ b, γα,β(b − a,d) ≤ γα,β(c,d) and all sigγ(a) are equal, then the attribute a with the missing values for all objects is selected and 310 yong liu and yi lin:variable precision rough set model based on set pair homeopathic ... let b = b − {a} ; step 5: if for some a ∈ b, γα,β(b − a,d) ≤ γα,β(c,d) and all sigγ(a) are equal, then go to step 6; otherwise go to step 3; and step6:output a reduction of the initial incomplete information system. 5 case analysis suppose that the decision-making information system s = (u, v,a, f) is about the initial accessories of the military aircraft engine system, where u = {u1, u2, ..., u10} is the set of initial accessories, c = {a1, a2, a3, a4, a5} the set of condition attributes, where ,a1, a2, a3, a4, and a5 stand respectively for importance, consumable, convertibility, procurability, and economical efficiency, and d = {d}. for va1 = {2, 1, 0} , 2 represents key, 1 represents more important, 0 represents general; for va2 = {2, 1, 0} , 2 represents consumable, 1 represents vulnerable, 0 represents general; for va3 = {1, 0} , 1 represents substitution, 0 represents cannot be replaced;, for va4 = {2, 1, 0} , 2 represents longer for the order cycle, 1 represents general for the order cycle, 0 represents immediate purchase; for va5 = {2, 1, 0} , 2 represents excellent, 1 represents general, 0 represents bad; for vd = {1, 0} , 1 represents yes, 0 represents no. due to the fact that there exist missing attribute values, the decision-making information system s = (u,a, v, f) is an incomplete decision-making information system s∗ = (u,a, v, f), the details of shown are given in table 1. (1)when α = 0.2, based on the set pair homeopathic similarity relationship, table 1 the decision-making information system for the initial accessories selection of the military aircraft engine system order# a1 a2 a3 a4 a5 d 1 2 1 0 * 1 1 2 1 0 1 2 2 1 3 2 * 0 1 1 1 4 0 2 * 2 0 0 5 * 1 0 1 1 0 6 2 * 1 0 2 1 7 0 2 1 1 * 0 8 0 1 0 0 1 0 9 2 0 1 2 2 1 10 2 * 1 1 2 0 the universe on the condition attribute set c can be divided into: u/c = {x0.2 1 , x0.2 2 , x0.2 3 , x0.2 4 } advances in systems science and application (2015) vol.15 no.4 311 where x0.2 1 = {u1, u3, u5, u8}, x0.2 2 = {u2, u6, u9, u10} and x0.2 3 = {u4, u7}. according to the decision attribute set d , the universe can be divided into: u/d = {y1, y2} where y1 = {u1, u2, u3, u6, u9} and y2 = {u4, u5, u7, u8, u10}. when β = 1 , the lower and upper approximations of the decision-making classes y1 and y2 can be computed as follows: aprα,β b (y1) = ∅, aprα,βb (y1) = {u1, u2, u3, u5, u6, u8, u9, u10}, aprα,β b (y2) = {u4, u7}, aprα,βb (y2) = {u1, u2, u3, u4, u5, u6, u7, u8, u9, u10}, and γα,β(b,d) = |{u4,u7}| |{u1,u2,u3,u4,u5,u6,u7,u8,u9,u10}| = 2 10 = 0.2 when β = 0.7 , the lower and upper approximations of the decision-making classes y1 and y2 can be produced as follows: aprα,β b (y1) = {u2, u6, u9, u10}, aprα,βb (y1) = {u1, u2, u3, u5, u6, u8, u9, u10}, aprα,β b (y2) = {u4, u7}, aprα,βb (y2) = {u1, u3, u4, u5, u7, u8}, and γα,β(b,d) = ||{u4,u7}|∪|{u2,u6,u9,u10}|| |{u1,u2,u3,u4,u5,u6,u7,u8,u9,u10}| = 6 10 = 0.6 (2) when α = 0.4 , based on the set pair homeopathic similarity relationship, the universe on the condition attribute set c can be divided into: u/c = {x0.4 1 , x0.4 2 , x0.4 3 , x0.4 4 , x0.4 5 , x0.4 6 } where x0.4 1 = {u1}, x0.4 2 = {u2, u9, u10}, x0.4 3 = {u3, u5}, x0.4 4 = {u4}, x0.4 4 = {u6}, x0.4 5 = {u7} and x0.4 6 = {u8}. according to the decision attribute set d , the universe can be divided into: u/d = {y1, y2} where y1 = {u1, u2, u3, u6, u9} and y2 = {u4, u5, u7, u8, u10}. when β = 1 , the lower and upper approximations of the decision-making classes y1 and y2 can be produced as follows: aprα,β b (y1) = {u1, u6}, aprα,βb (y1) = {u1, u2, u3, u5, u6, u9, u10}, aprα,β b (y2) = {u4, u7, u8}, aprα,βb (y2) = {u2, u3, u4, u5, u7, u8, u9, u10}, and γα,β(b,d) = ||{u1,u6}|∪|{u4,u7,u8}|| |{u1,u2,u3,u4,u5,u6,u7,u8,u9,u10}| = 5 10 = 0.5 when β = 0.65 , the lower and upper approximations of the decision-making classes y1 and y2 are given as follows: aprα,β b (y1) = {u1, u2, u6, u9, u10}, aprα,βb (y1) = {u1, u2, u3, u5, u6, u9, u10}, aprα,β b (y2) = {u4, u7, u8}, aprα,βb (y2) = {u3, u4, u5, u7, u8}, and γα,β(b,d) = ||{u1,u2,u6,u9,u10}|∪|{u4,u7,u8}|| |{u1,u2,u3,u4,u5,u6,u7,u8,u9,u10}| = 8 10 = 0.8 based on the attribute reduction algorithm in section 4.3, when α = 0.4 and β = 0.65, a reduction {a1, a5} can be acquired. from the reduction {a1, a5}, some probabilistic decision-making rules set can be generated as shown in table2. 312 yong liu and yi lin:variable precision rough set model based on set pair homeopathic ... table 2 probabilistic decision-making rules generated by the reduction {a1, a5} with respect to the threshold values (α = 0.4, β = 0.65) rules# supprot number confidence(%) a1 = 2 and a5 = 2 66.7%−−−→ d = 1 3 66.7 a1 = 1 and a5 = 2 100%−−−→ d = 1 1 100 a1 = 0 and a5 = 0 100%−−−→ d = 1 1 100 a1 = 0 and a5 = 1 100%−−−→ d = 1 1 100 a1 = 0 and a5 = 2 100%−−−→ d = 1 1 100 according to table 2, we can see that eight objects in the object set can be correctly classified. that is, the classification quality is 80%. from the reduction {a1, a5} , it follows that the company, which manufactures military aircraft engine systems, pays more attention to importance and economical efficiency of the initial accessories. from the set of probabilistic decision-making rules, it follows that if the importance of the initial accessories is general, then the initial accessories should not be selected; if the importance is important and the economical efficiency is excellent, then the initial accessories should be selected. based on the previous calculation and analysis, by adjusting the set pair homeopathic parameter α and the confidence threshold parameter β , the classification ability of the model cam be improved; and the model can be made to possess some fault tolerant ability so that it can materialize the correct classification, and effectively extract the rules for decision making. 6 conclusions due to the complexity and uncertainty of the objective world, as well as the limitation of human understanding, available information systems often contain incomplete information, and the omission of attribute values in the systems increases the uncertainty and the noise of the system. it has been difficult for the traditional rough set method, developed for handling complete information systems, to deal with incomplete information systems. by taking advantage of the set pair analysis methodology, in this paper we constructed the variable precision rough set based on set pair homeopathic similar relationship. by adjusting the confidence threshold parameters (α, β) , we can make the model possess the fault tolerant ability so that it can effectively deal with information systems with noise data, ambiguity, and other kinds of uncertain information. it is shown that our model can materialize the extraction of decision-making rules and knowledge discovery out of a given incomplete information system. for any incomplete information system that contains preference information, grey information, noise data, advances in systems science and application (2015) vol.15 no.4 313 and other kinds of uncertain information, how to construct the grey dominance variable precision rough set model based on the set pair homeopathic similar relationship needs to be investigated, while its potential scope of applications needs to be explored. acknowledgement this work is partially funded by a marie curie international incoming fellowship within the 7th european community framework programme (grant no. fp7-piif-ga-2013-629051); national natural science foundation of china (71301061; 71503103), national social science fund project (12azd111); ministry of education humanities and social sciences youth fund(13yjc630120); natural science foundation of jiangsu province (bk20150157); jiangsu province social science fund project(14glc008); the research base of chinese iot development strategy(133930), the fundamental research funds for the central universities(jusrp11583; jusrp1507zd). references [1] pawlak, z. 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(1999). “rough sets, fuzzy sets and knowledge discovery”, singapore: springer, pp. 1-98. corresponding author yong liu can be contacted at:clly1985528@163.com adv syst sci appl 2018; 03; 111-122 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/581 maximum power point tracking technique for pv system by using golden section search method sonvir singh1 1) department of computer science & engineering, visvesvaraya national institute of technology, nagpur, india e-mail: sonvirsingh036@gmail.com abstract: importance of non-conventional source of energy (ncse) is upsurging day by day to tackle the increased load, energy consumption, and environmental pollution related problems. out of non-conventional source of energy (ncse), solar energy, mainly solar photovoltaic (spv) energy, is one of the most promising renewable source of energy (res) because of pollution-free, no rotating part, an abundance of solar energy, low-cost energy, and high efficiency. the main thing is how to extract maximum power from the solar pv panel. power extract from the pv system depends upon the solar radiation (isolation), ambient temperature and terminal voltage. regulation of terminal voltage is required to extract the maximum power at the operating point. the technique of extracting maximum power by changing its terminal voltage by the various methods is called as mppt. this paper takes up the problem of finding the mppt power point under changing environmental conditions, developing efficient mmpt and changing the operating point of the system very fast according to the environmental condition. in this paper, a new mppt technique is developed for tracking maximum power point which is efficient to existing techniques with the help of the golden section search optimization method. this paper is intended to serve an efficient method and reliable to find the maximum power point at which maximum power is extracted from the pv system. keywords: maximum power point tracking (mppt) techniques, non-conventional source of energy (ncse), solar photovoltaic (spv) energy, photovoltaic (pv) system 1. introduction to extract maximum power from the photovoltaic (pv) systems under different environmental conditions, the maximum power point techniques are used. this principle is generally used for those sources which have variable power outcome, for example, optical power transmission, thermophotovoltaic etc. pv system has a different configuration in respect to their relationship to the inverter, grid system, batteries or electrical load. but the ultimate destination is that to transfer more power from the pv cell, which depends upon the three factors-falling of sunlight on the solar panels, the temperature of the pv panel and electrical load characteristic. temperature remains constant by using heat sink material at the backside of the panel. as the load characteristic, when sunlight intensity changes the maximum power transfer changes accordingly then the load characteristic that gives maximum transfer efficiency also changes. maximum power point technique (mppt) is a technique, which includes the electronics controller used for the extracting maximum available power from the pv panel under various weather conditions. the maximum power extraction from the pv panel varies with solar radiation, ambient temperature, solar cell temperature. mmpt helps in finding the point at which power transfer from the panel to load is maximum. pv cells have a complicated relationship between temperature and resistance that produce non-linear characteristic. which can be analyzed based on the i-v plot. the main function of mppt is to change the output mailto:sonvirsingh036@gmail.com 112 s. singh copyright ©2018 assa. adv. in systems science and appl. (2018) voltage of the pv cells and provide a proper load to obtain maximum power for any given condition [1]. fig. 1 block diagram of the proposed model. an mppt is used for adjusting the voltage level according to the varying output of pv panel for extracting the maximum power for the pv and transferring that power to the variable or fixed the load [2] [3]. a converter and controlling unit as an interface between the load and pv module. by changing the load voltage, the load impedance seen by the source is varied and matched at the point of peak power with the source so as to transfer the maximum power[3]. therefore mppt technique is needed to operate the pv system at the maximum power point. many mppt techniques have been proposed in the literature, examples p&o methods, incremental conductance(ic) methods etc [4] [5]. the main advantage of the p&o mppt algorithm is its low cost and easy implementation but, it may fail to track the maximum operating point under varying weather conditions [6]. the step size of the duty cycle change required at the initial stage and final stage to find the maximum power point [7]. value of step size of the duty cycle should be taken large for points far away from the maximum power point and small for points near the mppt [8]. different methods of finding the maximum power point have different outcomes. some are better in finding the maximum power point at varying environmental condition and some are fast in the respect of time, accuracy and efficiency [9,10]. in this research paper, the system performance is optimized by golden section search method. by changing the interval or selection of section without using a buck-boost converter, the source impedance can be matched to adjust the load impedance to improve the efficiency of the system. the proposed model is shown in fig. 1. 2. photovoltanic system. the system which converts solar energy into electrical energy is called a photovoltaic (pv) system. the basic or smallest unit of pv system is pv cell. these cells are arranged in parallel, series or series-parallel as per requirement. the voltage and currently available at the terminal of pv system, it is directly fed to small loads like lighting, dc motors or connected to the distribution grid by using energy conversion devices [2]. main parts of pv system are pv module, charger, battery, inverter, and load. 2.1. equivalent model a photovoltaic cell is a device used to convert solar radiation into electrical energy. the equivalent circuit diagram of the photovoltaic cell shown in fig. 2 pv cell is a current source, it is produced by breaking of bonds and generation of electron and hole which causes the flow of current in the cell. the equivalent circuit consists current source, diode, shunt resistance, series resistance and load. shunt resistance represents the electron-hole combination before its maximum power point tracking technique 113 copyright ©2018 assa. adv. in systems science and appl. (2018) reach to load. the generation currently in the pv panel depends upon the physical and chemical properties of the material, the age of the solar cell, irradiation per unit area and temperature of the cell, environmental conditions, spectral characteristics of sunlight, dirt, and shadow and so on [11,12]. fig. 2 equivalent single pv cell model the diode current is given by the equation l l lq(v -i r ) nkt d oi = i e -1       (1) the load current which flows in the load is a difference of the photocurrent, diode current and shunt resistance current is given by. l ph d shi i i i   (2) shunt resistance (rsh) represent the electron-hole combination before it reaches to load (for simplifying ignore rsh), ish current becomes zero. ( ) 1 l l lq v i r nkt ol phi i ei           (3) where,  il is the load current or cell current (a).  n is the ideality factor.  q is the charge of the electron (coulomb).  k is the boltzmann's constant (j/k).  t is the temperature of the cell.  iph is the photocurrent (a).  rs ,rsh are series and shunt resistance of the cell (ohms).  vl is the cell output voltage (v). after solving equation no.-3, the load voltage is ln 1 ph l l l s o i inkt v i r q i         (4) in the case, open circuit, il=0 than the corresponding voltage is open circuit voltage (voc). ln 1 ph oc o inkt v q i        (5) 114 s. singh copyright ©2018 assa. adv. in systems science and appl. (2018) this is the order of 10-5. it should be noted that voc will not reduce as much as iph change. in the case of short circuit, vl=0 and impedance are low tags corresponding current is short circuit current which can be approximated as:  sc n pi qg l l  (6) where g is the rate of generation of electrons and holes, ln and lp are the diffusion length of electrons and holes respectively. above equation is not true for the most of cell because of several assumptions. the above equation shows that the short circuit current depends upon the generation rate and diffusion rate. fig.3 i-v and p-v characteristics of pv panel the current-voltage characteristic curve of a pv cell for a certain irradiance at fixed cell temperature is shown in the fig no.-3. the current of pv cell depends upon the external applied voltage and amount of sunlight sticking the surface of the cell. the power vs voltage curve showed in fig no.-3.here p is the power extracted from the pv cells. the curve mainly depends upon current insolation and temperature when insolation increases the power output of pv panel increases whereas when the temperature of the cell increases the power output of pv panel is decreases [13]. constant power generated by the pv panel may also be used to provide the energy to the main grid and micro-grid. energy distribution priority is decided by the priority factor, load and prize forecasting [14]. 2.2. temperature effect. in this paper, consider only two factor falling sunlight on the solar panel and electrical load characteristics. make temperature as a constant throughout the process by using temperature controller like heat sink material system [15]. heat sink material put in between glass cover of solar panel and cells. increased in temperature of solar panel decreased the output of the solar panel. to improve the efficiency of the solar panel, consider the temperature as constant in this paper with the help temperature control system. for controlling the temperature of the solar panel the temperature sensor is used to detect the temperature and heat sink material sink the heat if the temperature is more than the pre-set value and this heat is transfer outside by maximum power point tracking technique 115 copyright ©2018 assa. adv. in systems science and appl. (2018) the help of small cooling fans. so that temperature remains constant throughout the day during processing. systematic diagram is shown in figure no.-4. fig.4 side view of solar panel 3. golden section search technique. the golden search technique is a technique to find out the minimum and maximum (extremum) of strictly increasing and decreasing curve for particular interval or say for unimodal function by successively narrowing the range of values inside in which extremum point know to exist. here, this technique is used to find out the maximum power point in the pv curve [16]. 3.1 unimodal function. a function f(x) is unimodal on [a,b] interval if for some point x* on [a,b],f(x) is strictly increasing on [a,x*] and strictly decreasing on [x*,b].concave and convex shape type functions are unimodal. unimodal type function make the search easier to find maximum point in the given interval.start with an interval of uncertainty (the interval in which maximum point must lie) equal to [a,b],whose length is difference of end point of interval i.e b-a. consider two points in the interval [a,b] and evaluate the function at these points a b x1 x2 if f(x1) < f(x2) , function is increasing in the range [x1,x2]. therefore, the function value must be greater than f(x1). since the function is unimodal, then maximum cannot be lying in the range of (a, x1]. thus concluding that the maximum is in the range of ( x1, b]. if f(x1) > f(x2), since function is unimodal, the function’s maximum must be greater than x2 .therefore the maximum must lie in the range [a, x2) .if f(x1) > f(x2), then the maximum must lie in the range of (x1,x2). since the points, x1 and x2 have to be on either side of the maximum. according to fibonacci method 21   nnn fff (7) 116 s. singh copyright ©2018 assa. adv. in systems science and appl. (2018) dividing the above equation by fn-1. 1 2 1 1     n n n n f f f f (8) golden ratio ( ) 1 1 2 ( 1) lim lim lim n jn n n n nn n n j ff f f f f            (9) take limit on both side of equation (2) and replace with the golden ratio. 1 1    , 2 1 0    (10) after solving the equation no.-4 1 5 2    , consider the only positive root of  1.618  (golden ratio value) [17]. r = 1 0.618   r is conjugate root or conjugate golden ratio. 3.2 iterative process for finding maxima. step-1: given initial of uncertainty = [a, b],whose length is (b-a), stopping tolerance ( ). step-2: )(* 1 abrl  , * 1 1 0lim n n n f l l f     2 1 0 1 *lim n n n n n f f l f f     now, * 1 1x a l  and * 2 1x b l  a x2 bx1 l2 * l2 * f(x2) f(x1) points on x-axis are x1 and x2, their corresponding values of function f(x1) and f(x2) respectively if f(x1) > f(x2), the interval of uncertainty is [a, b] but now the new interval become [a, x2) l1=r(b-a) , check the condition, if l1<  than stop the iteration process and maxima must lie in [a,x2). if not, then goes to next step step-3: generate x3. maximum power point tracking technique 117 copyright ©2018 assa. adv. in systems science and appl. (2018) a x1 x3 x2 f(x3) f(x1) now, x2=b )( 2 * 2 axrl  now, x3= x2l2 * and x1= a+ l2 * points on x-axis are x1 and x3, their corresponding values of function f(x1) and f(x3) respectively if f(x1) > f(x3) , discard (x3, x2] and new interval become [a,x3]. l2=r2 (b-a), check the condition, if l2 <  than stop the iteration process and maxima must lie in [a, x3]. if not, then goes to the next step this process is stopped when the distance between the outer points is smaller than stopping tolerance. if lk <  maxima are found, stop the iteration. 4. the golden search method. consider the voltage range in which the possibility of finding the maximum power point is (v, v). normally in the case of pv curve v equal to zero and v is open circuit voltage. after each iteration, the voltage range is changing accordingly the range length also reducing after execution of each iteration. searching will stop when range length less than or equal to the tolerance. length of uncertainty interval. )( oo k k vvrl  (11) here k represents no. of iteration to find out the voltage at which power transfer is more. solve above education by taking natural logarithmic both side to find the value of k r vvl k ook ln ))/(ln(   (12) algorithm: golden section search algorithm 1: define the interval of uncertainty as (vo – vo) =(v,v) 2: calculate v1k= vk r(vk -vk) 3: calculate v2k = vk + r(vk vk) 118 s. singh copyright ©2018 assa. adv. in systems science and appl. (2018) 4: if (p(v1k) < p(v2k)) then. 5: p* must be in (v1k, vk) 6: vk+1 = v1k 7: vk+1 = vk 8: else 9: p* must be in (vk, v2k) 10: vk+1 = vk 11: vk+1 = v2k 12: if lk = (vk vk) < then 13: stop 14: else 15: k = k + 1 16: go to line 2 k shows the no. 12 of iteration which required to find the voltage at which maximum power will be extracted from the pv panel. firstly define the interval of uncertainty, vo normally zero and vo equal to open circuit voltage of the pv panel than calculated v1k and v2k by the above mention formula of the golden search method. after the calculation of v1k and v2k , calculate power at v1k and v2k and compare these power if (p(v1k) < p(v2k)) than maximum power point must be lie in (v1k, vk). now voltage range will be a change from vk+1 = v1k and vk+1 = vk otherwise maximum power point must be lie in (vk, v2k) and voltage range will change from vk+1 = vk and vk+1 = v2k. check the stopping condition, lk = (vk vk) <  if it is true then stop the process otherwise increase the value of k by 1 and repeat the whole process till the finding of the voltage at which maximum power is extracted from the pv panel. 5. simulation result in this paper, the model is developed with matlab/simulink. the proposed model is shown in fig.-4. the pv module parameter considers for the proposed model are: voc (open circuit voltage) =21.8 v. isc (short circuit current) =3.11 a. vmp (voltage at mpp) =17.44. imp (current at mpp) =2.86 a. pmp (power at mpp) =50 w. the outputs of the pv panel are current and voltage and it depends upon the solar radiance and temperature of the panel. the temperature of pv panel is maintained constant at 25 degree celsius and solar intensity is varied up to the rated value of 1000 w/m2. the current fed to the multiplier to obtain the power in respect to the maximum power point tracking technique 119 copyright ©2018 assa. adv. in systems science and appl. (2018) fig.5 i-v and p-v characteristics of pv panel table 1. iteration process chart. iteration vo vo v1 v2 p(v1) p(v2) p* in interval lk=(vk-vk) 1. 0 21.8 8.33 13.47 25.91 41.76 (8.33,21.8) 21.8 2. 8.33 21.80 13.48 16.65 41.79 49.60 (13.48,21.8) 13.47 3. 13.48 21.80 16.66 18.62 49.61 47.26 (13.48,18.62) 8.32 4. 13.48 18.62 15.44 16.66 47.25 49.61 (15.44,18.62) 5.14 5. 15.44 18.62 16.65 17.41 49.6 50.02 (16.65,18.62) 3.18 6. 16.65 18.62 17.41 17.86 50.02 49.99 (16.65,17.86) 1.97 7. 16.65 17.86 17.11 17.40 49.99 50.02 (17.11,17.86) 1.21 8. 17.11 17.86 17.40 17.57 50.02 49.94 (17.11,17.57) 0.75 9. 17.11 17.57 17.29 17.39 50.03 50.02 (17.11,17.39) 0.46 10. 17.11 17.39 17.21 17.28 50.02 50.03 (17.21,17.39) 0.28 11. 17.21 17.39 17.28 17.32 50.03 50.03 018 output voltages of voltage interval changing system and output of pv panel is fed to the voltage interval changing system.v1 and v2 voltages are changed in each iteration. powers are calculated in respect to voltages v1 and v2 respectively by the multiplier. side by side length is compared by tolerance value if its length equal to or less than the tolerance value than iteration is stopped. at voltage, the iteration stop is voltage at which maximum power will be extracted. no. of iteration requires to finding the maximum power point at different weather condition show in table 1. no. of iteration is more compare to other methods but the time required in each iteration is very less. therefore this method has high speed to trace the maximum power point compare to others. the simulation circuit is shown in the fig-4. in this proposed circuit is without the buck-boost converter, the output voltage level is changed according to the load as well as radiation of sunlight and it is decided or change by golden section search method. during weather changing condition, the maximum power is not extracted so adjustment of the output voltage is required to extract maximum power. changing of the load is not in hand because it changes according to demand of consumers. in fig 5 the various parameter of the proposed method is shown and in fig-6 and fig-7 comparison of the proposed method is done with existing methods in all respect. the units of power and length are watts and voltage respectively. 120 s. singh copyright ©2018 assa. adv. in systems science and appl. (2018) iteration iteration fig.6 changing of parameters with an iteration of the proposed method. fig.7 comparisons of the proposed method with others table 2. iteration process chart. 6. conclusion in the present work, the maximum power point tracking is successfully carried out by using a golden search method. golden search method improves the efficiency and optimized the photovoltanic system without using a buck-boost converter compare to others methods of maximum power point technique. by changing the interval of voltage after each iteration, the length is decreasing by discarding the intervals in where the possibility of obtaining maximum power point is nil. this method is good in respect of accuracy and performance. in this paper, the temperature is removed by making it constant throughout the process by using the heat sink material layer at the back side of the pv panel. the performance is studied by the matlab/simulink. in future, the maximum power point tracking carried out by golden search method without using the buck-boost converter in order to reduce cost, improve response and efficiency, reduce complexity in making and complication of hardware can be removed. references [1] reisi, a. r., moradi, m. h. & jamasb, s. (2013.) classification and comparison of maximum power point tracking techniques for photovoltaic system: a review, renewable and sustainable energy reviews, 19, 433–443. [2] maxwell, j. c. (1881). a treatise on electricity and magnetism. clarendon press. mppt technique classification complexity tracking accuracy tracking speed efficient for partial shading application voltage-based mppt offline simple low slow no off grid current based mppt offline simple low slow no off grid perturb and observe online simple medium medium no both inc online complex very high fast yes both inr online complex high fast yes both rcc online complex high fast yes grid intelligent based hybrid complex very high fast yes both parasitic capacitance online complex very high medium yes off grid gradient descent hybrid medium high medium no off grid golden section search method (proposed) hybrid simple very high fast yes both maximum power point tracking technique 121 copyright ©2018 assa. adv. in systems science and appl. (2018) [3] chin, s., gadson, j., & nordstrom, k. (2003). maximum power point tracker. tufts university department of electrical engineering and computer science, 1-66. [4] ahmed, j., & salam, z. (2015). an improved perturb and observe (p&o) maximum power point tracking (mppt) algorithm for higher efficiency. applied energy, 150, 97-108. [5] subudhi, b., & pradhan, r. (2013). a comparative study on maximum power point tracking techniques for photovoltaic power systems. ieee transactions on sustainable energy, 4(1), 89-98. [6] ilyas, a., ayyub, m., khan, m. r., husain, m. a., & jain, a. (2018). hardware implementation of perturb and observe maximum power point tracking algorithm for solar photovoltaic system. transactions on electrical and electronic materials, 19(3), 222-229. [7] husain, m. a., & tariq, a. (2018). transient analysis and selection of perturbation parameter for pv-mppt implementation. international journal of ambient energy, (just-accepted), 1-8. [8] husain, m. a., jain, a., & tariq, a. (2016). a novel fast mutable duty (fmd) mppt technique for solar pv system with reduced searching area. journal of renewable and sustainable energy, 8(5), 054703. [9] husain, m. a., khan, a., tariq, a., khan, z. a., & jain, a. (2018). aspects involved in the modeling of pv system, comparison of mppt schemes, and study of different ambient conditions using p&o method. in system and architecture (pp. 285-303). springer, singapore. [10] ilyas, a., ayyub, m., khan, m. r., jain, a., & husain, m. a. (2017). realisation of incremental conductance the mppt algorithm for a solar photovoltaic system. international journal of ambient energy, 1-12. [11] agrawal, j., & aware, m. (2012, december). golden section search (gss) algorithm for maximum power point tracking in photovoltaic system. in power electronics (iicpe), 2012 ieee 5th india international conference on , 1-6. [12] ishaque, k., & salam, z. (2013), a review of maximum power point tracking techniques of pv system for uniform insolation and partial shading condition, renewable and sustainable energy reviews, (19), 475–488. [13] suresh, k. s., & prasad, m. v. pv cell based five level inverter using multicarrier pwm. international journal of modern engineering research, 1(2), 545-551. [14] funde, n., dhabu, m., deshpande, p., & patne, n. r. (2018). sf-oeap: starvationfree optimal energy allocation policy in a smart distributed multi-microgrid system. ieee transactions on industrial informatics. [15] husain, m. a., khan, z. a., & tariq, a. (2017). a novel solar pv mppt scheme utilizing the difference between panel and atmospheric temperature. renewable energy focus, 19, 11-22. [16] patrón, r. s. f., botez, r. m., & labour, d. (2012). vertical profile optimization for the flight management system cma-9000 using the golden section search method. in iecon 2012-38th annual conference on ieee industrial electronics society (54825488). [17] koupaei, j. a., hosseini, s. m. m., & ghaini, f. m. (2016). a new optimization algorithm based on chaotic maps and golden section search method. engineering applications of artificial intelligence, 50, 201-214. adv syst sci appl 2018; 4:13–38 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/624 a unified approach to reliability, availability, performability analysis based on markov processes with rewards valentina s. viktorova1∗, nikolai v. lubkov1 armen s. stepanyants1 1institute of control sciences, russian academy of sciences, moscow, russia received july 29, 2018; revised december 17, 2018; published december 31, 2018 abstract: this paper discusses a unified approach to reliability, availability and performability analysis of complex engineering systems. theoretical basis of this approach is continuous-time discrete state markov processes with rewards. from reliability modeling point of view complex systems are the systems with static and dynamic redundancy, imperfect fault coverage, various recovery strategies, multilevel operation and varying severity of failure states. we propose a unified method of calculating the reliability, availability and performability indices based on the definition of special forms of reward matrix. this method is proved to be effective in calculating both cumulative and instantaneous measures in steady-state and transient cases. we describe special analytical software which implements suggested method. we demonstrate the flexibility of the proposed method and software by analyzing multilevel process unit with protection and demand-based warm standby system. keywords: availability, markov process, markov reward model, multilevel engineering systems, performability, reliability. 1. introduction at the present state of development of reliability theory the models of analysis can be subdivided into two categories (classes): the dynamic models and the static models. all changes of system states in the class of dynamic models are considered as the processes developing in time. in the class of the static models, states of the system are determined by the sets of states of system elements at the time moment t. markov [1–4], semi-markov processes, asymptotic methods of the renewal theory and regenerative processes [5–7], monte carlo simulation techniques [8] are used within the framework of dynamic models. dynamic models allow to calculate all the main dependability measures both for repairable and non-repairable systems. these measures are: instantaneous indices (e.g. availability at the time instant); interval indices (e.g. reliability during the time interval); time-independent stationary indicators (e.g. mean time between failures). known drawbacks inherent in markov models are the size and stiffness of transient solution of a system of kolmogorov-chapman equations (or ill-conditionality for stationary case). possible ways to solve these problems are discussed in [9–11]. monte carlo simulation of modern high reliable systems may require large amount of simulations to obtain results with desired accuracy. these shortcomings can be eliminated with the help of special acceleration techniques. there is no limitation to type of distribution of waiting times between the changes in the class of monte carlo models. but the problem of creating a universal monte carlo ∗corresponding author: vsviktorova@gmail.com 14 v.s. viktorova, n.v. lubkov, a.s. stepanyants model describing the complex reliability behavior of different kind of systems remains unsolved. static models use two main groups of methods – combinatorial-probabilistic [3] and logical-probabilistic [12]. combinatorial-probabilistic methods use the basic formulas of combinatorics and probability theory (probability of the sum and product of events, the formula of total probability). these formulas are used mainly for serial-parallel and ”m out of n” redundant schemes. logical-probabilistic methods, which are based on the construction of a boolean function relating the state of the system with the states of its elements. the resulting boolean function is transformed to a form that allows to replace logical variables by corresponding probabilities. classic failure trees and reliability block diagrams are the main methodology of logical-probabilistic methods [13, 14]. classic static models for the case of repairable systems allow us to calculate only the differential (instantaneous) reliability indicators determined at the time instant t (e.g. availability, failure frequency, average efficiency at the time instant t). in the last decade, combined approaches to reliability models construction are successfully developing. these approaches are based on the implementation of dynamic properties into static models. dynamic fault trees are the most popular practical implementation of these approaches [15–19]. modern complex engineering systems with high demands on reliability are characterized by various features, which include: • multiple levels of operation efficiency (e.g., performance) and the graceful degradation of performance in the event of failures [20–23]; • a variety of redundancy implementation techniques (hardware, software, functional, temporal redundancy) [24–26], types of redundant schemes (parallel, standby, hybrid), modes of reserve components (cold and warm standby, shared) [3, 4]; • smart recovery strategies, restricted repair resources [27–29]; • imperfect built-in test equipment leading to the presence of undetected failures and false alarms [30, 31]; • implementation of special multiphase error handling procedures with classification on permanent and transient faults (mainly for fault tolerant computers) [32–34]; • the possibility of the occurrence of mutually exclusive failures that, with a certain multiplicity and sequence, can lead to different consequences at the system level [13,14]. markov modeling allows taking into account the above-mentioned features of the reliability behavior of systems and calculating most part indicators of reliability, safety, efficiency. at the design stage, we don’t have, as a rule, objective information about the distribution functions of random time variables of the model. therefore, we believe that the assumption of exponentiality is entirely permissible. the problem of large size of the markov model can be solved by decomposition and aggregation of the model parts and by automation of the model construction. in this paper, we consider the markov reward models as common tools for analyzing complex reliability behavior of technical systems. we propose the extension of markov reward modeling for the reliability, abailability, performability (rap) analysis of the systems at the design stages. we suggest the method of calculating reliability, availability and performability indicators based on a special definition of the reward matrix. this method is quite efficient and well suited for software implementation. it makes calculation both cumulative and point indices possible only by solving systems of differential or algebraic equations and does not require the usage of additional numerical integration operations. the remainder of this paper is organized as follows. a brief summary of the mathematical foundations of markov reward model is given in section ii. in section iii, firstly we give definitions of interval, point, steady-state rap measures. next, we present the method for for calculation of these measures in the framework of markov reward model. section iv presents analytical software implementing the proposed method. the flexibility of described copyright c© 2018 assa. adv syst sci appl (2018) a unified approach to rap analysis based on markov processes with rewards 15 techniques and software is illustrated by investigations of two practical examples in section v. in section vi, we give some concluding remarks. 2. mathematical foundations a system state at the time moment t can be described by stochastic process z(t) of transitions in system state space ω. we assume that the state space ω is discrete, finite and of sizen . ω is defined by states of system components (failed or operational). we define a quality function φ on z(t). changing system states leads to a change in the quality function φ. z(t) is stochastic process, therefore, φ(z(t)) = φ̃(t) are also stochastic functions. due to the discrete nature of ω, the functions φ̃(t) are the step functions, whose values are quality levels (for example, system performability) corresponding to the states. g are indices of system performability. g can be defined as a measure on the trajectories of the φ̃(t) . general representation of g takes the form g = m{f [φ(z(t)]}, (2.1) where m represents mathematical expectation; f is a functional, which is determined by the type of performability index. thus, the model of performability contains three components ω, z(t),φ. it is natural and obvious to present the model of performability by means of transition graph (state graph). the elements of the state space correspond to the vertices. possible transitions in the state space correspond to the arcs of the graph. φ is defined as a function of the system state si. this function takes the value of wi corresponding to the performance level of the system in this state. that is φ(si) = wi. (2.2) in addition, this model can display effects occurring at the transition of the system from one state to another. this is accomplished by determining reward matrix w = [wij] n n , (2.3) where wij – is an effect on the system that arises at the transition from state i to state j; wii = wi. random transition process z(t) is most simply set in the case where the value of time of stay in each state have an exponential distribution, i.e. when the process is a continuous-time markov chain (ctmc). the process z(t) is determined by transition rates, which correspond to arcs of the markov graph. similarly to the case of matrix w , we can create a matrix of transition rates λ = [λij] n n , (2.4) where λij – is the rate of transition from state i to state j; λii = − ∑ j,j 6=i λij . 2.1. construction of rap indices thus, the markov model of performability is defined by two matrices λ and w , i.e. {r, z(t),φ} ∼ {λ,w}. model {λ,w} is called markov reward model (mrm). markov reward model describes the behavior of systems with exponential distribution of elements failure and repair time. states of the model represent the states of the system, which correspond to different sets of failed and operable items. if criterion of system failure is specified, then expression (2.1) defines also the indices of reliability and availability. copyright c© 2018 assa. adv syst sci appl (2018) 16 v.s. viktorova, n.v. lubkov, a.s. stepanyants let failure criterion is {φ̃(t) < φcr} ⇒failure, where φ̃(t) is the current value of the quality function; φcr is limit value of the quality, which is allowed by operating requirements. then, by setting a particular type of the functional f the following indices can be determined from equation (2.1): • reliability over the time interval (0, t)r(t): r(t) = p{φ̃(τ) > φcr, ∀τ ∈ (0, t)} under f [·] = { 0, if ∃τ ∈ (0, t) : φ̃(τ) < φcr 1, otherwise (2.5) • mean time to first failure (mttf) tff : tff = ∞∫ 0 r(t)dt underf [·] = min{τ : φ̃(τ) < φcr} (2.6) • availability a(t) at the time instant t: a(t) = p{φ̃(τ) ≥ φcr} under f [·] = { 0, φ̃(τ) < φcr 1, φ̃(τ) ≥ φcr (2.7) • probability of being at the ith level of performance at the time instant t: p (t) = p{φ̃(t) = φi} under f [·] = { 1, φ̃(t) = φi 0, otherwise (2.8) • average accumulated time spent at the ith level of performance: under f [·] = ∫ t:φ̃(t)=φi dt (2.9) • average number of transitions from the ith to the jth performance level during the time interval (0, t): under f [·] = ∑ l : φ̃(tl−) = φi φ̃(tl+) = φj 1, (2.10) where tl the time moment of transition from ith level to jth performance level. • time-averaged accumulated reward for the time interval (0, t): f [·] = 1 t t∫ 0 φ̃(τ)dτ (2.11) 2.2. description of mrm basic relations the consept of reward in markov reward model is a generalized concept. this can be any effect, loss, costs. according to the provisions of previous section we define the model state space as a discrete set {si}, i = 1, n . also we define reward matrix. the elements of the reward matrix are interpreted as follows: wii is a reward at time unit while the system is copyright c© 2018 assa. adv syst sci appl (2018) a unified approach to rap analysis based on markov processes with rewards 17 in state si (reward rate); wij is impulse reward received in the system under the transition from state i into state j; n the number of states of the system. we define the total (or accumulated [20]) reward obtained by the system for a given time t as the functional f [φ(z(t))]: g̃(t ) = n∑ i=1 τiwii + ∑ i,j nijwij, (2.12) where τi is the residence time in state si during the time interval (0, t ); nij is the number of system transitions from state i into state j during the time interval (0, t ). accumulated reward g̃(t ) is a random variable, since τi and nij are random values. average accumulated reward is defined as the mathematical expectation of g̃(t ) y (t ) = m{g̃(t )}. (2.13) according to (2.1), y (t ) is an estimate of the performability index value. if we consider that members of (2.12) τi and nij depend only on the specific implementation of process z(t), it is obvious that the type of index is completely determined by the type of matrix w . the system of howard differential equations describes the behavior of the average accumulated reward: dyi(t)/dt = wii + ∑ j,j 6=i λijwij + ∑ j λijyj(t); i, j = 1, n, yi(0) = 0, (2.14) where yi(t) is the average accumulated reward for the time interval (0, t), given that the initial state is si(z(0) = si); λij the rate of transition from state si into state sj . the matrix form of equation (2.14) is dy (t)/dt = λy (t) +r; y (0) = 0, (2.15) where λ = ‖λij‖ matrix of transition rates; r is a column vector of constant terms: ri = wii + ∑ j,j 6=i λijwij; i, j = 1, n. (2.16) we assume that for absorbing states sr of the markov process valuewrr = 0. the absorbing state is a state that once entered, can not be left. this assumption is quite natural, since the absorbing states are usually identified with the non-operational states of the system. 3. method of calculating rap indices based on markov reward model the expression (2.1) of expected reward will determine the rap indices in accordance with (2.5) – (2.11), if the value of the reward for a particular process realization (2.12) coincides with the functional f[.] in (2.5) – (2.11). it requires to select the appropriate reward matrix w and optionally to correct the transition rate matrix λ. correction of matrix λ is necessary in case when the index reflect operation of the system till some event e, for example, till the system failure. since all performability indices in the model are interpreted through the reward, it is obvious that reward should be dismissed after event e has occurred. formally, this can be achieved by identification of the event e with a transition to an absorbing state. the reward associated with this state is assumed to be zero. the correction of matrix λ is that we treat the states corresponding to e as absorbing states. to do this correction, it is enough copyright c© 2018 assa. adv syst sci appl (2018) 18 v.s. viktorova, n.v. lubkov, a.s. stepanyants to equate to zero the elements of rows of matrix λ corresponding to these states. to carry out further calculations, we partition the set of model states ω into ωg, the set of operational system states, and ωf , the set of failed states. all rap measures can be subdivided into three groups: 1. interval measures these metrics are dependent on time and are determined on finite time interval [0, t ], for example, reliability r(0, t ). 2. point measures these metrics are dependent on time and are determined in time point t, for example, availability a(t). 3. steady-state measures these metrics are independent on time and are determined on an unlimited time interval (0,∞), for example, mttf. calculation of steady-state measures is reduced to finding the stationary solutions of the system (2.15), i.e., to solving a system of linear algebraic equations: λy +r = 0. (3.1) we mention that all indices calculated by the expressions (2.15) and (3.1) are vectors. the ith element of this vector is the estimated value of the index for the initial state si. 3.1. unreliability system unreliability over the time interval (0, t)(q(t)) can be calculated if we define elements of matrix w as wij = 0 ∀ i, j ∈ ωg;wij = 1 ∀ i ∈ ωg and j ∈ ωf : wij = { 1, if i ∈ ωg, j ∈ ωf 0, otherwise . (3.2) besides we must treat states in ωf as absorbing states. namely we should to delete in markov graph all arcs leading from states in ωf to states in ωg. the calculation is performed by (2.15). complementary index is reliability: r(t) = 1−q(t). 3.2. mean time to first failure mean time to first failure (tff ) can be calculated if we define diagonal elements of matrix w corresponding to the operational states as wii = 1 ∀ i ∈ ωg. wij = { 1, if i = j, i, j ∈ ωg 0, otherwise . (3.3) besides we must treat states in ωf as absorbing states. calculation is performed by (3.1). for the given type of reward matrix expression (3.1) is reduced to λ∗tff =  −1 ... −1  , (3.4) where λ∗ − k × k matrix derived from matrix λ by deleting rows and columns corresponding to nonoperational states; k total number of operational states. copyright c© 2018 assa. adv syst sci appl (2018) a unified approach to rap analysis based on markov processes with rewards 19 3.3. standard deviation of random time to first failure standard deviation (σ) and variance (σ2) of mean time to first failure can be calculated in accordance with the expression σ = √ m(t2)−m2(t) = √ t (2) ff − t 2 ff . to calculate vector t (2) ff , we have to find the solution of algebraic equations λ∗t (2) ff =  −2tff1 ... −2tffk  , (3.5) where tffi is mean time to first failure given the process starts from the ith state. 3.4. average accumulated time spent in the operational states average accumulated time spent in the operational states during the time interval (0, t) (t∑(t)) can be calculated if we define diagonal elements of matrix w corresponding to the operational states as wii = 1 ∀ i ∈ ωg. wij = { 1, if i = j; i, j ∈ ωg 0, otherwise . (3.6) here, we should not artificially create absorbing states. calculation is performed by the equation (2.15). we can also use this index for analysis of multilevel systems. in this case we should partition the set of system’s states ω into m sets ω1, . . .ωr, · · · ,ωm, where ωr the set of operational system states with rth performance level. average accumulated time spent in ωr during the time interval (0, t) (t r∑) can be calculated at wii = 1 ∀ i ∈ ωr. 3.5. availability availability a(t) is the probability that the system is operational at the time instant t. a(t) is the point measure. taking into account that a(t) = dtς(t)/dt we can calculate a(t) after calculation of tς(t) as follows: a(t) = λtς +r. (3.7) the formation of reward matrix is carried out in accordance with (3.6). unavailability u(t) = 1− a(t). 3.6. time-averaged availability general expression of time-averaged availability over the time interval (0, t) (aav(t)) has the form aav(t) = 1 t t∫ 0 a(τ) dτ. (3.8) for calculation of this indicator, it is necessary to take the reward matrix, in which the diagonal elements of the operational system states, are equal to 1/t and all other elements are zero: wij = { 1/t, if i = j; i, j ∈ ωg 0, otherwise . (3.9) artificial creation of absorbing states is not required. calculations are carried out by (2.15). copyright c© 2018 assa. adv syst sci appl (2018) 20 v.s. viktorova, n.v. lubkov, a.s. stepanyants 3.7. average failures number average (expected) failures number during the time interval (0, t)(n(t)) can be calculated if we define elements of matrixw columns corresponding to the failed states aswij = 1 ∀ i 6= j. values of other elements of matrix w are zero: wij = { 1, if i ∈ ωg, j ∈ ωf 0, otherwise . (3.10) n(0, t) is failure measure for repairable systems so we should not artificially create absorbing states. calculation is performed by (2.15). 3.8. failure frequency failure frequency (ω(t)) is a differential index with respect to n(t). after calculating of vector n(t) for each initial state, we put it into the right side of (2.15). the resulting vector dy (t)/dt will be the vector (ω(t)): ω(t) = λn(t) +  λ1 σ ... λnς  , (3.11) where λiς is total transition rate from ith operational state to failure states. for the failure state, it is zero. 3.9. average accumulated reward average accumulated reward over the time interval (0, t)(et (t)) integrates reward (loss) of the system during the time interval (0, t) proportionally to the time of stay in the states and the quantity of transitions between the states. reward matrix w has the most general form. the elements wii, wij are real rewards (losses) measured in terms of units of system performance. calculations are carried out by (2.15). 3.10. average reward average reward at the time instant t(eav(t)) is a point measure. this indicator characterizes the multi-level systems. therefore, system states set is subdivided into classes, in accordance with level of system performance. states of the ith class are characterized by performance (ei) per time unit (for example, throughput). the first step of eav(t) calculation is performed by the equation (2.15). as a result of the first calculation step, we get the vector of the average accumulated reward et (t). eav(t) is a differential index with respect to et (t). so the second calculation step is to put the vector et (t) into right side of equations (2.15): eav(t) = λet (t) +r. (3.12) 3.11. performability ratio this index is the ratio of the average accumulated reward to the nominal reward during the time interval (0, t): kp(t) = et (t) en(t) , (3.13) where en(t) nominal (maximum) accumulated reward generated by the system in the absence of failures. en(t) = emaxt. usually emax = e1. copyright c© 2018 assa. adv syst sci appl (2018) a unified approach to rap analysis based on markov processes with rewards 21 4. analytical software based on markov reward model the above uniform approach to calculating performance, availability and reliability measures was used by us in the development of analytical software. the software is operated under windows os and is written in c] programing language. it has an advanced graphical user interface (fig.4.1) and uses ole automation for generation reports in ms word formats. this software allows you to take into account all features of reliability behavior of complex technical systems, mentioned in sectoion 1: • multiple levels of operation performance and the graceful degradation of performance in the event of failures; • different types of redundancy implementation techniques; • variety of redundant schemes and backup components load levels; • recovery strategies and restrictions on repair resources; • imperfect built-in test; • possibility of transient faults occurrence; • common cause failures; • and so on. the user of this software can calculate all the rap indicators described in section 3: • reliability (unreliability) for a given time interval (0, t) ; • mean and standard deviation of random time to first failure; • average and accumulated time spent in the selected subset of states during the time interval (0, t); • availability (unavailability) at given time instant t; • average failures number for a given time interval (0, t); • failure frequency at given time instant t; • average reward at given time instant t; • average accumulated reward for a given time interval (0, t); • performability ratio for a given time interval (0, t). the software execution process is divided into three steps: creation of the mrm, setting up of the mrm, numerical solution of the system of howard equations. the composition and interrelationships of these processes are shown in fig.4.2-fig.4.4. mrm creation is executed by the user in interactive mode with the help of built-in graphical editor. the created model can be saved in an external xml file. a saved model can be reloaded from the xml file (fig.4.2). the model’s setting up is performed in the matrix configuration module in accordance with equations of section 3. setting up operations are performed automatically after the user chooses the metric for calculation (fig.4.3). calculation of values of user-selected rap indices is made in the numerical solution module (fig.4.4). in this module, numerical procedures for solving systems of differential and algebraic howard equations are realized. different approaches and methods of computational mathematics with regard to their software implementation outlined in [35 37]. detailed consideration of numerical evaluation of markov models transient behavior was presented in [9], [10]. three numerical techniques for finding the transient solution of large markov models were examined. they are uniformization, an explicit differential equation solution method (runge-kutta) and special stable implicit method (tr-bdf2). for the software implementation of numerical solution of the mrm we used efficient method based on evaluation of matrix exponential at small step. this method had been proposed and described in [38]. for a stationary case, a numerical solution is obtained by solving the system of algebraic equations by the gauss method or the rotation method [35,36]. the rotation method works well for bad conditioned systems, which are often generated by rap analysis models. copyright c© 2018 assa. adv syst sci appl (2018) 22 v.s. viktorova, n.v. lubkov, a.s. stepanyants fig. 4.1. screenshot of main form of the analytical software. copyright c© 2018 assa. adv syst sci appl (2018) a unified approach to rap analysis based on markov processes with rewards 23 fig. 4.2. execution process of the analytical software (step 1). copyright c© 2018 assa. adv syst sci appl (2018) 24 v.s. viktorova, n.v. lubkov, a.s. stepanyants fig. 4.3. execution process of the analytical software (step 2). copyright c© 2018 assa. adv syst sci appl (2018) a unified approach to rap analysis based on markov processes with rewards 25 fig. 4.4. execution process of the analytical software (step 3).copyright c© 2018 assa. adv syst sci appl (2018) 26 v.s. viktorova, n.v. lubkov, a.s. stepanyants 5. case studies to demonstrate the effectiveness of the software implemented mrm, we present two examples of reliability, availability, performability evaluation of complex systems. in first example we investigate technological object with protection system. the second one is demand-based warm standby system (db-wss), described in article [23]. 5.1. markov reward model of multilevel process unit we consider process unit consisting of technological object (to) with protection system (ps). the protection system includes a diagnostic unit (du) and the actuator (a) (fig.5.5). fig. 5.5. technological object with protection. protection systems are designed to generate control actions to the protected object in order to prevent transition from process equipment failures to accident. control actions may be different, such as change of the operating mode, reduction in productivity, emergency shutdown of faulty elements and elements belonging to the same processing chain. these actions prevent the development of accident. here, we consider the case of technological process shutdown. however, proposed approach is also suitable for the control actions that lead to poor performance or mode change. the main function of protection system is performed sporadically at the time instants of the occurrence of the object failures, so that the protection system operates in standby temporary mode. let us define rewards of operational states and losses of failed states. in this example, all possible system states can be subdivided into four classes. the first class includes such technical states in which the object is functioning normally and earns specific rewards per unit of time of stay in these states (e.g., these states correspond to the nominal performance of technological object). let us note that this group includes states with latent failures such as failure of diagnostic unit. the second class consists of the states of accident-free shutdown of the technological object. the losses associated with these states are only related to the downtime of the object. the reward in these states is either zero or negative, if the idle leads to additional losses per unit time. the third and fourth classes of states are catastrophic failures of the object. the transition to this class of states from the states of the first class brings a onetime damage (negative reward) associated with the occurrence of an accident (death of people, equipment breakdowns, emissions into the atmosphere, etc.). in this model, we consider two types of catastrophic failures differing in the severity of the consequences – accident i and accident ii. thus, the normal functioning of the technological object is accompanied by a linear increase in the accumulated reward in proportion to the time spent in the first class states. idle time leads to the preservation of the achieved level of accumulated reward (at zero values of the reward rate in each state) or to its descent (at negative values of the reward rates) in proportion to the time spent in states of second, third and fourth classes. when impulse rewards associated with the transitions between the states are not zero, there is an abrupt change (more often a decrease) in the accumulated reward. negative impulse rewards are due to the costs of recovery from failures or accidents, purchase of equipment, payment of fines or insurance, etc. copyright c© 2018 assa. adv syst sci appl (2018) a unified approach to rap analysis based on markov processes with rewards 27 we suppose that all accidents occur only due to failures of the technological equipment. let protection system in the event of critical situation instantly shut down the technological process by making the necessary equipment control (for example, switching off). let the protection system processes critical situation with coverage β(0 ≤ β ≤ 1). failures of the protection system which occur during interval of normal to operation can lead to different consequences. let us single out ps failures of two types: latent failures (no operation) and explicit failures (false alarm). latent failures of ps are manifested in the form of absence of protection actions in the event of critical situation which entails an accident. false alarms lead to the undesired protection actions in the absence of to failures. parameters of the markov reward model of the technological object with protection system are: wij impulse losses (negative impulse reward) under the transition from state i into state j; wii reward rate (negative or positive) in state i; β coverage probability for to, determined as conditional probability of processing of to catastrophic failure by a protection system, provided that the failure has occurred; αdu fraction of the protection system latent failures such as non operation of du; αa fraction of the protection system latent failures such as non triggering of actuator; 1− αdu fraction of the protection system explicit failures such as du false alarm; 1− αa fraction of the protection system explicit failures such as actuator false triggering; ηi fraction of technological object catastrophic failures of the first kind (accident i); ηii fraction of technological object catastrophic failures of the second kind (accident ii); λto, λdu , λa – failure rates of technological object, diagnostic unit, actuator respectively; µ repair rate of technological object after shutdown; µa repair rate of technological object after falling into accident ii state. the transition graph of markov reward model of the technological object with protection is shown in 5.6. we partition the set of the mrm states into four subsets: • normal operation (op) (states 1,3,4) • accident-free shutdown (sd) (states 2,6) • accident i (ai) (state 5) • accident ii (aii) (state 7) transition rates matrix λ of the model is λ =  λ11 λ12 λ13 λ14 λ15 0 λ17 λ21 λ22 0 0 0 0 0 0 λ32 λ33 0 λ35 0 λ37 0 0 0 λ44 λ45 λ46 λ47 λ51 0 0 0 λ55 0 0 λ61 0 0 0 0 λ66 0 0 0 0 0 0 0 λ77  . (5.1) the diagonal elements of the matrix λii are given by − ∑ j,j 6=i λij . transition rates between states of the graph are determined based on the model parameters as follows: λ12 = [1− (1− β)(ηi + ηii)]λto + (1− αdu)λdu + (1− αa)λa; λ13 = αduλdu ; λ14 = αaλa; λ15 = (1− β)ηiλto; λ17 = (1− β)ηiiλto; λ21 = λ61 = µ; λ32 = (1− ηi − ηii)λto + (1− αa)λa; copyright c© 2018 assa. adv syst sci appl (2018) 28 v.s. viktorova, n.v. lubkov, a.s. stepanyants λ35 = λ45 = ηiλto; λ37 = λ47 = ηiiλto; λ46 = (1− ηi − ηii)λto; λ51 = µa; fig. 5.6. markov graph of to with protection. by specifying the form of reward matrix, it is possible to obtain equations for calculating various indices. we want to determine the cumulative effect of operation of technological object taking into account the positive and negative impact of the protection system. therefore, it is advisable to calculate the following indices: 1. q(t) unreliability over (0, t). q(t) can be calculated from equation (2.15). to calculate q(t), we must equate to zero λ21, λ51, λ61 and define w as w =  0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  . (5.2) 2. tς(t) -average accumulated time spent in the operational states over the time interval (0, t). tς(t) is determined by (2.15), setting r as copyright c© 2018 assa. adv syst sci appl (2018) a unified approach to rap analysis based on markov processes with rewards 29 r = [ 1 0 1 1 0 0 0 ]t . (5.3) 3. a(t) system availability at the time instant t. a(t) is calculated by (3.7) after calculation of vector tς(t). 4. tff mean time to first failure. tff is determined by (3.4), where matrix λ∗ is λ∗ = [ λ11 λ13 λ14 0 λ33 0 0 0 λ44 ] . (5.4) 5. σ standard deviation of random time to first failure. σ is obtained from equation (3.5), where vector of free terms is [ −2tff1 −2tff2 −2tff3 ]t . 6. ta mean time to first catastrophic failure (accident i or ii). ta is determined by equation (3.4), where matrix λ∗ is 5× 5 matrix derived from matrix (5.1) by deleting rows and columns corresponding to accident i and accident ii states (states 5 and 7). 7. n(t) expected failures number. n(t) can be calculated from (2.15) where matrix w takes the form (5.2). 8. ω(t) failure frequency. ω(t) can be calculated from (3.11) after n(t) calculation and setting free terms vector as [ λ12 + λ15 + λ17, 0, λ32 + λ35 + λ37, λ45 + λ46 + λ47, 0, 0, 0 ]t . 9. na(t) expected catastrophic failures number. na(t) can be calculated from (2.15) where matrix w has unit elements only in the transitions to catastrophic failures (states 5 and 7). similarly, we can find separately nai(t) and naii(t). 10. tσsd(t) –average accumulated time spent in the states of technological object shutdown over time interval (0, t). tσsd(t) is determined by equation (2.15), setting r as [ 0, 1, 0, 0, 0, 1, 0 ]t . 11. nsd(t) average number of transitions to states of shutdown. nsd(t) can be calculated from equation (2.15) where matrix w has unit elements only in the transitions to shutdown states (states 2 and 6). 12. et (t) average accumulated reward over the time interval (0, t). calculation of et (t) is performed by (2.15). matrix of reward w is w =  w11 w12 w13 w14 w15 0 w17 w21 w22 0 0 0 0 0 0 w32 w33 0 w35 0 w37 0 0 0 w44 w45 w46 w47 w51 0 0 0 w55 0 0 w61 0 0 0 0 w66 0 0 0 0 0 0 0 w77  . (5.5) 13. eav(t) average reward at the time instant t. in accordance with (3.12) eav(t) is determined by solving the system of algebraic equations  eav1(t) ... eav7(t)  = λ  et1(t) ... et7(t) +  w11 ... w77(t) +  ∑ j 6=1 λ1jw1j ...∑ j 6=7 λ7jw7j  . (5.6) copyright c© 2018 assa. adv syst sci appl (2018) 30 v.s. viktorova, n.v. lubkov, a.s. stepanyants the numerical results of above indices calculation for different values of parameter β are summarized in table 5.1. all time-dependent metrics were calculated on one year period (t = 8760hours). we note once again that the solution, obtained by the mrm, is a vector whose ith component corresponds to the start of the markov process from state i. the table and diagrams include the first components of solution vectors, i.e the system starts from completely operational state s1. the functions of et (t), eav(t) for different values of catastrophic failure coverage parameter β are shown in fig.5.7 and fig.5.8 respectively. calculation of indices and plotting are performed with the following system parameters: λto = 0.0001(/hour);λdu = 0.00001(/hour);λa = 0.00005(/hour). µ = 0.1(/hour);µa = 0.004(/hour). αdu = 0.5;αa = 0.9. ηi = 0.4; ηii = 0.05. numerical values of impulse rewards are w15 = −105;w17 = −108;w12 = w13 = w14 = 0; w21 = 0;w32 = 0;w35 = −105;w37 = −108; w45 = −105;w46 = 0;w47 = −108; w51 = w61 = 0. numerical values of reward rates are w11 = 15(/hour);w33 = 10(/hour);w44 = 10(/hour); w22 = w55 = w66 = w77 = 0(/hour). fig. 5.7. to average accumulated reward et (t) versus time t for different coverage values β. the first family of curves (fig.5.7) shows the significant dependence of the average accumulated reward on the catastrophic failure coverage β. for the model parameters under consideration β should be greater than 0,9. the second family of curves(fig.5.8) shows a sharp decline in the growth of the average accumulated reward in time for all coverage values. it occurs due to the high rate of ps latent failures. to prevent such a recession, it is advisable copyright c© 2018 assa. adv syst sci appl (2018) a unified approach to rap analysis based on markov processes with rewards 31 fig. 5.8. to average reward eav(t) versus time t for different coverage values β. table 5.1. calculation results over one year period. catastrophic failure coverage β index 0.8 0.85 0.9 0.95 1 q(t) 0.612 0.612 0.612 0.612 0.612 tς(t) 8670.6 8682.4 8694.3 8706.2 8718.1 a(t) 0.9814 0.9836 0.9858 0.988 0.99 tff 9360.1 9360.1 9360.1 9360.2 9360.1 σ 9555.6 9555.6 9555.6 9555.6 9555.6 ta 4774.7 5143.4 5573.8 6082.8 6694.1 n(t) 0.942 0.943 0.944 0.946 0.947 na(t) 0.124 0.108 0.091 0.075 0.058 nai(t) 0.11 0.096 0.081 0.0663 0.0516 naii(t) 0.014 0.012 0.01 0.0083 0.0064 tσsd(t) 8.16 8.34 8.52 8.7 8.88 nsd(t) 0.817 0.835 0.853 0.871 0.889 et (t) -15489 3146 21834 40575 59368 eav(t) -6.068 -4.202 -2.33 -0.443 1.45 to carry out preventive maintenance of the process unit (to+ps), during which latent failures of the protection system are detected. 5.2. reliability and performability investigation of db-wss we take demand-based warm standby system (db-wss), described in section iv.a of article [23] as the second object of study. this db-wss is composed of two components a1 and a3 (we keep the numbering made by the authors of [23]). the components a1 and a3 can be in four degradation states, which differ in capacity. the components capacity levels are 5, 4, 2, 0 and 5,4,1,0, respectively. component a1 is online. component a3 can be online or in warm standby state depends on the system demand. we copyright c© 2018 assa. adv syst sci appl (2018) 32 v.s. viktorova, n.v. lubkov, a.s. stepanyants consider scenario b from [23], when demand determined as 5. so initial state of a3 is warm standby. the transition time distributions for the degradation processes for each component are exponential distributions with parameters λ1 = 1/100, λw3 = 1/300, λ3 = λo3 = 1/150 (/day), where (w) indicates warm standby state and (o) – operating online state. first, we will construct a markov model completely corresponding to the degradation process described in [23]. namely, we will construct a model of a binary-state system consisting of multistate components. the transition markov graph of this db-wss is shown in fig.5.9. the states s1 and s10 are operational states of the system that differ in the capacity of components. the state (j, k) corresponds to the jth capacity level of element a1(ca1 j ) and the kth capacity level of element a3(ca3 k ). state s11 is failed system state. ca1 j + ca3 k ≥ 5 for operational states; ca1 j + ca3 k < 5 for failed state. fig. 5.9. markov graph of binary-state db-wss. for the first case, we calculate the following reliability indicators: 1. r(t) reliability during (0, t). unreliability q(t) can be calculated from equation (2.15), where vector r = [ 0 0 0 0 0 0 λ3 λ1 + λ3 λ1 + λ3 λ1 0 ]t . reliability r(t) = 1−q(t). table 5.2 shows the results of our calculation of r(t) and values of r(t) taken from [23]. for the model under study (fig.5.9), an exact analytical solution was obtained (see appendix). the results of r(t) calculations using (a.2), (a.3) are given in the last column of table 5.2. 2. tff -mean time to first failure. copyright c© 2018 assa. adv syst sci appl (2018) a unified approach to rap analysis based on markov processes with rewards 33 table 5.2. calculation results of system reliability. r(t) time [day] mrm method proposed in [23] monte carlo from [23] exact analytical solution 50 0.99317 0.9917 0.9931 0.993170 100 0.93879 0.9333 0.9385 0.938794 200 0.67104 0.6610 0.6730 0.671043 300 0.38090 0.380899 400 0.18958 0.189584 500 0.08858 0.0864 0.0884 0.088584 the tff is determined in accordance with (3.4) tff=284.61 [day]. 3. σ standard deviation of random time to first failure. the σ is obtained from (3.5), where vector of free terms is [ −2t1 −2t2 −2t3 −2t4 −2t5 −2t6 −2t7 −2t8 −2t9 −2t10 ]t . σ = 154.05 [day]. next, we assume the possibility of a multi-level system operation. we assume that the states of the system with a capacity less than 5 can be subdivided into groups (levels) with capacity {4, 3, 2, 1, 0}. the model of the multistate system consisting of multistate components is shown in fig.(5.10). the properties of the levels are given in table 5.3. table 5.3. the groups properties and measures for multistate model levels the group capacity t i σ 1 s1,s2,s3,s4,s5,s6,s7,s8,s9,s10 5 273.10 2 s11,s13 4 63.48 3 s12 3 17.75 4 s15 2 16.38 5 s14 1 52.43 6 s16 0 76.86 for the second case, we calculate the following reliability and performability indicators: 1. t iς average total residence time of the system in the states of the ith group during (0, t). the t iς is determined in accordance with section 3.4 . for example, t 2 σ is calculated by (2.15) with vector r = [ 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 ]t . last column of table 5.3 shows the results of calculation of t iς for t = 500[day]. 2. kp(t) performability ratio. the kp(t) is calculated in accordance with (3.13). to calculate performability ratio, we should determine the reward matrix. for this system, the reward matrix will consist only of diagonal elements. we will define reward rates through the capacities, corresponding to the states of operation and degradation as wii = ci/(cmaxt). for cmax = 5 and t = 500 [day] we have w1 = · · · = w10 = 0.002;w11 = w13 = 0.0016;w12 = 0.0012;w14 = 0.0008;w15 = 0.0004;w16 = 0. as a result of solving the system of equations (2.15), we obtain kp(500) = 0.70315. this performability index is useful in choosing the best project version of a multi-level system, provided that the reliability requirements are met. in this case, the reliability of the system during the time interval (0,500) [days] is extremely small (see last raw of table 5.2). however, the value of the performability index is quite satisfactory, which confirms the correctness of the choice of the redundancy scheme. copyright c© 2018 assa. adv syst sci appl (2018) 34 v.s. viktorova, n.v. lubkov, a.s. stepanyants fig. 5.10. markov graph of multistate db-wss. copyright c© 2018 assa. adv syst sci appl (2018) a unified approach to rap analysis based on markov processes with rewards 35 6. conclusion this paper presents a unified approach for the estimation of reliability, availability, performability of engineering systems. the approach uses markov process with rewards to model transient and steady-state behavior of the systems. by using markov reward models, we can derive various instantaneous and cumulative measures and estimate wide range of indices. we present straightforward calculation technique for evaluation of reliability, availability and performability indices based on special definition of reward matrix. the technique implementation requires only solution of system of differential equations describing the behavior of the average reward. any additional calculations, such as numerical integration, are not required. based on this approach a software for reliability, availability, performability analysis was created. the software is operated under windows os and is written in c] programing language. it has an advanced graphical user interface, uses effective numerical method for solving mrm and applies ole automation for generation reports in ms word formats. the efficiency of the proposed approach and accuracy of numerical solution have been shown by the case studies of the technological system with protection and demand-based warm standby system. 7. appendix in this section, we present analytical solution of the model of a binary-state system consisting of multistate components (fig.5.9). the column vector p(t) of the state probabilities is calculated by solving a system of kolmogorov-chapman equations: p′1(t) = −p1(t) · (λ1 + λw3 ); p′2(t) = p1(t) · λ1 − p2(t) · (λ1 + λ3); p′3(t) = p1(t) · λw3 − p3(t) · (λ1 + λw3 ); p′4(t) = p2(t) · λ1 − p4(t) · (λ1 + λ3); p′5(t) = p2(t) · λ3 + p3(t) · λ1 − p5(t) · (λ1 + λ3); p′6(t) = p3(t) · λw3 − p6(t) · (λ1 + λw3 ); p′7(t) = p4(t) · λ1 − p7(t) · λ3; p′8(t) = p4(t) · λ3 + p5(t) · λ1 − p8(t) · (λ1 + λ3); p′9(t) = p5(t) · λ3 + p6(t) · λ1 − p9(t) · (λ1 + λ3); p′10(t) = p6(t) · λw3 − p10(t) · λ1; (7.1) the solution is copyright c© 2018 assa. adv syst sci appl (2018) 36 v.s. viktorova, n.v. lubkov, a.s. stepanyants p1(t) = −e−(λ1+λw3 )t; p2(t) = λ1 λ3 − λw3 (e−(λ1+λw3 )t − e−(λ1+λ3)t); p3(t) = λw3 te −(λ1+λw3 )t; p4(t) = ( λ1 λ3 − λw3 ) 2 (e−(λ1+λw3 )t − e−(λ1+λ3)t)− λ2 1 λ3 − λw3 e−(λ1+λ3)t; p5(t) = λ1 λ3 − λw3 (e−(λ1+λw3 )t − e−(λ1+λ3)t)− λ1λ3t λ3 − λw3 e−(λ1+λ3)t + λ1λ w 3 t λ3 − λw3 e−(λ1+λw3 )t; p6(t) = (λw3 t) 2 2 e−(λ1+λw3 )t; p7(t) = λ1 λ1 + λw3 − λ3 e−λ3t + λ2 1 + λ1(λ3 − λw3 ) (λ3 − λw3 )2 e−(λ1+λ3)t − λ3 1 (λ3 − λw3 )2(λ1 + λw3 − λ3) e−(λ1+λw3 )t + λ2 1t λ3 − λw3 e−(λ1+λ3)t; p8(t) = 2λ2 1 (λw3 − λ3)2 e −(λ1+λw3 )t + λ2 1(λw3 − 2λ3)t (λw3 − λ3)2 e−(λ1+λ3)t − (λ1t) 2λ3 λ3 − λw3 e−(λ1+λ3)t + λ2 1λ w 3 t (λw3 − λ3)2 e −(λ1+λw3 )t − 2λ2 1 (λw3 − λ3)2 e −(λ1+λ3)t; p9(t) = λ1λ w 3 t λw3 − λ3 e−(λ1+λ3)t − λ1(λ3t) 2 2(λ3 − λw3 ) e−(λ1+λ3)t − λ1 λw3 − λ3 e−(λ1+λw3 )t − λ1λ w 3 t λw3 − λ3 e−(λ1+λw3 )t − λ1(λw3 t) 2 2(λw3 − λ3) e−(λ1+λw3 )t + λ1 λw3 − λ3 e−(λ1+λ3)t; p10(t) = e−λ1t − (λw3 t) 2 + 2λw3 t+ 2 2 e−(λ1+λw3 )t. 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(1979) numerical methods for solving stiff systems. moscow: nauka publ., 81–112. copyright c© 2018 assa. adv syst sci appl (2018) adv syst sci appl 2022; 01:15–34 published online at https://ijassa.ipu.ru. direct torque control-fuzzy type 2 for direct current link voltages balancing of the five-level cascade converters used in a wind energy conversion system abdelhafidh moualdia1*, saleh boulkhrachef1, patrice wira2 1lrea laboratory, faculty of technology, university of medea, ain d’heb medea, algeria 2irimas laboratory, university of haute alsace, mulhouse, france abstract: the available power of a wind system depends mainly on the wind speed. in addition, the wind system will give a power output that varies according to the speed of its generator which is a double feed asynchronous machine in our case. in other words, there is an optimal operating point that makes the most of the power available. in this work an innovative technique of capturing maximum power based on type-2 fuzzy systems. the principle of this maximum power point tracking algorithm is to look for an optimal operating relationship at maximum power and then track the maximum power based on this relationship. as part of the variable speed conversion of wind energy, this article proposes a simplified power electronics for injecting the energy produced in the network, the conversion chain includes a variable speed double feed asynchronous generator, two (back-to-back) converters five-level neutral point clamped type operating in grid side rectifier and rotor-side inverter mode. the main objective of this article is to develop a new stabilization strategy of direct control compatible with voltage inverters at five levels, more particularly neutral point clamped structure. this strategy allows flow and torque control of the double feed asynchronous machine and stabilizes the input capacitor voltages of the inverter. the response of the system obtained with this algorithm makes it possible to validate the soft solution proposed and show, during variation of the wind speed, a fast and precise adaptation of the speed of the double fed induction generator. keywords: double fed induction generator, direct current link voltage, fuzzy type 2, direct torque control, wind energy, five level converter cascade 1. introduction from a general point of view, regardless of the topology, multilevel conversion structures offer huge advantages over a conventional solution, based on a two-level converter [1, 2]. these advantages are visible, on the one hand from a technological point of view and on the other hand from a functional point of view. first of all, the quality of the output signal of the inverter can be improved thanks to the additional degree of freedom which is the number of voltage level [3–6]. the switched voltage is of reduced amplitude and the switching is therefore easier to manage. despite the advantages of multi-level inverters, the instability of capacitor voltages on the dc side remains the major disadvantage of multi-level npc (neutral point clamped) inverters [7, 8]. as a result, the imbalance of these voltages leads to the failure of power components and deformation of the output voltage. for this, several solutions are proposed. these solutions include methods based on vector modulation techniques, where the concept of redundant ∗corresponding author: amoualdia@gmail.com 16 a. moualdia, s. boulkhrachef, p. wira voltage vectors has been applied to balance the electrical charge between capacitors [9–11]. however, for high levels, the number of voltage vectors increases considerably and thus the control becomes complex. other solutions, based on the addition of auxiliary circuits for the balancing of these dc voltages of the inverter, are proposed in the literature . to be able to replace the dc drive and enjoy the advantages of the asynchronous motor, the control must be more and more efficient. dtc (direct torque control) control strategy has emerged as competitive with vector control techniques. this dtc command was invented by i. takahashi in themid− 1980s. it is based on the separate regulation of the rotor flux and the electromagnetic torque of the double-feed asynchronous generator. in order to provide better control of npc-structured five-level inverter voltages and achieve better performance, the wind energy conversion system is connected to the power grid using rectifiers controlled by the width modulation. pulse (pwm). these rectifiers can provide a low harmonic distortion in the input currents, a grid-side adjustable power factor, and a constant dc output voltage. control of rectifiers based on fuzzy systems is also possible [12]. in particular, the fuzzy controller methodology appears useful when information sources are considered unclear or uncertain. in an ordinary fuzzy system, the membership functions, once determined, are completely precise, and therefore unable to take into account the uncertainty of the linguistic terms used in the premises and consequences of the rules. to solve this limitation, the fuzzy set type-2 was introduced as an extension of the fuzzy set type-1, where each degree of membership of each element is itself a fuzzy set in [0, 1]. [13–15]. first, we first present a description of the dfig-based wind energy conversion system. the second part of this work will be dedicated to the synthesis of innovative technique of maximum power capture based on type-2 fuzzy systems. the principle of this maximum power point tracking (mppt) algorithm is to look for an optimal operating relationship at maximum power and then track the maximum power based on this relationship. in the third step, the development of a stabilizing dtc control, to fig. 1.1. wind energy conversion system based on five-level npc cascaded converters solve the problem of the imbalance of the input voltages of the 5-level converter with npc structure on the rotor side. taking advantage of the redundancies of the state of the rotor side converter, producing the same voltage vector, but opposite effects on the voltages of the capacitors, an algorithm is developed. this allows us to balance the capacitor voltages in addition to controlling the torque and flux of the double feed asynchronous generator. the copyright © 2022 assa. adv syst sci appl (2022) dtc-fuzzy type 2 for dc link voltages balancing of the five-level cascade 17 wind energy conversion system description based on five-level npc cascaded converters is shown in figure 1.1. nomenclature β blade pitch angle; v wind speed (m/s); ρ air density; λ speed ratio; cp power coefficient; pm mechanical power (kw); tm mechanical torque ; vsdq, vrdq dq axis stator and rotor voltages ; isdq, irdq, dq axis stator and rotor current ; ωs, ωr, stator and rotor pulsation (rd/s) ; φsdq,φrdq, dq axis stator and rotor flux ; m, mutual inductance ; ps, qs generator active and reactive powers ; p1,p2 logical function ; j, f moment of inertia and coefficient of friction ; e1,e2,e3 load conditions ; k phase number (k=1, 2, 3); c1; c2; c3; c4 dc-link capacitors; uc1; uc2; uc3; uc4 dc bus voltages; ic1; ic2; ic3; ic4 capacitors currents; irec1; irec2; irec3; irec4 output rectifier currents; iref netk reference network phase current; iref rec reference network phase current; vsk; isk stator phase voltage and current; vsαβ, isαβ stator voltage and current in the stationary α− β plane ω,ωn rotor speed and speed nominal value; tem,tref electromagnetic torque and reference value; φs stator flux magnitude; φsα,φsβ stator flux magnitude in α− β plane fnet network frequency; vdi discrete voltage level of vector vs; cfl output hysteresis flux controller; ctr output hysteresis torque controller; abbreviations used npc : neutral point clamped hvdc : high voltage direct current dc : direct current dtc : direct torque control pwm : pulse width modulation csr : current source rectifier im : induction machine svm : space-vector modulation flc : fuzzy logic controller t1fls : type-1 fuzzy sets t2fls : type-2 fuzzy sets pi : proportional integral controller dfig : double fed induction generator mppt : maximum power point tracking rsc : rotor-side converter gsc : grid-side converter copyright © 2022 assa. adv syst sci appl (2022) 18 a. moualdia, s. boulkhrachef, p. wira 2. description of the wind conversion system 2.1. wind turbine model the mechanical power available on the shaft of a wind turbine can be expressed as [22, 26]: pm = 0.5cp (λ) πρr2v 3 1 , (2.1) for the variables speed wind turbines, approximate expression of the power coefficient can be described by the following expression: cpf (λ, β) = c1 ( c2 λi − c3 − c4 ) exp ( −c5 λi ) +c6λ, (2.2) where: 1 λi = 1 λ+ 0.08β − 0.035 β3 + 1 (2.3) where, c1 = 0.5176, c2 = 116, c3 = 0.4, c4 = 5, c5 = 21, c6 = 0.0068.the torque produced by the turbine is expressed in the following way: tm = pm ωm = ( 0.5πρr3v 2 1 ) cp(λ, β)) λ (2.4) no wind turbine could convert more than 59 of the kinetic energy of the wind into mechanical energy turning the rotor [16]. this is known as the betz limit and it’s the cpmax theoretical maximum coefficient of power (for any wind turbine): cpmax = 16/27 ≈ 0.593. in practice, for the good turbines it’s in the range of 0.45 to about 0.50. the tip speed ratio (λ) for wind (a) (b) fig. 2.2. (a): for pitch angle effect on the aerodynamic coefficient of power and (b): for mechanical power output. turbines is the ratio between the rotational speed of the tip of a blade and the actual velocity of the wind, see figure 2.2. copyright © 2022 assa. adv syst sci appl (2022) dtc-fuzzy type 2 for dc link voltages balancing of the five-level cascade 19 table 2.1. inference matrix de\e ng nm np ez pp pm pg ng ng ng nm nm np np ez nm ng nm nm nm np ez pp np ng nm np np ez pp pm ez ng nm np ez pp pm pg pp nm np ez pp pp pm pg pm np ez pp pm pm pm pg pg ez pp pp pm pg pg pg fig. 2.3. tip speed ratio control (mppt). 2.2. maximum power tracking via fuzzy-type 2 systems at a given wind speed, the maximum turbine energy conversion efficiency occurs at an optimal tsr figure 2.3. therefore, as wind speed changes the turbine’s rotor speed needs to change accordingly in order to maintain the optimal tip speed ratio tsr and thus to extract the maximum power from the available wind resources. the structure of the fuzzy type-2 regulator is shown in figure 2.4. in order to have the desired performances, the normalization gains at the input and at the output of the regulator are determined by adjustment. for input and output variables consisting of seven fuzzy sets type-2 interval, the conventional anti-diagonal inference matrix of a fuzzy system is given in table 2.1. copyright © 2022 assa. adv syst sci appl (2022) 20 a. moualdia, s. boulkhrachef, p. wira fig. 2.4. structure of the fuzzy type-2 controller. (a) (b) fig. 2.5. membership functions.(a):membership functions of input variables,(b):membership functions of output variables 2.3. modelling of the double fed induction generator in the rotating field reference frame of park, the model of the dfig is given by the following equations [30, 31]: vsd = rsisd + d dt φsd − ωsφsq vsq = rsisq + d dt φsq + ωsφsd vrd = rrird + d dt φrd − ωrφrq vrq = rrirq + d dt φrq + ωrφrd, (2.5) stator and rotor voltages components: φsd = lsisd +mird φsq = lsisq +mirq φrd = lrird +misd φrq = lrirq +misq. (2.6) copyright © 2022 assa. adv syst sci appl (2022) dtc-fuzzy type 2 for dc link voltages balancing of the five-level cascade 21 double fed induction generator electromagnetic torque: cem = cr + j dω dt + fω, (2.7) generator active and reactive powers at the grid side are:{ ps = vsdisd + vsqisq qs = vsqisd − vsdisq. (2.8) 2.4. five-level npc converter multilevel converters are power-conversion systems composed by an array of power semiconductors and capacitive voltage sources that, when properly connected and controlled, can generate a multiple-step voltage waveform with variable and controllable frequency, phase, and amplitude. the stepped waveform is synthesized by selecting different voltage levels. the numbers of levels of a converter is defined as the number of steps that can be generated by the converter between the output terminal and any reference node within the converter, is usually denoted by n and called neutral. to be called a multilevel converter, each phase of the converter has to generate at least three different voltage levels. this differentiates the classic tw o-level voltage source converter (2l− v sc) from the multilevel family. the neutral clamped inverter, also known as diode clamped inverter. the basic architecture of this inverter discussed in references [17, 18]. the neutral clamped inverter obtained the fig. 2.6. the five-level npc converter. staircase output voltage. if m is the number of level, then the number of capacitors required on the dc bus are (m -1), the number of power electronic switches per phase are 2(m -1) and the number of diodes per phase are 2(m− 2). the dc bus voltage has three levels using two capacitors c1 and c2, for five levels using four capacitors c1, c2, c3 and c4 as shown in figure 2.6. table 2.2 lists the voltage levels and their corresponding switch states. state 1 means that the switch is on, and 0 means that the switch is off. we suppose that all the dc voltage sources are the same and equal to nominal value . the output voltage vector is defined copyright © 2022 assa. adv syst sci appl (2022) 22 a. moualdia, s. boulkhrachef, p. wira table 2.2. switching states and output voltage of the first leg of the five-level npc inverter state v1m s11 s12 s13 s14 s15 s16 s17 s18 +2 +2uc 1 1 1 0 0 0 0 0 +1 +uc 1 1 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 -1 −uc 0 0 1 1 1 0 0 1 -2 −2uc 0 0 0 1 1 1 0 0 as: vs = vame j0 + vbme −j2π/3 + vcme −j4π/3 = vα + jvβ. (2.9) it can be observed that 24 vectors can be generated by a unique switching state, 18 vectors can be generated using two switching states each (2 redundancy), 12 vectors can be generated using three switching states each (3 redundancies), 6 vectors can be generated using four switching states each (4redundancies), and one vector can be generated using five switching states (5redundancies). [15, 16] 3. dtc for five-level rotor side converter (rsc) in recent years, a new control strategy based on direct control of flux and torque has been proposed. this technique, known as dtc, enables the induction motor to deliver a very quick and accurate torque response.the instantaneous values of flux and torque are calculated from measured variables (voltages and currents) and then controlled directly by selecting optimum inverter switching modes.the objective of this section is to present the dtc of induction machine fed by a five–level npc inverter. the schematic diagram of the proposed dtc system is shown in figure 3.7. as in the original dtc principle the α− β plane will be divided into several sectors [26]. in a five-level inverter the number of discrete voltage vectors is more important than those obtained with a two-level inverter. thus, the α− β plane will be divided into 12 sectors rather than six. at each one of these sectors, an appropriate voltage vector will be assigned to keep flux and torque references as needed.the speed of the stator flux vector is given by the modulus of the applied space voltage vector. thus, the space voltage vectors will be chosen according to the rotor speed. voltage vectors with low amplitude will be chosen for low speeds, and vectors with greater amplitude will be chosen for higher speeds. in the five-level inverter, four tables have been used, according to a specific speed range, as shown in table 2.2 . three different states are used to identify torque status and two states for flux. 4. control strategy of the balancing capacitor voltage we will present the proposed dtc algorithm to balance dc bus voltages using redundant configurations of the five-level inverter. the, each discrete voltage level can be obtained by more than one switching state. as the voltage evolution for a given capacitor will be different for each state, this redundancy let’s to control the capacitors voltages while the requested space vector voltage is supplied. based on this property, a control strategy will be copyright © 2022 assa. adv syst sci appl (2022) dtc-fuzzy type 2 for dc link voltages balancing of the five-level cascade 23 fig. 3.7. schematic diagram of the proposed dtc system. table 4.3. six groups of redundant vectors group 1 vp1 vp4 vp7 vp10 vp13 vp16 group 2 vp2 vp6 vp8 vp12 vp14 vp18 group 3 vp3 vp5 vp9 vp11 vp15 vp17 group 4 vp19 vp21 vp23 vp25 vp27 vp29 group 5 vp20 vp22 vp24 vp26 vp28 vp30 group 6 vp31 vp32 vp33 vp34 vp35 vp36 presented and applied to a five-level inverter. to do so, we firstly study the effect of different redundant vectors on capacitor voltages. in table 5.4, all the redundant vectors of the space vector diagram and the corresponding capacitors currents according to the load (i1, i2, i3) are presented. depending on the forms of relationships (equation 2.9, 4.10 and 4.11 in table 4.3), we distinguish six groups of redundant vectors: 4.1. effect of redundant vectors on capacitor voltages redundant vectors of each group can increase or decrease capacitor voltages, depending on load conditions sign of (equation 2.9, 4.10 and 4.11. for groups with one equation 2.9 (groups 1, 4 and 6), we have two possibilities of load conditions, each one is associated with a logical function: p1 = 1 if e1 > 0 else p1 = 0 p2 = 1 if e1 ≤ 0 else p2 = 0. (4.10) copyright © 2022 assa. adv syst sci appl (2022) 24 a. moualdia, s. boulkhrachef, p. wira for groups with three (equation 2.9, 4.10 (groups 2, 3 and 5), we have six possibilities of load conditions, associated with six logical functions: p1 = 1 if e1 < 0, e2 < 0, and e3 > 0, else p1 = 0; p2 = 1 if e1 < 0, e2 > 0, and e3 < 0, else p2 = 0; p3 = 1 if e1 < 0, e2 > 0, and e3 > 0, else p3 = 0; p4 = 1 if e1 > 0, e2 < 0, and e3 < 0, else p4 = 0; p5 = 1 if e1 > 0, e2 < 0, and e3 > 0, else p5 = 0; p6 = 1 if e1 > 0, e2 > 0, and e3 < 0, else p6 = 0. (4.11) 4.2. choice of redundancies for each case of redundancy, the vector which reduces the fluctuation voltages in capacitors will be selected. the diagram of the control algorithm is shown in figure 4.8. we select the vector which charge the undercharged capacitors, and discharge the overcharged ones. to do so, we must measure capacitor voltages and calculate their deviation case. each deviation case is characterized by a logical function cj (table 5.4.). for example, the first case : uc1 <uc2 < uc3 < uc4 is associated to the function c1, defined as: for each group of redundancies, given the deviation case (cj). and load state (pi), the vector that will decrease the largest capacitor voltage and increase the smallest capacitor voltage is selected. in table 5.4, the choice of redundancies (a, b, c, d) according to the load states and capacitor voltages deviation case is presented. fig. 4.8. capacitor voltages balancing algorithm 5. control of five-level grid side converter (gsc) in this section, a current control strategy of three-phase five-level pwm rectifier applying a type-2 fuzzy system is proposed. figure 5.10 shows the schematic of the control of the 5-level rectifier. this block diagram is based on the control of the average value of the output voltages, using a type-2 fuzzy controller. the purpose of servo-control is to determine the network reference currents applied to the hysteresis comparators. this control is done with a copyright © 2022 assa. adv syst sci appl (2022) dtc-fuzzy type 2 for dc link voltages balancing of the five-level cascade 25 unit power factor, requiring zero reactive power, if imposing a zero phase shift between the voltage and current per phase of the power grid. by hysteresis current control, the switching of the transistors ensures that the input phase currents track the reference currents within the tolerance band. the three-phase reference currents are given by the following expressions: iref = √ 2ireg sin ( ωt− ϕ− (j − 1)2π 3 ) ) ; j = 1, 2, 3 (5.12) ϕ is the desired phase shift given to the currents with respect to the corresponding voltages. in our case, ϕ is set to zero for unity power factor operation. 5.1. type-2 fuzzy logic controller design figure 5.9 depicts the structure of a type-2 flc. it is quite similar to a type-1 flc. the major difference being that at least, one of the fuzzy sets are type-2 and a type-reducer is needed to convert the output of the fuzzy inference engine into type-1 fuzzy set called type-reduce set. these type-reduced set is defuzzified to obtain the crisp output direg . throughout of fig. 5.9. structure of a t2 flc this paper, an interval typ2 flc with singleton input fuzzification is considered. indeed, the interval type-2 fuzzy sets are simple to use and can simplify effectively the computational process of type-reduction. the controller output direg is calculated by the t2flc of figure 5.9, which uses the rule base of table 2.1. this table is commonly used to construct fuzzy controller. the next step is to construct the fuzzy rules for processing the fuzzy input. assume that there are m rules in the type-2 fuzzy system, where the ith rule has the following form: ri : if e is f̃ i de then direg is g̃ i = [uil, u i r] ; i = 1, .....,m. (5.13) where f̃ i e and f̃ i de are antecedent linguistic terms modelled by the interval type-2 triangular fuzzy sets, direg is the output and the consequent term gi is an interval type-1 set which can be called weighting interval set [uil, u i r] . uil and uir are the singleton lower and upper control actions of the consequent part. copyright © 2022 assa. adv syst sci appl (2022) 26 a. moualdia, s. boulkhrachef, p. wira ta bl e 5. 4. se le ct io n of re du nd an ci es g ro u p s 1 2 3 4 5 6 c j lo ad st at e p i\ \ st at es ca p ac it or s vo lt ag es p 1 p 2 p 1 p 2 p 3 p 4 p 5 p 6 p 1 p 2 p 3 p 4 p 5 p 6 p 1 p 2 p 1 p 2 p 3 p 4 p 5 p 6 p 1 p 2 1 u c 2 < u c 1 < u c 3 < u c 4 a b b a b a b a b a b a b a a c c a c a c a d a 2 u c 2 < u c 1 < u c 4 < u c 3 a b b b b a a a b a b a b a a c c a c a b a c a 3 u c 2 < u c 3 < u c 1 < u c 4 a b b a b a b a b a b a b a a c c b b a c a d a 4 u c 2 < u c 3 < u c 4 < u c 1 a b b a b a b a b b b a a a a b b b b a a a b a 5 u c 2 < u c 4 < u c 1 < u c 3 a b b b b a a a b b b a a a a b b c c a b a c a 6 u c 2 < u c 4 < u c 3 < u c 1 a b b b b a a a b b b a a a a b b c b a a a b a 7 u c 1 < u c 2 < u c 3 < u c 4 a b b a b a b a b a a b b a a c c a c b c a d b 8 u c 1 < u c 2 < u c 4 < u c 3 a b b b b a a a b a a b b a a c c a c b b a c b 9 u c 1 < u c 3 < u c 2 < u c 4 a b b a a b b a b a a b b a b c c a a b c b d b 10 u c 1 < u c 3 < u c 4 < u c 2 b a a a a b b b a a a b b b b c c a a b c b a b 11 u c 1 < u c 4 < u c 2 < u c 3 b a a b b a a b a a a b b b a c c a c b b a c b 12 u c 1 < u c 4 < u c 3 < u c 2 b a a b a b a b a a a b b b b c c a a b b b a b 13 u c 3 < u c 2 < u c 1 < u c 4 a b b a a b b a b a b a b a b a a b b c c b d c 14 u c 3 < u c 2 < u c 4 < u c 1 a b b a a b b a b b b a a a c a a b b c a c b c 15 u c 3 < u c 1 < u c 2 < u c 4 a b b a a b b a b a a b b a b a a b a c c b d c 16 u c 3 < u c 1 < u c 4 < u c 2 b a a a a b b b a a a b b b b a a b a c c b a c 17 u c 3 < u c 4 < u c 2 < u c 1 b a a a a b b b a b b a a b c a a b b c a c b c 18 u c 3 < u c 1 < u c 4 < u c 2 b a a a a b b b a b a b a b b a a b a c a c a c 19 u c 3 < u c 4 < u c 1 < u c 2 b a a b b a a b a b b a a b c b b c c a b c c d 20 u c 4 < u c 2 < u c 3 < u c 1 b a a b b a a b a b b a a b c b b c b a a c b d 21 u c 4 < u c 1 < u c 2 < u c 3 b a a b b a a b a b a b a b c b b c c b b c c d 22 u c 4 < u c 1 < u c 3 < u c 2 b a a b a b a b a b a b a b c a a c a b b c a d 23 u c 4 < u c 3 < u c 2 < u c 1 b a a b a b a b a b b a a b c a a c b c a c b d 24 u c 4 < u c 3 < u c 1 < u c 2 b a a b a b a b a b a b a b c a a c a c a c a d copyright © 2022 assa. adv syst sci appl (2022) dtc-fuzzy type 2 for dc link voltages balancing of the five-level cascade 27 fig. 5.10. grid side five levels npc converter control the inference engine combines the fuzzy rules in order to map the crisp inputs to interval type-2 fuzzy output sets. based on the input and the antecedents of the rules, it calculates a firing interval for each rule and then applies these firing levels to the consequent fuzzy sets. the firing interval [f i, f i ] of the ith rule is an interval type-1 set, which is determined by its left-most and right-most points f i and f i such that : f i = min ( µ f̃ i e (e) , µ f̃ i e (de) ) (5.14) f i = min ( µ f̃ i e (e) , µ f̃ i e (de) ) (5.15) where µ f̃ i e (x) and µ f̃ i e (x) represent the grade of the lower and the upper membership function of the crisp input x to the type-2 fuzzy set f̃ i x in ith rule. the output corresponding to the fired rule is a type-2 fuzzy set which must be type-reduced before the defuzzifier can be used to generate a crisp output. there are several methods for type reduction, such as center-of-sums type-reduction, height type-reduction, modified height type-reduction, and center-of-sets type-reduction. in this paper, the centre-of-sets type-reducer is used which can be represented by diref = [ di irefl, di i refr ] .the output is an interval type-1 set. therefore, we only need to compute its two endpoints di irefl and di irefr as follows: direfl = ∑m i=1 f i l u i l∑m i=1 f i l (5.16) direfr = ∑m i=1 f i ru i r∑m i=1 f i r (5.17) in order to compute diref , we need to compute direfl and direfr . this can be done using the procedure given in [25], [26]. without loss of generality, assume that the pre-computed uir are arranged in ascending order,i.e.u1r ≤ u2r ≤ ..... ≤ umr • step 1 : compute direfr in equation 5.17 by initially setting f i r = (f i+f i) 2 , for i = 1, 2, ....,m where f i and f i have been computed using equation 5.15 respectively,and let direfr ′ = direfr. • step 2 : find k (1 ≤ k ≤m − 1) such that ukr ≤ direfr ′ ≤ u (k+1) r . copyright © 2022 assa. adv syst sci appl (2022) 28 a. moualdia, s. boulkhrachef, p. wira • step 3 : compute diref in equation 5.17 with f i r = f i for i ≤ k and f i r = f i for i > k, then set direfr ′′ = direfr . • step 4 :if direfr ′′ 6= direfr ′, then go to step 5. if direfr ′′ = direfr ′ then set direfr = direfr ′′ and go to step 6. • step 5 :set direfr ′ = direfr ′′ and return to step 2. • step 6 : end the procedure for computing direfl is very similar, only two changes need to be made: in step 2, we need to find k (1 ≤ k′ ≤m − 1) such that uml ≤ direfl ′ ≤ uml , m=k’+1 and in step 3, let f i l = f i for i ≤ k′ and f i l = f i for i > k′ . from the type-reduction stage, we have for each output a type-reduced set. the crisp output of the type-2 fuzzy controller can be obtained by using the average value of direfr and direfl . hence, the defuzzified crisp output becomes: direfk = direfkl + direfkr 2 (5.18) fig. 5.11. wind speed profile 6. simulation results in order to demonstrate the feasibility of the proposed control method, simulation testing are undertaken on the structure shown in figure 1.1. the most commonly encountered disturbances in drive applications are changes in the load torque or changes in the speed. it is important thereby to show if the proposed cascade is able to handle aforementioned transients and ensure the stability of dc-bus voltages. the wind speed profile applied to energy conversion system is shown with in figure 5.11. the multilevel dtc strategy has been tested by simulation in different level of speed control loop.the reference speed is obtained by the mppt block. the control objective in this section is to show the performance in both interest operation regions (i and iii) of the turbine characteristic. in the region iii, one can see that the stator power is limited at its copyright © 2022 assa. adv syst sci appl (2022) dtc-fuzzy type 2 for dc link voltages balancing of the five-level cascade 29 fig. 6.12. mppt result maximum value by the pitch angle control,as shows figure 6.12. this is approved by the waveforms illustrated in figure 6.17 of the tip speed ratio (λ) and the power coefficient (cp). the mechanical speed is kept constant, at its limit value, as plotted in figure 6.12. fig. 6.13. response of electromagnetic torque and rotor flux however, in region i the power is maximized by the mppt algorithm. in this case, as expected, tip speed ratio and the power coefficient are maintained at constant values see figure 6.12 (β = 0,λ = λopt = 8.1, cp = cp−max = 0.49). the reference tracking reflecting the robustness of the proposed control under random behavior of wind speed. figure 6.13 shows the rotor flux waveform, it is circular and kept constant at 1.6 wb. the random evolution of the rotor current magnitude is related to the electromagnetic torque variations and their pulsation is depending on the slip variations. the operation as a generator is illustrated by the negative sign of the electromagnetic torque cem < 0, with a variable amplitude, which depends on the wind speed evolution, as presented in figure 6.13 where we can see that the operation with constant power is clear within the over-speed zone. in figure 6.14, the copyright © 2022 assa. adv syst sci appl (2022) 30 a. moualdia, s. boulkhrachef, p. wira capacitor voltages with the proposed dtc balancing strategy are shown . it can be seen that the balancing of the capacitor voltages is achieved sufficiently over the full speed state and for all load torque of induction machine, which prove that the stability of dc voltages with proposed strategy is independent of machine operating points. in the studystate condition, the maximum of each capacitor voltage ripple is less than 4v (2%) , which shows the effectiveness of the dtc balancing strategy. consequently, the different between the voltages capacitor tends to zero, as shown in figure 6.15. fig. 6.14. response of dc-link voltages with proposed dtc control strategy. fig. 6.15. error dc-link voltages. 7. conclusion in this work, the application of direct torque control in a wind energy conversion system supplied by a five level npc back-to-back converter has been presented. the choice of this copyright © 2022 assa. adv syst sci appl (2022) dtc-fuzzy type 2 for dc link voltages balancing of the five-level cascade 31 fig. 6.16. response of stator and rotor currents structure allows a bidirectional flow of power, and provides regeneration capability.another advantage of the back-to-back system is that it can control easily the input power factor. the effectiveness of the proposed dtc-based voltage balancing strategy it demonstrated under various operating conditions of double fed induction generator. is may therefore be concluded that the proposed multilevel direct torque control not only has the advantage of reducing the undesirable torque ripple, but also has the additional advantage of preventing the voltage drift phenomen on of the dc-link capacitors of the back-to-back system, longer life of dc-link capacitors can be achieved. furthermore, the employment of multilevel topology improves the stator voltage quality, reducing electromagnetic interference and isolation stress problems of windings. this results in the reduction of radiated emissions, which makes this kind of drives a less polluting system than the traditional two-level drive. in addition, this study has successfully demonstrated the application of type-2 fuzzy systems to control the dc-link voltage, and the maximization of power tracking. it found that the type-2 fuzzy control scheme could achieve good performance in terms of overshoot, steady-state error, torque disturbance, and variable speed tracking. copyright © 2022 assa. adv syst sci appl (2022) 32 a. moualdia, s. boulkhrachef, p. wira fig. 6.17. response of stator active and reactive power acknowledgements the research is part of a project prfu’2020, realized respectively in the laboratory of electrical engineering and automatic lrea research, university of medea, and irimas laboratory, university of haute alsace, mulhouse, france. references 1. polinder, h., f. f. a. van der pijl, g. j.de vilder and p. j. 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(2009) high-order sliding mode control of a dfig-based wind turbine for power maximization and grid fault tolerance, ieee international electric machines and drives conference, miami,fl,usa, 183—189. copyright © 2022 assa. adv syst sci appl (2022) introduction description of the wind conversion system wind turbine model maximum power tracking via fuzzy-type 2 systems modelling of the double fed induction generator five-level npc converter dtc for five-level rotor side converter (rsc) control strategy of the balancing capacitor voltage effect of redundant vectors on capacitor voltages choice of redundancies control of five-level grid side converter (gsc) type-2 fuzzy logic controller design simulation results conclusion microsoft word 892 article source, copyedited.doc adv syst sci appl 2020; 02; 82-97 published online at https://ijassa.ipu.ru. original russian text © yu. v. mitrishkin, n. m. kartsev, a. e. konkov, m. i. patrov, 2019, published in problemy upravleniya / control sciences, 2019, no. 3, pp. 3–15. plasma control in tokamaks. part 3.1. plasma magnetic control systems in iter yuri mitrishkin1,2*, nikolay kartsev2, artem konkov1,2, mikhail patrov3 1) m.v. lomonosov moscow state university, faculty of physics, moscow, russia 2) v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia 3) a.f. ioffe institute of russian academy of sciences, st. petersburg, russia e-mail: yvm@mail.ru e-mail: n.kartsev@yandex.ru e-mail: konkov@phys.msu.ru e-mail: michael.patrov@mail.ioffe.ru abstract: the plasma magnetic control systems for iter (international thermonuclear experimental reactor) are presented. the systems comprise original engineering solutions for plasma position, current and shape control for the two versions of iter: iter-1 and iter-2, including those designed in v. a. trapeznikov institute of control sciences of the ras. it is noted that in iter-1 the plasma position and shape were controlled by all pf-coils and robust h∞-controllers, while to decrease peaks of control power for suppressing minor disruptions the additional nonlinear circuit was used without significant changes in displacements of the gaps between the plasma separatrix and the first wall. in iter-2 the special circuit with a fast voltage rectifier was used for plasma vertical speed stabilization about zero, while for plasma current and shape control the special cascade control systems were designed with and without the control channels decoupling, with robust h∞-controllers and predictive model, and with adaptive stabilization of the plasma vertical position as well. to increase the plasma controllability region in the vertical direction the additional horizontal field coils were introduced into the iter-2 vacuum vessel and the capabilities of the system with the new circuits to control the plasma vertical position at the noise presence were investigated. keywords: tokamak, plasma, plasma magnetic control, iter introduction currently, the flagship in solving the problem of controlled thermonuclear fusion is iter – international thermonuclear experimental reactor [1-3]. tokamak reactor iter is being built in france (cadarache) by an international consortium consisting of the european union, the usa, japan, russia, china, india, and south korea. all the most advanced tokamaks work in support of iter to provide a basic physical understanding of its reliable operation capabilities, including through magnetic and kinetic plasma control systems. on the russian side, a number of magnetic control systems for the plasma position, current, and shape have been developed for iter, which have been modelled on plasmaphysical codes dina (triniti, troitsk, russia) and pet (d.v. efremov research institute of electrophysical equipment, st. petersburg, russia). due to the peculiarity of the iter poloidal system associated with superconductivity of the poloidal field coils, in v.a. trapeznikov institute of control sciences of ras (moscow, russia) the robust magnetic * corresponding author: yvm@mail.ru plasma control in tokamaks. part 3.1. plasma magnetic control systems in iter 83 copyright ©2020 assa. adv. in systems science and appl. (2020) plasma control systems were developed. these systems are capable of working in conditions of uncertainty caused by the unmodeled dynamics of the plasma as a complex not fully studied plant. the developed systems work in the general hierarchical structural scheme of the plasma control system in iter, taking into account the structural scheme of controlling the position, current and gaps between the plasma separatrix and the first wall of the vacuum vessel (fig. i.1) [4]. fig. i.1. block diagram of the hierarchical system of magnetic plasma control in iter in the block diagram of magnetic plasma control in iter (fig. i.1) the poloidal field control device (pf-controller, pf – poloidal fields) will presumably consist of three separate units, namely: a supervisor located at the upper (adaptive) level of control, a feedforward controller and a feedback controller of the lower (basic) level. the basic level (control loop) of the poloidal field control system includes a controlled plant consisting of a plasma, coils of a poloidal magnetic field, passive metal structures, sensors, a measuring (diagnostic) unit, an actuator device representing power sources for the central solenoid and poloidal field coils; a feedback controller and a feedforward controller. the supervisor (fig. i.1), interacting with the plasma control system and the interlock system, controls the operation of the basic level. it generates program signals at the rate of observation and performs adaptive correction of the feedback controller about 10 times slower than the basic circuit. at present, magnetic plasma control systems are also being developed for the first thermonuclear power plant demo (demonstration power plant) [5]. it is important to develop a poloidal system, which would not be forced to introduce additional plasma position control coils inside the vacuum vessel, as was done in iter-2 which reduces the reliability of the thermonuclear reactor in stationary mode. coils for effective control of the plasma position, providing a sufficient controllability region in vertical direction with a given limit on the supply voltage at the vertical instability of the plasma, can be installed between the vacuum vessel and the coil of the toroidal magnetic field, as is done in the asdex upgrade tokamak and the t-15md tokamak project [2]. in the japanese variant of demo cs & pf coil power supplies disturbance diagnostics actuator plant measuring unit supervisor feedforward controller feedback controller ref y v ff pf controller interlock system plasma control system (codac) y v fb + + + adaptive level basic level i coils y b coils metalllic structures plasma sensors 84 y.v. mitrishkin, n.m. kartsev, a.e. konkov, m.i. patrov copyright ©2020 assa adv. in systems science and appl. (2020) [5] such magnetic coils are not provided, which requires further development and research of the demo poloidal system. 1. the iter project variants and scenarios in the iter project the iter-1 version with self-sustaining thermonuclear reaction and q = ∞ initially was developed but since 1998 the iter-2 version with q = 10 was begun (fig. 1.1, table. 1.1) [4], where q = pout/pin is the ratio of the output power of thermonuclear fusion pout to the input power pin. in iter, it was proposed to control the plasma boundary in the divertor configuration (lower x-point) by controlling the distances (called gaps) between the separatrix and the first wall at six different points, including the intersection points of the separatrix with the divertor plates. the plasma cross section, the location of the nine poloidal field coils (pf1 – pf9) in iter-1 and six pf coils (pf1 – pf6) in iter-2, the solid central solenoid (cs – central solenoid) in iter-1 and the six-section cs in iter-2, and the locations of the controlled gaps between the plasma boundary and the surface of the surrounding elements are shown in fig. 1.1. table 1.1. the plasma parameters in two versions of iter plasma parameters iter-1 iter-2 plasma current, ma 21 15 major radius, m 8.1 6.2 minor radius, m 2.8 2.0 elongation 1.6 1.85 thermonuclear fusion power, gw 1.5 0.5 q ¥ >10 burning time, с 1000 400 vertical instability time constant, sec 1.1 0.1 separatrix deviations from the given location must be within ±10 cm, which is less than 5% of the plasma major radius. the gap locations shown in the figures are chosen to provide a reasonably good overall performance of plasma shape control and at the same time for precise control at key points such as divertor strike points (g1, g2) and the plasma boundary in front of the ion-cyclotron radio frequency heating antenna (g3). all poloidal field coils are superconducting with a relatively small cross-sectional area. an ac/dc converter controls each coil. in iter-1, all the pf coils and the central solenoid control the unstable vertical position of the plasma. in iter-2, a special circuit is provided to control the plasma vertical position or speed, and in the subsequent version for this purpose, additional horizontal field coils are introduced into the vacuum vessel. plasma control in tokamaks. part 3.1. plasma magnetic control systems in iter 85 copyright ©2020 assa. adv. in systems science and appl. (2020) a b fig. 1.1. poloidal systems and arrangements of controlled gaps g1 g6 between the separatrix and the first wall: iter-1 (a) and iter-2 (b) the iter-2 scenario includes the following sequence of phases [3]: plasma start and growth of the plasma current, divertor configuration (x-point) formation, heating to ignition, controlled continuous burning, controlled reduction of fusion power and plasma current, termination of the x-point, plasma shutdown (the end of a discharge). the required evolution of the plasma configuration is created by changing of the poloidal currents (feedforward components) and adding of the feedback control. two main phases that can be identified are the most different in terms of control requirements: • limiter phase characterized by the plasma restricted by a limiter with a relatively low thermal and magnetic energy content: the control requirements are not very strict during this phase, and it is necessary basically only to suppress the vertical instability of the plasma; • divertor phase, during which the high amount of thermal and magnetic energy is a potential danger to the preservation of the outer layer of the plasma and its surrounding elements; this phase requires precise control of the plasma shape and current. 2. synthesis and modeling of plasma control systems in iter-1 the pf-controller (fig. i.1) should be able to control with given performance the gaps between the plasma and the first wall under the action of specified disturbances of the type of minor disruption (md). in doing so, the controller must guarantee to stabilize the vertical unstable position of the plasma in iter-1 or to stabilize the vertical velocity of the plasma around zero value in iter-2. the h∞ robust plasma current, position and shape control system was developed in iter-1 at q = ∞ [4, 6, 7] with an ncf-controller in the feedback (ncf – normalized coprime factorization). the robust system is weakly sensitive to errors and uncertainties of the tokamak plasma model. it remains stable at a large set of iter magnetic plasma configurations in comparison to classical systems, for example linear-quadratic control. for the first time by the method of mathematical modelling on the plasma-physical code dina it was shown that a larger robust stability margin of the multivariable control system provides a greater number of magnetic configurations from the iter database. for these configurations the control system stabilized the position of the separatrix under the action of mdperturbations on the divertor phase of the discharge [6, 7]. the study of the effect of perturbations of the elm type (edge located modes) showed a slight increase in the 86 y.v. mitrishkin, n.m. kartsev, a.e. konkov, m.i. patrov copyright ©2020 assa adv. in systems science and appl. (2020) amplitude of the currents in the poloidal field coils compared to their natural amplitude without feedback. in addition to the linear h∞ ncf robust controller in the feedback system, a nonlinear correction unit of the total control power (power management system – pms) was developed, which allowed to reduce power peaks by 30-40% in the simulation of the system in the presence of md-type perturbations without significant changes in the behavior of gaps [4, 6, 7]. fig. 2.1 shows the block diagram of closed-loop control system with the linear h∞ ncf controller in the feedback and the nonlinear correction unit. the principle of operation of the correction unit is as follows. the total control power signal, which is the sum of the products of the currents in and the voltages of the control coils un where ieqn is the current in the poloidal field coil, which specifies the plasma equilibrium (see fig. 2.1). the vector of corrective (additional) voltages is created as a product of the power module and voltages on the control coils in accordance with δu = –|p|u. this vector is introduced into the feedback to reduce the input voltages at a strong increase in power, which reduces the power peaks under the action of perturbations. fig. 2.1. control system with the correction unit of the full control power the developed controllers were modeled in closed-loop control systems on linear (create-l, pet-l, tsps-l, corsica-l) and non-linear (tsc, dina, corsica, pet, maxfea, tsps) models of the plant under the action of specified md-type and elm-type disturbances. modeling has shown that systems with developed controllers met technical requirements. as an example in fig. 2.2 for the system with the h∞ ncf controller at the point of the sob (start of burn) scenario on the pet-l linear model, graphs of transients after the md are given for variations of gaps, voltages, poloidal coil currents, total control power, its derivative, variations of plasma current. 10 1 ,n n n eqn n n p i u i i i = = = + då saturation -st e (1+st) delays planт t cs & pf coil power supplies feedback controller plasma current variation cs&pf coil current variations gap displacements hinfncf scalar product modulus product power management system power i nom + + _ _ u i plasma control in tokamaks. part 3.1. plasma magnetic control systems in iter 87 copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 2.2. transient responses with h¥ ncf controller in case of md-type disturbance (pet linear model, sob) 3. synthesis and modeling of plasma control systems in iter-2 the new principles and plasma control systems for iter-2, which were developed in solving of problems of modeling magnetic plasma control systems in iter were proposed and applied. the focus of the development of the systems is associated with an increase in their reliability, survivability and performance of control (speed and accuracy). (1) synthesis of h∞ control system based on the scheme of external disturbance rejection (fig. 3.1–3.3) [4, 8, 9]. the block diagram of the control system is shown in fig. 3.1a [3]. it consists of two control loops: the first scalar loop serves to stabilize the vertical plasma velocity relative to zero, and the second mimo loop is designed to control the plasma current and shape: the six gaps between the plasma separatrix and the first wall (fig. 1.1, b). to stabilize the vertical plasma velocity, a special scheme is applied (see fig. 3.1, b), consisting of parallel connection of pf2 pf5 coils with appropriate directivity and slow 88 y.v. mitrishkin, n.m. kartsev, a.e. konkov, m.i. patrov copyright ©2020 assa adv. in systems science and appl. (2020) rectifiers, to which a fast rectifier vs (vertical stability) is connected, directly acting on vertical plasma velocity. first, the loop shaping of the open-loop system [10] and the mixed sensitivity [11] were used to synthesize robust controllers and by means of μ-analysis the controllers were compared with each other, as well as with lqg-, lead-lagand p-controllers. the loopshaping controller showed the smallest peak of µ, that is, the greatest robust stability margin [8]. a b fig. 3.1. iter double-loop magnetic control system: a) vsc is a vertical stabilization converter, mc is a main converter, kbd is a block-diagonal controller, fd is a differentiating filter, f is a filter; b) connection diagram of a fast voltage converter for suppressing the vertical plasma velocity, vs is a vertical plasma stabilization thyristor rectifier, m2 – m5 are main rectifiers for pf2 – pf5 coils then two block-diagonal controllers were synthesized to control the speed of plasma vertical movement, current, and shape with a cascade of currents in the coils of the poloidal field (fig. 3.2, a) and without this cascade (fig. 3.2, b) [4, 9]. the systems were tested at various points of the iter scenario on linear models obtained from a plasma-physical pet code under the action of the md perturbation. a typical transient with such testing is shown in fig. 3.3 for a controller without a cascade with pf currents, when, after a splash of gaps, they enter the specified tube, and the plasma position in the vertical and horizontal reach certain levels consistent with the plasma shape, when the vertical plasma velocity is stabilized about zero. the currents in the central solenoid and in the coils of the poloidal field come to some new level, corresponding to the compensation of disturbances. this controller showed the best robust properties when comparing it with other controllers for which the lqg approach and decoupling of control channels were used [4, 9]. a b fig. 3.2. block diagrams of controller synthesis: a) currents in cs & pf coils, b) plasma current and shape plasma control in tokamaks. part 3.1. plasma magnetic control systems in iter 89 copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 3.3. variations of the output signals of the control system in case of md-type disturbance at the point of the sof (start of flattop) scenario: a gaps, vertical and horizontal plasma positions; b cs & pf currents (2) minimization of the h∞-norm of the mixed sensitivity function for the linearized plasma model while stabilizing the vertical plasma velocity about zero and simultaneously controlling the plasma shape and current for synthesizing robust scalar and multivariable controllers (see fig. 3.1) [12]. at the next stage, a siso and mimo controllers were synthesized while minimizing the h∞-norm of mixed sensitivity , where s(s) is the sensitivity transfer function of the closedloop system, k(s) is the controller. the block-diagonal controller was tested both on a linear model obtained by linearizing a plasma-physical dina code that implemented partial differential plasma equations, and on the nonlinear code itself, which distinguishes this work from the previous results. the plots of the processes obtained are similar to those shown in fig. 3.3. (3) cascade control (fig. 3.4 –3.6) [13–15]. further progress is related to the transition to cascade control and the solution of the problem of tracking the scenario values of gaps. fig. 3.4. structural scheme of the internal cascade (the lower level) for cs/pf coil current control ( ) ( ) ( ) ( ) ( ) ( ) 1 2 min k s w s s s w s k s s s ¥ ® 90 y.v. mitrishkin, n.m. kartsev, a.e. konkov, m.i. patrov copyright ©2020 assa adv. in systems science and appl. (2020) fig. 3.5. structural scheme of the external cascade (the upper level) for plasma shape and current control fig. 3.6. tracking of gaps g1–g6 here, in the inner loop, the original decoupling of control currents is carried out (see fig. 3.4). in the outer loop, the moore-penrose pseudo-inverse matrix establishes a connection between the displacements of the gaps between the first wall and the separatrix and the variation of the plasma current with control currents through the diagonal pii controller. at the same time, in order to avoid saturation of control currents, a nonlinear quadratic programming problem was solved (increasing system survivability) (see fig. 3.5). fig. 3.6 shows the operation of the cascade control system for the position, current and shape of the plasma with a decoupling of channels in the tracking mode for gaps on the dina code with a sufficiently high accuracy. (4) model predictive control (mpc). this control is attractive because the synthesis of the controller takes into account the restrictions on the input and output signals, which eliminates the development of specialized approaches, in particular, to prevent the saturation of plasma control currents. therefore, the proposed approach has no separation on two control loops of the plasma vertical velocity and plasma current and shape (fig. 3.7) [16]. this configuration takes into account limitations on the control currents. numerical modeling of the system with a predictive model on the non-linear dina code showed that, due to the natural consideration of constraints and the solution of the optimal problem at each control step, the system with a predictive model gives smaller deviations in the gaps when a md disturbance occurs [16]. plasma control in tokamaks. part 3.1. plasma magnetic control systems in iter 91 copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 3.7. block diagram of a magnetic plasma control system in iter with a predictive model the predictive model approach for plasma control in tokamaks continues to evolve. thus, a scalar system for stabilizing the vertical position of the plasma in iter is developed and simulated relative to zero with prediction [17]. the method with a predictive model involving the singular decomposition of matrices [18] is used to control the current and shape of the plasma in iter. prediction is used to control the profile of safety factor q in iter with variable constraints on the raptor code [19], as well as to control the vertical position of the plasma on the t-15md tokamak model with a variable parameter [20]. (5) a cascade control system with decoupling of channels in the inner cascade for controlling currents in the poloidal field coils and robust controller in the outer cascade (fig. 3.8) [21]. cascade control of plasma shape and current leads to a trend towards increasing margines of robust system stability, which is associated with an increase in system reliability. to achieve this goal, the outer cascade was synthesized as a robust when solving the minimization problem of the h∞-norm of mixed sensitivity. then, on a linear model, the synthesized system was compared with the system of issure (3) according to the criteria of margins of robust stability and robust performance. the result showed that the robust system surpasses the system from issue (3) by about five times according to robust criteria. simulation of the robust system on the dina code in the tracking mode for gaps showed the result on tracking accuracy close to the system of issue (3) (see fig. 3.6). fig. 3.8. block diagram of cascade control system of plasma shape and current in tokamak: the keys provide the ability to switch contrtollers in the outer cascade (6) hierarchical plasma control with mimo robust control loop for plasma current and shape, synthesized by macfarley-glover method, and adaptive control circuit for the vertical position of the plasma with predictive model (fig. 3.9) [22, 23]. in the 92 y.v. mitrishkin, n.m. kartsev, a.e. konkov, m.i. patrov copyright ©2020 assa adv. in systems science and appl. (2020) previous approaches, the speed of the plasma vertical position was stabilized around zero, and not the position itself. this approach has two drawbacks. one of them is connected with the fact that this approach does not allow the plasma to maintain its position in a limited region of vertically controllability, since the limitation of this region arises due to the instability of the plasma and the restriction on the control voltage. another disadvantage is associated with the fact that with such control the system is not strictly stable, since the speed and not the position of the plasma in the vertical direction is stabilized. for these reasons, a system was developed to control the position, current, and shape of the plasma, in which the vertical unstable position of the plasma is stabilized. to avoid contradiction between the plasma shape and its position, an adaptive contour with a controller based on a predictive model was added, which adapts the position of the plasma to its shape (see fig. 3.9). рис. 3.9. block diagram of hierarchical system of magnetic plasma control with adaptation in iter: mpc controller with a predictive model a brief comparison of the applied approaches. iter used various principles and structures of control systems for magnetic control of plasma (1) (6). this is because the plasma in iter is a complex plant with distributed and time-varying parameters, having parametric and structural uncertainties, which makes it necessary to search for the most effective methods of multivariable hierarchical plasma control. first, to counter uncertainties of plasma models, robust control systems based on the h∞optimization theory (1), (2), (5) and (6) were developed. various system structures were proposed: (1) reflection of external disturbances; (2) minimization of the h∞-norm of the mixed sensitivity function; (5) decoupling of current control channels and h∞-controller for the shape of the plasma. all these schemes were used with the stabilization of the vertical plasma speed relative to zero, in order to eliminate contradiction between the position of the plasma and its shape. these schemes led to approximately the same perormance at control of current and plasma shape, providing certain robust stability margins due to h∞-controllers. systems with control of the vertical speed of the plasma, rather than its vertical position, are flawed: they are not strictly stable, and the vertical position can change uncontrollably in transients. for this reason, it was proposed to directly control the vertical position of the plasma in order to overcome this drawback in the vertical stabilization circuit (6). therefore, an adaptation of the vertical position of the plasma to its shape (6) was developed. at the same time, the systems with decoupling of control channels both in the inner cascade of currents control in the coils of the poloidal field and in the outer cascade of plasma current and shape (3) were developed. this is due to the fact that when decoupling the channels, it is easier to tune each control channel, while in comparison with the h∞theory of optimization a multivariable controller is synthesized while minimizing one h∞criterion, which makes it difficult to understand how the whole system works, although it gives large robust stability margins. a current, position, and shape control system with a predictive model has been applied (4). this approach is promising in that with it it is possible to take into account the restrictions on the input and output values of the controlled plant, since these restrictions are laid into the algorithm with prediction at the input and output horizons. in addition, a method plasma control in tokamaks. part 3.1. plasma magnetic control systems in iter 93 copyright ©2020 assa. adv. in systems science and appl. (2020) with a predictive model can be applied to create a multivariable adaptive plasma current and shape control system, if there is an opportunity to identify a linear plasma model at each time instant. 4. synthesis and modeling of plasma control systems in iter-2 with internal coil for controlling the vertical plasma position it has been established in [24] that the plasma controllability region on the vertical in iter is limited to a relatively small value namely 4 cm, which is of 2% of the minor iter radius. the idea of controllability region estimation is presented in fig. 4.1. it consists in the fact that under the same conditions and the same saturation voltage supplied to the coils of the horizontal iter field according to the scheme shown in fig. 4.1, b, the plasma was released from different initial conditions vertically, and the voltage sign was chosen such that it counteracted the increase in vertical displacement. if the initial condition was within the controllability region, then the voltage after a certain time interval caused the vertical displacement to change in the opposite direction. this study was conducted, in particular, on the corsica code (usa). four proposals were put forward and investigated [25] on how to increase the region of vertical plasma controllability: increase in voltage vs of the power supply from 6 to 9 kv; insertion of a second vs-circuit using two cs sections: cs2l and cs2u; adding stabilizing rings inside the vacuum vessel; insertion of special control coils inside the vacuum vessel. these proposals were studied and compared with each other in [25]. the most effective means turned out to be coils inside the vessel. for this reason, at the end of 2013, special coils were installed inside the vacuum vessel in the iter project in order to bring them closer to the plasma as much as possible, thereby expanding the controllability region vertically and increasing the reliability of the plasma magnetic control system (see fig. 4.1, b) [24]. a b fig. 4.1. investigation of the vertical controllability region in iter-2 and its expansion: a) experimental method of estimation of the controllability region dzmax [24]; b) coils of the horizontal field which are located inside the vacuume vessel to increase the controllability region; the power supply vs3 is connected to these coils that is controlled by the controller in the feedback [25]. two systems for stabilizing the plasma vertical speed with respect to zero with external and internal coils were investigated [26]. the authors used the jin-trac (jet) transport 94 y.v. mitrishkin, n.m. kartsev, a.e. konkov, m.i. patrov copyright ©2020 assa adv. in systems science and appl. (2020) code and the equilibrium create-nl code with the free boundary, which were connected together for modeling. in the first case, the control law was chosen as follows where vs1(s) is the laplace transform of the voltage function of the external coil, zdot(s) is the laplace transform of the function of the vertical speed movement of the plasma current center, and in the second case the control law in the laplace transform domain was chosen as follows: where vs3(s) is the laplace transform of the voltage function on the inner coil, is3(s) is the laplace transform of the current function in the inner coil. the case of scenario 1 was chosen, l-mode, sof (start of flattop: the beginning of the flat phase), ip = 15 ма, li = 0,88, bpol = 0,06. the simulation results are summarized in table 4.1 where vdemax (vertical displacement event: the phenomenon of vertical displacement) is the maximum possible initial position of the plasma in the vertical direction, which can be stabilized (controllability region), mj is the phase stability margin, wt is the cut-off frequency, mgu and mgl are the upper and lower limits of the stability margin in amplitude. the results showed that the area of control vdemax significantly more for the case of an inner coil, about 5-6 times. table 4.2 shows the values of vdemax in the presence of noise with a limited bandwidth of 1 khz, zero mean, standard deviation of σ = 210 mm/s and σ = 430 mm/s. the machine configuration is rather insensitive to noise: at the noise level in question, there is a negligible degradation in performance in terms of vdemax. table 4.1. characteristics of the control system obtained in the analysis vdemax, mm mj wt rad/s mgu, db mgl, db outer coil |34 22o 13 5.6 –4.0 inner coil > 200 65o 78 13 –18 table 4.2. the best performance in the presence of noise obtained in the analysis configuration vdemax, mm noise level s, mm/s 210 430 vs1 (6 kv) 28 24 vs3 (2,3 kv) 148 141 in iter scenarios are continued to be developed and refined using plasma magnetic control systems on plasma-physical codes, taking into account only external coils for controlling the vertical position of the plasma [27], and taking into account the internal coil [26, 28]. fig. 4.2 shows the block diagram of the plasma shape control system, which was used in [26] to simulate transients during the transition from mode l (15 ma) to mode h (15 ma) and vice versa on the linear model create-l. the paper [26] presents the evolution of the internal g6 and external g3 gaps (see fig. 1.1, b) for the cases: the direct link is updated at t = 1 s; direct link is updated at t = 5 s; the direct link is updated at t = 1 s exactly after the transition and then at t = 9 s; the direct link is updated at t = 1 s and a transport delay of 0.2 s; direct link is calculated as a function of bpol. ( ) ( ) ( )1 2 1 /1815000 1 / 60 dot svs s z s s + = + ( ) ( ) ( )5 3 3 1 / 408000 1,2 10 1 / 6 dot s svs s z s i s s -+ é ù= ´ ´ë û+ plasma control in tokamaks. part 3.1. plasma magnetic control systems in iter 95 copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 4.2. block diagram of the plasma shape control system, similar to the extreme shape controller scheme in the jet tokamak (uk) studies of the plasma vertical controllability region for iter according to the method in [24] were continued in [27] on the pet code in more detail for various plasma variants. the results are summarized in table. 4.3, which shows that when using only external coils and vs1 power supplies, the range of the controllability region is in the range of 1.5–13.7 cm. if both external vs1 coils and internal vs3 coils are involved, then the range increases noticeably up to 16–25.5 cm. table 4.3. performance indicator max (z0) and instability time constant [26] plasma 15 ма plasma 1 li(3) = 1.2, βp = 0.1 plasma 2 li(3) = 1.0, βp = 0.1 plasma 3 li(3) = 0.73, βp = 0,6 maxz0, vs1 15 30 137 vs3+vs1 160 175 255 instability time constant, ms 56 78 171 the systems of magnetic plasma control have been investigated taking into account noise and various combinations of coils for stabilizing the vertical position of the plasma in iter on the dina code [27]: external coils vs1; vs3 internal coils; simultaneous use of external and internal coils. the studies were based on limited data analysis from c-mod, jet, and asdex-u tokamaks, which made it possible to specify for plasma activity in iter a model of vertical plasma speed noise dz/dt as a random signal evenly distributed over the frequency interval [0, 1] khz with a standard deviation of 0.6 m/s. the simulation was performed with low-frequency noise injection into the diagnostic signal dz/dt, which was used in the feedback of the vertical plasma stabilization. at the same time, the standard deviation of the uniformly distributed noise was increased until the vertical stabilization system lost stability or one of the engineering parameters reached its design limit. the best result was given by the combination of the windings of the internal and external coils. in this case, two signals were applied to the input of the vertical stabilization controller: dz/dt and ivs3. the maximum allowed standard deviation for noise in this case was 3 m/s, the standard deviation of the ivs3 current was about 10 ka, the standard deviation of the vertical plasma displacement during the flat phase of the plasma current was about 44 mm. the plasma control system was synthesized using the standard lqg method [29]. conclusion in this part of the review, attention is paid to the development and modeling of plasma position, current, and shape control systems in iter, and the contribution of v.a. trapeznikov institute of control sciences of russian academy of sciences to this work is highlighted. the results show a trend in the development of these systems, associated with an increase in accuracy and speed when tracking scenario signals and rejecting disturbances such as minor disruption. the trend reflects increase in the robust stability margin, which in turn leads to increase in reliability and survivability of plasma control systems in iter. 96 y.v. mitrishkin, n.m. kartsev, a.e. konkov, m.i. patrov copyright ©2020 assa adv. in systems science and appl. (2020) the introduction of horizontal field coils inside the iter vacuum vessel expands the controllability region in the vertical direction of plasma movement, which, at the given the control coil voltage limits, specifications for minor disruption disturbances and vertical plasma instability, significantly moves away the closed-loop system of magnetic plasma control from the boundaries of stability loss. magnetic plasma control systems in operating tokamaks, as well as for iter, continue to evolve in different ways. one of these areas is related to the development of highprecision methods for solving inverse plasma diagnostic problems, the data of which serve as input data for plasma control systems for internal plasma parameters, and approbation of control systems based on these data, rather than direct data, when all plasma parameters are known accurately enough [30, 31]. in section 43.2, experimental processing of scenarios for iter on diii-d (usa) and west (france) tokamaks, approaches in modeling and implementing plasma control systems in iter, preparation of the plasma control system for launch and operation in iter will be presented. road maps of the development and creation of the first demo fusion power station (following the iter step) will be shown, which indicate two demo development directions: on traditional tokamaks with relatively large aspect ratios and modular-type spherical tokamaks that will significantly reduce the creation time of demo. it will be also shown the main trends in the development of demo poloidal systems, as well as the initial version of the magnetic plasma control system in demo. acknowledgements this work was supported by the russian science foundation (rsf), grant no 17-1901022 (sections 2-5) and the russian foundation for basic research (rfbr), grant no 1708-00293 (introduction, section 1). references 1. mitrishkin, y. v., korenev, p. s., prokhorov, a. a., et al. (2018). plasma control in tokamaks. part 1. controlled thermonuclear fusion problem. tokamaks. components of control systems, advances in systems science and applications, 18 (2), 26-52. 2. mitrishkin, y. v., kartsev, n. m., pavlova, e. a., et al. (2018). plasma control in tokamaks. part. 2. magnetic plasma control systems, advances in systems science and applications, 18(3), 39-78. 3. iter. – url: https://www.iter.org. 4. mitrishkin, y. v. (2006). plasma control in experimental thermonclear devices: adaptive self-oscillations and robust control systems. (upravlenie plazmoy v ehksperimentalnyh termoyadernyh ustanovkah: adaptivnye avtokolebatelnye i robastnye sistemy upravleniya). – moscow: krasand, 400 p. [in russian]. 5. takase, h., utohab, h., sakamoto, y., et al. (2016). analysis of plasma position control for demo reactor, fusion engineering and design, 109–111, 1386-1391. 6. mitrishkin, y. v., lukash, v. e., khayrutdinov, r. r. (2005). plasma current, position and shape robust control system in iter, problems of atomic science and technology. series: thermonuclear fusion, 1, 61-81 [in russian]. 7. mitrishkin, y. v. (2004). comprehensive design and implementation of plasma adaptive selfoscillations and robust control systems in thermonuclear installations, proc. of 8th world multi-conference on systemics, cybernetics and informatics, orlando, fl, usa, xv, 247-252. 8. mitrishkin, y. v. & kimura, h. (2001). plasma vertical speed robust control in fusion energy advanced tokamak, proc. of 40th ieee conf. on decision and control, florida, usa, 1292– 1297. 9. mitrishkin, y. v., kurachi, k., kimura h. (2003). plasma multivariable robust control system design and simulation for a thermonuclear tokamak-reactor, international journal of control, 76 (13), 1358-1374. 10. mcfarlane, d., glover, k. (1990). robust controller design using normalized coprime factor plant descriptions. lecture notes in control and information sciences, springer-verlag, 212 p. 11. safonov, m. (1999). robust control / encyclopedia of electrical and electronics engineering (ed. j.g. webster), vol. 18. ny: wiley, 592-602. plasma control in tokamaks. part 3.1. plasma magnetic control systems in iter 97 copyright ©2020 assa. adv. in systems science and appl. (2020) 12. dokuka, v. n., kadurin, a. v., mitrishkin, y. v., khayrutdinov, p. p. (2007). synthesis and modeling of the h∞-system of magnetic control of the plasma in the tokamak-reactor, automation and remote control, 68(8), 1410-1428. 13. mitrishkin y. v., korostelev a. y., sushin i. s., et al. (2009). plasma shape and current tracking control system for tokamak, proc. of 13th ifac symposium on information control problems in manufacturing, c7.1, 2133–2138. 14. mitrishkin, y. v., korostelev, a. y. (2010). a cascade system of tracking plasma current and shape of a tokamak with decoupling control channels (kaskadnaya sistema slezheniya za tokom i formoy plazmy v tokamake s razvyazkoy kanalov upravleniya), vestnik bmstu, 79(2), 21-38 [in russian]. 15. mitrishkin, y. v., korostelev, a. y., dokuka, v. n., khayrutdinov, r. r. (2011). synthesis and simulation of a two-level magnetic control system for tokamak-reactor plasma. plasma physics reports, 37(4), 279-320. 16. mitrishkin, y. v., korostelev, a. y. (2008). system with predictive model for plasma shape and current control in a tokamak, control sciences, 5, 19-25. [in russian]. 17. veremei, e. i., sotnikova, m. v. (2011). plasma stabilization on the base of model predictive control with the linear closed-loop system stability, vestnik s.-petersburg univ. ser. 10. prikl. mat. inform. prots. upr, 1, 116-133. [in russian]. 18. gerkšič, s., pregelj, b., perne, m., et al. (2018). model predictive control of iter plasma current and shape using singular value decomposition, fusion engineering and design, 129, 158-163. 19. maljaars, e., felici, f., baar, m. r., et al. (2015). control of the tokamak safety factor profile with time-varying constraints using mpc, nuclear fusion, 55(2). 20. mitrishkin, y. v., golubtsov, m. p. (2018). a hybrid control system for an unstable nonstationary plant with a predictive model, automation and remote control, 79(11), 2005-2017. 21. kadurin, a. v., mitrishkin, y. v. (2011). multidimensional system of cascaded control of plasma form and current in tokamak with channel decoupling and h∞-controller, automation and remote control, 72, 20-53. 22. mitrishkin, y. v., kartsev, n. m. (2011). hierarchical plasma shape, position, and current control system for iter, proc. of the 50th ieee conf. on decision and control and european control conf., orlando, fl, usa, tua13.2, 2620-2625. 23. kartsev, n. m., mitrishkin, y. v., patrov, m. i. (2017). hierarchical robust systems for magnetic plasma control in tokamaks with adaptation, automation and remote control, 78(4), 700-713. 24. humphreys, d. a., casper, t. a., eidietis, n., et al. (2009). experimental vertical stability studies for iter performance and design guidance, nuclear fusion, 49(11). 25. albanese, r., ambrosino, g., ariola, m., et al. (2009). iter vertical stabilization system, fusion engineering and design, 84, 394-397. 26. cavinato, m., ambrosino, g., figini, l., et al. (2014). preparation for the operation of iter: eu study on the plasma control system, fusion engineering and design, 89, 2430-2434. 27. casper, t., gribov, y., kavin, a., et al. (2014). development of the iter baseline inductive scenario, nuclear fusion, 54(1). 28. gribov, y., kavin, a., lukash, v., et al. (2015). plasma vertical stabilisation in iter, nuclear fusion, 55(7). 29. belyakov, v., kavin, a., kharitonov, v., et al. (1999). linear quadratic gaussian controller design for plasma current, position and shape control system in iter, fusion engineering and design, 45, 55-64. 30. zaitsev, f. s., shishkin, a. g., lukianitsa, а. а., et al. (2018). the basic components of software-hardware system for modeling and control of the toroidal plasma by epsilon-nets on heterogeneous mini-supercomputers, commun. comput. phys., 24 (1), 1-26. 31. shishkin, a. g., stepanov, s. v., suchkov, e. p., et al. (2018). distributed access to the resources of plasma modelling and control complex hasp cs, 2018 ieee international conference on computer and communication engineering technology (ccet), beijing, china, august 18-20, 193-197. microsoft word 761 enhancing hadoop performance in homogenous and heterogeneous big data environment adv syst sci appl2020; 01; 13-26 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/761 enhancing hadoop performance in homogenous and heterogeneous big data environments by dynamic slot configuration ekhlas k. hamza 1) control and systems engineering department, university of technology, iraq e-mail: 100374@uotechnology.edu.iq, drekhlaskadhum@gmail.com, abstract: hadoop is one of the most famous platform solutions for processing large volume and scale of data in parallel processing in cloud computing. a hadoop system can be characterized based on three main factors: cluster, workload and user. each of these factors can be described as either heterogeneous or homogenous, which reflects the heterogeneity degree of the hadoop systemthe objective of this proposed research work is to investigate the degree of influence of heterogeneity for each of these factors on the performance of hadoop based on different schedulers. three schedulers are considered with different levels of hadoop heterogeneity and are tested and analyzed: the first algorithm considered is the fifo (first in first out), the second is the fair sharing, and the final is the coshh (classification and optimization based scheduler for heterogeneous hadoop). performance issues are related to hadoop schedulers and comparative performance analysis between different cases of jobs submission. these jobs are processed in different homogenous or heterogeneous data environments and under fixed or reconfigurable slot between map and reduce tasks for hadoop mapreduce java programming clustering model. the results showed that when assigning tunable knob between map and reduce tasks under certain schedulers like fifo algorithm, the performance enhanced significantly especially in cases of heterogeneity environment where the workload decreased significantly and the utilization of computational resources increase was obvious. keywords: hadoop, mapreduce, big data, slot configuration, scheduling algorithms, makespan, resources utilization. 1. introduction recently, big and parallel data processing has become a main concern for modern analysis and knowledge extraction for feasible prediction and decision making in many fields of businesses. data has recently become more complicated when generated in huge volumes and in many cases mixed between structure and unstructured data. thus, analyzing it exposes many challenges. moreover, modern data is acquired from different server sources and from different users in real-time, which add more complexity to the analysis processes [1],[2]. analyzing these big complicated and parallel data is always beneficial in terms of mining and extraction of useful information and for knowledge discovery purposes [3]. recently, hadoop has been considered as the most promising open source platform for the big and parallel data processing in real-time [4]. understanding the hadoop architecture and how it works is a challenge by itself. the target of utilizing hadoop is always to process the big and parallel data thereby reducing the computational costs of the usage of the processors and memory resources [5] as well as clustering efficiently the data in homogeneity and heterogeneity classifications. the accuracy of the data classification is aimed at researching 14 ekhlas k. hamza copyright ©2020 assa adv. in systems science and appl. (2020) this topic, as well as reducing the latency time when answering the end users’ enquiries for the purposes of data mining or discovery of certain knowledge [6]. from the beginning, this research has been motivated from certain challenging factors, which are as follows: firstly, simulating the real-time hadoop homogenous and heterogeneous environments in one desktop. secondly, to simulate different users’ enquiries and even if they are heterogeneous users it may mean loading the data to hadoop from different distributed file systems and servers. besides that, the makespan or alternatively the workload should be determined especially for job enquiries submitted to hadoop from different distributed file systems and the data obtained from different heterogeneous clusters [7]. the focus here is to analyze the volume of workload assigned to each job, which is assigned to the hadoop system by determining the consumption of the processors and memory sources as well as the time taken for completing the job. most important of all is how to reduce the makespan or the workload in hadoop mapreduce architecture, especially for obtaining data from heterogeneous clusters loaded from different distributed file systems (servers). mainly fixing the number and the ratio between map and reduce processes by using traditional scheduling algorithms such as fifo and fair sharing algorithms seems to have some difficulties as some jobs maybe stuck sometimes, especially in case of fifo scheduling [8]. further, the job convergence is not optimized especially in the case of dealing with intensive heterogeneous data environments. this will lead to increased workload (make span), which consumes more computational resources and resulting in more time processing. the proposed system implements a mechanism to dynamically allocate slots for map and reduce tasks or alternatively it means optimally adjusting the ratio between map and reduce processes by using more efficient scheduling algorithms for optimized classification in a heterogeneous data environment. this, as the results show, will decrease the make span and more efficiently utilize the processors and memory resources. 2. research background and literature review in this section, the main concepts of this research field are defined and explained to understand the system architecture of the proposal and how it works in processing and analyzing big and medical data in real time. at this point it is important to understand from the beginning the characteristics and definition of the modern big data. big data can be identified mainly in three parameters, which are complexity, volume and dynamicity. as regards the complexity parameter, this will make sense when it is understood that the generated data in recent modern scenarios are coming from different sources and formats. this can be text, audio, videos, symbols or numbers, which are different extensions and formats. it is clear that dealing with such unstructured data will impose enormous complexity for purpose of analysis and data mining. there should be efforts during pre-processing to make such data more uniformed in order to make pattern recognition feasible. with reference to the volume parameter, the volume and the size of the data are increased rapidly, dynamically and spontaneously in real-time. this can become clear if we imagine that in modern scenarios of medical data collection to support personal health records and monitoring health status remotely, the data are collected by different sensors for different health parameters such as heart rate, blood pressure, body temperature and glucose levels. all these data are collected simultaneously and in parallel in one database center for further analysis so as to send certain pattern recognition or trend analysis as results for end users to allow them to monitor their health conditions. in this case the data is growing exponentially in size, volume and dynamically in real-time with different formats and structures. in the centralized management efforts of the big data analysis, another added complexity factor is represented as the data come from different distributed sources and in many cases enhancing hadoop performance in homogenous and heterogeneous big data environments by dynamic slot configuration 15 copyright ©2020 assa adv. in systems science and appl. (2020) from heterogeneous sources. hadoop recently appeared as the most interesting open-source software platform utilized for distributed storage and parallel processing of big data by using the mapreduce programming model [9]. the main parts of the hadoop architecture consist of the storage part, which is known as hadoop distributed file system (hdfs) and the processing part, which is the mapreduce programming model. hadoop’s main job is to split large blocks of data and distribute them across nodes in different clusters in parallel and simultaneously [10]. the simplest cluster contains one master node and multiple slave nodes. the master node keeps tracking the tasks acomplished in the slave nodes by the job tracker in each master node.some researchers have discussed scheduling techniques in [ 12-15]. they have provided scheduling-aware data, prefetching and eviction mechanisms based on spark, alluxio and hadoop. the main advantage of the mapreduce programming model in hadoop is scaling the data processing over multiple computing nodes. first, the map process takes a set of data and converts it to another form by simply breaking the data into (key/value pairs) format. thereafter, the reduce process takes the output from the map task as input and then shuffles the data combined, as shown in figure 1. the jobs for assigned jobs of hadoop mapreduce are scheduled from the beginning based on certain scheduling algorithms such as first input first output (fifo), fair sharing and others. fig. 1. hadoop map and reduce processes [10] it is always desirable to reduce the number of maps and processes so as to decrease the workload in hadoop processing and that depends on the power of the scheduling algorithms as well as the ratio between maps, which reduces implementation. this research work focuses in reducing the workload of the hadoop processing jobs (referred to as same meaning as make span) by adjusting the ratio between maps and reducing the optimal minimum number to achieve the different tasks even in homogenous or heterogeneous data clustering to answer different end-users’ enquiries. the incoming jobs submitted to the hadoop system can be considered heterogeneous with respect to certain factors such as the number of tasks, data volume, required computation, arrival rates and the execution times. it is worthy to notice that hadoop resources are different in capabilities of data storage as well as the processing units. this leads to different priorities and minimum requirements for sharing between different users taking into consideration that the number and type of jobs assigned to each user can be different as well. the mean execution time for certain jobs is considered as indicator of the heterogeneity degree for both the workload and clusters in the hadoop system. mainly fixing the number and the ratio between map and reducing processes by using the traditional scheduling algorithms such as fifo and fair sharing algorithms, seem to have some difficulties as some jobs maybe stuck sometimes, especially in case of fifo scheduling. moreover, the job convergence is not optimized especially in case of dealing with intensive heterogeneous data environment. this will lead to increase in workload 16 ekhlas k. hamza copyright ©2020 assa adv. in systems science and appl. (2020) (makespan), which consumes more computational resources and resulting in more time processing. table 1 shows the comparisons of various scheduling techniques. table 1. show the comparative a of various scheduling techniques disadvantages advantages environment heterogeneous or homogenous idea of implementation preemp tion working taxonomy scheduler 1-considered only for singlekind of work. 2. low performance when run manykinds of works. 3. poor response periods for short worksrelated to large works. .1-rate of entire cluster arrangement method is less 2. simple to implement andcapable homogenous schedule works based on their significances in first-come first out. no better work with small clusters nonadaptive fifo .1-does not consider thework weight of each nodule. 1-less difficult. 2. mechanisms well when both lesser and large clusters. 3. it can offerfast responseperi ods for small jobs mixed with larger works. homogenous do a equal spreadingof calculatepropertie s among the users/jobs in the system yes better work with small clusters adaptive fair scheduling 1 . improve the overall system per formance. 2. addresses the fairness and the minimum share requirements. heterogeneous planned to improve the mean completion time of jobs yes better work with large clusters adaptive coshh 3. architecture of the proposed system in both the homogeneous and heterogeneous hadoop environments, the two modules are initially added to the job tracker, which is described below.  workload monitoring (wm)  slot assignment (sa) the function of wm is to collect previous workload information asexecution times are needed for completed tasks. based on this collected information, the workload of the current map and reduced tasks are estimated. thereafter, the role of sa is to adjust the slot ratio between map and reduce tasks by using estimated information received from wm. this adjustment is performed on each slave node. based on adjusting the slot ratio as well as the current slot status, the job tracker is used to assign tasks to slave nodes by both the workload monitor and slot assigner functions. in addition, the task tracker module along with the task manager function for checking the number of individual maps, reduce tasks running on the node. the performance of checking is done based on the new slot ratio received from the job tracker by running the task on each slave node.fig. 2 shows the architecture of the proposed method a batch of jobs is submitted from a user to the job tracker. the three modules involved in the job tracker are workload monitor, slot assigner and scheduler. the status of slot enhancing hadoop performance in homogenous and heterogeneous big data environments by dynamic slot configuration 17 copyright ©2020 assa adv. in systems science and appl. (2020) assignment and task assignment is communicated between job tracker and task tracker in order to perform the data communication with large-scale information. the task is split into several parts, which are functioned by different slots. thus, various slot configurations are performed and each slot contains separate task tracker and task manager. the four functions involved in the proposed architecture are:  the present workloads and their functions are estimated.  the best slot assignment of each node is decided in order to communicate between the job tracker and task tracker.  the task is assigned to each slave node.  constant monitoring of the task execution as well as the slot occupation situation. fig. 2. architecture of the proposed method [11] the modules involved in the proposed method to minimize the makespan are as follows:  homogenous environment  heterogeneous environment  makespan analysis  slot configuration prediction  evaluation 3.1. homogenous environment the homogenous system is defined as the homogenous characteristics of both workload and cluster. based on the job sizes,the homogenous system can be classified into either homogenous-small or homogenous-large. if all the jobs are small, then the system is defined as homogenous-small. if all the jobs are large, then the system is classified as homogenouslarge. mostly, the job size parameters affect the performance of the hadoop schedulers. the average completion time of all schedulers is almost equal. as both the cluster and workload are homogenous, the coshh algorithm suggests all resources as the best choice for all job classes. the two conditions used to develop homogenous computing environment are as follows:  the same storage representations along with the same results are considered as correct in hardware and software module of processor for operations on floating point numbers. 18 ekhlas k. hamza copyright ©2020 assa adv. in systems science and appl. (2020)  the correct floatingpoint value is transmitted and guarantee is provided to the communication layer during the process of floating point number between processors. the module involved in the homogenous phase is described as a flow diagram, as shown in figure 3. the five representations of data-analyzing hadoop benchmarks are described in the heterogeneous phase as a module for makespan and slot configuration prediction. the benchmarks are derived from mapreduce benchmarks suite, which are described below.  inverting the index task: generate certain list of the words as document indexing when entering a text file as an input.  rating the frequency to generate the histogram: as example, calculating the histogram from certain input of movie rating data.  counter for certain word tasks: this job can be described as counting the occurrence of each word when entering input text-file.  classifier: when inputting the movie rating data then, classify these movies into predefined clusters.  grepping task: to find certain patterns on the files from certain input text files. fig. 3. flow diagram of homogeneous environment the implementation of the proposed system architecture is achieved by simulating the medical unstructured big data in cloud environment developed in a single desktop using java programming in netbeans ide and wamp server services all working together, as depicted in figure4. enhancing hadoop performance in homogenous and heterogeneous big data environments by dynamic slot configuration 19 copyright ©2020 assa adv. in systems science and appl. (2020) fig. 4. developed cloud computing platform for medical big data of homogenous tasks 3.2. heterogeneous environment based on the workload and cluster factors, the user is either classified as homogenous or heterogeneous. here, these factors are considered as heterogeneous, thus, the user is also selected from a heterogeneous system.the challenging issue for the schedulers in this system is the arrival rate of different size jobs. based on the challenges,the system is classified into three such as heterogeneous-small, heterogeneous-equal and heterogeneous-large. if the arrival rate is high for small jobs, then the system is classified as heterogeneous-small. if the arrival rate is equal for all job sizes, then the system is classified as heterogeneous-equal. if the arrival rate is high for large jobs, then the system is classified as heterogeneous-large. due to the resolving resource and job mismatch problem schedulers algorithm, the average completion time is intended to be improved in all three heterogeneous systems using the proposed architecture. in the proposed method, the modules are divided into three sectionsas follows:  query process  three different queries  sub-query the processes involved in the proposed heterogeneous platform of medical big data is illustrated in the figure below. 20 ekhlas k. hamza copyright ©2020 assa adv. in systems science and appl. (2020) fig. 5. processes flow diagram of heterogeneous environment enhancing hadoop performance in homogenous and heterogeneous big data environments by dynamic slot configuration 21 copyright ©2020 assa adv. in systems science and appl. (2020) the implementation of the heterogeneous part of the proposed architecture methodology is depicted from the clients’ enquiry side as well as the distributed servers’ side, as shown infigures 6 and 7. fig. 6. client query from heterogenous distributed server in medical big cloud environment fig. 7. processed data from heterogenous distributed servers in medical big cloud environment 3.3. makespan prediction the proposed algorithm tunable slot configurations for reduced makespan (tuscrd) is used to improve the performance of a batch of mapreduce jobs by adjusting a basic system parameter with the objective of the proposed system. the significant challenge available in the cloud is a prediction of makespan. the three modules involved in the makespan prediction are described below:  launch a hadoop cluster  tuning the slot assignment  tunable slot configurations for reduced makespan (tuscrd) under fifo a classic hadoop cluster consists of multiple slave nodes and a single master node. the master node executes the jobtracker section, which acts for the scheduling jobs along with 22 ekhlas k. hamza copyright ©2020 assa adv. in systems science and appl. (2020) coordination between each job’s executions of tasks. each slave node executes the tasktracker part from the architecture for hosting the execution of mapreduce jobs. the task is stored as input data in the data node, which is derived as local whereas the remote is denoted by storing all the tasks in another node. if the data nodes are unknown, then the reduce tasks are considered as remote on all nodes and map tasks are considered as scheduler on the input node. every task is non-preemptable and shares the same deadline and release time as its job. initially, the system model is considered without failure and with pipeline/speculative execution. then, jobs are submitted continuously to the cluster and the master scheduler needs to schedule their tasks on the slaves without prior knowledge of future job arrivals. fig. 8. flow diagram of slot configurations 4. performance analysis of homogenous hadoop submitted jobs makespan and resource utilization are calculated once any homogeneity tasks such as line indexer, word counting and classification are clicked or the heterogeneity task of enquiring about medical data from different distributed servers. figure 9 shows an example of the makespan and cpu processing time as well as the memory space consumed during processing. fig. 9. evaluation of the inverted index homogenous task in term of makepsan and processing time enhancing hadoop performance in homogenous and heterogeneous big data environments by dynamic slot configuration 23 copyright ©2020 assa adv. in systems science and appl. (2020) table1explains thecomparative values of makespan between the inverted index under tunable slot configuration and the existing system. this shows that the inverted index is slightly higher compared tothe existing system as the medical big data are loaded from the same file. table 1. performance of tunable slot solution in hadoop homogenous inverted task on the other hand,tables 2 and 3 show significant enhancements of reducing the makespan workload where the medical big data are loaded from different distributed files. table 2. performance of tunable slot solution in hadoop classification table 3. performance of tunable slot solution for word count it is clear even in the homogenous hadoop big data environment but when loading the data from different files the makespan load is reduced significantly. in the second part of the implementation of the tunable slot configuration of the mapreduce hadoop, the environment considered is heterogeneous medical big data environment. in these cases, the end-clients can have different specific enquiries and the data are loaded and processed in different distributed servers.thereafter, the makespan workload and resources utilization in terms of cpu processing and memory space are determined, as shown in the following figure. 24 ekhlas k. hamza copyright ©2020 assa adv. in systems science and appl. (2020) fig. 10. client query from heterogenous distributed server workload and resources utilization table4 below shows that the workload significantly reduces, in case of loading the data from heterogeneous distributed files from distributed servers with the proposed technique of tunable slot configuration for the hadoop mapreduce platform of processing the big data. table 4. makespan work load analysis of answering end-client enquires from heterogeneous hadoop big data environment. in brief from the presented data analysis, it clearly shows that allowing the slot between map and reduce processes to be tunable under certain scheduling algorithm in hadoop big data environment, will result in significant workload reduction and computational resources utilization, especially when the heterogeneity level increased when loading the data from different distributed files and servers. recent researchers [16-21] have been presented relating to the big data mobile network. they have discussed the ultimate research of wireless classifications for big data in the structure of a smart factory, which are a) mobile nodes and b) animatedly changing heavy traffic load. these can cause differences in traffic and energy consumptions of cluster head nodes in categorized networks. however, supporting the program big data transmission over 5g wireless systems forces many new and open challenges because the program big data services are both time-subtle and bandwidth-intensive over time-varying wireless channels with controlled wireless resources. the new generations of mobile devices have high enhancing hadoop performance in homogenous and heterogeneous big data environments by dynamic slot configuration 25 copyright ©2020 assa adv. in systems science and appl. (2020) processing power and storage but they lag behind in terms of software systems for big data storage and processing. hadoop is a accessible platform that arrange for spread storage and computational capabilities on clusters of commodity hardware. 5. conclusion and future works the problem of high workload and therefore, increased computational resources in hadoop mapreduce platform for big data processing, is an attempt to mitigate the optimized hadoop solution by allowing the slot configuration between maps and reduce the tunable according to certain job priority or homogeneity as well as the available resource spontaneously. the simulation of the proposed system shows that a high percentage of reduced workload and increased utilization of the computational resources can simply be observed by the lesser number of map and reduce tasks needed to accomplish certain jobs. this is shown in the proposed solution for three jobs submitted in a homogenous hadoop environment where the data in some cases are loaded from different distributed files in different servers as well as for different job queries in heterogeneous hadoop clustering environment as the data is loaded as well from distributed files. it is recommended to test as well the other scheduling algorithms such as fair sharing and classification and optimizationbased scheduler for heterogeneous hadoop (coshh). it is expected that the performance will be more optimized specially for the coshh scheduling algorithm with dynamic slot configuration. references 1. bhosale, h. s., gadekar, d. p. (2014). a review paper on big data and hadoop. international journal of scientific and research publications, 4(10), 1. 2. big data. (2015). new delhi: springer, india, private ltd. 3. hilbert, m. (2016). big data for development: a review of promises and challenges. development policy review, 34(1), 135-174. 4. che, d., safran, m., peng, z. (2013, april). from big data to big data mining: challenges, issues, and opportunities. in international conference on database systems for advanced applications (pp. 1-15). springer berlin heidelberg. 5. o’driscoll, a., daugelaite, j., sleator, r. d. (2013). ‘big data’, hadoop and cloud computing in genomics. journal of biomedical informatics, 46(5), 774-781. 6. suthaharan, s. (2014). big data classification: problems and challenges in network intrusion prediction with machine learning. acm sigmetrics performance evaluation review, 41(4), 70-73. 7. raju, r., amudhavel, j., pavithra, m., anuja, s., abinaya, b. (2014, march). a heuristic fault tolerant mapreduce framework for minimizing makespan in hybrid cloud environment. in green computing communication and electrical engineering (icgccee), 2014 international conference on (pp. 1-4). ieee. 8. rasooli, a., down, d. g. (2014). coshh: a classification and optimization based scheduler for heterogeneous hadoop systems. future generation computer systems, 36, 1-15. 9. shvachko, k., kuang, h., radia, s., chansler, r. (2010, may). the hadoop distributed file system. in mass storage systems and technologies (msst), 2010 ieee 26th symposium on (pp. 1-10). ieee. 10. mctaggart, c. (2014). hadoop/mapreduce. object-oriented framework presentation, csci5448. 26 ekhlas k. hamza copyright ©2020 assa adv. in systems science and appl. (2020) 11. dwivedi, k., & dubey, s. k. (2014, september). analytical review on hadoop distributed file system. in confluence the next generation information technology summit (confluence), 2014 5th international conference(pp. 174-181). ieee. 12. frederic nzanywayingoma, yang yang (2018) task scheduling and virtual resource optimisingin hadoop yarn-based cloud computing environment int. j. cloud computing, vol. 7, no. 2, 2018 13. ibrahim abaker, targio hashem, et al (2018) mapreduce scheduling algorithms: a review. the journal of supercomputing 10 december 2018 14. sa'ed abed, duha s. shubair (2018) enhancement of task scheduling technique of big data cloud computing 2018 international conference on advances in big data, computing and data communication systems (icabcd) 15. xiaomin li, di li, song li, shiyong wang, chengliang liu (2017) exploiting industrial big data strategy for load balancing in industrial wireless mobile networks ieee access (volume: 6) 29 december 2017 16. xi zhang, qixuan zhu (2019) information-centric virtualization for software-defined statistical qos provisioning over 5g multimedia big data wireless networks ieee journal on selected areas in communications (volume: 37, issue: 8, aug. 2019) 17. johnu george, chien-an chen, radu stoleru, geoffrey g. xie (2019) hadoop mapreduce for mobile clouds. ieee transactions on cloud computing (volume: 7, issue: 1, jan.-march 1 2019) 18. sudip misra, ayan mondal, swetha khajjayam (2019) dynamic big-data broadcast in fat-tree data center networks with mobile iot devices ieee systems journal ( volume: 13, issue: 3, sept. 2019) 19. muhammad rehan raza, ahmad rostami, lena wosinska, paolo monti (2019) a slice admission policy based on big data analytics for multi-tenant 5g networks journal of lightwave technology (volume: 37, issue: 7, april1, 1 2019) 20. cristo suarez-rodriguez, ying he, eryk dutkiewicz (2019) theoretical analysis of rem-based handover algorithm for heterogeneous networks ieee access (volume 7) 17 july 2019 advances in systems science and application (2015) vol.15 no.2 148-163 living and banking systems comparison: prisoners’ dilemma “win-win” is not the solution pierre bricage international academy for systems and cybernetic sciences, iascys, vienna, austria abstract to survive that is ‘to eat and not to be eaten’ so as to live on. whatever its spatial and temporal level of organization, every living system owns 7 invariant qualitative degrees of freedom. any living system is formed by embedments and juxtapositions of pre-existing systems. the same goes for man banking systems! how are the local quantitative laws of the spatial-temporal structuring and functioning of banking systems associated with the basic law of survival of living systems? how do the local actors become mutually integrated into their global whole? and reversely (systemic constructal law), why and how is the global whole reciprocally integrating the local parceners? is victory a strategic success? what are the roots of interdependence, conflicts and strategic order challenges?how is emerging a new power balance? can banking systems survive as parasitic systems? like a “food chain” is a “money chain” a way of violence escalade? the evolution of living systems is often seen as a “cooperative evolution” resulting from altruist behaviours. it could be modelled and simulated using games such as the prisoners’ dilemma game, a game that shows why 2 individuals might not cooperate, even though it appears to be in their best interests to do so. is the prisoners’ dilemma game justifying extortion? what can we learn from reinforcement learning dynamics in social dilemmas? in reality, humans display a systematic bias towards cooperative behaviour, much more so than predicted by models of “rational” self-interested action. models based on different kinds of payoffs and driving forces (where people forecast how the game would be played if they formed coalitions to maximize their forecasts) are shown to make better predictions which resemble reality. keywords parasitism, pareto equilibrium, ponzi pyramid, prisoners’ dilemma game. 1 introduction for living systems to survive that is first ‘to eat and not to be eaten’ [1]. in a predator-prey relationship to survive the predator must eat preys but not too much. the predator’s survival is limited by its prey’s limitations of survival [2]. soon or late the preys are to be eaten. but sometimes preys can win and even eat predators. that is the case of bacterial preys which eat predator amoebas, mycobacteria or cancer cells which destroy their predator immune cells. whatever its level of organization, to live on, any living system has ‘to be lucky’ so advances in systems science and application (2015) vol.15 no.2 149 as ‘to be at the right place at the right time’ [3]. whatever is its spatial and temporal level of organization, to survive and live on it owns 7 invariant qualitative characteristics [1]: figure 1. to survive every living system is a particular place for mobilization of matter and energy. entering flows (inputs) are used to make products and wastes which are then used, stored (throughputs) or excreted (outputs). if the outputs/inputs balance allows an internal accumulation of matter and energy, the system may grow in mass. growth (quantitative increase) is not a goal in itself but it is always a prerequisite phase for development (acquisition of new qualitative capabilities) and reproduction (figure 1). formed by embedments and juxtapositions of pre-existing systems in a new whole (endophysiotope), a living system is always part of a food chain; it eats and is eaten, within an ecoexotope of survival it shares with other living systems (fig.1). ‘soon or late it is impossible not to be eaten.’ man is not an exception [2, 4-5]. man species is a champion for enhancing growth of domestic plants fig. 1 living systems structuring and functioning: interaction is construction [1, 6, 10] and animals but for its own growth [4-6]! man species is ensuring its survival through the increase of the hosting capacity of its ecoexotope of survival (figure 1). the same goes for man societies which are often guided by quantitative economic considerations such as “saving more and more money” [7-8] rather than 150 pierre bricage: living and banking systems comparison. prisoners dilemma “win-win” is ... by qualitative ethic considerations: “enough food for everyone everywhere”. living systems are food producing systems. we do know how they work [5, 9]. the modularity of living systems allows both a partial location and a global recycling of matter and energy [1]. the pleiotropy of the structures and functions, allowing ‘to kill two birds with one stone’, is a mechanism of exaptation. within any ecoexotope, the agoantagonistic relations balance soon or late ends in the disappearance of predators and a reduction of biodiversity [6]. only the merging into associations for the reciprocal and mutual sharing of advantages and disadvantages (armsada) allows the emergence of a new biodiversity [6,10]. banking systems are money producing systems. we do know how they work [11]. can their comparison with living systems permit to understand the origin of the current economic and living crises? and thus suggest adequate solutions [4, 12]? 2 money flows modelling workers exchange their labour against money [7]. this money input allows a person to buy food (to survive that is first to eat) and goods (in order not to be eaten by infectious diseases or to enhance her/his capacity of reproduction): figure 2.the two goals for our society to make money are health and beauty! if money inputs are exceeding their needs working consumers (or consuming workers) may put their money into a bank (or not). money flows can be analysed in terms of interactions, but biologically sounding! in food chains, into an ecosystem, in a commensalism situation only one partner receives benefits. the same goes in a parasitic situation but here the other partner is harmed! only a mutualistic situation allows both partners to have some benefits: table 1. if the bank takes your money and if you can get it back as you want, when you want and freely (without charge), it is a commensalism situation. you and the bank are “eating” the same money at the same table [7]. but, without your money, the bank has no money and cannot make money with your money. what does that mean in terms of advantages and disadvantages? if the bank has enough money -because a lot of people are storing their money in itthe bank can lend you some money. it is an advantage for you. you can buy something you could not otherwise. but there are never advantages without disadvantages. you must pay off your loan with interests. it is a disadvantage for you but an advantage for the bank. all that is an advantage for you is a disadvantage for the bank and reciprocally. how is that situation balanced? if enough deposits the bank has enough money to make loans, to give consumers credits that allow the bank to make money with your money:fig.3. it is a store-take-make situation (fig.2 & 3) [5-7]. to increase cash-flow [13], to have more money to loan, a bank can give you interests for the money you will put in its stock for a while. obviously, the inadvances in systems science and application (2015) vol.15 no.2 151 terest you will earn will be less than the money the bank will store with its loans interests. it is a mutualistic situation. you are at the same table, in the same food chain (the same money chain). you are nourishing the bank and the bank is nourishing you. of course the bank is eating more than you are. of course you can earn money only if other people in the same bank (the same food chain) have not enough money and if you have too much (figure 2). with increasing money stocks, the bank can keep on making more money and is growing. depending on the situation you are, either your money is growing too (if you are loaning money to the bank) or your debt is if you are a borrowing money from the bank. only those associated with the bank growth see their money increase too [13]. whatever its spatial and temporal level of organisation, to live on, any living system,owns 7 invariant qualitative capabilities (top left scheme): c1 mobilisation of matter and energy flows ,c2 growth (=accumulation), c3 reaction to stimulations, c4 organisation into space and through time, c5 integration (from integer: “to make one”) into an ecoexotope (exo: external, tope: spacetime, eco: of inhabitation) of survival, shared with other forms of life, c6 reproduction of its self, and c7 movement. these 7 capacities allow its endophysiotope (endo: internal, tope: space-time, physio: of functioning) to be hosted by an ecoexotope which furnishes the endophysiotope a hosting capacity, but only if the endophysiotope owns an adapted capacity to be hosted in it (integration). a system is always made of 3 kinds of entities: actors, interactions and the whole (top right model). it is always more and less than the sum of its parts. but whatever the system, actors are interacting and each action is a cause of an effect which may also be a cause (feedback) [14]: interaction is construction and construction is interaction (systemic constructal law). a living system is always a system of systems (down left scheme) made by embedments and juxtapositions of previous systems. an endophysiotope at a i level is an ecoexotope of survival for a i-1 level endophysiotope. it is an iteration process (a “matryoshka race”). formed by embedments and juxtapositions of pre-existing systems in a new whole (endophysiotope), a living system is always a part of a food chain -it eats and is eaten, within an ecoexotope of survival shared with other living systems-. growth (quantitative mass increase) is not a goal in itself but it is always a prerequisite phase (larval stage) before the endophysiotope development (acquisition of new qualitative capacities) and its reproduction (number growth). during the mass growth eco-phase, a mass threshold must be passed to reach the minimum volume so as to acquire the adult stage.“the capacity of reproduction has a cost paid by growth.” but a minimum duration (generation time) is required in order to gain it. to survive every living system is a particular place for mobilisation of matter and energy (down right scheme). entering flows (inputs) are used to make products and wastes which 152 pierre bricage: living and banking systems comparison. prisoners dilemma “win-win” is ... are then used, stored (throughputs) or excreted (outputs). if the outputs/inputs balance allows an internal accumulation of matter and energy, the system’s mass may grow [1,9]. there are different kinds of description of interactions between living systems. fig. 2 banking structuring and functioning: from synergism to commensalism and mutualism. wether a species/a partner (species a or species b) receives (or not) benefits from another one, we usually distinguish 3 patterns of interactions: commensalism (e.g. “to eat at the same table”, but unequally!), mutualism (e.g. “to share benefits”, though maybe unequally in quality or quantity), parasitism (e.g. “to eat another system” which is harmed). depending on the kinds of interactions (+ benefit, no benefit but harmed effects, 0 no positive or negative effect) we can add up neutralism, competition or injury situations. some people even add up the notions of amensalism or proto-cooperation [15]. these interactions build up a network (which can be graphed) between the different actors of a system of systems (figure 1). yet whatever the interaction (for example here: mutualism or competition, each actor’s action among every couple of actors is both a cause and an effect: systemic constructal law. usually, and particularly in economadvances in systems science and application (2015) vol.15 no.2 153 table 1. ago-antagonistic interactions between 2 locally isolated species. ic models, only a simplified point of view is taken into account with 2 actors (prisoners’ dilemma “game”), which results in 3 situations: “win-win” (the 2 actors cooperate.), “lose-lose” (the 2 actors defect.), “lose-win” or “win-lose” (one cooperates and loses, the other one defects and wins) [8]. of course, due to feed-backs [11, 14], reality is far more complex [16]! 3 the current crisis situation today in france, after the 2009 economic crisis (due to a lack of growth from states and banks), by the law, you must put your money in banks. you can use cash only for small payments. you cannot be paid with cash. your salary must be deposited into a bank. you must use credit cards, checks or money orders. you must pay for all of that! when you trust banks with your money you pay a bank service management to be able to use your money or to give your money back. you will never get back all the money you shall deposit! and when you get your money back you have to pay again some money to the bank. the bank is always making money with every money deposit. you get no advantage in putting your money into the bank. the bank always has advantages when you put your money into the bank. the bank takes a part of your money for you to use your money. and the bank freely uses your money to make loans and money. services banks were freely giving to you a century ago in exchange for them to 154 pierre bricage: living and banking systems comparison. prisoners dilemma “win-win” is ... fig. 3 banking systems structuring and functioning as parasites[6-7]. use your money, you must now pay for! it is a parasitic situation. the bank is eating your money! in less than a century, we ran from a synergetic situation to a parasitic one (table i). every day you are gaining and using money, the bank grows and you have less money. the body (money mass) of the bank grows allowing the bank to have more and more money and to create new shelters— in order to store more money—just like parasitic living systems are laying more and more eggs! more money entering the bank, more growth for the bank (figure 2) and increased lifetime for the bank [7]. that is exactly what we see in a ponzi scheme (figure 3), and that is exactly how modern states are growing and extending their area of growth and lifetime. looking back to 2007, the 2009 and 2012 crises were crises for consumers not for banks. for banks, it was an extraordinary growth increase [13]. their cashflow is now bigger than ever— with a 3 fold increase in 5 years. “you cannot eat the pie and have the pie” but banks can! not only banks but also states are growing, adding more and more “layers” to a ponzi pyramid [7-8]. but, soon or late, and faster and faster, a limit comes up! and the limitation for restoring growth is harder and harder to break. crises were exploding when the u.s.a. local growth stopped and the u.s.a. government had to extend its market outside, and worldwide, to a more and more global world market. china did the same but better. the european union did the same but with less efficiency [13]. in a ponzi pyramid the higher you are the more money you get. and the lower advances in systems science and application (2015) vol.15 no.2 155 you are the less you get. it is a pareto situation: 20% people get 80% of money or goods (and power), the remaining 80% get only 20%. you take money from the poorest to make money for the richest. it is typically a prisoners’ dilemma game situation [7-8, 17]. 4 is crisis a problem or a solution? ecosystems are often graphed as a pyramid. and just like in a ponzi pyramid the survival of the highest level depends on the survival of the largest down one [5, 10]. but in a food chain, the living pyramid is built from the bottom up. the ponzi one is built from top to bottom. indeed, food chains are not pyramids but mixed networks: “all eggs are not in the same basket”, “diversity is the rule”, “reciprocal exchanges are the law”. only reciprocal rewards can stabilize cooperation [18-19] and, soon or late, allow the merging of all the actors into an armsada [5-6,9]. that situation of reciprocal and mutual sharing of advantages and disadvantages can be modeled: figure 5. economic systems are neither machineries nor mechanistic linear systems but complex non-linear dynamic systems of systems as are ecosystems. thus, quality (rather than quantity), development (rather than) growth, creation and variety (rather than accumulation) are the keys. sharing limited resources could be equally done but parts amounts are decreasing with number in an hyperbolic way (figure 4) and “you cannot eat the pie and have the pie” [4,8]. the 2009 crisis is the cause (or the effect?) of a double increase of households debts [13]. in the loads of interests, commissions, financial services and administrative costs are voluntarily and artificially excluded, resulting in a false double reduction of these interests. through debt negotiation loans durations could be twice as long resulting in at least a double increase of loans costs, and interests are paid first! more debts are more money [7-8]! recession periods intervals (figure 3) are shorter and shorter: 1993-2002 (9 years), 2003-2009 (6 years), 2009-2012 (3 years), and each year now? that is the same sign of extension as for a pandemic infectious disease like flu! flu intervals were shorter and shorter: 1918-1957 (39 years), 1957-1968 (11 years), 1968-1977 (9 years), before -as in a ponzi pyramida new host was invaded (as a new food chain basis for the pyramid) [7]! and, as in a ponzi pyramid, regrowth resulted in a calm of 20 years (1977-1997). but then, flu crisis intervals bursted again shorter and shorter: 1997-2003 (6 years), 20032005 (2 years); in 2014, 4 different influenza viruses simultaneously were at the origin of epidemics in the popular republic of china. (-from top left to down right-) works people do allow them to gain money they use to buy foods and goods (economic exchange). if some money is not spent it can be stored (accumulation) through bank deposit (or not!). storing 156 pierre bricage: living and banking systems comparison. prisoners dilemma “win-win” is ... more money allows banks to have enough money flow control (take) and thus to loan money to people who need it to buy foods or goods. but the bank takes “guarantees” to ensure that the money returns and with interests. money flow is always balanced in favour of the bank (money creation), but without any cost for other people (local commensalism). to have a credit is an advantage for people if they can pay the interests and give the money back. money is neither their property nor that of the bank but the deposits/property of other people (global synergism). to have more money deposits, bigger amounts and for longer durations, banks can give (or not...) money interests paid by the bank to “money-sharers” (“i can put your money to work for you. don’t ask me how. just let me show you”). but the interests paid by banks are always smaller that the interests the banks earn through money loans. there are never advantages without disadvantages. for consumers, disadvantages are greater than advantages. but for banks, advantages are greater than disadvantages. of course banks have taxes and salaries to pay, but they do so with money they make from debts and with money they get from people who do not have debts but “stored” money (pareto situation [16]: figure 4). nowadays money is virtual. only banks know how much money there is! the first upturn was an advantage for asia and latin america, but there was no second upturn. after each break, when the upturns came, only countries from the o.p.e.c. (organization of petroleum exporting countries) still had the same growth they had before. in europa, from the first to the second relative upturn, in 6 years unemployment doubled and no job was created. using a ponzi pyramid process at the world scale, only the u.s.a. stabilized their re-growth, but that was an advantage only for u.s.a. and a disadvantage for all the other countries. only japan enters into a no-inflation new developmental stage [7, 13]. inputs (money deposits) are always greater than outputs (take: figure 2) because the interests paid by banks are always smaller that the interests the banks earn through money loans (accumulation flow). but if you must always pay a fee to put your money into a bank and pay again to get your money back from the bank, you will never have the “availability” of your money— the bank is eating your money! it is a parasitic situation! and when states say “you must put all your money into a bank”, banks can easily make money with your money and take the percentage they want from your money, when they want! it is a take-make situation (figure 1) in favour of banks. for each money flow a part is “rapt” by the bank, banks are growing but not your money! if we model this situation, with t time, with a money amount of a bank, or debt of a consumer (debt makes money), we typically have a ponzi pyramid graph. crisis situations, r (shaded areas), are promoting the slope increase of money accumulation and crises are nearer and nearer [13]. this graph is exactly that of the advances in systems science and application (2015) vol.15 no.2 157 evolution of the us federal debt (as published by the us department of treasury financial management service, from 1965 to 2013, with debt increasing from a factor 1 to 20, in 50 years -from research.stlouisfed.org-). more debts make more money only for banks: “banks can eat the pie and have the pie.” hyperbolic causes and consequences correlations are common laws in living systems (figure 4), as it is for example with the metabolic rate of the nitrogen/phosphorus ratio (n/p). for living systems, development x growth = constant, quality x quantity = constant. what about economic systems? -figure 4just like in economic supply/demand models, hyperbolic graphs can be linearized, by changing the variables or the representation used (logarithmic scale), but the dilemma remains the same... growth or development? quantity or quality? an hyperbolic graph typically is the mark of a pareto phenomenon (20% of actors having 80% of rewards, the remaining 80% getting only 20%), but is this an optimum or an equilibrium [20-22]? in a pareto optimum [16], in the case of the prisoners’ dilemma game [21], gamers/prisoners may cooperate and have equal rewards. in this case any other outcome gives a worse outcome for at least one player. one is rewarded: the bank, the other is threatened: the consumer. and the reward is the biggest banking systems can have. in a nash equilibrium, each player’s strategy is the best response to all other players’ strategies. but it has a cost: rewards or outcomes are lesser and equally shared! that is never the case... except in the benevolent voluntary and united sector of social and solidarity economy [8]! armsadas are emerging from such processes [4-5]: the maximum value of the whole is superior to the sum of the maximum and minimum values for an individual actor [21]. -leftthe winners-losers prisoners’ dilemma game (table 1) fig.4banking/living systems structuring and functioning: limits and limitations. [6,9] 158 pierre bricage: living and banking systems comparison. prisoners dilemma “win-win” is ... as a pareto equilibrium example [16]: simplified example of the greek bailout. -rightexample of biological hyperbolic laws xy=k of growth for 2 different legumes species, with y=n/p (where n is a limiting factor for both protein and nucleic acid syntheses, with protein synthesis limiting both mass and number growths, and p a limiting factor for both nucleic acid synthesis and energy storage, with energy storage limiting both growth and development) and x=relative growth rate. for living systems, development x growth = constant, quality x quantity = constant [4-7,9]. the higher the growth x, the smaller the developmental step [4]. -middlethe systemic constructal law: “interaction is construction, construction is interaction”. actors from adjacent levels of organization are interacting, causes getting effects and effects being new causes getting other effects and so on. growth is a prerequisite for development [9]. but just like in a predator-prey relationship, banks or states tell us only what they want us to hear! it is a sort of take-make-waste phenomenon like there is in dying living systems1. the hosting capacity, of any market or ecoexotope, is limited because of a supply and production interaction between the hosting capacity (of an ecoexotope) and the capacity to be hosted (of an endophysiotope). for living systems, growth is durable only if it is sustainable and sustained by each partner, for a benefit for their whole, as is it for an armsada. for a partner to survive all the partners must survive first (and their whole too). that is not the case in hidden banking networks2.the loss of resources among a trophic chain causes big and small animals to be less common and less productive higher up in the food chain. but medium ones are preserved with an increase in biodiversity. in ponzi pyramids the opposite happens. biggest ones will be bigger, poorest ones poorer and medium ones will disappear... the pareto 20%-80% situation! only reciprocal rewards may stabilise a system of systems, particularly in ecoexotopes of hard survival. capitalism must be redefined in terms of an e1if a living system does not stop its functioning because the concentration threshold of a toxic waste is under that of substrates for minimal activity, its growth shall stop quickly. the low hosting capacity of its ecoexotope lowers its endophysiotope growth. but if its endophysiotope has a high capacity to be hosted because of a low threshold of demand, its growth can be durable, even in presence of toxic wastes (figure 5). 2in the u.s.a., in 1950, a unit of tax paid for 1 social security recipient was supported by 16.5 workers but only by 3.0 in 2009, e.g. a load increase of 5.5 fold in 60 years, a duration which equals the time before retirement, and with an increase of payment durations of 10 years (from the age of 68 to 78!). this is a hidden ponzi pyramid sponsored by banks, insurances and states (source:social security administration, the 2010 annual report of the board of trustees, cdc, us life tables. if you are 55 today, current law will pay you 75 cents on the dollar: you paid 100 at 45 and will get 75 at 65 (source:the us social security trust fund, may 2011 report to the congress. payable benefits as percent of scheduled benefits). where are the missing 25? advances in systems science and application (2015) vol.15 no.2 159 cosystem. growth (quantitative increase) is only a prerequisite and a tool for development (qualitative acquisition). development is quality creation, step by step, using growth [1,5-6]. “quantity or growth is the problem” and “quality or development is the solution” [4]. modelling with biological concepts [9,15-16] supports input-output-recycling processes [4, 6,21-23]. -top, from left to righthyperbolic law qq=k [4]: the higher the number of parts q, the smaller the equal amount of each q (not a pareto situation here), just like in an economic situation (with the demand and production relationship); in an ecological situation the bigger the amount of food consumed, the bigger the wastes. but food is limited, under a minimal food threshold or above a maximal supported wastes threshold growth stops. -down, from left to rightecoexotope hosting capacity limitation and endophysiotope capacity to be hosted [6]. if a living system does not stop its functioning due to a concentration threshold of toxic waste (q toxicity) being lower than the concentration threshold of substrates for its minimal activity (q activity), its growth will stop very rapidly and eventually it shall die (down left). the low hosting capacity of its ecoexotope lowers the duration of its endophysiotope growth (figure 1). but if its endophysiotope has a high capacity to be hosted, “thanks for” a low threshold of demand (q activity is low) the system growth can last, even in presence of toxic wastes (q toxicity above q activity) [4]. armsada emergence, as a whole, results (down right) from complementing effects of the different local capacities to be hosted of the various partners (k and k). globally they share the same ecoexotope of survival but using different local parts of its global hosting capacity [6,9]. depending on the local mutual changes of the hosting capacity all that is an advantage for a partner is a disadvantage for another one and reciprocally [5]. 5 discussion-conclusion for banking systems to grow and survive that is ’to eat money and have their money not to be eaten’ just like for living systems it is to eat food and not to be eaten [1]. but with living systems we know that in a predator-prey relationship to survive the predator must eat the matter of preys but not too much [2, 4]. the survival of the predator is limited by the limitations of survival of its preys! isn’t the same ecologic law valuable for economic systems [7]? in a century, banking systems have gone from a mutualistic functioning to a parasitic one [7]. inputs are used only for banks profits, to make products that earn money for people having money. waste products, that is to say all that has a cost for banks, are paid by consumers (figure 3). services are servitudes now! the outputs/inputs balance allows an internal accumulation of money but only for bank shareholders. banking systems are growing and reproducing! and growth 160 pierre bricage: living and banking systems comparison. prisoners dilemma “win-win” is ... fig. 5 living systems structuring and functioning: limitations and limits [1,4,67] (quantitative increase) is their goal! is this functioning a cooperative one [15, 22-23] or an extortive one [8]? with living systems we know that growth is only, and always, a prerequisite phase for acquisition of new qualitative capabilities (development) [1, 9]. we also know that growth, development and survival are durable only if they are sustainable for and sustained by all the partners [1, 14]. banking system has to learn that to survive all the consumers must survive first! that is not the case nowadays—if banking systems evolution obeys the same law living systems evolution does, only systems with a high capacity to be hosted -because of a low threshold of demand (no cost of services, no extra fees)will last, particularly in crisis and even in presence of toxic products such as toxic sub-primes. if win-win situations may exist locally (chinese growth was 8.1% in 2011 but 7.5% in 2012 and is decreasing, taiwan growth was only 3.6% in 2013, 2.4% is expected for great-britain in 2014, but uk is playing his local game not the european global one!), with globalization no win-win situation can persist at a world scale. you can never always be a winner, soon or late you will be a loser. only armsadas are lasting! like ecology, economy obeys a cyclic organization of life [16], with growth and differentiation phases, with entrepreneurial and managerial behavior phases that are replacing one another all the time. each phase change is an emergency situation, as is metamorphosis for living systems [9]. macroeconomic disasters fit a power-law model [24] as fit living systems evolutionary changes due to ecological disasters [4,7,10]. you can never have too much money! it is only by coupling insights from ecology and economy [22] that we can begin to model and understand the complex dynamics which underlie the advances in systems science and application (2015) vol.15 no.2 161 generation of poverty [12] and bank profits [8, 13], through growth but not for development [4]! references [1] p.bricage. 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(2014), “survival management by living systems. a general system theory of the space-time modularity and evolution of living systems: armsada”, world conference on complex systems, pp.16, see https://hal.archives-ouvertes.fr/hal-01065974. [6] p.bricage. (2011), “thinking and teaching systemics: bio-systemics inhigher education”, iascys g.a., chengdu, p.r.china, pp.1-14, see http://tinyurl.com/biosystemics. [7] p.bricage. (2014), “alive and banking systems comparison: prisoners dilemma”, in globalization and crisis. systems complexity and governance, methodologies and tools for governance of world system, no.7, pp.301303. [8] j.-p.delahaye. (2014), “prisoners’ dilemma game and the illusion of extortion.” (the original is in french:“le dilemme du prisonnier et l’illusion de l’extorsion”), pour la science, no.435, pp.78-83. [9] p.bricage. 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(1993), prisoner’s dilemma, anchor, see http:// en.wikipedia.org/wiki/prisoner’s dilemma. [22] z. wang, a. szolnoki and m. perc. (2013), “optimal interdependence between networks for the evolution of cooperation”, scientific reports, no.3, pp.2470. [23] z. wang,a. szolnoki and m. perc. (2012), “if players are sparse social dilemmas are too: importance of percolation for evolution of cooperation”, scientific reports, no.2, pp.369. advances in systems science and application (2015) vol.15 no.2 163 [24] r.j.barro and t. jin. (2011), “on the size distribution of macroeconomic disasters”, econometrica, vol.79(3), pp.434-455. corresponding author pierre bricage can be contacted at: pierre.bricage@univ-pau.fr adv syst sci appl 2019; 02; 63-79 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/667 study and characterization of thermal comfort in a desert climate oudrane abdellatif1*, aour benaoumeur1,2 1) faculty of science and technology, university el-wancharissi of tissemsilt (cut), road bougra ben hamouda, 38004, tissemsilt, (algeria). e-mail: abdellatif.habadat@gmail.com 2) laboratory of applied biomechanics and biomaterials (labab), polytechnic of oranmaurice audin (enpo-ma), bp 1523 el mnaour, 31000, oran, (algeria). e-mail: ben_aour@yahoo.fr received february 12, 2018; revised december 9, 2018; published december 31, 2018 abstract: the objective major of this numerical study is the characterization of thermal comfort in new habitable architectures located in a completely desert area. this numerical characterization is intended to determine the parameters that affect the thermal comfort for the occupants of these architectures in this region. to achieve this objective, a numerical model describing the thermal exchanges taking place in a model of a habitable envelope has been developed. this model is based on thermal balances established at the level each wall of the habitat. the numerical models developed have been validated using climatic data recently measured in the renewable energy research unit of the saharan medium at adrar ‘’urer'ms‘’. a well detailed analysis of the some parameters that influence the thermal comfort in this architecture was raised and discussed. the fundamental equations governing thermal exchanges have been concretized by an implicit finite difference method, based on the nodal procedure. the system of algebraic equations obtained was solved by the iterative gaussian method. the results of the numerical simulation have shown that the material currently used in the construction of this architecture of adrar region, as well as the current climatological conditions, are the main causes of the thermal discomfort. keywords: finite difference, numerical models, adrar region, characterization, thermal discomfort, heat exchange. 1. introduction environmental issues and energy consumption are increasingly alarming global concerns today. it is essential to adopt solutions to obtain more energy-efficient and sustainable buildings, both new and existing. the use of solar energy is a means of improving the use of natural energy, which can reduce energy consumption [1]. the energy performance of a habitat can be estimated by performing thermodynamic simulations that take into account different conventional assumptions such as: the weather, occupancy, temperature set points, and uses of the window and shutter by the occupants and even the architecture of the habitats. the experience feedbacks in the habitable envelope with high energy performance highlight significant differences in energy consumption between forecasts and summer overheating [2]. ryzhov and al [3] in his recent study, he approached the importance of renewable energy exploitation, more specifically the solar energy in buildings. he attributed the strategy that promotes reduced energy consumption and environmentally friendly techniques. the development of a good energy concept for a house involves the search for a balance between reinforced insulation, compactness, passive solar gains, thermal inertia and comfort, * corresponding author: abdellatif.habadat@gmail.com mailto:ben_aour@yahoo.fr 64 a. oudrane, b. aour copyright ©2019 assa adv. in systems science and appl. (2019) indoor air quality and investment cost. this has been the subject of specific attention since the pre-project phase and most often requires the use of premature studies [4]. the essential function of a house is to ensure an interior ambiance well suited to our needs and our comfort. the inhabitant often places his comfort before saving energy. it is therefore necessary to plan the construction and installations to consume less energy while ensuring adequate comfort [5]. saheli and al [6] studied the influence of climatic conditions on the local heating efficiency of a building installed in the south algeria. furthermore, benhammou and al [7,8] analyzed the thermal behavior of a housing submitted to periodic solicitations under hot and dry climate. this analysis was made for the hottest july in the summer season, in order to examine the effect combined of the heat insulation and the passive cooling by eahe on the thermal performances of the livable envelopes for desert regions in the south algeria. this numerical study consists of visualizing not only the impact of climatic conditions and building materials used in adrar region's architecture on thermal comfort, but also its effects on the main heat transfer mechanisms related to the thermal comfort process. in addition, from the results of digital investigation of different variables such as: the internal temperature, external temperature and the density of the solar flux, we show ourselves such an approach which allows improving the thermal comfort in this architecture. 2. domestic architecture the raw earth is a material available everywhere in the planet. it is the universal material the privileged and the easiest handled by the human being to make a shelter while several millennia. globally, earthen houses today house more than a third of the world's population [9]. this construction material, respects the man and the environment, is perfectly recyclable. it ensures thermal comfort by its characteristics and offers advantageous economic convenience. algeria, this vast territory has experienced diversified land architecture according to the diversity of climatic zones. the domestic conception of adrar region, south-west of the sahara, differs from that of algiers, or that of m'zab valley at ghardaïa, at the lower sahara. the model of earth allows a real diversity of architectural language [9]. unfortunately, since the floods of 2009, the use of this material is banned, and replaced by cement (reinforced concrete). this material seen as modernity symbol and social promotion. in figure 2.1, we present the new architecture of adrar region which is based on reinforced concrete without total control of thermal insulation. fig. 2.1. new reinforced concrete architecture in adrar region. study and characterization of thermal comfort in a desert climate 65 copyright ©2019 assa. adv. in systems science and appl. (2019) on the other hand, in figure 2.2 we present the old architecture of this region. this architecture allows a thermal insulation based on the raw earth. the latter played the role of an insulating material against thermal discomfort. fig. 2.2. the old architecture with raw earth in adrar region. 3. climatological characteristics adrar has a typical hot desert climate of the hyper-dry saharan zone. it's the heart of sahara, with a hot, very long summer and a hot short, moderate winter. the annual average of precipitation in this region reaches hardly 14 15 mm, falling essentially in autumn or to spring [10]. the maximal average temperatures are 46 48 °c in july (the hottest month), what makes of adrar one of the world hottest cities. as an example: the peak of record temperature was established on monday, july 9th, 2018 with a temperature of 65 °c [11,12,13]. 3.1. solar data of the typical day chosen in adrar region table 3.1 presents the astronomical data for the designated typical day. noting that these data will be used to calculate the different densities of solar flux influencing heat exchange in habitable envelope. these data were recently measured by a radiometric station to in the renewable energy research unit in the saharan medium of adrar ‘’urer'ms’’. this autonomous radiometric measuring station was realized at the end of 2010. furthermore, it measures the global radiation parameters on three plans (horizontal, inclined at the latitude of the place and optimal monthly) and the ambient temperature. figure 3.1 present a local image of this radiometric station in adrar site. 66 a. oudrane, b. aour copyright ©2019 assa adv. in systems science and appl. (2019) fig. 3.1. local picture of adrar radiometric station. table 3.1. astronomical data of the typical day [12]. typical day (july 17, 2014) maximum ambient temperature in (°c) 47,70 minimum ambient temperature in (°c) 32,50 average ambient temperature in (°c) 40,70 maximum flux in (w/m2) 1051,00 average flow in (w/m2) 323,00 average wind speed in (m/s) 5,80 duration of the day in (h) 14,00 the time sunrise in (h) 5,00 the time sunset in (h) 19,00 the sun declination in (°) -13°,12’ the time correction in (minute) 17’,97’’ the hourly distribution of the wind speeds represents an indicator for the wind potential. his knowledge allows the estimation of the wind energy available on the site. the figure 3.2 represents the frequency distribution of average speeds measured for day given in percentage term for adrar site. the analysis of these curves shows that site of adrar has important wind energy potential and which is more favorable to exploitation of this energy type for electricity production. the reached average speed is from 5 to 6m/s in july. study and characterization of thermal comfort in a desert climate 67 copyright ©2019 assa. adv. in systems science and appl. (2019) fig.3.2. hourly evolution of the average wind speed [12]. 4. description of physical model the physical model used in this study is a parallelepiped-type habitable envelope composed of a single four-facade room, see figure 4.1. the construction is located on a surface of 20m2. this room is built according to the following standards:  the walls of the room are constructed of a light structure usually, 15 cm of block full of cement.  the floor slab is placed on a flat earth. it is cast directly onto a 4cm layer of thermal insulation.  the roof consists of a slab of reinforced concrete. fig.4.1. sketch of the piece studied. 5. implementation of model and resolution in this numerical study, we have opted for the nodal method use, which makes it possible to establish a thermal network equivalent to the modeling element. conductive, convective and radiative processes are considered in global form (conservation of the heat flux). according to on the complexity of the network adopted, one can to incorporate all or part of the heat transfer phenomena, which allows predicting, with more precision, the thermal behavior of 68 a. oudrane, b. aour copyright ©2019 assa adv. in systems science and appl. (2019) the studied part [14]. the phenomenological equation of heat relative to an infinitesimal volume makes it possible to write [15,16,17]: p.dτ).ddiv(dτ t t ρ.cp     (5.1) with; t t   : is the rate of variation of stored energy in dτ ; )div( : is the outgoing flow in dτ ; p.dτ : is the energy flow generated. by integrating this expression on a volume vi at temperature ti, the energy balance is written [15-18,19,20]: iradianceconvectionairconduction i i pφφφφ t t .ρ.cp    (5.2) or; cpi : is the calorific capacity of the volume vi pi : is the source of flux dissipated in the volume vi conductionφ , airφ , convectionφ and conductionφ : are the flows exchanged between vi and its environment by conduction, fluidic transport, convection and radiation. figure 5.1 shows the different modes of heat transfer in the study room. fig.5.1. description of the different modes of heat transfer for the studied room. the analog schematization, we allow to propose an electric circuit amounting to the thermal system to be able to apply the law of ohm [21,22,23,24]. study and characterization of thermal comfort in a desert climate 69 copyright ©2019 assa. adv. in systems science and appl. (2019) fig.5.2. electrical map of the studied room. and: extt : external ambient temperature in (° c); skyt : temperature of the celestial vault in (° c); soilt : soil temperature in (° c). as a condition the limits of the room studied, consideration was given to the ground temperature equal to the external ambient temperature. in addition, the temperature of the celestial vault is calculated by the following formula [25,26]: 5,10552,0 ambsky tt  (5.3) indeed, the coefficient of external convection is given by [25,26]: speed wind8,37,5 vh extc  (5.4) the law enforcement of ohm, in every knot of the studied room, gives the system of following equations [17-20]:  thermal balances of external facades of the studied room                                            absorpeesolpeevc b ext pee pe b pe absorpfpevc b extc pfpe pfp bpfp absorpoesolpoevc b extc poe po bpo absorpnevc b exctc pne pn bps absorpsesolrpsevc b extc pse ps b ps pt.hrt.hrt. e λ t.hr δt dt . s .cpm pt.hrt. e λ t.h δt dt . s .cpm pt.hrt.hrt. e λ t.h δt dt . s .cpm pt.hrt.hrt. e λ t.h δt t . s .cpm pt.ht.hrt. e λ t.h δt dt . s .cpm pnesol (5.5) 70 a. oudrane, b. aour copyright ©2019 assa adv. in systems science and appl. (2019)  thermal balance of interior facades of the studied room                                                                               t..sht..sht..sh t..sht..sht..sh δt dt ..cpm t.ht.ht.h t.ht.ht. e λ t.h δt dt . s .cpm t.ht.ht.h t.ht.h.δδ e λ t.h δt dt . s p .cm t.ht.ht.h t.ht.ht. e λ t.h δt dt . s .cpm t.ht.ht.h t.ht.ht. e λ t.h δt dt . s .cpm t.ht.ht.h t.ht.ht. e λ t.h δt dt . s .cpm pipc6pfp5cpe4c po3cpn2cps1c int a a pippeirpfpipeirpoipeir pnipeirpsipeir b intc pei pe bpe pippfpirpeipfpirpoipfpir pnipfpirpsipfpir b intc pfpi pfp bpfp pippoirpfpipoirpeipoir pnipoirpsipoir b intc poi po bpo pippnirpfpipnirpoipnir peipnirpsipnir b intc pni pn b pn pippsirpfpipsirpeipsir poipsirpnipsir b intc psi ps bps (5.6) the algebraic equations systems resulting from the discretization of transfer equations in both habitat environments (external and internal) are expressed in the form of matrices that can be written as: a . x = b. this equations system is solved by using the iterative numerical method of gauss.                                                                                                                    11 10 9 8 7 6 5 4 3 2 1 211111110118116114112 10111010109108106104102 91099 8118108887868482 7877 6116106866656462 5655 4114104846444342 3433 2112102826242221 1211 00000 0000 000000000 0000 000000000 0000 000000000 0000 000000000 0000 000000000 b b b b b b b b b b b t t t t t t t t t t t aaaaaa aaaaaaa aa aaaaaaa aa aaaaaaa aa aaaaaaa aa aaaaaaa aa tt a tt pei tt pee tt pfpi tt pfpe tt poi tt poe tt pni tt pne tt psi tt pse (5.7) or:            psersolpservc b cext ps pspstt pse hh e h ts cpm ta 1 . . 11 (5.8) study and characterization of thermal comfort in a desert climate 71 copyright ©2019 assa. adv. in systems science and appl. (2019)         e ta btt psi  12 (5.9)           ps t solpsersol t vcpservc t extcext t pse ps bps fsgththth t t s cpm b .... . 1 (5.10)            piprpsipfpirpsipeirpsipoirpsipnirpsi b c ps bpstt psi hhhhh e h ts cpm ta  int22 1 . . (5.11)  pnirpsi tt pni hta   24 (5.12) the temperature of the thermal comfort (tc) for a habitable one makes it possible to estimate the heating needs [27]. however, based on the bibliographic synthesis that was conducted in order to find a suitable model for the calculation of the comfort temperature, we opted for a relationship that was established between the comfort temperature (tc), internal temperature (tint) and mean radiant wall temperature (tpi), as follows [28,29]: 2 tt t 6n 1n piint c     (5.13) ppipfpipoipeipnipsi 6n 1n pi ttttttt    (5.14) taking into account the air as a transmission fluid between the ambiance and the room facades. table 2 shows the physico-thermal properties of the air that were taken during the numerical simulation. table 5.1. the air thermal properties [29]. λ thermal conductivity in (w.m-1.k-1) 0,0262 cp specific heat in (j.kg-1.k-1) 1006 ρ density in (kg.m-3) 1,177 μ dynamic viscosity in (kg.m-1.s-1) 1,85.10-5 υ kinematic viscosity in (m2.s-1) 1,75.10-5 pr number of prandtl 0,708 6. flow chart of numerical modeling the calculation of the different heat transfer parameters within the habitable envelope under consideration is based on the following steps:  introduction of climatic data of the considered region for the selected typical day.  calculation of the geometrical and astronomical parameters of the sun in the considered typical day.  introduction of climate data such as: the average amount of the ambient solar flux and the ambient temperature of the considered day.  determination of solar radiation incidence angles for each facade of the habitat as a function of the hour angle and the position of the sun in the sky. 72 a. oudrane, b. aour copyright ©2019 assa adv. in systems science and appl. (2019)  computation of the amount of solar flux density for of all the house facades. the numerical tests carried out during the development of the computer code in fortran language led to the retention of a time step of 300 seconds. in figure 6.1, we show the computational steps developed in the numerical simulation code. fig.6.1. flow chart of calculation steps in the habitat performed by the developed fortran code. 7. results and discussion 7.1. internal evolution of temperature of the room figure 7.1 shows the hourly variation of the temperature of different internal facades of the room during the typical day. it can be seen that the thermal inertia of the concrete plays a very important role in the heat transfer at the walls. indeed, the ceiling temperature (tpfp) is greater compared to the temperatures of the other internal faces of the south walls (tpsi), west (tpoi), north (tpni) and east (tpei). this is due to the conjugation of two essential factors that led to this increase in temperature: in the first place, the walls thickness of the room which involves the elevation of the thermal inertia and secondly, the inclination angle of the roof facade which is equal to 0 °. study and characterization of thermal comfort in a desert climate 73 copyright ©2019 assa. adv. in systems science and appl. (2019) fig.7.1. internal evolution of air temperature in the room, according to the local time. 7.2. internal evolution of temperature according to building materials to use construction materials wisely, it is essential to know their thermal properties. the thermo-physical properties of the materials used in this study are shown in table 7.1. the thermal capacity of a wall is especially useful if it is placed in the interior of the room and isolated from the external climatic conditions. to build in strong inertia, it is therefore to use heavy materials inside the room in order to store the solar heat and to attenuate the internal temperature variations [30]. conversely, the temperature of a room with low inertia increases rapidly at the slightest ray of sun without the possibility of storing solar heat. internal temperature differences will be significant and the risk of overheating will be high. strong inertia is especially useful in case of permanent occupation. low inertia can be interesting for habitable environments with intermittent use [30,31]. table.7.1. thermophysical properties of building materials [30,31]. heavy concrete lightweight concrete concrete stone light wood heavy wood thermal conductivity  (w.m-1. k-1) 1,75 1,00 1,40 0,14 0,20 density  (kg.m-3) 2200 1500 1895 540 800 specific heat cp (j.kg-1. k-1) 1000 1000 1000 2400 2700 emissivity  0,54 0,54 0,54 0,86 0,86 absorption coefficient  0,6 0,6 0,6 0,07 0,07 figure 7.2 illustrates the thermal capacity effect of solar energy storage in the room wall of for different building materials (light concrete, heavy concrete, stone concrete, light wood and heavy wood). it can be noted that for all habitable rooms built with high thermal capacity materials such as lightweight concrete (see fig. 7.2(a)), heavy concrete (see fig. 7.2(b)) and concrete stone (see fig 7.2(c)), the temperature of its internal space is optimal in the month of july because of the thermal inertia of these materials. indeed, it is well observed in figures (see fig. 7.2(a), fig.7.2(b) and fig.7.2(c)) that the temperature of the internal ambiance varies between 37°c and 38,5°c. in addition, if the construction of habitable rooms is based on low heat capacity materials, the temperature of the internal space varies around 34°c and 35°c because of the thermal inertia of this building materials type (heavy wood (see fig. 7.2(d)) and light wood (see fig. 7.2(e)). furthermore, it is noted that the temperature of the internal facades for building 74 a. oudrane, b. aour copyright ©2019 assa adv. in systems science and appl. (2019) materials with high thermal capacity is very high compared to the internal facades of buildings with low heat capacity materials. study and characterization of thermal comfort in a desert climate 75 copyright ©2019 assa. adv. in systems science and appl. (2019) fig.7.2. evolution of temperature of the room internal facades for different building materials. 7.3. evolution of the thermal comfort temperature figure 7.3 shows the hourly change in temperature of thermal comfort of different building materials as a function of local time. from this illustration, it can be seen that the choice of construction material has a significant influence on the hourly evolution of the thermal comfort temperature. indeed, when one uses for example the concretes selected categories in this study as building material with these dry climatological conditions, and without control of thermal insulation, the comfort temperature will exceed the conventional norms. that is to say, undesirable overheating of the habitable 76 a. oudrane, b. aour copyright ©2019 assa adv. in systems science and appl. (2019) environment with a temperature of 39°c at 15h00 in the afternoon as illustrated in figure 8 for the three categories of concrete (heavy, light and stone). on the other hand, the exploitation of other building materials such as the categories of wood (heavy and light) in this region with the same climatic conditions will help to get closer to the conventional values of the comfortable temperature in the environment. habitable, since the maximum temperature obtained using the wood does not exceed 37°c (value obtained at 15h00 pm). fig.7.3. evolution of the thermal comfort temperature in the room for different building materials in a typical day. 7.4. overlay on the psychrometric chart figure 7.4 shows the superimposition on the psychometric chart of temperature of thermal comfort within the room for different building materials in the typical day (july 17th, 2014). it can be seen that the thermal comfort temperature of concrete structures (heavy, light and stone) is positioned in the area of thermal discomfort and the need for evaporation. while for both types of wood (heavy and light) is positioned in the same area as that of the concrete categories, except that the comfort temperature for a concrete construction varies between 38°c and 39°c and the temperature of a wood construction varies between 34°c and 36°c. noting that these thermal comfort temperatures were calculated for the typical day of july 17th, 2014, which is the hottest day of the summer season in adrar region. fig.7.4. super-positioning temperatures of thermal comfort of the room for different building materials during the typical day. study and characterization of thermal comfort in a desert climate 77 copyright ©2019 assa. adv. in systems science and appl. (2019) 8. conclusion thermal comfort is an essential element for the well-being of the occupant in his built environment. taking it into account that the habitat implies there different aspects. however, after the numerical examination of the temperatures of different internal facades of the habitable room, the change effect of the building materials on the evolution of the internal temperature and the superposition of the thermal comfort temperatures on the psychometric chart, it has been found that:  the use of bare concrete as a building material in the current architecture of adrar region, involves undesirable overheating. in addition, the same effect has been noted for wood in this region, but with temperatures a little closer to conventional temperatures of thermal comfort.  the severe climatic conditions of this region contribute in a direct way to the thermal discomfort. on the basis of this study, it is recommended that before beginning the construction of a habitable envelope in this desert region, it is imperative to undertake an overall revision of conventional standards for thermal comfort, in particular the respect for bioclimatic concepts. on the other hand, we suggest:  ventilation and evaporative cooling during the summer season.  insulation and shading of the walls the most requested for excess solar radiation. nomenclatures abbreviation e the material thickness m pe east wall hc coefficient of internal heat exchange by convection -1-2 .kw.m pn north wall mps the south wall mass kg po west wall pabsor amount of heat absorbed w ps south wall spl south wall surface m2 pfp the ceiling wall tamb ambient temperature °c pni internal north wall tsol temperature of soil °c psi internal south wall tair air temperature °c pfpi internal wall of the ceiling t temperature difference °c pse external south wall tint full air temperature in the habitat °c poe external west wall text outside air temperature °c pip internal floor wall tb temperature of the concrete slab °c pee externe east wall tc thermal comfort temperature °c pfpe external wall of the ceiling fsg global solar flux w/m2 m'zab region north of sahara algerian references [1] dengjia w., yanfeng l., jing j., and jiaping l., (2015). the optimized matching of passive solar energy supply and classroom thermal demand of rural primary and secondary school in northwest china, procedia engineering vol. 121, 1089 – 1095. 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[31] mavromatidis l. e., m. el mankibi, (2012). numerical estimation of time lags and decrement factors for wall complexes including multilayer thermal insulation, in two different climatic zones, applied energy vol. 92, 480–491. adv syst sci appl 2019; 02; 120-133 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/593 optimal rotor design of claw-pole alternator for performance improving n.brahimi*1, s.tahi2, e. boudissa1, m.bounekhla1 1) laboratoire de système electrique et télécommande, université blida 1, blida, algeria. e-mail: nouzha_m74@yahoo.fr 2) electrical and industrial systems laboratory, usthb/fei, algiers, algeria. e-mail: stahi@usthb.dz received may 17, 2018; revised june 21, 2019; published july 10, 2019 abstract: the demand in electric energy, in vehicles, has experienced a remarkably rapid growth for a long time and this trend is expected to remain during the incoming years. the claw pole alternator is designed to provide the necessary power for all the electrical accessory loads over the entire operating speed, ranging from an idle speed of 1800rpm to a top speed of 6000 rpm. because of the relationship between the speed and the output power, producing enough power at idle speed becomes a challenging problem. the aim of this paper is to optimize the rotor design of conventional claw pole alternator in order to improve its performance during the idle speed. to derive the optimal rotor structure, cyclic coordinate method linked with magnetic equivalent circuit (mec) model is used, including magnetic saturation effect and leakage flux. an increase of up to 70% in output power at idle speed is demonstrated and significant improvements in performance over the whole speed range were observed. keywords: claw pole alternator, reluctance network, mec model, electric generator, shape optimization, cyclic coordinate method. 1. introduction automotive alternators are responsible for the generation of electricity in almost all vehicles using an internal combustion engine. they must be capable of powering all the relevant systems the vehicle requires during operation. until now, the claw pole alternator constitutes the unique device responsible for the generation of electricity within a vehicle [4]. the main advantages of the claw pole alternator are its low cost and construction simplicity. therefore, it is well adapted for industrial and mass production. however it suffers from low efficiency caused by high leakage flux [19] and poor output power [22]. performance and efficiency vary widely depending on the alternator speed and load conditions [12]. commonly, during idle speed (1800 rpm), the claw pole alternator isn’t able to sustain active electrical loads for a reasonable amount of time without undue drain on the battery. therefore, obtaining sufficient output power at idle speed is required. recently there has been a considerable amount of work into developing higher power alternators with improved performance. some of this work has focused on designing a new model of alternators such as the new claw pole alternator where the dc-excited winding is located in the stator [10]. another example is the doubly excited brushless claw pole alternator, where the hybrid excitation is achieved by ferrite permanent magnets suitably mounted on both the rotor and the stator in one hand, and a field winding in the stator on the other hand [5, 18]. most of these new designs of alternators need a lot more development before they become a * corresponding author: nouzha_m74@yahoo.fr mailto:stahi@usthb.dz optimal rotor design of claw-pole alternator 121 mature technology with possible integration in the vehicle power generating system. improved performances are also obtained by, simply, adding surface or interior permanent magnet to the conventional claw pole alternator. several studies have been carried out on hybrid excitation claw pole alternators [13, 24, 25]. in hybrid excitation, at least two magnetization sources are present in the rotor. in addition to the dc excited wound field, permanent magnets are placed into the rotor to compensate leakage flux and to boost rotor flux. permanent magnet increases the power density and efficiency [21], however their higher price undermines their usage [16]. furthermore, current regulations and industry marketing policies push the development process cycle energetically less expensive and environment friendly. this tight context implies finding new technical solutions to keep the same efficiency without permanent magnets. for these reasons a lot of efforts are being spent to improve performance and efficiency by optimizing the design of an existing traditional (conventional) claw-pole alternator [8, 23]. this paper describes the rotor design optimization of an existing conventional claw pole alternator while keeping same stator design and the same alternator footprint, in an attempt to increase the output power under different load conditions at the idle speed. the design improvement is achieved by using the magnetic equivalent circuit (mec), which is widely considered in literature [2, 7, 9]. it has been shown in [15] that mec model helps to predict performance of claw pole alternator with high accuracy and minimum cpu computation-time. optimal rotor is designed by using cyclic coordinate method. the optimization results which provide the optimized rotor geometrical dimensions are discussed. the performance and the output power of the optimal claw pole alternator over the entire speed range from idle speed of 1800rpm to top speed of 6000 rpm are presented. 2. 3d nonlinear magnetic equivalent circuit model claw-pole alternator mathematical modeling is complicated because the paths of the magnetic flux are 3d. therefore, the analytical method of magnetic equivalent circuit (mec) based on reluctance networks theory is used [6, 7]. the claw pole mec model employed in this work has been developed by rakotovao [17] and used for optimization by albert et al [2]. this mec model is less time consuming compared with numerical model based on finite element theory [9]. thus, the mec model is particularly suitable for optimization purpose [15]. in the mec method, each flux path section is characterized by magnetic reluctance. the good estimate of reluctances is essential because it guarantees the precision of the model and its coherence during optimization. accuracy of the mec model also depends on the topology of the network [14]. 2.1 expression of reluctances when material saturation is taken into account, the reluctance is non-linear function of the flux, as shown in equation (2.1) [9]: 𝑅 = 𝐿 ф 𝐻 ( ф 𝑆 ) (2.1) where, 𝐿 is the average flux path length, 𝑆 is the cross-section area and ф is the magnetic flux. h(b) is the analytical expression of the magnetic field versus the flux density 𝐵, which is expressed as follows [9]: 𝐵 = ф 𝑆 (2.2) 122 n.brahimi, s.tahi, e. boudissa, m.bounekhla 2.2 resolution method of the mec referring to the literature [9, 11], it has been reported that numerical resolution procedure of the claw pole mec is based on newton raphson algorithm and reluctance network theory. 2.2.1 mec resolution in no load condition mec resolution enables the evaluation of the fluxes circulating in the different parts of the magnetic circuit. in the case of no load condition, alternator’s back electromotive force (backemf) is a function of the direct flux component only. the computation of the no-load backemf is expressed as follows [9]: 𝐸 = 1 √2 𝑁𝑎 𝜔 ф𝑚 (2.3) where, 𝑁𝑎 is the number of turns per armature phase, 𝜔 is the angular frequency and ф𝑚 is the maximum flux crossing a phase which is evaluated using the mec resolution. no-load back-emf is plotted in fig. 2.1, where it can be noticed that the magnetic saturation occurs when the field current is greater than 1a. fig. 2.1. no-load back-emf emf versus field current 2.2.2 mec resolution under load condition claw pole alternator is a salient-pole synchronous generator with high saturated areas. therefore, including the effect of the armature magnetic reaction in both q-and d-axis is required. within the q-axis, magnetic saturation is low, resulting in a constant stator transverse inductance, while in the d-axis, saturation effect is important [2, 4]. then, mec accounting for the d-axis armature reaction is consider in order to carry out the flux linkage, especially, the one which allows the determination of the d-axis component of the back-emf ( 𝐸 𝑑 ), given by equation (2.3). under load condition, the back-emf phase is expressed in terms of its d and q components, as follows [9, 11]: 𝐸 = 𝐸 𝑑 + 𝐸 𝑞 (2.4) with { ‖𝐸 𝑑‖ = 𝐸𝑑(𝐼𝑓 , 𝐼𝑑) ‖𝐸 𝑞‖ = 𝑋𝑞. 𝐼𝑞 (2.5) where 𝐼𝑑 and 𝐼𝑞 are the d and q components of the armature current. 𝐼𝑓 is the field current. 𝑋𝑞 is the q-axis reactance. moreover, for resistive load and adopting blondel model illustrated in fig.2.2, the backemf: 𝐸 is given by equation (2.6) [2, 4]: 𝐸 = 𝑉 + 𝑅𝑠𝐼 + 𝑗𝑋𝜎𝐼 (2.6) optimal rotor design of claw-pole alternator 123 where: 𝑅𝑠 and 𝑋𝜎 are the phase resistance and leakage reactance respectively. rl is the resistive load. 𝑉 and 𝐼 are voltage phasors and the armature current respectively. (a) (b) fig.2.2. (a) blondel phase diagram in the case of resistive load. (b) claw pole alternator phase equivalent circuit. procedure for an operating point calculation is illustrated in fig 2.3. at each step, the change in the solution is monitored. the process continues either until the change is less than the newton raphson tolerance of 0.1%. the solution is obtained, in several seconds, after maximum ten iterations. fig.2.3. operating point calculation algorithm. 3. alternator performance in the following section the alternator performances are evaluated across the whole speed range. the 1800 rpm speed corresponds to idle speed and 6000 rpm corresponds to cruising speed. figure 3.1 (a) shows the calculated alternator output voltage versus the field current, parameterized by the load current. the alternator generated voltage increases along with 124 n.brahimi, s.tahi, e. boudissa, m.bounekhla alternator speed and the field current. at idle speed, the maximum current drawn from the alternator is 20.62 a, with field current and output voltage are equal to 5a and 12vac respectively. at this operating point, the saturation is well considered. it can be observed that when the load current increases the output voltage drops. 3.1 armature current characteristic the calculated armature current versus alternator speed is illustrated in fig.3.1 (b). the field current is set to 5 a. the armature current curve is characterized by three operating point. the first one is the generation starting speed (pgs) or 0-ampere-speed at which the alternator reaches its rated voltage without delivering power. the second one is the maximum output current at cruising speed (pc), which corresponds approximately to the shortcircuit current of the alternator. the third one is the output current at idle speed (pi)[12]. (a) (b) (c) fig. 3.1. (a) stator phase voltage versus field current. (b) armature current versus speed and the three operating points. (c) output power versus speed at fixed output voltage 3.2 alternator output power at fixed output voltage: particular attention is paid to the output power requirement at idle speed, at which, the alternator must deliver at least the power needed for long-term consumers. no output power is required below the idle speed. the calculated output power curve versus alternator speed is shown in fig.3.1. (c). 3.3 maximum output power of the alternator. the calculated maximum output power of the alternator is given in fig. 3.2. the field current is set to 5a. figure.3.2 (a) shows the load current that corresponds to the maximum power delivered from the alternator. figure.3.2 (b) shows the output voltage that corresponds to the maximum power. these curves show the amount of maximum available output power of the alternator. optimal rotor design of claw-pole alternator 125 (a) (b) fig.3.2.calculated alternator output power parameterized by the alternator speed, versus (a) armature current, (b) stator phase voltage. 4. optimization design procedure the aim is to optimize rotor structure of claw pole alternator at idle speed, including the magnetic saturation effect, while maintaining the same initial alternator footprint. the stator design parameters remain unchanged. table 4.1 reports the alternator parameters, with assigned constant values that do not change during the optimization process. two optimization procedures are described, namely: first optimization procedure in the first optimization procedure, the objective function is to maximize the output power at idle speed while maintaining the output voltage at 12vac. in this case, maximizing the output power leads to maximizing the delivered armature current. second optimization procedure in the second optimization procedure, the objective function is to maximize the output power without imposing the output voltage or armature current. table 4.1. main fixed parameters of the cpa. fixed parameter value number of phases 3 number of pairs of poles 6 number of stator slots 36 outer stator diameter (mm) 127 inner stator diameter (mm) 88.6 shaft diameter (mm) 18.1 height of plateau (mm) 11.95 length of rotor (mm): 52 4.1 rotor parameters optimization  first optimization procedure five crucial geometrical optimization parameters are selected, as shown in table 4.2 and illustrated in fig 4.1. some of these geometrical parameters determine the shape of the claw pole. determining the shape of the claw pole is the most important obstacle in the claw pole alternator designing process. the claw pole has an important role in closing the magnetic field lines because it ensures the route from the rotor excitation towards the air gap. the parameter, related to rotor core (core length) is, also, very important, especially as it conditions the space booked for the field winding.  second optimization procedure in this case, besides the five geometrical optimization parameters, the armature current comes to be added as the sixth optimization parameter. 126 n.brahimi, s.tahi, e. boudissa, m.bounekhla table 4.2. design of the sixth optimization parameters of the claw pole rotor optimization variable optimization parameter base value (mm) geometrical optimization parameters x1 width of the base of the claw 24.7 x2 height of claw tip 4.7 x3 core length 29 x4 air gap length 0.65 x5 width of claw tip 6 — x6 armature current — in addition to these optimization parameters, there are other five geometrical implicit parameters, listed in table 4.3 and illustrated in fig 4.1. to ensure the geometrical coherence of the rotor, these parameters are subject to change according to optimization parameters evolution during the optimization process. fig.4.1. geometrical parameters of rotor pole structure table 4.3.implicit optimization parameters of the claw pole rotor implicit variable implicit parameter base value (mm) x7 distance between claw 6.6 x8 claw pole length 29 x9 claw side plate thickness 11.5 x10 core radius 41.4 x11 outer rotor radius 87.3 the optimization of the rotor claw pole is a multivariable non-linear problem. in this investigation, the objective functions cannot be expressed as closed forms thus a derivative free optimization method is used for the objective function evaluation. therefore, cyclic coordinate method is applied. 4.2 cyclic coordinate method a cyclic coordinate method [3, 20, 26] using the unknown parameters as the search directions, is applied to maximize the objective function successively along each coordinate. more specifically, the method searches along the directions , where is a vector of zeros except for a one at the j th position. thus, along the search direction , the design variable is changed while all other variables are kept fixed. this technique is described in fig. 4.2 for the first iteration and in the case of two coordinate axes x1 and x2. starting from p0 (x10, x20), the maximization is performed successively along x1 and x2 and leads, respectively, to p1 (x11, x20) and p2 (x11, x21). the iterative process is repeated until the error test is satisfied. the algorithm shown in fig. 4.3, ndd ,.......,1 jd jd jx optimal rotor design of claw-pole alternator 127 estimates the vector parameters p that maximizes the objective function which increases at each iteration. fig.4.2. 2-d illustration of the cyclic algorithm fig.4.3. flowchart of the cyclic coordinate algorithm in this study, the alternator output power is chosen as objective function, which is expressed as follows [22]: 𝑃 = 3 ∗ 𝑉0 ∗ 𝐼𝑠 (4.1) where, 𝑃 is the output power, 𝑉0 and 𝐼𝑠 are the rms value of output voltage alternator and armature current respectively. both speed and field current are kept constant and equal to 1800 rpm and 5a respectively. orientation of the optimal solution toward a technically correct solution requires us to limit the search space. therefore, the optimization variables are constrained to vary in a range, between a lower and an upper limit, as shown in table4.4. the armature current is introduced, only in the second optimization procedure, as a sixth variable. table 4.4. optimization variables (mm or a). optimization variable lower limit upper limit x1 0.3 0.65 x2 1 5.5 x3 25 29 x4 6 38.02 x5 3 10 x6 10 80 128 n.brahimi, s.tahi, e. boudissa, m.bounekhla 5. optimization results and commentaries as expected, using mec coupled with cyclic coordinate method yields results, a few iterations (about 1h30) are required to achieve the maximum of the objective function. cyclic coordinate is a deterministic method, thus it is appropriate to secure the obtained results. this is done by using different sets of initial points and to check if the same maximum output power is obtained. figure.5.1 (a) and fig.5.1 (b) show the evolution of the objective function. the results of the optimal geometrical dimension of the rotor parameters are summarized in table 5.1 and table 5.2.  first optimization procedure : the output power, at idle speed, increases from 0.742 kw to around 1.27 kw which constitutes 71.15% improvement. the armature current increases from 20.62 a to 35.27 a.  second optimization procedure: at idle speed, the maximum output power reaches 1.3kw which constitutes 53.68% improvement. table 5.1.first optimization procedure, base and optimized values of design variables optimization parameter value before optimization (mm) value after optimization (mm) air gap length 0.65 0.3 height of claw tip 4.7 1 core length 29 25 width of the base of the claw 24.7 23.13 width of claw tip 6 4.35 implicit parameter distance between claw 6.6 8.7 claw pole length 29 25 claw side plate thickness 11.5 13.5 core radius 41.4 42.1 outer rotor radius 87.3 88 table 5.2.second optimization procedure, base and optimized values of design variables optimization parameter value before optimization (mm or a) value after optimization (mm or a) difference in parameter value (between 1st and 2nd optimization) (mm) air gap length 0.65 0.3 0 height of claw tip 4.7 1 0 core length 29 25 0 width of the base of the claw 24.7 23.94 -0.81 width of claw tip 6 4.17 0.18 armature current 29.08 32.09 — implicit parameter distance between claw 6.6 8.4 0.3 claw pole length 29 25 0 claw side plate thickness 11.5 13.5 0 core radius 41.4 42.2 0.1 outer rotor radius 87.3 88 0 optimal rotor design of claw-pole alternator 129 (a) (b) fig.5.1. evolution of output power versus number of iteration for the (a) first optimization procedure. (b) second optimization procedure.  impact of the air gap length: the air gap length can affect the output power considerably. in theory, to maximize the output power of an alternator, the air gap length should be designed as small as possible. for a typical alternator, the nominal air gap is approximately 0.4 mm but it can be larger or smaller. when the air gap is reduced, the output power is substantially improved [1]. however, the air gap of a claw-pole alternator should not be designed too small. when running at very high speeds, the alternator rotor poles will deflect due to centrifugal forces and the pole tip will touch the stator. in this investigation, the optimization process, can give solutions for very small values of the air gap, but the technically feasible solution selected is the one which corresponds to the air gap lower limit given above. the air gap flux is increased from 2.627 10-4 wb to 3.333 10-4 wb.  impact of the width of the base of the claw and the width of claw tip: these two parameters, determine the exchange surface between rotor and stator through the air gap. it is important to recall that, the magnetic radial flux is more concentrated at the basis of the claw pole than at its extremity. hence, the radial fluxes increases when the width of the base of the claw pole increases. however, the leakage fluxes between two adjacent claw poles, which are the most important leakage fluxes, increase when the width of the base of claw increases. the optimization process gives the optimal value of these two variables which lead to the maximum output power. in table 5.3, one can observe the reduction of the leakage flux between two adjacent claw poles and leakage flux between claw pole tip and opposite plate.  impact of the height of claw tip: the leakage fluxes between the claw pole and rotor winding can be affected by this parameter. to reduce these leakage fluxes, the distance between rotor core and the bottom of the claw should be designed as far as possible. this explains the fact that the optimal value of this variable tends to the lower limit given above. table 5.3 shows the reduction of the leakage flux between the claw pole and rotor winding.  impact of the core length: the optimization process assigns the lower limit to this variable. however, what is lost in length will be gained in claw side plate thickness. as a result, the junction who binds the core to the claw side plate is larger which enables more axial flux to be captured. table 5.3. leakages fluxes before and after optimization leakage fluxes base value (*10-4 wb) value after optimization (*10-4 wb) between adjacent claw pole fingers 1.1286 0.6855 130 n.brahimi, s.tahi, e. boudissa, m.bounekhla 6. alternator performance after optimization after optimization process, the output performances of the optimal claw pole alternator are evaluated. in fig. 6.1(a) and fig. 6.1 (b) the armature current and output power, are respectively plotted versus the rotational speed. it can be seen that, at idle speed of 1800rpm, optimized rotor design leads a higher armature current and hence to a higher output power. figure 6.1 (c) illustrates that only 1a field current is needed to deliver current at idle. in fig.6.2, (a) and (b) the maximum output power reaches 1.3 kw at idle speed. it can be observed that output performances of the optimazed alternator has been improved in the whole speed range from idle speed of 1800rpm to the cruising speed of 6000rpm. at the cruising speed of 6000 rpm the maximum output power reaches 4.5 kw. (a) (b) (c) between claw pole and rotor winding 0.04016 0.02449 between claw pole tip and opposite plate 0.02933 4.5953 10-6 optimal rotor design of claw-pole alternator 131 fig. 6.1. (a) stator phase current versus alternator speed. (b) alternator output power versus alternator speed. (c) stator phase voltage versus field current. (a) (b) fig. 6.2. (a) output power versus armature current. (b) output power versus stator phase voltage. 7. conclusion in this paper, the optimization method based on cyclic coordinate method coupled with magnetic equivalent circuit (mec) model, has been applied successfully for the design of rotor claw pole alternator. the main objective was to maximize the alternator output power for given stator dimensions and footprint at the idle speed of 1800rpm. the output power increased significantly at idle speed, it was demonstrated that output power and armature current of the optimized claw-pole alternator increased from 0.742 kw to 1.27 kw and from 20.62 a to 35.27a, which constitute 71.15% and 71.04% improvement, respectively. this is predominantly due to change in reluctance of geometric parameters, consequently affecting the flux throughout the alternator. at idle, the maximum output power delivered from the optimized alternator achieved 1.3 kw which constitutes 53.68% improvement. the obtained results indicate significant performance improvement of the optimized alternator over the whole speed range (1800 rpm to 6000 rpm), whereas the geometry optimization process has been fully concentrated at the idle speed. at high speed of 6000 rpm, the maximum available output power increased from 2.9 kw to 4.5 kw, a 50.90% improvement future investigation would consider the use of the mec model in the design optimization of various generators taking advantage of the claw-pole structure in harvesting energy systems. references [1] alaeddini,a., darab, a., & tahanian, h., (2015). influence of various structural factors of claw-pole transverse flux permanent magnet machines on internal voltage using finite element analysis. serbian journal of electrical engineering, 2 (12), 129-143. doi: 10.2298/sjee1502129a [2] albert, l., chillet, c., jarosz, a. & wurtz, f., (2003). analytic modeling of automotive claw-pole alternator for design and constrained optimization. 10th eur. conf. power electronics and applications, in processing, toulouse, france. 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[25] zhenyang zhang, huijuan liu & tengfei song, (2016), optimization design and performance analysis of a pm brushless rotor claw pole motor with fem, machines, mdpi, 3 (4). [26] zaïm m. e., dakhouche k. & bounekhla m., (2002). design for torque ripple reduction of a three-phase switched reluctance machine, ieee transactions on magnetics, 38(2): 1189–1192. https://doi.org/10.1109/ever.2013.6521547 https://doi.org/10.1109/indcon.2013.6725997 https://doi.org/10.1109/elma.2017.7955390 https://ieeexplore.ieee.org/xpl/mostrecentissue.jsp?punumber=4753968 adv syst sci appl 2021; 03:63–74 published online at https://ijassa.ipu.ru. the model of the production side of the russian economy nikolay pilnik1, stanislav radionov1* 1financial research institute of the ministry of finance of the russian federation, moscow, russia abstract: we consider the model of the production side of the russian economy. this model is derived as the solution of the nonlinear dynamic optimization problem of the macroeconomic agent we call producer. this agent maximizes his discounted profit flow under technologic, demographic and financial constraints. we use the method of relaxation of complementary slackness conditions to transform the model to the more regular one and evaluate its parameters on the russian macroeconomic data. we show that this model can successfully replicate the large set of russian macroeconomic indicators such as gross domestic product, loans of producers, volume of fixed assets etc. keywords: dynamic optimization, complementary slackness conditions, production of gdp 1. introduction and literature review we present the model of production side of the economy, which consists of all economic agents who produce the added value. in the context of general equilibrium models these agents can be considered as elements of macroeconomic agent producer, similarly to agents consumer, bank, government and others. this agent generates a supply of good and demand for labor and capital, which is typical for macroeconomic models, but also has connections with banking system in terms of loans both in national and foreign currency. to our knowledge, this is the first model which describes the process of production and interactions of producers with the banking system on such a detailed level. we consider the maximization problem of discounted profit flow of this macroeconomic agent under technologic, demographic and financial constraints. it leads to the nonlinear dynamic optimization problem. we solve it using lagrange method and transform the solution by relaxation of the complementary slackness conditions (see [22]). this transformation leads to the nonlinear dynamic system. we estimate its parameters on the russian macroeconomic data and find a set of parameters that allows to reproduce it with rather high accuracy. in the modern literature, the works which are closest to ours are devoted to the estimation of production functions. this topic is considered as one of the most important in the empiric economic studies, because it deals with the fundamental economic process of formation of goods. the description of this process in terms of mathematical models is thus essential for the understanding of the functioning of the economy. apart from theoretical interest, it also has crucial policy implications. a large body of econometric literature is devoted to the estimation of production functions on the microdata (cobb-douglas function is usually used). the problem of obtaining reliable estimations turns out to be rather challenging because of simultaneity – if firm knows its productivity level when choosing inputs, it leads to the correlation between errors and ∗corresponding author: saradionov@edu.hse.ru 64 n. pilnik, s. radionov regressors. another problem is that firms with low productivity levels are harder to observe because they cease to exist sooner. several methods were developed to overcome these difficulties. probably the most popular ones were proposed in [20] and [17], but they were questioned in [2]. other methods were proposed, among others, in [7] and [15]. modelling of gross domestic product in terms of production functions is also widely used in the literature, especially in the works dealing with output gaps and potential growth rates, see for example [11], [12], [14], [16]. these works adopt a rather straightforward methodology which could potentially also suffer from simultaneity problem. in the context of gdp estimation this problem was addressed, for example, in [23]. description of gdp production in terms of production function is also ubiquitous in the modern macroeconomic dynamic stochastic general equilibrium (dsge) models, however they usually use a very simple cobb-douglas production functions. it should be noted that over the last years, a ces production function becomes more and more popular tool of production side of the economy, because empirical studies reject the hypothesis of elasticities of substitution between capital and labor being equal to one (the case of cobb-douglas function). the constant elasticity of substitution production function was derived in [4], and the numerous articles with estimation of ces production function on us data soon followed. in the first estimates the elasticity of substitution was indeed close to one (e.g., [10]), but the methodology used for these estimations was criticized. use of more rigorous econometric methods led to estimates well below unity (e.g., [18], [19], [9], [13]). new arguments in favor of cobb-douglas function were presented in [6], but they were again challenged in [3]. it should also be noted that more flexible production function than cobbdouglas is required in many growth models to generate the plausible dynamics (e.g. [1]), and in policy analysis (e.g. [8], [5]). a thorough discussion of ces production function is presented in the special issue of journal of macroeconomics in june 2008. 2. the model: statement we consider the model of the whole sector of the economy, which generates the added value, as a single agent. we will reduce it to the usual dynamic model that determines the supply of produced product, the demand of producer for loans, investments and labor, taking prices, wages, interest rates and other external factors on the market into account. there are several non-standard features in the proposed model. first, we divide investments into two parts: investments in the maintenance of fixed assets and investments in building up fixed assets. this approach allows to explain the fluctuations of output with higher accuracy, perceiving these investments as transaction costs and capital costs respectively. secondly, the production function depends on the volume of used fixed assets, adjusted on the level of investment in the maintenance of fixed assets. thirdly, we use a similar scheme to describe the contribution of labor in output – similarly to the fixed assets, we include human capital in the production function, which can be increased due to special wage investments. the formation of production capital (fixed assets) by the producer is described by the following relations: d dt m (t) = jm (t) − δam (t)m (t) , (2.1) 0 ≤ jm (t) . (2.2) in this equation m (t) are the fixed assets, jm (t) are the investments in building up fixed assets, δam (t) is the depreciation coefficient. this equation allows to calculate the series jm (t) based on statistics on the level of fixed assets and depreciation. the balance of investments j (t) = jm (t) + ju (t) , (2.3) copyright © 2021 assa. adv syst sci appl (2021) the model of the production side of the russian economy 65 where j (t) is the overall level of investment (gross fixed capital formation) in base year prices, allows to calculate the investments in the maintenance of fixed assets ju (t). further we assume that the prices of these investments may be different, therefore the balance of investments in nominal volumes can be written as pj (t) = pm (t) jm (t) + pu (t) ju (t) , (2.4) where pj (t) is the overall level of investments (gross fixed capital formation) in the current prices, pm (t) is the deflator of investments in building up fixed assets, pu (t) is the deflator of investments in the maintenance of fixed assets. the formation of human capital is described in a manner similar to the description of fixed assets: d dt h (t) = jh (t) rmax (t) − δah (t)h (t) , (2.5) 0 ≤ jh (t) , (2.6) where h (t) is the human capital of the economically active (but not necessarily employed) population per capita, jh (t) is the investments in building up human capital, δah (t) is the depreciation coefficient. everywhere further in the model we will distinguish the total economically active population rmax (t) and the number of employed r (t). we assume that human capital is distributed among all participants of the economically active population, but only the part corresponding to the employed workers is used. the statistical data provides information on the total amount of payments made by the producer to its employees – wage and mixed income w (t). further, we assume that this flow (as well as investment costs) consists of two parts: payments to employees wr (t)r (t) and investments in human capital wh (t) jh (t). hence, investment in human capital actually leads to the productivity increase of the economically active population in the economy, but does not lead to its growth. the most important element of the producer’s problem is the production function, which describes the relationship between the use of factors of production and the resulting output. when modeling the russian economy, linear production function can be used. the procedure its estimation is quite simple and can be is carried out by standard econometric tools. moreover, when instead of the indicator of fixed capital, the indicator of investments in fixed capital is used, it demonstrates a fairly high accuracy on the russian data. however, it cannot be used in the optimization problem because investments will not enter the first order conditions of the producer and it is impossible to obtain an expression for it. another approach is using the cobb-douglas function, which is popular, for example, in dsge models. but another problem arises here: on the russian data, the use of capital in such a production function leads to a poor model accuracy, and using of investments instead of capital is debatable from a theoretical point of view. that is why the presented producer model separates the long-term process of capital replacement and the short-term process of changing the load of this capital. such a technique, as will be shown below, allows to obtain a sufficiently high accuracy of the data reproduction without restrictions of the linear production function and with explicit use of capital in the model. thereby we use the following production function: y (t) = a (um (t)m (t))α (uh (t)h (t)rmax (t))β , (2.7) where um (t) and uh (t) are the utilization rates, respectively, of fixed assets and human capital. we define these indicators in the following non-linear way: um (t) = ( ju (t) m (t) )b , uh (t) = ( r (t) rmax (t) )a . (2.8) copyright © 2021 assa. adv syst sci appl (2021) 66 n. pilnik, s. radionov we substitute (2.8) into (2.7) and also substitute a with several normalization coefficients in order to avoid the dimensionality problem. as a result, the production function takes the form y (t) = y0 ( ju (t) j0 )bα( m (t) m0 )α (1−b)( h (t) h0 )β ( r (t) r0 )aβ ( rmax (t) rmax0 )(1−a)β . (2.9) depending on the market conditions, however, the volume of production 0 ≤ y (t) and the volume of products sold 0 ≤ yp (t) may vary. the producer can thus form the stock of the product in size 0 ≤ z (t), which can be calculated as d dt z (t) = y (t) − yp (t) . (2.10) denote the loans attracted by the producer by l (t). the average terms for which loans are attracted will be denoted by (βl (t)) −1. the variable βl (t) will be referred below as the inverse duration and interpreted as the average frequency of the loans return. then the dynamics of loans is described by the equation d dt l (t) = k (t) − βl (t)l (t) , (2.11) where 0 ≤ k (t) is the flow of newly attracted loans. the producer pays interest on borrowed funds rl (t)l (t), where rl (t) is the effective interest rate on loans. in addition to loans in national currency (rubles), the producer also attracts loans vl (t) in dollars. the dynamics of foreign currency loans and deposits is described by the equation d dt vl (t) = vk (t) − βvl (t) vl (t) . (2.12) here βvl (t) is the reverse durations of currency loans. newly attracted foreign currency loans are denoted by 0 ≤ vk (t). the effective interest rate on foreign currency loans is denoted by rvl (t). during its activities, the producer pays four types of taxes: value added tax, labor tax (social contributions), property tax and income tax. the rates of these taxes are denoted by τy (t), τr (t), τtm (t), τpr (t). for servicing operations related to loans and investments, the producer uses current account n (t). it is assumed that its volumes in the balance sheet are proportional to ruble and foreign currency loans, fixed assets and human capital. these proportions are defined by the coefficients νl, νvl, νm, νh: n (t) ≥ νll (t) + νvlwvl (t) vl (t) + νmpm (t)m (t) + + νhwh (t) (1 + τh (t))rmax (t)h (t) , (2.13) where wvl (t) denotes the exchange rate. other expenses of the producer are denoted by oc o (t) and are considered as the exogenous variable. thus, the financial balance of the producer can be written as d dt n (t) = k (t) − βl (t)l (t) − rl (t)l (t) − (1 + τpr (t)) pr (t) + wvl (t) (vk (t) − βvl (t) vl (t) − rvl (t) vl (t)) −oc o (t) + (1 − τy (t)) py (t)yp (t) − pj (t) ju (t) − pm (t) jm (t)− τtm (t) ptm (t)m (t) − (1 + τr (t))wr (t)r (t) − (1 + τh (t))wh (t) jh (t) , (2.14) copyright © 2021 assa. adv syst sci appl (2021) the model of the production side of the russian economy 67 where pr (t) is the profit after taxation. the above relationships represent the limitations imposed on the producer’s ability to choose the values of its planned variables (controls): h (t) , jh (t) , jm (t) , ju (t) , k (t) , l (t) ,m (t) , n (t) ,pr (t) , r (t) ,yp (t) , z (t) , vk (t) , vl (t) , y (t) . (2.15) according to the principle of rational expectations, when planning its control variables, the producer can rely on an accurate forecast of information variables: oc o (t) , rmax (t) , βl (t) , βvl (t) , δah (t) , δam (t) , pj (t) , pm (t) , ptm (t) , py (t) , rl (t) , rvl (t) , τh (t) , τpr (t) , τr (t) , τtm (t) , τy (t) , wh (t) , wr (t) , wvl (t) . the planned variables of the producer are thus a function of current and future values of information variables. the goal of the producer in the model is to maximize the total discounted utility from the undistributed profit after taxation with discount rate being equal to the deflator: ∫ t t0 e−∆ t 1 − β ( pr (t) py (t) )1−β dt. (2.16) the problem is supplemented with terminal condition, which can be interpreted as a growth condition for some linear combination of the phase variables: ω (t0 ) γ ≤ ω (t ) , (2.17) where ω (t) = ah (t)h (t) + al (t)l (t) + am (t)m (t) + an (t)n (t) + az (t)z (t) + avl (t) vl (t). this is an analogue of no ponzi condition written in terms of the producer’s own capital which is a difference between its assets and liabilities. to derive the solution of the maximization problem of the function (2.16) under constraints (2.1) – (2.14) and the terminal condition (2.17) with respect to the variables (2.15), we write down a lagrange functional and find its saddle point. the dual variables are denoted by φ2 (t) ,φ4 (t) ,φ7 (t) ,φ9 (t). the obtained system, which is a set of sufficient conditions for optimality, is presented below divided into several groups. the first group are equations for primal variables, including equations from the original problem statement: d dt h (t) = jh (t) rmax (t) − δah (t)h (t) , d dt m (t) = jm (t) − δam (t)m (t) , d dt l (t) = k (t) − βl (t)l (t) , d dt vl (t) = vk (t) − βvl (t) vl (t) , d dt pr (t) = ( ρ (t) − ∆ η − d dt τpr (t) (1 + τpr (t)) η + d dt py (t) py (t) ( 1 − η−1 )) pr (t) , n (t) = νll (t) + νvlwvl (t) vl (t) + νmpm (t)m (t) + νhwh (t) (1 + τh (t))rmax (t)h (t) , copyright © 2021 assa. adv syst sci appl (2021) 68 n. pilnik, s. radionov the second group are differential equations for dual variables d dt φ4 (t) = ( βvl (t) + ρ (t) − d dt wvl (t) wvl (t) ) φ4 (t) + (1 − νvl) ρ (t) − rvl (t) − d dt wvl (t) wvl (t) , (1 + τr (t))wr (t)r (t) = aβ (1 − τy (t)) py (t)y (t) , d dt φ7 (t) = ( δam (t) + ρ (t) − d dt pm (t) pm (t) ) φ7 (t) − (1 + νm) ρ (t) δam (t) − τtm (t) + d dt pm (t) pm (t) + α (1 − b) (1 − τy (t)) py (t)y (t) pm (t)m (t) , d dt φ2 (t) = (βl (t) + ρ (t)) φ2 (t) + (1 − νl) ρ (t) − rl (t) , pj (t) ju (t) = bα (1 − τy (t)) py (t)y (t) , d dt φ9 (t) = ( δah (t) + ρ (t) − d dt τh (t) 1 + τh (t) − d dt wh (t) wh (t) − d dt rmax (t) rmax (t) ) φ9 (t) − δah (t) − ( νh + νh d dt τh (t) + 1 1 + τh (t) ) ρ (t) + d dt wh (t) wh (t) + d dt rmax (t) rmax (t) + β (1 − τy (t)) py (t)y (t) (1 + τh (t))wh (t)rmax (t)h (t) , third group are complementary slackness conditions: [φ9 (t)][jh (t)], [φ7 (t)][jm (t)], [φ4 (t)][vk (t)], [φ2 (t)][k (t)], [ ( −τy (t) ρ (t) + ρ (t) + d dt τy (t) ) py (t) + ( d dt py (t) ) τy (t) − d dt py (t)][z (t)], where [a][b] means 0 ≤ a, 0 ≤ b, ab = 0. the last condition is the transversality condition that defines terminal condition for conjugate differential equations: ω (t0 ) γ = ω (t ) , where ω (t) = ah (t)h (t) + al (t)l (t) + am (t)m (t) + an (t)n (t) + + az (t)z (t) + avl (t) vl (t) . we transform this system according to the method described in detail in [22]. first, we transform it to the discrete time by replacing derivatives with increments. note that backward increments are used for direct variables and forward increments are used for dual variables. second, some direct variables are replaced by their values in the previous period. third, differential equations for the dual variables are replaced by expressions of corresponding separatrices. forth, the complementary slackness conditions are replaced by their more regular approximations (the relaxation of complementary slackness conditions). after all these transformations, we proceed to the dynamic system which can also be presented by several groups of expressions. copyright © 2021 assa. adv syst sci appl (2021) the model of the production side of the russian economy 69 first group consists of several expressions defining growth rates of different exogenous variables (in different forms) and one more auxiliary variable: gpm (t) = pm (t) − pm (t− 1) pm (t− 1) , gwh (t) = wh (t) − wh (t− 1) wh (t− 1) , gpy (t) = py (t) − py (t− 1) py (t− 1) , gtaupr (t) = τpr (t) − τpr (t− 1) τpr (t− 1) + 1 , gtauh (t) = τh (t) − τh (t− 1) 1 + τh (t− 1) , grmax (t) = rmax (t) −rmax (t− 1) rmax (t− 1) , gttauy (t) = −τy (t) − τy (t− 1) −τy (t− 1) + 1 , b (t) = −oc o (t) + y (t) (1 − τy (t)) py (t) − pr (t) (1 + τpr (t)) . second group consists of expressions defining gdp and its components included in the model – different kinds of investments, change of stocks and profit: y (t)(−aβ−bα+1) = y0 ( bα (1 − τy (t)) py (t) pj (t) j0 )bα( m (t− 1) m0 )α (1−b) × × ( h (t− 1) h0 )β ( aβ (1 − τy (t)) py (t) wr (t) (1 + τr (t))r0 )aβ ( rmax (t) rmax0 )(1−a)β , (1 + τr (t))wr (t)r (t) = aβ (1 − τy (t)) py (t)y (t) , pj (t) ju (t) = bα (1 − τy (t)) py (t)y (t) , jh (t) = b1 (grmax (t) + gtauh (t) + gwh (t) − δah (t) − ρ (t))rmax(t)h(t− 1)− − a1 ((−νh − 1) ρ (t) + grmax (t) + gtauh (t) + gwh (t) − δah (t))rmax (t)h (t− 1)− a1 β (τy (t) − 1) (gtauh (t) + 1) py (t)y (t) (1 + τh (t))wh (t) + cc1b (t) wh (t) (1 + τh (t)) , jm (t) = b2 (δam (t) − gpm (t) + ρ (t))m (t− 1)− a2 ((νm + 1) ρ (t) + δam (t) + τtm (t) − gpm (t))m (t− 1)− α (b− 1) (τy (t) − 1) py (t)y (t) pm (t) + cc2b (t) pm (t) , z (t) = z (t− 1) + (b5 − a5 (−gpy (t) − gttauy (t) + ρ (t)))z (t− 1) + cc5b (t) py (t) (1 − τy (t)) , pr (t) = ( −∆ − gtaupr (t) + ρ (t) η + ( −η−1 + 1 ) gpy (t) + 1 ) pr (t− 1) . hereinafter the expressions without time dependence such as y 0, a, α etc. are the parameters which will be estimated in the next section. next group are the variables defining stocks of physical and human capital: m (t) = jm (t) − δam (t)m (t− 1) +m (t− 1) , h (t) = jh (t) rmax (t) − δah (t)h (t− 1) +h (t− 1) . copyright © 2021 assa. adv syst sci appl (2021) 70 n. pilnik, s. radionov the last group are the variables defining financial variables, both stocks and flows. vk (t) = b3 (βvl (t) − gwvl (t) + ρ (t)) vl (t− 1) − a3 ((νvl − 1) ρ (t) + rvl (t) + gwvl (t)) vl (t− 1) − cc3b (t) wvl (t) , k (t) = (b4 (βl (t) + ρ (t)) − a4 ((νl − 1) ρ (t) + rl (t)))l (t− 1) + + (−1 + cc1 + cc2 + cc3 + cc5)b (t) , n (t) = νll (t− 1) + νvlwvl (t) vl (t− 1) + νmpm (t)m (t− 1) + + νhwh (t) (1 + τh (t))rmax (t)h (t− 1) , l (t) = k (t) − βl (t)l (t− 1) + l (t− 1) , vl (t) = vk (t) − βvl (t) vl (t− 1) + vl (t− 1) . 3. the model: identification and results the model is identified on the russian macroeconomic data from 2005q1 to 2021q1. the data used in the model can be divided into four groups. first, it is gdp series in the current and fixed prices and its elements by use (gross fixed capital formation and changes of stocks) and incomes (wages and profits). this data is published by rosstat, but it is currently presented by several partially overlapping series. we compose unified time series of gdp and its components using the assumption of preserving growth rates. gdp and its components have significant seasonalality, so the seasonality elimination procedure described in [21] is used. this procedure is chosen because it performs rather well for the short time series. second, several indicators from the balance of the russian banking system published by the bank of russia are used. these indicators include volumes of loans to producers in rubles and in foreign currency, its interest rates and durations. moreover, we use current accounts of non-financial organizations as the indicator of liquidity available to producer. all these indicators do not have a significant seasonal component, so no additional transformation is required. third, the tax data published by the federal treasury is used. the taxes related to producer include value added tax, property tax, profit tax and social payments included in the consolidated budget of russia and budgets of non-budgetary funds. the treatment of this data is the most complicated because it has not only seasonal component but also in-year irregularity – the taxes not collected in the current quarter are paid in the next one. the algorithm similar to the one described in [21] is used for the effective tax rates calculated as the ratio of the collected tax to its base. finally, forth group consists of all the other indicators such as dollar to ruble exchange rate, volume of fixed assets and amortization rate. the last two indicators are published on a yearly basis, so they were decomposed into a quarterly data using the assumption of the fixed growth rate during the year. the parameters of the model are estimated using the following approach. we consider growth rates of the following eight variables: y ,r,m , l, vl,n , pr,w to the corresponding quarter of the previous year. for example, in case of gdp we have the following growth rates from the model and from data: gry mod(t) = y (t) − y stat(t− 4) y stat(t− 4) , gry stat(t) = y stat(t) − y stat(t− 4) y stat(t− 4) , (3.18) where y stat is the statistical value of the real gdp, y is the value calculated via the model. the function of errors is the sum of squares of gry mod(t) − gry stat(t) for every t and copyright © 2021 assa. adv syst sci appl (2021) the model of the production side of the russian economy 71 similar expressions for other variables. it should be noted that flow variables such as k and vk are not considered because of their exceptionally high volatility. this leads to the problem of unconstrained optimization, because every parameter of the model can be an arbitrary real number, which is solved using lsqnonlin command from matlab optimization package. we use the monte carlo method – initial points are generated in a rather wide region based on our a priori assumptions on the values of the model parameters and, after large number of iterations, we find the set of parameters which provides a good fit of the data and is also economically meaningful. mean absolute percent errors of the model variables are presented in the table 3.1. table 3.1. mean absolute percent errors of the model variables growth rates y r m l vl n pr w 2.46 0.82 1.62 2.96 6.43 6.86 3.03 2.85 moreover, the accuracy of data fit is shown on the plots below. (a) gdp growth rate (b) number of employed growth rate (c) fixed assets growth rate (d) ruble loans growth rate copyright © 2021 assa. adv syst sci appl (2021) 72 n. pilnik, s. radionov (e) foreign currency loans growth rate (f) current accounts growth rate (g) profit growth rate (h) wages growth rate as we can see in the plots above and the table 3.1, the model performs rather well for all model variables. the percent error for vl is relatively high because the values of this variable are very small, so small errors in absolute terms may lead to high relative errors. the percent error for n is relatively high because of the volatility of this variable. 4. conclusion the model of the macroeconomic agent producer is presented. to our knowledge, this is the first model which describes not only the technologic and demographic, but also financial constraints of firms and other agents who produce the added value. application of the relaxation of complementary slackness conditions method allows to transform the model to a nonlinear dynamic system. the model is estimated on the russian macroeconomic data, and the set of parameters was found which allows to reproduce the dynamics of russian macroeconomic indicators with rather high accuracy. acknowledgements copyright © 2021 assa. adv syst sci appl (2021) the model of the production side of the russian economy 73 this work is supported by the russian science foundation under grant 21-18-00482. references [1] d. acemoglu, s. johnson, j. robinson, and y. thaicharoen. institutional causes, macroeconomic symptoms: volatility, crises and growth. journal of monetary economics, 50(1):49–123, 2003. 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[23] a. swamy and b. fikkert. estimating the contributions of capital and labor to gdp: an instrumental variable approach. economic development and cultural change, 50:693– 708, 02 2002. copyright © 2021 assa. adv syst sci appl (2021) introduction and literature review the model: statement the model: identification and results conclusion adv syst sci appl 2019; 04; 58-65 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/763 resource allocation with a probabilistic amount of resource vladimir burkov*1, irina burkova1,2, larisa rossikhina2,3 1) trapeznikov institute of control sciences, russian academy of sciences, moscow, russia e-mail: vlab17@bk.ru 2) academy of management, the ministry of internal affairs of the russian federation, moscow, russia 3) voronezh institute of the federal penitentiary service of the russian federation, voronezh, russia received july 7, 2019; revised december 14, 2019; published december 31, 2019 abstract: the allocation problem of a limited resource under the probabilistic uncertainty over its amount is considered. the principal has a certain amount of resources to be allocated by him/her among consumers (agents). each agent submits the request for the resource to the principal. the principal allocates the resource in accordance with a specified resource allocation mechanism. in the theory of active systems, the priority-based resource allocation mechanisms were proposed and investigated. with these mechanisms, the resource is allocated proportionally to the values of the agent’s priority functions. three types of the priority-based mechanisms were identified, namely, the mechanism of absolute priorities, the mechanism of straight priorities and the mechanism of reverse priorities. previously, the priority-based mechanisms were considered under the assumption that the amount of available resource to be allocated by the principal is known. however, in many real resource allocation problems arising in practice this amount is often unknown. in this paper, the priority-based mechanisms are studied for the case in which the agents know the principal’s resource allocation function. the mechanisms of resource allocation based on the principle of reverse priorities are studied. keywords: resource allocation; mechanisms of reverse priorities; probabilistic uncertainty; nash equilibrium. 1. introduction the allocation problems of limited resources are widespread in practice [1, 2, 3]. the classical resource allocation scheme is as follows [4]. the principal has a certain amount of resources to be allocated by him/her among consumers (agents). each agent submits the request for the resource to the principal. the principal allocates the resource in accordance with a specified resource allocation mechanism [5, 6, 7]. in the theory of active systems, the priority-based resource allocation mechanisms were proposed and investigated [8, 9]. with these mechanisms, the resource is allocated proportionally to the values of the agent’s priority functions. three types of the priority-based mechanisms were identified, namely, the mechanism of absolute priorities, the mechanism of straight priorities and the mechanism of reverse priorities. in absolute priority mechanisms, resource allocation is proportional to the priorities set by the principal. these mechanisms are non-manipulated. it means that submit a request that * corresponding author: vlab17@bk.ru resource allocation with a probabilistic amount of resource 59 copyright ©2019 assa adv. in systems science and appl. (2019) reflects the true needs of the resource is beneficial for agents. their disadvantage is the fact that agents do not actually affect resource allocation. in the mechanisms of direct priority resources are allocated proportionally to the priorities that grow with the request on the principle of "more you ask – more you get". this principle leads to a trend of increasing of the request. it is significant disadvantage of the principle of direct priorities. however, it is still widely used in practice. in the mechanisms of reverse priority resource allocation is also proportional to functions of priority. however, these functions are decreasing functions of the request value on the principle of "more you ask – less you get". the principle of reverse priorities encourages resource savings. this is its significant advantage. it is experimentally tested in the allocation of water resources [2]. of the three described mechanisms of priority, the reverse priority principle is the most preferred. previously, the priority-based mechanisms were considered under the assumption that the amount of available resource to be allocated by the principal is known. however, in many real resource allocation problems arising in practice this amount is often unknown [10, 11]. in this paper, the mechanism of reverse priorities is investigated for the case in when agents know the principal’s resource allocation function. 2. problem statement consider an active system consisting of the principal and agents. the principal has a resource to be allocated among the agents in accordance with their requests. while submitting the requests, the agents have information about the principal’s allocation function f(r) only, where r is the amount of available resource to be allocated. the goal function of each agent i is an increasing function of the received resource xi. denote by si the request for the resource submitted by agent i. the principal allocates the resource using the mechanism of reverse priorities. the problem is to determine the nash equilibrium situation for three cases. in the first case the amount of resource available to the principal is known to the agents. in the second case, the agents know the distribution function f(r) of the resource available to the principal. in the third case, the distribution function of the resource available to the principal is discrete (more precisely, the resource takes two values with corresponding probabilities). 3. mechanism of reverse priorities: deterministic case consider the deterministic case in which the amount of resource available to the principal is known to the agents. define the priority function of agent i by   is ia isi  , ni ,1 , where ai is a parameter that restricts the agent’s priority. the resource allocation mechanism has the form   min , a rix s si i s yi           , (3.1) where aj y sj j   . assume the goal functions of the agents are increasing in xi. the deterministic statement of the problem yields the nash equilibrium 60 v. burkov, i. burkova, l. rossikhina copyright ©2019 assa adv. in systems science and appl. (2019) 2 1 y aj r j         , a ri si a j j    . in this case, isix  for all ni ,1 . now, consider another modification of the priority function given by   isiaisi  , ni ,1 . first, analyze the deterministic case:           r y isia issix ,min , where    j jsjay . in the nash equilibrium, r y isia is   , which gives ry ira is   and ry yia isia   . from the condition      j j ja yr y jsjay it follows that ray  * , r j ja ia is   * . the nash equilibrium exists only if the condition a > r is satisfied. 4. mechanism of reverse priorities: probabilistic case consider the probabilistic case in which the agents choose their requests for the resource using the information about the distribution f(r) of the available amount r only. consider the priority function   is ia isi  , ni ,1 , and the resource allocation mechanism   min , a rix s si i s yi           . denote by f(r) a distribution function of the available amount of the resource and let it be continuously differentiable with respect to r. the expected amount of resource requested by agent i is resource allocation with a probabilistic amount of resource 61 copyright ©2019 assa adv. in systems science and appl. (2019)      1 0 ri a rim s df r s f ri i s yi        , where 2 s yiri ai   , ni ,1 . find the maximum of this value over si under the hypothesis of weak contagion (the influence of si on y is small). in other words, while choosing their requests for the resource, the agents neglect the influence of si on y, considering y to be just a parameter. note that        ir ir drrfirfirrrdf 0 0 . calculate     2 0 riadm i r f r f r dri i ds s yi i                 ' ' 1 a dr dri i ir f r f r s f ri i i i i s y ds dsi i i      . a series of trivial transformations finally give     1 1 2 0 ridm f r f r dri ds ri i     . the resulting expression is independent of ai. assume the function м(s) is convex and therefore has a maximum point. the equilibrium value ri is the same for all agents, i.e., ri=r* for all i. the maximum point r* can be determined from the first-order optimality condition     * 1* 2 1 0 r f r f r dr ri   , where y ria is * *   . further, calculate  j ja r y j js ja y ** * , * * r j ja y   . finally, the nash equilibrium is ** r j ia ia is    . now, consider the priority function   isiaisi  , ni ,1 and the resource allocation mechanism           r y isia issix ,min . by analogy with the previous case, 62 v. burkov, i. burkova, l. rossikhina copyright ©2019 assa adv. in systems science and appl. (2019)             ir irfisrrdf y isia ism 0 1 , where s yiri a si i   ,  2isia iya ids dr   . calculate               irf ir drrfirfir yids dm 1 0 1           ir drrf y irf y ir 0 1 11 ,             irf y irf y ir irf ydr md 1' 1 1 2   0 ' 1        irf y ir . hence, the function  ism is concave, and the maximum point can be determined from the first-order optimality condition (the same for all agents)        r ydxxfrfyr 0 . 5. mechanism of reverse priorities: discrete case suppose the available resource r is q1 with a probability p1 and q2 with the probability 2 11p p  . adopt the mechanism of reverse priorities with the priority function   is ia isi  , ni ,1 . establish how the expected amount of the agent’s resource depends his/her request (for simplicity, the agent’s number i is omitted). calculate y aq d 1  and y aq d 2  . then three situations are possible as follows: 1) ds  . in this case , with the mechanism of reverse priorities the minimum will be achieved on the request, if 1qr  and if 2qr  . therefore,   ssm  . 2) dsd  . in this case, the minimum point ys aq x   1 will be achieved if 1qr  , and the minimum point sx  will be achieved if 2qr  . the expected amount of the resource will be   sp ys aq psm 2 1 1    . 3) ds  . in this case, the minimum point ys aq x   1 will be achieved if 1qr  , and the minimum point ys aq x   2 will be achieved if 2qr  . the expected amount of the resource will be     r sy a sy a qpqpsm     2211 . resource allocation with a probabilistic amount of resource 63 copyright ©2019 assa adv. in systems science and appl. (2019) the graph of the function  sm can be seen in fig 4.1. fig. 4.1 graph of function  sm . on the interval  dd , ,  sm is a convex function of s. therefore, it achieves a maximum at the points d or d. note that, if 1ds  , then dm  ; with 1qq  or 2qq  , the agent will receive the same amount dx  . if ds  , then dpp yd aq m 21 1    . the maximum of m is given by                 221 2 1 ;1max2 1 1;max pqp q q q y a dp yd q pd if the maximum is achieved on 1q , the agent will choose the strategy d; if on 22 2 1 1 qp q q p  , then the strategy d. define * 1p from the equation   2121 2 1 1 pqp q q q  . after simple calculations, 21 2* 1 qq q p   . so, if * 1 pp  , then each agent will choose the strategy d; if * 1 pp  , then the strategy d. in both cases, in the nash equilibrium the resources will be allocated as follows: q b ia ix  , where  i iab . next, consider the priority function   isiaisi  , ni ,1 . by analogy with the previous analysis,           y qsa sx ;min , where    j jsjay . using the equation   y qsa s   , find 64 v. burkov, i. burkova, l. rossikhina copyright ©2019 assa adv. in systems science and appl. (2019) 1 1 qy aq d   and 2 2 qy aq d   . if ds  , then   dxm  . if dsd  , then              2 11 2 1 1 p y qp s y a rsp y qsa pxm . if ds  , then       2211 qpqp y sa xm    . obviously, the maximum of  xm is achieved either at the point d or at the point d. hence,              dp y qda pdxm 2 1 1;             2 2 2 1 2 1 ; 1 1 max p yq q p yq q yq q a . the boundary value * 1p can be defined from the equation 2 2 2 1 2 1 1 1 p yq q p yq q yq q      . trivial calculations yield yq y p   1 * 1 . in contrast to the previous case, the value * 1p depends on y. if jdjs  , then    j bjdjay , 1qby  , b q p 1 1 * 1  . if idis  , ni ,1 , then b q p 2 1 * 1  . again, three cases are possible as follows. 1) b q p 1 1 . in this case, all agents will choose the strategy id , ni ,1 . 2) b q p 2 1 . in this case, all agent will choose the strategy id , ni ,1 . 3) b q p b q 1 1 2 1  . this situation has uncertainty. if all agents choose the strategy id , then b q p 1 1 * 1  ; therefore, * 1pp  and they will benefit more from the strategy id . if they choose the strategy id , then b q p 2 1 * 1  ; therefore, * 1pp  and the agent will benefit more from the strategy id . the situation is difficult to predict. resource allocation with a probabilistic amount of resource 65 copyright ©2019 assa adv. in systems science and appl. (2019) 6. conclusions in this paper, a study of the mechanism of inverse priorities for resource allocation in the deterministic and probabilistic case has been presented. for different priority functions the nash equilibrium has been obtained under the assumption that the goal functions of the agents are monotonically increasing in their amounts of resource. references 1. burkov v. n., gorgidze i. i., novikov d.a & yusupov b. s. (1997). models and mechanisms of distribution of costs and income in the market economy. moscow: institute of control sciences 2. burkov v. n., danev b. d. & enaev, a. k. (1989). large systems: modeling of organizational mechanisms. moscow: science. 3. burkov v. n., korgin n. a. & novikov d. a. (2009). introduction to the theory of control of organizational systems. moscow: librocom. 4. novikov, d. (2013), theory of control in organizations, new york: nova science publishers. 5. control mechanism (2013). organization management: planning, organization, stimulation, control, ed. by d. a. novikov. moscow: lenand. 6. mas-colell, a., whinston, m. d., & green, j. r. (1995). microeconomic theory (vol. 1). new york: oxford university press. 7. maschler, m., solan, e., & zamir, s. (2013). game theory (translated from the hebrew by ziv hellman and edited by mike borns). cambridge university press. 8. salanié, b. (2005). the economics of contracts: a primer. mit press. 9. goubko, m., burkov, v., kondrat’ev, v., korgin, n., & novikov, d. (2013). mechanism design and management: mathematical methods for smart organizations. new york: nova science publishers 10. burkov v. n., & kondratyev v. v. (1981) mechanisms of functioning of organizational systems. moscow: science. 11. ponomarev, v. a. & rossikhina , l.v. (2018) applied modeling of the mechanism of inverse priorities in the tasks of resource allocation using active experiment. bulletin of the voronezh institute of the federal penitentiary service of russia, 2, 94–99. 12. ponomarev v. a. (2018) a game-theoretic models of resource allocation. bulletin of the voronezh institute of the federal penitentiary service of russia, 4, 98–105. adv syst sci appl 2018; 1; 1-11 published online at http://ijassa.ipu.ru. copyright ©2018 assa. adv. in systems science and appl. (2018) algorithms for prosodic discourse feature interpretation in case of its processing using low-speed codecs. maxim a. bessonov 1 , natalia a. bessonova 2 , mais p. farkhadov 3 1) peoples' friendship university of russia, moscow, russia e-mail: bessonovma@gmail.com 2) peoples' friendship university of russia, moscow, russia e-mail: nbesson@yandex.ru 3) v.a. trapeznikov institute for control sciences of russian academy of science, moscow, russia e-mail: mais.farhadov@gmail.com abstract: in this article we propose two algorithms for discourse prosodic feature interpretation. the first algorithm based on wide phonetic categories and second algorithm based on audio signal melodic cross-correlation functions and short-timed energy series – as well as methodical recommendations for their use are proposed as a part of the problem of audio signal language identification based on a prosodic approach. an experimental evaluation of both algorithms is proposed. neural networks are used as a decision rule. wide phonetic categories were pause, pitch, noise. we have expanded wide phonetic categories to pause, pitch, noise, five levels of pitch, sites of decreasing energy, main maximum, adverse maximum. the total number of categories was 14. these algorithms can be applied for language identification or speaker identification. at the same time there is no requirement to restore the speech signal after processing it by low-speed codec. certainly, frames of the speech codec must contain such parameters as pitch, tone-noise parameter, energy. the base of speech signals consists of 10 languages 10 speakers per language. total time of the speech per speaker is 100 minutes. this time takes into account statistical regularities of languages. tests for evaluation of the algorithms were carried out with a multilayer perceptron. keywords: language identification, neural networks, discourse prosodic feature, wide phonetic categories. 1. introduction as numerous on-line human-machine interfaces are being developed, the problem of language identification stays unsolved. moreover, those systems are often required to support numerous languages. there are four methods for language identification: acoustic, phonotactic, lexical and prosodic. in one or more of their aspects, the first three methods are based on the discourse signal parameters: acoustic, mel-frequency cepstral coefficients, mixed mel-frequency cepstral coefficients and others. the prosodic approach uses parameters such as discourse melody, rhythm, tone and others [5–9]. prosodic parameters are difficult to describe and to interpret. therefore, in the present article two algorithms are proposed for discourse prosodic feature description in order to use them in automatic audio signal language assessment systems. the first algorithm is based on wide phonetic categories [4]. the second is based on discourse melody cross-correllation function and short-timed energy series. the main difference between the proposed algorithms and the ones described in the literature lays in their utilization for language assessment of audio signals that have been processed using low-speed codecs. this is based on the fact that low-speed codecs transmit mailto:nbesson@yandex.ru 2 m. bessonov, n. bessonova, m. farkhadov copyright ©2018 assa. adv. in systems science and appl. (2018) in communication channels such parameters as basal tone frequency, tone-noise signal and amplification of quasi-periodic fragments. 2. algorithms for prosodic features interpretation 2.1. algorithm based on wide phonetic categories let l={l1,l2,…,ln} be an ensemble of languages on which the language assessment procedure is performed, with n being the overall language number. let every language be represented by an ensemble li={l1,l2,…,lmi} of audio recordings from different speakers of this specific language, with mi being the overall number of audio recordings for a given language li. an audio recording is divided in quasi-stationary segments si(m) , each one of which having a duration of k samples, with i being the segment number of a given discourse signal: i = 1, 2, …, p, and p being the overall number of segments for a given discourse audio signal recording m=1,…,k-1. features are computed on each segment i, depending on its nature: vocal, non vocal or break    1 2i ia t s m ,i , , ..., p  (2.1) t being the operation allowing to define the type of segment. the evaluation of the segment's short-timed energy is denoted as    1 2 ik ie e s m ,i , , ..., p  (2.2) with e being the computation of the segment's short-timed energy. if the algorithm is used without reconstructing the source vocal signal waveform, then the parameters аi and ike are computed from the vocal transmission. thus series  paaaa ...,,, 21 and   pkkkk eeee ...,,, 21  are formed. if the segment is classified as a break, then ai=0. if the segment is classified as nonvocal, then ai=1. for each vocal segment the basal tone frequency (btf) is computed as   0 1 2i if f s m ,i , , ..., p  (2.3) with f being the operation of basal tone computation. afterwards the series  pffff 0...,,0,00 21 are formed. if the algorithm is used without reconstructing the source vocal signal waveform, the parameter f0i is computed from the vocal transmission. the basal tone frequency's range of variations is then divided into 5 intervals. each vocal segment is attributed a number, depending on which btf interval its frequency corresponds to  00 1 2 iuf uf ,i , , ...,f p  (2.4) f0ui being the btf level, uf the operation of btf change computation and segment encoding with numerical values  pufufufuf 0...,,0,00 21 . this allows the formation of btf values series for audio signal segments. afterwards segments during which the discourse short-timed energy increases or decreases are computed as   1 2 iu ke ue e ,i , , ..., p  (2.5) the encoding eui=(+/–)1 depends on whether the energy variation is increasing or decreasing correspondingly, ue being the operation of audio signal short-timed energy computation. the series  peueueueu ...,,, 21 are formed. if the short-timed energy decreases for a given segment, its btf value is multiplied by (-1). principal and lateral btf maxima on a segment between two breaks are used in order to algorithms for prosodic discourse feature interpretation 3 copyright ©2018 assa. adv. in systems science and appl. (2018) assess the principal and lateral accents. if the btf and short-timed energy maxima correspond in time and are maximum for a given fragment, then this segment is considered a principal maximum. if the maxima do not correspond in time, then the fragment is considered a lateral maximum ),0( euufmax i  with θ being the operation of principal and lateral maxima computation for the btf and short-timed energy series. this allows the constitution of the series  1 2 pmax max ,max ,...,max (2.6) therefore the final series  pxxxx ,...,, 21 of wide phonetic categories for a given audio record is constituted by elements xi, where 0, if break, 1, if non-vocal, 2,if interval1, 2, if interval1, 1, 3, if interval 2, 3, if interval 2, 1, 4, if interval 3, 4, if interval 3, 1, 5, if interval 4, 5, i 0 0 0 0 0 0 0 0f i i i i i i i i i i i u u u u u u u u u i u i u i f f e f f e f f a x e f f a                      interval 4, 1, 6, if interval 5, 6, if interval 5, 1, 7 0 0 lateral maximum principal maximum , if , 8, if . i i i i u u u u i i e f f e max max                                (2.7) figures 2.1 and 2.2 show the algorithm diagram that implements the discourse signal encoding. the autocorrelation function )(xr   is then computed on the series of wide phonetic categories x , with ψ being the operation of autocorrelation function computation. if the algorithm is used without reconstructing the source vocal signal waveform, the btf values are computed from the vocal transmission. if the algorithm is used with the source vocal signal waveform being reconstructed, then an algorithm for btf evaluation is required. there are numerous algorithms for btf evaluation [2]. this article presents the comparison of already implemented algorithms: the algorithm sift, based on the autocorrelation function; the algorithm amdf, based on short-time average difference function; and the algorithm for btf evaluation of the melp language encoding algorithm. table 2.1 shows the percentage values for correct btf computation p(от), erroneous assumption that a vocal fragment is non-vocal р(нв/в) and erroneous assumption that a non-vocal fragment is vocal р(в/нв). 4 m. bessonov, n. bessonova, m. farkhadov copyright ©2018 assa. adv. in systems science and appl. (2018) table 2.1. evaluation of btf correct evaluation algorithm sift amdf melp р(от), % 87±1 89±1 95±1,5 р(нв/в), % 7±1 6±1 3±0,5 р(в/нв), % 0,5 0,5 0,5 the algorithm melp was used for btf computation as its experimental evaluation proved it to be the most effective. fig. 2.1. algorithm diagram for vocal signal segments encoding algorithms for prosodic discourse feature interpretation 5 copyright ©2018 assa. adv. in systems science and appl. (2018) fig. 2.2. algorithm diagram for vocal signal segments encoding (continued) 2.2. algorithm based on basal tone melody cross-correlation functions and short-timed energy series the prosodic classification may be realized using basal tone melody cross-correlation function and short-timed energy of audio recordings. each audio recording is divided in quasi-stationary segments si(m) of k samples where i is the vocal signal number: i = 1, 2, …, p, р is the overall number of segments in a vocal signal m=1,…,k-1. for each segment i features are computed in accordance to the segment's nature: vocal, non vocal or break    1 2i ia t s m ,i , , ..., p  (2.8) t being the operation of segment type computation and the computation of the short-timed energy for one segment being    1 2 ik ie e s m ,i , , ..., p  (2.9) e being the operation of short-timed energy computation. the series  paaaa ...,,, 21 and  pekekekke ...,,, 21 are formed correspondingly. if a segment is classified as a break then ai=0. if a segment is classified as non-vocal, then ai=1. for each vocal segment the btf is computed   0 1 2i if f s m ,i , , ..., p  (2.10) f being the operation of btf computation. afterwards series  1 20 0 , 0 , ..., 0pf f f f are formed. if the algorithm is used without reconstructing the source vocal signal waveform, the 6 m. bessonov, n. bessonova, m. farkhadov copyright ©2018 assa. adv. in systems science and appl. (2018) parameter аi, eki, and f0i is computed from the vocal transmission. the cross-correlation function is computed on the btf series and the short-timed energy series ),0( ekfb  (2.11) ф being the operation of btf melody cross-correlation function computation and shorttimed energy series. the vector formed by the cross-correlation function values and series of wide phonetic categories is then given at the first layer of the neural network, that is used to infer the language group to which the presented vector corresponds. the feature computation algorithm is presented in figure 2.3. fig. 2.3. diagram of the algorithm of vocal segment encoding 3. methodical recommendations for the use of algorithms for prosodic feature interpretation algorithms as a part of the audio signal language identification methodical recommendations were developed in order to apply the aforementioned algorithm. they contain a succession of phases. phase 1. discourse dataset formation for training. the training dataset must fulfill the following conditions: if n is the overall number of languages, dm i the number of male speakers for a given language i, dm i the number of female speakers for a given language i, then vi(dm i , df i ) = vj(dm j , df j ), where i, j are the numbers of languages i, j = 1, …, n, meaning that all age groups must be represented in equal proportion among male and female speakers or that the volume of voice data for all age groups must be equal. the volume of voice data must be sufficient from a statistical standpoint in order for all the pronunciation variations to be described. the overall data volumes must be equal for every languages. step 1. reception from the source of a digital signal under the form st(fd, m, p, fr) , having the following characteristics: format "wav", sampling frequency fd = 8khz, regime m = mono, datadepth p = 16 bits, t being the audio signal number. step 2. filtering of the audio signal st(fd, m, p, fr) for unwanted noise suppression. this allows to receive the filtered signal st f (fd, m, p, fr) = ρ[st(fd, m, p, fr)], where p is the filtering operation. algorithms for prosodic discourse feature interpretation 7 copyright ©2018 assa. adv. in systems science and appl. (2018) step 3. training and control dataset formation. a set of audio signals zli{s1 f li(fd, m, p, fr), s2 f li(fd, m, p, fr), …, smi f li(fd, m, p, fr)} is formed for each language li, where mi is the overall number of audio signals for a given language li. the complete set of audio signals is then denoted by z = {zl1, zl2, …, zln}. step 4. treatment of all audio signals in every language made available by the vocal transmission. z vok = vok(z), vok where denotes the processing of the voice transmission dataset , z vok = {z vok l1, z vok l2, …, z vok ln}. step 5. audio signal parameter computation using the presented algorithms. afterwards a set of parameters z vok li mod1 = mod1(z vok li), z vok li mod2 = mod2(z vok li) is created, where mod1 and mod2 are the operation of parameter computation using the presented algorithms for prosodic parameters description. phase 2. neural network training. the neural network training operations allow the fine tuning of its parameters. neural networks with different topologies are described by different mathematical models, therefore in every specific situation a neural network is described by a specific formula. neural networks are defined for language groups. their number is equal to the number of combinations of 2 elements out of n. phase 3. neural network performance evaluation step 1. reception from the source of a digital signal under the form st(fd, m, p, fr) , having the following characteristics: format "wav", sampling frequency fd = 8khz, regime m = mono, data depth p = 16 bits, t being the audio signal number. step 2. filtering of the audio signal st(fd, m, p, fr) for unwanted noise suppression. this allows to receive the filtered signal st f (fd, m, p, fr) = ρ[st(fd, m, p, fr)], where p is the filtration operation. step 3. neural network’s testing. at the neural network's input, for each language pair li and lj, audio signals in the language i and j are given. the neural network's output gives an evaluation of the audio signal’s language identity, given t: the audio signal’s number:   f t t d rl net s fˆ ,m, p, f (3.1) step 4. evaluation of the number of correctly assessed audio signal for each language pair. this forms the vector d = (d12, d21, d13, d31, …, dn(n–1), d(n–1)n), where dij is the number of correctly assessed audio signal for a given language pair lilj, i ≠ j. step 5. hierarchical language tree construction based on the agglomerative hierarchical algorithm     k i l j min i j l x ,x ω k ω ρ ω ,ω min d xx ,    (3.2) where ωi and ωj are the languages li and lj, and ρ(ωi,ωj) is the distance between li and lj. the hierarchical language tree is the base upon which language groups are formed. 4. discourse database formation an audio signal dataset was formed in order to conduct test according to the presented methodical recommendation. its content is summarized in the table 4.1. audio records were taken from internet translation resources: television and radio, which implies that the discourse were processed using different codecs. in order to exclude the influence of the dataset's constitution on the experiment results, the number of speakers in each language was chosen equal. the overall duration of the audio signals was as well equal. the dataset was split equally in training and validation subsets. samples from the validation set were not present in the training set. for experimental purposes, all audio records both in the training and validation datasets were divided in 10seconds long fragments. the speakers' age repartition was approximated: men and women aging from 20 to 50 years old. 80% of each speaker's audio signal time was used for training, 20% for validation. the separation into training and validation was performed randomly. 8 m. bessonov, n. bessonova, m. farkhadov copyright ©2018 assa. adv. in systems science and appl. (2018) table 4.1. contents of the dataset used for prosodic feature models experimental validation language number of speakers overall audio signal duration for each speaker, min speaker sex (m-male, f-female) repartition in the training/testing sets, % chinese 10 100 5m/5f 80/20 english 10 100 5m/5f 80/20 finnish 10 100 5m/5f 80/20 french 10 100 5m/5f 80/20 german 10 100 5m/5f 80/20 japanese 10 100 5m/5f 80/20 farsi 10 100 5m/5f 80/20 portuguese 10 100 5m/5f 80/20 russian 10 100 5m/5f 80/20 spanish 10 100 5m/5f 80/20 5. neural network definition and tuning pattern recognition tasks are in most cases solved using statistical methods. however in the case of vocal data in different languages, it proves difficult to build the repartition function of the considered parameters. therefore in the present article neural networks were used for vocal segments classification. as stated in literature, for classification-type problems, the number of neurons in the network's first layer is equal to the number of elements in the feature vector that is given at the entrance [3]. the number of output neurons depends on the type of problem and the output number interpretation rules [3]. the number of neuron in the intermediate layers is given by the formula [3]     2 1 1 1 y p p w y x y y xp n n n n n n n n nlog n             (3.3) where ny is the neural network's (nn) output vector's number of element, np is the number of elements in the test dataset, nх is the number of element in the input vector and nw is the overall number of neurons. the choice of the nn's class and architecture is a non-trivial problem for which exact solutions do not exist [3]. in order to choose the number of neurons, one can highlight two methods: the more neurons, the more reliable the network will be and the more neurons, the worse the network will approximate the transfer function. the neural networks were implemented in matlab, using the environment's built-in functions. the following architectures were experimentally evaluated: kohonen maps, cascadeforward nn, elman networks, multilayer perceptron, hopefield networks, probabilistic networks, networks with radial basis functions (rbf), counter-propagation network with learning vector quantization. the networks were trained with built-in matlab functions [1]: quasi-newton algorithm, levenberg-marquardt algorithm with bayes regularization, fletcher-reeves conjugate gradient method, polak-ribiére conjugate gradient method, powell-beale conjugate gradient method, gradient descent, gradient descent with variable learning rate, levenberg-marquards algorithm, scaled conjugate gradient method, gradient descent with momentum, gradient descent with momentum and variable learning rate, one step secant method, random increment method and elastic error backpropagation algorithm. for the first phase, in order to build the limited groups of 10 languages, experiments were performed with each separated language pair, accounting for 45 neural networks in total. the best results have been obtained using a multilayer perceptron. therefore this architecture has been selected for fine tuning. since the language given at the nn's entrance is a priori unknown, it was decided to use a algorithms for prosodic discourse feature interpretation 9 copyright ©2018 assa. adv. in systems science and appl. (2018) unified architecture for each language pair. 6. evaluation of the wide phonetic categories algorithm according to the formula and the starting conditions for nn testing ny = 2, np = 600, the number of neurons in the hidden layers is then 117 ≤ nw ≤ 2015 for the wide phonetic categories model. since nw is between 117 and 2015, at the moment of the nn architecture definition the number of neuron was chosen from 100 to 2000, correspondingly to the number of neurons from 1 (one layer from 100 to 2000 neurons) to 20 (20 layers of 100 neurons) in the following configurations: from 100 to 1000 neurons with a 10 neuron step in a given layer or from 1000 to 2000 with a 100 neuron step. the maximal number of neurons in one given layer was 800. in order to build the different multilayer perceptron architectures for the 45 language pairs, a vector d=(d1,2, d2,1, d1,3, d3,1, di,j, dj,i, dn,n–1, dn–1,n) of goal indicators for assessment confidence was built, with n being the overall number of languages in the automatic language assessment system. therefore the length of the vector is d=90. each element di,j, dj,i = 100. the vector dk=(d k 1,2, d k 2,1, d k 1,3, d k 3,1, d k i,j, d k j,i, d k n,n–1, d k n–1,n) of goal indicators for assessment confidence for the current architecture has as well 90 elements. the distance between d and dk is defined as       2 2 2 1 2 1 2 2 1 2 1 k k k r , , , , i , j i , jd d d d d d d             2 2 2 i i 1 1 1 1 k k k j , j , n ,n n ,n n ,n n ,n... d d d d d d         (3.3) thus, the lower the distance dr, the better the nn tuning. at the end it was found that dr lays in the interval from 59.1861 to 532.4106. the best value, dr = 72.5358, was obtained for a nn with 1400 neurons overall, organized in one layer of 800 neurons and 2 layers of 600 neurons. the results of language assessment are presented in the table 6.1. table 6.1. average confidence values for language identification chinese english finnish french german japanese persian portugu ese russian spanish chinese 94.5 95.1 96.2 95.9 97.5 96.6 95.2 94.4 97.9 english 93.8 97.4 92.8 93.8 93.6 98.1 94.5 94.0 97.8 finnish 93.8 93.7 93.2 93.4 93.9 93.9 96.1 93.7 94.3 french 94.2 93.6 93.2 93.9 93.4 94.0 94.8 93.8 94.4 german 94.5 92.6 93.7 92.5 94.6 94.0 97.5 96.3 93.9 japanese 83.6 94.1 74.0 98.3 93.3 94.0 84.9 94.4 98.0 persian 84.4 94.0 74.6 93.3 93.8 83.6 92.7 84.3 93.2 portuguese 94.2 93.6 93.5 93.9 94.2 94.5 93.5 93.9 98.4 russian 94.4 95.1 94.1 95.3 93.4 94.0 94.4 94.3 94.5 spanish 93.9 94.3 93.4 93.2 94.2 93.8 94.1 94.5 93.2 the languages were used to form groups using the agglomerative algorithm. language pairs were used in quality of patterns to be recognized. the average confidence value for fixed first and second order error rates was used as a measurement of the distance between the two languages of one pair. the distance between classes is defined by the distance to the nearest neighbor:     k i l j min i j l x ω ,x ω kρ ω ,ω min d xx ,    (3.5) 10 m. bessonov, n. bessonova, m. farkhadov copyright ©2018 assa. adv. in systems science and appl. (2018) where ωi,ωj are the languages li and lj, and ρ(ωi,ωj) is the distance between li and lj. this allows the creation of a graph of hierarchical classification. this graph can then be used to assess groups of languages close to each other. 7. evaluation of the basal tone melody cross-correlation function and short-timed energy series algorithm according to the formula and the nn test starting conditions, ny = 2, np = 600, nx = 797, the number of neurons in the hidden layer is 117 ≤ nw ≤ 2806 for the cross-correlation function and btf and short-timed energy series model. since nw is comprised between 117 and 2086, at the time of the nn architecture definition the number of neurons in a layer was varied from 100 to 3000, correspondingly layers having from 1 (1 layer from 100 to 3000 neurons) to 20 (30 layers of 100 neurons each). the following configurations were used: from 100 to 1000 neurons with a 10 neuron step by layer, from 1000 to 3000 with a 100 neuron step. the maximum number of neurons was 800, dr = 89.1449. the results of language identification are presented in the table 7.1. table 7.1. average confidence values for language chinese english finnish french german japanese persian portuguese russian spanish chinese 97.7 94.7 92.8 97.8 97.9 93.8 91.7 93.1 92.1 english 91.2 91.4 92.3 92.9 94.8 92.7 97.7 90.3 92.0 finnish 90.9 91.5 95.8 94.7 94.6 95.4 90.9 93.6 95.9 french 92.1 92.9 92.4 93.9 96.7 97.5 92.1 91.8 91.8 german 92.5 90.2 91.4 90.4 91.8 92.2 92.4 93.0 95.4 japanese 80.6 91.8 90.1 82.3 71.9 90.7 90.5 94.7 97.2 persian 71.1 91.5 82.3 91.6 82.6 78.2 97.5 92.5 91.5 portuguese 90.7 91.0 92.0 92.0 93.2 93.4 92.1 94.5 92.5 russian 91.0 91.7 90.6 92.6 92.3 92.4 91.7 91.6 96.2 spanish 90.5 92.9 90.9 92.8 91.2 93.1 91.4 92.1 93.6 8. conclusion the algorithms presented in the article aim at a complex description of discourse prosodic feature for their usage in special data processing tasks, in particular audio signal language assessment. as seen in the presented tables, prosodic feature description using wide phonetic categories allows for high-confidence language identification. however this performance insignificantly surpasses cross-correlation function. distance indicators for current results of language assessment with respect to the goal indicator dr scored at dr = 72.5358 for the autocorrelation model from wide phonetic categories and dr = 89.1449 for the signal's crosscorrelation function model from basal tone value and short-timed energy series. the presented algorithms demark themselves from others in that they are used for audio signal language identification, after vocal transmission, but without reconstructing the source vocal signal waveform. references 1. diakonov v. & kruglov v. 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(2010) razrabotka metodiki ispolzovania shirokih foneticheskih kategoriy v zadachah verifikazii diktora [development of method of using broad phonetic categories in speaker identification task]. moscow, russia: miit [in russian]. 5. ambikairajah e., li h., wang l. & yin b. (2011) language identification: a tutorial. ieee circuits and systems magazine. 11(2), 82–108. 6. bhattacharjee u. & sarmah k. (2013) language identification system using mfcc and prosodic features. int. conference on intelligent systems and signal processing (issp), gujarat, 194–197. 7. lee r., leung c.-c. & ma b. (2013) spoken language recognition with prosodic features. ieee trans. on audio, speech, and language processing. 21(9), 1841–1853. 8. martinez d., jeida e. & ortega a. (2013) prosodic features and formant modeling for an ivector-based language recognition system. ieee int. conference on acoustics, speech and signal processing (icassp), vancouver, canada, 6847–6851. 9. martínez d., burget l., ferrer l. & scheffer n. (2012) ivector-based prosodic system for language identification. ieee int. conference on acoustics, speech and signal processing (icassp), kyoto, japan, 4861–4864. adv syst sci appl 2019; 02; 101-119 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/673 modeling the management of the economies of developing countries† s. bayzakov*1, jeffrey yi-lin forrest3, n.a. baizakov2 1) institute of mathematics and mathematical modelling, almaty,kazakhstan; e-mail: baizakov37@mail.ru; 2) institute of mathematics and mathematical modelling, almaty,kazakhstan; e-mail:nauryz.b@mail.ru; 3) school of business, slippery rock university, slippery rock, pa 16057, usa; e-mail: jeffrey.forrest@sru.edu received november 28, 2018; revised march 4, 2019; published july 10, 2019 abstract: in the present, the issues of algorithmization of the tasks of sustainable development of the economies of developing countries of the world are considered. the final result of the work is represented by a system of models for analyzing the development of the country's economy and the development of the well-being of its population, which is compared with fao models to ensure the productive security of the countries of the world proposed by the un world bank, as well as the oecd management criteria system. the basis for constructing the proposed system of models was actually the achieved level of human capital development. as a result, a new base has been created for assessing the contribution of the country's scientific and technological potential to the development of the productive power of labor, which is determined by the difference between labor productivity by income and labor productivity by the total labor input. and the total financial and economic capacity of a country is measured by the product of the number of man-hours spent in a calendar year by the achieved level of growth of the productive power of labor in the country's economy. keywords: potential, productivity, strength, labor, income, fao , model, technology. 1. a new eco-economic system of commodity-money management strategic planning for the development of market forces for developing countries relies on the old interpretation of the direct costing system. [1] thus, the process of determining net present value (npv) is based on the direct costing system, which was designated for accounting direct production expenditure by american scholar d. harris. according to this system, direct production expenditure is divided into two parts: that which is fixed and that which is variable. [2] in kazakhstan, with the help of the system developed here fixed expenditure includes expenses related to administrative and management activities aimed at the sale of output products. here, market research, commercial and general-administrative expenses are † this study was financially supported by grant no. ap05131044 from the scientific and technical programs and projects of the ministry of education and science of the republic of kazakhstan. * corresponding author: baizakov37@mail.ru mailto:baizakov37@mail.ru mailto:nauryz.b@mail.ru mailto:jeffrey.forrest@sru.edu modeling the management of the economies of developing countries 102 copyright ©2019 assa. adv. in systems science and appl. (2019) included. the variable expenditure includes expenses related to changes in the output volume of production: materials, energy and fuel, salaries for employees and engineers. as a result, the sum of direct and fixed costs represents the prime operating costs of the output. the amount of prime operating costs and depreciation of fixed assets of the enterprise, including taxes, determine the total key cost of production. as a result, the cash flow in terms of enterprise development years is calculated according to the formula: 𝑁𝑃𝑉 = 𝑅𝐸𝑉 − (𝑂𝑃𝐸𝑅 + ∑ 𝑇𝐴𝑋 + ∑ 𝐶𝑅), (1) where 𝑅𝐸𝑉 stands for the revenue from the sale of commodity ore in the reference year; 𝑂𝑃𝐸𝑅 the operating costs (filled by the prime cost) in the reference year; ∑ 𝑇𝐴𝑋 the tax paid in the reference year; and ∑ 𝐶𝑅 the credit repayment in the reference year (including the interest on credit). hence, the net present value 𝑁𝑃𝑉о(𝑖) is calculated according to the next formula: 𝑁𝑃𝑉о(𝑖) = 𝑁𝑃𝑉(𝑖) (1+ii1)∗(1+ii2)∗(1+ii3)∗…∗(1+iiн+1) , (2) where 𝑁𝑃𝑉(𝑖) stands for the cash flow of the reporting year i in million dollars, ii𝑖 the inflation index of year 𝑖 in %. finally, the present value, considering the discount coefficient, is calculated according to the formula: 𝑁𝑃𝑉(𝑖) = 𝑁𝑃𝑉𝑂1 ∙ (1 + 𝑟)1 + 𝑁𝑃𝑉𝑂2 ∙ (1 + 𝑟)2 + ⋯ + 𝑁𝑃𝑉𝑂𝑛 ∙ (1 + 𝑟)𝑛, (3) where r is the discount rate (money devaluation rate), indices 1, 2, 3, ..., n the discount intervals: years, quarters, months, days. in real-life applications, accounting and analysis based on the direct costing system revealed great opportunities to set tight connections between production volumes, gross and net profits, prime costs and gross revenues. they allowed practical use in the works of mathematical economists, production function methods, econometrics and regression analysis. however, any initiative becomes “a dead dogma”, if it does not continuously develop and does not find new arguments for the development of its viability. this point of view, focused on the continuous development of human capital, objectively determines the strict correspondence of the methods of market relations management with the level of development of labor and capital productive forces in the market-oriented economy of any country in the world. hence, in determining the break-even point of development according to the direct costing system, variable costs include salaries for workers and engineers, and fixed costs include salary costs for personnel employed in marketing, general and administrative divisions of an enterprise. it means that this system, which was developed in the 1940s under the conditions of tough competition between the economies of the soviet union and other developed countries of the world, did not take into account the development of human capital. such theoretical deficit hampers opportunities for the development of the working man himself, his creative, scientific and technological potential: the rich grow richer, the poor become poorer. in general, the direct costing system specifically limits the creative initiatives of employees' participation in the production of goods and services. it determines employment, 103 s. bayzakov, jeffrey yi-lin forrest, n.a. baizakov copyright ©2019 assa. adv. in systems science and appl. (2019) income and pricing indicators by including in the fixed and variable costs of production, the remunerations of people employed by an enterprise in the economy. however, the main drawback of this system is that it contradicts the macroeconomic approach of the quantitative theory of money, which describes the development of the market forces of the modern world economy and the globalization. that is, there is an objective need for us to develop management methods appropriate for the ecological and economic systems of developing countries of the world. such methods need to be adequate for the current level of forces underneath the productivity development. they need to ensure a mutual correspondence between monetarism macroeconomic indicators and the microeconomic indicators that measure the use efficiency of the local ecological and economic resources. in addition, the research results of the post-harris period, especially those related to the development of input-output tables by l. kantorovich and t. ch. koopmans, nobel laureates in 1975 [3], as well as the fundamental work of michael porter “competition” in 2001 [4], allow for consistency and assess the competitiveness of firms and enterprises on the productivity of local environmental and economic resources. both these works are unidirectional, are consistent with milton friedman’s [5] quantitative theory of money, and ensure that microeconomic indicators are comparable with macroeconomic ones. additionally, more related works should be added. for example, u. nordhaus and p. romer, who made a step back towards accounting of the use effectiveness of local ecological and economic resources, initiate an evaluation of scientific achievements[6]‡. in recent years, thanks to the initiatives of astana economic forum, large-scale global researches have been carried out in kazakhstan. among them, the works of s. baizakov, ye. utembayev, n. akimov, n. sagadiyev, m. amrenov, and k. berentayev [7] are considered important. these works are supported and responded by analysts from the developed western countries. ragnar bentzel was right in his presentation of the works of kantorovich and koopmans, as a representative of the royal swedish academy of sciences, by noting the obvious that “the main economic problems can be studied in a purely scientific sense, regardless of the political organization of the society in which they are studied” [3]. in general, new horizons are opened up for the development of the scientific and technological bases that will be useful for recording and analyzing production, employment, incomes and prices. therefore, the time is ripe for us to overcome the shortcomings of the direct-costing system in terms of the development of human capital and the initiative of the busiest person in the economy. to this end, firstly, it is necessary to ensure that the methods developed for managing the evolution of human capital, as a key component within the advance of the productive forces of developing countries of the world, are in line with the conditions for the development of human capital in the developed countries. secondly, it is recommended that in any strategic perspective the objectives of an operational planning are consistent with the long-term, sustainable development goals. thirdly, there is a need to identify key operators that allow the integration of commodity and financial markets through the evolutionary development of keynesian and monetarist theories. and fourthly, there is a need for a new multiplier developed for scientific and technological progress so that capital development models in their commodity and monetary forms can be integrated into the system of market relations. ‡ the swedish academy, in its message on the laureates, explains that u. nordhaus and p. romer created economic models which explain how the merket economy interacts with nature and knowledge. thus, p. romer in his works showed how knowledge could serve as an engine for long-term economic growth. while the works of u. nordhaus describe the interaction between society and nature. modeling the management of the economies of developing countries 104 copyright ©2019 assa. adv. in systems science and appl. (2019) 2. setting the task of digitization of sustainable development goals and the main operators of the algorithm for its implementation at the last astana forum (astana, june 2017), president of the state n.a. nazarbayev stated “the need for new methods for calculating indicators that measure the country's wealth and the well-being of citizens, as the frequently used indicator of gdp has a number of significant flaws. gdp does not reflect the long-term nature of economic activity, does not take into account damage to the environment, including the depletion of natural resources. in addition, it does not reflect the quality of life in a particular country, gdp per capita does not show the real well-being of citizens, does not consider the income stratification of the population. what i say is, the world community can adopt an updated methodology for calculating gdp on the basis of “green” gdp and such indices as the human development index, the oecd better life index. it should adequately reflect the need for a balanced development of countries” [8]. back in 2009, the president of the state wrote about this in the articles “keys to the crisis” and “the fifth way” [9-10], and in his address to the people of kazakhstan in 2018, he indicated specific ways to implement his project on a new model of economic growth [11]. algorithmizing the task of sustainable development of developing countries at the economic research institute is carried out by a limited number of operators. it allows policy makers to effectively manage the mutually agreed movements of indicators of economic and financial markets up to the implementation of their goals. setting up the task of achieving the goals of sustainable development and the operator tool for its algorithmization are aimed at implementing a transfer of national moneys of developing countries to their dollar equivalents. it uses, on a parity basis, the whole arsenal of analysis and forecasting methodology developed by international organizations, such as un and oecd. such a formulation of the task and the chosen tool for its implementation, according to the developers, proceeds from the objective need of ensuring parity of the national moneys of developing countries with the national currencies of developed countries that effectively interact with the world reserve currency, represented by the u.s. dollar. the main tool chosen for solving this problem allows developing countries to ensure the commensurability of their moneys with their dollar equivalents. this warrants the alignment of the development methodology of the managerial economics of all countries in the world. that is, each country on a parity basis can use innovative technologies proposed by international organizations and other tools for the effective and sustainable management of its national economies and finances. at the same time, a condition is created for the mutual coordination of the economic interests of local populations, businesses and regions of each country based on the development of human capital and effective use of local ecological and economic resources. in such a case, high rates of economic growth and intended goals of increasing the well-being of citizens are ensured in the most efficient way. 3.key operators that integrate product and financial markets through developing theories of keynesianism and monetarism our proposed operators are based on the qualitative theory of money developed by s. baizakov [12]. this theory was built by the evolutionary development of economic laws, identified by supporters of the macroeconomic theory of j. keynes [13] and the quantity theory of money of m. friedman [5]. and the corresponding system of models of the qualitative theory of money 105 s. bayzakov, jeffrey yi-lin forrest, n.a. baizakov copyright ©2019 assa. adv. in systems science and appl. (2019) is built on the basis of the conceptual idea of the president of kazakhstan n.a. nazarbayev, set out in “the fifth way”[10]. the algorithm for solving the problem of the sustainable economic development of developing countries ensures the commensurability of national moneys of developing countries with those of developed countries by establishing their true dollar equivalences. this algorithm consists of the following operators, accompanied by the functions they perform in accordance with their economic purposes. the microeconomic operator for determining the multiplier of the productivity of resource costs at the enterprise level is 𝜇 = 𝑁𝐺𝐷𝑃/𝑄𝑃, (4) where 𝑁𝐺𝐷𝑃 is nominal gdp, 𝑄𝑃 the local ecological and economic resources consumed for the production of nominal gdp. the macroeconomic operator for determining the multiplier of scientific and technological progress at the country level is 𝑐 = 𝜇/(1 + 𝜇). (5) the macroeconomic operator for determining the gdp deflator (inflation multiplier) is 𝑝𝑏. according to the monetary policy methodology, 𝑝𝑏 = 𝑁𝐺𝐷𝑃/𝑅𝐺𝐷𝑃, (6) where 𝑅𝐺𝐷𝑃 is real gdp determined by the same monetarism methodology. the microeconomic operator for determining the value multiplier of the u.s. dollar (𝑝𝑝) as a reserve currency is 𝑝𝑝 = 𝑐 𝑝𝑏⁄ . (7) the macroeconomic operator for determining the purchasing power of the national money, which is expressed as a multiplier of the dollar equivalent of the aggregate of the true prices of goods and services of the developing country is 𝑝𝑐 = 1 𝑝𝑝⁄ . (8) the inter-sectoral operator for determining the true price of each product or service in a developing country is given by calculating the savings in working time, determined by the labor equivalent 𝑡𝑖, the direct costs of local ecological and economic resources for producing a product 𝑖, and labor equivalents 𝑇𝑗, and total production costs of the same product. the macroeconomic operator for determining the goods and services actually consumed in a developing country for consumption and accumulation in physical units is calculated on the dollar equivalent of their money 𝐹𝐺𝐷𝑃 = 𝑐 × 𝑅𝐺𝐷𝑃, (9) where according to the current monetarism model 𝑅𝐺𝐷𝑃 = 𝑁𝐺𝐷𝑃/𝑝𝑏. the macroeconomic operator for determining the goods and services actually consumed in a developing countries for consumption and accumulation is calculated in national money in their dollar equivalent 𝐹𝐺𝐷𝑃 = 𝑝𝑝 × 𝑁𝐺𝐷𝑃, (10) where according to the current model of monetarism 𝑁𝐺𝐷𝑃 = 1 𝑝𝑝 × 𝑀, the mass of national money in circulation. modeling the management of the economies of developing countries 106 copyright ©2019 assa. adv. in systems science and appl. (2019) 4. system of operators of developing countries as a tool for further liberalization of the world economy based on the above system of operators, a developing country can fully establish the dollar equivalent of its money. as a result, it can apply the existing international experience and positive practices of other countries on a par with the developed countries in managing its economy. moreover, to analyze and forecast the development of any developing country, such international organizations as the un and the oecd countries can use analogues to assess key indicators determined by the multipliers of scientific and technological progress and the productivity of local ecological and economic resources. thus, the first operator for determining the productivity of local ecological and economic resources was used by representatives of the un’s world bank and consultants working in kazakhstan through the tacis of the economic commission for europe in 1999-2004. the ginsim models (united kingdom) for analyzing the economics of enterprises were particularly valuable in kazakhstan [14]. the results of these works were published in kazakhstan in 2002 under the scientific editorship of s. baizakov, and are used in the practical work of the economic research institute [15]. according to the results of research presented by kateryna schroeder (world bank), the world bank is still currently using the multiplier of the productivity of local ecological and economic resources. so, according to k. schroeder, this multiplier is developed in its original form by assessing the effectiveness of domestic spending on the resources of developing countries in central asia when exporting agricultural products to china (drc) [16]: 𝐷𝑅𝐶𝑖𝑗 = 𝑐𝑖𝑗 𝑑 (𝑝𝑖𝑗−𝑐 𝑖𝑗 𝑓 ) , (11) where 𝑐𝑖𝑗 𝑑 and 𝑐𝑖𝑗 𝑓 represent, respectively, the domestic and foreign costs of input products “𝑖” in the country. in the case of export of meat, for example, from the countries of central asia to a specific country (china), if the multiplier 𝐷𝑅𝐶𝑖𝑗 < 1, this exporting country “𝑖” has a “comparative advantage” in the production of good “𝑗” compared to china; that is, its good (meat) costs less than the same good in china (in the production of one dollar). accordingly, the smaller the indicator 𝐷𝑅𝐶𝑖𝑗 in central asian countries, the greater the comparative advantage of these countries will be. as can be seen, the constant for evaluating the efficiency of exports of developed countries is the dollar, not the dollar equivalent of the national moneys of developing countries. in the economies of developing countries, not all goods and services aim at fulfilling the purpose of export. and, not all of their goods consumed domestically are imported from other countries. for example, the entire consumption fund and gross accumulation of these countries turn around by means of national moneys, which are the same goods as other goods – in our case, the tenge (not an international reserve currency) is itself a commodity. this means that national moneys of developing countries become lightly liquid commodities and determine the value of the nominal volume of production of the final product. this nominal volume is the nominal gross domestic product (ngdp) in the current prices of national money, determined in terms of the dollar. at the same time, its physical volume, that is, the real volume of the final product, called the real gross domestic product (rgdp), is determined by means of the gdp deflator (inflation). according to the common practice of the world, in analyzing the development of economies of developing countries, world bank experts use this indicator of real gdp (rgdp). however, real gdp (rgdp) is not able to assess continuously the productivity of 107 s. bayzakov, jeffrey yi-lin forrest, n.a. baizakov copyright ©2019 assa. adv. in systems science and appl. (2019) local ecological and economic resources used to produce the nominal gdp (ngdp). without such an assessment, as shown by the multiplier of k. schroeder, a representative of the world bank, it is not possible to determine the effectiveness or impact of any innovative solution. so, the reciprocal of the multiplier of k. schroeder, represents the value (productivity) of a country's local ecological and economic resources used to produce one-dollar equivalent of this commodity (1/𝐷𝑅𝐶𝑖𝑗): 1/𝐷𝑅𝐶𝑖𝑗 = ( 𝑝𝑖𝑗−𝑐𝑖𝑗 𝑓 ) (𝑐𝑖𝑗 𝑑 ) = 𝑝𝑖𝑗 𝑐𝑖𝑗 𝑑 − 𝑐𝑖𝑗 𝑓 𝑐𝑖𝑗 𝑑 . (12) this indicator 1/𝐷𝑅𝐶𝑖𝑗 is in fact an indicator “𝜇” – that is, an indicator of the productivity of ecological and economic resources in the regions of this country. for this reason, it suffices, for example, to replace in the last multiplier 𝑐𝑖𝑗 𝑓 𝑡𝑜 𝑐𝑖𝑗 𝑑 . as a result of this replacement, we get an analogue of the multiplier "𝜇" for any country in the world that can be employed to conduct a comparative assessment of the effectiveness of its exports: 𝜇 = ( 𝑝𝑖𝑗−𝑐𝑖𝑗 𝑓 ) (𝑐𝑖𝑗 𝑑 ) = 𝑝𝑖𝑗−𝑐𝑖𝑗 𝑑 𝑐𝑖𝑗 𝑑 . (13) the above model of the multiplier of the un representative k. schroeder is a very valuable tool for analysis, as it expresses the production and economic relationship between producers of goods and ecological and economic factors of its production. the most important thing about it is that it reflects the relationship between microeconomic indicators and macroeconomic indicators. moreover, this model has been used in assessing the comparative advantages of more than 30 commodity positions of enterprises in kazakhstan. this work had been done by a group of kazakhstan experts under the methodological guidance of european expert jean michel, now an employee of the world bank of the un. the results of this work are also presented in the book [14, p. 41-78]. along with j. michel, the methodological basis of s. baizakov’s model can be seen in the ginsim model (uk), the software product of which was developed by the maxwel stamp company [ibid., p. 111-144]. in particular, the nominal rate of protection (nrp), determined by using both the schroeder multiplier and the ginsim project model, shows how much the price of local goods (pd) increases compared to the world price (pw) for the same product due to establishing custom duties [ibid, p. 133]: 𝑁𝑅𝑃 = [(𝑃𝑑 − 𝑃𝑤)/𝑃𝑤] × 100. (14) despite the possibility of integrating microeconomic indicators with macroeconomic ones, individual representatives of the world bank, when analyzing the sustainability of transition countries, limit themselves to assessing fluctuations of real gdp without considering studies, in particular, k. schroeder’s work on the cost intensity of the obtained final product or the productivity of ecological and economic resources. to this end, in table 1, the analysis of kazakhstan’s data was conducted by non-authors of the article having cited this table, the authors refer to the work of the representative of the world bank in kazakhstan, who studied the state of kazakhstan's economy in 2016. footnote to table 1 indicates this source. modeling the management of the economies of developing countries 108 copyright ©2019 assa. adv. in systems science and appl. (2019) table 1. contribution to real gdp growth, 2013-2016 (in percentage points, unless otherwise indicated) 2013 2014 2015 2016 estimate real gdp growth (in percent) 6.0 4.2 1.2 1.0 domestic demand 6.9 3.8 2.7 1.3 private consumption 5.1 0.7 1.0 -0.3 public consumption 0.2 1.0 0.3 0.3 gross capital formation 1.6 2.1 1.5 1.3 net export -1.0 0.1 -1.2 -0.2 exports of goods and services 1.1 -1.1 -1.2 -1.0 imports of goods and services -2.1 1.1 0.0 0.8 statistical discrepancy 0.1 0.3 -0.3 -0.1 source: world bank calculations based on the data published by the committee on statistics (some figures may not be accurate due to rounding) table 1 reflects only the abstract contribution of all participants, first of all, the real and monetary and financial sectors of the economy of kazakhstan, without detailing the contribution of each of them. that is, there is a need for such details on the basis of identifying the causes. according to w. leibniz, whose name was used in the name of the famous institute of hannover, “reality can be understood for its reasons”. therefore, the establishment of causes will be the basis for making new managerial decisions that improve the health of the economy. the second operator, according to s. baizakov, seems to be a multiplier of scientific and technological progress. it is essentially macroeconomic; and by its nature, scientific and technological progress is the main engine of innovation in managing not only the development of the real and monetary and financial sectors, but also in the managerial economic sector [17]. in the western economic literature, the stp multiplier is known by the name of “tfp” only as an indicator of the level of technological progress. according to k. marx, “the first most important of the innate properties of matter is motion, not only as a mechanical and mathematical motion, but even more as aspiration, life spirit, tension or, using the expression of jacob boehme, suffering [qual] of matter” [18]. the driving force – the vital spirit of an economy – is any desire of the three aforementioned sectors of the economy. it is called scientific and technological progress. however, the most commonly used models of econometrics and the concept of production functions do not fully reflect the scientific and technological progress (stp). such progress is scientific and technological because technology is being improved not only in the real sector of economy, but also in all three managerial sectors: real, financial and managerial. a piece of research work is considered fruitful only if all three sectors as the ten-year work of the astana economic forum (aef) showed, such is, for example, the concept of the “fifth way” of the president of kazakhstan n. a. nazarbayev [9-10], which, by the level of debatability, relates to world problems. to assess the applicability of this concept of a three-economy, you can refer to the work of the frenchman lionel guy stoléru, which serves as its counterpart and is connected with a specific example of cars, where the market value of the car production increases 90 times, the price of one car increases 3 times, and the overall index prices increased 5 times. then the final results of the three prices differ so much that one cannot ignore these differences and not recognize their objectivity[19]: 1. the growth index of the nominal value of products 90 00 1  jj jtjt qp qp i ; 109 s. bayzakov, jeffrey yi-lin forrest, n.a. baizakov copyright ©2019 assa. adv. in systems science and appl. (2019) 2. product growth index in physical terms 30 3 90 000 0 2  j jt jj jtj q q qp qp i ; 3. price increase index in constant french francs at relative value: 18 5 3 *30 00 3  jj jtjt q q i   , where jtp and jtq accordingly, the price and quantity of goods j during the period t , and j all the many benefits, the dynamics of which are studied in time 0 before t . t good j relative price. indeed, l. stoléru is right, it would be a gross mistake to measure and assess the level of “real” growth on the basis of two, nominal and real growth indices. it is the third index of balanced economic growth, which expresses the inter-branch nature of the sphere of production, and is the carrier of feedback between the sectors of the real economy. thanks to the third index of assessing the real value of goods and services, it will be possible to assess the gaps in the development of the real and financial sectors of the national economy. in fact, this third index is the main key to relieve economic and financial shocks. the book by p. sraffi “production of goods through goods” (1999) also explored the triple connection between individual product prices and market prices for goods and services. to explore the relationship between the two, sraffa introduces a new measure of the value “standard output” and “standard product” [20]. thus, taking the produced annual cumulative product for a standard equal to one, it determines the share of wages in the value of the “standard output”: c+v+m=1. (15) and, having accepted the produced national income as a standard product equal to one, he studies in it the ratio of wages and profits: v+m=1. (16) the first of these expresses the basis of the cost method, defined by the labor theory of value (x = c + v + m), and the second, the income method, determined by the quantity theory of money (y = v + m). the cost method, except for wages (v) and profits (m), takes into account the total costs of labor, representing the real costs of production (c). another standard in his theory is determined by equality according to work: la + lb + ... + lk= 1, (17) where la, lb, ...., lk the annual amount of labor, respectively, employed in industries producing products a, b, ..., k important is the result of the work of sraffa in determining the productivity of intermediate consumption goods by dividing the annual income generated (y = v + m) by the total labor input used for its production (x = c + v + m). in this context, alisher tleubayev, a doctoral candidate at the leibniz institute of agricultural development in transition economies (germany), presented the denominator of the tfp model, not in the former two-factor form of the production function, but in a qualitatively new form, namely in the form of the stp multiplier [21]: 𝑇𝐹𝑃 = 𝑌 𝐴(𝑡)×𝐾𝛼×𝐿𝛽×𝑀𝛾. (18) modeling the management of the economies of developing countries 110 copyright ©2019 assa. adv. in systems science and appl. (2019) this multiplier of technological progress (in his other article, jointly with i. bobojonov, one of the leaders of the leibniz institute [21]), in fact, is the multiplier of scientific and technological progress, given in the second operator of the qualitative theory of money of s. baizakov. now, the multiplier of scientific and technological progress, which is supported by such an authoritative institute named after w. leibniz (represented as the tfp model in terms of the interpretation of tleubayev-bobojonov), confirms the emergence of s. baizakov's new qualitative theory of money. in fact, this new qualitative theory of money has been developed through the evolutionary development of the quantitative theory of money of m. friedman, that is, by complementing it instead of denying it. thus, in calculations and theoretical studies of bobojonov and tleubayev, the numerator of the tfp technological progress indicator is expressed by the “𝑌” indicator, which is the nominal gdp of the second baizakov operator (𝑌 = 𝑁𝐺𝐷𝑃), and the denominator of the tfp indicator is determined not by two, but by a three-factor production function in the approach of bobojonov-tleubayev. in this case, the information base for the forecast of the tfp indicator itself is the dynamics of the relationship of the final product 𝑌 to the total cost of ecological and economic resources 𝑋, and the integrator of three factors (𝐿-labor, 𝐾-capital and 𝑀current cost of local ecological and economic resources). in general, the tfp technological progress indicator can be predicted based on the release of the system of national accounts. respectively, parameters 𝛼, 𝛽 and 𝛾 represent the elasticities of the above three factors. thus, the bobojonov-tleubayev’s model can be used in predicting the system of the un sustainable development goals [21]. the advantage of their model is that technological progress is expressed by the production attitude not of two factors, but of three factors: labor, capital and natural matter. due to the lack of such a three-factor approach, which is systemic, kazakhstan and other developing countries suffer a shortage of analytical and forecasting tools for long-term forecasting. as far as short-term forecasting is concerned, kazakhstan’s interpretation of the multipliers of scientific and technological progress was previously described in the article by baizakov and others (2014) [22]. unlike the multiplier 𝑇𝐹𝑃, by baizakov and others forecast model (2014) is designed for short-term planning of up to three-five years. and it is implemented according to the reported data of the statistical authorities of each country. therefore, for the model of forecasting the economic performance for the long-term perspective up to 2025–2050, it is proposed, mainly, to be based on the dynamics of the stp multiplier and other related progress multipliers. in general, this baizakov’s and others forecast model, as well as s. baizakov’s multiplier of scientific and technological progress, are being developed in the name of the bobojonovtleubayev’s (germany) long-term model and other models of developed countries and international organizations such as the human development index and oecd indicators. 5. stp multiplier: the key to integrate development patterns of capital in their commodity and monetary forms into a system of market relations as is known, according the definition of the value of a dollar using a macroeconomic textbook (reprinted in russia more than 13 times), “value of money is the amount of goods and services that can be exchanged per unit of money (dollar); and the purchasing power of a monetary unit is the reciprocal of the price level” [21]. this means that the purchasing power of the us dollar (pp), as the price of a reserve currency is determined by the formula: 111 s. bayzakov, jeffrey yi-lin forrest, n.a. baizakov copyright ©2019 assa. adv. in systems science and appl. (2019) 𝑝𝑝 = 𝑐 𝑝𝑏 , (19) where 𝑝𝑏 is the growth rate of the gdp deflator, 𝑐 is the growth rate of the multiplier of scientific and technological progress. this relation shows that the growth rate of the gdp deflator is decomposed into two indicators: 𝑝𝑏 = 𝑐 𝑝𝑝 = 𝑐 × 𝑝𝑐, (20) which implies 𝑝𝑐 = 1 𝑝𝑝 , (21) where 𝑝𝑐 is the true price of a national money in circulation, the market price of which is determined in the currency market of the country (m). but how is the supply of money in circulation determined? according to the basic economic law of the quantity theory of money, the following equality takes place: 𝑣 = 𝑁𝐺𝐷𝑃/𝑀, (22) where 𝑣 is the velocity of money circulation. according to this theory, not only the cost of nominal gdp (ngdp) is circulated, but also the cost of the volume of total costs for its production (𝑋): 𝑣𝑥 = 𝑋/𝑀, (23) where 𝑣𝑥 is the velocity of money circulation. that is, we have [21]: 𝑣 = 𝑁𝐺𝐷𝑃 𝑀 = 𝜇 1+𝜇 × 𝑋 𝑀 = 𝑐 × 𝑣х, (24) where 𝜇/(1 + 𝜇) = 𝑐 is the multiplier of scientific and technological progress, and 𝑣х is the multiplier of the total costs of producing the nominal gdp. since 𝑣 = 𝑐 × 𝑣х, the money supply 𝑀 is determined by the formula: 𝑁𝐺𝐷𝑃 = (𝑐∗𝑣х) × 𝑀, (25) where 𝑝𝑐 = 1/𝑝𝑝 = (𝑐 × 𝑣х) is the true price of deficiency of the dollar equivalent of a national money, as a liquid commodity, in relation to reserve currencies. that is, in the case of kazakhstan, the policy of the national bank should not focus on inflation targeting, but on reducing the deficit of the supply of money in circulation. it is this multiplier “𝑐∗𝑣х” that should become a guideline for setting the discount rates of the central banks of developing countries around the world. modeling the management of the economies of developing countries 112 copyright ©2019 assa. adv. in systems science and appl. (2019) as a result, the nominal gdp in the current national money will grow indefinitely, and the national money itself will depreciate indefinitely. this conclusion in relation to kazakhstan, as an intensively developing country, with the intention to enter the 30 best economic performing countries in the world, according to its economic development, makes it possible to argue that the current focus on the nominal gdp indicator is necessary but not sufficient. and, therefore, relying directly on the real gdp indicator obtained from the nominal one distorts the true rates of economic growth. the world bank report (2017) mentions that the development of the political system of kazakhstan indicates the transfer of activity on economic policy management from the administration of president to the structures of the government and parliament and the expansion of powers of local executive bodies. kazakhstan has announced an increase in the competitiveness of its economy – modernization 3.0 – and the implementation of five priorities for technical modernization, the business environment, macroeconomic stability, improving the quality of human capital, strengthening security institutions and measures to combat corruption [23]. however, the implementation of the super-program of modernization and the five priorities faces an imbalance on the part of the readiness of state (heads of the economic bloc of the government, the national bank) and market institutions to synchronize their actions with the problem of human capital development. this imbalance has led to a rather unexpected for outsiders’ view, but an absolutely adequate response on the part of the authorities to the demands of society, namely focusing on one of the five priorities the development of human capital. this was announced by the president of the state this year, noting the need to implement the five social presidential initiatives (mortgage “7-20-25”, student housing, etc.) [11]. at the same time, it should be assumed that the potential of these five initiatives will be difficult to assess and maximize in all spheres of the country's activities without an appropriate methodology. such a methodology should be based both on the existing traditional indicators and on the new three multipliers associated with the implementation of science and technology in the life of every citizen of the country. as a result, they are aimed at improving citizens’ well-being by increasing human capital and harmonizing relations in the following three chains: “man-science” (stp/tfp multiplier), “man-man” (multiplier of socio-economic progress), as well as “man-society” (multiplier of socio-political progress). according to the comparative analysis, the baizakov’s model is not aiming at increasing the money capital, which, unfortunately, is happening now in kazakhstan. but it is aiming at the development of human capital, which is the most important among all tasks related to the well-being of humanity. it also is confirmed by the un human development index, and is being transferred to the oecd countries. therefore, the use of the baizakov’s model means that in developing countries the passion for speculative capital ceases to exist, and it is not the subjective propositions of the quantitative theory of money that is working at present, but the objective laws of the qualitative theory of money. 6. experimental calculations for the analysis of the comparative effectiveness of innovative projects initial conditions of the comparative analysis. the initial base was the information of the consolidated balance of ssgpo for 2016, as an example for carrying out experimental calculations. it is supplemented with a forecast for 2017 and an expected estimate of data for 2018. for the forecast of the development of ssgpo for 2019–2030, a database of project data for its operational and investment development, exclusively defined in conventional numbers, has been used. consequently, the analysis presented below has only an experimental character for testing the algorithm for solving the problem according to the model of s. baizakov [24]. 113 s. bayzakov, jeffrey yi-lin forrest, n.a. baizakov copyright ©2019 assa. adv. in systems science and appl. (2019) the analysis was carried out by identifying the causes of risks and risk management in the real sector. the level of innovativeness of technology in each version of the real sector is determined by the cost effectiveness of current material resources (which are used to produce the final product). and the function of scientific and technological progress is defined as the result of the productivity of material resources. thus, table 2 shows the calculation scheme for determining the function of scientific and technological progress in one of two options by year of the analytical period, where its last columns express the indicators of the second option for 2018 (idle) and 2028 (forecast) years. according to a given database of design data, the baizakov’s calculation work algorithm is able to assess any prevailing macroeconomic or microeconomic situation. table 2. evaluation of the scientific technological potential of ssgpo algorithm the name of indicators unit. 2016 (fact) 2028 (forecast) nx = ngdp + qp scientific and technological potential of ssgpo in value terms thousan d.tg 183 963 219,0 302 054 905,9 µ = ngdp / qp productivity of variable costs excluding payroll components tg/tg 2,85 0,98 c = ngdp / nx = µ/(1+µ) the multiplier of scientific and technological progress on the productivity of variable costs without wages tg/tg 0,74 0,49 same percentage % 100,00 66,74 ngdp=µ/(1+µ)* nx the algorithm for calculating the ntp multiplier: 1% nx provides µ / (1 + µ)% ngdp thousan d.tg 136 140 105,9 149 191 474,1 nx=tw+tr+qp total consumption of resources for the production of rational consumption in ssgpo thousan d.tg 183 963 219,0 302 054 905,9 ngdp=tw+tr ngdp – gva ssgpo, analogue of the nominal gdp of the country thousan d.tg 136 140 105,9 149 191 474,1 ngdp=µ*qp ngdp – gva ssgpo, analogue of the country's nominal gdp thousan d.tg 136 140 105,9 149 191 474,1 ngdp= =µ/(1+µ)*nx ngdp – gva ssgpo, analogue of the nominal gdp of the country nominal gdp thousan d.tg 136 140 105,9 149 191 474,1 7. analysis of the innovativeness of projects (and operational plans) in the financial sector and the assessment of the cost effectiveness of their implementation. the analysis was performed by identifying the causes, as sources of risk and risk management in the financial sector of the economy. the innovativeness of technology in the economy of the financial sector is determined by the productivity of investment costs and the development of human capital (tw) in the form of normal profits (tr). as you know, the normal profit is determined relative to the wage fund, and it serves as one of the main sources of savings and, consequently, gross accumulation. in a civilized world focused on the development of human capital, the spiritual and creative potential of the person himself, all other conditions being the same, the level of the human development index should constantly increase and, at least, should not decrease in general, the function of socio-economic progress is defined as the result of the productivity of expenses for normal profit, defined as an indicator of rational savings. thus, table 3 shows the calculated scheme for determining the function of socio-economic progress by the years of the analytical period in our conditional example, in ssgpo, where the last modeling the management of the economies of developing countries 114 copyright ©2019 assa. adv. in systems science and appl. (2019) columns express changes in the indicators for 2018-2028 according to the same option 2. table 3 as table 2 is compiled by method of determining the multiplier s. baizakov, and the sources of the data used are the same. table 3. assessment of the socio-economic potential of ssgpo algorithm the name of indicators unit. 2016 (fact) 2028 (forecast) η = tw/tr productivity of expenses in ssgpo on normal profit in the short-term period tg/tg 0,39 0,23 q=η/(1+η) the multiplier of socio-economic potential in ssgpo, tg/tg 0,28 0,19 q=η/(1+η) same percentage % 100,00 67,33 tw=η/(1+η)*ngdp algorithm for calculating the effect of the multiplier of socio-economic potential in ssgpo thousand .tg 38 044 699,7 28 070 475,5 ngdp/l=φ average annual labor productivity per worker in ssgpo thousand .tg 7 318,9 11 387,7 8. analysis and assessment of socio-political progress in the development of the management economy sector the innovation of technology in the economy of the management sector is determined by the product of the multipliers of scientific and technological progress and socio-economic progress (c * q). innovation in the present means both the innovation itself, the innovation, and the process, that is, the potential capable of introducing, say, the economy of the real, financial and managerial sectors of something new, spiritually enriched with the desire not only of a human entrepreneur, but also as simple worker to progress. for example, new technology, or new technologies, patent inventions, or ideas for making management decisions in all the above areas of their activities. but as a potential, not every innovation contributes to an increase in the efficiency of production of goods and services, which are currently in demand and are supplied by demand from consumers. that is, innovation does not necessarily provide a comparative or absolute increase in the efficiency of production of the final results of intellectual activity, its imagination, creative process, discoveries, inventions and rationalization. the fruit of this aspiration of a person of labor and management is the synergistic effect associated with the productivity of the country's local ecological and economic resources and the multiplier of scientific and technological progress, determined on its basis. but the socio-political progress in the development of the managerial economy sector is connected not only with the harmonization of the development of commodity and financial capital. it is also connected with the development of the main component of the development of the productive forces of the country, the spiritual and material development of its human potential. such a three-level system means that the corresponding function of socio-political progress is harmonized with the scientific, technological and socio-economic progress. thus, the following table shows the calculated scheme for determining the function of socio-political progress over the years of the analytical period, and the last columns express the indicators for 2016-1028. table 4, as tables 3-4 compiled by the algorithm for determining the multiplier s. baizakov, on the basis of the previous information of option 2. table 4. assessment of the social and political potential of the ssgpo algorithm the name of indicators unit. 2016 (fact) 2028 (forecast) 115 s. bayzakov, jeffrey yi-lin forrest, n.a. baizakov copyright ©2019 assa. adv. in systems science and appl. (2019) tw/l= w the basic level of the average annual wage per worker in ssgpo thousand tenge 2 045,30 2 142,62 ngdp/l=φ basic annual average labor productivity per worker in ssgpo. thousand tenge 7 318,97 11 387,79 w=μ/(1+μ)*η/(1+η)*ѱ basic level of average annual wage per one worker at full costs in ssgpo thousand tenge 2 045,30 2 142,62 ѱ basic annual average labor productivity at full cost, per worker in ssgpo thousand tenge 9 889,96 23 055,87 с= φ/ ѱ=µ/(1+µ) the basic level of the multiplier ntp and ssgpo tg/tg 0,74 0,49 same percentage % 100,00 66,74 c*q=μ/(1+μ)*η/(1+η)* multiplier of social and political potential in ssgpo tg/tg 0,21 0,09 comparative analysis of two project options for innovative development of ssgpo (with all other conditions being the same). table 5. comparative analysis of innovative development projects of ssgpo algorithm the name of indicators unit. 2028 (option 1) 2028(option 2) nx = ngdp + qp scientific and technological potential of ssgpo in value terms thousand.t g 302 054 905,9 302 054 905,9 µ = ngdp / qp productivity of variable costs excluding payroll components tg/tg 0,98 0,98 c = ngdp / nx = µ/(1+µ) the multiplier of scientific and technological progress on the productivity of variable costs without wages tg/tg 0,49 0,49 same percentage % 68,39 66,74 ngdp=µ/(1+µ)* nx the algorithm for calculating the ntp multiplier: 1% nx provides µ / (1 + µ)% ngdp thousand.t g 136 140 105,9 149 191 474,1 l increase in the number of employed person -53 ѱ=nx/l productivity gains at full cost t/person +95,5 npv gain net present value thousand.t g + 926939.0 npv gain net present value thousand. dollars + 1806.4 as can be seen from the comparative analysis, the baizakov model is not aimed at increasing the money capital, which, unfortunately, is happening now in kazakhstan. and on the development of human capital, which is the most important of the tasks of the well-being of humanity, which is confirmed by the un human development index, and which are being transferred to the oecd countries. therefore, the application of the baizakov model means that in developing countries the passion for speculative capital ceases to exist, and it is not the subjective propositions of the quantitative theory of money that begin to work, but the objective laws of the qualitative theory of money. in conclusion, it should be noted that the theory of preserving the quality of money without applying the gold standard and in the context of globalization and digitalization is called the qualitative theory of money by analogy to the quantitative theory of money by m. friedman. the obtained practical results on the qualitative theory of money make it possible to bridge the gaps between: modeling the management of the economies of developing countries 116 copyright ©2019 assa. adv. in systems science and appl. (2019) micro and macroeconomic indicators; the theory of marginal utility and labor theory of value, short-term budget management plan and long-term strategic planned management tool; tools for assessing the effectiveness of investments in the accounting system of cost accounting of the “direct costing” resources of microeconomics and the productivity of current environmental and economic resources in applied models of trade and investment policy; using cobb-douglas-type macroeconomic production functions and estimating payback period based on net present value (npv) at the microeconomic level; criteria for managing human development in the real and financial sectors of the economy. 9. conclusions on the quality of national money and on the assessment of their impact on economic growth 1. the proposed methodology for the qualitative theory of money, which further develops the theory of keynesianism and the quantitative theory of m. friedman and is successfully applied in the practice of analyzing macroeconomic indicators of production, exchange, distribution and consumption. hence, the basic equation of a balanced economic growth assessment of the real final product fgdp fully has the form of a single cube: 1*fgdp =pp*ngdp= c*rgdp, (26) where ngdp and rgdp are respectively nominal and real gdp, pp and c are respectively indicators of the quality of national money (ngdp), which are in circulation and real gdp (rgdp). thus, a comparison of the growth rates of russia and kazakhstan with the corresponding indicators of four developed countries of the world is given in table 6. table 6. dynamics of changes in the parameters of balanced growth and equilibrium in the economy of russia and kazakhstan compared to the developed countries of the world in 2010-2015. ngdp rgdp fgdp fgdp/ ngdp fgdp/ rgdp 1*fgdp=0.76*ngdp=1.02*rgdp russian 526.9 107 294.5 0.56 2.75 1*294.5=0.56*526.9=2.75*107=294.5 kazakhstan 1572.5 92.7 224.2 0.14 2.42 1*224.2=0.14*1572.5=2.42*92.7=224. 2 germany 173.1 160 183 1.06 1.14 1*183=1.06*173.1=1.14*160=183 france 179 167 168 0.94 1.01 1*168=0.94*179=1.01*167=168 usa 176.2 160 149 0.85 0.93 1*149=0.85*176.2=0.93*160=149 great britain 175.1 130 133 0.76 1.02 1*133=0.76*175.1=1.02*130= 133 source: world development bank database, the analysis was specially conducted over the three five-year plans (2000-2015), on the basis of which, according to the technical task of the project program of the science committee of the republic of kazakhstan, a forecast for 2020 will be made. 2. taking into account the coefficients of bringing them into a single standard form of fgdp, these six countries of the world will be ranked in the order indicated in figure 1. russia 2000-2015 (2000=100%) kazakhstan 2000-2015 (2000=100%) 117 s. bayzakov, jeffrey yi-lin forrest, n.a. baizakov copyright ©2019 assa. adv. in systems science and appl. (2019) fig. 1. rating assessment of the growth rate of the actually used final product (fgdp) in six countries of the world. 3. based on the calculations made for these six countries of the world, the following conclusions can be made: 3.1. three-dimensional development figures of six countries of the world are constructed by reducing the three-dimensional measurement parameters to a one-dimensional standard measurement (fgdp). in terms of the growth rate of this key parameter of the economy, over the past five years, russia (294%), kazakhstan (224%), germany (183%), france (168%) ranked second, and the united states (149%) ranked second. ), united kingdom (133%). 3.2. the growth rates of the scientific and technological potential of each country in the world allow deciphering the growth rates of the socio-economic potential in the simplest form, as is the case for calculating tw = q * ngdp. but the actual final consumption product (ftw) created in the country is ftw = c * tw, and the actually created accumulation fund is equal to ftr = c * tr. 3.3. finally, the final result of the development of the economy of each country in the world is actually created by its socio-political potential, which, thanks to its system of sustainable development models, allows us to build a human development index. references 1. rasskazova-nikolaev, s. 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(1975). economic equilibrium and growth (vol. 1). amsterdam: northholland publishing company; new york: american elsevier. https://dixinews.kz/articles/ekonomika/28238/ https://www.zakon.kz/147878-statja-prezidenta-rk-pjatyjj-put.html 119 s. bayzakov, jeffrey yi-lin forrest, n.a. baizakov copyright ©2019 assa. adv. in systems science and appl. (2019) 21. tleubaev, a., bobozhonov, i., gets, l., hockmann, h. & glauben, t. (2017) determining the productivity and efficiency of grain production in kazakhstan: an approach to stochastic boundaries, discussion paper no. 160, leibniz institute for agricultural development in transition economies. http://hdl.handle.net/10419/155329 22. baizakov, s., dzhunusova, d., shuneev, s. & baizakov, n. (2014), towards a methodology of macroeconomic analysis and economic expertise. economics and statistics, 1, 4-13. 23. world bank annual report (2017, june 30) [online]: http://documents.worldbank.org/curated/en/143021506909711004/world-bank-annualreport-2017 24. baizakov, s., khambar, b., baizakov, n (2018) economic and mathematical foundations of digitization of kazakhstan, economics and statistics, 3, 18-26. adv syst sci appl 2020; 02:56–70 published online at https://ijassa.ipu.ru. numerical methods for constructing solutions of functional differential equations of pointwise type armen beklaryan1* 1national research university higher school of economics, moscow, russia abstract: the article discusses construction of traveling wave type solutions for the frenkelkontorova model on the propagation of longitudinal waves. for the first time, based on the existence and uniqueness theorem of traveling wave type solutions, as well as the approximation theorem, a complete family of traveling wave type solutions is constructed in the form of subfamilies of bounded solutions (horizontals) and unbounded solutions (verticals). keywords: traveling waves, functional differential equations, equations of mathematical physics, splines, forward-backward differential equation 1. introduction the theory of differential equations with delay, which has developed rapidly in recent decades [1–3], has in many ways acquired a complete form and is now actively used in modeling various objects. numerical algorithms for solving equations with delay of various types were also developed [4, 5]. at the same time, numerical methods for equations with advanced arguments (and, moreover, with mixed type deviations) are practically not studied, although references to them have been encountered for a long time, mainly in connection with the classification of equations with deviating argument [6]. as a rule, there is a parsing of equations of a particular kind with further obtaining the existence and uniqueness theorems based on the use of the properties of the right-hand side [7] and application of methods, such as the study of the roots of the characteristic quasi-polynomial [8], collocation methods and finite element scheme (expansion of a solution in terms of basis functions of some finitedimensional space) [9–11] or through the construction of a hilbert space of the reproducing kernels on the basis of boundary conditions [12]. for equations of a delayed type, as a rule, a well-known initial problem is considered when the initial moment of time coincides with the left end of the interval of equation’s domain. such a statement is characterized by the fact that the initial-boundary conditions are local in nature, and the equation itself can be integrated by the step method. in all other cases, and even more so in the case of functional differential equations of pointwise type (fdept) with mixed-type deviations, the initial-boundary conditions are nonlocal in nature. one of the approaches to the construction of numerical methods for fdept is the variational method based on solving the induced optimal control problem (ocp) as an unconditional optimization problem for the residual functional. as noted earlier, the initialboundary conditions for fdept are not local, therefore, the solution of the variational problem for the residual functional cannot be obtained by simple local improvements and requires global optimization. note that for problems of finite-dimensional optimization there ∗corresponding author: abeklaryan@hse.ru numerical methods for functional differential equations 57 are many approaches to finding a global extremum, sufficiently efficient algorithms are built, there are a large number of publications that present precedents for successfully solved problems [13, 14, for example]. at the same time, the task of finding global extremum for ocp remains one of the most acute tasks in the extreme problems theory. among classical approaches to this task one cannot fail to mention the works [15] and [16]. some works that apply search and genetic algorithms are also well-known [17, 18]. but unfortunately these efforts have not yet brought about effective algorithms that would be able to solve a wide spectrum of practical problems [19]. one of the ideas in this area is the idea of reducing the optimal control problem to a low-dimensional problem on the reachability set of a controlled system [20]. unfortunately, the problem of approximating the reachable set in the numerical solution is not much simpler than the problem of finding a global extremum. nevertheless, the properties of the reachable set known from theoretical studies can be used to construct specialized optimization algorithms for ocp [21]. in particular, the connectivity property of the reachability set allows one to construct a scheme of continuous control variation that leads to the solution of the problem. the idea of using the connectivity property of the reachability set when constructing numerical optimization algorithms belongs to a.g. chentsov [22]. in the majority of well-known approaches to the construction of non-convex optimization methods, the solution of the problem is divided into two stages: “global”, where a wide scan of the variable space is performed, and “local”, aimed on local refinement of the obtained solution [14]. the combination of different methods at each stage, as well as the order of the alternation of stages, determines the specific computational algorithm. for mentioned class of problems, we consider a idea of combining both stages in one algorithm package and creating a method that allows one to select and change both the “global scan” procedure and local improvement of the obtained approximate solution at each iteration [23]. 2. statement of the initial boundary-value problem the most important goal in fdept is the study of the basic initial-boundary value problem ẋ(t) = f(t, x(q1(t)), . . . , x(qs(t))), t ∈ br, (2.1) ẋ(t) = ϕ(t), t ∈ r\br, ϕ(·) ∈ l∞(r,rn), (2.2) x(t̄) = x̄, t̄ ∈ r, x̄ ∈ rn. (2.3) where f : r× rns −→ rn is a mapping of the c(0) class, qj(·), j = 1, . . . , s, represent diffeomorphisms of the line preserving orientation, andbr is either the closed interval [t0, t1], or closed half-line [t0,+∞), or the whole line r. in the case of a delayed-type equation, for br = [t0, t1], [t0,+∞), t̄ = t0, the initialboundary-value problem becomes a well-known initial problem with the initial function ζ(t) = x̄− ∫ t̄ t ϕ(τ)dτ, t ≤ t̄. the statement of the initial-boundary value problem (2.1)-(2.3) is correct without any restrictions on the type of argument deviations. definition 2.1: the solution to the basic initial-boundary value problem is any absolutely continuous solution of the equation (2.1) satisfying the boundary condition (2.2) and the initial condition (2.3). the aim of the fde research is to study the solution space of the boundary value problem (2.1)-(2.2) for each given boundary function ϕ, as well as to describe the obstacles due to which the solutions of the boundary value problem (2.1)-(2.2) do not inherit the properties of solutions of ordinary differential equations. copyright © 2020 assa. adv syst sci appl (2020) 58 a. beklaryan if we introduce notation for a fixed function ϕ(·) fϕ(t, z1, . . . , zs) = { f(t, z1, . . . , zs), t ∈ br, ϕ(t), t /∈ br, the boundary value problem (2.1)-(2.3) becomes a cauchy problem ẋ(t) = fϕ(t, x(q1(t)), . . . , x(qs(t))), t ∈ r (2.4) x(t̄) = x̄, t̄ ∈ r, x̄ ∈ rn. (2.5) obviously, in the case br = r, the equality fϕ = f holds. in the study of the cauchy problem (2.4)-(2.5), the role of the group < q1, . . . , qs > is very important. often, instead of such a group, it is useful to consider some wider finitely generated group q of diffeomorphisms of the line, that is, < q1, . . . , qs >⊆ q. moreover, we assume that the condition hq = supt∈r |q(t)− t| < +∞ is satisfied for all elements q of the group q. let’s formulate a system of restrictions on the right-hand side f : r× rn·s 7−→ rn of fdept (2.1), and on diffeomorphisms qj(·), j = 1, . . . , s: (a) f(·) ∈ c(0)(r× rn·s,rn); (b) for any t, zj, z̄j, j = 1, s, there is a quasilinear growth ‖f (t, z1, . . . , zs) ‖rn ≤m0(t) +m1 s∑ j=1 ‖zj‖rn , m0(·) ∈ c(0)(r,r), and a lipschitz condition ‖f (t, z1, . . . , zs)− f (t, z̄1, . . . , z̄s) ‖rn ≤ lf s∑ j=1 ‖zj − z̄j‖rn ; (c) there exists µ∗ ∈ r+ such that m0(·) ∈ lnµ∗c(0)(r); (d) the values hqj = sup t∈r |t− qj(t)|, j = 1, . . . , s, are finite; (e) with the constant µ∗ ∈ r+ from condition (c) the family of functions f̃q,z1,...,zs(t) = f(q(t), z1, . . . , zs)(µ ?)hq , q ∈ q, z1, . . . , zs ∈ rn, is equicontinuous on any finite interval. the continuity condition, growth conditions with respect to the phase variables and time variable, and the lipschitz condition (conditions (a)− (b)) are standard in the theory of ordinary differential equations. in fact, in item (b), the first inequality in the form of the growth condition with respect to the phase variables and time variable is a consequence of the second condition in the form of the lipschitz condition. under the lipschitz constant lf , the minimum value among the possible values of these constants should be understood. accordingly, we can assume that m1 = lf . moreover, we wrote the first inequality out separately in order to formulate condition (c) for functionm0(·). condition (c) for function f is related to the study of solutions on the half-line and line, which requires certain restrictions copyright © 2020 assa. adv syst sci appl (2020) numerical methods for functional differential equations 59 on the time growth of the right-hand side. it should be noted that we can always satisfy condition (d) by making a time change, but we may break condition (c). the last condition (e) is necessary for the right-hand side of the induced infinite-dimensional ordinary differential equation, with the phase space in a suitable banach space, to easily establish the fact of its bochner integrability. in fact, condition (e) can be removed, but this leads to further technical complications, as the right-hand side of the induced infinite-dimensional ordinary differential equation will only be measurable, and it will be necessary to establish the fact of its bochner integrability [24]. the right-hand side f(·) of fdept will be considered as an element of the banach space vµ∗(r× rns,rn) with lipschitz norm vµ∗(r× rns,rn) = { f(·) : f(·) satisfies the conditions (a)-(d) } , ‖f(·)‖lip = sup t∈r ‖f(t, 0, . . . , 0)(µ∗)|t|‖rn + + sup (t,z1,...,zs,z̄1,...,z̄s)∈r1+2ns ‖f(t, z1, . . . , zs)− f(t, z̄1, . . . , z̄s)‖rn∑s j=1 ‖zj − z̄j‖rn , where the parameter µ∗ ∈ r+ coincides with the corresponding constant from the condition (c). obviously, for the function f(·) ∈ vµ∗(r× rns,rn) the smallest value of the constant lf from the lipschitz condition (the condition (b)) coincides with the value of the second summand in the definition of the norm f(·). in what follows, speaking of the lipschitz condition, by the constant lf we mean exactly its smallest value. the right-hand side of the fdept (2.1) uniquely defines a pair (f(·), h), where f(·) is an element of the banach space vµ∗(r× rns,rn), and h = (hq1 , . . . , hqs), hqj ≥ 0, j = 1, . . . , s, are maximum deviation of the argument from the condition (d), which we will consider as parameters. therefore, such a right-hand side of the equation will uniquely determine the pair (lf ;h). we will seek a solution to the fdept with a quasilinear right-hand side in a oneparameter family of banach spaces of functions with a given exponential growth. the exponent is the parameter of the selected family of functions, which is defined as follows lnµc(k)(r) = { x(·) : x(·) ∈ c(k) (r,rn) , max 0≤r≤k sup t∈r ‖x(r)(t)µ|t|‖rn < +∞ } , ‖x(·)‖(k) µ = max 0≤r≤k sup t∈r ‖x(r)(t)µ|t|‖rn , k = 0, 1, . . . , µ ∈ (0,+∞). 2.1. existence and uniqueness theorem for the basic initial-boundary value problem in terms of the parameter µ ∈ (0, 1) and the space lnµc(0)(r) we formulate a condition guaranteeing the existence and uniqueness of a solution for the initially-boundary problem. theorem 2.1 ( [25]): let function f and diffeomorphisms qj(·), j = 1, . . . , s, satisfy the conditions (a)− (e) from the section 2. if for some µ ∈ (0, µ∗) ∩ (0, 1) the inequality lf s∑ j=1 µ−hqj < lnµ−1, (2.6) is satisfied then for any fixed initial-boundary conditions x̄ ∈ rn, ϕ(·) ∈ l∞(r,rn) there exists a solution (absolutely continuous) x(·) ∈ lnµc(0)(r) of the basic initialboundary value problem (2.1)-(2.3). such a solution is unique and, as an element of the copyright © 2020 assa. adv syst sci appl (2020) 60 a. beklaryan space lnµc(0)(r), depends continuously on the initial-boundary conditions x̄ ∈ rn, ϕ(·) ∈ l∞(r,rn) and the right-hand side of the equation (function f(·)). the condition (2.6) is exact and not improvable: there are differential equations for which violation of this condition leads to a violation of either the existence of the solution or uniqueness. note that if the inequality (2.6) has a solution, then there are µ1(lf ;h), µ2(lf ;h) such that for each solution µ the following condition holds µ1(lf ;h) < µ < µ2(lf ;h). 3. variational approach to constructing solutions to the basic initial-boundary value problem we study absolutely continuous solutions of the system fi(t, ẋ(t), x(t+ τ1), . . . , x(t+ τs)) = 0, i = 1, k, t ∈ [tl, tr], (3.7) under boundary conditions outside the system definition interval ẋ(t) = ϕl(t), t ∈ [tll, tl], (3.8) ẋ(t) = ϕr(t), t ∈ [tr, trr], (3.9) as well as initial-boundary conditions on the system definition interval km(ẋ(ξ), x(ξ1), . . . , x(ξp)) = 0, (3.10) where fi : r× rn × rns −→ rn, i = 1, k, is a mapping of the c(0) class; τj ∈ z, j = 1, s; tl, tr ∈ r; tll = tl + min{0, τ1, . . . , τs}, trr = tr + max{0, τ1, . . . , τs}; ϕl, ϕr : rn −→ rn are mappings of thec(0) class, defining fixed boundary vector functions;km : rn × rnp −→ rn,m = 1, q, is a mapping of the c(0) class; ξ, ξi ∈ [tl, tr], i = 1, p, is a fixed set of points on the system definition interval. we give a formal statement of the optimization problem induced by the initial-boundaryvalue problem. problem a. find the trajectory x̂(t), which provides a minimum of residual functional i(x̂(t)) = v(n) ( k∑ i=1 ∫ tr tl f 2 i (t, ˙̂x(t), x̂(t+ τ1), . . . , x̂(t+ τs))dt+∫ tl tll [ ˙̂x(t)− ϕl(t)]2dt+ ∫ trr tr [ ˙̂x(t)− ϕr(t)]2dt ) + v(k) q∑ m=1 k2 m( ˙̂x(ξ), x̂(ξ1), . . . , x̂(ξp)), (3.11) where v(n), v(k) ∈ r+ are weighting factors. the following statement holds. proposition 3.1: each solution to the initial-boundary value problem (3.7)-(3.10) is a solution to the optimization problem a. in particular, under the conditions of theorem 2.1, a solution to the initial-boundary value problem (2.1)-(2.3) exists and is a solution to the corresponding problem a. in the general copyright © 2020 assa. adv syst sci appl (2020) numerical methods for functional differential equations 61 case, the converse statement to proposition 3.1 is false, but, nevertheless, taking into account the uniqueness theorem and also the stability theorem of the solution, in the case of successful construction of a numerical solution, we can speak with high confidence about constructing a solution close to analytical one. the proposed variational approach to constructing solutions of the basic initial-boundary value problem lies at the basis of the numerical implementation of such solutions. the numerical methods themselves are based on the ritz method and spline collocation constructions and were implemented in [23, 26]. in order to solve the problem of the class under consideration, the trajectories of the system are discretized on a grid with a constant step, and a generalized residual functional is formulated that includes both the weighted residual of the original differential equation and the residual of the boundary conditions. a spline differentiation technique is used to evaluate the derivatives of the desired trajectories of the system, based on two spline approximation designs: using cubic natural splines and using a special type of spline whose second derivatives at the edges are also controlled using optimized parameters. 3.1. software package optcon-f a set of algorithms for local and global optimization was implemented for solving the stated finite-dimensional problems. the used technology includes: an algorithm for sequentially increasing the accuracy of approximation by multiplying the number of nodes in the grid of the discretization; algorithms for the difference evaluation of the derivatives of the functional from the first to the sixth degree of accuracy inclusive; method of successively increasing the precision of spline differentiation. the corresponding software complex (sc) optcon-f was implemented in the language c under the control of operating systems os windows, os linux and mac os using compilers bcc 5.5 and gcc. sc was designed to obtain a numerical solution of boundary value problems, parametric identification problems and optimal control for dynamical systems described by fdept [27]. among the local algorithms included in the sc optcon-f there are: 1) partan method; 2) powell-brent’s method; 3) gradient method of confidence intervals; 4) barzilaiborwein method; 5) newton’s method with the difference estimate of the hessian matrix; 6) generalized quasi-newtonian method; 7) direct-dual method of gradient descent; 8) differential euler optimization method of the 2nd order; 9) differential adams optimization method of the 4th order and others. as algorithms of “closers” there are: 1) adaptive modification of the hooke-jeeves method; 2) stochastic search methods in random subspaces of the indicated (2, 3, 4, or 5) dimension; 3) local version of the curvilinear search method. non-local algorithms that form the basis of the sc are: 1) “parabolic” method – a combination of coordinate-wise descent with a periodic multistart and a non-local onedimensional search by the parabola algorithm; 2) non-local method of curvilinear search; 3) luus-jaakola method; 4) “forest” method – multivariant adaptive method of random multistart and others. as part of the work on the sc, taking into account the specifics related to the qualitative properties of fdept, the following key results were obtained: • algorithm of construction of “controllable splines” has been developed. the problem of obtaining a high-precision approximation to the derivative of a function of one variable over a set of values of the function itself, defined on a fixed grid, was investigated. the performed computational experiments showed that in order to achieve good accuracy in estimating derived trajectories for fdept systems, it is not possible to use known types of splines (“natural”, with additional boundary conditions, akima splines, etc.). the proposed algorithm is based on the use of derivatives of cubic spline functions. • a technology has been developed for approximation of general fdept systems using a finite-dimensional unconditional minimization problem. by analyzing the copyright © 2020 assa. adv syst sci appl (2020) 62 a. beklaryan behavior of homeomorphisms on the initial (main) time interval, an extended time interval is calculated, for which the discretized grid is constructed. to approximate the initial continuous problem on a fixed grid over time, the ritz method is used: the trajectories are approximated using controlled spline functions, the coefficients of which are selected by searching for the minimum of the residual functional. • a specialized global optimization algorithm was developed, based on the idea of curvilinear search. to search for the minimum of the non-convex residual functional, a specialized global optimization algorithm has been developed, based on the idea of curvilinear search on the reachable set of a control system produced using a pair of randomly generated “support” controls. using the property of connectedness, variations in the space of control functions that do not violate the existing constraints are constructed at iterations of the algorithm. this is achieved by direct projection on a parallelepiped. to improve the current approximation, a modification of a non-local onedimensional search algorithm based on the “parabolic” method is applied [28]. also, as a globalizing mechanism, a non-local search in random directions is used, repeated many times at each iteration of the algorithm. to solve auxiliary non-convex problems of one-dimensional search, a modification of the stochastic p -algorithm, proposed by a. zhigljavsky and a. žilinskas [14], is implemented [29]. to enhance reliability of the proposed method, a periodic random multi-start was provided in the algorithm’s construction. • an algorithm for the numerical integration of fdept systems based on the sequential discretization technique has been developed. to improve the efficiency of calculations, tools have been implemented to build a sequence of approximative finitedimensional optimization problems with a growing number of variables. at the same time, solutions obtained at the previous stages of calculations performed on the current discretization grid are projected onto a new grid with an increased number of nodes while preserving the qualitative and quantitative characteristics of the trajectories. • the proposed numerical algorithms were tested on a wide range of tasks [27] with using the principle of “the best of known solutions” [30]. in all considered problems, the proposed algorithm allowed us to find the best known solution. the calculation experiments that have been carried out demonstrate considerably high fidelity of the proposed algorithms. the above heuristic search algorithm for the solution x̂(t) can be justified on the basis of the existence and uniqueness theorems for initial-boundary value problems for the investigated fdept, as well as theorems on approximating solutions of such equations on the whole line by solutions of the initial-boundary value problem on a sequence of expanding intervals. a description of such equations and corresponding results are presented in the following sections. 4. existence and uniqueness theorem for soliton solutions in the problem of longitudinal vibrations of an infinite homogeneous rod we consider a problem from the theory of plastic deformation, in which solutions of the traveling wave type for the following system are studied mÿi(t) = yi+1(t)− 2yi(t) + yi−1(t) + φ(yi(t)), i ∈ z, yi ∈ r, t ∈ r, (4.12) yi(t+ τ) = yi+1(t), τ ≥ 0. (4.13) we will study such a system under the most weak conditions on the potential φ(·) in the form of the lipschitz condition. the lipschitz constant will be denoted by lφ. we formulate a theorem on the existence and uniqueness of a traveling wave type solution (soliton solution), copyright © 2020 assa. adv syst sci appl (2020) numerical methods for functional differential equations 63 τ µ 1 µ̂ τ̂ µ1(τ ) µ2(τ ) 0 fig. 4.1. graphs of functions µ1(τ), µ2(τ) based on the correspondence of traveling-wave type solutions of the infinite-dimensional system (4.12)-(4.13) to the solutions of the induced fdept [24], reduced to a first-order system ẋ1(t) = x2(t), x ∈ r2, x = (x1, x2)′, t ∈ r, ẋ2(t) = m−1 ( x1(t+ τ)− 2x1(t) + x1(t− τ) + φ(x1(t)) ) , (4.14) as well as the existence and uniqueness theorem for solutions to such equations (theorem 2.1). to do this, we consider a transcendental equation with respect to two variables τ ∈ [0,+∞) and µ ∈ (0, 1) cτ ( 2µ−1 + 1 ) = lnµ−1, (4.15) where c = max{1; 2m−1 √ l2 φ + 2}. the set of solutions of the equation (4.15) is described by functions µ1(τ), µ2(τ) given in fig. 4.1. if we denote the right-hand side of the system (4.14) by f , then relations µ1(lf ;h) = τ √ µ1(τ) and µ2(lf ;h) = τ √ µ2(τ) will be valid, where lf = c, h = (τ, τ). for each µ ∈ (0, 1) we define the phase space of the infinite-dimensional equation (4.12)as a hilbert space of sequences with the corresponding norm k2 z2µ = {κ : κ = {zi}+∞ −∞, zi ∈ r2, i ∈ z, ∑ i∈z ‖zi‖2 r2µ2|i| < +∞}. theorem 4.1 ( [24]): let the potential φ satisfy the lipschitz condition with constant lφ. then, for any initial values ī ∈ z, a, b ∈ r, t̄ ∈ r, and characteristics τ > 0 satisfying the condition 0 < τ < τ̂ , copyright © 2020 assa. adv syst sci appl (2020) 64 a. beklaryan for the initial system of differential equations (4.12) there exists a unique solution of the traveling wave type {yi(·)}+∞ −∞ with characteristic τ such that it satisfies the initial conditions yī(t̄) = a, ẏī(t̄) = b. for any parameter µ ∈ (µ1(τ), µ2(τ)) the vector function ω(t) = {(yi(t), ẏi(t)) ′}+∞ −∞ belongs to the space k2 z2µ for any t ∈ r, and the function ρ(t) = ‖ω(t)‖2µ belongs to the space l1 τ √ µc (1)(r). such a solution depends continuously on the initial values a, b ∈ r, as well as on the mass m. theorem 4.1 not only guarantees the existence of a solution but also determines the limitation of its possible growth both in time t and in coordinates i ∈ z (over space). it is obvious that for each 0 < τ < τ̂ the space k2 z2(µ2(τ)−ε), for small ε > 0, is much narrower than the space k2 z2(µ1(τ)+ε). the theorem guarantees the existence of a solution in narrower spaces and uniqueness in wider spaces. 5. approximation of the solution of a functional differential equation defined on the line by solutions of an initialboundary value problem with expanding intervals of the definition next, we consider soliton solutions of the system, which will be implemented as solutions of the induced functional differential equations defined on the whole line. for the numerical integration of such an equation, one should be able to approximate the solutions of such an equation by the solutions of initial-boundary value problems defined on an expanding family of finite intervals. this section is devoted to this problem. let’s consider the initial boundary value problem ẋ(t) = f(t, x(t+ τ1), . . . , x(t+ τs)), t ∈ br, (5.16) ẋ(t) = ϕ(t), t ∈ r\br, ϕ(·) ∈ l∞(r,rn), (5.17) x(t̄) = x̄, t̄ ∈ r, x̄ ∈ rn, (5.18) where τj ∈ r. we formulate again the theorem on the continuous dependence on the initialboundary conditions. we define a banach space of functions with weight lnµl∞(r) = { ϕ(·) : ϕ(·) ∈ l∞ (r,rn) , sup t∈r vrai‖ϕ(t)µ|t|‖rn < +∞ } , µ ∈ (0, 1) and norm ‖ϕ(·)‖µ = sup t∈r vrai‖ϕ(t)µ|t|‖rn . if in the boundary condition (5.17), starting with some sufficiently large k ∈ z+ with the condition br ⊂ [−k, k], the boundary condition ϕ(·) ∈ l∞(r) replace outside the interval br ⊂ [−k, k] so that the new boundary function ϕ̃(·) satisfies the condition ϕ̃(·) ∈ lnµl∞(r), then the corresponding solution x̃(·) of the boundary value problem on br ⊂ [−k, k] will match the original solution x(·). we are going to formulate a proposition on the approximation of solutions of an initialboundary value problem defined on the whole line by solutions of the initial-boundary value copyright © 2020 assa. adv syst sci appl (2020) numerical methods for functional differential equations 65 problem defined on the interval [−k, k] as k → +∞. we consider the initial-boundary value problem on the whole line br = r ẋ(t) = f(t, x(t+ τ1), . . . , x(t+ τs)), t ∈ r, (5.19) x(t̄) = x̄, t̄ ∈ r, x̄ ∈ rn, (5.20) where τj ∈ r, j = 1, . . . , s, j = 1, . . . , s. for each k ∈ z we consider the initial-boundary value problem on a finite interval br = [−k, k] ẋ(t) = f(t, x(t+ τ1), . . . , x(t+ τs)), t ∈ [−k, k], (5.21) ẋ(t) = ϕ(t), t ∈ r\[−k, k], ϕ(·) ∈ ln1l∞(r), (5.22) x(t̄) = x̄, t̄ ∈ r, x̄ ∈ rn. (5.23) theorem 5.1 ( [31]): let the map f(·) satisfies the conditions (a)− (d). if for µ ∈ (0, µ?) ∩ (0, 1) the inequality lf s∑ j=1 µ−|τj | < lnµ−1 (5.24) holds, and (µ1(lf ;h), µ2(lf ;h)) is the maximum solution interval for the inequality (5.24), then for any x̄ ∈ rn, ϕ(·) ∈ ln1l∞(r), the solution x̂(·) of the initial-boundary value problem (5.19)-(5.20), as an element of the space lnµc(0)(r), is approximated by solutions x̂k(·) of the initial-boundary value problem (5.21)–(5.23) as k → +∞. moreover, for any arbitrarily small ε, 0 < ε < µ2 − µ1 there is cfϕε such that the following estimate is true ‖x̂(·)− x̂k(·)‖(0) µ ≤ cfϕε ( µ1(lf ;h) µ2(lf ;h)− ε )k . from the estimate it follows that the convergence rate will be the higher, the more the µ1(lf ;h) and µ2(lf ;h) values will differ from each other, the difference between them is inversely proportional to the value of lf max1≤j≤s |τj|. 6. construction of the complete family of traveling wave type solutions in the frenkel-kontorova model considering periodic functionals for the model from the section 4, we obtain the frenkelkontorova model from the theory of plastic deformation. in this case, taking the functional φ(y) = a sin(by), a,b ∈ r mÿi(t) = yi+1(t)− 2yi(t) + yi−1(t) + asin(byi(t)), i ∈ z, yi ∈ r, t ∈ r, (6.25) yi(t+ τ) = yi+1(t), τ > 0, (6.26) we construct solutions of the traveling wave type with characteristic τ . as noted earlier, the space of traveling wave type solutions for such a system coincides with the space of solutions of the fdept system ẋ1(t) = τx2(t), t ∈ r, ẋ2(t) = τm−1[x1(t+ 1)− 2x1(t) + x1(t− 1) + a sin(bx1(t))]. (6.27) moreover, there is a correspondence of solutions according to the rule yi(t) = x(τ−1t+ i). we will construct numerical solutions of the system (6.27) using the approximation theorem 5.1. copyright © 2020 assa. adv syst sci appl (2020) 66 a. beklaryan 6.1. numerical experiments next, the results of the computational experiments on the study of initial-boundary value problems for systems of fdept using optcon-f software will be presented. before demonstrating the examples, it is necessary to make a number of significant observations: (a) in the sc optcon-f, the possibility of satisfying the condition of uniform boundedness of the derivative is realized. the maximum deviation from zero is determined by the lipschitz constant of the right side of the equation, by the parameter µ, and also by the deviations of the argument; (b) numerical integration on an interval with initial-boundary conditions is realized as an integration process with given boundary conditions at the left end and procedures for minimizing deviations from the boundary conditions at the right end for the solution constructed with observance of the restrictions from the preceding item; (c) in the obtained theorem on approximating solutions of the original equation on the whole line by solutions on expanding finite intervals [−k, k], the only restriction on the boundary conditions themselves is the condition for their uniform boundedness. in particular, we choose the zero boundary condition; (d) we note that the obtained condition for the existence of a solution of the traveling wave type is just a sufficient condition. therefore, solutions of the traveling wave type can also be numerically constructed for τ > τ̂ . nevertheless, many of the central conditions of the presented theory are “exact”, that is, there are examples of equations for which violation of the indicated conditions leads to the lack of solutions; (e) in the sc optcon-f there is the possibility of sequential application of various algorithms within the framework of constructing a solution for one task. thus, the constructed intermediate solution in the previous step becomes the starting solution (“baseline”) for the following algorithm. in this case, such an implementation does not prevent the global search algorithms from “popping out” of the local solution. separately, we note the presence of a programming module that allows predetermining the order of application of algorithms, as well as the construction of complex chain of steps (conditional statements, loops, etc.) depending on the current or historical values of a number parameters (for example, error estimation or number of iterations). for the examples presented below, in addition to the basic global optimization algorithm described in section 1, the following scheme was used in cycle: the generalized quasinewtonian and powell-brent’s methods (with bi-directional line search along each dimension) were used alternately, and after the error changed by less than 10−p the adaptive modification of the hooke-jeeves method was used l times (p and l are computable functions on the basis of the loop iteration number, as well as the lipschitz constants of the equation itself and a number of other technical characteristics). the stopping criterion depended on the number of iterations in the first part of the cycle, as well as the current error estimate and its dynamics; (f) in view of all the points listed above, as well as stochastic elements in the applied algorithms, the presented value of the residual functional (rf, i.e. error of a numerical solution) can not be used to estimate the theoretical rate of convergence. we consider dynamical system in the following form:{ ẋ1(t) = 0.15x2(t), ẋ2(t) = 0.15× 100−1(x1(t+ 1)− 2x1(t) + x1(t− 1) + 500 sin(0.1x1(t))), t ∈ r, initial conditions{ x1(0) = c, x2(0) = 0. (6.28) copyright © 2020 assa. adv syst sci appl (2020) numerical methods for functional differential equations 67 (a) trajectories of x1(t) for the system (6.29) (b) trajectories of x1(t) for the system (6.28) fig. 6.2. trajectories of x1(t) at different c here, with respect to the system (6.27), we have a = 500, b = 0.1, τ = 0.15,m = 100, and the equality (4.15) takes the form 0.003 √ 2502(2µ−1 + 1) = lnµ−1 and has on the interval (0, 1) two solutions with approximate values µ1(0.15) = 0, 22 and µ2(0.15) = 0, 424191 (the exact values are expressed in terms of the lambert w -function and can’t be written out in quadratures). taking into account the impossibility of considering the numerical solution of the system on an infinite interval, we introduce the parameter k and the corresponding family of expanding initial-boundary value problems{ ẋ1(t) = 0.15x2(t), ẋ2(t) = 0.15× 100−1(x1(t+ 1)− 2x1(t) + x1(t− 1) + 500 sin(0.1x1(t))), t ∈ [−k, k], boundary conditions{ ẋ1(t) = 0, ẋ2(t) = 0, t ∈ (−∞,−k] ∪ [k,+∞), initial conditions{ x1(0) = c, x2(0) = 0. (6.29) according to the theorem 5.1, convergence is guaranteed under any given essentially bounded boundary conditions. in the system (6.29), the simplest boundary function is chosen, the identity zero. thus, the solution of the system (6.29) converges (according to the metric of the space lnµc(0)(r) for µ ∈ (µ1(0.15), µ2(0.15))) to the solution of the system (6.28) as k →∞. since the equation (6.27) is autonomous, the solution space of such equation is invariant with respect to time-variable shifts. on the other hand, from the periodicity of the right-hand side with respect to the x1(t) it follows that the solution space of such equation is invariant copyright © 2020 assa. adv syst sci appl (2020) 68 a. beklaryan (a) trajectories of x1(t) at different k (b) trajectories of x1(t) at different c fig. 6.3. unbounded trajectories of x1(t) with respect to a shift in x1(t) for a period equal to 2π b . therefore, it suffices to consider a family of solutions of the initial system (6.28) with a value of initial conditions x1(0) from zero to the value of the period. nevertheless, the stationary solutions x1(t) are repeated each half-period π b . since the right-hand side of the equation is an odd function with respect to the x1(t), the solution space of such equation can withstand the reflection transformation with respect to the axis t. hence, it is sufficient to construct trajectories x1(t) in the strip from zero to the half-period. figure 6.2 shows the graphs x1(t) for different values of the parameter c = x1(0) for both the system (6.29) and the initial system (6.28) (the values of c are reduced to half-period). note that fig. 6.2 shows the complete family of bounded graphs x1(t) (up to the above transformations) from the space lnµc(0)(r) for µ ∈ (µ1(0.15), µ2(0.15)). at the same time, for some initial conditions, there are only unbounded x1(t). figure 6.3a demonstrates the evolution of such an unbounded graph x1(t) with increasing k for the initial conditions x1(0) = π b , x2(0) = 50. figure 6.3b, in turn, shows a one-parameter family of unbounded graphs x1(t) for x1(0) = π b and different values of x2(0) = c. 7. conclusion the construction of numerical solutions of the traveling wave type for the frenkel-kontorova model on the propagation of longitudinal waves using the developed software package was demonstrated. for this model, a complete family of traveling wave solutions was constructed. acknowledgements the reported study was funded by rfbr according to the research project 19-01-00147. references 1. bellman, r. & cooke, k. l. 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(2017) traveling waves and functional differential equations of pointwise type. what is common? in proc. of the viii international conference on optimization and applications (optima-2017), petrovac, montenegro, october 2-7, ceur-ws.org. copyright © 2020 assa. adv syst sci appl (2020) https://dx.doi.org/10.1007/s10898-015-0346-8 https://dx.doi.org/10.1134/s0005117915030030 https://dx.doi.org/10.1007/3-540-53032-0 introduction statement of the initial boundary-value problem existence and uniqueness theorem for the basic initial-boundary value problem variational approach to constructing solutions to the basic initial-boundary value problem software package optcon-f existence and uniqueness theorem for soliton solutions in the problem of longitudinal vibrations of an infinite homogeneous rod approximation of the solution of a functional differential equation defined on the line by solutions of an initial-boundary value problem with expanding intervals of the definition construction of the complete family of traveling wave type solutions in the frenkel-kontorova model numerical experiments conclusion мягкая вероятность ошибки при передаче информации adv syst sci appl 2017; 4; 14-21 published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/482 selective attention compensates detrimental effect of auditory noise on performance in posner experiment borzou alipourfard1, kamyar arbabifard2, fatemeh bakouie3*, somayeh sadat hashemi kamangar3, shahriar gharibzadeh4 1 department of bioengineering university of texas at arlington, arlington, united states 2 department of computer engineering, amirkabir university of technology, tehran, iran 3 institute for cognitive and brain sciences, shahid beheshti university, tehran, iran 4 department of biomedical engineering, amirkabir university of technology, tehran, iran abstract: environmental noise has a degrading effect on individual’s capacity to pay attention. in this paper, we revealed that the individual’s impaired performance under noise can be counterpoised by recollecting and recalling his attention. we studied the effect of auditory white noise on individual’s performance in posner experiment; and examined if actively reorienting the individual’s attention to stimuli would mitigate the effects of auditory noise. our subjects, ten undergraduate students, participated in the classical posner experiment, while auditory white noise was being played for them through earphone. we varied the intensity of the auditory noise and studied subject’s performance in three conditions: neutral, valid cued, and invalid cued trials. our results showed that the presence of auditory noise degraded subject’s performance in neutral trials. however, subject’s performance was not affected by auditory noise in valid cued trials. keywords: auditory white noise, attention, posner experiment. 1. introduction most people prefer a quiet place for work; they commonly believe that in the presence of environmental noise, their performance is negatively impacted; and they have had experiences that seem to prove this point. thus, studying why environmental noise may affect performance on a task and how we can mitigate its effects is not only of theoretical importance, but is also of practical application and importance. there have been many studies concerned with various effects of auditory noise. some have considered the effects of noise in residential areas. for instance, auditory noise, such as road traffic, has been proved to have an unhealthy effect on residents sleeping patterns or children learning capabilities [1]. there are also studies concerning the detrimental effects of noise in working environments and the physiological and psychological consequences of long exposure to noise on workers [2]. at the physiological level, noise has been seen to cause increased heart rate and blood pressure [3]. furthermore, the neural correlation of the effects of background noise have been reported as increased direct-coupled potential, as induced decrease in brain alpha wave, or as increased dopamine turn over in pfc [4,5]. in experimental settings, the immediate effects that auditory noise may have on an individual’s performance have been examined. in most of these studies, subject’s performance under noise condition was compared with those recorded in standard condition; and impacts of noise, at a psychological and behavioral level, have been described in terms of an overload framework, arousal model, inverted-u hypothesis and stochastic resonance [6-10]. interestingly, the effects of noise on performance have been seen to be consistent with that of performance under induced arousal [8, 9]. that is, under the effect of noise, a subject’s * corresponding author: f_bakouie@sbu.ac.ir http://ijassa.ipu.ru/ojs/ijassa/article/view/482 mailto:f_bakouie@sbu.ac.ir 15 b. alipourfard, k. arbabifard, f. bakouie, s. sadat, h. kamangar, s. gharibzadeh copyright ©2017 assa. adv. in systems science and appl. (2017) performance in low priority task components is degraded while his performance in high priority task components is unaffected; and the subject also shows increased selectivity in utilizing cues [6-9]. although most studies have observed that noise induces arousal, it is to mention that not all the studies have observed such effect: there has been cases in which noise has induced simply a vegetative stress state, or has even improved a subject’s accuracy and sensitivity to stimuli [10, 11]. in this paper, we aimed to study why environmental noise may affect an individual’s performance on a task and how we can decrease this effect. to find an answer for the proposed question, we noticed that many of the previous studies have attributed changes in subject’s performance (under noise) to changes in attentional mechanisms (such as narrowing of attention) [7]. an intuitive thinking suggests that this conjecture is consistent with our personal experience: most of us find it more difficult to focus in a noisy environment. if such a hypothesis is true and that noise disrupts subject’s attentional mechanisms then we expect that with the aid of some other mechanism that captures, recalls and reorients individual’s attention, the impaired performance under noise can be recovered. a suitable experimental setting to determine if recollecting an individual’s attention would mitigate the effects of auditory noise on his performance is posner experiment. in posner task, the experimenter is able to orient and manipulate subject’s attention with the aid of visual cues of his own design. in such an experimental setting, we can study if reorienting an individual’s attention, with the aid of posner cues, compensates the detrimental effect of noise on his performance. but before that, we first have to show that noise does have a detrimental effect on a subject’s performance in posner experiment. posner experiment was one of the early demonstrations of selective attention. in a visual detection task, posner, prior to the onset of the detection stimuli (a black dot), used a cue (an arrow) to indicate the stimuli location to subjects [12]. posner showed that orienting subject’s attention to stimuli location (pre-cueing) decreased subject’s response time, facilitating his perceptual processing. furthermore, posner task has since provided a strong framework for many computational models of attention [13]. dosher and lu studied a subject’s performance in posner experiment, when the detection stimulus (visual stimulus, a dot) was embedded in different levels of background visual noise. through this paradigm they showed that visual noise had a detrimental effect on a subject’s performance and orienting subject’s attention to stimuli location mitigated this detrimental effect; hence they discovered a new attentional mechanism, which they called external noise exclusion. it is to point out that in such an experimental setting we would expect the noise to be harmful to a subject’s performance as it affects the visual information he requires to complete the task. on the other hand, it seems that auditory noise does not affect any sensory information an individual requires to complete the posner task. now we are encountered with two questions: 1) does the auditory noise affect subject’s performance in posner experiment detrimentally? 2) does recalling and reorienting individual’s attention mitigate the detrimental effect of auditory noise on subject’s performance in posner experiment? in this study we design an experiment to answer these two questions that leads the two novelties in comparison with dosher and lu study, i.e. i) the presence of auditory noise degraded subject’s performance; ii) orienting of attention disrupts the detrimental effect of auditory noise on subject. it should be mentioned that in the study of dosher and lu, both the stimulus and noise are visual; while in this study, we designed an experiment with visual stimulus and auditory noise. in this study, our subjects participated in the classical posner experimental setting while different levels of auditory white noise were played through an apple earpods earphone. we varied the intensity of the noise and studied subject’s performance in three conditions; namely: neutral, valid cued and invalid cued trials. through this methodology, we examined interactions between auditory noise and oriented attention, and we sought to learn whether actively reorienting the individual’s attention to stimuli location would mitigate the effects of auditory noise or not. with selective attention compensates detrimental effect of auditory noise 16 the intuitive thinking we mentioned before, we expected to observe an antagonistic interaction between noise and oriented attention in subject’s performance. 2. methods ten undergraduate students, all male, aged between 19 and 22, and with no observable medical conditions that could affect their performance in the task, participated voluntarily in this experiment. the subjects each sat on a chair placed in front of a lcd monitor and at a distance that each found to be comfortable. each subject was instructed to fix his gaze at the center of the fourteen-inch monitor, which had a white background. we asked each subject to press a key as quickly as possible upon detecting the stimulus. the detection stimulus, a black dot, occurred at either the left or the right of the fixation point. the detection stimulus was offset with respect to the fixation point; the offset was set at 600 pixels. our experiment consisted of 5 blocks of trials for five different levels of auditory noise intensity (specifically, no noise, 70 db, 85 db, 100 db, 115 db). white noises of different intensities were generated using matlabr2010a; the noise intensity levels were measured by an hm-303-6 voltage meter. in the study of arnsten & goldman-rakic, it is mentioned that the background noise intensity is 60-70 db; the loud noise intensity is more than 95 db and the noise intensities more than 115 db is annoying. as we wanted to cover all the noise intensities, from background noise to loud and annoying one, we chose the intensity levels of 70 db, 85 db, 100 db, 115 db. the generated noise was then fed to an apple earpods earphone to produce the desired level of auditory noise intensity for each block of trials. to prevent noise intensity adjustments from affecting the subject’s decisional criteria, the order in which these blocks took place in the experiment was designed to be random. each block of trial then consisted of ten cued trials and ten neutral trials. so, the total number of collected dataset for each subject is 5×10×2=100. at the beginning of each trial, subjects were presented with either a plus sign (neutral trials) or an arrow pointing to the right or left (cued trials). as shown in fig. 1, if the plus sign was presented for the trial, then the detection stimuli occurred with equal probability to the left or right of the fixation point. for the cued trials, the detection stimuli occurred at the indicated location with the probability of 0.8 (valid trials) and occurred at the opposite side with the probability of 0.2 (invalid trials). detection stimuli occurred 400 ms after the onset of either the plus sign or the arrow. the time between trials was set at three seconds. the subject’s performance was measured in each trial as indicated by his response time. the described framework was developed virtually in java environment. before the official experiment commenced, subjects were instructed on the paradigms and methods of the experiment; and each subject was allowed to practice the experiment for as many times as he desired, until he found himself comfortable with the process. finally, in the actual testing process, care was taken to provide each subject with a comfortable and a stress-free environment. all subjects were tested during the period between 14:00 and 18:00. 3. results the results of our experiment are shown in fig. 2. the upper plots show the statistical distribution of subjects’ response time (rt). in the lower plots of this figure, the average response time is shown as a function of noise intensity. the leftmost plots in this figure show the results for neutral trials; the middle plots are for the valid cued trials and the rightmost plots show the response time for invalid cued trials. data is plotted using logarithmic scale for noise intensity (x axis) and linear scale for response time (y axis). to check the accuracy of the collected data, we measured the difference in the average response time when the stimuli occurred at the left of the fixation point and when it occurred at its right; in an accurate data set, we would expect this difference to be very close to zero. in our experiment, this difference accumulated to only 2 ms. 17 b. alipourfard, k. arbabifard, f. bakouie, s. sadat, h. kamangar, s. gharibzadeh copyright ©2017 assa. adv. in systems science and appl. (2017) fig. 1. organization of posner task. if the plus sign was presented for the trial, then the detection stimuli occurred with equal probability to the left or right of the fixation point. for the cued trials, the detection stimuli occurred at the indicated location with probability of 0.8 and occurred at the opposite side with the probability of 0.2. a preliminary examination of the data in fig.2 shows that valid cue facilitates performance in all noise conditions while invalid cue does not facilitate performance. this observation is more prominent at higher noise intensities: in 115 db intensity condition, invalid cueing degrades performance while accurate cueing improves performance. as the data in fig. 2 shows, effects of noise on performance in neutral trials and valid cued trials are unalike. on the one hand, for neutral trials, the response time is an increasing function of noise intensity; in neutral trials, as the leftmost plots in fig. 2 show, the response time increases from 721 ms at no noise condition to 738 ms at noise intensity of 115 db. f-test was conducted by r3.1.1 software to compare subject’s response time in these conditions. there was a significant effect of noise on subject’s response time, f(1,9) = 5.64, mse = 1896, p <0.05. on the other hand, for the valid cued trials, as shown in the middle plot of fig. 2, the response time is seemingly independent of noise intensity, exhibiting a total variation of only 9ms. interestingly, in valid cued trials, by conducting f-test to compare subject’s response time in no noise condition and 115db noise condition, we see that there was no significant effect of noise on subject’s response time, f(1,9) = 0.33, mse = 2938 ns. as demonstrated in the right most plot of fig. 2, the response time for invalid cued trials is also an increasing function of noise intensity. specifically, in these instances, the increase in response time is greater as compared with neutral trials. the ample increase in subject’s response time in the 115db noise condition shows that the degrading effect that noise may have on subject’s performance can be as detrimental as the effect of removing valid cues: in the no-noise condition, we see that valid cueing improves performance by 34.2 ms and 115 db noise condition increases subject’s response time by 29.8 ms. in addition, the detailed data of fig. 2 is presented in table 1. in this table, we have provided the exact values of means and sds (std) of cued and neutral trials. on average, the difference between the means of cued and neutral trials as calculated from the table below is 48.50 ms. to further expand upon our hypothesis, we calculated differences in subject’s rt in neutral and valid cued trials and plotted them against noise intensity. the result is presented in fig. 3. selective attention compensates detrimental effect of auditory noise 18 fig.2 recorded response time is plotted against noise intensity level. statistical distributions of the collected rts are shown in the upper plots of this figure. the average rt is plotted in the lower figure. plotted data in the leftmost figures corresponds to the data collected in the neutral trials, while the middle figures and the rightmost figures show the results for the valid cued trials and invalid cued trials respectively. the data is plotted using logarithmic scale for noise intensity (x axis) and linear scale for response time (y axis) table 1. the exact values of means and sds (std) of cued and neutral trials noise level 1 2 3 4 5 mean rt in cued trials (std) 686.82(51.4) 684.77(48.8) 684.86(45.5) 678.68(44.6) 681.28(56.7) mean rt in neutral trials (std) 721.83(39.5) 724.05(46.5) 734.79(50.5) 734.66(49.8) 743.60(51.4 ) we then built a simple linear regression model (least square regression) using r3.1.1 software to study the relation between noise and differences in subjects' performance in valid cued and neutral trails. the value of r2 was calculated to be 0.0177. the results are shown in table 2. note the significant p-value for the noise level coefficient. the “intercept” of the calculated linear model shows that cueing, on average, facilitates performance by 31.09 ms. table 2 the coefficient for the regression model. coefficients estimate std. error tvalue pr(>|t|) intercept 31.09 6.32 4.898 1.56e-06 noise level 6.68 2.56 2.606 0.009 19 b. alipourfard, k. arbabifard, f. bakouie, s. sadat, h. kamangar, s. gharibzadeh copyright ©2017 assa. adv. in systems science and appl. (2017) fig.3 the difference in subject’s rt between valid cued and neutral trials as function of noise intensity. the blue line is a least square linear regression model that shows the dependency of the aforementioned data on the intensity level of the auditory noise 4. discussion in this study, our goal was to examine whether the underlying mechanisms by which environmental noise impacts performance, were related to human attentional mechanisms; we examined if the individual’s impaired performance under noise could be counterpoised by recollecting and recalling his attention. our subjects participated in the classical posner experiment, with the addition that different levels of auditory white noise were played through an apple earpods earphone. we varied the intensity of the noise, studied the changes in subject’s response time, and examined interactions between environmental noise and oriented attention. we found that the presence of noise was degrading to subjects’ performances in neutral trials. however, subjects’ performances were not affected by noise in valid cued trials. we have specifically compared our design with that of the dosher & lu 2000 [13]. in order to better capture our experimental setting, we examined a similar experimental setting proposed by dosher and lu. they studied a subject’s performance in posner experiment, when the detection stimulus (visual stimulus, a dot) was embedded in different levels of background visual noise. through this paradigm they showed that visual noise had a detrimental effect on a subject’s performance and that orienting subject’s attention to stimuli location mitigated this detrimental effect. hence, they discovered a new attentional mechanism, which they called external noise exclusion. it is to point out that in such an experimental setting we would expect the noise to be harmful to a subject’s performance as it affects the visual information he requires to complete the task. on the other hand, in our experimental setting, auditory noise does not affect any sensory information an individual requires to complete the visual posner task. hence, we would not expect it to be harmful to his performance. however, presence of noise has been shown to have a degrading effect. the main question that remains is that why noise may affect a subject’s performance. in neutral trials, our results show that noise impacts a subject’s performance negatively; this degrading effect of noise has also been seen in previous researches [8, 9]. on the other hand, for selective attention compensates detrimental effect of auditory noise 20 the valid cued trials we found that a subject’s response time was independent of noise intensity. this suggests that orienting attention to the stimuli location somehow counteracted the detrimental effects of noise. interestingly, we see that this interaction is present at all noise levels. and irrespective of the intensity, orienting the subject’s attention fully counteracts the effect of noise. this persistent antagonistic relation not only suggests that these two events are counteracting each other at some cognitive level [2], but is also an evidence to our main hypothesis since recalling subject’s attention fully negated the detrimental effect that noise had had on his performance. to further expand upon our hypothesis, we calculated differences in subject’s rt in neutral and valid cued trials and plotted them against noise intensity (see figure.3). the “noise level” coefficient can be interpreted as the average increase in a subject’s rt per noise level in neutral trials that was compensated by cueing in valid cued trials. if there was no interaction between noise and oriented attention on a cognitive level, we would expect the “noise level” coefficient to be zero and for cueing to decrease response time on average by 31.09 ms regardless of noise intensity (see table 1). however, our results suggest a very strong interaction (pr(>|t|)=0.009 ) between noise and oriented attention. this result is a strong evidence for the cognitive level presence of auditory noise that is fully nullified when orienting attention. another interesting aspect of the collected data set is concerning a subjects’ performance in invalid cued trials; however, we want to note that the results here are more qualitative than quantitative, since the small collected data for the invalid cued trials does not allow for a statistical inference. our results show that the presence of noise was the most hurtful to a subject’s performance in these trials (see fig. 2). in cued trials, we then have a task component in which performance is continuously worsened when increasing noise intensity (invalid cued trials) and a task component that is not affected by the presence of noise (valid cued trials). previous studies recognized that under noise conditions, a subject’s performance is continuously degraded in low priority task components, while the performance in high priority task components is not affected [8, 9]. in other words, the amount of noise effect on subject’s performance depends on the priority of the task; if the task has high priority, negative noise effect on subject’s attention would be less [8, 9]. taking into consideration the results of such studies, we may conclude that in our experiment, valid cueing attracts attention of the subject and make a priority for the task. subjects’ performance in valid cued trials represents their performance in the high priority component of the task, and in invalid cued trials represents their performance in the low priority task component. hence, we hypothesize that, pre-cueing had a prioritizing effect on subject’s surrounding sensory events. therefore, visual events on the cued side became his sensory events with high priority, while the visual events on the opposite side were given a lower priority. this prioritizing effect favors the hypothesis that pre-cueing orients subjects' attention and the decrease in his response time in valid cued trials is a result of the orientation of attention rather than a change in response criteria. 5. conclusion in this paper we used posner task to study why environmental noise may affect an individual’s performance on a task. we studied the accuracy of the common belief that the presence of noise makes it more difficult to focus and to pay attention; and examined if recollecting and recalling an individual’s attention would counterpoise his impaired performance under noise. our results supported our main hypothesis. they showed that orienting a subject’s attention to stimuli location mostly negated the detrimental effect that noise had on his performance. our study suggests that, it would be possible to decrease the negative effect of environmental noise even if its modality differs from the stimuli's. this would certainly raise the efficiency of individuals and helps them work well and accurately in noisy workplaces. 21 b. alipourfard, k. arbabifard, f. bakouie, s. sadat, h. kamangar, s. gharibzadeh copyright ©2017 assa. adv. in systems science and appl. (2017) references [1] hygge, s., boman, e., & enmarker, i. (2003). the effects of road traffic noise and meaningful irrelevant speech on different memory systems. scandinavian journal of psychology, 44(1), 13-21. [2] broadbent, d. e. (1971). decision and stress. london, uk: academic press. [3] hsu, s.-m., ko, w.-j., liao, w.-c., huang, s.-j., chen, r. j., li, c.-y., & hwang, s.-l. (2010). associations of exposure to noise with physiological and psychological outcomes among post-cardiac surgery patients in icus. clinics, 65(10), 985-989. [4] arnsten, a. f., & goldman-rakic, p. s. (1998). noise stress impairs prefrontal cortical cognitive function in monkeys: evidence for a hyperdopaminergic mechanism. archives of general psychiatry, 55(4), 362-368. [5] briggs, f., mangun, g. r., & usrey, w. m. (2013). attention enhances synaptic efficacy and the signal-to-noise ratio in neural circuits. nature, 499, 476–480. [6] bell, p. a. (1978). effects of noise and heat stress on primary and subsidiary task performance. human factors: the journal of the human factors and ergonomics society, 20(6), 749-752. [7] woodhead, m. m. (1966). an effect of noise on the distribution of attention. journal of applied psychology, 50(4), 296. [8] o'malley, j. j., & gallas, j. (1977). noise and attention span. perceptual and motor skills, 44(3), 919-922. [9] hygge, s., & knez, i. (2001). effects of noise, heat and indoor lighting on cognitive performance and self-reported affect. journal of environmental psychology, 21(3), 291-299. [10] manjarrez, e., mendez, i., martinez, l., flores, a., & mirasso, c. r. (2007). effects of auditory noise on the psychophysical detection of visual signals: cross-modal stochastic resonance. neuroscience letters, 415(3), 231-236. [11] hasenfratz, m., michel, c., nil, r., & bättig, k. (1989). can smoking increase attention in rapid information processing during noise? electrocortical, physiological and behavioral effects. psychopharmacology, 98(1), 75-80. [12] posner, m. i. (1980). orienting of attention. quarterly journal of experimental psychology, 32(1), 3-25. [13] dosher, b. a., & lu, z.-l. (2000). noise exclusion in spatial attention. psychological science, 11(2), 139-146. adv syst sci appl 2018; 03; 39-78 published online at https://ijassa.ipu.ru/index.php/ijassa/article/view/645 original russian text © yu. v. mitrishkin, p. s. korenev, a. a. prokhorov, n. m. kartsev, m. i. patrov, e.a. pavlova, 2018, published in problemy upravleniya / control sciences, 2018, no. 2, pp. 2–30 plasma control in tokamaks. part. 2. plasma magnetic control systems yuri v. mitrishkin 1,2 *, nikolay m. kartsev2, evgeniya a. pavlova1, artem a. prohorov1,2, pavel s. korenev1,2 , m.i. patrov 3 1) m. v. lomonosov moscow state university, faculty of physics, moscow, russia 2) v. a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia 3) ioffe institute, russian academy of sciences, st. petersburg, russia abstract: various systems of magnetic control of plasma position, current, and shape are considered in vertically elongated tokamaks including spherical tokamaks being in operation. the systems of resistive wall modes suppression are described. tokamaks constructions and crosssections, structural schemes of plasma control systems are given; heed is paid to operating principles of plasma control systems; experimental results of plasma control in tokamaks are given. various engineering realizations of plasma magnetic control systems in tokamaks are presented. keywords: tokamak, plasma, plasma magnetic control, plasma position control, plasma current and shape control, resistive wall modes suppression, plasma control systems realization. introduction the first part [1] of the survey was devoted to the general problem of controllable thermonuclear fusion. it covers the key features of tokamaks and components of plasma control systems, describes the constructions of tokamaks. in particular, the diagnostics system of the globus-m tokamak is considered. the experimental data obtained from this tokamak were used in m.v. lomonosov moscow state university and v.a. trapeznikov institute of control sciences to develop original plasma position, current, and shape control systems. the second part of the survey considers plasma magnetic control systems (plasma is a completely ionized gas) in tokamaks, which are distributed dynamic plants with a complex structure and uncertainties subjected to uncontrollable disturbances. plasma in a magnetic field is not in thermodynamic equilibrium, and, as a consequence, is liable to various instabilities, which are the main cause of a relatively slow approaching of plasma parameters to the lawson criterion. the scientific direction related to research, development, and improvement of tokamaks has been significantly advanced in former soviet union under the leadership of academician l. a. artsimovich [2] and then has spread across the world [3, 4]. the first tokamaks had the circular cross section and were meant to wide-ranging high temperature plasma physics research. a general trend toward enlargement of such tokamaks was observed. these tokamaks included devices which were located in the i.v. kurchatov institute of atomic energy (moscow, russia): т-3, т-4, т-7, то-1, т-10 и т-15; in the a.f. ioffe institute (st. petersburg, russia): tuman-3 (toroidal device with plasma magnetic adiabatic heating); and also * corresponding author: yvm@mail.ru 40 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov et al. copyright ©2018 assa adv. in systems science and appl. (2018) foreign devices: plt, tftr (test fusion tokamak reactor) (usa), tore-supra (france), textor-94 (germany), ft-u (italy), etc. a specific feature of the next generation of tokamaks consists in the vertical elongation of the cross section. this feature enables to increase the plasma pressure and to raise the heating by the own plasma current [5]. a “price to pay” for these advantages is vertical instability of the plasma caused by the magnetic fields required for elongation. nevertheless, the vertically elongated tokamaks are now the main experimental base for researches on the problem of controlled thermonuclear fusion. these are jet (joint european torus), united kingdom, jt60u, japan, asdex upgrade, germany (max planck institute for plasma physics), diiid, c-mod, usa, tcv, switzerland, compass (compact assembly), czech republic; east (experimental advanced superconducting tokamak), china, kstar (korean superconducting tokamak reactor), south korea. the spherical tokamaks with a small aspect ratio have also appeared: mast (mega-amp spherical tokamak), united kingdom; nstx (national spherical torus experiment), usa; globus-m/m2, ioffe institute, russia. these tokamaks allow to even further increase in the gas kinetic plasma pressure at a specified magnetic field and can lead to an additional reduction in cost of a reactor as a result. plasma as an automatically controlled plant has the following typical characteristics, which create difficulties of both fundamental and technical nature: plasma is a distributed system with an infinite number of degrees of freedom; imperfection of the theoretical models and the fact that processes proceeding in a plasma have not been sufficiently explored lead to considerable uncertainties in the structure and parameters of plasma models; plasma is a non-stationary plant which means that plasma parameters may change by several orders of magnitude for a short period of time when plasma is produced and heated at one operational cycle or experiment; plasma can be non-minimal-phase plant, since the transfer functions on certain control channels, under the assumption that the parameters are replaced by the fixed values, may have both poles and zeros with a positive real part; plasma is also subjected to uncontrollable disturbances, which in some cases can be estimated in re al time by observing the inputs and outputs of the plant; plasma is a source of broadband noises which are not sufficiently explored. this leads to difficulties in plasma parameters identification; plasma is a nonlinear dynamic plant by its nature; large values of natural frequencies of plasma oscillations require high speed of response and significant power of control systems; actuators, which are generating the control signals sent to the plasma, may contain energy converters with nonlinear characteristics (often discontinuous), dead-zones, and transport delays. this makes it significantly difficult to design and analyze closed loop plasma control systems. the complexity of dynamics and nonlinearity of the actuators serve as an additional source of uncertainties when building a controlled plant (plasma in a tokamak) model; diagnostic units in thermonuclear devices, in many cases, have uncertainties when identifying plasma, which also provide contribution to the total uncertainty of plasma models. despite the presence of peculiarities of plasma, which characterize it as one of the most complex controlled plants in nature, the automatic feedback control systems started being used in the 1960s for plasma confinement in the magnetic traps, and then started to play the significant role in controlled thermonuclear fusion. the investigations along this direction were begun in 1967-1968 by doctors in physics and mathematics v.v. arsenin and v.a. chuyanov in experiments on the trap with magnetic mirrors ogra-2 in the i.v. kurchatov institute of atomic energy. in the experimental device ogra-2 the flute and ion-cyclotron (kinetic) plasma instabilities have been suppressed and then the control systems have obtained 41 plasma control in tokamaks. part 2. copyright ©2018 assa adv. in systems science and appl. (2018) a spread to suppress other instabilities: drift, ionization, helical instabilities in tokamaks, to stabilize -pinches etc. in [6] a detailed overview of the results of researches and experiments on suppressing plasma instabilities is presented. initially, the main control problem in tokamaks was to stabilize the position of a plasma ring along the major radius by poloidal (lying in a meridional plane) magnetic field. the first experiments aimed to solve this task were carried out on the to-1 tokamak using an impedance controller in 1971 jointly by the staff of the i.v. kurchatov institute of atomic energy (l.n. artemenkov, i.n. golovin, et al.) and the v.m. glushkov institute of cybernetics of academy of sciences of the ukraine soviet socialist republic (yu.i. samoilenko, v.f. gubarev et al.) [7]. combined plasma equilibrium control is used in modern tokamaks, namely: preprogrammed reference control provides a scenario and correction of the plasma location is performed by a feedback control system. these systems became standard components of tokamaks and found application for the joint stabilization of orthogonally decoupled stable horizontal and unstable vertical plasma position in tokamaks. then this approach has started to be used for plasma shape control by means of poloidal field coils and in fact the controlled plant started to belong to a class of multivariable systems, these include the plasma in iter [8, 9]. the use of automatic control methods to ensure the stability and equilibrium of plasmas in thermonuclear devises with magnetic confinement became a generally recognized necessity. experiments on tokamaks have shown that the main parameters of plasma, which directly provide creation of the conditions for initiation of thermonuclear reaction, are solely sensitive even to slight displacements of the external magnetic surface of plasma column in relation to the vacuum vessel or diaphragm limiting the column. therefore, the accuracy of the plasma equilibrium control enables to reduce the rate of harmful impurities inflow into the plasma as well as loss of particles that makes it possible to increase attainable values of plasma density, temperature, and energetic lifetime [2, 7-9]. the position of the plasma boundary is stabilized as close as possible to the first wall in order to ensure the efficient use of the internal space of the vacuum vessel and also to reduce the increments of unstable displacements of the vertically elongated plasma. the smallest failures in the plasma control system can cause the melting of the vacuum vessel. uncontrollable contacts of the plasma with the vacuum vessel lead to the plasma powerful energy release outward, and also result in excessive mechanical loads and damage of the thermonuclear device. these failures are totally unacceptable for the thermonuclear reactor. apart from the plasma magnetic control systems [8, 9], plasma kinetic control systems are also being developed [10, 11] that enables to control the profiles of plasma parameters: plasma current, safety factor, temperature, density, and pressure, and also the burning power during thermonuclear reaction. these systems are necessary for obtaining the most profitable (optimal) operation modes of future thermonuclear reactors. the research work aimed at developing and implementing of plasma control systems in the soviet union was commenced in the laboratory of professor, doctor of engineering sciences lev n. fitsner in v.a. trapeznikov institute of control sciences since 1973. the introduction of developed plasma control systems to physical experiments and their numerical investigation on the distributed parameter models of the plasma were done in cooperation with collaborators of the i.v. kurchatov institute of atomic energy (moscow), troitsk institute for innovation & fusion research (triniti) (troitsk, moscow region), a.f. ioffe physicaltechnical institute (st. petersburg), the d.v. efremov institute of electro-physical apparatus (st. petersburg). general requirements for plasma control systems in tokamaks and general design methodologies of new plasma control systems have not been developed so far. each scientific group who works on operating tokamak designs plasma control systems depending on device functionalities and the type of tasks to be solved. since tokamaks have different configurations of poloidal systems and energy resources of power supplies, different plasma control systems 42 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov et al. copyright ©2018 assa adv. in systems science and appl. (2018) are obtained. presently, at the international conference on decision and control (cdc) the sections devoted to plasma control systems have been organized on the initiative of american specialists. there are also the papers on that subject presented at the ifac world congress in 2014 (references are in the survey). the papers have been published by the v.a. trapeznikov institute of control sciences of the russian academy of sciences at the cdcs [12–18] and ifac congresses [19–21]. 1.control systems of plasma position in all operating tokamaks there are control systems of a plasma position, namely, stabilization systems of the plasma magnetic axis or the center of the plasma current in various implementations. several russian and foreign developments will be showed in greater depth. 1.1. the t-14, tuman-3 and tvd tokamaks (russia) in the early national projects a number of plasma position control systems were developed: t14 (tokamak with a strong field, triniti, troitsk), tuman-3 (toroidal installation with magnetic adiabatic heating, ioffe institute, st. petersburg), tvd (elongated tokamak with a divertor, kurchatov institute, moscow). control systems were developed for the t-14 and tuman-3 devices, modeled and implemented in practice of physical experiments on tuman-3: control system that evaluates and compensates the external disturbances during stabilization of a major radius of the plasma column [22-25], as well as an adaptive self-oscillating system for stabilization of the horizontal plasma position, that minimizes the amplitude of self-oscillations at each quasi-period under variable parameters of the controlled plant [26-30]. the additive perturbation and two variable parameters of the model were estimated on line by the adaptive kalman filter [31]. a two-loop orthogonally decoupled self-oscillating system for stabilization of the horizontal and vertical plasma positions with thyristor voltage inverters as actuators was developed and applied in experiments [32-35]. 1.2. the globus-m tokamak (russia) the block diagram of the stabilization system for the plasma vertical position of the globusm tokamak is shown in fig. 1 [36]. the control system includes an actuator based on a current inverter [37], loaded on the control coil of the horizontal field. the control coil generates a radial magnetic field that is proportional to the current passing through it, and has an impact on the plasma column (controlled plant) and shifts it in the vertical direction z. the feedback loop of the system is closed through a controller that implements a proportionally-differentiate control algorithm (see fig. 1 for designations) α(ε ε)contr du t   , where (t)=z(t)–zref(t) is the error, zref(t) is the vertical displacement of the plasma. the error  is formed as  ε p ref p r ref pzi z i k t z i     , where r(t) is the radial flux, ip is the plasma current, k is the proportional gain. the results of the work of this version of the stabilization system of the plasma vertical position are shown in fig. 2 (the case of plasma stabilization in the equatorial plane of the tokamak at zref = 0). it can be seen from the oscillogram r(t) that plasma vertical stabilization system holds the plasma in the equatorial plane (r=0) with good accuracy until the end of the discharge pulse. the calculated elongation k of the plasma, that is determined by the efit code, at the end of the discharge pulse equals to two: k = 2. 43 plasma control in tokamaks. part 2. copyright ©2018 assa adv. in systems science and appl. (2018) br ÷ u + ccci actuator pdcontroller ucontr plasma in tokamak × kψr z zref ip ɛ fig. 1. the block diagram of the stabilization system for the plasma vertical position of the globus-m tokamak: cc – control coil; ci – current inverter; ψr – radial flux [36]; br – induction of a radial magnetic field fig. 2. oscillograms ip(t), i(t), r(t): discharge pulse № 10446 [36] 1.3. the jet tokamak (united kindom) the jet (joint european torus) tokamak [38] is one of the largest operating machines in the world. its magnetic configuration is close to the architecture of the iter tokamak project. in the jet tokamak magnetic control system there is a plasma adaptive vertical stabilization system, which stabilizes the vertical plasma velocity about zero. it consists of three main subsystems: the vertical stabilization controller, the controller of the actuator current and the adaptive controller. four poloidal coils connected to a fast frfa amplifier, that are capable of switching between nine output voltage modes during time of the order of 200 μs according to the non-linear law with hysteresis zones, are used as an actuator. the measuring system (speed observer) calculates the plasma vertical speed on the basis of the ampere's law and the approximation of the vertical moment of the total current as a weighted sum of measurements of magnetic field outside the plasma [38–40]. the controller of vertical stabilization is a proportional unit. however, the behavior of the system closed by such controller will be similar to the case of a relay controller due to the presence of dead zones and hysteresis in the model of the actuator: as soon as the plasma velocity becomes greater than the threshold value, a voltage appears at the output of the actuator, a current creating a force acting on the plasma in the opposite direction increases rapidly in the coil. when the plasma velocity passes through zero, the voltage at the output of the actuator disappears. further, when the opposite threshold value is reached, the reverse voltage appears. thus, an oscillating process arises. the use of a single closed stabilization loop of the plasma vertical velocity around zero cannot guarantee the stability of the plasma in the vertical direction [40]. the process will oscillate around zero, however, a drift of the average values of the plasma vertical position and current in the control coils is possible, since the control is carried out by speed, and not directly ε 44 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov et al. copyright ©2018 assa adv. in systems science and appl. (2018) by position. the problem of vertical stabilization is solved jointly with the plasma shape control system, which gives a guaranteed stable closed control system. fig. 3. stabilization system of the vertical plasma velocity around zero on the jet tokamak: frfa is fast radial field amplifier [38] it is also necessary to avoid exceeding the current limit value in the control coil and the output of the frfa amplifier in a saturation mode. a slow pi controller, which stabilizes the current in the control coil around zero, is used to solve this problem. due to the non-minimal phase nature of the vertical plasma motion model [40] the gain in this loop is negative. the risk of overheating of the frfa amplifier and the release of heat on it depend on the switching frequency, these determine the limit of the permissible gain in the stabilization loop of speed around zero. this limitation is further enhanced by the fact that the vertical speed is calculated from the magnetic measurements, i.e. the sensors are connected to the field of the control coil current, which creates additional oscillations. the model of the vertical plasma velocity varies substantially during the discharge. in addition, a significant uncertainty of the model arises when linear approximation of the actuator, working on complex nonlinear principles, is done. thus, it is impossible to use one set of stationary parameters of the controllers throughout the whole discharge, so the adaptation of the parameters of the controller of vertical stabilization is used on the jet tokamak. there is an approximation of the dependence of the frfa switching frequency fsw on the gain kvs in the stabilization loop of the vertical velocity and on the module  of a single unstable pole of its linear model [38-40]: fsw=γf(kvs), where f is a monotone function. an adaptive controller operates on the basis of this ratio, leading away the frfa mode from saturation. the switching frequency of the frfa in the jet tokamak is about 500 hz, which ensures that there is no overheating and provides small amplitude of the plasma oscillations over the entire range of the module of the unstable pole γ of the linear vertical plasma motion model. 1.4. the east tokamak (china) stabilization of the vertical position and plasma velocity around zero on the east tokamak (experimental advanced superconducting tokamak) is given serious attention, since vertical instability is a risk for the operation of tokamaks. in the east, an algorithm called rzip [41] is used to control the current and position of the plasma. the block diagram of the plasma position and current control system on the limiter phase is presented in fig. 4. the value of the plasma current is measured by the rogowski loop, and the estimations of the vertical and horizontal positions of the magnetic axis are reconstructed from the signals of the magnetic diagnostics outside the plasma. the feedback is closed via pid controllers and matrices m_matrix(rp, zp) and m_matrix(ip) [41]. the control is carried out by means of currents in poloidal field coils, for which the target values consist of the sum of the scenario signals and signals of the closed-loop control system. the coils of the central solenoid are used 45 plasma control in tokamaks. part 2. copyright ©2018 assa adv. in systems science and appl. (2018) to control the plasma current, the vertical and horizontal positions of the plasma are controlled by the coils of the poloidal field. additional coils connected to high-speed power supplies and located inside the vacuum vessel are used to suppress the vertical plasma instability, as well as in the iter project. pf error east command to pf power supply pf current control pf measured current pf current trajectories м-matrix ip м-matrix rp, zp δpf1~ δpf14 δpf11~ δpf14 pid pid ip error + + ip target ip measured current command to ic power supplypidfilter rp, zp error zp error rp, zp target e-matrix rp, zp magnetic diagnostic data fast z control ic current + + fig. 4. the block diagram of the plasma current and position control system on the east tokamak on the limiter phase of the discharge [41] (ic is an active feedback coil inside the vacuum vessel) the results of tracking the vertical and horizontal position of the plasma in the east for the target in the form of triangular pulses in the real experiment, discharges 10112 and 10113 respectively, are shown in fig. 5. the control error for the vertical position does not exceed 1 mm. the control contour of the horizontal position tracks the target noticeably more slowly due to the effect of the field penetration into the conducting structures of the vessel during horizontal displacement. the control error does not exceed 2 mm after the end of the transient process. fig. 5. tracking the vertical and horizontal position of the plasma on the east tokamak [41] the italian create group and chinese specialists during the experimental campaign of 2016 on east developed and applied the stabilization system of the plasma vertical velocity around zero, that was frequency decoupled with the plasma current and shape control system [42]. the control directly of the plasma position was conducted using the plasma shape control system. the controller scheme of this system is shown in fig. 6, it is similar to the plasma control system used on the jet tokamak [38-40]. the vertical plasma position control system in the east tokamak is modeled on the tsc (tokamak simulation code, usa) plasma-physical code. it is claimed in [43] that the results of modeling on the main parameters, such as plasma current, plasma shape and position, flux contours and magnetic measurements, coincided well with the experimental data. 46 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov et al. copyright ©2018 assa adv. in systems science and appl. (2018) plasma shape and position control isoflux control mode rzip control mode pf coils current controllerplasma current controller 12 siso pidcontrollers ipf + iff m(·, ip)pid ipf, ref1 ipf, ref ip, ref ip + m(·, zp; rp) 2 siso pidcontrollers rp, ref zp, ref rp, zp + m(·, seg1÷13) 13 siso pidcontrollers δψcp m(·, zx1; rx1) 2 siso pidcontrollers rx1, ref zx1, ref rx1, zx1 + m(·, zx2; rx2) 2 siso pidcontrollers rx2, ref zx2, ref rx2, zx2 + m(·, sym)pidsymref sym + + ipf, ref2 control mode selection fig. 6. the simplified scheme of the position, current and plasma shape controller on the east tokamak: m matrices are used to distribute signals between 12 current control loops in poloidal field coils; the target currents are monitored by the current controller in pfc coils [42] 2.plasma position, current, and shape control systems all tokamaks given in table 2 of the first part of this survey [1] have a common similarity consisting in the vertical elongation, but all of them differ from each other by the poloidal systems, that are the magnetic coils systems creating poloidal fields. the jet tokamak has an iron core that other tokamaks do not have. all the rest of tokamaks have an air central solenoid. the iron core makes additional problems associated with its nonlinear magnetization curve. tokamaks diii-d, nstx, jt-60u, and tcv have the poloidal field coils inside the toroidal field coil; other tokamaks on the contrary have the poloidal field coils outside the toroidal field coil. the east and iter tokamaks have all superconductive coils. these tokamaks have the horizontal field coils inside the vacuum vessel in order to stabilize the unstable plasma vertical position, which significantly expands the area of controllability and stability on the vertical coordinate with limited power supply sources of these coils. in all tokamaks, the coils of the poloidal field are located differently in the space around the vacuum vessel. such difference of the poloidal systems of known tokamaks leads to the different configurations of the plasma position, current, and shape control systems. 2.1. diii-d tokamak (usa) a general view of the diii-d tokamak is shown in fig 7, a and the magnetic configuration of the plasma in it is shown in fig. 7, b [44]. on the diii-d tokamak the magnetic flux is controlled at 13 points on the separatrix (fig. 7, b). the block diagram of the magnetic plasma control in the diii-d tokamak is shown in fig. 8, a, fig. 8, b shows the result of tracking the upper and lower gaps between the separatrix and the first wall, as well as the x-point coordinates. 47 plasma control in tokamaks. part 2. copyright ©2018 assa adv. in systems science and appl. (2018) a b fig. 7. diii-d tokamak: а – general view (internet resource), б – poloidal flux contour lines (configuration with lower х-point) [43] a b fig. 8. plasma control on the diii-d tokamak [45]: a – block diagram of the isoflux-control at the 13 separatrix points (see fig. 7, b); rt – real time; b – system response to control of the upper and left gaps between the separatrix and the first wall and x-point position coordinates. 48 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov et al. copyright ©2018 assa adv. in systems science and appl. (2018) on the diii-d tokamak, the following ideology of developing plasma position, current, and shape control system is adopted [44]. the vertical plasma instability is suppressed through a separate circuit, which stabilizes the plasma vertical velocity around zero (see fig. 8, a). a set of coils is used to control the plasma current. the 18 poloidal field f-coils are used to control the plasma shape. the isoflux-control method is applied: magnetic flux at 13 points on the plasma separatrix, as well as vertical and horizontal position of the x-point are calculated in real time using signals of magnetic diagnostics of currents, fluxes and fields by the efit algorithm. the error signals between the specified (reference) values of magnetic flux on the separatrix and the x-point coordinates are fed to the multivariable isoflux controller (mimo controller). the isoflux controller also receives signals of currents in the control coils. the output signals of the controller come to the inputs of the actuators, thereby closing the multivariable feedback loop. the multivariable plasma shape controller itself is designed on the basis of linear models, which are generated by a special software package named toksys. the set of points is given, which determines the desired location of the plasma separatrix and in which control will be carried out. currents in the poloidal field coils are tuned so as to keep equal magnetic poloidal flux at the x-point and at all the other boundary points. let the difference between the flux at the controlled points and the given flux references at these points be defined as δψ. the relation between the change of currents in the control coils δi and δψ can be represented as i = m–1. here m–1 is a control matrix, which is the inverse matrix to the matrix m composed of the values of the green's function for the grad–shafranov equation. the elements of the matrix m are poloidal flux values at each control point while control currents equal to unit. 2.2.asdex upgrade tokamak (germany) a general view of the asdex upgrade tokamak (modernized axially symmetric divertor experiment) is shown in fig. 9, a, and its cross section is shown in fig. 9, b. the control of the vertical and horizontal position of the plasma, as well as the coordinates of the divertor strike-points is carried out on this tokamak (fig. 10, 11). 2.3.jet (uk) tokamak on the jet tokamak (fig. 12) various control modes are used in conjunction with the vertical stabilization of the plasma, they are the horizontal position of the plasma, the currents in the poloidal fields coils, the plasma current, the coordinates of the divertor strike-points control modes, and the control mode of the plasma shape by the gaps between the first wall and the separatrix [38]. the structure of the controller in jet is founded on physical principles and based on the kirchhoff’s vector equation pf pf pf di v m ri dt   , where vpf and ipf are voltage vectors on the coils and the measured currents in them, m and r are matrices of mutual inductance and resistance, respectively, control of currents in the poloidal field coils is carried out in absolute values. not all coils are used for different control modes, but the most efficient individual coils or sets of coils. control modes can be combined: for example, most discharges can be performed with simultaneous use of modes of the currents in the poloidal field coils control, the plasma current control, the divertor strike-points coordinates control, and plasma shape control by values of the gaps. the control system has a cascade structure [49] shown in fig. 13, where there are controllers for plasma shape, current, and position of divertor strike-points in the block "shape and plasma current controller". 49 plasma control in tokamaks. part 2. copyright ©2018 assa adv. in systems science and appl. (2018) a b fig. 9. tokamak asdex upgrade: a – general view [46]; b –cross section with poloidal field coils [47]; the control coil for the vertical position of the plasma is placed between the vacuum vessel and the toroidal winding 50 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov et al. copyright ©2018 assa adv. in systems science and appl. (2018) references position shape current controllercontroller controller ffc ffc asdex upgrade experiment plasma equilibrium determination ffc icol iv ioh load compensation fig. 10. block diagram of the magnetic plasma control system of the asdex upgrade tokamak [48] fig. 11. plasma position and divertor strike-points location control response on the asdex upgrade tokamak [48] the general control law for controllers [38] is vpf = rest ipf + k (yref – y), where rest is the estimation of the resistance matrix, k is the matrix gain, yref and y are the vectors of the reference inputs and the measured outputs. in the control law structure there is the compensation element rest ipf of the voltage on the resistance rest that makes the behavior of the closed-loop control system of currents in jet coils close to the devices with superconducting coils and eliminates the need to introduce integral components for increasing the degree of astatism and achieve the required tracking error 1-2% [38]. the matrix gain is synthesized in the form of k = h*(mest t –1c –1), where h is the matrix of inputs/outputs choice, mest is the estimation of the mutual inductance matrix used for decoupling of the currents in the poloidal field coils, t is the static gain matrix linking the variations of the currents in the coils with variations of output variables: y = t ipf, and c is the matrix of the desired values of time constants in the closed-loop system. the plasma model in the vessel is taken into account as a coil with a distributed current. in doing so, the plasma resistance and mutual inductance of the plasma with all coils [50], except the coil p1, are neglected, that is justified by the slow dynamics of the plasma current. in practice, only during disruption the plasma can induce a significant voltage on the poloidal field coils. the values of magnetic fluxes at the points of the equatorial plane on the outer and inner circumference are used for the plasma horizontal position control mode. since the plasma separatrix is a line of the equal flux, it is enough to make the difference of fluxes in these points equal to zero by the coil p4 in order to stabilize the horizontal position (fig. 12, b, 14, a). 51 plasma control in tokamaks. part 2. copyright ©2018 assa adv. in systems science and appl. (2018) а b fig. 12. jet tokamak: a – general view (internet resource), b – cross-sectional view with control coils and central solenoid [38] shape controller system pf current controller ipf (t) + iff (t) shape and plasma current controller iref (t) plasma current and shape references + + plasma / circuits system vpf (t) vertical stabilization system + + zp (t), irfa (t) . ip (t), shape(t) fig. 13. block diagram of the plasma position, shape, and current on jet [49] the xloc algorithm [51] is used to calculate the gaps values between the vessel first wall and the plasma separatrix in the mode of plasma shape control. the gaps response to the test signals in the closed-loop control system on the jet tokamak is shown in fig 14, b when a discharge duration of at least 2 seconds. the poloidal field coils d2 and d3 are used to control the locations of the divertor strikepoints (see fig. 12, b). the signals of location of the intersection points of the separatrix with two vertical or horizontal axes (the specific option is set by the operator) are used as feedback signals. an oscillating mode of control for the location of the divertor strike-points is used for energy scattering, which allows to distribute the energy over a larger area of the divertor. it is implemented by introducing the reference saw-edged signal at frequency of 4 hz while the control loop bandwidth has a value of 10 hz. 52 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov et al. copyright ©2018 assa adv. in systems science and appl. (2018) a b fig. 14. tokamak jet: a – vertical cross-section, b – gaps response to the test signals [38] 2.4.tcv tokamak (switzerland) the control method of the plasma shape in the tcv tokamak (variable configuration tokamak) [52] by the values of the magnetic flux and field at certain points on the plasma separatrix is illustrated in fig. 15 [53]. the method is based on the fact that the separatrix is a line of a constant magnetic flux, and the x-point in the divertor configuration is the point of zero field. thus, having chosen the plasma shape scenario in advance, it is necessary to align the flux values at the selected points, reducing the field to zero at the expected x-point. a b fig. 15. tokamak tcv: a – cross-section; b – block diagram of the plasma shape control by flux and field values on the separatrix of the tcv tokamak [53] 53 plasma control in tokamaks. part 2. copyright ©2018 assa adv. in systems science and appl. (2018) the control scheme (fig. 15, b) consists of four main elements. one of them is the element k, it serves to calculate the deviations of the magnetic flux and field on the separatrix from the scenario values as a linear combination of the measurements of the sensors outside the plasma. another element is a block of pid controllers, one for each deviation signal, which reduces the values of controlled signals to zero. the next element 1m  calculates the derivatives of currents in the poloidal field coils, in which the current deviations will be reduced to zero in the finite time. this calculation is carried out for the model of magnetic configuration without plasma and currents in the passive structures of the vessel. finally, the element l calculates on the basis of the kirchhoff’s equations the required values of the voltages on the poloidal field coils as a function of the currents and their derivatives, as well as the voltage on the plasma ohmic heating coil. the given algorithm does not take into account the model of the plasma in the vessel of the tokamak and, therefore, can be used throughout the discharge; however, it does not allow achieving a high degree of decoupling of control channels. this approach was applied to the kstar tokamak [54], and its modified versions are being used in diii-d, east and alcator c-mod [55]. a b fig. 16. tokamak east: a – construction; b – cross-section [42] 2.5.east tokamak (china) the complete reduced copy of the iter is the chinese east tokamak with superconducting coils (sc) (fig. 16), in which there are control coils inside the tokamak vessel. the system of plasma shape magnetic control for the east, that was developed jointly by the chinese and american specialists, is one of the most advanced. the two algorithms for the plasma shape control on the isoflux-control principle (control of the magnetic flux and field values at certain points on the plasma separatrix) are used. these 54 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov et al. copyright ©2018 assa adv. in systems science and appl. (2018) algorithms are iso-elong and iso-dnull to control the elongated limiter configuration and the divertor configuration with two x-points, respectively. fig. 17 shows the basic block diagram of the control system for these algorithms. pf error eastcommand send pf current control pf measured current pf current trajectories pid pid error + + control points x-point error ip, zp ip, zp target e-matrix magnetic diagnostic data ic current м-matrix magnetic diagnostic data rtefit parameter calculation shape trajectories fig. 17. block diagram of the plasma current, shape, and vertical position control systems on the east tokamak for the divertor discharge phase (isoflux control): rtefit (real time equilibrium fitting) [41] a b fig. 18. plasma shape and position stabilization on the east tokamak for discharge 10618: a – magnetic configuration of east at the moment of 4.958 s; b – tracking errors at the reference points and the x-points [41] the positions of characteristic points on the separatrix, such as x-points or points of contact with the limiter, are calculated during the plasma shape reconstruction. the maximum value of the magnetic flux from the characteristic points is selected as the reference for all points. then, the differences of the magnetic flux values are used to close the feedback through the pid controllers (see fig. 17). one or two nearest poloidal field coils control the value at each 55 plasma control in tokamaks. part 2. copyright ©2018 assa adv. in systems science and appl. (2018) reference point, the plasma current is controlled by the central solenoid coils. the reference currents in the coils are calculated using the matrices of the control channel decoupling, scenario signals are added to them, and they are fed to the multivariable control loop of the currents in the poloidal field coils. the typical value of the unstable pole module on the east tokamak is 200-300 s–1; it can exceed 1000 s–1 for configurations with greater elongation or non-standard plasma parameters. it is necessary to quickly calculate the vertical position of the plasma and use high-speed power supplies as actuators in order to suppress vertical instability with such dynamics. an analogue of the rzip algorithm is used in the control system to calculate the estimation of the plasma vertical position, which is controlled by individual coils located inside the vessel. fig. 18, b shows the signals for discharge 10618 [41] with an elongation of 2.0 and a plasma current of up to 250 ka. algorithm iso-dnull starts working at the moment of 2.7 s and by 3.0 s the differences of values of the magnetic flux are less than 0.001 v·s/rad, and the error of xpoints position control is less than 1 cm. 3.spherical tokamaks 3.1.advantages of spherical tokamaks for successful operation of a thermonuclear reactor the plasma parameters must satisfy the numerous constraints imposed by the magnetohydrodynamics theory. one of the most important constraints is the maximum achievable β which characterizes the efficiency of plasma confinement and is defined as the ratio of the plasma pressure to the magnetic field pressure. numerical calculations [56] show that as the aspect ratio a decreases, the maximum acceptable β increases as 1/ a for conventional tokamaks ( 3a  ) and faster for spherical tokamaks ( 1.5a  ). thus, the best plasma confinement can be achieved on spherical tokamaks. another advantage of spherical tokamaks is the high value of the safety factor q at the plasma boundary. the safety factor increases as 2 3/21/ (1 )a a with decreasing aspect ratio, allowing to suppress the kink instability and enabling higher plasma current than on conventional tokamaks with the same magnetic field strength and small radius of the plasma. finally, relatively small size of the spherical tokamaks makes their creation and operation less costly, and allows for higher values of magnetic and electric fields to be attained with the same currents as in conventional tokamaks. these advantages mark the spherical tokamaks as promising candidates to be the future commercial thermonuclear power plants, which gives more relevance and significance to the plasma control problems on spherical tokamaks. in this context, the results of plasma control for the operating spherical tokamaks mast, nstx, and globus-m are given below. 3.2.mast-u tokamak (uk) the cross-section and structure of the mast-u tokamak (mega ampere spherical tokamak) are shown in fig. 19. the vertical position control on mast-u is implemented via the pd controller [57], while not the plasma vertical coordinate is taken as controlled variable but the product of plasma vertical coordinate and its current (fig. 20, a). like on east and diii-d tokamaks, the isoflux control method is applied for the shape control on the mast-u tokamak, with the real-time rtefit plasma equilibrium reconstruction code calculating flux values at control points [59]. 56 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov et al. copyright ©2018 assa adv. in systems science and appl. (2018) a b fig. 19. mast-u tokamak: a – tokamak structure, b – cross-section [57] a b fig. 20. plasma shape and position stabilization on mast-u: a – vertical position control [56]; b – inner and outer plasma radii control, x-point position control [59] 57 plasma control in tokamaks. part 2. copyright ©2018 assa adv. in systems science and appl. (2018) a b fig. 21. nstx-u tokamak: a – structure; b – cross-section [58] 3.3.nstx-u tokamak (usa) the structure and cross-section of the nstx-u tokamak (national spherical torus experiment) are shown in fig. 21. the isoflux control method [58] is used to control the plasma shape on the nstx-u tokamak. magnetic fluxes in the set of control points are calculated in real time using the rtefit code and via pid controllers are equated to a given flux that is determined by the flux at the x-point or the flux at the contact point of the plasma with the vessel. a b fig. 22. plasma shape stabilization on nstx-u [58]: a – resulting gaps on the outer side of the plasma; b –xpoint position control 58 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov et al. copyright ©2018 assa adv. in systems science and appl. (2018) pid controllers are also used to control the position of x-points and strike points. a control contour that minimizes the horizontal distance between the flux isolines of the upper and lower x-points is used for configurations with two x-points. fig. 22a shows the resulting outputs of the control system for gaps between control points on the outer side of the plasma and the walls of the vessel, the result of the x-point position control is shown in fig. 22, b. 3.4.globus-m tokamak (russia) spherical tokamak globus-m (a.f. ioffe physical-technical institute of the russian academy of sciences, st. petersburg) [1] is the only tokamak with a vertically elongated cross-section in the russian federation, fig. 23 [60, 61]. feedback systems are applied on the globus-m tokamak to control the horizontal and vertical position of the plasma with high-speed thyristor current inverters as actuators [37]. there is a set of poloidal field coils on the tokamak that is incorporated into the control loops of currents in these coils with multiphase thyristor rectifiers [72] and pd-controllers. so these loops enable to use program control to control plasma magnetic surfaces in each discharge. that has made it possible to collect a database of plasma discharges, which was used to develop and simulate hierarchical control systems for the position, current, and shape of the plasma with the equilibrium reconstruction codes in the feedback. two codes for the plasma equilibrium reconstruction from magnetic measurements outside the plasma have been developed in the matlab environment: one using picard iterations to solve the grad-shafranov equation by means of the green's functions [18, 62] and one using non-iterative moving filaments method to approximate the plasma current distribution [21]. the plasma equilibria reconstructed from the experimental data were used to construct arrays of linear plasma models from which linear models with variable parameters were created by means of linear interpolation [18, 21]. a b c fig. 23. globus-m tokamak: a – cross-section: – limiter; – magnetic loops; – vacuum vessel; – pf coils; – central solenoid; b – magnetic configuration reconstructed by the moving filaments method [21]; c – magnetic configuration reconstructed by the picard iterations method with the measuring displacements directions of gaps between the tokamak first wall and separatrix[18, 58] systems with time-varying robust h∞ controllers with switching for isoflux control (fig. 24) [21] and linear interpolation for controlling gaps between the first wall and separatrix (fig. 25) [18] were designed for these time-varying plasma models and modeled, using a new technique with simultaneous application of the linear models, experimental scenario signals, and reconstruction codes in the feedback [18, 21, 63, 64]. the combination of the moving filaments reconstruction code with the flux control on the separatrix showed the highest 59 plasma control in tokamaks. part 2. copyright ©2018 assa adv. in systems science and appl. (2018) computational speed of the system with the possibility of implementing it in real time on the globus-m tokamak using the speedgoat industrial computers (https://www.speedgoat.com) with the simulinkrt operating system from mathworks [21]. lpv model of plasma in globus-m tokamak current inverter current inverter thyristor rectifiers analog controller of the plasma vertical position analog controller of the plasma horizontal position pidcontrollers δz δr ihfc ivfc uhfc uvfc δupf δipf, δics δip switching robust mimo controller kmimo(s) br, bz, δψb ipf, ics, ip,ψ 0 plasma equilibrium reconstruction algorithm ipf, ics, ip, ψ δψ matrix exchanging at tk moment x(tk) vector state matching x(t) controller matrix database hierarchical level of controller switching pid-controllers of plasma current 0 0 0 a b fig. 24. time-varying system with controllers switching and moving filaments equilibrium reconstruction code for poloidal flux on separatrix and magnetic field in x-point control on the globus-m tokamak: a – block-diagram with lpv (linear parameter varying) model, b – deviations δb of magnetic field in x-point and deviations δψ of the poloidal flux difference between points on the separatrix during the shift from limiter to divertor phase of the discharge [21]. lpv model of plasma in globus-m tokamak δz δr ihfc ivfc uhfc uvfc δupf δipf, δics δip h∞ lpv gap g controller δg ipf, ics, ip, ψ δψ δgref _ δg switch s kmimo pid-controller of plasma current thyristor rectifiers mimo pidcontroller current inverter current inverter analog controller of the plasma vertical position analog controller of the plasma horizontal position plasma equilibrium reconstruction code ipf, ics, ip,ψ 00 0 0 a b fig. 25. system with the time-varying controller based on linear interpolation of controllers array and picard iterations equilibrium reconstruction code for gaps control on the globus-m tokamak: a – block diagram, b – deviations of gaps between the plasma separatrix and vacuum vessel [18] it is worth noting that the poloidal flux control in vacuum vessel without plasma was studied prior to the development of shape control system for the globus-m tokamak [64]. 60 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov et al. copyright ©2018 assa adv. in systems science and appl. (2018) stabilization of the plasma vertical position with fast contour is considered in [16], where the method for the plasma vertical position adaptation to its shape is proposed. this method was applied to the globus-m tokamak [65]. the block diagram of this application is shown in fig. 26, and the operation of the plasma position, shape and current control system is shown in fig. 27. fig. 26. block diagram of the hierarchical control system for the plasma position, current and shape with the adaptation of the vertical position of the magnetic axis on the globus-m tokamak [61] fig. 27. modeling results for plasma control system on globus-m tokamak: a – plasma vertical position deviation; b – gaps deviation; c – plasma current variation [61] 3.5.t-15m tokamak (russia) during the developing of plasma control system for the t-15m tokamak (national research center “kurchatov institute”) a part of the work was carried out at the v.a. trapeznikov institute of control sciences of the russian academy of sciences [66]. an important result of the conducted investigations is the transfer of the horizontal magnetic field coil, intended for control of an unstable vertical position of the plasma, from the position outside of the toroidal field coil to the position between the vacuum vessel and the toroidal field coil [17]. this is due to the fact that the horizontal field coil in its initial position was shielded by the neighboring poloidal field coils needed for the plasma shape control. because of that the control system acquires the internal instability property. this meant that the plasma position control system required unlimited increase of the current in the horizontal field coil to stabilize the vertical position of the plasma under the minor disruptions. this instability made the system, and, consequently, the entire plant, unworkable. this property of the system was discovered during its development and modeling [67, 68]. while the coil was transferred close to the vacuum 61 plasma control in tokamaks. part 2. copyright ©2018 assa adv. in systems science and appl. (2018) vessel, but inside the toroidal field coil, the control system became able to achieve internal stability [69] and operability with the unstable plant under control. the model of the vertical plasma motion in the t-15 tokamak for the poloidal system with the transferred horizontal field coil [71] was obtained by the identification of the plasmaphysical code dina (src rf triniti) [70]. after that the plasma vertical position control systems with different actuators have been developed: a multiphase thyristor rectifier and a transistor voltage inverter [72]. in the first case, a modal control system was synthesized with pole placement of the closed-loop system at single point in negative real part of the complex plane for maximum convenience of system tuning, in the second case the system was put into a sliding mode, and revealed its weaker robust properties. a system with an adaptive predictive model for the variable parameter of the plasma model was developed [73], as well as the system with the placement of the poles of the system in the lmi (linear matrix inequalities) -regions to reject the minor disruptions perturbations was designed. the latter system showed the advantages over the modal system in terms of rejection of the external disturbance and accuracy [74]. it has been shown that the plasma shape is controlled in the t-15 tokamak without the use of the horizontal field coil. with the multivariable controller with a state estimator [75] the shape control requires only poloidal field coils and the sections of the central solenoid, like in the iter version of 1995-1997 [9]. for the t-15 tokamak, the plasma shape control system was modeled on the dina code with stabilization of the vertical plasma velocity around zero and the lqg controller in the feedback [76]. 4.magnetic control systems of resistive wall modes let us consider the resistive wall modes (rwm) and methods of their suppression [77]. the general trend seems to be toward an increase of the plasma parameter β = 2μ0<p> / b0 2 (<p> is the average pressure, b0 is the toroidal magnetic field, μ0 is the vacuum permeability) and normalized parameter βn = aβ / (b0 ip) (iр is the plasma current, a is the minor radius) in modern tokamaks. the instability related to the rwm is one of the factors, which limits the growth of  and often leads to disruption of plasma discharge. thus, the low toroidal modes n = 1, n is a toroidal wave number, which might arise with the growing pressure, are of particular interest from the viewpoint of control systems design (the modes of the form ( )( ) i t m nr e     , m is the a poloidal wave number, are considered when examining the rwm, see § 5 of part 1 of this survey [1]). the tokamak conductive wall located close enough to the plasma could provide rwm modes suppression. however, it may only merely slow down the growth of the modes, since the wall with a finite conductivity could suppress the mode only for a certain time roughly the same as the time of magnetic field penetration into the wall. therefore, these modes are called the resistive wall modes. the following system of equations describes the dynamics of rwm using a simple cylindrical model approximation and the assumption of a rigid mode structure and ignoring the effect of plasma rotation [79]: 0, 0, , eff p pw w pc c wp p w w wc c w w cp p cw w c c c c c l i m i m i m i l i m i r i m i m i l i r i v            where , , p w ci i i are the current on the plasma surface, the current in surrounding passive (wall) structures, and the control coil current, respectively, abm represents the mutual inductance between conductors a and b, ar is the resistance in conductor a, al is the self inductance of conductor a, , { , , }a b p w c , effl is the effective self inductance, cv is the 62 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov et al. copyright ©2018 assa adv. in systems science and appl. (2018) voltage applied to the control coil. the model has been obtained for a tokamak where the control coils are inside the vacuum vessel (fig. 28). the degree of interactions between currents on the plasma surface, in the surrounding wall, and control coil are characterized by mutual inductances abm . plasma vessel wallcontrol current in coil mpc mwc mpw fig. 28. cross section of a cylindrical model of the rwm dynamics [78] the interconnection of the plasma rotation and the appearance of rwm is not well studied, however, the model for describing this phenomenon was derived in [80] using an argument based on the exchange of energy between the plasma mode and external conductors:   ,iw p pw ww i d b c b    where , p wb b are changes in the field at the plasma surface and at the vessel wall when the plasma is rotating;  is the plasma toroidal rotation frequency; 1 pw pwс m  ; iww is the value of the coupling of rwm energy transferred through the field pb to the toroidally in phase component of the field wb ; φd represents the energy coupled to the component wb that is 90o toroidally advanced, d represents dissipation mechanism. there are two primary approaches to suppressing the rwm. the first one is that of applying feedback magnetic control systems in order to suppress the system’s instabilities. the growth rate of oscillations decreases by an order of magnitude because of the tokamak vessel conductive wall, making it possible virtually to realize the feedback control by the control coils. the second approach is that of rotation stabilization of the plasma. in modern tokamaks a beam of neutral atoms is injected into the system, which transmits an angular momentum that is sufficient for keeping the plasma rotation, which in turn results in rwm suppression. the combination of these two approaches to control the rwm is considered as the most prospective for future thermonuclear reactors. the rwm control systems [81] for the diii-d tokamak (fig. 29) [82] were created using different design methods of controllers. the feedback control underlies in most of the proposed methods. the structural scheme of the closed loop control system is presented in fig. 30. the pd [83], lqg [83], h∞ncf [84], dk [84], etc. controllers are used as controllers in the feedback loop. the functional   0 ( ) t tj u x qx u ru dt    , where 0, 0 q r  are weighting matrices, is minimized when designing an lqg optimal controller (u = –kx is the control law, x is the state). the kalman filter (the observer) was applied to estimate the state [83]. dk-iteration also can be applied to synthesize the controller that is a combination of the н∞-synthesis and μ-analysis. in this case the controller k meets the criterion  1min min ( ) k d dn k d  , where 63 plasma control in tokamaks. part 2. copyright ©2018 assa adv. in systems science and appl. (2018) n(k) is the transfer matrix of the closed loop system,  is the set of matrices commuting with  ( i   , ( ) 1   ), d d   , where  is the maximum singular value of the matrix. the results of numerical simulation when the rwm growth rate changes are presented in fig. 31. it is shown that the lqg controller effectively suppresses the rwm and allows to increase the stability margin of the system [83, 84]. fig. 29. diii-d tokamak [78] plasma model reference outputfilter controller actuator fig. 30. structural scheme of the rwm control system fig. 31. the outputs of the rwm control system of the diii-d tokamak when the rwm growth rate changes [83,84] the tokamak diii-d construction provides the neutral beam injectors (fig. 32) which allow giving additional energy and momentum to the plasma in the direction opposite to its movement, or transmitting up to 10 mw of power without giving any additional momentum to the plasma [85]. the possibility of decoupling of the channels of transmission of power and momentum makes it possible to suppress the rwm and to slow down the plasma rotation considerably at .n n no wall   , where .n no wall is the value of the n coefficient without the wall. the high values of n are also achieved by two series of control coils, located inside (icoils) and outside (c-coils) of the vacuum vessel. fig. 33, a-e shows that the rwm increase without the i-coils feedback (i-coils are switched off), which results in the plasma disruption. however, the rwm are successfully suppressed in case of the i-coils feedback (i-coils are switched on) (fig. 33, f-j). 64 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov et al. copyright ©2018 assa adv. in systems science and appl. (2018) fig. 32. the location of the neutral beam injectors in the diii-d tokamak [85] fig. 33. a comparison of systems: а – e –without feedback control system (i-coils are switched off); f – j – with feedback control system (icoils are switched on) [85] on the other tokamaks, such as nstx [82], iter [83, 84] et al., similar principles are being applied to control the rwm taking into account their specific design features. 5.implementation of plasma control systems 5.1.real time test beds for tokamaks currently, the real time test beds are applied in various fields of engineering, industry, and science that allow to model control systems in real time, enable them to be tuned and debugged and then to be switched to a real controlled plant [89]. such an approach is also applied to the tokamaks as real controlled plants (fig. 34) [90]. for instance, this approach was applied to the tuman-3 tokamak and was based on the analog-digital controllers and analog plant model [9, p. 204, fig. 5.16]. a controlling computer on the diii-d [91] and east [92] tokamaks enables switching from the plant model to the tokamak or vice versa through a corresponding switch. this allows saving time during physical experiment and thoroughly tuning plasma control systems in real time on the tokamak plasma models. tokamak model of the tokamak controller switch k1 k21 2 1 2 fig. 34. the concept of the real time computer test bed: к1 и к2 are switches from the plant model to the tokamak and vice versa [9] http://context.reverso.net/%d0%bf%d0%b5%d1%80%d0%b5%d0%b2%d0%be%d0%b4/%d0%b0%d0%bd%d0%b3%d0%bb%d0%b8%d0%b9%d1%81%d0%ba%d0%b8%d0%b9-%d1%80%d1%83%d1%81%d1%81%d0%ba%d0%b8%d0%b9/design+features 65 plasma control in tokamaks. part 2. copyright ©2018 assa adv. in systems science and appl. (2018) 5.2.software realization of plasma shape control system in the jet tokamak the interface of the plasma shape control system in the jet tokamak [38] enables the user to divide the time of the experiment into segments, referred to as time windows, where it is possible to specify different combinations of the control regimes. for each time window the user may activate the specific controllers and set the programmed signals to track them. the user also can choose a shape control (sc) scenario containing the prepared set of controllers and programmed signals, the interface will adjust the signals shape for the specified initial and final values. the xsc scenarios became available when the extreme shape controller [93] system had been implemented, allowing to specify the cross section of the plasma visually and interactively update the programmed currents and gaps signals. the difference between the defined values for adjacent windows may cause substantial spikes of signals during the transition. to exclude this effect, the interface automatically introduces the additional smoothing time windows for the smooth transition from measured values at the end of the first window to defined values at the beginning of the next window. measurement => physics real-time diagnostic (time synchronization) evaluator ip, pos, shape, β, q, w, prad, regimedirect input · scaling · offset correction 1 2 control 3 i monitor: · operation limits, · generator speed feedback controller: · plasma current; · position; · shape; · confinement; · radiation; · profiles iv 4 5 reference generator: · waveforms interpolator; · intelligent waveform generator iiipulse scheduler: · waveforms; · transition rules segment scheduler: · watchdog; · conditional branching ii v command => actuator actuator adaptor: · load; · distribution; · modulation command output: · clipping; · scaling; · offsetting; · packaging vi diagnostic systems i actuator systems vi coils heating fuelling protection time event generators fig. 35. general scheme of the digital plasma control system for the asdex upgrade tokamak [94] 5.3.plasma control system for the asdex upgrade tokamak the digital dcs (discharge control system) for the asdex upgrade tokamak [94] (fig. 35) comprises the following elements: diagnostic system i, which includes the system of direct input 1, the real-time diagnostic 2 for time synchronization of the system, and evaluator for all measuring outputs 3, the control unit iv, that contains monitor, checking the operation limits and generator speed 4, and feedback control algorithms 5, which allow controlling the outputs of the system through actuators vi, pulse scheduler ii, reference generator iii, and segment scheduler v, containing watchdog and conditional branching. the plasma current, position, and shape control systems, plasma magnetic confinement system, radiation and plasma profiles control systems are integrated into the dcs. the control actions are formed on the 66 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov et al. copyright ©2018 assa adv. in systems science and appl. (2018) basis of the data of the diagnostic system. the control signals are fed to the actuators (control coils, additional heating system, fueling system, protection system, timing system of the events, generators), furthermore, a preliminary load, distribution and modulation and subsequently clipping, scaling, offsetting, and packaging of signals are carried out. the dcs provide the functional capabilities that allow one to guide the plasma discharges, coordinate measuring and actuating devices, and optimize the plasma parameters in the tokamak. 5.4.plasma control system of the tcv tokamak the closed-loop control system for the tcv (tokamak à configuration variable) [95] tokamak is presented in fig 36. the output signals from tcv tokamak (currents in coils, plasma density, magnetic characteristics, x-ray, etc.) come to the diagnostic system 1, and then are archived in the tcv database, and come to the input of the control system. the control signals are formed in blocks scd (système de contrôle distribué) 2 and hybrid control system 3, and are outputted to the programmable adder/switch 4, and then through actuators 5 (electron cyclotron resonance heating (ecrh), electron cyclotron current drive (eccd), toroidal (tf) and poloidal (pf) field coils, gas valve) are fed to the tokamak. the development of control systems for the tcv tokamak is carried out in a special interface of the scd host computer, that has access to the tcv database, can receive and transmit signals to the control unit, and realizes cooperation among all blocks of the digital system through the tcvpc computer. scd hybrid control system + programmable adder / switch actuators: ecrh/eccd, tf coils, pf coils, gas valve. diagnostics: coil currents, magnetics, density, x-ray, etc. archived in tcv database timer cards: trigger signals, clock signals. signals to program hardware tcvpc computer plant control & supervision shot-prepare parameters scd state, timer parameters scd host computer shot-prepare parameters post-shot data тсv database post-shot data algorithm code + waveforms signals to program hardware from tcvpc 5 1 3 4 2 fig 36. general scheme of the plasma control system for the tcv tokamak [95]. 67 plasma control in tokamaks. part 2. copyright ©2018 assa adv. in systems science and appl. (2018) 6.approaches to development of magnetic control systems for plasma in tokamaks the review of magnetic control systems that have been undertaken makes it possible to classify them into different groups depending on approaches to the plasma vertical position and shape control. there are two approaches to plasma vertical movement control. the stabilization of the plasma vertical speed around zero is the first one. this approach is being applied to many devices such as jet, diii-d, east, kstar, nstx, iter project, where there are a wide range of plasma shape control systems. the stabilization of plasma vertical speed around zero makes it possible to avoid a contradiction with the plasma shape control problem, however, at the same time the system is not strictly stable and may have low stability margins. this approach also leads to the necessity of switching the controller in the stabilization loop of an unstable plant, since the plasma position stabilization system have to be applied before turning on the plasma shape control system (the stabilization of the plasma vertical speed around zero can be used only in combination with the plasma shape control system). the stabilization of plasma vertical position is the second one. the plasma vertical position control is applied directly during the limiter phase of the discharge on the asdex upgrade, jt-60sa, globus-m, etc. in some cases, there is a possibility to use control of the plasma shape during the divertor phase of the discharge without using the additional control loops for an unstable vertical position of plasma or plasma vertical speed around zero. such technical solution was used in an earlier draft version of the iter-1998 project [9] and during the analysis of such opportunity using mathematical modeling for the t-15 tokamak [75] that is under construction. the values of controllable parameters in real time, which are impossible to obtain by direct measurement, are necessary to solve the plasma shape control problem. the parameters of plasma shape may be reconstructed from the magnetic measurements outside the plasma by indirect methods, which are, in fact, a complex computational task [1]. today, there are a variety of approaches, using the values of different parameters of plasma, which are applied for plasma shape control in modern devices with a vertically elongated magnetic configuration. the plasma shape control using the values of gaps between the separatrix and the first wall (gap control). this approach is being used in the jet, asdex upgrade tokamaks, iter project. the control of the gaps between the separatrix and the first wall (vacuum vessel or blanket) has direct physical significance since indeed the safety parameters, namely the distances between the plasma boundary and the first wall, are controlled. the disadvantages of this approach include the computational complexity of the algorithms of plasma equilibrium reconstruction and the difficulty of their implementation in real time. the plasma shape control using the values of the magnetic flux in a set of points on the separatrix (isoflux control) is used in the diii-d, tcv, east, kstar tokamaks. this approach lessens the computational complexity compared to the previous one since it applies the control on the basis of indirect parameters and the full plasma shape reconstruction is not required. conclusion control of plasma dynamics is one of the central fundamental problems of the theoretical and experimental study of thermonuclear fusion and the project of transition to thermonuclear energy. however, the control methods, especially with respect to the internal plasma parameters (see later the fourth part of this survey), remain insufficiently developed due to the complexity of the construction and the variety of tokamaks, the need to use complex mathematical models, and the solution of complicated ill-posed problems in plasma diagnostics (see the first part [1] of the review), the development of voluminous science68 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov et al. copyright ©2018 assa adv. in systems science and appl. (2018) intensive software and the use of high-performance computing. in practice, this leads to a long and costly work on the experimental selection of control system parameters and a large number of premature discharge disruptions during research and development campaigns. therefore, the survey, systematization and classification of the real control systems for toroidal plasma are important and relevant. the given overview of the magnetic control systems of the position, shape, and current of the plasma on modern vertically elongated tokamaks, also including spherical tokamaks, shows that the problem of selecting (developing) an efficient and reliable structure of the plasma magnetic control system has not been finally solved. in the world practice, there are competing approaches to vertical stabilization of the plasma and various approaches to controlling the shape of the plasma, which have their advantages and disadvantages that cause their use on specific devices. hence, in particular, there is a lack of standards for the development of plasma control systems in tokamaks. nevertheless, it is possible to note the main trends both in the development of the tokamaks themselves and in the systems of magnetic control of the plasma in them. the vertically elongated tokamaks develop in the direction of decreasing the aspect ratio, which leads to spherical tokamaks, that have better physical and technical characteristics for the purposes of creation of future thermonuclear power plants based on tokamaks. in this regard, the work on the development, research, optimization, and simplification of the engineering implementation of plasma control systems is especially relevant for spherical tokamaks, in particular, primarily for the russian spherical globus-m2 tokamak, since it exceeds in the plasma parameters the known foreign analogs like mast and nstx (see subsection 3.4). on the other hand, the analysis of plasma magnetic control systems done allows to mark out the general features of the configurations of such systems, namely: multi input multi output (mimo);; multiple-loop; cascade control; hierarchy; robustness; adaptability. а tendency has been outlined for the multivariable control of the plasma shape and current to decouple the control channels and to use the simplest pid controllers in them (see subsections 1.3, 1.4, 2.3-2.5). it should be noted that in addition to control systems for the plasma position, current, and shape, the development of control systems for the resistive wall modes takes place as well, which must be suppressed when the plasma density increases by means of special additional coils and feedback control. similar properties and trends may appear in these systems. acknowledgements this work was supported by the russian science foundation (rsf), grant no 17-1901022 (sections 2-6) and the russian foundation for basic research (rfbr), grant no 17-0800293 (introduction, section 1). references 1. mitrishkin, y.v., korenev, p.s., prokhorov, a.a., kartsev, n.m., et. al. 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(2018) advances in systems science and applications (2014) vol.14 no.4 361-377 evaluation of users’ behavior using adaptive fuzzy prediction technique joan m., john, a. and shajin nargunam computer science & engineering department, noorul islam university, tamilnadu, india-629180 abstract user behavior is a considerable determinant of a products ecological blow as manufacturing advances authorize enlarged effectiveness of product purpose, the consumers alternatives and habits ultimately enclose a major outcome on possessions utilized by the product. a variety of design methods developed in diverse contexts propose chances for engineers and other stakeholders functioning in the ground of sustainable improvement to involve consumers actions at communication with the result or system, in result building the user more competent. approaches to varying users actions from varied amount of fields are evaluated and discussed. but it fails to predict the user behaviors accurately. in this work, adaptive fuzzy prediction technique is presented to interpret and predict the users behavior towards product demand, quantity of products, features of the product/services, annual sales with the multi-clustered web usage data. the fuzzy model works on the feedback pattern to arrive at predictive feature of the web usage data value moments, across different time periods, and adapt to the recent consumer behavior and liking of the products and service. experimentation conduct with real data sets extracted from web log files collected from websites. performance evaluation is made to show the effectiveness of the proposed technique. keywords users behavior, product, behavior-patterns, feedback pattern, multiclustered data, adaptive fuzzy prediction technique. 1 introduction web data mining is a type of approaches that proficiently processes the tasks of providing the required information from the internet, enhancing the web site framework to offer better internet service quality and identifying the informative expertise from the internet for enhanced web appliances. usually, web data mining are of three types, web content, web structure and, web usage mining. in addition to identifying the requirements of the customers, organization also require to recognize what motivates them to buy, and how can processes the purchasing procedures to make sure that the products or services are on the purchasing list. understanding the customers will assist, to enhance and provide the product, in addition to obtaining the right price crossing points and enhancing successful promotional behaviors. the psychology of the purchasing procedures has been greatly noted and no matter what size organization business, proce362 joan m.: evaluation of users’ behavior using adaptive fuzzy prediction technique dures that assist company turn into be more successful. both organizations and consumers provide patterns of purchasing actions. the business representation is less process to debate as the business consumers will almost dominantly encompass some generalized procedures of buying in place. the organization task is to recognize the procedure and match the marketing behaviors to the diverse stages of the procedure. this means that the consumer will obtain the right type of contact at the precise time. to process insights into areas for instance indicators of consumer defection, crossing points sensitivity, clustering, and consumer needs examination, to process a few. most marketers identify the value of sensing consumer data, but also act as the endeavors of leveraging this expertise to produce intelligent, proactive routes rear to the consumer. data mining approaches and techniques for identifying and monitoring patterns among data assists businesses sift during layers of seemingly unspecified data for purposeful relationships, where they could processes, rather than merely route to, consumer requirements. defined and precise predictive representations are very significant in viewing schemes. the requirement for novel techniques and attitudes in representing calculation and vulnerability are prejudiced by the current proceeds in soft computing also the problematic accurateness and inapplicability to person calculation of formerly required after geometric analysis methods. thus founding accurate analytical representations become gradually harder for multivariable projecting representations. conventionally, such troubles have been lectured by geometric logistic deterioration methods for binary reliant variables. therefore, it can be accomplished that the web is an assorted, energetic and amorphous data depository, which gives huge quantity of valuable information and also demonstrates the difficulty of managing enormous amount of information from the diverse perspectives, users, web repair providers, commerce analysts. the efficient search tools are very greatly essential to recognize significant and practical information accurately. the growth has enthused the web service suppliers to predict the users web usage behaviors so that, they can • personalize the information offered to them • construct the websites more users sociable • decrease the traffic load • make or adapt their website to outfit diverse group of people in this work, predict the users behaviors based on their selection over product demand, quantity of products, features of the product/services, annual sales, etc., by employing the proposed adaptive fuzzy prediction technique. advances in systems science and applications (2014) vol.14 no.4 363 2 related work world wide web is an enormous depository of web pages and associations. it presents great quantity information for the internet users. data mining has developed as a ground of essential and functional study in computer science. in [1], an adaptive technique for predicting cancer susceptibility was designed by combining fuzzy concept with statistical logistic regression which was tested on cancer dataset. as to the authors suggestion regarding the generalization of the algorithm into prediction problem involving other types of intrinsic linear functions, and adaptive fuzzy prediction technique is used to evaluate users behavior. the development of web is unbelievable as it can be observed in current days. users admittance is traced in web logs. from the users viewpoint, it is very complicated to haul out practical knowledge from the enormous quantity of information and secondly, it is also hard to take out for the users to admit pertinent information professionally. web logs extracted and purpose of this exposition is to assess, suggest and progress the exercise of a quantity of the current approaches, architectures and web mining methods (gathering individual information from consumers) are the earnings of operating data mining techniques to persuade and take out helpful information from web information and service where data mining has been practiced in the pastures of e-commerce and e-business (that funds users actions) [2]. one method to solve such problem is the application of web usage mining as in [3]. the evaluation of users behavior using adaptive fuzzy prediction technique deploys the multi-clustered web usage data in order to derive the behavior pattern. a developing clustering algorithm is planned for cluster production. clusters are recognized and customized supported on restraint measure of mapping consistence and companionable dimension. auspiciously, online social networks and social standard have completed it simple for users to point to whom they reliant and whom they achieve not. nevertheless, this does not resolve the crisis since every user is only possible to recognize an insignificant portion of other users; the author must encompass techniques for deducing reliant and disbelieve among users who do not recognize one another. in [4], the authors present a novel technique for calculating both reliant and non-reliant (i.e., optimistic and pessimistic trust). the authors supply to this region a novel algorithm for efficiently forecasting reliance and mistrust in web-based communal systems. the authors in [5] unite a path possibility reliance inference algorithm with a new method employing spring-embedding to deduce system distance. the product of the classifiers is evaluated with offered techniques which exercise considerably diverse techniques. when a user needs admin contact, a communal conversation page is associate for users to converse and cast 364 joan m.: evaluation of users’ behavior using adaptive fuzzy prediction technique your vote on whether to disclose the mediator. optimistic and pessimistic votes are calculated as optimistic and pessimistic trust ratings [6]. the methods in [7] have exposed an enormous potential for humanizing classification accuracy. this revision is concerned with the assessment of training data allotment and its strength on the arrangement of frequent classifier schemes. temporal data clustering presents establishment techniques for shaping the intrinsic association and concentrating information in excess of chronological data. the author in [8] calculated techniques for communication seek out in associated, high-dimensional data sets, which are resultant surrounded by a clustering organization. the author takes you back that indexing by vector processes (va-file), which was anticipated as a way to match the disfigurement of dimensionality, uses scalar qualification, and so fundamentally ignores dependences obliquely magnitude, which represents an origin of sub-optimality [9]. researches on position-based service (lbs) have been promising in modern years remaining to a large cycle of probable applications [10]. one of the vital subjects is the removal and computation of mobile actions and associated communication. feature clustering is a scheming method to reduce the dimensionality of distinctive vectors for content tagging. in this paper, the author in [11] suggests fuzzy similarity-based self-constructing practices for characteristic grouping [12]. the paper [13] urbanized a novel resolution structure for adaptive production response fuzzy direct schemes aimed at efficiently selling with nonlinear schemes with numerous input-multiple production (mimo) setback and in the attendance of dynamics and normal function suspicions. a multiple delay fuzzy scheme calculation representation is consequent and its scheme possessions are elucidated. such a calculation representation allows the exercise of a model-based technique for fuzzy control. the paper [14] investigated the possibility of concerning a comparatively new neural system method, i.e., great learning mechanism (elm), to understand a neuro-fuzzy takagi-sugeno-kang (tsk) fuzzy supposition scheme. the tsk technique is an enhanced report of the standard neuro-fuzzy tsk fuzzy deduction scheme. at the equivalent time, the consequential fraction of the fuzzy rules is attained by numerous elms. finally, the estimated calculation value is dogged by a load calculation system. in [15], the author are disturbed with a technique for building quantum-based adaptive neuro-fuzzy systems (qanfns) by means of a takagi-sugeno-kang (tsk) fuzzy category supported on the fuzzy granulation from a specified effort production data set. for this reason, the author urbanized a methodical technique in creating habitual fuzzy rules supported on fuzzy subtractive quantum grouping. the clustering method is not only an addition of thoughts intrinsic to scale-space and support-vector grouping but also symbolizes an efficient sample that shows advances in systems science and applications (2014) vol.14 no.4 365 definite uniqueness of the objective scheme to be represented from the fuzzy subtractive technique. through the precedent few years, the primary deduction for corporeal repossession is being anticipated employing regression methods from sounder comments. the current system employs fuzzy reason and data clustering to institute an association among replicated sounder comments and impressive outlines. this association is more reinforced employing the adaptive neuro-fuzzy supposition scheme (anfis) by fine-tuning the offered fuzzy-rule stand [16]. the paper [17] presents an appliance of fuzzy-logic methods to the reversible firmness of grayscale images. with suggestion to a spatial degree of difference pulse code intonation (dpcm) system, calculation might be skilled in a space-varying manner moreover as adaptive, i.e., with predictors recalculated at every pixel, or as confidential, in which image chunks or pixels are tagged in a number of lessons. a novel method to the devise and employ of inferential sensors in the procedure trade is proposed in [18], which is supported on the lately introduced idea of developing fuzzy models (efms). they speak to the confront that the current procedure industry faces nowadays, that is, to expand such adaptive and selfcalibrating online inferential sensors that decrease the preservation costs while charging the elevated accuracy and interpretability/transparency. kalman filter (kf) is the mainly regularly employed assessment method for incorporating signals from short-term elevated presentation schemes, similar to inertial directionfinding schemes (inss), with position systems showing long-term constancy, similar to the comprehensive positioning scheme (gps) [19]. the paper [20] presents completion consequences using lately pioneered discrete-time adaptive calculation and direct methods employing online task approximators. a fuzzy time series [21] has been functional to the calculation of employment, hotness, stock directories, and other areas. associated studies mostly center on three issues, specifically, the separation of conversation, the pleased of forecasting system, and the techniques of defuzzification, all of which really control the calculation accurateness of forecasting representations. the presentation of an adaptive neurofuzzy assumption scheme (anfis) considerably falls when improbability lives in the data or scheme process. prediction period (pis) can enumerate the indecision connected with anfis point forecasts. the paper [22] first nears a method to acclimatize the delta method for the creation of pis for conclusions of the anfis representations. in [23], a developing fuzzy system (efs) is urbanized for scheme condition forecasting. 366 joan m.: evaluation of users’ behavior using adaptive fuzzy prediction technique 3 proposed adaptive fuzzy prediction technique to evaluate users behavior the process of identifying the users behavior based on their demands over the products and its quality is done by accomplishing the adaptive fuzzy prediction technique. the adaptive fuzzy prediction model is used to predict the users behaviors efficiently across different time periods and adapt to the recent user behavior and liking of the products and service. here the adaptive fuzzy prediction model is combined with the representation of fuzzy assumption with user behavior predictor. the fuzzy prediction technique is applied to the multi-clustered web usage data. the architecture diagram of the proposed evaluation of users behavior using adaptive fuzzy prediction technique [eubafp] is shown in fig. 1. from the fig.1, it is being observed that the web log file data is given as input to the system to estimate the performance. before fuzzification, the web log file data is being pre-processed with the corresponding method and normalized to the self-organizing maps. with these, web usage data was multi-clustered in an appropriate manner based on the similarity based clustering. with the web usage multi-clustered data, fuzzy prediction model is applied to classify the users data based on their products demands over the market. the fuzzy inference model will form a set of rules at first as a base for classifying the rules. the adaptive fuzzy prediction model of the website could be processed in the constructing phase under which it collects web browsing log. the browsing behaviors of the users are modeled and compared with the representation of the prediction model to improve the browsing performance of the users. fig.1 architecture diagram of the proposed eubafp 3.1 design of adpative fuzzy prediction technique fig.2 shows the block diagram of an adaptive fuzzy prediction technique. it comprises of two subsystems: a fuzzy prediction system, and behavior predictor. advances in systems science and applications (2014) vol.14 no.4 367 at first, in the fuzzy prediction model, the users in the web log data file are clustered based on their behaviors. the fuzzy prediction system assumes the possibility of the active time of the user in the cluster based on the behaviors of the user. next, in order to identify the active time of the user in the group, behavior predictor is used. the behavior predictor identifies the possibility of number of users visited the web page. the expected possibility information is employed in web usage data management to handle users’ behavior. fig.2 adaptive fuzzy prediction technique the browsing behaviors of users are predicted based on the prediction model. since the browsing behaviors are identified, the predictable patterns should be combined or the prediction pattern should be built, correspondingly. the browsing behavior of a user is compared with all matching visited web pages in prediction model for expecting the future traveling path. the design of the subsystems is given below. normally, there is a greater chance of the people to process the inaccurate information obtained from the web usage data. at first, before deriving the set of fuzzy rules, the clustering process is done based on users, session and the web pages they viewed. after forming the user groups, fuzzy inference system is presented. the fuzzy prediction follows the fuzzy ifthen rules for identifying the user behaviors. the fuzzy prediction model to identify the user behavior is processed based on subsequent steps. • choose appropriate input and output variables. • form a group based on the web pages, users and sessions they entered • select a appropriate kind of fuzzy prediction method • design a set of fuzzy if-then rules (information base) • predict the behavior of the user 368 joan m.: evaluation of users’ behavior using adaptive fuzzy prediction technique fuzzy prediction-based user behavior. the fuzzy prediction subsystem is a unique authority scheme. it utilizes an information support, articulated in terms of fuzzy implication rules, and a suitable implication engine to identify the active time of the user in the respective group. the facts based on the identification of the users behavior is considered as, • identify the relation between the user visited web pages and their behaviors • predict the rules based on their behaviors the primary step is to obtain the inputs and decide the amount to which they fit in to each of the suitable fuzzy sets by means of association functions. after the efforts are fuzzified, we recognize the extent to which every part of the visited web pages are used for every rule. the fuzzy rules are then formed once the groups are generated, based on user, web pages visited and sessions. in each set of fuzzy rule, the subsequent steps have to be followed. • assign each rule contain a binary weight (0 and 1). • consider weight of each rule is 0 which has no outcome of the behavior of the user. • assign appropriate weight to each of the rule formed to identify the behavior of the user. • apply inference technique to the corresponding rules design process of fuzzy prediction-based user behavior.the design process of fuzzy prediction-based user behavior comprises of two steps. • fuzzification • fuzzy assumption engine: a. fuzzy implication rules; b. fuzzy implication engine. in the first step of fuzzy prediction-based user behavior, fuzzification is applied where numerical value is converted to qualitative value based on the user input called as linguistic variables. these linguistic variables are explained using membership function which has a value between zero and one. let linguistic variable lm,j be the received input from the user at time tm with probability pm,i. for our study, three linguistic variables called triangular membership function are selected and stored in ulm,j which consists of elements, short (s), normal (n), and large (l) given as follows. triangle(y, s,n,l) =  0, y < s y − s/n − s, s ≤ y ≤ n l− y/l−n,n ≤ y ≤ l 0, l ≤ y (1) after the fuzzification process is completed the results are fed to fuzzy assumption engine which comprises of fuzzy implication rules and fuzzy implication engine. the fuzzy implication rules consist of a set of if-then statements known as linguistic rules which describes the behavior of the user for particular set of inputs advances in systems science and applications (2014) vol.14 no.4 369 at a given time. the rule set for deriving user behavior using fuzzy assumption engine for ith rule is given below: ri = { f (lm,0isa0,i) and (lm,1isa1,i) and (lm,2isa2,i) then (pm,0ispo,i) and (pm,2isp1,i) and (pm,2isp2,i) (2) where i = 1,2,...,i, is the total number of fuzzy rules for predicting user behavior using fuzzy adaptive preference techqnique. (lm,0, lm,1, lm,2) ∈ ulm,0, ..., and (pm,0, pm,1, pm,2 ∈ upm,0,.... denote linguistic variables, (a0,ia1,ia2,i, po,ip1,ip2,i) denotes the fuzzy set present in ulm,0 and upm,0 respectively. the fuzzy implication engine then evaluates the set of rules to compute a qualitative output result and matches with the visited web page which generates nine rules for analyzing the user behavior. the procedure below describes the process of the proposed adaptive fuzzy prediction technique shown below: input:set of parameters like product demand, quantity of products, features of products step 1: recognize the parameters that finest suits the fuzzy system for inputs, positions, and the outputs step 2: divide the discourse or the period extent by every variable step 3: significance into a number of fuzzy subsets step 4: conveying all a linguistic label step 5: assign or decide an association utility for every fuzzy subset step 6: determine the fuzzy relationships among the inputs, states fuzzy subsets step 7: determine the fuzzy subsets for outputs step 8: form the rule base step 9: standardize the constraint variables step 10: select suitable scaling features for the input and output variables step 11: fuzzify the inputs to the checker step 12: use fuzzy logic to deduce the output contributed from every rule step 13: amassed the fuzzy outputs suggested by every rule step 14: concern defuzzification to shape a crisp output the above procedure determined the process of the fuzzy sets to identify the set of users behavior based on their activities over the demand and qualities of products. the next section will describe about the experimental evaluation of the proposed scheme. 4 experimental evaluation experimentations are conduct at all the web usage data obtained from web server logs with real data sets extracted from websites. performance evaluation is 370 joan m.: evaluation of users’ behavior using adaptive fuzzy prediction technique made to show the effectiveness of the proposed evaluation of users behavior using adaptive fuzzy prediction technique [eubafp] compared to the existing one referred from different contributors of web usage mining. a sample set of data has been taken for experimental evaluation by taking five sets of attributes. the attributes used here are product demand, quantity of products, features of the product/services, annual sales, etc., to identify the behavior of the user based on their actions. table 1 web server logs dataset details the attributes are analyzed with the multi-clustered web usage data generated in the first phase. the predictive feature of the web usage data value is generated based on the fuzzy model across different time periods. the performance of the proposed evaluation of users behavior using adaptive fuzzy prediction technique [eubafp] is measured in terms of • implementation cost • prediction accuracy • execution time implementation cost.implementation cost for adaptive fuzzy prediction technique refers to the cost incurred building the procedure based on the number of users in clustered parts, pattern, and running, testing, and building essential changes. the implementation cost is evaluated using the formula with simulated time (st) and the simulation time (snt), to predict the user behavior for t interval of time is derived below: advances in systems science and applications (2014) vol.14 no.4 371 ic (n) = snt (n) /st (3) prediction accuracy.prediction accuracy defines the accuracy rate for predicting the users behavior based on adaptive fuzzy prediction technique towards product demand, quantity of products, features of the product/services, annual sales. the prediction accuracy is evaluated using the formula given below: pm,j ∗ = n ∗ pm,j 1 + (n − 1) ∗ pm,j (4) where n denotes the total number of test conducted for a specific user at varied interval of time and pm,j denotes the reliability of adaptive fuzzy prediction technique. execution time.the parameter execution time using adaptive fuzzy prediction technique refers to the time taken to execute, towards predicting the behavior of users product demand in network environment. exect ime = (insncount)i ∗ (clockcyclet ime)i (5) where execution is evaluated by the products of number of instructions to be executed for n users (i=1,2,3..,n) and clock cycle time for the specific ith user (i = 1,2,3...,n). 5 results and discussion in this work, we have seen that the proposed technique identified the behavior of the user based on their activities present in the network environment. the predictive feature of the web usage data are identified based on the fuzzy prediction model across different time periods. the below table and graph describes the performance of the proposed technique and compared the results with the existing adaptive fuzzy regression model (afr) [1] and user navigation pattern discovery using fast adaptive neuro-fuzzy inference system (unpd) for mining user profiles referred from different contributor s of web usage mining. fig. 4 describes the implementation cost required to handle the fuzzy inference systems based on the number of users in the clustered parts. the implementation cost for the fuzzy inference system in the proposed eubafp is low in the sense that adaptive fuzzy prediction technique is denoted by binary weighted model that results in less implementation cost than compared to the other two models afr and unpd. when the number of rules generated is small, its implementation cost is also less. as a result, the eubafp proposed system is realistic even when the number of users is large with a variance of 50-60% when compared to afr. 372 joan m.: evaluation of users’ behavior using adaptive fuzzy prediction technique table 2 number of users vs. implementation cost. the implementation cost required to build the fuzzy inference system based on the number of users present no. of users implementation cost proposed eubafp existing afr existing unpd 5 7 25 28 10 11 30 34 15 15 38 40 20 19 40 43 25 22 46 49 30 26 50 53 35 30 54 58 fig.4 no. of users vs. implementation cost table 3 number of users vs. prediction accuracy. the behaviors of users are identified by implementing the fuzzy inference systems by diverse set of approaches no. of users prediction accuracy (%) proposed eubafp existing afr existing unpd 5 65 55 40 10 70 62 45 15 72 64 51 20 78 68 54 25 80 70 59 30 82 71 62 35 84 74 64 advances in systems science and applications (2014) vol.14 no.4 373 fig.5 describes the prediction accuracy of users behavior based on the number of users present in it. from the figure it is evident that an increase in number of users also results in increased rate of prediction accuracy. when compared to the other two techniques afr and unpd, the proposed adaptive fuzzy prediction technique as shown in figure 5, achieves higher prediction accuracy. the higher prediction accuracy in adaptive fuzzy prediction technique is due to combining of both fuzzy prediction system and behavior predictor model which results in variance of 10-15% when compared to afr. fig.5 number of users vs. prediction accuracy the proposed eubafp construction presents the benefit of execution time. compared to the existing works like adaptive fuzzy regression model (afr) [1] and user navigation pattern discovery using fast adaptive neuro-fuzzy inference system (unpd), the proposed eubafp provides an improvement in execution time in predicting users behavior. fig. 6 describes the efficiency of fuzzy prediction systems in terms of execution to identify the behavior of users in the network environment. rather than using other existing works for user behaviors, the proposed eubafp achieves better efficiency in predicting the users behavior in terms of time with a variance of 1015% when compared to two other methods, using defuzzificaiton, center of area. as eubafp consumes less execution time and proceed with a set of rules, the efficiency in predicting the user product demands is also being high. finally it is being observed that the proposed eubafp performs well in terms of prediction accuracy and efficiency with less implementation cost. 374 joan m.: evaluation of users’ behavior using adaptive fuzzy prediction technique table 4 number of users vs. efficiency. the efficiency of the fuzzy prediction in terms of execution time for diverse set of approaches is illustrated no. of users execution time (sec) proposed eubafp existing afr existing unpd 5 32 40 50 10 40 43 55 15 46 52 65 20 50 58 70 25 55 65 75 30 60 70 80 35 62 75 85 fig.6 number of users vs. execution time 6 conclusion an adaptive fuzzy prediction system is developed to foresee the possibilities that the behavior user will be dynamic in the close by clusters at prospect instants employing the real-time dimension data established at the user. the merits of the adaptive fuzzy prediction system recline in • simplicity it is a one-way build-up method that does not need timeconsuming on-line accessing, • usefulness the possibilities are serious for balancing proficient operation of the web usage data of users, and • low cost the expected possibilities are attained supported on the appropriate demands of the user profiles. experimental results reveal that the performance of the adaptive fuzzy predicadvances in systems science and applications (2014) vol.14 no.4 375 tion system depends on the extent of behaviors of the user based on their product demands, and on the accessibility of information which confines the number of probable behaviors of the user. taking into report that the overall inference accuracy can be appreciably increased with value multiplexing, the fuzzy prediction scheme presents a fine resolution to getting users information. references [1] pankaj nagar and sumit srivastava, (2008), “adaptive fuzzy regression model for the prediction of dichotomous response variables using cancer data: a case study”, journal on applied mathematics, statistics and 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[14] sun zl, au kf, choi tm , (2007), “a neuro-fuzzy inference system through integration of fuzzy logic and extreme learning machines”, ieee transactions on systems, man, and cybernetics, part b: cybernetics. [15] kim ss, kwak kc, (2010), “development of quantum-based adaptive neuro-fuzzy networks”, ieee transactions on systems, man, and cybernetics, part b: cybernetics. [16] ajil, k.s. , lulea, sweden ,shukla, m.v. , pal, p.k., (2010), “a new technique for temperature and humidity profile retrieval from infraredsounder observations using the adaptive neuro-fuzzy inference system”, ieee transactions on geoscience and remote sensing. [17] aiazzi, b. , baronti, s. , (2002), “fuzzy logic-based matching pursuits for lossless predictive coding of still images”, ieee transactions on fuzzy systems. [18] angelov, p. , kordon, a., (2010), “adaptive inferential sensors based on evolving fuzzy models”, ieee transactions on systems, man, and cybernetics, part b: cybernetics. 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[23] wang, w. , vrbanek, j., (2008), “an evolving fuzzy predictor for industrial applications”, ieee transactions on fuzzy systems. corresponding author author can be contacted at: joanmjohn123@yahoo.co.in adv syst sci appl 2020; 04; 27-35 published online at https://ijassa.ipu.ru. determining the vertical force when steering nguyen tuan anh1*, hoang thang binh2 1) automotive engineering department, thuyloi university, 175 tay son, dong da, hanoi, vietnam e-mail: anhngtu@tlu.edu.vn 2) school of transportation engineering, hanoi university of science and technology, 1 dai co viet, hai ba trung, hanoi, vietnam e-mail: binh.hoangthang@hust.edu.vn abstract: when the vehicles move at high velocity and quickly steer, the vehicles can be rollover. the first sign of this phenomenon is that two wheels on the same side are completely separated from the road surface. the typical value for this sign is the vertical force fz at each wheel. if this value gradually decreases to zero, the wheel runs the risk of separating from the road surface that may lead to rollover. therefore, the value of the vertical force fz in different conditions needs to be determined to detect the imminent limit of the rollover phenomenon. this research established the spatial dynamic model of the vehicle to determine the vertical force fz based on the simulation method. besides, the equation describing the relationship between vertical force fz, velocity v, and acceleration steering  corresponding to the value of steering angle  is also established by the calculation process. from this equation, the value of the vertical force fz can be simply calculated with relatively high accuracy based on determining conditions. the results of this research are the basis for determining and establishing the vehicle's rollover limit. keywords: dynamic vehicle model, vertical force, function, steering acceleration, rollover 1. introduction 1.1. the instability problem of the vehicle automobiles are a popular vehicle in everyday life. however, the number of vehicle accidents is also very large. when the vehicles move at high velocity and quickly steer, the vehicles often encounter the situation: side slip or rollover. the side slip phenomenon usually occurs when the vehicles go on slippery roads, wheels are not able to contact the road surface. when the side slip occurs, the driver could not control the trajectory and direction of the motion of the vehicles. the sideslip problem is usually less dangerous than the rollover problem. the rollover phenomenon occurs when two wheels on the same side are completely separated from the road surface. if there is only one wheel separated from the road surface, the vehicle will be in an unstable state and at risk of rollover [15, 17]. the main cause of this phenomenon is due to the lateral acceleration ay appeared suddenly when quickly steering, especially in the case of fishhook steering [11, 18]. the parameter that warns the risk of rollover is the vertical force at wheels fz. if this value is sufficiently close to zero, the wheel runs the risk of separating from the road surface. therefore, if it is possible to determine the vertical force value fz, the timing of the rollover of the vehicle can be predicted. some solutions have been proposed to improve this situation such as the use of the active stabilizer bar, the electronic power steering, the active suspension, the electronic stability * corresponding author: anhngtu@tlu.edu.vn 28 n.t. anh, h.t. binh copyright ©2020 assa. adv. in systems science and appl. (2020) program, [etc.] [1, 16, 19]. the above systems use parameters of input signals such as the lateral acceleration ay, the roll angle , the vertical force at wheels fz [5, 20]. 1.2. literature review the topic of "rollover vehicle" has been researched by many authors. several authors have introduced a rollover index r [6, 8, 9, 10, 14]. this index is determined based on the difference in the vertical force of the wheels. if |r| <1, the car operates stably. in the case of |r| = 1, both wheels on one side are separated from the road surface, the vehicle rollovers completely. however, this index only really makes sense in case both wheels of the same side are separated from the road surface. if only one of the wheels is lifted off the road surface, the rollover index r cannot be determined. besides, the value of lateral acceleration ay is often used to determine the rollover threshold of the vehicles [3, 7, 13]. the value of the lateral acceleration ay can be calculated and simulated through link equations or experiments. however, this value is only really meaningful in case of ignoring the influence of the dimensions of the vehicles. some other studies use the roll angle of the vehicle  to determine the vehicle's rollover limited [2, 4, 12]. the value of the roll angle of the vehicles  is calculated by the lateral acceleration ay, which includes the influence of the dimensions. to accurately determine the limits of the vehicle instability, it is necessary to find the time when the wheels are separated from the road surface (fz = 0). this research focuses on identifying the limits at which the wheels are separated from the road surface. at the same time, the research also established the equation showing the relationship between vertical force fz, steering angle , and steering acceleration . therefore, it is easy to determine the vertical force fz at different times and conditions. 2. methodology 2.1. nomenclature : pitch angle, rad : roll angle, rad : steering angle, rad : yaw angle, rad ay: lateral acceleration, m/s2 b: half of the base width, m cij: damping coefficient, ns/m fcij: force of the damper, n fkij: force of the spring, n fktij: force of the tire, n fxij: longitudinal force, n fyij: lateral force, n fzij: vertical force, n h: distance from center to roll axis, m iz: moment of inertia of the z-axis, kgm2 ix: moment of inertia of the x-axis, kgm2 kij: stiffness of the spring, n/m ktij: stiffness of the tire, n/m l1: distance from the center to the front axle, m l2: distance from the center to the rear axle, m m: sprung mass, kg mij: unsprung mass, kg determining the vertical force when steering 29 copyright ©2020 assa. adv. in systems science and appl. (2020) uij: bump on the road, m vx: longitudinal velocity, m/s vy: lateral velocity, m/s z: displacement of the sprung mass, m zij: displacement of the unsprung mass, m 2.2. double-track dynamic vehicle model the motion of the vehicle is set based on the model of 10 degrees of freedom. the doubletrack model is set as below (fig. 2.1). assuming that the vehicle is moving at a constant velocity, the steering angle is small. ignore the influence of other factors. the motion determination matrix is described as follows [2]: 1 2 z ij i; j = 1 y xy1 y2 2 2 z i;j = 1 l1 im + m (v ψ) = (f f ) v (ψ 0) l1 im + m ij                 (2.1) where: y1 y11 y12f = f + f y2 y21 y22f = f + f fig. 2.1. the double-track model 2.3. spatial dynamic model 7 dof to determine the oscillation of the vehicle, it is necessary to establish the spatial dynamic model with 7 degrees of freedom as fig. 2.2. 30 n.t. anh, h.t. binh copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 2.2. dynamic vehicle model 7 dof the matrix of vehicle oscillation is given as follows [2]: 22 yx 1 2 2 x 2 x 1 b m (a cos φ + gsinφ)mhi + mh (z φ) = (f f ) + 0 1 b i + mh m i + mh                    (2.2) where: 1 c11 k11 c21 k21f = f + f + f + f 2 c12 k12 c22 k22f = f + f + f + f to be able to identify the above matrix, use the corresponding link equations. the vertical displacement of the unsprung mass: ij ij ktij cij kij m z = f f f (2.3) the elastic force of the suspension system: kij ij ij f = k (z z ± bφ) (2.4) the damping force of the suspension system: cij ij ij f = c (z z ± bφ) (2.5) the elastic force of the tire: ktij tij ij ij f = k (u z ) (2.6) the vertical force fz is calculated by using the equation below: zij ktij cij kij f = f f f (2.7) when the vertical force fz becomes close to zero, the wheel tends to separate from the road surface, the vehicle is in an unstable state. determining the vertical force when steering 31 copyright ©2020 assa. adv. in systems science and appl. (2020) 3. result and discussion 3.1. simulation conditions the established model is used for most of the common vehicles. to conduct simulation and review, the specifications of the reference vehicles are given in table 3.1. table 3.1. vehicle specifications symbol description value unit m sprung mass 2000 kg mij unsprung mass 45 kg l1 distance from center of gravity to front axle 1250 mm l2 distance from center of gravity to rear axle 1550 mm b half of the base width 750 mm ix moment of inertia of the x–axis 800 kgm2 iz moment of inertia of the z–axis 2400 kgm2 h distance from the center to roll axis 650 mm at the velocity v = 70 km/h, the steering angle  = 0–50 (linear increase in 1 second), the graph in fig. 3.1 shows the value of the vertical force fz at the wheels at the same time. fig. 3.1. the value of the vertical force fz from fig. 3.1 it can be seen that the value of the fz21 is the smallest, this wheel tends to separate from the road surface first. therefore, simulation is concentrated at this position to determine the time of separating the wheels from the road surface. the research will simulate the oscillation of the vehicle with two cases: case 1: steering angle  = 0–40 (linear increase), corresponding to the different values of steering acceleration and velocity of the vehicle. case 2: steering angle  = 0–50 (linear increase), corresponding to the different values of steering acceleration and velocity of the vehicle. 32 n.t. anh, h.t. binh copyright ©2020 assa. adv. in systems science and appl. (2020) 3.2. results using the parameters of the reference vehicle and the two proposed cases, the graph in fig. 3.2 shows the relationship between steering acceleration , velocity v, and vertical force fz21 of the wheel at position (21). the graph in fig. 3.2 indicates that: + at the same value of the velocity, if the steering acceleration increases, the vertical force at the wheel will decrease. in case the velocity value is small, this decrease is not much. otherwise, if the velocity value is large, this attenuation will be significant. + at the same value of steering acceleration, if the velocity increases, the vertical force will drop sharply. this decrease is almost linear (in case the value of steering acceleration is large). + at the same value of velocity and steering acceleration, if the steering angle is larger, the value of vertical force will be smaller. the relationship between the parameters in fig. 3.2 has the form of irregular planes. therefore, it is possible to set up many plane equations to determine the vertical force value fz when the velocity v and steering acceleration  are known. the equation for determining the value of the vertical force fz is as follows: z 1 2 3f = a + a ε + a v (3.1) where: a1, a2, and a3 are the coefficients of the equation for each specific determination interval. based on the results obtained from the simulation process, the above coefficients are given in table 3.2. the coefficients a1, a2, and a3 correspond to different values of steering angle and steering acceleration. fig. 3.2. the relationship between steering acceleration, velocity, and vertical force table 3.2. the coefficients of the equation steering angle (deg) steering acceleration (deg/s2) the coefficients of the equation a1 a2 a3 40   0.14 4989.07 – 71.94 – 43.73 40 0.07   < 0.14 5064.91 – 913.04 – 43.57 40  < 0.07 5099.01 – 2571.43 – 42.80 50   0.14 5023.14 – 136.69 – 54.07 50 0.07   < 0.14 5127.43 – 1434.78 – 53.63 50  < 0.07 4704.57 – 2714.29 – 48.03 determining the vertical force when steering 33 copyright ©2020 assa. adv. in systems science and appl. (2020) the vertical force value fz can easily be determined through the coefficients of the equation. table 3.3 shows the difference between the results of calculating vertical force fz by two different methods (simulation and equation). table 3.3. the value of the vertical force fz velocity (km/h) steering acceleration (deg/s2) steering angle  = 0–40 steering angle  = 0–50 fz simulation (n) fz equation (n) tolerance (%) fz simulation (n) fz equation (n) tolerance (%) 60 0.279 2345 2345 0.0 1741 1741 0.0 60 0.140 2354 2355 0.0 1757 1760 0.2 60 0.093 2366 2366 0.0 1776 1776 0.0 60 0.070 2377 2387 0.4 1784 1810 1.4 60 0.056 2387 2387 0.0 1787 1771 0.9 60 0.047 2388 2411 1.0 1791 1797 0.3 60 0.040 2389 2428 1.6 1795 1815 1.1 60 0.035 2391 2441 2.0 1796 1828 1.8 70 0.279 1877 1908 1.6 1165 1200 2.9 70 0.140 1884 1918 1.8 1179 1219 3.3 70 0.093 1896 1930 1.8 1199 1240 3.3 70 0.070 1909 1951 2.2 1216 1273 4.5 70 0.056 1919 1959 2.0 1224 1191 2.8 70 0.047 1926 1983 2.9 1231 1216 1.2 70 0.040 1931 2000 3.5 1237 1234 0.2 70 0.035 1939 2013 3.7 1241 1248 0.6 80 0.279 1440 1471 2.1 635 659 3.6 80 0.140 1448 1481 2.2 649 678 4.3 80 0.093 1461 1494 2.2 674 704 4.3 80 0.070 1477 1516 2.6 705 737 4.3 80 0.056 1493 1531 2.5 722 711 1.5 80 0.047 1506 1555 3.2 736 736 0.0 80 0.040 1516 1572 3.6 750 754 0.5 80 0.035 1525 1585 3.8 760 767 0.9 90 0.279 1033 1033 0.0 119 119 0.0 90 0.140 1043 1043 0.0 138 138 0.0 90 0.093 1059 1059 0.0 167 167 0.0 90 0.070 1080 1080 0.0 200 201 0.5 90 0.056 1103 1103 0.0 230 230 0.0 90 0.047 1123 1127 0.4 253 256 1.2 90 0.040 1142 1144 0.2 273 274 0.4 90 0.035 1157 1157 0.0 287 287 0.0 34 n.t. anh, h.t. binh copyright ©2020 assa. adv. in systems science and appl. (2020) the data in the table show that there is a difference between the two methods, but the tolerance is quite small. in the case of steering angle  = 0–40, the biggest tolerance is only 3.8%. in the case of steering angle  = 0–50, this value does not exceed 4.5%. in general, this difference is not much and this equation can be used in many different cases. 4. conclusion the stability and safety of the vehicle are expressed through the value of vertical force at the wheel fz. the velocity, steering angle, and steering acceleration have a great influence on this problem. when the steering angle increases or the steering acceleration increases or both factors increase, the value of the vertical force fz will decrease significantly. if the value of fz gets close to zero, the wheels tend to separate from the road surface. at this time, the vehicle is in an unstable situation and very dangerous. to determine the value of the vertical force fz at different velocities, steering angle, and steering acceleration, the established dynamic vehicle model can be used. however, the setup and simulation process is complicated and inconvenient. this research gave the function of determining the value of the vertical force fz at specific conditions. therefore, it is easy to identify fz without using the dynamic vehicle model to simulate. the function is not perfectly accurate, there is a difference in the vertical force value fz when determined by this function and simulation process. however, this difference is quite small (less than or equal to 3.8% when the steering angle is from 00 to 40 and less than or equal to 4.5% when the steering angle is from 00 to 50). the results of the research can be used in many different survey cases. this research has only focused on the theoretical calculation process and establish corresponding equations, it is necessary to experiment to accurately evaluate the results. the results of this research will be the basis for the development of other research in the future. references 1. anh n.t. & binh h.t. 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(2015). the study for anti-rollover performance based on fishhook and j turn simulation, proceedings of the 3rd international conference on material, mechanical and manufacturing engineering. guangzhou, china, 2084–2093. doi: 10.2991/ic3me-15.2015.401. 19. yu f., li d.f. & crolla d.a. (2008). integrated vehicle dynamics control, proceedings of the 2008 ieee vehicle power and propulsion conference. harbin, china, 1– 6. doi: 10.1109/vppc.2008.4677809. 20. zulkarnain n., imaduddin f., zamzuri h. & and mazlan s.a. (2012). application of an active anti-roll bar system for enhancing vehicle ride and handling, proceedings of the 2012 ieee colloquium on humanities, science and engineering, kota kinabalu, malaysia, 260–265. doi: 10.1109/chuser.2012.6504321. 1. introduction 1.1. the instability problem of the vehicle 1.2. literature review 2. methodology 2.1. nomenclature 2.2. double-track dynamic vehicle model 2.3. spatial dynamic model 7 dof 3. result and discussion 3.1. simulation conditions 3.2. results 4. conclusion adv syst sci appl 2021; 02:58–70 published online at https://ijassa.ipu.ru. case study: influence of muscle fatigue and perspiration on the recognition of the emg signal narek unanyan1*, alexey belov1* 1v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow 117997, russia abstract: emg data processing and muscle activity recognition has become the most popular method for upper limb prosthetics. the high sensitivity of emg sensors with respect to external disturbances and other factors prevent from accurate muscle activity recognition. the aim of the paper is to investigate robustness of window recognition method with respect to muscle fatigue and perspiration of the forearm skin. the current experiment was carried out using arduino nano microcontroller connected to emg sensors. the subject under study is a healthy man of 26 years old with an average build. the subject was asked to do physical exercises, thereby loading the muscles of the fingers of the hand to achieve partial or complete fatigue and perspiration. during the whole process, emg sensors have installed on the subject and transmitted the signal to the computer using arduino. all signal processing is done directly on the computer with a pre-recorded signal. experimental results have been shown that with the appearance of external factors during prosthesis operation recognition accuracy may degrade to unsatisfactory. false positives occur with perspiration of skin surface and complete muscle fatigue. an algorithm for automatic self-correction of the boundaries of motion detection zones has been introduced. instead of identification of causes that leads to performance degradation, we use correction scheduling started by timer. experimental results have shown that proposed automatic adaptive correction is effective. despite higher recognition delay, proposed auto-tuning method provides satisfactory muscle activity identification and feature extraction in real-time. keywords: sensors, electromyogram, myoelectric control, prosthetics, pattern recognition 1. introduction in recent decades, and especially in recent years, much effort has been made to implement efficient control algorithms based on processing of electromyographic (emg) signals [1]. emg signals provide easy and non-invasive access to the physiological processes that cause muscle contraction. that is why emg technique is extensively used in control of upper limb prostheses. since the first attempts in the late 1940s [2], several emg-based algorithms have been developed to improve the functionality and ease of use of hand prostheses [3]. currently, emg signal processing is the most common approach used to manipulate prosthetic hands. limitations in prosthetic mechanics and emg data processing are currently the main restriction to the development and implementation of a complete bioelectric prosthesis [4–7]. moreover, when it comes to recognizing myoelectric patterns, the big problem is that statistical properties of emg signal change over time. this leads to the fact that control systems become unstable or difficult to use after a certain period of time [8]. in practice, it is very important to maximize emg data efficiency. to do this, it is necessary to assess the condition of the end user of the prosthesis, observing the condition of ∗corresponding author: n.unanyan@mail.ru case study: influence of muscle fatigue and perspiration 59 the skin, tissues, skeletal anatomy, muscle strength and activity, as well as the range of motion. also, an important factor is monitoring the shape, amplitude, frequency of the signal. only after a complete study of the user it is possible design the system for correct receiving and recognizing of emg signal [9]. the high sensitivity of emg sensors with respect to external disturbances and other factors prevent from accurate muscle activity recognition. such factors are investigated in [5]. among them: 1. muscle fatigue during work [10, 11]. 2. perspiration of the skin. 3. displacement of sensitive elements [12–14]. 4. physiological characteristics of a person [15, 16]. factors such as muscle fatigue and perspiration can lead to incorrect muscle activity recognition and feature extraction. the researchers argue that the accumulation of sweat under the sensor leads to a decrease in amplitude and filtering of high frequency components [17–20]. however, little is known about the amount of sweat above which a number of problems can arise. many authors also study the effect of muscle fatigue on recognition. [21] argue that with muscle fatigue, both the amplitude and frequency of the emg signal change. it was found that muscle fatigue is usually quantified as a decrease in maximum muscle strength or power, resulting in different recordings of signals from emg electrodes over time [22]. various traits were examined to assess muscle fatigue, such as the wavelet transform [23,24], the number of zero crossings [25], and autoregressive coefficients [26]. it is important to note that muscle fatigue identification commonly is carried out in frequency domain. this is cased by emg spectrum shift towards lower frequencies [5]. in this paper, we investigate influence of muscle fatigue and perspiration at feature selection accuracy performed by window method developed and presented in [27]. additionally, fail-safe correction to prevent false recognition is introduced. in the proposed method fatigue or perspiration identification technique is not used. this is caused by low computational requirements and real-time implementation of the algorithm. instead of identification of causes that leads to performance degradation, we use correction scheduling started by timer. the rest of the paper is organized as follows. sect. 2 describes experimental details and some necessary information on hardware and software equipment. sect. 3 presents main result of the paper. some conclusive remarks and future works are emphasized in sect. 4. 2. preliminaries 2.1. hardware description the current experiment was carried out using arduino nano microcontroller connected to emg sensors. arduino nano was selected due to its low cost and low power consumptions. the microcontroller itself is connected to the computer. all signal processing is done directly on the computer with a pre-recorded signal. matlab software is used for processing and analysis received data. to read the muscle activity we use emg sensors from df robotics. these sensors combine the filter circuit and the amplifier circuit. emg sensor amplifies the minimum electron diffraction signal within 1.5 mv by a factor of 1000 and suppress noise (especially frequency interference) using a differential input and an analog filter. the output signal is analog, which takes 1.5 v as the reference voltage. output voltage range 0–3 v. the signal level depends on the intensity of muscle activity. the output signal indicates muscle activity and contributes to the analysis and study of the emg signal. for laboratory experiment, sensor surface has been cleaned and degreased at each sensor. after that, 5 electrodes are placed on the upper and lower parts of the forearm in the correct order. most of the electrodes were located near the forearm (see fig. 2.1), since the flexors copyright © 2021 assa. adv syst sci appl (2021) 60 n. unanyan, a. belov fig. 2.1. location of emg sensors. and extensors of all fingers except the thumb are mainly located in this area. these muscles are oriented mainly parallel to the axis between the elbow and the wrist. unlike wearable band or sleeve such kind of electrode location provides better emg signal recording. 2.2. software tools before evaluating the influence of various factors on recognition, it is worth to briefly explain an emg signal recognition window method software. this algorithm is given and described in [27]. we try to recognize finger movements as close to its physiological behavior as possible. to this end we selected three levels of finger states. they are defined as follows: g = 1, the muscle is relaxed, 2, the muscle is in half tense, 3, the muscle is in full tense. in last expression, value g = 1 corresponds to straightened finger while values g = 2 and g = 3 correspond to half and fully bent finger respectively. briefly, this algorithm can be summarized as follows: 1. preparation of the sensor and signal reading. 2. signal preprocessing and normalization. preprocessing consists of sliding mav calculation within the window of length n . here n is chosen experimentally to provide satisfactory accuracy. normalized signal is calculated as absolute difference between sliding mav and current measurement. 3. boundary values ai and bi identification. 4. depending on the area in which the processed signal is located, a decision is made on the value of g as: g(xmax) = 1 if a1 < xmax ≤ b1, 2 if a2 < xmax ≤ b2, 3 if a3 < xmax ≤ b3, (2.1) copyright © 2021 assa. adv syst sci appl (2021) case study: influence of muscle fatigue and perspiration 61 fig. 2.2. emg data processing: a.) raw signal. b.) results of muscle activity recognition using window method. where xmax is a maximum value of normalized signal at the window of length n counts. the result of muscle activity recognition using window method is depicted in figure 2.2. window algorithm has been tested for quality and correct operation under normal conditions. also, the window method was compared with artificial neural networks, which made it possible to conclude about benefits of this method [27]. 2.3. methods identification algorithm, described in [27], was tested at 28 healthy people both males and females from 20 to 55 years old. identification error for both hands is depicted in figure 2.3. analyzing the data obtained during the experiment, it was found that there are no substantial differences in identification depending on age. female muscle activity is defined a little bit better than male. in view of this, it was decided to select a healthy man of 26 years old with an average build for further experiments. the subject was asked to do physical exercises, thereby loading the muscles of the fingers of the hand. during the whole process, emg sensors have installed on the subject and transmitted the signal to the computer using arduino. we note that emg sensor data for different sensors have the same waveform. so, the further investigation and illustrations are carried out at one selected sensor dataset. the subject, due to physical exertion, led the muscles of the finger to partial fatigue. after that, a signal was recorded with three types of finger states. when the subject led to complete fatigue signal values for three types of finger states was also recorded. after that these signals were analyzed. another experiment was to test the effect of perspiration on the emg signal. for this, the subject was placed in a special room with a high temperature where he also performed physical exercises. after that emg data recording was carried out similar to previous experiment. all the results of these experiments are discussed below. 3. main result copyright © 2021 assa. adv syst sci appl (2021) 62 n. unanyan, a. belov fig. 2.3. identification error statistics. 3.1. muscle fatigue the term muscle fatigue is used to describe a temporary decrease in one’s physical capacity of performing motions. several researchers investigate the impact of muscle fatigue using time domain or frequency domain characteristics of emg signal such that maximum amplitude or power spectrum density [23]. many efforts have been made to identify the level of fatigue. in order to mitigate for the effects of fatigue research focuses on collecting data from multiple levels of fatigue and utilizes the abundance of information. this requires recording more data than a simple classification case and creates different computational requirements to the system. window method described in previous section also may suffer from muscle fatigue. further we investigate effect of muscle fatigue on feature extraction by window method. in this experiment, the subject performed physical exercises with chest expander to load the muscles with work until muscle would be tired. experiment is stopped when muscle has reached a complete fatigue. during all of the process, emg recording was carried out. we are interested in states of partially tired and completely tired muscles. emg pattern of all three finger positions is depicted in figure 3.4. figure 3.5 shows the raw emg signal on an enlarged scale. one can see the difference between two states, namely untired hand and completely tired one. both signals indicate relaxed muscles. it is noticeable that noises appear in the channel with accumulation of fatigue. however, it can also be concluded that noises are not significant and can be neglected. the root mean square (rms) value is also not much different. rms for ideal conditions is equal 1.53 while partially tired and tired muscles gives rms value equal to 1.51 and 1.53 respectively. the number of crossings of the mean square value is also the same. however, the number of changes in the sign of the function and its amplitude are significantly different due to the appearance of noise. we should note that the best recognition accuracy is reached only when muscle is untired. partially tired muscles also provides satisfactory recognition. however, activity of tired muscle cannot be recognized correctly. in figure 3.9 one can see that half tensed muscle recognized with incorrectly almost in 50% cases. 3.2. perspiration now let’s check how the system will behave if, during operation, the subject in the places where the sensing element is placed is wet due to perspiration. for this, emg data were collected from the subject, after which he performed a series of physical exercises for perspiration. the measurement result can be seen in figure 3.7. copyright © 2021 assa. adv syst sci appl (2021) case study: influence of muscle fatigue and perspiration 63 fig. 3.4. comparison of muscle fatigue charts: a.) untired muscle; b.) partially tired muscle; c.) tired muscle. fig. 3.5. comparison of emg shapes of untired and tired muscles. it is shown that emg signal has sufficiently distorted. in particular, the amplitude decreased and the number of changes in the signal sign increased. the rms has not changed much. it is equal to 1.52. however, the number of rms crossings has been increased. with such strong interference, there is a possibility of false triggering of the recognition algorithm. figure 3.8 illustrates recognition results after perspiration. one can see that algorithm provides false recognition during operation if parameters ai and bi are not corrected. copyright © 2021 assa. adv syst sci appl (2021) 64 n. unanyan, a. belov fig. 3.6. results of muscle activity recognition for muscle fatigue: a.) untired muscle; b.) partially tired muscle; c.) tired muscle. fig. 3.7. comparison of waveforms with perspiration. results of muscle activity recognition under sweat skin surface has shown high probability of false recognition. figure 3.8 clearly shows that the boundaries that were copyright © 2021 assa. adv syst sci appl (2021) case study: influence of muscle fatigue and perspiration 65 fig. 3.8. results of muscle activity recognition operation for perspiration. calculated before perspiration are no longer suitable and during operation, if the boundaries are not automatically corrected, the user will lose control of the prosthesis. 3.3. malfunction correction experimental results have been shown that with the appearance of external factors during prosthesis operation recognition accuracy may degrade to unsatisfactory. therefore, it is required to develop correction algorithm that performs correction of boundaries ai and bi to provide fail-safe recognition. the most interesting case is automatic recalculation to perform fail-safe recognition without manual mode even during active operating of the prothesis. we propose the following boundary values correction strategy. the program code for recalculation of new boundaries starts every 5 minutes in automatic mode without user interaction. the 5 minutes time interval has been selected empirically after several experiments. time interval can be changed individually for each patient. the correction process is a cycle in which the system inspects the user’s activity. after initialization of the boundary correction, the system is switched to waiting mode and wait for muscle activities. after activity detected, the correction algorithm collects about 150 measurements for each type of muscle activity. when data for three types of muscle activity are collected, correction step is carried out between contiguous movements, for example, between relaxed and half tensed muscle. new boundaries are defined by the following relations: copyright © 2021 assa. adv syst sci appl (2021) 66 n. unanyan, a. belov a1 = 0, a2 = b1 = n∑ i=1 xrelaxed i + n∑ i=1 xhalf i 2n , a3 = b2 = n∑ i=1 xhalf i + n∑ i=1 x full i 2n , b3 = n∑ i=1 x full i n + 2.5, (3.2) where xrelaxed i , xhalf i , and x full i are maximum window value for relaxed, partially tensed and fully tensed muscle respectively, n is a number of counts for calculating the average value (n = 150). table 3.1. recognition accuracy of window method. muscle activity relaxed half tensed tensed fatigue no correction, % 12 52 28.8 fatigue with correction, % 3.5 8.2 2.5 perspiration no correction, % 24 19.5 6 perspiration with correction, % 12 9.8 4.5 results of muscle activity recognition before and after correction are given in table 3.1. it is easy to see that correction algorithm allows to improve accuracy of window method. also we should note errors correspond to the transient process between states of muscle activity. it has been experimentally obtained that overall delay in recognition between two states of muscle activity may be increased up to 100 ms. this is satisfactory value for realtime operations. additionally, we derive boundaries for ideal conditions (untired muscle and dry skin surface) and corrected values for each type of deflection. boundaries for ideal conditions are defined in (3.3), corrected values for tired muscle are defined by (3.4), and corrected values for sweaty skin are given by (3.5). g = 1 if 0 < xmax 6 0.11, 2 if 0.11 < xmax 6 0.42, 3 if 0.42 < xmax 6 3.3. (3.3) g∗ fatig. = 1 if 0 < xmax 6 0.03, 2 if 0.03 < xmax 6 0.3, 3 if 0.3 < xmax 6 3. (3.4) g∗ persp. = 1 if 0 < xmax 6 0.12, 2 if 0.12 < xmax 6 0.45, 3 if 0.45 < xmax 6 3.5. (3.5) copyright © 2021 assa. adv syst sci appl (2021) case study: influence of muscle fatigue and perspiration 67 fig. 3.9. algorithm operation after border correction for muscle fatigue. fig. 3.10. algorithm operation after border correction for perspiration. one can see that for tired muscle corresponding boundary values are less than ideal ones while corrected boundaries for sweaty skin surface are larger than ideal. the operation of the algorithm after border correction can be seen in figures 3.9 and 3.10. it is shown that proposed correction procedure provides improved recognition without substantial fails during operation. copyright © 2021 assa. adv syst sci appl (2021) 68 n. unanyan, a. belov 4. conclusion the paper presents an experimental investigation of the influence of muscle fatigue and perspiration on the recognition of muscle activity using emg signal. a number of experiments have been carried out. during study of the perspiration and muscle fatigue, it was found that complete fatigue may increase identification error to unsatisfactory (over 50%). perspiration as well as fatigue leads to performance degradation. to improve identification accuracy, automatic correction method has been proposed. experimental results has shown that proposed solution allows to improve working precision of the amplitude-based window identification under perturbing factors. it has been shown that when deflection from ideal conditions occurs, the algorithm adapts to changes. the future works is focused on: 1. study of other factors for emg signal recognition. 2. complex tests on several subjects with different parameters of external disturbance. 3. development of rehabilitation strategy for amputees to provide prosthetic hand operation. compliance with ethical standards conflict of interest the authors declare that there is no conflict of interest. ethical standard the participant signed an informed consent before the experiment. the study was conducted in accordance with the declaration of helsinki. the protocol was approved by the v.a. trapeznikov ics ras institutional review board. acknowledgements the reported study was funded by rfbr, project number 19-38-90293. references 1. prakash, a., sharma, s. & sharma, n. 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(2014) assessment of muscle load and fatigue with the usage of frequency and time-frequency analysis of the emg signal. acta bioeng biomech, vol. 16, no. 2, 31–39. 24. camata, t.v., dantas, j.l., abrao, t., brunetto, m.a., moraes, a.c. & altimari, l.r. (2010) fourier and wavelet spectral analysis of emg signals in supramaximal constant load dynamic exercise. annu int conf ieee eng med biol soc, vol. 2010, 1364–1367. 25. masuda, t., miyano, h. & sadoyama, t. (1982) the measurement of muscle fiber conduction velocity using a gradient threshold zero-crossing method. ieee trans copyright © 2021 assa. adv syst sci appl (2021) 70 n. unanyan, a. belov biomed eng, vol. 29, no. 10, 673–678. 26. al-mulla, m.r. sepulveda, f., colley, m. & kattan, a. (2009) classification of localized muscle fatigue with genetic programming on semg during isometric contraction. annu int conf ieee eng med biol soc, vol. 2009, 2633–2638. 27. unanyan, n. & belov, a. (2020) a real-time fail-safe algorithm for decoding of myoelectric signals to control a prosthetic arm. proceedings of the 21th international carpathian control conference iccc, 1–6. copyright © 2021 assa. adv syst sci appl (2021) introduction preliminaries hardware description software tools methods main result muscle fatigue perspiration malfunction correction conclusion  2017 г adv syst sci appl 2020; 03; 1-23 published online at https://ijassa.ipu.ru. mathematical modeling of antitumor viral vaccine therapy: from the experiment to the clinic nina a. babushkina1*, ekaterina a. kuzina1 1) v.a. trapeznikov institute of control sciences of russian academy of sciences, 117997, profsoyuznaya street, 65, moscow, russia e-mail: babushkina_na@mail.ru, kate_k93@mail.ru abstract: the paper presents the model developed to identify efficient strategies of antitumor viral vaccine introduction. these strategies are able to produce complete suppression of the tumor growth. the model was developed in matlab-simulink. three efficient strategies of viral vaccine introduction were produced. it was found that the choice of the strategy depends on the tumor size at the start of the treatment, and the range of the tumor sizes for each of the strategies was identified. for the small tumors, elimination of the tumor can be achieved through single-shot vaccine administration in dosages that lead to the death of tumor cells caused directly by the virus. for the big tumors that are within the threshold size, elimination of the tumor can be achieved through repeated vaccine administrations with stepwise reduction of time periods between them. for the tumors of any size, the strategy of repeated administration of the virusbased vaccine that allows stabilizing the tumor size as per the start of the treatment was defined. keywords: mathematical model, experimental oncology, tumor cells, kinetic growth curves, vaccine therapy, virus, immune response, antibodies, allometric proportions 1. introduction effective treatment of cancer remains one of the key challenges of contemporary healthcare. addressing this challenge requires extensive research based on experimental studies that are further translated into clinic. one of the promising directions for combating oncological diseases is the development of different treatment methods within the domain of immunotherapy. cancer immunotherapy research by j. allison and t. honjo was awarded 2018 nobel prize in medicine. significant progress in cancer treatment was produced by breakthroughs in molecular biology and immunology that made it possible to understand the reasons behind tumor degeneration and the development of tumor process [1-3]. immune system can resist the emergence and development of the tumor process, and the aim of immunotherapy is to stimulate the immune system to fight against the tumor cells. there are two ways to stimulate the immune system: specific and non-specific. specific antitumor vaccines are based on dendritic cells that carry the information on the antigens specific for each type of tumor [4-8]. non-specific virus-based antitumor vaccines can induce the immune response against the tumor by producing new protein formations on the surface of the tumor cells [913]. such vaccines are based on the viruses that are not dangerous for the humans but are able to identify and destroy the tumor cells [14-17]. the virus is able to induce the process of tumor cells’ death that has two stages. the first stage is produced by the virus itself. settling down on the tumor cell, the virus penetrates * corresponding author: babushkina_na@mail.ru mailto:babushkina_na@mail.ru mailto:kate_k93@mail.ru n.a. babushkina, e.a. kuzina copyright ©2020 assa. adv. in systems science and appl. (2020) 2 inside, multiplies itself and destroys the cell. meanwhile the immune system reacts at the virus intrusion and destroys it together with the tumor cell. the second stage of the tumor cells’ death evolves later, due to the immune system’s reaction to the population of infected tumor cells. when settling down on the surface of the cells, the virus leaves specific protein formations on the membrane surface. this produces a population of infected tumor cells that are detected by immune system as foreign. in response, immune system produces antibodies that destroy the infected tumor cells [17-19]. thus the virus acts as a specific marker of the tumor cells and helps to overcome the irresponsiveness of the immune system against uninfected tumor cells. one of the key advantages of immunotherapy is the absence of toxic effects associated with chemotherapy and radiation therapy. since there is no need for additional treatment to mitigate the negative side effects, this makes immunotherapy a more harmless and costeffective method as compared to other approaches to cancer treatment. low toxicity makes it possible to expand the range of dosages – however, this requires additional experimental studies and therefore implies higher research costs. computational experiments based on the mathematical model of vaccine therapy make it possible to define the optimal dosages, assess the effectiveness of immune response depending on the vaccine dosage, and describe the dynamics of the tumor process for different treatment strategies. data obtained through computational experiments may be used to define optimal treatment strategies that produce complete tumor regression. given the high costs of experimental and clinical studies, computational experiments based on mathematical models can produce a lot of useful information based on the limited amount of experimental data. modelling makes it possible to expand the study of the range of applicable dosages and treatment strategies when exploring the new methods of antitumor therapy, thus reducing time and labor costs and the number of animals required for conducting an experiment in vivo. moreover, research in molecular biology, cell biology, biophysics and immunology has produced a large amount of data that also require processing and analysis with the use of specialized computer technologies based on mathematical modelling [8-11]. however, despite its advantages, modelling is still not widely used in experimental and clinical oncology. this paper aims to demonstrate that computational experiments based on mathematical modelling may produce meaningful results for defining cancer treatment strategies, complementing in vivo experimental approaches and enabling more efficient translation of experimental results to the clinic, building on the study published in mathematical biology & bioinformatics earlier in 2019 [19]. 2. problem statement this study aims to develop an algorithm for finding strategies for the use of antitumor viral vaccine to completely suppress tumor growth. to achieve this aim, a software package was developed in the matlab-simulink system, which includes several mathematical models: mathematical model of vaccine therapy, describing the mechanism of tumor cell death as a result of the immune response to the injection of the virus [10,18,19], mathematical model of infectious disease by g. i. marchuk, describing the dynamics of the formation of antibodies against the virus by the immune system [20-22], mathematical model by h. f. skipper, describing how the proportion of the fast proliferating tumor cells decreases in the growing tumor [10,23], mathematical model of antitumor therapy with discontinuous trajectories, evaluating the effectiveness of therapeutic effects on the experimental trajectories of tumor growth after the injection of a viral vaccine [10,18,19]. the values of the parameters of the complex mathematical models are given in the table 1 [19, p. 44]. the complexity and nonlinearity of differential equations of mathematical models does mathematical modeling of antitumor viral vaccine therapy… copyright ©2020 assa. adv. in systems science and appl. (2020) 3 not allow to obtain the solution analytically without the use of computational software. on the basis of the developed software package, a computational experiment was conducted, which allowed to investigate the effectiveness of various strategies for the injection of viral vaccines depending on the dose and size of the tumor at the time of its injection. 3. mathematical model of vaccine therapy with virus val the mathematical model of vaccine therapy describes the mechanism of tumor cells’ death under the influence of two factors – the virus itself and antibodies against infected tumor cells, on the basis of experimental data on the growth of erlich tumor in experimental animals after a single administration of the vaccine with venezuelan equine encephalitis (vee) virus [17]. the experiments were carried out on 2-month-old female mice of the line balb/c, с57в1/6 and dba/2. in total, 690 animals were used in the experiments [17]. the process of tumor cell growth without treatment (control) is described by a simple differential equation [19, p. 38]: ( ) ( ) ( ) dn t t n t dt =   , with 0 0( )n t n= , (1) where n(t) is the number of tumor cells, t denotes time,  (t) is the parameter characterizing the growth rate of tumor cells, n0 is the initial number of tumor cells at time t = 0. the type of function describing tumor growth without treatment was determined by experimental curves of erlich adenocarcinoma growth by regression analysis in matlab (fig. 1). the experimental curve of tumor growth without treatment is most accurately described by the gompertz function, which is the solution of the differential equation (1) for  (t) = n n exp(– n t). fig. 1. approximation of experimental data on the growth of ehrlich adenocarcinoma in control by the gompertz function the gompertz function is: 0( ) exp( exp( )) exp( (1 exp( ))n n n n nn t n t n t= −  − =  − − , (2) where 0 exp( )nn n =  is the maximum tumor size for t →  . calculated values of the parameters of the gompertz function and the sum of squared deviations (sко = 0.21) are shown in table 1 [19, p. 44]. analysis of experimental data on tumor growth after the introduction of a viral vaccine (fig. 2, lower curve) allows us to distinguish two periods of intensive death of tumor cells. n.a. babushkina, e.a. kuzina copyright ©2020 assa. adv. in systems science and appl. (2020) 4 the first period lasts from 6 to 8 days, and the second from 13 to 16 days. we can assume that the first stage is associated with the reaction of the immune system to the virus and the formation of antibodies specific to this virus. and the second stage is associated with the appearance of dead tumor cells infected with a virus, and the formation of antibodies specific to these tumor cells [15-17]. fig. 2. experimental data on the growth of ehrlich adenocarcinoma without treatment and after a single injection of the vaccine [17-19] as a result of the interaction of the virus with tumor cells, three populations of tumor cells emerge (fig. 3). fig. 3. interaction of the virus with tumor cells and immune system the population of tumor cells carrying the virus is part of the tumor cells on which the virus is absorbed. penetrating inside the cell and multiplying there, the virus makes the tumor cell die. mathematical modeling of antitumor viral vaccine therapy… copyright ©2020 assa. adv. in systems science and appl. (2020) 5 the population of infected tumor cells has specific protein formations on the cell membrane which are perceived as alien by the immune system. thus this population dies under the action of the antibodies produced by the immune system. the population of uninfected tumor cells remains alive and is capable of multiplying and resuming tumor growth. a mathematical model of vaccine therapy describing the dynamics of two-stage tumor cell death is presented in the form of a system of differential equations (1–10) [18–20]. the dynamics of tumor growth before the introduction of the vaccine is described by a differential equation of the form:  0, , ( ) ( ) ( ), vt dn t t n t t dt =   with 0 0( )n n t= , (3) where ( )n t is the population of tumor cells before administration of the vaccine, 1v cvt z=  + is the beginning of the immune response against viruses. 1 is the moment of injection of the vaccine, cvz is the period of the delay of immune response against the virus. the dynamics of growth and death of infected tumor cells after the introduction of the virus is described by the differential equation of the form: ( , , ( ) [ ( ) ( ) ( )] ( ), v v av v v v n dn t t k v t k a t n t t t t dt =  − −   with 0 ( )v vn n t= (4) where ( )vn t is the population of infected tumor cells following injection of the vaccine, ( )v t is the number of viruses, ( )va t is the number of antibodies against the virus, vk is the coefficient of the rate of reproduction of the virus in a tumor cell, avk is the coefficient of the rate of death of viruses as a result of interaction with antibodies ( )va t . the dynamics of growth and death of infected tumor cells under the action of antibodies at the second stage is described by the differential equation of the form:  , , ( ) [ ( ) ( )] ( ), v an n v n l dn t t k a t n t t t t dt =  −   with 0 ( )v nn n t= , (5) where 1n cnt z=  + is the moment of the immune response against infected tumor cells, cnz is the delay time of the immune response against infected tumor cells, ( )na t is the number of antibodies against infected tumor cells, ank is the dimensional coefficient. the death of tumor cells at each of the two stages occurs as a result of the development of the body's immune response to the introduction of a viral vaccine. the first stage is associated with the formation of antibodies against the virus ( )va t , and the second stage is associated with the formation of antibodies against infected tumor cells ( )na t . graphs of antibody formation dynamics ( )va t against the virus and against infected tumor cells ( )na t are shown in fig. 4 [19, p. 39]. n.a. babushkina, e.a. kuzina copyright ©2020 assa. adv. in systems science and appl. (2020) 6 fig. 4. graph of the dynamics of the number of antibodies against the virus ( )va t (the first stage of the immune response) and against infected tumor cells ( )na t (the second stage of the immune response)) for the experimental dose 0 0.015v = on τ1 = 1day the mechanism of their formation is calculated on the basis of the model of infectious disease by g. i. marchuk [20-22]. the parameters of this model are adapted for experimental curves of tumor growth after the introduction of viral vaccines. the dynamics of the number of viruses according to the mathematical model of an infectious disease [20, 22] is described by the equation of the form: ( ) ( ) ( ) ( ),v v v dv t v t a t v t dt =  − (6) where 0 1( )v v=  is the initial dose of virus vaccine, 1 is the moment of the first introduction of a viral vaccine, v is the rate of reproduction of the virus inside the cell, v is the rate of death of viruses in their interaction with antibodies ( )va t [19, p. 39]. the initial condition for the solution of equation (4) is taken as a control parameter characterizing the introduced dose of the viral vaccine 0 1( )v v=  . the first stage of the body's immune response to the introduction of the virus is determined by the number of antibodies ( )va t , which is calculated from the following equations: ( ) ( ) ( ) ( ) ( ),v a v v av v v v da t c t t a t v t a t dt =  − − − (7) where a is the rate of formation of antibodies from one plasma cell, av is the rate of loss of antibodies due to the interaction with viruses ( )va t , v is the rate of reduction of the number of antibodies due to natural destruction, 1v cvt z=  + is the moment of the beginning of immune response against viruses [19, p. 40]. due to the fact that the time of virus reproduction in the experimental tumor was not recorded in the available experimental data [15-17], when constructing the model, it was assumed that the period of virus reproduction inside the tumor cell, leading to its death, can be considered with sufficient accuracy equal to the time of delay of the immune response against the virus cvz . then in equations (7) and (8) the time of introduction of the virus was taken into account as a parameter tv = 1 +zcv, which records the moment of the beginning of the immune response against viruses. mathematical modeling of antitumor viral vaccine therapy… copyright ©2020 assa. adv. in systems science and appl. (2020) 7 number of plasma cells ( )vc t is determined from the equation: ( ) ( ) ( ) ( )v vc v v cv v vn dc t v t t a t t c t c dt      =  − − − − , with ,( )v v vnc t c= (8) where с is the rate of formation of plasma cells, cv is the constant coefficient, cvz is the period of delay in the development of antibodies against the virus (first stage of immune response). the second term of this equation reflects the maintenance of the initial number of plasma cells in the norm vnc [19, p. 40]. the second stage of the immune response of the body to the formed infected tumor cells was determined by the number of antibodies, which was calculated from the following equations: ( ) ( ) ( ) ( ) ( ),n an n n an n v nn n da t c t t a t n t a t dt =  − − − (9) where an is the rate of formation of antibodies, an is the coefficient that describes the rate of reduction of the number of antibodies ( )na t due to their interaction with infected tumor cells ( )vn t , nn is the rate of reduction of the number of antibodies due to natural destruction [19, p. 40]. number of plasma cells ( )nc t is determined from the equation: , ( ) ( ) ( ) ( )n cn v n n n cn n nn dc t n t t a t t c t c dt      =  − − − − with ,( )n n nnc t c= (10) where cn is the rate of formation of plasma cells, cn is the constant coefficient, сnz is the period of delay in the development of antibodies against the infected tumor cells (second stage of immune response), nnc is the initial number of plasma cells in the norm [19, p. 40]. to adequately describe the effectiveness of viral vaccines, it is necessary to take into account that one of the factors of high selectivity of viruses in relation to tumor cells is the high rate of their division in comparison with normal body tissues [4-7]. the measured tumor volume contains fractions of rapidly and slowly proliferating tumor cells, which is described in detail in the mathematical model of tumor growth by skipper [23]. as the size of the tumor increases, the fraction of rapidly dividing tumor cells decreases, while the fraction of slowly dividing cells and temporarily non-dividing cells increases. to describe how the tumor size is related to the efficiency of the vaccine, ( )np t function is included in the model. this function describes the dynamics of the decline in the share of rapidly proliferating cells with increasing size of the tumor [19]: 2 2 21 ( ) 1 ( ) , 1 p p p p t p t arctg k t     = −    −    (11) where p and pk are constant parameters, t denotes time (in days), *1/p t = , where *t is the moment when numbers of the rapidly and slowly proliferating cells are equal [19, p. 41]. then the number of infected cells in the measured tumor volume is calculated as ( ) ( ) ( )v n n nn t n t p t= , where 1n cnt z=  + is the moment when the immune response against infected tumor cells begins. as a result of the interaction of the virus with tumor cells, three populations of tumor n.a. babushkina, e.a. kuzina copyright ©2020 assa. adv. in systems science and appl. (2020) 8 cells develop in the tumor volume ( )vn t (fig. 5) [19, p. 41]. the structure of the equations of the model is such that the existence, uniqueness and nonnegativity of solutions in this work under nonnegative initial conditions are satisfied, which was proved and stated in the work of n. v. pertsev [21, p. 156-157]. fig. 5. calculation curves of the dynamics of infected cells ( )vn t (dotted line) and the kinetic trajectory of tumor growth ( )n t after dose injection 0 0.015v = at τ1 = 1 day, v and n are duration of the delay tumor growth after each stage of tumor cells’ death. experimental data is denoted by + figure 5 shows two calculated curves. the dotted line describes the dynamics of growth and death of infected tumor cells in accordance with the model of vaccine therapy (1) – (11). the solid line shows the dynamics of the number of surviving tumor cells after the first stage and the second stage of immune response in accordance with the mathematical model of anticancer therapy with discontinuous trajectories [19]. 4. a mathematical model of anticancer therapy with discontinuous trajectories to predict the dynamics of tumor growth when constructing a mathematical model of antitumor therapy with discontinuous trajectories, several assumptions were made to describe the death and subsequent growth of the experimental tumor [10, 18]. 1. tumor growth without the introduction of a viral vaccine (control) and after its introduction is described by the gompertz function with the preservation of the values of the parameters of the function. 2. the death of tumor cells occurs instantly, causing a sudden decrease in the size of the tumor at the beginning of the immune response at each stage of cell death. 3. the trajectory of tumor growth after the death of tumor cells is shifted in time for the duration of the delay in tumor growth 0( )v after each stage of their death. 4. duration of tumor growth delay 0( )v is the time interval between the start of immune response (i.e. when the number of tumor cells starts to decline affected first by the virus at vt and then by the antibodies at nt ) and the moment when the tumor, resuming its growth, reaches the same size as at vt and nt respectively [10, 18]. the mathematical model of anticancer therapy with discontinuous trajectories is used to construct dynamic trajectories of tumor growth after two-stage death of tumor cells after the mathematical modeling of antitumor viral vaccine therapy… copyright ©2020 assa. adv. in systems science and appl. (2020) 9 introduction of a viral vaccine (fig. 5 – ( )n t ) [19, p. 41]. the dynamics of tumor growth before the injection of the vaccine is described by a differential equation (12), which is similar to the equation of tumor growth in the control (1):  0, , ( ) ( ) ( ), vt dn t t n t t dt =   with 0 0( )n n t= , (12) where ( )n t is the size of the tumor before vaccine injection,  (t) = n n exp(– n t) is tumor growth rate in control, 1v cvt z=  + is the moment when tumor cells start to die in response to the virus (1st stage of immune response), 1 is the moment of vaccine injection, cvz is the period of delay of the immune response against the virus. the dynamics of tumor growth after the first stage of tumor death is described by a differential equation of the form:  , , ( ) ( ) ( ), v v v nt t t dn t t n t t dt = − −  with ) )( ( ) ( ) (r v v v v vtn n t s t t t=  − (13) where ( )vt t − is pulsed dirac function, describing the instantaneous death of tumor cells in the first stage of the immune response, ( )v vs t denotes the proportion of dying tumor cells infected with the virus at the first stage of the immune response, which is determined from the equation: ( ) ( ) ( ) v v v v v v n t s t n t  = (14) where ( )v vn t denotes the population of infected tumor cells at the start of the immune response, ( )v vn t is the number of dying tumor cells in the first stage of the immune response, which is calculated as the difference between the maximum and minimum number of infected cells in the period from the beginning to the end of the 1st stage of immune response according to the equation: 1 2( ) ( ) ( )v v v vv vn t n t n t = − (15) where 1 vt and 2 vt are the moments when the number of infected cells during the 1st stage of the immune response against the virus reaches maximal and minimal level (fig. 5, dotted line), ( )r vn t is the number of remaining tumor cells that continue to produce tumor growth, which is calculated from the equation: ) ( )( ) (r v v v vn tn t n t −= (16) the trajectory of tumor growth after the first stage of tumor cell death is described by the gompertz equations with a time shift for the duration of tumor growth delay (fig. 5): 0 0( ) exp( (1 exp( ( ( ))))),v n n v vn t t n t v− =  − − − (17) where 0( , )v vv t is the delay in tumor growth after the first stage of the immune response. the dynamics of tumor growth after the second stage of tumor cells’ death is described n.a. babushkina, e.a. kuzina copyright ©2020 assa. adv. in systems science and appl. (2020) 10 by the differential equation of the form:  , , ( ) ( ) ( ),n n n lt dn t t t n t t t dt = − −  with ) )( ( ) ( ) (r n v n n ntn n t s t t t=  − (18) where 1n cnt z=  + is the moment when the immune response against virus-infected tumor cells begins, cnz is the period of delay of immune response against virus-infected tumor cells, ( )nt t − is the pulsed dirac function, describing the instantaneous death of tumor cells in the second stage of the immune response, ( )n ns t is the proportion of dying tumor cells infected with the virus at the second stage of the immune response, which is determined from the equation: ( ) ( ) , ( ) v n n n n n t s t n t  = (19) ( )v nn t denotes the number of dying tumor cells in the second stage of the immune response. it is calculated as the difference between the maximum and minimum number of infected cells in the period from the beginning to the end of the 2nd stage of immune response according to the equation: 1 2( ) ( ) ( ),n n v vv nn t n t n t = − (20) where 1 nt and 2 nt are the moments when the number of infected cells during the 2nd stage of immune response against the virus reaches maximal and minimal level (fig. 5, dotted line), ( )r nn t is the number of remaining tumor cells that continue to produce tumor growth, which is calculated from the equation: ) ( )( ) (r n n v nn tn t n t −= (21) then the trajectory of tumor growth after the second stage of tumor cell death is described by the gompertz equation with a time shift for the duration of tumor growth delay (fig. 5): 0 0( ) exp( (1 exp( ( ( )))))n n n n nn t t n t v− =  − − − (22) where 0( )n v is the delay in tumor growth after cells’ death in the second stage of the immune response. the duration of the delay of tumor growth 0( )v v and 0( )n v was determined as the time interval between the start of immune response (i.e. when the number of tumor cells starts to decline affected first by the virus at vt and then by the antibodies at nt ) and the moment when the tumor, resuming its growth, reaches the same size as at vt and nt respectively. therefore: 0( ) ( ( ))v v vn t n t v= +  , (23) 0( ) ( ( ))n n nn t n t v= +  . (24) the values of the parameters of the vaccine therapy model and the model of anticancer therapy with discontinuous trajectories are given in appendix 1. mathematical modeling of antitumor viral vaccine therapy… copyright ©2020 assa. adv. in systems science and appl. (2020) 11 5. finding effective strategies of viral vaccine introduction evaluation of the effectiveness of various strategies of tumor growth after the injection of the viral vaccine was carried out on the basis of a mathematical model of anticancer therapy with discontinuous trajectories according to the equations (12) (24). the aim of efficient strategy is to reduce the number of tumor cells below the lethality threshold. in this case tumor will not be able to resume its growth. this threshold is denoted let 0n . if the number of surviving cells is below the threshold after the first stage of immune response ( 0( )r let vn t n ) or after the second stage of the immune response ( 0( )r let nn t n ), this is interpreted in the model as the complete destruction of tumor cells. in this case, the life expectancy of treated animals will be equal to the average life expectancy of experimental animals without tumors tl = 3 years. 5.1. strategy of stabilization of tumor growth with multiple injection of the viral vaccine modeling of multiple injections of a viral vaccine within the framework of the constructed model allowed us to determine the algorithm for finding a strategy for stabilizing tumor growth with multiple injections of a constant dose of a viral vaccine v0 stab (τ1). results indicate that the strategy of stabilization is possible for a narrow range of dosage. this dosage should be able to produce an equal number of antibodies at each of the two stages of the immune response stab stab 0 1 0 1( , ) ( , )v na v a v =  (fig. 6, a,b) and (fig. 7, a,b). а b fig. 6. strategy to stabilize the tumor size with the injection of a viral vaccine on τ1 = 3 days with an interval between injections ∆t = 18.4 days: а shows the trajectories of growth and death of tumor cells; b shows the dynamics of antibody formation 0 1( , )va v  at the 1st and 0 1( , )na v  at the 2nd stage of immune response n.a. babushkina, e.a. kuzina copyright ©2020 assa. adv. in systems science and appl. (2020) 12 а b fig. 7. strategy to stabilize the tumor size with the injection of a viral vaccine on τ1 = 20 days with an interval between injections ∆t = 25.7 days: а shows the trajectories of growth and death of tumor cells; b shows the dynamics of antibody formation 0 1( , )va v  at the 1st and 0 1( , )na v  at the 2nd stage of immune response the strategy of tumor growth stabilization is performed in the range of doses from stab 0v =0.04 to stab 0v =0.06 (fig. 8,b) at a constant value of the duration of the interval between administration of the vaccine. duration of intervals between repeated introductions of the vaccine ∆t increases with the size of the tumor (fig. 8,a). а b fig. 8. the dependence of the values of the parameters under the strategy of stabilization of tumor growth on the size of the tumor at the beginning of treatment: а shows the intervals between vaccine introductions stab 0 1( , )t v  , b shows the value of the introduced dose stab 0 1( )v  this strategy requires sufficiently large dosage which will be able to produce equally big number of antibodies at the 1st and the 2nd stages of immune response. thus, this strategy allows to transfer the course of the disease into a chronic state by restraining the growth of the tumor. within the framework of the constructed model, it is shown that it is possible to restrain the growth of a tumor of any size for an unlimited time. however, if vaccine is not introduced any longer, the tumor growth could resume. the strategy of tumor growth stabilization is similar to the effect of inoculation from own tumor cells in the period between two successive injections of the vaccine. mathematical modeling of antitumor viral vaccine therapy… copyright ©2020 assa. adv. in systems science and appl. (2020) 13 5.2. strategy of tumor growth suppression with multiple injection of the viral vaccine to solve the problem of reducing the number of tumor cells below the lethality threshold with repeated injection of the vaccine for all tumor sizes, agorhythm was developed, based on the strategy of stabilizing tumor growth, in which the duration of intervals between repeated injections of the viral vaccine was reduced step by step. figures 9,а and 10,а show the trajectories of tumor growth when the vaccine is injected 3 and 4 times, and the time intervals between consecutive injections are reduced. this allows to drive the number of tumor cells below the lethality threshold. a b fig. 9. the strategy for multiple injection of the vaccine by reducing the interval between the doses 0v = 0,056 at the beginning of the treatment on τ1 = 10 day with the initial interval between injections ∆tstab = 20.1 days: а shows the trajectories of growth and death of tumor cells; b shows the dynamics of formation of antibodies 0 1( , )va v  at the 1st and 0 1( , )na v  at the 2nd stage of the immune response (solid curve denotes av(t), dashed curve denotes an(t)) а b fig. 10. the strategy for multiple injection of the vaccine by reducing the interval between the doses 0v = 0,06 at the beginning of the treatment on τ1 = 20 day with the initial interval between injections ∆tstab = 25.7 days: а shows the trajectories of growth and death of tumor cells; b shows the dynamics of formation of antibodies 0 1( , )va v  at the 1st and 0 1( , )na v  at the 2nd stage of the immune response (solid curve denotes av(t), dashed curve denotes an(t)) the dose of the viral vaccine was determined depending on the size of the tumor to implement the strategy of stabilizing the tumor size at the beginning of treatment v0 stab( τ1) (fig.8 b). the interval between the first and second introduction of the vaccine ∆t1 stab (τ1) was also determined depending on the size of the tumor at the beginning of treatment based n.a. babushkina, e.a. kuzina copyright ©2020 assa. adv. in systems science and appl. (2020) 14 on the condition of stabilization of the tumor size (fig. 8,а). the moment of the second injection of the vaccine was defined as τ2 = τ1 + ∆t1 stab. beginning from the third moment of the vaccine injection τ3 and further τi, the moment of the subsequent injection of the vaccine was defined as τi = τi-1 + ∆ti-1. the duration of the interval between injections was reduced by the step value and was calculated as ∆ti-1 = ∆ti-2 – step. in fact, the step value is the control parameter for the implementation of the strategy of gradual reduction of tumor size with each subsequent injections of a viral vaccine. the number of repeated injections of the vaccine to achieve regression of tumor growth depends on the step value. in the framework of the constructed model, this occurs when the number of tumor cells decreases below the lethality threshold (fig. 9,a-10,а). this strategy allows to achieve a reduction in the number of tumor cells below the lethality threshold only if the duration of tumor growth does not exceed 20 days (fig. 10,a). for large tumors it is necessary to carry out surgery, or use a strategy to stabilize the growth of the tumor (fig. 11,a). а b fig. 11. the strategy for multiple injection of the vaccine by reducing the interval between the doses 0v = 0,06 at the beginning of the treatment on τ1 = 25 day with the initial interval between injections ∆tstab = 28.9 days: а shows the trajectories of growth and death of tumor cells; b shows the dynamics of formation of antibodies 0 1( , )va v  at the 1st and 0 1( , )na v  at the 2nd stage of the immune response (solid curve denotes av(t), dashed curve denotes an(t)) 5.3. strategy of single injection of the viral vaccines testing the effectiveness of various doses of viral vaccine in a single injection at different points in tumor growth was carried out on the developed program module. it was determined that even with a single injection of a viral vaccine in certain doses, it is possible to obtain a reduction in the number of tumor cells below the v0let threshold, i.e. achieve complete destruction (fig.12a, 13a). at the same time, the number of antibodies produced at the first stage of the immune response exceeds their number at the second stage (fig. 12b, 13b). mathematical modeling of antitumor viral vaccine therapy… copyright ©2020 assa. adv. in systems science and appl. (2020) 15 а b fig. 12. the strategy of a single lethal dose injection on let 0v = 0.048 on τ1 = 3 day; а shows the trajectories of growth and death of tumor cells; b shows the dynamics of formation of antibodies 0 1( , )va v  at the 1st and 0 1( , )na v  at the 2nd stage of the immune response [19, p. 48, fig. 9] а b fig. 13. the strategy of a single lethal dose injection on let 0v = 0.075 on τ1 = 8 day; а shows the trajectories of growth and death of tumor cells; b shows the dynamics of formation of antibodies 0 1( , )va v  at the 1st and 0 1( , )na v  at the 2nd stage of the immune response thus, the death of tumor cells with a single injection of lethal doses occurs immediately in the first stage of the immune response as a result of the formation of large amounts of antibodies against the virus 0( )va v and a slight formation of antibodies against infected tumor cells 0( )na v . as a result of the analysis of the obtained results, a dependence was constructed, indicating that the magnitude of lethal doses leading to the destruction of all tumor cells at the first stage of the immune response depends on the size of the tumor at the time of its introduction (fig. 14). however, to achieve the complete destruction of tumor cells is possible only for small tumors, the growth of which does not exceed 8 days. n.a. babushkina, e.a. kuzina copyright ©2020 assa. adv. in systems science and appl. (2020) 16 fig. 14. the graph of lethal doses from the beginning of treatment ( )let 0 1v  with a single injection of a viral vaccine [19, p. 49, fig.11] 6. predicting the duration of the periods between the repeated injections of virus vaccine for the clinic to predict the duration of the periods between repeated injections of the viral vaccine for the clinic according to the results obtained in the experiment on mice, it is proposed to use the well-known in biodynamics allometric ratio, which connects the rate of metabolic processes in the body with the body weight in mammals [24-26]: bm a s=  , (25) where m is the body weight of the mammal, s is the rate of metabolic processes in mammals, a and b are constant parameters. allometric relations connect the body weight of animals and humans with a variety of other biological parameters that reflect the temporal characteristics of the processes of vital activity of the organism, including the duration of its life [24-26]. then the ratio (25) can connect the body weight of the animal with any other time parameter. in this case, such a time parameter is considered the duration of the periods between successive injections of the viral vaccine in the implementation of the strategy of stabilizing the tumor size ∆t : ( )bm a t=   . (26) for two species of mammals, which are human and experimental mouse, we can write the allometric ratio of the form: ( )human human bm a t=   and ( )mouse mouse bm a t=   , (27) where humant and mouset denote the duration of the periods between the injections of viral vaccines for human being and for mouse. to define the duration of the period between injections for the human being we can use the ratio: ( ) human human b mouse mouse m t m t  =  based on the fact that the average body weight values are known for a human being and a mouse, and the interval between successive injections of the vaccine is determined, the mathematical modeling of antitumor viral vaccine therapy… copyright ©2020 assa. adv. in systems science and appl. (2020) 17 duration of the period between injections of the vaccine for a person can be calculated from the equation: 1/( ) human human mouse b mouse m t t m  =  (28) the value of the parameter b was calculated as the slope of the straight line formed by the logarithmation of the allometric ratio, which relates the average body weight of different mammals and humans with the average life expectancy lifet (fig. 15) [10]: ln ln ln lnlife lifem b t a b t c=  + =  + (29) as a result of the regression analysis, the calculated value of the parameter b was obtained as b = 2,4981, which characterizes the slope of the regression line. fig. 15. dependence of the life expectancy of animals on their body weight for ○ mouse, □ rats, + guinea pigs, х rabbits, ◊ dogs, and * humans [10] taking in equation (28) the mass of the mouse equal to mousem = 0,025 kg and the weight of a human being is equal to humanm = 70 kg, with b = 2,4981 = 2,5 and 1/b = 0,39 we can calculate the time conversion factor from mouse to human. 0.470 ( ) 23.9 0.025 k = = then the duration of the period between injections of the vaccine for a human being can be calculated from the expression: 1/( ) human human mouse b mouse mouse m t t t k m  =  =   (30) the results of the evaluation of the duration of the periods between repeated injections of viral vaccines in the implementation of the strategy of tumor size stabilization for humans and mouse are shown in the table 1. table 1. intervals between repeated injections of viral vaccines for humans and mouse obtained on the basis of allometric ratios dose of viral vaccine 0v and when the vaccine was first interval between injections to stabilize tumor size in the experiment on mouse 0 1( , )mouset v  interval between injections to stabilize the size of the tumor in the clinic for humans 0 1( , )humant v  n.a. babushkina, e.a. kuzina copyright ©2020 assa. adv. in systems science and appl. (2020) 18 introduced 1 0v = 0.04 on 1 =1 day 0 1( , )mouset v  = 17.2 day 0 1( , )humant v  =1,14 year 0v = 0.047 on 1 =3 days 0 1( , )mouset v  = 18.4 day 0 1( , )humant v  =1,22 year 0v = 0.05 on 1 =5 days 0 1( , )mouset v  = 19.0 day 0 1( , )humant v  =1,26 year 0v = 0.056 on 1 =10 days 0 1( , )mouset v  = 20.1 day 0 1( , )humant v  =1,33 year 0v = 0.06 on 1 =15 days 0 1( , )mouset v  = 22.8 day 0 1( , )humant v  =1,5 year 0v = 0.06 on 1 =20 days 0 1( , )mouset v  = 25.7 day 0 1( , )humant v  =1,7 year 0v = 0.06 on 1 =25 days 0 1( , )mouset v  = 28.9 day 0 1( , )humant v  =2 year the method of allometric ratios is indirect and cannot guarantee 100% accuracy and reliability of the assessment of the duration of the periods between repeated injections of viral vaccines for their use in the clinic. however, this period of repeated injections of the viral vaccine may be a recommendation for the patient and for the doctor about the timing of the control examination. according to the results of the examination, it is necessary to decide on the condition of stabilization of the tumor size, which should not exceed the size of the tumor at the time of treatment. the results of the medical inspection should clarify the dose and adjust the strategy of repeated injections of the viral vaccine. 7. conclusion the results of the computational experiment conducted within this study indicate that the choice of cancer treatment strategy depends on the tumor size at the start of the treatment. the model also indicates that it is possible to reduce the number of tumor cells beyond the lethality threshold which guarantees that the tumor does not resume its growth. for the small tumors (less than 8 days’ growth as per our model), elimination of tumor cells can be achieved through single-shot virus-based vaccine administration. in this case, the death of tumor cells is caused by their destruction by the virus. however, in these conditions the number of antibodies produced against tumor cells is limited, which can result in recurrence of the tumor growth in future. for bigger tumors (within 20 days’ growth as per our model) elimination of tumor cells is possible if the vaccine is administered repeatedly with stepwise reduction of time periods between administrations. the model developed within this study also makes it possible to define treatment strategy that may restrain the tumor development, i.e. lead to cancer chronification. dosages and time intervals between vaccine administrations for this stabilization strategy were defined in the present study. an advantage of this strategy is that it can be used for the tumors of any size. thus it can be beneficial for the patients who cannot be subjected to surgical treatment, and creates new treatment possibilities for far-advanced cancer. this study also proposes a method of transferring the results of the computational experiment to the treatment of humans, based on allometric relationships between human and animal body weights, enabling clinicians to calculate the appropriate intervals between repeated vaccine administrations. this method makes it possible to plan the return visit to the therapist to make a decision on subsequent vaccine administration. mathematical modeling of antitumor viral vaccine therapy… copyright ©2020 assa. adv. in systems science and appl. (2020) 19 finally, results of the computational experiment based on mathematical modelling may substantially and meaningfully complement in vivo experiments for defining efficient cancer treatment strategies. development of advanced methods of cancer treatment is highly dependent on the costs of conducting experiments. active implementation of mathematical modelling approach on different stages of experimental research and clinical studies may contribute to the reduction of these costs as well. references 1. baryshnikov, a. yu. (2003). vzaimootnoshenie opukholi i immunnoy sistemy organizma [interaction between tumor and immune system]. prakticheskaya onkologiya, 4(3), 128–130 [in russian]. 2. grinevich, yu. a., khranovskaya, n. n. 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[dynamics of hematopoiesis]. moscow: medizina [medicine], [in russian]. mathematical modeling of antitumor viral vaccine therapy… copyright ©2020 assa. adv. in systems science and appl. (2020) 21 25. monichev, a. yu. (1987). a mathematical model of the spatial structure of bone marrow in hemopoietic dynamics. cybernetics and systems analysis, 23(2), 274–280. 26. kuzina, e.a., babushkina, n.a. (2016). programmnaja realizacija metoda prognozirovanija jeffektivnosti vakcinoterapii ot jeksperimenta v kliniku [software implementation of the method for predicting the efficacy of vaccine therapy from experiment to the clinic]. proc. of the 12th int. conf. “physics and radioelectronics in medicine and ecology” phreme-2016. vladimir: vgu, 125–130, [in russian]. n.a. babushkina, e.a. kuzina copyright ©2020 assa. adv. in systems science and appl. (2020) 22 appendix 1. values of model parameters equation parameter description ( )dn t dt – (1-2) n = 3.3613, n = 0.0332 n = 23 0n = 0.75 the parameters of the function of gompertz approximating the experimental curves of growth of population of tumor cells without vaccination (control) ( )dn t dt – (3) 1 = 1 1v cvt z=  + cvz = 4.5 the moment of the first injection of a viral vaccine the moment the beginning of the immune response against the virus delay time of the immune response against the virus ( )vdn t dt – (4-5) 1n cnt z=  + the moment the beginning of the immune response against infected tumor cells сnz = 10.5 delay time of immune response against infected tumor cells vk = 0.25 avk = 0.8 ank = 0.8 the constant coefficients of the equation of dynamics of the number of cells after a single injection of the vaccine ( )dv t dt – (6) v = 0.1 the reproductive rate of the virus v = 15 the rate of death of viruses in their interaction with antibodies 0v = 0.015 the initial condition of equation (2), which reflects the dose of the viral vaccine ( )vda t dt – (7) a = 100 the rate of formation of antibodies from the single plasma cells av = 70 the rate of loss of antibody due to the interaction with viruses v = 5 the rate of reduction of the number of antibodies due to natural destruction max va = 1.05 the maximum calculated number of the antibodies ( )vdc t dt – (8) с = 100 the rate of formation of plasma cells cv = 4.5 the size factor vnc = 0.001 the initial number of plasma cells ( )nda t dt – (9) an = 30 the rate of formation of antibodies from the single plasma cells an = 6.2 the rate of decline of antibodies due to the interaction with the tumor cells nn = 6.3 the rate of the reduction of the number of antibodies due to the natural destruction max na = 4.64 the maximum calculated number of antibodies ( )ndc t dt – (10) cn = 76.677 the rate of formation of plasma cells cn = 38 the size factor nnc = 0.0001 the initial number of plasma cells ( )p t – (11) p = 0.3 pk = 0.95 *1/p t = the parameters of the p(t) function describing the dynamics of the decrease in the proportion of rapidly proliferating cells as the tumor size increases the moment at which the number of fractions of rapidly and mathematical modeling of antitumor viral vaccine therapy… copyright ©2020 assa. adv. in systems science and appl. (2020) 23 *t = 35 суток slowly proliferating cells are equal ( )dn t dt – (13) v = 4.5 сут. the delay in the growth of tumor cells as a result of their death under the action of the virus v = 9.8 сут. the delay in the growth of tumor cells as a result of their death under the action of antibodies against infected tumor cells 1 vt = 5.82 1 nt = 12.32 the moments of reaching the maximum number of infected tumor cells by the beginning of the immune response 2 vt = 7.13 2 nt = 14.29 the moments of reaching the minimum number of infected tumor cells by the end of the immune response ( )1 v v vn t = 1.31, ( )1 n n vn t = 1.15 the maximum values of the number of infected tumor cells before the immune response at each of the two stages of their death ( )2 v v vn t = 0.84, ( )2 n n vn t = 0.1 the minimum values of the number of infected tumor cells at the end of the immune response at each of the two stages of their death ( )1 v v vn t = 0.47, ( )1 n n vn t = 1.04 the number of infected cells killed in the first and second stages of immune system stimulation 1. introduction 2. problem statement adv syst sci appl 2021; 01:86–94 published online at https://ijassa.ipu.ru. enhanced results on stability criteria for linear time delay systems with distributed delay via relaxed double integral inequality r. jeetendra1∗, b. jeevanandan1,2 1assistant professor,department of mathematics, kongu engineering college, erode-638 060, india. 2assistant professor,department of mathematics, kongu engineering college, erode-638 060, india. abstract: this paper investigates the matter of stability criteria for linear time delay systems with distributed delay. firstly, a relaxed double integral inequality is established to estimate the double integral terms appearing within the derivative of lyapunov-krasovskii functionals (lkfs) with a triple integral term. unlike the recently introduced jensen’s inequalities, wirtinger based integral inequalities, refined jensen’s inequalities and therefore the auxiliary function based integral inequalities the proposed relaxed integral inequality provides large feasible solution region and fewer conservative results. secondly, by constructing an augmented lyapunov-krasovskii functional with a triple integral term, the robust stability criteria for linear time delay systems with distributed delay are given in terms of linear matrix inequalities (lmis), which may be easily computed by the lmi toolbox of matlab. finally, two numerical examples are performed to indicate the effectiveness of the proposed criterion. keywords: robust stability, delay-dependent stability, lyapunov functional, linear matrix inequalities, time-delay systems, distributed delay 1. introduction many dynamic systems in the real world inevitably have time delays, and such delays often cause poor performance, oscillation or even instability of the system. consequently, the stability issue of time-delay systems has attracted researchers for many years. the main attention is paid to determine the admissible delay region, for which the systems remain stable, by developing effective delay-dependent stability criteria via the lyapunov-krasovskii stability theory. the issue relies on the handling of the integral terms arising within the derivative of the lyapunov-krasovskii functionals (lkfs). the development of new methods for this problem has always been a very important consideration. so as to scale back conservatism of stability criteria, variety of techniques are presented, including for example, jensen’s integral inequality [1], the wirtinger-based integral inequality [2], the various forms of wirtinger-based double integral inequality [3-5], bessel-legendre inequality [6], auxiliary function-based integral inequalities [7,8], the free-weighting matrix approach [4,9-11], reciprocally convex method [12,13] and free matrix based multiple integral inequality [14]. the well-known jensen’s inequality is often adopted because it could lead on to a stability test with fewer matrix variables. recently, wirtinger integral inequality introduced in [2] is shown more powerful than jensen’s inequality. later, another forms of integral inequalities are reported in [6,7,15,16-19] to further reduce the conservatism. ∗corresponding author: jee4maths@gmail.com enhanced results on stability criteria for linear time delay systems... 87 this paper presents a new relaxed double integral inequality to estimate the double integral term within the derivative of lyapunov-krasovskii functionals(lkfs). a replacement stability criterion is established by applying the newly proposed inequality which provides less conservatism and fewer number of decision variables. to indicate the effectiveness of the proposed criterion, two numerical examples are provided. notations: throughout this paper, rn is the n-dimensional euclidean space and rn×n is the set of all n× n real matrices. x > 0(x ≥ 0) means that the matrix x is a real symmetric positive definite matrix (positive semi definite). i denote the identity matrix with appropriate dimensions; col {·} means a column vector. ∗ in a matrix represents the elements below the main diagonal of a symmetric matrix. the superscript t denotes the transpose of the matrix. 2. problem formulation consider the following system with state and distributed delays: ẋ(t) = ax(t) + a1x(t− h) + a2 ∫ t t−h x(s)ds, (2.1) x(t) = φ(t), t ∈ [−h, 0] where, x(t) ∈rn is the state vector, a,a1, a2 ∈rn×n are constant matrices, h is a constant time delay satisfying h > 0 and φ(t) is a continuous vector-valued initial condition. 2.1. lemma[20]: for a given matrix m > 0 , the following inequality holds for all continuously differentiable functions x : [a, b]→ rn: (b− a) ∫ b a ẋ(s)tmẋ(s)ds ≥ ωt 1mω1 + 3ωt 2mω2 + 5ωt 3mω3 + 7ωt 4mω4 where ω1 = x(b)− x(a) ω2 = x(b) + x(a)− 2 b− a ∫ b a x(s)ds ω3 = x(b)− x(a) + 6 b− a ∫ b a x(s)ds− 12 (b− a)2 ∫ b a ∫ b θ x(s)dsdθ ω4 = x(b) + x(a)− 12 b− a ∫ b a x(s)ds+ 60 (b− a)2 ∫ b a ∫ b θ x(s)dsdθ − 120 (b− a)3 ∫ b a ∫ b σ ∫ b θ x(s)dsdθdσ relaxed double integral inequality 2.2. lemma: for symmetric positive definite matrix z ∈ rn×n, scalars α < β, and vector φ : [α, β]→ rn such that the integration concerned is well defined, the following inequality holds:∫ β α ∫ β u ϕt (s)zϕ(s)dsdu ≥ 2 (β − α)2 ωt 5zω5 + 16 (β − α)2 ωt 6zω6 (2.2) copyright c© 2021 assa. adv syst sci appl (2021) 88 r. jeetendra, b. jeevanandan where ω5 = ∫ β α ∫ β u ϕ(s)dsdu ω6 = − ∫ β α ∫ β u ϕ(s)dsdu+ 3 β − α ∫ β α ∫ β u ∫ β θ ϕ(s)dsdθdu proof: for a function λ(s) = k1 + k2s, integration by parts, we have∫ β α ∫ β u λ(s)ϕ(s) dsdu = λ(a) ∫ β α ∫ β u ϕ(s) dsdu+ 2k2 ∫ β α ∫ β u ∫ β θ ϕ(s)dsdθdu by setting λ(a) = −1, 2k2 = 3 β−α , the above equality is rewritten as∫ β α ∫ β u λ(s)ϕ(s) dsdu = ω6 then the following equality is obtained for any vector ω0 and any matrix m > 0:∫ β α ∫ β u λ(s)ωt 0mϕ(s) dsdu = ωt 0mω6 similarly, the following equalities are derived:∫ β α ∫ β u ωt 0lϕ(s) dsdu = ωt 0lω5∫ β α ∫ β u ωt 0lr −1ltω0 dsdu = (β − α)2 2 ωt 0lr −1ltω0∫ β α ∫ β u ωt 0lr −1mtλ(s)ω0 dsdu = 0∫ β α ∫ β u λ2(s)ωt 0mr−1mtλ(s)ω0 dsdu = (β − α)2 16 ωt 0mr−1mtω0 using the above equalities and the schur complement derives the following equality: ∫ β α ∫ β u [ ω0 λ(s)ω0 ϕ(s) ]t [ lz−1lt lz−1mt l ∗ mz−1mt m ∗ ∗ z ][ ω0 λ(s)ω0 ϕ(s) ] dsdu = ∫ β α ∫ β u ϕt (s)rϕ(s)dsdu+ sym { ωt 0lω5 + ωt 0mω6 } + (β − α)2 2 ωt 0 { 8lz−1lt +mz−1mt 8 } ω0 ≥ 0. where ωt 0 = [ ωt 5 ωt 6 ] , l = −2 (β − α)2 [z 0]t and m = −16 (β − α)2 [0 z] copyright c© 2021 assa. adv syst sci appl (2021) enhanced results on stability criteria for linear time delay systems... 89 that is, ωt 0l = −2 (β − α)2 ωt 5z and ωt 0m = −16 (β − α)2 ωt 6z which leads to (2.2). this completes the proof. 2.3. remark: the proposed relaxed double integral inequality provides the tightest estimation value of the double integral term ∫ b a ∫ b θ xt (s)zx(s)dsdθ > 0, compared with the widely used jensen’s integral inequality and wirtinger-based integral inequality. moreover, the additional positive term 16 (β−α)2 ωt 6zω6 reduces the estimation gap. therefore, the proposed relaxed double integral inequality will cause less conservative than the prevailing ones within the literature. by setting ϕ(s) = ẋ(s), the subsequent lemma are often obtained from the above lemma 2.2. 2.4. lemma: for symmetric positive-definite matrix z ∈ rn×n, scalars α < β, and vector ẋ : [α, β]→ rn such that the integration concerned is well defined, the following inequality holds:∫ β α ∫ β u ẋt (s)zẋ(s)dsdu ≥ 2χt1zχ1 + 16χt2zχ2 where χ1 = x(β)− 1 β − α ∫ β α x(s)ds χ2 = −1 2 x(β)− 1 β − α ∫ β α x(s)ds+ 3 (β − α)2 ∫ β α ∫ β u x(s)dsdu 3. main results in this section, delay dependent stability criteria for the system with distributed delays are derived interms of lmi as follows: 3.1. theorem: given h > 0, the system (2.1) is assymptotically stable if there exists positive definite matrices p ∈ r4n×4n, q, s, z ∈ rn×n, such that the following lmi holds: ξ = γpυt + υpγt + ψ < 0 (3.3) where γ = [e1 e3 e4 e5] υ = [ e0 e1 − e2 he1 − e3 h2 2 e1 − e4 ] copyright c© 2021 assa. adv syst sci appl (2021) 90 r. jeetendra, b. jeevanandan ψ = e1qe t 1 − e2qet2 + h2e0se t 0 + h2 2 e0ze t 0 − (e1 − e2)s(e1 − e2)t − 3 ( e1 + e2 − 2 h e3 ) s ( e1 + e2 − 2 h e3 )t − 5 ( e1 − e2 + 6 h e3 − 12 h2 e4 ) s ( e1 − e2 + 6 h e3 − 12 h2 e4 )t − 7 ( e1 + e2 − 12 h e3 + 60 h2 e4 − 120 h3 e5 ) s ( e1 + e2 − 12 h e3 + 60 h2 e4 − 120 h3 e5 )t − 2 ( e1 − 1 h e3 ) z ( e1 − 1 h e3 )t − 16 ( −1 2 e1 − 1 h e3 + 3 h2 e4 ) z ( −1 2 e1 − 1 h e3 + 3 h2 e4 )t e0 = ae1 + a1e2 + a2e3 and ei ∈ r5n×n are elementary matrices, for example et2 = [0 i 0 0 0]. proof: consider a lyapunov-krasvoskii canditate as v (t) = v1(t) + v2(t) + v3(t) + v4(t) where v1(t) = ηt (t)pη(t), v2(t) = ∫ t t−h xt (α)qx(α)dα v3(t) = h ∫ t t−h ∫ t β ẋ(α)tsẋ(α)dαdβ and v4(t) = ∫ t t−h ∫ t β ∫ t σ ẋ(α)tzẋ(α)dαdσdβ. where η(t) = col [ x(t), ∫ t t−h x(α)dα, ∫ t t−h ∫ t β x(α)dαdβ, ∫ t t−h ∫ t β ∫ t σ x(α)dαdσdβ ] the time derivative v (t) along the trajectories of system can be computed as follows: v̇1(t) = 2ηt (t)p η̇(t) = 2ξt (t)γpυt ξ(t) v̇2(t) = xt (t)qx(t)− xt (t− h)qx(t− h) v̇3(t) = h2ẋt (t)sẋ(t)− h ∫ t t−h ẋ(α)tsẋ(α)dα v̇4(t) = h2 2 ẋt (t)zẋ(t)− ∫ t t−h ∫ t β ẋ(α)tzẋ(α)dαdβ where ξ(t) = col [ x(t), x(t− h), ∫ t t−h x(α)dα, ∫ t t−h ∫ t β x(α)dαdβ, ∫ t t−h ∫ t β ∫ t σ x(α)dαdσdβ ] copyright c© 2021 assa. adv syst sci appl (2021) enhanced results on stability criteria for linear time delay systems... 91 and it can be rewritten as v̇ (t) = ξt (t) { γpυt + υpγt + e1qe t 1 − e2qet2 + h2e0se t 0 + h2 2 e0ze t 0 } ξ(t) − h ∫ t t−h ẋ(α)tsẋ(α)dα− ∫ t t−h ∫ t β ẋ(α)tzẋ(α)dαdβ. applying lemma 2.1 and 2.4 to the above integrals leads to −h ∫ t t−h ẋ(α)tsẋ(α)dα ≤ −ξt (t) { (e1 − e2)s (e1 − e2)t + 3 ( e1 + e2 − 2 h e3 ) s ( e1 + e2 − 2 h e3 )t + 5 ( e1 − e2 + 6 h e3 − 12 h2 e4 ) s ( e1 − e2 + 6 h e3 − 12 h2 e4 )t + 7 ( e1 + e2 − 12 h e3 + 60 h2 e4 − 120 h3 e5 ) s ( e1 + e2 − 12 h e3 + 60 h2 e4 − 120 h3 e5 )t } ξ(t) − ∫ t t−h ∫ t β ẋ(α)tzẋ(α)dαdβ ≤ −ξt (t) { 2 ( e1 − 1 h e3 ) z ( e1 − 1 h e3 )t + 16 ( −1 2 e1 − 1 h e3 + 3 h2 e4 ) z ( −1 2 e1 − 1 h e3 + 3 h2 e4 )t } ξ(t) hence, we have v̇ (t) ≤ ξt (t) { γpυt + υpγt + ψ } ξ(t) v̇ (t) ≤ ξt (t)ξξ(t). this completes the proof. 4. numerical examples in this section, two examples are used to illustrate the effectiveness of the proposed method. 4.1. example: consider the following system with distributed delay: ẋ(t) = [ 0.2 0 0.2 0.1 ] x(t) + [ 0 0 0 0 ] x(t− h) + [ −1 0 −1 −1 ] ∫ t t−h x(s)ds copyright c© 2021 assa. adv syst sci appl (2021) 92 r. jeetendra, b. jeevanandan the purpose is to match the utmost allowable upper bounds of h that guarantees the asymptotic stability of the above system. table 4.1 lists the computed maximum allowable upper bounds and also the number of decision variables which keep the system stability by different methods. from table 4.1, it’s clear that the proposed approaches can provide higher upper bounds than those within the existing results. it should be noted that our method provides maximum allowable boundary which is adequate to the analytical bound with fewer number of decision variables. table 4.1. upper bounds on h obtained for example 4.1 methods maximum h allowed nodv chen and zheng 2007 1.6339 85 seuret and gouaisbaut 2013 1.877 16 park et al. 2015 1.9504 59 zeng et al. 2015 2.0395 75 trinch 2015 2.0395 27 zhao et al. 2017 2.0402 45 park et al. 2018 2.0412 42 3.1 theorem 2.0412 39 analytical bound 2.0412 4.2. example: consider the following system with distributed delay: ẋ(t) = [ −2 0 0 −0.9 ] x(t) + [ −1 0 −1 −1 ] x(t− h) + [ 0 0 0 0 ] ∫ t t−h x(s)ds table 4.2 lists the computed upper bounds by different methods and it shows that our method provides an upper bound which is quite close to the analytical bound. table 4.2. upper bounds on h obtained for example 4.2 methods maximum h allowed nodv zhao et al. 2017 6.1663 45 zeng et al. 2015 6.1664 75 gu et al. 2003 (n=3) 6.171 67 chen et al. 2016 6.1719 106 park et al. 2018 6.1719 42 3.1 theorem 6.1719 39 analytical bound 6.1725 5. conclusion in this article, delay dependent stability criteria for linear time-delay system with distributed delay are proposed by the employment of the lyapunov method . by the development of augmented lyapunov functional and relaxed double integral inequality, the delay dependent stability criterion has been proposed. compared to the recently proposed integral inequalities the obtained ones could provide more accurate estimations on the handling of the integral terms arising within the derivative of the lyapunov-krasovskii functionals(lkfs) the obtained stability condition provides larger feasible solution region and fewer conservatism with fewer number of decision variables than the present ones within the literature. two numerical examples are presented to indicate the effectiveness of the proposed approach. copyright c© 2021 assa. adv syst sci appl (2021) enhanced results on stability criteria for linear time delay systems... 93 acknowledgements the authors sincerely thank the editors and anonymous reviewers for their careful reading, constructive comments and suggestions to improve the quality of the manuscript. references 1. gu, k., chen, j. & kharitonov, v.l. (2003) stability of time-delay systems. springer science bussiness media. 2. seuret, a., & gouaisbaut, f. (2013) wirtinger-based integral inequality: application to time-delay systems. automatica 2013, 49(9), 2860–2866. 3. park, m., kwon, o., park, j.h., lee, s., & cha, e. (2015) stability of time-delay systems via wirtinger-based double integral inequality. automatica 2015, 55, 204–208. 4. zeng, h.-b., he, y., wu, m., & she, j. (2015) new results on stability analysis for systems with discrete distributed delay. automatica 2015, 60, 189–192. 5. zhao, n., lin, c., chen, b., wang, & q.-g. (2017) a new double integral inequality and application to stability test for time-delay systems. applied mathematics letters 2017, 65, 26–31. 6. seuret, a., & gouaisbaut, f. (2015) hierarchy of lmi conditions for the stability analysis of time-delay systems. systems control letters 2015, 81, 1–7. 7. park, p., lee, w.i., lee, & s.y. (2015) auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems. journal of the franklin institute 2015, 352(4), 1378–1396. 8. park, p., lee, w.i., lee, & s.y. (2016) auxiliary function-based integral/summation inequalities: application to continuous/discrete time-delay systems. international journal of control, automation and systems 2016, 14(1), 3–11. 9. he, y., wang, q.-g., xie, l., & lin, c. (2007) further improvement of freeweighting matrices technique for systems with time-varying delay. ieee transactions on automatic control 2007, 52(2), 293–299. 10. chen, w.-h., zheng, & w.x. (2007) delay-dependent robust stabilization for uncertain neutral systems with distributed delays. automatica 2007, 43(1), 95–104. 11. zeng, h.-b., he, y., wu, m.,& she, j. (2015) free-matrix-based integral inequality for stability analysis of systems with time-varying delay. ieee transactions on automatic control 2015, 60(10), 2768–2772. 12. park, p., ko, j.w., & jeong, c. (2011) reciprocally convex approach to stability of systems with time-varying delays. automatica 2011, 47(1), 235–238. 13. lee, w.i.,& park, p. (2014) second-order reciprocally convex approach to stability of systems with interval time-varying delays. applied mathematics and computation 2014, 229, 245–253. 14. chen, j., xu, s., & zhang, b. (2016) single/multiple integral inequalities with applications to stability analysis of time-delay systems. ieee transactions on automatic control 2016, 62(7), 3488–3493. 15. trinh, h. (2015) refined jensen-based inequality approach to stability analysis of timedelay systems. iet control theory applications 2015, 9(14), 2188–2194. 16. kim, j.-h. (2016) further improvement of jensen inequality and application to stability of time-delayed systems. automatica 2016, 64, 121–125. 17. lee, t.h., park, j.h., park, m.-j., kwon, o.-m., jung,& h.-y. (2015) on stability criteria for neural networks with time-varying delay using wirtinger-based multiple integral inequality. journal of the franklin institute 2015, 352(12), 5627–5645. 18. zhang, c.-k., he, y., jiang, l., wu, m., zeng,& h.-b. (2016) stability analysis of systems with time-varying delay via relaxed integral inequalities. systems control letters 2016, 92, 52–61. copyright c© 2021 assa. adv syst sci appl (2021) 94 r. jeetendra, b. jeevanandan 19. park, m.-j., kwon, o., & ryu, j. (2018) generalized integral inequality: application to time-delay systems. applied mathematics letters 2018, 77, 6–12. 20. zhang, s., & qi, x (2017) improved integral inequalities for stability analysis of interval time-delay systems. algorithms 2017, 10(4), 134. copyright c© 2021 assa. adv syst sci appl (2021) introduction problem formulation lemma[20]: lemma: remark: lemma: main results theorem: numerical examples example: example: conclusion adv syst sci appl 2020; 03:153–165 published online at https://ijassa.ipu.ru. control model of the floodplain territories structure alexander a. voronin1*, alexander v. khoperskov1, inessa i. isaeva1, anna yu. klikunova1 1volgograd state university, volgograd 400062, russia abstract: we have developed a model of an integrated hydrological, environmental and socioeconomic structure of the floodplain area, which is the subject of strategic control based on the observational data, the results of numerical hydrodynamic and geoinformation modeling. the state assessment of each structural element is determined by the degree of correspondence between the hydrological type and the socio-economic type of the floodplain area. the goal of control is to implement a territorial structure that maximizes the value of the aggregated criterion for a stable state of the system. the control mechanism is a complex of hydraulic engineering projects (multiproject) in various channels in the floodplain area. the simulations results of the implementation of the sustainable development strategy of the environmental and socio-economic system of the volga-akhtuba floodplain northern part are presented. keywords: simulations, development management, hydrological regime, environmental and socio-economic systems, hydraulic engineering projects, volga-akhtuba floodplain 1. introduction the problem of sustainable development (sd), which until recently was in the focus of management/control theory and practice, has largely lost its popularity in the last decade. in our opinion, the reason for this is the existence of a difficult gap between existing conceptual and formal models, on the one hand, and the development management practice of real environmental and socio-economic systems (eses), on the other. for example, in [1, 2] the authors developed the concept of sd control based on game-theoretic models and the active systems theory. the main parameters of these models are the “system compatibility indices” calculated on the basis of the objective functions of its actors and ecosystem sustainability criteria. however, justifying these values that underlie these and many other sd models is an almost insurmountable difficulty in most cases. indeed, the shortest periods of degradation and, moreover, the restoration of regional ecosystems are decades, so the “compatibility indices” should characterize conflicts between generations to a greater extent than conflicts between its actors. in the absolute majority of cases, the development of modern eses occurs within the framework of the “weak sustainability” concept, according to which natural capital is exchanged for human and productive capital with varying degrees of efficiency. a possible transition to sd in a state of degrading ecosystems requires both a study of the conditions for their restoration and a readiness for the necessary self-restraint in the context of economic competition. in addition, the level of modern theoretical and empirical knowledge about real ∗corresponding author: a.voronin@volsu.ru 154 a.a. voronin, a.v. khoperskov, i.i. isaeva, a.yu. klikunova ecosystems at the regional level does not allow us to confidently identify the their stability thresholds and predict their dynamics. the floodplains of regulated large rivers with a developed eses are among the territories in which, on the one hand, this gap manifests itself in the most acute form, on the other, it can potentially be bridged more easily than on other areas. the high conditionality of their territorial structure by the unstable nature of floods is an objective factor of weak environmental sustainability, which, as a rule, is destroyed in the process of their socioeconomic development. the development of high-performance supercomputers and effective methods of computational fluid dynamics in recent years makes it possible to effectively simulate a realistic flood regime, which is a system-forming factor in floodplain areas, allowing to significantly reduce the complexity of the constructed sd models. an example of a model successful construction for sustainable development of a regional environmental system is the work [3] in which the authors created the model of ecological and economic control of the biological resources extraction of the azov sea with sustainable reproduction. their complex numerical simulation model successfully combines models of hierarchical differential games, biological kinetics, hydrodynamics using high performance computational algorithms for supercomputers. however, we note that the ecological criterion in this model (the stability of the biosystem) essentially coincides with the long-term economic criterion (sustainable extraction of biological resources). the work [4] is aimed at creating computer technology for effectively assessing the reliability of meeting the demands of volga water users and, in particular, for the lower volga including projects to improve environmental conditions. the model is based on the multicriteria analysis methods and the compromises theory. integrated decision support systems (dss) allow solving the problems of planning water resources management in river basins, scarcity of river water resources for socio-economic needs, rational balance between socio-economic and environmental needs. let us point out some good examples of dss for the haihe river basin [5, 6], the northern part of the volga-akhtuba floodplain [7], the illinois river [8], the pinhao river basin [9], the biosphere reserve elbe river landscape in lower saxony [10], the pecos river [11]. this work and our research use similar methods and technologies, such as interdisciplinary modeling, gis technologies, multi-criteria assessments, peer review. our research is distinguished by a more active use of direct numerical hydrodynamic modeling. the objective of this study is to solve the problem of identification and realizability of the values of the territorial structure parameters, necessary for sd. our model is based on a simplified version of the “indices of system compatibility” (see [1]), implemented as a vector for assessing the self-compatibility of the territorial structure of the eses, the components of which are calculated by territorial aggregation of expert consistency indices between the hydrological (h) type and the environmental and socio-economic (ese) type for each site of the territory. the practical implementation of the proposed research concept is the identification and study of the attainability of the target h-structure of the northern part of the volga-akhtuba floodplain within the volgograd region (hereinafter called vaf) in the context of the sustainable development of its eses through hydrotechnical multiproject, based on the construction of hydraulic structures (canals and dams) with calculated characteristics. these characteristics should stabilize the h-structure of the floodplain area and thereby make it possible to implement the mechanisms of ecological and economic control by its economic entities, which is described in [12]. the set of software tools presented in this paper includes the decision support system (dss), which is an extension of the dss for river water resources distribution. this set of tools for control sustainable development of floodplain areas is an implementation of the general model of the sd information-analytical system proposed by the authors of works [1, 2]. copyright © 2020 assa. adv syst sci appl (2020) control model of the floodplain territories structure 155 2. research methods, tools and technologies the hydrological regime of the floodplain area is determined by three main factors: the river hydrograph (q(t) is the volume of water flowing per unit of time or the variation of discharge), the channel structure and the area topography [13–15]. natural changes in the last two factors, as a rule, are insignificant for several tens of years, and the first factor has high variability at short time intervals. if the distribution function of flooding volumes for the period t is determined, then the frequencies of flooding of interfluve fragments calculated for this period remain constant over time t (t � t ). in other words, the h-structure of the floodplain area z(h) exists stably during the time interval t . this structure is described as a set of maps territories 〈m(h) i (i = 1, ..., n)〉, flooded by water with frequencies from the specified ranges, and the distribution function of the relative flooded area at the floods peak ψ(x) : p (s(h)/s < x) gives the aggregated state of the system (s is the total area of the vaf). the time interval t is determined experimentally from the observations data of the parameters of flood river hydrographs over a long period t � t . the choice of the value of t affects the existence conditions, the type and accuracy of determining the stable hydrological structure z(h). in turn, the stability of the hydrological structure is the main factor in the formation of the territorial environmental and socio-economic (ese) systems of the floodplain area z(ese), characterizing the spatial sites distribution by functionally homogeneous ese-types. the term “territorial natural structure” in this work means a biotopes system of the main structural elements of a floodplain ecosystem. ese-structure can be built on the basis of the corresponding cadastral maps. hand esestructures define the complex structure z(c) = z(h) ∩ z(ese), which is given by flood maps of functionally homogeneous territories or ese maps of hydrologically homogeneous territories m(c) = ∪ijm(c) i,j , wherem(c) i,j =m(h) i ∩m(ese) j (i = 1, ..., n; j = 1, ...,m), (2.1) where ‖ s(c) ij ‖, s (c) ij = s(m(c) ij ) is the land area. the complex structure z(c) is also determined by the distribution functions of the relative area of the flooded terrain at the floods peak ψj(x) : p (s (h) j /s (c) j < x), s (c) j = n∑ i=1 s (c) ij , (2.2) where the number of frequency intervals n and their sizes, the number of ese-types m are determined by the purpose and accuracy of modeling. the construction of a hydroelectric dam and / or abrupt climatic changes are often the reasons for the formation of a new h-structure and an adaptive transformation of the esestructure of the floodplain. but these factors can also be the reason for the activation of slow changes in rivers channel topography, disrupting the stability of the h-structure, and, as a result, causing a decrease in the degree of self-compatibility between the hand esestructures, which leads to ese-degradation. ese-degradation counteraction requires periodic fine tuning of hand ese-structures. within the framework of the weak stability concept, the control task is the implementation of a constant or variable h-structure, which ensures high efficiency and safety of a constant or variable territorial ese-structure. the objective h-structure should first of all ensure the sustainability of the ecosystem in the case of a sustainable development concept. this in turn requires the stabilization of the h-structure itself with parameters that ensure stable flooding of its biotope. in the absence of a floodplain eses model, we estimate the potential efficiency of z(c) based on the expertly constructed vector of “disharmony” δ = (δ1, ..., δm), δj = 1− copyright © 2020 assa. adv syst sci appl (2020) 156 a.a. voronin, a.v. khoperskov, i.i. isaeva, a.yu. klikunova kj (j = 1, ...,m), in which the functions kj = k(ψj) ∈ [0; 1] characterize the degree of correspondence between hydrological regime of functionally homogeneous territory and its ese-type j. the complete uncertainty of the kj functions for some j, their exact coincidence for different j, the insensitivity of the functions k(ψj) for all j to certain ranges of flooding frequencies allow us to reduce the dimension of the actual structures z(h),z(ese),z(c) and the dimension of the problem (2.3). we can rewrite the general control problem for the structure of the floodplain area as a multi-criteria optimization problem: δτ+θ(u) −→ min u∈u , z(c) τ+θ∈ωz , (θ = 1, 2, ...,θ) , (2.3) where ωz is the set of admissible structures, depending on the chosen concept of development stability (weak or strong stability), u is the set of realizations of a hydraulic engineering multiproject, θ is planning horizon, the τ index corresponds to the calculation of hydrological structures for the period [τ − t , τ ]. the lack of effective methods for solving inverse problems of hydrodynamics allows using only heuristic branch-and-bound methods to solve the problem (2.3), while the exact methods can be used at some stages in (2.3). special particular problems of (2.3) and methods for their solution are described in the works [7, 16] using hydrodynamic, geoinformation and game-theoretic simulations. 3. tools and simulations results for volga-akhtuba floodplain vaf is located in the lower reaches of the volga river (the area of the northern part of the floodplain is approximately 870 km2, the number of small natural channels exceeds 200 with total length of about 1000 km). figure 3.1 shows a diagram of the general hydrological system of vaf, which can be conditionally divided into the western part, which is flooded mainly from the volga river. the hydrological regime in the eastern part is determined by the water dynamics in the akhtuba river and in the large middle trunk water system of natural channels of the floodplain. the structure of our software package for solving the problem (2.3) for eses of vaf is shown in the figure 3.2. individual parts of this software are described in detail in [7,16,17]. the key component is a numerical hydrodynamic model of the area flooding, which takes into account all the main physical and meteorological factors [13, 17, 18]. this mathematical model is based on 2d shallow water equations, which has been successfully applied to similar problems, see, for example, [19, 20]. the existence, type and accuracy of determining the stable hydrological structure of the vaf under the condition of the stability of the channel system and topography in the period 1962–2018 are established as a result of computational experiments. we have generated sequences of all possible samples of fragments of the time series v (t) (t = 5, 6, ..., 57), containing t data units, using the observations array of normalized volumes of v for flood hydrograph through the hydroelectric dam. for these fragments, we have constructed a set of experimental distribution functions ψ(t)(x) : p (v (t)/vmax < x) (vmax is the maximum volume for the entire observation period). the performed analysis indicates that the error in determining the magnitude of the flood volume realized in the vaf with frequency of nflood ≥ 0.85 for t ≥ 20 does not exceed 0.05. initial hydrological data are based on the official primary statistical information on the operation of the volga hydroelectric power station (see also [21]). the regression model for describing the flow of flood waters into the akhtuba river in the period 1962-2018 was built based on the results of measurements of flood water levels at the gauging station in the volgograd city [7]. the modeling results and the constructed retrospective hydrological copyright © 2020 assa. adv syst sci appl (2020) control model of the floodplain territories structure 157 fig. 3.1. scheme of the complex of hydraulic engineering projects. the blue lines show the system formed by the erik gniloj and the erik pahotnyj in the northern part and by the kashirskij water tract in the eastern part of the vaf. the yellow lines show only some of the small channels associated with this system. light purple channels are filled directly from the volga river. red lines indicate canals from the akhtuba river. magenta diamonds show possible positions of dams of different types. magenta dashed line delimits the conditional “western” zone from the “eastern” zone. structure z(h) 1982 show that the main factor of the loss of stability of this hydrological structure and the drying up of the interfluve is a gradual decrease in the average volume of low-flow and flood waters in the akhtuba river due to the progressive degradation of the volga riverbed after the construction of the volga hydroelectric power station in 1962. figure 3.3 shows the distribution functions ψ1982(x) and ψ2018(x). comparative analysis of curves 1 and 2 shows both the general reduction of the flooded territory and the qualitative change in its structure. the share of sustainably flooded territory is reduced from 85 percent to 47 percent. the northern territory of the vaf cadastral map contains 607 cadastral parcels that belong to 35 cadastral types. moreover, cadastral types have not been established for 35 percent of the vaf. such areas and recreational areas are not evaluated and the k(ψj) functions are undefined for them. we combine the remaining types into four groups based on the proximity of the estimated coefficients. the first two are wetlands (j = 1) and flood-meadows (type of farmland) (j = 2) and they collectively represent the floodplain ecosystem biotope for which non-flooding is damage. the other two are areas of residence of the population (j = 3) and forests + economic areas (j = 4), for which flooding is a damage. very small floods copyright © 2020 assa. adv syst sci appl (2020) 158 a.a. voronin, a.v. khoperskov, i.i. isaeva, a.yu. klikunova fig. 3.2. the structure of the software package for our model of the floodplain territory control. on the volga river were observed every 10–15 years in the period 1880–1930 before the construction of the volga system of hydroelectric power plants [21]. this allows us to relate the stability boundary of the floodplain ecosystem with the value of the threshold frequency of flooding of the floodplain biotope nflood = 0.85. table 3.1. evaluative coefficients of the efficiency of typical elements of the volga-akhtuba floodplain. nflood 0 (0; 0.25] (0.25; 0.50] (0.50; 0.85) [0.85; 1.0] ki,1 0 0 0 0.25 1 ki,2 0 0 0.25 0.5 1 ki,3 1 0 0 0 0 ki,4 1 0.1 0 0 0 copyright © 2020 assa. adv syst sci appl (2020) control model of the floodplain territories structure 159 fig. 3.3. model integral distribution functions of the flood area are shown by the following curves: 1 — ψ1982(x); 2 — ψ2018(x); 1e is eastern zone for 1962–1982; 1w is western zone for 1962–1982; 2e is eastern zone for 1998–2018; 2w is western zone for 1998–2018. the symbols 2wp and 2ep denote the corresponding curves, taking into account the implementation of hydraulic projects. the horizontal dashed line corresponds to flooding of the floodplain with a frequency of at least 0.85. evaluation of the effectiveness of the territorial structure of vaf is carried out using special functions of the form k(ψj) = 1 sj n∑ i=1 kij(nflood)sij (j = 1, ...,m) , (3.4) where kij(nflood) are the dependencies of the coefficients on the flooding frequency (see table 3.1). thus, the current complex (integrated) structure of vaf is characterized by n = 5, m = 4. the presence of a significant part of the territory with undefined evaluation functions k(ψj) is a source of great uncertainty in the task (2.3). the expert estimation of the threshold value of the area of the floodplain ecosystem biotope is approximately 0.5s, which noticeably exceeds both the area of modern stable flooding (0.3s) and the value of s1 + s2. to remove this uncertainty, we included in the first group of ese structures (j = 1) a part of the territory with indefinite evaluation functions k(ψj), which was steadily flooded in the period 1962– 1982. we estimate the error in determining the this territory area as 0.02, while it is 0.07 for the period 1998-2018. the reason for this is the stable flooding in the past of all the plains, while the modern border of the stable flooded area is within the plains. figure 3.4 shows both the target areas of stable flooding and areas where flooding causes damage. for the ese structure z(ese) 2018 changed in this way, we calculated the “disharmony” vectors for the complex and virtual structures using a series of computational experiments. z(c) 2018 = z(h) 2018 ∩ z (ese) 2018 : δ2018 = (0.45; 0.53; 0; 0.11) , z(c) virt = z(h) 1982 ∩ z (ese) 2018 : δvirt = (0; 0.06; 0.02; 0.17) . the first vector shows a high level of disharmony of the vaf eses due to catastrophic drying up of the interfluve, the second vector is the objective vector for solving the problem (2.3). the main part of the multiproject is the channel from the volgograd reservoir to the akhtuba river bypassing the volga river and a dam that closes the entrance to akhtuba from the volga river (see “barrier dam on the akhtuba river” on the figure 3.1). the parameter of the flood hydrograph is the maximum discharge through the canal (qch). the product of qch by the duration of the flooding τf (which is equal to 15 days in all experiments) gives copyright © 2020 assa. adv syst sci appl (2020) 160 a.a. voronin, a.v. khoperskov, i.i. isaeva, a.yu. klikunova fig. 3.4. upper panel: the target area for stable flooding of the vaf. bottom panel: areas where flooding causes damage. the volume of flood waters that determine the flooded territory map and its area. since the amount of flood water entering the akhtuba river is less than 10 percent of the total spring water (v (flood) tot ), the project allows for stable flooding of almost any target area in the eastern part of the vaf, limited by the condition of non-flooding of socio-economic territories (curve 2ep in figure 3.3) even in conditions of high natural variability of v (flood) tot . considering that qch is a small value of the total water discharge in the river volga during the flood period, we restrict ourselves to the case of a fixed qch. we determine the location of the dams by heuristic analysis of the directions of water flows across the boundary between the east and west zones (see dashed line in figure 3.1) during the flooding stage. the designed dams in the channels should prevent the flow of water from the western zone to the eastern zone. features of the real topography limit the number of copyright © 2020 assa. adv syst sci appl (2020) control model of the floodplain territories structure 161 fig. 3.5. maps of the stably flooded area of the vaf in the period 1998-2018. up panel: model without bypass channel with qch = 0. bottom panel: model with bypass channel qch = 2000 m3 sec−1. positions of such dams. we have found the dam locations that have a noticeable effect using hydrodynamic simulations. the results of hydrodynamic modeling and optimization of projects for deepening small channels and installing additional dams on the vaf territory (see works [22–24]) showed the possibility of reducing the discussed negative effect by 8 percent. figure 3.1 demonstrates the hydrological system of the floodplain and the complex of hydraulic projects. we distinguish copyright © 2020 assa. adv syst sci appl (2020) 162 a.a. voronin, a.v. khoperskov, i.i. isaeva, a.yu. klikunova the eastern zone (“e”), adjacent to the akhtuba river, and the western zone (“w”), for which the hydrological regime is determined directly by the volga river. we have highlighted in red small channels in the western part of the vaf, for which deepening of the channels is also advisable. the optimal locations of dams on small canals are marked by rhombuses in figure 3.1. fig. 3.6. dependencies of the vector components of “disharmony” δm on qch for the eses. figure 3.3 shows the distribution functions of flooded areas for two time intervals (1962– 1982 and 1996–2018) separately for the western zone (curves 1w, 2w, 2wp) and for the eastern zone (curves 1e, 2e, 2ep), which are highlighted in figure 3.1. the horizontal dashed line corresponds to stable flooding of the interfluve. the post-design distribution function for the western zone is denoted as 2wp. a comparison of the 2w and 2wp curves shows an approximately 25 percent reduction in stable flooded area. distribution function x = const = 0.47 (see vertical line 2ep) corresponds to the annual flooding with qch = 2000 m3 sec−1. the arrow in figure 3.1 indicates the position of a possible large bypass channel from the volgograd reservoir to the akhtuba river, the project of which is being discussed. such construction should include a barrier dam on the akhtuba river. figure 3.5 shows the results of our spring flood simulations with and without such a bypass channel. the bypass canal can significantly improve the hydrological regime for the eastern area of the vaf. small projects are based on the work to clear and deepen small channels (they are indicated in yellow, purple and red in figure 3.1), as well as the use of special dams to regulate the water flow in the middle channels. figure 3.6 shows the curves δm(qch), which determine the dependence of the objective functions on the main project parameter qch. the values δ(1) 1 and δ (1) 2 correspond to the project with a bypass canal and a barrier dam, and the values δ(2) 1 and δ (2) 2 additionally include hydraulic engineering projects on the territory of the vaf (deepening of channels, dams on small channels). the case qch = 0 corresponds to the pre-design state and δ2018 = (0.45; 0.53; 0; 0.11). the increasing sequence of qch values determines the sequence of nested flood maps of the vaf at the peak of floods. there are two critical values of the parameter qch that define the set of feasible solutions to problem (2.3). the value qch1 determines the largest value of qch, at which the equalities δ3 = δ4 = 0 are satisfied. this value corresponds to the largest flood map for the peak of high water included in the stable flood target map. the value qch2 defines the largest flood map by inclusion, at which the equality δ3 = 0 is fulfilled. thus, the segment [qch1;qch2] is the “compromise zone” that contains the value of qch. a decrease in copyright © 2020 assa. adv syst sci appl (2020) control model of the floodplain territories structure 163 the total disharmony index δ1 + δ2 + δ4 (δ1 = δ1e + δ1w, δ2 = δ2e + δ2w) in this segment is possible if some economic zones are added to the multiproject or if the target area of stable flooding is reduced. 4. conclusion we present an approach to the control of the spatial structure of the floodplain territory, which is the basis for creating strategies for their development. the consistency functions between the hydrological type of each spatial structural element and its ese-type are used as objective functions. the hydrotechnical multiproject for the interfluve channels is a control tool. the implementation of this approach for the northern part of the volga-akhtuba floodplain makes it possible to determine the parameters of the hydraulic engineering multiproject, which provide the conditions for its sustainable development. the analysis of expenses and ese-effects of the implementation of the investigated multiproject are beyond the scope of our task. we also do not discuss here game-theoretic models of the environmental and economic control mechanisms of the economic entities of the vaf. three such mechanisms (the control of the economic entity of the volga hydroelectric power station, the control of the urbanization process, the mechanism of co-financing for the project of deepening small channels) were investigated in our previous works [16, 24, 25]. modeling a complex of projects and mechanisms for sustainable development of the vaf territory, as well as a comparative analysis with a similar complex that implements the strategy of weak sustainability, are the subject of our future research. undoubtedly, each floodplain territory has a unique structure, therefore, the combination of hydraulic projects and mechanisms of ecological and economic control in each of them is also individual. however, we believe that the proposed approach to the control of the territorial structure of floodplain environmental and socio-economic systems will be useful both for theoretical research and for the practical implementation of strategies for their sustainable development. acknowledgements the authors were supported by the ministry of science and higher education of the russian federation (government task no. 0633-2020-0003). references 1. ougolnitsky, g. a. 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(2018). program complex for simulation modelling of hydrotechnical projects in floodplain terrains. mathematical physics and computer modeling, 21(2), 59–74. 24. vasilchenko, a., voronin, a., svetlov, a., antonyan, n. (2016). assessment of the impact of riverbeds depth in the northern part of the volga-akhtuba floodplain on the dynamics of its flooding. international journal of pure and applied mathematics, 110(1), 183–192. 25. voronin, a. a., eliseeva, m. v., khrapov, s. s., pisarev, a. v., & khoperskov, a. v. (2012). the regimen control task in the eco-economic system ”volzhskaya hydroelectric power station–the volga-akhtuba floodplain’”. ii. synthesis of control system. problemy upravleniya, 6, 19–25. copyright © 2020 assa. adv syst sci appl (2020) introduction research methods, tools and technologies tools and simulations results for volga-akhtuba floodplain conclusion advances in systems science and applications (2014) vol.14 no.3 199-229 the theory of parametric control of macroeconomic systems and its applications(iii) a. ashimov, zh. adilov, r. alshanov, yu. borovskiy and b. sultanov kazakh national technical university named after k. satpayev, 22 satpaev street, almaty, 050013, kazakhstan abstract this work consists of three parts and presents the recent results of development of the theory of parametric control of macroeconomic systems and some its applications for solving a number of concrete problems. keywords mathematical model, structural stability, parametrical identification, parametric control part 3. applications of the theory of parametric control of macroeconomic systems 3.1 macroeconomic analysis and parametric control of macroeconomic system based on an econometric model of small open economy 3.1.1 building an econometric model of a small open economy general view of the model for small open economy of kazakhstan, that describes equilibrium conditions in macroeconomic markets of goods, money, labor and of capital, taking into account its interaction with the russian federation and the rest of the world, is presented by the following relations [1-2]. equilibrium in the goods market of the republic of kazakhstan is presented by the formula : y d = y s (1) where y s real supply of goods in the republic of kazakhstan, in billions of tenge (national currency of the republic of kazakhstan) ; y d = c + i + g + nefull-real demand for goods in the republic of kazakhstan, in billions of tenge ; nefull = ne + neru real volume of goods net export from the republic of kazakhstan, in billions of tenge ; neru = qru ex − erq ru im real net export of goods from the republic of kazakhstan to the russian federation, in billions of tenge (indicator, that takes into account the terms of cooperation of the republic of kazakhstan within the regional customs union) ; qex real volume of exports of goods from the republic of kazakhstan to the russian federation, in billions of tenge ; qru im real volume of imports of goods to the republic of kazakhstan from russian federation, in billions of tenge ; er = epz/p real exchange rate in the republic of kazakhstan, tenge/us dollar ; e exchange rate of national currency in the republic of kazakshtan, tenge/us dollar ; pz the general price level in the outside world ; p the general price level in the republic of kazakstan ; ne = qw ex − erq w im real net export of goods from the republic 200 a. ashimov : the theory of parametric control of macroeconomic systems and ... of kazakshtan to the rest of the world, in billions of tenge (indicator that takes into account the terms of cooperation between the republic of kazakhstan and the rest of the world) ; qw ex real volume of exports of goods from the republic of kazakhstan to the rest of the world, in billions of tenge ; qw im real volume of imports of goods to the republic of kazakhstan from the rest of the world, in billions of us dollars ; g real volume of government expenditures in the republic of kazakhstan, in billions of tenge ; i real volume of the republic of kazakshtan investments to the basic capital, in billions of tenge ; c real volume of consumption by households in the republic of kazakhstan, in billions of tenge. all real data are presented for the year of 2000. equilibrium in the money market of the republic of kazakhstan is presented by the following relation : m/p = l (2) where l is real cash balances in the republic of kazakhstan, in billions of tenge ; m nominal money supply in the republic of kazakhstan (in billions of tenge). equilibrium in the labor market of the republic of kazakhstan : pdy/dn = w (3) where w nominal wage rate in the republic of kazakhstan, in thousands of tenge ; dy/dn marginal productivity of labor in the republic of kazakhstan ; y real gross domestic product (hereinafter gdp) in the republic of kazakhstan, in billions of tenge ; n number of employed people in the republic of kazakhstan, in thousands of persons. equilibrium in the capital market : pnefull = nke (4) where nke net export nominal volume from the republic of kazakhstan, in billions of tenge. let’s introduce additional notations for economic indicators used in development of a model : mru real money supply in russian federation (in billions of rubles) and gru real volume of government expenditures in russian federation (in billions of rubles), indicators allowing for conditions of operating of a country as part of regional customs union ; i the average interest rate of banks for loans in the republic of kazakhstan ; iz interest rate of the outside world (market yield on us, treasury securities at 1-year constant maturity, quoted on investment basis) ; pavg = 0.6p + 0.4epz/e2000 weighted average price level in the republic of kazakhstan ; expected exchange rate in the republic of kazakhstan (tenge/us dollar) ; êe = (ee− e)/e expected growth rate of the exchange rate in the republic of kazakhstan ; p oil average oil price (in thousands of tenge for a advances in systems science and applications (2014) vol.14 no.3 201 barrel) ; ∆ operator of first difference for the series : ∆x = x −x−1;x−1lag variable. preliminary econometric analysis showed the possibility for evaluation of macroeconomic indicators c, l, w, i, y, qru ex , qru im , qw ex , qw im, nke of equilibrium conditions in macroeconomic markets as a function of the regression on the basis of the following set of time series : y, c, l, i, nefull, w, n, pavg, i, qru ex , p oil, mru , gru , qru im , qw ex , er, qw im, nke, iz , êe for the years of 2000-2011 according to statistical data of national economies of the republic of kazakhstan and russian federation. in order to build non-spurious regression functions the considered sets were checked for stationary with the help of augmented dickey-fuller method (adf)and were decomposed [3]. according to the results of a check-up for stationarity of time series : regression functions of the following type c = c(y ), l = l(y, i,nefull), w = w (n,pavg), nke = nke(iz , êe, i), i = i(i), y = y (n), qru im = qru im (y,gru ), qw im = qw im(y, er) were built by the method of ordinary least squares on the basis of the following non-stationary series and the results of their analysis on the statistical significance are presented respectively in table 1 : table 1 regression functions for non-stationary series consumption of domestic products in the : c= 558.3 + 0.38y republic of kazakhstan(r2=0.99) (0.00) (0.00) demand for real cash balances in the l=0.7y -82.6 i -0.8 nefull republic of kazakhstan(r2=0.92) : (0.00) (0.02) (0.09) price of labor supply in the republic w =-0.15 n + 1174.8 pm+0.4ez/2000 of kazakhstan, where pm = 0.6p (r2 = 0.98) : (0.00) (0.00) net capital export from the republic of nke = 291iz − 6090ẽe − 25.6i kazakhstan (r2 = 0.65) : (0.03) (0.02) (0.35) investments of the republic of kazakhstan i = 5885.5-291.5 i to the basic capital (r2 = 0.33) : (0.01) (0.05) production function in the republic of y =-25255+4.27n kazakhstan (r2=0,95) : (0.00) (0.00) import of goods from the russian qru im = 0.07y + 0.89∆gru federation (r2=0.89) : (0.00) (0.00) import of goods from the rest of the world qw im = 0.22y + 3.75er (r2 = 0.79) : (0.00) (0.01) regression functions of the type qru ex = qru ex (er, p oil,mru , gru ), qw ex = qw ex(er, p oil) were built by the method of least squares on the basis of corresponding stationary series or stationary with respect to determined trends and the results of the analysis for statistical significance are presented respectively in table 2 : 202 a. ashimov : the theory of parametric control of macroeconomic systems and ... table 2 regression functions for stationary series import of goods from the republic of : qru ex = 4.4∆er + 31.4p oil kazakhstaninto the russian federation −0.066mru + 0.128gru (r2=0.72) : (0.03) (0.04) (0.02) (0.00) exportof goods from the republic of kazakhstan to the rest of the world qw ex = −11.2er + 278.0p oil + 1830.4 (r2=0.99) : (0.01) (0.00) (0.01) regressions functions (where corresponding time series are stationary with respect to the determined trends) were checked on spuriousness by the t-test [4]. spurious regressions functions (where time series were non-stationary) were checked on spuriousness by the engle-granger cointegration test [5]. the model of small open economy of the republic of kazakhstan based on the equilibrium conditions in macroeconomic markets of goods, money, labor and capital (1)-(4) and based on the built regression functions (tables 1, 2) has the following form :  y d = 3201.8 + 0.84 m p − 17.63 epz p + 4.01 e−1p z −1 p−1 + 281.8p oil − 0.06mru − 0.694gru + 0.811gru −1 y s = −25525 + 19967.4p + 13392.7pz e e2000 y zb0 = 6311.72 + 1066.9p oil − 66.72 epz p + 15.17 e−1p z −1 p−1 − 0.2276mru + 3.07gru −1 − 1003.45 iz p + 21000 ee e − 19369.8 1 p − 0.221 m p 2 − 1.04 epz p 2 + 0.205 e−1p z −1 pp−1 + 19.77 p oil p − 0.034 mru p − 0.04 gru p + 0.0478 gru −1 p y s = y d = y zb0 i = 18.5− 0.0025 m p − 0.0117 ep 2 p + 0.0023 e−1p z −1 pp−1 + 0.19p oil − 0.00003mru + 0.0005gru −1 (5) here y zb0 is the function of zero balance of payments of the republic of kazakhstan. advances in systems science and applications (2014) vol.14 no.3 203 3.1.2 estimating stability indicators of the model for a small open economy the quality of the researched econometric model of the small open economy is estimated by the stability indicator β (section 1.3.2), which characterizes a change in the equilibrium model solutions by small deviations of the input parameters used. if the stability indicator of the model takes a small value for a small deviation of input variables used, it is considered that the model is qualitative in the sense of stability indicator. estimation of stability indicators is made by the algorithm 5, where the vector x = {m,g,pz , iz , ee, p oil,mru} has been considered as the vector of input parameters, and the vector z = {y, e, i} has been considered as the vector of output variables. conducted computing experiments show that deviations of equilibrium solutions up to 1 % correspond to the deviations of input factors within 1%. this confirms the fact that the considered econometric model (5) is a qualitative model in sense of stability indicator β. 3.1.3 parametric control of the country’s export depending on uncontrollable factors based on the fact of dependence of the solution of algebraic equations on its coefficients, we propose an approach to parametric control of national economy evolution taking into account the requirements for equilibrium on macroeconomic markets, which comes down to making recommendations based on the optimal values of economic tools in the form of solutions of mathematical programming problems based on the econometric model of a small open economy. let us consider the possibility of estimating the optimal values of m and g tools of economic policy for the given values of the uncontrollable input parameters p oil, iz , pz , ee, mru and gru that represent the values of these factors in the framework of the customs union by example of one country and the rest of the world in 2011 within the framework of the model is-lmzb0 (built on statistics for 2000-2011) in sense of the maximum criterion : qex = qw ex +qru ex → max (6) here qex is the function of total exports of goods. the stated estimate can be obtained by solving the following problem of mathematical programming. problem 3.1. based on the mathematical model (5) find values (m, g), that provide maximum to the criterion (6) under the constraints (7) here m * and g* are accepted values of money supply and government expenditures respectively, for the years of 2008-2011 ; y ∗, p ∗, e∗, i∗ basic equilibrium solutions of the system (5) ; y , p , e, i optimal equilibrium solutions of the system (5). 204 a. ashimov : the theory of parametric control of macroeconomic systems and ...  |m −m∗| ≤ 0.1m∗, |g−g∗| ≤ 0.1g∗, |p − p ∗| ≤ 0.1p ∗, |e− e∗| ≤ 0.1e∗, |i− i∗| ≤ 0.1i∗, |y − y ∗| ≤ 0.1y ∗, (7) the proposed approach to the parametric control of national economy evolution consists in realization of the following algorithm : 1. choice of mathematical model based on statistical analysis of the regression functions and estimation of stability indicator of the econometric model of economic general equilibrium for the open economy of the republic of kazakhstan ; 2. statement of the mathematical programming problem ; 3. prediction of uncontrollable factors pz , iz , p oil, mru , gru and ee for the period of choosing the recommendations on economic policy ; 4. solution to the mathematical programming problem based on the selected mathematical model to forecast values of uncontrollable factors ; 5. making recommendations on values of the m and g tools based on the analysis of the results of the mathematical programming problem for predicted values of the uncontrollable factors and possible additional information on the economic conjuncture. below we present an illustration of the proposed approach of parametric control of the national economy evolution for 2012. 1. let the model of a small open economy be a mathematical model selected on the basis of estimation of stability indicators (less than 1%) in 2011. 2. as a statement of the optimization problem for the model of a small open economy in 2011 we take the statement of the problem 3.1. 3. forecasted values of uncontrollable factors, obtained on the basis of the models built taking into consideration the results of time series decomposition into components, took the following values for 2012 : pz = 1.32 ; p oil = 14.75 thousandtenge per barrel ; iz = 0.17% ; ee = 148.68 tenge for one us dollar ; mru = 27949.1 billion rubles and gru =10898.3 billion rubles. 4. solution to the mathematical programming problem on the basis of the model for a small open economy by example of the republic of kazakhstan and the predicted values of uncontrollable factors for 2012 are : m = 6733.0 billion tenge ; g = 1444.4 billion tenge ; the value of the criterion max(qex w + qex ru ) = 3465.1 + 335.6 = 3800.7 billion tenge ; 5. the following can be proposed as a recommendation : solutions obtained during the experiment m = 6733.0 billion tenge and g = 1444.4 billion tenge or advances in systems science and applications (2014) vol.14 no.3 205 some correcting values, those can be obtained on the basis of the additional data analysis on economic conjuncture. 3.2 macroeconomic analysis and parametric control of cyclical dynamics the major section of modern macroeconomic theory is propositions on market cycles, in which the factors generating them are considered and different mathematical models for their analysis are proposed [1,6-8]. suppression of market cycles is the major field of stabilization policy [9-10]. 3.2.1 macroeconomic analysis and parametric control of cyclical dynamics based on kondratiev cycle model model description this model combines descriptions of non-equilibrium economic growth and nonuniform scientific and technological advancement [11]. the model is described by the following system of equations, including two differential and one algebraic equation :  n(t) = ay(t)a, dx/dt = x(t)(x(t)− 1)(y0n0 − y(t)n(t)), dy/dt = n(t)(1− n(t))y(t)2(x(t)− 2 + µ+ l0 n0y0 ), n0 = ayα0 . (8) here t is the time (in months) ; x is the efficiency of innovations ; y is the capital productivity ratio ; y0 is the capital productivity ratio corresponding to the equilibrium trajectory ; n is the rate of savings ; n0 is the rate of saving corresponding to the equilibrium trajectory ; µ is the coefficient of withdrawal of funds ; l0 is the job growth rate corresponding to the equilibrium trajectory ; a and a are some model constants. estimation of the model parameters is carried out based on statistical information from the republic of kazakhstan for the years 2001-2005 [12]. the deviations in the observed statistical data and the calculated data do not exceed 1.9% within the considered period. as a result of solving the problem of parametric identification, the following values of the exogenous parameters are obtained : α = −0.0046235, y0 = 0.081173, n0 = 0.29317, µ = 0.00070886, l0 = 0.00032161, x(0) = 1.911144. a retrospective prediction for 2006 and 2007 are characterized by errors equal to 6.1% and 12.1%, respectively, for the capital productivity ratio, and 2.3% and 11%, respectively, for the rate of savings. the respective cyclic phase trajectory of the kondratiev cycle model is presented in fig.1. the period of cyclic trajectory corresponding to the statistical information of the republic of kazakhstan for the given years is estimated to be 206 a. ashimov : the theory of parametric control of macroeconomic systems and ... 232 months. fig.1 cyclic phase trajectory of the kondratiev cycle model fig.2 chain-recurrent set for the kondratiev cycle model estimating the robustness of the kondratiev cycle model without parametric control the estimation of structural stability (robustness) of the mathematical model advances in systems science and applications (2014) vol.14 no.3 207 is carried out according to the 4th component of the parametric control theory (section 1.1) in the chosen compact set of the model phase space. fig.2 presents an estimate of the chain-recurrent set r(f,n) obtained by the application of the chain-recurrent set estimation algorithm for the region n = [1.7; 2.3]×[0.066; 0.098] of the phase plane oxy of system (8). since the set r(f,n) is not empty, one can draw no conclusion about the weak structural stability of the kondratiev cycle model in n on the basis of robinson’s theorem. however, since there is a non-hyperbolic singular point in n , namely, the center (x0 = 2− µ+l0 n0y0 , y0), then system (8) is not weakly structurally stable in n. parametric control of the evolution of economic system based on the kondratiev cycle model choosing the optimal parametric control laws is carried out in the environment of the following four relations : 1) n0(t) = n0 ∗ + k1 y(t)− y(0) y(0) ; 2) n0(t) = n0 ∗ − k2 y(t)− y(0) y(0) ; 3) n0(t) = n0 ∗ + k3 x(t)− x(0) x(0) ; 4) n0(t) = n0 ∗ − k4 x(t)− x(0) x(0) . (9) here ki is the scenario coefficient ; n∗ 0 is the value of the exogenous parameter n0 obtained as a result of the estimation of parameters. the problem of choosing the optimal law of parametric control at the level of the econometric parameter n0 can be formulated as follows. on the basis of mathematical model (8), find the optimal parametric control law in the environment of the set of algorithms (9), ensuring reach of optimal values of the following criterion : k = 1 t t∑ t=1 (( x(t)− x0 x0 )2 + ( y(t)− y0 y0 )2 ) → min (10) (here t = 232 is the period of the cycle) under the constraints 0 ≤ y(t) ≤ 1, 0 ≤ n(t) ≤ 1, 0 ≤ x(t) (11) the base value of the criterion (without parametric control) is as follows : k = 0.0307. the value of criterion k = 0.007273 for the control law, that is optimal in 208 a. ashimov : the theory of parametric control of macroeconomic systems and ... fig.3 capital productivity ratio without parametric control and with use of law 4, optimal in the sense of criterion k fig.4 efficiency of innovations without parametric control and with use of law 4, optimal in the sense of criterion k the sense of the criterion (10) of the 4th law, from the set (9) represented before is obtained by solving the problem formulated above through application of the parametric control approach to the evolution of the economic system. corresponding value of adjustable coefficient of this law is . the values of the model’s endogenous variables without applying parametric control and with use of the optimal parametric control law for criterion k are presented below in graphic form (fig.3 and fig.4). estimating the structural stability of the kondratiev cycle mathematical model with parametric control advances in systems science and applications (2014) vol.14 no.3 209 to carry out this analysis, the expressions for optimal parametric control laws (11) with the obtained values of the adjustable coefficients are substituted into the right-hand side of the second and third equations of system (1) for the parameter n0. then, by using a numerical algorithm for estimating the weak structural stability of the discrete-time dynamical system for the chosen compact set n determined by the inequalities 1.7 ≤ x ≤ 2.3, in the state space of the variables (x, y), the estimation of the chain-recurrent set r(f,n) as the empty (or onepoint) set is obtained. this means that the kondratiev cycle mathematical model with optimal parametric control law is estimated as weakly structurally stable in the compact set n . analysis of the dependence of the optimal value of criterion k on the parameter for the variational calculus problem based on the kondratiev cycle mathematical model let us analyze the dependence of the optimal value of criterion k on the exogenous parameters µ (share of withdrawal of capital production assets per month) and a for parametric control laws (11) with the obtained optimal values of the adjusted coefficients ki, where the values of the parameters (µ, a) belong to the rectangle a = [0.00063; 0.00147]× [−0.01; 0.71] in the plane. plots of dependencies of the optimal value of criterion k (for parametric control laws 0 and 2, yielding the maximum criterion values) on the uncontrollable parameters (see fig.5) were obtained by computational experimentation. the projection of the intersection line of the two surfaces in the plane (µ, α) consists of the bifurcation points of the extremals of the given variational calculus problem. 3.2.2 macroeconomic analysis and parametric control of cyclical dynamics based on dynamic stochastic general equilibrium model for the economy of kazakhstan in nonlinear dynamical stochastic general equilibrium (dsge) model is presented on the base of given composition and behavior of agents, their inter action in stochastic conditions and of taking the principle of rational expectations [13]. this nonlinear dsge model of the economy consists of : first-order aggregate conditions of optimization problems of agents (house hold sand intermediate product producers) [14] ; description of government activity rules ; and rules of shocks specifying in terms of either first-order auto regression s or gaussian white noises. first-order aggregate conditions involve equilibrium conditions in market of labor, capital, intermediate and final goods. the nonlinear dsge model that was built involves both the model of actual economy, and the model of potential economy. the model of potential economy is similar to the model of actual economy by its composition, except that potential economy functions under flexible prices and wages (in the model of actual economy the se prices are not flexible), and also 210 a. ashimov : the theory of parametric control of macroeconomic systems and ... fig.5 plots of the dependencies of the optimal value of criterion k on exogenous parameters µ, a in absence of “extra charge” shocks (those are in the model of actual economy). the nonlinear dsge model in question has the following vector form : etf θ(xt−1, xt, xt+1,h σh t ) = 0 (12) here et is sign of conditional mathematical expectation given information available at the point of time t (t = 1, 2, ...) ; f θ is known vector function ; θ is parameters set, consisting of structural parameters of the model and auto regression parameters of shocks ; xt is vector, consisting of endogenous variables and shocks, determined by first-order auto regressions ; x0 is given ; hσh t is vector, consisting of gaussian white noises, σh is corresponding diagonal covariance matrix. according to the technique taken for dsge model, linear approximation of nonlinear dsge model (12) was built in neighborhood of its stationary point x [13]. mentioned point is found by solving vector equation (15) obtained from (14) by dropping time subscripts and nulling white noises : f θ(x,x,x, 0) = 0 (13) log-linearization of dsge model of f . s mets and r. wouters in neighborhood of its estimated stationary point gives linear dsge model of the following form : aθx̂t−1 +bθx̂t + cθetx̂t+1 +dθhσh t = 0, t = 1, 2, 3... (14) advances in systems science and applications (2014) vol.14 no.3 211 here the sign ≪∧≫ corresponds to linearized variable, aθ, bθ, cθ, dθ are matrices of corresponding dimensions. in this example we consider the case of implementing state economic policy by the taylor rule [15], describing behavior of national bank insetting interest rate, and rules for determining the size of government spending as well. in the framework of linear model (14) the taylor rule for determining governmental bonds yield is presented in the following form : r̂t = ρr̂t−1 + (1− ρ) ( π̄t + rπ(π̂t−1 − π̄t−1) + ry (ŷt − ŷ p t ) + r∆π(π̂t−1 − π̂t−1) + r∆y ( ŷt − ŷ p t − ( ŷt−1 − ŷ p t−1 )) + ηrt (15) and the rule of government spending in the following form : ĝt = ρgĝt−1 + ηgt (16) here r̂t, ŷt, ŷ p t , ĝt, π̂t are variables, corresponding to : governmental bondsyield (1 + interest rate), output, potential output, government spending and inflation. ηrt is interest rate shock, given in terms of gaussian white noise ; π̄t is inflation shock ; ηgt is government spending shock, ρ, rπ, r∆π, r∆y , ρg are the parameters of the equations (15), (16). estimating parameters of linear dsge model on the basis of statistical data of the economy of the republic of kazakhstan model (16) solution was obtained by the blanchard-kahn algorithm [16-17]. this solutionis presented in the form of first-order vector auto regression : x̂t r̂t ĝt  = qθ  x̂t−1 r̂t−1 ĝt−1 + f θhσh t , t = 1, 2, 3... (17) hereinafter x̂t is vector-column consisting of all endogenous variables in the model (including shocks determined in terms of auto regression), excluding state policy tools of governmental bonds yield r̂t, and the size of government spending ĝt. vectors x̂0, r̂0, ĝ0 are given ; qθ, f θ are matrices of corresponding dimensions. estimating parameters of the model in question (14) (using (17)) was made by the bayesian estimation method (the metropolis-hastings algorithm with the number of simulations of 4 000 000) using the kalman filter [18]. as observations were used quarterly data for seven macroeconomic indicators of kazakhstan (gdp, investments, consumption, employment, average wage, refinancing rate, and inflation) from 2002i till 2010iii. we found the logarithm of mentioned statistical indicators and linearly detrended them. for using the kalman filter within 212 a. ashimov : the theory of parametric control of macroeconomic systems and ... the bayesian approach the model (17) was supplemented with vector equation of the dimension : ŝt = m  x̂t r̂t ĝt  (18) here m is matrix, each row of which contains one unity, all of the rest its elements are equal to 0. as the results of measuring of observed variables were taken log-deviations from its linear trends of macroeconomic indicators values (consumption, investments, gdp, inflation, average wage, employment, refinancing rate), corresponding to observed variables. statistical data for the republic of kazakhstan from 2002i till 2011iii was used in this study. for using the bayesian approach there were given a priori density distribution p = p0(θ ′,σh) of parameters θ′,σh . the form and probabilistic characteristics of this distribution from were used in the research [13], with the exception of mathematical expectations of a priori distribution of parameters σh . mentioned mathematical expectations were increased 2.5 times relative to corresponding values from s mets f . and wouters r. in connection with large sampled standard deviations of economic indicators of kazakhstanin comparison with eurozone. according to the bayesian approach method [18], using like lihood function, obtained on the basis of the model(17), (18) using the kalman filter, and a priori distribution of parameters p0(θ′,σh) as well,was found posterior joint density distribution of initial estimates of parameters : p = p1(θ ′,σh). then using the metropolis-hastings algorithm with density was p1(θ ′,σh) generated a sample, consisting of 4 000 000 sets of parameters θ′,σh . finally, as required estimates of parameters were taken corresponding sampled averages. quality of applied method for finding the parameters estimates was tested by retro prognosis. for this purpose there were made predictions for mentioned observed economic indicators for four periods from 2010 iv till 2011 iii. root mean square deviations of obtained expected predicted values of economic indicators from corresponding statistical data were about 3%. analysis of shocks effectson gdp and inflation using estimating of impulse responses on disturbances within the framework of internal shocks of the economy of the republic of kazakhstan fig.6 and fig.7 present relatively impulse responses of real gdp and inflation on (unit positive) shocks. each diagram presented in figures is obtained by calculation of linear model(17) for initial zero values of all endogenous variables of the model and the value of chosen shock equal to its standard deviation for zero period. under this all of the values of this shock for non-zero time values, and also the values of all other shocks of the model for all of time values were taken as the null. advances in systems science and applications (2014) vol.14 no.3 213 fig.6 impulse responses of real gdp on singular positive shocks fig.7 impulse responses of inflation on singular positive shocks analysis of the diagrams of impulse response of real gdp presented in fig.6 shows the following : • given positive shocks of productivity (εat ), labor supply (εlt ) and investments (εit ) gdp increases. • positive shocks of preferences (εbt ) and government spending (ηgt ) also in214 a. ashimov : the theory of parametric control of macroeconomic systems and ... crease gdp (since these shocks increase,respectively, consumption and government spending) • positive shock of extra charge for goods (ηpt ) decreases gdp, and shock of extra charge for wage (ηwt ) increases gdp. • positive monetary shock (ηrt ) results in production decline (because of interest rate growth). analysis of the diagrams of impulse response of inflation presented in the fig.7 shows the following : • given positive shocks of extra charge for goods and extra charge for wage inflation increases • given positive shock of productivity inflation negligible decreases. the rest of shocks do not practically have an effect on inflation. these responses of the model to shocks correspond with theoretical propositions. decomposing indicators evolution to shocks effect parts in retrospective period indicators decomposition in retrospective period, presented in fig.8 and 9,illustrates the contribution (in percentage) of each shock effect of the estimated model (17) to deviations of actual values of gdp and inflation indicators from corresponding equilibrium values for the period 2002i-2012i. presented diagrams show how deviations of gdp and inflation from their corresponding trends in retrospective period (from 2002 i till 2012 iii) emerged according to positive and negative effects of the shocks in question. for instance, deviation of gdp from trend in 2009iiiequal to -4.93% is the sum of positive summands : 1. shock effects of labor supply (εlt ) equal to 3.02% deviation of gdp from trend, 2. shock effects of interest rates (ηrt ) equal to 1.51% deviation of gdp from trend, 3. shock effects of extra charge for goods (ηpt ) equal to 0.85% deviation of gdp from trend and negative summands : 4. shock effects of preferences (εbt ) equal to -3.88% deviation of gdp from trend, 5. shock effects of extra charge for wage (ηwt ) -3.63% deviation of gdp from trend, 6. shock effects of extra charge for capital (ηqt ) -1.16% deviation of gdp from trend, 7. shock effects of government spending (ηgt ) -1.12% deviation of gdp from trend, 8. shock effects of productivity (εat ) -0.69% deviation of gdp from trend (rest advances in systems science and applications (2014) vol.14 no.3 215 shock effects of investments (εit ) and shock effects of inflation (π̄t) are negligible). that is deviation of gdp from trend equal to -4.93% is the sum of all shock effects of the model for mentioned period. for instance, deviation of inflation from trend in 2007iv equal to 6.32% is the sum of only positive summands (all of the rest effects are negligible) : 1. shock effects of extra charge for goods equal to 4.87% deviation of inflation from trend ; 2. shock effects of extra charge for wage 1.43% deviation of inflation from trend. analysis of the diagram in fig.8 shows also that break-neck growth of gdp during the period 2004-2007 was mainly because of extra charge shocks (for wage, good, capital), lessening of gdp growth in 2008-2011 was mainly because of negative shock effects of preferences, extra charge for capital and wage. analysis of the diagram in fig.9 shows that inflation deviation from the equilibrium level in the period 2002-2011 was almost fully because of extra charge shock for good and wage. in other words, all of the rest shocks of the model do not practically have an effect on inflation values in mentioned period. fig.8 shocks effects on gdp rate fig.9 shocks effects on the quarterly inflation 216 a. ashimov : the theory of parametric control of macroeconomic systems and ... prediction of shocks effects on economic indicators and suppression of their effects based on dsge model of smets-wouters in the paper by estimated model (17) were obtained predicted values of macroeconomic indicators (gdp and inflation) for 1, 4, 10, 20, 30 and 40 quarters (i.e. correspondingly for 2011iv, 2012iii, 2014i, 2016iii, 2019i, 2021iii). for estimating shock effects on error variances of economic indicators predictions were used the standard technique for defining decomposition of error variance of predictions for the models of vector auto regressions [19]. the results obtained by the software dynare matlab toolbox [http ://www.dynare.org] are presented in tables 1 and 2. there are no shocks, in these tables, which effects on variance less than by 0.01%. analysis of the tables 3 and 4 shows the following. error variances of prognosis of gdp generally are determined by preference shocks, government spending shock, and extra charge shocks on the cost of capital. error variances of prognosis of inflation generally are determined by extra charge shocks on goods and extra charge shocks on wages. table 3 prognosis (billion tenge in average prices of 1994) and decomposition of variance of the quarterly gdp prognosis time prognosis decomposition of variance (in %) hori mathematical standard ηat ηbt ηgt ηlt ηit ηrt ηqt ηpt ηwt-zon expectation deviation 1 271.413 8.17 2.25 22.90 17.57 1.47 0.09 1.07 45.32 6.13 3.19 4 292.166 10.960 4.04 21.73 15.30 3.25 0.15 1.56 41.99 8.27 3.71 10 327.034 13.229 4.04 21.73 15.30 3.25 0.15 1.56 41.99 8.27 3.71 20 385.810 16.067 4.09 20.69 14.20 3.70 0.14 1.51 39.84 7.90 7.92 30 456.450 19.136 4.03 20.39 13.98 3.67 0.14 1.49 39.47 8.01 8.82 40 542.395 22.751 4.02 20.37 13.97 3.67 0.14 1.49 39.45 8.03 8.86 table 4 prognosis (in %) and decomposition of variance of the quarterly inflation prognosis time prognosis decomposition of variance (in %) horizon mathematical standard ηat ηqt ηpt ηwtexpectation deviation 1 1.22% 0.75% 0.29 0.01 92.24 7.47 4 1.36% 0.78% 0.30 0.06 72.31 27.33 10 1.62% 0.80% 0.28 0.15 62.65 36.92 20 1.78% 0.81% 0.27 0.16 60.71 38.85 30 1.83% 0.81% 0.28 0.17 60.55 39.00 40 1.85% 0.81% 0.28 0.17 60.53 39.01 advances in systems science and applications (2014) vol.14 no.3 217 in realization of the state policy for minimizing shock effects on economic indicators in the capacity of its tools we choose additive summands ηrt , η g t in the expressions (14), (15), desired values of which are searched in terms of deterministic values instead of respective shocks. the parametric control approach for minimizing shock effects consists in the following. let t be number of quarter, starting from which the state realizes parametric control policy for minimizing shock effects. at each time point t = t, t + 1, t + 2, ... on the basis of estimated model (17), written in the form x̂t+i r̂t+i ĝt+i  = qθ  x̂t+i−1 r̂t+i−1 ĝt+i−1 + f θhσh t+i , i = 1, 2, 3, ..., 40 (19) such deterministic values of tools ηrt+1, ..., ηrt+40, η g t+1,..., η g t+40 are defined, those give the minimum for criterion lt = et 40∑ i=1 βi(π̂2 t+i + λy ŷ 2 t+i),min(ηrt+1, ..., η r t+40, η g t+1, ..., η g t+40)lt (20) (characterizing expected discounted total deviation of gdp and inflation values from respective equilibrium values (trend)) under the following constraints on endogenous variables of the model. mathematical expectations for inflationand bond yield in this time horizon should not deviate from respective equilibrium values more than by 0.5% : |etπ̂t+i| ≤ 0.005, i = 1, 2, 3, ..., 40 (21) |etr̂t+i| ≤ 0.005, i = 1, 2, 3, ..., 40 (22) and mathematical expectation for the size of government spending by 5.0% from their trend values : |etĝt+i| ≤ 0.05, i = 1, 2, 3, ..., 40 (23) moreover, at each specified time point t, the value of current state of economy for the time point t−(x̂t, r̂t, ĝt) is known. after receiving a new information (the values of variables (x̂t+1, r̂t+1, ĝt+1) at the next time point (t + 1)), the values of tools ηrt+2, ..., ηrt+41, ηgt+2,..., ηgt+41 are calculated again by solving the problem (19)-(23) for respective period. here discount factor, is some weight coefficient. introduce a new minimization criterion : l̃t = 40∑ i=0 βi ( (etπ̂t+i) 2 + λy (etŷt+i) 2 ) ,min(ηrt+1, ..., η r t+40, η g t+1, ..., η g t+40)l̂t (24) 218 a. ashimov : the theory of parametric control of macroeconomic systems and ... it is not difficult to check, that this criterion differs from by the value, independent from variables ηrt+1, ..., η r t+40, η g t+1, ..., η g t+40. from the relation (19), by taking mathematical expectations for both of its parts, we get et  x̂t+i r̂t+i ĝt+i  = qθet  x̂t+i−1 r̂t+i−1 ĝt+i−1 + f θ′′ [ ηrt+i ηgt+i ] , i = 1, 2, 3, ..., 40 (25) here f θ ′′ is matrix, comprised by two corresponding columns of the matrix f θ consequently, optimal values of variables of the problems (19)-(23)and (21)-(25) coincide between each other. derived optimization problem (21)-(25) applies to classical (deterministic) type of variational calculus problems, which for each time point t is solved by the linear and quadratic programming method (with matlab application). below in the paper it is estimated the effectiveness of application of formulated above parametric control in assumption that the state will implement this policy during 30 years (120 periods). that is, assume that at each time point t (t = t, t +1, t +2, ..., t +119, t is number of quarter, corresponding to 2011iii) the stated etermines the values of parameters ηrt+1, η g t+1 by solving above mentioned optimization problem(19)-(23) (or, that is the same, (21)-(25)). in computing experiment it is assumed that the economy is precisely described by estimated model :  x̂t+i r̂t+i ĝt+i  = qθ  x̂t+i−1 r̂t+i−1 ĝt+i−1 + f θhσh t+i , i = 1, 2, 3, ..., 160 where x̂t is known. in the paper, for estimating the effectiveness of application of formulated above parametric control approach the monte-carlo method was used with estimate of 100 development scenarios of economy. let us present aggregative algorithm for estimating application of the parametric control approach. 1. generation of the sample, consisting of 100 elements-sets of values of vector gaussian random values (white noises) {h ′σh t+1,h ′σh t+2, ..., h ′σh t+120}j where j = 1, ..., 100, hσh t+i = [h ′σh t+i , η r t+i, η g t+i] t with known probabilistic characteristics of noises σh . 2.for each element of the sample {h ′σh t+1,h ′σh t+2, ...,h ′σh t+120}1, {h ′σh t+1,h ′σh t+2,..., h ′σh t+120}2,..., {h ′σh t+1,h ′σh t+2, ..., h ′σh t+120}120 : 2.1 the calculation of the model with parametric control : we solve optimization problem (21)-(25) for period t = t i.e. we find respective advances in systems science and applications (2014) vol.14 no.3 219 ηrt+1, ..., η r t+40, η g t+1, ..., η g t+40 ; from obtained set of valueswe take ηrt+1, η g t+1 (we drop rest values ηrt+2, ..., η r t+40, η g t+2, ..., η g t+40. 2.2. the model is calculated for 1 step with shocks values hσh t+1, which consist of tools values ηrt+1, η g t+1 determined by the state and shocks values h ′σh t+1,which were realized by economy independently (exogenously) from the state policy. x̂t+1 r̂t+1 ĝt+1  = qθ  x̂t r̂t ĝt + f θhσh t+1 2.3 the steps 2.1 and 2.2 are iteratedfor values t = t+1, t+2, t+3, ..., t+120. 3. on the bases of obtained 100 trajectories of gdp and inflation is built average trajectory (expected prognosis value) and standard deviations of prognosis for the period t = t, t +1, t +2, t +3, ..., t +120. obtained values are compared with basic prognosis. realization results of formulated algorithm show that for used sample of shocks the parametric control of suppressing shocks effects provides diminishing predicted standard deviations of gdp by 58.3% at the average in prognosis horizon from 2011iii till 2021iii (see fig.10). the parametric control of suppressing shocks effects provides diminishing predicted standard deviations of inflation by 32.0% at the average in prognosis horizon from 2011iii till 2021iii and diminishing samples tan dard deviation of inflation by 47.8% in comparison with actual data during the period2002itill 2011iii (see fig.11) 3.3 macroeconomic analysis and parametric control of the economic growth based on computable general equilibrium model for the economic sectors presentation of computable general equilibrium model non-autonomous computable general equilibrium model (cge model) in general form is presented by the following system of relations [2], [20]. 1) subsystem of differential equations, connecting endogenous variables values for two successive years : x1(t+ 1) = f1 (x1(t), x2(t), x3(t), µ(t), a(t)) (26) here t = 0, 1, ..., n−1 is number of year, discrete time ; x(t) = (x1(t), x2(t), x3(t)) ∈ rm is vector of endogenous variables of the system ; xi(t) ∈ xi(t) ⊂ rmi , i = 1, 2, 3 (27) here the variables x1(t) involve the values of capital assets of the sectors-producers, budgets of 220 a. ashimov : the theory of parametric control of macroeconomic systems and ... fig.10 prognostic values of real gdp for the basic scenario and the parametric control approach fig.11 prognostic values of inflation for the basic scenario and the parametric control approach (in %) economic agents and so on ; x2(t) involve the values of demand and supply of agents in different markets and so on ; x3(t) are different kinds of market prices and budget parts in markets with state-set prices for various economic agents ; m1 +m2 +m3 = m ; advances in systems science and applications (2014) vol.14 no.3 221 u(t) ∈ u(t) ⊂ rq is vector function of controllable (adjustable) parameters. coordinate values of this vector correspond to various state economic policy tools, for instance, such as state budget parts and budget parts of economic agents, various tax rates, governmental bonds yield and so on ; a(t) ∈ a ⊂ rs is vector function of uncontrollable parameters (factors). coordinate values of this vector characterize various external and internal social and economic factors depending on time : export and import goods prices, population size of the country, production functions parameters and so on ; x1(t), x2(t), x3(t), u(t) are compact sets with non-empty interiors ; xi =∪n t=1xi(t), i = 1, 2, 3 ; x = ∪3 t=1xi ; u = ∪n−1 t=0 u(t), is open connected set ; f1 : x × u ×a → rm1 , is continuous mapping. 2) subsystem of algebraic equations, describing behavior and interaction of agents in various markets within sampled year, these equations allow expressing the variables by exogenous parameters and rest endogenous parameters : x2(t+ 1) = f2 (x1(t), x3(t), x3(t), u(t), a(t)) (28) here f2 : x1 ×x3 × u ×a → rm2 is continuous mapping. 3) subsystem of recurrence relations for iterative calculations of equilibrium values of market prices in various markets and budget parts in markets with state-set prices for various economic agents : x3[q+ 1] = f3 (x2(t)[q], x3(t)[q], l, u(t), a(t)) (29) here q = 0, 1, ... is number of iteration ; l is the set of positive numbers (adjustable constants of iterations, when their values decrease,economic system comes faster to its equilibrium condition, however,at the same time the risk of the case when prices go to negative range increases ; f3 : x2×x3× (0,+∞)ms ×u ×a → rm2 is continuous mapping (that is compressing at fixed t ; x1(t) ∈ x1(t) ; u(t) ∈ u(t) ; a(t) ∈ a and some fixed l. in this case the mapping f3 has the only fixed point, to which converges the iterative process (28), (29). computable model (26), (28), (29) under fixed values of functions u(t) and a(t) for each time point t determine the value of exogenous variables x(t), corresponding to price equilibrium of demand and supply in the markets of goods and services of agents in the framework of the following algorithm. 1) in the first step it is assumed that t=0 and it is determined the initial values of variables x1(0). 2) in the second step for current the initial values of variables x3(0)[0] are determined in various markets and for various agents ; the values x2(t)[0] = f2 (x1(t), x3(t)[0], x3(t), µ(t), a(t)) (the initial values of demand and supply of agents in the markets of goods and services) are calculated by (28). 3) in the third step for current t it is run iterative process (28), (29). in this, 222 a. ashimov : the theory of parametric control of macroeconomic systems and ... for each value q current values of demand and supply are found from (29) : x2(t) = f2 (x1(t), x3(t)[q], x3(t), u(t), a(t)) by improvement of market prices and budget parts of economic agents. condition for stopping iterative process is equality of demand and supply values in various markets accurate within 0.01%. consequently,there are determined equilibrium values of market prices in each market and budget parts in markets with state-set prices for various economic agents. we omit the index q for such equilibrium values of endogenous variables. 4) in the following step on the basis of obtained equilibrium solution for the time point using differential equations (26) we define the values of variables x1(t+ 1). the value increases by unity. transition to the step 2. quantity of iterations of steps 2, 3, and 4 are determined in accordance with the parametric identification problems, prognosis and control in chosen in advance periods. considered cge model can be presented in the form of continuous mapping f : x × u × a → rm, determining transformation of the values of endogenous variables of the system for zero year to respective values of the next year according to presented above algorithm. here the compacts x(t) = x1(t)×x2(t)×x3(t), determining the compact x in the space of endogenous variables are defined by the set of possible values of variables x1 and respective equilibrium values of variables x2 and x3 calculated by relations (30)-(31). we will assume that for chosen point x1(0) ∈ int(x1) and corresponding, calculated by (28), (29) points x(0) = (x1(0), x2(0), x3(0)), the inclusion x(t) = f t(x(0)) ∈ int(x(t)) is true under some fixed u(t) ∈ int(u(t)), a(t) ∈ a for t = 0, ..., n. (n-fixed positive integer). this mapping f defines discrete dynamical system in the set x, on the trajectory of which imposed appropriate initial condition : {f t, t = 0, 1, ...}, x|t=0 = x0 (30) based on this conception specific cge model of economic sectors is considered below. parametric identification of cge model of economic sectors the model in question by statistical data of the republic of kazakhstan is presented by the following 19 economic agents. economic agent no.1. agriculture, hunt and forestry ; economic agent no.2. fishery, fish breeding ; economic agent no.3. mining ; economic agent no.4. manufacturing ; economic agent no.5. production and distribution of electricity, gas and water ; economic agent no.6. construction ; economic agent no.7. trade ; automobile and house articles maintenance ; advances in systems science and applications (2014) vol.14 no.3 223 economic agent no.8. hotels and restaurants ; economic agent no.9. transportation and communication ; economic agent no.10. financial activities ; economic agent no.11. transactions with real estates, lease and services to enterprises ; economic agent no.12. public administration ; economic agent no.13. education ; economic agent no.14. public health and social services ; economic agent no.15. other municipal, social and personal services ; economic agent no.16. housekeeping services ; economic agent no.17. aggregate consumer, combining households ; economic agent no.18. government, presented by the sum of central, regional and local governments, and non-budget funds as well. government determines tax rates and amount of subsidies for agents-producers and the size of social transfers for households. moreover, this sector includes non-commercial organizations, serving households (political parties, labor unions, social associations, etc.) ; economic agent no.19. bank sector, involving national bank and commercial banks. economic agent no.20. outside world. here,economic sectors no.1-16 are agents-producers. the considered model is presented as general expressions of relations of (26), (28), (29) respectively m1 = 67, m2 = 597, m3 = 34 by expressions, which help to calculate values of its 698 endogenous variables. this model contains also 2045 estimated exogenous parameters. in the result of combined solution of the problems a and b as per the constructed algorithm of parametric identification (section 1.2) using statistical data on evolution of the economy of the republic of kazakhstan. the relative value of deviations of estimated values of variables used mostly as criteria of corresponding observed values was equal to less than 0.63%. further the calculation of the model outside the period of parametric identification (forecast calculation) using extrapolated for the forecast period values of functions u(t), a(t) will be called as basic calculation. the results of calculation and retrospective basic calculation of the model for 2008 partially presented in the table 5 show estimated values, observed values and deviations of estimated values of main output variables of the model from corresponding observed values. here, the period 2000-2007corresponds to the period of parametric identification of the model ; 2008 is the retroprognosis period ; y is gross output (in prices of 2000) ; yg is gdp (in prices of 2000) ; p is consumer price index in percentages relative to the previous year ; sign ≪ ∗ ≫ corresponds to observed values, sign ≪ a ≫ corresponds to deviations (in percentages)of esti224 a. ashimov : the theory of parametric control of macroeconomic systems and ... table 5 observed, calculated values of output variables of the model and corresponding deviations indicator year 2000 2001 2002 2003 2004 2005 2006 2007 2008 y ∗(t) 5.44 6.32 6.47 6.86 7.72 8.52 9.25 9.69 9.84 y (t) 5.38 6.32 6.47 6.86 7.72 8.52 9.27 9.64 9.82 ∆y (t) -1.22 -0.02 0.00 0.00 0.05 0.08 0.21 -0.51 -0.26 y ∗ g (t) 2.45 2.78 3.05 3.36 3.72 4.09 4.55 5.01 5.18 yg(t) 2.47 2.78 3.05 3.35 3.72 4.09 4.55 5.01 5.20 ∆yg(t) 0.88 0.07 -0.04 -0.02 -0.02 -0.02 -0.04 -0.15 0.38 p ∗(t) 106.4 106.6 106.8 106.7 107.5 108.4 118.8 109.5 p (t) 107.6 106.8 106.9 106.7 107.3 108.2 118.6 109.4 ∆p (t) 1.13 0.18 0.08 -0.05 -0.23 -0.22 -0.24 -0.05 mated values from respective observed values. analysis of sources for economic growth based on computable general equilibrium model of economic sectors now, let us analyze sources for economic growth of economic sectors based on cge model of economic sectors and based on retrospective data for 2000-2009. for this, using expressions of production functions of economic agents of the model, we estimate the effect of change of arguments of these functions on the rates of growth of gav of sectors v y _i[t + 1] in assumption about constancy of coefficients ca_z_ji[t] under consumed by sector intermediate products v d_pj_iz[t], coefficients ca_k_i[t] under capital (v k_i[t] + v k_i[t + 1])/2 and coefficients ca_l_i[t] under labor v d_pi_il[t]. here i, j = 1, ..., 16 are numbers of economic agents ; t is number of year ; power (x,y ) corresponds to xy , exp(x) corresponds to ex ; ca_r_i is coefficient, characterizing technical progress in i-th sector. v y _i[t+ 1] = ca_r_i× exp(v d_p1_iz[t]× ca_z_1i)× exp(v d_p2_iz[t] × ca_z_2i])× exp(v d_p3_iz[t]× ca_z_3i)× exp(v d_p4_iz[t]× ca_z_4i) × exp(v d_p5_iz[t]× ca_z_5i)× exp(v d_p6_iz[t]× ca_z_6i) × exp(v d_p7_iz[t]× ca_z_7i)× exp(v d_p8_iz[t]× ca_z_8i) × exp(v d_p9_iz[t]× ca_z_9i)× exp(v d_p10_iz[t]× ca_z_10i) × exp(v d_p11_iz[t]× ca_z_11i)× exp(v d_p12_iz[t]× ca_z_12i) × exp(v d_p13_iz[t]× ca_z_13i)× exp(v d_p14_iz[t]× ca_z_14i) × exp(v d_p15_iz[t]× ca_z_15i)× exp(v d_p16_iz[t]× ca_z_16i) × power(((v k_i[t] + v k_i[t+ 1])/2), ca_k_i)× × power(v d_pi_il[t], ca_l_i) (31) advances in systems science and applications (2014) vol.14 no.3 225 after finding the logarithms of both parts (31) and total increment of the function ln (v y _i) and dropping members of the highest order infinitesimal, we get the following estimate of the growth rate yi of real gav of i-th sector depending on growth of arguments of production function : ca_r_i, v d_pj_iz, kim = (v k_i[t] + v k_i[t+ 1])/2, and v d_pi_il[t]. yi = ∆v y _i v y _i = ∆ca_r_i ca_r_i + 16∑ j=1 (ca_z_ji× v d_pj_iz) ∆v d_pj_iz v d_pj_iz + ca_k_i ∆kim kim + ca_l_i ∆v d_pi_il v d_pi_il (32) denote by ai = ∆ca_r_i ca_r_i the rate of technical progress in i-th sector ; zij = ∆v d_pj_iz v d_pj_iz is the rate of intermediate products consumed by i-th sector, and produced by j-th sector ; ki = ∆kim kim is the rate of capital accumulation in i-th sector ; li = ∆v d_pi_il v d_pi_il is the growth rate of labor inputs in i−th sector, where the sign ”∆” means change of variable ; time values in (32) were dropped for short. coefficients on the right-hand-side of formula (32) at stated above rates and characterize degree of effect of the factors in question on economic growth and allow comparing their effect with effect of technical progress, coefficient at which is equal to 1. by denoting these coefficients by aij = ca_z_ji × v d_pj_iz, βi = ca_k_i, γi = ca_l_i, from (32) we get its reduced writing : yi = ai + 16∑ j=1 αijzij + βiki + γili (33) let us present the values of coefficients, defining contributions of sources of economic growth of sectors based on the model in question for 2008. (see table 6). coefficients in the table show how much the growth rate of gav would increase given the one percent increase of growth factors (fixed assets, labor or demand for intermediate goods of economic agents). analysis of coefficients table βi, γi, αs i = ∑16 j=1 αij of the table 4 shows that if we exclude the rate of technical progress, effect of which on the rate of growth of all sectors in given model is the same, then from rest three factor rates of economic growth, the largest effect on the rate of real output of sectors 1, 5, 7, 12, 13, 16 of the economy has the rate of labor inputs ; for sectors 4, 6, 8, 10, 15-the rate of capital accumulation ; and for other sectors 2, 3, 9, 11, 14 -the rate of consumed by the sector intermediate products, produced by all of the sectors. also note that for sectors 5, 7 12, 16 the rates of capital accumulation practically do not have effect on corresponding rate of output growth ; the rates of labor inputs haven on-zero effect on the rates of output growth of all sectors ; for sectors 226 a. ashimov : the theory of parametric control of macroeconomic systems and ... 6, 16 the rates of consumed intermediate good shave no effect on corresponding rate of output growth. the results of analysis allow choosing the following budget parts of 16 economic sectors in the capacity of tools for solving the problems of economic growth. oij-budget part of i-th sector, which is for payment for goods and services, bought from j-th sector ; ol i-budget part of i-th sector, which is for payment for labor ; on i -budget part of i-th sector, which is for payment for investment goods. table 6 coefficients, characterizing an impact of economic growth factors number of βi γi αi1 αi2 αi3 αi4 αi5 αi6 αi7sector i 1 0.3089 0.9051 1.345· 9.480· 2.897· 2.171· 1.602· 2.028· 1.345· 10−18 10−02 10−14 10−13 10−14 10−14 10−12 2 0.2426 2.4964 1.590· 7.308· 7.087· 9.884· 1.390· 1.120· 1.590· 10−16 10−01 10−16 10−15 10−15 10−15 10−16 3 0.9650 0.6886 2.886· 0.000 2.970· 8.269· 9.894· 1.478 2.886· 10−15 10−12 10−13 10−14 10−15 4 1.2900 0.0805 2.227· 1.343· 8.634· 9.989· 9.029· 2.641· 2.227· 10−13 10−1 10−13 10−13 10−03 10−16 10−13 5 1.0· 2.4083 3.199· 5.537· 8.913· 6.720· 5.406· 5.300· 3.199· 10−10 10−17 10−17 10−14 10−4 10−3 10−4 10−17 6 0.9343 0.7721 2.078· 9.186· 3.313· 3.940· 2.467· 1.324· 2.078· 10−16 10−18 10−14 10−13 10−15 10−14 10−16 7 1.0· 1.8792 8.133· 1.061· 3.115· 2.660· 4.124· 2.508· 8.133· 10−10 10−16 10−15 10−14 10−13 10−14 10−13 10−16 8 1.0691 0.4706 2.003· 1.498· 4.343· 6.171· 3.375· 7.288· 2.003· 10−02 10−15 10−17 10−14 10−15 10−15 10−02 9 0.8660 0.2153 4.289· 2.265· 1.666· 8.753· 3.057· 4.005· 4.289· 10−16 10−13 10−15 10−14 10−14 10−16 10 0.6702 0.5492 0.000 0.000 0.000 4.397· 5.602· 8.811· 0.000 10−15 10−15 10−15 11 1.2022 0.1006 2.929· 2.267 5.064· 1.030· 1.664· 5.308· 2.929· 10−05 10−14 10−12 10−04 10−13 10−05 12 1.0· 2.5822 5.415· 0.000 0.000 4.255· 2.573· 0.000 5.415· 10−10 10−14 10−13 10−14 10−14 13 0.2635 1.7177 2.370· 0.000 0.000 8.847· 1.142· 5.739· 2.370· 10−14 10−13 10−13 10−15 10−14 14 0.0227 1.7814 6.736· 2.018· 6.280· 1.288· 1.919· 4.809· 6.736· 10−14 10−1 10−15 10−12 10−13 10−13 10−14 15 0.9304 0.2173 1.041· 1.408· 7.681· 3.712· 3.926· 1.579· 1.041· 10−15 10−3 10−15 10−13 10−14 10−13 10−15 16 0 1.9372 0 0 0 0 0 0 0 continuation of table 6. advances in systems science and applications (2014) vol.14 no.3 227 number of αi8 αi9 αi10 αi11 αi12 αi13 αi14 αi15 αi16sector i 1 4.325· 4.595· 1.807· 1.177· 0 1.681· 1.178· 9.304· 0 10−15 10−2 10−14 10−14 10−16 10−15 10−18 2 8.054· 1.087· 2.616· 2.280 0 0· 0 0 0 10−17 10−14 10−17 3 8.054 2.221· 1.436· 4.626· 0 1.581· 4.991· 1.571· 0 10−14 10−2 10−13 10−12 10−15 10−4 10−14 4 8.038· 1.444· 4.353· 6.338· 0 1.247· 3.379· 5.445· 0 10−15 10−13 10−14 10−16 10−16 10−03 10−16 5 1.642· 2.760· 6.132· 2.512· 0 3.273· 2.026· 5.494· 0 10−15 10−1 10−3 10−14 10−5 10−17 10−4 6 7.977· 1.117· 6.525· 2.014· 0 8.436· 1.554· 1.089· 0 10−15 10−2 10−16 10−14 10−17 10−17 10−16 7 6.067· 4.548· 3.407· 1.505· 0 5.394· 8.576· 4.404· 0 10−14 10−1 10−13 10−12 10−16 10−5 10−16 8 1.528· 8.267· 3.053· 1.507· 0 2.339· 1.338· 2.244· 0 10−15 10−15 10−14 10−14 10−17 10−17 10−16 9 3.346· 1.002· 3.203· 4.339· 0 3.657· 4.803· 1.091· 0 10−14 10−12 10−13 10−13 10−15 10−16 10−15 10 1.991· 3.270· 2.862· 5.408· 0 1.347· 0 8.805· 0 10−14 10−01 10−13 10−15 10−14 10−18 11 6.664· 6.596· 2.965· 2.646· 0 9.260· 1.661· 9.599· 0 10−14 10−13 10−13 10−12 10−14 10−03 10−06 12 4.185· 1.548· 9.506· 7.964· 0 2.126· 1.408· 2.886· 0 10−1 10−1 10−13 10−13 10−14 10−15 10−15 0 13 4.820· 5.807· 1.794· 1.331· 0 3.939· 7.738· 1.836· 0 10−14 10−1 10−14 10−14 10−15 10−17 10−14 14 1.891 2.118· 4.749· 2.743· 0 2.219· 1.782· 6.646· 0 10−13 10−17 10−13 10−16 10−14 10−14 15 6.285· 2.139· 9.739· 8.436· 0 9.538· 1.416· 1.222· 0 10−14 10−1 10−14 10−14 10−17 10−17 10−13 16 0 0 0 0 0 0 0 0 0 finding the optimal parametric control laws on the basis of cge model of economic sectors in computational experiments with cge model of economic sectors,the criterion below was used as maximization criterion k = 1 6 2015∑ t=2010 v y [t] (34) here k is average value of country’s gross output for 2010-2015 in prices of 2000. in experiments with optimization criterion (34), the constraints on growth of consumer price index of the following form were used : v pr[t] ≤ 1.0 228 a. ashimov: the theory of parametric control of macroeconomic systems and ... here is calculated consumer price index of the model without parametric control, is consumer price index with parametric control. in computational experiments control was performed for 1536 exogenous parameters j -th agent-producer’s budget parts for purchasing goods and services, produced by i -th agent-producer for 2000-2015 : oj i [t] ; t=2010,...,2015 ; i,j=1,...,16. here ∑16 i=1o j i (t) ≤ 1 for mentioned values of t. basic values of mentioned parts, obtained by solving the parametric identification problem of the model on data of 2000-2008, we will denote by ōj i ; i,j=1,...,16. the following problem of finding optimal values of adjustable parameters vectors was considered. on the basis of cge model of economic sectors to find mentioned values of budget parts of agents-producers oj i [t], which provide the maximum of criterion k under additional constraints on these parts of the following form : 0.5 ≤ oj i [t]/ō j i ≤ 2; i, j = 1, ..., 16; t = 2010, ..., 2015. solutions of these optimization problems were made using the nedler-mead algorithm. after using parametric control of budget parts of the model, criterion value turned out k = 1.6283 ·1013, criterion value increased by 33.14% in relation to the basic variant. references [1] tarasevich l.s., grebennikovp.i., leusskiy a.i. (2006), “macroeconomics”, moscow: vysshee obrazovanie, in russian. [2] ashimov a.a., sultanov b.t., adilov zh.m., borovskiy yu.v., novikov d.a., alshanov r.a. ashimov as.a. (2013), “ macroeconomic analysis and parametrical control of a national economy”, new york: springer. [3] e.s. said, d.a. dickey. (1984), “testing for unit roots in autoregressivemoving average models of unknown order”, biometrika, vol.71, no.3, pp.599607. [4] r.c. fair. (2004), “estimating how the macroeconomy works”, cambridge, mass.: harvard university press. [5] r.f. engle, c.w.j. granger. (1987) “co-integration and error correction: representation, estimation, and testing”, econometric, vol.55, no.2, pp.251276. [6] turnovsky s.j. (2000), “methods of macroeconomic dynamic”, cambridge: mit press. [7] kolemaev v.a.matematnqeskaya. (2002), “konomnka”, moskva: yuniti. [8] kydland f.e., prescott e.c. (1977), rules rather than discretion: the inconsistency of optimal planes, springer, journal of political economy, vol.87. advances in systems science and applications (2014) vol.14 no.3 229 [9] dornbusch r., fischer s. (1990), “macroeconomics”, new york: mcgraw-hill. [10] tumanova e.s., shagas n.a. (2004), “macroeconomics”, moscow: infra-m. [11] dubovskiy s.v. (1995), “the kondratiev cycle as the simulation object”, mathematical modelling, vol.7, no.6, pp.65-74, russian. [12] k.s. abdiev. (2001-2009), “statistical yearbook of kazakhstan”, astana: agency on statistics of the republic of kazakhstan. [13] smets f., wouters r (2003), “an estimated dynamic stochastic general equilibrium model of the euro area”, journal of the european economic association, vol.1, no.5, pp.1123-1175. [14] sims c. (2002), “random lagrange multipliers and transversality”, eco 504, spring, pp.1-7. [15] orphanides a. (2007) , taylor rules, federal eeserve board, working paper, no.18, washington, d.c.. [16] fernández-de-córdoba g., torres j.l. (2009), “forecasting the spanish economy with an augmented var-dsge model”, malaga economic theory research center working papers, no.1. [17] blanchard o.j., kahn c.m. (1980), “the solution of linear difference models under rational expectation, econometrica”, comput. math. math. phys, vol.24, pp.1812. [18] fernández-villaverde j. (2010), “the econometrics of dsge models, series”, vol.48, no.5, pp.1305-1312. [19] hamilton j.d. (1994), “time series analysis”, new jersey: princeton. [20] makarov v.l., bahtizin a.r., and sulakshin s.s. (2007), “application of computable models in state administration”, moscow: nauchniy expert. corresponding author a. ashimov can be contacted at: ashimov37@mail.ru advances in systems science and application(2016) vol.16 no.3 33-51 depth reduction factor assessment for evaluation of cyclic stress ratio based on site response analysis farzad farrokhzad department of civil engineering, babol university of technology abstract earthquake properties that affect the liquefaction of a soil are described with one parameter known as the cyclic stress ratio (csr). the stress reduction coefficient parameter accounts for the flexibility of the soil profile. the previous proposed equations could be used in routine engineering practice so long as the simplified (rd) was used to assess csr, but could be unconservative if used in sites with complex soil layers or in design of vital facilities such as dams, hospitals, bridges and . . . . this research presents the results of studies in babol city to develop, depth reduction factor (rd) based on equivalent linear analysis of study area for evaluation of cyclic stress ratio. as a part of microzonation study for the babol city a total of 35 boreholes have been drilled in 35 km2 of the research area. the depths of these boreholes ranged about 25 to 35 m. spt blow counts were taken in each 2 m depth. many geophysical investigations, generally 35 downhole logging surveys, are carried out in 35 mentioned boreholes for generation and measurement of shear wave velocity. based on these analyses, the (rd) diagrams and equations are developed for each mentioned zone. keywords depth reduction factor (rd), liquefaction, site response analysis, cyclic shear stress, earthquake return period, microzonation 1 introduction in geology, a fault is a planar fracture or discontinuity in a volume of rock, across which there has been significant displacement along the fractures as a result of earth movement [1]. an earthquake is caused by a sudden slip on a fault [2]. stresses in the earth’s outer layer push the sides of the fault together [3]. stress builds up and the rocks slips suddenly, releasing energy in waves that travel through the earth’s crust and cause the shaking that we feel during an earthquake [4]. the site safety during earthquakes is related with geotechnical phenomena such as amplification, liquefaction, landsliding and fault movements [5]. loose sand and silt that is saturated with water can behave like a liquid when shaken by an earthquake [6]. soil liquefaction describes a phenomenon whereby a saturated or partially saturated soil substantially loses strength and stiffness in response to an applied stress, usually earthquake shaking or other sudden change in stress condition, causing it to behave like a liquid [7]. evaluation of liquefaction potential requires comparison of the anticipated level of loading imposed on a 34 h. santhi, n. jaisankar: load-aware congestion adaptive multipath multicasting soil profile with the inherent resistance of the soil profile to liquefaction [8]. that procedure essentially compares the cyclic resistance ratio (crr) [the cyclic stress ratio required to induce liquefaction for a cohesionless soil stratum at a given depth] with the earthquake-induced cyclic stress ratio (csr) at that depth from a specified design earthquake [defined by a peak ground surface acceleration and an associated earthquake moment magnitude] [9]. during an earthquake, the soils will be subject to cyclic shear stresses induced by the ground shaking. the average cyclic stress ratio (csr) during an earthquake may be estimated by the following equation 1 [10-14]: csr = τave σ0′ = 0.65( amax g )( σ0 σ0′ ).rd (1) where amax = maximum acceleration at the ground surface, σ0 = total overburden pressure at depth under consideration, σ0 ′ = effective overburden pressure at depth under consideration and rd = stress reduction coefficient. seed and idriss considered a soil column as a rigid body [15]. in reality, soil behaves as a deformable body instead of as a rigid body. hence, the rigid body shear stress should be reduced with a correction factor to give the deformable body shear stress (τmax)d [16-18]. this correction factor is called the stress reduction coefficient (rd) and can be computed as follows: values of rd are commonly estimated from the diagram which is proposed by seed and idriss in 1971 (fig. 1). this chart was determined analytically using a variety of earthquake motions and soil conditions. average rd values in the diagram can be estimated using the following functions (equation 2) [19]. rd = 1− 0.00765z for z ≤ 9.15(m) rd = 1.174− 0.0267z for 9.15(m) < z ≤ 23(m) rd = 0.774− 0.008z for 23(m) < z ≤ 30(m) (2) also, equation 3 shows the suggestion of blake [20] rd = 1− 0.4113z0.5 + 0.04052z + 0.001753z1.5 1− 0.4177z0.5 + 0.05729z − 0.006205z1.5 + 0.00121z2 (3) regarding to the research of golesorkhi , statistically based rd curves were developed for different earthquake magnitude ranges [21]. the initial study of golesorkhi has been further extended by idriss and golesorkhi. advances in systems science and application(2016) vol.16 no.3 35 fig. 1 rd results from response analyses for 2153 combinations of site conditions and ground motions, superimposed with heavier lines (seed and idriss 1971) the proposed correlation for estimation of rd as a function of depth, magnitude, intensity of shaking, and site stiffness is presented in equation 4. in(rd) = α(z) + β(z).mw α(z) = −1.012− 1.126.sin( z 38.5 + 5.133) β(z) = 0.106 + 0.118.sin( z 37.0 + 5.142) (z = depth in feet) (4) rd(d,mw , amax, v ∗ s,12m) = for d < 65(ft)[ 1 + −23.013−2.949amax+0.999mw+0.016v ∗ s,40′ 16.258+0.201e 0.104(−d+0.0785v ∗ s,40′ +24.888) ] [ 1 + −23.013−2.949amax+0.999mw+0.016v ∗ s,12m 16.258+0.201e 0.104(0.0785v ∗ s,40′ +24.888) ] ± σεrd rd(d,mw , amax, v ∗ s,12m) = for d ≥ 65(ft)[ 1 + −23.013−2.949amax+0.999mw+0.016v ∗ s,40′ 16.258+0.201e 0.104(−d+0.0785v ∗ s,40′ +24.888) ] [ 1 + −23.013−2.949amax+0.999mw+0.016v ∗ s,12m 16.258+0.201e 0.104(0.0785v ∗ s,40′ +24.888) ] − 0.0014(d− 65)± σεrd σεrd = { 0.0072d0.850 d < 40ft 0.007240d0.850 d ≥ 40ft (5) 36 h. santhi, n. jaisankar: load-aware congestion adaptive multipath multicasting cetin et al recognized that rd is a function of site response, and developed a new correlation for estimation of rd. the proposed correlation is presented in equation 5 [22]. numerous seismic events around study area, such as manjil-rudbar earthquake on june 21, 1990, mazandaran earthquake on may 28, 2004, tabas earthquake on september 16, 1978 and bam earthquake on december 26, 2003 have demonstrated the relevance of subsurface soil layers and geotechnical conditions on seismic ground response [23-25]. equivalent-linear ground response modeling is by far the most commonly utilized procedure in practice [26]. equivalent-linear soil material modeling is widely used in practice to simulate true nonlinear soil behavior for applications such as ground response analyses [27-28]. the advantages of equivalent-linear modeling include small computational effort and few input parameters. in equivalent linear model, the soil stiffness g is modified in response to computed strains and g and ξ degradation curves. ground motion is computed for selected g and ξ pair at each layer; in particular, strain histories are calculated; from above effective shear strain γeff is calculated; from obtained effective shear strain, new pair of g(γ) and ξ(γ) are selected using the degradation curves; these steps are repeated until the maximum difference between computed shear modulus and damping ratio values in two successive iterations be less than ∼5% [29-31]. in the research of ishibashi and zhang, equivalent shear moduli and damping ratios for sandy soils were collected and equation 6 was proposed to best fit data points [32]. g gmax = k(γ, ip)σ̄ m(γ,ip)−m0 0 k(γ.ip) = 0.5 { 1 + tanh [ ln( 0.000102+n(ip) γ )0.492 ]} m(γ, ip)−m0 = 0.272 { 1− tanh [ ln(0.000556γ )0.4 ]} e−0.0145i13p n(ip) =  0 for ip = 0 sandy soils 3.37× 10−6i1.404p for 0 < ip ≤ 15 low plastic soils 7× 10−7i1.976p for 15 < ip ≤ 70 medium plastic soils 2.7× 10−5i1.115p for ip > 70 high plastic soils (6) 2 study area babol is located in the in the north of iran, between the northern slopes of the alborz mountains and approximately 20 kilometers south of caspian sea on the west bank of babolrud river and receives abundant annual rainfall. iranian tectonic plate in middle east affects the seism tectonic condition of babol (figure 2) [33-34]. the tectonic environment near babol city is unusually complicated. khazar and north alborz faults are the most significant faults around the study advances in systems science and application(2016) vol.16 no.3 37 area (figure 3). these faults are directed e-ne and w-sw. the khazar fault is the boundary between the caspian plain and alborz mountain (figure 4, figure 5). table 1 presents a list of active faults affecting the babol city [35]. regarding to seismicity of mazandaran, it can be assumed that for large earthquakes, the faulting process primarily involves repeated breaking of the same fault segment rather than creation of a new fault surface. table 2 shows recent major earthquakes that are felt in the zone (648584 e, 4049660 n) and (652032 e, 4043901 n). this zone closely bounds babol city. also, magnitude distribution of earthquakes in the range of 200 km around the babol city are presented in figure 6. fig. 2 tectonic of study area fig. 3 alborz and khazar faults fig. 4 surface section of khazar fault zone fig. 5 satelite map of north alborz fault 38 h. santhi, n. jaisankar: load-aware congestion adaptive multipath multicasting table 1 active faults around the study area (babol city) fault name type of fault distance from fault length babol city (km) (km) north alborz thrust fault 44 300 khazar thrust fault 16 550 firouzabad thrust fault 85 112 atari thrust fault 91 85 astaneh thrust fault 93 75 kandovan thrust fault 100 64 mosha thrust fault 91 400 north of tehran thrust fault 115 108 firouzkooh thrust fault 84 40 bayejan thrust fault 60 45 damghan thrust fault 136 100 orim thrust fault 72 44 table 2 recent major earthquakes around the study area (babol city)) earthquake magnitude depth (km) year chalus, mazandaran 6.3 17 2004 babol, mazandaran 5.1 15 2012 damghan, semnan 5.7 7 2010 roudbar and manjil 7.4 10 1990 (a) earthquakes of babol from 800 to 1900 (b) earthquakes of babol from 1900 to 2014 fig. 6 magnitude distribution of earthquakes in the range of 200 km around the babol city babolrud river originates in the alborz mountains and is one of the major rivers in iran. it is located on the left side of babol city. the study area is conadvances in systems science and application(2016) vol.16 no.3 39 stantly filled with the new alluvial sediments of the babolrud river. following paragraphs provide an overview of the site investigation conducted at 35 km2 of babol city and present the results of both geotechnical and geophysical investigations. also, supplemental activities related to babol seismic microzonation project are illustrated too. regarding to mentioned project, a total of 35 new boreholes have been drilled in study area. also, the results of previous investigations (consist of 60 borholes) were collected too (figure 7). the depth of boreholes ranged from 25 to 40 m. downhole logging surveys and spt tests carried out through new 35 boreholes, were performed at every 1.5 m and 2 m of depth. the location of drilled boreholes over babol city, a sample of geotechnical test results and shear wave velocity profile obtained by downhole test are shown in figure 8 and figure 9. the general topographic gradient in the study area is constant. during the investigation, the site soils were observed to consist of silty clay, silty sand and sandy clay. the depth of the water table as measured during drilling should be fig. 7 location of drilled boreholes over babol city 40 h. santhi, n. jaisankar: load-aware congestion adaptive multipath multicasting (a) sample of borehole log report (0-15 m) (b) sample of borehole log report (15-30 m) fig. 8 results of laboratory and in-situ tests (geotechnical investigations) advances in systems science and application(2016) vol.16 no.3 41 (a) sample of shear wave velocity profile (b) sample of s wave data obtained during downhole test fig. 9 results of geophysical (downhole) surveys fig. 10 location of drilled boreholes over babol city 42 h. santhi, n. jaisankar: load-aware congestion adaptive multipath multicasting carefully evaluated. it is always necessary to wait for at least 24 hours to check on the stabilized water table for the final measurement. groundwater levels in areas of the babol city is presented figure 10 [36]. 3 results and discussion in reality, soil behaves as a deformable body instead of as a rigid body. as a result, the actual peak shear stress induced at each depth is less than that predicted in surface. therefore, the rigid body shear stress should be reduced with depth reduction factor (rd), to give the deformable body shear stress. the depth reduction factor (rd) can be computed as follows (equation 7) [37-38]: rd = (τmax)depth (τmax)rigidboady = (amax)depth (amax)surface (7) amplification refers to the increase in the amplitudes of seismic waves as they propagate through the soil layers near the surface of the earth. this is because the ground under these districts is relatively soft. soft soils usually, amplify ground shaking. influence of the soil response on the seismic motion at the ground surface, is considered through the equivalent linear or nonlinear one-dimensional response of a soil column. these methods of analysis require selection and scaling of ground motions appropriate to design hazard levels. it is necessary to select empirical recordings of ground motion and scale these ground motions to the level of the design spectrum. in order to develop the depth reduction factor (rd), the max acceleration at surface and each depth should be calculated [39]. in this research, site response analyses, performed by the geostudio software, were carried out for 35 boreholes using 4 scaled earthquakes with 109, 475, 975 and 2500-year return period. the details of mentioned scaled accelerograms are presented in table 3 and figure 11. finally the (rd) diagrams for each zone of study area is drown and a new empirical correlation is proposed. table 3 details of scaled earthquakes earthquake name a max (g) at bedrock return period (year) kareh-bas 0.108 109 bam 0.215 475 northridge 0.302 950 san fernando 0.364 2500 advances in systems science and application(2016) vol.16 no.3 43 (a) scaled time history records of the kareh-bas earthquake (109-year return period) (b) scaled time history records of the bam earthquake (475-year return period) (c) scaled time history records of the northridge earthquake (950-year return period) (d) scaled time history records of the san fernando earthquake (2500year return period) fig. 11 time history records of scaled earthquake for performing site response analyses fig. 12 35 zones of study area (site response analyses are performed for each zone) 44 h. santhi, n. jaisankar: load-aware congestion adaptive multipath multicasting advances in systems science and application(2016) vol.16 no.3 45 46 h. santhi, n. jaisankar: load-aware congestion adaptive multipath multicasting fig. 13 variations of the stress reduction coefficient (rd) with depth for various zones of babol city advances in systems science and application(2016) vol.16 no.3 47 in order to define the depth reduction factor for babol city, 35 zones were each analyzed using scaled input motions (figure 12). then the rd diagrams were developed for each zone. figure 13 shows the depth reduction factor for each zone based on different input motions. most researchers today agree that there is no correlation for assessment of depth reduction factor which can be applicable for different soil types and all regions. different site conditions such as soil and sediment erosion, geological situation, soil structure and fabric, seismicity of area and ... are the main factors of mentioned uncertainty. so it can be said that the best empirical correlations for a specific region should be assessed regarding to in-situ tests and site response analyses which are performed in that region. in this research, an empirical correlation is suggested based on more than 200 site response analyses for different zones of babol city, using 4 scaled earthquake motions. the proposed new correlation for estimation of rd as a function of depth is presented in equation 8. table 4 details of scaled earthquakes rd = −0.0425z + 1.0333 0 ≤ z ≤ 6 equation 8 rd = 0.0006z2 + 0.0048z + 0.7865 6 < z < 12 rd = −0.02z + 1.06 12 ≤ z ≤ 14 rd = 0.0025z2 + 0.072z + 0.261 14 < z ≤ 20 4 conclusion liquefaction in soil is one of the major problems in geotechnical earthquake engineering. the simplified method of seed and idriss (1971) is the most common procedure used for evaluation of liquefaction potential. in mentioned simplified method, an empirical correlation was developed for estimation of rd based on the 2153 site response analyses. previous developed rd functions can be unconservative in comparison to actual case-specific seismic response analysis. therefore, in places with high potential of liquefaction, the csr is recommended to be calculated using a modified dynamic response analysis. this research proposed the values of depth reduction factor (rd) based on the results of several extensive site response analyses of babol city at 35 zones. firstly, 4 modified and scaled acceleration records (kareh-bas earthquake with 109-year return period, bam earthquake with 475-year return period, northridge earthquake with 975-year return period and san fernando earthquake with 2500year return period) were used as input motions at the bedrock of the 35 zones. soil layers were idealized as horizontal layers. engineering properties of each 48 h. santhi, n. jaisankar: load-aware congestion adaptive multipath multicasting layer, ground water level and other necessary data collected during geotechnical investigations and geophysical surveys were used to develop the soil column extending form the ground surface to bedrock. then, local site effects of study area were evaluated numerically with equivalent linear method and finally the depth reduction factor (rd) diagrams and correlation were prepared for study area. in some cases, depending on geotechnical and geological situation of a site or seismicity of a study area, the csr calculated by the simplified method could be incapable, overestimated or underestimated. to summarize it can be said that, the variation of csr obtained by site response analysis could be considered more reliable. also in complex and unusual sites or during strong shaking levels, it can be recommended that the more comprehensive alternative is to develop depth reduction factor based on site response analysis. references [1] nguyen kh and gatmiri b. 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[36] choobbasti aj, shooshpasha e and farrokhzad f (2013), “3-d modeling of groundwater table using artificial neural network-case study of babol”, indian journal of geo-marine science,vol. 42, no. 7, pp. 903-906. [37] seed hb, idriss im and arango i. (1983), “evaluation of liquefaction potential using field performance data”,journal of geotechnical engineering (asce), vol. 109, no. 3, pp. 458-482. [38] seed hb and lee kl (1966), “liquefaction of saturated sands during cyclic loading”, soil mechanics and foundations division (asce), vol. 92, pp. 105-134. [39] farrokhzad f, choobbasti aj and barari a (2011), “determination of liquefaction potential using artificial neural networks”, gradevinar,vol. 63, pp. 837-845. corresponding author farzad farokhzad can be contacted at: farzadfarokhzad@mit.ac.ir adv syst sci appl 2019; 01; 141-149 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/594 copyright ©0000 assa. adv. in systems science and appl. (0000) the application of graph decomposition to development of large-scale agent-based economic models v.l. makarov1, a.r. bakhtizin1, e.d. sushko1, g.b. sushko1 1) central economics and mathematics institute cemi ras, moscow, russia e-mail: albert.bakhtizin@gmail.com received may 17, 2018; revised march 13, 2019; published april 15, 2019 abstract: in this work we describe the application of the graph decomposition algorithms for the development of a scalable high-performance agent-based model of population of russia described in terms of demography, migration and transport flows. the simulated system consists of agents representing individuals and sets of links to other agents, which represent the social interactions of individual. individual agents in the model participate in several independent processes, for which different sets of social links is important such as family and neighbors. to perform a load balancing of agents between cluster computer nodes the metis graph decomposition algorithm was used. these algorithms allow to split the graph of agents and links into parts of similar size with least possible number of links between them. a number of numerical experiments was carried out for test model to estimate the influence of the parameters of the model on scalability. keywords: agent-based modelling, numerical modelling, parallel computing, graph decomposition. 1. introduction the simulation of complex economic processes by means of agent-based approach requires a large number of agents to be used. one can expect the number of agents to be comparable to the population of the modelled society such as the city or a country i.e. up to 109 agents. high number of agents requires the use of supercomputer clusters to perform the simulation and requires a special programming technique to be applied in order to be able to run the model on such computer. a typical supercomputer nowadays is a cluster of multiprocessor multicore nodes with the shared memory inside the node and distributed memory between the nodes of the cluster. that distributed memory model requires the data describing agents to be evenly split between the nodes of the cluster. the paper describes the application of the metis[1,2,3] spatial decomposition algorithm for conducting multi-agent simulations of the behavior of society using supercomputers. there is a number of technologies and frameworks for development of abm (agentbased modeling) simulation software such as microsoft axum [4] and repast hpc [5] and swages [6]. these frameworks implement mechanisms of exchange of messages between agents but the problem of optimal distribution of agents between processes should be solved by the simulation code taking into account all types of interactions in the model. for example, a similar approach using metis algorithm for a large-scale epidemiologic abm using repast hpc and metis was implemented in paper [7]. another example is the application of the metis decomposition for modeling of railway scheduling in paper [8]. e ach of these codes used its own implementation of the decomposition code to transform the objects in the model into graph representation for metis. mailto:albert.bakhtizin@gmail.com 142 v.l. makarov, a.r. bakhtizin, e.d. sushko, g.b. sushko copyright ©2019 assa. adv. in systems science and appl. (2019) the simulated system consists of agents which represent individuals and a set of links to other agents, which represent the social interactions of individual. the individual agents in the model participate in several independent processes, for which different sets of social links is important such as family and neighbors. the agents of the system participate in two processes: 1) the process of reproduction of the population, and 2) the process of migration. in the first process they use messages exchange to search for the partner to form a family. in the second process the message exchange mechanism is used to obtain information about available jobs in different regions to determine the direction of migration. to conduct agent modeling of demographic processes with a large number of agents, it is necessary to create an initial state of the population of agents with a given distribution by region, age, sex, income and other relevant parameters. in the test case, reading of the data by region from the text file and the geometry of regions from the image file was realized. the data is used to construct a rectangular grid, each cell of which is connected to a certain region and parameters of agents in the cell are determined by the parameters of the region. a random distribution of the number of resident agents between grid cells (uniform distribution of population within each region with the fixed total number of agents in the system) and calculation of grid cells decomposition by processors was implemented. 2. the model description the most common way of writing simulation programs for cluster computers is to use c++/fortran language and mpi library which are available on all supercomputers. to simplify the development of the model the distributed abm framework [9] was developed for use with the high-level microsoft .net platform. the use of the high-level programming language for model description requires the implementation of wrapper code to system-level functions for inter-process exchange using native mpi library. as most of modern supercomputers run on linux operating system, we have decided to use microsoft .net core and mono [10] implementations of .net platform available for this os. the choice of these technologies was determined by the following criteria: 1. the system has to be scalable across multiple computational cluster nodes (i.e. use resource of multiple nodes for speedup) therefore the multithreading calculation model was not suitable as it is limited to single cluster node. 2. the model should be easy to develop and maintain and therefore the high-level programming language c# was used. 3. the system should be efficient and therefore the native mpi library was used instead of tcp/ip sockets or .net libraries like windows communication foundation as these technologies are not optimal for supercomputers and hpc applications. mpi libraries installed on each cluster computer are usually tuned for particular proprietary network system which is used on the cluster such as infiniband. 4. the program should be portable and should be compatible with any operating system i.e. running on both developer workstations and on cluster computers. 5. the results of the simulation should not depend significantly on the number of processors used for the calculation. all interactions between agents on different processes on each simulation step should be taken into account. the simulation solution calculated on different number of processors can still be different due to the difference in the order of operations (due to the limited float point numbers precision) and the difference in the random number sequence (each process has its own independent random number sequence). an efficient mechanism of message exchange between agents was implemented by means of message queue and native mpi collective operations. the message queue accumulates a buffer of messages to different processes and then uses mpi alltoall exchange operation to deliver contents of messages. this operation delivers the buffers of the implementation of the scalable modelling framework 143 copyright ©2019 assa. adv. in systems science and appl. (2019) arbitrary size from each mpi process to all other processes in most efficient way by splitting the buffer into chunks of optimal size for network transfer and hiding the latency of network operations by performing simultaneous several send and receive operations. to use native operations with managed c# objects operations of binary serialization and deserialization of objects were implemented and c# wrappers for native functions were written. 2.1 the system decomposition algorithm the modeled system consists of a large number of agents connected with each other by the number of social links. to perform an efficient simulation of the system it is necessary to split the set of agents between mpi processes taking into account their links with other agents. this distribution of agents between processes determines the parallel scalability of the program. the quality of the decomposition is determined by the balance of the agents count on different processes and the number of links between agents assigned to different processes. the decomposition of a graph of agents with links can be performed by the application of graph decomposition algorithms such as metis [1], scotch [11] or jostle [12]. these algorithms provide a computationally efficient way of decomposition of large graphs into parts of equal size with minimal borders between them. such approach leads to a best possible system decomposition with a fine distribution of agents but it also leads to a number of drawbacks: 1) it requires the formation and a decomposition of a sparse matrix of size n, where n is the total number of agents, 2) each change of a number of agent due to agent’s birth or death leads to a recalculation of the decomposition. in order to calculate the decomposition in a more efficient way the system was split by a rectangular grid covering all regions. the decomposition of cells is much faster due to smaller matrix and doesn’t require recalculation of the decomposition after each time step as the distribution of population changes slowly with time. due to the small size of the input matrix the decomposition can be performed using sequential version of metis algorithm without sensible impact on the parallel performance of the simulation. to calculate the decomposition of the grid, a graph metis algorithm with a weighting was used (metis_partgraphrecursive). the metis algorithm takes an input on a graph specified through the constraint matrix and an array of weights of the nodes of the graph and returns the optimal distribution of the graph by a given number of parts with minimizing the links between the parts. the link matrix is given in csr [13] format i.e. by the number of rows (n) and two arrays ia and ja. the array ia contains a sequence of partial sums of the number of nonzero elements in the matrix. this array is used to separate the elements of the ja array by strings. the ja array contains index lists of non-zero elements in all rows. in the test example, the size of the matrix corresponded to the number of non-empty grid cells, the adjacent grid cells (not diagonally) were considered to be related. as the weight of the cell, the number of agents that should be created in the corresponding cell was used. the calculation of the most optimal decomposition of such matrix is very computationally expensive. the metis library implements a multilevel recursive coarsegrained algorithm for calculation of the reasonably good decomposition. on each level the original graph is transformed to a smaller graph using the coarse-graining algorithm, the decomposition of the smaller graph is performed and then a special refinement of the decomposition is performed. as a result of calling the metis algorithm for a given coupling matrix and weights and a given number of processors, an array of numbers is created that describes the optimal binding of the grid cells to the processors. these data are used to distribute cells and agents 144 v.l. makarov, a.r. bakhtizin, e.d. sushko, g.b. sushko copyright ©2019 assa. adv. in systems science and appl. (2019) by processors during the calculation, but do not affect the aggregation procedure of the received calculation results. the results of the calculation are aggregated by region and should not depend on the decomposition used. to test the decomposition algorithm a sample system was constructed using the data on the population of russia. russia can be a good example of a country with very different regions in terms of population. the total population of russia is 144 million people and the average density of population is 8.58 person per square kilometer ranging from 4626 people/km2 in moscow to 0.07 people/km2 in chukotka region. large cities such as moscow and st. petersburg were considered as separate regions in order to distinguish the parameters of the region and parameters of the enclosing region because of the principal difference in population density, income and other parameters. the figures below show the results of automatic grid decomposition using the metis algorithm for 8 processors in two variants: without taking into account the number of agents in cells (the weight of each cell = 1) and taking into account the number of agents (cell weight equals to number of agents). the implementation of the scalable modelling framework 145 copyright ©2019 assa. adv. in systems science and appl. (2019) fig. 1. the decomposition of a territory of russia by 8 processors taking into account only the number of cells (top) and the number of agents in cells (bottom). one can notice much smaller regions in the western part of russia on a second plot due to much higher population density. different shades of red show areas assigned to different processors. in the first case, the algorithm equalizes the number of grid cells on different processors with the minimum length of the boundary between them. in the second case, the number of agents attributed to the cells of a specific processor is equalized, while minimizing the length of the border. one can see that in the second case the regions have different area because of the difference in population density. the difference in density between european part of the country (27 people per km2) and asian part (3 people per km2) leads to much smaller regions in european part in terms of area and cell count. another effect associated with the formulation of criteria for optimizing distribution is that the cells located on islands and exclaves are isolated and not directly connected to the nearest regions and in terms of decomposition can be linked to any processor. 146 v.l. makarov, a.r. bakhtizin, e.d. sushko, g.b. sushko copyright ©2019 assa. adv. in systems science and appl. (2019) the use of the sparse matrix of links between cells allows the further development of decomposition criteria which should in principle include the connectivity of cells which have no common border. these connections may arise due to air communication, railways and other ways of traveling which have to be taken into account for correct simulation of migration. in the test example presented above, the total number of agents was set at 8400000. at the same time, the share of agents in each region was determined by the share of this region in the population of the country. within the region, agents were distributed uniformly across cells. such a calculation scheme allows us to change the total number of agents for test purposes, keeping their correct geographical distribution. using the algorithm described above, the distribution of the source system by the processors was obtained. in the implementation, it is important that such a distribution must be done before the agents are created, since the agents created must by initially correctly distributed among the nodes of the cluster to meet the limitations of the system memory. to analyze the obtained distribution of agents by processors, one can use amdahl's law, which connects the maximum achievable acceleration in parallel computations (sp) with the number of processors (p) and the fraction of consecutive computations (a). fig. 2. the illustration of amdahl’s law for description of parallel speedup on the number of processors and the value of parallel portion. in our case, sequential computations arise due to an imbalance in the number of agents on the nodes, i.e. if we have two nodes and the first 400 agents, and on the second 300, then we can say that the first 300 agents are processed by each node in parallel, and the second 100 agents are processed sequentially by the second one, because the second node can’t perform calculations at this time. the figure below shows the dependence of the parallel speedup on the number of mpi processes. the comparison of amdahl’s law estimates for decomposition with and without cell weights show that decomposition with cell weights leads to a very balanced distribution of agents and potentially to a very good parallel speedup. the real parallel speedup of the test simulations is significantly lower than amdahl’s law estimates but it still shows rather reasonable speedup. the implementation of the scalable modelling framework 147 copyright ©2019 assa. adv. in systems science and appl. (2019) fig. 3. the dependence of the parallel speedup on the number of mpi processes. the scalability of the test simulation is compared to the ideal speedup curve, and to the estimate speedup based on the amdahl’s law for the cases of decomposition with and without cell weights. for the test calculation, a system consisting of 8400000 agents was selected. for each agent the number of neighbor agents was selected randomly (1..10), the neighbor agents were selected randomly within the same cell on the map and it’s direct neighbors. the table shows the time of calculation of the step and the average number of agents on one processor. table. 1. the results of the simulation of 8.4 million of agents for different number of processors. number of cores time step simulation time, s parallel speedup number of agents per core fraction of remote message sends, % 1 55.683 1 8417590 0 2 28.872 1.92 4208795 0.15 4 16.114 3.45 2104397 0.39 8 10.262 5.42 1052198 0.68 12 7.006 7.94 701465 0.91 16 5.268 10.56 526099 1.10 24 3.796 14.66 350732 1.72 48 2.161 25.75 175366 2.26 96 1.378 40.38 87683 3.41 192 0.846 65.78 43841 5.56 148 v.l. makarov, a.r. bakhtizin, e.d. sushko, g.b. sushko copyright ©2019 assa. adv. in systems science and appl. (2019) the results shown in the table above show that the distribution of agents by processors as a result of the decomposition was fairly uniform, but the overall parallel speedup in this version is limited to a several dozen times. the table also shows the dependence of the fraction of remote message sends which were delivered to agent on another processor. the increase of the number of processors leads to increase of the number of borders between areas attributed to different processors, the number of remote sends and the ratio between network exchanges and local calculations within one processor which is another factor which affects the scalability of the simulation. uniform distribution of the number of agents by processors allows you to evenly distribute the load of ram of each node of the cluster. as a test case, a multi-agent system was calculated with 148 million agents on 192 cluster processors. the main difficulty in this case was that with the total amount of ram of the allocated nodes in 192gb on each node of the cluster there was only 8gb of ram. thus, it was necessary to distribute the agents evenly, so that on none of the cluster nodes the simulation processes exceed the memory consumption limit. fig. 4. decomposition of the computational grid into 192 processors taking into account population density. the figure above shows the distribution of the calculated grid for 192 processors, built taking into account the population density. in the distribution, the average number of agents per processor was 772000. the number of agents per processor ranged from 743711 to 787266, i.e. the spread of the number of agents on one processor was ~ 6% of the average. the calculation of one step in this simulation took about 15 seconds. in the future, the spread of the number of agents can be reduced by using a smaller calculation grid, which will improve the accuracy of the decomposition. 3.conclusion in this work a new framework for parallel calculations of agent-based models was presented and tested. the framework uses metis graph decomposition algorithm to perform natural spatial distribution of agents in abm-simulations to achieve high level of balance and parallel scalability. the decomposition algorithm based on cell distribution allows us to compute a balanced distribution of agents efficiently. which can be used for both initial distribution of agents and redistribution of agents during simulation. the provided results of the implementation of the scalable modelling framework 149 copyright ©2019 assa. adv. in systems science and appl. (2019) the test simulations show good scalability of the program across multiple computational nodes and possibility to perform abm simulations with number of agents comparable to population of large countries and regions. acknowledgements this work was supported by the russian science foundation (grant # 14-18-01968). the possibility to perform computer simulations at the mvs-100k joint supercomputer center and tianhe 2 supercomputer is gratefully acknowledged. references [1] karypis, g. & kumar, v. (1995). metis-unstructured graph partitioning and sparse matrix ordering system, version 2.0. [2] karypis, g. & kumar, v. (1999) parallel multilevel k-way partitioning scheme for irregular graphs. // siam review, vol. 41, no. 2, pp. 278 300 [3] lasalle, d. & karypis, g. 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(1967) direct solutions of sparse network equations by optimally ordered triangular factorization. // proceedings of the ieee, 55(11):1801–1809 https://en.wikipedia.org/wiki/axum_(programming_language https://www.bsc.es/computer-applications/pandora-hpc-agent-based-modelling-framework https://www.bsc.es/computer-applications/pandora-hpc-agent-based-modelling-framework http://www.mono-project.com/ https://en.wikipedia.org/wiki/digital_object_identifier https://doi.org/10.1137%2fs1064827598337373 adv syst sci appl 2018; 03; 29-38 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/654 situational awareness formation for large network-systems viacheslav abrosimov1* 1) smart solutions, samara, russia e-mail: avk787@ yandex.ru abstract: an approach to situational awareness formation as a necessary condition for the operation of large network-centric systems is considered. the basis is information acquisition from all system elements according to the “initial data-parameter-event-situation-solution” scheme. the situational awareness model is constructed as a neural network with an ensemble organization and associative-projective structure. the model is learned using precedent examples. keywords: network, situation, awareness, neuron, network, ensemble 1. introduction large systems include a large number of elements and have a complex hierarchical structure and, therefore, a complex character of connections. it is necessary to take into account the existence of random factors from the environment and also the elements of the system itself. the characteristic feature of modern large systems is network centricity, which consists in the network nature of systems and use of various information for decision making. situational awareness is a necessary condition for the operation of large network-centric systems. large systems differ from complex systems, as the operator in the control loop is included in system. the situational awareness systems are actively developed for ensuring the efficiency and safety of large systems in manned aircraft, transport and other fields (e.g., see [1, 2]). the basis of such systems is the high operational efficiency of information interaction. in defence applications, situational awareness models are developed as organizational models for knowledge management on the battlefield and also for the accumulation of incoming information about the current situation in special-purpose databases. these data come to situational awareness system from intelligence agents, advanced units and from higher-level governmental structures including a combat command control information network. managing the processes of data collection allows to perform a single visualization of the combat space, depending on the level of management and the access rights of government officials [3]. 2. situational awareness problems for large systems according to the endsley model [4], the situational awareness concept distinguishes between three basic elements: a) information about the spatial and temporal environment of the situation b) situation analysis and prediction in terms of scenarios and c) the actions of other participants. obviously, to ensure a high degree of situational awareness, all elements of a large system should be considered as information providers. analysis shows that the main problem of modern large-scale economic systems is the lack of properly organized information and communication network between the elements of the system that implements the principles of situational awareness, i.e., a common information environment and its continuous * corresponding author: avk787@ yandex.ru 30 v. abrosimov copyright ©2018 assa adv. in systems science and appl. (2018) updating using the largest possible information sources (large system inputs). but the major unresolved problems in the situational awareness fields is decision uncertainty caused by the abundance of redundant information. events and situations objectively occur in large systems. an event in a large system is understood as the realization of a random process associated with predicted or previously neglected factor demonstration. it is clear that events in large systems are not independent. moreover, in large systems, the emergence of an event from outside (the influence of the external environment) and the event generated by the system are both observed. often the source of an event remains undetermined, and its impact on large system functioning is critical. a situation in a large system is understood as a group of events that lead to the result of system functioning and requires decision making. a precedent in a large system is understood as a situation that arose as a result of the onset of an events aggregate caused by external factors or as a result of setting the aggregate parameters of a large system, according to which a) appropriate decisions were made and b) the consequences of such decisions are known. consider an example. in the large economic and ecological system "a forest production and extraction" described by various parameters (total area, volume of felling, percentage of deadwood, etc.), the event of abandoned fire occurs, leading to the situation of fire and causing a precedent (annual fires on the territory of the russian federation). it is impossible to study the influence of all factors determining events and situations in large systems. in this case, it is necessary to apply analytical methods allowing, on the one hand, to carry out estimation with maximum complete consideration of all features/ each element of a large system determines and accumulates information, assesses the situation in accordance with its limited capabilities, thereby being able to act only in its narrow area of responsibility. but in a properly constructed large network-centric system, there is the gradual accumulation of experience and knowledge. all system elements participate in this process with the course of time. however, such phenomena as "the of situational awareness impulse " may arise. this reflects the fact that previously missing information is suddenly detected by observing any of system elements. such impulses are associated with events occurring in large organizational and technical systems. in practice, also there may be a “gap in situational awareness”. it occurs as a result of limited or even lost communication and, accordingly, interaction between the elements of the system. with a high degree of situational awareness, the elements of the system may function as autonomous ones. however, in future system life, the accumulated underdevelopment can lead to dangerous errors. the elements of the system must know, predict and respond to potentially possible situations related to the performance of their own functions. they should also know the strategies of the other elements that are a part of the larger system to coordinate their actions. this means that some information resources should be constantly filled by current information: initial data, events and situations that arise during large system functioning. large system situational awareness formation is the process of collective accumulation of information that describes the events and situations generated by the environment and system initial conditions, which is carried out in parallel with system functioning and implemented by all large system elements. based on the mathematical model of neural networks, this paper suggests a methodological approach for investigating the problems of situational awareness formation. 3. basic principles let us formulate some basic principles. the principle of recurrence. it means that situations that occur during large system functioning in general way are repetitive, similar. however, situations could never be the same. but, according to our approach, a difference comes from a number of inessential background situation awareness formation for large network systems 31 copyright ©2018 assa. adv. in systems science and appl. (2018) details that do not affect the situation, while the situation’s similarity appears due to the coincidence of a few main features. the application of methods based on the principle of analogies for the study of events and situations is exposed to certain criticism in scientific papers. however, this principle is widely and successfully used by many researchers. the principle of hierarchy. the situational awareness formation can be conditionally described as a certain hierarchical structure that allows to design situational awareness components interacting with each other. the feasibility of such formalizations is obvious. the problem consists not so much in decomposition, but in the representation of constituent elements, the definition of appropriate interconnections between different-level elements of the hierarchy and also in the completeness of these connections and the degree of their mutual influence. the principle of comparability. the situational awareness entities can be compared using various criteria. in practice, such a comparison is made frequently and, depending on criteria, has either the character of compared integrated numbers or fuzzy statements. the principle of importance. the situational awareness information can be described by a limited set of most important parameters, events and situations. other parameters can be considered as a certain background. 4. approach to situational awareness formalization among the rich variety of possible mathematical approaches to the formalization of situational awareness, we suggest to use neural networks. application of neural networks for solving different problems was mentioned in literature. artificial neural networks are currently actively used in forecasting, pattern recognition, and control. first of all, the architecture of neural networks is very easy to redesign; this is extremely convenient for adding elements for structure when new influencing factors appear. secondly, the idea of future events and situations is always objectively incomplete and fuzzy. an advantage of neural networks is that they have ability to restore information from its separate parts. thirdly, neural networks have learning ability, which allows to make decisions by analogy in case of situations’ similarity. fourthly, neural networks enable to consider complex situations formed under the influence of factors of various nature and dimension. they can use heterogeneous information that is also essential for analysis models. with such networks, statistical data and expert estimations can be used together and the information of various probability can be processed. a neural representation is based on the concept of a neuron and neural ensemble [6]. as usually a neuron is a formal element whose output activity ranges from 0 to 1 and is determined by the level of input excitation. the excitement level is determined by the sum of input excitations transmitted from other similar neurons taking into account the appropriate weight functions between the neurons. a neural ensemble is a set of neurons united by mutual exciting connections so that the whole ensemble becomes excited if a given number of neurons in it are excited. neural ensembles have unique properties, which allow working with them as with a single unit and when necessary to split them up on small parts and to create new compound structures. the ensembles are intended for the formalization of neural image concepts with various degree of complexity. our model is based on the hypothesis about the separation of a neural network into individual hierarchical levels. at every hierarchical level, neurons are connected to each other and create neural ensembles. there are special operations over ensembles, which allow to add, subtract, create, and destroy ensembles, as well as to establish (destroy) connections between them, and so on. the ensembles can be united into groups (fields). the opportunity to obtain various fields of ensembles is provided, such as: associative (for creation of ensembles), buffer (for storage of ensembles) and difference (for comparison of ensembles among themselves). ensembles and 32 v. abrosimov copyright ©2018 assa adv. in systems science and appl. (2018) ensemble groups are linked to each other by the so-called “connections” transferring the excitation image from one level of hierarchy to another. through such “connections,” the excitement of ensembles, e.g., at a lower level of hierarchy causes the adjustable excitation of the higher-level ensembles. high reliability is an important feature of neural models. practical experiments have shown that the failure of 10% and even more elements of a neural network does not lead to significantly different results. it happens because each element in neural networks with the ensemble structure is responsible for many functions, not only for one. the failure of such an element deteriorates some functions, but owing to the presence of many elements and their interrelations this deterioration is so small that it becomes insignificant for the whole network. in conventional approaches, the factor of uncertain input information (which can be incomplete, fuzzy, doubtful etc.) may cause errors in decision making, as the elimination of initial uncertainty in prediction models can appear impossible in principle. in neural models, an image can be created using partially uncertain information. this image does not completely fit the reality but nevertheless is very "close" to it. by defining an excitement edge, the most significant factors can be studied. in other words, the influence of uncertain (and very often doubtful) input factors on the result can be reduced. 5. hierarchical neural network for describing large system situational awareness the basic idea of using neural networks to describe the situational awareness formation processes is to employ an excited ensemble as a semantic image of a parameter, characteristic, object, factor, event, situation, precedent, solution and other concepts of large systems operation [7]. from the author’s point of view, the idea to use a neural network for describing situational awareness has certain novelty. let us construct a multilayer hierarchical neural network and ensembles exciting by such neurons. the first level will be composed of neurons and ensembles of “input parameters,” corresponding to the initial data on various factors of the large system and its elements environment. the second level consists of neurons and ensembles of “events” that will semantically describe the events of the influencing environment or the facts occurring in the course of large system functioning. the third level consists of neurons and ensembles of “situations” that simulate situations arising in a large system. and, finally, at the fourth level, we will place the ensembles of “precedents,” which are associated with the decisions on such precedents. between the levels, we create projective connections. they will transmit excitation from one level to another. the excitation of lower-level ensembles in the hierarchy by projective connections leads to the controlled excitation of the upper-level hierarchy ensembles. the connections between neurons and neural ensembles will be established and changed according to the results of learning. they can be either exciting (increasing the activity of neurons and ensembles) or inhibitory (reducing activity) as a result, each concept is formalized by means of neural ensemble. at each level of hierarchy, the ensembles are identical in terms of structure (neurons which are linked by mutual exciting connections), yet different in terms of their meaning. the level of excitement of each ensemble depends on the importance of its influence on the ensembles in each consecutive level of hierarchy. connections between neurons and neural ensembles can be established and changed according to the results of training. they can be stimulating (increasing the activity of neurons and ensembles) or breaking (reducing the activity of neurons). the training procedure is as follows. “teacher” forms a variety of source data for neurons at the first hierarchical level. the excitation of the inputs of neurons and neural ensembles (the “parameters” of the first layer of situation awareness formation for large network systems 33 copyright ©2018 assa. adv. in systems science and appl. (2018) the hierarchy) through the ensemble and projective connections initiates the process of excitation for the neural network. the excitement of the projective links of the first and second layers is transmitted to the “event” ensembles. due to the connections within the ensembles and between the ensembles, the ensembles of the second layer are excited, and then, through the projective connections of the second and third layers, the “situation” ensembles of layer are excited. to each such ensemble at the third layer, the “precedent” ensemble is aligned at the fourth layer of the model hierarchy. the “precedent” ensembles are put in line with the decisions taken under given conditions and the consequences of such decisions. initiating the excitation of the corresponding neurons and ensembles in the course of training, “teacher” can set new non-standard events that form previously unknown situations, create the necessary precedents and point out the right solutions in such situations with the aim of shaping the behavior of a large system in the required effective direction. 6. situational awareness: filling of information the neural model is located and functions in a certain "internet-cloud". all elements of a large network-centric system have access to this source. situational awareness formation here is the excitation process of a multilayer neural network based on the “parameter-characteristic-eventsituation-precedent” principle, which runs sequentially as soon as information is received. each element of a large system at each time transmits its current data to the model. the "current situation" ensemble, which is excited in the way described above under the "parameter" "event" "situation" scheme, is compared with ensembles that are in the "precedents" field and built on the learning outcomes or mark during the experience of system development. the type and name of the ensemble that is closest to the emerged situation are then determined. the decisions that were made in similar cases and the consequences of such decisions are also determined. the analysis of these consequences makes it possible to estimate the occurrence of problems for a given realization of the set of input parameters and initial data. this information is used for making the decision. the neural model of situational awareness formation allows us to seek a solution in a neighborhood of the already known effective ones. for this purpose, an ensemble of "precedents" and the most effective solutions and satisfying consequences obtained in the process of learning is selected. a comparison between the characteristics of the input data vectors applied to the inputs of 1-st neurons level is made. the process of excitation again propagates through the neural network but with the modified input vector. a new “situation” ensemble is formed, and the ensembles are again compared. it is assumed that the necessary knowledge of precedents and effective solutions will be developed as the knowledge base of situational awareness during large system functioning. 7. conclusion the paper has reflected only the general principles of neural model functioning for the analysis of relations between different states. unfortunately, the limited format of a conference paper does not allow to give more details on many other important aspects of modeling. the problems of model training and other difficulties in its realization, such as the principles of coding for input information and decoding for output information, the issues of interpretation of modeling results, etc., have not been touched. the idea of neural models application for situational awareness formation has become an efficient tool to describe the situational awareness processes, to choose most effective strategies and to predict the consequences of decisions based on situational awareness information. 34 v. abrosimov copyright ©2018 assa adv. in systems science and appl. (2018) acknowledgements this work was supported in part by the russian foundation for basic research, project no. 16-08-00832-a. references [1] fedunov, b.e. (2002). bortoviye operativno sovetuyuschie ekspertnie sistemy takticheskih samoletov piatogo pokoleniya (obzor po materialam zarubezhnoy pechati) [onboard advisory expert systems of tactical aircraft of the fifth generation (review of foreign press materials)] moscow, sic gosniias, [in russian]. [2] ermakov, a.n.,merkulov, а.а., panfilov, s.a., raikov a.n. (2015). podderzhka reshenij v avarijnyh situacijah na zheleznoj doroge na osnove setevoj jekspertno-analiticheskoj sistemy [support of solutions in emergency situations on the railway road on the basis of the network expert-analytical system], intellektual'nye tehnologii na transporte, 2, 5-9 [in russian]. [3] rubtsov, i.v., boslyakov a.a.,lapshov,v.s., mashkov k.y, et.al. (2015). problemy i perspektivy razvitija mobil''noj robototehniki voennogo naznachenija [problems and prospects for the development of mobile robotics for military use]. inzhenernyj zhurnal: nauka i innovacii, 5(41), 26-31 [in russian]. [4] endsley, m.r. (1995) toward a theory of situation in dynamic systems. human factors, 37 (1), 32-64. [5] callan, r. (1999). the essence of neural networks. the essence of computing series. prentice hall europe, london. [6] kussul e.m. (1992). associativnye nejropodobnye struktury [associative neural structures]. kiev, naukova dumka [in russian]. [7] abrosimov, v.k. (2014). nejronnaja prostranstvenno-vremennaja model' dvizhenija ob"ektov upravlenija [neural space-time model of the motion of objects], nejrokomp''jutery: razrabotka, primenenie ,3, 26-35 [in russian]. adv syst sci appl 2021; 03:31–39 published online at https://ijassa.ipu.ru. convolution neural network based covid-19 screening model ashish nainwal*, gorav kumar malik, amrish jangra dept. of electronics and communication engineering, fet gurukul kangri (deemed to be) university, haridwar, uttarkhand, india abstract: coronavirus disease 2019 (covid-19) is a high death rate respiratory condition that requires easy-to-reach markers for prediction. the electrocardiograph (ecg) alterations that may occur after covid-19 hospitalization have not been fully studied yet. covid-19 also affects heart function, which can be seen on an ecg. as a result, ecg can be used to detect virusinfected individuals. the database consists of ecg images. in this scenario, a convolution neural network (cnn) is utilized to classify covid-19 ecg. the model is made up of eight layers, including a convolution layer, a max-pooling layer and a dense layer. the ecg image is fed into a cnn model, which classifies the covid-19 ecg. the model provides us with 98.11% accuracy, 98.6% sensitivity and 96.40% specificity. although 100.00% of the categorization of normal images and covid-19 ecgs were not accurately determined by the proposed cnn model, this is the first cnn model to categorize ecg images into normal and covid-19 classes from the ecg database and provide additional diagnostic to medical experts. keywords: ecg, convolution neural network, covid-19 1. introduction coronavirus disease 2019 (covid-19) is the clinical sign of contamination with the severe acute respiratory syndrome coronavirus-2 (sars-cov-2). more than 170,000,000 persons in more than 180 nations or regions worldwide suffered as a result of the 2019 coronavirus disease (covid-19) global pandemic [1]. the infection’s clinical course is characterized by respiratory symptoms (fever, cough, and tiredness), which can progress to pneumonia, acute respiratory distress syndrome (ards) and shock [2]. the modern world is in the midst of a never-before-seen health disaster. the covid-19 outbreak is wreaking havoc on hospitals and medical professionals worldwide. according to epidemiological statistics, coronavirus patients with prior cardiovascular irregularities are at a higher danger of pre and post covid19 related issues and mortality [3]. covid-19 has the potential to affect the heart and lungs. according to several articles, covid-19 has been related to myocarditis, acute coronary syndromes and decompensated heart failure [4] [5] [6]. the pandemic also emphasizes the importance of preparing cardiac healthcare for covid-19 as well as acute and chronic cardiac therapy in individuals where patient and medical delays could be hazardous. lecun et al. introduced cnn in 1990 and in recent years, it has become one of the most potent methods of machine learning [7]. cnn offers benefits both in accuracy and performance in imaging recognition, sound classification and semantic identification with the support of fast-growing graphics process unit (gpu) technology [8] [9] [10]. in addition, recent research has demonstrated the considerable promise of cnn with biological applications including categorization of animal behavior, the prediction of protein structures ∗corresponding author: ashishnainwal86@gmail.com 32 a. nainwal, g k malik, amrish and pattern recognition (emg) [11] [12]. recent research has also identified intriguing uses of cnn in bio-signals such as ecg for time series [13]. the cnn framework benefits from using the huge training data set to overall improvements of classification parameters. degerli et al. have developed a unique methodology to the composite location, gradation and detection of covid-19 from 15495 cxr images with the creation of the ”infection mappings” capable of precisely detecting and grading covid-19 seriousness with a precision of 98.69% [14]. kesim et al. suggested a new cnn model for chest radiography data (xray images) categorization [15]. several in-depth learning models, including chest x-ray pictures and computer ct scans have been proposed in recent research to detect anomalies of covid-19 in patient healthcare [16]. in this article, the covid-19 patient ecg and normal ecg images were put to the proposed convolution neural network. the proposed cnn model has eight layers, comprising three convolution layers, three pooling layers and two dense layers. the approaches yielded good results for classifying covid-19 patients based on their ecg. 2. method and material 2.1. database the collection of data is in image data form [17]. ch. pervaiz elahi institute of cardiology, nishtar medical university and punjab institute of cardiology are the data sources, all of which are situated in pakistan. this image data collection is collected with 12 lead-based edan series equipment. the rate of sampling is 500hz. several medical professionals have marked all ecg images. figure 2.1 and 2.2 shows the image of normal ecg and covid-19 patient ecg respectively. s. no type of ecg no of images sampling rate 1 covid-19 250 500 2 normal 859 500 table 2.1. data set details all ecg devices used to collect data have been set up as ”on” for important warnings. ecg technicians are taught to respond to all forms of edan ecg device alerts so that ecg technicians are able to carry out all precautionary actions during ecg operations. this step is crucial to more precisely capture ecg images, the data created by highly qualified professionals actively involved for many years, collected ecg images were manually reviewed by medical experts and supervised by senior physicians with ecg interpretation expertise. 2.2. preprocessing the ecg interpretation is done by image, so we need to improve image quality; we used a non-linear method gamma correction for enhancement of images its improve the interpretability or view of data and give ’better’ contribution to input preparation for image processing techniques. for normalization of images several operations are performed like scalar multiplication, addition and subtraction gamma correction alludes to the image enhancement on contrast by changing the unique range of pixel intensity distribution and conveys a non-linear technique on the source image pixels and can cause immersion of the picture is changed. moreover, if the gamma value is too large or small shows a low contrast image, from the outset, gamma revision appears to either obscure or light up a picture, however, this is a gross misrepresentation. figure 2.3 shows gamma-corrected image on two different gamma values 0.47 and 1.83. we would already be able to change the normal brightness of a picture by some adjusted standardization calculation, the easiest of which copyright © 2021 assa. adv syst sci appl (2021) convolution neural network based covid-19 screening model 33 fig. 2.1. normal ecg fig. 2.2. ecg of covid-19 patient would be basically enhancing every pixel force esteem, viably ”moving” the mean pixel power esteems across the whole picture. pixel (p ) of the image defined in a range between 0-255, δ is a representation of angle value, grayscale image of ecg signal represented by k of the pixel (kεp ) choose a midpoint km of the range [0, 255]. the linear map from pixel group p to group δ represents as: φ : p → δ = {ω|ω = φ(k)},φ(k) = πk/2km (2.1) the δ is mapped to γ (gamma value symbol) h : δ → γ,γ = {γ|γ = h(k)} (2.2) h(k) = 1 + f1(k) (2.3) f1(k) = a cos(φ(k)) (2.4) where a lies between 0 to 1 as a weighted factor and h is the histogram intensity level. according to map, group p and gamma group pixel values are interrelated. gamma number helps to find the arbitrary pixel value. let γ(k) = h(k), and gamma correction function is defined as: t (k) = 255 ( k 255 )1/γ(x) (2.5) copyright © 2021 assa. adv syst sci appl (2021) 34 a. nainwal, g k malik, amrish fig. 2.3. : input image: (i)ecg trace image original (ii)gamma corrected image when γ=.47 (iii) gamma corrected image when γ=1.83 where the output pixel correction value is represented by t(k) in grayscale. after gamma correction, the dataset is treated so that the ecg images scale to meet the cnn network input picture size needs. the z-score normalization was performed with the average and standard deviation of images. 2.3. convolution neural network cnn is the popular sort of neural artificial network that includes algorithms for supervised learning tasks. it is also an essential tool for deep learning with many hidden layers and parameters. in several areas like image processing, design recognition and other cognitive tasks, the cnn was widely used. a standard cnn is made up of three-layer types: the convolutions layer, the fully connected layer and the pooling layer. the convolution layer conducts overlapping operations throughout the entire image spatially to create characteristics. for downsampling maps, the pooling layer is responsible. the fully connected layer classifies depending on the characteristics learned. one-dimensional signal evolution is referred to as 1d convolution or simply convolution. if the convolution occurs between two signals covering two perpendicular dimensions, the convolution shall be called a 2d convolution. figure 2.4 shows the 2d convolution operation. this notion may be expanded to include multi-dimensional signals via which multi-dimensional convergence is possible. polling is used to simplifying or reducing the information acquired from feature maps spatial dimensions. the most frequent usage of pooling is the maximum of pooling because of its speed and increased convergence. this is what makes a filter (usually the size 2x2) and a step the length is the same. figure 2.5 shows the polling operation. each input is connected to each output and hence has the phrase ”fully connected. ”it is the last layer, usually following cnn’s last pooling layer. fully linked layers act like a typical neural network and contain around 90% of cnn parameters. this copyright © 2021 assa. adv syst sci appl (2021) convolution neural network based covid-19 screening model 35 fig. 2.4. 2d convolution operation fig. 2.5. operation of max polling layer essentially enters the last pooling layer output and produces a n-dimensional vector where n is the number of classes to pick from. as the function for neuron activation, non-linear transfer functions are employed in ann. for example, the most frequent activation functions are sigmoid f(x) = 1/(1 + exp(−x)) and hyperbolic tangent f(x) = tanh(x). sigmoid and hyperbolic tangents are both non-linear saturating functions, which with an increasing input decrease to almost zero. a recent study has shown the growth in cnn applications’ rate of speed and rating performance, as well as the classification performance of such a linear rectified function f(x) = maximal(0,x)(relu). the activation function relu is in the dense layer of our cnn model. the suggested 2-dimensional (2d) cnn model is shown in figure 2.6. the eight layers of the proposed model consist of three convolution layers, three max-pooling layers, and two dense layers(fully connected). the layers are converged to the corresponding kernel size for each convolution layer (64, 32 and 64). the max-pooling layer also called a downsample layer, should be utilized to the features maps after every two convolution layers. it was used to copyright © 2021 assa. adv syst sci appl (2021) 36 a. nainwal, g k malik, amrish fig. 2.6. cnn model decrease computer complexity and override controls. the stride is set to 1 and 2, respectively, for the convolution and the max-pooling layer. two activation functions relu and softmax are used in dense layer to improve the performance of the model. 3. result and discussion this section discusses the performance of the ecg image classification network. we suggested an efficient approach to extract and detect characteristics from each ecg image to examine the performance of the proposed system. the cnn model is built in python by using the keras library in colab.research (open source) google platform. the filtered image is sent into the cnn model, which classifies it. the image is filtered during preprocessing to increase system accuracy. different measures have been used to evaluate the system’s performance. three key indexes are used to assess the classifier’s performance, namely accuracy(acc), sensitivity(se) and specificity(sp). se = (tp/tp + fn))× 100 sp = (tn/(tn + fp )× 100 acc = ((tp + tn)/(tp + tn + fp + fn))× 100 predicted normal covid-19 accuracy sensitivity specificity original normal 847 12 98.11 98.60 96.40 covid-19 9 241 98.11 96.40 98.60 table 3.2. confusion matrix for complete data set predicted normal covid-19 accuracy sensitivity specificity original normal 243 7 97 97.2 96.8 covid-19 8 242 97 96.8 97.2 table 3.3. confusion matrix for balanced data set copyright © 2021 assa. adv syst sci appl (2021) convolution neural network based covid-19 screening model 37 fig. 3.7. training and validation accuracy and loss plot for complete data set and balanced data set where the number of impostor acceptations is false positive (fp), true negative (tn) is an impostor refusal number; false negative (fn) is a valid denial number and true positive (tp) is a valid acceptance. table 3.2 shows the confusion matrix for complete data set with 859 samples of normal ecg and 250 samples of covid-19 patient ecg.proposed convolution neural network used 60% of data for training and 40% data for testing purposes. it demonstrates that 98.60% of normal ecg segments are accurately classified in the normal class, while 96.40% of covid19 images are accurately classified in the covid-19 class. only 1.40% and 3.6%of ecg images are misclassified as covid-19 and normal class, respectively. the model incorrectly detects 12 normal ecg images as covid-19 ecg and correctly detects 847 images. out of 250 covid -19 ecg images, the network incorrectly classifies 9 of them, while the rest are correctly identified. with the entire data set, the accuracy is 98.11%. table 3.3 illustrates the confusion matrix for a balanced data set with 250 samples of each class. out of 250 images, 7 normal ecg images and 8 covid-19 ecg images are classified copyright © 2021 assa. adv syst sci appl (2021) 38 a. nainwal, g k malik, amrish wrong. in this data set 2.8 % of normal ecg images are incorrectly labeled as covid-19. furthermore, the misclassification rate of the covid-19 ecg image is approximately 3.2 %. the accuracy of this scenario is 97%. figure 3.7 shows the graph between accuracy and loss versus no of epochs. the first two graphs show the training and validation accuracy vs. epochs and loss vs. epochs, respectively for the complete data set. the following two plots are for a balanced data set. it can also be shown that after a few epochs, the networks achieve and stable with the highest accuracy and lowest loss. 4. conclusion this article presents an approach for the screening of covid-19 based on the convolution neural networks using data-set features. the suggested approach provides links between multiple layers of the original cnn architecture using convolution blocks, which produce dynamic layer combinations of different layers. the suggested approach is used to examine two scenarios of the classification task. the accuracy, sensitivity and specificity are used to analyze outcomes. the model gives 98.11% accuracy for complete data set and 97% accuracy for balanced data set. since the suggested approach leverages the ecg trace image that smartphone acquired and widely accessible facilities in low resource nations, this study helps to diagnose covid-19 and other heart defects as a second opinion by computer-aided methods. references 1. dong, e., du, h., & gardner, l. (2020) an interactive web-based dashboard to track covid-19 in real time. the lancet infectious diseases, 20, 533–534. 2. grasselli, g., et al. (2020) baseline characteristics and outcomes of 1591 patients infected with sars-cov-2 admitted to icus of the lombardy region, italy. jama, 323, 1574– 1581. 3. bergamaschi, l., et al. (2021) the value of ecg changes in risk stratification of covid-19 patients. annals of noninvasive electrocardiology, p. e12815. 4. doyen, d., moceri, p., ducreux, d., & dellamonica, j. (2020) myocarditis in a patient with covid-19: a cause of raised troponin and ecg changes. the lancet, 395, 1516. 5. zheng, y.-y., ma, y.-t., zhang, j.-y., & xie, x. (2020) covid-19 and the cardiovascular system. nature reviews cardiology, 17, 259–260. 6. angeli, f., spanevello, a., de ponti, r., visca, d., marazzato, j., palmiotto, g., feci, d., reboldi, g., fabbri, l. m., & verdecchia, p. (2020) electrocardiographic features of patients with covid-19 pneumonia. european journal of internal medicine, 78, 101–106. 7. lecun, y., bottou, l., bengio, y., & haffner, p. (1998) gradient-based learning applied to document recognition. proceedings of the ieee, 86, 2278–2324. 8. lee, h. & kwon, h. (2017) going deeper with contextual cnn for hyperspectral image classification. ieee transactions on image processing, 26, 4843–4855. 9. hershey, s., et al. (2017) cnn architectures for large-scale audio classification. in 2017 ieee international conference on acoustics, speech and signal processing (icassp), 131– 135, ieee. 10. pei, z., li, c., qin, x., chen, x., & wei, g. (2019) iteration time prediction for cnn in multi-gpu platform: modeling and analysis. ieee access, 7, 64788–64797. 11. zhai, x., jelfs, b., chan, r. h., & tin, c. (2017) self-recalibrating surface emg pattern recognition for neuroprosthesis control based on convolutional neural network. frontiers in neuroscience, 11, 379. 12. cheng, j., liu, y., & ma, y. (2020) protein secondary structure prediction based on integration of cnn and lstm model. journal of visual communication and image copyright © 2021 assa. adv syst sci appl (2021) convolution neural network based covid-19 screening model 39 representation, 71, 102844. 13. li, y., pang, y., wang, j., & li, x. (2018) patient-specific ecg classification by deeper cnn from generic to dedicated. neurocomputing, 314, 336–346. 14. degerli, a., ahishali, m., yamac, m., kiranyaz, s., chowdhury, m. e., hameed, k., hamid, t., mazhar, r., & gabbouj, m. (2021) covid-19 infection map generation and detection from chest x-ray images. health information science and systems, 9, 1–16. 15. kesim, e., dokur, z., & olmez, t. (2019) x-ray chest image classification by a smallsized convolutional neural network. in 2019 scientific meeting on electrical-electronics & biomedical engineering and computer science (ebbt), 1–5, ieee. 16. qiblawey, y., et al. (2021) detection and severity classification of covid-19 in ct images using deep learning. diagnostics, 11, 893. 17. khan, a. h., hussain, m., & malik, m. k. (2021) ecg images dataset of cardiac and covid-19 patients. data in brief , 34, 106762. copyright © 2021 assa. adv syst sci appl (2021) introduction method and material database preprocessing convolution neural network result and discussion conclusion adv syst sci appl 2019; 01; 31-43 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/649 activity of russian companies of the agri-food sector in regional industrial value-added chains veronika yu. chernova 1*, inna v. andronova 2, ekaterina a. degtereva 1,3,4, alexander m. zobov 1, vasily s. starostin 5 1) department of marketing, peoples' friendship university of russia, russian federation, moscow, russia 2) department of international economic relations, peoples' friendship university of russia, russian federation, moscow, russia 3) institute of europe of the russian academy of sciences (ras), moscow, russia 4) mgimo-university, moscow, russia 5) institute of marketing, state university of management, moscow, russia received september 24, 2018; revised march 12, 2019; published april 15, 2019 abstract: this article is concerned with the study of the factors, underlying the decision of russian companies to relocate production units to foreign countries and form global value-added chains (gvacs). it is proposed by the authors to consider the emergence of economic interests in relocation from the standpoint of synthesis of the concept of integration based on the natural advantages and the concept of gvac. a high potential for the development of cooperation by russian companies is noted herein, in particular, in the agro-industrial complex, and at the same time, there is a limited interest of russian business in the integration processes in the eaeu format. the factors that hinder the expansion of russian business to the partner countries in the eaeu, as well as the risks of deepening the cooperation ties, are singled out. keywords: industrial value-added chains, eaeu, industrial integration, cooperation ties. 1. introduction the problem of the prospects for the development of economies of the eaeu member states in the format of their participation in the eurasian integration project [1] to create four freedoms: the freedom of movement of goods, services, finance, and labor, is one of the most controversial nowadays [2-3]. the eaeu is a young integration project, which has been operating in the customs union environments since 2011, and in the economic union environments – since 2015. however, the key theme, from the point of view of enhancing macroeconomic stability and implementation of the competitive potential, is the formation of a coherent industrial, transport, energy and agrarian policy, and the deepening of industrial cooperation [4]. in the context of increasing global risks, the deepening of integration and the development of russia's cooperation with the eaeu countries are becoming ever more urgent. the eaeu was established to realize more completely the economic potential within the regional economic ties of the former single union state in the creation of the conditions for improving the competitiveness of the participating countries. currently, russia is aimed at modernization of the economy based on the technological breakthrough, the diversification, and expansion of the exports to world markets, the integration into global value-added chains along with the * corresponding author: veronica.urievna@mail.ru 32 v.yu. chernova, i.v. andronova, e.a. degtereva, a.m. zobov, v.s. starostin copyright ©2019 assa adv. in systems science and appl. (2019) partners in the eaeu. the obvious advantages of the international fragmentation of production, which are pushing the enterprises of the developed countries to participate in gvacs, carry quite significant risks in themselves. in this regard, it seems relevant to analyze the sources of the emergence of the economic interests of russian enterprises of the agri-food sector in the formation of and participation in regional value-added chains in the long term. 2. literature review the problem of the prospects for the development of economies of the eaeu member states in the format of their participation in the eurasian integration project [1] to create four freedoms: the freedom of movement of goods, services, finance, and labor, is one of the most controversial nowadays [2-3]. the eaeu is a young integration project, which has been operating in the customs union environments since 2011, and in the economic union environments – since 2015. however, the key theme, from the point of view of enhancing macroeconomic stability and implementation of the competitive potential, is the formation of a coherent industrial, transport, energy and agrarian policy, and the deepening of industrial cooperation [4]. according to the calculations presented [5], two-thirds of the cumulative integration effect in the form of gdp growth are potentially due to direct production effects resulting from the expansion of cooperation, technical and technological ties between the subjects of the eaeu, and the formation of intraregional value-added chains [6-7]. therefore, according to tkachuk, the coordinated economic policies of the eaeu member countries should be directed to the creation of such structures. under the value chain is understood the full range of operations for the creation and consumption of a product – from developing ideas to after-sales service. in the paper [8], the author has distinguished between producer-oriented value chains and buyer-oriented value chains. in later studies of value chains in agribusiness [9], it is stated that retailers and branded marketers are not the only buyers, and international traders and processors play a similar role in the value chains. buyers in different market segments have different requirements and organize different types of chains. in addition, not all chains have explicit coordinators. as gereffi [10] puts it, the type of value chain is determined by various combinations of three factors: the difficulty of transferring information and knowledge, especially with regard to product specifications and the process for making an inter-firm transaction; the extent to which this complexity can be reduced, and information and knowledge can be effectively transferred without making significant investment; the ability of actual and potential suppliers to fulfill customer requirements. value chain types include a wide range of variations from low levels of explicit coordination and power asymmetry between buyers and suppliers in case of market relations to high levels of explicit coordination and power asymmetry between buyers and suppliers in case of hierarchy, and intermediate chain types of relational, modular and subordinate types. humphrey [11] notes that in agribusiness, coordination through market relations is increasingly being replaced by coordination through the direct exchange of information between firms involved in the creation of the final product. the leading coordinating company should be able to provide instructions and monitor their implementation, make key decisions on the structure of manufacturing, inclusion in the chain and exclusion from it of particular suppliers, distribution of specific types of activities between the various participants in the chain. the fulfillment of such functions is connected with the presence of a certain market power in the leading firm. the asymmetry of market power in value chains identifies the links of the profit concentration [12] and, consequently, the concentration of resources for innovation and growth. activity of russian companies of the agri-food sector 33 copyright ©2019 assa adv. in systems science and appl. (2019) increasing concentration in individual links in the value chain and increasing quality and safety requirements for food, according to humphrey [11], are the basic trends in value chains in agribusiness. the participation in gvacs is facilitated by such factors as favorable business climate and the level of economic development. the international organizations define the tariffs and other trade restrictions as the barriers to participation in gvacs. in addition to the tariffs, the wto and the oecd identify three additional important barriers to participation in value-added chains: the inadequate infrastructure, the limited access to financing and non-compliance with the world standards. the costs associated with the introduction of new standards and the tightening of existing requirements have a significant impact on small firms included in the value chain. ollinger and moore [13] show that for small enterprises, this means a decrease in profitability and the need to either leave the industry or switch to other products. thus, food safety and quality requirements contribute to concentration in the value chains in agribusiness. concentration growth is marked in all links of the value chain. thus, in the paper [14], the concentration in the agrochemical and seed industries of the agro-industrial complex was studied, which according to the author is related to the protection of intellectual property. there are some works [15] which show concentration at the stage of making agricultural products and the stage of their processing, the main reasons for which are achieving the economy of scale, stricter food safety and quality requirements, the need for continuous development and implementation of innovation. moreover, the institutional environment, the business environment, the transport infrastructure and the qualification of the employees are the significant constraints. a key role in this is played by transportation costs. until recently, it was believed that distance was an important factor in the effective functioning of gvacs and the choice of suppliers. stephenson argues that the distance factor can be overcome by greater efficiency of the transport and logistics system [16]. preigerman [17] puts that the development of transport infrastructure and technologies, but not the geography of the country as before, determine the opportunities and the trajectory of the socio-economic development of countries. the benefits of participation in value chains have been studied by many scholars. thus, the research conducted by the uk companies [18] confirms the influence of participation in gvacs on the increase in productivity: a 10% increase in the external presence in the industry increases by 0.5% the overall factor productivity of the national producer of this industry. the studies of the influence of the integration of the companies in gvacs [19] showed that in the first year after integration in gvacs, the analyzed companies achieved +5% of the productivity advantage and +9% in four years. such significant differences in productivity growth can be associated with both the modernization of the company with its integration into the value chain and differences in the location of a link in the chain, ceteris paribus. at the same time, the companies that had left the global chains lost 1% of productivity during the first year, and the cumulative loss of performance over four years was a loss of productivity of 8%. according to baier & bergstrand [20], for international fragmentation of production, a significant obstacle is transport, tariff and non-tariff barriers. they showed that a 7.5% reduction in the tariff rates combined with a 5% reduction in the transportation costs contributes to vertical specialization (offshoring) by almost one-third. based on the study of german companies making decisions on offshoring, marin [21] noted the importance of not only the low wages and closeness to germany but also the importance of a lower level of corruption, better conditions for contracting. a study of more than 16,000 german companies [22] made it possible to identify the following factors (in the descending order) as very significant and important: linguistic and cultural barriers; institutional and administrative barriers; the cost-benefit ratio; distance to production sites; budgetary issues (taxes, etc.); interests of employees; business ethics; uncertainty regarding international 34 v.yu. chernova, i.v. andronova, e.a. degtereva, a.m. zobov, v.s. starostin copyright ©2019 assa adv. in systems science and appl. (2019) standards; risk of non-compliance with patent law; distance to major markets; the lack of local suppliers that meet the requirements of the company. according to heifets [23], the mainstreaming factors influencing the decision of russian companies to expand into foreign countries are the benefit and convenience of development. tkachuk [4] justifies the creation of sectoral regional clusters with export potential by the identification of “convergence zones” on the basis of analysis of sectoral priorities of the national economies, which are of interest at least for eaeu, which have competitive advantages in these sectors and correspond to the trends of the world economy. natural advantages and the available development potential can allow the eaeu agro-industrial sector to compete for the consumer in the world market [24] while maintaining the principle of security [25]. 3. materials and methods the aim of this paper is to study the possibility of developing cooperative ties between enterprises of the eaeu countries in the agri-food sector from the standpoint of the formation of regional value chains. in the study, the author relies on the concept of value chains in the agri-food business. the author distinguishes the main types of value chains: hierarchy, chains of subordinate, modular relational type and chains with market relations. the trends of their development in the agroindustrial sector detail the growing concentration in the individual links of the value chain and the increasing requirements for the quality and safety of food products, as well as the high significance of the efficiency of transport and logistics. the author wishes to show which types of chains are predominantly formed in the eaeu, who is the initiator of the formation of chains, how much the eaeu countries are involved in value chains and what obstacles exist for the development and deepening of cooperative ties between enterprises of different eaeu countries. the study is based on relevant empirical data, statistics and publications of the eec, edb cis, and the world bank for the analysis of which traditional methods and techniques of economic analysis were used in the form of the comparison of absolute, relative and average values, index method, and grouping methods. the use of mathematical analysis tools for research is difficult due to the lack of statistical information over a long period of time, the inability to level the impact of political factors on the integration processes and the development of cooperative ties, as well as the presence of factors distorting statistical data, such as illegal imports from third countries to the eaeu (for example, illegal imports from china to kazakhstan in 2017 amounted to more than $6 billion, in 2018 – $4 billion and exceeded legal imports [26]) and in the intra-union trade, the use of various tax evasion schemes by agri-food organizations. in this connection, in the study of the concentration of markets, the author used expert estimates, which in this situation are the most accurate. 4. results the agro-industrial complex is one of the most important strategic sectors of the economy of the eaeu countries. functioning in the eaeu format provides each of the five member states a number of advantages of a general economic nature [27]. 4.1. eaeu food safety requirements there is a unified regulatory and legal framework in the territory of the eaeu. the main legal instruments that establish mandatory requirements for products and processes of their life cycle are technical regulations [28]. in total, 47 technical regulations of the union were adopted, most of which have already entered into force, and unified requirements for safety and product quality cover already about 85% of all products traded in activity of russian companies of the agri-food sector 35 copyright ©2019 assa adv. in systems science and appl. (2019) the eaeu market. the requirements of the technical regulations of the eaeu are in many ways wider than those laid down in european standards. they can combine several dozen of such standards or european directives. technical regulations of the eaeu are applied not only to protect human life and/or health or to protect the environment. given the degree of risk of harm, they may contain special requirements for products, as well as related processes. currently, as shown by the global food security index [29], calculated by economist intelligence unit, of the three eaeu countries, for which the food safety index was calculated, the highest quality and food safety index (75.2 points and the 25th place) is in russia [29]. belarus and kazakhstan are far behind in terms of quality and safety, as well as in terms of accessibility, availability of resources and efficiency (fig. 1). source: [29] fig. 1. global food security index of russia, kazakhstan and belarus 4.2. concentration russia, being the largest economy of the union, produces the majority (about 85%) of food products of the eaeu, but the share of belarus (about 10%) and kazakhstan (4.27%) is also noticeable, the republic of armenia and kyrgyzstan together produce less than one percent (table 1). table 1. share of food production in the eaeu in 2017, % indicator armenia belarus kazakhstan kyrgyzstan russia eaeu food production, including: 0.58 9.98 4.27 0.31 84.86 100 processing and preserving of meat and making of meat products 0.21 10.42 2.29 0.13 86.95 100 processing and canning of fish, crustaceans and mollusks 0.13 7.58 0.88 0.02 91.38 100 processing and canning of fruits and vegetables 1.10 7.21 11.52 0.21 79.96 100 manufacture of vegetable and animal oils and fats 0.02 3.06 5.45 0.01 91.45 100 dairy production 0.55 18.83 4.02 0.43 76.17 100 making of bakery and flour products 2.01 5.32 5.54 0.74 86.39 100 beverages 2.13 6.82 6.60 0.67 83.79 100 source: compiled and calculated by the author according to [30]. 67 71 61 75 73 66 68 63 67 63 58 66 51 58 68 40 45 50 55 60 65 70 75 80 total accessibility availability quality and safety natural resources and efficiency russia belarus kazakhstan 36 v.yu. chernova, i.v. andronova, e.a. degtereva, a.m. zobov, v.s. starostin copyright ©2019 assa adv. in systems science and appl. (2019) in the eaeu countries in agriculture, the share of private farms of the population, which are small producers, is traditionally high. in kyrgyzstan and armenia, the share of commercial organizations is less than 5%, in kazakhstan – less than ¼, in the republic of belarus in agriculture the share of the public sector in the form of collective farms is very high (about 80%) and the sector of commercial organizations is completely absent (fig. 2). source: compiled by the author according to [30]. fig. 2. the structure of agricultural production by business types (as a percentage of the total, at current prices), 2017 taking into account the above, the concentration on the eaeu market is considered in the case study of russia. the russian meat market is characterized by a high concentration of production facilities mainly in the european part of russia; the meat industry is mainly represented by diversified holdings that produce poultry, pork, and beef and build a vertically integrated value chain. in the meat market, the share of the largest players continues to increase, which together produced 3 million tons of poultry meat, 1.7 million tons of pork and over 100 thousand tons of beef, which is 60% of the total national agricultural production and 46% of the total national meat production in the country. at the same time, more than a quarter of the national production of all kinds of meat and about 37% of poultry accounted for the five largest meat producers [31]. the highest concentration of production facilities and the highest competition is observed in the poultry industry, although, according to experts, compared to brazil, the united states or europe, the share of leading players in the russian poultry market is not high enough being 10-12% [31]. in the pork market, the 20 largest companies account for just over 60% of the total russian industrial output. according to experts, competition in the pork market will intensify in the coming years due to the implementation of large projects in this area. the beef market is significantly different from the pork and poultry markets. the largest producer accounts for about 8% of the total national output, and more than 60% of this type of meat is produced by private farms. at the moment, beef cattle are not attractive to investors, even if there is a demand for this type of meat due to the long payback periods over 12 years. vertical integration along the value chain from manufacturing to sales is a general development trend in vegetable production, especially greenhouse vegetables. high concentration is noted in other markets of the agro-industrial sector. thus, in the sugar market, a quarter of sugar beet processing and sugar production accounts for the 10 largest sugar factories in terms of daily output. high concentration is also observed in those parts of the value chain that are close to the buyer. for the past 15 years, large modern retail chains have steadily increased their share in 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% armenia belarus kazakhstan kyrgyzstan russia agricultural organizations commercial organizations private farms activity of russian companies of the agri-food sector 37 copyright ©2019 assa adv. in systems science and appl. (2019) the russian market, gradually crowding out small business and open markets, especially in large cities in the european part of the country. in 2016, retail chains accounted for 27.7% of the total turnover in the sector, in the food category – 33.1%. currently, large russian retail chains are beginning to develop value chains and invest in food production and agriculture. 4.3. transport and logistics of the eaeu the formation and development of value chains and the fragmentation of production at the level of regional specialization objectively lead to the need for the uninterrupted flow of raw stock, semi-finished products and services through customs territories along the gvacs on the way to the creation of finished products, increasing the requirements for the transport and logistics system. an important characteristic of the state of development of transport and logistics is the length of the existing communication lines. according to the eec, at the end of 2017, the total length of the existing eaeu communication lines is 1,712.8 thousand km of public roads (about 2.5% of the world total), 109.7 thousand km of railway tracks (about 7.8% of the world total) and 287.8 thousand km of pipelines. at the same time, the increase in the length of communication lines in 2013-2017 was provided mainly by the increase in the length of the communication lines of the russian federation. for example, the increase in the length of roads in russia in 2013-2017 (111.3 thousand km) exceeded the current length of roads in kazakhstan (102 thousand km) and belarus (95.4 thousand km). a generalized indicator of the effectiveness of the national transport and logistics systems is the logistics performance index (lpi). from 2007 to 2018, all the countries of the eaeu improved the position of their transport and logistics in the international lpi global ranking [32-33]. in general, the eaeu transport and logistics are characterized by the low lpi index, which testifies to low competitiveness and insufficient efficiency in comparison with other countries (fig. 3). the lack of a unified transport and logistics market of the union as such [34] is the result of the effect of the emerging consolidation of transport and logistics markets in these countries. source: compiled by the author based on the world bank data [33]. 2,0 2,5 3,0 3,5 4,0 4,5 5,0 customs infrastructure international shipments logistics competence tracking & tracing timeliness kazakhstan russian federation armenia belarus kyrgyz republic china germany 38 v.yu. chernova, i.v. andronova, e.a. degtereva, a.m. zobov, v.s. starostin copyright ©2019 assa adv. in systems science and appl. (2019) fig. 3. the logistics performance index of the eaeu countries, china and germany in 2018 currently, the issue of modernization of transport and logistics is a priority for the development of the eaeu. in russia, for example, the volume of investment in “transportation and warehousing” and “information and communications” in 2017 amounted to 18.3% and 3.5% of the total investment in the economy, in kazakhstan – 14.4% and 0.9%, in belarus – 10% and 3.2%, in armenia – 12.2% and 3.5%, in kyrgyzstan – 12.6% and 2.5%, respectively [35]. in general, the business conditions in the eaeu member countries are approximately the same, with the exception of the kyrgyz republic. so, according to the world bank, in the eastern europe-central asia region (24 countries), in 2018 russia ranks 4th, being inferior to georgia, latvia, macedonia. kazakhstan and belarus are the 5th and the 6th, respectively, armenia is the 12th and the kyrgyz republic is the 21st (fig. 4). source: [36] fig. 4. business conditions in the eaeu member countries and the rf in comparison 4.4. investment russia and kazakhstan are the main investors of mutual direct investments in the eaeu ($1,162 and $375 million, respectively), including in transport and logistics. at the end of 2017, russia accounted for about 75.6% of the exported volume of mutual investments in the eaeu countries. during the period from the 2nd quarter of 2017 to the 2nd quarter of 2018, the volume of mutual investments in the eaeu decreased by $1.2 billion, and the share of russia increased to 95.34%. mutual investments of the eaeu countries in the agro-industrial sector remain low. the agri-food sector of the eaeu countries is not interesting for russian investors. so, mutual (mainly russian) investments in the agro-industrial sector of kazakhstan make up 1.5% of the total investments, in the agro-industrial sector of belarus – 0.8%, while 0 5 10 15 20 25 30 st ar ti n g a b u si n es s d ea li n g w it h c o n st ru ct io n p er m it s g et ti n g e le ct ri ci ty r eg is te ri n g p ro p er ty g et ti n g c re d it p ro te ct in g m in o ri ty i n v es to rs p ay in g t ax es t ra d in g ac ro ss b o rd er s e n fo rc in g c o n tr ac ts r es o lv in g in so lv en cy kazakhstan belarus armenia kyrgyz republic russian federation activity of russian companies of the agri-food sector 39 copyright ©2019 assa adv. in systems science and appl. (2019) investments of the eaeu countries in the russian agro-industrial sector exceed 15% of the total investments (fig. 5). source: [37] fig. 5. the structure of mutual investments in the eaeu countries in 2017 such a result is explained not only by the large volume of the russian market and its unsaturation but also by the policy of import substitution and support for the agro-industrial sector in russia. the agri-food sector of russia is in the second place (15.8%) in terms of the attractiveness for the eaeu investors after the chemical sector (35.1%). the main volume of investments was provided by kazakhstan companies specializing in crop and dairy production [37]. 5. discussion as a result of the policy of import substitution and significant governmental support for domestic manufacturers, the russian food markets are currently saturated with a number of products (for example, poultry, pork). further production growth of the largest manufacturers is possible only at the expense of exports growth, since the volume of production already exceeds consumption. in addition, the growth of production is constrained by the increase in production cost against the background of higher prices for imported fodder components [31]. increasing competition, according to experts, will encourage manufacturers to look for ways to reduce costs, including via transferring part of their production facilities to the eaeu countries with cheaper labor and more suitable climatic conditions for agriculture. in addition, for russian manufacturers already interested in exports development, it may be interesting to place enterprises closer to end users – the countries of southeast asia. the development of vegetable markets, especially greenhouse vegetables using photoculture technology in the winter, is unprofitable for a number of vegetable crops, especially in the central, northwestern and eastern regions of russia due to high heating and lighting costs for greenhouses. cultivation of vegetable crops in the eaeu countries with a warmer climate and a greater number of sunny days would be economically justified subject to the development of the eaeu transport and logistics infrastructure. according to pak [34], 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% fdi to russia from the eaeu fdi to belarus from the eaeu fdi to kazakhstan from the eaeu fdi to armenia from the eaeu fdi to kyrgyzstan from the eaeu agri-food sector fuel ferrous metallurgy nonferrous metallurgy machinery and equipment infrastructure chemistry construction transport communications and it wholesale and retail finance tourism other industries 40 v.yu. chernova, i.v. andronova, e.a. degtereva, a.m. zobov, v.s. starostin copyright ©2019 assa adv. in systems science and appl. (2019) the low efficiency of the infrastructure component of eurasian integration, including transport, warehousing, customs, management, telecommunications, etc., in the long run can have a no less divisive role in eurasian integration than the preservation of non-tariff barriers in mutual trade. in the analysis of the domestic market of the eaeu on the basis of the market research of meat products [38], it was found that veterinary, sanitary and phytosanitary measures are the main obstacles to the access of belarusian, kazakh and kyrgyz exporters to the russian market. at the same time, the population of the eaeu countries is quite loyal to the import of goods of russian origin, especially in kazakhstan, tajikistan, and kyrgyzstan, while the republic of belarus is the most interested in foreign investment. in general, more than a third of the population of the eaeu countries supports the policy of rapprochement with russian business. russia is also the most attractive country on the part of the eaeu countries for cooperation in the field of science and technology, in joint development and partnership with which about half of the population of tajikistan, belarus and kyrgyzstan are interested [39]. the edb analysts believe that the deepening of eurasian integration in the eaeu and the establishment of common markets will gradually change the situation of corporate interaction. in the meantime, the benefits of membership in the eaeu are mainly used by large investors, for which cross-border barriers are less painful and their resources allow them to effectively overcome these barriers. the medium business tends not to go abroad so far [40]. in the opinion of the edb cis, the general factors preventing the integration and development of cooperation between the russian business and the partners in the eaeu is the presence of non-tariff barriers in the mutual trade of the eaeu countries, the excessive administrative and tax burden on business, the low level of financial support for small and medium-sized businesses, the interest rates on loans, as well as underdevelopment of cooperation between large business and small and medium-sized enterprises. for russian enterprises, the main risks related to the formation of regional value-added chains in the eaeu format are the following: national protectionism, trade wars between the individual member countries, currency conflicts, the macroeconomic instability and the existing system of economic and political relations both within the eaeu and between the individual countries [23]. the value chains within the eaeu format are currently not sufficiently studied. there is a very small amount of studies in this area. in particular, pobyvayev [41] in his research came to a conclusion that value chains, being a tool for increasing the degree of economic integration of regional economies, would benefit russian-belarusian integration, and the most effective way to participate in global value chains at this stage would be to embed small and medium businesses of the two countries into them. 6. conclusion the use of the eaeu format for the formation of regional value-added chains seems to be economically profitable for russian business. the agribusiness sector has a high potential for the development of cooperation ties, in a number of which the cooperation is a necessity and a condition for further development. despite this, russian business does not currently demonstrate an increased interest in the development of joint projects and cooperation ties, which is hampered by a number of factors. this study showed that regardless of some criticism of the very possibility of forming value chains in a national territory, they have already been formed. the predominant type of value chains is a hierarchical type (vertically integrated agri-holdings), which is associated with the high cost or impossibility of coordinating the manufacturing, monitoring, and control with a different organization of the production process in conditions of stricter safety requirements activity of russian companies of the agri-food sector 41 copyright ©2019 assa adv. in systems science and appl. (2019) and product quality. however, the author has not revealed widespread value chains with any enterprises of the eaeu countries. from the position of value chains, the factors hindering the development of cooperative ties and the formation of joint value chains in the agri-food sector of the eaeu are:  discrepancy between the quality and level of food safety of belarus, kazakhstan, kyrgyzstan and armenia with the requirements of technical regulations of the eaeu and the russian federation, which, if enterprises of these countries are included in the value chains of russian agricultural holdings, will significantly increase their costs on monitoring and controlling the quality of products and manufacturing processes;  mainly private farms’ economy in the structure of agricultural production in armenia, kyrgyzstan, kazakhstan – very small commodity producers and an extremely low share of commercial agricultural organizations, which, from the standpoint of the value chain management concept, also causes the growth of costs of agricultural holdings on monitoring and control;  missing sector of commercial organizations in the structure of agricultural production in belarus with a high proportion (about 80%) of state organizations (collective farms of the old soviet type) with an inefficient management system;  insufficient efficiency of the existing transport and logistics of the eaeu, which could ensure timely delivery of perishable products from the place of manufacturing to the processing sites or end users. acknowlegements the publication has been prepared with the support of the “rudn university program 5-100” in the frame of the project “improvement of marketing tools to support and expand the import of consumer goods in the real sector of the russian economy”. references [1] zhiltsov, s.s. 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(2016). global value chains and their potential role in the development of russian-belarusian integration. the world of new economy, 4. retrieved february 17, 2019, from https://cyberleninka.ru/article/n/globalnyetsepochki-stoimosti-i-ih-potentsialnaya-rol-v-razvitii-rossiysko-belorusskoy-integratsii на рассматриваемом множестве приоритетных проектов после нахождения решения выпуклой задачи распределения ресурсов в соответствии с общим объёмом ассигнований , достигнутой вероятностной оценкой реализуемости вместе с экспертными приоритетами adv syst sci appl 2019; 04; 66-78 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/827 a planning and scheduling method for large-scale innovation projects vladimir topka*, anatoliy d. tsvirkun trapeznikov institute of control sciences, russian academy of sciences, moscow, russia e-mail: topka3@mail.ru, tsvirkun@ipu.ru received may 16, 2019; revised december 12, 2019; published december 31, 2019 abstract. we consider the problem of uniform allocation according to the minimax criterion of a non-renewable resource of an innovative project's activities. a lexicographic method is used for ordering these minimax criteria to solve the problem one objective function developed a greedy heuristic for finding the maximum path on an acyclic digraph with double weights on arcs. the greedy heuristic gives approximate solution of the problem mentioned. the obtained solution allows building a project resources plan and schedule with four types of temporary slacks with inaccurate initial data. the proposed approach allows performing an analysis of the project risk of performance parameter, for which has not developed the apparatus for its analysis. these include the risks that the project when complete fails to perform as intended or fails to meet the mission or business requirements that generated the justification for the project. performance risks are estimated on the probability distribution function of a favorable/unfavorable outcome. proposed lexicographic disintegration is a decomposition procedure and is suitable for planning and scheduling large-scale projects. practical calculations for the convex problem in the initial statement may be performed with the help of ready-made software, and the obtained greedy solution of the problem-consequence has theoretical significance in graph theory. keywords: large-scale innovative project, planning, scheduling, reliability index of project activities, greedy algorithm, the maximum path with double weights on arcs. 1. introduction in existing project management software systems, some modules perform risk of cost and risk of schedule assessment. on the other hand, an approach based on the concepts of the theory of reliability can be proposed for modeling uncertainty and risk of innovative projects. in the theory of reliability, the indicator of network reliability is determined by the probability of its connectivity. this requires that at least one path is found connecting the beginning and the end of the network. however, in managing projects in the pert network, in order to ensure reliable execution of the project, it is necessary to perform all activities included in the project. in this case, the task is to assess not only the connectivity of the beginning and the end of the network, but also the completeness of the whole project. in present project management software, a quantitative assessment of risk of cost and risk of schedule is made on the basis of monte carlo simulation. the proposed approach allows one the analysis of the risk of performance [1] of a project for which the apparatus of its analysis has not been developed. the execution risk, upon completion, accomplish the required mission to achieve the specified technical characteristics, is based on the probability distribution function of a favorable/unfavorable outcome. for her: the probability of success of the project is the probability that the execution of this project will not fail, and the probability of reliable execution is taken as an index of the reliabil * corresponding author: topka3@mail.ru mailto:tsvirkun@ipu.ru a planning and scheduling method for large-scale innovation projects 67 copyright ©2019 assa adv. in systems science and appl. (2019) ity of implementation. the advantage of the proposed approach is that it provides a tool for working in an unexplored area. for activities of innovative projects with a high degree of uncertainty, to obtain the characteristics of random variables of their implementation the likelihood of achieving the goal or probability of technical success, that is, to build the distribution function of such a random variable, you can use empirical data or data obtained as a result of an active experiment in which the monte carlo method obtained the probability distribution function of the technical success of the activity, the degree of achievement of the specified performance characteristics of this development, in the form of a monotonously increasing with the saturation of the function of expended resources [2]. the likelihood of performing the activity by a particular time characterizes the possibility of attaining specific technical indicators of the result of the activity, provided that the previous activity, ensuring its beginning, has been completed. this probability depends on the amount of those or other resources spent, which have a value expression as cost. the approach developed in the article is aimed at developing quantitative methods for assessing and optimizing the project parameters, taking into account the criterion of uniform allocation of a non-renewable (stored) resource with a restriction on the assessment of project reliability index. 2. indicators of the reliability of activities and the project as a whole previously, integer, linear and some non-linear models were used to describe innovative projects execution [3]. we will model [4] the dependence of the probability of technical success (reliability index) of the project’s activities on the homogeneous non-renewable resource spent by 2-parameter power-concave functions of the form     α 0 1/ α 0 ( ) ε, 1 ,ε 0, ε, , 1,..., , j j j j j j j j u p u u u u j n          (2.1) where 0 α 1j  is the form parameter, 0 ju > 0 is the scale parameter. which at the late phases of an innovation project, when it is advisable to use quantitative methods to assess the uncertainty of the project, quite visually and precisely approximates the practical [2; 5] functions of the activities reliability of the innovation project. let an acyclic directed graph g (e, γ) be given, where e is the set of vertices corresponding to the project’s activities (activity-on-node representation), and г is the set arcs defining partial order relations of direct technological activity precedence. the network g (e, γ) has one final and one initial vertex; n final vertex the last number among all the vertices of the project and the dummy vertex 0, which means the launch of the project. thus, we have a network g (e, γ), at the vertices of which a development reliability index is given that satisfies the relation (2.1). we will set the reliability of the project by the probability of its technical success (the degree of achievement of the specified performance characteristics of the project) by a certain allowable period, the assessment of the reliability of the project in the most unfavorable case. in the theory of reliability, such an assessment is determined by the weakest link, i.e. the worst of the technological chains from the vertex of the lower level to the final vertex. 68 v. topka, a. d. tsvirkun copyright ©2019 assa adv. in systems science and appl. (2019)   α 0μ ( , ) μ 0 0 ( ) min ε, 1 , ε 0, α (0,1), [ε , 1], 0; μ , j i i j n jk j k j j jk j i u p u w u w u          (2.2) where 0 0 [ε , 1], ε 0jkw   is arc (j, k) reliability coefficient, (from the matrix of paths along the arcs of the network g (e, г)) transfer the result of the j-th activity by the arc (j, k) to perform the k-th activity (like the transfer function of connecting links in theory of automatic control). we denoteμ i the path with the number 1,...,i m on the network from the initial vertex to the final vertex n. when solving problems of project planning in terms of a power model [6], the time variable is not taken into account. in this article we will consider a deterministic network model of the project with disjunctive input arcs or, or with conjunctive input arcs and having the property that for the project reliability indicator is performed assessment of lower bound (2) [6]. thus, we will speak about pi from (2.1) as a reliability index of activity, and about pn from (2.2) as an assessment of indicator of guaranteed reliability of project performance. in the model for representing network constraints, we will use the matrix of paths (along arcs) of the network g (e, г): which is based on the list of arcs of the network as the characteristic function of the constraint system. in the network model of the project g (e, г), each path from the initial vertex to the final one is uniquely represented by a line of a specially constructed matrix of paths (along arcs)    , , μj k i l j k  , in which its elements  arcs stand in the i-th line only if (j, k) arc belongs to i -th path. all other cases when  , μ i j k  there are empty cells. 3. the problem of successive uniform allocation of nonrenewable resource by chebyshev criterion let  1 2u= , , ..., nu u u be the vector of a non-renewable resource allocated for the execution of all activities of the project g (e, г). as an objective function for the allocation of resources, we will use uniform, or chebyshev criterion in the form u max inf , 0, 1,..., .j j uj e u u j n     (3.1) a problem of the form (3.1) is usually called a discrete minimax problem. therefore, this discrete minimax problem is considered in the form of the following problem of smooth conditional minimization 0 0 ( , ) inf , ju u u u       0 α 00 ( , ) μ 1/ α 0 0 , 1,..., , ε, 1 , μ , 1,..., , ε, , 1,..., . j i j j j jk i j k j j j u u j n u u w p i m u u u u j n                       (3.2) by logarithmizing the second group of constraints of the problem (3.2) and taking into account the first one, we get a planning and scheduling method for large-scale innovation projects 69 copyright ©2019 assa adv. in systems science and appl. (2019) 0 0 ( , ) min , ju u u u   (3.3) 0 0 ( , ) μ ( , ) μ0 ( , ) μ 0 ln ln ln ln , α . i i i j jk j k j k j j k j u w p u u u u                 (3.4) let the constraints of the problem be consistent, both here and in the future .u  due to inequality (3.4), the consequence of the original formulation, taking into account the network specifics of the problem, we will seek its solution in the form of the maximum path with double weights, when the optimal solution of the problem (3.3), (3.4) is realized on a path (possibly a set of paths) with a value 0λ such that 0 0 ( , ) μ ( , ) μ0 μ ( , ) μ ln ln ln λ max α i i i i j jk j k j k j j k u w p          ( , ) μ μ ( , ) μ σ max . α i i i jk j k j j k      wherein, 0 0 0 ( , ) min expλ ju u u u   . therefore, it is necessary to find (possibly, a set) of maximal paths with double weights on arcs in an acyclic directed graph g (e, γ).     0* μ * 0 0 0* 0* μ max λ μ , exp λ exp λ μ , μ i i j arg u u j      we assume that the parameters of the problem are such that here and in the future 0 u u it is satisfied. received resource critical path, with the maximum consumption of a non-renewable resource. let  0* μ μ max λ μ i iarg are found. the optimal solution of (3.3), (3.4) we fix:   * 0 0* 0 0* fixed, μ , μ fixed. ju u j e j j       and we consider the problem of allocating a non-renewable resource according to a successively applied minimax criterion, varying uncommitted variables. in this case, the task of the second stage is: 0 u max inf , 0, 1,..., .j j uj e e u u j n      or     1 1 ( , ) 1 α α* 00 ( , ) μ 0 inf , , 1,..., , ε, 1 , , μ , 1,..., . j j j i u u u j j j jk j k j j i u u u j n u u u w p u u u i m                    70 v. topka, a. d. tsvirkun copyright ©2019 assa adv. in systems science and appl. (2019) by logarithmizing the second group of constraints of the problem and taking into account the first one, we get 0* 0 * 0 ( , ) μ ( , ) μ μ1 ( , ) μ ln ln ln α ln ln α i i i j jk j j j k j k j j j k u w p u u             0* 0 * 0 ( , ) μ ( , ) μ μ1 μ ( , ) μ ln ln ln α ln λ max α i i i i j jk j j j k j k j j j k u w p u                1 ( , ) μ 1* 1 μ μ ( , ) μ σ max , μ max λ μ . α i i i i jk j k i j j k arg       we also fix the optimal solution to the problem of the second stage:  1* 1 1 1* 1* 0* 1 exp λ μ fixed, μ \ μ = fixed. ju u j e      at the same time, we gradually expand the viewer set  1 0j e e e    . next, we consider the minimax problem of the third stage, the optimal solution of the previous stages is fixed, and the unfixed variables vary.  1 0 u max inf , 0, 1,..., .j j uj e e e u u j n       we continue to allocate resources according to the minimax objective function successively until at some stage q (the set indicator) we get the domain of definition of the problem  0... ... q re e e e     and   * * * 1 * 0 exp λ μ fixed, μ \ μ fixed, q q q q j q q q r u u e j j               0 q re e . (3.5) the overall problem is solved by successive optimization of the remaining sub graph by the minimax objective function at each stage, as in the lexicographic method of organizing the solution of multi-criteria optimization problems. when the whole set of vertices is covered in optimal ways, this will solve the problem of a successive, monotonically expanding the scanned area, uniform allocation of a non-renewable resource according to a minimax criterion with a restriction on the assessment of reliability index of the project in the worst case. this procedure converges in a finite number of stages. such a lexicographic disintegration, in essence, is a decomposition procedure and is suitable for the planning of large-scale projects. to find the path of maximum efficiency in [7, p. 13], an algorithm of a search type was proposed (with exponential complexity) that reduces to finding the maximum path in the network. in [8, p. 192], to find in the graph with double weights of the cycle with the minia planning and scheduling method for large-scale innovation projects 71 copyright ©2019 assa adv. in systems science and appl. (2019) mum value, the detection algorithm in the graph of the negative weight cycle is used (provided that for all cycles the sum of weights in the denominator is positive). the solution of this problem with double weights using the proposed algorithm requires 3 1 log η o e       operations, where η the magnitude of the error (weakly polynomial algorithm), e is the number of vertices in the network. in [9], a general method was proposed for solving the average problem in linear spaces. from it, as a special case, follows a strong polynomial algorithm for finding the minimal average contour in a weakly connected oriented graph, which complexity is  3 o e г . to solve the problem of finding the maximum path with double weights in the acyclic digraph g (e, г), we propose a greedy heuristic, for which, as will be shown below, the estimate  2 o г of its computational complexity is valid. the advantage of which, compared with the above, is lower computational complexity. 4. greedy algorithm of finding the maximum path in digraph with double weights on arcs construction of a set of paths  sm  , with weights ji and j on arcs lji , (j,i)  г of the digraph g (e, γ) covering all its vertices e. the initial step. watched: arcs :l  , vertices :e  , path number s: = 1. 1). selection of an arc on a set of unvisited arcs г =г\l. in the acyclic digraph g (e, γ), we choose an arc jil , where  ,j i г is optimal by the local criterion ( , ) σ max . α ji ji j i г j l arg   in the array l we bring the arc jil : : jil l . 2). build a path. 2.1). adding new arcs. construct a path s from the initial dummy vertex j: 1 jг   to the final vertex j = n: jг  by local criterion, i.e. for an arc jil , adjacent arcs kjl and itl are selected from conditions 1 ( , ) ( , ) σ σ max , α α σ σ max , α α j i kj ji j i l kj k г k j j l ji it j i l it t г j i j l l arg l arg                  which are added to the existing arcs : jil l , forming a path  ... ...s kj itl l l    with double weights. where ( , ) σ ji j i l  and α j j l  values for the path found in the previous step. 72 v. topka, a. d. tsvirkun copyright ©2019 assa adv. in systems science and appl. (2019) putting μ : 1s  , for the constructed in such a way the track μ s , we define the value  λ μs s , s = 1,2, ...  ( , ) μ μ μ ( , ) μ σ max λ μ α ji j i s s s j j i       . list of viewed arcs  : ... ...kj itl l l l   ; for viewed vertices – es= {j|j  s}; e' := e'  es. 2.2). avoiding algorithm loops on scanned paths. in order to prevent the algorithm from looping along the scanned path μ s , in the future we will calculate the indicator value 1 μ s . if for non-viewed vertices \e e  , then let \k e e we put 1, if , δ 0, if \ . k k e k e e     then μ μ 0 0 δ 1, ; δ 0, if μ : \ . k k k k k e k k e e             in this case, we set μ1 μ μ , if δ 1, 1, if δ 0, k ks k k           and then in the future, we will sequentially determine  ( , ) μ 1 1 μ 1 μ ( , ) μ σ max λ μ α ji j i s s s j j i           1 1λ μs s     only for new paths 1μ s that do not coincide with μs , s = 1,2, ... 3). cycle if for unvisited arcs г   , then cycle through : 1s s  and go to step 1. otherwise: viewing the constructed set of paths  1μs and determining the optimal path(s) with double weights on arcs 0*μ argmaxλ (μ )s s s  , s = 1,2, ... for the first stage (3.3), (3.4) of the problem:  0 0*λ λ μ , and for the subsequent ones  *λ λ μr r , r = 1, ..., q. in the algorithm, the review of arcs for constructing a set of paths is carried out from г to  , which allows you to avoid a complete search of the paths, and the introduction of a counter μ s   for paths that have at least one unwatched arc eliminates looping the algorithm on the same old path. described algorithm is, in essence, greedy heuristic and so it gives approximate solution of the above problem. the complexity of the algorithm described is estimated as follows. the selection of the initial locally optimal arc lji requires г operations, building a path  from the initial vertex to the final vertex is also requires the order of d г operations, where d is the maximum degree of the vertices of the graph g (e, γ). the construction of all such paths for each of the г arcs is repeated about  о г d г г   time plus the successive selection of the λ-optimal path, which will require order г operations. therefore, the complexity makes a planning and scheduling method for large-scale innovation projects 73 copyright ©2019 assa adv. in systems science and appl. (2019) up    2 о ог d г г г г    of operations. and for a complete vertex coverage, you will need to repeat such an algorithm one more г time, then the total will be required  3 о г operations. for clarity, we illustrate (table 1) finding the first maximal path with double weights on arcs by calculating all the available routes and choosing the maximal one. table 1. finding the first maximal values with double weights on the paths for 0lnu path number 0lnu 1 4.0628 2 2.3828 3 2.7416 4 2.3891 5 2.9082 6 2.1519 7 2.3016 8 2.1389 9 2.8276 10 2.5899 11 2.2212 12 2.2322 13 2.2010 14 2.5093 15 2.3978 16 2.2619 17 2.7578 18 3.1007 19 2.3034 20 2.2053 a test network consisting of 32 vertex-activities, from the well-known library of network models of projects from http://www.om-db.wi.tum.de/psplib/ provide precedence relations of the selected network project as in the table 2 below. table 2. precedence relations of the test network project. jobnr. activity number; #modes the number of execution modes (does not matter); #successors the number of vertices at which the edges of the relations of precedence go out of the vertex of this work; successors numbers of activities that go to the edges of the relations of the preceding from this vertex. table 2. precedence relations of the test network project. jobnr. #modes #successors successors 1 1 3 2 3 4 2 1 3 6 11 15 3 1 3 7 8 13 4 1 3 5 9 10 5 1 1 20 6 1 1 30 7 1 1 27 http://www.om-db.wi.tum.de/psplib/ 74 v. topka, a. d. tsvirkun copyright ©2019 assa adv. in systems science and appl. (2019) 8 1 3 12 19 27 9 1 1 14 10 1 2 16 25 11 1 2 20 26 12 1 1 14 13 1 2 17 18 14 1 1 17 15 1 1 25 16 1 2 21 22 17 1 1 22 18 1 2 20 22 19 1 2 24 29 20 1 2 23 25 21 1 1 28 22 1 1 23 23 1 1 24 24 1 1 30 25 1 1 30 26 1 1 31 27 1 1 28 28 1 1 31 29 1 1 32 30 1 1 32 31 1 1 32 32 1 0 above information from table 2 about relations of technological precedence of selected network project will be organized then with the help of matrix "columns vertices" and "rows paths by vertices". which specifies the characteristic function of the constraint system, and the parameters are specified in the last two columns of table 3 and table 4. table 3. upper the border ub for 0lnu № j= upper the border ub for 0lnu ub = form parameter j  scale parameter 0 ju  0 0.3000 0.5100 1 4.3944 0.2500 0.6000 2 4.1650 0.2200 0.5000 3 0 0.2400 0.5000 4 0 0.2500 0.5000 5 0 0.2800 0.4800 6 4.3565 0.2300 0.4780 7 0 0.2330 0.4750 8 0 0.2350 0.4600 9 0 0.2380 0.4550 10 0 0.2400 0.4500 11 0 0.3000 0.4480 a planning and scheduling method for large-scale innovation projects 75 copyright ©2019 assa adv. in systems science and appl. (2019) 12 0 0.3200 0.4460 13 0 0.3500 0.4450 14 0 0.3800 0.4420 15 0 0.4000 0.4400 16 0 0.4200 0.4300 17 0 0.4300 0.4200 18 0 0.4400 0.4000 19 0 0.4420 0.3800 20 0 0.4450 0.3500 21 0 0.4460 0.3200 22 0 0.4480 0.3000 23 0 0.4500 0.2400 24 0 0.4550 0.2380 25 0 0.4600 0.2350 26 0 0.4750 0.2330 27 0 0.4780 0.2300 28 0 0.4800 0.2800 29 0 0.5000 0.2500 30 4.4742 0.2800 0.7000 31 0 0.5000 0.3200 32 4.1759 0.3000 0.7000 table 4. numerical solution to the problem (3.2) as a whole № j= ln ju = restrictions c = upper the border ub = form parameter j  scale parameter 0 ju  0 3.1203 25.000 3,1203 0,300 0,510 1 4.3944 4.1021 4,3944 0,250 0,600 2 4.1650 3.7607 4,1650 0,220 0,500 3 3.8179 3.4393 3,8179 0,240 0,500 4 3.6652 3.2914 3,6652 0,250 0,500 5 0.2924 0.0444 3,1267 0,280 0,480 6 0.2924 0.0444 4,3565 0,230 0,478 7 0.2924 0.0394 3,7124 0,233 0,475 8 0.2924 0.0444 3,5443 0,235 0,460 9 0.2924 0.1276 3,4537 0,238 0,455 10 0.2924 0.0444 3,3789 0,240 0,450 11 0.2924 0.0444 2,6883 0,300 0,448 12 0.2924 0.0444 2,5063 0,320 0,446 13 1.7583 1.4512 2,2850 0,350 0,445 14 0.2924 0.0444 2,0868 0,380 0,442 15 0.2924 0.0444 1,9711 0,400 0,440 16 0.2924 0.0444 1,8225 0,420 0,430 17 1.2483 0.9752 1,7254 0,430 0,420 18 1.5753 1.2756 1,5753 0,440 0,400 76 v. topka, a. d. tsvirkun copyright ©2019 assa adv. in systems science and appl. (2019) 19 0.2924 0.0444 1,4522 0,442 0,380 20 1.2576 0.9652 1,2576 0,445 0,350 21 0.2924 0.0307 1,0538 0,446 0,320 22 0.9051 0.6127 0,9051 0,448 0,300 23 0.4052 0.1128 0,4052 0,450 0,240 24 0.3823 0.0899 0,3823 0,455 0,238 25 0.3506 0.0582 0,3506 0,460 0,235 26 0.3215 0.0291 0,3215 0,475 0,233 27 0.2924 0 0,2924 0,478 0,230 28 0.7010 0.4086 0,7010 0,480 0,280 29 0.4463 0.1539 0,4463 0,500 0,250 30 4.4742 4.1818 4,4742 0,280 0,700 31 0.9400 0.6476 0,9400 0,500 0,320 32 4.1759 3.8835 4,1759 0,300 0,700 0u 1.3396. if we solve the problem in question (3.2) as a whole, as the problem of mathematical programming using ready-made software, we obtain the numerical solution presented in table 4 above. for the 0 0 [ , 1], 0jkw    – arc reliability coefficient (j, k), (from the matrix of paths along the arcs of the network g (e, г):) transfer the result of the j-th activity through the arc (j, k) to perform the k-th activity (like the transfer function of the connection links in the theory of automatic control) we have the following table 5 values. table 5. arcs reliability coefficient 0г arcs value jkw 1 (5,20) 0.95 2 (11,20) 0.95 3 (18,20) 0.95 4 (16,22) 0.95 5 (17,22) 0.95 6 (18,22) 0.95 7 (10,25) 0.95 8 (15,25) 0.95 9 (20,25) 0.95 for all other arcs of the network 1 01 \ .jk jkw w г г г    5. determination of the schedule with inaccurate source data since the values of the scheduling model parameters 0 0 0 (0,1), [ , 1], 0, 0, , 1,..., , ( , ) ( , ) j jk jw u j k n j k г g е г          are known with a certain error, the resource allocation algorithm, as well as the physical volume of work v [usd  hour] and the amount of a non-renewable resource, found as a result of solving the considered problem, which has the value expression u [usd], are approximate, the project’s time parameters the duration of the activity * */ r j j jt v u , r = 0,1, ..., a planning and scheduling method for large-scale innovation projects 77 copyright ©2019 assa adv. in systems science and appl. (2019) q 1,...,j n ; and the activities slacks are determined by processing the numbers known with an error. the obtained solution of problem (3.1), (3.5) together with reliability constraints, vector       * * *u , 0, 1,..., , 0,1,..., , , r r j ju u j n r q r j q n      such that (omitting the index r)  * * * * * 0 0( ) ( )j j j j ju u u u u          . (5.1) by denoting inaccurately set time values as  δt , using actions on approximate numbers [5], one can obtain approximate parameters of the optimal schedule. to do this, we calculate the deterministic project schedule for one pass in the forward and reverse directions based on the obtained optimal (under given error) duration of the project * 0( )it  , starting from the initial project activities. the critical path method, with values, specified inaccurately (5.1), using actions on approximate numbers [5], also allows you to calculate late deadlines for performing project activities in a back pass through the network, starting from the project completion date (calculated by direct passing through the network). and also to calculate slacks of activities [5]: total (full) slack, free (local) slack, safe slack, independent slack. the total (full) slack tsl of the project the period of time by which you can postpone the activity without violating the limitations and deadlines for the project t ( ) ( ) ( ) ( ) ( ).lb eb lc ec i i i i isl t t t t         free (local) slack fsl of the project the period of time by which you can postpone the activity without violating the deadline for the subsequent work – f ( ) min( ( ) ( )) ( ). i eb ec i j ij i j г sl t d t         the safe slack ssl of the project activity is the period of time for which activity j can be prolonged when all its predecessors 1 ji г  started working at the latest date so that the shortest possible time to complete the project does not increase – 1 s ( ) ( ) max ( ). j lb lc j j i i г sl t t       an independent slack i jsl of the project activity j is equal to the time for which the duration of activity j can be prolonged regardless of the time of completion of its predecessors 1 ji г  and the time of the beginning of its followers jk г  1 i *( ) max 0;min ( ) max ( ) ( ) . j j eb lc j k i j k г i г sl t t t               the following relations are valid (under 0  ): t f i t s i( ) ( ) ( ); ( ) ( ) ( ). j j j j j jsl sl sl sl sl sl          conclusion the article discusses a deterministic network model of a project in which, for a project's overall reliability indicator, it is fair its assessment from below in the form of the weakest link – in the worst technological chain from the initial vertex of the project to the final one. 78 v. topka, a. d. tsvirkun copyright ©2019 assa adv. in systems science and appl. (2019) for the indicator of the reliability of the activity-vertex of the project, a power-concave model is used depending on the spent non-renewable (stored) resource. the peculiarity of the model is weighted network arcs, which set the arc reliability factor (j, k), transferring the result of the j-th activity by the arc (j, k) to perform the k-th activity. under these conditions, the problem of planning and scheduling projects is considered on the basis of a successively applied minimax criterion according to the type of lexicographic ordering of criteria in multi-criteria optimization problems. such a lexicographic dissociation, in essence, is a decomposition procedure and is suitable for planning large-scale projects. to solve the problem of uniform allocation of a non-renewable resource, a greedy heuristic has been developed for finding the maximum path with double weights on arcs of an acyclic digraph, computational complexity that is quadratic in the number of arcs. the greedy heuristic gives approximate solution of the problem under consideration. which allows you to get a resource critical path, with the maximum consumption of a non-renewable resource. after that, the project schedule is determined, including its temporary critical path, with inaccurate data. practical calculations for the considered convex problem (3.2) can be performed using ready-made software, and the resulting greedy solution of the problemconsequence (3.3), (3.4) has theoretical significance in graph theory. references 1. the owner’s role in project risk management. (2005) washington, d.c.: the national academies press, www.nap.edu. 2. elkjaer m. (2000). stochastic budget simulation, int. j. project management. 18(2), 139-147. 3. rabbani m., tavakkoli moghaddam r., jolai f. & ghorbani h.r. (2006). a comprehensive model for r&d project portfolio selection with zero-one linear goalprogramming, ije transactions a. basics.19(1), 55-66. 4. tsvirkun a.d., akinfiyev v.k. & konovalov ye.n. (1991). modelirovaniye i upravleniye innovatsionnymi programmami v krupnomasshtabnykh tekhnicheskikh sistemakh. [modeling and management of innovative programs in large-scale technical systems] – moscow, russia: (working paper / ics). [in russian]. 5. topka v.v. (2014). lexicographic solution of two-objective project planning problem under constrained reliability index, j. computer & systems sci. int. 53(6), 877-895. 6. topka v.v. (2012). minimization of project time and cost under constrained reliability index in the disjunctive project model, automation and remote control. 73(7), 11731180. 7. burkov, v.n., zalozhnev, a.yu. & novikov, d.a. (2001). teoriya grafov v upravlenii organizatsionnymi sistemami. [graph theory in the management of organizational systems.]. moscow, russia: sinteg, [in russian]. 8. christofides n. (1975). graph theory: an algorithmic approach (computer science and applied mathematics). n.y.: academic press. 9. karzanov a.v. (1985). o minimal'nykh po srednemu vesu razrezakh i tsiklakh oriyentirovannogo grafa [on minimal cuts and cycles with respect to average weight of an oriented graph] / in: kachestvennyye i priblizhennyye metody issledovaniya operatornykh uravneniy. (pp. 72-83). [qualitative and approximate methods for the investigation of operator equations.] yaroslavl', ussr: yar su, [in russian]. http://www.nap.edu/ https://dl.acm.org/author_page.cfm?id=81100484367&coll=dl&dl=acm&trk=0 adv syst sci appl 2018; 3:1–16 published online at http://ijassa.ipu.ru. person re-identification based on sad and histogram in a non-overlapping camera network abdullah salem baquhaizel ∗, belal alshaqaqi, meriem boumehed, mokhtar keche laboratoire signaux et images, département d’électronique, université des sciences et de la technologie d’oran mohamed boudiaf, usto-mb. oran, bp 1505, el m’naouer. 31000, algeria abstract: in this paper, we present the conception and implementation of a system for person re-identification in a camera network, based on the appearance. this system aims to associate an identifier to each detected person, which keeps this identifier in the same camera and in other cameras even if he or she disappears and then reappears again. our system comprises an improved moving objects detection step that is implemented by combining the mixture of gaussians method (mog) and a proposed difference method, to improve the detection results. then localization and verification step to eliminate false detection. it also comprises a tracking step that is implemented using the sum of absolute differences algorithm (sad), with an acceleration strategy to reduce calculation complexity. the re-identification stage is realized using three steps: the tracking for the temporal association, the histogram and the new smart exploitation of the interpolation technique for comparison. our proposed system build an online database that contains the history of every person that enters the field of view of the cameras, it does not require any previously collected training data, it integrates both the detection and the tracking so it can be used in real applications, and another advantage is the simplicity of the used techniques in term of implementation and calculation rapidity, without affecting the quality of results. the global system was tested on a real data set collected by three cameras. the experimental results show that our approach gives very courageous results. keywords: person re-identification, mog, sad, histogram, interpolation 1. introduction in recent times, video surveillance has grown more and more. this resulted in an increase of cameras installed in different places (public or private), making their exploitation and monitoring extremely difficult for a human being. consequently, much research has been done to create intelligent vision systems that can help the human being, in interpreting scenes and reacting with alarms in case of any anomaly. currently, there are several types of video surveillance systems (people recognition, access control in sensitive locations, control of traffic congestion, ...etc.). in this paper, we are particularly interested in the problem of person re-identification in a camera network. person re-identification in computer vision systems aims to follow a person, associate an identifier to him, and store it in a database. when the person leaves the scene then reappears in the field of view of any camera, it will be assigned the same identifier. in a crowded and uncontrolled environment observed by cameras from unknown distances, person re-identification relying upon conventional biometrics, such as face recognition, is neither feasible nor reliable, due to insufficiently constrained conditions and insufficient ∗corresponding author: abdullah.baquhaizel@{univ-usto.dz ; gmail.com} 2 a.s.baquhaizel, b.alshaqaqi, m. boumehed, m.keche image details for extracting robust biometrics [31]. alternatively, visual features based on the appearance of people, determined by their clothing and objects carried or associated with them, can be exploited more reliably for person re-identification. the main contributions of this paper are: our proposed system is able to build a fully automated online database that contains the history of every person captured by the cameras, based on their appearance. the proposed system does not require any previously collected training data, so no specific reference for a person is stored a priori. compared to most of the other state-of-the-art approaches, our proposed system is a self-contained system that is ready to use in real applications since it integrates both the detection and the tracking phases. the simplicity of used techniques. furthermore, the improvement in each block, object detection by combining the mog method and the difference method, the tracking and the re-identification. the rest of this paper is organized as follows: section 2 presents some related works from the literature. in section 3, we describe the proposed system and give the details for its blocks. the experimental results and their discussion are presented in section 4. finally, some conclusions are drawn in section 5. 2. related works in the literature, the approaches of person re-identification can be grouped into several classes, according to several criteria [20]: 1. the number of images per person: this class comprises two families. the first family is the family of mono-sample methods, where the signature of a person is extracted from a single image as in [4, 9, 12, 26, 30, 39]. for example, in[4], authors project the gallery and the probe data into the regularized canonical correlation analysis (rcca) subspace then the reference descriptors (rds) of the gallery and probe are constructed by measuring the similarity between them and the reference data. in [12], a metric learning framework is used to obtain a robust metric for large margin nearest neighbor classification with rejection. the second family is the family of multi-sample methods, where multiple images are used to calculate the signature of a person as in [10, 11, 13, 19, 22, 29, 34, 36]. for example, authors in [10] propose custom pictorial structure (cps) for re-identification. in [22] they use the implicit shape model (ism) and sift features for the person reidentification. in [29] spatiotemporal person features are extracted using multi-frame twin-channel descriptor based on a gabor filter, then mahalanobis distance metric learning algorithm is used for matching. in [36] the discriminating nature of the sparse representation is exploited in order to perform people re-identification task. 2. the type of representation: the first family in this class is the family of global approaches, where the whole information in the image is exploited for calculating the person’s signature, as in [4, 5, 7, 17, 21, 27]. for example, in [21], the jensen-shannon kernel is proposed to learn nonlinear distance metrics. in [5] non-articulated 3d body models are exploited to spatially map appearance descriptors (color and gradient histograms) into the vertices of a regularly sampled 3d body surface. in[17], the appearance by a set of region covariance descriptors is modeled, then a discriminative model is learned using boosting for feature selection. in [27], authors extract kdes (kernel descriptor) from human roi for person classification. the second family is that of local approaches, which represent the image by several feature vectors, each vector describes a region or an interest point locally detected, such copyright c© 2018 assa. adv syst sci appl (2018) person re-identification based on sad and histogram in a non-overlapping camera network3 as in [11, 14, 15, 42]. for example, in [11] interest points are detected with fast-hessian detector, then they are described in color and surf description. the approach in [15] uses matching of signatures based on interest points descriptors detected by a variant inspired from surf called camellia key-points. in [42], authors use the face to video retrieval, they analyzed four local features for describing face images: harris corner operators, sift descriptors, surf descriptors, and eigenfaces. 3. the existence of a set of images mapped a priori: this class includes supervised approaches like in [7, 9, 22, 25, 28]. for example, two structured learning based approaches are proposed in [25], they explore the adaptive effects of multiple low-level visual features with an optimal ensemble of their metrics. in [28], a data driven approach is proposed for learning color patterns, they model color feature generation by jointly learning a linear transformation and a dictionary to encode pixel values. in the other hand, we have the unsupervised approaches as in [18, 23, 38, 40]. for example, a video representation, called spatio-temporal pyramid sequence (stps) is developed by [23] to encode space-time information, and they formulate a novel time shift dynamic time warping (ts-dtw) model and its multi-dimensional extension named mdts-dtw for selective matching between sequences. in [40] a dynamic graph matching (dgm) method is proposed; a graph for samples in each camera is constructed, and then graph matching scheme is introduced for cross-camera labeling association. a very nice survey of people re-identification approaches is presented in [37]. they are therein grouped as a multidimensional taxonomy according to camera setting, sample set cardinality, adoption of a body model, signature, machine learning techniques, and application scenario. 3. description of the proposed system we present the conception and implementation of a system for person re-identification in a camera network, based on the appearance. in this section, we describe the different blocks of the proposed system and how all these blocks are connected. these blocks are: the detection of moving objects, their localization and verification, their tracking and their re-identification. the detailed flowchart of the proposed system is shown in figure 3.1. in the beginning, we initialize the number of found objects to zero. then for each frame, we start detecting moving objects, next we perform their localization, then a verification and confirmation stage is applied to improve the results by eliminating the false detection. in parallel to the detection and for each frame also, we have the tracking process of any found objects from the previous frame. after that, the obtained results from the tracking are fused with those of the detection and verification stage, by using the technique of intersecting and overlapping. then comes the feature extraction step where the histogram for each found object is calculated. for the re-identification, we use the tracking result for the temporal association, the histogram and the trajectory interpolation for the comparison. the re-identification and association block tries to associate the found objects with objects already seen or add them as new objects if they appear for the first time, thus, building an online database which contains the history of every person that appears in front of the camera. in the coming subsections, we will discuss in detail the techniques and the tools used in each block. 3.1. detection of moving objects this initial phase is performed by combining the mixture of gaussians (mog) method [35] and a proposed method we called the difference method. copyright c© 2018 assa. adv syst sci appl (2018) 4 a.s.baquhaizel, b.alshaqaqi, m. boumehed, m.keche 11 initiation: n° found objects=0 image capture intersection differencemog or morphological operations update db with new id *1 trajectory interpolation *3 histogram *3 update db with matched id *2 if found objects from update db with id of tracking *2 detection only intersection or tracking only localization verification yes no retrieved detected objects n° found objects >0 tracking yes yes retrieved tracked objects no noyes yes no no update db database db test db *3:comparison *1: new id *2: existed id update db with interpolated id *2 histogram fig. 3.1. the detailed flowchart of the proposed system the mog is one of the most used and successful methods in surveillance systems, because it is adaptive, and can handle multimodal backgrounds [8]. it is the most common method to build a background[32]. furthermore, it is robust to slow lighting changes, periodic motions from a cluttered background, slow moving objects, long term scene changes, and camera noises [41]. in this method, the recent history of each pixel, {x1, ..., xt}, is modeled by a mixture of k gaussian distributions. the probability of observing the current pixel value is given by: copyright c© 2018 assa. adv syst sci appl (2018) person re-identification based on sad and histogram in a non-overlapping camera network5 p (xt) = k∑ i=1 ωi,t ∗ η(xt, µi,t,σi,t) (3.1) where ωi,t is the portion of the data accounted by the ith gaussian in the mixture at time t and η(xt, µi,t,σi,t) is a gaussian probability density function with mean value µi,t and covariance matrix σi,t of the ith gaussian in the mixture at time t , where η is given by: η(xt, µ,σ) = 1 (2π) n 2 |σ| 1 2 e− 1 2 (xt−µt)t σ−1(xt−µt) (3.2) k is the maximum number of gaussian components, depending on the available memory. currently, this parameter ranges from 3 to 5. at each pixel xt in a new frame, the following update equations for the k distributions are performed: ωk,t = (1− β)ωk,t + β (3.3) β is the learning rate of adaptation. this value ranges from 0.0 to 1.0, determines the importance of the previous observation. the first b distributions are chosen as the background model, where: bt = argminb ( b∑ k=1 ωk > t ) (3.4) the value of t determines the minimum portion of the data that should be accounted by the background. when higher values for t are chosen, a multi-model background model is formed which can handle repetitive background motion [35]. in the other hand, we have the difference method, we first take the difference between two successive images in gray-scale ig(t) and ig(t−1), as in eq. (3.5), and then we compare the resulting difference image idiff to a threshold to detect pixels in movement. idiff = ig(t) − ig(t−1) (3.5) the combination of the detection resulting from the mog and the difference methods is performed using the logical or operation. after that, we apply morphological operations as erosion, dilatation, and fill the holes. the holes of a binary image corresponding to the set of its regional minima, which are not connected to the image border [33]. 3.2. localization and verification to localize the detected objects, we use the labeling technique [16]. it consists in separating the areas in the mask obtained from the detection step. we associate with each area an integer value (label) by using an 8-connected neighborhood. then we propose a verification phase to eliminate false detection. we first calculate some proprieties for each localized area, e.g. x and y coordinates, height, width and sum of foreground pixels. then to be validated, each object has to verify the following three conditions: • the height to width ratio has to lie between min and max thresholds. • the surface of the rectangle containing the object (surface = height x width) has to lie between min and max thresholds. this is to reject very small and very big objects due to false detection. • the ratio of the sum of foreground pixels to the surface also has to be bounded. copyright c© 2018 assa. adv syst sci appl (2018) 6 a.s.baquhaizel, b.alshaqaqi, m. boumehed, m.keche 3.3. tracking the tracking process is done by template matching using the sum of absolute differences algorithm (sad), like in our previous proposed driver drowsiness detection systems[2, 3]. this algorithm is widely used for image compressing and object tracking in real-time application [1], it can give high localization rate when the image is with high illumination variation [24]. in digital image processing, the sad is a measure of the similarity between image blocks. it is calculated by taking the absolute difference between each pixel in the original block x (a portion from the current frame) and the corresponding pixel in the y block being used for comparison (model from the previous detection). these differences are summed to create a simple metric of block similarity as in eq. (3.6), zero means that the two blocks are identical. we sweep all the positions in the frame, then the block with the smallest metric is the tracked block. the sad value for two blocks x and y is calculated by: sad = m∑ i=1 n∑ j=1 |x(i, j)− y (i, j)| (3.6) for a given y model, the most similar block x is the one that minimizes the sad. 3.4. re-identification after the stages of detection, localization, verification, and tracking, we have the stage of reidentification and online construction of database db containing the history of each person that appeared in the view field of the cameras. the lower part of figure 3.1 presents a detailed flowchart of this stage. this stage deals with the moving objects obtained from the detection and tracking stages, which are called ’found objects’. first, we proposed to perform the intersection between the found objects resulting from the detection and tracking, then we calculate the percentage of the intersection if it exceeded a predefined threshold we merge the objects from detection and tracking into one object, otherwise, they stay separated objects. the intersection (a ∩b) of two rectangles a and b is the rectangle that contains all elements of a that also belong to b. now, we test each found object, if it resulted from the detection only, the tracking only or from both (intersection). if the found object comes from the intersection or tracking only, we update the database with the identifier of tracked object. on the other hand, if that found object comes from detection only, then we calculate its histogram. an image histogram is a type of histogram that acts as a graphical representation of the tonal distribution in a digital image. it plots the number of pixels for each tonal value. the histogram of the found object is compared to the histograms of identified objects stored in the database. if there is a match, then we update the database by associating that object with the matched identifier, otherwise, we make a trajectory interpolation. the interpolation is a method of constructing new data points within the range of a discrete set of known data points. if there is an interpolation, we update the database with the interpolated identifier, otherwise, we consider this object as a new one and assign to it a new identifier that is added to the database. 4. experimental results in this section, we present the material and the database used, the experimental results, and their discussion. copyright c© 2018 assa. adv syst sci appl (2018) person re-identification based on sad and histogram in a non-overlapping camera network7 fig. 4.2. layout of the cameras (a) (b) (c) fig. 4.3. fields of view of the 3 cameras: (a) camera 1, (b) camera 2, (c) camera 3 4.1. system development environment the material that was used for the development of our application is: 1. a laptop with: • processor: intel r©coretm i7 4702mq cpu @ 2.20 ghz 2.20 ghz. • ram memory: 8.00 go. • operating system: windows 8.1, 64-bits • hard drive: 1 tb. 2. digital video recorder dvr. 3. camera with characteristics: • 1 /3 sony hr ccd • 420 tv lines • 0.2 lux • adjustable focal between (3mm and 8 mm). to test our system we build our own database [6], composed of sequences of images recorded on the third floor of the department of electronics at usto university. three cameras, set to a height of (2.30m) and with an angle of ( −30◦), were used to take these images. each sequence contains from one to three people who walk in the fields of view of the three cameras. the cameras were placed as shown in the layout presented in figure 4.2. copyright c© 2018 assa. adv syst sci appl (2018) 8 a.s.baquhaizel, b.alshaqaqi, m. boumehed, m.keche to fulfill the condition of a non-overlapping camera network, the database was realized so that a person lies in the field of view of only one camera, at a given instant. figure 4.3 shows the fields of view of the three cameras. 4.2. experimental results, and discussion in this section, we will present and discuss the results of each step of the proposed system. 4.2.1. results of the detection the mixture of gaussian gives us raw results of detection from each camera, after having defined suitable settings according to some criteria, like: indoor or outdoor environment, people movement speed and lighting changes. figure 4.4 (b) presents an example of these results. to improve these raw results, we combine them with the results of the difference method (figure 4.4 (c)), which allows for the detection of the edges of moving objects, then we apply morphological operations as erosion, dilatation and proceed to a holes filling of the resulting image to minimize false detection and obtain better results as illustrated in figure 4.4 (d). (a) (b) (c) (d) fig. 4.4. example of moving objects detection. (a) original image, (b) results of detection by mog, (c) results of detection by difference, (d) results of the holes filling of the or between b and c 4.2.2. results of the verification to evaluate the verification phase, we compare the results before and after the verification. we consider the true detection from the detection before verification as positive and the false detection as negative. table 4.1 presents for videos from camera 1, camera 2 and camera 3, the obtained evaluation measures, namely the tp (true positive), the fn(false negative), the fp (false positive), and the tn (true negative). to understand these measures we present figure 4.5, let us assume that we have four objects; copyright c© 2018 assa. adv syst sci appl (2018) person re-identification based on sad and histogram in a non-overlapping camera network9 1 2 3 4     (a) 1 2 3 4     (b) fig. 4.5. before and after verification evaluation, (a) detection results (before the verification), (b) results after the verification table 4.1. evaluation results of the proposed verification technique camera 1 camera 2 camera 3 tp 411 399 341 fn 23 8 7 fp 2 2 3 tn 19 30 73 objects 1 and 2 are not people, however, objects 3 and 4 are people. figure 4.5 (a) is the results before the verification, so objects 1 and 2 are false detection, however objects 3 and 4 are true detection. in figure4.5 (b), we have the results after the verification, objects 1 and 3 are kept, but objects 2 and 4 are removed. now we can assign each measure of evaluation to the corresponding object: • object 1 is a false positive (fp). • object 2 is a true negative (tn). • object 3 is a true positive (tp). • object 4 is a false negative (fn). from these measures, we can define the following rates: tp tpfn % = tp/ (tp + fn): % of true person kept. fn tpfn % = fn/ (tp + fn): % of true person removed. fp tnfp % = fp/ (tn + fp): % of false person kept. tn tnfp % = tn/ (tn + fp): % of false person removed. the obtained evaluation rates are given in table. 4.2, for videos from camera 1, camera 2 and camera 3. from this table one can observe, that the proposed verification technique performs well, since it manages to remove 90.48%, 93.75% and 96.05% of false detection in camera 1, camera 2 and camera 3, respectively, while removing only 5.30%, 1.97% and 2.01% of true detection, in camera 1, camera 2 and camera 3, respectively. table 4.2. evaluation rates of the proposed verification technique camera 1 camera 2 camera 3 tp tpfn% 94.70 98.03 97,99 fn tpfn% 5.30 1.97 2.01 fp tnfp% 9.52 6.25 3.95 tn tnfp% 90.48 93.75 96.05 copyright c© 2018 assa. adv syst sci appl (2018) 10 a.s.baquhaizel, b.alshaqaqi, m. boumehed, m.keche in figure 4.6, we give the localization and verification results. after the localization by the labeling technique, we apply the verification procedure to each object. in figure 4.6 (a), only the objects that pass the conditions of verification are kept (the person in green rectangle), the others in red are ignored. figure 4.6 (b) shows the results of detection . (a) (b) fig. 4.6. example of localization and verification, (a) localization and verification results on the original image, (b) results of detection 4.2.3. results of the tracking the tracking step is run in parallel with the detection step and it is realized by the sad. to accelerate its execution we decided to apply it only to a limited region of interest, instead of searching in the whole frame. this region is determined by the coordinates of the model to track. the obtained results of tracking are satisfactory. figure 4.7 shows the tracking results, figure 4.7 (a) is a detection in frame 137 and figure 4.7 (b) is its tracking in frame 194, it means that we still maintain this object even after about 57 frames. (a) (b) fig. 4.7. tracking results, (a) detection in frame 137, (b) the tracking of detection of frame 137 in frame 194 4.2.4. results of the re-identification the re-identification stage is realized using three techniques, the tracking for the temporal association, the histogram and the interpolation technique for comparison. in figure 4.8, we present the multiplication of the detected object with its mask to extract the silhouette only. then, we calculate histograms of red, green, blue channels and gray-scales as shown in figure 4.9. we use the histogram of the silhouette to avoid the effect of the background. these different histograms are used for comparison with the models stored in the database. if there is a match, we associate the matched identifier to copyright c© 2018 assa. adv syst sci appl (2018) person re-identification based on sad and histogram in a non-overlapping camera network11 the actual object, otherwise, we use the trajectory interpolation technique, as shown in figure 4.10 (a), to determine if this object is in the trajectory of an existing object. if so, we associate the actual object to this object, otherwise, we consider that the actual object is new and add it to the database with a new identifier. from figure 4.10 (b), we can deduce that in the period between the frames 220 and 270, mentioned by two verticals lines in red, the two person appear together and figure 4.10 (c) is a sample from that period. note that the appearance of person 1 starts from the frame 120, and that of person 2 from the frame 180. the miss in the appearance of the person are due to the elimination of certain items by the verification stage. the performance of our re-identification system is evaluated using two criteria: 1. number of created ids, 2. number of appearances and associations of each id, evaluation based on the number of created ids: we can evaluate our proposed reidentification system by comparing the number of assigned ids that created by the system to the number of real ids. the ideal is when these two numbers are equal. in table. 4.3, these numbers are compared, using videos from the different cameras. the number of real ids is obtained by the ground truth of used dataset, and their appearance is how many time they appear. however, the number of created ids is obtained by the system, and their appearance is how many time they appear. it can be observed that the number of created ids is larger than the number of real ids; this is due to the false detection. it can also be observed that by eliminating most of the false detection using the verification stage, we reduce the number of created ids to a level close to the number of real ids. table 4.3. comparison of the number of created ids (nb c ids) to the number of real ids (nb r ids), and their number of appearance (nb a) real created nb r ids nb a nb c ids nb a camera 1 3 434 7 515 camera 2 3 407 5 541 camera 3 3 348 8 494 table 4.4. number of appearances (nb a) and person-ids associations of each id(p) in camera 1, 2 and 3 id camera 1 camera 2 camera 3 nb a p nb a p nb a p 1 175 p1 181 p1 98 p1 2 36 p2 27 131 p2 3 31 172 p2 29 4 41 p3 57 40 p1 5 117 p1 104 p3 30 6 27 33 7 88 p3 102 p3 8 31 p3 evaluation based on the number of appearances and associations of each id: another way to evaluate our proposed system is the number of appearances of each id,(see table. 4.4). from this table, we can determine the number of ids for each camera and the number of appearances of each id. for example, we can see that for camera 1, seven copyright c© 2018 assa. adv syst sci appl (2018) 12 a.s.baquhaizel, b.alshaqaqi, m. boumehed, m.keche ids were created, and id 5 was associated 117 times. in addition, from this table, we can determine the ids associated with each person. for example, it can be seen from table. 4.4, that with camera 3 the ids 1 and 4 were associated with the person 1, named p1. (a) (b) (c) fig. 4.8. the multiplication of detected object with its mask, (a) detected object, (b) mask of detected object and (c) results of multiplication (a) (b) (c) (d) fig. 4.9. different histograms of the silhouette, (a), (b), (c) and (d) are the histograms of red, green, blue channels and gray-scales respectively 5. conclusion in this paper, we proposed the conception and implementation of a system for person reidentification in a camera network, based on the appearance. this system is able to assign an identifier to each detected person, that it keeps everywhere in the fields of view of the cameras and even if he or she disappears and then appears again. our system implements an improved detection technique that combines the mixture of gaussian method with the difference method. the sad algorithm with an acceleration copyright c© 2018 assa. adv syst sci appl (2018) person re-identification based on sad and histogram in a non-overlapping camera network13 time ( frames ) (a) time ( frames ) (b) (c) fig. 4.10. trajectory interpolation, (a) the trajectories of the two objects through time, (b) the trajectories of the two objects along the y-axis (c) the two objects that appear in this sequence strategy is used for the tracking step, whereas the re-identification stage is realized using three techniques: the tracking for the temporal association, the histogram and the trajectory interpolation technique for comparison. our proposed system does not require any previously collected training data to build an online database that contains the history of every person that enters the field of view of the cameras. it can be used in real applications because it integrates both the detection and the tracking. another advantage is the simplicity of the used techniques in term of implementation and calculation rapidity, without affecting the quality of results. a real data set is collected by three cameras to test the global system. the experimental results show that our approach leads to very courageous results with an opportunity for improvement in the re-identification stage, by using local histograms instead of using the global one. also as a future work, we plan to evaluate our method quantitatively and compare it with other methods. references [1] alshaqaqi, b., baquhaizel, a. s., ouis, m., & boumehed, m. 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(2017). a comparative study of local features in face-based video retrieval. journal of computing science and engineering, 11(1), 24–31. doi:10. 5626/jcse.2017.11.1.24 copyright c© 2018 assa. adv syst sci appl (2018) https://dx.doi.org/10.1007/978-3-662-05088-0 https://arxiv.org/abs/1011.1669 https://arxiv.org/abs/1011.1669 https://dx.doi.org/10.1109/cvpr.1999.784637 https://arxiv.org/abs/cvpr.1999.784637 https://dx.doi.org/10.1109/ipta.2010.5586809 https://dx.doi.org/10.1109/ipta.2010.5586809 https://dx.doi.org/10.1145/2543581.2543596 https://dx.doi.org/10.1145/2543581.2543596 https://dx.doi.org/10.1007/978-3-319-10593-2_45 https://dx.doi.org/10.1109/iccv.2007.4409019 https://dx.doi.org/10.1109/iccv.2017.550 https://dx.doi.org/10.1109/iccv.2017.550 https://arxiv.org/abs/1709.09297 https://dx.doi.org/10.1109/cvpr.2005.304 https://dx.doi.org/10.5626/jcse.2017.11.1.24 https://dx.doi.org/10.5626/jcse.2017.11.1.24 introduction related works description of the proposed system detection of moving objects localization and verification tracking re-identification experimental results system development environment experimental results, and discussion results of the detection results of the verification results of the tracking results of the re-identification conclusion paper title (use style: paper title) adv syst sci appl 2018; 02; 59-62 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/578. specifics of long-term forecasting for global gas markets vyacheslav kulagin1,*, anna galkina1 1) department of world and russian energy complex research energy research institute of the russian academy of sciences moscow, russia abstract: the article presents a methodology for developing long-term forecasts for global gas markets using optimization modeling. this approach can be effectively used for an integrated analysis of the global gas market conjuncture, and assessment of management decisions in the gas industry in both the short and long term. keywords: gas market; modeling; forecast; demand; production; prices 1. introduction the article covers the specifics of forecasting for global markets employing optimization models. these models are based on the principle of the monge-kantorovich problem of large dimension, in which the world’s total costs of satisfying the global gas demand are minimized: 𝑓(𝑥) → 𝑚𝑖𝑛 where 𝑓(𝑥) stands for the total global costs of gas production, including taxes, global transportation costs, gas storage, as well as liquefaction and regasification in the case of lng transportation. 2. input and output parameters in forecasting of world gas markets development the main results of calculations in regarded optimization models are the optimal volumes of global gas trade by route, production by countries and gas prices. the key modeling input parameters and assumptions are as follows:  volumes of gas supply with corresponding production costs;  gas demand volumes;  capacity of the existing and planned infrastructure for piped gas and lng, including corresponding transportation costs;  volumes and pricing conditions of the concluded contracts for piped gas and lng supply;  prices of alternative energy resources, taking into account prices of co2 emissions and the extent of demand switching for such energy resources. the level of detail in the global supply curve presentation, which contains information on the opportunities for gas production by fields and groups of fields per each forecast year, including corresponding costs, largely determines whether the model reflects the real market with due accuracy. a number of market factors and assumptions about future production volumes and costs (for instance, the rate of decline in gas production in europe, the cost of extracting shale gas and coalbed methane in asia, the prospects of gas production in east * corresponding author: vakulagin@yandex.ru 60 v. kulagin, a. galkina copyright ©2018 assa adv. in systems science and appl. (2018) africa, etc.) greatly affect the relative competitiveness of gas producers. thus, when the development of individual national or regional gas markets, it is necessary to incorporate changes in the balance of supply and demand across all the global markets, even without any need for receiving such a detailed forecast. gas production costs are not a fixed parameter, either: in particular, they are influenced by the development of technologies for the production of conventional and unconventional gas, devaluation/revaluation of currencies, changes in oil and gas market prices and tax burden. thus, since h2 2014, the costs of hydrocarbon production have significantly decreased (fig. 1). fig. 1. indices of operational (uoci) and capital (ucci) costs in the oil and gas production sector. source: ihs cera [1]. in this regard, an important assumption in plotting the gas supply forecast curve, which directly influences the forecast level of gas prices, is the rate of production cost escalation. the global gas demand drivers have an impact on modeling results that is no less important than that of the assumptions for gas supply. the demand for gas, as part of energy demand, shall be determined by both the level of domestic energy consumption (as a whole and by sectors) and the conditions of inter-fuel competition. forecasting energy consumption of countries (nodes) by types of energy resources shall be carried out separately and shall not be deemed the subject-matter of the study described. but the demand for gas in the model shall be refined further in order to consider inter-fuel competition, the opportunities for switching the demand for alternative energy resources are added when certain threshold levels of prices are reached in the process of calculations. here, the coefficients reflecting the assumptions for co2 emission prices are added to the prices of carbon fuels. countries (more precisely, nodes, since in the detailed analysis of certain countries, especially big ones, it is expedient to allocate several nodes within a single country) with the given production capacities and the volume of gas demand in the models under consideration are connected with one another by gas transportation and storage infrastructure. this includes gas pipelines, gas storage facilities, lng plants and lng regasification terminals. the main parameters of gas transportation infrastructure are capacity, transportation costs, their escalation rates, expected service life, and directions of routes (dispatch nodes and destination nodes). thus, the possibilities of gas transportation between the nodes with surpluses and deficit of supply are formed. in view of a large share of transportation costs in the market price of gas, manipulating the possibilities can significantly distort simulated conditions of competition. in particular, in the short run, when analyzing a volatile market environment with price wars, scenarios can become realistic in which only operational costs for gas production and transportation are considered while the correct option for long-term forecasts is to incorporate long-term marginal costs. specifics of long-term forecasting for global gas markets 61 copyright ©2018 assa adv. in systems science and appl. (2018) a part of supply and demand for gas is withdrawn from the free competition environment through the introduction of gas supply contracts (piped gas and lng). gas supplies under contracts can differ from optimum competition alternatives by volumes, routes and price. in the real market swap deals take place, and it is possible to overestimate future demand when signing the long-term contracts. however, long-term contracts, first and foremost, ensure the security of supply through the guarantee of supply and demand and reduction of transaction costs. the increased flexibility of gas supply conditions under long-term contracts also contributes to their role in gas markets. however, on the background of market development the volume of gas supplies under short and medium-term contracts has markedly increased: in 2016, almost 30 % of global lng supplies were made under contracts valid for up to 4 years [2]. in the models, the input parameters characterizing gas supplies under long-term contracts include annual volumes of supplies, contract duration, gas price formula, level of quarterly and annual minimum volumes. gas prices in contracts can be directly indexed to oil prices, the price of the japanese crude cocktail (jcc), prices of petroleum products (gasoil, fuel oil), as well as to spot prices (for example, henry hub, gaspool, nbp, ttf). accordingly, the projected oil prices and spot gas prices to be indexed in contracts are exogenous variables. these prices are also used to calculate the cost of the fuel required to transport lng. the optimization results are the volumes of gas supplies in all directions (can be grouped into import and export forecasts by countries), the gas production required for these supplies (in the same level of detail as gas supply input in the model), gas storage volumes, gas prices as long-term marginal costs of supplying gas to each node and refinement of gas consumption volumes, with consideration of the switching of part of the demand for alternative fuels. based on the calculation results per each node, the following condition is fulfilled: gas production + piped gas and lng import + withdrawal from gas storage gas consumption piped gas and lng export injection into gas storage = 0. in general, the scheme of input parameters and assumptions and output data appears as follows (fig. 2). fig. 2. inputs and outputs in optimization models of global gas market development. source: eri ras. 3. sensitivity analysis and results as the studies and results of the calculations show [3, 4], currently, gas prices are more influenced by oil prices in the asian and european markets, but in the long term, the role of gas-on-gas competition and hence the role of gas production and transportation costs will increase in all key global markets. in the us market, gas production costs already directly affect market gas prices.the prospects for changing directions and the level of global gas trade in the next 25 years will largely be determined by the demand in asia. russian exports to the east will mainly depend on the opportunities for production, the projects implemented and contracts signed. in the european and the cis market the dynamics of demand and the possibilities of competition will 62 v. kulagin, a. galkina copyright ©2018 assa adv. in systems science and appl. (2018) play a key role, while the production and transportation capacities will have sufficient reserves for increasing supplies when needed. the considered approach allows for the shaping of complex scenarios, exploring interactions between market participants and analyzing the sensitivity of national gas markets to various local and global factors (for example, to changes in global or regional demand, the costs and volumes of supply of conventional and unconventional gas, changes in taxation, terms of supply contracts), as well as assessing the competitiveness of projects for gas production and transportation under various conditions. references [1] indices of operational (uoci) and capital (ucci) costs in the oil and gas production sector. ihs indexes [online] available https://www.ihs.com/info/cera/ihsindexes/ [2] the lng industry annual report 2017, giignl. [3] makarov , a. grigoriev, t. & mitrova t. (2016) global and russian energy outlook. moscow, russia: eri ras – acrf. [4] kulagin, v., & mitrova, n. (2015). gazovyj rynok evropy: utrachennye illyuzii i robkie nadezhdy [the european gas market: lost illusions and glimmers of hope]. moscow: national research university higher school of economics. moscow, russia. energy research institute of the russian academy of sciences. [in russian]. adv syst sci appl 2018; 04; 64-73 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/555 solution of one global problem by approach of the parametric control theory abdykappar ashimov1,*, yuriy borovskiy1 1)kazakh national research technical university named after k. satpayev, almaty, kazakhstan e-mail: ashimov37@mail.ru, yuborovskiy@gmail.com received february 12, 2018; revised december 9, 2018; published december 31, 2018 abstract: purpose of this paper is to show the potential of parametric control theory approach for the optimal solution of global problems (in the case of the food problem) and to substantiate the benefits of joint action to solve them on the basis of the developed global multi-country computable general equilibrium model (hereinafter model). the main results of the paper: model verification (testing the possibility of its practical application) of the constructed and calibrated model was successfully carried out by methods of parametric control theory: using sustainability assessments and assessments of sustainability indicators of smooth mappings, which are defined by the model; as well as through conducting of series of counterfactual and forecast scenarios. based on the model a number of parametric control problems were solved in finding the optimal values of the instruments of public policy, which provide economic growth, reducing the gap in economic development between rich and poor regions, as well as increasing the production of agricultural products in the regions. value consists in the developed model and in showing approaches of parametric control theory on providing guidance to address global problems and to substantiate the benefits of joint action within the framework of their decision. keywords: global problem, parametric control theory, computable general equilibrium (cge) model, model verification. 1. introduction the formed world economy (after independence of former colonies, collapse of the soviet union, emergence of new states and economic unions) for many years keeps the imbalances in the economic development of individual countries and regions of the planet. these imbalances formulate global and regional problems (including individual countries problems), without their solution further humanity progressive advance along the path of economic progress is impossible. the most urgent problems are the "north-south" problem the gap in economic and social development between rich and poor countries and the associated food problem, which lies in inability of poor countries to support them completely with even vital food. different mathematical models are widely used to assess the scenarios of food products output in various regions of the world for different periods of time. for example, the econometric model which has been proposed in [1] is used for short-term assessment of grain yield in different parts of the us. the paper [2] contains a description of the international food security assessment model, which is designed to assess food consumption, food access, and food gaps (previously called food needs) in low-income countries through 2025. let us * corresponding author: ashimov37@mail.ru mailto:ashimov37@mail.ru mailto:yuborovskiy@gmail.com solution of one global problem by approach of the parametric control theory 65 copyright ©2018 assa. adv. in systems science and appl. (2018) give as example the basic characteristics of the most ambitious impact model [3]. it is partial equilibrium, multi-commodity, multi-country model which generates projections of global food supply, demand, trade, and prices. impact covers over 46 crops and livestock commodities and it includes 115 countries/regions where each country is linked to the rest of the world through international trade and 281 food producing units. the forecast data until 2050 for the three scenarios (baseline, yield increase, energy shock) are presented in the mentioned paper. it should be noted, that the above and other discussed models in the available literature are mainly used to obtain projections, and they have not been used for evaluation of optimal agreed measures of economic policies of different regions to address the food problem in these regions. however, it is generally recognized that for solution of these global problems it is required to have all humanity coordinated efforts and, in particular, efforts of all countries concerned states. this interest in coordinated efforts can be improved if based on the model tested for the possibility of its practical application (that is, tested for transferring the computational experiments results into the subject area of macroeconomic analysis and the choice of the values of economic policy tools), will: 1) reasonable optimal steps will be offered (steps which will be defined by economic policy instruments) which should be taken in order to solve these global challenges; 2) the justified assumptions that coordinated actions of all states (or at least states of the countries within regional economic union) will give a greater effect for each country than optimal uncoordinated action of this country. in this paper, based on the macroeconomic model developed and tested on the possibility of its practical application, the possibilities of the parametric control theory [4] approach were demonstrated to solve a number of formulated problems of parametric control; and the benefits of joint action to address them are justified. the considered parametric control problems are aimed to evaluate the optimal values of economic instruments that provide economic growth, the convergence of macroeconomic indicators and the growth of regions’ agricultural output. note that the main difference between the parametric control theory and other well-known macroeconomic theories is in using only such mathematical models that have been tested for the possibility of their practical application through a series of original and classical methods. at the same time model verification is carried out not only for its basic calculation, but for optimal scenarios resulting from solving these parametric control problems. 2. the model 2.1. general features of the model in this paper, a static computable general equilibrium model globe 1 [5] is developed to describe the dynamics of functioning of interacting economies of 9 regions: five countries (kazakhstan, russia, belarus, armenia, kyrgyzstan) forming the eurasian economic union (eaeu); european union (eu, as one country); usa; china and the rest of the world (as one country). the economy of each region of the model includes the following 16 industries: 1 – agriculture, forestry and fisheries; 2 – production of crude oil and gas; 3 – metal-working production and machine-building; 4 – metallurgical industry; 5 – education, health, and public administration; 6 – production and transmission of electricity, gas, and hot water; 7 – production of food, beverages, and tobacco; 8 – professional, scientific and technical activities; 9 – other industries; 66 a. ashimov, yu. borovskiy copyright ©2018 assa. adv. in systems science and appl. (2018) 10 – other services; 11 – mining (except for oil and gas production); 12 – construction; 13 – production of textiles, apparel, and leather and related products; 14 – financial services; 15 chemical and petrochemical production; 16 – transport. in addition to the specified producers, present in each region of the model, there are such consumer agents as the households and the state. in the output of products, the industries utilize two production factors which are owned by the households in their region: the labor and the capital. in the model there is also yet another special agent (a.k.a. region) the globe, importing transport services from the rest of the regions and providing these services to other regions for the import of goods. the industries, the households, and the government (within the framework of productions consumption) in each region (except the globe) are composed of a large number of representative price-taker agents, annually solving the appropriate optimization problems. the model, as compared to the base case of the globe 1, was developed as follows. the static globe 1 model was developed to become a dynamic model by describing the following variables by using dynamic equations: technological factors of production functions for the gva of all the industries in the regions, and the supply of factors by the households in the regions. 2.2. conceptual description of the economy and the mathematical model it is assumed that producer agents, household agents and government agents are assumed to be perfect rationality agents. here we list their basic functions. producer agent in its activities annually: • produces one, corresponding to the name of the industry, type of production (subject to the condition of minimizing the costs); • produces gva gross value added (based on the use of factors as the labor and the capital of the households); • exports a part of produces production (subject to the condition of maximizing the profits); • imports intermediate productions from other regions and consumes intermediate production; • pays net tax payments to their government. producer agents solve following two pairs of the nested optimization problems: • minimization costs for the purchase of the intermediate productions and gva costs for a given production output; • minimization costs for the purchase of the production factors at the given output of the final production; • profit maximization from sales within the region and beyond, at the given production output; • profit maximization from exports to various regions at the given level of production exports. household agent in its activities annually: • gains income from the producers in their region on the basis of demand for the belonging to them factors; • consumes the productions from all the regions (according to the solution of the problem of maximizing their utility functions under the budget constraints); • perform savings in the form of investment products based on their income and consumption; solution of one global problem by approach of the parametric control theory 67 copyright ©2018 assa. adv. in systems science and appl. (2018) • pays net tax payments to their government. government of each region (except for globe) in its activities annually: • determines the effective tax rates and receives income in the form of net tax revenue (including revenue from customs duties); • consumes the final productions (government spending); • performs savings in the form of investment products based on its income and expenses. industry, household, and government agents each year jointly solve the following optimization problems: • determination of the optimal share of imports in the consumption of each type of production subject to minimum of costs of domestic and imported components of this production; • determination of the optimal regional structure of each type of imported production subject to minimum of costs of this type of imported production. the conceptual description of the model economy contains the statements of the above optimization problems with the corresponding first order conditions, other equations describing the functions of the agents, balance ratios for prices and quantities (indicators, measured in the prices of the producer), the internal balances in the accounts of the government and the external balances of trade accounts. in the model is used the system of endogenous prices for all the 16 types of productions of each region, including the buyer's and seller's prices, the prices of the exporter and importer, and etc. the calculated values of the prices ensure the performance of the annual balance sheet ratios, providing for: • the equilibrium in factor (labor and capital) markets; • the equilibrium in the markets for each type of production; • bilateral current account balances for each pair of the regions; • the equilibrium of savings (households, governments) and their investments in the regions’ industries. the mathematical model (which has been built on the basis of its conceptual description) was obtained as the result of combining into a single set of equations describing: the first order conditions of all the agents’ optimization tasks and other rules of the agents’ activity, including the equations describing the dynamics of technological factors of production functions and the factors supply. the balance and the auxiliary equations are also included to the mathematical model. 2.3. source database, calibration, and calculation of the model the core of the database of the model is the set of concordant social accounting matrices (sam) of the regions for each year under review (2004-2021), which are demonstrating how the productions flows (in monetary terms) and financial flows are distributed among the producers, households, governments, importers and exporters. the mentioned sam sets for 2004, 2007, and 2011 were extracted with a special converter from the [6]. note that in the database there are no data for the remaining years of the considered historical 2004-2014 period. therefore, for 2005, 2006, 2008-2010, and 2012-2014 the desired sets of sam were calculated using the developed algorithm (algorithm 1) on the basis of the available statistical sources, containing input-output tables (see, e.g., [7], indicators of mutual trade [8], using the basic ratios calculated using the known sam of the nearest last year (2004, 2007, or 2011). for the forecast period (2015-2021), algorithm 2 was used, which allows to calculate the mentioned sam sets on the basis of the following lookahead indicators of the regions provided by the [9]: gdp, total investment, imports of goods, imports of services; exports of goods, exports of services, total government revenues, and total government expenditures. at the same time were used the basic ratios, calculated using the obtained sams for 2014. 68 a. ashimov, yu. borovskiy copyright ©2018 assa. adv. in systems science and appl. (2018) in addition to the obtained sam sets, the source database of the model includes the values of replacement coefficients of various factors in the production function of producers; replacement coefficients of various types of productions in the output functions of producers, utility functions of households and aggregation functions describing the consumption by the agents as well as the initial values of the dynamic equations of the corresponding model. these replacement coefficients were obtained from the gtap 9 database for 2004, 2007, and 2011 and then extrapolated for the remaining years of calculation period of the model (20042021). the calibration is carried out at each start-up of the model. at this stage, on the basis of the formed database using special expressions, computation is performed of all the values of the exogenous variables in the model for the time from 2004 to 2021. the solution of the calibrated equation set of the model (model calculation) is performed with the help of the software which was implemented in the integrated design environment of the gams [10] with the use of the built-in solver path [11]. the results of calculation of the considered baseline scenario of the calibrated model precisely reproduce the statistical and forecast data used in the building of sams for the baseline database of the model. 3. verification (testing the possibility for practical application) of the model verification of the calibrated model was performed by three methods, the first two of which were proposed by the authors within the framework of the development of the parametric control theory. 3.1. assessing the stability of smooth mappings defined by the model the availability of the stability properties of mapping baf : , transferring the values of exogenous parameters ap into the solutions (the values of endogenous variables) suggests preserving the qualitative properties of such mapping at small (in some sense) changes of the mapping. to assess the stability of smooth mapping of the specified type, based on the numerical evaluation of the fulfillment of its conditions, the authors proposed the corresponding set of numerical algorithms for the cases of immersion, submersion, and submersion with a fold [12]. in the experiments, basic mappings of types with 5dim a and 9dim b were considered, whereas the arguments of f were taken the value-added tax rates of the five eaeu countries (kazakhstan, russia, belarus, armenia, and kyrgyzstan) for 2015. as the output variables of f were taken of values gdp in all the 9 regions of the model for 2021. a five-dimensional a box (with the center at the point  51 ,, ppp  , corresponding to the baseline values of the specified tax rates) boundaries are distanced from the values ip to the value of ip5.0 . it should be noted that the time to calculate the implemented mappings stability algorithms will increase approximately exponentially with increasing the dimension of a box ( adim ). this limits the use of this approach, so to obtain a reasonable calculation time the set (of the most important factors used in the solution on the basis of the model of specific problems for macroeconomic analysis and parametric control) was selected. the results of the specified numerical experiments demonstrated the absence of singular points of the f mapping in a box and the stability of this immersion. 3.2. estimation of stability indicators of smooth mappings defined by the model the ),(  pf stability indicator of (defined by the model) mapping baf : at the ap point and for the selected positive  number is the diameter of the image (when f mapping) of the ball with its radius  and with its center at the point p (in relative terms). if solution of one global problem by approach of the parametric control theory 69 copyright ©2018 assa. adv. in systems science and appl. (2018) for all the ap points the numerical assessment of the ),(lim)( 0   pp ff value is uniformly close to zero, then the f mapping, defined by the tested model is assessed on the a set as continuously dependent on exogenous values. in the experiments with the model, as a set, a ball was considered centered at p point corresponding to the baseline values of all the tax rates in all the regions in 2004, whereas endogenous variables tb set – gdp, exports, and imports of all the regions of the model for the fixed computational year t (from 2004 to 2021). as an example, table 1 shows the calculated values of the model stability indicators (in percent) for the base point p and 01.0 . table 1. values of stability indicators for the basic calculation of the model year 2004 2005 2006 2007 2008-2021 )01.0,(pf 0.4536 0.0813 0.0097 0.0060 0.0000 all the specified in table 1 assessments of stability indicators do not exceed 0.4536, which characterizes the stability of the model (in terms of the considered stability indicators) for calculations up to 2021 as sufficiently high. in particular, the acquired for the year 2004 value of the )01.0,(pf indicator means that the image of a sphere centered at the point p (corresponding to the basic values of all the 2004 tax rates) and having the radius of 0.01 (in relative terms) in the calculation of the model is transformed into a set with the diameter of 0.4536 (in relative terms) for the output variables values (gdp, exports, and imports of all the regions in 2021). various options of the calculated values of the limit indicators )(pf were close enough to zero at 0001.0 , which evaluates the tested mapping as a continuous one in a box. 3.3. implementation of counterfactual and forecast scenarios according to the well-known macroeconomic theory, the reduction of taxes levied on producers and consumers, as well as the increased state demand for consumer products will increase the country’s output and gdp. as part of its testing, based on the model counterfactual and forecast scenarios were calculated to assess the implementation of the provision of the theory. in particular, the scenario was performed featuring a 10% decrease in the effective tax rates of value added tax, tax on the producers' income and 10% increase in government consumption in each eaeu country. the results of the calculation of this scenario demonstrated an changes in gva in each sector in the relevant country (ranging from -3.85% to 6.16%) and increases in gdp in each region ranging from 0.0279% in 2009 for global economy to 0.7715% in 2012 for eaeu, compared with the observed data. the above results of the three test methods make it possible to draw the conclusion of a successful verification of the tested model. 4. solutions to parametric control problems on the basis of the model as part of the proposal of optimal measures to reduce the gap in economic development among the regions, as well as of the action to increase agricultural output, a series of parametric control problems were solved on the basis of the model. in those problems, the values of all the uncontrolled exogenous variables of the model match the baseline forecast of the variables. hereinafter indices correspond to the region number ( 9,,1i – the regions of the model): 1 – kazakhstan, 2 – russia, 3 – belarus, 4 – armenia, 5 – kyrgyzstan, 6 – european union, 7 – usa, 8 – china, 9 – the rest of the world; 10 – eurasian economic union, and 11 – world economy. 70 a. ashimov, yu. borovskiy copyright ©2018 assa. adv. in systems science and appl. (2018) statement of the ip problem ( 11,,1i ): to identify on the basis of model the values of the control parameters u(t) (effective tax rates for producers revenues, sales and customs duty taxes differentiated by production and industry sector: shares of public spending for consumption) that provide the maximum value of the ik criterion (2) (4) with constraints on control instruments of )()( tutu  type and with restrictions (1) for some of the endogenous variables. here, for 2021,,2015t ; u(t) – is a box with the center at the point of base values u(t) and boundaries spaced at ± 10% of the baseline values. for ip , ( 9,,1i ) problems the u(t) control parameters are the specified public policy tools only for i-th region, for the 10p problem – those of the five eaeu countries, and for the 11p problem – those of all of the nine regions of the model in the aggregate. the restrictions for the endogenous variables of the model in all the ip problems look like: )()( tqvaptqvap rr  , 9,,1r , 2021,,2015t . (1) here: )(tqvapr is gdp per capita in the region r, )(tqvapr are the basic values of this indicator. meeting the conditions (1) will guarantee that subject to the solving by any region (or by the 5 countries making up the eaeu, or by all the regions of the model) of their ip problems, the values )(tqvapr will not decrease (compared to the baseline forecasts), both in their specific regions and also in all the other regions of the model. in each of the first formulated ip problems, ( 9,,1i ), pursuance of the optimal fiscal policy aimed at the economic growth is supposed and the growth of agricultural output during 2015-2021 in their specific regions is supposed, whereas the behavior of the rest of the regions of the model is characterized by the baseline values of the exogenous variables. therefore, in these problems the ik criterion is recorded as:     2021 2015 )()( t iii ttqxattqvak , 9,,1i (2) where )(ttqvai is gdp per capita rate, )(ttqxai is the rate of output of industry (agriculture, forestry, and fisheries) per capita in the region i in year t (in current usd) and  is adjusted factor ( 3.0 ). it is assumed that in the 10p problem carrying out a coordinated optimal fiscal policy within the framework of the five eaeu countries in 2015-2021 as aimed at the economic growth, the growth of eaeu agricultural output, and the approximation of the per capita gdp values in each of the five eaeu countries to the respective values in the united states. (usa, (r = 7) is the region having the highest value of gdp per capita of all the regions according to the calculation in the model until 2021). therefore, the problem criterion 10k was selected as follows:                      5 1 2021 2015 7 7 5 1 2021 2015 101010 )( )()( )()( r t r r r r t tqvap tqvaptqvap ttqxattqvak . (3) here )(10 ttqva is the rate of gdp per capita of eaeu in the year t; is the rate of output of agriculture, forestry, and fisheries per capita in the eaeu in year t; )(tqvapr is gdp per solution of one global problem by approach of the parametric control theory 71 copyright ©2018 assa. adv. in systems science and appl. (2018) capita in the region r in year t; r is the weighting factor, its value equal to 1.0r for medium-developed regions (kazakhstan and russia), and 1r for all other regions eaeu members;  is adjusted factor ( 1 ). in the 11p problem it is considered the hypothetical possibility of coordinated optimal fiscal policy within the framework of all regions of the model in 2015-2021, aimed at the global economic growth, the growth of world agricultural output, and the approximation of the gdp values per capita in each region to the respective values in the united states. therefore, the problem 11k criterion was selected as follows:                      9 1 2021 2015 7 7 9 1 2021 2015 111111 )( )()( )()( r t r r r r t tqvap tqvaptqvap ttqxattqvak . (4) here )(11 ttqva is the rate of the world gdp per capita in the year t; )(11 ttqxa is the rate of the global output of agriculture, forestry, and fisheries per capita in year t; the value of factor 1.0r for  8,6,2,1r , 07  and 1r for all other r values. the formulated problems were solved numerically using the optimization algorithm provided by gams [13]. the results of their resolve as the increments of the average gdp value of and the increments of the average output of agriculture, forestry and fisheries in the regions for 2015-2021 (per capita, in percent relative to the baseline scenario) are shown in tables 2 and 3. table 2. the average value increments of the output in agriculture, forestry, and fisheries in the regions following the solutions of the 11 ip problems problem number of region r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 1p 3.8219 0.0026 0.0224 0.0007 0.7509 0.0009 0.0012 0.0028 0.0007 2p 0.0353 6.7404 0.5553 0.8582 0.8786 0.0059 0.0078 -0.0008 -0.0056 3p 0.0107 0.0121 6.8270 0.0056 0.7580 0.0014 0.0011 0.0010 0.0013 4p 0.0047 0.0011 0.0018 2.8709 0.0076 0.0000 0.0001 0.0001 0.0000 5p 0.0095 0.0011 0.0065 0.0002 1.3197 0.0000 0.0001 0.0008 0.0001 6p 0.4617 0.4226 0.8165 0.9414 0.6130 10.1658 0.3555 0.1951 0.4326 7p 0.0582 0.0856 0.0863 0.0586 0.7361 0.1427 5.7370 0.0741 0.1958 8p 0.0767 0.0818 0.0397 0.0370 0.6741 0.0415 0.2032 6.7602 0.1166 9p 0.2160 0.2357 0.1609 0.2441 0.0277 0.1398 0.3250 0.0760 6.3946 10p 4.0196 6.8185 11.6397 3.7374 1.3891 0.0084 0.0082 0.0002 -0.0054 11p 4.9816 7.8776 13.1665 5.3487 4.5170 10.9244 7.0752 7.6066 6.8962 table 3. the average values of increments of gdp of the regions following the solutions of the 11 ip problems problem number of region r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 1p 0.5949 0.0008 0.0089 0.0005 0.0000 0.0005 0.0000 0.0007 0.0003 2p 0.0298 1.4875 0.4111 0.6758 0.0574 0.0000 0.0000 0.0000 0.0000 3p 0.0073 0.0073 0.7963 0.0042 0.0000 0.0005 0.0000 0.0000 0.0009 4p 0.0002 0.0001 0.0001 1.4413 0.0000 0.0000 0.0000 0.0000 0.0000 5p 0.0002 0.0001 0.0000 0.0000 1.5198 0.0000 0.0000 0.0000 0.0000 6p 0.3606 0.4084 0.4073 0.9281 0.8454 1.9219 0.1035 0.1062 0.2595 7p 0.0310 0.0498 0.0448 0.0656 0.0000 0.0611 0.7446 0.0265 0.1107 8p 0.0678 0.0256 0.0210 0.0374 0.8326 0.0390 0.0402 1.1463 0.0467 9p 0.1206 0.1123 0.0630 0.2134 0.2164 0.0885 0.0558 0.0939 0.9169 72 a. ashimov, yu. borovskiy copyright ©2018 assa. adv. in systems science and appl. (2018) 10p 0.7659 1.4433 4.3875 2.1344 3.0802 0.0000 0.0000 0.0003 0.0000 11p 1.4083 2.0763 5.2096 3.7368 5.9613 2.1219 0.8123 1.4071 1.1225 the obtained optimal values of economic policy tools of the discussed above parametric control problems were tested for the possibility of their practical implementation as follows. the scenario options of the calibrated model for the specified optimum values of the tools were tested by the three methods referred to in section 3. in all cases, the results were obtained similar to the above ones specified in section 3 for the baseline scenario, namely: • permissible values of assessments of stability indicators; • absence of singular points of the considered mappings in their appropriate domains and the stability of these mappings and • coincidence of the results of the 2015-2021 forecast scenarios to the main provisions of the macroeconomic theory. analysis of bold type numbers in each column of tables 2 and 3 shows that within problems ip , 11,,1i , the approach of parametric control at the level of all regions (the problem 11p ), as well as at the level of the five countries of the eaeu (the problem 10p ), for the most part gives greater effects for separate region in comparison with parametric control at the level of separate region (problems ip , 9,,1i ). additionally, as a result of solving the problem 11p was reached the alignment of regions economic development, which is characterized by a decrease ratio of the maximum to minimum per capita gdp of all regions to 4.86% in 2021 as compared to the option without control; 4.17% to 2021 compared to 2015. the growth rate of per capita gdp in 2021 was also achieved in comparison with 2015 at 29.32%, 30.86% and 20.10% respectively for kyrgyzstan, armenia, and rest of the world (less developed regions). generally, across the world, as a result of solving the problem 11p the growth of gdp per capita in 2021 is 1.31%, while industry output growth in agriculture, forestry and fisheries per capita is 7.89% compared to the base case. at that growth of global gdp per capita in 2021 is 22.38%, and industry global output growth in agriculture, forestry and fisheries per capita is 38.41% compared to 2015. analysis of the problems ip solving results and the results of relevant tests shows the high potential of parametric control approach to make recommendations for optimal coordinated government economic policy at the global level and at the level of regional economic union. 5. conclusion 1. the results of the development of the dynamic global multi-country computable general equilibrium model are presented. the model is calibrated using the sams, as extracted from gtap data base, and obtained by extrapolation of extracted sams. 2. the results of the model verification with using of the three approaches are presented. 3. there are presented problem statements and the solution results of the 11 parametric control problems which are aimed at economic growth, output growth of agricultural production and reducing the gap in economic development between the rich and poor regions. these results demonstrate the effectiveness of the approach of the parametric control theory to develop recommendations to address global problems. solution of one global problem by approach of the parametric control theory 73 copyright ©2018 assa. adv. in systems science and appl. (2018) references [1] sakamoto, t., gitelson, a., & arkebauer, t. (2013). modis-based corn grain yield estimation model incorporating crop phenology information, remote sensing of environment, 131, 215-231. [2] rosen, s., meade, b., & murray, a. (2015) international food security assessment, 2015-2025. united states department of agriculture: economic research service. [3] rosegrant, m., tokgoz, s., & bhandary, p. (2012). the new normal? a tighter global agricultural supply and demand relation and its implications for food security. american journal of agricultural economics, april, 1-7. [4] ashimov, a., sultanov, b., adilov, zh., borovskiy, yu., novikov, d., alshanov, a., & ashimov, as. (2013) macroeconomic analysis and parametrical control of a national economy. new york, ny: springer. [5] globe 1. (2016). applied general equilibrium modelling. [online]. available www.cgemod.org.uk/globe1.html [6] gtap 9 data base. (2017). gtap data bases. [online]. available https://www.gtap.agecon.purdue.edu/databases/v9/default.asp [7] world input-output database. (2017). [online]. available http://www.wiod.org/home [8] world integrated trade solution. (2017). [online]. available http://wits.worldbank.org [9] world economic outlook databases. (2017). international monetary fund. [online]. available http://www.imf.org/external/ns/cs.aspx?id=28 [10] gams. (2017). [online]. available www.gams.com [11] ferris, m. & munson, t. (2015). path 4.7. [online]. available https://www.gams.com/latest/docs/s_path.html [12] ashimov, a., adilov, zh., alshanov, r., borovskiy, yu., & sultanov, b. (2014). the theory of parametric control of macroeconomic systems and its applications (i), advances in systems science and application, 14(1), 1-21. [13] nlpec. (2015). gams. [online]. available https://www.gams.com/latest/docs/s_nlpec.html http://www.cgemod.org.uk/globe1.html http://www.wiod.org/home http://wits.worldbank.org/ http://www.imf.org/external/ns/cs.aspx?id=28 http://www.gams.com/ adv syst sci appl 2020; 02:119–130 published online at https://ijassa.ipu.ru. structural change in multisector monopolistic competition model igor pospelov1,2, stanislav radionov1,3,4* 1frc csc of the ras, moscow, russia 2national research university higher school of economic, moscow, russia 3lebedev physical institute of the ras, moscow, russia 4financial research institute of the ministry of finance of russia, moscow, russia abstract: we present a natural generalization of the dixit-stiglitz monopolistic competition model (dsm) — we assume that there is a continuum of industries, each of them described as in dsm, and each characterized with its own elasticity of substitution. although firms in all industries share the same level of productivity and costs, exogenous technological progress leads to non-trivial reallocations of labor and production to industries with lower elasticities of substitution. thus the model, despite is simplicity and the absence of additional assumptions about industry structure, generates the structural changes described in the economic growth literature. keywords: dixit-stiglitz model, monopolistic competition, structural change, market reallocations. 1. introduction in modern economic growth, simon kuznets wrote: “we identify the economic growth of nations as a sustanined increase in per capita or per worker product, most often accompanied by an increase in population and usually by sweeping structutal changes. in modern times these were changes in the industrial structure within which product was turned out and resources employed — away from agriculture toward nonagricultural activities, the process of industrialization...” existence of these structural changes was considered by kuznets as one of the main stylized facts of development. indeed, the loss of relative importance of agricultucal sector in favor of industrial sector and then to services sector is well documented, see for example [3], [5], [10]. a number of economic models were developed to describe these structural changes and their relation to economic growth. the simplest mechanism, which generates structural changes, was proposed in [4] — different economic sectors grow at different rates because they have different rates of technological progress. this idea was developed in [13], [15] and [2] among many others. our work contributes to another strand of literature based on the idea that structural changes are driven not by differences in production technologies but by demand factors, namely by non-homotheticity of consumer preferences. for example, in [12] authors assume that household consumption consists of agricultural, manufacturing and services goods. a special form of utility function generates reallocations of consumption: after the consumption of certain amount of agricultural good, the household starts to demand other goods, moreover, it consumes service goods only after some level of manufacturing ∗corresponding author: saradionov@edu.hse.ru 120 i.g. pospelov, s.a. radionov consumption is reached. the model in [8] is based on the idea that new goods are introduced in economy as a luxury (a good with high income elasticity), but with time income elasticity decreases and good becomes necessity. there are also several models incorporating, for example, financial development, demographic transition, urbanization, migration, production organization, human capital, fertility, see [1] for a survey. as we can see, all these models use rather special assumptions about industry structure. the model we present in this paper, on the other hand, is very simple and free of special assumptions, but nevertheless is able to generate the reallocations described by kuznets. the key idea of our work is the same as in [8] — the main difference between goods is income elasticity, and structural change is a decline of industries producing goods with low income elasticities and rise of the ones producing goods with high income elasticities. but the model in [8] is highly stylized — there is not much said about the production side of the economy, so it remains unclear what kind of industry structure generates new goods with high income elasticities and why income elasticities of old goods decrease with time. we, on the other hand, describe both consumption and production sides of economy, thus making our model more microfounded. we present a simple monopolistic competition model, based on the constant elasticity of substitution utility function, proposed in [7]. we assume there is a continuum of industries each of them described as in [7]. the only difference between industries is intra-industry elasticity of substitution. to the best of our knowledge, this description of production side of economy, although rather straigthforward, is new in the literature. similarly to [8], we assume that “simpler” the commodity, higher the elasticity of intra-industry substitution. this assumption is in line with empirical studies on product differentiation, see for example [11] for the calculations and also [9] for the discussion. as in [8], industries with high elasticities of substitution may be interpreted as agricultural, with middle elasticities — as manufacturing, with low elasticities — as services. technological progress in our framework is modeled naturally as a decrease of variable costs of a firm (an increase of workers’ productivity) at the expense of an increase of fixed costs (which may be interpreted as investments). although firms in all industries share the same levels of productivity and costs, labor and production flows from less differentiated (higher elasticity of substitution) to more differentiated (lower elasticity of substitution) goods. thus our model, despite its simplicity and with no additional assumptions on industry structure, generates kuznets structural changes. 2. model 2.1. model setup there is a continuum of industries indexed with ρ ∈ (0, 1). for every industry, preferences of aggregate consumer are given by a ces function: v (ρ) = n(ρ)∑ i=1 ci (ρ) ρ  1 ρ , (2.1) where n(ρ) is the number of firms working in this industry, ci(ρ) is the quantity of consumed good produced by i-th firm. elasticity of substitution between goods in a given industry is σ = 1/(1− ρ). so, if ρ is high then the elasticity of substitution in this industry is high and goods in this industry are weakly differentiated and vice versa. preferences of the aggregate consumer over the goods in the continuum of industries are given by the function copyright © 2020 assa. adv syst sci appl (2020) structural change 121 u = 1 ν ∫ 1 0 a (ρ)v (ρ)ν dρ, (2.2) where a(ρ) ≥ 0 may be interpreted as consumer’s preference for the products of industry ρ. this is also needed for the dimension correctness, because goods in different industries can have different units of measurement. firms are assumed to have both fixed and variable costs, so in order to produce ci(ρ) units of good, i-th firm has to use li (ρ) = α ci (ρ) + f (2.3) units of labor. productivity level 1/α and fixed costs f are assumed to be equal across firms. the model becomes highly complicated without this assumption (see, for example [6], who deal with heterogeneous markets). full stock of labor in the economy is denoted by l, the wage, the same for all workers, is w. so the aggregate consumer has wl units of income. 2.2. market equilibrium and social welfare consider the aggregate consumer’s problem of maximizing utility given budget constraint: u = 1 ν ∫ 1 0 a (ρ)  n(ρ)∑ i=1 ci (ρ) ρ  1 ρ  ν dρ→ max, ∫ 1 0 n(ρ)∑ i=1 ci (ρ) pi (ρ) dρ ≤ wl. to solve it, we form a lagrange function in the following form: l = 1 ν ∫ 1 0 a (ρ)  n(ρ)∑ i=1 (ci (ρ)) ρ  1 ρ  ν dρ+ ξν−1 wl− ∫ 1 0 n(ρ)∑ i=1 ci (ρ) pi (ρ) dρ  . first of all, we should check if the problem has a solution, for this the integrand must be concave. applying sylvester’s criterion, we conclude that the function is concave if the two following conditions are valid: ν < 1 and ν < ρ. so we have two possibilities: if ν ∈ (0, 1), then the set of possible values of ρ must be restricted to (ν, 1), or we can assume that ν < 0. we will consider only the second case, because, as we will see later, the first case leads to some very unpleasant degeneracies at the point ν = ρ. from an economic point of view, the first case is problematic because we have to assume a relationship between parameters which does not have an economic meaning. so, from now on, ν is assumed to be negative. fixing some j ∈ {1 . . . n} and equalizing the partial derivative of lagrange function with respect to cj(ρ) to zero, we get the optimum level of consumption of cj(ρ): cj (ρ) = ( pj (ρ) ξ ν−1 a (ρ)v (ρ)ν−ρ ) 1 ρ−1 . (2.4) we involute both sides of this equality to the power ρ and find a sum over j = 1 . . . n(ρ), and after some calculation we get copyright © 2020 assa. adv syst sci appl (2020) 122 i.g. pospelov, s.a. radionov v (ρ) = ξ a (ρ)− 1 ν−1 p (ρ) 1 ν−1 , (2.5) where p (ρ) ρ ρ−1 = n(ρ)∑ i=1 pi (ρ) ρ ρ−1 (2.6) is the price index associated with the goods index v (ρ) as in [7] or [14]. taking (2.5) into account, (2.4) may be rewritten in the following form: cj (ρ) = pj (ρ) 1 ρ−1 v (ρ)p (ρ)− 1 ρ−1 . (2.7) now consider the behavior of a firm. first of all, it is obvious that due to the fact that all firms have the same levels of fixed and variable costs, and the symmetry of consumer’s preferences over the goods in a fixed industry, all firms in the industry face the same consumer’s demand and set the same price. taking (2.3) into account, the problem of firm in the industry ρ has the form π (ρ) = p (ρ) c (ρ)− (c (ρ)α + f)w → max . (2.8) substituting (2.7) and finding maximum with respect to p(ρ), we find the price set by firm: p (ρ) = αw ρ , (2.9) like, again, in [14]. now we will impose a free entry condition and demand firm profit to be equal to zero. substituting (2.5), (2.6), (2.9) into (2.8), after some calculations we get the following expression for the number of firms in the industry ρ: nm(ρ) = ( (1− ρ)α ν ν−1 ξ w 1 ν−1 ρ ν ν−1fa(ρ) 1 ν−1 ) (ν−1)ρ ν−ρ (2.10) substituting (2.10) to consumer’s budget constraint, we get the equation defining ξ: ∫ 1 0 a (x)− x ν−x w x ν−xf− ν (x−1) ν−x α ν x ν−x ξ (ν−1)x ν−x ( x− ν ν−1 − x− 1 ν−1 ) ν (x−1) ν−x x− ν ν−1dx = l. (2.11) unfortunately, ξ can be found from (2.11) only numerically. substituting (2.9) into (2.8) and setting firm profit to zero, we find that firm output may be rewritten in the following form: cm (ρ) = fρ α (1− ρ) , (2.12) which is again in line with [7]. now consider the problem of the benevolent social planner, who maximizes consumer utility with respect to technological limitation. copyright © 2020 assa. adv syst sci appl (2020) structural change 123 u = 1 ν ∫ 1 0 a (ρ)  n(ρ)∑ i=1 ci (ρ) ρ  1 ρ  ν dρ, α ∫ 1 0 n(ρ)∑ i=1 ci (ρ) dρ+ f ∫ 1 0 n (ρ) dρ ≤ l. form the lagrange function in the following form: l = 1 ν ∫ 1 0 a (ρ)  n(ρ)∑ i=1 ci (ρ) ρ  1 ρ  ν dρ+ λ l− α ∫ 1 0 n(ρ)∑ i=1 ci (ρ) dρ+ f ∫ 1 0 n (ρ) dρ  . differentiating with respect to c(ρ) and n(ρ), after some calculations we get nw (ρ) = ( (1− ρ)α ν ν−1λ ρfa(ρ) 1 ν−1 ) (ν−1)ρ ν−ρ , (2.13) cw (ρ) = fρ α (1− ρ) . substituting (2.12) into the planner’s technological constraint, we get the formula defining λ:∫ 1 0 α ν x ν−xa (x)− x ν−x λ (ν−1)x ν−x f− ν (x−1) ν−x x− (ν−1)x ν−x (1− x) ν (x−1) ν−x dx = l. (2.14) similarly to (2.11), it can be solved only numerically. as we can see, outputs in market equilibrium and in the social welfare problem are the same, but the numbers of firms are different. this proves the following proposition 1. for any ν ∈ (−∞, 0) and any parameters of the economy l, f, α, w market equilibrium is inefficient. thus, in the market equilibrium consumer cannot optimally distribute her expenses across industries, but can do it within an industry. recall that in one-sector dixit-stiglitz model the equilibrium is efficient, but in the presence of second market of homogenous product (as it was proposed in the original paper), it is not. this result is disappointing, but rather expectable — market efficiency is rare thing in the presence of monopolists. it is also worth noting that this result cannot be considered as a trivial corollary of general theorems about the efficiency in monopolistic competition models proven, for example, in [6] and [16], because utility function (2.1)–(2.2) does not belong to the class of variable elasticity of substitution utility functions analyzed in these papers. figure 1 shows the distribution of the number of firms in the market equilibrium and in social welfare state for arbitrarily chosen parameters of the economy α = 0.01, f = 0.5, l = 30, a(ρ) = 1, w = 1 and consumer’s preferences characterized with ν = −1. figure 2 shows the distribution of firms in the same economy and for consumer’s preferences characterized with ν = −5. copyright © 2020 assa. adv syst sci appl (2020) 124 i.g. pospelov, s.a. radionov fig. 2.1. ν = −1. fig. 2.2. ν = −5. as we see, smaller ν, closer the market equilibrium to the social welfare state. note that as ν tends to −∞, utility function (2.2) converges to leontieff-like minimum function: u−∞ = min ρ∈(0,1) v (ρ). (2.15) this observation leads us to proposition 2. for consumer’s preferences defined by (2.15), market equilibrium is efficient. proof. as ν tends to −∞ in (2.10) and (2.11), we get n∗m (ρ) = ( fρ ξ (1− ρ)α )−ρ , n∗w (ρ) = ( fρ λ (1− ρ)α )−ρ ,∫ 1 0 ξxf−x+1αxx−x (−x+ 1)x−1 dx = l,∫ 1 0 λxf−x+1αxx−x (−x+ 1)x−1 dx = l. obviously, λ = ξ and n∗m(ρ) = n∗w (ρ). � 2.3. effects of technological progress and population growth consider the effects of technological progress in this model. we assume that in our framework technological progress means an increase of workers’ productivity 1/α. this, however, doesn’t come without cost — we assume that fixed costs f also increase. thus, progress is due to the increase in capital expenditures. figures 3–6 show the effect of technological progress in economy with parameters α = 0.01, f = 0.5, l = 30, w = 1 and consumer’s preferences defined by parameters a(ρ) ≡ 1, ν = −∞, on the distribution of the number of firms, output and labor. technological progress is modeled by decreasing α 3 times and increasing f 3 times. the solid line is before progress, dashed — after. conclusions are summed up in the following copyright © 2020 assa. adv syst sci appl (2020) structural change 125 fig. 2.3. number of firms. fig. 2.4. number of workers. fig. 2.5. output. fig. 2.6. ratio of outputs. proposition 3. with decrease of variable costs and increase of fixed costs: α′ = 1 k α, f ′ = kµf, k > 1, µ > 0, for any parameters of economy and consumer’s preferences of the type (2.1)-(2.2) with a(ρ) = 1 for any ρ, 1. for µ ≤ 1 output increases in all industries ρ ∈ (0, 1), 2. number of workers increases in industries with ρ ∈ (0, ρ∗) and decreases for industries with ρ ∈ (ρ∗, 1) for some ρ∗ ∈ (0, 1). 3. if µ+ ν ≤ 0, consumer’s utility increases. proof. 1. obviously, with the change of costs, ξ will change as well: ξ′ = θξ, where θ can depend on µ. we need to prove that c (ρ)n (ρ) < c′ (ρ)n′ (ρ). substituting the expressions from (2.11) and (2.12), we get ρ f ′ α′ (1− ρ)w ( (1− ρ)α ν ν−1 ξ w 1 ν−1 ρ ν ν−1fa(ρ) 1 ν−1 ) (ν−1)ρ ν−ρ = k− ν(µ+1)(ρ−1)+ρ ν−ρ θ (ν−1)ρ ν−ρ × × c (ρ)n (ρ) > c (ρ)n (ρ) . (2.16) copyright © 2020 assa. adv syst sci appl (2020) 126 i.g. pospelov, s.a. radionov note that −ν(µ+1)(ρ−1)+ρ ν−ρ > 0 and (ν−1)ρ ν−ρ > 0 for all ν < 0 and ρ ∈ (0, 1). so if we prove that θ > 1, inequality (2.16) will be proven. in order to prove that θ > 1, consider budget constraint (2.11) of economy after technological progress, which may be rewritten in the following form: ∫ 1 0 k −ν(µx−µ+x) ν−x θ (ν−1)ρ ν−ρ (1− x) ν (x−1) ν−x x− ν x ν−1a (ν−1)x ν−x dx = l f , (2.17) where a is a combination of parameters of the model. budget constraint for economy before technological progress is then∫ 1 0 (1− x) ν (x−1) ν−x x− ν x ν−1a (ν−1)x ν−x dx = l f . (2.18) note that if we prove that∫ 1 0 k −ν(µx−µ+x) ν−x (1− x) ν (x−1) ν−x x− ν x ν−1a (ν−1)x ν−x dx < l f , (2.19) that will mean exactly that for (2.17) to be true, θ must be greater than 1. due to (2.18), inequality (2.19) may be rewritten in the following form:∫ 1 0 ( 1− k− ν(µx−µ+x) ν−x ) (1− x) ν (x−1) ν−x x− ν x ν−1a (ν−1)x ν−x dx > 0. (2.20) denote the integrand in (2.20) as g(x). figure 6 shows the graph of g(x). this function is positive for ρ > µ 1+µ and negative otherwise. fig. 2.7. g(x) for µ = 0.7. we will prove that if µ ≤ 1, than g(x) > −g(1− x) for all x ∈ ( µ 1 + µ , 2 µ 1 + µ ) , (2.21) which proves (2.20). note that for µ > 1 it doesn’t have to be true. consider the function h(x) = − g(x) g(1−x) . analysis of this function shows that limx→ µ 1+µ h(x) = 1 and it is copyright © 2020 assa. adv syst sci appl (2020) structural change 127 monotonically increasing. so h(x) > 1 for ρ ∈ (0.5, 1) and hence (2.21) holds, hence (2.20) holds, hence θ > 1, which concludes the proof. 2. we will prove that l′(1)n′(1) < l(1)n(1), l′(0)n′(0) < l(0)n(0) and l · n is a monotonic function of ρ. limρ→1 l(ρ)n(ρ) = α ν ν−1 ξ w 1 ν−1 , so l′(ρ)n′(ρ) l(ρ)n(ρ) = k− ν ν−1 θ. we need to prove that θ < k ν ν−1 . (2.22) assume otherwise: θ ≥ k(ν/(ν−1)) and substitute it to the budget constraint in the form l = ∫ 1 0 k− ν(µx−µ+x) ν−x θ (ν−1)ρ ν−ρ (1− x) ν (x−1) ν−x x− ν x ν−1a (ν−1)x ν−x dx ≥ ≥ ∫ 1 0 k νµ(1−x) ν−x (1− x) ν (x−1) ν−x x− ν x ν−1a (ν−1)x ν−x dx >∫ 1 0 (1− x) ν (x−1) ν−x x− ν x ν−1a (ν−1)x ν−x dx = l this contradiction proves (2.21) and the fact that l′(1)n′(1) < l(1)n(1). limρ→0 l(ρ) = f , so obviously l′(0) > l(0). finally, analysis of the derivative of l shows that is indeed monotonically increasing. 3. consumer’s utility may be written in the following form: u ′ = 1 ν ∫ 1 0 ( c′ (ρ)n′ (ρ) 1 ρ )ν dρ = 1 ν ∫ 1 0 ( k− µρ−µ+ρ ν−ρ θ ν−1 ν−ρ c (ρ)n (ρ) 1 ρ )ν dρ > > 1 ν ∫ 1 0 k ν(µρ−µ−ν+ρ) ν−ρ ( c (ρ)n (ρ) 1 ρ )ν dρ > 1 ν ∫ 1 0 ( c (ρ)n (ρ) 1 ρ )ν dρ = u. here we used (21) in the first inequality and the assumption ν ≤ −1 in the second one (µρ− µ− ν + ρ > −µ− ν > 0). � there are few things regarding these results we should point out. first, the decrease of the number of firms in all industries, which can be seen in figure 3, is expectable result of increasing fixed costs and may be interpreted as creation of big corporations. the second part of proposition 3 is the most important to us: it means that because of technological progress, labor flows from less differentiated to more differentiated industries. this reallocation is exactly of the type described in the citation in the beginning of the article and the main feature of this model. figure 6 indicates another good property of output: the rate of output growth is higher in more differentiated industries, which is in line with what we observe in the modern economy. we do not include a proof because it can be done trivially by analyzing the derivative of the ratio. note that another important indicator, output per worker, does not show nontrivial dynamics in our framework: it can be easily calculated that c(ρ)/l(ρ) = ρ/α, so with technological progress this ratio increases in all industries equally. now consider the effect of population growth. figures 7, 8, 9 show the effects of population growth of 150% in the same economy as in the previous figures, but everything is now per capita. proposition 4. with the population growth, for any parameters of economy and consumer’s preferences of the type (2.1)-(2.2) with a(ρ) = 1 for any ρ ∈ (0, 1), 1. output per capita decreases in industries with ρ ∈ (0, ρ1) and increases in industries with ρ ∈ (ρ1, 1) for some ρ1 ∈ (0, 1). 2. labor per capita decreases in industries with ρ ∈ (0, ρ2) and increases in industries with ρ ∈ (ρ2, 1) for some ρ2 ∈ (0, 1). copyright © 2020 assa. adv syst sci appl (2020) 128 i.g. pospelov, s.a. radionov fig. 2.8. number of firms per capita. fig. 2.9. output per capita. fig. 2.10. number of workers in industry per capita. proof. 1. denote l′ = kl, k > 1, ξ′ = δξ. first, we shall prove that δ > 1. assume otherwise, δ ≤ 1, then l′ = kl = ∫ 1 0 δ (ν−1)x ν−x (1− x) ν (x−1) ν−x x− ν x ν−1a (ν−1)x ν−x dx ≤ ≤ ∫ 1 0 (1− x) ν (x−1) ν−x x− ν x ν−1a (ν−1)x ν−x dx = l, which contradicts to the assumption k > 1. next, we prove that δ > k. to do so, consider budget constraint in the form (2.17): l′ = kl = ∫ 1 0 δ (ν−1)x ν−x (1− x) ν (x−1) ν−x x− ν x ν−1a (ν−1)x ν−x dx < < δ ∫ 1 0 (1− x) ν (x−1) ν−x x− ν x ν−1a (ν−1)x ν−x dx = δl. copyright © 2020 assa. adv syst sci appl (2020) structural change 129 limρ→0 c(ρ)n(ρ) l = 0, so we have to analyze derivatives: limρ→0 ( c(ρ)n(ρ) l )′ = f l and f l′ < f l , hence c(ρ)n′(ρ) l′(x) < c(ρ)n(ρ) l(x) for small ρ’s. limρ→1 c(ρ)n(ρ) l = α 1 ν−1 ξ w 1 ν−1 l and xi′ l′ = δξ kl > ξ l , hence c(ρ)n′(ρ) l′(x) > c(ρ)n(ρ) l(x) for big ρ’s. analysis of derivative of the function c(ρ)n(ρ) l(x) shows that it is monotonically increasing. 2. similarly to 1: limρ→0 c(ρ)n(ρ) l = f l , limρ→1 c(ρ)n(ρ) l = α ν ν−1 ξ w 1 ν−1 l . � as we can see, and as it might be expected, population growth leads the economy to, in a sense, opposite direction compared to economic growth, consumer now cares more about “simple” homogenous goods. this behavior may be interpreted in the following way: as the population increases, it becomes harder and harder to “feed” them, so people start to care less about luxury (differentiated goods) and more about simple material goods. 3. conclusion we developed a simple and natural model, which generalizes the dixit-stiglitz monopolistic competition model. it provides a natural way to describe technological progress, which leads to non-trivial labor and product reallocations. these reallocations may be interpreted as transfers from “simple” homogeneous to “more complicated” differentiated goods (from agricultural to manufacturing goods and then to services). under some assumptions, it leads to an increase in consumer utility. so, despite its simplicity, our model is able to reproduce essential features of modern economies, described in economic growth literature. references 1. d. acemoglu. introduction to modern economic growth. princeton university press, 2007. 2. d. acemoglu and v. guerrieri. capital deepening and nonbalanced economic growth. journal of political economy, 116(3):467–498, june 2008. 3. r. c. allen. economic structure and agricultural productivity in europe, 1300–1800. european review of economic history, 4(1):1–25, 2000. 4. w. j. baumol. macroeconomics of unbalanced growth: the anatomy of urban crisis. economic journal, 57(3):415–426, june 1967. 5. t. boppart. structural change and the kaldor facts in a growth model with relative price effects and non-gorman preferences. econometrica, 82(6):2167–2196, 2014. 6. s. dhingra and j. morrow. monopolistic competition and optimum product diversity under firm heterogeneity. technical report, london school of economics, 2012. 7. a. k. dixit and j. e. stiglitz. monopolistic competition and optimum product diversity. american economic review, 67(3):297–308, 1977. 8. r. foellmi and j. zweimüller. structural change, engel’s consumption cycles and kaldor’s facts of economic growth. journal of monetary economics, 55(7):1317–1328, 2008. 9. h. gray and j. martin. on the meaning and measurement of product differentiation in international trade: a reply. review of world economics (weltwirtschaftliches archiv), 118(2):335–337, june 1982. 10. b. herrendorf, r. rogerson, and a. valentinyi. growth and structural transformation. in handbook of economic growth, volume 2, pages 855–941. elsevier, 2014. 11. g. hufbauer. the impact of national characteristics & technology on the commodity composition of trade in manufactured goods. in the technology factor in international copyright © 2020 assa. adv syst sci appl (2020) 130 i.g. pospelov, s.a. radionov trade, nber chapters, pages 143–232. national bureau of economic research, inc, august 1970. 12. p. kongsamut, s. rebelo, and d. xie. beyond balanced growth. review of economic studies, 68(4):869–82, october 2001. 13. k. matsuyama. agricultural productivity, comparative advantage, and economic growth. journal of economic theory, 58(2):317–334, december 1992. 14. m. j. melitz. the impact of trade on intra-industry reallocations and aggregate industry productivity. econometrica, 71(6):1695–1725, 2003. 15. l. r. ngai and c. a. pissarides. structural change in a multisector model of growth. american economic review, 97(1):429–443, 2007. 16. e. zhelobodko, s. kokovin, m. parenti, and j.-f. thisse. monopolistic competition: beyond the constant elasticity of substitution. econometrica, 80(6):2765–2784, 2012. copyright © 2020 assa. adv syst sci appl (2020) introduction model model setup market equilibrium and social welfare effects of technological progress and population growth conclusion microsoft word 977 article text, copyedited.doc adv syst sci appl 2020; 03; 166-177 published online at https://ijassa.ipu.ru. a differential stackelberg game theoretic model of the promotion of innovations in universities mukharbek malsagov1*, gennady ougolnitsky2, anatoly usov3 1) ingush state university, nazran, russia e-mail: mmm1956@bk.ru 2) southern federal university, rostov-on-don, russia e-mail: gaugolnickiy@sfedu.ru 3) southern federal university, rostov-on-don, russia e-mail: abusov@sfedu.ru abstract: a two-level dynamical game theoretic model "federal state universities" in openloop strategies is built and investigated. a refinement of the electronic learning courses, and their differentiation using modern methods of information technologies made by competing a la cournot universities (agents) are treated as innovative investments. the algorithms of building nash (in the non-coalitional game of agents) and stackelberg (in the hierarchical game of the principal with the grand coalition of agents) equilibria are proposed and implemented. for the solution of the respective dynamic control problems the pontryagin maximum principle and simulation modeling are used. a comprehensive analysis of the received results is given. keywords: nash equilibrium, stackelberg equilibrium, stackelberg differential games, innovations, method of qualitatively representative scenarios in simulation modeling 1. introduction a review of applications of the game theoretic models to the analysis of innovations is presented in [5]. the authors divide three levels of these applications: (1) intra-organizational level within a firm, where players are innovators, project managers, and resource administrators; (2) inter-organizational level where the players are competitive firms; (3) meta-organizational level, where players are a social planner (innovation policy maker, government, a social or government institution, e.g., a research foundation) and an aggregate innovative entrepreneur. a traditional approach to the building and investigation of the differential game models is exposed in [1,4]. an original method of solution of the stackelberg differential games based on building of a mutually benefit program of actions and punishment in the case of deviations, is described in [9]. in our papers [13,16] the approach of [9] is extended for the case of several followers with consideration of the requirements of sustainable development of the controlled dynamic system [12]. ougolnitsky and usov [14] have proposed a method of qualitatively representative scenarios in simulation modeling. this method allows for a good enough qualitative forecast of the dynamics of a controlled system by means of few scenarios of the computer simulation. an interesting dynamic game theoretic model of the oligopolistic investment to the product differentiation is studied in [2, 3]. the paper [2] considers open-loop strategies, and the paper [3] analyzes closed-loop ones. cellini and lambertini [2, 3] have found that in the case of open-loop strategies the nash stable volume of investments to the product differentiation (innovations) increases with a number of firms, i.e. the level of market * corresponding author: mmm1956@bk.ru a differential stackelberg game theoretic model… 167 copyright ©2020 assa. adv. in systems science and appl. (2020) competition (arrow hypothesis). in contrary, in the case of closed-loop strategies the dependence is inverse (schumpeter hypothesis). the authors notice that innovation activity in the product differentiation can be considered as a contribution of private investors to the production of a public good. this fact embeds their paper to the context of public goods economics. interesting models of stackelberg oligopoly are proposed and investigated in [68]. this paper develops the papers [2, 3] with consideration of other mentioned sources in the following directions. first, its problem domain is a development of the electronic learning courses in universities, where a modification of the courses is treated as an innovative activity. there is no doubt in the actuality of this problem. on march 25, 2020 the it holding talenttech and on-line universities "netology" and edmarket have presented the research results about the on-line education market†. as the authors notice, a volume of the russian market of on-line education in the b2c segment was 38,5 billion rubles in the end of 2019. in the end of the year 2023, according to that forecast, its value will be equal to 60 billion rubles a year. the global market of on-line education up to the year 2023 tends to the amount $282,62 billion. an even higher estimate is given by the interfax academy. according to its report, the russian market of the on-line education after 2019 equals to 45–50 billion rubles, and in 2020 will be equal to the amount 55–60 billion rubles, the annual growth is 20–25%. the global on-line education market is estimated by the value $74 billion (about 4,8 trillion rubles) after the year 2019, so the potential for russia is great‡. it is evident that the growth of the distant forms of education due to covid-19 pandemics will essentially enforce the noticed trends. second, we propose a hierarchical setup of the problem with the federal state as a leader (principal), and competitive universities as followers (agents). a description of a university as active system [11] is given in [10]. the hierarchical impact of the principal to the agents may be administrative (compulsion) or economic one (impulsion). in the former case the principal impacts to the sets of feasible strategies of the agents, and in the latter case to their payoff functionals [12]. thus, the contribution of the paper has four aspects. first, we consider a combination of the aggregative non-cooperative oligopolistic game of the agents with the stackelberg game of the type "principal-agents". second, the parameter of the demand function varies in time, and the character of this variation depends on the agents' actions (strategies) in the form of a differential equation. third, the agents choose both outputs and investments, i.e. an agent's strategy includes a parameter of its cost function called the constant cost. fourth, the following approach is used. from the point of view of the agents, their interaction is modeled as a game in normal form where the nash equilibrium is built. from the point of view of the principal it is supposed that the agents form the grand coalition, and respectively a stackelberg two-person game "principal coalition of agents" arises. this permits to avoid a challenging question about what should be considered as a best response of several agents to the principal's strategy. the rest of the paper is organized as follows. in the section 1 the setup of a dynamic problem of hierarchical control is given. in the section 2 the nash equilibrium for two symmetrical agents with administrative and economic impact of the principal in open-loop strategies is built. algorithms of building the stackelberg equilibrium by means of the method of qualitatively representative scenarios in simulation modeling are presented. the section 4 describes a numerical simulation in the problem of building the stackelberg equilibrium. the section 5 contains a comparative analysis of the received results, and the conclusions are formulated. †http://neorusedu.ru/news/rossijskij-rynok-onlajn-obrazovaniya-ozhidaet-burnyj-rost ‡https://www.kommersant.ru/doc/4275439 168 m. malsagov, g. ougolnitsky, a. usov copyright ©2020 assa. adv. in systems science and appl. (2020) 2. the problem setup let us consider the following hierarchical modification of the model proposed by cellini and lambertini [2, 3]. several universities (agents) competing a la cournot develop electronic learning courses for sale. a refinement of the courses, and its differentiation using modern methods of learning and information technologies are treated as innovative investments. the differentiation of the courses may be considered as a public good, and the respective investments as a private production of the public good [3]. on the higher (relative to the agents) control level a principal (federal state or its authorized bodies) is situated. the principal tends to increase a public good (with possible additional consideration of its own interests) by means of administrative or economic control methods. in the case of administrative impact (compulsion) the principal bounds from below the contributions of the agents to the innovative development (production of the public good) that incurs control cost. in the case of economic impact (impulsion) the principal grants the agents on the base of an available budget. in both cases the principal and the agents use open-loop strategies. the game is played on a finite interval of time in the case of compulsion the model for n agents has the form: the principal's payoff functional (2.1) the principal's control constraints ; (2.2) the agents' payoff functionals (2.3) the agents' control constraints ; ; (2.4) the equation of system dynamics [9, 10] (2.5) the current payoff function of the i-th agent (2.6) in the principal's payoff function the profit πi reflects an approximate estimate of the positive externality from the universities' activity, namely, the gnp growth due to increasing of the educational level of the society; the inverse demand function [2,3]; (2.7) the payoffs of the principal and the agents in the moment of time t respectively the total payoff functionals of the principal and the agents; a symmetrical degree of substitutability between any pair of courses. if then the courses are completely homogeneous. if then the courses are unique and each agent is a monopolist [2,3]; an output level of the i-th agent at constant returns to scale, then total operative cost per period are ; individual investment of ].,0[ t max)()())()(( 0 0 1 1 0 ®+ú û ù ê ë é ÷ ø ö ç è æ--= ò å å = = tgdttyztstej t n i n i iii t pr max)(0 ktyi ££ ;max)())()(( 0 ò ®++= t iii t i tgdttstej pr max)()( ktkty ii ££ max)(0 qtqi ££ );( )(1 )( td tk tk dt dd + -= ;)0();()( 1 bdtktk n i i ==å = )()(])([)( tktqctpt iiiii --=p å ¹ --= ij jii tqtdtbqatp )()()()( ;,...,2,1),(])()()([)( nitqctqtdtbqatg ii ij jii =---= å ¹ .)(])()()([)()( 11 0 å åå = ¹= ---== n i ii ij ji n i i tqctqtdtbqatgtg ijj ,0 ],0[)( btd î btd =)( 0)( =td )(tqi ),0(),()( iiiii actqctc î= )(tki a differential stackelberg game theoretic model… 169 copyright ©2020 assa. adv. in systems science and appl. (2020) the i-th agent to the innovative development, the overall industry expenditure, an upper bound for any ; an upper bound for any ; functions are the principal's grants to the i-th agent (in the case of compulsion they are given); t – the length of the game. then, are the principal's lower bounds established for a discount factor; demand parameters; a convex increasing administrative control cost function, the function is supposed to be linear the equation of dynamics (5) is treated as a production function with the output created by the input . this technology can be shown to exhibit decreasing returns to scale w.r.t. . thus is non-increasing function of time which tends to zero when [3]. in the case of impulsion the model has the form: the principal's payoff functional (2.8) the principal's control constraints ; (2.9) the agents' payoff functionals (2.10) the agents' control constraints ; . (2.11) here are the principal's control to be determined; is a total volume of the principal's grants (with consideration of possible savings). all input functions of the model are supposed to be continuous, and the controls of the agents and the principal belong to the class of piecewise continuous functions. we investigate the model (2.1)–(2.7) for compulsion and (2.5)-(2.11) for impulsion from the point of view of different control agents. from the point of view of the agents there is a noncooperative n person game which solution is supposed to be a nash equilibrium. from the point of view of the principal there is a stackelberg game. let us assume in this case that the agents cooperate (create the grand coalition) and have the summary payoff functional in the form (2.12) other model relations do not change. thus, we have a stackelberg game between the principal and the grand coalition of agents. this game has the following information structure: 1. the principal chooses its open-loop strategies for impulsion or for compulsion, . 2. given these strategies for impulsion or for compulsion, the agents within the grand coalition choose their open-loop strategies , using (2.12). as an integrand function in (2.12) depends continuously on its arguments, and the domains of feasible controls of the agents (2.4) or (2.11) are non-empty closed sets, the problem of determination of the agents' best response to any principal's strategy is resolvable. )(tk constmax =k )(tki constmax =q )(tqi )(tsi )(tyi );(tki )1,0(îr 0,0 >> ba )(×z ;0)0( =z )(×z .cons;)( teexxz == )(/ tdd!)(tk )(tk )(td ¥®)(tk max)()]()([ 0 0 1 0 ®+-= ò å = tgdttstej t n i ii t pr å = £³ n j ji ststs 1 )(;0)( ò ®++= t iii t i tgdttstej 0 max)())()((pr max)(0 ktki ££ max)(0 qtqi ££ )(tsi s .max)())()(( 1 01 ®÷÷ ø ö çç è æ ++== å òå = = n i i t ii t n i ia tgdttstejj pr )(tss ii = )(tyy ii = nitt ,...,2,1],,0[ =î )(tss ii = )(tyy ii = ),(),( tqqtkk iiii == nitt ,...,2,1],,0[ =î 170 m. malsagov, g. ougolnitsky, a. usov copyright ©2020 assa. adv. in systems science and appl. (2020) 3. the principal maximizes its payoff functional (2.1) or (2.8) for the worst best response of the coalition of agents to its strategy or . 4. the received set of strategies for impulsion or for compulsion is the stackelberg equilibrium. 3. building the nash equilibrium let us first investigate the model from the point of view of the agents when they use openloop strategies. the principal's strategies are supposed to be given. this interpretation corresponds to the case of an indifferent principal without its own objectives. then we receive a differential n-person game (2.3)-(2.7) in the case of compulsion and (2.5) – (2.7), (2.10), (2.11) in the case of impulsion. its solution is assumed to be a nash equilibrium. to build it we use the pontryagin maximum principle [15]. the hamilton function of the -th agent both for compulsion and impulsion has the form: where is a conjugate variable (as function of time). from the necessary condition of extremum in the case of symmetrical agents ( ) we receive the system of equation for determination of their control variables (3.1) thus . (3.2) besides, we have the system of differential equations (3.3) from (3.1) we receive therefore, the following proposition is proved. proposition (3.1): the formulas (3.2), (3.3) determine the point of maximum of the hamilton function for some value if the system (3.2), (3.3) has a solution and the values (3.2) belong to the domains of feasible controls (2.4) for compulsion or (2.11) for impulsion. if these conditions are not satisfied for some t then the hamilton function attains its maximum for this at one of the bounds of the segments (2.4) or (2.11). 0j ),...,( 1 nssr ),...,( 1 nyyr ),...,,,...,,,...,( ** 1 ** 1 ** 1 nnn qqkkss ),...,,,...,,,...,( ** 1 ** 1 ** 1 nnn qqkkyy i ,))(1 ,)( )()()()()()),(),(),(),(( 1 1 1 å å å = = ¹ + ++÷ ÷ ÷ ø ö ç ç ç è æ ---= n j j n i i iiiii n j ij jiiiii tk tkd ttsktqctqdtbqattdttqtkh ll )(til ï ï î ïï í ì = = ¶ ¶ = ¶ ¶ ni q h k h i i i i ,...,2,1, 0 0 ;cci º niiggqqkksshh iiiiii ,...,2,1;;;,;;; ===ººººº ll ;0 )1( 1 2 =+ --= ¶ ¶ nk d k h l .0)()1()(2 =----= ¶ ¶ ctqndtbqa q h ))1(2 )(;1)( -+ = -+= ndb сatq n dtk l ;11 2 2 ÷ ø ö ç è æ -+÷ ø ö ç è æ + -== ¶ ¶ ddb ca dt d d h l ll ; )1)((2 ))(1( )( )( -+ -= ¶ ¶ = ntdb aсn td gtl .)0(;11 bd ddt dd = +-= l .0;0)1(2; )1( 2 2 2 2 32 2 = ¶¶ ¶ <---= ¶ ¶ + = ¶ ¶ qk hndb q h nk dn k h l t t a differential stackelberg game theoretic model… 171 copyright ©2020 assa. adv. in systems science and appl. (2020) an analytical investigation of the system of equations (3.3) is impossible due to the form of the equation of dynamics (2.5) and presence of the third summand in (2.7) that is principal for the considered problem setup. thus the system (3.3) was analyzed numerically by the method of shooting. the system of equations (3.3) has a solution not for all values of the model input parameters. the picard theorem of existence and uniqueness of solution of the system of differential equations is not satisfied, the right hand sides are defined only for the negative values of λ(t)d(t). besides, in the neighborhood of zero the lipschitz conditions on λ and d are not satisfied. therefore, the range of input model parameters for which the system (3.3) has the unique solution was determined numerically. a numerical identification of the model has a testing character and provides a reasonable relation of the model parameters and variables that allows for acceptable qualitative conclusions about the comparative analysis of the results of numerical modeling. thus, the professors of southern federal university develop annually about 100 new learning courses, therefore the maximal possible value is taken an average cost of the development is equal to 50-80 thousand rubles, so (thousand rubles per year). the value of parameter b varied in the range (thousand rubles per year). we divided the cases of small investment of the principal ( ), middle investment ( ), and considerable investment ( ), as well as different cases of the agents' control by the principal: namely, the value of z varied from 2 thousand rubles per year (soft control) till 10 thousand rubles per year (hard control). the discount factor was estimated as according to the annual inflation rate. the period of modeling was equal to 3 years, i.e. days for two universities ( ). other parameters were evaluated by the experts (table 3.1). table 3.1. test values of the model parameters parameter value 100 50 900 5 3000 0.04 600 2 dimension thousand rubles per year thousand rubles per year thousand rubles per year thousand rubles per year thousand rubles per year for determination of the range of model parameters in which the system (3.3) has the only solution we have implemented about 80 numerical simulations with variation of the values: (thousand rubles per year); n from 2 till 100. as a result it was established that for any input data there is a moment of time exists a solution of (3.3): . the moment belongs to the range from 950 till 1000 days and depends not essentially on the parameters . its value slightly increases with n or b, decreases with a, and does not depend on c. some of the results are presented in table 3.2. table 3.2. dependence of the moment of time on model parameters 2 110 100 10 997 2 110 100 1 999 2 110 100 25 989 2 110 100 250 980 2 200 100 10 999 .100max =q ,3000max =k ,900=а 50=c 151-=b 10/maxkns = 5/maxkns = maxkns = 04.0=r 1095=t 2=n maxq cci º a b maxk r ssi º z ]200,10[îc ]250,1[; îb ]1000,100[; îa :01 >t ),[0)(;0)( 1 tttanyfortdt î³<l 1t ncba ,,, 1t n a c b 1t 172 m. malsagov, g. ougolnitsky, a. usov copyright ©2020 assa. adv. in systems science and appl. (2020) 2 1000 100 10 1000 2 300 10 10 1000 2 300 200 10 1000 2 210 200 10 995 10 200 100 10 999 10 110 100 25 1000 100 110 100 25 953 analysis of compulsion from the point of view of the agents is reduced to (3.2), (3.3) in the case when the functions , are identically equal to zero. example 3.1. for the values from table 3.1 for impulsion and : (thousand rubles), i=1,2. for the values from table 3.1 and (thousand rubles per year) and (thousand rubles) respectively. for the values from table 3.1 and (thousand rubles per year) we receive and (thousand rubles), and in the case (thousand rubles per year) the payoffs are and thousand rubles respectively. the growth of the total operative cost (value c) or demand parameter (b) implies decreasing of the agents payoffs and value of output (q), and it is profitable for the agents to develop homogeneous non-unique courses. in the majority of investigated numerical examples the agents' investment to the development of new courses (k) is absent. increasing of the principal's grants tends expectably to the increase of the agents' payoffs (not very big), and their strategies practically not change. for compulsion with the data from table 3.1 consider the case of soft ( ) and hard ( ) control of the agents' investment by the principal. here the agents are obliged to invest in the development of new courses but this only diminish their payoffs. thus, the values of data from table 3.1 give the payoffs thousand rubles for the soft control, and thousand rubles for the hard control respectively. in the case of an indifferent principal and absence of its control of the agents they do not invest for a broad class of model parameters. thus, farther we consider the case of an interested principal. suppose that the agents form the grand coalition that have the summary payoff functional (2.12). it is difficult to find analytically the stackelberg equilibrium in the game of the principal with the grand coalition of agents. so, we have calculated it numerically by means of the method of qualitatively representative scenarios in simulation modeling [14]. 4. building the stackelberg equilibrium in the case of grand coalition of agents here we consider a model (2.1), (2.2), (2.4) – (2.7), (2.12) for compulsion and (2.5)-(2.9), (2.11), (2.12) for impulsion. its solution is the stackelberg equilibrium given that the agents form the grand coalition. we use the qrs method in simulation modeling. let be the set of possible outcomes of the stackelberg game with n agents where are sets of the feasible strategies of the principal and the agents respectively. in this case nitsi ,...,2,1),( = 2=n 33=ij 8,2=b 5.81=ij 2.21 100,10=c 36=ij 29 300,30=is 31=ij 32 4/or8/ maxmax kkyyi =º 2/or8/3 maxmax kkyyi =º 28and31=ij 20and24 nn uuvv ´´´´´=w ...... 11 niuv ii ,...,2,1,, = ; compulsion,0: impulsion,,0: max 1 ïþ ï ý ü ïî ï í ì ïî ï í ì ££ £³== å = kyy ssssvv ii n i iii ii a differential stackelberg game theoretic model… 173 copyright ©2020 assa. adv. in systems science and appl. (2020) and . denote an empirical qrs method is based on the hypothesis that , where the sets contain qualitatively representative strategies of the -th agent and of the principal in relation to the -th agent. it is supposed that the cardinality of the sets is small: where numbers are small. then is the qrs set of the game, and its cardinality is equal to . definition 4.1 [8]: a set is called the qrs set of a difference stackelberg game with precision if: (a) for any two elements it is true that ; (b) for any element there is an element such that . here are the principal's payoffs; , . the reason of the qrs set is that the strategies from it imply essential difference between the principal's payoffs, while this difference between any other scenario and one of the scenarios from qrs set is not essential. thus, instead of deliberately impossible complete enumeration or even enumeration of many scenarios it is sufficient to consider a few scenarios from qrs set. evidently, the constant should be small enough, say, not greater than 10% of the typical values of payoffs. for the evaluation we use an indicator . because the qrs method is empirical, it is important to choose appropriate initial scenarios as candidates to the qrs set. let us choose the following scenarios. in the case of compulsion the principal bounds from below the values of investment of the agents to the innovative development. suppose that the principal's strategies do not vary with time, and . the case corresponds to the absence of control; describes a soft control, and describes a hard control of the agents by the principal. in the case of impulsion the principal determines the grants allocated to the agents, and the total amount of all grants cannot exceed the whole budget . the principal's strategies are taken so that . the case corresponds to the absence of grants, the case describes a middle level of granting, and the case reflects a considerable amount of the grants. the strategies of the agents describe the outputs and the investments { }maxmax 0;:),( qqkkyqkuu iiiiiii ££££== nivvvvv iii ,...,3,2),...,,( 121 == { } { }.:),...,,(...;:),...,,(... 211211 iinniinn uuuuuuuuvvvvvvvv î=´´=î=´´= qrs ii qrs ii uuvvni ===" ;,,...,2,1 qrs i qrs i uv ; i i niuv qrs i qrs i ,1; = ,; 21 kukv qrs i qrs i == 21,kk qrsuuvv qrs n qrsqrs n qrs =´´´´´ ...... 11 õ = === n i nnqrs i qrs i kkuvqrsm 1 21 { })()2()1( ),(,...,),(,),( muvuvuvqrs = d qrsuvuv ji î)()( ),(,),( d>|| )( 0 )( 0 ji jj qrsuv l ï)(),( qrsuv j î)(),( d£|| )( 0 )( 0 jl jj )( 000 ,, )()( ljjj ji ),...,,,...,( )( 1 )( 1 )()( 10 )( 0 sss n ss uuvvjj = 0;,, >d= ljis d ),( ),(),( max 000 0000 ),(),( ),( 00 vuj vujvuj vuvu qrsvu ¹ î }, 2 ,0{)( maxmax n k n kyty ii îº 0=iy n kyi 2 max= n kyi max= s },,0{)( )2()1( iiii sssts îº åå == == n i i n i i ssss 1 )2( 1 )1( ; 2 0=is )1( ii ss = )2( ii ss = }, 2 ,0{ max max qqqi î 174 m. malsagov, g. ougolnitsky, a. usov copyright ©2020 assa. adv. in systems science and appl. (2020) . for simplicity it is assumed that the strategies do not vary with time. then we receive for small values the computer simulations are quite implementable. with consideration of the proposed strategies from qrs set the state variable is found analytically as 5. numerical calculation of the stackelberg equilibrium the computer simulations were implemented on personal computer with microprocessor a10 intel pentium g4620 with operative memory 4 gb using an object oriented programming language c# according to the proposed algorithm. an average time of one simulation for the construction a qrs set was less than three seconds. the received results were analyzed by the following criteria: (a) summary discounted payoff of the principal calculated by formulas (2.1) or (2.8); (b) index of system compatibility [12]: , where is the principal's payoff in the stackelberg equilibrium. this index demonstrates how necessary is principal's presence in a control system. the closer is the value of to one, the more the system is compatible, and the less is a need in the hierarchical control by the principal. for an initial qrs set the conditions (a) and (b) from the definition 1 are checked. the value of is selected so that a difference between the principal's payoffs for two any strategies from the initial qrs set does not exceed 10%. then given the model parameters the second condition from the definition 1 is checked. if necessary, the initial qrs set is extended or narrowed. in the case of extension the initial qrs set is added by new strategies. they correspond to the values situated between the previous ones (the strategies are supposed to be constant in time). then the computer simulations are implemented. we have realized about 150 numerical calculations for three agents that form the grand coalition. the model parameters varied in the following range: for compulsion; for impulsion. in all cases the index of system compatibility is equal to one, and the system is completely compatible. besides, in the case of impulsion on a short period of modeling (up to 4 years) both for the principal and for the agents it is not profitable to invest to the development of innovative technologies (learning courses). below the results of computer simulations for several sets of input data are presented. let the grand coalition includes three agents, and days (4 years). example 5.1: the values of model parameters are (thousand rubles per year). we varied the values the results for impulsion in the case are given in table 5.1. in the majority of calculations . table 5.1. results of numerical simulations (example 2) for impulsion }, 2 ,0{ maxmax n k n kki î .27nqrsm == 6<n .;)0()( 1 1 å = + == n i i k kt kkedtd 0j max * 0 / jji = ),(),,(maxmax ** 0 * 00max vujjvujj vvuu == îî i d ;2005 ££ c ;252 ££ b ;1000100 ££ a ;20020 max ££ q ;3000100 max ££ k 151 ££ z 90050 ££ s 1460=t ;501 =c ;1002 =c ;103 =c ;6000=s 900=a ,, maxqb .maxk 1,2,3)(0 == iyi btd º)( 2 100 3000 207587 b maxq maxk 0j a differential stackelberg game theoretic model… 175 copyright ©2020 assa. adv. in systems science and appl. (2020) for small values of the parameter b (less than 17 thousand rubles per year) it is profitable for the agents to develop new non-unique courses. when the value of parameter b decreases (it is less than 1 thousand rubles per year), the number of developed courses is growing up to the maximal value . when the value of parameter b increases, the agents stop the development. for a broad class of model parameters the payoff of the grand coalition of agents coincide with the principal's payoff. to create an incentive for development of new unique courses it is necessary to increase the considered period of time or introduce additional requirements of sustainable development of the system controlled by the principal. the condition of sustainable development is taken in the form where is a given constant that determines a minimal acceptable level of investment. in the case of compulsion the calculations considered (a) middle ( thousand rubles per year) and (b) considerable ( thousand rubles per year) principal's grants allocated to the agents, notice that in this setup the interests of the principal and the agents coincide. for all studied examples , and the results are similar to the case of impulsion. the only essential distinction is a difference between the payoffs of the players that is determined by the principal's grants allocated to the agents. as the interests of the principal and the agents coincide, the principal does not constraint the agents' investment from below. 6. conclusion a two-level control system of innovation in the universities is investigated. the nash equilibrium in the game of agents in normal form is built analytically for a specific class of input functions by means of the pontryagin maximum principle. an algorithm of building the stackelberg equilibrium in the game of the principal with the grand coalition of agents is proposed and implemented numerically by means of the method of qualitatively representative scenarios in simulation modeling. the following conclusions are received. 1. a demand parameter b in this model characterizes a degree of substitutability between two different courses in the initial moment of time. for the considered period of time (4 years) it is not profitable for the agents to develop new courses for big values of the parameter b. 2. for the chosen form of payoff functionals a need in control of the agents by the principal is absent. the interests of the principal and the agent coincide, and any compulsion is not required. 3. for successful promotion of the innovative learning courses in the universities it is necessary to formulate a condition of sustainable development as an obligatory one for the maxq å = ³ n i i qq 1 lim limq 30=is 600=is .2,1=i niyi ,..,2,1,0 =º 5 100 3000 117313 8 100 3000 73703 12.5 100 3000 398515 5 300 3000 63138 5 50 3000 96271 5 100 1000 117317 5 100 5000 117317 5 300 1000 63309 5 50 1000 96271 176 m. malsagov, g. ougolnitsky, a. usov copyright ©2020 assa. adv. in systems science and appl. (2020) principal. after that, the interests of the principal and the agents cease to coincide, and the principal should provide the conditions of innovations. the setup of dynamic problem of hierarchical conflict control in open-loop strategies is quite complicated, and many research directions stay to be open for further deep investigation. first of all, it concerns a theoretical substantiation of the existence of nash equilibrium. it is also useful to study the solution's behavior on the bound of the domain of feasible controls where the switching points are possible. the situation is complicated by the game theoretic setup due to which the mentioned peculiarities could differ for different players, and the right hand sides of their optimal control problems can be discontinuous. at last, other forms of the inverse demand function can be used. the main research tool, especially in a stackelberg game, was the method of qualitatively representative scenarios in simulation modeling. its idea is to receive a qualitatively acceptable description of the dynamics of a controlled system by means of the enumeration of a few impact scenarios found from reasonable considerations. in a stackelberg game a set of scenarios is considered to be qualitatively representative if the principal's payoffs for these scenarios differ essentially (more than a constant ), and for any other scenario exists a scenario from this set such the difference of the respective payoffs is less than (i.e. the difference is not essential). these conditions are based on the well known notions of internal and external stability of a set and seem to be natural. however, any objective estimates of the value of are still absent. from engineering and physical analogies it seems sufficient that the value of does not exceed 5-10% from the typical values of payoffs received in numerical simulations. certainly, this hypothesis needs to be refined and substantiated. acknowledgements this work was supported by the russian science foundation, project #17-19-01038. references 1. basar t., olsder g.j. (1999). dynamic non-cooperative game theory, siam, 1999. 520 p. 2. cellini r., lambertini l. (2002). a differential game approach to investment in product differentiation, journal of economic dynamics and control, 27(1), 51-62. 3. cellini r., lambertini l. (2004). private and social incentives towards investment in product differentiation, international game theory review, 6(4), 493–508. 4. dockner e., jorgensen s., long n.v., sorger g. (2000). differential games in economics and management science. cambridge university press, 2000. – 396 p. 5. dubina i. (2011). foreign investigations in the field of game-theoretic analysis of innovations, automation and remote control, 72(8), 1753-1765. 6. geras'kin m. i. (2018). modeling reflexion in the non-linear model of the stackelberg three-agent oligopoly for the russian telecommunication market, automation and remote control, 79(5), 841–859. 7. geraskin m. i. (2020). reflexive games in the linear stackelberg duopoly models under incoincident reflexion ranks, automation and remote control, 81(2), 302–319. 8. geraskin m. i. (2020). the properties of conjectural variations in the nonlinear stackelberg oligopoly model, automation and remote control, 81(6), 1051–1072. 9. gorelov m.a., kononenko a.f. (2015). dynamic models of conflicts. iii. hierarchical games, automation and remote control, 76(2), 264-277. 10. malsagov m. kh. (2019). sustainable management in universities, advances in systems science and applications, 9(3), 65-79. 11. mechanism design and management: mathematical methods for smart organizations (2013). ed. by d. novikov. new york: nova science publishers. – 163 p. 12. ougolnitsky g.a. (2011). sustainable management. new york: nova science publishers, 2011. – 288 p. d d d d a differential stackelberg game theoretic model… 177 copyright ©2020 assa. adv. in systems science and appl. (2020) 13. ougolnitsky g.a., usov a.b. (2016). solution algorithms for differential models of hierarchical control systems, automation and remote control, 77(5), 872-880. 14. ougolnitsky g.a., usov a.b. (2018). computer simulations as a solution method for differential games. in m.d. pfeffer and e. bachmaier (eds.). computer simulations: advances in research and applications (pp. 63-106). n.y.: nova science publishers. 15. pontryagin l. s., boltyanskii v. g., gamkrelidze r. v., mishchenko e. f. (1962). the mathematical theory of optimal processes. new york/london. john wiley & sons. – 369 p. 16. ugol’nitskii g.a., usov a.b. (2014). equilibria in models of hierarchically organized dynamic systems with regard to sustainable development conditions, automation and remote control, 75(6), 1055-1068. adv syst sci appl 2020; 03; 36-49 published online at https://ijassa.ipu.ru. transformation of industries in the conditions of new technological challenges ol'ga p. smirnova1*, lidiya m. averina1, alena o. ponomareva1 1) institute of economics of the ural branch of russian academy of sciences, ekaterinburg, russia email: olygsmirnova95@gmail.com abstract: the paper substantiates the significance of the problem of developing new branches of regional specialization. the mechanisms of transformation of regional specializations are investigated. the productive role of the concept of “smart specialization” for socio-economic and industrial development is noted. the empirical base of the study was russia’s statistics for 20102018. the method of correlation and regression analysis identifies the influence of the way of using digital technologies on the development of traditional industries. the factors affecting the functioning of the industrial sector of russia were identified. the current requirements and approaches to identifying the priorities of specialization of a region are given. as a result of the study, it was concluded that in order to transform traditional specialization industries and increase their competitiveness in modern conditions, their active interaction (integration and associated participation in production chains) with high-tech fast-growing industries is paramount. as expected, the introduction of new digital technologies will contribute to the growth of traditional industries, which will lead to an increase in their economic sustainability. keywords: industrial sector, industry 4.0, economic sustainability, digitalization, industries of regional specialization 1. introduction at the present stage of the development of the world economy, the role of the regional component is significantly increasing. oecd documents [27] and publications of leading scholars [7,10,32,36,37] emphasize that the global competitiveness of countries is currently largely ensured by the concentration in some territories of high-tech companies, research centers, modern production and innovation infrastructure. the most important issue in achieving sustainability is the difficulty of cooperative optimization of goals. the most important problem in achieving a stable level of economic sustainability of the industrial sector in the context of dynamic transformations is the inability to foresee, develop and implement an industrial policy that ensures joint optimization or mutual reinforcement of social goals (economic well-being, environmental quality and increasing production volumes) [22]. conceptual provisions regarding the role of regional economies are generally summarized by karpenko et al. [15]: “in the context of global integration and the transition to a ‘new economy’, the effective and sustainable development of the country’s general mega-system can be ensured only on the basis of the competitiveness of its internal regional economic systems”. the term “competitiveness” in relation to individual countries and regions began to be widely used in economic science at the end of the 20th century. michael porter [30] proposed the concept of a country’s competitiveness, and subsequently, individual elements of the concept are reflected in the calculation of the corresponding competitiveness index proposed by robert huggins associates consulting group [11]. the * corresponding author: olygsmirnova95@gmail.com transformation of industries 37 copyright ©2020 assa. adv. in systems science and appl. (2020) ideas of porter contributed to the development of theoretical and methodological foundations of regional competitiveness: the theory of industrial areas of marshall; the italian theory of industrial districts; the swiss theory of territorial production systems; the american clusters theory by porter and enright; the french academic doctrine – the theory of “poles of competitiveness” and a number of other scientific doctrines and directions. the competitiveness of the economy, in accordance with the provisions of modern economic theory, is determined by the advantages in the development of a particular industry or group of industries, individual economic entities or their combination [38], it can be stated that the competitiveness of a region is ensured by the effective functioning of the specialization industries. for russia, the problem of transforming regional specialization is extremely relevant, because the industrial sector has been a backbone industry. currently, ¼ of russia’s gdp is created in the industry. with this in mind, it seems appropriate to study the branches of the industrial sector of russia and assess the impact on their development of modern technological trends. in this context, it is important to understand that new rates of economic growth are ensured to a large extent due to a significant transformation of industry specialization, since it is the industries with higher growth rates that give impetus to accelerate economic development. the stated position led the authors to the formulation of the goal of this study, which consists in assessing the impact of new technological trends, including digital transformation of the manufacturing industry, on changing regional specializations of sectors. to this end, a hypothesis was put forward that there is a direct relationship between the volume of production in the industry and the level of development of its it infrastructure. that is, the created information and communication infrastructure in the industry becomes a determining factor in its transformation into the region’s industry of specialization. a set of academic tasks includes conducting an econometric analysis of the impact of the way of using digital technologies on the development of traditional industries; identification of the conditions for the formation of new industries, as well as consideration of problems and prospects for their development. 2. literature review regional specialization has an essential theoretical basis. a fundamental contribution to the substantiation of the concept of regional specialization was made by the theory of absolute advantages of adam smith [34] and the theory of comparative advantages of david ricardo [31]. later, the idea of specialization was developed by heckscher and olin in relation to trade [13]. in the 1980s, issues of regional specialization and the conditions for its transformation in french scientific literature were considered from the perspective of the theory of “poles of competitiveness”. aglietta and bouillet [1] note that the ability to aggressively transform industries depends primarily on the availability of competitive poles, their strengthening and renewal. at the present stage, issues of regional specialization are disclosed in the theory of spatial competition, new economic geography, industry markets, positioning the region in a competitive environment [15]. the ongoing global technological changes, the introduction of digital technologies have formed a request for a new model of regional development. in this model, “the ability to generate new ideas, recombine existing assets and knowledge, identify promising technological trajectories becomes the main competitive advantage” [35]. the new regional development model is based on the concept of smart specialization formulated by foray, david and hall. the concept of smart specialization draws on a number of previous studies [8-9]. these ideas were further developed, resulting in more than a hundred publications [7, 14,18,21,23-24] etc. 38 o.p. smirnova, l.m. averina, a.o. ponomareva copyright ©2020 assa. adv. in systems science and appl. (2020) an analysis of the main provisions of the concept of smart specialization allows making the following conclusion: in a generalized form, the concept identifies the potential for internal regional growth by finding unique types of economic activity that together provide the development of the region’s competitive advantages in the markets for products and advanced technologies. thus, smart specialization contributes to the activation of long-term structural changes in the economy with the transformation of regional specializations through the formation of new sectors. at the present stage, issues of regional specialization are disclosed in the theory of spatial competition, new economic geography, industry markets, and positioning the region in a competitive environment [15]. ultimately, based on research dated 2007-2009, foray et al. [8-9] formulated the concept of “smart specialization”, which consists in identifying and developing unique industries for the regional economy. scholars propose various approaches to the definition of unique fields of specialization, one of these options – the differentiated approach – is developed by the authors [17]. the continuous updating of regional specializations based on technological innovation in the search for new competitive specialization is a global trend in the global economy [29]. in these conditions, the search for methods and mechanisms for the transformation of regional specializations and the formation of new specializations, taking into account the smart specialization opportunities, are particularly relevant. there are a significant number of convincing arguments in this favor. under these conditions, the search for methods and mechanisms for the transformation of regional specializations and the formation of new specializations are of particular relevance. according to pilyasov, in the classical sense, regional specialization is the competitiveness (resilience) of one industry proven at the national level or a technologically coupled combination of industries that form the economic profile of a given region. based on current trends in the development of the world economy, pilyasov considers it necessary to revise the traditional definition of regional specialization (and, accordingly, the methods for calculating it). first of all, this is because of accounting for the global trend – a transition from macro-specialization to specialization as a result of a decrease in spatial scale to the level of part of the region. this transition is caused by the fact that the emerging need for a higher intelligence of production is being met due to the greater localization of the desired section of the process chain. another factor that should be taken into account is the need to assess not only national but also global competitiveness in order to identify regional specialization. diversification of a large branch of specialization leads to an increase in the number of regional specializations, or, at a minimum, to the emergence of micro-specializations. the russian experience shows that the transformation of regional specializations is carried out in the course of parallel processes: the consistent diversification of existing regional branches of specialization with a change in their structure; the emergence in the regions within the framework of the existing regional specialization of new highly efficient industries not in the regional but in the territorial format (new industries in technology parks, special economic zones, clusters, etc., activities of which, under certain conditions, contribute to the formation of a new economic competitive specialization of the region) [33]. the study by kutsenko et al. [19] highlights the shortcomings of the existing methodology for calculating regional specialization. this is mainly due to shortcomings in the system of state statistical reporting. the indicators used in some cases do not reflect the real situation, and, therefore, distort the picture of regional specialization. recently, in the literature, preference has been given to indicators of revealing comparative advantages and economic complexity [12,14]. a review of studies in the field of regional specialization showed that there is a prevailing opinion about the need to revise the classical definition of regional specialization and, accordingly, adjust its calculation methods. transformation of industries 39 copyright ©2020 assa. adv. in systems science and appl. (2020) 3. materials and methods the aim of this study is to assess the impact of new technological trends, including digitalization of the industry on the transformation of regional specializations of sectors. in identifying regional promising specializations, an approach should be taken based on accounting for the potential place of the regional industrial sector in the national market of goods and technologies, and for a number of commodities – positioning in the system of international markets. this approach is complemented by taking into account new technological trends in the sectors of current and potential regional specialization; diagnostics of promising technologies with a potential to form new sectors of economic growth; with a reference to the formation of new segments of the knowledge economy. as part of the study, it is proposed to use a number of indicators of the economic sustainability of industrial enterprises. these indicators are analyzed in dynamics and segmented into five groups: market potential; production potential; innovation potential; financial potential; hr potential (fig. 1). initial data on the indicators of each segment were collected from rosstat for the period from 2010 to 2018. fig. 1. indicators for assessing the economic sustainability of the industrial sector one of the main methods for assessing economic sustainability is the method of comparing the actual values of indicators with their threshold values. for convenience, assessing the degree of remoteness of indicators from their threshold values, indicators are reduced to a dimensionless form using normalizing methods and displayed in a single coordinate system. the choice of normalizing determines, as a rule, the dynamic range of visualization of the results [25]. normalizing was done according to mityakov [16]: – for the ratio of “not less than the threshold value”: (1 / )/ ln(10/3) log10/3( / ) , if / 1;2 , if / 1.2 a x a x x a x x a − −  =   (1) – for the ratio not exceeding threshold value: 40 o.p. smirnova, l.m. averina, a.o. ponomareva copyright ©2020 assa. adv. in systems science and appl. (2020) (1 / )/ ln(10/3) log10/3( / ) , if / 1;2 , if / 1.2 a x a x x a x x a − −  =   (2) where x is the actual value of the indicator; a is its threshold value; 𝑥 ̅ is normalized value. the interpretation of the comparison of the obtained normalized estimates is as follows: 𝑥 ̅=1 corresponds to the equality of the indicator and its threshold value, 𝑥 ̅<1 indicates a threat to economic stability (the indicator has not reached its threshold value), 𝑥 ̅>1 corresponds to the indicator reaching its threshold value, i.e. in this case, the indicator is in a stable zone. one of the main indicators of the operational stability of the industrial sector is considered to be the market component, namely the growth of production output. to this end, a hypothesis was put forward that there is a direct relationship between the volume of production in the industry and the level of development of its it infrastructure. that is, the created information and communication infrastructure in the industry becomes a determining factor in its transformation into the region’s industry of specialization. to test the hypothesis, the linear regression modeling method is applied, the significance of the obtained equations is established using the coefficient of determination (r2). the equation of the linear regression has the form: (3) where y is the average value of the effective indicator; x is the influencing factor; a and b is regression coefficients. the coefficients of the regression equation can be interpreted in this case as follows. if a> 0, then with increasing the coefficient x, fiscal security increases, and if a <0, then with increasing the coefficient x it decreases. to check the adequacy of the equation, the coefficient of determination is calculated: 2 2 2 )( 1 ( ) ii i a by x r yy − − = −  − (4) where yi is the level of socio-economic sustainability in the i-th year; xi is factor assessment in the i-th year; is the average level of socio-economic sustainability over several years. to verify the adequacy, the fisher test was used: 2 2 ( 2) (1 ) r f n r = − − (5) where r2 is the coefficient of determination; n is the sample size. if the value of f exceeds the critical value in absolute value, then the equation is adequate and can describe existing patterns. the obtained regression models are used to predict possible prospects for the development of the industrial sector depending on the expected values of sectoral technological development. in this forecast, the point type of the forecast was conventionally applied to visually indicate the trend of the digital transformation of industry. 4. results as part of the study, development indicators for the industrial sector of russia, including enterprises of processing and construction industries (appendix a), were selected. the indicators are defined from the perspective of the industry’s characteristics for five development potentials – market, production, innovation, financial, and hr. from appendix a, it is seen that for many indicators there is a negative trend of change. to calculate the economic sustainability of industrial enterprises, it is necessary to bring all indicators to a single normalized value. the calculations on normalizing the indicators for assessing the economic sustainability of the industrial sector make it possible to form curves transformation of industries 41 copyright ©2020 assa. adv. in systems science and appl. (2020) of indicators of changes in the economic stability of the industrial sector of russia for the five considered potentials as shown in fig. 2. 0,250 0,300 0,350 0,400 0,450 0,500 0,550 0,600 0,650 0,700 2010 2011 2012 2013 2014 2015 2016 2017 2018 market potential production potential innovation potential financial potential hr potential economic security (integral indicator) fig. 2. the dynamics of indicators of economic sustainability of the industrial sector of russia for 2010-2018 fig. 2 shows that the sustainability of the industrial sector is most determined by the market potential of economic security. it is important to note that the production potential of the industry is more stable and less affected by the crisis of 2014, in contrast to indicators in the context of finance and innovation (innovation and financial potential). as is known, the introduction of innovations and modern technologies is one of the main components of economic growth [6,20,26]. one of the challenges of technological development at the moment is digitalization [2,4,5]. to assess the level of digitalization in the industrial sector, the authors consider the dynamics of a number of indicators characterizing this process. appendix b provides information on the use of information and communication technologies. in 2017, the share of organizations using personal computers in construction is 88.9%, and in processing – 95.5%. currently, digitalization processes affect the mineral wealth sector of the economy to the greatest extent and the construction sector to the lowest. it is important to note that in terms of the use of personal computers, there is a negative trend in all considered sectors. in this study, the authors will identify the trend component that describes the impact of the digitalization level of industries on the development of the industrial sector. for this purpose, paired linear regression equations are constructed that describe the time series of the main growing trend under consideration (figs. 3-6). as this level, the authors determined the average value of the indicator “the use of information and communication technologies in organizations” as a percentage of the total number of examined organizations of the corresponding type of activity, which is an exogenous (influencing) variable. the endogenous (dependent) variable (y) is the value of the volume of own-produced shipped goods, own works and services provided. 42 o.p. smirnova, l.m. averina, a.o. ponomareva copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 3. the dynamics of the digitalization trend for the industrial sector as a whole, 2011-2018 fig. 4. the dynamics of the digitalization trend for the mineral wealth industry, 2011-2018 fig. 5. the dynamics of the digitalization trend for the processing industry, 2011-2018 fig. 6. the dynamics of the digitalization trend for the construction industry, 2011-2018 analyzing the obtained equations, one can conclude that the relationship between the level of production and the level of digitalization of the industry is close. the highest value of the coefficient of determination was found in the processing industry of the economy, a weaker value is observed in the construction industry. this conclusion can also be drawn by examining the values of the coefficients of the dependent variable: β = 1,437.7 in processing, β = 522.2 in mineral wealth mining, to the lowest degree the level of digitalization affects construction (β = 166.8). the variability of the development of the industrial sector of russia from the level of digitalization is shown in table 1. table 1. variability of the digitalization status of the industrial sector of russia indicator regression equation determination coefficient fisher test predictive critical total y=2,731x – 130,987 0.9739 2.1629 2.0048 mineral wealth mining y=522.09x – 26,503 0.7043 0.8758 2.7764 processing y=1,437.7x – 78,052 0.9289 1.6503 4.3026 construction y=166.81x – 6,480 0.5519 0.6211 3.1824 the significance of the obtained equations was checked using the fisher test. based on the calculated critical value of the fisher test, the authors accept an alternative hypothesis and conclude that there are statistically significant differences in the frequency of outcome depending on the impact of the factor. according to the established pattern, there is a rather strong dependence of the volumes of own-made products on equipping the production environment of it infrastructure. transformation of industries 43 copyright ©2020 assa. adv. in systems science and appl. (2020) 5. discussion the advent of new technologies leads to the modernization of existing industries and the emergence of new ones. traditional industries of regional specialization of the industrial sector are supplemented by new ones such as ict, industries using biotechnology, photonics, robotics, artificial intelligence, etc. the authors present the results of a rating of the costs of russian regions for ict. in 2013, the cost of ict in russia as a whole amounted to 69.5 billion rubles. by 2019, this figure increased by 2.3 times. note that in industrialized regions the growth rate of this indicator is even higher, for example, in the sverdlovsk region over the same period, ict expenses increased by 2.5 times (from 1.06 to 2.73 billion rubles); in the chelyabinsk region, ict expenses increased by 4.5 times (from 0.26 to 1.14 billion rubles). 40000 60000 80000 100000 120000 140000 160000 180000 2013 2014 2015 2016 2017 2018 2019 million rubles fig. 7. dynamics of costs for information and communication technologies in russia these trends suggest support for the continued use of ict. the presence of an information barrier in the growth of companies has been repeatedly noted in studies [39]. the dependencies obtained earlier allow comparing the growth of the real sector of the economy with the growth rate of using ict in these sectors. here is the forecast for the development of the russian industry as a whole, as well as in mineral wealth mining, processing and construction (figs. 8-11). this forecast is based on the obtained relationships of the influence of the digitalization level on the development of the industrial sector. the digitalization trend, as mentioned above, in this study is described by the average of indicators such as the share of enterprises in the relevant industry using personal computers, servers, local area networks, global information networks, and the share of organizations that have a web page. the forecast for the development of the manufacturing industry as a whole shows that, while maintaining the average growth rate of the digitalization in the industry (2.2% per year), the predicted volume of industrial goods and services by 2023 may amount to more than 76.9 billion rubles, and the average growth of this indicator will be more than 8% a year. 44 o.p. smirnova, l.m. averina, a.o. ponomareva copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 8. forecast of the impact of the digitalization trend on the industrial sector as a whole for the period of 2019-2023 fig. 9. forecast of the impact of the digitalization trend for the mineral wealth industry for the period of 2019-2023 fig. 10. forecast of the impact of digitalization trend for the processing industry for the period of 2019-2023 fig. 11. forecast of the impact of the digitalization trend for the construction industry for the period of 2019-2023 the digitalization trend of the mineral wealth industry has somewhat slower dynamics of changes and a lower connection with the resulting volume indicator. while maintaining the existing average growth rate (about 1.5% per year), by 2023 the volume of output in the mineral wealth mining industry will amount to more than 14.24 billion rubles per year, and the average growth rate in the industry is about 7% per year. the highest degree of digitalization is noted in the processing industry, and here the predicted growth values will be about 2.2% per year. this will allow the industry to reach a production level of over 52 billion rubles per year by 2023, provided that 90% of organizations in this industry will use digital technology (under this condition, the industry will show average growth rates at almost 9%). the construction industry has shown the least degree of digitalization. the predicted share of organizations using digital technologies in the industry by 2023 will be less than 70%. the output volume will approach 5 billion rubles, and the average growth rate in the construction industry will be about 2.8% per year. the obtained data on the dynamics of industrial development make it possible to formulate a forecast for the structure of the real sector of the national economy (fig. 12). transformation of industries 45 copyright ©2020 assa. adv. in systems science and appl. (2020) 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 mineral wealth mining processing construction forecast fig. 12. real and forecast structure of the real sector of the economy fig. 12 shows a gradual decrease in the share of the construction industry, as well as of mineral wealth mining, which is caused by lower development rates of these industries compared to the processing industry. that is, the influence of new technologies and the digital transformation of the manufacturing industry contributes to positive structural changes in the economy. an increasing degree of influence of new technologies, including ict, will inevitably contribute to the transformation of the manufacturing industry. industries of regional specialization will also be transformed. the study showed that existing technological trends must be taken into account, as they have a significant impact on the development of the economy. the results of this study suggest that the feature of new industries is rapid growth, accumulation and combination of various types of resources, of great importance for their development is scientific and industrial cooperation, interregional interaction and scientific collaboration [3]. one of the popular tools for the development of new technologies is such a kind of strategic planning and management of the regional economy as smart specialization. this concept makes it possible to single out individual specialized activities that do not correspond to the profile type of the region, but represent a unique highly specialized niche in the markets of high-tech products, services and technologies, i.e. creation of effective promising regional specializations [28]. 6. conclusion this paper identifies the factors that influence the transformation of regional specializations, taking into account the global challenges of technological development. the role of digitalization in increasing the competitiveness of industries with promising economic specialization of the region was identified, which is due, inter alia, to cooperative interindustry interaction and the level of development of innovative infrastructure. the results obtained confirm the hypothesis that the created information and communication infrastructure in the industry is becoming a determining factor in its transformation into the region’s industry of specialization. that is, digitalization in the aspect of regional development is a mechanism that allows certain sectors to achieve competitiveness. therefore, there is an objective need to take into account the level of digitalization in the methods for identifying regional specialization. these methods should account for not only traditional approaches based on the inter-regional division of labor and the rational 46 o.p. smirnova, l.m. averina, a.o. ponomareva copyright ©2020 assa. adv. in systems science and appl. (2020) distribution of productive forces, but also the priority of high-tech industries and their digitalization level. thus, the modern methodological approach to identifying the specialization of the region should be guided to a greater extent by the qualitative indicators of development, their technological transformation of the economy and the current market situation. acknowledgements the paper was prepared in accordance with the research plan for the laboratory for spatial development of the territory at the institute of economics of the ural branch of the russian academy of sciences for 2019-2021. references 1. aglietta, m., & boyer, r. (1982). poles de competitivite, strategie industrielle et politique macro-economique. paris. couverture orange cepremap, no. 8223. 2. akberdina, v., kalinina, a., & vlasov, a. 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(2018). the information problem on the way to becoming a “gazelle.” in proceedings of the european conference on innovation and entrepreneurship, ecie (vol. 2018-september, pp. 394–401). appendix а table. indicators of the potential of the industrial complex of russia (for enterprises of processing and construction industries) indicator 2010 2012 2014 2016 2018 market potential dynamics of income, % 100.00 111.11 108.30 104.15 105.00 ratio of the production index and the index of changes in numbers (in % to the previous year) 107.6 104.2 103.5 101.3 102.9 index of changes in labor intensity and index of changes in the working time in % of the production index (in % to the previous year) 110.5 96.4 96.7 99.4 99.2 costs of production and sale of products per 1 ruble of manufactured products, kopecks 75.3 77.2 80 80.1 80 production potential share of investments aimed at reconstruction and modernization in the total volume of investments in fixed assets 18.8 19.5 17.4 16.3 15.5 index of physical volume of investments in fixed assets aimed at reconstruction and modernization, % 103.5 107.1 92.5 95.1 101.6 average age of machinery and equipment available at the end of the year 11.1 11.5 11.2 11.3 12.2 index of changes in capital productivity, % 100.9 104.2 88.7 101.2 97.0 innovation potential volume of innovative goods, works, services, billion rubles 1.244 2.873 3.580 4.364 4.516 index of physical volume of investments in fixed assets aimed at reconstruction and modernization, % 103.5 107.1 92.5 95.1 92 number of advanced manufacturing technologies used, units 236 350 439 548 524 number of new technologies (technical achievements) acquired by organizations, units 119 63 28 14 48 financial potential return on sales, % 10 8.6 7.3 7.6 12.3 return on assets % 6.7 6.1 2.5 5.9 6.4 equity to total assets ratio 0.52 0.48 0.4 0.42 0.48 current ratio 1.34 1.25 1.21 1.24 1.01 hr potential average annual number of employees in organizations, million people 16.901 17.166 17.111 16.941 17.06 labor productivity index 105.2 104.8 102.5 100.2 95.9 index of changes in capital-labor ratio, % 102.2 99.3 113.5 100.9 103.00 average monthly nominal accrued wages per employee, thousand rubles 9.078 24.512 29.511 34.592 40.722 source: rosstat. transformation of industries 49 copyright ©2020 assa. adv. in systems science and appl. (2020) appendix b table. digitalization indicators of the industrial sector of russia (at the beginning of the year) indicator 2012 2014 2016 2018 pcs total 94.1 94.0 92.3 92.1 mineral wealth mining 94.6 95.6 93.1 90.7 processing 97.3 97.2 97.1 95.5 construction 96.0 94.3 92.9 88.9 servers total 19.7 19.7 47.7 50.6 mineral wealth mining 30.0 30.4 69.9 69.1 processing 25.9 25.2 67.4 74.5 construction 20.7 19.4 61.2 58.0 local area networks total 71.3 73.4 63.5 61.1 mineral wealth mining 85.1 86.3 78.3 73.3 processing 84.2 85.2 76.6 76.2 construction 82.7 81.6 68.3 59.9 global information networks total 85.6 88.7 89.0 89.7 mineral wealth mining 91.8 92.9 91.8 89.0 processing 94.3 95.2 96.0 94.5 construction 92.5 92.3 91.4 87.1 web pages total 33.0 41.3 42.6 47.4 mineral wealth mining 30.0 36.8 37.2 39.7 processing 53.3 57.9 57.5 63.8 construction 34.3 38.7 40.1 38.7 source: rosstat. 1. introduction 2. literature review 3. materials and methods 4. results 5. discussion 6. conclusion acknowledgements appendix а appendix b paper title (use style: paper title) adv syst sci appl 2019; 04; 87-93 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/830 exchange rate of the ruble modeling: macroeconomic level anton kuzmin department of data analysis, decision-making and financial technologies the financial university under the government of the russian federation moscow, russian federation e-mail: a_kuzmin@rambler.ru received may 16, 2019; revised december 12, 2019; published december 31, 2019 abstract: as underlying determinants of the author’s model are used capital flows, import and export operations, elasticities of cross-country trade, terms of foreign trade, indexes of real gdp, indexes of domestic and export prices, solutions of microagents. the model was verified on data from the crisis period in russia 2013-2015. the simulation results are applied to the analysis of contemporary monetary policy of russia. keywords: exchange rate of the ruble, modelling, balance of payments, currency crisis. 1. introduction the monetary approach to exchange rate modeling is important: the models of frenkel [1], mussa [2, 3], hooper-morton [4] and others. model with sticky prices was created by dornbush [5, 6]. the two-country model with maximization of individual preferences was created by stockman [7]. representative of another class of dynamic stochastic general equilibrium (dsge) models is the monopolistic competition model obstfeld-rogoff [8] of exchange rate of two open economies. it should be noted that economic fundamentals takes an important place both in these models and in the model clark-macdonald [9, 10]. a number of researches was devoted to the dynamics of the ruble: strizhkova [11], mirkin [12] and others. however, to construct a mathematical fundamental determinants model of the russian ruble new developments are necessary. we have to base on the study of the fund movements of the russian balance of payments. we also need to engage in modeling the terms of foreign trade as one of the most important fundamental determinants. as a result, it's necessary to investigate the equilibrium medium-term dynamics of the ruble in the context of the currency crisis in 2013-15. in this paper the proposed dynamic model of the equilibrium exchange rate of russian ruble is a development of previous author’s research [13, 14]. it takes into account in the medium-term period capital flows, import and export operations, elasticity of cross-country trade, terms of foreign trade, indexes of real gdp, indexes of export and domestic prices, solutions of micro agents. the constructed exchange rate will balance the balance of payments. as a consequence, it will balance the offer of national and foreign currencies on the cross-country monetary market. the main objective of the modeling was to identify the dynamics of the ruble exchange rate from the system of major macroeconomic factors. this system must necessarily include the factors of the monetary sector, as well as the factors of the real sector. modeling will be mailto:a_kuzmin@rambler.ru 88 a. kuzmin copyright ©2019 assa adv. in systems science and appl. (2019) based on the principles of the international flows equilibrium exchange rate (ifeer), developed in the works of the author. but in this article we will present a new model. 2. a dynamic model of the exchange rate of the ruble the exchange rate policy of the bank of russia is the regime of independent floating according to the classification of the international monetary fund. russia is also characterized by special public attention to the dynamics of the nominal exchange rate of the national currency. as a result, the importance of the bank of russia's exchange rate policy is typical in modern conditions. initially, we consider all real market transactions at nominal exchange rates , (1, )ie i n on the domestic currency market that occurred in a certain period of time. then iii rde ,, in i-th transaction are: the nominal exchange rate, the amount in the determined foreign currency and the amount in the national currency respectively. these variables are linked by ratios: iii rde  , and therefore, iii dre / . in the russian foreign exchange market, a significant part of transactions takes place in us dollars. transactions in other currencies, such as the euro, the british pound are directly linked to these prevailing rates through the system of cross-rates in the international and local market. thus, we further consider the us dollar as a foreign currency and the direct quotes of us dollar to russian ruble. it should be noted that the contribution of each market transaction is different. it depends on the volume of the transaction. therefore, we define the synthetic value of the exchange rate for the certain period as the sum of exchange rates, weighted by amounts in determined foreign currencies: 1 1 n i in i j j d e e d     , (2.1) using the definition of the exchange rate in i-th transaction, we get from (2.1):          n j j n i in i i i n j j i d r d r d d e 1 1 1 1 so, on a conceptual level, we can disaggregate the country's balance of payments flows:          cbkca cbkca ddd rrr e (2.2) in this formula (2.2) we consider flows in the respective currencies that came to the domestic currency market: as the supply of the foreign and national currencies in accordance with current account operations amount of funds with index ca, as the supply of the foreign and national currencies in accordance with capital operations amount of funds with index к, as a result of currency interventions of the bank of russia amount of funds with index св. this formula takes into account such important components as export revenue, demand for imports, demand for foreign assets and so on. exchange rate of the ruble modeling 89 copyright ©2019 assa adv. in systems science and appl. (2019) conceptually, formula (2.2) can be applied not necessarily to analyze the dynamics of the russian ruble, but in other cases as well. in the work of the author [6], its adapted version was used to model the dynamics of equilibrium exchange rates under the equality of economies. however, in this paper we will conduct a modeling of the ruble exchange rate in the conditions “open economy rest of the world”. further two-period model will be considered in periods t, t-1. it should be noted that russian exports affect the economic situation in the country. of course, the dynamics of actual export prices will directly determine the state of the domestic foreign exchange market. amount of foreign currency in the market at period t will be determined by the volume of exports decisions of exporters at period t-1. and the most important factor in increasing the physical volume of exports is the real terms of international trade. we define these conditions in this situation as the division of exports and domestic prices adjusted to the nominal exchange rate: 1 * 1 11     t t tt r p p ee , where pt the internal price index (e.g., cpi), * 1tp the actual export price index. here the terms of international trade can be structurally correlated with the real exchange rate, used in the author's work [13]. thus, the amount in foreign currency at period t that came to the domestic foreign exchange market as part ek of export earnings in actual export prices  tp is equal to zr t x x t x tet ca eqqkpd )()( 1 1 1 1 1 1        (2.3) where te qconstk , the index of total real output (e.g., real gdp). let’s give some explanations to the formula (2.3). index z values response to the state of the terms of international trade at period t-1. the part 1 11 1 1( ) x x x e t tk q q    reflects that the physical export is a part of averaged total real output not only in current period, but also in the previous time. at the same time averaging method in 1 11 1 1( ) x x x e t tk q q    shall not have a significant influence on the final result from economic positions. this is due to the insignificance in the real output changes, compared to possible fluctuations of price indices in the medium term. rate  shows a "slightly higher" non-negative growth of national exports in relation to imports as a function of real output. the main reason is the restriction of domestic demand and thus the need to realize the growing domestic output due to the growth of exports. the parameter x=(z-y) associates parameters y and z in the dependencies (2.3) and (2.4). next, make an assumption about the import dependencies: the consumption of imports at time t by residents with proportion ik of their income )( 1 1 1 1    x x t x tt qqp is represented both current income and income in the previous period as follows yr t x x t x tti ca eqqpkr ))(( 1 1 1 1 1     (2.4) the parameter у also values response to the state of the real terms of international trade. 90 a. kuzmin copyright ©2019 assa adv. in systems science and appl. (2019) of course, as it can be seen from formula (2.2), capital flows directly determine the dynamics of the exchange rate. at the same time, a significant number of exchange rate models ignore this problem, while remaining at the level of the trade balance. however, within the framework of the proposed ifeer modeling the capital flows can be involved in the process. we accept the hypothesis for the functional dependence of capital outflows: this is the demand for international assets of russian residents for savings purposes and it is part of total income in domestic prices: 1 1 1 1 1 ( ) ( )( ) k cb x r yx x t t t tk r r p k q q e          , (2.5) where constk k  . it will also assumes that in (2.5) and further in (2.6) transactions of the bank of russia are considered. for the functional dependence of capital inflows, we accept that it reflects the inflows of portfolio and direct investments. according to this international speculators and investors want to buy a part of the real national product in international prices. another important factor is the terms of international trade because the changes in them will certainly affect the investment climate: 1 11 1 1 1 ( ) ( ) ( ) k cb x r zx x t t t tk d d p k q q e            (2.6) where constk k  , 0 . dependence (2.6) suggests that the impact of national product growth is increased more than proportionally due to psychological factors and the implemented policy of import substitution, which is carried out by the government of the russian federation in recent years. in this dependence it is expressed by the parameter  in part     1 1 1 1 1 )( x x t x tk qqk . we will directly substitute dependencies (2.3) (2.6) into the formula (2.2):                                        1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 )()( )()( ))(( ))(( zr t x x t x ttk zr t x x t x tte yr t x x t x ttk yr t x x t x ttit eqqpk eqqpk eqqpk eqqpke   exchange rate of the ruble modeling 91 copyright ©2019 assa adv. in systems science and appl. (2019) 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 ( )( ) ( ) ( ) ( ) ( ( ) ) . x r yx x t t t t i k x r zx x t t t t x x x e t t k p q q e k k p q q e k q q k                                   (2.7) based on economic logic, we can assume that ( ) 0   and the part )( 1 1 1   õ t õ õ t qq is stable, comparing to other model variables in the medium term. this makes it possible to consider the component as a constant in (2.7): constk kqqk kk x k x x t x te ki           1/ 1 1 1 1 )( ))(( )(  . thus as you can see, this component k' is the more stable, the smaller the contribution of the current balance flows compared to the capital flows. formula (2.7): / 1 1 1 1 1 1 / 1 1 * 1 1 1 1 1 1 ( ) ( ) ( ) ( ) . ( ) ( ) x t t x r õx x t t t t x t x õx x t t t t t t k p e p q q e k p p p q q e p                    then: x x t t tx t t tx tt q p p kq p p kee                         1 1 1 1/1/ 1)(  (2.8) and then after intertemporary separation of variables and notation 1 x  : ./1/        t t tx t t t t q p p kq p p ke (2.9) 3. verification of the model of course, to verify the model most appropriate would be the crisis period 2013 2015, associated primarily with the significant depreciation of the ruble. the starting point of the research period is december 2013. at the moment, both the domestic macroeconomic situation and the international environment were characterized by stability. the exchange rate of the ruble according to the bank of russia at that time was 32,73 to the us dollar. 92 a. kuzmin copyright ©2019 assa adv. in systems science and appl. (2019) we used the cpi (consumer price index) as the model variable p [15], the prices of blend crude oil as the model variable p* (us dollars per brent-mix barrel, intercontinental exchange, [16]), the index of the real gdp as the model variable q [17]. our setting used the least squares method with normalization of the investigated variables. thus, parametric minimization problem            t t tt e ee 2 )nominal( )nominal()( min   (3.1) was solved here under the conditions 0 . the exchange rate e(nominal) is the nominal exchange rate usd/rur [17], according to the bank of russia calculations monthly at the end of the each period. solution of parametric optimization (3.1) in accordance with numerical simulation is 5,0 . here we use the updated data compared with the data of work [14]. in figure 3.1 (author's calculations, monthly updated data) presents the dynamics of the exchange rate of the russian ruble against the us dollar in accordance with the result formula of our research (2.9) e(theor) and the nominal exchange rate e(nominal). fig. 3.1. the nominal and calculated exchange rates of russian ruble usd/rur (2013 2015, monthly updated data). as results, the average of absolute normalized deviations of the calculated and nominal exchange rates is 0.3% and average of normalized deviations is 3%. this allows us to speak about the high quality of the developed mathematical model. 4. conclusion the results of the modeling generally confirm the conclusions made in the media about the causes of the currency crisis in this period. a serious drop in oil prices and closely correlated the prices of natural gas and processed products was the main reason for the actual double devaluation of the ruble. however, it should be noted that the internal situation in the form of accelerated inflation and gdp decline has made a serious contribution to this process. of course, a statement of this fact cannot say about the adequacy of the situation. in the transition to a floating exchange rate such a rigid dependence of the ruble exchange rate from the foreign economic environment requires increasing degrees of intervention by the bank of russia on the domestic forex market. references 1. frenkel j. (1976). a monetary approach to the exchange rate: doctrinal aspects and empirical evidence, the scandinavian journal of economics, 78(2), 200 – 224. exchange rate of the ruble modeling 93 copyright ©2019 assa adv. in systems science and appl. (2019) 2. mussa m. (1976). the exchange rate, the balance of payments and monetary and fiscal policy under regime of controlled floating, scandinavian journal of economics, 78(2), 229 – 248. 3. mussa m. (1984). the theory of exchange rate determination, in exchange rate theory and practice. in j.e.o. bilson and r.c. marston, eds. (pр. 13 – 78), chicago: university of chicago press, nber. 4. hooper p. & morton j. (1970). fluctuations in the dollar: a model of nominal and real exchange rate determination, international finance discussion paper (board of governors of the federal reserve system), october, №168. 5. dornbush r. (1976). expectations and exchange rates dynamics, journal of political economy, 84, 1161-1176. 6. dornbush r. (1976). capital mobility, flexible exchange rate and macroeconomic equilibrium, in recent developments in international monetary economics, e. claasen and p. salin, eds., (pр. 261-278). north-holland 7. stockman a.c. (1980). a theory of exchange rate determination, journal of political economy, 88, 673-698. 8. obstfeld m. & rogoff k. (1995). exchange rate dynamics redux, journal of political economy, 103(3), 624-660. 9. clark p.b. & macdonald r. (2000). filtering the beer a permanent and transitory decomposition / imf working paper 00/144 washington: international monetary fund. 10. clark p.b. & macdonald r. (1998). exchange rates and economic fundamentals: a methodological comparison of beers and feers, imf working paper 98/67, washington: international monetary fund, march. 11. strizhkova l.a. (2017). the relationship between inflation, exchange rate and parameters of economic policy (on example of russia), bulletin of the institute of economics of the russian academy of sciences, 5, 156-176. [in russian] 12. mirkin y.м. (2018). future dynamics of russian ruble exchange rate, finance, money, investment, 67(3), 3-7. [in russian] 13. kuzmin a. (2018). equilibrium exchange rate modeling, in eleventh international conference on management of large-scale system development (mlsd 2018), https://ieeexplore.ieee.org/document/8551843/metrics#metrics, publisher: ieee, electronic isbn: 978-1-5386-4924-4, doi: 10.1109/mlsd.2018.8551843. 14. kuzmin a. (2015). exchange rate modeling: the case of ruble, review of business and economics studies, 3(3), 39-48. 15. federal service of state statistics of russia [www.gks.ru]. 16. information agency *bloomberg*. 17. bank of russia [www.cbr.ru]. https://ieeexplore.ieee.org/document/8551843/metrics#metrics microsoft word 897 article text, copyedited.doc adv syst sci appl 2020; 03; 91-104 published online at https://ijassa.ipu.ru. classifying amorphous polymers for membrane technology basing on accessible surface area of their conformations oleg miloserdov1* 1) institute of control sciences, russian academy of sciences, moscow, russia e-mail: oleg_milos@mail.ru abstract: almost 400 amorphous polymer materials used in membrane gas separation technology are clustered on the basis of the shape of their polymer chains conformations. obtained clusters, which rely solely on the geometry of polymer chains and not on the chemical class (polyamides, polyacetylenes, etc.), are shown to discriminate polymers with respect to their transport properties, in particular, the coefficient of diffusion. the method proposed consists of several steps. firstly, realistic conformations of large polymer macromolecules are constructed using the program code developed in the rdkit environment for python. then, polymer conformations are characterized by a curve that relates the “accessible surface area” (i.e., the contact surface between the spherical model of a macromolecule and a spherical “probe”) to the radius of this probe, and also seven similar curves, which relate the polarized (neutral, positively or negatively charged, etc.) accessible surface area to the radius of the spherical probe that represents the variety of penetrant gases. an improved algorithm for surface area calculation maps out the outer surface of the macromolecule to eliminate its influence. the curves are averaged between ten polymer conformations to obtain more robust figures. finally, agglomerative clustering is used to separate different polymers in the space of these curves that align their accessible-surface-area-related quantities against the probe radius. the proposed classification of polymers can be used to develop more precise predictive models of polymers’ transport properties for the theory-guided and computer-aided materials design. keywords: machine learning, agglomerative clustering, qspr, molecular modeling, polymer membranes, gas separation 1. introduction membrane technologies of gas separation are extensively used all over the world in hydrogen and oxygen production, natural gas purification and carbon dioxide separation. despite the great variety of materials used in liquid, metal, ceramic and polymer membranes, there is still a high need for a scientifically-based search for new materials, especially for polymer membranes. the development of mathematical models to predict properties of polymers from their structure allows saving financial and time resources during the development of new membrane materials. unfortunately, the mathematical models for transport properties of polymeric materials so far have either insufficient predictive power or are limited to specific polymers or their classes. primarily, this is due to the fact that transport properties of polymers used for membrane gas separation are determined by the geometry of their polymer chains, characterized by a wide variety of chemical structures and methods of their organization (spatial isomerism, statistical sequence of polymer units, etc.). several approaches are known from the literature to predict transport properties of polymers. those most popular and well-founded are based on computer simulation of atoms and their interactions [19,8]. first of all, they include molecular dynamics (md) and the grand canonical monte carlo method (gcmc) [13] that are used to predict solubility and * corresponding author: oleg_milos@mail.ru 92 o. miloserdov copyright ©2020 assa. adv. in systems science and appl. (2020) diffusion coefficients. while md is used to predict diffusion, gcmc models have shown their effectiveness in predicting solubility coefficients. one of the common shortcomings of both md and gcmc is their high computational complexity. they require a lot of computing resources and machine time. therefore, these methods seem to be too timeconsuming for mass prediction of the properties (screening) of hypothetical (previously not synthesized) polymeric materials. at the same time, there is an alternative approach to the prediction of sorption properties of polymers and the prediction of their transport parameters. this is a typical task for qspr methods [17,21]. their goal is to establish statistically significant correlations between the numerical values of the physicochemical properties of materials and molecular descriptors. earlier, in [11], the method was proposed and then improved in [10] for predicting the sorption parameters of polymers (gas solubility coefficient at infinite dilution s and the solubility constant of henry's law kd in the model of double sorption) for a number of light gases in glassy polymers. the approach [11,10] is based on calculating the dependence of a number of geometric indices on the effective penetrant radius for conformations of a polymer chain segment (200-600 atoms length), obtained from computer modeling of a polymer chain segment. in this article this approach to molecular modeling and data analysis is further developed and applied to building a classification of glassy polymers used in membrane technology. the isolated clusters turn out to be closely related to the physicochemical properties of substances that are important for the membrane technology. in the future, it can be used to construct refined regressions that can predict the main transport properties of polymeric materials — solubility, diffusion, and permeability coefficients. 2. molecular modeling 2.1. molecular mechanics modeling problems of quantitative property forecasting from statistics and those of massive polymer classification impose some requirements on the underlying approach to molecular mechanicsmodeling (hereinafter mmm). 1. conformations (macromolecule atoms’ positions on a scene) obtained from mmm should be as realistic as possible (conform to real positions of atoms in a polymer membrane). 2. manual tuning of the mmm algorithm should be avoided, and the algorithm should be as automated as possible. 3. the mmm algorithm should be able to parallelize on a server and have an acceptable calculation time for one polymer. 4. the method should ensure the stability of the results and their reproducibility. 5. the method should be applicable to specific polymers used in membrane gas separation. 6. the method should use the free software. in this paper, we further improve the mmm method from [11,10], which is based on the modeling of conformations of the polymer macromolecule of about several hundred atoms size. in [11], for each polymer, an oligomer (200–600 atoms length) consisting of several polymer monomer units, was created in the instantjchem molecular simulation environment [4]. for this chain, using conformer plugin [6] of the instantjchem chemaxon package, several conformations of a short chain segment of the polymer under consideration (typically including several monomer units and from 200 to 600 atoms) are generated by relaxation in the empirical dreiding field from random initial positions of atoms. further calculation of the indices necessary for constructing the regression model was also performed using instantjchem tools. despite the fact that, using the obtained conformations, one of the transport parameters of polymer membranes, s solubility, was successfully predicted, this approach did not meet many criteria, listed above. the most important drawback is the classifying amorphous polymers for membrane technology … 93 copyright ©2020 assa. adv. in systems science and appl. (2020) unrealistic nature of the many conformations obtained, as well as the difficulty in expanding this solution to server platforms. in [10], the problem of the unreality of the obtained conformations was solved by performing their modeling in a program in the perkinelmer chem3d package [5] using a similar technique. to assign a random initial position of atoms to the simulated polymer chain, molecular mechanics simulation was performed at a temperature of 300 k to 3000 k for 1000 iterations, after which the resulting structure was optimized by free energy in the empirical field mm2. this approach showed good results in terms of realistic conformations, but made it impossible to automate the mmm process and did not allow the efficient use of multiprocessor systems. during the search, many different mmm solutions were tested, for example, lammps [12], openmm [7], hoomd-blue [3,9] and others. some solutions did not allow automation of the process (lammps), the others (e.g., openmm) were developed for biopolymers and had no empirical force fields relevant for glassy polymers used in membrane gas separation. rdkit package [16] an open-source toolkit – is widely used by the scientific community to solve various problems in chemoinformatics and machine learning. the main data structures and rdkit algorithms are written in c++, which ensures high performance. rdkit also has shells in python, java, and c #, which makes it easy to use. to arrange the molecule in 3d space in the rdkit environment, the flexible embedmolecule procedure is used [15]. the initial coordinates of atoms positions in molecule can be determined both through the eigenvalues of the distance matrix, and through a random positioning of atoms, and the value of the random number generator can be fixed, which will later make it possible to obtain the same molecule. then mmm is performed in an empirical force field uff (universal force field [14]). it is important that rdkit tools allow to fix the part of a molecule during mmm. the polymer chain is built by a simple algorithm, which uses the embedmolecule function of rdkit and roughly mimics the real process of polymerization. at the beginning, the monomer unit of the polymer is located in the space, then a similarly processed unit is added to it, but with other random coordinates. all atomic coordinates, except the atomic coordinates of the last two added monomer units, are fixed, and two free units undergo energy minimization in the uff field, which simulates the process of sequential polymerization of a macromolecule. in this case, the algorithm takes into account the experimental values of the torsion angles and uses the base of universal rules for the mutual arrangement of atoms, for example, the mandatory presence in the same plane of the atoms of the benzene ring. in less than half of the samples, due to the peculiarities in their structure, attempts to construct a macromolecule in this way failed after several iterations. in this case, the procedure of placing the atoms of a molecule in space was performed with the use of random initial coordinates for the entire macromolecule, and not its individual monomer units. examples of the obtained conformations are presented in fig. 2.1. 94 o. miloserdov copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 2.1. examples of three polymer repeating units are shown above, and the corresponding macromolecules of about 700-1200 atoms size obtained with mmm are shown below 2.2. calculation of geometric indices as in [10], the curves that represent the value of various accessible-surface-area-based geometric indices of polymers’ conformations as functions of the radius of a spherical “probe” are used to characterize polymers, since the main hypothesis of this method is that transport properties of a polymer (e.g., solubility coefficient and the similar measures) to some extent depend on the characteristics of the contact surface between the polymer macromolecule and the penetrant gas molecule. the basic and the simplest index of this family is the accessible surface area (asa). its calculation is illustrated by fig. 2.2. asa is the surface area circumscribed by the center of the spherical "probe" of a given radius in all possible positions of its contact with the van der waals surface of this molecule. fig. 2.2. examples of calculating the accessible surface area of a molecule the whole family of the employed indices is presented in table 2.1. classifying amorphous polymers for membrane technology … 95 copyright ©2020 assa. adv. in systems science and appl. (2020) table 2.1. geometric indices index index description asa standardized (per cm3 of the sample) accessible surface area (å2⋅mol/cm3) asa+ accessible surface area where contact occurs at a point on the surface with a partial positive (q > 0) charge* (å2⋅mol/cm3) asa– accessible surface area where contact occurs at a point on the surface with a partial negative (q < 0) charge* (å2⋅mol/cm3) asah accessible area of hydrophobic (with low, |q| < 0.125, level of partial charge*) surface (å2⋅mol/cm3) asap accessible area of polarized (with high, |q| ≥ 0.125, level of partial charge*) surface (å2⋅mol/cm3) dpsa3 dpsa3 = ∑i asai⋅qi, where asai – is the contribution of the i-th atom to the specific accessible surface area of the molecule and qi is the partial charge* of i-th atom (å2 e моль/см3) * ppsa3 ppsa3 = ∑i asai⋅qi, where the summation is limited to atoms with a positive partial charge*: asai⋅ qi > 0 (å2e mol/cm3) pnsa3 pnsa3 = ∑i asai⋅qi where the summation is limited to atoms with a negative partial charge: asai⋅qi < 0 (å2e mol/cm3) * partial charges according to gasteiger–marsili were calculated using the rdkit.chem.rdpartialcharges module. ** here “e” means the electron charge unit. compared to the previous versions of this method, the new tools of mmm presented in this article allowed to build bigger molecules and more complex conformations, which address not only the interaction of adjacent atoms in a polymer chain but also that of atoms came close to each other due to the bending of a macromolecule chain (see fig. 2.1). also, a new improved index calculation algorithm was implemented, which excludes from consideration the outer surface of the oligomer, and thus takes into account interactions of a penetrant with the polymer macromolecule. the macromolecules in fig. 2.1 are rather short polymer chains, and their outer shell has considerable surface area. realistic polymer matrices are much bigger and have no such outer shell. the contribution of atoms that form these outer shells distorts the value of surface-area-based indices from table 2.1, and their contributions must be excluded from index calculation. therefore, based on the center-of-mass position of the molecule and the value of the current “probe” radius r, the indices from table 2.1 were calculated only for atoms of the macromolecule located at no more than d = 10 – r angstrom from its center of mass. hence, the atoms that belong to the outer shell of the conformation were excluded. to draw the dependences asa(r), asa+ (r), etc., the index value was calculated for each r from 0å to 0.2å with a step of 0.05å, from 0.2å to 1å with a step of 0.1å and from 1å to 2å with a step of 0.2å. typical dependencies of a number of geometric indices from table 2.1 from the “probe” radius for one of the polymers, calculated for different d, are presented in fig. 2.3. one can see from fig. 2.3 that curves for d = 7å – r resemble those for d = 10å – r, so the adopted approach is robust with respect to small deviations of the selection rule for “internal” atoms. to justify the sufficient chain length and the number of generated conformations for the stability of the results obtained below, 6 conformations of different lengths (200, 300, 400, 600, 800, atoms without hydrogen) were generated for 40 different polymers. for them, indices from the entire set were calculated for a number of r values from the range from 1 å to 3 å. in fig. 2.4 presents, as an example, the obtained dependences of three indices (at different r) on the macromolecule size for 16 random polymers from 40 calculated. it can be seen from the figure that the indices begin to stabilize when the length of the macromolecule exceeds 600 atoms, which was taken as the base length. 96 o. miloserdov copyright ©2020 assa. adv. in systems science and appl. (2020) the same logic shows that averaging over 10 conformations is enough to obtain stable index values (see fig. 2.5). fig. 2.3. typical dependencies of a number of geometric indices from table 2.1 from the “probe” radius for one of the polymers, calculated for various d fig. 2.4. asa, asa+ and dpsa3 indices vs the macromolecule size for 16 random polymers classifying amorphous polymers for membrane technology … 97 copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 2.5. asa, asa+ and dpsa3 indices averaged over conformations vs the number of conformations being averaged for 16 random polymers 3. classification of conformational structures in machine learning problems correctly collected and pre-processed data play a key role in obtaining a qualitative result. the database “gas separation parameters of glassy polymers” (hereinafter referred to as the database) was created in 1998 at the laboratory of membrane gas separation at the topchiev institute of petrochemical synthesis, russian academy of sciences. the database is a unique source of information and a tool for predicting gas separation properties of polymers. since its inception, the database is being continuously updated. the main transport characteristics of polymer gas separation membranes are coefficients of permeability, diffusion, and solubility. the permeability coefficient p can be written as a product of the solubility coefficient s and the diffusion coefficient d. the first (s) describes the driving force of the process of transporting gas molecules, while the second (d) corresponds to the kinetic component of the process. to solve the clustering problem, 397 unique polymers were selected from the database, each having several records containing experimental data on the interaction of the gaspolymer pair under different conditions (2629 records in total). while the database covers 98 o. miloserdov copyright ©2020 assa. adv. in systems science and appl. (2020) more unique polymer structures, for the clustering problem it is necessary to maintain a balance in the data on their chemical classes. therefore, the structures were selected according to the rule: • if more than 42 structures of a certain chemical class exist in the database, then records that have experimental data for p, d, s were selected, • otherwise, all structures of this chemical class were selected. as justified in section 2 above, for each polymer in a sample ten conformations were generated of size at least 600 atoms (not counting hydrogen atoms). see fig. 2.1 for the examples of conformations built. then, for each conformation and for each r in the range from 0å to 3å, the indices from table 2.1 are calculated using the selection rule d = 10å – r, which rules off the atoms of the outer hull of the conformation. typical dependences of geometric indices asa, asa+ and dpsa3 on the “probe” radius for one of the polymers are shown in fig. 3. in contrast to [10], where the optimal range of [r–, r+] linearization of the obtained dependencies was searched for, the obtained slope and bias coefficients were used as an explanatory regression variables, in this article the entire curve is used for classification. for the clustering of polymers, the agglomerative method (a sort of hierarchical classification) was used [1]. the agglomerative clustering method was launched for the number of clusters from k = 2, ..., 30, since the optimal number of clusters was unknown. according to the values of the silhouette coefficient, the calinski–harabasz index (c-h) and the davies–bouldin index (dbi) for the clustering constructed, the value k = 15 has been chosen. the data supplied to the input of the algorithm was previously standardized. to illustrate the quality of the resulting clustering using the tsne algorithm [20], the placement of points in 2d space was constructed. it is clearly seen that the tsne algorithm, being different from agglomerative clustering in its nature, distributes the data points in accordance with the obtained clustering. the embedding also allows to evaluate the distance between clusters (see fig. 3.1). table 3.1. the distribution of polymers of different chemical classes over clusters chemical class cluster number sum by class 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 polyacrylates 1 3 1 9 1 15 polyesters 2 18 1 1 4 4 4 8 42 polyethers 1 8 12 4 2 2 5 3 37 polyphosphasenes 3 3 polyacetylenes 1 4 3 11 19 polynorbornenes 5 7 2 2 2 9 1 6 3 37 polysulphones 19 1 18 2 40 other n-containing 4 1 5 3 4 9 26 polyamides 11 1 16 3 2 4 37 polystyrenes 2 3 1 2 1 13 22 vinylic polymers 1 3 2 1 5 1 1 14 polycarbonates 14 1 1 3 19 polyimides 3 3 5 6 9 16 42 polyamidoimides 2 2 1 1 11 23 40 other carbo-chained 1 2 3 other hetero-chained 1 1 sum by cluster 4 15 64 34 63 20 6 34 6 31 28 11 64 4 13 397 classifying amorphous polymers for membrane technology … 99 copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 3.1. data embedding in 2d with tsne algorithm (clusters are shown with different signs and colors) 4. clustering and transport properties the clusters constructed turn out to be closely related to the transport properties of polymers that are important for membrane technology, first of all, permeability, diffusion, and solubility coefficients of a polymer material with respect to different penetrant gases. the, so-called, robeson diagram [18] is a conventional representation of transport properties for a collection of polymers. the robeson diagram is constructed for a pair of penetrant gases (for example, oxygen-nitrogen) in the coordinates αp = po2/pn2 — separation selectivity of a pair of gases (oxygen-nitrogen) depending on po2 — the permeability coefficient for a more permeable gas (oxygen) and in coordinates αd = do2/dn2 — diffusion selectivity of a pair of gases (oxygen-nitrogen) depending on do2 — the diffusion coefficient (see fig. 4.1 and fig. 4.2, respectively). 100 o. miloserdov copyright ©2020 assa. adv. in systems science and appl. (2020) robeson diagrams are often used by membrane gas separation specialists for comparative analysis of different groups of polymers. hence, it is quite natural to represent the clusters of polymers constructed in the previous section as the points on a robeson diagram to analyze the relation between the obtained classification of polymers and their transport properties. the experimental data on the coefficient of permeability and diffusion were taken from the database. since the experimental data were obtained at various temperatures, they were brought to a single standard value of 308k using the algorithm from [2]. then, the diagrams were constructed using the experimental data on the permeability (see fig. 4.1) and diffusion (see fig. 4.2) coefficients available from the database. for the convenience of the analysis, along with the points of an individual cluster the center of this cluster and the center of the entire data sample (calculated without the explicit outliers) are added to the robeson diagram. the clusters turn out to be closely related to the transport properties of polymers. the location of the center of mass of the cluster relative to the center of mass of the entire data sample, the shape of the cluster and its composition demonstrate these relationships. the first thing worth noting is that the points of most clusters are rather crowded on the robeson diagrams. consequently, materials with similar transport properties are collected in one cluster. it is important that clustering was built only on the basis of the shape and geometry of conformations of polymer molecules without using any hint like the chemical class of a polymer. now consider the clusters separately. the most revealing relationships can be found on the robeson diagram for the diffusion selectivity in figure 4.2. so, the centers of mass of large clusters 3, 5, 13 are shifted to the upper left quadrant of the general diagram. this indicates a high selectivity for d with a low diffusivity. moreover, in combination with the information from fig. 4.1, polymers from these clusters are a good combination of permeability selectivity and permeability itself. also, cluster 11 can be attributed to them if there were more experimental data on the diffusion coefficient. for the most part, polymers from clusters 6, 8, 12 turned out to be low-selective in terms of permeability and diffusion. however, they were highly permeable and highly diffuse. from table 3.1 it can be noted that, in general, clustering does not depend on the chemical class of polymers. so, the above clusters 3, 5, 6, 8, 13 consist of polymers of various chemical classes. however, there are exceptions. for example, all highly diffuse and highly permeable polymers with a low selectivity from 12 cluster are polyacetylenes. all polyphosphasenes entered cluster 8, most polycarbonates entered cluster 3, and polysulphones were divided into two clusters 5 and 10. polyamidoimides and polyimides, being often similar in structure and properties, were mainly divided into clusters 11 and 13. let us turn to the structure of polymers themselves. cluster 6 consists only of polymers with bulky fluorine substituents, which in this case belong to different chemical classes. in combination with the information that the polymers of this cluster are highly permeable and highly diffuse and, at the same time, low selective, one can judge the consequences of adding bulky fluorine substituents. moreover, cluster 7, which contains mainly polymers with a pentafluorophenyl group, is rather the opposite of low selectivity and low diffusion. this suggests that not always the presence of a large number of fluorine molecules leads to properties similar to polymers of 6. it is also interesting to compare cluster 6 and cluster 10. cluster 10 also has bulky fluorine substituents; however, monomer units are often several times longer and most polymers belong to polyamidoimides and polyimides, which are never met in cluster 6. classifying amorphous polymers for membrane technology … 101 copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 4.1. polymer clusters on the robeson chart for permeability. the center of the whole data sample is depicted with the red circle, while the center of the cluster is depicted with the green triangle 102 o. miloserdov copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 4.2. polymer clusters on the robeson chart for diffusion. the center of the whole data sample is depicted with the red circle, while the center of the cluster is depicted with the green triangle classifying amorphous polymers for membrane technology … 103 copyright ©2020 assa. adv. in systems science and appl. (2020) the above analysis reveals the deep relation between the transport properties of polymers and the shape and geometry of conformations of their macromolecules. the resulting clustering allows one to identify the signs and characteristics of polymer molecules with various extreme properties, which will undoubtedly be useful in the search and synthesis of new promising polymers. 5. conclusion based on a large sample consisting of almost 400 amorphous polymers from 16 different chemical classes used in membrane gas separation, clustering was constructed only on the basis of data on the shape and geometry of the conformations of polymer molecules. obtained 15 clusters are closely related to transport parameters important for membrane gas separation, such as permeability coefficient p and diffusion coefficient d. the method proposed consists of several steps. at the beginning, 10 realistic conformations of polymer macromolecules are constructed. for these purposes, a program code is written in the rdkit environment for python. it satisfies all the requirements listed in chapter 2: the realism and reproducibility of the resulting structures, fast automated calculation, the possibility of parallelization, and free distribution of software. then, the dependencies were calculated of 8 accessible-surface-area-based indices (see table 2.1) on the radius of a spherical probe that represents the variety of penetrant gases. an improved algorithm for calculating indices eliminates the influence of the outer shell of the macromolecule, which allows focusing on the processes occurring inside the polymer membrane. the obtained 8 curves are averaged between ten polymer conformations to obtain more robust figures. then, these dependences characterizing the polymer are used as predictors for the agglomerative clustering method. the plans for further research include constructing separate regressions for the resulting clusters in order to more accurately predict transport properties (p, d, and s coefficients). the methods developed allow us to approach the solution of the problem of obtaining substances with predetermined properties. for example, the prediction of transport parameters for a collection of hypothetic polymers, followed by the manual selection of several most promising polymers, could guide the synthesis experiments of novel polymers. acknowledgements the reported study was funded by rfbr according to research projects 18-37-00265 and 1937-90004. references 1. agglomerative clustering. recursively merges the pair of clusters that minimally increases a given linkage distance [online]. available https://scikitlearn.org/stable/modules/generated/sklearn.cluster.agglomerativeclustering.html 2. alent'ev a.ju. (2003) prognozirovanie transportnyh svojstv stekloobraznyh polimerov: rol' himicheskoj struktury i svobodnogo ob'ema. doctor of science thesis, moscow [in russian]. 3. anderson j. a., lorenz c. d., travesset a. (2008) general purpose molecular dynamics simulations fully implemented on graphics processing units, journal of computational physics, 227(10), 5342-5359. 4. chemaxon. software solutions and services for chemistry & biology [online]. available https://www.chemaxon.com 5. chemoffice professional. chemical communications software [online]. available http://www.perkinelmer.com/product/chemoffice-professional-chemofficepro 6. conformer plugin. walk-through manual on how to use the conformer plugin [online]. available https://docs.chemaxon.com/display/docs/conformer+plugin 104 o. miloserdov copyright ©2020 assa. adv. in systems science and appl. (2020) 7. eastman p., et al. (2017) openmm 7: rapid development of high performance algorithms for molecular dynamics, plos comp. biol, 13(7), e1005659. 8. fried, j. (2006) materials science of membranes for gas and vapor separation, ed. by yu. yampolskii, i. pinnau, b.d. freeman, wiley, chichester, p. 95. 9. glaser j., et.al. (2015) strong scaling of general-purpose molecular dynamics simulations on gpus, computer physics communications, 192, 97-107, https://doi.org/10.1016/j.cpc.2015.02.028 10. goubko m., miloserdov o., yampolskii yu., alentiev a., ryzhikh v. (2019) prediction of solubility parameters of light gases in glassy polymers on the basis of simulation of a short segment of a polymer chain, polymer science, 61(5), 718– 732. 11. goubko m., miloserdov o., yampolskii yu., alentiev a., ryzhikh v. (2016) a novel model to predict infinite dilution solubility coefficients in glassy polymers, journal of polymer science part b: polymer physics, 55(3), 228-244. 12. lammps. molecular dynamics simulator [online]. available https://lammps.sandia.gov/ 13. norman, g.e.; filinov, v.s. (1969) high temp, 7, 216. 14. rappe a. k. et.al. (1992) uff, a full periodic table force field for molecular mechanics and molecular dynamics simulations, journal of the american chemical society, 114 (25), 10024-10035. 15. rddistgeom module. module containing functions to compute atomic coordinates in 3d using distance geometry [online]. available https://www.rdkit.org/docs/source/rdkit.chem.rddistgeom.html 16. rdkit. open-source cheminformatics software [online]. available http://www.rdkit.org 17. robeson l.m. et. al. membr. sci., 1997, 132, 33. 18. robeson, l.m. (1991) correlation of separation fac-tor versus permeability for polymeric membranes, journal of membrane science, 62 (165), 165–185. 19. theodorou, d. (2006) materials science of membranes for gas and vapor separation, ed. by yu. yampol-skii, i. pinnau, b.d. freeman, wiley, chichester, p. 49. 20. tsne. t-distributed stochastic neighbor embedding [online]. available https://scikit-learn.org/stable/modules/generated/sklearn.manifold.tsne.html 21. yampolskii yu., shishatskii s., alentiev a., loza k. (1998), j. membr. sci. 148, 59. adv syst sci appl 2018; 1; 110-131 published online at http://ijassa.ipu.ru. models of adaptation in dynamical contracts under stochastic uncertainty mikhail. v. belov1, dmitry. a. novikov2 1) ibs, moscow, russia e-mail: mbelov59@mail.ru 2) trapeznikov institute of control sciences, russian academy of sciences, moscow, russia e-mail: novikov@tushino.com abstract: in this paper, we establish new sufficient conditions for the optimality of lump-sum and compensatory incentive schemes (contracts) under stochastic uncertainty. in addition, we suggest and analyze dynamical models of principal’s and agents’ adaptation to changes in statistical characteristics of an external environment. keywords: contract theory, incentive problem, stochastic uncertainty, adaptive behavior. 1. introduction incentive problems in organizational systems, i.e., stimulation of controlled subjects to choose certain actions for the benefit of a control subject, are studied in organizational control [12, 13] and contract theory [21, 28]. within the latter, much attention is paid to the situations in which the results of activity of controlled subjects depend on their own actions and also on external random factors. if the participants of an organizational system have a repeated interaction with the course of time, then dynamical contracts arise naturally. these contracts are described using the methods of repeated games (in discrete time [10, 24, 27] or continuous time [22, 29]), see the surveys in [11, 23, 27]. however, in some situations the characteristics of an external uncertainty change with time, and hence it is necessary to develop and analyze models for a proper consideration of such effects by organizational system participants, including their efficient response. this paper is organized as follows. in section 2, we introduce a classification system for the models of contracts in organizational systems and consider the static models of contracts, particularly the model under additive uncertainty (subsection 2.4) and the simple agent model (subsection 2.5). for these models, we establish new sufficient conditions for the optimality of lump-sum and compensatory incentive schemes (contracts). section 3 is dedicated to contracts in multiagent systems while sections 4 and 5 to the dynamical models of earned value and adaptation processes of organizational system participants to changeable characteristics of an external environment. 2. static model consider an organizational system (os) [[13]] that consists of a single control subject (principal) and a single controlled subject (agent). the agent chooses a nonnegative action y ≥ 0. in combination with a realized value of an uncertain parameter (known as the state of nature   [0, ∆]), the action uniquely defines the result z = y –  of his activity. this setup is called the additive uncertainty model, as the uncertainty distorts the agent’s action models of adaptation in dynamical contracts 111 copyright ©2018 assa. adv. in systems science and appl. (2018) additively. assume the agent’s cost c(y, r) depends on his action y and type r > 0 (a parameter that reflects the efficiency of the agent’s activity). let c(y, r) be a strictly monotonically increasing smooth convex function in y that vanishes in the origin and also a strictly monotonically decreasing function in r. the agent’s cost representation as a function of two variables is standard in organizational control (e.g., see [5–13]). in most cases considered below, the agent’s type r is fixed and/or conceptually insignificant. for the sake of compactness, we will omit r in the cost function, using the notation c(y) whenever no confusion occurs. the principal offers the agent to conclude a contract σ(z) that determines a nonnegative reward (and conditions to obtain it) depending on the achieved result. the agent’s goal function is the difference between the incentive and cost functions, i.e., ).,()(),),(( ryczzyf −=  (1) the principal obtains an income h(z) from the agent’s activity (where h(∙) is a continuous nondecreasing function) and also bears incentive cost. in other words, the principal’s goal function is the difference between the income and incentive functions: ).()()),(( zzhz  −= (2) accept the following sequence of moves in this system, which is traditional for the incentive problems in organizational control [[2], [12], [13]] and contract theory [[21], [26], [28], [30]]. first, the principal offers the agent a contract; second, the agent chooses his action in response to this offer and the payments are made accordingly. note that the agent may reject the contract, choosing the zero action. in this case, he obtains no reward but bears no cost. assume both participants of this organizational system seek to maximize their goal functions. since the result of the agent’s activity depends on his action and also on the state of nature, the principal and agent have to use all available information in order to eliminate the external uncertainty. there are the following types of uncertainty depending on the awareness of the system participants. – no uncertainty (the deterministic case in which ∆ ≡ 0, and this is common knowledge for the principal and agent); – complete awareness (both subjects know the true value of the state of nature); – interval uncertainty (both subjects know merely a set of admissible values for the state of nature, i.e., a range [0, ∆]); – probabilistic uncertainty (both subjects know a probability distribution over the set of admissible values for the state of nature or for the result of the agent’s activity); – fuzzy uncertainty (both subjects know a membership function for an uncertain parameter defined on the set of admissible values). denote by <f(σ(∙), y)> and <φ(σ(∙), y)> the deterministic goal functions of the agent and principal, respectively; these functions are obtained after elimination of the existing uncertainty for the state of nature. as a rule, an interval uncertainty is eliminated using the principle of maximum guaranteed result (see a survey of uncertainty elimination methods for incentive problems in [[12]]); a probabilistic uncertainty, using the principle of expected utility; a fuzzy uncertainty, the principle of maximally undominated alternatives. designate as p(σ(∙)) the set of agent’s optimal actions that are implemented by the principal within a contract σ(∙), i.e., )}.),(({maxarg))(( 0 yfp y =   similarly, designate as m the set of admissible contracts. accept the hypothesis of benevolence [[12], [13]], stating that the agent chooses the most profitable action for the principal from the set of implementable actions. then the incentive problem is to find an optimal contract σ*(∙), i.e., an admissible contract that maximizes the principal’s goal function: 112 m.v. belov, d.a. novikov copyright ©2018 assa adv. in systems science and appl. (2018) )}}.),(({max{maxarg)( )( * y pym     (3) if the hypothesis of benevolence is not satisfied, the problem is to find an incentive scheme (contract) that yields the maximum guaranteed efficiency: )}}.),(({min{maxarg)( )( * y pym g     even for the model with one uncertain parameter, there are many possible combinations of the principal’s and agent’s awareness (more specifically, the number of such combinations is 17 = 1 + 42, as the cases of nontrivial mutual awareness [[14]] are beyond the scope of this paper). let us study some of them. 2.1. deterministic case here z ≡ y, and the solution of problem (3) is a lump-sum incentive scheme based on the principle of cost compensation for the agent [[1], [3], [13]]. it has the form 0 0 0 0 ( ) , , , ( ) 0, , c c x y x y x x  y       =    (4) where the optimal plan (the agent’s action desired by the principal) is )}.()({maxarg 00 zczhx z −=  (5) substituting (4) and (5) into (2) gives the principal’s optimal payoff )}.()({max 00 zczhk z −=  (6) the corresponding value of the agent’s goal function is 0. if the hypothesis of benevolence fails, the solution is the ε-optimal incentive scheme 0 0 0 ( ) , , ( , ) 0, ;gc c x y x x y y x   +  =   (7) where ε > 0 represents an arbitrarily small constant. 2.2. complete awareness of principal and agent assume that the agent and principal know the realized value of the state of nature before choosing the action and incentive scheme. then the principal can use the so-called flexible planning mechanism [[9]] in which the optimal plan x*() = arg 0 max y [h(y – ) – c(y)] (8) and the incentive scheme * * * * ( ( )), ( ) , ( ( ), ) 0, ( ) , c c x z x x z z x          − =   − (9) both explicitly depend on the state of nature . the value of the agent’s goal function is 0 while the principal’s optimal payoff takes the form k() = 0 max y [h(y – ) – c(y)]. (10) obviously, as the agent’s cost function is nondecreasing, we obtain k() ≤ k0 0  . this means that the uncertainty has a negative effect on the principal’s payoff. 2.3. interval uncertainty if there is incomplete awareness in the system, we should discriminate between the cases of symmetrical (identical) and asymmetrical awareness of the principal and agent. (a standard hypothesis is that the agent has at least the same awareness of the uncertain parameters as the principal [[13]]. hence, considering the case of asymmetrical awareness below, we assume that the agent knows the realized value of the state of nature while the principal makes his decision under uncertainty). note that the agent may report information to the principal [[2], [13]], this setup is not studied here. models of adaptation in dynamical contracts 113 copyright ©2018 assa. adv. in systems science and appl. (2018) asymmetrical awareness. in this case, the principal knows merely the admissible range [0, ∆] for the state of nature and hence is forced to guarantee cost compensation for the agent. xmgr = arg 0 max y [0 ] min ;  [h(y – ) – c(y)] = (11) = arg 0 max y [h(y – ∆) – c(y)]. the principal uses the incentive scheme σc(xmgr, z) = mgr mgr mgr ( ), , 0 . c x z x , z x  −    −  (12) the agent knows the realized value for the state of nature  and chooses the action y*() = xmgr – ∆ + , (13) which leads to the expected result (xmgr – ∆) of his activity. the agent benefits from plan fulfilment, which is easy to check directly by comparing his payoffs. for any state of nature, the principal’s optimal payoff k∆ = 0 max y [h(y – ∆) – c(y)] (14) under asymmetrical awareness is not higher than under complete awareness (see (10) and (14)). the agent’s goal function takes the value f(σc(xmgr, xmgr), y*(), xmgr) = c(xmgr) – c(xmgr – ∆ + ) ≥ 0 . (15) value (15) is called informational rent, i.e., the payoff obtained by a subject (in our model, agent) owing to better awareness in comparison with other subjects (the principal). symmetrical awareness. in this case, neither the agent nor the principal know the realized value of the state of nature at the moment of decision-making. the only available information is the admissible range [0, ∆]. thus, the principal uses the incentive scheme (12) and obtains payoff(14), whereas the agent is forced to choose the action guaranteeing a nonzero reward to him: ymgr = xmgr . (16) as a result, the agent has zero payoff. note that the organizational system suffers from the overproduction (∆ – ) ≥ 0. 2.4. probabilistic uncertainty: additive model let the uncertain state of nature  be a random variable with a continuous distribution function f̂ (∙): [0; ∆] → [0; 1], which has a density function p(∙). in the sequel, we will use a distribution function fθ(∙): (–∞; +∞) → [0, 1] of the form fθ(ξ) = 0, 0, ( ), [0, ], 1, . f̂            assume there is asymmetrical awareness of this uncertain parameter in the system. for a given agent’s action y, the result of activity z = y –  is a random variable with the distribution function fz(∙, y): [y – ∆, y] → [0, 1] defined by fz(q, y) = 1 – f(y – q) . (17) let the principal’s choice be limited to the parametric class of lump-sum incentive schemes σc(π, z) = , , 0, , z z       (18) where π ≥ 0 and λ ≥ 0 denote some parameters, π is a planned result (the result of the agent’s activity desired by the principal). (as demonstrated in [[6], [12]], in the case under study the lump-sum incentive schemes can be not optimal, see the discussion below; nevertheless, they are simple and widespread in applications). under an action y ≥ π chosen by the agent, the expected value of his reward (18) is 114 m.v. belov, d.a. novikov copyright ©2018 assa adv. in systems science and appl. (2018) ez σc(π, z) = λf(y – π) (19). suppose the agent seeks to maximize his expected utility [[12], [13], [21]]. by the firstorder optimality conditions, the agent’s action y*(π, λ) ≥ π satisfies the equation λ p(y *(π, λ) – π) = c(y*(π, λ)). (20) the principal’s problem is to choose the parameters π ≥ 0, λ ≥ 0 of the incentive scheme (contract) in order to maximize his expected utility: .max)),(()()),(( 0,0 * 0 *   →−−−    yfdpyh (21) example 1. assume the principal has a linear income function h(z) = γz, where γ > 0 is a given constant; the agent has a quadratic cost function c(y, r) = y2/2r, where r > 0 is his type [[13]], which describes the efficiency of the agent’s activity; the distribution f(∙) is uniform, i.e., f(v) = v/∆, v  [0, ∆]. according to formula (19), the expected reward of the agent is λ(y – π)/∆. hence, for any action y ≥ π the agent receives the expected payoff ez f(σc(π, z), y, z) = λ (y – π)/∆ – y2/2r. (22) maximizing his expected payoff (22), the agent chooses the optimal action (also, see expression(20)) * for , 2 ( , ) 0 for . 2 r r y r            =      (23) the expected value of the principal’s goal function is ez φ(σc(π, z), z) = γ(y – ∆/2) – λ(y – π)/∆. (24) substituting (23) into (24), we obtain the following optimization problem for the parameters of the incentive scheme (18) (see (21)): γ (λr/∆ – ∆/2) – λ(λr/∆ – π)/∆ → 0, /(2 ) max r     . (25) the solution of problem (25) has the form λ* = γ∆, π* = γr/2. then the expected payoff of the principal is γ(γr – ∆)/2 while the expected payoff of the agent is 0. •* let us design the optimal incentive scheme for the probabilistic additive uncertainty model. the general solution procedure of probabilistic incentive problems is as follows [[12]]. first, for each agent’s action x, find a minimal incentive scheme min(x, ) in terms of the expected incentive cost of the principal that implements this action, i.e., the scheme stimulating the agent to choose x  p(min(x, )). second, find the action that is most beneficial to implement in terms of the principal’s goal function, i.e., the one maximizing his expected payoff (also see expression (3)): x* = arg 0 max x eφ(min(x, ), x), (26) where e means the expectation operator. denote x** = arg max x [ 0 ( ) ( )h x p d    − – c(x)]. lemma 1: in the probabilistic incentive problem, for any agent’s action x ≥ 0 there does not exist an incentive scheme implementing this action with the principal’s expected incentive cost strictly smaller than the agent’s cost, i.e., min(x, ) ≥ c(x). * throughout the paper, the symbol “•” indicates the end of an example or proof. models of adaptation in dynamical contracts 115 copyright ©2018 assa. adv. in systems science and appl. (2018) proof. assume on the contrary that there exists an agent’s action 0x  and an incentive scheme ( )z such that ez ( )|z x = 0 ( ) ( )zz p v, x dv +  < ( )c x , (27) and the incentive scheme ( )z implements the action 0x  , i.e., for any y  0, ez ( )|z x – ( )c x ≥ ez ( )|z y – c(y). (28) for y = 0, by c(0) = 0 inequality (28) takes the form ez ( )|z x ≥ ( )c x + ez ( )|0z . this result contradicts (27), as the incentive and its expected value are nonnegative. • we will establish sufficient conditions for the optimality of an incentive scheme of form (18), namely, the contract σc(x, z) = ( ), , 0, c x z x z x .  −    −  (29) proposition 1: for all x  [x** – , x**], let ** ** ( ) ( ) ( ) c x p x x c x   −  . (30) then in the probabilistic additive uncertainty model the incentive scheme (29) implements the agent's action x** with the minimum expected incentive cost c(x**) of the principal and is optimal. proof. calculate the expected value of the agent’s reward from choosing an action y given a plan x: ez σc(x, z)|y = 0 c ( ) ( ), px  y d    −  − . it follows from (19) that ez σc(x, z)|y = c(x) f(y – x + ∆). the agent’s expected utility is ez σc(x, z)|y – c(y) = ( ), , ( )  ( ) ( ), [ , ], ( ) ( ), c y y x c y y x x c x c c x f y  x y y x     .  − −  −   −  −  −  +   given the plan x = x**, by condition (30) the maximum value of this expected utility is achieved at the zero action or at the action coinciding with the plan x (condition (30) guarantees that the agent’s expected utility is a nondecreasing function in his action on the interval [x** – , x**]). on the strength of the hypothesis of benevolence, the agent chooses the action x**, which makes the expected incentive cost of the principal equal to the agent’s cost. hence, by lemma 1 the incentive scheme (29) is optimal. • obviously, if the hypothesis of benevolence fails, then under condition (30) the ε-optimal incentive scheme is σcg(x, z) = ( ) , , 0, c x z x z x . +  −    −  example 2. for the data of example 1, condition (30) takes the form γr ≥ 2∆. • to proceed, study sufficient conditions for the optimality of a compensatory incentive scheme. to this end, find an incentive scheme ( )ˆ z that nullifies the agent’s expected utility for any actions, i.e., ( ) ( ) ( ) y y ˆ z p y z dz c y − − = . (31) proposition 2: if there exists a contract ( ) 0ˆ z  satisfying relationship (31) then it is optimal in the probabilistic additive uncertainty model. 116 m.v. belov, d.a. novikov copyright ©2018 assa adv. in systems science and appl. (2018) the validity of proposition 2 follows from property (31) of the incentive scheme ( )ˆ z and lemma 1. therefore, an incentive scheme ( )ˆ z is optimal if there exists a positive solution of the integral equation (31). let us formulate the existence conditions of such a solution. proposition 3: if the functions ( )yc y and ( )p v  are continuous on the domains of definition, then the integral equation (31) has a unique solution, which can be calculated using the sequential approximations        = − −  =  = + ....,2,1, )0( )( )(ˆ )0( )( )(ˆ , )0( )( )(ˆ 0 1 0 idu p uzp u p zc z p zc z z ii       (32) for this solution to be positive, a necessary condition is ....,2,1,)()(ˆ)( 0 =−  iduuypuyc y i  (33) proof. differentiation of the integral equation (31) yields ).()()(ˆ)(ˆ)()(ˆ)0( zcduuzpuzpzp z z =−+−−  −   (34) first, solve (34) for z  : ).()()(ˆ)(ˆ)0( 0 zcduuzpuzp z =−+    write this expression as . )0( )( )(ˆ )0( )( )(ˆ 0 du p uzp u p zc z z     − −  =  (35) equation (35) is a volterra integral equation of the second kind. as is well-known (e.g., see [[17]]), under the accepted hypotheses it has a unique solution, which can be calculated using the sequential approximations (32). condition (33) is necessary for the positive solution. thus, we have completely solved the problem for z  . based on (34), it is possible to obtain a similar solution for z  [j; (j + 1)], j = 1, 2, …: 0 ˆ{ ( ) ( ) ( )} ( ) ˆ ( ) ( ) . (0) (0) z c z p z p z u z u du p p         +  − − = −  note that it suffices to prove the positivity of ( )ˆ z on the first interval only, as c'(z) + p()(z – )  c'(z) for the subsequent intervals. example 3. consider the uniform density function )(p defined by 0 1 0 1 1 for [ , ], ( ) 0 for [ ; ], a a p a a       =    where 0 < a0 and a1 = a0 + . then (31) can be written as  − − = 0 1 ).,()( az az rzcdzu differentiation of both sides leads to the functional equation (z – a0) – (z – a1) = с'(z). since c(y) = 0 and c (y) = 0 for y  0, the solution of this equation is given by the functional series   = −=− 00 ).()( i izcaz models of adaptation in dynamical contracts 117 copyright ©2018 assa. adv. in systems science and appl. (2018) recall that )(c is continuously differentiable and convex. hence, the sum of the functional series is positive and increasing, which agrees with the main requirement to the incentive scheme (). particularly, for the data of example 1, c(y, r) = y2/2r and c(y, r) = y/r for y > 0; then   = −−=− 00 ),)(()( i iziz r az  where )( is the heaviside step function 1 for 0, ( ) 0 for 0. u u u   =   • 2.5. probabilistic uncertainty: the simple agent model an alternative to the additive uncertainty model considered in subsection 1.4 is the socalled simple agent model [[3], [5], [6], [12]], in which the distribution function of activity results has the form ( ), , ( , ) 1, . z g q q y f q y q y  =   (36) here g(∙): [0; +∞) → [0; 1] is a given distribution function such that g(0) = 0, with a density g(∙). like in the additive model, the agent’s action defines the maximum possible result while the distribution g(∙) does not explicitly depend on the action. denote by pz(q, y) the density function associated with distribution (36). if the principal uses the lump-sum incentive scheme (18) and the agent chooses an action y ≥ π, the expected reward of the agent is λ(g(y) – g(π)). for the simple agent model, an analog of the first-order optimality condition (20) has the form λ g(y*(π, λ)) = с(y*(π, λ)). as proved in the book [[12]], compensatory incentive schemes are optimal in the simple agent model with a risk-neural agent. for this class of models, the following optimal incentive schemes were constructed in [[6]]: – for a risk-averse agent, the compensatory incentive schemes σk(z) = 0 ( ) 1 ( ) z c v dv g v  − ; (37) – for a risk-seeking agent, the lump-sum incentive schemes σc(x, z) = ( ) , , 1 ( ) 0, . c x z x g x z x   −   (38) clearly, the incentive scheme (37) is nonnegative, increasing and convex. by analogy with [[6], lemma 1], we may establish the following properties of the incentive schemes (37) and (38). proposition 4: in the simple agent model, for any y  0 the incentive schemes satisfy the relationships 1) 0 ( ) ( , ) ( )k zq p q y dq c y + = , 2) 0 ( ) ( , ) ( )c zq p q y dq c x + = . at the conceptual level, relationship 1) of proposition 4 means that for any action of the agent his expected reward (37) coincides with his cost to choose this action. hence, under the incentive scheme (37) used by the principal, any actions of the agent yield zero utility for him and make his choice indifferent. by the hypothesis of benevolence, the agent will choose the action that is most profitable for the principal. 118 m.v. belov, d.a. novikov copyright ©2018 assa adv. in systems science and appl. (2018) if the principal adopts the incentive scheme (38), then the agent is indifferent between the zero action (reject of the contract) and plan fulfilment. for the agent’s expected utility to reach a unique maximum as the result of plan fulfilment, the principal has to increase payments for plan fulfilment by an arbitrarily small positive value ε. note that this incentive scheme is not optimal but ε-optimal. the next property can be verified directly. proposition 5: in the simple agent model, for any x  0 the incentive scheme c  (x, z) = ( , ) , , 1 ( ) 0, , c x r z x g x z x +  −   (39) is ε-optimal, i.e., it implements the agent’s action x with the minimum expected incentive cost of the principal. regardless of the scheme used by the principal (compensatory or lump-sum), his optimal plan for the agent is given by x* = arg 0 max y [ 0 ( ) ( ) y h z g z dz + (1 – g(y)) h(y) – c(y, r)]. (40) the first-order optimality condition for (40) has the form h(x*) + (1 – g(x*)) h'(x*) = c(x*, r). (41) example 4. for the data of example 1, let g(z) = z/(1 + z). then it follows from (37) that 2 2 ( ) 1 2 3 k z z z r    = +    ; using formula (41) we calculate         − − + = 1 1 31 2 1* r r x   . • this paper considers the simple agent model with a risk-neural agent. hence, choosing between the incentive schemes (37) and (38), we should give preference to the lump-sum incentive scheme, as (a) it is simpler and (b) its ε-optimal analog stimulates the agent to fulfill the plan (see proposition 5). the main results of section 1 dedicated to static problems of contract theory are the analytical relationships (31) and(37), which allow to formulate and solve complex problems (particularly, the dynamical ones with changeable characteristics of agents and/or state of nature, e.g., the parameters of distribution). first, we will extend the model with one agent and additive uncertainty to the multiagent case (section 3). then, we will generalize the static simple agent model to the case of several sequential action periods (section 4). 3. multiagent model consider an organizational system composed of the principal and n subordinate agents with simultaneous and independent decision-making. denote by n = {1, …, n} the agent set and by ci(yi, ri) = c(yi, ri) the cost function of agent i; as before, yi ≥ 0 specifies the action of agent i and ri > 0 is his type. designate as y = i i n y   the total action of all agents. assume the principal is interested in a total result x ≥ 0 of the activity of all agents with a probability not smaller than a given threshold α [0; 1]. the value α is called contract reliability [[4]]. for the additive uncertainty model, this condition takes the form y ≥ x + n 1f − (α). (42) the value n 1f − (α) can be treated as payment for uncertainty in terms of agents’ activity. consider the following problem. what are the optimal plans for actions? using proposition 1, we obtain that for each agent the expected incentive cost of the principal coincide with the agent’s cost to choose a corresponding action (under the incentive scheme models of adaptation in dynamical contracts 119 copyright ©2018 assa. adv. in systems science and appl. (2018) (31) used by the principal, each agent receives a constant expected payoff regardless of his action; hence, by the hypothesis of benevolence each agent prefers plan fulfilment). since the cost functions of the agents are nondecreasing, in the optimal solution condition (42) holds as equality. hence, optimal plan calculation is reduced to the constrained optimization problem      += →    −   ni i ni x ii nfxx rxc i ).( min),( 1 }0{  (43) using the lagrange method of multipliers, we easily establish the following result. proposition 6: in the additive uncertainty model, the optimal plans {xi *} in the contract yielding a total result x ≥ 0 with a reliability α are given by * ix = c'–1(, ri), i  n, (44) where μ > 0 is the solution of the equation   −− += ni ii nfxxrc ).(),( 11   (45) as the distribution function is strictly monotonic, a direct analysis of problem (43) leads to proposition 7: in the additive probabilistic uncertainty model, the principal’s minimum cost to implement a given total result of agents’ activity does not decrease for higher contract reliability. example 5. consider the cobb–douglas cost functions for the agents, i.e., c(y, r) = 1  yμr1–μ, μ > 1. it follows from (44) and (45) that * ix = i j j n r r   (x + n 1f − (α)), i  n . (46) using the optimal plans (46), calculate the optimal value of the goal function: ( )    −  −         + = ni nj j ii r nfx rxc . )( ),( 1 1 *      (47) the right-hand side of (47) is not decreasing in α (see proposition 5). the payment for uncertainty (the difference between (47) and the value of the goal function in the corresponding deterministic problem) constitutes 1 1 ( ( )) ( )j j n x nf x r       − −  + −  , not decreasing for higher contract reliability. • 4. earned value model as a matter of fact, the earned value model is widespread in project management, both in theory and applications. in this section, we show a connection between the suggested models of contracts and the earned value model. consider the interaction of the principal and one agent within the scope of a certain project (a sequence of discrete periods). by a period t0 (called the project completion time), the principal has to implement a given total result x0 ≥ 0 of activity. let the states of nature { t}t=1,2,… in different periods be independent random variables obeying the same distribution f(∙). assume the principal concludes the optimal contract ( )tˆ z with the agent 120 m.v. belov, d.a. novikov copyright ©2018 assa adv. in systems science and appl. (2018) (see proposition 2) that satisfies (31) and specifies the agent’s reward depending on the result of his activity in period t, where t = 1, 2, … . the agent’s type and cost function are independent of periods. hence, under a given reliability α of each single-period contract, in each period the principal has to allocate the same plan to the agent given by (48)x0 = x0/t0 + 1f − (α) (compare with formula (42)). by (31), the agent benefits from plan fulfilment. the total plan for the agent’s activity by period t makes up 0 tx = t x0 – t 1f − (α) = tx0/t0. (49) in terms of the earned value approach of modern project management [[7], [16]], sequence (49) is called the budgeted quantity of work scheduled (bqws). since the result z = x0 –   of the agent’s activity in period  is a random variable, the total result xt achieved by a period t is also a random variable of the form xt = tx0 – 1 t    =  = t(x0/t0 + 1f − (α)) – 1 t    =  = = 0 tx + t 1f − (α) – 1 t    =  .(50) sequence (50) is called the actual quantity of work performed (aqwp). now, introduce other indices of the earned value approach subject to the additive uncertainty model (t = 1, 2, …, t) [[7]] as follows. – the expected planned cost of the principal, or the budgeted cost of work scheduled (bcws), defined by 0 tc = tс(x0/t0 + 1f − (α), r); (51) – the actual cost of the principal, or the actual cost of work performed (acwp), defined by ct = 1 0 )( t ˆ x        = − ; (52) – work underrun (in terms of time, positive or negative), defined by δ(t) = min {δ | 0 t tx x− = }; (53) – earned value (ev), or the budgeted cost of work performed (bcwp) as the planned cost of actually performed work, defined by t ec = ( ) 0 t tc − ; (54) – the current forecast t(t) of project completion time defined by t(t) = t0 + ε(t); (55) – total planned cost, also called the budget at completion (bac) or the budget cost (bc), defined by c0 = t0 с(x0/t0 + 1f − (α), r); (56) – the current linear estimate of total cost, defined by c(t) = t(t) ct / t; (57) – the actual project completion time, defined by t’ = min {t ≥ 0 | xt ≥ x0}; (58) – the difference between the actual and budgeted cost, or cost overrun (co), defined by ∆сe(t) = ct – ct e; (59) – schedule performance index (spi), defined by at = t ec / 0 tc ; (60) – cost performance index (cpi), defined by bt = t ec /ct. (61) models of adaptation in dynamical contracts 121 copyright ©2018 assa. adv. in systems science and appl. (2018) the budgeted cost indices (48)–(61), which are traditionally divided into primary (48)– (52) and derived (53)–(61), are efficient tools for project management, at the stages of planning and implementation. example 6. for the data of example 1, let r = 1, t0 = 100, x0 = 100, ∆ = 1, and α = 0.2. the trajectories of the total plan (49), the total result (50) and the expected result xt = 0 tx + t( 1f − (α) – e θ)) are illustrated fig. 1. simulation was performed in rds software complex [[15]]. fig. 1. dynamics of results in example 6 the dynamics of the planned and actual cost and also of earned value are shown in fig. 2. the calculations yielded t’ = 145, bac = 72, and ∆сe(t’) ≈ 60%. • fig. 2. dynamics of cost and earned value in example 5 5. adaptation model incentive problems in dynamical organization systems can be classified using different bases such as the relationship between periods, the foresight of system participants, the mode of decision-making and others, see [[11]]. in this section, we first introduce a classification of dynamical incentive problems with a single-time change of a model parameter, e.g., the principal’s goal function, the agent’s cost function or the distribution function of activity results. such a change occurs at a time td and will be further called a discord. by assumption, the system participants are short-sighted: in each period, they make decisions for this period only and do not consider the consequences in future periods. then we study a model in which a discord occurs for the distribution function while the time of discord is unknown to the principal and agent. the changes in the behavior of the system participants after total plan (black), expected result (blue), and total result (red) periods planned cost (black), actual cost (red), and budgeted cost (blue) periods 122 m.v. belov, d.a. novikov copyright ©2018 assa adv. in systems science and appl. (2018) detection of the discord can be treated as their adaptation to the new operating conditions [[8], [18]]. 5.1. a classification of dynamical incentive problems in table 1 we present all possible cases for the awareness of the principal and agent about the new functions (income, cost, distribution) after a discord occurs. these functions will be denoted using appropriate symbols with tilde. assume that, before choosing their decisions for a period t, the principal knows the history zt–1 = (z1, …, zt–1) while the agent the histories zt–1 and yt–1 = (y1, …, yt–1). the awareness of the principal and agent are shown in columns 3 and 4 of table 1. 1-2. a change of the principal’s income function ( )( ~ )( zhzh → ) is considered in rows 1 and 2 of table 1. if the principal knows the new income function and the discord time td (row 1 of table 1), then the problem is reduced to a set of typical static incentive problems studied in section 1, which are solved for each period independently. however, if the new income function )( ~ zh or the discord time td are not available for the principal, the problem makes no sense: the principal does not have enough information for decision-making. 3-9. a change of the agent’s cost function ( )(~)( ycyc → ) is considered in rows 3–9 of table 1. if the agent and principal both know the new cost function and the discord time (row 3 of table 1), then the problem again is reduced to a set of typical static incentive problems, which are solved for each period independently. assume the principal is aware of the new cost function of the agent but has no information about the discord time (row 4 of table 1). in this case, his rational behavior is to offer the agent a menu of optimal contracts for a set of possible cost functions. this is the screening principle used under asymmetric awareness [[13], [21]]). if the agent knows the new cost function and the discord time while the principal neither of them (row 5 of table 1), the problem makes no sense due to the following. for obtaining a positive payoff himself, the principal has to stimulate the agent’s activity by offering a contract with a nonnegative payoff of the latter. yet, being unaware of the agent’s cost function, the principal cannot form such a contract. for the same reasons, the problems with the awareness described by rows 8, 12, and 15 of table 1 are ill-posed: the principal does not know the new cost function )(~ yc or the distribution function ),( ~ yzf . models of adaptation in dynamical contracts 123 copyright ©2018 assa. adv. in systems science and appl. (2018) table 1. classification of dynamical incentive problems n o. discord agent knows principal knows problem 1 )( ~ )( zhzh → no matter td; )( ~ zh typical 2 nothing or td ill-posed 3 )(~)( ycyc → td; )(~ yc td; )(~ yc typical 4 )(~ yc solvable by screening 5 nothing ill-posed 6 )(~ yc td; )(~ yc typical 7 )(~ yc typical 8 nothing ill-posed 9 nothing no matter ill-posed 1 0 ),( ~ ),( yzfyzf  → td; ),( ~ yzf td; ),( ~ yzf typical 1 1 ),( ~ yzf solvable by screening 1 2 nothing ill-posed 1 3 ),( ~ yzf td; ),( ~ yzf typical 1 4 ),( ~ yzf d1 1 5 nothing ill-posed 1 6 nothing no matter ill-posed let the principal know the new cost function of the agent and the discord time while the agent merely this function (row 6 of 124 m.v. belov, d.a. novikov copyright ©2018 assa adv. in systems science and appl. (2018) table 1). this situation leads to a set of typical problem. really, using complete information the principal will offer a contract with c(y) before the discord and with )(~ yc after it; in turn, the agent can identify the discord time by the changing offers of the principal and then respond optimally. again, we obtain a set of typical static incentive problems. now, consider the case in which the principal and agent are both aware of the new cost function but the discord time is uncertain (row 7 of table 1 models of adaptation in dynamical contracts 125 copyright ©2018 assa. adv. in systems science and appl. (2018) table 1). recall that the agent’s cost function is continuously differentiable and strictly monotonic. hence, observing his actual cost, the agent can detect the fact of discord (if any); more specifically, the agent reliably detects the change of the cost function at the end of the period after the discord, in which the agent chooses some action y such that )(~)( ycyc  . however, in this period the agent chooses his action without any knowledge of the cost function. dealing with the short-sighted agent, the principal has to offer a contract associated with the worst-case cost function for the agent, i.e., with the function ( ) max{ ( ); ( )}с̂ y c y c y= , before the discord and after it including detection by the agent. thus, we arrive at a typical problem with additional cost of the principal, which can be assessed by both players. if the agent does not know the new cost function, the problem is ill-posed regardless of the principal’s awareness (row 9 of table 1). in this case, he cannot estimate possible loss in several periods until the new cost function is identified. the agent prefers zero action accordingly. 10-16. a change of the distribution function of activity results or of the distribution function of the state of nature ( ),( ~ ),( yzfyzf  → ) is considered in rows 10–16 of table 1. assume the principal knows the new distribution function ),( ~ yzf and the discord time while the agent at least ),( ~ yzf (rows 10 and 13 of table 1). then the problem again is reduced to a set of typical static incentive problems. let the agent be aware of both the new distribution function ),( ~ yzf and the discord time and let the principal be aware of ),( ~ yzf only. here a sequential screening problem arises naturally (row 11 of table 1). if the agent does not know the new distribution function ),( ~ yzf , the problem is ill-posed regardless of the principal’s awareness (row 16 of table 1). in this case, he cannot estimate possible loss or payoff in several periods until the new distribution function ),( ~ yzf is identified. the agent prefers zero action accordingly. model d1 (row 14 of table ) is a multiperiod contract model with a change of the distribution function ),( yzf at some time. in this model, the principal and agent both know the new distribution function ),( ~ yzf and also expect a single change of this function. but they are a priori unaware of the discord time. before period 1 of their interaction, the agent and principal have no information about the discord except the prior. hence, they have to act under the hypothesis that in period 1 the result corresponds to ),( yzf or ),( ~ yzf . dealing with the short-sighted agent, the principal has to offer a contract associated with the worst-case cost function for the agent (otherwise, the agent rejects the contract and both players do not receive new information about the state of nature, which makes further interaction unreasonable). if the principal forms such a contract and the agent is rational, then the former can predict the action y of the latter. the result z is observed by the principal, and hence the principal has the same posterior information as the agent. this fact can be generalized as the principle of transparent stimulating contract and formulated in the following way. assume the principal can form a stimulating contract while the agent is rational and does not have strategic behavior; then the principal can reliably predict the agent’s actions and use the same complete information as the agent. hence, after period 1 the principal can design a contract for period 2 using the prior knowledge about the functions ),( yzf , ),( ~ yzf and also using the observations z1, y1. in subsequent periods t, the principal will act in the same way, forming contracts t(z) based on the available functions ),( yzf , ),( ~ yzf and the observations zt–1, yt–1. 126 m.v. belov, d.a. novikov copyright ©2018 assa adv. in systems science and appl. (2018) for the short-sighted principal, an alternative is to use an optimal contract given the distribution f(z, y) until he detects the discord (this situation is studied in detail below for model d1). note that, owing to identical awareness, the principal and agent detect the discord “simultaneously.” depending on the availability of additional prior information for players, there may exist several contract design statements. if both players have prior information about the distribution of the discord times, it is possible to obtain a sequential optimal bayesian algorithm to design the contract t(z). finally, if both participants possess no additional information about the possible discord times and their interaction can be terminated in any period, then the minimax approach is optimal. 5.2. discord problem (problem d1) consider the multiperiod simple agent model with unconnected periods [[10]] and let the principal use the optimal incentive scheme (38) with the optimal plan (40). assume that initially the principal and agent have the same information about the distribution function g(∙). at some time td > 0 a discord occurs, which changes the distribution g(∙) to ( )g  . the new distribution ( )g  is a priori known to the principal and agent, but none of them have information about the discord time. as the result of this discord, in a single period the expected utility of the agent varies by the value ∆f(g(∙), ( )g  ) = c(x*) * * * ( ) ( ) 1 ( ) g x g x g x − − . (62) choosing in each period the action x*, the agent (as well as the principal) observes the sequence of results zt (also the agent observes the sequence yt = (x*, …, x*)). both players have to decide whether a discord occurs or not. the sequential problem is therefore decomposed into single-period problems with additional information (owing to independent periods). for period t, the agent and principal renegotiate the contract using information about the possible distributions )(g , )( ~ g and the additional observations zt–1. define the value lt = ln( ( , ) ln( ( , ))t t t tg z y g z y− to formulate the optimal sequential maximal likelihood rule for discord detection as follows. in each period t > 0, calculate lt (l0 = 0) by 1 1 1 0, if 0, , if 0. t t t t t t t l l l l l l l − − − +  =  + +  (63) if for some period lt > , then the discord takes place. here the value  describes the error characteristics of the first and second kind. as is well-known [[19]], in comparison with other statistics the maximal likelihood statistic yields the most efficient decision rule in terms of the following criterion. one of the errors is detected with a probability not smaller than a given threshold while the other error is optimized. the threshold δ is chosen using the error characteristics of the first and second kind. example 7. for the data of examples 1 and 4, let g(z) = z/(β1 + z) and ( )g z = z/(β2 + z). then ez(y) = βi ln (1 + y/βi). using condition (41), we find x* = 4 1 1 2 ( -1) r r r        − − −     . expression (38) takes the form models of adaptation in dynamical contracts 127 copyright ©2018 assa. adv. in systems science and appl. (2018) σc(x*, z) = * 2 * * * ( ) ( ) , , 2 0, . x x z x r z x    +     (64) according to the principle of transparent stimulating contract, the agent always chooses the action         − − −−== 1 )1( 4 1 2 1 1*     yr r rxy . before the discord time, the result of the agent’s actions obeys the distribution 2 1 1* * 1 1* )( )(),( z xz x xzg + +− + =      for z  [0, x*]; after the discord time, the distribution 2 2 2* * 2 2* )( )(),(~ z xz x xzg + +− + =      for z  [0, x*], where () denotes the delta-function. then *12 1 2 * *2 1 * 1 2 ln 2ln for [0, ), ln 2ln for . t t t t z z x z l x z x x            + +      +     =     + + =    +    take t = 500, td = 200, r = 1, γ = 10, β1 = 100, and β2 = 60. the dynamics of the planned, expected and actual results (in cumulative sums) are shown in fig. 3 (dashed lines correspond to the dynamics without discord detection). the dynamics of the cumulative cost of the agent and the principal’s incentive cost are illustrated in fig. 4 (dashed lines correspond to the dynamics without discord detection). fig. 3. dynamics of cumulative results in example 7 planned result(black), expected result (blue), and actual result (red) periods 128 m.v. belov, d.a. novikov copyright ©2018 assa adv. in systems science and appl. (2018) fig. 4. dynamics of cumulative cost (agent, principal) in example 7 in the cumulative sum method, we may use the discord indicators defined by 1 1 ( ) t *ts t ezz x  = = − (65) or * * 2 1 ( , ) ( , ) t t cs x z tc x r   = = − . (66) their dynamics in our example can be observed in fig. 5. let us apply the maximal likelihood rule (63). the corresponding dynamics are presented in fig. 6. the means of the statistic lt before and after the discord are –0.04 and +0.04; the root-mean-square deviations are –0.29 and 0.27, respectively. note that, before the discord, the statistic lt takes the value * 2 1 * 1 2 ln ln 0.29 x x        + + = −    +    with the probability 1 * 2 0.5 x   = + ; after the discord, with the probability 2 * 2 0.38. x   = + fig. 5. dynamics of discord indicators s1 and s2 in example 7 in our example, for δ = 2 the discord is detected after 10 periods since its occurrence. • agent’s actual cost (black) and principal’s incentive cost (red) periods discord indicators: s1 (red) and s2/100 (blue) periods discord indicator models of adaptation in dynamical contracts 129 copyright ©2018 assa. adv. in systems science and appl. (2018) fig. 6. dynamics of discord indicator in example 7 6.conclusions in this paper, we have considered contracts between the short-sighted principal and agents that operate under external probabilistic uncertainty (knight’s measurable uncertainty [[25]]) with changeable characteristics (a reflection of knight’s true uncertainty). a proper response to true uncertainty is a basic function of control subjects for the adaptive behavior of subordinate structural elements of activity [[1], [20]]. among the promising lines of future research, we mention other descriptive methods for the influence of external uncertainty on activity results, contract renegotiation conditions for the longsighted principal and agents, and discord problems in multiagent dynamical organizational systems. references [1] belov, m. & novikov, d. (2017). strukturnye modeli kompleksnoi deyatel’nosti [structural models of complex activity]. moscow, russia: fizmatlit [in russian]. [2] novikov, d. (ed.) 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(1997). lectures on the theory of contracts and organizations. chicago: univ. of chicago. http://restud.oxfordjournals.org/ adv syst sci appl 2018; 03:17–28 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/583 visual odometry approaches to autonomous navigation for multicopter model in virtual indoor environment alexander v. abdulov1, alexander n. abramenkov1 ,∗andrey a. shevlyakov1 1 institute of control sciences of the russian academy of sciences, moscow, russia abstract: using simulators and virtual environments can simplify the development and testing of various control, behavior and navigation scenarios significantly. in this article we describe our experience using the ros environment and the gazebo simulator to develop an autonomous multicopter uav that can navigate an unknown building and locate an exit using computer vision techniques. keywords: virtual simulation, visual odometry, uav, autonomous navigation 1. introduction as more powerful embedded computers appear, along with smaller cameras, autonomous indoor navigation for mobile robots becomes a sufficiently interesting and tractable problem, inciting new research [8, 13]. since global positioning signal is usually unavailable indoors, mobile robots face hard problems of determining their position in reference to some external beacons, route planning and actuating movement. the essential difficulty is not navigation per se, but locating beacons using available sensors. autonomous mobile robots may face additional challenges in dynamic environments. today’s technology allows for relatively easy development of autonomous mobile robots that can navigate using only onboard sensors [1, 9]. however, there are no universal algorithms for navigation in unknown scenes with dynamic obstacles. simulation software exists to facilitate the development and lower the entry threshold of knowledge (e.g. diy robotics). well-designed simulators (see gazebo [11]) make rapid testing possible, as well as robot model building and their deployment in realistic scenarios. in this article we share our experience if solving a problem of autonomous navigation in virtual environment using widespread visual odometry methods. we proceed with the problem and scenario description, which is considered as a test ground for two different approaches. we discuss strong points and limitations of the methods, as well as evaluation of their applicability and overall difficulties of this type of research. ∗corresponding author: aash29@gmail.com 18 a.v. abdulov, a.n. abramenkov, a.a. shevlyakov 2. problem statement as an example of a navigation problem, we chose the following mission: a multicopter of a specified model must autonomously locate a window in an unknown building and leave via this window, without colliding with obstacles or using any kind of external positioning system. onboard sensors include imu (accelerometers and gyroscopes) and computer vision sensors (lidar and video cameras). building in question is an enclosed virtual hangar with one open window (see fig. 2.1). the hangar has a rectangular plan. texture, size and color of walls may vary. obstacles are represented by cylindrical pillars of varying height, diameter and color. pillars are static, but their position is random. they may also be combined to form more complicated structures. the window is rectangular, although its position and dimensions are not known. as the mathematical models of sensors and whole uavs are usually available, we shall not pay too much attention to the hardware. for our research we chose a quadrotor scheme (fig. 2.2) as the most widespread and small enough (less than 0.5 m in dia) to operate indoors. a multitude of virtual sensors can be combined and installed onboard. virtual multicopter movement is controlled by feeding an embedded controller the reference values of linear and angular velocity. the make of embedded controller is fixed, thus we implement the autonomous movement with higher-level pid controllers that have to be tuned separately. fig. 2.1. top view of the virtual hangar fig. 2.2. multicopter model copyright c© 2018 assa. adv syst sci appl (2018) visual odometry approaches to multicopter autonomous navigation 19 3. proposed solution 3.1. technology simulation is run on ubuntu 16.04 os with ros kinetic installed. ros (robot operating system) is a framework for robot programming. it is based on graph architecture, where data processing is performed in the nodes that can receive and transmit messages to other nodes. there are many useful packages for ros, which can also be written from scratch using standard interface. c++ and python are supported. we should mention separately the mavros package that provides the data exchange interface between ros and autopilot firmware using mavlink (micro air vehicle link) protocol, which is designed for uav communication. as a simulator we chose the gazebo software due to the availability of necessary uav and sensor models, as well as ros integration. additionally, we use opencv library for image processing and dlib library for numerical optimization. 3.2. flight strategy high-level strategy for a chosen uav model is described as a finite state machine and contains the following steps (see fig. 3.3a). 1. climb to a given altitude. 2. survey the area. on this step the multicopter rotates (up to 360 degrees). two tasks are accomplished – searching for window and building a local obstacle map. 3. if a window is found, multicopter changes heading to face the window, and next step is performed; otherwise, multicopter flies forward in the direction chosen according to the algorithm and surveys again (step 2). 4. head to the window avoiding obstacles and exit the building. local obstacles are mapped using frontal laser lidar (see fig. 3.3b). for each heading we save the distance to the nearest obstacle and add a safety margin which allows to avoid collisions. a new direction of movement is chosen as the clearest one and closest to the course set. on a lower level, we must control and stabilize the uav model by flight altitude, yaw angle and linear velocity. (a) flowchart of states change (b) obstacles are grey, safety margins are pink fig. 3.3. flight strategy and local obstacles map copyright c© 2018 assa. adv syst sci appl (2018) 20 a.v. abdulov, a.n. abramenkov, a.a. shevlyakov 3.3. angle stabilization orientation estimate obtained via mems gyroscopes can accumulate error due to “zero drift” problem. algorithms exist for fusing different sensors (accelerometers, gyroscopes and magnetometers) to refine estimates. for example, madgwick ahrs [12] algorithm or extended kalman filter [14]. we should note that the model firmware used imu readings to interpret the control signals. in case of yaw angle error this feature did not allow to control the uav correctly and had to be accounted for while specifying control signal. since in case of simulation the magnetometer readings were relatively accurate, they were used to rectify the yaw estimate. 3.4. altitude stabilization it is reasonable to use a downward-facing range-finder (altimeter) to stabilize altitude. an ultrasound range-finder can be used, but a more narrow-bear laser range-finder is preferable. additionally, orientation of the aircraft must be accounted for. on the basis of this estimate, it is relatively simple to tune the coefficients for a pid controller which can adequately stabilize the reference altitude (see fig. 3.4). here and below, we denote: target – reference value; real – ground truth, obtained from simulator; pid – sensor estimate, fed to the pid controller. 3.5. velocity stabilization in gps-denied indoor spaces we need to use other information to obtain current velocity estimate. in principle, imu can be used, but due to large errors it needs to be amended by other means. visual odometry is a promising approach [3]. a sequence of images captured by camera can be a main or auxiliary source of position information. depending on number of cameras and their configuration, various methods can be used. we’ll consider two approaches, mono and stereo visual odometry, that necessitate installing one or two calibrated cameras respectively. fig. 3.4. flight altitude stabilization by pid copyright c© 2018 assa. adv syst sci appl (2018) visual odometry approaches to multicopter autonomous navigation 21 fig. 3.5. perspective projection camera model 3.6. camera model visual odometry operates on a basis of a camera model. often this is a relatively simple perspective projection model (see fig. 3.5), represented by a matrix (3.1) of intrinsic parameters (focal length and principal point). in more complicated cases, distortion transformations are also accounted for. c = ∣∣∣∣∣∣ fx 0 cx 0 fy cy 0 0 1 ∣∣∣∣∣∣ (3.1) usually the intrinsic parameters are determined by camera calibration. for a virtual camera we chose image center as the principal point (cx, cy) and obtain focal length f = fx = fy from equation (3.2), knowing resolution s and field of view (fov) angle α. α = 2 · arctan s 2 · f (3.2) equation (3.3) gives us the relation of point (x, y, z) in 3d scene and its projection (u, v) on image plane up to a scale coefficient w. this is a composition of rotation matrix r and translation vector t . w · ∣∣∣∣∣∣ u v 1 ∣∣∣∣∣∣ = c · |r t | · ∣∣∣∣∣∣∣∣ x y z 1 ∣∣∣∣∣∣∣∣ (3.3) 3.7. monocular visual odometry consider a case of one camera, laser altimeter and 6-axis imu. if a camera is downward-facing, then we can assume that most of the observed feature points belong to one plane. this is a reasonable assumption if the flight altitude is low, and it was almost always valid in our simulation. knowing altitude h and camera orientation quaternion qimu , we can compute floor plane equation a · x+ b · y + c · z + d = 0 coefficients, as indicated in (3.4). first three coefficients are obtained by rotation operation ◦ a unit vector pointing along an axis z. (a, b, c) = q−1 imu ◦ (0, 0, 1); d = −h · c (3.4) copyright c© 2018 assa. adv syst sci appl (2018) 22 a.v. abdulov, a.n. abramenkov, a.a. shevlyakov (a) keypoints optical flow with homographic filtering between two consequent images of floor; matched points in green, filtered points in red (b) keypoints flow projection on the floor plane fig. 3.6. monocular odometry approach to velocity estimation taking coordinates (u, v) of floor image feature points and substituting (3.5) in (3.6), we reconstruct their position (x, y, z) in camera frame. since frame used in aircraft firmware does not match the camera frame, some axes and signs are inverted in equations for x and y. û = u− cx fx ; v̂ = v − cy fy ; (3.5) z = d a · û+ b · v̂ + c ; x = −z · v̂; y = −z · û (3.6) as a result, linear velocity estimate is carried out using feature keypoints position (fast [5], good features to track [7]) from to consequent measurements. correspondence between feature points can be found, for example, by optical flow or by comparing descriptors [4]. we used a well-established lucas-kanade algorithm [2], due to its modest computational requirements and adequate performance in case of small movements. the implementation of this algorithm is available in opencv library. it is useful to apply additional filters to reject outliers: mean velocity, direction, etc. an assumption of all points belonging to one plane allows us to successfully use the filter based on homography matrix (see fig. 3.6a), obtained on ransac [10] method. this approach is schematically depicted on fig. 3.6b. in this case, multicopter speed is towards negative direction of axis y . thus, projections of feature points are moving in the opposite direction. red line is normal distance to the floor, blue line is its estimate by altimeter and imu readings, magenta plane is the observed floor plane, cyan markers are feature points and their movement. in general, monocular approach allows us to estimate velocity while flying at low altitude (less than 1 m), and is computationally inexpensive. flying higher leads to poor performance. this is due to two factors: 1. aircraft rotation at higher altitude leads to more displacement for optical flow to handle, which leads to more errors and outliers. 2. flying higher increases the probability of obstacles in view field, which invalidates the assumption of all points belonging to the floor plane. here the task of odometry is not fully solved. only velocity is estimated, and then stabilized by a suitably tuned pid controller, without trajectory construction. copyright c© 2018 assa. adv syst sci appl (2018) visual odometry approaches to multicopter autonomous navigation 23 fig. 3.7. velocity stabilization via monocular odometry approach therefore, the accuracy of this method makes it possible to successfully use it. speed estimates are adequate, as can be seen on fig. 3.7. stabilization time is acceptable, although overshooting is present. 3.8. stereo visual odometry stereo visual odometry is an alternative approach to estimating velocity. in this work we used a stereoscopic method based on reprojection error minimization [6]. it consists of several steps: 1. for each feature point p 3d need to find correspondence between its projections on current left and right frames of a stereo camera (pl t, p r t ) and on one of the previous frames (pl t−1). this can be seen in fig. 3.8. as in monocular case, we use lucas-kanade optical flow. 2. using the established correspondence between the projections of all feature points on the left (ul, vl) and right (ur, vr) frames of a stereo camera with base b, the triangulation problem (3.7) is solved to restore their position (x, y, z) in 3d space (see fig. 3.9). (x, y, z) = b ul − ur · (ul − cx, vl − cy, f) (3.7) 3. knowing current position on each feature point p3d i,t in 3d space, and its projection pl i,t−1 at a previous time step, linear displacement t and rotation r are estimated by minimizing (3.8) reprojection error. argmin r,t ∑ i ‖pl i,t−1 − ĉ(r · p3d i,t + t )‖ (3.8) equation (3.8) contains a projection operator ĉ that maps a point from 3d space to the image plane according (3.3). we used the euclidian distance ‖·‖ as a norm. this approach is more versatile, and does not demand additional hardware apart from two properly fixed cameras (downward or forward facing). the main limitation is the necessity of multiple observed feature points. their allowed distance depends on stereo camera base. copyright c© 2018 assa. adv syst sci appl (2018) 24 a.v. abdulov, a.n. abramenkov, a.a. shevlyakov fig. 3.8. schematic representation of 3d point projection on stereo camera matrix during movement. cl t, cr t – left and right cameras at time t. cl t−1, cr t−1 – left and right cameras at time t− 1 fig. 3.9. triangulating p for a stereo camera with base b stereo visual odometry approach is also more computationally expensive – for the same video parameters (648x486, 36fps) processing was 2-3 times longer. in our case, monocular odometry allowed for real time computation, while stereo odometry had to skip a half of available frames. it is possible, however, to decrease the precision in order to speed up the computations. in the examples considered, forward facing camera did not always perform well due to lack of texture on pillars which yielded few feature points. downward facing camera was more stable, although there was a minimum necessary altitude that allowed both cameras to see the same objects. copyright c© 2018 assa. adv syst sci appl (2018) visual odometry approaches to multicopter autonomous navigation 25 fig. 3.10. velocity stabilization via stereo odometry approach fig. 3.11. errors of calculated velocities for monocular and stereo approaches, error = |pid −real| figure 3.10 shows the velocity stabilized with tuned pid controllers via the estimate obtained by this approach. velocity estimation is somewhat better than in monocular case – the plots of real and pid nearly match (see fig. 3.11). 4. search for window to detect a window, we used color segmentation. assuming that the background visible through the window was a different color than the rest of the scene, we needed to find a suitably colored shape (see fig. 4.12a). furthermore, shape filtering, e.g. rectangle detection (see fig. 4.12b), can be used. at a last stage, when the aircraft approaches a window, it can become disoriented as number of feature points decreases or they disappear entirely. in this case, we solve a ballistics problem – aim for the window and fix a exit speed, disabling regular velocity controllers. copyright c© 2018 assa. adv syst sci appl (2018) 26 a.v. abdulov, a.n. abramenkov, a.a. shevlyakov (a) color segmentation and threshold binarization (b) specifically-colored rectangle searching fig. 4.12. searching for window (a) simple (b) difficult fig. 4.13. examples of the scenes 5. simulation results as a result of several simulation trials, we established that the described approach solves a problem of exiting the building through the window. two scenes of different layouts were explored. simple scene dimensions are 40x50x10 m (see fig.4.13a). obstacles are represented by cylindrical pillars of different radii. aircraft could detect the window directly from the starting point and approached it while avoiding the obstacles. difficult scene dimensions are 50x90x15 m, obstacles are simple pillars and composite objects of several different-sized cylinders, imitating trees. (see fig. 2.1, 4.13b). in this case, obstacles blocked the window completely, thus search strategy had to be activated. the parameters chosen resulted in a fairly conservative behavior – there was no collisions, but search required a sufficiently long time. thus, a relatively simple search algorithm, that didn’t map the scene or localize of the aircraft, could accomplish the task of exiting the building. 6. technical difficulties we encountered some complications while using the gazebo simulator, which should be accounted for. notably, high computational requirements, which resulted in unstable work. sometimes, a kind of “numerical wind” was apparent, a seemingly random perturbations that forced the aircraft to crash. copyright c© 2018 assa. adv syst sci appl (2018) visual odometry approaches to multicopter autonomous navigation 27 the effect could be reproduced by running both client and server parts of the simulator on the same medium-powered computer. for all our experiments we used an intel core i7-4510u cpu @ 2.00ghz, 8gb ram pc. additional disturbances were created by other applications running in the background. practically, gazebo is not the most comfortable software to use in case of multiple test runs. each run required restarting the whole virtual environment and took a lot of time, which combined with the “numerical wind” made our work more difficult – especially the tuning of pid controllers. we encountered various other difficulties, including the necessity to convert between every coordinate frame involved (world, aircraft, sensor). this, however, is anticipated by mavlink protocol which offers the possibility of specifying control in several coordinate frames (ned, enu, etc.) 7. results numeric simulation using the ros framework, the gazebo simulator and the rest of the corresponding ecosystem allowed us to test the applicability of the chosen algorithms in specified conditions. use of this class of simulators and frameworks can be of much help while developing real hardware and software platforms for uavs. references [1] a. weinstein, a. cho, g. loianno, & v. kumar. (2018). visual inertial odometry swarm: an autonomous swarm of vision-based quadrotors. ieee robotics and automation letters, 3 (3), 1801–1807. [2] b.d. lucas, & t. kanade. (1981). an iterative image registration technique with an application to stereo vision. international joint conference on artificial intelligence, 647–679. [3] c. cadena, l. carlone, h. carrillo, y. latif, d. scaramuzza, j. neira, . . . j.j. leonard. (2016). past, present, and future of simultaneous localization and mapping: towards the robust-perception age. ieee transactions on robotics, 32 (6), 1309–1332. [4] d. bekele, m. teutsch, & t. schuchert. (2013). evaluation of binary keypoint descriptors. in 2013 ieee international conference on image processing (pp. 3652–3656). doi:10.1109/icip.2013.6738753 [5] e. rosten, & t. drummond. (2006). machine learning for high speed corner detection. 9th european conference on computer vision, 1, 430–443. [6] h. badino, & t. kanade. (2011). a head-wearable short-baseline stereo system for the simultaneous estimation of structure and motion. iapr conference on machine vision applications (mva),nara, japan. [7] j. shi, & c. tomasi. (1994). good features to track. proceedings of ieee conference on computer vision and pattern recognition, 593–600. [8] k. mo, h. li, zh. lin, & j. lee. (2018). the adobeindoornav dataset: towards deep reinforcement learning based real-world indoor robot visual navigation. arxiv preprint arxiv:1802.08824. [9] l. silvestri, l. pallottino, & s. nardi. (2017). design of an indoor autonomous robot navigation system for unknown environments. in international conference on modelling and simulation for autonomous systems (pp. 153–169). springer. [10] m.a. fischler, & r.c. bolles. (1981). random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. commun. acm, 24 (6), 381–395. doi:10.1145/358669.358692 copyright c© 2018 assa. adv syst sci appl (2018) https://dx.doi.org/10.1109/icip.2013.6738753 https://dx.doi.org/10.1145/358669.358692 28 a.v. abdulov, a.n. abramenkov, a.a. shevlyakov [11] n. koenig, & a. howard. (2004). design and use paradigms for gazebo, an open-source multi-robot simulator. proceedings of 2004 ieee/rsj international conference on intelligent robots and systems, 2149–2154. [12] s. madgwick. (2010). an efficient orientation filter for inertial and inertial/magnetic sensor arrays. retrieved from http : //x io . co .uk/ res/doc/ madgwick internal report.pdf [13] sh. yang, s.a. scherer, x. yi, & a. zell. (2017). multi-camera visual slam for autonomous navigation of micro aerial vehicles. robotics and autonomous systems, 93, 116–134. [14] x. li, m. chen, & l. zhang. (2016). quaternion-based robust extended kalman filter for attitude estimation of micro quadrotors using low-cost mems. control conference (ccc), 2016 35th chinese, 10712–10717. copyright c© 2018 assa. adv syst sci appl (2018) http://x-io.co.uk/res/doc/madgwick_internal_report.pdf http://x-io.co.uk/res/doc/madgwick_internal_report.pdf introduction problem statement proposed solution technology flight strategy angle stabilization altitude stabilization velocity stabilization camera model monocular visual odometry stereo visual odometry search for window simulation results technical difficulties results adv syst sci appl 2018; 03; 79-89 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/653 simulation modelling as a tool for design and development in large-scale automated systems: smart city application in terms of lack of statistical information dmitri razumov1*, prof., dr. (eng.) vladimir aleshin2 1) dept. school of it management russian academy of national economy (ranepa) moscow, russia e-mail: dmitrirazumov@yandex.ru 2) dept. school of it management russian academy of national economy (ranepa) moscow, russia e-mail: aleshin_vladimir@mail.ru abstract: design and development of large-scale automated systems (as) is always associated with overcoming a large number of uncertainties, which structuring often through the using of practice standards and modeling at the systems & software engineering. methods of analysis of large systems (ls) involve their decomposition in the context of structural and/or functional paradigms, which allow us to consider the obtained components as objects for modelling: functional modelling, simulation etc. design the automated control system (acs) of region (smartcity project) need to optimize the decisions. in this paper we propose the new methods of determining the parameters of project in terms of lack statistical information based on the simulation model. keywords: acs automated control system, life cycle, modeling, smart city. 1. introduction already at the initial stages of systems engineering, different models and methods of mathematical science are used. these are: mathematical statistics, discrete mathematics, game theory etc. advantages in area of information technology has allowed to bring to the practice of creating ls such powerful tools as simulation. in many cases simulation is the only one way to get any notion of the behavior of a complex system and to conduct its analysis. the acs in the crisis and emergency situations is a big complex system (bs), in the nodes of which are located decision-making centers, related with the aid of information flows with operating forces and resources from the state and municipal institutions. the complexity is due to the fact that the organization of such systems does not have a straightforward structural implementation and is determined by many factors (availability of communication channels, level of automation, duty services, interaction between departments, the level of financial support, etc.). therefore the design of such bs is connected with the decomposition of the general problem at the level of paradigm life cycle (lc) [1] in the context of structural-functional component hierarchy, which have a nodes (fig. 1) as the duty dispatcher service (dds) or command control centers(ccc). the problem of simulation of the overall structure of the system is the lack of sufficient statistical * corresponding author: dmitrirazumov@yandex.ru 80 d. razumov, v. aleshin copyright ©2018 assa adv. in systems science and appl. (2018) information for all dds. the authors propose a method, which ensures a sufficiently adequate model in these limits. 2. functional and structural analysis for designing acs of region the function "to manage the life cycle stage" on the stage of the lifecycle "design” solves a non-trivial problem of building a hierarchy of command control centers and decision-support centers. the situation with organization of this structure differs from region to region. as a rule the hierarchy of divisions in the area is formed from the newly created command control centers (ccc) at the level of municipal districts from the unified duty dispatcher service (udds) and existing ccc (utilities and other emergency services, fig. 1). fig. 1. structural analysis of system often due to lack of technical capacity information can be transmitted not to the main samples but in any of the duty services. the task of designing the structures contains many elements of uncertainty. the requirements of operators, resources, emergency response number, power and performance of computer systems and throughput of communication channels in the project document do not have a clear quantitative and scientific basis and developed at best on the basis of comparative assessments of the operational histories of the existing call centers (cc). however it is not a sufficient basis for parameters definition of the designed system. underestimation of requirements for operators, resources, the quantity and quality of technical means cannot lead to lower response time, but rather its can increase it. such approaches may lead to waste of millions of dollars during the construction of such systems. 3. task definition for simulation modelling we need to offer a science-based method of assessment requirements for the resources ccc/udds, guaranteeing the fulfillment of key performance indicators (kpi) the system functioning in framework the main operational process, under limited amount of statistically reliable data and uncertain structure that represents a hierarchy of decision-support centers, communicate links and the management links. simulation modelling as a tool for design and development 81 copyright ©2018 assa. adv. in systems science and appl. (2018) 3.1. building the base model class of ccc as already noted, ccc is defined as the basic element of the system in context of structuralfunctional decomposition. the ccc model is used also to model the main operational process which uses a discrete event simulation model on the basis of libraries of standard classes of the system modeling anylogic. anylogic allows presenting the ccc functionality in an object-oriented paradigm. discrete event simulation model on the basis of libraries of standard classes anylogic is used for modelling the main process, which implements the basic blocks at the level of mechanisms described in the theory of mass service: event sources, the queue, processing servers, etc. you can pick up an optimal set of characteristics for the compilation of parameters for the specifications a particular system by manipulating the parameters of the objects based on the implementation of these classes, as well as analyzing the statistical dependencies. the designed system, as already noted, is a hierarchical structure of control centers interacting with each other to coordinate control of forces and means of response. 4. methods for evaluation of the stochastic characteristics of the model as the main stochastic characteristics of the basic simulation model of the considered functional unit (ccc) allocated: • intensity of input flow requests; • the length of the intervals between user requests of call centers automated dispatch control system (adcs) one of the ccc; • the temporal characteristics of requests processing; • actions of emergency response services. for each of these parameters were applied a method involving the following steps: • pre-selection of a family of theoretical distribution; • estimation of parameters (position, scale, and shape) of distributions; • testing of hypotheses about the correctness of the choice one; • verification of the results obtained by comparing the estimated and real data, using statistics on one of the nodes (ccc). 4.1. estimation the intensity of input flow request at the fig. 2 the first result obtained in the evaluation of the intensity the input flow of requests ccc. it needs the separation the original time series from non-random components. fig. 2. lengths of the intervals between the requests ones of the ccc 82 d. razumov, v. aleshin copyright ©2018 assa adv. in systems science and appl. (2018) at first we accept for an initial hypothesis about the probability distribution of the lengths of the intervals between the receipts of requests in the ccc in accordance with the theoretical recommendations. this assumption is the exponential character of this distribution. the program package statistica was used for the test and analyze these hypotheses based on the data for 2013 year. the result being analyzed with the kolmogorov-smirnov criterion histogram of distribution density of time intervals between receipts of bids (fig. 3). fig. 3. the gistogramm analyze in the programm statistica thus we obtain the average value of the intervals between receipt of bids is equal to 7.8 min., the dispersion of the intervals of the arrivals of 10.12, the value of the kolmogorov-smirnov 0,05335. hypothesis of an exponential distribution is confirmed. 4.2. estimation the temporal characteristics request processing the gamma distribution is the most "suspicious" law from the perspective of better applicability for estimating parameters of a stochastic variable of type "duration of service". scale parameter b and shape parameter a of this distribution are connected by a system of equations. simulation modelling as a tool for design and development 83 copyright ©2018 assa. adv. in systems science and appl. (2018) fig.4. the histogram analysis of duration time bids service (4.1), thus m(x) – expected value ( ); d(x) – distribution (s2); the solution of system equations (4.1) allows you to determine the most plausible values of the parameters of the gamma distribution. accordingly, the parameter estimates have the form: , (4.2) the data for 2013 and the criterion of kolmogorov-smirnov have been used to test the hypothesis about the applicability of the gamma distribution for estimating the processing time of applications in the executive division of ccc (fig.4). when we checked these parameters on the model (fig.5) we get the deviation from the actual number of filings for the same period in 2014 year of no more than 12%. fig.5 clearly shows that the model graphs of the probability density the service requests and interval requests are almost identical with the statistics (fig.3, fig.4). 5. development the base ccc model on the data of the input streams queueing system modeling each of the call centers should be provide for: classification of bids according degree of service complexity (which service responds, how many services respond etc.); possibility of information exchange according applications between ccc, both vertically and horizontally (fig.1). 84 d. razumov, v. aleshin copyright ©2018 assa adv. in systems science and appl. (2018) fig. 5. the test of ccc simulation model 5.1. partitioning the original input stream requests on the substreams in order to be able to draw conclusions about the all participating ccc, for which we have no statistics, and the capacity of inter-node relationships, you have to divide the available input stream on certain types: • 01 – only required response of forces and means of dds 01, • 02 – ccc 02, • 03 – ccc 03, • 0103 – ccc 01 and 03, respectively etc. on this base being made the conclusion about the streams of 01, 03 and then created the simulation model of common ccc. thus "traces" of the joint activity is possible to estimate the parameters of the "missing statistics". fig. 6 shows the identification of distribution laws for each of the "substream". way above is also the identity of the distribution laws for the service interval for each of the selected types substream. the experiment confirms that the splitting of the flow of requests into classes with the corresponding parameters of the distribution does not change the parameters of the distribution of initial flow. thus the basic universal template (class complexsource) simulates a typical ccc. simulation modelling as a tool for design and development 85 copyright ©2018 assa. adv. in systems science and appl. (2018) fig. 6. identification of distribution laws for each of the "substream" fig. 7. the common universal simulation model integrated dds/ccc with distributed input streams in terms of anylogic 5.2. development the base simulation model ccc standard model of ccc (base class) with split input and output, generating a request to ccc 01, 02, 03 (fig. 7), allows testing the hypothesis that the model command control center by splitting the input stream will be adequate in terms of the formation the final steam model without splitting (fig. 8). 86 d. razumov, v. aleshin copyright ©2018 assa adv. in systems science and appl. (2018) fig.8. experiment 6. development the model of regional government system based on the base (universal) model for local ccc as node objects being used the entities which classes are inherited from the base class of the model local ccc/dds (fig.7). fig. 9. graphic interpretation the common simulation model on base universal ccc(complexsource) class simulation modelling as a tool for design and development 87 copyright ©2018 assa. adv. in systems science and appl. (2018) fig. 10. the parameters of local model ccc 03. fig. 11. parameters of the model ccc 02 service in figures 10 and 11 it is seen that the input streams of an object that models behavior of some local ccc 03 differ from the object of the same class modeling of local ccc 02. the specificity of 03 is reflected by the fact that is dominated by flows associated with the receipt of events including type 03 (for more details see above) which is reflected by the intensity of the exponential distribution. streams which do not fit the specifics of ccc 03 are ignored because their intensity is negligible. such techniques are used to describe the specifics of 02 ccc too (see the parameters of the object complexsource_02 fig. 9). here you can see that are significant the streams that have type of applications including reaction units 02. the highest intensity has a flow of "clean" applications 02. graphics system allows you to choose the optimal resource values that manipulating it is easy to identify the processes mutual influence in the work of each of the nodes on other subsystems (fig.12 ). 88 d. razumov, v. aleshin copyright ©2018 assa adv. in systems science and appl. (2018) fig. 12. the identification of optimal resources for each local ccc through simulation model. 5. conclusion thus the authors proposed the formulation of the problem, which served as the basis for the construction of a simulation model. this allows you to determine the required amount of smartcity system resources to perform key performance indicators within the existing constraints. the presented material can be useful for designers and developers of large automated systems which include the acs of the smartcity, as well as the acs of the regional and municipal level, large sports events of international and state scale, situation centers of regional and municipal governments including such a specific area as public authorities and management. the following articles will discuss the problems of system design from the point of view of functional modeling in the life cycle including obtaining an adequate structure of the model as well as approaches related to multivariate analysis. acknowledgements we thank the members of the organizational committee of mlsd’2017 for their kind invitation to contribute the paper presented at the conference to the journal and andrey shevlyakov, the secretary of advances in systems science and applications, for technical assistance. references [1] blanchard, benjamin s. (2004) system engineering management. john wiley & sons. [2] gost r iso/mehk 12207 (1999) «informacionnaya tekhnologiya. processy zhiznennogo cikla programmnyh sredstv», identichnyj [iso/iec 12207(1995) system and software engineering — software life cycle processes]. simulation modelling as a tool for design and development 89 copyright ©2018 assa. adv. in systems science and appl. (2018) [3] malyshev v.v. (2010). metody optimizacii v zadachah sistemnogo analiza i upravleniya. [optimization methods in system analysis and management problems] moscow: maiprint [in russian]. [4] razumov d.a., aljoshin v.d. (2014). modelirovanie v zhiznennom cikle avtomatizirovannyh sistem upravleniya v krizisnyh i chrezvychajnyh situaciyah. prikladnaya informatika. [modeling in the life cycle of automated control systems in crisis and emergency situations. applied informatics] 6(54), 102-116. [in russian]. advances in systems science and applications (2013) vol.13 no.4 355-368 technique of constructing predictive inferences for future outcomes with applications to inventory management problems konstantin n. nechval1 and nicholas a. nechval2 1applied mathematics department, transport and telecommunication institute lomonosov street 1, lv-1019, riga, latvia 2statistics department, evf research institute, university of latvia raina blvd 19, lv-1050, riga, latvia abstract predictive inferences (predictive distributions, prediction and tolerance limits) for future outcomes on the basis of the past and present knowledge represent a fundamental problem of statistics, arising in many contexts and producing varied solutions. in this paper, new-sample prediction based on a previous sample (i.e., when for predicting the future outcomes in a new sample there are available the observed data only from a previous sample), within-sample prediction based on the early data from a current experiment (i.e., when for predicting the future outcomes in a sample there are available the early data only from that sample), and new-within-sample prediction based on both the early data from that sample and the data from a previous sample (i.e., when for predicting the future outcomes in a new sample there are available both the early data from that sample and the data from a previous sample) are considered. it is assumed that only the functional form of the underlying distributions is specified, but some or all of its parameters are unspecified. in such cases ancillary statistics and pivotal quantities, whose distribution does not depend on the unknown parameters, are used. in order to construct predictive inferences for future outcomes, the invariant embedding technique representing the exact pivotal-based method is proposed. furthermore, this technique can be used for optimization of inventory management problems. an illustrative example is given. keywords future outcomes, inventory management, predictive inferences 1 introduction prediction of future outcomes based on the past and current data is the most prevalent form of statistical inference. predictive inferences for future outcomes are widely used in risk management, finance, insurance, economics, hydrology, material sciences, telecommunications, and many other industries.practical problems often require the computation of predictive distributions, prediction and tolerance limits for future values of random quantities. consider the following examples : 1) a consumer purchasing a refrigerator would like to have a lower limit for the failure time of the unit to be purchased (with less interest in distribution of the population of units purchased by other consumers) ; 2) financial managers in 356 konstantin n. nechval : technique of constructing predictive inferences for future... manufacturing companies need upper prediction limits on future warranty costs. a large number of problems in inventory management, production planning and scheduling, location, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty. most of the inventory management literature assumes that demand distributions are specified explicitly. however, in many practical situations, the true demand distributions are not known, and the only information available may be a time-series of historic demand data. when the demand distribution is unknown, one may either use a parametric approach (where it is assumed that the demand distribution belongs to a parametric family of distributions) or a non-parametric approach (where no assumption regarding the parametric form of the unknown demand distribution is made). under the parametric approach, one may choose to estimate the unknown parameters or choose a prior distribution for the unknown parameters and apply the bayesian approach to incorporating the demand data available. scarf [1] and karlin [2] consider a bayesian framework for the unknown demand distribution. specifically, assuming that the demand distribution belongs to the family of exponential distributions, the demand process is characterized by the prior distribution on the unknown parameter. further extension of this approach is presented in [3]. parameter estimation is considered in [4]. liyanage and shanthikumar [5] propose the concept of operational statistics and apply it to a single period newsvendor inventory control problem.in this paper we consider the case, where it is known that the demand distribution function belongs to a parametric family of distribution functions. however, unlike in the bayesian approach, we do not assume any prior knowledge on the parameter values. conceptually, it is useful to distinguish between “new-sample” prediction, “within-sample” prediction, and “new-within-sample” prediction. for new-sample prediction, data from a past sample are used to make predictions of future outcomes from the same process or population. for within-sample prediction, the problem is to predict future outcomes in a sample or process based on early data from that sample or process. for new-within-sample prediction, the problem is to predict future outcomes in a sample or process based on early data from that sample or process as well as on a past sample data from the same process or population. various solutions have been proposed for the prediction problem, that is, the problem of making inferences on a random sample {yj ; j = 1, ...,m} given independent observations {xi; i = 1, ..., n} drawn from the same distribution. the y ′ j s and the x ′ is are commonly featured as “future outcomes” and “past outcomes” respectively. inferences usually bear on some reduction z of the y ′ j s-possibly a minimal sufficient statistic and consist of either prediction intervals or likelihood or predictive distribution for z, depending on different authors. kaminsky and nelson [6] discussed point and interval prediction of order statistics. best linear unbiased predictors advances in systems science and applications (2013) vol.13 no.4 357 based on location-scale family of distributions are reviewed. prediction intervals based on such predictors as well as those based on pivotals are studied. a brief discussion of bayesian prediction is also given. predictive distributions are found in the bayesian framework (see aitchison and sculthorpe [7]). lawless [8] applied the conditional method, which was first suggested by fisher [9] and promoted further by a number of others (nechval et al. [10] ; murthy et al. [11]), to different problems relating to the weibull and extreme value distributions. in practice the proposed methods have limited applications and it is the purpose of this paper to obtain predictive inferences concerning z via the simple invariant embedding technique [12-16]. the obtained results are given below. 2 constructing predictive inferences for future outcomes let us assume that the random variable x follows the exponential distribution with the probability density function fσ(x) = σ−1 exp(−x/σ), σ > 0, (1) and the cumulative distribution function fσ(x) = 1− exp(−x/σ), (2) where σ is the scale parameter (σ >0). 2.1 new-sample prediction theorem 1. let x1 ≤ ... ≤ xk be the first k ordered observations from the previous sample of size n from the exponential distribution (1) and yr be the rth order statistic in a set of m future ordered observations y1 ≤ ... ≤ ym also from the distribution (1).then the probability distribution functionof the ancillary statistic yr/sk is given by pr { yr sk ≤ η } = m∑ j=r ( m j ) j∑ i=0 ( j i ) (−1)i 1 [1 + (i+m− j)η]k , (3) where sk = ∑k i=1xi + (n− k)xk. proof. the joint density of x1 ≤ ... ≤ xk is given by fσ(x1, ...,xk) = n! (n− k)! k∏ i=1 1 σ exp ( −xi σ ) exp ( −(n− k) xk σ ) (4) it is known that sk = k∑ i=1 xi + (n− k)xk (5) 358 konstantin n. nechval : technique of constructing predictive inferences for future... is the sufficient statistic for σ. then vk = sk/σ (6) is the pivotal quantity, the probability density function of which is given by f(vk) = 1 γ(k) vk−1 k exp(−vk), vk ≥ 0 (7) using the invariant embedding technique [12-16], we reduce pr {yr ≤ yr} = m∑ j=r ( m j ) [fσ(yr)] j [1− fσ(yr)] m−j = m∑ j=r ( m j )[ 1− exp ( −yr σ )]j[ exp ( −yr σ )]m−j (8) to pr { yr sk ≤ η|vk } = m∑ j=r ( m j ) j∑ i=0 ( j i ) (−1)i exp (−(i+m− j)ηvk) (9) where η = yr/sk (10) it follows from (9) that pr { yr sk ≤ η } = e  m∑ j=r ( m j ) j∑ i=0 ( j i ) (−1)i exp (−(i+m− j)ηvk)  (11) = ∫ ∞ 0 m∑ j=r ( m j ) j∑ i=0 ( j i ) (−1)i exp (−(i+m− j)ηvk) f(vk)dvk = m∑ j=r ( m j ) j∑ i=0 ( j i ) (−1)i 1 [1 + (i+m− j)η]k (12) this ends the proof. corollary 1.1. a lower one-sided new-sample α prediction limit h on the rth order statistic yr in a set of m future ordered observations y1 ≤ ... ≤ ym is given by h = ηsk, (13) advances in systems science and applications (2013) vol.13 no.4 359 where η satisfies the equation m∑ j=r ( m j ) j∑ i=0 ( j i ) (−1)i 1 [1 + (i+m− j)η]k = α. (14) (observe that an upperone-sided 1− α prediction limit h may be obtained from a lower one-sided α prediction limit by replacing α by 1− α.) 2.2 within-sample prediction theorem 2. let y1 ≤ ... ≤ yl be the first lordered observations (order statistics) in a sample of size m from a continuous distribution with some probability density function fθ(x) and distribution function fθ(x), where θ is a parameter (in general, vector). then the joint probability density function of y1 ≤ ... ≤ yl and the rth order statistics yr(1 ≤ l < r ≤ m) is given by gθ(y1, ..., yl, yr) = gθ(y1, ..., yl)gθ(yr|yl), (15) where gθ(y1, ..., yl) = m! (m− l)! l∏ i=1 fθ(yi)[1− fθ(yl)] m−l, (16) gθ(yr|yl) = (m− l)! (r − l − 1)!(m− r)! [ fθ(yr)− fθ(yl) 1− fθ(yl) ]r−l−1[ 1− fθ(yr)− fθ(yl) 1− fθ(yl) ]m−r · fθ(yr) 1− fθ(yl) = (m− l)! (r − l − 1)!(m− r)! r−l−1∑ j=0 ( r − l − 1 j ) (−1)j [ 1− fθ(yr) 1− fθ(yl) ]m−r+j fθ(yr) 1− fθ(yl) (17) represents the conditional probability density function of yr given yl = yl. proof. the joint density of y1 ≤ ... ≤ yl and yr is given by gθ(y1, ...,yl, yr) = m! (r − l − 1)!(m− r)! l∏ i=1 fθ(yi)[fθ(yr)− fθ(yl)] r−l−1· fθ(yr)[1− fθ(yr)] m−r = gθ(y1, ..., yl)gθ(yr|yl) (18) it follows from (18) that gθ(yr|y1, ..., yl) = gθ(y1, ...,yl, yr)/gθ(y1, ..., yl) = gθ(yr|yl), (19) 360 konstantin n. nechval : technique of constructing predictive inferences for future... i.e., the conditional distribution of yr, given yi = yi for all i = 1,...,l, is the same as the conditional distribution of yr, given only yl = yl, which is given by (17). this ends the proof. corollary 2.1. the conditional probability distribution function of yr given yl = yl is pθ {yr ≤ yr|yl = yl} = 1− (m− l)! (r − l − 1)!(m− r)! · r−l−1∑ j=0 ( r − l − 1 j ) (−1)j m− r + 1 + j [ 1− fθ(yr) 1− fθ(yl) ]m−r+1+j (20) corollary 2.2. let y1 ≤ ... ≤ yl be the first lorder statistics in a sample of size mfrom the exponential distribution (1). then the conditional probability distribution function of yr given yl = yl is pθ {yr ≤ yr|yl = yl} = 1− (m− l)! (r − l − 1)!(m− r)! · r−l−1∑ j=0 ( r − l − 1 j ) (−1)j m− r + 1 + j [ exp ( −yr − yl σ )]m−r+1+j (21) theorem 3. let y1 ≤ ... ≤ yl be the first lordered observations (order statistics) in a sample of size m from the exponential distribution (1), where the parameter α is unknown.then the probability distribution function of the ancillary statistic yr/yl is given by pr { yr yl ≤ yr yl } = 1− m! (r − l − 1)!(m− r)! r−l−1∑ j=0 ( r − l − 1 j ) (−1)j m− r + 1 + j · (22)( l−1∏ i=0 [( yr yl − 1 ) (m− r + 1 + j) +m− l + 1 + i ])−1 proof. we reduce (20) to pr{yr ≤ yr |yl = yl} = 1− (m− l)! (r − l − 1)!(m− r)! r−l−1∑ j=0 ( r − l − 1 j ) · (−1)j m− r + 1 + j [ exp ( −yl σ ( yr yl − 1 ))]m−r+1+j , (23) where yr/yl is the ancillary statistic whose distribution does not depend on the parameter σ, yl/σ is the pivotal quantity. using the probability density function advances in systems science and applications (2013) vol.13 no.4 361 of yl/σ, we eliminate the unknown parameter σ from the problem as pr { yr yl ≤ yr yl } = ∫ ∞ 0 pr{yr ≤ yr |yl = yl}g(yl/σ)dyl/σ = 1− m! (r − l − 1)!(m− r)! r−l−1∑ j=0 ( r − l − 1 j ) (−1)j m− r + 1 + j · ( l−1∏ i=0 [( yr yl − 1 ) (m− r + 1 + j) +m− l + 1 + i ])−1 (24) where g(yl/σ) = m! (l − 1)!(m− l)! [ 1− exp ( −yl σ )]l−1 exp ( −yl σ (m− l) ) exp ( −yl σ ) = m! (l − 1)!(m− l)! l−1∑ i=0 ( l − 1 i ) (−1)i exp ( −yl σ (m− l + 1 + i) ) , (yl/σ) ∈ (0,∞) (25) represents the probability density function of the pivotal quantity yl/σ. this ends the proof. corollary 3.1. a lower one-sided within-sample α prediction limit h on the rth order statistic yr in a set of m future ordered observations y1 ≤ ... ≤ ym is given by h = ηyl, (26) where η satisfies the equation 1− m! (r − l − 1)!(m− r)! r−l−1∑ j=0 ( r − l − 1 j ) (−1)j m− r + 1 + j( l−1∏ i=0 [(η − 1)(m− r + 1 + j) +m− l + 1 + i] )−1 = α. (27) theorem 4. let y1 ≤ ... ≤ yl be the first lordered observations (order statistics) in a sample of size m from the exponential distribution (1), where the parameter α is unknown.then the probability distribution function of the ancillary statistic (yr − yl)/sl is given by pr { yr − yl sl ≤ yr − yl sl } = 1− 1 b(r − l, (m− r + 1) r−l−1∑ j=0 ( r − l − 1 j ) (−1)j m− r + 1 + j [ 1 + (m− r + 1 + j) yr − yl sl ]−l (28) 362 konstantin n. nechval : technique of constructing predictive inferences for future... where sl = l∑ i=1 xi + (m− l)yl (29) is the sufficient statistic for σ. proof. using the technique of invariant embedding [12-16], we reduce (20) to pr {yr ≤ yr|yl = yl} = 1− (m− l)! (r − l − 1)!(m− r)! r−l−1∑ j=0 ( r − l − 1 j ) (−1)j m− r + 1 + j [ exp ( −sl σ yr − yl sl )]m−r+1+j (30) where (yr−yl)/sl is the ancillary statistic whose distribution does not depend on the parameter σ ; sl/σ is the pivotal quantity. since the probability density function of sl/σ is known,we eliminate the unknown parameter σ from the problem as pr { yr − yl sl ≤ yr − yl sl } = ∫ ∞ 0 pr{yr ≤ yr |yl = yl}g(sl/σ)dsk/σ = 1− 1 b(r − l,m− r + 1) r−l−1∑ j=0 ( r − l − 1 j ) (−1)j m− r + 1 + j [ 1 + (m− r + 1 + j) yr − yl sl ]−l (31) where f(sl/σ) = (γ(l))−1(sl/σ) l−1 exp(−sl/σ), (sl/σ) ≥ 0. (32) this ends the proof. corollary 4.1. a lower one-sided within-sample α prediction limit h on the rth order statistic yr in a set of m future ordered observations y1 ≤ ... ≤ ym is given by h = yl + ηsl (33) where η satisfies the equation 1− 1 b(r − l,m− r + 1) r−l−1∑ j=0 ( r − l − 1 j ) (−1)j m− r + 1 + j [1 + (m− r + 1 + j)η]−l = α. (34) advances in systems science and applications (2013) vol.13 no.4 363 2.3 new-within-sample prediction theorem 5. let x1 ≤ ... ≤ xkbe the first kordered observations from a previous sample of size n from the exponential distribution (1) and y1 ≤ ... ≤ yk be the first lordered early observations from a new sample of size m also from the distribution (1), where the parameter α is unknown. then the probability distribution function of the ancillary statistic (yr − yl)/(sk + sl) is given by pr { yr − yl sk + sl ≤ yr − yl sk + sl } = 1− 1 b(r − l, (m− r + 1)) r−l−1∑ j=0 ( r − l − 1 j ) (−1)j m− r + 1 + j × [ 1 + (m− r + 1 + j) yr − yl sk + sl ]−(k+l) (35) proof. for the proof we refer to theorems 1 and 4. corollary 5.1. a lower one-sided within-sample α prediction limit h on the rth order statistic yr in a set of m future ordered observations y1 ≤ ... ≤ yk is given by h = yl + η(sk + sl), (36) where η satisfies the equation 1− 1 b(r − l,m− r + 1) r−l−1∑ j=0 ( r − l − 1 j ) (−1)j m− r + 1 + j [1 + (m− r + 1 + j)η]−(k+l) = α. (37) 3 inventory management froblem and predictive inference in this section, we consider a single-period newsvendor model. single-period stocking decisions often occur in practice ; these require the decision maker to choose the stocking level of an item for which demand exists for only a single period. several factors affect this decision : the distribution of demand, the cost and price of the item, the salvage value of the item, and the loss of customer goodwill due to stockouts. the newsvendor model addresses this problem and develops a formula for what is usually called the ‘critical-fractile’. the optimum order quantity is calculated using the critical-fractile of the distribution of the demand for the period. underlying the mathematical simplicity of the critical-fractile formula is a powerful and intuitively appealing insight for the determination of the order quantity. the order quantity depends only on the optimum balance between two types of costs. the first is the cost per unit associated with the unavailability of stock to meet the manifest demand (i.e., underage cost). the second is the unit cost associated with excess inventory at the end of the period for which there is no demand (i.e., overage cost). following hadley and whitin [17], we review 364 konstantin n. nechval : technique of constructing predictive inferences for future... the single-period newsvendor model and provide a broader interpretation to the structure of its solution. the notation, we use for the newsvendor model, is given below. y random variable for single-period demand fθ(y) probability density function of single-period demand fθ(y) probability distribution function of single-period demand θ parameter (in general, vector) c1 unit selling price c unit procurement cost, which is independent of the procured amount g unit salvage value for unsold items remaining at the end of the period c2 unit stockout penalty cost (over and above any lost profit) u variable representing the order quantity u∗ optimal order quantity q(u) expected profit as a function of the order quantity co unit overage cost cu unit underage cost different versions of the problem may equivalently consider expected opportunity cost minimization or expected profit maximization. we examine the latter and write the expected profit as q(u) = c1 ∫ u 0 yfθ(y)dy + c1u ∫ ∞ u fθ(y)dy − cu+ g ∫ u 0 (u− y)fθ(y)dy − c2 ∫ ∞ u (y − u)fθ(y)dy (38) assuming that c1 + c2 > g (which is generally true in most situations), we can show that the expected profit function q(u) is concave. we, therefore, set the first derivative equal to zero to find a maximizing solution. the value of u that maximizes (38) is the one that satisfies fθ(u ∗) = (c1 − c+ c2)/(c1 − g + c2). (39) clearly the lost contribution margin (c1-c) plus the stockout penalty (c2) represents the ‘underage cost’. similarly the item cost (c) minus the salvage value (g) represents the ‘overage cost’. if we refer to the underage cost as cu and the overage cost as co, we may rewrite (39) as fθ(u ∗) = cu/(cu + co). (40) we should choose the order quantity u∗ such that the cumulative distribution function (cdf ) of u∗ equals the ratio of the underage cost to the sum of the advances in systems science and applications (2013) vol.13 no.4 365 underage and overage costs. a relatively high underage cost results in a higher order quantity, whereas a relatively high overage cost leads to a lower order quantity, as one would expect. if the single-period demand y follows the exponential distribution (1) then q(u) = σ [ (c1 − g)− (c− g) u σ − (c1 − g + c2) exp ( −u σ )] (41) u∗ = σ ln ( 1 + c1 − c+ c2 c− g ) (42) and q(u∗) = σ [ c1 − c− (c− g) ln ( 1 + c1 − c+ c2 c− g )] (42) parametric uncertainty. consider the case when the parameter σ is unknown. let x1 ≤ ... ≤ xn be the past observations (order statistics) of single-period demands from the exponential distribution (1).then s = n∑ i=1 xi, (44) is a sufficient statistic for σ ; sis distributed with gσ(s) = [γ(n)σn]−1sn−1 exp(−s/σ)(s > 0) (45) to find the best invariant decision rule µbi , we use the invariant embedding technique [12-16] to transform (41) to the form, which is depended only on the pivotal quantity v = s/σ and the ancillary factor η = u/s. in statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters whose probability distribution does not depend on unknown parameters. note that a pivotal quantity need not be a statisticąłthe function and its value can depend on parameters of the model, but its distribution must not. if it is a statistic, then it is known as an ancillary statistic. transformation of q(u) is given by q(η|v) = σ [(c1 − g)− (c− g)ηv − (c1 − g + c2) exp (−ηv)] (46) we find the expected profit for the statistical decision u = ηs as e{q(η)} = ∫ ∞ 0 q(η|v)g(v)dv =σ [ (c1 − g)− (c− g)ηn− (c1 − g + c2)(η + 1)−n] (47) where g(v) = [γ(n)]−1vn−1 exp(−v)(v > 0). (48) 366 konstantin n. nechval : technique of constructing predictive inferences for future... the value of η that maximizes (47) is given by η∗ = [1 + (c1 − c+ c2)/(c− g)]1/(n+1) − 1. (49) thus, ubi = η∗s = [ [1 + (c1 − c+ c2)/(c− g)]1/(n+1) − 1 ] s. (50) comparison of decision rules. for comparison, consider the maximum likelihood decision rule that may be obtained from (42) as uml = σ̂ ln [1 + (c1 − c+ c2)/(c− g)] = ηmls, (51) where σ̂ = s/n is the maximum likelihood estimator of σ, ηml = ln [1 + (c1 − c+ c2)/(c− g)]1/n. (52) since µbi and µml belong to the same class c = {u : u = ηs}, (53) it follows from the above that µml is inadmissible in relation to µbi . if, say, c= 2, c1=490, c2=2, g=1 (in terms of money), and n=1, we have that rel.eff.e{q(η)}{uml, ubi, σ}=e{q(ηml)} / e{q(η∗) = (c1 − g)− (c− g)ηmln− (c1 − g + c2) 1 (ηml+1)n (c1 − g)− (c− g)η∗n− (c1 − g + c2) 1 (η∗+1)n = 0.93. (54) thus, in this case, the use of µbi leads to a growth in the expected profit of about 7% as compared with µml. the absolute expected profit will be proportional to σ and may be considerable. predictive inference.it will be noted that the predictive probability density function of the single-period demand y, which is compatible with (38), is given by f(y|s) = n+ 1 s ( 1 + y s )−(n+2) (y > 0). (55) using (55), the predictive profit is determined as q(p)(u|s) = c1 ∫ u 0 yf(y|s)dy + c1u ∫ ∞ u f(y|s)dy − cu+ g ∫ u 0 (u− y)f(y|s)dy − c2 ∫ ∞ u (y − u)f(y|s)dy = s n [ (c1 − g)− (c− g) u s n− (c1 − g + c2) (u s + 1 )−n ] , (56) advances in systems science and applications (2013) vol.13 no.4 367 which can be reduced to q(p)(η|s) = s n [ (c1 − g)− (c− g)ηn− (c1 − g + c2) 1 (η + 1)n ] . (57) it follows from (57) that eσ{q(p)(η|s)} = ∫ ∞ 0 q(p)(η|s)gσ(s)ds = σ [ (c1 − g)− (c− g)ηn− (c1 − g + c2)(η + 1)−n] = e{q(η)} (58) thus, ubi can be found immediately from (56) as ubi = argmax u q(p)(u|s). (59) 4 conclusion the technique proposed in this paper represents a simple and computationally attractive statistical method based on the constructive use of the invariance principle in mathematical statistics. the main advantage of this technique consists in that it allows one to eliminate unknown parameters from the problem and to use the previous and current data of observations for obtaining predictive inferences as completely as possible. we have illustrated the technique for the exponential distribution. applications to other log-location-scale distributions could follow directly. références [1] scarf h. (1959), “bayes solutions of statistical inventory problem”, ann. math. statist, vol.30, pp.490-508. [2] karlin s. (1960), “dynamic inventory policy with varying stochastic demands”, management sci, vol.6, pp.231-258. [3] azoury k.s. (1985), “bayes solution to dynamic inventory models under unknown demand distribution”, management sci, vol.31, pp.1150-1160. [4] conrad s.a. (1976), “sales data and the estimation of demand”, oper. res. quart, vol.27, pp.123-127. [5] liyanage l.h., shanthikumar j.g. (2005), “a practical inventory control policy using operational statistics”, oper. res. lett, vol.33, pp.341-348. [6] kaminsky k.s., nelson p.i. (1998), “prediction of order statistics”, in : handbook of statistics-17 : order statistic : applications, n. balakrishnan and c. r. rao (eds.), elsevier science, pp.431-450. 368 konstantin n. nechval : technique of constructing predictive inferences for future... [7] aitchison j., sculthorpe d. (1965), “some problems of statistical prediction”, biometrika, vol.52, pp.469-483. [8] lawless j.f. (1982), statistical models and methods for lifetime data, new york : john wiley. [9] fisher r.a. (1934), “two new properties of mathematical likelihood”, proc. roy. statist. soc, a, vol.144, pp.285-307. [10] nechval n.a., nechval k.n., vasermanis e.k. (2003), “statistical models for prediction of the fatigue crack growth in aircraft service”, in : fatigue damage of materials 2003, a. varvani-farahani and c. a. brebbia (eds.). southampton, boston : wit press, pp.435-445. [11] murthy d. n. p., xie m., jiang y. (2004), weibull models, new york : john wiley and sons inc. [12] nechval n.a., nechval k.n., vasermanis e.k. (2003), “effective state estimation of stochastic systems”, kybernetes (an international journal of systems & cybernetics), vol.32, pp.666-678. [13] nechval n.a., berzins g., purgailis m., nechval k.n. (2008), “improved estimation of state of stochastic systems via invariant embedding technique”, wseas transactions on mathematics, vol.7, pp.141-159. [14] nechval n.a., nechval k.n., purgailis m. (2011), “prediction of future values of random quantities based on previously observed data”, engineering letters, vol.9, pp.346-359 [15] nechval n.a., purgailis m., nechval k.n., strelchonok v.f. (2012), “optimal predictive inferences for future order statistics via a specific loss function”, iaeng international journal of applied mathematics, vol.42, pp.40-51. [16] nechval n.a., purgailis m., cikste k, berzins g., nechval k.n. (2010), “optimization of statistical decisions via an invariant embedding technique”, in : lecture notes in engineering and computer science : proceedings of the world congress on engineering 2010, wce 2010, london, 30 june-2 july 2010, pp.1776-1782. [17] hadley g., whitin t.m. (1963), analysis of inventory systems, new jersey : prentice-hall inc. corresponding author nicholas a. nechval can be contacted at : nechval@junik.lv microsoft word 1111-article text-7257-1-6-20220131 adv syst sci appl 2024; 03; 104-113 published online at https://ijassa.ipu.ru. the approximation matrix method and its comparison with the analytical hierarchy process by t. saaty viktor p. korneenko v.a. trapeznikov institute of control sciences of the russian academy of sciences, moscow, russia e-mail: vkorn@ipu.ru abstract: the article presents an optimization method for the formation of quantitative weights of objects (importance of criteria, priorities of alternatives) according to the initial expert judgment matrix in multi-criteria selection problems. since the matrix of pairwise comparisons can be considered as some perturbation of the multiplicative matrix, the proposed method is based on the approximation of the original matrix of pairwise comparisons by the multiplicative matrix according to the matrix criterion of minimum distances between matrices. there is a one-to-one mapping between the elements of the weight vectors and the elements of the multiplicative matrix. for the first time, using a specific example using the matrix criterion, a relative estimate of the approximate solution of the analytical hierarchy process by t. saaty concerning the optimal solution obtained by the approximation matrix method is given. on account of the approximation matrix method being mathematically justified and due to the simplicity of finding optimal solutions, it can be recommended instead of the analytical hierarchy process by t. saaty. keywords: multi-criteria choice, normalized object weights, expert judgment matrix, multiplicative matrix, matrix criterion. 1. introduction when solving applied problems of multicriteria choice on a set of objects (alternatives, management decisions, options) presented in the form of preference relations, the problem of their expert measurement in the quantitative scale of relations arises. to date, many approaches and methods have been proposed to solve this problem, based on the resulting preference relations to narrow the set of non-dominant alternatives, as well as on paired comparisons of objects (solutions, criteria) [1, 2], which do not always allow us to identify a single alternative or management solution without attracting additional information. however, when solving applied problems related to the measurement of objects in expert scales, as well as the formation of local weights of criteria presented in the form of a hierarchical tree of the importance of criteria, expert methods of evaluating and ranking objects are usually used [3]. direct methods of expert evaluation of criteria weights have found application in the planning methodology through relative indicators of technical evaluation (pattern). experts are asked to evaluate the normalized local weights of criteria at each level of the hierarchy on a quantitative scale, and then the global weights are found by multiplying local weights along the branches of a multi-level criteria tree [4]. another expert approach based on the matrix of paired comparisons to assign "weights" to a finite set of compared objects is the analytical hierarchy process (ahp) by t. saaty which is now firmly established in the theory and practice of multicriteria selection problems [5–7]. following the analytical hierarchy process, experts form a so-called matrix of paired comparisons (judgments) of objects 𝑉 = [𝑣 ], 𝑖, 𝑗 = 1, 𝑛, in the scale of relations, and then find the approximation matrix method and its comparison… 105 copyright ©2024 assa adv. in systems science and appl. (2024) the right eigenvector �⃗� = (𝑤 , . . . , 𝑤 ) of this matrix, corresponding to the maximum eigenvalue. the desired weight vector is a vector whose elements are normalized by the sum of the elements of the right eigenvector. since the calculation of the vectors of weights of objects (criteria and alternatives) is performed by a numerical (approximate) method, t. saaty aware of this, introduced a special numerical indicator: consistency index the compatibility index of the judgment matrix 𝑉 and the multiplicative matrix 𝑊 obtained based on the values of the eigenvector, in the form of the hadamard product [9]: 𝑆. 𝐼. = 𝑒 𝑉°𝑊 𝑒, 𝑒 = (1,1, … ,1), where v is the original judgment matrix, and 𝑊 = [𝑤 ] = is a multiplicative square matrix whose elements are determined from the normalized elements of the right eigenvector �⃗� of the judgment matrix v. in this case, the ratio takes place: 𝑤 = 𝑤 𝑤 = 𝑤 / ∑ 𝑤 𝑤 / ∑ 𝑤 = 𝑤 𝑤 , 𝑖, 𝑗 = 1, 𝑛. the s.i. index it characterizes the degree of confidence in the results obtained with the help of ahp and is interpreted as a kind of measure of the deviation of the initial perturbed judgment matrix 𝑉 from the multiplicative one w. in the work of t. saaty [8, p. 76], it is shown that if we perform the hadamard matrix product, then the compatibility index takes the form 𝑆. 𝐼. = > 1, where λ is the maximum eigenvalue of the matrix 𝑉. with a sufficiently close approximation to the unit value of the index, the matrix of paired comparisons 𝑉 is "close" to the multiplicative matrix w. if the consistency index exceeds a certain "threshold" value, then it is impossible to conclude the proximity of these matrices, and therefore it is not recommended to use ai in such cases. the hierarchy analysis method and its applications are described in many reviews, monographs, scientific articles, as well as works popularizing this method [10–15]. however, it can be clearly stated that the method of hierarchy analysis by t. saaty is approximate since the method of determining the weight vector is based on numerical (approximate) methods for calculating the roots of a polynomial. the problem of finding the roots of polynomials of degree 𝑛 ≥ 5 is unsolvable in radicals. therefore, in t. saaty's method, for the number of objects at least five, the procedure for finding the eigenvalues of the matrix of degrees of the superiority of the importance of criteria or preferences of alternatives is carried out using numerical approximate methods for finding the roots of a polynomial implemented in the package expert choice [16]. v.d. noghin states that "the value of the compatibility index can only indirectly judge the magnitude of the final "model" error: it can never be precisely determined by anyone. this is the specificity of this heuristic approach" [10, p. 1194]. the article suggests a more efficient method of the approximation matrix (mam) for forming optimal object weights based on the matrix criterion of distance to the original matrix of judgments than the method of analyzing hierarchies of t. saaty. 2. statement of the problem of approximation of matrices of judgments in multi-criteria applied problems the aggregation mechanism is usually represented in the form of an additive candle of object ratings (alternatives, variants) 𝑎 ∈ 𝐴 = {𝑎 |𝑙 = 1, 𝑛 }, according to criteria with weights of importance in the form [1]: 𝐹(𝑎 , 𝑤 , … , 𝑤 ) = ∑ 𝑤 𝑓 (𝑎 ), ∑ 𝑤 = 1, 106 v.p. korneenko copyright ©2024 assa adv. in systems science and appl. (2024) where 𝑓 (𝑎 ) is the object score 𝑎 in the resulting scale according to the criterion 𝑓 , 𝑗 = 1, 𝑛; 𝑤 = 𝑤( 𝑓 ) is the quantitative (normalized) weight 𝑓 of the criterion. let us consider the formulation of the formation of object weights by the criterion of proximity to the original matrix of paired comparisons of a multiplicative matrix. let us have as initial data: 𝑉 = [𝑣 ] – the initial expert matrix of judgments about the relative importance of objects (criteria) 𝑓 ∈ 𝐹, 𝑖, 𝑗 = 1, 𝑛; 𝑊 – the set of multiplicative square matrices n of the nth order over the field of real numbers. as a measure of proximity 𝑑(𝑉, 𝑊) between the original 𝑉 = [𝑣 ] and the multiplicative 𝑊 = 𝑤 matrix, where 𝑊 ∈ 𝑊 , we take the square of the euclidean l2-norm equal to the difference of these matrices [9]: 𝑑(𝑉, 𝑊) = ‖𝑉 − 𝑊‖ = (𝑣 − 𝑤 ) . (2.1) then the mathematical formulation of the problem of choosing a multiplicative matrix w that approximates the original matrix 𝑉 = [𝑣 ], i.e., the closest approach to the original expert judgment matrix, is reduced to minimizing the indicator (2.1) in the form (v − w ) → 𝑚𝑖𝑛 ∈ , (2.2) provided that the matrix elements are multiplicative 𝑊 = [𝑤 ]: w = w w , ∀ i, j, k = 1, n. (2.3) due to condition (2.3), it is not analytically possible to obtain a solution to the original problem using one of the classical optimization methods, for example, the lagrange multiplier method. let us use the properties of the multiplicative matrix and solve this problem by reducing the original problem to an equivalent problem. 3. special linear properties of a multiplicative matrix to solve the original problem (2.2)–(2.3), it is necessary to find elements of the multiplicative 𝑊 = 𝑤 , ∀𝑖, 𝑗 = 1 , 𝑛, matrix, that provides a minimum for the quadratic criterion 𝑑(𝑉, 𝑊) (2.1) under condition (2.3). it turns out that for a multiplicative matrix, the relationship between the elements of the columns of the matrix and the elements of the right eigenvector is valid. to do this, consider two statements. the work of b.g. mirkin [17, p. 183–184] provided that the matrix 𝐵 = 𝑏 , ∀ 𝑖, 𝑗 = 1, 𝑛 is over traditional if there exists a positive vector 𝑥 = (𝑥 , . . . , 𝑥 ) such that that 𝑏 = and the vector 𝑥 is a point of equilibrium process: 𝑞 = 𝐵𝑞 , 𝑡 = 1, 2, . . ., which in the limit leads to a private vector 𝑞 = lim ⟶ 𝑞 . we show that the multiplicative matrix has several other properties that will be useful in reducing the original, analytically unsolvable problem (2.2)–(2.3) in the framework of classical optimization methods to an equivalent one. 3.1. the relationship between normalized column elements of a multiplicative matrix and elements of the right eigenvector theorem 3.1: 1. between the elements 𝑤 of a multiplicative matrix 𝑊 = 𝑤 , ∀𝑖, 𝑗 = 1, 𝑛, and any pair (𝑤 , 𝑤 ) component of the right eigenvector �⃗� = (𝑤 , . . . , 𝑤 ) true bijective mapping ↦ 𝑤 , whose every attitude to one mapping element 𝑤 of the matrix w and back the approximation matrix method and its comparison… 107 copyright ©2024 assa adv. in systems science and appl. (2024) 𝑤 ↦ 𝑤 𝑤 , provided this is true equality: 𝑤 = 𝑤 𝑤 , ∀ 𝑖, 𝑗 = 1, 𝑛. (3.4) 2. the normalized elements 𝑤 of the columns �⃗� = (𝑤 , … , 𝑤 ) , 𝑗 = 1, 𝑛, coincide with each other and are equal to the normalized right eigenvector, i.e. �⃗� = 𝑤 … 𝑤 = 𝑤 … 𝑤 = �⃗�, ∀ 𝑗 = 1, 𝑛, (3.5) where 𝑤 = ∑ is the normalized element 𝑗 of the j-th column of �⃗� ; 𝑤 = ∑ is the normalized element of the i-th row of the right eigenvector �⃗�. 3. the multiplicative matrix has one basis row. proof. 1. since by hypothesis the matrix 𝑊 = 𝑤 multiplicative then will provide that, if for any pair of numbers from {𝑤 , . . . , 𝑤 } the validity of the equation (3.4), then in this case the condition of multiplicative between the elements of the matrix 𝑊 = , i.e. thus there exists a one-to-one mapping between the elements: 𝑤 ↔ . indeed, if 𝑤 = and 𝑤 = , then we have 𝑤 𝑤 = × = = 𝑤 , that is, for all 𝑖, 𝑗, 𝑘 = 1, 𝑛 , the multiplicativity condition is satisfied. 2. let the elements of the vector �⃗� = (𝑤 , … , 𝑤 ) satisfy equality (3.4). we show that �⃗� = (𝑤1,. . . , 𝑤 𝑤𝑛) is the right eigenvector of the matrix 𝑊 = , ∀𝑖, 𝑗 = 1, 𝑛. let us verify that 𝑊�⃗� = λ�⃗� is valid: 𝑊�⃗� = 𝑤 /𝑤 𝑤 /𝑤 … 𝑤 /𝑤 𝑤 /𝑤 … 𝑤 /𝑤 … … … 𝑤 /𝑤 … 𝑤 /𝑤 𝑤 /𝑤 … 𝑤 /𝑤 𝑤 𝑤 … 𝑤 = 𝑛𝑤 𝑛𝑤 … 𝑛𝑤 = 𝑛 𝑤 𝑤 … 𝑤 = 𝑛�⃗�, where λ = 𝑛 is the maximum eigenvalue. for arbitrary columns �⃗� и �⃗� 𝑤𝑘 𝑎𝑛𝑑 𝑤𝑞 (1 ≤ 𝑘, 𝑞 ≤ 𝑛) of the matrice w, whose elements satisfy the multiplicativity condition 𝑤 (2.3), we normalize the elements. since 𝑤 = 𝑤 𝑤 , then 𝑤 = 𝑤 /𝑤 , whence for any column numbers 𝑘, 𝑞 we have: 𝑤 = ∑ = / ∑ = ∑ = ∑ = 𝑤 , ∀ 𝑖 = 1, 𝑛, i.e., the normalized components of the columns of the matrix coincide with each other. on the other hand: 𝑤 = 𝑤 ∑ 𝑤 = 𝑤 /𝑤 ∑ 𝑤 /𝑤 = 𝑤 ∑ 𝑤 = 𝑤 , ∀ 𝑖 = 1, 𝑛. thus, the normalized components of the matrix columns are also equal to the normalized elements of the right vector of the multiplicative matrix. 3. since the matrix 𝑊 = by linear transformations is reduced to the form: 108 v.p. korneenko copyright ©2024 assa adv. in systems science and appl. (2024) 𝑤 /𝑤 𝑤 /𝑤 . . . 𝑤 /𝑤 𝑤 /𝑤 . . . 𝑤 /𝑤 . . . . . . . . . 𝑤 /𝑤 . . . 𝑤 /𝑤 𝑤 /𝑤 . . . 𝑤 /𝑤 ~ ∏ 𝑤 1/𝑤 1/𝑤 . . . 1/𝑤 1/𝑤 . . . 1/𝑤 . . . . . . . . . 1/𝑤 . . . 1/𝑤 1/𝑤 . . . 1/𝑤 ~ ∏ 𝑤 1/𝑤 0 . . . 0 1/𝑤 . . . 0. . . . . . . . . 0. . . 1/𝑤 0 . . . 0 , the rank of the multiplicative matrix is equal to one: rg 𝑊 = 1, i.e., the multiplicative matrix has one basic row. the theorem is proved. ∎ 3.2. reducing the elements of the matrix columns to integer values and equality to the right eigenvector theorem 3.2: let the elements 𝑞 , 𝑖, 𝑘 = 1, 𝑛, of the multiplicative matrix 𝑊 = [𝑞 ], ∀ 𝑖, 𝑘 = 1, 𝑛, be represented in general by integers and rational numbers in the form of irregular fractions. then, if the elements of 𝑞 any column �⃗� , 𝑘 = 1, 𝑛, of the matrix is multiplied by the least common multiple 𝑛 denominators of the rational elements of the column, obtained in the result of this procedure, the integer elements 𝑧 = 𝑛 𝑞 , ∀ 𝑘 = 1, 𝑛, the columns of the matrix coincide with each other on any line number, and will be equal to the integer elements 𝑤 right eigenvector �⃗� = (𝑤 , . . . , 𝑤 , … 𝑤 ) corresponding to its own maximum eigenvalue 𝜆 = 𝑛: z … z = 𝑤 … 𝑤 , 1 ≤ 𝑘 ≤ 𝑛. (3.6) proof. by theorem 1, the elements of any multiplicative inverse-symmetric matrix can be brought to mind (3.4), namely: 𝑞 = , where 𝑤 are positive integer numbers – the elements of the right eigenvector, the least common multiple of positive integers to rational denominators of the elements of such a multiplicative matrix is equal to 𝑛 = 𝑤 , ∀ 𝑘 = 1, 𝑛. as a result, for k-th column �⃗� = , . . . , , we find 𝑛 �⃗� = 𝑛 × 𝑤 /𝑛 … 𝑤 /𝑛 = 𝑤 × 𝑤 /𝑤 … 𝑤 /𝑤 = 𝑤 … 𝑤 = �⃗�, ∀ 𝑘 = 1, 𝑛, that is, we proved the correctness of (3.6). it is easy to verify that �⃗� is the right eigenvector of the matrix 𝑊, which was required to prove. the theorem is proved. ∎ example 1. consider a multiplicative matrix 𝑊 = ⎝ ⎜ ⎜ ⎜ ⎛ 1 1 2 1 3 1 5 2 1 2 3 2 5 3 3 2 1 3 5 5 5 2 5 3 1⎠ ⎟ ⎟ ⎟ ⎞ (3.7) for the columns of the matrix, the lowest common multiples are: 𝑛 = 𝑤 = 2 × 3 × 5 = 30, 𝑛 = 𝑤 = 3 × 5 = 15, 𝑛 = 𝑤 = 2 × 5 = 10, 𝑛 = 2 × 3 = 6. from here we come to the matrix: 𝑊 = 30 15 10 6 30 15 10 6 30 15 10 6 30 15 10 6 . it is easy to verify that the right the approximation matrix method and its comparison… 109 copyright ©2024 assa adv. in systems science and appl. (2024) eigenvector �⃗� = (30, 15, 10, 6) of the matrix w (3.7), corresponding to the eigenvalue 𝜆 = 4 and the normalized elements of the columns of the matrix coincide with each other and are equal to the elements of the normalized right vector of the matrix: �⃗� = 30 61 , 15 61 , 10 61 , 6 61 . the original matrix can also be represented as 𝑊 = : 𝑊 = 30/30 15/30 10/30 6/30 30/15 15/15 10/15 6/15 30/10 15/10 10/10 6/10 30/6 15/6 10/6 6/6 . 4. reducing the original problem to an equivalent one since, following (3.5), the elements of the 𝑖-th rows of the normalized vector columns of the multiplier and vector matrix are equal to the 𝑖-th normalized component of the right proper vector, namely: 𝑤 = ⋯ = 𝑤 = ⋯ = 𝑤 = 𝑤 , (4.8) where ∑ 𝑤 = 1, to we have: 𝑤 = = / ∑ / ∑ = , ∀𝑖, 𝑗 = 1, 𝑛. we show that the multiplicativity condition (2.3) for the normalized row elements (4.8) of the matrix is satisfied: 𝑤 𝑤 = 𝑤 𝑤 × 𝑤 𝑤 = 𝑤 𝑤 = 𝑤 , 𝑖, 𝑗, 𝑘 = 1, 𝑛. thus, finding the elements of the approximating matrix 𝑊 = [𝑤 ] of the problem (2.2)– (2.3) is equivalent to finding the elements of a normalized column vector: �⃗� = (𝑤 , … , 𝑤 , … , 𝑤 ) . as the target indicator of the approximation problem, we take the square of the difference between the normalized elements of the original and multiplicative matrices in the form: 𝜌 𝑉, 𝑊 = 𝑣 − 𝑤 (4.9) then the mathematical formulation of the original problem is reduced to finding the normalized right column vector of the approximating matrix and providing the minimum criterion (4.9): 𝑣 − 𝑤 → min ( ,…, ) , (4.10) where 𝑣 = ∑ are the normalized elements of the matrix 𝑉 = [𝑣 ] of judgments, 𝑗 = 1, 𝑛. the reasonableness of the transition from the original perturbed matrix 𝑉 = [𝑣 ] to the normalized 𝑉 = [𝑣 ] is based on the fact that the rank and magnitude of the relation between the elements of the same vector column are preserved for the original and normalized matrix. 110 v.p. korneenko copyright ©2024 assa adv. in systems science and appl. (2024) 4.1. theorem on the optimal solution: theorem 4.3: the optimal solution to the problem (4.10) is a normalized vector �⃗�∗ = 𝑤∗ … 𝑤∗ , the components of which are taken as the coefficients of the importance the criteria 𝑤∗ = 𝑤(𝑓 ) and calculated by the formula: 𝑤∗ = 1 𝑛 𝑣 , ∀ 𝑖 = 1, 𝑛, while delivering a minimum of the indicator (4.9). the elements of the optimal approximation matrix 𝑊∗ = 𝑤∗ are determined by the formula: 𝑤∗ = 𝑤∗ 𝑤∗. as an estimate of the approximation matrix to the original judgment matrix, we take the euclidean matrix norm [9]: 𝑑(𝑉, 𝑊∗) = ‖𝑉 − 𝑊∗‖ = 𝑣 − 𝑤∗ (4.11) the proof is trivial and is based on necessary and sufficient conditions for the existence of an optimal solution of a function of many variables. 4.2. algorithm for generating optimal weights for criteria step 1. normalization of elements of the original matrix of pairwise comparisons of the importance of criteria by the sum ∑ 𝑣 of elements columns of the original matrix of judgments: 𝑣 = ∑ are the normalized elements of the judgment matrix 𝑉 = [𝑣 ]. step 2. the calculation for each row of the normalized matrix 𝑉 = [𝑣 ] of judgments of its average value, we take it as the normalized weights of the criteria, namely: 𝑤∗ = 1 𝑛 𝑣 , 𝑤∗ = 𝑤(𝑓 ) ∀ 𝑖 = 1, 𝑛. step 3. recovery of the elements of the multiplicative matrix by the optimal normalized weights of the criteria: 𝑊 = ∗ ∗ . step 4. estimation of the proximity between the elements of the original matrix of pairwise comparisons and the optimal multiplicative matrix by the formula 𝑑(𝑉, 𝑊∗) (4.11). 5. comparison of the effectiveness of the method with the analytical hierarchy process by t. saaty let us compare the error of calculating the weights of the importance of objects (criteria, alternatives) by t. saaty's method with the optimization method of the approximation matrix for the approximation matrix method and its comparison… 111 copyright ©2024 assa adv. in systems science and appl. (2024) the formation of object weights in multi-criteria problems. as multiplicative matrices consider the square matrix 𝑊 = , , . and 𝑊 = ∗ ∗ , , . , in which the elements are determined by the vector of priorities, found by the method of analysis of hierarchies and using approximate matrices. to do this, we will use the data from the example of buying a house by a family with average incomes, given in the work of t. saaty (see [6, p. 41–44]). the problem consists in choosing one house from three available alternatives {a, b, c} based on eight factors that serve as criteria for the multi-criteria selection problem. table 5.1 shows expert estimates of pairwise comparison of the importance of criteria and the priority vector calculated from the maximum eigenvalue following the method of hierarchy analysis by t. saaty (𝜆 = 8,811, compatibility index 𝑆. 𝐼. = , ≈ 1,101). the summed elements of the judgment matrix and the vector of weights of the importance of criteria found from these data using the approximation matrix method are presented in table 5.2. table 5.1. initial matrix 𝑉 of judgments and priority vector according to ahp [6, p. 42] factor (criteria) 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 priority vector, 𝑤 size, 𝑓 1 5 3 7 6 6 1/3 1/4 0,175 transport, 𝑓 1/5 1 1/3 5 3 3 1/5 1/7 0,062 environment, 𝑓 1/3 3 1 6 3 4 1/2 1/5 0,103 age, 𝑓 1/7 1/5 1/6 1 1/3 1/4 1/7 1/8 0,019 yard, 𝑓 1/6 1/3 1/3 3 1 1/2 1/5 1/6 0,034 facilities, 𝑓 1/6 1/3 1/4 4 2 1 1/5 1/6 0,041 state, 𝑓 3 5 2 7 5 5 1 1/2 0,221 finance, 𝑓 4 7 5 8 6 6 2 1 0,348 table 5.2. normalized matrix 𝑉 = ∑ of judgments and a vector of priorities by mam factor (criteria) 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 priority vector, 𝑤 size, 𝑓 0,11 0,23 0,25 0,17 0,23 0,23 0,07 0,10 0,174 transport, 𝑓 0,02 0,05 0,03 0,12 0,11 0,12 0,04 0,06 0,068 environment, 𝑓 0,04 0,14 0,08 0,15 0,11 0,16 0,11 0,08 0,108 age, 𝑓 0,02 0,01 0,01 0,02 0,01 0,01 0,03 0,05 0,021 yard, 𝑓 0,02 0,02 0,03 0,07 0,04 0,02 0,04 0,07 0,038 facilities, 𝑓 0,02 0,02 0,02 0,10 0,08 0,04 0,04 0,07 0,047 state, 𝑓 0,33 0,23 0,17 0,17 0,19 0,19 0,22 0,20 0,212 finance, 𝑓 0,44 0,32 0,41 0,20 0,23 0,23 0,44 0,39 0,333 to evaluate the accuracy, we restore multiplicative matrices of pairwise relations by priority vectors: �⃗� = (0,175; 0,062; 0,103; 0,019; 0,034; 0,041; 0,221; 0,348), (5.12) �⃗� = (0,174; 0,068; 0,108; 0,021; 0,038; 0,047; 0,212; 0,333), (5.13) obtained by the method of calculating the eigenvector and the method of approximating the matrix of pairwise comparisons using the minimum distance criterion, and compare the results. tables 5.3 and 5.4 present multiplicative matrices �⃗� , �⃗� , with priority vectors �⃗� (5.12) and �⃗� (5.13) formed from normalized weights. table 5.3. multiplicative matrix w , formed from normalized weights of the priority vector 𝑤 factor (criteria) 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 size, 𝑓 1,00 2,82 1,70 9,21 5,15 4,27 0,79 0,50 transport, 𝑓 0,35 1,00 0,60 3,26 1,2 1,51 0,28 0,18 environment, 𝑓 0,59 1,66 1,00 5,42 3,03 2,51 0,47 0,30 age, 𝑓 0,11 0,31 0,18 1,00 0,56 0,46 0,09 0,05 yard, 𝑓 0,19 0,55 0,33 1,79 1,00 0,83 0,15 0,10 112 v.p. korneenko copyright ©2024 assa adv. in systems science and appl. (2024) facilities, 𝑓 0,23 0,66 0,40 2,16 1,21 1,00 0,19 0,12 state, 𝑓 1,26 3,56 2,15 11,63 6,50 5,39 1,00 0,64 finance, 𝑓 1,99 5,61 3,38 18,32 10,24 8,49 1,57 1,00 table 5.4. multiplicative matrix 𝑊 formed from normalized weights of the priority vector 𝑤 factor (criteria) 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 size, 𝑓 1,00 2,54 1,62 8,37 4,63 3,70 0,82 0,52 transport, 𝑓 0,39 1,00 0,64 3,30 1,83 1,46 0,32 0,21 environment, 𝑓 0,62 1,57 1,00 5,17 2,86 2,29 0,51 0,32 age, 𝑓 0,12 0,30 0,19 1,00 0,55 0,44 0,10 0,06 yard, 𝑓 0,22 0,55 0,35 1,81 1,00 0,80 0,18 0,11 facilities, 𝑓 0,27 0,69 0,44 2,26 1,25 1,00 0,22 0,14 state, 𝑓 1,22 3,10 1,98 10,21 5,65 4,52 1,00 0,64 finance, 𝑓 1,91 4,86 3,10 16,02 8,86 7,08 1,57 1,00 we find the values of the norm of the difference between the original matrices and the multiplicative matrices of relations formed from the values of the priority vector (the importance of objects): 𝑑 = ‖𝑉 − w ‖ = √200,66 ≈ 14,2; 𝑑 = ‖𝑉 − 𝑊 ‖ = √139,61 ≈ 11,8. let us determine the accuracy of the solution 𝑑 by the method of t. saaty concerning the optimal 𝑑 obtained in the framework of the optimization problem by the criterion (4.11). using the ε-approximation formula, we find that: 𝜀 = |𝑑 − 𝑑 | 𝑑 × 100 % = 14,2 − 11,8 11,8 × 100 % ≈ 20,3 %. comparison of methods for the accuracy of obtaining the weights of objects for the original perturbed matrix of the 8th order is shown in fig. 5.1. fig 5.1. comparison of methods based on the accuracy of obtaining object weights it follows that the error estimate is 20,3 % and the ahp solution that differs from the optimal one by this value cannot be considered satisfactory. thus, t. saaty's method of finding priorities for the importance of criteria and objects based on the eigenvector of the matrix of pairwise comparisons should be attributed to approximate, and not to exact, as the author of ahp declares. 7. conclusion the paper explores the problem of forming quantitative weights of objects based on the matrix of pairwise comparisons in the relationship scale. since in the hierarchy analysis process of t. saaty, the method of determining the weight vector is carried out using polynomials, the problem of finding the roots of polynomials of degree 𝑛 ≥ 5 is unsolvable in radicals. therefore, in t. saaty's method, for the number of objects at least five, the procedure for finding the eigenvalues of the matrix of degrees of the superiority of the importance of criteria 11,8 14,2 10,0 11,0 12,0 13,0 14,0 15,0 the approximation matrix method the analytical hierarchy process the distance of multiplicative matrices to the original matrix of judgments the approximation matrix method and its comparison… 113 copyright ©2024 assa adv. in systems science and appl. (2024) or preferences of alternatives is carried out using numerical methods for finding the roots of a polynomial implemented in the package expert choice [17]. on the other hand, and because of the inverse symmetry of the elements of the matrix of judgments 𝑉 = [𝑣 ], 𝑖, 𝑗 = 1, 𝑛, obtained by sequential comparison of all pairs of objects, the expert has to answer ( ) questions about the values 𝑣 . in this regard, the t. saaty's method is justified only for a small number of criteria and objects. in this article, using a concrete example, it is shown that the analytical hierarchy method is approximate and at the same time, its error is estimated relative to the optimal solution obtained by the method of the approximation matrix for the formation of object weights in multi-criteria problems. since the presented method is mathematically justified, and also because of the computational simplicity of forming object weights, it can be recommended instead of the method t. saaty in solving applied problems. references 1. figueira, j., greco, s., & ehrgott, m. (2005). multiple criteria decision analysis: state of the art surveys multiple criteria decision analysis: state of the art surveys. berlin, germany: springer. 2. noghin, v. d. (2018). reduction of the pareto set: an axiomatic approach. berlin, germany: springer. 3. korneenko, v. p. (2019). optimizatsionniy metod vybora rezuliruyushego ranzhirovaniya obiektov, predstablennyh v rangovoy shkale izmereniya [optimization method for selecting the resulting ranking of objects represented in the rank scale of measurement], large-scale systems control, 82, 44–60, [in russian], doi: 10.25728/ubs.2019.82.3 4. sigford, s. v., & parvin, r. h. (1995). project pattern a methodology for delernining relevance in complex decision-making, ieee trans., 12(1), 9–13. 5. saaty, t. l., & vargas, l. g. (1984). comparison of eigenvalue, logarithmic least squares and least squares method in estimating ratios, mathematical modeling, 5, 309–324. 6. saaty, t. l. (1986). axiomatic foundation of the analytic hierarchy process, management science, 32(7), 841–855. 7. saaty, t. l. (1990). the analytic hierarchy process: planning, priority setting, resource allocation. pittsburgh, pa: rws. 8. saaty, t. (2008). prinyatie resheniy pri zavisimostyah i obratnyh svyazyah: analiticheskiye seti [decision-making under dependencies and feedback: analytical networks]. moscow, lki, 360 p. [in russian]. 9. horn, r. a., & johnson, c. r. (2013). matrix analysis. cambridge, uk: cambridge university press. 10. noghin, v. d. (2004). a simplified variant of the analytic hierarchy processes based on a nonlinear scalarizing function, computational mathematics and mathematical physics, 44(7), 1194–1202. 11. zahedi, f. (1986). the analytic hierarchy process – a survey of the method and its applications, interfaces, 16, 96–108. 12. takeda, e. (1990). the analytic hierarchy process: an overview, systems, control and information, 34, 669–675. 13. forman, t. y., & gass, s. i. (2001). the analytic hierarchy process – an exposition, operations research, 21, 469–486. 14. bodin, l., & gass, s. i. (2003). on teaching the analytic hierarchy process, computer and operations research, 30, 487–1497. 15. vaidia, j. s., & kumar, s. (2006). analytic hierarchy process: an overview of applications, european journal of operational research, 168, 1–29. 16. expert choice. (2021). [online]. available: www.expertchoice.com. 17. mirkin, b. g. (1974). problema gruppovovo vybora [the problem of group selection]. moscow, ussr: nauka [in russian]. adv syst sci appl 2019; 03; 80-92 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/689 influence of information sharing, partnership, and collaboration in supply chain performance; study on apples agroindustry alfredo tutuhatunewa*1,2, surachman1, purnomo b. santoso1, imam santoso1 1) brawijaya university, mt haryono street 167 malang, indonesia e-mail: alfredo.tutuhatunewa@hotmail.com 2) pattimura university, ir. m. putuhena street poka, ambon, indonesia received january 11, 2019; revised august 26, 2019; published october 1, 2019 abstract: supply chain plays an essential role in the development of business organizations to achieve competitive advantage. supply chain studies are continuing mainly to improve supply chain performance. many of the literature write about the role of information sharing, partnership processes and collaboration in improving the performance of the supply chain. this research was conducted to find the relationship and influence of information sharing, supply chain partnership, and collaboration on supply chain performance on smes. the object of research is apple agroindustry, in east java, indonesia. the analysis was conducted with partial least square (pls). the results indicate only an indirect effect of information sharing on supply chain collaboration. also, indirect effect of information sharing on supply chain performance of smes. keywords: information sharing, partnerships, collaboration, partial least square, small and medium enterprises, supply chain performance. 1. introduction apple is an annual fruit plant originating from the west asian region with a sub-tropical climate. in indonesia, apples can grow and bear good fruit in the highlands. the centers of apple production in east java are malang (batu, pujon, and poncokusumo) and pasuruan (nongkojajar). in addition to apple cultivation, the processed industries of apples continue to be developed. processed industries are conducted to increase the added value of apples into various food and beverage products. the majority of the processed apple industry is a micro and small business unit (smes), which is a home industry. the apple agroindustry supply chain is dynamic because it involves the flow activity, among others: raw materials, finished products, ordering, shipping, payment, and information among the parties involved. it causes the overall management of these streams to be challenging to perform effectively. it is hard to meet the production targets due to the absence of raw materials at certain times, due to failure to share information and establish coordination and collaboration between actors in the supply chain of apple agroindustry. the supply chain (sc) is defined as the management of upstream and downstream relationships with suppliers and customers in order to deliver superior customer value at less cost to the supply chain as a whole [10]. sc has become an essential focus for business organizations to enhance competitive advantage. companies must implement the right supply chain management strategy to compete at the sc level. this strategy needs to be integrated and coordinated throughout the sc to produce the performance of sc members [14,18]. supply chain management (scm) studies emphasize how to maximize the overall value of a * corresponding author: alfredo.tutuhatunewa@hotmail.com influence of information sharing, partnership 81 copyright ©2019 assa. adv. in systems science and appl. (2019) company by using and sharing resources across the company better. its become apparent for many organizations that are assessing their performance is essential to succeeding efficient and effective sc [1]. due to globalization, outsourcing, customization, time to market, and pricing pressure have compelled enterprises to adopt efficient and effective scm [40]. therefore, it is crucial to coordinate decisions and actions among partners in sc to improve the performance of the sc [47]. a suitable coordination mechanism between actors in sc through an online information network plays an essential role in increasing the effectiveness of material flow, information, and money [34]. that is, if companies want to improve collaborative capabilities, then companies need to prepare themselves by building a network of information technology to support the ability to share information first. furthermore, the ability to share information and collaborative capabilities jointly affects supply chain performance [46]. the benefits of information sharing include improving inventory management, increasing sales, and knowing the demand better [19]. failure to share information in the supply chain causes a bullwhip effect [11], i.e., amplification of the demand flow variance, which is flowing in the entire sc, from customer to factory. sc collaboration is also influenced by the practice of partnerships among actors taking place in it [50]. 2. materials and methods information sharing information sharing refers to the extent to which critical information is communicated to other supply chain partners [27]. many researchers have emphasized the importance of sharing information in scm practice. moreover, yu et al. [49] suggest that the adverse effects of the bullwhip effect on the supply chain can be reduced or eliminated by sharing information with trading partners. the empirical findings of childerhouse & towill [9] reveal that a simplified flow of materials, including streamlining and making all the information flows through the chains, is an integrated and effective supply chain. currently, the companies do not operate alone; they are now connected to many other partners [29]. information sharing can help supply chain members establish partnerships for better supply chain system performance [49]. information sharing is found to impact operational performance [13]. meanwhile, information sharing is one of the characteristics of collaboration [35]. information sharing also plays an important role in supporting collaborative capabilities [46]. h1a: information sharing has a positive impact on sc partnership. h1b: information sharing has a positive impact on sc collaboration. h1c: information sharing has a positive impact on sc performance. partnership the partnership is defined as purposive strategic relationships between independent firms having common goals, striving for mutual benefit, and recognizing a high degree of interdependence [25]. in the context of cooperation, the relationship between the company and its suppliers can take many forms. the first, such as joint ventures or strategic alliances, involves negotiating and maintaining explicit contracts that explain expectations and deliveries and sometimes revenue-sharing [7] and they have a legal structure that sets the boundaries [45]. supply chain partnerships, on the other hand, tend to operate without formal contracts [22]. the type of relationship between buyers and suppliers might vary from hostilities to cooperatives [6]. research from zhang et al. [50] has concluded that partnership management 82 a. tutuhatunewa, surachman, p. b. santoso, i. santoso copyright ©2019 assa adv. in systems science and appl. (2019) had a significantly positive influence on supply chain collaboration. moreover, the partnership has an importance role to supply chain performance [20,33,49]. h2a: partnership has a positive impact on sc collaboration. h2b: partnership has a positive impact on sc performance. collaboration collaboration is a process of participation of a group of people or organizations, working together to achieve mutually desired outcomes, and building interconnected systems to address problems and opportunities. collaboration is defined by several researchers among others, [26], which refer to long-term conditions, win-win conditions, and open information exchange agreements, in which both parties are engaged in joint efforts to improve performance and commit to quality, cooperation, and conflict resolution. meanwhile, [2] defines collaboration as two or more companies sharing responsibility in planning, management, execution, and performance measurement information. from the above understanding, it can be concluded that collaboration is a form of cooperation, interaction, and compromise of some individuals, or organizations, involved directly or indirectly, over a long period, receiving consequences and benefits. collaboration is based on values among others, the common goal, the similarity of perception, the willingness to process, and mutual benefit. collaboration involves multiple shared resources and responsibilities, in planning, implementing, and evaluating all activities to achieve common goals. all parties involved must be willing to share their vision, mission, resources, and strengths. in supply chains, collaboration involves designing a set of strategies where two or more different actors, with complementary capabilities, achieve shared aspirations and goals in a competitive environment, which cannot be achieved individually [21]. from this perspective, collaboration in the supply chain becomes an essential strategy for achieving competitive advantage. the escalation of competition, the flow of globalization, and the increasing demands of customers led to the idea that companies cannot compete on their own in the marketplace. therefore, companies look beyond their boundaries and establish cooperation with other complementary firms to minimize potential risks [32]. the literature suggests that collaboration is associated with increased performance, regarding increased visibility, increased service levels, increased flexibility, improved end-customer satisfaction, and reduced cycle time [12,37,43]. h3a: collaboration has a positive impact on sc performance. supply chain performance some experts and practitioners recommend several methods that accommodate all the dimensions of supply chain performance [39], namely: • total supply chain cost. fulfillment costs as a percentage of revenues or fulfillment costs per order case. • service level. it includes the level (availability the ratio of the number of items ordered by the customer and the number of items sent to the customer). • asset management. it focuses on capital utilization of investments in facilities and equipment and working capital invested in inventories. • customer accommodation. it aims to capture the size of the request flawlessly. • cash-to-cash cycle time. it is time it takes to convert the costs spent on inventory into profits that are collected from the proceeds of the sale. • benchmarking. it makes management aware of state-of-the-art business practices. it includes: internal benchmarking, competitor benchmarking, and benchmarking limited. influence of information sharing, partnership 83 copyright ©2019 assa. adv. in systems science and appl. (2019) the global performance of the supply chain can be enhanced by exchanging information between its members at different decision levels [28]. the performance of the supply chain is strongly influenced by two things, namely information sharing and collaboration capabilities [28,38,46,48]. the research framework is shown in fig. 1. fig. 2.1. research framework methodology the research was conducted in the supply chain of apple agroindustry, from apple farmers, suppliers of food additives, suppliers of plastic cups and bottles, packaging suppliers, apple processing agroindustry, distribution, and retailers, in batu city, east java. data collection was done by distribution the questionnaire, which was compiled to know how the process of share the information, partnership and collaboration in the supply chain of apple agroindustry, as well as supply chain performance. research variables and measuring instruments have described in table 2.1. data were collected from 36 smes, with product classification as in table 2.2. table 2.1. variable and its measuring instruments variable measuring instrument ref. information sharing (x1) • smes only shares inventory data with supply chain partners (x1-1); • smes shares inventory and demand data with other supply chain partners. (x1-2) • smes shares inventory, demand, and capacity or production data with other supply chain partners. (x1-3). [8] partnership (y1) • trust (y1-1) • commitment (y1-2) [5] [16] collaboration (y2) • resource sharing (y2-1) • decision synchronization (y2-2) • incentive alignment (y2-3) [4] supply chain performance (y3) • supply chain flexibility (y3-1) • the extent of co-operation (y3-2) • customer responsiveness (y3-3) [42] [17] [51] information sharing partnership collaboration sc performance h1a h1b h1c h2a h2b h3a 84 a. tutuhatunewa, surachman, p. b. santoso, i. santoso copyright ©2019 assa adv. in systems science and appl. (2019) data analysis and structural equation modeling using partial least square (pls), are as follows: 1. measurement model (outer model). the measurement model defines how each indicator block corresponds to its latent variable. the measurement model design determines the indicator properties of each latent variable, whether reflexive or formative, based on the operational definition of the variable. 2. structural model (inner model). the structural model describes the relationship between latent variables based on substantive theory. the latent variables in this study are information sharing, partnership, collaboration, and supply chain performance. 3. path diagram. 4. evaluation of the goodness of fit q-square can be calculated by the equation: 𝑄2 = 1 − (1 − 𝑅1 2)(1 − 𝑅2 2). . (1 − 𝑅𝑝 2) (2.1) where: 𝑅1 2, 𝑅2 2 ... 𝑅𝑝 2 are r-square of the endogenous variable in the model. 5. hypothesis testing (resampling bootstrapping) table 2.2. product classification of respondents product number of respondents apple cider 14 apple chips 9 apple vinegar 2 another apple processed 11 3. results and discussion evaluation of measurement model convergent validity is used to measure the validity of a reflexive indicator as a latent variable measure, which can be seen from the outer loading of each variable indicator. an indicator is said to have excellent reliability if the value of outer loading above 0.70. results of outer loadings can be seen in table 3.1. for information sharing variables (x1), the three indicators show values greater than 0.7. thus, it can be concluded that the three indicators are capable of measuring information sharing variables well. in partnership variable (y1), two indicators, also have outer loading value more than 0.7, which means that both indicators capable of measuring partnership variables well. for collaboration variable (y2), which is an endogenous latent variable, capable explained by three indicators. so is the performance variable (y3), which has outer loading value greater than 0.7. table 3.1. outer loadings variables x1 y1 y2 y3 x1-1 0.901 x1-2 0.935 x1-3 0.923 y1-1 0.914 y1-2 0.916 influence of information sharing, partnership 85 copyright ©2019 assa. adv. in systems science and appl. (2019) y2-1 0.919 y2-2 0.946 y2-3 0.910 y3-1 0.918 y3-2 0.897 y3-3 0.822 the criteria for measuring discriminant validity can be seen on the cross-loading between the indicator and the construct. the result of cross-loading is shown in table 3.2. table 3.2. cross loading variables x1 y1 y2 y3 x1-1 0.901 0.587 0.528 0.559 x1-2 0.935 0.641 0.711 0.646 x1-3 0.923 0.586 0.599 0.577 y1-1 0.633 0.914 0.674 0.637 y1-2 0.574 0.916 0.715 0.676 y2-1 0.563 0.648 0.703 0.919 y2-2 0.636 0.703 0.835 0.946 y2-3 0.598 0.637 0.702 0.910 y3-1 0.661 0.753 0.918 0.759 y3-2 0.589 0.622 0.897 0.831 y3-3 0.509 0.625 0.822 0.509 the result of cross-loading shows that the correlation of the construct between each variable with the indicator has a higher value than the correlation with other indicators. it is concluded that the latent variables predict the indicator on it block better than the indicator on the other block. construct reliability can be measured by cronbach's alpha. this value reflects the reliability of all indicators in the model. the minimum value of 0.7 is ideally 0.8 or 0.9. results of data processing can be seen in table 3.3. table 3.3. construct reliability and validity cronbach's alpha rho_a composite reliability information 0.909 0.917 0.943 partnership 0.806 0.806 0.912 performance 0.855 0.876 0.911 collaboration 0.916 0.923 0.947 from table 3.3, it can be seen that all variables have cronbach's alpha more than 0.7 with the lowest value in the partnership variable. thus, it can be concluded that no reliability or unidimensionality problems were found in the established model. evaluation of the structural model the structural model analysis was conducted to examine the effect of information sharing, partnership, and collaboration on supply chain performance of apple agro-industrial. the analysis is done by using r-square. 86 a. tutuhatunewa, surachman, p. b. santoso, i. santoso copyright ©2019 assa adv. in systems science and appl. (2019) r-square indicates the extent to which a construct can describe the model as a whole, or by indicating the magnitude of a particular influence of latent variables on the latent dependent variable. r-square value can be seen in table 3.4. table 3.4. r-square original sample (o) sample mean (m) standard deviation (stdev) t statistics p values partnership 0.434 0.441 0.153 2,835 0.005 performance 0.733 0.754 0.075 9,758 0,000 collaboration 0.569 0.594 0.096 5.904 0,000 from table 3.4., it can be summarized as follows:  r-square of partnership variables indicates that information sharing gives 43.4% influence on the partnership.  r-square of performance variable indicates that information sharing (x1), partnership (y1) and collaboration (y2) give the effect of 73.3% to performance.  r-square of the collaboration variable indicates that the information sharing (x1) and partnership (y1) give 56.9% influence to the collaboration variables (y2). path diagram fig. 3.1 shows the path diagram obtained. the diagram shows the relationships among information sharing, partnerships, collaboration and supply chain performance of apple agroindustry. fig. 3.1. path diagram evaluation of the goodness of fit evaluation of the goodness of fit is done by calculating a q-square value for the constructive model. q-square measures how well the model and its parameter estimation generate the observation value. the q-square value > 0 indicates that the model has predictive relevance, otherwise if the q-square value ≤ 0 indicates the model lacks predictive relevance. q-square can be calculated by equation (1): 𝑄2 = 1 − (1 − 𝑅1 2)(1 − 𝑅2 2)(1 − 𝑅3 2) information sharing collaboration performance partnership influence of information sharing, partnership 87 copyright ©2019 assa. adv. in systems science and appl. (2019) where: 𝑅1 2 = 0.434; 𝑅2 2 = 0.569; and 𝑅3 2 = 0.733 (see table 6) so, 𝑄2 = 0,935 from the calculation of q-square, it can be concluded that the resulting model has a very good predictive relevance of 93.5%. hypothesis testing the impact of information sharing on supply chain partnership. path diagram results in fig. 3.1 shows that at 5 percent significant level, information sharing affects sc partnership, with path coefficient 0.659 (p-value = 0.000). thus, this result support h1a is information sharing has a positive impact on sc partnership. the impact of information sharing on supply chain collaboration. path diagram results in fig. 3.1 shows that at 5 percent significant level, information sharing has no impact on sc collaboration, with path coefficient 0.311 (p-value = 0.077). thus, this result does not support h1b is information sharing has a positive impact on sc collaboration. the impact of information sharing on supply chain performance path diagram results in fig. 3.1 shows that at 5 percent significant level, information sharing has no impact on sc performance, with path coefficient 0.150 (p-value = 0.188). thus, these results not support h1b is information sharing has a positive impact on sc performance. the impact of partnership on supply chain collaboration path diagram results in fig. 3.1 shows that at 5 percent significant level, the partnership has a positive impact on sc collaboration, with path coefficient 0.513 (p-value = 0.005). thus, this result support h2a is partnership has a positive impact on sc collaboration. the impact of the partnership on supply chain performance path diagram results in fig. 3.1 shows that at 5 percent significant level, the partnership does not affect sc performance, with path coefficient 0.306 (p-value = 0.110). thus, these results do not support h2b is partnership has a positive impact on sc performance. the impact of collaboration on supply chain performance path diagram results in fig. 3.1 shows that at 5 percent significant level, collaboration has a positive effect on sc performance, with path coefficient 0.494 (p-value = 0.003). thus, this result support h3a is collaboration has a positive impact on sc performance. indirect effect testing table 3.5 shows the indirect effects of the research variables. indirect effect on table 7 show that at 5 per cent significant level, information sharing has indirect effect to the sc performance (t-statistic = 4.636; p-value = 0.000), information sharing has indirectly effect to the sc collaboration (t-statistics = 2.643; p-value = 0.008) and partnership has indirectly effect to the sc performance (t-statistics = 2.025; p-value = 0.043) 88 a. tutuhatunewa, surachman, p. b. santoso, i. santoso copyright ©2019 assa adv. in systems science and appl. (2019) table 3.5. indirect effects original sample (o) sample mean (m) standard deviation (stdev) t statistics p values information -> partnership information -> performance 0.522 0.522 0.113 4,636 0,000 information -> collaboration 0.338 0.316 0.323 2,643 0.008 partnerships -> performance 0.253 0.237 0.125 2,025 0.043 partnerships -> collaboration collaboration -> performance 4. discussion this study uses pls to analyze the correlation between information sharing, partnerships, and collaboration on sc performance. the research was conducted at the apple agroindustry supply chain, batu city, indonesia. the results of the study did not find the direct effect of information sharing on sc performance, nor the direct effect of partnerships on sc performance. likewise, the direct effect of information sharing on sc collaboration was not found. the results of the study only indicate that there is an indirect effect of information sharing on sc collaboration, information sharing on sc performance, and partnerships on sc performance. this result is in line with the study from baihaqi and sohal [3], which states that information sharing does not have a direct relationship with organizational performance. this relationship is mediated by the practice of collaboration between supply chain actors. the absence of such correlations is typical for energy consumption since power grids possess constructive constraints and, hence, other tools have to be developed to improve performance of energy supply chains [30]. the present result is in line with the findings of zhao et al. [51], which states that information sharing significantly affects the supply chain performance, both in total costs and service levels. zhou & benton [52] also state that supply chain practices will be effective when the level of information sharing increases, which in turn will also improve supply chain performance. this result also supports the results of research by lin et al. [23], which states that information sharing can reduce demand uncertainty, which will improve supply chain performance. information sharing provides a large number of benefits, which in turn will increase the efficiency of supply chain performance in the manufacturing sector [24]. the effect of partnership on sc performance was following the results proposed by khan et al. [20] that buyer-supplier partnerships in the supply chain have a positive effect on supply chain performance. results of gallear et al. [15] also show that there is a positive relationship between partnership management and supply chain performance. wibowo and sholeh [44] also state that supplier partnerships are one of the supporting factors for supply chain performance in construction projects. the effect of collaboration on sc performance was following the findings of vereecke & muylle [41], who found empirically the relationship between sc collaboration and improved performance obtained. these empirical findings support the statement that proper collaboration with suppliers and customers provides benefits for improving performance. singhry et al. [36] found a significant relationship between scc and sc performance, which has been tested through covariance structural equation modeling. ramanathan and gunasekaran [31] also found that collaborative alliances improve sc performance, however, there are no results of research that show an indirect relationship of the effect of information sharing on sc performance. this condition can be caused by the population, which is mostly smes with limited resources. the process of information sharing is not carried out as done by large companies. influence of information sharing, partnership 89 copyright ©2019 assa. adv. in systems science and appl. 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(2007) supply chain practice and information sharing. journal of operations management 25(6), 1348–65, doi: 10.1016/j.jom.2007.01.009. microsoft word 992 article text, copyedited.doc adv syst sci appl 2021; 02; 29-41 published online at https://ijassa.ipu.ru. cooperation between smes and large industrial enterprises: russian case olga a. romanova1, evgeny a. kuzmin1,2*, marina v. vinogradova3, olga s. kulyamina3 1) institute of economics of the ural branch of russian academy of sciences, yekaterinburg, russia 2) ural state university of economics, yekaterinburg, russia 3) russian state social university, moscow, russia abstract: in the modern global economy, the system of functional cooperative relations that lead to the formation of hybrid structures – production networks has become actively spread. however, the strength and scale of cooperation are not uniform. using the example of russia, the authors consider the effectiveness of the production network of cooperation between small and mediumsized enterprises with large companies. we tested a number of hypotheses based on common ideas about the effects of cooperation. empirical results make it possible to clarify the mechanism of formation and features of interfirm production chains in russia. the “anchor” role of large enterprises with state participation as centers of cooperation formation is noted. in the course of the study, 14 enterprises were selected, distributed across key sectors of the russian economy. statistical and correlation analysis methods were used to evaluate the effects of cooperation. the results showed that the orders placed by large manufacturing enterprises with small and mediumsized enterprises increased over the period of 2015–2019. “anchor” enterprises, as a rule, reduce the production localization degree. however, this does not have a significant impact on improving the profitability of their activities, and also does not depend on the share of state participation. besides, placing orders with small and medium-sized enterprises does not allow them to reduce the number of employees. many of the expected internalities that are characteristic of cooperative relations in developed countries are not reflected in the specifics of the russian economy, or their manifestation is limited. the russian experience clearly demonstrates the weakness of cooperative partnership, although with positive trends of change. there is a need to further improve the mechanisms for supporting small and medium-sized enterprises in the production sector, aimed at creating sustainable networks. the proposed approach can be applied to assess inter-firm production chains in other countries. a comparative study will determine the strength of the formation of production networks across countries, which will expand the understanding of the economic processes of networkization. keywords: cooperation, globalization, small and medium-sized businesses, production networks, subcontracting, specialization, production chains 1. introduction transformations in the economy that occur under the influence of globalization processes encourage enterprises to look for effective forms of organization of production activities. such forms of integration as subcontracting, franchising, leasing, venture financing, technology parks, joint ventures, tolling, etc. are becoming more and more popular [1]. the system of functional production of cooperative relations has become widely used. bringing a small number of specialized companies in private manufacturing businesses provides the flexibility to respond to market conditions. one of the main factors determining cooperation is the creation of useful communication mechanisms, which are usually carried out in the form of subcontracting processes. * corresponding author: kuzmin.ea@uiec.ru 30 o.a. romanova, e.a. kuzmin, m.v. vinogradova, o.s. kulyamina copyright ©2021 assa. adv. in systems science and appl. (2021) the closeness of ties and the depth of cooperation between industrial enterprises are not uniform. in russia, this is facilitated by the procurement system of state-owned companies, which obliges large enterprises to purchase from small and medium-sized enterprises (smes), which in turn can be carried out by transferring part of the production cycle to a subcontractor (subcontract). the volume of purchases from smes, including purchases in which the contractor must engage the sme as a subcontractor, must be at least 20% of the total annual value of contracts concluded by customers based on the results of purchases. at the same time, at least 18% of the total annual value of contracts should be allocated to procurement involving only smes [2]. however, these measures have not yet significantly changed the role of smes in the economy as a whole. according to the analytical center for the government of the russian federation, in comparison with foreign countries, the share of public procurement by smes is low is about 1–5% against 20%. the reasons for these discrepancies are both institutional and structural [3]. it is obvious that for sustainable economic development, it is necessary to search for optimal mechanisms for expanding cooperation between industrial enterprises of various levels. therefore, the purpose of the study is to assess the effects of cooperation between smes and large industrial enterprises in russia. to do this, the authors analyzed the purchasing activities of large enterprises with state participation in key russian sectors: oil, gas, minerals and related activities (petroleum products transportation); industrial production, and electric-power supply industry. preliminary data indicate that the degree of cooperation between enterprises is relatively low. in this regard, the authors put forward several assumptions that should explain this feature. the identified impact factors will allow understanding better the mechanism of forming production chains through cooperation and assessing its potential in the russian economy. 2. literature review the relationship between competition and cooperation is one of the key issues in the strategic management theory. this is reflected in the variety of approaches that explain the pros and cons of various strategies for organizing the production process of enterprises in their interaction with the external environment [4-7]. cooperation develops along the entire value chain. preference is given to all possible contractual forms in comparison with intra-company integration. the intensive growth of industrial cooperation raises questions about the blurring of lines of the economic agent, the formation of hybrid structures, which are increasingly referred to as networks. the phenomenon of intercompany network relations, which has become widespread in recent years, attracts researchers who are trying to explain the reasons for its occurrence [8]. in the most general terms, intercompany networks are perceived as a way to regulate the interdependence between companies. it should be taken into account that initially, the definitions of intercompany networks differ both in the terminology used and in the emphasis [9-11]; the objective and research direction are the decisive factors. the development of network production cooperation with the participation of smes is presented in the works of many scientists [12-14]. mins and schneider [15] define the transformation of the world economy and the principles of business introduction as metacapitalism. justifying this idea, among the reasons, along with globalization, integration of global capital markets, the spread of information and communication technologies and ebusiness, they pointed to the fundamental restructuring of companies, which led not only to their reengineering of business processes but also contributed to the creation of transnational production networks [15]. as the modern world practice shows, it is necessary to use production cooperation in its newest forms for the sustainable development of industrial enterprises. petrishcheva [16] cooperation between smes and large industrial enterprises 31 copyright ©2021 assa. adv. in systems science and appl. (2021) attempts to set the concept of industrial cooperation and points out the potential for its development. it is generally agreed [17-21] that one of the most promising organizational forms of integration of small, medium, and large enterprises is subcontracting. this form of cooperation is designed to use a wide network of suppliers [16, 22-23]. more generally, subcontracting refers to a specific aspect of the organization of industrial production, in which large and small firms coexist (with a high degree of specialization) in production, and sometimes in making investment decisions [24]. in fact, a large enterprise transfers part of its production functions, which ultimately reduces inventory and optimizes the production process, focusing on the assembly of the final product and quality control [25]. the use of this mechanism leads to a reduction in capital investment in the means of production and a reduction in the number of people employed in production [26]. as a result, subcontracting helps to diversify business risks and regulate production levels more flexibly [27-28]. some scientists [29-31] have suggested that cost minimization is the main explanation for subcontracting production. at the same time, tijun et al. [32] stated that the main idea of subcontracting went beyond cost minimization. they justified the idea that the key factors influencing the use of this production strategy were the focus on the core business, access to the professional and technological capabilities of partners, and the release of internal resources. according to handfield and nichols [33], manufacturers can match future product needs with existing resources through close collaboration with key suppliers. in developed countries, industrial cooperation is a tool for improving the efficiency of industrial production and ensuring overall economic growth [34-36]. since smes are the initiators of many innovations and provide the basis for sustainable economic development [37], they must be adequately protected in order to survive in the industrial market [38-39]. this can be achieved through government policies that encourage industrial sectors to increase the pace of production cooperation with smes. one of the mechanisms is the regulation of public procurement. the topic of supporting small and medium-sized businesses in the procurement system is very popular [40]. some scientists [41-43] consider the mechanism of granting preferences through the introduction of special procedures involving smes to be economically unjustified. support methods include several mechanisms that differ from state to state. in the united states, information and consulting support are provided, procurement quotas are set, some contracts are subcontracted to small and medium-sized businesses, and innovative developments are supported. many eu countries use a simplified procedure for purchasing goods, works, and services at the expense of budget funds, and provide for the transfer of part of contracts to small businesses, but do not carry out procedures in which only small and medium-sized businesses participate. the question of the effectiveness of public procurement procedures, in which only smes can participate, attracts the attention of many scientists, but their conclusions are ambiguous [42, 44]. the review has shown that the economy is no longer contrasting small and medium-sized businesses with large businesses, and their relations are transformed, moving to a new stage of development. modern forms of cooperation are gradually changing the philosophy of inter-company communications. the target function of cooperation is to expand external economic relations – to create a production network. 3. materials and methods the initial analysis of trends in industrial cooperation and the expected effects of its implementation in practice allowed forming the following main hypotheses, which will be tested in the course of the study: h1. despite the legal requirements for the share of public procurement from smes, the development of cooperation between small enterprises and large state-owned companies is 32 o.a. romanova, e.a. kuzmin, m.v. vinogradova, o.s. kulyamina copyright ©2021 assa. adv. in systems science and appl. (2021) inefficient. in this regard, it is assumed that the share of orders transferred to the sme sector within the framework of production cooperation does not depend on the share of state participation in the capital of large industrial enterprises. h2. “anchor” industrial enterprises that are more actively involved in industrial cooperation have higher indicators of production efficiency, in particular, the indicator of gross profitability and profitability of outlay. h3. placing orders with smes allows large industrial enterprises with state participation to reduce the number of employees. to assess the degree of development of cooperation between industrial enterprises and smes in russia, “anchor” large enterprises were selected, divided into key industries: oil, gas, minerals and related activities (petroleum products transportation); industrial production, and electric power (see appendix a). the list includes such companies as bashneft, vankorneft, gazprom, rosneft, russian helicopters, ngo almaz, united aircraft corporation (uac), united engine corporation (uec), eastern energy company (eec), mosenergo, rosseti moscow region (moesk), rosseti, rushydro, and transneft. the choice of “anchor” companies is determined by their contribution to the russian economy. all these companies are among the top 100 largest companies in russia, their revenue for the last report in 2019 varies from 39 billion rubles to 4.8 trillion rubles. besides, all industrial enterprises have a share of state participation. this feature is also an area of research restrictions. in addition to the selection of industrial enterprises with state participation, the analysis of the degree of cooperation with smes is carried out only among legal entities (companies), individual entrepreneurs and individuals are ignored (although they perform work, provide services and produce products for “anchor” companies). the study used the following indicators of enterprises: revenue, cost of production, works (services); the amount and share of revenue of large industrial enterprises attributable to smes; the amount and share of the cost of production, works (services) of large industrial enterprises attributable to smes. the data panel was supplemented with information on the volume of purchases from smes, the share of state participation in the capital of industrial enterprises, and the number of employees. the sources of information were the news agencies spark-interfax and interfax corporate information disclosure center [45]. statistical and correlation analysis methods were used for a comprehensive assessment of the effects of sme cooperation with large industrial enterprises in russia. the production localization degree (a1) was calculated as part of the statistical approach: , (1) where is the orders of large industrial enterprises placed with smes; is the cost of products, works (services) of large industrial enterprises. the share of revenue (a2) and the share of cost (a3) of large industrial enterprises attributable to smes were determined using the formulas: , (2) , (3) where is the cost of large industrial enterprises attributable to smes; is the revenue of large industrial enterprises; is the revenue of large industrial enterprises attributable to smes. in order to compare the performance of enterprises, the authors used the profitability indicators – gross profit margin (gpm) and outlay (po): smes 1 orders lie a pc = smesorders liepc smes 2 re relie a = smes 3 lie pca pс = smespc relie smesre cooperation between smes and large industrial enterprises 33 copyright ©2021 assa. adv. in systems science and appl. (2021) , (4) , (5) where gp is the gross profit; re is the revenue; oi is the operating profit; pc is the production, works (services) cost. the pearson correlation ratio was calculated to determine the strength of the statistical relationship between the indicators: , (6) where n is the sample size, , are the mean values of parameters; , are the variances of parameters; , are the root-mean-square (standard) deviations of parameters. 4. results and discussion during the observation period from 2015 to 2019, all selected large industrial enterprises with state participation increased the volume of orders placed with smes by several times from 2.15 (rosneft) to 210.8 (transneft). data on order dynamics is provided in appendix b. at the same time, the level of internal production localization (autonomy) for all the considered enterprises decreased (table 1), and enterprises increased the share of orders placed with smes in the cost price. however, the degree of localization varied heterogeneously over the period under review. table 1. production localization degree of large industrial enterprises with state participation in 2015–2019, % enterprise 2015 2016 2017 2018 2019 bashneft 8.42 9.38 6.89 2.92 79.53 vankorneft 4.09 61.35 0.92 0.35 74.85 russian helicopters 10.97 3.13 8.52 33.22 134.75* gazprom 0.34 0.57 0.31 0.03 5.50 eec 0.55 0.65 0.63 0.78 36.01 mosenergo 1.68 1.56 1.72 0.30 75.15 moesk 13.63 7.639 8.99 23.76 78.22 ngo almaz 1.94 1.13 1.13 0.11 73.15 uac 8.76 3.60 5.07 0.60 67.96 uec 1.85 1.80 3.99 0.59 51.99 rosneft 5.43 1.55 0.26 0.02 6.40 rosseti 16.41 9.86 10.26 1.62 735.63* rushydro 17.61 15.44 8.37 4.27 157.79* transneft 0.04 0.71 0.32 0.25 7.01 note: * – for holding companies, the cost of production, works (services) has a heterogeneous distribution within the group. source: [45]. based on the collected data, the share of revenue and the share of the cost of “anchor” large industrial enterprises accounted for by smes was calculated – separately for mediumsized, small, and micro enterprises according to the classification adopted in russia [46]. the criteria for a medium-sized enterprise are that the average number of employees is not more gpgpm re = oipo pc = 1 ( )( ) ( 1) n i i i xy x y x x y y r n s s = = å x y 2 xs 2 ys xs ys 34 o.a. romanova, e.a. kuzmin, m.v. vinogradova, o.s. kulyamina copyright ©2021 assa. adv. in systems science and appl. (2021) than 250, and the annual income is not more than 2 billion rubles. the share of organizations in the capital of medium-sized enterprises that are not related to smes should not exceed 49%, the share of the state, regions, or non-profit organizations shall not exceed 25%. the small business criteria are that the average number of employees is not more than 100, and the income is not more than 800 million rubles. the micro-enterprise criteria are the average number of employees no more than 15 and annual income no more than 120 million rubles. restrictions on the structure of the authorized capital are similar. the largest share of orders for medium-sized enterprises in the revenue of large industrial enterprises in revenue in 2019 was observed in rosseti (48.48%), uac (39.6%), rushydro (39.69%). the largest share of orders for medium-sized enterprises in the cost of large industrial enterprises was observed in russian helicopters (36.75%), rushydro (31.38%), and moesk. gazprom, transneft, and rosneft placed the smallest share of orders with medium-sized businesses (fig. 1). fig. 1. shares of revenue and cost of large industrial enterprises attributable to medium-sized enterprises in 2019, % source: [45, 47]. a similar situation is observed for small businesses. the largest share of orders for small businesses in the revenue of large industrial enterprises in 2019 was recorded in russian helicopters (39.12%), rushydro (44.79%), rosseti (33.28%). mosenergo (36.95%), rushydro (32.33%), and bashneft (26.16%) accounted for the largest share of orders for small enterprises in the cost of large industrial enterprises. gazprom, rosneft, and transneft demonstrated the worst work with small enterprises (fig. 2). cooperation between smes and large industrial enterprises 35 copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 2. share of revenue and cost of large industrial enterprises attributable to small enterprises in 2019, % source: [45, 47]. gazprom (0.30%), rosneft (0.19%), and transneft (0.32%) showed a low share of the revenue from large industrial enterprises attributable to micro-enterprises. such a low share of micro-enterprise participation is also observed in the cost of these companies (fig. 3). rushydro (10.99%), ngo almaz (7.04%), and eec (6.16%) show the best positions in working with micro-enterprises in terms of the revenue share. in terms of cost, the degree of participation of these companies is close – rushydro (7.32%), mosenergo (6.66%), and ngo almaz (5.11%). fig. 3. shares of revenue and cost of large industrial enterprises attributable to micro-enterprises in 2019, % source: [45, 47]. 36 o.a. romanova, e.a. kuzmin, m.v. vinogradova, o.s. kulyamina copyright ©2021 assa. adv. in systems science and appl. (2021) in general, the leaders in placing orders for the sme sector in 2019 were such enterprises as rushydro (95.46% of revenue), rosseti (86.90% of revenue), russian helicopters (80.76% of revenue), mosenergo (68.21% of revenue), moesk (68.22% of revenue), ngo almaz (66.64% of revenue), and uac (65.91% of revenue). the smallest share of orders attributable to smes in revenue is observed in the “anchor” companies in the oil and gas sector. the average order amount placed with one small enterprise in 2019 was 249,114.9 thousand rubles, with one medium-sized enterprise – 943,719.6 thousand rubles, and with one micro-enterprise – 30,243.6 thousand rubles. thus, the structure of the distribution of orders for smes is dominated by medium-sized enterprises. recall that medium-sized enterprises are characterized by the presence of an average number of employees up to 250 people and an annual income (revenue) of no more than 2 billion rubles. the calculated correlation ratio between the parameter of the share of state participation in large industrial enterprises and the share of orders attributable to the sme sector is -0.04. on a chedoke scale [48], the reverse relationship is weak; therefore, the hypothesis h1 was confirmed. this means that the largest industrial state-owned companies prefer not to place orders with smes, which is due to both the low level of development of the sme sector in russia and the industry specifics of large businesses (the largest russian production companies belong to the oil and gas industry), while smes are mainly concentrated in the services and trade sectors. the correlation ratio between the indicator of the share of orders from smes of large industrial enterprises in the cost price and the gross profit margin is 0.1469. the data on profitability and the share of cost attributable to smes are given in appendix c. the results obtained allow concluding that the correlation is direct and weak. the first part of the h2 hypothesis about the manifestation of greater activity in production cooperation on the part of “anchor” enterprises was not confirmed. this means that large industrial enterprises do not sufficiently use the advantages of production cooperation with smes to increase efficiency by reducing production costs. the correlation ratio between the indicator of the share of orders from smes of large industrial enterprises in the cost and profitability of outlay is 0.2924. the correlation is also direct and weak, and the second part of the h2 hypothesis is not confirmed. the h3 hypothesis implements the assumption that placing orders with smes allows “anchor” industrial enterprises with state participation to reduce the number of employees. the analysis of the growth rate of the average number of employees of the majority of large industrial enterprises under consideration for 2015/2019 leads to the conclusion that despite the increase in the level of placing orders with smes, the average number of employees has increased. the largest growth rate in the number of employees is observed in the uec (40 times), which is due to the reorganization of the enterprise. taking the uec observation as a statistical outlier, the average rate of growth in the number of employees of large enterprises is 107.61% (appendix c). the correlation ratio between the indicator of the share of orders from smes of large industrial enterprise in the cost price and the growth rate of the average number of employees is -0.2023. the relationship is inverse and weak, so the h3 hypothesis is not confirmed. based on the study, it can be concluded that production cooperation in russia between “anchor” large industrial enterprises with state participation through the use of subcontracting with the sme sector is not carried out effectively. many of the expected internalities that are characteristic of cooperative relations in developed countries, both for contractors and subcontractors, are not reflected in the specifics of the russian economy or their manifestation is limited. cooperation between smes and large industrial enterprises 37 copyright ©2021 assa. adv. in systems science and appl. (2021) 5. conclusion the development of specialization and cooperation of small, medium and large enterprises in the modern conditions of the global market is becoming an economic necessity and is a consequence of the new competitiveness paradigm [49]. this statement finds convincing arguments in world practice. production cooperation is formed along the entire value chain and leads to the emergence of a new phenomenon of intercompany relations – production networks. however, the strength and scale of cooperation are not uniform. the russian experience clearly demonstrates the weakness of cooperative partnership, although with positive trends of change. the results indicate that the level of internal production localization (autonomy) of large industrial enterprises with state participation in russia in 2015–2019 decreased; all these enterprises increased the share of orders placed with smes. therefore, there is an expansion of subcontracting as a form of industrial cooperation. the largest share of orders placed with smes is observed in large industrial enterprises in the electric power supply industry, and the smallest – in the oil and gas industry. in terms of the volume of orders placed by “anchor” large industrial enterprises, the leaders are medium-sized enterprises; the number of orders placed with such enterprises is 3.8 times higher than the number of orders placed with small businesses, and 31.2 times – with micro-enterprises. at the same time, the hypothesis that placing orders with smes allows large industrial enterprises with state participation to reduce the number of employees has not been confirmed, as well as the hypothesis that “anchor” industrial enterprises, which are more actively involved in industrial cooperation, have higher production efficiency indicators. this allowed concluding that cooperation is inefficient among large companies with state participation in russia. references 1. asaul, a.n., skumatov, e.g., & lokteeva, g.e. 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(p. 174). appendix a selected large industrial enterprises of russia for the analysis of industrial cooperation with smes enterprise activity type by okved* share of state participation in the capital, % average number of employees, people revenue in 2019, thousand rubles cost of sales in 2019, thousand rubles 1. production of oil, gas, and minerals bashneft 06.10.1 60.5 9,183 703,150,528 514,466,674 vankorneft 06.10.1 0.01 1,600 383,329,128 308,750,775 gazprom 46.71 50.0 26,691 4,758,711,459 2,657,654,354 rosneft 06.10.1 40.4 4,553 6 827 526 407 4 782 222 071 2. production russian helicopters 30.30.3 85.71 427 39,853,657 23,884,842 ngo almaz 26.30.17 1.16 11,387 101,586,138 92,535,194 united aircraft corporation (uac) 30.30.3 8.99 661 54,734,083 53,083,289 united engine corporation (uec) 30.30.13 87.45 14,297 94,038,976 63,187,526 3. electric-power supply industry eastern energy company (eec) 35.14 0.08 3,962 97,746,207 90,981,227 mosenergo 35.11.1 26.4 7,922 189,781,589 172,256,268 rosseti moscow region (moesk) 35.12 88.4 14,377 160,375,521 139,860,598 rosseti 35.12 88.4 642 39,434,924 4,658,385 rushydro 35.11.2 62.2 5,396 155,180,091 93,884,445 4. other activities transneft 49.50.1 78.55 1,257 960,811,881 787,367,559 note: * is the russian national classifier of economic activities (okved); 06.10.1 is the extraction of crude oil; 46.71 is the wholesale of solid, liquid and gaseous fuels and related products; 30.30.3 is the production of helicopters, planes and other flying vehicles; 26.30.17 is the production of radio and television transmitting equipment; 30.30.13 is the manufacture of jet engines and their parts; 35.14 is the power-supply trade; 35.11.1 is the electricity production by thermal power plants, including activities for ensuring the operability of power plants; 35.12 is the transmission of electricity and technological connection to distribution power grids; 35.11.2 is the production of electricity by hydroelectric power stations, including activities for ensuring the operability of power plants; 49.50.1 is the transportation via pipelines of crude oil and petroleum products. source: [45, 47]. appendix b volume of orders of large industrial enterprises with state participation from smes in 2015–2019, thousand rubles enterprise 2015 2016 2017 2018 2019 change 2015/2019 bashneft 26,710,063 29,465,010 26,231,483 14,708,667 409,149,836 +1,532% vankorneft 8,113,407 114,701,623 2,325,058 1,127,552 231,096,848 +2,848% russian helicopters 668,099 531,842 1,154,521 6,176,591 32,185,779 +4,818% gazprom 7,646,833 12,769,846 7,773,306 816,985 146,174,290 +1,912% cooperation between smes and large industrial enterprises 41 copyright ©2021 assa. adv. in systems science and appl. (2021) enterprise 2015 2016 2017 2018 2019 change 2015/2019 eec 249,589 311,905 331,445 620,313 32,758,637 +13,125% mosenergo 2,509,753 2,586,024 2,800,891 509,229 129,458,616 +5,158% moesk 15,489,675 9,472,757 12,216,046 33,163,019 109,405,277 +706% ngo almaz 495,657 555,598 534,157 59,048 67,692,533 +13,657% uac 4,262,882 1,795,264 2,608,787 310,220 36,075,422 +846% uec 492,931 531,626 546,298 191,144 32,849,933 +6,664% rosneft 141,847,000 44,426,939 8,977,148 1,021,700 306,224,474 +216% rosseti 712,467 424,912 461,972 75,382 34,268,268 +4,810% rushydro 11,263,353 8,701,764 7,017,781 4,135,669 148,137,273 +1,315% transneft 261,718 4,993,648 2,317,868 2,022,262 55,168,592 +21,079% source: [45]. appendix c data on gross profit margin, the profitability of outlay and the share of cost attributable to smes for selected large industrial enterprises in russia for 2019, % enterprise profitability share of cost attributable to smes growth rate of the average number of employees 2015/2019 gross profit outlay bashneft 26.69 11.03 55.96 121.12 vankorneft 19.46 19.06 50.08 35.56 russian helicopters 40.07 33.84 65.15 66.51 gazprom 44.15 15.23 2.24 107.45 eec 6.92 2.61 25.41 116.09 mosenergo 9.23 10.17 63.39 99.99 moesk 12.79 4.71 34.18 95.48 ngo almaz 8.91 9.59 54.12 248.57 uac 3.02 -4.66 45.14 105.42 uec 32.81 20.85 22.72 4016.01 rosneft 29.86 12.51 3.29 111.51 rosseti 88.19 746.54 62.09 98.17 rushydro 39.5 65.29 71.03 95.76 transneft 18.05 12.24 4.19 97.29 source: [45]. adv syst sci appl 2021; 03:75–90 published online at https://ijassa.ipu.ru. applications of sine-cosine wavelets method for solving drinfel’d–sokolov–wilson system naser azizi1, reza pourgholi1* 1school of mathematics and computer science, damghan university, damghan, iran abstract: in this article, we use the sine-cosine wavelets (scws) method to numerically solve the drinfel’d–sokolov–wilson (dsw) system. for this purpose, we use an approximation of functions with the help of scws, and we approximate spatial derivatives using this method. the operational matrix based on scws has a large number of zero components, which ensures good system performance and provides acceptable accuracy even with fewer collocation points. in the end, to show the effectiveness and accuracy of the method in solving this system one numerical example is provided. keywords: drinfel’d–sokolov–wilson system, numerical method, sine-cosine wavelets method, operational matrix 1. introduction nonlinear coupled partial differential equations (pdes) are very significant in a type of scientific field, especially in fluid mechanics, solid-state physics, plasma waves, plasma physics, and chemical physics. since many nonlinear physical phenomena can be explained by the exact and numerical solutions of nonlinear equations, the attempt for finding the exact and numerical solutions to these phenomena is important. in this article, our main goal is to solve numerically the dsw system. a generalized form of the dsw system is given by:{ ψt + αφφx = 0, φt + βφxxx + γψφx + δψxφ = 0, (1.1) where α, β, γ, and δ are some nonzero parameters. system (1.1) plays an important role in fluid dynamics [8,12] and is originally introduced by drinfel’d and sokolov [7] and wilson [24] as a model of dispersive water waves. many researchers have devoted considerable efforts by successfully implementing various methods to extract solitary wave solutions and other solutions of dsw system [1, 10, 17, 23, 25]. one way to solve equations numerically is to use wavelets. the basic idea of wavelets goes back to the early 1960s [4,5]. there are developments concerning the multiresolution analysis algorithm based on wavelets [6] and the construction of compactly supported orthonormal wavelet bases [16]. so far, several problems have been solved numerically using different wavelets, for example, we can refer to references [2, 3, 9, 13, 18, 19, 26, 27]. in this paper, we consider system (1.1) by using the scws method to find numerical solutions. scw has been used and showed efficiency to solve various problems. to indicate this, we can refer to some ∗corresponding author: pourgholi@du.ac.ir 76 n. azizi, r. pourgholi works. razzaghi and yousefi in [20] have employed a scw to solve variational problems. tavassoli kajani et al. [15] for solving integro-differential equations have presented a method based on scws. a numerical evaluation of hankel transform for seismology has been given in [14] using the scws approach. amir and umer saeed in [22] have used scws to solve the fractional nonlinear oscillator equations. in the present article, we intend to use the scws method to numerically solve the dsw system (1.1) with the initial conditions ψ(x, 0) = f1(x), φ(x, 0) = f2(x), x ∈ [0, 1], (1.2) and the boundary conditions ψ(0, t) = g1(t), φ(0, t) = g2(t), t ∈ [0, tfin], φ(1, t) = k2(t), φx(0, t) = w2(t), t ∈ [0, tfin], (1.3) where tfin represents the final time. the differentiable functions fi(x), gi(t), for i = 1, 2, k2(t), and w2(t) are known. the structure of this article is as follows: in section 2, we describe the properties of the scws. in the following, expanding functions into the scws series and operational matrix of them for the numerical solutions are discussed. in section 3, the procedure of implementation of the scws method, for system (1.1) with specified initial and boundary conditions (1.2) and (1.3) is presented. the numerical performance of the method is made in section 4, and finally, concluding remarks are given in section 5. 2. properties of scws wavelets are useful mathematical functions constructed from the dilation and translation of a single function called the mother wavelet, which can be denoted by ω. assuming that the expansion parameter η and the translation parameter ν are considered, we have the continuous wavelets family as follows [11]: wη,ν(x) = |η|− 1 2ω (x− ν η ) , η, ν ∈ r, µ 6= 0. if the parameters η and ν are restricted to take values η = η−κ0 and ν = rν0a −κ 0 , a family of discrete wavelets is obtained as: wκ,r(x) = |η0| κ 2ω(ηκ0x− rν0), (2.4) where η0 > 1, ν0 > 0, and r and κ are positive integers. the set {wκ,r(x)} in (2.4), forms a wavelet basis for l2(r). especially, if η0 = 2 and ν0 = 1, the set {wκ,r(x)} forms an orthonormal basis. scws are defined on interval x ∈ [0, 1) as [14]: wr,s(x) = 2 κ+1 2 gs(2 κx− r)χ[ r 2κ , r+1 2κ ), (2.5) where κ = {0} ∪ n, r = 0, 1, 2, . . . , 2κ − 1, and χ[ r 2κ , r+1 2κ ) denotes the characteristic function given as χ[ r 2κ , r+1 2κ ) = { 1, x ∈ [ r 2κ , r+1 2κ ), 0, elsewhere. (2.6) also, gs(x) =  1√ 2 , s = 0, cos(2sπx), s = 1, 2, . . . , `, sin(2(s− `)πx), s = `+ 1, `+ 2, . . . , 2`, (2.7) copyright © 2021 assa. adv syst sci appl (2021) applications of sine-cosine wavelets method for solving drinfeld-sokolov-wilson 77 where ` is any positive integer. scws have compact support and are an orthonormal basis forl2([0, 1)). the orthonormal basis functions for scws by assuming κ = 1 and ` = 1 are obtained as follows: for 0 ≤ x < 1 2 =⇒ w0,0(x) = √ 2, w0,1(x) = 2 cos(4πx), w0,2(x) = 2 sin(4πx), for 1 2 ≤ x < 1 =⇒ w1,0(x) = √ 2, w1,1(x) = 2 cos(2π(2x− 1)), w1,2(x) = 2 sin(2π(2x− 1)). (2.8) so, with the collocation points xm = 2m− 1 2n , m = 1, 2, . . . ,n = 2κ(2`+ 1), (2.9) the graphs of wr,s(x) for κ = ` = 1, are shown in fig. 2.1. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 w0,0(x) w0,1(x) w0,2(x) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 w1,0(x) w1,1(x) w1,2(x) fig. 2.1. the graphs of wr,s(x) for κ = ` = 1. 2.1. expanding functions into the scws using the set of scws, any function υ(x) ∈ l2([0, 1)) can be approximated as an infinite series of these functions as follows: υ(x) = ∞∑ r=0 2∑̀ s=0 ar,swr,s(x), (2.10) where ar,s =< υ,wr,s >= ∫ 1 0 υ(x)wr,s(x) dx. by truncating the infinite series (2.10) at levels r = 2κ − 1 and s = 2`, we obtain an approximate representation for υ(x) as υ(x) ' 2κ−1∑ r=0 2∑̀ s=0 ar,swr,s(x) = atγ(x), (2.11) where a and γ are (n × 1)-vectors and are introduced as follows: a = [ a0,0, a0,1, . . . , a0,2`, a1,0, a1,1, . . . , a1,2`, . . . . . . , a2κ−1,0, a2κ−1,1 . . . , a2κ−1,2` ]t , γ = [ w0,0,w0,1, . . . ,w0,2`,w1,0,w1,1, . . . ,w1,2`, . . . . . . ,w2κ−1,0(x),w2κ−1,1 . . . ,w2κ−1,2` ]t . (2.12) copyright © 2021 assa. adv syst sci appl (2021) 78 n. azizi, r. pourgholi the scws matrix γn×n at the collocation points (2.9), is given as follows: γn×n = [ γ ( 1 2n ) ,γ ( 3 2n ) , . . . ,γ (2n − 1 2n )] , in other words γn×n =  w0,0( 1 2n ) w0,0( 3 2n ) . . . w0,0(2n−1 2n ) w0,1( 1 2n ) w0,1( 3 2n ) . . . w0,1(2n−1 2n ) ... ... . . . ... w0,2`( 1 2n ) w0,2`( 3 2n ) . . . w0,2`( 2n−1 2n ) w1,0( 1 2n ) w1,0( 3 2n ) . . . w1,0(2n−1 2n ) w1,1( 1 2n ) w1,1( 3 2n ) . . . w1,1(2n−1 2n ) ... ... . . . ... w1,2`( 1 2n ) w1,2`( 3 2n ) . . . w1,2`( 2n−1 2n ) ... ... . . . ... ... ... . . . ... w2κ−1,0( 1 2n ) w2κ−1,0( 3 2n ) . . . w2κ−1,0(2n−1 2n ) w2κ−1,1( 1 2n ) w2κ−1,1( 3 2n ) . . . w2κ−1,1(2n−1 2n ) ... ... . . . ... w2κ−1,2`( 1 2n ) w2κ−1,2`( 3 2n ) . . . w2κ−1,2`( 2n−1 2n )  . in particular, for κ = ` = 1, the scws matrix γ6×6 is given as follows: γ6×6 =  √ 2 √ 2 √ 2 0 0 0 1 −2 1 0 0 0√ 3 0 − √ 3 0 0 0 0 0 0 √ 2 √ 2 √ 2 0 0 0 1 −2 1 0 0 0 √ 3 0 − √ 3  . 2.2. the operational matrix of scws due to the vector form (2.12), the integration of γ(x) can be calculated as follows:∫ x 0 γ(t) dt = qγ(x), where q is n ×n operational matrix given by q = 1 2κ+ 1 2  f s · · · s 0 f · · · s ... ... . . . ... 0 0 · · · f  , where s and f are (2`+ 1)× (2`+ 1) matrices (see [15]). copyright © 2021 assa. adv syst sci appl (2021) applications of sine-cosine wavelets method for solving drinfeld-sokolov-wilson 79 3. application of the method in this section, we use the scws method for finding the approximate solutions of system (1.1) with specified initial and boundary conditions (1.2) and (1.3). for this, dividing the interval [0, tfin] into m equal parts of length ~t = tfin m and denoting tn = (n− 1)~t, n = 1, 2, . . . , (m+ 1), we expand ψ̇′ and φ̇′′′ in terms of scws as, ψ̇′(x, t) ∼= 2κ−1∑ r=0 2∑̀ s=0 ar,swr,s(x) = atγ(x), (3.13) φ̇′′′(x, t) ∼= 2κ−1∑ r=0 2∑̀ s=0 br,swr,s(x) = btγ(x), (3.14) where prime and dot mean differentiation concerning x and t, respectively. now, we consider the following two cases: case 1: considering the equation (3.13): by integrating this equation once concerning t from tn to t and once concerning x from 0 to x, we have ψ′(x, t) = (t− tn)atγ(x) + ψ′(x, tn), (3.15) ψ̇(x, t) = atqγ(x) + g′1(t), (3.16) now, integrating equation (3.16) once concerning t from tn to t, we obtain ψ(x, t) = (t− tn)atqγ(x) + [g1(t)− g1(tn)] + ψ(x, tn). (3.17) case 2: considering the equation (3.14): by integrating this equation once concerning t from tn to t and three times concerning x from 0 to x, we obtain φ′′′(x, t) = (t− tn)btγ(x) + φ′′′(x, tn), (3.18) φ′(x, t) = (t− tn)btq2γ(x) + φ′(x, tn) + [w2(t)− w2(tn)] + x[φ̇′′(0, t)− φ̇′′(0, tn)], (3.19) φ(x, t) = (t− tn)btq3γ(x) + φ(x, tn) + [g2(t)− g2(tn)] + x[w2(t)− w2(tn)] + x2 2 [φ̇′′(0, t)− φ̇′′(0, tn)], (3.20) φ̇(x, t) = btq3γ(x) + g′2(t) + xw′2(t) + x2 2 φ̇′′(0, t). (3.21) by using the boundary condition φ(1, t) = k2(t) equations (3.19)-(3.21) are changed as follows: φ′(x, t) = (t− tn)btq2γ(x) + φ′(x, tn) + (1− 2x)[w2(t)− w2(tn)] + 2x[k2(t)− k2(tn)] − 2x[g2(t)− g2(tn)], (3.22) φ(x, t) = (t− tn)btq3γ(x) + φ(x, tn) + (1− x2)[g2(t)− g2(tn)] + x(1− x)[w2(t)− w2(tn)] + x2[k2(t)− k2(tn)], (3.23) φ̇(x, t) = btq3γ(x) + (1− x2)g′2(t) + x(1− x)w′2(t) + x2k′2(t). (3.24) copyright © 2021 assa. adv syst sci appl (2021) 80 n. azizi, r. pourgholi discretizing the results (3.15)-(3.17) and (3.18) and (3.22)-(3.24), by assuming x→ xm and t→ tn+1, we have ψ′(xm, tn+1) = ~tatγ(xm) + ψ′(xm, tn), (3.25) ψ̇(xm, tn+1) = atqγ(xm) + g′1(tn+1), (3.26) ψ(xm, tn+1) = ~tatqγ(x) + [g1(tn+1)− g1(tn)] + ψ(xm, tn), (3.27) φ′′′(xm, tn+1) = ~tbtγ(xm) + φ′′′(xm, tn), (3.28) φ′(xm, tn+1) = ~tbtq2γ(xm) + φ′(xm, tn) + (1− 2xm)[w2(tn+1)− w2(tn)] + 2xm[k2(tn+1)− k2(tn)]− 2xm[g2(tn+1)− g2(tn)], (3.29) φ(xm, tn+1) = ~tbtq3γ(xm) + φ(xm, tn) + (1− x2 m)[g2(tn+1)− g2(tn)] + xm(1− xm)[w2(tn+1)− w2(tn)] + x2 m[k2(tn+1)− k2(tn)], (3.30) φ̇(xm, tn+1) = btq3γ(xm) + (1− x2 m)g′2(tn+1) + xm(1− xm)w′2(tn+1) + x2 mk ′ 2(tn+1), (3.31) where, xm’s are the collocation points that are introduced in (2.9). to linearized the nonlinear terms φφx, ψxφ, and ψφx in system (1.1), we use the linearization form given by rubin and graves [21] as follows: φφx = φx(x, tn)φ(x, tn+1)− φx(x, tn)φ(x, tn) + φ(x, tn)φx(x, tn+1), (3.32) ψxφ = φ(x, tn)ψx(x, tn+1)− φ(x, tn)ψx(x, tn) + ψx(x, tn)φ(x, tn+1), (3.33) ψφx = φx(x, tn)ψ(x, tn+1)− φx(x, tn)ψ(x, tn) + ψ(x, tn)φx(x, tn+1). (3.34) using linear expressions (3.32)-(3.34), the discrete form of system (1.1) considering xm and tn+1 is as follows: ψ̇(xm, tn+1) + αφ′(xm, tn)φ(xm, tn+1) + αφ(xm, tn)φ′(xm, tn+1) = αφ′(xm, tn)φ(xm, tn), φ̇(xm, tn+1) + βφ′′′(xm, tn+1) + δφ(xm, tn)ψ′(xm, tn+1) + δψ′(xm, tn)φ(xm, tn+1) +γφ′(xm, tn)ψ(xm, tn+1) + γψ(xm, tn)φ′(xm, tn+1) = δφ(xm, tn)ψ′(xm, tn) +γφ′(xm, tn)ψ(xm, tn). (3.35) now, by using equations (3.25)-(3.31), system (3.35) leads to{ at θ1 + bt θ2 = h1(xm, tn), at θ3 + bt θ4 = h2(xm, tn), (3.36) where the matrices θi, i = 1, 2, 3, 4 are matrices with dimensions n ×n as follows: θ1 = qγ(xm), θ2 = [ αφ′(xm, tn)~tq3γ(xm) + αφ(xm, tn)~tq2γ(xm) ] , θ3 = [ δφ(xm, tn)~tγ(xm) + γφ′(xm, tn)~tqγ(xm) ] , θ4 = [ q3γ(xm) + β~tγ(xm) + δψ′(xm, tn)~tq3γ(xm) + γψ(xm, tn)~tq2γ(xm) ] , copyright © 2021 assa. adv syst sci appl (2021) applications of sine-cosine wavelets method for solving drinfeld-sokolov-wilson 81 and the matriceshi, i = 1, 2 are matrices with dimensions n × 1 as follows: h1(xm, tn) =− αφ(xm, tn)φ′(xm, tn)− g′1(tn+1)− αφ′(xm, tn) [ (1− x2 m)[g2(tn+1)− g2(tn)] + xm(1− xm)[w2(tn+1)− w2(tn)] + x2 m[k2(tn+1)− k2(tn)] ] − αφ(xm, tn) [ (1− 2xm)[w2(tn+1)− w2(tn)] + 2xm[k2(tn+1)− k2(tn)] − 2xm[g2(tn+1)− g2(tn)] ] , h2(xm, tn) =− βφ′′′(xm, tn)− δφ(xm, tn)ψ′(xm, tn) − γφ′(xm, tn)[g1(tn+1)− g1(tn) + ψ(xm, tn)] − [ (1− x2 m)g′2(tn+1) + xm(1− xm)w′2(tn+1) + x2 mk ′ 2(tn+1) ] − δψ′(xm, tn) [ (1− x2 m)[g2(tn+1)− g2(tn)] + xm(1− xm)[w2(tn+1)− w2(tn)] + x2 m[k2(tn+1)− k2(tn)] ] − γψ(xm, tn) [ (1− 2xm)[w2(tn+1)− w2(tn)] + 2xm[k2(tn+1)− k2(tn)]− 2xm[g2(tn+1)− g2(tn)] ] . the matrix-vector form of system (3.36) is as follows:[ θ1 θ2 θ3 θ4 ] 2n×2n [ a b ] 2n×1 = [ h1 h2 ] 2n×1 (3.37) from (3.37), the coefficients vector a and b can be calculated. with these coefficients and using the equations (3.27) and (3.30), the approximate solutions are successively obtained. 4. numerical experiments in this section, we apply the scws method to obtain the numerical solutions of the dsw system (1.1). to compare the obtained numerical results, we use the following solutions that obtained by arnous et al. ( [1]): ψ(x, t) = 6c γ+2δ sech2 (√ c βk2 (k(x− ct)− ξ0) ) , φ(x, t) = ± √ 12c2 α(γ+2δ) sech (√ c βk2 (k(x− ct)− ξ0) ) , where c and k are arbitrary real constants. to show the effectiveness and accuracy of the proposed method, we considered an example with α = β = γ = δ = 1, tfin = 1, ~t = 0.01. remark 4.1: for describing the error, we introduce the infinity-norm of absolute error and the root mean square (rms) error norm as follows: lψ ∞ = ||ψ(xm, t)−ψ∗(xm, t)||∞ = max 1≤m≤n |ψ(xm, t)−ψ∗(xm, t)|, rmsψ = [ 1 2n 2n∑ m=1 ( ψ(xm, t)−ψ∗(xm, t) )2 ] 1 2 , (4.38) copyright © 2021 assa. adv syst sci appl (2021) 82 n. azizi, r. pourgholi where ψ∗ is the approximate solution of ψ. similarly, the lφ ∞ and rmsφ are obtained according to formulas (4.38). the numerical results for ψ(x, t) and φ(x, t) at time t = 1 when ` = κ = 1 are reported in table 4.1. for different values of κ, table 4.2 stated the calculated errors (4.38) at time t = 0.5, also, the execution times for these values are given in table 4.3. difference between exact and numerical solutions ψ and φ at time t = 1 are shown in figs. 4.2-4.4 and 4.5-4.7, respectively. table 4.1. the numerical results for ψ(x, t) and φ(x, t) at t = 1 when ` = κ = 1. xm ψ(xm, 1) ψ∗(xm, 1) |ψ(xm, 1)−ψ∗(xm, 1)| φ(xm, 1) φ∗(xm, 1) |φ(xm, 1)− φ∗(xm, 1)| 0.0833333 0.153514 0.153517 2.358647e− 06 0.159955 0.159953 2.434473e− 06 0.25 0.157382 0.157384 2.526247e− 06 0.161958 0.161949 8.477983e− 06 0.416667 0.160643 0.160646 2.873385e− 06 0.163627 0.163619 7.684178e− 06 0.583333 0.163242 0.163244 2.279712e− 06 0.164945 0.164933 1.248603e− 05 0.75 0.165133 0.165128 5.211826e− 06 0.165898 0.165906 7.514665e− 06 0.916667 0.166281 0.166280 1.685403e− 06 0.166474 0.166457 1.740170e− 05 table 4.2. the calculated errors (4.38) at time t = 0.5 with ` = 1. ψ(x, 0.5) φ(x, 0.5) l∞ κ = 1 1.333830e− 06 8.630715e− 06 κ = 2 6.055654e− 08 1.078673e− 06 κ = 3 2.810132e− 09 7.847945e− 08 κ = 4 8.968863e− 11 5.304455e− 09 κ = 5 3.105035e− 12 3.172111e− 10 κ = 6 9.736049e− 14 2.018670e− 11 rms κ = 1 7.661924e− 07 5.199718e− 06 κ = 2 3.315228e− 08 5.115186e− 07 κ = 3 1.000311e− 09 2.933696e− 08 κ = 4 3.271928e− 11 2.026272e− 09 κ = 5 9.981293e− 13 1.217716e− 10 κ = 6 3.154537e− 14 7.797156e− 12 table 4.3. the execution times for different values of κ with ` = 1. κ = 1 κ = 2 κ = 3 κ = 4 κ = 5 κ = 6 cpu time (s) 102.835546 189.641444 373.585049 753.232887 1519.095320 3163.873643 copyright © 2021 assa. adv syst sci appl (2021) applications of sine-cosine wavelets method for solving drinfeld-sokolov-wilson 83 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -6 -5 -4 -3 -2 -1 0 1 2 3 ψ (x ,1 ) − ψ ∗ (x ,1 ) ×10 -6 (ℓ = 1, κ = 1) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 ψ (x ,1 ) − ψ ∗ (x ,1 ) ×10 -7 (ℓ = 1, κ = 2) fig. 4.2. difference between exact and numerical solutions ψ at time t = 1, when ` = 1 and κ = 1, 2. copyright © 2021 assa. adv syst sci appl (2021) 84 n. azizi, r. pourgholi 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -12 -10 -8 -6 -4 -2 0 2 4 6 8 ψ (x ,1 ) − ψ ∗ (x ,1 ) ×10 -9 (ℓ = 1, κ = 3) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -4 -3 -2 -1 0 1 2 3 ψ (x ,1 ) − ψ ∗ (x ,1 ) ×10 -10 (ℓ = 1, κ = 4) fig. 4.3. difference between exact and numerical solutions ψ at time t = 1, when ` = 1 and κ = 3, 4. copyright © 2021 assa. adv syst sci appl (2021) applications of sine-cosine wavelets method for solving drinfeld-sokolov-wilson 85 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -1.5 -1 -0.5 0 0.5 1 ψ (x ,1 ) − ψ ∗ (x ,1 ) ×10 -11 (ℓ = 1, κ = 5) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -4 -3 -2 -1 0 1 2 3 ψ (x ,1 ) − ψ ∗ (x ,1 ) ×10 -13 (ℓ = 1, κ = 6) fig. 4.4. difference between exact and numerical solutions ψ at time t = 1, when ` = 1 and κ = 5, 6. copyright © 2021 assa. adv syst sci appl (2021) 86 n. azizi, r. pourgholi 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -2 -1.5 -1 -0.5 0 0.5 1 φ (x ,1 ) − φ ∗ (x ,1 ) ×10 -5 (ℓ = 1, κ = 1) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 φ (x ,1 ) − φ ∗ (x ,1 ) ×10 -6 (ℓ = 1, κ = 2) fig. 4.5. difference between exact and numerical solutions φ at time t = 1, when ` = 1 and κ = 1, 2. copyright © 2021 assa. adv syst sci appl (2021) applications of sine-cosine wavelets method for solving drinfeld-sokolov-wilson 87 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -2 -1.5 -1 -0.5 0 0.5 1 φ (x ,1 ) − φ ∗ (x ,1 ) ×10 -7 (ℓ = 1, κ = 3) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -12 -10 -8 -6 -4 -2 0 2 4 6 8 φ (x ,1 ) − φ ∗ (x ,1 ) ×10 -9 (ℓ = 1, κ = 4) fig. 4.6. difference between exact and numerical solutions φ at time t = 1, when ` = 1 and κ = 3, 4. copyright © 2021 assa. adv syst sci appl (2021) 88 n. azizi, r. pourgholi 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -8 -6 -4 -2 0 2 4 6 φ (x ,1 ) − φ ∗ (x ,1 ) ×10 -10 (ℓ = 1, κ = 5) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -5 -4 -3 -2 -1 0 1 2 3 4 φ (x ,1 ) − φ ∗ (x ,1 ) ×10 -11 (ℓ = 1, κ = 6) fig. 4.7. difference between exact and numerical solutions φ at time t = 1, when ` = 1 and κ = 5, 6. given the approximation function υ(x) expressed in section 2, the solutions of the system (1.1), can be expanded as: υ(x) = ∞∑ r=0 2∑̀ s=0 ar,swr,s(x). (4.39) copyright © 2021 assa. adv syst sci appl (2021) applications of sine-cosine wavelets method for solving drinfeld-sokolov-wilson 89 in the investigated example, we approximate the solution of this equation as follows: υ(x) ' 2κ−1∑ r=0 2∑̀ s=0 ar,swr,s(x), (4.40) which is the truncating the infinite series (4.39). by substituting the solutions ar,s in (4.40), we get the error function e (x) as follows: e (x) = ∣∣∣∣∣υ(x)− 2κ−1∑ r=0 2∑̀ s=0 ar,swr,s(x) ∣∣∣∣∣. therefore, as κ increases, the series (4.40) becomes larger and closer to the series (4.39), in other words, e (x) approaches zero. the obtained numerical results for ψ and φ confirm this. 5. conclusion in this article, using the scws method and using the initial and boundary conditions (1.2) and (1.3), we solved the dsw system (1.1) numerically. considering the obtained numerical results in tables 4.1 and 4.2, figs. 4.2-4.7, and also comparing these results with the exact solutions, it can be concluded that the presented method for solving the dsw system (1.1) is an efficient and high accuracy method. the strength of this method is the simplicity of calculations with low storage space. references 1. arnous, a., mirzazadeh, m., & eslami, m. 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(2019). legendre wavelets approach for numerical solutions of distributed order fractional differential equations. applied mathematical modelling, 70, 350–364. copyright © 2021 assa. adv syst sci appl (2021) introduction properties of scws expanding functions into the scws the operational matrix of scws application of the method numerical experiments conclusion microsoft word _done_1116-article text-5520-1-6-20220524 adv syst sci appl 2023; 01; 91-98 published online at https://ijassa.ipu.ru. nowcasting gdp of major economies during the crisis: does energy matter? ivan a. kopytin1, nikolay p. pilnik1, ivan p. stankevich1* 1) primakov national research institute of world economy and international relations, russian academy of sciences, moscow, russia abstract: in this article we compare the accuracy on nowcasts obtained with different models and different sets of indicators used as predictors for a set of 19 major economies. we compare the performance of mixed-frequency bayesian var models, dynamic factor models and unrestricted midas models with l1 regularization. we test different groups of commodity prices as possible predictors: energy indicators, agricultural commodities, precious metals and industrial metals prices. we find that among all the indicator groups tested energy commodities prices yield the highest average nowcasting accuracy, even though the accuracy of models utilizing all the indicators available remains slightly higher. among all the models tested, the highest quality is yielded by mixed-frequency bayesian var models. we also emphasize the importance of manual selection of predictors for non-diversified economies, where it can significantly improve the accuracy of nowcasts compared to the models with a wide set of predictors keywords: nowcasting, bayesian vector auto regression, gdp, oil price. 1. introduction one of the significant problems with macroeconomic statistics is the publication lag: the most important macroeconomic data is often published with several month lag (or longer). this problem becomes especially crucial in circumstances such as the recent crisis, caused by the new coronavirus infection: the necessity of stimulus is evident, but the exact amount is hard to determine without all the relevant statistics. one of the possible solutions for this problem is nowcasting of macroeconomic indicators. nowcasting is an estimation of the current level of an indicator that is published with a significant lag and is not observable at the moment the calculations are made, using a set of indicators that are published faster. the most typical nowcasting problem is gdp nowcasting as the gdp data, on the one hand, is usually published with a several month lag after the end of the corresponding period, and on the other hand, gdp is crucial in policy making and decision making of many agents. in this paper we investigate gdp nowcasting based on a set of commodity prices and pmis. from a technical point of view, there are several main approaches to gdp nowcasting. first, these are coupling equations based on predicting a high-frequency series using standard methods, then aggregating it into lower frequency data and using the resulting indicator as an explanatory variable in the low-frequency equation for the indicator of interest. this approach was presented, for example, in [11], and still retains considerable popularity, see [17, 18]. another very common class of econometric models is the so-called midas (mixed data sampling) mixed data frequency). this approach is based on using in the equation for a lower frequency series (for example, quarterly) as explanatory variables several values of a higher frequency series related to the current period (for example, three monthly values of another indicator related to this quarter). depending on the specification, a number of * corresponding author: ivanstankevich0@gmail.com 92 i. kopytin, n. pilnik, i. stankevich copyright ©2023 assa adv. in systems science and appl. (2023) restrictions on the values of the coefficients of the model can be introduced: models without restrictions are usually called unrestricted midas models (u-midas). presented in [8, 9], models of this class are widely used for the nowcasting of macroeconomic indicators, see, for example, [2, 7, 14]. from technically more complex approaches, one can single out mixed frequency vector autoregression (mfvar), based on the description of the joint dynamics of high-frequency indicators and the unobserved decomposition of low-frequency indicators into higher frequency data, see, for example, [12] or [16] in the application to nowcasting. there are also bayesian generalizations of this approach [15]. when building a nowcasting system, one has to choose between different model types and different indicators that can be used as explanatory variables in a nowcasting model. this choice becomes especially important for tasks of building a nowcasting system for a set of countries with different structures of the economy. in this paper, we test a set of widely used nowcasting models (midas, mixed frequency bayesian vars, dfm) on data for 19 major economies. we also compare different sets of operative indicators: a set of pmi indicators and several sets of world commodity prices: prices for energy commodities, metals (both precious and industrial) and agricultural commodities. pmis are a common choice in nowcasting and are often found to increase the nowcasting accuracy in real-time settings (see, e.g., [13] for the us and [5] for the euro area). the most popular choice among commodity prices is oil price, see, e.g., [2,4], and our main hypothesis is in line with this choice: among all the commodity groups, prices of energy indicators (including oil and gas prices) can yield the highest nowcasting accuracy. 2. methods we employ a relatively standard for nowcasting literature set of models: midas (mixed data sampling) models, dynamic factor models and mixed-frequency bayesian vector autoregressions. midas models are based on the utilization of higher-frequency indicators as explanatory variables. this class of models was presented [8,9] and is widely used in nowcasting of macroeconomic indicators (see [2, 7, 14]). in the general case, a midas model assumes that ( ) ( ) 1 0 0 i i p k i i t j t j j tm j t i m j j y y x u          , where ty is lower-frequency data (in our case – quarterly series of gdp growth rate); ( )i tx factors of higher frequency (in our case – monthly series of pmi of energy commodities prices); im defines the number of ( )ix observations in one observation of dependent variable (3 months in a quarter in our case). midas models also often imply restrictions on ( )i j , but in our case, we focus on unrestricted midas models without autoregressive component. instead, we use lasso regularization so that our resulting model can be written as: ( ) ( 1 0 ) 0 i i k i i t j tm j t i j m y x u        with target function  ( ) 2 ( ) 1 1 0 ( ) min i i j t k i t t j t i j m y y           . regularization for u-midas models can prevent overfitting while allowing for more complex dependencies and yielding higher forecasting accuracy (see [19]). mixed-frequency bvars describe the joint dynamics of variables of different frequencies in a single var model. they are widely used in nowcasting literature: see [15, 16] for one of the first applications, or more recent [3] shows that nowcasting performance of nowcasting gdp of major economies during the crisis: does energy matter? 93 copyright ©2023 assa. adv. in systems science and appl. (2023) mfbvars matches the performance of state-of-the-art dfm while having a more general structure and allowing for a greater flexibility. mfbvar assumes that all the processes (of different observed frequencies) evolve at higher frequency (monthly in our case), but are observed at lower frequency. for the variables observed in a lower frequency we assume that observed values ,q ty are obtained from the original monthly process ,q tx with intra-quarterly averaging (here and further when describing mfbvar model we follow the notation of [1]) , , 1 , 2 , 1 ( ), { , , , } 3 , q t q t q t q t x x x t march june september december y else         the same scheme is used in [16] for a very similar task of short-term forecasting of macroeconomic variables, including gdp. for the tx vector process we assume a standard var(p) model 1 1 , ~ n(0, )t t p t p t tx x x        т т , where 1,..., p  are coefficient matrices. by combining ,...,t t px x  into a single vector tz , we can get a companion form of var(p) model 1 , ~ n(0, )t t t tz z u u     . the observation equation for the original ty process is t t ty m z  , where tm is a selection matrix (that determines at which periods quarterly indicators are observed),  is an aggregation matrix based on weighting scheme employed (intra-quarterly averaging in our case). both these matrices are known and determined by the data structure and assumptions made.  and  are coefficients of companion form var model. we also utilize a minnesota-style prior which assumes for a model written in matrix form x w e  , where 1( , , )tw w w  • , 1( , , ,1)t t t pw x x  • • • ,  is the coefficients matrix and e errors such that vec( ) | ~ n(vec( ), )    ,  [( 1) 1]( ) diag( ) 0n p     • , 3 2 1 2 2 4 , lag l of variable r, i =(l-1)n+r ( ) , 1 ri l s i np           , where ,  parameters are prior beliefs specified by the researcher, i are diagonal elements of  , all  parameters control the “tightness” of the prior and are tuned when estimating a mfbvar model, 2 is are the residual variances from auxiliary ar(4) models. in our case, we utilize widely-used minnesota-style prior, where prior means for all the parameters except ar(1) parameters are chosen to be 0 and prior means for ar(1) parameters are  . hence, we imply that processes for all the series analyzed are ar(1) processes until the evidence (taken from the data) is enough for the estimation procedure to make the model more complex. priors for  in our case are taken to be 0 in order not to restrict them to any given value because our target variable (gdp growth rate) is stationary. another popular choice 1  is usually applied in non-stationary dependent variable cases. for the error covariance matrix we assume the quite common inverse wishart prior: 94 i. kopytin, n. pilnik, i. stankevich copyright ©2023 assa adv. in systems science and appl. (2023) 2 2 1 ~ ( , ) ( 1) ( ,..., ) 2 n iw s s n diag s s n          where 2 is are the residual variances from auxiliary ar(4) models. for the calculations we use the r package mfbvar of [1]. dynamic factor model implies that all the series observed are a combination of unobserved common factors (the number of factors is lower than the number of variables studied), that can be identified and used to make forecasts and nowcasts of indicators analyzed. the general specification of a dynamic factor model can be given by t t ty f    , 1 , ~ (0, ) p t i t i t t q i f a f bu u iid n i    , where ty is a vector of variables investigated, vector  and matrix  are parameters to be estimated, unobserved common factors tf are assumed to follow a var(p) process. we follow two-stage estimation approach as in [10], where parameters of  matrix and unobserved factors tf are estimated with principal components based on a standardized and balanced panel of explanatory variables and then re-estimated on an unbalanced set of regressors using kalman smoothing. the resulting factors are used in the model. estimation is performed in r using nowcasting package, [6] 3. data and research methodology we use the data on quarterly gdp growth rates for the period of 2001 – 2020 for the following countries: usa, russia, china, india, argentina, australia, brazil, britain, canada, france, germany, indonesia, italy, japan, mexico, saudi arabia, turkey, korea, south africa. the countries selected represent a significant part of the world gdp and in the same time belong to different geographical regions and have different structures of the economy – it can be important to test the stability of results obtained. we use 4 groups of monthly commodity prices (obtained from world bank’s pink sheet – [20]) as explanatory variables:  energy commodities (crude oil brent, dubai, wti; coal, australian, south african, natural gas us, europe; liquefied natural gas, japan),  agricultural commodities (wheat, us srw; rice, thai 5%; maize, palm oil, soybeans),  industrial metals (aluminum, iron ore, copper, nickel),  precious metals (gold, platinum, silver) we also use a set of monthly pmi series for the countries presented in the sample as a benchmark we compare the nowcasting accuracies over the last 10 points (quarters) for different groups of indicators. during the testing phase for each data point tested we restrict the sample to the information that replicates the actual nowcasting practice: all the explanatory variables for the current quarter are known (and are not known for the subsequent quarter), the indicator nowcasted is not known. we measure the accuracies using mean absolute errors (mae) for quarterly gdp growth rates nowcasting gdp of major economies during the crisis: does energy matter? 95 copyright ©2023 assa. adv. in systems science and appl. (2023) 4. results and discussion for each country we choose the model with the lowest mae. tables 4.1-4.3 present the types, factor groups and mae for the best model for each of 19 countries and the best model with mae for the case when all the indicators were used as a reference. accuracies in the table 4.1 are estimated on the whole sample, in the table 4.2 – without 2020, to measure the quality during a relatively stable period. we see that not in all the cases the best nowcast is obtained using the bigger sample of indicators. for the countries with economies focused on a particular group of products, such as saudi arabia, the accuracy of nowcast using only the most appropriate group of prices is almost twofold higher than for the model with all the indicators (even though this set still includes energy commodities prices). it allows us to conclude that in at least some cases the more is not always the better and manual selection of predictors can boost the performance of the model table 4.1. types of best models and their nowcasting mae for models with one groupof factors and models with all factors, % gdp one group all indicators type factor mae type mae usa midas_l1 indmetal 1.56 mfbvar 1.77 russia dfm pmi 1.74 midas_l1 1.60 china midas_l1 precmetal 1.86 mfbvar 1.71 india midas_l1 energy 3.89 dfm 4.34 argentina mfbvar energy 3.85 mfbvar 2.64 australia midas_l1 pmi 1.44 midas_l1 1.35 brazil dfm pmi 2.04 mfbvar 1.48 britain midas_l1 pmi 3.17 midas_l1 2.96 canada mfbvar pmi 1.13 mfbvar 1.49 france midas_l1 indmetal 2.82 mfbvar 3.04 germany dfm pmi 1.79 mfbvar 1.15 indonesia midas_l1 precmetal 1.30 midas_l1 1.45 italy dfm pmi 2.60 midas_l1 2.79 japan midas_l1 indmetal 1.63 mfbvar 1.55 mexico mfbvar energy 3.16 mfbvar 2.79 saudi arabia mfbvar energy 1.19 dfm 2.26 turkey mfbvar energy 2.00 mfbvar 2.97 korea mfbvar indmetal 1.31 mfbvar 0.96 south africa midas_l1 agriculture 0.57 midas_l1 0.60 the first table answers the question of the best way to predict the economic growth of each of the countries, especially in the specific conditions of 2020. the first thing to pay attention to is the question of whether one group of factors can be distinguished, or whether models with the entire set of variables work best. moreover, the group of those countries for which models with the entire set of indicators work better includes, for example, russia, china, germany and the united kingdom. on the other hand, for example, for countries such as the united states, india and france, it is possible to single out one group of variables that work best. we also see that, among all the indicator groups, the most frequent “winners” are energy commodity prices and pmi. these results hold even for the more stable sample without 2020. 96 i. kopytin, n. pilnik, i. stankevich copyright ©2023 assa adv. in systems science and appl. (2023) table 4.2. types of best models and their nowcasting mae for models with one group of factors and models with all factors without 2020, % gdp one group all indicators type factor mae type mae usa midas_l1 indmetal 0.39 mfbvar 0.40 russia mfbvar agriculture 0.58 midas_l1 0.69 china midas_l1 precmetal 0.37 midas_l1 1.40 india mfbvar agriculture 0.57 mfbvar 0.66 argentina mfbvar energy 2.40 mfbvar 2.46 australia mfbvar pmi 0.41 mfbvar 0.34 brazil mfbvar indmetal 0.49 midas_l1 0.47 britain midas_l1 pmi 0.52 midas_l1 0.49 canada midas_l1 pmi 0.44 mfbvar 0.63 france midas_l1 indmetal 0.34 midas_l1 0.53 germany dfm pmi 0.68 mfbvar 0.68 indonesia midas_l1 precmetal 0.21 midas_l1 0.23 italy midas_l1 energy 0.52 midas_l1 0.41 japan midas_l1 indmetal 0.63 midas_l1 0.68 mexico mfbvar pmi 0.84 mfbvar 1.15 saudi arabia mfbvar energy 0.88 dfm 1.38 turkey mfbvar energy 2.03 mfbvar 2.49 korea mfbvar pmi 0.42 mfbvar 0.46 south africa midas_l1 agriculture 0.43 midas_l1 0.70 comparing the results from the two previous tables is the most interesting. this approach actually allows us to draw meaningful conclusions about how radically the mechanisms that determine a significant part of the economic growth of the world's largest economies have changed. all countries can be divided into three groups depending on what composition of factors is used, and whether this composition of factors changes in models with and without 2020. the first group of countries includes australia, brazil, the uk and germany. the most working models for these countries both without and taking into account 2020 include all variables. the second group of countries includes the usa, canada, france, indonesia, saudi arabia, turkey and south africa. the most working models for these countries contain only one group of variables, regardless of whether 2020 is included in the models. for the us and france, industrial metal prices are the best leading indicators. for saudi arabia and turkey, prices for energy products. that is, a factor that is a good leading indicator can be both the main export product and the product on whose import the economy is critically dependent. finally, the third group includes russia, china, india, argentina, italy, japan, mexico and korea. for them, models with the addition of data for 2020 differ significantly from models without this period. almost all of these countries (the only exceptions are italy and india, which at different times experienced a very strong shock from the spread of the coronavirus) have the same trend. with the addition of the 2020 data, the model that used only one group of indicators changes to a model that includes all groups of factors. it seems to us that this is a rather significant effect associated with the crisis events of 2020. during this period, the interrelationships in the economy become more complex, and it is impossible to say in advance which variable will make the greatest contribution. therefore, models that take into account all the analyzed factors begin to work more accurately. nowcasting gdp of major economies during the crisis: does energy matter? 97 copyright ©2023 assa. adv. in systems science and appl. (2023) table 4.3 present the mean mae for all the models and predictor groups with and without 2020. in both cases the best nowcasting quality is obtained using mfbvar and in both cases the best indicator group is energy commodities prices. however, for the whole sample mfbvar with all the indicators performs better. table 4.3. average errors (mae) across all country models, % gdp energy pmi precmetal indmetal agriculture all indicators all time mfbvar 2.66 3.34 3.13 2.76 4.38 2.37 dfm 2.96 2.74 3.10 3.03 3.07 3.02 midas_l1 3.03 3.30 3.23 3.36 3.23 2.55 without 2020 mfbvar 1.14 1.26 1.29 1.16 1.23 1.17 dfm 1.63 1.54 1.66 1.61 1.64 1.62 midas_l1 1.63 1.94 2.12 1.95 1.70 1.17 5. conclusion in this article we compare the accuracy on nowcasts obtained with different models and different sets of indicators used as predictors for a set of 19 major economies. we compare the performance of mixed-frequency bayesian var models, dynamic factor models and unrestricted midas models with l1 regularization. we test different groups of commodity prices as possible predictors: energy indicators, agricultural commodities, precious metals and industrial metals prices. we found that energy really matters in nowcasting tasks: especially for non-diversified economies and during stable periods of time. energy commodities prices used as predictors for nowcasting models yield lower mean nowcasting errors than any other group of indicators, including widely-used pmis. we also found that mixed-frequency bayesian vars tend to have the highest accuracy in most of the cases analyzed. acknowledgements the article was prepared within the project "post-crisis world order: challenges and technologies, competition and cooperation" supported by the grant from ministry of science and higher education of the russian federation program for research projects in priority areas of scientific and technological development (agreement № 075-15-2020-783). references 1. ankargren, s., & yang, y. 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[online] available https://www.worldbank.org/en/research/commodity-markets microsoft word 792-article text-3045-1-18-20200630.doc adv syst sci appl 2020; 02; 20-31 published online at https://ijassa.ipu.ru. global stability analysis of typhoid fever model olumuyiwa james peter1*, adebisi ajimot folasade2, michael oyelami ajisope3, fidelis odedishemi ajibade4, adesoye idowu abioye5, festus abiodun oguntolu6 1,5 department of mathematics, university of ilorin, ilorin, nigeria 2 department of mathematics, osun state university, oshogbo, osun state, nigeria 3department of mathematics, federal university oye-ekiti, oye-ekiti, nigeria 4department of civil engineering, federal university of technology, akure, nigeria 6department of mathematics, federal university of technology, minna, niger state, nigeria e-mail: peterjames4real@gmail.com abstract: we analyze with four compartments a deterministic nonlinear mathematical model of typhoid fever transmission dynamics. using the lipchitz condition, we verified the existence and uniqueness of the model solutions to establish the validity of the model and derive the equilibria states of the model that is, disease-free equilibrium (dfe) and endemic equilibrium (ee). the computed basic reproductive number r0 was used to establish that the disease-free equilibrium is globally asymptotically stable when its numerical value is less than one, the disease will be under control. in addition, the lyapunov function was applied to investigate the stability property for the (dfe). the model was numerically simulated to validate the results of the analysis. keywords: typhoid fever, equilibria, stability, nonlinear mathematical model 1. introduction typhoid fever is an infection caused by salmonella typhi bacteria. typhoid is usually triggered by the ingestion of food or water contaminated with feces or urine of infected individuals and is, therefore, a typical illness in regions with poor sanitation. (brooks [1]; roumagnac et al., [2]). in developing economies, typhoid fever outbreaks occur from time to time in overwhelmed areas and refugee camps with high population density. the disease causes high morbidity in children below ten years of age, with at least seventeen million new cases globally and nearly 600 000 deaths annually. the disease is not rampant in north america: in the us, an estimated four hundred cases are reported every year; seventy percent of the cases are traced to those who return from endemic regions. the mortality of typhoid fever is ten percent, but with adequate treatment, it can be limited to one percent. (lin et al., [3]; sinha et al., [4]; hyman, [5]). intestinal fever treatment is anchored on the blood culture condition of the patients. if the species is sensitive, the oral antibiotic is used. typhoid fever is becoming an increasingly common illness worldwide, making antibiotic treatment more expensive and harder. enteric fever signs vary and are similar to the signs of different microbial infections. symptoms of typhoid fever are as follows: variable level of high fever in 75 cases, body aches and muscle pain, chills, shriveled loss of appetite, abdominal pain in 20 to 40 cases, nose bleeds, headache, dizziness, rashes on the skin, bowel constipation or looseness, weakness and fatigue, sore throat and cough (lifshitz [6]). some mathematical models have been formulated on the transmission of typhoid fever. (adetunde [7]; lauria et al., [8]; * corresponding author: peterjames4real@gmail.com global stability analysis of typhoid fever model copyright ©2020 assa. adv. in systems science and appl. (2020) 21 kalajdzievska [9]; mushayabasa [10]; cvjetanovic et al.,[11]; moffact [12]; pitzer et al., [13]; date et al.,[14]; muhammad, et al.,[15]; watson & edmunds [16]; nthiiri [17]; moatlhod & gosaamang [18]; mushayabasa [19]; peter & ibrahim [20]; peter et al.,[21]; peter and ibrahim [22]). the aim of this study is to extend and complement previous works by formulating a model that captures the following controls; vaccination and education. 2. materials and methods the model comprises four compartments: susceptible class; represents the proportion of those who are prone to typhoid. infected class; represents the population of individuals who have been infected with typhoid fever and capable of transmitting the infection to susceptible populations through interaction. carrier-class; represents the population of individuals who have been infected with typhoid fever but without signs of infectiousness in them. recovered compartment; represent the number of individuals infected with typhoid fever but are now cured of typhoid as a result of treatment. recruitment into susceptible populations is by birth or immigration at the rate . it is assumed that a certain proportion in the susceptible class moved to the carrier infectious class at rate while the complement moved to the infectious compartment. we also assume that the rate of disease transmission of carrier individuals will be higher than the disease transmission rate of infected individuals this is because they are more likely to be unaware of their infection. carrier's disease symptom is noticeable at the rate . contagious individuals may receive treatment and recuperate at the rate . susceptible individuals receive vaccination to protect themselves from contracting the disease at the rate . is an educational parameter that serves as the limitation for carriers and indicative persons from transmitting typhoid fever. this parameter lies within the range . when it implies that educational campaign not in position so that the vulnerable population are oblivious of typhoid fever and when it denotes that all vulnerable persons are well informed of the causes of typhoid fever, that is, they explicitly understand what brings about the diseases, how the disease is being transmitted and how to safeguard themselves from being infected by the disease. figure 1. pictorial description of the model q r r-1 b g a y f-1 10 ££ f 0=f 1=f o. j. peter, a. f. adebisi, m. o. ajisope, f.o. ajibade, a. i. abioye copyright ©2020 assa adv. in systems science and appl. (2020) 22 (1) where (2) table 1. variables and parameters interpretation variables description vulnerable populations carrier contagious populations contagious populations recovered populations parameters interpretation recruitment rate in vulnerable class mortality rate of vulnerable class mortality rate for carrier infected class mortality rate for infected class mortality rate for recovered class the rate at which individual carriers develops symptoms parameter for education rate of vaccination the probability of newly infected persons becoming asympt omatic or carrier rate of transmission of infection for carrier individuals rate of transmission of infection for infectious individuals force of disease infection rate at which individual in the infectious class recovered ï ï ï ï þ ïï ï ï ý ü -+ +--+-------ris dt dr iis dt di iis dt dic sss dt ds c cc 4 3 2 1 = )()(1)(1)(1= )(1)(1= )(1= µdy dµfaflr faµfrl yflµq iic gbl += ï ï ï ï þ ïï ï ï ý ü -+ +--++-----+ --+-ris dt dr iiiis dt di iiiis dt dic siiss dt ds cc ccc c 4 3 2 1 = )()(1)())(1(1= )(1))(1(= ))(1(= µdy dµfagbfr faµfgbr yfgbµq ( )s t ( )ic t )(ti )(tr q 1µ 2µ 3µ 4µ a f y r b g l d global stability analysis of typhoid fever model copyright ©2020 assa. adv. in systems science and appl. (2020) 23 3. solution of the model 3.1. existence and uniqueness of the model the validity of any mathematical model is a function of the solution of the model, provided the solution is unique. we verify the singularity of the solution of the model in equation (2) via the popular lipchitz condition. given the system of equation (2) be as follows (3) theorem 3.1 assuming the region is denoted by (4) and suppose that r’ meets the lipchitz condition on each occasion, the pairs and , where is a positive constant. hence, there is a constant such that there exists a unique continuous vector solution of the system in the interval . it is vital to mention that the requirement is fulfilled by the condition that is continuous and bounded in . considering the model equation (2), we are of the interest in the region we look for the solution that is bounded in the region and whose partial derivatives satisfy where and are positive constants. theorem 3.2 let j represent the region then equation (2) has a unique solution if it is established that are continuous and bounded in r for , for , ï ï ï þ ïï ï ý ü -+ +--++-----+ --+-risj iiiisj iiiisj siissj cc ccc c 44 33 22 11 = )()(1)())(1(1= )(1))(1(= ))(1(= µdy dµfagbfr faµfgbr yfgbµq â ),........,(=),........,,(=1,, 20102100 noon yyyyyyyyyyacc £-£2121 ),(),( yykycfycf -£( )1, yc ( ) jyc î2, 0³d )(cy d£0cc ..1,2,=,, ji x f j i ¶ ¶ â .0 ⣣a 0££ af a d 1,2,3,4=,, ji x j j i ¶ ¶ 1j ¥--+-¶ ¶ <))(1(= 1 1 yfgbµ ii s j c ¥-¶ ¶ <)(1=1 fbs i j c ¥-¶ ¶ <)(1=1 fgs i j ¥ ¶ ¶ <0=1 r j 2j ¥-+ ¶ ¶ <))(1(=2 fgbr ii s j c ¥---¶ ¶ <)(1)(1= 2 2 faµfbrs i j c o. j. peter, a. f. adebisi, m. o. ajisope, f.o. ajibade, a. i. abioye copyright ©2020 assa adv. in systems science and appl. (2020) 24 , for , , for , , , the partial derivatives exist, continuous and bounded, so the model has a unique solution that completes the proof of theorem 3.1. 3.2. feasible region and equilibrium from equation (2) we have that and thus, along with each solution. also from (2), we see that where . therefore, we have omitted r because r does not appear in other equations. this reveals that the model can be studied in the feasible region; . is positively invariant in relation to (2). once the dynamics are understood, those of r can then be determined from the equation . the first stage of our analysis is to find the disease-free equilibrium states from the equations by setting the right-hand side to zero i.e., the model always has a disease-freeequilibrium . and the endemic equilibrium satisfies from the equilibrium equations we can show that a unique exist with where ¥¶ ¶ <)(1=2 fgrs i j ¥ ¶ ¶ <0=2 r j 3j ¥+-¶ ¶ <))()(1(1=3 ii s j c gbfr ¥-+-¶ ¶ <)(1))(1(1=3 fagfr s i j c ¥+--¶ ¶ <)())(1(1= 3 3 dµgfr s i j ¥ ¶ ¶ <0=3 r j 4j ¥ ¶ ¶ <=4 y s j ¥ ¶ ¶ <0=4 ci j ¥ ¶ ¶ <=4 d i j ¥¶ ¶ <= 4 4 µ r j s dt ds )( 1µyq +-£ 1 )(suplim µy q + £ ¥® ts t ncrcicicsc dt dn c -£----£ qq 4321 }{ 4321 ,,.min ccccc = c tn t q £ ¥® )(suplim î í ì þ ý ü £++ + £âî=g + c iissiis cc q µy q ,:),,( 1 3 g ),,( iis c risr 4 ' µdy --= ),,( iis c 0=))(1(1 siiss c yfgbµq --+-0=)(1))(1( 2 ccc iiiis faµfgbr ----+ 0=)()(1)())(1(1 3 iiiis cc dµfagbfr +--++-ú û ù ê ë é + ,0,0= 1 0 µy qa ),,( **** iisa c= .0,, *** >iis c *a )( ))(( = 223 23* gµrgµrbµbdrag aµdµ +-++ ++ kk k s )(1= f-k global stability analysis of typhoid fever model copyright ©2020 assa. adv. in systems science and appl. (2020) 25 for to exist in the feasible region , the necessary and sufficient condition requires , or equivalently, define (4) then a threshold parameter determines the actual number of equilibria. in the next section, we shall explain the basic reproduction number. proposal 3.1 if then is the only equilibrium in ; if so, two equilibria exist, , and a unique endemic equilibrium, 3.3. basic reproduction number the basic reproductive number measures the number of secondary cases that the bacterium is able to introduce into the whole population of fully susceptible individuals in a stable demographic state (kalajdzievska [9]). the basic number of reproduction is a significant quantity in epidemiology since it sets the threshold for the analysis of an outbreak of disease and the evaluation of its control strategies. the value of the reproductive number, therefore, indicates whether a disease becomes endemic or dies out in a population. if , it indicates that each contagious person will not cause an infection and therefore, the disease will always disappear but when the basic reproduction number is greater than one i.e., each contagious person will cause at least one secondary infection which will subject the entire population to the attack of the disease. it is obtained by taking the largest eigenvalue(spectral radius) of the matrix where is the rate of new infections in infectious class, is the transfer of persons out of the compartment by another means, is the disease-free equilibrium. applying the next generation matrix approach, the basic reproduction number for the model is given as (5) in equation (5), the expression in the big square brackets is the per capita mean number of secondary infections. this expression is multiplied by , the number of susceptible individuals in the absence of infection to come about the basic reproduction number 3.4. disease-free equilibrium’s stability in order to check the local stability in the absence of infection a0, the jacobian matrix will be evaluated at *a g 1 *0 µy q + £< s .1 )( * 1 ³ + sµy q [ ] )])([( )1()(1 231 23 1 *0 k kk s r aµdµyµ rgµdµbragq µy q +++ -+++ = + = 10 <r 0j g ,10 >r 0a *a 10 <r ,10 >r 1 0 )()(= ú ú û ù ê ê ë é ¶ ¶ ú ú û ù ê ê ë é ¶ ¶ j oi j oi x xv x xfr if iv ox ú û ù ê ë é ++ + + + +++ = ))(( )1( )())(( 23 2 2231 0 kkk kkr aµdµ rgµ aµ br aµdµ ag yµ q yµ q +1 or ú û ù ê ë é + ,0,0= 1 0 µy qa o. j. peter, a. f. adebisi, m. o. ajisope, f.o. ajibade, a. i. abioye copyright ©2020 assa adv. in systems science and appl. (2020) 26 the following stability results indicate that is a sharp threshold. proposal 3.2 a0 locally asymptotically stable if and is unstable if . proof one eigenvalue of is < 0. the other two eigenvalues are matrix. we want to show that whenever then the routh-hurwitz conditions hold, that is, and . going by the assumption that and therefore, and this shows that and the whenever . this establishes the proposition. ú ú ú ú ú ú ú ú ú ú û ù ê ê ê ê ê ê ê ê ê ê ë é +-÷÷ ø ö çç è æ + ---+÷÷ ø ö çç è æ + -÷÷ ø ö çç è æ + --+-÷÷ ø ö çç è æ + ÷÷ ø ö çç è æ + --÷÷ ø ö çç è æ + --+)()1)(1()1()1)(1(0 )1())1(()1(0 )1()1()( =)( 3 11 1 2 1 11 1 0 dµ yµ q gfrfa yµ q bfr yµ q frgfaµ yµ q frb yµ q fg yµ q fbyµ aj 10 <r 10 >r )( 0aj )( 11 yµl +-= 22´ ú ú ú ú ú ú ú ú ú û ù ê ê ê ê ê ê ê ê ê ë é +-÷÷ ø ö çç è æ + ---+÷÷ ø ö çç è æ + -÷÷ ø ö çç è æ + --+-÷÷ ø ö çç è æ + )()1)(1()1()1)(1( )1())1(()1( =)( 3 11 1 2 10 dµ yµ qgfrfa yµ qbfr yµ qfrgfaµ yµ qfrb dj 10 <r 0)( <atra 0)det( >a ú ú ú ú ú û ù ê ê ê ê ê ë é + ÷÷ ø ö çç è æ + -++ ú ú ú ú ú û ù ê ê ê ê ê ë é -+ ÷÷ ø ö çç è æ + -+= 1 )( )1(1( )(1 )1( )1( ))1(()( 3 1 3 2 1 2 dµ yµ qgfr dµ faµ yµ qfrb faµatra 1 ))(( )1( )())(( 23 2 2231 0 <ú û ù ê ë é ++ + + + +++ = kkk kkr aµdµ rgµ aµ br aµdµ ag yµ q )1( )1( 2 1 faµ yµ qfrb -+ ÷÷ ø ö çç è æ + 1 )( )1(1( 3 1 < + ÷÷ ø ö çç è æ + -dµ yµ qgfr )1( )1( 2 1 faµ yµ qfrb -+ ÷÷ ø ö çç è æ + 01 )( )1(1( 3 1 <+ ÷÷ ø ö çç è æ + -dµ yµ qgfr 0)( <atra 0)det( >a 10 <r global stability analysis of typhoid fever model copyright ©2020 assa. adv. in systems science and appl. (2020) 27 3.5. global stability of disease-free equilibrium point (dfe) theorem 3.3 a0 is globally stable in the region if . proof. to establish the global asymptotic stability of (dfe) we adopt the method of lyapunov functions. define where , substituting the values of both x and y as in above and simplifying, we shall obtain using the condition we have so, if furthermore, if therefore, the largest invariant set in where is the singleton . hence, by lasalle’s principle [23], is globally asymptotically stable in the region . table 2. values of parameters for the model parameter value source assumed assumed mushayabasa [10] assumed mushayabasa [10] assumed assumed assumed mushayabasa [10] assumed g 10 £r yixil c += ))(( )1( )( 23 2 2 kk x aµdµ rgµ aµ rb ++ + + = ))(( 23 k ky aµdµ ag ++ = dt diy dt di x dt dl c += [ ] [2( )(1 ) (1 ) (1 )(1 ) ( )c c c cx s i i i i y s i ir b g j µ a j r j b g+ + + ]3(1 ) ( )ci ia j µ d+ + ))(1( 0 iisr dt dl c gb +-= yµ q + << 1 *0 s 0))(1( 0 £+-£ iisr dt dl c gb 0£ dt dl 10 £r 0= dt dl 00 == ii g 0= dt dl { }0a 0j g 2µ 0.20 y 0.30 1µ 0.1420 3µ 0.2 4µ 0.142 a 0.3 r 0.5 b 0.02 g 0.01 d 0.75 o. j. peter, a. f. adebisi, m. o. ajisope, f.o. ajibade, a. i. abioye copyright ©2020 assa adv. in systems science and appl. (2020) 28 estimated 106 lauria, et al. [8] 4. results and discussion we performed numerical simulations to describe graphically the long-term impact of early treatment on the dynamics of typhoid fever. table 2 indicated the various set of parameter values adopted to validate the outcome of the analysis. graphical illustrations showed time graphs of each state variable. figure 2 described the time-sensitive graph. with high vaccination achievement, both the effective contact rate and the prevalence rate were low, thereby limiting the pool of susceptible populations over time. on the other hand, the population of susceptible individuals increased drastically when the rate of vaccination was low. figure 3 showed the diagram of infective carriers against time. the number of infectious carriers was rising because they are not aware of their conditions. however, as the symptoms of the disease began to manifest, the population began to fall as a result of early treatment. figure 4 depicted the chart of infective against time. the population of the infected grows continuously which might be attributed to late treatment. in figure 5, the graph for the recovered individuals continued to grow as a result of treatment and regular medical check-ups. figure 2. susceptible population f 0.3 q )(ti c )(ti )(tr global stability analysis of typhoid fever model copyright ©2020 assa. adv. in systems science and appl. (2020) 29 figure 3. infected carrier population figure 4. infected population figure 5. recovered population o. j. peter, a. f. adebisi, m. o. ajisope, f.o. ajibade, a. i. abioye copyright ©2020 assa adv. in systems science and appl. (2020) 30 5. conclusion we analyzed a deterministic model for the management of typhoid. the conditions for the existence and stability of the equilibria described by the basic reproduction number of the model were successfully derived. the analysis showed that there is a disease-free equilibrium that is locally and globally asymptotically stable as long as an unstable if . the model revealed that the transmission of typhoid fever depended largely on the contact rate with the infected individuals in a population. thus, timely detection and early treatment could reduce the infection rate. references 1. j. brooks (1996). the sad and tragic life of typhoid mary, the canadian medic assoc j., 154(6), 915-916. 2. p. roumagnac, f.x weill, c. dolecek, s. baker, s. brises, n.t. chinh et. al. (2006). evolutionary history of salmonella typhi, science, 314, 1301-1304, doi: 10.1126/science.1134933. 3. f.y. lin, a.h. vo, v.b. phan, t.t. nguyen, d. bryla & c.t. tran (2000). the epidemiology of typhoid fever in the dong thap province, mekong delta region of vietnam, amer. j. trop. med. hygiene, 62, 644-648. 4. a. sinha, s. sazawal, r. kumar, s. sood, v.p. reddaiah, and b. singh (1999). typhoid fever in children aged less than 5 years, lancet, 354, 734-737. 5. j.m. hyman (2006). differential susceptibility and infectivity epidemic models, math. biosci. eng., 3, 88-100. 6. e.l lifshitz (1996). travel trouble: typhoid fever-a case presentation and review, j. amer. college health, 45(3), 99-105. 7. i.a. adetunde (2008). mathematical models for the dynamics of typhoid fever in kassenanankana district of the upper east region of ghana, j. modern math stat, 2, 4549. 8. d.t. lauria, b. maskery, c. poulos, and d. whittington (2009). an optimization model for reducing typhoid cases in developing countries without increasing public spending, vaccine, 27(10), 1609-1621. 9. d. kalajdzievska (2011). modeling the effects of carriers on the transmission dynamics of infectious diseases, math. biosci. & eng., 8(3), 711-722. 10. s. mushayabasa (2011). impact of vaccines on controlling typhoid, j. modern math stat, 5(2), 54-59. 11. b. cvjetanovic, b. grab, k. uemura (2014). epidemiological model of typhoid fever and its use in the planning and evaluation of antityphoid immunization and sanitation programs, bulletin of the who, 45(1), 53-75. 12. n. moffact (2014). mathematical model and simulation of the effects of carriers on the transmission dynamics of typhoid fever, trans. on comp. sci. eng. and its applic, 2(3), 14-20. 13. v.e. pitzer, c.b. cayley, b. stephen, k. gagandeep, b. veeraraghavan et. al. 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(2020) 31 16. c.h. watson & w.j. edmunds (2015). review of typhoid fever transmission dynamic models and economic evaluations of vaccination, vaccine, 33(3), 42-54. 17. j.k. nthiiri (2016). mathematical modeling of typhoid fever disease incorporating protection against infection, british j. math comp. sci, 14(1), 1-10. 18. k. moatlhodl & r. gosaamang (2017). mathematical analysis of typhoid infection with treatment, j. math sci: adv. applic., 40(1), 75-91. 19. s. mushayabasa (2017). a simple epidemiological model for typhoid with saturated incidence rate and treatment effect, int. j. sci., basic appl. res. (ijsbar), 32(1), 151168. 20. o.j. peter. & m.o. ibrahim (2017). application of differential transform method in solving a typhoid fever model, int. j. of math. anal. and optim.: theo. and appli., 1(1), 250-260. 21. o.j. peter, m.o. ibrahim, o. b. akinduko & m. rabiu (2017). mathematical model for the control of typhoid fever. iosr j. of math. (iosr-jm) 13(4), 60-66. 22. p. o. james and m. o. ibrahim (2019). application of variational iteration method in solving typhoid fever model, big data, knowledge and control systems engineering (bdkcse), sofia, bulgaria, 2019, 1-5, doi: 10.1109/bdkcse48644.2019.9010598. 23. j.p. lasalle (1976). the stability of dynamical systems, regional conference series in appl. math., siam, philadelphia. microsoft word 1104 article text, copyedited.doc adv syst sci appl 2021; 04; 1-15 published online at https://ijassa.ipu.ru. simulating events in requirements engineering by using pre-conceptual-schema-based components from scientific software domain representation paola a. noreña c.1,2*, carlos m. zapata j.1 1) facultad de minas, universidad nacional de colombia, medellín, colombia e-mail: paolanorena@itm.edu.co 2) facultad de ingenierías, instituto tecnológico metropolitano, medellín, colombia e-mail: cmzapata@unal.edu.co abstract: event simulation is a process for analyzing the behavior of a natural or automated system. such behavior is produced by events–phenomena or something that happens at any moment. events are simulated in scientific software domains (ssd) for finding scientific results and developing scientific software, which allow making critical decisions. usually, scientists use scientific models with simulation components for simulating events and requirements engineering (re) models for representing ssd in a software development process. however, the separation of simulation components in scientific models and software components in re models causes inconsistency in elements of the system domain, especially in events. consistency is required for a near solution to the needs of the scientific stakeholders. business analysts also use re models and know software development processes, but they lack scientific knowledge for describing elements of a ssd. therefore, scientists and business analysts require a computing model, which integrates common knowledge by using scientific and software components for simulating events and producing scientific software. pre-conceptual schemas (pcs) are computing models used on re for representing a domain, which is near to stakeholders. several simulation components are found in pcs, i.e., events. thus, we propose pcs components for simulating events in re from ssd. after, we validate the pcs understand and usage in ssd. scientists and business analysts can take advantage of the pcs for representing and simulating events and developing scientific software from the same model. such a proposal contributes to reducing the gap between both engineering and science into scientific software development. keywords: event simulation, pre-conceptual schemas, requirements engineering, scientific software domains, scientific software 1. introduction event simulation is a process used for studying the behavior of several complex systems by using events [21]. such a simulation decreases cost and risk for analyzing natural or automated systems [6]. events are instantaneous occurrences or phenomena of such systems [15]. events are responsible of system behavior by changing the state of processes and triggering other events in a time sequence, e.g., volcano erupts, patient suffers heart attack, signal decreases, and earthquake arrives [20]. commonly, a scientific software is developed in scientific software domains (ssd) for making critical decisions and solving real-life problems by using event simulation, e.g., making weather predictions based on climate models [10]. examples of scientific software domains are mathematics, engineering, medicine, and sciences [11–13]. consequently, the event simulation presents new challenge to scientific software development processes. * corresponding author: panorenac@unal.edu.co 2 p.a. noreña, c.m. zapata copyright ©2021 assa. adv sys sci appl (2021) scientific software is characterized by including a complex knowledge based on event simulation. rus et al. [22], brosig et al. [2], gonzález et al. [7], gonzález et al. [8], bonaventura and castro [1], song et al. [23], camejo et al. [3], laurindo et al. [12] use scientific simulation models for describing the complex knowledge of a ssd and analyzing alterations and impacts of the system from events. simulation component set of a model comprise clock, initial state, variables, parameters, events, activities, attributes, event generator, and random variable generator [6]. such authors also use requirements engineering (re) models with software components for representing the domain in a software development process. software component set of a re model comprise structural (entities and relationships) and dynamic (the system flow, processes, and events) features for representing a domain. gonzález et al. [7] recognize the growing need for simulation and programming complex systems. but both simulation and software components are separated into two or more scientific and re models, generating inconsistency in the element representation of the scientific software domains. since several components in different notations cause some elements may disappear or be confused with other elements in the development process, interfering with the understanding of the domain. in addition, scientists create scientific applications without a re process and business analysts lack scientific knowledge, taking them more time for understanding a ssd [9]. thus, scientists and business analysts require a computing model, which allows integrating a common and consistent knowledge with elements of simulation and software for simulating events in ssd from the same model. pre-conceptual schemas (pcs) are conceptual models based on object-oriented computing used in requirements engineering for representing a domain containing events. the pcs are models near to natural language of stakeholders for easily understanding the complete view of software systems by using dynamic and structural features [25]. the notation in pcs involves scientific and simulation components for representing events and mathematical equations, i.e., initial conditions, variables, events, parameters, etc. [20], which are used in event representation of ssd as: electronics [16], chemistry [20], environmental engineering [5], geology [18], industry [19], epidemiology [26] and oil engineering [24]. consequently, in this paper we propose simulation and software components integrated into pcs for simulating events in requirements engineering of a ssd. re is used for understanding problems of stakeholders and solving them by applying software engineering methods previously to the software development phase, which can be used in ssd [13]. after, we validate the pcs understanding and usage in scientific software domains with scientists and engineers. such a solution is proposed for business analysts and scientists can use the pcs as a computing model integrating common knowledge from software and science fields by using software and simulation components for recognizing, analyzing, designing, simulating, and coding the functionality of involved events to ssd as an advance to reducing the gap between both fields. this integration between simulation and software components offers a greater understanding of real-world events in a domain and a greater consistency among the modeling, simulation, and implementation phases. this paper is structured as follows: in section 2 we define the conceptual framework; in section 3 we state the problem; in section 4 we propose a solution; in section 5 we validate the proposal. finally, we discuss conclusions and future work in section 6. simulating events in requirements engineering by using pre-conceptual-schema… 3 copyright ©2021 assa. adv sys sci appl (2021) 2. conceptual framework 2.1. events occurrences that happen in the dynamic view of complex systems [14]. events are important elements because their information is used in the functional requirements during the software development life cycle for controlling the system behavior. also, events are used for changing the state of processes [15]. such changes occur by accomplishing constraints, which are conditions for processing events, activities, and data [20]. a trigger is an event used for initiating processes and other events in the system in time sequence; such a sequence is used for controlling the execution of event flow in the system [17]. trigger event types are none or statement (instruction for triggering a process, e.g., earthquake arrives); conditional (constraint for specifying a requirement, e.g., if seismometer alert = “active”); and timer (time for controlling events and processes, e.g., the statement event “time passes” with a cycle for increasing seconds from 0 to 28800 seconds [20]). 2.2. event simulation process used for analyzing structural and functional behavior of different complex systems, which are controlled by using events. also, event simulation is used for studying alterations and impacts of a system by keeping the real system intact [6]. events are simulated by using models, which are used for representing the system by using mathematical, logical, structural, and dynamic structures. such a simulation is applied for predicting the effects of changes in the system by using events and significantly reducing their risks and costs [7]. components are the elements and modules used from event simulation. elements. clock (counter for saving changes in the system, which are produced by events in discrete times for the beginning and the end of a simulation); initial state (set of variables and parameters for describing the system state); entity (concepts for describing roles or resources of the system, e.g., patient, student, room); attribute (concept for describing features of an entity, e.g., name, age); variable (number or concept for representing a variable value in a simulation process, e.g., door is opened); parameters (constant value, e.g., pi = 3.1416); event (occurrence for changing states of the system, e.g., volcano erupts); and activity (concept for representing an action e.g., engineer measures water lever) [6, 21]. modules. initialization (module for starting the system by using the initial state); clock generation (module for updating the clock simulation); random variable generation (module for creating the system behavior); and event generation (module for changing the state system) [6, 21]. 2.3. pre-conceptual schemas (pcs) conceptual model used in requirements engineering for computationally representing and recognizing main elements of a domain as events in the software engineering process. pcs link structural and dynamic features for describing the complete view of a software system [25]. also, elements with linguistic, mathematical, and graphical structures are included in the pcs components for representing events and mathematical notation [16, 20] (see fig. 1). components. structural relationship (verb for relating dependence among classes or entities and leaf concepts or attributes), dynamic relationship (verb for representing processes or activities), and eventual relationship (verb for representing events); concept (node for representing class and leaf concepts); specification (gatherer for assigning values or integrating operations), constraint (gatherer for operations with conditions and derived 4 p.a. noreña, c.m. zapata copyright ©2021 assa. adv sys sci appl (2021) values), and frame (gatherer for including reports or operations, which are same types); implication (connection for linking dynamic relationship and event flow), connection (connection for linking concepts and relationships), concept-note (connection for linking specification, constraint, and value-note; [25]), and operation (connection for linking operator, concept, variable, vector, and parameter); value-note (gatherer for representing value of a variable, parameter, and leaf concept), class-leaf concept (node for summarizing structural relationship among class and leaf concepts); operator (node for representing mathematical equations), i.e., + (sum), sin (sine), sqr (square root); timer (event for controlling time of other events and processes [17]); event (occurrence for triggering dynamic relationships and other events, which can be have a specification or a constraint); vector (node for saving information of the system); variable (node for representing a variable value); parameter (node for representing a constant); and initial conditions (gatherer for starting variables and parameters of the system [20]. fig. 1. pcs components based on [20] 2.4. scientific software domain (ssd) ssd are focused on solving the needs of scientific studies and research by using scientific software systems. such systems are used as tools for understanding phenomena, which can be analyzed from real systems avoiding risks and impacts, e.g., biologic, chemical, physical, and environmental systems. such software systems play an important role in critical decision making and solution of real-life problems in fields as mathematics, engineering (e.g., electronic, electrical, mechanic, etc.) medicine, and sciences (e.g., geography, chemistry, meteorology, geology, etc.) [11, 13]. according to kanewala and bieman [10], scientific software types are industrial (a system developed for complex processes), research (a system developed for specific research in a scientific context), and small-scale (a system developed for specific solutions of science students). 3. problem statement rus et al. [22] represent four models from re as a process model, cause-effect diagram, uml class diagram, and a simulation model (a discrete event model) for arriving at the implementation phase, the information of the several models is integrated into the simulation model. although they check consistency among such diagrams, the models have different elements. brosig et al. [2] use uml diagrams for representing re with parameters, services and actors, and petri nets for representing activities and transitions in a simulation process. discrete event system specification (devs) is the most general formalism for modeling systems with discrete events. however, scientists have little experience in requirements and software project practices [1,9]. gonzález et al. [7] use devs for executing models using simulation protocols, they recognize growing need for simulation dynamic relationship concept + specification { constraint } structural relationship eventual relationship implication connection operation operator concept-note value-note class leaf concept frame volcano passes parameter sin sqr volcano erupts initial conditions variable time wave starts signal events simulating events in requirements engineering by using pre-conceptual-schema… 5 copyright ©2021 assa. adv sys sci appl (2021) and programming complex systems. so, they use uml state machine diagrams with simulation elements (events, activities, constrains and states) in an automatic teller machine (atm) case. however, the diagrams lack a description logic of the system since it is inconsistent with the code. gonzález et al. [8] indicate the modeling and simulation for design and prototyping of systems are widely used techniques, they agree in the usage the model driven development (mdd). therefore, the authors use uml state machine diagrams for describing structural and dynamic features of a software system with an objectoriented paradigm with simulation elements as: events, activities, states, and transitions, but such a representation lacks several elements for a complete representation and simulation of events in the same atm case. bonaventura and castro [1] propose a methodology for complex software projects based on modeling and simulation (m&s) formal, devs, and software project practices in a scientific project about questions of the origin of the world. the methodology is a very close solution, but such a project is represented in a high level of data networks. then, the simulation only is applied to the network and both the methodology, and the project lack requirements specifications. song et al. [23] propose a grey box testing method for availability simulation software based on the event tree model. this model allows event simulating and modeling with elements as: events, parameters, initial conditions, logic and sequence events, and relationships among elements. but the graphical representation applied in the software testing phase, only includes nodes with numbers for the events and edges for the relationships, logic, and sequence, which. camejo et al. [3] propose a computational simulation model for milk production process, which is developed in the programming language arena, a simulation software based on events and processes. so, they use elements as entities, attributes, activities, events, and mathematical models. however, the model is used by a flow diagram with general processes. laurindo et al. [12] present a communication mechanism between an event simulation software and a dynamic software, using a and a conceptual model (a graph with events, processes, and time) and a similar simulation model in the software simulation. nevertheless, the integration lacks other elements of domain necessary in the requirement engineering. in summary, the authors use separately two or more kinds of models with several notations for integrating event simulation and software components related to scientific software development process: (i) a conceptual model from requirements engineering (e.g., uml diagrams and other models from complex systems) including elements as: entities, attributes, events, and activities from ssd (ii) a simulation model with elements as: time, events, activities, initial conditions, parameters, variables, and complex notation in mathematical models. such separation between event simulation and software components generates inconsistency in the modeling, simulation, and implementation in a scientific software development process as: some elements may disappear or be confused with other elements in the development process causing interferences in the understanding of the domain. (iii) scientists usually create scientific applications without a requirement engineering process and business analysts lack scientific knowledge, taking them more time for understanding the ssd [9]. consequently, a computing model is required, which allows integrating a common and consistent knowledge with components of simulation and software for simulating events from re process in ssd. the pcs notation integrates (i) software components from re and computational linguistics as structural and dynamic features for representing the complete view of any domain [25]; (ii) scientific components as linguistic, graphical, and mathematical structures for representing events are included from ssd, which can be used in the modules for simulating events [20], (iii) pcs are used in event representation of signals [16], chemical 6 p.a. noreña, c.m. zapata copyright ©2021 assa. adv sys sci appl (2021) mixtures [20], noise environmental [5], geology [18], industry [19], epidemiology [26] fluid pressure of the petroleum [24], and simulation [20] for developing scientific software. so, we select the pcs as a computing model for proposing a solution. 4. proposal solution 4.1. identifying pcs components (elements) for simulating events from ssd we observe the functionality of components in event simulation, and we compare it to the functions of pcs components (see fig. 1) for identifying common elements; then, we find the elements of the table 1, which have same function. such elements also are related to mathematical notation into the model. table 1. pcs elements of event simulation based on [6, 20] simulation component pcs (software) component function initial state initial conditions integrating initial variables and parameters clock timer controlling the simulation time entity class concept relating main concepts, which have leaf concept attribute leaf concept featuring concepts of the classes event event triggering dynamic relationships events, and state changes of the system activity dynamic relationship relating processes to the system variable variable saving variable values or data parameter parameter saving a constant value or data 4.2. defining pcs components (modules) for simulating events from ssd initialization. module for automatically starting the system behavior by using the initial conditions of the pcs. such conditions contain initial values of parameters and variables for simulating events, e.g., the parameters simulation time = 28800 seconds, threshold = 0,7, and rock rigidity = 7 x 107 kg/meters2 and the variables timestamp = “next”, time = 0 seconds, random value = 0,0, tectonic plate state = “position”, wave acceleration = 0,0 meters/seconds2, seismometer alert = “inactive”, and seismograph state = “off”; which are initial values for analyzing the behavior system about an earthquake. clock generation. module for automatically updating the simulation time and controlling other modules by using a timer of the pcs (see fig. 1). such an event is used for generating the beginning and end of a simulation by increasing time variable (a sum); we can use second-minute-hour units and digital format of time. timer types are cyclical time (when the event has a cycle for increasing the time, e.g., time passes has a cycle from 0 to 28800 seconds) and specific date (when the time is a parameter used in a condition, [19]). also, we include timestamps, which are variables used for saving time moments when events happen; thus, they can start or stop the time [14–15], e.g., timestamp = “stop” when a seismometer alert = “active”. random variable generation. module for automatically changing system behavior of the pcs. random variables are used in simulation and statistical probability for giving a value and analyzing their effects. random function rand () is integrated to the pcs for generating random values between 0 to 1 (random variable = rand, see fig. 2a), random values between an upper limit and a lower limit (random variable = rand () * upper limit + lower limit, see fig. 2b), and a random variable between two values (random variable = rand () linking two simulating events in requirements engineering by using pre-conceptual-schema… 7 copyright ©2021 assa. adv sys sci appl (2021) values, see fig. 2c); however, if another function for a random variable is required can be represented by using the pcs notation (see fig. 1). fig. 2. random variable generation event generation: module for automatically operating and changing states of variables of the system by using event sequence in the pcs. such a sequence allows for generating autonomous processes by using initialization, clock generation, and random variable generation modules. then, events are executed from their internal function for changing the system behavior by using their specifications and constraints, which include dynamic relationships for inserting and updating data by accomplishing conditions of time values, states of variables, and initial values, e.g., earthquake arrives, seismometer alert emerges, and station measurement starts and ends. 4.3. simulating events in requirements engineering from ssd by using pcs we propose the pcs usage steps for simulating events in requirements engineering from ssd, which are exampled by representing an early earthquake alert generation system as a lab study in the ssd of geology. scientists use such systems for simulating events, predicting, and avoiding risks of damages and disasters. representing elements of ssd. scientists and business analysts should represent the elements of the system domain by using pcs components (elements) as classes with its attributes (leaf concepts), e.g., seismologist (with code and name), tectonic plate (with code and name), earth zone (with code, name, and tectonic plate), station (with code, name, location, latitude, longitude, and altitude), measurement (with local time, slip distance, and fault area), and earthquake (with date moment time, focus, epicenter, latitude, longitude, final slip, distance, final fault, area, seismic moment, magnitude, and intensity) are classes of the system identified and represented in the fig. 3. representing the flow of ssd. scientists and business analysts should describe the complete logic and flow the system. such description will be translated to a programming language in the development phase of the scientific software. first, they should use the pcs components (modules) as initialization, clock generation, and random variable generation (if other variables or ranges are required, then the variable component should be used). then, they should include pcs components (elements) as initial conditions, timer, events, dynamic relatioships (processes or activities), variables, and parameters. values, processes, and events can be detailed by using especifications, which include mathematical equations or internal conditions. finally, the logic and flow of processes and events is related by using the implication link if a implication is required (from an event or condition to a process, from a process to other process, and an event to other event). example, the initialization module is represented in initial conditions for starting parameters: simulation time, threshold, and rock rigidity and variables: simulation time, time random value, tectonic plate state, wave acceleration, seismometer alert, and seismograph = rand random variable = rand random value upper limit * = randrandom variable “connected” “disconnected”lower limit + (a) (b) (c) 8 p.a. noreña, c.m. zapata copyright ©2021 assa. adv sys sci appl (2021) state (see fig. 3.) the clock generation module is represented in fig. 3. with the time passes timer, which start the system flow by using a cycle from 0 seconds to the simulation time (28800 seconds according to the initial conditions) if timestamp = “next” and time <= simulation time. fig. 3. simulation event model by using pcs earthquake arrives event is random event that happens only when a random value fulfils the condition if random value >= threshold (threshold = 0,7 according to the initial conditions); when a tectonic plate collides, another random variable wave acceleration in seismometer alert event is generated (random variable generation and event generation modules are applied). if wave acceleration > 0,05 m/s2 (minimum value of risk) then seismometer alert is activated, also the time is stopped by using timestamp = “stop” else seismometer alert = “inactive.” such an event triggers the measurement station starts event when seismometer alert = “active” for changing the seismograph state to “on” and automatically inserting the measurement data (local time, slip distance, fault area); in this { } { } { } { } code code initial conditionsearth zone has earthquake arrives { } = rand >= yes threshold = “collision” random value = “position” no random value station earthquare date epicenter focus seismologist code name has magnitude emerges has inserts earthquake local time = time name measurement local time has location slip distance manages inserts updates deletes searchs plate tectonic state plate tectonic state time = 0 seconds threshold = 0,7 random value = 0,0 has moment time starts station measurement wave acceleration > 0,05 m/s2 seismometer alert = “active” seismometer alert = “inactive” wave acceleration time simulation = 28800 seconds yes no rock rigidity = * * plate tectonic has code name seismometer alert latitude longitude name altitude fault area seismograph state “on” “active” seismometer alert = seismograph state “off” ends station measurement { } earthquake magnitude / = 3 log10 * 10.7 2 final slip distance final fault area seismic moment intensity { } = + 1 seconds time time passes time earthquake seismic moment earthquake seismic moment earthquake slip distance earthquake fault area wave acceleration = 0,0 m/s2 seismometer alert = “inactive” rock rigidity = “position” plate tectonic state = seismograph state “off” = = 7 x 107 kg/m2 = micro light moderate strong -very strong violent -extreme wave acceleration * = “next”timestamp <= time time simulation = rand measurement timestamp “next” = and <= time time simulation timestamp “next” = and = “collision” plate tectonic state = “next”timestamp = “stop”timestamp >= time time simulation seismometer alert “inactive” = or simulating events in requirements engineering by using pre-conceptual-schema… 9 copyright ©2021 assa. adv sys sci appl (2021) case, we use the implication link for obligatorily relating to the cause of other event. thus, timestamp = “next” for increasing the time and returning to seismometer alert emerges, comparing the wave acceleration value, and saving measurement to the end of the earthquake. if seismometer alert = “inactive” or time >= simulation time, then station measurement ends. when measurement ends, then seismologist manages—updates, inserts, searches, deletes, which are dynamic relationships—the earthquake class. such a class has leaf concepts time moment, date, epicenter, focus, final slip distance, final fault area, magnitude, intensity, and seismic moment. magnitude and seismic moment are derived attributes (values calculated by applying mathematical equations with leaf concepts inserted, numbers, and parameters; magnitude = 2/3 * (log 10 seismic moment – 10,7) and seismic moment = rock rigidity * slip distance * fault area * wave acceleration; the intensity is micro, light, moderate, strong, very strong, violent, and extreme according to the magnitude.) simulating events from ssd by using pcs. the complete logic allow performing the event simulation with data required in the system and the database. thus, both scientists and business analysts should validate the requirements with scientific stakeholders with representation, then they can understand the complete system, simulating events and its data previously to the development phase of the scientific software. example, we simulate events in an earthquake occurred in santander, colombia. such a phenomenon is analyzed by using the pcs of the fig. 3. we show the epicenter and station in fig. 4, where the earthquake was measured and detected. we use the real information of variables, parameters, classes, and leaf concepts for simulating the earthquake. fig. 3. earthquake occurred in santander, colombia [4] values (leaf concepts) of the tectonic plate, earth zone, seismologist, and station classes are stored in the tables of the database tectonic plate, earth zone, seismologist, and station (see tables 2, 3, and 4). values (local time, slip distance, fault area, and station code) of the 10 p.a. noreña, c.m. zapata copyright ©2021 assa. adv sys sci appl (2021) measurement class are automatically inserted by using the event station measurement starts (see fig. 3) and stored in the table measurement of the database (see table 6; wave acceleration random value assigned is ‘1,2 m/s2’ for applying the mathematical equation of the seismic moment, its unit is newton (n) meters. values (date, moment time, earth zone code, station, focus, epicenter, latitude, longitude, final slip distance, final fault area, seismic moment, magnitude, and intensity) of the earthquake class are inserted by the seismologist when the station measurement ends event happens (see fig. 3) and stored in the table earthquake of the database (see table 7). earthquake had an intensity “strong” and a magnitude ‘5,5 n meters’, value summarized in ‘5,5’ by excluding units in richter scale. some fictional information is added to the real data in order to complete the simulation, e.g., the seismologist. table 2. tectonic plate. the authors code name earth zone code ptc1 cocos 101 ptc2 south american 101 ptc3 nazca 101 ptc4 caribbean 101 table 3. earth zone the authors code name 101 colombia table 4. earth zone the authors code name 104521452 juan manuel gaviria table 5. station. the authors code name location latitude longitude altitude earth zone code seismologist code 62 barranca santander 7,101o -73,7o 132 101 104521451 table 6. measurement. the authors local time (s) slip distance (m) fault area (m2) station code 30000 0,5 33 62 31000 1,5 34,1 62 32000 2 36,3 62 33000 2,8 38,5 62 34000 3 39 62 35000 3,35 40,2 62 simulating events in requirements engineering by using pre-conceptual-schema… 11 copyright ©2021 assa. adv sys sci appl (2021) table 7. earthquake. the authors d at e m om en t ti m e( s) ea rth z on e co de st at i o n co de fo cu s ( m ) ep ic en te r la tit ud e lo ng itu de fi na l s lip d ist an ce (m ) fi na l f au lt ar ea (m 2 ) se ism ic m om en t ( m ) m ag ni tu de in te ns ity 30 /0 5/ 20 18 30 00 10 1 62 14 0. 00 0 sa nt an de r 6, 83 ° 7 3, 11 ° 3, 35 40 ,2 94 26 9x 1 05 5, 5 st ro ng 5. validation we evaluate the understanding and the usage the pcs components by the expert opinion. we collect the data with a survey as instrument. we select two independent variables: current role and area. the sample size is 28 experts (10 scientists: 6 in geology, 1 in energy, 1 in petroleum, and 2 in simulation; 11 developers, 3 professor, and 4 analysts in re and software engineering) and four dependent variables: pcs representation. first, experts observed the pcs in fig.3. after, they answered the question q1. describe with your words, what is being represented in the model? for analyzing the pcs representation variable. we observe 61% of experts described phrases related to an earthquake early warning system, simulation, detection, and automated measure of an earthquake, i.e., a system to measure earthquakes and tectonic activity, tectonic plate movement simulation, a system to detect and collect information on earthquakes. in addition, the 28% of experts indicated phrases related to an earthquake, a tectonic activity, a seismic event, and events of an earthquake and a seismological station, i.e., a seismologist takes measurements of an earthquake, a conceptual model of earthquakes, their properties, and measurements. only 11% of experts described different phrases as: relational model, a system, and remembers. the results indicate the 89% (25) of experts perform a near description to the system domain (see fig. 4). fig. 4. pcs representation component integration. experts answered the question: q2. does the pcs integrate scientific, software, and simulation components? for analyzing of the component integration variable. the 86% (24) of experts and affirm the model integrate such components and the 11% (3) indicates perhaps integrate the components. the results confirm the pcs include 12 p.a. noreña, c.m. zapata copyright ©2021 assa. adv sys sci appl (2021) scientific, software and simulation components, which allow represent a ssd and simulating events (see fig. 5). fig. 5. component integration pcs understand. experts answered the question: q3. is the pcs understandable? for analyzing the pcs understand variable. from the point of view of the experts, 20 experts are strongly agreed and agreed with the pcs is understandable, 7 experts are neutrals, and 1 expert is disagreed (see fig. 7). the majority opinion of the results in show acceptation in the understanding of the schema. fig. 6. pcs understand pcs usage. experts answered the question: q4. would you use the pcs in scientific software development? e.g., in modeling, simulation and requirements elicitation for analyzing the pcs usage variable. the 72% (20) experts affirm they could use the pcs in scientific software development processes, the 21% (6) experts indicate perhaps would use it and the 7% (2) answer they would not use it (see fig. 7). the results present a positive position in the pcs usage for modeling ssd, event simulation, and requirements elicitation in scientific software development processes. simulating events in requirements engineering by using pre-conceptual-schema… 13 copyright ©2021 assa. adv sys sci appl (2021) fig. 7. pcs usage 6. conclusion initial conditions, timers, class concepts, leaf concepts, events, dynamic relationships, variables, and parameters are identified in pre-conceptual schemas as components (elements) for simulating events. such a representation was refined by defining initialization, clock generation, random variable generation, and event generation modules; such modules were proposed for completing components of event simulation in requirements engineering. pcs allow for modeling and simulating events in ssd by integrating simulation and software components in the pcs notation. such components are applied in a lab study of an early earthquake alert generation system in the ssd of geology. event simulation by using the pcs is a new approach for integrating scientific and software components in the same model. thus, scientists and software analysts can use the pcs as a computing model for modeling and simulating events in re for the scientific software development process: they can validate the requirements with scientific stakeholders by using pcs, understand the elements of the complete system, and simulate events and data previously to the development phase of the scientific software. we also evaluate the understanding and the usage the pcs components by the expert opinion was favorable and both scientists and software experts understanded the model. this approach offers a better understanding of real-world events in any domain and consistency among the modeling, simulation, and implementation phases reducing the gap between science and software engineering. we suggest as future work the automation of pre-conceptual schemas for simulating events, which allows for automatically generating results and reports from a simulation software. acknowledgements this paper is a product of the ph.d. research project an extension to pre-conceptual schemas for refining event representation and mathematical notation with hermes code 39886 at universidad nacional de colombia. this work is sponsored by minciencias (ministry of science, technology, and innovation) from colombia in the program ph.d. students in colombia, bid 727 of 2015. 14 p.a. noreña, c.m. zapata copyright ©2021 assa. adv sys sci appl (2021) references 1. bonaventura, m. a., and castro, r. d. 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(2020). conversión de eventos desde esquemas preconceptuales en código pl/pgsql: simulación de software en la cuarta revolución industrial. revista ibérica de sistemas e tecnologias de informação, (39), 18–34. microsoft word 3. 872_done adv syst sci appl 2023; 02; 11-23 published online at https://ijassa.ipu.ru. __________________________________ *corresponding author: ssg.mech.official@gmail.com an overview of multiple criteria decision making techniques in the selection of best laptop model shankha s. goswami*, dhiren k. behera indira gandhi institute of technology, dhenkanal, sarang, india, 759146 abstract: this research article presents a detailed study of a laptop selection problem by analyzing through three (multi-criteria decision making) mcdm techniques namely, (weighted sum model) wsm, (weighted product model) wpm and (weighted aggregated sum product assessment model) waspas. the main goal of this study is to propose the best laptop model among a group of seven alternative models based on 10 selection criteria. for this purpose, (analytic hierarchy process) ahp is used to determine the criteria weightages whereas, wsm, wpm, waspas techniques are applied individually to determine the best model and to rank the alternatives. the best model is proposed based on the output results of the three mcdm techniques which suggest that alternative 3 and alternative 2 are the optimum and the foulest choice respectively. all the rankings given by the three methodologies are also compared with the previous result which shows that all the three methods are giving the same outcome and the rankings are more or less same with very minor alterations. keywords: laptop selection, multi-criteria decision making, analytic hierarchy process, alternative ranking, weighted sum and product model 1. introduction mcdm have become a trending topic for the decision makers to utilized it as a judgmental tool that helps to make proper selection of products or processes. for any sector, it is very important to make right decision at the right time. the primary problem associated in decision making process is the selection criteria that creates confusion among the buyers while purchasing some items like mobile [15, 16, 21], refrigerator, car [42], air-conditioner [1] etc. let us take an example of a cell phone [21] to explain it clearly, suppose someone wants to buy a cell phone having descent specifications and obviously within the budget, now there are lots of phones available in the market from different brands that create confusion among the buyers which one is the appropriate model to buy and the buyers also have some choices and preferences regarding the phone specifications (i.e. criteria’s) like processor, camera, battery capacity, screen size, etc. ideally, it is not possible that the buyers may get all of their preferable specifications in one model but still they have to select one best model so that most of their requirements can be fulfilled [21]. to get rid from such type of hard situations that arises in our everyday life while taking any decisions, mcdm can guide us to take the right path and provide solution to these types of problems. proper selection of laptop model [14, 17, 23, 27, 35, 44] is such type of problem that came under the category of mcdm, since its selection is based on various criteria like screen resolution, ram, cost, hard disk capacity, brand, color, etc. so, it is quite difficult for the buyers to choose the appropriate laptop model among thousands of available models based on their budget and specifications that perfectly suits their personality. therefore, mcdm would be the suitable tool for investigating the present problem having various conflicting factors. although, 12 s.s. goswami, d.k. behera copyright ©2023 assa adv. in systems science and appl. (2023) many other techniques like artificial neural network (ann), linear programming problem (lpp), goal programming etc. are present, but none of the mentioned optimization tools can ever reach the level of mcdm in terms of simplicity and easiness. mcdm techniques are very simple and easy to understand, it requires less computational time, and mcdm necessitates some simple mathematical steps for its operation. on the other hand, alternative optimization process like ann is quite difficult to grab and full of complex calculations. a detailed analysis of the best laptop model selection is presented in this article from which the buyers can develop some idea about the laptop model while purchasing it. not only that, but also the laptop manufacturing companies will be benefitted from the present market scenario that helps to reconstruct their future production according to the present market demand. the ongoing problem is adopted from the paper presented by adali and isik [2], where the selection of laptop is carried out by applying multi-objective optimization on the basis of ratio analysis (moora) [7], full multiplicative form of moora (multimoora) [8] and multiobjective optimization on the basis of simple ratio analysis (moosra) [22,26] methods. a panel of five expert members having high market experience of more than 10 years was formed. the panel board members are higher officials and executives who are associated with some of the reputed electronic firms and laptop manufacturing companies. after going through a massive brainstorming session, they [2] considered 10 selection parameters as follows: processor clock speed (pcs) (in ghz), cache memory (cm) (in mb), storage space (ss) (in gb), display card storage (dcs) (in gb), ram (r) (in gb), screen resolution (sr) (in pixel), screen dimension (sd) (in inch), brand trustworthiness (bt), weight (w) (in kg), and price (p) (in turkish liars). according to the experts, these are the most important factors for a laptop that a customer usually looked for before purchasing it. the seven laptop models were chosen after studying the present market demand and rigorous research of the past laptop sales history. from the records and statistics of some electronic stores and verbal communication with some of the laptop users, the board members sorted out the seven most suitable laptop models for this analysis. the number of alternative laptop models is restricted to 7 because, with the increasing of alternatives and selection criteria, the order of the decision matrix increases; this makes the calculation portion time-consuming and complicated. the selection process was carried out on the basis of 10 selection parameters and the best model was proposed among the seven chosen alternative models. the criteria weightages were determined by adali and isik [2] using ahp [34,38] and the same weightages values are reused in this analysis. the decision makers put their own views and opinions for constructing the ahp pair-wise comparison analysis matrix followed by the criteria weightage’s determination which are as follows: wc1 = 0.297, wc2 = 0.025, wc3 = 0.035, wc4 = 0.076, wc5 = 0.154, wc6 = 0.053, wc7 = 0.104, wc8 = 0.017, wc9 = 0.025, wc10 = 0.214. table 4.1 presents the seven chosen alternative laptop models and their respective specifications (criteria’s). the screen resolution is measured in 1-3 judgment scale which denotes that: 1 – worst (1366 × 768 pixels), 2 – medium (1600 × 900 pixels), 3 – best (1920 × 1080 pixels) and similarly, brand reliability is also measured in 1-10 judgment scale which denotes that: 1 worst and 10 best [2]. a problem similar to the stated above is further extended by implementing three other mcdm techniques, i.e. wsm [11, 30, 33, 45], wpm [32, 45] and waspas [48–50] which are discussed in this article. the output results are also compared with the previous results to check whether the outcomes from all the methods are same or not. an overview of multiple criteria decision making techniques… 13 copyright ©2023 assa adv. in systems science and appl. (2023) 2. literature review many potential researchers have on various mcdm techniques applied in wide variety of areas including industrial sector [9,13,18,19,24], banking and finance sector [25,39], energy sector [20], health and educational sector [6,36], environmental management [37,43], etc. afshari et al. [3] applied simple average weighting (saw or wsm) method to solve the personnel selection problem. men et al. [31] used technique for order preference by similarity to ideal solution (topsis) method to analyze the harmonius society development of east china. zhang et al. [51] established a technology project credit evaluation index system model, where ahp is used to determine the weights of each evaluation index and the evaluation results was analyzed through fuzzy comprehensive evaluation method (fcem). gong et al. [12] applied fuzzy analytic hierarchy process (fahp) for determining the customer requirements in software quality function deployment, and the vitality degree was analyzed using trapezoidal fuzzy function. wang et al. [47] used anp and fuzzy assessment method to assess and rank the risk factor strategies using triangular fuzzy numbers. zavadskas et al. [50] measured the accuracy of wsm and wpm methods and proposed a joint waspas methodology for evaluation of accuracy, based on initial criteria values. bagocius et al. [5] developed a deep-water sea port in the klaipeda region by a combination of waspas and entropy methods. dejus and antucheviciene [10] applied the entropy and waspas method for the assessment and selection of appropriate solutions for occupational safety. siozinyte and antucheviciene [40] proposed a rational solution to improve daylighting in vernacular buildings by applying ahp to determine the weightages and copras, waspas, topsis to rank the alternatives. staniunas et al. [43] evaluated the greenhouse emissions and aimed at decreasing emissions significantly by applying complex proportional assessment (copras), waspas and topsis methodologies. zavadskas et al. [48] selected the most preferable façade’s alternative for public or commercial building by waspas and the results was also validated by moora and multimoora. zolfani et al. [52] used stepwise weight assessment ratio analysis (swara) method to calculate the relative weightages of the criteria and waspas method to evaluate the alternatives for selecting the perfect location to establish a shopping mall. chakraborty and zavadskas [9] explored eight manufacturing decision making problems such as arc welding process, machinability of materials, cutting fluid selection, electro-discharge micro machining process parameters, industrial robot, electroplating system, milling and forging condition by the application of waspas method. liu and lin [29] assessed audit risk judgment by constructing a dominant if rough set based mcdm methodology. karande et al. [24] used couple of mcdm methods, i.e. multimoora, waspas, wpm, moora and wsm to investigate two real time industrial robot selection problems. li et al. [28] proposed a new decision method to address the challenge on criteria reduction in mcdm problems with numerical values based on the rough set theory and the relation of criteria values. venkateswarlu and sarma [46] selected the best supplier for implementing the spring manufacturing industry by incorporating saw and višekriterijumsko kompromisno rangiranje (vikor) methods. apart from this, many researchers also adopted different mcdm methods to choose the best alternatives used for domestic purposes like, srichetta and thurachon [41] evaluated the notebook selection problem by applying fuzzy-ahp. srikrishna et al. [42] used topsis technique for a new car selection problem. lakshmi et al. [27] and tampi et al. [44] selected the best laptop model by using topsis and ahp methodology. adali and isik [1] executed an airconditioner selection problem with copras and additive ratio assessment (aras) methods. kalyani et al. [23] implemented extended topsis method for selecting the best laptop model. 14 s.s. goswami, d.k. behera copyright ©2023 assa adv. in systems science and appl. (2023) agajie [4] adopted ahp to select the best personal computer on the basis of three criteria i.e. cost, user-friendliness and software availability. from the above scenario of the literatures it is noticeable that, there are very few research works have been recorded till date where wsm, wpm and waspas methods are used for solving household decision making problems rather than the industrial applications. so, an initiative is taken in this article to utilize these three methods as a decision-making tool for solving a laptop selection problem. moreover, mcdm is a quite straightforward 3. theoretical analysis all the three methods are explained clearly in details step by step under this section. there is not much difference between these methods and the initial steps of all the three methods starts with a decision matrix which is shown below by equation (3.1) followed by the normalization as shown in equations (3.2) and (3.3) [24]. [𝐴 ] × = ⎣ ⎢ ⎢ ⎢ ⎡ 𝑎 𝑎 𝑎 … 𝑎 𝑎 𝑎 𝑎 … 𝑎 𝑎 𝑎 𝑎 … 𝑎 … … … … … 𝑎 𝑎 𝑎 … 𝑎 ⎦ ⎥ ⎥ ⎥ ⎤ , (3.1) where ‘m’ is the number of alternatives and ‘n’ is the number of criteria (i = 1, 2, 3…, m; j = 1, 2, 3…, n). now, the normalization of the decision matrix is done by using equation (3.2) or (3.3) according to the criteria nature. for beneficial criteria (maximum criteria) whose higher values are expected, nij = (3.2) for non-beneficial criteria (minimum or cost criteria) whose smaller values are expected, nij = (3.3) 3.1. weighted sum model (wsm) wsm [11,30] is the most widely used mcdm tool [24] due to its simplicity and less complex calculations. the best option and the ranking are evaluated based on the overall weighted sum of the alternatives which is determined by using equation (3.4) shown below. 𝑊 = ∑ 𝑁 𝑊 (3.4) 3.2. weighted product model (wpm) wpm [32,45] is very similar to wsm [24] but instead of addition, multiplication is done and the criteria weightages are used as the power to the normalized values. overall weighted product is determined for every alternative and the ranking is done based on these values. the weighted product values are determined by using equation (3.5). 𝑊 = ∏ 𝑁 , (3.5) where ‘𝑊 ’ are the ahp computed weightages of criterion in equation (3.4) and (3.5). an overview of multiple criteria decision making techniques… 15 copyright ©2023 assa adv. in systems science and appl. (2023) 3.3. weighted aggregated sum product assessment model (waspas) waspas [48-50] is the combination of wpm and wsm where the aggregated sum product weightage is determined to rank the alternatives. a joint generalized criterion of the weighted summation and the multiplication methods is proposed by zavadskas et al. [48,49] which is shown by equation (3.6) [24]. qi = 0.5𝑊 + 0.5𝑊 = 0.5∑ 𝑁 𝑊 + 0.5∏ 𝑁 , (3.6) where, where qi is the aggregated sum product weightage of the ith alternative (i = 1, 2, 3…, m; j = 1, 2, 3…, n). in order to increase the effectiveness and the ranking accuracy, a more generalized equation was developed by zavadskas et al. [50] which is given by equation (3.7) [24]. qi = λ𝑊 + (1-λ)𝑊 = λ∑ 𝑁 𝑊 + (1-λ)∏ 𝑁 , (3.7) where, where λ = 0, 0.1, 0.2, 0.3…, 1. from equation (3.7), if λ = 0 then it is converted to wpm and if λ = 1 then it is converted to wsm. 4. research methodology adali and isik [2] solved a laptop selection problem by using moora, multimoora and moosra method and the same problem is adopted in this present research work and further extended by analyzing through wsm, wpm and waspas method. in [2], they determined the criteria weightages using ahp which are kept constant in this analysis. the three methods are applied to the above stated problem and the output ranking of the alternatives are compared to the previous results. all the calculation details are covered in this section and the decision matrix as created by adali and isik [2] according to equation (3.1) is shown in table 4.1 which is the initial step of the analysis. table 4.1. selected alternative models and their specifications criteria nature maxi maxi maxi maxi maxi maxi maxi maxi mini mini models pcs cm ss dcs r sr sd bt w p lm1 3.5 6 1256 4 16 3 17.3 8 2.82 4100 lm2 3.1 4 1000 2 8 1 15.6 5 3.08 3800 lm3 3.6 6 2000 4 16 3 17.3 5 2.9 4000 lm4 3 4 1000 2 8 2 17.3 5 2.6 3500 lm5 3.3 6 1008 4 12 3 15.6 8 2.3 3800 lm6 3.6 6 1000 2 16 3 15.6 5 2.8 4000 lm7 3.5 6 1256 2 16 1 15.6 6 2.9 4000 max or min value 3.6 6 2000 4 16 3 17.3 8 2.3 3500 (source: adali and isik, 2017) now the normalization of the decision matrix shown in table 4.1 is done by using equations 3.2 and 3.3 based on the nature of the criteria. the maximum criteria (i.e. beneficial criteria) is normalized using equation 3.2 whereas, the minimum criteria (i.e. non-beneficial) is normalized using equation 3.3. the normalized decision matrix is shown in table 4.2. table 4.2. normalized matrix weights 0.297 0.025 0.035 0.076 0.154 0.053 0.104 0.017 0.025 0.214 16 s.s. goswami, d.k. behera copyright ©2023 assa adv. in systems science and appl. (2023) pcs cm ss dcs r sr sd bt w p lm1 0.972 1 0.628 1 1 1 1 1 0.816 0.854 lm2 0.861 0.667 0.5 0.5 0.5 0.333 0.902 0.625 0.747 0.921 lm3 1 1 1 1 1 1 1 0.625 0.793 0.875 lm4 0.833 0.667 0.5 0.5 0.5 0.667 1 0.625 0.885 1 lm5 0.917 1 0.504 1 0.75 1 0.902 1 1 0.921 lm6 1 1 0.5 0.5 1 1 0.902 0.625 0.821 0.875 lm7 0.972 1 0.628 0.5 1 0.333 0.902 0.75 0.793 0.875 (source: author’s own elaboration) the overall weighted sum (𝑊 ) for each alternative is calculated by using equation (3.4). the calculated weighted sum is shown in table 4.3 below. table 4.3. overall weighted sum of the alternatives pcs cm ss dcs r sr sd bt w p weighted sum lm1 0.289 0.025 0.022 0.076 0.154 0.053 0.104 0.017 0.020 0.183 0.943 lm2 0.256 0.017 0.018 0.038 0.077 0.018 0.094 0.011 0.019 0.197 0.743 lm3 0.297 0.025 0.035 0.076 0.154 0.053 0.104 0.011 0.020 0.187 0.962 lm4 0.248 0.017 0.018 0.038 0.077 0.035 0.104 0.011 0.022 0.214 0.783 lm5 0.272 0.025 0.018 0.076 0.116 0.053 0.094 0.017 0.025 0.197 0.892 lm6 0.297 0.025 0.018 0.038 0.154 0.053 0.094 0.011 0.021 0.187 0.897 lm7 0.289 0.025 0.022 0.038 0.154 0.018 0.094 0.013 0.020 0.187 0.859 (source: author’s own elaboration) the overall weighted product (𝑊 ) for each alternative is calculated by using equation (3.5). the calculated weighted sum is shown in table 4.4 below. table 4.4. overall weighted product of the alternatives pcs cm ss dcs r sr sd bt w p weighted product lm1 0.992 1 0.984 1 1 1 1 1 0.995 0.967 0.938 lm2 0.957 0.990 0.976 0.949 0.899 0.943 0.989 0.992 0.993 0.983 0.712 lm3 1 1 1 1 1 1 1 0.992 0.994 0.972 0.959 lm4 0.947 0.990 0.976 0.949 0.899 0.979 1 0.992 0.997 1 0.755 lm5 0.974 1 0.976 1 0.957 1 0.989 1 1 0.983 0.885 lm6 1 1 0.976 0.949 1 1 0.989 0.992 0.995 0.972 0.879 lm7 0.992 1 0.984 0.949 1 0.943 0.989 0.995 0.994 0.972 0.831 (source: author’s own elaboration) by using equation (3.7) a joint weightage of wsm and wpm is determined for every alternative. now, λ value ranges from 0 to 1 at an interval of 0.1. so, eleven joint weightages are proposed for each and every alternative based on eleven λ values i.e. λ = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1 in order to observe that, whether there will be some differences in the final output results and ranking for different λ values. the joint generalized criterion [24,50] weightages for different λ values are calculated and shown in table 4.5. table 4.5. joint generalized criterion weightages for different λ values λ = 0 λ = 0.1 λ = 0.2 λ = 0.3 λ = 0.4 λ = 0.5 λ = 0.6 λ = 0.7 λ = 0.8 λ = 0.9 λ = 1 an overview of multiple criteria decision making techniques… 17 copyright ©2023 assa adv. in systems science and appl. (2023) lm1 0.938 0.939 0.939 0.940 0.940 0.941 0.941 0.941 0.942 0.942 0.943 lm2 0.712 0.715 0.718 0.721 0.724 0.727 0.730 0.733 0.737 0.740 0.743 lm3 0.959 0.959 0.959 0.959 0.960 0.960 0.960 0.961 0.961 0.961 0.962 lm4 0.755 0.758 0.761 0.764 0.766 0.769 0.772 0.775 0.777 0.780 0.783 lm5 0.885 0.885 0.886 0.887 0.888 0.888 0.889 0.890 0.891 0.892 0.892 lm6 0.879 0.881 0.882 0.884 0.886 0.888 0.890 0.891 0.893 0.895 0.897 lm7 0.831 0.833 0.836 0.839 0.842 0.845 0.848 0.850 0.853 0.856 0.859 (source: author’s own elaboration) from the above table 4.5 it can be clearly seen that, for λ = 0, the joint weightages are exactly same as the weighted product shown in table 4.4 since the first part of the equation (3.7) is eliminated in this case and on the other hand, when λ = 1, the second part of the equation (3.7) got eliminated so the joint weightages exactly matches with the weighted sum shown in table 4.3. 5. results and discussions the weighted sum, weighted product and the joint generalized weightages are determined for every alternative by applying wsm, wpm and waspas method which are presented in table 4.3, table 4.4 and table 4.5 respectively. now according to the decreasing magnitude of these weightages, three ranking order of the alternatives can be done for each and individual methods which are presented in table 5.1 (wsm and wpm) and table 5.2 (waspas). table 5.1. ranking of the alternatives by wsm and wpm wsm wpm alternatives weighted sum rank weighted product rank lm1 0.943 2 0.938 2 lm2 0.743 7 0.712 7 lm3 0.962 1 0.959 1 lm4 0.783 6 0.755 6 lm5 0.892 4 0.885 3 lm6 0.897 3 0.879 4 lm7 0.859 5 0.831 5 (source: author’s own elaboration) table 5.2. ranking of alternatives by waspas alternatives λ = 0 rank λ = 0.1 rank λ = 0.2 rank λ = 0.3 rank λ = 0.4 rank λ = 0.5 rank lm1 0.938 2 0.939 2 0.939 2 0.940 2 0.940 2 0.941 2 lm2 0.712 7 0.715 7 0.718 7 0.721 7 0.724 7 0.727 7 lm3 0.959 1 0.959 1 0.959 1 0.959 1 0.960 1 0.960 1 lm4 0.755 6 0.758 6 0.761 6 0.764 6 0.766 6 0.769 6 lm5 0.885 3 0.885 3 0.886 3 0.887 3 0.888 3 0.888 3 lm6 0.879 4 0.881 4 0.882 4 0.884 4 0.886 4 0.888 4 lm7 0.831 5 0.833 5 0.836 5 0.839 5 0.842 5 0.845 5 alternatives λ = 0.6 rank λ = 0.7 rank λ = 0.8 rank λ = 0.9 rank λ = 1 rank lm1 0.941 2 0.941 2 0.942 2 0.942 2 0.943 2 lm2 0.730 7 0.733 7 0.737 7 0.740 7 0.743 7 lm3 0.960 1 0.961 1 0.961 1 0.961 1 0.962 1 18 s.s. goswami, d.k. behera copyright ©2023 assa adv. in systems science and appl. (2023) lm4 0.772 6 0.775 6 0.777 6 0.780 6 0.783 6 lm5 0.889 4 0.890 4 0.891 4 0.892 4 0.892 4 lm6 0.890 3 0.891 3 0.893 3 0.895 3 0.897 3 lm7 0.848 5 0.850 5 0.853 5 0.856 5 0.859 5 (source: author’s own elaboration) from the above two tables 5.1 and 5.2, it can be clearly seen that all the three mcdm techniques are giving the same output results and suggest that model 3 is the best laptop model among the group followed by model 1 in the 2nd position whereas, model 2 is the worst choice. all the rankings are more or less same with very little alteration. there is a very tough competition between model 5 and model 6 for the 3rd and 4th place. from table 5.1, wsm is suggesting that model 6 should come in the 3rd place whereas, wpm is suggesting that model 5 should occupy that place. to remove this confusion waspas is applied and the variations in ranking for all the values of λ is observed. it is noticeable in table 5.2 that the ranking matches with the wpm ranking as the λ value varies from 0 to 0.4, since the second part of the equation (3.7) is given more importance in the favor of wpm. similarly, as the λ value varies from 0.6 to 1 the ranking shifted towards the wsm, since the second part of the equation (3.7) slowly tends to zero and took the favor of wsm. so, in waspas method equal importance should be given to each wsm and wpm method by keeping the value of λ as 0.5 and it is suggested to use equation (3.6) while calculating the joint generalized weights. hence, the ranking given by waspas method when λ = 0.5 can be accepted as the final ranking. it should be noted, that the joint weights for model 5 and model 6 are given as same i.e. 0.888 in table 5.2 under the column λ = 0.5 is due to round off error. if up to five decimal places are considered then for model 5 and model 6 the joint weights would be 0.88850 and 0.88775 which clearly shows that model 5 should occupy the 3rd position and model 6 should come in the 4th place. the present three rankings are also compared with the previous researchers results in table 5.3 and graphically in fig. 5.1 as proposed by adali and isik [2] using moora, multimoora and moosra methods. table 5.3. ranking comparisons of different mcdm methods alternatives wsm wpm waspas moora multimoora moosra final ranking lm1 2 2 2 2 2 2 2 lm2 7 7 7 7 7 7 7 lm3 1 1 1 1 1 1 1 lm4 6 6 6 6 6 6 6 lm5 4 3 3 3 3 3 3 lm6 3 4 4 4 4 4 4 lm7 5 5 5 5 5 5 5 (source: adali and isik, 2017; author’s own elaboration) an overview of multiple criteria decision making techniques… 19 copyright ©2023 assa adv. in systems science and appl. (2023) fig. 5.1. graphical comparisons of the proposed rankings (source: author’s own elaboration; created using ms word chart option) 6. conclusion from the above research study, it can be concluded that model 3 is the best laptop model followed by model 1 among these chosen seven alternatives and model 2 is the worst model to buy for the customers. the best and the worst choice of the alternatives suggested by all the six methods are exactly the same but, there is a very minor alternation in the wsm ranking regarding the 3rd and 4th places. since, the ranking of the rest five mcdm methods shown in table 5.3 exactly matches with each other so, the final ranking of the laptop models can be proposed as follows: model 3 > model 1 > model 5 > model 6 > model 7 > model 4 > model 2 limitations: there are lots of mcdm tools available whose implementation to the same problem may generate different output results and ranking. moreover, if different subjective or objective weightage calculation methods are used other than ahp like, criteria importance through inter criteria correlation (critic), entropy, best worst method (bwm) etc. it may also alter the final results. future scope: different mcdm methods can be applied to the above problem and the ranking can be compared with these proposed rankings. the criteria weightages can also be determined by other methods and the variation in results can be noted. not only these but also, mcdm tools can also be utilized as a decision tool in the selection of other electronic gadgets and household appliances e.g. television, washing machine, camera etc. acknowledgement we are grateful to igit sarang for providing us such a wonderful research environment. we are thankful to bput rourkela for their immense cooperation and support. we are also obliged to aicte india for their monthly financial assistance. 0 1 2 3 4 5 6 7 8 lm1 lm2 lm3 lm4 lm5 lm6 lm7 wsm wpm waspas moora multimoora moosra final ranking 20 s.s. goswami, d.k. behera copyright ©2023 assa adv. in systems science and appl. 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(2013). decision making on business issues with foresight perspective; an application of new hybrid mcdm model in shopping mall locating, expert systems with applications, 40(17), 7111–7121, https://doi.org/10.1016/j.eswa.2013.06.040. adv syst sci appl 2020; 04:36–44 published online at https://ijassa.ipu.ru. statistical properties of vanet-based information spreading imre varga1∗, gergely kocsis1 1university of debrecen, department of it systems and networks, debrecen, hungary abstract: in this work, we present a robust model of vehicular ad hoc networks (vanet) in order to study information spreading on such topologies. vehicles are moving along the fastest routes between their starting points and their destinations on a map derived from a real urban topology. vehicles can exchange information through the use of short-range wireless communication devices. the source of information is a roadside unit which provides packets for the nearby vehicles. these agents can carry and forward messages for others. as a result of the data dissemination, the major part of the system becomes informed quite quickly. presented results include the investigation of information spreading in the system, e.g. the time evolution of the average awareness, the age distribution of information owned by separate vehicles and the statistical properties of time intervals between information exchange events. it has been shown how the effectiveness of this complex system depends on the density of intelligent devices. scalefree behavior was found by time series analysis. our computer simulation results can help to design smartcity applications of the future. keywords: agent-based simulation, vanet, information diffusion, traffic simulation, time series analysis 1. introduction the spreading of information in vehicular networks plays a key role in many smart city services. because of this, the topic is in the focus of research in the last decade. the aim of these applications is to make urban traffic safer and comfortable. previous studies have analyzed the topological properties of urban road maps [1, 2] and the traffic flow was measured and studied [3] in some other works in order to increase the efficiency of these intelligent transportation systems. several different algorithms and methods were developed to simulate the motion of vehicles and generate traffic in urban or in highway environment [4–6]. the communication of moving wireless devices (possibly carried by members of these vehicular networks) may be described by using standardized communication protocols, like dedicated short range communication (dsrc) or ieee 802.11p standard [7–9]. in vanets (vehicular ad hoc network) both the routing [10–12] and the broadcasting [4,13] are actively investigated fields. distribution of information was also studied in different wireless systems, e.g. in mobile peer-to-peer systems [14] or self-organized sensor networks [15]. some questions related to the statistical properties of the general spreading processes in vanets are however still open. the goal of our research is to introduce and analyze a new framework in order to be able to answer some of the following questions. what are the limits of the information spreading? can we reach all actors of the traffic system based only on self-organization? do all vehicles own up-to-date information? similar questions have been appeared and already answered in ∗corresponding author: varga.imre@inf.unideb.hu statistical properties of vanet-based information spreading 37 social networks [16,17], but due to the continuously changing topology, the characteristics of spreading can be very different. in section 2 the construction of the realistic urban topology is shown. section 3 presents the details of the simulation of vehicular motion. information spreading based on carry-andforward and multihop broadcast dissemination schemes is presented in section 4 and then the first results of our studies are shown in section 5. at the end we close by some conclusions. 2. underlying map topology of simulations we applied a real city map, namely the map of our home city as the underlying network topology to reach a realistic simulation environment. the dataset describing the map was gained and is available at the page of the openstreetmap project [18]. since in this work we would like to use this map for agent-based simulation a much more simplified topology is needed. because of this the source was reduced keeping the topology of the crossroad network and the distances between junctions, but losing the real geographical locations of road sections. according to the original osm format any crooked road can be built up from shorter straight segments and the geographical coordinates of their endpoints. in this way, a road section between crossroads can be described by a list of internal nodes with degree 2. in our approach, the shape of a road section is negligible and only the length of the section is important. this was the base of our topology simplifying method. in case of any two road segments between nodes a−b and nodes b − c, node b was eliminated if it has no other neighbors than a and c, merging the segments to only one longer segment between nodes a− c with a distance equal to the sum of lengths of the previous two segments. after the above reduction process the resulting network can be analyzed from two different points of view. (i) taking a look to the network as an abstract topology (undirected graph) we can find that there are 3422 nodes (junctions) connected by 4812 links (road sections). it was found that 84% of these nodes have a degree of 3 or 4. in the unit of link number (ignoring road length) the diameter of the network is 96. (ii) from the geographical aspect however our network still has some spatial properties. taking into account the distances, the average distance between two crossroads is 121.5m, however, the distribution is quite wide, there are almost 3 orders of magnitude difference between the shortest and the longest road section. the average distance between two randomly chosen nodes is 4.1± 1.9km. (more details are available in [19, 20].) 3. motion of vehicles on the above described map vehicles are moving from their randomly chosen starting node toward their randomly chosen destination node along the fastest path. even though today many different navigation options are available to be considered, practice shows that drivers usually use a route with the shortest travel time instead of the shortest distance route, or the smallest number of left turns [21]. the original data set contains information about the rank of all road segments (e.g.: primary, secondary, residential, living street, etc.). the average speeds of cars depend on the rank of the road. based on the speed prediction/offer of the google maps [22], different average speed is applied in case of different road ranks. thus the shortest and the fastest route can be different. vehicles move with constant speed between two neighboring nodes, at a crossroad they turn according to their route (and perhaps change speed). traffic jams, traffic lights or the finite size of vehicles are not taken into account during the simulation because from the point of the later spreading process the short-term fluctuations of the speed of cars are negligible. thus, in this kind of mean-field approach, the velocity of vehicles is not influenced by the copyright c© 2020 assa. adv syst sci appl (2020) 38 i. varga, g. kocsis actual traffic situations. even though, the source and destination nodes are random the density of the traffic is really diverse due to the topology (connectivity, ranks). we assumed that the number of moving cars in the system at a given time can be constant since the simulated (typically few 10 minutes long) time interval is small compared to the daily life cycle of a city or the duration of rush-hour traffic. in this way, different scenarios (e.g. rush-hours or off-peak time) can be simulated separately using a distinct number of cars. at the beginning of the simulation, there are n vehicles in the system. later, when a vehicle arrives to its destination, it is removed and immediately a new one is initialized and started. at the beginning of the simulation, all the cars are just departed. it is easy to understand that in order to avoid artificial transient effects the measurement related to spreading is started only later (t = 0) when the system becomes randomized, however, the simulation of the traffic is started at t = −t0. the length of the randomization time interval (−t0 ≤ t < 0) is longer then most of the trips (t0 = 750s, average travel time is 459± 261s), so when the scientific observation is started all the initial cars have been arrived and others are launched in different time moments. the simulation is stopped at t = tmax. the system evolves in discrete time steps. the time step ∆t is small enough to move only a few meters, so it is tiny compared to the whole simulation time ∆t� t0 + tmax. the time interval of the analysis (0 ≤ t ≤ tmax) is long enough to cover several generations of vehicles. we found that in order to get reliable results the total number of simulated cars (nt) has to be at least five times greater than number of cars at a given moment (nt > 5n ). this approach of macroscopic traffic is quite simplified. the motion of vehicles is much slower than, spreading of information (detailed in the next section) due to the consecutive quick message forwarding. this fact allows us to neglect more details of the micro-level traffic. in addition to the approximate spatial distribution of agents, motion is only required to regularly change the communication topology. 4. spreading of information in the system, smart vehicles are represented by agents able to interact by short-range communication. if the distance of two vehicles at a given time moment is less than the range r of the wireless communication, they can exchange information. if agent i can receive information from agent j, communication to the opposite direction is also possible technically. based on this, in our model the agents can have two different states. on the one hand agent i can be uninformed, so it has not received any data (denoted by si = 0). on the other hand, it can be informed, so it has already got some data (denoted by si = 1). beside this vehicle-to-vehicle (v2v) communication, there is infrastructure-to-vehicle communication (i2v) as well. in the latter case the on board units (obu) of smart vehicles can receive information from road side units (rsu). in our model initially all agents are in uninformed state and only one rsu is present, playing the role of an information source. when an agent passes close enough to the rsu it receives new up-to-date public information (e.g. traffic or weather alert). the actual content of messages is negligible. the agent stores it together with the actual timestamp and later it shares with others within the communication range. if one of these neighboring agents is uninformed it becomes informed. if both agents that are in contact have been already informed, the agent with an older timestamp will update its knowledge storing the newer information with its given timestamp. thus information can spread in this dynamically changing network from the rsu to any vehicle even if they have never passed by the rsu. in order to characterize agent i in detail we introduce the quantity ti which is the latest/newest timestamp of information owned by the informed agent i or ti = −1 if agent i is uninformed. (so ti > 0 is the simulation time when the given information unit carried by agent i entered into the system by the rsu.) the behavior of the system is shown in fig. 4.1. a vehicle (agent i) proceeds from node a to node d. it goes by the rsu in node b receiving new information at t = t . an other vehicle (agent j) moves from node e towards node f . both of them are copyright c© 2020 assa. adv syst sci appl (2020) statistical properties of vanet-based information spreading 39 fig. 4.1. the behavior of the system. in the vicinity of the node c at the same time. since they are within the range r, agent i can transmits the information to agent j. between nodes a and b agent i is uninformed, but between b and d it is in an informed state, having timestamp t . agent j becomes informed at node c and possesses also timestamp t between nodes c and f . at simulation time t, an informed agent i has information with age ai = t− ti. the average age of information 〈a〉 owned by agents in a given time moment can be written as 〈a〉 = ∑ i tisi n inf , (4.1) where n inf is the number of informed agents, defined as n inf = ∑ i si. a large value of n inf/n indicates extensive information spreading. when the average age of information 〈a〉 is low, it means that our smart traffic system is in an up-to-date phase. thus the number of informed agents n inf and the average age of information 〈a〉 are good measures of the effectiveness of information spreading in vanet. 5. results since during the simulation an si (susceptible-infected) model [23] is applied, initially more and more agents become informed. however, the system never reaches a fully informed state, because during the simulation new, uninformed agents appear in the system, while informed ones disappear as they reach their destination. investigating the time evolution of the agents it was found that the system reaches a steady state described by saturating functions. in fig. 5.1 a, one can observe that at t = 0 (when the rsu is just activated) there are no informed agents in the system, but soon some agents pass by the information source of the infrastructure. then the vehicles carry the information during their motion to different places of the city meanwhile they behave as secondary information sources speeding up the spreading of information so leading to increasingn inf (t)/n function with a significant slope. after a quite short time period, spreading slows down resulting in saturation of the number of informed agents. the average movement of vehicles during a simulation step ∆t is the half of the applied range of communication r. (of course, increasing range r speeds up the spreading.) the reason why the system reaches this almost steady state is the fact that the propagation of information can be faster than the motion of vehicles. the saturation level depends on the number of agents (the density of smart vehicles in the city) and in most cases the n inf (t)/n curves never reach 1.0. this is shown in fig. 5.1 b. as one can observe the information coverage of vanet can be effective only if the number of smart vehicles copyright c© 2020 assa. adv syst sci appl (2020) 40 i. varga, g. kocsis 0.0 0.2 0.4 0.6 0.8 1.0 n in f /n 0 500 1000 1500 2000 t n=100 n=1000 n=10000 a) 0.0 0.2 0.4 0.6 0.8 1.0 n in f (t m a x )/ n 10 10 2 10 3 10 4 n b) fig. 5.1. a) number of informed agents (vehicles) as a function of time for different numbers of agents. after a short time period a saturation is achieved at a quite high value. b) the previous saturation level depends on the number of vehicles in the system (of course more smart vehicle leads to higher level of awareness). exceeds a given threshold (about few hundreds of vehicles in the case of the medium-sized city debrecen). even though in most cases the number of informed agents n inf is proved to be relatively high in the system, the really important questions are the following ones. how old is the average information? is the system in an up-to-date phase continuously? the average information age as a function of time 〈a〉 (t) can give the answers. as it is shown on fig. 5.2 a, most of the agents have relatively young information. recent information from rsu overwrites the system very quickly without any outer control. of course the saturation level of 〈a〉 (t) (far from the opening time period) is determined by the number of agents. more smart vehicles lead to a more up-to-date system. (see the fig. 5.2 b.) the average age of information is even less than the length of the time period needed to reach the saturation of the number of informed agents. 0 100 200 300 400 500 600 < a > 0 500 1000 1500 2000 t n=1000 n=10000 a) 0 200 400 600 < a > (t m a x ) 10 10 2 10 3 10 4 n b) fig. 5.2. a) the average age of information owned by the vehicles as a function of time. it shows saturation for different system size. b) the average age of information in the saturation phase decreases logarithmically with the number of vehicles, so a denser vehicle park in the city results in a more up-to-date system. the above averages describe the system on a macro scale. for microscale characterization, we analyzed the time intervals between information exchanges for all agents. the average information age can be low only if the time interval ∆trecv between two subsequent information receive events is short for each vehicle. as one can see in fig. 5.3 a, the copyright c© 2020 assa. adv syst sci appl (2020) statistical properties of vanet-based information spreading 41 1 10 -1 10 -2 10 -3 10 -4 p ( t r e cv ) 10 10 2 10 3 trecv n=5000 n=10000 n=15000 a) 1 10 -1 10 -2 10 -3 10 -4 p ( t f o rw ) 10 10 2 10 3 tforw n=5000 n=10000 n=15000 b) fig. 5.3. the distribution of the average message receiving (a) and forwarding (b) time interval of vehicles. they obey a power law with exponents around 3.4 (a) and 2.5 (b) respectively, independently of the vehicle density. distribution of ∆trecv has a power-law form: p(∆trecv) ∼ ∆trecv −γrecv . this means that most of the agents frequently receive new information packets, but some of them keep old information for relatively long time. the density of vehicles has no effect on the value of the exponent which is close to γrecv ≈ 3.4. the distribution of the time intervals between message forwards ∆tforw also proved to be scale-free, however its exponent γforw is definitely lower, it is around 2.5. (see fig. 5.3 b.) one can ask why this difference in the distributions appear even though the number of receive and forward events are the same. this can be explained however by conditions of information exchange. when vehicles approach each other within the communication range, only the up-to-date agent can forward its recent information. in this way, an agent, who has a quite old information packet has to wait a long time to get a chance to forward it. contrarily old information can be overwritten by almost any other message, so the high receiving chance leads to very few old information to overwrite. on micro scale, one can follow the spreading of each information holding any given timestamp t . agents receive new information from the rsu at time t = t . then it is transmitted to other vehicles, so the number of agents nt having the same timestamp t is increasing. sooner or later they will be updated so nt (t) is decreasing, finally all of these information units will disappear at t = t + tl, where tl denotes the lifetime of the given information. note that tl is not the lifetime of a given information packet, but the time interval when the given information (with a given timestamp) is present somewhere in the system. the time evolution of nt (t) is qualitatively similar for all timestamps, however quantitatively they are very different. the general form of thent (t) function can be obtained by rescaling and averaging all the separate curves. the result is illustrated in fig. 5.4 a. it shows that at the beginning information spreads very quickly but after te time the expansion reaches its maximum, where m = nt (t + te) is the maximal number of agents carrying the information originated at time t . after the expansion, we found a dying out phase with duration td = tl − te . the whole average lifetime 〈tl〉 of information in a system depends on the number of vehicles proceeding in the town. see fig. 5.4 b. the curve has a minimum. in case of small systems, more agents result in lower lifetime tl due to the increasing number of updates (younger information) related to fig. 5.2 b. in order to understand the increasing regime, we have to study the maximum of nt (t− t ) denoted by m . figure 5.4 c illustrates that for large system 〈m〉/n increases linearly with n , so an increasing proportion of agents is carrying the given timestamp in the whole population. if there are more smart vehicles in the city due to the fast spreading a few information timestamps dominate. there can be hundreds of different information in the system, however a dominant part of agents carries copyright c© 2020 assa. adv syst sci appl (2020) 42 i. varga, g. kocsis 0 20 40 60 80 n t 0 40 80 120 t-t te tl m a) 0 100 200 300 400 500 600 < t l > 0 3000 6000 9000 12000 15000 n 0 3000 6000 9000 12000 15000 n b) 0.0 0.02 0.04 0.06 < m > /n 0 7000 14000 n c) 0.0 0.2 0.4 0.6 0.8 < t e > / < t l > 0 3000 6000 9000 12000 15000 n d) fig. 5.4. a) the average time evolution of the number of vehicles carrying information with a given timestamp (t ) for n = 5000. a shorter expansion period te is followed by a longer dying out period separated by a maximum m at time t = t + te . b) the average lifetime of information with a given timestamp as a function of the number of cars. increasing vehicle density first leads to decreasing lifetime, but above a certain number of vehicles, given information can present in the system for a longer time interval. c) the maximal ratio of vehicles carrying given information with timestamp t at t = t + te is increasing linearly for large systems. solid, gray line just guides the eyes indicating linear dependence. d) the ratio of the expansion period and the lifetime as a function of the number of agents n . the expansion phase is shrinking by increasing the number of agents (indicating speeding up of the spreading since m is increasing). only a few timestamps. in case of a given timestamp the number of agents carrying it can be a few percent of the population, so hundreds of vehicles. the time needed to overwrite all of them is long, thus the average lifetime of timestamps 〈tl〉 can increase with n . the expansion phase is usually shorter than the dying out phase, but their ratio is not fixed. the 〈te〉/〈tl〉 ratio as a function of the n number of smart vehicles can be characterized by an almost linear decreasing curve in the studied systems. (see fig. 5.4 d.) in case of n = 103 agents, more than 40% of the lifetime of the average timestamp is in the expansion phase, while in case of n = 104 only 13% of the lifetime is spent in the expansion period. a crowded traffic system results in a short and really intensive spread of new information and it is followed by a long disappearing section with the presence of a few vehicles having old (maybe no longer valid) information. copyright c© 2020 assa. adv syst sci appl (2020) statistical properties of vanet-based information spreading 43 6. summary in this work, we presented an agent-based model of information spreading on vehicular ad hoc networks. the time-dependent network topology of agents was based on the motion of communicating smart vehicles. vehicles are moving based on shortest travel time paths between the randomly selected starting and destination points of a real city map. due to the short-range communication moving vehicles can receive public information from each other or from a fixed road side unit. in this ad hoc network, the statistical properties of information spreading were investigated. above a certain number of smart vehicles in the system information spreads very fast, and a dominant part of the system can be in an almost homogeneous informed state. this efficient dissemination of information does not require considerable computational power of devices because there is no addressing or routing process and the devices store only the latest packages. however, we found that some agents can stay in an out-of-date state for a quite long time. the number of smart vehicles has a huge effect on spreading, only a large self-organized system can be effective. on micro scale, we analyzed the time intervals between information exchanges for all agents. we found that the distribution of these intervals has a power-law form where the density of vehicles has no effect on the exponents. the evolution of the spreading of each information holding any given timestamps was also analyzed. we found out that the lifetime of information can be separated into two periods: an expansion period and a dying out period. we found that the number of agents in these complex systems may affect the information spreading and the ratio of these two periods in significant and well describable ways. all these results claim appropriate treating of out-of-date vehicles. the presented results have shown that the system has many interesting features, although real-life applications require more realistic simulations. in our further research, we try to find answers to essential, practical questions. what happens if the rsu is removed (turned off) or more than one such units are placed to the system? how to avoid the presence of old (out-of-date, fake) information? what is the effect of the introduction of a susceptible-infected-susceptible (sis) model [23] (forgetting old information)? how to optimize spreading reducing the number of information exchanges (for energy efficiency), but keeping the system in an up-to-date phase? what is the topology of this ad hoc communication network? acknowledgements the publication is supported by the efop-3.6.1-16-2016-00022 project. the project is co-financed by the european union and the european social fund. this work was supported by the construction efop-3.6.3-vekop-16-2017-00002. the project was supported by the european union, co-financed by the european social fund. map data copyrighted openstreetmap contributors and available from https://www.openstreetmap.org. references 1. porta, s., crucitti, p., latora, v. (2006). the network analysis of urban streets: a dual approach, physica a, 369, pp. 853–866. 2. jiang, b. (2007). a topological pattern of urban street networks: universality and peculiarity, physica a, 384, pp. 647–655. 3. yan, y., zhang, s., tang, j., wang x. (2017). understanding characteristics in multivariate traffic flow time series from complex network structure, physica a 477, pp. 149–160. copyright c© 2020 assa. adv syst sci appl (2020) 44 i. varga, g. kocsis 4. zeadally, s., hunt, r., chen, y. s., irwin, a., hassan, a. (2012). vehicular ad hoc networks (vanets): status, results, and challenges, telecommunication systems, 50, pp. 217–241. 5. fiore, m., härri, j., filali, f., bonnet, c. (2007). vehicular mobility simulation for vanets, 40th annual simulation symposium (anss ’07), norfolk, pp. 301–309. 6. bátfai, n., besenczi, r., mamenyák, a., ispány, m. 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(2020). a complex sis spreading model in ad hoc networks with reduced communication efforts, advances in complex systems (accepted). 21. stephan winter (2002). modeling costs of turns in route planning, geoinformatica, 6 (4), pp. 345-361. 22. google maps (2018). [online]. available https://maps.google.com. 23. m. e. j. newman, (2010). networks an introduction, oxford university press, pp. 627–675. copyright c© 2020 assa. adv syst sci appl (2020) introduction underlying map topology of simulations motion of vehicles spreading of information results summary adv syst sci appl 2021; 04:65–86 published online at https://ijassa.ipu.ru. dynamics of covid-19 outbreak and optimal control strategies: a model-based analysis naba kumar goswami1*, b. shanmukha2 1department of mathematics, pet research foundation, mysuru, india 2department of mathematics, pes college of engineering, mandya, india abstract: covid-19 is an infectious disease caused by the sars-cov-2 virus, which spreads so fast in the inhabitants. the virus is transmitted through direct contact with respiratory droplets of an infected individuals through coughing and sneezing or indirect contact through contaminated objects or surface. in this article, a non-linear mathematical model is proposed and analyzed to manifest the impact of transmission dynamics of the covid-19 pandemic based on indian condition by considering asymptomatic and symptomatic infections. it is assumed that the transmission rates due to asymptomatic and symptomatic individuals are different. the basic reproduction number of the model is computed and studied the stability of different equilibria of the model in detail. the sensitivity analysis is presented to identify the key parameters that influence the basic reproduction number, which can be regulated to control the transmission dynamics of the disease. also, this model is extended to the optimal control model and is analyzed by using the pontryagin’s maximum principal and solved numerically. it has been observed that the optimal control model gives better result as compacted to the model without optimal control model as it reduces the number of infectives significantly in a desired interval of time. keywords: covid-19, basic reproduction number, stability analysis, sensitivity analysis, optimal control 1. introduction the ongoing covid-19 outbreak has put mathematical models in the limelight. in 1960, the human coronavirus was first identified and in 21st century, a large number of people in this world have been affected by the three outbreaks such as sars, mers, and 2019-ncov. in 2003, the severe acute respiratory syndrome(sars) outbreaks, especially in the chinese mainland, hong kong, taiwan, and canada of the world [1] and in 2012 and 2015, the middle east respiratory syndrome(mers) outbreak in saudi arabin [2] and south korea [3] respectively. coronaviruses belonging to the family of coronaviridae and order of nidovirales, enveloped, non-segmented, single-stranded positive-sense rna viruses [4]. all coronavirus are zoonotic. they start in animals and can then, following mutation, recombination, and adaptation, be passed on to humans. in the human-to-human transmission of covid-19 can occurs via respiratory droplets directly (through droplets from coughing or sneezing) or indirectly (touching surfaces or objects contaminated with virus and touching their mouth, nose, eyes) [5]. the incubation period for the new coronavirus from 2 to 14 days in human to human transmission [6]. the common symptoms of covid-19 are fever, cough, difficulty of breathing, and fatigue. at present, there are no specific drugs for the disease to protect the people, only hygiene measures can reduce the ∗corresponding author: nabakrgoswami@gmail.com 66 n.k. goswami, b. shanmukha rate of transmission. according cdc covid-19 vaccination is a safer way to help build protection measure. covid vaccines can stop or reduce most of the people from getting sick, but not everyone. despite taking all recommended doses of vaccines and waits a few weeks for immunity to build up, still there is a chance that people can get infected. so far there is no evidence, however, that human coronaviruses can be transmitted by animals. in december 2019, a new outbreak of pneumonia of unknown cause has been identified in wuhan city, the capital of china’s hubei province [9]. this outbreak has some other potential causes like influenza, avian influenza, adenovirus, and sars, but there are no symptoms like mers [7]. later on 7th january, 2020, the causative pathogen was identified as a novel coronavirus (2019-ncov) [8]. as the virus is very closely related to sars and mers, so the name 2019-ncov can distinguish the virus from both [9]. on 30th january 2020, the world health organization (who) declared that the outbreak is a public health emergency of international concern(pheic) [10] as it is worldwide spread. as the number infective is increasing rapidly and the epidemiological evidence of human-to-human (doorplates, direct contact, etc.) transmission suggests that 2019-ncov is more contagious than both sars and mers [11–13]. on 11th february 2020, the world health organization announced a new name of coronavirus disease as covid-19 [14]. on 11th march 2020, the world health organization officially declares the covid-19 outbreak as a pandemic as it spread worldwide. the most affected countries in this world are the usa, india, brazil, italy, spain, france, uk, germany, iran, china, belgium, netherlands, etc. in this ongoing outbreak, in the usa more than 43,107,628 people are infected and more than 694,619 people have died. presently more than 229,382,253 infected cases have been reported across 223 territories, where more than 206,052,168 infected cases are recovered and more than 4,707,336 infected cases have been died till 20th sept 2021. in india, the first novel coronavirus case was reported on 30th january 2020 [15] in the state of kerala. due to the crises of coronavirus pandemic, prime minister of india declared 21 + 19 = 40 days nationwide lockdown from 25th of march to 3rd of may 2020 as a preventive measure for the covid-19 [16]. the ministry of health and family welfare of india has suggested various precautionary measures to prevent the spread of viruses such as washing hands frequently, physical/social distancing, wearing mask, avoiding touching face, nose, and eyes [17, 18]. apart from lockdown the government of india performing many awareness programs about preventative measures through media and social networks (tv, radio, newspaper, facebook, twitter, etc.). the effect of lockdown and social distancing play an important role to reduce the coronavirus infection. in 2020, total number of infected cases are recorded in india are 10,266,674 and january to september 2021, number infected cases are 23,211,745. in 2021, also statewide or locality wise lockdown declared according to basis of active cases in that particular state or region. on 20th september, 2021, more than 33,478,419 infected covid-19 cases have been reported in 32 states in india, while 32,715,105 cases are recovered and 445,165 many cases have been died. most affected states in india are maharashtra, karnataka, kerla, delhi, gujarat, rajasthan, madhya pradesh, uttar pradesh, tamil nadu, andhra pradesh, telangana, west bengal, jammu and kashmir. among the cities in india, mumbai, bangalore and delhi are the badly effected by covid-19 outbreak. the primary aimed to study this model to study indian conditions and will forecast future pandemic by using information available. in [1] authors presented a deterministic model and simplified from the seijr model, which is adapted to analyze the important parameters of the model of sars epidemic; in [19] authors constructed a seqijr model of epidemic disease transmission which includes immunization and varying population size is studied; in [20] authors studied a mathematical about early transmission dynamics of the infection and evaluating the effectiveness of control measures; in [21] authours proposed and studied a copyright © 2021 assa. adv syst sci appl (2021) dynamics of covid-19 outbreak and optimal control strategies: a model-based... 67 model on the various impact of the intervention on the spread of covid-19 in india; in [22] authors discussed a data-driven analysis in the early phase of the outbreak. they estimated the basic reproduction number of novel coronavirus (2019-ncov) in china; in [23] authors develop a mathematical model for the spread of the coronavirus disease 2019 (covid-19); in [24] authors developed a bats-hosts-reservoir-people transmission network model for simulating the potential transmission from the infection source to the human infection. in [31] authors proposed a mathematical model and they focuses on the impacts of face mask, hospitalization of symptomatic individuals and quarantine of asymptomatic individuals on the transmission dynamics of covid-19 pandemic in india. this paper is organized as follows: in section 2 presents the model; in section 3 discussed existence of equilibria and basic reproduction number; in section 4 presents the stability analysis of the model; in section 5 deal with sensitivity analysis of basic reproduction number; in section 6 illustrate the effects of parameters on disease outbreak ; in section 7 we extend the model to optimal control model and analysis it. demonstrates the numerical simulation results of the optimal control model; finally, section 7 we conclude the paper. 2. the model in this section, a dynamic model for covid-19 pandemic is presented and discussed based on india condition. the model divides the total population n(t) = s + e + ia + is +h +r into six different compartments according to the nature of the disease such as susceptible individuals (s), exposed individuals (e), asymptomatic infective individuals (ia), symptomatic infective individuals (is), hospitalized individuals (h) and recovered individuals (r). it is assumed that the total population is varying and homogeneously mixed i.e., all people are equally likely to be infected by the infectious individuals if they come into contact. the individuals are employed in the province at a constant rate λ and join the susceptible class. the natural birth and deaths in the population are also considered in the model. it is assumed that susceptible individuals after being exposed to the covid19 infection can progress to asymptotic infective and symptomatic infective at the rates β. assume that an asymptomatic individual joins the symptomatic populations class at the rate ρ. further, both asymptomatic and symptomatic infectious individuals will progress to the hospitalized or quarantine compartment with clinical symptoms of covid-19 at the rate δ1 and δ2 respectively. also, some asymptotic and symptomatic individuals may recover without hospitalized or quarantine at the rates γ1 and γ2 respectively. hospitalized individuals may recover and after recovery, it progresses to recovered class at the rate γ3. however, the rates of recovery may vary from one compartment to another. the natural mortality rate of each individuals class is µ. due to critical illness of covid-19 disease some of the symptomatic and hospitalized individuals class have additional mortality rates µ1 and µ2, respectively. the flow diagram of the model is given in figure 1, and biological interpretations of parameters are shown in table 1. keeping the above facts/assumptions in mind, a mathematical model covid-19 is proposed as follows: ds dt = λ− βasia − βssis − µs de dt = βasia + βssis − (κ+ µ)e dia dt = ξκe − (ρ+ γ1 + µ+ δ1)ia dis dt = (1− ξ)κe + ρia − (γ2 + µ1 + µ+ δ2)is (1) copyright © 2021 assa. adv syst sci appl (2021) 68 n.k. goswami, b. shanmukha dh dt = δ1ia + δ2is − (γ3 + µ2 + µ)h dr dt = γ1ia + γ2is + γ3h − µr fig. 1. flow diagram of the model. table 1. biological interpretations of parameters parameter biological interpretations λ : rate of recruitment in the susceptible class, βa : rate of infection of susceptible with asymptomatic individuals βs : rate of infection of susceptible with symptomatic individuals κ : rate of incubation ρ : rate of progression from asymptomatic to symptomatic ξ : fraction of exposed individuals not showing symptoms δ1 : hospitalized/quarantine rate of asymptomatic individuals δ2 : rate of hospitalized symptomatic individuals γ1 : recovery rate of asymptomatic individuals γ2 : recovery rate of symptomatic individuals γ3 : recovery rate of hospitalized individuals µ : natural mortality rate of human µ1 : disease related mortality rate for symptomatic individuals µ2 : disease related mortality rate for hospitalized individuals copyright © 2021 assa. adv syst sci appl (2021) dynamics of covid-19 outbreak and optimal control strategies: a model-based... 69 3. analysis of the model system from the system (1), we have ds dt ∣∣∣ s=0 = λ > 0, de dt ∣∣∣ e=0 = βasia + βssis ≥ 0, dia dt ∣∣∣ ia=0 = ξκe ≥ 0 dis dt ∣∣∣ is=0 = (1− ξ)κe + ρia ≥ 0, dh dt ∣∣∣ h=0 = δ1ia + δ2is ≥ 0, dr dt ∣∣∣ r=0 = γ1ia + γ2is + γ3h ≥ 0 here, all the rates are non-negative on the bounding planes. so, if we start in the interior of the 6-dimensional closed hyperoctant r6 +, we will always remain there, in view of the fact that the direction of the vector field is inward on all the bounding planes. thus, non-negativity of all the solutions of the model system (1) is guaranteed. further, from the model system (1), we note that the total human population n = s1 + s2 + i +h +q+r satisfies, dn dt = λ− µn − µ1is − µ2h this gives lim sup t→∞ n ≤ λ µ therefore, all the solutions s(t), e(t), ia(t), is(t), h(t), r(t) are bounded by λ µ . hence, the biologically feasible region for the system (1) is given by the following positively invariant set: ω = {(s,e, ia, is, h,r) ∈ r6 + : 0 ≤ s + e + ia + is +h +r ≤ λ µ } 3.1. basic reproduction number the disease-free equilibrium for the model (1) as e0 = (s0, e0, i0 a , i 0 s , h 0, r0) = ( λ µ , 0, 0, 0, 0, 0). we find the basic reproduction number r0 by using the next generation matrix method [26]. the new infection terms of the matrix f and the transition terms of the matrix v of the system (1) are respectively, as follows: f =  βasia + βssis 0 0 0  , v =  (κ+ µ)e −ξκe + (ρ+ γ1 + δ1 + µ)ia −(1− ξ)κe − ρia + (γ2 + δ2 + µ1 + µ)is −δ1ia − δ2is + (γ3 + µ2 + µ)h  now, we find the matrices f (of new infection terms) and v (of the transition terms) as copyright © 2021 assa. adv syst sci appl (2021) 70 n.k. goswami, b. shanmukha f =  0 βas βss 0 0 0 0 0 0 0 0 0 0 0 0 0  , v =  κ+ µ 0 0 0 −ξκ ρ+ γ1 + δ1 + µ 0 0 −(1− ξ)κ −ρ γ2 + δ2 + µ1 + µ 0 0 −δ1 −δ2 γ3 + µ2 + µ  it follows that fv −1 =  m11 m12 m13 0 0 0 0 0 0 0 0 0 0 0 0 0  where, m11 = κξβaλ µ(κ+ µ)(ρ+ γ1 + δ1 + µ) + βsλκ(ξρ+ (1− ξ)(ρ+ γ1 + δ1 + µ)) µ(κ+ µ)(ρ+ γ1 + δ1 + µ)(γ2 + δ2 + µ1 + µ) m12 = βaλ µ(ρ+ γ1 + δ1 + µ) − ρβsλ µ(ρ+ γ1 + δ1 + µ)(γ2 + δ2 + µ1 + µ) , m13 = βsλ µ(γ2 + δ2 + µ1 + µ) the basic reproduction number is same as the spectral radius of the next-generation matrix fv −1. thus, from above, we obtain the expression for r0 as r0 = κλ µ(κ+ µ)(ρ+ γ1 + δa + µ) [ ξβa + βs{ξρ+ (1− ξ)(ρ+ γ1 + δ1 + µ)} γ2 + δ2 + µ1 + µ ] the quantity r0 is known as basic reproduction number, the expected number of secondary cases produced in completely susceptible population, by a typical infective individual for the system (1). 3.2. existence of endemic equilibrium point the endemic equilibrium of the system (1) satisfies the following algebraic equations such as ds dt = 0, de dt = 0, dia dt = 0, dis dt = 0, dh dt = 0, dr dt = 0 the system (1) realize a unique positive solution e1 = (s∗, e∗, i∗a , i ∗ s , h ∗, r∗) s∗ = κ+ µ (βad1 + βsd2) , e∗ = λ(βad1 + βsd2)− µ(κ+ µ) (βad1 + βsd2)(κ+ µ) , i∗a = ξκe∗ ρ+ γ1 + δ1 + µ = d1e ∗, i∗s = (1− ξ)κe∗ + ρd1e ∗ γ2 + δ2 + µ1 + µ = d2e ∗, h∗ = (δ1d1 + δ2d2)e∗ γ3 + µ2 + µ = d3e ∗, r∗ = (γ1d1 + γ2d2 + γ3d3)e∗ µ where, d1 = κξ ρ+ γ1 + δ1 + µ , d2 = (1− ξ)κ+ ρd1 γ2 + δ2 + µ1 + µ , d3 = (δ1d1 + δ2d2) γ3 + µ2 + µ copyright © 2021 assa. adv syst sci appl (2021) dynamics of covid-19 outbreak and optimal control strategies: a model-based... 71 4. stability analysis of the model theorem 4.1: for model system (1), the disease-free equilibrium e0 is locally asymptotically stable if r0 < 1 and unstable if r0 > 1. 4.1. global stability of disease-free equilibrium to prove the global stability of disease-free equilibrium, we are using the theorem by castillochavez et al. [25] theorem 4.2: if the given mathematical model can be written in the form: dx dt = f (x, y ), and dy dt = g(x, y ), g(x, 0) = 0 (2) where x = s , y = (e, ia, is, h)t , denoting the number of uninfected and denoting the number of covid-19 infected people respectively. then the disease-free equilibrium is represented here by e0 = (x0, 0) = ( λ µ , 0) for the global asymptotically stable, the condition (h1) and (h2) given below must be satisfied. h1 : for dx dt = f (x0, 0), h2 : g(x, y ) = ay − ĝ(x, y ), ĝ(x, y ) ≥ 0, here a = dyg(x0, 0) is m-matrix (in m-matrix, all the off diagonal element of matrix are non-negative) . if the given system of differential equation in mathematical model satisfies the given condition in (2) then the point e0 = (x0, 0) is a global asymptotically stable equilibrium of given mathematical model provided r0 < 1. and for the given mathematical model, the result is shown in the next theorem, as given below. theorem 4.3: the point e0 = (x0, 0) of the system (1) is global asymptotically stable (g.a.s.), provided r0 < 1. and the condition given in (2) are satisfied. proof by using theorem (2.1) to our model system (1), we get f (x0, 0) = λ− µs, g(x, y ) = ay − ĝ(x, y ) where, a = f − v =  −(κ+ µ) βas βss 0 ξκ −(ρ+ γ1 + δ1 + µ) 0 0 (1− ξ)κ ρ −(γ2 + δ2 + µ1 + µ) 0 0 δ1 δ2 −(γ3 + µ2 + µ)  copyright © 2021 assa. adv syst sci appl (2021) 72 n.k. goswami, b. shanmukha then g(x, y ) = ay − ĝ(x, y ) = ay −  ĝ1(x, y ) ĝ2(x, y ) ĝ3(x, y ) ĝ4(x, y )  = (s0 − s)(βaia + βsis) 0 0 0  where ay =  −(κ+ µ)e + s0(βaia + βsis) ξκe − (ρ+ γ1 + δ1 + µ)ia (1− ξ)κe + ρia + (γ2 + δ2 + µ1 + µ)is δ1ia + δ2is + (γ3 + µ2 + µ)h  and ĝ(x, y ) =  (κ+ µ)e + s(βaia + βsis) ξκe − (ρ+ γ1 + δ1 + µ)ia (1− ξ)κe + ρia + (γ2 + δ2 + µ1 + µ)is δ1ia + δ2is + (γ3 + µ2 + µ)h  here, we can easily see s0 ≥ s, hence g(x, y ) ≥ 0 for all (x, y ). also by the defination of m matrix we can say that the matrix a is m matrix. hence, disease-free equilibrium (e0) is global asymptotically stable. 4.2. global stability of endemic equilibrium theorem 4.4: the endemic equilibrium e1 = (s∗, e∗, i∗a , i ∗ s , h ∗, r∗) of the given mathematical model is globally asymptotically stable. proof for the global stability result, we will use the method discussed in korobeinikov and wake [27], li and muldowney [28, 29]. here we consider the following lyapunov function: l = c1 ( s − s∗ − s∗ln s s∗ ) + c2 ( e − e∗ − e∗ln e e∗ ) + c3 ( ia − i∗a − i∗a ln ia i∗a ) +c4 ( is − i∗s − i∗s ln is i∗s ) then the time derivative of l is given by dl dt = c1 ( 1− s∗ s ) ds dt + c2 ( 1− e∗ e ) de dt + c3 ( 1− i∗1 i1 ) di1 dt + c4 ( 1− i∗2 i2 ) di2 dt now from the mathematical model we put the expressions for ds dt , de dt , dia dt , dis dt , in the above equation, which gives dl dt = c1 ( 1− s∗ s ) {λ− βaias − βsiss − µs} + c2 ( 1− e∗ e ) {βaias + βsiss − (κ+ µ)e} + c3 ( 1− i∗a ia ) {ξκe − ρia − (γ1 + µ)ia − δ1ia} copyright © 2021 assa. adv syst sci appl (2021) dynamics of covid-19 outbreak and optimal control strategies: a model-based... 73 + c4 ( 1− i∗s is ) {(1− ξ)κe + ρia − (γ2 + µ1 + µ)is − δ2is} (3) the mathematical model system satisfies the following relation at the equilibrium point. λ = βai ∗ as ∗ + βsi ∗ ss ∗ + µs∗, κ+ µ = βai ∗ as ∗ + βsi ∗ ss ∗ e∗ , ρ+ γ1 + δ1 + µ = ξκe∗ i∗a , γ2 + δ2 + µ1 + µ = (1− ξ)κe∗ + ρi∗a i∗s putting all the above expressions in (3) we get, dl dt = c1 ( 1− s∗ s ) [βai ∗ as ∗ + βsi ∗ ss ∗ + µs∗ − βaias − βsiss − µs] + c2 ( 1− e∗ e )[ βaias + βsiss − ( βai ∗ as ∗ + βsi ∗ ss ∗ e∗ ) e ] + c3 ( 1− i∗a ia )[ ξκe − ( ξκe∗ i∗a ) ia ] + c4 ( 1− i∗s is )[ (1− ξ)κe + ρia − ( (1− ξ)κe∗ + ρi∗a i∗s ) is ] then, dl dt = −c1 ( (s∗ − s)2 s ) µ+ c1 [( 1− s∗ s ) {βai∗as∗ + βsi ∗ ss ∗ − β1ias − β2iss} ] + c2 ( 1− e∗ e )[ βaias + βsiss − ( βai ∗ as ∗ + βsi ∗ ss ∗ e∗ ) e ] + c3 ( 1− i∗a ia )[ ξκe − ( ξκe∗ i∗a ) ia ] + c4 ( 1− i∗s is )[ (1− ξ)κe + ρia − ( (1− ξ)κe∗ + ρi∗a i∗s ) is ] dl dt = −c1 ( (s∗ − s)2 s ) µ+ g(x1, x2, x3, x4) where, s s∗ = x1, e e∗ = x2, ia i∗a = x3, is i∗s = x4, βas ∗i∗a = p, βss ∗i∗s = q, ξκe∗ = r, (1− ξ)κe∗ = s now, g(x1, x2, x3, x4) = c1(p+ q − px1x3 − qx1x4)− c1p 1 x1 − c1q 1 x1 + c1px3 + c1qx4 + c2(px1x2 + qx1x4 − ax2 − qx2)− c2p ( x1x3 x2 ) − c2q ( x1x4 x2 ) copyright © 2021 assa. adv syst sci appl (2021) 74 n.k. goswami, b. shanmukha + c2(p+ q) + c3(rx2 − rx3 − r x2 x3 + r) + c4(sx2 − sx4 − s x2 x4 + s) = x2(−c2p+ c3r + c4s) + x3(c1p− c3r) + x4(c1q − c4s) + x1x3(−c1p+ c2p) + x1x4(c1q + c2q) + c1(2p+ q) + c2(p+ q) + c3r − c3r ( x2 x3 ) + c4s− c4s ( x2 x4 ) − c1(p+ q) ( 1 x1 ) − c2(px1x3 + qx1x4) ( 1 x2 ) to get the values of c1, c2, c3, c4 we take the coefficients of x1x3, x1x4, x2, x3, x4 equal to zero and solve the algebraic equations in c1, c2, c3, c4. this gives c1 = c2; c3 = c1p r ; c4 = c1q s choosing c1 = c2 = 1 and n = 1, we get g(x1, x2, x3, x4) = p ( 3− 1 x1 − x1x3 x2 − x2 x3 ) + q ( 3− 1 x1 − x1x4 x2 − x2 x4 ) since the arithmetic mean is greater than or equal to geometric mean, we have 1 x1 + x1x3 x2 + x2 x3 ≥ 3 and 1 x1 + x1x4 x2 + x2 x4 ≥ 3 hence, dl dt = − ( (s∗ − s)2 s ) µ+ p ( 3− 1 x1 − x1x3 x2 − x2 x3 ) + q ( 3− 1 x1 − x1x4 x2 − x2 x4 ) thus it is easy to observe that dl dt ≤ 0 and the equality dl dt = 0 hold on;y for x1 = x2 = x3 = x4 = 1 for which s = s∗, e = e∗, ia = i∗a , is = i∗s . from the lasalle’s invariance principle [30], the equilibrium e1 of the given system is globally asymptotically stable for r0 > 1. 5. sensitivity analysis of reproduction number in this section, we also perform sensitivity analysis for the parameters involved in reproduction number r0, which reflects that increase or decrease in these parameter causes increase or decrease in r0. the sensitivity of r0 to different parameters is shown in figure 3. it is used to discover the parameters that have a high impact on r0 and should be targeted by intervention strategies. sensitivity indices allows to measure the relative change in a variable when parameter changes. for that we use the forward sensitivity index of a variable, with copyright © 2021 assa. adv syst sci appl (2021) dynamics of covid-19 outbreak and optimal control strategies: a model-based... 75 respect to a given parameter, which is defined as the ratio of the relative change in the variable to the relative change in the parameter. if such variable is differentiable with respect to the parameter, then the sensitivity index is defined using partial derivatives. the normalized forward sensitivity index of r0, which is differentiable with respect to a given parameter α, is defined by y r0 α = α r0 ∂r0 ∂α the above formula can be used to compute the analytical expression for the sensitivity of r0 to each parameter that it includes. accordingly, the sensitivity indexes of the model (1) are illustrate in figure 2. consequently, the value of r0 increases with increase in the values of all positive indices parameters λ, βa, βs, κ, and ξ with r0. also, the parameters γ1, γ2, δ1, δ2, ρ, µ, and µ1 have negative index with r0. it is clearly observed that the effect of the parameter λ is the maximum and hence it is the most sensitive parameter of r0. it means that small change (increase or decrease) in the parameters (λ) will significant change in the value of r0 by 100%. it is obvious that phenomenon of a lower value of r0 will boost to prevent the disease prevalence. thus, to control the disease from the population, we have to control the increase of parameters having positive indices with r0, whereas parameters with negative indices should be maintained. 6. effects of parameters on disease outbreak for the numerical simulation of the model, we consider all the parameters are in per day basis. first we consider the following set of parameters which corresponds to disease-free equilibrium. λ = 2; βa = 0.00085; βs = 0.00029; γ1 = 1/14; γ2 = 0.00245; γ3 = 0.002;κ = 1.19; µ1 = 0.1;µ2 = 0.015;µ = 0.00425; ρ = 0.001; ξ = 0.0015; δ1 = 0.16; δ2 = 0.01; for the above set of parameters we get r0 = 0.0932 < 1 and the disease-free equilibrium point e0(462.52, 0, 0, 0, 0, 0) is stable. this fact is demonstrated in figure 4 (a). later, we did few changes for endemic equilibrium as follows: λ = 40; βa = 0.00085; βs = 0.00029; γ1 = 1/14; γ2 = 0.000245; γ3 = 0.002; κ = 0.89; µ1 = 1.05; µ2 = 0.015; µ = 0.0125; ρ = 0.69; ξ = 0.95; δ1 = 0.012; δ2 = 0.019; from the above set of parameter we get r0 = 2.0075 > 1, and the endemic equilibrium e1(801.56, 26.82, 25.04, 24, 43, 38.75, 208.6, ) is stable. the stability of the equilibrium point e1 is shown in figure 4(b). the effect of different values of recovery rate from asymptomatic (γ1), symptomatic (γ2) and hospitalized (γ3), which corresponds to infective human is demonstrated in figure 4(c), 4(d), and 4(e) respectively. it is clear that the parameter of recovery rate γ1, γ2, and γ3 increases, simultaneously the infected population decreases. the effect of s, h, and r also shown in the figure 4(f) with respect to endemic equilibrium point. copyright © 2021 assa. adv syst sci appl (2021) 76 n.k. goswami, b. shanmukha a s 1 2 1 1 2 parameters -1.5 -1 -0.5 0 0.5 1 f or w ar d s en si tiv ity o f r 0 fig. 2. forward sensitivity of r0 7. the optimal control model in this section, the model (1) is extended to formulate optimal control problem by incorporating two time-dependent optimal control parameters, namely u1(t), and u2(t). if u1, and u2 equal zero, no effort is placed in these controls at time t, and if they equal one, maximum effort is applied. thus, optimal control variables are given, as follows: the control variable u1(t) represents the reduction in the transmission between humanto-human via using surgical face masks, social distancing, self-isolation, sensitization and awareness of transmission of the disease. the control variable u2(t) represents the increase in the testing facility and treatment, which can lead to fast detection of covid infected cases and recovery and add additional time-dependent parameter ηu2(t) in the rate of direction δ2. keeping in view of the above assumptions, the optimal control model is formulated as follows: ds dt = λ− (1− u1)βaias − (1− u1)βsiss − µs de dt = (1− u1)βaias + (1− u1)βsiss − (κ+ µ)e dia dt = ξκe − (ρ+ γ1 + µ+ δ1)ia dis dt = (1− ξ)κe + ρia − (γ2 + µ1 + µ)is − (δ2 + ηu2)is (4) dh dt = δ1ia + (δ2 + ηu2)is − (γ3 + µ2 + µ)h dr dt = γ1ia + γ2is + γ3h − µr copyright © 2021 assa. adv syst sci appl (2021) dynamics of covid-19 outbreak and optimal control strategies: a model-based... 77 0 1 1 1 r 0 0.8 s 0.5 0.6 a 2 0.4 0.20 0 0 0.5 1 1.5 (a) 0 1 1 1 r 0 0.80.5 0.6 a 2 0.4 0.20 0 0.5 1 1.5 (b) 0 1 1 1 r 0 0.80.5 0.6 s 2 0.4 0.20 0 0 2 4 6 8 10 12 (c) 0 1 1 1 r 0 0.80.5 0.6 a 2 0.4 0.20 0 0.5 1 1.5 (d) 0 1 1 1 r 0 0.80.5 0.6 2 0.4 0.20 0 0 0.5 1 1.5 (e) 0 1 1 1 r 0 0.80.5 0.6 3 2 0.4 0.20 0 0 0.5 1 1.5 2 (f) fig. 3. (a) influence of βa and βs on r0, (b) influence of βa and ρ on r0, (c) influence of βs and ξ on r0 (d) influence of βa and ξ on r0, (e) influence of λ and ξ on r0, (f) influence of γ3 and κ on r0 copyright © 2021 assa. adv syst sci appl (2021) 78 n.k. goswami, b. shanmukha 0 200 400 600 800 1000 time(in days) 0 200 400 600 800 1000 p op ul at io ns s e i a i s h r fig. 4. variation of s, e, ia, is, h , r performing the stability of disease-free equilibrium point with r0 = 0.0932 < 1. 0 200 400 600 800 1000 time(in days) 0 200 400 600 800 1000 1200 p op ul at io ns s e i a i s h r fig. 5. variation of s, e, ia, is, h , r performing the stability of endemic equilibrium point with r0 = 2.0075 > 1. 0 50 100 150 200 250 300 350 400 450 500 time(in days) 0 50 100 150 in fe ct ed p op ul at io ns 1 =0.526 1 =0.0526 1 =0.00526 fig. 6. variation of ia with time performing different values of γ1 copyright © 2021 assa. adv syst sci appl (2021) dynamics of covid-19 outbreak and optimal control strategies: a model-based... 79 0 50 100 150 200 250 300 350 400 450 500 time(in days) 0 50 100 150 in fe ct ed p op ul at io ns 2 =1.5 2 =0.5 2 =.005 fig. 7. variation of is with time performing different values of γ2 0 50 100 150 200 250 300 350 400 450 500 time(in days) 0 50 100 150 in fe ct ed p op ul at io ns 3 =0.2 3 =0.02 3 =0.0002 fig. 8. variation of h with time performing different values of γ3 7.1. the optimal control problem in this section, we study the behavior of the proposed model by using optimal control theory. the objective functional for fixed time tf if given by j(u1, u2) = ∫ tf 0 [ a1ia + a2is + 1 2 (b1u 2 1 +b2u 2 2) ] dt (5) subject to the model system (4). the parameter a1 ≥ 0, a2 ≥ 0, b1 ≥ 0, b2 ≥ 0 are the weight and balancing constants, which measure the respective cost involvement over the interval [0, tf ]. in order to find an optimal control, u1 ∗, and u2 ∗ such that j(u1 ∗, u2 ∗) = min (u1,u2)∈ω j(u1, u2), (6) where ω is the control set and is defined as ω = {(u1, u2) : 0 ≤ u1, u2 ≤ 1, t ∈ [0, tf ]} here, all the controls are bounded and measurable. 7.1.1. existence and characterization of optimal controls here, we shall first establish the existence of such control functions that minimizes the cost functional j . the lagrangian l copyright © 2021 assa. adv syst sci appl (2021) 80 n.k. goswami, b. shanmukha of this problem is defined as: l(ia, is, u1, u2) = a1ia + a2is + 1 2 b1u1 2 + 1 2 b2u 2 2 now, we shall use pontryagin’s maximum principle [32, 33] for necessary conditions for optimal controls system (4). for that by choosing x = (s,e, ia, is, h,r) ,ω = (u1, u2) and λ = (λ1, λ2, λ3, λ4, λ5, λ6), the associated hamiltonianh can be written as h(x,ω, λ) = l(ia, is, u1, u2) + λ1 ds dt + λ2 de dt + λ3 dia dt + λ4 dis dt + λ5 dh dt + λ6 dr dt (7) since u∗1, and u∗2 are solutions to the control problem (4), there exists the adjoint variables λ1, λ2, λ3, λ4, λ5, λ6 satisfying the following conditions. dx dt = ∂h(t, x, u∗1, u ∗ 2, λ1, λ2, λ3.λ4, λ5, λ6) ∂λ 0 = ∂h(t, x, u∗1, u ∗ 2, λ1, λ2, λ3.λ4, λ5, λ6) ∂u dλ dt = −∂h(t, x, u∗1, u ∗ 2, λ1, λ2, λ3.λ4, λ5, λ6) ∂x (8) theorem 7.1: for the objective functional (5) and the control set (8) subject to control system (4) there exists an optimal control u∗ = (u1 ∗, u2 ∗) ∈ ω such that j(u1 ∗, u2 ∗) = min ω j(u1, u2). proof to establish this result, we follow the theorem 4.1 mentioned in [40] for the existence of optimal controls. as, we have discussed above that all the state variables (population) are bounded for each bounded controls coming from the control set ω. furthermore, lipschitz condition with respect to state variables is satisfied by the right hand part of the model system (4). the control variable set ω is also convex and closed by the definition and the model system (4) is linear in control variables. thus, all the conditions for the existence of controls are fulfilled (for more details one can follow [28, 29]). hence the result. theorem 7.2: for optimal controls measures u∗1, u ∗ 2 and the state solutions s∗, e∗, i∗a , i ∗ s , h ∗, r∗ of the state system (4), there exists adjoint variables λ = (λi) tf ∈ r6, i = 1, 2, 3, 4, 5, 6 such that dλ1 dt = (1− u1)βaia(λ1 − λ2) + (1− u1)βsis(λ1 − λ2) + µλ1 dλ2 dt = (κ+ µ)λ2 + ξκ(λ4 − λ3)− κλ4 dλ3 dt = −a1 + (1− u1)βas(λ1 − λ2) + ρ(λ3 − λ4) + γ1(λ3 − λ6) + (µ+ δ1)λ3 dλ4 dt = −a2 + (1− u1)βss(λ1 − λ2) + γ2(λ4 − λ6) + (µ1 + µ)λ4 + (δ2 + ηu2)(λ4 − λ5) dλ5 dt = γ3(λ5 − λ6) + (µ2 + µ)λ5 (9) dλ6 dt = µλ6 copyright © 2021 assa. adv syst sci appl (2021) dynamics of covid-19 outbreak and optimal control strategies: a model-based... 81 with transversality conditions λ1(tf ) = λ2(tf ) = λ3(tf ) = λ4(tf ) = λ5(tf ) = λ6(tf ) = 0 (10) proof let u∗1, u ∗ 2 be the optimal control functions and s∗, e∗, i∗a , i ∗ s , h ∗, r∗ are the corresponding state variables. then, pontryagin’s maximum principle ensures the existence of the following adjoint variable λi(i = 1, 2, 3, 4, 5, 6) ∈ r6, which satisfies the following canonical equations: dλ1 dt = −∂h ∂s , dλ2 dt = −∂h ∂e , dλ3 dt = −∂h ∂ia , dλ4 dt = −∂h ∂is , dλ5 dt = −∂h ∂h , dλ6 dt = −∂h ∂r with transversality conditions (10) and the hamiltonian (7). the adjoint system (9) can be obtained. in the following result, we shall state the analytical forms of the optimal controls. theorem 7.3: the optimal controls (u∗1, u ∗ 2) which minimizes j over the region ω are given by u∗1 = min{1, max(0, ũ1)} u∗2 = min{1, max(0, ũ2)} where ũ1 = (β1ia + β2is)s(λ2 − λ1) b1 , ũ2 = ηis(λ4 − λ5) b2 , proof using optimally condition, we have ∂h ∂u1 = 0, ∂h ∂u2 = 0 we have ∂h ∂u1 = b1u1 + (βaia + βsis)s(λ1 − λ2) = 0 this gives u1 = ((βaia + βsis)s)(λ2 − λ1) b1 := ũ1 similarly, ∂h ∂u2 = b2u2 + ηis(λ5 − λ4) = 0 this implies u2 = ηis(λ4 − λ5) b2 := ũ2 moreover, lower and upper bounds of these control are 0 and 1 respectively. thus, if ũ1 > 1, ũ2 > 1, then u1 = u2 = 1. copyright © 2021 assa. adv syst sci appl (2021) 82 n.k. goswami, b. shanmukha also, ũ1 < 0, ũ2 < 0 then u1 = u2 = 0. otherwise, we have u1 = ũ1, and u2 = ũ2 hence, for these controls u∗1, u ∗ 2 we get optimum value of the function j . 8. simulation of optimal control problem in this section, we simulate our optimal control model using matlab. the parameter values are keeping same the parameters corresponding to stability of endemic equilibrium point e1 of the model (1). the weight constants for the optimal control problem are taken as a1 = 1, a2 = 1, b1 = 45, b2 = 65. we solve the optimality system by iterative method with the help of forward and backward difference approximations [32, 34–36]. we consider the time interval as [0,180]. first we solve the state equations by the forward difference approximation method then we use the backward difference approximation method to solve the adjoint equations. we consider different types of strategies to see the impact of optimal control in the total number of human infectives. 8.1. strategy a: employing hygiene promotion, social distancing and self-isolation (u1), only. here, only control measure u1(t) is used to optimize the objective function j , while control intervention u2(t) = 0, were not employed. the influence of u1(t) is demonstrated in figure 5(a), to minimize the objective function, the optimal control u1(t) is maintained at the maximum level. a single preventive measure can influence the spread of the covid-19 in the population. maintaining social distancing and self-isolation leads to control significant number reduction in asymptomatic and symptomatic cases in the population. from the figures, it is clear that the optimal control u1(t) is a little more effective compared to other types of controls but we need to maintain it to one for a longer period which is not easy to achieve. this control strategy is for using surgical face masks, social distancing, selfisolation, awareness of the transmission of disease, and sensitization. 8.2. strategy b: increase testing facility and treatment of the symptomatic individuals (u2) only. here, only control measure u2(t) is used to optimize the objective function j , while control intervention u1(t) = 0, were not employed. in figure 5(b), we present the plots of population and the effects of the increase in testing facility and treatment are demonstrated to minimizing the cost and reducing the number of coronavirus infections in the population. 8.3. strategy c: employing both the control interventions (u1, u2). here both the control interventions (u1(t), u2(t)) are used to optimize the objective function j. from figure 5(c), it is easy to say that by combining both optimal controls u1(t) and u2(t), the total number of infectious individuals decreases significantly. the simulation result indicates the effectiveness of optimal control strategies in reducing the number of infectives. it is observed that combined controls are more useful in reducing the number of infected cases significantly. finally, it observed that from figure 5(d), the optimal control model gives a better result as compacted to the model without the optimal control model as it reduces the number of infectives significantly in a desired interval of time. copyright © 2021 assa. adv syst sci appl (2021) dynamics of covid-19 outbreak and optimal control strategies: a model-based... 83 0 20 40 60 80 100 120 140 160 180 time(in days) 0 0.2 0.4 0.6 0.8 1 u 1(c o n tr o l p ro fi le ) u 1 , when u 2 = 0 fig. 9. influence of u1, when u2 = 0 0 20 40 60 80 100 120 140 160 180 time(in days) 0 0.2 0.4 0.6 0.8 1 u 2(c o n tr o l p ro fi le ) u 2 , when u 1 = 0 fig. 10. influence of u2, when u1 = 0 0 20 40 60 80 100 120 140 160 180 time(in days) 0 0.2 0.4 0.6 0.8 1 u 1(c o n tr o l p ro fi le ) u 1 u 2 fig. 11. influence of u1, u2 0 20 40 60 80 100 120 140 160 180 time(in days) 0 50 100 150 200 in fe ct ed p op ul at io ns without control with control fig. 12. variation of infected human populations against time with and without optimal control copyright © 2021 assa. adv syst sci appl (2021) 84 n.k. goswami, b. shanmukha 9. conclusion dynamics of covid-19 pandemic and its optimal control strategies are discussed in this present study through a mathematical model. the host population of the model is subdividing into six different compartment according to nature of the disease such as susceptible, exposed, asymptomatic infective,symptomatic infective, hospitalized and recovery. this non-linear mathematical model is proposed based on indian condition by considering asymptomatic and symptomatic infections populations. it is assumed that the transmission rates due to asymptomatic and symptomatic individuals are different. the epidemiological threshold (basic reproduction number) is computed by using next generation matrix method and discussed different equilibria of the model in details. the model is globally asymptotically stable for the global stability of disease-free equilibrium when basic reproduction number is less than unity using the castillo-chavez theorem. also the theoretical analysis is carried out the global stability of endemic equilibrium is globally asymptotically stable using lyapunov method when basic reproduction number is greater than one. sensitivity analysis of the model performed to identify the key parameter that influence the basic reproduction number, which will regulate to control transmission dynamics of the disease. it emphasized that λ and µ are the most sensitive parameter, followed by asymptomatic transmission coefficient β1 and ξ. furthermore, the mathematical model is extended to optimal control problem by incorporating two time-dependent optimal control parameters to reducing the burden due to covid-19 using pontryagin’s maximum principal. introduced control parameter u1 in the model as surgical face masks, social distancing, self-isolation, sensitization and awareness of transmission of the disease in asymptomatic and symptomatic invective individuals, which effectively reduce transmission rate. social distancing should always implement at a higher percentage than self-isolation. the optimal control drives a significant reduction in the asymptomatic and symptomatic infected populations. the control parameter u2 included reducing the cost infrastructure like testing facility, which can lead to fast detection of covid infected cases. the optimal control suggests that unless the cost is very high, the social distancing should implement at the maximum level throughout the time examined. the optimal control model provides a more reliable result as compacted to the model without the optimal control model. this control strategy reduces the number of infectives significantly in a desired interval of time. acknowledgements the authors would like to thank the editor and anonymous referees for their valuable comments and suggestions which led to an improvement of our original manuscript. references 1. guanghong, d.,chang,l.,jianqiu,g.,ling, w., cheng, k., and zhang, d. 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(2020)mathematical modeling and optimal cost-effecting control of covid-19 transmission dynamics. eur phys j plus.135(11). copyright © 2021 assa. adv syst sci appl (2021) introduction the model analysis of the model system basic reproduction number existence of endemic equilibrium point stability analysis of the model global stability of disease-free equilibrium global stability of endemic equilibrium sensitivity analysis of reproduction number effects of parameters on disease outbreak the optimal control model the optimal control problem existence and characterization of optimal controls simulation of optimal control problem strategy a: employing hygiene promotion, social distancing and self-isolation (u1), only. strategy b: increase testing facility and treatment of the symptomatic individuals (u2) only. strategy c: employing both the control interventions (u1,u2). conclusion microsoft word 641 article text, copyedited.doc adv syst sci appl 2021; 02; 1-7 published online at https://ijassa.ipu.ru. smart control mechanisms in digital technologies of decision-making vladimir n. burkov1, irina v. burkova2*, viktor g. zaskanov3 1) trapeznikov institute of control sciences, russian academy of sciences, moscow, russia e-mail: vlab17@bk.ru 2) trapeznikov institute of control sciences, russian academy of sciences, moscow, russia e-mail: irbur27@mail.ru 3) samara national research university, samara, russia e-mail: zaskanov@mail.ru abstract: this paper considers the problems of adopting digital technologies in decision-making. two types of digital decision technologies are described, namely, the direct (or traditional) technology with human decision-making based on computer-aided advisor systems and also the inverse technology with independent computer decision-making in which a human merely monitors the decision process. these technologies are compared with each other, and some applications are discussed. keywords: digital economy, correct mechanisms, decision support system, the kantorovich– glushkov conditions, direct and inverse technologies 1. introduction the development of digital economy as well as a wide adoption of digital technologies, not only in data processing and electronic documentation but also in solid decision-making, require a proper review of decision methodology. the attempts to use digital technologies in decision-making were undertaken long ago [1]. in this context, note the role of soviet academicians l.v. kantorovich and v.m. glushkov. in the wake of the 1970s large-scale automation in the ussr, glushkov suggested the idea of a computer-aided planning system for the national economy. kantorovich designed a mechanism to coordinate national economic plans with the interests of enterprises based on his original methodology of duality. the optimal dual estimates of associated planning problems were considered as a tool of such coordination. unfortunately, the attempt failed because the real interests of economic agents poorly fit the strict framework of duality. kantorovich and glushkov formulated two prerequisites for an efficient use of digital technologies in decision-making as follows: (1) all information necessary for decisionmaking must be reliable and (2) the economic agents must benefit by implementing given decisions (fulfilling their plans). in the sequel, these prerequisites will be called the kantorovich–glushkov conditions. in the 1970s, they were difficult to satisfy [1] but the situation has dramatically changed in the recent time, following the design of smart control mechanisms: contract theory [2, 3, 5, 8], the theory of mechanism design, where the incentive compatibility conditions and the revelation principle [6, 7], as well as the work [4, 7] and many others. * corresponding author: irbur27@mail.ru 2 v.n. burkov, i.v. burkova, v.g. zaskanov copyright ©2021 assa. adv. in systems science and appl. (2021) note that the incentive compatibility conditions, and the revelation principle were originally proposed in 1971 in article [9], where the revelation principle was called the "fairplay principle". let us discuss this class of mechanisms in detail. any changes in social being, particularly, control mechanisms, lead to fast consequences in human behavior. smart control mechanisms are the control mechanisms that change human behavior for public interests, e.g., stimulate truth-telling (the provision of reliable information), decision implementation, efficient development, etc. of crucial importance for digital technologies are the smart mechanisms that guarantee truth-telling and decision implementation: decision-making with these mechanisms satisfy the kantorovich–glushkov conditions. it is possible to identify two types of digital decision technologies as follows. the first type, known as the direct technology (traditional for modern society), relies on human decision-making with computer-aided decision support systems. the development of such technologies is connected with the design of active advisor systems: after the implementation process, the decision of a human decision-maker (dm) is compared with the one suggested by an advisor system using the so-called recalculation models. the technology of the second type (inverse) is based on independent computer decisionmaking while a human merely monitors the decision process without any interference (except for critical situations and force majeur). in such technologies, a decision support system (dss) becomes a decision-making system (dms) while a dm a person who assesses decisions (decision analyzer, da). however, control mechanisms are designed by managers and other interested persons who are responsible for the results of functioning. 2. direct (traditional) decision technology fig. 1. direct decision technology the structural diagram of the direct decision technology is shown in fig. 1. as mentioned above, the power of decision belongs to a decision-maker (dm), who uses information about an object and external environment as well as his/her experience and dss recommendations. figure 1 has the following notations: u as the principal’s decision (in this case, the principal is the dm); v as the dss recommendation; j as the available information about the object; finally, x(u) as the result of decision implementation. this technology demonstrates high efficiency if the principal well knows the object, seeks for its successful functioning and has operative tools to guarantee decision implementation. otherwise, the dm encounters unreliable information from the object, the non-implementation of required decisions and a possible occurrence of corruption ties. consider the direct technology on elementary examples of order allocation. dss object dm x(u) j v u smart control mechanisms in digital technologies of decision-making 3 copyright ©2021 assa. adv. in systems science and appl. (2021) example 1. the principal needs products in an amount r. there are n suppliers of these products. denote by xi the production plan of supplier i. let the product cost be given by , where a parameter ri describes the production efficiency of supplier i. the principal’s problem is to find the plans xi so that (1) (the supply satisfies the demand) and the total cost ф(х) are minimized, i.e., . (2) for the exact values r = (ri), the optimal plan has the form , , (3) where h is the total efficiency of all suppliers, . however, the principal often knows only approximate values of the parameters ri. in this case, the information s = (si) about these parameters is reported by the suppliers. the interest of supplier i is determined by the profit , (4) where λ denotes the product price established by the principal. he/she makes some decision using a control mechanism x = π(s) and λ(s), where π(s) are λ(s) are the functional relationships between the plan (x), control (λ) and the information s reported by the suppliers. the revelation mechanism [2] suggested in the theory of active systems is correct, i.e., guarantees truth-telling and plan fulfilment. in example 1, the revelation mechanism has the form (5) the direct technology suffers from several drawbacks as follows. first, it is difficult to construct an adequate model of an object, and even more difficult to design an adequate correct decision mechanism. second, this technology is susceptible to corruption––possible collusions between the principal and suppliers and also between suppliers themselves. so, the direct technology was further refined to the two-channel control mechanisms (active advisor systems) based on recalculation models (rms) [3]. an rm yields a more accurate estimate of the object’s parameters using additional information about the results of decision implementation. this approach allows comparing the economic effects from the decisions made by the principal and the ones suggested by a dss. the structural diagram of decision-making processes in two-channel mechanisms is presented in fig. 2. 21 2 ii irz x= i i x r=å ( ) 21min 2 = å ix i i ф x x r i i rx r h = 1, ,= !i n j j h r=å ( ) 21, 2i i i i i y x x x r l l= , 1, , , . = = = å !i i i i x s i n r s l l 4 v.n. burkov, i.v. burkova, v.g. zaskanov copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 2. decision-making in two-channel mechanisms figure 2 has the following additional notations: rm as a recalculation model; δ as the difference between the economic effects from the dm’s decision and dss recommendation. if δ > 0, the dm obtains an incentive; if δ < 0, a penalty. the two-channel mechanisms stimulate the principal’s interest in correct decisions and also reduce the corruption component. example 2. for the order allocation mechanism of example 1, a recalculation model can be easily constructed in the following way. let zi be the real cost of supplier i ( ). then , , and hence , (6) where wi denotes the plan recommended by the dss. consider a collusion between the dm and some supplier (without loss of generality, supplier 1). using his/her power of decision, the dm overrates the price λ1 for supplier 1. to this end, supplier 1 pays the dm a bribe––α% of his/her profit. the profit of supplier 1 makes up , and his/her beneficial plan is x1 = (1 – α)λ1 r1. without corruption, supplier 1 obtains the profit . the collusion makes sense for him/her if . so the beneficial corruption condition takes the form , or . let , where k > 1 is some parameter adjusted by the dm if he/she enters collusion with supplier 1. then the beneficial corruption condition can be written as 21 2 ii irz x= 2 2 i i i xr z = 1, ,= !i n 2 2 i i i i i w z z x æ ö d = -ç ÷ è ø å ( ) 2 1 1 1 1 1 2 xx r a l( )21 12 r h r ( ) 2 2 2 1 1 1 1 11 2 2 rr r h a l æ ö> ç ÷ è ø ( ) 11 r h a l> ( ) 1 1 1r r r h a l> > 1 r hkl = dss object dm y(u) j v u rm δ smart control mechanisms in digital technologies of decision-making 5 copyright ©2021 assa. adv. in systems science and appl. (2021) . note that, the higher is the supplier’s efficiency, the smaller is his/her gain from collusion. moreover, under a sufficiently large penalty for any deviations from the recalculation model-based cost, corruption becomes unbeneficial to the supplier. the major difficulty for a practical use of two-channel mechanisms is the design of adequate recalculation models. 3. inverse decision technology consider the inverse decision technology in which the principal’s role is played by a computer (digital decision support system, ddss) while the principal’s supervisor monitors the decision process without any interference (except for critical situations and force majeure). actually the supervisor and ddss change their places, which explains the term “inverse technology”. for making this technology efficient, the mechanism should be discussed with all interested persons and then enacted as a law. in this sense, the ddss performs a purely computational function. the whole responsibility for decisions lies on the designers of the mechanism (in the first place, the principal’s supervisor), not on the ddss. if he/she concludes that the mechanism is inefficient (e.g., in new conditions of functioning), then the supervisor considers possible corrections. of course, as noted earlier, the mechanism must satisfy the kantorovich–glushkov conditions, i.e., be correct. this forms the main problem for adopting digital technologies in decision-making and management: the correctness of mechanisms is not so trivial to guarantee. let us describe a possible design of correct mechanisms. consider a system of a single principal and n agents (enterprises, organizations, etc.). the interests of each agent i, , are determined by his/her goal function yi(l, xi), where l denotes control and xi is the plan established by the ddss. the sequence of functioning is as follows. 1. the principal announces the set l of admissible controls. 2. for each λ ∈ l, the agents report their beneficial plans to the principal: . 3. the principal chooses λ ∈ l and the corresponding plans xi(λ), , that maximize his/her goal function ф(x, λ). this mechanism is correct under the hypothesis of weak contagion [2], stating that all agents do not consider the influence of their messages on the control l. indeed, for any values l, the agents obtain beneficial plans and hence are interested in truth-telling and plan fulfillment. remark. if the set of admissible controls is large, the principal does not need to announce all its elements. he/she may organize an iterative procedure, adding new controls at each step to increase the value of his/her goal function. such a procedure resembles decomposition methods. example 3. consider the order allocation problem as described in example 1. assume the principal announces the set of admissible prices l (step 1) while the agents report their beneficial plans . the principal chooses l to minimize the total cost subject to the constraint . for this problem, the hypothesis of weak contagion was justified in the theory of active systems in the case of sufficiently many agents [2]. ( ) 1 1 1h k r a> > 1, ,= !i n ( ), 1, , ,= î!ix i n ll l 1, ,= !i n ( ) , 1, ,= = !i ix r i nl l ( )i i x rl =å 6 v.n. burkov, i.v. burkova, v.g. zaskanov copyright ©2021 assa. adv. in systems science and appl. (2021) an important advantage is that the inverse technology considerably reduces corruption effects. really, the principal’s supervisor has no right to interfere into the ddss and hence cannot affect decision-making. so collusions between the principal and agents take no place. however, corruption may occur when the agents report information to the principal. for struggling against this corruption component, the inverse technology should be augmented by a recalculation model. the main stages of adopting digital technologies in economy include the following. 1) choosing a situation that can be formalized as a model; 2) designing a smart control mechanism that guarantees truth-telling and decision implementation as well as coordinates the interests of all persons involved; 3) developing a software product, rules and regulations in accordance with this mechanism; 4) implementing this mechanism, analyzing its efficiency and making periodic corrections subject to new conditions of functioning. 4. some applications of digital technologies consider some applications of digital technologies [3]. 1. the automated quantitative complex assessment of activity results (akkord). this system was designed at trapeznikov institute of control sciences in the 1980s for assessing the efficiency of all industrial enterprises within the ministry of instrumentation, automation and control systems of the ussr (minpribor), following the initiative of minister m.s. shkabardnya. the general assessment method and procedure were discussed at the scientific and engineering council of the ministry and also at the board of the ministry and then approved by shkabardnya. calculations were performed using a special software complex without human interference. 2. cost-effective taxation systems. in 1990–1991, trapeznikov institute of control sciences participated in an experiment on new taxation systems for research activities organized by the state committee for science and technology of the ussr. the experts of the institute suggested the cost-effective taxation system––a smart incentive mechanism that stimulates any organizations (even monopolies) to reduce cost and prices. the corresponding rules, regulations and software were developed after debates at the scientific council of the institute. for two years, the institute was functioning in experimental conditions, and the results confirmed theoretical expectations—it was not beneficial to overrate the cost of contract work. 3. the reverse priority mechanism for limited resource allocation. this smart mechanism stimulates customers to submit objective requests for a certain resource (truthtelling). the mechanism was adopted in the water resource management system of bulgaria and highly appreciated later (awarded by the golden mark for high technological advance). 5. conclusion the article discusses two smart decision-making mechanisms based on digital technologies – direct and inverse. the question arises why such mechanisms are few in practice. we give four reasons: 1. the complexity of the smart mechanisms design. it is not by chance that several nobel prizes in economics were received for the development and implementation of one or another smart mechanism. 2. weak interest of officials and managers. indeed, the introduction of managerial innovations in an organization, region or industry is a considerable risk. drawing an analogy with medicine, we can say that this is an operation on a living organism, which is carried out smart control mechanisms in digital technologies of decision-making 7 copyright ©2021 assa. adv. in systems science and appl. (2021) by the patient himself. not every leader will dare to do it. moreover, an effective system of estimation and motivation of managers and officials for the result has not yet been created. 3. the problem of compatibility, the essence of which is that an excellent mechanism is often rejected by the organization because it is incompatible with its mentality, traditions, etc. 4. lack of trained personnel capable of developing and implementing smart mechanisms. the theory of smart mechanisms design and the theory of active systems are studied in a small number of universities. nevertheless, the authors are convinced that the future belongs to smart mechanisms. note the most promising directions to adopt smart digital technologies. 1. assessment systems for the activity of managers and officials at any level, with stimulation based on activity results. 2. allocation systems for limited resources (finances, etc.). 3. efficiency improvement programs for organizations, municipalities, regions, etc. acknowledgements this work was partially supported by the russian foundation for basic research, project no. 18-07-01258. references 1. ivanov, v. & malinetskii, g. (2017). strategicheskie prioritety tsifrovoi ekonomiki [strategic priorities of digital economy], strategicheskie prioritety, 3, 54–95. 2. burkov, v. (1977). osnovy matematicheskoi teorii aktivnykh sistem [fundamentals of mathematical theory of active systems]. moscow, russia: nauka. 3. goubko, m., burkov, v., kondrat’ev, v., korgin, n. & novikov, d. (2013). mechanism design and management: mathematical methods for smart organizations. new york, usa: nova science publishers. 4. mas-colell, andreu, michael dennis whinston, and jerry r. green (1995) microeconomic theory. vol. 1. new york: oxford university press. 5. bolton p., dewatripont m. contract theory. – cambridge: mit press, 2005. – 740 p. 6. harsanyi j. (1967, 1968) games with incomplete information played by "bayesian" players // management science. part i: 1967. vol. 14. № 3. p. 159 – 182. part ii: 1968. vol. 14. № 5. p. 320 – 334. part iii: 1968. vol. 14. № 7. 486–502. 7. maschler m., solan e, zamir s. (2013) game theory, cambridge: cambridge university press, 1008 p. 8. salanie b. (2005) the economics of contracts. 2nd edition, massachusetts: mit press. – 224 p. 9. burkov v.n., lerner a.j. (1971) fairplay in control of active systems. differential games and related topics, amsterdam: north-holland publishing company, 325–344. adv syst sci appl 2021; 03:1–11 published online at https://ijassa.ipu.ru. simultaneously robust chaos synchronization and identification of unknown parameters for nonlinear gyro system abolghasem daeichian1,2*, shahram aghaei3 1department of electrical engineering, faculty of engineering, arak university, arak, 38156-8-8349, iran 2institute of renewable energy, arak university, arak, 38156-8-8349, iran 3department of electrical engineering, yazd university, yazd, iran abstract: this paper concerns robust synchronization and parameter identification for nonlinear gyroscope systems. gyros are widely utilized in navigational applications where synchronization plays a vital role. a system of nonlinear dynamical equations with some parameters presents a model of gyro systems. the parameters of gyro can vary in time, which can lead to desynchronization of the gyro systems. in this paper, we assume that a gyro system has bounded time-varying unknown parameters and the synchronization problem is considered in two situations. first, the synchronization of two gyroscopes with identical dynamical model and, second, the synchronization of a gyroscope with the rössler system. the lyapunov stability theory with control terms is employed to cope with the problem. also, the identification of timevarying unknown parameters is the side goal of the paper. the proposed scheme synchronizes chaotic nonlinear systems in both situations appropriately. in addition, the slave parameters converge to the nominal values of master parameters despite uncertainty. simulation results illustrate the superiority of the proposed method. keywords: robust synchronization, chaos, parameter identification, gyroscope, rössler system 1. introduction numerous studies confirm that complex and chaotic behaviors are observed in physics, mechanics, engineering, etc. [1–4]. the chaotic behavior of dynamical systems is interested in many researches [5,6]. also, chaos synchronization in applied systems is studied [7]. the intent of chaos synchronization is to synchronize the states of at least two chaotic systems despite the difference in initial conditions, the difference in model parameters, and even the difference in dynamical models. how to reach synchronization under such distinctions is a challenging crux [8]. the identification of unknown system parameters is also the side goal of many studies. many schemes have been proposed to synchronize two chaotic systems, such as adaptive control [9–11], sliding mode control [12, 13], robust methods [14, 15], intelligent base methods [16–19] and lyapunov base methods [20–23]. the gyroscopes are widely employed in navigational, aeronautical, and space engineering. those have complex dynamical motion including periodic, sub-harmonic, quasi-periodic, and chaotic [24]. gyro synchronization for a couple of chaotic systems with one-way linking is considered in [25] and it is extended in [26] by applying an active control. fuzzy sliding mode control is employed to synchronize uncertain chaotic nonlinear gyros in [27]. a variable structure control approach using neural network with multi-quadratic radial basis ∗corresponding author: a-daeichian@araku.ac.ir, a.daeichian@gmail.com 2 a. daeichian, s. aghaei function is introduced in [28] where gyros with known and unknown system parameters are synchronized. the existence of external disturbance is investigated in [29], where fast nonsingular terminal sliding mode surface is employed to derive an adaptive finite-time control in order to synchronize the gyro systems. synchronization and parameter identification for the symmetric gyroscope system with constant parameters have been done in [30]. it is not robust with respect to parameter’s variation and does not converge to the nominal value of parameter. this paper concerns developing robust synchronization based on the lyapunov theorem with control terms for nonlinear gyroscope systems with unknown time-varying parameters. in addition, unknown parameter identification is expected. the idea is to find the control input u such that the parameters’ uncertainty eliminates in the lyapunov function. two situations are considered for the slave system. first, a similar gyro dynamical system, and second, the rössler system which has another dynamical equation of master gyro. the parameters of slave system or control input converge to the nominal value of master system parameters by a parameter updating law in spite of the uncertainty. it is proved by the lyapunov stability theorem that the proposed controller can synchronize chaotic systems with time-varying unknown parameters. simulation results show that method in [30] has a larger margin of error than the proposed robust method here. section 2 states the problem. controller and the parameter identifier are designed in section 3. simulation results verify achieving synchronization in section 4. finally, section 5 concludes the paper. 2. problem statements fig. 2.1 shows a gyro which is established on a vibrating base. the dynamical equations of the system are expressed by the euler’s angles θ, φ and ψ and a multiple harmonic equation∑n k=1ak sinωkt which represents base vibration. if we define x1 = θ, x2 = θ̇ and x3 = φ̇, the dynamical equations of the system in the state space framework are [30, 31]: ẋ1 = x2 ẋ2 = −(βφ − βϕ cosx1)(βϕ − βφ cosx1) i2 sin3 x1 − c1(t) i x2 + mgl i sinx1 − mg i n∑ k=1 ak sin (ωkt) sinx1 ẋ3 = −2 cosx1 sinx1 x2x3 + βϕx2 i sinx1 (2.1) where i , mg, and l denote the polar moments of gyro inertia, the gravity force, and the distance between origin and the center of gravity. βφ and βϕ are constants of the motion. c1(t) is a parameter vector which is unknown, time varying, and uncertain. it worth noting that x1 = θ, which is known as nutation angle, is the angle between thexy z (fixed) axis and xyz (body) axis. it is shown in fig.(11-16) of [31] that θ(t) is limited by two extreme values 0 < θ1 < θ(t) < θ2 < π which correspond to the turning points of the central-force. this fact prevents eq. (2.1) to become singular. for more details on the derivation of this model we refer to chapter 11 of [31]. however, c(t) has the nominal value c with uncertainty or disturbance ∆c(t), that is to say c1(t) = c + ∆c(t). it is assumed that the uncertainty is bounded, namely: assumption 2.1: (uncertainty boundedness) ‖∆c‖ ≤ k ∀t ∈ r+ (2.2) copyright © 2021 assa. adv syst sci appl (2021) simultaneously robust chaos synchronization and identification of unknown 3 where k is a known, positive, and real number. fig. 2.1. schematic of a gyroscope on a vibrating base eq. (2.1) is the master (drive) system model. the major goal is that the states of a slave (response) system synchronize with the states of the master system. here, we consider two different dynamical systems as slave system. first, a gyro with state space representation similar to the master system: ẋs1 = xs2 + u1 ẋs2 = −(βφ − βϕ cosxs1)(βϕ − βφ cosxs1) i2 sin3 xs1 − c2(t) i xs2 + mgl i sinxs1 − mg i n∑ k=1 ak sin (ωkt) sinxs1 + u2 ẋs3 = −2 cosxs1 sinxs1 xs2x s 3 + βϕx s 2 i sinxs1 + u3, (2.3) second, the rössler system as slave system: ẋs1 = −xs2 − xs3 + u1 ẋs2 = xs1 + b1x s 2 + u2 ẋs3 = a1 + xs3 (xs1 − c1) + u3 (2.4) where a1, b1, and c1 are the rössler system parameters. three control inputs u1, u2 and u3 are augmented to the slave dynamical equations to control the synchronization of the systems. c2(t) is the parameter vector of the slave system. the aim is to not only synchronize the states of two gyros but also c2(t) converges to c (nominal value of the unknown parameter), simultaneously. considering assumption 2.1, any robust controller has to satisfy the following conditions: 1. limt→∞ ‖e(t)‖ = 0 2. limt→∞ ‖c̃(t)‖ = 0 where e(t) = xs(t)− x(t) and c̃(t) = c2(t)− c. copyright © 2021 assa. adv syst sci appl (2021) 4 a. daeichian, s. aghaei 3. design of controller and identifier 3.1. gyro system as slave if gyro dynamical equations (2.3) are considered as a slave system, then the error dynamic can be written in the form: ė1 = e2 + u1 ė2 = −(βφ − βϕ cosxs1)(βϕ − βφ cosxs1) i2 sin3 xs1 + (βφ − βϕ cosx1)(βϕ − βφ cosx1) i2 sin3 x1 −c2 i xs2 + c1 i x2 + mgl i (sinxs1 − sinx1) −mg i n∑ k=1 ak sin (ωkt)(sinx s 1 − sinx1) + u2 ė3 = −2 cosxs1 sinxs1 xs2x s 3 + 2 cosx1 sinx1 x2x3 + βϕx s 2 i sinxs1 − βϕx2 i sinx1 + u3 (3.5) where e1 = xs1 − x1, e2 = xs2 − x2, and e3 = xs3 − x3. the following theorem introduces control inputs u1, u2, u3, and parameter identification updating law such that limt→∞ ‖e(t)‖ = 0 and limt→∞ ‖c̃(t)‖ = 0. theorem 3.1: consider the master system (2.1) with unknown parameter c1(t) which satisfies assumption 2.1 and the slave system (2.3). then, there exist the control laws u1 = −e2 − e1 u2 = (βφ − βϕ cosxs1)(βϕ − βφ cosxs1) i2 sin3 xs1 − (βφ − βϕ cosx1)(βϕ − βφ cosx1) i2 sin3 x1 +( c2 i − 1)e2 − mgl i (sinxs1 − sinx1)− ke2x2 i‖e2x2‖ x2 + mg i n∑ k=1 ak sin (ωkt)(sinx s 1 − sinx1) u3 = 2 cosxs1 sinxs1 xs2x s 3 − 2 cosx1 sinx1 x2x3 − βϕx s 2 i sinxs1 + βϕx2 i sinx1 − e3 (3.6) and the parameter updating law ċ2 = e2x2 i (3.7) which leads to master-slave synchronization. moreover, the unknown parameter of the slave system converges to the nominal value of the master one. proof consider the following lyapunov function: v (e1, e2, e3, c̃) = 1 2 (e21 + e22 + e23 + c̃2) (3.8) copyright © 2021 assa. adv syst sci appl (2021) simultaneously robust chaos synchronization and identification of unknown 5 where c̃(t) = c2(t)− c. the first derivation of the lyapunov function along the error dynamic eq.(3.5) is: v̇ = e1(e2 + u1) + e2(− (βφ − βϕ cosxs1)(βϕ − βφ cosxs1) i2 sin3 xs1 + (βφ − βϕ cosx1)(βϕ − βφ cosx1) i2 sin3 x1 − c2 i xs2 + c1 i x2 + mgl i (sinxs1 − sinx1)− mg i n∑ k=1 ak sinωkt(sinx s 1 − sinx1) + u2) +e3(− 2 cosxs1 sinxs1 xs2x s 3 + 2 cosx1 sinx1 x2x3 + βϕx s 2 i sinxs1 − βϕx2 i sinx1 + u3) + c̃ċ2 = −e21 − e22 − e23 + ∆c i x2e2 − ke22x2 2 i‖e2x2‖ (3.9) taking into account the inequality ‖a+b‖ ≤ ‖a‖+ ‖b‖, we obtain the inequality: v̇ ≤ −‖e‖2 + ‖∆c i ‖‖e2x2‖ − ‖ k i ‖‖e2x2‖ (3.10) finally, assumption 2.1, gives: v̇ ≤ −‖e‖2 (3.11) so, v̇ < 0 for ‖e‖ 6= 0. this lyapunov function and its derivation demonstrate that the error dynamic eq.(3.5) is asymptotically stable, that is to say, the master-slave synchronization is achieved. it also guarantees the convergence of the slave system parameter c2(t) to the nominal value of the master system parameter c. 3.2. rössler system as slave now, the rössler system is considered as a slave. the error dynamic in this situation is: ė1 = −xs2 − xs3 − x2 + u1 ė2 = xs1 + b1x s 2 + c1 i x2 − mgl i sinx1 + (βφ − βϕ cosx1)(βϕ − βφ cosx1) i2 sin3 x1 + mg i n∑ k=1 ak sin (ωkt)(sinx1) + u2 ė3 = a1 + xs3(x s 1 − c1) + 2 cosx1 sinx1 x2x3 − βϕx2 i sinx1 + u3 (3.12) the control inputs and parameter identification updating law are given as the following theorem. theorem 3.2: consider the master system (2.1) with unknown parameter c1(t) which satisfies assumption copyright © 2021 assa. adv syst sci appl (2021) 6 a. daeichian, s. aghaei 2.1 and the slave system (2.4). then, there exist the control laws u1 = xs2 + xs3 + x2 − e1 u2 = −xs1 − b1xs2 − c2 i x2 − ke2x2 i‖e2x2‖ x2 + mgl i sinx1 −(βφ − βϕ cosx1)(βϕ − βφ cosx1) i2 sin3 x1 − mg i n∑ k=1 ak sin (ωkt)(sinx1)− e2 u3 = −a1 − xs3(xs1 − c1)− 2 cosx1 sinx1 x2x3 + βϕx2 i sinx1 − e3 (3.13) and the parameter updating law, ċ2 = e2x2 i (3.14) which leads to master-slave synchronization. moreover, the unknown parameter of the slave system converges to the nominal value of the master one. proof the proof can be written down in the same way as the proof of theorem 1 by considering the error dynamic eq.(3.12). so, the two gyro systems synchronize despite initial state and parameter mismatches. the next section shows the simulation results of the proposed method. furthermore, the outcomes of the proposed method are compared with the results of [30]. 4. simulation consider the dynamical model parameters βφ = 2, βϕ = 5, i = 1,mg = 4, l = 0.25, ω1 = 1, and a1 = 12.1. the rössler system is chaotic where a1 = 0.2, b1 = 0.2, and c1 = 5.7. also, we consider different initial conditions for the master gyro as x1(0) = −0.5, x2(0) = −1.2, x3(0) = 10, slave gyro as xs1(0) = xs2(0) = xs3(0) = 0.1, and slave rössler system as xs1(0) = −0.1, xs2(0) = 0.2, xs3(0) = 0.5. assuming the unknown, time varying, and uncertain parameter c1(t) = 0.5 + 0.7 sin t where the nominal value is c = 0.5 and ∆c = 0.7 sin t. thus, the maximum norm of ∆c can be set as k = 0.7. the initial value of slave system parameter is also assumed as c2 = 0. moreover, the simulation time step is chosen as 0.0001 (sec). the states of two master and slave gyros as well as the error of their synchronization by the method of [30] are depicted in fig. 4.2. the mean square errors (mse) of the states synchronization are 6.617× 10−4, 1.151× 10−2, and 1.802× 10−1 for the e1, e2, and e3, respectively. since the unknown parameter is in the dynamical equation of x2, the steady state synchronization error is observed in e2. this can be quantified by neglecting transient time, i.e. considering mse for data obtained after 10 seconds from the beginning which are 1.721× 10−12, 7.749× 10−03, and 4.686× 10−10 for e1 to e3, respectively. it can be seen in fig. 4.2 that e2 does not converge to zero and it has fluctuations between −0.120 and 0.265. this fact is due to the steady state error in the parameter identification algorithm which is shown in fig.4.5. the mean value of the identified parameter by the method of [30] after initial transient time is 0.447 which suffers approximately 10.6% steady state error. theorems 1 and 2 assert that control inputs (3.6) and (3.13) with the parameter-updating laws (3.7) and (3.14) not only synchronize the master and slave systems but also guarantee the identification of the unknown parameter of the master system correctly. figs. 4.3 and 4.4 illustrate the states and synchronization error for gyro-gyro and gyro-rössler systems by the proposed method, respectively. the simulation results show that synchronization copyright © 2021 assa. adv syst sci appl (2021) simultaneously robust chaos synchronization and identification of unknown 7 0 50 100 150 -3 -2 -1 0 x 1 0 50 100 150 -5 0 5 x 2 0 50 100 150 0 10 20 x 3 0 50 100 150 -3 -2 -1 0 x 1s 0 50 100 150 -5 0 5 x 2s 0 50 100 150 0 10 20 x 3s 0 50 100 150 time (sec) -0.5 0 0.5 1 e 1 0 50 100 150 time (sec) -0.5 0 0.5 1 1.5 e 2 0 50 100 150 time (sec) -10 -5 0 e 3 fig. 4.2. method of [30]: row 1: master system states, row 2: slave gyro system states, row 3: synchronization error 0 50 100 150 -3 -2 -1 0 x 1 0 50 100 150 -5 0 5 x 2 0 50 100 150 0 10 20 x 3 0 50 100 150 -3 -2 -1 0 x 1s 0 50 100 150 -5 0 5 x 2s 0 50 100 150 0 10 20 x 3s 0 50 100 150 time (sec) -0.5 0 0.5 1 e 1 0 50 100 150 time (sec) -0.5 0 0.5 1 1.5 e 2 0 50 100 150 time (sec) -10 -5 0 e 3 fig. 4.3. proposed method (gyro-gyro): row 1: master system states, row 2: slave gyro system states, row 3: synchronization error error asymptotically tends to zero for all states regardless of uncertainty in parameters. in the case of gyro-gyro synchronization, the mse of the synchronized x2 reduces to 1.019× 10−3 which shows a decrease of more than 91%. omitting initial transient time give mse equal to 3.45× 10−5, which is almost 99.5% more accurate than the previous method. synchronization mses for the gyro-rössler situation are 2.941× 10−4, 1.202× 10−3, and 1.659× 10−1 related to e1, e2, and e3, respectively. the mses decrease to 0.765× 10−12, 0.399× 10−4, and 4.315× 10−10 if neglecting the initial transient time. the gyro-rössler copyright © 2021 assa. adv syst sci appl (2021) 8 a. daeichian, s. aghaei 0 50 100 150 -3 -2 -1 0 x 1 0 50 100 150 -5 0 5 x 2 0 50 100 150 0 10 20 x 3 0 50 100 150 -3 -2 -1 0 x 1s 0 50 100 150 -5 0 5 x 2s 0 50 100 150 0 10 20 x 3s 0 50 100 150 time (sec) -0.5 0 0.5 1 e 1 0 50 100 150 time (sec) -0.5 0 0.5 1 1.5 e 2 0 50 100 150 time (sec) -10 -5 0 e 3 fig. 4.4. proposed method (gyro-rössler): row 1: master system states, row 2: slave gyro system states, row 3: synchronization error 0 100 200 300 time (sec) -0.2 0 0.2 0.4 id e n ti fi e d v a lu e 0 100 200 300 time (sec) 0 0.2 0.4 0.6 id e n ti fi c a ti o n e rr o r fig. 4.5. gyro-gyro synchronization: left: parameter identification, right: identification error; dashed-dotted line (black): nominal value, dashed line (blue): proposed method, solid line (red): method of [30] 0 100 200 300 time (sec) -0.2 0 0.2 0.4 id e n ti fi e d v a lu e 0 100 200 300 time (sec) 0 0.2 0.4 0.6 id e n ti fi c a ti o n e rr o r fig. 4.6. gyro-rössler synchronization: left: parameter identification, right: identification error; dasheddotted line (black): nominal value, dashed line (blue): proposed method, solid line (red): method of [30] synchronization also ensures convergence to zero of all errors. the mses for all situations are written in table 4.1. the unknown parameter is also identified perfectly by the developed method. figs. 4.5 and 4.6 indicate the convergence of the unknown parameter to the nominal value c as well as the error of estimation for the proposed method and that of [30] in the situation of gyro-gyro and gyro-rössler systems, respectively. it can be seen that the fluctuations in the e2 decrease as time goes by and the parameter identification error converges to zero in the proposed method while the method of [30] suffers lasting fluctuations as well as steady state error. copyright © 2021 assa. adv syst sci appl (2021) simultaneously robust chaos synchronization and identification of unknown 9 table 4.1. mse of synchronization mse mse for time>10 (sec) method e1(×10−4) e2(×10−3) e3(×10−1) e1(×10−12) e2(×10−4) e3(×10−10) method of [30] 6.617 11.51 1.802 1.721 77.49 4.686 proposed gyro-gyro 6.617 1.019 1.802 1.721 0.345 4.686 method gyro-rössler 2.941 1.202 1.659 0.765 0.399 4.315 5. conclusion the proposed synchronization of gyro system has been founded based on the elimination of dynamical equations’ nonlinearity by the controller. simultaneously, a parameter identification algorithm estimates the unknown, time-varying, and uncertain parameters. the proposed method not only guarantees convergence of synchronization error to zero but also ensures the convergence of unknown parameter estimation to the nominal value. considering a typical gyroscope, the synchronization error in order of magnitude 10−12, 10−5, and 10−10 for e1, e2, and e3, respectively, are obtained in the situation of gyro-gyro and gyro-rössler synchronization. moreover, the unknown parameter is estimated at time 300 second with less than 2.7% of error. however, the proposed algorithm requires the upper bound of the parameter uncertainty to be determined. another point is that the control inputs are a complex nonlinear function of states that impose high computation load. nomenclature parameter definition x, y, z inertia base vectors (fixed) x, y, z body axis θ, φ, ψ the euler’s angles i the polar moments of gyro inertia l the distance between origin and the center of gravity m the mass of gyro g gravitational acceleration constant ωk k-th harmony of the base vibration frequency ak amplitude of base vibration with frequency ωk βφ, βϕ the constants of the motion a1, b1, c1 the rössler system parameters xi i-th state of the master system (i=1,2,3) xsi i-th state of the slave system (i=1,2,3) references 1. rakkiyappan, r., latha, v. p., zhu, q., & yao, z. 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(2004) classical dynamics of particles and systems. brooks/cole—thomson learning. copyright © 2021 assa. adv syst sci appl (2021) introduction problem statements design of controller and identifier gyro system as slave rössler system as slave simulation conclusion adv syst sci appl 2020; 04; 11-26 published online at https://ijassa.ipu.ru. frequency-geometric identification of magnetization characteristics of switched reluctance machine laila kadi1*, adil brouri1, abdelmalek ouannou1 1) imsm team, l2mc laboratory, ensam, moulay ismail university, meknes, morocco e-mail: lailakadigi@gmail.com, a.brouri@ensam-umi.ac.ma, abdelmalekouannou@gmail.com abstract: accurate modeling of electrical drives for online testing is a relevant problem. switched reluctance machine (srm) has lately attracted significant attention because it has several advantages compared to conventional engines. it is simple, free of rare-earth and faulttolerant machine. an analytical model of a srm has not been reported yet due to the dynamic and strong nonlinearity of srm. the most srm control and applications are based on several assumptions and simplifications. therefore, it is convenient to develop an accurate approach to identify the srm characteristics. in this paper, an analytical modeling and identification method of magnetization characteristics of switched reluctance machine (srm) is proposed. presently, an exact mathematical model of srm is established. unlike several previous studies, in this approach the system nonlinearities of srm are allowed to be hysteresis (i.e. the hysteresis effect is considered) and taking account the inherent magnetic nonlinearity. indeed, the srm is considered as highly nonlinear which makes the modeling of these machines difficult to achieve. then, it is convenient to develop an accuracy model of srm because it is always operated in the magnetically saturated mode to maximize the energy transfer. the developed model can be used in control, simulation and design development. furthermore, an identification method, at standstill test, based on frequency technics is developed allowing the identification of srm nonlinearities (considering the saturation and the hysteresis effects). in this respect, it is shown that the nonlinear behavior of srm can be exactly described by a block-oriented nonlinear structure. specifically, the srm can be described by a wiener nonlinear model. compared to the existing methods, the proposed study gives good accuracy of flux-linkage value characteristics and enjoys the simplicity of implementation. keywords: switched reluctance machine, hysteresis effect, saturation effect, nonlinear identification, nonlinear modeling, wiener model, frequency-geometric method 1. introduction the world is now focused on how to reduce the cost of equipment used in renewable energy, electrical vehicles (ev), as well as in other applications. for wind energy conversion, many research has been done to evaluate the generator that is characterized by low cost, easy to maintain, high performance, and able to work under variable speeds [1, 2]. moreover, the choice of motor drives for ev is very critical. in this context, the srm is typically selected as a generator or motor because of its low cost, simplicity, robust structure, small moment of inertia rotor and fault-tolerant capability. then, it is easy to maintain. note that the majority of losses show up in the stator because the srm has no rotor winding, no brushes, and does not have a magnet permanent [3] (see fig. 1), there are also others benefits of srm. in view of these significant remarks, the srm is relatively easy to cool and is insensitive to high temperatures [4]. the srm can be a suitable choice for wind energy and ev applications. in this respect, the srm becomes more competitive compared to the classical machines such as induction machines. however, its major defects are torque ripples and acoustic noise [4]. note that several studies are still ongoing to mitigate these effects [5, 6]. * corresponding author: lailakadigi@gmail.com 12 l. kadi, a. brouri, a. ouannou copyright ©2020 assa. adv. in systems science and appl. (2020) even though the several advantages of the srm, the strong nonlinearity makes the modeling of srm not an easy task. despite all efforts in modeling and identification of srm, finding an analytical model, which can accurately represent the nonlinear behavior of srm, is widely open for scientific research. as commonly used in the literature, the modeling and identification methods of srm flux linkage or self-inductance characteristics are classified into 4 categories: experimental measurements [7, 8], numerical solutions [9, 10], intelligent methods [11, 12], and analytical approaches [13, 14]. roughly, the experimental methods are costly, require considerable time, can damage the machine and possibly create a safety hazard. therefore, this method can be used to validate the obtained results. previous research has shown that two magnetization characteristics have been generally looked for srm modeling. the first method is established using the inductance characteristics based model. the second solution is based on the flux-linkage [4]. at this stage, it is worth emphasizing the difficulty to establish a model that can accurately describe the behavior of the machine. this study proposes the development of an exact analytical model for the srm, which is featured by a simple structure and reduced computational time. then, an identification method is proposed for determining the flux linkage magnetization characteristics. lately, researchers have shown an increased interest in using block-oriented nonlinear models based on the input and output systems [16–18]. in this paper, it is proved that the srm behavior can be accurately designed using block-oriented nonlinear model. specifically, it is shown that the srm at standstill can be exactly described by the wiener model (fig. 2). this later is given by the cascade connection of a linear dynamic system followed by a static or dynamic nonlinearity block [17, 18] as shown in fig. 2. to the author’s knowledge, for the first time, it has been shown that the srm can be exactly described by the wiener model. as soon as a model of srm is obtained, the identification method of the system parameters using this model constitutes also an important problem. in this article, a new approach is suggested for the identification of the magnetization characteristics often called flux linkage characteristics are much dependent on current and rotor position of the motor. note that the problem of identifying the srm nonlinearities will be carried out including saturation and hysteresis effects. presently, a frequency-domain solution is designed to determine the srm nonlinearities. the frequency identification methods have been widely used to identify the nonlinear systems parameters [16, 19, 20]. it is shown that the srm can be analytically described by a nonlinear system of wiener model. the proposed frequency approach consists of applying a sine input voltage ( ) sin( )v t v t= to the phase winding, for any given rotor position. then, by recording the current measurement at the winding terminal, the system nonlinearities of the model describing the srm can be identified using a simple frequency-geometric method. then, the srm can be modeled analytically using the data acquisition obtained by the finite element method (fem). note that this study is quite different from previous works [21–24]. the first key step in the development of this identification approach is the determination of an accurate model describing the nonlinear behavior of srm. another key step consists of designing an identification method using an appropriate excitation signal to estimate the parameters and the characteristics of srm. then, the system nonlinearities of srm are allowed to be static or hysteretic operator. furthermore, the model and the identification solution take into account the inherent magnetic nonlinearity. then, using the developed nonlinear model of srm, an identification method at standstill test is developed. the identification approach is based on frequency technics using simple sine input voltage. it is shown that the inner signal of the srm model can be analytically determined at any time t. in the first stage, the hysteresis effect is not considered. a filtering algorithm of the current at the winding terminal is suggested. accordingly, the system nonlinearities of the model describing the srm can be easily identified using the recording of the phase current https://www.linguee.fr/anglais-francais/traduction/commonly+used.html 13 frequency-geometric identification of magnetization characteristics copyright ©2020 assa. adv. in systems science and appl. (2020) (the filtered signal) over one half-period of time. then, the flux-linkage characteristics can be easily obtained. furthermore, these results are compared with those obtained using magnetostatic method. in the second stage, the saturation and the hysteresis effects of srm nonlinearities are considered. for any given rotor position, the srm phase winding is excited using also a sine input voltage. then, using the proposed filtering algorithm of the phase current, the system nonlinearities of the model describing the srm can be accurately estimated using the recorded data over one period of time. accordingly, the flux-linkage characteristics can be provided without any other experiment (using uniquely the phase voltage and the recorded current). compared to the magnetostatic method, it is shown that the proposed study gives good accuracy of flux-linkage value characteristics, enjoys the simplicity of implementation and characterized by a reduced computational time. to the authors’ knowledge, no previous study has found an exact model simulating the behavior of srm. in addition, very few published works have attempted to identify the hysteresis operator. the paper is organized as follows: the principle of operation and the equations describing the srm are stated in section 2. then, the mathematical model and the description of the identification method of srm nonlinearity are dealt within section 3. section 4 presents examples of obtained results using the proposed study. fig. 1.1. example of a 8/6 srm ξ v x i y fig.1.2. wiener model 2. operation principle and equations describing srm srm is a double saliency device in which the stator comprises a salient magnetic circuit provided with coils supplied by a current i, and the rotor is also a salient ferromagnetic circuit but without any conductor or magnet [24], [25] as shown in fig.1.1. in such machine, torque is produced by the tendency of its rotor to move to a position where the reluctance of the excited winding is minimized (aligned position) [24], [25]. then, the machine rotates the magnetic field by synchronizing continuously and consecutively the supply of different phases with the rotor position by means of the firing angles (turn-on and turn-off). srm can operate as a motor or generator. in order to obtain motoring mode, a stator phase is excited when the rotor is moving from an unaligned position towards the aligned position. likewise, rotor (does not contain any coil) stator coils ( )ig s h(.) 14 l. kadi, a. brouri, a. ouannou copyright ©2020 assa. adv. in systems science and appl. (2020) by exciting a stator phase when the rotor is moving from aligned towards the unaligned position, a generating mode will be attained [26]. to feed each stator phase of the srm, we generally use an asymmetric half-bridge converter [27], [28] (fig.2.3). the magnetic flux created by the ampere-turns ni oscillates between two extreme values correspond to [29]: unaligned position in which the magnetic circuit has a maximum reluctance (minimum inductance) as illustrated by fig. 2.4a. aligned position in which the magnetic circuit has a minimum reluctance (maximum inductance) as explained by fig. 2.4b. the structure of the studied machine is of four phase 8/6 srm, where 8 designates the number of poles in the stator (ns) and 6 is the number of poles in the rotor (nr). the choice of ns and nr is important since they have significant implications on the torque. the material of the srm stator and rotor is steel m36 with the nonlinear b-h characteristic. the mutual coupling between adjacent phases is very small and can be neglected[25], [29]. let i, r, l, ω, θ, and v denote the phase current, phase resistance, inductance, rotor speed, rotor position angle, and phase voltage respectively. then, the voltage equation of a phase winding can be given as: ( ) ( ) ( , )d i i dt v t r t   = + () where λ is the flux-linkage per phase. under the hypothesis of nonmagnetic coupling between phases (at any given time, only one phase is excited), the flux-linkage can be expressed as follows: ( , ) ( , )i l i i  = (2.2) due to the nonlinear behavior of the ferromagnetic material, the flux in the stator phases varies nonlinearly according to the rotor position θ and the phase current i. accordingly, through direct derivation of (2.2) in (2.1), one immediately gets the following expression of the phase voltage: ( ) ( ) ( ) ( ) ( ), , ,i i l i l idi di v t r t l i dt i dt      = + +    +     () the last term in (2.3) is the back electromotive force (emf) voltage that is expressed by: ( ), e i l i   =   () fig.2.3. asymmetric half bridge converter 15 frequency-geometric identification of magnetization characteristics copyright ©2020 assa. adv. in systems science and appl. (2020) (a) (b) (c) fig.2.4 (a) unaligned position; (b) aligned position; (c) midway position 3. modeling and identification approach of srm at standstill test, the back electromotive force (emf) voltage is nil because it is proportional to the shaft speed. accordingly, the expression of phase voltage in (2.3) becomes: ( ) ( ) ( ) ( ), ,i i l idi di v t r t l i dt i dt  = + +   () the rotor of srm is blocked at specific positions relative to the phase to be tested, e.g. at aligned, unaligned, and a midway between the above two positions. at blocked rotor (for any given rotor position  ), the nonlinearity ( )l i  (i.e. the inductance according to the current phase) can be described using a finite orthogonal function, e.g. polynomial decomposition [30], [31]: 0 ( ) n k k k θ a il i = =  () where n is the polynomial function degree and ka its coefficients. then, it is readily follows from (3.1) and (3.2) that the phase voltage can be expressed as: ( )k+1 k=0 ( ) n k d b i dt v t =  (3.3a) where: 1 1 2 1 k k r a for k b a for k + = =      (3.3b) this result means that the phase voltage can be obtained by deriving a polynomial function according to the phase current. then, let introduce the following polynomial function: k+1 k=0 ( ) n kf i b i=  (a) then, in view of (3.4a), it follows that the phase voltage ( )v t in (3.3a) can be rewritten as follows: ( ) ( ) df i dt v t = (3 .4b) remark 3.1: 16 l. kadi, a. brouri, a. ouannou copyright ©2020 assa. adv. in systems science and appl. (2020) it follows from the formulae (3.2)-(3.4) show that the nonlinear system having, the phase current i(t) as input and the phase voltage v(t) as output, can be described by a hammerstein model, i.e., nonlinear element (f(i)) followed by a linear block (derivative term). at this point, it is worth emphasizing that the derivative action can be modeled by a linear block. it is readily follows that the analytical nonlinear system (3.4a-b) (the phase voltage v(t) according to the phase current i(t) for any given rotor position) can be modeled by a nonlinear subsystem followed by a linear dynamical element as shown by fig.3.1. this nonlinear model is called hammerstein system. fig. 3.1. hammerstein nonlinear model the aim presently is to develop an identification approach allowing to estimate the inductance nonlinearity ( )l i  , for any given rotor position  , and the flux-linkage ( , ) ( , )i l i i  = . furthermore, the hysteretic shape of the flux linkage can be seen. then, it is worth emphasizing that, from practical point of view, the input signal of srm is not a current excitation, but rather a phase voltage and the output signal of srm is the phase current. indeed, the srm is generally excited by a phase voltage and the phase current can be viewed as the output measurement. then, the output electrical signal of srm (i.e. the phase current i(t)) according to the input excitation (i.e. the phase voltage v(t)) can be modeled by a nonlinear system of wiener model (fig.1.2), where h(.) is the system nonlinearity, ( )ig s is the linear subsystem and ξ(t) is the measurement noise. specifically, the linear element is an integrator block ( ( ) 1/ig s s= ). the above model is analytically described by the following equations: ( ) ( )* ( )ix t g t v t= () ( )( ) ( )i t h x t= () ( ) ( ) ( )y t i t t= + () where: 0 ( ) ( ) st i ig t g s e ds + =  (the inverse laplace transform of ( )ig s ); the symbol '*' refers to the convolution operation; v(t) and y(t) represent the (measurable) input and output of the system, respectively. in this respect, note that the internal signal x(t) is not accessible to measurement. at this stage, an identification approach is proposed based on frequency method. then, the srm characteristics and the parameters determination can be dealt using frequency-geometric tools. accordingly, the following sine signal is being applied to the system input (phase voltage): ( ) sin( )v t v t= () in the steady state, the internal signal x(t) is also sinusoidal (of the same period of v(t)) and can be written as [15], [16], [18]: ( ) ( / ) cos( )x t v t = − () f(.) g(s) i v f ( . ) 17 frequency-geometric identification of magnetization characteristics copyright ©2020 assa. adv. in systems science and appl. (2020) commonly, the main complexity of wiener model identification problem is due to the fact that the internal variable x(t) is not accessible to measurement. fortunately, in our nonlinear model, it follows (3.9) the inner signal x(t) can be easily estimated at any time t. presently, a frequency domain method is proposed to identify the system nonlinearity h(.) using the data obtained from finite element method. in this respect, the sinusoidal voltage ( ) sin( )v t v t= is being applied to the phase winding and the resulting phase current is recorded. then, the srm nonlinearities characteristics can be accurately determined using only the system input (i.e. the phase voltage v(t)) and the measurable output signal (i.e. the phase current i(t)). it is interesting to bear in mind that, if the input excitation v(t) is t=2π/ω-periodic, then the steady state of inner signal x(t) and output system i(t) are also periodic of the same period t as v(t) [15], [18]. this result is a quite interesting achievement as it shows that, the curve of i(t) according to x(t) can be easily obtained using uniquely the set of points ( ) ( )( ), ( ) -( / ) cos( ), ( )x t i t v t i t = , in the steady state, over one period t. the determination of the functions ( ) ( )( ), ( ) ( / ) cos( ), ( )x t i t v t i t = − constitutes another key challenge of this work, because it makes possible the estimation of nonlinearity h(.) of srm model (fig. 1.2) at any given rotor position  . practically speaking, the useful information (the phase current i(t)) is not directly accessible to the measurement. indeed, the measurement signal y(t) is the phase current with additive noise (please see (3.7)). let ˆ ( ) m i t denotes the phase current estimate. in this respect, thanks to periodicity of i(t) (t=2π/ω), an accurate estimate of the phase current can be obtained using the following estimator: ( )  ) 1 0 1ˆ ( ) 2 / f 0 2 /or m mdef k i t y t k t m     − = = +  (a) ˆ ˆ( 2 / ) ( ) m m i t k i other iset w + = (b) where m is an integer, preferably of large value. then, the identification of the system nonlinearity h(.) can easily be achieved using the collected data over the interval  )0 2 /m  . for convenience, let us summarize the identification steps of system nonlinearity h(.): firstly, using the set of recorded data  ) ( ); 0y t for t mt , the estimate of the current phase i(t) can be obtained using the estimator (3.10a-b). then, we generate the fictive internal signal ( ) ( / ) cos( )x t v t = − over one period t. therefore, the estimate ˆ (.) m h of nonlinear element h(.) can be simply performed by constructing the set of points: ( )  ) ˆ( ), ( ) ; 0 m x t i t for t t . finally, an accurate estimate of srm nonlinearity, for different rotor positions, can be obtained using the above frequency-geometric method. the above results lead to the following proposition. proposition 3.1: let consider the identification problem of srm model at standstill (fig. 1.2), where the noise ( )t is supposed to be a zero-mean ergodic stochastic process. this latter is excited by the sine input ( ) sin( )v t v t= . accordingly, one has with probability 1 (w.p.1): 1. the current estimate ˆ ( ) m i t converges to the true phase current signal i(t). 18 l. kadi, a. brouri, a. ouannou copyright ©2020 assa. adv. in systems science and appl. (2020) 2. the system nonlinearity estimate ˆ (.) m h converges to the true function h(.). proof. to prove the first part of the proposition, let us combining (3.7) and (3.10a-b), one thus gets: ( ) 1 1 0 0 1 1 ˆ ( ) 2 / 2 /( ) m m m k k i t i t k t k m m     − − = = = + + +  () bearing in mind that the phase current i(t) is t=2π/ω -periodic, (3.11) can thus be rewritten as: 1 0 1 ( ) ( ) ( 2 / )ˆ m m k t i t t k m i    − = = + + () using the fact that the noise is a zero-mean ergodic stochastic process, the last term in (3.12) converges to zero as m tends to infinity (w.p.1). this completes the first part of proposition. to prove the second part of proposition, let us recall that for the input ( ) sin( )v t v t= (fig.1.2), the inner signal x(t) belongs to the interval / /v v   − . accordingly, the system nonlinearity h(.) can be described, within the interval / /v v   − , as the set of all points ( ) ( )( )  ) ( )( ), ( ) ( ), ; 0x tx t i t x t h t t=  . therefore, the nonlinearity estimate ˆ (.) m h can be featured by plotting the set of points ( )  ) ˆ( ), ( ) ; 0 m x t i t t t . it follows from the first part of proposition that: ( ) ( )  )ˆ( ), ( ) ( ), ( ( )) 0lim   m m x t i t x t h x t t tfor any → =  () this means that the nonlinearity estimate ˆ (.) m h converges to the true function h(.) w.p.1 for any  )0t t . this completes the proof of proposition. this result is a quite interesting achievement as it shows that, it is possible to obtain an accurate estimate of srm characteristics for any given rotor position . at this stage, other significant results can be observed. on one hand, if the system nonlinearity h(.) is static, the estimate ˆ (.) m h of h(.) can be done by using simply the set of points ( )ˆ( ), ( ) m x t i t over one-half period  )0 / 2t t . on the other hand, in the case where the nonlinear element h(.) of srm is hysteresis, the system nonlinearity estimate ˆ (.) m h can be featured using the set of points ( )ˆ( ), ( ) m x t i t over one period of time (i.e. ( )  ) ˆ( ), ( ) ; 0 m x t i t for t t ). these results lead to the following proposition. proposition 3.2: let us reconsider the above identification problem of srm model (fig. 1.2). this machine is excited by the phase voltage (3.8), where the amplitude v and the frequency ω (rd/s) of v(t) are kept constant. let’s assume that, in steady state, ( ) ( )( ) (( 1) )h hx kt x k t= + . then, the following results are obtained: 1. if the hysteresis effect of srm magnetization characteristics is not considered, for / /x v v    − the nonlinear element h(.) (in the steady state) can be constructed by fitting the set of points (x(t), i(t)) over one half-period (  )0 / 2for t t ). 19 frequency-geometric identification of magnetization characteristics copyright ©2020 assa. adv. in systems science and appl. (2020) 2. now, the hysteresis effect of srm magnetization characteristics is considered. then, for any fixed v and  , the hysteretic nonlinearity h(.) (in the steady state) can be obtained by plotting the set of points ( )  ) ( ), ( ) ; 0x t i t t t (i.e. over one period of time). proof. first, one immediately gets using (3.6) -(3.9) for any  )0t t : ( ) ( )( ) ( ) ( / ) cos( )i t h x t h v t = = − () then, if the srm system nonlinearity h(.) is static (does not contain the hysteresis effect), then for any  )0 / 2t t there exists 't t t= − such that: ( ) ( )  )' / 2( ) ( ') ( )   t t tani t h t h x t dx = = () accordingly, the static nonlinear function h(.) can be described uniquely by the set of points ( )  ) ( ), ( ) ; 0 / 2x t t t ti  . this completes the first part of proposition. at this stage, the function h(.) is hysteretic nonlinear operator. accordingly, using the assumption ( ) ( )( ) (( 1) )h x kt h x k t= + one immediately gets that all points of coordinates ( )  ) ( ), ( ) ; 0x t t t ti  build a closed locus [16], [18]. furthermore, for any given amplitude v and frequency ω, a single closed locus is obtained. finally, the shape of hysteretic nonlinearity h(.), for fixed values of (v, ω), can be described by plotting the points of coordinates (x(t), i(t)) for  )0t t . presently, the aim is to present a method of modeling and identification of the fluxlinkage ( , )i  . the above obtained achievement is useful to determine the flux-linkage features. accordingly, it is readily follows from (2.1) that, the flux-linkage ( , )i  can be expressed as: ( )( )( )( , ) tv t ti r i d   −= (3.16) then, it follows from (3.9) and (3.16) that, the flux-linkage can be rewritten as: ( )( , ) cos( ) t v i t r i dt    = − −  (3.17) ( )( )( )( , ) x t h x ti r dt  = −  (3.18) where the first term part on the right side corresponds to x(t), given by (3.9). therefore, it readily follows from (3.17) that, the flux-linkage characteristics can be achieved using the resistor measurement r, the values of x(t) and the phase current measurement ˆ ( ) m i t . in this respect, it is worth emphasizing that no other information (experiment) is required to determine the flux-linkage characteristics. indeed, the latter can be estimated using only the available information’s in the first stage (i.e. ˆ ( ) m i t and x(t)). 20 l. kadi, a. brouri, a. ouannou copyright ©2020 assa. adv. in systems science and appl. (2020) 4. simulation results & discussion to show the efficiency of this approach, the machine 8/6 srm is considered in simulations. let us choose the rotor position 0 =  as the aligned position of phase a (fig. 2.4a). accordingly, 30 =  corresponds to the unaligned position of phase a (fig. 2.4b) and 15 =  corresponds to the midway position. simulations were carried out using a fem software by collecting data acquisition at several specific positions (e.g. at aligned, midway and unaligned positions). firstly, the srm is being excited by the sinusoidal phase voltage ( ) sin( )v t v t= , where the amplitude 400v v= and the frequency 314 / srd = . according to the proposition 3.2, note that the amplitude v and frequency  (rd/s) of the input signal are kept constant. accordingly, the steady state of phase current i(t) is also periodic of the same period of v(t). furthermore, note that a noise signal ( )t is added to the useful information (i.e. i(t)). this noise is supposed to be a zero-mean ergodic stochastic process. then, the system output y(t), i.e. the undisturbed phase current added to the measurement noise, is collected on the interval  )0t mt , where m=10. fig.4.1 illustrates y(t) for three rotor positions. the plot confirms that the phase current measurement y(t) is also periodic of the same period of v(t). then, using the estimator (3.10a-b), the phase current estimate ˆ ( ) m i t can be obtained for any given rotor position. this latter is plotted in fig. 4.2 over one period of time. it is clearly seen that, as the rotor gets closer to the aligned position, the saturation effect becomes more significant and notably distorts the shape of the phase current (the black curve in fig. 4.2). bearing in mind that x(t) is not accessible to measurement, but it can be easily determined at any time t using the expression (3.9) (fig. 1.2). then, the nonlinearity estimate ˆ (.) m h can be featured by the set of points ( )  ) ˆ( ), ( ) ; 0 m x t i t t t (using the phase current estimate ˆ ( ) m i t ). at this stage, it is worth emphasizing that two cases can be observed according to the system nonlinearity h(.) (proposition 3.2). fig. 4.1. output signal ( )y t fig. 4.2. phase current estimate signal ˆ ( )mi t 4.1. srm magnetization characteristics without considering the hysteresis effect (static curves) presently, we suppose that h(.) is static nonlinearity. then, the shape of h(.) can be described using the phase current estimate and the inner signal x(t) over one half-period (proposition 21 frequency-geometric identification of magnetization characteristics copyright ©2020 assa. adv. in systems science and appl. (2020) 3.2). accordingly, an accurate estimate ˆ (.) m h of the system nonlinearity h(.) can be obtained using the set of points ( )  ) ˆ( ), ( ) ; 0 / 2 m x t i t t t (i.e. by plotting the phase current estimate ˆ ( ) m i t in respect to x(t)). then, the nonlinearity estimate ˆ (.) m h for different rotor positions, e.g. aligned, midway and unaligned, can be achieved by repeating the same experiment. in this respect, the obtained nonlinearity estimate ˆ (.) m h for aligned, midway, and unaligned positions are shown by fig.4.3. the plots confirm that the function ˆ (.) m h presents a severe nonlinearity at aligned position due to the saturation effect. then, another important feature of srm is the flux-linkage, which is very interesting to be estimated. for convenience, let us consider the expression of flux-linkage ( , )i  in (3.17). accordingly, this latter can be determined by integrating the estimated current and using the signal x(t) (given by (3.9)). in this respect, it is worth emphasizing that the determination of flux-linkage features can be obtained using uniquely the results of experiments carried out in the first stage. in the case where the phase current is unidirectional, the first quadrant of (λ, i) plane is considered. then, the obtained flux-linkage characteristics are shown in fig.4.4. these results show that, using the proposed approach, the flux-linkage characteristics are acquired with good accuracy compared to those obtained using magnetostatic model. fig. 4.3. the estimate of the function h(.) without taking into account the hysteresis effect 22 l. kadi, a. brouri, a. ouannou copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 4.4 flux-linkage characteristics without considering hysteresis effect 4.2. srm magnetization characteristics considering the hysteresis effect presently, the nonlinear elements are allowed to be hysteretic. to this end, a frequencygeometric tool is used as shown in section 3. in this respect, the determination of electrical nonlinearities and flux-linkage characteristics of srm can be found using the same experiment suggested in the first case (without considering the hysteresis effect). then, it follows proposition 3.2 and using phase current estimate ˆ ( ) m i t , the hysteretic nonlinearity estimate ˆ (.) m h can be featured by the set of points ( )ˆ( ), ( ) m x t i t for  )0 2 /t   . so, the curves of system nonlinearity estimate ˆ (.) m h for three key positions (aligned, midway and unaligned positions) are shown by fig.4.5. as can be seen, the obtained curves of ˆ (.) m h are not static and are closer to closed locus (in steady state). these locus will be referred to the hysteretic behaviors of srm. furthermore, note that the surface of hysteretic curve increases according to the position rotor. accordingly, the hysteretic losses are small at the aligned position and become high at the unaligned position. . fig. 4.5. the estimate of the function h(.) considering the hysteresis effect 23 frequency-geometric identification of magnetization characteristics copyright ©2020 assa. adv. in systems science and appl. (2020) in the case where the phase current is unidirectional, the first quadrant of ( , )i plane is considered. accordingly, the obtained flux-linkage ( , )i  characteristics are shown in fig.4.6. the achieved flux-linkage characteristics show that ( , )i  is not static, but it is close to a closed locus. moreover, the hysteretic surface is considerably narrow at the aligned position and increases according to the position rotor. note that, the validation of the identification method was performed using magnetostatic technic as can be seen in fig .4.4. let us recall that, the magnetostatic method is a pure numerical solution which requires a full set of data obtained from fem [25], [32]. this method differs substantially from our approach, which is based only upon the input-output measurement data. as can be seen in fig.4.4, the achieved flux-linkage shapes using the proposed method and those obtained using magnetostatic give very close results. in the case where the hysteresis effect is considered, the present analysis confirms the existence of hysteresis effect, which is not too discussed in the literature. fig. 4.6. flux-linkage characteristics with hysteresis effect 5. conclusion this paper proposes an analytical modeling and identification method of magnetization characteristics of srm. one key step in the design of such method is the construction of the nonlinear block-oriented model modelling the behavior of the srm. specifically, it is shown that the srm can be described by a wiener model. to our knowledge, no previous work has been dealt with this technic. then, an analytical identification method is developed to obtain the magnetization characteristics of the srm. the identification method is dealt based on a simple frequency-geometric method by applying a sine input voltage to the phase winding, for any given rotor position. then, the inner signal x(t) of the nonlinear wiener system (the srm model) can be easily estimated at any time t. accordingly, all the electrical parameters of srm can be accurately estimated using the phase voltage and the recording of the current measurement at the winding terminal. on one hand, if the hysteresis effect of srm nonlinearities is not considered, the magnetization characteristics can be estimated using data-acquisition over one-half period. the obtained results are compared with other achieved data using the fem method. on the other hand, the identification of srm magnetization features can be performed even if the hysteresis effect is considered using only data-acquisition over one period. in this respect, it 24 l. kadi, a. brouri, a. ouannou copyright ©2020 assa. adv. in systems science and appl. (2020) is worth emphasizing that very few available works have attempted to identify the hysteresis effects of srm. furthermore, it is shown that the hysteretic losses are generally small at the aligned position and can be of high value at the unaligned position. this result is confirmed by the area of the obtained hysteresis curve. another originality of the present study lies in the fact that the machine model can be analytically represented using block-oriented systems. the proposed wiener model still be very simple in use and the magnetization characteristics can be easily achieved using the frequency-geometric method. another feature of this work is the fact that the modeling design and the identification scheme require less computational time, unlike several previous studies. the accuracy of the model has been verified by comparing the obtained characteristics using this approach with those using the magnetostatic method. despite all efforts in modeling and identification of srm, finding an accurate analytical model, which represents the nonlinear behavior of srm, is wide open for scientific research. in terms of future work, it would be interesting to use the same model in real time. then, this model can used to obtain the dynamical model of srm. the hysteresis measurement (surface of the hysteretic curve) can offer some suggestions to handle the losses. this approach could be tested on some applications of srm using the obtained model. references 1. r. cardenas, r. pena, m. perez, j. clare, g. asher, and p. wheeler, “control of a switched reluctance generator for variable-speed wind energy applications,” ieee transactions on energy conversion, vol. 20, no. 4, 781–791, 2005. 2. d. a. torrey, “switched reluctance generators and their control,” ieee transactions on industrial electronics, vol. 49, no. 1, 3–14, 2002. 3. t. j. e. miller, switched reluctance motors and their control. magna physics , oxford : clarendon press , new york : oxford university press., 1993. 4. a. emadi, handbook of automotive power electronics and motor drives. taylor & francis., 2017. 5. n. saha, a. k. panda, and s. panda, “speed control with torque ripple reduction of switched reluctance motor by many optimizing liaison technique,” journal of electrical systems and information technology, vol. 5, no. 3, 829–842, dec. 2018. 6. d. marcsa and m. kuczmann, “design and control for torque ripple reduction of a 3phase switched reluctance motor,” computers & mathematics with applications, vol. 74, no. 1, 89–95, jul. 2017. 7. t. a. s. barros, p. j. s. neto, m. v. paula, a. b. moreira, p. s. n. filho, and e. r. filho, “automatic characterization system of switched reluctance machines and nonlinear modeling by interpolation using smoothing splines,” ieee access, 1–1, 2018. 8. m. v. terzic, h. li, b. bilgin, and a. emadi, “comparison of experimental methods for electromagnetic characterization of switched reluctance motors,” in 2018 xiii international conference on electrical machines (icem), 2018, 1881-1888. 9. b. parreira, s. rafael, a. j. pires, and p. j. costabranco, “obtaining the magnetic characteristics of an 8/6 switched reluctance machine: from fem analysis to the experimental tests,” ieee transactions on industrial electronics, vol. 52, no. 6, 1635–1643, dec. 2005. 25 frequency-geometric identification of magnetization characteristics copyright ©2020 assa. adv. in systems science and appl. 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(2020) switched reluctance machine,” 1–6, 2019. 26. a. arifin, “switched reluctance generator drive in the low and medium speed operation : modelling and analysis,” massey university, 2013. 27. e. darie and c. cepişcǎ, “the use of switched reluctance generator in wind energy applications,” 2008 13th international power electronics and motion control conference, epe-pemc 2008, no. october 2008, 1963–1966, 2008. 28. s. vukosavic and v. stefanovic, “srm inverter topologies: a comparative evaluation,” ieee transactions on industry applications, vol. 27, no. 6, 1034–1047, 1991. 29. r. krishnan, switched reluctance motor drives : modeling, simulation, analysis, design, and applications. crc press, 2001. 30. y. sofiane, a. tounzi, and f. piriou, “a non linear analytical model of switched reluctance machines,” applied physics, vol. 172, 163-172., 2002. 31. s. w. zhao, n. c. cheung, c. k. lee, x. y. yang, and z. g. sun, “survey of modeling methods for flux linkage of switched reluctance motor,” in 2011 4th international conference on power electronics systems and applications, 2011, 1-4. 32. l. kadi and a. brouri, “numerical modeling of a nonlinear four-phase switched reluctance machine,” in 2017 international renewable and sustainable energy conference (irsec), 2017, 1-6. frequency-geometric identification of magnetization characteristics of switched reluctance machine 1. introduction 2. operation principle and equations describing srm 3. modeling and identification approach of srm 4. simulation results & discussion 5. conclusion microsoft word 1165 article text, copyedited.doc adv syst sci appl 2021; 04; 100-114 published online at https://ijassa.ipu.ru. assessment of impact of trade wars on production and exports of the russian federation using the agent-based model aleksandra mashkova 1,2*, albert bakhtizin2 1) orel state university named after i.s. turgenev, orel, russia e-mail: aleks.savina@gmail.com 2) central economics and mathematics institute russian academy of sciences, moscow, russia e-mail: albert.bakhtizin@gmail.com abstract: in this article we consider methods of simulating consequences of trade wars using the agent-based model. the presented model simulates dynamics of trade relations between russia, the united states, china, the european union, and the rest of the world. we present event structure of the model which reflects interaction among different types of agents in the model: countries, organizations and residents. international trade is simulated at micro-level, as a set of supplies (purchases and sales) of organizations in different countries. volume and structure of trade flows is changed annually, based on the algorithm that takes into account existing and newly imposed trade restrictions and the expected change in final demand. for information support of the model we use official statistical sources of the countries included in the model. before loading these data to the model we process it to the similar sectoral structure and time period. for scenario calculations we consider optimistic and pessimistic scenarios for the world economy dynamics in the context of epidemiological risks and three possible options for world trade policy: preservation, cancellation or imposition of new trade restrictions. simulation results for the russian federation show that current sanctions against it affect exports in a number of industries, but do not have a significant impact on the dynamics of value added, while imposition of new restrictions could slow down the economic growth rate by 0.3-0.5% annually. keywords: agent-based model, trade wars, trade restrictions, import, export, scenario calculations 1. introduction various sanctions measures have increasingly become an instrument of world economic policy in recent years. imposed restrictions affect various sectors of the economy, access to financial markets, imports and exports of countries. despite the fact that a number of studies have shown that all participants suffer to a greater or lesser extent from participation in a trade war [4,5,18], restrictions continue to operate and be updated. in particular, since 2014, the russian federation has been under sanctions from the united states and the european union in relation to the financial sector, mining and transportation of minerals. in this regard, the question arises of assessing how much damage the imposed sanctions cause to the industries that fall under their influence, how they affect the output of these industries and the volume of export of their products to various countries. the subject of this study is trade flows among countries involved in trade conflicts, as well as changes in their volumes and structure under the influence of imposed trade restrictions. this task implies taking into account a large number of factors influencing trade flows, such as the final demand for products, availability of raw materials, utilization of fixed * corresponding author: aleks.savina@gmail.com assessment of impact of trade wars on production and exports of the russian… 101 copyright ©2021 assa. adv. in systems science and appl. (2021) assets, access to investments and fluctuations in prices for goods, therefore, to solve it, it is proposed to develop of a computer model for assessing consequences of trade wars. agent-based modeling was chosen as the main research method, which makes it possible to evaluate the dynamics of the global system as a result of the interaction of different agents: countries, organizations and residents. in the proposed model interaction among these agents determines the direction and structure of the trade flows among countries and their change under the influence of demand and government regulation. the developed model is based on arrays of real data on the economy and population of its member countries and trade exchange among them, which makes it possible to make reasonable forecasts of trade relations dynamics in various political conditions for each participating country. 2. literature review in the early 2000s most of the scientific publications devoted to analysis of trade wars referred to reviews of events and econometric analysis of retrospective data series [3,11]. later in 2010s the researcers turned to developing models including sets of abstract countries that do not reflect the characteristics of real countries [2,8,18]. in the last decade the task of forecasting the economic consequences of trade wars for the participating countries is most often solved using computable general equilibrium (cge) models, in which statistical data for different countries are loaded.the most famous project in the field of developing tools for quantifying trade wars between real countries is the global trade analysis project (gtap), which brings together scientific researchers from all over the world [1,7]. gtap was initiated in 1992 and proposed a unified methodology based on the cge approach. the model complexes developed using the gtap methodology include groups of countries (sometimes the whole world) and all sectors of the economy.within gtap a set of simulation complexes was developed: 1. worldscan, which consists of general equilibrium models of 29 products, trading in 30 countries, having the largest weight in world gdp, and consolidated regions that include several countries. in the model, supply and demand for a specific product in a given country are formed taking into account supply and demand of this product in other countries, and the price depends on substitution possibilities, transport costs, trade barriers, and other factors [4]. 2. globe multisector model [16], developed by specialists of the hohenheim university and the us naval academy, was used to assess the consequences of trade wars between the countries of the north american free trade zone, regulated by north american free trade agreement. 3. miragrodep multicountry multisector model, developed at the international research institute for food policy (washington, usa), based, in addition to the gtap methodology, on the more general mirage model (modeling international relationships under applied general equilibrium). the focus of this model is on goods trade between the us, china and mexico [5]. 4. the center for international trade and economics and the institute of world economy and politics of the chinese academy of social sciences have developed a global model to assess the consequences of the trade war between the united states and china [13]. the largest international organizations have developed their multi-country models: organization for economic co-operation and development (oecd) и international monetary fund (imf). the oecd’s new global model considers 6 groups of oecd countries and 3 groups of non-oecd countries. oecd [12]. the imf’s global macrofinancial model belongs to the class of dynamic stochastic general equilibrium models (dsge) and considers more than 40 of the largest economies in the world [20]. the listed models are aimed at considering the world's largest players, most often the usa and china, sometimes also eu countries. even though in some of the pre-sented model 102 a. mashkova, a. bakhtizin copyright ©2021 assa. adv. in systems science and appl. (2021) complexes russia stands out as a separate participant, the available publications do not provide an assessment of the consequences of economic sanctions for russia, which have been regularly introduced against it since 2014. thus, developing a model of trade wars, which would include russia as one of the key participants, remains an urgent task. 3. materials and methods 3.1. structure of the model of trade wars within the agent-based model of trade wars, we study dynamics of trade relations between russia, the united states, china, the european union, and the rest of the world (fig. 3.1). the model is programmed as a software simulator which represents interactions among three types of agents: organizations, residents and states. fig. 3.1. concept of the model resident agents act as employees of organizations, on the one hand, and consumers of goods and services, on the other. agents' income goes to the accumulator, from which purchases of the final product are then paid. trade relations in the model are simulated at the level of individual organizations, and states regulate the processes of commodity exchange by introducing and canceling restrictions and changing tariffs on imports and exports of products. for agent-organizations, the current volume of output, the price in national currency, the number of workers and the volume of fixed assets are set. organizations interact with each other through supplies that reflect purchases and sales, including international ones. the suppliers in the model are agent-organizations, and the buyers can be other agent-organizations, the state, or the agentsresidents. supplies are divided into three types: 1. intermediate. intermediate supplies include supplies of raw materials, components and services to other organizations. for the convenience of further calculations, intermediate assessment of impact of trade wars on production and exports of the russian… 103 copyright ©2021 assa. adv. in systems science and appl. (2021) supplies are divided into basic (directly dependent on the dynamics of the organization's output) and additional (general business needs and accompanying services). 2. investment supplies, which include various types of fixed assets. investment supplies are also divided into basic (equipment and buildings) and additional (other supplies of products that are put on the balance sheet of organizations as fixed assets). 3. final products supplies purchased by resident agents. for each supply, the identifier of the supplier and the buyer is specified, as well as volume of supply in standard units, selling and purchase prices. the selling price is set in the currency of the state in which the supplier agent is located, excluding taxes. the purchase price consists of the sales price, sales taxes, export and import taxes (for international shipments), and is set in the currency of the country in which the agent-buyer is located. the states in the model regulate tax rates, benefits to the population, amount of financing of public sector organizations and subsidies, and also adjust the existing sanctions on commodity exchange with other countries. trade restrictions on import and export are set as the share of the turnover of industry j that falls under their influence, while ; . interaction of various agents in the model within its main events is presented on fig. 3.2. in the event “benefits payment”, the state transfers benefits to resident agents, while the corresponding accumulators (state budget and residents' funds) are changed. the event "public sector financing" implements transfer of funds from the budget to public sector organizations to pay their expenses. also, the state makes payments on the federal debt and may impose sanctions to other countries in the form of trade restrictions. fig. 3.2. event structure of the model the event "purchase of materials" simulates supply of intermediate products among production organizations, including import and export of products. when the product is sold, the payment is credited to the account of the agent-supplier at the selling price in the currency of sale and debited from the account of the agent-buyer in the currency of purchase. 104 a. mashkova, a. bakhtizin copyright ©2021 assa. adv. in systems science and appl. (2021) trade organizations act in the same way in the event "wholesale purchase", resident agents in the event "retail purchase", and public sector organizations in the event "state procurement". after the completion of sales, taxes are transferred to the budgets of countries: internal taxes are charged on supplies where the supplier and the buyer were from the same country, and import / export taxes are charged on the international supplies. the event "wage payment" is implemented by organizations of all types, these funds are transferred to the accumulators of resident agents. for commercial organizations, the financial result is calculated. profit is defined as the difference between sales and costs of intermediate supplies, salaries and taxes. the profit is divided into two parts: the first is distributed as personal income from entrepreneurial activities, the second is used to pay for investments, along with subsidies received from the state. the production and sales forecast is carried out taking into account the existing and newly imposed trade restrictions and the expected change in final demand, which, in turn, entails a change in the output of raw materials and components. 3.2. algorithms the actions of organizations in countries, which became subject to trade restrictions, are based on a number of assumptions: supplies from countries that have imposed restrictions may be replaced by domestic supplies or supplies from other countries of products of a similar industry. purchasing organizations of various industries have equal priority in the procurement of raw materials, that is, the resulting shortage of some materials and components would be proportionally distributed among all buyers. substitution of supplies that have fallen under trade restrictions for supplies from available sources is carried out in proportion to the share of available sources in the initial distribution of supplies. that is, if the supply of products of industry j from country b accounted for 60% of the supply of this industry to country a, then in replacing the missing supplies from other countries, suppliers from country b would also account for 60% of the deficit in industry j. the algorithm below shows the sequence of recalculation of supplies under the influence of trade restrictions in accordance with the introduced assumptions: 1. calculation of permitted supplies for organizations in industry j, for products of which trade restrictions has been imposed: (3.1) where – volume of supply of organization-buyer b available from organization-supplier s of industry j in the next modeling year (y+1), – volume of supply of organization-buyer b delivered from organization-supplier s in the current modeling year y; – trade restriction on import of products of industry j, imposed in the modeling year (y+1); , – trade restriction on export of products of industry j, imposed in the modeling year (y+1). 2. calculation of lacking supplies for organizations: (3.2) where – volume of lacking supply of organization-buyer b from organizationsupplier s in the next modeling year (y+1); – volume of supply of organization-buyer b available from organization-supplier s in the next modeling year (y+1), – volume of 𝑉𝑏𝑠 𝑦+1 = 𝑉𝑏𝑠 𝑦 ∗ )1 − 𝑟𝑗 𝑖𝑚𝑝 0 ∗ (1 − 𝑟𝑗 𝑒𝑥𝑝 ) 𝑉𝑏𝑠 𝑦+1(𝑙𝑎𝑐𝑘) = 𝑉𝑏𝑠 𝑦 − 𝑉𝑏𝑠 𝑦+1 assessment of impact of trade wars on production and exports of the russian… 105 copyright ©2021 assa. adv. in systems science and appl. (2021) supply of organization-buyer b delivered from organization-supplier s in the current modeling year y. 3. grouping of lacking supplies by industry j across all organizations: (3.3) where – volume of lacking supply of organization-buyer b from organizationsupplier s in the next modeling year (y+1); – total volume of lacking supply of organizations-buyers from organizations-suppliers of industry j in the next modeling year (y+1). 4. choice of organizations for increase of supplies of industry j from countries that have not imposed trade restrictions. 5. calculation of increase of supplies of organizations from countries that have not imposed trade restrictions: (3.4) where – volume of additional supply of organization-buyer b available from organization-supplier s of industry j in the next modeling year (y+1), – total volume of lacking supply by industry j in the next modeling year (y+1); – share of organization s in supplies of products of industry j; – share of organization b in purchase of products of industry j. 6. recalculation of supplies of organizations from countries that have not imposed trade restrictions: (3.5) where – volume of supply of organization-buyer b delivered from organizationsupplier s of industry j in the next modeling year (y+1), – volume of supply of organization-buyer b delivered from organization-supplier s in the current modeling year y; – volume of additional supply. the change in trade flows between countries is also greatly influenced by the change in demand for the products of industries from organizations and consumers. to model this process, the following assumptions are made in the model: the starting point for changing the output and supplies of organizations is the final consumer demand, and the intermediate demand for materials and components is considered as a derivative of the final one. the industry structure of supply organizations is considered unchanged throughout the modeling period. intermediate supplies of organizations are divided into basic and additional, and within the framework of the sales adjustment algorithm, only basic intermediate supplies are changed. supplies are divided in such a way so that in the terminal sectors of the algorithm (agriculture and mining), all intermediate supplies would be additional. this assumption makes it possible to avoid the infinite recursion of the algorithm, since the organizations of industries that do not have basic supplies are processed at last. the algorithm below shows the sequence of recalculation of supplies due to change in final demand is carried out in accordance with the introduced assumptions: the work of the algorithm begins with assessing of the expected change in final demand on the wholesale purchases of the final products (fig. 3.3). for this, the demand for the final product is recalculated: 𝑉𝑗 𝑦+1(𝑙𝑎𝑐𝑘) = --𝑉𝑏𝑠 𝑦+1(𝑙𝑎𝑐𝑘) 𝑚 𝑠=1 𝑛 𝑏=1 𝑉𝑏𝑠 𝑦+1(𝑎𝑑𝑑) = 𝑉𝑗 𝑦+1(𝑙𝑎𝑐𝑘) ∗ 𝑃𝑠 𝑗 ∗ 𝑃𝑏 𝑗 𝑉𝑏𝑠 𝑦+1 = 𝑉𝑏𝑠 𝑦 + 𝑉𝑏𝑠 𝑦+1(𝑎𝑑𝑑) 106 a. mashkova, a. bakhtizin copyright ©2021 assa. adv. in systems science and appl. (2021) (3.6) where – final demand for products of industry i in the next modeling year (y+1); – final demand for products of industry i in the current modeling year y; – yearly dynamics of final demand. fig. 3.3. algorithm for changes in sales due to the dynamics of final demand then there is a sequential processing of organizations supplying the final product, within which the values of their sales are adjusted due to changes in final demand for their product: (3.7) where – total sales of products of industry i in the next modeling year (y+1); – final demand for products of industry i in the next modeling year (y+1); – intermediate demand for products of industry i in the current modeling year y; – investment demand for products of industry i in the current modeling year y. the coefficient of change of intermediate supplies is calculated: (3.8) where – yearly dynamics of sales of products of industry i; – total sales of products of industry i in the next modeling year (y+1); – total sales of products of industry i in the current modeling year y. 𝐹𝐷𝑖 𝑦+1 = 𝐹𝐷𝑖 𝑦 ∗ (1 + ∆𝑑) 𝑆𝑖 𝑦+1 = 𝐹𝐷𝑖 𝑦+1 + 𝐼𝑛𝑡𝐷𝑖 𝑦 + 𝐼𝑛𝑣𝐷𝑖 𝑦 ∆𝑠𝑖 = 𝑆𝑖 𝑦+1 𝑆𝑖 𝑦 − 1 assessment of impact of trade wars on production and exports of the russian… 107 copyright ©2021 assa. adv. in systems science and appl. (2021) the values of intermediate deliveries of organizations-suppliers of final products are recalculated: (3.9) where – intermediate demand for products of industry j from organizations of industry i in the next modeling year (y+1); – intermediate demand for products of industry j from organizations of industry i in the current modeling year y; – yearly dynamics of sales of products of industry i. for each organization-supplier of intermediate products, the values of their sales are adjusted, due to the cumulative change in final and intermediate demand for their product: (3.10) where – total sales of products of industry i in the next modeling year (y+1); – final demand for products of industry i in the next modeling year (y+1); – intermediate demand for products of industry i in the next modeling year (y+1); – investment demand for products of industry i in the current modeling year y. the corresponding coefficients of change in intermediate supplies and their new values are calculated by the formulas (3.8) and (3.9). the procedure for checking supplier organizations is determined by their industry affiliation: first, organizations that produce final products (light industry), then intermediate (production of fuel, materials and chemical products), and at the end raw materials (agriculture and mining). 3.3. information support of the model initial modeling data is based on official sources presented on the portals of the russian federation federal state statistics service [19], bureau of economic analysis of the united states department of commerce [6], national bureau of statistics of china [17] and eurostat [9]. from these sources tables of gdp structure, input-output tables, structure of import and export between different countries is used. sectoral classifiers of the economy differ significantly in different countries, therefore, in the model, aggregated industries were identified, representing commodity groups that are significant for international trade: 1. agriculture & food production. 2. mining. 3. fuel production. 4. public sector. 5. chemical production. 6. production of materials. 7. production of equipment and transport. 8. light industry. 9. service. 10. trade. 11. construction table 3.1 presents industries in different countries included in the aggregated industry “production of materials”. the time periods for which production information is available also differ. the most relevant input-output tables for the eu and the united states are presented for 2019, for china and russia for 2017, therefore, for the last two countries, the tables were recalculated in proportion to the change in value added in the respective industries in 2018 and 2019. 𝐼𝑛𝑡𝐷𝑖𝑗 𝑦+1 = 𝐼𝑛𝑡𝐷𝑖𝑗 𝑦 ∗ (1 + ∆𝑠𝑖) 𝑆𝑖 𝑦+1 = 𝐹𝐷𝑖 𝑦+1 + 𝐼𝑛𝑡𝐷𝑖 𝑦+1 + 𝐼𝑛𝑣𝐷𝑖 𝑦 108 a. mashkova, a. bakhtizin copyright ©2021 assa. adv. in systems science and appl. (2021) table 3.1. structure of aggregated industry “production of materials” aggregated industry in the model industries in the eu industries in russia industries in the usa industries in china production of materials manufacture of wood and of products of wood and cork, except furniture; manufacture of wood and of products of wood and cork, except furniture; wood products manufacture of wood and wood products, paper and paper products; publishing and printing manufacture of pulp, paper and paper products manufacture of paper and paper products paper products publishing, printing and reproduction of recorded media publishing, printing and reproduction of recorded media printing and related support activities manufacture of basic metals and fabricated metal products manufacture of basic metals manufacture of basic metals primary metals manufacture of fabricated metal products, except machinery and equipment manufacture of fabricated metal products, except machinery and equipment fabricated metal products initial data was loaded into the agent-based model of trade wars in the form of excel spreadsheets. 3.4. program realization the presented model of trade wars was programmed in microsoft visual studio using c# and postgresql as the database management system. this instrument was chosen instead of specified multi-agent simulation platform, since the model works with large arrays of data and quite a large number of agents (about 800 thousand), and programming in c# allows to reach high-performance computing. fig. 3.4 shows the software architecture of the trade wars model, which includes the model database and two main procedures: generation of model objects based on source data and simulation procedure that reproduces the dynamics of trade flows between countries in accordance with the algorithms described above. fig. 3.4. software architecture of the model generation procedure is the link from aggregated statistical data to the objects of the agent-based model, stored in the model database. within generation procedure population and organizations are created and interconnections among them are set. population in both models is reproduced taking into account gender-age structure, as discussed in [15]. one agent in the model stands for 10000 of residents in the real world, which results in 779400 agents living in 5 countries in the model (table 3.2). assessment of impact of trade wars on production and exports of the russian… 109 copyright ©2021 assa. adv. in systems science and appl. (2021) table 3.2. count of agents in countries of the model country resident count in 2020, mln agents count russia 146,75 14675 the usa 331,45 33145 the european union 447 44700 china 1411,78 141178 the rest world 5457,1 545710 total 7794 779400 organizations in the model of trade wars are aggregated: one organization in the model responds to a set of organizations of one aggregated industry in one country. thus, in the developed model, there are 55 organizations of 11 industries in 5 countries, which create and maintain more than 2000 interconnected trade flows. agents of working age are assigned to organizations through workplaces. these issues are discussed in more detail in [14]. the generated objects are stored in the model database and used for multivariate calculations using the procedure for simulation of the dynamics of trade flows between countries under the conditions of imposed restrictions, which are set as the control parameters of the model. the obtained results are also stored in the database, and are available for further analysis via the postgresql database management system interface. 4. results and discussion based on the developed software model, a series of calculations was carried out to assess the impact of trade restrictions on the output and export of organizations in countries under sanctions restrictions. in the first year of modeling, corresponding to 2020, the fall in the countries' gdp was reproduced, corresponding to the official estimates of their statistical offices. in particular, for russia, the federal state statistics service estimates the decline in gdp at 3.1% [19]. based on the algorithms of the model, a corresponding recalculation of supplies among organizations was made. thus, the forecast simulation period starts from 2021. to make forecasts of dynamics of production and commodity exchange, a medium-term period of 5 years was chosen, which is associated, first of all, with the significant uncertainty of the global epidemiological situation. in the context of this uncertainty, two scenarios are considered: an optimistic one, assuming a global economic recovery in 2021-2022, and a pessimistic one, in which a repetition of waves of coronavirus infection would lead to slowdown in the recovery of production and demand. within each scenario, three options for trade relations among countries are considered: 1. cancellation of all existing restrictions on the import and export of products. under these conditions, the dynamics of commodity exchange would be determined only by speed of economic recovery. 2. preservation of existing restrictions. currently, there are eu and us restrictions packages against russia, russian restrictions against the eu and the united states, and reciprocal sanctions between the united states and china. for russia, the imposed restrictions affect sectors of agriculture and food production (aggregated industry #1 in the model), aircraft and rocketry (aggregated industry #7) [10]. the eu, and especially the united states, restrict the construction of the nord stream 2, which affects industries of mechanical engineering, mining and fuel production (aggregated industries #2, #3 and #7 in the model). 3. imposition of new restrictions since 2022 affecting about 5% of trade volume in key industries. within the framework of scenario calculations, the model was used to compare the dynamics of russia's gdp and export volumes under various variants of world trade policy. consider the results obtained under the optimistic scenario. as the data in fig. 4.1 show, the cancellation or preservation of existing trade restrictions does not have a significant 110 a. mashkova, a. bakhtizin copyright ©2021 assa. adv. in systems science and appl. (2021) effect on the dynamics of gdp, while imposition of new restrictions leads to slowdown in annual economic growth rates by 0.5%, and in the first two years of after imposition of restrictions, their influence is more noticeable. fig. 4.1. dynamics of russian gdp within alternative trade policy (optimistic scenario) fig. 4.2 shows the forecast for the dynamics of export volumes under the optimistic scenario. here, the comparatively weak influence of the existing restrictions is also noticeable, however, the effect of imposition of new ones is more noticeable: from 3-3.5% growth, the dynamics of exports falls to 1-2% after imposition of new restrictions. fig. 4.2. dynamics of russian exports within alternative trade policy (optimistic scenario)) in general, the results presented in fig. 4.1 and 4.2 indicate a fairly high degree of stability of the russian economy and a relatively weak impact of the current sanctions on its development. the results obtained as a result of computer calculations can also be presented in a more detailed form, for separate exporting industries (table 4.1). the obtained data demonstrates that, while maintaining the existing restrictions, the export of the industry "production of machinery and equipment" suffers insignificantly, while growth rates of products of other industries stay at the same level. after imposition of new restrictions in 2022, the export of mining industry changes the growth rate from positive (3-4%) to negative (about -1%), while fuel sales continue to grow at a slower pace (12% instead of 4%). also the export of the "manufacturing of machinery and equipment" sector shows a slowdown in growth rates by about 0.5% annually. assessment of impact of trade wars on production and exports of the russian… 111 copyright ©2021 assa. adv. in systems science and appl. (2021) table 4.1. dynamics of export by industry, percents relative to the previous year (optimistic scenario) industry year 2020 2021 2022 2023 2024 2025 cancellation of restrictions agriculture & food production -5,3 5,1 4,6 5 5,6 5,9 mining -3,9 4,7 3,5 3,6 3,6 3,7 fuel production -4,3 4,9 3,9 4,1 4,3 4,5 chemical production -3,4 3,9 3,6 3,7 3,9 4 production of materials -1,1 3 2,9 3,1 3,2 3,4 production of machinery and equipment 0 3 3,1 3,3 3,4 3,5 miscellaneous manufacturing -8,8 7,4 6 6,2 6,3 6,5 preservation of restrictions agriculture & food production -5,3 5,1 5,3 4,9 5,9 5,6 mining -3,9 4,7 3,5 3,6 3,6 3,7 fuel production -4,3 4,9 3,9 4,1 4,3 4,5 chemical production -3,4 3,9 3,6 3,7 3,9 4 production of materials -1,1 3 2,9 3,1 3,2 3,4 production of machinery and equipment 0 2,7 2,9 3 3,2 3,4 miscellaneous manufacturing -8,8 7,4 6 6,2 6,3 6,5 imposition of restrictions agriculture & food production -5,3 5,1 4,6 5 5,4 5,7 mining -3,9 4,7 -1,2 -1 -0,7 -0,3 fuel production -4,3 4,9 1,1 1,5 1,9 2,2 chemical production -3,4 3,9 3,6 3,7 3,9 4 production of materials -1,1 3 3 3,1 3,3 3,4 production of machinery and equipment 0 2,7 2,4 2,6 2,8 3,1 miscellaneous manufacturing -8,8 7,4 6,1 6,2 6,4 6,5 in the context of the pessimistic scenario, there is also no significant effect of cancellation of trade restrictions on the dynamics of gdp compared to their preservation, while imposition of new ones slows down the economic growth rate by 0.3% annually (fig. 4.3). fig. 4.3. dynamics of russian gdp within alternative trade policy (pessimistic scenario) in 2022, the preservation of restrictions leads to even greater growth in russian export volumes compared to their cancellation, which is associated with restrictions to other countries resulting in rise of orders to suppliers from russia (fig. 4.4). imposition of new restrictions in the context of the pessimistic scenario leads to a drop in export growth rates to 0.8-1.2%. 112 a. mashkova, a. bakhtizin copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 4.4. dynamics of russian gdp within alternative trade policy (pessimistic scenario) calculations based on the presented agent model of trade wars for the russian federation show that the imposed sanctions affect agriculture, mining and fuel production to a greater extent, however, only the mining industry has a negative dynamic with the introduction of new restrictions (up to -1,2%). the rest of the industries maintain a positive growth rate, losing only a little in its speed (from 4,5% to -2,2% for the fuel production). as for the general economic situation, sanctions do not have a significant impact on the dynamics of value added, and the income of the population. the results obtained as a result of modeling reflect possible changes in trade turnover between countries in the sectoral context, as well as the impact of these changes on the overall economic indicators of countries, in particular, gdp. this structure of the modeling output data coincides with the structure of the results of the models based on the gtap methodology. the results obtained are directly related to the assumptions of the model, the most important of which are the following: we consider goods from one industry are interchangeable and do not take into account preferences of the customers to goods from a particular country. such assumptions are widespread in economic models [3,4]. investment in fixed assets is considered elastic, directly proportional to the change in output, which makes it possible to neglect the delays associated with expansion of production. in the calculations performed, the change in trade volume in prices of the base year is estimated, excluding inflation and fluctuations in the exchange rates. direct bans on the import of certain categories of products are considered as sanctions measures, and the increase in customs tariffs that affect the price of products for buyers is not taken into account. the assumptions made generally exclude price competition as a factor influencing international trade, and shift the main emphasis to the fundamental availability of supplies of products of certain industries in the required quantity. the assumption of the interchangeability of goods from different countries makes it possible not to take into account such a problem as the complete unavailability of raw materials and materials of a certain type. thus, the model does not assess the effects of trade wars, such as reduced output or production interruptions due to lack of required materials. the assumption that there are no delays associated with the expansion of production is supposed to be removed in subsequent works to take into account the impact of investment processes in different countries on international trade. assessment of impact of trade wars on production and exports of the russian… 113 copyright ©2021 assa. adv. in systems science and appl. (2021) 5. conclusion this article examined issues of developing a model of trade wars, including as the main participants russia, the european union, the united states and china, as well as the united rest of the world. for the development, an agent-based approach was chosen, which distinguishes the presented model from known analogs based on the principles of computational general equilibrium. the chosen method makes it possible to study international trade relations at the micro level, in the context of supplies of individual organizations, purchases of residents and state procurements. the article presents a number of results obtained, among which the most significant are: 1. the event structure of the model of trade wars, reflecting interaction of state, public sector organizations, manufacturers, trade agents and residents. 2. the algorithm reflecting annual change in the volume of sales and supplies of organizations, which, in turn, determines change in the volume of imports, exports and added value of industries. 3. the sectoral structure of the model, leading to a general view of sectoral classifiers in different countries. 4. parametrization of scenario calculations, including two scenarios for the restoration of the world economy in the context of epidemiological risks (optimistic and pessimistic) and three options for world trade policy (preservation, cancellation or imposition of new trade restrictions). 5. forecasts of the dynamics of gdp, volumes and sectoral structure of russian exports in the context of various scenarios and a comparative assessment of the consequences of trade restrictions. as a direction for further research in this area, a more detailed analysis of the impact of investing in expanding production on the consequences of trade restrictions for different countries was chosen. acknowledgements the reported study was funded by russian science foundation according to the research project № 21-18-00136 “development of a software and analytical complex for assessing the consequences of intercountry trade wars with an application for functioning in the system of distributed situational centers in russia”. references 1. aguiar, a., narayanan, b., mcdougall, r. (2016). an overview of the gtap 9 data base. journal of global economic analysis, 1, 181-208. https://doi.org/10.21642/jgea.010103af 2. alvarez, f. & lucas, r. (2005). general equilibrium analysis of the eaton–kortum model of international trade. journal of monetary economics, 54(6), 1726-1768. https://doi.org/10.1016/j.jmoneco.2006.07.006 3. baier, s.l. & bergstrand, j.h. (2001). the growth of world trade: tariffs, transport costs, and income similarity. journal of international economics, 53 (1), 1–27. https://doi.org/10.1016/s0022-1996(00)00060-x 4. bollen, j. & rojas-romagosa h. (2018, july). trade wars: economic impacts of us tariff increases and retaliation: an international perspective. cpb background document. cpb netherlands bureau for economic policy analysis. [online]. available https://www.cpb.nl/sites/default/files/omnidownload/cpb-background-documentnovember2018-trade-wars-update.pdf 114 a. mashkova, a. bakhtizin copyright ©2021 assa. adv. in systems science and appl. (2021) 5. bouët, a. & laborde, d. (2017, august). us trade wars with emerging countries in the 21st century. make america and its partners lose again. ifpri discussion paper. [online]. available http://ebrary.ifpri.org/cdm/ref/collection/p15738coll2/id/131368. 6. bureau of economic analysis of the united states department of commerce. [online]. available https://www.bea.gov/. 7. corong, e., hertel, t., mcdougall, r., tsigas, m., van der mensbrugghe, d. (2017). the standard gtap model, version 7. journal of global economic analysis 2, 1-119. https://doi.org/10.21642/jgea.030101af 8. dixon, p., jerie, m., rimmer, m. (2016). modern trade theory for cge modelling: the armington, krugman and melitz models. journal of global economic analysis 1(1),1– 110. https://doi.org/10.21642/jgea.010101af 9. eurostat. [online]. available https://ec.europa.eu/eurostat. 10. federal law no. 127-fz. (2018, june 4) on measures of influence (counteraction) on unfriendly actions of the united states of america and other foreign states. [online]. available http://publication.pravo.gov.ru/document/view/0001201806040032. 11. glick, r. & taylor, a. (2005). collateral damage: trade disruption and the economic impact of war. nber working paper series. working paper no. 11565. [online]. available http://www.nber.org/papers/w11565 12. hervé, k., pain, n., richardson, p., sédillot, f., beffy, p. (2010). the oecd’s new global model. economic modelling, 28, 589-601 (2011). https://doi.org/10.1016/j.econmod.2010.06.012 13. li, c., he, c., lin, c. (2018). economic impacts of the possible china-us trade war. emerging markets finance and trade, 54, 1557-1577. https://doi.org/10.1080/1540496x.2018.1446131 14. mashkova, a.l., nevolin, i.v., savina, o.a., burilina, m.a. & mashkov, e.a. (2020). generating social environment for agent-based models of computational economy. communications in computer and information science, 1349, 291-305. https://doi.org/10.1007/978-3-030-67238-6_21 15. mashkova, a.l., novikova, e.v., savina, o.a. & mashkov, e.a. (2021). generating synthetic population for the agent-based model of the russian federation spatial development. advances in social simulation (essa 2019). springer proceedings in complexity. 1349, 183-187. springer nature switzerland. https://doi.org/10.1007/978-3030-61503-1_17 16. mcdonald, s., thierfelder, k.: globe v2: a sam based global cge model using gtap data. [online]. available http://cgemod.org.uk/globev2_2014.pdf 17. national bureau of statistics of china. [online]. available http://www.stats.gov.cn/english/. 18. ossa, r. (2014). trade wars and trade talks with data. american economic review 104(12), 4104–4146. doi: 10.1257/aer.104.12.4104. 19. russian federation federal state statistics service. [online]. available http://www.gks.ru. 20. vitek, f. (2018, april). the global macrofinancial model. international monetary fund working paper, 18/81. [online]. available https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3182506. adv syst sci appl 2020; 01; 114-118 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/864 a data management model for proactive risk management in healthcare dmitry meshkov, lyudmila bezmelnitsyna, sergey cherkasov v.a. trapeznikov institute of control sciences, russian academy of sciences, russia e-mail: dmitrymeshkov@mail.ru, blyu18@gmail.com, cherkasovsn@mail.ru abstract: the delphi investigation made possible to create and present a model of data flows and expertise for decision making in healthcare. the model structures and visualizes data flows including population health risks assessment and healthcare interventions value. most elements of this model are used in practice and are regulated by international or local documents. the investigation also indicates the need for the creation and utilization of additional level of expertise including integrated analysis and modeling of the results of management decisions. structuring of data management in accordance with this model provides an opportunity to implement proactive risk management in healthcare there to increase healthcare effectiveness significantly. keywords: model, proactive, healthcare, management, data, health technology assessment, guidelines, clinical. 1. introduction decision making and management in healthcare always involve the redistribution of budgets and existing major healthcare models are traditionally based on the description of financial flows [4]. the emergence of innovative technologies and their costs not only affected the quantitative changes in financial flows, but also the change in information quality, quantity and distribution in healthcare decision-making. creation in majority of countries health technology assessment agencies responsible for expertise of clinical economic characteristics of healthcare interventions indicates the need for novel supportive models for healthcare decision making based on development and rational use of appropriate methods of prophylactics, diagnostics and treatment [7]. it is evident now that achieving both the expected results in population health as well as in rational spending of funds requires appropriate conditions for use of the intervention within healthcare system such as decrease of accessibility barriers (technological; administrative; and psychological); availability and affordability. our early research demonstrated correlations between the number of new drugs (inn) available in the market and the reduction of patient mortality in the example of cancer. this effect was more probably induced by additional opportunities and proper healthcare management rather than the increase of clinical efficacy of these drugs [5]. these results indicate the need for a novel model of healthcare management formed around population health risks and medical technologies, which meets modern requirements and would make it possible to increase the efficiency of decision-making in healthcare independently or when used in conjunction with other models. 2. objectives the objective of the study was to identify the structural model for proactive healthcare management corresponding to the modern challenges of population health and health mailto:dmitrymeshkov@mail.ru mailto:blyu18@gmail.com mailto:cherkasovsn@mail.ru a data management model for proactive risk management in healthcare 115 copyright ©2020 assa adv. in systems science and appl. (2020) technologies and applicable to any healthcare system regardless of funding sources and health care model as well. oncology was chosen for the case study since this area is of great importance for population health and a significant number of new technologies allowed us to clearly identify the challenges and the processes occurring in this area [5]. 3. method modified delphi investigation was used to determine the nodal elements of the model and their hierarchical relationships [3]. the group of analysts included 7 people. all of them had experience of practical work as doctors and healthcare management as well. two of them had also an economic high education and experience in solving economic issues in healthcare. the other two experts had experience of data generation of medicinal products starting from laboratory investigations to healtheconomic assessments and inclusion of these products in reimbursement lists. the group of experts included 64 physicians specialized in oncology who had also practical experience in management of oncology units and hospitals (finance/management/staff load). they were presented in databases of clinicians participating in international industry sponsored clinical trials and thus were well aware about worldwide experience of oncology diseases treatment. the study was conducted in several stages: 1. analyze of the source data and creation a questionnaire for the expert group to fill in. 2. conducting a survey of the expert group. 3. analysis of the responses received and development of proactive risk-management model in healthcare. the questionnaire consists of 33 questions covering different aspects of changes during last 10 years (to link with assessment of statistical data provided by national oncology register) of factors influencing accessibility, efficacy, efficiency, effectiveness and safety of medical care and interventions to oncology patients and to express the impact of each factor in points from 1 to 5. overall, each expert had to respond 462 questions covering 14 groups of oncology diseases according to icd-10 who fic. 4. results the proposed structural model for assessing and proactively risks managing in healthcare is presented at fig.1. it consists of node elements or blocks most of which have already been developed and applied in practice for expertise and decision making in healthcare. the model is based on world health organization (who) recommendations and country commitments to use the who family of international classifications (who fic). the who fic contains three basic classifications making possible to form a global terminology base for discrete description of the presence or absence of diseases (international classification of diseases (icd), degrees of functional body or organ impairment (international classification of functioning (icf) and international classification of healthcare interventions (ichi). these classifications allow to achieve comparability of data between different geographical regions and countries, as well as to observe the comparability of data over time, determine trends and make a forecast of population health. fic makes possible to monitor population healthcare and healthcare technologies as well as making forecast in both areas [5]. these classifications allow us to achieve uniform representation of diverse data and to build predictive models according to format acceptable for regulatory decision making. next level of model is the assessment of disease burden summarizing the complex of medical, social and economic consequences of a particular disease. it is the basis for decision 116 d. meshkov, l. bezmelnitsyna, s. cherkasov copyright ©2020 assa adv. in systems science and appl. (2020) on the need for medical technologies to reduce the impact of the disease and for the successive data generation on the value of this medical technology in the treatment of the disease [1,6]. next sequential levels include data generation on the value of healthcare intervention in laboratory conditions (“preclinical studies”) and studies with patients participation (“clinical studies”) assessing the likelihood of an outcome, systematic reviews, and clinical guidelines (in accordance with who recommendations) that summarize all available information on diseases and their treatment options. clinical guidelines are systematically developed provisions designed to help a doctor make decisions about medical tactics and the use of appropriate medical technologies in certain clinical situations. in clinical recommendations, there is a link between each statement and scientific data, and scientific facts take precedence over expert opinion. these documents do not have formal legal force, but are a tool that helps doctors make optimal therapeutic choices. the use of clinical recommendations developed on the basis of international clinical studies in specific economic conditions and the health system is the task of the government and health managers, and is carried out with the help of other formalized documents that have legal force. the development of a prognosis focused on the needs of practical health care will also provide a timely evidence base for the creation of clinical recommendations that meet the requirements of who and serve as the basis for treatment standards and procedures for providing medical care. [2]. the obtained information is used for the “integrated assessment and scenario models” developing with the purpose of the issue resolution, as well as for predicting of medical technologies development (“horizon scanning”) [8]. it is obvious that market authorization of the new intervention and overcoming barriers requires time which delay the availability of medicines n market and decrease the effectiveness of healthcare system. the earlier the preparation for introduction of new interventions begins the better result can be achieved on population level. horizon scanning deliver data for clinical and economic evaluation of drugs (“health technology assessment”) and for the “integrated assessment and scenario models” block which summarizes and present information for administrative solutions. it turned out that this last element of the model is practically not represented in the process of forming an expert assessment and making a decision while that the requirements for the final expert report depend on the requests of the regulatory authorities and the tasks they face. local legislation and formal regulatory procedures in each country form the landscape influencing on regulatory solutions which must lead to appropriate proactive risk management aimed at eliminating the predicted population health issues by accelerated development of appropriate healthcare structure and regulations as well as introduction of medicines into healthcare practice. proper healthcare management is also linked to timely provision of information about population health threats and healthcare interventions value by physicians education at high medical schools and within in the framework of continuous medical training and professional development. providing information to specialists and relationships with medical schools will allow them to prepare specialists beforehand, taking into account the forecast of the dynamics of public health indicators and making them familiar with innovative healthcare interventions aimed at improving these indicators. a data management model for proactive risk management in healthcare 117 copyright ©2020 assa adv. in systems science and appl. (2020) fig.1. node elements and data flow of the proactive risk management healthcare model. regulatory control according to international requirements world health organization family of international classification (who fic) international statistical classification of diseases and related health problems (icd) international classification of functioning disability and health (icf) international classification of healthcare interventions (ichi) population health monitoring healthcare interventions monitoring preclinical studies clinical studies systematic reviews and clinical guidelines health technology assessment regulatory control according to local legislation medical education and training regulations and procedures healthcare organization and management population health forecast horizon scanning disease burden integrated assessment & scenario models 118 d. meshkov, l. bezmelnitsyna, s. cherkasov copyright ©2020 assa adv. in systems science and appl. (2020) 5. discussion and conclusions the model of data generation and distribution for proactive healthcare management is presented. a comprehensive evidence-based approach linked to the population health forecast provides an opportunity to improve the quality of healthcare management as well as the effectiveness of medical care. most elements of the presented model are already existing and are used in practice. there are international or local documents regulating data proceeding for each blocks and data submission to the regulatory authorities for making management decisions. the survey indicated that the only block “integrated assessment & scenario models” is not used in practice. regulatory decisions are made on separate assessment of clinical efficacy and safety (market authorization), clinical guidelines (education and training, regulatory procedures) and economic effectiveness (inclusion into reimbursement lists) while a comprehensive assessment of these data and modeling of outcomes under different scenarios would be the most effective solution of expert assessment and regulatory decisions in healthcare continuity. references 1. cutler rl, fernandez-llimos f, frommer m, benrimoj c, garcia-cardenas v (2018) economic impact of medication non-adherence by disease groups: a systematic review. bmj open. 8(1), 1-13. doi: 10.1136/bmjopen-2017-016982. 2. guidelines for who guidelines. who practice guidelines: recommended process. global programme on evidence for health policy (2003). world health organization. geneva, switzerland eip/gpe/eqc/2003,1. 1-24 available: http://apps.who.int/iris/bitstream/10665/75146/1/9789241548441_eng.pdf 3. jandhyala r. (2020) a novel method for observing proportional group awareness and consensus of items arising from list-generating questioning. curr med res opin. mar 11:1-11. doi: 10.1080/03007995.2020.1734920. available: https://www.tandfonline.com/doi/full/10.1080/03007995.2020.1734920 4. katarina bohm, achim schmid, ralf gotze, claudia landwehr, heinz rothgang (2013) five types of oecd healthcare systems: empirical results of deductive classification/ health policy 113(2013), 258-269 5. khabriev r.u., isaeva a.v., bezmenitsyna l.yu., meshkov d.o., berseneva e.a., cherkasov s.n. (2016) dostupnost innovacionny`x lekarstvenny`x preparatov i snizhenie pokazatelej smertnosti ot onkologicheskix zabolevanij [the availability of innovative drugs and a reduction in cancer mortality rates] bulletin of the national public health research institute, 6, 5-19 [in russian] 6. mcdonald sa, nijsten d, bollaerts k, bauwens j, praet n, et. al. (2018) methodology for computing the burden of disease of adverse events following immunization. pharmacoepidemiol drug saf. 2018 jul; 27(7):724-730. doi: 10.1002/pds.4419 7. proposal for a regulation of the european parliament and of the council on health technology assessment and amending directive (2011) 2011/24/eu. brussels, 31.1.2018 com (2018) 51 final 2018/0018 (cod); 1-48 8. the international information network on new or emerging, appropriate use and reassessment needed health technologies (2020) [online] available: https://www.euroscan.org 9. world health organization. classifications (2020) [online] available: https://www.who.int/classifications/en/ https://www.tandfonline.com/doi/full/10.1080/03007995.2020.1734920 https://www.euroscan.org/ https://www.who.int/classifications/en/ adv syst sci appl 2020; 04:83–104 published online at https://ijassa.ipu.ru. on optimal therapy protocols in the mathematical model of prostate cancer progression marie-christin litzinger1, y. todorov1, miriam föller-nord1, manitjaiswal k. chaudhary2, alexander s. bratus3,4∗ 1university of applied sciences, paul-wittsack-str. 10, 68163, mannheim, germany 2nikolsky mathematical institute, rudn university, miklukho-maklaya str.6, moscow, 117198, russia 3russian university of transport, obraztsova 15, moscow, 127994, russia 4moscow center of fundamental and applied mathematics, msu, gsp-1, leninskie gory, moscow, 119991, russia abstract: a mathematical model for the treatment of prostate cancer regarding androgen ablation is considered. the effect of the therapy on healthy cells is taken into account and modelled as a second order state constraint. a mathematical overview about possible therapy strategies is presented. keywords: prostate cancer, androgen ablation, therapy strategies, optimal control, state constraint 1. introduction regarding prostate cancer progression and development testosterone and its role in the biological mechanisms of epithelial prostate cells are of particular interest for developing therapy strategies. therefore the main steps of testosterone procession are now outlined briefly. after entering the cells testosterone is either able to bind to androgen receptors (ar) straight away or it is first converted to dihydrotestosterone (dht). in contrast to testosterone, dht and ar form a more stable complex. after being phosphorylated and dimerized that complex binds to the dna, where it causes a higher transcription rate of with proliferation, survival and differentiation associated genes [11][10]. consequently, there are several points in the mechanism where the therapy can intervene, for example through inhibiting the testosterone production or blocking androgen receptors [2]. after some time prostate cancer cells are able to grow even in the absence of androgen. due to the following therapy resistance an intermittent androgen therapy can be applied [9, 19]. it is important that healthy and cancer cells both produce prostate-specific antigene (psa). in noncancerous cells only a limited amount of psa enters into the bloodstream due to natural barriers. in the presence of cancer these barriers break down and more psa leaks into the blood. the psa-level in the blood serum therefore acts as an indicator for the stadium and the presence of prostate cancer [11]. due to the complex mechanisms of cancer the cooperation of different scientific areas is necessary and mentioned in [3] and [6]. since the last fifty years the application of ∗corresponding author: alexander.bratus@yandex.ru 84 m.-c. litzinger, y. todorov, m. föller-nord, m.k. chaudhary, a.s. bratus mathematical approaches in order to analyze the dynamics of cancer is therefore common and discussed in [3]. furthermore mathematical models and methods for the analysis of the dynamics of tumor-immune interaction are considered to evolve research perspectives [4]. however, not only the competition between cancer and healthy cells, but also the toxicity of the therapy, which damages healthy cells, are negative side effects. these are essential for developing effective medicine and therapy strategies. consequently, the mathematical theory of optimal control processes is needed to develop successful therapy protocols [17][22]. due to the adverse effects caused by the ablation of androgen, for example the decrease of quality of life [5], it is convenient to introduce a limitation of the healthy epithelial prostate cells during androgen ablation therapy. in the present work three approaches – an alternative, suboptimal and optimal one – in the mathematical model for prostate cancer progression in response to androgen ablation therapy are introduced. in all three approaches the side effects of the androgen ablation therapy are considered as a second order pure state constraint on the amount of the healthy epithelial prostate cells. the goal is to minimize the amount of cancer cells while keeping the amount of healthy cells above a given critical limit. the resistance of prostate cancer cells during the androgen ablation therapy is also taken into account. in the following a brief description of each therapy strategy is given. the alternative therapy strategy is based on the dynamical analysis of the system and its critical points. this approach circumvents the classical optimal control theory. in combination with a numerical improvement algorithm it provides feasible results which are similar to the optimal one. pontraygin’s maximum principle without state constraints is used to develop a therapy protocol at which the conditions of the states on the constraint boundary are analyzed. the considered approach always guarantees a feasible therapy process if there exists such possible one. the results are used to obtain the so called suboptimal therapy strategy. an analytical determination of an optimal therapy strategy in the presence of state constraints of order two is considered by means of analytical and numerical techniques. therefore the software tool bocop is used. the results of the presented approaches are compared and discussed. 2. mathematical model of prostate cancer in this article a mathematical model of prostate cancer based on [11] is considered. copyright c© 2020 assa. adv syst sci appl (2020) on optimal therapy protocols in the math. model of prostate cancer progression 85 de(t) dt = µee(t) ( 1− e(t) ηe ) − εe(t)(m(t) +n(t)) ηe − δee(t) dn(t) dt = (1− αm)µnn(t) ( 1− n(t) +m(t) η ) − δnn(t) dm(t) dt = (µmm(t) + αmµmn(t)) ( 1− n(t) +m(t) η ) − δmm(t) dr(t) dt = αr − λrr(t)− kdf r(t)d(t) + kdr a(t)− ktf r(t)t (t) + ktr a(t) dd(t) dt = βt t (t) kt + t (t) − λdd(t)− kdf r(t)d(t) + kdr a(t) dt (t) dt = f(ts)− λtt (t)− βt (t) t (t) kt + t (t) − ktf r(t)t (t) + ktr at(t) da(t) dt = −λaa(t) + kdf r(t)d(t)− kdr a(t) dat(t) dt = −λatat(t) + ktf r(t)t (t)− ktr at(t) dp (t) dt = βp(a(t) + at(t) + ϕ)− αpp (t)efrac − γpp (t) (m(t) +n(t))2 kp + (m(t) +n(t)) − λpp (t) t ∈ [0, t ], e(0), n(0), m(0), r(0), d(0), t (0), a(0), at(0), p (0) ∈ r+, (2.1) where efrac = vc e pv ol , pv ol = αvt dv bdvv + tdv . in (2.1) it is assumed that the healthy cells e have the proliferation rate µe , the death rate δe and a limit ηe . the magnitude of the competition between healthy and cancer cells is modelled by means of the parameter ε. the androgen-dependent cancer cells n proliferate at rate µn , decease with rate δn and have the limit ηn . with the probability αm ∈ (0, 1) cancer cells n mutate to castrationresistant cancer cells m where the parameters µm and δm represent the proliferation and dead rate respectively. because of the character of this article the remaining state variables of the dynamic system (2.1) which represent specific hormone levels in the organism are only mentioned: t testosterone concentration, p tissue psa concentration, r free androgen receptor concentration, d dihydrotestosterone concentration, a dihydrotestosterone-activated androgen receptor concentration,at testosterone-activated androgen receptor concentration. in this article all cell types are estimated in billions. detailed information, the biological background of the variables and the associated parameters are given in [?]. assuming µn = µm = µ, δn = δm = δ, n(t) and m(t) can be substituted in order to represent the total number of the prostate cancer cells. from the addition n(t) +m(t) = z(t) follows dz(t) dt = µz(t) ( 1− z(t) η ) − δz(t). (2.2) the following figures illustrate the behaviour of healthy and cancer cells as well as the psa-level in the absence of therapy. the parameters of the model which were used in this copyright c© 2020 assa. adv syst sci appl (2020) 86 m.-c. litzinger, y. todorov, m. föller-nord, m.k. chaudhary, a.s. bratus article describe a scenario in which the amount of healthy cells is not strongly influenced by cancer cells. the set of parameters and their description is given (see a.1). 0 50 100 150 non-dimensional time 0 50 100 150 200 250 300 fig. 2.1. time response of healthy e(t) and cancer cells z(t) 0 50 100 150 non-dimensional time 0 1 2 3 4 5 6 7 8 9 10 fig. 2.2. time response of the psa function p (t) copyright c© 2020 assa. adv syst sci appl (2020) on optimal therapy protocols in the math. model of prostate cancer progression 87 as mentioned before intermittent therapy is one way to slow down the development of therapy resistance. the following figures show the dynamics of the system when such therapy is applied. 0 50 100 150 non-dimensional time 0 1 2 3 4 5 fig. 2.3. amount of the administered drug u(t) 0 50 100 150 non-dimensional time 0 50 100 150 200 250 300 fig. 2.4. time response of healthy e(t) and cancer cells z(t) copyright c© 2020 assa. adv syst sci appl (2020) 88 m.-c. litzinger, y. todorov, m. föller-nord, m.k. chaudhary, a.s. bratus 0 50 100 150 non-dimensional time 0 1 2 3 4 5 6 7 8 9 10 fig. 2.5. time response of the psa function p (t) in the following sections different therapy strategies are determined and used to minimize cancer cells without violating the constraint boundary on healthy cells. 3. an alternative therapy strategy consider the optimal control problem after the substitution (2.2) min u j = ∫ t 0 z(t)dt→ min (3.3) copyright c© 2020 assa. adv syst sci appl (2020) on optimal therapy protocols in the math. model of prostate cancer progression 89 de(t) dt = µee(t) ( 1− e(t) ηe ) − εe(t)(m(t) +n(t)) ηe − δee(t) dz(t) dt = µ(h)z(t) ( 1− z(t) η ) − δz(t) dr(t) dt = αr − λrr(t)− kdf r(t)d(t) + kdr a(t)− ktf r(t)t (t) + ktr a(t) dd(t) dt = βt t (t) kt + t (t) − λdd(t)− kdf r(t)d(t) + kdr a(t) dt (t) dt = f(ts)− λtt (t)− βt (t) t (t) kt + t (t) − ktf r(t)t (t) + ktr at(t) da(t) dt = −λaa(t) + kdf r(t)d(t)− kdr a(t) dat(t) dt = −λatat(t) + ktf r(t)t (t)− ktr at(t) dp (t) dt = βp(a(t) + at(t) + ϕ)− αpp (t)efrac − γpp (t) (m(t) +n(t))2 kp + (m(t) +n(t)) − λpp (t) dh(t) dt = −γh(t) + u(t) 0 ≤ u(t) ≤ umax, t ∈ [0, t ], t -fixed e(0), z(0), r(0), d(0), t (0), a(0), at(0), p (0), h(0) ∈ r+, (3.4) and the state constraint on the amount of healthy cells g(t) = e(t)− ec ≥ 0 ∀t ∈ [0, t ]. (3.5) the last equation in (3.4) describes the pharmacokinetic and the functions µ(h) = µ0 ( 1− kh h+ 1 ) and µe(h) = µ0 ( 1− keh h+ 1 ) the pharamcodynamic of the administered medicine. the parameters γ, k, ke, µ0, µ0 e ∈ r+, where γ stands for the dissipation rate of the administered drug respectively, k and ke describe the effect of the drug on cancer and healthy cells and µ0 and µ0 e are the replication rates of the two cell populations. the equation (3.5) has to guarantee that the amount of the healthy cells does not undercut a given critical limit ec during the therapy process. because of the mathematical complexity of the optimal control problem (3.3)-(3.5) an alternative approach, which will further be called alternative therapy strategy, is considered at first. in order to analyze whether small changes of the initial conditions of the considered system lead to a different behaviour of the corresponding trajectories its critical points are analyzed (see a.2). due to the practical irrelevance of the trivial critical point (ē, z̄, h̄) = (0, 0) only z̄ = ( 1− δ µ(h) ) η, ē = ηe − ε(µ(h)− δ)η µe(h)µ(h) − δeηe µe(h) , h̄ = u γ (3.6) is taken into account. from the dynamical analysis of the considered system (see a.3) it is obvious, that for realistic parameters of the model the critical point (3.6) is a stable node. copyright c© 2020 assa. adv syst sci appl (2020) 90 m.-c. litzinger, y. todorov, m. föller-nord, m.k. chaudhary, a.s. bratus assuming now that u is a constant parameter such that 0 ≤ u ≤ umax we consider the objective function j̃(u) := j̃(ē, z̄) = z̄ + κ(ē − ec)2 → min, (3.7) where with the scale of the weighting factor κ ∈ r+ the state constraint on the amount of healthy cells is taken into account. it follows from the implicit function theorem that ē and z̄ can be considered as functions of u and consequently the minimization of the functional (3.7) can be approached as a mathematical programming problem. due to the theorem of weierstrass a solution of this problem always exists. assuming that u = ū ∈ [0, umax] is the value for which the minimum of the functional (3.7) is achieved. now consider the function u(t) = umax, 0 ≤ t ≤ t̂. here umax and t̂ are two constants such that the solution of the last equation in (3.4) with initial condition h(0) = 0 reaches the value h = ū γ at the moment t = t̂. since h(t) = umax γh (1− e−γt) (3.8) the condition h(t̂) = ū γ is fulfilled iff t̂ = 1 γ ln ( 1− ū umax ) . (3.9) the last equality provides the required time of the function h(t) to reach the value of ū γ if the control function u(t) = umax in the time interval 0 ≤ t ≤ t̂ , which is called intensive therapy time. in the remaining time interval t̂ < t ≤ t the control function u(t) has to keep the value ū γ which minimizes the objective function j̃(u). this therapy interval time is called relaxation therapy time. the determined strategy is called alternative therapy strategy and can be defined as u(t) = umax, 0 ≤ t ≤ t̂, ū γ , t̂ < t ≤ t. (3.10) according to the considerations above the alternative control strategy consists out of two stages: the stage of the intensive therapy and the stage of relaxation. in order to avoid disadvantages in the approach considered above, the following modification, presented in [22], will be used. this requires the introduction of the shifting variable s ∈ r+ in the penalty term of the objective function (3.7) j̃(u) := j̃(ē, z̄) = z̄ + κ(ē − (ec + s))2 → min. (3.11) in the numerical procedure some iterations are usually necessary to obtain a good result and the new value of the shifting variable after every iteration has to be calculated as follows: s(q+1) = s(q) + max{ec − e(t)| e(t) < ec, t ∈ (0, t )}, where q is the number of the last iteration. if the maximal number of iteration qmax or the desired accuracy θ ≥ |s(q) − s(q−1)| is reached the procedure has to be terminated. the following numerical results illustrate the time response of healthy and cancer cells as well as the psa-level and the control function using the alternative therapy strategy. copyright c© 2020 assa. adv syst sci appl (2020) on optimal therapy protocols in the math. model of prostate cancer progression 91 0 50 100 150 non-dimensional time 44 46 48 50 52 54 56 58 fig. 3.6. time response of healthy cells e(t) using the alternative therapy strategy 0 50 100 150 non-dimensional time 115 120 125 130 135 140 145 150 fig. 3.7. time response of cancer cells z(t) using the alternative therapy strategy cancer and healthy cells decrease (fig. 3.6), (fig. 3.7). healthy cells reach the constraint boundary without violating it. copyright c© 2020 assa. adv syst sci appl (2020) 92 m.-c. litzinger, y. todorov, m. föller-nord, m.k. chaudhary, a.s. bratus 0 50 100 150 non-dimensional time 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 fig. 3.8. amount of the administered drug u(t) using the alternative therapy strategy 0 50 100 150 non-dimensional time 0 1 2 3 4 5 6 7 8 9 10 fig. 3.9. time response of the psa function p (t) using the alternative therapy strategy the maximal dose of therapy is used for a short amount of time and then a constant amount is administered (fig. 3.8). due to the reduction of cancer cells less psa is produced in the tissue and can consequently be found in the blood serum (fig. 3.9). copyright c© 2020 assa. adv syst sci appl (2020) on optimal therapy protocols in the math. model of prostate cancer progression 93 4. suboptimal therapy strategy in this section a suboptimal therapy strategy for the problem (3.3)-(3.5) is introduced. in order to minimize the objective function (3.3) it is natural to apply the maximal admissible amount of the drug so long as possible. however, it has to be guaranteed that the amount of healthy cells does not violate the constraint (3.5). i.e. that the time derivation ė on the constraint boundary has to be non negative. from ė(tj) ≥ 0 follows µe(h)ec ( 1− ec ηe ) − εecz(tj) ηe − δeec ≥ 0 ⇒ µ0 e ( 1− keh(tj) h(tj) + 1 ) ≥ εz(tj) + δeηe ηe − ec ⇒ keh(tj) ≤ ( 1− εz(tj) + δeηe (ηe − ec)µ0 e ) (h(tj) + 1) (4.12) ⇒ h(tj) ( ke − ( 1− εz(tj) + δeηe (ηe − ec)µ0 e )) ≤ ( 1− εz(tj) + δeηe (ηe − ec)µ0 e ) ⇒ h(tj) ≤ ( 1− εz(tj) + δeηe (ηe − ec)µ0 e ) ke − ( 1− εz(tj) + δeηe (ηe − ec)µ0 e ) , where tj := {t ∈ (0, t )| e(t) = ec}. due to the continuity of e the strong inequality in the last expression can not appear. the last expression in (4.12) gives the necessary condition for h on the constraint boundary, which guarantees an admissible process. therefore the following analysis is needed. from the last equation of the system (3.4) holds for u(t) = umax h(tj) = e−γtj ∫ tj 0 eγtumaxdt = umax γ ( 1− e−γtj ) . (4.13) thus if the maximal amount of the drug, u(t) = umax, t ∈ [0, tj), is administered until the critical boundary of the healthy cells is reached, the following relation has to be fulfilled ( 1− εz(tj) + δeηe (ηe − ec)µ0 e ) ke − ( 1− εz(tj) + δeηe (ηe − ec)µ0 e ) = umax γ (1− e−γtj). (4.14) note, that this is a special case which can only appear with a given set of parameters of the model and initial conditions. in this case the therapy strategy can be changed when the amount of the healthy cells reaches the critical value without violating the therapy process. the therapy strategy stated below is in this case optimal (see a.4). copyright c© 2020 assa. adv syst sci appl (2020) 94 m.-c. litzinger, y. todorov, m. föller-nord, m.k. chaudhary, a.s. bratus u(t) =  umax, 0 ≤ t ≤ tj, γ 1− εz(t) + δeηe (ηe − ec)µ0 e  ke− 1− εz(t) + δeηe (ηe − ec)µ0 e  , tj < t ≤ t. (4.15) in general the change of the administered amount of the drug can not induce the change of the slope of the development of the healthy cells immediately. this is caused by the inertia of the dynamic system. therefore investigations of the more realistic cases have to be done. the case in which the maximal admissible amount of the drug has to be changed before the critical boundary of the healthy cells is reached, is considered. i.e.,( 1− εz(tj) + δeηe (ηe − ec)µ0 e ) ke − ( 1− εz(tj) + δeηe (ηe − ec)µ0 e ) < umax γ (1− e−γtj). (4.16) the admissible process can be guaranteed if the maximal admissible amount of the drug is reduced at the point of time t′ < tj so that h(t′) = ( 1− εz(tj(umax)) + δeηe (ηe − ec)µ0 e ) ke − ( 1− εz(tj(umax)) + δeηe (ηe − ec)µ0 e ) holds, where z(tj(umax)) is the amount of the cancer cells on the constraint boundary using the maximal dose of medicine (u(t) = umax, t ∈ [0, tj]). thus the suboptimal therapy strategy can be given by u(t) =  umax, 0 ≤ t ≤ t′, γ 1− εz(tj(umax)) + δeηe (ηe − e(t))µ0 e  ke− 1− εz(tj(umax)) + δeηe (ηe − e(t))µ0 e  , t′ < t ≤ t. (4.17) the following figures present the results of the suboptimal control approach. cancer and healthy cells decrease and the healthy cells do not reach the constraint boundary (fig. 4.10), (fig. 4.11). since cancer cells are decreasing continuously the quantity of psa in the tissue and therefore in the blood serum also decreases (fig. 4.13). 5. optimal therapy strategy in this section the therapy strategy related to the optimal control problem (3.3)-(3.5) is investigated with pontryagin’s maximum principle, which is the necessary condition for an optimal process. the restriction on the minimal admissible amount of healthy cells (3.5) has to be interpreted as a second order state constraint. because of the high complexity of optimal copyright c© 2020 assa. adv syst sci appl (2020) on optimal therapy protocols in the math. model of prostate cancer progression 95 0 50 100 150 non-dimensional time 44 46 48 50 52 54 56 58 fig. 4.10. time response of healthy cells e(t) using the suboptimal therapy strategy 0 50 100 150 non-dimensional time 120 125 130 135 140 145 150 fig. 4.11. time response of cancer cells z(t) using the suboptimal therapy strategy control problems with higher order state constraints the introduced approach is a composition of analytical investigations and numerical results. copyright c© 2020 assa. adv syst sci appl (2020) 96 m.-c. litzinger, y. todorov, m. föller-nord, m.k. chaudhary, a.s. bratus 0 50 100 150 non-dimensional time 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 fig. 4.12. time response of the quantity of the u(t) using the suboptimal therapy strategy 0 50 100 150 non-dimensional time 0 1 2 3 4 5 6 7 8 9 10 fig. 4.13. time response of the psa function p (t) using the suboptimal therapy strategy because of the nature of the maximum principle it is essential to primarily investigate the optimal therapy strategy without state constraints. the optimality of this process till the state variable e reaches the constraint boundary is determined. copyright c© 2020 assa. adv syst sci appl (2020) on optimal therapy protocols in the math. model of prostate cancer progression 97 the investigations given in a.3 prove the following theorem. theorem 5.1: the optimal therapy strategy in the absence of the state constraint (3.5) is given as u∗(t) = umax ∀t ∈ [0, t ]. from the practical point of view this result is equivalent to the situation in which the test results of a patient are acceptable during the therapy. therefore the therapy has to be executed with the maximal admissible amount of the drug in order to eradicate the cancer cells. in the opposite case the amount of healthy cells e reaches the critical value ec. because of the complexity the strict mathematical investigation of the constraint problem is not considered. instead of that numerical results using the software tool bocop are presented. the numerical results show the development of healthy cells, cancer cells and the optimal control. 0 50 100 150 non-dimensional time 40 42 44 46 48 50 52 54 56 58 60 fig. 5.14. time response of healthy cells e(t) using the optimal therapy strategy healthy and cancer cells decrease (fig. 5.14), (fig. 5.15). healthy cellse(t) then progress along the boundary. after a period where the maximal amount of drug is administered follows a short relaxation phase. then the maximal admissible amount of medicine which minimizes the objective function is applied. the following table presents the values of the objective function, the healthy cells and cancer cells at the end of the therapy process (t = 150). note that in the results achieved by the alternative therapy strategy the modification (3.11) is used. therefore the results of the alternative strategy are better than the ones obtained with the suboptimal one. copyright c© 2020 assa. adv syst sci appl (2020) 98 m.-c. litzinger, y. todorov, m. föller-nord, m.k. chaudhary, a.s. bratus 0 50 100 150 non-dimensional time 100 110 120 130 140 150 160 fig. 5.15. time response of the cancer cells z(t) using the optimal therapy strategy 0 50 100 150 non-dimensional time 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 fig. 5.16. amount of the administered drug u(t) using the optimal therapy strategy 6. summary a simplified mathematical model for the therapy of prostate cancer regarding androgen ablation is considered. the effect of the therapy on cancer and healthy cells is assumed to copyright c© 2020 assa. adv syst sci appl (2020) on optimal therapy protocols in the math. model of prostate cancer progression 99 therapy strategy j(t ) e(t ) z(t ) alternative therapy strategy 1.988 · 104 46.222 117.88 suboptimal therapy strategy 2.029 · 104 47.414 123.14 optimal therapy strategy 1.195 · 104 46.000 114.98 table 5.1. comparison of the terminal values of the three considered approaches reduce the birth rate of the populations of the considered cell types. an optimization problem with a constraint on the amount of healthy cells during the therapy process is introduced. three therapy strategy approaches of the problem are illustrated and solved by combinating analytical and numerical procedures. the presented results of the therapy strategies using the suboptimal and alternative control demonstrate that the mathematical challenges of the classical optimal control problem with state constraints can be circumvented and the approaches can be used for medicine. moreover the suggested mathematical model produces possibilities for further investigations regarding the physical condition of a patient which can be modelled by means of different parameters of the model. from the mathematical point of view the considered problem covers theoretical investigations on the necessary and sufficient conditions of optimal control problems with state constraints. acknowledgements this research is supported by the russian scientific foundation grant 19-11-00008 and grant rfbi 20-04-60157 and ministry of science of higher education grant 075-15-20191621. references 1. agus, d.b., et al. (1990), prostate cancer cell cycle regulators: response to androgen withdrawal and development of androgen independence, j natl cancer inst, 91, 1869– 1876. 2. balk, s.p. (2002), androgen receptor as a target in androgen-independent prostate cancer, urology, 60, 132–138. 3. byrne, h.m. (2010), dissecting cancer through mathematics: from the cell to the animal model, nature reviews cancer, 10, 221–230. 4. bellomo, n. & preziosi, l. (2000), modelling and mathematical problems related to tumor evolution and its interaction with the immune system, mathematical and computer modelling, 32, 413–452. 5. donovan, k.a., walker, l.m., wassersug, r.j., thompson, l.m. & robinson, j.w. (2015), psychological effects of androgen-deprivation therapy on men with prostate cancer and their partners, cancer, 121, 4286–4299. 6. gatenby, r.a. and maini, p.k. (2003), mathematical oncology: cancer summed up, nature, 421, page 321. 7. gann, p.h., hennekens c.h., ma j., longcope, c., stampfer, m.j. (1996), prospective study of sex hormone levels and risk of prostate cancer, j natl cancer inst, 88, 1118– 1126. 8. heinlein, c.a., chang, c. (2004), androgen receptor in prostate cancer, endocr rev, 25, 276–308. 9. hirata, y., tanaka, g., bruchovsky, n. and aihara, k. (2012), mathematically modelling and controlling prostate cancer under intermittent hormone therapy, asian journal of copyright c© 2020 assa. adv syst sci appl (2020) 100 m.-c. litzinger, y. todorov, m. föller-nord, m.k. chaudhary, a.s. bratus andrology, 270–277. 10. hussain, m., et al. (2009), prostate-specific antigen progression predicts overall survival in patients with metastatic prostate cancer: data from the southwest oncology group trials 9346 (intergroup study 0162) and 9916, j clin oncol, 27, 2450–2456. 11. jain, h.v., clinton, s.k., bhinder, a. & friedman, a. (2011), mathematical modelling of prostate cancer progression in response to androgen ablation therapy, proc. natl acad. sci. usa, 108, 19701–19706. 12. ledzewicz, u., schättler, h. (2006), application of optimal control to a system describing tumor anti-angiogenesis, proceedings of the 17th international symposium on mathematical theory of networks and systems (mtns), kyoto, japan, 478–484. 13. ledzewicz, u., schättler, h. (2005), a synthesis of optimal controls for a model of tumor growth under angiogenic inhibitors, proceedings of the 44th ieee conference on decision and control, 945–950. 14. miyamoto, h., messing, e.m., chang, c. (2004), androgen deprivation therapy for prostate cancer: current status and future prospects, prostate, 61, 332–353. 15. murray, j.m. (1990), optimal control for a cancer chemotherapy problem with general growth and loss functions, math biosci., 98(2), 273–287. 16. pontryagin, l.s., boltyanskii, v.g., gamkrelidze, r.v. & mishchenko ef (1962), the mathematical theory of optimal processes, interscience publishers. 17. rubinow s.i., lebowitz j.l. (1976), a mathematical model of the chemotherapeutic treatment of acute myeloblastic leukemia, biophys. j., 1257–1271. 18. seidenfeld j., et al. (2000), single-therapy androgen suppression in men with advanced prostate cancer: a systematic review and meta-analysis, ann intern med, 132, 566–577. 19. suzuki, y., sakai, d., nomura, t., hirata, y. and aihara, k. (2014), a new protocol for intermittent androgen suppression therapy of prostate cancer with unstable saddle-point dynamics, journal of theoretical biology, 350, 1–16. 20. swanson, k.r., true, l.d., lin, d.w., buhler, k.r., vessella, r. and murray, j.d. (2001), a quantitative model for the dynamics of serum prostate-specific antigen as a marker for cancerous growth: an explanation for a medical anomaly, the american journal of pathology, 158(6), 2195–2199. 21. swan, g.w. (1990), role of optimal control theory in cancer chemotherapy, math biosci., 101(2), 237–284. 22. todorov, y., nuernberg, f. (2014), optimal therapy protocols in the mathematical model of acute leukemia with several phase constraints, ocam, 35(5), 559–574. copyright c© 2020 assa. adv syst sci appl (2020) on optimal therapy protocols in the math. model of prostate cancer progression101 a. appendixes a.1. model parameter values the following values were used for the model parameters (taken from [11]). parameter value parameter value parameter value ηe 110.24 δe 0.0081 δ 0.00405 γ 0.1 µe0 0.22 µ0 0.1 ke 0.889 k 0.99 ats 1.00 bts 22.23 cts 0.23 ktf 0.14 ktr 0.07 αp 0.001515 ε 0.009 λd log(2) 9 λp log(2) 12.3 λat log(2) 3 βm 0.003 βn 0.002 βt 4.569 pn 10 pm 10 kp 7 γp 0.6 104 kdf 0.018 kdr 0.053 100 kt 0.104 aet 1.62 λr log(2) 3 λa log(2) 3 λt log(2) 3 av 18.397 bv 67.753 cv 28.4 dv 12 vc 5.56 106 r0 e 90 0.002622 β table a.2. parameters of the model more detailed information is given in [11]. a.2. dynamical analysis the motivation for the construction of the alternative control strategy rests upon the asymptotically behaviour of the considered system (3.4). a relocation of an asymptotic stable critical point in order to minimize the functional (3.3) provides the alternative therapy strategy. therefore the type of the nontrivial critical point given in (3.6) is analyzed. define the jacobian j = ∂f ∂x =  ∂e(t) ∂e ∂e(t) ∂z ∂z(t) ∂e ∂z(t) ∂z  = µe(h)− 2eµe(h) ηe − εz ηe − δe −εe ηe 0 µ(h)− 2zµ(h) η − δ  at the considered critical point (3.6) holds j |(ē,z̄) = −µe(h̄) + ēµe(h̄) ηe + δe −εē ηe 0 −µ(h̄) + δ  . copyright c© 2020 assa. adv syst sci appl (2020) 102 m.-c. litzinger, y. todorov, m. föller-nord, m.k. chaudhary, a.s. bratus in order to analyze the behaviour of the considered critical point it is necessary to analyze the sign of the eigenvalues of the jacobian. i.e., to find the roots of the polynomial defined by det(j − λē) = ( − µe(h̄) + ēµe(h̄) ηe + δe − λ )( − µ(h̄) + δ − λ ) . the eigenvalues λ1, λ2 of j are λ1 = −µ(h̄) + δ < 0 and λ2 = −µe(h̄) + ēµe(h̄) ηe + δe. for the given set of parameters (see a.1) the critical point (ē, z̄) is always stable. a.3. optimal control without state constraint constructing the pontryagin-hamilton function [16] to the problem (3), (4) h := h(e,z,r,d, t,a,at, p, h, ψ0, ψ1, ψ2, ψ3, ψ4, ψ5, ψ6, ψ7, ψ8, ψ9, u) = + ψ0(z) + ψ1 ( µee ( 1− e ηe ) − εez ηe − δee ) + ψ2 ( µz(t) ( 1− z(t) η ) − δz(t) ) + + ψ3 ( αr − λrr− kdf rd + kdr a− ktf rt + ktr a ) + + ψ4 ( βt t kt + t − λdd − kdf rd + kdr a ) + + ψ5 ( f(ts)− λtt − βt t kt + t − ktf rt + ktr at ) + + ψ6 ( − λaa+ kdf rd − kdr a) + ψ7(−λtat + ktf rt − ktr at ) + + ψ7 ( −λatat(t) + ktf r(t)t (t)− ktr at(t) ) + + ψ8 ( βp(a+ at + ϕ)− αppefrac − γpp (n +m)2 kp +n +m − λpp ) + ψ9 ( − γh+ u ) the optimal control law is given as u∗(t) = umax, ψ9(t) > 0, 0, ψ9(t) < 0, us ∈ (0, umax), ψ9(t) = 0 ∀t ∈ (t1, t2), (t1, t2) ⊂ [0, t ], where ψ9(t) is the so called switching function. note, that the optimal control function u∗(t) maximizes h in all points t ∈ [0, t ]. copyright c© 2020 assa. adv syst sci appl (2020) on optimal therapy protocols in the math. model of prostate cancer progression103 from pontryagin’s maximum principle the system of adjoint variables is given by ψ̇1(t) = −∂h ∂e = −ψ1 ( +µe(h(t))− 2e(t)µe(h(t)) ηe − εz(t) ηe − δe ) , (a.18) ψ̇2(t) = −∂h ∂z = −ψ0 + ψ1 e(t)ε ηe + ψ2 ( −µ(h(t)) + 2µz(t) η + δ ) , (a.19) ψ̇9(t) = −∂h ∂h = −ψ1 dµe(h(t)) dh(t) e(t) ( 1− e(t) ηe ) − ψ2 dµ(h(t)) dh(t) µz(t) ( 1− z(t) η ) + +ψ9γ, ψ1(t ) = ψ2(t ) = ψ9(t ) = 0. lemma a.1: in the optimal control problem (3.4) holds ψ0 = −1. proof from ψi(t ) = 0, 1 ≤ i ≤ 9, and the condition (ψ0, ψ1, ..., ψ9) 6= 0 follows that ψ0 = −1. from lemma a.1 follows ψ1(t) = 0 ∀t ∈ [0, t ], ψ2(t) = ew2(t) (∫ t 0 e−w2(s)ds− ∫ t 0 e−w2(t)dt ) , ψ9(t) = eγt[− ∫ t 0 e−γsψ2(s) dµ(h(s)) dh(s) z(s) ( 1− z(s) η ) ds+ + ∫ t 0 e−γtψ2(t) dµ(h(t)) dh(t) z(t) ( 1− z(t) η ) dt], where w2(t) = ∫ t 0 ( −µ(h(s)) + 2µ(h(s))z(s) η + δ ) ds. ψ9(t) depends on ψ2(t) and therefore this function is analysed for the mathematical solution of this problem. note, that ψ9(t) also depends on ψ1(t) but as shown before ψ1(t) ≡ 0. proof of theorem 5.1 lemma a.2: ψ2(t) is a negative monotone increasing function on [0, t ]. proof in order to analyse the behaviour of ψ2(t) the auxiliary function ψ̃2 = e−w2(t)ψ2(t) which has the same sign and zeros as ψ2(t) is used. it holds ψ̃2(t) < 0, t ∈ [0, t ) due to t∫ 0 e−w2(t)dt > t∫ 0 e−w2(s)ds ∀t ∈ [0, t ) and ψ̃2(t ) = 0. copyright c© 2020 assa. adv syst sci appl (2020) 104 m.-c. litzinger, y. todorov, m. föller-nord, m.k. chaudhary, a.s. bratus because t∫ 0 ew2(s)ds is a strictly increasing function ψ̃2(t) has no zeros on [0, t ) and therefore ψ2(t) is a negative monotone increasing function with ψ2(t) ≤ 0 ∀t ∈ [0, t ]. lemma a.3: ψ9(t) is a positive decreasing function on [0, t ]. proof from ψ9(t) = eγt ( ψ90 − ∫ t 0 e−γsψ2(s) µ(h(s)) dh(s) z(s) ( 1− z(s) η ) ds ) and ψ9(t ) = 0 ψ90 = ∫ t 0 e−γtψ2(t) µ(h(t)) dh(t) z(t) ( 1− z(t) η ) dt from the definition of the given problem follows z(t) ≥ 0 ∀t ∈ [0, t ] and( 1− z(t) η ) ∈ (0, 1). moreover it holds ψ2 ≤ 0 and dµ(h) dh < 0 ∀t ∈ [0, t ]. therefore it follows∫ t 0 e−γtψ2(t) dµ(h(t)) dh(t) z(t) ( 1− z(t) η ) dt > ∫ t 0 e−γsψ2(s) dµ(h(s)) dh(s) z(s) ( 1− z(s) η ) ds ∀t ∈ [0, t ]. thus ψ9(t) > 0 ∀t ∈ [0, t ) and u∗(t) = umax. a.4. proof of the statement proof from theorem 5.1 follows u∗(t) = umax for t ∈ [0, tj). because of the minimization of the objective function (3.3) the maximal admissible value of u on the constraint boundary has to be chosen. the limit of this value is given by u(t) = ( 1− εz(tj) + δeηe (ηe − ec)µ0 e ) ke − ( 1− εz(tj) + δeηe (ηe − ec)µ0 e ) , t ∈ [tj, t ]. copyright c© 2020 assa. adv syst sci appl (2020) introduction mathematical model of prostate cancer an alternative therapy strategy suboptimal therapy strategy optimal therapy strategy summary appendixes model parameter values dynamical analysis optimal control without state constraint proof of the statement adv syst sci appl 2019; 02; 33-43 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/709 potential scales of structural transformation of the russian electric power system based on low-carbon distributed generation fedor veselov1, irina erokhina1, andrey khorshev1* 1) the energy research institute of the russian academy of sciences, moscow, russia received march 3, 2019; revised may 5, 2019; published july 10, 2019 abstract: this article considers conditions and scenarios of transformation of the traditional structure of generating capacity in the russian electric power system in the next 10-15 years considering the mass decommissioning of operating thermal power plants (tpps) and active development of low-carbon distributed generation (dg) technologies. for this purpose, the ultimate development potential of russia's key distributed cogeneration technology was assessed, taking into account the forecast of capacity balance situation in the national electric power system. in addition to it, the possible scales of the involvement of renewable power (res) plants, as well as the potential impact of consumer’s demand response (dr) programs to replace large-scale generating facilities are considered. keywords: electric power system, distributed generation, low-carbon technologies, cogeneration, renewable power plants, demand response, capacity balance. 1. introduction for many decades, the development of the electric power industry in most countries has mainly relied on large power plants that produce power to high voltage transmission grid. however, in recent years, the development of small power sources† connected to the distribution network has been more and more intensive, and this is becoming one of the key trends in the transformation of the electric power industry of the 21st century worldwide. distributed generation (dg) becomes significant, and in some countries, such as denmark, [1] – it is also the dominant component of the national electric power system. the rapid growth of this segment is facilitated (to varying degrees in different countries) by several drivers that are of regulatory, technological and market nature. the most significant regulatory drivers are formed by purposeful public actions to implement the priorities of the national energy policy. many countries, having raised the level of energy security, increase in the energy efficiency and environmental acceptability of the electric power industry as political goals, have chosen to maximize the involvement of local (primarily renewable) energy resources and the intensive development of other low-carbon cogeneration technologies using gas, biomass and biofuel, as one of the main ways to achieve these goals. the regulatory framework and economic (tariff-based, tax-related or credit-based) mechanisms for supporting such projects have made the new sector attractive for large-scale investments. technological drivers that have provided for a significant improvement in the dg competitiveness in recent years, based on the fold reduction in the cost of wind and solar power as a result of technological improvements and the scale effect due to the multifold equipment production growth [2-4]. according to recent estimates, in a number of countries, renewable * corresponding author: epos@eriras.ru † in this article, distributed generation includes all types of power plants with an installed capacity of up to 25 mw and that are connected to a distribution grid 34 f. veselov, i. erokhina, a. khorshev copyright ©2019 assa adv. in systems science and appl. (2019) energy technologies have equalized or even surpassed traditional technologies for levelized cost of electricity (lcoe), even without special incentive measures provided by the state (for example, [5]) or a higher price of carbon; for other countries, this parity is expected in the next decade. finally, a significant role in dg development is played by market drivers associated with the development of competitive electricity markets, due to which distributed energy supply often becomes more attractive to consumers. the opportunity to choose the supplier, the differentiation of the final price of electricity with the allocation of competitive and tariff component, the transition to hourly and even shorter pricing periods stimulate consumers to a wider, more active influence on market conditions. the demand curve for electricity is becoming more elastic, and it is distributed power that offers consumers additional options for choosing power supply conditions, based on a combination of price, reliability and quality of services. often, consumers' own generation sources are combined with storage devices, as well as other mechanisms for price-dependent demand response (dr). it is necessary to note one more important feature of the dg, making it a more attractive investment solution in the context of slowly growing demand for electricity. unlike power plants with large power units, distributed sources due to small power gains can more accurately balance supply and demand for electricity and capacity both in amount and territory. this significantly reduces the risks of over-investment and excess capacity. finally, in conditions where competitive market mechanisms are limited mainly by the wholesale level, the public regulatory actions in the retail market, the rules and validity of approval of network tariffs, the practice of cross-subsidizing in the interests of certain consumer groups (usually, population) is another incentive for consumers to the abandonment of the traditional format of energy supply through a common grid and full or partial switching to own energy sources. this driver is, for example, the main incentive for the development of the dg in the russian energy system, since distortions in prices introduced by non-economic factors (in particular, keeping energy prices low for the population while having higher prices for other consumers) are the main economic reason for industrial and commercial consumers to choose an alternative model of energy supply [6]. combinations of causes and incentives that contribute to the dg development in different countries are unique and affect both the technological priorities and the overall scale of this new segment of the electric power industry. one of the countries, where the influence of regulatory and technological drivers is most significant, is germany. the active policy on the structural transformation of the electric power industry introducing renewable energy sources (res) led to the fact that in 2015 the installed capacity of the latter exceeded 90 gw. and the majority of res plants with a capacity below 25 mw connected to the distribution grid approached 79 gw [7]. together with 6 gw of small thermal power plants, this amounts to more than 40% of the total installed capacity of the national electric power system. a significant dg segment was created in the uk (25 gw, or 25%, of the installed capacity) [8]. unlike these countries, for example, in the usa, the active development of renewable energy is mostly linked with large wind farms and solar farms [9]. with a total res plants capacity of more than 100 gw, only 18 gw have a capacity below 25 mw. most of the dg sources in the us are thermal, cogeneration plants (17.5 gw), and the total share of the dg in the installed capacity of the national electric power system is still insignificant and does not exceed 5%. as we shall show below, compared to other countries, the dg role in the energy system of russia is still negligible; however, the growth rate of the small-scale power sources is noticeably higher than that of large power plants. the purpose of this article is to analyze key drivers, priorities and possible scales of the dg development for the russian electric power industry, based on current forecasts of growth in demand for electricity, the status of existing capacities, price factors in the wholesale and retail markets. potential scales of structural transformation 35 copyright ©2019 assa. adv. in systems science and appl. (2019) 2. distributed generation in the electric power system of russia and the potential for its development the unified energy system (ues) of russia is one of the largest in the world, occupying the fifth place in the world in terms of the installed capacity and the volume of electricity production in 2016, according to eia [10]. staged ues formation in the 20th century took a traditional way of generating capacities concentration, construction of large power plants (up to thousands of megawatts) and development of a high voltage network (330-750 kv). at present, thermal power plants (67.7%), nuclear power plants (11.7%) and large hydro (20.3%) play the main role in the structure of installed capacity. one feature of the ues of russia is the significant spread of cogeneration technologies (chp), which constitute about 50% of the total capacity of all thermal power plants (tpp). another feature is the extremely low level of renewable energy – the total capacity of wind, solar, geothermal and small hydro power plants in 2016 was about 0.5 gw or 0.3% of the total volume in the ues. at present, russia is pursuing a rather cautious policy to stimulate renewable energy. in accordance with the current decisions of the regulator, until 2024, about 5 gw of wind and solar power will receive tariff support, but subsequently the volume of res plants that receive support will be reduced due to the tariff growth limitations for the final consumers. according to state statistics, in 2016, russia had 36,000 power plants with a capacity below 25 mw (that are considered as dg in this article), and their total capacity was about 13.0 gw. over the past 10 years, the total installed capacity of dg power plants in russia increased by 30% (table 1), which is twice as much as the increase in installed capacity of large power plants (15.5%). however, to date, the dg is only a small, albeit an actively growing segment of the electric power industry: for 10 years, its share in the total capacity of russia's power plants increased from 4.5 to 4.9%, and in electricity production – from 2.0 to 2.6%. table 1. assessment of the scale of the dg development in russia in 2006 and 2016 2006 2016 installed capacity, gw production of electricity, twh quantity, pcs. installed capacity, gw production of electricity, twh quantity, pcs. dg – total, including: 9.97 19.76 23625 12.96 27.85 36002 % of total values for russia 4.5 2.0 4.9 2.6 in the zone of centralized power supply 2.76 8.38 261 4.47 13.65 445 tpp 2.39 7.05 196 3.38 11.18 256 res 0.37 1.33 65 1.09 2.48 189 in the zone of autonomous power supply (tpp) 7.21 11.38 23364 8.50 14.19 35557 the share of total indicators for the russian electric power industry 4.5% 2.0% 4.9% 2.6% source: russian state statistical service in the dg structure of russia, the dominant role is played by thermal power plants used for autonomous power supply of settlements isolated from the national electric power system (ues). approximately 8.5 gw (that is, about 2/3 of the entire dg capacity) is operated outside any connection to the ues of russia. the remaining 4.5 gw are used by consumers connected to the ues, to reserve supplies from the grid and to provide for themselves with electrical energy and heating. the most common types of dg are diesel, gas turbine, gas piston units, which are used as backup or peak, but mostly as cogeneration sources. in remote and isolated from power system areas distributed generation is a natural and nonalternative investment solution, whereas within the ues boundaries its development takes place in conditions of economic competition with the traditional schemes of electricity (and also heating) supply from the grid. in recent years, the growth of the dg in russia in the 36 f. veselov, i. erokhina, a. khorshev copyright ©2019 assa adv. in systems science and appl. (2019) absence of special measures of their regulatory support is due to the insufficiency of market mechanisms in the retail market, because of which the cost of supplying electricity from the grid (and for new consumers – also the level of connection fee) is higher than the cost of their own production. an important role is played by the imperfection of regulation in the heating market, where heat tariffs are less attractive for many consumers compared to the cost of heating produced from an own boiler house or a small chp. the planned transition to the heating price formation model based on long-term marginal costs envisages that the cost of heating will be determined by a competing technology – a new boiler house. such a change in the heating market will also contribute to the growth of its price and commercial attractiveness of distributed cogeneration power plants. this will reduce the risks of unprofitable sales of heating and will reduce the cost of electricity for consumers, based on the lcoe for chp based on the calculation of total discounted costs for electricity and heat production. thus, market drivers are key factors in the dg growth in russia, and the pace of this development depends on government regulatory actions in the retail electricity and heating markets that can make distributed generation an effective alternative to the traditional supply chain. however, speaking about the restructuring of the electric power system, it is necessary to assess the technically possible potential for changing the proportions between large and distributed sources in the context of a possible change in the balance of supply and demand in the ues. at present, the ues of russia has a capacity surplus, due to a suboptimal configuration of incentive mechanisms for investment into new power plants in conditions of stagnant demand. at present, it is estimated at 32 gw, and the actual power reserve in the power system is 38% and more than 2 times higher than the normative (17%). despite the extremely moderate forecast of demand growth, already in the period until 2025 this excess capacity may disappear, and in the subsequent period the capacity deficit in the ues will rapidly increase. the reason for this is the deterioration of a significant proportion of the existing thermal power plants, which equipment has already reached or will soon reach the service life limit. accordingly, investment decisions on reconstruction, replacement of equipment with more efficient or final dismantling and replacement with new capacities, should be made for these objects. currently, the wholesale market prices of electricity and capacity are insufficient to pay back investments in the reconstruction of existing large thermal power plants. the special mechanism to support investments in such projects will be launched in 2019 but any case it will not cover the entire volume of capacities that require replacement. under these conditions, there are high risks that significant volumes of the tpp’s existing capacity will leave the market. this opens up good competitive prospects for a distributed generation to replace the retiring capacity of large power plants. first of all, this refers to the replacement of large chps, many of which, as a result of a sharp decline in industrial demand for heat, are forced to contain excess heat capacity. the substitution of such power plants with smaller units, which heating capacities are adequate to actual heating demand, is one of the important directions for the dg development (the quantitative estimates of such substitution are given below). in order to assess the growing capacity deficit in conditions when the existing tpps are shutting down, it is necessary to compare the capacity requirements of the ues of russia with the volume of the guaranteed installed capacity of all types of power plants (table 2). with the macroeconomic expectations of the ministry of economic development of russia‡ the demand for electricity in the ues of russia will increase by an average of 1.2% per year and by 2035 it will amount to more than 1300 billion kwh. taking into account the normative ‡ the assessment of the change in demand for electricity and the demand for installed capacity was performed on the basis of the basic version of the long-term forecast of the socioeconomic development of the russian economy for the period until 2035, submitted to the government of russia by the ministry of economy in may 2017. potential scales of structural transformation 37 copyright ©2019 assa. adv. in systems science and appl. (2019) reserve margin, as well as various restrictions on the use of installed power plant capacity, the demand capacity requirement by 2035 corresponding to this forecast will exceed 250 gw. the value of the guaranteed installed capacity of all types is determined by the following components: the capacity of the existing tpps that have not yet reached the service life limits and do not require investment decisions (table 2), which by 2035 will decrease by 71 gw and will amount to about 55% of the 2016 level. the main reduction in the size of this segment will have to occur already in the period until 2025; the capacity of the tpps already under construction, the commissioning of which will be implemented in the coming years (until 2020) in the amount of 9 gw, based on the mid-term ues development plans for 2017-2023, approved by the system operator and by the ministry of energy of the russian federation. [11]; capacity of all types of non-carbon power plants (hydro, nuclear and res), adopted in accordance with the long-term ues development plans approved by the government of the russian federation in 2016 [12]. the capacity of nuclear power plants in the ues will increase by 7.3 gw by 2035 (by 26% from the level of 2016), the growth of large hydro plants will be about 4 gw (8% from 2016), mainly in siberia and the far east. the growth in the res capacity is expected to be about 6.5 gw, mainly in the european part of the country. table 2. characteristics of the forecast balance situation in the ues of russia in the period until 2035, gw 2016 2020 2025 2030 2035 demand for electricity 1048.5 1096.4 1165.7 1234.8 1308.0 demand for centralized heating 1235 1260 1285 1242 1310 demand for installed capacity 205.0 216.1 226.8 239.6 252.0 guaranteed installed capacity of power plants – total 236.6 245.8 188.4 186.8 189.5 operating tpp 157.2 153.1 93.6 89.2 86.2 tpps under construction 2.9 9.0 9.0 9.0 9.0 nuclear 27.9 30.6 29.7 30.5 35.2 hydro 48.1 50.7 51.2 52.1 52.1 res plants 0.5 2.4 5.0 6.0 7.0 excess capacity 31.6 29.7 capacity deficit after 2020 – total -38.6 -52.7 -62.5 including due to the growth in demand -10.7 -23.5 -35.9 thus, the amount of the guaranteed generating capacity in the ues of russia by 2025 will be reduced by almost 50 gw, remaining approximately at this level until 2035. taking into account the expected in demand for electricity and capacity requirements, the need for additional (newly commissioned) capacity, which at least partially can be provided by dg sources, can reach 39 gw by 2025, and 63 gw by 2035. based on these estimations of the potential needs for the dg, several scenarios for the mix of distributed energy technologies will be further considered, including: i. distributed cogeneration technologies; ii. res based microgeneration iii. demand management technologies. 3. scenarios for the development of the dg based on cogeneration sources from the perspective of the end-users, the problem of efficient heating supply in russian climatic conditions is no less (and perhaps more) relevant than reliable and high-quality power supply. that is why the development of distributed energy now and in the future will, first of all, focus on new small-scale cogeneration (and in the southern regions – also trigeneration) 38 f. veselov, i. erokhina, a. khorshev copyright ©2019 assa adv. in systems science and appl. (2019) sources. in this study, when assessing the development potential of the dg based on cogeneration, three groups were identified. the first group consists of the dg units, which substitute the heating supply in the service area of the existing chp plants, while the latter's capacity is reduced due to the decommissioning of the deteriorated equipment. with the introduction of market mechanisms supporting the reconstruction of existing tpp that do not take into account the low efficiency of their operation in the heating market, the reduction in their capacity as a whole in the ues of russia by 2025 may be about 28 gw, and by 2035 – more than 30 gw. accordingly, heating supply to consumers will decrease relative to 2016 by 30% by 2035. if this reduction in supplies is compensated for by dg cogeneration sources with a full load in the heating mode, the capacity of such plants can be about 20 gw by 2025-2030. the second group consists of the dg units which can respond to the additional demand for heating. given the continuing decline in the heat intensity of the russian economy, the increase in demand for heating relative to 2016 is estimated to be a modest amount – only 2% by 2025 and about 6% by 2035. taking into account the fact that cogeneration, as the most energy efficient way of electricity and heat supply, is a priority for the national energy policy, its share in the heat balance will increase, replacing traditional boiler houses. at the same time, heating output from chp will grow faster than the total demand for heating, and will increase by 7% by 2025 and by 26% by 2035. in the event that all this increase in heat supply from the chp will be provided exclusively by small sources of combined generation of electricity and heating, this will make it possible to additionally use about 5 gw of cogeneration plants by 2025 and about 18 gw by 2035. in total, two groups of dg considered above can provide input of more than 23 gw of cogeneration plants by 2025 and 39 gw by 2035. however, this value is less than the potential balance capacity deficit that may arise in the ues during this period (table 2). one of the ways to eliminate this deficit is an even more intensive development of distributed cogeneration. in this connection, the third group of dg is identified in this assessment, which will have to substitute the existing boiler houses in the heat balance, instead of their capital intensive reconstruction. one of the options for this replacement of boiler houses is their reconstruction into small chp plants with the installation of gas turbine or gas piston units. according to russian experts, the economically feasible potential of such a reconstruction of boiler houses is estimated at 60-70 gw [13]. however, based on the capacity requirements, the reasonable substitution scales should be 2-3 times smaller and will result to 17 and 25 gw of additional small chp capacity in 2025 and 2035 (table 3, scenario 1), respectively. this will lead to a serious structural rearrangement of the centralized heat balance with a sharp increase in the share of cogeneration (totally large and distributed chp) from 45% in 2016 to 70% by 2035. taking into account the high fuel utilization factor at chp plants (up to 8090%), even partial implementation of such scenario and structural reorganization of not only the electric power industry, but also the district heating system is attractive from the point of view of energy efficiency and ecological compatibility. another concomitant effect with such an active development of the chp plants may occur in the electricity market. in general, according to the ues, the share of distributed generation can reach 21% in 2025 and more than 29% in 2035. the high efficiency of cogeneration sources, mainly based on the heating load profile, will result in their annual capacity factor (50-57%) being significantly lower than the annual load factor of the ues (about 75%). to bridge this gap in the electricity balance, it will be necessary to increase the capacity factor of the most efficient large tpps with the lowest fuel costs, from 48% in 2016 to 64% by 2035 (table 3, scenario 1). in addition to the increased volumes of electricity generation at the chp in the cogeneration mode with minimal fuel costs, this change in the electricity balance structure will help reduce the price of the spot market, reshaping the profile of the electricity supply curve [14]. potential scales of structural transformation 39 copyright ©2019 assa. adv. in systems science and appl. (2019) table 3. changes in the structure of installed capacity and electricity production in the ues under various scenarios for the dg development dg development scenarios 2016 2020 scenario 1 (chp) scenario 2 (chp and res) scenario 3 (chp and demand management) 2025 2035 2025 2035 2025 2035 capacity required – total, gw 204.1 214.5 227.0 252.0 227.0 252.0 227.0 252.0 guaranteed installed capacity total, including: 236.4 246.0 188.4 189.5 188.4 189.5 188.4 189.5 existing dg 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 excess (+) / deficit (-) 32.3 31.5 -38.6 -62.5 -38.6 -62.5 -38.6 -62.5 dg capacity additions i. cogeneration, incl. 40.3 64.0 23.3 38.8 23.3 38.8 substitution of existing chps 18.6 20.7 18.6 20.7 18.6 20.7 ensuring the increase in demand for heating 4.8 18.1 4.8 18.1 4.8 18.1 substitution of boiler houses 16.9 25.1 ii. micro-res 16.9 25.1 additional capacity for res reservation 13.5 / 0§ 20.1 / 0 iii. demand management 16.9 25.1 electricity production – total, twh, incl. 1048.5 1096.4 1165.7 1308.0 1165.7 1308.0 1165.7 1308.0 nuclear 196.4 205.8 222.7 245.3 222.7 245.3 224.1 274.4 hydro 176.5 186.5 188.8 195.1 188.8 195.1 188.8 195.1 res plants – total 1.8 6.2 12.0 18.0 62.8 93.5 12.0 18.0 incl. res microgeneration 50.8 75.4 large tpps 662.6 686.7 529.6 518.6 563.5 568.9 604.2 598.5 capacity factor, % 48 49 61 64 57 / 65 58 / 71 69 74 distributed cogeneration 11.2 11.2 212.5 331.0 127.9 205.3 138.0 222.0 capacity factor, % 38 38 55 56 55 56 60 60 decrease in annual со2 emissions relative to the case without dg, mln tons -26 -44 -40 -64 -18 -44 share of dg in total installed capacity,% 2 3 21 29 20 / 21 27 / 29 15 22 share of dg in electricity generation,% 1 2 19 27 16 23 13 18 4. combined scenarios for the dg development with res and dr components the development of the distributed cogeneration plants can be combined with other dg technologies, including wind and solar microgeneration, as well as demand response options. each of these scenarios has its positive sides and risks and requires more detailed analysis, which is presented below. in the scenario of dg development as a mix of cogeneration and res units (table 3, scenario 2) there are additional risks associated with ensuring the reliability of energy supply. § the first value when reserving additional res with traditional tpps, the second – for the case of combination of res with accumulators 40 f. veselov, i. erokhina, a. khorshev copyright ©2019 assa adv. in systems science and appl. (2019) one risk factor is the stochastic operation of res plants and the need to reserve their capacity. in conditions of expected capacity deficit, this will require, along with commissioning of new res plants, to put in operation additional peaking (gas turbine or gas piston) units or to develop a lot of power storage systems that are still expensive for such mass use, including the pumped hydro storage plants. another risk factor is the low capacity factor of res plants (about 20-25% for onshore wind and 10-15% for solar plants on average for the conditions in russia). therefore, in addition to their reservation with peaking capacities to balance the growing electricity demand in the ues, it will be necessary to increase the electricity production of thermal power plants and their capacity factor to 58% (table 3, scenario 2). in this scenario, the replacement of large plants by a combination of distributed cogeneration and renewable sources can potentially provide for the whole ues, about 16% of the total electricity production in 2025 and about 23% by 2035. an alternative option is a combination of res plants with storage systems to provide controlled output of generated (and stored) electricity to the power system for a long time (4 to 6 hours). such a solution, as in the case of the reserve capacity additions, will require more capital and maintenance costs (system integration costs), and also involves losses of about 10% of the electricity in the storage systems. as a result [15-16], the levelized cost of electricity (lcoe) at renewable energy plants in russia, already exceeding these indicators of traditional generation (gas and coal-fired tpp and nuclear), increases by another 1.5 times (table 4). table 4. relative competitiveness of thermal, nuclear and res plants in russia (relative to lcoe of ccgt – 100%) 2015 2035 coal-fired tpp 137% 126% nuclear 171% 103% wind (capacity factor 23%) without system integration costs 327% 199% with cost of peaking plants reservation 412% 258% combined with storage systems 605% 322% solar (capacity factor 17%) without system integration costs 501% 248% with cost of peaking plants reservation 615% 346% combined with storage systems 879% 442% the results for scenario 2 provided in table 3 show that integration of res plants with storage systems will not require additional capacity reserves in the ues, but for the compensation of losses in storage process, the tpps capacity factor will increase to 65% in 2025 and up to 71% by 2035. another scenario, in contrast to the two previously discussed, provides that the remaining capacity deficit is eliminated not by capacity additions, but by reducing the peak load of consumers through demand response programs. at this it is assumed that in spite of the load profile changes the forecasted electricity demand will remain the same as in other scenarios, i.e. the impact of the energy saving programs is not considered. in this scenario, to balance the electricity demand it will be required to strongly increase the utilization of practically all types of generating capacities (table 3, scenario 3), including not only large tpps (capacity factor will grow to 74%), but also nuclear power plants (capacity factor growth to 89%) and the sources of distributed cogeneration commissioned to compensate the retirement of large chp plants and the increase in demand for heating (capacity factor growth from 56 to 60%). these estimates show that intensive demand response measures can in extremis lead to a tense balance situation in terms of marginal (and effective) modes for the operation of generating capacities and, if possible, should be accompanied by energy saving measures, resulted to the demand reduction, but not only reshaping of its hourly and seasonal profile. potential scales of structural transformation 41 copyright ©2019 assa. adv. in systems science and appl. (2019) otherwise, especially given the large share of chp in russian electric power industry, this will lead to an increase in the marginal costs of electricity generation and, as a result, an increase in electricity prices in the spot market. taking into account the possible implementation of dr programs, the contribution of dg to the required capacity will be the same as in previous versions, but due to the smaller volume of electricity production, its share in the total electricity production in the ues of russia will be slightly lower: up to 15% in 2025 and about 22% in 2035. 5. conclusion the estimates of the potential for transformation the russian electric power industry based on the use of dg sources, provided in the article, can be considered as the upper limit of its impact on the structure of the installed capacity and electricity production in the ues. clarification of these estimates, determination of cost-effective potential will require more detailed analysis of costs and benefits, comparison of additional costs for the construction and operation of different types of dg sources, costs savings of developing large power plants, also taking into account the possible increase in environmental charges. despite the significant differences in the mix of technological solutions, all three scenarios considered for restructuring the ues on the basis of distributed generation sources (low-carbon gas power plants, wind and solar power plants) are united by the fact that they provide a significant reduction in greenhouse gas (ghg) emissions compared with the "case as usual" option, within which the existing technological structure of generating capacities on the basis of large thermal power plants is preserved. the main reason for this is a gain in the fuel savings of cogeneration plants having a higher efficiency in the contrast with existing steam turbine thermal power plants (on average 38.5% for gas and 34% for coal-fired tpp), which in the "case as usual" scenario will remain in operation after equipment replacement with the same one. fuel savings makes it possible to reduce ghg emissions by 2035 by more than 40 million tons of со2, and the growth of microgeneration based on res will increase this effect by about 1.5 times (table 3). given the possible development of economic mechanisms for constraining ghg emissions to meet russia's national obligations under the paris protocol, this effect creates an additional competitive advantage for distributed generation, which can significantly facilitate the implementation of the scenarios considered. the economic effects in the grid will be ambivalent: on the one hand, the dg development allows reducing the costs of strengthening the transmission and distribution network within the traditional “vertically-oriented” (or “up-down”) scheme of energy supply. on the other hand, the rapid growth of the dg will require additional investments into the strengthening of “horizontal” electrical connections in the distribution grid, as well as the adaptation of the electric power system to bi-directional electricity flows. another important fact is that a significant part of the dg is the investment projects of electricity consumers, mainly industrial and commercial. their willingness and financial ability to invest in electricity generation assets that are not related to their main type of economic activity is a separate research task that requires microand macroeconomic modeling. at the same time, the state itself, as a regulator of the electricity market, can both stimulate and restrain the growth of the dg, primarily by liberalizing the rules of the retail market and reducing (albeit gradual) the amount of cross-subsidies that distort the economically feasible cost of electricity for consumers, influencing their investment choices. on the other hand, the state can ensure effective competition between the dg and large generation in the framework of the new economic mechanism for renewing large thermal power plants, which will enable them to identify the most effective locations for their replacement by new technologies. 42 f. veselov, i. erokhina, a. khorshev copyright ©2019 assa adv. in systems science and appl. (2019) acknowledgements the study was carried out with the financial support of the russian science foundation (project no. 17-79-20354). references 1. ropenus, s., & klinge jacobsen, h. (2015). a snapshot of the danish energy transition: objectives, markets, grid, support schemes and acceptance. study. berlin, germany: agora energiewende. [online]. available https://www.agoraenergiewende.de/fileadmin/projekte/2015/integration-variabler-erneuerbarer-energiendaenemark/agora_snapshot_of_the_danish_energy_transition_web.pdf 2. irena (2016). the power to change: solar and wind cost reduction potential to 2025. [online]. available https://www.irena.org/publications/2016/jun/the-power-to-changesolar-and-wind-cost-reduction-potential-to-2025 3. iea (2014). world energy investment outlook 2014. [online]. available https://www.iea.org/publications/freepublications/publication/weio2014.pdf 4. frankfurt school-unep centre/bnef (2018). global trends in renewable energy investment. [online]. available http://www.iberglobal.com/files/2018/renewable_trends.pdf 5. eu/irena (2018). renewable energy prospects for the european union. [online]. available https://www.irena.org//media/files/irena/agency/publication/2018/feb/irena_remap_eu_2018.pdf 6. pankrushina, t.g., solyanik, a.i., & zolotova, i.y. (2018). impact of grid tariffs on the competitiveness of distributed generating sources in the regions of russia. proceedings of the international scientific conference "far east con" (iscfec 2018). vladivostok, russia, 633-637, https://doi.org/10.2991/iscfec-18.2019.155 7. bundesnetzagentur (2016). kraftwerksliste der bundesnetzagentur. stand 10.05.2016 [power plant list of federal network agency as of 10.05.2016], [in german]. [online]. available https://www.bundesnetzagentur.de/shareddocs/downloads/de/sachgebiete/energie/untern ehmen_institutionen/versorgungssicherheit/erzeugungskapazitaeten/kraftwerksliste/kraftw erksliste_2015.xlsx 8. beis (2016). digest of united kingdom energy statistics 2016. [online]. available https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/577712/duk es_2016_final.pdf 9. eia (2017). eia form 860. [online]. available https://www.eia.gov/electricity/data/eia860/ 10. eia (2017). international energy statistics. [online]. available https://www.eia.gov/beta/international/rankings/#?cy=2015&aid=7&pid=2&tl_id=2-a 11. prikaz minenergo rossii №143 ot 01.03.2017 "ob utverzhdenii skhemy i programmy razvitiya ees rossii na 2017 – 2023 gody [the order of the ministry of energy of the russian federation dated march 1, 2017 no. 143 "on the approval of the scheme and program for the development of the unified energy system of russia for 2017-2023"]. [online]. available https://minenergo.gov.ru/system/download-pdf/8170/72659 12. generanaya skhema razmeshcheniya obyektov elektroenergetiki do 2035 goda. utverzhdena rasporyazheniyem pravitelstva rossiyskoy federatsii ot 12.06.2017 № 1209-r [general scheme of the placement of electric power facilities in russia until 2035. approved https://www.agora-energiewende.de/fileadmin/projekte/2015/integration-variabler-erneuerbarer-energien-daenemark/agora_snapshot_of_the_danish_energy_transition_web.pdf https://www.agora-energiewende.de/fileadmin/projekte/2015/integration-variabler-erneuerbarer-energien-daenemark/agora_snapshot_of_the_danish_energy_transition_web.pdf https://www.agora-energiewende.de/fileadmin/projekte/2015/integration-variabler-erneuerbarer-energien-daenemark/agora_snapshot_of_the_danish_energy_transition_web.pdf https://www.irena.org/publications/2016/jun/the-power-to-change-solar-and-wind-cost-reduction-potential-to-2025 https://www.irena.org/publications/2016/jun/the-power-to-change-solar-and-wind-cost-reduction-potential-to-2025 https://www.iea.org/publications/freepublications/publication/weio2014.pdf http://www.iberglobal.com/files/2018/renewable_trends.pdf https://www.irena.org/-/media/files/irena/agency/publication/2018/feb/irena_remap_eu_2018.pdf https://www.irena.org/-/media/files/irena/agency/publication/2018/feb/irena_remap_eu_2018.pdf https://doi.org/10.2991/iscfec-18.2019.155 https://www.bundesnetzagentur.de/shareddocs/downloads/de/sachgebiete/energie/unternehmen_institutionen/versorgungssicherheit/erzeugungskapazitaeten/kraftwerksliste/kraftwerksliste_2015.xlsx https://www.bundesnetzagentur.de/shareddocs/downloads/de/sachgebiete/energie/unternehmen_institutionen/versorgungssicherheit/erzeugungskapazitaeten/kraftwerksliste/kraftwerksliste_2015.xlsx https://www.bundesnetzagentur.de/shareddocs/downloads/de/sachgebiete/energie/unternehmen_institutionen/versorgungssicherheit/erzeugungskapazitaeten/kraftwerksliste/kraftwerksliste_2015.xlsx https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/577712/dukes_2016_final.pdf https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/577712/dukes_2016_final.pdf https://www.eia.gov/electricity/data/eia860/ https://www.eia.gov/beta/international/rankings/#?cy=2015&aid=7&pid=2&tl_id=2-a https://minenergo.gov.ru/system/download-pdf/8170/72659 potential scales of structural transformation 43 copyright ©2019 assa. adv. in systems science and appl. (2019) by the order of the government of the russian federation dated june 9, 2017, 1209-r]. [online]. available http://static.government.ru/media/files/zzvuuhfq2f3ojik8azkvsxrgibw8engp.pdf 13. filippov, s.p., & dilman, m.d. (2014). perspektivy ispolzovaniya kogeneratsionnykh ustanovok pri rekonstruktsii kotelnykh [prospects for the use of cogeneration plants in the reconstruction of boiler houses]. industrial power engineering, 4, 7-11, [in russian]. 14. veselov, f.v., novikova, t.v., & khorshev, a.a. (2015). technological renovation of thermal power plants as a long-term check factor of electricity price growth. thermal engineering, 62, 843-852. https://doi.org/10.1134/s0040601515120137 15. veselov, f.v., makarova, a.s., novikova, t.v., tolstoukhov, d.a., & atnyukova, p.v. (2017). konkurentnyye perspektivy aes v formirovanii nizkouglerodnogo profilya rossiyskoy elektroenergetiki [competitive prospects of nuclear power plants in building the low-carbon profile of the russian electric power industry], energeticheskaya politika, 3, 6877, [in russian]. 16. makarov, a.a., veselov, f.v., makarova, a.s., novikova, t.v., & pankrushina, t.g. (2017). strategic prospects of the electric power industry of russia. thermal engineering. 64, 817828. https://doi.org/10.1134/s0040601517110064 http://static.government.ru/media/files/zzvuuhfq2f3ojik8azkvsxrgibw8engp.pdf https://doi.org/10.1134/s0040601515120137 https://doi.org/10.1134/s0040601517110064 adv syst sci appl 2020; 04:70–82 published online at https://ijassa.ipu.ru. multi-valued neural networks ii: a robot group control dmitry maximov1∗ 1v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia abstract: for the new concept of a multi-valued neural network introduced earlier, an analogue of the t-norm in fuzzy mathematics is considered. in the multi-valued neural network, all variables are elements of the lattice of linguistic variables, i.e., they are all only partiallyordered. the lattice operations are used to build the network output by inputs. however, a lattice elements’ multiplication may also be used to determine such operations in the case when not all of them are allowed by the lattice construction. in this paper, the lattice is assumed to be residuated, and the residual construction gives the analogue of a t-norm. the lattice elements’ multiplication determines the implication which is used, together with other lattice operations, in output determining of the neural network. though, such a construction determines a multi-valued associative memory similar to the brouwer lattice case considered earlier, this variant is more natural to use in the kohonen-like networks that we demonstrate with the example of a robot group management. keywords: multi-valued neural networks, fuzzy neural networks, robot group management, associative memory, linguistic variable lattice, system tasks’ lattice 1. introduction the concept of multi-valued neural networks was introduced in [1], [2], and “multi-valued neural networks i” by d. maximov, v. i. goncharenko, and yu. s. legovich. it continues a line of research to assess the system state using elements of a lattice. the lattice may be a lattice of sets as in [3], [4] or, even, a lattice of graphs of system state configurations [5], [6]. such an approach is used in different tasks when the situation is estimated by a set of linguistic variables along with degrees of significance/certainty of the values of these variables. usually, in fuzzy systems, confidence levels take values in a numerical interval. however, the degree of fuzziness is determined by experts. in [7] it was demonstrated that it is not necessary to use numbers: linguistic variables (but already partially-ordered), which do not require a mandatory numerical evaluation, can again serve as an estimate. such assessments are determined by the situation itself and do not need an expert opinion or, at least, the experts assessment of the situation is greatly facilitated. in this case, to compare the valuations and obtain control solutions in [7], the concepts of multi-valued logic are used in which the scale of truth values is a brouwer lattice of a general form. such scales of truth values generalize a linearly ordered scale (in particular, in fuzzy logic) and naturally define the implication as a lattice operation. in the case of a complicated lattice, the calculation of implications becomes a laborious task. in [1] and [2], it is proposed to use a multi-valued neural network to calculate implications and a multi-valued associative memory to obtain a control solution quickly. the ∗corresponding author: jhanjaa@ipu.ru, dmmax@inbox.ru multi-valued neural networks ii 71 associative memory generalizes a fuzzy one of [8] to the case of using multi-valued logic operations instead of fuzzy ones. in “multi-valued neural networks i” the results of [8] on fuzzy associative memory with thresholds to the case of multi-valued one were expanded: the authors introduced the concept of such a network, investigated the properties of the associative memory and gave a generalization of the learning algorithm of fuzzy associative memory with thresholds to the multi-valued case. the inputs, outputs and connection weights of the network are linguistic variables (partially ordered), not numbers, which, however, made it possible to use such a network for processing control and diagnostic information of complex dynamic objects. however, a brouwer lattice was used as the scale of truth value in both these cases, which restricts application possibilities. in [8], a general fuzzy associative memory with a t-norm is also considered. in residuated lattices, properties in the definition of the t-norm coincide with the same as in the residuated construction definition. in this paper we use this fact to expand the results of [8] to associative memories with variables taking values in residuated lattices that may not be brouwerian in this case. we investigate the properties of such a multi-valued associative memory and give a generalization of the learning algorithm of fuzzy associative memory to the multi-valued case similarly to “multi-valued neural networks i” and [1]. however, the application of such a construction is natural in kohonen-like neural networks rather than in associative memories due to the residuals’ intuitive meaning. we demonstrate such an application — with the example of a robot group management. 2. backgrounds feedforward fuzzy neural networks in which internal operations are based on fuzzy operations of joins and meets ∨ − ∧, were proposed in [9]. such networks are called ∨ − ∧ fuzzy associative memory, and fuzzy information in them represented by the elements of the interval [0, 1]. we will use elements of a general lattice in multi-valued associative memory instead of elements of the interval [0, 1]. 2.1. lattices [10] definition 2.1: a lattice is a partially-ordered set having, for any two elements, their exact upper bound or join ∨ (sup, max) and the exact lower bound or meet ∧ (inf, min). definition 2.2: the exact upper bound of a subset x ⊆ p of a partially-ordered set p is the smallest p element a, larger than all the elements of x: min(a) ∈ p : a ≥ x, ∀x ∈ p . definition 2.3: the exact lower bound is dually defined as the largest p -element, smaller than all the elements of x . definition 2.4: a complete lattice is a lattice in which any two subsets have a join and a meet. this means that in a non-empty complete lattice there is the largest “>” and the smallest “0” elements. if we take such a lattice as a scale of truth values in a multi-valued logic, then the largest element will correspond to complete truth (true), the smallest to complete falsehood (false), and intermediate elements will correspond to partial truth in the same way as the elements of the segment [0,1] evaluate partial truth in fuzzy logic. in logics with such a scale of truth values, implication can be determined by multiplying lattice elements, or internally, only from lattice operations. copyright c© 2020 assa. adv syst sci appl (2020) 72 dmitry maximov definition 2.5: lattice elements, from which all the others are obtained by join and meet operations are called generators of the lattice. definition 2.6: a lattice is called atomic if every two of its generators have null meets. definition 2.7: a brower lattice is the lattice that has internal implications. definition 2.8: in such a lattice, the implication c = a⇒ b is defined as the largest c : a ∧ b = a ∧ c. definition 2.9: the implication ¬a = a⇒ 0 is called the pseudo-complement of a. distribution laws for join and meet are satisfied in brouwer lattices. the converse is true only for finite lattices. 2.2. residuated lattices [11] in non-distributive lattices, the implication cannot be defined. however, we may introduce a multiplication of the lattice elements and use it to define an external implication. definition 2.10: a residuated lattice is an algebra (l,∨,∧, ·, 1,→,←) satisfying the following conditions: • (l,∨,∧) is a lattice; • (l, ·, 1) is a monoid; • (→,←) is a pare of residuals of the operation ·, that means ∀x, y ∈ l : x · y 6 z ⇔ y 6 x→ z ⇔ x 6 z ← y in this case, the operation · is order preserving in each argument and for all a, b ∈ l both the sets {y ∈ l|a · y 6 b} and {x ∈ l|x · a 6 b} each contains a greatest element (a→ b and b← a respectively). the monoid multiplication · is distributive over ∨: x · (y ∨ z) = (x · y) ∨ (x · z). also, x · 0 = 0 · x = 0. a special case of residuated lattices is a heyting algebra, when the monoid multiplication coincides with ∧. in non-commutative monoids, residuals → and ← can be understood as having a temporal quality: x · y 6 z means “x then y entails z,” y 6 x→ z means “y estimates the transition had x then z,” and x 6 z ← y means “x estimates the opportunity if-ever y then z.” you may think about x, y, and z as bet, win, and rich correspondingly (wikipedia). though, the temporal quality refers to residuals, we refer the same meaning to any multiplication x · w 6 z. however, only the residual y estimates the transition x→ z. definition 2.11: a residuated lattice a is cancellative if it satisfies the equations x · y = x · z =⇒ y = z and y · x = z · x =⇒ y = z [12]. a residuated lattice is cancellative if, and only if, it satisfies the equations x→ xy = y and yx← x = y [12]. copyright c© 2020 assa. adv syst sci appl (2020) multi-valued neural networks ii 73 definition 2.12: a residuated lattice a is said to be integrally closed if it satisfies the equations x · y 6 x =⇒ y 6 1 and y · x 6 x =⇒ y 6 1, or equivalently, the equations x→ x = 1 and x← x = 1 [13]. every cancellative residuated lattice is an integrally closed one, and any upper or lower bounded integrally closed residuated latticel is integral, i.e., a 6 1, ∀a ∈ l [13]. therefore, a finite calcellative lattice is integral. we will assume that the lattices used in a multi-valued neural network are complete, finite and residuated. we can compare the definition of a residuated lattice and the definition of a fuzzy operator and a t-norm [14], [8]: definition 2.13: we call the mapping t : [0, 1]→ [0, 1] a fuzzy operator, if the following conditions hold: • t (0, 0) = 0, t (1, 1) = 1; • if a, b, c, d ∈ [0, 1], then a 6 c, b 6 d, ⇒ t (a, b) 6 t (c, d); • ∀a, b ∈ [0, 1], t (a, b) = t (b, a); • ∀a, b, c ∈ [0, 1], t (t (a, b), c) = t (a, t (b, c)). if t is a fuzzy operator, and ∀a ∈ [0, 1], t (a, 1) = a, we call t a t-norm. for a, b ∈ [0, 1], let us define a→ b ∈ [0, 1] : a→ b = sup{x ∈ [0, 1]|t (a, x) 6 b}. we see that the latter definition coincides with the residual construction definition in the case of a commutative monoid and l = [0, 1]. excluding the condition of commutativity and the requirement l = [0, 1], we obtain the construction of a complete and residuated lattice (we demand also its finiteness). 3. multi-valued ∨ − ∗ associative memory as in the case of fuzzy neural networks [8], a multiplication of variables is used together with lattice operations in ∨ − ∗ multi-valued associative memory, where ∗ stands for the multiplication. let us suppose, an input signal is x ∈ ln, and an output signal is y ∈ lm, where l is the residuated lattice used. in this case, the input-output relationship in a twolayer multi-valued associative memory can be written as y = x ~ w, where ~ stands for the ∨ − ∗ composition operation, and w = (wij)n×m ∈ µn×m is the n×mmatrix of the weights of connections with elements from l, i ∈ n = {1...n}, j ∈m = {1...m} (fig. 3.1). in [1], several of the simplest composition variants have been suggested for ∨ − ∧ multi-valued associative memory and the next two of them are considered: yj = ∨ i {xi ∧ wij}; (3.1) yj = ∧ i {xi ⇒ wij}. (3.2) in the theory of the simplest fuzzy associative memory, only option (3.1) is considered. more complex models use a combination of ∨ and t-norms [8], [15] and a combination with implication [15]. however, they all use a linear number segment as a set of variables, unlike this paper where a general non-number lattice with multiplying elements is used. copyright c© 2020 assa. adv syst sci appl (2020) 74 dmitry maximov fig. 3.1. bilayer associative memory thus, we consider the composition 3.1, but with the monoid multiplication ∗ instead of ∧ (fig. 3.1), as in [8], however with all variables taking values in the residuated lattice l: yj = ∨ i {xi ∗ wij}. (3.3) in the vector notation, it can be written as: y = x ~ w. (3.4) we denote (x, y ) = {xk, yk | k ∈ p}, p = {1...p} — the family of pairs multi-valued patterns with xk = (xk1, ..., x k n) and yk = (yk1 , ..., y k m). we establish the connection weight matrix w ∗ = (w∗ij)n×m: w∗ij = ∧ k∈p (xki → ykj ). (3.5) the definition generalizes the similar one from [8]. then, let us define the sets [8]: sij(w∗, x, y ) = {k ∈ p | ykj 6 xki ∗ w∗ij}. mw = {w ∈ µn×m | ∀k ∈ p, xk ~ w = yk}. this is the set of the pattern pair family and the connection matrix, which satisfy the equation 3.3. the next property follows from the residual operation definition and holds both in fuzzy [8] and residuated [11] cases: a ∗ (a→ b) 6 b. (3.6) also, the following properties hold in residuated lattices [11]: c→ ( ∧ y ) = ∧ {c→ y|y ∈ y }; (3.7) b→ (a→ c) = (a ∗ b)→ c. (3.8) the property (3.6) is used in the following theorem proof exactly in the same way as in [8] due to the fact that the t-norm is a special case of the residuated lattice construction. however, such a theorem in [8] contains a necessity condition in point (iii) which does not hold in the case of a general non-chain lattice. in the general case, it may be ∨ixi = y, though ∀xi < y that contradicts the necessity. copyright c© 2020 assa. adv syst sci appl (2020) multi-valued neural networks ii 75 theorem 1: let w∗ = (w∗ij)n×m ∈ µn×m ∈ lm. then: (i) ∀k ∈ p, xk ~ w∗ ⊂ yk, and if the matrix w satisfies: ∀k ∈ p, xk ~ w ⊂ yk, then, w ⊂ w∗; (ii) if mw 6= ∅, then, w∗ ∈mw, and ∀w ∈mw, w ⊂ w∗, i.e., ∀i ∈ n, j ∈m, wij 6 w∗ij; (iii) the set mwcd 6= ∅ if ∀j ∈m, ⋃ i∈n sij(w∗, x, y ) = p . theorem 1 tells us that w∗ is the largest solution of the equation (3.4), if such a solution exists, and we will use this fact in the next section when we go down from the largest lattice element to w in a learning algorithm. 3.1. learning algorithm the iteration scheme for learning the connection weight matrix w of the multi-valued associative memory is similar to [8] with changes as in [1] and “multi-valued neural networks i”. it generalizes the dynamic δ-learning algorithm of fuzzy associative memory, introduced in [16], [17], [18] for different fuzzy cases, to the ∨ − ∗ multi-valued case. however, we are bound by atomic cancellative (hence, integrally closed) residuated lattices in the following algorithm, since theorem 2 is proved only in this case. hence, 1 = > in the lattice used since it is finite. thus, we get an algorithm of wij iteration for i ∈ n and j ∈m : step 1. initialization: for i ∈ n , j ∈m let us put wij(0) = >, t = 0; step 2. let w(t) = (wij(t)); step 3. let us calculate the resulting output: y = x ~ w, i.e., ∀k ∈ p, j ∈m, ykj (t) =∨ i[x k i ∗ wij(t)]. step 4. weights selection. denotation 1: from this place, we denote by {ykj } the set of generators contained in the lattice element ykj : {gl|gl 6 ykj } and by {w}ij the matrix of such sets. the matrix elements are the sets of generators of the weight matrix elements. matrices wij of weights and {w}ij are one-to-one correspondent to each other in distributive lattices. in non-distributive lattices, not one set {w}ij may correspond to one wij . in this case, we take max{w}ij as corresponding set to wij .the set difference will be denoted by a minus sign. then, {w}ij(t+ 1) = ⋂ k  {w}ij(t)− sym({ykj (t)} − {ykj }), if k : xki ∗ wij(t) > ykj ; {w}ij(t), otherwise. (3.9) here operation sym means the symmetrized set difference: sym(a−b) = { a−b, a ∩b ⊂ a; b − a, a ∩b = a. (3.10) step 5. for i ∈ n , j ∈m , let us check {w}ij(t+ 1) = {w}ij(t)? if this is true, then the algorithm stops, otherwise t = t+ 1 and goes to step 2. copyright c© 2020 assa. adv syst sci appl (2020) 76 dmitry maximov theorem 2: let the matrix sequence {w(t) | t = 1, 2...} is obtained by the learning algorithm. then, (a) {w(t) | t = 1, 2...} is non-increasing sequence; (b) {w(t) | t = 1, 2...} converges; (c) {w(t) | t = 1, 2...} converges to w ⊆ w∗, where w∗ij is defined in (3.5). proof (a) for i ∈ n , j ∈m , k ∈ p we get from (3.9) that {w}ij(t+ 1) 6 {w}ij(t). therefore, for i ∈ n , j ∈m : w(t+ 1) ⊂w(t). thus, {w(t) | t = 1, 2...} is a non-increasing sequence. (b) {w(t) | t = 1, 2...} converges, since the sequence wij(t) is bounded below by the smallest lattice element 0 ∀t = 1, 2.... (c) let us prove ∀t ∈ {0, 1, ...}, wij(t) 6 w∗ij . if t = 0, then wij(0) = > ≡ 1. if wij(t) = > for t = 1, 2, ..., then xki ∗ wij(t) 6 ykj , or wij(0) = > is a decision, by (3.9). by the definition of the residual, ∀k ∈ p, xki → ykj = >, or wij(0) = w∗ij by theorem 1. therefore, w∗ij = > ≡ 1 = wij(t), ∀t ∈ {0, 1, ...}. let us suppose limt{wij}(t) 6 w∗ij . let t0 is the first step such that wij(t0) 6 w∗ij . then, xki ∗ wij(t0) 6 xki ∗ w∗ij = xki ∗ ∧ k′∈p (x k′ i → yk ′ j ) 6 xki ∗ (xki → ykj ) 6 ykj by (3.6). thus, by (3.9), we get wij(t0 + 1) = wij(t0) 6 w∗ij . let limtwij(t) = wij(t0) = lij 6 w∗ij . then, lim t ykj (t) = ∨ i∈n (xki ∗ lij) 6 ykj . (3.11) therefore, by (3.9) and (3.11), we get: lim t {wij}(t) = {lij} = = ⋂ k [lim t {wij}(t)− sym({lim t ykj (t)} − {ykj })] = = ⋂ k [{lij} − ({ykj } − {lim t ykj (t)})]. (3.12) thus, limt y k j (t) = ykj for ∀k ∈ p , and lij = wij(t0) = limtwij(t) 6 w∗ij is a decision of (3.4), or {lij} ⋂ ({ykj } − {limt y k j (t)}) = ∅. the latter case means, at least, that {lij} is incomparable with ({ykj } − {limt y k j (t)}). thus, {ykj } ⋂ {limt y k j (t)} ⊇ {ykj } ⋂ {lij}. then, {ykj } ⋂ {limt y k j (t)} = {ykj } ⋂ {lij}, since xki ∗ lij 6 lij in integral lattices, and, therefore, lim t ykj (t) = ∨ i∈n (xki ∗ lij) 6 lij. (3.13) then, let us suppose limt{wij}(t) = lij > xki → ykj for ∀k ∈ p , hence lij > w∗ij . then, we get by the definition of the residual: xki ∗ lij > ykj and, therefore, lim t ykj (t) = ∨ i∈n (xki ∗ lij) > ykj . (3.14) hence, we obtain similarly (3.12) with (3.13) that limt y k j (t) = ykj that contradicts (3.14). thus, limt{wij}(t) = lij ≯ xki → ykj for ∀k ∈ p that means limt{wij}(t) 6 {w∗ij}, or limt{wij}(t) is incomparable with {w∗ij}. copyright c© 2020 assa. adv syst sci appl (2020) multi-valued neural networks ii 77 however, limt{wij}(t) may not be incomparable with w∗ij . indeed, let t0 is the first step such that wij(t0) is incomparable with w∗ij . then, if xki ∗ wij(t0) is incomparable with ykj , we get limt{wij}(t) = lij = wij(t0) by (3.9), and limt y k j (t) = ∨ i∈n(x k i ∗ lij) is incomparable with ykj or limt y k j (t) = ∨ i∈n(x k i ∗ lij) > ykj . however, the latter case contradicts (3.12) with (3.10) similarly to (3.14) and (3.13). the same holds if xki ∗ wij(t0) > ykj , also similarly to (3.14) and (3.13). let limt y k j (t) = ∨ i∈n(x k i ∗ lij) is incomparable with ykj . however, (3.12) and (3.10) demand then {limt y k j (t)} − {ykj } = ∅ since (xki ∗ lij) 6 lij , and, hence, {limt y k j (t)} ⊆ {lij}. now, let us limt{wij}(t) = lij is incomparable with w∗ij , and xki ∗ lij 6 ykj . then, xki ∗ lij is also incomparable with xki ∗ w∗ij , since the lattice is cancellative (otherwise, e.g., xki ∗ lij 6 xki ∗ w∗ij =⇒ lij 6 xki → xki ∗ w∗ij = w∗ij). hence, it should be xki ∗ lij < ykj and xki ∗ w∗ij < ykj , otherwise, they are comparable. let us consider lij → w∗ij = lij → ∧ k x k i → ykj . thus, lij → w∗ij = ∧ k lij → (xki → ykj ) = ∧ k(x k i ∗ lij)→ ykj by (3.7) and (3.8). therefore, (xki ∗ lij)→ ykj = 1 in integral lattices, since (xki ∗ lij) < ykj [13]. hence, lij → w∗ij = 1, and lij 6 w∗ij which contradicts their incomparability. thus, we obtain the unique result: limtwij(t) 6 w∗ij . 4. example of a robot group management such a network may be used as an associative memory or pattern classifier in the same manner as ∨ − ∧ networks in [2] and “multi-valued neural networks i”. it may be useful in the case of a non-distributive lattice of linguistic variables. however, a systematic method to introduce the monoid multiplication in such a lattice is absent, and we might use heuristics in every particular case. moreover, the residual intuition treats the multiplication as action: we had something in the past, have something else now, that entails a third thing as a result of the operation. all these reasons lead us to the fact that the networks are more suitable for using, not with patterns (their storing or classifying), but in some leader output detecting. thus, we consider here the problem of task distribution in a group of janitor robots. the group is fulfilling a set of tasks. then, a new task arises. robots must decide which one (or several) of them will perform the new task. such a problem was considered in [5] based on linear logic. however, the structure of linear logic is rather intricate and difficult to implement programmatically . the residuated construction is much simpler and allows the task lattice to be determined more adequately. let us consider a group of robots that can perform the following tasks: x1 — trash search, e — taking out the trash, x3 — sawing of the found garbage when it cannot be taken out, x2 — the stupor: it is the state of a robot when it tries, but cannot perform a regular task, e.g., the robot cannot pick up the trash, or cannot saw it, or can not move. all these main tasks have subtasks (fig. 4.2): trash search x1 includes d1 — a move, and d4 — a video camera operation; trash removing e includes d1 and d3 — grabbing of the debris; sawing the garbage x3 includes d3 and d2 — sawing which is not depicted for simplicity. the stupor x2 includes only the grip: a robot has grabbed a thing, but can do nothing with it. elements x2, x3, d1, and d4 may be taken as generators. all others are their meets and joins. such tasks are, e.g., c42 — a robot in a stupor operates with a video camera, or c43 — the same during sawing. in this case, we consider the joined tasks as fulfilling in parallel. however, in other cases, we may consider joined tasks as performed sequentially. the lattice in fig. 4.2 differs from the similar one in [5]: now, we include all joined tasks copyright c© 2020 assa. adv syst sci appl (2020) 78 dmitry maximov fig. 4.2. a lattice of system task estimations which have meanings. however, including all of them may be unnecessary. then, the lattice may be non-distributive. thus, we consider, as in previous papers, the task lattice as the lattice of linguistic variables denoting tasks. though, the tasks may also be associated with the sets of operations required for a task to be performed. as usually, the lattice partial order orders lattice elements by their values: the higher an element lies in the lattice diagram fig. 4.2, the greater the value it has. thus, maximal active system behaviour has the top value >, and activity absent has the bottom value 0. these values estimate transitions a→ b = c: the value c estimates the transition from the task a to the task b. or, once more, “had a then c (arising) entails b.” 4.1. monoid definition there is no a general method to define the monoid operation in a residuated lattice. every time we should use some heuristics. in our case, we demand c42 = 1, since we want to change the stupor x2 into the grip d3 if we obtain the stupor in two consequent iterations: x2x2 = d3 †, though in other cases, e.g., d1d1 = d1 ‡. therefore, it may be x2 < 1. subtasks d3 and d4 arising should not usually change the current task. thus, we consider c42 = 1 as the only right neutral element of the monoid multiplication (correspondingly, we consider only the right multiplication and left residuals x→ y), since one can hardly give equally meaningful meaning to a left neutral element. then, we want ex2 = d3, i.e., if we get stupor during debris pick up because the robot cannot lift the trash, only the grab remains from the trash remove. therefore, d1x2 ∨ d3x2 = d3, and, hence, d1x2 = d3 or d1x2 = 0, and d3x2 = 0 or d3x2 = d3. we chose d1x2 = 0 (that is evident: the stupor during the move entails an activity cessation) and d3x2 = d3, because it should be for all lattice elements x1 = x: d11 = d1x2 ∨ d1d3 ∨ d1d4 = d1, and, therefore, †we omit the multiplication sign in this section for simplicity. ‡we may also demand x2x2 = 0 — stupor changes to activity absent. copyright c© 2020 assa. adv syst sci appl (2020) multi-valued neural networks ii 79 d1x2 6= d3 §. thus, d1d3 = 0 and x2d3 = d3 by monotonicity, and, hence, d1d4 = d1. also, we demand x1x2 = d4, i.e., if a robot cannot move during a debris searching, only camera functioning remains. then, d4x2 = d4 — the stupor does not influence camera functioning. similarly, x3x2 = d3 and x1d3 = d4d3 = d4. then, evidently, x1d1 = x1, d4d1 = x1 (since, a move does not cancel the previous camera operation), x1d4 = x1 (since, the robot may stop during the search), and d4x1 = x1x1 = x1. hence, x1e = x1, and ex1 = d3x1 = e since we suppose d3d1 = d1 and d3d4 = d3 — from d31 = d3. thus, e.g., c42 = 1 = d4 → d4 and c1e = x1 → x1. note, that the robot in search does not normally switch to trash removing, as in ants’ colonies, because x1e = x1 — “had x1, now e arises, that entails x1”. this result was obtained in [3] [3] from the linear logic structure definition, while here the conclusion is the consequence of our multiplication operation choice. now, let us x2d1 = d1. therefore, x2x1 = c2e, because x2d4 = x2 from x21 = x2. then, let it be x3d1 = d1 and x3d4 = x3 — from x31 = x3 and evidently x3d3 = d3. then, x3x1 = c3e. at least, let us consider x3 multiplication: d1x3 = x3, d3x3 = x3, d4x3 = c43. the first two are evident, in the latter case we establish that sawing does not cancel the camera operation. also, we suppose x2x3 = > since we do not know if the stupor can switch to sawing at this place, or transition to any task (or again to the stupor) is possible. then, u23e = x2 → >. thus, we obtain the following basic definitions from the consideration above, though, our choice was sometimes not unique and, therefore, may be changed: d1d1 = d1 d3d3 = d3 d4d4 = d4 x2x2 = d3 or 0 x3x3 = x3 d1d3 = 0 d3d1 = d1 d4d3 = d4 x2d1 = d1 x3d1 = d1 d1d4 = d1 d3d4 = d3 d4d1 = x1 x2d4 = x2 x3d4 = x3 d1x2 = 0 d3x2 = d3 or 0 d4x2 = d4 x2d3 = d3 x3x2 = d3 d1x3 = x3 d3x3 = x3 d4x3 = c43 x2x3 = > x3d3 = d3 it is easy to verify that the (right) monoid definition satisfies these equations. 4.2. neural network operation a variant of the kohonen network may be used to determine the leading output or outputs in the problem of task distribution in a robot group. the kohonen net [19] is a bilayer neural network in which the output layer consists of n neurons with m inputs in each neuron. the output of the j-th neuron is obtained according to the following formula: yj = dj + ∑ i∈[0;m] wijxi, where x = (x1, ..., xm) is an input vector, and dj is a threshold. the network operates by the rule “winner takes it all” which determines the leading output. residuated lattice operations may be used in the formula as in ∨ − ∗ ((3.3)) and ∨ − ∧ [2], [1] multi-valued associative memories. here, we consider only a variant without a threshold, with wij = 0 if i 6= j, and with a unique input x (this is the arising task in the robot group). the formula for neuron outputs is also different then: yi = ti ∨ x > ti(ti → (ti ∨ x)) (4.15) in the expression, ti are tasks which the i-th robot supposes to perform (its intentions), and ti ∨ x denotes the i-th robot intention to fulfil task x together with the other robot tasks. §if we chose d3x2 = 0, then ex2 = 0 that is also possible. copyright c© 2020 assa. adv syst sci appl (2020) 80 dmitry maximov fig. 4.3. the neural network topology for the system task choice the right part of (4.15) is a consequent of (3.3) and corresponds to the usual logic rule: [a ∧ (a⇒ b)]⇒ b. that means we obtain b if we have a, and a entails b. the formula has the same meaning in the residuated case: ac 6 b, where c 6 a→ b. hence, we take ti ∨ x as an output and we might have an extremal principal to choose the necessary output. we take here mini[ti ∨ x] as the principal, because it is naturally to chose a robot with a minimal set of tasks which it is supposed to perform. clearly, we take several such outputs in the absence of the minimum. thus, we consider the neural network depicted in fig. 4.3, in which the input variable x is an arising task in the robot group which is fulfilling tasks ti at the current time. an arising task may be here x1, e, and x3, since stupor x2 refers to only one concrete robot, but not to all of them. we might choose now the necessary output for different input cases and tasks performed. here, we will use an extended form for outputs instead of (4.15): tix 6 ti ∨ x (“there was ti, now x arises, entails ti ∨ x”), in the case of x 6 ti → (ti ∨ x). let us suppose a standard situation: robots are looking for a trash and remove it. then, if a search task x1 arises, some of the robots already performing x1 will include this new task into their intention set. indeed, x1x1 = x1 is the minimal intention of joined tasks. a robot fulfilling subtask d4 of x1 (the camera operation) may also include the new task into its intentions, since d4x1 = x1. such a state d4 may happen if a searching robot falls into a stupor: x1x2 = d4. a robot moving without any other task (subtask d1, e.g., which return after the trash remove) may also switch to this new one, since d1x1 = d1 < x1. if a task e of taking out the trash arises, then it will be performed by a robot already removing the debris: ee = e is the minimal value in the standard situation. though, a moving robot (in d1) or a grabbing one (in d3) may also switch to taking out: d1e = d1 < e, d3e = e. the latter case arises after sawing when only grabbing remains. thus, sawn-off pieces may be removed by the same robot that has sawn them off, or one robot can saw, and another remove the sawn-off pieces. if a task of garbage sawing x3 arises, then it will be performed by one of robots already fulfilling sawing (x3x3 = x3), or by a robot in grabbing: d3x3 = x3 — these values are minimal. the latter situation arises when the robot picking up the debris can not lift it: ex2 = d3 — only the grabbing remains. if the robot cannot saw the trash, it switches to grabbing: x3x2 = d3. however, in this situation, no regular task can be performed, and collective removing is required: d3e = e, where the task e is a part of the collective action. but this decision is external to the neural network application. also, such an external decision copyright c© 2020 assa. adv syst sci appl (2020) multi-valued neural networks ii 81 is needed if the robot cannot grab the trash: ex2 = d3, then, d3x2 = d3, and we obtain in two iterations x2x2 = d3 or x2x2 = 0. thus, we see that the system of robots behaves quite reasonable, and the behaviour is due only to our choice of basic multiplications in the residuated lattice. 5. conclusion a new concept of multi-valued neural networks with variables and weights taking values in a residuated lattice was introduced in this study. the concept continues a row of studies in which the state of a system is estimated not by numbers, but by elements of a partially ordered set, namely, the lattice. in our case the lattice is finite and residuated, and it may consist of some linguistic variables. such an approach can facilitate the assessment of the situation by experts in cases requiring their participation. we have expanded the results on fuzzy neural networks to such a ∨ − ∗multi-valued case in which variables may take values in a non-distributive lattice. we found out the conditions under which it is possible to store in the multi-valued associative memory given pairs of network variable patterns. we also gave the learning algorithm that generalizes the fuzzy one for the multi-valued case. however, the algorithm is suitable in the special case of the residuated lattice: it 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(1989) self-organizing maps, berlin-new-york: springer-verlag. copyright c© 2020 assa. adv syst sci appl (2020) introduction backgrounds lattices residuated lattices multi-valued associative memory learning algorithm example of a robot group management monoid definition neural network operation conclusion microsoft word 1115-source texts-6538-1-18-20231219 adv syst sci appl 2023; 04; 148-155 published online at https://ijassa.ipu.ru. comparison to the proposed hybrid model and machine learning techniques for survival prediction of corona, infected patients md. asadullah1*, md. murad hossain1, md. matiur rahman molla2, md. matiur rahaman1 1 bangabandhu sheikh mujibur rahman science and technology university, gopalganj, bangladesh 2 islamic university, kushtia, bangladesh abstract: sars-cov-2, a novel coronavirus discovered in wuhan, china is spreading quickly and has a high incidence rate around the globe. as a result, everyone on the planet is having difficulty adjusting to the effects of corona and is unable to foresee the devastation and disaster caused by covid-19. in this work, we predict the survival status of patients infected with coronavirus using three distinct machine learning (ml) techniques: random forest (rf), support vector machine (svm), and logistic regression (lr). we also assess the classification performances of these algorithms. here, we put out a hybrid model and evaluated it against the three previously discussed machine learning techniques. the outcomes demonstrated the 97.85% prediction accuracy of our suggested hybrid model. aside from our suggested hybrid model, the random forest machine learning method demonstrated the highest accuracy of 94.62% among the three. nonetheless, the prediction accuracy of the hybrid model outperforms that of the random forest and is significantly better than that of the other three ml techniques. the classification performances were assessed using the f-score, sensitivity, specificity, and precision metrics. using 10-fold cross-validation, roc assessments and confusion matrices produced by these machine learning algorithms were provided and examined. to assess the effectiveness of the classification. these machine learning algorithms' roc assessments and confusion matrices are shown and examined by 10-fold cross-validation. keywords: covid-19, machine learning, hybrid model, random forest, roc curve 1. introduction the entire world is struggling to adjust to corona's effects and financial losses. they were unable to foresee the disaster and loss brought on by covid-19. numerous research investigations have been carried out to characterize the various models, strategies, innovations, and levels of awareness about the transmission of covid-19. the categorization of verified covid-19 cases associated with the epidemic diseases identified a significant obstacle to sustained advancement. the neural network's group method of knowledge handling (gmdh) form, which is linked to fake intelligence techniques, provided an explanation for binary classification modeling [1]. a deep-learning neural network concept is used to provide a verification method. using the concept of a deep-learning neural network, this system employs long short-term memory (lstm) and a gated recurrent unit (gru) for the last step of dataset training [2]. the xgboost machine learning method was assisted by building a prognostic prediction model and emphasizing emphasis. as a result, 29 patients who were cleared after february 19th were tested [3]. ai-driven techniques assist in classifying covid-19 outbreaks in a way that makes sense for their global propagation. the original goal of the research is to identify active learning-based cross-population train/test * corresponding author: asadullahstat@gmail.com word template for assa manuscript 149 copyright ©2023 assa. adv. in systems science and appl. (2024) models that use multitudinal and multimodal data in order to detect covid-19, similar to other healthcare difficulties [4]. washkaro combines real sources of data with daily news and presents it in hindi using nlp techniques, machine learning, and m-health to provide an awareness-raising solution. additionally, the appliance provides community-focused audiovisual content in local languages that has been hand-picked and verified by humans [5]. when implementing a mobile phone-based web survey, recommend utilizing machine learning techniques to be prepared to improve potential covid-19 case identifications more quickly. additionally, this will lessen the spread among those who are vulnerable [6]. an early screening model to distinguish covid-19 pneumonia from influenza was created using deep learning techniques within a virus infection and healthy cases with lung ct pictures [7]. using chest x-ray radiographs, three distinct convolutional neural networkbased models are proposed for the identification of individuals afflicted with coronavirus pneumonia. using 5-fold cross-validation, roc assessments and confusion matrices using these explained models are provided and examined [8]. described the machine learningbased classification of the extracted deep feature on chest x-ray images using patients with covid-19 and pneumonia using resnet152. it is also possible to anticipate the spread of the novel coronavirus (covid-19) in likely patients using non-invasive and early forecasting of the virus by examining chest x-rays [9]. this study [10] reveals and uses machine learning-based ct radionics models to predict hospital stays for individuals with pneumonia associated to sars-cov-2 illness. uses the publicly available covid-19 chest x-ray dataset to illustrate how bayesian convolutional neural networks (bcnn) can quantify the uncertainty in deep learning solutions to improve the diagnostic performance of the human-machine combination and describe the uncertainty in prediction is strongly associated with the accuracy of the prediction [11]. this research assumes that artificial intelligence's deep learning methods could afford to extract covid-19's specific graphical features and supply a clinical diagnosis, thus saving critical time for disease decline, to do the pathogenic test, through the support of covid-19 radiographical changes in ct images [12]. utilizing the recently developed vaxign-ml machine learning technique in conjunction with vaxign reverse vaccinology, covid-19 vaccine candidates are predicted. six proteins, including the five non-structural proteins (nsp3, 3cl-pro, and nsp8-10) and s protein, were predicted to be adhesins after experiments with the entire proteome of sarscov-2 [13]. the previous study suggested three well-known techniques: enhancing the currently limited data, using a panel selection method to select the most straightforward forecasting model among multiple models, and optimizing a personal prediction model's parameters for optimal accuracy. however, this study uses a method that supports the three benefits of knowledge mining from a small, poor dataset [14]. to ascertain the scope, duration, and end date of covid-19 in china rather than using epidemiological models to analyze the dynamics of covid-19 transmission in china. this work suggests real-time covid-19 forecasting techniques influenced by fake intelligence (ai) [15]. try testing the model's forecasting power in this article by training it on data from the 24th of january to the 3rd of march window, matching the forecasts up to the 1st of april. a one-week buffer window (march 25–april 1) was provided even for italy and the republic of korea to validate the model's forecast. in the instance of the us, the model accurately depicts the progression of the infected curve and projects that the infection will stop by april 20, 2020 [16]. a case study utilizing fuzzy rule induction and cmc—which is improved by deep learning networks—is examined in order to gain more accurate stochastic insights on the evolution of the pandemic. fuzzy rule induction approaches are used in conjunction with a deep learning-based cmc, as opposed to relying solely on uniform and uncomplicated assumptions for an mc [17]. throughout this work, a novel bio-inspired metaheuristic simulates the transmission and infection of healthy individuals by the coronavirus. to replicate coronavirus behavior as closely as feasible, relevant parameters such as superspreading rate, traveling rate, and re-infection probability are added to the model [18]. 150 i. petrov, j.f. doe copyright ©2023 assa adv. in systems science and appl. (2023) incorporating city-to-city links, a mathematical framework is constructed to determine the total number of secondary cases caused by the imported cases, as well as the number of imported cases of the novel virus from epidemic resources. additionally, to replicate outbreaks in several cities, a meta-population compartmental model was constructed and backed by a classical sir technique [19]. this study compares the prediction outcomes of three distinct mathematical models based on many parameters and multiple locations. while the gompertz model's fitting effect is also superior to the bertalanffy model's, the logistic model's fitting effect is also the best [20]. three biomarkers were chosen by machine learning methods in order to show the survival of specific patients by identifying susceptible predictive biomarkers of the severity of the disease. in order to prioritize patients at the highest risk and maybe lower the rate, this research proposes a straightforward and workable method [21]. studies have shown that patients with cardiovascular and metabolic problems were more likely to contract covid-19 and that the infection worsened. this investigation aims to determine the relationship between metabolic and cardiovascular illnesses using covid-19 [22]. nonetheless, the aforementioned studies discuss diverse mathematical strategies, artificial intelligence models, neural network models, and verification and validation methods for the prediction of patients impacted by corona. some articles even used a statistical, stochastic, and mathematical model to describe how corona transmission occurs. however, a small number of studies—particularly those using hybrid models—used machine learning methods. no such noteworthy research paper attempted to use machine learning tools to create a survival model for covid-19 patients. in this work, we developed a hybrid model and focused on forecasting the survival of coronavirus-infected individuals utilizing currently available machine learning devices. we have made an effort to more closely evaluate our suggested hybrid model with the available machine learning tools. in this study, we also attempt to quantify the effectiveness of different machine learning technologies in terms of classification, validation, and accuracy. lastly, we try to demonstrate that the employed data set is well described by our suggested hybrid model, demonstrating strong performance in terms of accuracy, precision, sensitivity, specificity, f1 score, and auc. 2. materials and methods 2.1 dataset collection and processing we used metadata from "https://github.com/ieee8023/covid-chestxray-dataset" in this paper. there are 312 patients in our dataset, and each patient has additional data (referred to as a variable). the following unique characteristics are included in our dataset: patient id, finding/type of pneumonia, age (years of the patient), sex (male or female), survival status (yes/no), offset (number of days), went_icu (yes (y) if the patient was in the intensive care unit (icu) or critical care unit (ccu) at any point during this sickness or no (n) or blank if unknown), iintubated (yes (y) if the patient was intubated (or ventilated) at any time during this illness, etc. there are 109 female patients and 203 male patients among the 312 patient records in this collection. this is a list of all the metadata fields along with covid-19related explanations. we forecast the survival status of patients infected with corona based on the characteristics. version 3.6.3 of the r program is used for data processing and analysis. 2.2 proposed hybrid model the idea behind hybrid or ensemble techniques is that a combination of several single models can often produce effective discriminatory rules. we are going to build ensembles of machine learning algorithms. we integrate predictions from four caret models via stacking. this may indicate that the models are proficient in multiple domains, enabling a novel word template for assa manuscript 151 copyright ©2023 assa. adv. in systems science and appl. (2024) classifier to determine how to extract the best performance and accuracy from each model. in this instance, we will utilize the random forest method to combine the predictions, which is why the hybrid model shows the highest accuracy when compared to all or any machine learning software alone. 2.3 experimental setup figure 2.1 outlines the several stages that our investigation covered. we pre-processed the dataset for our predictive model in the first stage. the dataset is then divided into training and testing datasets as the next phase. three well-known classification methods as well as a hybrid model were used in our investigation. as a result, the most accurate prediction model gets approved to be used as a future model. fig.2.1. flowchart for proposed hybrid model 2.4. machine learning technique 2.4.1 support vector machines (svm) support-vector machines (svms, support vector networks) are supervised learning models with corresponding learning algorithmic programs used for regression analysis and compartmentalization through simulated data in machine learning techniques. in addition to performing linear variety, support vector machines (svms) can also do nonlinear classification by implicitly transforming their inputs into high-dimensional feature spaces, a technique known as the kernel trick. the support vector machine category was created by vladimir vapnik and alexey chervonenkis in an effort to transfer a linearly separable hyperplane and split the dataset into two groups. assuming perfect data separation, let's proceed. after that, we can maximize the subsequent: minimize 2, subject to: 152 i. petrov, j.f. doe copyright ©2023 assa adv. in systems science and appl. (2023) the last two constraints can be compacted to: a quadratic algorithm called linear svm works effectively with datasets that are easily divided into two sections by a hyper-plane. however, datasets can occasionally be complex and challenging to categorize with a linear kernel. 2.4.2 random forest model even without hyper-parameter adjustment, the random forest method effectively employs a machine-learning algorithm that produces excellent results most of the time. because of its ease of use and versatility, it's also one of the most used algorithms (used for classification and regression jobs). the random forest has a great advantage when used to classification and regression tasks, which make up the majority of machine learning systems in use today. the hyperparameters of random forests are identical to those of decision trees and bagging classifiers. we may also use the regressor algorithm to carry out regression operations for random forest. the irregular forest classifier is made up of a combination of tree classifiers. each tree casts a unit vote to identify the most prevalent input vector. each classifier is constructed using an arbitrary vector that may be freely examined from the input vector. 2.4.3 logistic regression there are significant similarities between logistic regression and linear regression. nevertheless, their intended application remains the primary distinction. while logistic regression is utilized for classification tasks, linear regression methods are employed for value prediction. let be a binary outcome from bernoulli with . the logistic regression model can be defined as: 2.5 performance evaluation we evaluate the machine learning methods for classification based on the following criteria. technique names formula accuracy = precision = sensitivity/recall = specificity = f1-score = 3. results and discussion to test the three machine learning classification algorithms and one hybrid model to predict the survival status of patients infected with coronavirus, we conducted several analyses. the prerequisites for machine learning algorithms' specialized performance evaluation are shown in table 3.1. table 3.1. performance measurements for classification technique algorithm accuracy sensitivity specificity precision f1 score auc rf 0.9462 0.6923 0.9875 0.9875 0.8432 0.9514 word template for assa manuscript 153 copyright ©2023 assa. adv. in systems science and appl. (2024) svm 0.9355 0.7692 0.9625 0.9625 0.8449 0.8683 lr 0.7634 0.1538 0.8625 0.8625 0.2611 0.8587 hybrid 0.9785 0.8462 1.0000 1.0000 0.9167 0.9793 the results of the performance evaluations for the three machine learning methods used to predict survival are shown in table 3.1. it also explains how a suggested hybrid model was developed. out of the three machine learning methods, random forest performs the best with an accuracy of 94.62%, while logistic regression has the lowest accuracy at 76.34%. we evaluate the performance of three machine learning algorithms using the current standards, and the random forest model consistently outperforms the other two. however, compared to random forest, our hybrid model produced superior accuracy (97.85%). out of all the machine learning techniques, our suggested hybrid model produced better results when compared to the other evaluation criteria. we also use the roc curve to illustrate the performance of the various ml methods. the discriminant power in four distinct algorithms is shown in graph 2 for various colors. the color black represents our suggested hybrid model. our suggested model has a strong ability to discriminate. the suggested model has an auc value of.9793, which indicates that the predictions of the proposed hybrid model are 97.93% accurate, according to table 3.1's auc value. if we take into account logistic regression as well, the auc=.8587 indicates that logistic regression accurately predicts 85.87% of patients with corona infection who do not survive. fig. 2. roc curve for different machine learning algorithm compare to hybrid model 4. conclusion after examining the trial data, we came to the conclusion that our suggested hybrid model achieves the best accuracy, 97.85%, and auc, 97.93%. as a result, our suggested approach can identify corona-infected patients who do not survive more precisely. in terms of classification accuracy or area under the roc curve, our hybrid model performed the best. the suggested hybrid model's auc score is close to 1, indicating a more accurate prediction. our suggested hybrid model appears to perform better across the board for all performance assessment factors, according to the experimental results. we observe that the suggested 154 i. petrov, j.f. doe copyright ©2023 assa adv. in systems science and appl. (2023) model with the lowest error has the highest accuracy. as the literature study has shown, we conclude that the criteria of a predictive model for corona-infected patient survival prediction has only been partially met. in conclusion, we can state that, when compared to the three machine learning approaches, our suggested hybrid model is the most predictive model for all machine learning methods and has a lower classification error. acknowledgement we thank all the co-authors who contributed to preparing this manuscript – especially the author md. asadullah has does analysis and contributed to interpretation. secondly md. murad hossain prepared an introduction and contributed to overall manuscript writing. remaining co-authors help us reviewing all write-ups. references [1] asadullah, m., hossain, m. m., rahaman, s., amin, m. s., sumy, m. s. a., et al. 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(2020). a machine learning-based model for survival prediction in patients with severe covid-19 infection, medrxiv: 2020.02.27.20028027, [online]. available: https://www.medrxiv.org/content/10.1101/2020.02.27.20028027v3. [25] yan, l., zhang, h.-t., xiao, y., wang, m., guo, y., et al. (2020). prediction of criticality in patients with severe covid-19 infection using three clinical features: a machine learning-based prognostic model with clinical data in wuhan, medrxiv: 2020.02.27.20028027, [online]. available: https://www.medrxiv.org/content/ 10.1101/2020.02.27.20028027v2. microsoft word 1001-article text-5834-1-11-20221223 adv syst sci appl 2023; 01; 99-114 published online at https://ijassa.ipu.ru. numerical investigation of fluid-structure, thermal coupling for a heated slab oudrane a.1*, aour b.2, hamouda m.1, chesneau x.3, zeghmati b.3, balti j.4 1) laboratory of sustainable development and informatics (lddi), faculty of science and technology, ahmed draya university of adrar, algeria 2) laboratory of applied biomechanics and biomaterials (labab), bp 1523 el mnaouer, national polytechnic school of oran-maurice audin (enpo-ma), 31000, oran, algeria 3) laboratory of mathematics and physics groups of energy mechanics (lamps), university of perpignan via domitia, 52, avenue paul alduy, 66860 perpignan cedex france 4) department of physics, faculty of science, university of carthage, 7021 jarzouna, tunisia abstract: this work focuses on a numerical analysis of fluid-structure thermal coupling in a heating slab. the latter consists of a rectangular cross-section duct located in the concrete slab of a discredited habitat. we modeled the thermal transfers of fluid flow in the pipe. in fact, the navier-stokes equations that govern this flow have been solved numerically. these equations were by an implicit method of finite differences. the systems of algebraic equations thus obtained were solved by the algorithms of gauss and thomas. the equation of conduction in the concrete slab was solved using the same methodology as that of flow. in this work, we based on an algorithm that interacts non stationary solid medium with a fluid medium consisting of permanent a state by ensuring equal flows and temperatures on the common interface between the two mediums at every moment. the numerical simulation of heat transfers and the thermal behavior of the heating slab were analyzed for various parameters influencing thermal diffusion. the results obtained show that the numerical methodology adopted for the control of fluid-structure coupling is acceptable in comparison with the literature. keywords: heated slab; thermal coupling; fluid flow; heat transfer; not stationary processes 1. introduction one of the renewable energy sources is solar energy, which is the most used within cities in the mediterranean. passive solar is a process of generating thermal energy by converting solar radiation into heat. solar energy is the most developed compared to other renewable energies [1–3]. however, the behavior of conversion systems for this energy type is highly dependent on variations in climatic parameters such as temperature, solar irradiation and storage means [4]. in the context of thermal coupling, giles and al. [5] have studied the numerical stability and procedures of fluid-structure thermal coupling. the aim of this study is to analysis the thermal diffusion phenomenon with a continuity of temperature and a heat quantity at the interface. birken and al. [6] highlighted the importance of fluid-structure, thermal coupling in industrial cooling processes for the steel heat treatment [6]. this numerical study is devoted to the thermal interaction study between fluid and structure in the heating slab, also called conjugated heat transfer. monge and birken [7] have considered two areas with jumps in the material conductivity coefficient through the connection interface. heuzé and al. [8] * corresponding author: abdellatif.mebarek@univ-adrar.edu.dz 100 oudrane a. et al. copyright ©2023 assa adv. in systems science and appl. (2023) have developed a digital tool to simulate the thermomechanical coupling procedure based on a fluid-structure coupling to describe the state of the structure matter throughout. currently, numerical simulation of fluid-structure interaction problems is one of the biggest challenges for modern scientific computing. typical examples are found in aeronautics, where air flow around an elastic aircraft or air-sheet oscillations in air flow are well presented in the work of dowell and al. [9]. in addition, this interaction is important in the in-turbomachinery field, where the energy transfer will take place between a rotor and air [10]. furthermore, in the field of biomechanics, the elastic behavior of micro-pump or artificial membranes in blood flow is affected by this type of interaction by referring to the work of scotti and al. [11] and then tezduyar and al. [12]. underfloor heating is a technique that provides good comfort while minimizing energy consumption. in this context, the aim of this study is to characterize the variable heat exchange between a laminar flow of fluid in forced convection and a concrete slab of considerable thickness, the top of which is subjected to a constant ambient temperature of 28°c. the modelling is based on the thermal balance calculation at the level for system elements: fluid-structure. model validation was performed using the results obtained by andreo and al [13]. the latter used the same heating system with a heat supply provided by solar energy. the second step in this work is to test the parameters influencing the heat transfer within the heating slab, analysing the system thermal behavior. as part of this work, we propose to exploit this energy potential in the habitat through the use of a closed-circuit heating floor. studies on floor heating technology are numerous in the north african region. we are interested in the case of a solar heating system for a singlezone space in a dry climate similar to that studied by mokhtari and al [14]. the system is equipped with a concrete slab to store and produce heat from the floor within a habitable envelope. the principle of operation is to circulate directly into the concrete slab a fluid heated by solar collectors [15]. in fact, this slab will have the diffuser role of a soft and homogeneous heat throughout the house. 2. physical description of the slab studied the element of the numerical modelling is a hydronic heating floor which consists of three layers with a coil tube (figure 1). the insulation material is a plate of expanded polystyrene, which is the most used in this kind of habitable constructions. the insulation layer height is 5 cm. above it, we have a concrete screed 10 cm high, 1m long and 1m wide. in the latter are arranged the cross-linked polyethylene tubes [16,17]. they are very often used for the heated floors realization. in fact, these semi-rigid pipes are flexible, and they do not need welding to be carried out like those of copper. the tubes are arranged in (u) shape with a diameter (d) of 20 mm and a 10 cm for spacing [17]. a layer of concrete coating is superimposed on the heating grid. fig.1. physical description of the slab studied numerical investigation of fluid-structure… 101 copyright ©2023 assa. adv. in systems science and appl. (2023) 3. mathematical formulation 3.1. simplifying hypotheses a set of assumptions is retained in this study to simplify the mathematical modelling for thermal transfer model. these assumptions are derived from the physical properties of fluid flow in a horizontal pipe embedded in a concrete slab. the main assumptions taken into account in this study are as follows:  the fluid is newtonian, assumed to be viscous and incompressible;  the flow is transient with a laminar regime;  viscous dissipation is negligible: the diffusion of purely mechanical energy is neglected because the water speed and viscosity are low;  flow has only two velocity components: one longitudinal speed and the other transverse;  no internal heat sources 0s  ;  the physical properties (µ, cp, ρ, λ) are constant;  low pipe thickness is neglected in numerical calculations;  the fluid-structure interface is in thermodynamic equilibrium;  the solid medium to be isotropic. 3.2. fluid flow modelling in the pipe we initially opted for a flow study in a rectangular cross-section duct. this involves the flow of viscous fluid between two long plates (l), parallel and separated by a small distance (d). both plates are fixed, and the fluid is moved by a pressure gradient (figure 2). the solution governing a flow of poiseuille for maximum speed (u0) in the pipe medium is as follows [18]:              2 0 41)( d y uyu f (1) fig.2. physical description of fluid flow within the pipe it should be noted that a rigorous treatment of the boundary layer would require the complete solution to navier-stokes equations. their complexity prompted prandtl to simplify them to retain only the most important terms. the main idea is to neglect the axial gradients (∂/∂x) in front of the transverse gradients (∂/∂y). thus we obtain the prandtl equations for boundary layer which govern a laminar flow in the heating slab pipe as follows [19,20,21]:  equation the amount of movement 0 1 2 2                  y p and y u v x p y u v x u u t u ff f f f f f f f f  (2)  equation of mass conservation 0      y v x u ff (3) 102 oudrane a. et al. copyright ©2023 assa adv. in systems science and appl. (2023)  energy conservation equation 2 2 y t cy t v x t u t t f fpf f f f f f                (4) 3.3. mathematical model of thermal diffusion the floor slab is regarded as a homogeneous solid to which the classical equation of heat diffusion is applied [22]. heat diffusion is defined as a heat transmission mode in a solid caused by a temperature difference between two regions of this solid medium [23,24]. the numerical modelling is based on the two-dimensional study of heat conduction within the heating slab. the equation of thermal conduction is:                 2 2 2 2 y t x t ct t bb bb bb p  (5) this equation is discredited by the finite difference method using a three-point forward scheme in the contact interface. after the discretization, we get a tridiagonal algebraic system, the resolution of which is done by the thomas algorithm. 3.4. initial and boundary conditions  at the moment t = 0, the field of calculation under consideration is initialized by: 0)0(;0)0(;0)0(  fff tvu (6) 0)0(  b t (7)  the fluid velocity at the channel inlet is given by x = 0 and 0 ≤ y ≤ d;              2 0 41)( d y uyu f (8)  at the canal outlet we have x = l and 0≤ y ≤ d; 00;0          x t and x v x u fff (9)  the coupling interface (fluid-structure) is governed by the following expression 0<x< l and y=d; dy y t dy y t f f f b b b b f         (10) where fb and  are respectively the thermal conductivities of the solid and the fluid.  at the ambient air-slab interface we have 0< x < l and y = d;  airbc dbyb b b tth y t       (11) where hc convective coefficient of exchange with ambient air in (w.m-2.k-1), b  is the concrete thermal conductivity in (w.m-1.k-1).  at the left edge of the slab (adiabatic walls) x = 0 and d ≤ y ≤ d+ep; 0   dby b y t (12)  at the right border of the slab (adiabatic walls) x = l and d ≤ y ≤ d+ep; 0   dby b y t (13)  at the bottom surface of the tube (adiabatic walls) y = 0 and 0 ≤ x ≤ l; numerical investigation of fluid-structure… 103 copyright ©2023 assa. adv. in systems science and appl. (2023) 0   lfx f y t (14) 4. coupling methodology steady-state fluid-structure coupling has been treated many times. it involves coupling a navier-stokes solver to a conduction solver and searching for a stationary state in fluid and structure medium. it is to highlight that that this type of problem has already been studied digitally by errera and al. [25]. while using the same tools, the work we present here is of a different nature. indeed, we are interested in the non-stationary coupling and more precisely in the detailed description of the thermal exchanges in a concrete slab subjected to convective flow due to laminar flow. 4.1. fluid-structure interface figure 3 shows the interface between two mediums with different thermal conductivities. the distance between the node (j) and the neighbouring nodes (j-1 and k+1) is calculated according to the space pitch [26,27,28]. fig. 3. spatial discretization at the fluid-structure interface. a finite difference formulation based on the galerkin method was used to solve the equations that govern the fluid-structure interaction model in the heating slab [29]. in perfect contact, let us consider the case where the parietal transfer is entirely convective-conductive in nature. regardless of the mode of convection, the transfer of energy between the surface of a solid body at temperature (tb) and the fluid is done by thermal conduction since the fluid velocity is zero at the solid body surface. the density continuity the energy flow at the surface thus allows writing [30,31,32]: 00 )2.(         fyf f f byb b bmw y t y t  (15) the discretization of the energy flow, density terms of the lower surface of the solid using an implicit method with finite differences to be a three-point forward scheme is given by the following expression: b bbb b byb b b y kitkitkit y t        .2 )2,()1,(.4),(.3 0  (16) on the other hand, the density terms discretization for energy flow at in upper surface of the fluid using of an implicit method with finite differences with a three-point back pattern is given by the following expression: f fff f fyf f f y jitjitjit y t        .2 )2,()1,(4),(.3 0  (17) 104 oudrane a. et al. copyright ©2023 assa adv. in systems science and appl. (2023) we pose: f b a f a a a y y yand y c b          , (18) or ay is the relationship between the fluid spatial step and the solid spatial step, a is the ratio between the solid thermal conductivity and the fluid thermal conductivity, ft is the fluid temperature and bt is the solid temperature. after an equality and simplification between the two equations (16) and (17), we find the expression, of the temperature of fluid-structure coupling for the interface nodes which is given as follows: )1.(3 ))1,(.4)2,(())2,()1,(.4.( )1,(),( c jitjitkitkitc itnjit ffbb bf    (19) 5. dimensional variables of local equations to put back the equations governing the flow of the simpler fluid. a set of reduced variables (table 1) were introduced to produce an exploitable non-dimensional form. table.1. non-dimensional variables of local equations [33]. sizes time length velocity pressure temperature ladder t l u p t associated non-dimensional sizes t u t d  0 dd y yet x x  0 u u u  2 0 0 . ).(2 u pp p     0 t t t  d the pipe reference diameter m u0 the maximum reference velocity at the pipe entrance m.s-1 t0 = d/u0 non-dimensional reference time s t0 reference ambient temperature °c ρ.u0 2 the reference pressure pa 5.1. non-dimensional equations in order to generalize the results, the equations governing for physical phenomenon are written in dimensionless forms based on table 1. non-dimensional variables are pipe diameter and input sizes (velocity and temperature). in the channel, for a two-dimensional case in cartesian coordinates, the equations system of heat and mass transfers by forced convection are formulated in the following dimensionless form [20]:  quantity equation of the fluid movement; 2 2 * . re 1 ** * * * *           y u x p y u v x u u t u fff f f f f (20)  the conservation equation of fluid mass; 0 **     y v x u ff (21)  the conservation equation of energy in the fluid; 2 2 * . re.pr 1 * * * * *         y t y t v x t u t t ff f f f f (22)  equation of energy conservation in the solid domain (the slab); numerical investigation of fluid-structure… 105 copyright ©2023 assa. adv. in systems science and appl. (2023)                     2 2 2 2 * * * * re 1 * * y t x t pct t bbb fbb b   (23)                                                                                 2* 2 2* 2 * 2* 2 2* 2 0 * 2* 2 2* 2 0 * ** . re 1 . pr 1 . . * ** . . . 1 . . . . * ** .. . 1 . . . . * y t x t cpt t y t x t udcpcpt t y t x t udcpcpt t bb a a b bb f f ff f a a b bb f f ff f a a b              (24) or (ρ.cp)a is the ratio between the fluid heat capacity and that of the solid and df : height of the rectangular pipe. 5.2. initial conditions and non-dimensional limits in order to write the conditions at the boundary of two mediums in non-dimensional forms, we enter the variables in table 1 in the equations (6 to 14) and we obtain the following expressions: 5.2.1. non-dimensional initial conditions  at the moment (t*), the domain of computation considered is initialized by; 0 * 0 * 0 * )0(;)0(;)0( t t t u v v u ff f f f u u  (26) 0 * )0( )0( t t t b b  (27) 5.2.2. non-dimensional boundary conditions  the fluid velocity at the channel input is given by x* = 0 and 0 ≤ y* ≤ 1;               2* * 0 * 1 41)( y uy f u (28)  at the channel output, we have x * = (l / d) and 0 ≤ y* ≤ 1; 00 * * * *       x v and x u ff (29)  at the fluid-structure coupling interface is governed by 0 <x*<(l / d) and y* = 0; 11 * * * * * *        f f f b b b a y y t yy t  (30)  at the top surface of the slab [27], we have 0 <x* < (l/d) and y* = 1 + (ep/d);  ** )/(1 * 1 * * airb depy b a tt y t b b       (31) with: (h* c=hc/h0 ) is the convective exchange coefficient with the ambient air. 106 oudrane a. et al. copyright ©2023 assa adv. in systems science and appl. (2023)  at the slab left border (adiabatic walls) x* = 0 and 1 ≤ y ≤ 1+(ep/d); 0 * * *    dby b y t (32)  at the slab right boundary (adiabatic walls) x* = l and 1≤ y ≤ 1+ (ep/d); 0 * * *    dby b y t (33)  at the tube bottom surface (adiabatic walls) y* = 0 and 0 ≤ x* ≤ l/d; 0 * * *    lfx f x t (34) 6. flow chart of fluid-structure thermal coupling the flowchart in figure 4 shows the procedure for calculating the temperature at the slab top surface by following the following steps:  requisitions of the physico-thermal properties for fluid-structure system;  initialization of the system at room temperature (28 °c);  resolution of navier-stokes equations in the field of fluid;  solving the of the conduction equation in the slab;  coupling temperature test;  test on the calculation overall time;  recording the results obtained. 7. data from the numerical simulation table 2 shows the physico-thermal properties of air, concrete and water, which are used in numerical simulation. table.2. physico-thermal properties of the studied domain [14,13,34,35]. physico-thermal properties air water(ref) concrete cp specific heat in (j.kg-1.k-1) 1006 4182 880  density in (kg.m-3) 1,177 1000 2000  dynamic viscosity in (kg.m-1.s-1) 1,85.10-5 10-3  kinematic viscosity in (m2.s-1) 1,75.10-5 10-6  thermal conductivity in (w.m-1.k-1) 0,0262 0,597 1,75 pr number of prandtl 0,708 7,01 the constants used for numerical resolution of the problem studied are [13];  heat exchange coefficient: hc=8,12 w.m-2 .k-1;  ambient air temperature: tair=28°c ;  fluid inlet temperature: te = 60 ° c;  reynolds number: re = 500;  step time of time: δt = 10s. numerical investigation of fluid-structure… 107 copyright ©2023 assa. adv. in systems science and appl. (2023) fig.4. flowchart of the numerical modeling of fluid-structure thermal coupling. the data of the calculation grid in the two fluid-structure subdomains are presented in table 3: table. 3. calculation grid data in the fluid-structure domain. length (m) height (mm) ni j nk ∆x ∆y time step ∆t (s) fluid 1 20 101 41 10-2 5x10-4 10 structure 10 101 81 1,25x10-3 ni node indices on the x-axis of the two calculation domains. nj node indices on the y-axis of the fluid domain. nk node indices on the y-axis of the solid domain. ∆x step space on the x-axis of the two calculation domains. ∆y step space on the y-axis of the two calculation domains. ∆t calculation increment time. 8. results and discussions 8.1. validation of the numerical model figure 5 shows the temperature evolution of a point on the slab upper surface as a function of time. it is noted that there is an acceptable agreement between the results obtained by the developed model and those of andreo and al [13], so mokhtari and al [14] with relative errors equal to 5,04% and 1,77% respectively. 108 oudrane a. et al. copyright ©2023 assa adv. in systems science and appl. (2023) fig.5. evolution of the temperature profile at a point in the slab medium as a function of the overall time. 8.2. evolution of the concrete slab temperature according to the spacing step figure 6 shows the variation in the heating slab temperature as a function of the space pitch ratio for four times. this variation has been highlighted for three different positions in the system (at the inlet, the medium and at the outlet). we can notice that:  the evolution of thermal diffusion in the concrete slab is almost similar for the three positions considered (inlet, medium and outlet);  there is a remarkable temperature difference between the positions in the system for the four selected global calculation times;  for a ratio of space varying between 0 and 0.12 with a fluid inlet temperature in the pipe at 60°c, the heat propagation in the solid decreases almost to temperature values (in the medium) of about 38°c for t = 3600s (fig. 6.a), 47°c for t = 10800s (fig. 6.b), 48°c for t = 14400s (fig. 6.c) and 49°c for t = 43200s (fig. 6.d). numerical investigation of fluid-structure… 109 copyright ©2023 assa. adv. in systems science and appl. (2023) fig.6. comparison between the three profiles (input, medium and output) of the temperature as a function of the slab space pitch ratio for (a) t = 3600s, (b) t = 10800s, (c) t = 14400s and (d) t = 43200s 8.3. evolution of the system temperature for different thicknesses of the slab figure 7 illustrates the change in temperature in the medium slab as a function of time for different thicknesses of the concrete top layer. it is clear that the heat propagation in the slab decreases considerably with the increase in the thickness of the upper layer the concrete. that is to say, when the concrete thickness is decreased (for example 10 cm), the evolution of the temperature increases rapidly as a function of time and vice versa. in other words, the heat diffusion in the room will be even lower than the concrete mass is important. fig.7. evolution of the temperature profile in the slab as a function of time for different thicknesses of the concrete 8.4. effect of variation of the convective exchange coefficient with the ambient air figure 8 shows the evolution of heating slab temperature as a function of time for different convective coefficients with ambient air. it can be observed that when the value of the convective exchange coefficient increases, the thermal diffusion increases slightly and vice versa. it should also be noted that the variation for convective exchange coefficient with ambient air has a notable influence on the thermal diffusion of heat within the heating slab. in addition, we will note that the evolution of the temperature will start from 28°c, the atmosphere temperature , and it will increase over time to the highest value around 60°c, which is the temperature at which the fluid inlet the slab pipe. 110 oudrane a. et al. copyright ©2023 assa adv. in systems science and appl. (2023) fig.8. evolution of the heating slab temperature as a function of time for different convective coefficients with ambient air 8.5. evolution of the velocity profile of the fluid in the pipeline the evolution of fluid velocity within the pipe as a function of the space step for different reynolds numbers is illustrated in figure 9. in this figure, it can be seen that the shape of the fluid velocity profile in the pipe is parabolic. in addition, it keeps the same shape, but its maximum value increases with the increase in the number of reynolds or it reaches a maximum value equal to 0,0175m/s for re = 800. this evolution shows that the variation influence of the reynolds number on the fluid flow is important. fig. 9. profiles of fluid velocities within the pipe versus space step for different reynolds numbers 9. conclusion the objective of this digital study is the adaptation of a numerical methodology in order to be able to control and characterize the thermal exchanges within a heating slab intended for heating by the ground. in fact, we analyzed the temperature evolution at the inlet, medium and outlet of the system according to the space step. in addition, we analyzed the evolution of the slab temperature for different thicknesses of the concrete and also the effect of variation of the convective exchange coefficient with the ambient air on the surface temperature of the slab. on the other hand, we have highlighted the effect of the reynolds number variation on the evolution of the fluid velocity profile in the pipe. the validity of this numerical approach is confirmed based on the work done by mokhtari and andreo. the methodology we have presented makes it possible to analyses the thermal numerical investigation of fluid-structure… 111 copyright ©2023 assa. adv. in systems science and appl. (2023) transient phase over very short periods of time using the current numerical model of fluidstructure thermal coupling. nomenclature cp heat capacity. 11 ..  kmw d pipe diameter. m ep concrete thickness. m hc convective coefficient with ambient air. 12 ..  kmw 0 h convective coefficient reference. 12 ..  kmw i index of the abscissa axis incrementation. l pipe length. m j index of the ordinate axis incrementation in the fluid domain. k increment index of the y-axis in the solid domain. pr number of prandtl. f p fluid pressure. pas s  the internal heat quantity. 2. mw re number of reynolds. air t air temperature. k e t inlet temperature of the fluid in the pipe. k milieu t temperature in the slab middle. k entrée t temperature at the pipe entrance. k sortie t temperature at the pipe outlet. k *t dimensional temperature. f t fluid temperature. k b t concrete temperature. k t time. s 0 u initial longitudinal velocity of the fluid. 1. sm *u dimensional longitudinal velocity of the fluid. f v initial transverse velocity of the fluid. 1. sm *v initial transverse velocity of the fluid. y directional normal of the fluid diffusivity. greek symbols  thermal conductivity. 11 ..  kmw a report between of the fluid thermal conductivity and the solid.  dynamic viscosity of the fluid. 11..  smkg  volumic mass. 3. mkg references [1] mehdary a., naamane a., m'sirdi k. 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(2018), ‘’contribution à la modélisation et au développement des systèmes de chauffage solaire à usage individuel’’, p.h.d thesis d’ecole nationale polytechnique d’oran maurice audin, algérie. adv syst sci appl 2020; 04; 105-112 published online at https://ijassa.ipu.ru. automation of cluster large-scale production systems valentin chadeev1, natalya aristova1* 1) v.a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia e-mail: avtprom@ipu.ru abstract: the paper covers the рroduction of capital goods of various types of industrial robots in the context of industrial clusters. it uses a model of production, invariant with the types of technological operations (tos). the paper examines algorithms for three automation options: full рroduction of capital goods, partial рroduction of capital goods, use of robots purchased outside the cluster. keywords: the рroduction of capital goods, industrial robots, level of automation, technological operation, operation performance time, production theory invariant with the types of technological operations, cost dynamics 1. introduction today, active research is being carried out in russia to build cluster models in different areas of the economy, in different geographical regions, as well as for the purpose of determining the structure, components and growth potential. the main purpose of the cluster policy is to ensure high rates of economic growth and economic diversification by improving the competitiveness of enterprises, suppliers of equipment, components, specialized production and after-sales services of research and educational organizations that form territorial production clusters [1]. an economic cluster is a geographically concentrated group of interconnected companies, specialised suppliers, service providers, firms in their respective industries and related organisations (e.g. universities, standardisation agencies and trade unions) in certain areas that compete but also work together" [2]. according to m. porter, the competitiveness of a region or a country should be considered in terms of the competitiveness of clusters associations of organisations and enterprises in various industries that are able to use internal resources effectively. a cluster takes into account the market mechanism as much as possible and is a flexible form of organisation [3]. this is its main difference from the territorial and industrial complex and the sectoral approach. as an example of a modern cluster established in the russian federation, we should mention the petrochemical enterprises cluster of the republic of tatarstan [4]. to describe any production system within a cluster, it is necessary to have: a list of all manufactured products (including robots); a description of the production procedure for each type of product; a description of the production procedure for each type of robot; specifications of robots as production elements; a list and specifications of all types of technological operations (tos) required to produce all types of products and robots; the total number of people involved in the production process. the paper's objective: using the above data, to calculate a production automation scheme within a cluster where human costs are minimal. * corresponding author: avtprom@ipu.ru 106 v. chadeev, n. aristova copyright ©2020 assa. adv. in systems science and appl. (2020) 2. parameters of the products and robots of the cluster system parameters of the products to be manufactured within the system are determined by the following matrices [5]: ijnn = , of size ms, knn = , of size 1s, kdd = , of size 1s, where nij is the number of i-type tos that is needed to manufacture one k-type product, s is the number of types of products, nk is the number of k-type products to be manufactured, dk is the cost of purchased products needed to manufacture one k-type product. parameters of the robots are determined by the following matrices [5]: ijbb = of size mf, where bji is the time during which i-type tos are performed by a j-type robot, f is the total number of types of robots used in production, m is the number of types of tos needed to manufacture a product, ijhh = of size mf, where hij is the number of i-type tos needed to manufacture one j-type robot. the service life of robots t, the cost of robots c and the cost of the working time unit of robots  are input by the following matrices, where j-index stands for the type of robot: jtt = , of size 1f, jcc = , j = . production capacities of people are contained in the matrix [5]: igg = , of size 1m, where gi is the time needed for a person to perform an i-type to. to shorten the formulae in some places we will regard a person as a 0-type robot — the element ibb 00 = , of size 1m, of the matrix ijbb = . a person's working time t0(t) will be an aggregate of the person's working time while working at the station t (for a year, for instance). 3. automation matrix the planning of automation of autonomous production is aimed to distribute the operations between people and robots, so that people could save on costs as much as possible. this requires the use of an automation matrix as a management tool: ji= of size mf, where αij is the level of automation of i-type tos performed by a j-type robot. the level of automation must meet the following conditions: 10  ji ; mi ,...,2,1= ; mj ,...,1,0= . if all i-type technological operations are performed by an automaton then the level of automation αi = 1, and if by a person αi = 0. if a particular type of tos is to be performed while manufacturing a product, it must be done by either a person or one of the types of robots. if this condition is not met, the assembly of the product will not be completed and, as a result, the product will not be manufactured, but will be rejected. product manufacture cost с(а) is a function of the automation matrix. the optimal automation matrix a that minimizes the production cost contains only 0 and 1. 4. robot cost calculation cost of robot automated manufacture by a system of robots with the use of automation matrix a is calculated according to the formula [5]: 0 0 0 1 1 1 1 1 1 ( ) m m m k i i ki i i ki fi fi f ki i i i c b h b h b h       = = = = + + +   , (4.1) automation of cluster large-scale production systems 107 copyright ©2020 assa. adv. in systems science and appl. (2020) where m is the number of types of tos required to manufacture a product, bij is the time needed for a j-type automaton to perform i-type tos, j is the cost of the working time unit of a j-type automaton, gi =b0i is the time that a person needs to perform i-type tos, f is the total number of the types of robots available for automation, hij is the number of identical itype tos needed to manufacture a product ck(). the first amount stands for the human costs, the second amount stands for the first-type robot costs and the last amount stands for the f-type robot costs. 5. dynamic equations for the cost of a рroduction of capital goods initial data required for the calculations should contain information on the robots that will be used in the cluster, on the technology of their production and on the available means that will make their production automated. cost of manufacturing a robot in case of automated production consists of three components: ( ) ( )c b d g  = + + , (5.1) where g() is the direct cost of a person to assemble the robot, d is the cost of purchased parts, b() is the cost of robot time,  is the level of production automation, which is a vector whose components are the level of automation of each specific type of technological operations,  is the cost of the working time unit of the robot involved in the manufacture of the robot. let's define the cost of the robot's working time unit as the ratio of its cost c to the robot's lifetime (service time) t: /c t = . (5.2) given (5.3), let's rewrite (5.2) as follows: ( ) / ( )c сb t d g = + + . (5.3) equation (5.3) describes the process of manufacturing a robot by a person and a robot of the previous generation. the parameters of the manufacturing robot may differ from those of the manufactured one. to take this into account, let's rewrite (5.3) as follows: ( ) ( 1) ( ) / ( )c n с n b t d g = − + + , (5.4) where c(n) is the cost of the robot of the n-th generation, c(n-1) is the cost of the robot of the (n-1)-th generation. the solution of the difference equation (5) looks like (assuming that the automation vector  is constant): ( ) ( ) 1 / ( ) (0) / 1 / n nb tg d с n с b t t b t −+ =  + − , (5.5) automation of robot production with the help of robots is only feasible if the cost of the manufacturing robot b() is less than its service life t. in this case, the ratios included in the equation b()/t < 1, and the asymptotic solution will be equal to: ( ) g d с t b +  = − . (5.6) provided that b()/t<1, this solution does not depend on the initial conditions, but only on the automation vector . costs of the person g() and the robot b() in (5.1) are described by the formulae: 108 v. chadeev, n. aristova copyright ©2020 assa. adv. in systems science and appl. (2020) 1 ( ) (1 ) m i i i i g g h  = = − , 1 ( ) m i i i i b b h  = = . (5.7) where m is the number of types of tos required to manufacture a robot. this index in the sums will be omitted, assuming, however, that the summation is done for all types of operations, both automated and manual, gi is the time needed for a person to perform an itype to, bi is the time needed for a robot to perform an i-type to, hi is the number of identical i-type tos needed to manufacture a robot, i is the level of automation of the i-type to, and the vector , determines the production automation scheme. 6. cooperation of plants plant no. 1 manufactures robots of the n-th generation r(n), using structurally similar robots of the previous generation r(n-1) to make their production automated. plant no. 2 manufactures hydraulic drives m(0). let's consider cooperation of plants when the robot plant sells its products to the hydraulic drive plant to make its production automated, and the hydraulic drive plant reduces the price for its products purchased by the robot plant (fig. 6.1). fig. 6.1. scheme of cooperation between plants. on the scheme m(n) hydraulic drive of the n-th generation; r(n) robot of the n-th generation. in the figure, the upper line produces generations of hydraulic drives; the lower line produces robots. cost of automated manufacture of the robot is calculated according to the formulae (5.5) and (6.1). the whole system of cooperation of plants without automation of hydraulic drives production will be described by a system of equations: ( ) ( 1) ( ) / ( ) r fc n c n b t g d d = − + + + , (6.1) ( ) ( 0) ( 0) (0)f f ff n g b d f = = + = +  , (6.2) ( ) (1 )i i ig g h = − , ( ) i i ib b h = , (6.3) ( 0)f i ig g n = = , ( 0) 0fb  = = . (6.4) where c(n) is the cost of the robot of the n-th generation, gi is the time needed for a person to perform an i-type to, hi is the number of i-type tos needed to manufacture a robot, ni is the number of i-type tos needed to manufacture a hydraulic drive, bi is the time needed for a robot to perform an i-type to, dr is the cost of purchased parts needed to manufacture a robot, df is the cost of purchased parts needed to manufacture a hydraulic drive, f(0) is the cost of purchased hydraulic drives that have been manually assembled. m(0) m(1) m(n) . . . m(n+1) r(0) r(1) r(n) r(n+1) . . . automation of cluster large-scale production systems 109 copyright ©2020 assa. adv. in systems science and appl. (2020) 7. initial data for modeling let's consider how the cost of robots changes across generations. we will assume that manufacturing a robot requires purchased parts and the performance of four types of technological operations (tos). initial data. the robot's lifetime t = 1000. the time needed for a person to perform tos: g1=4, g2=3, g3=3, g4=23. the time needed for a robot to perform tos: b1=5, b2=12, b3=240, b4=4. the number of identical tos when manufacturing a hydraulic drive: n1=n2=n3=4, n4=6. the number of identical tos when manufacturing a robot: h1=12, h2=20, h3=10, h4=8. the cost of purchased parts for hydraulic drives for all tos df=10, the cost of purchased parts for robots for all tos dr = 80. the cost of the working time unit of each robot λ(0)=0.86. first, we will calculate the cost of the manual assembly of a hydraulic drive. using (6.2) and (6.4), we will get 4 1 (0) i i f i f g n d = = + . (7.1) using the initial data, we will get (0) 218f = . to calculate the cost of the manual assembly of a robot, using (6.1) and (6.3), we will get the formula 4 1 (0) (0)i i r i с g h d f = = + + . (7.2) using the initial data and (8.1), we will finally get: c(0)=860. the cost of the robot's working time will therefore be equal to λ(0)=0.86. 8. robot cost dynamics from generation to generation the change in the robot's cost from generation to generation at a fixed cost of purchased hydraulic drives is based on the following equation ( ) ( 1) ( ) ( ) (0)rc n n b g d f  = − + + + . (8.1) to solve this equation, it is first necessary to determine the optimal automation vector (1, 2, 3, 4) which would allow getting the minimum cost of the robot. for this purpose, we will make table 8.1 to determine the optimal automation vector for production of the 1st generation robots, using the 0-th generation robots. table 8.1. the automation vector for production of the 1st generation robots а b c d e f g h j i z(1) и (1) с(1)=z(1)*h+f(0)+ dr то g b λ(0)b z(1) (1) h f(0) dr 1 4 5 4,3 4 0 12 26 80 154 2 3 12 10,32 3 0 20 22 80 162 3 3 240 206,4 3 0 10 22 80 132 4 23 4 3,44 3,44 1 8 148 80 255,52 c(1) 703,52 λ(1) 255,52 first, we will calculate the cost of performing tos by a robot (column d). then, from columns d and b we will select the minimum cost of performing tos by a person or a robot and draw a vector z(1) of minimum values and the corresponding vector (1). the cost of any (e.g. the n-th) generation robots will be calculated according to (8.1), but each time λ(n1) and z(n) must be determined beforehand. after that, we will calculate the cost of the 110 v. chadeev, n. aristova copyright ©2020 assa. adv. in systems science and appl. (2020) manual assembly of a hydraulic drive (column h). the cost of a robot will be calculated in column i. making such calculations for the next generations, we will see robot cost dynamics at a fixed cost of purchased hydraulic drives. the results are given in table 8.2. table 8.2. the robot cost dynamics at a fixed cost of purchased hydraulic drives generation number n 0 1 2 3 4 c(n) 860 704 693 692 692 λ (n) 0,860 0,704 0,693 0,692 0,692 let's consider the automated production of hydraulic drives with the help of the 0-th generation robots. using (6.2) and (6.4), we will get: (1) (1 ) (0)i i i i i i ff g n b n d  = − + +  . (8.2) since the types of tos used to make the production of both robots and hydraulic drives automated are the same, so are the automation schemes. therefore, using the initial data, we will calculate the cost of the 1st generation hydraulic drive (table 8.3). table 8.3. the cost of the 1st generation hydraulic drive то z(1) (1) n df f(1)=z*n+df 1 4 0 4 10 26 2 3 0 4 10 22 3 3 0 4 10 22 4 3,44 1 6 10 30,64 f(1) 100,64 9. exchange strategy between plants plant no. 2 has decided to make the production of hydraulic drives cheaper by automating their production with the help of robots manufactured by plant no. 1. in general, the automation of hydraulic drive production can be either beneficial or not. it all depends on the capacities of robots and their cost. let's consider a case when automation has led to a decrease in the cost of hydraulic drives: f(0)>f(1). in the next generation, plant no. 1, using cheaper hydraulic drives, will reduce the cost of manufacturing robots: the cost of a robot with cheaper hydraulic drives will be reduced by the amount of reduction in the cost of hydraulic drives and will be equal to c(1)<c(0). accordingly, the cost of the working time unit of the next generation robot is (1) = c(1)/t, which is less than (0). we should keep in mind that there is no change in the product design when the production gets automated. plant no. 2 always produces the same hydraulic drive design, plant no. 1 always produces the same robot model. the hydrodrives of different generations m(0), m(1), m(2) do not differ from each other structurally they are twins. similarly, the models of robots of different generations r(0), r(1), r(2) transferred to the hydraulic drive plant do not structurally differ from each other they are also twins. symbiosis between the two plants leads to a successive reduction in the cost of both hydraulic drives and robots. the design does not change and the cost is reduced! we should note that it involves a change of generations. if plant no. 2 has produced a new (e.g. n-th) generation robot, and plant no. 1 is still using the (n-1)-th generation robot whose lifetime is not over yet, it makes no sense to change it. but when the (n-1)-th generation robot dies, it should be replaced with a cheaper robot of the n-th generation. thus, the evolution rate of a production automation system depends on the lifetime of the robot: the shorter the lifetime of the robot is, the faster the evolution can be. there is a gradual displacement of people by cheap robots from the production sector. that's what automation is all about. automation of cluster large-scale production systems 111 copyright ©2020 assa. adv. in systems science and appl. (2020) 10. cost dynamics let's examine the dynamics of hydraulic drives cost and the robot's working time unit cost in the process of production automation and a change of robot generations. the initial data for the calculations are given above. we have calculated the cost of hydraulic drives and the 0-th generation robots: c(0)=860; f(0)=218; λ(0)=0.860. difference equations (8.1) and (8.2) allow calculating future values of the robot cost on the basis of the parameters of the previous generation, i.e. the cost of the (n+1) generation robots (table 10.1). table 10.1. the cost of the 1st generation hydraulic drive то calculate z(n+1) and (n+1) calculate the cost of the robots c(n+1) g b λ(n) b z(n+1)  (1) h f(n) dr c(n+1) 1 4 5 4,3 4 0 12 26 80 154 2 3 12 10,32 3 0 20 22 80 162 3 3 240 206,4 3 0 10 22 80 132 4 23 4 3,44 3,44 1 8 148 80 255,52 c(n+1) 703,52 t 1000 λ(n+1) 0,70352 the modeling results are shown in fig. 10.1. fig. 10.1 shows a significant reduction in the cost of рroduction of capital goods. fig. 10.1. cost dynamics. the dashed line shows the change in the cost of a рroduction of capital goods without a component supplier's cooperation, the thick line — with a supplier's cooperation references 1. oskanyan, a.a. (2014). a cluster as an organizational form of ensuring innovation competitiveness of the region. peculiarities of state regulation of foreign trade activities under present-day conditions. proc. of the research-to-practice conference. rostov-ondon. the russian customs academy. rostov branch. 157-167. [in russian]. 2. porter, michael e. (1998). on competition. boston: harvard business school press. 3. lizunov, v.v. (2016). production clusters as a key element of regional social and economic system. specification of regional industrial policy using elements of cluster approach (on materials of omsk region). novosibirsk .164-202. [in russian]. 112 v. chadeev, n. aristova copyright ©2020 assa. adv. in systems science and appl. (2020) 4. lubnina, l.a., galimulina, f.f., bronskaya, v.v., aminova, g.a. (2020). on the development of petrochemical clusters. a case study of tatarstan republic. automation in industry. 1. 47-50. [in russian]. 5. chadeev, v.м., aristova, n.i. (2019). automation of autonomous largescale production systems. proc. of the 12 international conference "management of largescale system development" (mlsd2019). ieee. 1-4. 1. introduction 2. parameters of the products and robots of the cluster system 3. automation matrix 4. robot cost calculation 5. dynamic equations for the cost of a рroduction of capital goods 6. cooperation of plants 7. initial data for modeling 8. robot cost dynamics from generation to generation 9. exchange strategy between plants 10. cost dynamics adv syst sci appl 2019; 02; 90-100 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/701 utility dependency reduction by multi-agents participation in smart grid sonvir singh* department of computer science & engineering, visvesvaraya national institute of technology, nagpur, india e-mail: sonvirsingh036@gmail.com received february 17, 2019; revised may 31, 2019; published july 10, 2019 abstract: the electric vehicle has a battery to store electric energy in the form of chemical energy as the energy stored in invertor’s batteries. these sources are used as a self-supporting energy source at the time of outage of electricity due to a fault, equipment’s failure and other not working conditions if they have stored energy. in this paper, a multi-agent system is consisting of microgrid agent, home agent, control distribution agent, and supply source. the supply source agents include those agent who controls the storage; it also called as energy storage agents which are controlled by house agents. if the utility is not providing the demand energy to the consumers due to fault or some other reasons in this case stored energy in batteries and other resources like a solar panel, small wind plant of the microgrid and provide the energy within the microgrid and outside the microgrid. in this paper main focus is given on the priority index, cost optimization, and load optimization algorithms. by doing this it helps in providing the energy during the outage at the lower cost and dependency on the main grid may be reduced. it also boosts the principle of bidirectional flow of power which make consumer as a supplier also. this will reduce the electric bill charges and increases the reliability of supply electricity. keywords: electric vehicles, multi-agent systems, smart grids, centralized control, decentralized control, energy management. 1. introduction in today's world, the problem of air pollution and a shortage of fossil fuel has increased day by day. more attention is directed toward combustion type vehicles, which are the major source of air pollution and fuel consumption. the concentration on the hybrid vehicle and electric vehicle in the last decade is to reduce pollution and make the vehicle environment-friendly. the electric vehicle is of many types like a hybrid, fully electric which have the capability to connect with the grid to charge the batteries, known as a plug-in vehicle. plug-in vehicles can be a threat and opportunity for the power grid. while they put a huge load on the grid, this is a major threat but on the other side, responsive load and storage of electrical energy are an opportunity for the power grid. therefore, by proper control of them, it helps the power grid in various aspects like reliability improvement, frequency control, voltage control and reactive power control [1]. renewable energy recourses and ev leads to the emergence of the smart grid revolution in the past decade. self-healing is the ability to diagnose and repair the system after any fault. self-supporting sources make the grid to work in the island mode by helping each other in respect of energy and data sharing. hence, this plays an important role in a smart grid. this feature may help in that area where there is no backup plan to supply the loads [2,3]. when a fault arises in the main grid, the consumers are disconnected from the main grid. usually, the duration of repair time to diagnose and repair the fault is not more than one or two hours. in * corresponding author: sonvirsingh036@gmail.com mailto:sonvirsingh036@gmail.com 91 s. singh copyright ©2019 assa. adv. in systems science and appl. (2019) this condition, the microgrid work in island mode and they exchange their stored energy with each other. the stored energy may be taken from the batteries which are charged by the main power supply, solar panel, windmill, invertors, ev. main grid distributed network management system island mode home agent other agent (ev, in, wm, sp) microgrid-01 microgrid-02 microgrid-03 fig.1. proposed energy scheduling mechanism. mas is the framework that is good for the smart distribution grid system with sharing and exchanging the energy between the microgrid in order to reduce the operation cost and avoid power quality and transformer life issue [4]. nomenclature. mas multi-agent system. voll value of lost load. ev electric vehicle. esbde energy served by battery dg distributed generator. during emergency. mg microgrid. soc state of charge. dnms distributed network ens energy not served. management system. dod death of discharge. in inverter. r repair time wm windmill. data sharing data sharing data sharing utility dependency reduction by multi-agents participation in smart grid 92 copyright ©2019 assa. adv. in systems science and appl. (2019) in a fuzzy based mas in the smart distribution system with dg designed with the three types of agents: feeder agent, regulator agent, load agent [5]. this proposed system can eliminate the feeder congestion, control the reactive and active power, maintain the voltage level and reduce the burden of the transformer. there are other proposed methods also for self-healing property to remove the fault [6,7,8]. in this paper, a common mas is proposed which include agents: house agent, ev agent, sp, and wm agent, in agent, microgrid agent and main grid agents. the function of each agent is discussed at the normal and emergency condition. furthermore, the main focus is given on the self-healing and self-supporting system. the proposed energy scheduling mechanism is mentioned in the fig.1. results of the multi-agent system show that the imposed cost of the outage can be decreased significantly by the proposed optimization, while the comfort of suppliers is also considered. the rest of the paper is organized as follow1. cost optimization. 2. scheduling of energy on the basis of pi. 3. the self-supporting mechanism at the microgrid level. 2. self dependant and supporting multi-agent system in the fig-2., the functioning of various microgrid during the fault or outage is described through the flowchart. house agents are introduced in this paper for making the consumer selfdependent in respect of energy during the outage and also contribute to microgrid if have more energy than load demand. 93 s. singh copyright ©2019 assa. adv. in systems science and appl. (2019) start grid information ( capacity, topology, load curve, supply condition.....) operation (self-healing, demand response ...) fault occurred? no yes yes no main utility power? detection of the type of fault send command to microgrids and restoration process if possible fault removed yes supply remain data collection provide by main grid no stop send island mode operation information from the supplier command to microgrid with the outage regarding the access energy optimize the cost and send commands supply energy to demanding microgrid alarm the fault and active island mode fig.2. micro grid algorithm. 2.1. cost determination of the home agent. value of lost load (voll) is the estimated amount that customer receiving electricity with firm contracts would be willing to pay to avoid a disruption in their electricity service [9]. the value of these losses can be expressed as a customer damage function [10]. the cdf define as loss (rs/kw) = f (duration, season, time of day, notice) utility dependency reduction by multi-agents participation in smart grid 94 copyright ©2019 assa. adv. in systems science and appl. (2019) (rs/kw) loss small & medium c/i large c/i residential outage of duration fig. 2a the cost of load outage can be calculated by multiplying the loss(volllosses) and ens (energy not served). the value of losses is different for residential, large and small commercial /industrial consumers. therefore, the cost of an outage of i-th load can be calculated from the following: rpens ii  (1) ii i ensvollc 1 (2) the cost of an outage is reduced for the resident if the energy of ev batteries used to supply some load. in this case, another cost that we should consider is the cost of discharging of the batteries. this cost contains the cost of charging the batteries, batteries degradation due to use to give power to the microgrid. the cost of energy is calculated based on the price of electricity in the given time and the efficiency of the charger [11]. if ev is present to give power in outage condition, the cost of the energy can be calculated from 3. ingch iceyelectricitesbde c arg pr 2    (3) soc(state of charge) shows the state of the battery in use. while dod(depth of discharge) is most often when discussing the life of battery after repeated use. )()( argarg ingchingdischout ttrt  (4) invertorbattery final c esbde socsoc   (5) min min 3 ))97.0/()97.0(( socsoc socsoc taxiprice taxipricesocsoc c final finalfinalfinal        (6) battery jj j csocsocenergyachi  )(. min (7) the total cost can be estimated with the following:    m j jj n i i total cccc 1 32 1 1 )()( (8) 95 s. singh copyright ©2019 assa. adv. in systems science and appl. (2019) this cost is imposed on the consumers and user of home and ev. therefore, it should be optimized to reduce cost. the control variable to optimize the cost is connectivity of load and active power of the ev and inverter. main constraints of evs are the energy of batteries and the nominal power of batteries and inverter. comparison of costs with supplied energy is shown in fig.3. fig.3. different costs plots vs supplied energy. 2.2. self-backing action during the outage in the other microgrid. during the outage, mg agent communicates with the sp and wm agent other active microgrid and take the data in respect of power and cost then decide that the communicated mg is acted as buyer or seller if mg has more load then its acts as seller otherwise it acts as a buyer. if microgrid acts as a buyer than mg agent communicates with the distributed network management system for the requirement of power. dnms have data of all microgrid, on the basis those data which mg have more power than load can be shared with the buyer mg but it will be done on the basis of priority index. priority index pi play a very important role in the energy training mechanism . while deciding the pi of buyer mgi, the three-factor are taken into consideration: 1. the amount of energy is provided by the mg in the past. 2. the amount of load is a demand by the mg in the present. 3. the rate mg provides the energy in past and present willingness. average i total i total i i cost cost load load oncontributi oncontributi pi  (9) in equation (9), contributions is the contribution by the buyer mg till now and contribution total is the contribution by the all mg in past. the load is the load demanding by the buyer mg and loadtotal is load demanding by all buyer mg in the present time [12]. similarly, costi is the cost at which buyer mg want to purchase the energy and costaverage is the average cost of all seller mg in the present time [13]. cost kwh utility dependency reduction by multi-agents participation in smart grid 96 copyright ©2019 assa. adv. in systems science and appl. (2019) 2.3. self-backing action during the outage at smart homes level. if a fault occurs in the microgrid, now faulty microgrid is disconnected from the other mgs. now home agent play a role to overcome the problem and also help in providing the power to the house by using the batteries stored energy. batteries were charged by the invertors, solar panels, windmills etc. the cost optimization is done by the various agents of the grid and power is controlled by the dnms. dnms manage the power between the mg and the main grid on the basis of the priority factor. fig.4. functioning of various agents and mechanism during the outage. 3. energy scheduling at house, microgrid and main grid level. we have three levels of energy scheduling, i.e. house level, microgrid level, main grid level in a distributed system with the multiple microgrids. house level mainly deals with the house agent and sp & wm agent. microgrid level deals with the microgrid agents of other microgrids to share the power between each other as per requirements of the microgrids. it is decided on the basis of the priority index. in main grid level scheduling, it comes in a scenario when the requirement is more than the generation of the microgrid. in this condition dnms take the power from the main grid to fulfill the requirements of a microgrid in the respect of power and energy. it is shown in fig.5. 97 s. singh copyright ©2019 assa. adv. in systems science and appl. (2019) fig.5. four microgrids with various agents. 3.1. energy scheduling at house level. the information regarding the load demand and energy generated by the house by ev during the outage is available with the house agent. the house agent gets the data from the ev. on the basis of data, a house agent takes a decision. if ev has accessed energy more than the basic requirement then it communicates to the house agent. the house agent act as a manager firstly fulfills the energy requirement of itself if there some energy left than the house agent will share the access energy with the microgrid agent. it is the responsibility of the house agent to provide the energy to own house first than to give to another microgrid. if house agent has no energy in this condition house agent take energy from the sp & wm agent if they have, otherwise asked to dnms. 3.2. energy scheduling at the microgrid level. the microgrid level is basically used for maintaining the fair distribution of energy between the microgrid with the help of dnms. the function depends on the demand load and amount of energy microgrid have. those microgrids have access energy than the load they share with the help of dnms. dnms give the energy to others microgrid on the basis of priority index to those who have less generation than load in that period of time. at this level energy that generated by the microgrids are shared with others as per the supply and demand scenario. cost of energy generated at the microgrid level is low compared to energy purchased from the main grid because the source of energy through which it generated is a renewable source of energy. 3.3. energy scheduling at the main grid level. there are three cases in the main grid level on the basis of demand load and generation at the microgrid level. in this level, dnms will represent itself as buyer or seller its depend on the conditions below. case-i: when ii loadgeneration  the extra energy can be bought or taken from the main grid. case-ii: when ii loadgeneration  the balanced energy can be sell to the main grid. case-iii: when ii loadgeneration  it acts as a neutral. demand and supply is balanced in this case. utility dependency reduction by multi-agents participation in smart grid 98 copyright ©2019 assa. adv. in systems science and appl. (2019) 4. results and discussion. for simulation purpose, a distributed system with four microgrids is considered. which shown in table-1. in block 1, mg-1 is having the load of 7 mw and generation by the sp and wm is 8 mw. its also having the power generated by ev and in which should use within the microgrid, it is managed by ev agent. but in the simulation we do not use this generation or due to less value. now mg-1 will act as a seller and will try to sell its excess energy (1 mw) to other microgrids. in block 1, mg-2 is having a load of 5 mw and a generation of 5 mw. in this block generation and the load is the same therefore the microgrid does not act as seller or buyer. in block 1, mg-3 is having the load of 10 mw and generation of 4 mw. there is a deficiency of 6 mw. mg-4 also need 3mw. the roles assigned to various microgrids and energy requirement is mentioned in table-1. since mg-3 and mg-4 registered themselves as buyers, the dnms calculate the priority index of both microgrid on the basis of past data. if we consider the power generated by the ev and in than the overall requirement will be decreased by some value. however, in the first block, the contribution to another microgrid by block-1 is zero because block-1 is acting as a buyer. block 2, the main grid is not present, in this case, mode of the distributed system is island mode. in this block, mg-1 and mg-4 act as a buyer and other two act as a seller. in this case, the main grid is also not present so the power will be shared between the microgrids. in the second block, the contribution to another microgrid by block-2 is zero. similarly, block-3 all microgrid act as a neutral. there is no requirement of power from the main grid but in block-4, the total generation of power by all microgrids is more than the requirement. in this case, block-4 have access power to supply to another block at own rate of price. table 1. functioning of various microgrids block main grid mode micro grid total load (l)(m w) generation by sp and wm (g)(mw) generation by ev and in (e) (mw) g-l (mw ) role power supplied by main grid remark 01. connected 1 7 8 0.01136 1 seller 8 2 5 5 0.02235 0 neutral 3 10 4 0.01242 -6 buyer 4 8 5 0.01156 -3 buyer for selfsupporting. 02. not connected (fault condition) island mode 1 8 4 0.02156 -4 buyer 0 requirement 2 mw but some power provided by ev and in within mg the share of power decide on the basis of the priority index 2 6 10 0.03142 4 seller 3 6 10 0.01562 4 seller 4 10 4 0.02549 -6 buyer 03. not connected (fault condition) island mode 1 9 8.8 0.21500 -0.20 neutral 0 no need to share the power 2 7 6.5 0.5600 -0.50 neutral 3 9 8.9 0.1250 -0.10 neutral 4 10 9.9 0.1265 -0.10 neutral 04. connected 1 15 16 0.0125 1 seller 0 4 mw access power 2 10 9 0.2541 -1 buyer 3 12 14 0.2314 2 seller 4 10 12 0.0564 2 seller 99 s. singh copyright ©2019 assa. adv. in systems science and appl. (2019) 5. conclusion in this paper, a self-supporting mas, cost optimization, energy scheduling in the distributed system having multiple microgrids for the smart grid is proposed. a novel method of finding the priority index for microgrid on the basis of past and present data is also presented. the system can supply loads of a smart home or the grid via evs batteries to minimize the overall cost during the outage. in this paper, a house agent pays an important role in providing the energy within the house first then it will think for sharing with others microgrid if it's having more energy than a requirement. it will share with microgrid agents at the lower cost compared to the cost at which the main grid provides. this will help in reducing the cost and dependency on the main grid. the implementation procedure of the proposed mechanism in real life is discussed in details. the proposed mechanism is stable and simple for real-time implementation which may also increase electricity reliability. references [1] yilmaz, m., & krein, p. t. (2013). review of the impact of vehicle-to-grid technologies on distribution systems and utility interfaces. ieee transactions on power electronics, 28(12), 5673-5689. [2] alagoz, b. b., kaygusuz, a., & karabiber, a. (2012). a user-mode distributed energy management architecture for smart grid applications. energy, 44(1), 167-177. [3] venayagamoorthy, g. k., sharma, r. k., gautam, p. k., & ahmadi, a. (2016). dynamic energy management system for a smart microgrid. ieee transactions on neural networks and learning systems, 27(8), 1643-1656. [4] haidar, a. m., muttaqi, k. m., & sutanto, d. (2014). technical challenges for electric power industries due to grid-integrated electric vehicles in low voltage distributions: a review. energy conversion and management, 86, 689-700. [5] elmitwally, a., elsaid, m., elgamal, m., & chen, z. (2015). a fuzzy-multiagent selfhealing scheme for a distribution system with distributed generations. ieee transactions on power systems, 30(5), 2612-2622. 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[11] habibidoost, m. and bathaee, s.m.t., 2018. a self-supporting approach to ev agent participation in smart grid. international journal of electrical power & energy systems, 99, pp.394-403. utility dependency reduction by multi-agents participation in smart grid 100 copyright ©2019 assa. adv. in systems science and appl. (2019) [12] jadhav, a. m., & patne, n. r. (2017). priority-based energy scheduling in a smart distributed network with multiple microgrids. ieee transactions on industrial informatics, 13(6), 3134-3143. [13] funde, n., dhabu, m., deshpande, p., & patne, n. r. (2018). sf-oeap: starvation free optimal energy allocation policy in a smart distributed multi-microgrid system. ieee transactions on industrial informatics adv syst sci appl 2019; 03; 118-130 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/556 assessment of measures to combat poverty based on a global multi-country hybrid econometric model abdykappar ashimov1*, yuriy borovskiy2, dauren aidarkhanov3 1)kazakh national research technical university named after k. satpayev, almaty, kazakhstan e-mail: ashimov37@mail.ru 2)kazakh national research technical university named after k. satpayev, almaty, kazakhstan e-mail: yuborovskiy@gmail.com 3)kazakh national research technical university named after k. satpayev, almaty, kazakhstan e-mail: aidarkhanov@inbox.ru received february 12, 2018; revised august 23, 2019; published october 1, 2019 abstract: a hybrid global multi-country econometric model describing the interacting economies of 62 countries and the rest of the world is presented. the hybrid model was built by developing and combining two macroeconomic models: the fair's quarterly model and the imf’s annual macroeconomic balance model (using two developed interaction interfaces between these models). the possibility of transferring the results of computational experiments based on the model to practice (the model verification) using two approaches is discussed: parametric control theory methods and known methods of testing the results of econometric models. a number of parametric control problems was set and solved, aimed at maximizing gross domestic product per capita at purchasing power parity at the levels of individual countries and their associations. one such problem additionally takes into account the conditions of convergence of economic growth of five countries of the eurasian economic union. the optimizable tools of these problems are sets of instruments of monetary, exchange rate and fiscal policy with appropriate restrictions. the solution results of the mentioned problems were used to estimate the rate of change in poverty levels in the respective countries. these results demonstrate the advantages of coordinated macroeconomic policies at the levels of country groupings in comparison with the levels of individual countries that are members of such groupings. keywords: poverty rate, convergence, global multi-country hybrid econometric model, coordinated macroeconomic policy, theory of parametric control. 1. introduction the problem of overcoming hunger and poverty is being addressed by the international community in major development objectives in the declaration for the new millennium [17] and in the subsequent international agreement of 2015 with the new sustainable development goals. poverty reduction has become one of the performance criteria of social and economic systems. the significance of the problems associated with poverty and income distribution in terms of economic policy-making necessitates that the social consequences of economic shocks and economic policies are considered [4, 15]. * corresponding author: ashimov37@mail.ru mailto:ashimov37@mail.ru mailto:yuborovskiy@gmail.com evaluation of measures to combat poverty 119 copyright ©2019 assa. adv. in systems science and appl. (2019) in [14] poverty is measured by the incomes of the poorest 20% and 40% of the population, showing that an increase in gdp growth rates leads to a direct increase in the growth rates of average incomes of the poorest 40% of the population in a one-to-one ratio. in [13] the following empirical model is obtained by evaluating the relationship between the change of poverty rate and the economic growth rate for 62 countries using the nonlinear least squares method: rate of poverty reduction = [-9.33 × (1 g)3] × economic growth rate. (1.1) here g is the gini index, economic growth rate = growth rate of gdp per capita. in (1.1), the rate of change in the proportion of people with a standard of living below the poverty line was used as the rate of poverty reduction. for example, in a country with a low level of inequality, where the gini index is 0.30, the proportion of people living below the poverty line will decrease by half in 10.5 years if the annual growth rate of gdp per capita is 2%. in contrast, in a country with high inequality, where the gini index is 0.60 with the same economic growth, this result will take 57 years. in [3, 6-7] it is proposed to implement an impact assessment of economic policy measures on poverty based on the mathematical models of macroeconomic systems. however, the mentioned papers do not consider the simultaneous impact assessment of monetary, exchange rate and fiscal policy tools on poverty rates. in [10], on the basis of a multi-country econometric model, the exchange rate of a country’s national currency per unit of us dollar is estimated as a regression function of the interest rate ratio and price level ratio of the national economy and anchor economy. in the paper, fair considers the impact of monetary policy measures, government consumption (for the us) and fiscal policy measures on some macro indicators (gdp, investments, etc.), but does not consider a simultaneous impact assessment of measures in the fields of monetary, exchange rate and fiscal policy instruments on poverty rates. in [12] only the assessment of effective equilibrium exchange rates of the national currencies of the countries under review is carried out on the basis of relevant statistical data. in this paper, a global multi-country hybrid econometric model based on the developed quarterly fair multi-country macroeconomic model [10] (hereinafter fair model), the annual international monetary fund multi-country econometric model [12] (hereinafter imf model) and the development of two interaction interfaces for the advanced fair model and the imf model is proposed. the conditions for transferring the computational experimental results based on the built model to practice are evaluated. in this paper, gdp per capita at purchasing power parity is taken as the main optimality criterion of optimization problems. the simultaneous impact assessment of measures of tools in the areas of monetary, exchange rate and fiscal policies on poverty and the convergence of the adopted poverty rate at the regional union level and global level by setting and solving a number of extreme problems using methods of parametric control theory is carried out. 2. building a global multi-country hybrid econometric model a global multi-country hybrid econometric model (hereinafter referred to as model) was built to assess the measures of influencing the tools of monetary, exchange rate (as part of effective exchange rate evaluation), and fiscal policy on the adopted poverty rate and convergence of the indicator. the model was built in accordance with the following steps. step 1 selection of the basic structures of the econometric models for building the model. step 2 development of the basic structures of econometric models. 120 a. ashimov, yu. borovskiy, d. aidarkhanov copyright ©2019 assa. adv. in systems science and appl. (2019) step 3 development of the interaction interfaces of the developed basic structures of the econometric models for building the model. step 4 preparing and testing time series and panel data for use in the reconstruction of the relevant regression functions and the interaction interfaces in the developed basic structures of the econometric models. step 5 parameter estimation of the regression functions of the developed fair model by the two-step least squares method. step 6 parameter estimation of the panel regression functions of the imf model by the generalized method of moments [5]. step 7 parameter estimation of the regression function of the first output interface in the imf model to the input of the fair model to transfer annual effective exchange rate to input of the fair quarterly model. step 8 estimation of smoothing parameters in the second interface of the outputs of the fair quarterly model to the inputs in the annual imf model. step 9 building the model. building the model within the framework of the proposed algorithm is as follows. in accordance with step 1 and the designation of the model, based on the analysis of the available literature for multi-country econometric models, the fair multi-country econometric model [10] and the imf multi-country econometric model [12] were selected as the source of basic structures. the fair model was selected as the initial basic structure to build the model, because it provides its designation if we use effective equilibrium exchange rates as an estimate of exchange rates. the imf model was also selected as the initial basic structure for building the model under the following conditions: the values of the fundamental factors to calculate the effective equilibrium rates, providing for the internal and external equilibrium [12] based on the fair model were used as basic data; the imf model should be used together with the fair model. thus, while creating relevant interfaces of their interaction, the hybrid of the fair model and the imf model will provide the designation of the model. in accordance with step 2 of model building, the fair model was developed as follows. the basic identities for each i-th country were supplemented by the descriptions of the process of formation of the state budget and public debt. the influence of the level of the world crude oil price on the economies of the countries under consideration was taken into account. the descriptions of conditions of economic integration in the areas of inflation state budget deficit and public debt, with due account of their effects on the behavior of the economic sectors (i.e., households, firms, financial and foreign sector) were further included into the descriptions of the countries of the eurasian economic union (hereinafter, eaeu) consisting of kazakhstan, russia, belarus, kyrgyzstan and armenia. in order to improve the accuracy of prediction, the gdp deflator in the fair model was presented by disaggregating through the following deflators: household consumption; fixed capital accumulation; government consumption; and exports and imports. the fair model’s basic identities were also supplemented by the corresponding expressions for the fundamental factors [12] of the imf model based on endogenous variables. in accordance with step 3, as the basic structure of interface 1, which converts the annual effective equilibrium exchange rate of the national currency of i-country into the corresponding observable quarterly exchange rate, the ru-midas model was adopted [11]. indicators of monetary and exchange rate policy as well as an indicator of a cyclical factor were added as explanatory variables to the basic structure in the model of the first interface adopted in the paper. evaluation of measures to combat poverty 121 copyright ©2019 assa. adv. in systems science and appl. (2019) in our work, the following model of the moving average filter was adopted as the model of the 2nd interface that converts the output (xi,t) the i-th quarterly indicator of the developed fair model to the corresponding annual indicator (�̃�𝑖,𝑇) of the imf model [2]: �̃�𝑖,𝑇 = ∑ 𝜑𝑖,𝑗𝑋𝑖,𝑡+𝑗 2 𝑗=−2 , (2.1) here, each calculated year t corresponds to the second quarter of this year t, 𝜑𝑖,𝑗 is the filter smoothing parameter. at step 4, the preparation and testing of time series and panel data to build the corresponding regression functions is carried out in the eviews econometric environment. in doing so, testing and preparation of each time series is limited to estimation of the presence of seasonality, emissions and the subsequent correction of the considered series. preparing and testing panel data are reduced to the formation of the corresponding panel data by gluing together time series, and their testing is connected with the test results of the subsequent regression functions. if the test results for the corresponding regression functions meet the requirements, they can be used to judge the adopted panel data for their subsequent use. in step 5 and 6 of building the model, the parameters of the corresponding regression functions are estimated using the methods specified in the step 5 and 6 i.e. the regression functions for the fair model are restored by the 2-step least squares method, and the regression functions of the imf model are restored by the generalized method of moments. when restoring the developed structures of the fair model and imf model, the conditions of the parity of the trade balance elasticities in the nominal exchange rate in the developed fair model and imf model are met. these elasticities were taken from [15]. according to step 7, the parameters of the developed model of interface 1 were assessed by the ordinary least squares method. in step 8, the model’s (1.1) parameters φij of interface 2 were estimated by the method proposed in [2]. in step 9, the building of the model is carried out using the developed fair model, the imf model, and models of interface 1 and 2 for further launching by programming. the hybrid model built using this algorithm consists of 2 blocks: the fair block, which describes the functioning of the relevant sectors (i.e., households, firms, financial sector, government and the external sector, without the exchange rate formation process) and the imf model block, which describes the formation of effective equilibrium exchange rates of national currencies based on the output data generated by the second interface of the fair model block. the obtained estimates of the equilibrium exchange rates from the imf model block are transmitted using the first interface to the input of the fair model block. the hybrid model describes the interacting major economies of 44 countries, taking into account the exogenous impact of the indicators of the economies of the other 18 countries and the rest of the world. real and nominal gdp, gdp at purchasing power parity, state budget deficit, government debt, nominal exchange rate of the national currency against the us dollar, short-term and medium-term interest rates in the country, the rate of 3-month us treasury bonds, unemployment rate, consumer price inflation, prices of export and import products in the country, world price for crude oil, fiscal position normalized to nominal gdp, relative income in relation to other countries, the terms of trade of the country, and many other variables are the main endogenous variables of the hybrid model. real government consumption, indirect tax rate, effective corporate and individual income tax rates, effective import duty rate, central bank foreign exchange reserves and many other variables are the main exogenous variables of the hybrid model. 122 a. ashimov, yu. borovskiy, d. aidarkhanov copyright ©2019 assa. adv. in systems science and appl. (2019) 3. testing the model for practical application testing the model for possible practical application, i.e., testing the model for the conditions of the possibility of transferring the results from computational experiments into the subject area was done by: 1) testing within the stages of building and developing the model (including building and development of the fair model, the imf model and the first interface model) at the level of time series testing and regression functions; 2) testing of the built model within the framework of the cowles commission approach [9]; 3) testing of the built model within the framework of parametric control theory. the testing of time series and regression functions, respectively, was reduced to testing on outliers, level shifts and seasonality in time series, as well as cleaning of these components; evaluation of significance of the regression functions’ variables; significance of the regression function itself. this involved testing the following the autocorrelation of regression residuals; the normality of the regression residual distribution; the stability of regression function parameters, and also the verification of coincidence of signs of regression function coefficients with the macroeconomic theory provisions; the over-identifying restrictions for the selected 2 least square method tools (eliminating the bias of the parameter estimation) and the conditions of the rational expectations hypothesis [8,18]. while testing the time series and regression functions, we were able to establish the following. out of the tested time series containing 4955, seasonality was revealed in 3235, while outliers were identified in 564 we identified outliers. the analysis of the results of the regression function testing (total of 2118) demonstrated their ability to be used as components in the model. testing the model within the framework of the cowles commission approach was carried out by estimating the signs of changes in a number of endogenous variables of the model when conducting a scenario with changes in some exogenous parameters and checking the compliance of these signs with the main provisions of macroeconomic theory. table 3.1 below presents the results of estimating the impact of a 10 percent increase in the values of these exogenous variables on total gdp at purchasing power parity in the period 2015-2020 compared to the baseline forecast using the example of three countries: the republic of kazakhstan (rk), the russian federation (rf) and the republic of belarus (rb). here, the baseline prediction was obtained using the model calculations, the exogenous variables of which were predicted using autoregressive models. since some of the variables in the basic version of the model indicated in table 3.1 are endogenous (i.e., nominal exchange rate, short-term interest rate, export price, import price, world crude oil price, and rate of 3-month us treasury bonds), they were exogenized in advance respectively, by disabling the corresponding equations of the model describing the dynamics of these variables. table 3.1. changes in total gdp at ppp in the period 2015-2020 with an increase in the value of the exogenous variables by 10 percent in the same period (as a percentage compared with the baseline calculation) exogenous variable country rk rf rb real government consumption in the country 1.32 0.83 0.35 indirect tax rate in the country -0.34 -0.41 -0.05 personal income tax rate in the country -0.11 -0.20 -0.02 nominal exchange rate in the country 1.66 1.41 0.39 evaluation of measures to combat poverty 123 copyright ©2019 assa. adv. in systems science and appl. (2019) short-term interest rate in the country -0.44 -0.28 -0.06 export prices in the country -1.85 -0.88 -2.16 import prices in the country 0.30 1.02 -0.30 world crude oil price 0.92 0.23 -0.02 rate of 3-month us treasury bonds -0.15 -0.12 -0.09 as can be seen from table 3.1, the real growth of the belarusian economy, unlike the cases of kazakhstan and the russian federation, negatively reacts to the increase in import prices. the analysis of the transmission of changes in import prices on real gdp and the resulting changes in other endogenous variables in the model showed the following. the increase in import prices leads to an increase in the level of consumer prices and the gdp deflator, which eventually increased by 8.04 and 7.00 percent, respectively. an increase in the level of consumer prices directly reduces household consumption and increases the short-term interest rate. an increase in the short-term interest rate leads to an increase in the medium-term interest rate. growth of the medium-term interest rate directly reduces household consumption and investment in fixed capital in the country. a decrease in household consumption and investment in fixed capital negatively influence the value of domestic sales. the decline in domestic sales as a representative of domestic demand in the country leads to a decrease in real gdp. on the other hand, the increase in import prices leads to a decrease in real imports, which eventually decreased by 1.04 percent. however, this did not improve the situation of the trade balance, as real exports in belarus also fell by 1.03 percent. real exports fell for the following reason: an increase in the gdp deflator leads to an increase in export prices (which amounted to 2.77 percent). the growth of export prices leads to a fall in real exports due to the loss of price competitiveness. as a result, the trade balance fell by more than 1.00 percent, and household consumption and fixed investment fell by 0.13 and 0.10 percent, respectively, which led to a drop in real gdp by 0.30 percent. the analysis of the signs given in table 3.1 of the values shows that they correspond to the main provisions of macroeconomic theory. testing within the framework of parametric control theory was carried out by evaluating the values of the mapping stability indicators that convert the values of exogenous variables of the model into the values of its endogenous variables [1]. the stability indicator βf(p,α) defined by the model of the mapping f: a→b at point p ∈ a and for selected positive number α is the diameter of the image (with mapping f) of the ball of radius α and the center at point p (in percentage deviations). if for all points p ∈ a the numerical estimate of βf(p) = limα→0βf(p,α) is evenly close to zero, then the mapping f, given by the model under study is estimated on the set a as continuously dependent on exogenous values. in experiments with the model, a ball with a center at point p corresponding to the baseline values of the following exogenous variables was considered as the set a that belong to the selected region for 2015– 2020government consumption, indirect tax rate (vat and excise), individual income tax rate, and central bank currency reserves. possible values of all endogenous variables belonging to the same region for 2015-2020 were considered as set b. estimates of some indicators of stability βf(p,α), obtained using the monte carlo method are given in the following table 3.2. these estimates are obtained by specifying 1000 pseudo-random points xi evenly distributed in a ball of radius α with center at point p and determining the percentage deviations of the coordinates of their images f(xi) from the corresponding coordinates of point f(p). in the case of assessing the stability indicator for the entire eaeu, the values of the input factors in the specified set were simultaneously perturbed for all the eaeu countries. stability indicator values are calculated on the basis of all endogenous variables of eaeu countries. 124 a. ashimov, yu. borovskiy, d. aidarkhanov copyright ©2019 assa. adv. in systems science and appl. (2019) the analysis of the testing results by the above methods shows the feasibility of using the model for short-term and medium-term forecasting, for macroeconomic analysis, and for assessing the effective versions of economic policies. table 3.2. stability indicator values βf(p,α) for different values of the radius of the ball α region radius α of the ball value of βf(p,α) eaeu 1.0000 0.0037 eaeu 0.5000 0.0013 eaeu 0.2500 0.0005 eaeu 0.1250 0.0003 4. solving parametric control problems on the basis of the model in this section, gdp per capita at purchasing power parity is taken as the main criterion for various extreme parametric control problems at the levels of individual countries and their groupings. figure 4.1 presents the results of the comparative analysis of the gdp per capita in purchasing power parity terms on the average over the period 2015-2020 for a number of countries whose economies are described in the model. the analysis of figure 4.1 demonstrates that in the period 2015-2020, among the countries which are considered in the model, the highest living standards will remain in the european union countries (norway being the forerunner, followed by ireland, etc.) and in the usa. the countries with the lowest quality of life include kyrgyzstan, syria, pakistan and india. the gap between the richest and the poorest countries, even in the considered sample of the model, is more than 60.00 thousand international us dollars. the average value of the first 22 countries is about 49.02 thousand international us dollars, while the average of the remaining countries is about 15.96 thousand international us dollars (i.e., the gap is approximately 33.06 thousand international us dollars). compared with the period of 2010-2014, this gap amounted to 28.00 thousand us dollars, which indicates the aggravation of the problem of convergence of countries in this indicator. evaluation of measures to combat poverty 125 copyright ©2019 assa. adv. in systems science and appl. (2019) fig. 4.1. gdp per capita in ppp terms in 44 countries in the model the model calculations (which are close to the imf weo october 2017 forecasts) showed that, in addition to the problem of convergence in terms of the adopted poverty indicator, low growth in the world economy and slow growth in world trade (see figure 4.2) are challenges that need to be faced by the international community. kyrgyz republic syria pakistan india philippines armenia jordan peru south africa colombia venezuela brazil china belarus thailand mexico argentina chile russia kazakhstan greece portugal malaysia spain italy new zealand japan korea france finland united kingdom belgium canada denmark germany austria australia netherlands sweden saudi arabia ireland switzerland united states norway thousand international dollars c o u n tr y 126 a. ashimov, yu. borovskiy, d. aidarkhanov copyright ©2019 assa. adv. in systems science and appl. (2019) fig. 4.2. estimated values (ex-post and ex-ante analysis) of global economic growth and growth of world trade based on the model (year to year) based on the model and parametric control theory, a number of extreme problems for maximizing gdp per capita at purchasing power parity were formulated and solved, taking into account the results of the analysis of figures 4.1 and 4.2. problem 4.1 is the extreme problem at the country level, problem 4.2 is the extreme problem of coordinated parametric control at the level of the selected set of countries, and problem 4.3 is the extreme problem of coordinated parametric control at the level of the 5 countries of the eaeu, taking into account the condition of convergence of gdp per capita at the ppps of these countries. the extreme problem at the country level has the following form: problem 4.1. based on the model for a given country i (𝑖 ∈ {1, … ,44}) and taking into account the adopted monetary policy in each country, to find the values of the monetary policy instruments (rsi(t)) and fiscal policy (gi(t), taxr1i(t), taxr2i(t), taxr3i(t)), where t ∈ {2015, ..., 2020}, providing the criterion maximum: 𝐾𝑖 1 = ∑ 𝑌𝑃𝑃𝑃𝑖(𝑇)2020 𝑇=2015 , (4.1) under restrictions (𝑇 ∈ {2015, … , 2020}): |𝑅𝑆𝑖(𝑇) − 𝑅𝑆𝑖 ̅̅ ̅̅ (𝑇)| ≤ 0.1 × 𝑅𝑆𝑖 ̅̅ ̅̅ (𝑇) , |𝐺𝑖(𝑇) − 𝐺�̅�(𝑇)| ≤ 0.1 × 𝐺�̅�(𝑇) , |𝑇𝑎𝑥𝑅1𝑖(𝑇) − 𝑇𝑎𝑥𝑅1𝑖 ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ (𝑇)| ≤ 0.1 × 𝑇𝑎𝑥𝑅1𝑖 ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ (𝑇) , |𝑇𝑎𝑥𝑅2𝑖(𝑇) − 𝑇𝑎𝑥𝑅2𝑖 ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ (𝑇)| ≤ 0.1 × 𝑇𝑎𝑥𝑅2𝑖 ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ (𝑇) , (4.2) |𝑇𝑎𝑥𝑅3𝑖(𝑇) − 𝑇𝑎𝑥𝑅3𝑖 ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ (𝑇)| ≤ 0.1 × 𝑇𝑎𝑥𝑅3𝑖 ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ (𝑇) , 𝑌𝑃𝑃𝑃𝑖(𝑇) ≥ 𝑌𝑃𝑃𝑃̅̅ ̅̅ ̅̅ ̅̅ 𝑖(𝑇). in addition, for the eaeu countries (i ∈ {1, ..., 5}), additional restrictions of sustainable growth are used: 𝑃𝐼𝑖(𝑇) ≤ min 𝑟∈{1,…,5} 𝑃𝐼𝑟(𝑇) + 0.05, 𝐺𝐷𝐸𝐹𝑖(𝑡) ≤ 0.03 ∗ 𝑌𝑖(𝑡) ∗ 𝑃𝑌𝑖(𝑡), (4.3) 0 2 4 6 8 10 12 14 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 p er ce n t ch an g e year world economic growth world trade growth evaluation of measures to combat poverty 127 copyright ©2019 assa. adv. in systems science and appl. (2019) 𝐺𝐷𝐸𝐵𝑇𝑖(𝑡) ≤ 0.5 ∗ 𝑌𝑖(𝑡) ∗ 𝑃𝑌𝑖(𝑡). here, t is the number of the year, 𝑌𝑃𝑃𝑃𝑖(𝑇) is the gdp per capita at ppp, 𝑅𝑆𝑖(𝑇) is the shortterm annual interest rate, 𝐺𝑖(𝑇) is government consumption, 𝑇𝑎𝑥𝑅1𝑖(𝑇) effective corporate income tax rate, 𝑇𝑎𝑥𝑅2𝑖(𝑇) effective individual income tax rate, 𝑇𝑎𝑥𝑅3𝑖(𝑇) effective import duty rate, 𝑃𝐼𝑖(𝑇) inflation of consumer prices; 𝑌𝑃𝑖(𝑇) − nominal gdp in national currency; 𝐺𝐷𝐸𝐹𝑖(𝑇) − state budget deficit in national currency; 𝐺𝐷𝐸𝐵𝑇𝑖(𝑇) the size of public debt in national currency. the symbol ̅ corresponds to the basic calculation of the specified variable. in this problem, the choice of restrictions on the changes of instruments within 10% relative to their baseline values was made for reasons of clarity in order to demonstrate the possibility of solving problems with similar restrictions. as the limitation intervals increase, the optimal value of the criterion of the corresponding problem can only increase, but the risk associated with the adaptation of the economy and society to more significant changes in fiscal and monetary policy (which is not covered by the model) also increases. in the case of the practical implementation of the proposed approach to develop recommendations for conducting macroeconomic policy, the limitations of extreme problems are subject to agreement with the decision maker. the problem of coordinated parametric control at the level of a selected set of l countries with the numbers 𝑖 = 1, … , 𝐿 is formulated as follows. problem 4.2. for the selected set of 𝐿 ∈ {2, … , 44} countries based on the model and taking into account the adopted monetary policy in each country, to find the values of the monetary policy instruments (rsi(t)) and fiscal policy (gi(t), taxr1i(t), taxr2i(t), taxr3i(t)), where 𝑇 ∈ {2015, … , 2020}, 𝑖 ∈ {1, … , 𝐿} providing the criterion maximum: 𝐾𝐿 2 = ∑ ∑ 𝑌𝑃𝑃𝑃𝑖(𝑇)2020 𝑇=2015 𝐿 𝑖=1 , (4.4) with restrictions (4.2) for all countries of this set and restrictions (4.3) for eaeu countries from this set. here, in the case of l = 5, the specified set of countries is the eaeu; in the case of l = 44, the specified set of countries is considered quasi for the world economy. problem 4.3. for the selected set of l = 5 of the eaeu countries based on the model and taking into account the adopted monetary policy in each country, to find the values of the monetary policy instruments (rsi(t)) and fiscal policy (gi(t), taxr1i(t), taxr2i(t), taxr3i(t)), where 𝑇 ∈ {2015, … , 2020}, 𝑖 ∈ {1, … ,5}, providing the criterion maximum: 𝐾3 = ∑ ∑ (𝑌𝑃𝑃𝑃𝑖(𝑇) − 𝛽𝜀𝑖 ( 𝑌𝑃𝑃𝑃(𝑇)−𝑌𝑃𝑃𝑃𝑖(𝑇) 𝑌𝑃𝑃𝑃(𝑇) ) 2 )2020 𝑇=2015 5 𝑖=1 , (4.5) with restrictions (4.2), (4.3). here, 𝑌𝑃𝑃𝑃(𝑇) − is the aggregated gdp per capita at ppp of the eaeu countries, 𝜀𝑖 is the adjustable weights of the countries, and 𝛽 is the adjustable weight of the component responsible for the convergence in the criteria. the use of the empirical model (1.1) allows transforming the growth rates of gdp per capita at ppp in each of the reviewed countries to the corresponding rates of change in the poverty level (the proportion of the population living below the poverty line) obtained as a result of solving the formulated problems 4.1-4.3. the world bank's gini index estimates for the eaeu countries were used (link). some results of solving the above formulated extremal problems are presented in table 3 (in percentage changes of the specified indicators of the country on average for the period 2015–2020 as a result of solving the problem compared to the baseline calculation). here, gdp means the gdp at ppp per capita and convergence 128 a. ashimov, yu. borovskiy, d. aidarkhanov copyright ©2019 assa. adv. in systems science and appl. (2019) indicator ( 𝑌𝑃𝑃𝑃(𝑇)−𝑌𝑃𝑃𝑃𝑖(𝑇) 𝑌𝑃𝑃𝑃(𝑇) ) 2 , participating in the definition of the criterion (4.5). the country codes in table 4.1 are the following: kz kazakhstan, ru russian federation, by belarus, kg kyrgyzstan, am armenia, pa pakistan and id india. comparative data analysis on the example of the eaeu countries shows that the effectiveness of coordinated optimization measures aimed at maximizing the sum of the criteria of effectiveness of the functioning of the countries of this regional union is higher than the effectiveness of the same measures of each individual eaeu country. analysis using the example of india and pakistan also shows that the effectiveness of coordinated optimization measures aimed at maximizing the sum of performance criteria of the country groupings functioning is higher than the effectiveness of the same measures of a particular country within these groupings. it should be noted that for the eaeu countries, the effectiveness of measures for coordinated optimization at the level of the quasi-world economy is higher than at the level of the regional union. in addition, the results of solving problem 4.3 in table 4.1 show that monetary, exchange rate and fiscal policy measures provide a noticeable convergence of gdp per capita at ppps of the five eaeu countries. table 4.1. percentage change in indicators compared with the baseline forecast as a result of solving parametric control problems problem indicator country kz ru by kg am pa id 4.1 gdp 2.99 2.63 2.00 0.69 2.04 1.15 2.33 4.1 poverty -10.6 -5.9 -7.7 -2.5 -5.6 4.2 (l = 5) gdp 3.75 2.65 5.68 0.96 2.64 4.2 (l = 5) poverty -13.3 -6.0 -22.0 -3.4 -7.2 4.2 (l = 44) gdp 6.93 5.84 7.37 1.95 4.02 2.86 3.93 4.3 gdp 2.99 2.63 5.78 1.62 3.46 4.3 convergence -1.03 -0.31 -2.13 -0.64 -1.21 the optimal values of economic policy instruments found as a result of solving the corresponding extremal problems were tested for their practical implementation by estimating the values of sustainability indicators for the scenario corresponding to the optimal values of economic policy instruments found. the results of this testing are illustrated below using the example of the scenario obtained as a result of solving problem 4.2. in assessing the βf(p,α) indicator, the following set of exogenously given parameters for all eaeu countries for 2015– 2020 was considered as a vector p. this set includes the following parameters potential gdp, foreign exchange reserves of the central bank, and indicators of government budget accounts. the following endogenous macroeconomic indicators for each eaeu country in the specified period were considered as output variable of mapping f real gdp, real household consumption, real imports of goods and services, real exports of goods and services, export prices, import prices, gdp deflator, nominal exchange rate of national currency to us dollar, and medium-term interest rate. the results of computational experiments to estimate the sustainability indicators, given below in table 4.2, show that when the radius α tends to zero with a given pace, the value of the stability indicator tends to zero with the same pace. evaluation of measures to combat poverty 129 copyright ©2019 assa. adv. in systems science and appl. (2019) table 4.2. the values of stability indicators within the optimal values of the economic policy instruments that were found from problem 4.2 with l = 5 radius α of the ball value of stability indicator βf(p,α) 1.0000 0.0248 0.5000 0.0124 0.2500 0.0062 0.1250 0.0031 data analysis of table 4.2 demonstrates the fulfillment of the necessary conditions for the possibility of practical implementation of the results of solving problem 2.2. 5. conclusion 1. a global multi-country hybrid econometric model was built. 2. estimates of the necessary conditions for transferring the results of computational experiments based on the model into the subject domain were obtained. 3. ex-post and ex-ante state snapshot analysis of the considered economies was carried out. 4. a number of extreme problems aimed at economic growth were formulated and solved, taking into account the conditions of economic convergence. based on the solutions to these problems, estimates of the rates of change in poverty levels in the respective countries were obtained. 5. the effectiveness of the proposed approaches in evaluating versions of optimal economic policy at the levels of individual countries and their groupings was presented. references [1] ashimov, a.a., sultanov, b.t., adilov, z.m., borovskiy, y.v., novikov, d.a., alshanov, r.a. & ashimov, a.a. (2013). macroeconomic analysis and parametric control of national economy. new york, ny: springer. [2] baxter, m. & king, r. (1999). measuring business cycles: approximate band-pass filters for economic time series, the review of economics and statistics, 81, 575– 593. [3] bénassy, j.p. (2002). the macroeconomics of imperfect competition and nonclearing markets: a dynamic general equilibrium approach. the mit press. [4] blejer, m. & chu, k.y. (1989). fiscal policy, stabilization and growth in developing countries. [5] cashin, p. & beidas-strom, s. (2011). are middle eastern current account imbalances excessive?, imf working papers, 11195. [online]. available https://doi.org/10.5089/9781462305247.001 [6] devarajan, s., lewis, j. & robinson, s. (1990). policy lessons from two-sector models, journal of policy modeling, 12, 625–657. [7] essama-nssah, b. (2005). the poverty and distributional impact of macroeconomic shocks and policies: a review of modeling approaches, world bank working paper, 3682. [online]. available https://openknowledge.worldbank.org/bitstream/ handle/10986/8614/ wps3682.pdf [8] fair, r.c. (1993). testing the rational expectations hypothesis in macroeconometric models, oxford economic papers, 45, 169–190. [9] fair, r.c. (2004). estimating how the macroeconomy works. london: harvard university press. 130 a. ashimov, yu. borovskiy, d. aidarkhanov copyright ©2019 assa. adv. in systems science and appl. (2019) [10] fair, r.c. (2013). macroeconometric modeling. [online]. available https://fairmodel.econ.yale.edu/mmm2/ mm.pdf [11] foroni, c., guerin, p. & marcellino, m.g. (2015). using low frequency information for predicting high frequency variables, ssrn electronic journal. [online]. available https://doi.org/10.2139/ ssrn.2695385 [12] lee, j., milesi-ferretti, g.m., ostry, j., prati, a. & ricci, l.a. (2008). exchange rate assessments: cger methodologies, imf occasional papers, 261. [13] ravallion, m. (2004). pro-poor growth, world bank working paper, 3242. [online]. available https://openknowledge.worldbank.org/bitstream/handle/10986/14116/ wps3242growth.pdf [14] roemer, m. & gugerty, m.k. (1997). does economic growth reduce poverty?, harvard institute for international development caer discussion paper, 5. [online]. available http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.525.7199&rep= rep1&type=pdf [15] scarf, h. & shoven, j. (1984). applied general equilibrium analysis. cambridge university press. [16] tokarick, s. (2014). a method for calculating export supply and import demand elasticities, the journal of international trade & economic development, 23(7), 1059–1087. [online]. available https://doi.org/10.1080/09638199.2014.920403 [17] united nations. (2000). united nations millennium declaration. resolution 55/2. [online]. available http://www.un.org/millennium/declaration/ares552e.pdf [18] wooldridge, j. m. (2000). introductory econometrics. a modern approach. boston: south-western college publishing. мягкая вероятность ошибки при передаче информации adv syst sci appl 2019; 03; 65-79 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/705 sustainable management of universities m. kh. malsagov* ingush state university, nazran, russia e-mail: mmm1956@bk.ru received february 2, 2019; revised september 2, 2019; published october 1, 2019 abstract: this paper considers the problems of sustainable management of universities. a university as an active system is analyzed, and some sustainable development criteria are suggested and formally described using the integrated assessment approach. a series of sustainable management models at the levels of department, faculty, and university are presented as follows: a dynamic incentive model for the participants of a research project; a static reward distribution model as a game in the characteristic function form; a dynamic working time distribution model for department staff as a differential game in the normal or characteristic function form based on compulsion; dual discrete programming problems with sustainable development constraints for determining an optimal staff structure of a faculty, with assessing the efficiency of different paired unions of departments; dynamic coordination models for the social and private interests of university staff in the course of innovations promotion, as differential games in the normal and characteristic function form based on impulsion; finally, dynamic models of anti-corruption drive. major emphasis is placed on problem statements and possible solution methods. keywords: active systems, dynamic games, integer programming problems, universities, sustainable management. 1. introduction the theory of sustainable management of active systems was introduced in [8–13, 18]. the main postulates of this theory can be applied to the sustainable management of higher education institutions (universities), which forms a topical problem. the approach suggested below has the following distinctive features. first, we study management processes of universities as organizations [7], without considering, e.g., the interaction of lecturers and students in the course of learning. second, we are mostly concerned with the sustainable management problem of universities as active systems. third, our analysis employs mathematical models, predominantly the dynamic models of conflict and cooperative control (hierarchical differential games) [1, 15] and their solution methods (simulation modeling). and fourth, the ultimate goal of our research is to develop and test practical procedures for solving some sustainable management problems of universities based on the corresponding mathematical models and software. a significant contribution to this field of investigations was made by the monograph [6], which specified the general methodology of organizational control to educational systems. in particular, the author [6] proposed the main principles of education improvement (in terms of quality, availability and efficiency), which leads to the corresponding control problems; concluded that the backbone element of the theory of control of educational systems is the category of an organization; identified six hierarchical levels of educational systems; specified a series of general concepts of organizational control (the types, principles and mechanisms of control) to educational systems and gave illustrative examples. * corresponding author: mmm1956@bk.ru 66 m. kh. malsagov copyright ©2019 assa adv. in systems science and appl. (2019) the game-theoretic framework that will be developed below treats the sustainable development of universities as the simultaneous satisfaction of the homeostasis and motivation conditions. homeostasis means that the key indicators of the university’s operation (the characteristics of research, educational, and innovative and business activities) must take values within certain ranges. motivation implies that the private interests of university staff are properly considered and coordinated with the social interests of the whole organization. it seems reasonable to solve the sustainable management problems of universities by constructing and examining separate sustainable management models at the levels of department, faculty, and university. these models include the following: a dynamic incentive model for the participants of a research project; a static reward distribution model as a game in the characteristic function form; a dynamic working time distribution model for department staff as a differential game in the normal or characteristic function form based on compulsion; dual discrete programming problems with sustainable development constraints for determining an optimal staff structure of a faculty, with assessing the efficiency of different paired unions of departments; dynamic coordination models for the social and private interests of university staff in the course of innovations promotion, as differential games in the normal and characteristic function form based on impulsion; finally, dynamic models of anti-corruption drive. we suggest the method of qualitatively representative scenarios of simulation modeling as the main tool for solving dynamic games [13]. in addition, we adopt traditional methods such as pontryagin’s maximum principle, dynamic programming, and discrete optimization methods. therefore, this paper considers sustainable management models for universities, with major emphasis on problem statements and possible solution methods. the remainder of this paper is organized as follows. in section 2, we introduce the definition of a university as an active system. in section 3, we present sustainable development criteria for universities. in section 4, we propose separate sustainable management models at the levels of department, faculty, and university. in section 5, we describe the models of anti-corruption drive for universities. in the conclusions, we summarize the outcomes of this paper and outline further research. 2. university as an active system universities are complex multilevel dynamic systems with specific goals and interests of different agents at all levels and also with a complex system of relations with an environment. consider a university as an active system in detail in accordance with the following scheme: elements; subsystems; functions; environment; problems. elements. the upper level of the organizational structure of a university is occupied by administrative staff. the university’s rector, deans (directors of institutes) and heads of departments (laboratories) implement line management––general direction in all fields of activity. vice-rectors, vice-deans, and the heads of support auxiliary units (education and science, accounting, human resources, maintenance, etc.) are responsible for functional management (separate fields of activity). line managers often combine administrative load with research and/or teaching. lecturers and researchers are a controlled subsystem for administrative staff and a control subsystem for students (underand postgraduates). in other words, each lecturer of a modern university has to be engaged in research in order to know the current state of affairs in his/her specialty area and be prepared for teaching. students, including postgraduates working for the degrees of candidate or doctor, have dual role. on the one hand, they form a controlled subsystem taught by lecturers and researchers; on the other, they are not passive at all. first, a commonly encountered idea in pedagogical science dictates that the subject–object impact on a student must be replaced by the subject–subject interaction with proper consideration of the student’s interests and his/her active involvement in educational process. second, the active system concept implies that a controlled subsystem is intensively affecting the control process. students play major role in the university’s life, sustainable management of universities 67 copyright ©2019 assa adv. in systems science and appl. (2019) including management and decision-making. this is done through student bodies and unions, in particular, by integrating students into the academic councils of different institutes and departments, by collecting students’ assessments of different lecturers (feedback), by analyzing the students’ choice of different courses, etc. subsystems. as natural subsystems the structure of a university consists of faculties and institutes as well as functional units. this determines the traditional line-functional management structure. a series of universities, however, use project management (e.g., the management of educational programs and research projects); in this case, project groups can be treated as temporal subsystems, in addition to the permanent (line-functional) ones. functions. historically, universities were mostly in charge of education although university lecturers always included prominent scientists. at the junction of the 18th and 19th centuries, brothers alexander and wilhelm von humboldt suggested the concept of a new university: education must rest on firm foundation of research, and research must play key role. in the second half of the 20th century, the classical “research+education” formula pioneered by the von humboldts was supplemented with the third component, which is difficult to characterize by a single term. the matter concerns innovations, entrepreneurship (business activities), and the contribution of universities into regional development. in the recent two decades the terms “entrepreneurial university” and “academic capitalism” have become firmly entrenched in the literature [2, 17]. thus, a modern university has a triune function that can be described as “research+education+application.” note that the key role in this triad is played by research. precisely research creates prerequisites for implementing educational programs of all levels (especially for master programs) and also for developing regional and industry-oriented applications, including new products and technologies, additional education, consultation with regional authorities, etc. environment. the following elements make up the environment for universities: the ministry of education and higher education of the russian federation; regional authorities; state science foundations; other universities of a given and other regions; regional population; finally, regional enterprises and organizations. as indicated by the above analysis, there exist the following basic problems that cause conflicts in universities as active systems. 1. the distribution of limited resources among different units and also among the participants of research projects; 2. the distribution of working time among the main activity at a university and external activities for higher gain (combining jobs at different universities, tutoring, consulting, etc.) by university staff; 3. the contradictions between the aspiration of separate departments (their staff and heads personally) for their preservation and the objective norms of controllability in combination with the interests of faculties; 4. the contradiction between the need for technological and managerial innovations and the aspiration of university staff for minimum effort and risk; 5. the contradiction between the well-established idea that the university’s managers (from rector to the heads of departments) have to be researchers, the leaders of scientific schools and methodologists, and the modern trend to give these positions to “effective managers” often far from research and education; 6. the aspiration of auxiliary units for imposing their “rules of play” on lecturers and researchers; 7. corruption at universities. these problems can be solved by coordinating the interests of all active agents for the sustainable development of a university using suitable mathematical models––in the first place, the dynamic game-theoretic models that describe the conflict and cooperation of players and guarantee a reasonable compromise between them. however, in a series of cases, (simpler) static optimization models seem a more appropriate choice. 68 m. kh. malsagov copyright ©2019 assa adv. in systems science and appl. (2019) 3. sustainable development criteria for universities the sustainable development of a university as an active system consists in the homeostasis condition and the motivation condition of active agents, which must be satisfied simultaneously. the original theory of sustainable development defines homeostasis using the concept of “three pillars,” meaning that ecological, economic, and social goals must be considered and balanced all together. in the course of modeling, the state vector must include variables associated with some economic, ecological, and social indexes. for universities, the three pillars are research activity (ra), teaching and methodological activity (tma), and also innovative and entrepreneurial activity (iea), which is of equal importance. indexes by these three groups have to be defined and analyzed at the level of department, faculty, and university using some aggregation procedures. therefore, the university’s homeostasis will be verified using an integrated assessment procedure that rests on the integrated assessment approach introduced in [6]. this procedure includes several steps as follows: 1) compile the list of indexes by groups ra, tma and iea at the level of department; 2) to each index assign numerical scores }5,4,3,2{,, ijkijkijk zyx , with the following notations: 2, 3, 4, and 5 as “poor,” “fair,” “good,” and “excellent,” respectively; x as ra; y as tma; z as iea; i as faculty number; j as department number; finally, k as index number. then ||||||,||||,|| ijkijkijk zzyyxx  are the numerical score matrices of the initial indexes of all departments; 3) calculate the indexes of ra, tma and iea at the level of department as )(),(),( zhzygyxfx ijijijijijij  ; for the sake of simplicity, let , : ij ij iji j f g h f    (uniform aggregation); 4) calculate the indexes of ra, tma and iea at the level of faculties as 1 1 , in i ij ji x x n    where in denotes the number of departments at faculty i; ii zy , are calculated by analogy, with standard rounding rules; 5) calculate the indexes of ra, tma and iea at the level of university as 1 1 , m i i x x m    where m denotes the number of faculties at the university; zy, are calculated by analogy, with standard rounding rules; 6) rank the departments in the faculty’s three-dimensional space, the faculties in the university’s three-dimensional space, and the universities in the country’s (or world’s) threedimensional space in the following way: a) find the pareto set in the index space; b) calculate a single homeostasis index. note that the same index for assessing the university’s homeostasis at any of the three levels is an oversimplification: a more adequate approach is to consider the three-dimensional space of qualitatively different criteria of ra, tma and iea. nevertheless, sometimes a single homeostasis index seems appropriate, and it can be calculated using a similar aggregation procedure as described above. it should be emphasized that steps 1–3 of this procedure inevitably have subjective character. one would hardly suggest any objective grounds for choosing indexes, their numerical scores (to say nothing of more complicated assessments!) and further aggregation into a single index. general principles of systems analysis, the experience and knowledge of experts and further testing of assessment procedures in practice are the only possible remedies here. thus, consider an integrated assessment procedure as an illustrative example. its specifics consist in steps 1–3, because the indexes at the levels of faculty and university are obtained by sustainable management of universities 69 copyright ©2019 assa adv. in systems science and appl. (2019) simple averaging of the corresponding values calculated at the lower level. pareto set calculation is also a standard operation, although it can be implemented in different ways [5]. 1. the list of initial homeostasis indexes at the level of department is given in table 1. basically, they correspond to standard reporting, with some supplements and modifications. for fixed numbers of department and faculty, it suffices to number indexes only. table 1. homeostasis indexes by groups, level of department ra tma iea х1––papers published by department staff per annum (regular journals indexed by web of science/scopus), in total y1––textbooks and teaching aids with official stamp published by department staff per annum, in total z1––attracted funding х2––papers published by department staff per annum (regular journals listed by the state commission for academic degrees and titles of the russian federation), in total y2––electronic educational resources developed by department staff per annum, in total z2––contracts with third parties (basic departments at enterprises, joint projects, etc.) х3––papers at international conferences published by department staff per annum, in total y3––master’s dissertations supervised by department staff per annum, in total х4––dissertations (cand. sci., dr. sci.) defended or supervised by department staff within three years, in total y4––average results of students’ groups taught by department staff х5––members of editorial boards of scientific journals, organizing committees of conferences, and dissertation councils among department staff, in total х6––reviews of submitted draft papers, dissertations and abstracts of dissertations per annum, in total 2. of course, assigning numerical scores {2,3,4,5} to the chosen indexes has the highest complexity here. the rules suggested below are not unique and seem somewhat conditional; but they rest on experience and certain plausible arguments. the indexes of ra are as follows: 5, u1 ≥ 1; 4, 0.5 ≤ u1 < 1; х1 = 3, 0.25 ≤ u1 < 0.5; 2, u1 < 0.25, where u1 denotes the papers of type 1 per one department’s employee; 70 m. kh. malsagov copyright ©2019 assa adv. in systems science and appl. (2019) 5, u2 ≥ 2; 4, 1 ≤ u2 < 2; х2 = 3, 0.5 ≤ u2 < 1; 2, u2 < 0.5, where u2 denotes the papers of type 2 per one department’s employee; 5, u3 ≥ 1; 4, 0.5 ≤ u3 < 1; х3 = 3, 0.25 ≤ u3 < 0.5; 2, u3 < 0.25, where u3 denotes the papers of type 3 per one department’s employee; 5, d ≥ 1 and c ≥ 3; 4, c ≥ 2; х4 = 3, c ≥ 1; 2, c = d = 0, where d and c denote cand. sci. and dr. sci. dissertations; 5, u4 ≥ 1; 4, 0.5 ≤ u4 < 1; х5 = 3, 0.25 ≤ u4 < 0.5; 2, u4 < 0.25, where u4 denotes the members of the corresponding type per one department’s employee; 5, u5 ≥ 2; 4, 1 ≤ u5 < 2; х6 = 3, 0.5 ≤ u5 < 1; 2, u5 < 0.5, where u5 denotes reviews per one department’s employee. the indexes of tma and iea are assessed by analogy. 3. for fixed faculty and department, the numerical score matrices x, y and z turn into vectors. hence, the indexes of ra, tma and iea of a department are constructed by specifying an aggregation function ppf ik : , where }5,4,3,2{p and ik is the number of all indexes in group i=1,2,3 (table 1). denote by i i k k pppp  ),...,( 1 the value set of all indexes in group i. possible aggregation procedures are as follows. sustainable management of universities 71 copyright ©2019 assa adv. in systems science and appl. (2019) 1) }{max 1 max j kj pf i  , determining the extremely soft assessment; 2) }{min 1 min j kj pf i  , determining the extremely hard assessment; 3) majmaj pf  , where 2/|}{| imaj kp  is the majority assessment; 4) medmed pf  , where medp is the median (the medium term of the sequence p in the ascending order), determining the median assessment; 5)    ik j jjw pwf 1 , determining the weighed assessment (with subjective weights jw such that    ik j jj ww 1 1,0 ). note that the third and fourth aggregation procedures generally require an odd number of initial indexes. the choice of an aggregation procedure is a matter of principle because the initial indexes from table 1 have different significance. consider an illustrative numerical example (table 2). assume department staff consists of 10 employees. obviously, the first and second aggregation procedures are not completely adequate because of the extreme assessments that coincide with one or two components among the six ones. the other procedures yielding the same index value should be treated adequate (in the case of weighted assessment, after rounding up or down to the nearest integer). table 2. indexes of ra initial value of index specific value (if applicable) assessment of index index of ra 6 u1 = 0.6 x1 = 4 fmax = 5 21 u2 = 2.1 x2 = 5 fmin = 3 7 u3 = 0.7 x3 = 4 fmaj = 4 c = 2, d = 0 c = 2, d = 0 x4 = 4 fmed not defined due to even number of indexes 4 u4 = 0.4 x5 = 3 fw = [4.1] = 4, w=(0.3,0.3,0.1,0.1,0.1,0.1) 6 u5 = 0.6 x6 = 3 now, consider the second component of sustainable development––the motivation of active agents (university’s employees). as a quantitative measure of motivation choose the system concordance indexes 0 max 0 jjsci  , (1) where j0 is the principal’s maximal guaranteed payoff for different informational rules of the games and max 0j is the global maximum of this value. then the system is concordant if 0sci . the condition 0sci indicates that the system is almost concordant. assume a department, a faculty (an institute) and a university have a tree-like hierarchical structure in which the principal’s role is played by its head, dean (director) and rector, respectively and agents are their subordinates. in formula (1) the value j0 is calculated using different pairs of classification attributes (openand closed-loop strategies, strategies without/with agents’ control feedback, compulsion/impulsion), which gives eight different system concordance indexes. the efficiency of different sustainable management methods is analyzed using the values of these indexes. 72 m. kh. malsagov copyright ©2019 assa adv. in systems science and appl. (2019) 4. sustainable management problems for universities and solution methods dynamic incentive model with homeostasis requirements. this model has the form      t ni i t dttxtustxtuhej 0 0 max]))(),(())(),(([ , (2) ;,0)( nisi  (3)    t ii t i dttxtuhtxtusej 0 max))](),(())(),(([ , (4) ;,1)(0 nitui  (5) 0)0(,))(),(( xxtutxfx ni i     . (6) the model describes relations in the “principal–agent” tree-like control system. the principal can be the manager of a research project while the agents the executors. the notations are the following: n as the agents set; h , 0)0,0( h , as an increasing and convex function that characterizes the principal’s income; ih , 0)0,0( ih , as an increasing and convex function that characterizes the cost of agent i; iu as the labor effort of agent i (e.g., his/her working time within the scope of the project); is as the compensation of the labor effort of agent i paid by the principal (an incentive mechanism representing a nonnegative function); x as the state variable (e.g., the papers published within the scope of the project); ),...,( 1 nuuu  as the labor effort vector of all agents (all these variables are considered as time-dependent functions); ]1,0[ as the discount factor; ijj ,0 as the payoff functionals of the principal and agents, respectively; finally, t as a finite horizon. the static theory of incentive mechanisms was introduced in [7]. resting on the same considerations, for model (2)–(6) the optimal incentive mechanism is expected to have the form * * * ( ( ), ( ), ( )) if ( ) ( ), ( ( ), ( )) 0 ( ), i i i i i i h u t u t x t u t u t s u t x t otherwise i n       (7) where 1 1 1( ,..., , ,..., ),i i i nu u u u u   and      t ni i t tu dttxtuhtxtuheargtu 0 )( ]))(),(())(),(([max)(*  . (8) the first result in this field of investigations was obtained in the paper [16]; more specifically, the above hypothesis was confirmed for the dynamic discrete-time infinite-horizon version of the sustainable management of universities 73 copyright ©2019 assa adv. in systems science and appl. (2019) “principal–agent” incentive problem with the principal’s and agent’s payoff functionals defined in terms of expectations. now, supplement model (2)–(6) with the homeostasis requirement *)( xtxt  (9) and also introduce the sets * { ( ) : ( ) *}and *smu u t t x t x u u ne     , where by assumption  smuu ,* , and ne denotes the set of nash equilibria in the differential normal-form game (4)–(6) of the agents. consider the incentive mechanism ( ( ), ( ), ( )) if ( ) ( ), ( ) , ( ( ), ( )) 0 ( ). sm sm sm sm sm i i i i i i h u t u t x t u t u t u t u s u t x t otherwise i n        (10) then the problem is to construct the set smu and compare the incentive mechanisms (7) and (10). besides traditional methods, the set smu can be constructed (and the optimal control problem (8) can be solved) using the method of qualitatively representative scenarios from simulation modeling [13]. dynamic working time control model with compulsion. this model has the form max))](())(()([ 0 00    dttqdtxctsej t t , (11) ;,1)(0 nitqi  (12)    t iii t i dttxctstupej 0 max))](()())(1([ , (13) ;,1)()( nitutq ii  (14)    n i ii xxtxtukx 1 0)0(),()(  , (15) subject to the homeostasis requirement (9). like model (2)–(6), this model describes the relations in the tree-like “principal–agent” control system. the agent’s control variable iu is, e.g., the share of working time spent on a research project. similar to model (2)–(6), the state-space variable x can be interpreted as the papers published or another index of ra. the dynamic equation (15) is assumed to be linear for the sake of simplicity; in this equation, ik denotes the agent’s working time-to-paper conversion coefficient and  the “depreciation” factor (less papers published without any labor efforts of the agents). the specifics of model (11)–(15) are fully determined by the integrand of formula (13). by assumption each agent distributes his/her working time between a research project (the share iu ) and other types of activity (the share ( iu1 )). participating in other types of activity, each agent obtains additional income described by an increasing and concave function ip . the aggregate participation of all agents in a research project produces an income )(xc (an increasing and 74 m. kh. malsagov copyright ©2019 assa adv. in systems science and appl. (2019) concave function), which is allocated among them by the principal using given time-varying coefficients nitsi ,...,1,0),(  . such models are called social and private interests coordination engines models (spice-models) [4]. the principal’s control variable iq restricts the agent’s “selfishness” from below, which corresponds to compulsion [8]. in this case, the principal’s administrative control cost )(qd (an increasing and convex function such that 0)0( d ) have to be considered for avoiding the trivial solution 1iq , i =1,...,n. model (9), (11)–(15) can be analyzed for different informational rules. fixing the values )(tqi leads to the differential normal-form game (13)–(15) of the agents with the state-space constraints (9), with nash equilibrium as a natural solution. in the hierarchical setup (11)–(15), (9), the solution is a stackelberg equilibrium in which the principal uses open-loop strategies )(tqi , closed-loop strategies ))(,( txtqi , without any control feedback or with the control feedback ))(,( tutqi . the efficiency of these control methods can be compared using the system concordance indexes (1). moreover, from the differential normal-form game (13)–(15) it is possible to pass to the corresponding differential game in the characteristic function form, for further analysis of optimality principles to distribute the grand coalition’s payoff among the players as well as their time consistency [15]. dynamic innovations promotion model with impulsion. this model has the form max))(()( 0 00    dttxctsej t t , (16) ;,...,1,0,1)(0,1)( 0 nitsts i n i i   (17)    t iii t i dttxctstupej 0 max))](()())(1([ , (18) ;,...,1,1)(0 nitui  (19)    n i ii xxtxtukx 1 0)0(),()(  , (20) subject to the homeostasis requirements (9). like model (11)–(15), this is a dynamic spice-model but with impulsion used instead of compulsion. the principal chooses the control variables )(tsi , i.e., the total income shares allocated to different agents, and report them to the latter. this model implies no compulsion; hence, the control cost is 0 and the agents’ control variables )(tui (the shares of working time or another resource spent on innovations promotion at a university) are not bounded from below. the informational rules are the same as in the previous model. dual discrete programming problems for faculty staff optimization. let the staff structure of some faculty j be described by a collection  m jaj m jdj m jpjadp nnnnnnn 111 }{,}{,}{},,{  , sustainable management of universities 75 copyright ©2019 assa adv. in systems science and appl. (2019) where adp nnn ,, are the sets of professors, associate professors and senior lecturers (including assistants), respectively; m denotes the number of departments at faculty j . then the following dual discrete programming problems arise naturally:                  znnn m j k i k i k i ijijijijijij adp zyx ,, 1 1 1 1 max 1 2 3  , (21) rnsnsns aaddpp  ; (22)   znnn aaddpp adp nsnsns ,, min , (23) .,...,1;2,1,;4,...,1,;6,...,1, *** mjizziyyixx iijiijiij  (24) here the additional condition is if ( , , ) , ( , , ) { , , } ( 1,..., ); h h i pj dj aj i ij i pj dj aj ij ij ij ijl i l w n n n n w f n n n w x y z w otherwise i k       , (25) which determines the pairs of problems (21)–(22) and (23)–(24) as dual. the other notations are the following: ijijij zyx ,, as the indexes of ra, tma and iea (table 1); *** ,, iii zyx as their homeostatic values; ijijij  ,, as the corresponding weight coefficients; i as index number; 321 ,, kkk as the number of indexes in each group; adp sss ,, as the wages of the corresponding categories of lecturers; r as the total wage fund of the faculty; ,h l i iw w )( l i h i ww  as the high and low values of the indexes },,{ ijijij zyx ; h in as the set of the values ),,( adp nnn under which h iw is reached; },,{ 321 kkkkl  ; finally, },,{|,| adpznn zz  . in particular, this setup can be used for assessing the efficiency of different paired unions of faculty’s departments, a topical problem that arises in the conditions of teaching staff optimization. to this end, just consider 2 mc potential unions of departments k and mlkl ,...,1,,  , that lead to the teaching staffs alakdldkplpk nnnnnn  ,, , and solve problems (21)–(22), (23)– (24) subject to (25). note that the major difficulty consists in the identification of the function if in formula (25), which requires subjective considerations and expert appraisals. after that, the problems can be solved by standard dynamic programming methods [14] and computer simulations. reward distribution problem for participants of research project as static game in characteristic function form. the main task of the theory of cooperative games (games in the characteristic function form) is to distribute the grand coalition’s payoff among separate players. hence, this mathematical model seems natural for studying different reward distributions among the participants of a research project. denote by n the set of research project executors and write  jiadp nnnnnn , , where adp nnn ,, are its nonintersecting subsets of professors, associate professors, and senior lecturers (including assistants). assume each executor mni can independently publish imi aa  papers in regular journals indexed by web 76 m. kh. malsagov copyright ©2019 assa adv. in systems science and appl. (2019) of science/scopus and also imi bb  papers in regular journals listed by the state commission for academic degrees and titles of the russian federation, where },,{ adpm . then the executor’s characteristic function can be defined as nicbcaiv ii  ,0,)( . (26) this leads to the following analysis problems: completely define the superadditive characteristic function rv n 2: in different possible ways nknnnnnfkv kkk a k d k p  |,|),,,()( ; for the resulting game in the characteristic function form, find the solutions using different optimality principles (the core, the neumann–morgenstern solution, the shapley value, the nucleolus, etc. [15]); compare the results obtained for different values of the parameters imim banvc ,),(, ; verify the homeostasis requirements **,     ni ni ii bbaa . 5. models of anti-corruption drive in universities the author’s concept of corruption modeling was described in detail in the monograph [3]. as an illustrative example of struggling against administrative corruption consider a simple corruption model of an examination described an extensive-form game [15]. this game involves lecturer and student. in the course of an examination, lecturer is finding out the student’s qualification: whether he/she deserves positive assessment (h––high qualification) or not (l–– low qualification). if student has high qualification, then lecturer chooses between two strategies: e––extortion (wringing money out of student under threat of negative assessment) and h––honesty (giving the well-deserved positive assessment). if student has low qualification, then lecturer also chooses between two strategies: c–– capture (offering positive assessment for money) and h––honesty (sending to a resit). if lecturer chooses honesty, then the game ends; in this case, the lecturer’s payoff is 0 while the student’s payoff is either 5 (high qualification) or -2 (low qualification). if lecturer suggests any of the corruption schemes, then student chooses between two strategies: b––agreeing with a bribe and a––making an appeal. assume that, in case of appeal, lecturer is fired out while student receives the well-deserved assessment minus some cost. the resulting payoffs and the complete graph of this game are shown in fig. 1. this game will be solved by the backward induction method [15]. obviously, with the above payoffs, lecturer prefers honesty in the case of high qualification of student and corruption scheme in the case of low qualification (and student prefers to agree). instead of the extensive-form game, this conflict can be considered from the lecturer’s view under the assumption that he/she may assess the conditional probabilities of student’s choice (bribe or appeal) depending on qualification. for the same parameter values as before (see fig. 1), this approach leads to the following lecturer’s payoffs: under extortion, je = 10pb|h 100pa|h = 10(pb|h 10pa|h); under capture, jc = 10pb|l 100pa|l = 10(pb|l 10pa|l), where р with appropriate subscripts denotes conditional probabilities. sustainable management of universities 77 copyright ©2019 assa adv. in systems science and appl. (2019) in both cases, corruption is not beneficial for lecturer (homeostasis) if pb < 10pa. in the general case, the homeostasis condition takes the form wbpb < wapa , where wa and wb are the lecturer’s payoffs under appeal and bribe, respectively. the probability distributions can be estimated on the students set n using empirical data. as an illustrative example of struggling against economic corruption consider a dynamic model of research funding. introduce the following notations: },...,1{ nn  as the set of competing participants (authors of research projects) who prefer bribery; },...,1{ mm  as the set of grant competitions. any project ni can participate in any grant competition mj . the funding of each grant competition mj can be used to support zk j  projects with a fixed grant jg . n h l lecturer lecturer e h c h student 0 student 0 b a 5 b a -2 10 -100 10 -100 3 4 1 -3 fig. 1. corruption model during examination as extensive-form game at the beginning of each planning year tt ,...,1 , 1) the participants ni choose their strategies t ijb (kickback shares) ;,...,1,,,0,10 1 ttmjnibb t ij m j t ij   (27) let ;,...21 mjbbb t nj t j t j  2) integers mjkk t j t j  ],,0[ , are chosen by a random uniform draw; the participants t jk,...,2,1 receive their grants; the residue grants t j t j kk  are distributed among ni and are not considered at subsequent steps (years). the payoff functionals of the participants have the form       t t m j t ij t ij t i rbej 1 1 max)1( , (28) where , 0 . jt ij g if participant i receives grant in competition j r otherwise     78 m. kh. malsagov copyright ©2019 assa adv. in systems science and appl. (2019) the values of the ra indexes 41 xx  (see table 1) are calculated from the dynamic difference equation .1,...,1,0;;4,3,2,1,,)1( 0 0 1 1                ttnikxxrbfxx kiki m j t ij t ijk t ki t ki (29) the homeostasis conditions have the form .,...1,0;;4,3,2,1,* ttnikxx k t ki  (30) in the basic setup, model (27)–(29) can be studied by the method of qualitatively representative scenarios t t m j t j t t m j n i t ij kb 11111 }{,}{  , with a proper verification of conditions (30). in addition, a game-theoretic setup with a simultaneous choice of a strategy t ijb by each agent, or a hierarchical setup with a strategy t jk chosen by the principal are also possible. 6. conclusions in this paper, a systems approach to the sustainable management of universities has been introduced. several setups of sustainable management problems for universities have been considered in which the sustainable development of universities is treated as the simultaneous satisfaction of the homeostasis and motivation conditions. a university as an active system has been analyzed, and some sustainable development criteria have been suggested and formally described. a series of game-theoretic sustainable management models at the levels of department, faculty, and university have been presented as follows: a dynamic incentive model for the participants of a research project; a static reward distribution model; a dynamic working time distribution model for department staff; an optimal staff structure model of a faculty; an innovations promotion model considering the social and private interests of university staff; finally, dynamic models of anti-corruption drive. as a basic tool for solving the dynamic gametheoretic models the method of qualitatively representative scenarios of simulation modeling [13] has been suggested. further research will be focused on a detailed study of these models and their testing for real universities. references 1. basar, t. & olsder, g.y. (1999). dynamic non-cooperative game theory. siam. 2. clark, b.r. (1998). creating entrepreneurial universities: organizational pathways of transformation, in issues in higher education. paris: iau press, pergamon, elsevier science. 3. gorbaneva, o.i., ougolnitsky, g.a., & usov, a.b. (2016). modeling of corruption in hierarchical organizations. new york: nova science publishers. 4. gorbaneva, o.i. & ougolnitsky, g.a. (2018). static models of coordination of social and private interests in resource allocation, automation and remote control, 79(7), 1319–1341. 5. miettinen, k. (1999). nonlinear multiobjective optimization. springer. 6. novikov, d.a. (2009). vvedenie v teoriyu upravleniya obrazovatel’nymi sistemami [introduction to theory of control of educational systems]. moscow, russia: egves [in russian]. 7. novikov, d. (2013). theory of control in organizations. new york: nova science publishers. 8. ougolnitsky, g. 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(2018). computer simulations as a solution method for differential games, in computer simulations: advances in research and applications, eds. m.d. pfeffer and e. bachmaier. new york: nova science publishers, 63–106. 14. parker, r.g. & rardin, r.l. (1988). discrete optimization. academic press. 15. petrosyan, l.a. & zenkevich, n.a. (1996). game theory. singapore: world scientific publishing. 16. rokhlin, d.b. & ougolnitsky, g.a. (2018). stackelberg equilibrium in a dynamic stimulation model with complete information, automation and remote control, 79(4), 691– 702. 17. slaughter, s. & leslie, l.l. (1997). academic capitalism. politics, policies, and the entrepreneurial university. baltimore and l.: the johns hopkins university press. 18. ugol’nitskii, g.a. & usov, a.b. (2014). equilibria in models of hierarchically organized dynamic systems with regard to sustainable development conditions, automation and remote control, 75(6), 1055–1068. adv syst sci appl 2019; 2:23–32 http://ijassa.ipu.ru/index.php/ijassa/article/view/725 machine learning algorithms for automatic anomalies detection in data storage systems operation mikhail hushchyn1,2∗, andrey sapronov1,3, andrey ustyuzhanin1,2 1national research university higher school of economics, myasnitskaya 20, 101000, moscow, russia 2moscow institute of physics and technology, institutskiy per. 9, dolgoprudny, moscow reg., 141700, russia 3joint institute for nuclear research, joliot-curie 6, dubna, moscow region, 141980, russia received april 9, 2019; revised june 6, 2019; published july 10, 2019 abstract: data storage reliability and availability play important role for a wide range of services and business processes. manufacturers provide data storage systems that resistant to hardware and software failures but not for all cases. well-timed detection of these failures helps to recover the system faster and prevent the failures before they occur. in this work a range of machine learning and time series analysis algorithms for failures detection for data storage systems is considered. the algorithms are applied and compared on the real system. preliminary results show that binary classification methods demonstrate high failure detection and low false alarm rates. time series prediction based approach shows similar results and outperforms one-class classification methods. keywords: machine learning, time series analysis, anomaly detection, data storage systems 1. introduction modern data storage systems are built from hundreds of solid-state drives (ssds) and hard disk drives (hdds), networks and controllers to handle and process incoming data. a lot of data generated every day has to be properly saved for further usage. storage systems have to provide high availability and reliability. any failure in these systems will negatively affect any processes and applications that use the data and lead to significant revenue loss. any anomalous system behaviour can potentially be the result of a failure of its components and data loss. it requires from the storage system operators to explore it and, if necessary, take actions to resolve the problem. timely anomaly detection allows to react on system failures faster decreasing shutdown time and preventing data loss. anomalies can be the result of the system failures that have already happened or will happen in the near future. in the first case anomalies detection allows to reduce time delay for corrective actions. in the second case it gives time to take actions to prevent the failures and to reduce undesirable effects. the paper has the following structure. overview of related works is considered in sec. 2. data storage system that is used in this work for anomalies detection is described in sec. 3. discussion of different anomalies detection algorithms with examples and results is provided in sec. 4. finally, conclusions of this work are presented in sec. 5. ∗corresponding author: hushchyn.mikhail@gmail.com 24 2. related works there are a variety of anomaly detection methods [1, 2] applied in different domains of systems. these methods are the most widely used in computer networks to detect cyber attacks [3–7] and web-traffic anomalies [8–10]. there are a set of works where system logs of online service systems [11], supercomputers and distributed systems [12, 13] are analysed to detect various failures and anomalies. several approaches were presented to predict failure of hard drives [14, 15] based on their smart (self-monitoring, analysis and reporting technology) data. interesting results were demonstrated at the european organization for nuclear research (cern) where anomaly detection methods are used in data quality system for monitoring quality of data generated in high energy physics detectors [16]. the most promising methods of failures detection are based on machine learning and time series analysis algorithms. these approaches use clustering, one-class and binary classification algorithms for anomalies detection. clustering methods [1, 2] suppose that normal and anomalous system behaviours form different clusters in some parameter space. the methods learn boundaries of these clusters and use them to distinguish different system states. this approach works well when the clusters are separable from each other. there are variety of clustering algorithms [17]: k-means, hierarchical clustering, gaussian mixtures, dbscan and others. similarly, one-class methods learn system behaviour in the normal state. they suppose that normal states form dense clusters in system parameter space and recognize boundaries of these clusters. everything that is beyond the boundaries is considered as an anomaly. the most used algorithms are isolation forest [18], elliptical envelope [19], one-class svm [20], asvdd [21] and wsvdd-cba [22]. binary classification [1, 2] is used when anomalies are known. it takes normal behaviour and anomalies as two classes and learns separation rule between them in parameters space. then this rule is used to classify the system states. the most popular classifiers are naive bayes, logistic regression, svm, decision tree, random forest, gradient boosting over decision trees and artificial neural networks that are described in [17]. time series analysis based methods [1, 2] are based on prediction of system parameters values in time. they use previous values of the parameters to predict the current one. large deviations from the predictions indicate anomalies. there are a lot of predictive models for time series [23]: ets models, autoregression (ar) and moving average (ma) models, arma, arima, artificial neural networks and others. in this work several approaches of anomaly detection for data storage systems are considered. 3. tatlin storage description the goal of this work is to develop the algorithm of automatic failure detection for tatlin [24] storage system. the system contains up to 4 storage controllers, peripheral component interconnect express (pcie) fabric controller and up to 16 drive enclosures as it is shown in fig. 3.1. storage controllers are based on yadro vesnin [25] hardware platform. they provide user access to data, perform computations and run the tatlin software. each storage controller consists of 4 power8 turismo scm processors, 256 gb ram, pcie fabric connection adapter for data transfer and 2 gigabit ethernet (gbe) switches for internal network. fabric controller integrates all components of the storage system and provides high data operation performance using fast ssds. it has up to 96 ssds with capacity of 2 tb each, 2 tb ram cache, shared among storage controllers, 4 pcie fabric adapters, gigabit ethernet switch for internal network and drive enclosure connection module. drive enclosures host up to 96 serial attached scsi (sas) disks with capacity of 12 tb each to provide storage space for data that does not require high speed of operations, compared with ssd. the storage reliability is provided by policies based on reed-solomon codes with minimal redundancy. it uses 25% redundancy that supports simultaneous failure of 2 drivers in 8 data copyright c© 2019 assa. adv syst sci appl (2019) 25 fig. 3.1. tatlin storage system. + 2 parity drives configuration. the fabric controllers provide shared access to the entire storage space for all storage controllers. shared ram cache gives hardware improvement by reducing cpu load in storage controllers for intensive data operations. the storage design excludes one point failure. 4. anomalies detection 4.1. collected data behaviour of each component of the storage system is described by a set of parameters, that are used for anomalies detection. examples of these parameters are cpus and ram usage, cpus temperatures and fan speed for storage controllers, input and output traffics for network interfaces, smart data for ssds and sas drives and others. except hardware components information about logical data volumes inside storage pools is collected. each volume is described by storage pool id, capacity, state, read and write speed for each storage controller, number of read and write operations, average request processing time, average request queue size etc.. several of known hardware failures for the all components are generated to test algorithms of anomalies detection. parameters are measured approximately every 20 seconds and saved for the further analysis. for each measurement true state is recorded. there are two possible states: normal and failure. measurements of a component parameter are represented as a time series {(ti, xi)}ni=1, where xi is a measured parameter value and ti is a timestamp of the measurement. each time series is divided into time bins with defined width. all measurements inside one bin are averaged. this aggregation helps to reduce noise of the measurements and to create timeregular observations of the parameters for further steps of the analysis. an example of time series after aggregation with width = 60s is demonstrated in fig. 4.2. copyright c© 2019 assa. adv syst sci appl (2019) 26 anomaly anomaly anomalyanomaly fig. 4.2. (top) time series of percentage of cpu usage for one of the storage components of the storage system after aggregation. (middle) residual errors for the time series prediction. (bottom) cusum test values for the anomalies detection. 4.2. time series analysis consider time series analysis methods to detect anomalies. time series analysis is used to predict parameter values of a component. to do this vector autoregression (var) model [23] is used. the principle of the model is following. suppose a group of m time series is given: {yi1, yi2, ..., yit}mi=1, where yit is t-th measurement of i-th time series. measurements of all time series with the same t are grouped into a vector yt = (y1t y2t · · · ymt) t . var model supposes that a vector of predicted parameters values ŷt is a function of previous k measurements: ŷt = f(yt−1, yt−2, ..., yt−k) (4.1) where ŷt is a predicted vector of parameters values; yt is a vector of measured values of the parameters. for simplicity each time series has its own prediction model: ŷit = fi(yt−1, yt−2, ..., yt−k) (4.2) where ŷit is a predicted value for i-th time series. to approximate the functions fi different machine learning algorithms [17] for the regression problem are used: linear regression model, shallow neural networks, random forest and gradient boosting over decision trees regressions. these algorithms are selected because they demonstrate high quality of the prediction and work well with multivariate time series. the models are trained with the following loss function: l = ∑ t (ŷit − yit)2 (4.3) copyright c© 2019 assa. adv syst sci appl (2019) 27 parameters of the models are optimized using grid search. the model with the lowest loss function value on the validation sample wins and used for anomalies detection. for each time series residual errors rit are calculated: rit = ŷit − yit (4.4) where rit a residual error of a predicted value for i-th time series on t-th measurement. example of a time series with two anomalies is shown in fig. 4.2. it corresponds to the percentage of cpu usage for one of the storage controllers. the time series is aggregated as it is described in sec. 4.1. to predict the current value ŷit the previous 30 observations of this time series are used in this example. there are about 2500 observations that correspond to about 150000 seconds. the first 60000 seconds are used to train var model and the next 25000 seconds are used for the validation. residual errors of the prediction are shown in fig. 4.2. the figure shows larger residuals for the anomalies. to detect them the cumulative sum (cusum) [26] test is used. the test calculates new time series {tt}nt=1, where the first value t1 = 0 and the next values are estimated by the recurrent formula using residual errors: tt = max(0, tt−1 + εt) (4.5) εt = log f0(rit) f∞(rit) (4.6) where f∞(rit) and f0(rit) are residual distributions for normal and anomalous states respectively. normal distributions for the residuals are considered: f∞(rit) = 1√ 2πσ2 ∞ e − (rit−r∞)2 2σ2∞ (4.7) f0(rit) = 1√ 2πσ2 0 e − (rit−r0) 2 2σ20 (4.8) where r∞, σ∞ are mean and standard deviation of residuals for normal states; r0, σ0 are mean and standard deviation of residuals for anomalous states; residuals are not necessary have normal distributions. however, it was shown in [27] that even with strong deviations from the normal distribution cusum works well. an anomaly is detected when the tt > talarm. the distribution parameters and talarm are estimated during calibration on the validation sample without anomalies, where tt has to take values close to 0 without growing trend. this approach allows to detect mean and variance changes of the residuals. an example of the cusum test is demonstrated in fig. 4.2. the test value increases for anomalies and falls when the anomaly disappears. to avoid the test values relaxation after the anomaly tt values can be reset right after the anomaly detected or by operator’s demand. 4.3. one-class classification one-class classification algorithms are used for anomalies and novelty detection in data. these algorithms learn boundaries of a normal class and everything that outside that boundary is considered as an anomaly. isolation forest [18] is one of the best one-class algorithms. suppose a sample with n objects is given. each object has d parameters. the idea of the isolation forest in the following: step 1 select a subsample with 0 < n < n objects to build a new isolation tree. step 2 randomly select a parameter. step 3 randomly select an object from the current tree node. use the selected parameter value of this object as a splitting rule for the node. copyright c© 2019 assa. adv syst sci appl (2019) 28 step 4 repeat steps 3 and 4 for the left and right children of the node. stop the process when the termination conditions are satisfied. step 5 repeat steps 1-4 to build forest of isolation trees. according to [18], anomalies have shorter decision paths in isolation trees than normal objects. for an object x anomaly score ŝ is estimated as: s(x) = 2− e(h(x)) c(n) (4.9) where h(x) is a length of s decision path for the object in an isolation tree; e(h(x)) is an average length of the decision path for the object in the forest; c(n) is an average length of the decision path for all n objects in the forest. the score takes values in [0, 1]. objects with the score close to 1 are anomalous. on the other hand, objects with small score are normal. in case, when for all objects in a sample s(x) ≈ 0.5, the sample has no anomalies. similar to the time series analysis approach, isolation forest uses vectors yt of measured parameters values to estimate the anomaly score ŝit for a measurement: ŝit = fi(yt, yt−1, ..., yt−k) (4.10) an example of working of isolation forest is demonstrated in fig. 4.3. the first 60000 seconds are used to train a classifier. the figure shows a slightly higher anomaly score for anomalies. all measurements with ŝit ≥ 0.5 can be considered as anomalies. results can be improved using cusum test similar to var models and will be considered further. 4.4. binary classification isolation forest binary classification fig. 4.3. (top) time series of percentage of cpu usage for one of the storage components of the storage system after aggregation. (middle) anomaly score predicted by isolation forest. (bottom) anomaly score predicted by binary classifier. binary classification is a powerful tool to detect known anomalies. normal and anomalous measurements are considered as two classes. the classifier learns the separation surface copyright c© 2019 assa. adv syst sci appl (2019) 29 between them and is using it for anomalies detection for new measurements. similarly to the var models, several binary classifiers [17] are considered: logistic regression, naive bayes classifier, shallow neural networks, random forest and gradient boosting over decision trees classifiers. they are used to predict anomaly scores: ŝit = fi(yt, yt−1, ..., yt−k) (4.11) where ŝit ∈ [0, 1] is a predicted anomaly score. parameters of the classifiers are optimized using grid search to minimize the logistic loss function: l = − ∑ t (sit log ŝit + (1− sit) log (1− ŝit)) (4.12) where sit ∈ {0, 1} is a true anomaly score for t-th measurement of i-th time series; the classifier with the lowest value of the loss function wins and is used for the anomalies detection. the example of the binary classification work is shown in fig. 4.3. the first 110000 seconds are used to train the classifiers. 4.5. models comparison 0.00 0.01 0.02 0.03 0.04 0.05 false alarm rate 0.95 0.96 0.97 0.98 0.99 1.00 fa ilu re d et ec tio n ra te binary classification var (a) 0.0 0.1 0.2 0.3 0.4 0.5 false alarm rate 0.5 0.6 0.7 0.8 0.9 1.0 fa ilu re d et ec tio n ra te isolation forest + cusum isolation forest (b) fig. 4.4. dependencies of failure detection rate from false alarm rate for different anomaly detection algorithms: (a) binary classification and var; (b) isolation forest and isolation forest with cusum methods. we collected data with induced hardware failures in different storage system components. a failure is considered as detected when at least one parameter of the system demonstrates anomalous behaviour. time series for all parameters corresponded to one component are united into a group and used to estimate anomaly scores as described in previous sections. a measurement is anomalous when its anomaly score ŝt > τ , where τ is a threshold value. for a set of τ values failure detection rate (fdr) and false alarm rate (far) are calculated to measure the failures detection quality: fdr(τ) = 1 nanomalies ∑ i i[ŝi ≥ τ |si = 1] (4.13) far(τ) = 1 nnormal ∑ i i[ŝi ≥ τ |si = 0] (4.14) wherenanomalies in the total number of anomalous measurements;nnormal is the total number of normal measurements. dependencies of failure detection rate from false alarm rate copyright c© 2019 assa. adv syst sci appl (2019) 30 roc auc fdr (far = 5%) fdr (far = 1%) binary classification 0.999 0.997 0.991 var 0.997 0.987 0.975 isolation forest 0.755 0.0 0.0 isolation forest + cusum 0.956 0.852 0.836 table 4.1. quality metrics for the anomalies detection algorithms. for different anomaly detection algorithms are shown in fig. 4.4. these dependencies are also known as roc-curves. area under the roc-curve (roc auc) is widely used in machine learning to measure quality of classification. roc aucs, failure detection rates that correspond to false alarm rate values of 5% and 1% for different anomaly detection algorithms are presented in tab. 4.1. the results demonstrate that the binary classification method has the best quality. it detects larger than 99% of all anomalous states with false alarm rate below 5%. these results can be explained by the learning procedure of the method. during this procedure it recognizes separation surface between normal and anomalous states in the system parameters space. then it uses this surface to distinguish the system states on new measurements. var model shows similar results. it recognizes about 98% of all anomalous states with false alarm rate below 5%. however, unlike the binary classification it does not use any information about anomalies. the model takes only the system parameters measurements for normal states to learn normal behaviour of the system. any significant deviation from this behaviour considered as anomaly. this allows var to detect unknown anomalies. isolation forest demonstrates high false alarm rates. due to the noise in the parameters measurements isolation forest detects some normal measurements as anomalous and generates false alarms. it does not use any information about anomalies during the training procedure, that makes it harder to detect anomalies compared with binary classification. cusum test helps to reduce a number of false alarms and improves quality of the anomalies detection in a small far region as it is shown in fig. 4.4. 5. conclusion several approaches for automatic anomalies detection for data storage systems were tested and compared. these approaches are based on binary and one-class classification algorithms as well as on var models of time series analysis. the binary classification method has demonstrated the best detection quality. it allows to detect about 99% of anomalous states with false alarm rate of about 1%. the main limitation of this method is that it requires information about anomalies to learn how to detect them. this provides the high detection quality but makes it impossible to use the method for detection previously unseen anomalies. to resolve this limitation two additional methods were considered. these methods do not use information about anomalies during their training procedures. they are able to detect previously unseen anomalies. isolation forest shows very small failures detection rate in a region of false alarm rate below 5%. however, var model of time series analysis in combination with cusum test demonstrates promising results. it detects about 97% of anomalous states with false alarm rate is about 1%. the method learns normal behaviour of the system and detects any changes of it. in our future works we are going to continue developing methods of anomaly detection for data storage systems based on time series analysis methods. the goal is to investigate the methods that are able to recognize any anomalies without training procedure that uses information about these anomalies. copyright c© 2019 assa. adv syst sci appl (2019) 31 6. acknowledgements the research was carried out with the financial support of the ministry of science and higher education of russian federation within the framework of the federal target program research and development in priority areas of the development of the scientific and technological complex of russia for 2014-2020. unique identifier rfmefi58117x0023, agreement 14.581.21.0023 on 03.10.2017. 7. reference references 1. a. patcha and j.-m. park, “an overview of anomaly detection techniques: existing solutions and latest technological trends,” computer networks, vol. 51, no. 12, pp. 3448 – 3470, 2007. 2. v. chandola, a. banerjee, and v. kumar, “anomaly detection: a survey,” acm comput. surv., vol. 41, pp. 15:1–15:58, july 2009. 3. m. aiello, m. mongelli, e. cambiaso, and g. papaleo, “profiling dns 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http://ijassa.ipu.ru/index.php/ijassa/article/view/558 multiagent model of people evacuation from premises while emergency andrey samartsev1, vladimir ivaschenko1, alexander rezchikov*1, vadim kushnikov1 , leonid filimonyuk1, aleksey bogomolov1 1) institute of precision mechanics and control of ras, saratov, russia e-mail: iptmuran@san.ru received may 17, 2018; revised march 13, 2019; published april 15, 2019 abstract: the multi-agent model of emergency evacuation process in case of limited space conditions is offered. a distinctive feature of the proposed model is the systematic consideration of physical interactions between agents and their reasonable behavior in conditions of evacuation from premises of a complex configuration. the model allows to investigate the state and behavior of people in emergency and determine the time of their evacuation. the proposed model can be used as a basis for determining ways and developing plans for evacuation from premises in emergency situations. keywords: multiagent model, intelligent agent, evacuation of people, premises, emergencies. 1. introduction currently, the problem of combating emergencies is becoming especially urgent. the statistics for 2016 presented by the ministry of emergency situations of russia [37] indicate significant human and material losses caused by emergencies. emergency evacuation of people from the premises as well as pre-emptive solutions [10, 26, 29-32, 40-45] are the most effective ways to reduce damage in case of accidents, catastrophes, fires and terrorist acts. in many situations, evacuation occurs spontaneously. this, among other things, is largely due to the inadequate development of existing organizational, technical and software tools for managing the evacuation process, the need to develop adequate models and algorithms to support decision making to ensure the effective evacuation of people in emergencies. in cases of spontaneous people evacuation from the premises, each person independently chooses a route and, on the way, makes decisions that they consider beneficial for achieving their goals. at the same time people decision making is influenced by the state and behavior of other people, the environment (walls and other obstacles), as well as such factors as panic [7, 8, 25], the failure of warning systems, unfamiliarity with the evacuation plan, the instinct of self-preservation, etc. therefore, the research of spontaneous evacuation is significant. in addition, the investigation of spontaneous evacuation makes it possible to take into account the behavior of people and identify bottlenecks, such as the maximum intensity of the flow of people through the exits from the premises, the places of their maximum accumulation, the capacity of the premises, etc., to organize the optimal control of the evacuation process [33, 34]. *corresponding author: rezchikov1939@mail.ru mailto:iptmuran@san.ru multiagent model of people evacuation from premises while emergency 99 copyright ©2019 assa. adv. in systems science and appl. (2019) thus, it is very important to ensure quick and unhindered evacuation in case of emergency. the quality of the evacuation process is usually checked by computer simulation, since the natural experiment in this situation can be dangerous and difficult to reproduce. in connection with the above, it is not surprising that the dynamics of the behavior of large masses of people, in particular during evacuation, has long been actively studied. the model of social forces, proposed by helbing and molnar in 1995, is widely used. its modifications have been actively proposed and are being investigated so far [2, 14-17, 24, 48]. in the classical interpretation of this model, people in the crowd are acted upon by forces on the one hand that attract them to the goal, and on the other hand forces that keep them away from each other and from obstacles at a certain distance. in addition, people can be affected by randomly changing force, designed to take into account the deliberate deviation of pedestrians from a given route. in general, the model is considered quite successful, although it is criticized for the fact that the underlying assumptions simplify the behavior of pedestrians. pedestrians in the model tend to adjust the speed of motion in inverse proportion to the distance from obstacles and other pedestrians. in fact, pedestrians can use better escape strategies, for example, "squeeze" between two other pedestrians to find a calmer route. in addition, the model is rather complicated in terms of the computation speed [47]. there are also fluid dynamic models [13, 27, 35, 47, 49]. they are quite accurate when modeling human flows of high (but not extremely high) density in the absence of panic. disadvantages of these models are that they give little information about the behavior of specific people, and only simulate the situation in general. models of this class are often difficult to understand intuitively, in addition, in practice, there are serious differences between the patterns of fluid movement and the panicking crowd. at the moment, the most promising are the agent models [3, 4, 7-9 12, 21, 23, 25, 35, 36, 39, 47, 49], in which the system is defined as a set of agents interacting according to certain rules. such models a priori determine the states of agents, their characteristics, as well as the rules of their interaction and behavior in various situations. agent behavior is understood as a function (or algorithm) that links the information that an agent receives about its local environment and the actions that it takes. interaction of agents in the process of evacuation can consist in avoiding collisions with each other or walls, in physical contact, in the spread of panic sentiments. note that the process of spontaneous evacuation is a vivid example of a decentralized system, when management is not carried out from the outside, and each person acts in accordance with their personal interests, and the global behavior of the system arises as a result of the many evacuees actions. in this case, the global behavior of the system may differ from the behavior of the elements of the system (this phenomenon is called emergence). for example, if each evacuation participant begins to hurry, the overall evacuation rate may be reduced due to a crush at the exit (the effect "faster is slower"). agent models make it possible to take into account the emergence effect to the full [47]. among the agent models that are suitable for describing the evacuation process, there are continuous and discrete ones. discrete models are based on the apparatus of cellular automata [1, 18-20, 35, 38, 47, 49]. such models are conceptually simple and computationally effective. however, in most cases, discrete models are an oversimplification of reality: pedestrians move discretely from their cells to their neighboring cells, the trajectories of their movement are angular and resemble the movements of a chess rook, collisions with each other and with walls are described rather roughly. nevertheless, by using such models, it is possible to explain some aspects of crowd behavior, for example, crushes at the exits and the effect "faster is slower". the most natural and convincing realization of the agent approach are continuous agent models. in such models, the characteristics of agents can take any value on some continuous segment. this approach requires more computational resources than discrete, but it allows to 100 a. samartsev, v. ivaschenko, a.rezchikov, v.kushnikov et al. copyright ©2019 assa. adv. in systems science and appl. (2019) achieve better modeling accuracy. continuous agent models are presented in [4, 6, 21, 22, 40, 46]. the model presented in this article belongs to that class either. the existing works of these classes are not without some drawbacks. some models are too complex in terms of the quantity of parameters [4] or the computational speed, for instance [2, 14-17, 21, 24]. many works do not consider evacuation from premises of complex shape [4, 14-17, 20, 21, 24, 38]. some models do not take into account the physical interaction of agents, their inertia [4, 21]. many models do not take into account the reasonableness of evacuated people, in particular the possibility of overtaking one agent by another [14-17]. a lot of models do not take into account the specifics of the evacuation process, but only model the behavior of "walking people" [21-24]. commercial models often do not reveal many details of the crowd behavior modeling process they use. the purpose of this article is to describe a model without all problems mentioned in previous paragraph, to develop software package based on the model and to simulate the evacuation process with use of the software package. the aim of simulation is to validate model by comparing model simulation results with ones of other models and statistic data, visually observing agents behavior and estimating of “reasonability” of their actions. 2. problem formulation without loss of generality let us consider the premises, the floor plan of which is shown in fig. 1. 1 people, 2 walls, 3 – exit zones fig. 1. premises map and people’s distribution inside it. in the xoy plane parallel to the floor of the premises, the following components are specified: 1. a finite set of walls, each of which is specified by parameters: (xw, yw) – coordinates of the lower-left corner of the wall, xw w and yw w – its width and length. we assume that the walls are perpendicular to the coordinate axes ox и oy, since this is the case in majority of the floor plans. 2. a finite set of exits from the premises specified by the parameters: (xe, ye) – coordinates of the lower left corner of the exit; xw e and yw e – its width and length, respectively. after a person got into one of the exit zones, it can be assumed that they multiagent model of people evacuation from premises while emergency 101 copyright ©2019 assa. adv. in systems science and appl. (2019) successfully evacuated. note here that the exit zone should not be directly in the doorway, but at some distance from it, since after the person leaves the room, they still influences on the evacuation process [5]. 3. a finite set of evacuated people, which we will consider as a set of their circle projections onto the plane xoy. we consider the centers of these circles as coordinates of people. 4. a finite set of evacuation zones, each of which is specified by the parameters: (xz, yz) – coordinates of the lower left corner of the evacuation zone, xw z и yw z – its width and length. at the very beginning of the evacuation people are concentrated within such zones. the presence of evacuation zones allows to take into account the possible absence of people in some rooms and in the space outside the premises at the first moment of time. parameters of people: – radius vector (coordinates) of the center of the circle around the projection of the person on the plane xoy – x(t); – radius of projection – r; – mass of a person – m; – vectors of its velocity – v(t) and acceleration – a(t); – the maximum possible speed – vmax and acceleration – amax. the speed and acceleration of the person at the beginning of the evacuation are assumed to be equal to zero. it is necessary to perform a simulation of the process of spontaneous people evacuation from the premises and to develop activities aimed at managing and speeding up this process. the activities include the possible creation, movement and expansion of certain building openings, adjustment of evacuation plans, and personnel instructions. these events must help to speed up people's movement along evacuation routes and avoid crushing. 3. multi-agent model 3.1. model construction in the proposed model, as in [2, 14-17, 22, 24], the law of agents’ motion is determined by the relations: .)()()( ;)()()( ttattt tttxttx     (1) where δt is model time step. the agents’ velocities change after collisions. the description of their physical interaction with each other and obstacles can be presented in various ways. it is permissible to regard collisions of agents as partially elastic [28]. during partially elastic collision some of the kinetic energy passes into thermal or other forms of energy. to describe the partially elastic collision, a recovery coefficient 0 ≤ ε ≤ 1 is introduced, which determines the nature of the interaction of the colliding bodies. for ε = 1 the collision is absolutely elastic, if ε = 0 it is absolutely inelastic, and for 0 < ε < 1 it is partially elastic. the normal components of the velocity to the shared tangent plane to the surfaces of the colliding bodies at the point of their contact (the collision plane) after impact at a partially elastic collision are calculated by the formulas: .)1( ;)1( 21 2211 22 21 2211 11 mm vmvm vu mm vmvm vu nn nn nn nn         (2) here v1n и v2n are the normal projections of the agent velocities to the collision plane before impact, u1n and u2n are the normal projections of the agent velocities to the collision 102 a. samartsev, v. ivaschenko, a.rezchikov, v.kushnikov et al. copyright ©2019 assa. adv. in systems science and appl. (2019) plane after impact, m1 and m2 are the masses of the colliding agents. tangential projections of the agent velocities to the collision plane after impact do not change [28]. when the agent collides with a wall, the projection of its velocity of motion parallel to the wall does not change, while the other projection reverses its sign and decreases its value according to the coefficient ε. for example, when an agent collides with a wall parallel to the ox axis, its speed is recalculated according to the relations ,; yyxx uu   where v = (vx, vy), u = (ux, uy) is the agent's velocity before the collision with the wall and after, respectively. let the direction of the acceleration vector a(t) of each agent, provided that each of them tends to get to the nearest exit from the room as fast as possible. to do this, we introduce the concept of the optimal vector (from the agent's point of view) of the velocity vopt(t), which approximates the agent to the chosen exit and allows, if possible, not to interfere with obstacles other agents and walls. the actual velocity of the agent v(t) may differ from the optimal one due to its collisions with other agents and walls. then we can assume that the agent moves with the acceleration a(t), whose modulus is determined from the physiological capabilities of the person (|a(t)| = amax), and the direction coincides with vopt(t) – v(t). if vopt(t) = v(t), then the agent moves without acceleration. fig. 2 shows a graphical interpretation of the relationship between agent parameters. here we should pay attention to the fact that the vectors a и vopt – v are collinear. fig. 2. graphical interpretation of connection between agent parameters the optimal speed module vopt(t) is bounded from above by the physiological capabilities of the agent vmax. if there are no obstacles ahead of the agent along the route, for example other agents or walls, then usually |vopt(t)| = vmax. to avoid collision with other agents along the route, the vopt(t) module may decrease. let us consider the approach to the choice of the vector vopt(t) in more detail. let e be a vector specifying the direction to the nearest exit, taking into account the layout of the room. the choice of vector e is determined solely by the location of the agent, walls and exit zones, but not by the location of other agents. thus, it is possible to determine the vector field e for a given premises map. in this case, the calculations of this field during the simulation are carried out only once, which is an advantage from the point of view of the computational speed. the field of the vector e for the premises map under study is shown in fig. 4. let lα – be the distance from the center of the agent to the nearest obstacle (human or wall) in the direction at an angle α to the vector e, l is the specified critical distance, and r is the radius of the person projection on the xoy plane. multiagent model of people evacuation from premises while emergency 103 copyright ©2019 assa. adv. in systems science and appl. (2019) then the modulus of the agent optimal velocity vector vαopt(α) in the direction at an angle α to the vector e can be calculated from the expressions: .,0)( ;,/)()( ;,)( rl rllrlrl rll opt maxopt maxopt          (3) people have different physiological capabilities, therefore it is necessary that each agent has individual values of vmax and аmax. let function        2 ; 2 -);cos()()(   optf (4) then the angle γ between the vectors vopt and e is defined as the angle at which the value of the function f(α) maximal. that is, the direction of the vector vopt of the agent's movement is chosen according to the criterion        2 ; 2 -and ifmax)(  f (5) the value of the vector vopt is determined by substituting α = γ into expression (3): |vopt |=vαopt(γ) (6) the presented way of choosing the value of vopt allows the agent to maneuver between other agents, trying firstly to avoid collision with them as much as possible, and secondly, to approach the nearest exit as quickly as possible. here we are assuming a good familiarity with the floor plan of the premises. thus, the choice of the optimal speed vector vopt takes into account both the layout of the room and the location of other agents around it. the vopt vector must be recalculated for each agent at each step of the model time. note that in the described model there are also attributes used in known models [4, etc.], such as: the angle and range of the agent's view of the surrounding space, the moment of inertia and the angle of rotation of the agent's head: 1. the direction that approximates the agent to the exit (taking into account the presence of walls in the room) is determined by the field of the vector e. the viewing angle of the agent can be considered equal to π, since in expression (5)        2 ; 2   , i.e. the agent analyzes all possible alternatives for movement in the plane in front of him. 2. since the agent calculates the optimal speed values based on the situation analysis within the critical zone, we can assume that the agent's viewing range is not less than l. 3. part of the energy in the impact goes into a rotational motion. in the model, these energy losses are taken into account exclusively by the recovery coefficient (formula (2)). 4. the ability of the agent to rotate the head is accounted for by the large viewing angle mentioned in 1. thus, in the model some parameters are set implicitly, which simplifies the model and increases the computational speed. 3.2 approach to model implementation 3.2.1 calculation optimizations as follows from the model (1) (6), for each agent for each step of model time, it is required to calculate distances to all other agents and walls of the room. proceeding from the calculated values, it is necessary to calculate the maximum possible velocities for a certain set of directions of motion of a given agent by the ratio (3) (16 possible directions are accepted with step π/8), and then by criteria (4) (6), determine the direction and the magnitude of the optimal speed vector, and then the acceleration of the agent. 104 a. samartsev, v. ivaschenko, a.rezchikov, v.kushnikov et al. copyright ©2019 assa. adv. in systems science and appl. (2019) the algorithm (for one step of model time) in this case will have computational complexity o(n2+nm), where n is the number of agents, and m is the number of walls. the quadratic complexity is not acceptable for systems with a large number of agents, so optimization is needed here, aimed to improve the asymptotic of the computational speed. let с = {cij | 0 ≤ i < n, 0 ≤ j < m} is the network that divides the space into cells with side h; n, m – number of cells horizontally and vertically, respectively; cij is a cell with coordinates (i, j), and w ⊂ c is the set of cells to which the wall corresponds. in the software implementation, the room was divided by the network c into cells of width h = 10 cm. this cell size is small enough for having centers of two agents inside at once. the following data structure is being supported during calculations: – each agent has the coordinates of the cell in which he is located; – each cell cij contains a reference to an agent whose center is in the given cell (in the absence of such an agent, the reference is null). each cij cell has the following attributes: – belonging to a wall or floor. – distance to the nearest exit dij; – vector eij, which specifies the direction to the exit. therefore, for a given agent, it is necessary to test a limited set of neighboring cells for the presence of walls and other agents, which makes it possible to improve the computational complexity of the algorithm to o(n). 3.2.2 calculation of the distances scalar field to the exit the attribute of the distance between exit and cij cell dij was calculated using the graph g = (v, e) given in the following way. suppose that for each cell cij  w here exists a vertex of the graph g vij  v, and eijkl  e is the edge of the graph between the vertices vij and vkl. then the set of edges e is defined as follows: e = {eijkl |∀a,b (a ∈ [min(i, k); max(i, k)] & & b ∈ [min(j, l);max(j, l)] → cab ∉ w) & (7) & (i, j) ≠ (k, l) & (i–k) ⊥ (j–l) & |i–k |≤ 2 & |j–l| ≤ 2)}. in addition, for each edge of the graph g, its length is given, which is proportional to the real distance between the cell centers corresponding to the vertices of the edge. the length lijkl of the edge eijkl is defined by the relation 22 )()( ljkilijkl  (8) consider the formula for defining the set of edges in more detail. 1. ∀a,b (a ∈ [min(i, k); max(i, k)] & b ∈ [min(j, l); max(j, l)] → cab  w) means that among cells cij and ckl, and also cells between them in a rectangle with a diagonal connecting cij and ckl, there are no cells corresponding to the walls. 2. (i, j) ≠ (k, l) means that the graph is not reflexive. there is no sense for loops since they can not change the value of the distance from the exit to any vertex. 3. (i–k) ⊥ (j–l) means that numbers (i–k) and (j–l) are coprime, i.e. these numbers have no common divisors except number 1. this relation allows to simplify the graph, since edges for which this relation is not satisfied, can not change the value of the distance from the exit to any vertex. 4. |i–k |≤ 2 & |j–l| ≤ 2 means that the distance vertically and horizontally between the cells is not greater than 2. thus, two vertices are connected by an edge only in cases where the corresponding cells: a) directly touch each other (they are vertical or horizontal neighbors); b) touch each other at one point (they are diagonal neighbors); c) are one cell far from each other by one dimension and two cell far from each other by the other (here an analogy with the l-shaped move of the chess knight can be drawn). multiagent model of people evacuation from premises while emergency 105 copyright ©2019 assa. adv. in systems science and appl. (2019) additionally, the connected cells and the cells that are between them should not correspond to the cells with the wall. fig. 3 shows a fragment of graph g superimposed on the part of the room shown in fig. 1 near the point with coordinates (15, 8). for clarity, in the picture edges, incident to two vertices are highlighted in red. one of these vertices is far from the walls and therefore has 16 incident edges. the second vertex is near the walls, so the number of edges incident to this vertex is less, and, in this case, is 9. fig. 3. a fragment of the graph g near the point with coordinates (15, 8) on a given graph, it is necessary to implement the algorithm for finding the shortest paths for finding the distance from the exit to all other cells. thus, for each vertex vij for which cij w, the distance dij to the nearest exit is determined. for cells cij∈w we assume that dij = ∞. in fig. 4 graphically shows the scalar field d of the distances to the exit zone: the darker areas of the room are located further away from the exit. also, fig. 4 shows the field of the vector e, which is directed toward decreasing values of the field d. 3.2.3 calculation of the vector field setting the direction to the nearest exit let us dwell in more detail on the vector e field calculation procedure. we introduce the graph g’, which differs from the graph g by the set of edges. the set of edges e' of the graph g' is defined as follows e’ = { eijkl | (i, j) ≠ (k, l) & (i–k) ⊥ (j–l) & |i–k |≤ 2 & |j–l| ≤ 2)}. (9) thus, when choosing a set of edges the presence of walls in the room is ignored. for the cell cij there are 16 possible directions of the vector eij with step π/8. these directions correspond to all the incident edges of the vertex vij in the graph g’. we number these edges from 0 to 15 in counterclockwise order starting from the edge parallel to the ox and denote them respectively by e0 … e15. for each of these directions, we calculate the ratio mk = dk/lk, where lk is the length of the edge ek, and dk is the difference between the values of the distance attribute to the exit between vertices incident to ek. thus, we obtain a vector m of 16 elements (m1 … mk). we introduce the vector m0 = (m01 … m0k), obtained from m by averaging the neighboring elements as follows: .10/)(5/)(5/2 16mod)18(16mod)14(16mod)17(16mod)15(0   kkkkkk mmmmmm (10) let the element m0s be minimal, m0s = min m0. then the angle between the vector e and the ox axis is defined as s * π/8. the field of the vector e calculated for the premises is shown in fig. 4. 106 a. samartsev, v. ivaschenko, a.rezchikov, v.kushnikov et al. copyright ©2019 assa. adv. in systems science and appl. (2019) fig. 4. scalar field d and vector field e 3.2.4 multiparticle collisions when the model is implemented in practice, the question of multiparticle collisions arises. two-particle collisions in the model are resolved by recalculating the velocities of the collided agents according to formula (2), provided that the distance between the centers of the agents is less than or equal to the sum of their radii, and the agents are approaching closer to each other. multiparticle collisions are resolved by recalculating the velocities of colliding agents according to formula (2) in pairs for all colliding agents in the order determined by the indexes of the colliding agents on the list. it should be noted that many-particle collisions, as a rule, arise only in the case of a crush. in its absence, such collisions are extremely rare, in view of the fact that in the model the time step is sufficiently small, and the collision of agents takes only one step of model time. 4. model usage results 4.1 program structure the software for implementing the multi-agent model is written in the java programming language using the swing gui library. here is the structure of the program for implementing the model: 1. set the initial data: the room dimensions, the coordinates of the walls, the exits from the room, the number and parameters of the agents, and the type of simulation results. 2. initializing of the network c: constructing the graph g or the network c, defining the scalar field d and the vector field e. 3. generation of agents within the evacuation zones. 4. the calculation of vopt for each agent (3 6). 5. the calculation of a for each agent. 6. the calculation of v for each agent (1). 7. the calculation of x for each agent (1). 8. collision check for each pair of agents. if the collision takes place, then the speed should be recalculated. multiagent model of people evacuation from premises while emergency 107 copyright ©2019 assa. adv. in systems science and appl. (2019) 9. check of collisions with walls for each agent. if there is a collision, then the speed should be recalculated. 10. verification of the exit condition for each agent. if the agent has reached the exit, then he is excluded from the list. it is considered that the agent has reached the exit if his center is in the exit zone. 11. collection and storage of statistics. if there are no people left in the room, go to step 14. 12. display the position of agents and walls on the screen in the program window. 13. go to the next step of the model time. go to step 4. 14. if the required number of intermediate experiments is not completed, then proceed to step 3. 15. displaying the results of the experiment (graphics of functions). 16. go to step 1 (user request). 4.2 modelling results computational experiments conducted on the software and information complex have shown that during evacuation agents demonstrate "reasonable" behavior. most vivid example of such behavior is overtaking of one agent by another (fig. 5). fig. 5. illustration of overtaking one agent by another (three stage of overtaking from left to right) in the situation there is no physical contact between the agents. the agent behind and having the maximum speed greater than the one of the agent moving in front of him, analyzes the situation around and chooses a route for overtaking. eventually he gets to the exit, not waiting for the agent in front and does not spend his energy on physical contact with the other agent. the other examples of “reasonable” behavior of the agent are avoidance of collision with walls, deceleration when approaching obstacles (walls or congestion of other agents), maneuvering between other agents and choice of the nearest exit from premises during evacuation. however, agents demonstrate quite poor level of strategic “reasoning”, for example they never try to choose other exit when the one they approached occasioned to be flooded by other agents. all types of that behavior could be visually observed via software package ui. when studying the evacuation process with a large number of evacuated, there was an accumulation of agents in the exit zone and a crush which were also observed in [2, 14-17, 20, 22]. as an example, we present the results of numerical simulation of the evacuation process from a premises measuring 20 × 10 meters, the plan of which is shown in fig. 1. the room contains 34 walls, each of which is specified by the parameters (x, y, xw, yw) in meters: (3.0; 3.0; 0.2; 10.0); (5.9; 3.0; 0.2; 1.0); (5.9; 5.0; 0.2; 2.0); (5.9; 5.0; 0.2; 2.0); (5.9; 108 a. samartsev, v. ivaschenko, a.rezchikov, v.kushnikov et al. copyright ©2019 assa. adv. in systems science and appl. (2019) 8.0; 0.2; 3.0); (5.9; 12.0; 0.2; 1.0); (7.9; 3.0; 0.2; 5.0); (8.9; 10.0; 0.2; 3.0); (11.9; 10.0; 0.2; 3.0); (15.9; 10.0; 0.2; 1.0); (15.9; 12.0; 0.2; 1.0); (17.9; 3.0; 0.2; 5.0); (17.9; 10.0; 0.2; 3.0); (19.9; 3.0; 0.2; 1.0); (19.9; 5.0; 0.2; 2.0); (19.9; 8.0; 0.2; 2.0); (20.9; 10.0; 0.2; 1.0); (20.9; 12.0; 0.2; 1.0); (22.9; 3.0; 0.2; 10.0); (3.0; 3.0; 3.5; 0.2); (7.5; 3.0; 11.0; 0.2); (19.5; 3.0; 3.5; 0.2); (3.0; 6.0; 3.0; 0.2); (20.0; 6.0; 3.0; 0.2); (7.9; 7.8; 5.1; 0.2); (15.0; 7.8; 3.1; 0.2); (3.0; 10.0; 4.0; 0.2); (8.0; 10.0; 2.0; 0.2); (11.0; 10.0; 1.0; 0.2); (16.0; 10.0; 0.5; 0.2); (17.5; 10.0; 1.0; 0.2); (19.5; 10.0; 3.5; 0.2); (3.0; 12.8; 10.0; 0.2); (15.0; 12.8; 8.0; 0.2). on the map, in 3 meters from the walls of the room the exit zones are set and specified by the parameters (x, y, xw e, yw e) in meters: (0; 0; 0.2; 16); (0.2; 0; 25.8; 0.2); (25.8; 0.2; 0.2; 15.8); (0.2; 15.8; 25.8; 0.2). the evacuation zone is defined by one element, which is specified by the parameters (x, y, xw z, yw z) in meters: (3; 3; 20; 10). the following numerical values of the evacuation process parameters were used: – number of evacuees (agents): 100; – their maximum speed vmax: random uniformly distributed quantity on a segment 1 – 2 m/s; – their maximum acceleration amax: randomly uniformly distributed quantity on the interval 1 – 2 m/s2; – the initial coordinates of each agent were chosen according to the uniform distribution law inside evacuation zones, proceeding from the condition that initially the projection of a person should not intersect the projections of other people and the walls; – radius of projection r: random uniformly distributed quantity on the segment 0,22 – 0,29 m; – critical distance: l = 2 m; – mass m: random uniformly distributed quantity on the segment 60 100 kg; – model time step: δt = 0,004 s; – coefficient of recovery: ε = 0,4 [22, 28]. the results of modeling the evacuation process are shown in fig. 6 9. the function hn(t) shows the dependence of the people inside premises number on time, provided that at the initial moment there are n people in the room. the function m nh is the result of averaging the function hn(t) over m implementations obtained during the simulation. the chart of function )( 100 100 th is shown on fig. 6. fig. 6. chart of function )( 100 100 th multiagent model of people evacuation from premises while emergency 109 copyright ©2019 assa. adv. in systems science and appl. (2019) the results showed that in the first few seconds of the evacuation process very few people leave the premises, since most of them are only at the beginning of their way to the exits. then, at the exits, crowds of people begin to form, at which time the intensity of the flow of people through the doors increases. by about the 17th second, the intensity of the human flow through the exits is weakening. this is due to the fact that at this time, agents with lower speed of movement begin to evacuate, as well as with uneven distribution of the load over the exits from the premises (evacuation through some exits by this time could have already stopped). the long "tail" on the chart of the function )( 100 100 th in the range of the argument values from 35 to 70 s is explained by the formation of crushes at the exits from the room for some realizations of the experiment. thus, with some initial arrangements of agents and the distribution of their parameters, evacuation can be delayed. dependency resemble to )( 100 100 th was also analyzed in [2, 11] and graphs of a similar shape were obtained both in computer simulation and in the analysis of statistical data obtained during the actual evacuation. the quantitative differences between our research and results in [2, 11] are explained by different initial conditions (premises map and the number of people). a computational experiment was also performed to analyze the dependence of the average evacuation time tm(n) from the number of people n in the room: dtth n nt nm )( 1 )( 0 100    (11) the simulation results are shown in fig.7. fig. 7. chart of function tm(n) as it can be seen from fig. 7, the time of evacuation of people increases with the number of people increase. some deviations from the monotonicity of the function tm(n) can be explained by the instability of the evacuation process in relation to the initial conditions, in particular, to the location of the agents in the premises at the very beginning of evacuation, and by the frequent formation of crushes. let the function vh(n), which is defined for integer values of the interval [1; h], here h is the number of people at the beginning of the evacuation, n is the agent number in the agent list sorted by the order in which they reach the premises exit, and the value of the function is the maximum speed vmax of the agent with the number n in the specified list. 110 a. samartsev, v. ivaschenko, a.rezchikov, v.kushnikov et al. copyright ©2019 assa. adv. in systems science and appl. (2019) let the function )(tv m h as the result of averaging the function vh(n) over the m implementations obtained during the simulation. chart of function )( 1000 50 tv is shown in fig. 8. fig. 8. chart of function )( 1000 50 tv as it can be seen in the fig. 8, agents with a higher maximum speed are usually evacuates faster. the maximum speed of agents, which managed to evacuate among the very first ones, is especially great, because the crushes have not yet formed in the openings, and such agents can use their speed advantage to the full extent. after that, the maximum agent speed is reducing, and, finally, the last agents with the lowest maximum speed are evacuated, who did not managed to take advantageous positions in the evacuation process and had to wait for the disappearance of crushes in openings. also, the analysis of the premises was carried out for the presence of places most dangerous in terms of quantity of the agents collisions with each other. in fig. 9 the heat map of the premises is presented where the most dangerous places are marked with a more saturated red color. as it can be seen, the most dangerous are positions before the openings and slightly to the left and to the right of the opening. this can be explained by agents attempts to wedge into the main flow before the opening and the intensity of this flow. multiagent model of people evacuation from premises while emergency 111 copyright ©2019 assa. adv. in systems science and appl. (2019) fig. 9. heatmap for collision possibility this map was constructed after modelling of 50 people evacuation from a given premises and averaging the results after 50 experiments. 5. conclusion the model of the people evacuation process from premises in emergency situations in conditions of limited space of complex configuration is offered, which allows to obtain a more adequate assessment of the evacuation process due to the system considering of physical interaction of people in the evacuation process and the procedure for reasonable correction of goals pursued by intellectual agents in the evacuation process. the simulation of evacuation process in case of emergency on the basis of the proposed model makes it possible to assess the state and behavior of people in the occurrence of emergencies in the premises, as well as the number of people evacuated from the premises with different floor plan and occupation, depending on the maximum possible values of speed and acceleration. the model also allows to research impact of agent parameters on the order of their evacuation and find the most insecure places in the premises, where agents collide with each other and crushes appear. strengths of the model include decent computational performance, account of agents’ physical interaction and complex premises map, focus on spontaneous evacuation process and possibility of computation parallelism. experiments have shown results close to the ones of known evacuation models, during them agents have demonstrated patterns of “reasonable” behavior. weaknesses of the current model implementation include absence of consideration of some factors, for example, agents’ ignorance of floor plan, agents’ verbal interaction, agents’ mutual assistance, other weakness is poor strategic “reasoning” in some cases. integration with emergency models is of scientific and practical interest, for that emergency influence rules on agents must be considered. the model can be used for buildings evacuation safety validation or as a part of simulator for emercom personnel training. 112 a. samartsev, v. ivaschenko, a.rezchikov, v.kushnikov et al. copyright ©2019 assa. adv. in systems science and appl. 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[49] zheng, x., zhong, t., & liu, m.: modeling crowd evacuation of a building based on seven methodological approaches. building and environment, 44(3), 437-445 (2009) adv syst sci appl 2019; 01; 12-30 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/651 evaluating environmental impacts of photovoltaic technologies using data envelopment analysis svetlana v. ratner*1, andrey v. lychev2 1) trapeznikov institute of control sciences of russian academy of sciences, moscow, russia, e-mail:lanarat@mail.ru 2) national university of science and technology misis, moscow, russia e-mail: lychev@misis.ru received september 29, 2018; revised february 20, 2019; published april 15, 2019 abstract. this study contributes to the literature by proposing a new method of complex evaluation of multiple life cycle environmental impacts of different pv technologies based on the data envelopment analysis (dea). the main advantage of dea as a non-parametric technique is that it does not require prior knowledge of underlying production functions. an empirical production technology frontier is estimated based on best-practice boundary of the input-output relationship. dea evaluates comparative or relative efficiency, which means the measurement with reference to some set of units we are comparing with each other. the proposed approach allows to aggregate disparate quantitative estimates of individual negative environmental effects from the literature and special databases in a transparent and easily understandable index or coefficient of ecology efficiency. the evaluation of environmental effects is performed on data from the ecoinvent database. the results of this study clearly show that from an environmental point of view it is more practical to prefer technologies, which are less resource and energy intensive in manufacturing and upstream activities. as of right now, this requirement is met by thin-film technologies: amorphous silicon (a-si), cadmium telluride (cdte), and copper-indiumdiselenide (cis); however, their ecologic efficiency evaluation may change as we obtain more data on the final stages of the lifecycle for pv modules of various types. our computational results show that 5 out of 19 pv technologies were identified as efficient compared to the other technologies against which they were assessed. cdte, a-si, and ribbon silicon laminated panels installed slanted-roof, cis panel mounted slanted-roof, and open ground installation m-si are demonstrated highest efficiency scores from an ecological point of view throughout the life cycle. based on results of this study a number of opportunities for improving existing government incentives and rationalizing the design of state programs under elaboration can be identified. keywords: environmental impact categories; solar power; photovoltaic system; life cycle assessment; data envelopment analysis. 1. introduction throughout the last decades, the global sales and installation of photovoltaic (pv) systems have grown rapidly. annual installations of pv systems reached a record 98 gw in 2017, while global total installations attained 402 gw [37]. although pv-technologies have very low environmental and human health impacts compared to conventional electricity generation, the processes of manufacturing, transportation, installation, and disposal of pv-modules are associated with significant energy consumption, usage of working fluids containing chlorates and nitrites, formation of sewage and other negative environmental effects that need to be considered. at present, several photovoltaic technologies are being developed in the world: monocrystalline silicon, polycrystalline (multicrystalline) silicon and thin-film. it is well known, * corresponding author: lanarat@mail.ru mailto:lychev@misis.ru s.v. ratner, a.v. lychev 13 copyright ©2019 assa adv. in systems science and appl. (2019) that silicon wafer manufacturing process is very energy intensive. mono-crystalline cells require up to 1000 kwh/kg-si and poly-crystalline cells’ production needs up to 700 kwh/kg-si [11]. manufacturing of silicon-based pv cells is accompanied by emissions of such hazardous materials as silica dust, silanes, diborane, phosphine, and solvents. heterojunction solar cells (shj) are produced from silicon wafers in a low-temperature process that does not exceed 200 °c, thus, the energy intensity of the manufacturing process and, consequently, its environmental impact should be lower. besides, a low-temperature coefficient of shj cells leads to higher energy yields compared to conventional c-si modules [29]. however, there is as of yet no precise data on the environmental effects of this technology over the whole life cycle due to its relative novelty. it is also known, that mining cadmium and manufacturing cdte solar cells can cause occupational health risks. some questions also remain regarding the toxicity of pv modules’ disposal [18,19]. therefore, the environmental issues of cadmium telluride pv cells and module production and utilization are still under discussion. tellurium is also considered as a rare earth element, so the problem of resource constraints must also be taken into consideration when choosing this technology [36]. the main negative environmental impacts of manufacture of copper indium gallium selenide (cigs) photovoltaic cells, as is reported in [2] are connected with up-stream activities including extraction and processing of primary materials. for example, the extraction and processing of silver, which is needed for stringer and screen printing processes, is accompanied with emissions of toxic heavy metals such as mercury, lead, and arsenic. the mining of copper (used in the cable for the balance of the system and in co-production of selenium, gallium, and indium) is potentially associated with the exposure of radioactive materials. such manufacturing processes as energy-intensive co-evaporation needed for to production of the cigs layer and water-intensive surface washing of the cell’s substrate also affect negatively on the environment. another issue that needs to be considered is the scarcity of indium, which can restrict the production process [34]. until recently, the choice between industrial developments of a particular technology was carried out, mainly, by two criteria: cost and efficiency (or energy conversion coefficient). however, with the growth of production capacities around the world, the ecological aspects, such as the emission of pollutants by production facilities into the air, water and soils, the consumption of rare earth metals, water, energy, etc., become more important when choosing a particular technology for development and support through multiple government policies. this problem become more relevant for russia with the booming developing of pv-industry in recent years [56, 33]. with the enactment in 2013 of new government’s renewable energy support scheme focusing on grid-connected power generation facilities more than 5 mw, the process of construction and connection of such facilities to the federal grid company has notably intensified. in 2013–2018 period, more than 100 solar generation projects of a total capacity of 1.852 gw were selected for a subsequent support (fig. 1). https://en.wikipedia.org/wiki/copper_indium_gallium_selenide_solar_cells https://en.wikipedia.org/wiki/copper_indium_gallium_selenide_solar_cells 14 evaluating environmental impacts of photovoltaic technologies copyright ©2019 assa adv. in systems science and appl. (2019) source: own calculations fig. 1. planed pv-installations on the wholesale market supported by the state program in the framework of the government decree no. 449 (28 of may, 2013) an important feature of the renewable energy support programs introduced in russia is the high production localization index, which must be achieved in the project in order to be selected for financial support. such support conditions are aimed at creating in the country own full-cycle production of photovoltaic modules and other types of equipment for renewable energy. the active development of new pv-manufacturing technologies and the current and future increase in production volumes boost the relevance of the problem of estimating and predicting the possible negative environmental effects of photovoltaics at all stages of the life cycle: production, operation, and disposal. currently, life cycle assessment (lca) followed the international standards organization (iso) 14040 series is a mature methodology and a valuable tool for providing a comprehensive “cradle-to-grave” view of the environmental loads of a technology. it is often used for comparative analysis of manufacturing alternatives and optimization of product system design. due to the fact, that a full life cycle assessment is usually a time-, energy-, and data-intensive process requiring sophisticated methodology, many scholars prefer to use a simplified approach and quantify only greenhouse gas (ghg) performance of the technology or product under consideration, though for each product this specific type of environmental influence is the most significant. current papers on life cycle assessment of pv technologies can be divided into three groups. the papers of the first group focus on detailed analysis of one or several types of negative ecologic effects of specific types of pv-technologies. for example, hsu et al. [22] shows an analysis of ghg emissions from solar c-si pv lca. kim [25] evaluates the same type of environmental impact from thin-film pv lca. the research [20] analyzes and obtains estimates for сo2 emissions and embedded energy of a hybrid photovoltaic–thermal module. fthenakis [19] evaluates atmospheric cd emissions from the life-cycle of cdte models. the [28] research shows a very wide set of environmental impacts for new marine pv-technologies, including besides ghg emissions such categories as acidification potential, eutrophication, ecotoxicity, human toxicity, depletion of abiotic resources, terrestric ecotoxicity potential, and photochemical ozone creation potential. results of these papers improve upon the existing knowledge on the influence of new technologies on the environment. the second group compares multiple pv-technologies between each other or between other renewable energy technologies based on a specific category on environmental influence. most of the time the researched category is the ghg emissions. for example, the study [2] compares in detail the environmental influence from the production of two thin-film photoelements based on 35,2 140 199 255 285 308 373 132 125 0 50 100 150 200 250 300 350 400 2014 2015 2016 2017 2018 2019 2020 2021 2022 m w s.v. ratner, a.v. lychev 15 copyright ©2019 assa adv. in systems science and appl. (2019) cu(in,ga)se2: in particular, whether the use of zinc oxysulfide (zn(o,s)) or cadmium sulfide (cds) minimizes the ghg emissions during the production of photoelements through screen printing and stringer. paper [3] summarizes the results of over 75 researches performed via life cycle assessment (lca) and focusing on ghg emissions for electricity and heat generation from photovoltaic, solar thermal, onshore and offshore winds, hydropower, marine technologies (wave power and tidal energy), geothermal, biomass, waste, and heat pumps. the results of these papers may be used to improve the production chain, choose better designs for production systems, as well as make decisions on regional and national government levels to support one type of technology over another. however, for a full comparison of the ecologic influence of different technologies one must evaluate not just one (perhaps important) category of environmental influence, but the entire range of negative ecologic effects. hence, in modern literature the amount of works, which compare pv-technologies against each other or other renewable energy tech with multiple environmental influence categories in mind, is constantly increasing. furthermore, these comparisons can be performed both separately and using various aggregate indicators. for example, [27] low-concentration pv and conventional pv are compared separately in the climate change (ghg emissions), acidification potential, eutrophication potential, human toxicity, and ozone layer depletion categories. the same approach is used in [30], where mc-si, ingap and ingap/mc-si solar modules are compared in climate change, abiotic resource depletion, acidification, human toxicity, and fresh water ecotoxicity separately. in [41] one can find a comparative analysis of environmental impacts of organic and conventional pv-technologies completed using recipe v1.0.5 midpoint (h) impact categories. celik et al. [7] shows an aggregate indicator of toxicity (toxicity for humans, ecotoxicity of sea and freshwater, ecotoxicity of sea and freshwater bottom sediments), which is used to compare conventional si pv technologies and perovskite solar cells. paper [34] investigates ghg emissions, embedded energy and presents an aggregate indicator for environmental influence: the normalized environmental impact (eco-points) of several popular silicon-based pv technologies. in the case that several environmental effects are taken into consideration for comparison of designs of products or manufacturing processes, another important research question arises: which of these negative environmental impacts are more important and how should it be accounted for? this study contributes to the literature by proposing a new method of complex evaluation of multiple life cycle environmental impacts of different pv technologies based on the data envelopment analysis (dea). dea is a methodology for evaluating the relative performance of a set of homogeneous units, which transform multiple inputs into multiple outputs [9]. the main advantage of dea as a non-parametric technique is that it does not require prior knowledge of underlying production functions. an empirical production technology frontier is estimated based on best-practice boundary of the input-output relationship. dea evaluates comparative or relative efficiency, which means the measurement with reference to some set of units we are comparing with each other. the efficiency score has a clear economic meaning. it shows how much the dmu should reduce its resources or increase its outcome to become efficient. since dea was first introduced, there have been a large number of papers written on dea or applying this methodology on various sets of managerial, operational and economic problems. today dea is not only a tool for technical-efficiency analysis but also an increasingly popular performance management tool for benchmarking in many areas of research including energy sector [15]. the evaluation of environmental effects is performed on data from the ecoinvent database which is currently one of the most reliable and complete information resources for a significant number of industrial products. the paper’s added value as compared to above-mentioned research consists in the following: a) the work takes into account a much wider range of negative environmental effects; b) the developed method of their complex comparison allows to rank the technology according to the degree of aggregated environmental impact, as well as to determine the target parameters for reducing negative environmental effects. first attempts to apply dea to the problem of choosing the most ecologically efficient solar power technologies has been made https://www.sciencedirect.com/science/article/pii/s0927024816300605#! https://www.sciencedirect.com/science/article/pii/s2212827117308831#! 16 evaluating environmental impacts of photovoltaic technologies copyright ©2019 assa adv. in systems science and appl. (2019) in an earlier work of one of the authors [35], and it is thanks to that we could prove the applicability of this approach to compare different photovoltaic technologies from the environmental perspective. however, during the last year data on photovoltaic technology has significantly expanded and updated thanks to progress both in r&d and in production. besides that, the aforementioned paper has shown a suboptimal choice of indicators of environmental impact: some impact categories haven’t been accounted for in the analysis. in [35] we also compared ecologic impact of photovoltaic technology without considering the method of solar panel installation which could have distorted the results of the analysis. this paper addresses and eliminates these issues, and considers the newly-attained results from a perspective of policy applications. the remainder of the paper is organized as follows: section 2 provides an overview of lca methods, used in the ecoinvent database, and explains the choice of datasets and impact categories, which we used in the evaluation of pv-technologies. it also provides insight into the methodology (dea). in section 3 we discuss the results of the dea-based evaluation of modern pv technologies and the possibilities to develop a proposed approach for management applications as well as its limitations. section 4 concludes and offers policy recommendations. 2. data and methodology in rest of the paper we use the following abbreviations for pv technologies and types of pvinstallations: pv technologies a-si amorphous silicon cdte cadmium telluride cis copper-indium-diselenide m-si multicrystalline silicon r-si ribbon silicon s-si singlecrystalline silicon pv-installations fi facade installation fri flat-roof installation li integrated laminate ogi open ground installations pm mounted panel sri slanted-roof installation in this study, we used the data from the ecoinvent database (a non-profit association of research organizations in switzerland) for assessment of the environmental impact of the photovoltaic life cycle. currently, ecoinvent is the world’s leading life cycle assessment (lca) database compliant with the iso 14040-14044 standards and contains life-cycle data sets of more than 12,800 products and services [31]. it is important to emphasize that the ecoinvent database is not simply a library of individual lca datasets. the data are interrelated in such a way that all semifinished products, input streams, electricity consumption, demand for raw materials and materials and equipment requirements depend on sub-processes of production and delivery of semi-finished products and services for product redistribution. thus, lca results are calculated in a matrix system, so that any update in one set of process data will affect the accumulated lca results in all other connected data sets. version 3.3 (2016) of the database contains 356 datasets on photovoltaic energy, from which 227 datasets are devoted to electricity production, 43 – to transportation of semi-finished products such as mounted systems, cells, panels, modules, wafers and single crystals of silicon, 3 – to cell and panel factory contraction, 32 – to installations, 10 – to cell/panel production, 12 – to laminate production, 2 – to single wafer production, 2 – to single crystal production, 10 – to panel production, 7 – to mounting system production and 4 – to treatment of waste form silicon cells and s.v. ratner, a.v. lychev 17 copyright ©2019 assa adv. in systems science and appl. (2019) panel production. each data set of environmental effects includes effects from all upstream activities under evaluation, therefore the consideration of last activity in a production chain is sufficient. there are no datasets in the database devoted to disposal and recycling of pv models and panels because even the earliest pv-installations considered in the database are still in operation. lifecycle ends with low voltage electricity produced with the 3 kwp module, assuming an average yield. the 3 kwp module has been chosen in most of the datasets devoted to electricity production as the basic module for building integrated pv electricity production, due to the fact that larger modules can easily be built with 3 kwp modules without producing a significant error in environmental impact calculations. the datasets represent several mature pv technologies: multicrystalline silicon (m-si), singlecrystalline silicon (s-si), amorphous silicon (a-si), cadmium telluride (cdte), copper-indium-diselenide (cis), and ribbon silicon (r-si). the modules made of m-si, s-si, a-si, and r-si can be installed of different parts of the building (facade installation (fi), flat-roof installation (fri) or slanted-roof installation (sri)) as an integrated laminate (li) of a mounted panel (pm). the modules made of cdte can be installed in the form of integrated laminate only while cis models only in the form of a mounted panel. besides this, the database contains several datasets for 570 kwp m-si open ground installations (ogi). thus, we have different production chains specified by the cells and module production technologies, form and place of installation. the referent product in all datasets on electricity production is low voltage 1 kwh1. the simplifying assumption was made in all datasets that all electricity is directly provided as low voltage electricity so that no transformation has to take place. the lifetime of all pv modules is supposed to be 30 years. in order to compare energy technologies, we need to choose the datasets, yield (or calculated by extrapolation of field data) for the same geographic location, which is characterized by the value of specific solar yield in kwh per kilowatt peak installed. since the most reliable statistical data in the database is collected for switzerland (the longest period of observation), we chose data on various options (19 total) for electricity production using photovoltaic installations in this geographic location (table 1). one should note that this dataset can be used for comparison of energy technologies only, but not for assessment of average production patterns in different geographic locations, for example, in russia. table 1. comparing electricity production options for switzerland option period of observation 3kwp fi m-si li 2005-2016 3kwp fi m-si pm 2005-2016 3kwp fi s-si li 2005-2016 3kwp fi s-si pm 2005-2016 3kwp fri m-si 2005-2016 3kwp fri s-si 2005-2016 3kwp sri a-si li 2005-2016 3kwp sri a-si pm 2005-2016 3kwp sri cdte li 2005-2016 3kwp sri cis pm 2005-2016 3kwp sri m-si li 2005-2016 3kwp sri m-si pm 2005-2016 3kwp sri m-si pm label-certified 2010-2016 3kwp sri r-si li 2005-2016 1 although it is known that large central photovoltaic power stations in the higher kilowatt to megawatt range can feed directly into the mediumor high voltage grid, in order to treat all photovoltaic installations the same way, low voltage electricity is assumed as a product in this dataset 18 evaluating environmental impacts of photovoltaic technologies copyright ©2019 assa adv. in systems science and appl. (2019) 3kwp sri r-si pm 2005-2016 3kwp sri s-si li 2005-2016 3kwp sri s-si pm 2005-2016 3kwp sri s-si pm label-certified 2010-2016 570kwp ogi m-si 2008-2016 life cycle impact assessment can be carried out using several indicator methods, which differ from each other in the spectrum of considered environmental impact categories. the latest version of ecoinvent uses 13 basic methods, of which, according to our opinion, cml2001 is the most complete and informative. this method was developed by center of environmental science of leiden university and published in 2001 as a new operational guide for implementation of iso environmental management standards [21]. in addition to such basic categories of environmental impact as greenhouse gas emissions, this method takes into account many other negative environmental effects. according to [8] it gives more accurate estimates of chemical impacts on human health. main impacts on the environment, according to cml2001 method can also be considered at different time horizons. table 2 summaries impact categories, covered in cml2001, their indicators and measure units. table 2. cml2001 environmental impact categories and their indicators impact category group name of the impact category in the method unit acidification acidification potential average europe acidification potential – generic kg so2-eq climate change climate change gwp100 climate change gwp20 climate change gwp500 climate change lower limit of net gwp100 climate change upper limit of net gwp100 kg co2-eq depletion of abiotic resources depletion of abiotic resources elements, economic reserve depletion of abiotic resources elements, reserve base kg antinomy-eq ecotoxicity freshwater aquatic ecotoxicity faetp infinitive freshwater aquatic ecotoxicity faetp100 freshwater aquatic ecotoxicity faetp20 freshwater aquatic ecotoxicity faetp500 freshwater sedimental ecotoxicity fsetp infinitive freshwater sedimental ecotoxicity fsetp100 freshwater sedimental ecotoxicity fsetp20 freshwater sedimental ecotoxicity fsetp500 marine aquatic ecotoxicity maetp infinitive marine aquatic ecotoxicity maetp100 marine aquatic ecotoxicity maetp20 marine aquatic ecotoxicity maetp500 marine sedimental ecotoxicity msetp infinitive marine sedimental ecotoxicity msetp100 marine sedimental ecotoxicity msetp20 marine sedimental ecotoxicity msetp500 kg 1,4-dcb-eq s.v. ratner, a.v. lychev 19 copyright ©2019 assa adv. in systems science and appl. (2019) terrestrial ecotoxicity tetp infinitive terrestrial ecotoxicity tetp100 terrestrial ecotoxicity tetp20 terrestrial ecotoxicity tetp500 eutrophication eutrophication generic eutrophication average europe po4-eq nox-eq human toxicity human toxicity htp infinitive human toxicity htp100 human toxicity htp20 human toxicity htp500 kg 1,4-dcb-eq ionising radiation radiation dalys land use land use land competition m2a odour odour (malodours air) m3 air ozone layer depletion ozone layer depletion odp steady state ozone layer depletion odp10 ozone layer depletion odp15 ozone layer depletion odp20 ozone layer depletion odp25 ozone layer depletion – odp30 ozone layer depletion – odp40 ozone layer depletion odp5 kg cfc-11-eq photochemical oxidation (summer smog) photochemical oxidation ebir (low nox) photochemical oxidation high nox photochemical oxidation low nox photochemical oxidation mir (very high nox) photochemical oxidation moir (high nox) kg formed ozone kg ethylene-eq kg ethylene-eq kg formed ozone kg formed ozone some of the environmental effects of photovoltaics presented in table 2 (such as ionising radiation, ozone layer depletion, and photochemical oxidation) are negligibly small (less than 1×10-8), therefore, in order to compare the various variants of electricity generation with solar cells, we chose the following impact categories: acidification potential, climate change, eutrophication potential, ecotoxicity (all types), land use, malodorous air and depletion of abiotic resources. the duration of the impact/accumulation of each negative ecological effect was set to 100 years (medium-term period). all input data for comparative analysis of different pvtechnologies is presented in table a.1, see appendix a. for complex evaluation of multiple life cycle environmental impacts of different pv technologies we use data envelopment analysis (dea). based on the work initiated by farrell [16], charnes, cooper and rhodes [9] developed linear programming technique now called dea to estimate a production frontier. it is a tool to evaluate the relative efficiency of a set of homogeneous decision making units (dmu). in traditional dea models, each dmu is assumed to be a black-box whose internal structure is not considered, i.e. dmu is regarded as the entity responsible for converting inputs into outputs and whose performances are to be evaluated. in the energy sector, dmus can be, for instance, manufacturing plants, power plants, power distribution divisions etc. in this paper we consider a number of selected combinations of pv technology and installation method having common input and environmental outputs as distinct dmus. in dea model, the relative efficiency score of any dmu is determined as a measure of the relative improvements in inputs and outputs between the dmu and its assigned target. in order to provide relative comparisons, a collection of dmus is used to evaluate each dmu against each other. the efficiency score can be assessed by minimization of input keeping its output levels constant (input-oriented model), or by maximization of output keeping the inputs at the same rate (output-oriented model) [40]. 20 evaluating environmental impacts of photovoltaic technologies copyright ©2019 assa adv. in systems science and appl. (2019) fig. 2. efficiency evaluation in dea approach fig. 2 explains the basic idea of dea approach for efficiency evaluation. points a-g are the set of dmus under assessment. units b-e are efficient, they form efficient frontier bcde against which other inefficient units are assessed. the efficiency score of inefficient unit a can be measured relative to the projection a’ obtained by simultaneous minimization of inputs. the ray from the origin shows the direction of radial projection of unit a onto efficient frontier. thus, the radial input efficiency score of unit a is equal to oa’/oa. selection of inputs and outputs plays important role in benchmarking studies using dea [12]. the inputs and the outputs should reflect the resources utilized by the dmus and its production, respectively. dea environmental assessment usually considers both desirable and undesirable outputs to assess the performance of dmus. see for example, sueyoshi and goto [39], zhou et al. [43], førsund [17]; wang et al. [42], etc. in this study, all environmental impact categories are treated as undesirable outputs, and desirable output is electricity production. we also assume that the maintenance costs per 1 kwh for all pv technologies are the same, and therefore input parameters are not considered. for this reason, we consider environmental impacts (undesirable outputs) as inputs in the efficiency evaluation process [39]. since all environmental impact indicators are provided per 1 kwh electricity production, no return to scale assumption is needed. therefore, the dea model chosen in this work to identify the efficient frontier is ccr model introduced by charnes et al. [9] based on constant return to scale assumption. the basic motivation of the paper is to compare the performance of pv technologies under minimization of environmental impacts, hence input oriented ccr model is chosen. the input-oriented ccr model is written as:               r i i m k ko ss 11 min  mksxx k n j jkjko ,,1,0 1      (1) riysy ioi n j jij ,,1, 1      njj ,,1,0  , where n is the number of dmus, s.v. ratner, a.v. lychev 21 copyright ©2019 assa adv. in systems science and appl. (2019) m is the number of inputs, r is the number of outputs, xj = (x1j,…,xmj) is the vector of inputs of dmuj, yj = (y1j,…,yrj) is the vector of outputs of dmuj, and (xo,yo) is the vector of inputs and outputs of dmuo under evaluation, θo, optimal objective function, is the relative efficiency score of dmuo, no 1 , j , nj ,,1 , are the coefficients of linear combination for assessing dmuo;  mss ,,1  and  rss ,,1  are slacks variables, ε is an infinitesimal constant (a non-archimedean quantity). we can avoid handling ε, but then problem (1) has to be solved in two stages [9]. next, we assume that each model is solved in this way. dea efficiency scores θo are evaluated as the distance from the dmus to the efficient frontier. from the model formulation, it follows that 0 < θo ≤ 1. if efficiency score θo = 1, and optimal slacks ),,( ** 1 *   msss  and ),,( ** 1 *   rsss  , then dmuo is efficient; if optimal solution satisfies θo = 1, then dmuo is weakly efficient; units with θo < 1 are inefficient. 3. results and discussion model (1) is solved in the present paper using frontiervision software [26] that calculates relative efficiency scores and target impacts. the efficiency scores reported in table 3 show that 5 out of 19 dmus were identified as efficient compared to the other dmus against which they were assessed. cdte, a-si, and r-si laminated panels installed slanted-roof, cis panel mounted slanted-roof, and open ground installation m-si are demonstrated highest efficiency scores from an ecological point of view throughout the life cycle. table 3. efficiency scores installation method, pv technology eff. score, % 3kwp fi m-si li 65.76 3kwp fi m-si pm 61.20 3kwp fi s-si li 64.20 3kwp fi s-si pm 59.19 3kwp fri m-si 92.77 3kwp fri s-si 90.61 3kwp sri a-si li 100.00 3kwp sri a-si pm 85.32 3kwp sri cdte li 100.00 3kwp sri cis pm 100.00 3kwp sri m-si li 98.75 3kwp sri m-si pm 80.79 3kwp sri m-si pm label-certified 90.31 3kwp sri r-si li 100.00 3kwp sri r-si pm 91.81 3kwp sri s-si li 97.49 3kwp sri s-si pm 77.81 3kwp sri s-si pm label-certified 87.42 570kwp ogi m-si 100.00 22 evaluating environmental impacts of photovoltaic technologies copyright ©2019 assa adv. in systems science and appl. (2019) table 3 shows that slanted-roof installations are more efficient than flat-roof installations. facade installations have the lowest efficiency scores. this result is quite expected and can be explained by the fact that a tilt angle of a slanted roof is close to the local latitude in switzerland (46n average) and thus allows to harvest the maximum amount of solar radiation on an annual basis [10] and, therefore, produce more electricity for its lifespan. also, it should be noted that laminated integrated panels are more ecologically efficient in comparison with mounted panels. it can be explained by the absence of aluminum frame and some other construction details due to integration straight into the roof, which leads to exclusion from the life cycle of several production activities. the higher efficiency of 570 kwp open ground pv plant is also an expected result due to the fact that the system on the ground has more airflow to cool down the panels and, thus, to keep up an optimal temperature for pv modules’ operation and to extend its carrying capacity and increase its overall electricity production [38]. comparing the ecological effectiveness of s-si, m-si and thin-film pv technologies (cdte, cis, a-si), we can see that the mean of efficiency scores of single-si modules for different installation methods is 79.45%, the mean of multi-si is 81.6% and the mean of thin-film pv is 96.33% (see tables 4-6). table 4. efficiency scores for s-si technology installation method, pv technology eff. score, % 3kwp fi s-si li 64.20 3kwp fi s-si pm 59.19 3kwp fri s-si 90.61 3kwp sri s-si li 97.49 3kwp sri s-si pm 77.81 3kwp sri s-si pm label-certified 87.42 mean score 79.45% table 5. efficiency scores for m-si technology installation method, pv technology eff. score, % 3kwp fi m-si li 65.76 3kwp fi m-si pm 61.20 3kwp fri m-si 92.77 3kwp sri m-si li 98.75 3kwp sri m-si pm 80.79 3kwp sri m-si pm label-certified 90.31 mean score 81.60% table 6. efficiency scores for thin-film technologies installation method, pv technology eff. score, % 3kwp sri a-si li 100.00 3kwp sri a-si pm 85.32 3kwp sri cdte li 100.00 3kwp sri cis pm 100.00 mean score 96.33% s.v. ratner, a.v. lychev 23 copyright ©2019 assa adv. in systems science and appl. (2019) it is important to understand that these results cannot be explained by pv cells or module efficiency, traditionally measured as percentage of the incident solar energy that the pv cell converts into electricity under the standard rating conditions (table 7), and correspond mainly to resource and energy intensity of production processes [10-11]. therefore, the bigger amount of energy produced by silicon-based pv systems in the exploitation period comparing the amount of energy produced by thin-film pv systems does not compensate the negative ecological effects of upstream processes. table 7. cell/module efficiency for different pv technologies [13, 31-32] pv technology cell efficiency (%) module efficiency (%) s-si 15.3 14.0 m-si 14.4 13.2 r-si 13.1 12.0 a-si 6.5 6.5 cdte 10.9 10.9 cis 10.7 10.7 comparing the results obtained by dea-approach with the results of simple ranking of pvtechnologies by one of the indicators of environmental or energy efficiency (see table 8), it can be noted that the technology performance indicator calculated by the dea-model, although correlated with the ratings for energy pay back time, ghg emissions and human toxicity, overall gives a more consistent and balanced result. table 8. ranks of pv-technologies according to dea model (own calculation), energy pay-back-time [23], ghg-emissions (ecoinvent, version 3.3) and human toxicity (ecoinvent, version 3.3.) installation method, pv technology rank on eff.score rank on epbt rank on ghg rank on ht 3kwp fi m-si li 12 9 16 17 3kwp fi m-si pm 14 9 17 19 3kwp fi s-si li 13 10 18 16 3kwp fi s-si pm 15 10 19 18 3kwp fri m-si 4 6 7 6 3kwp fri s-si 6 8 13 5 3kwp sri a-si li 1 n/a 2 12 3kwp sri a-si pm 9 5 10 15 3kwp sri cdte li 1 2 1 11 3kwp sri cis pm 1 3 4 7 3kwp sri m-si li 2 n/a 5 3 3kwp sri m-si pm 10 4 12 14 3kwp sri m-si pm label-certified 7 n/a 8 10 3kwp sri r-si li 1 n/a 3 2 3kwp sri r-si pm 5 1 6 8 3kwp sri s-si li 3 n/a 11 4 3kwp sri s-si pm 11 7 15 13 3kwp sri s-si pm label-certified 8 n/a 14 9 570kwp ogi m-si 1 n/a 9 1 24 evaluating environmental impacts of photovoltaic technologies copyright ©2019 assa adv. in systems science and appl. (2019) in addition, the dea solution also allows obtaining target values for each indicator of the environmental impact for each inefficient pv technology. the target impacts for each of the inefficient pv technology were calculated according to the following equation: * * 1 , 1, , n ko j kj j x x k m    , (3.1) where j * , j = 1,…,n are optimal -variables in problem (1). the values of the target impacts are presented in table a.2, see appendix. knowledge of the target parameters that each of the photovoltaic technologies needs to achieve in order to become efficient in environmental terms can significantly simplify the process of managing applied research aimed at improving these technologies, as well as developing new photovoltaic technologies, such as organic pv. table 9 presents a summary of peer frequency, i.e. number times each efficient pv technology is a peer for another. a sum of peer weights is calculated for each efficient dmuk as a sum of variables k * in optimal solutions of problem (1) for all inefficient dmus excluding dmuk. table 9. peer count summary installation method, pv technology peer frequency sum of peer weights 3kwp sri a-si li 6 1.96 3kwp sri cdte li 7 0.98 3kwp sri cis pm 0 0 3kwp sri r-si li 13 10.83 570kwp ogi m-si 13 0.23 note that dmus 3kwp sri r-si li and 570kwp ogi m-si are the peers for 13 inefficient technologies. however, first technology is more valuable because the sum of its weights is much greater. this means that this technology is closer to efficient targets of inefficient units than another. on the contrary, 3kwp sri cis pm technology is the self-evaluator peer according to the classification of edvardsen et al. [14], because it only references itself. since all types of studied pv technologies are still under research and development, it is reasonable to consider the different technical possibilities to reach target indicators and make prevalent technologies more environmentally efficient. it will require deeper analysis for each of the environmental impacts considered at various stages of the life cycle. it is also necessary to mention that all obtained results so far do not include the environmental impacts of pv-systems’ disposal due to the lack of primary data. the inclusion of such downstream stages of the life cycle as disposal and recycling can change the environmental efficiency estimates of pv-technologies in the future and proactive planning for a pv recycling infrastructure can contribute to the ecological efficiency of each technology. 4. conclusions and policy applications a number of opportunities for improving existing government incentives and rationalizing the design of state programs under elaboration can be identified based on results of this study. our evaluations of the complex ecologic efficiency of several commercially successful pv technologies may be used for developing various state programs for supporting pv equipment manufacturers in russia and in other countries, which are just starting their own production of pv modules. the currently existing approaches to stimulation do not differentiate between technologies, leaving a choice to the customer, and most customers prefer pv modules with a higher cell efficiency. a change is necessary in the current order for sustainable development. the results of this study clearly show that from an environmental point of view it is more practical to prefer technologies, which are less resource and energy intensive in manufacturing and upstream activities. as of right now, this requirement is met by thin-film technologies (a-si, cdte, and cis); s.v. ratner, a.v. lychev 25 copyright ©2019 assa adv. in systems science and appl. (2019) however, their ecologic efficiency evaluation may change as we obtain more data on the final stages of the lifecycle for pv modules of various types. considering the issue of the installation method of pv modules, open ground installations remain preferable. in this regard, the existing state support system for photovoltaics in russia, which focuses on large power plants with capacities above 5 mw, is logical and can be deemed successful. while developing new microgeneration state support programs, it is reasonable to prefer slanted-roof installations and laminated integrated panels. stimulating the population to use these specific installation types for pv systems can be done both via standardization and certification (for example, municipal standards for installation of solar panels), as well as via special educational events and training programs. r&d support programs in the field of solar energy can also be constructed in a way that stimulates not just research targeting the growth of cost-competitiveness and energy efficiency of photovoltaics, but also improvement of their environmental efficiency. in the near future, we can use target environmental impacts, obtained in this research as benchmarks for emerging each of the currently existing, commercially viable pv technologies, which are presented in this study. as technologies develop and new data on their environmental impacts is obtained, these target parameters may be re-calculated using the herein suggested dea approach. despite the fact that abovementioned practical conclusions have independent scientific value for environmental management, the main contribution of our research is the development of a method for the integrated assessment of the negative impact of renewable technology on the environment for the widest possible range of ecology effects taken into account. the combined lca and dea approach is proposed for a first time, therefore the paper expands the scope of practical applications of the popular decision-making methodology such as dea. the main advantage of proposed method of integrated assessment of the negative impact of pv-technologies on the environment during entire life cycle is its scalability: it is easy to integrate into the calculations both a greater number of new technologies and more identified environmental effects. even for a large number of studied technologies and a large number of categories of environmental impact, there is still the possibility of ranking alternatives. the main limitation of this study is the lack of empirical data on environmental impacts of new pv-technologies, which are only entering the stage of mass production: most importantly, heterojunction solar technologies, being developed by the biggest russian pv-manufacturer hevel group. this problem opens a further area of research, which, however, can be easily integrated into the proposed approach of relative complex environmental efficiency evaluation. another limitation is connected with the lack of discrimination capability for efficient dmus classical dea models. because of this, several technologies can have the highest rank in integrated environmental efficiency at the same time. as one can see in our study 5 technologies are classified as efficient ones. therefore, complete ranking is not possible with ccr model. in order to generate a complete ranking of dmus, many theoretical extensions of basic dea models have been proposed by various researchers. a recent survey on complete ranking methods in dea can be found in [1]. each approach has their own strength and weaknesses and detailed discussion on those methods is beyond the scope of the present paper. we assume that further choice between eco-efficient pv technologies can be made using one of dea ranking methods or with the help of traditional instruments of technical and economic analysis. acknowledgements dea calculations were conducted with the support of the russian science foundation (project no. 17-11-01353). 26 evaluating environmental impacts of photovoltaic technologies copyright ©2019 assa adv. in systems science and appl. 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(2019) appendix a table a.1. input data for comparative analysis installation method, pv technology so2-eq (gen) gwp100a nox-eq pox-eq faetp100a fsetp100a htp100a land air maetp100a msetp100a resources 3kwp fi m-si li 0.00080479 0.11721 0.00036814 0.00040589 0.27186 0.63365 0.18929 0.009284 1388.1 0.91367 1.13230 0.0007772 3kwp fi m-si pm 0.00082222 0.12094 0.00037988 0.00041474 0.30161 0.70768 0.19433 0.00945 1439.8 1.00800 1.26120 0.0008035 3kwp fi s-si li 0.00091488 0.13939 0.00042352 0.00044450 0.27868 0.64846 0.18877 0.009791 1410.9 0.93778 1.15750 0.0009270 3kwp fi s-si pm 0.00093127 0.14289 0.00043456 0.00045282 0.30666 0.71806 0.19351 0.009947 1459.5 1.02650 1.27870 0.0009517 3kwp fri m-si 0.00548520 0.081018 0.00025345 0.00027563 0.20446 0.48003 0.12667 0.006288 956.25 0.68352 0.85491 0.0005488 3kwp fri s-si 0.00062211 0.095803 0.00029034 0.00030144 0.20776 0.48676 0.12636 0.006626 970.21 0.69560 0.86628 0.0006479 3kwp sri a-si li 0.00047230 0.064313 0.00020003 0.00023177 0.18401 0.43101 0.13757 0.003955 318.82 0.61342 0.76251 0.0004385 3kwp sri a-si pm 0.00059623 0.085838 0.00026219 0.00027205 0.24332 0.57601 0.18127 0.004633 469.1 0.80436 1.01540 0.0005568 3kwp sri cdte li 0.00043030 0.049012 0.00017623 0.00025864 0.18300 0.42532 0.13686 0.003735 318.36 0.61210 0.75626 0.0003297 3kwp sri cis pm 0.00051273 0.074027 0.00024123 0.00031670 0.20854 0.48391 0.12810 0.004859 454.52 0.69280 0.85354 0.0004850 3kwp sri m-si li 0.00051316 0.074603 0.00023597 0.00026671 0.18190 0.42439 0.11706 0.006198 923.45 0.61103 0.75818 0.0005064 3kwp sri m-si pm 0.00062838 0.092526 0.00029008 0.00031474 0.22285 0.53606 0.14868 0.007206 1091.2 0.76381 0.95532 0.0006138 3kwp sri m-si pm label-certified 0.00055929 0.082354 0.00025819 0.00028014 0.20338 0.47712 0.13233 0.006414 971.26 0.67983 0.85029 0.0005463 3kwp sri r-si li 0.00050074 0.070164 0.00023003 0.00026025 0.18086 0.42209 0.11463 0.005542 718.65 0.60785 0.75494 0.0004699 3kwp sri r-si pm 0.00055134 0.078666 0.00025441 0.00027498 0.20443 0.47994 0.13138 0.005779 771.09 0.68332 0.85597 0.0005136 3kwp sri s-si li 0.00058886 0.089770 0.0002739 0.00029305 0.18654 0.43445 0.11733 0.006542 939.36 0.62744 0.77533 0.0006081 3kwp sri s-si pm 0.00071034 0.109050 0.00033121 0.00034343 0.23228 0.54382 0.14795 0.007578 1105.9 0.77763 0.96840 0.0007254 3kwp sri s-si pm label-certified 0.00063224 0.097058 0.0002948 0.00030567 0.20674 0.48403 0.13169 0.006744 984.31 0.69214 0.86193 0.0006456 570kwp ogi m-si 0.00049902 0.083707 0.00025718 0.00020105 0.099367 0.22837 0.10382 0.042508 948.81 0.34096 0.40829 0.0005503 30 evaluating environmental impacts of photovoltaic technologies copyright ©2019 assa adv. in systems science and appl. (2019) table a.2. efficient targets for pv technologies installation method, pv technology so2-eq (gen) gwp100a nox-eq pox-eq faetp100a fsetp100a htp100a land air maetp100a msetp100a resources 3kwp fi m-si li 0.00052923 0.077077 0.00024209 0.00026691 0.17878 0.41669 0.12448 0.006105 912.81 0.60083 0.74460 0.0005111 3kwp fi m-si pm 0.00054069 0.079530 0.00024981 0.00027273 0.19834 0.46537 0.12779 0.006215 946.81 0.66286 0.82937 0.0005284 3kwp fi s-si li 0.00060163 0.091663 0.00027851 0.00029230 0.18326 0.42643 0.12414 0.006438 927.81 0.61668 0.76117 0.0006096 3kwp fi s-si pm 0.00061240 0.093964 0.00028577 0.00029777 0.20166 0.47220 0.12725 0.006541 959.77 0.67503 0.84087 0.0006258 3kwp fri m-si 0.00360707 0.053277 0.00016667 0.00018125 0.13445 0.31567 0.08330 0.004135 628.83 0.44948 0.56219 0.0003609 3kwp fri s-si 0.00040910 0.063000 0.00019093 0.00019823 0.13662 0.32009 0.08309 0.004357 638.01 0.45743 0.56967 0.0004261 3kwp sri a-si pm 0.00039208 0.056447 0.00017242 0.00017890 0.16001 0.37878 0.11920 0.003046 308.48 0.52895 0.66773 0.0003662 3kwp sri m-si li 0.00033745 0.049059 0.00015517 0.00017539 0.11962 0.27908 0.07698 0.004076 607.26 0.40181 0.49858 0.0003330 3kwp sri m-si pm 0.00041322 0.060845 0.00019076 0.00020697 0.14655 0.35251 0.09777 0.004739 717.57 0.50228 0.62822 0.0004036 3kwp sri m-si pm label-certified 0.00036779 0.054156 0.00016979 0.00018422 0.13374 0.31375 0.08702 0.004218 638.70 0.44706 0.55915 0.0003592 3kwp sri r-si pm 0.00036256 0.051731 0.00016730 0.00018083 0.13443 0.31561 0.08640 0.003800 507.07 0.44935 0.56289 0.0003377 3kwp sri s-si li 0.00038723 0.059033 0.00018012 0.00019271 0.12267 0.28569 0.07716 0.004302 617.72 0.41260 0.50986 0.0003999 3kwp sri s-si pm 0.00046712 0.071711 0.00021780 0.00022584 0.15275 0.35762 0.09729 0.004983 727.24 0.51137 0.63682 0.0004770 3kwp sri s-si pm label-certified 0.00041576 0.063825 0.00019386 0.00020101 0.13595 0.31830 0.08660 0.004435 647.28 0.45515 0.56681 0.0004246 adv syst sci appl 2019; 03; 23-37 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/747 study of operational efficiency of large-scale publicly owned companies in russia valentina e. guseva1, evgeny a. kuzmin2,3*, andrey i. vlasov4 1) tyumen industrial university, tyumen, russia 2) institute of economics of the ural branch of the russian academy of sciences (ie ub ras), ekaterinburg, russia 3) ural state university of economics, ekaterinburg, russia 4) bauman moscow state technical university, moscow, russia received february 12, 2018; revised december 9, 2018; published december 31, 2018 abstract: a sustainable economic growth is partly a consequence of the best possible balance between public and private sectors. reasons for changed government efforts in economy are highly diverse and lie in the area of strategic development goals. a review of evidence-based studies shows that public ownership in a capital of companies influences some characteristics of their operational activities. the specific components, elaborated in the paper, contribute to this. based on data provided in financial statements of leading publicly owned companies in russia, the research aims at identification of causal relationships in composition and ownership structure of companies, as well as their impact on economic ratios. researchers verified the assumptions, put forward in this respect, using the correlation-regression analysis. the basis for the operational efficiency comparison included the return on equity (roe), return on assets (roa), return on sales (ros), equity ratio and current liquidity ratio. findings show that there are sufficient arguments in favour of the assumption that publicly owned companies are consistently less efficient against the private ones. obtained data made it possible to conclude that a change in the state’s share in the capital of large-scale companies in russia influences the return on assets and return on equity only. other ratios under consideration express poor correlations. the heterogenic impact on adjacent ratios of operational efficiency prevents from concluding in a rigorous manner that publicly owned companies are a priori less efficient. one needs to keep in mind sector affiliation and a capital concentration ratio. keywords: operational efficiency, publicly owned companies, private companies, capital concentration, economic growth, sustainable development. 1. introduction efficiency of publicly owned and private companies has not become less of an issue in both the developed and developing countries. in this discussion, there is a main problem of a search for the best possible ratio between incorporation forms in economy [30]. steady trends towards a growth of the public sector in russian economy are a clear example saying that the issue is pressing. at the same time, this sector has historically concentrated production facilities and capital resources. the transition to market economy in the 1990-ies had been marked by a declined government share (to 38%) in gdp by 2006 [51]. however, later, this trend did not develop and a balance between sectors started recovering. by 2016, the share of the public sector in russia's gdp had reached 46% [39]. these data find their confirmation in ownership structure statistics. the national government is the largest shareholder in the russian market. the share of publicly owned companies that disclose their ownership structure, in the * corresponding author: kuzminea@gmail.com 24 v.e. guseva, e.a. kuzmin, a.i. vlasov copyright ©2019 assa adv. in systems science and appl. (2019) moscow exchange index, is at least 45% [37]. 28 of 100 companies, ranking by ra expert in 2017, are the publicly owned companies ones (percentage holding is 25%) [55]. 5.5% of all the employees in the russian economy works for them in 2017. the share of proceeds of the top 100 public sector companies was 50% of their total revenue in 2017. according to riarating in 2018, the publicly owned companies account for more than 48% of top 100 largest companies capitalization [56]. it indicates that the bunching of publicly owned companies companies is significantly high. we have recently observed further strengthening of a role of the national government, not only as a participant in economic affairs, but also as their regulator. efficiency of the public sector is a key issue in definition and justification of its required existence. the component analysis might help to make aspects of this issue clearer. there are convincing evidence-based data saying that overall efficiency of factors of production at russian publicly owned companies is much lower than in the private sector [52]. labour productivity at publicly owned companies is more than 30% below the national average rate due to the fact that crucial decisions often serve political purposes instead of commercial ones. hence, efficiency criteria in state’s ownership management are not formal ratios (such as profitability, scope of tax deductions to a budget), but a quality of provided public goods and other institutional ratios [44]. however, followers of this approach exclusively view publicly owned companies one-sidedly, as an element of infrastructure support to the social and economic life. in the framework of the systematic view of the issue, researchers focus on an analysis of comparative efficiency of publicly owned companies as isolated agents. unsatisfactory management of public assets and an excessive size of the public sector indeed pose a serious risk to the economic growth [44]. in the countries where the public sector’s share is significant, they always serve for alignment of a business cycle and employment maintenance [1], often to the detriment of economic interests. the specifics available in the shift of interests manifests itself in the balance between the public and private capital in an ownership structure of companies. it makes sense in the discussion that this should cause a change in their performance, expressed in objective ratios. therefore, our research aims at identification of causal correlations in a composition and ownership structure of companies, their impact on economic ratios. the conclusions that we hope to make would allow us being closer to a search for the best possible ratio between public and private sectors in economy, which make it possible to achieve sustainable development. 2. review of literature boyett [6] believes that the process of public entrepreneurship appeared as a response to a transitional state in an economic system, as a new form of entrepreneurship in terms of an unstable external environment in response to uncertainty. there are various formal procedures, with which it is possible to calculate a share of the public sector in economy, its efficiency and various expert opinions, often contradictory. in the research, publicly owned companies refer to those companies where for various reasons, a national government exercises control being also a dominating owner. an ownership structure and management models might differ [46]. in any case, controlled parameters of companies' activities measure their financial standing, main characteristics of which affect competitiveness of an enterprise in a sector and in a market as a whole [13, 26]. they also affect a potential and opportunities of business cooperation with other companies [11, 19] and involvement of stakeholders [8, 17]. therefore, the analysis of corporate activities at runtime aims to show the extent, to which companies have properly consumed available resources [34]. the conventional view is that publicly owned companies are less efficient than private ones. some researchers explain this by the fact that, under market conditions, publicly owned operational efficiency of large-scale publicly owned companies 25 copyright ©2019 assa. adv. in systems science and appl. (2019) companies are usually established as a compensation for market failures [43]. if market failures are treated by joint efforts as ppp projects, the return on investment is maximized when the private capital prevails [3]. conceptual understanding of this phenomenon lies in the theory of public administration [54]. at the same time, in terms of institutional economy, a national government is unable to provide control over management in an efficient manner due to high information asymmetry (principal-agent theory). it is usual that for the companies with a high state’s share, external mechanisms of management monitoring do not actually perform their disciplining function [41]. borcherding, pommerehne and schneider [5] review findings from 50 evidence-based studies, in 40 of which authors concluded that private entrepreneurship had turned out to be much more efficient than public one. in 7 other cases, it was impossible to identify obvious advantages of one or another incorporation forms. in the similar comparison of functioning of private and publicly owned companies, boardman and vining [4] reviewed the results obtained earlier and compared them with own calculations. their main conclusion is as follows: when considering a wide range of factors that influence transaction efficiency, large-scale production companies with joint ownership and similar companies that are 100%owned by a national government are significantly less efficient against similar private companies. muller [31] comes to similar conclusions, summarising findings from 71 studies. in 56 cases, publicly owned enterprises were less efficient. specifics of publicly owned companies makes specific a number of components that affect operational and strategic efficiency [45, 49, 50]. they include politics-driven resource allocation, weak managerial mechanisms and limited managerial autonomy, poor coordination of activities, risk-evasive behaviour of managers, poor market discipline, strong social control, poor employment motivation, bureaucrats, lack of flexibility, etc. in general, the terms, in which publicly owned enterprises operate, are soft budget constraint conditions with no active incentives for development. at private companies with an insignificant state’s share, which has not yet become controlling, mentioned components act to a lesser extent. benefits from such cooperation come to the fore. there are reduction in a cost of investment attraction [9, 16, 29] and protectionism in commodity and resource markets [27]. private companies consider the presence of the national government as a factor to increase competitiveness in terms of growing globalization in economy. note that even the presence of a prevailing state’s share in a company's capital structure is not in itself a prerequisite of inefficiency. however, a circle of the papers that support this idea is narrow. it is noteworthy that such papers only cover a number of infrastructural sectors (4 out of the 5 cases, electric power production and distribution). there, for a number of reasons, competition is poor [39]. when analysing a wider market, there is lower efficiency of publicly owned enterprises against private enterprises operating in the same sector. in many cases, production costs at publicly owned enterprises are higher. a desire of managers at publicly owned enterprises to increase their budget/organizational size (niskanen model [33]) and bureaucrats largely explain inefficiency at publicly owned enterprises. findings from evidence-based research on an impact of public ownership on efficiency of russian companies are not categorical. kuznetsov and murav’yev think that the higher a proportion of outsiders is, the higher processing efficiency of companies is, measured as labour productivity. on the contrary, ownership of insiders and the national government causes lower productivity [25]. kapelyushnikov [20] shows that enterprises with a dominating state’s share in the authorized capital have the worst financial ratios. the higher school of economics’ review [53] includes offbeat findings in terms of the state's influence on efficiency of russian companies. authors of the review focused on the selective influence of the state on individual enterprises in conjunction with their modernization. 26 v.e. guseva, e.a. kuzmin, a.i. vlasov copyright ©2019 assa adv. in systems science and appl. (2019) ruzhanskaya and krutikov [40] believe that, as an owner, the state does not demonstrate efficient management of companies. they have nevertheless found a positive influence of the state’s share in the capital on productivity, investment attractiveness and profitability ratios. at the same time, the state’s share in ownership is not statistically related to the companies’ growth. in absence of the obvious economic growth, the highly concentrated ownership of russian companies has became at the same time a condition and consequence of the developed integration processes in the corporate segment [2, 35]. findings from various evidence-based research of russian production enterprises [24, 42] support such a conclusion [24, 42]. in the completed review, we emphasized that efficiency of publicly owned and private companies is a subject of an extensive discussion. there are sufficient arguments in favour of the assumption that publicly owned companies are comparatively less efficient than the private ones. the question has remained open, to what extent this influence is apparent, whether it is of a selective nature by sector affiliation and concentration level of state ownership in the capital. in the narration below, we will try to make these points clearer. 3. materials and methods one might approach company’s efficiency in terms of various aspects [18, 32]. experts usually distinguish operational and strategic efficiency [38]. this research focuses operational efficiency of companies exactly. its important characteristics include return on equity (roe), return on assets (roa), return on sales (ros), equity ratio and current liquidity ratio. research source data included data from financial statements of companies as reported by the interfax centre for corporate information disclosure [10]. recent 7 years was a period for the analysis (2012-2018). all the publicly owned companies for the research were grouped by the international monetary fund (imf) classification [14]: (1) 100%-publicly owned. the group includes the entities that are 100-percent ownership of the national government (e.g. russian railways (rr), united shipbuilding corporation (usc)), (2) joint companies. the group includes the companies jointly established by governmental and non-governmental entities, as well as the companies established by governmental entities and non-resident entities, in which a share of a non-resident entity does not exceed 10%, (3) private companies with participation of the state. for the research, we selected 12 companies from various sectors of economy. see their general characteristics in table 1. companies represent different branches of economy, their gross output share is extremely large. according to spark-interfax, over 2018, these 12 companies account for 16.5 trillion rubles of revenue, which is about more than 12% of the turnover of all the russian companies. undoubtedly, the limited sample has its disadvantages. meanwhile we rely on the sample importance in terms of the scale and share of participation in economy. table 1. description of the companies that have been selected for the research name sector public ownership, % rr transportation 100.00 usc shipbuilding 100.00 gazprom oil and gas production 50.23 rosneft oil and gas production 50.00 uac aircraft industry 96.95 transneft transportation of oil and oil products 20.86 inter rao electric power sector 44.30 rushydro electric power sector 75.40 rosseti power sector 88.89 rostelecom telecommunications 54.90 tatneft oil and oil product production 34.00 alrosa diamond mining 66.00 operational efficiency of large-scale publicly owned companies 27 copyright ©2019 assa. adv. in systems science and appl. (2019) source: compiled as reported by spark-interfax [47]. to study specifics of operational efficiency at publicly owned companies in russia, we put forward a number of assumptions. assumption testing followed methods of the correlation-regression analysis. correlating makes it possible to estimate a strength of a component influence on the process under consideration. the linear correlation coefficient takes values from -1 to +1. correlations between attributes might be poor and strong (close). their criteria are estimated with the cheddok scale [21]. r2 sample determination coefficient is a measure of an overall quality of the regression equation (correspondence of the made equation with statistical data). by definition, r2 takes values in the range [0;1]. the closer r2 is to one, the better the regression approximates empirical data. statistical significance of individual regression ratios is checked with student’s t-test, and the equation as a whole – using fisher’s f-test. 4. results and discussion 4.1. overall analysis of economic standing of companies based on financial statements, we calculated operational efficiency ratios for each company under consideration (annex a). see table 2 for summarized estimates for 2018. the current state of publicly owned companies provides the first illustration of their operating performance at the moment that is regarded further over the preceding period. cumulative data provides with comparing, first of all, the differences in branch specific nature; secondly, study their heterogeneity. the demonstrative example is labour productivity, which is significantly diversified. the explanation of this consists in performing a range of executive functions by publicly owned companies as united shipbuilding corporation (usc). table 2. operational efficiency ratios for sample companies in 2018 company/ ratio roa roe ros equity ratio current liquidity ratio labour productivity, thousand rub/ human conventional operational efficiency rr 0.30 0.42 7.83 0.70 0.47 2391 low usc 0.04 0.12 -3.12 0.31 10.33 126265 low gazprom 6.8 9.88 18.02 0.66 1.96 11113 high rosneft 3.94 24.83 5.08 0.16 1.43 22624 low uаc -6.63 -9.81 -0.72 0.66 1.76 4375 low transneft 0.98 5.46 5.75 0.18 0.79 8147 low inter rao 4.4 4.92 1.64 0.86 2.14 1113 high rushydro 3.64 4.38 40.52 0.82 7.63 2336 high rosseti -2.93 -3.2 5.83 0.92 36.22 125 low rostelecom 0.92 2.11 3.99 0.41 0.58 2372 low tatneft 24.81 31.33 31.49 0.78 3.14 37551 high alrosa 5.29 8.15 20.13 0.63 1.58 6280 high let us consider in more detail a standing of each company at runtime in 2012-2018. at rr, roa and roe were low in the period under consideration. thus, in 2017, the sector-specific average return on assets at rail transport enterprises engaged in intercity and international passenger traffic was 1.1% [22], whereas at rr, it was 0.30%. at the same time, the return on sales was comparable to the sector-specific average of 8.3% (at rr, it was 8.25%). in 2012-2018, at rr, dynamics of profitability ratios was unstable. by 2014, there had been a clear decline in values. a recovery in values to values of 2012 had only been recorded by 2017-2018. in 2012-2018, at rr, the equity ratio was higher than the statutory value (0.5%), with an obvious decline of this ratio from 0.8% to 0.7%. the current liquidity ratio was significantly lower than the statutory value (1.5-2.5) and had the steady decreasing dynamics over the period under consideration. it is possible to conclude that rr has relatively low operational efficiency. 28 v.e. guseva, e.a. kuzmin, a.i. vlasov copyright ©2019 assa adv. in systems science and appl. (2019) operational efficiency ratios of another publicly owned company, usc, assume that its profitability ratios are also low in spite of the positive dynamics in 2012-2015. as reported by the federal tax service of russia (fts), in 2017, the return on assets in production of other vehicles and equipment was 1.9% [22]. at usc, this ratio was at the level of 0.03%. the average sector-specific return on sales was 12.9%, while at usc, the value was negative (3.78%). at the same time, in 2018, the six-fold excess of the current liquidity ratio over the statutory value points out to an excess and inefficient use of funds. in 2012-2018 at gazprom, the return on assets and return on equity had a tendency towards an increase, while the return on sales significantly declined. according to the federal tax service, in the sector of crude oil and gas production, in 2017, the average return on assets was 11.1% [22]. at gazprom, this ratio was more than 2 times higher (25.2%). the company’s equity ratio is relatively high (share of equity capital is at the level of 66%), but there is a tendency towards its decline. the current liquidity ratio fully complies with statutory values. thus, at gazprom, operational efficiency ratios are quite high with the clear positive dynamics. next, let us analyse rosneft activities. the state’s share in the company’s capital was over 50%. calculated data assume the negative changes available in profitability ratios that are significantly lower than sector-specific average values. there are also low levels of the equity ratio. in 2018, the share of equity was only 16% of total liabilities. the current liquidity ratio follows this tendency being below the statutory ratio. it is obvious that at rosneft, operational efficiency is relatively low. with the state’s share of over 96%, united aircraft corporation (uac) was not distinctive in operational efficiency. as dynamics for 2012-2018 assumes (annex a), operational efficiency was low. negative profitability values were observed almost over the entire period under consideration. transneft was another company in the sample. in 2018, the company's return on assets, as well as return on equity sharply dropped, roa was 0.98%, and roe was 5.46%. at the same time, as early as in 2017, values were 5.40% and 31.53%, respectively. sector-specific average roa in the pipeline transportation sector is about 5.5% [22], while ros is 13.5% [22]. in 2018, the equity ratio was 18%. the current liquidity ratio also was below the normal level. at inter rao, the effective state’s share is 44.3%. its operational efficiency ratios show the positive dynamics, while the capital structure is highly sustainable. the current liquidity ratio was also within the standard statutory range. in 2012-2013, over the short period of time, profitability ratios had been negative. however, by 2016, they had already increased. the average return on sales in generation, transmission and distribution of electric power is at the level of 11.1% [22], whereas at inter rao, in 2017, this type of profitability was negative (0.19%). the return on assets was slightly lower than the sector-specific average value, 3.95% versus 4.8% in 2017. in general, inter rao typically has relatively high operational efficiency. rushydro is another representative of the electric power sector. minor fluctuations in roe and ros in 2012-2018 (annex a) confirm sustainability and rigidity of a company's business model. at the same time, values of profitability ratios are close to sector-specific average values of roa and much better in terms of ros. in 2018, the equity share in the capital structure was 82% and the current liquidity ratio was 5 times higher than the standard statutory rate. rosseti stands out from this list to some extent, which also operates in the electric power sector. its main activity includes the long distance electric power transmission and management of the energy system. in the structure of the shared capital at rosseti, the state’s share is 88.89%. the company has numerous subsidiaries, which, due to the high state’s share in the parent company, also have high dependence on government entities. operational efficiency of the company was low in terms of profitability ratios. at the same time, in 2018, operational efficiency of large-scale publicly owned companies 29 copyright ©2019 assa. adv. in systems science and appl. (2019) the equity share was 92%, the current liquidity ratio was more than 24 times more than the statutory value, which points out to excessive working assets. rostelecom is the largest telecoms provider in russia. in 2012-2018, its return on assets and profitability ratio (annex a) had negative dynamics being below the required threshold. thus, in 2017, in the information and communication sector, sector-specific average roa and ros were 9.1% and 14.2%, respectively [22], whereas, in 2017, rostelecom’s roa was at the level of 4.64%, ros was at the level of 4.37%. the equity ratio was also lower than the recommended rate (41% of equity). tatneft is another representative of publicly owned business. the state’s share at this company is 34%. the analysis completed showed that the company's operational efficiency is quite high as evidenced by the positive dynamics of profitability ratios (roa, roe and ros) (all ratios increased over the period and their values were higher than sector-specific average values). the equity ratio was 0.78 (against the statutory ratio of 0.50). in the period under review, the current liquidity ratio was also higher than the statutory value. alrosa was a final company in the research sample. as one can see from calculated data (annex a), all profitability ratios in 2012-2018 had tendencies towards a decline and their values were below average sector-specific values. in the mining industry, average roa is about 11.0% [22]. however, in 2017, the company achieved the profitability of almost 2 times below than this value is (4.18%). at the same time, average sector-specific ros is 25.9%. in 2012-2016, alrosa exceeded the average sector-specific ros, but in 2017-2018, it was significantly inferior in this aspect. the current liquidity ratio and equity ratio are in the statutory range. the analysis suggests that the company has relatively high operational efficiency. for the research, the dynamics of industry average of the russian companies operational efficiency is important to be noted. the federal tax service of russia regularly monitors average return on average assets (roa) and return on sales (ros). in 2018, roa for hc and gas production totaled 35.3; for railway sector, it totaled 7.6; for pipeline industry, it totaled 12.2; for electrical power distribution and transmission, it totaled 12.5; for information and communication service, it totaled 14.6; for mining operations if totaled 33.6; for other transport and equipment production it totaled 12.0. the average of ros for russia in 2018 totaled: for hc and gas production totaled 20.4; for railway sector it totaled 20.2; for pipeline industry it totaled 4.0; for electrical power distribution and transmission it totaled 5.0; for information and communication service it totaled 7.4; for mining operations if totaled 17.3; for other transport and equipment production it totaled 2.0. although, the industry average of operational efficiency resulted from both private and public companies. therefore, the comparison of the obtained values of efficiency with average ones will be incorrect and inconsistent. the research goal is to determine of efficiency dependency based on changing of government share in company’s assets. thus, observed heterogeneity in operational efficiency among publicly owned companies in the sample leads us to a need in a research of a component influence. 4.2. component influence on operational efficiency in order to study specifics of operational efficiency at publicly owned companies in russia, we put forward a number of assumptions. we tested assumptions using methods of the correlation-regression analysis. the study conceptual framework involves identification of a relationship between a value of the state's share in the authorized capital and a value of each operational efficiency ratio (profitability ratio, return on assets, and return on sales, equity ratio, and current liquidity ratio). numerous evidence-based studies and the findings obtained by us from the analysis of a financial standing of a number of publicly owned russian companies make it possible to put forward the idea that an ownership structure has an essential impact on operational efficiency of these companies. this assumption became the background of the research, which 30 v.e. guseva, e.a. kuzmin, a.i. vlasov copyright ©2019 assa adv. in systems science and appl. (2019) presumes the correlation between operational efficiency values and the government share in company’s assets. this idea found its expression in five independent assumptions as follows. meanwhile, this correlation must be inexplicit, as operational efficiency is influenced by many other factors, which are not included in evaluation further. besides, the dependency of operational efficiency on government share should be observed not on the whole sample, but on its part, since differential peculiarities of public administration are common for not all managers, mechanism of business operations takes place to be objectively, socio-economic conditions of work are heterogenic, etc. hypothesis 1: roa depends on the state's share in the company’s capital. the higher the state’s share is, the lower roa is. the calculations completed show that the pair-correlation coefficient between the state's share in the company's ownership and return on assets was -0.57151. in this case, the correlation is inverse and significant against the parameters under consideration. the linear regression equation looks like y = -0.162 x + 14.038. the regression coefficient (b = -0.162) shows that the change in the resulting ratio is elastic. in this case, with the 1-point growth, roa has an average decline of -0.162. the free constant (a = 14.038) formally shows an expected initial rate of the resulting ratio with x (state’s share) = 0. to check whether the linear regression equation is significant, determination coefficient r2 is used and its value is 0.327. next, there is a calculation of the actual value of f-test (4.85). the tabular value of the criterion with degrees of freedom k1=1 and k2=10, ftable = 4.96. as far as the actual value of f > ftabl, the determination coefficient is statistically significant (estimated value of regression equation is statistically reliable). the assumption does not lack support, roa declines with the growth in the state’s share in the company's capital. although, cause-effect relationship is not well-defined. the influence of other factors that are not taken into account in the framework of this regression model is not excluded. hypothesis 2: roе depends on the state’s share in the company’s capital. the higher the state’s share is, the lower roe is. the correlation coefficient between roe and the state's share in the company’s capital was 0.64310. this suggests a measurable correlation between attributes. the linear regression equation looks like y = -0.2715 x + 24.231. the regression coefficient (b = -0.271) shows that the change in the resulting ratio is elastic. in this case, with the 1point growth in the state’s share, roe declines on average by -0.271. with х (state’s share) = 0, initial expected roe is 24.231. determination coefficient r2 is 0.4136. the actual value of f-test is 7.05. the tabular value of the criterion with degrees of freedom k1=1 and k2=10, f tabl = 4.96. as far as the actual value of f > ftabl, the determination coefficient is statistically significant (estimated value of regression equation is statistically reliable). the assumption does not lack support, roa declines with the growth in the state’s share in the company's capital. as in case of hypothesis 1, cause-effect relationship of observed values is questionable. the correlation can be confirmed with expanding of sample and the scopes of observations. hypothesis 3: ros depends on the state’s share in the company’s capital; the higher the state’s share is, the lower ros is. the pair correlation coefficient between the state's share in the company’s capital and the return on sales was -0,23513. according to the typology, the correlation between attributes is poor. operational efficiency of large-scale publicly owned companies 31 copyright ©2019 assa. adv. in systems science and appl. (2019) the linear regression equation looks like y = -0.117 x + 19.001. the regression coefficient (b = -0.117) shows that the change in the resulting ratio is elastic. in this case, with the 1-point growth in the state’s share in the capital, ros declines on average by -0.117. the free constant a = 19.001 while х (state’s share) is 0. determination coefficient r2 is 0.0553. next, there is a calculation of the actual value of f-test (0,59). a tabular value of the criterion with degrees of freedom k1=1 and k2=10, ftable = 4.96. as far as the actual value of f > ftabl, the determination coefficient is not statistically significant (estimated value of regression equation is not statistically reliable). the assumption lacks support. ros does not decline with a growth of the state’s share in the company's capital. hypothesis 4: the equity ratio depends on the state’s share in the company’s capital. the higher the state’s share is, the lower the er is. the correlation between the equity ratio and the state’s share was 0.26209 saying of the poor direct correlation. the linear regression equation is y = 0.00255 x + 0.425. the regression coefficient (b = 0.00255) shows that the change in the resulting ratio is elastic. in this case, with the 1-point growth of the state's share in the capital, the equity ratio has an average increase of 0.00255. if х (state’s share) = 0, an initial expected rate of the equity ratio is 0.425. determination coefficient r2 is 0.0687. the actual value of f-test is 0.704. as far as the actual value of f < ftabl, the determination coefficient is not statistically significant (estimated value of regression equation is not statistically reliable). the assumption lacks support. the equity ratio does not decline with an increase in the state’s share in the company’s capital. hypothesis 5: the current liquidity ratio depends on the state's share in a company's capital. the higher the state’s share is, the lower the current liquidity ratio is. the correlation between the equity ratio and the state’s share in the company’s equity was 0.38002. its level is an evidence of a direct moderate correlation in place. the linear regression equation is y = 0.142 x -3.565. the regression coefficient (b = 0.142) shows that the change of the resulting ratio is elastic. in this case, with a 1-point growth of the state’s share, the current liquidity ratio has an average increase of 0.142. at the same time, the free constant takes a negative value (a = -3.565) when x (state’s share) is 0. determination coefficient r2 is 0.1444. next, there is a calculation of an actual value of f-criterion (1.688). as the actual value of f < ftab, the determination coefficient is not statistically significant (found estimate of regression equation is not statistically reliable). the assumption lacks support. the current liquidity ratio does not decline as the share of public ownership in a company’s capital grows. thus, the component analysis of operational efficiency at publicly owned companies in russia has showed that only return on assets and profitability ratio are dependent on the ownership structure. with the 1%-increase in public ownership, the ratios have a decline by 0.162% and 0.271% respectively. the other operational efficiency ratios that we have considered (return on sales, equity ratio and current liquidity ratio) do not depend on the share of public ownership. 5. conclusion the public sector is a key player in economy of russia. over the past few years, a number of publicly owned companies has had a steady growth, while a share of the public sector in russia's gdp has increased up to 46%. the completed component analysis based on correlation-regression methods shows that the size of the publicly owned share has an overall 32 v.e. guseva, e.a. kuzmin, a.i. vlasov copyright ©2019 assa adv. in systems science and appl. 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(2005). entrepreneurship in the public sector: a framework of analysis in european local governments, entrepreneurship & regional development, 17(1), 43-64. https://doi.org/10.1080/0898562042000310723 [55] ra expert. (2018). [online]. available https://raexpert.ru/rankingtable/top_companies/2018/main/ [56] ria rating. (2019). [online]. available http://riarating.ru/infografika/20190129/630115992.html https://doi.org/10.1057/9780230624955 https://doi.org/10.1080/0898562042000310723 36 v.e. guseva, e.a. kuzmin, a.i. vlasov copyright ©2019 assa adv. in systems science and appl. (2019) annex a table. operational efficiency ratios at runtime at sample publicly owned companies in 2012-2018 ratio research period ( )s x ( )var x 2012 2013 2014 2015 2016 2017 2018 russian railways (rr) roa 0.33 0.02 -0.93 0.01 0.12 0.30 0.30 0.02 0.17 roe 0.41 0.02 -1.25 0.01 0.17 0.41 0.42 0.03 0.30 ros 4.94 4.27 4.17 5.53 7.43 8.25 7.83 6.06 2.59 er 0.80 0.77 0.73 0.70 0.74 0.72 0.70 0.74 0.00 cr 0.65 0.67 0.74 0.76 0.57 0.56 0.47 0.63 0.01 united shipbuilding corporation (usc) roa -0.37 -0.19 0.19 0.56 0.12 0.03 0.04 0.05 0.09 roe -0.46 -0.28 0.34 1.29 0.34 0.10 0.12 0.21 0.32 ros -275.72 0.12 -42.61 -6.49 -40.49 -3.78 -3.12 -53.16 9960.67 er 0.76 0.59 0.52 0.38 0.35 0.36 0.31 0.47 0.03 cr 1.88 2.58 4.44 10.55 6.15 5.65 10.33 5.94 11.78 gazprom roa 5.69 6.01 1.64 3.20 3.07 2.66 6.80 4.15 3.91 roe 7.21 7.73 2.16 4.38 4.04 3.51 9.88 5.56 7.59 ros 27.14 24.45 23.09 18.73 8.46 2.33 18.02 17.46 81.02 er 0.79 0.77 0.74 0.72 0.80 0.72 0.66 0.74 0.00 cr 2.21 2.41 2.28 2.35 2.02 1.63 1.96 2.12 0.07 rossneft roa 13.06 3.65 7.85 2.78 1.02 1.32 3.94 4.80 18.35 roe 25.34 10.34 36.54 17.16 6.69 8.64 24.83 18.51 119.56 ros 9.40 5.61 3.62 3.05 0.35 3.25 5.08 4.34 7.84 er 0.50 0.28 0.17 0.15 0.15 0.15 0.16 0.22 0.02 cr 4.36 1.24 1.26 2.08 1.33 1.31 1.43 1.86 1.30 united aircraft corporation (uаc) roa -0.03 0.26 2.75 -2.49 -0.70 0.04 -6.63 -0.97 8.61 roe -0.04 0.36 4.11 -3.59 -0.96 0.06 -9.81 -1.41 18.86 ros -2.64 -7.37 -3.06 -3.32 -0.47 -0.62 -0.72 -2.60 5.91 er 0.74 0.69 0.65 0.73 0.73 0.69 0.66 0.70 0.00 cr 6.97 11.84 2.80 3.42 2.82 2.52 1.76 4.59 13.04 transneft roa 1.17 1.22 1.14 1.07 2.58 5.40 0.98 1.94 2.64 roe 7.56 7.61 7.68 7.98 17.55 31.53 5.46 12.20 88.09 ros 2.68 1.71 1.71 2.76 4.40 5.78 5.75 3.54 3.11 er 0.16 0.16 0.14 0.13 0.16 0.18 0.18 0.16 0.00 cr 0.97 0.26 0.19 1.28 0.83 0.87 0.79 0.74 0.15 inter rao roa -3.58 -13.58 0.12 1.05 24.60 3.95 4.40 2.42 132.90 roe -4.11 -15.03 0.13 1.12 25.51 4.10 4.92 2.38 149.38 ros -2.68 -4.70 -1.43 6.39 0.60 -0.19 1.64 -0.05 12.54 er 0.86 0.95 0.94 0.94 0.98 0.94 0.86 0.92 0.00 cr 1.57 3.29 2.56 4.49 9.99 4.49 2.14 4.08 8.04 rushydro roa 2.08 4.50 3.68 3.43 4.65 3.82 3.64 3.69 0.71 roe 4.44 5.80 4.58 4.11 5.54 4.53 4.38 4.77 0.41 ros 40.57 43.49 37.17 40.27 37.85 42.08 40.52 40.28 4.88 er 0.79 0.76 0.84 0.83 0.85 0.84 0.82 0.82 0.00 cr 3.04 3.67 6.11 5.49 7.40 3.80 7.63 5.31 3.42 rosseti roa -2.11 -141.67 -30.77 -10.78 72.97 -3.24 -2.93 -16.93 4079.08 roe -2.36 -150.73 -31.01 -12.12 82.46 -3.53 -3.20 -17.21 4763.67 ros 44.41 46.59 38.12 33.34 5.37 5.49 5.83 25.59 369.26 er 0.87 1.00 0.99 0.82 0.92 0.92 0.92 0.92 0.00 cr 1.91 18.56 10.99 5.11 9.17 17.47 36.22 14.20 130.64 rostelecom roa 6.21 6.43 5.44 3.87 1.78 1.48 0.92 3.73 5.52 roe 11.45 13.28 11.80 7.99 5.38 4.64 2.11 8.09 17.83 operational efficiency of large-scale publicly owned companies 37 copyright ©2019 assa. adv. in systems science and appl. (2019) ratio research period ( )s x ( )var x 2012 2013 2014 2015 2016 2017 2018 ros 17.87 16.32 14.42 11.63 4.30 4.37 3.99 10.41 37.20 er 0.52 0.44 0.48 0.49 0.19 0.46 0.41 0.43 0.01 cr 0.62 1.34 0.61 0.48 0.49 0.66 0.58 0.68 0.09 tatneft roa 13.80 12.30 14.78 13.98 15.34 13.52 24.81 15.50 17.76 roe 18.86 15.96 18.10 16.50 17.92 16.26 31.33 19.28 29.42 ros 29.61 26.89 23.37 25.80 18.76 21.48 31.49 25.34 20.12 er 0.75 0.79 0.84 0.85 0.86 0.81 0.78 0.81 0.00 cr 5.63 4.50 4.02 3.51 3.31 3.75 3.14 3.98 0.74 alrosa roa 11.65 8.68 5.09 3.84 23.24 2.73 5.29 8.65 50.67 roe 19.96 15.39 9.67 7.63 40.18 4.18 8.15 15.02 150.92 ros 40.05 38.46 42.64 45.67 19.64 14.76 20.13 31.62 166.08 er 0.56 0.56 0.49 0.51 0.64 0.67 0.63 0.58 0.00 cr 2.26 1.50 3.64 2.45 4.26 2.03 1.58 2.53 1.09 note: er is equity ratio, cr is current ratio, ( )s x is time average value; ( )var x is dispersion. adv syst sci appl 2021; 01; 95-112 published online at https://ijassa.ipu.ru. improving dcp haze removal scheme by parameter setting and adaptive gamma correction cheng-hsiung hsieh1*, yi-hung chang2 1) department of computer science and information science, chaoyang university of technology, taichung, taiwan e-mail:chhsieh@cyut.edu.tw 2) department of computer science and information science, chaoyang university of technology, taichung, taiwan e-mail: s10827616@ cyut.edu.tw abstract: recently, single-image haze removal based on the dark channel prior (dcp), originally proposed by he et. al., has attracted much attention in the image restoration community. this dehazing algorithm, called the dcp scheme here, is well-known to have four main problems in its dehazed images: artifacts, hue distortion, color over-saturation, and halos. in this paper, an improved dcp (idcp) is proposed to deal with the four aforementioned problems, by setting the model parameters, i.e. scaling factors and window size and smoothing factor of a guided image filter in the dcp scheme. note that a dehazed image is generally dim and low in contrast. an adaptive gamma correction (agc) is introduced for dehazed image enhancement. the proposed idcp and agc are used to create the idcp/agc scheme, in which the idcp scheme performs haze removal and the agc enhances the dehazed image. the idcp/agc scheme was justified through extensive experiments and compared with the dcp scheme, an optimization-based scheme, and two learning-based schemes on two datasets. the results indicated that the proposed scheme is subjectively and objectively superior to the comparison schemes. keywords: dark channel prior, single image haze removal, adaptive gamma correction 1. introduction image haze removal by far is an active research field in the image restoration and image enhancement community. the haze mainly resulted from the adversary weather condition degrades the visual quality of an image. a hazy image is generally of low contrast and visibility, which may affects the following computer visionary applications, such as video-based surveillance systems, automatic driver’s assistance systems, and object tracking/recognition systems. since haze removal is in great need, much work has been done in this field. a popular class of haze removal schemes relies on statistical observations and thus assumptions, such as contrast assumption [1], un-correlation assumption [2], dark channel prior [3], and color line [4]. besides, in [5], a pioneer work based on machine learning for single image haze removal was proposed. in [6], a multi-scale convolutional neural networks was presented to estimate parameters in the haze image model. in [7], an end-to-end framework based on a deep learning neural network for single image haze removal was introduced. in [8], another end-to-end framework, called all-in-one dehazing network, was proposed. in [9-10], generative adversarial networks were applied to image haze removal. for more other schemes, one may consult survey papers in [11-13]. this paper will concentrate on the popular scheme based on dark channel prior, which was originated by he et al. in [3]. the scheme will be called the dcp scheme hereafter. with its * corresponding author: chhsieh@cyut.edu.tw 96 c.-h. hsieh, y.-h. chang copyright ©2021 assa. adv. in systems science and appl. (2021) simplicity, the dcp scheme has attracted much attention recently. however, the original dcp scheme has several problems needed to be improved, i.e. computational cost, artifacts, color distortion, and halos. to reduce the computational cost, in [14] he et al. used a guided image filter (gif) to replace the soft matting algorithm in [3] for transmission map refinement. the artifacts and color distortion usually occur in the sky region of a dehazed image, while the problem of halos happens in large depth discontinuities. to improve the dcp scheme, many researchers have proposed a variety of solutions to relieve the aforementioned problems. some of them are listed below. in [15], with a soft segmentation, the transmission map estimation was divided into two part: one for the sky region and the other for the non-sky region. for the sky region, a luminance model was applied to estimate the transmission map, while the dark channel prior was used to estimate the transmission map in the non-sky region. then the two transmission maps were fused to obtain the final transmission map. in [16], a sky segmentation scheme based on quad-tree splitting was presented to deal with the problem of the dcp scheme in the sky region, where the transmission map was refined by an edge preserving gif. in [17], the sky region is segmented through a quad-tree decomposition and region-growing scheme. then the estimate of transmission map obtained from the dark channel was modified based on the segmented sky region. the atmospheric light was estimated through the segmented sky region. in [18], the transmission map was found through minimization of an energy function with a piecewise smooth assumption. in [19], a segmentation scheme based on the binary image from gradient image of input image was employed to separate sky regions, through which an estimate of atmospheric light was found. with the help of edge information, an initial transmission map was obtained and then refined by a gif. in [20], a light intensity reverse algorithm was proposed to deal with the artifact problem in the sky region or white objects. by a threshold, the region of interest was segmented, whose rgb pixel values were replaced with a constant manually specified by the user. then the dcp scheme was applied to find the dehazed image. in [21], the initial transmission map was found by the 5 × 5 minimum filter and then refined by the sobel filter and mean filter. to avoid color distortion, a pixel-based adaptive lower bound for the final transmission map was calculated through a constrained piece-wise linear function. in [22], a saliency detection was proposed to extract white objects according to superpixel intensity contrast. then the atmospheric light and transmission map were estimated with the preprocessed image. besides, an adaptive upper bound was given to avoid over-exposure in dehazed images. in [23], a two-stage transmission map estimation was employed. in the first stage, a dehazed image was found by the dcp scheme. in the second stage, the pixel-based transmission map was obtained from the dehazed image. then morphological operations were applied to find the final transmission map. finally, the image was restored by the final transmission map and atmospheric light estimated by the dcp scheme in the first stage. in the aforementioned schemes, most of them use a segmentation skill to separate sky and nonsky regions to avoid the artifacts and color distortion found in the sky region, since many researchers conjecture that the problem is due to the inappropriateness of dark channel prior for the sky region. moreover, many researchers attribute the halo problem to the large depth discontinuities in the input image. however, the two conjectures are arguable. though the dark channel prior is not suitable in the sky region or white objects, it may not suggest that the dcp scheme is inappropriate accordingly. in [3], it says that the sky and non-sky regions can be handled gracefully, even the dcp is not a good prior for sky regions. this paper will justify that the statement is correct, through the proposed improved dcp (idcp) scheme, where no segmentation is required for the sky and non-sky regions. in addition, problems of artifacts, hue distortion, color over-saturation, and halos in the dcp scheme will be relieved by the proposed idcp scheme through model parameter setting. section 3 will have the details. in general, a dehazed image is dim and of low contrast. conventional image enhancement methods, such as histogram equalization (he) and gamma correction (gc), are not suitable for dehazed image enhancement, since over enhancement and color distortion happen most of improving dcp haze removal scheme... 97 copyright ©2021 assa. adv. in systems science and appl. (2021) time. even so, some researchers have tried to apply he and gc to image haze removal. in [24], an adaptive gc was applied in the transmission map estimation. in [25], the haze image model and a gc were combined to form a model, called concise gamma-correction-based dehazing model. in their results, color distortion can be found in the given examples. in [26], a contrast limited adaptive he (clahe) was applied to visibility enhancement of the dehazed image. however, over enhancement is found, when sky region is large. this paper will propose an adaptive gamma correction, which can be applied to dehazed image enhancement without color distortion. this paper has four main contributions. first, an alternative solution is proposed to the problems of artifact and color distortion in sky regions, that is, by an adaptive scaling factor in the atmospheric light estimation. this is different from the sky and non-sky region segmentation schemes as described previously. second, artifacts in the sky region and color distortion in a dehazed image is relieved by an adaptive scaling factor in the estimation of initial transmission map. thus, no sky and non-sky region segmentation is required. this provides a way to mitigate this problem. third, the halo problem happened in large discontinuities is solved by a gif parameter setting in the transmission refinement. this doing gives a solution to the halo problem. fourth, an adaptive gamma correction (agc) is presented to alleviate the color distortion in the conventional gamma correction. then the proposed agc is employed to dehazed image enhancement. this brings for the possibility to apply gamma correction in the image haze removal field. this paper consists of two parts. the first part is to present an improved dcp (idcp) dehazing scheme and the second part is to introduce an adaptive gamma correction (agc) as a postprocessing to enhance the dehazed image. in section 2, the dcp scheme is briefly reviewed. in section 3, the proposed idcp dehazing scheme and the proposed agc for dehazed image enhancement are introduced. then the two schemes are combined as the idcp/agc scheme. next, the proposed idcp/agc scheme is extensively justified by two image datasets and compared with four dehazing schemes in section 4. finally, conclusion is made in section 5. 2. review of the dcp scheme the dcp scheme is based on the following haze image model, 𝑰(𝑥) = 𝑱(𝑥)𝑡(𝑥) + 𝑨[1 − 𝑡(𝑥)] (2.1) where 𝑰(𝑥) is the observed hazy image; 𝑱(𝑥) is the haze-free image; 𝑨 is the global atmospheric light or simply atmospheric light; 𝑡(𝑥) = 𝑒−𝛽𝑑(𝑥) is the transmission map which represents the portion of the non-scattered light to the camera; 𝛽 is the scattering coefficient of the atmosphere and 𝑑(𝑥) is the scene depth at position 𝑥. the dcp scheme is based on the following observation. in general, at least one of rgb components has very low intensity in the non-sky pixels of a haze-free image. this statistical observation is called dark channel prior in [3], which can be obtained through a block minimum filter. the result is called dark channel. fig. 2.1 is an example to justify the dark channel prior where the 15×15 minimum filter is employed. (a) (b) fig. 2.1. image forest (a) original (b) the corresponding dark channel the derivation to obtain the initial transmission map in [3] is briefly reviewed in the following. assume a hazy image 𝑰 is in the rgb color space and without sky regions or white objects. when considering one component of 𝑰, eq. (2.1) can be rewritten as 98 c.-h. hsieh, y.-h. chang copyright ©2021 assa. adv. in systems science and appl. (2021) 𝐼𝑐(𝑥) = 𝐽𝑐(𝑥)𝑡(𝑥) + 𝐴𝑐 − 𝐴𝑐𝑡(𝑥) (2.2) where 𝑐 ∈ {𝑅, 𝐺, 𝐵}. next, eq. (2.1) is normalized by 𝐴𝑐 and is obtained as 𝐼𝑐(𝑥) 𝐴𝑐 = 𝐽𝑐(𝑥) 𝐴𝑐 𝑡(𝑥) + 1 − 𝑡(𝑥) (2.3) assume the transmission map within the block ω(𝑥) is a constant. through a block minimum filter, the dark channel in eq. (2.3) is obtained as min 𝑦∈ω(𝑥) min 𝑐 [ 𝐼𝑐(𝑥) 𝐴𝑐 ] = �̃�(𝑥) min 𝑦∈ω(𝑥) min 𝑐 [ 𝐽𝑐(𝑥) 𝐴𝑐 ] + 1 − �̃�(𝑥) (2.4) where �̃�(𝑥) denotes as the initial transmission map. by the property of dark channel prior, the dark channel for haze-free image 𝐽𝑐(𝑥) approaches to zero and thus min 𝑦∈ω(𝑥) min 𝑐 [ 𝐽𝑐(𝑥) 𝐴𝑐 ] = 0 is considered in eq. (2.4). consequently, by eq. (2.4) the initial transmission map can be estimated from the input image 𝑰 as �̃�(𝑥) = 1 − min 𝑦∈ω(𝑥) min 𝑐 [ 𝐼𝑐(𝑥) 𝐴𝑐 ] (2.5) to avoid the halo problem, the initial transmission map �̃�(𝑥) is further refined by the soft matting algorithm in [3] or a guided image filter (gif) in [14]. for more details, one may consult [3, 14]. given image 𝑰 in the rgb color space, the implementation steps of the dcp scheme are given as follows. step 1. find the initial block dark channel through a block minimum filter as 𝐼ω 𝑑𝑎𝑟𝑘(𝑥) = min 𝑦∈ω(𝑥) min 𝑐 [𝐼𝑐(𝑦)] (2.6) where ω(𝑥) is a 𝑁 × 𝑁 window centered at 𝑥 and 𝑐 ∈ {𝑅, 𝐺, 𝐵}. in [3], 𝑁 = 15 is employed. step 2. estimate the atmospheric light 𝑨 = [𝐴𝑅 𝐴𝐺 𝐴𝐵] by 𝐼ω 𝑑𝑎𝑟𝑘(𝑥). find the 0.1% pixels of the highest values in 𝐼ω 𝑑𝑎𝑟𝑘(𝑥); trace back to the corresponding pixels in image 𝑰; and find the pixel with the highest intensity as the estimate of 𝑨. step 3. calculate the normalized block dark channel as 𝐼ω̅ 𝑑𝑎𝑟𝑘(𝑥) = min 𝑦∈ω(𝑥) min 𝑐 [ 𝐼𝑐(𝑦) 𝐴𝑐 ] (2.7) step 4. obtain the initial transmission map as �̃�(𝑥) = 1 − 𝜔 × 𝐼ω̅ 𝑑𝑎𝑟𝑘(𝑥) (2.8) where 0 < 𝜔 ≤ 1 is a scaling factor. in [3], 𝜔 is set to 0.95. step 5. refine the initial transmission map �̃�(𝑥) by the soft matting algorithm in [3] or by the gif) in [14] to obtain the final transmission map 𝑡(𝑥). when the gif is used, the settings given in [14] are as follows: the input image 𝑰 as the guidance image; the window size 𝑁 = 20; the smoothing factor 𝜖 = 0.001. step 6. recover the scene radiance as 𝐽𝑐(𝑥) = 𝐼𝑐(𝑥)−𝐴𝑐 max[𝑡0,𝑡(𝑥)] + 𝐴𝑐 (2.9) where 𝑡0 is a user-defined lower bound of 𝑡(𝑥). it is set to 0.1 in [3]. 3. the proposed idcp scheme and agc this section will introduce the proposed idcp haze removal scheme and the agc for dehazed image enhancement in this study. in section 3.1, the idcp scheme is presented to improve the dcp performance. then, the post-processing agc scheme is proposed to enhance the visual quality of the dehazed image by the idcp scheme in section 3.2. 3.1. the idcp dehazing scheme it is well-known that for better performance, at least four problems should be appropriately addressed in the dcp scheme: estimation of 𝑨, estimation of 𝑡(𝑥), the sky/non-sky region handling, and halos. the solutions in the idcp scheme are given below. improving dcp haze removal scheme... 99 copyright ©2021 assa. adv. in systems science and appl. (2021) 3.1.1. atmospheric light estimation in the dcp scheme, the hue distortion is often found in the dehazed images, especially in the sky region. to see the problem, image village is served as an example, which is shown in fig. 3.1(a). the dehazed image by the dcp is shown in fig. 3.1(b) where the sky region is of hue distortion. in [27], it has proven that the hue distortion in a dehazed image results from the estimation error of 𝑨. in other words, the problem of hue distortion in the dcp scheme is from an inappropriate estimation of 𝑨. fortunately, the problem can be relieved by introducing a scaling factor 𝛼 on 𝑨. in the dcp scheme, the scaling factor of 𝑨 can be considered as 𝛼 = 1 as in eq. (2.6). when 𝛼 = 0.85, the dehazed image is shown in fig. 3.1(c). obviously, the hue distortion is relieved by scaling 𝛼 value. thus, an adaptive scaling factor 𝛼𝑎 will be employed in the idcp scheme. by experiments, 𝛼𝑎 = min[(𝜇1)0.0975, 0.975] works well most of time, where 𝜇1 = mean[𝐼1 𝑑𝑎𝑟𝑘(𝑥)] and 𝐼1 𝑑𝑎𝑟𝑘 the pixel-based dark channel. besides, 𝐼ω 𝑑𝑎𝑟𝑘(𝑥) in step 2 of the dcp scheme is replaced by 𝐼1 𝑑𝑎𝑟𝑘 to find 𝑨 for efficiency. (a) (b) (c) fig. 3.1. image village (a) original (b) 𝛼 = 1 (c) 𝛼 = 0.85 3.1.2. initial transmission map estimation in the dcp scheme, the fixed scaling factor 𝜔 = 0.95, as in eq. (2.8), is used to find the initial transmission map �̃�(𝑥) according to the aerial perspective phenomenon. however, we find that artifacts in the sky region and color distortion in a dehazed image are caused by the inappropriate fixed scaling factor 𝜔 = 0.95 . thus, it introduces the related distortions accordingly. to see the problem, image dogs is given as an example, which is shown in fig. 3.2(a). after the dcp scheme, the dehazed image is depicted in fig. 3.2(b), which shows the artificial contours and color distortion due to 𝜔 = 0.95. fortunately, an adaptive scaling factor 𝜔𝑎 can help the situation. with 𝜔 = 0.65, the result is given in fig. 3.2(c). in the idcp scheme, an adaptive scaling factor 𝜔𝑎 will be employed to estimate initial transmission map �̃�(𝑥) . as a rule of thumb, 𝜔𝑎 = min [(𝜇0.9)0.325, 0.95] is employed where 𝜇0.9 = mean[𝐼ω̅ 𝑑𝑎𝑟𝑘(𝑥) ≤ 0.9] and 𝐼ω̅ 𝑑𝑎𝑟𝑘(𝑥) the normalized block-based dark channel. (a) (b) (c) fig. 3.2. image dogs (a) original (b) 𝜔 = 0.95 (c) 𝜔 = 0.65 3.1.3. sky and non-sky region handling in [3], it says that the sky and non-sky regions can be handled gracefully, even the dcp is not a good prior for sky regions. the statement can be verified as follows. in a bright sky region, the pixel intensity 𝐼𝑐(𝑥) → 1; the atmospheric light 𝐴𝑐 → 1; the normalized dark channel 𝐼ω̅ 𝑑𝑎𝑟𝑘(𝑥) → 1; the transmission map 𝑡(𝑥) → 1, and thus the dehazed scene can be found as 𝐽𝑐(𝑥) = 𝐼𝑐(𝑥)−𝐴𝑐 max[𝑡0,𝑡(𝑥)] + 𝐴𝑐 → 𝐴𝑐 (3.1) which is generally consistent with the real situation. consequently, by eq. (3.1) there is no need to handle sky and non-sky regions separately, though many published papers headed 100 c.-h. hsieh, y.-h. chang copyright ©2021 assa. adv. in systems science and appl. (2021) toward the way to the sky/non-sky region segmentation. in other words, the dcp scheme can be applied to the sky and non-sky regions equally well, if appropriate parameters are employed. by our observations, the problems of color over-saturation distortion, hue distortion, artifacts, and over-exposure found in the sky region come from inappropriate estimations of 𝑨 and 𝑡(𝑥). this can be verified by fig. 3.1 and fig. 3.2. in the idcp scheme, these problems can be relieved by setting parameters appropriately. 3.1.4. initial transmission map refinement in general, halos occur in the large depth discontinuities of a dehazed image. it is observed that the halo problem in the dcp scheme comes from an inappropriate parameter setting in the gif for transmission map refinement. in the gif, three parameters are guidance image 𝑰𝑔, window size 𝑁, and smoothing factor 𝜖. in the dcp scheme, 𝑰𝑔 is the input image 𝑰; 𝑁 = 20; 𝜖 = 0.001. by observations, we find that appropriately setting parameters 𝑁 and 𝜖 in the gif is able to deal with the halo problem. a large 𝑁 in the gif reduces the halo effect in a dehazed image. however, it introduces minor color over-saturation. fortunately, it can be solved by using a large 𝜖. a. the effect of 𝑁 here, image house, shown in table 3.1, is used as an example to investigate the effect of the gif parameters, 𝑁 and 𝜖, on a dehazed image. in the simulation, the dcp scheme is employed with the gif setting 𝑁 = 20 and 𝜖 = 0.001, as suggested in [14]. to investigate the effect of parameter 𝑁, 𝜖 is fixed at 0.001. the dehazed images with different values of 𝑁, i.e. 20, 40, and 55, are given in table 3.1, where the corresponding refined transmission 𝑡(𝑥) are also shown for comparison. in the case of 𝑁 = 20, the dehazed house has a severe halo problem at large depth discontinuities. as 𝑁 increases to 40, the halos diminish significantly. when 𝑁 = 55, the halo is not visible at all. besides, one may observe that the significant edges in 𝑡(𝑥) become more clear and more details appear, as 𝑁 varies from 20 to 55. it implies that a large window size may avoid the halo problem, as expected. consequently, 𝑁 = 55 will be used in the proposed idcp scheme. though the window size 𝑁 = 55 is able to deal with the halo problem, minor color over-saturation is found in the dehazed house, as shown in the fourth row of table 3.1. fortunately, the introduced distortion can be avoided by setting smoothing factor 𝜖. the following subsection has the details. table 3.1. the effect of 𝑁 in the gif on the dehazed image house. gif setting 𝑡(𝑥) 𝐽(𝑥) original image 𝑁 = 20 𝜖 = 0.001 𝑁 = 40 𝜖 = 0.001 𝑁 = 55 𝜖 = 0.001 improving dcp haze removal scheme... 101 copyright ©2021 assa. adv. in systems science and appl. (2021) b. the effect of 𝜖 the effect of smoothing parameter 𝜖 on the dehazed house is investigated here. in the simulation, 𝜖 is set to 0.001, 0.05, and 0.1, where 𝑁 is fixed at 55. the corresponding dehazed images and their corresponding 𝑡(𝑥) are given in table 3.2, respectively. by table 3.2, one can observe that the details of 𝑡(𝑥) has been smoothened more and more, as 𝜖 increases from 0.001 to 0.1. besides, the problem of color over-saturation gradually relieves, as 𝜖 becomes larger. it suggests that the problem of color over-saturation can be dealt with a large 𝜖, i.e. 0.1. thus, the smoothing factor 𝜖 = 0.1 will be employed in the proposed idcp scheme. table 3.2. the effect of 𝜖 in the gif on the dehazed image house. gif setting 𝑡(𝑥) 𝐽(𝑥) original image 𝑁 = 55 𝜖 = 0.001 𝑁 = 55 𝜖 = 0.05 𝑁 = 55 𝜖 = 0.1 3.1.5. implementation of the idcp scheme with the above discussion, the proposed idcp scheme is summarized in the following implementation steps, where input image 𝑰 is assumed in the rgb color space. step 1. find the pixel-based dark channel as 𝐼1 𝑑𝑎𝑟𝑘(𝑥) = min 𝑐 [𝐼𝑐(𝑥)] (3.2) where 𝑐 ∈ {𝑅, 𝐺, 𝐵}. step 2. find the maximum in 𝐼1 𝑑𝑎𝑟𝑘(𝑥) and its corresponding pixel in 𝑰, 𝒑𝑚𝑎𝑥. then estimate the atmospheric light as 𝑨 = [𝐴𝑅 𝐴𝐺 𝐴𝐵] = 𝛼𝑎 × 𝒑𝑚𝑎𝑥 , where 𝛼𝑎 = min[(𝜇1)0.0975, 0.975] and 𝜇1 = mean[𝐼1 𝑑𝑎𝑟𝑘(𝑥)]. step 3. calculate the normalized block-based dark channel as 𝐼ω̅ 𝑑𝑎𝑟𝑘(𝑥) = min 𝑦∈ω(𝑥) min 𝑐 [ 𝐼𝑐(𝑦) 𝐴𝑐 ] (3.3) step 4. obtain the initial transmission map as �̃�(𝑥) = 1 − 𝜔𝑎 × 𝐼ω̅ 𝑑𝑎𝑟𝑘(𝑥) (3.4) where 𝜔𝑎 = min [(𝜇0.9)0.325, 0.95] and 𝜇0.9 = mean[𝐼ω̅ 𝑑𝑎𝑟𝑘(𝑥) ≤ 0.9]. step 5. find the final transmission map 𝑡(𝑥) through refining �̃�(𝑥) by the gif with the guidance image 𝐼1 𝑑𝑎𝑟𝑘(𝑥), the window size 𝑁 = 55, and the smoothing parameter 𝜖 = 0.1. step 6. estimate the dehazed image as 𝐽𝑐(𝑥) = 𝐼𝑐(𝑥)−𝐴𝑐 max[𝑡0,𝑡(𝑥)] + 𝐴𝑐 (3.5) where 𝑡0 = 0.1 is a user-defined lower bound of 𝑡(𝑥). 102 c.-h. hsieh, y.-h. chang copyright ©2021 assa. adv. in systems science and appl. (2021) in comparison with the dcp scheme described in section 2, the proposed idcp scheme has at least four main differences. first, the pixel-based dark channel 𝐼1 𝑑𝑎𝑟𝑘(𝑥) is used to estimate the atmospheric light 𝑨 with an adaptive scaling factor 𝛼𝑎. second, 𝐼1 𝑑𝑎𝑟𝑘(𝑥) is employed as the guidance image in the gif to refine the initial transmission map �̃�(𝑥). this helps improve the efficiency. third, an adaptive scaling factor 𝜔𝑎 is applied in the estimation of initial transmission map �̃�(𝑥) to avoid the artifacts and color distortion happened in the sky region. this doing makes the proposed idcp scheme is able to deal with the sky and non-sky regions equally well, without segmentation. fourth, the gif setting with large 𝑁 = 55 is employed to relieve the halo problem, while large 𝜖 = 0.1 is to handle the color over-saturation. with those modifications, the proposed idcp scheme shows much better performance than the dcp scheme. this will be verified in section 4. 3.2. dehazed image enhancement with an agc a dehazed image generally is dimmer than its original image. consequently, an adaptive gamma correction (agc) is used to improve visual quality of dehazed images. the agc is modified from a conventional gamma correction (cgc). given image 𝑰𝑖, whose elements are denoted as 𝐼𝑖 𝑐(𝑥) and 𝑐 ∈ {𝑅, 𝐺, 𝐵}, a cgc transforms 𝐼𝑖 𝑐(𝑥) to a new value 𝐼𝑜 𝑐(𝑥) as 𝐼𝑜 𝑐(𝑥) = [ 𝐼𝑖 𝑐(𝑥) 𝐼𝐻 𝑐 ] 𝑔 𝐼𝐻 𝑐 (3.6) where 𝐼𝐻 𝑐 = max 𝑥 [𝐼𝑖 𝑐(𝑥)] is the maximum value in component 𝑐. when put the dynamic ranges of input image 𝑰𝑖 and output image 𝑰𝑜 into account, eq. (3.6) can be rewritten in a more general form as 𝐼𝑜 𝑐(𝑥) = [ 𝐼𝑖 𝑐(𝑥)−𝐼𝐿 𝑐 𝐼𝐻 𝑐 −𝐼𝐿 𝑐 ] 𝑔 [𝐼𝑜,𝐻 𝑐 − 𝐼𝑜,𝐿 𝑐 ] + 𝐼𝑜,𝐿 𝑐 (3.7) where 𝐼𝐿 𝑐 = min 𝑥 [𝐼𝑖 𝑐(𝑥)] and 𝐼𝐻 𝑐 = max 𝑥 [𝐼𝑖 𝑐(𝑥)]. notations 𝐼𝑜,𝐿 𝑐 and 𝐼𝑜,𝐻 𝑐 are user-defined lower limit and upper limit of 𝐼𝑜 𝑐(𝑥) for the output image 𝑰𝑜, respectively. the superscript 𝑔 in eq. (3.7) is a user-defined factor. it is well-known that the cgc suffers from the problem of color distortion. therefore, it hinders the application to dehazed image enhancement. an example, image women, is given in fig. 3.3(a), where the dehazed image, shown in fig. 3.3(b), is obtained by the proposed idcp scheme. the dehazed image, post-processed by the cgc, is shown in fig. 3.3(c) which has a noticeable color distortion. (a) (b) (c) (d) fig. 3.3. image women (a) original (b) after the proposed idcp scheme (c) with the cgc (d) with the agc to relieve the color distortion in the cgc, an adaptive gamma correction (agc) is introduced here. by our observations, the color distortion in the cgc results from the inappropriate set of upper and lower limits, [𝐼𝐿 𝑐 𝐼𝐻 𝑐 ]. consequently, the proposed agc replaces the set [𝐼𝐿 𝑐 𝐼𝐻 𝑐 ] with set [𝐼𝐿 𝐼𝐻] where 𝐼𝐿 = min 𝑐 [𝐼𝐿 𝑐] and 𝐼𝐻 = max 𝑐 [𝐼𝐻 𝑐 ]. with the set [𝐼𝐿 𝐼𝐻], eq. (3.7) is modified as 𝐼𝑜 𝑐(𝑥) = [ 𝐼𝑖 𝑐(𝑥)−𝐼𝐿 𝑐 𝐼𝐻−𝐼𝐿 ] 𝑔𝑎 [𝐼𝑜,𝐻 𝑐 − 𝐼𝑜,𝐿 𝑐 ] + 𝐼𝑜,𝐿 𝑐 (3.8) where 𝑔𝑎 is an adaptive factor. the way to determine parameter 𝑔𝑎 in eq. (3.8) is discussed below. note that the parameter 𝜔𝑎 for initial transmission map estimation affects the strength of haze removal significantly. that is, a larger 𝜔𝑎 results in a stronger dehazing effect and vice versa. a stronger haze removal makes the dehazed image dimmer in general. consequently, 𝑔𝑎 should be inversely improving dcp haze removal scheme... 103 copyright ©2021 assa. adv. in systems science and appl. (2021) related to 𝜔𝑎. to increase the brightness, 𝑔𝑎 is restricted to be less than 1 and greater than 0.7 to avoid artifacts. by experiments, 𝑔𝑎 = max[(1 − 𝜔𝑎)0.095, 0.707] is employed in the agc. with this 𝑔𝑎 , the proposed agc is applied to fig. 3.3(b), whose result is shown in fig. 3.3(d). as expected, the color distortion is avoided and visual quality is enhanced. since the proposed agc is a post-processing scheme, it can be added to the proposed idcp scheme after step 6 in section 3.1.5. that is, step 7. enhance the dehazed image �̂� by the agc. the combination of the idcp scheme and the agc will be call the idcp/agc scheme. 4. results and discussions in this section, the proposed idcp/agc scheme is justified by the datasets reside in [13] and kedema in [28]. the reside dataset is a large-scale benchmark for single image dehazing algorithms, which consists of synthetic and natural images. in the following experiments, the indoor training set (its) [29] is employed, that has 10,000 clear images and 100,000 synthetic hazy images generated by eq. (2.1) with various 𝐴 and 𝛽 . besides, the outdoor training set (ots) [30] is also used, which consists of 3,981 clear images and 136,160 synthetically generated hazy images. by the large amount of images, the proposed idcp/agc scheme is justified and compared with four recently reported haze removal schemes in section 4.1, where both objective and subjective evaluation are considered. the four compared schemes are the conventional dcp scheme in [14], the optimization-based scheme in [31], which will be called rro scheme, the learning based scheme in [32], which will be called cap scheme, and the deep learning based scheme dehazenet (dnet for short) in [33]. in section 4.2, the kedema dataset is used, which consists of 25 natural hazy images with different scenarios, to further verify for the proposed idcp/agc scheme, objectively and subjectively, and compared with the dcp, rro, cap, and dnet schemes. in addition to the subjective comparison, the performances of the related schemes are also justified objectively. to compare the related schemes in different aspects, the objective assessments include full-reference methods: the peak signal-to-noise ratio (psnr) and the structural similarity index measure (ssim) in [34]; half-reference method: the dehazing quality index (dhqi) in [35]; no-reference methods: the blind/reference-less image spatial quality evaluator (brisque) in [36] and the integrated local natural image quality evaluator (il-niqe) in [37]. the compared results are given below. 4.1 results and comparisons with the reside dataset to justify the performance of the proposed idcp/agc scheme, an extensive experiment is conducted with the reside dataset in this section, where the results from the four compared schemes are also given for objective and subjective comparisons. first, the proposed idcp/agc scheme and the compared schemes are run with the its dataset in section 4.1.1 and then ots dataset in section 4.1.2. 4.1.1. results with the its dataset in this subsection, the idcp/agc scheme is verified by the its dataset. the results for the four compared schemes are also given for comparison. the first part shows the objective results, where the performance indices psnr, ssim, dhqi are better for a higher value, while brisque, il-niqe are the less the better. in the second part, subjective results will give for visual comparison. a. objective comparison with the its dataset the objective comparisons with psnr, ssim, dhqi, brisque, il-niqe are shown in table 4.1, for the proposed idcp/agc scheme and the compared schemes, where the ranking is shown in parentheses. by the results, the dcp scheme has the best and second best in 104 c.-h. hsieh, y.-h. chang copyright ©2021 assa. adv. in systems science and appl. (2021) brisque and il-niqe, respectively. however, it has the worst ranking in psnr, ssim, and dhqi. that is, the dcp scheme lies on either the worst end or the best end with the average ranking 3.6. for the rro scheme, it ranks the second best in brisque; the third place in dhqi and il-niqe; the fourth place in psnr and ssim. it ends with the average ranking 3.2. as for the cap scheme, it has the second best psnr; the third place in ssim; but the fourth place in dhqi and brisque, and the worst ranking in il-niqe. the average ranking is 3.6. as for the dnet, it has the best psnr and ssim; the second best in dhqi; but the fourth place in il-niqe and the worst ranking in brisque. its average ranking is 2.6. the ranking of the proposed idcp/agc scheme falls within the top three among full-reference, semi-reference, and no-reference assessments. it suggests that the performance of the proposed idcp/agc scheme has a relatively stable performance among the compared schemes. overall, the proposed idcp/agc scheme has the best performance, then dnet, rro, cap, and dcp schemes, in terms of the average ranking (ar). table 4.1. objective comparison with the its dataset idcp/agc dcp rro cap dnet psnr 19.1669(3) 16.4216(5) 18.2363(4) 19.5537(2) 20.3097(1) ssim 0.8742(2) 0.8374(5) 0.84681(4) 0.8733(3) 0.8909(1) dhqi 59.8120(1) 55.6912(5) 57.3045(3) 56.9135(4) 59.2964(2) brisque 15.6748(3) 13.4386(1) 13.9044(2) 19.1409(4) 22.6275(5) il-niqe 32.6799(1) 33.4390(2) 33.5820(3) 33.9286(5) 35.3254(4) ar 2(1) 3.6(4) 3.2(3) 3.6(4) 2.6(2) b. subjective comparison with the its dataset in addition of objective comparison, the subjective comparison is also given for the its dataset. here, six images are selected to subjectively evaluate the proposed idcp/agc scheme and the compared schemes. the six images include two less hazy images (no. 1 and no. 2), two moderate hazy images (no. 3 and no. 4), and two heavy hazy images (no. 5 and no. 6), which are shown in table 4.2. for reference, the image filenames in its are also given in table 4.2 with the associated psnr for each selected image. by table 4.2, the dcp scheme tends to have hue distortion and color over-saturation in all six dehazed images. the rro scheme has hue distortion in images 2, 4, and 5; minor color over-saturation for images 1 and 3. the dehazing effect of the cap scheme seems not strong enough, especially for the heavy hazy images 5 and 6. basically, the dnet scheme performs well in the cases of less hazy and moderate images, but fails to remove haze in heavy hazy images 5 and 6. as for the proposed idcp/agc scheme, it provides better visual quality in the dehazed images. table 4.2. subjective comparison with the its dataset (6 selected images) no clear input idcp/agc dcp rro cap dnet 1 0025_08_0.7048 psnr=31.26 16.32 24.97 22.42 20.17 2 1075_03_0.7312 psnr=30.72 13.29 13.96 24.53 25.63 3 0499_05_0.8353 psnr=22.63 15.50 21.89 19.78 21.47 improving dcp haze removal scheme... 105 copyright ©2021 assa. adv. in systems science and appl. (2021) 4 0691_09_0.8315 psnr=24.53 19.06 18.08 20.01 23.91 5 0803_10_0.9055 psnr=22.97 13.28 13.83 19.42 19.44 6 0962_02_0.7819 psnr=22.56 18.43 15.69 14.01 14.77 next, three worse cases, shown in table 4.3, for the proposed idcp/agc scheme are given and discussed. for image 1, better visual quality is for the proposed idcp/agc scheme, even though the cap and dnet schemes have better psnr. it suggests that the psnr is not consistent with the subjective evaluation in some cases. similar results are found in images 2 and 3. the dehazed images by the proposed idcp/agc scheme are obviously better than the compared schemes, even some of their psnr are higher. the reason might be that the proposed idcp/agc scheme enhances the contrast and brightness in the dehazed images. thus, it increases the mean squared error between the dehazed images and their clear images, and lower psnr results. consequently, in this study five objective assessments are employed to have a more fair comparison among the proposed idcp/agc scheme and the compared schemes. table 4.3. subjective comparison with the its dataset (3 chosen cases) no clear input idcp/agc dcp rro cap dnet 1 0244_02_0.8649 psnr=25.28 16.14 15.09 33.10 34.54 2 0388_09_0.7129 psnr=20.84 16.78 30.50 25.11 29.11 3 0146_04_0.7433 psnr=15.72 16.07 27.77 24.16 21.88 4.1.2. results with the ots dataset to have more understanding the performance, the proposed imdcp/aglgc scheme is further justified by the ots dataset and compared with the dcp, rro, cap, and dnet schemes. the objective comparison is given first and then the subjective comparison follows. a. objective comparison with the ots dataset as in the its dataset, five objective assessments, psnr, ssim, dhqi, brisque, and ilniqe, for the proposed idcp/agc scheme, dcp, rro, cap, and dnet are given in table 4.4. the dcp scheme has the second best in il-niqe and the third place in brisque, but with the worst psnr, ssim, and dhqi. this ends up with average ranking 4. for the rro scheme, it has the second best in brisque, the third place in il-niqe, and the fourth place in psnr, ssim, and dhqi. its average ranking is 3.4. the cap scheme has the best ssim and the second best psnr, the third place dhqi, but the fourth place in brisque and ilniqe, which results in the average ranking 2.8. for the dnet, it has the best psnr and dhqi, the second best in ssim, but the worst brisque and il-niqe. the average ranking is 2.8. as for the proposed idcp/agc scheme, it has the best brisque and il-niqe, the second best dhqi, and the third place in psnr and ssim. its average ranking is 2. by the average 106 c.-h. hsieh, y.-h. chang copyright ©2021 assa. adv. in systems science and appl. (2021) ranking, the best one is the proposed idcp/agc scheme, and then dnet or cap, rro, and dcp at final. by table 4.4, it implies that the proposed idcp/agc scheme has more stable performance than the compared schemes, among the five objective assessments. table 4.4. objective comparison with the ots dataset idcp/agc dcp rro cap dnet psnr 22.5973(3) 15.9340(5) 19.0970(4) 23.3067(2) 23.7265(1) ssim 0.9111(3) 0.8456(5) 0.8779(4) 0.9236(1) 0.9161(2) dhqi 50.6845(2) 44.9371(5) 47.7001(4) 49.9232(3) 51.6284(1) brisque 16.1972(1) 16.9098(3) 16.3732(2) 16.9410(4) 19.0443(5) il-niqe 21.2804(1) 21.3985(2) 21.4482(3) 21.4876(4) 23.1376(5) ar 2(1) 4(4) 3.4(3) 2.8(2) 2.8(2) b. subjective comparison with the ots dataset for subjective comparison, six images are selected from the ots dataset, which are shown in table 4.5. the six images are with different haziness. two images (no. 1 and no. 2) are less hazy, two images (no. 3 and no. 4) are of moderate haziness, and two images (no. 5 and no. 6) have more haziness. the corresponding psnr for each image is also given in table 7 for reference. in the given images, the dcp scheme has artifacts in image 2, color over-saturation in images 3 and 4 , and hue distortion in images 5 and 6. the rro scheme has less dehazing effect in image 1, artifacts in image 2, hue distortion in images 3 and 5, and minor color oversaturation in images 4 and 6. the cap scheme has a dark chiffon in images 5 and 6. similarly, the dnet scheme has similar, but minor, results in images 5 and 6. for the rest of images, the dnet generally works better than the cap scheme, in terms of visual quality of the dehazed images. as for the proposed idcp/agc scheme, it gives better subjective results than the four compared schemes, even images 3, 5, and 6 have less psnr, which is caused by brightness and contrast enhancement. table 4.5. subjective comparison with the ots dataset (6 selected images) no clear input idcp/agc dcp rro cap dnet 1 1178_0.8_0.04 psnr=34.09 26.94 22.88 25.56 24.14 2 0014_1_0.04 psnr=23.21 14.52 19.62 21.63 19.14 3 0255_1_0.1 psnr=26.11 18.55 18.09 25.65 27.06 4 0250_0.85_0.12 psnr=25.72 15.90 17.88 23.07 24.06 5 0027_0.8_0.2 psnr=21.02 14.38 23.01 20.68 25.01 improving dcp haze removal scheme... 107 copyright ©2021 assa. adv. in systems science and appl. (2021) 6 0004_0.8_0.2 psnr=19.29 12.77 17.00 21.47 14.77 for the ots dataset, three worse cases in the proposed idcp/agc scheme are given and discussed here. the three images are shown in table 4.6, where their psnr are given as well. obviously, the visual quality of the dehazed images by the proposed idcp/agc scheme is better than the four compared schemes, even though some of psnr are lower. for images 1 and 2, the ground true image, i.e. clear image, is somewhat hazy, not really clear. thus, it affects the psnr calculation. in other words, the schemes have less dehazing effect, like the rro, cap, and dehazenet, achieve better psnr. for image 3, it is a night shot. the proposed idcp/agc scheme not only removes haze but also enhances brightness and contrast. consequently, it increases the mean squared error and thus less psnr results. it explains why the proposed idcp/agc scheme takes the third place in psnr, as shown in table 4.6. that is why this study adopts five different objective assessments and the average ranking is used to evaluate the overall performance for a fair comparison. table 4.6. subjective comparison with the ots dataset (3 chosen cases) no clear input idcp/agc dcp rro cap dnet 1 0267_0.95_0.16 psnr=25.10 14.97 20.02 30.58 26.96 2 0150_0.85_0.2 psnr=16.36 16.38 16.69 32.36 19.74 3 0113_0.85_0.12 psnr=15.94 29.45 19.40 18.13 26.31 4.2 results and comparisons with the kedema dataset in this section, the proposed idcp/agc scheme is applied to the natural images, where no ground true images are available. in the experiments, the kedema dataset, which contains 25 natural hazy images, is employed to verify the proposed idcp/agc scheme further, whose results are compared with the dcp, rro, cap, and dnet schemes as previously. 4.2.1 objective comparison with the kedema dataset three objective metrics, dhqi, brisque, and il-niqe, are calculated with the dehazed images obtained by the proposed idcp/agc and the compared schemes. note that the full reference metrics, psnr and ssim, are not applicable, since ground true images are not available. for those three metrics do not require ground true images, the results are shown in table 4.7. by table 4.7, the dcp scheme ranks between 3 and 5; the rro scheme lies within top three ranking; the cap scheme takes the best in brisque and the fourth place in dhqi and il-niqe; the dnet scheme has the worst ranking in brisque and il-niqe, and the third place in dhqi; the proposed idcp/agc scheme has the best dhqi, and the second best both in brisque and il-niqe. by the average ranking, the proposed idcp/agc scheme outperforms the compared schemes, and the rro, cap, dcp, and dnet schemes follow. again, the proposed idcp/agc scheme provides a better and more stable performance than the compared schemes. 108 c.-h. hsieh, y.-h. chang copyright ©2021 assa. adv. in systems science and appl. (2021) table 4.7. objective comparison with the kedema dataset idcp/agc dcp rro cap dnet dhqi 60.5390(1) 57.7240(5) 60.1320(2) 58.9870(4) 61.1010(3) brisque 11.8990(2) 12.5420(4) 12.3470(3) 10.9520(1) 13.3190(5) il-niqe 23.9100(2) 24.6400(3) 23.5640(1) 25.6000(4) 27.7350(5) ar 1.7(1) 4(4) 2(2) 3(3) 4.3(5) 4.2.2 subjective comparison with the kedema dataset for subjective comparison, the 25 dehazed images for the kedema dataset are shown in table 4.8, which are obtained from the proposed idcp/agc scheme and the four compared schemes. in general, the dcp scheme suffers from the problems of color over-saturation, halos, and artifacts throughout the given examples. the rro scheme has artificial contours in the sky region of image 1, hue distortion in image 5, color over-saturation in image 7, less dehazed results in images 9, 11. for the cap scheme, it has artificial contours in the sky region of image 1, and less dehazed results in images 3 to 6, 8, 18 to 23. generally, the cap scheme has a relatively weak dehazing performance. the dnet scheme has artificial contours in the sky region of image 1, less dehazed results in images 3, 4, 6, 11, 18, 20, 21, 22, and 23. as for the proposed idcp/agc scheme, it generally gives a stable and better result than the compared schemes in visual quality, which is demonstrated in the aspects of naturalness, brightness, and contrast, as shown in table 4.8. table 4.8. subjective comparison with the kedema dataset no input idcp/agc dcp rro cap dnet 1 2 3 4 5 6 7 improving dcp haze removal scheme... 109 copyright ©2021 assa. adv. in systems science and appl. (2021) 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 110 c.-h. hsieh, y.-h. chang copyright ©2021 assa. adv. in systems science and appl. (2021) 24 25 5. conclusion in this paper, an improved dcp (idcp) scheme was presented to solve the four problems in the dcp scheme: artifacts, hue distortion, color over-saturation, and halos. the dehazed image by the idcp was further enhanced by an adaptive gamma correction (agc). the overall dehazing scheme is called idcp/agc. the proposed idcp/agc scheme was extensively justified by the synthetic hazy images in the its and ots datasets, and the natural images in the kedema dataset. besides, the proposed idcp/agc scheme was compared with four recently reported haze removal schemes objectively and subjectively. to have a balanced comparison, five objective assessments, including full-reference, half-reference, and noreference methods, were employed. the results indicated that the objective evaluation was for the proposed idcp/agc scheme in terms of the average ranking, while the subjective comparison showed that the proposed idcp/agc scheme was relatively stable and able to provide a better visual quality in the given image datasets. in the future, an optimization algorithm will be investigated for the heuristic parameter setting in the proposed idcp/agc scheme. references 1. tan, r.t. 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"no-reference image quality assessment in the spatial domain, ieee transactions on image processing, 21(12), 46954708, doi: 10.1109/tip.2012.2214050 37. zhang, l., zhang, l., and bovik, a.c. (2015). a feature-enriched completely blind image quality evaluator, ieee transactions on image processing, 24(8), 2579-2591, doi: 10.1109/tip.2015.2426416 https://www.dropbox.com/sh/mrtguzk1a1l11o6/aaarckahsq-lz34psgm5muuza?dl=0 https://www.dropbox.com/sh/mrtguzk1a1l11o6/aaarckahsq-lz34psgm5muuza?dl=0 https://www.dropbox.com/s/86qp410gen5u2uk/its.zip?dl=0 1. introduction 2. review of the dcp scheme 3. the proposed idcp scheme and agc 3.1. the idcp dehazing scheme 3.1.1. atmospheric light estimation 3.1.2. initial transmission map estimation 3.1.3. sky and non-sky region handling 3.1.4. initial transmission map refinement a. the effect of 𝑁 b. the effect of 𝜖 3.1.5. implementation of the idcp scheme 3.2. dehazed image enhancement with an agc 4. results and discussions 4.1 results and comparisons with the reside dataset 4.1.1. results with the its dataset a. objective comparison with the its dataset b. subjective comparison with the its dataset 4.1.2. results with the ots dataset a. objective comparison with the ots dataset b. subjective comparison with the ots dataset 4.2 results and comparisons with the kedema dataset 4.2.1 objective comparison with the kedema dataset 4.2.2 subjective comparison with the kedema dataset 5. conclusion microsoft word 1378-article text-6924-1-18-20240328 adv syst sci appl 2024; 01; 142-162 published online at https://ijassa.ipu.ru. a new face swap detection technique for digital images rasha thabit 1,2*, heba mohammed fadhil 3, hassan falah fakhruldeen 4,5, akram hatem shather6, mohanad a. al-askari7 1) dijlah university college, baghdad, iraq 2) al-iraqia university, baghdad, iraq 3) university of baghdad, baghdad, iraq 4) imam ja’afar al-sadiq university, baghdad, iraq 5) university of kufa, kufa, iraq 6) al kitab university, altun kopru, kirkuk, iraq 7) university of al-anbar, al-anbar, iraq abstract: in recent years, the rapid development of deep learning-based face image manipulation algorithms and applications became one of the challenges that are facing information forensics and information security systems. using these applications, one can easily swap the face in a digital image with another face for different intentions where most of them are malicious intentions. different face swap detection techniques have been presented in recent years to check the authenticity of the face in a digital image. most of the available techniques are machine-learning or deep-learning based which makes them vulnerable to false detection results in addition to the time-consuming training process. in this paper, a new technique for face swap detection (fsd) is presented based on the image watermarking process. the proposed technique consists of two main algorithms called embedding and authentication algorithms. several experiments have been conducted to evaluate the performance of the proposed technique and to prove its efficiency in detecting fake faces. the proposed technique outperforms various deep-learning-based techniques because no training is required and the detection accuracy is 100 %. the performance of the proposed fsd technique is promising therefore it is applicable in different practical applications. keywords: information security, information forensics, deep-fake detection, face swap detection, face image manipulation detection. 1. introduction biotechnology and biometric information have been utilized in different verification systems such as security and financial industries [1]. the biometric information can be generated using extrinsic sources such as iris, fingerprint, and face or using intrinsic sources such as hand-vein, finger-vein, and palm-vein [2,3]. over the years, the facial recognition systems [4] have been widely used to identify individuals or their feelings using images or videos processing techniques, however, identity theft and delivering fake information have been considered as challenges to these systems [5,6]. face swap is a type of deep-fakes that refers to the process of replacing the face of one person with another in a digital image [7,8]. the face swap has several advantages in the movies and games industry [9], however, it has been considered as one of the most dangerous attacks that are required to be detected effectively especially when it is applied with malicious intentions. the fake face images can be used for fake videos and news to destroy the reputation of people, threaten them, identity theft, and many other malicious intentions [10–13]. over the years, different face-swap algorithms and applications * corresponding author: rashathabit@yahoo.com a new face swap detection technique for digital images 143 copyright ©2024 assa. adv. in systems science and appl. (2024) have been presented consequently different face-swap detection techniques have been implemented to serve the data security and digital media forensics fields [14–17]. the research community in the last few years witnessed an increased interest in face swap detection because these techniques became requested by many institutions and companies [17]. in [18], a face-swap detection technique based on machine learning has been presented. in this work, 83 landmarks for each face are extracted and used for training different classifiers. different machine learning techniques have been tested such as support vector machine (svm) [19], random forest (rf) [20], and multi-layer perceptron’s (mlp) [21]. the images are classified into two classes that are innocent class and swapped images class. two types of classifiers (i.e., linear and non-linear) have been tested to find the best. the results of the nonlinear classifier were better than that of the linear classifier, however, both types of classifiers have recorded false detection results and the best accuracy that have been obtained was around 92 % for only a specific images dataset. in [22], another face-swap detection technique has been presented based on deep learning and the error level analysis (ela) process. the main principle of this technique is related to the errors that are generated in the manipulated face images. when a face region is cut and another face is pasted in its place, an error level will acquire between the manipulated area and the surrounding area. the residual network (resnet-18) has been used with custom dense layers for the classification process. to reduce the time for training which may reaches days or weeks, the authors suggested the use of transfer learning. the results of this work were promising; however, false detection results have been also recorded and the best accuracy that have been obtained was around 97 % for only a specific images dataset. in [23], a face-swap detection technique based on a convolution neural network (cnn) has been presented. the method of detection consists of two stages that are preprocessing stage and the classifier stage. in the preprocessing stage, the features of the face are extracted and the alignment process is applied [24, 25]. in the classifier stage, mobilenet-like cnn pretrained on an imagenet has been used [26, 27]. this technique obtained better accuracy results compared to previous techniques based on mesonet and xception networks [28, 29], however, false detection results have been obtained and the best accuracy that have been obtained was between 98 % to 99 % based on the images dataset. the main problems of this technique are the use of a specific images’ dataset and the time complexity for training and testing processes. as explained before, there are some limitations in the machine-learning and deep-learningbased face swap detection techniques such as the false detection results, the high time complexity for training and testing, the need for high-quality images for training, the need for large datasets, and others. to avoid these limitations, this paper presents a new face-swap detection (fsd) technique based on image watermarking. the main idea of the proposed technique is inspired by the region-of-interest (roi) based medical image authentication techniques [30–32]. in this paper, the face area will be considered as roi and its authentication data will be extracted and embedded in the area outside roi. the data generation, embedding, extraction, and authentication processes are all based on the watermarking techniques that have been presented in [33–35]. to detect the face area two face detection algorithms are tested to choose the one with the best performance. the proposed technique can distinguish between innocent and fake faces in digital images without the need for training and with 100 % accuracy thus it outperforms the machine learning and deep-learning-based techniques. the proposed technique can be applied for any image regardless its quality which increased its ability in practical applications. the rest of the paper is organized as follows: in section 2, the related works are presented; in section 3, the algorithms of the proposed fsd technique are explained in details; in section 4, the experiments and their discussion are presented; and finally, section 5 illustrates the conclusions that have been drawn from this research. 144 r. thabit, h. m. fadhil, h. f. fakhruldeen, a. h. shather, m. a. al-askari copyright ©2024 assa adv. in systems science and appl. (2024) 2. related works the proposed fsd technique starts by detecting the face area in the digital image. to detect the face region two widely used face detection algorithms have been tested to choose the one with the best performance. the face detection algorithms are briefly explained in the following subsection. on the other hand, inspired by the medical image authentication technique presented in [34], the proposed fsd adopted slantlet transform (slt)-based watermarking. slt-based watermarking techniques [33], [35–39] proved their efficiency in various applications, therefore, this type of watermarking has been adopted as part of the implemented algorithms in the proposed fsd technique. the second subsection presents the slt-based embedding and extraction algorithms for a block of image. 2.1. face detection two well-known face detection algorithms are tested to choose the best one in order to be adopted as the first part of the proposed fsd technique. the adaboost-based algorithm [40] and multi-task cascaded cnn (mtcnn)-based algorithm [41] are tested for different face images. in [40], the detection algorithm starts by calculating the integral image for the input image followed by extracting the features using the harr-like filters. then a small number of the generated features is selected using the adaboost algorithm. to extract the promising regions of the image, the cascade structure is used for the complex classifiers. after several processing steps, the face regions are selected. in [41], a joint face detection and alignment algorithm is presented in which a shallow cnn algorithm is applied to generate the candidate windows. then a more complex cnn algorithm is used to reject the non-face windows. thereafter, another robust cnn algorithm is applied as the final touch-up to refine the result and generate the landmark positions. samples of the preliminary tests for the abovementioned algorithms are shown in fig. 2.1. the results proved that the mtcnn-based algorithm performs better in comparison with the adaboost-based algorithm where the latter has missed some of the faces. therefore, the mtcnn-based algorithm has been chosen to be applied as the first stage in the proposed fsd technique. 2.2. slt-based watermarking the slt-based watermarking algorithms have been applied in different applications and they proved their efficiency in comparison with different other algorithms in terms of visual quality, robustness, and time complexity [35], [42–44]. based on the previous studies, we suggested the used of slt-based watermarking in face swap detection algorithms. in the proposed fsd technique, the slt-based watermark embedding and extraction algorithms for a single image block have been adapted from [34]. the adopted slt-based embedding and extraction algorithms are explained in table 2.1 and table 2.2, respectively. a new face swap detection technique for digital images 145 copyright ©2024 assa. adv. in systems science and appl. (2024) fig. 2.1. preliminary test results for mtcnn-based and adaboost-based face detection algorithms. table 2.1. slt-based watermark embedding algorithm for a single block [34]. input: original image block (size 16×16 pixels) and binary sequence of 64 bits. output: watermarked image block (size 16×16 pixels). step 1 read the input image block 𝐵 and the binary sequence 𝐵𝑖𝑛 . step 2 transform 𝐵 using slt matrix as follows: 𝑇 = [𝑆𝐿𝑇 ] [𝐵] [𝑆𝐿𝑇 ] where 𝐵 and 𝑇 are the original and watermarked blocks, 𝑆𝐿𝑇 and 𝑆𝐿𝑇 are the slt matrix and its transpose both of size (16×16). step 3 divide the coefficients in 𝑇 into four subbands called (𝐿𝐿, 𝐻𝐿, 𝐿𝐻, 𝑎𝑛𝑑 𝐻𝐻). 𝐿𝐿 = 𝑇 (1: 8, 1: 8) 𝐻𝐿 = 𝑇 (1: 8, 9: 16) 𝐿𝐻 = 𝑇 (9: 16, 1: 8) 𝐻𝐻 = 𝑇 (9: 16, 9: 16) where 𝐿𝐿, 𝐻𝐿, 𝐿𝐻, 𝑎𝑛𝑑 𝐻𝐻 are low-low, high-low, low-high, and high-high subbands, respectively. step 4 for 𝑥 = 1 𝑡𝑜 64 𝑏 = 𝐵𝑖𝑛 (𝑥) for 𝑖 = 1 𝑡𝑜 8 for 𝑗 = 1 𝑡𝑜 8 𝐷1 = 𝐻𝐿(𝑖, 𝑗) − 𝐿𝐻(𝑖, 𝑗) if 𝑏 = 1 and 𝐷1 < 𝑇ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑 , then increase 𝐻𝐿(𝑖, 𝑗) and decrease 𝐿𝐻(𝑖, 𝑗) as follows: 𝑁𝑒𝑤𝐻𝐿(𝑖, 𝑗) = 𝐻𝐿(𝑖, 𝑗) + 𝑁𝑒𝑤𝐿𝐻(𝑖, 𝑗) = 𝐿𝐻(𝑖, 𝑗) − 146 r. thabit, h. m. fadhil, h. f. fakhruldeen, a. h. shather, m. a. al-askari copyright ©2024 assa adv. in systems science and appl. (2024) if 𝑏 = 1 and 𝐷1 ≥ 𝑇ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑, then do-nothing. 𝐷2 = 𝐿𝐻(𝑖, 𝑗) − 𝐻𝐿(𝑖, 𝑗) if 𝑏 = 0 and 𝐷2 < 𝑇ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑 , then increase 𝐿𝐻(𝑖, 𝑗) and decrease 𝐻𝐿(𝑖, 𝑗) as follows: 𝑁𝑒𝑤𝐻𝐿(𝑖, 𝑗) = 𝐻𝐿(𝑖, 𝑗) − 𝑁𝑒𝑤𝐿𝐻(𝑖, 𝑗) = 𝐿𝐻(𝑖, 𝑗) + if 𝑏 = 0 and 𝐷2 ≥ 𝑇ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑, then do-nothing. note: the 𝑇ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑 variable is used for controlling the visual quality and the robustness of the embedded watermark. the 𝑇ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑 value that has been adopted in [34] is 3 because it gives a good compromise between visual quality and robustness. step 5 replace the original 𝐻𝐿 and 𝐿𝐻 subbands in 𝑇 with 𝑁𝑒𝑤𝐻𝐿 and 𝑁𝑒𝑤𝐿𝐻 subbands. save the resultant matrix as 𝑁𝑒𝑤𝑇 . step 6 apply inverse slt on 𝑁𝑒𝑤𝑇 to obtain the watermarked image block as follows: 𝑊 = [𝑆𝐿𝑇 ] [ 𝑁𝑒𝑤𝑇 ] [𝑆𝐿𝑇 ] where 𝑊 is the watermarked image block that carries a binary sequence of 64 bits. table 2.2. slt-based watermark embedding algorithm for a single block [34]. input: original image block (size 16×16 pixels) and binary sequence of 64 bits.. output: watermarked image block (size 16×16 pixels). step 1 read the input watermarked image block 𝑊 . step 2 transform 𝑊 use the ng slt matrix as follows: 𝑇 = [𝑆𝐿𝑇 ] [𝑊 ] [𝑆𝐿𝑇 ] where 𝑇 is the transformed watermarked image block. step 3 divide the coefficients in 𝑇 into four subbands called (𝐿𝐿, 𝐻𝐿, 𝐿𝐻, 𝑎𝑛𝑑 𝐻𝐻). 𝐿𝐿 = 𝑇 (1: 8, 1: 8) 𝐻𝐿 = 𝑇 (1: 8, 9: 16) 𝐿𝐻 = 𝑇 (9: 16, 1: 8) 𝐻𝐻 = 𝑇 (9: 16, 9: 16) step 4 let 𝑥 = 1 for 𝑖 = 1 𝑡𝑜 8 for 𝑗 = 1 𝑡𝑜 8 𝑏(𝑥) = 1 𝑤ℎ𝑒𝑛 𝐻𝐿(𝑖, 𝑗) ≥ 𝐿𝐻(𝑖, 𝑗) 𝑏(𝑥) = 0 𝑤ℎ𝑒𝑛 𝐿𝐻(𝑖, 𝑗) > 𝐻𝐿(𝑖, 𝑗) 𝑥 = 𝑥 + 1 the loop continues until extracting a binary sequence of length 64 bits. 3. proposed fsd technique the proposed fsd technique consists of two main algorithms called embedding and authentication algorithms. the embedding algorithm is applied at the sender side to detect the face area, generate its authentication information and to hide this information in the region outside the face area. the authentication algorithm is applied at the receiver side in which the embedded information is extracted and compared with the information generated from the received face area to ensure the authenticity of the face in the digital image. when the compared information is not matched the face is classified as unauthentic and the manipulated region is localized in the received image. the following subsections illustrate the details of the proposed embedding and authentication algorithms. a new face swap detection technique for digital images 147 copyright ©2024 assa. adv. in systems science and appl. (2024) 3.1. proposed embedding algorithm the embedding algorithm of the proposed fsd technique can be summarized as illustrated in fig. 3.1. the algorithm starts by reading the original (red, green, and blue) rgb color face image 𝐼 (: , : , 𝑖) where 𝑖 = 1, 2, 3 which refers to the r, g, and b channels of the input image, respectively. in order to detect the face box, the mtcnn is applied to 𝐼 as explained in section 2.1. the output of the mtcnn algorithm is an array contains four readings (𝑦, 𝑥, 𝑤, ℎ) where (𝑥, 𝑦) is the top left corner of the detected face’s box, 𝑤 is the width of the box, and ℎ is the height of the box. to define the detected box in terms of pixels’ positions, the top left corner and the bottom right corner must be defined as integers. the readings (𝑦, 𝑥, 𝑤, ℎ) are rounded to their nearest integer numbers that are greater than or equal to the values. let the resultant array after round is (𝑦 , 𝑥 , 𝑤 , ℎ ), the top left corner of the face box is (𝑥 , 𝑦 ) and the bottom right corner of the face box (𝑥 + ℎ , 𝑦 + 𝑤 ). thus, the pixels’ positions of the face box can be defined as (𝑥 : 𝑥 + ℎ , 𝑦 : 𝑦 + 𝑤 ). a binary mask image 𝐼 is generated according to the obtained positions of pixels that are related to the face box. the following procedure is conducted to generate 𝐼 :  read the size of 𝐼 (𝑀 × 𝑁 × 𝐶) , where 𝑀 = ℎ𝑒𝑖𝑔ℎ𝑡 (𝐼 ) , 𝑁 = 𝑤𝑖𝑑𝑡ℎ (𝐼 ) , and 𝐶 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐ℎ𝑎𝑛𝑛𝑒𝑙𝑠 𝑖𝑛 (𝐼 );  generate a binary image of zeros with size (𝑀 × 𝑁);  convert the pixels at the positions (𝑥 : 𝑥 + ℎ , 𝑦 : 𝑦 + 𝑤 ) to ones;  save the resultant binary image as 𝐼 . after generating the mask image, each channel from 𝐼 can be watermarked using the following procedure:  read the channel image 𝐼 (: , : , 𝑖) where 𝑖 refers to the level of the channel.  divide the channel image and 𝐼 into non-overlapping blocks each of size (16×16) pixels.  classify the blocks of the channel image as shown in fig. 3.2 where the mean value of the pixels (𝜇) in each 𝐼 block is calculated to classify the blocks into two groups (i.e., face blocks and non-face blocks).  generate the authentication information from the ‘face blocks’ by calculating the mean value of the pixels for each block followed by rounding the result to the nearest integer value. the resultant values are converted to binary sequences and concatenated to generate one binary sequence which must be embedded in the ‘nonface blocks’.  to increase the robustness of the embedded sequence, apply bch (11,15) encoding. to embed binary sequence in ‘non-face blocks’, the sequence must be divided into non-overlapping subsequences each of length 64 bits. in order to prepare the binary sequence 𝑏𝑐ℎ for the embedding, the length of the sequence must be divisible by 64. the following steps are applied to prepare the binary subsequences for embedding: o 𝑅𝑒𝑚 = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑏𝑐ℎ /64; o if 𝑅𝑒𝑚 = 0 then 𝐵𝑖𝑛 = 𝑏𝑐ℎ ; o else 𝐸𝑥𝑡𝑒𝑛𝑑 = 64 − 𝑅𝑒𝑚, 𝐸𝑥𝑡𝑒𝑛𝑑 = 𝑧𝑒𝑟𝑜𝑠(1: 𝐸𝑥𝑡𝑒𝑛𝑑); o 𝐵𝑖𝑛 = [𝑏𝑐ℎ , 𝐸𝑥𝑡𝑒𝑛𝑑 ].  apply slt watermarking algorithm (explained in section 2.2) to embed the binary subsequence of 𝐵𝑖𝑛 in ‘non-face blocks’. the resultant watermarked ‘non-face blocks’ and the original ‘face-blocks’ are used to construct the watermarked channel image. 148 r. thabit, h. m. fadhil, h. f. fakhruldeen, a. h. shather, m. a. al-askari copyright ©2024 assa adv. in systems science and appl. (2024) fig. 3.1. the embedding algorithm of the proposed fsd technique. a new face swap detection technique for digital images 149 copyright ©2024 assa. adv. in systems science and appl. (2024) fig. 3.2. classification of channel image blocks into two groups. the procedure of watermarking one channel image is repeated to generate the watermarked channel images which are used to construct the resultant watermarked face image 𝐼 . the watermarked image is sent to the receiver side at which the authentication algorithm must be applied to ensure the safety of the face image and to reveal any manipulation in the face region if exists. the proposed fsd technique doesn’t need the original image or the authentication information at the receiver side which makes the technique completely blind and it can reveal manipulations using the received watermarked image only. 3.2. proposed authentication algorithm the authentication algorithm of the proposed fsd technique can be summarized as illustrated in fig. 3.3. the algorithm starts by reading the watermarked face image 𝐼 and applying the mtcnn algorithm to detect the face box as explained in the embedding procedure. the pixels specification, mask image 𝐼 generation, and blocks classification procedure are the same as those which have been explained at the embedding side. the following procedure is repeated to extract the embedded authentication data and to calculate the authentication data from the ‘face blocks’ in the received 𝐼 :  read channel image 𝐼 (: , ∶, 𝑖) from the received 𝐼 and read the generated 𝐼 .  divide 𝐼 (: , ∶, 𝑖) and 𝐼 into non-overlapping blocks of size (16×16).  classify the blocks into two groups ‘face blocks’ and ‘non-face blocks’ as explained in the embedding procedure.  calculate the new authentication data from the received ‘face blocks’.  extract the embedded authentication data using slt extraction algorithm (explained in section 2.2) from ‘non-face blocks’. as shown in fig. 3.3, the extracted authentication data and the calculated authentication data are compared to check the authenticity of the ‘face blocks’. the block is considered authentic when the compared data are identical. if the compared data are not identical, then the ‘face block’ is considered not authentic and a border is drawn on the block to localize it in the face image. the face image is considered authentic only when all the blocks in the face region are authentic. 150 r. thabit, h. m. fadhil, h. f. fakhruldeen, a. h. shather, m. a. al-askari copyright ©2024 assa adv. in systems science and appl. (2024) fig. 3.3. authentication algorithm of the proposed fsd technique. 4. results and discussion to test the performance of the proposed fsd technique, the experiments have been conducted for color face images with different sizes which have been collected using google image search from different websites such as [45–48]. samples of the test images are shown in fig. 4.1. a new face swap detection technique for digital images 151 copyright ©2024 assa. adv. in systems science and appl. (2024) fig. 4.1. sample test images. the first experiment has been conducted to ensure the accuracy of the proposed fsd technique in detecting the face’s pixels and generating the mask image. to ensure the safety of the face images after watermarking, the visual quality of the watermarked face images has been tested. the ability of the proposed fsd technique in detecting fake faces has been evaluated. experiments have been conducted to test the capacity and its relationship with the number of ‘non-face blocks’. the experiments also include test of payload and its relationship with the number of ‘face blocks’. the following subsections present the experimental results followed by a general comparison between the proposed fsd technique and the previous deep-learning based face swap detection techniques. 4.1. accuracy of mask image generation the mask image generation process depends on the accuracy of selecting the pixels that are related to the face region as illustrated in section 3. the accuracy of generating the mask image has been tested for different test images before proceeding to other tests. the experimental results proved that the proposed technique can accurately select the positions of face’s pixels and generate the mask image without errors. samples of the mtcnn results and their related mask images are shown in fig. 4.2. 152 r. thabit, h. m. fadhil, h. f. fakhruldeen, a. h. shather, m. a. al-askari copyright ©2024 assa adv. in systems science and appl. (2024) fig. 4.2. sample results of mask image generation test. 4.2. visual quality test the visual quality of the resultant watermarked images after watermark embedding using the proposed fsd technique must be tested to ensure that there are no visual artifacts in the image, no errors in rearranging the blocks, and no errors in constructing the watermarked images. the first test is conducted by displaying the original face image and its resultant watermarked image side by side to check differences. samples of this test are shown in fig. 4.3. the results proved that the watermarked images are unscathed and there are no visual artifacts in the image. the second test has been conducted to calculate the peak signal-to-noise ratio (psnr) in db and the mean squared error (mse) between the original image and the watermarked image. the experiments have been conducted for the test images shown in fig. 4.1 and the results are shown in table 4.1. the results proved that the visual quality of the watermarked image depends on the ratio of the face’s blocks to the non-face’s blocks and depends also on the contents of the image where some images require less changes to carry the authentication bits while others require more changes in the image contents. the psnr results shown in table 4.1, illustrate the efficiency of the proposed fsd technique in generating watermarked images with high visual quality. fig. 4.3. sample watermarked images. table 4.1. visual quality test results. image name image size size of face area mse psnr (db) image 1 3000×1987×3 668×533×3 0.313 +53.17 image 2 7952×5304×3 1452×1214×3 0.0181 +65.56 image 3 5472×3648×3 1206×944×3 0.0679 +59.81 a new face swap detection technique for digital images 153 copyright ©2024 assa. adv. in systems science and appl. (2024) image 4 3888×2592×3 1206×944×3 0.0496 +61.17 image 5 3456×5184×3 2029×1677×3 0.042 +61.90 image 6 5075×5760×3 2453×1955×3 0.0761 +59.32 image 7 3008×2008×3 409×338×3 0.003 +73.42 image 8 6240×4160×3 1037×885×3 0.0097 +68.28 image 9 2395×2395×3 1096×913×3 0.0763 +59.31 image 10 3027×2007×3 779×625×3 1.129 +47.60 image 11 5599×3733×3 1461×1277×3 0.4056 +52.05 image 12 3442×2295×3 1277×998×3 0.0426 +61.84 image 13 4090×7360×3 1291×985×3 0.1367 +56.77 image 14 6000×4000×3 1203×1024×3 0.0144 +66.56 image 15 2620×2096×3 1383×1076×3 0.4011 +52.10 4.3. fake faces detection test to test the ability of the proposed fsd technique in detecting fake faces in digital images, face swap attack has been imposed on the watermarked face images using ps adobe photoshop version (21.2.1). the results of this experiment proved the efficiency of the proposed fsd technique in revealing the fake face in the digital image and localizing the manipulated part in the face region. samples of the results are shown in fig. 4.4– fig. 4.9. the proposed fsd technique detects fake faces for all test images without error which makes the accuracy 100 % regardless the quality of the test images. fig. 4.4. fake face detection using the proposed fsd technique for ‘image 2’. 154 r. thabit, h. m. fadhil, h. f. fakhruldeen, a. h. shather, m. a. al-askari copyright ©2024 assa adv. in systems science and appl. (2024) fig. 4.5. fake face detection using the proposed fsd technique for ‘image 4’. fig. 4.6. fake face detection using the proposed fsd technique for ‘image 6’. fig. 4.7. fake face detection using the proposed fsd technique for ‘image 7’. fig. 4.8. fake face detection using the proposed fsd technique for ‘image 13’. a new face swap detection technique for digital images 155 copyright ©2024 assa. adv. in systems science and appl. (2024) fig. 4.9. fake face detection using the proposed fsd technique for ‘image 15’. 4.4. embedding capacity test the embedding capacity of the proposed fsd technique depends on the size of the image and the size of the face region. as mentioned in the embedding procedure, each (16×16) block from the ‘non-face blocks’ can carry 64 bits which is based on the adopted slt watermarking technique (explained in section 2). the number of the ‘non-face blocks’ in one channel is multiplied by 3 to calculate total number of the ‘non-face blocks’ in the image. then the total number of the ‘non-face blocks’ is multiplied by 64 bits to calculate the total embedding capacity of the image. the results of this test are shown in table 4.2 and the relationship between the total embedding capacity and number of ‘non-face blocks’ is illustrated in fig. 4.10. the results proved that the larger the number of ‘non-face blocks’, the higher embedding capacity and vice versa. fig. 4.10. relationship between total capacity and number of ‘non-face blocks’. table 4.2. embedding capacity test results. image name image size size of face area no. of nfb 1 in single channel no. nfb 1 in three channels total capacity (bits) image 1 3000×1987×3 668×533×3 21683 65049 4163136 image 2 7952×5304×3 1452×1214×3 157500 472500 30240000 image 3 5472×3648×3 1206×944×3 73416 220248 14095872 image 4 3888×2592×3 1206×944×3 36720 110160 7050240 image 5 3456×5184×3 2029×1677×3 56416 169248 10831872 image 6 5075×5760×3 2453×1955×3 95178 285534 18274176 0 5000000 10000000 15000000 20000000 25000000 30000000 35000000 t ot al c ap ac it y (b it s) number of non-face blocks in single channel relationship between total capacity and number of non-face blocks 156 r. thabit, h. m. fadhil, h. f. fakhruldeen, a. h. shather, m. a. al-askari copyright ©2024 assa adv. in systems science and appl. (2024) image 7 3008×2008×3 409×338×3 22928 68784 4402176 image 8 6240×4160×3 1037×885×3 97704 293112 18759168 image 9 2395×2395×3 1096×913×3 18141 54423 3483072 image 10 3027×2007×3 779×625×3 21625 64875 4152000 image 11 5599×3733×3 1461×1277×3 73957 221871 14199744 image 12 3442×2295×3 1277×998×3 25705 77115 4935360 image 13 4090×7360×3 1291×985×3 112278 336834 21557376 image 14 6000×4000×3 1203×1024×3 88810 266430 17051520 image 15 2620×2096×3 1383×1076×3 15437 46311 2963904 1 no. of nfb = number of ‘non-face blocks’. 4.5. payload test the payload in each channel from the input image refers to the total number of bits that are generated in 𝐵𝑖𝑛 as explained in (subsection 3.1). the generated payload depends on the size of the face region. as mentioned in the embedding procedure, the authentication information is generated from the ‘face blocks’, converted to binary, bch coded, and prepared for embedding in ‘non-face blocks’. the results of payload test are shown in table 4.3 and the relationship between the total payload and number of ‘face blocks’ is illustrated in fig. 4.11. the results proved that the larger the number of ‘face blocks’, the higher payload and vice versa. fig. 4.11. relationship between total payload and number of ‘face blocks’. table 4.3. payload test results. image name image size size of face area no. of fb in single channel length of 𝐵𝑖𝑛 in single channel total payload (bits) image 1 3000×1987×3 668×533×3 1505 16448 49344 image 2 7952×5304×3 1452×1214×3 7007 76480 229440 image 3 5472×3648×3 1206×944×3 4560 49792 149376 image 4 3888×2592×3 1206×944×3 2646 28928 86784 image 5 3456×5184×3 2029×1677×3 13568 148032 444096 image 6 5075×5760×3 2453×1955×3 18942 206656 619968 image 7 3008×2008×3 409×338×3 572 6272 18816 image 8 6240×4160×3 1037×885×3 3696 40384 120960 0 100000 200000 300000 400000 500000 600000 700000 t ot al p ay lo ad ( b it s) number of face blocks in single channel relationship between total payload and number of face blocks a new face swap detection technique for digital images 157 copyright ©2024 assa. adv. in systems science and appl. (2024) image 9 2395×2395×3 1096×913×3 4060 44352 133056 image 10 3027×2007×3 779×625×3 2000 21888 65664 image 11 5599×3733×3 1461×1277×3 7360 80320 240960 image 12 3442×2295×3 1277×998×3 5040 55040 165120 image 13 4090×7360×3 1291×985×3 5022 54848 164544 image 14 6000×4000×3 1203×1024×3 4940 53952 161856 image 15 2620×2096×3 1383×1076×3 5916 64576 193728 1 no. of fb = number of ‘face blocks’ 4.6. time complexity the execution time required for the embedding and authentication procedure is very important in practical applications, therefore, the run-time of the proposed fsd algorithms have been calculated for different test images. since the run-time is affected by the hardware and software used in the experimental tests it is useful to mention that the computer used in the experimental tests has 1.80 ghz intel® core tm i7 cpu and 16 gb memory. the software used is matlab (r2020a) and the commands used for this experiment are tic and toc commands. the results of this experiment are shown in table 4.4 which proved the efficiency of the proposed fsd technique in executing embedding and authentication algorithms in seconds. the execution time of the algorithms depends on the size of the face image and the total payload. the higher the payload, the longer the execution time. as shown in the results, the execution time for authentication is lower than that for the embedding which makes the proposed fsd technique suitable for the practical application that require face authentication before proceeding to other processing steps. the test for small size images gives much lower execution time, for instance an image of size (164×307×3) and payload (1024) required (0.31235) seconds for embedding and (0.2701) seconds for authentication. 4.7. comparison with previous techniques as mentioned in the introduction section, the false detection, high time complexity for training and testing, the need for high-quality images for training, and the need for large datasets are some of the limitations in the face swap detection methods that are based on machine learning and deep learning algorithms. since the proposed fsd techniques adopted digital watermarking, there is no need for training and it can be applied for any face image regardless its quality. the accuracy of the proposed technique is 100 % and there are no false detection results thus it outperforms the techniques in [18,22,23] which have recorded accuracies around [92 %, 97 %, 98 % to 99 %] for only specific datasets. the results of the techniques in [18,22,23] can be further degraded when the test images are different from the datasets used in the training process while the results of the proposed fsd technique are always accurate. table 4.4. time complexity test results. image name image size size of face area total payload (bits) embeddi ng time (sec.) extractio n time (sec.) image 1 3000×1987×3 668×533×3 49344 3.3191 1.8827 image 2 7952×5304×3 1452×1214×3 229440 24.6058 14.1453 image 3 5472×3648×3 1206×944×3 149376 12.1752 7.3814 image 4 3888×2592×3 1206×944×3 86784 6.4060 3.5111 image 5 3456×5184×3 2029×1677×3 444096 12.1816 8.7077 image 6 5075×5760×3 2453×1955×3 619968 15.9596 12.1469 image 7 3008×2008×3 409×338×3 18816 2.9516 1.0532 image 8 6240×4160×3 1037×885×3 120960 14.4640 7.1083 158 r. thabit, h. m. fadhil, h. f. fakhruldeen, a. h. shather, m. a. al-askari copyright ©2024 assa adv. in systems science and appl. (2024) image 9 2395×2395×3 1096×913×3 133056 5.4332 3.2164 image 10 3027×2007×3 779×625×3 65664 3.8354 2.4048 image 11 5599×3733×3 1461×1277×3 240960 12.1705 11.3076 image 12 3442×2295×3 1277×998×3 165120 5.4606 4.0822 image 13 4090×7360×3 1291×985×3 164544 17.5745 10.5852 image 14 6000×4000×3 1203×1024×3 161856 14.5385 7.9509 image 15 2620×2096×3 1383×1076×3 193728 3.6695 2.3317 5. conclusions the rapid development of new technology and the spread of easy-to-use applications can be considered as double-edged sword where these applications can be used for innocent or malicious intentions. recently, several applications have been introduced to swap the faces in digital images which can be adopted for identity theft, threatening, destroying the reputation, spreading fake news, and many others. to authenticate the face image, the research community presented different face swap detection techniques based on machine-learning and deeplearning. there are some limitations in these techniques such as the false detections results, the long time required for training and testing, the need for large datasets for training, the need for high quality images to obtain better detection results, etc. in this paper, a new face swap detection (fsd) technique is presented based on images watermarking technology. the proposed fsd technique have two main algorithms each of them starts by the proposed steps to detect the face region in the image. the embedding algorithm is applied to generate and hide the authentication information while the authentication algorithm is applied to extract and authenticate the face information in the received image. experiments have been conducted to evaluate the performance of the proposed fsd technique for different test images. the results proved that capacity and payload depend on the size of the image and the size of the face region. the larger the size of the face region, the higher the payload. the embedding capacity increases with the increment in the ratio of the number of blocks outside face region to the number of blocks belong to face region. the subjective and objective evaluation of the visual quality proved the efficiency of the proposed fsd technique where high quality watermarked images are generated without any visible distortions. the proposed fsd technique can effectively detect fake face in the image and the accuracy is 100 %. the execution time of the algorithms is low even for large size images which makes the proposed technique suitable for practical applications. for the future work, different watermarking techniques can be applied and compared with the proposed fsd technique in the aim of further improving the performance. funding this research received no external funding. acknowledgements the authors would like to thank their institutions for encouraging and supporting their scientific researches. conflicts of interest the authors declare no conflict of interest. a new face swap detection technique for digital images 159 copyright ©2024 assa. adv. in systems science and appl. 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[online]. available https://www.istockphoto.com/search/2/image?phrase=family adv syst sci appl 2021; 02:83–103 published online at https://ijassa.ipu.ru. stability analysis and optimal measure for controlling eco-epidemiological dynamics of prey-predator model jamiu a. ademosu, samson olaniyi*, sunday o. adewale department of pure and applied mathematics, ladoke akintola university of technology, ogbomoso, nigeria abstract: an eco-epidemiological model representing the interactions between prey and predator populations affected by a disease in an ecosystem is presented. the model is governed by a five-dimensional nonlinear system of ordinary differential equations coupling both ecological and epidemiological features of interacting populations. the well-posedness of the model is established with respect to positivity and boundedness of solutions. conditions for asymptotic stability of different equilibrium points are extensively investigated to determine the existence and coexistence of prey and predator species using local linearization and lyapunov functions techniques. additionally, the analysis of the model is extended to assess the effects of three timedependent control functions, such as disease prevention, treatment and alternative resource for predator, on the population dynamics of the prey-predator coexistence in the system. keywords: ecology, epidemiology, prey-predator model, stability, optimal control problem, simulations 1. introduction interaction between organisms of different species is a common phenomenon in an ecosystem. it is an intrinsic feature which cannot be downplayed in the study of ecological system. this interaction, especially between prey and predator species, often affects or changes the population sizes of organisms in the system [11]. since the advent of the classical lotka and volterra prey-predator models [20, 31], many ecosystem models, which can be defined as mathematical representations of ecological system of prey and predator interactions, have been studied in literature to better understand the real system (see, e.g. [5, 11, 13, 17, 27]). in [11], analysis of modified lotka-volterra model was carried out by taking into consideration an environmental case containing two related populations of prey and predator species having influence on the size of each other. kumar and kumari [17] studied the impact of fear on a chaotic model describing the interactions of the species in the food chain of one prey and two predators. prasad et al. [27] investigated the role of mutual interference between predators on the dynamics of additional food provided predator-prey system. in recent decades, a number of studies have considered ecosystems of interacting populations among which diseases spread. the resulting ecological systems are described by eco-epidemiological models [7–9, 21, 29]. das [9] studied the predator-prey model with parasitic infection transmitting among the predator population only. nandi et al. [21] developed a predator-prey system where prey species only are infected with a viral disease resulting into susceptible and two-stage infected classes. bera et al. [8] presented theoretical analysis supported by simulation of the dynamical behaviors of a prey-predator system where both populations are affected by diseases. ∗corresponding author: solaniyi@lautech.edu.ng 84 j.a. ademosu, s. olaniyi, s.o. adewale interest has been growing recently in the application of optimal control theory to the dynamics of eco-epidemiological models for prey-predator system. al-nassir [6] studied a two-dimensional continuous prey-predator system incorporating an optimal control variable with the aim of reducing the number of the predator density in the population. in [10], the authors extended a prey-predator mathematical model with infection and harvesting on prey to include one time-dependent control for preventing infection in the prey population only. in another development, hugo et al. [16] applied optimal control theory in minimizing the spread of newcastle disease among chickens (prey) and humans (predator) populations using three control variables, such as vaccination of prey, education campaign and treatment of infected humans. in this work, a five-dimensional nonlinear prey-predator model is studied to gain further insights into the dynamics of eco-epidemiological interactions between prey and predator populations. the analysis of the model is extended to investigate the impact of three optimal control variables, namely disease prevention in both prey and predator populations, treatment control for prey, and provision of alternative resources control for predator population. the rest of the work is organized as follows: the model is formulated and its well-posedness is established in section 2. in section 3, existence and stability of possible equilibrium points are investigated. in section 4, the model is extended to explore optimal control dynamics with numerical simulations. section 5 is devoted to concluding remarks. 2. model formulation the dynamic eco-epidemiological prey-predator system is formulated by sub-dividing the prey population at time, t, into susceptible xs(t), infected xi(t) and recovered xr(t). on the other hand, the predator population is sub-divided into two, namely susceptible ys(t) and infected yi(t). it is assumed that the susceptible prey population increases logistically with intrinsic growth rate r and carrying capacity k. following effective contact with the infected prey, the susceptible prey becomes infected at rate β1. to avoid extinction of the prey population due to disease, the infected prey is allowed to recover at per capita rate γ. the natural mortality rate for the prey population is given by µ1. further, it is assumed that the susceptible predator, being healthy, has the capacity to attack and consume both susceptible and infected prey, while the recovered prey is assumed to be under cover from predation. it is also assumed that predators depend on prey for sustenance. the susceptible predator contracts disease from infected predator at rate β2, while the infected predator, being unhealthy, has the capacity to attack and consume infected prey only as assumed in [8]. it has been assumed that only infected prey population recovers due to their accessibility to treatment. hence, recovered class for predators is not considered. this assumption is based on some eco-epidemiological cases, for examples: carnivorous animals (predator) and humans (prey), and hawks (predator) and chicks (prey). the predation and conversion rates for the predator population are given, respectively, by pi and ci, for i = 1, 2, 3. the natural mortality rate for predator population is given by µ2. concise description of the variables and the parameters of the prey-predator system is presented in table 2.1 and the nonlinear ordinary differential equations describing the interacting prey and predator populations is given by copyright © 2021 assa. adv syst sci appl (2021) stability analysis and optimal measure 85 dxs dt =rxs ( 1− xs k ) − β1xsxi − p1xsys − µ1xs dxi dt =β1xsxi − p2xiys − p3xiyi − (µ1 + γ)xi dxr dt =γxi − µ1xr dys dt =c1p1xsys + c2p2xiys − β2ysyi − µ2ys dyi dt =β2ysyi + c3p3xiyi − µ2yi. (2.1) table 2.1. the variables and parameters of the malaria model (2.1) variable description xs(t) susceptible (healthy) prey xi(t) infected (unhealthy) prey xr(t) recovered prey ys(t) susceptible (healthy) predator yi(t) infected (unhealthy) predator parameter description r intrinsic growth rate k carrying capacity β1 disease transmission rate in prey β2 disease transmission rate in predator pi, i = 1, 2, 3 predation coefficients ci, i = 1, 2, 3 conversion rates γ recovery rate of infected prey µ1 natural mortality rate of prey µ2 natural mortality rate of predator 2.1. well-posedness of the model the mathematical and eco-epidemiological relevance of the prey-predator system (2.1) depends on the well-posedness of the model. keeping in mind that all the parameters of the model are non-negative. here, the boundedness and positivity of solutions of the model are investigated to establish the well-posedness of the model. 2.1.1. boundedness of solutions the following result is required to establish the boundedness of solutions of the model (2.1). lemma 2.1: the susceptible prey population xs(t) is bounded. proof it is clear from the first equation of the model (2.1) that dxs dt ≤ rxs − x2 s k . (2.2) copyright © 2021 assa. adv syst sci appl (2021) 86 j.a. ademosu, s. olaniyi, s.o. adewale solving the nonlinear first order differential inequality of bernoulli type (2.2) yields xs(t) ≤ kxs(0) ke−rt +xs(0)(1− e−rt) . it follows that the lim supxs(t) < k as t→∞. theorem 2.1: the solutions of the system (2.1) are uniformly bounded. proof let the total prey and predator populations be represented, respectively, by x and y , so that the total population of both species be t = x + y . then it follows that dt dt ≤ rxs − µ1x − µ2y. (2.3) using lemma 2.1 and let µ = min{µ1, µ2}, then (2.3) becomes dt dt + µt ≤ rxs ≤ rk, which on integration gives t (t) ≤ rk µ (1− e−µt) + t (0)e−µt. consequently, lim supt (t) < (rk/µ) + ε, ∀ ε > 0 as t→∞. this ends the proof. 2.1.2. positivity of solutions theorem 2.2: the solution set {xs, xi, xr, ys, yi} of the eco-epidemiological model (2.1) with nonnegative initial conditions, xs(0), xi(0), xr(0), ys(0), yi(0) in ω, remain non-negative for all time t > 0. proof sincexs(t) < k as shown in lemma 2.1, then (1−xs/k) ≥ 0 with equality if the size of the susceptible prey is exactly its carrying capacity. hence, the first equation of the model (2.1) implies that dxs(t) dt + (β1xi + p1ys + µ1)xs(t) ≥ 0, (2.4) which on using integrating factor gives d dt [ xs(t) exp (∫ t 0 (β1xi(φ) + p1ys(φ))dφ+ µ1t )] ≥ 0. (2.5) further integration of (2.5) gives xs(t) ≥ xs(0) exp [ − (∫ t 0 (β1xi(φ) + p1ys(φ))dφ+ µ1t )] > 0, ∀t > 0. (2.6) all other variables can be proved to be positive in a similar approach. copyright © 2021 assa. adv syst sci appl (2021) stability analysis and optimal measure 87 hence, it is sufficient to consider the dynamics of the flow generated by the system (2.1) in a feasible region defined by ω = { (xs, xi, xr, ys, yi) ∈ r5 + : t < rk µ + ε } . in the region ω, the prey-predator model (2.1) can be said to be eco-epidemiologically wellposed. 3. analysis of the model in this section, the prey-predator model is analysed around the possible equilibrium points. 3.1. existence of equilibrium points the prey-predator model (2.1) has the following possible equilibrium points: 3.1.1. trivial equilibrium point (e0) this is a steady state in the absence of prey and predator populations. hence, no interactions exist. the equilibrium point is given by e0 = (0, 0, 0, 0, 0). (3.7) 3.1.2. axial equilibrium point (e1) this is a steady state where only the susceptible prey is present. in other words, it is the disease-free cum predator-free equilibrium point. it exists provided that the intrinsic growth rate r exceeds the natural mortality rate µ1. the equilibrium point is given by e1 = ( k [ 1− µ1 r ] , 0, 0, 0, 0 ) . (3.8) 3.1.3. disease-free equilibrium point (e2) this is a steady state where there is no disease in both prey and predator populations. it exists provided the inequality r ( 1− µ2 kc1p1 ) > µ1 holds. hence, it is obtained as e2 = ( µ2 c1p1 , 0, 0, 1 p1 [ r ( 1− µ2 kc1p1 ) − µ1 ] , 0 ) . (3.9) 3.1.4. predator-free equilibrium point (e3) this is a steady state where there is no predator in the system. it exists provided the inequality r ( 1− ( µ1+γ kβ1 )) > µ1 holds. the equilibrium is given as e3 = ( µ1 + γ β1 , 1 β1 [ r ( 1− ( µ1 + γ kβ1 )) − µ1 ] , γ β1µ1 [ r ( 1− ( µ1 + γ kβ1 )) − µ1 ] , 0, 0 ) . (3.10) 3.1.5. infected prey-free equilibrium (e4) this is a steady state where the infected prey does not exist. the equilibrium will exist if k > k r ( p1µ2 β2 + µ1 ) and c1p1 β2 [ k − k r ( p1µ2 β2 + µ1 )] > µ2. thus, it is given by e4 = ( k − k r ( p1µ2 β2 + µ1 ) , 0, 0, µ2 β2 , c1p1 β2 [ k − k r ( p1µ2 β2 + µ1 )] − µ2 ) . (3.11) copyright © 2021 assa. adv syst sci appl (2021) 88 j.a. ademosu, s. olaniyi, s.o. adewale 3.1.6. infected predator-free equilibrium (e5) this is a steady state where the infected predator is absent in the ecosystem. the equilibrium is given by e5 = (x∗ s , x ∗ i , x ∗ r , y ∗ s , 0), (3.12) such that x∗ s > 0, x∗ i > 0, x∗ r > 0, y ∗ s > 0, and where x∗ s = c2[p2(r − µ1)− p1(µ1 + γ)]− β1µ2 c2p2 ( 1 k + p1β1 p2 ) − c1p1β1 , x∗ i = µ2 c2p2 − c1p1 c2p2 c2[p2(r − µ1)− p1(µ1 + γ)]− β1µ2 c2p2 ( 1 k + p1β1 p2 ) − c1p1β1  , x∗ r = γx∗ i µ1 , y ∗ s = β1x ∗ s − (µ1 + γ) p2 . 3.1.7. interior equilibrium point (e6) this is a steady state where all the populations for prey and predator species are present in the system. the equilibrium is given by e6 = (x∗∗ s , x ∗∗ i , x ∗∗ r , y ∗∗ s , y ∗∗ i ), (3.13) such that x∗∗ s > 0, x∗∗ i > 0, x∗∗ r > 0, y ∗∗ s > 0, y ∗∗ i > 0, and where x∗∗ s = 1 k [ (r − µ1)−x∗∗ i ( β1 − c3p1c3 β2 )] , x∗∗ i = β2[(p2 − p3)µ2 + β2(µ1 + γ)− (r − µ1)(β1β2 − c1p1p3)] β2(p2p3c3 + c2p2p3) + (β1β2 − c1p1p3)(β1β2 − c3p1p3) , x∗∗ r = γx∗∗ i µ1 , y ∗∗ s = µ2 − c3p3x ∗∗ i β2 , y ∗∗ i = β1x ∗∗ s − p2y ∗∗ s − (µ1 + γ) p3 . 3.2. stability analysis the jacobian matrix of the prey-predator system (2.1) is given by j =  j1 −β1xs 0 −p1xs 0 β1xi j2 0 −p2xi −p3xi 0 γ −µ1 0 0 c1p1ys c2p2ys 0 j3 −β2ys 0 0 0 β2yi β2ys + c3p3xi − µ2  , (3.14) copyright © 2021 assa. adv syst sci appl (2021) stability analysis and optimal measure 89 where j1 = r − 2rxs k − β1xi − p1ys − µ1, j2 = β1xs − p2ys − p3yi − (µ1 + γ), j3 = c1p1xs + c2p2xi − β2yi − µ2. the stability of each of the equilibrium points is analysed by finding the eigenvalues of the jacobian matrix (3.14) evaluated at each point. 3.2.1. stability of e0 the jacobian matrix (3.14) is evaluated at the trivial equilibrium point (3.7). hence, solving the corresponding characteristic equation |j(e0)− λi5)| = 0, where λ is the eigenvalue and i5 is the identity matrix of order five, the following eigenvalues are obtained: λ1 = r − µ1, λ2 = −µ1, λ3 = −µ2, λ4 = −(µ1 + γ), λ5 = −µ2. it can be seen that all the eigenvalues are unconditionally negative except λ1. thus, e0 will be stable if r < µ1 and unstable if r > µ1. this result is theorized as follows. theorem 3.1: the trivial equilibrium point e0 of the prey-predator model (2.1), given by (3.7), is stable if r < µ1 and unstable if r > µ1. the implication of theorem 3.1 is that the coexistence of prey-predator population will not occur if the intrinsic growth rate r does not exceed the natural mortality rate µ1. in other words, stable equilibrium point e0 will result into extinction of prey and predator species in the ecosystem. 3.2.2. stability of e1 the characteristic equation of the matrix (3.14) evaluated at the axial equilibrium point e1 is given by |j(e1)− λi5)| = 0, which gives the following eigenvalues: λ1 = µ1 − r, λ2 = −µ1, λ3 = β1k ( 1− µ1 r ) − (µ1 + γ), λ4 = c1p1k ( 1− µ1 r ) − µ2, λ5 = −µ2. the stability result is theorized as follows. theorem 3.2: the axial equilibrium point e1 of the prey-predator model (2.1), given by (3.8), is stable if the inequalities: µ1 < r, β1k ( 1− µ1 r ) < (µ1 + γ) and c1p1k ( 1− µ1 r ) < µ2 hold. the implication of theorem 3.2 is that the stable e1 will support the existence of prey population only in the ecosystem provided these threshold conditions are satisfied: r01 < 1,r02 < 1 andr03 < 1, where r01 = µ1 r , r02 = β1k ( 1− µ1 r ) µ1 + γ , and r03 = c1p1k ( 1− µ1 r ) µ2 . now, suppose r0 = max{r01,r02,r03}, and considering the prey sub-model only such that the axial equilibrium point e1 becomes ẽ1 = (x0 s , 0, 0), where x0 s = k [ 1− µ1 r ] . the following global stability result is proved. theorem 3.3: the equilibrium point ẽ1 is globally asymptotically stable ifr0 < 1. proof since p1 = 0 in the absence of predation, thenr0 = max{r01,r02}. consider the lyapunov copyright © 2021 assa. adv syst sci appl (2021) 90 j.a. ademosu, s. olaniyi, s.o. adewale function (combination of goh-volterra [2, 3, 13, 15, 22] and linear types) given by l = xs −x0 s −x0 s ln xs x0 s +xi. (3.15) the time derivative of lyapunov function (3.15) along the trajectory of the prey only submodel is given by l̇ = ( 1− x0 s xs ) dxs dt + dxi dt . (3.16) using dxs dt = rxs ( 1− xs k ) − β1xsxi − µ1xs and dxi dt = β1xsxi − (µ1 + γ)xi in (3.16) with xs ≤ x0 s , it follows that l̇ = (xs −x0 s ) ( r [ 1− xs k ] − β1xi − µ1 ) + β1xsxi − (µ1 + γ)xi ≤ r [ 1− x0 s k ] − β1xi − µ1 + β1x 0 sxi − (µ1 + γ)xi. (3.17) since r [ 1− x0 s k ] = µ1 at the disease-free steady state of the prey only sub-model, then it follows from (3.17), by further simplifications, that l̇ ≤ −β1xi − (µ1 + γ) ( 1− β1x 0 s µ1 + γ ) xi. (3.18) sincer02 = β1x 0 s µ1 + γ , consequently (3.18) becomes l̇ ≤ −β1xi − (µ1 + γ) (1−r02)xi. therefore, l̇ ≤ 0 when r02 < 1 with equality, l̇ = 0, if and only if xi = 0. it follows from lasalle’s invariance principle [19] that the largest invariant set contained in {(xs, xi, xr) ∈ r3 + : l̇ = 0} is the singleton set {ẽ1}. hence, the equilibrium point ẽ1 = (k[1− (µ1/r)], 0, 0) is globally asymptotically stable. the implication of theorem 3.3 is that the presence of disease in the prey population can be eliminated if r02 < 1, irrespective of the initial sizes of the infected prey in the ecosystem. thus, every solution of the prey only sub-model will always converge to the point ẽ1 wheneverr02 is less than unity. this theoretical result is demonstrated in figure 3.1. 3.2.3. stability of e2 considering that there are only two disease classes in the prey-predator system (2.1), then using the notations in the next generation operator method [30], it follows that f = [ β1µ2 c1p1 0 0 β2 p1 m1 ] and v = [ p2 p1 m1 + µ1 + γ 0 0 µ2 ] , where m1 = r ( 1− µ2 kc1p1 ) − µ1. hence, the basic reproduction number of the full system (2.1), denoted by r∗ 0, is given by r∗ 0 = ρ(fv −1) = max  β1µ2 c1p1 ( p2 p1 m1 + µ1 + γ ) , β2m1 p1µ2  . (3.19) copyright © 2021 assa. adv syst sci appl (2021) stability analysis and optimal measure 91 0 5 10 15 0 10 20 30 40 50 60 70 time in fe ct ed p re y fig. 3.1. convergence of solution trajectories of prey sub-model to the equilibrium point ẽ1 at different initial sizes. the parameter values used are: r = 11.2, β1 = 0.04, µ1 = 0.85, γ = 0.75, k = 30, so thatr01 = 0.0759 andr02 = 0.6931, implying thatr0 = r02 < 1. therefore, by theorem 2 of [30], the following stability result is claimed. theorem 3.4: the disease-free equilibrium point e2 of the prey-predator model (2.1), given by (3.9), is stable provided that r∗ 0 < 1. the implication of theorem 3.4 is that the presence of few species around the equilibrium point e2 will support the coexistence of both prey and predator species in a disease-free ecosystem whenever the basic reproduction number, r∗ 0, is less than unity. the global dynamics of the prey-predator around the disease-free equilibrium point e2 is explored next. theorem 3.5: the disease-free equilibrium point of the prey-predator model (2.1), given by (3.9), is globally asymptotically stable if r∗ 0 > 1. proof the proof is based on using the lyapunov function defined by f = p1 p2m1 + p1(µ1 + γ) xi + 1 µ2 yi, (3.20) which when differentiated with respect to time, t, gives df dt = p1 p2m1 + p1(µ1 + γ) [β1xsxi − p2xiys − p3xiyi − (µ1 + γ)xi] + 1 µ2 (β2ysyi + c3p3xiyi − µ2yi). copyright © 2021 assa. adv syst sci appl (2021) 92 j.a. ademosu, s. olaniyi, s.o. adewale since xs ≤ β1µ2 c1p1 and ys ≤ 1 p1 [ r ( 1− µ2 kc1p1 ) − µ1 ] , noting that m1 = r ( 1− µ2 kc1p1 ) − µ1. therefore, it follows, after simplification, that df dt ≤  β1µ2 c1p1 ( p2 p1 m1 + µ1 + γ ) − 1 xi − p1p3 p2m1 + p1(µ1 + γ) xiyi + [ β2m1 p1µ2 − 1 ] yi + 1 µ2 (c3p3xiyi) ≤ max  β1µ2 c1p1 ( p2 p1 m1 + µ1 + γ ) , β2m1 p1µ2 − 1  (xi + yi) − [ p1 p2m1 + p1(µ1 + γ) − c3 µ2 ] p3xiyi = [r∗ 0 − 1](xi + yi)− [ p1 p2m1 + p1(µ1 + γ) − c3 µ2 ] p3xiyi. consequently, df/dt ≤ 0 if r∗ 0 < 1 and xi = 0 or yi = 0. thus, xs → µ2/(c1p1), xr → 0 and ys → m1/p1 as both xi → 0 and yi → 0. therefore, by lasalle’s invariance principle [19], the disease-free equilibrium point e2 is globally asymptotically stable. the implication of theorem 3.5 is that elimination of disease in the ecosystem is independent of initial sizes of infected prey and infected predator whenever r∗ 0 < 1. this result is demonstrated in figure 3.2(a) for infected prey and figure 3.2(b) for infected predator population, where every solution trajectory converges to the disease-free equilibrium. 0 5 10 15 20 25 30 35 40 0 1 2 3 4 5 6 7 8 9 10 time in fe c te d p re y (a) 0 5 10 15 0 5 10 15 20 25 time in fe c te d p re d a to r (b) fig. 3.2. convergence of solution trajectories of prey-predator model to the disease-free equilibrium point e2 at different initial sizes. the parameter values used are: mu1 = 0.8, r = 3, β1 = 0.4, p1 = 1.0, γ = 0.35, p2 = 0.2, p3 = 1.0, µ2 = 0.85, k = 30, β2 = 0.3, c1 = 0.25, c2 = 0.25, c3 = 0.15, so that r∗ 0 = 0.6565 < 1. copyright © 2021 assa. adv syst sci appl (2021) stability analysis and optimal measure 93 3.2.4. stability of e3 evaluating the jacobian matrix (2.1) at predator-free equilibriume3 leads to the characteristic equation |j(e3)− λi5| = 0. thus, the following eigenvalues are obtained: λ1 = −µ1 λ2 = −µ2, λ3 = c1p1 β1 (µ1 + γ)− c2p2 β1 (m2 + µ2), λ4 = −r 2 ( µ1 + γ kβ1 ) + 1 2 √ r2 ( µ1 + γ kβ1 )2 + 4m2(µ1 + γ), λ5 = −r 2 ( µ1 + γ kβ1 ) − 1 2 √ r2 ( µ1 + γ kβ1 )2 + 4m2(µ1 + γ), where m2 = r ( 1− µ1+γ kβ1 ) − µ1. the stability of e3 is theorized as follows theorem 3.6: the predator-free equilibrium point e3 of the prey-predator model (2.1), given by (3.10), is stable if the following conditions hold c1p1(µ1 + γ) < c2p2(m2 + µ2) and √ r2 ( µ1 + γ kβ1 )2 + 4m2(µ1 + γ) < r ( µ1 + γ kβ1 ) . this result implies that stable e3 will not be able to support the coexistence of both prey and predator populations. the presence of few species around the predator-free equilibrium point e3 will make predator population to go extinct in the ecosystem. 3.2.5. stability of e4 the jacobian matrix (3.14) is evaluated at the infected prey-free equilibrium point (3.11). hence, solving the corresponding characteristic equation |j(e4)− λi5)| = 0 gives one eigenvalue, λ1 = −µ1, and the remaining four eigenvalues are obtainable from λ4 +w1λ 3 +w2λ 2 +w3λ+w4 = 0, (3.21) where w1 = r − µ1 − p1 µ2 β2 , w2 = d1d2 − 2µ2(d1 + d2) + µ2 + c1p1d3 − c1p1 µ2 β2 with d1 = r − 2r d3 k − p1 µ2 β2 − µ1, d2 = β1d3 − p2 µ2 β2 − c1p1p3 d3 β2 − (µ1 + µ2 + γ) and d3 = k − k r ( p1µ2 β2 + µ2 ) . w3 = c1p1 [ (d2 − µ2) µ2 β2 − d3(d2µ2 + d1) ] − d2µ2(2d1 + µ2) and w4 = d2µ 2 2 ( d1 + p1 β2 ) . copyright © 2021 assa. adv syst sci appl (2021) 94 j.a. ademosu, s. olaniyi, s.o. adewale applying routh-hurwitz’s criterion, the roots of (3.21) will have negative real parts if w1 > 0, w1w2 > w3, w1w2w3 > w 2 1w4 +w 2 3 . the stability result is hereby theorized as follows. theorem 3.7: the infected prey-free equilibrium of the prey-predator model (2.1), given by (3.11), is stable if w1 > 0, w1w2 > w3 and w1w2w3 > w 2 1w4 +w 2 3 . the implication of theorem 3.7 is that the presence of few species around the stable infected prey-free equilibrium e4 will support the coexistence of both prey and predator populations in the ecosystem. 3.2.6. stability of e5 two of the eigenvalues of the jacobian matrix (3.14) evaluated at the infected predator-free equilibrium point e5 are given by λ1 = −µ1 and λ2 = −µ2, while the remaining can be obtained from the characteristic equation λ3 + e1λ 2 + e2λ+ e3 = 0, (3.22) where e1 = −(f1 + f2 + f3), with f1 = 2c1p1f̃1 − µ2(1 + c2p2) such that f̃1 = c2p2(r − µ1)− p1c2(µ1 + γ)− βµ2 c2p2 ( 1 k + p1β1 p2 ) − c1p1β1 , f2 = r − µ2 − [ 2r k + c2p2 + kp1β1(c2 − c1) k(c2p2 + kp1c2 − c1p1β1) ] f̃1 − β1 ( µ2 − c1p1 c2p2 f̃1 ) (1− p1) f3 = β1f̃1 − p2 [( c2p2 + kp1β1(c2 − c1) k(c2p2 + kp1c2 − c1p1β1) ) f̃1 − β1 ( µ2 − c1p1 c2p2 f̃1 )] − (µ1 + γ). e2 = f1f2 + f3(f1 + f2) + β2 1 f̃1 ( µ2 − c1p1 c2p2 f̃1 ) + p 2 1 c1 p2 f̃1(f3 + (µ1 + γ)− β1f̃1), e3 = 1 p2 (f3 + (µ1 + γ)− β1f̃1) [( µ2 − c1p1 c2p2 f̃1 ) (p2(c2[p1f̃1 − p2]− c1p1))− c1p 2 1 f̃1(f3 + f1f2) ] . by routh-hurwitz’s criterion, the roots of the characteristic equation (3.22) will have negative real parts if e1 > 0 and e1e2 > e3. this result is stated as follows. theorem 3.8: the infected predator-free equilibrium point of the prey-predator system (2.1), given by (3.12), is stable if e1 > 0 and e1e2 > e3. the result in theorem 3.8 implies that coexistence of prey and predator species in the ecosystem is possible in the absence of infected predator, provided that the given parametric conditions are valid. 3.2.7. stability of e6 recall from (3.19) that the basic reproduction number of the full system (2.1) is given by r∗ 0 = max  β1µ2 c1p1 ( p2 p1 m1 + µ1 + γ ) , β2m1 p1µ2  . the following global stability result is explored. copyright © 2021 assa. adv syst sci appl (2021) stability analysis and optimal measure 95 theorem 3.9: the interior equilibrium point e6 of the prey-predator model (2.1) is globally asymptotically stable if r∗ 0 > 1. proof let r∗ 0 > 1, such that the interior equilibrium point e6 exists. noting from (3.13) that e6 = (x∗∗ s , x ∗∗ i , x ∗∗ r , y ∗∗ s , y ∗∗ i ), then the following lyapunov function of quadratic type [28] is defined: v = 1 2 [(xs −x∗∗ s ) + (xi −x∗∗ i ) + (xr −x∗∗ r ) + (ys − y ∗∗ s ) + (yi − y ∗∗ i )]2 (3.23) the time derivative of v in (3.23) along the solution path of the full system (2.1) is given by v̇ = [(xs −x∗∗ s ) + (xi −x∗∗ i ) + (xr −x∗∗ r ) + (ys − y ∗∗ s ) + (yi − y ∗∗ i )] dt dt , (3.24) where t (t) = xs +xi +xr + ys + yi. recall from (2.3) of the boundedness result (theorem 2.1) that dt dt ≤ rk − µt, and since x∗∗ s +x∗∗ i +x∗∗ r + y ∗∗ s + y ∗∗ i = rk µ . it follows from (3.24) that v̇ ≤ [ t (t)− rk µ ] [rk − µt (t)] = µ [ t (t)− rk µ ] [ rk µ − t (t) ] = −µ [ t (t)− rk µ ]2 = −µ[(xs −x∗∗ s ) + (xi −x∗∗ i ) + (xr −x∗∗ r ) + (ys − y ∗∗ s ) + (yi − y ∗∗ i )]2. it can be seen that v̇ ≤ 0 with equality if and only if xs = x∗∗ s , xi = x∗∗ i , xr = x∗∗ r , ys = y ∗∗ s and yi = y ∗∗ i . hence, by lasalle’s invariance principle [19], the largest invariant set in {(xs, xi, xr, ys, yi) ∈ r5 + : v̇ = 0} is the only set {e6}. thus, the interior equilibrium point e6 = (x∗∗ s , x ∗∗ i , x ∗∗ r , y ∗∗ s , y ∗∗ i ) is globally asymptotically stable. this completes the proof. the implication of theorem 3.9 is that coexistence of prey and predator species in the ecosystem is possible while disease persists in the population. the presence of disease in the ecosystem, such that r∗ 0 > 1, is independent of initial population sizes of both infected prey and predator species. this global stability result is depicted in figure 3.3(a) for infected prey population and figure 3.3(b) for infected predator population. the phase portrait which illustrates the stable coexistence of prey and predator populations is presented in figure 3.4. 4. optimal control model in order to enhance the coexistence of both prey and predator species in the ecosystem, the prey-predator model (2.1) is extended by incorporating the following three time-dependent optimal control functions: copyright © 2021 assa. adv syst sci appl (2021) 96 j.a. ademosu, s. olaniyi, s.o. adewale 0 5 10 15 20 25 30 35 40 0 2 4 6 8 10 12 14 16 18 20 time in fe c te d p re y (a) 0 5 10 15 0 5 10 15 20 25 time in fe c te d p re d a to r (b) fig. 3.3. convergence of solution trajectories of prey-predator model to the interior equilibrium point e6 at different initial sizes. the parameter values used are:mu1 = 0.5, r = 11.2, β1 = 0.4, p1 = 1.0, γ = 0.35, p2 = 0.2, p3 = 1.0, µ2 = 0.65, k = 30, β2 = 0.65, c1 = 0.25, c2 = 0.25, c3 = 0.15, so that r∗ 0 = 9.7293 > 1. 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 prey p re da to r fig. 3.4. phase portrait of predator against prey showing stable coexistence of both populations around the interior (positive) equilibrium point. 1. the control function u1(t) ∈ [0, 1], which represents disease prevention measure for both prey and predator populations. 2. the function u2(t) ∈ [0, 1], which represents treatment control for prey population. 3. the control function u3(t) ∈ [0, 1] representing alternative source of food for predator survival. therefore, the prey-predator model (2.1) becomes a non-autonomous system of ordinary differential equations of the form: copyright © 2021 assa. adv syst sci appl (2021) stability analysis and optimal measure 97 dxs dt = rxs ( 1− xs k ) − (1− u1(t))β1xsxi − p1xsys − µ1xs dxi dt = β1xsxi − p2xiys − p3xiyi − (µ1 + γ)xi − u2(t)r0xi dxr dt = γxi + u2(t)r0xi − µ1xr dys dt = c1p1xsys + c2p2xiys − (1− u1(t))β2ysyi − µ2ys + u3(t)c0ys dyi dt = (1− u1(t))β2ysyi + c3p3xiyi − µ2yi, (4.25) where r0 and c0 are the rate constants for the optimal controls u2(t) and u3(t), respectively. since the goal is to reduce the risk of both prey and predator species going extinct, such that the populations of infected prey and predator are minimized at minimum costs possible, then the following objective functional is formed j = ∫ tf 0 ( a1xi + a2yi + 1 2 [b1u 2 1 +b2u 2 2 +b3u 2 3] ) dt, (4.26) where a1, a2, b1, b2 and b3 are the positive weight constants for balancing the terms in the objective functional over the time interval [0, tf ]. further, the cost function associated with the disease prevention measure for prey and predator populations is represented by the term 1/2b1u 2 1, while the treatment measure for prey population and the provision of alternative food source for predator are, respectively, represented by the terms 1/2b2u 2 2 and 1/2b3u 2 3. the objective functional is made of nonlinear cost on controls of quadratic type in line with the standards for optimal control problems (see, e.g. [1, 4, 14, 16, 24, 25]). therefore, the optimal control problem for the prey-predator system (4.25) is to seek a control triple u∗ = (u∗1, u ∗ 2, u ∗ 3, ), such that j(u∗) = min {j(u1, u2, u3) : u1, u2, u3 ∈ u} , (4.27) where u = {ui(t) : 0 ≤ ui(t) ≤ 1,lebesgue measurable, t ∈ [0, tf ], for i = 1, 2, 3} is a nonempty set of all admissible controls. 4.1. analysis of the control model the necessary conditions that must be satisfied by the optimal control triple u∗ for the minimization problem are derivable from pontryagin’s maximum principle [26]. this principle converts the problem in (4.27) with (4.26) subject to the state system (4.25) into a problem of minimizing pointwise a hamiltonian h, with respect to u1(t), u2(t) and u3(t). thus, the hamiltonian for the control problem is given by copyright © 2021 assa. adv syst sci appl (2021) 98 j.a. ademosu, s. olaniyi, s.o. adewale h = a1xi + a2yi + 1 2 [b1u 2 1 +b2u 2 2 +b3u 2 3] +η1 [ rxs ( 1− xs k ) − (1− u2(t))β1xsxi − p1xsys − µ1xs ] +η2 [β1xsxi − p2xiys − p3xiyi − (µ1 + γ)xi − u2(t)r0xi] +η3[γxi + u2(t)r0xi − µ1xr] +η4[c1p1xsys + c2p2xiys − (1− u1(t))β2ysyi − µ2ys + u3(t)c0ys] +η5[(1− u1(t))β2ysyi + c3p3xiyi − µ2yi], (4.28) where η1, η2, η3, η4 and η5 are the adjoint or co-state variables corresponding to the state variables xs, xi, xr, ys and yi, respectively. applying the existence results as given in [2, 12], the following result is obtained. theorem 4.1: for the optimal control triple u∗ that minimizes the objective functional (4.26) over u subject to (4.25), there exist adjoint variables η1, η2, η3, η4 and η5 satisfying the system governing the adjoint variables dη1 dt = 2rxs k η1 + (1− u1)β1xi(η1 − η2) + p1ys(η1 − c1η4) + (µ1 − r)η1 dη2 dt = (1− u1)β1xs(η1 − η2) + p2ys(η2 − c2η4) + p3yi(η2 − c3η5) +(γ + u2r0)(η2 − η3) + µ1η2 − a1 dη3 dt = µ1η3 dη4 dt = p1xs(η1 − c1η4) + p2xi(η2 − c2η4) + (1− u1)β2yi(η4 − η5) +(µ2 − u3c0)η4 dη5 dt = p3xi(η2 − c3η5) + (1− µ1)β2ys(η4 − η5) + µ2η5 − a2 (4.29) and with the transversality conditions η1(tf ) = η2(tf ) = η3(tf ) = η4(tf ) = η5(tf ) = 0. (4.30) moreover, the optimal control triple (u∗1, u ∗ 2, u ∗ 3) is characterized as follows u∗1 = max { 0,min { 1, 1 b1 [β1xsxi(η2 − η1) + β2ysyi(η5 − η4)] }} , u∗2 = max { 0,min { 1, 1 b2 [r0xi(η2 − η3)] }} , u∗3 = max { 0,min { 1, −c0ysη4 b3 }} . (4.31) proof the system (4.29) governing the adjoint variables is obtained by finding the partial derivative of the hamiltonian h (4.28) as follows: dη1 dt = − ∂h ∂xs , dη2 dt = − ∂h ∂xi , dη3 dt = − ∂h ∂xr , dη4 dt = − ∂h ∂ys , dη5 dt = − ∂h ∂yi copyright © 2021 assa. adv syst sci appl (2021) stability analysis and optimal measure 99 with the transversality conditions: η1(tf ) = η2(tf ) = η3(tf ) = η4(tf ) = η5(tf ) = 0. finally, differentiating the hamiltonian h partially with respect to each of the three controls u1, u2 and u3, so that ∂h ∂u1 ≡ b1u1 + β1xsxi(η1 − η2) + β2ysyi(η4 − η5) = 0, ∂h ∂u2 ≡ b2u2 + r0xi(η3 − η2) = 0, ∂h ∂u3 ≡ b3u3 + c0ysη4 = 0, which when solved on the interior of the control set u gives the required characterization (4.31). hence, the proof. 4.2. simulations of the control model here, the numerical simulations of the coupled system of state equations (4.25) with adjoint equations (4.29) are conducted. the standard iterative fourth order forward-backward rungekutta method is used to solve the optimality system due to difference in time orientations of the state and adjoint equations [18, 23]. the optimal control problem is simulated over the time interval [0, 15] in years using these parameter values: µ1 = 0.5, r = 11.2, β1 = 0.4, p1 = 1.0, γ = 0.35, p2 = 0.2, p3 = 1.0, µ2 = 0.65, k = 30, β2 = 0.65, c1 = 0.25, c2 = 0.25, c3 = 0.15. in addition, the balancing weight constants are given as a1 = 1, a2 = 0.1, b1 = 0.01, b2 = 0.0001 and b3 = 0.008. the rate constants for the controls are chosen as r0 = 0.4 and c0 = 0.15. thus, to minimize the objective functional (4.26) with disease prevention control only at low cost of implementation, the control u1 should be maintained at maximum (100%) effort for 8 years before being dropped to zero in final time as shown in the figure 4.5(a). the control profile shown in figure 4.5(b) indicates that single implementation of treatment control u2 requires maximum (100%) effort throughout the period of the control intervention. it is observed in figure 4.5(c) that alternative food source control u3 should be at the upper bound for 14 years before coming to the lower bound. the combined implementation of the three controls for minimizing the objective functional is depicted in figure 4.5(d), where it is can be observed that prevention control u1 should be maintained at the upper bound for 8 years before declining to the lower bound, treatment control u2 and provision of alternative food source u3 should be maintained at maximum (100%) for about 2 years and almost 1 year, respectively. the effects of combining all the three optimal controls on the dynamical behaviors of prey and predator populations are illustrated in figure 4.6. the size of susceptible prey population with control reduces due to predation when compared with the size without control, but the population does not go extinct due to the presence of the combined optimal control as shown in figure 4.6(a). this suggests the possibility of stable coexistence of predators with low density of prey population as indicated in [27]. the infected prey population with optimal control (figure 4.6(b)) decreases sharply when compared with the case without control. further, in figure 4.6(c), it can be seen that the size of susceptible predator population with combined optimal control is higher than the size without control. whereas, the presence of the combined optimal control reduces the population size of the infected predator in the ecosystem. copyright © 2021 assa. adv syst sci appl (2021) 100 j.a. ademosu, s. olaniyi, s.o. adewale 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 o p tim a l c o n tr o l u 1 time (a) 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 o p tim a l c o n tr o l u 2 time (b) 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 o p tim a l c o n tr o l u 3 time (c) 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time o p ti m a l c o n tr o l p ro fi le s u 1 u 2 u 3 (d) fig. 4.5. control profiles for single and combined implementations of u1, u2 and u3. 5. conclusion a nonlinear mathematical model of prey-predator interacting populations in an ecological system with disease spread is studied. the five-dimensional eco-epidemiological model is analyzed with a view to providing insights into the behavior of ecosystem in the presence of disease. seven possible equilibrium points, namely trivial, axial, disease-free, predatorfree, infected prey-free, infected predator-free and interior equilibrium points are analytically determined. asymptotic stability of the eco-epidemiological system is analyzed to investigate the behavior of the system around each of the obtained equilibrium points. some conditions that guarantee the existence of each prey and predator species, as well as coexistence of both prey and predator populations are given using local linearization and lyapunov functions techniques. in addition, the eco-epidemiological model is extended to include three time-dependent optimal controls, such as disease prevention in both species, treatment strategy for prey and provision of alternative food source for predator survival. hence, the optimal control preypredator model is analyzed using pontryagin’s maximum principle in order to enhance the coexistence of both prey and predator species by reducing the risks of their extinction in the ecosystem, such that the populations of infected prey and predator species are minimized at minimum costs possible. copyright © 2021 assa. adv syst sci appl (2021) stability analysis and optimal measure 101 0 5 10 15 0 5 10 15 20 25 30 time s u s c e p ti b le p re y u 1 ≠ 0, u 2 ≠ 0, u 3 ≠ 0 u 1 =u 2 =u 3 =0 (a) 0 5 10 15 0 0.5 1 1.5 2 2.5 time in fe c te d p re y u 1 ≠ 0, u 2 ≠ 0, u 3 ≠ 0 u 1 =u 2 =u 3 =0 (b) 0 5 10 15 0 2 4 6 8 10 12 14 time s u s c e p ti b le p re d a to r u 1 ≠ 0, u 2 ≠ 0, u 3 ≠0 u 1 =u 2 =u 3 =0 (c) 0 5 10 15 0 2 4 6 8 10 12 time in fe c te d p re d a to r u 1 ≠ 0, u 2 ≠ 0, u 3 ≠0 u 1 =u 2 =u 3 =0 (d) fig. 4.6. combined effects of optimal controls u1, u2 and u3 on the dynamics of the prey-predator model. the analysis of the eco-epidemiological model presented in this study can be applied to a wide spectrum of interactions between different kinds of living beings in the environment. acknowledgement the authors are thankful for the constructive comments and suggestions of the editor and the anonymous reviewers. initial discussion on the original manuscript with professor mini ghosh of vellore institute of technology, chennai, india is also acknowledged with thanks. references 1. abidemi, a. and aziz, n.a.b. 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(1926). variazioni e fluttaazioni of numero di individul in specie animali conviventi, mem. accd. linc., 2, 31–113. copyright © 2021 assa. adv syst sci appl (2021) introduction model formulation well-posedness of the model boundedness of solutions positivity of solutions analysis of the model existence of equilibrium points trivial equilibrium point (e0) axial equilibrium point (e1) disease-free equilibrium point (e2) predator-free equilibrium point (e3) infected prey-free equilibrium (e4) infected predator-free equilibrium (e5) interior equilibrium point (e6) stability analysis stability of e0 stability of e1 stability of e2 stability of e3 stability of e4 stability of e5 stability of e6 optimal control model analysis of the control model simulations of the control model conclusion adv syst sci appl 2022; 03:1–17 published online at https://ijassa.ipu.ru. stability and bifurcation of a delay cancer model in the polluted environment ahmed a. mohsen1*, raid k. naji2 1department of mathematics, college of education for pure science (ibn-al-haitham), university of baghdad. 2department of mathematics, college of science, university of baghdad. abstract: it is well known that the spread of cancer or tumor growth increases in polluted environments. in this paper, the dynamic behavior of the cancer model in the polluted environment is studied taking into consideration the delay in clearance of the environment from their contamination. the set of differential equations that simulates this epidemic model is formulated. the existence, uniqueness, and the bound of the solution are discussed. the local and global stability conditions of disease-free and endemic equilibrium points are investigated. the occurrence of the hopf bifurcation around the endemic equilibrium point is proved. the stability and direction of the periodic dynamics are studied. finally, the paper is ended with a numerical simulation in order to validate the analytical results. keywords: cancer, environment pollution, time delay, stability, the hopf bifurcation. 1. introduction cancer is an abnormal growth of cells that tend to proliferate in an uncontrolled way and in some cases to spread. in fact, cancer or tumor growth can be considered as not one disease but it is a group of more than one hundred different and distinctive diseases. cancer can involve any tissue of the body and have many different forms in all body areas [2]. according to the world health organization, the cancer is one of the global diseases that affect the human as much as 18 million new cases annually, and more than half of the mortalities in 2018, and the numbers are expected to nearly double in 2040. although there is many advanced experimental in developing interventional therapies for cancer such as immunotherapy, virotherapy, targeted drug therapies, and chemotherapy, in addition to the surgical resection, treatment options are still limited and the disease is considered fatal [5]. therefore, we can consider the mathematical models as one of the ways for studying this disease. for a brief review of mathematical models describing tumor-immune dynamics see tsygvintsev et al. [12]. lestari et al [6] proposed a mathematical model of the spread of cancer with chemotherapy and then studied their dynamical behavior. weerasinghe et al [14] discussed some models of plasticity, tumor progression, and metastasis using three broadly conceived mathematical modeling techniques: discrete, continuum, and hybrid, each with advantages and disadvantages. simmons et al. [9] presented a brief overview of breast cancer, focusing on its heterogeneity, and then explained the role of mathematical modeling and simulation in teasing apart the underlying biophysical processes. in [8], pillis and radunskaya proposed and investigated the immune response to tumor invasion. trisilowati et al. [11] proposed and analyzed the optimal control model of dendritic cell treatment of growing ∗corresponding author: aamuhseen@gmail.com 2 a.a. mohsen, r.k. naji tumors. moreover, wilson and levy [15] formulated a model to investigate the effect of immunotherapy on tumor growth. recently, abernathy et al [16], studied the long-term dynamics of a system of nonlinear differential equations that describes the role of virotherapy on tumors and the impact of immune response specific to fighting cancer. al-tuwairqi et al [1] presented a realistic mathematical model that describes the interaction between the innate immune system and uninfected tumor cells. this is based on the fact that both tumor and virus-infected cells are recognized by natural killer cells that are part of the innate immune system on the other hand, there have been several papers proposed to show the effect of the time delay of some epidemic disease models as seen in zuo et al. [17], which studied the impact of the media and delay on the spread of an epidemic. in [3], cooke et al. proposed an epidemic model with two delays. wang and wu [13], investigated an seir model with delay. recently, naji and majeed [7] studied the effect of delay on a stage structure prey-predator model. in this article, it is assumed that cancer start outbreak due to environmental pollution, and then the effect of delay in cleaning up this environment on the dynamic behavior of the proposed system is discussed. so that, the next section deals with the formulation of the model and presents its main properties. in section 3, the stability analysis of the diseasefree equilibrium point and endemic equilibrium point is discussed and the possibility of occurrence of the hopf bifurcation is also studied. section 4, discusses the stability and direction of the periodic dynamics that resulting from the hopf bifurcation as the delay starting increases. numerical simulations are used in section 5 to further understand the dynamics of the model. finally, the paper is ended with a discussion section as given in section 6. 2. the mathematical formulation of cancer model it is well known that one of the main reasons for the spread of cancer is the polluted environment. hence, in case of existence of such disease in a population of size n(t) at time t, then the population is divided into two compartments, namely susceptible compartment, which is denoted to their population’s size at time t by s(t) and cancer compartment, which contains all the individuals that infected by cancer and denoted to their population’s size at time t by c(t), such that n (t) = s (t) + c(t). moreover, it is assumed that the pollution level at time t in the environment is given by e(t). on the other hand, since the delay in clearance of the environment from their contamination has a vital effect on the spread of cancer. therefore, the effect of delay, with the amount of delay given by τ > 0, is considered in the formulation of the model. accordingly, the dynamics of the spread of cancer within such an environment can be represented in the following block diagram. fig. 2.1. the block diagram of cancer model. copyright © 2022 assa. adv syst sci appl (2022) stability and bifurcation of a delay cancer model ... 3 and the corresponding dynamic model has been formulated by the following system of nonlinear delay differential equations ds dt = ψ − β (1−m)s (t)e (t)− µs (t) dc dt = β (1−m)s (t)e (t)− (µ+ α)c (t) de dt = p − θe (t− τ)  , (2.1) where s (r) = s (0) > 0, c (r) = c(0) ≥ 0 and e (r) = e (0) ≥ 0 for r ∈ [−τ, 0). it is assumed that all parameters in system (2.1) have the following description: ψ > 0 is the natural birth of susceptible; µ > 0 represents the death rate of all populations; β > 0 represents the contact rate of the susceptible with the contaminated environment. however, the constant m represents the body resistance rate due to chromosomal fixed against environment effect such that 0 ≤ m ≤ 1. while p ≥ 0 is the pollution rate in the environment and θ > 0 is the clearance rate in the environment. note that, all the interaction functions on the right-hand side of the system (2.1) are continuous and have continuous partial derivatives and hence the system has a unique solution in r3 +. furthermore, the boundedness of the solution is shown in the following theorem. theorem 2.1: all the solutions of system (2.1) with initial conditions belong to r3 +, are bounded. proof let (s (t) , c (t) , e (t)) be any solution of system (2.1) with initial conditions belongs to r3 +. hence, by adding the first two equations in system (2.1) to each other we get: ds dt + dc dt = ψ − µn − αc. therefore, it is obtained that: dn dt + µn ≤ ψ. then, solving the above inequality by using the gronwall lemma we obtain for t→ ∞ that: n ≤ ψ µ . similarly, from the 3rd equation in the system (2.1) we have: de dt = p − θe. this implies that: de dt + θe = p. hence for t→ ∞, we have e(t) ≤ p θ . this completes the proof. now, since the variable c in the second equation of system (2.1) dose not appear in the other two equations, then the following system, say system (2.3), can be solved independently. then by substituting the obtained solution, say (sυ, eυ) in second equation of system (2.1), we obtain the following solution: copyright © 2022 assa. adv syst sci appl (2022) 4 a.a. mohsen, r.k. naji c (t) = β (1−m)sυeυ µ+ α , (2.2) where (sυ, eυ) is a solution of the system: ds dt = ψ − β (1−m)s (t)e (t)− µs (t) de dt = p − θe (t− τ)  . (2.3) 3. analysis of system 2.3 in this section, the existence and their stability analysis are carried out. it is easy to verify that system (2.3) has two equilibrium points and can be described as follows: 1. the first point is called the disease-free equilibrium point dfep, which is denoted by e0 = (s0, 0), where: s0 = ψ µ . (3.4) clearly, e0 exists provided that: p = 0. (3.5) 2. the other point is called the endemic equilibrium point eep and that denoted by e1 = (s1, e1), where: s1 = ψθ β(1−m)p+µθ e1 = p θ  . (3.6) obviously, e1 exists uniquely in the interior of the positive quadrant of the se−plane provided that: p > 0. (3.7) now, to analyze the stability of the above equilibrium points the general jacobian matrix for the system (2.3) at any point, say (s,e), can be written as: j (s,e) = [ − [β (1−m)e + µ] −β (1−m)s 0 −θe−λτ ] . (3.8) accordingly, the jacobian matrix of system (2.3) at the dfep that given in eq. (3.4), becomes: j (e0) = [ −µ −β (1−m)s0 0 −θe−λτ ] . (3.9) then, the characteristic equation of the matrix (3.9) can be written as: λ2 + µλ+ (θλ+ µθ) e−λτ = 0. (3.10) obviously, when τ = 0, eq. (3.10) becomes: copyright © 2022 assa. adv syst sci appl (2022) stability and bifurcation of a delay cancer model ... 5 λ2 + (µ+ θ)λ+ µθ = 0. (3.11) clearly, the eigenvalues in such a case can be written as: λ [τ=0] 1 = −µ, λ[τ=0] 2 = −θ. hence, e0 is always locally asymptotically stable for τ = 0. furthermore, by using the function h(s,e) = ( 1 se ) as the dulac function, which is a continuously differentiable unction in a simply connected ω ⊂ r2 + such that ∂(hf(s,e)/∂s + ∂(hg(s,e)/∂e dose not change the sign in ω and vanishes at most on a set of measure zero then the system (2.3) not have periodic orbits in ω. therefore, it is easy to verify that system (2.3) with τ = 0 has no periodic dynamics and hence e0 is always globally asymptotically stable. on the other hand, since the dfep exists under the condition (3.5), hence system (2.3) can be reduce to a single equation due to the fact é < 0. therefore, the system (2.3) has no periodic dynamics when τ > 0 and p = 0, and hence e0 is always globally absolutely stable [10] for all τ ≥ 0. now the stability of the eep of system 2.3 that given by (3.6), is investigated using the linearization technique. accordingly, the jacobian matrix that given in (3.8) of system (2.3) at the eep can be written as: j (e1) = [ − [β (1−m)e1 + µ] −β (1−m)s1 0 −θe−λτ ] . (3.12) hence, the characteristic equation of j(e1) can be written: λ2 +b1λ+ (b2λ+b3) e −λτ = 0, (3.13) where b1 = β (1−m)e1 + µ, b2 = θ, and b3 = θ [β (1−m)e1 + µ] . so for τ = 0, then eq. (3.13) becomes: λ2 + (b1 +b2)λ+b3 = 0. (3.14) it easy to see that the roots of the above equation have negative real parts. consequently, whenever the eep exists, it is always locally asymptotically stable (las). indeed, it is globally asymptotically stable as mentioned above with the help of the dulac criterion. now, for τ > 0, assume that eq. (3.13) has a pair of purely imaginary roots, say λ = ±iω with ω > 0, cross the imaginary axis. hence, by substituting λ = iω in eq. (3.13) we get: −ω2 + ib1ω + (ib2ω +b3) (cosωτ − isin ωτ ) = 0. therefore, by separating the real and imaginary parts we obtain: b1ω = b3sinωτ −b2ω cosωτ −ω2 = −b2ωsinωτ −b3cosωτ } . (3.15) clearly, squaring the equations in eq. (3.15) and adding to each other gives that: ω4 + ( b2 1 −b2 2 ) ω2 −b2 3 = 0, (3.16) set k = ω2, then eq. (3.14) becomes: k2 + ( b2 1 −b2 2 ) k −b2 3 = 0. (3.17) copyright © 2022 assa. adv syst sci appl (2022) 6 a.a. mohsen, r.k. naji clearly, by using descartes’ rule of signs, eq. (3.17) has a unique positive root, namely k = ω2 0 . hence, ω0 is a positive root of eq. (3.16) too. therefore, there is at least a pair of purely imaginary roots ±iω0 satisfying eq.(3.13). substituting ω0 in eq. (3.15) and then solving the resulting equation with respect to τ we get that: τj = 1 ω0 sin−1ω0 (b1b3 +b2ω0 2)( b2 2ω0 2 +b3 2 ) + 2jπ ω0 ; j = 0, 1, 2, · · · (3.18) now, define that τ0 = minj≥0 τj , then λ (τ) = γ (τ) + iω(τ) be a root of eq. 3.13, such that γ (τ0) = 0 and ω (τ0) = ω0. then we have the following theorem. theorem 3.1: assume that condition (3.7) holds. then the eep is conditionally stable. proof it is well known that the equilibrium point e1 is conditionally stable if it is asymptotically stable for τ ∈ [0, τ0). moreover, as shown above, the equilibrium point e1 is globally asymptotically stable for τ = 0 and the transcendental characteristic equation given by eq. (3.13) has at least a pair of purely imaginary roots ±iω0 at τ = τ0. consequently, if we assume that λ (τ) = γ (τ) + iω(τ) is an eigenvalue of eq. (3.13) with γ (τ0) = 0 and ω (τ0) = ω0 > 0, where τ0 is given by eq. (3.18) then the proof follows if we can show that the following transversality condition hold.[ d(reλ (τ)) dτ ] τ=τ0 ̸= 0. (3.19) by deriving eq. (3.13) with respect to τ we obtain that:[ 2λ+b1 +b2e −λτ − τ(b2λ+b3)e −λτ] dλ dτ = λ(b2λ+b3)e −λτ . (3.20) therefore, we obtain that:[ dλ dτ ]−1 = (2λ+b1) −λ(λ2 +b1λ) + b2 λ(b2λ+b3) − τ λ . (3.21) since λ = iω0 at τ = τ0, then eq. (3.21) can be rewritten as:[ dλ dτ ]−1 τ=τ0 = 2iω0 +b1 b1ω0 2 + iω0 3 + b2 −b2ω0 2 + iω0b3 − τ0 iω0 . now since sgn [ d(reλ) dτ ] τ=τ0 = sgn [ re ( dλ dτ )−1 ] λ=iω0 . (3.22) accordingly, from the fact: re [ 2iω0 +b1 b1ω0 2 + iω0 3 ] = 2ω0 2 +b1 2 ω0 2(b1 2 + ω0 2) , re [ b2 iω0b3 −b2ω0 2 ] = −b2 2 b2 2 ω0 2 +b3 2 , copyright © 2022 assa. adv syst sci appl (2022) stability and bifurcation of a delay cancer model ... 7 re [ τ0 iω0 ] = 0. hence, we obtain that:[ re ( dλ dτ )]−1 τ=τ0 = 2ω0 2 +b1 2 ω0 2(b1 2 + ω0 2) − b2 2 b2 2 ω0 2 +b3 2 > 0. therefore, d(reλ(τ)) dτ at τ = τ0 is not equal zero and does not change sign. clearly, the obtained result shows that the eigenvalue of characteristic equation eq. (3.13) crosses the imaginary axis from left to right as τ passes through τ0. hence system (2.3) losses its stability and undergoes the hopf bifurcation at τ = τ0. thus the proof is complete. 4. stability and direction of the hopf bifurcation in the previous section, we have shown that system (2.3) undergoes the hopf bifurcation near the eep at τ = τj . in this section, however, the direction of the hopf bifurcation and their stability at τ = τ0 is investigated. the normal form theory and center manifold theorem are used in this investigation, see [4]. let x1 = s − s1, x2 = e − e1, xi(t) = xi(τt) and τ = τ0+u, then we drop the bars for simplification of notations. therefore system (2.3) is transformed into functional differential equations in c = c ([−1, 0] , r2) as: ẋ (t) = lu (xt) +h (u, xt) , (4.23) where x (t) = (x1, x2) t ∈ r2, lu : c → r2 and h:r×c → r2 are given as: lu (γ) = (τ0 + u) [ − [β (1−m)e1 + µ] −β(1−m)s1 0 0 ] [ γ1(0) γ2(0) ] +(τ0 + u) [ 0 0 0 −θ ] [ γ1(−1) γ2(−1) ]  , (4.24) and h (u,γ) = (τ0 + u) [ −β(1−m)γ1(0)γ2(0) 0 ] , (4.25) here γ(v) = (γ1(v),γ2(v)) t ∈ c. now, by the riesz representation theorem, there exists a function of bounded variation, say η(v, u) for v ∈ [−1, 0], such that: luγ = ∫ 0 −1 dη (v, u) γ (v),γ ∈ c. (4.26) here we can choose: η (v, u) = (τ0 + u) [ − [β (1−m)e1 + µ] −β(1−m)s1 0 0 ] δ (v) − (τ0 + u) [ 0 0 0 −θ ] δ (v + 1)  , (4.27) where δ(v) is the dirac function, which define as follows, then eq. (4.26) is satisfied. copyright © 2022 assa. adv syst sci appl (2022) 8 a.a. mohsen, r.k. naji δ (v) = { 1 v = 0. 0 v ̸= 0. (4.28) for γ ∈ c1 ([−1, 0] , r2) , define: a (u) γ(v) = { dγ(v) dv − 1 ≤ v < 0.∫ 0 −1 dη (ς, u) γ (ς) v = 0. (4.29) and r (u) γ(v) = { 0 − 1 ≤ v < 0. h (u,γ) v = 0. (4.30) hence, system (4.23) can be transformed into an operator differential equation of the form: x́t = a (u)xt +r (u)xt, (4.31) where xt = x (t+ v), v ∈ [−1, 0]. now, we defined the adjoint operator a∗ of a, where ψ ∈ c1 ( [−1, 0] , (r2) ∗), by: a∗ (0)ψ(s) = { −dψ(s) ds 0 < s ≤ 1,∫ 0 −1 dηt (ς, 0)ψ (−ς) s = 0, (4.32) with a bilinear inner product form: ⟨ψ (s) ,γ(v)⟩ = ψ(0)γ (0)− ∫ 0 v=−1 ∫ v ξ=0 ψ t (ξ − v) dη (v) γ (ξ) dξ, (4.33) where, η (v) = η (v, 0). clearly, a (0) and a∗(0) are adjoint operators. now since ±iτ0ω0 are the eigenvalues of a (0) as we show in the previous section then they are also eigenvalues of a∗(0). furthermore, we need to find the eigenvector q of a corresponding to iτ0ω0 and the eigenvector q∗ of a∗ corresponding to −iτ0ω0 so that they satisfying the normalization conditions ⟨q∗, q⟩ = 1 and ⟨q∗, q⟩ = 0, as determined in the following theorem. theorem 4.1: the eigenvectors of a (0) and a∗ corresponding to iτ0ω0 and −iτ0ω0 are given by q (v) = (1, r1) t eivτ0ω0 and q∗ (s) = d(1, r2) t eisτ0ω0 respectively, where: r1 = − [iω0 + β (1−m)e1 + µ] β (1−m)s1 , r2 = β (1−m)s1 iω0 − θeiω0τ0 , d = [ 1 + r1r2 − τ0θr1r2e −iω0τ0 ]−1 . proof since q (v) = (1, r1) t eivτ0ω0 , is the eigenvector of a(0) corresponding to iτ0ω0, then: a (0) q (v) = iτ0ω0 q (v) . so, we obtain: copyright © 2022 assa. adv syst sci appl (2022) stability and bifurcation of a delay cancer model ... 9 a (0) q (0) eivτ0ω0 = iτ0ω0 q (0) e ivτ0ω0 . now according to the definition of a (0), we get τ0 ( iω0 + β (1−m)e1 + µ β (1−m)s1 0 iω0 + θe−iω0τ0 )( 1 r1 ) = ( 0 0 ) . consequently, we obtain that: r1 = − [iω0 + β (1−m)e1 + µ] β (1−m)s1 . similarly, since q∗ (s) = d(1, r2) t eisτ0ω0 is the eigenvector of a∗ corresponding to −iτ0ω0, hence: a∗q∗ (s) = −iτ0ω0q ∗ (s) . from the definition of a∗ we obtain: τ0 ( iω0 − [β (1−m)e1 + µ] 0 −β (1−m)s1 iω0 − θe−iω0τ0 )( 1 r2 ) = ( 0 0 ) . such that: r2 = β (1−m)s1 iω0 − θeiω0τ0 . to guarantee that ⟨ q∗ (s) , q (v) ⟩ = 1, then the parameter d should be determined, so by using eq. (4.33), we get that: ⟨q∗ (s) , q(v)⟩ = d (1, r2) ( 1 r1 ) − ∫ 0 v=−1 ∫ v ξ=0 d (1, r2) e −i(ξ−v)ω0τ0dη (v) ( 1 r1 ) eiξω0τ0dξ = d { 1 + r1r2 − ∫ 0 −1 (1, r2) ve ivω0τ0dη (v) ( 1 r1 )} = d { 1 + r1r2 + [ τ0 (1, r2) ( 0 −θr1 )] e−iω0τ0 } = d {1 + r1r2 − τ0θr1r2e −iω0τ0} . thus we can choose d so that: d = [ 1 + r1r2 − τ0θr1r2e −iω0τ0 ]−1 . in addition, it can be easily verify that ⟨q∗, q⟩ = 0 by applying the adjoint property ⟨φ,aγ⟩ = ⟨a∗φ,γ⟩, as follows: iτ0ω0 ⟨q∗, q⟩ = ⟨−iτ0ω0q ∗, q⟩ = ⟨a∗q∗, q⟩ = ⟨q∗, aq⟩ = ⟨q∗, iτ0ω0q⟩ = −iτ0ω0 ⟨q∗, q⟩ . then, we have ⟨q∗, q⟩ = 0. in the following, the technique given by hassard et al. [4] to compute the coordinates describing center manifold c0 at u = 0 is used. define copyright © 2022 assa. adv syst sci appl (2022) 10 a.a. mohsen, r.k. naji z (t) = ⟨q∗, xt⟩ w (t, v) = xt (v)− z (t) q (v)− z (t) q (v) = xt (v)− 2re {z (t) q (v)} } , (4.34) where xt (v) = x(t+ v) be the solution of eq. (4.24). also, on the center manifold c0, we have: w (t, v) = w (z (t) , z (t) , v) , (4.35) here w (z (t) , z (t) , v) = w20 (v) z2 2 + w11 (v) zz + w02 (v) z2 2 + . . . , such that, z(t) and z(t) are local coordinates of center manifold c0 in the direction of q∗ and q∗, respectively. clearly, we know that xt ∈ c0 of eq. (4.31), since u = 0, we know that ⟨φ,aγ⟩ = ⟨a∗φ,γ⟩, for ⟨γ, φ⟩ ∈ d(a)×d(a), then: ź(t) = ⟨q∗, x′(t)⟩ = ⟨q∗, a(0)xt +r(0)xt⟩ = ⟨q∗, a (0)xt⟩+ ⟨q∗, r (0)xt⟩ = ⟨a∗(0)q∗, xt⟩+ ⟨q∗, r (0)xt⟩ . thus ź (t) = iω0τ0z (t) + q∗ t (0)h(0, w (t, 0) + 2re {z (t) q(0)}) = iω0τ0z (t) + q∗ t (0)h0 (z, z) } , (4.36) here h0 (z, z) = h(0, z, z), now rewrite eq. (4.36) as: ź (t) = iω0τ0z (t) + g (z, z) , (4.37) where g (z, z) = 1 2 g20z 2 + g11zz + 1 2 g02z 2 + 1 2 g21z 2z + . . . so according to the eq. (4.34) with the eq. (4.31), we obtain: ẃ (t, v) = x′t (v)− ź (t) q (v)− ź (t) q (v) = a(0)xt +r(0)xt − iω0τ0z(t)q(v)− q∗ t (0)h0(z, z)q(v) +iω0τ0z (t) q (v)− q∗t (0)h0 (z, z) q (v) } = a(0)xt +r(0)xt − a(0)z(t)q (v)− a(0)z(t)q (v) −2re { q∗ t (0)h0(z, z)q(v) } } = a(0)w(t, v) +r(0)xt − 2re { q∗ t (0)h0(z, z)q(v) } . therefore, it is obtain that: ẃ (t, v) = a (0)w (t, v)− 2re { q∗ t (0)h0 (z, z) q (v) } for v ∈ [−1, 0) . a (0)w (t, 0) +h0 (z, z)− 2re { q∗ t (0)h0 (z, z) q (0) } for v = 0. (4.38) accordingly, we can rewrite eq. (4.38) for v ∈ [−1, 0) as follows: ẃ (t, v) = aw (t, v) +g (z, z, v) , (4.39) copyright © 2022 assa. adv syst sci appl (2022) stability and bifurcation of a delay cancer model ... 11 where g (z, z, v) = g20 (v) z2 2 +g11zz +g02 z2 2 + . . . so, differentiate the two sides of (4.35), gives that: ẃ = wz ź + wz ź. (4.40) then, using the eqs. (4.35), (4.36) and (4.39) give that: (a− 2iτ0ω0)w20 (v) = −g20 (v) , aw11 (v) = −g11 (v) , (a+ 2iτ0ω0)w02 (v) = −g02 (v) . (4.41) according to eqs. (4.36) and (4.37), we have that: g (z, z) = q∗ t (0)h0(z, z) = 1 2 g20z 2 + g11zz + 1 2 g02z 2 + 1 2 g21z 2z + . . . , (4.42) where q∗t (0) = d (1, r2). also from eq. (4.34), it gets that: x (t+ v) = (x1 (t+ v) , x2(t+ v))t = zq (v) + zq (v) + w (t, v) , here q (v) = (1, r1) t eivτ0ω0 . hence by eq. (4.35), it is obtained that: x1 (t+ 0) = z + z + w(1) (t, 0) = z + z + w (1) 20 (0) z 2 2 + w (1) 11 (0) zz + w (1) 02 (0) z 2 2 + . . . , x2 (t+ 0) = zr1 + zr1 + w(2) (t, 0) = zr1 + zr1 + w (2) 20 (0) z 2 2 + w (2) 11 (0) zz + w (2) 02 (0) z 2 2 + . . . , x1 (t− 1) = ze−iτ0ω0 + zeiτ0ω0 + w(1) (t,−1) = ze−iτ0ω0 + zeiτ0ω0 + w (1) 20 (−1) z 2 2 +w (1) 11 (−1) zz + w (1) 02 (−1) z 2 2 + . . . , x2 (t− 1) = zr1e −iτ0ω0 + zr1e iτ0ω0 + w2 (t,−1) = zr1e −iτ0ω0 + zr1e iτ0ω0 + w (2) 20 (−1) z 2 2 +w (2) 11 (−1) zz + w (2) 02 (−1) z 2 2 + . . . . (4.43) now, substituting eq. (4.25), when u = 0 and γ(v) ≡ x(t+ v) in eq. (4.42), gives that: g (z, z) = dτ0 (1, r2) ( −β (1−m)x1 (t+ 0)x2 (t+ 0) 0 ) . therefore, the following is obtained: g (z, z) = −dτ0β (1−m) [ r1z 2 + (r1 + r1) zz + r1z 2 + ( 2w (2) 11 (0) + w (2) 20 (0) + 2r1w (1) 11 (0) + r1w (1) 20 (0) ) z2z 2 + ( 2w (2) 11 (0) + w (2) 20 (0) + 2r1w (1) 11 (0) + r1w (1) 02 (0) ) z z2 2 + · · · ]  . (4.44) copyright © 2022 assa. adv syst sci appl (2022) 12 a.a. mohsen, r.k. naji so comparing the coefficients in eq. (4.44) with those in eq. (4.42) gives that: g20 = −2dτ0β (1−m) r1, g11 = −dτ0β (1−m) (r1 + r1) , g02 = −2dτ0β (1−m) r1, g21 = −dτ0β (1−m) ( 2w (2) 11 (0) + w (2) 20 (0) + 2r1w (1) 11 (0) + r1w (1) 20 (0) ) . (4.45) since eq. (4.45) contains w11 and w20, hence it needs to compute them. now, from eqs. 4.38 and (4.39), for v ∈ [−1, 0), we have: g (z, z, v) = −q∗t (0)h0 (z, z) q (v)− q∗t (0)h0 (z, z) q (v) . then by using eqs. (4.36) and (4.37), it obtains that: g (z, z, v) = −g (z, z) q (v)− g (z, z) q (v) . (4.46) by comparing coefficients in both sides gives: g20 (v) = −g20q (v)− g02q (v) , g11 (v) = −g11q (v)− g11q (v) , (4.47) now, substituting eqs. (4.47) into eqs. (4.41) respectively gives that: w′ 20 (v) = 2iτ0ω0w20 (v) + g20q (v) + g02q (v) , w′ 11 (v) = g11q (v) + g11q (v) . (4.48) now by solving eqs. (4.48), it is easy to verify that the solutions are written respectively as: w20 (v) = −g20q(0) iτ0ω0 eiτ0ω0v − g02q(0) 3iτ0ω0 e−iτ0ω0v +k1e 2iτ0ω0v, w11 (v) = g11q(0) iτ0ω0 eiτ0ω0v − g11q(0) iτ0ω0 e−iτ0ω0v +k2. (4.49) wherek1 = ( k (1) 1 , k2 1 ) andk2 = ( k (1) 2 , k2 2 ) are constants vectors to be determined in the following. from the definition of a when v = 0, and eqs. (4.41), it is obtained that:∫ 0 −1 dη (v)w20 (v) = 2iτ0ω0w20 (0)−g20(0). (4.50) ∫ 0 −1 dη (v)w11 (v) = −g11(0). (4.51) also from eqs. (4.38) and (4.39), it is easy to verify that: g20 (0) = −g20q (0)− g02q (0) + 2τ0 [ r1β (1−m) 0 ] . (4.52) g11 (0) = −g11q (0)− g11q (0) + 2τ0 [ β (1−m) (r1 + r1) 0 ] . (4.53) therefore by substituting eqs. (4.49) and (4.52) into eq. (4.50), and then using the results: copyright © 2022 assa. adv syst sci appl (2022) stability and bifurcation of a delay cancer model ... 13 ( iτ0ω0i − ∫ 0 −1 eiτ0ω0vdη (v) ) q (0) = 0,( −iτ0ω0i − ∫ 0 −1 e−iτ0ω0vdη (v) ) q (0) = 0. (4.54) it is obtained that:( 2iτ0ω0 i − ∫ 0 −1 e2iτ0ω0vdη (v) ) k1 = 2τ0 [ r1β(1−m) 0 ] . (4.55) similarly, by substituting eqs. (4.49) and (4.53) into eq. (4.51), and then using eqs. (4.54), it is obtained that: k2 ∫ 0 −1 dη (v) = −2τ0 [ β (1−m) (r1 + r1) 0 ] . (4.56) now, using eq. (4.28) with u = 0 into eqs. (4.55) and (4.56), gives that: [ 2iω0 + β (1−m)e1 + µ β (1−m)s1 0 2iω0 + θe−2iω0τ0 ] k1 = 2 [ r1β(1−m) 0 ] . (4.57) [ −β (1−m)e1 − µ −β (1−m)s1 0 −θ ] k2 = −2 [ β (1−m) (r1 + r1) 0 ] . (4.58) accordingly, solving the linear systems given by eqs. (4.57) and (4.58), gives that: k (1) 1 = 2r1β(1−m)[2iω0+θe−2iω0τ0 ] [2iω0+β(1−m)e1+µ][2iω0+θe−2iω0τ0 ] , k (2) 1 = 0. (4.59) and k (1) 2 = 2β(1−m)(r1+r1) [β(1−m)e1+µ] , k (2) 2 = 0. (4.60) consequently, all the values of gij; i = 0, 1, 2; j = 0, 1, 2 given in eqs. (4.45) can be determined by the parameters and delay. thus, it can calculate the following quantities: c1 (0) = i 2τ0ω0 ( g20g11 − 2|g11|2 − |g02|2 3 ) + g21 2 , µ2 = −re{c1(0)} re{λ′(τ0)} , β2 = 2re {c1(0)} , t2 = − im {c1(0)}+µ2im {λ′(τ0)} τ0ω0 . (4.61) these quantities are used to determine the direction of the hopf bifurcation and stability of bifurcated periodic solutions of system (2.3) at the critical value τ0 as shown in the following theorem. theorem 4.2: the direction of the hopf bifurcation is determined by the sign of µ2 at the critical value τ0, so that copyright © 2022 assa. adv syst sci appl (2022) 14 a.a. mohsen, r.k. naji 1. the the hopf bifurcation is supercritical if µ2 > 0 and the hopf bifurcation is subcritical if µ2 < 0. 2. the stability of bifurcated periodic solutions is determined by β2 so that the periodic solutions are stable if β2 < 0 and unstable if β2 > 0. 3. the period of bifurcated periodic solutions is determined by β2 so that the period increases if t2 > 0 and decreases if t2 < 0. 5. numerical simulation in this section, the obtained results are illustrated using numerical simulation. the following set of hypothetical parameters set is adopted throughout this section. ψ = 0.87, β = 0.010, m = 0.05, µ = 0.015, p = 0, α = 0.3, θ = 0.2, τ = 7.7 < τ0 ∼= 7.85. (5.62) all the obtained trajectories of the system (2.1) are drawn using the matlab of version 8. starting from different sets of initial data, system (2.1) is solved numerically using the parameters set given in eq. (5.62) and then the obtained trajectories are drawn in fig. (5.2). fig. 5.2. : the trajectories of the system (2.1) using data given by eq. (5.62) approach to dfep. (a) trajectories of susceptible. (b) trajectories of cancer. (c) trajectories of environment. (d) 3d phase plot for globally asymptotically stable dfep. clearly, fig. (5.2) illustrates that system (2.1) has a globally asymptotically stable dfep for that data (5.62). however, for the same data given by eq. (5.62) with p = 0.6, it is observed that, although that system (2.1) is solved with different sets of initial points, it has a globally asymptotically stable eep given by e1 = (17.9, 1.6, 3) as shown in fig. (5.3) below. copyright © 2022 assa. adv syst sci appl (2022) stability and bifurcation of a delay cancer model ... 15 fig. 5.3. : the trajectories of the system (2.1) using data given by eq. (5.62) with p = 0.6 approach to eep. (a) trajectories of susceptible. (b) trajectories of cancer. (c) trajectories of environment. (d) 3d phase plot for globally asymptotically stable eep. now, to show the effect of varying the body resistance rate (m) due to chromosomal fixed against environment effect on the system behavior, the system is solved numerically for different values, say m = 0.1, 0.7 respectively, keeping other parameters fixed as given in eq. (5.62) with p = 0.6, and then the solution of system (2.1) is drawn in figs. (5.4a) and (5.4b). fig. 5.4. : time series of the trajectory of the system (2.1) using data given by eq. (5.62) with p = 0.6 and different values of m. (a) trajectory approaches to eep given by e1 = (18, 1.5, 3) when m = 0.1. (b) trajectory approaches to eep given by e1 = (32, 0.92, 3) when m = 0.7. clearly, in fig. (5.4) as m increases, the trajectory of the system (2.1) still approaches asymptotically to the eep point. in fact, it is observed that as m increases the number of individuals with cancer decreases, and the number of susceptible increases without any effect of environment. copyright © 2022 assa. adv syst sci appl (2022) 16 a.a. mohsen, r.k. naji now, the dynamical behavior of the system (2.1) near the eep point under the effect of increasing the time delay is investigated. the system (2.1) is solved numerically for the set of parameter values given by eq. (5.62) with p = 0.6 and τ = 7.86 then the trajectory of the system (2.1) is drawn in figs. (5.5a-5.5e). fig. 5.5. : the existence of periodic solution near eep of the system (2.1) for data given by equation (5.62) with p = 0.6 and τ = 7.86. (a) trajectories of susceptible. (b) trajectories of cancer. (c) trajectories of environment. (d) 3d periodic solution. 6. conclusion in this paper, a mathematical model that describes the spread of cancer in a polluted environment incorporating delay in cleaning up the environment from the contaminated has been proposed and studied. the properties of the solution are discussed. it is observed that the proposed model has two equilibrium points namely dfep and eep. the stability analysis of the model shows that the dfep is globally asymptotically stable for all τ ≥ 0. while the eep is conditionally stable so that it’s globally asymptotically stable for all τ ∈ [0, τ0) and there is the hopf bifurcation at τ0. however, it is an unstable point for τ > τ0 and periodic dynamics occurred. the stability and direction of the periodic dynamics are also investigated analytically by finding the normal form using the center manifold theory as well as numerically. it is observed that for the hypothetical set of data given by eq. (5.62) with p = 0.6 the eep is still globally asymptotically stable for 0 ≤ ô < τ0 ∼= 7.853. while the hopf bifurcation occurs and periodic solutions bifurcate near the eep point as τ passes through the above critical value τ0. on the other hand, all the quantities, which are given in eqs. (4.61), are determined for the periodic dynamics drawn in fig. 5.5 as c1 (0) = 7.360− 77.426i, β2 = 14.72 > 0, µ2 = −0.294 < 0 and t2 = 51.873 > 0. therefore, according to theorem (4.2), the periodic copyright © 2022 assa. adv syst sci appl (2022) stability and bifurcation of a delay cancer model ... 17 resulting from the hopf bifurcation is subcritical, unstable, and the size of the period increases. acknowledgements the authors are thankful to the referees for their valuable suggestions. references 1. al-tuwairqi, s. m., al-johani n., o., & eman a. s., (2020). modeling dynamics of cancer virotherapy with immune response. advances in difference equations, 438. 2. chaffer, c. l. & weinberg, r. a., (2011). a perspective on cancer cell metastasis. science journal, 331, 1559–1564. 3. cooke, k. l., & driessche, v. d., (1996). analysis of an seirs epidemic model with two delays, j. math. boil., 35, 240–260. 4. hassard, b., kazarino d., & wan, y., (1981). theory and application of the hopf bifurcation, cambridge university press. 5. international agency for research on cancer. (16 october 2019). [online] available: https://gco.iarc.fr/tomorrow/home. 6. lestari, d., sari, e. r. & arifah, h., (2019). dynamical of a mathematical model of cancer cells with chemotherapy. journal of physics conf. series, 1320. 7. naji, r. k. & majeed, s. j., (2020). the dynamical analysis of a delay pray-predator model with a refuge-stage structure prey population. iranian journal of mathematical sciences ad informatics, 15(1), pp 135–159. 8. pillis, l. c. & radunskaya, a., (2003). a mathematical model of immune response to tumor invasion, computational fluid and solid mechanics, pp. 1661–1668, elsevier science, amsterdam. 9. simmons, a., burrage, p. m., nicolau, d.v., lakhani s.r. & burrage, k., (2017). environmental factors in breast cancer invasion: a mathematical modeling review. pathology, 49(2), 172–180. 10. tipsri, s. & chinviriyasit w., (2015). the effect of time delay on the dynamics of an seir model with nonlinear incidence. chaos solitons & fractals 75, 153–172. 11. trisilowati, t., mccue, s., & mallet, d., (2013). numerical solution of an optimal control model of dendritic cell treatment of growing tumour, anziam journal, 54, 664– 680. 12. tsygvintsev, a., marino, s. & kirschner, d. e., (2012). a mathematical model of gene therapy for the treatment of cancer, in mathematical models and methods in biomedicine, 357–373. springer-verlag. 13. wang, l., & wu, x., (2018). stability and the hopf bifurcation for an seir epidemic model with delay, advances in the theory of nonlinear analysis and its applications, 113–127. 14. weerasinghe, h. n., burrage, p. m., burrage, k. & nicolau d.v., (2019). mathematical models of cancer call plasticity. journal of oncology, 2019, 2403483. 15. wilson, s. & levy, d., (2012). a mathematical model of enhancement of tumor vaccine efficacy by immunotherapy, bulletin of mathematical biology, 74(7), 1485–1500. 16. zachary, a., kristen a., & jessica s., (2020). a mathematical model for tumor growth and treatment using virotherapy. aims mathematics, 5(5), 4136–4150. 17. zuo, l., liu, m. & wang, j., (2015). the impact of awareness program with recruitment and delay on the spread of an epidemic, mathematical problems in engineering, 235935. copyright © 2022 assa. adv syst sci appl (2022) introduction the mathematical formulation of cancer model analysis of system 2.3 stability and direction of the hopf bifurcation numerical simulation conclusion adv syst sci appl 2017; 3:42–48 published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/503 interval-valued data forecasting using dual-parametric neural networks yuliya. e. polozova1, pavel. v. saraev1 1lipetsk state technical university, lipetsk, russia abstract: this study is focused on the approach to modelling and forecasting interval-valued data using a dual-parametric neural network (dpnn). this concept was proposed as a subclass of interval neural networks that contains two types of parameters: real and interval ones. this approach makes it possible to get guaranteed inclusion of an exact (single value) solution into interval calculation results. in this paper we intend to give a theoretical overview of previous research on the subject and description of the new developed methods and algorithms for learning dpnn. the experiments demonstrate that interval calculation results obtained by using the proposed approach include of exact solution at least in 60% of cases. keywords: interval neural network, forecasting, dual-parametric neural network, intervalvalued data, parametric identification 1. introduction today neural networks are successfully applied to many different problems, particularly time series data forecasting. these applications usually use sets of single point values as input and output data. but in many applications it is more natural to use the input values and the predicted results in the form of intervals. an interval neural network (inn), which contains interval arithmetic, is used to calculate such interval-valued data. one of the most important features of interval analysis is the guaranteed inclusion of the exact solution into interval calculation results. previous studies show that this feature is not achieved in the case of inn. trying to eliminate this drawback the authors introduced dualparametric neural network (dpnn) as a new subclass of inn. models of this subclass contain two types of parameters: real and interval ones. in this paper, we generalize previously developed and new learning methods and algorithms of dpnn and apply them to forecasting interval-valued time series. 2. theoretical overview interval neural network (inn) is a neural network that contains at least one interval parameter, namely input, output, or weight. the inns can be used for the following reasons: • initial data are sometimes presented in the form of interval values rather than singlepoint ones; • training data sets are reduced in size due to the use of clustering analysis; • inns allow for making reliable accuracy estimates of calculation results. ∗corresponding author: julipolozova@yandex.ru http://ijassa.ipu.ru/ojs/ijassa/article/view/503 interval-valued data forecasting using dual-parametric neural networks 43 the existing training methods for interval neural networks are based on the backpropagation (bp) algorithm for interval data [1–5] with training error calculated as the quadratic function of quality: j(w) = 1 2 k∑ i=1 qi(w) = 1 2 k∑ i=1 { (yi(w)− ỹi)2 + (yi(w)− ỹi)2 } (2.1) where k is the number of examples in the training data set, w is the weight vector, qi(w) is the training error on example i of the training data set, yi(w) is the lower bound of the network‘s output interval on example i, ỹi is the lower bound of the training output interval on example i, yi(w) is the upper bound of the network‘s output interval on example i, ỹi is the upper bound of the training output interval on example i. the bounds of the output interval for neuron i (index i is already used for training samples. it would be more clear to use another letter, say, k or j for neurons) of layer l are computed as follows: y(l,i) = σ(net(l,i)) = σ ( nl−1∑ j=0,wj>0 w (l,i) j y(l−1,j) + nl−1∑ j=0,wj<0 w (l,i) j y(l−1,j) ) (2.2) y(l,i) = σ(net(l,i)) = σ ( nl−1∑ j=0,wj>0 w (l,i) j y(l−1,j) + nl−1∑ j=0,wj<0 w (l,i) j y(l−1,j) ) (2.3) where y(l,i) is the lower bound of the output interval for neuron i of layer l, σ(·) is the activation function for hidden layer neurons, net(l,i) the lower bound of the activation level for neuron i of layer l, nl−1 is the number of neurons in layer l − 1, w(l,i) j is weight j for neuron i of layer l, y(l−1,j) is the lower output bound for neuron j of layer l − 1, y(l−1,j) is the upper output bound for neuron j of layer l − 1, y(l,i) is the upper output bound for neuron i of layer l, net(l,i) is the upper bound of the activation level for neuron i of layer l. the gradient of the inn training quality function is calculated as follows: ∂qk(w) ∂w (l,i) j = { s(l,i)σ′net(net (l,i))y(l−1,j) + s(l,i)σ′net(net (l,i))y(l−1,j), w (l,i) j > 0, s(l,i)σ′net(net (l,i))y(l−1,j) + s(l,i)σ′net(net (l,i))y(l−1,j), w (l,i) j < 0. (2.4) where s(l,i), s(l,i) are the lower and upper bounds of the factor determined by the recurrent procedure s(l,i) = ( nm+1∑ j=1,wj>0 s(l+1,j)σ′net(net (l+1,j))w (l+1,j) i + nm+1∑ j=1,wj<0 s(l+1,j)σ′net(net (l+1,j))w (l+1,j) i ) s(l,i) = ( nm+1∑ j=1,wj>0 s(l+1,j)σ′net(net (l+1,j))w (l+1,j) i + nm+1∑ j=1,wj<0 s(l+1,j)σ′net(net (l+1,j))w (l+1,j) i ) (2.5) with the initial condition s(l,1) = y(w)− ỹ, s(l,1) = y(w)− ỹ, where l is the number of layers in the neural network model. copyright c© 2017 assa. adv syst sci appl (2017) 44 y.e. polozova, p.v. saraev 3. proposed approach 3.1. modification of error function one of the most important advantages of interval analysis methods is the possibility to get guaranteed estimates of single-point solution in the output interval. however, function (2.1) does not guarantee that output intervals of the training examples will be fully included in the output intervals of the network. to solve this problem, a training quality function for an inn model with one output was proposed in previous research: j([w]) = k∑ i=1 qi([w]) = k∑ i=1 pibi, (3.6) where bi = [ bi1 bi2 ] = [ ỹi − yi([w]) yi([w])− ỹi ] ; pi = [pi1 pi2] , piq = { 1, biq ≥ 0, − biq n , biq < 0; q = 1, 2. here k is the size of the training data set, [w] is the interval weight vector, qi([w]) is the training error on example i , pi is the row vector of weight coefficients for the deviation of the network‘s output interval bounds from the bounds of the example i of the training data set, bi is the vector of deviation between the network‘s output interval bounds and the bounds of example i, ỹi is the lower bound of the output interval on training example i, yi([w]) is the lower bound of the network’s output interval on sample i, yi([w]) is the upper bound of the network’s output interval on example i, ỹi is the upper bound of output interval of example i, piq is element q of row vector pi, biq is is element q of row vector bi, q is the number of the vector element, n is the tolerance level showing the admissible sequence of deviations of an interval bound (e.g. 0.1, 0.01, 0.001). to train the inn model, it was offered to use the following interval adaptive algorithm of global function optimization based on weight vector bisection [6] instead of interval bp algorithm. algorithm 1 input: the minimum width δ > 0 of the bar, required training error ε > 0. output: [w] is the weight vector, which is a minimum of function (3.6); j∗ is the minimum value of function (3.6). 1. initialization of the initial weights of [w]. 2. calculation of inn output y∗ and quality function j∗. 3. initialization of list l = [w], j∗. 4. initialization of bisection coordinate l = 0 and iteration counter c = 0 (which counts iterations needed for exiting the cycle). 5. cycle: as long as minwidni=1[wi] > δ and j∗ > ε. (a) calculation of bisection coordinate l = l + 1. if l exceeds the weight vector, then l = l. (b) bisection of [w] in coordinate l into [w′] and [w′′]. (c) calculation of inn outputs y∗ and quality functions j ′ and j ′′. (d) if j ′ > j∗ and j ′′ > j∗, then go to step 5e. otherwise, go to step 5f. (e) assignment c = c+ 1. if c is not equal to the weight vector value, then go to step 5a. otherwise, exit the cycle. (f) deletion of element ([w], j∗) from l. (g) addition of ([w′], j ′) and ([w′′], j ′′) to list l. (h) arrangement of list l in ascending order by the value of the second field. (i) the first record is denoted by ([w], j∗). thus, usage of the proposed error function and learning algorithm makes it possible to guarantee that output interval contains exact solution. copyright c© 2017 assa. adv syst sci appl (2017) interval-valued data forecasting using dual-parametric neural networks 45 it is known that a result of calculating a function in interval analysis depends on the way the variable is represented in the formal expression. to obtain a more accurate value the number of variables contained in the formal expression should be minimized. in the above formulation of the problem it means that the number of layers should be minimized too. this conclusion is confirmed by experimental results of previous research. thus, minimizing ofnn layers improves accuracy of training. but a too small number of layers reduces ability of inn to approximation. this conflict was overcome by introduction of the new subclass of inn. 3.2. dual-parametric neural network dual-parametric neuron network (dpnn) is a subclass of inn that contains two types of parameters: real and interval ones. a dpnn model with a single interval output contains: • n interval output neurons; • m hidden layers; • real neuron weights in all layers (from layer 1 to layer m− 1); • interval neuron weights in layer m. besides, input and output values are also interval. in the case when input or output values should be real it is possible to represent them as intervals whose left and right bounds are equal. real weights make it possible to get high quality of learning of the neural network model. interval weights guarantees that the intervals of the training examples will be fully included in the output intervals of the network. thus, it is possible to guarantee that output intervals contains single-point solutions. for the purpose of training the dpnn, it was proposed in previous research to use an algorithm combining the interval bp method and the global optimization algorithm for interval values. 1. at the first stage, the real weights are trained using the interval bp algorithm. 2. at the second stage, the resulting weights are only used to initialize the model, followed by the training of interval weights by means of the interval global optimization algorithm. besides, a structural identification algorithm was developed for dpnn models with one hidden layer. it allows for selecting the optimal number of neurons, which are consecutively added to the hidden layer until the training error is reduced. 3.3. learning algorithm based on the intervals extension procedure as we note earlier to train the second stage model of dpnn we use the interval global optimization algorithm based on bisection of weight vector. this approach has several drawbacks. one of them is that the initial intervals of the weight vector need to have enough width. this width should be enough to include training data outputs into the model outputs for all examples in a training set when the signal passes through the network at the first time. thus, there is a difficulty in generating initial weights to train the model at the second stage of dpnn learning algorithm. the second drawback is related to the first one. according to the traditional interval global optimization algorithm an efficient choise for the bisection is cut along a coordinate with the maximum width [7]. but if the widths of all intervals is the same there is no reason to consider any of them as the most appropriate. it is possible to generate random initial intervals with different widths. but the question remains how to achieve inclusion of training data outputs into model outputs at the first iteration if this is not satisfied. to eliminate these drawbacks we introduce a learning algorithm based on the intervals extension procedure instead of the bisection. copyright c© 2017 assa. adv syst sci appl (2017) 46 y.e. polozova, p.v. saraev algorithm 2 step 1. we initialize a model with one interval output, n input interval neurons, m hidden layers and weights assigned in real values to the neurons of all layers. step 2. we train the model using the bp algorithm. step 3. we initialize a model with one interval output, n input interval neurons, m hidden layers and weights assigned in real values to the neurons of all layers (from layer 1 to layer m− 1). the neuron weights in hidden layer m are assigned in interval values. step 4. we assign real values to the weights in the model trained at step 2, and the ones in the model defined at step 3 for all hidden layers. step 5. we train the model formed at step 4 in accordance with the algorithm of training inns with interval weights. the weights of neurons in all hidden layers (from layer 1 to layer m− 1) are regarded as constant values. only the weights of hidden layer m are subject to change. step 5.1. we select the increment value m of an interval. step 5.2. a loop starts along the coordinates of the weight vector. with each iteration m substracts from the lower bound of the weight vector interval. at each iteration, the network output and the value of quality function are calculated. if the received training error value is less than the error obtained in the previous step, the weights vector and training error are fixed as minimum ones. step 5.3. step 5.2 is repeated while the training error on each loop iteration is reducing the minimum one. step 5.4. steps 5.2 and 5.3 are executed in the same way for the upper bound of weight vector intervals. in this case, m is added to the upper bound of the interval at each loop iteration. 4. numerical experiments the goal of our experiments is to show a difference between forecasting results obtained using inn with the interval bp algorithm and the proposed approach with dpnn. in our experiments, for the learning of a dpnn we generate interval-valued time series: x(i) = 0.0, 0.1, 0.2, ...; y(i) = 0.2 sin(2πx(i)) + 0.1 ( x(i) )2 + 0.2 + 0, 005rnd[−1, 1]; y(i) = 0.2 sin(2πx(i)) + 0.2 ( x(i) )2 + 0.3 + 0, 005rnd[−1, 1]. here rnd[−1, 1] is a random real number in the closed interval [−1, 1]. training set includes 35 pairs ( y(i), y(i) ) . the next 5 pairs are used as a basis for projections. we use the proposed model of dpnn with a single output neuron, 9 input neurons and one hidden layer with 3 neurons in it. to compare proposed approach and interval bp algorithm we make projections using both of them. results are shown in fig. 4.1, 4.2 and in table 4.1. table 4.1. experimental results no. of the experiment 1 2 training time, sec 28 302 training error 0.07 4.13 average relative deviation of projected value bounds from actual ones 0.15 0.12 the experiments demonstrate that projected values obtained by using the proposed approach (experiment 2) include of actual values at least 3 of 5 values. thus, in 60% of cases we can guarantee that projected value includes the exact solution. projected results interval covers just a part of actual values interval because the forecasting error accumulates with copyright c© 2017 assa. adv syst sci appl (2017) interval-valued data forecasting using dual-parametric neural networks 47 fig. 4.1. forecasting using only interval bp algorithm (experiment 1). fig. 4.2. forecasting using the proposed approach with the learning algorithm based on the intervals extension procedure (experiment 2). each subsequent projected interval. for the same reason actual values lies out of the projected interval bounds in the last values and not in the first ones. but, nevertheless, projected values obtained in experiment 2 seem to be more informative for practical use than projected results in experiment 1. training time in table 4.1 is so different, because bp algorithm used in experiment 1 is a part of learning algorithm of experiment 2. training errors is so different, because algorithms in experiment 1 and 2 uses different error functions – (2.1) and (3.6) respectively. average relative deviation of projected value bounds from actual ones is calculated as follows: δ = 1 2n n∑ i=1 ( |yi − ỹi| |yi| + |yi − ỹi| |yi| ) , where n is the number of projected values; yi, yi are the lower and the upper projected value bounds; ỹi, ỹi are the lower and the upper actual value bounds. copyright c© 2017 assa. adv syst sci appl (2017) 48 y.e. polozova, p.v. saraev 5. conclusion this study was centered on the approach to modelling and forecasting addedof interval-valued data using dpnn models. in the article we provided a theoretical overview of the technology we use, and described the proposed approach that makes it possible to get guaranteed inclusion of exact (single value) solution into interval forecasting results. our experiments confirmed the efficiency of dpnn models in predicting intervalvalued data as at least in 60% of cases we can guarantee that projected values include the exact solutions. as a prospective investigation our research will focus on training of inn and dpnn models. we take the following training data set of interval values as the initial data: {[x̃i], [ỹi]}ki=1. the inn (dpnn) will be used to compute interval functions for interval arguments in the following manner: [y] = f([w], [x]), where [w] is the vector of interval parameters (weights) of the network. besides, we are going to consider the problem of computing such inn (dpnn) weight values that allow the model outputs to include all interval outputs comprised by the training data set. one more problem to be considered is the optimization of the training quality function q([w]) = max d([yi], [ỹi])→ min with the distance between intervals d calculated as follows: d([yi], [ỹi]) = { +∞, if yi < ỹi or yi < ỹi; max { |yi − ỹi|, |yi − ỹi| } , otherwise. references 1. patil r.b. (1995) interval neural networks. apic’95, el paso, extended abstracts, a supplement to the international journal of reliable computing, el paso, tx, 164. 2. belohlavek r. (1997) backpropagation for interval patterns. neural network world, 7(3), 335–346. 3. garczarczyk z.a. (2000) interval neural networks. 2000 ieee international symposium on circuits and systems. emerging technologies for the 21st century. proceedings (ieee cat no.00ch36353), 3, 567–570. 4. kim h.j. & ryu t.-w. (2005) time series prediction using an interval arithmetic fir network. neural information processing – letters and reviews, 8 (3), 39–47. 5. yang d., li z. & wu w. (2016) extreme learning machine for interval neural networks. neural computing and applications, 27(1), 3–8. 6. sharyi s.p. (2008) randomized algorithms in interval global optimization numerical analysis and applications, 1(4), 376389. https://doi.org/10.1134/s1995423908040083 7. jaulin l., kieffer m., didrit o. & walter e. (2001) applied interval analysis. london, uk: springer. copyright c© 2017 assa. adv syst sci appl (2017) https://doi.org/10.1134/s1995423908040083 introduction theoretical overview proposed approach modification of error function dual-parametric neural network learning algorithm based on the intervals extension procedure numerical experiments conclusion adv syst sci appl 2024; 01:69–81 published online at https://ijassa.ipu.ru. combined therapeutic strategies for cancer: integrating oncolytic viruses and inhibitors in a mathematical model majda el younoussi1*, khalid hattaf1,2, noura yousfi1 1laboratory of analysis, modeling and simulation (lams), faculty of sciences ben m’sick, hassan ii university of casablanca, p.o box 7955 sidi othman, casablanca, morocco 2equipe de recherche en modélisation et enseignement des mathématiques (ermem), centre régional des métiers de l’education et de la formation (crmef), 20340 derb ghalef, casablanca, morocco abstract: the greatest cause of death worldwide continues to be cancer, a complicated set of diseases characterized by uncontrolled cell development. although early identification and therapeutic approaches have improved, the incidence of the disease is still on the rise, demanding continued study into its underlying causes and cutting-edge treatment paradigms. for creative interventions and focused medicines, the variety of cancer kinds, which are influenced by genetics, way of life, and environmental variables, poses both difficulties and opportunities. in this paper, we present a mathematical model to treat cancer with combined therapies, oncolytic viruses and mitogen-activated protein kinase inhibitors. we demonstrate that our model is both biologically and mathematically well-posed through the existence, the non-negativity and the boundedness of solutions. furthermore, we study the equilibrium points as well as the stability of these equilibria. finally, we use numerical simulations to illustrate the effect of this combined therapy on tumor cells. keywords: mapk inhibitors, oncolytic viruses, mathematical modeling, stability, hopf bifurcation. 1. introduction a type of biological therapy called oncolytic viruses is made to target and eliminate cancer cells while sparing normal cells. these viruses are produced naturally or genetically modified in a lab to target only cancer cells. the virus replicates inside the cancer cells after it has infected them, leading to the cells’ bursting and dying. this process also releases new viral particles, which can infect nearby cancer cells and continue the cycle of destruction [2]. certain oncolytic adenoviruses are notably dependent on the coxsackie-adenovirus receptor (car), and variations in car expression levels within target cells could potentially impact the efficacy of viral infection and the resulting therapeutic advantages. car has been linked to numerous facets of cancer biology, including cell adhesion, signaling, and migration, in addition to its function in promoting viral entry, making it a viable therapeutic target [9]. additionally, mitogen-activated protein kinase, also referred to as mapk or mek, is a family of serine/threonine protein kinases that are important for cellular functions like cell growth, differentiation, proliferation, and promoting car expression. mapk has the potential to exploit the complex interplay between car and oncolytic viruses to enhance their cancer-killing capabilities and stimulate immune responses against tumor cells [1]. ∗corresponding author: majda.elyounoussi-etu@etu.univh2c.ma 70 the development of our knowledge of cancer biology and treatment, on the other hand, has been greatly aided by mathematical modeling, particularly the use of ordinary differential equations (odes). the intricate dynamics and interactions between various cellular processes, signaling pathways, and therapeutic interventions in cancer can be better understood using ode models. researchers can predict how different treatment modalities will affect cancer cells and their microenvironment by creating and analyzing ode-based models. they can also find potential targets for novel therapies. further, ode models can help integrate experimental and clinical data, allowing for the quantitative assessment of cancer progression and treatment outcomes. in 2007, zurakowski and wodarz [11] used an ode model to discuss the interactions between the populations of the average level of car expression on the surface of the cells, free virus populations, susceptible, uninfected and infected tumor cells. this model was generalized by youshan and qian in their work [10], the researchers constructed a mathematical model to simulate the impacts of both mek inhibitors and viruses on tumor cells. this model is a free boundary problem, which means that it takes into account the growth and shrinkage of the tumor as the therapies are applied. the researchers used the model to explore how the combined therapies could reduce the tumor size. recently, nono et al. [7] recently expanded upon a prior model, applying it to brain cancer and introducing optimal control techniques to optimize the combination of oncolytic virotherapy and mek inhibitors. incorporating delays in mathematical models for cancer treatment can play a crucial role in capturing the realistic behavior of biological systems and improving the accuracy of the model predictions. motivated by all that, we propose in this paper an ode model to study the dynamics of oncolytic viruses and their interaction with the coxsackie-adenovirus receptor and mapk inhibitors in the context of cancer therapy, taking into account the duration required by tumor cells that have been infected to generate fresh viruses following the entry of the virus. our paper is organized into several key sections to present our research cohesively. we introduce our mathematical model in section 2, and rigorously examine its well-posedness. moving to section 3, we delve into equilibrium points and their stability, including an exploration of the hopf bifurcation, shedding light on the system’s dynamic behavior. section 4 presents the results of our numerical simulations, providing practical insights into the behavior of the model under various conditions. finally, in section 5, we offer a comprehensive conclusion, summarizing our findings, discussing their implications, and highlighting the broader significance of our research. 2. presentation and well-posedness of the model within this section, we present the subsequent ordinary differential equation (ode) model: ds dt = r(1− u)s(t)(1− s(t)+i(t) k )− βw (t)s(t)v (t) 1+αv (t) − ds(t), di dt = βw (t−τ)s(t−τ)v (t−τ)e−mτ 1+αv (t−τ) − δ(1− u)i(t)− di(t), dv dt = nδ(1− u)i(t)− βw (t)s(t)v (t) 1+αv (t) − µv (t), dw dt = ηu(γ −w (t))− hw (t), (2.1) where s(t), i(t), v (t) and w (t) are the concentration of uninfected tumor cells, infected tumor cells, free oncolytic virus particles and the average level of car molecules on cell surfaces at the time t, respectively. the factor r is the rate of tumor growth per individual in a population, slowed down by the value (1− u), where u represents the intensity of mapk inhibitor and varies between 0 and 1. if u = 1, the mapk inhibitor has the maximum possible effect. if u = 0, every cell in the first phase continue to grow and the production of car copyright © 2024 assa. adv syst sci appl (2024) 71 molecule is stopped. moreover, for biological and mathematical reasons, we will suppose as in [11] that d < r(1− u), where d is the rate of natural death of cells. the parameter k signifies the maximal tumor size. the term βsvw 1+αv models the rate of tumor cells infection by the virus in presence of car receptor and the interaction between them on the uninfected cell, where α measures the saturation effect, and β is the rate of infection process. the factor δ represents the virus induced death rate while the rate µ represents the decay of the virus. the parameter denoted as n represents the quantity of newly released viruses following the lysis of an infected tumor cell. cells generate car molecules at a rate denoted as η and experience a loss of these molecules from their surface at a rate of h. the term γ − w characterizes the saturation of car expression. furthermore, τ represents the time required for the transition from tumor cell infection to new virus production, wherem denotes the death rate for infected cells prior to virus production, and e−mτ signifies the probability of survival during the time interval [t− τ, t]. to prove that the model is mathematically well-posed and biologically meaningful, it is important to demonstrate the existence, the boundedness and the non-negativity of solutions as time evolves. let c be the set of continuous functions from the interval [−τ, 0] to r4, with the supremum norm ||ϕ|| given by sup−τ≤ζ≤0 |ϕ(ζ)|, where ϕ ∈ c. applying the fundamental theory of functional differential equations [4], we conclude that a single solution exists (s(t), i(t), v (t),w (t)), where the initial condition (s0, i0, v0,w0) are in c and we suppose that: s0(ζ) ≥ 0, i0(ζ) ≥ 0, v0(ζ) ≥ 0,w0(ζ) ≥ 0, ζ ∈ [−τ, 0]. (2.2) theorem 2.1: let’s suppose that the initial conditions fulfill (2.2). then each solution of model (2.1) stays non-negative for all t ≥ 0. proof using (2.2), we derive the following: s(t) = s(0)e ∫ t 0 (1−u)(1−s(x)+i(x) k )−βw (x)v (x) 1+αv (x) −d dx, then for all t > 0, we get s(t) ≥ 0. the second equation of model (2.1) gives i(t) = i(0)e−αt + e−mτ−αt ∫ t 0 βs(x−τ)v (x−τ)w (x−τ) 1+αv (x−τ) eαxdx, where α = δ(1− u) + d. then i(t) ≥ 0 for every t ≥ 0. from the third equation of model (2.1), we obtain v (t) = (v (0)e− ∫ t 0 βs(x) 1+αv (x) dx +nδ(1− u) ∫ t 0 i(y)eµy− ∫ t y βs(x) 1+αv (x) dxdy)e−µt. thus, v (t) ≥ 0 for every t ≥ 0. utilizing the final equation from model (2.1), we acquire: w (t) = w (0)e ∫ t 0 ηuγ w (x) −ηu−h dx ≥ 0, for all t ≥ 0. hence, every solution of model (2.1) is non-negative for all t ≥ 0. theorem 2.2: each solution of the model (2.1), given non-negative initial conditions (2.2), remains bounded for all t ≥ 0. proof by the first equation of our model, we have copyright © 2024 assa. adv syst sci appl (2024) 72 ds dt ≤ r(1− u)s(t)(1− s(t)−i(t) k ). using the comparison principal, we get lim sup t→+∞ s(t) ≤ k. therefore, s(t) is bounded. let z(t) = s(t− τ)e−mτ + i(t), hence, we have dz dt = r(1− u)s(1− s + i k )e−mτ − dse−mτ − δ(1− u)i − di ≤ r(1− u)ke−mτ − (r(1− u) + d)se−mτ − (δ(1− u) + d)i ≤ r(1− u)ke−mτ − cz(t), where c = c ′ (1− u) + d and c′ = min{r, δ}. then, lim sup t→+∞ z(t) ≤ rk(1−u)e−mτ c , and we get lim sup t→+∞ i(t) ≤ rk(1−u)e−mτ c . hence, i(t) is bounded. from the third equation we deduce dv dt ≤ nδ(1− u)i − µv, by lim sup t→+∞ i(t) ≤ rk(1−u)e−mτ c , we obtain lim sup t→+∞ v (t) ≤ nδ(1−u)2e−mτ µc . thus, v (t) is bounded. by the fourth equation of (2.1) and u ∈ [0, 1], we get dw dt ≤ ηγ − (η + h)w, then lim sup t→+∞ w (t) ≤ ηγ η+h . thus, w (t) is bounded. 3. equilibria and stability analysis within this section, we explore the three equilibrium points of model (2.1) along with their stability characteristics. copyright © 2024 assa. adv syst sci appl (2024) 73 3.1. equilibrium points of the model when there is no virus, the model (2.1) admits two infection-free equilibrium. the equilibrium point e0 = (0, 0, 0,w0), which reflects the non-existence of cells and virus, and the equilibrium point e1 = (s1, 0, 0,w1) where s1 = k(1− d r(1−u) ) and w0 = w1 = ηuγ ηu+h . by r(1− u) > d, the equilibrium e1 exists. in the existence of the virus, there exists another equilibrium point called the endemic equilibrium e∗ = (s∗, i∗, v ∗,w ∗). we suppose that i∗ > 0, v ∗ > 0 and we put r0 = βnδw1s1(1− u)e−mτ (δ(1− u) + d)(µ+ βw1s1) . r0 is the reproduction number and represents the potential for the oncolytic virus to spread within the tumor cell population. by simple calculus, we show that the equilibrium e∗ exists if r0 > 1, this means that the virus can sustain its presence and spread within the tumor cells, leading to a persistent infection. (s∗, i∗, v ∗, w ∗) are the solution of this system: r(1− u)s∗(1− s∗ + i∗ k )− βw ∗s∗v ∗ 1 + αv ∗ − ds∗ = 0, (3.3) βw ∗s∗v ∗e−mτ 1 + αv ∗ − δ(1− u)i∗ − di∗ = 0, (3.4) nδ(1− u)i∗ − βw ∗s∗v ∗ 1 + αv ∗ − µv ∗ = 0, (3.5) ηu(γ −w ∗)− hw ∗ = 0. (3.6) by equation (3.6), we get w ∗ = ηuγ ηu+ h . (3.7) by adding the equation (3.4) to the equation (3.5), we get v ∗ = δ(1− u)(n − emτ )− demτ µ i∗. (3.8) by adding the equation (3.3) to the equation (3.4), we obtain i∗ = s∗(r(1− u)(k − s∗)− dk) r(1− u)s∗ +kemτ (δ(1− u) + d) . (3.9) obviously, by determining s∗ we will determine v ∗ and i∗, for that we will use (3.7), (3.8) and (3.9) to find s∗. for simplification, we put: p = δ(1−u)(n−emτ )−demτ µ and o = nδ(1− u). by equation (3.5), we get (1− αv ∗)oi∗ − βw ∗s∗v ∗ − µ(1 + αv ∗)v ∗ = 0, then, o − βw ∗s∗p − pµ+ (αpo − µαp 2)i∗ = 0. copyright © 2024 assa. adv syst sci appl (2024) 74 using (3.9), we obtain as2 + bs + c = 0, where, a = −βw ∗pr(1− u)− αopr(1− u) + αp 2µr(1− u), b = or(1− u)− pµr(1− u)− βw ∗p ∗k(1− u)emτ (δ + d) +αopk(r(1− u)− d)− αp 2µr(1− u)k + αp 2µdk, c = (oδ(1− u)− pµδ(1− u) +od− pµd)kemτ . then s∗ = −b− √ ∆ 2a , where ∆ = b2 − 4ac. thus s∗, i∗, v ∗ and w ∗ are defined. 3.2. stability analysis the characteristic equation at any equilibrium e = (s, i, v,w ) is given by∣∣∣∣∣∣∣∣∣∣∣∣∣∣ −r(1−u)s k − λ − r(1−u)s k − βws (1+αv )2 − βsv 1+αv βwv e−(m+λ)τ 1+αv −δ(1− u)− d− λ βwse−(m+λ)τ (1+αv )2 βv se−(m+λ)τ 1+αv − βwv 1+αv nδ(1− u) − βws (1+αv )2 − µ− λ − βsv 1+αv 0 0 0 −ηu− h− λ ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ = 0. (3.10) theorem 3.1: the equilibrium state e0 = (0, 0, 0,w0) exhibits instability. proof at e0 = (0, 0, 0,w0) we get the following equation: (r(1− u)− d− λ)(δ(1− u) + d+ λ)(µ+ λ)(ηu+ h+ λ) = 0, (3.11) then the roots are λ1 = r(1− u)− d, λ2 = −δ(1− u)− d, λ3 = −µ and λ4 = −ηu− h. since we have r(1− u) > d, we get λ1 > 0. thus, e0 is unstable. theorem 3.2: the equilibrium point e1 = (s1, 0, 0,w1) is locally asymptotically stable for every τ ≥ 0 if r0 < 1 and unstable if r0 > 1. proof at e1, (3.10) becomes( 2r(1− u) s1 k + d+ λ )( ηu+ h+ λ )( λ2 + (δ(1− u) + d+ βw1s1 + µ)λ +(δ(1− u) + d)(βw1s1 + µ)(1−r0e −λτ ) ) = 0. (3.12) clearly, λ1 = −2r(1− u)s1 k − d and λ2 = −ηu− h represent two of the roots of the aforementioned equation, with the remaining roots arising from solutions to the subsequent copyright © 2024 assa. adv syst sci appl (2024) 75 equation: λ2 + (δ(1− u) + d+ βw1s1 + µ)λ+ (δ(1− u) + d)(βw1s1 + µ)(1−r0e −λτ ) = 0. (3.13) if r0 > 1, let f(λ) = λ2 + (δ(1− u) + d+ βw1s1 + µ)λ+ (δ(1− u) + d)(βw1s1 + µ)(1−r0e −λτ ). we have f(0) = (δ(1− u) + d)(βw1s1 + µ)(1−r0) < 0 and lim λ→+∞ f(λ) = +∞. in this case, the equation f(λ) = 0 possesses at least one positive root. hence, if r0 > 1 e1 is unstable. if r0 < 1, we discuss two cases: when τ = 0, we get δ(1− u) + d+ βw1s1 + µ > 0 and (δ(1− u) + d)(βw1s1 + µ)(1− r0) > 0. then all the roots of (3.12) have negative real parts for τ = 0 and r0 < 1. when τ > 0, let iω be a purely imaginary root of (3.13) where ω > 0. then,{ −ω2 + (δ(1− u) + d)(βw1s1 + µ) = (δ(1− u) + d)(βw1s1 + µ)r0cos(wτ), (δ(1− u) + d+ βw1s1 + µ)ω = −(δ(1− u) + d)(βw1s1 + µ)r0sin(ωτ), thus we get ω4 + ( (δ(1− u) + d)2 + (βw1s1 + µ)2 ) ω2 + (δ(1− u) + d)2(βw1s1 + µ)2(1−r2 0) = 0. let x = ω2, then we obtain x2 + ( (δ(1− u) + d)2 + (βw1s1 + µ)2 ) x+ (δ(1− u) + d)2(βw1s1 + µ)2(1−r2 0) = 0. hence if r0 < 1, there is no positive solution. thus, e1 is locally asymptotically stable for r0 < 1. theorem 3.3: if r0 < 1, the equilibrium e1 is globally asymptotically stable. proof take into consideration the presented lyapunov function: l(t) = δ(1− u)emτi(t) + δ(1− u) + d n emτv (t) + δ(1− u) ∫ t t−τ βw (ξ)s(ξ)v (ξ) 1 + αv (ξ) dξ, then, we obtain dl dt = δ(1− u) βwsv 1 + αv − (δ(1− u) + d)emτ n ( βwsv 1 + αv + µv ) . we have v 1+αv ≤ v , lim sup t→∞ s(t) ≤ s1 and lim sup t→∞ w (t) ≤ w1. then, we get dl dt ≤ (δ(1− u) + d)(βw1s1 + µ)(r0 − 1)v n . hence, if r0 < 1 we obtain dl dt ≤ 0. clearly, dl dt = 0 if and only if s = s0, i = 0, v = 0 and w = w0. then the largest invariant set contained in {(s, i, v,w )|dl dt = 0} is the singleton {e1}. by lasalle’s invariance principale [5], we deduce that e1 is globally asymptotically stable when r0 < 1. copyright © 2024 assa. adv syst sci appl (2024) 76 the characteristic equation at the equilibrium e∗ can be expressed in the following manner: (ηu+ h+ λ) ( λ3 + p1λ 2 + p2λ+ p3 + (q1λ+ q2)e −λτ ) = 0, (3.14) where, p1 = (1− u) ( rs∗ k + δ ) + βw ∗s∗ (1 + αv ∗)2 + d+ µ, p2 = (1− u) ( βw ∗s∗ (1 + αv ∗)2 + µ )( rs∗ k + δ + d 1− u ) − β2s∗v ∗w ∗2 (1 + αv ∗)3 + r(1− u)s∗ k ( δ(1− u) + d ) , p3 = (δ(1− u) + d)s∗ ( r(1− u) k ( βs∗w ∗ (1 + αv ∗)2 + µ ) − β2v ∗w ∗2 (1 + αv ∗)3 ) , q1 = (1− u)βs∗w ∗ 1 + αv ∗ ( rv ∗ k − δn 1 + αv ∗ ) e−mτ , q2 = (1− u)βs∗w ∗ 1 + αv ∗ ( µrv ∗ k + nδ 1 + αv ∗ ( βv ∗w ∗ 1 + αv ∗ − r(1− u)s∗ k )) e−mτ . since the root λ1 = −ηu− h is negative, it remains to determine the roots of the following equation: λ3 + p1λ 2 + p2λ+ p3 + (q1λ+ q2)e −λτ = 0, (3.15) when τ = 0, the equation (3.15) becomes λ3 + p1λ 2 + (p2 + q1)λ+ p3 + q2 = 0. (3.16) we have p1 > 0, and by simple calculation we can find that p3 + q2 > 0. by routh-hurwitz criterion, we conclude the result bellow: lemma 3.4: suppose that r0 > 1 and p1(p2 + q1)− (p3 + q2) > 0. then in the absence of delay (τ = 0), all the roots of (3.15) have negative real parts. hence, the equilibriume∗ = (s∗, i∗, v ∗,w ∗) is locally asymptotically stable. when τ > 0, let iω (ω > 0) be a purely imaginary root of the equation (3.15). thus,{ p1ω 2 − p3 = q1ω sin(ωτ) + q2 cos(ωτ), −ω3 + p2ω = −q1ω cos(ωτ) + q2 sin(ωτ), (3.17) then, ω6 + ( p21 − 2p2 ) ω4 + ( p22 − q21 − 2p1p3 ) ω2 + p23 − q22 = 0. (3.18) for x = ω2, the equation (3.18) is reduced to x3 + (p21 − 2p2)x 2 + (p22 − q21 − 2p1p3)x+ p23 − q22 = 0. (3.19) we consider the following function: f(x) = x3 + c2x 2 + c1x+ c0, (3.20) where c0 = p23 − q22 , c1 = p22 − q21 − 2p1p3 and c2 = p21 − 2p2. obviously f ′(x) = 3x2 + 2c2x+ c1, and ∆′ = 4(c22 − 3c1) its discriminant. hence, we get the following result: copyright © 2024 assa. adv syst sci appl (2024) 77 lemma 3.5: (i) if c0 < 0, then the equation f(x) = 0 has at least one positive root. (ii) if c0 ≥ 0 and ∆′ ≤ 0, then the equation f(x) = 0 has no positive roots. (iii) if c0 ≥ 0 and ∆′ > 0, then the equation f(x) = 0 has a positive root if x1 > 0 and f (x1) ≤ 0, where x1 = √ c22−3c1−c2 3 is a root of f ′(x) = 0. by lemma 3.4 and the previous lemma we deduce the following theorem: theorem 3.6: assume that r0 > 1, c0 ≥ 0 and p1 (p2 + q1)− (p3 + q2) > 0. if any of the subsequent conditions are met, • ∆′ ≤ 0, • ∆′ > 0 and x1 ≤ 0, • ∆′ > 0 and f (x1) > 0, then the equilibrium pointe∗ is locally asymptotically stable for any non-negative time delay. on the other hand, we study the hopf bifurcation of model (2.1) at the equilibrium point e∗. the stability of the equilibrium point e∗ changes when the equation (3.15) has purely imaginary roots. so, we assume that ω1, ω2 and ω3 are these positive roots. moreover, we consider τ as a parameter of bifurcation. substituting ω = ωϵ and τ = τ ϵ in (3.17) where ϵ = 1, 2, 3, we get q2(p1ω 2 ϵ − p3)− q1ωϵ(−ω3 ϵ + p2ωϵ) = q22cos(ωϵτ ϵ) + q21ωϵ 2cos(ωϵτ ϵ), thus, we obtain τ ϵn = 1 ωϵ ( arccos ( q2(p1ω 2 ϵ − p3) + q1ω 2 ϵ (ω 2 ϵ − p2) q22 + q21ω 2 ϵ ) + 2πn ) , (3.21) where n ∈ n. obviously, ±iωϵ are purely imaginary roots of (3.15) with τ = τ ϵn. let τ0 = min ϵ∈{1,2,3} {τ ϵ0} and λ(τ) = ψ(τ) + iω(τ) be the root of the equation (3.15) where ψ (τ ϵn) = 0 and ω (τ ϵn) = ωϵ. differentiating equation (3.15) with respect to τ , we get( dλ dτ )−1 = 3λ2 + 2p1λ+ p2 + q1e −λτ λ (q1λ+ q2) e−λτ − τ λ . therefore, it is straightforward to deduce re ( dλ dτ )−1 ∣∣∣∣∣ τ=τϵn = 3ω4 ϵ + 2 (p21 − 2p2)ω 2 ϵ + p22 − q21 − 2p1p3 q21ω 2 ϵ + q22 = f ′ (ω2 ϵ ) q21ω 2 ϵ + q22 . by f ′ (ω2 1) > 0, f ′ (ω2 2) < 0 and f ′ (ω2 3) > 0, the transversality condition is verified, and we suppose these conditions: (a) c0 < 0, (b) c0 ≥ 0,∆′ > 0, x1 > 0 and f (x1) ≤ 0, and we get the following result: theorem 3.7: suppose r0 > 1 and p1 (p2 + q1)− (p3 + q2) > 0. if one of conditions (a)-(b) is satisfied, the equilibrium point e∗ is locally asymptotically stable for all time delays τ ∈ [0, τ0). furthermore, e∗ becomes unstable when τ > τ0. in addition, when τ = τ ϵn model (2.1) undergoes a hopf bifurcation at e∗ where ϵ = 1, 2, 3 and n ∈ n. copyright © 2024 assa. adv syst sci appl (2024) 78 4. numerical simulations this section presents numerical simulations of our system to demonstrate its dynamics and behavior. the system is evaluated with various initial conditions satisfying s0, i0, v0,w0 > 0, and the time interval is set from t = 0 to t = 2000. the following set of parameters, selected based on previous works [3, 6, 8, 11], are used: r = 0.5, u = 0.5, d = 0.1, k = 2× 109, α = 1.95× 10−10, β = 1.2× 10−10, δ = 0.5,n = 1000, µ = 20, η = 0.17, h = 0.07, γ = 7, τ = 2, m = 1. using these values, we obtain r0 = 3.8481 > 1, p1(p2 + q1)− (p3 + q2) = 11.2223 > 0, c0 = 0.1873 > 0, ∆′ = 6.7108× 105 > 0 and f(x1) = 0.1224 > 0. according to theorem 3.6, the equilibrium point e∗ is locally asymptotically stable for any τ ≥ 0. this is illustrated and validated in figure 4.1. fig. 4.1. dynamical behavior of system (2.1) around the equilibrium point e∗ when τ = 2, µ = 20 and γ = 7. we then consider the same parameter values, except that γ is changed to 2 and µ to 0.65. in this case, we obtain c0 = −9.7104× 10−4 < 0, which means that the condition of theorem 3.6 is not satisfied. therefore, the equilibrium e∗ is unstable, as shown in figure 4.2. when m = 0 and n = 100, we obtain r0 = 19.0795 > 1, p1(p2 + q1)− (p3 + q2) = 0.0054 > 0, and c0 = −9.0776× 10−4 < 0. therefore, the conditions for theorem 3.7 are satisfied. using equation (3.19) and (3.21), we obtain ω = 0.4186 and τ0 = 0.8567. in figure 4.3, when τ = 0.7 < τ0, the equilibrium point e∗ is locally asymptotically stable. however, when τ = τ0 = 0.8567, model (2.1) undergoes a hopf bifurcation at the equilibrium e∗. furthermore, when the parameter τ exceeds the critical threshold τ0 associated with the hopf bifurcation, the equilibrium at the point e∗ transitions from stable steady state to stable limit cycle oscillation, as illustrated in figure 4.4. moreover, all these results confirm our theoretical results stated in theorem 3.7 copyright © 2024 assa. adv syst sci appl (2024) 79 fig. 4.2. dynamical behavior of system (2.1) around the equilibrium point e∗ when τ = 2, µ = 0.65 and γ = 2. fig. 4.3. the stability of the equilibrium point e∗ when τ = 0.7 < τ0. copyright © 2024 assa. adv syst sci appl (2024) 80 fig. 4.4. the instability of the equilibrium point e∗ when τ = 5 > τ0. 5. conclusion this paper has presented a novel contribution by introducing an ordinary differential equation model with delay designed to comprehensively capture and analyze the intricate dynamics of tumor cells following the administration of oncolytic viruses. our model has carefully considered the role of the coxsackie adenovirus receptor (car), as well as the influence of mitogen-activated protein kinase inhibitors, factors that have played pivotal roles in the interaction between the virus and the tumor microenvironment. by incorporating these crucial elements, our research seeks to provide a deeper understanding of the underlying mechanisms governing the response of tumor cells to oncolytic virus treatment. through this innovative model, our aim has been to provide valuable information that can inform the development of more effective therapeutic strategies to combat cancer. for that, we have proven that our proposed model is mathematically and biologically meaningful through its existence, non-negativity, and boundedness of solution. we have explored the equilibrium points of the model and have identified three possible steady states. the first, e0, represents a state where there are no cells (uninfected or infected) and no virus particles. this equilibrium may not have biological relevance, as it reflects the non-existence of tumor cells and virus particles. the second, e1, describes where there are uninfected tumor cells at a constant concentration s1, but no infected tumor cells or virus particles, and the average level of car molecules on the surface of the cells is at w1. this equilibrium could represent a state where the virus fails to infect and spread in the tumor. the third, e∗, was an endemic equilibrium in which uninfected tumor cells, infected tumor cells, virus particles, and car molecules on the surface of the cells reached constant concentrations over time: s∗, i∗, v ∗, and w ∗, respectively. this equilibrium could represent a coexistence between the tumor cells, infected cells, and the virus. furthermore, an examination of the local stability of these three equilibrium points has been carried out employing the characteristic equation. moreover, the global stability of the equilibrium point e1 has been established by employing copyright © 2024 assa. adv syst sci appl (2024) 81 an appropriate lyapunov function. in addition to the stability analysis of the equilibria, we have also investigated the effects of time delay on the dynamics of the system. our analysis has revealed that the introduction of a time delay parameter can lead to a hopf bifurcation at the equilibrium point e∗, inducing a shift in equilibrium from a stable steady state to a stable limit cycle oscillation. our numerical experiments additionally have validated the theoretical findings, demonstrating the impact of time delay on the system’s behavior and the emergence of the hopf bifurcation. overall, our study has highlighted the importance of considering the time delay in modeling the dynamics of oncolytic virus therapy and has provided valuable information on the long-term behavior of the system. acknowledgements we would like to extend our appreciation to everyone who contributed their knowledge and ideas to this research, regardless of how small a way. the editors and anonymous referees are thanked by the authors for their insightful criticism and recommendations, which significantly increased the caliber of this work. references 1. cuschieri, j. & maier, r. v. (2005) mitogen-activated protein kinase (mapk), critical care medicine, 33, s417–s419. 2. everts, b. & van der poel, h. g. (2005) replication-selective oncolytic viruses in the treatment of cancer, cancer gene therapy, 12, 141–161. 3. friedman, a., tian, j. p., fulci, g., chiocca, e. a. & wang, j. (2006) glioma virotherapy: the effects of innate immune suppression and increased viral replication capacity, cancer research, 66, 2314–2319. 4. hale, j. & verduyn lunel, s.m. (1993) introduction to functional differential equations. new york, ny: springer. 5. huo, h., zhao, h. & zhu, l. (2015) the effect of vaccines on backward bifurcation in a fractional order hiv model, nonlinear anal., real world appl., 26, 289–305. 6. linsenmann, t., jawork, a., westermaier, t., homola, g., monoranu, c. m. & vince, g. h. (2019) tumor growth under rhgm-csf application in an orthotopic rodent glioma model, oncolytic letter, 17(6), 4843–4850. 7. nono, m. k. & ngouonkadi, e. m. (2020) synergistic effects of oncolytic adenovirus and mek inhibitors on glioma treatment dynamics: analysis and optimal control, applied mathematical sciences, 14(16), 781–800. 8. okamoto, k. w., priyanga, a. i. & petty, t. d. (2014) modeling oncolytic virotherapy: is complete tumor-tropism too much of a good thing?, journal of theoretical biology, 358, 166–178. 9. wunder, t., schmid, k., wicklein, d., groitl, p., dobner, t., lange, t., anders, m. & schumacher, u. (2013) expression of the coxsackie adenovirus receptor in neuroendocrine lung cancers and its implications for oncolytic adenoviral infection, cancer gene therapy, 20, 25–32. 10. youshan, t. & guo, q. (2008) a mathematical model of combined therapies against cancer using viruses and inhibitors, science in china series a: mathematics, 51(12), 2315–2329. 11. zurakowski, r. & wodarz, d. (2007) model-driven approaches for in vitro combination therapy using onyx-015 replicating oncolytic adenovirus, j theoret biol, 245, 1–8. copyright © 2024 assa. adv syst sci appl (2024) introduction presentation and well-posedness of the model equilibria and stability analysis equilibrium points of the model stability analysis numerical simulations conclusion adv syst sci appl 2018; 1; 41-58 published online at http://ijassa.ipu.ru. copyright ©2018 assa. adv. in systems science and appl. (2018) review of recent trends in coarse grain reconfigurable architectures for signal processing applications raghavachari ramya 1 , sridharan moorthi 2 1) vlsi systems research laboratory, department of electrical and electronics engineering, national institute of technology, tiruchirapalli, india e-mail:407114003@nitt.edu 2) vlsi systems research laboratory, department of electrical and electronics engineering, national institute of technology, tiruchirapalli, india e-mail: srimoorthi@nitt.edu abstract: coarse grained reconfigurable architecture got the attention of researchers working in designing computing architectures for processing massive streaming data associated with the multimedia applications in portable entertainment and communication electronics. the algorithms for processing audio, video, and graphics are very complex in nature. these data intensive computation algorithms belong to the domain of signal processing. as the complexity of algorithms increases, a matching improvement in speed performance of the hardware becomes essential to maintain the quality of service. the observed growth of algorithmic complexity is much higher than the growth rate of integration density governed by moore’s law. also, the constraints on memory bandwidth in the traditional von neumann architectures along with the slow growth in the battery capacity demands a paradigm shift in computer architecture design. reconfigurable hardware architecture is proposed as a possible alternative in this regard. the reconfigurable architectures are designed to exploit the regular and repetitive structure of signal processing algorithms and the coarse grained processing elements are designed to match with the word level granularity of these complex algorithms. the research shows that the coarse grain reconfigurable architectures with heterogeneous processing elements are a better option for system design in dsp applications, which exploit granularity matching between the algorithms and the processing hardware, and the inherent parallelism of dsp algorithms for the realization of low power dsp systems. keywords: reconfigurable architectures, cgra, dsp, fpga, asic 1. introduction the fastest growing segment of the electronic industry is the battery driven products of entertainment electronics and wireless communication systems. this growth is heavily indebted to the recent advancements in digital signal processing (dsp). but the computationally intensive dsp applications limit the battery life in portable devices such as smart phones, mp3 players, hearing aids etc., the hardware platform chosen for the implementation of mobile wireless multimedia applications decides the speed, flexibility and cost. the major attraction of the microprocessor based system development is that its ram based structure offers large scale flexibility such that products for various applications can be software based and in turn avoids the necessity of costly application specific integrated circuits (asics) for each and every application. but the sequential nature of processing by microprocessors is the major performance limiting factor. the challenging design criteria are extremely low power, high performance, flexibility and low cost. the growing performance gap among application complexity, vlsi technology, and battery technology is the 42 r. ramya, s. moorthi copyright ©2018 assa. adv. in systems science and appl. (2018) constraint in delivering satisfactory performance at low power and high speed. the flexibility of a system is also an important aspect to accommodate the rapidly changing consumer needs. limiting the applications to a specific domain is expected to provide better energy efficiency by compromising flexibility. the speed and power performance of applications pertaining to dsp domain can be further improved by suitable architecture designs which exploit the regular and repetitive structure of dsp algorithms. reconfigurable architecture is a probable architecture suitable for the said purpose. the idea of restructurable computing put forward by gerald estrin in the early sixties [1, 2].the restructurable or reconfigurable architecture is a deviation from the von neumann architecture with adjustable hardware processing elements. the general purpose processors like microprocessors have fixed structure but estrin’s proposal has provision for adjustable hardware. the adjustable hardware is called as reconfigurable fabric (rf) and the process of adjusting the hardware for different applications is called reconfiguration. there are two classes of reconfigurable architectures: fine grain and coarse grain. field programmable gate array (fpga) is a fine grain general purpose reconfigurable device that supports a broad range of applications [3, 4]. the processor architecture that uses bit oriented simple processing elements like look up table (lut) of an fpga as the fundamental building block of the processor are called fine grained. the architecture that is composed of word oriented complex logical blocks like alu is termed as coarse grain. the main disadvantage of fpga like fine grain systems when compared with asics are low or medium performance, high configuration overhead, large silicon area, propagation delay, and high power consumption. coarse grained reconfigurable processors have gained more popularity in the recent past, because they offer a new method for a dynamic and programmable execution similar to fpga and tend to achieve the performance of application specific hardware. the paradigm shift to meet the computing compulsions of real-time communication and multimedia processing is from instruction stream driven systems to data driven systems and fixed hardware systems to reconfigurable hardware systems. the coarse grained reconfigurable architecture (cgra) is a convenient platform for data streaming processing in multimedia dsp applications. these reconfigurable computing architectures have been surveyed extensively in [5-12]. the review discusses the design goals and architectural features of cgra in section 2. section 3 enumerates the architectural adaptations of cgra which makes it very suitable for the domain specific applications of dsp. section 4 focuses on the development of energy efficient coarse grain architectures for dsp applications. finally conclusions are presented in the section 5. 2. coarse grain reconfigurable architectures the coarse grain reconfigurable architectures compromise on the flexibility of fpga to match with the performance of asic by limiting themselves to a particular application domain [10, 13-16]. the performance improvement over fpga is obtained by the inherent word level configurability which is designed to match with the instruction and data granularity of algorithms results in the reduced number of cycles per instruction and hence good performance in the aspects of speed of operation. the common architecture of cgra is a combination of a control processor and a rf which is either tightly coupled to act as a co-processor or loosely coupled to act as an augmenting independent unit for processing dedicated domain specific instructions. generally the control processors execute the nonloop sequential code, control the mapping of configurations to the rf grid and supervise the execution activities. the basic architecture of cgra is shown in fig. 2.1. the rf is a combination of coarse grain processing elements (pes) or functional units (fu), word level data paths, and fast interconnects. the presence of coarse grain processing elements reduces review of recent trends in coarse grain reconfigurable architectures 43 copyright ©2018 assa. adv. in systems science and appl. (2018) the configuration data and this makes the devices to do the reconfiguration faster and reduces the area, delay, and power consumption of circuits. fig. 2.1. basic architecture of cgra the data driven processing elements are configured by configuration words stored in a dedicated memory called configuration or context memory. the context words stored in the context memory decides the functionality of each processing element and also the flow of data between pes. reconfiguration is effected by choosing another context word in context memory. the different reconfigurable options available to the designer are: reconfiguration can be designed to take place at the beginning of the execution and retains the context for the rest of the time (static reconfiguration) or it may occur at one or more times during the execution (dynamic reconfiguration). reconfiguration can affect all elements in the architecture (total reconfiguration) or only some of them (partial reconfiguration). another flexibility that the cgra architecture designers can exercise is the option to select either a homogeneous set of pes or a heterogeneous set of pes for the rf. the recent approach is to give preference to have reconfigurable system on chip which includes even general purpose processor (gpp), dsp, and fpga integrated along with cgra in to the chip [17-19]. heterogeneous architecture for cgra is preferred since, some algorithms run more efficiently on bit level configurable architecture and some perform optimal on word level reconfigurable platforms. an understanding of the algorithmic domain is essential for the design of domain specific applications in order to achieve asic like performance. the major application of cgra is in the domain of dsp. table 2.1. illustrates the important coarse grain reconfigurable architectures listed in the literature. table 2.1. different coarse grain reconfigurable architectures listed in literature year of publication name of the processor research group processing element(pe) processing element configuration data path granularity target application 1990 paddi[20] university of california, berkeley cluster of 8 exu; exu: rfile,mux, nano store crossbar 16-bit dsp 1995 kress array[21] university of kaiserslautern rdpu 2d-mesh family: select path width to implement computational datapaths 1996 rapid [22] university of washington multiplier,3 alus, 6 dpr, 3 local memories 1d-array 16-bit pipelining applications 1996 matrix [23] mit alu, multiplier, 256 × 8-bit memory, control logic 2d-mesh 8-bit, multi granular general purpose co-processor 44 r. ramya, s. moorthi copyright ©2018 assa. adv. in systems science and appl. (2018) 1997 pleiades [24 ] university of california, berkeley satellite processors crossbar multigranular multimedia 1997 raw [25] mit alu, registers, instruction & data memories, config. logic,prog.switch 2d-mesh 8-bit, multigranular general purpose 1998 piperench [26] carnegie mellon university alu & pass register file 1d-array 128-bit co-processor for streaming multimedia acceleration 1998 remarc[27] stanford university 16-bit nano processor 2d-mesh 16-bit multimedia 1998 morphosys [28] university of california, irvine alumultiplier, a shift unit, two input mux, & 32-bit context register 2d-mesh 8-bit/16bit multimedia & dsp 1999 chess array[29] hewlett packard labs interleaved alus and switchboxes hexogonal array multigranular multimedia 2000 dream[30] university of darmstadt 8-bit alu, 2 shifters,cu,cm 2d-mesh 8-bit/16-bit next generation wireless 2001 pact-xpp[31] pact informations technologie ag 32-bit alu,adder, router,cm,cu 2d-array 16-bit& 32-bit multimedia, dsp,telecommuni cation 2002 montium [32] university of twente five alu, two rfiles, sequencer tile array 16-bit streaming dsp application 2002 imagine [33] stanford university alu clusters, multi port stream register file, scratch pad memory stream model 16-bit media processing 2002 dart[34] university of rennes reconfigurable data path cluster 16-bit mobile communication 2003 trips[35] university of texas, austin 32-bit alu, routers,cu,op-reg, configuration cache 2d-array 32-bit general purpose 2003 adres[36] mec , belgium 32-bit alu, registers,router, cm,cu array 8-bit multimedia and mobile communication 2004 nec-drp[37] nec electronics 8-bit alu,8-bit dmu,16 8-bit rfu,8-bit ffu tile 8-bit streaming multimedia application 2007 mora[38] university of calabria multipliers, adders,registers, muxes,3:2 compressors linear array 8-bit multimedia 2008 rica[39] university of edinburgh heterogeneous pe array 32-bit dsp,viterbi decoding 2008 smartcell[40] worcester polytechnic institute 16-bit alu, i/o registers tile array 8-bit dsp, multimedia 2009 flora[41] seoul national university alu,rfile,mux, configuration memory 2d-array 24-bit dsp, multimedia 2009 drra[42] kth royal institute of technology, sweden morphable dpuarithmetic partition, logic partition & post processing partition 2d-array 16-bit dsp 2011 syscore[43] university college, dublin configurable functional unit systolic 24-bit biomedical signal processing 2013 bilrc[44] bilkent university, ankara, turkey alu,memory, multiplier 2d-array 16-bit multichannel fir filters,viterbi & turbo decoder 2014 fpca[45] university of california ces, lmus, alus, on-chip buffers, registers 2dmesh 32-bit dsp,medical imaging, image processing 2017 hycube [46] national university of singapore alu,memory, switches 2d-mesh 32-bit general purpose review of recent trends in coarse grain reconfigurable architectures 45 copyright ©2018 assa. adv. in systems science and appl. (2018) alu: arithmetic logic unit; cu: control unit; cm: configuration manager; ce: computation elements; config. logicconfigurable logic; dmu: data management unit; dpu: data path unit; exu: execution unit; ffu: flip flop unit; lmu: local memory unit; mux: multiplexer; op-reg: operand registers; prog.switchprogrammable switch; rfile-register file; rdpu: reconfigurable data path unit; rfu: register file unit 3. coarse grain reconfigurable architectures for dsp the paradigm shift happened in the ic design space is that it changed from a two dimensional problem of area and speed to a three dimensional problem of speed, power and complexity. conventional processing architectures such as gpp, dsp, asic and even fpga finds it difficult to satisfy the design constraint of high speed, low power performance to meet the requirements of highly complex algorithms for the current and future applications. but the industry needs processors to fill the gap among algorithmic complexity, vlsi technology and battery technology. coarse grain computing paradigm which got a boost in the past two decades is one specific answer to the above defined problem. the cgra addresses these constraints by limiting the design to domain specific applications. the larger chunk of applications addressed by cgra for high performance is from the application domain of dsp [5, 19, 47-52]. the basic concept used in the design is that hardware adapts to algorithm instead of adapting the algorithm to the hardware. the dsp algorithms are characterized by the regular and repetitive structures present in them. these regular and repetitive computational parts of dsp algorithms that accounts for large fraction of the execution time and energy consumption are called dsp kernels. these kernels are suitable for spatially distributed computing with word level processing rather than the sequential computing at a bit oriented fashion. for example the dsp kernels of fir filter contains a multiply-accumulate operation and the kernel of a fast fourier transform (fft) contains the fft butterfly which are very much suitable for word oriented processing if suitable functional units are implemented for their processing at word level in a cgra fabric. algorithms belonging to the same algorithmic domain have similar kernels and operate on similar data structures. therefore the same processing elements can be reconfigured without much control overhead for execution of different algorithms. a thorough understanding of the algorithm domain is important in the design of a power efficient reconfigurable architecture. considering the application domain of dsp, there is an excessive demand for streaming communication and computation for wireless protocol processing and multimedia processing. also, the understanding of the underlying vlsi technology plays a vital part in the design of low power systems. the dominant vlsi technology is the cmos technology with the major component of energy consumption is dynamic power consumption. a first order approximation to dynamic power consumption in cmos circuitry is given by the formula where pd is the power in watts, is the effective switching capacitance in farads, v is the supply voltage in volts, α is the switching activity factor and f is the frequency of operation in hertz. the above equation suggests that the power can be reduced by, reducing the capacitive load ceff, reducing the supply voltage v, reducing the switching frequency f, or reducing the switching activity α. since the technology already touched the 1v wall [53,54] and increase of frequency of operation also causes undesirable effects, some techniques should be exploited to reduce the switching activity and capacitance. thus it is explored to see how the concept of locality of reference is exploited to reduce the capacitance and switching activity in order to achieve better power performance in low power systems. locality of reference [19, 52, 55] is a major concept widely exploited to 46 r. ramya, s. moorthi copyright ©2018 assa. adv. in systems science and appl. (2018) reduce the power consumption. the switching activity factor α is also reduced by the locality of reference. spatial locality of reference refers to the fact that once a specified location is referenced, there is often exists a chance to refer a nearby location in the near future. accessing a large distant memory is less energy-efficient than accessing small and local memory. energy efficiency can be substantially improved by exploiting the locality of reference principle. due to the locality of reference principle communication within a tile dominates communication between tiles. the functional-level reconfiguration offers opportunities to improve energy efficiency of flexible architectures. minimizing the reconfiguration data volume reduces the energy wastage. the coarse-grain paradigm of computation optimizes storage as well as computation resources from an energy point of view. another important characteristic for data streaming applications is computing parallelism, which means the computational task has the potential to be distributed across multiple processing components. the temporal and spatial parallelism inherent in dsp algorithms can be exploited to extract better performance. in temporal parallelism, the pipeline technology is adopted at either the instruction level or at the task level, in which the instruction code or computing task is separated into multiple stages. several instructions or tasks are overlapped in the same pipeline at different stages to improve system throughput. on the other hand, spatial parallelism distributes the data and tasks onto different computational nodes to process in parallel. in data streaming applications, data-level parallelism (dlp) can often be exploited since there are no data dependencies between different input data blocks. in this case, single instruction multiple data (simd) computational style is widely used to apply the same kernel functions to different data elements. similarly, task-level parallelism (tlp) is usually exploited to execute different application threads in data streaming applications. in many cases, the computing task involved in stream processing can be decomposed into multiple stages. these stages can be overlapped into multiple computing resources to concurrently process different data sets through the pipeline. given plentiful parallelism, it is a key requirement that the computing architecture design for data streaming applications should be able to efficiently exploit and map the parallelism onto available hardware resources. as mentioned earlier, the configurability of cgra is due the reconfigurability of the functionality of the pes according to context words stored in the configuration memory. recently the cgra developers focusing on configurability of the communication network in place of the configurability of the pe. in syscore [43] a conceptual interconnect structure named round about interconnect (rai) is introduced. the interconnect configuration registers associated with the rai supports distant neighbor data transfer without the cost of area and power of commonly used mesh network interconnect or a mesh variation structure. the hycube [46] architecture goes even further with a reconfigurable interconnect to provide single cycle communication between distant neighbors. the reconfigurable interconnect network reduces the density of the interconnect network, hence reduces the power dissipation in the interconnect network. 4. energy efficient architectures for dsp the dsp algorithms are characterized by the regular and repetitive structures called kernels present in them. since the algorithms belonging to the same algorithmic domain have similar kernels and operate on similar data structures, the same processing elements can be reconfigured without much control overhead for execution of different algorithms. by executing dominant kernels of a given domain of algorithms on optimized processing elements, significant energy savings with minimum of energy overhead is achieved. this review of recent trends in coarse grain reconfigurable architectures 47 copyright ©2018 assa. adv. in systems science and appl. (2018) section deals with important energy efficient domain specific low power cgra processors intended for signal processing and multimedia applications. 4.1 montium tile processor [19, 51, 56] the computer architecture design and test for embedded systems (cadtes) group at the university of twente designed the montium tile processor and the processing core has been further developed by recore systems. this coarse grain architecture targets for 16-bit dsp applications. the montium tile is characterized by its coarse-grained reconfigurability, high performance and low energy consumption. the montium achieves flexibility through reconfigurability. the block diagram of a single montium processing tile is shown in fig. 4.1. the lower part contains the communication and configuration unit (ccu) and the upper part shows the reconfigurable tile processor (tp). the tile processor is the computational part which can be configured to implement a specific algorithm. it consists of five alus connected to 10 memory banks through a circuit switched network and these five processing parts together called the processing part array. a sequencer controls the operation of the processing part array. an array of configuration memory is provided for memory, register, alu and interconnects. the sequencer with the help of respective decoders selects appropriate context words during execution of an algorithm. the ccu implements the interface for off-tile communication. fig. 4.1. block diagram of montium processing tile [56] high performance is achieved by parallelism, because the montium has several parallel processing elements. the principle of locality of reference is exploited to reduce the energy wastage. the memory hierarchy of montium tiles allows different levels of local register, local memory, global memory and global register. the two layer decoding employed in the montium tile design simplifies the instruction fetch part, which reduces energy consumption and cost. montium is designed for dsp algorithms found extensively in mobile applications. such algorithms are usually regular and have high computational density. it has also be noted that thread-level parallelism is addressed by the multi-core approach as different tiles can run different tasks, data-level parallelism (dlp) is achieved by the montium processing tiles, which employ parallelism in the data path and instructionlevel parallelism (ilp) is addressed by the montium processing tiles as multiple data path 48 r. ramya, s. moorthi copyright ©2018 assa. adv. in systems science and appl. (2018) instructions can be executed concurrently. the architecture found to be highly flexible in mapping dsp kernels and energy efficient too. the main characteristics required for a cgra to act as a dsp core are low power, high speed and flexibility. the architecture is found to be highly flexible in mapping dsp kernels. the energy performance characteristic of the montium is compared with the characteristics of other state-of-the-art reconfigurable architectures in [56]. the fft algorithm is used to compare the energy consumption of a montium tile processor with an asic, a xilinx virtex-ii pro fpga, a silicon hive avispa and an arm920t processor. the results presented in the paper shows that montium processor provide an alternative for mobile devices for which energy-efficiency is an important factor. the montium tile processor satisfies the requirements mentioned above for a dsp core and can be used as a single dsp accelerator core or clustered in a large reconfigurable subsystem. 4.2 dynamic reconfigurable resource array (drra) [57, 58] drra is a heterogeneous coarse grain reconfigurable architecture capable of hosting multiple, complete radio and multimedia applications. the design template was developed at kth royal institute of technology, sweden. a fragment of drra fabric which consists of drra cells arranged in rows and columns fashion is illustrated in fig. 4.2. the architecture relies on high bandwidth, distributed memory integrated in 3d within the logic tile. any signal processing applications such as fir, iir, fft etc., can be mapped efficiently in the drra fabric. the drra fabric is composed of morphable data path units (mdpu) and register files (rfile) organized in a 4×n matrix. bottom and top layers are of rfiles and the inner layers are of mdpus. a sequencer, switch box, mdpu and rfile form the basic drra cell. fig. 4.2. dynamically reconfigurable resource array fabric [57] reconfigurability of mdpus and the interconnect that combine the multiple mdpus and rfiles is central to creating algorithmic building blocks. drra cells are connected together through interconnects using a sliding window 3-hop communication scheme. every resource input is connected to every resource output within three-column range, making the boundaries of the architecture flexible enough to implement most of the algorithmic building blocks. extended connectivity beyond three columns can be achieved by using intermediate review of recent trends in coarse grain reconfigurable architectures 49 copyright ©2018 assa. adv. in systems science and appl. (2018) buffers and this seamless connectivity offers flexibility in creating on the fly partition to create new algorithmic building blocks as and when required. the drra architecture is characterized by the large pool of resources for computational, storage, and interconnects functionalities. the architecture relies on clustering these resources at run time to serve individual applications and once an application is over, it is reclaimed and reused for next application. drra being a fabric, the computation is distributed across the chip. multiple threads, algorithms and applications are intended to run in parallel in the fabric. the distributed memory architecture (dimarch) found in drra is partitionable and the architecture is designed to keep the cost of partitioning and re-partitioning low both in terms of cycles and energy. the distributed nature of memory architecture and the concept of private execution environments enable a short distance between storage and computation, which in turn satisfies the locality of reference. the drra reduces the granularity mismatch at three levels. at the building block level, the dpu and the register file have been endowed with custom modes like macs, butterflies, programmable burst address generation, bit-reverse addressing modes etc., to efficiently implement commonly occurring dsp operations in the signal processing algorithms. this coarse grain mode reduces the silicon and bit-width mismatches. the drra cell is the basic unit when composing a drra fabric, but its components dpu, register file and sequencer can be combined individually to create hierarchical fsmds (fsms + data-paths) of arbitrary complexity; the combining happens in the drra interconnect fabric. this is the third level at which drra reduces the instruction granularity mismatch. drra’s interconnect scheme enables dynamic creation of an arbitrary wide vliw, where each issue can be an arbitrary instruction. the power performance of drra is computed and compared with asic, fpga in detail and presented in [57]. the performance is tested by simulating fir, fft and sorting algorithms. the static and dynamic power analysis shows that the drra achieves performance comparable with that of an asic processor. for comparison, the dynamic power dissipated in the clock net is on an average over all the algorithms is 79.47x that of asic, but for an fpga it is 255.57x. similarly the dynamic power spend for computation is 2.76x compared to 60.87x for fpga compared to that of an asic. the drra consumes 24.64x of the static power consumption of asic and fpga consumes 403.19x that of an asic. the performance of drra is evaluated further by implementing a 1024 fft. for a 1024-point fft, in terms of fft operations per unit energy, drra-1 and drra-2 outperforms all cgras by at least 2x and is worse than asic by 3.45x. the number of operations per second achievable by drra equals that of the asic. also considering the good flexibility of drra, it possesses all the essential features of a dsp core for mobile platforms. 4.3 smartcell [40, 59,60] smartcell is a coarse grain reconfigurable architecture designed for stream based applications. the block diagram of smartcell architecture is shown in fig.4.3. the architecture is made up of three main components: cell unit, reconfigurable interconnect fabric and data i/o. 50 r. ramya, s. moorthi copyright ©2018 assa. adv. in systems science and appl. (2018) fig.4.3. block diagram of smartcell architecture [60] the reconfigurable cell units are the basic components in smartcell. they are organized in a tiled structure. there are four identical pes in each cell block. multiple pes can be chained together to implement more complex algorithms. a 4-stage pipeline structure is developed in each processor. the reconfigurable interconnect fabric are used for inter and intra cell communication. the data flow can be dynamically reconfigured for different applications. the deep pipeline and ilp at single pe and the tlp among multiple cells arranged in a tiled mesh are exploited to realize high computing capacity and energy efficiency. an instruction controller is provided to control the data-path and functionality of each pe. the number of pes involved in an application task, function of each pe and the inner and inter cell connectivity are reconfigurable in real time. also, the dynamic power consumption is significantly reduced by the use of gated clock technology to turn off the inactive pes. the features of the smart cell [59] are listed below: (i) coarse-grained granularity: smartcell is designed to generate coarse-grained configurable system targeted for computation intensive applications. the processing elements operate on 16-bit input signals and generate a 36-bit output signal, which avoids high overhead and ensures better performance compared with fine-grained architectures. (ii) flexibility: due to the rich communication resources, versatile computing styles can be easily mapped onto the smart cell architecture, including simd, mimd, and 1d or 2d systolic array structures. this also expands the range of applications to be implemented. (iii) dynamic reconfiguration: by loading new instruction codes into the configuration memory through the spi structure, new operations can be executed on the desired pes without any interruption with others. the number of pes involved in the application is also adjustable for different system requirements. (iv) fault tolerance: in the smart cell system, defective cells, caused by manufacturing fault or malfunctioned circuits, can be easily turned off and isolated from the functional ones. (v) deep pipeline and parallelism: two levels of pipeline are achieved—the instruction level pipeline in a single processor element and the task level pipeline among multiple cells. the data parallelism can also be explored to concurrently execute multiple data streams, which in combine ensures a high computing capacity. review of recent trends in coarse grain reconfigurable architectures 51 copyright ©2018 assa. adv. in systems science and appl. (2018) (vi) hardware virtualization: distributed context memories are used to store the configuration signals for each pe. the cycle-by-cycle instruction execution supports hardware virtualization that is able to map large applications onto limited computing resources. (vii) smartcell provides explicit synchronization that eases the exploration of computing parallelisms. (viii) unique system topology: the cell units are tiled in a 2d mesh structure with four pes inside each cell. this topology was designed to meet different computational requirements. with the help of the hierarchical on-chip connections, the smartcell architecture can be dynamically reconfigured to perform in variant operational styles. it is experimentally shown that the smartcell offers significant improvement in energy and speed performance [60] compared to fpga, dsp and morphosys cgra. the smart cell is found to be 1.5 times faster compared to xilinx’s virtex ii pro xc2vp20 fpga in executing an fft block. it is found to be 3.6 times energy efficient than the fpga. a comparison with dsp for the same experimental setup shows that smartcell is about 20.8 times faster and about 28.9 time more energy efficient than tms320c6713. the smartcell satisfies the basic requirement of low power, high speed, and flexibility required for portable dsp processors and hence a good substitute as a dsp core. 4.4 fully pipelined composable architecture (fpca) [45] the architecture of fpca is also that of a 2d-mesh with neighbor-to-neighbor(n2n) connectivity. each tile is a cluster of pes. the architecture of the pe cluster is shown in fig.4.4. the cluster is a set of heterogeneous pes including computation elements(ces), local memory units(lmus) and register chain to act as onchip buffers and registers respectively. the architecture may be considered as a two level architecture where processing elements are first connected by a permutation matrix with high connectivity within a cluster and then by a global n2n network for more scalable connectivity. fig.4.4. internal structure of a processing element (pe) cluster [45] mostly, a single application fit into a single processing element cluster and this alleviates the challenges of routability and dynamic composition. a small part of the resources is used for a single application and hence the unused resources can shared for other applications. multiple copies of a single application can also be mapped to the unused resources. this process is called dynamic composition which will improve the overall throughput. a runtime scheduler is used to map the idle resources as mentioned to the incoming applications. the challenge of complex routing in dynamic composition is overcome by the programmable 52 r. ramya, s. moorthi copyright ©2018 assa. adv. in systems science and appl. (2018) interconnects provided with the fpca architecture. the processing element cluster approach introduces heterogeneity which support memory, arithmetic and logic operations altogether in a tile. this enables one to offload execution control to the pe cluster and hence aggressive pipelining made possible based on a modulo scheduling scheme. the full pipelining and dynamic composition of the architecture improve the energy and speed performance of the device. it is observed that fpca prototype achieve 1.5-3.4x speedup compared to the dual-core arm. also shown that a 2x improvement in speed performance is achieved by duplication based on dynamic composition. experiments show that fpca architecture can achieve a >50x energy savings compared to arm processor. 4.5 hycube [46] hycube architecture follows the 2d-mesh topology as shown in fig.4.5. the functional units (fus) are connected in a 2d-mesh structure. the architecture comprises of two types of tiles. tiles of the first type are connected to the data memory and capable of performing memory operations. tiles of the second type compute only tiles and both have an alu, a configuration memory and cross bar switch. an additional load-store unit is present in the memory tile enable them to interface with data memory. fig.4.5. a 4 × 4 hycube cgra [46] by introducing a reconfigurable interconnect hycube provides 1-cycle communication between distant tiles. the main circuit element in the programmable interconnect is the crossbar switch that is driven by a clockless repeater. the repeaters can be configured to bypass data to successive hops asynchronously or to receive data. since data can be passed to distant tiles by bypassing the entry of data into the in between tiles, 1-cycle communication can be realized between tiles that are not nearest neighbors. the hycube has a compiler controlled noc and hence there is no routing and flow control logic. the input and output ports has only single registers and facilitates multihop multicast path between tiles scheduled completely at compile time. given the reconfigurable single-cycle multi-hop interconnect, hycube scales very well for 4×4 cgra and offers 3x performance and 4x performance-per-watt compared to arm. the experimental results also show that average power efficiency of hycube is 1.5x and 3x compared to a cgra with standard noc and a n2n cgra, respectively. the experiments gives a demonstration of the improvements made with regard to the interconnect of the hycube enables it to improve the energy performance. unlike the other processors discussed hycube is not a dedicated dsp cgra architecture. but the idea of reconfigurable interconnect and reduction of total hardware cost due to interconnects are also seems to be welcome features for the design of future cgras for improving power performance as demonstrated by the hycube architecture. review of recent trends in coarse grain reconfigurable architectures 53 copyright ©2018 assa. adv. in systems science and appl. (2018) 5. conclusions in the post dennard scaling era, the supply voltage scaling is limited by exponential increase in leakage current and it touched the 1v wall. the computer architecture designers are thus exploring alternatives to circumvent the limitations imposed by the vlsi technology. the speed and power performance can be improved by exploiting the inherent word level parallelism of dsp algorithms to design suitable parallel computer architectures matching with the structure and granularity of the algorithms. another concept well exploited in the design of low power dsp system is the judicious use of locality of reference by providing distributed memory. the flexibility of cgra were restricted by limiting the applications to a specific domain and suitable control mechanism are provided to turn off unutilized hardware in run-time to get better energy efficiency. also heterogeneous architectures ensures better granularity matching which can be exploited for realizing low operational frequencies which is directly proportional to the power dissipated. the study reveals that coarse grained reconfigurable heterogeneous architecture with distributed memory is the future solution for the electronic industry to assure speed, power and cost performance. recent research shows that reconfigurable interconnect networks reduces the density of the interconnect network, and hence reduces the power dissipation in the interconnect network. a clever design choice of reconfigurable processing elements and interconnects may result in delivering asic like energy and speed performance by cgras especially in the application domain of signal processing. references [1] estrin, g. 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(2009). smartcell: an energy efficient reconfigurable architecture for stream processing (doctoral dissertation), wpi, worcester, ma. https://doi.org/10.1147/rd.462.0169 https://doi.org/10.1109/vlsid.2011.45 adv syst sci appl 2020; 01:119–127 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/866 local normal forms of autonomous quasi-linear constrained differential systems alexander m. kotyukov1*, stanislav o. nikanorov1, natalya g. pavlova1,2 1v.a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia 2moscow institute of physics and technology (state university), dolgoprudnyi, russia abstract: the paper presents a study of impasse (singular) points of autonomous quasi-linear constrained differential systems, also called differential-algebraic equations. the interest in such systems is motivated by their applications in various problems of pure and applied mathematics, including control theory, biology, and electric engineering. local normal forms of such systems in a neighborhood of their impasse points are established. keywords: impasse point, direction field, normal form, diffeomorphism, symmetry 1. introduction we consider systems of differential equations in the form a(ξ)ξ′ = b(ξ), ξ′ = dξ dt , ξ = (ξ1, . . . , ξn)∗ ∈ rn, (1.1) where a = (aij) is an n× n matrix and b = (b1, . . . , bn)∗ is a vector function. the components aij , bj are assumed to be c∞-smooth functions on ξ. there are many possible names of systems (1.1): differential systems of sobolev type, generalized vector fields, descriptor systems, quasi-linear constrained† differential systems, etc. the interest in systems (1.1) is motivated by their applications in various problems of pure and applied mathematics (including control theory and electric engineering). for instance, systems (1.1) describe the dynamic of electric circuit in nonlinear rlc-networks (networks consisting of a resistor, an inductor, and a capacitor). see [1]– [14] and the references therein. in a pioneer work [15], systems (1.1) appear as a tool for studying a systems of pdes of the mixed type, which describes the motion of a body filled with a viscous incompressible fluid. in a recent series of papers [16]– [20], systems (1.1) in dimension n = 3 describe geodesic lines in singular metrics. there exists several different approaches for studying systems (1.1) and, more generally, nonlinear systems of differential equations f (t, ξ, ξ′) = 0, ξ′ = dξ dt , ξ = (ξ1, . . . , ξn)∗ ∈ rn, (1.2) ∗corresponding author: amkotyukov@mail.ru †the reason for the this term will soon become clear. 120 a.m. kotyukov, s.o. nikanorov, n.g. pavlova which are often called implicit differential equations or differential-algebraic equations (daes). here f : r2n+1 → rn is assumed to be a c∞-smooth mapping. there exists several approaches to investigation of systems (1.1), (1.2). an analytical approach is based on the decoupling procedure. the idea is to rearrange terms within the given system so that it is decomposed in two subsystems of lower dimensions (separated as far as possible), where the first subsystem is equivalent to a standard ode of of maximum possible dimension and the second one is a dae having a special form. the highest goal is so-called complete decoupling, where the both subsystems are completely independent, that is, they do not have common unknowns. for linear daes with constant coefficients this method is comparable with the weierstrass–kronecker normal form of regular matrix pencils. this approach is presented in the book [21]. another geometric approach goes back to h. poincaré.‡ he used it for a single implicit differential equation f (x, y, y′) = 0, that is, the case n = 1. further development of this approach is given, e.g., in [22]– [34]. in the modern terminology, the main idea is the following. by jk denote the space of k-jets of vector functions ξ(t) : r→ rn. in the case of systems (1.2), consider a manifold mn+1 f ⊂ j1, j1 ' r2n+1, defined by the equation f = 0. moreover, dae (1.2) defines a direction field on mn+1 f , whose integral curves are 1-graphs (legendrian lifts) of integral curves of (1.2). in other words, we pass from the multivalued vector direction field given by dae (1.2) in j0 (i.e., the (t, ξ)-space) to a single-valued direction field on the manifold mn+1 f . the natural projection π : j1 → j0 sends integral curves of the field defined on mn+1 f to integral curves of dae (1.2). the restriction of the projection π : mn+1 f → j0 is a mapping of two (n+ 1)-dimensional manifolds. it has singularities at those points of mn+1 f where the matrix (∂f/∂ξ′) vanishes. integral curves of (1.2) have singularities (generically, cusps) at corresponding point of the (t, ξ)-space. in the case of systems (1.1), the above construction is essentially simplified: no need to consider the space j1. indeed, writing system (1.1) in the pfaffian form a11dξ1 + a12dξ2 + · · ·+ a1ndξn − b1dt = 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . an1dξ1 + an2dξ2 + · · ·+ anndξn − bndt = 0, (1.3) one can see that system (1.1) defines a direction field in the (t, ξ)-space: ξ̇1 = ∆1(ξ), . . . , ξ̇n = ∆n(ξ), ṫ = ∆(ξ), (1.4) where ∆ is the determinant of the matrix a, and ∆i is the determinant of the matrix obtained from a by replacing of its ith column with b. here the dot over a symbol means the differentiation by a new parameter playing the role of time. a principal difference between system (1.1) and usual autonomous odes is that system (1.1) (and consequently, field (1.4)) possesses so-called degenerate hypersurface γ = {ξ : ∆(ξ) = 0}, which is also called the criminant of the system (1.1). generically, there are no integral curves of (1.1) passing through γ; see [29]. therefore, γ is also called impasse hypersurface of the system, and points of γ are called impasse points. it is worth observing that if system (1.1) describes a model of a natural process in real time, the existence of impasse points in the applicability domain often means that the model is not adequate. ‡mémoire sur les courbes définies par les équations différentielles (1885). copyright © 2020 assa. adv syst sci appl (2020) local normal forms of differential systems 121 the germs of two systems (1.1) are called ck-equivalent, if there exists a local ckdiffeomorphism of the ξ-space that conjugates the germs of the corresponding direction fields (1.3). our goal is to simplify field (1.3) (and consequently, the corresponding system (1.1)) in a neighborhood of its impasse points using an appropriate local ck-diffeomorphism of the ξ-space. here k ≥ 1 is integer or∞ or ω. cω is the standard symbol for the class of analytic maps.§ the germs of two systems (1.1) are called orbitally ck-equivalent, if there exist a local ck-diffeomorphism of the ξ-space and ck-mapping of the independent variable t 7→ τ(t, ξ) that conjugate the germs of the corresponding direction fields (1.3). the classification of systems (1.1) at all impasse points is not observable. according to the general ideology of singularity theory, a number of classes of impasse points that admits relatively simple normal forms is determined. for the cω-equivalence, many results of this sort appear in [28], where systems (1.1) are considered in real, complex and even infinite dimensional (banach) complex spaces. for the ck-orbital equivalence, many results appear in [32]. in this paper, we present some new results aboutck-normal forms of the germs of systems (1.1) at their impasse points of a special type. 2. main results there are several geometric objects naturally connected with system (1.1). the firts object of thid sort is the criminant γ. other important object are the family of linear operators a(ξ) : tξrn → tξrn, their images ima(ξ) and kernels kera(ξ). the criminant γ is the locus of points ξ where dim ima(ξ) < n or, equivalently, dim kera(ξ) > 0. the simplest type of impasse points is so-called non-singular impasse points ξ ∈ γ that satisfy three following conditions: 1. d∆(ξ) 6= 0, that is, γ is a regular hypersurface. 2. dim kera(ξ) = 1 and the direction kera(ξ) is transversal to γ. 3. b(ξ) /∈ ima(ξ), that is, ima(ξ)⊕ 〈b(ξ)〉 = rn. theorem 2.1: the germ of system (1.1) at every its non-singular impasse point is c∞-equivalent (in the analytic category, cω-equivalent) to ξ′1 = 0, . . . , ξ′n−1 = 0, ξn ξ ′ n = ±1. moreover, in the category of orbital equivalence, one can reduce ±1 to 1. the proof of theorem 2.1 can be found in [27, 28, 32]. now consider a point ξ◦ ∈ γ such that the 1st and 2nd conditions hold true, but the 3rd condition fails. from dim kera(ξ◦) = 1 (the 2nd condition) and the known identity dim kera(ξ) + dim ima(ξ) = n it follows that dim ima(ξ◦) = n− 1, i.e., rank of the matrix a(ξ◦) = n− 1. then there exists i ∈ {1, . . . , n} such that rank of the matrix obtained from a(ξ◦) by eliminating the ith column is n− 1, and the germs of ∆, ∆1, . . . ,∆n belong to the ideal i = 〈∆,∆i〉 in the ring of smooth functions, and the set of singular points (equilibriums) of the corresponding §in speaking about cω-equivalence, we assume that the systems (1.1) are also analytic. copyright © 2020 assa. adv syst sci appl (2020) 122 a.m. kotyukov, s.o. nikanorov, n.g. pavlova field (1.4) is given by two equations: ∆ = ∆i = 0. (the detailed proof of this fact can be found in [31].) therefore, singular point of the field (1.4) are not isolated, they fill a submanifold w ⊂ γ of codimension 2. at every point ξ ∈ w in a neighborhood of ξ◦, the spectrum of the linear part of the field (1.4) is (λ1, λ2, 0, . . . , 0). here λ1,2 are complex numbers, generically reλ1,2 6= 0 at almost all points of w . lemma 2.1: assume that reλ1,2(ξ◦) 6= 0. then in a neighborhood of ξ◦, the following statements hold true: (i) the set w is the center manifold of the field (1.4). (ii) there exist i1, i2 ∈ {1, . . . , n} such that the ideal i = 〈∆i1 ,∆i2〉. proof the first statement is trivial. for the second statement, remark that it is possible to select as generators of i any two elements v, w ∈ i such that dv, dw at the point ξ◦ are linearly independent. consider the jacobi matrices j0 = ∥∥∥∥∥∂(∆1, . . . ,∆n,∆) ∂(ξ1, . . . , ξn, t) ∥∥∥∥∥, j = ∥∥∥∥∥∂(∆1, . . . ,∆n) ∂(ξ1, . . . , ξn) ∥∥∥∥∥. at the point ξ◦. since the functions ∆1, . . . ,∆n,∆ do not depend on t, the spectrums of matrices j0 and j have the form (λ1, λ2, 0, . . . , 0) with the same λ1,2. therefore, among the functions ∆1, . . . ,∆n there exist a couple v = ∆i1 , w = ∆i2 whose differentials at ξ◦ are linearly independent. therefore, v, w are generators of the ideal i . lemma 2.1 shows that using an appropriated renaming of the variables (ξ1, . . . , ξn) 7→ (x, y, z), z = (z1, . . . , zn−2), one can write the corresponding field (1.4) in the form ẋ = v, ẏ = w, żi = αiv + βiw, i = 1, . . . , n− 2, ṫ = av + bw, (2.5) here a, b, α, β, v, w are c∞ (resp., cω) functions on x, y, z such that v, w vanish at ξ◦. the ideal i = 〈v, w〉, and the center manifold w consisting of singular points is given by the equations v = w = 0. since all components of the field (2.5) do not depend on t, one can consider the projection of (2.5) to the ξ-space: ẋ = v, ẏ = w, żi = αiv + βiw, i = 1, . . . , n− 2. (2.6) the classification of fields (2.6) much simpler than the classification of fields with isolated singular points with the same spectrum, see [17, 18, 31]. it is explained by the fact that the non-trivial dynamics is connected with the center manifold of the field. in the case of (2.6), the center manifolds consists of singular points of the fields, therefore the non-trivial dynamics is identically zero. lemma 2.2: assume that reλ1,2(ξ◦) 6= 0 and there are no resonances p1λ1(ξ◦) + p2λ2(ξ◦) = 0, p1,2 ∈ z+, p1 + p2 ≥ 1. (2.7) then for any integer k ≥ 1, the germ of (2.6) is ck-equivalent to the germ ẋ = ṽ(x, y, z), ẏ = w̃(x, y, z), żi = 0, i = 1, . . . , n− 2, (2.8) copyright © 2020 assa. adv syst sci appl (2020) local normal forms of differential systems 123 at the origin, with some ṽ, w̃ ∈ 〈x, y〉. moreover, if reλ1,2(ξ◦) have the same sign, the above equivalence holds true with k =∞. without loss of generality, further we assume that |λ1(ξ◦)| ≤ |λ2(ξ◦)|. put λ◦ = λ1(ξ◦) : λ2(ξ◦), µ◦ = λ2(ξ◦) : λ1(ξ◦). lemma 2.3: assume that the conditions of lemma 2.2 hold true. then the germ of the field (2.6) is orbitally ck-equivalent to the germ (2.8), where ṽ, w̃ have the form (a) ṽ = x, w̃ = λ(z)y, if λ1,2(ξ◦) are real and µ◦ /∈ n ∪q−, (b) ṽ = λ(z)x, w̃ = y + α(z)xµ◦ , if µ◦ ∈ n \ {1}, (c) ṽ = α(z)x+ β(z)y, w̃ = −β(z)x+ α(z)y, if λ1,2(ξ◦) are complex. note that λ(0) = λ◦, µ(0) = µ◦ in the cases (a), (b). in the case (c), α(0), β(0) are respectively the real and imaginary parts of λ1,2(ξ◦). lemmas 2.2, 2.3 can be found in [17, 18, 31]. in the cases (a), (b), (c) enumerated in lemma 2.3, consider three families of local diffeomorphisms (rn, 0)→ (rn, 0) given by the following changes of the variables x, y: (a) x 7→ xϕ, y 7→ yϕλ(z), (2.9) (b) x 7→ xϕ, y 7→ yϕµ(z) + α(z)µ(z)xµ◦ ϕµ(z) − ϕµ◦ µ(z)− µ◦ , z 6= 0, y 7→ yϕµ(z) + α(z)µ(z)xµ◦ϕµ◦ lnϕ, z = 0, (2.10) with all smooth strictly positive functions ϕ = ϕ(x, y, z), and (c) x 7→ eα(z)ϕ(x cos β(z)ϕ+ y sin β(z)ϕ), y 7→ eα(z)ϕ(y cos β(z)ϕ− x sin β(z)ϕ), (2.11) with all smooth functions ϕ = ϕ(x, y, z). lemma 2.4: the families of diffeomorphisms (2.9), (2.10), (2.11) consist of symmetries of the direction field (2.8) in the cases (a), (b), (c), respectively. in other words, every diffeomorphism from the families (2.9), (2.10), (2.11) sends vector field (2.8) into a parallel vector field¶ in the cases (a), (b), (c), respectively. it is worth observing that families (2.9), (2.10), (2.11) are subgroups of the whole group of symmetries of the direction field (2.8) in the cases (a), (b), (c), respectively, but non of them coincide with the whole group. for instance, the family (2.9) does not contain the linear changes x 7→ c1x, y 7→ c2y with arbitrary constants c1,2 6= 0. the proof of lemma 2.4 is by direct calculations and omitted. now we can get the main result of the paper. theorem 2.2: let ξ◦ ∈ γ be an impasse point of the system (1.1) satisfying the above conditions. in addition, in the cases (a), (b), we assume that the eigenvectors with λ1,2(ξ◦) are transversal to the ¶two vector fields are parallel, if one of them is obtained from another by multiplication by a non-vanishing smooth function. in other words, two vector fields are parallel, if the corresponding direction fields coincide. copyright © 2020 assa. adv syst sci appl (2020) 124 a.m. kotyukov, s.o. nikanorov, n.g. pavlova criminant γ. then for any integer k ≥ 1, the germ of system (1.1) is ck-equivalent to the germ (x+ y)ψx′ = ṽ, (x+ y)ψy′ = w̃, z′1 = 0, . . . , z′n−2 = 0, (2.12) at the origin, where ṽ, w̃ are defined in lemma 2.3 and ψ = ψ(x, y, z) is a c∞-smooth function, ψ(0) 6= 0. under the same conditions, the germ of system (1.1) is orbitally ckequivalent to (2.12) with the same ṽ, w̃ and ψ ≡ 1. moreover, if reλ1,2(ξ◦) have the same sign, the above equivalences hold true with k =∞. proof by lemma 2.3, there exist a ck-smooth local diffeomorphism f : (ξ1, . . . , ξn) 7→ (x, y, z), z = (z1, . . . , zn−2), that sends the point ξ◦ to 0 and brings the germ of the direction field (2.6) at ξ◦ to the orbital normal form (2.8). then f transforms the corresponding direction field (2.5) into the form ẋ = ṽ, ẏ = w̃, żi = 0, i = 1, . . . , n− 2, ṫ = aṽ + bw̃, (2.13) where ṽ, w̃ are defined in 2.3 and a, b are smooth functions on the variables x, y, z. the criminant of the system (1.1) corresponding to (2.13) is given by the equation aṽ + bw̃. the condition that the eigenvectors with λ1,2(ξ◦) are transversal to the criminant yields a(0) 6= 0, b(0) 6= 0. let us show that there exists a diffeomorphism of the families (2.9), (2.10), (2.11) in the cases (a), (b), (c), respectively, that convert the criminant γ into a hyperplane, for instance, the hyperplane x+ y = 0. consider that case (a). then ṽ = x, w̃ = λ(z)y. by lemma 2.4, any diffeomorphism of the family (2.9) transforms the direction field (2.13) into ẋ = x, ẏ = λ(z)y, żi = 0, i = 1, . . . , n− 2, ṫ = axϕ+ bλ(z)yϕλ(z) = aϕ (x+ a−1bλ(z)ϕλ(z)−1y). (2.14) without loss of generality, assume that a−1(0)b(0)λ(0) > 0. otherwise one can make the change variables y 7→ −y, which preserves the first n− 1 components of the field (2.14) and brings its last component to the desirable form. thus, it suffices to establish the existence of a positive smooth function ϕ satisfying the equation a−1(xϕ, yϕλ(z), z) b(xϕ, yϕλ(z), z)λ(z)ϕλ(z)−1 = 1. (2.15) the derivative of the left-hand side of (2.15) by ϕ is not equal to zero. by the implicit function theorem, a function ϕ with the desirable properties exists in a neighborhood of the origin. the diffeomorphism (2.9) with ϕ founded above preserves the first n− 1 components of the direction field (2.14) and transforms its last component into ṫ = ψ(x, y, z) (x+ y), where ψ is a smooth function non-vanishing at the origin. the corresponding system (1.1) can be written in the form (x+ y)ψx′ = x, (x+ y)ψy′ = λ(z)y, z′1 = 0, . . . , z′n−2 = 0. (2.16) finally, if we deal with the orbital equivalence, one can make the change of the independent variable: t 7→ τ , where the new and old variables are connected with the differential equation dt dτ = ψ(x, y, z). copyright © 2020 assa. adv syst sci appl (2020) local normal forms of differential systems 125 the geometric sense this equation is that we make a parameterization integral curves of system (2.16), which depends on a point on every curve, so that the function ψ in (2.16) becomes identically 1. the case (b) is similar to (a) and omitted. consider that case (c). it is easy to see that (α− β)a− (α + β)b, (α + β)a+ (α− β)b do not simultaneously vanish at the origin. without loss of generality, we assume that the latter functions does not vanish at the origin. by lemma 2.4, any diffeomorphism of the family (2.11) transforms the direction field (2.13) with ṽ = α(z)x+ β(z)y, w̃ = −β(z)x+ α(z)y into the direction field whose first n− 1 components coincides with those in (2.13) and the nth component is ṫ = xeαϕ((αa− βb) cos βϕ− (αb+ βa) sin βϕ)+ yeαϕ((αa− βb) sin βϕ+ (αb+ βa) cos βϕ). thus, it suffices to establish the existence of a smooth function ϕ such that (αa− βb) cos βϕ− (αb+ βa) sin βϕ = (αa− βb) sin βϕ+ (αb+ βa) cos βϕ, (2.17) where α, β are given functions on z, while a, b depend on ϕ: a = a(eαϕ(x cos βϕ+ y sin βϕ), eαϕ(−x sin βϕ+ y cos βϕ), z), b = b(eαϕ(x cos βϕ+ y sin βϕ), eαϕ(−x sin βϕ+ y cos βϕ), z). equation (2.17) can be transformed to the equivalent form tan(βϕ) = (α− β)a− (α + β)b (α + β)a+ (α− β)b (2.18) the existence of a function ϕ satisfying equation (2.18) follows from the implicit function theorem. the remaining part of the proof in the case (c) repeats the reasonings carried out in the case (a). acknowledgements this paper was written with the financial support of rfbr (grants no. 19-01-00080, no. 20-01-00610). theorem 2.2 was obtained by the first author with the financial support of the russian science foundation (project no. 20-11-20131). references 1. brayton, r. & moser, j. 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(1992) typical singularities of differential 1-forms and pfaffian equations. math. monogr., 113, providence, rhode island: amer. math. soc. copyright © 2020 assa. adv syst sci appl (2020) introduction main results adv syst sci appl 2021; 01:139–149 published online at https://ijassa.ipu.ru. application of the minimum principle of a tikhonov smoothing functional in the problem of processing thermographic data eugeniy laneev1∗, natalia chernikova1, obaida baaj1 1peoples’ friendship university of russia, moscow, russia abstract: the paper considers a method for correcting thermographic images. mathematical processing of thermograms is based on the analytical continuation of the stationary temperature distribution as a harmonic function from the surface of the object under study to the heat sources. the continuation is performed by solving an ill posed mixed problem for the laplace equation in a cylindrical region of rectangular cross-section. the cylindrical area is bounded by an arbitrary surface and plane. the cauchy conditions are set on the surface-the boundary values of the desired function and its normal derivative. inhomogeneous conditions of the first kind are set on the side faces of the cylinder. the problem is the inverse of the corresponding mixed problem for the poisson equation. in this paper, an approximate solution of the problem is obtained that is stable with respect to the error in the cauchy data and inhomogeneity in the boundary conditions. in the course of constructing an approximate solution, the problem is reduced to the fredholm integral equation of the first kind, which is solved using the minimum smoothing functional principle. the convergence of the approximate solution of the problem is proved when the regularization parameter is matched to the error in the data. keywords: termogram, ill-posed problem, inverse problem, cauchy problem for the laplace equation, integral equation of the first kind, tikhonov regularization method 1. introduction digital technologies have penetrated into all branches of human activity and one of the urgent problems is to improve the quality and information content of representations of research results, in particular, the quality of images obtained from measurement data, through their mathematical (digital) processing. this applies, for example, to images obtained by thermal imaging methods using a thermal imager that registers thermal electromagnetic radiation from the surface of the object under study in the infrared range. in particular, in medicine, thermal imaging has become an effective means of early diagnostics [1]. the image on the thermogram, which is a map of the temperature distribution on the surface of the patient’s body, makes it possible to assess functional abnormalities in the state of his internal organs. at the same time, the image on the thermogram in some cases turns out to be somewhat distorted due to the processes of thermal conductivity and heat exchange. the paper proposes a method for correcting the image on a thermogram within a certain mathematical model. as a corrected thermogram, the image of the temperature distribution on the plane near the density of heat sources is considered as more accurately transmitting the image of heat sources. it is proposed to obtain this distribution as a result of the continuation (similar to the continuation of gravitational fields in geophysics problems [2]) of the temperature distribution from the surface from which the original thermogram is taken. the continuation is obtained by solving ∗corresponding author: elaneev@yandex.ru 140 e. laneev, n. chernikova, o. baaj the inverse problem to a certain mixed boundary value problem for the poisson equation. the considered inverse problem is ill-posed, since small errors in the initial data (the initial thermogram) may correspond to significant errors in the solution of the inverse problem. to construct its stable approximate solution, we use the tikhonov regularization method [3], based on optimization methods [4]. 2. statement of the problem let’s consider a physical and then a mathematical model, within which we will set the inverse problem. the physical model is a homogeneous heat-conducting body in the form of a rectangular cylinder, bounded by the surface s and containing heat sources with a time-independent density function that create a stationary temperature distribution in the body. we associate the density function of heat sources with the object under study. we assume that a given temperature distribution is maintained on the side faces of the cylinder, and on the surface s there is a convective heat exchange with the external environment of temperature u0, described by newton’s law, according to which the heat flux density at a point on the surface is directly proportional to the temperature difference inside and outside. let’s move on to the mathematical model. in the cylinder of rectangular cross section d∞ = {(x, y, z) : 0 < x < lx, 0 < y < ly, −∞ < z <∞} ⊂ r3 consider a cylindrical domain d(f,∞) = {(x, y, z) : 0 < x < lx, 0 < y < ly, f (x, y) < z <∞}, (2.1) bounded by the surface s = {(x, y, z) : 0 < x < lx, 0 < y < ly, z = f (x, y) < h}. (2.2) let γ be the sum of side faces of the domain d(f,∞). in the domain d(f,∞) consider the following mixed boundary value problem for the laplace equation ∆u(m) = ρ(m), m ∈ d(f,∞), ∂u ∂n ∣∣∣ s = h(u0 − u) ∣∣∣ s , u|γ = f1, u is bounded when z →∞. (2.3) the problem (2.3) corresponds to the steady-state temperature distribution created with heat sources of the distribution density function ρ, on the surface s – the third boundary condition is set, corresponding to convective heat exchange with the external environment of temperature u0 with the coefficient h, on the boundary γ the temperature is set as a function f1. we assume that the density carrier ρ is located in the domain z > h. we also assume that the functions ρ, f1 are such that the solution of the problem (2.3) exists in c2(d(f,∞)) ⋂ c1(d(f,∞)). in particular, the solution of the problem (2.3) gives the boundary value u|s . now let’s set the inverse problem. inverse problem 1. let within the model (2.3) be set the following functions f = u|s, f1 = u|γ. (2.4) we need to find a continuous function ρ. copyright c© 2021 assa. adv syst sci appl (2021) application of the minimum principle... 141 note that density recovery is associated with the same difficulties as solving the inverse potential problem [5], for which significant restrictions on uniqueness classes are known. therefore, to solve the inverse problem, we apply the approach [2] used in geophysics problems. source of information about the density of ρ we will consider the function u|z=h on the plane z = h, that is closer to the density carrier ρ than the surface s. since the carrier of the function ρ by the condition of the problem (2.3) is located in the domain z > h, then the solution of the problem (2.3) in the domain d(f,h) = {(x, y, z) : 0 < x < lx, 0 < y < ly, f (x, y) < z < h} (2.5) satisfies the laplace equation. the sum of side faces of the domain d(f,h) denote by γh . instead of the inverse problem 1, we will solve the following inverse problem inverse problem 2. let within the model (2.3) be set the following functions f = u|s, f1 = u|γh . (2.6) we need to find a solution u to the boundary value problem in the domain d(f,h) ∆u(m) = 0, m ∈ d(f,h), u|s = f, ∂u ∂n ∣∣∣ s = h(u0 − f) ∣∣∣ s , u|γh = f1. (2.7) we will consider the function u|z=h as a source of information about the density ρ. we assume that the functions f, f1 in (2.6), (2.7) are taken from the set of solutions of the direct problem (2.3), so the solution of the inverse problem exists in c2(d(f,h)) ⋂ c1(d(f,h)). we note that in the problem (2.7) on the surface s of the form (2.2), cauchy conditions are set, that is, the boundary values f of the desired function u and the values of its normal derivative are set, so the problem (2.7) has a unique solution. the boundary z = h of the domain d(f,h) of the form (2.5) is free and, thus, the problem (2.7) is unstable with respect to data errors, i.e. it is ill-posed. we will construct an explicit representation of the exact solution of the problem (2.7). 3. exact solution of the problem let’s construct an exact solution of the problem (2.7), following the scheme [6, 7]. consider the source function ϕ(m,p ) of the dirichlet problem in the cylinder d∞: ∆u(p ) = ρ(p ), p ∈ d∞, u|x=0,lx = 0, u|y=0,ly = 0, u→ 0 when |z| → ∞, (3.8) i.e., ϕ(m,p ) = 1 4πrmp +w (m,p ), (3.9) where rmp is the distance between points m and p and w (m,p ) is a harmonic function of point p . the source function can be obtained by the reflection method as a sum of point source functions with period 2lx in the variables x and period 2ly in the variable y, ϕ(m,p ) = 1 4π ∞∑ n,m=−∞ ( 1 r1,nm − 1 r2,nm − 1 r3,nm + 1 r4,nm ) , (3.10) copyright c© 2021 assa. adv syst sci appl (2021) 142 e. laneev, n. chernikova, o. baaj where r1,nm = [(xm − xp + 2lxn)2 + (ym − yp + 2lym)2 + (zm − zp )2]1/2, r2,nm = [(xm + xp + 2lxn)2 + (ym − yp + 2lym)2 + (zm − zp )2]1/2, r3,nm = [(xm − xp + 2lxn)2 + (ym + yp + 2lym)2 + (zm − zp )2]1/2, r4,nm = [(xm + xp + 2lxn)2 + (ym + yp + 2lym)2 + (zm − zp )2]1/2, and, in particular, r1,00 = rmp . let m ∈ d(f,h). then, applying the green formulas in the domain d(f,h) to the function u(p ), i.e., the solution of problem (2.7), and to the functions 1 4πrmp and w (m,p ) in (3.9), we obtain u(m) = ∫ ∂d(f,h) [∂u ∂n (p ) 1 4πrmp − u(p ) ∂ ∂np 1 4πrmp (m,p ) ] dσp , m ∈ d(f,h) (3.11) and 0 = ∫ ∂d(f,h) [∂u ∂n (p )w (m,p )− u(p ) ∂w ∂np (m,p ) ] dσp , m ∈ d(f,h). (3.12) summing (3.11) and (3.12) taking into account (3.9) we obtain u(m) = ∫ ∂d(f,h) [∂u ∂n (p )ϕ(m,p )− u(p ) ∂ϕ ∂np (m,p ) ] dσp , m ∈ d(f,h). (3.13) given homogeneous boundary conditions for ϕ and inhomogeneous ones for u on the side faces γh of the cylindrical domain d(f,h), we obtain u(m) = ∫ s [ h(u0 − f(p ))ϕ(m,p )− f(p ) ∂ϕ ∂np (m,p ) ] dσp− − ∫ γh [ f1(p ) ∂ϕ ∂np (m,p ) ] dσp + ∫ π(h) [∂u ∂n (p )ϕ(m,p )− u(p ) ∂ϕ ∂np (m,p ) ] dσp , where π(h) = {(x, y, z) : 0 < x < lx, 0 < y < ly, z = h}. (3.14) in the domain zm < h , we introduce the notation φ(m) = ∫ s [ h(u0 − f(p ))ϕ(m,p )− f(p ) ∂ϕ ∂np (m,p ) ] dσp − ∫ γh [ f1(p ) ∂ϕ ∂np (m,p ) ] dσp , (3.15) v(m) = ∫ π(h) [∂u ∂n (p )ϕ(m,p )− u(p ) ∂ϕ ∂np (m,p ) ] dσp , zm < h. (3.16) then we obtain the solution of the problem (2.7) in the form u(m) = v(m) + φ(m), m ∈ d(f,h), (3.17) copyright c© 2021 assa. adv syst sci appl (2021) application of the minimum principle... 143 where the function φ is calculated from known functions f and f1. if the solution of the problem (2.7) exists, then the function v of the form (3.16), harmonic in the domain d(−∞, h) = {(x, y, z) : 0 < x < lx, 0 < y < ly, −∞ < z < h}, can be represented in d(f,h) ⊂ d(−∞, h) according to (3.17) in the form v = u− φ and then it may be defined on the boundary of π(h) as a continuous function v |z=h= u |z=h −φ |z=h= vh . (3.18) thus, the function v can be viewed as a solution of the problem ∆v(m) = 0, m ∈ d(−∞, h), v|z=h = vh , v|x=0,lx = 0, v|y=0,ly = 0, v → 0 when z → −∞, (3.19) and the function v can be expressed in terms of the function vh by using the green function of problem (3.19) as follows: v(m) = − ∫ π(h) ∂g ∂np (m,p )vh(p )dxpdyp , m ∈ d(−∞, h), (3.20) where ∂g ∂np (m,p ) ∣∣∣ p∈π(h) = ∂g ∂zp (m,p ) ∣∣∣ p∈π(h) = = − 4 lxly ∞∑ n,m=1 exp { knm(−h + zm) } sin πnxm lx sin πmym ly sin πnxp lx sin πmyp ly , (3.21) knm = π ( n2 l2x + m2 l2y )1/2 . (3.22) it follows that if problem (2.7) has a solution, then (3.20) implies that the function v in the domain d(−∞, h) can be represented as the fourier series v(m) = v(x, y, z) = − ∞∑ n,m=1 (ṽh)nm exp { knm(z −h) } sin πnx lx sin πmy ly , (3.23) (ṽh)nm = 4 lxly lx∫ 0 ly∫ 0 vh(x′, y′) sin πnx′ lx sin πmy′ ly dx′dy′, (3.24) of a complete system of functions{ sin πnx lx sin πmy ly }∞ n,m=1 . (3.25) the series (3.23) uniformly converges in the domain d(−∞, h − ε) for any ε > 0, because∣∣(ṽh)nm exp { knm(z −h) } sin πnx lx sin πmy ly ∣∣ 6 ∣∣(ṽh)nm ∣∣ exp { − εknm } . copyright c© 2021 assa. adv syst sci appl (2021) 144 e. laneev, n. chernikova, o. baaj thus, it follows from the representation (3.17) of the solution of problem (2.7) and from (3.23) that, to obtain an explicit expression for the exact solution of problem (2.7), it suffices to express the function vh (3.18) in terms of the prescribed functions f and f1. let us show that the function vh satisfies a fredholm integral equation of the first kind. let m ∈ d(−∞, f ), where d(−∞, f ) = {(x, y, z) : 0 < x < lx, 0 < y < ly, −∞ < z < f (x, y)}. applying the green formula in the domain d(f,h) to the function u(p ), i.e., a solution of problem (2.7), and to a function ϕ(m,p ) of the form (3.9), we, by analogy with (3.11), (3.12), and (3.13), obtain the relation 0 = ∫ ∂d(f,h) [∂u ∂n (p )ϕ(m,p )− u(p ) ∂ϕ ∂np (m,p ) ] dσp , m ∈ d(−∞, f ). from this, with regard to the homogeneous boundary conditions for the function ϕ and to the inhomogeneous boundary conditions for u and notation (3.15) and (3.16), we obtain v(m) = −φ(m), m ∈ d(−∞, f ). (3.26) let a < min (x,y) f (x, y) andm ∈ π(a), where π(a) is a domain of the form (3.14) for z = a. then, by formulas (3.26) and (3.20), we obtain the integral equation of the first kind∫ π(h) ∂g ∂np (m,p )vh(p )dxpdyp = φ(m), m ∈ π(a). (3.27) from the equation (3.27) taking into account the decomposition (3.21) for zm = a we obtain the following relations between the fourier coefficients of the unique solution vh of this integral equation and the fourier coefficients of its right-hand side: − (ṽh)nm exp { − knm(h − a) } = φ̃nm(a), (3.28) where the φ̃nm(a) are the fourier coefficients of the function φ(m)|m∈π(a), φ̃nm(a) = 4 lxly ∫ π(a) φ(x, y, a) sin πnx lx sin πmy ly dxdy. (3.29) note that formula (3.28) characterizes the decrease in the fourier coefficients φ̃nm(a) with increasing n and m if, for the functions f and f1, there exists a solution of problem (2.7) and hence a function vh defined by (3.18). we express the fourier coefficients (ṽh)nm, substitute them into the series (3.23), and obtain the function v in the domain d(−∞, h) : v(m) = − ∞∑ n,m=1 φ̃nm(a) exp {knm(z − a)} sin πnx lx sin πmy ly , m(x, y, z) ∈ d(−∞, h). (3.30) the series (3.30), just as the series (3.23), uniformly converges in the domaind(−∞, h − ε) for any ε > 0 if there exists a solution of problem (2.7) for the given functions f and f1. formula (3.17), where the functions v and φ are given by (3.30) and (3.15), respectively, gives an explicit expression for the solution of problem (2.7). copyright c© 2021 assa. adv syst sci appl (2021) application of the minimum principle... 145 4. approximate solution of the problem let the functions f and f1 in problem (2.7) be given with an error; i.e., let, instead of them, functions f δ and f δ1 be given such that ‖f δ − f‖l2(s) 6 δ, ‖f δ1 − f1‖l2(γh) 6 δ. we construct an approximate solution of problem (2.7) converging to the exact solution as δ → 0. here the function φ defined by formula (3.15) can be obtained approximately as φδ(m) = ∫ s [ h(u0 − f δ(p ))ϕ(m,p )− f δ(p ) ∂ϕ ∂np (m,p ) ] dσp− − ∫ γh [ f δ1 (p ) ∂ϕ ∂np (m,p ) ] dσp . (4.31) we apply the cauchy-schwarz inequality to the difference of functions (4.31) and (3.15) for m ∈ π(a), a < min (x,y) f (x, y), and obtain an estimate of the right-hand side of integral equation (3.27), |φδ(m)− φ(m)| 6 h max m∈π(a) ( ∫ s ϕ2(m,p )dσp )1/2‖f δ − f‖l2(s)+ + max m∈π(a) ( ∫ s [ ∂ϕ ∂np (m,p ) ]2 dσp )1/2‖f δ − f‖l2(s)+ + max m∈π(a) ( ∫ γh [ ∂ϕ ∂np (m,p ) ]2 dσp )1/2‖f δ1 − f1‖l2(γh) 6 cδ. (4.32) for an approximate solution of eq. (3.27), we take the extremal of the tikhonov functional [3, p. 68] with zero-order stabilizer, mα[w] = ∥∥∫ s ∂g ∂n wdσ − φδ ∥∥2 l2(π(a)) + α‖w‖2 l2(π(h)), α > 0, (4.33) where π(a) and π(h) are domains defined by formula (3.14). the extremal can be obtained as a solution of the euler equation for the functional (4.33) which, in the fourier coefficients of the function w, has the form exp {−2knm(h − a)} w̃nm + αw̃nm = − exp {−knm(h − a)} φ̃δ nm(a), where φ̃δ nm(a) = 4 lxly ∫ π(a) φδ(x, y, a) sin πnx lx sin πmy ly dxdy (4.34) are the fourier coefficients of the function φδ(m)|m∈π(a). solving the equation for the fourier coefficients of the extremal and substituting the extremal wδα for vh into representation (3.23), we obtain an approximation vδα to the function copyright c© 2021 assa. adv syst sci appl (2021) 146 e. laneev, n. chernikova, o. baaj v in the domain d(−∞, h), vδα(m) = − ∞∑ n,m=1 φ̃δ nm(a) exp{knm(zm − a)} 1 + α exp{2knm(h − a)} sin πnxm lx sin πmym ly . (4.35) note that coefficients of the series (4.35) differ from corresponding coefficients of the series (3.30) in the factor (1 + α exp{2knm(h − a)})−1, and the series (4.35) converges uniformly. according to the representation (3.17), we obtain an approximate solution of problem (2.7) in the form uδα(m) = vδα(m) + φδ(m), m ∈ d(f,h), (4.36) where vδα and φδ are the functions defined by formulas (4.35) and (4.31) respectively. theorem. assume that there exists a solution of problem (2.7). then, for any α = α(δ) > 0 such that α(δ)→ 0 and δ/ √ α(δ)→ 0 as δ → 0, the function uα(δ) of the form (4.36) uniformly converges as δ → 0 to the exact solution of problem (2.7) on any compact k ⊂ d(f,h). proof. on any compactk ⊂ d(f,h), according to representations (4.36) and (3.17), we estimate the difference |uδα − u| 6 |vδα − v|+ |φδ − φ|. (4.37) obviously, there is ε > 0 such that k ⊂ d(−∞, h − ε). for the modul of the difference vδα − v in the domain d(−∞, h − ε) we obtain |vδα − v| 6 |vδα − vα|+ |vα − v|, (4.38) where vα is a function of the form (4.35) for exact functions f and f1, vα(m) = − ∞∑ n,m=1 φ̃nm(a) exp{knm(zm − a)} 1 + α exp{2knm(h − a)} sin πnxm lx sin πmym ly . (4.39) to estimate the difference vδα − vα on the right-hand side in inequality (4.38) for zm < h − ε, we use inequality (4.32) |vδα(m)− vα(m)| 6 ∣∣∣∣ ∞∑ n,m=1 exp{knm(zm − a)} 1 + α exp{2knm(h − a)} ∣∣∣∣ · 4 max p∈π(a) ∣∣φδ(p )− φ(p ) ∣∣ 6 6 c1δ ∞∑ n,m=1 exp{knm(h − ε− a)} 1 + α exp{2knm(h − a)} 6 6 c1δmax x [ ex 1 + αe2x ] ∞∑ n,m=1 exp{−knmε} 6 c2 δ√ α . (4.40) we estimate the difference vα − v in inequality (4.38) for zm < h − ε, |vα − v| 6 ∞∑ n,m=1 α exp{2knm(h − a)} exp{knm(h − ε− a)} 1 + α exp{2knm(h − a)} ∣∣φ̃nm(a) ∣∣. copyright c© 2021 assa. adv syst sci appl (2021) application of the minimum principle... 147 from this, using (3.28) and applying the cauchyschwarz inequality, we obtain |vα − v| = ∞∑ n,m=1 α exp{2knm(h − a)} exp{−knmε} 1 + α exp{2knm(h − a)} ∣∣ ˜(vh)nm ∣∣ 6 6 [ ∞∑ n,m=1 ( α exp{2knm(h − a)} 1 + α exp{2knm(h − a)} )2 exp{−2knmε} ]1/2 · 2√ lxly ||vh ||l2 . since the series depending on the parameter α is majorized by the converging numerical series with coefficients exp{−2εknm}, it is possible to pass to the limit in α, and hence |vα − v| → 0 when α→ 0. (4.41) it follows from (4.38), (4.40), (4.41), and the assumptions of the theorem that |vδα(δ) − v| → 0 when δ → 0. (4.42) the second difference on the right-hand side in inequality (4.37) can be estimated by analogy with (4.32). we apply the cauchy-schwarz inequality to this difference for m ∈ k, and obtain |φδ(m)− φ(m)| 6 hmax m∈k (∫ s ϕ2(m,p )dσp )1/2 ‖f δ − f‖l2(s)+ + max m∈k (∫ s [ ∂ϕ ∂np (m,p ) ]2 dσp )1/2 ‖f δ − f‖l2(s)+ + max m∈k ( ∫ γh [ ∂ϕ ∂np (m,p ) ]2 dσp )1/2 ‖f δ1 − f1‖l2(γh) 6 c3δ. from this relation, inequality (4.37), and formula (4.42) the assertion of the theorem follows. 5. numerical solution of the problem the effectiveness of the proposed method for solving the problem (2.7) is shown in the following model example. in the problem (2.3), let the surface s be the plane π(0), f1 = u0 = 24, h = 0.5, lx = 30, ly = 30, h = 1.4, and the function ρ corresponds to three point sources at points in the plane π(h) : (x1, y1) = (8.8), (x2, y2) = (10.8), (x3, y3) = (10.10). the boundary value of the solution of the model problem (2.3) in this case has the form f(x, y) = u0 + ∞∑ n,m=1 3∑ i=1 e−knmh knm + h sin πnxi lx sin πmyi ly sin πnx lx sin πmy ly , (5.43) where knm is calculated using the formula (3.22). to set the inverse problem (2.7), we consider that the function f, calculated by the formula (5.43), a known function. also f1 = u0 = 24, h = 0.5, lx = 30, ly = 30, h = 1.4 are known. copyright c© 2021 assa. adv syst sci appl (2021) 148 e. laneev, n. chernikova, o. baaj fig. 5.1. the initial data of the inverse problem (initial thermogram) fig. 5.2. the result of restoring the thermogram u|z=h to solve the inverse problem (2.7), we use the formulas (4.36), (4.35), (4.34), (4.31). in the formula (4.31) we use the representation for the fundamental solution ϕ(m,p ) = 2 lxly ∞∑ n,m=1 e−knm|zm−zp | knm sin πnxm lx sin πmym ly sin πnxp lx sin πmyp ly (5.44) when zm = a, zp = 0. the fourier coefficients in the formula (4.34) are calculated without calculating the function φ, similarly to [8]. when using the formula (4.34), integration is performed under the sign of the integral in (4.31) and under the sign of the sum in (5.44). taking into account the orthogonality of the system of functions (3.25), the calculation formulas for calculating the fourier coefficients φnm are significantly simplified. to obtain a numerical result, the problems (2.3), (2.7) are discretized. a uniform grid of 91x91 points is introduced on the rectangles π(a), a = −0.5 and π(h). the hamming algorithm [9, p.83] is used to sum discrete fourier series. the calculation results are shown in fig.5.1 and fig.5.2. fig.5.1 shows the initial data of the inverse problem – the function f calculated from the discrete analog of the formula (5.43). the relative magnitude of the added error is 0.28%. the three sources are perceived as a single whole. fig.5.2 shows the result of restoring the u|z=h function using the formulas (4.36), (4.35), (4.34), (4.31). three sources are clearly visible. regularization parameter α = 10−8. with the regularization parameter α = 0, the solution is destroyed. copyright c© 2021 assa. adv syst sci appl (2021) application of the minimum principle... 149 6. conclusion the inverse problem (2.7) and its stable solution can be used for mathematical processing of thermograms, in particular, in medicine [1], in order to correct the image. as already mentioned, a thermogram obtained using a thermal imager transmits an image of the structure of heat sources inside the body approximately. refinement of the image on the thermogram can be performed within the framework of the problem (2.7). in this case, the f function will be associated with the original thermogram, and the uh function will be considered as the result of processing the thermogram. since the function u|z=h represents the temperature distribution on a plane closer to the heat sources under study than the original surface s, we can expect a more accurate reproduction of the source image on the calculated thermogram u|z=h . the results of calculations, performed on the model example, show the effectiveness of the proposed method and algorithm based on the formulas (4.36), (4.35), (4.34), (4.31), which can be used for processing thermographic images. acknowledgements the research is supported by the russian science foundation, project n 21-11-00064. references 1. ivanitskii g.r. (2006). teplovideniye v meditsine [thermovision in medicine], vestnik ran, 76(1), 44-53, [in russian]. 2. tihonov a.n., glasko v.b., litvinenko o.k. & melihov v.r. (1968). o prodolzhenii potentsiala v storonu vozmushchayushchih mass na osnove metoda regulyarizatsii [on the continuation of the potential towards disturbing masses based on the regularization method], izvestiya an sssr. fizika zemli, 1, 30-48, [in russian]. 3. tihonov a.n. & arsenin v.ya. (1979). metody resheniya nekorrektnyh zadach [methods for solving ill-posed problems]. moscow, russia: nauka, [in russian]. 4. arutyunov a., jacimovic v.& pereira f. (2003). isecond order necessary conditions for optimal impulsive control problems ,journal on dynamical and control systems, 9(1), 131-153. 5. prilepko a.i. (1973). inverse problems of potential theory (elliptic, parabolic, hyperbolic, and transport equations), math. notes, 14(5), 990-996. 6. laneev, e.b. (2018). construction of a carleman function based on the tikhonov regularization method in an ill-posed problem for the laplace equation, differential equations, 54(4), 476–485, https://doi.org/10.1134/s0012266118040055 7. chernikova n.y., laneev e.b., muratov m.n.& ponomarenko e.y. (2020). on an inverse problem to a mixed problem for the poisson equation. in: pinelas s., kim a., vlasov v. (eds) mathematical analysis with applications. concord-90 2018. springer proceedings in mathematics & statistics, vol. 318, pp. 141–146. springer, cham. https://doi.org/10.1007/978-3-030-42176-2 14 8. laneev, e.b., mouratov, m.n. & zhidkov, e.p. (2008). discretization and its proof for numerical solution of a cauchy problem for laplace equation with inaccurately given cauchy conditions on an inaccurately defined arbitrary surface, physics of particles and nuclei letters, 5(3), 164–167. 9. hamming r.w. (1962). numerical methods for scientists and engineers. new york, usa: mcgraw-hill book company. copyright c© 2021 assa. adv syst sci appl (2021) introduction statement of the problem exact solution of the problem approximate solution of the problem numerical solution of the problem conclusion adv syst sci appl 2021; 02:104–116 published online at https://ijassa.ipu.ru. achieving angular superresolution of control and measurement systems in signal processing boris lagovsky1, evgeny rubinovich2* 1russian technological university (mirea), moscow, russia 2trapeznikov institute of control sciences, russian academy of sciences, moscow, russia abstract: algebraic methods of processing data obtained by control and measurement systems to achieve angular superresolution are presented. the efficiency of using the methods in the formation of approximate images of objects at low signal-to-noise ratios is shown. the results of numerical experiments demonstrate the possibility of obtaining images with a resolution exceeding the rayleigh criterion by 3-10 times. the robustness of the solutions obtained by the methods of algebraic exceeds many well-known approaches. the relative simplicity of the presented methods allows the use of inexpensive computing devices and perform real-time measurement processing. keywords: angular superresolution, inverse problem, integral equation, rayleigh criterion 1. introduction an important modern problem of improving control and measurement systems is increasing their information content based on new signal processing methods. one of the directions of its solution is to increase the angular resolution of angle-measuring systems, which makes it possible to detail the image of the object under study. in this regard, the problem of restoring the image of the object with an angular superresolution becomes highly significant. this paper is an extension of work in this direction originally presented in ieee 2020 7th international conference on control, decision and information technologies (codit-2020) [1]. due to the importance of the problem, hundreds of publications in many countries have already been devoted to the problems of achieving angular superresolution. we will highlight many articles that are general [2–5]. currently popular methods are music [6–8], esprit [9], the deconvolution method [10], the maximum entropy method [11, 12], the borgiottilagunas method [13], the capon method [14], the maximum likelihood method [15] and others [16]. all these methods are not universal and are not always effective. they begin to work successfully and allow us to increase the effective angular resolution at a signal to noise ratio (snr) not lower than 20-25 db. the developed algebraic methods [17–19] differ favorably from those mentioned above in that they have significantly higher noise immunity. in addition, they are relatively simple, which significantly reduces signal processing time and allows the use of relatively simple computing devices. as a result, unlike many other methods, it is possible to apply them in real-time. ∗corresponding author: rubinvch@hotmail.com achieving angular superresolution of control and measurement systems 105 2. problem statement let us first consider the one-dimensional case. the signal u(α) received by the goniometer system when scanning the area under study is an integral transformation u(α) = ∫ ω f (α − φ) i(φ) dφ, (2.1) where f (α) is the directional pattern (dp) of the system, i(α) is the unknown angular distribution of the amplitude of the signal reflected (or emitted) by the object, ω is the angular sector, in which the object under study is located. the problem of reconstructing the image of the source i(α) with the highest possible angular resolution exceeding the rayleigh criterion, i.e. with super-resolution, is posed. dp and measurement data in the form of u(α) are considered known. mathematically, the search for the angular distribution i(α) is reduced to the approximate solution of fredholm integral equations of convolution type (2.1) concerning the function i(·). it is known that of the three conditions for the correctness of hadamard problems (existence of the solution, uniqueness of the solution, stability of the solution), the fredholm equation (2.1) does not satisfy the second and third conditions. thus, the problem under consideration belongs to the class of inverses and is incorrect. because of this, attempts to increase the angular resolution and exceed the rayleigh criterion generally lead to instabilities in the solutions and, as a result, to significant errors. the main obstacle to obtaining solutions with super-resolution is the random components present in the received signal. when obtaining a resolution within the rayleigh criterion, their influence is usually negligible but increases dramatically (exponentially) when trying to obtain a higher resolution. achieving super-resolution, as studies show, is possible, but up to a certain limit, determined by the snr, measurement accuracy, and accuracy of the dp assignment. for a comparative assessment of the quality of the approximate solution of the problem (2.1), three indicators are usually used by different methods. the first one is the achieved degree of exceeding the rayleigh criterion. the second indicator is the value of the displacement of the found positions of objects from the true ones since the positions of the resolved objects and their elements found during signal processing do not always exactly correspond to the actual directions to the radiation sources. when observing a single radiation source, the bias is usually close to zero. in the case of two or more sources, the offset can be different from zero and increase as you increase the reach of effective solutions. the third indicator is the snr, at which the specified degree of excess of the rayleigh criterion is achieved. it is theoretically known and has been confirmed in the course of numerical experiments that the solutions of inverse problems are by their nature very sensitive to the presence of random components in the source data. because of this, the first and third indicators can be combined. this is the dependence of the degree of excess of the rayleigh criterion on the snr. this indicator was considered the main one in the development of methods and algorithms for achieving superresolution. 3. algebraic methods of solution to solve the inverse problem (2.1), we propose methods and algorithms for digital signal processing, which can be called algebraic. they consist in finding solutions in the form of decompositions over the given sequences of functions gm(α) [20], which are orthonormal in copyright© 2021 assa. adv syst sci appl (2021) 106 b. lagovsky, e. rubinovich the region of the source location ω. i(α) = ∞∑ m=1 bm gm(α) � n∑ m=1 bm gm(α), (3.2) where bm are unknown expansion coefficients. by using (3.2), instead of (2.1), we get a decomposition of the useful signal u(α) obtained by scanning over a system of non-orthogonal functions χm(α) u(α) � n∑ m=1 bm χm(α), (3.3) χm(α) � ∫ ω f (α − φ) gm(φ) dφ. (3.4) all the bm necessary to obtain the solution (3.2) are found from the condition of minimizing the root-mean-square deviation of the right-hand side (3.3) from u(α). as a result, the search for the vector b of unknown coefficients bm is reduced to the solution of slae: v = gb, (3.5) where the elements of the vector v and matrix g are: vm = ∫ θ u(α) χm(α) dα, gnm = ∫ θ χn(α) χm(α) dα, (3.6) n,m = 1, 2, . . . ,n. thus, the search for an approximate solution of i(α) in the form of a finite system expansion of the selected functions allows us to parametrize the inverse problem and reduce its solution to the slae solution [17–19]. the principal feature of the resulting systems is their ill-conditioned, which is a consequence of the attempt to solve the inverse problem. the larger the chosen area of integration θ, the higher the stability of the obtained approximate solutions. however, the received signal always contains random components. for this reason, to obtain an adequate solution to the inverse problem (2.1), the level of the useful signal must significantly exceed the level of the random components. with the growth of θ, the snr decreases in the areas close to the boundary, which limits the area of integration θ. therefore, the integration region should be defined as the sector of angles within which the snr value is sufficient to obtain stable solutions. it should be noted that the necessary snr for obtaining solutions should be provided not directly at the reception of the signal but after the primary signal processing. so, the presented algebraic method and its variants make it possible, by consistently increasing the number of functions used in (3.2), i.e., by increasing the effective angular resolution, to approach the limit resolution for each problem to be solved. the iterative procedure for finding an approximate solution continues as long as it is possible to obtain a stable solution. comparing the algebraic method with others, it should be noted that it potentially allows one to obtain an exact solution to the inverse problem (2.1) if the chosen system of functions gm(α) in (3.2) provides an exact representation of the solution i(α) using a finite number of terms. it is easy to prove that such a choice of the system gm(α) for a known distribution i(α) is always possible. thus, the algebraic method turns out to be at least as good as any other known method for solving the inverse problem (2.1). the question of choosing the optimal system of functions gm(α) for the desired i(α) remains, however, open. it is known that a significant improvement in the quality of solutions to inverse problems can be achieved by using a priori information about the solution. the presence of such copyright© 2021 assa. adv syst sci appl (2021) achieving angular superresolution of control and measurement systems 107 information allows, in particular, to use it when optimizing the choice of the system of functions gm(α) for constructing a solution. preliminary information about the solution, in addition, is realized by using algebraic methods in the form of: choosing the location, size, and shape of the studied area of localization of the signal source ω, introducing additional conditions in the form of equations and inequalities that connect the coefficients of expansion over sequences of functions and thereby regularize the problem. fig. 3.1 shows the solution to the problem of image reconstruction of two closely spaced smoothly inhomogeneous signal sources (dash line 1) in the form of a distribution of the reflected signal amplitude at a low noise level. for illustration, curve 3 shows a signal received by an angle-measuring system with a beamwidth of θ0.5 in the angular region of the source location. when constructing the solution (3.3)-(3.6), we used preliminary information fig. 3.1. the restoration of a smoothly inhomogeneous signal source about the smoothly inhomogeneous nature of the amplitude angular distribution of the signal reflected by the source. this predestined the use of trigonometric functions as a system of functions gm(α) in (3.2). the resulting approximate solution (solid curve 2) allowed us to resolve the sources and almost accurately determine their angular position and shape. due to the oscillating nature of the system of functions chosen to represent the solution, false sources with small amplitudes have emerged. fig. 3.2 shows the solution to the problem of image reconstruction of four closely located identical signal sources (dash line 1), close to the point at a very small snr of 13 db. the resulting approximate solution (solid line 2) allowed us to resolve all four sources and determine their angular position with good accuracy. for illustration, curve 3 shows the signal received and processed by the goniometer system. the usual methods of obtaining superresolution mentioned earlier do not allow us to obtain adequate solutions at such a low snr level. satisfactory solutions with the resolution of all four objects, but somewhat worse quality, can be obtained even at even lower snr values. copyright© 2021 assa. adv syst sci appl (2021) 108 b. lagovsky, e. rubinovich fig. 3.2. the restoration of the images of point sources at low snr as already noted, with an increase in the number of functions gm(α) used in the decomposition, the conditionality numbers of matrices g in (3.5) and similar ones, they increase sharply (according to the exponential law) and the solutions become less stable. the dimension of the matrices can be reduced by selecting the functions used. the purpose of selection is to select only those functions from the selected system that best represent the solution. selection can be carried out in several ways: 1. based on the analysis of the signal spectrum and function spectra, followed by their selection with the spectra closest to the spectrum of the signal under study. 2. based on the analysis of the mutual correlation functions of the signal and the functions from the selected family. 3. based on the selection of functions from the used family with the highest values of the expansion coefficients in the solution representation. fig. 3.3 shows the results of restoring the source image at a very high noise level, comparable to the useful signal (snr = 6-8 db). the dashed curve is the original intensity distribution, the solid bold curve is the reconstructed image using six wavelets. for comparison, a solid thin polyline is shown as a reconstructed image using the same six wavelets in the absence of noise. the obtained solutions provided a 2-3 fold excess of the rayleigh criterion, and accuracy of source localization θ 0.5/8, and very high noise immunity. most of the known methods are designed to obtain solutions to one-dimensional problems. their generalization to two-dimensional problems significantly complicates the algorithms and increases the instability of solutions. in addition, the signal processing time increases dramatically. to obtain satisfactory results, it is sometimes necessary to use parallel processors. generalization of algebraic methods for solving one-dimensional problems to two-dimensional ones does not lead to a serious complication of the algorithms. we present the desired angular two-dimensional distribution i(α, ϕ) as a decomposition over a finite system of orthogonal functions gnm(α, ϕ) in the two-dimensional domain ω with copyright© 2021 assa. adv syst sci appl (2021) achieving angular superresolution of control and measurement systems 109 fig. 3.3. restoring the source image using the selected images mhat wavelets at snr = 6 db unknown coefficients. it is convenient to use separable systems, which are represented as the product of one-dimensional systems gnm(α, ϕ) = gn(α)gm(ϕ). then: i(α, ϕ) � n∑ n,m=1 bnm gn(α) gm(ϕ). (3.7) and the received signal by (2.1) is obtained in the form: u(α, ϕ) � n∑ n,m=1 bnm ψnm(α, ϕ), ψnm(α, ϕ) = ∫ ω f (α − α′, ϕ − ϕ′) gn(α′) gm(ϕ′) dα′ dϕ′, n, m = 1, 2, . . . ,n. (3.8) the coefficients bnm, which provide the minimum root-mean-square deviation in the twodimensional region θ of the synthesized signal (3.8) from the received one, are found as slae solutions: ∫ θ u(α, ϕ) ψ jk(α, ϕ) dα dϕ = = n∑ n,m=1 bm ∫ θ ψ jk(α, ϕ) ψnm(α, ϕ) dα dϕ, j, k = 1, 2, . . . ,n. (3.9) numerical experiments have shown that the solution time and stability of solutions to twodimensional problems in comparison with one-dimensional problems at the same level of copyright© 2021 assa. adv syst sci appl (2021) 110 b. lagovsky, e. rubinovich superresolution vary slightly [21, 22]. thus, algebraic methods allow us to parametrize oneand two-dimensional inverse problems, and to reduce the solutions of integral equations to the slae solution, sometimes with additional conditions [23, 25, 27]. fig. 3.4 shows an approximate solution of problems (3.7)-(3.9). the original source was defined as four identical objects with small angular coordinates, located at the corners of a square with a side of 0.9θ 0.5, which were not resolved by direct observation. the width f (α, ϕ)θ 0.5 was assumed to be the same at both corners. when constructing the solution, we used preliminary information about the object under study as consisting of a group of small-sized sources. as gnm(α, ϕ), functions close to gauss functions were chosen to represent the solution, the maxima of which were located in the centers of squares with sides ∆α, ∆ϕ equal to θ 0.5/n, into which the region was successively divided. the best solution with minimal errors was obtained at n = 3, i.e. when dividing the area ω into 9 squares. step ∆α corresponded to the desired angular resolution. as a result, all small-sized sources were resolved and their position was found with good accuracy. since the inverse incorrect problem was solved, small errors appeared in the form of false sources with a small amplitude, which should be ignored when analyzing solutions. fig. 3.4. restored image of a two-dimensional source copyright© 2021 assa. adv syst sci appl (2021) achieving angular superresolution of control and measurement systems 111 4. two-beam method further increase of the degree of achieved super-resolution is possible based of of more complex measuring and signal processing. in [24] considered the use of interpolation to find solutions with superresolution, in [19] described the use of regression methods for the analysis of measurement data in [28] using extrapolation methods, in [26] described the application of the developed technique in the use of uwb signals. consider another possibility to increase the degree of superresolution when using active goniometer systems, i.e. systems that emit probing signals. let signal reception and radiation carried out by two independent scanning dptransmitting fe(β) and receiving dp fr(α). this, for example, can be achieved in the radio wave range using a smart antenna. then, the relationship of the values i, u, and the scanning dp is expressed as the convolution integral of two variables α and β, which correspond to the positions of the maxima of the transmitting and receiving rays: u(α, β) = ∫ ω fe(α − ϕ) fr(β − ϕ) i(ϕ) dϕ. (4.10) the amount of information obtained by scanning with two beams (4.10) significantly exceeds the amount obtained by single-beam scanning (2.1). this allows us to expect to obtain a more accurate solution of the inverse problem (3.2)-(3.5) with a higher achievable level of superresolution. one of the effective methods of signal processing (4.10) is the preliminary integration of the scanning angle β of the receiving or α emitting antenna. the larger the chosen area of integration λ, the higher the stability of the obtained approximate solutions. however as λ increases, the snr decreases in areas close to the boundary. therefore, the integration region should be defined as the sector of angles within which the snr value is sufficient to obtain stable solutions. usually, the area of λ is noticeably larger than the area of ω. integrating (4.10), for example, by the angle β we find v(α) = ∫ λ u(α, β) dβ = = ∫ ω fe(α − ϕ) i(ϕ) ∫ λ fr(β − ϕ) dβ dϕ = = ∫ ω fe(α − ϕ) i(ϕ) f(ϕ) dϕ = ∫ ω fe(α − ϕ) ĩ(ϕ) dϕ, (4.11) where f(ϕ) = ∫ λ fr(β − ϕ) dβ, ĩ(ϕ) = i(ϕ) f(ϕ). (4.12) the resulting integral equation (4.11) for the introduced function ĩ(ϕ) formally coincides with the original one (2.1). therefore, processing of the results may be carried out using all techniques and algorithms applicable to (2.1). the solution (4.11) in the form of ĩ(ϕ) allows using (4.12) to easily express the desired distribution i(α). numerical studies on models have shown that the conditionality numbers of matrices of type (3.4)-(3.5) obtained for the problem (4.11) are significantly smaller than those of matrices g from (3.4). this indicates increased stability of the resulting solutions. in addition, pre-integration (4.11) reduces the role of random rapidly oscillating noise components of the signal, and thus also increases the stability of solutions. increasing resiliency allows us to use more functions when presenting solutions (3.2) and, as a result, to get a higher resolution. copyright© 2021 assa. adv syst sci appl (2021) 112 b. lagovsky, e. rubinovich the disadvantage of pre-integration is the increased signal processing time. to increase the processing speed, instead of integrating by α and β, we can set a selection of angle values (αp, βq), p, q = 1, 2, . . . ,n, covering the entire scan area λ. then, instead of (4.11), using (3.2), we get v(α, β) = n∑ m=1 bm ∫ ω fe(αp − ϕ) fr(βq − ϕ) gm(ϕ) dϕ. (4.13) by entering the notation wpq = u(αp, βq), φpqm = ∫ ω fe(αp − ϕ) fr(βq − ϕ) gm(ϕ) dϕ, (4.14) we arrive at the slae with respect to the coefficients bm in the form wpq = m∑ m=1 φpqm bm, p, q = 1, 2, . . . ,n. (4.15) the system (4.15) is overridden. its solution is sought by standard algorithms in the rootmean-square approximation, which reduces the role of random components in the received signal. the solutions of the system (4.15), as shown by numerical experiments on a mathematical model, are not inferior in stability to solutions based on integration (4.11),(4.12). at the same time, the signal processing speed based on (4.13)-(4.15)is noticeably higher, which allows the algorithm to be used in real-time. fig. 4.5 illustrates the differences in the quality of solutions to inverse problems by the one-and two-beam method with a low snr of 11 db. the true source of the reflected signal was two separate small-sized sources with a distance between them of 0.7θ 0.5, which were not resolved by direct observation (the solid curve 1). step functions were chosen to represent the solution i(α). the solution based on the single-beam method is presented using the dashed curve 2. it should be considered inadequate: the image has little to do with the true sources. at the same time, the solution based on the two-beam method (the solid curve 3) allowed us to resolve the sources and correctly localize their location. the resulting false objects are characterized by a small amplitude of the reflected signal, and they should be ignored when solving such inverse problems. thus, the two-beam method of object image reconstruction reduces the role of random components, which ultimately manifests itself as an additional regularizing factor that increases the stability of solutions to the inverse problems under consideration. this is fully inherent in two-dimensional problems. when scanning the two-dimensional region θ with the transmitting and receiving beams at the angles (α, ϕ) and (γ, β), respectively, the received signal is obtained in the form: u(α, ϕ, γ, β) � n∑ n,m=1 bnm ψnm(α, ϕ, γ, β), ψnm(α, ϕ, γ, β) = = ∫ ω fr(α − α′, ϕ − ϕ′) fe(γ − α′, β − ϕ′)gn(α′)gm(ϕ′)dα′dϕ′. (4.16) copyright© 2021 assa. adv syst sci appl (2021) achieving angular superresolution of control and measurement systems 113 fig. 4.5. single and double-beam solutions at high noise levels using the representation (3.7), the coefficients bnm, which provide a minimum of the rootmean-square deviation of the sum in (4.16) from the received signal, are found as solutions of slae: ∫ θ u(α, ϕ, γ, β) ψnm(α, ϕ, γ, β) dα dϕ dγ dβ = = n∑ n,m=1 bnm ∫ θ ψ jk(α, ϕ, γ, β) ψnm(α, ϕ, γ, β) dα dϕ dγ dβ, j, k = 1, . . . ,n. to increase the processing speed, as previously for one-dimensional tasks, we can set a selection of angle values that cover the entire scan area instead of integrations. fig. 4.6,4.7 show the solution to the problem of restoring the image of a two-dimensional object. the rectangular coordinate system (α, ϕ) is used. a complex object consists of four identical small-sized objects located at the corners of a square with a side of 0.6θ 0.5. fig. 4.6 shows their location in the θ area. in the same figure, the u(α, ϕ) signal received during a single-beam scan is shown in the form of a grid. direct observation, as well as the processing u(α, ϕ), did not allow us to obtain a stable solution in the form of an image of an object with a super-resolution. the use of a two-beam method of measurement and signal processing provided an image of the object. fig. 4.7 shows the results of the mentioned signal processing in the region ω when using step functions. the decision based on of a two-beam method has allowed us to resolve all four sources. their angular position is correctly found and localized. the complex objects that arise when solving the inverse problem have a small amplitude and can be considered as angular noise. when solving problems of this kind, they should be filtered out, for example, using threshold devices. copyright© 2021 assa. adv syst sci appl (2021) 114 b. lagovsky, e. rubinovich fig. 4.6. the location of individual elements of a complex object and the angular distribution of the received signal 5. conclusion the developed algebraic methods of signal processing allow us to form approximate images of complex one-and two-dimensional objects with super-resolution. the achieved angular resolution increases when using a priori information about the solution reaches values 5-10 times higher than the rayleigh criterion. images are restored with relatively small errors in the values of the intensity and angular positions of individual elements of objects. the results of numerical studies have shown that the minimum required for obtaining a stable solution with a superresolution of snr is 12-16 db for simple algebraic methods, which is significantly less than for known methods. modifications of algebraic methods based on two-beam scanning, targeted selection of functions chosen to represent the solution, and extensive use of a priori information about the solution allow us to obtain a stable solution with a superresolution in some cases up to an snr of 7-9 db. in an alternative interpretation, the modified algebraic methods of constant snr allow increasing the level of the achieved superresolution. the created high-speed algorithms for algebraic methods allow to process of signals in a real-time mode. acknowledgment the reported study was partially supported by rfbr research project no. 20-07-00006. copyright© 2021 assa. adv syst sci appl (2021) achieving angular superresolution of control and measurement systems 115 fig. 4.7. restored image of the object in fig. 4.6 references 1. lagovsky b., rubinovich e. 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(piers 2015-prague). 1548–1551. copyright© 2021 assa. adv syst sci appl (2021) introduction problem statement algebraic methods of solution two-beam method conclusion microsoft word 1167 adv syst sci appl 2022; 04; 233-241 published online at http://ijassa.ipu.ru. estimation of the posterior probabilities of classes by the approximation of the anderson discriminant function valery zenkov v.a. trapeznikov institute of control sciences of ras, moscow, russia email: zenkov-v@yandex.ru abstract: the approximation of the anderson discriminant function at a given point in the feature space of two classes using a supervised training sample allows estimating the posterior probabilities of classes as easily as converting fahrenheit degrees to celsius degrees. these probabilities make it possible to further solve the classification problem with subjectively specified costs of classification errors and criteria. the training sample is simply converted to a regression analysis sample by replacing the class numbers with differences in error costs. a nonparametric method for approximating the discriminant function at a point is proposed. it does not require the specification of an approximation function at a point more complex than a linear one. examples are given. keywords: machine learning, anderson discriminant function, approximation, class posterior probability, weighted least squares. 1. introduction the posterior probabilities (pop) of the classes are exhaustive information for solving the problem of object classification. they allow attributing a classification object to a particular class, taking into account different matrices of the costs of classification errors, set by the user not only for the entire task as a whole, but also for some categories of classified objects separately, for example, by the status of the classified object. we solve the problem in two steps. at the first stage, for the object of classification, we find the pop of the classes objective data. at the second stage, we assign the object to one or another class, taking into account the subjective preferences of the user: different the classification error costs, the method of making a decision taking into account some restrictions, etc. in connection with the importance of the pop of the classes for solving the classification problem and for interpreting the decisions made when classifying objects, add-ons to some already existing classification methods are used. so, for the support vector machine, the platt calibrator is used [1], which estimates the pop of the classes by the distance of a point in the class feature space to the boundary between classes. in this case, a hypothesis is put forward about the form of the dependence of the pop of the classes on the distance of the point to the boundary between the classes, and then the parameters of the hypothetical dependence are found using the maximum likelihood method using the training set. the anderson discriminant function (adf), according to the sign of which a point in the class feature space belongs to one of two classes, is a regression dependence. it is convenient in that, according to its approximation, obtained by the weighted least squares method from a supervised training set and by the cost of classification errors used to determine it, we calculate pop classes as simply as we get degrees celsius from degrees fahrenheit. we define adf from the bayesian solution of the classification problem [2], obtained by theodore wilburne anderson. adf is the difference between the two functions of average losses from classification errors in the space of attributes of the two classes at given costs of losses from classification errors. 234 v. zenkov copyright ©2022 assa adv. in systems science and appl. (2022) we approximate the adf using a supervised training set of the classification problem, transforming it into a regression analysis set by replacing the class numbers (labels) with the corresponding differences in the cost of classification errors. when approximating the adf at a point, there is no need to specify the form of the approximating dependence in the class feature space. it is enough to construct a linear approximation of the adf by the closest points to a given point, or by the nadaray-watson method [3] to find the adf value at this point. the pop estimates of the classes at a point based on the adf approximation at this point do not depend on the choice of error costs used to determine the adf. you can set the error costs equal, for example, 1 and 1, or not equal, for example, 2 and 5. the adf and approximations at the point will be different, but the pop estimates at the point will be the same. this is the main property of adf. the classification error costs are used to transform the supervised training set into a regression analysis set. class numbers in the first case are replaced by -1 and +1 in the first case and by -2 and +5 in the second case. due to the independence of the estimates of the pop of the classes on the choice of the costs of classification errors, the classification error can always be set to 0.5 and 0.5. the method for obtaining pop estimates from the adf approximation does not depend on the class imbalance in the training sample. class imbalance in the training set occurs when the number of points of one class is much less than the number of points of another class. class imbalance makes it difficult to detect objects of a small class if classification methods are used that are not based on pop estimations and costs of classification errors. the method for solving the classification problem at a point in the feature space using a supervised training set, when we use the entire training set for each classified point, is nonparametric, and the resulting model is a nonparametric model. 2. anderson discriminant function we define adf as the difference of two functions of average losses of classification errors in a problem with two classes:        12 1 2 1 2|, , , –k x k kf x c g x c g x c m c c   , (2.1) where         1 12 2 21, 1 1 , , 1| |g x c c p x g x c c p x   there are average losses at the point, if the point is classified as class 1 or, respectively, class 2; ijc is the cost of the error when a point from a class j is mistakenly referred to a class i. 0, # , 0ij iic i j c  . p(k|x) is the pop of the class k at the point x,    1 2 1| |p x p x  ,       ,/| |kp k x p p x k p x    | k kp x p p x k  , pk is the prior probabilities of classes, p(x|k) is the conditional distributions of class features; k is the class numbers 1 or 2;  |k xm  is the mathematical expectation of at a point. if  12 , 0f x c  , then the point referred to class 1, otherwise to class 2. if  12 , 0f x c  , then the point referred to class 1, otherwise to class 2. 3. adf properties 3.1. adf is by definition a regression function. to transform the training set of the supervised classification problem into a regression analysis set, due (2.1), on the set, the class number 1 should be replaced by -c21 and the class number 2 by c12. estimation of the posterior probabilities of classes… 235 copyright ©2022 assa. adv. in systems science and appl. (2022) 3.2. the first class pop and the adf defined for given c12 and c21 are identically related       12 12 12 211 – , /|p x c f x c c c  . (3.1) corollary 3.1: if we set the cost of errors under the condition 12 21 1c c  , that it does not lead to a loss of generality, then the equal (3.1) is simplified  121 * , * ( | )p x p f x p  (3.2) and on the boundary between the classes, where    12 12, * | 0, * 1f x p p p x с   and 211 *p c  . the pop of the first class on the class boundary is p*. corollary 3.2: the pop of the classes at the point x are not addicts on for which * 0p  , or c12 and c21, the adf is defined. for the formal proof, it suffices to substitute expression (2.1) for adf in (3.1) or (3.2). corollary 3.3: when choosing the error costs c for classifying a point by the pop of the classes obtained at the first stage of solving the classification problem, at the second stage there will be indistinguishability of classes in the feature space if     12 12, 0 v )( , 0x xmin f x c max f x c  . (3.3) in this case, all points will need to be assigned to the same class. 4. adf approximation methods the adf can be approximated in at least three ways: – to approximate the adf at a point in the class feature space to obtain estimates of the pop of the classes at this point according to (3.1) or (3.2). this is a nonparametric model for solving a problem. – to approximate the adf in the entire area of the training set, having specified the type of the approximating function and choosing the approximation method. for this parametric model, a method is proposed for approximating the adf in the initially unknown neighborhood of its zero values, to solve more accurately the classification problem for the selected type of approximating dependence and the given costs of classification errors and not pursue the goal of evaluating the pop of the classes. – to approximate several adf for the given pop values of the first class at the boundary between the classes, for example, 0.1, 0.3,…, 0.9 and for a given type of approximating dependence. this is a variant of a series of adf approximations. for a given point, the pop of the classes is found by the method of interor extrapolation over neighboring approximations of the adf, between which the point is located. the nonparametric model is slower it uses the entire training set in the test stage. parametric ones work faster, but they require the choice of the type of the approximating function for adf in the training stage. 4.1. adf approximation at point this version builds a nonparametric model. its distinctive feature is the need to use all or part of the supervised training sеt to estimate the pop of the classes at each given point. in [4], to estimate the adf at a point, we select such costs of classification errors for which the adf approximation at a given point takes zero value. but more preferable, in our opinion, 236 v. zenkov copyright ©2022 assa adv. in systems science and appl. (2022) is the adf approximation by a plane (at least by a second-order polynomial) in the vicinity of a given point. the criterion for approximating the adf at a given set point xj is the weighted root mean square error. at the training stage, for each set point xj used as a test one, there is a vector of coefficients λj approximating the adf in the form (1, xj) λj for given values of the parameters w and s of the weight function, for example, an exponent: 11, 2 2( ) [ ) ] ( || || ), = min – – (1, exp j n nj k k n j n n n s jn n j q x c c x w xx      , (4.1) where the row vector (1, xn), in addition to the row vector of features xn of the training set of components, contains a component one. n is the number of rows in the set without row xj at the training stage, as required by the loo overfitting method (leave-one-out). kn is the class number 1 or 2 in the set line n. xn is the feature vector in the set line n. 0w  is a weighting factor that sets the rate of decay of the weighting function (in this case, the exponent is from the distance of the training set point to a given point). s is an exponent to which the distance from the sampling point to a given point is raised. 1 2 – n n k k c c is the estimate of the adf in the set row n, obtained by replacing the class number kn, 1 or 2, in the row by the difference in error costs determined by the given p*. having obtained the vector of coefficients λj by minimizing criterion (4.1) for the given parameters of the method w and s, we calculate the estimate of the adf at the point xj, and, using (3.1), (3.2), we find the estimate of the pop of the first class at the point xj. if the number of classes 2k  , then similarly we obtain the pop of the class at this point, using the method one class against all the others. the best values of the parameters w and s are selected, for example, according to the minimum criterion 1 12 2 21 1( )r n c n c n  (4.2) where n1 is the number of points of the first class, erroneously assigned to the second class, n2 is the number of points of the second class, erroneously assigned to the first class. often s = 1 and the value of the only parameter w is selected. the type of the weighting function is not necessarily exponential. 4.2. adf approximation in the neighborhood of zero values in some cases (a large volume of the training set, the presence of a priori information about the conditional distributions of features of classes, or about the type of surface separating the classes), it is more expedient to build a parametric model. it can be in the form of a discriminant function of a given form with unknown parameters determined from the training set. to classify a point, we can use the bayes formula to recalculate the reconstructed conditional distributions of class features to the pop of the classes at the specified point and, taking into account the costs of classification errors, assign the point to a particular class. we propose in [5] a heuristic method for constructing a parametric model of the classification problem based on a supervised training set, used for the approximation of the discriminant function in the vicinity of its zero values for given costs of classification errors. in this case, we usually do not talk about evaluating the pop of the classes. the supervised training set does not contain information about the position of the adf zero points. for given values of classification errors, there may be no boundary between classes and all points should be assigned to the same class. to solve the problem, a heuristic method [5] of approaching the boundary between classes is used. first, based on the given costs of classification errors, the training set is transformed into a set of the regression analysis problem by replacing the class numbers (labels) with the corresponding differences of the costs of classification errors (2.1). then, several iterations of solving the problems of approximation of adf by the weighted least squares method are estimation of the posterior probabilities of classes… 237 copyright ©2022 assa. adv. in systems science and appl. (2022) started. the weight function from the distance of the point to zero on the previously obtained adf approximation is used as the weights of the set points. the measure of the distance of a point to the zero of the previous approximation is the module of the previous approximation of the adf at this point. solution criterion at each step with an exponential weighting function: 2 11 2 1 min – ]– ' exp ( ) {[ ( ) ( | ( ) | )}' i n n s i k k i n i n i n n nq c c x w x         , (4.3) where i is the iteration number, 1,2,...,{ }i i . at the first step, the problem is solved without a weighting function  0 0w  ; at subsequent steps, the weighting function gives more weight to points closer to the zero region of the previous adf approximation. i is a given number of iterations i  . wi is a given weighting factor at a step s is a given exponent, usually 1s  . the weighting factor in iterations can be constant or change, for example * , {0,1,..., }iw w i i i  . for each iteration, the losses are found from the obtained adf approximation (4.3). the best value of λ is the vector to which the lower losses correspond (4.2). 4.3. estimation of pop of the class at a point from a series of adf approximations in this parametric method [6], based on a supervised training set, we construct several approximations of the adf specified for multiple pop of the first class at the boundaries between classes. this completes the training process and the training set is no longer used. the constructed series of approximations can be used to estimate the pop of the first-class at a given point in the feature space by two methods: the interpolation method and the smoothing method. in the interpolation method for estimating the pop of the first class at a given point by the signs of the adf approximations in the series, at this point, there is a pair of neighboring approximations, between which the point is located. if a pair is found, then the interpolation method yields an estimate for pop of the first class. a measure of the distance to neighboring boundaries and known by the construction of pop of the first class on these boundaries is the module of the values of neighboring adf approximations at a given point. if a given point is outside the limits of the adf approximations, then the estimate of the pop of the class is found by extrapolation from the first or last adf approximation in the series, checking the result for non-negativity and not exceeding one. in the smoothing method, the weighted pj* values used to obtain the j-approximations of the adf are used to estimate the pop of the classes at a given point. as a measure of the distance of a point to the boundary j between classes, the modulus of the value of the japproximation of the adf at a given point   , * j jf x p is used. as a weighting function, for example, the exponent is used * 1 ( ) ( | ,( exp f , ) ) |j j j j jw x w x p    , (4.4) where w is the parameter setting the smoothing properties of the method. the weighted estimated pop at a point using a series of adf approximation and (4.4) will be * ( | ) ( ) ( )1 / j j j j jp x p w x w x   . (4.5) the best value of w can be selected according to (4.2), performing classifications according to the pop of the classes (4.5). 238 v. zenkov copyright ©2022 assa adv. in systems science and appl. (2022) 5. examples 5.1. approximation of the adf at a point we present the case of solving the problem by minimizing (4.3) and (4.2) in fig. 5.1 with normal conditional distributions of two-dimensional features of three classes. one of them, class a, is located between classes b and c. class a is designated by the number 1, classes b and c will be combined into one class and designated by the number 2. class middles:       0, 0 , 3, 0 , 3, 0a b cm m m    .the covariance matrices of the classes are the same. the prior probabilities of the classes 0.5ap  , 0.25b cp p  . the set generated by this parameter contains 120 points. the figure shows the theoretical densities of conditional distributions of class features, theoretical adf and pop of the first class, as well as estimates of pop at given points, obtained from 1s  and 3w  (4.2). fig. 5.1 the root-mean-square errors (rms) in the estimation of pop of the first-class about with concern to the theoretical ones were 0.09 and 0.18, respectively, for the linear and polynomial approximation of the second-order of the adf. the linear approximation of adf turned out to be more accurate in this set. the estimates of pop of the first class at the given points are located near the curve of pop of the first class, obtained from the estimates of the parameters of the normal laws of conditional distributions of features (the dashed curve next to the solid theoretical curve of pop of the first class). since there are two features, the flat figure shows a section by a plane passing along the axis of the values of the first feature and the ordinate. estimation of the posterior probabilities of classes… 239 copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 5.2 fig. 5.2 shows an example with a one-dimensional feature distributed according to a uniform law of probability distribution. classes a, b and c have averages 0, 3, 3a b cm m m    . the limits of maximum deviations from the average in both directions are the same and equal to 2. the classes are intersected in the feature space. the prior probabilities of the classes are the same as in the previous case. there are 120 points in the training set, 1s  and 3w  . visually, the estimates of pop of the first class by a second-order polynomial in being more accurate. explanations for fig. 5.1 and fig. 5.2. the sampling points of the three classes a, b, and c are located at levels p* and 1 *p . the conditional distributions of features of classes 1 and 2 (one against combining the other two) are the dashed lines  |1p x and  | 2p x in the figures. the solid thin line is the distribution density of x, p(x). the inscription adf denotes the position of the theoretical adf (solid thick line with convexity downward).  |1p x is the theoretical pop of the first class (solid thick line with a convex upward). close to theoretical dashed lines is adf and pop constructed from the training set, for the known conditional distributions of class attributes and the given costs of classification errors, * 0.5p  . the adf constructed for the training set in fig. 5.2 is not shown. it is a mirror image of the pop constructed for the training set (dashed line). round points near pop of the first-class – estimates of pop by approximating adf at given points; crosses — estimates of pop using a second-order polynomial to approximate the adf. 5.2. approximation of the adf in the vicinity of zero values when choosing the type of adf approximating dependence, one can use the features, according to their correlation coefficients with the desired value. so in the example [5] with the dimension of the feature space of 217, the training set consisted of only 252 lines. three were selected with a correlation coefficient estimates of the desired value not lower than 0.64 and with a cross-correlation, not higher than 0.66. the classification error according to the adf approximation in the vicinity of the zeros according to the linear approximation was 6.8 %, according to the second-order polynomial – 4.8%. for comparison, the classification 240 v. zenkov copyright ©2022 assa adv. in systems science and appl. (2022) method based on the hypothesis of conditional normal distributions of the selected features of the classes, according to the polynomial and according to the linear model, was 10.3% and 9.9%, respectively. we made [7] a comparison on 15 examples of the quality of solving classification problems by the adf approximation method in the vicinity of zeros and the support vector machine method (svm) for linear adf approximation and approximation by a second-order polynomial at equal costs of classification errors. in 14 examples, the first method gave a lower classification error than the second. a few examples may not be a sufficient basis for judging the advantages of one method over another. 5.3. estimation of the pop of the first class at a point from a series of the adf approximations as a model example, fig. 5.3, [6] the classification problem with two classes with normal conditional distributions of class attributes in a two-dimensional space with different covariance matrices was considered. the average values of the features of the first class  1 0, 0m  . the covariance matrix is 1 ((1, 0); (0,1))s  . for the second class:  2 0, 2m  , 2 ((4,1); (1,1))s  . the prior probabilities of the classes are 1 0,6p  , 2 0,4p  . fig. 5.3 based on the initial data and the generated set of 500 points, the pop of the first class was calculated in the following ways: 1. exact according to the known distributions according to the bayes formula; 2. sample based on sample estimates of distribution parameters using bayes' formula for calculating pop of the first class; 3. inter-extrapolation for the series of the adf approximated by a second-order polynomial; 4. by the method of smoothing according to the series of adf approximations. a series of the adf approximations is built for p* values from 0.95 to 0.05 with a step of 0.1. circles are points of the first class, crosses are of the second. the root-mean-square error estimate of the pop of the first class for the training set by the interpolation method for a series of adf approximations was 0.055. the error in its estimation by the smoothing method was 0.06. the error in estimating pop of the first class based on sample estimates of feature distributions and using bayes' formulas was 0.02, i.e. three times less. but in justification of estimation of the posterior probabilities of classes… 241 copyright ©2022 assa. adv. in systems science and appl. (2022) other methods, it should be remembered that they do not require knowledge of the laws of distribution of the features of classes. 6. conclusion approximation of the anderson discriminant function based on a supervised training set allows you to simultaneously find estimates of the posterior probabilities of classes of classified objects. the posterior probabilities of the classes are comprehensive information for solving the problem of classification according to various criteria. the supervised training set of the classification problem is transformed into a training set of the regression analysis problem by replacing the class numbers in the training set with the corresponding differences with the costs of classification errors, set arbitrarily at the training stage to determine the approximations of the anderson discriminant function. at the second stage of solving the problem, according to the estimates of the objective posterior probabilities of classes, the actual classification of the object is performed according to the subjectively specified classification criteria. methods for solving the problem related to the use of nonparametric and parametric approaches are proposed. with the nonparametric approach, to approximate the anderson discriminant function at a point in the class feature space representing the classified object, a training set is used for each classified point, which may be unacceptable for larger set sizes and with certain performance requirements. for a non-parametric approach a rather linear approximating function. in parametric approaches, weighted least squares are used to obtain a more accurate approximation of a function in the vicinity of its zero values. in another parametric approach, a series of approximations of the anderson discriminant functions are constructed from a given set of a posteriori probabilities on the class boundaries. then, for the classified point, the posterior probabilities of the classes are found using the interpolation method or the smoothing method. methods are demonstrated by examples. references 1. platt, j. (2000). probabilistic outputs for support vector machines and comparisons to regularized likelihood methods, advances in large margin classifiers, 10(3), 61–74. 2. anderson, t.w. (2003) an introduction to multivariate statistical analysis, 3 rd. ed. new york, ny: john wiley & sons. 3. hardle, v. (1993) applied nonparametric regression. moscow, russia: mir. 4. zenkov, v.v. (2018). estimating the probability of a class at a point by the approximation of one discriminant function, autom remote control, 79(9), 1582–1592, https://doi.org/10.1134/s0005117918090047. 5. zenkov, v.v. (2017). using weighted least squares to approximate the discriminant function with a cylindrical surface in classification problems, autom. remote control, 78(9), 1662–1673, https://doi.org/10.1134/s0005117917090107. 6. zenkov, v.v. (2019). estimation of the posterior probability of a class from a series of discriminant anderson functions, autom remote control, 80(3), 447-458. https://doi.org/10.1134/s0005117919030056. 7. zenkov, v.v. (2020). applying an approximation of the anderson discriminant function and support vector machines for solving some classification tasks, autom remote control, 81(1), 118–129, https://doi.org/10.1134/s0005117920010105. microsoft word 988 article text, copyedited.docx adv syst sci appl 2021; 02; 20-28 published online at https://ijassa.ipu.ru. using patent landscapes for technology benchmarking: a case of 5g networks dmitry kochetkov 1*, marat almaganbetov 2 1) ministry of science and higher education of the russian federation, moscow, russia e-mail: kochetkovdm@hotmail.com 2) lexisnexis eastern europe, moscow, russia e-mail: marat.almaganbetov@lexisnexis.com abstract: patent landscape reports (plr) have long been used in business, science and r&d. they enable data-driven decision making and minimizing the risks of strategic technology choices. the authors analyzed the patent landscape for 5g wireless communications using the patentsight® software product. among the companies, the clear leader by the patent portfolio index is samsung electronics; among countries, it is usa and china. however, patents of high technological relevance and/or commercial value may appear in small tech companies, or as a “by-product” of other corporate activities. the research results are of interest to researchers and practitioners involved in the development and implementation of 5g technologies. keywords: patent landscape, 5g, patent assets index, technological relevance, market coverage, competitive impact 1. introduction the patent landscape is a systematic study of patent documents that identifies and visualizes trends in business, science and technology. patent landscape reports usually focus on one industry, technology, or geographic region. patent landscape reports (plrs) support informed decision making and are designed to effectively address high-stakes challenges in various areas of technology, thereby increasing trust. through patent analytics and plr, these critical decisions can be made using evidence that provides informed choices and mitigates decision-making risks [13]. this patent study focuses on 5g technology. 5g stands for fifth-generation cellular wireless communications, and the initial standards for it were established at the end of 2017. since 2019, mobile operators have begun rolling out 5g networks around the world. the main advantage of the new networks is that they will have higher bandwidth, providing higher data transfer rates. thanks to the increased network bandwidth, mobile operators will be able to compete with isps providing cable internet services. besides, latest generation wireless networks open up a whole range of new applications in the internet of things (iot), augmented/virtual reality, m2m systems [9]. the study used patent analytics provided by lexisnexis (founded in 1977, is part of the largest transnational information holding relx group). lexisnexis ip is a global leader in information solutions and services to meet the needs of the intellectual property market, government agencies, and academia. the company aggregates data from 111 patent offices around the world [1]. * corresponding author: kochetkovdm@hotmail.com using patent landscapes for technology benchmarking: a case of 5g networks 21 copyright ©2021 assa. adv. in systems science and appl. (2021) 2. theoretical background 5g publications began to appear in 2014, and since then their number has grown steadily. we used the web of science® database, which is one of the most widely used reference databases for scientific periodicals. for information retrieval, we used the '5g' query with the category constraint 'telecommunications.' the request returned 7093 publications. fig. 2.1 shows the dynamics of the number of publications from 2014 to 2019. fig. 2.1. dynamics of 5g-related publications. data source: clarivate analytics web of science®. we visualized the 5g scientific landscape using vosviewer® as in [7] (fig. 2.2). it is a software that enables creation, visualization and analysis of network maps based on bibliographic data. vosviewer® also offers text mining functionality that can be used to construct and visualize co-occurrence networks of important terms extracted from a body of scientific literature [18]. vosviewer® pays special attention to the visualization of bibliometric maps. vosviewer's functionality is especially useful for displaying large bibliometric maps in a simple and efficient way. it is especially useful for maps containing at least a moderately large number of items (e.g., at least 100 items) [15, 16]. the software is based on the idea of association strength: (1) where sij is the similarity measure, cij is the co-occurrence of i and j, and ciсj the expected number of co-occurrences of i and j under the assumption that co-occurrences of i and j are statistically independent. sij takes values in the range [0, 1]. vosviewer® applies the link strength [14] by default. then, the program uses the mapping technique detailed by [17]. finally, vosviewer® assigns the network nodes to clusters; the clustering technique is presented in [19]. vos determines the locations of items in a map by minimizing , (2) subject to: . (3) 22 d. kochetkov, m. almaganbetov copyright ©2021 assa. adv. in systems science and appl. (2021) xi and xj denote the location of i and j respectively. bibliometric maps have two dimensions and rely on the euclidean distance. for the purposes of this study, we mainly used text mining functionality for constructing co-occurrence networks of terms extracted from metadata. fig. 2.2. co-occurrence analysis of 5g-related publications. data source: clarivate analytics web of science®. built with vosviewer®. the red cluster covers general issues of 5g network management, including network architecture, green radiotechnological aspects of network functioning, yellow management of network power parameters and the blue one relates to optimization of resource allocation. however, the picture would be incomplete without the patent landscape because scientific inventions enter everyday life precisely through intellectual property objects and their commercialization. patent landscapes are a widespread tool for analyzing the evolution of technological fields and identifying hot topics [2]. patent landscapes (also patent landscape reports (plrs) enable informed business&technology decisions [13]. however, there is no single definition of the patent landscape; it could be an overview of patent information for a specific technology or geographic area. the landscape usually strives to answer specific practical questions and present complex information about patenting activities in a transparent and intelligible form. industry has long used the patent landscapes to make strategic decisions about investments, research and development (r&d), and competitors. högskolan i halmstad. & niamba (2016) provide a good overview of patent indicators, subdividing them into four categories [5]: • performance indicators deal with the patent output of the analyzed entities (inventors or applicants) that are used to monitor the technological performance of companies/institutions and inventors/researchers [10]. • technology indicators analyze patent classifications, classifying each patent into one or more classes according to its technology area. • patent value indicators give an idea of the economic value of a project, which is determined by various factors such as family size, citation, and geographic coverage. using patent landscapes for technology benchmarking: a case of 5g networks 23 copyright ©2021 assa. adv. in systems science and appl. (2021) • collaboration indicators provide information about collaboration patterns between entities. technology benchmarking is used to compare the performance of organizations in terms of technology applications. benchmarking focuses on analyzing your own processes and then comparing your own performance with the performance of others [8]. patents are reliable kpis for comparison [6]. at the same time, patents are a lagging indicator of the company's research, since it is at least 18 months old (most patent offices around the world have a secrecy period of 18 months). in 2011, a new technology benchmarking methodology based on patent information [3] was presented. the main indicator in this methodology is the patent asset index, which is constructed based on the 'portfolio size', 'market coverage' and 'technology relevance' metrics. 'market coverage' is a measure of the extent of patent protection in global markets. 'technology relevance' is a citation-based indicator to assess the technological impact of patents, which eliminates systematic distortions of existing citation-based patent indicators. authors argued that the patent asset index offered a more accurate assessment of a firm's patent than other similar methods. we used this technique in our research, so we describe it in more detail in the next section. 3. data and methods the study used data on 4069 patent families associated with 5g technology. the search was carried out using the keywords “5g” and “fifth generation” with a filter by domain (ipc) h04w (wireless communication networks [2009.01]). patent family is a set of patents obtained in different countries for the protection of one invention (when the first application in the country priority then extends to other patent offices) [11, p.60]. the data source is patentsight®(“patent research and analytics products | lexisnexis patentsight®,” n.d.) from lexisnexis. this tool provides objective metrics of global technological power and influence, developed by german scientists headed by n. omland and h. ernst [3] (fig. 3.1). fig. 3.1. the patent asset index structure. source: [12]. 24 d. kochetkov, m. almaganbetov copyright ©2021 assa. adv. in systems science and appl. (2021) the patent asset index ™ (pai) takes into account both the quantity of actively protected inventions and their quality. the method was developed and tested in scientific research and has been used by leading companies in many industries for several years. the patent assets index is the overall strength of the patent portfolio. it is based on the technology relevance ™ and market coverage ™ of the company's patents to analyze the competitive impact of a portfolio. technology relevance is the worldwide citation derived from later patents, taking into account age, patent office practice, and technology area. market coverage is the size of the market protected by active patents and pending patent applications for a particular invention. finally, competitive impact ™ measures the strength of an individual patent, i.e. the relative commercial value of the patent. 4. results fig. 4.1 presents the analysis of portfolio size, patent assets index, and competitive impact or relative business value. the undisputed market leader is samsung electronics, which has the largest 5g patent portfolio with the highest patent assets index. the other side of the coin is the low portfolio selectivity. at the same time, we observed two companies with small patent portfolios, but extremely high value (at&t and interdigital). fig. 4.1. portfolio size (horizontal axis), competitive impact (vertical axis), and patent assets index (bubble size). created with patentsight®. samsung leads not only in the area as a whole, but in all sub-areas (level 4 of the ipc classification) (fig. 4.2). using patent landscapes for technology benchmarking: a case of 5g networks 25 copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 4.2. the size of the patent portfolio by level 4 of the ipc classification. created with patentsight®. however, if we look at the indicator of technological relevance, then we will not see companies with the largest patent portfolios in the list (fig. 4.3). often, the most cited patents appear in small tech companies or as a "by-product" of other businesses (eg toyota). fig. 4.3. technological relevance of the patent portfolio. created with patentsight®. finally, in terms of the origin of patent families, two jurisdictions are leading by a noticeable margin usa and the prc (fig. 4.4). 26 d. kochetkov, m. almaganbetov copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 4.4. geographical breakdown of patent families by jurisdiction of origin. created with patentsight®. 5. conclusion patent landscape reports (plr) have long been used in business, science and r&d. they enable data-driven decision making and minimizing the risks of strategic technology choices. in this study, we analyzed the patent landscape for 5g wireless networks. among the companies, the clear leader by the patent portfolio index is samsung electronics; among countries, it is usa and china. however, patents of high technological relevance and/or commercial value may appear in small tech companies, or as a ‘by-product’ of other corporate activities. the analysis shows that the 5g family of technologies has entered a phase of maturity, i.e. further developments will be incremental. at this stage, the market is already formed, the emphasis is shifting to commercialization and implementation. the new generation of mobile communication systems is already here [4]. however, the new generation of communication networks (6g) will be built on the existing scientific groundwork related to understanding both target applications and the most promising candidate technologies. historical analysis of technology development based on scientometric analysis and patent landscapes enables making informed strategic decisions for technological breakthroughs in the future. acknowledgements the study was financially supported by the russian foundation for basic research within the framework of the research project no. 18-00-01040 komfi ‘the impact of emerging technologies on the urban environment and the quality of life of urban communities.’ using patent landscapes for technology benchmarking: a case of 5g networks 27 copyright ©2021 assa. adv. in systems science and appl. (2021) references 1. about us. (2020). retrieved july 14, 2020, from lexisnexis website: https://www.lexisnexisip.com/about-us/ 2. clarke, n. s., jürgens, b., & herrero-solana, v. (2020). blockchain patent landscaping: an expert based methodology and search query. world patent information, 61(march). https://doi.org/10.1016/j.wpi.2020.101964 3. ernst, h., & omland, n. (2011). the patent asset index – a new approach to benchmark patent portfolios. world patent information, 33(1), 34–41. https://doi.org/10.1016/j.wpi.2010.08.008 4. giordani, m., polese, m., mezzavilla, m., rangan, s., & zorzi, m. (2019). towards 6g networks: use cases and technologies. 5. högskolan i halmstad., f., & niamba, c. n. (2016). patent bibliometrics and its use for technology watch. journal of intelligence studies in business, 6(1), 17–33. retrieved from https://ojs.hh.se/index.php/jisib/article/view/152/0 6. jain, r., tripathi, m., agarwal, v., & murthy, j. 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(2014). visualizing bibliometric networks. in measuring scholarly impact. https://doi.org/10.1007/978-3-319-10377-8_13 17. van eck, n. j., waltman, l., dekker, r., & van den berg, j. (2010). a comparison of two techniques for bibliometric mapping: multidimensional scaling and vos. journal of the 28 d. kochetkov, m. almaganbetov copyright ©2021 assa. adv. in systems science and appl. (2021) american society for information science and technology, 61(12), 2405–2416. https://doi.org/10.1002/asi.21421 18. vosviewer visualizing scientific landscapes. (n.d.). retrieved august 28, 2019, from https://www.vosviewer.com/ 19. waltman, l., van eck, n. j., & noyons, e. c. m. (2010). a unified approach to mapping and clustering of bibliometric networks. journal of informetrics, 4(4), 629–635. https://doi.org/10.1016/j.joi.2010.07.002 microsoft word 1130 article text, copyedited.doc adv syst sci appl 2021; 04; 31-44 published online at https://ijassa.ipu.ru. the role government policy and supports play to stimulate economic growth jeffrey yi-lin forrest1*, abdou k. jallow1, chao wen2, juehui shi3, huan guo4 1) slippery rock university, slippery rock, pa, usa e-mail: jeffrey.forrest@sru.edu; abdou.jallow@sru.edu 2) ohio northern university, ada, oh 45810, usa e-mail: c-wen@onu.edu 3) angelo state university, san angelo, tx, usa e-mail: juehui.shi@angelo.edu 4)jianghan university, wuhan, china e-mail: guohuan.2007@163.com abstract: this paper studies the reason why policy tools might work, if they do, what the functional mechanism is, and why government’s policies and supports are necessary for stimulating economic growth. by employing systems thinking and methods and the logical reasoning as that commonly used in mathematics and natural science, this paper establishes 6 formal and generally true propositions on these related issues. at the conclusion, we provide a whole list of recommendations for policy makers and government officers in terms of when and how their implemented policies will lead to their desired outcomes. at the conclusion, this paper provides directions and open problems for future research. keywords: economic agent, feedback, market, momentum, policy tool, system 1. introduction fast development of communication technology in recent decades has prompted a score of leading nations to introduce policies and provide supports in order to transform their industries. to maintain their leadership in the globalizing world, these nations challenge themselves to elevate their industries from automated manufacturing to intelligent manufacturing, such as industry 4.0 [1]. hence, the following questions arise naturally: why are policy tools expected to work in an economic system? why are government policy tools and supports fundamentally necessary for stimulating economic growth? answers to these questions are both theoretically and practically important for all nations from around the world. developed nations, at least some of them, attempt to preserve their leading positions in the increasingly globalizing world economy. developing and underdeveloped nations with their miserable failures of modernization experienced in the past one hundred plus years [2] face the likelihood of tumbling further behind the developed nations economically, socially, and politically. although many scholars have worked on these and other related questions since before the time of adam smith, the relevant debates have come and gone in waves without achieving anything definite other than producing inconsistent and uncompromising suggestions [3.4]. other than filling a gap existing in the literature on whether or not governmental policies and supports are necessary in economic development, by addressing these questions and related issues, this work also tempts to answer rostow’s [4] call for an * corresponding author: jeffrey.forrest@sru.edu 32 j.y-l. forrest, a.k. jallow, c. wen, j.h. shi, h. guo copyright ©2021 assa. adv. in systems science and appl. (2021) appropriate methodology for such investigations in order to actually produce scientifically sound and dependable results. in particular, in terms of how policies actually work, wieczorek and hekkert [5] consider how policy instruments can improve the function of an innovation system. by considering which channel is likely to amplify the effects of monetary policy, auclert [6] empirically finds that the following three channels work the earnings heterogeneity channel from unequal income gains, the fisher channel from unexpected inflation, and the interest rate exposure channel from real interest rate changes. boeckx et al. [7] examine the effectiveness and transmission mechanism of the euro-system’s credit support policies. boeckx et al. [7] find successes of policies in stimulating credit flows from banks to the private sector. rebei [8] empirically looks at the crowding-in effect of government spending on private consumption and finds that in early times government spending increasingly crowded in private consumption, while this relation was reverted in recent times. cúrdia and woodford [9] investigate consequences of a variable credit spread optimal policy responses and find that a simple target criterion provides a good approximation to optimal policy. jaehrling et al. [10] provides insights on negotiations and outcomes of labor clauses across different stages of policy process, demonstrating the importance of alliances among local politicians, unions and employers. enriching the literature, this study demonstrates why governmental policies are important, when a government is able to align business objectives of a large proportion of individual economic agents, and what policy tools can help accomplish the goal. as for why governmental policies and supports are necessary in economic development, the literature is naturally divided into two camps: (i) yes, they are necessary; (ii) no, they don’t work generally. for example, andreoni and chang [3] provide a framework for strategically coordinating packages of interactive industrial policy measures. howell et al. [11] argue that the future of us competitiveness in international markets depends on realistic federal, industry, and company policies with regard to microelectronics. conversely, jomo [12] studies eight asian economies (i.e., japan, south korea, taiwan, singapore, hong kong, malaysia, thailand, indonesia), which achieved a statistically unlikely rapid economic growth between 1965 and 1990. although government interventions played a role, he notes that the gains from industrial policies are ambiguous. regarding how well the advanced manufacturing initiative for america’s future, a concerted us government effort, would harmonize with the obama administration, hemphill [13] finds that the answer is not well. comparing to what is summarized above, this study contributes to the literature by establishing several generally true results. for example, it shows why a nation needs to provide policy supports to maintain an extant economic growth momentum, when an implemented policy will positively affect the economic performance of manufacturing enterprises, and why different policies are needed to promote economic growth in different geographical regions. epistemologically speaking, all results established in this paper do not suffer from the constraints of dataand anecdote-based approaches, as commonly employed in the literature, and avoid the methodological pitfall experienced by rounds of debate on whether or not governmental policies are necessary for economic development since the time even before adam smith [3] and by rounds of studies on what was really underneath the occurrence of the industrial revolution [4]. in particular, the former respectively produced inconclusive and noncompromising reasons for and against the use of policies based on anecdotes and statistics. and the latter led to questionable lists of factors that were underneath the occurrence of the industrial revolution [2,4]. in comparison, results developed in this work are scientifically definite. for the following reasoning to flow smoothly, assume that each business firm is established to satisfy a market niche so that the firm is operationally maintained by a positive cash flow as a consequence of its business conducts in the marketplace. and by consumers, it means end users of market offering; by customers those firms that employ their inputs to the role government policy and supports play to stimulate economic growth 33 copyright ©2021 assa. adv. in systems science and appl. (2021) produce their outputs. when both consumers and customers coexist, they are jointly known as customers. the rest of this paper is organized as follows. section 2 studies the functional mechanism of policy tools. section 3 investigates why governmental policies and supports are necessary in economic development. section 4 concludes the paper with practical recommendations and open questions for future research. 2. functional mechanism of policy tools this section investigates why policy tools are expected to work in the focal economic system? 2.1. a systemic modeling of the economy by system, it means such a notion that models an organization or a structure as a collection of components or objects and associations among the objects. the components have been conventionally treated as isolated in classical sciences, while the associations help make the components into an organic whole (or system) [14]. hence, the concept of systems really appears everywhere in life. for example, each family is a system; each business firm is a system; … as a matter of fact, a main characteristic of the world is the systemness of various kinds of organizations (and structures). this realization explains why the concept of systems can be appropriately applied in investigations of business-related matters and issues, see [4,15,16]. symbolically, a system is an ordered pair with containing all the isolated objects of system and relations that connect the objects in into a whole [14]. for example, each business organization consists of a set of components, such as employees, properties, equipment, etc. and, these objects are connected together through particular relations, because of which the whole is acknowledged as an organization. in this junction, one should note that each object in can also be a system again, just as in the case that a family consists of people as its objects, and each person is a biological system, made up of various organs; each organ is again a system of even smaller parts, … at the same time, a relation in does not have to be numerical. for example, it will not be appropriate to express the relationship that two people and are brothers, unless one only likes to capture this relationship partially. given system , another system is said to be a partial system of , if is a subset of and for any relation there is relation such that that is, the relation of the system is the restriction of a relation in the system . a system is multi-leveled, if there is at least one object such that is a system, and there is at least one object such that is a system, …, where is known as a first-level object system, a second-level object system, … a system is said to be centralized if each object in is a system and there exists a system such that and for any distinct objects x, y ∈ m, say and , and , where and . the system c is called a center of the centralized system s. speaking in non-mathematical terms, a system is centralized, if and only if there is another system that is a partial system of every object system in . to address the question stated above, let us model the economy of the focal nation as a system that consists of all economic agents as component parts, such as consumers, families, 34 j.y-l. forrest, a.k. jallow, c. wen, j.h. shi, h. guo copyright ©2021 assa. adv. in systems science and appl. (2021) business firms, economic sectors, and markets, and associations that connect the component parts into a functional economy. in this model, the associations reflect how money, information and knowledge flow within the economy, how the magnitudes and types of supplies and demands constantly change, how consumer’s preference and tastes evolve, etc. each family is a system that consists of family members; each business firm is a system that is made up of members of various families; each economic sector is composed of many firms; and each market connects individual consumers and firms through exchanges of goods, information, money, etc. that is, our focal economy can be well modeled as a multileveled system. within this systemic model of the economy, the government represents a formal structure that supports the stable existence and smooth operation of the economic system with the top level of the government serving as the center, as seen in the concept of centralized systems. any slight change in the center affects most areas of the economic system. the formal structure of the government can be imagined as the three-dimensional hierarchy of individuals, such as consumers on the bottom, whose consumption drives all other layers of the economy, business firms as the next layer upward, each of which serves a segment of consumers, … and markets on the top, where eventual exchanges of information, knowledge, money, goods, etc., take place. the government guarantees the economy operates smoothly through laws, regulations, and related reinforcements. 2.2. the importance of governments as a multi-leveled, large-scale complex system, a nation’s economy surely satisfies the following theoretical fact: theorem 1: under zfc let cardinality be infinite and regular, satisfying that for any , . if s = (m,r) is a system with each object also a system , satisfying • , • , for each object in s, and • there exists such an element g that belongs to at least objects in m, then there exists a partial system b of s satisfying that • the object set of b is of cardinality ≥ , • g belongs to each object of b, and • the partial system b forms a centralized system. (a) eddy motion of a general system (b) meridian field of the yoyo model (c) typical trajectory of how matters return fig. 1. the yoyo model for a general system for the proof of this result, see appendix. this theorem implies that within a nation’s economy, although there might be areas the government has either no or little influence, there is at least one portion of the economy that is of roughly the same scale of the entire economy, over which the government can influence. to better understand this explanation, the following yoyo model helps us intuitively imagine what a system is and how systems the role government policy and supports play to stimulate economic growth 35 copyright ©2021 assa. adv. in systems science and appl. (2021) evolve and possibly interact with each other. in particular, every system can be seen as an abstract yoyo in fig. 1 [14]: each system is a multi-dimensional entity that spins about its axis. if we fathom such an entity in our 3-dimensional space, such a structure, as shown in fig. 1(a), appears. the input side pulls in ‘things’ (e.g., materials, information, investment, and human talents). after funneling through the “neck”, things are spit out in the form of outputs (e.g., products, services, etc.). some things, spit out as outputs, never return to the other side and some will, fig. 1(b), where fig. 1(c) depicts the trajectory of how things return. due to its general shape, such a structure is known as a yoyo. from this systemic model, it can be seen that the government plays a non-negligible role in the nation’s economy as indicated by the center arrow in the yoyo body that provides the orientation for the system and influences component elements (industries/companies) of the abstract systemic model of the economy. speaking differently, when the government advocates and seeks after the goal of advancing the economy by increasing the degree of marketization, deepening political reform, and encouraging dramatic rise of private enterprises, its use of policy tools will be most likely successful. the reason why this approach practically works is that the government can effectively impose its will and desired outcome on a large segment of the economy through employing policy tools, as guaranteed by theorem 1 that characterizes centralized systems. so, the following conclusion follows: proposition 1: each government’s commitment stands for a process of social influence and support in which relevant governmental officials can gather the assistance and support of a large number of economic agents in the accomplishment of a determined national objective. such governmental commitment eventually creates a harmonious way for individual economic agents to work jointly and collectively to accomplish the desired outcome. a similar but more narrowly stated conclusion than this proposition has been empirically confirmed by [17] in the name of leadership and by [18]. by goal orientated system, it means such a system that focuses on a certain determined collection of tasks and consequences. the study on goal-oriented systems is voluminous. typical examples of goal-oriented systems are [19]: regulation, control, self-organization, learning, autopoiesis, self-reproduction, self-correction, adaptation, evolution, etc. by combining this concept and the systemic study of leadership [20], one has: proposition 2: the importance of the government is reflected in its ability to adjust the nation’s underlying organizational structure as a goal-oriented system so that most of individual economic agents will be able to adjust their orientations and operations without much difficulty. systemically speaking, there naturally exist development unevenness and imbalances among different regions of any nation, as shown by the evolution of flow patterns in the dishpan experiment [21]. in particular, a dishpan, filled with fluid, can model a nation’s economy with the “fluid” modeling the movement of such things as “money, information, goods, and others” within the economy. then the flow pattern, seen from either the input or output side, alternates between a uniform one, fig.1(a), and a chaotic one, fig. 2, demonstrating the fact that imbalances in economic development are commonly seen phenomena. fig. 2 depicts why this proposition holds true, where the entire economy is modelled as a spinning yoyo with fields around. the large dark-colored arrows through the middle stands for the orientation of the overall system, as desired by the government. if the orientation of a 36 j.y-l. forrest, a.k. jallow, c. wen, j.h. shi, h. guo copyright ©2021 assa. adv. in systems science and appl. (2021) local spin field, say a1, is not in agreement with that of the overall yoyo body, then a1 will be either dissolved under the joint effects of its adjacent fields or marginalized by forced to the periphery of the overall yoyo field, as indicated by theorem 1. so, a1 will not be any part of the centralized partial system b of the original economic system s. 2.3. conditions when policy tools will work proposition 3: if a government can employ its organizational influence by using policy tools to obtain aids and supports of other economic agents, be they large or small, in accomplishing a determined national objective that will benefit a large number of economic agents, there will appear a joint ambition that potentially involves a major portion of the economy. fig. 2. the structure of a goal-oriented system this is a restatement of theorem 1. the if-condition matches that the system s = (m,r) satisfies. therefore, the conclusion follows from the existence of the centralized partial system b. for a relevant but different study, see [17]. proposition 4: if a government is effective in terms of implementing its policies, then it will be able to align the business objectives of a large proportion of individual economic agents, even though they follow their respectively different and often opposing business strategies. hence, a united front of effort will be formed. this follows from theorem 1 and fig. 2, where the local fields located in the adjacent areas between a1, a2, a3, and a4, spin in directions opposite to those of a1, a2, a3, and a4. differences in directions reflect that businesses, even employing opposing strategies, can peacefully coexist. proposition 5: the initial concept of further advancing the nation’s economy can become an extraordinary reality through implementing appropriate policy tools. this result follows jointly from propositions 2 – 4 above. the importance of these propositions is that no matter what policy tool(s) are adopted and which implementation approaches are used, they need to be rooted in the outcome of benefiting a large economic segment for the policy and approach to actually work. such consistent, conceptual and practical specifications provide a common goal for a large proportion of economic agents to aim at and a cheering point for the citizens of the nation to feel excited about. the role government policy and supports play to stimulate economic growth 37 copyright ©2021 assa. adv. in systems science and appl. (2021) generalizing theorem 3 in [22] produces the following, which describes how a market signals its need for innovations and additional competition. the proof is given in appendix. theorem 2: in nash equilibrium, if the consumer surplus of the market described below is larger than the loyal-customer base of one incumbent company, then the market calls for new innovation and additional competition. if a company answers the call by entering the market with its version of substitute offer, then its expected profit can be potentially larger than that of at least one incumbent company. here, the market satisfies the following conditions: • it is served by a number of companies with their horizontally differentiated offers; • its operation is only affected by market forces, such as demand and supply, and consumers’ forever evolving preferences and tastes; • each incumbent company has a base of loyal customers if the price is not more than their reservation price. • there are switchers who make purchases based on whose price/value is lower; they are collectively known as consumer surplus; and • all companies’ pricing strategies are known to the incumbent companies who respond by playing nash equilibria through untainted self-analyses. for the market considered above, it generally means that the technology and management used in production have been standardized. hence, for a company to profitably enter the market, it must have introduced a more efficient technology and/or managerial routine that can significantly reduce its business expenditure in making its product of increased sophistication and functionality, as christensen et al. [23] confirms empirically. 3. the necessity of governmental policies and supports 3.1. linkage mappings within a supply-chain ecosystem to address why government policy tools and supports are fundamentally necessary for stimulating economic growth, theorem 2 indicates that when recognizing a market signal innovatively, a firm is in a competitive position over its competitors. however, in practice, the consequent enhanced performance of the firm really depends on other players in its supply-chain ecosystem [24]. in particular, to produce the offering to satisfy an innovatively recognized demand, a firm has to obtain its needed supplies from its suppliers. that can be a challenge to the supply-chain ecosystem. even after having met such challenge and created the expected value, the firm can still be hampered in its capturing of value. that might well depend on infrastructures necessary for the firm to reach consumers in the marketplace [24]. the supply-chain ecosystem of any firm consists of upstream components, such as suppliers, and/or downstream complements, such as customers, supporters and assistants, known as complementors, who help to make the firm’s product usable by consumers [24]. although outside the firm’s direct supply chain, complementors help connect the firm’s offering and consumers by constructing infrastructures. for example, governments, regulators, etc., are complementors that help build and maintain, for instance, roads necessary for transportation or specifications of new safety procedures, etc. the systemic structure of this supply-chain ecosystem is given in fig. 3. the focal firm utilizes inputs, called components, from n supplies and delivers its outputs to customers with the support and assistance of m complementators, m, n = 1, 2, 3, … other than the shown first-tier components and complements, this structure in real life needs to extend leftward and rightward along the value-creation chain to include other tiers, such as suppliers' suppliers, customers' customers, complementors’ complementors, etc. by including the full 38 j.y-l. forrest, a.k. jallow, c. wen, j.h. shi, h. guo copyright ©2021 assa. adv. in systems science and appl. (2021) range of different tiers, the systemic characteristics of this ecosystem can be vividly seen, where the focal firm internally focuses on providing organizational supports so that its employees can innovatively integrate components into market offerings. for similar systemic constructs, see, for example, [24,25]. fig. 3. the supply-chain ecosystem of a focal firm let denote the supply chain of a firm. model firm in this chain as an abstract system . assume the index set of these firm systems is , satisfying that for any , there is a unique firm system in the supply-chain ecosystem; and for any firm system in the supply-chain ecosystem, there is a unique index that labels the system . then, can be modeled as a hierarchy of firm systems , for . let , i = 1, 2, be systems and h: → a mapping. for each relation , define . without confusion, write h: → . a mapping h: → is said to be partial, if for some object , is not defined. the identity mapping is defined by , for . theorem 3: assume that the hierarchy does not include such systems and that some outputs of are used as inputs of , while some outputs of are employed by as inputs. then the supply-chain ecosystem can be made into a partially linked hierarchy of systems by a family of partial linkage mappings of . the proof is given in appendix. 3.2. maintenance of an existing momentum of economic growth to investigate how an existing momentum of economic growth can be maintained, let us first look at the concept of feedback systems [14]. assume that and are two linear spaces and and and and linear functions from to and from to , respectively. then the feedback system of s by is defined as the input-output system such that the following expression holds true in our current context, theorem 3 means that the existence of a large magnitude consumer surplus, as provided in the assumptions of theorem 2, represents the fact that the incumbent firms of the market can no longer satisfy the forever changing consumer preference and tastes. in real life, of course, this fact (or market call) can be understood in many different ways. if our focal firm innovatively deciphers the market call by introducing its original product, then that implies that each tree-like subset of the partially ordered set t in terms of the order relationship contains minimal elements, consisting of various market demands, and that every player in the supply-chain ecosystem of the focal firm will be the role government policy and supports play to stimulate economic growth 39 copyright ©2021 assa. adv. in systems science and appl. (2021) challenged to provide their correspondingly innovative supplies in order to jointly answer the market call. even though the initial market call might not be answered adequately by a single firm’s reply with its new product, the aggregate of all relevant original products offered by various entrepreneurial firms will definitely satisfy the market demand. so, this discussion combined with the concept of feedback systems leads to the following fact regarding the importance of government’s policies. proposition 6: to maintain the extant economic growth momentum, assuming that such a momentum already exists, a nation has to constantly provide “fuels” through policy supports for the prevailing feedback system, which reinforces the horse race between market demands and manufacturing production so that the race will continue to intensify, to continuously function. in fact, what is discussed in the previous paragraphs indicates that market exchange stimulates manufacturing production, while the production encourages consumers to alter their preferences and tastes. this cyclical evolution continues and helps both market exchange and manufacturing production enter into a horse race against each other. that is, an operational feedback system appears. in this feedback system, forever changing market demands provide stimuli for manufacturers to produce more products and make better products. and the growing magnitude of manufacturing production forces firms to hire more employees with rising salaries. such mutual reinforcements further strengthen the population’s purchasing power, which in turn pushes market demand onto a higher level; … to keep the prevailing feedback system functional smoothly, the economy (or the government) has to encourage as many entrepreneurial firms as possible to answer various market calls by providing their original products. however, many of these original products will not be readily usable by the consumers of the marketplace unless adequate infrastructures are constructed and maintained by forces supported either directly or indirectly by the government. speaking differently, the reason why policy supports are needed to fuel the prevailing feedback system in order for it to function smoothly is because the system is too colossal for any individual economic enterprise to handle, while the government can reach and mobilize a large proportion of the economic system (theorem 1). hence, the government can at least help coordinate many individual firms in very large scales to join their efforts, resources, etc., by utilizing policy tools and supports. for example, the literature shows that to stimulate economic growth the government can use policy tools to • promote the ideology of commercialization at various societal levels, especially in rural areas [2,26]; • encourage entrepreneurship and risk taking [27]; and • provide the freedom for people to move geographically and professionally and to make career choices [28], and can provide various supports [2,4,16,29,30], such as • development of the necessary domestic and international markets of sufficient depth and distribution networks of appropriate sophistication; • smooth operations of these markets and networks; • financial and politically stability; • construction of necessary infrastructures (such as roads, irrigation systems, energy supplies, etc.); • smooth flows of knowledge and information that promote coordination and specialization of labor; 40 j.y-l. forrest, a.k. jallow, c. wen, j.h. shi, h. guo copyright ©2021 assa. adv. in systems science and appl. (2021) • educational programs for necessary skill trainings; • resources and needed coordination of resources; • and others. moreover, the discussion in the previous sections elucidates why further development of the focal nation’s economy needs to pay equal attention to increasing degree of marketization, continuously deepening political reform, and dramatically raising the magnitude of market sector of private enterprises. in particular, by increasing the degree of marketization, market demands can be freely and adequately signaled through market phenomena and forces. by dramatically raising the magnitude of the market sector of private enterprises, entrepreneurs can individually and differently comprehend market signals and take their respectively appropriate actions by designing and offering their varied original products and services to the marketplace. by continuously deepening political reform, the government’s way of conducting its business can be brought closer to the current state of the accelerating horse race between the market demands and manufacturing productions. only by synchronizing the political system that is underneath the composition and operation of the government with the prevalent working state of the feedback system of the economy, which reinforces the horse race between market exchanges and manufacturing productions through raising purchasing power of the population, the overall organizational structure of the focal nation, seen as an entity that is simultaneously political, economic, social, etc., will stay as a stable, smoothly functional system. 4. conclusion because of the use of systems science and methodology, this work is able to scientifically address: (1) why are policy tools expected to work in an economic system? (2) why are government policy tools and supports fundamentally necessary for stimulating economic growth? and (3) several other relevant issues. historically, there had been three rounds, plus the current round for a total of four rounds, of debate that directly attempted to address these questions from different angles. however, due to the fact that only data/anecdote-based approaches were employed, what achieved turned out to be theoretically inconsistent and uncompromising suggestions that are practically unreliable [3]. the entire situation is similar to that reflected in the literature on the industrial revolution [2,4] but with such a major difference that for the latter rostow recognized the need for introducing an appropriate methodology in order for relevant studies to escape from the trap of not being able to produce consistent conclusions. by comparing these two lines of related but different studies, this paper develops generally-true conclusions by innovatively employing logical reasoning based on results established rigorously. in particular, among others, the following main results are established: • if a government is able to introduce and implement appropriate policy tools, then the determination of further advancing the nation’s economy can become an extraordinary reality. it is because (propositions 2 – 5) in this case the government is able to adjust the economy’s underlying organizational structure as a goal-oriented system within which most individual economic agents will be able to adjust their orientations and operations without much difficulty. • to maintain an existing momentum of economic growth, a nation needs to constantly provide policy supports for the prevailing feedback system, which reinforces the horse race between market demands and manufacturing production so that the race will continue to intensify, to continuously function (proposition 6). the role government policy and supports play to stimulate economic growth 41 copyright ©2021 assa. adv. in systems science and appl. (2021) because of the particular methodology employed here, the established theoretical conclusions provide policy makers with dependable bases to make their decisions. for example, the general recommendations below for a national government follow naturally: (1) it needs to constantly maintain its capability to make adjustment(s) to the underlying organizational structure of its economy. only with such capability, the government is able to fine-tune and to redirect the development and evolution of the economy (proposition 2). (2) it needs to frequently identify objectives of economic development that potentially benefit most economic agents. by accomplishing such a goal, the policy makers of the nation will be able to develop a national ambition supported by a majority of the nation (proposition 3). (3) if all possible, it needs to find effective ways to implement its policies in order to practically materialize the goal of further advancing the nation’s economy from the present state (propositions 4 and 5). (4) it needs to create and maintain a functional feedback mechanism that intensifies the horse race between market demands and manufacturing production (proposition 6). first, by creating such a feedback mechanism from scratch, an originally impoverished nation can potentially evolve into an economy of proto-industrialization [16,30], where local handicraft productions, alongside commercial agriculture, will be developed to such a level that, beyond local markets, can also satisfy external markets. secondly, by maintaining such an existing feedback mechanism, a developed economy will be carried to a higher level of economic development [2]. there are also some limitations to this work. first, although conclusions of this paper depend heavily on the methodology of systems science, we only applied one of the many tools available that are developed for analyzing organizations, their evolutions and interactions [31]. so, when other tools of systems science are employed one by one in studies of business-related issues and problems, we will be able to establish finer and reliably applicable conclusions. second, all reasonings used in this work assume why a firm exists – it attempts to satisfy a particular market niche with its operation financially maintained by a positive cash flow as a consequence of its business conducts. although this assumption is conventionally true in the past [32], it is no longer the case in the present business landscape. for example, some e-business operations have focused on pumping up their future promises and potentials to continuously attract sufficient venture capitals by placing emphasis on increasing their market shares, although they have been losing money year after year since their inception [33]. hence, any violation of this assumption can realistically stand for why applications of our general conclusions developed in this paper fail to work in practice, if the firms of concern exist for purposes other than what is assumed. these limitations, along with others not listed here, of this current work provide directions for future research. appendix: proofs of theorems for related terminologies and symbolic expressions used in these proofs, see [14]. proof of theorem 1. for convenience, assume that and that there is a common element in all the object systems in m, which implies (1) without loss of generality, for each object ∈ m, assume (2) so, for each , the order type < of is a subset of . because is regular and , there exists a so that = {z ∈ m: has order type } has cardinality . let us fix such a and deal only with the partial system of s, where is the restriction of the relation set r on . 42 j.y-l. forrest, a.k. jallow, c. wen, j.h. shi, h. guo copyright ©2021 assa. adv. in systems science and appl. (2021) for each , implies that less than objects of the partial system have object sets as subsets of and is cofinal in . if z ∈ and , let (ξ) be the ξth element of . because is regular, there is some ξ such that { (ξ): z ∈ } is cofinal in θ. let be the least such ξ. then the condition that there exists a common element in each system in implies that > 0. let (3) then < and < for all z ∈ and all . by transfinite induction on , pick so that and (4) let . then and whenever , are distinct objects. because for each , there exists an and a with and for each , , forms a centralized system, where is the restriction of the relation set r on b. proof of theorem 2. assume that the market is occupied by incumbent companies, the, individually different boundary conditions of companies can be normalized so that the production cost is 0, customer’s reservation price 1, price satisfies , loyal-customer bases of the incumbent companies are of the same percentage scale with being the scale of the consumer surplus. assume that the entering company randomizes its price between its cost 0 and consumer reservation price 1. to protect their established territories, each incumbent sets its price by considering all competitors. so, the equilibrium indifference condition of incumbent company is (5) where is the price distribution of company ; and in nash equilibrium, the incumbents do not have any pure pricing strategy [22]. hence, for the incumbents, their symmetric equilibrium pricing is (6) the assumption implies that equation (6) defines a mixed strategy for each incumbent for . because experiences a jump of at p = 1, the expected profits of the entrant are: (7) (8) the first term on right-hand side of equation (7) is equal to the entrant’s expected profits when it charges a price lower than the incumbents and captures all switchers, and the second term the entrant’s expected profits when it directly competes with the incumbent companies. the expected profits of an incumbent are the role government policy and supports play to stimulate economic growth 43 copyright ©2021 assa. adv. in systems science and appl. (2021) (9) because , and when , , there is so that when , . that is, the entrant can actually expect to make more profits in the said market than some incumbent. proof of theorem 3. in fact, the input-output relationship among the firms in implies that the index set can be ordered partially: for , if, and only if, some outputs of firm system are used as inputs of firm system . for , if , define a partial (linkage) mapping from firm into customer firm by (10) for any such that system ’s output is applied as a component in ’s product . because for any , satisfying , we have (11) this systemic modeling implies that we can treat the supply-chain ecosystem of any firm as a partially-linked hierarchy of firm systems . references 1. hermann, m., pentek, t., & otto, b. (2016) design principles for industry 4.0 scenarios. the 49th hawaii international conference on system sciences (hicss), koloa, hi, 3928–3937. 2. wen, y. 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(2015) complexity analysis of dual-channel game model with different managers’ business objectives, communications in nonlinear science and numerical simulation, 20: 199–208. microsoft word 1049 article text, copyedited.doc adv syst sci appl 2022; 01; 1-14 published online at https://ijassa.ipu.ru. plasma magnetic control systems in d-shaped tokamaks and imitation digital computer platform in real time for controlling plasma current and shape yuri v. mitrishkin* lomonosov moscow state university, faculty of physics, moscow, russia v.a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia e-mail: y_mitrishkin@hotmail.com abstract: the direction of developing plasma magnetic control systems in d-shaped tokamaks with its actuality and scientific novelty is presented. shortly the poloidal systems of iter and demo are characterized. the new imitation computer digital platform for modeling plasma magnetic control systems in real time for application in physical experiments is described. the preliminary results of testing the plasma magnetic control system for the globus-m2 tokamak (ioffe institute, s-petersburg, russia) on the target computer of the platform are given. the digital twin of the imitation platform is characterized. the methodology of risk minimization is presented. keywords: tokamak, plasma control, imitation digital platform, real time, globus-m2, iter, demo 1. introduction the world challenge of the controlled thermonuclear fusion (ctf), researches on which have been begun in the beginning of 50th years of the last century, is one of central problems in science and engineering. the solution of this problem will open a new, safe, almost inexhaustible source of energy from the synthesis of nuclei of light elements, as well as eliminate carbon dioxide emissions into the atmosphere from the combustion of natural resources and stop climate change on the earth in an unfavorable and dangerous direction for mankind. fig. 1. vertically elongated tokamak without an iron core: 1 is the vacuum vessel; 2 is the toroidal field coil; 3 is the poloidal field inner and outer coils; 4 is plasma and helical magnetic lines (© iter project center, russia) * corresponding author: y_mitrishkin@hotmail.com 2 y.v. mitrishkin copyright ©2022 assa. adv. in systems science and appl. (2022) the most perspective direction of the ctf is tokamaks with d-shaped cross-section (fig. 1), on the basis of which it is planned to create thermonuclear power plants. confinement and heating a plasma in tokamaks to provide a self-sustaining thermonuclear reaction are ensured by magnetic and kinetic plasma control systems with feedback. the plasma in the magnetic field of a tokamak vacuum vessel represents extremely difficult for diagnostics and control a nonstationary nonlinear dynamical plant with distributed parameters, with structural and parametric uncertainties, subject to the influence of uncontrolled disturbances in the presence of broadband noises. that's why one of the main tasks of the ctf has not been solved yet specifically prolonged confinement of the high-temperature plasma in tokamaks with set parameters and prevention of plasma major disruptions. the paper is structured as follows. section 2 highlights the significance of plasma magnetic control systems in tokamaks. in section 3 the scientific novelty of the direction presented in the paper is shown and concentrated on the imitation digital computer platform in real time for controlling plasma current and shape in tokamaks. section 4 is devoted to some important features of tokamaks in whole, iter, and demo. in section 5 the short introduction into practice of physical experiments of the globus-m2 tokamak of the digital plasma control system is given. section 6 presents the basic new idea of the paper specifically the imitation digital computer platform structure in real time having new design for application to plasma control in tokamaks. in section 7 the digital twin of the plasma magnetic control system is shown in the digital imitation platform. the results of the preliminary testing of one of the plasma current and shape control systems for the globus-m2 tokamak on the target computer is given in section 8. section 9 describes the methodology of risk minimization of application of the direction suggested. the conclusion summarizes the main new features of the direction of the plasma control systems development. 2. significance of plasma magnetic control systems in tokamaks in this situation the urgency of continuing the development and research of magnetic plasma control systems in tokamaks only increases [2, 3]. it is explained first of all by the fact that the systems of magnetic plasma control have not yet reached a proper level of reliability and survivability for their application in thermonuclear reactors, where increased reliability of systems operation in conditions of presence of the thermonuclear reaction in the plasma of reactors is required. it is especially important for round-the-clock operation of thermonuclear power plants. moreover, modern experiments on tokamaks require more and more advanced plasma control systems, which would allow deeper and more accurate study of plasma processes during generated discharges, could guarantee rejection of minor disruptions and also prevent major disruptions, so that they do not lead to emergency situations and destruction of fusion power plants. in doing so, at present nobody knows models of the plasma with the thermonuclear reaction and so nobody can guarantee the operability of the plasma magnetic control systems in thermonuclear reactors like iter (international thermonuclear experimental reactor) and demo (demonstration fusion power plant). design of plasma magnetic control systems in tokamaks should have the feature of automatic adaptation to fusion plasma new models. 3. scientific novelty of direction of control systems design further development of the plasma magnetic control systems in the stated direction in the continuation of the achievements in the grant of the russian science foundation № 17-19-01022 (2017-2019) “hierarchical automatic control systems of plasma position, current, and shape in toroidal axial-symmetric magnetic configurations in a wide range of aspect ratios” will be carried out in real time with their high scientific novelty by the team of [4]. plasma magnetic control systems in d-shaped tokamaks and imitation digital 3 copyright ©2022 assa. adv. in systems science and appl. (2022) the novelty will be provided primarily by the originality of the imitation digital computer platform created in real time. installation of new plasma control algorithms obtained on the digital twin of the control system is planned to be practiced over the internet. an application for a patent of the russian federation is being made for this digital imitation platform. the platform will allow setting up plasma control systems with an algorithm for plasma equilibrium reconstruction in real time. the russian federation patent for invention no. 2702137 was received in 2019 for the method of modeling such setting [5]. the industrial computer will allow replacing all analog controllers in the control loops of plasma position and currents in the coils of the central solenoid and poloidal field with digital controllers (8 controllers) on the globus-m2 tokamak itself, and will also allow applying the plasma shape control system with feedback in experiments. digitalization of the whole system of magnetic plasma control on the tokamak through the third platform computer will allow to set and solve a new task of optimization of joint coordinated operation of all plasma control circuits in real time. as it follows from the surveys on the systems of magnetic plasma control in tokamaks [6, 7], such a task was not set and was not solved by anyone. the solution of this problem is necessary for an optimal integration of the plasma magnetic control system with the kinetic control system of plasma parameters profiles (current density, safety factor q, density, temperature, pressure) in the long term. fig. 2. general configuration of three levels hierarchical control system [8] novelty of the plasma control systems will be provided by the fact that the control systems will be developed in the class of digital hierarchical control systems (fig. 2) for multivariable dynamic plants with time varying parameters [8]. this scientific direction allows to obtain various practically unlimited new combinations of different control and identification algorithms at different hierarchical levels. sensor networksactuators controlled process baseline controller adaptation algorithm selforganization algorithm disturbance noise i. basic control level ii. adaptive level iii. intelligent level general feedback loop decision-making self-learning, self-configuring self-optimizing level function adapting basic controller to time-varying plant level name _ r stabilizing of plant outputs y or tracking of references r yv e 4 y.v. mitrishkin copyright ©2022 assa. adv. in systems science and appl. (2022) the main directions in development of hierarchical plasma control systems will be as follows: ø identification (building models based on experimental data of observation of inputs and outputs of the controlled plant), ø robust control, ø adaptation, ø elements of artificial intelligence. in the control directions it is supposed to be developed: • control systems with a predictive model with adaptation of parameters to changing parameters of the controlled plant [9]; • robust systems with multivariable pid controllers adjusted by means of linear matrix inequalities [10]; • decoupling of control channels with subsequent adjustment of control channels by means of the method of quantitative feedback theory [11]; • decoupling of control channels by means of relative gain array approach and h∞optimization theory [12]; • robust and adaptive systems with coordination of control loops of position and shape of the plasma [13]; • methods of picard iterations, moving filaments, and neural networks for reconstruction of plasma equilibrium on the basis of the magnetic measurement outside the plasma [14-16]. at the same time, the development and application of new linear and non-linear plasma models will be continued. linear models are designed for synthesis of linear magnetic plasma control systems without taking into account slow plasma processes related to the transfer of matter and energy. non-linear models take into account the transfer processes and are aimed at future development and implementation of kinetic plasma control systems. novelty of plasma control systems development directions of our team is confirmed by three reports at the ifac world congress in 2020 (ifac wc 2020), which was held remotely in berlin (germany) [9, 10, 16]. 4. tokamaks, iter, and demo europe is a leader in solving the ctf problem. at the european tokamaks, in particular, jet (major radius r=3 m, england), asdex upgrade (r=1.65 m, germany), tcv (r=0.88 m, switzerland), the experimental researches of the high-temperature plasma are intensively conducted, new plasma control systems are developed, applied, and investigated. in europe, the first iter thermonuclear reactor with r=6.2 m is under construction in cadarache (france) [17, 18], where the assembly of the reactor itself has already been started. there are two roadmaps for the creation of demo on tokamaks: (a) with large aspects ratio [19] and (b) spherical tokamaks [20, 21]. the first roadmap leads to the giant demo having the very large major radius of about 9-10 m. the second roadmap leads to the modular demo on spherical tokamaks as modules with the lower aspect ratio and relatively small major radius specifically less than 2 m. the cost of electricity from giant tokamaks should be very high but the cost of electricity of the modular demo is very competitive and is of about 0.06 $/kwh [21]. plasma magnetic control systems in d-shaped tokamaks and imitation digital 5 copyright ©2022 assa. adv. in systems science and appl. (2022) a b c fig. 3. vertical cross-sections of (a) iter [22], (b) asdex upgrade [23], (c) globus-m2 [24] there are a set of poloidal systems (poloidal field coils around the vacuum vessel) for the giant demo as copies of the poloidal system of iter with the rough mistake in the design of this system [22]. this original system led to the nonsense: at the major radius of iter of 6.2 m the controllability region of the plasma unstable vertical position was about 3-4 cm. because of that the internal extra coils to control plasma unstable vertical position have been installed into the vacuum vessel of the iter tokamak (fig. 3a) [22]. the engineering solution of including the pf-coils inside the vacuum vessel of demo is not the best idea from the following points of view: • the thermonuclear plasma may burn these coils, • the space for the plasma became less inside the vacuum vessel and the plasma is located farther from the first wall in comparison with tokamaks which do not have the pf-coils inside the vacuum vessel. this essentially decreases the effectiveness of acting on the plasma of the eddy currents induced in the first wall. it is much effective and reliable to locate the pf-coils for the plasma vertical position control between the vacuum vessel and the toroidal field coil like for instance in the asdex upgrade tokamak (fig. 3b) [23] or design the whole poloidal system like in the spherical globus-m2 tokamak (fig. 3c) [24]. 5. introduction into practice of physical experiments of the globus-m2 tokamak of the digital plasma control system the digital simulation platform will provide an opportunity for the first time to implement the developed new plasma magnetic control systems in real time in the physical experiments of the operating globus-m2 spherical tokamak. solution of off-line identification tasks will be carried out on the existing globus-m2 tokamak equipment, which is planned in the direction for successful guaranteed application of new plasma control systems in the experiments. the multivariable plasma magnetic control system which is operating at present on the spherical globus-m2 tokamak in experiments is shown in fig. 4a [13]. it contains 2 closed control loops for plasma position, 1 control loop for the current in the central solenoid (cs), and 5 control loops for the currents in the poloidal field (pf) coils. to generate plasma discharge scenarios the program reference signals are fed to these loops. 6 y.v. mitrishkin copyright ©2022 assa. adv. in systems science and appl. (2022) a set of plasma magnetic control systems for the globus-m2 tokamak was developed and modelled in line with the new methodology presented in [25] for plasma current and shape control with feedback in the matlab/simulink environment. 6. description of the imitation digital platform there is the basic approach to model plasma control systems in real time by means of test bed containing two target computers connected by feedback [26]. for control of plasma current and shape with feedback in real time on the globus-m2 tokamak it is proposed to create and apply a new real-time imitation digital computer platform, which has a feedback bunch on two high-tech computers with high efficiency. these are target computers such as " plant simulator ̶ controller", where the switching in the “controller” of the control algorithm from the internal model to the “plant simulator” and back takes place to configure the control system with subsequent transfer of a control algorithm through remote access to the target computer with its internal model connected to the tokamak (fig. 4b). the real-time platform realizes real-time plasma simulation in the tokamak with diagnostics, actuators, control loops for the horizontal and vertical position of the plasma and currents in the pf-coils as well as plasma current and shape loops on three industrial computers (ic) with communication devices with the controlled plant, which is the plasma in the tokamak. the ic1 simulates only the plasma in the tokamak with acting control loops of plasma position and cs/pf-currents (fig. 4a), while the other ic2 and ic3 simulate the plasma in the tokamak with acting control loops, and the plasma current and shape controller connected to each other by internal feedback with the plasma equilibrium reconstruction code at the controller input. the ic2 and ic3 use two multivariate switches which switch the c controller of the ic2 from the plasma internal model (min) to the plasma external model (mext) or the tokamak and back to adjust the control system on the min and then apply it out on the mext or the tokamak. that allows to simulate by the c controller of the ic2 connected to the mext real physical experiments on the tokamak. host computers (hc) for the development and modeling of plasma control systems in computer time with a display d for visualization of modeling of plasma control processes are connected to each of ic1, ic2, and ic3. these hcs are used to carry out loading from them the developed controllers and plasma models into ic1, ic2, and ic3. as well as displays d are connected to ic1, ic2, and ic3 to visualize internal control processes in real time. a database server (s) is connected to each of the three hcs, through which the data are downloaded for all hcs to develop plasma control systems in computer time and further to be downloaded to ic1, ic2, and ic3 for the purpose of modelling plasma control systems in real time in line with the simulation method with plasma reconstruction code in the feedback [5] and future plasma control in real time. the ic3 with hc and d is installed and connected directly to the tokamak, the internal controller c with the reconstruction code at its input is set up on the min in real time, and then is switched to the tokamak. as a result, the controller c previously adjusted on the min controls the plasma current and shape in real time during plasma discharges. for the progress of plasma control systems, the internal controller c in the ic3 may be changed by means of remote access to a new internal controller. then this new controller is adjusted on the min and is switched to plasma control in the tokamak in real time by means of two multivariate switches. plasma magnetic control systems in d-shaped tokamaks and imitation digital 7 copyright ©2022 assa. adv. in systems science and appl. (2022) a b fig. 4. (a) block diagram of the plasma control system operating on the globus-m2 tokamak: cz, cr, ccs, cpf1, cpf2_top, cpf2_bottom, cpf3, and ccc are analogue controllers, ahfc, avfc, acs, apf1, apf2_top, apf2_bottom, apf3, and acc are actuators [13]. (b) structural scheme of the imitation computer platform: c is a controller, min, mout are internal and external plant models, d is a display, hc is a host computer, ic is an industrial computer, s is a server, tok is a tokamak 7. digital twin of real control sytem in imitation platform digital twins of control systems are being developed in industry when designing control systems because control processes can be simulated in real time on the digital twin, they can be optimized or new control algorithms can be synthesized and the optimal system results found are then applied to the real plant under control. this is much cheaper than solving the same problems at once on the real plant. at the same time, digital twins are also used to help to control complex dynamic plants, for example, in experimental physics, in order to find optimal control solutions for these plants independently of the physical experiments. in the imitation platform (fig. 4b) two industrial computers namely ic1 and ic2 constitute the digital twin and the third industrial computer ic3 is planned to be connected to the real plant specifically the globus-m2 tokamak to organize the real plasma digital control system fig. 5. digital twin of feedback control system of real dynamical plant [ifac 2020 wc] cz ahfc ― ccs acs tokamak ― ics reference magnetic flux (21 loops) ip z reference z ihfc ivv+p cr avfc ― r reference uhfc uvfc ivfc ics cpf1 apf1 ipf1 ipf1 reference cpf3 apf3 ipf3 reference ipf3 ccc acc icс reference icc ipf2_top ipf2_bottom r ucs upf1 upf3 ucc cpf2_top apf2_top ipf2_top reference cpf2_bottom apf2_bottom ipf2_bottom reference upf2_top upf2_bottom ― ― ― ― ― mest ― ic2 мin hc c ic1 dhc d ― ic3 мinc dhc d s msu/ics ras tok ― ic3 мinc dhc d ioffe inst. d plant controlled modelcontrol unit digital twin real time data virtual space feedback real controlled plantcontrol unit real control system real space feedbackrecommendations 8 y.v. mitrishkin copyright ©2022 assa. adv. in systems science and appl. (2022) 8. plasma control system tests in real time the preliminary tests of the control system of plasma current and shape as “isoflux control” (fig. 6) [15, 27] on the target ic of speedgoat type (www.speedgoat.com) with simulinkrt operation system were carried out in real time, which showed sufficient speed of response for physical experiments on the globus-m2 tokamak. from the mhd theory of tokamak plasma configurations it is known that the magnetic field at the x-point for any discharge is zero and the plasma boundary is an isolevel line of the poloidal flux. thus, by minimizing |br|, |bz| the x-point is placed at the desired location and by minimizing the plasma separatrix is placed at the points 1 and 2 (fig. 7). to test the operation of the control system in real time, the entire control system was transferred to discrete time and tested on the performance real-time computer of speedgoat (www.speedgoat.ch) with a processor intel(r) core(tm) i7-3770k cpu 3.50ghz. the main time step was 50 μs for all subsystems, except for the equilibrium reconstruction algorithm: for it the time step was 500 μs. as a result, a 50 μs task is performed on average in 24 μs and a 500 μs task is processed on average for 347 μs for 5 filaments. this shows that the system, taking into account computer power, is quite applicable in real experiments. the results of the simulation are shown in figs. 8 and 9. fig. 6. block diagram of multiloop and multivariable hierarchical plasma magnetic control system of the globus-m2 tokamak [15] plasma magnetic control systems in d-shaped tokamaks and imitation digital 9 copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 7. the idea of the isoflux plasma shape control [15] fig. 8. deviations of the magnetic field in the location of x-point for the shot № 31648 in discrete time [15] fig. 9. deviations of the flux in the desired location of the plasma separatrix (fig. 4) for the shot № 31648 in discrete time [15] 10 y.v. mitrishkin copyright ©2022 assa. adv. in systems science and appl. (2022) 9. methodology of risk minimization of research any new scientific work is subject to different kinds of risks, which may not lead to the desired result, especially as in this project, when the research is associated with a complex technique, a large field of research of a complex controlled plant, such as the plasma in tokamaks, the development of new methods and systems of plasma control, the introduction of real-time platforms in the computer network, organizational difficulties in solving problems with foreign partners, etc. therefore, it is important to foresee in advance the possibility of appearance of different types of risks and methods of their minimization or complete elimination: ü when developing new tools, there may be the risk of their ineffectiveness or their application and refinement may take too long. this risk can be reduced by the creative abilities of team members. in this case, the most talented and persistent professionals get into the team who have abilities to minimize this risk. ü the risk that the purchased equipment in the form of speedgoat performance computers is not fast enough to solve the problems of magnetic plasma control in the globus-m2 tokamak. in order to minimize this risk, it is necessary to continue the work on studying these computers. for this purpose, it is necessary to consider variants, when delays in communication devices with the plant can have even less delay, than 8 µs. it can be fpga or use of the hdl coder. this issue is part of the study for the applicability of plasma position control channels of the imitation platform. for plasma current and plasma shape control, these delays are quite small and do not interfere with the control process, as our already conducted studies show. in order to reduce the risk of applying real-time plasma control platform to control the plasma in the globus-m2 tokamak, it is advisable to first purchase only two speedgoat performance computers, mount, configure, and test the platform on two "plant-controller" computers connected by the feedback, simulate on it the available plasma position, current, and shape control systems, to understand what needs to be changed in this platform (if necessary) in order to adapt it to solve plasma control tasks on the globus-m2 tokamak. after that, one can purchase one speedgoat performance computer with corrected characteristics to be applied on the tokamak directly. ü the risk from withdrawal from the team of some of its members. such risk exists, but it can be minimized by attracting the most talented young people to the team. for this purpose, the consortium has a leading higher education institution of russia namely lomonosov moscow state university, which can solve this problem because it is the source of young talents in their specialty. ü the risk from unacceptable input-output controllability of the controlled plant [28]. it means that it is impossible to achieve the set control goals on this plant because of its properties, which were obtained when creating this plant. for example, such situation arose with iter, when the poloidal system was already designed and the project development was in progress, it was found that at the major radius of tokamak of 6.2 meters, the vertical region of plasma controllability is only 3-4 cm. in this case it was necessary to change the controlled plant, which was done by introducing additional horizontal magnetic field coils inside the vacuum vessel. this led to expansion of the vertical controllability region by order of magnitude, which is enough for this machine. in the demo project such mistake should not be made: fusion electric power plant should work in twenty-four-hour mode and with increased reliability level, and presence of coils inside the vessel essentially reduces this reliability. therefore, at the next stage in the world project it is proposed for demo, first of all, to develop a poloidal tokamak system, so that the controlled plant has an acceptable level of input-output controllability for the design of a sufficiently reliable control system of the plasma position, current, and shape. ü the risk from feedback controller synthesis, which does not provide sufficient stability margin and does not provide the required performance. in this case, to minimize this risk, the developers of the control system should first analyze the sources of poor operability of the controller. such sources may be the following: plasma magnetic control systems in d-shaped tokamaks and imitation digital 11 copyright ©2022 assa. adv. in systems science and appl. (2022) o insufficient accuracy of the plant model on which the control system is synthesized; o insufficient accuracy of reduction of either the plant model or of the controller itself; o failure to take into account all the properties of the plant during synthesis of the controller, which affect its operability, for example, failure to take into account all disturbances, which affect the controlled plant, plant varying parameters, nonlinearities, saturated input and output signals, etc. in this case, the developers must identify the cause of the unsuccessful synthesis of the controller, redesign the controller, again simulate the entire system. synthesis of the system should be iterative and cover not only modeling, but also application to the controlled plant. fig. 10 shows the controller design cycle, which covers a number of stages [26]. in general, it should be noted that in thermonuclear tokamak reactors an explosion is impossible, because in case of emergency it is possible to close the valve of gas supply to the vacuum vessel and the process development will stop. therefore, fusion electric power plants are safe sources of energy in comparison with nuclear power plants. fig. 10. design cycle of dynamic plant control systems to achieve the required stability margins and performance of systems [26] 10. conclusion as follows from the indicated novelty of the direction and plenary reports at the ifac 2020 world congress, the magnetic plasma control systems in the globus-m2 tokamak will be developed in accordance with the world trends in the development of automatic control systems: • digitalization, automation, and elements of artificial intelligence lead to the autonomy of systems, which may give the human independent systems. the direction involves the use of artificial neural networks to reconstruct the plasma equilibrium, which are elements of artificial intelligence; • creation of digital twins of control systems to be created on the digital simulation platform; • the increasingly widespread use of machine learning, since the amount of information processed, is constantly increasing, for example, in iter, the codac information management system will process about 1 million signals. in this direction, similar methods will be used to 12 y.v. mitrishkin copyright ©2022 assa. adv. in systems science and appl. (2022) configure neural networks both on evolutionary plasma models and according to the data from real experiments. nobody knows the models of the plasma with thermonuclear reaction and so nobody can guarantee the operability of any known plasma magnetic control system. so, one should design plasma magnetic control systems for iter and demo having a feature of automatic adaptation to a fusion plasma new model. the design of such control systems is in line with our basic research direction which corresponds to the world development of control systems: combination of automation, digitalization, artificial intelligence giving autonomy. acknowledgements the author thanks the admins of v.a. trapeznikov institute of control sciences of the russian academy of sciences and lomonosov moscow state university for purchasing for two industrial speedgoat computers to design imitation digital computer platform in real time to develop plasma current and shape control system on the globus-m2 tokamak. this work was partially funded by the grant of the russian science foundation № 17-1901022 to develop plasma magnetic control systems in tokamaks. the plasma control systems developed in the framework of that grant (see section 3) are supposed to be installed into the digital imitation platform (see section 6) to operate in real time. references 1. wesson, j., campbell, d. 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(2005) multivariable feedback control: analysis and design 2nd ed., john wiley & sons, ltd. adv syst sci appl 2021; 02:71–82 published online at https://ijassa.ipu.ru. algorithm for optimal two-link trajectory planning in evasion from detection problem of mobile vehicle with non-uniform radiation pattern andrey galyaev1, pavel lysenko1,2*and victor yakhno1 1institute of control sciences of ras, moscow, russia, 117997 2moscow institute of physics and technology, dolgoprudny, russia, 141701 abstract: the problem of an optimal pp/tp (path/trajectory planning) evasion of a mobile vehicle from the detection is considered. the objective is to minimize the risk of a moving vehicle being detected by a static sensor when moving between two specified points on the plane. detection is based on the primary acoustic field emitted by a vehicle with an inhomogeneous radiation pattern. an algorithm for finding a two-link optimal trajectory is proposed. the optimal trajectory and the law of speed of a mobile vehicle, as well as the value of the criterion, are found. keywords: uuv path/trajectory planning; non-detection probability; non-uniform radiation pattern, evasion from detection 1. introduction the problems of path/trajectory planning of autonomous and controlled vehicles are currently interesting due to their wide usage. for example, in the evasion problem, the moving object must remain hidden for the detector system [1]. these problems are extremely relevant for modern military applications. technological advances allow the onboard algorithms of unmanned mobile vehicles to be non-trivial and use solutions from broad scientific fields, such as optimal control and differential games. the problem of evading radar detection, in particular, is considered in [2] as a problem of automated planning of the trajectory of combat unmanned aerial vehicles (uavs) in the presence of radar-guided surface-to-air missiles. it turns out that the solution of the problem significantly depends on the assumption of the isotropic capabilities of the radar (i.e., the independence of the reflected signal from the spatial orientation of the uav). in case of isotropic characteristics, an analytical solution for the problem is possible, which facilitates the construction of an onboard control algorithm. homogeneous radiation capabilities of mobile vehicle are explored in papers [3–8]. the paper considers the problem of the covert infiltration of an unmanned underwater vehicle (uuv) into a given area under the supervision of a stationary passive sonar conducting a search on the primary hydro-acoustic field as in [9]. in contrast to previous works dealing with path planning problems, the acoustic signal emitted by a mobile vehicle has a non-uniform pattern. the task of the uuv optimal route planning from the starting point of space to the end point can be formulated as a problem of calculus of variations. the optimization criterion in this problem is the uuv non-detection probability, e.g. the chance of the sonar to make a decision about uuv presence in the region under study during ∗corresponding author: pashlys@yandex.ru 72 a. galyaev, p. lysenko uuv’s passing along the route. the decision rule in threshold statistics is examined to make the detection decision [10]. let the uuv trajectory be represented by a piecewise linear trajectory with the duration of a movement on each rectilinear section equal to the duration of the observation ”tact”. for each clock cycle, we denote the judgment ”uuv absent” by 0, and the judgment ”uuv present” by 1. then the results of making decisions on the uuv trajectory can be represented by a sequence of zeros and ones. uuv will not be detected on the path if there are no ones in the corresponding sequence of zeros and ones on the selected path. the optimal trajectory is the one for which the probability of such an event is maximum. articles [5, 11, 12] are devoted to the problems of optimal planning of the uuv route under threat conditions, in which other probabilistic and energy criteria are implemented. at codit 2020 conference results for optimal one-link trajectories of mobile vehicle, evading from one sonar, were presented [13]. the detection of sonar is based on primary hydro-acoustic field signals. but later in [14] it was shown, that sufficient optimal conditions can brake and then multi-link trajectory becomes optimal. in that article problem statements for two-link optimal trajectories were discussed. the current article considers the algorithm for solving of the supplementary problem, needed for two-link optimal trajectories constructing. firstly, in the article the non-detection probability of uuv on the route is derived. secondly, a path planning problem is formulated and analytically solved. later in next chapter the special case of two-link optimal trajectories is explored and the algorithm for finding such trajectories is obtained. finally, several examples are presented. 2. non-detection probability of uuv under passive surveillance the uuv covertness on the selected trajectory with a given speed law change can be characterized by the probability pnd that during the passage of the route it will not be detected at any tact. this probability is denoted pnd and will be called the non-detection probability of uuv on the trajectory. in the case of one sonar this probability is [3] pnd = ∏ j fn  χ2 1−α,n 1 + σ2 0(υj/υ0) µr20 σ2 nr 2 j γ  , (2.1) where fn is χ2 is a probability distribution with n degrees of freedom, α is a false alarm probability, γ is an attenuation coefficient, rj is the distance from uuv to sonar at j tact, starting from the moment of appearance of uuv on the trajectory, and υj is a constant speed of uuv at this tact, σ0, σn, µ, r0, υ0 are some model parameters. the number of factors in the product is equal to the number of tacts when moving uuv along a trajectory. in the case of several sonars, the product of expressions of the form (2.1) over all available sonars is taken [1]. thus the formalization of the uuv covertness concept is reduced to the formula (2.1). now the task is to construct the optimal trajectory and the optimal law of the uuv velocity variation, maximizing the probability pnd. based on actual algorithms for information processing, using decisive threshold rules, in the article [13] a formula was obtained for calculating the risk of mobile vehicle detection. it is represented as an integral, which we call the threat functional [6], [13] in the problem of uuv route planning for the case of passive sonar r = ∫ t0 0 σ2 s(t) σ2 n(t) dt. (2.2) copyright © 2021 assa. adv syst sci appl (2021) algorithm for optimal two-link trajectory planning in evasion 73 this conclusion coincides with the result given in [3] for σ2 s = σ2 0 (υj/υ0) µ r20 r2 γ, and σ2 n(t) = const. (2.3) moreover, for the path planning task, it is not required to know the exact values of the information processing parameters because they are not included in (2.2). 3. path planning problem 3.1. risk functional to formulate the path planning problem as the problem of calculus of variations let us rewrite the risk functional (2.2). due to (2.3), after omitting the constants the expression in integral can take a general form as a power model σ2 s(t) σ2 n(t) ∼ vµ rk . (3.4) as well as in [1] from the physical point of view this function is the instantaneous level of signal s, received by the sensor. it depends on the current distance between sensor and evading object r, for some types of physical fields – on absolute instant velocity of the object v too and can be represented as s = vµ rk . (3.5) the exponents k and µ characterize the physical field used for detection. depending on values of k and µ this can be magnetic, thermal, acoustic or electromagnetic fields. moreover, this signal depends on the receiving diagram of the sensor and the radiation pattern of the object s = vµ rk a0(ϕ)g(ψ, ϕ), (3.6) where multipliera0(ϕ) is responsible for diagram of sensor antenna and g(ψ, ϕ) is a radiation pattern of the moving object. the risk r is the integral value of this signal, so the criterion of optimization (2.2) is a function of phase coordinates of the object and qualities of the sensor and the object itself: r = ∫ t0 0 ( vµ rk a0(ϕ)g(ψ, ϕ) ) dt. (3.7) we consider sensor antenna diagram to be homogeneous, so a0(ϕ) ≡ 1. the geometric meaning of angles ψ and ϕ is as follows. here ψ is the angle of rotation of object’s velocity vector and ϕ is the angle of rotation of radius vector as shown in fig. 3.1. we study the case of k = 2 and µ = 2 for acoustic field. non-uniform radiation pattern of the object can be described as g(ψ, ϕ) = g(β). thus signal expression (3.6) has a form s = v2 r2 g(β). copyright © 2021 assa. adv syst sci appl (2021) 74 a. galyaev, p. lysenko fig. 3.1. the moving object in the cartesian coordinate system with sensor s. fig. 3.2. rectangular triangle of velocity vectors ~v,~vr and ~vϕ. 3.2. mathematical statement of the problem therefore we can formulate the optimal path planning problem. problem 3.1: it is required to find such trajectory (r(t), ϕ(t)), which minimizes the functional r = ∫ t0 0 v2 r2 g(β)dt, (3.8) where v is a velocity of the object, r is the distance between the sensor and the object. boundary conditions are r(0) = xa, r(t0) = rb, ϕ(0) = ϕa, ϕ(t0) = ϕb. (3.9) time t0 of moving on route from point a to b is fixed. the detection system consists of one static sensor placed in the origin of cartesian coordinate system with x axis passing point a. the polar coordinate system is used for solution simplicity. it is easy to see that ψ − ϕ = β is the angle in triangle built from radial vr and transversal vϕ velocities of the object, e.g. the angle between velocity of the object and its projection on radius vector as shown in fig. 3.2. next lemma is valid. lemma 3.1: substitution of variable ρ = ln r brings functional (3.8) to the form r(ρ(·), ϕ(·)) = ∫ t0 0 sdt = ∫ t0 0 (ρ̇2 + ϕ̇2)g ( arctan ϕ̇ ρ̇ ) dt. (3.10) copyright © 2021 assa. adv syst sci appl (2021) algorithm for optimal two-link trajectory planning in evasion 75 a proof for lemma 3.1 can be found in [13]. instead of solving the original problem 1 according to lemma 2 it is necessary to solve a two point boundary value variational problem on the minimum of the functional (3.10). problem 3.2: it is required to find the trajectory (ρ∗(t), ϕ∗(t)), which minimizes the functional r(ρ(·), ϕ(·)) = ∫ t 0 ( ρ̇2 + ϕ̇2 ) g ( arctan ϕ̇ ρ̇ ) dt→ min ρ(·),ϕ(·) . (3.11) with boundary conditions ρ(0) = ρa, ρ(t ) = ρb, ϕ(0) = ϕa, ϕ(t ) = ϕb. 4. solution of the path planning problem 4.1. the necessary optimality conditions theorem 4.1: suppose that 0 < g1 < g(β) < g2 for all β ∈ [0, 2π] is a twice continuously differentiated function of β, where g1, g2 are some constant values, and ρ̈(t), ϕ̈(t) exist and are continuous functions of t. then the extremal trajectory satisfies the following system of equations{ ρ̇ = const, ϕ̇ = const. (4.12) thus the extremal trajectory, which can be the solution for problem 3.2 can be represented in the parametric form of logarithmic spiral r(t) = ra exp ( t t0 ln rb ra ) , ϕ(t) = ϕa + ϕb − ϕa t0 t. (4.13) on the plane (r, ϕ) this line has a form r(ϕ) = ra exp ( ϕ− ϕa ϕb − ϕa ln rb ra ) . (4.14) next two lemmas define the velocity law of the vehicle on the extremal trajectory and the risk value on it. lemma 4.1: the velocity law on the extremal trajectory equation (4.14) is represented as follows v(t) = ra t exp ( t t ln rb ra )√ ln2 rb ra + (ϕb − ϕa)2. (4.15) lemma 4.2: the explicit dependence risk (3.11) from boundary conditions on the extremal trajectory equation (4.14) has the form r∗ = (ρb − ρa)2 + (ϕb − ϕa)2 t g(β0) = l2 t g(β0), (4.16) copyright © 2021 assa. adv syst sci appl (2021) 76 a. galyaev, p. lysenko where β0 = arctan ϕb − ϕa ρb − ρa , and l = √ (ρb − ρa)2 + (ϕb − ϕa)2 is the length of the straight line segment between points a and b in the space (ρ, ϕ). the proofs of lemmas 4.1 and 4.2 remain valid as shown in reference [9]. 4.2. the sufficient optimality conditions the sufficient optimality conditions for path planning problem 3.2 and the hessian matrix explicit form are presented in [9] as follows lemma 4.3: let s(ρ, ρ̇, ϕ, ϕ̇, t) = ( ρ̇2 + ϕ̇2 ) g (β), g(β) – thrice continuously differentiated function of β, then hessian matrix h equals h = ( h11 h12 h21 h22 ) where h11 = ∂2s ∂ρ̇2 = 2g (β)− 2ϕ̇ρ̇g′ (β)− g′′ (β) ϕ̇2 ϕ̇2 + ρ̇2 , h12 = ∂2s ∂ρ̇∂ϕ̇ = ( ρ̇2 − ϕ̇2 ) g′(β)− g′′(β) ϕ̇2 + ρ̇2 , h21 = ∂2s ∂ϕ̇∂ρ̇ = ( ρ̇2 − ϕ̇2 ) g′(β)− g′′(β) ϕ̇2 + ρ̇2 , h22 = ∂2s ∂ϕ̇2 = 2g (β) + 2ϕ̇ρ̇g′ (β) + g′′ (β) ρ̇2 ϕ̇2 + ρ̇2 , (4.17) and the hessian itself is the determinant of the matrix deth = 4g2(β) + 2g(β)g′′(β)− g′2(β). (4.18) theorem 4.2: assume that the conditions of theorem 4.1, lemma 4.3 are satisfied, and the inequality deth > 0 is valid for all values β. then the optimal trajectory given by equation (4.12) brings the strong minimum to the risk functional equation (3.11). 5. algorithm for finding two-link optimal trajectories if conditions of theorem 4.2 are not fulfilled, e.g. deth ≤ 0, then optimal trajectory can consist of many segments of logarithmic spirals. the current article explores the case of two segments, however it can be shown that any optimal multi-link trajectory is constructed from segments of two base directions. as explained above, these segments in (ρ, ϕ) space transform into straight lines. this fact is illustrated in figure 5.3. copyright © 2021 assa. adv syst sci appl (2021) algorithm for optimal two-link trajectory planning in evasion 77 0 ρ− ρ0 ϕ− ϕ0 l0 l1 l2 β0 β1 β2 fig. 5.3. the mobile vehicle in the (ρ, ϕ) coordinate system. next lemma allows to choose two optimal angles β1 and β2 from the whole set of such angles, fulfilling the boundary conditions of the problem. a corresponding to these angles optimal risk is denoted as r∗(β1, β2) and is calculated according to lemma 4.2. lemma 5.1: angles β∗1 , β ∗ 2 for optimal trajectory can be found from system cos(β2 − β1) = cos(ξ(β1)− ξ(β2)), g(β2) g(β1) = cos2 ξ(β2) cos2 ξ(β1) , (5.19) where function ξ(β) = arctan 1 2 g′(β) g(β) . we introduce two new functions χ(β) = β + ξ(β), η(β) = g 1 2 (β) cos ξ(β) (5.20) and rewrite the system (5.19) as { χ(β1) = χ(β2), η(β1) = η(β2). (5.21) notice that cos ξ(β) > 0 according to definition function ξ(β) in lemma 5.1. thereby the search of optimal two-link trajectories is reduced to the solution of the system (5.21). unfortunately, it can not be solved analytically, so we have developed a specific algorithm for finding solutions of system (5.21). first of all, let us investigate each of the functions χ(β) and η(β) by extreme’s and find their first derivatives dχ(β) dβ = 4g2(β)− (g′(β))2 + 2g′′(β)g(β) 4g2(β) + (g′(β))2 . the numerator of this fraction is a part of expression for deth, so dχ(β) dβ = 0 ⇐⇒ deth = 0. copyright © 2021 assa. adv syst sci appl (2021) 78 a. galyaev, p. lysenko as for η(β) function we have dη(β) dβ = g 1 2 (β)g′(β) 4g2(β) √ 4g2(β) + g′2(β) deth from last expression follows dη(β) dβ = 0 ⇐⇒ deth = 0 or g′(β) = 0. the illustration of the theorem 4.2 implementation is as follows. if deth > 0 for all β, then dχ(β) dβ > 0, i.e. the function χ(β) increases monotonically and therefore there is no solution for the first of the equations (5.21) and one-link trajectory is optimal. now we propose algorithm for finding system (5.21) solution. we consider β ∈ [0◦, 450◦] and denote all intervals β ∈ (a2i−1, a2i), i = 1, .., n , where deth > 0. algorithm 1: algorithm for finding optimal values of β1, β2 result: β∗1 , β∗2 initialization β1 = a1, β2 = a3, r∗ = r(β1, β2) while i < n , i = i+ 1 do k = i+ 1; while k < n , k = k + 1 do find system (5.21) single solution (β1, β2), β1 ∈ [a2i−1, a2i], β2 ∈ [a2k−1, a2k]; if such solution exists then calculate r(β1, β2); if r(β1, β2) < r∗ then r∗ = r(β1, β2); (β∗1 , β ∗ 2) = (β1, β2); else continue end else continue end end end the proposed algorithm has the following advantages. it looks for solutions on reduced β domains only where deth > 0 and additionally where solution is possible. all possible optimal directions are found, regardless of the initialization values of the algorithm, as will be shown in the example. the optimal value of the criterion depends on the directions β1 and β2 and the length of the corresponding trajectory segments. moreover, that is an efficient algorithm. the only other way to find optimal angles β1, β2 is a brute force approach of sorting through all possible two-link trajectories, which is a much more time-consuming process, compared to the algorithm above. 6. example this section demonstrates the use of proposed algorithm in one model case. the considered algorithm has been implemented in matlab. also matlab scripts have been developed to validate and illustrate obtained results. copyright © 2021 assa. adv syst sci appl (2021) algorithm for optimal two-link trajectory planning in evasion 79 figures 6.4 and 6.5 show a fairly simple radiation pattern of a mobile vehicle g(β) = 1 + 0.3 cos(4β) 1.3 . (6.22) fig. 6.4. radiation pattern of the mobile vehicle on cartesian plane fig. 6.5. radiation pattern of the mobile vehicle as a function of β figure 6.4 shows a radiation pattern on a cartesian plane. this pattern is symmetrical in respect to both cartesian axes. the solid line represents the function g(β) level line. the value g(β) is a radius-vector modulo directed from zero point to point lying on the shown level line. figure 6.5, on the other hand, presents a radiation pattern as a dependence from β. figures 6.6 and 6.7 demonstrate functions χ(β) and η(β) respectively. blue color solid line segments on those graphs are assigned to intervals β ∈ (a2i−1, a2i), i = 1, .., n , where deth > 0. we see that in the example n = 5. fig. 6.6. function χ(β) copyright © 2021 assa. adv syst sci appl (2021) 80 a. galyaev, p. lysenko fig. 6.7. function η(β) let us introduce notation a′i for further exploration. we declare that for β ∈ [a2i−1, a2i] and fixed k a′2i−1 = max(a2i−1, arg(χ(β) = χ(a2k−1))), a′2i = min(a2i, arg(χ(β) = χ(a2k))). this notation takes place because χ(β) is monotonically increasing function on β ∈ [a2i−1, a2i]. further we need to narrow down the exploration area on β of functions χ(β), η(β). figures 6.8 and 6.9 correspond to the first pair of narrowed intervals of hessian positivity [a′1, a ′ 2] and [a′3, a ′ 4]. black dots show found at this iteration of the algorithm 1 solutions β1, β2. fig. 6.8. enlarged segment of function χ(β). fig. 6.9. enlarged segment of function η(β). so the domain of values of function χ(β) when β belongs to the first interval [a1, a2] do not cross with the same one when β belongs to the third, forth and fifth interval [a5, a6], [a7, a8], [a9, a10], then the algorithm continues searching the solution comparing intervals two and three on β, then three and four, and so on. candidates of optimal pairs of (β1, β2) are approximately (59◦, 121◦), (149◦, 211◦), (239◦, 301◦), (329◦, 391◦). these directions and trajectories are presented in figure 6.10. if the direction β0 lies in the domain where deth > 0 then the optimal trajectory is one-link one. on the contrary if the direction β0 lies in the domain where deth < 0 then two-link trajectory is optimal. found by algorithm 1 optimal pairs (β1, β2) in every domain where deth < 0 are the same as shown on figure 6.10. copyright © 2021 assa. adv syst sci appl (2021) algorithm for optimal two-link trajectory planning in evasion 81 fig. 6.10. optimal trajectories for different β0. 7. conclusions the problem of minimizing the risk of uuv detection by a static sensor while moving between two given points on a plane is solved. the detection is based on the primary acoustic field radiated by the vehicle with a non-uniform radiation pattern, which differs this work form the previous researches in the area. in the first part of the article, the non-detection probability is derived. further it is used as an optimization criterion in the path planning problem. the optimal trajectory and velocity law of the uuv are found, as well as the risk value. a case of the two-link optimal trajectories is studied, when sufficient optimal conditions are not fulfilled. a special algorithm has been developed for finding optimal angles of logarithmic segments. it allows to solve the whole problem and find optimal angles β1, β2. in practical applications this algorithm allows to preprocess these angles for all initial conditions of β0 for a particular radiation pattern g(β) of the mobile vehicle, store it in memory and use later this information for onboard calculations. this approach saves a lot of computational power, which is very important for resource-efficient missions. acknowledgements the work of a.a.g. was partially supported by programm of basic research of ras; the work of p.v.l. was partially supported by russian foundation for basic research grant 2038-90215; the work of v.p.y. was partially supported by programm of basic research of ras. the authors declare that there is no conflict of interest. references 1. galyaev, a. a. & maslov, e. p. 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(eds.), proceedings of the 13th biennial computational techniques and applications conference, ctac-2006, vol. 48 of anziam j., 293–306. 13. galyaev, a. a., lysenko, p. v., & yakhno, v. p. (2020) evasion from detection of the moving object with non-uniform radiation pattern. in 2020 7th international conference on control, decision and information technologies (codit), vol. 1, 463–468. 14. galyaev, a. a., lysenko, p. v., & yakhno, v. p. (2021) 2d optimal trajectory planning problem in threat environment for uuv with non-uniform radiation pattern. sensors, 21. copyright © 2021 assa. adv syst sci appl (2021) introduction non-detection probability of uuv under passive surveillance path planning problem risk functional mathematical statement of the problem solution of the path planning problem the necessary optimality conditions the sufficient optimality conditions algorithm for finding two-link optimal trajectories example conclusions adv syst sci appl 2021; 01:113–138 published online at https://ijassa.ipu.ru. software for testability analysis of aviation systems valentina s. viktorova1∗, armen s. stepanyants1 1v. a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia abstract: this paper describes models, methods and software tool for testability analysis of aviation systems. it comprises analytical and programing aspects of calculations of main testability, reliability and availability indices. it presents general description of the software, xml schema of input data and technique of their mapping to the database structure. the procedure for generating the initial data for testability analysis based on the line replaceable units failure modes report is described. fault tree model for analysis of the built-in test conformity is suggested. markov models have been created for analyzing reliability and availability, taking into account the features of the built-in test and the specifics of the aircraft operation. an approach to the construction of trends in the operative availability of aviation systems in the inter-maintenance interval is proposed. keywords: testability, built-in-test, fault coverage, failure detection ratio, built-in-test equipment conformity, failure mode and effects analysis, markov reliability models, operative availability trend 1. introduction reliability, availability, testability, maintainability are the main properties of aviation items that determine the safety of an aircraft flight [1]. these properties are interrelated (chapter 6 in [2] ), an improvement (worsening) of one leads to a corresponding changes in the others. testability is a property of an item that characterizes its adaptability to be controlled by specified diagnostic systems. these systems include both internal and external diagnostic means. internal diagnostic means integrated on-line referred to as built-in-test (bit). in one of the first regulatory documents in this area [3], the targets of testability analysis are defined as assessing the ability to detect and isolate system faults to the faulty replaceable assembly level. our research focuses on design stage advanced testability analysis, including investigation of impact of bit performance on the reliability and availability of aircraft systems. analytical models that consider imperfect bit performance are known as imperfect fault coverage models. many articles are devoted to these models and associated reliability analysis techniques [4–8]. we have developed a reliability model considering imperfect fault coverage and aircraft-specific recovery strategies for the latent failures. when constructing the reliability models, a technical inspection and maintenance program [9, 10] typical for civil aviation was taken into account. at the aircraft design stage, it is mandatory to carrying out a qualitative and preliminary quantitative failure modes and effects analysis (fmea) [11]. if the fmea table records are supplemented with information on the failure detection methods [13–15], then they can be used as input data for calculating the testability indices and conducting reliability and availability analysis taking into account the bit characteristics. testability-oriented fmea ∗corresponding author: vsviktorova@gmail.com 114 v.s. viktorova, a.s. stepanyants analysis is performed for all functional aircraft systems on the level of the line replaceable units (lru) [16]. storage, selection, sorting and filtering of data of such complexity and volume as aircraft fmea reports can be carried out only with the involvement of modern database management systems. implementation of code of definition and solving reliability and testability analysis models is only possible through the advanced programming languages with high precision numeric data types and support of dynamic data structures. we chose oracle dbms and pl/sql programming language. in this paper, we present software for joint testability and reliability analysis of civil aircrafts. the models and analysis methods used in this software reflect the experience we gained during the project testability analysis of the sukhoi superjet and mc-21 aircraft family. this software provides the following options • testability indices calculation; • construction of distributions of testability indices by levels of criticality of failures; • construction of distributions of testability indices by fault detection methods and by impact on departure delay; • evaluation of built-in-test equipment conformity; • automatic construction of markov reliability models of redundant lru assemblies with imperfect fault coverage and aviation-specific recovery scenarios; • investigation of the impact of bit functional features on system reliability and availability; • creation of operative availability trends within the specified time frame between the aircraft maintenance periods. the rest of this paper is organized as follows. in section 2, we give the software description including meta definition of testability data and their mapping to the database structure. section 3 presents mathematical background of the testability&reliability analysis. in section 3.1, we derive the formulas for calculating single testability measures. in section 3.2, we define logical-probability model of bit conformity. in section 3.3, we suggest the parametric markov reliability model of redundant lru assemblies with imperfect fault coverage by bit. section 3.4 describes an approach to trending of aircraft operative availability. some results and graphical illustrations of the analysis are summarized in section 4. in the appendix, we provide the xml schema of testability data. 2. testability analysis software description 2.1. testability metadata structure the actual practice of database design is the preliminary creation of a platform-independent meta-definition of the data. in accordance with the recommendations of international reference books [17–19] on technical publications in aerospace industries, the xml language is used for the meta definition of information items of this software. special language referred to as xml schema definition (xsd) [20] are used for description of the structure, data types, elements and attributes of an xml document. we have developed the xml schema that defines the xml image of testability data testability data xml schema definition (txsd). the pilot version of the txsd was described in [21]. current txsd is presented in appendix. it defines four-level hierarchical structure with the help of four global complex types tprojects, tsystems, tlrus, tfmeas. each of them is an unbounded sequence of elements of type tproject, tsystem, tlru, tfmea, respectively. the complex type tproject defines project. project is a generic term. an aircraft as a whole, arbitrary functional systems, for example, air conditioning, fire protection, integrated modular avionics can be considered as a project. subsystems and assembles, for example, a landing gear steering or an engine interface unit of the central computer module of the indicating/recording system also may be represented as a project. thus, the level of copyright c© 2021 assa. adv syst sci appl (2021) software for testability analysis of aviation systems 115 detail of the analyzed object is determined by the user. the tproject context is divided into three parts descriptors (codes and names of the project), time parameters (maintenance periods, recovery times...), set of functional systems (tsystems). the tsystem context also is divided into three parts descriptors (codes and names of the system), technical specifications for the system reliability and testability indices, set of the line replaceable units (tlrus). the tlru context is divided into six parts descriptors (codes and names of the lru), lru part and serial numbers, lru identifier in the centralized maintenance system (cms), redundancy schema, technical specifications for the reliability and maintainability indices, set of failure entries (tfmeas). the tfmea type describes output data of design stage analysis of failure modes and effects of the lru. tfmea has complex structure. the elements of this structure define various properties of failure, including failure descriptors in cms, failure rate, failure severity level, detection method, phase of flight, flag of departure delay, bit resolution features and so on. txsd contains global simple types which define • string type of ata code [22] (tcodata) of system and lru; • decimal types of time parameters and reliability indices (trindex), normalized testability indices (tratio); • enumeration types of lru properties lru redundancy schema (trschema), lru mode (tlrutype); • enumeration types of failure properties detection method (tfdm), severity level (tseverity), flight profile [23] (tphases), mode (tlrufailure), departure delay flag (tdelay). txsd includes a specialized complex datatype, namely resolution list (tlruchain). it is an unbounded sequence of tcodata elements. tlruchain defines a list of replaceable units that will be isolated by bit when a failure is detected. a global element named ”projects” specifies the root of any txsd-based xml document. the complete composition of the txsd is given in the appendix. 2.2. testability database structure the mapping of txsd to a relational database schema is carried out according to the following rules: • global complex types tproject, tsystem, tlru, tfmea define the presence and structure of four main database tables storing information about projects, functional systems, lru, fmea; • simple type elements define the data types, identity, check constraints of columns of these tables; • elements of the sequence type tsystems, tlrus, tfmeas contained in complex elements project, system, lru respectively define referential integrity within the database; • nesting order of txsd elements determines relations by foreign key between parents and childes tables of the database; • enumeration type elements are embodied in look-up, reference tables. fig.2.1 presents graphical interpretation of the mapping procedure. 2.3. testability software features testability analysis software is a web application that runs on an oracle database (fig. 2.2). this application is built using oracle database’s native low-code development platform apex. this choice enables to build high-performance, scalable, secure apps. oracle apex uses a 3-tier architecture where requests are sent from the browser, through a web server, to the database. within the database, the request is processed by oracle apex. once the copyright c© 2021 assa. adv syst sci appl (2021) 116 v.s. viktorova, a.s. stepanyants processing is complete, the result is sent back to the browser. processes of requesting and submitting pages are realized through a modern implementation of the oracle net listener called oracle rest data service (ords). user interface of the software is mobile friendly. pages have smart layout and include forms, charts, reports and other user interface components that can work across varying screen resolutions. database includes main tables (see subsection 2.2), reference tables and views. views contains data from join of fmea and reference tables. specifications and bodies of analytical procedures and functions are contained in four pl/sql packages testpckg, relpckg, apckg, numpckg. testpckg, relpckg, apckg includes procedures and functions of testability, reliability, availability analysis respectively. numpckg implements numerical methods for mathematical computations . reports of testability and reliability analysis can be download into external files of csv, xlsx, rtf, xml, pdf formats. fig. 2.1. txsd to database schema conversion (fk – foreign key, pk – primary key). . 3. mathematical background of computational algorithms the following subsections describe models, methods and calculation procedures applied in the software for quantification and analysis of various testability, reliability and availability indicators. we use the generally accepted in aviation reliability analysis assumption about an exponential distribution of random times to failure [12]. in this case, the reliability indicator called the failure rate is equal to the constant parameter of the distribution. 3.1. single testability indicators single testability indicators (sti) are used to specify bit functionality or reliability. these indicators are used to assess a single bit property. when calculating sti, a preliminary partition of the entire set of the lrus failure modes into disjoint subsets is carried out. sti can obviously be estimated as the ratio of the number of selected failures modes (ns) to the total number (nt ) of failure modes copyright c© 2021 assa. adv syst sci appl (2021) software for testability analysis of aviation systems 117 fig. 2.2. testability analysis software structure. . sti = ns nt . (3.1) however, for joint reliability modeling of the object under test and bit, sti should be set as the ratio of some probabilistic measures. the expediency of such definition is explained by the fact that when modeling reliability behavior, it will be possible to split the total flow of the object failures into several components interesting for researcher. for example, detected failures and hidden (latent) failures. in this case, sti can be defined as the conditional probability of the failure modes of interest, provided that the failure of the object has occurred: sti = (1− exp−λ̃st) (1− exp−λ̃t t) , (3.2) where λ̃t = 1 t ∫ t 0 λt (τ)dτ – average failure rate of the object; λ̃s = 1 t ∫ t 0 λs(τ)dτ – average rate of selected failure modes of the object. if we consider the exponential distribution of random time to failure, then sti = (1− exp−λst) (1− exp−λt t) ≈ λt�1 λs λt . (3.3) if we assume the incompatibility of the object failure modes copyright c© 2021 assa. adv syst sci appl (2021) 118 v.s. viktorova, a.s. stepanyants sti = λs λt (1− exp−λt t) 1− exp−λt t = λs λt . (3.4) most testability indicators given in reference books [1, 3] are the ratios similar to (3.1), (3.4). the main single testability indices are: • fault detection ratio (fdr); • fault isolation ratio (fir); • bit efficiency; • bit unreliability factor (uf); • bit false alarm factor (faf). they are formed based on summing the failure rates (failure numbers) fetched from the fmea table: ∑ i∈ω1 λi/ ∑ j∈ω2 λj or ∑ i∈ω1 ni/ ∑ j∈ω2 nj . the software executes sql queries to fmea table and generates the following subsets of the set of failure modes: • ω – total set of the object failures; • ωb, ωm, ωc – disjoint subsets of the object failures detected by bit, maintenance, crew, respectively (see appendix, tfdm type); • ωn – subset of the object hidden (latent) failures; • ωb1 – subset of the object failures detected by bit and isolated to single faulty lru; • ωi , ωii , ωiii , ωiv , ωv – disjoint subsets of the object failures related to severity levels i, ii, iii, iv, v, respectively (see appendix, tseverity type); • ωbi , ωbii , ωbiii , ωbiv , ωbv – disjoint subsets of the object failures detected by bit and related to severity levels i, ii, iii, iv, v, respectively; • ωnop, ωfa – disjoint subsets of bit failures of type no operation and false alarm (see appendix, tfailuretype); • ωff– subset of functional failures; • ωd– subset of failures impacting on departure delay (see appendix, tdelay type). table 3.1 shows how sets of failure modes ω1,ω2 are formed when calculating single testability indices. table 3.1. definition of ω1, ω2 index ω1 ω2 fault detection ratio ωb ω fault isolation ratio ωb1 ωb fdr in flight ωb ∪ ωc ω bit efficiency ωb ωb ∪ ωm ∪ ωc bit unreliability factor (uf) ωnop ∪ ωfa ω bit false alarm factor (faf) ωfa ∪ ωfa ωnop ∪ ωfa fdr for severity level i(fdri) (i=i,ii,iii,iv,v) ωbi ωi construction of distribution of fault detection ratio by levels of failures criticality are based on fdri calculation. in the reliability theory, in particular, when solving maintenance problems, the index of average failures number (afn) is used. this indicator is the mathematical expectation of the number of the object failures during a given time interval (n(t)). at exponentially distributed with parameter λ random times to failures, afn is calculated by the formula [2, pp. 89–90]: n(t) = λt. (3.5) afn is not a regulated index of testability. but when it is calculated in relation to the failure modes covered by different detection methods, it becomes an obvious measure of testability. in this software, the afn is calculated over flight and different maintenance periods. copyright c© 2021 assa. adv syst sci appl (2021) software for testability analysis of aviation systems 119 3.2. bit conformity for an integral assessment of bit quality we use a composed probabilistic indicator called the conformity of bit. bit conformity is a complex characteristic that takes into account both functional and reliability properties of bit. in this case, two incompatible types of bit failures are considered non-operation and false alarm. let us derive an expression for bit conformity under the following initial data and assumptions: • event a means the controlled object is in operable (good) state ; event a means the controlled object is in inoperable (failure) state; prob(a) = pa, prob(a) = qa are the probabilities of these events; • bit has two failure modes, the first mode is non-operation (i) and the second one is false alarm (ii); event k is a good (failure-free) bit state, event kno is occurrence of nonoperation bit failure, event kfa is occurrence of false alarm bit failure; prob(k) = pk , prob(kno) = qno, prob(kfa) = qfa are the probabilities of occurrence of corresponding events; • events k, kno, kfa are a complete group of mutually exclusive events, so pq + pno +qfa = 1; • b is the event that bit recognizes the object state as operable, b bit recognizes the object state as failed; • η is a fault detection ratio of bit. we define the event of bit conformity c as the correct recognition of the state of the object c = (a ∧b) ∨ (a ∧b). (3.6) to find the probability of a truth of this event prob{c = 1}, we use the fault tree model. one of the techniques for constructing fault trees is to apply the theorem of decomposition into inconsistent hypotheses in some logical variables, which ensures the decomposition of the logical model and simplifies the construction of individual tree branches [24, pp. 165– 172], [25, pp. 103–115]. to construct a fault tree with a top eventc, we use the decomposition with respect to the object states (events a and a ). and in the tree branches with a we apply the decomposition with respect to bit coverage, viz. insertion the inhibit gates with conditional events η and 1− η. the inhibit gate is used to indicate that the output occurs when the input events (bottom events) occur and the input condition (right side event) is satisfied. fig. 3.3 shows the resulting fault tree. the fault tree contains repeated basic events (event 1 (a), event 2 (k), event 4 (kfa)). repeated events are colored green. the logical expression corresponding to the constructed fault tree model is given below. c = a ∧ (k ∨kno) ∨ a ∧ η ∧ (k ∨kfa) ∨ a ∧ (1− η) ∧kfa = (3.7) a ∧ (k ∨kno) ∨ (a ∧kfa) ∨ (a ∧ η ∧k). let’s write down the probability function for a truth of the found logical expression. since the decomposition into inconsistent hypotheses was used when creating fault tree, we obtained logical expression in orthogonal form. therefore, one can directly substitute the corresponding probabilities instead of logical variables, and replace logical operations with arithmetic ones [26, pp. 39,278–336]. the final expression for bit conformity prob(c = 1) = pc = pa(pk +qno) +qaqfa +qaηpk . (3.8) copyright c© 2021 assa. adv syst sci appl (2021) 120 v.s. viktorova, a.s. stepanyants fig. 3.3. fault tree model of bit conformity. . 3.3. reliability measures analytical modeling of the systems reliability behavior taking into account bit fdr and redundancy is implemented on continuous-time markov chains with discrete finite state space of size n. markov reliability model (mrm) allows to model complex transient and steadystate reliability behavior and to estimate wide range of reliability and availability indices [25, pp. 147–199]. mrm is formalized by the system of kolmogorov differential equations that describes the behavior of the state probability vector p (t) as a function of time t: dp (t)/dt = p (t)λ; p (0) = p0, (3.9) where λ = ‖λij‖ – n× n infinitesimal matrix, λij, i 6= j is the rate of transition from state i to state j, λii = − ∑ i 6=j λij . the ith component pi(t) of the row vector p (t) is the probability the system is in state i at time t. the software performs automatic generation of the markov model. the mrm describes reliability behavior of the redundant assembly of line replaceable units (ral) between maintenance periods such as heavy checks. the model generation algorithm is based on the following provisions • a set of non-redundant line replaceable units are combined into assembly based on functional or design features; • both recovery and redundancy are implemented at the assembly level; • redundancy type is parallel operating k out of m, where m total quantity of identical assemblies, k quantity required for success operation (m ≥ 1, k ≤ m); copyright c© 2021 assa. adv syst sci appl (2021) software for testability analysis of aviation systems 121 • incomplete coverage of lru failures occurs due to functional imperfections in the bit (0 ≤ η ≤ 1); • two ral recovery scenarios between checks are considered – the ral operability is restored in case of occurrence of the hidden failures of all assemblies (scenario i) (for example, engine control system); – the ral operability is not restored in case of occurrence of the hidden failures of all assemblies (scenario ii) (for example, a fire protection system). • the presence of at least one failure detected by the bit in the assembly leads to a complete assembly recovery, regardless of whether there are other (including hidden) failures in the assembly. when building the model, the states are aggregated. all detected (hidden) failures of the assembly elements are combined into one state of a detected (hidden) assembly failure. each mrm state is encoded with a m-character code d1d2. . .di. . .dm. the symbol di of the code is determined by the rule di =  1 if the assembly is operable d if a failure detected by the bit has occurred in the assembly h if a failure undetected by the bit (hidden) has occurred in the assembly hd(dh) if both detected and hidden failures have occurred in the assembly to ensure the correctness of the aggregated mrm, it is necessary to operate with the mean recovery times of the assembly mttrassembly = ∑ i λi µi∑ i λi , (3.10) where λi, µi are the failure and recovery rates of the ith lru of the assembly. the mrm state set is partitioning into two subsets of operational (good) states ωg and non operational (failure) states ωf . the formal condition of belonging of the model state si to one of these subsets is{ si ∈ ωg if total number of symbol ”1” in state code ≥ k si ∈ ωf if total number of symbol ”1” in state code < k the parameters of the procedure for generating the infinitesimal matrix of this mrm are m, k, η, recovery scenario number. values of parameters m, k, η are determined based on sql queries to the database. the markov graph corresponding to the generated matrix at m = 3, k = 1, scenario i is shown in fig. 3.4. the software calculates on the mrm the following reliability measures 1. interval measures reliability r(t) and unreliability u(t) on time interval (0, t ). 2. point measure availability a(t) in time instant t. 3. steady-state measure mean time to first failure mttff which is defined on unlimited time interval (0,∞). to calculate r(t), it is necessary to solve the system (3.9), having previously made all failure states absorbing. this requires equating to zero the elements of rows of matrix λ corresponding to failure states. r(t) = ∑ i∈ωg pi(t), u(t) = ∑ j∈ωf pj(t) = 1−r(t). (3.11) copyright c© 2021 assa. adv syst sci appl (2021) 122 v.s. viktorova, a.s. stepanyants fig. 3.4. graph representation of the mrm (m = 3, k = 1, scenario i). the calculation of a(t) is performed in a similar way, but in this case the failure states should not be forced to be absorbing. calculation of mttff is reduced to solving a system of linear algebraic equations: − p ∗(0) = tλ∗, (3.12) where λ∗ − r × r matrix derived from matrix λ by deleting rows and columns corresponding to nonoperational states; r – total number of operational states (r < n); p ∗(0) = [ p1(0) p2(0) . . . pr−1(0) pr(0) ]. provided that the system starts from a completely healthy state p ∗(0) = [ 1 0 . . . 0 0 ]. mttff = ∑r i=1 ti. the solutions of systems of differential (3.9) and algebraic (3.12) equations are implemented using numerical methods focused on the stiffness and bad conditioning [27], inherent in the reliability models of the repaired systems [28, pp. 48–59]. 3.4. operative availability model aviation systems are classified as hybrid recovery systems [29]. these systems have the cyclic mode of operation whose duty cycle consists of a target operation phases (top), repair phases (rp), inspection, maintenance and correction phases (imcp). recovery actions after failures are not possible on-line (top) and can be conducted only off-line during rp and imcp. when the system is online (top) information about failures detected by the bit is recorded in memory unit of cmcs (central maintenance computer system). cmcs collects and reports lru faults data in order to aid maintenance personnel in recovery and maintenance procedures. during rp, only faults detected by the bit can be eliminated. full recovery of system operation is carried out only at stage imcp where latent faults also eliminated. thus, due to the imperfection of the built-in test, between imcp stages there is a consistent decrease in the system availability at the beginning of each subsequent top. the main indicator that can be used to assess the reliability behavior of the cyclic mode systems, in particular aviation systems, is operative availability. the operative availability a(t, t+ t0) is index depending on two variables – the operating time t (usually counted from 0), when recovery from failure states is possible, and the operative time t0 (counted from t), when copyright c© 2021 assa. adv syst sci appl (2021) software for testability analysis of aviation systems 123 recovery from failure states is impossible. in general, this index is calculated as a(t, t+ t0) = ∑ i∈ωg ai(t)r(t, t+ t0)/s(t)=i, (3.13) wherer(t, t+ t0)/s(t)=i – the reliability of the system on the interval (t, t+ t0), provided that the system is in state i at time t. this software builds a trend of operative availability for a given number of phases of target operation (qstep). the construction of the trend is carried out by a program that performs a step-by-step calculation of reliabilityr(ti) (ti is the duration of the ith top). the calculation is performed on the mrm shown in fig. 3.5. a step-by-step calculation of the reliability r(ti) is carried out each time with new initial conditions, which makes it possible to take into account the fact of a decrease in availability arising from incomplete coverage of failures at the previous i− 1 recovery phases (rp). the vector of initial conditions for the ith top c(i) are formed according to the following rule c(i) = [ c (i) 1 c (i) 2 . . . c (i) j . . . c (i) n ] , (3.14) where c (i) j = { p1(ti−1) + ∑ k∈ωb pk(ti−1) if j = 1 pj(ti−1) if j ∈ ω\ωb 0 if j ∈ ωb fig. 3.5. graph representation of the mrm over target operation phase (1 ≤ m ≤ 4, 1 ≤ k ≤ m). copyright c© 2021 assa. adv syst sci appl (2021) 124 v.s. viktorova, a.s. stepanyants 4. case studies of the software six cases below illustrates the software capability. the initial data when performing the calculations of testability and reliability indices are the output of the design stage analysis of modes and effects of lru failures. a constant failure rate is specified in fmea tables for each type of failures. the failure rates are calculated using the methods described in various reference books and standards [30] or/and processing statistics on failures. the calculation of interval indicators of testability and reliability are carried out for the periods of time regulated by the scheduling maintenance program referred as a b c d checks [10, pp. 75–84]. the complexity of service tasks grows from check a to check d. the actual periods of these inspections depend on the type of aircraft, but the lighter checks a and b are carried out more often than the heavier checks c and d. we use the following times between checks: tchecka = 600h, tcheckb = 3500h, tcheckc = 7000h, tcheckd = 70000h. flight duration tflight = 5h. we demonstrate software capability on sample system consisting of three lru. screenshot of the software page with the fmea report is in fig. 4.6. fig. 4.6. screenshot of the fmea report. copyright c© 2021 assa. adv syst sci appl (2021) software for testability analysis of aviation systems 125 4.1. case 1. failure modes classification the software implements the classification of the lru failure modes by following criteria: • detection methods – bit – failure is detected by built-in-test – maint – failure is detected in ground maintenance – crew – failure is detected by crew – none – failure is not detected until check d • failure types – ff – functional failure – bit nop – no operation built-in-test failure – bit fa – false alarm built-in-test failure • impact on departure delay the percentage of each group is calculated. the percentage is calculated by summing failure rates (estimation by frequency) fig. 4.7, 4.8, 4.9 (a), or by the number of failures (quantification) fig. 4.7, 4.8, 4.9 (b). fig. 4.7. failure detection methods. a estimate by frequency, b quantification. 4.2. case 2. sti calculation the software implements calculation of the following single testability indices: • total fault detection ratio including bit, crew, maintenance (tfdr) • bit fault detection ratio (fdr) • bit fault isolation ratio (fir) • automated fault isolation capability (afic = fdr× fir) • bit efficiency calculations of tfdr, fdr, fir, afic, bit efficiency and creation of fdr distribution by failure mode severity levels are implemented in accordance with the definitions set out in section 3.1. fig. 4.10 presents the screenshot with results of the analysis. full coverage of high severity (catastrophic, critical) failures is desirable when designing bit. comparison of frequency (i) and quantity (ii) estimates can be useful in assessing the quality of testability solutions. for example, if the fdr i < fdr ii, then the design must be changed, since the most frequently failure modes are not covered by the bit. copyright c© 2021 assa. adv syst sci appl (2021) 126 v.s. viktorova, a.s. stepanyants fig. 4.8. failure mode types. a estimate by frequency, b quantification. fig. 4.9. failures impact on departure delay. a estimate by frequency, b quantification. 4.3. case 3. afn calculation the average failures number over time interval is calculated by (3.5) and the distribution of this indicator by failure severity is constructed. the calculation is performed for time intervals: flight, check a, check b, check c. distributions are constructed for the following types of failures: • failures detected by built-in test (afn bit); • failures detected during maintenance process (afn maint); • hiden (latent) failures not covered by bit, crew and maintenance (afn none). screenshots of the afn summary page and respective charts are presented in fig.4.11 and fig.4.12. 4.4. case 4. analysis of bit conformity in the framework of bit conformity analysis the following procedures are implemented: • calculation of bit conformity based on the technical specifications of bit reliability and functional properties (conformity i); copyright c© 2021 assa. adv syst sci appl (2021) software for testability analysis of aviation systems 127 fig. 4.10. screenshot of the report of sti calculations. . • calculation of bit conformity based on the fmea data concerning rates and coverage of failures modes (conformity ii); • comparison of the values of conformity i and conformity ii and issuing recommendations for bit modernization if conformity ii < conformity i; • study of behavior of the conformity index in the space of bit reliability parameters. an approach to comparing identical indicators calculated on the basis of different type of initial data (technical specifications and fmea) is proposed in [12]. fig. 4.13 is a screenshot of the report of calculating bit conformity i and ii and their constituents. the conformity indicator is calculated in accordance with the definitions set out in section 3.2. the plots in fig. 4.14 (a) and (b) show the results of calculation of bit conformity as a function of bit unreliability factor uf (see table 3.1). uf characterizes unreliability of bit equipment relative to unreliability of object under test. the set of curves is formed by varying the parameter kfa. kfa determines the ratio of the failure modes of bit and is calculated as faf (see table 3.1). plot of fig. 4.14 (a) shows small decrease of conformity appears linear copyright c© 2021 assa. adv syst sci appl (2021) 128 v.s. viktorova, a.s. stepanyants fig. 4.11. screenshot of summary of the system average failures number. in uf on short time interval (flight). plot of fig. 4.14 (b) demonstrates limit behavior of conformity on long time interval (check a). (limt→∞c(t) = kfa). 4.5. case 5. reliability analysis. the calculation and study of reliability indicators are conducted on models described in section 3.3. the formation of the elements of matrix λ is carried out by executing sqlqueries to fmea table to select the failure modes rates and average recovery times of the lru. the curves of reliability (unreliability), availability (unavailability), mttff are plotted as a function of fdr. a family of curves is plotted for each indicator. a separate curve corresponds to one of three standard redundancy schemes duplicated (1 out of 2), tripled (1 out of 3), majority (2 out of 3). the fourth curve corresponds to a non-redundant assembly configuration (1 out of 1). the chart in fig. 4.15 (a) demonstrates a strong dependence of mttff on fault detection ratio. a sharp jump in the mttff is observed at fdr values close to 1. the chart in fig. 4.15 (b) demonstrates a weak dependence of the reliability on fdr at short time intervals. with an increase in the time interval, this dependence increases (fig. 4.15 (c). fig. 4.15 (c) also shows that a 2 out of 3 redundancy schema can only be used in conjunction with high fault detection ratio. otherwise, this option turns out to be worse than the non-redundant scheme. on long time intervals, high bit fdr and redundancy can dramatically increase the assembly reliability (fig. 4.15 (d)). fig. 4.16 (a) and (b) show availability as a function of fdr. the curves of this chart show that bit fdr is a factor affecting the efficiency of using redundancy at recovery systems. copyright c© 2021 assa. adv syst sci appl (2021) software for testability analysis of aviation systems 129 fig. 4.12. distribution of the system average failures number by severity and detection methods. . copyright c© 2021 assa. adv syst sci appl (2021) 130 v.s. viktorova, a.s. stepanyants fig. 4.13. screenshot of the bit conformity analysis report (conformity i vs conformity ii) . fig. 4.14. bit conformity vs unreliability factor. a. t = tflight, b. t = tchecka . copyright c© 2021 assa. adv syst sci appl (2021) software for testability analysis of aviation systems 131 fig. 4.15. reliability indices vs bit fault detection ratio a. mttff b. r(0, tflight) c. r(0, tchecka) d. r(0, tcheckb). . fig. 4.16. availability vs fault detection ratio a. a(tflight) b. a(tchecka) . . copyright c© 2021 assa. adv syst sci appl (2021) 132 v.s. viktorova, a.s. stepanyants 4.6. case 6. operative availability analysis. the software provides the ability to build an operative availability trend for an arbitrary time frame. the time frame is defined by the user by choosing the start time counted from the last maintenance cycle with 100% renewal and quantity of the tops. the trend construction is based on models of operative availability described in section 3.4. trend analysis enables the aircraft designer to define bit functionalities and maintenance periods during which latent failures are detected and eliminated. this avoids a decrease of the operative availability below the acceptable level. fig. 4.17 (a,b) shows the trends in operative availability at fdr values equal to 0.7 and 0.9. the curves clearly show that an increase in the fdr leads to increase a probability of starting the system from a failure-free state. the limiting value of this probability, equal to one, is achieved at 100% failure coverage by bit fig. 4.17 (c). fig. 4.17. trend of operative availability (qstep = 20) a). fdr=0.7 b). fdr=0.9 c). fdr=1.0. . copyright c© 2021 assa. adv syst sci appl (2021) software for testability analysis of aviation systems 133 5. conclusion in this paper, we presented software implemented analytic techniques of testability evaluation and joint testability and reliability analysis of the aircraft systems [31, 32]. the approach to testability evaluation relies on parametric analysis of fmea data, including parameters as failure severity, fault detection method, failure type, impact on departure delay. we have presented two models for joint testability and reliability analysis. the first model is the fault-tree of bit conformity considering functional properties and unreliability of built-intest equipment. the second one is markov reliability model considering imperfection of bit fault coverage and cyclic operation mode specific for aviation systems. we have proposed a method for calculating operative availability taking into account the recovery scenario inherent in aviation. these models and methods are implemented in the web application of the oracle database. the architecture of the database of system characteristics and fmea reports was developed. extended preliminary effort of developing database architecture was devoted to creation of meta specification of testability data as xml schema. the codes of programs defining and solving the models were implemented in the pl sql language. the software pages with user interface components have smart grid layout adopted to varying screen resolutions both pc and mobile. we have demonstrated the software capability on the sample system analysis including testability evaluation, reliability analysis of redundant assemblies with imperfect fault coverage, modeling of operative availability trend on inter-maintenance interval. the software could be of practical value to designers of built-in-test systems and analysts of testability departments. references 1. gost r 56081 (2014) izdeliya aviatsionnoy tekhniki. bezopasnost’ poleta, nadezhnost’, kontroleprigodnost’, ekspluatatsionnaya i remontnaya tekhnologichnost’. poryadok normirovaniya i kontrolya pokazateley. 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(2011) flight control system reliability study based on hidden markov model imperfect fault coverage model-hidden markov model. int.l conf. on quality, reliability, risk, maintenance, and safety engineering. xi’an, 126–131, doi: 10.1109/icqr2mse.2011.5976582. 6. xiong, x., zhang, p. (2012) reliability analysis of flight control system for large civil aircraft with imperfect fault coverage model. proc. ieee prognostics and system health management conf. (phm-2012 beijing). beijing, 1–5, doi: 10.1109/phm.2012.6228935. 7. lubkov, n.v., spiridonov, i.b., stepanyants, a.s. (2016) vliyaniye kharakteristik kontrolya na pokazateli nadezhnosti sistem. [influence of control characteristics on system reliability indicators.] // trudy mai, 85, [in russian]. [online]. available: http://www.mai.ru/science/trudy/published.php?id=67501. 8. rani, p., pahuja, g. (2018) reliability analysis of flight control system under perfect and imperfect fault coverage. 3rd ieee int. conf. on recent trends in electronics, copyright c© 2021 assa. adv syst sci appl (2021) 134 v.s. viktorova, a.s. stepanyants information & communication technology (rteict). 759-763. 9. a4a msg-3. (2018) operator/manufacturer scheduled maintenance development. revision 2018.1. 10. chekryzhev, n.v. (2015) osnovy tekhnicheskogo obsluzhivaniya vozdushnykh sudov: ucheb. posobiye. [aircraft maintenance fundamentals:study. manual]. samara: izd. sgau, [in russian]. 11. mil-std-1629a (1984) procedures for performing a failure mode, effects and criticality analysis. notice 2. 12. arp4761 (1996) guidelines and methods for conducting the safety assessment process on civil airborne systems and equipment. sae international. 13. bidokhti, n., loeser, m. (2012) does your system have sufficient diagnostics coverage? proc. an. reliability&maintainability symp.. reno, nv, 1–6, doi: 10.1109/rams.2012.6175510. 14. liu, d., zeng, z., huang, c., & li, f. (2012) the testability modeling and model conversion technology based on multi-signal flow graph. proc. ieee prognostics & system health management conf. (phm-2012 beijing). beijing, 1–8, doi: 10.1109/phm.2012.6228919. 15. long, w., yue, l., yanling, q. tengfei, q. & minhao, w. (2017) a method of testability analysis and design based on fmea extension. 13th ieee int. conf. on electronic measurement & instruments (icemi), yangzhou, 361–367, doi: 10.1109/icemi.2017.8265968. 16. viktorova, v.s., stepanyants, a.s. (2010) proyektnyy analiz kontroleprigodnosti tekhnicheskikh sistem (teoriya, metody rascheta, programmnoye obespecheniye). 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(2005). ata 100 chapter and section headings. [online]. available: http://www.s-techent.com/ata100.htm 23. molina, m., carrasco, s., martin, j. (2014). agent-based modeling and simulation for the design of the future european air traffic management system: the experience of cassiopeia. j.m. corchado et al. (eds.): paams workshops, ccis 430, 22–33 https://doi.org/10.1007/978-3-319-077673 3. 24. henley, e.j., kumamoto, h. (1996). probabilistic risk assessment and management for engineers and scientists (2nd ed.) nj.: ieeepress. 25. viktorova, v.s., stepanyants, a.s. (2016) modeli i metody rascheta nadezhnosti tekhnicheskikh sistem [models and methods of reliability analysis of technical systems]. m.: lenand, [in russian]. 26. ryabinin, i. (1976) reliability of engineering systems. principles and analysis. english translation. m.: mir publishers. 27. ustinov, s.m. , zimnitskiy, v.a. (2009) vychislitel’naya matematika [computational mathematics]. spb.: bkhv-peterburg, [in russian]. copyright c© 2021 assa. adv syst sci appl (2021) software for testability analysis of aviation systems 135 28. viktorova, v.s., stepanyants, a.s. (2020) analiz nadezhnosti i effektivnosti mnogourovnevykh tekhnicheskikh sistem [analysis of reliability and efficiency of multilevel technical systems]. m.: lenand [in russian]. 29. stepanyants, a.s., viktorova, v.s. (2019) reliability analysis of systems with hybrid recovery and imperfect built-in-test. proc. of the 22nd int. conf. on distributed computer and communication networks: control, computation, communications (dccn-2019, moscow). cham: springer, 413–423, [in russian]. 30. petrukhin, b.p. (2014) osobennosti prognozirovaniya nadozhnosti sistem i ustroystv po metodike 217 plus tm chast’ 1. modeli intensivnosti otkazov komponent [features of predicting the reliability of systems and devices according to the 217 plus tm method. part 1. models of component failure rate]. datchiki i sistemy, 6, 37–43 [in russian]. 31. viktorova, v.s., lubkov, n.v., stepanyants, a.s. (2017) sowtware for testability analysis of aircraft functional systems: certificate of state registration of computer programs 2017660269 rf; dated 20.09.2017. 32. viktorova,v.s., stepanyants, a.s. (2019) analysis of the trend of operational availability of aircraft systems: certificate of state registration of computer programs 2019618035 rf; dated 26.06.2019. 6. appendix. testability data xml schema definition <?xml version=”1.0” encoding=”utf-8”?> <xs:schema xmlns:xs=”http://www.w3.org/2001/xmlschema” xmlns:mstns=”http://tempuri.org/testabilityxmlschema.xsd” xmlns=”http://tempuri.org/testabilityxmlschema.xsd” elementformdefault=”qualified” targetnamespace=”http://tempuri.org/testabilityxmlschema.xsd”> <xs:annotation> <xs:appinfo>testability software</xs:appinfo> <xs:documentation xml:lang=”en”> this schema defines a structure and properties of testability xml data </xs:documentation> </xs:annotation> <!-global simple types-> <xs:simpletype name=”trindex”> <!--time parameters and reliability indices--> <xs:restriction base=”xs:decimal”> <xs:mininclusive value=”0”/> </xs:restriction> </xs:simpletype> <xs:simpletype name=”tratio”> <!--testability indices normalized from 0 to 1--> <xs:restriction base=”xs:decimal”> <xs:mininclusive value=”0”/> <xs:maxinclusive value=”1”/> </xs:restriction> </xs:simpletype> <xs:simpletype name=”tcodata”> <!--six or eight character code--> <xs:restriction base=”xs:string”> <xs:pattern value=”[0-9]{6}|[0-9]{6}-[0-9]{2}”/> </xs:restriction> </xs:simpletype> <xs:simpletype name=”tlrutype”> <!--lru type--> <xs:restriction base=”xs:string”> <xs:enumeration value=”lru”/> <!--line replaceable unit--> copyright c© 2021 assa. adv syst sci appl (2021) 136 v.s. viktorova, a.s. stepanyants <xs:enumeration value=”pl”/> <!--pipeline--> <xs:enumeration value=”w”/> <!--wires--> </xs:restriction> </xs:simpletype> <xs:simpletype name=”trschema”> <!--redundancy schema--> <xs:restriction base=”xs:string”> <xs:enumeration value=”1out1”/> <!--single module--> <xs:enumeration value=”1out2”/> <!--double module redundancy--> <xs:enumeration value=”1out3”/> <!--triple module redundancy--> <xs:enumeration value=”2out3”/> <!--majority redundancy--> </xs:restriction> </xs:simpletype> <xs:simpletype name=”tfdm”> <!--failure detection methods--> <xs:restriction base=”xs:string”> <xs:enumeration value=”bit”/> <!--built-in-test--> <xs:enumeration value=”crew”/> <!--by flight crew--> <xs:enumeration value=”maint”/> <!--by repair team (check a)--> <xs:enumeration value=”none”/> <!--by repair team (check b)--> </xs:restriction> </xs:simpletype> <xs:simpletype name=”tseverity”> <!--failure severity level--> <xs:restriction base=”xs:string”> <xs:enumeration value=”negligible”/> <!--severity level v--> <xs:enumeration value=”minor”/> <!--severity level iv--> <xs:enumeration value=”marginal”/> <!--severity level iii--> <xs:enumeration value=”critical”/> <!--severity level ii--> <xs:enumeration value=”catastrophic”/> <!--severity level i--> </xs:restriction> </xs:simpletype> <xs:simpletype name=”tphases”> <!--flight profile--> <xs:restriction base=”xs:string”> <xs:enumeration value=”allphases”/> <xs:enumeration value=”takeoff”/> <xs:enumeration value=”climb”/> <xs:enumeration value=”cruise”/> <xs:enumeration value=”descent”/> <xs:enumeration value=”landing”/> </xs:restriction> </xs:simpletype> <xs:simpletype name=”tfailuretype”> <!--type of lru failure--> <xs:restriction base=”xs:string”> <xs:enumeration value=”ff”/> <!--functional failure--> <xs:enumeration value=”bit nop”/> <!--no operation bit failure--> <xs:enumeration value=”bit fa”/> <!--false alarm bit failure--> </xs:restriction> </xs:simpletype> <xs:simpletype name=”tdelay”> <!-departure delay flag--> <xs:restriction base=”xs:string”> <xs:enumeration value=”no”/> <!--failure does not impact on departure delay--> <xs:enumeration value=”yes”/> <!--failure impacts on departure delay--> </xs:restriction> </xs:simpletype> <!-global complex types-> copyright c© 2021 assa. adv syst sci appl (2021) software for testability analysis of aviation systems 137 <xs:complextype name=”tfmdescriptor”> <xs:sequence > <!-failure mode descriptor (uid and names)--> <xs:element name=”fid” type=”xs:string”/> <!--failure mode identifier--> <xs:element name=”fmname” type=”xs:string”/> <!--failure mode name--> <!-failure mode descriptor in centralized maintenance system (cms)--> <xs:element name=”fidcms” type=”xs:string”/> <xs:element name=”fmnamecms” type=”xs:string”/> <!--crew alerting system (cas) message--> <xs:element name=”casmsg” type=”xs:string” minoccurs=”0” /> </xs:sequence> </xs:complextype> <!--resolution list--> <xs:complextype name=”tlruchain”> <xs:sequence > <xs:element name=”codata” type=”tcodata” minoccurs=”0” maxoccurs=”unbounded” /> </xs:sequence > </xs:complextype > <!--definition of fmea type--> <xs:complextype name=”tfmea”> <xs:all > <xs:element name=”fmdescriptor” type=”’tfmdescriptor”/> <xs:element name=”fr” type=”trindex”/> <!--failure rate--> <xs:element name=”fdm” type=”tfdm”/> <xs:element name=”severity” type=”tseverity”/> <xs:element name=”phase” type=”tphases”/> <xs:element name=”failuretype” type=”tfailuretype” /> <xs:element name=”delay” type=”tdelay” /> <xs:element name=”resolution” type=”xs:positiveinteger” /> <xs:element name=”lruchain” type=”tlruchain” /> </xs:all> </xs:complextype> <!--definition of lru type--> <xs:complextype name=”tlru”> <xs:all > <xs:element name=”lruname” type=”tdescriptor”/> <xs:element name=”partnumber” type=”xs:string”/> <xs:element name=”serialnumber” type=”xs:string”/> <!--uid in centralized maintenance system--> <xs:element name=”idcms” type=”xs:string”/> <xs:element name=”lrutype” type=”tlrutype” /> <xs:element name=”rschema” type=”trschema” /> <!--mean time to first failure by specification--> <xs:element name=”mttff” type=”trindex” /> <!--mean time to repair by specification--> <xs:element name=”mttr” type=”trindex” /> <xs:element name=”fmeas” type=”tfmeas” /> </xs:all> </xs:complextype> <!--definition of system type--> <xs:complextype name=”tsystem”> <xs:all > <xs:element name=”sname” type=”tdescriptor”/> copyright c© 2021 assa. adv syst sci appl (2021) 138 v.s. viktorova, a.s. stepanyants <!--mean time to first failure by specification--> <xs:element name=”mttff” type=”trindex”/> <xs:element name=”fdr” type=”tratio”/> <!--fault detection ratio--> <xs:element name=”fir” type=”tratio”/> <!--fault isolation ratio--> <xs:element name=”brf” type=”tratio”/> <!--bit reliability factor--> <xs:element name=”faf” type=”tratio”/> <!--false alarm factor--> <xs:element name=”lrus” type=”tlrus” /> </xs:all> </xs:complextype> <!--definition of project type--> <xs:complextype name=”tproject”> <xs:all > <xs:element name=”prname” type=”tdescriptor”/> <xs:element name=”tf” type=”trindex”/> <!--flight duration--> <xs:element name=”tc” type=”trindex”/> <!--a check interval--> <xs:element name=”tb” type=”trindex”/> <!--b check interval--> <xs:element name=”tp” type=”trindex”/> <!--c check interval--> <xs:element name=”tr” type=”trindex”/> <!--d check interval--> <xs:element name=”systems” type=”tsystems” /> </xs:all> </xs:complextype> <!--set of fmeas of the lru--> <xs:complextype name=”tfmeas”> <xs:sequence > < xs:element name=”fmea” type=”tfmea” minoccurs=”1” maxoccurs=”unbounded” /> </xs:sequence > </xs:complextype > <!--set of lrus of the system--> <xs:complextype name=”tlrus”> <xs:sequence > < xs:element name=”lru” type=”lru” minoccurs=”1” maxoccurs=”unbounded” /> </xs:sequence > </xs:complextype > <!--set of systems of the project--> <xs:complextype name=”tsystems”> <xs:sequence > < xs:element name=”system” type=”tsystem” minoccurs=”1” maxoccurs=”unbounded” /> </xs:sequence > </xs:complextype > <!--definition of the set of projects--> <xs:complextype name=”tprojects”> <xs:sequence > < xs:element name=”project” type=”tproject” minoccurs=”1” maxoccurs=”unbounded” /> </xs:sequence > </xs:complextype > <!--global element/root of xml document--> <xs:element name=”projects” type=”tprojects” /> </xs:schema> copyright c© 2021 assa. adv syst sci appl (2021) introduction testability analysis software description testability metadata structure testability database structure testability software features mathematical background of computational algorithms single testability indicators bit conformity reliability measures operative availability model case studies of the software case 1. failure modes classification case 2. sti calculation case 3. afn calculation case 4. analysis of bit conformity case 5. reliability analysis. case 6. operative availability analysis. conclusion appendix. testability data xml schema definition adv syst sci appl 2023; 03:177–190 published online at https://ijassa.ipu.ru. on bi-laminar neural field models of electrical activity in the primary visual cortex evgenii burlakov1,2*, ivan malkov1 1tyumen state university, tyumen, russian federation 2 derzhavin tambov state university, tambov, russian federation abstract: we investigate the modelling framework for studying electrical activity in the primary visual cortex of the brain based on a bi-laminar neural field equation. the deep layer of the neural field models the orientation-independent electrical activity, whereas the orientation-dependent superficial layer captures the selectivity to spatially oriented stimuli of the orientation columns in the primary visual cortex. we verify the solvability of a cauchy problem for the bi-laminar neural field equation with both sigmoidal and heaviside-type neuronal activation. we also construct connections between the solutions that correspond to these types of neuronal activation, which justifies the use of the heaviside-type neuronal activation functions that is crucial in the problems of computer simulations involving vast ensembles on neurons. we prove the possibility of a correct approximation of the bi-laminar neural field model with a two-layer neuronal network. we also highlight some perspectives opened by the results of the present research related to the studies of travelling waves of evoked electrical activity in the visual cortex as well as the neural activity control problems in the framework of the neurofeedback paradigm. keywords: mathematical models of primary visual cortex, bi-laminar neural field models, twolayer neuronal nerwork models, well-posedness, heaviside activation function 1. introduction mathematical models of macroand mesoscopic neuronal activity of the human brain cortex involve the description of electrical activity of vast ensembles of neuronal elements, which can be registered using electroand magnetoencefalography (eeg and meg) [1, 2] and indirectly observed in fmri recordings [3], are usually presented in the form of neural field equations (see e.g. the pioneering work [4] and the review [5]). the most well-known neural field model is the amari neural field equation (see [4]) ∂tu(t, x) = −τu(t, x) + ∫ ω ω(x, y)g(u(t, y))dy. (1.1) here u(t, x) represents the level of electrical activity in the neural field ω at time t and position x, the so-called connectivity function ω defines the strengths of interneuronal connections in the neural field, the value g(u) determines the probability of activation (firing) of a neuron with electrical activity level u. in the mathematical neuroscience community, the connectivity ω is typically assumed to be an exponentially decaying function symmetric with respect to the vertical axis or a sum of such functions, and the activation function f is taken to be a continuous sigmoidal-shaped function. ∗corresponding author: eb @bk.ru 178 e. burlakov, i. malkov several extensions of (1.1) relying on the fact that the neural media is not spatially uniform formalize the heterogeneity of the brain cortex in general forms (see e.g. [6–8]). such mathematical models are rather well-studied, including the issues of well-posedness [8, 9], construction and justification of the schemes for numerical simulations [8–10], studies of special types of solutions that are physiologically relevant [6, 7, 11, 12]. however, none of these neural field models could capture the characteristical features of primary visual cortex, whose elements of microstructure are selective with respect to perceptions of visual stimuli of certain directions, before the introduction of the following model (see [13]): ∂tud(t, x) = −τdud(t, x) + ∫ ω ωd(x, y)gd(u(t, y))dy +νd π 2∫ −π 2 gs(us(t, x, ψ))dψ, ∂tus(t, x, φ) = −τsus(t, x, φ) + ∫ ω π 2∫ −π 2 ωs(x, φ, y, ψ)gs(u(t, y, ψ))dψdy +νsgd(ud(t, x)). (1.2) here ud(t, x) defines the level of orientation-independent activity in the deep layer and us(t, x, φ) is the orientation-dependent activity in the superficial layer. the connectivities in the two layers are denoted by ωd and ωs, respectively, and the corresponding time constants are given by τd and τs. we also include vertical inputs from the deep to the superficial layer with the strength νd and back from the superficial to the deep layer with the strength νs (the latter is averaged with respect to the orientation preference of neurons in the superficial layer). functions gd, gs are probabilistic functions of firing (neuronal activation functions) for the deep layer and the superficial layer, respectively. we refer the reader to the work [13] for more details on the biophysical justification of the modeling framework (1.2). to the best of our knowledge, the work [13] is the only published study capturing the orientation selectivity in the visual cortex in the framework of the neural field setting. however, several important mathematical issues were overlooked in this study. the aim of the present paper is to fill in these mathematical blank spots in the justification of the modelling framework (1.2). namely, in section 2 we verify the solvability of a cauchy problem for (1.2) and prove the possibility of correct spatially discretized approximation of (1.2). section 3 deals with the solvability problem for (1.2) in the case when the probabilistic activation functions are replaced with the instant-activation heaviside-type functions. in section 4 we establish connections between the modelling approaches of the two previous sections. in section 5 we highlight some close perspectives opened by the results of the present research. 2. bi-laminar neural field model with continuous activation functions for convenience of the presentation of the forthcoming material, we introduce the following notations. for any metric space λ, any λ0 ∈ λ, s ⊂ λ, and r > 0, we define bλ(λ0, r) to be the ball in the space λ of the radius r > 0 centered at λ0, s to be the closure of s in λ. we denote by rm the m-dimensional real vector space with the norm | · | and by ω – a compact subset of r2. denote by lp(ω× (−π 2 , π 2 ],r2) and cp(ω× (−π 2 , π 2 ],r2) the banach spaces of integrable and continuous, respectively, functions from ω× (−π 2 , π 2 to r2), which are πperiodic in the second variable. for all t > 0, we denote ξt = [0, t ]× ω× (−π 2 , π 2 ] and copyright © 2023 assa. adv syst sci appl (2023) on bi-laminar neural field models 179 definebcp(ξt ,r2) to be the banach space of bounded continuous functions ϑ : ξt → r2 πperiodic in the third variable with the norm ||ϑ||bcp(ξt ,r2) = max x∈ξt |ϑ(x)| and l∞,p(ξt ,r2×2) – to be the banach space of 2× 2 matrix functions from ξt to r with essentially bounded components π-periodic with respect to the third variable. in this section we consider the solvability of a cauchy problem for the bi-laminar neural field equation (1.2) in the case of continuous activation functions (i.e. gd = fd, gs = fs) and justify the possibility to approximate the bi-laminar model (1.2) with a two-layered neuronal network. we start out with the system ∂tud(t, x) = −τdud(t, x) + ∫ ω ωd(x, y)fd(ud(t, y))dy +νd π 2∫ −π 2 fs(us(t, x, ψ))dψ, ∂tus(t, x, φ) = −τsus(t, x, φ) + ∫ ω π 2∫ −π 2 ωs(x, φ, y, ψ)fs(us(t, y, ψ))dψdy +νsfd(ud(t, x)) (2.3) together with the initial condition ud(0, x) = ûd(x), us(0, x, φ) = ûs(x, φ) (2.4) where ûd : ω → r, ûs : ω× (−π 2 , π 2 ] → r are continuous and lim φ→−π 2 ûs(x, φ) = ûs(x, π 2 ). assume that (aω) the neuronal connectivity functions ωd, ωs : ω× ω → rn are continuous; (af ) the neuronal activation functions fd, fs : r2 → [0, 1] are lipschitz continuous; theorem 2.1: let assumptions (aω) and (af ) be satisfied. then there exists a unique solution of the problem (2.3), (2.4), which is a continuous function from [0,∞)× ω× (−π 2 , π 2 ] to r2. proof the system (2.3) can be written as ∂tu(t, x, φ) = −τu(t, x, φ) + ∫ ω π 2∫ −π 2 v (x, y, φ, ψ)f (u(t, x, φ))dψdy, (2.5) where f (u) = ( fd(ud) fs(us) ) , v (x, y, φ, ψ) = ( ωd(x, y)/π νsδ(x− y) νdδ(x− y)/π ωs(x, φ, y, ψ) ) . we will consider the problem of solvability of the system (2.3), (2.4) in a more general setting that reads as follows: u(t, x, φ) = û(x, ψ) + t∫ 0 ∫ ω π 2∫ −π 2 w (t, s, x, y, φ, ψ)f (u(s, y, ψ))µ1(dψ)µ2(dy)ds (2.6) copyright © 2023 assa. adv syst sci appl (2023) 180 e. burlakov, i. malkov where u(t, x, φ) = ( ud(t, x) us(t, x, φ) ) , û(x, φ) = ( ûd(x) ûs(x, φ) ) , w (t, s, x, y, φ, ψ) = ( exp(−τd(t−s)) 0 0 exp(−τs(t−s)) ) v (x, y, φ, ψ), and µ1 and µ2 are complete σ-additive measures defined on r and r2, respectively, and finite on bounded subsets of their ranges of definition. introduce the following two conditions: (aw ) for any t > 0 and any (t, x, φ) ∈ ξt , the function w (t, ·, x, ·, φ, ·) belongs to l∞,p(ξt ,r2×2), the function (t, x, φ) 7→ ∥w (t, ·, x, ·, φ, ·)∥l∞,p(ξt ,r2×2) is bounded, and for any measurable set i ⊂ ξt and any (t0, x0, φ0) ∈ ξt , it holds true that lim (t,x,φ)→(t0,x0,φ0) ∫ ∫ ∫ i∩ ( [0,t]×ω ) w (t, s, x, y, φ, ψ)µ1(dψ)µ2(dy)ds = ∫ ∫ ∫ i∩ ( [0,t0]×ω×(−π 2 ,π 2 ] ) w (t0, s, x0, y, φ0, ψ)µ 1(dψ)µ2(dy)ds. (af ) the function f : r2 → [0, 1]2 is lipschitz continuous (hereinafter, we denote [0, 1]2 = [0, 1]× [0, 1]). choose t > 0. we define ut ∈ bcp(ξt ,r2) to be a t -local solution to (2.6) if ut satisfies the equation (2.6) on the set ξt . we consider a continuous function u∞ : [0,∞)× ω× (−π 2 , π 2 ] → r2 to be a global solution to (2.6) if for any t > 0, the restriction of u∞ to ξt is a t -local solution to (2.6). let us formulate the statement on solvability of the integral equation (2.6) in terms of the definitions given above. lemma 2.1: let assumptions (aw ) and (af ) be satisfied. then for any t > 0, the equation (2.6) has a unique t -local solution that is the restriction to ξt of the unique global solution to (2.6). proof of lemma 2.1 we choose arbitrary t > 0 and take any two functions u1, u2 ∈ bcp(ξt ,r2). we estimate ∥itntu1 − itntu2∥bcp(ξt ,r2) = max (t,x,φ)∈ξt |(itntu1)(t, x, φ)− (itntu2)(t, x, φ)|, where for any ξ ∈ bcp(ξt ,r2), the operator it is defined by the relation (it ξ)(t, x, φ) = t∫ 0 ∫ ω π 2∫ −π 2 w (t, s, x, y, φ, ψ)ξ(s, y, ψ)µ1(dψ)µ2(dy)ds (2.7) and, due to condition (aw ), has the image in bcp(ξt ,r2) (see e.g. [14], chapter 3, § 5.5); the operator nt defined as (nt ξ)(t, x, φ) = f (ξ(t, x, φ)) acts from the space bcp(ξt ,r2) to itself due to (af ). by the virtue of (af ), we have max (t,x,φ)∈ξt |(itntu1)(t, x, φ)− (itntu2)(t, x, φ)| copyright © 2023 assa. adv syst sci appl (2023) on bi-laminar neural field models 181 = max (t,x,φ)∈ξt ∣∣∣∣∣ t∫ 0 ∫ ω π 2∫ −π 2 w (t, s, x, y, φ, ψ)f (u1(s, y, ψ))µ 1(dψ)µ2(dy)ds − t∫ 0 ∫ ω π 2∫ −π 2 w (t, s, x, y, φ, ψ)f (u2(s, y, ψ))µ 1(dψ)µ2(dy)ds ∣∣∣∣∣ ≤ max (t,x,φ)∈ξt t∫ 0 ∫ ω π 2∫ −π 2 |w (t, s, x, y, φ, ψ)||f (u1(s, y, ψ))− f (u2(s, y, ψ))|µ1(dψ)µ2(dy)ds ≤ max (t,x,φ)∈ξt t∫ 0 ∫ ω π 2∫ −π 2 |w (t, s, x, y, φ, ψ)|lf |u1(s, y, ψ)− u2(s, y, ψ)|µ1(dψ)µ2(dy)ds ≤ lf max (t,x,φ)∈ξt t∫ 0 ∫ ω π 2∫ −π 2 |w (t, s, x, y, φ, ψ)|µ1(dψ)µ2(dy)ds ∥u1(s, y, ψ)− u2(s, y, ψ)∥ where lf > 0 is the lipschitz constant of f . by choosing t = t1 > 0 in a way that lf max (t,x,φ)∈ξt1 t1∫ 0 ∫ ω π 2∫ −π 2 |w (t, s, x, y, φ, ψ)|µ1(dψ)µ2(dy)ds < 1 and applying banach fixed point theorem (see e.g. [15], chapter 2, § 14) to the problem u = it1nt1u (2.8) in the space bcp(ξt1 ,r2) we prove the existence of a unique solution ut1 ∈ bcp(ξt1 ,r2) of (2.8), which is a unique t1-local solution of the integral equation (2.6). we introduce a new time-variable t′ = t− t1. now we consider the problem (2.6) with t = t′. we will refer to this new problem as (2.6)′ (with the initial problem u′d(−t1, x) = û′d(x), u ′ s(−t1, x, φ) = û′s(x, φ)). we apply the procedure described above and prove the existence of a t ′ 1-local solution ut ′ 1 to (2.6)′ for some t ′ 1 > 0. we thus obtain a t2-local solution ut2 to (2.6) where t2 = t1 + t ′ 1, u = { ut1 t ∈ [0, t1] ut ′ 1 t ∈ [t1, t2] . the t2-local solution ut2 is continuous at (t1, x) for any x ∈ ω, that is, ut2 ∈ on the next step, we choose any t2-local solution ut2 ∈ to the equation (2.6) we introduce a new time-variable t′′ = t − t2 and repeat the procedure we thus obtain a strictly increasing sequence {ti}, i = 1, 2, . . . and the corresponding sequence of of local solutions uti , i = 1, 2, . . . such that for any i1 < i2, uti1 is the restriction of uti2 ∈ bcp (ξti2 ,r2) to the set ξti1 . we find lim i→∞ ti = t̂ . take any t∗ ∈ (0, t̂ ). for some number i, t ∈ (ti−1, ti) and ut∗ therefore is a t∗-local solution. we have constructed the mapping t∗ 7→ ut∗ . prove that {ti}, i = 1, 2, . . . is not bounded. indeed, assuming the contrary, we get ti < t ∗ for some t ∗ <∞ and all i = 1, 2, . . ., so that the norms of the copyright © 2023 assa. adv syst sci appl (2023) 182 e. burlakov, i. malkov corresponding solutions satisfy the relation lim t→t ∗−0 ∥ut∥bcp(ξt ,r2) = ∞ which contradicts to (aw ). we thus proved that {ti}, i = 1, 2, . . . is not bounded and, hence that the sequence of local solutions constructed in the proof has a unique limit that is a global solution to (2.6). thus, lemma 2.1 is proved. note that the validity of conditions (aω) and (af ) naturally provides the fulfillment of (aw ) and (af ). now, applying lemma 2.1 to (2.6) in the case when µ1(dψ) is the lebesgue measure on r µ2(dy) is the lebesgue measure on r2, we prove the theorem. the validity of the following statement is implied by lemma 2.1. remark 2.1: for any t > 0, the restriction of the unique (global) solution obtained in theorem 2.1 to the set ξt is the unique solution to the problem (2.3), (2.4) on the set ξt . consider now a two-layer neural network ∂tv i d(t, n)=−τdvid(t, n)+ n∑ j=1 ωij d(n)fd(v j d(t, n))+νd m∑ l=1 fs(v il s (t,m)), ∂tv ik s (t, n,m)=−τsviks (t, n,m)+ n∑ j=1 m∑ l=1 ωijkl s (n,m)fs(v jl s (t, n,m))+νsfd(v i d(t, n)) (2.9) having n ”spatial” elements in each of the layers and m ”directions” in the orientation columns layer, and parameterized by the dimensions n and m of its layers. for any natural n and m, the values vid(t, n), v ik s (t, n,m) correspond to the neuronal activity of the deep layer and the superficial layer in the network, the constants ωij d(n), ω ijkl s (n,m) define the strengths of connections inside each of the layers, and the constants νd, νs define the strengths of connections between the layers. the following statement establishes correspondence between the bi-laminar neural field model (2.3) and the two-layer neuronal network (2.9). proposition 2.1: let assumptions (aω), (af ) be satisfied. for each naturalm and n, let {δ1k (m), k = 1, ...,m} and {δ2i (n), i = 1, ..., n} be finite families of open subsets of r and r2, respectively, satisfying the conditions m⋃ k=1 δ1k (m) = (−π 2 , π 2 ], n⋃ i=1 δ2i (n) = ω, lim m→∞ max k=1,...,m mes(δ1k (m)) = 0, lim n→∞ max i=1,...,n mes(δ2i (n)) = 0, where mes(·) denotes the lebesgue measure. let ψk(m) and yi(n) (k = 1, ...,m, i = 1, ..., n) be arbitrary points in δ1k (m) and δ2i (n), respectively. then for each natural n and m and any α(n,m) ∈ rnm+n, there exists a unique continuous solution to the system (2.9) considered together with the initial conditions( vid(n) ) (0) = αi d(n), ( viks (n,m) ) (0) = αik s (n,m), (2.10) which is a function v(n,m) = (vd(n), vs(n,m)) from [0,∞) to rnm+n. copyright © 2023 assa. adv syst sci appl (2023) on bi-laminar neural field models 183 moreover, the sequence of solutions vik(n,m) = (vid(n), v ik s (n,m)) to the initial value problem (2.9), (2.9), where ωij d(n) = mes(δ2i (n))ωd(yi(n), yj(n)), ωijkl s (n,m) = mes(δ2i (n))mes(δ1k (m))ωs(yi(n), yj(n), ψk(m), ψl(m)), αik d(n) = ûd(0, yi(n)), α ik s (n,m) = ûs(0, yi(n), ψk(m)) (2.11) converges to the solution u(t, x, φ) (t ≥ 0, x ∈ ω, φ ∈ (−π 2 , π 2 ]) of the initial value problem (2.3), (2.4), as n,m→ ∞, in the following sense: lim n→∞ m→∞ max t∈[0,t ] ( sup 1≤i≤n 1≤k≤m ( sup x∈δ2i (n) φ∈δ1k (m) |u(t, x, φ)− ( vik(n,m) ) (t)| )) = 0 (2.12) for any t > 0. proof by a reasoning similar to the one applied in the proof of theorem 2.1, we can conclude that for any natural n and m, the problem (2.9), (2.10) is equivalent to the equation (2.6) with µ1 = µ1 1/m, µ2 = µ2 1/n, where µ1 1/m and µ2 1/n are the sums of m dirac point measures concentrated at the points ψk(m) ∈ (−π 2 , π 2 ] and n dirac point measures at the points yi(n) ∈ ω, respectively. we define µ1 0 and µ2 0 to be the lebesgue measures on r and r2, respectively. therefore, solvability of (2.9), (2.10) for each natural n and m follows from theorem 2.1. the proposition conditions imply that the measures µ1 (·) and µ2 (·) are weakly right-continuous at 0 on the sets (−π 2 , π 2 ] and ω, respectively. indeed, for any continuous function υ(x, φ) : (−π 2 , π 2 ]× ω → r2, ∫ ω π 2∫ −π 2 υ(y, ψ)µ1 1/m(dψ)µ 2 1/n(dy) = n∑ i=1 m∑ k=1 υ(yi(n), ψk(m))mes(δ1k (m))mes(δ2i (n)) = ∫ ω π 2∫ −π 2 υ(y, ψ)µ1 0(dψ)µ 2 0(dy) (2.13) as n,m→ ∞. finally, assumptions (aω), (af ), together with the relations (2.11) and the property (2.13) provide the convergence (2.12). copyright © 2023 assa. adv syst sci appl (2023) 184 e. burlakov, i. malkov 3. bi-laminar neural field model with heaviside-type activation functions in this section we derive conditions for solvability of (1.2) in the case of discontinuous heaviside-type activation functions (i.e., gd = hd, gs = hs): ∂tud(t, x) = −τdud(t, x) + ∫ ω ωd(x, y)hd(ud(t, y))dy +νd π 2∫ −π 2 hs(us(t, x, ψ))dψ, ∂tus(t, x, φ) = −τsus(t, x, φ) + ∫ ω π 2∫ −π 2 ωs(x, φ, y, ψ)hs(us(t, y, ψ))dψdy +νshd(ud(t, x)) (3.14) where (ah) the components of the neuronal activation function h = (hd, hs), h : r2 → {0, 1} are heaviside-type activation functions: hk(u) = { 0, u ≤ hk, 1, u > hk, where hk is the threshold of activation, k = d, s. following the procedure described in the proof of theorem 2.1 we rewrite the problem (3.14), (2.4) as follows: u(t, x, φ) = û(x, ψ) + t∫ 0 ∫ ω π 2∫ −π 2 w (t, s, x, y, φ, ψ)h(u(s, y, ψ))dψdyds (3.15) where h = ( hd hs ) . using the ideas of a.f. filippov (see [16], chapter 2, § 4), we address the problem of solvability of the integral equation (3.15) and, hence, the problem (3.14), (2.4), in the sense of the so-called generalized solutions. we define a generalized solution to the problem (3.14), (2.4) to be a continuous function from [0,∞)× ω× (−π 2 , π 2 ] to r2, which satisfies the inclusion u(t, x, φ) ∈ û(x, ψ) + t∫ 0 ∫ ω π 2∫ −π 2 w (t, s, x, y, φ, ψ)h(u(s, y, ψ))dψdyds (3.16) where (ah) the set-valued function h : r2 ⇒ [0, 1]2 is defined as h = (hd,hs), hk(u) = { 0, u < hk, [0, 1], u = hk, 1, u > hk, k = d, s (hk are the same as in (ah)). copyright © 2023 assa. adv syst sci appl (2023) on bi-laminar neural field models 185 theorem 3.1: let assumptions (aω) and (ah) be satisfied. then there exists a generalized solution to the problem (3.14), (2.4), which is a continuous function from [0,∞)× ω× (−π 2 , π 2 ] to r2, proof we start out with proving the solvability of the inclusion (3.16) in the following sense. for any t > 0, we define ut ∈ bcp(ξt ,r2) to be a t -local solution to (3.16) if ut satisfies the inclusion (3.16) on the set ξt . we say that a continuous function u∞ : [0,∞)× ω× (−π 2 , π 2 ] → r2 is a global solution to (3.16) if for any t > 0, the restriction of u∞ to ξt is a t -local solution to the inclusion (3.16). lemma 3.1: let assumptions (aw ) and (ah) be satisfied. then, for any t > 0, the inclusion (3.16) has a t -local solution. any t -local solution can be extended to a global solution to (3.16). proof of lemma 3.1 we choose arbitrary t > 0 and represent the mapping on the right-hand side of (3.16) as follows: (itntu)(t, x, φ) = t∫ 0 ∫ ω π 2∫ −π 2 w (t, s, x, y, φ, ψ)h(u(s, y, ψ))dψdyds where it : bcp(ξt ,r2) → bcp(ξt ,r2) is defined by (2.7) and for any ξ ∈ bcp(ξt ,r2), (nt ξ)(t, x, φ) = h(ξ(t, x, φ)). by the virtue of (aw ) we have that it is a continuous operator. the operator h is upper-semicontinuous due to (ah). as h is upper-semicontinuous and it is a linear continuous operator, for mξ r = bc(ω×(−π 2 ,π 2 ],r2)(ξ, r), the composition itnt ξ : mξ r → c(ω× (−π 2 , π 2 ],r2) is a convex valued mapping. show that the composition itnt ξ is uppersemicontinuous and closed. choose ui and ϑ such that ϑ ∈ inui and ∥ui − u0∥bcp(ξt ,r2) → 0, ∥ϑi − ϑ0∥bcp(ξt ,r2) → 0 where u0 ∈ bcp(ξt ,r2), ϑ0 ∈ bcp(ξt ,r2) are some limit points and vi → v0 ∈ bcp(ξt ,r2). choose also wi ∈ n vi such that ϑi = iwi. consider the sequence {wi} ⊂ l as the sequence of bochner integrable mappings wi : [0, t ] → l(ω× (−π 2 , π 2 ]). by the virtue of condition (ah) and convergence vi → v0 we can apply kolmogorov-riesz compactness theorem [17] and consequently proposition 4.2.1 [18], and conclude that wi → w0 weakly, w0 ∈ l. further, using mazur’s lemma (see [19], section 5.1, theorem 2) we have ŵi = ∞∑ j=i βijwj such that ∥ŵi − w0∥ → 0 (3.17) where the coefficients βij satisfy the following conditions: • ∞∑ j=i βij = 1 for all i = 1, 2, ...; • one can find a number j0 such that βij = 0 for all j > j0 and for each i = 1, 2, .... the relation (3.17) implies (see section 41, theorem 4 [15]) the existence of a subsequence of the sequence ŵi converging to w0 almost everywhere. copyright © 2023 assa. adv syst sci appl (2023) 186 e. burlakov, i. malkov due to upper semicontinuity of h, for almost all (t, x, φ) ∈ ξt and for any ε > 0 there exists a number i0 = i0(t, x, φ, ε) such that for all i > i0 it holds true that h(t, x, φ, vi) ⊂ b((h(t, x, φ0))(t, x, φ), ε) we thus have wi ∈ b((h(t, x, φ0))(t, x, φ), ε). as an ε-neighborhood of a convex set is convex, we obtain ŵi ∈ b((h(t, x, φ0))(t, x, φ), ε). by the virtue of the closedness of h, the latter relation implies that w0 ∈ b((h(t, x, φ0))(t, x, φ), ε) so that w0 ∈ nu0. putting ϑ̂i = it ŵi = it ∞∑ j=i βijwj = ∞∑ j=i βijitwj = ∞∑ j=i βijϑj we obtain ∥ϑ̂i − ϑ∥ → 0, which due to the continuity of it implies that ϑ = itw0 ∈ itntu0. thus, the closedness and, consequently, the upper semicontinuity of the composition itnt is proved. now we choose some sufficiently large t > 0 and put r = 2max(û). using (aw ) and (ah), find the maximal t1 ∈ (0, t ] such that max (t,x,φ)∈ξt1 ∣∣∣∣∣ t∫ 0 ∫ ω π 2∫ −π 2 w (t, s, x, y, φ, ψ)ξr(s, x, ψ)− û(x, φ)dψdyds ∣∣∣∣∣ ≤ r. thus, we have it1nt1(mξ r) ⊂ mξ r, we apply bohnenblust–karlin theorem and prove the existence of a fixed point ut1 ∈ ξt1 that is a t1-local solution to the problem (3.16). choose any t1-local solution ut1 to problem (3.16). we introduce a new time-variable t1 = t− t1. now, we consider the problem (3.16) with t = t1. we apply the procedure described above and prove the existence of a t ′ 1-local solution u′t ′ 1 to (3.16) for some t ′ 1 > 0. we thus obtain a t2-local solution ut2 to (3.16), where t2 = t1 + t ′ 1. u ={ ut1 t ∈ [0, t1] u′t ′ 1 t ∈ [t1, t2] . the t2-local solution ut2 is continuous at (t2, x, φ). in the next step, we select any t2-local solution ut2 to the problem (3.15), (2.4). we introduce a new time-variable t2 = t− t2 and repeat the procedure above. we thus obtain a strictly increasing sequence {ti}, i = 1, 2, . . . and the corresponding sequence of of local solutions uti , i = 1, 2, . . . such that for any i1 < i2, uti1 is the restriction of uti2 ∈ bcp (ξti2 ,r2) to the set ξti1 . we find lim i→∞ ti = t̂ . take any t∗ ∈ (0, t̂ ). for some number i, t ∈ (ti−1, ti) and ut∗ therefore is a t∗-local solution. we have constructed the mapping t∗ 7→ ut∗ . prove that {ti}, i = 1, 2, . . . is not bounded. indeed, assuming the contrary, we get ti < t ∗ for some t ∗ <∞ and all i = 1, 2, . . ., so that the norms of the corresponding solutions satisfy the relation lim t→t ∗−0 ∥ut∥bcp(ξt ,r2) = ∞ which contradicts to (aw ). we thus proved that {ti}, i = 1, 2, . . . is not bounded and, hence that the sequence of local solutions constructed in the proof has a unique limit that is a global solution to (2.6). thus, lemma 3.1 is proved. lemma 3.1 implies that the problem (3.14), (2.4) possesses a generalized solution, which proves the theorem. copyright © 2023 assa. adv syst sci appl (2023) on bi-laminar neural field models 187 4. continuous dependence of solutions to bi-laminar neural field model under the transition from continuous to heaviside-type neuronal activation we introduce here the following parameterized version of (2.3) ∂tud(t, x) = −τdud(t, x) + ∫ ω ωd(x, y)f i d(ud(t, y))dy +νd π 2∫ −π 2 f i s(us(t, x, ψ))dψ, ∂tus(t, x, φ) = −τsus(t, x, φ) + ∫ ω π 2∫ −π 2 ωs(x, φ, y, ψ)f i s(us(t, y, ψ))dψdy +νsf i d(ud(t, x)) (4.18) with a natural parameter i. the next statement presents the main result of this section. theorem 4.1: let assumption (aω) be fulfilled and for all natural i, the functions f i d, f i s : r → [0, 1] satisfy assumption (af ). then for any natural i, there exists a unique solution, say ui, to the equations (4.18) with the initial condition (2.4), which is a continuous function from [0,∞)× ω× (−π 2 , π 2 ] to r2. moreover, if for any ε > 0, one can find a number iε such that |f i k(u)−hk(u)| < ε, u ∈ r \br(hk, ε), i > iε, k = d, s, (4.19) a continuous function u0 : [0,∞)× ω× (−π 2 , π 2 ] → r2, u0 = (u0 d, u 0 s ), obtained from the relation lim t→∞ lim i→∞ max (t,x,φ)∈ξt |ui(t, x, φ)− u0(t, x, φ)| = 0 in the case if it satisfies the condition mes ( {(t, x) ∈ [0,∞)× ω, u0 d(t, x) = hd, u 0 s (t, x, φ) = hs} ∪{(t, x, φ) ∈ [0,∞)× ω× (−π 2 , π 2 ], u0 d(t, x) = hd, u 0 s (t, x, φ) = hs} ) = 0, (4.20) is a generalized solution to the problem (3.14), (2.4). proof similarly to the previous sections we can rewrite (4.18) as u(t, x, φ) = û(x, ψ) + t∫ 0 ∫ ω π 2∫ −π 2 w (t, s, x, y, φ, ψ) ( f(u, 1 i ) ) (s, y, ψ)dψdyds, (4.21) where f(u, 1 i ) = ( f i d(ud) f i s(us) ) . copyright © 2023 assa. adv syst sci appl (2023) 188 e. burlakov, i. malkov unique solvability of the problem (4.18), (2.4) for all natural i follows from theorem 2.1. choose arbitrary t > 0. assumptions (aω) and (af ) imply that the set ∞⋃ i=1 t∫ 0 ∫ ω π 2∫ −π 2 w (t, s, x, y, φ, ψ) ( ft (bcp(ξt ,r2), 1 i ) ) (s, y, ψ)dψdyds and, hence, the set of solutions to the problem (4.18), (2.4) defined on the set ξt , say ui t , (for all natural i) is relatively compact in bcp(ξt ,r2). choose any limit point, say u0 t ∈ bcp(ξt ,r2), of the set ∞⋃ i=1 ui t . denoting ft (u, 0) = ( hd(ud) hs(us) ) we notice that the relations (4.19), (4.20) and the properties of set-valued integral (see e.g. [20], §1.5.1) provide that for any ε > 0, there exists a number iε such that itft (u i, 1 i ) ∈ bbcp(ξt ,r2)(itf(u0, 0), ε) for all i > iε. the latter allows to get the following relation: bbcp(ξt ,r2)(u 0, ε/2) ∋ ui = itf(ui, 1 i ) ∈ bbcp(ξt ,r2)(itf(u0, 0), ε/2), which means that u0 is a limit point of itft (u 0, 0) = itntu 0 (see section 3). noticing that the values of itft are closed (see e.g. the proof of lemma 3.1), we prove that u0 ∈ bcp(ξt ,r2) is a generalized solution of the problem (3.14), (2.4) on the set ξt . taking into account the arbitrary choice of t > 0, we finalize the proof. 5. conclusions and outlook in this research we established fundamental properties of the mathematical model for the macroand meso-scale electrical activity in the primary visual cortex based on the neural field concept. this opens several perspectives of further applications of the modelling framework (1.2). in recent studies of the human brain travelling waves, which are the most well-identified phenomenon of the brain electrical activity, underlying the brain functioning in both normal regime and in various pathological states (such as e.g. epilepsy and parkinson’s disease), the neural field equation (1.1) was successfully used in the simulations of travelling waves in the sensorimotor cortex [21]. the model (1.1) was supplemented by an extra component formalizing a slow negative feedback in the neural media due to the presence of the inhibitory neurons (see e.g. [22]). a much more intriguing problem from the point of view of neurophysiology is to investigate and model the interaction of travelling waves in the visual cortex with the orientation columns under a presentation of spatially oriented stimuli. in a similar way to (1.1), the models (2.3) and (3.14) can be equipped with the corresponding negative feedback components for mathematical modelling in the framework of the aforementioned studies. moreover, the presentation of spatially oriented stimuli can be treated as an impulse control problem for the modelling system (1.2), which can be studied e.g. based on the ideas developed in [23]. copyright © 2023 assa. adv syst sci appl (2023) on bi-laminar neural field models 189 the models of the form (1.2) can be used in the framework of the neurofeedback paradigm, where the target characteristics of the brain activity are converted into humaninterpretable visual, auditory or tactile stimuli [24, 25]. for example, a person undergoing a neurofeedback session learns to regulate the activity of his own central nervous system by performing task of maintaining the stimulus in a certain state trying to keep the angle of a rotating arrow displayed on the monitor screen within certain limits, which corresponds to maintaining the target characteristic of brain activity in the required range. this paradigm is used both for the correction of the psycho-emotional state, and for training to improve the efficiency of cognitive functions, as well as the treatment of a wide range of neurodegenerative diseases, including epilepsy [26]. another paradigm involves stimulation and alteration (transcranial magnetic, using direct or alternating current) of brain activity, depending on the current state of the central nervous system [27, 28]. it is aimed at suppressing pathological activity or inducing a specific behavioral response, which is used e.g. to suppress tremor in patients with parkinsonism or reduce the probability of an epileptic seizure. the paradigms described can be related to minimization problems constructed based on the mathematical framework (3.14), as it is more suitable for computer simulations compared to the model (2.3). in these minimization problems, the discontinuity in the nonlinear activation functions implies difficulties in using the standard theory that relies on the smoothness of the mappings involved. however, we conjecture that due to the presense of the heaviside-type activation functions in (3.14), one can construct isotone operators acting in an appropriate ordered space and apply the results on minimization of functionals in partially ordered spaces developed in [29] to prove the solvability of the minimization problem. acknowledgements the present research is supported by the russian science foundation (grant no. 22-2100756). references 1. muller, l., chavane, f., reynolds, j. & sejnowski, t. 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(2016). on well-posedness of generalized neural field equations with impulsive control, russian mathematics (iz vuz), 60(5), 66–69. 24. evans, n., gale, s., schurger, a. & blanke, o. (2015). visual feedback dominates the sense of agency for brain-machine actions, plos one, 10(6), e0130019. 25. hady, a. el. (2016). closed loop neuroscience. elsevier. 26. sitaram, r., ros, t., stoeckel, l., haller, s., scharnowski, f. et. al. (2016). closed-loop brain training: the science of neurofeedback, nature reviews neuroscience, 18, 86–100. 27. mcintyre, c.c. & warren, m.g. (1999). excitation of central nervous system neurons by nonuniform electric fields, biophys. j., 76(2), 878–888. 28. zrenner, c., belardinelli,p. , mueller-dahlhaus, f. & ziemann, u. (2016). closedloop neuroscience and non-invasive brain stimulation: a tale of two loops, front cell neurosci., 00092. 29. arutyunov, a.v., zhukovskiy, e.s. & zhukovskiy, s.e. (2019). caristi-like condition and the existence of minima of mappings in partially ordered spaces, j. optimis. theory app., 180(1), 48–61. copyright © 2023 assa. adv syst sci appl (2023) introduction bi-laminar neural field model with continuous activation functions bi-laminar neural field model with heaviside-type activation functions continuous dependence of solutions to bi-laminar neural field model under the transition from continuous to heaviside-type neuronal activation conclusions and outlook microsoft word 803 a novel enriched lasso based compression technique for energy efficient wireless sensor networks adv syst sci appl 2020; 01; 66-82 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/803 a novel enriched lasso based compression technique for energy efficient wireless sensor networks anitha k1, jaison b2, nalini m3, shiny irene d4 1) department of computer science and engineering, saveetha school of engineering, chennai, india email: anithak.rameshr@gmail.com 2) department of computer science and engineering, r.m.k engineering college, chennai, india email: bjn.cse@rmkec.ac.in 3) department of computer science and engineering, saveetha school of engineering, chennai, india email: nalinitptwin@gmail.com 4) department of computer science and engineering, saveetha school of engineering, chennai, india email: dshinyirene@gmail.com abstract: the capability of transmission in wireless sensor networks (wsn) is circumscribed due to the precincts in energy utilization, controlled resources of transmission devices and network components. information compression is taken into account as the best option, because the major part of energy consumed is for transmission of information. habitually lossy compression is adopted, since wsn abides some error in the reconstructed signals subjected to some acceptable tolerance. lasso based models have been ascertained their capability to effectually compress both multivariate and univariate data. traditional lasso considers ℓ1-norm regularization for learning in multi-dimensional data sets and assumes sparsity as model parameters. lasso prominence on sparsity and deal with the correlation between the data points. however, model sparsity may be constricting and not essentially the foremost applicable assumption in several problem domains. to eliminate this limitation, an enriched lasso (mlasso) is proposed for compression bearing in mind both sparsity and correlation. in specific the strategy can select data that are having strong features to reconstruct the data and are less correlated between each other. furthermore, an efficient alternating direction method of multipliers (admm) is adopted to resolve the ensuing sparse non-convex optimization problem. extensive experiments on diverse datasets provides the proof that mlasso outperforms other similar algorithms for signal compression. thus the proposed method ensures less energy consumption, decreases power loss and improves the operational life and reliability of network components. keywords: multidimensional data, compression, energy efficiency, senor networks, lossy compression 1. introduction in current scenario, there is an immense development in usage of mobile technologies and remote sensing devices. progresses in compact hardware and sensing devices aided to impart them into every object leading to internet of things (iot) [1]. large scale wireless sensor networks (wsns) are employed to acquire and collect the data from different envia novel enriched lasso based compression technique for energy efficient wireless sensor networks 67 copyright ©2020 assa adv. in systems science and appl. (2020) ronments and objects. they are communicated to a data acquisition center for monitoring and surveillance. the application of wsns embrace environmental monitoring [2], remote patient monitoring in healthcare [3], animal behavior classification [4], smart grid [5] and structural health monitoring for infrastructures [6]. the efficiency and the results of the applications depend on the amount of data and the acquisition speed of sensors. more data at high speed will help the application to perform efficiently. the same can be realized by higher sampling speed that ends up with large volume of raw data from sensors. thus the network needs more energy for transmitting and huge memory for storage. each battery supplying energy to the sensor nodes have limited stored energy and it is challenging to recharge or replace. as per kimura and latifi [7] each remote sensor consumes 80% of battery capacity for data transmission. the results presented by barr and asanovic [8] provides the evidence that the energy consumed by the network for transmitting a single bit of information is almost same as that required by the processing unit for performing thousands of computing operations. also more energy consumption lead to more heating of sensor nodes and thus reduce the life. thus it becomes a big challenge to manage equipment life, operate and maintain the data acquisition and storage systems in application scenarios. the energy requirements can be regulated by adopting procedures at various sections of wsn protocol stack [9-12] by energy efficient routing [13] and battery saving media access control [14]. a packet forwarding protocol is presented by l. zhang and y. zhang [15] and proved as energy efficient. the other approaches involved in improving the life of networks includes sample data reduction, battery replacement and data compression [16]. the perspective of this work given in this paper is to reduce transmission of data by introducing effective data compression. in general there are two methods of data processing that will result in reduction [17] data aggregation and data approximation. aggregation methods use statistical data such as minimum, maximum and average and effectively reduces the data size but fails to capture the important trends and changes in data required for arriving results. data approximation is a model based method and will not disturb the actual information, if the data feed contains large amount of redundancy. based on the method and purpose of application, the approximation method are categorized as time series analysis [18-20], data mining model [21], probabilistic model [22, 23] and data compression [24-26]. data compression proves to be the more favourable technique that provides good data quality, best of system performance and significantly reduces the energy consumption of wsns. embracing data compression algorithm in data storage and transmission devours some energy to compute the compression but it will be much less than the energy required for storage and transmission. in literature numerous algorithms and techniques have been offered and their capability to compress the time series data is proven and established. however many of them are derived and tested for compression of single variate sensor data such as temperature, relative humidity, etc. in-spite of vast applications of multivariate signals only few literature works are available addressing the issues in systems used to acquire, analyse and store multivariate signals. the proposed work adopts an algorithm based on lasso approximation, enriched lasso (mlasso) that can compress both multivariate and univariate sensor data. lasso has proved itself very much suitable for multidimensional regression. lasso assumes that there is no correlation in input data. but in wsns, the data is captured using different sensor units in a node. these sensors will at the same time acquire many types of data, like sound intensity, acceleration, temperature, humidity, light intensity, and video. these data used to have certain correlation. lasso happens to select only one feature among the correlated features and ends up in system underperformance [27]. this issue is addressed in the enriched lasso (mlasso) technique. in addition to discovering the correlation between the data, it also better 68 anitha k., jaison b., nalini m., shiny irene d. copyright ©2020 assa adv. in systems science and appl. (2020) discriminates similar data. this property of dlasso helps in achieving high reconstruction accuracy. the proposed algorithm in particular creates optimum predictions of multi variate data and thus achieves efficient energy utilization. to demonstrate the capability and suitability of the proposed algorithms in comparison with the competing algorithms available in literature, the system is evaluated and tested with benchmark data sets from different application domains. the experimental results of proposed system performed significantly better when tested with smooth multivariate data sets. this makes the system most suitable in applications such as behaviour monitoring. the work also includes development, optimizing and computing compression consuming less running time. in addition guidance for selection of parameters influencing the algorithm performance is also described. the organization of this paper is as follows: in section 2, a brief about the related work is given. the univariate and multivariate lasso compression algorithm and the methodology of implementation is proposed in section 3. the description of real world publicly available data sets and the comparison of several compression algorithms available in literature is done in section 4. finally, section 5 briefs few concluding remarks. 2. related work in sensors that are intricate in long term real time monitoring, efficient consumption of energy is a conspicuous feature for their satisfactory performance. several available compression algorithms are not directly malleable for sensor nodes. only explicitly formulated and derived compression algorithms are appropriate and implemented for sensor nodes applications [28]. multivariate data measured from one sensor node are correlated. each original time series is not linear, however deligiannakis [17] proposed an algorithm keeping base signal as an independent variable. then the other series data is approximated using regression model. the base signal values are directly extracted from the actual sensor raw data and replaced during changes occur. the algorithm is more suitable for data having larger multivariate correlation. on the other hand the algorithm doesn’t consider error constraints. race algorithm [24] is developed for data acquired from single sensor node. few of the other techniques developed includes distributed source coding (dsc) [29], compressed sensing (cs) [30], distributed source modelling (dsm) [31] and distributed transform coding (dtc) [32]. there are other compression techniques grouped as temporal compression techniques classified as: lossless and lossy compression algorithms. they are characterized based on different principles. data accuracy [8] is preserved in lossless compression. this accuracy is preserved by eliminating redundant data during compression and decompression process. on the other hand, lossy compression techniques are derived to achieve better compression ratio at the cost of sacrificing the accuracy. a number of classic lossless compression methods have been derived and analysed. few of the algorithms suitable and used in sensors is sensor lempel-ziv-welch (s-lzw) algorithm and a distinct variant of previous technique lsz [33] made suitable for sensors. it is specially developed to storage and energy constraints in sensor networks. lossless entropy compression (lec) algorithm [33] is a lossless compression approach proved very efficient is based on traditional information encoding. the lossless compression techniques are not well suited for sensor applications. in some cases they require large memory and in other cases their assumptions about entries of static dictionary are not appropriate. lossy compression relaxes the accuracy and accepts deviations from the original to reach the flexibility to achieve less energy consumption and reconstruction accuracy for higher compression ratio, in order this will increase the lifetime of wireless sensors. schoell hammer has presented a less complex lightweight temporal compression (ltc) technique. a a novel enriched lasso based compression technique for energy efficient wireless sensor networks 69 copyright ©2020 assa adv. in systems science and appl. (2020) small value of error is added with each reading, controlled by knob [34,35]. ltc technique is simple and less complex method and can be used for temporal data compression [36]. the performance of the algorithm decays for the fluctuating sensor readings. piecewise linear approximation based scheme with less number of line segments (plamlis) algorithm exploits and estimates minimum variety of segments to approximate the given statistic. the error limit [37] is fixed during the compression process and the difference between any approximation value and its actual value is maintained less than the fixed limits. the temporal lossy compression algorithms explained above are well suitable for slowly and gradually varying signals such as temperature and not for signals such as vibrations since they use piecewise linear representation of time series. also ltc and plamlis are compression methods that are not effective for compression of multi-dimensional or multivariate time series. the work presented in this paper presents adoption of enriched lasso approximation algorithm for data compression. the algorithm is best suitable to compress both univariate and multivariate signals. 3. the compression algorithm let the one dimensional signal be 1 2( , ,....... )ny y y y observed or sensed at times 1 2 1 2( , ,....... ) ...n nt t t t where t t t � � � . that is the signal iy is measured at a time it , by the sensor. we can say that the reading of accelerometer y used to measure the acceleration of an equipment during a seismic test in the time duration 1 0 30nt s and t s . it can be seen that the values measured by y will be varying during every time instant in a given time limit t, every values of y captured by the sensor in the time interval t should be acquired for monitoring and stored for further studies, analysis, modelling and development. furthermore, it can be for a sequence of instances the change in values of y can be of less significance. in this case, if the data read in during the time between p qt and t is such that 1 ......p p qy y y� for some p q� . in this scenario, we could eliminate the data observed during the time p and q and store the reading , 1(... ....)p qy y y � in the data base corresponding to instances , 1(... ....)p qt t t � . thus data sensed in the time frame p qt and t is not varying vastly, we can be able to store the space required to store p q� recorded sensor values and also the time stamps. thus reducing the storage requirements also reduces the energy required for transmitting the data. in addition a very significant economic benefit in developing the sensor network can also be achieved. in practical installations, the values read by the sensors may be with slight variations and will not be constant in a particular time period ( , )p qt t . in such cases, the size cannot be reduced just by eliminating the data, to compress the data that are varying continuously, an efficient algorithm derived from lasso regularization [38] is presented. previously lasso approximation has been applied to order the features in a meaningful sequence in applications such as comparative prostate cancer analysis [39], genomic hybridization [40] and timevarying networks [41]. from the literature it is evident that the basic lasso-based algorithms assume restricted freedom among the variables, and their idea is to perform regression separately for each response vector rather than performing it jointly for all the response vectors. as a result they perform only data approximation and representation. compression of signal y is attained by eradicating the small differences in the signal. the signal x a smoother approximation of signal y is computed by minimizing the differences between data at adjacent time intervals. many algorithms have been proposed and much improvement has been established in the basic lasso, but the achieved response may not be the optimum. the lasso based algorithms 70 anitha k., jaison b., nalini m., shiny irene d. copyright ©2020 assa adv. in systems science and appl. (2020) considers the correlations between the data points and computes an approximate signal x of signal y . the lasso algorithm performs well if the sensor data has less variance and its performance decays with increased variance between the data points in sensor data set. to improve and achieve a generalized results, it is necessary to consider the correlation between the data points in the data set and correlation between data points and response. in view of above mentioned issues in existing lasso-type approximation methods, an enhanced lasso (referred to as mlasso) is proposed for data compression. this algorithm not only discovers the correlations between the data points and the response, in addition also discriminates similar data points. this characteristics of the algorithm differentiates the proposed algorithm from the rest of lasso based algorithms. the proposed mlasso method uses a novel graph regularizer on the data points which simultaneously considers the ‘response-data point correlation’ and the ‘data point-data point correlation’ in the data. in this work alternating direction method of multipliers (admm) is employed to optimize the mlasso algorithm method and does the data compression effectively. 3.1. lossy compression for sensor signal to achieve the requirement of compressing an original signal, it shall be sparse in some possibly transformed domain. compressive sampling theory or compressed sensing developed for data transmission has become an active investigation area. the basis of this theory is that the true data can be represented using reduced samples and the same shall be recoverable from this samples with an allowed error bound. on this basis the true sensor signal e can be recuperated from approximated fewer samples of signal derived by the solution of the problem: 1 min w e e (1) subject to yie where the original signal e is a 1pu matrix to be compressed and compression theorem w is a p pu known matrix. the n pu sensing matrix φ, contains in rows, n bases for the measurements needed to be sampled from the sensed signal. the general choice to initialize φ is n randomly generated data points. finally 1nu vector y will contain the sampled data points from the original signal e such that n p� . the data acquired by the sensor in the field may contain noise. considering the measurements in practical cases the problem given in (1) is generally rewritten as: 1 min w e e (2) 2 2 subject to y ie h� d where h represents the noise in the measured signal. as mentioned in (2) the 1l norm of e under the compression transform w is minimized, such that reconstructed signal *e from n measurements of y (compressed signal) of true signal shall be consistent. the optimization problem in eq. (2) with constraints shall be rewritten in the following form: 2 2 1 min y w e ie o e� � (3) a novel enriched lasso based compression technique for energy efficient wireless sensor networks 71 copyright ©2020 assa adv. in systems science and appl. (2020) the values of h in eq. (2) and o in eq. (3) can be determined empirically. the eq. (3) can be found that it is similar to the lasso eq. (4) 2 2 1, min 1y x d e d e o e� � � (4) a main difference between eq. (3) and eq. (4) is the compression transform w available in the second term of eq. (4) called penalty term. 3.2. enriched lasso for data compression since the introduction of lasso 1l regularization has been adopted for learning in highdimensional databases. extending lasso eq. (4), the learning compressible models can be formulated as follows: 1, min ( ,1 )l y x w d e d e o e� � (5) the proposed data compression method is motivated by the objective to support the selected samples to correlate more with the reconstructed signal due to low redundancy between them. therefore, equations eq.4 and eq.6, are combined and propose the mlasso method for data compression and formulated as: 2 1 22 1 min ty s e ie o e o e e� � � (6) where 1o , 2o ≥ 0 are tuning parameters. note that t se e is a non-convex constrain. our mlasso disparities with previous lasso-type data compression methods, which use convex methods and which may be suboptimal in terms of the accuracy. here, the proposed mlasso method enforces stricter non-convex constrictions, i.e., ‘data point-response correlations’ and ‘data point-data point correlations’, in locating the optimal regression e . once the solution e of eq.7 is obtained, the selected samples can be easily recuperated. 3.2.1 learning compressible models let x is a n pu matrix with constant p , y x e h � is the sensed data with noise. the lasso refers to: 2 2 min y xe� such that 1 le d (7) and in lagrangian it becomes 2 2 1min 2 n n y x n e o e� � (8) the proposed mlasso algorithm solve the non-convex problem in eq. (7) by using alternating direction method of multipliers (admm). the concept of admm approach is to bifurcate a very complicated problem into a set of simpler and solvable problems. the best features of augmented lagrangian methods and dual decomposition are combined to solve the constrained optimization problem [42]. by introducing an auxiliary variable j into the objective function eq.7, the problem solved by admm takes the following form: 2 1 212 1min ( ) ( ) 2 t t tf g y x s e e j e o j o e e� � � � (9) 72 anitha k., jaison b., nalini m., shiny irene d. copyright ©2020 assa adv. in systems science and appl. (2020) 0subject to e j� which is found as same as the problem in eq.(7) and j in the eq. (8) shall be considered as a proxy for e . the augmented lagrangian associated with the constrained problem in eq. (8) is given by 22 2 2 1 1 2 1( , , ) ( ) ( , ) 2 2 t t tl z y x s z ue j e o e e o j e j e j � � � � � � � (10) here u is a positive penalty factor and z is a lagrange multiplier corresponding to the equality constraint e j . the objective function in eq. (7) is simplified by decoupling the function by introducing an additional variable j and a constraint 0e j� . the original eq. (6) is solved by iteratively minimizing ( , , )l ze j overe andj , and then z is updated. the learning rule adopted is as given below: 1 arg min ( , , ) d i k k r l z e e e j� � (11) 22 2 2 1 1 2 1 ( ) ( , ) 2 2 t t ty x s z ue o e e o j e j e j e e w w ª º � � � � � � �« »w w ¬ ¼ =0 2 ( ) 0t k kxy xx s ze o e u e j� � � � � 1 1 2 ( ) 0 2 ( 2 ) t k k k k t k t k k xy xx s z xy z xx s xx s i xy z e o e u e j uj e o e ue e o u uj� � � � � � � � � � � � � � � 1 1arg min ( , , ) d k k k r l z e j e j� � � (12) by applying 1k ie � and the multipliers , 1,2......k iz i d are fixed in the lagrangian, the minimization problem of 1, 1,2......k i i dj � is: � � � �21 1 1 1 1 min [ . ] 2i d d d k k i i i i i i i i i z j uo j j e j� � � �¦ ¦ ¦ (13) derivative of equation with respect to ij and equating it to zero, we get: 1 1 ( ) 0 ( ) i i k k i i i i i i k k i i i i z z o j u j e j o j u j e j � � w � � � w w � � w � � � � 1 1 1 1 1 1 1 1 0 [ , ] k k k k i i i i i i i k k k k i i i i i i k k i i i z if z z if z z ue o ue o u j ue o ue o u ue o o � � � � � � ­ � � � !° ° ° � � � � �® ° ° � � � ° ¯ (14) 1 1 1( )k k k k i i i iz z u e j� � � � � (15) a novel enriched lasso based compression technique for energy efficient wireless sensor networks 73 copyright ©2020 assa adv. in systems science and appl. (2020) the learning algorithm is considered converged and stops if the primal and dual residuals meets the predefined stopping criterion mentioned as absolute tolerance and relative tolerance. the penalty parameter u affects the primal and dual residuals, and hence affects the termination of the algorithm. a large ρ tends to produce small primal residuals, but increases the dual residuals. a fixed 0u is commonly used. but there are some schemes for varying the penalty parameter which achieve better convergence. 4. experimental results in this section, we brief and report the results to illustrate the performance of lasso approximation algorithm on both multivariate and univariate datasets. we implemented both algorithms in java with eclipse as front end. all experiments were performed on a hp with an intel processor with access to 5gb of ram. 4.1. experimental data the compression capability of mlasso algorithm is verified with datasets provided by sensor scope in collaboration with hes-so, clim-arbes project and microphone provided by cmu. sensor scope: the real time data is collected using a small sensor network [26]. the data collection consists of 1472 recordings of temperature and humidity sampled at 2 seconds. it is noticed that the recorded values of humidity is higher than the recorded temperature values. the recorded values are smooth i.e. there will not be any sudden changes. the data is represented as fn_t (temperature) and fn_h (humidity). microphone (cmu): the dataset contains 2887 data sampled at an interval of 4-9 seconds done at cmu room nsh 4622. the univariate data is discontinuous or non-smooth due to presence of lot of noise and surges. the dataset is denoted as sc_m. mobile health (mhealth): among the several multivariate data sets available for public, the mobile health (mhealth) three dimensional data set from uci machine learning repository [25] is the first of its kind. the dataset comprises of 483840 data with a sampling rate of 0.02 second. the collection contains body motion and vibrant signs from ten volunteers while performing twelve physical activities using shimmer wearable sensors. the multiple sensors used measures the motion experienced by the body parts namely acceleration, the rate of turn and the magnetic field orientation. from the mhealth data set, two-lead ecg (mh_ecg), three-axis acceleration of right wrist (mh_ar), three-axis acceleration of left ankle (mh_al), and three-axis magnet data (mh_mg) are selected for experimentation. crawdad: the second multivariate data set contains the three axis acceleration readings (cj_a) of vehicles. the data set is collected by the motion of different vehicles using the mobile phone of its drivers. the collection contains 16060 data sampled at 0.0625 second intervals. in addition to the above dataset the third multivariate data set is provided by samuel madden et al. (http://db.csail.mit.edu/labdata/labdata.html). the data set contains 2.30 million data and were collected by 54 mica2dot nodes at the same time. four attribute data humidity, temperature, voltage and light intensity were collected from each node. each real 74 anitha k., jaison b., nalini m., shiny irene d. copyright ©2020 assa adv. in systems science and appl. (2020) number such as sampling data, wavelet coefficient, regression coefficient, which is stored in mica2nodes, needs 2 bytes for storage. 2 bytes are enough for storing an integer such as start number start and the count number length of data.. to understand the effectiveness of the proposed algorithm, the results achieved are compared with the below mentioned well-established algorithms available in literature: � lightweight temporal compression (ltc), is a low-complexity piecewise linear approximations lossy compression algorithm. it fits the consecutive measurements as a straight line within the desired error margin. the greater compression ratio is obtained, while the larger error bound is given. � piecewise linear approximation with minimum number of line segments (plamlis), takes an n-length sequence of measurements and finds the minimum number of line segments required to represent the sequence within an error bound. � lasso : compression of signal y is achieved by removing the small differences between the data points in the signal. the signal x a smoother approximation of signal y is computed by minimizing the differences between data at adjacent time intervals 4.2. performance evaluation 4.2.1. correlation length for a discrete time series � �x n with n = 1, 2. . . , n, xp is the mean and 2 xv is the variance the correlation length of � �x n can be defined as the smallest value *n subjected to the autocorrelation function of signal is lesser than a threshold w (predefined). the autocorrelation function is given by: 2 [( ( ) )( ( ) )( ) x x x x e x m x m nn p pu v � � � (16) and * 0 arg min{ ( ) }x n n nu g ! � (17) 4.2.1. compression ratio (k ) if the time series � �x n with n = 1, 2. . . , n, needs ( )bn x bits to store and its compressed version � �x̂ n needs ˆ( )bn x bits to store then k is estimated using: ˆ( ) ( ) b b n x n x k (18) 4.2.2. total energy consumption a novel enriched lasso based compression technique for energy efficient wireless sensor networks 75 copyright ©2020 assa adv. in systems science and appl. (2020) the total energy consumed is the sum of energy needed to compress the signal and to transmit the compressed signal expressed in joule. the number of operations (addition, subtractions, divisions, multiplications and comparisons) required for compression is estimated and mapped into number of clock cycles and then to energy consumption. energy required for transmission and receiving the data depends on the energy consumed by equipment to transmit a bit of data and number of bits. 4.2.3. total error if the compressed signal � �x̂ n is reconstructed to its original form and is given by � �y n . the error is the difference between the reconstructed and original signal. the total error is calculated using: � � 2 0 ˆ( ) ( ) n i error x i x i �¦ (19) 4.4. experimental results to evaluate the proposed algorithm, the synthetic signals of correlation length *n with lengths {1, 10, 20, 50 . . . 500} time slots are used to test the system. w = 0.05 have been adopted for the results presented in this paper. additionally, a gaussian noise with standard deviation noisev = 0.04 has been included in the signal. acceptable tolerance in error between actual signal and reconstructed signal has been set to noiseh [v where 0[ t . the same signal ( )x n has been used for all the compression methods for each simulation run and value of *n . the plot in fig. 1 between provides the compression ratio achieved for different correlation lengths *n for four compression methods. form these results, it can be found that compression performance of all the four methods are poor for small values of *n and improves with the increased correlation length. this ensures that the correlation length *n is a key factor that decides the performances of the algorithm. the energy required for compression is presented in fig. 2. we can achieve better compression values for higher values of *n . the same is not true in the case of energy consumption among all scheme. fig. 1. illustration of changes in compression ratio for various correlation length for univariate data. 76 anitha k., jaison b., nalini m., shiny irene d. copyright ©2020 assa adv. in systems science and appl. (2020) fig. 2. compression ratio vs energy consumption for compression methods for fixed 4 noiseh v we shall look at the dependence of energy required for compression on correlation length *n . the energy consumption depends on the operational model of a particular algorithm adopted for compression. in case of ltc, the input signal ( )x n is incrementally, initially starts the first sample and including one sample at a time. thus correlation length *n has weak influence on the consumed energy. plamlis divides and reiterates based on the correlation length and thus the method performs lesser iterations if the correlation length increases. thus the number of operations required decreases, consequently decreasing the energy consumption. for lasso and mlasso, the energy requirements increases during the addition of new sample to the model. the error tolerances have to be verified after updating the model. thus energy consumption by these models rises with increased correlation length of the input time series. in general the reconstruction error increases with the increased compressed ratio. a method can be considered as superior if it compresses the dataset and if the reconstructed dataset from the compressed data is most similar(less reconstruction error) to the original dataset. in order to explore the capabilities of the methods in compressing the signals, error of the reconstructed signals from signals of different compression ratios are presented in fig.3. it can be found that the reconstructed signal from the compressed data using mlasso is more similar to the original signal(less error). also it can be observed that the proposed method for compression provides generalized performance for data of different types. the error remains almost same for a range of compression ratio. a novel enriched lasso based compression technique for energy efficient wireless sensor networks 77 copyright ©2020 assa adv. in systems science and appl. (2020) fig. 3. the change in error at different compression ratio for different datasets this performance of the proposed lasso based method mlasso allows to adopt the method to compress signals with more variance between the signal samples. also by adopting the method it is possible to compress the signal to very smaller sizes, since the original signal can be reconstructed using small error. the plots in fig.4 shows the total energy gain for different correlation lengths. the total energy gain, defined as the ratio between the energy spent for transmission in the case with no compression and the total energy spent for compression and transmission using the selected compression techniques. the proposed method mlasso, produces the highest energy gain. in wsn scenario the total energy is highly contributed by the energy required for computations required for compression. thus, lightweight methods, ltc and plamlis, also performs better. in the case of uwn, the energy required for computations of compression is negligible compared to transmission energy requirements. in this scenario, mlasso, whose performance is the best will support more for energy savings. in this case energy savings by methods such as ltc, is limited. it can be noted that till now the compression ratio has been considered as a parameter to evaluate the system performance. and the compression ratio depends on error tolerance noiseh [v , so the compression algorithm takes ξ as an input from the user. the relative error ξ can be related with the compression ratio. here we relate mathematically ξ as a function of compression ratio. in the same way cn can also be expressed with respect to compression ratio. these relationships have been derived. 78 anitha k., jaison b., nalini m., shiny irene d. copyright ©2020 assa adv. in systems science and appl. (2020) fig. 4. energy gain vs correlation length through synthetic signals, extensive simulations and numerical fitting. the relative error tolerance can be related to compression ration through the following formulae: 2 * 4 3 2 1 2 3 1 ( , ) 1 2 3 4 5 1 p p p lasso q n p p p p p mlasso q k k k [ k k k k k k ­ � � ° �° ® � � � �° ° �¯ (20) the fitting parameters p1, p2, p3, p4, p5 and q1 depends on the correlation length. the fitting formulae are validated against the carwad dataset. the temperature and humidity values in the dataset are utilized for validation. the experimental relationships of eq. (14) for mlasso and lasso are provided in fig. 5(a) and 5(b). the plot with marks provide the performance obtained to compress temperature and humidity dataset using lasso and mlasso. from these plots, it can be noted that the numerical fittings obtained with the synthetic signals also suitable for representing the signals from datasets. it can be also observed for decreasing values of *n , the shape of the curves related to ξ and compression ratio remain similar but shifts towards the right. lastly, it is noticed that the influence of *n on the performance is mainly noticeable at lesser values of *n , and for correlation lengths larger than 100, the plots be likely to converge. it was also found that the computational cost cn depends linearly on compression ratio and the correlation length ( *n ) and can be expressed as in eq.(15): * *( , ) ( )cn n cr cr nd j e � � (21) from the experiments it is ensured that cn more depends on compression ratio and less on correlation length. hence eq.(15) takes a simple form as in eq.(16) ( ) ( )cn cr crd e � (22) fig. 6 provides the results of eq. (16). the results ensure results obtained for experimental signals against the results obtained on real world signals. a novel enriched lasso based compression technique for energy efficient wireless sensor networks 79 copyright ©2020 assa adv. in systems science and appl. (2020) fig. 5. relationship between compression ratio and fitting functions *( , )n[ k for mlasso and lasso fig. 6. relationship between compression ratio and no. of cycles per bit 5. conclusion in this paper we have systematically compared the lasso and a mlasso compression algorithms for saving the energy in sensor networking. the intension was to explore the better method for energy savings are possible depending on the statistics of sensor algorithm, performance of compression and the type of hardware used to construct the network. the results in the paper reveal that there is considerable energy savings when signal compression is done by using lasso based algorithms, since they need lesser computational cost. but it can be noticed that in the case of underwater transmission of data more energy shall be required and in these case algorithms that can produce higher compression performance will be more suitable. at the end formulas for parameters that are suitable for lasso based compression methods that relates computational requirements, reconstruction error and compression ratio are 80 anitha k., jaison b., nalini m., shiny irene d. copyright ©2020 assa adv. in systems science and appl. 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(2009). the composite absolute penalties family for grouped and hierarchical variable selection, the annals of statistics, pp. 3468–3497. 40. amr ahmed and eric p xing (2009), recovering time-varying networks of dependencies in social and biological studies, in proc. of the national academy of sciences, 106(29):11878–11883. 41. boyd, s.,parikh, n., chu, e.,peleato, b., eckstein, j (2011). distributed optimization and statistical learning via the alternating direction method of multipliers, foundations and trends in machine learning 3(1), 1–122. microsoft word 859-article text-2988-1-6-20200617.doc adv syst sci appl 2020; 02; 44-55 published online at https://ijassa.ipu.ru. influence assessment of intelligent unmanned ground vehicles on the transport network state andranik akopov1,2, nerses khachatryan1,2, fedor belousov1,2* 1) national research university higher school of economics, moscow, russia e-mail: aakopov@hse.ru, nkhachatryan@hse.ru, fbelousov@hse.ru 2) central economics and mathematics institute ras, moscow, russia e-mail: nerses@cemi.rssi.ru abstract: this article is devoted to econometric analysis of the results of experiments conducted with two agent-based models, which describe the movement of ground vehicles. there are two types of road users in these models: manned ground vehicles (mgv) and unmanned ground vehicles (ugv). in the first model, the main difference between ugv and mgv is an ability to exchange massages between ugv for transmitting information about extreme situations, which allows them to adjust speed and direction of movement. in the second model, in addition to the above differences, ugv have an additional advantage, namely, the ability to intelligently assess density of traffic flow for efficient maneuvering. in these models, at a given roundabout, traffic characteristics such as output stream traffic and the number of traffic accidents are analyzed. the main task of the econometric analysis is to study dependence of these traffic characteristics on the model parameters such as average vehicle speed, input flow rate, message exchange rate between ugv, and the impact of the effect obtained from the implementation into ugv ability of intelligent estimation of traffic flow density. keywords: unmanned ground vehicles, manned ground vehicles, agent-based model, experimental results, econometric analysis 1. introduction currently, the task of developing new intelligent control systems for an ensemble of ground-based ugv is being actualized in order to ensure high-speed and safe traffic, maximize the capacity of the transport system and minimize the number of factors (for example, emerging traffic accidents) that threaten other road users. works [1-7] are devoted to development of ugv control systems. in particular, the paper [1] proposes an algorithm based on predictive tracking control, designed to reduce time for decision-making. a very promising direction in the design of ugv control systems is an approach based on the use of machine learning methods, in particular, reinforced learning [2]. such methods make it possible to ensure an effective response of ugv to the occurrence of certain situations on the road (for example, the sudden appearance of a pedestrian). at the same time, even the use of multilayer neural networks with a large number of neurons and synoptic connections (deep learning) [8, 9], the use of ultraprecise neural networks [10], etc. does not allow for zero learning error in pattern recognition and classification of moving objects with complex characteristics. therefore, the task of developing ugv control systems based on simple rules that take into account the most common road situations for making effective decisions about maneuvering, changing lanes [4, 7], adjusting the speed and direction of traffic, etc. is urgent. such a task can be solved using agent-based simulation methods [6, 7, 11, 12], fuzzy clustering [7, 13, 14, 15] and genetic optimization algorithms [16]. so, earlier we developed and implemented two agent-based models of mgv and ugv motion in anylogic simulation system [6, 7]. * corresponding author: fbelousov@hse.ru influence of unmanned vehicles on the state of the transport network 45 copyright ©2020 assa. adv. in systems science and appl. (2020) the first model, using the previously proposed phenomenological approach [11], is designed to assess the influence of various parameters (such as average initial speeds, input flow intensity, data exchange rate between ugv, etc.) on the behavior and condition of unmanned and manned ground vehicles in a dense flow [6]. the spatial dynamics of mgv and ugv is defined by a system of difference equations with a variable structure. the model takes into account the effects of "turbulence" and "traffic congestion" caused mainly by high vehicle density and occurrence of road accidents. at the same time, effective interaction is carried out between the ugv, in particular, periodic exchange of messages about the traffic situation (for example, about the places where accidents occur), in order to timely adjust speed and direction of movement. an important feature of this model is the proposed concept of the agent's personal space. the behavior of agents (mgv and ugv) in flow depends on density of the surrounding space. as the flow density increases, the radius of the agent's personal space is compressed (i.e., the transport flow is compacted). however, after reaching a certain threshold value of density, the radius of the agent's personal space increases significantly in manned ground vehicles, due to the occurrence of panic, and partially increases in ugv, due to a predetermined desire to avoid collision in dense traffic flow. fig. 1.1. configuration of road space for mgv and ugv the second model is designed for effective management of ugv, in particular, by maneuvering when changing lanes, the algorithm of which is based on the proposed fuzzy clustering algorithm [7]. the movement of the mgv and ugv ensemble in a certain twodimensional space is considered. the space consists of combination of circular motion, two entrances and two exits from the circle (fig. 1.1). at the same time, ugv makes individual decisions about trajectory adjustment based on simple rules. if there is less dense traffic in one of the adjacent lanes, this ugv is tuning to the corresponding lane. in this case, the flow density is estimated using fuzzy clustering methods for each of the available alternatives (lanes) across the entire ensemble of vehicles (both ugv and manned ground vehicles that do not have all the information about the density of the surrounding space). as a result, adaptive ugv management is supported, which minimizes the risks of accidents (accidents involving ugv) and maximizes traffic (total output stream) in conditions of heavy traffic flow. software implementation in anylogic of the developed simulation model was performed and numerical experiments were carried out. modes that ensure safe and highspeed movement of vehicles in dense traffic flow are found (fig. 1.2). the developed system (fig. 1.2) allows us to evaluate influence of multiple control parameters (for example, average speeds of mgv and ugv, intensities of input flows of mgv and ugv, etc.) on the total number of accidents and traffic of the output stream. 46 a. akopov, n. khachatryan, f. belousov copyright ©2020 assa adv. in systems science and appl. (2020) fig. 1.2. software implementation of the model in anylogic simulation system this work is devoted to the analysis of the results of experiments conducted with these models. the main task of this analysis is to study dependence of the output stream traffic and the number of accidents on model parameters such as average vehicle speed, input stream intensity, message exchange frequency between ugv, and the impact of the effect obtained from the implementation into ugv ability of intelligent estimation of traffic flow density. to do this, a pool of identical experiments was conducted within each model. a set of experiments is formed as a result of multiple runs of the model, in each of which the specified model parameters take values in a given range with a given step. finally, a database consisting of the results of more than one hundred thousand experiments is formed for each model. based on these results, an econometric analysis was carried out: the dependence of the output stream traffic and the number accidents on the parameters in both the first and second models was studied. also the effect obtained as a result of the introduction of ugv with the ability to intelligently assess the density of road traffic, consisting in increasing the output stream traffic and reducing the number of accidents, was estimated. 2. source data, primary data analysis this paper uses data obtained from a pool of experiments with each of the two agent-based models [6, 7] that describe the movement of vehicles, both manned and unmanned, in a given section of circular motion. in these experiments, such traffic characteristics as the output stream traffic and the number of traffic accidents over a certain period of time, which act as dependent variables, were calculated. the following parameters of agent-based models are explanatory variables: • intensityofunсrewedvehicles – intensity of entry of ugv to the specified section of circular motion (in units of model time); • intensityofusualagents – intensity of entry of mgv to the specified section of circular motion (in units of model time); • speedofuncrewedvehicles – average ugv speed (km/h); • speedofusualagents – average mgv speed (km/h); • intensityofconnections – intensity of messaging between ugv (in units of model time). as noted in the introduction, lot of experiments are formed as a result of multiple runs of the model implemented in anylogic using the special option “sensitivity analysis”. in this case, influence of unmanned vehicles on the state of the transport network 47 copyright ©2020 assa. adv. in systems science and appl. (2020) the specified model parameters take values in the specified range with the specified step (table 2.1). table 2.1. variation of model parameters № name of parameter range of variation variation step number of variations 1 intensityofuncrewedvehicles 0.005 – 0.05 0.005 10 2 intensityofusualagents 0.005 – 0.05 0.005 10 3 speedofuncrewedvehicles 40 – 140 10 11 4 speedofusualagents 40 – 140 10 11 5 intensityofconnections 0 – 1 0.1 11 it is easy to see that the total number of experiments in each pool is 133100. the table 2.2 shows the main descriptive statistics of dependent variables. table 2.2. descriptive statistics average median standart deviation minimum maximum output stream traffic (model 1) 56.5 56 25.4 3 149 output stream traffic (model 2) 67.8 65 35.3 3 207 number of accidents (model 1) 16 10 18.6 0 148 number of accidents (model 2) 5.7 1 12.7 0 145 according to table 2.2, introduction of intelligent ugv leads to an increase in output traffic by 20% and a decrease in the average number of accidents by 2.8 times. it is also worth noting that the median number of accidents is reduced from 10 to 1, i.e. in model with the ability to intelligently estimate the traffic density, approximately half of the experiments do not have an accident. in fig. 2.1 and fig. 2.2 histograms of the output stream traffic distribution are presented, and in fig. 2.3 and fig. 2.4 – histograms of the distribution of the number of accidents in model 1 and model 2, respectively. fig. 2.1. histogram of the output stream traffic distribution in model 1 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 10 0 10 5 11 0 11 5 12 0 12 5 13 0 13 5 14 0 14 5 15 0 n um be r o f e xp er im en ts output stream traffic 48 a. akopov, n. khachatryan, f. belousov copyright ©2020 assa adv. in systems science and appl. (2020) fig. 2.2. histogram of the output stream traffic distribution in model 2 fig. 2.3. histogram of the number of accident distribution in model 1 fig. 2.4. histogram of the number of accident distribution in model 2 from fig. 2.3 and fig. 2.4 it follows that the number of experiments without an accident in model 2 is three times less than in model 1. we finish the initial data analysis with the graph 0 10000 20000 30000 40000 50000 60000 70000 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 10 2 10 8 11 4 12 0 12 6 13 2 13 8 14 4 n um be r o f e xp er im en ts number of accidents 0 5000 10000 15000 20000 25000 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 10 2 10 8 11 4 12 0 12 6 13 2 13 8 14 4 n um be r o f e xp er im en ts number of accidents 0 200 400 600 800 1000 1200 1400 1600 0 7 14 21 28 35 42 49 56 63 70 77 84 91 98 10 5 11 2 11 9 12 6 13 3 14 0 14 7 15 4 16 1 16 8 17 5 18 2 18 9 19 6 20 3 21 0 n um be r o f e xp er im en ts output stream traffic influence of unmanned vehicles on the state of the transport network 49 copyright ©2020 assa. adv. in systems science and appl. (2020) below (fig. 2.5). it shows how the number of experiments in which the number of accidents is not less than the specified number (between the minimum and maximum number) decreases in both the first and second models. fig. 2.5. number of experiments with number of accidents not less than a given value 3. econometric model specification selection let's proceed to the study of the influence of the model parameters specified in table 2.1 on the output stream traffic and the number of accidents, i.e., to construct and evaluate the corresponding regression equations. at the initial stage of this study, we will try to determine the model specification. since the set of explanatory variables is defined in advance in the framework of building the above agent-based models, we are trying to determine the type of assumed dependency. we restrict ourselves to two types of functions that are most often encountered in econometric studies: linear and power-law. in this regard, we present matrices of pairwise correlations of all variables (both explanatory and dependent), as well as their logarithms for both models (tables 3.1, 3.2, 3.3, 3.4). the following notation is used in these tables: • x1 – intensityofuncrewedvehicles; • x2 – intensityofusualagents; • x3 – speedofuncrewedvehicles; • x4 – speedofusualagents; • x5 – intensityofconnections; • y1 – output stream traffic; • y2 – number of accidents. thus – x1, x2, x3, x4, x5 – explaining variables, а y1, y2 – dependent. table 3.1. correlation matrix of variables in model 1 x1 x2 x3 x4 x5 y1 y2 x1 1.00 -0.20 -0.008 0.03 -0.0002 0.07 0.47 x2 -0.20 1.00 0.002 0.03 0.0005 0.15 0.46 x3 -0.008 0.002 1.00 -0.02 -0.0006 0.40 0.11 x4 0.03 0.03 -0.02 1.00 0.002 0.74 -0.21 x5 -0.0002 0.0005 -0.0006 0.002 1.00 0.0003 -0.003 y1 0.07 0.15 0.40 0.74 0.0003 1.00 -0.16 y2 0.47 0.46 0.11 -0.21 -0.003 -0.16 1.00 0 20000 40000 60000 80000 100000 120000 1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 10 3 10 9 11 5 12 1 12 7 13 3 13 9 14 5 n um be r o f e xp er im en ts number of accidents model 1 model 2 50 a. akopov, n. khachatryan, f. belousov copyright ©2020 assa adv. in systems science and appl. (2020) table 3.2. correlation matrix of logarithms of variables in model 1 log(x1) log(x2) log(x3) log(x4) log(x5) log(y1) log(y2) log(x1) 1.00 -0.22 -0.008 0.02 -0.0007 0.09 0.47 log(x2) -0.22 1.00 0.001 0.03 0.0005 0.08 0.44 log(x3) -0.008 0.001 1.00 -0.015 -0.0003 0.37 0.10 log(x4) 0.02 0.03 -0.015 1.00 0.002 0.75 -0.17 log(x5) -0.0007 0.0005 -0.0003 0.002 1.00 -0.0003 -0.001 log(y1) 0.09 0.08 0.37 0.75 -0.0003 1.00 -0.10 log(y2) 0.47 0.44 0.10 -0.17 -0.001 -0.10 1.00 table 3.3. correlation matrix of variables in model 2 x1 x2 x3 x4 x5 y1 y2 x1 1.00 -0.26 -0.02 0.12 -0.001 0.23 0.19 x2 -0.26 1.00 0.04 -0.008 -0.002 0.13 0.39 x3 -0.02 0.04 1.00 -0.05 -0.001 0.37 0.12 x4 0.12 -0.008 -0.05 1.00 -0.0009 0.74 -0.41 x5 -0.001 -0.002 -0.001 -0.0009 1.00 -0.003 0.001 y1 0.23 0.13 0.37 0.74 -0.003 1.00 -0.37 y2 0.19 0.39 0.12 -0.41 0.001 -0.37 1.00 table 3.4. correlation matrix of logarithms of variables in model 2 log(x1) log(x2) log(x3) log(x4) log(x5) log(y1) log(y2) log(x1) 1.00 -0.23 -0.02 0.10 -0.002 0.20 0.14 log(x2) -0.23 1.00 0.04 -0.03 -0.0006 0.07 0.45 log(x3) -0.02 0.04 1.00 -0.05 -0.001 0.35 0.10 log(x4) 0.10 -0.03 -0.05 1.00 -0.0006 0.77 -0.47 log(x5) -0.002 -0.0006 -0.001 -0.0006 1.00 -0.001 0.001 log(y1) 0.20 0.07 0.35 0.77 -0.001 1.00 -0.40 log(y2) 0.14 0.45 0.10 -0.47 0.001 -0.40 1.00 based on the data from these tables, we can draw the following conclusions: 1. the pairwise correlation both between explanatory variables and between their logarithms is low in both models, which, when evaluating the corresponding regression equations, cannot become a source of multicollinearity. this is important given that we are interested in the contribution of each explanatory variable to changes in dependent variables. 2. comparing pairwise correlations between dependent and explanatory variables does not allow us to determine which of the functions (linear or power-law) most adequately describes the dependence of output stream traffic and the number of accidents on the model parameters. therefore, in the future we will conduct econometric analysis using both types of functions. in connection with the use of power functions, which will be logarithmized before estimation, we discard the results of experiments in which a variable assumes a zero value. thus, to construct the regression equations for the output stream traffic, the data of 102448 observations were used, and for the number of accidents – the data of 65134 observations. 4. econometric analysis of traffic characteristics in the framework of linear regression equations we proceed to evaluate the dependences of the output stream traffic and the number of accidents in both models on the indicated parameters of the model in the framework of linear regression equations. so, we will build and study linear regression equations with the following dependent variables: • output stream traffic in model 1; • output stream traffic in model 2; • number of accidents in model 1; • number of accidents in model 2. the coefficients were estimated in the framework of the classical multiple regression model using the least squares method. since the white test confidently rejects the null hypothesis of influence of unmanned vehicles on the state of the transport network 51 copyright ©2020 assa. adv. in systems science and appl. (2020) homoskedasticity of regression residues, standard errors are estimated by using the white procedure. 4.1. results of an econometric analysis of the output stream traffic in the framework of linear regression equations here are the results of evaluating the dependence of the output stream traffic (table 4.1) on the above set of explanatory variables. table 4.1. dependent variable – output stream traffic, linear regression explanatory variables model 1 model 2 constant -37.76*** (0.27) -79.16*** (0.40) intensityofuncrewedvehicles 147.22*** (3.72) 499.05*** (5.29) intensityofusualagents 284.38*** (4.56) 446.01*** (6.70) speedofuncrewedvehicles 0.34*** (0.002) 0.48*** (0.002) speedofusualagents 0.60*** (0.002) 0.88*** (0.003) intensityofconnections 0.40 (0.28) -0.12 (0.25) r2 0.73 0.76 note. in parentheses are the values of standard errors. ***, **, * – significance at the 1, 5 and 10% level, respectively. as follows from table 4.1, all explanatory variables in both models except intensityofconnections, which describe the output stream traffic, are significant at a 1% level. we also note that in both models the values of the determination coefficients are quite high. comparing the coefficients for significant variables in both models, we can draw the following conclusions: 1. an increase in the ugv input stream intensity by 0.005 units of model time (variation step, table 2.1) leads to an increase in model 2 output stream traffic compared to model 1 output stream traffic by about 1.76 units. for mgv, the indicated difference is 0.81 units. 2. an increase in the average ugv speed by 10 km / h (variation step, table 2.1) leads to an increase in the output stream traffic of model 2 compared with the output stream traffic of model 1 by 1.4 units. for mgv, the indicated difference is 2.8 units. 4.2. results of an econometric analysis of the number of accidents in the framework of linear regression equations here are the results of evaluating the dependence of the number of accidents on the above set of explanatory variables (table 4.2). table 4.2. dependent variable – number of accidents, linear regression explanatory variables model 1 model 2 constant -22.46*** (0.20) -6.34** (0.24) intensityofuncrewedvehicles 797.77*** (2.98) 414.23*** (3.96) intensityofusualagents 810.13*** (3.14) 546.11*** (4.00) speedofuncrewedvehicles 0.07*** (0.001) 0.04*** (0.001) speedofusualagents -0.15*** (0.001) -0.22*** (0.002) intensityofconnections -0.16 (0.13) 0.12 (0.16) r2 0.62 0.45 note. in parentheses are the values of standard errors. ***, **, * – significance at the 1, 5 and 10% level, respectively. 52 a. akopov, n. khachatryan, f. belousov copyright ©2020 assa adv. in systems science and appl. (2020) from table 4.2 it follows that, as in the previous case, the intensityofconnections variable is not significant. all other explanatory variables describing the number of accidents in both models are significant at a 1% level. the positive values of the coefficients in the variables intensityofuncrewedvehicles and intensityofusualagents are quite natural. the average speed of ugv has a positive effect on the number of accidents, and the average speed of mgv is negative. this is explained as follows: an increase in the speed of mgv, in contrast to an increase in the speed of ugv, is a result of enough free traffic and leads to a decrease in road congestion, and an increase in the speed of ugv can lead to unpredictable actions by mgv. comparing coefficients for explanatory variables in both models, we can draw the following conclusions: 1. an increase in ugv input stream intensity by 0.005 units of model time (variation step, table 2.1) leads to a decrease in the number of accidents in model 2 compared to the number of accidents in model 1 by about 1.92 units. for mgv, the indicated difference is about 1.6 units. 2. an increase in the average ugv speed by 10 km / h (variation step, table 2.1) leads to a decrease in the number of accidents in model 2 compared with the number of accidents in model 1 by 0.3 units. for mgv, the indicated difference is 0.7 units. in conclusion, we note that evaluating the dependence of the output stream traffic and the number of accidents on the specified model parameters using the linear function gives generally good results. only an assessment of the dependence of the number of accidents on the studied parameters in model 2 can be called not very successful due to the rather low value of the determination coefficient equal to 0.45. we proceed to the estimation of these dependences using a nonlinear function. 5. econometric analysis of traffic characteristics in the framework of nonlinear regression equations let's proceed to estimate the dependences of the output stream traffic and the number of accidents in both models on the indicated parameters of the model in the framework of nonlinear regression equations. so, we will build and study power regression equations with the following dependent variables: • output stream traffic in model 1; • output stream traffic in model 2; • number of accidents in model 1; • number of accidents in model 2. after logarithming of the equation, the coefficients were estimated in the framework of the classical multiple regression model using the least squares method, and standard errors were estimated using the white procedure. here are the results of evaluating the dependence of the output stream traffic (table 5.1) and the number of accidents (table 5.2) on the indicated set of explanatory variables. table 5.1. dependent variable –. output stream traffic, nonlinear regression explanatory variables model 1 model 2 constant -2.65*** (0.03) -3.79*** (0.02) intensityofuncrewedvehicles 0.08*** (0.002) 0.17*** (0.002) intensityofusualagents 0.08*** (0.002) 0.13*** (0.002) speedofuncrewedvehicles 0.55*** (0.003) 0.68*** (0.003) speedofusualagents 1.07*** (0.004) 1.35*** (0.003) intensityofconnections 0.002 (0.002) -0.0000013 (0.002) r2 0.72 0.77 influence of unmanned vehicles on the state of the transport network 53 copyright ©2020 assa. adv. in systems science and appl. (2020) table 5.2. dependent variable – number of accidents, nonlinear regression explanatory variables model 1 model 2 constant 11.77*** (0.06) 13.86*** (0.07) intensityofuncrewedvehicles 1.11*** 0.005 0.66*** (0.006) intensityofusualagents 1.10*** (0.004) 1.12*** (0.006) speedofuncrewedvehicles 0.30*** (0.008) 0.20*** (0.009) speedofusualagents -0.61*** (0.008) -1.61*** (0.009) intensityofconnections -0.07*** (0.005) 0.004 (0.005) r2 0.58 0.52 note. in parentheses are the values of standard errors. ***, **, * – significance at the 1, 5 and 10% level, respectively. comparing table 5.1 and table 5.2 with table 4.1 and table 4.2, respectively, we can conclude that power regressions give qualitatively the same conclusions as linear ones. this is especially true for the description of these dependencies for output stream traffic. as for the description of the dependence for the number of accidents, here we can identify some advantage of power regression. firstly, the value of the determination coefficient for model 2 increases from 0.45 to 0.52, and secondly, in contrast to the linear dependence, the intensityofconnections variable is statistically significant in the first model at a 1% level. the corresponding coefficient value in table 5.2 (-0.07) allows us to state that an increase in message intensity between ugv by 14% allows reducing the number of accidents by 1%. moreover, in the second model, this variable is insignificant. this is explained by the fact that in the second model, ugv have ability to estimate density of traffic flow, which allows them to maneuver effectively. in such a situation, the exchange of messages about the presence of an accident becomes irrelevant for the ugv. 6. conclusion in [6, 7] two agent-based models of movement of ugv and mgv, which were developed and implemented in anylogic simulation system, are described. the main difference between the second model and the first is that ugv is endowed with an additional capability the ability to intelligently assess the density of traffic flow. the main purpose of this article is to evaluate the effect obtained from the implementation of these ugv, which is to increase output stream traffic and reduce the number of accidents. for this purpose a pool of identical experiments was carried out within each model, the results of which became the initial basis for conducting an econometric analysis. for both the first and second models, the dependence of the output stream traffic and the number of accidents on a given road section in a given period of time on a number of model parameters such as the average vehicle speed, input stream intensity, and the frequency of exchanges between ugv was studied. the study of these dependencies was carried out both in the framework of linear and power regressions. evaluation of these dependencies made it possible to determine an increase in the output stream traffic and a decrease in the number of accidents in the second model relative to the indicated characteristics in the first model depending on the model parameters. acknowledgements the reported study was funded by rfbr, project number 19-29-06003. 54 a. akopov, n. khachatryan, f. belousov copyright ©2020 assa adv. in systems science and appl. 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(2019). parallel multiagent real-coded genetic algorithm for large-scale black-box single-objective optimization. knowledge-based systems, 174, 103–122, https://doi.org/10.1016/j.knosys.2019.03.003 adv syst sci appl 2023; 01:8–21 published online at https://ijassa.ipu.ru. on codimension 3 singular points of first-order implicit differential equations yana s. agakhanova1, alexander s. nikachev1*, irina n. oblasova2 1moscow institute of physics and technology (state university), dolgoprudnyi, russia 2north-caucasus federal university, stavropol, russia abstract: we investigate local phase portraits of first-order implicit differential equations in a neighborhood of their singular points of codimension 3. namely, we consider singular points where the lifted field of the equation on a surface is non-singular, but the projection of the surface to the phase plane has a singularity of one of three types: lips, beaks, swallowtail. we also consider generic bifurcations in one-parameter families of such equations. keywords: implicit differential equations, vector fields, singularities of mappings, fold, cusp, swallowtail 1. introduction the work in this paper is a part of an ongoing research on understanding singular points of first-order implicit differential equations f (x, y, p) = 0, p = dy/dx, (1.1) that is, points where fp vanishes (see, for example, [1] – [19] and the references therein). here x ∈ r1, y ∈ r1, and f : r3 → r1 is a smooth (c∞) function. since a single point of the (x, y)-plane may correspond to several different values of p such that f (x, y, p) = 0, equation (1.1) determines a multi-valued direction field on the (x, y)-plane. for studying this equation, we shall use the legendrian lift of the equation to a surface in the 1-jet space, which goes back to poincaré. the idea of this method is quite similar to the riemann surface for multi-valued functions of the complex variable. it is used in the most of modern works devoted to implicit differential equations, including the works mentioned below. moreover, this method naturally generalizes the case of multi-dimensional systems of implicit equations. the latter leads to an interesting phenomenon: vector fields with non-isolated singular points, see [12, 16]. the recent paper [20] should be especially remarked. 1.1. lifting of equation let j1 be the 1-jet space of functions y(x), i.e., the space with coordinates (x, y, p). equation (1.1) defines a surface f in j1, which is regular at points where ∇f ̸= 0. further, we shall always assume that this condition holds true, and the tangent plane is defined at every point of ∗corresponding author: nikachev441@gmail.com singular points of implicit differential equations 9 f . the intersection of the field of contact planes pdx− dy = 0 with the fields of the tangent planes defines a direction field of f , which is called the lifted field of equation (1.1). integral curves of equation (1.1) are obtained by the projection π of trajectories of the lifted field from the surface f to the (x, y)-plane parallel to the p-direction. the p-direction in the space j1 we shall call vertical. thus, a study of the phase portrait of equation (1.1) is reduced to the projection of the phase portrait of the lifted field from the surface f to the (x, y)-plane. the projection π has singularity at point of the surface f where fp = 0. singularities of the germs of smooth mappings are very well studied, the first result in this direction belongs to h. whitney. see, for example, [21–23] and [24, 25]. the intersection of the contact planes pdx− dy = 0 with the fields of the tangent planes to f can be written in the pfaffian form: fxdx+ fydy + fpdp = 0, pdx− dy = 0. (1.2) which yields the field ẋ = fp, ẏ = pfp, ṗ = −g, g = fx + pfy, (1.3) where the dot stands for differentiation by a new parameter t, which plays a role of time. formula (1.3) presents the lifted field of equations (1.1). we remark that the lifted field is defined at all points of the surface f except points where the contact plane coincide with the tangent plane, that is, fp = g = 0. the equality fp = 0 defines a subset of f , which is called the criminant of equation (1.1). points of the criminant are called singular points of equation (1.1), they are critical points of the projection π. the equality g = 0 defines a subset of f , which is called the inflection curve of equation (1.1), since its projection on the (x, y)-plane is the locus of inflection points of integral curves of (1.1). generically, the criminant and the inflection curve are curves on the surface f intersecting at isolated points, which are singular points of the lifted field (1.3). 1.2. classification of singular points remind that we excluded from consideration points where ∇f = 0 and we call a point of f a singular points of equations (1.1), if fp = 0. here we shall use terminology from [16]: definition 1.1: a singular point of equation (1.1) is called proper, if g ̸= 0 and improper otherwise. for every proper singular point there exists a unique integral curve of the field (1.3). the projection of this integral curve form f to the (x, y)-plane has a singularity. for simplicity, assume that the proper singular point is the origin of the space j1. if the germ of f at this point has a finite multiplicity by p, that is, ∂f ∂p (0) = 0, . . . , ∂n−1f ∂pn−1 (0) = 0, ∂nf ∂pn (0) ̸= 0 (1.4) with an integer n ≥ 2, then the germ of the corresponding integral curve of equation (1.1) has the form x = tnφ(t), y = tn+1ψ(t), φ(0)ψ(0) ̸= 0, (1.5) with smooth functions φ, ψ, see [3]. from the geometric viewpoint, here there are two different cases: if n is even or odd. in fig. 1.1 we present integral curves (1.5) with n = 2 (left) and n = 3 (right). a more interesting and complicated problem is a study of local phase portraits of equation (1.1) in a neighborhood of its singular point. in a neighborhood of a proper singular point the copyright © 2023 assa. adv syst sci appl (2023) 10 y. s. agakhanova, a. s. nikachev, i. n. oblasova fig. 1.1. two types of integral curves passing trough a proper singular point phase portrait of the lifted field on f is diffeomorphic to a family of parallel lines (the flow box theorem), whence the problem is reduced to the study of its projection. the phase portraits of equation (1.1) near a proper singular point where the mapping π has a fold or a cusp (pleat), are well studied; see [9, 10] or survey in [16]. for the convenience of the reader, we shall briefly describe the related results below. the phase portraits of equation (1.1) near an improper singular point where the mapping π has a fold (codimension 2) are studied in [9,10]. the phase portraits of equation (1.1) near an improper singular point where the mapping π has a pleat (codimension 3) are studied in [7]. the cases mentioned above cover singularities of equation (1.1) of codimensions 1 and 2, but not all singularities of codimension 3, see the table below. the columns correspond to all types of singularities of the projection π : f → r2 up to codimension 3 called fold, pleat, lips, beaks, swallowtail. the first row presents the corresponding left-right normal forms (see, for example, [24, 25]). the second (third) row contains proper (respectively, improper) singular points of equation (1.1) with the corresponding singularity of the mapping π. the dash stands in the cases if the codimension of singularity is greater than 3. in other cases we write new if the given type of singularity is not studied before, otherwise we give the references to related sources. it is worth observing that the left-right normal forms mentioned above are obtained by independent c∞-diffeomorphisms (changes of variables) in the image and preimage. therefore, they cannot be directly applied to simplification of equation (1.1), since a differential equation is connected with another group of transformations. fold pleat lips beaks swallowtail (u, v) → (z, w) z = u2 z = u3 + uv z = u3 + uv2 z = u3 − uv2 z = u4 + uv w = v w = v w = v w = v w = v g ̸= 0 codim = 1 codim = 2 codim = 3 codim = 3 codim = 3 [1, 2] [2, 9, 10] new new new g = 0 codim = 2 codim = 3 codim = 4 codim = 4 codim = 4 [2, 9, 10] [7] − − − the paper presents a study of three cases marked as new in the table above. for convenience of the reader, we start with a brief description of results for proper singular points with the fold and pleat of π. after that, we consider proper singular points with the swallowtail, lips and beaks of π, which are not considered before. together with [7], the obtained results cover all singularities of equation (1.1) of codimensions 1, 2, 3. copyright © 2023 assa. adv syst sci appl (2023) singular points of implicit differential equations 11 2. preliminary results without loss of generality, we shall assume that the considered singular point of equation (1.1) is the origin of the space j1. then the condition g(0) ̸= 0 is equaivalent to fx(0) ̸= 0, and the germ of f is the graph of a smooth function x = f(y, p), where f(0) = 0 and fp(0) = 0. therefore, we shall further assume that f (x, y, p) = f(y, p)− x. by the hadamard lemma, we have f (x, y, p) = f(y, p)− x = f0(y) + f1(y)p+ f2(y, p)p 2 − x, where fi are smooth functions, f0(0) = 0 (follows from f (0) = 0) and f1(0) = 0 (from fp(0) = 0). after the change of variables x̃ = f0(y)− x, we get f0(y) ≡ 0, and we brought equation (1.1) to the form f (x, y, p) = f(y, p)− x = αp2 + pyg(y) + p3h(y, p)− x = 0, (2.6) where g, h are smooth functions, α = fpp(0)/2 is a real number. 3. fold and pleat from now on, we shall deal with equation (2.6). if α ̸= 0, that is, fpp(0) ̸= 0, then the germ of π has a fold. if α = 0, but g(0) ̸= 0 and h(0) ̸= 0, that is, fpp(0) = 0, fppp(0) ̸= 0, fpy(0) ̸= 0, then the germ of π has a pleat (cusp). in fig. 3.2, we present the surface f and the projections of integral curves of the lifted field to the (x, y)-plane for the both cases. fig. 3.2. two singularities of the mapping π : f → r2. fold (on the left) and pleat (on the right). theorem 3.1: the germ of equation (1.1) at every proper singular point where π has a fold, is equivalent to the simple equation p2 − x = 0. (3.7) in a neighborhood of such singular point, the phase portrait of equation (1.1) is a family of semicubic parabolas, whose cusps fill the discriminant curve of the equations, the projection of the criminant to the (x, y)-plane; fig. 3.2 (left). theorem (3.1) states that an appropriate local diffeomorphism of the (x, y)-plane brings this family to the standard form y = 2 3 x3/2 + const, which corresponds to equation (3.7). the proof can be found in [1]. copyright © 2023 assa. adv syst sci appl (2023) 12 y. s. agakhanova, a. s. nikachev, i. n. oblasova consider the germ of equation (2.6) at a proper singular point where π has a pleat; see fig. 3.2 (right). using topological arguments, in [9] it is proved that the classification of the germs of equations (2.6) with pleat has functional invariants. moreover, functional invariants do not disappear even if changes of variables are homeomorphisms. however, the geometric classification of possible phase portraits is quite simple. consider integral curves of the corresponding lifted field on the surface f , which has the form ẏ = pfp(y, p), ṗ = 1− pfy(y, p). (3.8) let us present f in the form f(y, p) = apy + bp3 + py2h1(y, p) + p4h2(y, p), ab ̸= 0. (3.9) then, substituting (3.9) into (3.8), one can see that the criminant of equation (2.6) is given by the equation y = kp2 + o(p2) with k = −3b/a. in a neighborhood of the origin, the criminant is similar to the parabola y = kp2, which is tangent to the direction (3.8) of zero. this point is the cusp of the discriminant curve, which has the form x = f(kp2 + o(p2), p) = −2bp3 + o(p3), y = kp2 + o(p2). this yields the following result: theorem 3.2: there exist only two geometrically different types of phase portraits of the field (3.8) corresponding to different signs of k. these phase portraits and their projections to the (x, y)plane are presented in fig. 3.3 (up and down, respectively). fig. 3.3. down: two geometrically different types of phase portraits of equation (2.6) at a proper singular point where π has a pleat. up: the corresponding lifted fields. in the both cases, there exists a narrow domain on the (x, y)-plane, where integral curves constitute a 3-web. 4. singularities of codimension 3 in this section, we present the main results of the paper: a study of the local phase portraits of equation (1.1) at its proper singular points, where the mapping π has a singularity of one of three following types: swallowtail, lips, beaks. copyright © 2023 assa. adv syst sci appl (2023) singular points of implicit differential equations 13 4.1. swallowtail consider the germ of equations (1.1) at its proper singular points, where the mapping π has a swallowtail (not to be confused with a surface with the same name). this means that fp(0) = fpp(0) = 0, fpy(0) ̸= 0, fppp(0) = 0, fpppp(0) ̸= 0. (4.10) under conditions (4.10), the germ of equation (2.6) has the form f (x, y, p) = f(y, p)− x = 0, f(y, p) = apy + bp4 + py2h1(y, p) + p2yh2(y, p) + p5h3(y, p), ab ̸= 0. (4.11) the surface of such equation is presented in fig. 4.4, together with the criminant of the equation and its discriminant curve. the lifted field on f has the form ẏ = pfp(y, p), ṗ = 1− pfy(y, p). (4.12) the above formulas show that the criminant of (4.11) has the form y = kp3 + o(p3), where k = −4b/a. in a neighborhood of the origin, the criminant is similar to the cubic parabola y = kp3, which is tangent to the field (4.12) at the origin. the tangency point correspond to a singular point of the discriminant curve: x = f(kp3 + o(p3), p) = −3bp4 + o(p4), y = kp3 + o(p3). (4.13) fig. 4.4. left: the surface f of equations (1.1) whose mapping π has a swallowtail; the bold curve on f is the criminant of the equation, and the bold curve on the plane is its discriminant curve. right: intersection of the surface s (blue) with the surface swallowtail (red) in the space of the coefficients. lemma 4.1: locally, the discriminant curve of equation (4.11) splits the (x, y)-plane into two open domains, in which equation (4.11) has two real roots p or zero real roots p, respectively. at point of the discriminant curve (except the origin) it has one double real root p, while at the origin the multiplicity of p is equal to 4. proof by the division theorem [22, 23], the germ of equation (2.6) satisfying conditions (4.10) is equivalent to f̃ (x, y, p) = p4 + 3∑ i=0 ai(x, y)p i = 0, p = dy/dx, (4.14) where ai(x, y) are smooth functions, ai(0) = 0, and da0, da1 at zero are linearly independent. using an appropriate change of variables (x, y), one can kill the monomial p3, that is, bring copyright © 2023 assa. adv syst sci appl (2023) 14 y. s. agakhanova, a. s. nikachev, i. n. oblasova (4.14) to the form p4 + a(x, y)p2 + b(x, y)p+ c(x, y) = 0, p = dy/dx, (4.15) with smooth functions a, b, c, a(0) = b(0) = c(0). form the independence of da0, da1 in (4.14) it follows the independence of db, dc in (4.15). then the statement of the lemma can be interpreted in the following way. in the space with cartesian coordinates a, b, c consider the discriminant surface of the polynomial m(p) = p4 + ap2 + bp+ c, (4.16) that is, a surface that consists of points (a, b, c) that the polynomial m(p) has multiple roots. this surface w defined by the equations m(p) =m ′(p) = 0 is called the swallowtail, it is presented in fig. 4.4 (right) in red. it separates the whole space into three open domains u4, u2, u0, where m(p) has 4, 2, 0 real roots, respectively. in the (a, b, c)-space equation (4.15) defines a surface s a = a(x, y), b = b(x, y), b = c(x, y), (4.17) it is presented in fig. 4.4 (right) in blue. since the differentials of the functions b, c at zero are linearly independent, in a neighborhood of the origin the surface s is regular and it intersects the swallowtail w as presented in fig. 4.4 (right). this proves all statements of the lemma. theorem 4.1: there exist only two geometrically different types of local phase portraits of equation (4.11) corresponding to different signs of k, which is equivalent to those of ab. these phase portraits are presented in fig. 4.5. fig. 4.5. down: two different types of local phase portraits of equation (1.1), whose mapping π has a swallowtail, which correspond to different signs of ab in (4.11). up: the corresponding phase portraits of the lifted field. the dotted lines present the criminant of the equation (up) and its discriminant curve (down). proof consider local phase portraits of the lifted field (4.12) for equation (4.11). in a neighborhood of the origin integral curves of the field (4.12) are similar to straight lines parallel to the paxis, and the criminant is similar to the cubic parabola y = kp3. this proves the statement of the theorem. copyright © 2023 assa. adv syst sci appl (2023) singular points of implicit differential equations 15 4.2. lips and beaks consider the germ of equations (1.1) at its proper singular points, where the mapping π has a singularity of the type lips or beaks. then, after an appropriate linear change y 7→ αy equation (2.6) can be brought to the form f (x, y, p) = f(y, p)− x = 0, f(y, p) = p3 + h1(y, p)yp 2 + h2(y, p)y 2p+ h3(y, p)p 4, (4.18) where the hessian of the function fp(y, p) at the origin is not zero: h[fp](0) = 4(3h2(0)− h21(0)) ̸= 0. (4.19) the mapping π has a singularity of the type lips (respectively, beaks) if h[fp] > 0 (respectively, h[fp] < 0). the corresponding surfaces of the equation (1.1) are presented in fig. 4.6. fig. 4.6. the surface f of equations (1.1) whose mapping π has a singularity of the type lips (left) or beaks (right). the bold curve on f is the criminant and the bold curve on the plane is its discriminant curve. by the division theorem, the germ of the function f from (4.18) has the form f (x, y, p) = φ(x, y, p) ( p3 + a(x, y)p2 + b(x, y)p+ c(x, y) ) , where a, b, c,φ are smooth functions, a(0) = b(0) = c(0) = 0, and φ(0) ̸= 0. therefore, the germ of equation (4.18) is equivalent to p3 + a(x, y)p2 + b(x, y)p+ c(x, y) = 0, p = dy/dx. (4.20) lemma 4.2: in (4.20), the coefficient c has the form c(x, y) = xc̃(x, y) with a smooth function c̃. the condition g ̸= 0 for equation (1.1) is equivalent to c̃(0, 0) ̸= 0. proof formula (4.18) shows that the surface f given by the equation x = f(y, p) contains the y-axis (x = p = 0). this means that the substitution x = p = 0 into (4.20) gives a certain identity by y. this yields c(x, y) ∣∣ x=0 ≡ 0, and the statement follows from the hadamard lemma. lemma 4.3: in (4.20), the equality by(0) = 0 holds true. copyright © 2023 assa. adv syst sci appl (2023) 16 y. s. agakhanova, a. s. nikachev, i. n. oblasova proof substitution x = f(y, p) with the function f from (4.18) brought equation (4.20) to a certain identity by y, p. the left hand side of this identity contains the monomial by(0)yp, while the right hand side is identically zero. this proves the statement. lemma 4.4: hessian of the function fp at zero is equal to h[fp](0) = 4φ2(0)∆, where ∆ = 3 2 byy(0)− a2y(0), whence the type of singularity of π is determined by the sign of ∆: lips if ∆ > 0 and beaks if ∆ < 0. proof from (4.18) we have: fp(y, p) = fp(x, y, p) = φp(x, y, p)(p 3 + a(x, y)p2 + b(x, y)p+ c(x, y))+ +φ(x, y, p)(3p2 + 2a(x, y)p+ b(x, y)), where x is replaced with f(y, p). let us calculate the quadratic term of the obtained function on y, p. from lemmas 4.2 and 4.3 it follows that the 2-jet of the function p3 + a(f(y, p), y)p2 + b(f(y, p), y)p+ c(f(y, p), y) is zero and the 1-jet of 3p2 + 2a(f(y, p), y)p+ b(f(y, p), y) (4.21) is zero as well. therefore, the quadratic part of the germ fp(y, p) coincides with the quadratic part of (4.21) multiplied by φ(0). taking into account 4.3, a simple calculation shows that the quadratic part of (4.21) is 3p2 + 2ay(0)py + 1 2 byyy 3. this yields the statement of the lemma. theorem 4.2: the local phase portrait of equation (1.1) at proper singular point whose mapping π has a singularity lips is presented in fig. 4.7. here the discriminant set of the equation is a single point, and integral curves are regular at all points except for this point. theorem 4.3: the local phase portrait of equation (1.1) at proper singular point whose mapping π has a singularity lips is presented in fig. 4.8. here the discriminant set of the equation is a pair of two curves tangent at a single point, which separate the (x, y)-plane into four open domains with different stricture of integral curves: 1-webs or 3-webs. proof to prove theorems 4.2 and 4.3, consider the left hand side of equation (4.20) as a polynomial on the variable p with coefficients smoothly depending on x, y: m(p) = p3 + a(x, y)p2 + b(x, y)p+ c(x, y). (4.22) in the space with cartesian coordinates a, b, c consider the discriminant surface of the polynomial m(p) = p3 + ap2 + bp+ c, that is, a surface that consists of points (a, b, c) that the polynomial m(p) has multiple roots. this surface w defined by the equations m(p) =m ′(p) = 0 is called the cuspidal edge, it is presented in fig. 4.9 in orange. it copyright © 2023 assa. adv syst sci appl (2023) singular points of implicit differential equations 17 fig. 4.7. left: local phase portrait of equation (1.1) whose mapping π has lips. right: the corresponding phase portrait of the lifted field. here the criminant and the discriminant set are isolated points. fig. 4.8. left: local phase portrait of equation (1.1) whose mapping π has beaks. right: the corresponding phase portrait of the lifted field. the criminant and the discriminant set of the equation are depicted with dotted lines. separates the whole space into two open domains u1, u3, where m(p) has 1, 3 real roots, respectively. the domain u3 is more narrow. in the (a, b, c)-space equation (4.15) defines a surface s defined by (4.17). the proof of theorems 4.2, 4.3 is based on the intersection of the surfaces s and w . using lemmas 4.2 – 4.4, it is not hard to show that the mutual position of s and w is exactly as it is presented in fig. 4.9 for lips and beaks. here the surfaces s and w are depicted in blue and orange, respectively. in the case ∆ < 0 (lips) the surface s belongs to the open domain u1 except for a point, which corresponds to the unique singular point of the mapping π. in the case ∆ > 0 (beaks) the surface s is cutted by w into four parts, two of which belong to u1 and two belong to u3. this proves the theorems. fig. 4.9. the mutual position of surfaces w and s for singularities lips (left) and beaks (right). copyright © 2023 assa. adv syst sci appl (2023) 18 y. s. agakhanova, a. s. nikachev, i. n. oblasova 5. bifurcations on one-parameter families 5.1. bifurcations of the swallowtail consider a generic one-parameter perturbation of equation (4.11): p4 + a(x, y, µ)p2 + b(x, y, µ)p+ c(x, y, µ) = 0, p = dy/dx, (5.23) where µ is a small parameter. in the space with cartesian coordinates a, b, c consider the family of surfaces sµ defined by the coefficients a(x, y, µ), b(x, y, µ), c(x, y, µ) of equation (5.23). in the same space consider the discriminant surface w of the polynomial m(p) = p4 + ap2 + bp+ c, which is the swallowtail. the bifurcation of the phase portrait of equation (5.23) is determined by the intersection of the family sµ with w . assume that ∂a ∂µ (0, 0, 0) ̸= 0. (5.24) then the function a(0, 0, µ) changes its sign when the parameters µ passes through zero. without loss of generality we shall further assume that the sign of the function a(0, 0, µ) coincides with the sign of µ. then the mutual position of sµ and w is presented in fig. 5.10, and the corresponding bifurcation of the phase portrait of equation (5.23) is presented in fig. 5.11. it is worth observing that for all µ < 0 the exists a narrow domain on the (x, y)plane where integral curves constitute a 4-web. as µ→ −0 this domain tends to a point, and for µ ≥ 0 integral curves have no quadruple intersections. fig. 5.10. the surfaces sµ and w for a generic family (5.23) depicted in blue and red respectively. from the left to right: µ < 0, µ = 0, µ > 0. fig. 5.11. bifurcation of a generic family (5.23): the phase portrait on the (x, y)-plane, the dotted line is the discriminant curve. from the left to right: µ < 0, µ = 0, µ > 0. copyright © 2023 assa. adv syst sci appl (2023) singular points of implicit differential equations 19 5.2. bifurcations of lips and beaks consider a generic one-parameter perturbation of equation (4.18) with singularity of the type lips or beaks: f (x, y, p) = p3 + a(x, y)p2 + b(x, y)p+ c(x, y), p = dy/dx, (5.25) where µ is a small parameter. in the space with cartesian coordinates a, b, c consider the family of surfaces sµ defined by the coefficients a(x, y, µ), b(x, y, µ), c(x, y, µ) of equation (5.25). in the same space consider the discriminant surface w of the polynomial m(p) = p3 + ap2 + bp+ c, which is the cuspidal edge. the bifurcation of the phase portrait of equation (5.25) is determined by the intersection of the family sµ with w . assume that ∂b ∂µ (0, 0, 0) ̸= 0, then the function b(0, 0, µ) changes its sign when the parameters µ passes through zero. without loss of generality we shall further assume that the sign of the function b(0, 0, µ) coincides with the sign of µ. the mutual position of sµ and w for a generic family (5.25) with singularity of the type lips is presented in fig. 5.12. the corresponding bifurcation of the phase portrait is presented in fig. 5.13. for all µ > 0 there exists a narrow bounded domain on the (x, y)-play (which looks like lips), where integral curves constitute a 3-web. as µ→ +0 this domain tends to a point, and for µ ≥ 0 integral curves have no triple intersections. fig. 5.12. the surfaces sµ and w for a generic family (5.25) with lips depicted in blue and orange respectively. from the left to right: µ > 0, µ = 0, µ < 0. fig. 5.13. bifurcation of a generic family (5.25) with lips. from the left to right: µ > 0, µ = 0, µ < 0. the closed bold curve in the left image is the discriminant set, which degenerates into a point when µ = 0. the mutual position of sµ and w for a generic family (5.25) with singularity of the type beaks is presented in fig. 5.14. the corresponding bifurcation of the phase portrait is presented in fig. 5.15. copyright © 2023 assa. adv syst sci appl (2023) 20 y. s. agakhanova, a. s. nikachev, i. n. oblasova fig. 5.14. the surfaces sµ and w for a generic family (5.25) with beaks depicted in blue and orange respectively. from the left to right: µ > 0, µ = 0, µ < 0. fig. 5.15. bifurcation of a generic family (5.25) with beaks. from the left to right: µ > 0, µ = 0, µ < 0. the dotted curve is the discriminant set. 6. conclusion in this paper, we studied local phase portraits of first-order implicit differential equations, that is, equations in the form f (x, y, y′) = 0, in a neighborhood of their proper singular points of codimension 3. this study, together with previous works, completes qualitative investigation of singularities of such equations up to codimension— 3. moreover, we investigated generic bifurcations in one-parameter families of such equations. it is worth observing that an interest to such equations is motivated by their applications in various mathematical and technical problems; see, for example, [6] and the lists of references in [11, 14]. references 1. arnol’d, v. i. (1988) geometrical methods in the theory of ordinary differential equations, berlin, germany: springer. 2. arnol’d, v. i. & ilyashenko, yu. s. (1988). ordinary differential equations, dynamical systems i. encycl. math. sci. 1, 1–148. 3. bruce, j. w. (1984). a note on first-order differential equations of degree greater than one and wavefront evolution. bull. london math. soc., 16, 139–144. 4. bruce, j. w., fletcher, g. j., & tari, f. (2000). bifurcations of binary differential equations, proc. royal society of edinburg, 130 a, 485–506. 5. bruce, j. w., fletcher, g. j., & tari, f. (2004). zero curves of families of curve congruences, contemporary math., 354, 1–18. 6. barlukova, a. m. & chupakhin, a. p. (2012). partially invariant solutions in gas dynamics and implicit equations, j. appl. mech. tech. phys., 53(6), 812–824. 7. chertovskih, r. a. & remizov, a. o. 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(2021). smooth local normal forms of hyperbolic roussarie vector fields, moscow math. j., 21(2), 413–426. 13. pavlova, n. g. & remizov, a. o. (2022). oscillating and proper solutions of singular quasi-linear differential equations, adv. syst. sci. appl., 22(4), 51–64. 14. pazij, n. d. & pavlova, n. g. (2022). local analytic classification for quasi-linear implicit differential systems at transversal singular points. j. dyn. control syst., 28(3), 453–464. 15. remizov, a. o. (2002). implicit differential equations and vector fields with nonisolated singular points, sb. math., 193(11), 1671–1690. 16. remizov, a. o. (2008). multidimensional poincaré construction and singularities of lifted fields for implicit differential equations, j. math. sci., 151(6), 3561–3602. 17. seiler, w. m. & seiss, m. (2021). singular initial value problems for scalar quasi-linear ordinary differential equations, j. differ. equations, 281, 258–288. 18. tari, f. (2007). geometric properties of the integral curves of an implicit differential equation, discrete contin. dyn. syst., 17(2), 349–364. 19. sotomayor, j. & zhitomirskii, m. (2001). impasse singularities of differential systems of the form a(x)x′ = f (x), j. differ. equations, 169(2), 567–587. 20. medrado, j. c. r. & silva, l. a. (2022). codimension-one singularities of constrained systems on r3, bull. sci. math., article id 103142, 22 p. 21. arnol’d, v. i., gusein-zade, s. m., & varchenko, a. n. (1985). singularities of differentiable maps. volume i: the classification of critical points, caustics and wave fronts, monogr. math., 82, boston-basel-stuttgart: birkhauser. 22. broker, th. & lander, l. (1975). differential germs and catastrophes, cambridge, uk: cambridge university press. 23. golubitsky, m. & guillemin, v. (1973). stable mappings and their singularities, grad. texts math., 14, new york-heidelberg-berlin: springer-verlag. 24. rieger, j. h. (1987). families of maps from the plane to the plane, j. london math. soc., ii, 36(1–2), 351–369. 25. saji, k. (2010). criteria for singularities of smooth maps from the plane into the plane and their applications, hiroshima math. j., 40(2), 229–239. copyright © 2023 assa. adv syst sci appl (2023) introduction lifting of equation classification of singular points preliminary results fold and pleat singularities of codimension 3 swallowtail lips and beaks bifurcations on one-parameter families bifurcations of the swallowtail bifurcations of lips and beaks conclusion adv syst sci appl 2019; 03; 131-139 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/718 modeling and estimating the impact of the opec agreement on oil production in russia valery akinfiev * v.a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia e-mail: akinf@ipu.ru received march 27, 2019; revised august 28, 2019; published october 1, 2019 abstract: the shale revolution in the united states dramatically changed the situation on the global oil market. the response to the shale boom of 2013-2014 was the opec + agreement, which aims to regulate the market in order to stabilize oil prices. the article analyzes the impact of the opec+ agreement on the global oil market, taking into account the dynamics of us oil shale production. a simulation dynamic game model is proposed, which describes the relationship between the oil demand, the strategy of oil supplying by the players and the dynamics of oil price. the model allows, by setting different strategies for the behavior of players, to simulate the movement of oil prices, which in turn affect the choice of players in their decisions on oil production and its supply to the market. the model includes three players: us shale companies, opec+ countries (including russia), and non-opec countries (including conventional us oil production). the research methodology is based on scenario modeling and analysis of opec+ strategies for managing the supply and demand balance (decrease or increase in production) for targeting oil prices at specified target levels. we tried in this study to answer the question: is the opec agreement favorable for russia? calculations show that for russia, in terms of maintaining market share, the option of targeting oil prices at lower levels is the most profitable, but at the same time, gross oil revenues of russia are noticeably reduced. keywords: opec agreement, oil market models, supply and demand balance. 1. introduction the emergence of fracking shale oil production technology in the united states has changed the outlook for the global petroleum industry in recent years. it was the us shale industry that became the main reason for the sharp increase in oil supply in the market in 2013-2015, which led to a sharp drop in oil prices to 30 usd/bbl in mid-2014. fig. 1. us oil production dynamics. * corresponding author: akinf@ipu.ru mailto:akinf@ipu.ru 132 v. akinfiev copyright ©2019 assa adv. in systems science and appl. (2019) figure 1 plots the dynamics of the us crude oil production, for 1985-2015 (in million barrels per day). source: u.s. energy information administration (eia), 2015. the explosive growth of shale oil production in the united states in 2013-2015 came as a surprise to the market. in 2015, the volume of shale oil produced in the united states almost reached the level of traditionally produced oil more than 4.5 million barrels per day. for reference, it is almost half of the volume of oil production in saudi arabia or russia. if in the period up to 2013, the increase in oil supply on the market was offset by an increase in demand, in 2014 oil consumption grew at a lower rate, and the difference between supply and demand reached 1.5-2.0 million barrels per day. such an imbalance of supply and demand led to a shock reduction in the price of oil. opec could reduce the level of oil production and thus eliminate the excess supply, which would help the price to return to a higher level. however, in 2014, opec abandoned the use of such a strategy. this decision is aimed at maintaining its share by lowering oil prices. shale oil producers reacted to the decline in oil prices by lowering drilling volumes at the least profitable fields: the number of active drilling rigs almost halved by 2016. but by 2016, the world oil market has moved from the phase of competition and the price war of 2014–2015 to market regulation aimed at stabilizing prices at the desired levels. therefore, in november 2016, opec agreed to reduce oil production by 1.2 million barrels per day. in december, another 11 non-opec countries, including russia, joined the agreement. the parties to the agreement agreed to reduce production by 1.75 million barrels per day compared with the level of october 2016. a similar decision was made in december 2018. according to it, the opec + countries, including russia, decided to reduce oil production by 1.2 million barrels per day. the target for the oil price set by the agreement was 70 + usd/bbl. saudi arabia and other opec members in the current situation are counting on cooperation with oil-producing countries led by russia, which will help reduce the negative effects of the american shale revolution on the world oil market. opec + solutions aimed at restoring the balance of supply and demand in the oil market can be quite effective. the reduction in production of opec + leads to an increase in reserve production capacity, which reduces price volatility. nevertheless, thanks to the implementation of the opec + strategy to regulate the market, shale oil companies can develop and increase production at high rates. according to the opec forecast, the annual growth of shale oil production is expected on average by 1.4 million barrels per day in 2019-2020 [10]. then, after 2023, a slowdown in the growth of shale oil production in the united states is predicted. in the period 2027-2028 peak shale oil production is projected at 14.3 million barrels per day (or approximately 13.5% of the market share). then, by 2040, production will drop to 12.1 million barrels per day (approximately 10.5% of the market share). for comparison, in 2018, shale oil production in the united states reached 7.5 million barrels per day, which accounted for about 7.5% of the market share [10]. opec also expects growth in global oil demand in the medium term by 1.2 million barrels per day to 104.5 million barrels per day by 2023. then, a “gradual slowdown in growth rates” of demand is projected, which will reach a maximum in 2040 at the level of 110.0–111.5 million barrels per day. this forecast is consistent with the iea forecast, which also implies a slower growth in global oil demand after 2040. leading global oil companies, such as royal dutch shell and bp, expect a peak in global demand in the years 3035–2040 at 110.3–110.5 million barrels per day. the important question is how profitable is the opec + agreement for russia? the opec forecast was made on the condition that oil prices remain at 70 usd/bbl. in this situation, the share of shale oil in 7-9 years can almost double. at the same time, the share of the remaining producing countries, including russia, will decrease. it is clear that it is desirable for russia to maintain its share in the global market, which in 2018 was 11.8%. but, clearly, this possibility occurs at lower price levels. modeling and estimating the impact of the opec agreement 133 copyright ©2019 assa. adv. in systems science and appl. (2019) it should be noted that the growth rate of shale oil production largely depends on the price of oil on the global market. of practical interest is the study of scenarios in which opec + will support other price ranges, for example, (60-70), (50-60), (40-50) usd/bbl. everywhere else, if it is not specifically stated, the price of oil refers to the brent price. what is oil price russia needs to fulfill budget commitments and economic development, as well as to save its share in the global oil market? the article is devoted to the discussion of these important issues. 2. model of behavior of u. s. shale producers the us shale revolution has changed the parameters of the global oil market model: the supply of oil by producers has become more elastic. us shale companies increase oil production in response to price increases, and, conversely, reduce production if oil becomes cheaper. a quick response from the us shale industry to the changing situation in the oil market is capable of keeping prices within a narrow range, which also depends on the oil production strategy of other of other producers. before the us shale revolution, oil price shocks were common due to the inelastic supply of conventional oil. an important question arises how can we describe the behavior of us shale companies depending on the dynamics of world oil prices? currently, the shale oil companies are drilling and producing at five shale plays, which differ significantly among themselves, including production profiles, well productivity, well flow rate growth dynamics, transport costs and selling price. the main indicator that affects the profitability of investments in shale projects and their ability to generate positive cash flow is the breakeven price. figure 2 shows the distribution of wti breakeven prices for all 2q 2018 horizontal oil completions for the major u.s. liquids plays (source: rystad energy shale well cube, august 2018). the breakeven price calculation is based on the traditional dcf model, which includes all investments, operating and fiscal costs (excluding land value) plus a 5% yield. fig. 2. breakeven oil prices distribution for main shale plays it should be borne in mind that due to the exploitation of shale basins, companies will have to move from areas with a lower break-even price to areas with a higher break-even price. for this reason, in the future, the average break-even price for the main us shale basins will rise, despite improvements in production technology and lower costs for some of the cost items. another important factor that can influence the rate of u.s. shale oil production is also the size of the proven reserves. according to the latest data provided by the sec, reserves in the permian basin (the most promising in terms of development) can reach only 3.8 billion barrels. the eagle ford and bakken pools can hold 5 billion barrels each. a number of publications are devoted to the problem of forecasting us shale oil production [6, 8, 9]. the most popular is the approach associated with the use of various modifications of the autoregressive models. in [6], to estimate future supplies of shale oil (using the example of the bakken basin), it is proposed to use a non-linear and linear forecasting model based on the auto regressive integrated moving average method 134 v. akinfiev copyright ©2019 assa adv. in systems science and appl. (2019) (arima). in [8], a new hybrid model nmgm-arima was proposed to forecast u.s. shale oil production, in which the arima method was supplemented with a non-linear model to correct the residual term of the sequence. empirical results show that the nmgm-arima method can improve forecast efficiency [8]. these methods are designed to forecast the horizon for several quarters. another approach is to use models in which the relationships between a large numbers of factors are modeled and expert information is widely used. these methods allow you to make forecasts with a horizon of several years. note that in these methods, the market price of oil is also exogenous and is set in the form of several scenarios. in [9], the results of studies obtained using the medium-term model of the us shale industry are presented. the model generates forecasts for 6 key indicators for each of the five shale basins (bakken, eagle ford, anadarko, niobrara and perm). the model allows building forecasts of production dynamics depending on the wti oil prices, possible growth of drilling equipment productivity and other technological and resource constraints. the simulation results show that, if the wti oil price is at the level of usd 70 per barrel and higher, then annual production growth in the next two to three years may reach 2.0-2.5 million barrels per day [9]. if the price of wti oil is in the range of 60-70 us dollars per barrel, then production growth is projected at the level of global demand growth, that is, at the level of 1.2-1.4 million barrels per day. when the price of wti crude oil is about $ 50 per barrel, production growth will be negligible or even zero. with prices below $ 50 a barrel, shale projects will no longer be attractive to investors. it should be borne in mind that the price of wti crude oil is on average 10% lower than the price of brent crude oil. we will use these results and the results of other analysts when building a model for the behavior of us shale producers. to implement the proposed approach, it is necessary to build the dependence of u.s. shale oil production on two parameters: the market price of oil and the forecast period. the solution to this problem depends on a large number of factors, many of which can only be evaluated expertly. when constructing these dependencies in fig. 3, the author used the results obtained in [9], as well as research results and forecasts from other sources. let tgshale the increase in shale oil production in period t. tgshale depends on the dynamics of oil prices and the time elapsed from the beginning of the forecast period. then ),(g 1t tpt shale  , where tp is the price of oil on the market. this dependence can be defined as a series of graphs depending on the period of time and the dynamics of production growth in previous periods. the averaged dependences of the increase in u.s. shale oil production on the wti oil price for the periods (2019-2020), (2023-2024) and (2026-2027) are shown in figure 3. fig. 3. the dependence of u.s. shale oil production growth on the wti oil price -4,00 -3,00 -2,00 -1,00 0,00 1,00 2,00 3,00 30 40 50 60 70 80 90 м ill io n b ar re ls p e r d ay wti oil price, usd per barrel the annual production growth 2019 2020 2023 2024 2026 2027 modeling and estimating the impact of the opec agreement 135 copyright ©2019 assa. adv. in systems science and appl. (2019) then the production of shale oil is given by the following recurrent formula: ),(ss 11-tt tpt shaleshale  (1) please note that the price of oil )( tp is not known in advance and is determined based on the market pricing mechanism, depending on the ratio of supply and demand. we will discuss this relationship in detail in the next section. 3. the model of the oil market based on the supply and demand balance in [1, 4, 5, 7], the problem of modeling the oil market and forecasting oil prices was considered, taking into account the dynamics of u.s. shale oil production. in [4], a game approach was proposed for analyzing the strategies of opec +, taking into account the dynamics of u.s. shale oil production. various cournot equilibrium models were used, in which the pricing model in the market is defined as a function of linear demand. it is assumed that the oil demand on the part of consumers depends linearly on its market price. this approach does not take into account the low elasticity of oil demand from its price. based on calculations and comparison of the obtained results with historical data for the period 2014–2016, the authors concluded that the proposed approach does not explain the real data and needs to be improved. in [5], using the structural vector auto regression (var) model, factors affecting the appearance of oil market shocks is analyzed. the study made it possible to draw a number of interesting conclusions regarding the importance of the demand and supply balance factor in explaining fluctuations in the oil market, as well as the effect of various assumptions regarding the elasticity indicators of the oil market. the relationship between oil prices and macroeconomic indicators is analyzed in [7] based on the general equilibrium model, which takes into account technological shocks depending on the development of new technologies, oil supply shocks and forecasts for future oil supplies. in [1] the problem of choosing the investment strategies of oil companies with production of conventional and shale oil is considered. a mathematical model is proposed that describes the relationship between the investment strategies of companies and the price of oil, which depends on the ratio of supply and demand on the world oil market. the solution is reduced to the analysis of the bimatrix game, in which the payoff matrix is formed as a result of numerical simulation. we present a simulation dynamic game model, which is shown in figure 4. the model describes the relationship between oil demand, the strategy of oil supply by producers and the dynamics of changes in the price of oil. the model allows, by setting various strategies of producers 'behavior, to calculate the movement of oil prices, which, in turn, affects the choice of producers' decisions regarding the oil supply. the model includes three producers (let's call them players): us shale companies, opec+ countries (including russia) and nonopec countries (including conventional us oil producers) the research methodology is based on scenario modeling and analysis of the opec + management strategies to maintain the supply and demand balance (decrease or increase in production) for targeting oil prices at specified target levels. when modeling each scenario, two criteria are assessed: gross oil revenues for a period and the dynamics of changes in market share. 136 v. akinfiev copyright ©2019 assa adv. in systems science and appl. (2019) fig. 4. integrated model let t opecx control by opec + in the form of a decrease or increase oil production in period t. the value t opecx is determined depending on the ratio of market oil prices in the previous period t-1 and the boundaries of a given interval. the choice mechanism is denoted by , ),,(x вн tt opec ppp , where нp and вp are the boundaries of the target range. for example, this mechanism may be as follows: tttt opec ksd    )(x 11 , where the coefficient tk , which depends on the degree of deviation 1tp from the target level )10(  tk . t kd the scenarios of global oil demand. it is assumed that the demand t kd is unknown in advance to market participants. the pricing model on the market is given by the recurrence formula [2]:   ) s ()(1( 2 2 t 2*       t t t k t t t t t k t d d ttpp  (2) here ts is the total oil supply on the market, t nonopec t opecshale ss   tt ss . in accordance with [2], the coefficient of price elasticity in the model   )(t , if 0st t kd and   )(t otherwise. the medium value  is equal to 9.75 and, accordingly,  is equal to 28.7. the model uses the information on the dynamics of supply and demand for the two previous periods to calculate the forecast price. the value  tp* is calculated as follows:    0** ptp  , where  0*p is the oil price in the initial period. if at some time "t    tstd  , then  tp* "tp for all "tt  . the model takes into account the "hysteresis" property when calculating the price. the new equilibrium value of the price can be determined at a level different from the initial one. in [2], it was shown that the coefficient of elasticity in the zone of oil deficiency is less than the coefficient of elasticity in the zone of surplus. this is due to the fact that in a period of high oil prices even with a shortage of oil (demand exceeds supply), it is more profitable for oil companies to increase production and not raise prices too high, since this may lead to an irreversible decrease in demand due to market adaptation to new conditions. during the period of oil surplus on the market (supply exceeds demand), it is more profitable for oil companies to reduce the oil price and thereby stimulate oil demand. moreover, companies with a low break-even point have the advantage of reducing the oil model behavior us shale companies market model demand scenarios opec +: production and supply of oil scenarios of production of non-opec countries modeling and estimating the impact of the opec agreement 137 copyright ©2019 assa. adv. in systems science and appl. (2019) price without losing a positive profitability. as a result, some players leave the market and, accordingly, the oil supplies are reduced [3]. figures 5 and 6 show the results of testing the proposed model on historical data on demand, supply and oil price in the period 2q 2013 4q 2018. source: international energy agency (https://www.iea.org/oilmarketreport/omrpublic). fig. 5. oil demand and supply balance fig. 6. price and balance on the oil market the results show that the oil price obtained using the model is quite consistent with historical data. the deviation of the forecast price from the historical one in some periods can be explained by the increasing role of factors unrelated to the balance of supply and demand in the market. for example, in the period 2q 2013 4q 2018, the oil price was also influenced by foreign factors, mainly related to the events in venezuela and the us sanctions against iran. the model described in this section makes it possible, using data on demand options, production scenarios of nonopec countries, the opec quota formation mechanism, to calculate the oil price dynamics based on model (2). the problem is to choose t opecx in each period t the value at which the oil price will remain in a given target range. the software simulation model (fig. 4) and the optimization and parameter selection methods are implemented using ms excel. 138 v. akinfiev copyright ©2019 assa adv. in systems science and appl. (2019) 4. analysis of the results the oil demand is not elastic, that is, it does not depend on changes in the market price and is set exogenously in the model in the form of a set of scenarios. the main scenario for modeling is the opec forecast of global demand in the period 2019 2028 [10]. table 1 presents the results of calculations of the dynamics of oil production and market share for major producers. it also presents the results of the calculation of the decrease / increase in oil production by opec countries to maintain the price at the target level of usd 65 per barrel. table 1. forecast of the dynamics of oil production and market share in the period 2019 2028 1 2019 2020 2021 2022 2023 2024 2025 2026 2027 2028 2 101,2 102,2 103,2 104,1 105,0 105,9 106,7 107,6 108,4 109,2 3 7,40 9,20 10,64 11,79 12,71 13,45 14,04 14,51 14,89 15,08 4 7,3% 9,0% 10,3% 11,3% 12,1% 12,7% 13,2% 13,5% 13,7% 13,8% 5 48,4 47,6 47,1 46,9 46,9 47,0 47,3 47,7 48,1 48,7 6 47,8% 46,6% 45,7% 45,1% 44,6% 44,4% 44,3% 44,3% 44,4% 44,6% 7 0,0 -0,8 -0,5 -0,2 0,0 0,2 0,3 0,4 0,4 0,6 8 45,4 45,4 45,4 45,4 45,4 45,4 45,4 45,4 45,4 45,4 in the first column of table 1, the following notation is used: 1 periods, 2 demand, million barrels per day (opec forecast), 3 us shale oil production, million barrels per day, 4 the us shale market share, 5 opec + oil production, million barrels per day, 6 opec + market share, 7 the decrease / increase in oil production by opec (year-over-year), 8 non-opec + production, million barrels per day. in addition, two more scenarios were considered: the opec forecast for growth global oil demand plus 15% and the opec forecast for growth global oil demand minus 15%. the results of the analysis of the market share and currency income of russia from the sale of oil, depending on the target level of the price are presented in table. 2 table 2. market share of russia depending on the target oil price level the target oil price level 40 50 50 60 60 70 70+ opec forecast of growth in world oil demand in 2019-2028 market share by 2028, % 12,0% 11,1% 10,5% 10,1% gross oil revenues of russia, trillion usd 13,338 15,280 17,334 19,443 opec forecast of growth in world oil demand in 2019 2028 plus 15% market share by 2028, % 12,1% 11,2% 10,7% 10,3% gross oil revenues of russia, trillion usd 13,596 15,595 17,706 19,873 opec forecast of growth in world oil demand in 2019 2028 minus 15% market share by 2028, % 11,8% 10,9% 10,4% 10,0% modeling and estimating the impact of the opec agreement 139 copyright ©2019 assa. adv. in systems science and appl. (2019) gross oil revenues of russia, trillion usd 13,083 14,968 16,965 19,017 conclusion the calculations (table 1) confirm that opec +, by reducing production in order to maintain the oil price at a high target level, creates a free niche in the market. this niche is occupied by producers outside the agreement, mainly shale oil producers, who can continue to increase production. with lower target oil prices set by opec +, oil production growth by shale companies will be limited and, under certain growth scenarios for global demand, opec + quotas can be set to increase production, which will help maintain their market share. however, with the current strategy of opec + (the target price level of oil in the range of 6070 usd per barrel), the united states will remain the leader in production growth. calculations carried out on the model show that by 2028, us shale oil production compared to 2018 could almost double and reach 15.1 million barrels per day. us shale oil will occupy 13.8% of the market, while the market share of russian oil will drop to 10.5%. calculations show that for russia, in terms of maintaining market share, the option of targeting oil prices at lower levels is the most profitable, but at the same time, gross oil revenues of russia are noticeably reduced. on the whole, it can be concluded that russia's participation in the opec agreement, as well as the participation of other countries in it, under current conditions is a reasonable strategy. references [1] akinfiev, v. (2017) a model of competition between oil companies with conventional and unconventional oil production. large-scale systems control. issue 67. moscow: ics ras. 52-80, [in russian]. [2] akinfiev, v. (2018) an analysis and forecasting volatility of crude oil market. proceedings of the 11th international conference "management of large-scale system development" (mlsd). moscow: ieee, https://doi.org/10.1109/mlsd.2018.8551767 [3] akinfiev, v. (2018). modeling competition in the oil market. herald of computer and information technologies. № 2. 18-27, [in russian], https://doi.org/10.14489/vkit.2018.02. [4] ansari, d. (2017). opec, saudi arabia, and the shale revolution: insights from equilibrium modeling and oil politics. energy policy, volume 111, december, 166-178. [5] caldara, d., cavallo, m., & iacoviello, m. (2019). oil price elasticities and oil price fluctuations. journal of monetary economics, 103, 1-20. [6] smith, jl. (2018) estimating the future supply of shale oil: a bakken case study energy economics, elsevier energy economics 69, 395–403. [7] olovsson, c. (2019). oil prices in a general equilibrium model with precautionary demand for oil. review of economic dynamics, volume 32, april, 1-17. [8] wang, q., song, x. & li, r. (2018) a novel hybridization of nonlinear grey model and linear arima residual correction for forecasting u.s. shale oil production. energy 165, 1320-1331. [9] salikhov, м. & kurilov, v. (2018) new wave of growth of shale oil production in the usa: scenario forecasts. institute of energy and finance, analytics, october. [in russian]. [online]. available https://fief.ru/img/files/slanceva__otrasl__20181031.pdf [10] world oil outlook 2018. (2018). the annual opec forecast. [online]. available https://woo.opec.org/. https://ieeexplore.ieee.org/author/37086244644 https://doi.org/10.1109/mlsd.2018.8551767 https://www.sciencedirect.com/science/journal/03014215 https://www.sciencedirect.com/science/journal/03014215/111/supp/c https://scholar.google.ru/citations?user=yc2mxdaaaaaj&hl=ru&oi=sra https://www.sciencedirect.com/science/article/pii/s0140988317304188 https://www.fief.ru/employee/detail.2.htm https://www.fief.ru/employee/detail.2.htm adv syst sci appl 2024; 02:203–218 published online at https://ijassa.ipu.ru. solving the energy consumption barrier in brackish water reverse osmosis desalination plants: a genetic algorithm and energy recovery approach moumni mohammed*, massour el aoud mohamed, moumni fatima zahra sultan moulay slimane university, khouribga, morocco abstract: reverse osmosis desalination is an effective technology for supplying potable water to regions facing water stress. however, this process consumes a significant amount of energy, limiting its widespread adoption globally. this study aims to analyze the variations in specific energy consumption (sec) in the reverse osmosis desalination process for brackish water at a plant in morocco, considering the feed water parameters. additionally, the study examines energy consumption with and without the implementation of energy recovery devices at this plant, which produces 10 million cubic meters of water annually. a genetic algorithm is utilized to identify the optimal combination of design and operational parameters to achieve the lowest sec. the findings indicate that incorporating energy recovery devices in the future design of the plant could reduce the sec by up to 30%. keywords: desalination, brackish water, reverse osmosis, energy recovery 1. introduction at present, increases in the global population, coupled with drought and desertifcation due to climate change, will undoubtedly aggravate water security [1]. desalination technologies have been emerged to supply water from unconventional water sources, and reverse osmosis systems account for a large share of desalination facilities [2]. reverse osmosis (ro) process is an important filtration process that is used extensively for the desalination of sea and brackish water all over the world [3]. reverse osmosis is generating growing interest due to its energy efficiency and versatility compared to other desalination technologies. the need to supply drinking water to populations in regions suffering from water shortages has made the development of this technology essential. [4]. reducing the specific energy consumption (sec) poses a significant challenge, driving technological innovation and research in the desalination industry. the energy cost of the desalination process (ro) can account for up to half of the total cost of producing one cubic metre of drinking water. [5]. nevertheless, compared with other desalination technologies, such as multi-stage flash (msf) [7], [8], multiple-effect distillation (med) [9], membrane distillation (md) [ [10], [11], [12], [13],and electrodialysis (ed) [15], the ro process consumes relatively little energy. consequently, most large-scale seawater and brackish water desalination plants have been designed to use the ro method. according to several research studies, the swro process consumes between 2 and 5 kwh/m³ depending on the feed characteristics [24]. on the other hand, the total energy consumption of the bwro process, including electrical energy, is between 1.5 and 2.5 kwh/m³ [20]. many researchers and engineers have dedicated significant effort to finding innovative ways ∗corresponding author: moumni.mohammed@gmail.com 204 m. moumni, m. el aoud mohamed, m. fatima zahra to reduce the energy consumption of the reverse osmosis process. they have focused on several key areas and methods to achieve this objective, including the following: • improvement of ro membranes: developing and utilizing high-performance membranes to enhance filtration efficiency and reduce energy requirements. • reduction of membrane fouling: implementing prevention and cleaning techniques to decrease membrane fouling, maintaining optimal efficiency and lowering energy consumption. • efficiency of energy-consuming equipment: enhancing the efficiency of pumps and other energy-intensive equipment to minimize energy use. • energy recovery devices (erds): using energy recovery devices to capture and reuse energy, thereby reducing the overall energy demand. • innovative membrane module designs: introducing new membrane module configurations to reduce pressure drops in the membrane channel, thus improving energy efficiency. • process optimization: applying optimization algorithms to dynamically adjust operational parameters and maximize energy efficiency in real-time. • integration of renewable energy: utilizing renewable energy sources, such as solar or wind power, to supply desalination plants, reducing reliance on traditional energy sources. first, advances in ro membranes have contributed to energy savings. high-performance reverse osmosis membranes, characterized by increased water flow and salt rejection rates, mitigate the excess pressure required above the osmotic pressure of seawater. researchers have successfully developed an efficient ro system for low sec water desalination by adjusting membrane type and size [25] and [26], respectively. furthermore, low fouling propensity is critical for lowering energy consumption in the swro process because membrane fouling reduces water permeability, which requires higher operating pressure. surface modification has been used to develop fouling-resistant ro membranes by improving hydrophilicity, reducing surface roughness, and decreasing concentration polarization at the membrane’s surface [16] and [17] . moreover, to achieve an optimal combination for a reverse osmosis system with the lowest sec, some researchers are focusing on the design of single-stage or two-stage ro systems. they analyze the performance of these systems with varying feed parameters, membrane permeabilities, and recovery rates. ( [1, 27–30]). two-stage reverse osmosis can achieve a recovery rate of over 50% ( [31]).mingheng [32]. it has been demonstrated that, without inter-stage booster pumps, single-stage is more energy efficient than two-stage in bwro due to lower retentate pressure drop. li et al. [33] investigated the validation of a model-based optimization for bwro operations. in addition, energy-intensive equipment such as high-pressure pumps (hpps), booster pumps (bps), and erds have been developed and improved to reduce the overall energy consumption of the swro process. the efficiency of hpp has increased by about 90%, which is a practical limit for centrifugal pump efficiency. large pumps are recommended for improving hpp efficiency, with pump efficiency in a large-scale desalination plant reaching up to 85% [18], [22]. in other areas, the integration of photovoltaic (pv) and wind power in desalination processes has been evaluated, with promising results in regions with high solar and wind potential. for example, yahiaoui et al. optimized a pv-diesel generator-battery hybrid system for the city of djanet, algeria, using the grey wolf optimizer [23]. in the last 20 years, advancements in erd have played a significant role in reducing the energy consumption of the swro process. the development of erd equipment, including francis turbines (fts) and pressure exchangers (pxs), has significantly reduced energy consumption [18], [19]. the efficiency of px can exceed 95% [18]. as a result, advancements in energy-consuming units have nearly copyright © 2024 assa. adv syst sci appl (2024) solving the energy consumption barrier... 205 reached their practical limits. many research investigations have been conducted to compare erds for saline waters. alexander et al. analyzed the choice between two types of erd (turbocharger and isobaric erd) for brackish water in his study [35], taking into account the conversion rate parameter and the life cycle cost analysis. in addition to its efficiency of between 94 and 98 percent for a discharge rate of up to 110 m3 per hour, which is close to the discharge rate of the plant in this case study, (alexander et al. concluded that the isobaric erd is more energy efficient than the turbocharger, particularly at a conversion rate of 75 to 81 percent, which is also the case for the plant in this study. as a result, the isobaric erd will be considered an energy recovery system for the ro process used in this plant, which lacks erd. furthermore, because the organization intends to expand this plant by adding more ro trains, an analysis of the sec with and without erd will be conducted to justify the feasibility of implementing this erd at this plant. this analysis will determine whether implementing an energy recovery device (erd) in the expanded plant is a viable option. to achieve this, a genetic algorithm will be applied to identify the optimal erd configuration, maximizing energy efficiency while considering the specific parameters of the reverse osmosis process. the genetic algorithm, known for its ability to solve complex optimization problems, will help explore a wide range of possible solutions and converge on the best configuration. 2. description of bwro desalination plant the bwro desalination plant consists of three production lines, as depicted in fig 2.1. each production line is structured in two stages. in this setup, the concentrate from the first stage serves as the feed water for the subsequent stage, allowing for additional permeate production. the permeate collected from the first stage is combined with the permeate from the second stage. the hpp increases the pressure of the pre-treated brackish water to a suitable value for fig. 2.1. schematic diagram the membrane. the pressure required depends on the concentration and temperature of the feed water. osmotic pressure increases with concentration, so the operating pressure must be higher than the osmotic pressure corresponding to the concentration of the rejected brine flow rate at the membrane rack outlet, as well as membrane fouling. this plant’s feed water concentration varies from 1g/l to 3g/l throughout the year, as does its temperature, which ranges from copyright © 2024 assa. adv syst sci appl (2024) 206 m. moumni, m. el aoud mohamed, m. fatima zahra table 2.1. equipment characteristics number of production line 3 number of membrane by line 600 membrane sectional area 35m2 nominal flow of hp pump 550m3/h nominal pressure of hp pump 20 bar nominal speed of hp pump 3000 rpm 7c° to 32c°. to manage the changing characteristics of brackish water and keep within the operating range of this plant, the recommended conversion rate for each production line with its two stages is 75%, with a production flow rate of about 390 m3/h. to ensure that the above instructions are followed, each hp feed pump is equipped with a vfd that allows the pump speed to be varied, which changes both the flow rate and the pressure of the membrane feed water. the article [36] used fuzzy logic to effectively control these parameters in the station object of this study. table 2.1 shows the equipment used in this station’s ro process. 3. materiels and methods before evaluating the specific energy consumption (sec) of the (ro) process implemented in this station, it is essential to first model the ro process and develop a simulation model. this initial step was thoroughly addressed in the work presented in article [37].using matlab/simulink software tools, a numerical model was developed and validated against real plant values, as well as against the model used by arun-joseph and vasanthi damodaran in table 3.1. ro process equations definition equation osmotic perssure (bar) π = i.cf .r.t (3.1) permeate flux (m.s−1) jw = aw(∆p −∆π) (3.2) solvent permeability (m.s−1.pa−1) aw = a0exp(6.433− 1885 t ) (3.3) permeat flow (m3/h) qp = s.jw (3.4) drop pressure across membrane (bar) ∆p = ( pf + pc 2 − pp) (3.5) ∆p = ∆π + ( qp s.aw ) (3.6) conversion rate y = qp qf (3.7) their study [38]. the equations used for this modeling are summarized in table 3.1. with this model in place, the current sec of the ro process can be evaluated both with and without an energy recovery device (erd). this comparison will demonstrate the potential benefits of installing an erd in the plant. following the conclusion to install the erd, it is necessary to determine the optimal parameters for the ro process, taking into account the parameters of copyright © 2024 assa. adv syst sci appl (2024) solving the energy consumption barrier... 207 the erd, which will also be modeled using equations. this optimization will be carried out using a genetic algorithm to ensure the most efficient operation of the expanded plant. 4. evaluation of sec at bwro electricity consumption is a major obstacle in the development of the ro process, with highpressure pumps (hpps) being the primary energy consumers. these pumps need substantial energy to overcome the osmotic pressure of saline water. optimizing energy use involves lowering the feed’s osmotic pressure and selecting the most efficient pump. hpps account for about 75% of total specific energy consumption, with the remainder coming from membrane operations [26]. this study, therefore, focuses on the energy consumption of hpps in the ro process for freshwater production. the equation below shows a mathematical method for estimating the sec of the ro membrane without the need for an erd system: es = pf .qf qp.η (4.1) the model generated in matlab/simulink to evaluate the fluctuation of sec in the station under study can be used to get an understanding of the predicted energy consumption and losses at the discharge. three additional scenarios are added to the three cases given in the article [37] to study the energy fluctuation, as shown in table 4.1. furthermore, the energy assumed to be lost during discharge can be calculated by adding the flow rate and pressure of the concentrates. given that the organization intends to expand this station with more trains, and based on the results, which show that the energy lost to create a cubic meter of permeate can average 0.3 kwh, a erd for the new entity must be installed. 5. modeling erd as previously mentioned, an isobaric energy recovery device (erd) will be utilized for the ro process implemented in this plant. rotary isobaric devices use a small rotor to recover hydraulic energy from the concentrate stream. the rotor contains ducts that alternately fill with high-pressure brine and low-pressure feed water. as the rotor turns, these ducts are exposed to highand low-pressure zones alternately, effectively replacing the high-pressure brine with saltwater on a 1-to-1 basis. the timing of the water exchange ensures that the chamber is never completely empty, creating a static water piston that prevents the two streams from mixing. [39]. the equations presented below are used to determine the best design for the ro process with an erd. the installation of an erd in an existing ro process necessitates knowledge of its operating parameters. figure 5.1 shows the erd system’s inputs and outputs. fig. 5.1. the erd system’s inputs and outputs copyright © 2024 assa. adv syst sci appl (2024) 208 m. moumni, m. el aoud mohamed, m. fatima zahra 5.1. flow calculation the erd requires for its adequate choice the preknowledge of the inlet flows(qic, qif ) and outlet flows (qoc,qof ) of this equipment. water flows from the concentrate of ro process and the low pressure pumps are the two types of water flows that enter the erd. qic can be determined as is defined, the concentrate flow rate, qc which is a function of the ro process feed flow qf and its conversion rate y : qic = qf .(1− y ) where : qf = qp y = qc +qp (5.1) each erd is is characterised by brine flow loss or leak (l) and overflush (of) [34] where : qif , qof and qoc can be defined using the following equations: flow balance: qic +qif = qof +qoc (5.2) overflush (of): overflush range is provided by isobaric erds manufactures: of = qif −qof qof (5.3) table 4.1. simulation values and actual values comparaison cnd unit case1 case2 case3 t c° 20 13,5 24 cf g/l 2 1.6 1.26 pa st* mb** er % st mb er % st mb er % pf bar 15,9 15,2 4,4 18,1 17,8 1,65 12,9 14,1 9,3 pc1 bar 13,9 13,2 5,03 15,2 15,7 3,28 10,5 12,1 15,2 pc2 bar 11,9 11,2 5,88 12,9 13,7 6,2 8,95 10,1 12,84 qp1 m3/h 305,4 299,04 2,1 296,9 323 8,7 298,9 287,3 3,88 qp2 m3/h 76,05 74,4 2,16 91,3 80,4 11,9 80 71,5 10,6 qc1 m3/h 230,45 225,6 2,1 215 243 13 226,6 216,6 4,4 qc2 m3/h 154,4 151,15 2,1 140,2 163,1 16,33 146,6 145.1 1 cp1 g/l 0,04 0,02 50 0,01 0,01 0 0,02 0,02 0 cp2 g/l 0,09 0,06 33,3 0,02 0,03 50 0,06 0,04 33,3 y % 71,2 72,7 2,1 73,5 71,1 3,2 72,1 71,2 1,2 es kwh/m3 0,8 0,76 5 0,85 0,89 4,4 0.68 0.71 4,4 ec kwh/m3 0,17 0,15 7,5 0,16 0,2 23,5 0,15 0,16 6 cnd unit case4 case5 case6 t c° 24 10 12 cf g/l 1 2 1.2 pa st* mb** er % st mb er % st mb er % pf bar 13,4 13,95 4,1 19,1 19,55 2,35 17,8 18,09 1,6 pc1 bar 11,6 11,95 3 17,7 17,55 1,1 15,9 16,09 1,1 pc2 bar 10,2 9,95 2,4 15,9 15,5 2,5 14,3 14,09 1,4 qp1 m3/h 291,04 285,62 1,8 345,2 338,1 2 330,2 325,28 1,4 qp2 m3/h 72,05 71,11 1,3 96,3 84,17 12,5 82,1 80,98 1,3 qc1 m3/h 220,45 215,47 2,2 257,2 255,07 0,8 243,6 245,39 0,7 qc2 m3/h 149,4 144,36 3,3 173,2 170,9 1,3 167,6 164,41 1,9 cp1 g/l 0,04 0,02 50 0,05 0,04 20 0,02 0,02 0 cp2 g/l 0,07 0,05 28,5 0,11 0,09 18,1 0,07 0,06 14,2 y % 71,19 72,1 1,2 71,1 71,1 0 72,1 71,2 1,2 es kwh/m3 0,66 0,69 4,5 0,92 0,98 6,5 0,88 0,91 3,4 ec kwh/m3 0.15 0,13 13,3 0,25 0,24 4 0,22 0,21 3 copyright © 2024 assa. adv syst sci appl (2024) solving the energy consumption barrier... 209 brine flow loss or leak (l): is provided by isobaric erds manufactures as a function of temperature and brine input flow per isobaric erds unit [40]: l = qic −qof qic (5.4) therefore the output flow rate from the side of feed membranes qof can be expressed as : qof = qic.(1− l) (5.5) and qif : qif = (of + 1).qof (5.6) finally by using flow balance: qic +qif = qoc +qof (5.7) the output flow rate from erd to the rejection can be expressed as : qoc = qic +qif −qof (5.8) 5.2. salinity calculation after defining the inlet and outlet flows of the erd system, the salinity of the flows in the erd sides can be determined. cicis the salinity of the concentrate of the ro process cc and cif is the salinity of feed water with low-pressure flow which is equal to the salinity cf . it remains to determine the concentration of outflows from the erd system coc and cof which are respectively the salinity of reject flow and the salinity of pressured flow by the erd. each isobaric erd has its own mixing mx ratio [41] which is the ratio of the volume of brine that transfers into a volume feed water and can be calculated with the following equation independent of the pressure exchanger high and low pressure flow balance: mx = cof − cif cic − cif (5.9) using the equation (5.9) cof can be expressed as a function of the previously known parameters: cof = mx(cic − cif ) + cif (5.10) according to the salinity balance : qic.cic +qif .cif = qoc.coc +qof .cof (5.11) coc can be expressed as: cic = (qic.cic +qif .cif )−qof qoc (5.12) 5.3. pressure calculation the erd’s input pressures pic and pif are respectively, the one of the from the concentrate flow rate of the ro process pc, and the second is the pressure of low pressure flow rate coming from the low pressure pump, which is generally the circulation water pressure taken at approximately 2 bar. moreover, the erd outlet pressure may be estimated as follows: the output pressure recovered by erd pof : pof = pic −∆p1 (5.13) the pressure of the discharge from the erd poc: poc = pif −∆p2 (5.14) where : ∆p1 and ∆p2 are the losses across the isobaric erd. copyright © 2024 assa. adv syst sci appl (2024) 210 m. moumni, m. el aoud mohamed, m. fatima zahra 6. booster pump a multi-stage brackish system without an interstage boost can be designed in a similar manner to a single-stage system. in this scenario, the previous stage’s concentrate is used to pressurize a feedwater stream for the first stage. the circulation pump compensates for pressure losses in the membrane stages, pipework, and erd. because the brackish and seawater ro processes are not the same, the implementation of isobaric erds in brackish water ro systems must be handled differently. the small amount of pressure loss caused by membranes, friction in the erd, and the piping circuit necessitates the use of a booster pump in isobaric erd. this pump is used as the erd’s output in single-stage seawater systems. the erd booster pump, on the other hand, can play two important roles in a two-stage brackish water system by being installed between stages one and two. the erd booster pump acts as an inter-stage booster pump in this configuration, lowering the required pressure from the main high-pressure feed pump and balancing flux between stages 1 and 2 [35]. in this study, the booster pump will function as a circulation pump to recover pressure lost in the circuit (membranes, erd, and pipes). 7. the sec of bwro with erd in order to evaluate the sec by a bwro equiped with a erd, firstly it is necessary to express the energy recovered in the concentrate part of the process, which is function of the ro brine pressure pc , flow rate of the concentrate qc, and the erd efficiency ηerd as shown : eerd = pc.qc.ηerd (7.1) by introducing a booster pump (bp) with efficiency ηbp , the energy consumed by this pump must be considered in order to evaluate the energy saving envisaged in the new design. in fact, this pump takes the water flow qof coming from the erd with the pressure pof in order to achieve the required pressure at the ro process inletpf , therfore, the energy consumed by(bop) can be expressed as follows: ebp = (pf − pof )qof ηbp (7.2) the pressure provided by the booster pump pbp : pbp = pf − pof (7.3) the sec by the bwro with a erd will be the combined energy consumed by the hp pump and the booster pump [?]: sec = (qf −qof ).pf ηhp .qp + pbp .qof ηbp .qp (7.4) 8. results and discussions the matlab/simulink model developed in the article [37] and validated by the real values of the studied station makes it possible to evaluate the sec by the process ro and especially the hp pumps. furthermore, the values considered in the above study are taken at the start of exploitation of the said plant. however, after two years of operation, it was discovered that feed pressure values had increased for the same feedwater properties (salinity and temperature). this is due to membrane fouling and its life span. following the development copyright © 2024 assa. adv syst sci appl (2024) solving the energy consumption barrier... 211 of the operating parameters of the ro process in this plant, it was observed that feed pressure and concentrate pressure rise by 2 to 3 bars. as a result, the sec increases, proving the significance of the current study. using the erd modeling equations and the ro model developed in matlab/simulink, the erd has been implemented into this model while taking into account the inclusion of a booster pump, which is essential for an ro process with an erd. the proposed design is illustrated in figure 8.3. then, the specific energy consumption (sec) of the three reverse osmosis (ro) trains implemented in the plant under study is evaluated by considering various states of the saltwater feed, specifically by varying its salinity and temperature. these two parameters have a direct impact on the sec of an ro process, as indicated in several studies ( [6,20,21]), regardless of the system’s design and structure. the graphs below illustrate the evolution of sec for the ro process in this plant with and without energy recovery device (erd). when the feed water salinity is set at 1.85 g/l and its temperature is varied within the recorded range at the station, the sec of the ro process with erd decreases from 2.1 kwh/m³ to 1.45 kwh/m³ as the feed water temperature increases (figure 8.1). in contrast, the sec for the current design without erd decreases from 3.05 kwh/m³ to 2.15 kwh/m³ under the same conditions (figure 8.2). in the same way, the feed water temperature was set to 22°c, and the salinity was adjusted within the station’s operating range. the sec with an erd ranges from 1.55 kwh/m3 to 1.72 kwh/m3 (figure 8.5), while the sec without an erd ranges from 2.27 kwh/m3 to 2.53 kwh/m3. implementing the erd at this station can reduce the sec by an average of 30%, even with the installation of a booster pump, according to the study’s findings. in addition, the sec formula (7.4) takes into account the energy used by this pump. to evaluate the sec fig. 8.1. sec with t(with erd) fig. 8.2. sec with t(without erd) table 8.1. algorithm genetic parameters parameter number of variables population size mutation migration crossover iteration value 6 50 for five 0,71 0,41 0,7 155 definition of variables xi x1 x2 x3 x4 x5 x6 variable y l qif of qf qp lower value lb 0.7 0.003 136 0 490 370 upper value ub 0.75 0.013 162 0.05 550 400 copyright © 2024 assa. adv syst sci appl (2024) 212 m. moumni, m. el aoud mohamed, m. fatima zahra fig. 8.3. the proposed design provided by the ro process in the three production lines, several salt water scenarios that closely resemble reality were captured and then simulated on the matlab platform. it is clear that the inclusion of an erd has an important effect on energy saving, and according to the 3d multivariate representation (figure 8.6 and figure 8.7), it can be observed that the temperature variation has a significant effect on the sec more than the salinity variation fig. 8.4. sec with cf (without erd) fig. 8.5. sec with cf (with erd) table 8.2. results of genetic algorithm pressure (bar) 10 14 17 20 22 25 sec(kwh/m3) 1,07 1,45 1,73 2,01 2,29 2,48 y 0,7 0,7 0,7 0,7 0,7 0,7 l 0,03 0,03 0,03 0,03 0,03 0,03 qif (m3/h) 136,64 136,92 136,16 136,66 136,99 136,65 of (m3/h) 0,04 0,04 0,04 0,04 0,04 0,04 qf (m3/h) 549,54 549,88 549,85 549,45 549,99 549,58 qp(m3/h) 412,99 412,95 412,68 412,78 412,99 412,93 copyright © 2024 assa. adv syst sci appl (2024) solving the energy consumption barrier... 213 fig. 8.6. sec without erd fig. 8.7. sec with erd fig. 8.8. sec compared to pf fig. 8.9. energy saving and qp because its range of variation in this station is limited. as concluded in various studies, including the article [37], changes in saltwater parameters (temperature and salinity) cause variations in the feed pressure. the relationship between sec and feed pressure is illustrated in figure 8.8, which shows that sec increases with rising feed pressure. figure 8.9 depicts the energy savings, defined as the difference in sec between the ro process with erd and the process without erd. the curve indicates that energy savings increase with the produced flow (permeate flow). the reduction rate of sec with the erd remains approximately 30%, regardless of changes in permeate flow. therefore, deploying the erd is beneficial for ro stations with high energy consumption, especially for saline water within the same variable range of salinity and temperature. this advantage can be further supported by a techno-economic analysis. by incorporating the genetic algorithm-based optimization approach, significant improvements in energy efficiency can be achieved for the ro process in water desalination plants. the observed 30% reduction in the sec by adding an erd in the ro process highlights the potential benefits of using this approach. additionally, the genetic algorithmbased optimization can identify the most suitable erd configuration based on the specific parameters of the ro process, leading to further improvement in energy efficiency. the copyright © 2024 assa. adv syst sci appl (2024) 214 m. moumni, m. el aoud mohamed, m. fatima zahra fig. 8.10. best individual combination of erd deployment and genetic algorithm-based optimization could provide a more sustainable and cost-effective approach to water desalination. in order to optimize the specific energy consumption (sec) of the ro process using a genetic algorithm, the formula (7.4)for sec it’s developed as a function of the parameters that contribute to the choice of the erd(overflush: of and leak: l). the equation can be expressed as follows: sec = (1− (1− y )(1− l)).pf .qf ηhp .qp + pbp .qif (of + 1).ηbp .qp (8.1) the optimal values of the sec were obtained by using the genetic algorithm on the given formula, considering the feed pressure values ranging from 7 to 25 bar. it is important to note that the feed pressure values were calculated based on the salinity and temperature of the water. the parameters of the applied genetic algorithm are detailed in table 8.1 and the obtained optimization results are presented in table 8.2. upon analyzing the values obtained through the genetic algorithm, it can be concluded that the specific energy consumption (sec) of the implemented ro process in the studied station ranges from 0,73 to 2,3. this means that, at the maximum feed pressure values considered in this station, energy consumption can be reduced by over 40%. table 8.2 summarizes the optimal values of the variables xi required for the operation of the ro process in this station, taking into account the parameters that characterize the erd. for the process to function optimally, an appropriate erd should have an overflush value of 0,04 and a leak value of 0,03. the figure 8.10 shows that variables x3,x5 and x6 are the parameters that have the most influence on energy optimization in this ro process. 9. conclusions the analysis of specific energy consumption (sec) in the moroccan plant’s reverse osmosis desalination process for brackish water reveals significant potential for energy savings. the study shows that by using a genetic algorithm to optimize design and operational parameters, the incorporation of energy recovery devices can reduce sec by up to 30%. these findings highlight the significance of energy optimization in increasing the feasibility and sustainability of reverse osmosis desalination. implementing these technologies can make the process more cost-effective and environmentally friendly, encouraging widespread copyright © 2024 assa. adv syst sci appl (2024) solving the energy consumption barrier... 215 adoption in water-scarce regions. future research will focus on implementing intelligent control systems for the operating parameters to ensure consistent production under optimal conditions. this approach aims to maintain the desired output quality while adapting to varying feed water characteristics, thus ensuring the process remains efficient and reliable. 10. abbreviations [erd] energy recoverey device [ro] reverse osmosis [hp] high pressure [bw] brackish water [bwro] brackish water reverse osmosis [swro] sea water reverse osmosis [hpp] high pressure pump [bp] booster pump [lpp] low pressure pump [vfd] variable frequency drive [sec] specific energy consumed (kwh/m3) [pf ] feed pressure (bar) [pc] concentrate pressure (bar) [pp] permeat pressure (bar) [qf ] feed flow rate (m3/h) [qc] concentrate flow rate (m3/h) [qp] permeat flow rate (m3/h) [cf ] feed water salinity (g/l) [cc] concentrate salinity (g/l) [cp] permeat salinity (g/l) [t ] temperature (c°) [es] specific energy consumed without erd (kwh/m3) [ec] energy lost on the rejection (kwh) [eerd] energy recovered by erd (kwh) [ebp ] energy consumed by bp (kwh) [ηhp ] hpp efficiency [ηbp ] bp efficiency [ηerd] erd efficiency [pif ] erd input pressure from lpp (bar) [pic ] erd input pressure from the concentrate (bar) [pof ] erd output pressure from erd to bp (bar) [poc] erd output pressure from erd to the rejection (bar) [qif ] erd input flow rate from lpp (m3/h) [qic] erd input flow rate from the concentrate (m3/h) [qof ] erd output flow rate from erd to bp (m3/h) [qoc] erd output flow rate from erd to the rejection (m3/h) [cif ] erd input salinity of flow rate from lpp (g/l) [cic] erd input salinity of flow rate from the concentrate (g/l) [cof ] erd output salinity of flow rate from erd to bp(g/l) [coc] erd output salinity of flow rate from erd to the rejection (g/l) [l] erd flow loss [of ] erd over flush range copyright © 2024 assa. adv syst sci appl (2024) 216 m. moumni, m. el aoud mohamed, m. fatima zahra [mx] mixing ratio [∗st ] station values [∗ ∗mb] matlab/simulink values [er] margin of error between st and mb [i] number of ions dissociated in the case of an electrolyte [r] ideal gas constant r = 8.314 (j .mol−1.k−1) [s] surface of the membrane (m2) references 1. shah abedi, m., hashemi, s. h. & fazeli, m. 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(2020). design, selection and application of energy recovery device in seawater desalination: a review, energies, 13, 4150. doi:10.3390/en13164150 copyright © 2024 assa. adv syst sci appl (2024) introduction description of bwro desalination plant materiels and methods evaluation of sec at bwro modeling erd flow calculation salinity calculation pressure calculation booster pump the sec of bwro with erd results and discussions conclusions abbreviations microsoft word 1077 article text, copyedited.docx adv syst sci appl 2021; 02; 117-132 published online at https://ijassa.ipu.ru. mathematical modeling of color printing devices: color gamut visualization and colorimetric measurements regularization dmitry v. tunitsky* v.a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia e-mail: dtunitsky@yahoo.com abstract: the article concerns a piecewise linear modeling of a vast range of color printing devices: various printers of different kind and nature, presses, etc. the central purpose is not a presentation of ready to use computational algorithms, or, moreover, accomplished software solutions. the point is to reveal some aspects of mathematical modeling, which could serve as both a guideline for creation of practical and robust engineer solutions and an example of nontrivial application of piecewise linear topology. keywords: colorant space, reference space, piecewise linear model, singular faces, proper printing device, non-degenerate printing device, regular printing device 1. preliminaries consider a stimuli space ℝ! of m colorants and a reference space ℝ" of n colors. in color management practice, the dimension of reference space n is always equal to three, n = 3, since any color can be described exactly by three real numbers who are the coordinates of this color in some color space, e.g., xyz, lab, etc. the dimension m of stimuli space can vary in accordance with the number of colorants the considered printing device uses. this dimension is called a dimension of a printing device. in most practical cases the dimension of printing device is either three, m = 3, or four, m = 4. in the first case we deal with a three-dimensional printing device and in the second case we deal with a four-dimensional printing device. not all the combinations (x1,…,xm) of real numbers represent combination of colorants, because the amount xk, 1£k£m, of the k-th colorant is measured in percents and can vary from zero percents (do not put the colorant at all) to one hundred percents (put as much colorant as possible). a printing device renders a combination (x1,…,xm) of colorants into the corresponding color (y1, y2, y3). it means that a printing device can be described by a map f: wm ® ℝ#, where wm = {( x1, … , xm), 0£ x1, … , xm£100} is the m-dimensional colorants cube in the stimuli space ℝ! and f(x1, … , xm) = (y1, y2, y3) is the point, whose coordinates correspond to the color rendered by mixture of the colorants combination. we will call such a map a printing device. to find the value of this function we should with the printing device under consideration reproduce on a sheet of paper the patch formed by the colorants combination (x1,…,xm) and measure the color coordinates (y1, y2, y3) of this patch by a colorimetric measurement tool. the number of colorants combinations (x1,…,xm) in the colorants cube is infinite and of cause it’s impossible to print and measure all of them, since actually we can print and measure only the finite subset of such combinations. consider a finite set {wi} ì wm, i.e., a * corresponding author: dtunitsky@yahoo.com mw 118 d.v. tunitsky copyright ©2021 assa. adv. in systems science and appl. (2021) mesh of fixed points wi, i=1,…,m, inside the colorants cube and a set {pi} î ℝ# = {( y1, y2, y3), ¥ < y1, y2, y3 <+ ¥} of the corresponding values pi, f(wi) = pi, i=1,…,m, in three-dimensional color space. call this mesh function a measurement data for the printing device under consideration. hence, a measurement data is a discrete map f: {wi} ® {pi} such that f(wi) = pi = f(wi) for i=1,…,m. for simplicity, we will restrict ourselves to the case of a regular mesh only. recall its definition. definition. let wm = [a1, b1] ´ … ´ [am, bm] be an m-dimensional rectangular parallelepiped with a1 < b1, …, am < bm. for k = 1,…,m consider finite sets zk = {xk0 , …, xkm(k)}, ak = xk0 < … < xkm(k) = bk, of m(k)+1 real numbers. the product mesh {wi} = z1 ´ … ´ zm ì wm of m points, m = ( m(1)+1)…(m(m)+1), is called regular inside the m-dimensional parallelepiped . there are two principal problems of mathematical modeling of a printing device. direct problem. to find a continuous map f: wm ® ℝ# being a satisfactory approximation to a given discrete function f, i.e., to the measurement data of the printing device under consideration. we will call the solution of this problem a model of the printing device or for conciseness just a printing device described by the corresponding measurement data (see section 1). inverse problem. to find a continuous map g: f(wm) ® wm, being an inverse map to f, i.e., the composition of the maps g and f should be the identical map of the set f(wm), . the problem, naturally, involves description of the gamut, i.e., the image f(wm) to the map f. moreover, in case m>3 an inverse map should satisfy to some additional conditions, for example, the maximal (minimal) possible amount of the m-th colorant component xm. we will restrict our examination of these problems to the class of piecewise linear maps. remind the necessary notions. a k-dimensional simplex is a convex hull of k+1 affinely independent points of an m-dimensional space ℝ! , where 0£k£m. the boundary of a kdimensional simplex consists of faces with different dimensions: k-1, k-2, …, 0. onedimensional faces are called edges, and zero-dimensional faces are called vertices. definition. suppose the colorants cube wm is decomposed into a union of n, n>0, sets dj, . this decomposition is called simplex if for j=1,…, n the set dj, is an m-dimensional simplex and the intersection of any two simplexes dj and dk is either empty, dj ç dk = æ, or is a union of whole (m–1)-, (m–2)-, …, and 0-dimentional faces to these simplexes. definition. a continuous map f: wm ® ℝ#, is called piecewise linear if there exists a simplex decomposition of the mdimensional colorant cube such that all the restrictions f½dj : dj ® ℝ# of the map f to tetrahedrons dj are linear maps. in other words, f½dj (x) = cj + bjx, where x = ( x1, … , xm)t mw mw )( mwfidgf =! ! nj j mw ,...,1= d= ! nj j mw ,...,1= d= mw mathematical modeling of color printing devices: color gamut visualization… 119 copyright ©2021 assa. adv. in systems science and appl. (2021) is an m-dimensional vector of colorants combination, bj is a 3´m matrix, and cj is a threedimensional vector, cj î ℝ#, for j=1,…, n (cf. [1, sections 1.4 and 2.3]. remark. generally speaking, for real applications the formulated above notions of simplex decomposition of the colorants cube wm and its piecewise linear map is not sufficient. indeed, rather often a colorants limitations, for example, , where , should be applied to the colorants cube . such limitations have practical sense. in particular, they allow to avoid putting too much of colorants to a sheet of paper. moreover, sometimes several color limitations should be applied to the colorants cube. as a result in general case we will get not a cube decomposed into simplexes but a convex polyhedron decomposed into intact and truncated simplexes. nevertheless all the constructions below actually are through in this general case. therefore the reader, especially the one who is mostly interested in ideas rather than in technical details, can easily content himself with the case of cubes and simplexes only because in general case all the same ideas and methods are practiced. 2. direct problem for three-dimensional printing devices firstly examine a problem of approximation for a three-dimensional printing device. let a finite set {wi} ì w3 of points wi, i=1,…, m, inside the colorants cube w3 be a regular mesh. consider a discrete map f: {wi} ® {pi}, of measurement data, where pi = f(wi) = f(wi) for i=1,…, n. to approximate the given discrete map f by a continuous one f: w3 ® ℝ#, use a piecewise linear or, what is the same in three-dimensional case, a tetrahedral interpolation. we will mostly follow the text-book [2] in our constructions. by definition of a regular mesh, for k=1,2,3 there exist the one-dimensional meshes zk = {xk0, … , xkm(k)}, ak = xk0 < … < xkm(k) = bk, of m(k)+1 real numbers such that {wi} = z1 ´ z2 ´ z3 ì w3 and m = (m(1)+1)(m(2)+1)(m(3)+1). it means that the three-dimensional colorants cube w3 can be decomposed into the union of the mesh parallelepiped cells pi,j,k = [x1i-1, x1i] ´ [x2j-1, x2j] ´ [x3k-1, x3k], i=1,…,m(1), j=1,…,m(2), k=1,…,m(3). inside each of these parallelepiped cells the continuous approximation f of the measurement discrete map f is constructed by the following way. consider an arbitrary three-dimensional rectangular parallelepiped p = [a1, b1] ´ [a2, b2] ´ [a3, b3] = {(x1, x2, x3), a1£ x1£ b1, a2£ x2£ b2, a3£ x3£ b3}, where a1, b1, a2, b2, a3, and b3 are fixed real numbers. there is an obvious one-to-one correspondence of the 8 vertices to the rectangular parallelepiped p and the 8 vertices (0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), (1,1,1) to the unit three-dimensional cube p1 = {(x1, x2, x3), 0£ x1£ 1, 0£ x2£1, 0£ x3£ 1}. cx m i i £å =1 mc 100< mw ! )3(,...,1 )2(,...,1 )1(,...,1 ,, 3 mk mj mi kjiw = = = p= 120 d.v. tunitsky copyright ©2021 assa. adv. in systems science and appl. (2021) numerate all the 8 vertices of the rectangular parallelepiped p by means of the corresponding vertices of the unit cube p1, x000, x001, x010, x011, x100, x101, x110, x111. apply the same numeration to the values of the discrete map f, i.e., put pijl = f(xijl) for i,j,l=0,1. define the map f inside the rectangular parallelepiped p, yn = fn(x1, x2, x3) = pn000 + rn1dx1 + rn2dx2 + rn3dx3, where n, n=1,2,3, is the number of component of the map f in three-dimensional color space ℝ# and dxi = (xi xi0)/(xi1 xi0) for i=1,2,3. the coefficients rni, i=1,2,3, are determined in correspondence with the following table (cf. [2, p. 70–72]). table 1. coefficients rni, i=1,2,3 the interpolation under consideration has pure geometrical sense. we decompose a threedimensional rectangular parallelepiped into six tetrahedrons. these tetrahedrons are defined by the conditions in the second column of the table above. inside each tetrahedron the map f is constructed by linear interpolation of the values pijl, i,j,l=0,1, of the discrete map f at the vertices to the tetrahedrons. thus we have constructed the piecewise linear map f, being a solution to the direct problem of mathematical modeling for a three-dimensional printing device. in other words, we have constructed a piecewise linear model of a three-dimensional printing device. 3. direct problem for four-dimensional printing devices now consider a four-dimensional printing device, which is modelled in a way similar to the three-dimensional case. let a finite set {wi} ì w4 of points wi, i=1,…,m, of the colorants cube w4 be a regular mesh. the measurement data pi = f(wi) = f(wi), i=1,…, m, determines a discrete map f: {wi} ® {pi}, to approximate the discrete map f by a continuous map f: w4 ® ℝ#, use a piecewise linear or, which in four-dimensional case is the same, pentahedral interpolation. by definition of a regular mesh, for k=1,2,3,4 there exist the one-dimensional meshes zk = {xk0 ,…, xkm(k)}, ak = xk0 < … < xkm(k) = bk, of m(k)+1 real numbers such that {wi} = z1 ´ z2 ´ z3 ´ z4 ì w4 and m = (m(1)+1)(m(2)+1)(m(3)+1)(m(4)+1). it means that the fourdimensional colorants cube w4 can be decomposed into the union of the mesh of parallelepiped cells: , pi,j,k = [x1i-1, x1i] ´ [x2j-1, x2j] ´ [x3k-1, x3k] ´ [x4k-1, x4k]. inside each parallelepiped cell pi,j,k the continuous approximation f of the discrete map f is constructed by the following way. no conditions rn1 rn2 rn3 1 dx1 ³ dx2 ³ dx3 pn100pn000 pn110pn100 pn111pn110 2 dx1 ³ d x3 ³ dx2 pn100pn000 pn111pn101 pn101pn100 3 dx3 ³ dx1 ³ dx2 pn101pn001 pn111pn101 pn001pn000 4 dx2 ³ dx1 ³ dx3 pn110pn010 pn010pn000 pn111pn110 5 dx2 ³ dx3 ³ dx1 pn111pn011 pn010pn000 pn011pn010 6 dx3 ³ dx2 ³ dx1 pn111pn011 pn011pn001 pn001pn000 ! )4(,...,1 )3(,...,1 )2(,...,1 )1(,...,1 ,,, 3 ml mk mj mi lkjiw = = = = p= mathematical modeling of color printing devices: color gamut visualization… 121 copyright ©2021 assa. adv. in systems science and appl. (2021) consider an arbitrary four-dimensional rectangular parallelepiped p = {(x1, x2, x3, x4), a1£ x1£ b1, a2£ x2£ b2, a3£ x3£ b3, a4£ x4£ b4}. there is an obvious one-to-one correspondence between the vertices of the parallelepiped p and the vertices (0,0,0,0), (0,0,0,1), (0,0,1,0), (0,0,1,1), (0,1,0,0), (0,1,0,1), (0,1,1,0), (0,1,1,1), (1,0,0,0), (1,0,0,1), (1,0,1,0), (1,0,1,1), (1,1,0,0), (1,1,0,1), (1,1,1,0), (1,1,1,1) of the unit four-dimensional cube p1 = {(x1, x2, x3, x4), 0£ x1£1, 0£ x2£1, 0£ x3£1, 0£ x4£1}. numerate the vertices of the rectangular parallelepiped p by the corresponding vertices of the cube p1: x0000, x0001, x0010, x0011, x0100, x0101, x0110, x0111,x1000, x1001, x1010, x1011, x1100, x1101, x1110, x1111. apply this numeration also to the values of the map f, i.e., put pijkl = f(xijkl) for i,j,k,l=0,1, and define the components of the map f on the parallelepiped p: yn = fn(x1, x2, x3, x4) = pn000 + rn1d x1 + rn2d x2 + rn3d x3 + rn4d x4, where n=1,2,3, dxi = (xi xi0)/( xi1 xi0) for i=1,2,3,4, and rni are determined by the table: table 2. coefficients rni, i=1,2,3,4 no conditions rn1 rn2 rn3 rn4 1 dx1 ³ dx2 ³ dx3 ³ dx4 pn1000-pn0000 pn1100-pn1000 pn1110-pn1100 pn1111-pn1110 2 dx1 ³ dx2 ³ dx4 ³ dx3 pn1000-pn0000 pn1100-pn1000 pn1111-pn1101 pn1101-pn1100 3 dx1 ³ dx4 ³ dx2 ³ dx3 pn1000-pn0000 pn1101-pn1001 pn1111-pn1101 pn1001-pn1000 4 dx4 ³ dx1 ³ dx2 ³ dx3 pn1001-pn0001 pn1101-pn1001 pn1111-pn1101 pn0001-pn0000 5 dx1 ³ dx3 ³ dx2 ³ dx4 pn1000-pn0000 pn1110-pn1010 pn1010-pn1000 pn1111-pn1110 6 dx1 ³ dx3 ³ dx4 ³ dx2 pn1000-pn0000 pn1111-pn1011 pn1010-pn1000 pn1011-pn1010 7 dx1 ³ dx4 ³ dx3 ³ dx2 pn1000-pn0000 pn1111-pn1011 pn1011-pn1001 pn1001-pn1000 8 dx4 ³ dx1 ³ dx3 ³ dx2 pn1001-pn0001 pn1111-pn1011 pn1011-pn1001 pn0001-pn0000 9 dx3 ³ dx1 ³ dx2 ³ dx4 pn1010-pn0010 pn1110-pn1010 pn0010-pn0000 pn1111-pn1110 10 dx3 ³ dx1 ³ dx4 ³ dx2 pn1010-pn0010 pn1111-pn1011 pn0010-pn0000 pn1011-pn1010 11 dx3 ³ dx4 ³ dx1 ³ dx2 pn1011-pn0011 pn1111-pn1011 pn0010-pn0000 pn0011-pn0010 12 dx4 ³ dx3 ³ dx1 ³ dx2 pn1011-pn0011 pn1111-pn1011 pn0011-pn0001 pn0001-pn0000 13 dx2 ³ dx1 ³ dx3 ³ dx4 pn1100-pn0100 pn0100-pn0000 pn1110-pn1100 pn1111-pn1110 14 dx2 ³ dx1 ³dx4 ³ dx3 pn1100-pn0100 pn0100-pn0000 pn1111-pn1101 pn1101-pn1100 15 dx2 ³ dx4 ³ dx1 ³ dx3 pn1101-pn0101 pn0100-pn0000 pn1111-pn1101 pn0101-pn0100 16 dx4 ³ dx2 ³ dx1 ³ dx3 pn1101-pn0101 pn0101-pn0001 pn1111-pn1101 pn0001-pn0000 17 dx2 ³ dx3 ³ dx1 ³ dx4 pn1110-pn0110 pn0100-pn0000 pn0110-pn0100 pn1111-pn1110 18 dx2 ³ dx3 ³ dx4 ³ dx1 pn1111-pn0111 pn0100-pn0000 pn0110-pn0100 pn0111-pn0110 19 dx2 ³ dx4 ³ dx3 ³ dx1 pn1111-pn0111 pn0100-pn0000 pn0111-pn0101 pn0101-pn0100 20 dx4 ³ dx2 ³ dx3 ³ dx1 pn1111-pn0111 pn0101-pn0001 pn0111-pn0101 pn0001-pn0000 21 dx3 ³ dx2 ³ dx1 ³ dx4 pn1110-pn0110 pn0110-pn0010 pn0010-pn0000 pn1111-pn1110 22 dx3 ³ dx2 ³ dx4 ³ dx1 pn1111-pn0111 pn0110-pn0010 pn0010-pn0000 pn0111-pn0110 23 dx3 ³ dx4 ³ dx2 ³ dx1 pn1111-pn0111 pn0111-pn0011 pn0010-pn0000 pn0011-pn0010 24 dx4 ³ dx3 ³ dx2 ³ dx1 pn1111-pn0111 pn0111-pn0011 pn0011-pn0001 pn0001-pn0000 the interpolation under consideration has pure geometrical sense. we decompose a fourdimensional rectangular parallelepiped into 24 pentahedrons. these pentahedrons are defined by the conditions in the second column of the table above. inside each tetrahedron the map f is constructed by linear interpolation of the values pikjl, i,j,k,l=0,1, of the discrete map f at the vertices to the pentahedrons. 122 d.v. tunitsky copyright ©2021 assa. adv. in systems science and appl. (2021) thus we have constructed the piecewise linear map f, being a solution to the direct problem of mathematical modeling for a four-dimensional printing device. in other words, we have constructed a piecewise linear model of a four-dimensional printing device. 4. gamut description for three-dimensional printing devices consider a piecewise linear model f: w3 ® ℝ# of a given three-dimensional printing device. for practical applications it is important to find in color space the gamut of the printing device, i.e., the image f(w3) of the piecewise linear map f. in particular, we need it to go on with the inverse problem. by definition of a piecewise linear map f, we have the simplex decomposition of the three-dimensional colorants cube w3 into the union of n, n>0, tetrahedrons dj: . each tetrahedron has four two-dimensional faces. these faces are triangles and each triangle belongs either to one or several tetrahedrons of the set {dj}. in a standard way the faces of these tetrahedrons can be divided into two classes. definition. fix a tetrahedron dl, l=1,…,n, and consider it’s two-dimensional face d, which is a triangle. the face d is called boundary if it doesn’t belong to any other tetrahedron of the set {dj}. in other words, d ì dl, and d ë dk for k=1,…,l-1,l+1,…,n. the face d is called internal if there is a tetrahedron dk from the set {dj} such that d belongs to both dl and dk, i.e., d í dl ç dk. denote the set of all the boundary faces of the colorants cube w3 by q. remark. the set q of all the boundary faces doesn’t depend on the choice of the threedimensional printing device, i.e., on the choice of the corresponding piecewise linear map f. indeed, the union of all these faces always coincides with the boundary ¶w3 of the threedimensional colorants cube: . suppose the printing device under consideration is non-degenerate, i.e., the corresponding piecewise linear map f is non-degenerate. by definition, it means that all the restrictions f½dj : dj ® ℝ#, of the map f to tetrahedrons dj are non-degenerate linear maps f½dj (x) = cj + bjx (see section 1). in other words, the determinant of the corresponding matrix bj is either positive, det bj > 0, or negative, det bj < 0. definition. fix a number l, l=1,…,n, and consider two-dimensional internal face d of the tetrahedron dl, which is a triangle. the internal face d is called singular if there exists a tetrahedron dk from the set {dj} such that d belongs to both dl and dk, d í dl ç dk, and the determinants of the corresponding matrixes bl and bk have different signs, i.e., det bl × det bk < 0. denote the set of all the singular faces of the given three-dimensional printing device by s. remark. on the contrary to the set q of all the boundary faces, the set s of all the singular faces essentially depends on the choice of the three-dimensional printing device, i.e., on the choice of the corresponding piecewise linear map f. for example, for some printing devices this set is empty and for some it is not (see section 6). it is possible to describe the gamut boundary of a non-degenerate printing device in terms of boundary and singular faces. the following theorem is through. theorem. for any non-degenerate three-dimensional printing device the boundary of the gamut is a subset of the images of all the boundary and singular faces, i.e., ! nj jw ,...,1 3 = d= 3w¶= qî ! d d mathematical modeling of color printing devices: color gamut visualization… 123 copyright ©2021 assa. adv. in systems science and appl. (2021) ¶f(w3) í f(q) è f(s). this theorem can seem to be a pure abstract mathematical proposition. and, of course, it really is. but nevertheless, it does significantly more because it gives a strict mathematical ground for creation of a wide range of computational algorithms for practical approximation of color gamut boundaries of various three-dimensional printing devices. in particular, the author designed a variant of such an algorithm, which constructs a three-dimensional gamut boundary for any real color printing device with three colorants. as an input it takes a file with colorimetric measurements data of the printing device under consideration (see section 1), and as an output it constructs the color gamut boundary, i.e., the corresponding boundary of the three-dimensional body in a reference space of three colors. as it was said in the preface, the same details of computational algorithms are not presented in this article. therefore, we will just give an example of what the designed algorithm does in practice. the picture below visualizes a piecewise linear approximation of the gamut boundary of a rather standard printing device with three colorants, which means that the stimuli space is three-dimensional in this case. in reference space the lab coordinate system is chosen, and one colorants limitation is applied. fig. 1. a piecewise linear approximation of the gamut boundary for a three-dimensional printing device the next picture gives another example of the output of the same algorithm, which is a visualized color gamut boundary. here we handle exactly the same printing device as in the previous case but the colorants limitation is changed to the tighter one, namely to . 250 3 1 £å =i ix 150 3 1 £å =i ix 124 d.v. tunitsky copyright ©2021 assa. adv. in systems science and appl. (2021) the way of visualization is also exactly the same, including angle of view, the scale, and the coordinate system. fig. 2. results of the algorithm for the same printing device with tighter colorants limitation 5. gamut description for four-dimensional printing devices consider a piecewise linear model f: w4 ® ℝ#, of a four-dimensional printing device. definition. if the image of the boundary ¶w4 of the four-dimensional colorants cube coincides with the image of the whole cube w4, i.e., f(w4) = f(¶w4), then the fourdimensional printing device is called proper. remark. from technological point of view, the assumption of a four-dimensional printing device to be proper is reasonable for most real four-dimensional printing devices. by definition of the piecewise linear map f, all the restrictions f½dj : dj ® ℝ#, of the map f to pentahedrons dj are linear maps, i.e., f½dj (x) = cj + bjx, where bj is a 3´4 matrix for j=1,…,n. let bji be the 3´3 matrix obtained by throwing away the i-th column from the 3´4 matrix bj and put cj = ( det bj1, det bj2, det bj3, det bj4 ) for j=1,…, n. definition. consider the four-dimensional printing device corresponding to the piecewise linear map f. a vector field c on the colorants cube w4 is called the characteristic vector field of the printing device under consideration if c½dj = cj for j=1,…, n . the four-dimensional printing device is non-degenerate if the corresponding characteristic vector field c is nondegenerate, i.e., cj ¹ 0 for all j=1,…,n. mathematical modeling of color printing devices: color gamut visualization… 125 copyright ©2021 assa. adv. in systems science and appl. (2021) remark. by definition, the characteristic vector field of any four-dimensional printing device is a four-dimensional piecewise constant vector field on the four-dimensional colorants cube w4. in this section we will describe the gamut of a proper non-degenerate four-dimensional printing device, i.e., the image f(w4) of the corresponding piecewise linear map f in color space. by definition of a piecewise linear map, we have the simplex decomposition of the fourdimensional colorants cube w4 into the union of n, n>0, pentahedrons dj, . each pentahedron has five three-dimensional faces. these faces are tetrahedrons and each tetrahedron either belongs to one or several pentahedrons of the set {dj}. definition. fix a pentahedron dl, l=1,…,n, and consider it’s three-dimensional face, which is a tetrahedron d. the face d is called boundary if it doesn’t belong to any other pentahedron of the set {dj}. in other words, d ì dl, and d ë dk for k=1,…,l-1,l+1,…,n. denote the set of all the boundary faces of the colorants cube w4 by q. remark. the set q of all the boundary faces doesn’t depend on the choice of the fourdimensional printing device, i.e., on the choice of the corresponding piecewise linear map f. the union of all these faces always coincides with the boundary ¶w4 of the four-dimensional colorants cube, . on the boundary ¶w4 of the four-dimensional colorants cube w4 there exists the normal vector field n to this cube. let dj, j=1,…,n, be a boundary face of the four-dimensional colorants cube w4 belonging to the pentahedron d j. denote by nj the restriction of the normal vector field n to this face: nj = n½dj. let dk and dl be boundary faces of the four-dimensional colorants cube w4 such that dk ì dk, dl ì dl for some pentahedrons dk and dl, k,l=1,…,n. by definition, these boundary faces are tetrahedrons. suppose they have a two-dimensional face, which is a triangle d, in common, d = dk ç dl. definition. the triangle d is called a singular face of a non-degenerate four-dimensional printing device that is defined by the piecewise linear map f if the inner products (nk, ck) and (nl, cl) of the normal vector field n and the characteristic vector field c have different signs: (nk, ck) × (nl, cl) < 0. denote the set of all the singular faces of the given four-dimensional printing device by s. remark. the set q of all the boundary faces doesn’t depend on the choice of a printing device. on the contrary, the set s of all the singular faces essentially depends on the choice of a four-dimensional printing device, i.e., on the choice of the corresponding piecewise linear map f. moreover, a boundary face is a three-dimensional simplex, i.e., a tetrahedron, whereas a singular face is a two-dimensional simplex, i.e., a triangle. there is also a serious difference between threeand four-dimensional cases of printing devices. indeed, in three-dimensional case the boundary faces are two-dimensional and in four-dimensional case – four-dimensional. an other important difference is the following: for most three-dimensional printing devices the set s of all the singular faces is empty, while for any four-dimensional printing devices the set s of all the singular faces is not empty. it is possible to describe the gamut boundary of a proper non-degenerate printing device in terms of singular faces only. the following theorem is through. theorem. for any proper non-degenerate four-dimensional printing device the boundary of the gamut is a subset of the images of all the singular faces, i.e., ¶f(w4) í f(s). all the remarks that were made in the previous section after the theorem about the gamut boundaries of the three-dimensional printing devices are entirely correct in the case described by the theorem under consideration. in particular, a wide range of algorithms that approximate ! nj jw ,...,1 4 = d= 4w¶= qî ! d d 126 d.v. tunitsky copyright ©2021 assa. adv. in systems science and appl. (2021) the color gamut boundary of any real color printing device with four colorants can be designed grounded on it. as in the case of three-dimensional stimuli space an input for this algorithm is a file with colorimetric measurements data of the printing device under consideration (see the previous section). and as an output this algorithm produces piecewise linear approximation of the color gamut boundary, i.e., the corresponding three-dimensional body boundary in a reference space of three colors. the following picture gives an example of such approximation. it shows the color gamut boundary in reference space of a rather standard printing device with four colorants (cyan, magenta, yellow, and black), which means that the stimuli space is four-dimensional. to be definite, call the measurement file of this device cmyk.dat. in reference space the lab coordinate system chosen, and one colorants limitation is applied. fig. 3. the color gamut boundary for a four-dimensional printing device the next picture gives another example of the algorithm output. here we have exactly the same printing device as in the previous picture, which is described by measurement data from cmyk.dat file, but the colorants limitation is changed to a tighter one, namely to . the way of rendering is exactly the same, including angle of view, the scale, and the coordinate system. 250 4 1 £å =i ix 150 4 1 £å =i ix mathematical modeling of color printing devices: color gamut visualization… 127 copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 4. results of the algorithm for the same printing device with tighter colorants limitation 6. three-dimensional regular printing devices consider a piecewise linear model f: w3 ® ℝ# of a three-dimensional printing device. definition. the three-dimensional printing device is called regular if the piecewise linear map f is an injection. lemma. let a topological space w be compact and a map f, f: w ® f(w), be a continuous injection. then there exists the unique continuous inverse map g = f-1: f(w) ® w. in other words, then the map f is a homeomorphism. proof. see [3, ch. 2, § 8]. remark. since the three-dimensional cube w3 is a compact topological space the lemma under consideration gives a satisfactory approach to solution of the inverse problem of modeling of three-dimensional regular printing devices (see section 1). important to note, that most of the three-dimensional printing devices are regular though sometimes singular printers are met. 128 d.v. tunitsky copyright ©2021 assa. adv. in systems science and appl. (2021) by definition of a piecewise linear map, we have the simplex decomposition of the threedimensional colorants cube w3 into the set of n, n>0, tetrahedrons dj, , such that all the restrictions f½dj : dj ® ℝ#, of the map f to tetrahedrons dj are linear maps, i.e., f½dj (x) = cj + bjx, where bj is a 3´3 matrix, and x, cj are three-dimensional vectors for j=1,…, n. definition. a three-dimensional printing device is called strictly non-degenerate if all the determinates of the matrixes bj are of the same sign, i.e., det bj × det bj > 0 for all the indexes i,j=1,…,n. remark. by definition of a singular face (see section 4), a three-dimensional printing device is strongly non-degenerate if and only if the set s of all its singular faces is empty, s = æ. any three-dimensional strongly non-degenerate printing device is non-degenerate. the inverse statement is false because there exist three-dimensional non-degenerate printing devices that are not strongly non-degenerate. there is an effective criterion of a three-dimensional printing device to be regular. theorem. let f: w3 ® ℝ# be a piecewise linear model of a three-dimensional printing device. this printing device is regular if and only if it is strongly non-degenerate and the restriction f½¶w3 : ¶w3 ® ℝ# of the map f to the boundary ¶w3 of the three-dimensional colorants cube w3 is an injection. by this theorem, the necessary condition of a three-dimensional printing device to be regular is its strong non-degeneracy. by definition, it means that all the determinates of the matrixes bj of the piecewise linear map f have the same sign. describe the scheme of an algorithmic approach to forcing a three-dimensional printing device to become strictly nondegenerate. at the first step count the number n+ of positive determinants and the number nof negative determinants. for clarity, assume that n+ > n-. at the second step define a positive threshold e, e > 0, which is usually a small real number, and construct the error functional r, , where rj = rj(p1, … , pm) = 0 if det bj ≥ e, and rj = rj(p1, … , pm) = (e det bj)2 if det bj < e, j=1,…, n. here p1, … , pm are the three-dimensional vectors in color space, forming the measurement data of the three-dimensional printing device under consideration (see section 1). by construction of the direct problem solution, all the determinants det bj of the piecewise linear map f are third order polynomials with respect to measurement data p1,…, pm for j=1,…,n (see section 2). hence, all the functions rj are smooth for j=1,…,n and the error functional r = r(p1, … , pm) is smooth with respect to measurement data p1,…,pm too. at the third step minimize the error functional r with respect to measurement data p1,…,pm, i.e., r(p1, …, pm) ® min, by some minimization method. the resulting argument (p10, … , pm0) of the minimal value is the measurement data for regularized three-dimensional printing device. remark. there are m three-dimensional vectors in measurement data. therefore the total dimension of the space is 3m. thus we have a 3m-dimensional non-convex minimization problem. by construction, the error functional r is not convex and can have more than one minimal point. ! nj jw ,...,1 3 = d= å = == n j mjm pprpprr 1 11 ),...,(),...,( mathematical modeling of color printing devices: color gamut visualization… 129 copyright ©2021 assa. adv. in systems science and appl. (2021) 7. four-dimensional regular printing devices consider a piecewise linear model f: w4 ® ℝ# of a four-dimensional printing device. definition. the four-dimensional printing device is called regular if the following three conditions hold for the piecewise linear map f. (1) the color gamut f(w4) is homeomorphic to closed three-dimensional disk d3. (2) for any internal point p of the color gamut f(w4), p î int f(w4), the pre-image f-1(p) is homeomorphic to a segment [a, b], a<b, and the intersection f-1(p) ç ¶w4 of this pre-image and the boundary ¶w4 of the colorants cube w4 consists exactly of the two boundary points of the pre-image f-1(p). (3) for any boundary point p of the gamut f(w4), p î ¶f(w4), the pre-image f-1(p) consists of exactly one point. remark. if a four-dimensional printing device is regular then it is non-degenerate and proper. of cause, the inverse statement is false because there exist four-dimensional nondegenerate proper printing devices that are not regular. let c be the characteristic vector field of the four-dimensional printing device under consideration. by definition, it is a piecewise constant vector field such that c½dj = cj, where cj = ( det bj1, det bj2, det bj3, det bj4 ) for j=1,…,n (see section 5). definition. a four-dimensional printing device is called strictly non-degenerate if it is non-degenerate and at any point x of the four-dimensional colorants cube w4 all the four coordinates of the characteristic vector field c have the same sign. in other words, for all j=2,…,n the i-th coordinate cji = (-1)i+1det bji of the characteristic vector field c at the j-th simplex has the same sign as the i-th coordinate c1i = (-1)i+1det b1i of the characteristic vector field c at the first simplex for i=1,2,3,4. remark. it is possible to show that for a strictly non-degenerate four-dimensional printing device the set s of all its singular faces is homeomorphic to two-dimensional sphere s2. there is a sufficient condition of a four-dimensional printing device to be regular. theorem. suppose the four-dimensional printing device under consideration is strictly non-degenerate and the restriction f½s of the piecewise linear map f to the set s of all its singular faces is an injection. then this printing device is regular. describe the scheme of an algorithmic approach that allows forcing a four-dimensional printing device to become strictly non-degenerate. at the first step count the number ni+ of positive i-th coordinates and the number niof negative i-th coordinates of the characteristic vector field c, i=1,2,3,4. for clarity, assume that ni+ > nifor i=1,2,3 and n4+ < n4. at the second step define a positive threshold e, e > 0, which is usually a small real number, and construct the error functional r, . here rji = rji(p1, …, pm) = 0 if (-1)i+1det bji ≥ e and rj = rj(p1, … , pm) = (e det bj)2 if (-1)i+1det bji < e for i=1,2,3. for i=4 rj4 = rj4(p1, …, pm) = 0 if det bji ≥ e and rj = rj(p1,…, pm) = (e det bj)2 if det bji < e for j=1,…,n. in both cases p1, …, pm are the three-dimensional vectors in color space that form the measurement data of the four-dimensional printing device under consideration (see section 1). by construction of the piecewise linear map f, all the determinants det bji are the third order polynomials with respect to measurement data vectors åå = = == 4 1 1 11 ),...,(),...,( i n j m i jm pprpprr 130 d.v. tunitsky copyright ©2021 assa. adv. in systems science and appl. (2021) p1,…,pm for j=1,…,n and i=1,2,3,4 (see section 3). thence all the functions rji are smooth for j=1,…,n, i=1,2,3,4, and the error functional r = r(p1, …, pm) is smooth with respect to measurement data vectors p1,…,pm too. at the third step minimize the error functional r with respect to measurement data p1,…,pm, i.e., r(p1, …, pm) ® min, by some minimization method. the resulting argument (p10, … , pm0) of the minimal value is the measurement data for regularized four-dimensional printing device. remark. the dimension of the space of measurement data is 3m. thus we obtain a 3mdimensional non-convex minimization problem. by construction, the error functional r is not convex and can have more than one minimal point. consider the measurement file cmyk.dat that we have used in the example of section 5. direct calculations show that this printing device is not strictly non-degenerate. indeed, the table below indicates that some coordinates of the characteristic vector field are positive whereas some are negative: fig. 5. distribution for coordinates of the characteristic vector field from geometrical point of view, it means that we have too many singular faces that lead to formation of redundant faces or singularities inside the color gamut. visualization of these singularities can be seen in the picture below: fig. 6. singularities inside the color gamut mathematical modeling of color printing devices: color gamut visualization… 131 copyright ©2021 assa. adv. in systems science and appl. (2021) in particular, the singularities are the polyhedrons in white, magenta, yellow, and red. therefore, it makes sense to implement the described above algorithmic scheme and apply the resulting tool to the measurement data from cmyk.dat file. the author have implemented such a tool and successfully applied it to force the data to become strictly non-degenerate. the table below shows that after the work of this tool the data became really strictly non-degenerate: fig. 7. distribution for coordinates after applying the algorithm from geometrical point of view, it means that all the redundant singular faces have been removed and, therefore, no singularities exist anymore. visualization of color gamut of the printing device after forcing it to become strictly non-degenerated is given in the picture below: fig. 8. the color gamut without singularities after applying the algorithm indeed, no singularities are present anymore. 132 d.v. tunitsky copyright ©2021 assa. adv. in systems science and appl. (2021) acknowledgements this research was carried out with the support of the russian foundation for basic research under grant no. 20-01-00610 a and the russian scientific foundation under grant no. 19-1100223. references 1. rurk c.p., sanderson b.j. (1982) an introduction to piecewise linear topology. springer–verlag, berlin–new york. 2. kang h. (1997) color technology for electronic imaging devices. spie: bellingham, washington. 3. seifert h., trelfall w. (1980) a textbook in topology. algebraic press, new york. microsoft word 1080-source texts-5245-1-11-20220930.docx adv syst sci appl 2022; 03; 18-35 published online at https://ijassa.ipu.ru. models of industrial risk control systems mikhail geraskin1, elena rostova2* 1) samara national research university, samara, russia e-mail: innovation@ssau.ru 2) samara national research university, samara, russia e-mail: el_rostova@mail.ru abstract: we investigate a problem of searching for pareto equilibrium sets of an insurance rate and an industrial damage utilization price. we consider a system, which, in the case of industrial accidents, arises around an industrial firm. an industrial firm, a waste utilization firm, and an insurance company are considered as the system’s agents. we develop profit functions for the agents, and we determine compromise prices on waste utilization and insurance, which provide the system’s stability. we analyze the set of industrial risk control systems with a various number of the agents and the agent’s relations. a problem of determining an optimal solution is solved on the basis of maximizing agents’ profit functions. the sets of an equilibrium industrial damage utilization price and an equilibrium insurance rate are defined as pareto equilibrium. a problem of determining the set of an insurance rate is solved taking into account constraints according to requirements of an industrial firm and an insurance company. a problem of determining the set of an industrial damage utilization price is solved taking into account constraints according to requirements of an industrial firm and a waste utilization firm. we consider the following models of industrial risk control systems: agents have a strong relation and a weak relation, additionally, one agent of each type and of many agents of the same type. keywords: industrial risk, insurance, waste utilization, optimization, risk control 1. introduction the industrial risk control is an important problem for every firm because the influence of different external market factors. the risk control problems cover a financial risk, human errors, a non-fulfillment of contracts, an industrial risk, an environmental risk, etc. these problems were solved by means of the following methods: the scenario method [17], the multi-agent systems [1, 7, 19], the multi-criteria models [6, 27, 31]. additionally, this problem was analyzed on various levels: the world market risk [11], the regional economic system risk [20, 24, 26], the firm’s risk [2, 28], the technology operation risk [3, 8]. the risk control problems are related to various aspects, in particular, an assessment of the risk factors, a choice of the risk management method, a prediction of the damage, etc. rasmussen and svedung emphasized that «risk management can no longer be based on responses to past accidents and incidents, but must be increasingly proactive» [21]. therefore, an importance of developing measures to prevent risks prevails over minimizing the damage from accidents. wu, olson, and choi indicated that «optimization and risk minimization inherently run counter to each other» [30]. consequently, a choice of the risk management method should be based on an assessment of the preventive measures economic efficiency. these problems were solved on the basis of the multicriteria decision making (mcdm) methods [6, 27, 31], and the biconvex models and algorithms [25]. * corresponding author: first@ras.ru models of industrial risk control systems 19 copyright ©2022 assa. adv. in systems science and appl. (2022) for mcdm, heller [27] proposed using a pair wise comparison of the risk competing objectives. the following criteria were analyzed: a power outage, a fire, a flood, an earthquake, a hurricane, a destruction of buildings, network failures, etc. as a result, a matrix of the risk criteria assessments was formed, which is used in a qualitative risk analysis for three buildings that differ in qualitative features. abla et al. [6] used mcdm to derive an aggregated risk score on the basis of the fuzzy logic. for assessing development scenarios of the risk situation, the decision-making process was considered under the following criteria: a technical and functional efficiency, an economic sustainability, a social sustainability, an institutional environmental sustainability, an overall sustainability. the risk management strategy was selected based on a weighted sum of these criteria. in the case of the flood risk, this decision making technique was applied to select the most sustainable strategy under uncertainty. yazdani et al. [31] proposed mcdm type for assessing risks of the cultivated areas’ flooding; they ranked various agricultural projects, which can mitigate the flood risks. dudin m. n. et al. [5] investigated the risks of an industrial enterprise and calculated external factors influence weights and a likelihood of unforeseen events for political, macroeconomic, social, and technological factors. on the basis of these weights, the external risk average level of russian industrial enterprises was calculated. internal risks of the enterprise were assessed according to the following criteria: a fulfillment of the production plan, an economic security of current obligations, an economic security of supply contracts, a human factor, and a likelihood of success in innovations commercialization. the authors examined hedging and insurance methods for the risk management of an industrial enterprise, however, the insurers and other related organizations were not considered as separate agents, and their interactions were not investigated. krokhina j.a. et al. [16] explored the environmental risks of industrial enterprises using a tree method and assessed the integral risk of an industrial facility as a result of its negative impact on an environment, a human health, and an enterprise’s economy. the authors applied the tree method to assessing the risk of an accident on the main pipeline and assessed an economic efficiency of the measures to reduce the environmental risk of industrial enterprises, but did not take into account an interaction of the enterprise with other economic agents. in contrast to the aforementioned literature, multi-agent systems (mas) models expanded a range of the risk management techniques. mas models were studied in production design and development systems, a production planning and management, and a supply chain management (scm) [19]. in a set of agents, mas models described the participants in the production process supply chain. in scm, an influence of small and medium-sized enterprises in the interaction with large firms [7] and buyers with suppliers [1] on the risk level was considered. in the case of different-scale enterprises in scm, finch [7] emphasized varying degrees of the supply chain disruption risk, because large and small enterprises are subject to different degrees of the risks and their sensitivity to the risk factors is different. ahn and park [1] studied the information exchange processes between participants in the supply chain as mas agents and assessed an impact of an agent’s awareness on scm. in general, mas model was applied for the risk management in systems, which consist of an industrial enterprise and its suppliers, i.e., the participants in the supply chain. finally, it should be emphasized that the aforementioned studies were carried out for specific industries. for example, the problems of risk in the chemical and oil industries were solved [3, 8] and technical, human and organizational factors of the industrial facility risk were taken into account; on the basis of the fuzzy logic, the risk of the industrial facility in the oil and gas industry was analyzed using the criteria of frequency, detectability and damage value [30]. therefore, the results of these studies cannot be applied to all industries. thus, we demonstrate the following research gap in the problem framework of the industrial risk management. on the one hand, the industrial researchers pointed to a need for 20 m. geraskin, e. rostova copyright ©0000 assa adv. in systems science and appl. (0000) the insurance and the technical measures to prevent or eliminate consequences of industrial accidents. on the other hand, they did not investigate the interaction mechanisms of an enterprise with insurers and waste utilization firms as the specific agents. additionally, they did not generalize the results for the universal industrial enterprise; they were limited to the specific industry. at the same time, mas researchers did not study the principles of applying mas in the process of industrial risk management. hence, we can formulate the following research question of the industrial risk management problem: to describe the industrial risk management process for the universal industrial enterprise within the system of interconnected economic agents and calculate the equilibrium prices for services that circulate within this system. our study aims at calculating the price equilibrium in the system, which appears as a result of the preventive measures to minimize the consequences of technical accidents in the industry. 2. problem framework in this paper, the risk is considered at the firm’s level, and it includes an internal damage and an external damage. the internal damage causes a reduction in the firm’s assets. the external damage is the property wastes of other firms, individuals and the environment. additionally, the fiscal penalties (the ecology payment, the penalty for a damage to health, and a property of other firms and individuals, the compensation caused by the non-fulfillment contracts) depend on a value of the external damage. the industrial firm can reduce the internal/external damage by means of additional expenses on the risk reduction. these costs expenses are named the voluntary risk costs (vrc). we analyze the problem of the industrial risk control for a system with three agents: the industrial firm, the waste utilization firm, and the insurance company. these agents are in various relationships in the system. we consider the following problem: to search for a compromise price of waste utilization and a compromise insurance rate, which are compliant with all agents of the system. we assume that each participant in the system is intended to increase its profits, and he chooses the optimal price. if these optimal prices are different, then the participants may not agree to conclude a contract, then the industrial risk management will not be implemented. therefore, in this case, we determine the set of possible values of the insurance rate and the price of waste utilization, at which the participants in the system will agree. we introduce the following assumptions, which determine the applicability limits of the model. assumption 1. the product price is an exogenous constant, that is, the firm does not affect the price ,0 dq dp (1) where p is the price of the production, q is the production volume. the waste utilization firms and the insurance companies are in the monopolistic competition market, that is, the following conditions are fulfilled: ,0,0  u y u y dx dp dy dp (2) ,0,0       ss y t x t (3) models of industrial risk control systems 21 copyright ©2022 assa. adv. in systems science and appl. (2022) where py is the price of the utilization of a conventional waste unit, t is the insurance rate, xu and yu are the internal and external utilized damage, xs and ys are the internal and external insured damage. assumption 2. the production growth leads to a decreasing return: ,0qqc (4) where c is a value of the firm’s costs. assumption 3. an increase in the production assets leads to an increasing in the possible damage; the internal damage and the external damage are reduced with an increase in vrc; the internal damage is limited from above due to technology features and the production volume 0],,0(,0,0 maxmax       xxx f x q x . (5) where xmax is the maximum possible internal damage, x is the internal damage, f is vrc. assumption 4. the external damage y is proportional to the internal damage x: 0   x y . (6) assumption 5. the voluntary combination insurance is considered, the wear is not included. the insurance indemnity w is proportional to the insured damage xs and ys, the indemnity does not exceed the damage: .,0,0 ss ss yxw y w x w       (7) assumption 6. the cost of the utilization of a conventional waste unit cy is a constant. cy = const. (8) assumption 7. the firm’s external damage y=ys+yu+yres consists of the insured external damage ys= s y, the utilized external damage yu= u y, and the residual external damage yres= res y. the firm’s internal damage x=xs+xu+xres consists of the insured external damage xs= s x, the utilized internal damage xu= u x, and the residual internal damage xres= res x. 0,,,1  resusresus  , (9) 0,,,1  resusresus  . (10) the production costs function, according to assumption 2, has the following form [4], [29]. bqqсq )( , 0],2,1(],,1( maxmax  b , (11) where b and β are the parameters of the production costs function, βmax is the maximum possible parameter value. the internal damage function satisfies assumption 3, and it has the following form: feqfqx   )(),( , ],1,0(],,0( maxmax   0)(  q . (12) this function х(q) expresses an exponential distribution of the damage, which corresponds to man-made accidents, φ(q) is the dependence of the damage on the production volume q,  is the parameters of the internal damage function, max is the maximum possible parameter value. 22 m. geraskin, e. rostova copyright ©0000 assa adv. in systems science and appl. (0000) the external damage function satisfies assumption 4: 0,)(  xxy . (13) the coefficient of the accident consequences expansion μ expresses the ratio of the external damage and the internal damage, taking into account the specifics of the industrial complex in a region, geographical features, etc. the insurance indemnity satisfies assumption 5: ,10),(),(   ssss yxyxw (14) where α is the coefficient of the insurance indemnity. the penalty function has the following form: ,0,  aхaayh  (15) where a is the parameter of a relationship between the penalty and the external damage. we consider the systems, which include the agents of three types: the 1st agent is the industrial firm, the 2nd agent is the waste utilization firm, and the 3rd agent is the insurance company. the industrial firm (the 1st agent) produces the production volume q, and it sells the product at the price p. we introduce the following notation: cq is the production costs function, x is the internal damage, y is the external damage, f is vrc, h(y) is the penalty function, f(х,y) is the value of waste utilization costs, v(x, y) is the insurance premium, w(x, y) is the insurance indemnity, qmax is the maximum possible production volume, fmax is the maximum possible vrc. the revenue function of the 1st agent is wqpr  . (16) the total costs function of the 1st agent is fhvxfcс res q   . (17) the profit function of the 1st agent is пi=r – c∑. (18) we formulate the problem of the firm’s choice as follows: to search for the production volume and vrc function, which maximize the profit of the 1st agent, that is: .maxarg*}*,{ , i aqaf пqf qf   (19) }0,:{ maxmax   qqqrqaq (20) )},0(,)(:)({ maxmax if rfffrfa   (21)                     ).( ),( , ),( , , ,)( ss ss res uu y q f yxtv yxw ayh xypf bqc xy eqx         (22) the waste utilization firm (the 2nd agent) has the following parameters. the utilized damage uu xy   does not exceed the level y , the price py does not exceed the level yp , where yp is the maximum possible price, y is the maximum possible waste utilization. consequently, the inverse demand function of the 2nd agent, according to assumption 1, has the form: )( uuy yy xy y p pp   . if 0yp , then yxyyx uu  :, models of industrial risk control systems 23 copyright ©2022 assa. adv. in systems science and appl. (2022) the profit function of the 2nd agent is ))(( uu yyii xycpп   . (23) we formulate the problem of the choice of py : to search for the price of the utilization of a conventional waste unit, which maximizes the profit of the 2nd agent, that is: ii rp y пp y   maxarg* (24) ).( uuy yy xy y p pp   (25) the insurance company (the 3rd agent) has the following parameters. the insurance premium depends on the insurance rate t and the insured damage ss xy   ; t is the maximum possible insurance rate, x is the maximum possible insured damage. consequently, the inverse demand function of the 3rd agent, according to assumption 1, has the form: x t xytt ss )(   . if 0t , then xxyyx ss  :, . the profit function of the 3rd agent is wvпiii  . (26) we formulate the problem of the choice of the insurance rate: to search for the insurance rate, which maximizes the profit of the 3rd agent, that is: iii tst пт )1,0( maxarg*   (27)             . ),( ,)( ),( xxy xyw x t xytt xytv ss ss ss ss     (28) we consider the following problem of the agent’s optimal control: to search for the pair (q*, f*), which is optimal according to criterion (19), to search for the price py*, which is optimal according to criterion (24), and to search for the rate t*, which is optimal according to criterion (27). this system consists of three agents, and, if we vary the parameters q, f, py, t, then three agents achieve maximums of their profits. we consider the system of the industrial damage control of the following types. the agents have a strong relation, if they have the collective criterion function, and their costs are not separable. the agents have a weak relation, if the costs are separable, and each agent has the individual criterion function. we investigate the types of the system in the following models. model 1: the 1st agent is the customer of the waste utilization and the insurance, the 2nd agent is the contractor of the waste utilization, the 3rd agent is the insurer of the internal damage and the external damage of the 1st agent. this system has the weak relation type. (fig. 1) 24 m. geraskin, e. rostova copyright ©0000 assa adv. in systems science and appl. (0000) fig. 1. the agent’s interaction schema in model 1. the 1st agent pays the sum )( uu y xypf   to the 2nd agent, and the agents achieve the contract, if the price py complies with everyone. we formulate the problem with two criteria to search for the compromise price com yp , under which the system is stable. by using an analogy with the previous case, the 1st agent pays the sum )( ss xytv   to the 3rd agent, and the agents achieve the contract, if the insurance rate t complies with everyone. we formulate the problem with two criteria to search for the compromise rate comt , under which the system is stable. thus, the system of three agents is stable, if the compromise price com yp and compromise rate comt are indicated in the contracts. consequently, we formulate the following problems: to search for the compromise price com yp and the compromise rate comt , which satisfy the following conditions: ii gp i gp пп yy   maxmax , (29) iii t i t пп  maxmax , (30) }0)(0)(|{  yiiyiy pпpпpg , (31) }0)(0)()1,0(|{  tпtпtt iiii . (32) in formulas (29), (30), the symbol of the conjunction “ “ means that the maximums are determined according to both criteria, taking into account the pareto optimal principle. model 2: the 1st agent and the 2nd agent have the strong relation; the 3rd agent is the insurer of the internal damage and the external damage of the 1st and the 2nd agents; the 3rd agent has the weak relation to other agents. (fig. 2) models of industrial risk control systems 25 copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 2. the agent’s interaction schema in model 2. the aggregate profit function of the 1st and the 2nd agents in the system is resuu yqiiiiii xxychvfcwqpппп   )(, . (33) the problem of searching for the optimal production volume and vrc for the 1st agent, and, additionally, the optimal rate of the 3rd agent, has the following form: iii aqaf пqf qf , , maxarg*}*,{   , (34)                     ).( ),( , , , , ,)( ss ss res uu q f yxtv yxw ayh yxy bqc xy eqx         (35) these variables influence on the waste utilization, the insured damage, the penalties, and the insurance rate. we formulate the problem of searching for the compromise rate accounting to the following conditions: iii t iii t пп 11 maxmax ,   , (36) }0)(0)()1,0(|{ ,1  tпtпtt iiiiii . (37) model 3: two agents of the 1st type are customers of the waste utilization and the insurance; the 2nd agent is the contractor of the waste utilization; the 3rd agent is the insurer of the internal damage and the external damage of the 1st agent. this system has the weak relation. (fig 3) 26 m. geraskin, e. rostova copyright ©0000 assa adv. in systems science and appl. (0000) fig. 3. the agent’s interaction schema in model 3. the problem of searching for vrc and the optimal production volume of the 1st type agents is iii res iiiiqiiiii fhvxfcwpqп   , 2,1i , (38) i aqaf ii qifi qf maxarg , *}*,{   , (39)                     ).( ),( , ),( , , ,)( s ii s iii s ii s iiii res iii u ii u iiyi iiiq ii f iii yxtv yxw ayh xypf qbc xy eqx i i         2,1i (40) where }2,1,{  iп iii is the vector of criteria in model 3. the profit function (23) of the 2nd agent is the sum of the revenues, which are provided by two agents of the 1st type, and it has the following form:    2 1 )()( i u ii u iiyyii xycpп  . (41) the problem of searching for the price py* according to maximization of the 2nd agent’s profit function is ii rp y пp y   maxarg* , (42) .2,1),(  ixy y p pp u ii u ii y yy  (43) by using an analogy with the previous case, the profit function (26) of the 3rd agent is    2 1 )( i iiiii wvп . (44) the problem of searching for the insurance rate t according to the maximization of the 3rd agent’s profit function is iii tst пт )1,0( maxarg*   (45) models of industrial risk control systems 27 copyright ©2022 assa. adv. in systems science and appl. (2022)                .)( ),( ,)( ),( 2 1 xxy xyw x t xytt xytv i s ii s ii s ii s iiii s ii s ii s ii s iii     (46) we formulate the problem of searching for the compromise price com yp and the compromise rate comt , accounting to the following conditions: ii gp i gp i gp ппп yyy 111 maxmaxmax 21   . (47) }0)(0)(0)(|{ 211  yiiyiyiy pпpпpпpg . (48) iii t i t i t ппп 222 maxmaxmax 21   (49) }.0)(0)(0)()1,0(|{ 212  tпtпtпtt iiiii (50) 3. results assertion 1. the function |)(|ln 1 * kqf    , u y u y resssresss attk   and the value q*, which is calculated from the equation 0 )( )(1     q q qbp    , are the solution of the problem (19 – 22) for the continuously differentiable functions )( and under conditions 0))(())()1(()( 222   keqkeqqbkeq fff   and k > 0, 0)()()( 2  qqq  . the maximal profit of the 1st agent is   1 ****  fbqpqпi . proof. the profit function of the 1st agent (18) is   resresfss i ayeqfbqyxtqpп   )())(( )( uu y yxp   . we search for the partial derivatives of this function, which are equal to zero: 0)]())([()(1     uu y resresssfi pateq f п   0)]())([()(1     uu y resresssfi pateqqbp q п   we solve these equations as follows: 1)]())([()(  uu y resresssf pateq   28 m. geraskin, e. rostova copyright ©0000 assa adv. in systems science and appl. (0000) we introduce the denotation )())(( uu y resresss patk   , then we can write the optimal function vri as follows: ))(ln( 1 * kqf    . we write the equation 0)(1   keqqbp f  , and, for the aforementioned symbols k and f*, we get 0 )( )(1     q q qbp    . we solve this equation, and we search for q*. we consider a function )( )( )( 1 q q qbpqh       . this function is continuous function qaq . due to the features of continuous functions, the equation h(q)=0 has a solution, because 0)( qh for )( )( | 1 q q qbpaq q       , 0)( qh for )( )( | 1 q q qbpaq q       , and 0 )( )()()( )1()( 2 2 2     q qqq qbqh    if 0)()()( 2  qqq  . we check a fulfillment of the maximum sufficient condition for the function пi(q, f) at f=f* and q=q*. for this purpose, we define the sign of 22 2 2 2 2                 qf п q п f п iii . keq f п fi     )(2 2 2 , keqqb q п fi       )()1( 2 2 2 , keq qf п fi     )( 2 . 0))(())()1(()( 222   keqkeqqbkeq fff   . if k>0, then 2 2 f п i   <0 and 2 2 q пi   <0, consequently, the pair (f*,q*) is the maximum point according to sylvester’s criterion. █ assertion 2. the value 2 * yy y pc p   is the solution to the problem (24 – 25) for the continuously differentiable function y(py). the maximal profit of the 2nd agent is пii*=пii( * yp )= ))(( 4 1 2 yy y cpy p  . proof: we write the profit function of the 2nd agent (23), which is subjected to the condition (25): models of industrial risk control systems 29 copyright ©2022 assa. adv. in systems science and appl. (2022) .))(( y yyyyii p y ppcpп  we transform this expression as follows: .2 yy y y y yii cyc p y yp p y pп              this expression is the second-order power function, and it has the parabola graph with the maximum at . 2 * yy y cp p   . we search for the maximum of the profit function *)(* yiiii pпп  of the 2nd agent:                            yy y yy y yy ii cyc p y y cp p ycp п 22 * 2 2)( 4 yy y cp p y  .█ assertion 3. the insurance rate 2 *   t t is the solution to the problem (27 – 28) for the continuously differentiable functions x(t) and y(t). the maximal profit of the 3rd agent is 2)( 4 1 *)(*  tx t tпп iiiiii . proof: the profit function of the 3rd agent is t tt xtxytwvп ss iii   )())((  . we transform this function as follows: x t x xt t x tп iii               2 . this function has the parabola graph with the maximum point at 2 *   t t . we determine the maximum of the profit function of the 3rd agent as follows: 2 2 )( 422 *)(*                             t t x x t x x t t xt tпп iiiiii .█ assertion 4. if 2 y y p c  , then     2 ; y y com y p cp is the solution of the problem (29), (31) for the continuously differentiable functions )( , else com yp ø. proof: the profit function of the 1st agent is )()( uu y res qyi yxphvxfcwqppп   the utilized damage uu yx   corresponds to the demand function: . y yuu p p yyyx   then the profit function of the 1st agent is 1 2 1 1)( kyp p y p p p ypkpп y y y y y yyi        , 30 m. geraskin, e. rostova copyright ©0000 assa adv. in systems science and appl. (0000) where 1k is hvxfcwqp res q   . the maximum point of this function is 2 y y p p  , and the maximum of the profit function of the 1st agent )( yi pп is 142 k ypp п yy i       . the profit function пii(py) is analyzed in the proof of assertion 2. fig. 4. compromise set of the 1st and the 2nd agents the value 0 yp is yc . the figure 4 shows that the 1st agent prefers price     2 ,0 y y p p , the 2nd agent prefers price . 2 ,0       yy yy pc pp if 2 0 y y p p  then     2 ,0 y y com y p pp else         2 ,0 2 ,0 yyy y ppc p ø █ the value com yp is the solution of the problem (29), (31), and it enables us to establish the price yp , which complies with the 1st and the 2nd agents. if one of the agents changes this price, then the profit of other agent decreases, therefore, this is pareto optimal equilibrium set for 1st and the 2nd agents’ prices according to profit function (18) as the criterion of the 1st agent and profit function (23) as the criterion of the 2nd agent. assertion 5. if 2 t  , then        2 ; t t com  is the solution of the problem (30), (32) for the continuously differentiable functions )( , else comt ø. proof: the profit function 1st agent is )()( ssres qi yxtfhxfcwqptп   . models of industrial risk control systems 31 copyright ©2022 assa. adv. in systems science and appl. (2022) the insured damage ss yx   corresponds to the demand function in the insurance market: . t t xxyx ss   we transform the profit function of 1st agent as follows: .1)( 2 2 2 kxt t x t t t xtktпi        the value 2k is hfxfcwqp res q   . the minimum point of profit function )(tпi is 2 t t  and the minimum of the function is 242 k xtt пi       . the profit function пiii(t) is analyzed in the proof of assertion 3. the graphs of the 1st and the 3rd agents’ profits for problem (30), (32) are demonstrated in fig. 5. fig. 5. compromise set of the 1st and the 3rd agents the acceptable set comt is interval       2 ; t . these values enable us to transact of the insurance contract, because the value comt complies with the 1st and the 3rd agents. the deviation of the insurance rate relative to comt leads to a decrease in the profit of one of the agents. thus, we identify the set of the possible values of the insurance rate comt and the price of waste utilization com yp , at which the system agents interact, and the process of the industrial risk management is implemented. similar to the conclusion from assertion 4, we establish the pareto optimal equilibrium set for the 1st and the 3rd agents’ prices according to profit function (18) as the criterion of the 1st agent and profit function (26) as the criterion of the 3rd agent. 32 m. geraskin, e. rostova copyright ©0000 assa adv. in systems science and appl. (0000) 4. discussion the currently accepted risk management model is based on the standards [13] [15] and, in fact, implements the intra-firm management [3], [4], [6], [8], [16], [27], [31]. in comparison with the aforementioned literature, we investigate the process of the industrial risk management in the system of several organizations interconnected within the framework of this process. our system consists of three agents (the industrial firm, the insurer, and the waste utilization firm) and provides a comprehensive examination of the risk management process. in contrast to use of mas for scm, we consider the mas model, which includes the agents that are not linked in a supply chain. we add the insurers and the waste utilization firm to this model; therefore, our results extend the studies [18], [23]. in addition, our results are industry invariant; consequently, it can be used for any industrial production. this generality of results distinguishes our study from authors who considered risks for a particular industry, for example [3], [8]. we derive an analytical form of the preventive risk costs function for an industrial enterprise and prove that these costs depend logarithmically on the production volume of an enterprise. this pattern means that the preventive risk costs increase more slowly in comparison with a growth of the production. this feature encourages enterprises to take the preventive measures and develops the concept of the proactive risk management [21]. therefore, we prove that the proactive risk management strategy is optimal, i.e., it corresponds to the minimum costs. we calculate the optimal values of the waste utilization price and the insurance rate, which maximize the objective functions of the waste utilization firm and the insurer. in comparison with the approach [4], [16], which is based on the exogenously specified insurance rate, in our model, the insurance rate is determined endogenously. in other words, our model is based on such values of the waste utilization price and the insurance rate that induce these firms to participate in contracts with the industrial enterprise. consequently, in these conditions, the decentralized decision-making system is configured according to the mechanism of the centralized system. in addition, we find the conditions of the compromise domain, i.e., the ranges of the waste utilization price and the insurance rate, within which the system participants are interested in concluding contracts. therefore, we expand the mas approach [1, 7, 19] related to the revenue sharing, and reformulated it into the mechanism of the price sharing contract. next, we prove that the contract prices within the specified ranges are pareto efficient. therefore, when the price varies within the compromise domain, the profit of one participant grows, and the profit of the counterparty decreases, i.e., the price sharing contract corresponds to the profit sharing contract. this is an important advantage of our approach: our model not only allows us to estimate the damage from an industrial accident, but also to calculate the economic effects of all participants in the risk management process and choose the preventive costs sum that corresponds to the optimal solutions for all participants. the results of this article can be used by industrial enterprises, insurance companies, and waste utilization firms to determine the insurance rate and the utilization price. the resulting compromise values of the rate and the price demonstrate the set of acceptable values at which a contract will be concluded. finally, we briefly outline the directions for further research within our version of the mas model. our results are obtained under certain restrictions on the type of a market (assumption 1), the production function (assumption 2), and the damage function (assumption 3). in the future, we plan to expand the study to consider other types of markets and other production and damage functions. models of industrial risk control systems 33 copyright ©2022 assa. adv. in systems science and appl. (2022) 5. conclusion the optimization problems of searching for the firms’ risk prevention costs were investigated in our previous articles [9], [22]. in particular, the optimal vrc function in the case of the internal damage prevention according to the condition of the firm’s profit maximization was derived. the problem of the external damage control was analyzed, and the optimal vrc function with regard to fiscal penalties for the environmental damage and the civil penalties for individuals’ property damages was proved. the problem of searching for the optimal risk costs, taking into account the reinvestment of the firm’s profit, was considered, and the optimal vrc function for a firm’s activity in the consequent periods was obtained. in this paper, the problem of the industrial risk control in different system’s structures is investigated. we consider a risk control system of an industrial firm, which includes the insurance company and the waste utilization firm. this system enables us to determine the conditions of the insurance contract and the waste utilization contract. the problem of the industrial risk control is a problem of the agents’ interests congruence. the calculated values of the insurance rate and the waste utilization price are determined as the pareto equilibrium set for price and insurance rate. we obtain the following results. the problem of the firm’s choice of the product volume and vrc function in the system, which includes an industrial firm, a waste utilization firm, and an insurance company, is solved in assertion 1. the problem of searching for the price of the utilization, which maximizes the profit of a waste utilization firm, is solved in assertion 2. the problem of searching for the insurance rate, which maximizes the profit of an insurance company, is solved in assertion 3. the problem of searching for the compromise utilization price and the compromise insurance rate, taking into account pareto optimal principle for this variable, is solved in assertion 5. the final result provides the set of acceptable values at which a contract will be concluded. references 1. ahn, h.j., park, s.j. 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(1963). production and cost functions: an econometric survey. econometrica. the econometric society, econometrica. 30. wu, d.d., olson, d.l., choi t.m. (2017) guest editorial special issue on risk analytics in industrial systems, ieee systems journal. vol 11, №. 3, 1476-1478. 31. yazdani, m., gonzalez, e.d.r.s. and chatterjee, p. (2019), a multi-criteria decision-making framework for agriculture supply chain risk management under a circular economy context, management decision, vol. ahead-of-print no. ahead-of-print. https://doi.org/10.1108/md-10-2018-1088 microsoft word 868_done adv syst sci appl 2023; 02; 1-11 published online at https://ijassa.ipu.ru. pretreatment performance evaluation of the seawater desalination plant of beni saf bwc mourad berrabah1*, hassiba bouabdesselam1 , noreddine ghaffor2 1) lte research laboratory, national polytechnic school of oran, algeria 2) king abdullah university of sciences and technology, thuwal, saudi arabia abstract: the beni saf seawater desalination plant is one of the largest projects undertaken by the algerian state for supplying drinking water, with a capacity of 200,000 m3 per day. the plant uses the reverse osmosis technique as a desalination process. the performance of such systems requires the production of pretreated, good-quality feed water. moreover, a number of large-scale experiments have shown that pretreatment of seawater before reverse osmosis (ro) desalination is key to retard fouling in osmosis membranes. this work aims to study the performance of the pretreatment selected by the beni saf desalination plant. to achieve this, we determined the physico-chemical and bacteriological parameters of the raw seawater used by the plant. variation of sdi, which is a key parameter in controlling fouling potential, was monitored during the year 2019. the pretreatment performance was investigated by monitoring the efficiency of each stage of the pretreatment process. the results obtained show the efficiency of the pretreatment adopted by this desalination plant. keywords: desalination; fouling; pretreatment; retarding. 1. introduction to address water shortage, algeria has reinforced its water resources by waste water treatment and seawater desalination plants. the desalination plant of beni saf produces 200,000 m3/d of water to meet the needs of the region of ain temouchent and oran. located on the mediterranean coast of algeria in the wilaya of ain temouchent, the plant covers an area of 65700 m2. the desalination process adopted for this plant is the reverse osmosis (ro) technique. a major challenge experienced by the plant is the poor quality of the raw seawater and therefore the rapid fouling of the reverse osmosis membranes. feed water is characterised by turbidity, bacterial content and tds (total dissolved solids), which causes fouling and results in the degradation of the membrane performance. when a membrane is affected by fouling, there is deterioration of its basic functions; this includes salt passage, decreased permeate flow and pressure drop across the membrane. on the other hand, inorganic scaling caused by exceeding the solubility limit of soluble salts is considered less problematic since it can be controlled by ph and addition of antiscalants [7]. deposits on the membrane or on the pores leads to a decrease in the performance of the membrane [5], and to prevent rapid formation of these deposits, the seawater desalination plant of beni saf adopted a conventional physico-chemical treatment. the objectives of the pretreatment are as follows: * improving the quality of water, enhancing the performance of the membranes (reducing membrane fouling). pretreatment has proved to be efficient in improving the performance of the membrane and reducing the cost of its replacement since it eliminates turbidity, bacteria and tds (total dissolved solids). without pretreatment, ro * corresponding author: berrabahmourad534@gmail.com membranes are prone to rapid fouling which results in frequent replacement with the consequential increase in operational costs. to address this issue, a pretreatment method 2 m.berrabah, h.bouabdesselam copyright ©2023 assa adv. in systems science and appl. (2023) must be integrated before feed water enters the ro system to help reduce membrane fouling and improve thereby its performance [1]. 2. materials and methods 2.1. physico-chemical and biological characteristics of the raw seawater from the desalination plant of beni saf the raw seawater of the beni saf plant was characterized by taking a sample of raw seawater and analyzing the physico-chemical and bacteriological parameters. water sampling is a sensitive operation. it allows defining the quality of water at a given time or over a more or less long specified period. therefore, it should be handled with the greatest care [3]. table 1 shows the results of the physico-chemical and bacteriological analyses performed on a sample taken on 27/11/2019. table 1. results of the physico-chemical and bacteriological analyses of the raw seawater from beni saf plant physico-chemical analysis of raw water turbidity (ntu) 1.76 temperature (°c) 19 ph 8.12 conductivity ms.cm-1 54.85 tds (mg.l-) 39.55 dbo5 5.00 dco 10.00 alkalinity (mg.l-de caco3) 142.7 hardness (mg.lde caco3) 7158.4 ionic balance ca++ (mg.l-) 417.20 mg++ (mg.l-) 1487.93 na++ (mg.l-) 12165 k+ (mg.l-) 420.71 sr++ (mg.l-) 11.25 fe++ (mg.l-) 0.03 ag+ (mg.l-) 0.11 so4 2(mg.l-) 2,950.00 cl(mg.l-) 21,950.12 hco3 (mg.l-) 158.92 f(mg.l-) 1.25 co3 2(mg.l-) 8.62 bacteriological analyses total califorms 3/100ml fecal califorms 0 total streptococci 0 fecal streptococci 0 aerobic germs22°c 2/100ml salmonella abs pretreatment performance evaluation of the seawater dessalination plant… 3 copyright ©2023 assa adv. in systems science and appl. (2023) compared to other parameters such as turbidity or suspended solids, the silt density index (sdi) is key in controlling the fouling potential of seawater, and therefore, requires continuous monitoring [2]. to this end, the evolution of the sdi of the seawater from this desalination plant was monitored over a certain period. sdi tests were performed as per astm d 4189-95 standard by filtrating a water sample through a membrane of 0.45 μm (microfiltration) and 47mm of diameter at a constant transmembrane pressure (2.1 bars). the sdi is determined by comparing filtration times, t1 and t2 required to achieve a fixed filtration volume at time 0 and after time t respectively. sdi15 (t = 15 minutes) is defined by the astm as the time required for accurate and standardized testing. sdi15 (t = 15 minutes). figure 1 represents the variation of the raw seawater sdi3 of the desalination plant of beni saf as a function of time for 2019. fig.1. variation of the raw seawater sdi of the desalination plant of beni saf as a function of time for 2019 from the sdi3 versus time plot, one can see that the average sdi3 in autumn and winter is low with little evolution but then increases in spring and summer. 2.2. selection of pretreatment for the desalination plant of beni saf knowledge of the physico-chemical characteristics of the raw seawater requires the selection of the most adapted pretreatment technology to produce good quality supply water, and delay fouling of ro membranes. the pretreatment selected by the desalination plant of beni saf is a conventional physical and chemical treatment. figure 2 illustrates the different pretreatment stages used by the plant. 4 m.berrabah, h.bouabdesselam copyright ©2023 assa adv. in systems science and appl. (2023) fig .2. the pretreatment process used by the desalination plant of beni saf 2.2 performance evaluation of the pretreatment adopted by beni saf plant to monitor the performance of the pretreatment used in this plant, the pretreatment process was subdivided into three systems. then samples were taken downstream of each system (sampling points represented in the diagram fig.2) at regular time intervals, and the parameters most influenced at this stage were measured (sdi15, turbidity, temperature, ph). samples were taken during the daily the production over a one-year period (2019). system 1: description of the system injection of sodium hypochlorite into seawater (pre-chlorination) for pre-chlorination, the plant uses commercial sodium hypochlorite as a biocide. hypochlorite is dosed automatically and continuously into the seawater collection tank to remove organic matter. this operation is handled by three dosing pumps with a rate of 25%, i.e., 187.51 l/h under a pressure of 2 bars. sulfuric acid dosing concentrated sulfuric acid (98%) is dosed upstream of the sand filters, to set the ph and prevent precipitation of carbonates and bicarbonates on the ro membranes. this operation is handled by three dosing pumps with 25% rate, i.e., 83.75l/h under a pressure of 6 bars. ferric chloride dosing (coagulation) coagulation is the most popular treatment process used for the elimination of potential impurities such as colloidal and particulate matter. the purpose of coagulation is to combine small particles into larger aggregates/flocs (i.e. large groups of loosely bound suspended particles) by neutralizing the charges of the particles [6] . performed upstream of the sand filters, the coagulation serves to destabilize colloids and suspended solids so that these substances are retained during filtration. beni saf desalination plant uses 40% commercial iron chloride as coagulant. ferric chloride is automatically dosed into the discharge manifold of the seawater intake pump upstream of the sand filters at a rate that depends on the seawater flow rate. five dosing pumps with 25% rate, i.e., 83.75 l/h are dedicated to this operation. pretreatment performance evaluation of the seawater dessalination plant… 5 copyright ©2023 assa adv. in systems science and appl. (2023) sand filtration a filtration bed constituted of sand and silica with two different sizes (dual layer) is employed in this plant. this dual layer filtration is sufficient to achieve an sdi < 4 and effectively eliminate the pigments of algae if the installation is fed with good quality raw seawater [4]. the layer of coarse sand (base layer) at the bottom of the filter is topped by a finer sand layer (filtration layer). the 1000 mm thick filtration layer (upper layer) with its low granulometry serves to trap particles. the filtration sand has the following characteristics: table 2. characteristics of the filtration sand (upper layer) sio2 content >99,5 effective size (mm) 0.9 uniformity coefficient 1.4 absolute density (kg/m3) 2650 bulk density (kg/m3) 1600 grain size (micron) 139 the thickness of the coarse sand bed (base layer) is ~300 mm. the sand has the following characteristics: table 3. characteristics of the filtration sand (base layer) the turbidity analyzer console of each train includes a sampling device (sampling point 1) for the measurement of sdi and turbidity together with other parameters. system 2: description of the system anthracite filtration was added to this system. the anthracite filtration system consists of two filter trains comprising seven modules or filter pairs, making up a total of twenty-eight filters. each module comprises two identical filters operating as a single filtration unit. in normal operation, the flow rate required to feed the anthracite filter system is 18,000 m3/hour. the desalination plant of beni saf employs anthracite filters with a filtration bed of 1000 mm. the anthracite used has the following characteristics: table 4. the characteristics of the anthracite used in the desalination plant of beni saf rate of charcoal % >90 effective size (mm) 1.2 uniformity coefficient 1.4 absolute density 1400 bulk density (kg/m3) 1050 grain size (micron) 186 the turbidity analyzer console of each train comprises a sampling device, intended for measuring the rate of sdi matter, turbidity together with other parameters (sampling point 2). sio2 content >99,5 effective size (mm) 2 uniformitycoefficient 1.4 absolute density 2650 bulk density (kg/m3) 1600 grain size (micron) 309 6 m.berrabah, h.bouabdesselam copyright ©2023 assa adv. in systems science and appl. (2023) system 3: description of the system physico-chemical treatments were added to this system antisclants dosing this chemical treatment is carried out upstream of the cartridge filters. a commercial dispersant with a concentration of 95% diluted to 10% is dosed using 5 pumps with a flow rate of 25%, i.e., 56.75l/h under a pressure of 6 bars. sodium bisulphite dosing chemical treatment is performed downstream of the cartridge filters using commercial sodium bisulphate 61% diluted to 10%. the operation is handled by five pumps with a flow rate of 25%, i.e., 133.51 l/h under 6 bars of pressure for dosing sodium bisulphite cartridge filtration this is the first pretreatment stage aimed at retaining particulate <5 µg and ensuring the protection of the reverse osmosis installation. the filter cartridges used are made of polypropylene because of its low water absorption. each filter contains 380 cartridges type pp-5, with an outer diameter of 61 mm and a length of 1270 mm with a rating of 5 microns. at the end of this process, a connection for taking samples is placed on the filtered water outlet line to measure the sdi fouling index, turbidity and other parameters (sampling point 3). 3. results and discussion 3.1. study of the pretreatment temperature, ph, turbidity and sdi measurements were taken after each pre-treatment system for the year 2019 by computing the monthly average. the results are consolidated in the following tables. table.5. results of analyses after sand filtration at bwc plant for 2019 physical characteristics ph temperature (°c) sdi15 (%/min) turbidity standard 7-8 16-25 5-6 0.7-1 january 7.45 18.77 5.2 0.76 february 7.61 17.3 5.25 0.76 march 7.71 15.73 5.6 0.77 april 7.51 17.28 5.7 0.8 may 7.45 18.85 5.73 0.86 june 7.40 21.03 5.63 0.87 july 7.67 24.5 5.95 0.85 august 7.52 24.4 5.75 0.88 september 7.63 22.3 5.5 0.78 october 7.76 23.2 5.4 0.77 november 7.61 19.44 5.3 0.75 december 7.64 16.9 5.1 0.71 pretreatment performance evaluation of the seawater dessalination plant… 7 copyright ©2023 assa adv. in systems science and appl. (2023) figure 3 shows the evolution of these parameters as a function of time for the first system fig. 3. evolution of temperature, ph, sdi15, turbidity after the 1st filtration system of the plant for 2019. by comparing the results found and those of the raw seawater, a significant reduction in turbidity and sdi15 was observed with a difference of 10-15% and 0.7-1 ntu for the sdi15 and turbidity respectively, which shows the effectiveness of the system (coagulation+ sand filtration). the sand-based filtration aims at retaining suspended solids present in the seawater and flocs that formed at the preliminary stage of coagulation. the filter medium used by the plant is of two sizes (dual layer); a layer of coarse sand (base layer) located at the bottom of the filter is topped by a layer of finer sand (filtration layer) that serves to retain finer particles. an increase in sdi15 and turbidity was observed. this increase is caused by the fouling of the filtration bed by trapped particles. therefore, the filters must be washed when the head loss reaches a limit value, which results in a drop in filters output. table 6. analyses results after anthracite filter at bwc plant physical characteristics ph temperature (°c) sdi15(%/min) turbidity standard 7-8 16-25 44,5 0.70,5 january 7.45 18.77 44 0.65 february 7.61 17.3 4.45 0.6 march 7.71 15.73 4.3 0.53 april 7.51 17.28 4.25 0.55 may 7.45 18.85 4.15 0.50 june 7.40 21.03 4.13 0.52 july 7.67 24.5 4.1 0.51 august 7.52 24.4 4 0.5 september 7.63 22.3 4.2 0.58 october 7.76 23.2 4.39 0.6 november 7.61 19.44 4.5 0.68 december 7.64 16.9 4.45 0.7 8 m.berrabah, h.bouabdesselam copyright ©2023 assa adv. in systems science and appl. (2023) figure 4 shows the evolution of these parameters as a function of time for the 2nd filtration system. fig. 4. evolution of temperature, ph, sdi15, turbidity after the 2nd filtration system of bwc plant for 2019 figure 4 shows the filtration performance of the 2nd system in the elimination of sdi15 and turbidity. the sdi15 recorded after filtration by anthracite filters fares better than that recorded after a single filter (sand filter only). a discrepancy of 0.5-1.5 was observed between the two configurations (system 1 & 2).the second system shows that anthracite filtration operates as an adsorbent. the anthracite fulfils a number of functions among which the elimination of residual concentrations of oxidant agents such as chlorine and ozone, together with other carcinogenic derivatives resulting from the treatments. anthracite adsorbs these substances or reduces them by catalysis to benign forms it also traps organic matter, algae, pesticides and generally any compounds that may alter the odour or taste of the drinking water. therefore, the anthracite filters are necessary after the sand filters in order to produce water with good quality. table7. results of the analyses after the cartridge filters (pretreated water) at bwc plan for 2019 physical characteristics ph temperature (°c) sdi15(%/min) turbidity standard 78 1625 <4 <0.5 january 7.45 18.77 3.6 0.35 february 7.61 17.3 3.4 0.38 march 7.71 15.73 3.45 0.4 april 7.51 17.28 3.2 0.42 may 7.45 18.85 3.1 0.35 june 7.40 21.03 3 0.3 july 7.67 24.5 3.12 0.35 august 7.52 24.4 3.15 0.30 september 7.63 22.3 3.25 0.38 october 7.76 23.2 3.3 0.35 november 7.61 19.44 3.4 0.4 december 7.64 16.9 3.48 0.38 pretreatment performance evaluation of the seawater dessalination plant… 9 copyright ©2023 assa adv. in systems science and appl. (2023) figure 5 below shows the evolution of these parameters as a function of time for the 3rd system. fig. 5. evolution of temperature, ph, sdi15, turbidity after the 3rd filtration system of bwc plant for 2019. the recommended limit values for sdi and turbidity, upstream of the ro membranes, are < 4 (95 % of the time), and<0.5 ntu respectively [5]. figure 5 shows variation in turbidity and sdi15 as a function of time. as can be seen, the recorded sdi15 and turbidity values are below the recommended limits upstream of the reverse osmosis system, confirming thereby the efficiency of the pretreatment used by the desalination plant. filtration by cartridge filters is the final pretreatment stage; it serves to protect the highpressure unit as well as the ro membranes, and retain smaller particles that may cause damage to equipment. it also serves to reduce subsequently the sdi15 and turbidity to produce water ready for passage through the ro membranes. upstream of the cartridge filters, antiscalant dosing allows for avoiding salts precipitation on the ro membranes. down the cartridge filters, the dosing of sodium bisulphite allows for the elimination of residual chlorine resulting from the sodium hypochlorite employed for the disinfection of seawater. 3.2. study of the influence of the pretreatment used by the desalination plant of beni saf on the performance of reverse osmosis the desalination process is carried out by passing pretreated seawater through semipermeable membrane modules (hydranautics swc5). these modules require the following conditions (table 8). table 8. operating conditions conditions recommended values maximum pressure applied 82,73 bars maximum chlorine concentration < 0.1 ppm maximum operating temperature 113 °f (45 °c) ph range, continuous (cleaning) 2-11 (1-13)* maximum feed water turbidity 1.0 ntu maximum feed water sdi (15 min) 5.0 maximum feed flow 17.0 m3 /h maximum pressure drop for each element 1,03 bar 10 m.berrabah, h.bouabdesselam copyright ©2023 assa adv. in systems science and appl. (2023) these requirements must be fulfilled by the plant to ensure stable membrane performance. to investigate the influence of pretreatment on the performance of ro, variation of the pressure drop of both stages of line no. 1 together with its permeate flow rate were monitored. the following figures show the variation of permeate flow and head losses as a function of time (line 1). les measurements were taken during the daily production. fig .6. evolution of permeate flow as a function of time fig.7.evolution of head losses as a function of time as can be seen from figure 3, which illustrates permeate flow variation (line 1), there is a quasi-stability of the permeate flow during for january-march. for april, one can observe a decline in flow rate (~10%) attributable to the fouling phenomenon, which necessitates a chemical cleaning of the membranes. as shown in figure 7, the variation curve of head losses illustrates the stability of both the first and second stages for jan-march, which can be explained by the absence of the fouling phenomenon. by contrast, april sees an increase in head loss due to fouling. the results found show that albeit the seawater of the desalination plant of beni saf is properly pretreated, there is yet a persistent fouling. currently, a very noteworthy trend includes the use of membrane pretreatment (microfiltrationultrafiltration), to improve the quality of the pretreated water and consequently the performance of the ro installation. pretreatment performance evaluation of the seawater dessalination plant… 11 copyright ©2023 assa adv. in systems science and appl. (2023) 4. conclusion the main challenges faced by the desalination plant of beni saf is the rapid fouling of ro membranes associated to particulate/colloids (deposit), organic compounds (scaling) and to biological growth (biofilm). results of physico-chemical and bacteriological analyses on the raw seawater of the desalination plant show that this seawater necessitates a pretreatment to eliminate the materials responsible for fouling.poor feed water leads inevitalby to shorter life cycle of the ro membrane. for this reason, the desalination plant of beni saf employs a conventional pretreatment (physico-chemical treatment) to fight the formation of deposits on the ro membrane. our study shows that each stage of pretreatment used by the plant leads to significant improvement in the quality of the feed water. the first system shows that pre-chlorination, coagulation, and sand filtration reduces sdi15 by 10-15% /min whereas the second pretreatment system shows the effectiveness of the anthracite filtration, and a discrepancy in sdi15 between the two systems ( 1 & 2) of 0.5-1.5. whereas the last system shows the effectiveness of the filtration based on cartridges (0.5 µm) and addition of antiscalants.sdi15 and turbidity values were found to be in the range < 4 (95 % of the time), and <0.5 ntu, respectively. these results demonstrate the effectiveness of the pretreatment used by the desalination plant of beni saf. in spite of the effectiveness of the pretreatment employed by the plant, the fouling phenomenon persists, which requires an additional pretreatment process such as membrane pretreatment (desalination plant of maktaa) to improve the quality of pretreated seawater and increase the life cycle of ro membranes. references 1. hashlamon, a., ahmad, a. & choonhong, l. (2015). pre-treatment methods for seawater desalination and industrial wastewater treatment: a brief review. int. j. sci. res. sci. eng. technol., 1, 2394–4099. 2. kader, g. (2011). a large review of the pre treatment, expanding issues in desalination. [online] available http://www.intechopen.com/books/expandingissuesin-desalination/a-large-review-of-the-pre-treatment 3. legrand, c., solerieu, m., & goglio, e. (2006). traitement des sites et sols pollués [treatment of polluted sites and soils]. voiron, france: territorial ed., 85, [in french]. 4. leparc, j., rapenne, s., courties, c., lebaron p., croué, j. p., et al. (2007). water quality and performance evaluation at seawater reverse osmosis plants through the use of advanced analytical tools. desalination, 203, 243–255. 5. mosqueda, j., daniella, b. & huck, p. m., (2009). effect of biofiltration as pretreatment on the fouling of nanofiltration membranes. desalination, 245(1–3), 60– 72. 6. sinha, s., yoon, y., amy, g. & yoon, j. (2004). determining the effectiveness of conventional and alternative coagulants through effective characterization. schemes. chemosphere, 57, 1115–1122. 7. yoon, j., yoon, y., amy, g. & her, n. (2005). determination of perchlorate rejection and associated inorganic fouling (scaling) for reverse osmosis and nanofiltration membranes under various operating conditions. j. environ. eng., 131, 726–733. adv syst sci appl 2017; 3:1–8 published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/502 auv thrust allocation with variable constraints v.v. kostenko1, a.yu. tolstonogov1,2∗ 1institute for marine technology problems feb ras, sukhanova st. 5a, vladivostok, 690091, russia 2 far eastern federal university, sukhanova st. 8, vladivostok, 690091, russia abstract: modern multipurpose autonomous underwater vehicles (auvs) represent the next generation of robotic systems with new technological tasks faced by researchers. one method of extending the functionality of the vehicle is installing tunnel thrusters in addition to the stern propulsion system. thereby the vehicle becomes able to undertake both survey-style missions and low speed interactions with the environment. but the efficiency of the tunnel thruster depends strongly on the vehicle’s velocity, due to hydrodynamic aspects. the usual solution to this problem uses different control models and thrust allocation methods for these types of mission. a unified approach to thrust allocation is presented in this paper. the approach is based on solving the allocation problem by quadratic programming with variable constraints, depending on the velocity of the auv and the thrusters’ angle of attack. the well-known implementation of active set method was used to model the proposed allocation method. keywords: auv, propulsion system, thruster hydrodynamics, control allocation problem, quadratic programming, interior point method, active set method 1. introduction the design of a control algorithm for an underwater vehicle is often divided into several levels (fig. 1.1) [1]. first, a high-level motion control algorithm is designed to compute a vector of virtual unbounded inputs to the vehicle fc (eq. (1.1)) from the target and current vehicle states and the control type. fig. 1.1. control system structure. fc is the vector of commanded inputs, f is the virtual vector of inputs, τ is the vector of allocated thrusts, u is the vector of low-level actuator inputs. ∗corresponding author: tolstonogov.anton@gmail.com http://ijassa.ipu.ru/ojs/ijassa/article/view/502 2 v.v. kostenko, a.yu. tolstonogov ẋ(t) = a(t)x +b(t)f y(t) = c(t)x (1.1) here, x(t) ∈ rn is the system state vector, t is time, y(t) is the vector of outputs, f ∈ rm is the vector of virtual inputs, which should be equal to the command vector fc of the high level motion algorithm, anda(t), b(t), c(t) are the coefficient matrices of the mechanical system. the virtual inputs are usually chosen to have a number of forces and torques that equals the number of degrees of freedom, m, that the motion control system wants to control. second, the control allocation algorithm is designed in order to map the vector of virtual input forces and torques fc to individual thruster forces τ such that the total forces and torques generated by all thrusters f amount to the commanded virtual input fc. f = b(x, t)τ (1.2) here, b(x, t) is the thruster configuration matrix. it contains the location and orientation of all thrusters, in the vehicle body-fixed frame. τ ∈ a ⊂ rp is the vector of actuator thrusts generated by the vehicle propulsion system, a are constraints from the saturation of the thrusters or other physical constraints, and p is the number of actuators. third, there is a separate high-frequency low-level controller for each actuator, that controls the desired thrust τi by a low-level control input ui. for each actuator, τi = hi(x, t, ui) (1.3) where h is a function, τi is the thrust of actuator i, and ui ∈ u ⊂ r is the low-level control input of actuator i. usually, the effector model is linear in u: τi = hi(x, t, ui) = g(x, t)ui (1.4) but the relation between the actuator thrust and the low-level control input is not considered in this article. we assume that each low-level controller maintains the desired thrust τi with sufficient quality. this modularity allows the high-level motion control algorithm to be designed without detailed knowledge about the vehicle propulsion system. in addition to coordinating the effects of the different thrusters in the system, issues such as thruster/fault tolerance, redundancy, and control constraints are typically handled within the control allocation module. in the case of an over-actuated propulsion system, when the number of thrusters is greater than the number of degrees of freedom (dof) controlled by the vehicle (p > m), the control allocation module solves an optimization problem to achieve a minimal power consumption of the propulsion system. there are different approaches to the high-level motion control of a vehicle. the pid (proportional-integral-derivative) is a widespread control method, but new methods such as linear quadratic gaussian (lqg) [2], the h∞ control method [3], or fuzzy logic control, are being rapidly developed. an interesting method is model predictive control [4], which solves a suboptimal motion control problem at each iteration, with the ability to combine the first and second levels of the motion, with the consequent increase in the computational complexity. in this paper, the problem of high-level control is assumed to be solved and the only the problem of optimal thrust allocation in the case of an over-actuated vehicle is considered. this problem is well studied. there is a survey devoted to this problem [1]. different approaches to this problem, including linear iterative approach as well as the quadratic programming approach satisfying optimality criteria, are treated in that survey. there are also papers published more recently [5–7]. these papers devoted to control allocation are focused on reducing the computational complexity of the optimal thrust allocation, but there is no mention of the fact that the thruster constraints can drift dynamically due to thruster hydrodynamics. a new approach, taking into account the vehicle speed and the thrusters’ angle of attack, is proposed in the present article. copyright c© 2017 assa. adv syst sci appl (2017) auv thrust allocation with variable constraints 3 this approach allows controlling both survey-style missions and low speed interactions with the environment for a vehicle equipped with tunnel thrusters. 2. problem formulation let the vector of virtual inputs computed by the high-level motion control or vehicle operator be denoted by the generalized force vector f = (fx, fy, fz,mx,my,mz) t , here fi is the projection of the force onto the axis i (i equals to the x, y and z axis) and mi is the projection of the torque onto the axis i in the body-fixed coordinate frame. the body-fixed reference frame [8] is used in this work. the x axis is directed along the longitudinal vehicle axis from the vehicle’s stern to the fore, the y axis is directed along the latitudinal vehicle to the starboard, and the z axis completes the frame to a right-handed coordinate system. assume that the system is equipped with p thrusters with control thrust τi(i = 1, . . . , p). this leads to the following relation between τ = (τ1, τ2, . . . , τp) and the virtual inputs f : bτ = f (2.5) according to [9], the optimization problem can be written as min τ,s 1 2 (τtqτ + strs) bτ = f + s (2.6) τmin ≤ τ ≤ τmax where s is the vector of slack variables used to penalize |bτ− f |, q and r are the weight diagonal matrices for the thrusters and dofs, respectively, τmin and τmax are the vector of thrust constraints according to bollard pull tests. the thrust allocation of an overactuated auv with tunnel vertical thrusters is a velocitydependent problem. for example, the pitch motion of the vehicle on zero velocity is better created by tunnel thrusters due to their large thrust arm. but the effectiveness of a tunnel thruster tends to zero at high velocities (fig. 2.2). we propose a variable constraints method for the thrust allocation problem. this method allows reallocating the thrust depending on the vehicle velocity. the method considers thrust constraints that depend on the vehicle velocity, as well as the thrusters’ angle of attack: τmin(v, θ) ≤ τ ≤ τmax(v, θ) where τmin(v, θ) and τmax(v, θ) are the variable thrust constraints depending on the vehicle velocity and the thrusters’ angle of attack. 3. the model of thruster constraints the thrust and torque generated by a thruster can be described by the following formulas: τ = kτ(j0)ρω |ω|d4 m = km(j0)ρω |ω|d5 where ρ is the density of water, ω is the rotational speed of the thruster propeller, andd is the diameter of the propeller.kτ(j0), km(j0) are the coefficients of thrust and torque determined copyright c© 2017 assa. adv syst sci appl (2017) 4 v.v. kostenko, a.yu. tolstonogov (a) v < 0.7ms−1 (b) v > 0.7ms−1 fig. 2.2. thrust allocation for pitch motion depending on vehicle velocity. by the form of the propeller and the characteristics of the motor. j0 = v/ωd is the advance ratio (where v is a velocity of thruster input flow, this velocity equals to velocity of the vehicle in case of the absence of currents). the function kτ(j0) can be fitted as a linear function of j0 at j0 > 0 and v > 0 [10]: k+ τ (j0) = k0 τ (0)− a1j0 (3.7) where k+ τ (j0) is the thrust coefficient kτ(j0) at j0 > 0 and equi-directional state of ambient flow and axial thruster flow. k0 τ is the thrust coefficient at j0 = 0 and a1 is a coefficient determined by the shape of the propeller. the equation may be rewritten as a function of velocity by fixing the value of ω: kτ(v) = k0 τ − b1v (3.8) where b1 = a1/dω. the thrusters’ efficiency also depends on the thrusters’ angle of attack θ. at small angle of attack (θ < θ∗1, where θ∗1 is the first critical incoming angle [10]), kτ(v, θ) can be written as kθ τ (v, θ) = k0 τ + ( k0 τ −k+ τ (v) ) [ sin ( θ θ∗1 π 2 ) − 1 ] (3.9) where kθ τ (v, θ) is the thrust coefficient at vehicle velocity equals to v and the thrusters’ angle of attack equals to θ,k+ τ (v) is determined by equation 3.8, and θ∗1 is the first critical incoming angle: θ∗1 = π/2− a1v. during bollard pull tests (at v = 0), the thrust of the actuators can be obtained as a function of the input code or the current. the maximum thrust of the thruster can be written as a function of kτ: copyright c© 2017 assa. adv syst sci appl (2017) auv thrust allocation with variable constraints 5 τmax = k0 τρωmax ∣∣ωmax ∣∣d4 (3.10) where τmax is the maximum thrust generated by the thruster during the bollard pull test and ωmax is the number of revolution per second of the propeller at maximum thrust. hence, the maximum thrust τmax for a stern thruster at vehicle velocity v and the thrusters’ angle of attack θ can be obtained from equations (3.7), (3.9) and (3.10): τmax(v, θ) = τmax − cv [ 1− sin ( θ θ∗1 π 2 )] (3.11) where τmax(v, θ) is the maximum thrust generated by the thruster at vehicle velocity v and the thrusters’ angle of attack θ, c = b1/ρωmax ∣∣ωmax ∣∣d4. more complicated cases, when j < 0 or v < 0, can be obtained from the appropriate equations [10]. the steady state performance of a tunnel thruster can be fitted as an exponential function of the velocity ratio v/vjet [11]: τ(v) = τ(0)exp [ −c ( v vjet )2 ] where τ(v) is the the resulting force of the tunnel thruster for a vehicle velocity of v, τ(0) is the thruster’s performance in the bollard pull test, and vjet is the axial velocity of flow generated by the thruster: vjet = √ τ(0) ρs . (3.12) here, ρ is the density of water and s is the cross sectional area of the tunnel thruster. hence, the tunnel thruster constraints can be written as τ(v)max = τ(0)maxexp [ −c ( v vjet,lim )2 ] (3.13) where τ(v)max is the maximum thrust when the vehicle’s motion has velocity v and τ(0)max is the thruster constraint provided by the bollard pull tests of the thruster. 4. propulsion system setup the model of the auv “mt-2012” [12] propulsion system was used for algorithm simulation. the propulsion system consists of five thrusters: four thrusters located at the stern of the vehicle at an angle of α to the longitudinal axis (x axis), and a vertical tunnel thruster located at the forward part of the vehicle (fig. 4.3). the propulsion system of the vehicle can be described by the thrusters’ configuration matrix b [13]: b = [bb, bl, bu, br, bf ] where bd, bl, bu, br are column vectors for (down, left, up, right) the stern thrusters and bf is the column vector for the forward tunnel thruster. the column vectors in four dofs (surge, heave, pitch and yaw) take the form described below. copyright c© 2017 assa. adv syst sci appl (2017) 6 v.v. kostenko, a.yu. tolstonogov fig. 4.3. propulsion system of “mt-2012”. for the up and down stern thrusters: bu =  cosα − sinα 0 luz cosα− lux sinα  , bd =  cosα sinα 0 ldz cosα− ldx sinα  . for the left and right stern thrusters: bl =  cosα 0 lly cosα− llx sinα 0  , br =  cosα 0 lry cosα− lrx sinα 0  . for the vertical tunnel thruster: bf =  0 1 −ltunnelx 0  where ri = [lix, l i y, l i z] is the arm vector of thruster i (where i = u, d, l, r, f ). for this propulsion system, for reasons of symmetry, luz = lry = −ldz = −lly = lsternzy and lux = ldx = llx = lrx = lsternx . the parameters of the propulsion system are shown in table 4.1. table 4.1. parameters of the auv “mt-2012” propulsion system lsternx , m 1.88 α, grad 22.5◦ lsternzy , m 0.23 f sternlim , n 120.1/-68.2 ltunnelx , m 1.20 f tunnellim , n 122.1/-122.0 5. simulation setup the proposed allocation control method with variable constraints was tested on depth maneuvering at different velocities, 0.3, . . . , 1.8 ms−1 (corresponding to forward forces 3.6, . . . , 129.6 n, and 50 nm of pitch torque. copyright c© 2017 assa. adv syst sci appl (2017) auv thrust allocation with variable constraints 7 the optimal thrust allocation problem for different velocities was solved by matlab optimization toolbox. since q > 0 and r > 0 (equation (2.6)) this is a convex quadratic problem in τ, so the interior point method or active set method for convex optimization problem can be used. the results of the thrust allocation for different vehicle velocities are shown in fig. 5.4. fig. 5.4. thrust allocation for different velocities. there are some implementations of quadratic programming solvers in the c/c++ languages. these implementations are faster than the matlab version, and are easy to integrate in vehicle software. there are different types of quadratic programming solvers: qpoases implements the active set method [14], which allows using a hot start to reduce the calculation time for each new iteration using the data of the previous one, and there are several libraries with implementations of the interior point method: [15] and [16]. 6. conclusion to create a multi-purpose auv capable of undertaking both survey-style missions and low speed interactions requires an over-actuated propulsion system and a sophisticated thrust allocation algorithm. one method of extending the functionality of the vehicle is installing tunnel thrusters to the vehicle propulsion system. a new approach for thrust allocation algorithm is proposed in the present article. the method based on the quadratic cost optimal problem with constraints. the main feature of the method is that thrust constraints depend on the vehicle velocity, as well as the thrusters’ angle of attack. this method takes into account the decrease in the thrust of tunnel and stern thrusters at high velocities, and reallocates the thrust accordingly. the proposed algorithm was tested with a simulation model of the real auv “mt-2012” propulsion system. the propulsion system consists of five thrusters: four thrusters located at the stern of the vehicle and the vertical tunnel thruster located at the forward part of the vehicle. the result shows the adaptive thrust reallocation depending on vehicle velocity in case of depth maneuvering. copyright c© 2017 assa. adv syst sci appl (2017) 8 v.v. kostenko, a.yu. tolstonogov references 1. fossen, t. i. & johansen, t. a. (2013) control allocation—a survey. automatica, 1, 1087–1103. 2. naeem, w., sutton, r., & chudley, j. (2006) soft computing design of a linear quadratic gaussian controller for an unmanned surface vehicle. in 14th mediterranean conference on control and automation, palermo, italy, 1–6. 3. moreira, l. & guedes soares, c. (2008) h2 and hinf designs for diving and course control of an autonomous underwater vehicle in presence of waves. ieee journal of oceanic engineering, 33, 69–88, https://dx.doi.org/10.1109/icsmc.2012.6378129. 4. yan, z., chung, s. f., & wang, j. (2012) model predictive control of autonomous underwater vehicles based on the simplified dual neural network. in 2012 ieee international conference on systems, man, and cybernetics (smc), seoul, korea, 2551–2556, https://dx.doi.org/10.1109/icsmc.2012.6378129. 5. pham, c. d., spiten, c., & from, p. j. (2015) a control allocation approach to haptic control of underwater robots. in 2015 ieee international workshop on advanced robotics and its social impacts (arso), lyon, france, 1–6, https://dx.doi.org/10.1109/arso.2015.7428209. 6. saback, r., cesar, d., arnold, s., lepikson, h., santos, t., & albiez, j. (2016) fault-tolerant control allocation technique based on explicit optimization applied to an autonomous underwater vehicle. in oceans 2016 mts/ieee monterey, monterey, ca, 1–8, https://dx.doi.org/10.1109/oceans.2016.7761251. 7. huang, h., zhang, g., yang, y., xu, j., li, j., & wan, l. (2016) thrust optimal allocation for broad types of underwater vehicles. in advances in swarm intelligence: 7th international conference, icsi 2016, bali, indonesia, 491–502, https://dx.doi.org/10.1007/978-3-319-41009-8 53. 8. ben, m. & lee, t. h. (2011) unmanned rotorcraft systems. berlin, germany: springerverlag. 9. johansen, t., fossen, t., & tøndel, p. (2005) efficient optimal constrained control allocation via multiparametric programming. journal of guidance, control, and dynamics, 28, 506–515. 10. kim, j. (2009) thruster modeling and controller design for unmanned underwater vehicles (uuvs). in inzartsev, a. v. (ed.), underwater vehicles, (pp. 235–250), intechg, https://dx.doi.org/10.5772/6705. 11. palmer, a., hearn, g., & stevenson, p. (2008) modelling tunnel thrusters for autonomous underwater vehicles. ifac proceedings volumes, 41, 91–96. 12. matvienko, y. v., boreyko, a. v., lvov, o. y., & vaulin, y. v. (2015) kompleks robototehnicheskih sredstv dlja vypolnenija poiskovyh rabot i obsledovanija podvodnoj infrastruktury na shel’fe [complex robotic tools to perform searches and surveys of underwater infrastructure on the shelf]. podvodnye issledovaniya i robototekhnika, 1, 4–15, [in russian]. 13. fossen, t. (2011) handbook of marine craft hydrodynamics and motion control. chichester, uk: john wiley & sons, ltd. 14. ferreau, h., kirches, c., potschka, a., bock, h., & diehl, m. (2014) qpoases: a parametric active-set algorithm for quadratic programming. mathematical programming computation, 6, 327–363. 15. gertz, e. m. & wright, s. j. (2003) object-oriented software for quadratic programming. acm transactions on mathematical software, 1, 58–81. 16. yan, y. (2014), c++ implementation of the interior point methods (cpipm). [online]. available https://github.com/yimingyan/cppipm. copyright c© 2017 assa. adv syst sci appl (2017) https://dx.doi.org/10.1109/icsmc.2012.6378129 https://dx.doi.org/10.1109/icsmc.2012.6378129 https://dx.doi.org/10.1109/arso.2015.7428209 https://dx.doi.org/10.1109/oceans.2016.7761251 https://dx.doi.org/10.1007/978-3-319-41009-8_53 https://dx.doi.org/10.5772/6705 https://github.com/yimingyan/cppipm introduction problem formulation the model of thruster constraints propulsion system setup simulation setup conclusion microsoft word 729 concept of a universal smart machine with protection against spark discharge adv syst sci appl 2020; 01; 1-12 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/729 concept of a universal smart circuit breaker with protection against spark discharge andrey i. vlasov1*, sergei s. filin2, alexey i. krivoshein1,2 1) bauman moscow state technical university, moscow, russia 2) llc “konnekt”, moscow, russia abstract: the concept of a universal device that can provide diagnostics and monitoring of individual and collective power supply systems, the construction of life support systems, the state of household appliances, the internet of things devices and housing and communal services devices is considered. this paper proposes new design solutions able to improve the existing technologies for assessment of the quality of the energy flow by two orders of magnitude in terms of overall characteristics relative to the existing analogues. the developed smart microcontroller is designed to operate as an intelligent unit, to analyze the parameters of the ac voltage network and control the electromechanical elements of the smart circuit breaker, as well as to send data through a standard interface. the implementation of protection against spark discharge is proposed. the proposed solution is based on the application of cloud and boundary computing technology. the possibilities for improvement of the efficiency of design technologies and construction of energy-efficient buildings, provision of the industry and households with a new type of information management services (monitoring energy infrastructure, etc.) are presented. keywords: smart circuit breaker, monitoring, control, smart networks, power supply systems, internet of things. 1. introduction the creation of a technological basis for the development of innovative housing infrastructure that provides diagnostics and monitoring of the energy status of the infrastructures, individual and collective energy supply systems, building life support systems, monitoring of the state of household appliances, internet of things (iot) devices and housing and utilities infrastructure is a very relevant task and corresponds to the goals of the energy.net roadmap [12, 18]. this process is accompanied by the development of technological solutions in the field of energy status monitoring and predictive maintenance and repair [2, 3, 5, 14, 27-29]. one of the directions for the implementation of the roadmap action plan is the transition of the electrical support systems to the iot concept [20-22, 25, 26, 28]. at the moment, there are many solutions in the market aimed at protecting the household network against various malfunctions [1, 10, 12, 18, 23]. however, in general, they implement only certain types of protection. there are practically no solutions to implement comprehensive protection, including the following functions: overload and short circuit protection; current leakage protection; spark protection; overvoltage or overload protection; determining the cause of the shutdown after the automatic triggering; sending data to an external device; ability to shut off the circuit breaker at the command of the remote user. some companies have the solutions, partially performing these functions, in the form of several individual devices. for example, the solution of the company abb (https://new.abb.com) has the form of 5 separate devices, with a total width of 11 standard * corresponding author: vlasov@iu4.ru 2 a.i. vlasov, s.s. filin, a.i. krivoshein copyright ©2020 assa adv. in systems science and appl. (2020) modules. the solution based on schneider electric devices (https://www.se.com) represents 5 separate devices, with a total width of 7 standard modules, and legrand (https://legrand.ru/) has 4 separate devices in 7 standard modules. the installation of a system of several devices imposes additional requirements on the qualifications of electricians. it is obvious that modern requirements for the qualitative characteristics of energy consumption require wider use of intelligent controls and predictive services, which can be distinguished by the complexity of their functions and would be implemented in a publicprivate partnership [6, 8]. to solve this problem, the goal was set to develop a universal smart circuit breaker with protection against spark discharge, the concept of which is presented in this study. 2. power protection requirements power quality takes into account all aspects of electromagnetic compatibility, but characterizes only the electrical network. the permissible levels of electromagnetic compatibility established for it are called power quality indicators (pqi). the regulatory values of the pqi and the list of them are set by the standards (in the european union – en 50160, iec 61000-4-30: 2008; in russia – gost r 54149, gost 13109-97, etc.), which are the guidance for the developers of the electrical equipment connected to the network, in terms of their noise immunity, on the one hand, and the level of interference, on the other. if the noise immunity level of these technical devices is higher than the maximum permissible pqi values for the network, electromagnetic compatibility (emc) will be ensured [10]. the actual pqi values should be monitored using specialized measuring instruments under operating conditions, and the corresponding characteristics of emc – by the necessary tests during their development and production. indicators of power quality can be divided into three groups. the first group includes frequency deviations and voltage deviations, which are associated with the peculiarities of the process of production and transmission of electricity. the quality of frequency and voltage deviations regulation determines their level in the electric power system. the second group can be attributed to the pqis, characterizing the non-sinusoidal shape of the voltage curve, the asymmetry and voltage fluctuations. the sources of these distortions (emitters) are mainly electrical receivers. in order to take into account the impact of electromotive forces generated by such electrical receivers, it is necessary to take technical measures both at the stage of development and production and during their operation. the third group includes the pqis characterizing random electromagnetic phenomena and electrical processes, which are inseparably connected with the technological process of production, transmission, and consumption of electricity. these include voltage dips, overvoltage, and voltage pulses. the pqis of the first two groups are standardized, and two permissible levels are set for them: normal and limit. the pqis of the third group are not standardized, but the statistical information about them is of great importance for the normal operation of the electric power system. the frequency f is a system-wide parameter of the sos mode and is determined by the active power balance. in the event of a shortage of generated power in the system, the frequency is reduced to a value at which a new balance of generated and consumed power is established. on the contrary, with an excess of generated power, the frequency increases. the voltage deviation from the nominal value in the mode of the maximum (δu2 max) and minimum (δu2 min) load may differ from the allowable values. in the electrical installation rules, it is recommended to maintain the voltage in the cpu at a level not lower than 105% of the nominal under the highest load and not higher than 100% – at the lowest load. this requirement meets the principle of counter voltage regulation. the means of voltage regulation are used for the implementation. if voltage deviations are created under the influence of relatively slow load changes determined by its schedule, then rapid load changes create voltage fluctuations. voltage fluctuations are determined by the envelope of the concept of a universal smart circuit breaker with protection against spark discharge 3 copyright ©2020 assa adv. in systems science and appl. (2020) effective or amplitude voltage values and are characterized by the span δut and the frequency of repetition of voltage changes or the intervals between voltage changes. a significant share of the load in the electrical network is represented by the eds with nonlinear current-voltage characteristic. such eds consume current, the form of which differs significantly from the sinusoidal one. 3. development features of the smart circuit breaker 3.1. concept of a universal smart circuit breaker due to the lack of an integrated solution that implements the listed functions, a conceptual solution of a universal smart circuit breaker is proposed (fig. 1). when designing the circuit breaker, modern requirements for the protection of consumer and industrial electronics are taken into account. the functions of the circuit breaker include:  thermal overload protection;  electromagnetic protection against overcurrent and short circuit;  protection against leakage (triggered when a difference in phase and neutral currents occurs);  protection against the occurrence of an arc on the line (triggered when an arc of more than 0.15 seconds in duration appears);  shutdown by external user command (via rs-485 interfaces);  measurement of current and power consumed by the load;  measurement of voltage on the line, as well as the fundamental frequency and frequency spectrum of the network;  saving information about the reasons for shutting down the circuit breaker and sending data to external devices (via the rs-485 interface, since the device is designed to be used in conjunction with automation controllers, which typically have an rs-485 interface; if necessary, it is possible to transfer data from a smart circuit breaker via the wi-fi interface through the automation and monitoring controller). fig. 1. 3d model of a smart circuit breaker fig. 2. 3d model of the internal system of the smart circuit breaker the internal system of the smart circuit breaker, according to the principle of operation, is conditionally divided into two functional units (fig. 2): a) an electromechanical unit, including electromagnetic and thermal protection mechanisms, as well as transformers acting as current sensors; and b) an electrical unit including an analysis board with an intelligent controller installed on it (intelligent control unit). 4 a.i. vlasov, s.s. filin, a.i. krivoshein copyright ©2020 assa adv. in systems science and appl. (2020) in the electromechanical unit during development it is advisable to implement the functions of thermal overload protection, electromagnetic short-circuit protection, current leakage protection function, an electrical circuit disconnection system, and a current sensor for measuring the phase current. the electrical unit must perform the functions of a smart circuit breaker that are impossible or impractical to implement in the form of electromechanical automation. the intelligent electronics unit must perform data processing functions from the phase current sensor and transformer to measure the differential current, measure the voltage at the input and output of the smart circuit breaker, determine the network parameters (such as power consumption, frequency, sine wave distortion level), switch off the circuit breaker in the case of overvoltage or arc striking on the line, send data to an external device and control the display. a block diagram of a universal smart circuit breaker is shown in fig. 3. fig. 3. block diagram of the smart circuit breaker: 1 – 220 v mains; 2 – electromechanical unit; 3 – analysis module; 4 – electromechanical module; 5 – 200 v input; 6 – thermal protection unit; 7 – short circuit protection unit; 8 – differential current sensor; 9 – current in phase sensor; 10 – relay with arc suppression; 11 – output 220v; 12 – load; 13 – voltage divisor with offset; 14 – adc; 15 – power supply; 16 – uart; 17 – pwm outputs; 18 – inputs; 19 – control output; 20 – uart-rs485 converter; 21 – rgb-indication; 22 – control buttons; 23 – amplifier; 24 – rs-485 interface. considering that the data processing and decision-making functions in the electronics unit are implemented by the microchip, the functions of the analysis board are reduced to ensuring the functioning of the chip, converting signals from the electromechanical unit to acceptable levels for the microcontroller, and converting the control signal of the microcircuit to shutting off the electromechanical unit. in addition, on the analysis board, the indication elements of the state of the circuit breaker should be installed, and buttons for manual control of the smart circuit breaker. the necessary functions of the analysis board include: 1. conversion of an alternating analog signal from the ca input (peak voltage is 400v) to a proportional analog signal in the range of 0-uadc (with an offset of 0.5uadc), where uadc is the maximum voltage of the microcontroller adc; 2. the conversion of the ac analog signal from the ca output (peak voltage is 400v) to the analog signal in the range of 0-uadc (with an offset of 0.5uadc); concept of a universal smart circuit breaker with protection against spark discharge 5 copyright ©2020 assa adv. in systems science and appl. (2020) 3. conversion of an alternating analog signal from a phase current sensor (peak voltage is 7v) to an analog signal in the range of 0-uadc (with an offset of 0.5uadc); 4. conversion of an alternating analog signal from a differential current sensor (peak voltage is 7v) to an analog signal in the range of 0-uadc (with an offset of 0.5uadc); 5. amplification of the control signal of the ic to a power sufficient to trigger the electromagnetic pusher and turn off the smart circuit breaker; 6. the presence of an indication that allows displaying the status of the circuit breaker and the reason for shutdown; 7. the presence of a button for manual control of the mode of operation of the smart circuit breaker, and a button for resetting the electrical unit of the smart circuit breaker; 8. the presence of a converter interface data output of the microcontroller to the rs-485 interface. such a set of functions imposes increased requirements on the element base, which makes it expedient to create a specialized “smart controller” as a special ic. 3.2. approach to the implementation of protection against spark an important component of the electrical protection of the housing infrastructure is protection against the occurrence of a spark or arc, both longitudinal and transverse. both types of arc result in heating of the conductor, melting of the insulation, and may cause a fire. at the same time, none of the standard means of automatic protection, such as a circuit breaker, or a differential circuit breaker, can detect such a fault. there are specialized means of protection against the occurrence of an arc in a line, the so-called afdds (arc fault detection devices). unlike standard circuit breakers based exclusively on electro-mechanics, almost all afdd devices are designed as a combination of an electrical board analyzing network parameters and a network disconnecting switch triggered by command of an electrical board [9, 17]. recently, relatively cheap afdd devices have begun to appear, but their sensitivity remains controversial. based on literature review [15, 16, 17, 26, 27, etc.], one can state that currently, most developers use simplified mechanisms to generate sparks: interrupted circuits based on mechanical contact opening, twisted wires, etc. in some cases, a spark generation unit with the graphite-copper contact is used to demonstrate the operation of the spark detection device. however, the sparking of graphite-copper contact is very different from the sparking of the copper-copper contact under real-life conditions. this paper proposes a new approach based on a special device that provides automatic spark generation at the copper-copper contact; an experimental test bench was built with a spark generating device between conductors (fig. 4). 6 a.i. vlasov, s.s. filin, a.i. krivoshein copyright ©2020 assa adv. in systems science and appl. (2020) fig. 4. device generating a spark in the composition of the experimental stand the proposed solution provides for fixing two wires and ensuring their contact using a rotary mechanism. then, voltage is applied and the adaptive spark control algorithm is initiated, which controls the interruption and restoration of the contact based on the current sensor readings in order to ensure maximum possible sparking. the experiments with a spark under various types of load have shown that when the arc occurs, a sinusoid of alternating current is distorted and the noise-like signals are generated. spark detection is based on analyzing the network spectrum of a signal by current and voltage. the occurrence of a spark generates a noise-like distortion of the network, often of small amplitude, especially when a longitudinal spark occurs. due to the high electrical capacitance and inductance of household wiring, the amplitude of the signal generated by the spark weakens with increasing distance from the point where the spark originated. also, the signal weakens when the active or impulse load in the network is connected between the location of the spark and the location of the afdd device [4, 15, 16]. the determination of the presence of the arc on the line is carried out through the analysis of the spectrum of the network, as well as the analysis of the shape of the sinusoid ac current. when the arc occurs, the breakdown occurs at a certain phase of the sinusoid of the mains voltage, then the load increases sharply and the sinusoid is distorted. this causes the occurrence of high-frequency harmonics, which can be detected by continuously analyzing the spectrum of the network. in this case, another problem arises due to the absence of a difference between signal distortions caused by the appearance of an arc on the line before and after the circuit breaker. the solution to this problem is the introduction of a filtering element into the circuit of an automaton that changes the amplitude of high harmonics. by the difference of the signal before and after such an element, it will be possible to determine which side of the circuit is the source of high harmonics. if the arc originated from the voltage source, the level of high harmonics from the load side will be lower. in the case of an arc on the load side, the level of harmonics will be lower on the side of the voltage source. in the simplest case, such an element could be an inductor. the measurement of the amplitude of high harmonics before and after the inductance, taking into account the unpredictability of their composition, is made by measuring the amplitude of the signal passed through the hf filter and the half-wave rectifier. the calculation of the inductance of the input coil, necessary for the possibility of determining the difference of the amplitudes of the hf signal before and after the filter element, is carried out as follows. since the voltage difference at the outputs of the rectifiers concept of a universal smart circuit breaker with protection against spark discharge 7 copyright ©2020 assa adv. in systems science and appl. (2020) is actually measured, it is first necessary to determine the minimum step of the adc of the microcontroller. the resolution of the microcontroller adc is 12 bits, the range of the measured voltage is 0-1v. calculate the minimum step by the formula (1): n 12 u 1 0.244 2 2 adc stp v u mv   (1) where stpu is the minimum measurement step for the adc, uadc is the upper limit of the adc measurement range, n is the precision of the adc. the noise level at the output of the rectifier, determined empirically, does not exceed 0.1 mv, the error level of the adc is 0.5 minimum step. i.e. the useful step of the adc is approximately 0.5 mv, or 0.05% of the range measured by the microcontroller voltage. the arc can occur both in the presence of a load and in its absence. in this case, the arc itself will be the load, the value of which will depend on the distance between the conductors on which the arc originated. the drop in the amplitude of the rf signal at the coil will depend on the value of the coil impedance and the load, i.e. on the load level and inductance value. the greater the impedance of the load, the smaller the drop in rf on the coil, and hence the difference in the signal level. therefore, one can calculate the required inductance value at the lowest possible load. the occurrence of the arc, according to the source, is possible at a voltage not lower than 10 v and a current of at least 0.1 a. based on these parameters, one can calculate the required resistance of the filter element (2-4): 0.05%induct induct r x x x   , (2) 0.0005 0.9995induct xr x  , (3) 0.0005induct rx x  (4) with a minimum current of 0.1 a and a voltage of 250 v, the total resistance of the arc is (formula 5): 2500 max r min u x om i   (5) then 1.25 inductx om the higher the average frequency of the arc spectrum, the greater the inductance resistance and the greater the voltage difference after rectification. let the minimum average frequency of harmonics f = 1 khz. the resistance of inductance inductx is determined by the formula 6: 2inductx fl (6) then the inductance is calculated by the formula 7: 200 2 ix l µh f    (7) thus, in order to determine the position of the source of the arc, it is necessary to include in the circuit of the automaton an inductance coil at least l = 200 µh. 8 a.i. vlasov, s.s. filin, a.i. krivoshein copyright ©2020 assa adv. in systems science and appl. (2020) 4. results modern comprehensive power grid protection should include: cable overheating protection, short circuit protection, sparking protection, protection against significant voltage deviation, and current leakage protection. the monitoring function is designed to control parameters such as voltage, current strength, power, and electricity consumption. in addition, monitoring in smart security systems should make it possible to determine the state of the system and the cause of the shutdown. the concept of implementing an intelligent smart circuit breaker for the intellectualization of the energy sector infrastructure includes two main functional blocks: an integrated power grid protection unit and a unit for monitoring the power grid parameters. fig. 5 shows the interaction of components of the integrated power system of the russian federation [31]. individual retail companies interact with end-users; they ensure the accounting and management of end consumers’ operation modes. the aim of the work is to develop a concept of implementing universal, intelligent smart circuit breakers, interacting with specific consumers, which provide additional functions to monitor the state of the power grid and provide recommendations for predictive repair. fig. 5. functional scheme of integration of smart circuit breakers into intelligent energy infrastructure by the example of the russian federation the proposed smart circuit breaker can be used anywhere, both in private households and in the manufacturing sector. this device provides comprehensive power grid protection and occupies a small part of the electrical control panel. in addition to the basic protection functions, the smart circuit breaker makes it possible to facilitate keeping records of electricity consumption and the quality of power supply, as well as to reduce the number of emergencies due to deeper diagnosis. the advanced functionality of the proposed smart circuit breaker is ensured by using a new specialized microcontroller (smc). concept of a universal smart circuit breaker with protection against spark discharge 9 copyright ©2020 assa adv. in systems science and appl. (2020) the use of the proposed smc ensures the functioning of an intelligent unit for analyzing the parameters of the ac voltage network and controlling the electromechanical elements of the smart automat, as well as sending data through a standard interface. the smc as part of the smart circuit breaker allows measurement and analysis of the following parameters:  ac input voltage;  current in phase;  difference of phase and zero currents;  power consumed by the load connected via a smart circuit breaker;  frequency spectrum of alternating voltage (analysis of occurrence of a spark). switching off the smart circuit breaker on a command from the smc occurs through a signal to its control outputs connected to the electromechanical controls of the smart circuit breaker. the logic of switching off the automaton is given by the algorithm of its operation. the use of a special microcontroller allows reducing the dimensions of the analysis board. also, the use of an smc makes it possible to reduce the prime cost of a smart circuit breaker in the case of mass production. 5. conclusion the proposed solution is based on the application of cloud and boundary computing technology, and fully allows for remote diagnostics and monitoring of the energy status of the light and digital infrastructures, individual and collective energy systems, life support of buildings, the state of household appliances, iot devices and housing and communal services. this underlines the ability of the smart circuit breaker to be widely used [7, 11, 13, 19, 24, 27, 30]. as part of the technological solution, this device provides comprehensive protection of the electrical network, while occupying a small part of the electrical panel, and surpasses the existing analogs. in addition to the basic protection functions, the smart circuit breaker will make it easier to record data on electricity consumption and the quality of power supply, and reduce the number of emergency situations by simplifying their diagnosis. it can be stated that innovative smart circuit breakers, the operation of which is based on the principles of the internet of things, become an important aspect of social infrastructure. the creation of specialized controllers should ensure at the minimum expense the implementation of monitoring data analysis functions, the transfer of smart circuit breaker data through the service interface to external automation controllers. this paper proposes an implementation concept and recommendations on the configuration of a specialized controller for a smart circuit breaker, designed to analyze the parameters of an ac network. according to the results of the implementation of the technological basis for the development of innovative social infrastructure based on smart circuit breakers using a specialized microcontroller, it can be argued that this approach simplifies development by reducing energy and resource consumption, and the selected controller configuration is the most efficient and fully complies with the requirements imposed on innovative smart circuit breakers. directions of further research will be focused on improving the computational efficiency of monitoring algorithms and forecasting the state of the power grid with the possibility of implementing the predictive repair functions. acknowledgments the research was conducted with the support of the ministry of science and education of russia within the framework of the project under the agreement no. 14.579.21.0158, id rfmefi57918x0158. 10 a.i. vlasov, s.s. filin, a.i. krivoshein copyright ©2020 assa adv. in systems science and appl. 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(2019). sustainable energy systems in the digital economy: concept of smart machines. entrepreneurship and sustainabilility issues, 6(4): 1975-1986. adv syst sci appl 2018; 2:11–25 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/604 surrogates for the matrix `0-quasinorm in sparse feedback design: numerical study of the efficiency alexey bykov1, pavel shcherbakov1,2∗ 1institute of control sciences, russian academy of sciences, moscow, russia 2institute for systems analysis, federal research center “computer science and control”, russian academy of sciences, moscow, russia abstract: some formulations of the optimal control problem require the resulting controller to be sparse; i.e., to contain zero elements in the gain matrix. on one hand, sparse feedback leads to the drop of performance as compared to the optimal control; on the other hand, it confers useful properties to the system. for instance, sparse controllers allow to design distributed systems with decentralized feedback. some sparse formulations require the gain matrix of the controller to have a special sparse structure which is characterized by the presence of zero rows in the matrix. in this paper, various approximations to the number of nonzero rows of a matrix are considered and applied to sparse feedback design in optimal control problems for linear systems. along with a popular approach based on using the matrix `1-norm, more complex nonconvex surrogates are proposed and discussed, those surrogates being minimized via special numerical procedures. the efficiency of the approximations is compared via numerical experiments. keywords: sparse control, `1-optimization, linear systems, optimal control, linear matrix inequalities 1. introduction optimal control problems in linear theory may have a requirement imposed on the structure of the controller along with the necessity of optimizing the objective function. one example of such requirement would be sparsity of the controller, which is understood as having many zero elements in its gain matrix; e.g., see [12–14,22]. elementwise and block sparsity allows to design distributed systems with decentralized control. this additional sparsity requirement naturally leads to a drop in performance, which is explained by the fact that the set of sparse controllers is contained in the feasible set for the corresponding classical formulation. nevertheless, in some formulations of control problems, a reasonable drop in performance is allowed for the sake of having a sparse control. for instance, the objective function in the standard lqr problem has the form of a quadratic functional, which is related to energy consumption in the system; e.g. fuel consumption of an aircraft. on one hand, a sparse control leads to worse (higher) values of the objective function, i.e., fuel consumption is increased; on the other hand, it augments the system with various useful properties. for example, decentralized control makes the system more reliable and fail-safe; decreasing the number of active actuators allows to slow down an equipment deterioration, thereby increasing the lifetime of the system. in [15], a special kind of sparsity was introduced, which is different from a typical elementwise sparsity. the authors considered a controller to be sparse if its gain matrix has ∗corresponding author: alexey.bykov.mipt@gmail.com, cavour118@mail.ru http://ijassa.ipu.ru/ 12 a. bykov, p. shcherbakov many zero rows or columns. such controllers facilitate ease of hardware implementation of the control systems, such as a reduction of the number of actuators or sensors and the amount of information transmitted via control channels. a direct minimization of the number of nonzero rows or columns of a matrix is known to be np-hard, since it involves combinatorial search. as the dimensions of the problem increase, the direct approach becomes inapplicable due to its exponential time complexity. in [15, 16], an approach was proposed which is based on solving convex surrogate problem instead of the original nonconvex one. this substitution is implemented using special matrix norms which are convex approximations (also called surrogates) to the number of nonzero rows/columns of a matrix (nonconvex matrix `0-quasinorms). this heuristic, however, does not guarantee an occurrence of zero rows/columns in the gain matrix, and by all appearances strict results cannot be obtained due to nonconvexity of the original problem. nevertheless, the exploited heuristic is shown to be efficient and the approach seems to work for numerous examples. such non-strict approaches require computer simulations for proving their efficiency, and in the current paper we put emphasis on the numerical study. apart from convex surrogates, some papers [2–4, 13] suggest using nonconvex approximations for more efficient detection of sparse solutions in case of vector and matrix variables. the main goal of this paper consists in comparing the efficiency of different approximations to the matrix `0-quasinorm, which can be used for sparse feedback design in optimal control problems. also, a general scheme for gaining trade-off between optimality and sparsity of the solution is proposed which allows to use various approximations in a similar way. the key aspects of this paper are the formulation of several surrogates for the matrix `0-quasinorm, which can be used for promoting matrix row-sparsity, designing the scheme of numerical study, and the analysis of the results of experiments. both models of simple systems and linearized models of real systems were chosen as test cases for conducting the numerical experiment. also, various software for numerical computing [5, 18, 19], and algorithms for nonconvex optimization [3, 21] were used. numerical study is the central part of this paper due to the following reason. although the approach based on convex surrogates heuristic is said to be efficient, one cannot be sure about getting zero rows in the gain matrix. thus, different surrogates demonstrate different efficiency for different problems; this paper aims to study the efficiency of various approximations. our paper essentially exploits linear matrix inequality (lmi, [1]) technique, which allows to formulate many classical problems from control theory in the form of semidefinite programs (sdp, [20]). 2. sparse feedback design in optimal control problems consider the following control system: ẋ = ax+bu, x(0) = x0, (2.1) where x(t) ∈ rn is the state vector, u(t) ∈ rp is the control input,a ∈ rn×n,b ∈ rn×p, and the pair (a,b) is assumed to be controllable. if system (2.1) is not stable, a stabilizing control is to be found, which also has to minimize an objective function along the system’s trajectories. various formulations differ in the form of the control input; in this paper we exploit a static state linear feedback form: u = kx, k ∈ rp×n. (2.2) from this point, we consider the lqr (linear control regulation) problem, the approach and the numerical experiment being applicable to other optimal control problems. in the lqr copyright c© 2018 assa. adv syst sci appl (2018) surrogates for the matrix `0-quasinorm 13 problem, the objective function has the form of the following quadratic functional: j = ∞∫ 0 ( x>rx+ u>su ) dt, (2.3) where r ∈ rn×n and s ∈ rp×p are positive–definite matrices. it is worth emphasizing that minimizing functional (2.3) automatically leads to stabilizing system (2.1). indeed, if the system was unstable, integral (2.3) would be divergent; however, since the pair (a,b) is controllable, there exists a controller, which makes the functional j finitely valued. the standard way of solving the lqr problem consists in finding a solution to the associated algebraic riccati equation; however, in this paper we adhere to another approach (e.g., see [7]) based on the linear matrix inequality technique. in [7], the classical lqr problem is reduced to the following sdp problem: problem 1: let popt, yopt be the solution of the sdp trp −→ max s.t.( ap + pa> +by + y >b> p y > ∗ −r−1 0 ∗ ∗ −s−1 ) 4 0, (2.4) then the controller (2.2) with gain matrix kopt = yoptp −1 opt (2.5) stabilizes system (2.1), the objective functional (2.3) being minimized, and the optimal value of the functional is equal to jopt = x>0 p −1 optx0. (2.6) this approach involving lmis can be very useful, for instance, for robust formulations where the matrices of the system contain uncertainty. but what’s more important for the current paper, it is the lmi-based approach that allows to solve the problem in sparse formulations. as it was mentioned, the sparse formulation that we consider, imposes an additional requirement on the structure of the gain matrix; namely the intention is to have as many zero rows in matrix k as possible. clearly, due to the structural constraint on the matrix k, the feasible set of controllers becomes smaller, so that a drop in performance of the sparse controller is unavoidable, and we do not want it to be significant. the problem of comparison of the two controllers with respect to a quadratic performance index is not as straightforward as it may seem; e.g., see [8]. if one compares the values of the functional j(p ) = x>p−1x corresponding to the two different stabilizing controllers k1 and k2 with the corresponding matrices p1 and p2, the sign of the inequality j(p1) ≶ j(p2) may change depending on the initial condition x0. in this paper we compare sparse controllers with the optimal ones, which, by construction, yield the optimal value of the functional j for any initial condition x0. in other words, the sparse controller with matrix psp will always lose the optimal controller with matrix popt in terms of the value of the quadratic functional j . the magnitude of the loss (performance drop) for a given initial condition x0 is jsp jopt = x>0 p −1 sp x0 x>0 p −1 optx0 ; copyright c© 2018 assa. adv syst sci appl (2018) 14 a. bykov, p. shcherbakov this value, obviously, can change significantly for various x0. there is a well-known approach [11] based on averaging the values of j over initial conditions uniformly distributed on the unit n-dimensional sphere. it is not hard to show that the mathematical expectation of j is equal to e(x>0 p −1x0) = 1 n trp−1; thus the magnitude of the sparse-to-optimal control loss “on the average” equals jsp jopt = trp−1sp trp−1opt . (2.7) in this paper we adhere to the described approach based on “averaging”; i.e., we use relation (2.7). in [16], a procedure is proposed to design sparse controllers in optimal control problems, in particular, in the lqr problem. the key stage of the procedure, which is responsible for the detection of the sparse structure of the controller, consists in minimizing the 1∞-norm of the gain matrix. this norm plays the role of a convex surrogate for the number of nonzero rows of a matrix and is defined as follows: ‖x‖1∞ = n∑ i=1 max 16j6p |xij|, x ∈ rn×p. (2.8) the three-step procedure from [16], generalized to the case of arbitrary approximation to the matrix `0-quasinorm, was used as a basis for the numerical study in this paper. the key aspects of the procedure are given below. step 1 (optimal control design). the classical lqr problem is solved (problem 1) and the value trp−1opt is fixed. this value corresponds to the optimal value jopt of the quadratic functional j . step 2 (detection of sparse structure). let α > 1 denote the maximum admissible loss in the quadratic performance index that appears due to sparsity; i.e., the following condition is imposed: trp−1sp trp−1opt 6 α, α > 1. so, α is the coefficient of the allowed performance drop “on average” and it acts as the parameter of the following sdp problem: problem 2: ‖y ‖(sp) −→ min s.t. (2.4),( x i i p ) < 0, and trx 6 α trp−1opt . (2.9) note that, similarly to problem 1, the gain matrix k does not explicitly appear in the constraints or in the objective function of the sdp. thus, the variables are positive–definite matrices p and x and an auxiliary matrix variable y = kp ; the latter was introduced to transform the constraints to the form of linear matrix inequalities. the minimized function in problem 2 is denoted by ‖·‖(sp) and it plays the role of an approximation to the matrix `0-quasinorm. the 1∞-norm is one of such approximations and copyright c© 2018 assa. adv syst sci appl (2018) surrogates for the matrix `0-quasinorm 15 it allows to solve convex problem instead of the nonconvex one. there are more complex and efficient surrogates, some of which are described in the next section. let the matrix y0 be the solution of problem 2 and assume it contains rows with all elements being equal or close to zero. these rows are memorized and kept as strict zeros, which automatically means that the gain matrix k0 = y0p −1 will also contain the same zero rows. the latter fact follows from the rules of the matrix multiplication. note that the value of the objective function does not exceed αjopt “on average”. step 3 (optimization over sparse-structured controllers). the lqr problem is again solved but with sparse structure of the controller being fixed as found at step 2. in other words, problem 1 is solved with additional constraints on the matrix variable y . to sum up, the procedure described above gives a way to design a controller which may have a sparse structure at the expense of a reasonable drop in performance; thus, a trade-off between optimality and sparsity of the control can be achieved. 3. approximations to the matrix `0-quasinorm so, the key step in the sparse feedback design procedure is the sparsity detection step. the detection is performed via minimizing a surrogate for the number of nonzero rows of a matrix. below we consider four various approximations to the matrix `0-quasinorm, efficiency of which is compared via numerical experiments. all approximations are described as functions of ri, where ri qualifies the magnitude of the values of the ith row of the matrix y . if a row of the matrix is treated as a vector, then ri can be, e.g., one of the standard vector norms: l1-, l2-, or l∞-norm. a matrix row is called a zero row if and only if all of its elements are zeros. also, if any of the mentioned norm approaches zero, then all of the row elements approach zero. numerical experiments show that the choice of the particular norm is not essential, so, to be specific, we chose l∞-norm, the maximum of the absolute values of the components. thus, we define ri as follows: ri = max 16j6n |yij|, y = ‖yij‖ ∈ rp×n. (3.10) 3.1. 1∞-norm the 1∞-norm (2.8) was introduced in [16]; in terms of ri (3.10) it can be written as: ‖y ‖1∞ = p∑ i=1 ri. (3.11) this function is convex, therefore its minimization subject to constraints (2.4), (2.9) is an sdp problem, which can be efficiently solved by means of convex optimization tools [20]. 3.2. weighted 1∞-norm in [3], the authors developed the ideas of `1-optimization for promoting sparsity in case of vector variables. they introduced a new iterative procedure which consists in solving a sequence of weighted `1-optimization problems. at each iteration, the weight coefficients are updated due to the solution received at the previous step. the weighted 1∞-norm is defined as follows: ‖y ‖w1∞ = p∑ i=1 wiri. (3.12) the iterative procedure from [3] can be schematically written down as follows: algorithm 1: copyright c© 2018 assa. adv syst sci appl (2018) 16 a. bykov, p. shcherbakov 1. initial estimate: l = 0, w (0) i = 1, i = 1, . . . , p. 2. the convex problem with currently accepted weights is solved: r(l) = argmin r [ p∑ i=1 w (l) i ri ] s.t. (2.4), (2.9). 3. the weights are updated: w (l+1) i = 1 |r(l)i |+ ε , ε > 0. 4. quit the loop if the procedure converged or the maximum number of iterations is exceeded. if both conditions are false, go back to step 2. the authors of the original paper [3] confirmed via numerous examples that the new approach involving the minimization of the weighted `1-norm performs much better than the “standard” non-weighted approach. this can be explained by the fact that the sequence of convex problems can better approximate the original nonconvex problem of minimizing the matrix `0-quasinorm. 3.3. nonconvex sparsity detector, nsd in [2], a nonconvex sparsity-promoting surrogate was introduced: ‖y ‖nsd = p∑ i=1 ri ∏ j 6=i rj rj + 1 . (3.13) the authors provide some motivation for this particular approximation and demonstrate its efficiency as compared to the 1∞-norm. the function (3.13) is shown to be a dc-function (difference of convex, see [6]) and for its minimization, the concave-convex procedure (abbr. cccp, e.g., see [21]) is proposed. let the function (3.13) be represented as a dc-function: nsd(r) = u(r)− v (r). then the procedure for its minimization is given below: algorithm 2: 1. initial estimate: l = 0, r (0) i = max 16j6n |yij|, i = 1, . . . , p, where yopt = ‖yij‖ is the solution of problem 1. 2. the following convex problem is solved: r(l+1) = argmin r [ u(r)− r>∇v ( r(l) )] s.t. (2.4), (2.9). 3. quit the loop if the procedure converged or the maximum number of iterations is exceeded. if both conditions are false, go back to step 2. in [9], the results on the convergence of the cccp procedure are presented; in the general case, global convergence is not guaranteed, since the problem is not convex. however, for many applications this procedure yields an appropriate solution. copyright c© 2018 assa. adv syst sci appl (2018) surrogates for the matrix `0-quasinorm 17 3.4. nonconvex approximation log-sum in [3, 4, 13], the function log-sum (see definition below) is shown to present a more precise approximation to the `0-norm of a vector than convex approximations. indeed, let us consider the following scalar functions (see fig. 3.1) which can be used as penalty functions for nonzero values: f0(x) = 1[x 6=0], f1(x) = |x|, flog,ε(x) = cε log ( 1 + |x| ε ) ; the factor cε is chosen in a such way that the equality flog,ε(1) = 1 = f0(1) = f1(1) holds. -1 0 1 x 0 1 f 0 (x) f 1 (x) f log, ǫ (x) fig. 3.1. approximations to the `0-quasinorm the derivative of the function flog,ε(x) in the positive vicinity of the origin grows approximately like 1 ε as ε −→ 0. this leads to a relatively big amount of penalty for small but not exactly zero values. moreover, note that flog,ε(x) −→ f0(x) as ε −→ 0. unfortunately, as ε tends to zero, the minimization of the function log-sum becomes difficult; for heuristic of choosing ε see [3]. the function log-sum adapted for promoting matrix row-sparsity can be written as ‖y ‖log-sum = p∑ i=1 log (1 + ri) . (3.14) since the function (3.14) is concave, it cannot be minimized via convex optimization framework. however, it can be treated as a special case of dc-function, so, it can be minimized via concave-convex procedure (cccp) just like the nsd function. 4. results of numerical experiments in the previous section, several well-known and relatively new approximations to the number of nonzero rows of a matrix were considered. among these surrogates are the 1∞-norm copyright c© 2018 assa. adv syst sci appl (2018) 18 a. bykov, p. shcherbakov exploited in [16], and the nsd function introduced in [2]. both approximations, however, have not yet received enough approval from numerical experiments; therefore, they were chosen as contestants for the experiment in this paper. also, two other approximations (the weighted 1∞-norm and the function log-sum), which were earlier applied in the case of vector variables, were adapted for promoting matrix row-sparsity. all these surrogates were used at step 2 of the three-step procedure to design sparse controllers in the numerical experiment. all the examples described below were solved with matlab using the cvx framework [5] for solving convex optimization problems and the sdpt3 solver [19]. it was checked that using other solvers such as sedumi [18] did not give any specific benefits or drawbacks. the main goal of the experiment was to compare the efficiency of approximations (3.11), (3.12), (3.13), (3.14) used for getting zero rows in the matrix y0 at step 2 of the threestep procedure, which allows to design sparse controllers in the lqr problem. during the experiments, both models of simple systems and linearized models of real systems were considered. the latter were taken from the compleib, a popular collection of test problems from control system design and related fields. the general scheme of the experiment is as follows: algorithm 3 (scheme of experiment): 1. the classical lqr problem is solved (problem 1) and the value trp−1opt is fixed. this value corresponds to the optimal value jopt of the quadratic functional j . 2. the initial value of the maximum admissible loss in the quadratic performance index is chosen: αmin = 1 + ε, ε > 0. 3. the magnitude of α is increased inside the loop until the most sparse control can be acquired. for every value of α the following steps are executed: (a) the detection of zero rows is performed (problem 2) for a given α̂. the zero structure obtained from the solution is fixed; the number of zero rows denoted by nnz is memorized. (b) the lqr problem (problem 1) is solved with the sparse structure of the controller being fixed as found at the previous step. the performance drop is determined by the relation αsp = trp−1 sp trp−1 opt . hence, every iteration of the loop gives us two points at the plane with coordinates the number of nonzero rows and the performance drop. these points are (nnz, α̂) and (nnz, αsp), where α̂ is the value of the maximum admissible loss which leads to the detection of zero rows, and αsp is the actual loss in performance, which corresponds to the detected sparse structure of the controller. it is worth noting that αsp might be drastically less than α̂. 4. the brute-force enumeration of all possible combinations of zero rows in the gain matrix is performed and for every combination and the corresponding sparse structure the lqr problem (problem 1) is solved. note that such a brute-force search is performed exceptionally within the experimental setup for illustration purposes only; it is not needed in the respective theorems. for the combination with index k ∈ [1; 2p − 2] the performance drop is determined by the relation αbf,k = trp−1 bf,k trp−1 opt . note that we consider 2p − 2 combinations of zero rows instead of 2p, since two cases are excluded. the first one corresponds to the absence of zero rows, i.e., to the classical formulation without the sparsity requirement. the second obvious case corresponds to the situation where all rows of the gain matrix are zero rows. the points (nnz, αsp) obtained as an output of the algorithm 3 are related to the pareto optimality, a well-known concept from the theory of multiobjective optimization. namely, a copyright c© 2018 assa. adv syst sci appl (2018) surrogates for the matrix `0-quasinorm 19 point is said to be pareto-optimal if none of the criteria can be improved without worsening the others. also, note that the values of αsp might not be optimal for a given number of nonzero rows nnz, since we solve the surrogate for the original nonconvex problem and the global convergence is not guaranteed. nevertheless, varying the importance of one or another criterion we can gain a necessary trade-off between optimality and sparsity. 4.1. examples from compleib the experiment was conducted in accordance with the described scheme for several problems from the compleib collection: “ac1”, “ac9”, “ac12”, “he3”, “he4” (linearized models of aircrafts and helicopters). these examples are of interest in terms of the dimension of the control input. the behaviour of various approximations to the matrix l0-quasinorm for all five models is quite similar, so it is reasonable to present the results for just one example – model “he4”. this eight-order model represents a twin-engine, multi-purpose military helicopter. let us recall that the behaviour of the system is described by equations (2.1). the values of the entries of the matrices a and b are not presented for space considerations; they can be found in the compleib documentation. the weight matrices r and s in the quadratic functional (2.3) were set to identity. the results of the experiment conducted in accordance with algorithm 3 are presented in figs. 4.2, 4.3, and 4.4. 2 3 4 the number of nonzero rows 1 1.002 1.004 1.006 1.008 1.01 1.012 1.014 1.016 1.018 r e la x a ti o n c o e ff ic ie n t α results for problem he4 1∞-norm w1∞-norm nsd sum-log fig. 4.2. “he4”. the detection of zero rows for various α̂ copyright c© 2018 assa. adv syst sci appl (2018) 20 a. bykov, p. shcherbakov figure 4.2 shows the points (nnz, α̂) obtained for the functions (3.11), (3.12), (3.13), (3.14) in accordance with algorithm 3. this plot demonstrates how big one should set the admissible loss in performance in order to get zero rows in the gain matrix of the controller. one can see that using approximations (3.12), (3.13), (3.14) leads to the detection of sparse structure for smaller values of α̂. it turned out that for any surrogate except for the 1∞-norm, in order to detect two zero rows out of the total four, we needed to set α̂ slightly less than 1.005, which corresponds to the performance drop of just 0.5%. the transition from one zero row to two zero rows happens at close values of α̂, therefore using an insufficiently dense grid of α̂ values can mislead into believing that both zero rows appear simultaneously. such simultaneous appearances of two zero rows were not noticed during the experiment, and they obviously should not happen due to the following reason. every next zero row of the gain matrix makes the feasible set of controllers smaller, so the optimal value of the quadratic functional for the controller with n+ 1 zero rows is a priori worse than the corresponding value for the controller with n zero rows. 2 3 4 the number of nonzero rows 1 1.002 1.004 1.006 1.008 1.01 1.012 1.014 1.016 1.018 a c tu a l lo s s results for problem he4 1∞-norm w1∞-norm nsd sum-log fig. 4.3. “he4”. the actual loss in performance index figure 4.3 shows the points (nnz, αsp) obtained from the algorithm 3, which correspond to the actual sparse-to-optimal loss in performance. for the 1∞-norm we had α̂� αsp, while using the other surrogates yields a sparse structure for α̂ close to αsp, which is surely an advantage of these approximations. indeed, if one had to set α̂� αsp in order to get a sparse control, the interpretation of α̂ as a maximum admissible performance loss would be incorrect. for instance, using the 1∞-norm in example “he4”, in order to detect sparse control which yields 0.5% of the actual performance drop, we had to allow for a maximum admissible performance loss of 15− 20% during the detection step. moreover, using the copyright c© 2018 assa. adv syst sci appl (2018) surrogates for the matrix `0-quasinorm 21 1∞-norm in some examples can lead to a large difference between α̂ and αsp. using other approximations usually allows to detect a sparse structure for the values α̂ which are close to the corresponding values of αsp; i.e., α̂ is a more accurate estimate of αsp. 2 3 4 the number of nonzero rows 10 0 10 1 10 2 a c tu a l lo s s results for problem he4 1∞-norm w1∞-norm nsd sum-log fig. 4.4. “he4”. the actual loss in performance index. all possible combinations of zero rows figure 4.3 depicts the points marked as black crosses which were obtained at step 4 of the algorithm 3 during the brute-force enumeration. it turned out that for our example the threestep procedure yielded the optimal sparse structures for every number of zero rows. figure 4.3 is just a part of the whole picture presented in fig. 4.4, where all possible combinations of zero rows of the gain matrix can be seen. as we can see, a poor choice of the sparse structure can lead to a dramatic loss in the performance index. it is important to understand that the occurrence of large values of performance drop in fig. 4.4 is explained by the specific properties of the system itself rather than by sparsity detection methods; some systems just do not tolerate zeroing out specific control inputs. this fact also leads to an interesting “sideeffect” which consists in the ability to filter out sparse controllers with “poor” structure a priori by choosing appropriate values for α̂. similar results were obtained for other examples from compleib such as “ac1”, “ac9”, “ac12”, “he3”. therefore, all the observations and conclusions made for model “he4” are also valid for all listed examples. 4.2. mass spring system consider the system which consists of n masses mis connected via springs with stiffness coefficients kis; the masses can slide frictionless along the line (see fig. 4.5). let us denote copyright c© 2018 assa. adv syst sci appl (2018) 22 a. bykov, p. shcherbakov the displacement of the ith mass from its reference position as pi and let the state variables be x1 = [p1 . . . pn ] > and x2 = ẋ1. fig. 4.5. model “mass spring system” similarly to the previous section, the behaviour of the system is described by the equation (2.1). in this example we used identity weight matrices r and s in the quadratic functional (2.3). 2 3 4 5 6 7 8 9 10 the number of nonzero rows 1 1.5 2 2.5 3 3.5 4 r e la x a ti o n c o e ff ic ie n t α results for problem ms1 10 1∞-norm w1∞-norm nsd sum-log fig. 4.6. “ms”. the detection of zero rows for various α̂ for the sake of simplicity we considered the system with the following parameters: m1 = · · · = mn = 1, k1 = · · · = kn = 1, copyright c© 2018 assa. adv syst sci appl (2018) surrogates for the matrix `0-quasinorm 23 so that the matrices of system (2.1) has the form a = [ o i t o ] , b = [ o i ] , where t ∈ rn×n is a tridiagonal toeplitz matrix with −2 on its main diagonal and 1 on its first superand sub-diagonal, i ∈ rn×n is the identity matrix, and o ∈ rn×n is the zero matrix. this model is useful, since it allows to choose arbitrary values for the number of masses n , i.e., to vary the dimension of the problem. the experiment was conducted for several values of n , the observed results being quite similar. we present the results for n = 10, since even for n = 10 the number of all possible combinations of zero rows is equal to 210 − 2 = 1022, thus, the brute-force approach is still possible yet challenging. 2 3 4 5 6 7 8 9 10 the number of nonzero rows 1 1.5 2 2.5 3 3.5 4 a c tu a l lo s s results for problem ms1 10 1∞-norm w1∞-norm nsd sum-log fig. 4.7. “ms”. the actual loss in performance index the results of the experiment for this example are presented in figs. 4.6 and 4.7. the figures show that use of the 1∞-norm in this problem did not lead to the successful detection of a sparse structure even for large values of α̂. this reminds us of the example “he4” from the previous section, where we noticed that using the 1∞-norm makes α̂ a poor estimate of αsp. this observation is valid for all values of n tested during the experiment. it is hard to explain such an inefficiency of the 1∞-norm. in this particular example one of the possible reasons might be a homogeneous internal structure of the system, which probably makes the choice of the control inputs to be zeroed out a challenging task. copyright c© 2018 assa. adv syst sci appl (2018) 24 a. bykov, p. shcherbakov although the weighted 1∞-norm succeeded in yielding sparse controllers, it is less efficient than the nonconvex approximations (3.13) and (3.14). nevertheless, algorithm 1 for minimizing the weighted 1∞-norm is quite straightforward in implementation and allows to achieve better results than the standard minimization of the 1∞-norm. there is an interesting detail about the process of the detection of zero rows. this process is sequential, i.e., the number of zero rows grows step-wise or one by one. if the values in the row approach zero, say become less than 10−10, then, according to the algorithm, the weight for this row becomes very large, since it is inversely proportional to the maximum among the absolute values of the elements of this row. for a row with the weight 1010 it is hardly possible to gain nonzero values. thus, if at any iteration some row becomes zero, it will stay zero till the end of the algorithm. implementation of the algorithm to minimize approximations (3.13), (3.14) (cccp) requires calculating a gradient of the corresponding functions, its iterative nature being similar to algorithm 1. in most cases, the nonconvex surrogate nsd performs as good as the log-sum approximations, and sometimes nsd happens to be better. figure 4.7 shows that using functions (3.13), (3.14) yielded sparse controllers which are close or equal to optimal sparse controllers. in terms of optimality, among sparse controllers with the fixed number of zero rows the weighted 1∞-norm (3.12) is less efficient than nonconvex surrogates. this detail is important due to the fact that global convergence is not guaranteed. on the ground of the results of the numerical experiments described above we arrived at the following conclusions: • using the 1∞-norm can lead to situations where the actual sparse-to-optimal performance drop αsp is significantly smaller than the maximum admissible loss in performance α̂. other surrogates usually do not suffer from this drawback. • using approximations to the matrix `0-quasinorm usually yields a sparse controller which is optimal among all controllers with the same number of zero rows. • in some problems (e.g., see the example from section 4.2) the 1∞-norm does not succeed in the detection of sparse structure even for large values of α̂. • the nsd function appears to be effective similarly to the nonconvex surrogate log-sum, and in some cases the nsd happens to perform better. 5. conclusion in this paper we considered various approximations to the matrix `0-quasinorm, which can be applied to sparse feedback design in optimal control problems. the results of the numerical experiments show that to some extent all the surrogates analyzed can be applied to the detection of zero rows of the gain matrix, but nonconvex surrogates perform better. future efforts to be made in this area include the analysis of examples where one or another approximation to the matrix `0-quasinorm happens to be inefficient, and studying the alternative numerical procedures which can be applied to sparse feedback design, e.g., admm, alternating direction method of multipliers. acknowledgements this work is supported by the russian science foundation through grant no. 16-1110015. references copyright c© 2018 assa. adv syst sci appl (2018) surrogates for the matrix `0-quasinorm 25 1. boyd, s., ghaoui, l.e., feron, e. & balakrishnan, v. 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(2013) performance and design of cycles in consensus networks, systems and control letters, 62(1), 85–96. copyright c© 2018 assa. adv syst sci appl (2018) introduction sparse feedback design in optimal control problems approximations to the matrix 0-quasinorm 1-norm weighted 1-norm nonconvex sparsity detector, nsd nonconvex approximation log-sum results of numerical experiments examples from compleib mass spring system conclusion adv syst sci appl 2019; 02; 8-22 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/727 models of uncertain-random programming georgy veresnikov1*, ludmila pankova1, valeriya pronina1 1) v.a. trapeznikov institute of control sciences of russian academy of scienses moscow, russian e-mail: veresnikov@mail.ru received february 12, 2018; revised december 9, 2018; published december 31, 2018 abstract: the article proposes the models of optimization with constraints under conditions of parametric mixed uncertainty ‒ aleatory and epistemic. we model parameters with aleatory uncertainty by random values with probability distribution functions obtained from statistical data. we model parameters with epistemic uncertainty by uncertain values introduced in the uncertainty theory of liu b. experts define the uncertainty distribution functions. we model a function of random and uncertain parameters by uncertain-random value, interpreted as epistemic value parameterized by random values. optimization criteria (deterministic duplicates of objective functions) are combination of different characteristics of random and uncertain values, which allows both to average objective functions and to take into account risks or reliability arising from the variability of random and uncertain values. using the proposed models of uncertain-random programming, we formalized as a two-criterion optimization problem with constraints and solved the task of preliminary aerodynamic design in the conditions of parametric mixed uncertainty ‒ calculation of aircraft weight parameters. the uncertainty theory makes possible under certain conditions (for sufficiently wide class of functions) to obtain analytical expressions for characteristics of uncertain functions, that significantly reduces computational costs. to calculate weight parameters of aircraft, we use multicriteria genetic algorithm and statistical modeling. we investigate the dependence of the optimization result on the given probability levels for random values and the expert belief degree for epistemic values reflecting the reliability of the obtained solution. keywords: aleatory uncertainty, epistemic uncertainty, uncertain-random quantity, deterministic duplicate, uncertain-random programming, mixed uncertainty. 1. introduction aleatory (objective) and epistemic (subjective) uncertainties are two types of uncertainty, reflecting nondeterminism. in the context of modeling technical objects and decision making aleatory uncertainty occurs when information about a stochastic parameter is accumulated in statistical data and parameters are modeled by random variables with certain distributions. epistemic uncertainty arises when information about a parameter is obtained from experts, while the parameter may be either stochastic, but there are no or insufficient statistical data, or deterministic, but its value is unknown to date. the parameters with epistemic uncertainty are modeled by fuzzy, possibilistic, uncertain [1], and others values. decision making in the design of technical objects, as a rule, occurs under conditions of mixed uncertainty, when there are parameters both aleatory and epistemic. the existing methods and design tools do not take into account the presence of parameters with epistemic uncertainty. however if we consider the epistemic parameters as random variables, it can lead to errors, which is associated with the nonadditivity of expert judgments (the measure of uncertainty is nonadditive in contrast to the probabilistic measure). * corresponding author: veresnikov@mail.ru mailto:veresnikov@mail.ru models of uncertain-random programming 9 copyright ©2019 assa. adv. in systems science and appl. (2019) it is important to note that in construction of optimization models in conditions of uncertainty, transition from nondeterministic objective functions and restrictions to their deterministic duplicates is necessary. deterministic duplicates of objective functions are their numerical characteristics (mean, variance, quantile, etc.). let ),(f  be a function of epistemic values and aleatory values , i.e. a value with mixed uncertainty. there are two known approaches to modeling a value with mixed uncertainty and its characteristics (deterministic duplicates) based on different interpretations of a value with mixed uncertainty [1-4]. the first approach considers ),(f  as random value parameterized by epistemic values. we determine the characteristic of function ),(f  in two stages. first, we calculate the numerical characteristic of the function as a random quantity for each implementation of epistemic quantities. this characteristic does not depend on random quantities and, as a function of epistemic quantities, is an epistemic quantity. then we calculate the characteristic of this epistemic quantity, which is the characteristic of the function with mixed uncertainty. the second approach considers ),(f  as epistemic value parameterized by random values. we determine the characteristic of function ),(f  in two stages. first, we calculate the characteristic of the function as an epistemic quantity for each implementation of random quantities. this characteristic does not depend on epistemic quantities and, as a function of random quantities, is a random quantity. then we calculate the numerical characteristic of this random quantity, which is the characteristic of the function with mixed uncertainty. calculating characteristics of mixed uncertainty values is very expensive, especially in absence of explicit formulas for epistemic and/or random characteristics [2-4]. in this regard, it is relevant to highlight conditions under which there are analytical expressions of characteristics. thus, in [4] in framework of the first approach, the authors investigate values with mixed uncertainty, where possibilistic values model epistemic values, while possibilistic-random values have a shift-scale representation: f (ξ, ω) =α(ω)+σ(ω)∙ β(ξ), where α(ω) and σ(ω) are random values, β(ξ) is a possibilistic value. the authors obtain formulas for calculating the characteristics of the weighted sum of the possibilistic-random values of the shift-scale form with a certain type distribution for the possibilistic and random values. uncertainty theory makes possible for wider class of functions to obtain analytical expressions for characteristics of uncertain functions. that significantly reduces computational costs. in this paper, we model epistemic values by uncertain values introduced in theory of uncertainty [1]. further, we will call uncertain only such values. chance theory [1] introduces uncertain-random value, where uncertain values model epistemic values. the chance is a measure of mixed uncertainty. uncertain-random value is a real function on uncertain-random space with chance measure and have distribution function of chance measure (definitions 10-12, appendix). characteristics of uncertain-random values, defined as the characteristics of the chance measure distribution function (for example, definition 13, appendix), will always be averaged epistemic characteristics, since chance measure is the mathematical expectation of uncertain measure (definition 1, appendix). characteristics are actually determined as in the second approach, where at the first stage the epistemic characteristic is calculated, at the second stage the mathematical expectation is always calculated [1]. thus, use of chance theory in solving optimization problems leads to models with averaging criteria, i.e. the solution will be effective only “on average”, while risk of unwanted solutions is not considered. when modeling constraints using a chance measure, fulfillment of constraints is required only “on average” and risk of not fulfilling the constraints is not considered [1]. the paper proposes the models of uncertain-random programming, that is, the optimization models with constraints under conditions of parametric mixed uncertainty within the 10 g.s. veresnikov, l.a. pankova, v.a. pronina copyright ©2019 assa adv. in systems science and appl. (2019) framework of the second approach, which makes it possible to use analytical expressions for sufficiently wide class of functions of uncertain values. the choice of duplicates of objective functions and constraints allows to take into account risk and reliability requirements. section 2 describes the models of uncertain-random programming. in section 3 the task of calculating the weight parameters of an aircraft under conditions of parametric mixed uncertainty is formalized as a multicriteria optimization problem with constraints. we use two proposed models: with averages and with quantiles. the method for solving the task of calculating weight parameters and the results of calculations are given. the appendix contains definitions and statements from the theory of uncertainty and chance theory. 2. models of uncertain-random programming let ),(  ,xf be the objective function, x is solution vector, ξi, i = 1, …, n, are uncertain values, ωj, j = 1, ..., m, are random values that is, ),(  ,xf is uncertain-random value for each x . we interpret ),x(f  , as the uncertain value parameterized by the random values in the framework of second approach. we determine the characteristic of the function ),x(f  , in two stages. first, we calculate the characteristic of uncertain function ),x(f  , , where random values are parameters. then, we calculate the characteristic of this random value, which is characteristic of the function ),x(f  , with mixed uncertainty. mathematical expectation/expected value, quantile (definition 9, appendix), variance, probability/ belief degree of non-exceedance of a given value by the function can be chosen as characteristics of random and uncertain values. using combination of characteristics, you can build different optimization models. the choice of the optimization criterion depends on the specific task, reliability requirements and is the prerogative of the decision maker. the optimization model with mathematical expectation/expected value criteria, averaging the objective function, gives an effective solution “on average”, while risk or reliability is not taken into account. in robust optimization, when it is necessary to ensure the least variability of the objective function, the variance is used. in tasks of optimal control and design of aircraft under conditions of aleatory uncertainty, quantile and the probability of not exceeding (while minimizing) the objective function of a given threshold value have become widespread, since they are aimed at making optimal decisions based on risk or reliability requirements. we consider some models of uncertain-random programming and reduce them to mathematical or stochastic programming models. let )x(f  ,, be the objective function, x be the solution vector, ξ1, ξ2, …, ξn be independent uncertain values with uncertainty distribution functions φ1, φ2, …, φn , having inverse distribution functions, ω1, …, ωm be independent random values with probability distribution functions ψ1, ψ2, …, ψm. let )x(f  ,, be a continuous strictly increasing function with respect to ξ1, …, ξk and a strictly decreasing function with respect to ξk+1, …, ξn. for the model 1 (ee), we will select expected value em as characteristic at first stage, and mathematical expectation as characteristic at second stage. then the model 1 is: ))).(((min  ,,xfee mp x expected value of function )x(f  ,, where random values  are parameters is (theorem 2, appendix):  dxf,,xfe nkk m )),1(),...,1(),(),...,(,())(( 1 0 11 1 11 1     . (1) models of uncertain-random programming 11 copyright ©2019 assa. adv. in systems science and appl. (2019) mathematical expectation of random value ))x(f(e m  ,, is: ....)),1(),...,1(),(),...,(,( )))((( 1 1 0 11 1 11 1 m r nkk mp dddxf ,,xfee m          the model 1 has the form: ....)),1(),...,1(),(),...,(,(min 1 1 0 11 1 11 1 m r nkk x dddxf m       thus, we have reduced the model 1 to a deterministic model of mathematical programming. for the model 2 (qe), we use expected value as characteristic at first stage, and quantile of the random variable as characteristic at second stage. then the model 2 has the form: ,10.5 ,))),,((( where ,min   rxfep r m x or (according to the formula (1)): .10,)))),1(),...,1(),(),...,(,((( where ,min 1 0 11 1 1 k 1 1      rdxfp r nk x thus, we reduce the model 2 to stochastic model with quantile criterion. for the model 3 (eq), we use quantile as characteristic at first stage, and mathematical expectation as characteristic at second stage. then the model 3 has the form: ))),(((infmin  ,xfe p x , where in accordance with theorem 2 (d) of the appendix: ).),1(),...,1(),(),...,(,()),(( 11 1 11 1      nkk xf,xfinf (2) thus, we have reduced the model 3 to a deterministic model of mathematical programming. for the model 4 (qq), we use quantile of uncertain value as characteristic at first stage, and quantile of random value as characteristic at second stage. then the model 4 has the form: ,min r x where p (infα( ),,( xf ) < r} > β, 1 0 . thus, we reduce the model 4 to stochastic model with quantile criterion (formula (2)). consider representation of duplicate for constraint 0,, )x(g  , where g is an uncertainrandom value, and x is the solution vector, ξ1, ξ2, …, ξn are independent uncertain values with uncertainty distribution functions φ1, φ2, …, φn, having inverse functions, ω1, …, ωm are independent random values with probability distribution functions ψ1, ψ2, …, ψm. 12 g.s. veresnikov, l.a. pankova, v.a. pronina copyright ©2019 assa adv. in systems science and appl. (2019) in the case of hard constraint, the constraint must be satisfied for any implementation of random and uncertain values, and duplicate value has form: 0)(max ,   ,,xg . in the case of soft constraint, the constraint performs with guaranteed belief degree and probability. we will build duplicate of soft constraint. we interpret )x(g  ,, as uncertain value parameterized by random values. then in the first stage, where random variables are parameters, the duplicate of constraint 0,, )x(g  has the form:   ))x(g(m 0,, , where m is the degree of expert confidence (see appendix), α is given by the designer. if )x(g  ,, is continuous strictly increasing function with respect to ξ1, …, ξk and a strictly decreasing function with respect to ξk+1,…, ξn, then the inequality   ))x(g(m 0,, is equivalent to the inequality (theorem 2, appendix): 0)),1(),...,1(),(),...,(,( 11 1 11 1      nkkxg . at the second stage, we replace this inequality containing random values by the probability that this inequality is satisfied: ,)0)),1(),...,1(),(),...,(,(( 11 1 11 1      nkkxgp where β is the confidence level of probability specified by the designer. thus, we reduced constraint with uncertain-random value to probabilistic constraint. reducing the model of uncertain-random programming to a deterministic model of mathematical programming greatly simplifies the solution. reducing model of uncertainrandom programming to model of stochastic programming simplifies solution, but in general case they remain quite complicated. this is due to complexity of finding the analytical form of probabilistic criterion and constraint, and, in absence of one, complexity of numerical solution methods. in particular cases, deterministic equivalents of probabilistic criteria and constraints can be found [5, 6]. 3. calculation of aircraft weight parameters 3.1. models we formalize the one of the tasks of calculating weight parameters of passenger aircraft as optimization problem with constraints under conditions of mixed uncertainty based on the method of calculating the weight report of passenger aircraft [7]. in the original form, the functions for calculating the basic weight parameters of aircraft at the preliminary design stage are: pepayloadraeecppchairframefuel mmmmmmmmm  0 pepayloadpcaa mmvvm  )(0  wetsmairframe aqkm )1(  aaqsqss vvkkq lg)lg( 10  eqairframech kmm 3.0 )( 10 wet a engengengpp a v kktm   models of uncertain-random programming 13 copyright ©2019 assa. adv. in systems science and appl. (2019) where 0m is takeoff mass; airframem is airframe mass; chm is equipment mass for control and hydraulics; ppm is power plant mass; fuelm is fuel mass; ecm is weight of crew and equipment; raem is mass of board radioelectronic equipment; payloadm is commercial load; pem is passenger equipment; a is the average density of aircraft; av is aircraft volume; pcv is volume of the passenger compartment; mk is composites utilization coefficient; sq is specific weight per square meter of airframe; weta is area of washed surface of aircraft; 0qsk is statistical coefficient; 1qsk is statistical coefficient; eqk is coefficient taking into account production technology; eng is technological coefficient, which shows the ratio of weight of engine to maximum thrust at h = 0, v = 0 (h, v are height and speed, respectively); t is required thrust at h = 0, v = 0; 0engk and 1engk are statistical coefficients. to calculate the optimal design parameters we select optimization criteria that affect the fulfillment of specified technical requirements: operating costs and flight range. we minimize takeoff weight to reduce fuel consumption, as the main part of operating costs consists of fuel costs. on the other hand, we maximize the fuel supply, which increases the range of the aircraft. in this regard, the task of weight calculation is formalized and presented in form of following optimization model:        landingfuel0 a fuel0 mmm m m 8.0 ,500400 ,max,min  (1) based on the study of technical requirements, the designer choices the design parameters. the design parameters are weta , av , payloadm , pem , raem . the available information about the parameters determines the separation of the parameters into random and uncertain. statistics provide information on random parameters. experts provide information on uncertain parameters. the parameters with epistemic uncertainty are mk , eqk , pcv , ecm , a . the parameters with aleatory uncertainty are 0qsk , 1qsk , 0engk , 1engk , eng , t. the expression landingfuel0 mmm  8.0 determines the constraint on landing weight of aircraft, associated with length of run during landing. let 1 a  , 1 pcv , 1 ecm , 1 eqk , 1 mk be inverse uncertainty distribution functions of uncertainty for parameters a , pcv , ecm , eqk , mk ; 0qsk , 1qsk , 0engk , , 1engk eng , t be probability distribution functions of uncertainty for parameters 0qsk , 1qsk , 0engk , 1engk , eng , t. to solve this problem, as well as to compare the results and study the models, we construct two two-criterion models based on the single-criterion models proposed in section 2. in model a, for each criterion, select the "averaging" model (ee). in model b, for each criterion we choose the "quantile" model (qq). model a: 14 g.s. veresnikov, l.a. pankova, v.a. pronina copyright ©2019 assa adv. in systems science and appl. (2019)              ],1,0[,))8.0(( ,)400( ,)500( ],[max],[min landingmlandingmlandingmlandingfuel0 aa aa fuel0 ppmmmmp m m meme      where m is measure of uncertainty (expert belief degree),  a  ,  a  , landingm  are levels of belief degrees that the inequalities in question are satisfied, landingm p is level of probability that belief degree in the implementation of inequality landingfuel0 mmm  8.0 will be greater than landingm  . to solve this task, first the designer sets  a  ,  a  , landingm  , . landingm p then the multicriteria optimization algorithm is applied. for each combination of design parameters varied in process of optimization, we calculate expected values of objective functions according to the formulas:    1 0 11 0 ,)))1()(((][  dmmvme pepayloadpcvaa ,)) )1(((][ 1010 1 0 1 0 tengengkengkqskqskpepayloadrae mr ecmppchairframe ' fuel dddddddmmm mmmmme        ;))1()(( 11 0 pepayloadpcvaa ' mmvm    );1(3.0 1    eqkairframech mm .))1(1( 1 wetsmkairframe aqm    model b:                  ],1,0[,))8.0(( ,)400( ,)500( ],1,0[,)][(sup ,max],[min landingmlandingmlandingmlandingfuel0 aa aa fuelmfuelmfuel fuelm 0 0m ppmmmmp m m pprmp rminf        where fuelm p is level of probability that rm fuel fuelm ][sup , fuelm is level of expert belief degree for quantile criterion. as the objective functions are used: pepayloadmvam mmvminf 0pc0a0m   ))1()((][ 11 0   , r, subject to models of uncertain-random programming 15 copyright ©2019 assa. adv. in systems science and appl. (2019) ;) )(( 1' 0 fuelmpepayloadrae fuelmecmppchairframe prmmm mmmmp     ;))()(1( 11' 0 pepayloadfuelmpcvafuelma mmvm    );(3.0 1 fuelmeqkairframech mm   wetsfuelmmkairframe aqm ))(1( 1   . note that the first criterion does not depend on random variables and is deterministic. we account for constraints for both models as follows. the constraint  aam  )500( is equivalent to 500)(1  aa   . the constraint  aam  )400( is equivalent to .400)1(1   aa   to check constraint on landing weight of aircraft, we calculate probability that belief degree in constraint performing will be greater than the specified level: landinglanding mmlandingfuel0 pmmmmp  ))8.0((  . this inequality is equivalent to inequality (theorem 2d, appendix): landingmlandingfuel0 pmmp(m  )08.0 , ,))1()(( 11 0 pepayloadlandingmpcvalandingma mmvm    ,)(1 0 pepayloadraelandingmecmppchairframefuel mmmmmmmm    ),(3.0 1 landingmeqkairframech mm   .))(1( 1 wetslandingmmkairframe aqm   to calculate weight parameters of aircraft, we use multicriteria genetic algorithm and statistical modeling. 3.2. methods and results we apply a genetic algorithm of multicriteria optimization with values  a  ,  a  , sbm  , sbm p given by the designer. we calculate the probabilistic characteristic of uncertain-random values based on the monte carlo statistical modeling method. as an example, we present the calculation algorithm for model a presented in section 3.1. for each combination of variable design parameters, we calculate the values of objective functions using following formulas:    1 0 11 0 ,)))1()(((][  dmmvme pepayloadva pca ,) )((][   dmmm 1mmmm n 1 me pepayloadrae n 1i 1 0 1 ecmppchairframe ' 0fuel      ,))()(( pepayload 1 pcva 1 a ' 0 mm1vm    ),1(3.0 1    eqkairframech mm ,))(( wets 1 mkairframe aq11m    16 g.s. veresnikov, l.a. pankova, v.a. pronina copyright ©2019 assa adv. in systems science and appl. (2019) ,lg)lg}({}{ aai1qsi0qss vvkkq  ).}{}({}{}{ 10 wet a iengiengiiengpp a v kktm   where 1 a  , 1 pcv , 1 ecm , 1 eqk , 1 mk are inverse uncertainty distributions for the parameters a , pcv , ecm , eqk , mk ; n is number of value combinations for parameter 0qsk , 1qsk , 0engk , 1engk , eng , t while we form the values of the parameters in accordance with the specified probability distributions; {●}i is the i-th element of value set for parameter {●} formed by statistical modeling. the constraint  aam  )500( is equivalent to 500)(1  aa   . the constraint  aam  )400( is equivalent to 400)1(1   aa   . to check the restrictions on the landing weight, an approximate probability value is calculated: ,].[)).((    n 1i fuel0landingmlandingfuel0 m80mi n 1 mm80mmp  where       иначе0 mm80m1 m80mi landingfuel0 fuel0 , ., ].[ , ,))()(( pepayloadlandingm 1 pcvalandingm 1 a0 mm1vm    ,)( pepayloadraelandingm 1 ecmppchairframe0fuel mmmmmmmm    ),(. landingm 1 eqkairframech m30m   ,))(( wetslandingm 1 mkairframe aq1m   the constraint is satisfied subject to landingmlandingmlandingfuel0 pmm50mmp  )).((  . data on uncertain and random parameters obtained from experts and statistics collection are approximated by uncertainty and probability distribution functions. normal distributions are most often used in the design of technical objects for approximation. we will conduct research for these types of distribution of uncertainty and probability. to generate the values of each random parameter, we used the standard normal distribution.                          2 2 ( ( ( 2 )( 1 2 1 )( ) ) ) x erfx     ,  x tdtexerf 2 0 1 )(  . for each epistemic parameter, we set the normal uncertainty distribution function: 1 ( ( ( 3 )( 1)(                           )e ) ) xe expx    ,           1 ln 3 )( ( ( 1 )e )e . models of uncertain-random programming 17 copyright ©2019 assa. adv. in systems science and appl. (2019) we used the nominal (mean) values of random and epistemic parameters as the )( and )e ( respectively. we set the standard deviations )(  and )e (  as a percentage of )( and )e ( . we investigate the influence of model parameters that reflect the characteristics of uncertainty and preferences of design maker. we present in fig. 1 and 2 the results of applying the optimization models a and b for different variants of the initial data: standard deviations of parameters )(  , )e (  , and values related to reliability landingm  , landingm p , 0m , fuelm , fuelm p ,  a  ,  a  . the upper line in fig. 1-3 is the result of applying the optimization model with deterministic values of uncertain and random parameters equal to their average values. the pareto-front, which is the result of the application of model a, practically coincides with the upper line, but narrows with increasing reliability. the values landingm  , landingm p , 0m , fuelm , fuelm p ,  a  ,  a  are equal and vary from 0.6 to 0.9 with a step of 0.1 (bottom four lines in fig. 1 and 2). set for all random parameters ))   (( 03,0   , for all epistemic parameters ))e e   (( 03,0 . then we obtained the pareto fronts shown in fig.1. fig. 1. pareto fronts for model b at ) )   (03,0 (   and )e e )   (03,0 (  (the parameters are random and epistemic) set for all random parameters )(05,0 )(     , for all epistemic parameters .05,0 (( ))e e   the parameters landingm p , fuelm p , landingm  , 0m , fuelm ,  a  ,  a  are equal and vary from 0.6 to 0.9 with a step of 0.1. then we obtained the pareto fronts shown in fig.2. 18 g.s. veresnikov, l.a. pankova, v.a. pronina copyright ©2019 assa adv. in systems science and appl. (2019) fig. 2. pareto fronts for model b at ) )   (05,0 (   and )e e )   (05,0 (  (the parameters are random and epistemic) the figures show that the pareto-fronts obtained as a result of performing optimization calculations using the model b with increasing landingm p , fuelm p , landingm  , 0m , fuelm ,  a  ,  a  are shifted to the region of the worst values of the objective functions. while ensuring high reliability of the solution (p and α close to 1) at fixed take-off weight the difference in fuel mass can reach more than 4000 kg. for comparison, consider the case where the parameters mk , eqk , pcv , ecm , a , 0qsk , 1qsk , 0engk , 1engk , eng , t are random. we use the following optimization model:                     .]1,0[,)8.0( ],1,0[,)400( ],1,0[,)500( ],1,0[,)( ],1,0[,)( ,max,min 2 0010 21 landingmlandingmlandingfuel0 aaa aaa fuelmfuelmfuel mm ppmmmp ppp ppp pprmp pprmp rr     let for all random parameters ))   (( 03,0   . the parameters 0m p , landingm p , fuelm p ,  a p ,  a p are equal and vary from 0.6 to 0.9 with a step of 0.1 (bottom four lines). then we obtained the pareto fronts shown in fig.3. the upper line in figures 3 is the result of applying the optimization model with deterministic values of uncertain and random parameters equal to their average values. models of uncertain-random programming 19 copyright ©2019 assa. adv. in systems science and appl. (2019) fig. 3. pareto fronts at ) )   (03,0 (   (all parameters are random) pareto fronts also move to the area of the worst values of the objective functions, but less than in fig. 1 and 2. the explorations show that the greater the degree of uncertainty and the greater the level of reliability, the more conservative obtained solutions. the obtained results do not contradict the experience and intuition of the designer, which confirms the adequacy of the models. 5. conclusion the models of the optimization problem with constraints under conditions of parametric mixed uncertainty are proposed. uncertain values introduced in the theory of uncertainty models parameters with epistemic uncertainty. under certain conditions (for rather wide class of functions), optimization models with mixed uncertainty are reduced to deterministic models or to stochastic programming models with probabilistic criteria. we formalize and solve the task of preliminary aerodynamic design in conditions of parametric mixed uncertainty using the proposed models. acknowledgements this work was supported by the russian foundation for basic research (grant 18-08-00822a) references [1] liu b. (2010). uncertainty theory: a branch of mathematics for modeling human uncertainty. springer-verlag, berlin. [2] eldred s., swiler t., tang g. (2011). mixed aleatory-epistemic uncertainty quantification with stochastic expansions and optimization-based interval estimation. reliability engineering and system safety. 96 (9), 1092-1113. 20 g.s. veresnikov, l.a. pankova, v.a. pronina copyright ©2019 assa adv. in systems science and appl. (2019) [3] yazenin a.v. (2016). osnovnye ponyatiya teorii vozmozhnostej [basic concepts of the theory of possibilities]. m., fizmatlit. [in russian]. [4] lockwood b., anitescu m., mavripilis d. (2012) mixed aleatory/epistemic uncertainty quantification for hypersonic flows via gradient-based optimization and surrogate models. 50th aiaa aerospace sciences meeting, aiaa-2012-1254. [5] vishnaykov v. b., kibzun a. i. (2006). deterministic equivalents for the problems of stochastic programming with probabilistic criteria. automation and remote control, 67 (6), 945–961. [6] geletu, a., kl¨oppel m. zhag h., li p. (2012). advances and applications of chanceconstrained approaches to systems optimization under uncertainty. international journal of systems science, 44 (7), 1209-1232. [7] kolokolova l.g. (1995). metod obobshchennykh modelej svojstv samoleta dlya etapa rannego proektirovaniya [method of generalized models of the properties of the aircraft for the early design stage]. tekhika vozdushnogo flota, № 5-6. [in russian]. appendix [1] definition 1: let γ be a nonempty set, let λ be a σ-algebra over γ, and let m be an uncertain measure. then the triplet (γ, l, m) is called an uncertainty space. a number m{λ} will be assigned to each event λ to indicate the belief degree with which we believe λ will happen. the measure of uncertainty m is dual, subadditive, the measure of the product of events is equal to the minimum of the measures of these events. definition 2: an uncertain variable is a function ξ from γ to the set of real numbers such that {γ |ξ(γ)∈ b}is an event for any borel set b of real numbers. definition 3: the uncertainty distribution of an uncertain variable ξ is defined by φ(x) = m {ξ ≤ x} for any real number x. definition 4: an uncertainty distribution φ(x) is said to be regular if it is a continuous and strictly increasing function with respect to x at which 0 < φ(x) < 1, and definition 5: let ξ be an uncertain variable with regular uncertainty distribution φ(x). then the inverse function φ−1 (α) is called the inverse uncertainty distribution of ξ. definition 6: the uncertain variables ξ1, ξ2, …, ξn are said to be independent if )()}({ 11 ii n k ii n i bmbm    .1)(lim,0)(lim   xx xx models of uncertain-random programming 21 copyright ©2019 assa. adv. in systems science and appl. (2019) for any borel sets b1, b2, …, bn of real numbers. theorem 1: let ξ1, ξ2, …, ξn be uncertain variables, and let f be a real-valued measurable function. then )...,,,( 21 nf  is an uncertain variable. definition 7: let ξ be an uncertain variable. then the expected value of ξ is defined by .}{}{][ 0 0 drrmdrrme      definition 8: let ξ be an uncertain variable with finite expected value e[ξ]. then the variance of ξ is v[ξ] = e[(ξ – e[ξ])]2. theorem 2: let f(ξ1, ξ2, …, ξn) be continuous, strictly increasing with respect to ξ1, ξ2, …,ξm continuous, strictly decreasing with respect to ξm+1, ξm+2,…, ξn. if ξ1, ξ2, …, ξn are independent uncertain variables with regular uncertainty distributions φ1, φ2, …, φn, respectively, then: a) ξ = f(ξ1, ξ2, …, ξn) is uncertain variable with inverse uncertainty distribution: )),1(...,),1(),(...,),(),(()(1 11 1 11 2 1 1      nmmf b) ,))1(),...,1(),(),...,(),((][ 11 1 11 2 1 1  dfe nmm     c) ,])[))1(...,),1(),(),...,(),(((][ 1 0 211 1 11 2 1 1  defv nmm     d) for any   [0, 1] m{f(ξ1, ξ2, …, ξn) < 0} > α is equivalent .0))1(),...,1(),(),...,(),(( 11 1 11 2 1 1      nmmf definition 9: a) the quantiles of the random variable ξ: supα[ξ ] = sup{r | pr { ξ > r} > α} – α-optimistic value, infα[ξ ] = inf{r | pr { ξ < r} > α} – α-pessimistic value,   [0, 1], b) the quantiles of the uncertain variable ξ: supα[ξ ] = sup{r | m{ ξ > r} > α} – α-optimistic value, infα[ξ ] = inf{r | m{ ξ < r} > α} – α-pessimistic value,   [0, 1]. definition 10: let (ω , a, pr) be a probability space and (γ, λ, m) be an uncertainty space. then the product (γ × ω, λ × a, m × pr) is called a chance space, where γ × ω = {(γ, ω)| γ ∈ γ, ω ∈ ω}. then the chance measure of an event θ of λ × a is: 22 g.s. veresnikov, l.a. pankova, v.a. pronina copyright ©2019 assa adv. in systems science and appl. (2019) x}dxθ}γ|(γ,ω)ω|m{γ{ω}сh{θ   1 0 pr . m{γ ∈ γ |(γ, ω) ∈ θ} is just the uncertain measure of cross section of θ at ω. since m{γ ∈ γ |(γ, ω) ∈ θ} can be regarded as a function from the probability space (ω , a, pr) to [0, 1], it is a random variable. thus the chance measure ch{θ} is just the expected value (i.e., average value) of this random variable. the chance-measure is monotonous, dual and subadditive. definition 11: an uncertain random variable is a function ξ from γ ×ω to the set of real numbers such that {(γ, ω)| γ ∈ γ, ω ∈ ω |ξ (γ, ω) ∈ b} is an event in l × a for any borel set b of real numbers. definition 12: let ξ be an uncertain-random variable. then its chance distribution is defined by φ(x) = ch{ξ ≤ x} for any real x. definition 13: the expected value of uncertain-random variable ξ is:     0 0 ][ x}dxch{dxx}ch{e  provided that at least one of the two integrals is finite. theorem 3: let η1, η2, … , ηm be independent random variables with probability distributions ψ1, ψ2, … , ψm, and let τ1, τ2, … , τn be independent uncertain variables with uncertainty distributions y1, y2, …, yn, respectively. if f(η1, η2, … , ηm, τ1, τ2, … , τn) is a measurable function, then has an expected value e[ξ] is: )(y)...dψ(ydψyyg),...,τ, τ, ...,ηf(ηe mmm mr nm 11111 ),...,(][  , where )],...,,,...,([) 111 nmm yyfe, ...,yg(y  − is the expected value of the uncertain variable ),...,,,...,( 11 nmyyf  for any real numbers ,,...,1 myy and is determined by y1, y2, …, yn. adv syst sci appl 2020; 01:104–113 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/860 comparison of some molecular structures by minimizing the comparison function natalia chernikova1*, eugeniy laneev1 1peoples’ friendship university of russia, moscow, russia abstract: the geometric difference in the relative arrangement of atoms in molecules with the same chemical formula significantly affects the properties of the crystal. for quantitative comparison of the spatial geometric structure of two molecules, we used the method of optimal superposition of molecules through the minimization of a certain comparison function by rotating the molecules after superposing their centers of mass. the search for the minimum of the comparison function over euler rotation angles is performed by the rosenbrock method. the assumed criterion for comparing molecules allows us to quantify the proximity in the spatial structure of molecules. the implementation of the method is shown by comparing the structure of eight molecules in four crystalline substances. keywords: conformation, euler angles, rotation matrix, optimization methods, rosenbrock method 1. introduction the paper deals with the mathematical problem that arises in structural chemistry, namely: the numerical study of the spatial structure of molecules with the same chemical formula, which may differ in the geometry of the mutual arrangement of atoms. the study of specific crystals was carried out in the framework of a mathematical model for comparing two molecules, studied in [1, 2]. the specified model is reduced to a comparison of two objects consisting of n ordered points with rigid geometry that behave like rigid bodies in r3. the comparison principle in this model is based on optimizing the alignment of these two objects by shifting and rotating. the optimal superposition of these objects involves the minimization over shifts and rotation angles of a certain comparison function of the geometry of the objects, which is the sum of the squares of the distances between the points of two objects with the same indices. it is proved that the minimum over shifts is achieved by superposing some characteristic points, conventionally called centers of mass of molecules. to minimize the [3–5] comparison function over rotation angles, the rosenbrock zero-order method [6] is used. the results obtained are used to study the geometry of molecules in real crystal structures. four organic crystalline substances (i, ii, iii, iv) are investigated. substances ii-iv consist of molecules with the same chemical formula and bond structure (same graphs), substance i is only a part of this graph. in addition, substances ii and iii contain symmetrically independent molecules (iia, iib, iic, iid, and iiia, iiib) that may have the same graph, but may differ in spatial structure, i.e., in the geometry of the arrangement of atoms. the problem arises of objectively comparing the geometry in the space of all these eight molecules and ∗corresponding author: cherni@list.ru comparison of some molecular structures... 105 quantifying the differences that exist in them, since the standard characteristics – bond lengths (interatomic distances) and valence angles (angles between valence bonds) do not always show differences in the geometry of molecules. in addition, it may be necessary to compare fragments of different molecules. for this purpose, [1, 2] provides a method for pairwise quantitative comparison of the geometry of molecules based on minimizing the comparison function by shifting and rotating the molecules. here is a brief description of the comparison method. 2. assumed mathematical model the study of molecules with the same chemical formula by comparing their geometries uses the concept of structure. here is the definition of the structure adopted in [1, 2] definition 2.1: a geometric structure, or in short – structure, will mean a rigid geometric construction of n ordered points in r3 with coordinates (xi, yi, zi), i = 1, ..., n , whose movement in r3 is performed like the motion of a rigid body. we assume that each i−th point of the structure has a weight coefficient wi > 0, and∑n i=1wi = w > 0. let ik, k = 1, ..., k – all indices corresponding to non-zero weight coefficients wi. then w = n∑ i=1 wi = k∑ k=1 wik > 0. (2.1) in accordance with [1, 2], we will associate the optimal criterion for superimposing two geometric structures with the minimum over shifts and rotation angles (euler angles [7]) of the comparison function of the form u(r1,0, r2,0, ϕ, θ, ψ) = n∑ i=1 wi|r1,i − r1,0 −q(r2,i − r2,0)|2, (2.2) where the vectors r1,i and r2,i determine the position of points in the first and second structure, the vectors r1,0 and r2,0 determine the offsets of the corresponding points of the first and second structure from the coordinate center, q = q(ϕ, θ, ψ) — the rotation matrix for euler angles (ψ — precession angle, θ — nutation angle, ϕ — self rotation angle): q = ( cosψ cosϕ− sinψ sinϕ cos θ − cosψ sinϕ− sinψ cosϕ cos θ sinψ sin θ sinψ cosϕ+ cosψ sinϕ cos θ − sinψ sinϕ+ cosψ cosϕ cos θ − cosψ sin θ sinϕ sin θ cosϕ sin θ cos θ ) (2.3) thus, the u function of the form (2.2) is the sum of the squares of the distances between “eponymous” points of two geometric structures with weights wi after matching points defined by vectors r1,0 and r2,0, with the coordinate center and rotation of the second structure relative to the first. by matching some characteristic points of two geometric structures, called “centers of mass” you can reduce the task of minimizing the comparison function over a full set of variables to finding the minimum of the u function over the rotation angles ϕ, θ, ψ. the following theorem is proved in [1]. theorem 2.1: the minimum of the u function of the form (2.2) is achieved at the point corresponding to copyright © 2020 assa. adv syst sci appl (2020) 106 n. chernikova, e. laneev “center of mass” of two geometric structures and is defined by vectors rj,0 = 1 w n∑ i=1 wirj,i, j = 1, 2. (2.4) so, the minimum of the function (2.2) corresponds to the shift of “centers of mass” of geometric structures defined by the formula (2.4) to the origin. the function (2.2) can now be considered as a function of rotation angles u(ϕ, θ, ψ) = n∑ i=1 wi|r1,i − r1,0 −q(ϕ, θ, ψ)(r2,i − r2,0)|2, (2.5) where r1,i, r2,i – given coordinates of geometric structures, and “centers of mass” r1,0 and r2,0 are calculated using the formula (2.4). the minimum of the function (2.5) is determined by the rotation angles ϕ, θ, ψ. let the minimum of the function u = u(ϕ, θ, ψ) of the form (2.5) be reached at the point (ϕ0, θ0, ψ0). in accordance with [1, 2], we will call the value s = (u(ϕ0, θ0, ψ0) w )1/2 . (2.6) the measure of proximity of two geometric structures. this value can be considered as a quantitative characteristic of the proximity of geometric structures, since it represents the average distance between points with the same indices in two structures after “superposing”. we will call two geometric structures equal if they can be superposed by points with the same indices by various displacements of these structures as solids. as shown in [1], this superposition is achieved by minimizing the comparison function (2.5) and corresponds to s = 0. we will call two structures approximately equal if s = ( 1 w min ϕ,θ,ψ u(ϕ, θ, ψ) )1/2 = (u(ϕ0, θ0, ψ0) w )1/2 6 s0, (2.7) where s0 is the specified value (in applications, it is determined by the specific practical situation). the inequality s 6 s0 will be called the criterion of proximity of structures. practical experience in studying the conformation of molecules, based on the results of comparing a significant number of structures [8, 9], resulted in the following conditional classification: s 6 s0 = 0, 1å – molecules are practically equal, 0, 1å < s 6 0, 2å – molecules are close, s > 0, 2å – molecules are different. note that if the minimum point is not unique and there is a point (ϕ1, θ1, ψ1) such that u(ϕ1, θ1, ψ1) = u(ϕ0, θ0, ψ0), then the value of s does not change and the non-uniqueness of the minimum point does not affect the proximity criterion of structures. the test for comparing the geometry of two geometric structures consists of three stages: 1) shifting the center of mass of each geometric structure to the coordinate center; 2) minimizing the function (2.5) over angles; 3) calculating the value of s using the formula (2.7) and inferring the proximity of the structures. additional information is provided by residual values ∆ri = |r1,i − r2,i|. when comparing two structures in accordance with the algorithm 1)-3), determination of the minimum of the comparison function u(ϕ, θ, ψ) over rotation angles after superposing the centers of mass of structures is performed numerically. copyright © 2020 assa. adv syst sci appl (2020) comparison of some molecular structures... 107 fig. 3.1. molecule c14h13no2 4,6-dimethyl-3-(phenylcarbonyl) pyridin-2 (1h) -one (i). 3. numerical study of crystal structures this section presents the results of the numerical study comparing chemically identical molecules in a crystal by minimizing the comparison function (2.5). in this case, we will consider molecules with an ordered structure ofn points-atoms as a geometric structure. the coordinates of the molecules in the crystal are obtained from x-ray analysis. in this study, the combined atoms in the molecules are assigned a weight of wi = 1. the remaining atoms are assigned a weight of wi = 0. the coordinates of the hydrogen atoms were not included in the calculations, since the accuracy of their determination is lower than that of other atoms and their presence is insignificant in the present task of superposing molecules, so they are assigned a weight of wi = 0. when comparing molecules, we will be interested in differences in the spatial structure of molecules with a single structural formula (graph). since the comparison function (2.5) is not convex, the rosenbrock zero-order method [6] is used to numerically minimize it. the program that implements the comparison algorithm 1)-3) uses the optimization library module [10] to minimize the comparison function (2.5) using the rosenbrock method. in this paper, we set the task of comparing the spatial structure of 4,6-dimethyl-3(phenylcarbonyl) pyridin-2 (1h) -one (i) molecules with the chemical formula c14h13no2 where n = 30 and in 3-benzoyl-4,6crystals-dimethyl-1-[2-(4-methylphenyl)-2-oxoethyl] pyridin-2 (1h)-one (ii, iii, iv) with the chemical formula c23h21no3, where n = 48. the composition, bonds and geometry of these molecules are schematically depicted in fig. 3.1 and fig. 3.2. the atomic coordinates were obtained by the method of x-ray analysis [11]. the characteristics of these substances in crystallography are described as follows: i c14h13no2 space group p21/c, z = 4 (molecule occupies 1 position); ii c23h21no3 space group pi, z = 8 (four independent molecules a, b, c, d); iii c23h21no3 · 12h2o (two independent molecules a, b); iv c23h21no3 · 12h2o space group c 2/c, z = 8 (molecule occupies one position). space group [12] describes all possible symmetries of an infinite number of periodically located points in three-dimensional space. the number z shows the number of molecules in the unit cell. in crystals i and iv, the molecules occupy one system of positions, that is, they are transformed into each other using symmetry operations of the space group. in crystals ii and iii, the molecules occupy independent positions, respectively, four (iia,iib,iic,iid) and two (iiia,iiib), that is, the molecules are not transformed into each other by symmetry operations of the space group of the crystal and therefore can be different in structure. copyright © 2020 assa. adv syst sci appl (2020) 108 n. chernikova, e. laneev fig. 3.2. molecule c23h21no3,3-benzoyl-4,6crystals-dimethyl-1-[2-(4-methylphenyl)-2-oxoethyl] pyridin-2 (1h)-one (ii, iii, iv) table 3.1. comparison of four independent molecules a, b, c, d in crystals ii the value of “residuals” ∆ri, characteristic s in crystals ii atom a− b a− c a− d b− c b− c b− d c− d n 1 0.20 0.06 0.03 0.23 0.05 0.18 0.08 c 2 0.19 0.16 0.04 0.34 0.08 0.15 0.20 c 3 0.06 0.15 0.05 0.20 0.02 0.03 0.19 c 4 0.05 0.10 0.06 0.09 0.03 0.10 0.16 c 5 0.02 0.08 0.05 0.08 0.08 0.07 0.11 c 7 0.05 0.06 0.01 0.04 0.04 0.04 0.04 c 8 0.19 0.03 0.03 0.17 0.13 0.16 0.02 c 9 0.03 0.10 0.03 0.08 0.06 0.08 0.08 o 10 0.27 0.22 0.04 0.47 0.07 0.29 0.23 c 11 0.07 0.24 0.04 0.30 0.14 0.04 0.26 o 12 0.22 0.35 0.08 0.56 0.30* 0.14 0.42 c 13 0.09 0.15 0.06 0.24 0.44* 0.13 0.12 c 14 0.30 0.35 0.06 0.65 0.69* 0.24 0.41 c 16 0.42 0.46 0.16 0.88 0.92* 0.26 0.63 c 18 0.37 0.41 0.15 0.78 0.96* 0.22 0.56 c 20 0.23 0.24 0.06 0.45 0.77* 0.18 0.30 c 22 0.12 0.09 0.07 0.19 0.53* 0.09 0.10 c 24 0.18 0.15 0.11 0.20 0.03 0.24 0.26 c 33 0.16 0.12 0.08 0.20 0.34* 0.15 0.12 o 34 0.31 0.28 0.15 0.39 0.56* 0.22 0.21 c 35 0.10 0.10 0.06 0.18 0.50* 0.14 0.14 c 36 0.04 0.29 0.04 0.27 0.40* 0.06 0.32 c 38 0.13 0.33 0.04 0.38 0.65* 0.12 0.30 c 40 0.24 0.14 0.06 0.37 0.91* 0.21 0.17 c 41 0.28 0.25 0.06 0.43 1.06* 0.24 0.26 c 43 0.19 0.24 0.08 0.35 0.88* 0.18 0.21 c 45 0.32 0.27 0.11 0.57 1.06* 0.35 0.22 s 0.21 0.23 0.08 0.40 0.08 0.18 0.27 copyright © 2020 assa. adv syst sci appl (2020) comparison of some molecular structures... 109 note that a comparison of bond lengths and bond angles does not reveal differences in the spatial structure of molecules. in eponymic bonds, the differences in length are no more than 0,02å, and in bond angles no more than 5◦. some differences are observed in individual torsion angles up to 19◦; however, how substantially this affects the general geometry of the molecules is not always clear. thus, it is not known in advance whether these molecules are similar in structure or whether their differences are significant. in this situation, there is a need for an objective, quantitative, comparison of the structure of molecules. for this purpose, the technique described in the previous section was applied to the study of these molecules. comparison of molecules was carried out by minimizing the comparison function (2.5). as already mentioned, the coordinates r1,i, r2,i, i = 1, ..., 48 (for substances ii iv), i = 1, ..., 30 (for substance i) for each pair of compared molecules were obtained on based on x-ray analysis data. all crystals (i iv) are centrosymmetric (both left and right molecules are present), therefore, to compare some pairs, one of the molecules was taken with inversion. the centers of mass were determined by the formula (2.4). the comparison function (2.5) was minimized numerically using the rosenbrock method over three euler angles after superposing the centers of mass of the two molecules being compared. the essential thing in our method of quantitative comparison of the spatial structure of molecules is the variation in the values of the weight factors wi. the ability to superpose molecules with separate fragments is achieved by setting the values of wi = 1 for compatible atoms and wi = 0 for other atoms. this allows you to change the center of gravity of the molecules, combine the molecules along the closest or most significant parts and thereby more clearly visualise certain differences in the structure of the molecules and their parts. as already mentioned, in our calculations we set the values wi = 0 for hydrogen atoms h. in table 3.1 we present the results of applying the comparison algorithm 1) 3) for matching 4 symmetrically independent molecules (a, b, c, d) that coexist in one crystalline substance ii. here, the residuals ∆ri = |r1,i − r2,i|, i = 1, ..., 48, i.e., the distances between the eponymic atoms in the two molecules under comparison are calculated after matching (at the point of minimum of the comparison function u ), as well as magnitude s – magnitude measure of proximity. results are shown for non-hydrogen atoms. the (*) indicates atoms for which wi = 0 was taken in this calculation. let us analyze the data obtained. in practice, only the pair of a-d molecules (s = 0.08å) are practically equal in their spatial structure; another pair of b− dmolecules (s = 0.18å) are close. other pairs of molecules have greater or lesser differences. the largest difference in the structure of molecules, is in the b− c pair (s = 0.40å). for this pair, another calculation was performed in which only for atoms of the central pyridine ring and atoms directly connected with it with indices i = 1, ..., 11, 24 and weight wi = 1. in this comparison, s = 0, 08å, which means the equality of these fragments and, in addition, this comparison highlighted the differences in the rotation of the phenyl rings. the result of this comparison is presented in fig. 3.3. as the calculations showed, two symmetrically independent molecules a and b in crystal iii (s = 0.04å), as well as pairs of molecules iv and iiia (s = 0.03å), iv and iiib (s = 0.04å) are almost identical. calculations comparing molecules from iv with four independent molecules from crystal ii indicated their significant differences in geometry. to identify differences for the same pairs of molecules, additional calculations were performed with varying weight factors. the ∆ri values for atoms with wi = 0 are also marked with (*). the calculation results are given in table 3.2. the results of the comparison point to parts of the molecules where there are differences and make us look for geometric characteristics to visually describe these differences. these characteristics were found as the differences in the angles between the planes (. 3.3) plane 1: through the atoms nc2c3c4c5c6, plane 2: through the atoms c13c14c16c18c20c22, plane 3: through the atoms (c35c36c38c40c41c43). copyright © 2020 assa. adv syst sci appl (2020) 110 n. chernikova, e. laneev fig. 3.3. the best combination of molecules iib and iic with wi = 1 for i = 1, ..., 11, 24. table 3.2. comparison of a molecule from crystal iv with a, b, c, d molecules in crystal ii. the value of “residuals” ∆ri, characteristic s atom iv &iia iv &iia iv &iib iv &iib iv &iic iv &iic iv &iid iv &iid n 1 0.43 0.04 0.38 0.03 0.41 0.02 0.41 0.02 c 2 0.31 0.01 0.31 0.03 0.38 0.05 0.29 0.04 c 3 0.22 0.02 0.16 0.04 0.33 0.02 0.17 0.05 c 4 0.35 0.06 0,40 0.04 0.43 0.04 0.29 0.02 c 5 0.44 0.06 0.47 0.06 0.47 0.08 0.40 0.05 c 7 0.45 0.01 0.41 0.04 0.40 0.06 0.44 0.04 c 8 0.49 0.06 0.46 0.13 0.50 0.03 0.51 0.06 c 9 0.57 0.10 0.57 0.09 0.55 0.13 0.58 0.07 o 10 0.43 0.04 0.50 0.03 0.40 0.06 0.44 0.05 c 11 0.13 0.08 0.17 0.06 0.22 0.17 0.13 0.13 o 12 0.08 0.29* 0.21 0.21* 0.36 0.50* 0.08 0.29* c 13 0.43 0.34* 0.34 0.29* 0.56 0.72* 0.45 0.34* c 14 0.27 0.17* 0.40 0.10* 0.48 0.69* 0.27 0.20* c 16 0.42 0.40* 0.27 0.27* 0.84 1.14* 0.32 0.26* c 18 0.99 0.92* 0.62 0.85* 1.40 1.66* 0.84 0.72* c 20 1.36 1.15* 1.22 1.21* 1.60 1.73* 1.32 1.04* c 22 1.06 0.85* 1.05 0.90* 1.15 1.25* 1.11 0.84* c 24 0.42 0.07 0.61 0.07 0.51 0.05 0.38 0.04 c 33 0.30 0.45* 0.31 0.54* 0.39 0.29* 0.36 0.37* o 34 0.35 0.46* 0.35 0.54* 0.60 0.09* 0.40 0.30* c 35 0.26 0.84* 0.20 1.05* 0.36 0.68* 0.27 0.80* c 36 0.18 0.83* 0.20 2.38* 0.35 0.93* 0.16 0.84* c 38 0.39 1.28* 0.35 2.54* 0.69 1.37* 0.41 1.37* c 40 0.78 1.80* 0.64 2.07* 0.88 1.53* 0.79 1.87* c 41 0.87 1.82* 0.67 3.37* 0.81 1.30* 0.83 1.81* c 43 0.58 1.36* 0.44 3.02* 0.55 0.88* 0.51 1.29* c 45 1.14 2.34* 0.96 2.65* 1.38 2.18* 1.22 2.50* s 0,60 0,05 0,54 0,06 0,72 0,08 0,59 0,05 table 3.3. angles (in degrees) between planes in molecules iia, iib, iic, iid and molecule in crystal iv planes iia iib iic iid iv planes 1 2 84.4 85.9 78.2 89.4 86.7 planes 1 3 86.5 87.8 71.7 85.7 71.2 planes 2 3 55.4 51.8 61.8 52.5 84.1 in addition, we present in table 3.4 the distances between the far points of the c18 and c45 molecules. the maximum distance is observed in molecules iic and iid, the minimum in iv. this illustrates the largest difference s = 0, 72å for the iic iv pair. fig. 3.4 shows the copyright © 2020 assa. adv syst sci appl (2020) comparison of some molecular structures... 111 table 3.4. distances (in angstroms) between the “far points” c18 and c45 of the iia, iib, iic, iid molecules in crystals ii and in the crystal iv molecule. iia iib iic iid iv distance c18c45 12.8 12.3 13.4 12.8 11.3 fig. 3.4. the best combination of molecules iib and iv with wi = 1 for i = 1, ..., 11, 24. result of combining iic iv with wi = 1 for the pyridine nucleus and the nearest atoms (nc2c3c4c5c7c9c11c24o2o10). the application of the comparison algorithm 1) 3) also works well for comparing molecules from i with the corresponding fragments of molecules iia, iib, iic, iid, iiia, iiib, iv. the results are presented in table 3.5. the table contains the calculated residuals ∆ri = |r1,i − r2,i|, i = 1, ..., 30. the results are shown for non-hydrogen atoms (except for h8 from crystal i). comparison of molecule i with fragments of molecules iv, iiia, iiib gave the same results, s = 0, 27å (there are some differences), which is explainable, since molecules iv, iiia, iiib are practically identical to each other in their geometry. the molecule in crystal i has the biggest differences with molecule iic, s = 0, 54å. to identify these differences, we use the same technique variation of weighting factors. the calculation by combining the planes of the pyridine rings and the directly bonded c and o atoms gave s = 0, 17å (parts of the fragments are close). this result indicates differences in the relative positions of the pyridine and phenyl rings. in this regard, the angles between plane 1 s = 0, 17å and plane 2 (c13c14c16c18c20c22) are calculated. for the molecule of their crystal i, the angle is 81, 1◦; the angle in iic is the most different from it, which is 78, 2◦. in addition, differences in the orientation of c = o bonds were found: differences in the torsion angles of c4c3c11o12, o12c11c13c22 and c3c11c13c22 are 19, 7◦, 19.9◦ and 12.5◦, respectively. 4. conclusion the calculations presented here show the effectiveness of the technique used to study the geometry of molecules in crystals using the comparison function. interesting facts were copyright © 2020 assa. adv syst sci appl (2020) 112 n. chernikova, e. laneev table 3.5. comparison of molecule i with corresponding fragments of molecules iia, iib, iic, iid, iiia, iiib, iv ((*) marked atoms for which wi = 0 is calculated) the value of “residuals” ∆ri, characteristic s atom i&iia i&iib i&iic i&iic i&iid i&iiic i&iiib i&iv n 1 0.28 0.27 0.46 0.05 0.24 0.19 0.20 0.20 c 2 0.37 0.39 0.45 0.07 0.36 0.30 0.31 0.31 c 3 0.20 0.23 0.26 0.04 0.19 0.15 0.13 0.13 c 4 0.33 0.33 0.45 0.04 0.29 0.15 0.14 0.15 c 5 0.37 0.37 0.45 0.08 0.34 0.24 0.24 0.24 c 7 0.17 0.24 0.21 0.10 0.18 0.13 0.15 0.14 () 8* 0.78* 0.68* 1.03* 0.52* 0.77* 0.77* 0.83* 0.69* c 9 0.45 0.53 0.44 0.10 0.44 0.44 0.44 0.45 o 10 0.70 0.71 0.83 0.10 0.61 0.56 0.56 0.57 c 11 0.38 0.37 0.55 0.06 0.37 0.17 0.18 0.19 o 12 0.78 0.75 0.96 0.73* 0.75 0.37 0.37 0.36 c 13 0.10 0.13 0.03 0.97* 0.05 0.05 0.05 0.04 c 14 0.31 0.41 0.15 1.19* 0.33 0.14 0.13 0.12 c 16 0.16 0.23 0.17 1.96* 0.19 0.23 0.23 0.23 c 18 0.55 0.55 0.63 2.57* 0.46 0.31 0.31 0.31 c 20 0.73 0.83 0.78 2.39* 0.73 0.25 0.26 0.25 c 22 0.40 0.47 0.41 1,55* 0.45 0,08 0.09 0.09 c 24 0.62 0.56 0.88 0.08* 0.49 0.28 0.27 0.27 s 0.45 0.47 0.54 0.17 0.42 0.27 0.27 0.27 obtained 3-benzoyl-4,6-dimethyl-1[2(4-methylphenyl) -2-oxoethyl] pyridin-2 (1h) one molecules have the same conformation in different crystals of iii and iv, and different conformation in one crystal ii. acknowledgements the above calculations were carried out by the authors using the program comparison in the language c], written by n.v. bakhtadze, to whom the authors express their sincere gratitude. the work was supported by the russian foundation for basic research (grant 17-5112064, grant 18-01-00106, grant 18-01-00590). references 1. laneev, e.b. & chernikova, n.yu. (2019). comparing the spatial structure of molecules by minimizing a comparison function, computational mathematics and mathematical physics, 59(1), 128-135. 2. chernikova, n.yu.& laneev, e.b. (2020). on method of nonlinear optimization in the study of molecules spatial structure yugoslav journal of operations research, 30(1), 91-99. 3. izmailov, a.f., & solodov, m.v. (2003) chislenniye metodi optimizatsii [numerical methods of optimization]. moscow, russia: fizmatlit, [in russian]. 4. arutyunov, a.v.& tynyanskii, n.t. (1985). the maximum principle in a problem with phase constraints. soviet journal of computer and system sciences, 23, 28-35. 5. arutyunov, a a.& mishchenko, a.s. (2019). a smooth version of johnson’s problem on derivations of group algebras, sb. math., 210(6) 756-782 6. himmelblau, d.m. (1972) applied nonlinear programming. new york, usa: mcgraw-hill book company. 7. lurie, a.i., (2002) analytical mechanics. berlin, germany: springer. copyright © 2020 assa. adv syst sci appl (2020) comparison of some molecular structures... 113 8. chernikova, n.yu., lavut, e.e. & zorkii, p.m. (1979) comparing the structure of symmetrically independent molecules in crystals of coordination compounds, russian journal of coordination chemistry, 5(8), 1265–1270. 9. lavut, e.e., zorkii, p.m., & chernikova, n.yu. (1982) contact conformers in crystals of coordination compounds, 2020 journal of structural chemistry 22(5), 715-718. 10. kniaz k. optimization.net, url: http://www.kniaz.net/software/rosnm.aspx. 11. feklicheva, e. m., rybakov, v. b., babaev, e. v., chernikova, n. yu. & ofitserov, e. n. (2020) the structure of products of phenacylation and subsequent (re)cyclizations of 3benzoyl-4,6-dimethylpyridin-2(1h)-one according to x-ray structural analysis, arabian journal of chemistry (in press). 12. international tables for crystallography. volume a: space-group symmetry / edited by m. i. aroyo. international union of crystallography, 2016. isbn 978-0-470-97423-0. copyright © 2020 assa. adv syst sci appl (2020) introduction assumed mathematical model numerical study of crystal structures conclusion adv syst sci appl 2020; 03:105–112 published online at https://ijassa.ipu.ru. estimating the response time of a cloud computing system with the help of neural networks anastasia v. gorbunova1*, vladimir m. vishnevsky1 1v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia abstract: the article presents a new approach to assessing the average response time of a cloud computing system and its dispersion. a fork-join system or a system with request splitting was chosen as a queuing model, and artificial neural networks were used as a method for estimating a variable of interest. the analysis showed that the estimates obtained were more accurate than those previously known. besides, the proposed approach allows expanding the analysis of the cloud system to the case of a model with a non-poisson input stream and non-exponential service time, as well as obtaining estimates for a larger number of performance indicators of the cloud system, which was not previously possible. keywords: cloud computing, parallel computing, queuing system, parallel service of requests, average response time, multilayer perceptron, artificial neural networks, machine learning methods 1. introduction cloud computing is a technology that allows a remote user to access various services that, as a rule, have significant computing resources upon request via the internet [2]. at the same time, even despite the presence of powerful computing infrastructure, the task of ensuring the required level of quality of the services provided and, at the same time, a logical interest in reducing the overall load on the equipment continue to be relevant for cloud providers [1, 9, 10, 23]. this is due to the growing popularity of the services they provide for many objective reasons, as well as the understandable need to reduce costs due to, for example, increased energy efficiency. one of the possible solutions to this problem is parallel data processing, which could be accomplished due to the use of virtualization technology, which results in a decrease in response time, and, accordingly, in maintaining or even improving the quality of the services provided in the cloud. thus, a cloud center consisting of several physical machines that can be used together to process complex user requests served in the order they arrive in the system is considered [5]. one of the most important indicators of the functioning of the cloud system is the response time, the evaluation of which has a large number of publications, for example, works [1,5,6,9,10,23] devoted to. nevertheless, the study of this parameter is still in demand due to the relevance of the practical task, and as a consequence of the need to improve the quality of existing estimates of this characteristic. the article is organized as follows: section 2 describes the mathematical model of the cloud center, section 3 discusses the possibility of applying a new approach to the analysis ∗corresponding author: avgorbunova@list.ru 106 a.v. gorbunova, v.m. vishnevsky of average response time, namely, artificial neural networks (anns), provides a numerical example, and finally, section 4 highlights the prospects for further research. 2. the mathematical model of a cloud computing system a queuing system (qs) of the fork-join type is considered as a model of a cloud system. systems of this type are also called “qs with parallel service of requests” [7, 8]. fork-join systems are models for many real physical systems in which the parallelization of the tasks is performed. for example, using the method of datagrams in computer networks, when a message arrives at the source node, it is divided into packets that are transmitted by different routes to the address node, in the buffer of which the message is assembled. other examples are the processes of assembling orders at the warehouse or the processes of functioning of high-performance applications in the production, medical, financial, scientific or any other areas, based on the concept of distributed or parallel computing [11, 15]. features of the functioning of the qs with the splitting of requests are as follows (fig. 2.1): 1) at the time the request enters the system, it is instantly split into k (k > 2) smaller components, i.e., sub-requests, each of which, in turn, becomes serviced by the device (if it is busy) or immediately begins to be serviced if the corresponding device is free. it is considered that each sub-request has its type, which must correspond to the number of the device on which it will be serviced; 2) after the service is ended the sub-request falls into the so-called synchronization buffer and remains there until all related sub-requests, that is, the sub-requests originally belonging to one request, finish their service. then the whole request is instantly assembled and only after this the request is considered served and may leave the system. since modeling a cloud center in the context of solving complex computational problems is discussed in this case, virtual machines (vms) will act as devices in the fork-join models. at the time of receipt, a complex user request is divided into k smaller components, each of which is processed on the corresponding vm. then it enters the synchronization buffer, where it remains until the last of the subtasks is processed, after which the user request is considered served. fig. 2.1. the model of the cloud center system of the fork-join type note that other sub-request servicing schemes are known. for example, when devices are blocked until all parts of the request are serviced or the sub-requests are indistinguishable in the sense that any k sub-requests are required to assemble the request, etc. a more detailed overview can be found, for example, in [7, 20]. one of the main performance indicators of any queuing system is its response time. the fork-join system in this sense is no exception. therefore, this concept is specified in the context of splitting requests. since the request is considered to be served only at the moment when the last sub-request is serviced, to calculate the residence time of the request in the copyright © 2020 assa. adv syst sci appl (2020) estimating the response time of a cloud computing system... 107 system, taking into account the fact that the moments of the appearance of all its sub-requests in the system coincide, it is sufficient to determine the maximum of all the times of the residence of its sub-requests. however, the task of calculating the exact value of the average response time, even despite the poisson nature of the incoming stream and the exponential service time on all devices in the case of splitting into more than 2 sub-requests, remains unresolved. the expression in closed form exists only for k = 2 [14], and for k > 2 only approximations of various degrees of accuracy were obtained [14, 21, 22]. this is explained by the complexity of the analysis of the residence times of the parts of the request in the system because of the existing relationship between them due to the general moments of receipt. the residence times of subrequests in the system are positively associated random variables. therefore, their maximum is stochastically no greater than the maximum of independent random variables with the same distribution. below are formulas that estimate the average response time in the system with splitting requests in the case of the same service intensities on all devices µk = µ, k = 1, k, and the incoming stream intensity equal to λ [14, 21, 22]. e[wk ] ≈ [ hk h2 + 4 11 ( 1− hk h2 ) ρ ] 12− ρ 8 1 µ− λ , (2.1) e[wk ] ≈ [ hk + ( k∑ i=1 ( k i ) (−1)i−1 i∑ m=1 ( i m ) (m− 1)! im+1 −hk ) λ µ ] 1 µ− λ , (2.2) e[wk ] ≈ 1 µ [ hk + ρ 2(1− ρ) ( k∑ k=1 1 k − ρ + (1− 2ρ) k∑ k=1 1 k(k − ρ) )] , (2.3) e[wk ] ≈ 1 µ− λ hk , (2.4) e[wk ] ≈ 1 µ− λ ( 1 + k − 1√ (2k − 1) ) , (2.5) where hk = ∑k i=1 1/i is the partial sum of the harmonic series, and ρ = λ/µ < 1 is the load factor. the first two of the above formulas are obtained due to the combination of the analytical approach with the empirical one, which consists in observing the behavior of the random variable under investigation through simulation and the subsequent use of the results of this observation in the preparation of the corresponding estimates. the third formula is derived by interpolation of high and weak input loads. approximation (2.4) is the result of analyzing the average response time for k independent m |m |1 qss operating in parallel, which is not only natural but also objectively justified assumption, since the expressions for the marginal probabilities of the number of k-type sub-requests (k = 1, k) in the original fork-join system coincide with the expressions for stationary probabilities in the m |m |1 system [5]. finally, the last expression (2.5) from the above is the upper limit of the average value of the maximum of order statistics for independent exponentially distributed parameterized random variables with (µ− λ). in addition to estimating the average response time, the approximations for variance, and higher moments of the studied value are worth noticing. these numerical characteristics make it possible to formulate a complete picture of the response time behavior. however, there is no longer such a variety of analytical expressions, and it is desirable to improve the existing estimates. so, in [6], an expression is presented for the variance of the average response time copyright © 2020 assa. adv syst sci appl (2020) 108 a.v. gorbunova, v.m. vishnevsky for both identical µ and different µk parameters of exponential service times, k = 1, k d[wk ] ≈ k∑ l=1 2 (µl − λ)2 − ∑ 16l<m6k 2 (µl + µm − 2λ)2 + + ∑ 16l<m<k6k 2 (µl + µm + µk − 3λ)2 + . . .+ (−1)k−1 2 (µ1 + µ2 + . . .+ µk −kλ)2 − − ( k∑ l=1 1 µ− λ − ∑ 16l<m6k 1 µl + µm − 2λ + ∑ 16l<m<k6k 1 µl + µm + µk − 3λ + + . . .+ (−1)k−1 1 µ1 + µ2 + . . .+ µk −kλ )2 , (2.6) d[wk ] ≈ 2 (µ− λ)2 k∑ i=1 ( k i ) (−1)i−1 1 i2 − ( 1 µ− λ k∑ i=1 1 i )2 . (2.7) 3. artificial neural networks and queuing systems with request splitting issues of using anns to study qs are currently poorly reflected in the world literature, although ann methods can be effectively used to study several queues that are not solved in theory. in particular, this refers to the problem of forecasting or regression, which comes down to the problem of approximating the function of one or more variables. according to the theorems given in [3, 19], as well as in the survey [16], any continuous function can be represented as a combination of linear operations and a single nonlinear element. thus, these theorems turn out to apply to the ann, and, accordingly, to the multilayer perceptron, in which the so-called activation function acts as a nonlinear element. in the theory of queuing, there are many models, for example, non-markovian models, for which classical analytical methods of solution are not applicable, and the numerical approach is rather laborious. in such conditions, addressing neural networks seems to be the best solution. nevertheless, until recently, there have been very few works devoted to the application of anns to solving problems of queuing theory, namely, the analysis of complex queuing models. the articles [17,18] were devoted to building a model of the classicm |m |1 queuing system using an ann and analyzing the adequacy of this simulation. the experiments showed that the values modeled using the ann coincide with calculated values obtained using the classic mathematical approach. one of the first works (if not the very first), which affected the application of neural network apparatus for the analysis of non-markovian qs models is the article [12], in which non-markovian qs with heating [13] is researched since even in the case of qs h2|m |m |3 or m |h2|m |3 numerical algorithms for calculating stationary probabilities of states turn out to be very laborious and resource-intensive. the use of neural networks, in this case, made it possible to significantly reduce the complexity without losing accuracy in the calculations. 3.1. ann model for fork-join the multilayer perceptron will be used to approximate the average value and variance of the response time. moreover, to increase the accuracy of the approximation, it is necessary to train not one perceptron with two output neurons corresponding to each of the indicated characteristics, but two perceptrons with one output neuron in each to estimate the mathematical expectation and standard deviation of the response time. copyright © 2020 assa. adv syst sci appl (2020) estimating the response time of a cloud computing system... 109 despite the sufficiency of one hidden layer according to the above theorems on the approximation of functions, let us stay with the ann with two hidden layers with the logistic activation function on each neuron ϕ(x) = 1/(1 + e−x) (fig. 3.2) from the estimate accuracy increase point of view. the load ρ and the k (k = 1, 20) number of sub-requests into which the request entering the system is split with a fixed value of λ = 2 (table 3.1) will be used as input data. fig. 3.2. three-layer perceptron scheme table 3.1. input data for calculating qs fork-join ρ 0.1 0.2 ... 0.9 0.1 0.2 ... 0.9 0.1 ... 0.9 k 3 3 ... 3 4 4 ... 4 5 ... 20 the vectors of the values of the training and test samples were obtained using simulation in a specialized gpss software environment (general purpose simulation system), and the construction of the neural network model itself and its training was carried out using the python programming language. in the process of training anns with the backpropagation method, the best neural network configuration was selected based on the mean square error (mse), mean absolute error (mae), and average absolute relative error (mape): mse = 1 n n∑ i=1 (yi − ŷi)2, mae = 1 n n∑ i=1 |yi − ŷi|, mape = 1 n n∑ i=1 ∣∣∣∣(yi − ŷi)yi ∣∣∣∣ · 100%, where yi is the desired (actual) output value for the i-th input element from the training set, ŷi is the real value produced by the system, n is the number of elements in the training set, i = 1, n . the choice of the latter of the measures presented here is since it was used by the authors to assess the quality of the approximate formulas obtained by them. since the best approximation for the average response time is given by formula (2.1), in particular, the error of this formula for k 6 32 does not exceed 5%, the results of the ann are compared precisely with the results of applying this formula. so, in table 3.2, the values of the above types of errors calculated already on the test sample in the case of using simulation, neural network, and formula (2.1) for calculations are presented. it is the trained perceptron that shows the best result even despite the relatively small number of vectors in the training sample, which is measured in hundreds and probably does not improve the quality of approximation in a general case. nevertheless, the obtained result speaks for itself and only confirms the potential of the proposed approach. in the case of estimating the standard deviation of the response time, the well-known formulas show a large error of approximation, in contrast to estimating the average value of this quantity. therefore, anns predictably give the best result (table 3.3). copyright © 2020 assa. adv syst sci appl (2020) 110 a.v. gorbunova, v.m. vishnevsky table 3.2. comparison of error values in calculating the average response time using formula (2.1) and ann. error type mse mae mape, % formula (2.1) 0,022164 0,081634 1,592066 neural network 0,000649 0,015875 0,739039 table 3.3. comparison of error values in calculating the standard deviation of the response time using formula (2.7) and ann. error type mse mae mape, % formula (2.7) 0,080494 1,696545 6,896426 neural network 0,000642 0,010495 0,363507 4. conclusion due to applying to a new approach to estimating the random value of the response time of a cloud computing system, a more accurate approximation is obtained for the mathematical expectation and variance of this characteristic. application of neural networks to the analysis of indicators of service quality in the forkjoin type qss allows expanding the class of models under consideration. the reason is the absence of restrictions on assumptions about the type of incoming flow or distribution of service times, which, as a rule, greatly complicates the analytical or numerical process of solving the problem, while there is no loss in estimation quality of required parameters. also, it becomes possible to increase the number of analyzed parameters. so, for example, in the future, it is possible to clarify the well-known estimate of the average time spent by the first sub-request in the synchronization buffer in anticipation of the last of its related components [6]. that is, in fact, the average residence time of related sub-requests in the buffer until exiting the system. information about this value is important, for example, for designing a cloud system when planning the size of the waiting buffer, because the longer this time, the larger the buffer should be. acknowledgements the authors are grateful to a.v. lebedev (moscow state university) for valuable recommendations and useful comments. references 1. alhamad, m., dillon, t., wu, ch. & chang, e. (2010) response time for cloud computing providers, proceedings of the 12th international conference on information integration and web-based applications & services, 603–606. 2. buyya, r., broberg, j. & goscinski a.m. (2011) cloud computing: principles and paradigms. new jersey, usa: john wiley & sons. 3. gorban’ a.n. (1998) obobshchennaya approksimacionnaya teorema i vychislitel’nye vozmozhnosti nejronnyh setej [generalized approximation theorem and computational capabilities of neural networks], sibirskij zhurnal vychislitel’noj matematiki [siberian j. of numer. mathematics], 1(1), 12–24, [in russian] 4. gorbunova, a.v. & lebedev, a.v. (2020) bivariate distributions of maximum remaining service times in fork-join infinite-server queues, problems of information transmission, 56(1), 73–90. 5. gorbunova, a.v., zaryadov, i.s., matushenko, s.i. & samuylov, k.e. 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(2009) service performance and analysis in cloud computing, 5th ieee world congress on services (services-1’09) proceedings, los angeles, 693– 700. copyright © 2020 assa. adv syst sci appl (2020) introduction the mathematical model of a cloud computing system artificial neural networks and queuing systems with request splitting ann model for fork-join conclusion adv syst sci appl 2022; 02:46–61 published online at https://ijassa.ipu.ru. graph methods for improving the invalid solution of the locomotive assignment problem under time constraints liudmila zhilyakova1*, vasily koreshkov1 1v.a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia abstract: finding the optimal assignment of locomotives to fulfill freight train transportation schedule under time constraints is a complex problem, even on a linear section of the railway. in previous studies, a graph model was proposed that allows solving the assignment problem using a static graph algorithm. it was proved that finding the optimal assignment without constraints is equivalent to finding the minimal path cover of a specific acyclic graph. however, due to time constraints, the optimal solution may not be valid. this paper presents methods for modifying the found solution in order for it to satisfy the time constraints on locomotives. two operations on path in the minimal path cover of a graph are introduced: crossover and changeover. they allow to rebuild the found solution without spoiling the admissible sequences of transportation, while the invalid ones are corrected. keywords: graph models, layered digraphs, minimal path cover with constraints, optimal assignment problem 1. introduction a wide range of methods and algorithms are used to solve problems of assigning locomotive and scheduling freight rail transportation. the papers [1–6] present different methods of solving the optimization problem for rail transportation. the paper [1] discusses graph methods and algorithms; in [2] a multi-commodity network and column generation model are considered; [3, 4] use dynamic programming methods; in [5] a mixed integer programming formulation of the problem is presented; in [6] genetic algorithms are applied. the article [7] presents an overview of the multimodal freight transportation optimization models. a model of the related problem, crew scheduling in railway transportation for a real world case, is introduced in [8]. this article proposes a two-step algorithm and investigates its efficiency. the present research is a continuation of the study [9] where the problem of optimal locomotives assignment to fulfill the freight traffic schedule on a linear railway section is solved using a static graph algorithm. this model, in turn, originates in studies [10, 11]. these papers present the concepts and definitions that formed the basis of the graph model constructed in [9]. if there are time constraints on locomotives (early completion, late start), the first stage of the search for a solution coincides with the solution of the problem without constraints. a special acyclic graph is constructed and the minimal path cover is sought for it. any isolated vertex is also considered a path, and therefore such a cover always exists. methods for constructing the minimal path cover of a directed acyclic graph are described in [12–15]. in these papers, it is shown that the problem of finding such a cover of an acyclic ∗corresponding author: zhilyakova.ludmila@gmail.com graph methods for improving the invalid solution of the locomotive... 47 digraph is equivalent to finding the maximum matching in a bipartite graph that is constructed from the given acyclic graph. an acyclic graph is constructed as follows. its vertices correspond to transportation that must be carried out at some time moments; the arcs connecting two vertices indicate that two transportations can be performed sequentially, one after the other by the same locomotive. the minimal path cover of such an acyclic graph is a solution to the original problem: it specifies locomotive assignments to trains over the entire planning horizon. in a problem with constraints, the resulting solution may not be valid: some locomotives may not be available for some of the assigned transportations. therefore, the solution found in the previous step is checked for validity. in [9], methods for correcting an invalid solution are proposed, taking into account the time constraints. however, some cases where these methods do not allow finding the existing optimal solution remained beyond the scope of consideration. these cases are rare exceptions, but they also need to be handled. in this paper, we propose two operations that change the found minimal path cover of the acyclic graph. the first one is called ”crossover” by analogy with the genetic recombination, following the terminology adopted in genetic algorithms [16, 17]. the second is called ”changeover”. they make it possible to construct new minimal covers corresponding to valid solutions to the problem with time constraints. 2. graph model of the problem 2.1. basic definitions let us consider freight transportation on a linear railway section with k stations. the schedule for the transportation of freight trains is given as follows. for each of the n trains, the station and time of departure and the station of arrival are specified. it is believed that locomotives carry trains at a constant speed. for each of the m locomotives, the start station and time constraints (late start, early end) are specified. the set of stations s = {s0, . . . , sk} is linearly ordered by their arrangement. time moments ti correspond to the departure and arrival of trains. the lengths of the intervals between moments may vary. the set of time moments is denoted as {t0, . . . , tfin} where tfin is a given planning horizon. the set of transportations (train trajectories) is denoted as tr = {tr1, . . . , t rn}. the set of locomotives is denoted as l = {l1, . . . , lm}. we will assume that the velocities of all locomotives are the same and equal to v. the trains also move at speed v. the schedule can be represented as a graph of the transportations on a plane where the abscissa axis indicates a section of the railway with stations {s0, . . . , sk}, and the ordinate axis indicates time interval [t0, tfin]. then train trajectories are represented as segments with angles to the y-axis v and −v. their ends have coordinates (tij1 , s i k1 ) and (til2 , s i p2 ), where indices 1 and 2 denote the start and destination points, i is the index of transportation thi. the initial positions of the locomotives are denoted by points on this plane (fig. 2.1). given locomotive li has coordinates (t0, sq) if it is available starting from time moment t0. otherwise it has time constraint ”on the left” and its coordinates are (tr, sq) where r > 0 (locomotives l10 − l12 in fig. 2.1). we consider a linear section of the railway, however, there may be several tracks on this section. thus, several parallel transportations are allowed, like tr4 and tr5 in fig. 2.1. 2.2. reachability graph consider two transportations tri and trj . they characterize by two pairs of coordinates: (tih1 , si k1 ), (til2 , s i p2 ) and (tjh1 , sj k1 ), (tjl2 , s j p2 ). suppose that transportation tri ends before copyright © 2022 assa. adv syst sci appl (2022) 48 l. zhilyakova, v. koreshkov s1 s2 s3 s4s0 0 4 7 2 8 tfin tr1 tr2 tr6tr4 tr5 1 tr7 tr3 3 tr8 tr9 s5 s6 s7 5 10 6 9 l6 l7 l8 l9l2l1 l5 l4l3 locomotives transportationsli trj l10 l11 l12 fig. 2.1. geometric transportation model. all locomotives and the earliest transportations are labeled transportation trj starts: til2 < tjh1 . these transportations can be carried out by the same locomotive if their coordinates meet the condition: sj k1 − si p2 ≤ v(tjh1 − til2); (2.1) sj k1 − si p2 ≥ −v(tjh1 − til2). (2.2) in general case, a transportation can be carried out after transportation tri if it is inside the ”cone of reachability” from the right end of transportation tri. s − si p2 ≤ v(t− til2); s − si p2 ≥ −v(t− til2). since all coordinates of transportations are known, it is possible to construct a reachability graph. its vertices are transportations, and arcs denote the reachability relation of the subsequent transportation from the previous (fig. 2.2). 2.3. transformations of reachability graph it was shown in [9] that the graph in the form represented in fig. 2.2 is difficult to treat ”as is.” for a problem without time constraints, it can be used to find a solution. however, in a problem with time constraints, the existing solution may not be found on such a graph. copyright © 2022 assa. adv syst sci appl (2022) graph methods for improving the invalid solution of the locomotive... 49 t0 t1 t2 tfin … transportations reachability relation fig. 2.2. reachability graph. vertices represent transportations, arcs denote the reachability relation. the time axis is directed from top to bottom. the time moments of departure and arrival of trains are marked. several sequential transformations were proposed, making it possible to obtain a regular structure graph. the conversion sequence is shown in fig. 2.3. fig. 2.3(a) contains the same graph as in fig. 2.2. dotted lines represent all potentially possible transportation sequences. 2.3.1. layer-by-layer structure. we define the layered structure as follows. the first layer contains those transportation j for which there are no transportations i satisfying inequalities (2.1) and (2.2). in other words, they cannot be transported after any other transportation by the same locomotive. the second layer is formed only by those transportations that are reachable from the first layer’s transportations, and from no others. that is, they have parents from only one layer. transportation of the third layer has parents in the first two layers (necessarily in both). then the process is repeated. the result of such transformation (with some more modifications) is demonstrated in fig. 2.3(b). note that the time sequence of transportations given in fig. 2.3(a) is violated. 2.3.2. elimination of transitive closure. in [9], it was proved that the arcs of transitive closure passing through one or several layers can be eliminated in a number of cases. all valid cases are described. in fig. 2.2 and fig. 2.3(a), most of these arcs are already removed in order to simplify the picture. note that not all arcs connecting vertices from non-adjacent layers can be deleted. 2.3.3. cloning vertices to all the layers. in a problem without constraints, such a procedure is superfluous. however, in a problem with time constraints, when the found solution is invalid and needs to be corrected, the clone vertices allow finding those solutions in copyright © 2022 assa. adv syst sci appl (2022) 50 l. zhilyakova, v. koreshkov which locomotives with time constraints can stop at an intermediate layer (or start from an intermediate layer). on intermediate layers, clone vertices are built so that each pair of adjacent layers can be considered separately. in fig. 2.3(c) cloned vertices and new arcs are indicated in red. fig. 2.3(d) demonstrates one of the solutions: minimal path cover with nine paths. it is easy to prove that the number of paths in the cover cannot be less than the number of vertices in the graph’s maximum layer (layer with a maximum number of vertices). (b) (d) (a) (c) fig. 2.3. sequential transformations of reachability graph. (a) graph coinciding with that shown in fig. 2.2; dotted lines are all possible interchanging of locomotives between transportations; (b) layer-by-layer graph organization; blue dashed lines are arcs between non-adjacent layers; (c) cloning vertices to the intermediate and bottom layers; (d) the minimal cover found 3. solving the problem without time constraints in [9], it is proved that solving the problem without time constraints on locomotives is equivalent to finding the minimal path cover of an acyclic graph. 3.1. finding the minimal path cover the minimal path cover of a directed acyclic graph can be found as the maximum matching in a corresponding bipartite graph where each ”intermediate” vertex of an original graph (a vertex that has an ancestor and a child) is divided into two halves. one half has only input arcs, the other only output arcs [12–15]. this is demonstrated schematically in fig. 2.3 (b)(d). copyright © 2022 assa. adv syst sci appl (2022) graph methods for improving the invalid solution of the locomotive... 51 the layered structure of a reachability graph shows that the vertices or half-vertices that form two adjacent layers can be considered as independent components of a bipartite graph. in this case, the maximum matching can be found between each pair of layers separately. this decomposition significantly reduces the size of the problem. after finding the maximum matching, back-gluing the half-vertices restores the solution: the minimal path cover. 3.2. formal definitions let g = (x,a) be a directed graph, where x is a set of vertices, a is a set of arcs. definition 3.1: the sequence of vertices and arcs (xi0 , ai0 , xi1 , ai1 , ..., xik−1 , aik−1 , xik) in graph g is called a path. here aij = (xij−1 , xij), j = 1, k. definition 3.2: a simple path is a sequence of vertices and arcs (xi0 , ai0 , xi1 , ai1 , ..., xik−1 , aik−1 , xik) such that all vertices in it are distinct. remark 3.1: note that in a directed acyclic graph, any path is simple. paths in the reachability graph will be denoted as {path1, ..., pathk}. pathi = (xi0 , ai0 , xi1 , ai1 , ..., xik−1 , aik−1 , xik) = (xi, ai). definition 3.3: a set of paths p = {path1, ..., pathk}, where pathi = (xi, ai), is called a path cover of a directed graph, if xi ∩xj = ∅ for i 6= j and ∪ki=1xi = x; accordingly, ai ∩ aj = ∅ for i 6= j and ∪ki=1ai ⊆ a. the cardinality of a path cover k = |p | is called the cover size. since any isolated vertex is a reduced case of a path, a path cover always exists. definition 3.4: a path cover p = {path1, ..., pathk} is minimal if there is no cover of size l, such that l < k. 3.3. optimal solution the number of paths in the minimal cover determines the minimum number of locomotives for the performing of all transportations. to solve the assignment problem, one more matching must be performed – between the found paths and the available locomotives. if there is a matching that involves all paths, then the unconstrained problem has a solution. let the solution of the unconstrained problem exist. we can assign at least one locomotive to each path (i.e. consequence of transportations). the question arises whether the found solution can be corrected if it does not satisfy the time constraints imposed on some locomotives. such a solution will be called invalid. 4. correcting invalid solution 4.1. crossover and changeover operations. distance between paths to describe the procedure for correcting the found minimal cover, consider the illustration in fig. 4.4. fig. 4.4(a) represents the locomotive assignment to the transportation sequences denoted by paths in the cover. consider the fragment of the cover with three paths and three locomotives with indices 5, 6, and 7. let locomotive 6 have a time constraint, and the last transportation in the path cannot be performed (fig. 4.4(b)). copyright © 2022 assa. adv syst sci appl (2022) 52 l. zhilyakova, v. koreshkov 1 2 4 3 5 6 7 9 8 5 6 7 5 6 7 (a) (b) (c)! ! ! fig. 4.4. correction of the invalid minimal path cover. (a) minimal path cover, coinciding with fig. 2.3(d), and the locomotive assignment for each sequence of transportations; (b) the highlighted fragment of fig. 4.4(a): locomotive 6 has a time constraint and cannot perform the last transportation; (c) the two paths exchange sections between the last two layers. obtaining the valid cover in [9], a method is described that allows changing the assignments of locomotives without changing the paths if such a new assignment exists. so, let us assume that it is impossible to obtain a valid cover by changing the locomotives’ assignment. in this instance, this means that locomotive 6 has no alternative connection to the first vertices of the other paths. the only way to correct the solution is to change the cover locally so that two paths swap the end sections. then the invalid path becomes valid. and the second path will remain valid if its locomotive has no constraints (fig. 4.4(c)). we introduce the procedures that will allow to change paths while maintaining minimal cover. first, let us introduce the notions of free arcs and vertices. definition 4.1: arcs and clone vertices of a reachability graph that do not participate in the current minimal path cover are called free. definition 4.2: a path consisting of free vertices and free arcs is called free. remark 4.1: obviously, the vertices corresponding to the transportations cannot be free, they always belong to the minimal cover. 4.1.1. crossover. following the terminology adopted in genetic algorithms [16], we will call the exchange of path sections a crossover. we introduce the concepts of simple and complex crossover. copyright © 2022 assa. adv syst sci appl (2022) graph methods for improving the invalid solution of the locomotive... 53 definition 4.3: a simple crossover, or 2-crossover, is the exchange of two paths by homologous sections using free arcs. homologous sections contain the same number of vertices located in the same layers. crossover can only be implemented between two adjacent layers. a simple crossover between paths is feasible if and only if there is at least a pair of adjacent layers in which the vertices of two paths form a biclique in a bipartite graph. then two free arcs are involved, which switches the paths’ ends. let us consider the matching formed by the arcs between adjacent layers as a permutation on a set of paths. the crossover of the paths corresponds to a transposition, or a permutation in which a pair of elements swaps (fig. 4.5 (a),(b)). (b) (c)(a) fig. 4.5. simple crossover: (a) the minimal cover; (b) the minimal cover and the free arcs, indicated by grey lines; the selected biclique is highlighted in red; (c) the simple crossover and the new minimal cover the cycle of length k is a permutation of a subset of a set x: {x1, x2, . . . , xk} ⊆ x mapping xk into x1, xi into xi+1, (i < k). the transposition is a cycle of length 2. definition 4.4: we define a cyclic exchange involving k ≥ 3 paths as a complex crossover. a complex crossover of the order k, or k-crossover, is a cyclic exchange of k paths by homologous segments. an example of a 3-crossover is shown in fig. 4.6. any permutation can be uniquely decomposed into a composition of disjoint cycles of length k ≥ 2, and the expression of the permutation is unique up to the order of the cycles. thus, the two introduced operations make it possible to construct any arbitrarily given permutation. theorem 4.1: the k-crossover (k ≥ 2) operation converts a valid path with a locomotive without constraints into a valid path. copyright © 2022 assa. adv syst sci appl (2022) 54 l. zhilyakova, v. koreshkov (b) (c)(a) fig. 4.6. complex crossover: (a) the minimal cover; (b) the minimal cover and the free arcs, indicated by grey lines; the selected arcs are highlighted in red; (c) the complex crossover and the new minimal cover proof if the crossover is possible, this means that the initial graph contains all the arcs necessary to form new paths. the arc in the graph denotes the reachability of the next transportation from the end of the previous one. thus, the new path section is reachable for the locomotive as well as the previous path section. and since the locomotive has no time constraints, the new path will be entirely valid. 4.1.2. changeover. in a crossover, two or more paths exchange their parts, while the number of paths in the cover always remains unchanged. there are other situations where the end of an invalid path needs to be switched from one vertex to another. usually, the new valid end is an isolated vertex at the bottom level (clone or original) or a path consisting of clone vertices (like the red paths in fig. 2.3(c),(d)). definition 4.5: changeover of a path pathi is switching the end of a path to a free vertex or a free path using a free arc (definitions 4.1, 4.2). in some cases a switch can be made to the original isolated vertex if it is a path. unlike crossover, which is a binary or, in general, n-ary operation, the changeover operation is unary. fig. 4.7 demonstrates an example of a changeover. in the reachability graph in fig. 4.7(a), the colors of vertices indicate the availability of locomotives. the blue locomotive has a time constraint and cannot be assigned to the transportation of the bottom level. in this case, the vertices of the previous level with blue parts are cloned (fig. 4.7(b)). in fig. 4.7(c), the invalid cover is presented; fig. 4.7(d) demonstrates the changeover operation. it often happens that a changeover makes free a vertex that can become a part of another path, i.e., the corresponding transportation can be carried out by another locomotive. in copyright © 2022 assa. adv syst sci appl (2022) graph methods for improving the invalid solution of the locomotive... 55 this case, one more switch occurs, which we will call a co-changeover. an example of cochangeover operations is shown in fig. 4.7(e). (a) (b) (с) (e)! ! ! !!(d) fig. 4.7. changeover and co-changeover: (a) the initial reachability graph; colors of vertices correspond to available locomotives; blue locomotive has a time constraint; (b) cloning vertices to the bottom level; (c) the invalid cover; (d) changeover from an invalid vertex to a valid clone vertex; (e) co-changeover from the clone vertex to the real transportation the figure 4.7 shows a changeover and a co-changeover that do not change the number of paths in the cover. however, there may be situations where the changeover operation generates a new path that requires one more locomotive. such a situation is demonstrated in fig. 4.8. note that if there were no time constraints, two locomotives would be sufficient to perform all the transportations. remark 4.2: the examples in figures 4.7 and 4.8 consider the case of early termination of locomotives. the second possible case is a late start. without loss of generality, we will consider time constraints corresponding to early termination. the problem with a late start is dual and is solved in a similar way. remark 4.3: as in the case of k-crossover, k-co-changeover is possible, when there is a cyclic replacement of the ends of the paths with the release of one clone or original vertex or a path. 4.1.3. distance between paths. let us number the paths from left to right according to their beginnings in the graph. the distance d(pathi, pathj) from the path pathi to the path pathj is a vector of length 2: d(pathi, pathj) = (d1, d2). the first component d1 is equal to the minimum number of crossover operations that must be performed in order for the path pathi to get the end of the path pathj . the second component takes the value 0 or 1, depending on whether a changeover operation is needed to reach the path pathj . if pathj is unreachable from pathi, we will assume that d(pathi, pathj) = (∞,∞). there cannot be more than one changeover operations when determining the distance, since a changeover can only be used at the final stage. copyright © 2022 assa. adv syst sci appl (2022) 56 l. zhilyakova, v. koreshkov (a) (b) (с) (d)! !! ! ! ! ! ! fig. 4.8. changeover and co-changeover: (a) the initial reachability graph; colors of vertices correspond to available locomotives; the red and blue locomotives have time constraints; (b) cloning vertex to the bottom level; (c) the invalid cover; (d) changeover (blue) from an invalid vertex to a valid clone vertex, co-changeover (red), and assigning a new locomotive (green) generally, the distance defined in this way is non-symmetric: the equality d(pathi, pathj) = d(pathj, pathi) may not hold. it depends on the topology of the graph. fig. 4.9 (b) demonstrates the symmetric distance between paths path1 and path2; fig. 4.9 (c) and (d) show that d(path1, path3) 6= d(path3, path1), since d(path1, path3) = (1, 0) and d(path3, path1) = (2, 0). remark 4.4: it should be taken into account when measuring the distance that each crossover is performed between a pair of adjacent layers. all permutations within a pair of layers are considered a single complex crossover (fig. 4.9 (c)). 4.2. changing locomotive assignment by constructing a layer-by-layer structure, we transformed the graph so that every two consecutive layers represent an independent bipartite graph. when searching for the minimal cover, the maximum matching is searched locally between each pair of layers. we will assume that the problem without time constraints has a solution and any consequence of transportations can be performed by at least one locomotive. before applying the algorithm presented below, let us try to correct not the cover, but the assignments of the locomotives. this procedure is described in detail in [9]. the idea is shown in fig. 4.10. we construct a three-layer graph as follows. the mid-level vertices are the condensation of the paths that make up the minimum cover of the reachability graph. top and bottom layers contain all available locomotives. the arcs between top and middle layers are all possible assignments of locomotives to the initial vertices of the paths. the arcs between middle and bottom layers indicate the correspondence of locomotives and the ends of each path. so, if the same locomotive is connected to the middle vertex from the upper and copyright © 2022 assa. adv syst sci appl (2022) graph methods for improving the invalid solution of the locomotive... 57 (a) (b) (d)(c) fig. 4.9. distance between paths: (a) the initial reachability graph coinciding with the graphs in fig. 4.5, fig. 4.6; (b) d(path1, path2) = d(path2, path1) = (1, 0) ; (c) d(path1, path3) = (1, 0); (d) d(path3, path1) = (2, 0) lower layers, then it can carry out the entire sequence of transportations. in terms of a threelayer graph, this means that we should remove arcs that are not symmetrical with respect to the middle layer, since they connect the beginning and end of the track with different locomotives. fig. 4.10(a) demonstrates the principle of construction of a three-layer graph. fig. 4.10(b) contains a reduced graph from fig. 4.10(a), on which only arcs are left that are symmetrical with respect to the middle layer. as it is easy to see, blue locomotive is not connected with any path. this cover cannot be corrected by reassigning locomotives. fig. 4.10(c) considers another case. here the invalid assignment of locomotives can be corrected. the reduced symmetric graph immediately offers a solution – fig. 4.10(d). the situation in fig. 4.10(a),(b) will be considered again after the description of the algorithm of correcting the solution fig. 4.11. 4.3. algorithm for improving invalid paths in the minimal cover 1. find the minimal path cover of a layered reachability graph. 2. construct a three-layer graph locomotives-paths-locomotives (fig. 4.10). 3. find an invalid path pathi in this graph (the path associated with disjoint sets of locomotives of the upper and lower layers). 4. find the locomotive vertex ls in the upper layer that is not assigned to any valid path and has an arc to the path pathi. since it is assumed that the problem has a solution for locomotives without constraints, such a locomotive ls necessarily exists. 5. find a valid path pathj that has a connection with the locomotive ls at the lower level. if there are more than one of such paths, rank the set of candidates in order of increasing the second coordinate of the distance vector – first all paths with d2(pathi, pathj) = 0, then with d2(pathi, pathj) = 1. ranking on the second coordinate is done to avoid the copyright © 2022 assa. adv syst sci appl (2022) 58 l. zhilyakova, v. koreshkov (a) (b) (c) (d) fig. 4.10. construction of a three-layer graph and two situations of possible locomotive assignments: (a) the three-layer graph; the middle layer is the condensation of the paths; (b) arcs symmetrical with respect to the middle layer; solution does not exist; (c) another possible situation of matching paths and locomotives; invalid assignment of the red locomotive; (d) the symmetric arcs and the found solution changeover operation, since it may increase the number of paths in the minimal cover (fig. 4.8). take as pathj the first path from the list. 6. if d1(pathi, pathj) = 1 and d2(pathi, pathj) = 0, then a single k-crossover is needed. if crossover is simple (k = 2). • if pathj assigned to a locomotive with constraints and the crossover makes this path invalid then delete pathj from the list; if there are other crossover candidates, go to step 5, else to step 3 (in this case, there is no valid cover, however, we will improve the cover as long as possible). • otherwise, perform a 2-crossover between paths pathi, pathj on corresponding layers. • if there are other invalid paths, go to step 3; otherwise a valid cover is found, exit. if k > 2. • if all the paths included in the k crossover, except pathi, have locomotives without constraints, make a k-crossover; if there are other invalid paths, go to step 3; otherwise, a valid cover is found, exit. • if some of the paths included in the k-crossover, other than pathi, assigned to locomotives with constraints, and the crossover makes at least one of the new paths pathj invalid, remove pathj from the list of candidates for exchange; if there are other candidates for exchange, go to step 5. • if we have iterated over all candidate paths, go to step 3. 7. if 1 < d1(pathi, pathj) <∞ and d2(pathi, pathj) = 0 find a path connecting the beginning of pathi with the end part of pathj . carry out d1 successive crossovers so that each of them contains one arc from the found path by performing step 6 d1 times. copyright © 2022 assa. adv syst sci appl (2022) graph methods for improving the invalid solution of the locomotive... 59 if there are other invalid paths, go to step 3; otherwise, a valid cover is found, exit. 8. if d2(pathi, pathj) = 1. • find a path connecting the beginning of the path pathi with the final part of path pathj . perform d1 crossovers in accordance with steps 6 and 7. • make the changeover. • check if co-changeover exists. • if co-changeover exists, make it. if there are other invalid paths, go to step 3; otherwise a valid cover is found, exit. • if no co-changeover exists, check if there is an available locomotive for the path resulting from the changeover. • if the locomotive exists, assign it. if there are other invalid paths, go to step 3; otherwise a valid cover is found, exit. 9. there is no complete valid cover. the found cover has been improved as much as possible. exit. if an invalid path pathi can be corrected by swapping homologous sections of one or more paths, it will be corrected. the same applies to the situation when the ends of the paths are switched to other sections (changeover and co-changeover). if such a procedure cannot be done, the algorithm switches to another invalid path. after correcting all the other invalid paths, the algorithm will return to pathi again and will try to make a crossover or changeover on a new set of paths. this procedure is repeated until all possibilities to correct the paths are exhausted. fig. 4.11 illustrates the operation of the algorithm in the case of d1 = 1, d2 = 0, k = 2. we find a biclique in a bipartite graph between a pair of adjacent layers and perform a crossover. in fig. 4.11(c) the crossover has been performed, in fig. 4.11(d) the resulting valid cover is shown. (a) (b) (c) (d) fig. 4.11. construction of a valid cover: (a) a three-layer graph for which there is no valid cover; (b) the selected biclique; (c) crossover; (d) valid cover; paths in it have exchanged homologous sections remark 4.5: since the algorithm certainly does not corrupt any path, it converges. either all the paths will copyright © 2022 assa. adv syst sci appl (2022) 60 l. zhilyakova, v. koreshkov be corrected, or there will be a set of invalid paths for which, under the given constraints on locomotives, there is no solution. remark 4.6: there should not be too many locomotives with time constraints to have a high probability of obtaining a valid cover. if there are many of them, the planning horizon should be adjusted. 5. conclusions a model of transportation on a linear section of the railway is represented as an acyclic graph. vertices in this graph correspond to transportations, and arcs indicate the possibility to perform one transportation after another by the same locomotive. such a graph was called a reachability graph. the problem of constructing optimal locomotive assignment without time constraints is equivalent to the problem of finding the minimal path cover of an acyclic graph. and this problem, in turn, is reduced to finding a perfect matching in a bipartite graph. in the absence of time constraints on locomotives, any minimal path cover of a given acyclic graph is an optimal solution to the original problem. due to time constraints, some transportation sequences cannot be completed. under such conditions, the problem becomes much more complicated. dynamic algorithms are often unable to find an existing solution [10]. heuristic algorithm [9] improves the invalid solution (finds more assignments), but also does not always find an optimal solution when it exists. to simplify the finding an optimal solution under time constraints, a topological sorting of the graph and a number of other transformations are performed. as a result, the structure of a reachability graph becomes regular and the dimension of the problem is reduced. to correct an invalid solution under time constraints, two operations are introduced: crossover and changeover. a crossover can be simple when a pair of paths exchange homologous sections, and complex when a permutation occurs between several paths. in crossover, two or more paths are exchanged with homologous sections containing the ends of these paths. a changeover switches the end of an invalid path to another free vertex or free path. the co-changeover operation allows in some cases to avoid the increase in the number of paths in the minimal cover, which occurs due to changeover. the algorithms is proposed that convert an invalid cover into a valid one. this algorithm always finds a solution if it exists. however, during its implementation, it is necessary to check a large number of conditions. acknowledgements this work was partly supported by the russian foundation for basic research (project no. 20-07-00190a). references 1. gainanov d. n., kibzun a. i., rasskazova v. a. 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(2019). genetic algorithm. evolutionary algorithms and neural networks. studies in computational intelligence, vol 780. springer, cham. 43 – 55. copyright © 2022 assa. adv syst sci appl (2022) introduction graph model of the problem basic definitions reachability graph transformations of reachability graph layer-by-layer structure. elimination of transitive closure. cloning vertices to all the layers. solving the problem without time constraints finding the minimal path cover formal definitions optimal solution correcting invalid solution crossover and changeover operations. distance between paths crossover. changeover. distance between paths. changing locomotive assignment algorithm for improving invalid paths in the minimal cover conclusions adv syst sci appl 2019; 03; 52-64 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/713 optimal values in an uncertain optimal control model with application to capital asset management tolulope latunde1* 1) department of mathematics, federal university, oye-ekiti, nigeria e-mail: tolulope.latunde@fuoye.edu.ng received march 21, 2019; revised september 6, 2019; published october 1, 2019 abstract: a new model of capital asset management was developed under the assumptions of hyperbolic absolute risk aversion, and employing the basic skills of mathematical modelling. the solution to the model was sought by formulating a continuous-time utility portfolio model satisfying some uncertainty criteria where the investment is continuous, the investor does not possess enough power to determine price and the investor can borrow money for a given period at a particular interest rate. the model was solved using analytical method and numerical method and optimal values of some input factors are derived. keywords: capital asset management, optimal control, uncertainty theory, optimal values. 1. introduction capital asset models have been subject to an enormous number of empirical studies since lintner and sharpe in the mid-1960. among the most notable early tests of the models are those in [1, 4]. other contributors are included in [3, 5, 13, 14]. the earliest researchers found the relationships between the risk-free rate and market risk premium. since merton developed and solve a portfolio selection model under uncertainty for the case of infinite lifetimes and finite lifetimes: the continuous-time case for risk and risk-free assets, capital asset management has adapted the portfolio theories, see [12]. this results to some researchers formulating asset management models using different mathematical approaches such that optimal values of controls and input factors are obtained. a stochastic optimal control approach was utilised to model debt crisis so as to evaluate debt crisis in international finance, thus, the optimal debt to preclude crisis was obtained [15]. the work [19] was based on the concepts of uncertainty theory where uncertain optimal control was applied to solve a portfolio selection model and obtained a fundamental result called equation of optimality for uncertain optimal control, thus obtain the optimal value of the control. formulation of two instances of uncertain multidimensional optimal control models with 𝑛 jumps based on uncertainty theory from the perspectives of agents ( consumers) and principal (government) was carried out in the work [2] the optimal control problems were applied to research and development fiscal policy and optimal control decisions were obtained. it is understood from existing works that the research carried out in [12,19, 2] examined the selection of risk-free asset and risk asset together in the formulation of the models. meanwhile, it was examined in [15], the selection of risky assets only with stochastic optimal control approach without considering depreciation and taxation as input factors. the choice of uncertainty theory over the conventional probability theory exists when the sample size is * corresponding author: tolulope.latunde@fuoye.edu.ng mailto:tolulope.latunde@fuoye.edu.ng t. latunde 53 copyright ©2019 assa. adv. in systems science and appl. (2019) small to estimate a probability distribution and degree belief are ascertained from experts to work in place of frequency since human beings always overweight unlikely events consequently, a new model of asset management based on uncertainty theory was formulated in [6] where the selection of capital assets, classified as risky assets is examined. thus, depreciation and taxation are considered as input factors in the formulation of the models. however, this work is an extension of the work [6, 7] such that we seek to provide solutions to a problem of capital assets using a real life situation. the proposed model deals with a case where a nation invests her wealth in capital assets, for a particular period of time. it is assumed that there is difficulty in deciding the fraction of the nation’s net worth to be incurred on the investments of capital assets, thus leading to the problem of how to optimize the expected present value of the utility of assets. 2. preliminary uncertainty theory is a branch of mathematics for modelling belief degrees established by liu, [8] and refined in [11]. the choice of uncertainty theory over the conventional probability theory exists when the sample size is small to estimate a probability distribution and degree belief are ascertained from experts to work in place of frequency since human beings always overweight unlikely events. for the sake of this work, the following concepts are utilized. let γ be a nonempty set and 𝐿 a 𝜎algebra over γ such that (γ, 𝐿) be a measurable space. each element λ ∈ 𝐿 is called an event. definition 2.1 [8]: a set function 𝑀 defined on the 𝜎-algebra over 𝐿 is called an uncertain measure if it satisfies the following axioms: axiom 1. (normality axiom): 𝑀{λ} = 1 for the universal set 𝛤. axiom 2. (duality axiom): 𝑀{λ} + 𝑀{λ𝑐} = 1 for any event λ. axiom 3. (subadditivity axiom): for every countable sequence of events, λ1, λ2, ⋯, we have 𝑀{⋃∞ 𝑖=1 λ𝑖} ≤ ∑∞ 𝑖=1 𝑀{λ𝑖} (2.1) axiom 4. (product axiom): let (γ𝑘, 𝐿𝑘, 𝑀𝑘) be uncertainty spaces for 𝑘 = 1,2, ⋯the product uncertain measure 𝑀 is an uncertain measure satisfying 𝑀{∏∞ 𝑘=1 λ𝑘} = min1≤𝑘≤∞𝑀𝑘{λ𝑘} (2.2) where λ𝑘 are arbitrarily chosen events from 𝐿𝑘 for 𝑘 = 1,2, ⋯, respectively. definition 2.2 [10]: an uncertain process 𝐶𝜎 is said to be a canonical liu process if (i) 𝐶0 = 0 and almost all sample paths are lipschitz continuous, (ii) 𝐶𝜎 has stationary and independent increments, (iii) every increment 𝐶𝑠+𝜎 − 𝐶𝑠 is a normal uncertain variable with expected value 0 and variance 𝜎2. the uncertainty distribution of 𝐶𝜎 is φ𝜎(𝑥) = [1 + exp ( −𝜋𝑥 √3𝜎 )] −1 , 𝑥 ∈ ℜ (2.3) and the inverse distribution is φ𝜎 −1(𝑦) = 𝜎√3 𝜋 ln 𝑦 1−𝑦 , 𝑦 ∈ ℜ (2.4) definition 2.3 [8]: let 𝜉 be an uncertain variable. then the expected value of 𝜉 is defined by 𝐸[𝜉] = ∫ +∞ 0 𝑀{𝜉 ≥ 𝑥}𝑑𝑥 − ∫ 0 −∞ 𝑀{𝜉 ≤ 𝑥}𝑑𝑥 (2.5) provided that at least one of the two integrals is finite definition 2.4 [9]: an uncertain process 𝑋𝑡 is said to have independent increments if 𝑋𝑡1 − 𝑋𝑡0 , 𝑋𝑡2 − 𝑋𝑡1 , ⋯ , 𝑋𝑡𝑘 − 𝑋𝑡𝑘−1 54 optimal values in uncertin optimal control model copyright ©2019 assa adv. in systems science and appl. (2019) are independent uncertain variables where 𝑡1, 𝑡2, ⋯ , 𝑡𝑘 are any times with 𝑡0 < 𝑡1 < ⋯ < 𝑡𝑘 that is, an independent increment process means that its increments are independent uncertain variables whenever the time intervals do not overlap. it is noted that the increments are also independent of the initial state. definition 2.5 [9]: suppose 𝐶𝑡 is a canonical liu process, and 𝑓 and 𝑔 are two functions. then 𝑑𝑋𝑡 = 𝑓(𝑡, 𝑋𝑡)𝑑𝑡 + 𝑔(𝑡, 𝑋𝑡)𝑑𝐶𝑡 (2.6) is called an uncertain differential equation. a solution is a liu process 𝑋𝑡 that satisfies (2.3) and (2.4) identically in 𝑡. definition 2.6 [9]: let 𝑋𝑡 be an uncertain process. then for each 𝛾 ∈ 𝛤, the function 𝑋𝑡(𝛾)is called a sample path of 𝑋𝑡. definition 2.7 [11]: an uncertain process 𝑋𝑡 is said to be sample-continuous if almost all sample paths are continuous functions with respect to time 𝑡. definition 2.8 uncertainty distribution of solution [16]: let 𝛼 be a number with 0 < 𝛼 < 1. an uncertain differential equation 𝑑𝑋(𝑡) = 𝑓(𝑡, 𝑋(𝑡))𝑑𝑡 + 𝑔(𝑡, 𝑋(𝑡))𝑑𝐶(𝑡) is said to have an 𝛼-path 𝑋(𝑡)𝛼 if it solves the corresponding ordinary differential equation 𝑑𝑋(𝑡)𝛼 = 𝑓(𝑡, 𝑋(𝑡)𝛼)𝑑𝑡 + |𝑔(𝑡, 𝑋(𝑡))|φ−1(𝛼)𝑑𝑡 (2.7) where 𝛷−1(𝛼) is the inverse uncertainty distribution of standard normal uncertain variable, that is, φ−1(𝛼) = √3 𝜋 ln 𝛼 1 − 𝛼 , 𝛼 ∈ ℜ theorem 2.1 extreme value of solution [18]: let 𝑋(𝑡) and 𝑋(𝑡)𝛼 be the solution and 𝛼-path of the uncertain differential equation (2.6). then, for any time 𝑡 > 0 and strictly increasing function 𝐽(𝑥), the supremum sup𝑡0≤𝑡≤𝑡𝑛 𝐽(𝑋(𝑡)) has an inverse uncertainty distribution ψ−1(𝛼) = sup𝑡0≤𝑡≤𝑡𝑛 𝐽(𝑋(𝑡)𝛼) (2.8) and the infimum inf𝑡0≤𝑡≤𝑡𝑛 𝐽(𝑋(𝑡)) has an inverse uncertainty distribution ψ−1(𝛼) = inf𝑡0≤𝑡≤𝑡𝑛 𝐽(𝑋(𝑡)𝛼) (2.9) theorem 2.2 [16]: let 𝑋(𝑡) and 𝑋(𝑡)𝛼 be the solution and 𝛼-path of the uncertain differential equation (2.6). then 𝑀{𝑋(𝑡) ≤ 𝑋(𝑡)𝛼, ∀𝑡} = 𝛼, 𝑀{𝑋(𝑡) > 𝑋(𝑡)𝛼, ∀𝑡} = 1 − 𝛼. (2.10) 3. model formulation a model of capital asset management is presented herein such that it is assumed that an investor invests his wealth in capital asset, 𝐴(𝑡), of a large business for time, 𝑡, from 𝑡0 to 𝑡𝑓. t. latunde 55 copyright ©2019 assa. adv. in systems science and appl. (2019) suppose he starts with a known initial net worth 𝑋0(𝑡). at time 𝑡, what ratio of his net worth, 𝜓, must he select to utilize on capital asset in the presence of liability such that the expected net present value of the utility of asset, 𝐽(𝜓), is optimized ? table 3.1. definition of parameters of the objective function to the model parameter description 𝑈 utility function 𝐴(𝑡) capital asset at time 𝑡 𝜂 subjective discount rate, e.g., 𝐴 𝜂+1 = presentvalue 𝜆 degree of relative risk, where (1 − 𝜆) is the risk aversion 𝜓 capital asset ratio (control) 𝜓 ∈ ℜ 𝑋(𝑡) net worth at time (state variable)𝑡 table 3.2: definition of parameters of the constraint to the model parameter description 𝜎𝑟(𝑡) diffusion volatility of liability (with variance 𝜎𝑟 2 per unit time) 𝜓 capital asset ratio (control) 𝜓 ∈ ℜ 𝜎𝑏(𝑡) diffusion volatility of asset (with variance 𝜎𝑏 2 per unit time) 𝜅(𝑡) capital gain on asset due to inflation at time 𝑡 𝜎𝑝(𝑡) diffusion volatility on asset price (with variance 𝜎𝑝 2 per unit time) 𝛽(𝑡) mean rate of return on asset 𝜔 mean interest rate of liability 𝐶(𝑡) uncertain process at time 𝑡 𝜇(𝑡) consumption level at time 𝑡 𝑗(𝑡) tax ratio at time 𝑡 𝑔(𝑡) depreciation ratio at time 𝑡 ℎ(𝑡) asset supplies ratio at time 𝑡 theorem 3.1 [12]: if 𝜆 > 0 and 𝑈 is such that the integral 𝐸𝐶 [∫ 𝑡𝑓 𝑡0 𝑈(𝐴, 𝑡)𝑑𝑡] is absolutely convergent, then the maximization or minimization of 𝐸𝐶 [∫ 𝑡𝑓 𝑡0 𝑈(𝐴, 𝑡)𝑑𝑡] is equivalent to the maximization or minimization of 𝐸0 ∫ 𝑡𝑓 𝑡0 𝑒−𝜂𝑡𝑈(𝐴, 𝑡)𝑑𝑡 where 𝐸𝐶 is the conditional expectation operator over all random variables excluding 𝜆. by theorem 3.1, an investor who faces an exponentially-distributed uncertain investment of capital asset invests as if there is no terminal period, but with a subjective rate of time preference equal to the investments terminals. 𝐽 = opt𝐸𝐶 [ ∫ 𝑡𝑓 𝑡0 𝑒−𝜂𝑡𝑈(𝐴(𝑡))𝑑𝑡] (3.1) where 𝐸𝐶 denotes conditional expectation, 𝜂 ∈ (0,1) is the arbitrary discount rate. in selecting the discount rate, the effective length of time is inversely proportional to the discount rate in the sense that a high discount rate implies a short time interval, [15]. utility function 𝑈(𝐴) measures satisfaction of an investor as a function of usage or efficiency of capital assets with respect to risk aversion. however, risk aversion is the behaviour of the investors when exposed 56 optimal values in uncertin optimal control model copyright ©2019 assa adv. in systems science and appl. (2019) to attempt to reduce the uncertainty in their investments. there are various measures of risk aversion under expected utility theory. thus, a special case of hyperbolic absolute risk aversion (hara) is considered as the model’s utility function which helps in focusing more on ratios and its assumption also lowers the dimension of dynamic system for the model to be effortlessly solved analytically unlike some other utility functions. that is 𝑈(𝐴) = ( 1 𝜆 𝐴𝜆, 0 < 𝜆 < 1 lna, 𝜆 = 0 (3.2) where 1 − 𝜆 > 0 is the investor’s relative risk aversion. the larger the 𝜆, the more reluctant to own a risky asset, [12]. the efficiency or performance of the capital asset, 𝜓, is expressed as the ratio of capital asset, 𝐴(𝑡), and net worth, 𝑋(𝑡), such that 𝑋(𝑡) ≠ 0, [15]. that is, 𝜓 = 𝐴(𝑡) 𝑋(𝑡) (3.3) 𝐴(𝑡) = 𝜓𝑋(𝑡) from the proceeding, the model of risky capital asset is an optimal control of the form. 𝐽(𝜓) = max𝜓𝐸𝐶 [ ∫ 𝑡𝑛 𝑡0 1 𝜆 𝑒−𝜂𝑡(𝜓𝑋(𝑡))𝜆𝑑𝑡] (3.4) subject to 𝑑𝑋(𝑡) = [(𝜅 + 𝛽)𝜓 − (𝜔(𝜓 − 1) + 𝜇 + ℎ − 𝑗 − 𝑔)]𝑋(𝑡)𝑑𝑡 +[𝜓𝜎𝑝 + 𝜓𝜎𝑏 − 𝜎𝑟(𝜓 − 1)]𝑋(𝑡)𝑑𝐶(𝑡) [6] (3.5) 4. application of the capital asset model here, the model is analysed using real life data in order to provide some optimal solutions satisfying the optimality criteria. utilizing the model in international finance, the revenue is taken to be the gross domestic product of a nation (gdp) or value added, the risky capital asset as the capital investment or the gross fixed capital formation, consumption as household and government consumption, depreciation is taken as the consumption of fixed capital. the debt is also taken as a study case of liability to be considered. table 4.1: definition of parameters according to the model application parameter description 𝐽 expected present value of utility of gfcf t. latunde 57 copyright ©2019 assa. adv. in systems science and appl. (2019) 𝜂 subjective discount rate, e.g., 𝐴 𝜂+1 = presentvalue 𝜆 degree of relative risk, where (1 − 𝜆) is the risk aversion 𝜓 gfcf ratio (control) 𝜓 ∈ ℜ 𝜏(𝑡) debt ratio (control) at time 𝑡, 𝜏 = 𝜓 − 1 𝑋(𝑡) net worth at time 𝑡 (gfcf minus debt ) 𝜅(𝑡) capital gain on gfcf with net worth at time 𝑡 𝛽(𝑡) mean rate of return on gfcf with net worth 𝜔(𝑡) mean rate of debt with net worth 𝜎𝑝(𝑡) diffusion volatility on asset price (with variance 𝜎𝑝 2 per unit time) 𝜎𝑏(𝑡) diffusion volatility of gfcf (with variance 𝜎𝑏 2 per unit time) 𝜎𝑟(𝑡) diffusion volatility of debt (with variance 𝜎𝑟 2 per unit time) 𝜎(𝑡) diffusion volatility of the whole process (𝜎𝑝 + 𝜎𝑏 − 𝜎𝜔), [15] 𝜇(𝑡) consumption level with net worth at time 𝑡 𝑠(𝑡) net foreign supplies net worth ratio at time 𝑡 𝑗(𝑡) tax-net worth ratio at time 𝑡 𝑔(𝑡) depreciation-net worth ratio at time 𝑡 𝐶(𝑡) liu canonical process at time 𝑡 in order to examine the debt crisis in nigeria and propose a warning signal, the data that are available after the paris debt forgiveness in 2006 are used. thus, tables 4.2 and 4.3 below represent the base parameter set for the case study. table 4.2: nigeria net worth profile year gdp debt gfcf net worth consumption indirect tax depreciation supplies 2007 166.451 22.330 15.396 -6.934 149. 152 2.553 3.738 5.089 2008 208.065 21.399 17.318 -4.081 161.035 3.436 3.853 30.988 2009 169.481 25.817 20.487 -5.330 147.601 3.180 2.952 0.445 2010 369.062 40.100 61.099 21.860 293.507 5.623 16.079 28.662 2011 411.744 47.898 63.960 16.062 323.540 4.516 18.815 38.719 2012 460.953 48.496 65.283 16.787 348.597 5.686 24.260 86.210 2013 514.966 64.510 72.964 8.454 453.699 7.929 23.857 26.280 2014 568.499 67.726 85.737 18.011 464.696 6.857 25.272 32.499 2015 481.066 65.429 71.329 5.900 417.560 5.362 23.097 0.000 2016 405.083 57.392 73.261 15.869 206.414 3.188 10.332 -3.341 source: columns 1 and 3. the world bank (http://data.worldbank.org/indicator) column 2. debt management office of nigeria (https://www.dmo.gov.ng/). columns 5, 6, 7, 8 and 9. national bureau of statistics (http://nigerianstat.gov.ng/) table 4.3 is derived from table 4.2. table 4.3: parameters for the nigeria net worth profile year 𝜅 𝛽 𝜔 𝜎𝑝 𝜎𝑏 𝜎𝑟 𝜎 𝜇 h j g 58 optimal values in uncertin optimal control model copyright ©2019 assa adv. in systems science and appl. (2019) 2007 -7.25 -51.72 -6.10 -3.78 -21.47 -2.46 -22.79 -21.51 -0.73 -0.37 -0.54 2008 -12.32 -87.88 -10.36 -6.42 -36.48 -4.17 -38.73 -39.46 -7.59 -0.77 -0.94 2009 -9.43 -67.29 -7.93 -4.92 -27.93 -3.20 -29.65 -27.69 0.08 -0.60 -0.55 2010 2.30 16.41 1.93 1.20 6.81 0.78 7.23 13.43 1.31 0.26 0.74 2011 3.13 22.33 2.63 1.63 9.27 1.06 9.84 20.14 2.14 0.28 1.17 2012 2.30 21.37 2.52 1.56 8.87 1.02 9.41 20.77 5.14 0.34 1.45 2013 5.95 42.42 5.00 3.10 17.61 2.02 18.69 53.67 3.11 0.94 2.82 2014 2.79 19.91 1.43 1.46 8.27 0.95 8.78 25.80 1.92 0.37 1.40 2015 8.52 60.79 4.37 4.44 25.23 2.89 26.78 70.77 0.00 0.91 3.91 2016 3.17 22.60 1.63 1.65 9.38 1.07 9.96 13.01 -0.21 0.20 0.65 measurements the measurements considered in obtaining data in tables 4.1 – 4.3 are described below. all the values of parameters are measured in billion us dollars except the following parameters: 𝐶(𝑡) measures the uncertainty process which exists in the interval 0 < 𝐶(𝑡) < 1; 𝜆 is used to measure risk which exists in the interval 0 < 𝜆 < 1; and 𝜂 measures discount rate which exists in the interval 0 < 𝜂 < 1. the debt is calculated as the total debt of the nation by summing the external debt stock (federal government and state) and domestic debt (federal government and state) together. the net worth is also calculated by deducting the debt from the gfcf. the cbn official exchange rate of 𝑈𝑆𝐷 at 31st december of each year is used while current market prices from the national account are used in the computations. 4.1 solution to the model here, the analytical and numerical solutions are derived. for the analytic solution, the required problem under consideration is 𝐽(𝜓) = min𝜓𝐸𝐶 [ ∫ 𝑡𝑛 𝑡0 1 𝜆 𝑒−𝜂𝑡(𝜓𝑋(𝑡))𝜆𝑑𝑡] subject to 𝑑𝑋(𝑡) = [(𝜅 + 𝛽)𝜓 − (𝜔(𝜓 − 1) + 𝜇 + 𝑠 − 𝑗 − 𝑔)]𝑋(𝑡)𝑑𝑡 +[𝜓𝜎𝑝 + 𝜓𝜎𝑏 − 𝜎𝑟(𝜓 − 1)]𝑋(𝑡)𝑑𝐶(𝑡) with 𝛼-path equation 𝑑𝑋(𝑡)𝛼 = [(𝜅 + 𝛽)𝜓 − (𝜔(𝜓 − 1) + 𝜇 + ℎ − 𝑗 − 𝑔)]𝑋(𝑡)𝛼𝑑𝑡 + |[𝜓𝜎𝑝 + 𝜓𝜎𝑏 − 𝜎𝑟(𝜓 − 1)]𝑋(𝑡)𝛼|φ−1(𝛼)𝑑𝑡. the analytical solution to the constraint is t. latunde 59 copyright ©2019 assa. adv. in systems science and appl. (2019) 𝑋(𝑡) = 𝑋0exp([(𝜅 + 𝛽)𝜓 − (𝜔(𝜓 − 1) + 𝜇 + 𝑠 − 𝑗 − 𝑔)]𝑡 + [𝜓𝜎𝑝 + 𝜓𝜎𝑏 − 𝜎𝑟(𝜓 − 1)]𝐶(𝑡)) and its inverse uncertainty distribution is ψ(𝑡)−1(𝛼) = 𝑋0exp([(𝜅 + 𝛽)𝜓 − (𝜔(𝜓 − 1) + 𝜇 + 𝑠 − 𝑗 − 𝑔)]𝑡 + [𝜓𝜎𝑝 + 𝜓𝜎𝑏 − 𝜎𝑟(𝜓 − 1)]𝑡√3 𝜋 ln 𝛼 1 − 𝛼 ) hence, by theorem 2.1, ψ(𝑡)−1(𝛼) = 𝐸(𝑋(𝑡)𝛼) numerical solutions are presented via trapezoidal rule for the objective functional and, euler method and fourth order runge-kutta method for solving uncertain differential equations due to its ability to yield more precise outcomes than other methods for the constraints, [17]. trapezoidal method: ∫ 𝑏 𝑎 𝑓(𝑥)𝑑𝑥 = ℎ 2 (𝑓0 + 2𝑓1 + 2𝑓2 + ⋯ + 2𝑓𝑛−1 + 𝑓𝑛) = ℎ( 𝑓0 + 𝑓𝑛 2 + ∑ 𝑛−1 𝑖=1 𝑓𝑖), where 𝑓(𝑥𝑘) ≡ 𝑓𝑘 the runge-kutta method for solving uncertain differential equations was designed in [17] with respect to the following definition and theorems. runge-kutta method is an effective method for solving ordinary differential equations. the generally used runge-kutta formula is a fourth-order formula. it should be noted that there is a wide range of fourth-order schemes and here, just one common structure is exhibited. for an ordinary differential equation with initial value 𝑋0 𝑑𝑋(𝑡) = 𝐹(𝑡, 𝑋(𝑡))𝑑𝑡. the scheme uses the following formula 𝑋(𝑡𝑛+1) = 𝑋(𝑡𝑛) + 1 6 (𝑘1 + 2𝑘2 + 2𝑘3 + 𝑘4) where 𝑘1 = ℎ𝐹(𝑡𝑛, 𝑋𝑛), 𝑘2 = ℎ𝐹(𝑡𝑛 + ℎ 2 , 𝑋𝑛 + 1 2 𝑘1), 𝑘3 = ℎ𝐹(𝑡𝑛 + ℎ 2 , 𝑋𝑛 + 1 2 𝑘2), 𝑘4 = ℎ𝐹(𝑡𝑛 + ℎ, 𝑋𝑛 + 𝑘3) and ℎ is the step size which is assumed to be constant for all steps. however, based on theorem 2.4, a runge-kutta method for uncertain differential equations was designed as 𝑋𝑖+1 𝛼 = 𝑋𝑖 𝛼 + 1 6 (𝑘1 + 2𝑘2 + 2𝑘3 + 𝑘4) where 60 optimal values in uncertin optimal control model copyright ©2019 assa adv. in systems science and appl. (2019) 𝑘1 = ℎ(𝑓(𝑡𝑖, 𝑋𝑖 𝛼) + |𝑔(𝑡𝑖, 𝑋𝑖 𝛼) |φ−1(𝛼)), 𝑘2 = ℎ(𝑓(𝑡𝑖 + ℎ 2 , 𝑋𝑖 𝛼 + 1 2 𝑘1) + |𝑔(𝑡𝑖 + ℎ 2 , 𝑋𝑖 𝛼 + 1 2 𝑘1)|φ−1(𝛼)), 𝑘3 = ℎ(𝑓(𝑡𝑖 + ℎ 2 , 𝑋𝑖 𝛼 + 1 2 𝑘2) + |𝑔(𝑡𝑖 + ℎ 2 , 𝑋𝑖 𝛼 + 1 2 𝑘2)|φ−1(𝛼)), 𝑘4 = ℎ(𝑓(𝑡𝑖 + ℎ, 𝑋𝑖 𝛼 + 𝑘3) + |𝑔(𝑡𝑖 + ℎ, 𝑋𝑖 𝛼 + 𝑘3)|φ−1(𝛼)). for the proposed optimal control model of net risky capital asset with an uncertain differential equation 𝑑𝑋(𝑡) = [(𝜅 + 𝛽)𝜓 − (𝜔(𝜓 − 1) + 𝜇 + ℎ − 𝑗 − 𝑔)]𝑋(𝑡)𝑑𝑡 + [𝜓𝜎𝑝 + 𝜓𝜎𝑏 − 𝜎𝑟(𝜓 − 1)]𝑋(𝑡)𝑑𝐶(𝑡) with initial value 𝑋0 and its 𝛼-path equation. i.e., 𝑑𝑋(𝑡)𝛼 = [(𝜅 + 𝛽)𝜓 − (𝜔(𝜓 − 1) + 𝜇 + ℎ − 𝑗 − 𝑔)]𝑋(𝑡)𝛼𝑑𝑡 +|[𝜓𝜎𝑝 + 𝜓𝜎𝑏 − 𝜎𝑟(𝜓 − 1)]𝑋(𝑡)𝛼|φ−1(𝛼)𝑑𝑡, this is solved using the algorithm below. 4.2. algorithm 4.1: runge-kutta method for solving the model step 1. given time interval 𝑡, [𝑎, 𝑏], iteration number 𝑁, step length ℎ = 𝑏−𝑎 𝑁 . set 𝑡𝑖 = 𝑎 + 𝑖ℎ, 𝑖 = 0,1, ⋯ , 𝑁 and 𝛼 > 0. step 2. compute the corresponding differential equation 𝑑𝑋(𝑡)𝛼 = [(𝜅 + 𝛽)𝜓 − (𝜔(𝜓 − 1) + 𝜇 + 𝑠 − 𝑗 − 𝑔)]𝑋(𝑡)𝛼𝑑𝑡 +|[𝜓𝜎𝑝 + 𝜓𝜎𝑏 − 𝜎𝑟(𝜓 − 1)]𝑋(𝑡)𝛼| 𝜎√3 𝜋 ln 𝛼 1−𝛼 𝑑𝑡, 𝑋0 𝛼 = 𝑋0, with the runge-kutta method for solving uncertain differential equations. step 3. set 𝑖 = 𝑖 + 1, repeat step 2 and step 3 for 𝑁 times, then 𝑋(𝑡)𝛼 is derived. go back to step 1 until 𝑡𝑖 = 𝑏, table 4.4: results of analytical solution to the model with ℎ = 0.05, 𝑎 ≤ 𝑡 ≤ 𝑏, 𝑎 = 0, 𝑏 = 1, 𝜂 = 0.9, 𝜆 = 0.1 and 𝑋0 = 18.011 𝜓 x j -9 -1175.785 16.657 -7 -28.783 11.209 -5 -0.277 6.812 -3 -0.118 5.941 -1 -1.612 6.916 1 19.050 8.854 3 95.377 11.609 5 91.492 12.166 7 77242.383 24.684 9 4.159 × 106 37.710 table 4.5: results of numerical solution to the model (ℎ = 0.05, 𝑎 = 0, 𝑏 = 1, 𝜂 = 0.9, 𝜆 = 0.1 and 𝑋0 = 18.011) 𝜓 x j -9 -1638.001 17.218 -7 -31.576 11.313 t. latunde 61 copyright ©2019 assa. adv. in systems science and appl. (2019) -5 -0.322 6.916 -3 -0.150 6.088 -1 -3.037 7.368 1 19.007 8.852 3 93.596 11.587 5 84.931 12.076 7 74991.214 24.612 9 3.524 × 106 37.090 furthermore, using the available data, the numerical and analytical solution to the asset-liability management problem was presented. using the net worth of the year 2007 to 2016 were to study the behaviour of the optimal control in each year and provide a control policy to the problem. let 𝜓𝑂 be the optimal control which implies the rate of capital asset such that the expected present value of the utility of assets, and 𝜓𝐴 be the actual control which is obtained from the given ratio of capital asset and net worth. the optimal control 𝜓𝑂 for each year was derived analytically using the equation of optimality proposed in [19] for uncertain optimal control problem, where 𝜓𝑂 = (𝜇+𝑗+𝑔+𝜔−ℎ)𝜆−𝜂 (1−𝜆)(𝜅+𝛽−𝜔) and the actual control is 𝜓𝐴 = 𝐴(𝑡) 𝑋(𝑡) . table 4.6: results on performance of capital asset year 𝜓𝐴 𝜓𝑂 𝜓𝐴 𝜓𝑂 2007 -2.22 -4.09 1.87 2008 -4.24 -5.88 1.64 2009 -3.84 -5.09 1.25 2010 2.80 0.67 2.13 2011 3.98 1.42 2.56 2012 3.89 1.22 2.67 2013 8.63 5.59 3.04 2014 4.76 2.01 2.75 2015 12.09 7.88 4.21 2016 4.62 0.74 3.88 the warning signal will be based on the difference between the actual liability ratio 𝜏𝐴 and the optimal liability ratio 𝜏𝑂. table 4.7: warning signal year 𝜏𝐴 𝜏𝑂 𝜏𝐴 𝜏𝑂 2007 -3.22 -5.09 0.87 62 optimal values in uncertin optimal control model copyright ©2019 assa adv. in systems science and appl. (2019) 2008 -5.24 -6.88 0.64 2009 -4.84 -6.09 0.25 2010 1.80 -0.33 1.13 2011 2.98 0.42 1.56 2012 2.89 0.22 2.67 2013 7.63 4.59 2.04 2014 3.76 1.01 1.75 2015 11.09 6.88 3.21 2016 3.62 -0.26 2.88 4.2 optimal u, 𝝀 and 𝜼 by hyperbolic absolute risk aversion utility function, 𝑈(𝐴) = ( 1 𝜆 𝐴𝜆 , 0 < 𝜆 < 1 lna, 𝜆 = 0 𝑑𝑈 𝑑𝜆 = 0 − 1 𝜆2 𝐴𝜆 + 1 𝜆 𝐴𝜆ln𝐴 = 0 1 𝜆 = ln𝐴 𝜆 = 1 ln𝐴 hence, the optimal 𝜆 is 𝜆∗ = 1 ln𝐴 . similarly, optimal 𝑈 is 𝑈∗ where 𝑈∗ = 1 𝜆∗ 𝐴𝜆∗ = (ln𝐴)𝐴 1 ln𝐴 from table 4.2, let 𝐴1 = 15.396 and 𝐴2 = 85.737 where 𝜆 = 𝜆1 and 𝐴 = 𝐴1. this implies 𝜆1 = 1 ln𝐴1 = 0.366 let 𝑈 = 𝑈1 𝑈1 = (ln𝐴1)𝐴1 1 ln𝐴1 = 7.432 and let 𝜆 = 𝜆2, 𝐴 = 𝐴2 𝜆2 = 1 ln𝐴2 = 0.225 𝑈2 = (ln𝐴2)𝐴1 1 ln𝐴2 = 12.100 therefore, 𝜆2 ≡ 𝜆∗ = 0.225 and 𝑈2 ≡ 𝑈∗ = 12.100 also, from equation (3.1) t. latunde 63 copyright ©2019 assa. adv. in systems science and appl. (2019) 𝐽 = max𝐸𝐶 [ ∫ 𝑡𝑓 𝑡0 𝑒−𝜂𝑡𝑈(𝐴(𝑡))𝑑𝑡] = max𝑈(𝐴)max ∫ 1 0 𝑒−𝜂𝑡𝑑𝑡 = 𝑈∗max 1 𝜂 (1 − 𝑒−𝜂) = 1 𝜂 𝑈∗ given in table 4.1 is the net present value 𝐽 as 𝐴 𝜂+1 this implies 𝐴2 𝜂 + 1 = 𝑈∗ 𝜂 thus, 𝜂∗ = 𝑈∗ 𝐴2 − 𝑈∗ ⇒ 𝜂∗ = 0.164 therefore, using the calculated optimal values, the following results are obtained. table 4.8: numerical result of expected present value of utility of asset 𝐽(𝜓)𝛼 with different 𝛼-paths 𝛼 𝐽(𝜓)𝛼 10−6 1.152 × 106 .1 25619.235 .24 6056.883 .38 448.529 .52 429.982 .64 3372.794 .8 15826.851 .999999 1.273 × 106 5. conclusion based on uncertainty theory, an optimal control model of capital asset was formulated by adapting the expected value operator to quantify the objective rewards in the model. furthermore, the model was solved by using some optimality criteria to derive the optimal values of some input factors, thus applied to a real life problem. in future work, the objective of the model may be viewed from the perspective of stakeholders or regulators other than that of the investors. references [1] black, f., jensen, m. c. & scholes, m. (1972). the capital asset pricing model: some empirical tests. in m. jensen (ed.), studies in the theory of capital markets (pp. 79121). newyork: praeger publishers. [2] deng, l. & zhu, y. (2012). an uncertain optimal control model with n jumps and application. comsis 9(4), 1453-1468. [3] douglas, g. w. (1968). risk in the equity markets: an empirical appraisal of market efficiency. university microfilms, inc., ann arbor, michigan. 64 optimal values in uncertin optimal control model copyright ©2019 assa adv. in systems science and appl. (2019) [4] fama, e. f. & macbeth, j. d. (1973). risk, return, and equilibrium: empirical tests. journal of political economy, 71, 607-636. [5] friend, i. & blume, m. (1970). measurement of portfolio performance under uncertainty. american economic review, 60(4), 607-636. [6] latunde t. & bamigbola o. m. (2016). uncertain optimal control model for management of net risky capital asset. iosr journal of mathematics (iosr-jm), 12(3), 2230. [7] latunde, t. & bamigbola, o. m. (2018). parameter estimation and sensitivity analysis of an optimal control model for capital asset management. advances in fuzzy systems, 2018, 1-11. https//doi.org/10.1155/2018/4756520 [8] liu, b. (2007). uncertainty theory (2nd ed.). berlin: springer. [9] liu, b. (2008). fuzzy process, hybrid process and uncertain process, journal of uncertain systems, 2(1), 3-16. [10] liu, b. (2009). theory and practice of uncertain programming (2nd ed.). berlin: springer verlag. [11] liu, b. (2016). uncertain theory (5th ed.). china: uncertainty theory laboratory. [12] merton, r. c. (1971). optimal consumption and portfolio rules in a continuous time model. journal of economic theory, 42(3), 373-413. [13] miller, m. & scholes, m. (1972). rates of return in relation to risk: a reexamination of some recent findings. studies in the theory of capital markets, michael c. jensen, ed., praeger, new york, 47-78. [14] stambaugh, r. f. (1982). on the exclusion of assets from tests of two-parameter model: a sensitivity analysis. journal of financial economics, 10(3), 237-268. [15] stein, j. l. (2003). stochastic optimal control modelling of debt crises. cesifo (working paper 1043), 1-23. [16] yao, k. & chen, x. (2013). a numerical method for solving uncertain differential equations. journal of intelligent fuzzy systems, 25(3), 825-832. [17] yang, x. & shen, y.(2015). runge-kutta method for solving uncertain differential equations, journal of uncertainty analysis and applications, 3(17), 1-12. [18] yao, k. (2013). extreme values and integral of solution of uncertain differential equation. journal of uncertainty analysis and applications, 1(2), 1-21. [19] zhu, y. (2010). uncertain optimal control with application to a portfolio selection model. cybernetics and systems: an international journal, 41, 535-547. adv syst sci appl 2021; 01:60–75 published online at https://ijassa.ipu.ru. formularization method for calculating the breakaway and break-in points and the corresponding gain of root locus graphs hassan shibly1∗, orwah h. shibly1,2 1robotics and mechatronics engineering technology program,school of engineering, science, and technology, ccsu university, new britain, ct, usa 2research and development, comanox corp, va, usa abstract: break points, break-away and break-in points, are an essential part in root locus technique for single input single output linear invariant control systems. the importance of break points comes from the fact that at the break points at least two roots of the characteristic equation of the closed loop control system change their type from real to a complex at the break away point, and from complex to real at break-in point. this change affects the response of the system which can be crucial for some of systems’ applications. the conditions for being a break point are analyzed and a new formulated systematic method for finding the break points and their corresponding gains is presented. an efficient algorithm was developed and can be solved analytically. there is no mathematical differentiation during calculation and the algorithm can be programmed easily. the developed algorithm is applicable for any order of transfer function of a linear invariant control system. this method is compared with other common methods to show its merits and effectiveness. keywords: formularization method for break points calculation, breakaway points’ calculation, breakaway formula, gain at breakaway point, break-in points’ calculation 1. introduction one of the common methods used in the analysis and design of a linear time invariant single input and single output of feedback control systems is the root locus method. the root locus technique in control systems was first introduced by evans in the years 1948 and 1950, [1,2]. the root locus method is a technique used to determine the locus of the closed loop poles which are the roots of the closed loop characteristic equation on the complex s-plane as a function of a parameter of interest, where s is the complex variable. usually the parameter of interest is the static gain of the open loop system. the parameter of interest value is varied theoretically from zero up to infinity. ∗corresponding author: hshibly@ccsu.edu formularization method for calculating the breakaway and break-in... 61 the root locus method developed by evans is a powerful and efficient tool used to examine stability, to analyze and to design a single input single output linear time invariant system. in addition to examining the stability of a control system, the proper value of the gain for a specific type of response can be found. to investigate the data obtained from the root locus plot of a single input single output linear invariant control systems neglecting noise as it is shown in figure 1.1 , there is a need to know the system open loop transfer function; g(s) = gc(s)gp(s), its feedback transfer function h(s), and its closed loop transfer function t(s), where the closed loop transfer function of such a system with a negative feedback is t (s) = gc(s)gp(s) 1 +gc(s)gp(s)h(s) = g(s) 1 +g(s)h(s) . (1.1) fig. 1.1. feedback control system block diagram. the characteristic equation (c.e.) of a control system is obtained by setting the denominator of the closed loop transfer function t(s) to zero, so it is: 1 +g(s)h(s) = 0, (1.2) where the mathematical expression of g(s)h(s) has a rational form of two polynomials with constant coefficients such as g(s)h(s) = k n(s) d(s) . (1.3) the degree of the numerator’s polynomial n(s) is m, and its roots are called zeros, while the degree of the denominator’s polynomial d(s) is n, and its roots are called poles. in general n ≥ m. the locus of the roots of the characteristic equation of a control system on the s-plane is obtained by varying the static gain k, theoretically 0 ≤ k <∞ . the path which connects the locus of the roots is known as the root locus of the system, and any point on the root locus path should satisfy two conditions: the angle condition, and the magnitude condition. it is known that the root locus begins at a pole when k = 0 and ends at zero when k →∞ , of the open loop transfer function. to satisfy the angle condition of the root locus on the real axis of the s-plane, the segments of the root loci on this axis are located to the left of an odd copyright c© 2021 assa. adv syst sci appl (2021) 62 h. shibly, o.h. shibly number of poles and zeros. therefore, the static gain k value of a control system defines the roots of the characteristic equation and consequently the loci of the roots on the root locus graph on the complex s-plane. as a result, it defines the system response. from laplace inverse transform the time domain response of any real root is exponential, while an oscillatory response occurs when there are at least one complex conjugate pair of roots of the characteristic equation [3–5]. the total response of a system is exponential with no oscillation if all its components have exponential responses. this occurs when all its roots are real. root loci leaving the real axis have complex conjugate pair of roots and consequently have an oscillatory response. in root locus method the root locus begins at poles and migrate toward zeros as a result of varying the static gain k value. if two root loci leave two poles on the real axis of the s-plane and move in opposite directions as shown in figure 2.2 they will collide at a point, the point of collision is called a breakaway point, and if a complex conjugate pair roots move in opposite directions toward the real axis they will collide at a point on the real axis is called break-in point. at a break point the characteristic equation has double roots for a certain value of k. geometrically the break point is on the root locus segments between two poles, or between two zeros. for a larger value of k than the value at the breakaway point the characteristic equation will have at least one complex conjugate pair of roots and consequently the two loci split and leave the real axis in opposite direction and continue to move in a symmetrical way, while for larger k than its value at the break-in point the characteristic equation will have two different real roots and consequently the root locus split and continue to move in different directions on the real axis.the process of the break in point is an opposite manner to the breakaway point. in this article breakaway points and break-in points are called just break points in this work a new formulated method for finding the real breakaway points and break-in points which are along the real axis is developed and investigated. there are a number of methods on how to find the break points and the corresponding gains, [3, 6]. the common method is based on finding a local extremum for the parameter of interest, usually the static gain. this is done by differentiating the mathematical expression of the parameter of interest which is a function of the complex variable s, and then setting the derivative to zero. then solving for s = σ to get various roots. in this method not all the roots are break points and then there is a need to continue to figure out which point is a break point. the roots at the break point are real and are along the root locus segment on the real axis between two poles or between two zeros. the mathematical expression of the open loop transfer function usually has a rational form of two polynomials, numerator and denominator, and its differentiation becomes cumbersome especially for a higher degree of the numerator. a second method was presented by remec [7], and it is based on the theory of equations. it is a tabulation method similar to routh-hurwitz array construction which is a long process. a third method is called the transition method by franklin [8]. this method is based on the fact that the natural logarithm has a zero derivative at the same point that the derivative of copyright c© 2021 assa. adv syst sci appl (2021) formularization method for calculating the breakaway and break-in... 63 the parameter of interest expression is also zero. in this method no need for differentiation but there is a need to do a number of polynomials factors multiplication to get the final form of the equation, and then to figure out what are the applicable roots out of all roots. it has the same numerator as the differentiation of the fractional form of the two polynomials. a fourth method was developed by krishnan [9] and it is based on successive differentiation of numerator and denominator to form arrays for an algorithm. it is a long process method. this work is divided into two parts. in the first part the development of a new method for calculating the break points, breakaway points and break in points, and their corresponding gain. the second part is a comparison with the common methods and the analysis of the proposed efficiency. 2. derivation of new method for break points calculation substitute equation 1.3 into the c.e., equation 1.2, to get 1 + kn(s) d(s) = d (s) +kn(s) d(s) = 0. (2.4) it means that the numerator is zero and the new equation is d (s) +kn (s) = 0. (2.5) where the numerator n(s) and the denominator d(s) in equation (1.3) are polynomials of order m and n respectively such as n (s) = bms m + bm−1s m−1 + bm−2s m−2 + · · ·+ b2s 2 + b1s+ b0. (2.6) d (s) = ans n + an−1s n−1 + an−2s n−2 + · · ·+ a2s 2 + a1s+ a0. (2.7) substitute equations (2.6) and (2.7) into equation (2.5) to get the characteristic equation c.e. of the control system as ans n + an−1s n−1 + an−2s n−2 + · · ·+ a2s 2 + a1s+ a0+ +k ( bms m+bm−1s m−1 + bm−2s m−2 + · · ·+ b2s 2 + b1s+ b0 ) = 0. (2.8) collect equal terms to obtain the general form of the characteristic equation, equation (2.9). this equation is a polynomial of order n for n ≥ m, and constant coefficients ci. cns n + cn−1s n−1 + cn−2s n−2 + · · ·+ c2s 2 + c1s+ c0 = 0. (2.9) where cq = aq +kbq for q = 1, 2, · · · , n , and bq = 0 for q > m. (2.10) it means that the numerator is zero and the new equation is copyright c© 2021 assa. adv syst sci appl (2021) 64 h. shibly, o.h. shibly s1 = s2 = σ. (2.11) equation (11) means that the characteristic equation at break points has the factor (s+ σ)2 = s2 + 2σs+ σ2. (2.12) if equation (2.12) is a factor of the characteristic equation, equation (2.9), the remainder of their division should be zero as shown in example 1. example 1. given the system open loop transfer function g(s) = k s(s+2) and its negative unit feedback is h (s) = 1. the characteristic equation of this system is 1 +gh (s) = 1 + k s (s+ 2) = s (s+ 2) +k s (s+ 2) = 0. (2.13) as a consequence of being equation (13) equal to zero, its numerator becomes equal to zero, and the new equation is s (s+ 2) +k = s2 + 2s+k = 0. (2.14) the two terms of remainder of equation (2.14) are set equal to zero to get . (1− σ) = 0, and ( k − σ2 ) = 0. (2.15) actually, equation (2.15) contains two equations of sigma as the variable. solving the two equations gives:σ = 1, k = σ2 = (1)2 = 1. then the breakaway point is at point s = −σ = −1 and the gain is k=1. these values are validated by the root locus plot shown in figure 2.2. the two terms of the remainder of the division result of the general form of characteristic equation of a second order system are −c1 + 2c2σ = 0, and c0 − c2σ2 = 0. (2.16) a similar results to equation (2.16) are obtained from the division of a higher order systems. by setting the two remainders of each order equal to zero, gives the two equations that are in table 2.1. those two equations are the two conditions that the second order polynomial of equation (2.12) is a factor of the characteristic equation of the system. substitute the coefficients of the equations of the two conditions of order n which are given in table 2.1 as in equation (2.10) to obtain n∑ q=1 (−1)qq (aq +kbq )σ q−1 = 0. (2.17) (a0 +kb0 ) + n∑ q=1 (−1)q−1 (q − 1) (aq +kbq )σ q = 0. (2.18) copyright c© 2021 assa. adv syst sci appl (2021) formularization method for calculating the breakaway and break-in... 65 fig. 2.2. root locus plot of a second order control system of example 1, showing the direction of roots migration and the breakaway point at s = −1. table 2.1. the two conditions for systems of orders= 2, 3, · · · , n. degree of c.e. first condition second condition 2 −c1 + 2c2σ = 0. c0 − c2σ2 = 0. 3 −c1 + 2c2σ − 3c3σ 2 = 0. c0 − c2σ2 + 2c3σ 3 = 0. 4 −c1 + 2c2σ − 3c3σ 2 + 4c4σ 3 = 0. c0 − c2σ2 + 2c3σ 3 − 3c4σ 4 = 0. 5 −c1 + 2c2σ − 3c3σ 2 + 4c4σ 3 − 5c5σ 4 = 0. c0 − c2σ2 + 2c3σ 3 − 3c4σ 4 + 4c5σ 5 = 0. ... ... ... n ∑n q=1 (−1) qqcqσ q−1 = 0. c0 + ∑n q=1 (−1) q−1 (q − 1) cqσ q = 0. rewriting the equations of the two conditions for the existence of break points, equations (2.17) and (2.18), so that each equation is a sum of two polynomials. the first polynomial is constructed based only on the coefficients of the numerator n(s), and the second polynomial is constructed based only on the coefficients of the denominator d(s) and the static gain k as a factor yields n∑ q=1 (−1)qqaqσq−1 +k n∑ q=1 (−1)qqbqσq−1 = 0. (2.19) a0 + n∑ q=1 (−1)q−1 (q − 1) aqσ q+k [ b0 + n∑ q=1 (−1)q−1 (q − 1) bqσ q ] = 0. (2.20) for simplicity equation (2.19) and equation (2.20) may written as follows copyright c© 2021 assa. adv syst sci appl (2021) 66 h. shibly, o.h. shibly p1 (σ) +kp2 (σ) = 0. (2.21) p3 (σ) +kp4 (σ) = 0. (2.22) where the ps polynomials are p1 (σ) = n∑ q=1 (−1)qqaqσq−1. (2.23) p2 (σ) = n∑ q=1 (−1)qqbqσq−1. (2.24) p3 (σ) = a0 + n∑ q=1 (−1)q−1 (q − 1) aqσ q. (2.25) p4 (σ) = b0 + n∑ q=1 (−1)q−1 (q − 1) bqσ q. (2.26) solve for k in equation (2.21) and in equation (2.22) to havet k = −p1 (σ) p2 (σ) . (2.27) k = −p3 (σ) p4 (σ) . (2.28) eliminate k to obtain p1 (σ) p2 (σ) = p3 (ó) p4 (σ) . (2.29) then a cross multiplication gives p (σ) = p 1 (σ)p4 (σ)− p2 (σ)p3 (σ) = 0. (2.30) equation (2.30) is called the break polynomial equation, and it is a polynomial in sigma. part of the polynomial roots, sigma values, are the break points. algorithm steps finding of the break points and the corresponding k gain for a given control system has the following steps: step 1. write the rational form of the system transfer function as shown in equation(1.3) . copyright c© 2021 assa. adv syst sci appl (2021) formularization method for calculating the breakaway and break-in... 67 step 2. write the four polynomials p1 (σ) , p2 (σ) , p3 (σ) , and p4 (σ) using the formulas given by equations (2.23)-(2.26) in the same polynomial’s terms order that is shown in equations (2.6) or (2.7) . step 3. substitute the four polynomials into equation (2.30) to obtain the break polynomial equation. step 4. solve the break equation (2.30) for sigma. the roots σi are the solution of the break polynomial equation. step 5. find the applicable values of sigma (break points) which satisfy the root locus technique condition. step 6. evaluate the four polynomials in equations (2.23)-(2.26) only for the applicable values of sigma. step 7. calculate the gain k. if p2 (σ) 6= 0 equation (2.27) is used, else equation (2.28) is used. note that in equations (2.23)-(2.26) the polynomials terms’ degree is from lower degree to higher degree, and when using matlab software the terms’ order in the polynomial array have to be flipped. the flowchart of the procedure is shown in figure 2.3. fig. 2.3. flowchart of the algorithm steps. copyright c© 2021 assa. adv syst sci appl (2021) 68 h. shibly, o.h. shibly 3. analysis and calculation procedure of the break points the proposed formularization method is used to obtain the break polynomial which is based on coefficients of the open loop transfer function’s polynomials. substitution of those coefficients into equations (2.23) – (2.26) yields the four polynomials p1-p4. then the four polynomials are substituted into equation (2.30) to obtain the break polynomial. then solving the break polynomial equation to obtain its roots. the break points are some of the roots which are located on the root locus segment that satisfy the angle condition of the root locus graph on the real axis. it means that the applicable roots (sigma values) are on the real axis and are located to left of an odd number of poles and zeros of the system open loop transfer function. specifically, they are between two poles or between two zeros on the real axis. by knowing the actual break points the corresponding gains are obtained by substituting the actual break points’ values into equation (2.27) or into equation (2.28). the calculated value of k is a “border” value where two roots at least alter their type between real and complex and vice versa for any increase in k. for breakaway point the two real roots become complex pair roots, and for break-in point a complex conjugate roots alter to real roots. to show the calculation steps of this formularization method and to compare it with other common methods it is used in the solution of two examples for a two control systems. the transfer function in the second example has two zeros and three poles, while the transfer function in the third example has two zeros and four poles. the objective of having two finite zeros is to enable the existence of both breakaway point, and break-in point. example 2. in this example the control system open loop transfer function is gh (s) = k (s+ 3)(s+ 5) s (s+ 2) (s+ 4) . (3.31) the polynomials of the numerator and denominator are d (s) = s3 + 6s2 + 8s. (3.32) n (s) = s2 + 8s+ 15. (3.33) the coefficients of both polynomials are: a3 = 1, a2 = 6, a1 = 8, a0 = 0, b3 = 0, b2 = 1, b1 = 8, b0 = 15. substitute into the p’s polynomials, equations (23-26), to have p1 (σ) = −a1 + 2a2σ − 3a3σ 2 = −8 + 12σ − 3σ2. (3.34) p2 (σ) = −b1 + 2b2σ − 3b3σ 2 = −8 + 2σ. (3.35) p3 (σ) = a0 − a2σ2 + 2a3σ 3 = −6σ2 + 2σ3. (3.36) copyright c© 2021 assa. adv syst sci appl (2021) formularization method for calculating the breakaway and break-in... 69 p4 (σ) = b0 − b2σ2 + 2b3σ 3 = 15− σ2. (3.37) or into equation (2.30) to get the break polynomial p (σ) = ( −8 + 12σ − 3σ2 ) ( 15− σ2 ) − (−8 + 2σ) ( −6σ2 + 2σ3 ) = 0. (3.38) simplify to obtain the break polynomial equation such as p (σ) = σ4 − 16σ3 + 85σ2 − 180σ + 120 = 0. (3.39) solve to get the break polynomial’s roots as σi = 1.2221 , 7.8311, 3.4734 + 0.6888i, 3.4734− 0.6888i. (3.40) the applicable roots are: [σ1,2 = 1.2221, 7.8311], and substitution these two values to obtain the two corresponding gains as: [k = 0.393, 12791]. then the break-away point is:[s1 = −σ1 = −1.222], and the break-in point is:[s2 = −σ2 = −7.831]. to validate the result the root locus plot is drawn using matlab software as shown in figure 3.4. the matlab root locus plot shows very close results. the insignificant differenc in the results is due to the graphical plot accuracy. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 -4 -3 -2 -1 0 1 2 3 4 root locus plot of gh(s)=(s+3)(s+5)/s(s+2)(s+4) real axis (seconds-1) im ag a xi s (s ec on ds -1 ) system: g gain: 0.393 pole: -1.24 damping: 1 overshoot (%): 0 frequency (rad/s): 1.24 system: g gain: 12.8 pole: -7.83 + 0.0462i damping: 1 overshoot (%): 0 frequency (rad/s): 7.83 fig. 3.4. root locus plot of the control system of example 2. copyright c© 2021 assa. adv syst sci appl (2021) 70 h. shibly, o.h. shibly example 3. in this example the open loop transfer function of the control system is gh (s) = k (s+ 3)(s+ 4) s(s+ 2)(s+ 5)(s+ 8) . (3.41) the polynomials of the numerator and denominator are d (s) = s4 + 15s3 + 66s2 + 80s. (3.42) n (s) = s2 + 7s+ 12. (3.43) the coefficients of both polynomials are: a4 = 1, a3 = 15, a2 = 66, a1 = 80, a0 = 0, b4 = 0, b3 = 0, b2 = 1, b1 = 7, b0 = 12 substitute in the ps polynomials, equations (23-26), to have p1 (σ) = −a1 + 2a2σ − 3a3σ 2 + 4a4σ 3 = −80 + 132σ − 45σ2 + 4σ3. (3.44) p2 (σ) = −b1 + 2b2σ − 3b3σ 2 + 4b4σ 3 = −7 + 2σ. (3.45) p3 (σ) = a0 − a2σ2 + 2a3σ 3 − 3a4σ 4 = −66σ2 + 30σ3 − 3σ4. (3.46) p4 (σ) = b0 − b2σ2 + 2b3σ 3 − 3b4σ 4 = 12− σ2. (3.47) then substitute into equation (2.30) to get the break polynomial p (σ) = ( −80 + 132σ − 45σ2 + 4σ3 ) ( 12− σ2 ) + . − ( −66σ2 + 30σ3 − 3σ4 ) (−7 + 2σ) = 0. (3.48) collect equal terms to obtain p (σ) = 2σ5 − 36σ 4 + 258σ3 − 922σ2 + 1584σ − 960 = 0. (3.49) the break polynomial’s roots are σi = 1.2436, 3.4909, 3.5875 + j2.2987, 3.5875 − j2.2987, 6.0906. (3.50) the applicable roots are: [σ1,2,3 = 1.2436, 6.0906, 3.4909] , and the gains for these three values are: [k = 4.9312, 8.0296, 141.7091]. then the break away points are: [s1,2 = −σ1,2 = −1.2436, − 6.0906], and the break-in point is: [s3 = −σ3 = − 3.4909]. it copyright c© 2021 assa. adv syst sci appl (2021) formularization method for calculating the breakaway and break-in... 71 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 -8 -6 -4 -2 0 2 4 6 8 root locus plot of gh(s)=[(s+3)(s+4)]/[s(s+2)(s+5)(s+8)] real axis (seconds-1) im a g a xi s (s e co n d s -1 ) system: g gain: 4.93 pole: -1.25 damping: 1 overshoot (%): 0 frequency (rad/s): 1.25 system: g gain: 142 pole: -3.49 damping: 1 overshoot (%): 0 frequency (rad/s): 3.49 system: g gain: 8.03 pole: -6.08 damping: 1 overshoot (%): 0 frequency (rad/s): 6.08 fig. 3.5. root locus plot of the control system of example 3. can be seen that the result obtained from matlab root locus plot is too close to the calculated result. 4. comparison with other common methods to examine the methods accuracy and calculation’s efficiency of finding the break points of the considered control system, the formulated method, and other common methods are used in solving two examples, examples 2 and 3. the accuracy, simplicity, and the amount of mathematical operations to obtain the final results are considered. 4.1. comparison with the root locus sketching method comparison between the break points’ values and the corresponding gains’ values of the two examples 2 and 3 in the previous section using root locus sketching method as shown in figures 3.4 and 3.5 and using the formulated method shows very small difference in the values. the difference is a result of the root locus sketching method being a semi-graphical method 4.2. comparison with the via differentiation method the characteristic equation of the control system is copyright c© 2021 assa. adv syst sci appl (2021) 72 h. shibly, o.h. shibly 1 +gh (s) = 1 +k n(s) d(s) = 1 +k (s+ 3)(s+ 5) s (s+ 2) (s+ 4) = 0. (4.51) solve for k expression to have k = −d (s) n (s) = −s (s+ 2) (s+ 4) (s+ 3) (s+ 5) = −s 3 + 6s2 + 8s s2 + 8s+ 15 . (4.52) via differentiation method is to differentiate k with respect to s to obtain the local extremums of k. dk ds = −d ′ n −dn ′ n2 = 0. (4.53) or d ′ n −dn ′ = 0. (4.54) differentiate both polynomials d(s) and n(s) and the substitute into equation (4.54) to get( 3s2 + 12s+ 8 ) ( s2 + 8s+ 15 ) − ( s3 + 6s2 + 8s ) (2s+ 8) = 0. (4.55) multiply and collect equal terms to obtain the polynomial s4 + 16s3 + 85s2 + 180s+ 120 = 0. (4.56) the roots of this polynomial are si = −1.2221 , −7.8311 , −3.4734 + 0.6888i , −3.4734 − 0.6888i. (4.57) as discussed before the applicable roots are [s1,2 = −1.2221, − 7.8311], and the corresponding gain k is calculated by using equation (4.52) as follows ki = − d (si) n (si) . (4.58) k1 = − (−1.2221)3 + 6(−1.2221)2 + 8 (−1.2221) (−1.2221)2 + 8 (−1.2221) + 15 = 0.393. (4.59) k2 = − (−7.8311)3 + 6(−7.8311)2 + 8 (−7.8311) (−7.8311)2 + 8 (−7.8311) + 15 = 12.791. (4.60) copyright c© 2021 assa. adv syst sci appl (2021) formularization method for calculating the breakaway and break-in... 73 4.3. comparison with franklin’s via transition method in this method the break points are determined by using: n∑ i=1 1 σ + pi = m∑ i=1 1 σ + zi . (4.61) applying equation (4.61) to the open loop transfer function of the same example gives an equation which a summation of a number of fractions such as 1 σ + 1 σ + 2 + 1 σ + 4 = 1 σ + 3 + 1 σ + 5 . (4.62) algebraic simplification gives (σ + 2) (σ + 4) + σ (σ + 4) + σ (σ + 2) σ (σ + 2) (σ + 4) = (σ + 5) + (σ + 3) (σ + 3) (σ + 5) . (4.63) then a cross multiplication gives [(σ + 2) (σ + 4) + σ (σ + 4) + σ (σ + 2)] [(σ + 3) (σ + 5)]+ − [(σ + 5) + (σ + 3)] [σ (σ + 2) (σ + 4)] = 0. (4.64) this result is the same as in the via differentiation method as given in equation (4.55). it can be noticed that this result, equation (4.64), is actually the differentiation result of the factored form of the denominator and numerator of the transfer function of the control system. after mathematical operations and algebraic simplification we get the same polynomial as in equation (4.56) σ4 + 16σ3 + 85σ2 + 180σ + 120 = 0. (4.65) and the roots are σi = −1.2221 , −7.8311 , −3.4734 + 0.6888i , −3.4734 − 0.6888i. (4.66) the applicable roots are [s1,2 = −1.2221,−7.831], and the corresponding gain k is calculated the same way as in the via differentiation method, but there is no need for differentiation along the calculation 5. conclusion the result of the comparison via solution of example 2 shows that the new method is accurate and efficient compared to the very common root locus sketching method. the results of the via differentiation method are the same as in the formulated method where s is equal negative sigma, however there is a need for polynomials differentiation, and it requires more mathematical operations. copyright c© 2021 assa. adv syst sci appl (2021) 74 h. shibly, o.h. shibly in principle the two most used methods in class rooms are the via differentiation method, and the transition method. they are very much similar. the difference is that in the via differentiation method the differentiation part is done by the user, and in the transition method the differentiation part was done by the author to obtain the used formula. in both methods the break polynomial can be found by using the following equation: [all combinations of the product of (n− 1) factors of d (s)] ·n (s)+ − [all combinations of the product of (m− 1) factors of n (s)] ·d (s) = 0. (5.67) as it can be noticed that equation (5.67) is based on the factored form of the numerator n(s), and the denominator textitd(s). the proposed formularization method is used to find the break polynomial based on the substitution of the coefficients of the two polynomials of the fractional form of the open loop transfer function of a control system. this method gives the breakaway points, break-in points, and the corresponding gains. the results of this proposed formularization method are the same as results of the other methods which shows that it is an accurate method and requires less mathematical operations. the calculation procedure shows its merits as a systematic method. there is no mathematical differentiation, and its algorithm can be programmed easily. the formulated method is applicable to transfer functions of any degree, and in addition to that, a constant numerator where all its coefficients are zeros simplifies and shorten the process of obtaining break polynomial. these features simplify the calculations for finding the points at which the characteristic equation roots change type. the use of this method in plotting the root locus of a control system for a design purpose becomes more accurate especially for a larger step size of the static gain. references 1. evans, w.r. (1948). graphical analysis of control system, aiee transactions, 67, 547–551. 2. evans, w.r. (1950). control system synthesis and root locus method, aiee transactions, 69, 66–69. 3. benjamin c. kuo (1975). automatic control systems. prentice hall, 3rd edition. 4. richard c. dorf, robert h. bishop (2016). modern control systems. pearson, 13th edition. 5. katsuhiko ogata (2010). modern control engineering. pearson, 5th edition. 6. norman s. nise (2015). control systems engineering. wiley, 7th edition. 7. remec m. j. (1965). saddle-points of a complete root locus and an algorithm for their easy location in the complex frequency plane, proc. natl. electronics conference, 21, 605–608. 8. franklin, g. f., powell, j.d., emami-naeini, a. (1991). feedback control of dynamic systems. addison-wesley, 2nd edition. copyright c© 2021 assa. adv syst sci appl (2021) formularization method for calculating the breakaway and break-in... 75 9. krishnan v. (1966). semi-analytic approach to root locus, ieee transaction, automatic control, ac-11, 102–108. copyright c© 2021 assa. adv syst sci appl (2021) introduction derivation of new method for break points calculation analysis and calculation procedure of the break points comparison with other common methods comparison with the root locus sketching method comparison with the via differentiation method comparison with franklin's via transition method conclusion adv syst sci appl 2022; 02:11–20 published online at https://ijassa.ipu.ru. investigation of reliability for information system for natural gas quality analysis ivan brokarev1*, sergei vaskovskii2 1national university of oil and gas gubkin university, moscow, russia e-mail: brokarev.i@gubkin.ru 2v. a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia e-mail: v63v@yandex.ru abstract: it is an extremely crucial task for oil and gas industry to assess how reliable gas quality analyzing systems are. the traditional approaches to assess the reliability of this class of systems, due to a number of conditions, do not seem to be applicable. to assess the reliability of such systems, we propose a probabilistic method to determine a distribution function of operating time, a reliability function, and mean time to failure. we build a structural diagram of the system, obtain basic formulae to determine the required functions and reliability indicators. we investigate various scenarios of system failure depending on the number of parameters for which accuracy decreases. for each of the scenarios, we obtain formulae to describe the required functions and reliability indicators. keywords: reliability, gas quality analysis system, probabilistic method, distribution function of operating time, reliability function, mean time to failure 1. introduction it is an extremely difficult task to determine and assess the reliability of traditional gas analysis systems to probe the component composition and energy characteristics of gas. one of the problems in full-scale and semi-natural tests of this class of systems is to ensure reliability. in addition, the life cycle of such systems is 25-30 years at a fairly high cost of equipment. when conducting semi-natural tests of gas analysis systems in special heat chambers at high temperature and humidity, it is difficult to obtain a realistic picture of the system’s reliability characteristics. in addition, the tests are expensive and time-consuming [15]. however, it is imperative to evaluate how reliable such systems are. such assessments exist and applicable. however, our system, where the energy characteristics of gas are analyzed, differs in structure and information-computational functionality from those already assessed systems. the existing methods are generally not suitable for our system. to close this gap, this work develops both an adequate model of the system’s reliability and a method to obtain the indicators necessary to do the job, for example, up-time, etc. for this purpose, we modify the probabilistic method to assess reliability of the gas analysis system. the developed method allows us to design specialized structural diagrams of gas analysis systems. the study of such structural schemes makes it possible to evaluate the reliability characteristics of systems by probabilistic methods. also, the schemes help to detect problem ∗corresponding author: brokarev.i@gubkin.ru 12 i. brokarev, s. vaskovskii areas of gas analysis systems in terms of reliability and maintenance characteristics. finally, the method helps to evaluate the developed gas analysis system in terms of the system’s reliability. to assess the reliability of the developed gas-analyzing information and computing system, it is necessary to determine the basic terms and reliability indicators to determine the level of reliability of this class of systems. according to the document [14], reliability is the property of an object to keep in time and within the established time intervals the ability to perform the required functions in specified modes and conditions of use, maintenance, storage, and transportation. the main terms and indicators of the reliability of our system are associated with how we assess its reliability. to assess the reliability of the system, it is necessary to determine the type of failures that occur in the system. in general case, a failure is an event when the system is not functioning. in our task, failures in the system are parametric (leading to the parameters going beyond the permissible time intervals). the main reliability indicators of a system are: the probability of a system failure (the probability of a failure occurring when the system operates); average time of failure-free operation of the system; failure rate; the distribution density of failure time. methods to assess system reliability make it possible to predict the quality of how systems function, as we take into account the errors that occur, the number and intensity of failures. we can also correct errors in the development and operation of systems [11]. many methods to assess system reliability are based on probabilistic markov processes. these methods make it possible to assess how reliable equipment, and devices function in different modes. analytical methods are also used to calculate the reliability indicators of technical systems [9]. these include methods of the random process theory, the expert estimate theory (heuristic forecasting), decomposition (equivalence), logical-probabilistic, asymptotic, analytical, and statistical methods. the methods of the random process theory are used to assess how reliable various systems and facilities with continuous technological processes are. also these methods help us to study the reliability of industrial enterprises systems, including gas complexes. usually, at the design stage of distributed systems, reliability calculations are based on failure rate data of the systems’ constituent elements; the rates are determined experimentally. but in order to draw a conclusion about the reliability of the system as a whole, it is necessary to take into account the system structure and the possibilities to improve the system. methods to calculate structural reliability based on failure rate data are investigated. examples of how to construct and optimize similar models for distributed computing systems and network management systems can be found in [5, 6, 8]. however, in relation to the gas analysis problem, this approach has disadvantages. the concept of structural reliability as ”the ability of a system to perform specified functions within a given time interval” is too general in the problem of gas analysis; it requires specifying what functions are important in this case. the primary task is to analyze the component composition and energy characteristics of natural gas. it is possible to parameterize how this system functions via a discrete finite state space of the system (for example, ”works” / ”does not work”). however, this does not allow us to construct a practical model of the system, since the concept of accuracy measurement is a characteristic that changes continuously and can vary depending on the application. therefore, in this chapter, we propose a new approach and an accuracy analysis-based model to determine the gas parameters. to calculate reliability, a distributed system is usually divided into elements. to calculate reliability indicators, structural logical reliability schemes are used. these schemes help us to graphically display how the elements are related and included in the system. also, it helps us to investigate how the elements impact on the system reliability as a whole. structural logic diagram is a set of elements connected to each other in series and / or in parallel. copyright © 2022 assa. adv syst sci appl (2022) investigation of reliability for information system for natural gas 13 methods to calculate structural reliability are well known and our approach works well under standard assumptions such as, for example, exponential failures, independent elements [2, 7]. the block diagram of our distributed data acquisition system is shown in fig. 1.1. in this version of the scheme, information enters the analysis system via two independent measuring channels. we use two independent measuring channels to verify the received measuring information. we can add additional measuring channels to increase both the reliability and accuracy characteristics of the distributed system, but in this work we investigate a system with just two. in this version of the system, the measuring devices (md) are connected in parallel, since it is assumed that if the measuring device fail, the system stay operational, but the system’s accuracy decreases. to calculate the structural reliability according to the structural diagram, we build a state graph. we take into account that all elements have different reliability indicators. the nature of the failure rate, and recovery rate for each element of the system are determined. based on these data, we determine the probability to find each element in a working or inoperative state at a certain moment of time t. fig. 1.1. structural design of our distributed gas analysis system. the reliability of a redundant system pr(t), where the system consists of l independent elements, is determined by: pr(t) = 1− l∏ k=1 ( λk λk + µk − λk λk + µk e−(λk+µk)t ) (1.1) where λk and µk – are failure and repair rates of element k. accordingly, the reliability (the probability of failure-free operation) of a series system with (n− 1) redundant m blocks can be calculated by the formula [1]: prm(t) = m∏ i=1 ( 1− n∏ j=1 ( λij λij + µij − λij λij + µij e−(λij+µij)t )) (1.2) copyright © 2022 assa. adv syst sci appl (2022) 14 i. brokarev, s. vaskovskii this scheme is applicable for standard approaches, but does not allow us to ensure the measurement accuracy. the standard scheme to calculate reliability contains drawbacks and is not suitable for our task. it is impossible to detect faults, if we follow the standard approach and apply formulas (1.1-1.2). failure to ensure the specified measurement accuracy is essentially a latent failure for an externally operating measuring device to transmit correct data. on the other hand, from the functional point of view, this is a critical failure, because the main function of the system – to determine the energy characteristics of the gas is not fulfilled. if a system fails to collect, process, and analyze information, this failure can be detected explicitly (failure of the computer). on the other hand, if measuring devices fail, this cannot be detected immediately. if there is a failure of measuring devices, then there may be no obvious external signs of this phenomenon. so, this phenomenon is not observable, and we can judge about it only by the output data: for this, all measuring devices are duplicated (see fig. 1.1). we will assume that a failure has occurred, if the data transmitted from the measuring devices differs greatly (exceeds some threshold) between the first and second measuring devices. this requires us to develop a new approach to assess the reliability of our measuring device subsystem. 2. study of a measuring devices reliability model for our case, we assume that to analyze the composition of natural gas for an n -component mixture, 2n measuring devices are needed. for each parameter, two measuring devices are needed the main and the backup; the two devices operate independently of each other. measuring devices independently transmit information via two different channels to an information collection system and then to an information processing system these components together form the measuring channel (mc). the hot standby mc also receives information from measuring devices (standby measuring devices). thus, the distributed system collects information from 2n measuring devices (n main and n backup). since the main task of the system is to determine the energy characteristics, the system will be considered reliable if it provides a given level of accuracy to determine the required gas characteristics. the reliability of the ”measuring channel computing system” subsystem is determined according to the standard scheme described, for example, in [7, 8]. and, since all subsystems function independently, the reliability of each of them can be assessed independently. let’s consider a model of the measuring device subsystem reliability. we assume that the subsystem to measure one parameter of the gas mixture is in working condition if both measuring devices (main and backup) give approximately the same readings for the same time instant of the same measurement point. if the readings do not differ greatly, then in this case we assume that the system is working. in the other cases, we assume that a failure has occurred. in this case, we can assume that the measuring device designed to determine the corresponding gas parameter is out of order, if at the output we get a large difference in readings (between the main and backup devices). in this situation, a failure of the measuring device subsystem for the corresponding gas parameter is recorded. now let’s move on to a more rigorous description of our model. for this, we introduce the following definitions. by the reliability of a measuring system we mean the system’s ability to perform specified functions within a given time interval. in this case, the ”given function of the system” is the system’s ability to measure gas parameters with a given accuracy. definition 2.1: as we measure the i− th gas parameter, we consider the measurement accurate, if the values obtained from the first and second measuring devices differ in a certain pres-elected range (−εi, εi), εi > 0. copyright © 2022 assa. adv syst sci appl (2022) investigation of reliability for information system for natural gas 15 when it is necessary to determine n parameters, then we will deal with a vector of positive values: ε = (ε1, ...εn), εi > 0,∀i = 1, ..., n, i.e. a certain range of admissible values (−εi, εi) is specified for each measuring device. definition 2.2: by system failure, we mean an accuracy indicator goes beyond the permissible time intervals. in this case, in practice, what we mean by failure must be clarified each time when we solve a specific problem. for example, in the case of an n -component mixture, the system fails when it goes beyond the specified measurement accuracy for subsystems of k measuring devices fromn , where k can take values from 1 up ton (depending on operational conditions and others, for example, economic factors and customer requirements). in case k = 1, the most stringent requirements are imposed on the measuring device subsystem, if the system is unable to accurately measure at least one gas parameter, it is already considered a failure. in the case k = n , for the device subsystem to work, it is sufficient to be able to measure at least one gas parameter. the essence of the reliability assessment method, where we monitor how the readings of the main and backup measuring devices deviate, is illustrated in figure 2.2 and in formula 2.3. |y (1) 1 − y (1) 2 | = ξ1 ... |y (n) 1 − y (n) 2 | = ξn (2.3) fig. 2.2. the essence of our reliability estimation method. we monitor how the outputs of the main and the backup devices differ from each other. copyright © 2022 assa. adv syst sci appl (2022) 16 i. brokarev, s. vaskovskii 3. subsystem reliability model to measure a single gas mixture parameter suppose that only one indicator is measured in the system. the readings are received at discrete moments of time from two measuring devices (main and backup). we assume that if the measuring devices operate normally, then the readings from the measuring devices give approximately the same result. we observe a discrete sequence of some random variables: ξi, i = 0, 1, 2... which we will interpret as the difference between the readings of measuring devices. it is natural to assume that if the readings of the measuring devices approximately coincide, then the values of this sequence fluctuate around zero. moreover, if we consider absolute measurements, then the observations would be dependent. but, since we observe the modulus of the difference, then, regardless of fluctuations in the real composition of the gas, we take that ξi, i = 0, 1, 2... is an independent and equally distributed random variables. also, we can interpret ξi as some measurement error. therefore, we assume that the sequence elements of are normal random variables: ξi ∼ n(0, σ2), i = 0, 1, 2.... without loss of generality, we can assume that σ2 = 1, i.e. the elements of the sequence are standard normal random variables. let us make the following assumptions regarding measuring devices: as the number of measurements increases, the measuring devices wear off more and more, and accordingly, the error (the difference between the indicators) grows. the growth rate can be considered variable in the general case; the rate depends on time and on the state of the system m(i,xi). therefore, as a model of how the system operates, we can consider a random process in discrete time, where the process characterizes the sum of accumulated errors: xi = m(i,xi)i+ i∑ k=0 ξk (3.4) where i = 0, 1, 2. . . ,m(i,xi) can be interpreted as the error increase rate, where the error is associated with how the measuring device wears off (error accumulation) due to additional factors, and ξi ∼ n(0, 1). without loss of generality, we can assume that measurements are made quite often, therefore, we go from (3.4) to a continuous model. in this case, a natural brownian motionbased model is relevant [12]. and such a model can be investigated using well-known results from the theory of random processes [10]. 4. preliminary investigation. the reliability of a measurement device subsystem as n=1 we observe a random process in continuous time x(t); the process can be described by a simple differential equation [12]: dx(t) = µ(t,x(t))dt+ σ(t,x(t))dw (t) (4.5) this approach can also be interpreted as follows: the error (difference) can be modeled as a one-dimensional brownian motion x(t) = w (t), or, formally, dx(t) = dw (t). generally, dx(t) = σ(t,x(t))dw (t), where the coefficient σ may not be constant in the general case, but depends on the state of the system. that is, we assume that there is some randomness in our measurements: either the main or the backup measuring device show a higher value, so the error increases and decreases. we also assume that the measuring device wears out over time, so the error grows on average with a certain rate µ. copyright © 2022 assa. adv syst sci appl (2022) investigation of reliability for information system for natural gas 17 intuitively, the model of how the error x changes is described by a random walk. when the time step tends to zero, the process tends to the one defined by equation (4.5). however, in what follows we assume that σ > 0 and µ ≥ 0 are constants: dx(t) = µdt+ σdw (t) (4.6) now, suppose that a certain permissible range of values (−a,a), a > 0 is specified for the measuring device; the range determines how the system operates normally [2]. definition 4.1: the system is workable if x(t) ∈ (−a,a). definition 4.2: the system fails if the process gets out of the set: x(t) /∈ (−a,a) : x(t) ≤ −a or x(t) ≥ a. definition 4.3: mttf is the first time when the trajectory reaches a: τa = inf(t ≥ 0 : |x(t)| = a), (if µ is strictly positive, p (x(t) = a)→ 0) definition 4.4: the system reliability function (in general case) is defined as: r(t) = p (τ > t), definition 4.5: the operating time distribution function of the system (in general case): f (t) = 1−r(t) = p (τ ≤ t), problem 4.1: for a system described in general case by equation (4.6), estimate the mean time between failures mtbf = eτ (in the simplest case, find it analytically). to select the threshold values ai for each of the measured parameters, it is necessary to specify how accurately we determine the physical parameters of the gas. most standards [4] regulate the reduced error of how we determine the required parameters. the values of the thresholds must be the values of the maximum absolute error, where the discrepancy of the device readings does not go beyond the permissible values. we take the value of our measurement repeatability as such values, since there are no regulated values of the absolute error. threshold values for all input and output parameters are shown in table 4.1. table 4.1. gas parameters threshold values physical gas parameter units of measurement threshold value ai methane concentration, xch4 % 0.1 propane concentration, xc3h8 % 0.1 nitrogen concentration, xn2 % 0.1 carbon dioxide concentration, xco2 % 0.1 speed of sound, c m/s 0.2 thermal conductivity, χ w/m*k 0.005 5. reliability of the measuring device subsystem a feature of the model is that if the measuring device fails, the system does not, but the system gets its accuracy characteristics reduced. in our diagram, the measuring devices are connected in parallel. however, if we calculate reliability as the structural reliability of a parallel subsystem, it will not work. indeed, since if the n − 1 subsystem fails, this does not lead to an actual failure of the system, but provides the ability to determine only one gas parameter. therefore, the reliability model “k out of n” (k ≤ n) is correct to take copyright © 2022 assa. adv syst sci appl (2022) 18 i. brokarev, s. vaskovskii into account how failures of individual measuring devices influence calculation accuracy; hereinafter, we will assume n = 5 everywhere. let’s consider several scenarios of how the system operates: scenario 5.1: if accuracy indicators of all parameters decrease to a given threshold, we take this as a system failure. scenario 5.2: if the accuracy indicator of at least 1 parameter decreases to a given threshold, we take this as a system failure. scenario 5.3: if accuracy indicators of at least k(k < 5) parameters decrease to a given threshold, we take this as a system failure. let’s investigate the first scenario. let f1(t) be a distribution function of the system operating time for this scenario. theorem 5.1: the distribution function of the subsystem operating time f1(t) for scenario 5.1 is described by the following formula: f1(t) = 5∏ i=1 ( 1− φ ( ai − µit σ √ t ) + e −2µiai σ2 φ ( −ai − µit σ √ t )) (5.7) demonstration 5.1: we assume that for each parameter (i = 1, 2, 3, 4, 5) in the gas mixture there is a model of error variation and the parameters ai, µi, σ. then the distribution function of the system operating time can be defined as: f1(t) = p (τ1 ≤ t, ..., τ5 ≤ t) = p ( sup 0≤s≤t x1(s) ≥ a1, ..., sup 0≤s≤t x5(s) ≥ a5) = = p ( sup 0≤s≤t µ1t+ ∫ t 0 σ dw1(t) ≥ a1, ..., sup 0≤s≤t µ5t+ ∫ t 0 σ dw5(t) ≥ a5) = = p ( sup 0≤s≤t w1(t) ≥ a1 − µ1t σ , ..., sup 0≤s≤t w5(t) ≥ a5 − µ1t σ ) (5.8) let us use the following well-known result for a wiener process w (t) to intersect a linear boundary as µ and σ are given [3, 13]. p ( sup 0≤s≤t w (t) ≥ a) ≈ 1− φ ( a− µt σ √ t ) + e −2µa σ2 φ ( −a− µt σ √ t ) (5.9) where φ is the distribution function for the standard normal distribution. from the previous formulas it is easy to demonstrate: f1(t) = 5∏ i=1 ( 1− φ ( ai − µit σ √ t ) + e −2µiai σ2 φ ( −ai − µit σ √ t )) (5.10) corollary 5.1: as a consequence of the theorem, the reliability function and mean time to failure for scenario 5.1 are: copyright © 2022 assa. adv syst sci appl (2022) investigation of reliability for information system for natural gas 19 r1(t) = 1− f1(t) (5.11) mttf1 = ∫ ∞ 0 r1(t) dt (5.12) corollary 5.2: the distribution function of the subsystem operating time f2(t), the reliability function and the mean time to failure for scenario 5.2 are: f2(t) = n∏ i=1 ( 1− φ ( ai − µit σ √ t ) + e −2µiai σ2 φ ( −ai − µit σ √ t )) (5.13) r2(t) = 1− f2(t) (5.14) mttf2 = ∫ ∞ 0 r2(t) dt (5.15) theorem 5.2: the distribution function of the subsystem operating time f3(t) for scenario 5.3 is: f3(t) = k∏ i=1 ( 1− φ ( ai − µit σ √ t ) + e −2µiai σ2 φ ( −ai − µit σ √ t )) (5.16) demonstration 5.2: we can find the probabilities that a system of n parameters can determine exactly k (all the parameters of the mixture are considered equivalent, therefore the accuracy is determined only by the number of the working blocks): pffblock = n∑ i=k p1(t) (5.17) 6. conclusion our method to assess the reliability of an information-computing system is based on a probabilistic method to determine a distribution function of operating time, a reliability function and mean time to failure. for the structural diagram of our system, we obtained basic formulas to determine the required functions and reliability indicators. we investigated various scenarios of system failure; the scenarios depend on the number of parameters for which accuracy decreases. for each of the scenarios, we obtained formulae for the required reliability functions. our method differs from the existing ones as our method takes into account several independent measuring channels and independently operating measuring devices. copyright © 2022 assa. adv syst sci appl (2022) 20 i. brokarev, s. vaskovskii references 1. andreev, a.v. (2018) theoretical foundations of technical system reliability. spb: izdatelstvo politekhnicheskogo universiteta, 164 p. 2. granig, w., faller, l.-m., zangl h. (2018) sensor system optimization to meet reliability targets. microelectronics reliability 87. doi: 10.1016/j.microrel.2018.06.005. 3. ibragimov, i.a., hasminskyr.z. (1981) statistical estimation. asymptotic theory, applications of mathematics. springer verlag, new york. 4. iso 6976: 1995 international standard. (1995) natural gas calculation of calorific value, density and relative density. 46 p. 5. kalimulina, e.y. (2013) a new approach for dependability planning of network systems, international journal of system assurance engineering, 4: 215, https://doi.org/10.1007/s13198-013-0185-2. 6. kalimulina, e.y. (2017) analysis of system reliability with control, dependent failures, and arbitrary repair times, international journal of system assurance engineering, 8: 180, https://doi.org/10.1007/s13198-016-0520-5. 7. kalimulina, e.y. (2017) analysis of unreliable open queueing network with dynamic routing, distributed computer and communication networks. dccn 2017. communications in computer and information science, 700, springer, cham. 8. kalimulina, e.y. (2018) math modeling of the reliability control and monitoring system of complex network platforms, intelligent systems design and applications. isda-2018 2018. advances in intelligent systems and computing, 941, springer. 9. konesev, s.g., hazieva, r.t. (2015) methods to estimate reliability of complex systems, modern problems of science and education, 1, part 1. [in russian]. 10. koralov, l., sinai, y. g. (2007) theory of probability and random processes, springer science & business media, mathematics, 358 p. 11. lavrisheva, e.m., pakulin, n.v., ryzhov, a.g., zelenov, s.v. (2018) analysis of methods to estimate equipment reliability. practical aspects, trudy isp ran, 30(3), 99– 120.[in russian]. 12. oksendal, b. (2003) stochastic differential equationsm, an introduction with applications. fifth edition, corrected printing, springer-verlag, berlin. 13. osipov, l.v., petrov, v.v. on how to estimate the residue of the central limit theorem, theory probab. appl., 12:2, 281–286. [in russian]. 14. state standard 27.002–89. (1990) reliability of equipment. basic terminology. m.: standard publisher, 40 p. [in russian]. 15. state standard 52350.29.1-2010 (iec 60079-29-1:2007) (2010) explosive environments, part 29-1 gas analyzers. general technical requirements to test gas analyzers of combustible gases. standartinfo. [in russian]. copyright © 2022 assa. adv syst sci appl (2022) introduction study of a measuring devices reliability model subsystem reliability model to measure a single gas mixture parameter preliminary investigation. the reliability of a measurement device subsystem as n=1 reliability of the measuring device subsystem conclusion adv syst sci appl 2019; 02; 44-62 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/708 the relaxation of complementary slackness conditions as a regularization method for optimal control problems nikolay pilnik1,2,3 and igor pospelov2 and stanislav radionov1,2,3* 1)national research university higher school of economics, moscow, russia; e-mail: saradionov@edu.hse.ru 2)federal research center “computer science and control”, moscow, russia; 3)financial research institute of the ministry of finance of the russian federation, moscow, russia received march 3, 2019; revised june 21, 2019; published july 10, 2019 abstract: a new approach to the transformation of solutions of optimal control problems is presented. it utilizes the approximation of differential equations for the dual variables, the transformation of continuous time equations to their discrete analogues, the relaxation of complementary slackness conditions. the proposed approach is tested on the russian banking system model, which is derived as a solution of a linear nonautonomous optimization problem with mixed constraints. it is shown that the use of this method regularizes the model in a sense it becomes applicable for the forecasting of the main russian banking indicators (the accuracy of forecasts is comparable with advanced econometric models) and the policy analysis. the stability of model parameters with respect to the change of the estimation time interval and the stability of the model with respect to small disturbances of parameters is confirmed. keywords: optimal control; complementary slackness conditions; banking system. 1. introduction and literature review in this paper, an approach to transforming solutions of optimal behavior problems of macroeconomic agents that constitute the blocks of general economic equilibrium models is presented. the features of this approach using the example of the russian banking system model are demonstrated, which is intended to be a block of a larger model of the russian economy. the essence of this method is a special transformation of some relationships (complementary slackness conditions) to a more regular ones of the special form. this allows to find the analytical expressions for all variables of the model, which greatly simplifies the procedure of parameter estimation. the presented methodology is substantially different from the approach applied in the dynamic stochastic general equilibrium (dsge) framework, which is a mainstream in a modern macroeconomic modeling nowadays. it is well known fact that dynamic stochastic general equilibrium (dsge) models cannot be solved exactly except for the simplest cases due to their exceptional mathematical complexity. hence, the approximation techniques should be used. the most common technique is called the perturbation method. the key idea is to approximate the model around the so-called steady state and then find the exact solution of the approximation, see for example [1]. * corresponding author: saradionov@edu.hse.ru the relaxation of complementary slackness conditions 45 copyright ©2019 assa adv. in systems science and appl. (2019) hence, the dynamics of the economy is considered as a deviation from its steady state caused by shocks, which can be interpreted as positive or negative depending on the sign of their impact. the steady state is a numeric vector which contains the steady values of all model variables. it can be calculated after some model transformations, usually the detrending (if the variables exhibit balanced growth), or the change of variables of the model to their stationary version. in the practical models the first order approximation is usually used (see for example [2, 3, 4]), because higher-order approximations of the models with more than 5-7 equations is prohibitively time-costly during the parameter estimation phase, and also lead to unreliable estimations (with high standard deviations). some modifications of perturbation method were proposed, for example, in [5, 6]. nevertheless, it is known that linear approximation can perform poorly for the regions not very close to the steady state. even worse, linear approximation can lead to wrong conclusions about the existence and uniqueness of equilibrium, see [7]. for these reasons, there a growing interest in more advanced quadratic and higher order approximation techniques, and some advances in this direction have been made, see for example [8, 9]. for this moment, however, the linear approach is still a mainstream due to the issue of unreliable coefficients described above. a brief review of the macroeconomic models incorporating banking sector as well as a review of papers dedicated to the russian banking system was presented in the accompanying paper [10]. the summary of this review is that the banking system gradually becomes an element of dsge models, but the description of this sector remains highly stylized due to limitations of the model size described above. the model presented in this paper describes an agent “bank” with substantially wider set of available controls. bank can lend to producers and consumers and can take savings from both. savings and loans can be nominated in national currency (russian rubles) and in a foreign one (actually in a currency basket containing us dollars and euros). in comparison with the version of the model presented in the accompanying paper [10], we now model loans of firms and households introduced as separate variables. we also include one more restriction on the banking system associated with a capital adequacy ratio. this work continues our research on the russian banking sector presented in [11, 12, 13]. the contribution of this article is threefold. first, a new methodology of the transformation of optimal control problem solution using the turnpike property and relaxation of complementary slackness conditions is presented. second, it is shown how this methodology allows to transform the system of equations of the model and obtain a complete analytical solution, which is used later on at the stage of estimating the parameters of the model. third, a wider set of macroeconomic variables whose trajectories are successfully replicated by our model is considered. 2. the model 2.1. statement of the model the model of the whole banking system as a single agent is considered. it is transformed to the usual dynamic model that determines the demand of the banking system for deposits and the supply of loans depending on the current balance of the bank, as well as on interest rates and other external market factors. the bank model is considered in continuous time on the interval  0 ,t t . using the sufficient conditions of the maximum in the lagrange form, the analytical solution of the bank problem is obtained. the subsequent use of the turnpike property and the transition to discrete time allows to use the technique of relaxation of the complementary slackness conditions. after that, the 46 n. pilnik, i. pospelov, s. radionov copyright ©2019 assa adv. in systems science and appl. (2019) expressions for all endogenous variables of the model are obtained, in the right-hand parts of which only exogenous variables and previous values of endogenous variables are included. the following tables contains the list of the model variables – the money stocks, the flows associated with it, its durations, interest rates and the model parameters. the relationships between these variables are assumed to follow equalities and inequalities presented below. table 2.1. the model variables variable meaning associated flow duration interest rate ( )lh t household loans in rubles ( )kh t ( )lh t ( )lhr t ( )sh t household deposits in rubles ( )vh t ( )sh t ( )shr t ( )la t firm loans in rubles ( )ka t ( )la t ( )lar t ( )sa t firm deposits in rubles ( )va t ( )sa t ( )sar t ( )vlh t household loans in foreign currency ( )vkh t ( )vlh t ( )vlhr t ( )vsh t household deposits in foreign currency ( )vvh t ( )vsh t ( )vshr t ( )vla t firm loans in foreign currency ( )vka t ( )vla t ( )vlar t ( )vsa t firm deposits in foreign currency ( )vva t ( )vsa t ( )vsar t ( )sc t deposits of the central bank ( )scr t ( )lc t loans of the central bank ( )lcr t ( )vsf t deposits of the foreigners ( )vsfr t  n t balances of settlement accounts  nn t supply of balances of settlement accounts  a t liquid funds in rubles  va t liquid funds in foreign currency  rc t ( )z t dividends paid by the bank  oop t other liabilities  ooc t table 2.1. the model parameters parameter meaning  ww t dollar to ruble exchange rate sh , sa , lh , la , vsh , vsa , vsf , vlh , vla parameters of liquidity constraints  sn t ,  vsn t parameters of reserve requirement  a t parameter of capital adequacy constraints  , ,  liq t parameters of utility function the relaxation of complementary slackness conditions 47 copyright ©2019 assa adv. in systems science and appl. (2019) consider the bank, which issued household loans nominated in rubles in the amount of ( )lh t by the time t . the average terms of loans (durations) will be denoted by 1 ( )lh t . the variable ( )lh t will be referred below as the inverse durations and interpreted as the average return frequency of household loans in rubles. then the dynamics of loans is described by the equation         d d lhlh t kh t t lh t t   , (2.1) where ( ) 0kh t  (2.2) is the flow of newly issued loans. note that in all the formulas given below, unless otherwise specified, it is assumed that the relationship is valid for any point in time on the interval  0 ,t t . with respect to loans granted, the bank receives interest payments ( ) ( )lhr t lh t , where ( )lhr t is the effective interest rate on household loans in rubles. denote the amount of household deposits in rubles the bank attracted by the time t by ( )sh t . the average terms for which deposits are attracted, will be denoted by 1 ( )sh t . the variable ( )sh t will be referred below as the inverse durations and interpreted as the average frequency of the return of deposits. then the dynamics of household deposits in rubles is described by equations         d d shsh t vh t t sh t t   , (2.3) where ( ) 0vh t  (2.4) is the flow of newly attracted deposits. the bank pays interest on borrowed funds ( ) ( )hr t sh t , where ( )shr t is the effective interest rate on household deposits in rubles. similarly, the bank interacts with firms, giving them loans and attracting deposits from them in rubles. these processes are described by the equations         d d lala t ka t t la t t   ,         d d sasa t va t t sa t t   , (2.5) where ( ) 0ka t  , ( ) 0va t  are the flows of newly issued firm loans in rubles and newly attracted firm deposits in rubles. variables ( )la t and ( )sa t describe the frequency of repayment of loans and deposits of firms. interest payments are made according to the effective interest rates ( )lar t and ( )sar t . in addition to operations in rubles, the bank conducts operations in foreign currency. to describe these processes, we duplicate the description of loans and deposits for individuals and firms given above with the prefix “v” added to each variable, which indicates that this variable is nominated in dollars. the equations which describe the currency transactions are as follows:         d d vlhvlh t vkh t t vlh t t   , ( ) 0vkh t  , (2.6)         d d vlavla t vka t t vla t t   , ( ) 0vka t  ,         d d vshvsh t vvh t t vsh t t   , ( ) 0vvh t  , 48 n. pilnik, i. pospelov, s. radionov copyright ©2019 assa adv. in systems science and appl. (2019)         d d vsavsa t vva t t vsa t t   , ( ) 0vva t  . (2.7) denote the dollar to ruble exchange rate by  ww t . interest payments in dollars are made at effective interest rates ( )vlhr t ( )vshr t , ( )vlar t и ( )vsar t . in addition to deposits, the bank also attracts funds in the form of interest free balances of settlement accounts  n t . there is no regulating value for the size of these balances, and the bank should simply focus on the supply from the clients. therefore, this value is considered to be bounded exogenously:    nn t n t , where  nn t is the supply of the balances of settlement accounts known to the bank. similar logic is used to describe two other external sources of funds: deposits of the central bank, which it invested in the banking system  sc t (in rubles) and deposits of foreign organizations attracted by the bank from abroad  vsf t (in dollars). the three variables introduced above  nn t ,  scsc t ,  vsfvsf t , in fact, describe the state of the other agents of the economy that interact with the bank. current accounts characterize the economic activity of households and firms, deposits of the central bank describe the monetary policy pursued by it (the amount of funds invested in the banking system), and deposits of foreign organizations show the intensity of interaction with the external world. these variables become especially important when describing the 2014–2015 crisis phenomena and the external constraints that appeared in the form of various sanctions. interest paid on deposits of the central bank and foreign organizations are defined by the rates ( )scr t and ( )vsfr t . the bank also has another channel of interaction with the central bank interest accounts, which in our terms means loans issued by the bank to the central bank. we denote them by  lc t . the bank receives interest payments on this account in rubles at the rate of ( )lcr t . the only restriction on this variable is the condition of non-negativity   0lc t  . (2.8) for servicing operations related to loans and deposits, the bank uses two types of liquid funds: the first one nominated in rubles  a t and the second one nominated in foreign currency  va t . it is assumed that their volumes in the balance sheet are proportional to ruble and foreign currency loans and deposits, respectively. these proportions are defined by the coefficients  with corresponding indices. these coefficients have a legislative counterparts. in fact, they are model analogues of liquidity adequacy ratios.          sh sa lh laa t sh t sa t lh t la t       , (2.9)            vsh vsa vsf vlh vlava t sh t sa t sf t lh tv la tv v v v         . the bank’s activity in attracting deposits is limited by the reserve requirement, which obliges the bank to keep part of the funds raised in the central bank accounts. we model this requirement as                     ,s vs wrc t n t sh t sa t n t w t vsh t vsa t vsf t     (2.10) where  sn t and  vsn t are the reserve requirement coefficients for the ruble and the foreign currency deposits. the relaxation of complementary slackness conditions 49 copyright ©2019 assa adv. in systems science and appl. (2019) taking into account all the operations described above, the financial balance of the bank can be written out as:                     d d lh lh la laa t kh t t r t lh t ka t t r t la t t          (2.11)                                                                                     d d d d d d d d d d w vlh vlh vla vla sh sh sa sa w vsh vsh vsa vsa sc w vsf lc w w t vkh t t r t vlh t vka t t r t vla t vh t t r t sh t va t t r t sa t w t vvh t t r t vsh t vva t t r t vsa t sc t r t sc t w t vsf t r t vsf t lc t r t lc t t t t n t w t va t t                                                  d d ( ) . d d o ot rc t op t oc t z t t t          the flow ( )z t is the amount of dividends paid by the bank. variable  oop t is “other liabilities” and means the difference between all other liabilities and bank balance assets that are not described in the model. the flow  ooc t includes tax payments, investments in fixed assets (including participation in property), as well as operating expenses. these flows (except, perhaps, taxes) are not directly related to the assets and liabilities of the bank, hence we will not delve into the  ooc t structure. finally, the last limitation of the bank's activity in the model is the capital adequacy ratio. equity is not explicitly included in the model, therefore it is written as the difference of assets and liabilities of the bank. the minimum value of equity capital is determined by the total volume of bank assets in the proportion set by the capital adequacy ratio  a t . in the model, this restriction looks like this:                                                           . w w a w t a t la t lh t lc t rc t w t va t vla t vlh t a t la t lh t lc t rc t w t va t vla t vlh t sa t sh t sc t n t op t w t vsa t vsh t vsf t                            (2.12) the above relationships represent the limitations imposed within the model on the bank’s ability to choose the values of its planned variables (controls):                                             , , , , , , , , , , , , , , , , , ( ), ( ), , , , , . kh t lh t ka t la t vh t sh t va t sa t vkh t vlh t vka t vla t vvh t vsh t vva t vsa t sc t lc t sf t z t n t rc t a t va t (2.13) according to the principle of rational expectations underlying the models of intertemporal equilibrium, when planning its control variables, the bank can rely on an accurate forecast of information variables:                                                       , , , , , , , , , , , , , , , , , , , ( ), , , , , , , , . lh sh la sa vlh vsh vla vsa lh sh la sa sc lc vlh vsh vla vsa vsf a s vs n o o sc sf w t t t t t t t t r t r t r t r t r t r t r t r t r t r t r t t n t n t n t op t oc t sc t sf t w t          (2.14) as a result, the choice of the planned variables by the bank actually determines the supply 50 n. pilnik, i. pospelov, s. radionov copyright ©2019 assa adv. in systems science and appl. (2019) of loans, as well as its demand for attracted and liquid funds as a function of current and future values of information variables (primarily interest). the goal of the bank is to maximize the total discounted utility from the received undistributed profit taking into account the given deflator  liq t . therefore, the functional of the bank can be written as    0 1 e d 1 t t t liq z t t t              , with respect to the variables (2.13) with constraints (2.1) – (2.11) on  0 ,t t , the values of phase variables specified at the initial time and the given trajectories of exogenous values (2.14). for the solvability of the constrained optimization problem it is necessary to supplement it with terminal conditions, which can be defined as a growth condition for some linear form of the phase variables:                                                                   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( ) ( ( ) . o w o w a t lh t la t lc t sh t sa t sc t n t rc t op t w t va t vlh t vla t vsh t vsa t vsf t a t lh t la t lc t sh t sa t sc t n t rc t op t w t va t vlh t vla t vsh t vsa t vsf t                                     (2.15) this is an analog of no ponzi condition written in terms of bank’s own capital which is a difference between its assets and liabilities. 2.2. optimal behavior in the model the formulated problem belongs to a very complex class of optimal control problems — it is a linear nonautonomous problem with mixed constraints. next, we describe the main stages of finding and transforming its solution. each of the constraints is assigned a dual variable (non-negative to inequality and with indefinite sign to equality), a lagrange functional is composed and the conditions of its saddle point are written out, which represent a system of sufficient optimality conditions, see for example [14]. these conditions fall into four groups:  differential equations, the initial conditions for which are assumed to be given;  conjugate differential equations for the dual variables;  transversality conditions that define terminal conditions for conjugate differential equations;  complementary slackness conditions for inequalities. in general, they consist of three conditions ( ) ( ) 0, ( ) 0, ( ) 0t x t x t t     , where ( )x t is the direct variables in the original inequality, and ( )t is the dual variable to this inequality. the system of such conditions we will denoted further by   ( ) ( )t x t . consider the technique of working with a system of sufficient conditions obtained after transforming variations of the lagrange functional in direct variables on the example of equations for variables describing the bank’s interaction with the consumer (the conditions for interacting with a firm look similar). start with the above-mentioned the relaxation of complementary slackness conditions 51 copyright ©2019 assa adv. in systems science and appl. (2019) conditions of complementary slackness, derived from inequalities for the positiveness of new deposits attracted (2.2) and loans issued (2.4) in rubles and foreign currency. this set of conditions is as follows:      [ 6 ][ ]t t kh t , (2.16)        [ 8 ][ ]wt t w t vkh t ,      [ 14 ][ ]t t vh t ,        [ 16 ][ ]wt t w t vvh t . (2.17) in this case,        2 , 4 , 6 , 8t t t t    is the normalized dual variables to the corresponding inequalities,  t is the dual variable to the financial balance. the dynamics of the dual variable  t turns out to be extremely important for describing the optimal behavior of the bank. for convenience, the following variable will be used later     ( ) d t dtt t      . the rate of decline of the dual variable to the financial balance ( )t can be interpreted as the profitability of the agent. transforming the variations of the lagrange functional with respect to stocks ( ), ( ), ( ), ( )lh t vlh t sh t vsh t and flows ( ), ( ), ( ), ( )kh t vkh t vh t vvh t allows us to obtain differential equations for normalized dual variables        2 , 4 , 6 , 8t t t t                 d 6 6 (1 , d )lh lh lc lht t t t r t r t t          (2.18)                         d d d8 8 d d d1 1 , w vlh w w vlh lc vlh vlh w w t tt t t t t w t w t tr t r t w t                                          d 1 14 14 d 1 ( ) ( ) , 1 ( ) sh s sh lc a a sh a t t t t n t r t t t t r t t t                                                  d ( )d d16 16 d 1 ( ) d 1 d1 . 1 ( ) w a vsh w a w vsh vs vsh vsh a w lc w t ttt t t t t t w t t w t tn t r t r t t w t                                    (2.19) variation of the lagrange functional with respect to the variable  z t leads to a differential equation for this variable 52 n. pilnik, i. pospelov, s. radionov copyright ©2019 assa adv. in systems science and appl. (2019)           d d 1 d . d liq liq tt tz t z t t t                  (2.20) relations (2.16) (2.20), restrictions on direct variables (2.1), (2.3), (2.5), (2.6) (2.7) and (2.11), as well as conditions (2.9) (2.10) transformed into equalities together with the conditions on the terminal values of dual variables, obtained by varying the lagrange functional with respect to phase variables at the time t , fully describe the solution of the optimal control problem. 2.3. transformation of the solution in addition to the standard methods developed for solving such problems, novel methods developed for the transition from problems in continuous time, like the one discussed above, to problems in discrete time, which can be used for matching with statistical data are also used. this method will be presented in this subsection. at the first stage, differential equations for dual variables (2.18) (2.19) are transformed, using the following property. all these equations are highly unstable in straight time. the only bounded solution on  0 ,t t is the one close to the quasistationary one, normalized in such a way to fulfill the terminal condition as equality. this solution is a zero order approximation of the abovementioned normalization coefficient (see [15]). therefore, a large part (actually almost the whole interval  0 ,t t for a given observed rates and durations) is spent around the separatrix, deviating sharply from it to fulfill the terminal condition. this property of the solution is called the turnpike property and is described in detail, for example, in [16]. using it, differential equations (2.18) (2.19) can be approximated by the following expressions after replacing the derivative of the normalized dual variable with zero according to the definition of separatrix.           (1 , ) 6 lh lc lh lh r t r t t t t         (2.21)                       d d1 1 8 , d d w vlh lc vlh vlh w w vlh w w t tr t r t w t t w t tt t w t                           ( ) 1 1 ( ) 1 ( ) 14 , a sh s sh lc a a sh t t r t n t r t t t t t t                    the relaxation of complementary slackness conditions 53 copyright ©2019 assa adv. in systems science and appl. (2019)                         ( ) 1 1 ( ) 1 ( ) d d1 16 . d d l a vsh vs a a w vsh vsh w c w vsh w t t n t r t t t w t tr t w t t w t tt t w t                          (2.22) note that using the above four relations for dual variables leads to elimination of the influence of the future (terminal conditions that determine their value at the moment of time t ) on the past (the entire trajectory until this last moment) typical for the dual variables. the terminal condition (2.15) ensures the solvability of the original problem and allows to obtain the values of the dual variables at the moment of time t . that is why the terminal conditions themselves will not be of further interest, since they only affect for a very small amount of time before the moment t . from the point of view of using models of this kind in applied calculations, it should be borne in mind that the statistics of the entire trajectory even for exogenous variables is not available. the data is fundamentally discrete and is calculated with a certain frequency: month, quarter, year, and so on. hence, the data either on average for this period or the value corresponding to its end is used. in order to adapt the model to this type of data, the sampling procedure is used the transition to discrete time (see [17] for more details). it is assumed that, although the bank acts as described in its optimization problem in continuous time on an interval  0 ,t t , it is possible to observe it at discrete moments. it should be noted that this technique is essential in the construction of the model. if a model is directly written in discrete time, then the complementary slackness conditions (2.16) (2.17) will have to be resolved in a uniform way: the direct variables in the right bracket will always have to be assumed larger than zero, and the dual variables in the left bracket will be considered equal to zero. the latter, by virtue of relations (2.21) (2.22), will lead to the appearance of rigid relations between rates, which, is not what is observed in the data. it is also incorrect to reject these relations both from a mathematical point of view, thereby violating the equality between the number of equations and the number of endogenous variables, as well as the essential one. dual variables        2 , 4 , 6 , 8t t t t    do not simply take into account the influence of individual interest rates that are in their relations (2.21) (2.22), but also link these effects within the entire model through a variable  t . this property allows the model to describe not only the direct effects of interest rates (for example, a change in the rate on ruble loans to the households on the volume of ruble loans issued to the households), but also crossimpact. the model takes into account the dependencies of all endogenous variables from all exogenous variables (which, in principle, is typical for general equilibrium models). in the process of deriving a system of equations describing optimality conditions, the type of functional form for each such dependence is determined. it seems that it is extremely difficult to obtain such dependencies in the framework of econometric models, because, firstly, in this case the functional form has to be guessed (the testing possibilities, although they exist, are very limited), and secondly, take into account the influence of all 54 n. pilnik, i. pospelov, s. radionov copyright ©2019 assa adv. in systems science and appl. (2019) exogenous variables even per one endogenous variable are practically impossible the number of estimated coefficients is critical for the available data set. further, every indicator in discrete time is associated with indicator in continuous time. within this approach, each derivative can be replaced by the difference between the current and the previous value. ( ) ( ) ( 1) d x t x t x t dt    , but at the same time, the variable itself, which occurs in the same equations without a differential, must be replaced by its value at the previous time. the result of applying this technique can be demonstrated for the balance of ruble loans issued to the public (2.1), which should be written as          1 1lhlh t lh t kh t t lh t     . (2.23) balances (2.3), (2.5), (2.6) (2.7) should be rewritten in a similar way. then the financial balance (2.11) should be rewritten in the form                       1 1 1 lh lh la la a t a t kh t t r t lh t ka t t r t la t                (2.24)                                                                                       1 1 1 1 1 1 1 1 1 1 1 w vlh vlh w vla vla sh sh sa sa w vsh vsh w vsa vsa sc n n w t vkh t t r t vlh t w t vka t t r t vla t vh t t r t sh t va t t r t sa t w t vvh t t r t vsh t w t vva t t r t vsa t sc t sc t r t sc t n t n t rc t rc t lc t lc t                                                                                1 1 1 1 1 . lc o o o w vsc r t lc t op t op t oc t z t w t va t va t vsc t vsc t r t vsc t                 expressions (2.9) (2.10) and (2.12), written as equalities, should be rewritten as            1sh sa lh laa t sh t sa t lh t la t d seas t        ,            vsh vsa vsf vlh vlava t sh t sa t sf t lh tv la tv v v v         ,                    s vs wrc t n t sh t sa t n t w t vsh t vsa t vsf t     ,                                                            w w a w t a t la t lh t lc t rc t w t va t vla t vlh t a t la t lh t lc t rc t w t va t vla t vlh t sa t sh t sc t n t op t w t vsa t vsh t vsf t                            . in turn, relation (2.20) can be rewritten as               11 1 1 . 1 liq liq liq t tt z t z t z t t                   (2.25) the complementary slackness conditions (2.16) (2.17) require a separate discussion. in fact, they are degenerate functional dependencies, the graph of which delineates the boundaries of the first quarter, and, therefore, the optimal solution of the system under study “jumps around the corners”. from the economic point of view, they describe the the relaxation of complementary slackness conditions 55 copyright ©2019 assa adv. in systems science and appl. (2019) infinitely elastic supply or demand functions of the agent. from the point of view of the subsequent model calibration, these relationships present a significant difficulty, substantially increasing the instability of the coefficient estimates, and, as a result, the forecasts calculated using the model. nevertheless, with the help of natural assumptions about switching modes determined by the method of resolving each of the conditions of the complementary slackness, it is possible to move to more regular and convenient ratios from the point of view of the model calibration. this technique is studied in more detail in [17]. the main idea behind this method is that although the bank itself works in continuous time, it can only be observed at specific points of time (at the end of each month). because of this, pure regimes of resolving the complementary slackness conditions are never seen. indeed, some mixtures are observable, which can be described as follows: "bank issued new loans x% time in this month, and did not issue them (1-x)% time in this month". the equations presented below answer the question of what is the proportion of time spent in this or that regime. using the previously obtained expressions for dual variables (2.21) (2.22) and the special rules for their normalization, the relaxation coefficients , , , ,i i i i ia b c cc cf are added, where 1,...,8i  is the number of the corresponding complementary slackness condition. consider the principles of relaxation of the complementary slackness conditions on the example of the households’ foreign currency deposits because the equation (2.22) is the most complicated one. according to our approach, (2.17) is replaced with the following relation                        1 1 1 1 1 1 16 1 1 . ( ) w w vsh w c f vvh w t w t vvh t b a t t t vsh t w t c t cc t cf t b t                      the direct variable included in the complementary slackness condition always appears as a dependent one. the first bracket on the right side is a linear function of the dual variable. in this case, there are two coefficients 1 1,a b that will be estimated later. this bracket actually determines the proportion of time during the month (discrete grid step) when new deposits are taken. however, it is important to remember that this bracket itself has no dimension, and the dependent variable clearly has a dimension. therefore, the first bracket is normalized to two more factors: to the corresponding phase variable of the previous period (variable of the “stock” type), and to the bracket that coincides with the denominator of the expression (2.22) for the dual variable. such a normalization allows us to analytically find all model variables in discrete time. the resulting first term in the right part actually describes the choice of banks for the volume of deposits taken depending on their level at the previous point in time, their interest rate, duration and exchange rate (only for currency indicators). in addition, information about the corresponding indicators of other bank-controlled variables is transmitted through the variable ( )t . to take into account the influence of exogenous variables, a second term containing coefficients 1 1 1, ,c cc cf is added to this expression, in which, for convenience, the following variables are introduced              1 1 ,n n o o ot n t n t op t op t oc t z t                  1 1 ,c sct sc t sc t r t sc t      56 n. pilnik, i. pospelov, s. radionov copyright ©2019 assa adv. in systems science and appl. (2019)           ( ) 1 1 .f w vsft w t vsf t vsf t r t vsf t      the function ( )vvhb t is specifically chosen to agree on the dimensionality of the components of the entire expression, and for the convenience of its further use when substituting into the financial balance. for example, for the variables  vvh t and  vkh t the similar function should contain the dollar exchange rate ( )ww t . preliminary analysis of the data showed that there is a clear seasonal component in some series, namely in ruble deposits of the households and firms and in ruble liquidity, showing clear peaks at the end of each year. no preliminary adjustment to this seasonality is made, but instead, if necessary, an amendment for a special dummy variable  seas t is introduced, which takes unit values every december and is zero at all other points. consider the expressions only for variables describing the interaction of the bank and the households. as a result of the described transformations, the following expressions are obtained                                        1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , 1 ( ) w w vsh vsh vsh w a vsh vsh lc a c f w a w t w t vvh t a r t b t a b w t u t n t a r t u t a b t vsh t c t cc t cf t w t u t                                (2.26)                                2 2 2 2 2 2 2 2 2 3 1 1 1 , 1 ( ) sh sh a sh sh lc a c f a a r t b t u t n t a r t vh t sh t u t a b t c t cc t cf t d seas t d seas t u t                                                               5 5 5 5 5 5 5 5 5 1 1 1 1 1 , w w vlh vlh vlh w c f vlh lc w w t w t vkh t a r t b t a b w t c t cc t cf t a r t b t vlh t w t                                                  6 6 6 6 6 6 6 1 1 4 5 1 , lh lh lh lc c f kh t a r t b t a r t b t lh t c t cc t cf t d seas t d seas t                   (2.27) where the following notation is used for convenience 1 ( ) . 1 ( ) a a u t t    now we describe the logic of the relationship between variables in the resulting dynamic system at a time t . suppose that its state is fully known at the moment of time 1t  (all variables from (2.13) are known), information variables from (2.14), and all coefficients are also known. in addition, for now assume that the value of ( )t is known. then, from the relaxed complementary slackness conditions (2.26) (2.27), the variables ( ), ( ), ( ), ( )kh t vkh t vh t vvh t and ( ), ( ), ( ), ( )ka t vka t va t vva t are determined. then from balances of type (2.23) ( ), ( ), ( ), ( )lh t vlh t sh t vsh t and ( ), ( ), ( ), ( )la t vla t sa t vsa t can be found. from (2.25) ( )z t can be found. as a result, all unknown variables at time t from the relaxation of complementary slackness conditions 57 copyright ©2019 assa adv. in systems science and appl. (2019) the financial balance (2.24) can be written as functions of profitability ( )t , parameters, information variables, and variables at time 1t  . after substituting such expressions in the financial balance, we get an expression for the variable ( )t .                                                                                 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 lh la vlh w vla w w a a a a vs vsh lc vsh w w a vsh a vsh a a sh s lc a r t lh t r t la t r t w t vlh t r t w t vla t g t u t b u t u t u t n t r t g t a w t vsh t u t b t u t a r t u t u t n t a r t u t t b                                                                                                                                    2 3 3 3 3 4 4 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 sh a sh a w a a a a vs vsa lc vsa w w a vsa a vsa a a s sa lc a sa a sa a t sh t u t a r t u t g t u t b u t u t u t n t r t g t a w t vsa t u t b t u t a r t u t u t n t a r t u t b t u t a r t u t                                                               1sa t                                                                           1 2 3 4 1 2 3 4 1 2 3 4 2 3 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 w lc a n op sc vsf a c a f a w w wa a a t w t va t rc t r t lc t u t n t op t r t sc t w t r t vsf t c c c c t cc cc cc cc u t t cf cf cf cf u t w t t u t d seas t d seas t b u t a w t vsh t b u t                                                                         2 3 3 4 4 , 1 1 1 1 1 a a aw u t a sh t b u t a w t vsa t b u t a sa t                 where         1 1 w w w w w t w t g t w t     . in fact, the expression for profitability is a weighted combination of interest rates, and the weights are determined, among other things, by the structure of bank assets and liabilities in the previous period, their durations, as well as the interest rate and other exogenous variables. finally, one more property of the system under consideration should be noted. with a special selection of parameters, the equation for ( )t can be simplified due to the fact that some combinations of exogenous variables with the corresponding coefficients in total are zero. therefore, to simplify the calculations carried out further, it is assumed that 58 n. pilnik, i. pospelov, s. radionov copyright ©2019 assa adv. in systems science and appl. (2019) 8 1 1i i с   , 8 1 1i i сc   , 8 1 1i i сf   . it should be noted that all the analytical transformations described above are performed via the ecomod support system developed under the guidance of i. pospelov [16]. it allows to check the information links of the model, correctness of the balances, and the dimensions. it also automatically writes down the sufficient optimality conditions in the lagrange form. 3. numerical results as in the accompanying paper [10], the main source of statistical data is the form 101 (turnover sheet) published by the bank of russia on a monthly basis starting from january 2004. to estimate this version of the model, this data on the period from january 2010 to november 2018 is used. the principles of aggregation of the form 101 accounts into the model variables, as well as calculations of the interest and duration rates are described in [10]. to estimate the parameters of the model, the same multistep forecasting methodology described in the accompanying paper [10] is applied. to put it simple, the parameters such that the model produces the most accurate forecasts for up to six months ahead from the current point are found. in other words, the error function is the sum of squared relative errors of forecasts for 1, 2, …, 6 months calculated for every moment from 0 to t-1. using monte-carlo method, a stable vector of parameters in a sense that algorithm (we use lsqnonlin command from matlab’s computing environment optimization package) converges to it from different regions is found. of course, the uniqueness of solution of this error minimization problem cannot be guaranteed, but another set of parameters with comparable accuracy of forecasts during rather extensive computations was not found. the average forecast errors of the model variables in per cents for a given length of forecasts are summarized in th e following table: table 3.1. forecast errors length of forecasts, months 1 2 3 4 5 6 mean la 1.1 1.5 1.7 1.9 2.3 2.5 1.8 sa 2.7 3.6 4.7 5.2 5.9 6.9 4.8 vla 2.0 2.7 3.3 4.0 4.8 5.4 3.7 vsa 3.0 4.5 5.5 6.1 6.8 7.5 5.6 lh 1.7 2.2 2.7 3.3 3.5 3.9 2.9 sh 1.4 2.0 2.4 2.8 3.2 3.7 2.6 vlh 1.6 2.1 2.6 3.3 3.6 4.0 2.9 vsh 1.9 2.7 3.2 3.2 3.3 3.6 3.0 a 5.5 5.4 5.8 6.0 6.2 6.1 5.8 rc 0.0 0.3 0.5 0.6 0.7 0.8 0.5 va 12.7 12.4 12.0 11.3 11.2 11.1 11.8 lc 92.4 87.3 85.3 82.5 88.6 110.2 91.0 the “mean” column contains average forecast errors for different forecasts lengths. as can be seen, the model successfully replicates the dynamics of all the model variables, the relaxation of complementary slackness conditions 59 copyright ©2019 assa adv. in systems science and appl. (2019) except for lc. the forecast quality of lc is poor because this variable is rather small in the forecasting region – under current economic conditions banks do not have extra funds to store them in the central bank. instead they prefer to use them in more profitable ways. nevertheless, we cannot ignore this variable, because the model is oriented on the analysis of banking system reaction on external influences. in the last 15 years the need of the banking system for this instrument was observed several times. the forecast accuracy presented in the table 1 is comparable (and for some variables outperforms) the models of the same type and also econometric models, see [3]. the presented version of the model contains, apart from the common variables such as volumes of loans and deposits, several variables related to capital adequacy ratio. the variables of this type are bank capital itself, “other liabilities” and bank deposits in the central bank. from the technical point of view, capital adequacy requirement complicates our model significantly, because it contains almost all model variables. however, the transformation method we applied allows us to find the analytic solution which contains the explicit formula for ( )t . we think this relation, together with a set of extra coefficients describing the connection between continuous and discrete time scales, allows us to preserve the forecasting quality of the model despite the fact it became significantly more complicated. the sensitivity analysis of the obtained set of parameters was performed with plausible results. first, the parameters of the model were reevaluated for several shorter time intervals and remained almost the same. this fact confirms the assumption that the model with the found set of parameters describes long-term relationships between the variables. second, the variables of the model were recalculated for a slightly distorted version of the parameters found and remained almost the same. this fact confirms the stability of the model, making it applicable for policy analysis and forecasting. the following figures 1-6 show an example of the model forecasting. the forecast is calculated from december 2017 for the next six months. figure 1. liquid funds in rubles (a), trillion rubles figure 2. liquid funds in foreign currency (va), trillion dollars 0 0,5 1 1,5 2 2,5 3 3,5 2 4 6 8 10 12 2 4 6 2017 2018 statistical data december 2017 forecast 0 0,005 0,01 0,015 0,02 0,025 0,03 0,035 0,04 0,045 2 4 6 8 10 12 2 4 6 2017 2018 statistical data december 2017 forecast 60 n. pilnik, i. pospelov, s. radionov copyright ©2019 assa adv. in systems science and appl. (2019) figure 3. firm loans in rubles (la), trillion rubles figure 4. household loans in rubles (lh), trillion rubles figure 5. firm deposits in rubles (sa), trillion rubles figure 6. household deposits in rubles (sh), trillion rubles 4. conclusions the model presented in this paper is based on the new transformation techniques of solutions of optimal control problems with inequality constraints. these techniques include the turnpike property-based approximation of differential equations for the dual variables, the special method of transformation of continuous time equations to their discrete analogues, the relaxation of complementary slackness conditions. this model can be used by policymakers for measuring the effects of regulation policies such as changes of key interest rate, deposits of central bank in the banking system or reserve requirements, and by the market participants as well. the proposed model successfully replicates the dynamics of 11 variables which describe the main indicators of the banking system (with 12 endogenous variables in the model in total). for comparison, the benchmark dsge model of the us economy described in [3] replicates 7 variables, and the model described in [18] replicates 10 variables. more advanced model of the czech economy contains 28 endogenous variables in the version of the model described in [19]. further extension of the presented work can 0 5 10 15 20 25 30 35 40 2 4 6 8 10 12 2 4 6 2017 2018 statistical data december 2017 forecast 0 2 4 6 8 10 12 14 16 2 4 6 8 10 12 2 4 6 2017 2018 statistical data december 2017 forecast 0 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 2 4 6 2017 2018 statistical data december 2017 forecast 0 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 2 4 6 2017 2018 statistical data december 2017 forecast the relaxation of complementary slackness conditions 61 copyright ©2019 assa adv. in systems science and appl. (2019) include the analysis of the applicability of the proposed approach to a wider class of optimal control problems and the development of more general-purpose approach. for example, presented approach can be used in modelling of other macroeconomic agents such as consumer, producer, trader, and in the development of general equilibrium models composed of the agents of this kind. further development of the presented banking system model (more detailed description of its structure with appropriate increase of number of variables) also seems promising. acknowledgements the reported study was funded by rfbr according to the research project № 18-3100353. references [1] judd, k.l. & guu, s.m. (1997). asymptotic methods for aggregate growth models. j. econ. dyn. con. 21 (6), 1025–1042. https://doi.org/10.1016/s01651889(97)00015-8 [2] pilnik, n., radionov, s. & yazykov, a. 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(2011). a dsge model of the czech economy: a ministry of finance approach, ecomod 3007. https://doi.org/10.1007/978-3-030-10934-9_30 adv syst sci appl 2018; 03; 123-143 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/543 an efficient approach to civil structures health monitoring using fog computing as clusters through 5g network environment divya sinha1, kavish doshi1, m. rajasekhara babu1 1) school of computer science and engineering vit university, vellore, 632014, india e-mail: divya.sinha2014@vit.ac.in,doshikavish.nilam2014@vit.ac.in,mrajasekharababu@vit.ac.in abstract: structural health monitoring (shm) is a process to trace real time information about the physical health of a civil structures at any given point of time. shm systems can also approximate the life expectancy of the structure. this constant monitoring can help during emergency situations such as earthquakes. the increasing need for shm is the result of deterioration of several set of structures over time. a proper maintenance plan is beneficial to not just keep the structure in a favorable state throughout its lifecycle, but to additionally minimize the relative costs required by the structures through resource optimization. various forms of shm have been implemented over the past three decades like traditional wired systems. recent development has resulted in wireless sensor networks (wsn) being designed for the same purpose. few publications have also included iot framework along with wsns. the introduction of wsn has led to easy deployment of shm compared to the traditional wired network. but wsn's have led to significant challenges. the main challenge that has been focused in this paper is the constraint of power consumption. the power consumption is most critical factor for the sensors deployed in high-rise structures, high density traffic bridges etc., since it is difficult to reach the places to replace the battery drained sensors and also involved high installation cost. this paper proposes a concept of a framework for shm with fog computing which hasn't been explored in the field of shm yet. in this model, the wsn is comprised of a piconet which transmits data through bluetooth low energy (le) and the fog transmits data to cloud by exploiting 5g network features in order to reduce the power consumption furthermore. thus, saving power at every level possible in the shm architecture. it also introduced a head sensor algorithm within clusters which maximizes the lifetime of the wsn. a mathematical model has been developed in support to evaluate how the power of the system can be minimized for the fog node and cloud servers in accordance with the 5g network environment. experiments have been carried out to validate proposed framework and demonstrate the results. keywords: fog computing, civil structural health monitoring (shm), civil structures, wireless sensor networks (wsn), internet of things(iot), sensors, cluster algorithms 1. introduction the internet of things (iot) has rightly been claimed as the next wave of technology after the internet of people. iot has transformed ubiquitous computing with a plethora of applications centered around embedded and sensory systems. [1] iot enables devices to talk to each other, sense what is going on in the environment and adjust the comfort level according to the needs of the users. the major part of the iot network consists of communication, computing and caching components and the even distribution of these resources. this network is believed to expand to 40-50 billion objects by 2020, with each object contributing exorbitant amounts of data. this size d. sinha, k. doshi, m.r. babu 124 copyright ©2018 assa. adv. in systems science and appl. (2018) of data can easily outsmart today’s database management systems. [2] to tackle this big data and integrate the needed resources in a wireless network architecture, cloud computing comes in the picture. cloud computing provides numerous on-demand services such as platforms, softwares and infrastructure. (paas, saas, iaas) [3] while the internet isn’t scalable enough to accommodate iot big data, the cloud does provide virtual storage and processing facilities that are sufficient to iot requirements. however, transferring such high volume of data is expensive and uses significant amount of bandwidth. [2,4] adding to that, the need to transfer all information to the cloud will bring forward high latency rates and delay the entire decision-making procedure. [5] this will severely hamper applications where prompt actions are to be taken by iot devices such as healthcare. furthermore, users often share their personal information through iot objects. this sensed private data needs to be kept secure which poses a great challenge in front of the vast cloud accessible by millions of devices. another challenge faced by iot devices is the great value of power consumption. for example, in smart grid application, an iot device is assigned to each home in a residential area, this smart meter collects user’s electricity usage log, and reports to the control unit periodically (e.g. every 15 minutes [6]). the control unit makes real-time decisions based on this reported data and takes suitable actions. in order to make the most of the information gathered by iot devices, the more precise and frequent the data transferred, the better real-time analysis followed by accurate decision making can be carried out. but to be able to send real-time data regularly to the cloud, it will cost humongous communication resources. so, the problems faced by cloud computing are high latency, big data, power consumption and privacy. to overcome these problems an intermediate architectural concept of “fog computing” was brought into picture by cisco. this concept introduces a node between the iot device and the cloud. this node supports the cloud computing platform in handling a part of the workload locally at the edge of the network instead of transmitting the whole workload to the cloud. [7]. the main factor that differentiates cloud computing from fog computing is the fog's localization and proximity to all of the nodes as compared to the cloud. the fog is an extension of the cloud to the network's edge. thus, making the sensory embedded iot devices closer to the fog than the cloud. this makes it easier and faster to access the local fog than the global cloud. in simpler terms, the fog is nothing but a descended cloud. [8] edge computing mainly corresponds to the edge network used as mobile edge. this concept of computing is also known as fog computing, is gathering attention in order to overcome the problems of cloud computing [9][11]. while cloud computing requires high computational capacity, edge computing requires medium to low computational capacity. unlike cloud servers that are very large in size and centralized, edge computing needs servers which are smaller in size and placed over many locations. while cloud computing is suitable for delay tolerant and computationally intensive applications, edge is used for applications demanding low latency and real-time operation. for the cloud computing to work, the devices need to be connected to the internet at all times but in the case of edge the information is cached in the servers and at particular intervals of time the information is sent to the cloud for storage. it is seen that cloud computing requires complicated deployment planning while in the case of edge computing there is a possibility of ad-hoc deployment with no or minimal planning. [12] 1.1 fog computing – architecture the generic architecture of fog computing comprises of the below 3 layers 1. cloud computing layer, 2. fog cluster layer, 3. wireless terminal layer. the main focus of the architecture is the cloud computing layer. it is the center of the whole network and end responsible for storage of resources. following the cloud computing layer are the fog clusters situated at the 125 an efficient approach to civil structures health monitoring copyright ©2018 assa. adv. in systems science and appl. (2018) edge. the fog clusters communicate with mobile devices at the network's edge. every fog cluster is responsible for a particular region's mobile devices which transmits data to the fog cluster. this is advantageous for mobile devices as they can easily achieve resources through the fog cluster without having to connect with any other mobile devices. this will in turn also decrease the power consumption. additionally, they don't have to store resources and can simply retrieve the wanted resources from the fog cluster as per their time constraints. they also will not face any high latency issues as the fog cluster is present right at the edge. thus, making it easy for them to exchange and transfer data with fewer hops. [13] a fog micro data centre(mdc) also enables context-aware computing. the gateway is that device which collects information from the underlying nodes. in particular scenarios, the gateway is required to perform pre-processingor interoperability-related tasks, the gateway cannot do these tasks on its own. hence, the gateway has to be made better equipped with the capabilities of the fog. this gateway is commonly known as the fog smart gateway (fsg). an fsg is feedback driven and highly context aware. it transfers data only when needed. [8] considering a network with millions of end users, will not only contain numerous iot devices transferring information but will also require the proper computation of the received data. so, just having fog nodes is not sufficient, we need to maintain certain standards alongside these nodes. some new standards need to be in order when it comes to certain domains for wide acceptance of fog such as data management, security and privacy, building of unified fog-cloud platforms etc. [14] 1.2 role of iot gateway the generic architecture discussed earlier may be effective for a single sensory iot device but it doesn’t account for millions and millions of iot devices scattered in different regions transferring big data to the fog and in turn to the cloud. this huge scalability also introduces the persistence problem of security as more and more devices are vulnerable to other devices. additionally, sensors have constrained capabilities when it comes to their networking abilities. adding to this is the managerial aspect, it’ll be very troublesome to monitor the operation of each iot device in the network individually. instead, if there is a single point of access for all iot devices. we will simply have to monitor that point of access in place of the several iot devices. the iot gateway can be this point of access between all iot devices and the fog. this flow has been shown in figure 1. owing to the above-mentioned issues in the generic architecture, a better architecture is suggested that comprises of the iot gateway. the gateway will help in streamlining the transfer of data in a coherent manner. the gateway ensures all data passing through it follows a certain set of rules or protocols that establishes uniformity and also encourages change in protocol for certain data when required while switching between networks. while the gateway is an intermediary between the iot devices and the fog, it shouldn’t be confused with a dumb proxy that simply transfers data from sensors to fog. it also undergoes the important task of pre-processing the data according to the requirement of cloud outlets or data centers. these pre-processing tasks can vary according to the needs of the system from message filtering to aggregation. [15] securing the network is probably the foremost concern today with respect to iot. adding a layer of security can be done relatively easily and effectively with a gateway as compared to an architecture that lacks the gateway. a pki infrastructure is followed to ensure security, wherein each device is assigned a unique identity, that is, a pair of cryptographic keys also referred as digital certificate which makes the communication secure via encryption. this can be a very tedious task to carry out with the absence of an iot gateway. [16] d. sinha, k. doshi, m.r. babu 126 copyright ©2018 assa. adv. in systems science and appl. (2018) figure 1: cycle showcasing the flow of different components of an iot architecture. now, after integration of the gateway-viewing the architecture as a whole, it can be seen as a 4-layer model constituting of the device layer, hub layer, fog layer and the cloud layer as shown in figure 2. the lowest tier i.e. the device layer consists of the iot devices with sensors. the next tierthe hub layer contains the gateway. above the hub layer, lies the fog layer which is formed by the edge and the fog nodes/clusters. the final layer is the cloud layer which as the name suggests has the main cloud. device layer: the lowermost layer of the architecture, which is used by the end users is known as the device layer. it mostly consists of different kinds of iot devices, which are used to collect data and send it over to the next layer. hub layer: this layer is sometimes called the gateway or an intermediate layer between the device and the cloud. this layer not only helps in maintaining protocols over different networks, it also takes care of the security and makes sure that the data reaches the right destination. the gsm module attached to some of the iot device can also be considered as a hub layer. fog layer: this additional layer between the gateway and the cloud has a series of benefits. this layer collects data from its nearby devices and then periodically sends all the data to the cloud for actual analysis and prediction. this not only helps in saving energy of the iot devices but also helps save computational time by computing the result locally. these features have given this layer another name which is "edge nodes", since they are located at the edge of a network. fog with the hub layer makes the network more stable since, the data would follow certain protocols in a network. while, a fog layer without a hub layer with too many devices would make it difficult to cope with it but could be manageable if the number of devices are sizeable. cloud layer: since, this layer is needed for storing data from all the devices and compute the results. it becomes the most crucial layer of all. architecture iot device gatewayfog cloud 127 an efficient approach to civil structures health monitoring copyright ©2018 assa. adv. in systems science and appl. (2018) figure 2: fog computing architecture 1.3 mapping the architecture to the functional diagram of iot the functional diagram of iot consists of 6 different functionalities, which are device, communication, services, management, security and the application. let us map our architecture to these functionalities as depicted in figure 3. the first block, the iot devices fall under the ‘device’ functionality. the gateway block has the principle duty of maintaining protocols while switching networks is responsible for the ‘management’ functionality. the primal duty of the fog is to compute data of a local area, and send the required data to the cloud in an encrypted and secure manner. the fog node comes under the functional category of ‘security’ and ‘communication’ along with some functionality of ‘services’. lastly, the cloud, whose main aim is to store data, compute data gathered globally and assist in proper decision making comes under ‘services’ and ‘application’. figure 3: mapping: iot devices account for device functionality, gateway accounts for the management functionality, fog accounts for the security, services and communications functionality and the cloud accounts for the application functionality. 2. head clustering algorithm let us take up a scenario, where there are many number of iot devices which are concentrated in a particular region. let's say they all respond/receive signals from/to a single fog node. this situation again becomes similar to the one we were facing with the cloud computing itself but in a smaller scale. so, is it possible to reduce these computations too? yes, it is possible. it would be d. sinha, k. doshi, m.r. babu 128 copyright ©2018 assa. adv. in systems science and appl. (2018) to define a set of iot devices as a cluster/region using the pre-defined sequence of conditions and after identifying this region, a cluster head is selected based on a condition. this situation will be even more favorable if these regions that are formed are dynamic in nature based on a fixed parameter or value. so, let us assume these regions are made, and now a cluster head needs to be assigned. a cluster head is an iot device among all iot devices which collects all the data of the region and then transfers it to the nearest fog node. the cluster head is decided based on the battery level and power consumption. the device which appears to be most sound in these two aspects i.e. high battery level and low power consumption is chosen as the cluster head. once, the battery of the cluster head is drained to a level below the desirable percentage, the cluster head of that region is changed to another iot device with a higher battery level and thus saving huge amount of power consumption. but, the only assumption in this theory is that, the fog node is the node which has continuous power supply. a brief outline of the above is shown in figure 4. figure 4: each cluster signifies a separate region. the differently colored mobile device in each cluster is the cluster head which takes data from the remaining mobile devices of the respective cluster and transfers it to the fog node which transfers it to the cloud. 3. structural health monitoring the natural disasters like earthquake, floods and landslides cause tremendous amount of damage to the health of an infrastructure which in-turn puts human life in danger. adding to this is corrosion, fatigue, scour etc. which is the outcome of organic aging of a structure. health of a structure deteriorates with the passage of time and gets catalyzed with the onset of these natural calamities'. [17] even though these detrimental disasters are not in our control, however the extent of damage brought on by these unfortunate happenings can be controlled by us to some extent. this can be done by carefully monitoring the health of structures thereby ensuring its robustness and stability in the wake of a climatic mishap. but that in itself isn't sufficient, this monitoring system in place needs to be fast enough to be allow consequent action to take place. it should be able to qualitatively as well as quantitatively give real time analysis of a structure's health and it's time to live. this purpose is served by a structural health monitoring (shm) model. shm aims to provide, a diagnosis of the materials present, of different parts, and the full assembly of parts constituting the structure as a whole at every moment during the life of a structure. shm has other functionalities too. it also involves the integration of sensors, possibly smart materials, data transmission, computational power, and processing ability inside the structures. it therefore, makes it possible to change or reconsider the design of the structure and the full management of the structure itself and of the structure considered as a part of wider systems. 129 an efficient approach to civil structures health monitoring copyright ©2018 assa. adv. in systems science and appl. (2018) according to [18], shm application can be seen as a four-step procedure. this four-step procedure has been illustrated in figure 6. the first step being detection, which self explanatorily gathers data pertaining to the presence of any damage in the structure. the localization step comes after the detection, where the location of the damage detected in the previous step is estimated. once we know that there is damage and the probable location of the damage, we carry out the assessment method which estimates the extent of this damage. the final step provides information about the remaining lifetime of the structure for maintenance purposes as well as disaster management. another step has been introduced into this process that lies between damage localization and assessment by the authors of [18] and [19]. as per these two publications, a new aspect i.e. the type of damage has been given importance by emphasizing that before estimating the extent of the damage, classification should be done to figure out the type of damage that exists. combining all these detailed steps of shm, it can be executed in the following manner in an orderly fashion :1) detection; 2) localization; 3) classification; 4) assessment; and 5) prediction. [20] shm application can be broken down into three components as mentioned in [19] and illustrated in figure 5. these components are: 1) sensing and gathering data: sensors gather data from structures under changing environmental conditions. this stage also deals with transmitting this data collected from sensors to the base station or server. this part consists of thinking about the type of sensor to deploy, where to deploy them and how many of them to be deployed and also their mode of transmission to the server. 2) data management: once the data reaches the server, it needs to be evaluated and accessed. this careful assessment and data processing comes under this component. at times, the data reaching the server is overwhelming and contains lot of noise and unwanted data. preprocessing methods need to be carried out to extract only the meaningful data out of the vast volumes of incoming data. the main processing deals with distinguishing data to tell whether building is damaged or not. there are a variety of techniques that are used to analyze the sensor network data ranging from machine learning to genetic algorithms. there are a broad number of algorithms that have been put forward but the one which has gained significant appreciation lately is novelty detection based on artificial neural networks [21][23]. 3) data access and retrieval: this component is the final outcome of our shm application. after data analysis and processing, results are examined for appropriate decision making. figure 5: the 3 components of structural health monitoring d. sinha, k. doshi, m.r. babu 130 copyright ©2018 assa. adv. in systems science and appl. (2018) figure 6: the four-step structural health monitoring process starting with detection of any damage on our structure and ending with the prediction phase i.e. the time remaining of a structure. 3.1 implementation of shm: traditional systems while we’ve established the need of shm and its components. the next thing which needs to be considered is how should shm be implemented such that it’s fast, reliable and actually serves the purpose of providing real time data about the health of a structure. let’s take a scenario to get a better understanding. let’s take the example of a building instead of considering any arbitrary structure. traditionally, a wired approach for shm was adopted. however, this traditional model increases latency due to prolonged deployment time, it is also not economical as a heavy cost goes into wire installation. [24] apart from this, the amount of data from numerous structural data sources are very vast which brings forward the idea of bigdata that cannot be handled by traditional database management systems. [20] 3.2 wsns within shm to overcome the challenges posed by the traditional wired approach, wireless sensor networks are now being adopted for shm. wsns have immense potential in the field of shm that is still being explored and bettered. wsns are far ahead of their counter traditional wired systems when it comes to shm mainly due to its simplicity and ease of installation. in wsns, sensors are strategically placed over the surface area of our building in consideration. the sensors are placed such that they are able to capture the building’s response (like vibrations) which are caused by external factors like the wind, earthquakes or dynamic loads. [25] while wsns reduce cost and deployment time in comparison to wired systems but it surfaces certain problems of its own. shm applications have a high sampling frequency (greater than 100s of hz) which leads to a very large amount of data which in turn reaches huge values for each sensor in one round of data elicitation. this makes the entire shm application data intensive. also, with so much data, there is often a good amount of data that is extra and not needed which also gets transmitted. [26] wsns also have the constraint of limited bandwidth, this low bandwidth again fails to handle the tremendous volume of data. for this, data compression techniques might help but they might also increase latency. a wsn network might contain multiple communication hops before the data is actually transmitted, this can delay the entire procedure. but the biggest limitation of wsns is that sensors have a fixed battery life and energy supply. additionally, if the sensor battery drains out, it is problematic to charge them over big structures such as our building. [27] 131 an efficient approach to civil structures health monitoring copyright ©2018 assa. adv. in systems science and appl. (2018) 4. our proposed model to go a step further after wsn in shm, some of these problems can be easily tapped by the internet of things. this will enable powerful data processing beyond the scope of wsn. the introduction of cloud will enhance storage and transmission of the big data. to go even beyond, comes the idea of the latest cloud technology i.e. fog computing. this is where our proposed model comes into play. so far, we’ve seen how shm has been implemented from wired to wsn and also using internet of things and cloud computing but the major challenge it continues to face is of battery consumption. our model proposes to tackle this problem by maximizing the network lifetime. this can be done by integrating the fog head cluster node algorithm discussed in the previous section. let’s showcase the algorithm specific to shm. different regions with buildings to monitor will be separated and identified as clusters. each of these clusters will comprise of few buildings and these buildings will contain sensors deployed on them. broadly speaking, a cluster will be a collection of sensors. each cluster will have its own cluster head. this cluster head will be a sensor from all the sensors of the cluster chosen dynamically depending on its battery level. the sensor having the highest battery level compared to all other sensors in its network will be assigned as the cluster head. initially, any one sensor will be set as the default head. every other sensor in the network will be responsible for acquiring data from the structure and sending this data to the head sensor. once the shm network starts functioning, different sensors will have varied battery consumption levels. this variation will be due to different communication hops to the head node and due to the variable euclidean distance [25] between sensors. besides transmission, coverage and data acquiring also takes up significant amount of energy. [28] with the battery of each sensor changing erratically, there will always exist a difference between the battery level of all the sensors i.e. some sensors may be closer to the head resulting in less power consumption and a higher battery level whereas some sensors might be far away from the head leading to higher power consumption and lesser battery levels. this change in battery levels will also cause change in head of the cluster. furthermore, the head cluster needs to additionally forward all this data incoming ahead for data processing and analysis leading to significant power consumption of the head itself. so, to ensure no rapid change in assignment of head sensor is taking place, the assignment of one head sensor to another will only take place if the difference in battery level is significant enough. once all the data for a cluster is collected by the head sensor. the head sensor is responsible for sending this data ahead to the fog node, the fog will act as a descended cloud where the data analysis and processing will take place leading to the decision making. the data sent over to the fog can be categorized into two categories: first, is the real-time data which detects damage in the building. in such a case, this real-time data in the fog can warn and create an alert to vacate the building and prevent the building from collapsing. in this case, no latency can be afforded, hence the data isn’t sent over to the cloud and dealt with at the fog level itself. the second category is the real-time data which doesn’t detect any immediate damage in the building but can be used to predict the lifetime of the building and can be used for the maintenance of the building. this data can be stored in the cloud instead of the fog node. this data can be checked remotely from a variety of devices not depending on the location of the cluster, this will help in the restoration of buildings spread out at different locations. now for this to implement successfully, lets closely discuss the kind of sensors that’ll make it possible. the network of sensors is known as a sensor network. each sensor in a sensor network notes measurement in accordance to the time-stamp along with the physical measurements like heat, sound, light, pressure, or even motion. these physical measurements are then represented as data and stored into its own storage. later when a query reaches this sensor though the sensor d. sinha, k. doshi, m.r. babu 132 copyright ©2018 assa. adv. in systems science and appl. (2018) network it sends the data, of what is asked in the query to the particular node in the network. the sensor itself doesn't have enough memory space to hold the continuous data it gets. since, these types of sensors get data very frequently. because of that the old data is compressed and most of the time erased to make space for the new data. so, the question arises that if we make one node as the head node and that too dynamic head node where is this data going to be stored? it is possible for each sensor to be connected to an external storage device. so, this solves the problem of storage. secondly, to save power we can turn on the power of this secondary storage device only when that sensor connected to it is made the head node. so, this head node will then send the query to all the other sensors in its cluster who will in turn send the data back to the head node. when the data is arrived from all the other sensors the data is sent to the fog node. the different sensors that are used in this are temperature sensors, strain gauges, accelerometers, anemometers, seismometers, load cells etc. [20]. this proposed mechanism tackles problems being faced by traditional systems and wsns head on. the problem of big data is solved by introducing the fog node and forming clusters. every sensor can handle the data within it’s cluster, hence solving the problem of big data. basically, the scalability of shm is reducing by breaking it down into clusters. data aggregation will happen efficiently with data being categorized at the fog node, which removes any noise and either keeps the data or sends it to the cloud where it’s due processing is done. the head sensor algorithm maximizes the lifetime of the entire wsn network not completely eradicating the problem of power supply but optimizing it to an extent. [24] moreover, the integration of shm, iot, cloud and fog computing will invite ubiquitous services and provide scope for powerful data analysis techniques which can process the data streams beyond the capabilities of previous systems. the iot architecture uses shm data intelligently for smart monitoring and actuation with smart devices. [25] combining all aspects, the final architecture of shm is shown in figure 7. 133 an efficient approach to civil structures health monitoring copyright ©2018 assa. adv. in systems science and appl. (2018) figure 7: the architecture of our shm proposed model algorithm 1 the algorithm for assigning head nodes input: a predetermined located set of sensors, n. a set of buildings in which shm has to be implemented b, a set of fog nodes located in each cluster, f → fi is the corresponding fog node for the cluster ci. output: a global cloud k with complete data of the structures in review from all clusters c1…. cn covering all buildings in b step 1: define the clusters c1…cn depending upon the area and buildings such that each building in b is covered by c and each sensor in n is present in any c step 2: divide n in sets → all sensors in a cluster form one set step 2: for each ci for i= 1 to n do step 3: assign a head sensor h with the maximum energy step 4: if (no of h >1) check euclidian distance of each n in ci with all h choose min dist. (all h) and mark it as real h end if step 5: define a query q asking for sensed data step 6: h sends q to all n in ci step 7: all n in ci send the required data back to h step 8: if (h receives all the data from each n in ci) send the gathered data to fi end if step 9: when (fi receives the data) calculate the damage d if (d is critical) issue alert else send the data to the cloud k end if end when step 10: k calculates the damage and the lifespan of the building. data stored in k is accessible for maintenance purposes end for this algorithm is depicted with the help of flowcharts in figure 8 and figure 9 and the change of the head sensor within a cluster is depicted in figure 10 d. sinha, k. doshi, m.r. babu 134 copyright ©2018 assa. adv. in systems science and appl. (2018) figure 8: the flowchart depicts how the fog decides whether or not to send data to the cloud depending on the emergency of the situation and the time left on hand by processing the data sent to it from the head sensor. figure 9 : data flow of the proposed system model 135 an efficient approach to civil structures health monitoring copyright ©2018 assa. adv. in systems science and appl. (2018) (a) (b) (c) figure 10: (a) (b) and (c) depict how the battery of different sensors are changing over time which in turn is changing the assignment of the head sensor. the red batteries depict low power left while the green ones depict high power left, the remaining are having moderate power left. (a) shows the network sensor in its initial stage where most sensors have high battery and the one having highest is assigned as head sensor, (b) and (c) showcase the same network over time where the battery of sensors drop and the head sensor changes depending on the one with highest power remaining. 5. system model (a) our model comprises of n sensors forming a piconet and communicating with each other via bluetooth le. the head sensor is present from one of these sensors and again uses bluetooth le technology to transmit data to the fog. the fog then performs processing tasks on the data and sends it to the cloud through a 5g heterogenous network. the fog node can take care of delay sensitive data and pass on the rest for the cloud to process. let’s first pay close attention to the 5g network between fog node and cloud server 1. power consumption of the fog node : our model comprises of multiple fog nodes,for a single fog node α,the power consumption covers the computation of data at the fog node as well as the transmission of data from the fog to cloud via 5g network. there are two basic properties that the power consumption function of the fog should adher to as mentioned in [29] : a) the computational power consumption is directly propotional to the computation amount b) marginal power consumption is increasing for every fog node. so,the power consumption function can be written as a function of the computation amount of fog node α i.e. x cbxaxpfog   2 , (1) where a > 0 and b,c ≥ 0 are pre-determined parameters. 2. transmission rate from fog to cloud : d. sinha, k. doshi, m.r. babu 136 copyright ©2018 assa. adv. in systems science and appl. (2018) this transmission is done via 5g network communication. considering fog node α,it can handle the data in 2 ways. it can either do computation at the fog node or it can pass on the data to the cloud server depending on the level of latency and criticality of the situation and allow the cloud node to take care of the data. let the data to be dealt with by fog node α be seen as a task t . we will use 1,, kjia to denote that fog i chooses to accomplish the task t by transmitting to cloud server j through the channel k. on the other hand, kjia ,, = 0 denotes that the fog does not pass on data to cloud and does computation on it’s own. j belongs to j ={1,2,3…j} which are the number of cloud servers. and i belongs to i={1,2,3…..i} which are the number of fog nodes. k belongs to k={1,2,…k}. both the fog nodes and cloud servers operate in the same frequency band for the efficient usage of the spectrum. this spectrum is categorized into k channels of identical bandwidth w. these different channels are denoted by k. [30] if kjia ,, = 1 the transmission rate from fog node α can be formulated through channel k :  2 ,2, /1log   kk igpwr p is the power of the fog node. g is the channel gain between the fog node and the cloud. ki , denotes the interference at the cloud on channel k, due to other fog nodes trying to transmit data to cloud. σ2 is the background noise power. the total transmission rate from a single fog node to it’s respective cloud server can be calculated as    k k kikii rar 1 ,,2, 3. power consumption of cloud server for the task t , d is the volume or size of the input data which needs to be computed. let c be the computing ability required for accomplishing this task. unit of measurement for the computing ability is the number of cpu cycles. the total power consumption by cloud can be formulated by energy e = r i crdpb  / (2) where r is the cost of energy required by the cloud for a single cpu cycle. the cloud will always have higher computation capacity as compared to the fog,which is r . the no. of channels that are utilized by fog node α to transmit data to the cloud is showcased using b    k k kiab 1 ,2, 5.1. constraints latency constraints are not specified as they vary depending on the data and the criticality of the damage of the civil structure. other constraints which have been taken into account are mentioned below: 5.1.1 fog-cloud workload balance both the fog nodes as well as the cloud servers have computational constraints due to which the workload or data that needs to processed should be balanced between the two. thus,not draining any one completely. for this the size of cluster needs to be managed i.e. the number of sensors 137 an efficient approach to civil structures health monitoring copyright ©2018 assa. adv. in systems science and appl. (2018) attached to a fog node should be in check. even if all the data that the fog node is receiving turns out to be critical and latency sensitive then all the data will have to be computed by the fog. the fog should be able to handle such a situation. the other extreme could be when all of the data needs to be processed by the cloud alone. in that case,the cloud should be well euipped to handle all the data. the workload allotted to the fog and cloud respectively is shown by x and y let l denote the total sensor input from all the sensors. the incoming data or traffic from all sensors to the fog node α can be denoted by t thus, yx t should hold good 5.1.2. fog area constraint every fog decive has a limit on its processing ability. clusters should be formed in such a way that the fog node can handle the incoming data and workload from all sensors belonging to that cluster. for any fog device α, maxp denotes the maximum computational capacity it carries out. also,the workload x on a fog node should not be more than the rate of incoming traffic t [25] therefore,we can say 0 ≤ x ≤ min{ maxp , t } 5.1.3. single channel constraint a single channel can be allocated to atmost one fog node    n i kia 1 ,2, 1 , i ∈n,j ∈{2,3} 5.2. problem : it is important and desirable to minimize the aggregated power consumption of all fog nodes and cloud servers. the complete power consumption of the 5g networked system can be represented as the combined power consumption of the fog node and the cloud server. in order to minimize the power consumption of the system,the individual power consumption of the fog and cloud needs to be minimized.      ni mj cloud j fog i total ppp 5.3. solution : the equation of power consumption of a fog node as in equation (1) is a quadratic function which is monotonically increasing and a strictly convex function. [25] this function can be plotted to find the relation between the power consumption with the computation amount x for a fog node α. this relation is plotted in figure 11,the minima for power consumption will be attained when abx 2/ as showcased in the graph. this is when power consumption of the fog node will be minimum. d. sinha, k. doshi, m.r. babu 138 copyright ©2018 assa. adv. in systems science and appl. (2018) figure 11: graph depicted the relationship between the power consumption of the fog and the computation amount of the same fog node. the minima is obtained at a point where the x coordinate i.e. the computation amount of the fog node= -b/2a where a and b are from equation (1) the second part is to minimize the power consumption of the cloud server as well. this is denoted by equation (2) , in order to do so reduction followed by applying eeco scheme illustrated in [30] which will get the minimum value. 6. system model (b) in our proposed model, the sensors are divided into clusters and each cluster has one fog node. each cluster has a head node and it dynamically keeps on changing based on the criteria that if the battery level of that sensor decreases by a significant margin. these sensors transmit data through bluetooth and so it is really important to predict the remaining battery life of the sensor to be able to assign the head node. here each cluster is known as "piconet" where the head node is the "master" and the other sensors are the "slaves". the following mathematical model helps in predicting the remaining battery life of the sensors. to estimate the remaining battery life in the sensors we have taken inspiration from the method used in [31] where cc2540 bluetooth development kit is used. each sensor goes through the following stages in a single cycle of data flow: stage 0: initial sleep mode-the slave component is in the sleep mode until the master sends a request to send the data stage 1: wake up mode -whenever the master sends the request for transmitting the required data there is a spike in current followed by slight decrease in the flow of current. this marks that the sensor is ready to send some data. stage 2: preprocessing mode-soon after the request is received by the slave from its master, the storage space for the data read by the sensor is made available and the radio is prepared to store the data in form of stack in the storage device and also to transmit the data to the master. stage 3: receiving mode-the radio receives the data from the sensor and stores it in the available space made during pre-processing. 139 an efficient approach to civil structures health monitoring copyright ©2018 assa. adv. in systems science and appl. (2018) stage 4: receiving mode to transmitting mode -the radio stops receiving data from the sensor, the data packet is made ready to be transmitted and the radio gets ready to transmit the data. stage 5: transmitting mode-the radio transmits the data packet to the master stage 6: post transmitting mode-sleep mode protocol is put into effect and the countdown is initiated stage 7: final sleep mode-the slave again goes back to sleep all the sensors would be in a connected state with a 1 second connection interval and zero slave latency. this means that each sensor would be connected to the master for an interval of 1 second and during that period of time it will undergo the above-mentioned stages. here, stage 0 and stage 7 would be known as the sleep modes of the cycle while stage 1-6 would be considered the awake modes of the cycle. this classification is necessary because maximum energy would get utilized when the sensors would be in awake mode. as the sensor has to send different amount of data and in different quantities in each cycle so, there are possibilities that the sensors will remain in awake mode for different interval of time. the cases of different awake time occur due to the possibility of sensor taking more time to read data, process data, transmit data, or form data packets etc. here is the distribution of amount of time spent on each stage when the awake mode interval was the shortest (2.4ms) and when the awake mode was the longest (2.8ms). table 1: time distribution in shortest awake time time [μs] current [ma] state 1 400 6.0 state 2 298 7.8 state 3 369 18.4 state 4 103 17.5 state 5 109 18.4 state 6 1177 7.8 table 2: time distribution in longest awake time time [μs] current [ma] state 1 400 6.0 state 2 329 7.8 state 3 393 18.4 state 4 116 17.5 state 5 150 18.4 state 6 1393 7.8 secondly, during the sleep mode of the cycle there will be 0.0009ma current flowing through the circuit. moreover, we assume that on an average 0.0001ma of current would be lost in the printed circuit board (pcb) due to unknown resistance and thus on an average around 0.001ma of current is used while in sleep mode. let this current be known as sleepi . the following steps would be necessary to estimate the amount of battery level in the sensor: d. sinha, k. doshi, m.r. babu 140 copyright ©2018 assa. adv. in systems science and appl. (2018) step 1: find the average current used while in awake mode. let us denote this current as avgawakei  . from above tables, we know that each state in the awake mode has different time period and different current so let us denote the time taken to complete state 1 as 1t , for state 2 as 2t and so on till state 6. similarly, let us denote the amount of current generated during state 1 as 1i , state 2 as 2i and so on until state 6. so, the total awake time awaket will be: awaket = 621 ..... ttt  now, let us find avgawakei  which will be given as: avgawakei  = awaketititit /)]*(.....)*()*[( 662211  so, avgawakei  for table i would be: 1awaket = (400 μs) + (292 μs) + (369 μs) + (103 μs) + (109μs) + (1177 μs) = (2456 μs)  1avgawakei [ (400 μs) *(6 ma) + (292 μs) *(7.8 ma) + (369 μs) *(18.4 ma) + (103μs) * (17.5 ma) + (109μs) *(18.4ma) + (1177 μs) *(7.8 ma)]/ (2456 μs) = 9.9576 ma the average current consumption during a single connection event with the shortest slot, a wake time of 2.456ms, is calculated to be approximately 9.9576 ma. similarly, avgawakei  for table ii would be: 2awaket = (400 μs) + (329 μs) + (393 μs) + (116 μs) + (150μs) + (1393 μs) = (2781 μs)  2avgawakei [ (400 μs) *(6 ma) + (329 μs) *(7.8 ma) + (393μs) *(18.4 ma) + (116 μs) *(17.5 ma) + (150μs) *(18.4ma) + (1393 μs) *(7.8 ma)]/ (2781 μs) = 10.0154 ma the average current consumption during a single connection event with the longest slot, a wake time of 2.781ms, is calculated to be approximately 10.0154 ma. step 2: now, to find the average current over the entire connection interval average current ( avgi ) is the amount of current dissipated in the entire connection interval ( ervaltint ). as discussed earlier ervaltint = 1000ms, and average current is given by: avgi = ( ervaltint awaket ) * sleepi + awaket * avgawakei  average current ( 1avgi ) for the values in table i would be: 1avgi = ( ervaltint 1awaket ) * sleepi + 1awaket * 1avgawakei  = = [ (1000 ms – 2.456 ms) *(0.001 ma) + (2.456 ms) *(9.9576ma)] / (1000 ms) = 0.0254 ma 141 an efficient approach to civil structures health monitoring copyright ©2018 assa. adv. in systems science and appl. (2018) average current ( 2avgi )for the values in table ii would be: 2avgi = ( ervaltint 2awaket ) * sleepi + 2awaket * 2avgawakei  = = [ (1000 ms – 2.781 ms) *(0.001 ma) + (2.781 ms) *(10.0154 ma)] / (1000 ms) = 0.0279 ma to find the average current of the sensor over any period of any awake mode interval, the number of times the shortest wake time and the number of times the longest wake time occurred would be noted and weighted average method would be used to find avgi of the sensor. let us say that the shortest wake time occurred for 45% of the time ( sp ) and the longest wake time for the other 55% ( ip ). so, by calculating the weighted average based on the above values we get: ]*)[(]*)[( 21 avgiavgsavg ipipi  therefore, taking values from above we get ]0279.0*)55.0[(]0254.0*)45.0[( mamaiavg  = 0.0268ma hence the average current consumption while the device is in a connected state is approximately 0.027 ma (27 μa). step 3: the final step would be to find out the battery life ( lifeb ) of the sensor. battery life of a sensor can be known by dividing the available battery or the given battery capacity ( capacityb ) with average current consumption ( avgi ). lifeb = capacityb / avgi the available battery or the battery capacity of the sensor was 260mah. therefore, lifeb = (260mah) / (0.027 ma) = 9630 hrs (approximately) thus, it can be confirmed that the battery of the sensor will last approximately another 9630 hrs which is equivalent to approximately 401 days, while continuously running in a connected environment with a 1 second connection interval and zero slave latency. from the above steps, it can be concluded that at any given moment, the available battery life of any sensor can be calculated. as more and more time slots are noted, the available battery life becomes more accurate. thus, this method helps in choosing the head node in the cluster dynamically. 7. conclusion this paper has introduced the concept of fog computing within structural health monitoring. the entire system model consists of a piconet among sensors. these sensors are divided into clusters where each cluster follows a head sensor clustering algorithm which enables lesser power consumption as compared to normal transmission by all the sensors of the cluster to the fog. the sensor with the maximum battery remaining is assigned as the head sensor. to be able to assign d. sinha, k. doshi, m.r. babu 142 copyright ©2018 assa. adv. in systems science and appl. (2018) the head sensor, we need to be able to find the battery remaining in a sensor at any given point while it is undergoing bluetooth transmission, this has been validated in system model (b). once the head sensor transmits data to the respective fog node of the cluster, the criticality of the data received at the fog is checked. if the data indicates immense structural damage and less time on hand. the fog does the decision-making part and issues an alert to vacate the building else the data is sent to the cloud via 5g network where it is processed and stored for structural maintenance. this transmission between the fog and node needs to be done such that minimum power consumption of the two takes place which is taken into consideration in our system model (a). references [1] ejaz, w., anpalagan, a., imran, m. a., jo, m., naeem, m., qaisar, s. b., & wang, w. (2016). internet of things (iot) in 5g wireless communications. ieee access, 4, 10310-10314. 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[32] icon credit in figures 2,10 and 5: https://thenounproject.com advsystsciappl 2019; 04; 1-13 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/778 advanced solow model as a tool for coordination of interests of spatial economic systems’ development (on the materials of the south russian macro-region) t.y.anopchenko1,o.i. gorbaneva1, e.i. lazareva1, a.d. murzin1, g.a. ougolnitsky*1 1) southern federal university, rostov-on-don, russia received august 12, 2019; revised september 24, 2019; published december 31, 2019 abstract: the paper presents the results of the initial stage of the study of the dynamic socioecological-economic model of the synergetic development of individual entities within the macroregion, aimed at coordinating the common and private interests of each region. the maximization of specific consumption is proposed as a target parameter. in relation to private interests for the region, in relation to common interests for the macro-region. the model provides the possibility of using the available resources of each region both in the tasks of its own development, and for the general development goals of other regions of the macro-region. the model is studied on the materials of the subjects of the southern federal district. the study came to several debatable conclusions. in particular, calculations show that in the current conditions it is not profitable for any region to develop its own production sector, the optimal strategy for each of them is only to increase consumption in the hope of production activity in neighboring regions (a free-rider problem). in view of the selection of this rational strategy by all regions at the same time, general degradation of the production sphere and stagnation of the regional economy are forecasted. at the same time, it was revealed that a further reduction in production becomes unprofitable for almost all lagging regions (republics of the southern federal district), while inaction remains profitable for leading regions (regions of the southern federal district). the results are debatable and require further discussion. keywords: modeling of the coordination of interests, management of regional development, interregional interaction, production activity of the region, macro-region of the southern federal district. 1. introduction a methodology for the socio-ecological and economic development of the region modeling based on the theory of optimal control was proposed in the works by gurman et al. [15]. analytical methods were used along with simulation modeling in the tasks of regional development research. in the monograph [7], the well-known solow macroeconomic model [14] is modified taking into account the spatial aspect and environmental pollution. a detailed review of models and decision support systems in the field of sustainable development management is presented in [16], models of economic growth are discussed in [5]. an important class of dynamic models of territories’ economic development is formed by the computable general equilibrium (cge) models [6,8,18]. these models have a solid microeconomic justification and ensure that the sectoral structure of the economy is fully taken into account, as well as the impact of changes in some economic sectors on others. the main disadvantage of cge models is that they are economic-mathematical and do not adequately *corresponding author: ougoln@mail.ru 2 t.y. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2019 assa adv. in systems science and appl. (2019) describe environmental and social aspects, as well as effects associated with dynamics and uncertainty; their identification is also difficult. in the works [8, 9, 11] the authors of this paper introduced an original integrated approach to the modeling of the processes of accumulation and productive use of tangible and intangible assets – resources for territorial sustainable development as well as to the modeling of coordination processes of social and private interests for resource allocation in hierarchical control systems. an implementation of this approach allowed to establish the system compatibility conditions for different control problems setups; to justify a control strategy design methodology for balancing the social and economic interests of national, regional (local) and global economic agents in the reproduction and utilization of welfare resources for the sustainable development of a regional system; to propose system coordination mechanisms and study their properties; to initiate application of the coordination models of social and private interests to real regional management problems, more specifically, to the design of administrative and economic coordination mechanisms (engines) for the interests of territorial subjects [2, 4, 8, 9, 10, 12, 17]. the multidimensional experience of applying various modifications of the solow model to solve economic problems demonstrates both efficiency and certain difficulties associated with the reflection of conditions and parametric characteristics adequate to the ongoing transformations. the “weakness” of the toolkit for reconciling the interests of spatial economic systems of various hierarchy levels is among the problems of the analytical support of the process of modern economic growth [1]. the proposed paper contains a detailed description of the approach to the process of coordinating the interests of these systems’ modeling on the basis of development, identification and software implementation of the advanced solow model. the solow model is supplemented by the optimization (game-theoretic) block as well as it is modified taking into account the spatial factor. the list of key assumptions of the model includes the endogenous nature of the model; threelevel structuring of the national spatial and economic system “subjects federal districts russian federation”; lack of parameters characterizing significant external financial flows (federal subsidies); insufficiently complete parametric characteristics of the comparative advantages of regional systems. despite these assumptions, testing the proposed model on a representative information base of the southern federal district showed its applicability in the management system of modern spatial economic systems’ sustainable development. the improvement of the model by leveling of simplifying assumptions will make it possible to use it as a tool for designing a strategy for reconciling the interests of spatial economic systems in the process of implementation of various types of territorial development projects. thus, the results of the initial stage of research on the application of models for reconciling public and private interests in managing regional development, where agents are not separate municipalities of one subject, but different regions, will be presented below. at this stage, the adequacy of the model in the constraints of a self-contained system is investigated and the highest level of the management hierarchy (federal center) is not considered. as an assumption, we study the optimal behavior of the regions in the macro-region in order to increase both their own specific consumption (living standards) and increase the specific consumption of the entire region. to achieve this goal, regions can spend their funds both for their development and the development of neighboring regions. the further structure of the article is as follows. section 2 describes a mathematical model of the interaction of regions within the macro-region. section 3 substantiates in detail the process of model parameters for the regions of the southern federal district identifying. section 4 presents the procedure for finding a nash equilibrium for the regions of the southern federal district in the region's investment in production. section 5 summarizes the findings of the study. dynamic model of general coordination and private interests 3 copyright ©2019 assa adv. in systems science and appl. (2019) 2. mathematical model regions in a macro-region are considered. the system that makes up the macro-region is considered to be closed. interaction with the external world for the system occurs due to the fact that the regions can spend their money on external goals for the macro-region. but this is reflected on the regional level only because the region theoretically does not manage all its means. feedback, or receiving funds by the macro-region from the outside, does not occur. the model of a regional social-ecological-economic system has the form: 𝑌𝑖(𝑡) = 𝐴𝑖(𝑡)𝐾𝑖 𝛼𝑖(𝑡)(𝑅𝑖𝐿𝑖)1−𝛼𝑖(𝑡); (2.1) 𝐼𝑖(𝑡) = 𝑠𝑖(𝑡)𝑌𝑖(𝑡); (2.2) 𝐶𝑖(𝑡) = [1 − 𝑠𝑖(𝑡)]𝑌𝑖(𝑡); (2.3) 𝑅𝑖(𝑡 + 1) = (1 + 𝜂𝑖)𝑅𝑖(𝑡); (2.4) 𝐾𝑖(𝑡 + 1) = (1 − 𝜇𝑖)𝐾𝑖(𝑡) + ∑ 𝑘𝑗𝑖(𝑡)𝐼𝑗(𝑡)𝑛 𝑗=1 ; (2.5) 𝐿𝑖(𝑡 + 1) = (1 + 𝑏𝑖 − 𝑚𝑖)𝐿𝑖(𝑡); (2.6) 𝑃𝑖 𝑎(𝑡) = [1 − 𝑐𝑖 𝑎𝑣𝑖 𝑎(𝑡)𝐼𝑖(𝑡)][𝐵𝐾𝑖 𝑎 𝐾𝑖(𝑡) + 𝐵𝐿𝑖 𝑎 𝐿𝑖(𝑡)]; (2.7) 𝑃𝑖 𝑤(𝑡) = [1 − 𝑐𝑖 𝑤𝑣𝑖 𝑤(𝑡)𝐼𝑖(𝑡)][𝐵𝐾𝑖 𝑤 𝐾𝑖(𝑡) + 𝐵𝐿𝑖 𝑤𝐿𝑖(𝑡)]; (2.8) 𝐾𝑖(0) = 𝐾𝑖 0; 𝐿𝑖(0) = 𝐿𝑖 0; 𝑅𝑖(0) = 𝑅𝑖 0; (2.9) ∑ 𝑘𝑖𝑗(𝑡)𝑛 𝑗=0 + 𝑣𝑖 𝑎(𝑡) + 𝑣𝑖 𝑤(𝑡) = 1; 0 ≤ 𝑠𝑖(𝑡) ≤ 1; 𝑣𝑖 𝑎(𝑡) ≥ 0; (2.10) 𝑣𝑖 𝑤(𝑡) ≥ 0; 𝑘𝑖𝑗 ≥ 0; 𝑖 = 1, … , 𝑛; 𝑗 = 0, … , 𝑛; 𝑡 = 0,1,2 … , 𝑇. (2.11) index i denotes a region within a macro-region. the model has discrete time t = 0, 1, 2, ... with a step of one year. the model also includes other variables and parameters as follows. 𝑌𝑖(𝑡) as the final output of agent i in year t (in financial terms); 𝐾𝑖(𝑡) as the agent’s basic production assets (capital) in year t; 𝐿𝑖(𝑡) as the agent’s labor resources in year t; 𝑅𝑖(𝑡) as the efficiency of the agent’s labor resources in year t; 𝐴𝑖(𝑡) as the influence function of the agent’s innovative activity on the final output in year t; 𝛼𝑖 as the parameter of the agent’s cobb-douglas production function; 𝐼𝑖(𝑡) as the agent’s production investments in year t; 𝐶𝑖(𝑡) as the agent’s nonproduction consumption in year t; 𝑠𝑖(𝑡) as the share of the agent’s production investments in its final output in year t; 𝜂𝑖 as the efficiency growth parameter of the agent’s labor resources; 𝜇𝑖 as the depreciation factor of the agent’s basic assets; 𝑘𝑖𝑗(𝑡) as the share of the investments of agent i in the activity of agent j (the cooperation coefficient of these agents); here index j=0 describes an external agent for the whole system (for example, the federal state); 𝑏𝑖, 𝑚𝑖 as the reproduction and retirement coefficients of the agent’s labor resources, respectively; 𝑃𝑖 𝑎(𝑡), 𝑃𝑖 𝑤(𝑡) as the agent’s pollutant emissions into air and water in year t, respectively; 𝑣𝑖 𝑎(𝑡), 𝑣𝑖 𝑤(𝑡)as the agent’s allocations to prevent air and water pollution in year t, respectively; 𝑐𝑖 𝑎, 𝑐𝑖 𝑤as the efficiency coefficients of environmental allocations; 𝐵𝐾𝑖 𝑎 , 𝐵𝐾𝑖 𝑤 as the specific rates of industrial pollution into air and water, respectively; 𝐵𝐿𝑖 𝑎 , 𝐵𝐿𝑖 𝑤 as the specific rates of human pollution (labor resources) into air and water, respectively; 𝐾𝑖 0, 𝑅𝑖 0, 𝐿𝑖 0as given initial values of the model variables. therefore, the agent’s state vector is: 4 t.y. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2019 assa adv. in systems science and appl. (2019) 𝑋𝑖(𝑡) = (𝑌𝑖(𝑡), 𝐼𝑖(𝑡), 𝐶𝑖(𝑡), 𝐾𝑖(𝑡), 𝐿𝑖(𝑡), 𝑅𝑖(𝑡), 𝑃𝑖 𝑎(𝑡), 𝑃𝑖 𝑤(𝑡)), (2.12) the control vector is 𝑈𝑖(𝑡) = (𝑠𝑖(𝑡), 𝑣𝑖 𝑎(𝑡), 𝑣𝑖 𝑤(𝑡), {𝑘𝑖𝑗(𝑡)}𝑗=0 𝑛 ), (2.13) and the parameter vector is 𝑍𝑖 = (𝛼𝑖, 𝜂𝑖 , 𝑏𝑖, 𝑚𝑖 , 𝜇𝑖 , 𝑐𝑖 𝑎, 𝑐𝑖 𝑤, 𝐵𝐾𝑖 𝑎 , 𝐵𝐾𝑖 𝑤 , 𝐵𝐿𝑖 𝑎 , 𝐵𝐿𝑖 𝑤). (2.14) the following quantities were taken as model parameters: 𝑌𝑖(𝑡) − gross regional product (grp); 𝐾𝑖(𝑡) − fixed assets (fa) in the economy; 𝐿𝑖(𝑡) − employed population (ep); 𝑅𝑖(𝑡) − gross output (ratio of manufactured products to labor costs grp / ep); 𝛼𝑖 − fixed assets elasticity; 𝜂𝑖 − increase (dynamics) of production; 𝜇𝑖 − wear of fa, i.e. share of the value transferred to the manufactured product (grp); 𝑏𝑖, 𝑚𝑖 −increase and decrease in the employed population, respectively; 𝑃𝑖 𝑎(𝑡), 𝑃𝑖 𝑤(𝑡) −agent emissions of pollutants into the atmosphere and water, respectively, in year t; 𝑣𝑖 𝑎(𝑡) + 𝑣𝑖 𝑤(𝑡) − the difference between the total expenditures of the region and the sum of national, national-economic, housing and communal and socio-cultural expenses (consolidated budget); 𝑐𝑖 𝑎, 𝑐𝑖 𝑤 − the share of neutralization of air pollutants, the share of reused resources in the total volume of water use, respectively, is used for calculating. cross-border interaction of municipalities is described using control variables 𝑘𝑖𝑗(𝑡) . parameter 𝜂𝑖 characterizes labor productivity; 𝜇𝑖 − resource-saving technologies; 𝑐𝑖 𝑎, 𝑐𝑖 𝑤, 𝐵𝐾𝑖 𝑎 , 𝐵𝐾𝑖 𝑤 − environmental technology; 𝑏𝑖 − population policy; 𝑚𝑖 − healthcare innovation; 𝐵𝐿𝑖 𝑎 , 𝐵𝐿𝑖 𝑤 − environmental awareness. it is natural to define the agent optimality criterion in model (2.1)-(2.11) as a function of a combination of general and private interests: 𝐽𝑖 = ∑ 𝑒−𝜌𝑡[𝑐𝑖(𝑡) + 𝑟𝑖(𝑡)𝑐(𝑡)]𝑇 𝑡=1 , (2.15) where 𝑐(𝑡) = 𝐶𝑖(𝑡) 𝐿𝑖(𝑡) is current specific consumption of the agent, reflecting the private interests of the region; 𝑐(𝑡) = ∑ 𝐶𝑖(𝑡𝑛 𝑖=1 ) ∑ 𝐿𝑖(𝑡)𝑛 𝑖=1 is specific consumption of the macroregion, which reflects the common interests of the regions in the macroregion; 𝜌 is a discount rate; 𝑟𝑖(𝑡) reflects the region’s interest in increasing the specific consumption of the macro-region (the specific gravity of the region in total consumption is used as this parameter). the agent’s interest in maximizing not only domestic specific consumption, but also all neighboring regions, and, consequently, the macro-region as a whole, means the desire to improve the general standard of living and well-being of residents. accordingly, the smaller the difference in the consumption parameters of agents, the better the social climate of neighboring regions, the lower the level of migration of unskilled labor, the more favorable the overall economic dynamics due to the matching cost of resources on the market, which determines the costs of implementing joint projects and development programs of the macroregion. practical testing of the model was carried out on the basis of cross-border interaction between the subjects of the southern federal district (sfd) of the russian federation, which include the rostov region (the index is assigned i = 1), volgograd region (i = 2), krasnodar region dynamic model of general coordination and private interests 5 copyright ©2019 assa adv. in systems science and appl. (2019) (i = 3), republic of adygea (i = 4), astrakhan region (i = 5), republic of kalmykia (i = 6), republic of crimea (i = 7). simulation scenarios in model (2.1)-(2.11) include defining trajectories of control variables from vector (2.13). to study the cross-border interactions of regions within the southern federal district, it is necessary to identify the parameters of the zi vectors for each region, and also to create a program for calculating the main indicators, specifically 𝑌𝑖 , 𝐶𝑖, 𝐼𝑖 , 𝐾𝑖, 𝐿𝑖 , 𝑅𝑖for the time period from 2017 to 2022. 3. parameter identification the model parameters 𝑍𝑖 = (𝛼𝑖, 𝜂𝑖 , 𝑏𝑖, 𝑚𝑖 , 𝜇𝑖, 𝑐𝑖 𝑎, 𝑐𝑖 𝑤, 𝐵𝐾𝑖 𝑎 , 𝐵𝐾𝑖 𝑤 , 𝐵𝐿𝑖 𝑎 , 𝐵𝐿𝑖 𝑤) and the ai(t) coefficient were identified according to the data of the federal state statistics service of the russian federation for 2005, 2010, 2015-2017 [19]. these time intervals are determined on the basis of criteria for stable positive dynamics of macroeconomic growth and in order to exclude the extreme impact of global pre-crisis economic trends in 2008 and the systemic consequences of the socio-political russian crisis of 2013. the model parameters for each agent are distinctive characteristics that show the level of development of individual areas of each region as part of the macro-region. they are formed in retrospective dynamics and reflect the historical factors of the sectoral structure of the economy. 1) the parameter 𝛼𝑖 of the cobb-douglas production function is not directly reflected in the statistical data. a formula to determine the elasticity of output yi(t) by fixed asset was used to find 𝛼𝑖: α𝑖 = 𝐾 𝑌 ∙ 𝜕𝑌 𝜕𝐾 . (3.1) calculations according to the formula (3.1) were made on the basis of information characterizing in dynamics the values of the volume of fixed assets ki(t) and grp of each region yi(t). after determining the elasticity coefficients α𝑖 the arithmetic mean values of these coefficients are calculated for each of the studied regions (table 3.1). table 3.1. arithmetic mean values of elasticity coefficients of output yi(t) by volume of fa in the region’s economy region arithmetic mean values of elasticity coefficients value rostov region α1 0,214360 volgograd region α2 0,107626 krasnodar region α3 0,144864 republic of adygea α4 0,236457 astrakhan region α5 0,078457 republic of kalmykia α6 0,145812 republic of crimea α7 0,150498 2) to calculate the difference between the reproduction rate and the labor retirement rate (𝑏𝑖 − 𝑚𝑖), we used the data of natural growth by region for the selected periods, from which the arithmetic average was formed. as a result, the following parameter values are obtained (table 3.2). 6 t.y. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2019 assa adv. in systems science and appl. (2019) table 3.2. parameter values (bimi) for the regions of the southern federal district region parameter (bimi) value rostov region 𝑏1 − 𝑚1 -0,0005 volgograd region 𝑏2 − 𝑚2 -0,01775 krasnodar region 𝑏3 − 𝑚3 0,00575 republic of adygea 𝑏4 − 𝑚4 -0,008 astrakhan region 𝑏5 − 𝑚5 0,00375 republic of kalmykia 𝑏6 − 𝑚6 -0,00375 republic of crimea 𝑏7 − 𝑚7 -0,007 3) to determine the parameter of labor efficiency 𝜂𝑖 dynamics we use data on labor productivity ri(t) for the corresponding periods, which is determined by the ratio of the gross product value yi(t) and the size of the able-bodied population li(t). then the annual growth is calculated. to determine the growth in the five-year interval 2005-2010 and 2010-2015 we use the formula η 𝑖 = √ 𝑅𝑖(𝑡+5) 𝑅𝑖(𝑡) 5 -1, and at further annual intervals η 𝑖 = 𝑅𝑖(𝑡+1) 𝑅𝑖(𝑡) -1. further, all the obtained parameters are averaged over the period (table 3.3). table 3.3.values of the parameters of labor efficiency for the regions of the southern federal district dynamics region the parameters of the labor efficiency dynamics value rostov region η 1 0,009251 volgograd region η 2 -0,017337 krasnodar region η 3 0,024048 republic of adygea η 4 0,001221 astrakhan region η 5 0,027663 republic of kalmykia η 6 0,002532 republic of crimea η 7 0,017375 4) depreciation ratio of fixed assets 𝜇𝑖 is taken from the statistics in direct form for each year, followed by averaging (table 3.4). table 3.4.the values of the coefficient of depreciation of fixed assets for the regions of the southern federal district region depreciation ratio of fixed assets value rostov region μ 1 0,4064 volgograd region μ 2 0,4064 krasnodar region μ 3 0,511 republic of adygea μ 4 0,511 astrakhan region μ 5 0,4064 republic of kalmykia μ 6 0,511 republic of crimea μ 7 0,711667 5) specific emissions of pollutants from industrial activities into the atmosphere 𝐵𝐾𝑖 𝛼 and water 𝐵𝐾𝑖 𝑤 are the parameters of environmental technology that are regulated by the state. specific dynamic model of general coordination and private interests 7 copyright ©2019 assa adv. in systems science and appl. (2019) emissions of pollutants 𝐵𝐿𝑖 𝛼 , 𝐵𝐿𝑖 𝑤 from human sources of activity are not controlled. to determine this data, we use the values of the region’s discharge of pollutants into the atmosphere and water 𝑃𝑖 𝑎(𝑡), 𝑃𝑖 𝑤(𝑡) for the indicated time periods, as well as the percentage of pollution of air and water, respectively, as a result of industrial activity and life population. we denote them by prka i(t), prla i(t), prkw i(t) and prlw i(t), where prka i(t) + prla i(t) = 1 and prkw i(t) + prlw i(t) = 1. all these parameters are determined on the basis of the data of the federal state statistics service of the russian federation for 2005, 2010, 2015-2017 [19]. the previously identified data on the volume of fixed assets ki(t) and the able-bodied population li(t) were also used for calculations. the desired values are calculated as follows: 𝐵𝐾𝑖 𝛼 = 𝑃𝑖 𝑎(𝑡)∙ prka𝑖(t) 100 𝐾𝑖(𝑡) ,𝐵𝐿𝑖 𝛼 = 𝑃𝑖 𝑎(𝑡)∙ prla𝑖(t) 100 𝐿𝑖(𝑡) ,𝐵𝐾𝑖 𝑤 = 𝑃𝑖 𝑤(𝑡)∙ prkw𝑖(t) 100 𝐾𝑖(𝑡) ,𝐵𝐿𝑖 𝑤 = 𝑃𝑖 𝑤(𝑡)∙ prlw𝑖(t) 100 𝐿𝑖(𝑡) . further, each parameter is reduced to the arithmetic mean value. as a result, the following values were obtained for the regions of the southern federal district (table 3.5). table 3.5.the values of environmental technology parameters for the regions of the southern federal district region environmental technology parameters value rostov region 𝐵𝐾1 𝛼 𝐵𝐾1 𝑤 𝐵𝐿1 𝛼 𝐵𝐿1 𝑤 8,34e-05 5,365e-05 0,0232252 0,0756389 volgograd region 𝐵𝐾2 𝛼 𝐵𝐾2 𝑤 𝐵𝐿2 𝛼 𝐵𝐿2 𝑤 9,585e-05 6,303e-05 0,0687909 0,0617981 krasnodar region 𝐵𝐾3 𝛼 𝐵𝐾3 𝑤 𝐵𝐿3 𝛼 𝐵𝐿3 𝑤 4,2518e-05 0,00013121 0,02400843 0,22684937 republic of adygea 𝐵𝐾4 𝛼 𝐵𝐾4 𝑤 𝐵𝐿4 𝛼 𝐵𝐿4 𝑤 2,35727e-05 8,18161e-05 0,028614278 0,10440849 astrakhan region 𝐵𝐾5 𝛼 𝐵𝐾5 𝑤 𝐵𝐿5 𝛼 𝐵𝐿5 𝑤 5,63e-05 2,27e-05 0,185544 0,085229 republic of kalmykia 𝐵𝐾6 𝛼 𝐵𝐾6 𝑤 𝐵𝐿6 𝛼 𝐵𝐿6 𝑤 2,222e-06 0,0001432 0,0220744 0,0896685 republic of crimea 𝐵𝐾7 𝛼 𝐵𝐾7 𝑤 𝐵𝐿7 𝛼 𝐵𝐿7 𝑤 3,64224e-06 9,41469e-07 0,02335533 0,006049158 8 t.y. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2019 assa adv. in systems science and appl. (2019) 6) to calculate the efficiency coefficients of environmental protection allocations 𝑐𝑖 𝛼and 𝑐𝑖 𝛼, respectively, with the index a for the atmosphere, and with the index w for water, data on the share of the neutralization of pollutants are used. direct statistics are taken for the air environment, and the proportion of reused water is used to estimate the costs of protecting the aquatic environment. we denote these quantities doa i(t) for air and dow i(t) for water. previously determined values of the volumes of pollutant discharges into the atmosphere and water 𝑃𝑖 𝑎(𝑡), 𝑃𝑖 𝑤(𝑡)are used for calculations. the amount of expenses for compensation of environmental damage from pollutants is determined on the basis of available official statistics as the difference between the total expenditures of the region and the sum of national, national-economic, housing and communal and socio-cultural (consolidated budget). we denote these costs by 𝜈𝑖(𝑡). then, the ratio of environmental costs to total costs 𝑣𝑖 𝑎(𝑡) + 𝑣𝑖 𝑤(𝑡) is calculated, i.e.𝑣𝑖 𝑎(𝑡) + 𝑣𝑖 𝑤(𝑡) = 𝜈𝑖(𝑡) 𝐼𝑖(𝑡) ), which gives a share of the cost of eliminating pollution. we assume that the costs of water and air purification are proportional to the volume of pollutant discharges, i.e. 𝑣𝑖 𝑎(𝑡) = 𝑝𝑖 𝑎(𝑡) 𝑝𝑖 𝑎(𝑡)+𝑝𝑖 𝑤(𝑡) (𝑣𝑖 𝑎(𝑡) + 𝑣𝑖 𝑤(𝑡)),𝑣𝑖 𝑤(𝑡) = 𝑝𝑖 𝑤(𝑡) 𝑝𝑖 𝑎(𝑡)+𝑝𝑖 𝑤(𝑡) (𝑣𝑖 𝑎(𝑡) + 𝑣𝑖 𝑤(𝑡)). then we calculate the desired values from (2.7) (2.8) by the formulas с𝑖 𝑎(𝑡) = 1− 𝑑𝑜𝑖 𝑎(𝑡) 100 𝑣𝑖 𝑎(𝑡)𝐼𝑖 (𝑡) ,с𝑖 𝑤(𝑡) = 1− 𝑑𝑜𝑖 𝑤(𝑡) 100 𝑣𝑖 𝑤(𝑡)𝐼𝑖 (𝑡) . each value is reduced to the arithmetic mean value. as a result, the following initial data were obtained for the regions of the southern federal district (table 3.6). table 3.6.the values of the efficiency ratios of environmental appropriations for the regions of the southern federal district region environmental appropriations effectiveness ratios value rostov region 𝑐1 𝛼 𝑐1 𝑤 0,000393 0,000104 volgograd region 𝑐2 𝛼 𝑐2 𝑤 0,000497 0,000528 krasnodar region 𝑐3 𝛼 𝑐3 𝑤 0,000635 9,42e-05 republic of adygea 𝑐4 𝛼 𝑐4 𝑤 0,016904 0,002975 astrakhan region 𝑐5 𝛼 𝑐5 𝑤 0,00085 0,001107 republic of kalmykia 𝑐6 𝛼 𝑐6 𝑤 0,029967 0,002841 republic of crimea 𝑐7 𝛼 𝑐7 𝑤 0,000772 0,001659 7) the agent’s interest rate in increasing the specific consumption in the macro-region ri(t) is calculated as a share of the region’s consumption in the total consumption in accordance with the formula (3.2). 𝑟𝑖 = 𝐶𝑖(𝑡) ∑ 𝐶𝑖(𝑡𝑛 𝑖=1 ) . (3.2) with subsequent reduction to the arithmetic mean value, we obtain the following values of the desired parameter (table 3.7). dynamic model of general coordination and private interests 9 copyright ©2019 assa adv. in systems science and appl. (2019) table 3.7.the values of the coefficients of the region’s interest in increasing the specific consumption in the macro-region region coefficients of the region’s interest in increasing the specific consumption in the macro-region value rostov region 𝑟1 0,278 volgograd region 𝑟2 0,18 krasnodar region 𝑟3 0,438 republic of adygea 𝑟4 0,05695 astrakhan region 𝑟5 0,05695 republic of kalmykia 𝑟6 0,00899 republic of crimea 𝑟7 0,01 8) we understand the totality of all phenomena that lead to an increase in production without the use of additional resources by innovative activity in this work. we believe that innovation affects the results of modeling through the introduction of more advanced equipment and technologies into production, or through special measures to improve the skills of the workforce. these improvements are set externally and take the form of a time-dependent function. indices of the innovative activity impact in the regions of the russian federation for 2005, 2010, 2015 were obtained as a result of previous research by d.s. lozovitskaya and e.i. lazareva based on official data from federal state statistics service of the russian federation [19] on the following indicators: the degree of depreciation of fixed assets, investment in fixed assets per capita, developed advanced production technologies (inventive activity rate). the index is built on the basis of regression models for each period (2005, 2010, 2015), in which these indicators are defined as statistically significant [3, 13]. the values of the calculated coefficients are given in table 3.8. table 3.8.the values of the indices of the impact of innovation on the production of the final product of the region year value of index 2005 0,308 2010 0,222 2015 0,27 by interpolating the available values, the following type of index of the influence of innovation activity is obtained: 𝐴𝑖(𝑡) = 0,00268 ∙ 𝑡2 − 10,7774 ∙ 𝑡 + 10835,33. 9) the initial values of the parameters of the model ki(0), li(0) and ri(0) are determined according to the official data of federal state statistics service of the russian federation [19] on the volume of fixed assets and the number of working population, as well as finding the ratio of production / grp to the number of labor resources in the region for 2017 year. the obtained parameter values are presented in the table 3.9. table 3.9. initial values of model parameters region model parameters value rostov region 𝐾1(0) 𝐿1(0) 2786870 1958,1 10 t.y. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2019 assa adv. in systems science and appl. (2019) 𝑅1(0) 687,984679 volgograd region 𝐾2(0) 𝐿2(0) 𝑅2(0) 2180917 1124,6 685,9694113 krasnodar region 𝐾3(0) 𝐿3(0) 𝑅3(0) 5937791 2599,1 856,4186449 republic of adygea 𝐾4(0) 𝐿4(0) 𝑅4(0) 202111 152,1 653,556213 astrakhan region 𝐾5(0) 𝐿5(0) 𝑅5(0) 1498692 487,6 863,3328548 republic of kalmykia 𝐾6(0) 𝐿6(0) 𝑅6(0) 203657 111,1 598,6642664 republic of crimea 𝐾7(0) 𝐿7(0) 𝑅7(0) 2212391 840,4 427,3089005 10) the discount factor ρ based on the average refinancing rate of the central bank of the russian federation for the period under consideration is taken equal ρ=0,094. 4. nash equilibrium regional strategies are characterized by the following parameters: percentage 𝑠𝑖(𝑡) of grp spent on production; shares 𝑣𝑖 𝑎(𝑡), 𝑣𝑖 𝑤(𝑡) from industrial investments that are used to eliminate the effects of air and water pollution; the shares 𝑘𝑖𝑗(𝑡) from production investments that are spent both on the general development of the macroregion and on their own development (for i = j). the share 𝑘𝑖0(𝑡)is also allocated by the region to goals external to the southern federal district (federal and inter-district programs and projects). when finding nash equilibrium, you need to find a set of agent strategies {𝑠𝑖(𝑡),𝑣𝑖 𝑎(𝑡), 𝑣𝑖 𝑤(𝑡), (𝑘𝑖𝑗(𝑡))𝑗=0 𝑛 }-i , delivering maximum of their objective function with fixed strategies of other regions. using simulation, nash equilibrium is sought by the method proposed in [16]: step 1. in the loop, we sort through all the agents and for each agent: {𝑠𝑖(𝑡),𝑣𝑖 𝑎(𝑡), 𝑣𝑖 𝑤(𝑡), (𝑘𝑖𝑗(𝑡))𝑗=0 𝑛 }-i ; 2) for a fixed strategy of other agents, we sort through the strategies of this agent according to a certain principle and remember the one that delivers the maximum of its objective function, we denote this strategy nei; step 2. the resulting nei strategies form the equilibrium outcome of the game ne=(ne1,ne2, …, nen). for the model described and investigated in the article, fixing the strategies of the environment of the agent i is as follows. if j<i then the optimal strategy nei has already been found. it should be defined as a fixed strategy of the agent i. if j>i then the optimal strategy nei has not yet been found, therefore, we fix a strategy in which the available resources are evenly distributed by the agent for all goals and tasks: i.e. 𝑠𝑖(𝑡) = dynamic model of general coordination and private interests 11 copyright ©2019 assa adv. in systems science and appl. (2019) 0,5 (equal parts go to investments in production and consumption), 𝑣𝑖 𝑎(𝑡) = 0,1, 𝑣𝑖 𝑤(𝑡) = 0,1, 𝑘𝑖𝑗(𝑡) = 0,1 for j=0,1,…,n. point 2) of step 1 causes significant difficulties, since for n = 7, t = 5, even if each strategy consisted of only two possible values, it is necessary to sort out 23+n+1=211 values in only one moment in time, and when t = 5 the number of iterations will be (211)t=(211)5=255. but really, there are much more than two options for each strategy, and to sort through strategies with a step of 0.1, 1155 operations are required. to reduce the number of operations, we apply the following unidirectional iteration method. we sequentially sort through each of the eleven agent strategies with the remaining ten strategies fixed. agent strategies are sorted in the following order: in the first place, 𝑠𝑖(𝑡) is moved, since in fact the volume of required resources for the development of regions and environmental costs depend on it; further strategies are a fraction of this value; then the shares 𝑣𝑖 𝑎(𝑡), 𝑣𝑖 𝑤(𝑡) from production investments that go to the elimination of air and water pollution are moved; then goes each of the 𝑘𝑖𝑗(𝑡) strategies. this reduces the number of steps to 11 * 5 * 11 = 605 for each value in increments of 0.1. simulation modeling showed that the scenarios with small values of 𝑠𝑖(𝑡) gave large values of the payoff function of the agent, compared with large 𝑠𝑖(𝑡). this can be explained as follows. since the values of specific consumption in each region and in the macro-region as a whole are subject to maximization, in fact, maximization also occurs in relation to the volume of regional consumption. and since the final product is distributed between production and consumption, while maximizing the volume of consumption, it is necessary to minimize the volume of production. therefore, the optimal value is 𝑠𝑖(𝑡) = 0. but since resources are not spent on production, all cost items, including the development of own production and the general production programs of neighboring regions, as well as environmental costs for environmental protection and the elimination of air and water pollution, are indifferent. therefore, the dominant strategy of each agent is 𝑠𝑖(𝑡) = 0. at first glance, it is surprising that the region’s dominant strategy is the lack of investment in production. indeed, in this case, the fixed assets are not replenished, but only retired, therefore, by virtue of (2.1), the final product is formed, but in a smaller quantity, i.e. production decreases over time. but the level of pollution of natural resources is becoming less. however, to maximize the volume of consumption, it is necessary to increase the volume of production, which in modern conditions is due to extensive forms of management and, accordingly, is highly dependent on the volume of fixed assets. the latter, in turn, require constant replenishment due to natural and economic depreciation. under such conditions, the behavior of each region can be interpreted as waiting for the help of neighboring agents. however, as can be seen from the calculation results, all regions choose similar strategies, reasoning rationally and not wanting to invest resources in the development of material production. as a result, the replenishment of fixed assets does not occur in any region, since all agents choose a production stagnation strategy. nash equilibrium is formed by the strategies {𝑠𝑖(𝑡) = 0, 𝑖 = 1, … , 𝑛} with indifferent values of other strategies of all agents. this outcome of the game, when all agents choose a free-rider strategy of economic inaction based on the production activity of neighboring regions, inevitably leads to a decrease in production due to a decrease in available resources and should, as it seemed, reduce the value of the target function of the regions. but, as calculations showed, this mode of behavior continues to be beneficial to the regions (rostov, volgograd, astrakhan, krasnodar), but at the same time it ceases to be beneficial to the republics (republic of adygea, republic of kalmykia, republic of crimea). a strategy to reduce production and increase consumption is not advisable for any of the regions in real economic conditions. the presented results show the need for further theoretical 12 t.y. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2019 assa adv. in systems science and appl. (2019) studies of the model, the introduction of additional parameters and limitations. in particular, we assume the importance of taking into account external financial flows from the federal center, the introduction of control variables to reflect the process of harmonizing common and private interests in the field of socio-economic development of regions. 5. conclusion the article presents the results of the initial stage of studies of a dynamic model of combining the common and private interests of regions in the macro-region using the example of the southern federal district. maximizing the region’s specific consumption is considered as the region’s private interests, and maximizing the specific consumption of the entire macro-region as general interests. each region of the southern federal district can invest in its own development, as well as in the development of other regions as part of the macro-region. it was revealed that the optimal strategy of each region in the current conditions is a complete rejection of investments in the development of production. calculations showed that the republics of the southern federal district immediately become unprofitable for the general reduction of production, while the regions of the southern federal district continue to benefit from this strategy in the future. it is not possible to recommend the obtained strategies to present specific practical recommendations in real economic conditions. further practical studies of the model and the introduction of additional parameters and limitations are required for practical reasons. acknowledgements this work was supported by the russian foundation for basic research: project no. 18-01000594. references [1] anopchenko, t., gorbaneva, o., lazareva, e., murzin, a., & ougolnitsky, g. 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[online]. available http://www.gks.ru/wps/wcm/connect/rosstat_main/rosstat/en/main/ adv syst sci appl 2023; 03:108–126 published online at https://ijassa.ipu.ru. exploratory data analysis and natural language processing model for analysis and identification of the dynamics of covid-19 vaccine opinions on small datasets alexander chkhartishvili1, dmitry gubanov2, vladislav melnichuk3,4*, vladislav sych3 1laboratory no. 57, v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia 2laboratory no. 11, v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia 3department of applied mathematics, faculty of fundamental sciences, bauman moscow state technical university, moscow, russia 4machine learning track, technopark vk education, moscow, russia abstract: in this study, the successful implementation of an active learning algorithm on small-scale datasets is demonstrated. the study also examines the dynamics of public opinions on covid-19 vaccinations using vk (social network) commentaries related to the covid19 vaccine and masks for opinion evaluation. the proposed methodology includes several stages such as natural language processing, classification with active learning, exploratory data analysis, and opinion dynamics. natural language processing is used for text preprocessing, tokenization, and feature extraction. a machine learning model with active learning is employed to identify opinions as positive, negative, or neutral/unknown. the model includes classical machine learning, machine learning and deep learning models. the results show that the highest classification accuracy is 69.1% and 73.1% without and with the active learning algorithm, respectively. the experimental results suggest that classifiers using active learning perform better than simple natural language processing classifiers on small-scale datasets. keywords: active learning, deep neural network, opinion dynamics, opinion analysis, bert, covid-19 vk opinion classification 1. introduction the covid-19 pandemic is a deadly virus that has affected many countries around the world, leading the world health organization to declare it a pandemic disease in march 2020 [40]. the virus has caused a great loss of life, and governments worldwide have employed various measures to combat it, including travel restrictions, vaccination, and facility closures. the covid-19 vaccination is one of the most significant techniques applied by governments to control its spread, and many countries have reported fewer covid cases due to a high percentage of vaccination. the russian government has launched a widespread immunization effort, and the vaccine is now available to the general population after several stages of vaccine trials. the success of any immunization campaign depends on its public approval rate and the speed of acceptance. however, there are many misconceptions and doubts about covid19 vaccinations among ordinary people. therefore, it is necessary to understand public opinion dynamics to ensure an effective immunization company. this study aims to develop ∗corresponding author: vlamelni@gmail.com assa latex template 109 an algorithm that can identify opinions and comprehend opinion dynamics from smallscale datasets with sufficient accuracy. specifically, the study focuses on the opinion analysis of vk messages on the russian language regarding the covid-19 vaccine and the identification of their opinions as positive, negative, or neutral/unknown. the results of this work can comprehend in improving machine learning (ml) models via active learning and in formulating policies to address people’s queries before mass vaccination [38]. active learning in machine learning involves an interactive process where a learning algorithm seeks feedback from a user or another information source to assign desired outputs to new data points, which helps the algorithm improve its efficiency. the article is organized as follows: sections 2, 3, and 4 discuss opinion dynamics, specifically the dynamics of opinions on covid-19. section 5 and 6 detail the bertbased algorithm and active learning, bert is bidirectional encoder representations from transformers. section 7 presents the experimental results, illustrations, and discussions, followed by the conclusions in section 8. 1.1. novelty of the proposed work the current research presents the practical implementation of an algorithm for opinion analysis of russian people’s opinions about covid-19 vaccines based on vk (social network) messages [13, 23] (see also the study on the dynamics of opinions regarding the wearing of medical masks [18, 19]). the methodology proposed consists of four main steps: natural language processing (nlp), opinion classification using active learning, exploratory data analysis (eda), and analysis of opinion dynamics. the nlp step involves data preprocessing, including data cleaning, removal of irrelevant words, data normalization, and tokenization [14,20]. further, preprocessed data is used as input in bert-based model, which is trained using active learning after feature extraction and data labeling. the bert-based model is utilized for opinion classification, with positive, negative, and neutral/unknown being the three categories. the eda and opinion dynamics are conducted on a labeled dataset of two million using the bert-based model with active learning. to conduct this study and build the bert-based model, five machine learning techniques were utilized, such as logistic regression, gradient boosting, extreme gradient boosting (xgb), transformers, and active learning [10, 15]. the application of active learning for small datasets in opinion dynamics tasks was analyzed in this study. fig. 1.1. the charts of relative and absolute values of ”for” and ”against” opinions copyright © 2023 assa. adv syst sci appl (2023) 110 d. gubanov, a. chkhartishvili, v. melnichuk, v. sych 2. preliminary data analysis for opinion dynamics in this work, data collected from the vk social network is used, and approximately 4gb of text data are used, containing users’ commentaries, their identifiers, time, and source, user ids and those with whom they interacted, information about likes on commentaries and posts. the data was collected from popular news communities from january 19, 2020, to january 19, 2021, reflecting the beginning of the covid-19 pandemic and covering the first two waves of infections. fig. 2.2. a graph of the density distribution of commenting frequency for all users first, each table is cleaned of junk data. for the commentaries tables, these were rows without the text of the commentary itself, and then they were checked for duplicates. for table 7.3, uniqueness was checked by h id comment, but during encryption, some commentaries received the same h id comment, so it was necessary to change this hash to avoid throwing away false duplicates. as a result, after cleaning, 2.2 million records remained. the first thing to look at is the opinions over the entire period, using table 7.3 as a basis, which was labeled by the bert-based model with an accuracy of 73.10%, trained on labeled samples. this will be described in the article below. first, relevant commentaries were taken from a large dictionary, and now the table has 400 000 records. then, records with likes of these comments from table 7.5 were added to this table of comments. the opinion of the liker is considered being like the opinion of the commentary on which the like was placed. as a result, the charts of relative and absolute values of ”for” and ”against” opinions are presented in figure 1.1. the analysis was conducted without considering comments and likes with the opinion ”unknown = 0”. according to the relative graph, we can see that the average ”for” (and ”against” as well) opinion remains unchanged and hovers around 30%. peaks of activity are clearly visible on the absolute value graph. comparing this graph with the 2020 disease incidence, these maximums coincide with the active phase of the first two waves of disease incidence growth. the plateau after the maximums coincides with the peak of disease incidence. the first wave was the most frightening for people, so the first peak is noticeably higher than the second, although in terms of infections, it was inferior to almost all the others, which indicates a trend towards a decrease in interest in the pandemic. 2.1. user commentaries distribution the distribution of user comments shows that 211 000 unique users wrote 2.2 million commentaries throughout the year, half of whom wrote only one commentary. this is a bad copyright © 2023 assa. adv syst sci appl (2023) assa latex template 111 sign for us because we cannot analyze their opinions over different periods of time, but we still need to monitor their influence on others. 90% of users wrote fewer than 15 comments in a year. a graph of the density distribution of commenting frequency for all users (on a logarithmic scale) is presented in figure 2.2. we were able to verify that this is a power-law distribution with a heavy tail, f(x) = 0.1357( x 4222 )−(0.1357+1). 2.2. distributions by groups and posts figure 2.3 shows a histogram for communities with over 1000 comments (20 in total). ”for” opinions range from 26–40% (table 7.4 is used). fig. 2.3. histogram for communities with over 1000 comments communities with polar opinions (names are hashed): 417...: 26% support (20% if likes are considered), 366...: 40% support (by comments only), f1d...: 40% support (by comments and likes). next, we will analyze opinions on posts from the selected communities (20 out of 62) in figures 2.4 and 2.5. half of the posts have only a few commentaries, so it was decided not to consider posts with less than 10 commentaries to avoid getting anomalous average opinions ”for” and ”against”. fig. 2.4. histograms without considering likes in the first histogram, the density distribution is shown for very active posts with up to 70 commentaries, compared to the second histogram, which only includes posts with average copyright © 2023 assa. adv syst sci appl (2023) 112 d. gubanov, a. chkhartishvili, v. melnichuk, v. sych activity. it is evident that the second graph is more biased towards lower support for covid measures. when taking into account the likes, historgrams changes (fig. 2.5). fig. 2.5. histograms with consideration of likes and commentaries on the histogram for regular activity posts, it can be seen that support for polarizing opinions, especially negative ones, has increased [17]. this means that provocative comments are more often written in a negative tone. posts with polarizing opinions are presented in table 2.1. table 2.1. posts with polarizing opinions label h post id -1 0 1 pros 379 8bd... 732 187 34 4.43 419 d0d... 129 59 6 4.44 ... ... ... ... ... ... 63 212... 53 52 240 81.91 210 6c0... 96 77 691 87.80 examples of posts with negative reviews contain the following news: 1. it’s now almost official: russia has the highest number of covid-19 deaths per capita. this results from the actions of the authorities in the summer of 2020. 2. the last tests for a new vaccine against covid-19 are underway in russia. the vaccine will be offered first to doctors and pensioners, and then to everyone else. 3. the authorities plan to issue vaccination passports through the government services website. 4. ”tsargrad” was the first media outlet to report that sberbank’s subsidiary company, ”immunotechnology”, which was created only in may of this year (it has no production or logistics facilities of its own), could become the sole supplier of vaccines to prevent covid-19. examples of descriptions of posts with positive user ratings: 1. mishustin contracted covid-19. 2. according to most doctors, covid-19 spreads fairly quickly but is not one of the most dangerous diseases the fatality rate is 3.6%. 3. the number of covid-19 deaths in italy has set a daily record. 4. the ministry of internal affairs named the regions with the highest number of violations of the isolation regime. copyright © 2023 assa. adv syst sci appl (2023) assa latex template 113 5. for the first time in russia, over 15 000 new cases of covid-19 were detected in one day. 3. identifying a network environment of users we need to determine the influence/trust of users (each to each) and predict their last opinions based on the cleaned data. then, we can compare the results with the true and random outcomes. figure 3.6 illustrates the structure of commentary threads and how we record interactions in a table. fig. 3.6. method for obtaining the influence table from a commentary thread the influence/trust table is compiled in several stages:counting comments within threads using (example: interaction between id2 and id4), counting replies to a parent commentary (example: id1, id2, id3), counting likes on comments (example: id5, id6, id7). the trust is calculated using the following formula [16]: trust = w1kcij + w2klij (3.1) where w1 and w2 are the weights assigned to comments and likes, cij and lij denote the number of comments and likes of agent i ∈ n towards agent j ∈ n , respectively. the initial weights are set as w1 = 0.1 and w2 = 1. from the interaction table, a directed graph is constructed and the adjacency matrix is calculated using the networkx library. let’s inspect the interaction graph. 3.1. visualization and characterization of the interaction graph the visualization is created using the gephi program with the forceatlas2 layout. the colors in figure 3.7 are determined by the modularity parameter, which measures the structure of the graph and indicates how well it is divided into dense communities. the modularity score ranges from –1 to 1, and in our case, the results for each time interval are around 0.7, indicating a clear division into communities. when calculating modularity in gephy, the algorithm groups nodes that are more densely connected to each other than to other nodes in the graph into the same community or cluster. this grouping leads to the division of the graph into subgraphs or clusters, each of which has numerous internal connections and relatively few external connections to other clusters. copyright © 2023 assa. adv syst sci appl (2023) 114 d. gubanov, a. chkhartishvili, v. melnichuk, v. sych in the graph, 10 clusters are clearly distinguished, which have been identified as subscribers to different vk communities. some users are subscribed to several similar communities, while others may have a different position and gather in smaller isolated groups. the ”asteroid belt” around the clusters represents non-communicative users with at least one interaction. fig. 3.7. the graph represents the interactions during the first week the large nodes represent opinion leaders whose statements are actively discussed or liked, while the size of the nodes is proportional to their degree, i.e., the number of connections. the small branches of isolated user islands from opinion leaders represent likers. it is possible that these likes were artificially inflated to promote a particular opinion, but more likely, these are users who do not express their own opinions but only agree with others. another informative parameter is the degree of assortativity in the graph, which represents the preference of network nodes to connect to other nodes that are in some way similar to them. in our case, the assortativity parameter for each week was –0.1, indicating a negative correlation between nodes of different degrees. positive assortativity values indicate a correlation between nodes of similar degrees, while negative values indicate relationships between nodes of different degrees. when examining graphs over larger time intervals than current ones, the grouping in the center of the graph is blurred due to the fact that many users show activity in the same communities. 4. the dynamics of opinions on covid-19 task in this part of the work describes the creation of a trust matrix from an interaction graph, a model, and an initial prediction. formula (4.2) will be used for further prediction [16]: b ′ = b ∗ a (4.2) where b is a vector of initial average user opinions for the selected interval, which is multiplied by the trust matrix a to obtain the prediction b ′ . the graph described above is read as an adjacency matrix, which will be referred to as the trust/influence matrix ”each with each”. this is a sparse matrix, whose elements are taken as the trust value, considering the quantity and weights of interactions from (3.1). copyright © 2023 assa. adv syst sci appl (2023) assa latex template 115 the next step is to modify the matrix: add a constant term w3 to the formula for matrix elements (3.1), add self-trust to the main diagonal with an initial weight of w4 = 0.5, ensure stochasticity by making sure that ∑n j=1 aij = 1 for each row, where users trust themselves with weight w4 and everyone else with weight 1− w4. the resulting dynamics of the opinions of agent i ∈ n (n the set of users) is given by the equation [16]: x ′ i = w4xi + (1− w4) ∑ j∈n\{i} aijxj (4.3) where aij ∈ [0; 1] represents the degree of trust that agent i has in agent j (i, j ∈ n ). the trust matrix a is constructed as follows. first, we form a matrix a′, where the elements are given by [16, 17]: a ′ ij = w1cij + w2lij + w3∑ j∈n\{i}(w1cij + w2lij + w3) (4.4) the denominator serves as a normalization factor to ensure stochasticity. we then account for self-trust (w4 ∈ [0; 1]) [16]: a = w4e + (1− w4)a ′ (4.5) further model complexity is possible but is not practical due to incomplete information, as shown by optimizing weights using scipy’s powell method to minimize the jensen-shannon divergence. starting from the initial weights described above, the optimizer converged to the following weights: w1 = 7.88, w2 = 5.48, w3 = 6.25, w4 = 0.99. the most significant result is the near-unity self-trust weight, indicating that each user’s opinion is influenced only slightly by others. people are more likely to change their opinions by exploring news sources and discussing offline rather than engaging in online communication. fig. 4.8. prediction of opinions on covid-19 however, the optimized weights allow us to predict the percentage of negative opinions about vaccines, which we present in figure 4.8. for the known period, the percentage is 61.85%, and for the forecast period, it is 73.72%. copyright © 2023 assa. adv syst sci appl (2023) 116 d. gubanov, a. chkhartishvili, v. melnichuk, v. sych 5. identifying opinions on vaccination the proposed methodology in this study for opinion analysis, classification and opinion dynamics involves several steps: 1. data gathering 2. data preprocessing 3. data labelling 4. opinion analysis 5. opinion classification using a bert-based model without active learning 6. opinion classification using the bert-based model with active learning 7. opinion dynamics research the flowchart, encoder and decoder of the bert-based model, along with the active learning algorithm, are shown in fig. 5.9, fig. 5.10, and algorithm 1, respectively. the details of each step of the proposed methodology are described in subsequent subsections. 5.1. data gathering in an era marked by unprecedented challenges, social media platforms have emerged as crucial instruments for understanding public sentiment and opinion [1, 2, 6]. among these platforms, vk, a popular social network, offers a rich dataset for analyzing public views on pressing issues such as covid-19 vaccinations. this study leverages data from vk to conduct a comprehensive opinion analysis on the subject [3, 5]. 5.1.1. data source and origins the data for this study was sourced from vk, specifically from popular news communities. a total of 2.2 million commentaries related to ”covid19 vaccination” were collected using vk’s api (application programming interface) [24]. these commentaries were part of various discussions, posts, and threads within these communities. importantly, the data spans a period from january 19, 2020, to january 19, 2021. this timeframe captures the beginning of the covid-19 pandemic and includes the first two waves of infections, thereby providing a dynamic view of evolving public opinion. 5.1.2. data anonymization and ethics to adhere to privacy and ethical standards, all collected commentaries were anonymized prior to preprocessing. identifiable information such as usernames, profile pictures, and any other personal identifiers were removed to maintain the anonymity of the individuals involved. 5.1.3. relevance to the study the vk dataset is particularly relevant to this study for several reasons: 1. volume: the large number of commentaries provides a robust sample for analysis. 2. diversity: the dataset includes opinions from individuals of various demographics, offering a more comprehensive view of public sentiment. 3. timeliness: the data covers a critical period in the covid-19 pandemic, making it highly current and relevant. by leveraging this dataset, the study aims to offer a nuanced understanding of public opinions on covid-19 vaccinations, thereby providing valuable insights for policymakers, healthcare providers, and the general public. 5.2. data preprocessing in this chapter, the data preprocessing process is explained, which includes four sub-steps: data cleaning, normalization of data, tokenization of commentaries, and vectorization. as the collected commentaries are noisy and unlabelled, only 7 000 commentaries out of 2.2 copyright © 2023 assa. adv syst sci appl (2023) assa latex template 117 million were labelled manually by three linguistics and medical science experts. the data cleaning process is applied to eliminate noises such as unnecessary words, emojis, additional characters in sentences, blank spaces, stopping words, and punctuation. to normalize the data, stemming and lemmatization techniques are used from the wordnetlemmatize package of natural language tool kit (nltk) for logistic regression (lr), gradient boosting, and xgb models. the stemming technique is used to determine the root form by removing terminators from words, while lemmatization groups similar words in various forms to reduce dimensionality. finally, the normalized dataset is tokenized by dividing the text into tokens and used as features for training, validation, and test datasets. fig. 5.9. flowchart of the proposed research 5.3. data labelling three experts in linguistics and medical science manually labelled a training dataset of sentences as negative, positive, or neutral/unknown, represented by the values 0, 1, and 2, respectively. the text data was then transformed into a vectorized form using the tf-idf (term frequency–inverse document frequency) vectorization technique for lr. this technique extracts features from labelled data based on the frequency of words in the text. the phrase frequency (tf ) measures the frequency of a word in a document, while the inverse frequency of words (idf ) measures the frequency of a word in the document set: tf (t, d) = ft,d∑ k fk,d (5.6) copyright © 2023 assa. adv syst sci appl (2023) 118 d. gubanov, a. chkhartishvili, v. melnichuk, v. sych where ft is the count of a term t in a commentary d. idf (t,d) = log |d| 1 + |{d ∈ d : t ∈ d}| (5.7) where d is infering to our document space (training dataset), d = d1, d2, . . . , dn, n is the number of commentaries in d, |d| = n. the denominator |{d ∈ d : t ∈ d}| represents the total count of occurrences of the term t appeared in a commentary d. the tf − idf score is calculated by multiplying the tf score with the idf score: tf − idf (t, d,d) = tf (t, d)× idf (t,d) (5.8) 5.4. opinion analysis different machine learning models including logistic regression, gradient boosting, extreme gradient boosting (xgb), bert models (transformers) and active learning are applied for opinion classification into positive, neutral, and negative/unknown categories. a brief description of each machine learning technique is provided in the following subsections. 5.4.1. logistic regression machine learning applies a logistic regression classifier, a statistical model that establishes the relationship between independent and dependent variables [27]. the classifier uses a logistical function to determine the input, set of weighted functions, and the correlation between classes. proper selection of features can improve the model’s accuracy and generalizability. feature vector i is classified as positive, neutral/unknown, or negative using a mathematical expression represented as: s(f = 1|i) = l(i) = 1 1 + hzwi (5.9) where, z means the feature weight, s is the probability of commentary i which belongs to class f . 5.4.2. gradient boosting in machine learning, the gradient boosting technique involves the combination of numerous weak models to form a robust predictive model that is suitable for categorizing extensive datasets [9]. this model is capable of reducing bias error, thus resulting in an accurate and efficient prediction model. by creating an approximation ĥ(b) of the function h∗(b), the gradient boosting model maps the input instances b to their output values z. this function approximation h∗(b) can be expressed as a weighted sum of functions, as shown by the mathematical expression: hc(b) = hb−1(b) + pcqc(b) (5.10) where, pc means the weight of the c-th function qc(b). 5.4.3. extreme gradient boosting xgb is a set of gradient boosting techniques designed for current data science challenges [11, 12]. it uses an ensemble model of classification and regression tree sets (cart). xgb is known for its high scalability, parallelizability, speed, and regularized approach to control over-fitting. the mathematical representation of the xgb model is expressed as: âl = p∑ p=1 qp(ml), qp ∈ q (5.11) where qp for the p-th tree denotes a function in functional space q, and p represents the total number of trees. the set of all possible carts is represented by q. copyright © 2023 assa. adv syst sci appl (2023) assa latex template 119 5.5. bert-based model applying an active learning algorithm to the bert-based model can increase its accuracy for training and testing [7, 10, 15, 29]. by cyclically adding labeled data to the training dataset, the active learning algorithm can improve the model’s predictive accuracy beyond that of a single bert-based model. fig. 5.10. bert-based model with active learning cycle. figure 5.10 shows a block diagram of the bert-based model with active learning. the active learning cycle includes the pre-training phase and the cycle phase. in the pre-training phase, the model is trained on a corpus of data with the original labeling. then, in the cycle phase, additional data is labeled and added to the training dataset. to create the bert-based model, a range of machine learning algorithms were evaluated, including logistic regression, gradient boosting, extreme gradient boosting, and multiple bert models, such as xlm roberta large, roberta, ruroberta, bert large uncased, rubert, gpt (generative pre-trained transformer), and distillbert (distilled bidirectional encoder representations from transformers) [25, 35]. xlm-roberta stands for ”cross-lingual language model robustly optimized bert pretraining approach,” combining multilingual capabilities with optimized pretraining for robust performance. the ”ru” prefix indicates that the model has been trained on a russian-language data corpus. the bert-based model constructed using xlm roberta large and active learning is shown in figure 5.10 [36,39]. the bert-based model operates as follows: 1. the dataset is split into three sets: training, validation, and testing. 2. the bert-based model is trained on the trainset and validated on the validation set. 3. the accuracy of the model is evaluated on the testing dataset. 4. the resulting data labelled by the active learning algorithm is added to the trainset. 5. steps 1–4 are repeated until the accuracy stabilizes. copyright © 2023 assa. adv syst sci appl (2023) 120 d. gubanov, a. chkhartishvili, v. melnichuk, v. sych 6. the final accuracy measurement is taken on the testing dataset. 7. 2.2 million commentaries are labeled by the trained model. fig. 5.11. second active learning cycle 6. bert-based classification algorithm with active learning algorithm 1. active learning a concept of active learning algorithm was implemented without a specific template [38]. the algorithm consists of two parts: pretrain and cycle. in the pretrain part, the model is trained on a corpus of data with initial labeling. the corpus in this task consists of 7 000 instances, with 5 670 for training, 630 for validation, and 700 for testing. pretrain can be considered as a well-trained version of the model for use in reference labeling. next, the cycle part involves repeating a set of specific actions. an unlabeled corpus of commentaries is taken, which in this case is 63 000 instances, and labeled with the model trained in the pretrain part (or in the previous step of the cycle part). then the recognition accuracy of each class is calculated from the test corpus (700 instances), and the class with the lowest recognition accuracy is selected. the reason for this is our objective to achieve average accuracy over all classes rather than focusing on accurate prediction of a single class [38]. various methods of adding instances to the training corpus are available, such as fixed selection, percentage selection, taking a percentage of the current selection volume, and percentage selection with accuracy (taking only those selections that exceed the classification accuracy threshold). a comparison of the methods is presented in table 6.2. in the next step, only the necessary number of added elements for the class with the lowest accuracy is taken from the labeled corpus of 63 000 instances. before this, our predictions are sorted in descending order of confidence in predictions by classes, and the necessary number of initial elements are taken to add to the training corpus. also, the instances already taken are marked to avoid taking them again. copyright © 2023 assa. adv syst sci appl (2023) assa latex template 121 table 6.2. performance evaluation of different methods for augmenting the training corpus way of adding instances average precision score percentage of the current selection volume 73.10 percentage selection with accuracy 73.02 percentage selection 72.78 fixed selection 72.15 if the number of instances is not sufficient, those available are taken, or none are taken, and the algorithm retrains the network to improve accuracy, giving the opportunity to take the necessary number of elements in the next iteration. while it is possible to retrain the network in each iteration of the cycle part, it should be understood that this may lead to overfitting, which can result in worse network training outcomes. testing with training from scratch and continuous retraining of the model in the cycle part showed that the model trained in the continuous retraining mode performed better despite overfitting [26]. it is recommended to create an additional test set for final evaluation at the end of the process to exclude any implicit overfitting. active learning improves network accuracy by an average of 3–4% in this task. the pretrain part is essential, and it is better to choose a larger number of epochs for training the model in this part, as the most accurate initial labeling has a greater impact than accurate labeling in the later iterations of the cycle part (the accuracy is higher by 0.5% on average) [23]. at the end of the cycle iteration, accuracy is calculated on the test corpus, and a training corpus is created with the added set of instances determined to be necessary for adding. the next iteration of the cycle part starts again on the updated training dataset. the active learning algorithm’s operation is presented in fig. 5.11 and fig. 6.12. fig. 6.12. third active learning cycle. copyright © 2023 assa. adv syst sci appl (2023) 122 d. gubanov, a. chkhartishvili, v. melnichuk, v. sych 7. experimental results and practical application this section covers the results of opinion analysis using natural language processing and the performance evaluation of machine learning techniques applied. the work collected a total of 2.4 million commentaries from russian people in 2020 and 2021 for opinion analysis on vaccines using the vk api. fifteen fields were extracted such as post source id, post id, comment id, user id, parent comment id, liker id, user subscriptions, date, text, comment label, which were shown in tables 7.3–7.6. table 7.3. unlabelled commentaries h owner id h post id h comment id h commenter id ... h addr comment id date text 3a4... d8b... 9d4... d41... ... nan 158... uzhas [horror] table 7.4. likes on posts and user commentaries h owner id h post id h comment id true label text 3a4... d8b... 9d4... 0.0 nuzhno lechit’ koronovirus [cure covid] table 7.5. information about posts h owner id h post id h comment id true label text 3a4... d8b... 9d4... 0.0 tol’ko 6% rossiyan... [only 6% of russians...] table 7.6. commentaries on vaccination h owner id h post id h comment id true label text 3a4... d8b... 9d4... 0.0 a gde budut vakcinirovat’... [where will they vaccinate...] 0ef... edc... 9b4... 1.0 spasibo nashim uchyonym [thanks to our scientists] 92e... fe4... 576... nan oni vrut [they lie] 2e1... e43... 76f... 2.0 smekh,vakcina pomozhet [jokes aside, the vaccine will help] table 7.7. distribution of opinions in commentaries after data labelling total number of commentaries positive negative neutral/unknown 2,285,620 911,269 (39.87%) 421,580 (18.44%) 952,771 (41.69%) the trained classifier was used to label the data, and the resulting labels are added as an additional column label to table 7.3. the distribution of opinion after labelling are shown in table 7.7. the majority of input commentaries were analyzed as neutral/unknown positive and followed by negative commentaries. the results indicate that 41.69% of input commentaries were classified as neutral/unknown, 39.87% as positive, and 18.44% as negative. inference (inf.) requires on average three times fewer ram resources. table 7.8. performance evaluation and resource utilization of different classifiers model precision score training time ram/hdd (train) inf. time ram/hdd (inf.) bert-based & active learning 73.10 2 hour(s) 15/32 gb 0.50 sec 5/8 gb bert-based classifier 69.17 0.4 hour(s) 15/32 gb 0.50 sec 5/8 gb logistic regression 56.14 0.1 hour(s) 5/32 gb 0.25 sec 3/8 gb xgb 51.83 0.1 hour(s) 15/32 gb 0.25 sec 5/8 gb gradient boosting 48.21 0.1 hour(s) 15/32 gb 0.25 sec 5/8 gb copyright © 2023 assa. adv syst sci appl (2023) assa latex template 123 7.1. active learning the precision score of different classifiers and bert-based models were presented in tables 7.8 and 7.9, with the bert-based model first trained in the pretrain part of active learning followed by the cycle part. the study found that active learning increased accuracy by 3–4% on small datasets on average. the parameters values initialized for the bert-based model and active learning are presented in source notebook on platform kaggle [23]. table 7.9. performance evaluation and resource utilization of different classifiers model precision score training time xlm roberta large with active learning 73.10 2 hrs xlm roberta large 69.17 0.4 hrs ruroberta 65.23 0.1 hrs roberta 63.58 0.1 hrs gpt 62.76 0.2 hrs distillbert 57.33 0.2 hrs rubert 52.40 0.1 hrs bert large uncased 41.72 0.1 hrs 8. conclusion this study analyzes the opinion of russian individuals towards various covid-19 vaccines. vk platform data was collected as a sample for analysis. the raw text messages received several natural language processing operations for text preprocessing. the opinion analysis classified 41.69%, 39.87%, and 18.44% of commentaries as neutral/unknown, positive, and negative, respectively. to classify the opinion, a proposed bert-based model with active learning was used, achieving a precision of 73.10%. the results indicate that active learning can improve precision by 3–4% on small datasets. future work will involve usage of graph neural networks (gnn) and all available data, including user subscriptions [4, 28, 37]. the majority of people’s opinions on social media regarding vaccinations and their effects are neutral or unknown. however, only 39.87% of people are optimistic, which is a worrying situation for policymakers and the government [8]. in order to make the vaccination program effective, the government must convince the majority of the population that the vaccine will have positive outcomes and consequences [30, 31]. therefore, policymakers and the government should focus on reducing vaccine anxiety before starting mass vaccination. according to recent research, although approximately 22 million russians have been infected with covid-19, a significant proportion of the population still believes that the pandemic has been exaggerated, based on our research [22, 32]. this has led to skepticism and opposition towards the covid-19 vaccine, driven by concerns such as vaccine safety, skepticism towards pharmaceutical companies, and questions about the effectiveness of the vaccines available [33, 34]. despite legitimate doubts, some unfounded conspiracy theories surrounding the virus have been dispelled. the government, pharmaceutical companies, and non-governmental organizations (ngos) should make a significant effort to educate the general public about the vaccine program and the importance of returning to normal life. special attention should be paid to addressing people’s fears and misconceptions about the vaccine to encourage vaccine uptake. using various models such as logistic regression, gradient boosting, xgb, and bertbased models with active learning, the research categorizes people’s feelings towards the vaccine as positive, neutral/unknown, or negative. the proposed bert-based model achieved 73.10% accuracy, outperforming other classifiers in terms of classification accuracy [21]. an analysis was also conducted to identify the dynamics of opinions. the predicted opinions are moving towards the negative side, indicating that, without any new external factors, people’s opinions about vaccines are deteriorating under the same news background. copyright © 2023 assa. adv syst sci appl (2023) 124 d. gubanov, a. chkhartishvili, v. melnichuk, v. sych acknowledgements the authors, vladislav melnichuk and vladislav sych, express their gratitude to their academic supervisors, dmitry gubanov and alexander chkhartishvili, for their support. 8.1. funding information this research did not receive any specific grants from public, commercial, or non-profit funding agencies. 8.2. data availability statement the data can be viewed in the dataset uploaded to kaggle [24]. references 1. alam k.n., khan m.s., dhruba a.r., khan m.m., al-amri j.f., et. al. 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[online]. available: https://www.who.int/publications/m/item/situation-report—55 copyright © 2023 assa. adv syst sci appl (2023) introduction novelty of the proposed work preliminary data analysis for opinion dynamics user commentaries distribution distributions by groups and posts identifying a network environment of users visualization and characterization of the interaction graph the dynamics of opinions on covid-19 task identifying opinions on vaccination data gathering data source and origins data anonymization and ethics relevance to the study data preprocessing data labelling opinion analysis logistic regression gradient boosting extreme gradient boosting bert-based model bert-based classification algorithm with active learning experimental results and practical application active learning conclusion funding information data availability statement microsoft word 931 article text, copyedited.docx adv syst sci appl 2020; 03; 113-135 published online at https://ijassa.ipu.ru. control, activity, personality dmitry novikov trapeznikov institute of control sciences, russian academy of sciences, moscow, russia e-mail: novikov@ipu.ru abstract: the paper presents the view of a control theoretician on the theories of personality and activity, in search of their results necessary for the formulation and solution of control problems for systems with active elements (man). the constructions proposed below can be treated as an “interface” that establishes a correspondence between the approaches and results of personality psychology, control methodology, and control science. the paper is intended for control theoreticians, who may read it for getting initial ideas about the theories of personality and activity as well as about their connection with control of social and organizational-technical systems. in turn, psychologists may observe not only the prospects of using their results in the above-mentioned branches of control theory, but also a number of new challenges. keywords: control theory, complex activity, personality psychology, structure of activity, structure of personality 1. introduction control1 is an impact on a controlled system with the goal of ensuring its required behavior. a control system (cs) consists of three components, namely, a controlled system, a subject of control, and relations between them (see fig. 1). as a rule, a controlled system has its own internal structure, including different (e.g., causal) relations between all elements of the system. knowing the cause-effect relations, one can influence some elements (“causes,” often called subject matters or objects of control) for changing the states of other elements (“effects”). the general control problem is to find admissible impacts on the objects of control that will ensure the required dynamics of the states (behavior) of a controlled system. fig. 1. basic structure of control system 1 main terms, when first mentioned, are marked by italic font. state disturbance control output goal of control subject of control controlled system feedback loop 114 d. novikov copyright ©2020 assa. adv. in systems science and appl. (2020) over a century and a half of its development, control theory has encompassed many classes of controlled systems [13]. for the effective development of the corresponding branches of control theory and the acquisition of knowledge about a controlled system, a close cooperation between mathematicians (experts in control) and the representatives of other relevant branches of science is required. for example, historically, “mechanical” systems (then technical systems) were the first classes of mass controlled systems studied in theory; moreover, precisely for these classes of systems, the most profound and extensive theoretical results on the solution of control problems were obtained in the 20th century, which found many applications in practice (control of moving objects, production systems, etc.). this was due to the progress of physics and mechanics in the 19–20th centuries, which yielded well-developed models of controlled systems. over the past half century, the mathematical theory of control has been actively considering new classes of controlled systems (from the 1950s–1960s, economic systems; a little bit later, ecological-economic, living, and other systems), following a current progress in the description of the corresponding control systems in economics, biology, and other sciences. in recent decades, social systems have increasingly been in the focus of this theory. any science is determined by its subject matter and methods [15]. man and social systems are the subject matter of many sciences and scientific fields. each of them highlights its own subject of research, including the one considered below (see fig. 2), as follows: the psychology of personality, the mental attributes of a man and the corresponding phenomena and processes; methodology (as the science of activity organization), activity [1, 15]; mathematical control theory, control processes.2 the objective reality, indicated by a dotted ellipse in fig. 2, is reflected differently in each of the sciences exploring this reality, which all form the “world of models”; see the dotted arrows there. each of the sciences highlights its own aspects of interest, abstracting from certain “details,” as well as introducing certain assumptions and hypotheses about the subject of research. in other words, each science (and each individual researcher) studies its own subject matter and constructs its own model of a controlled system. fig. 2. subjects of research developing in parallel, different sciences enrich each other, “sharing” their models with each other, introducing and extending the results of other sciences, and resulting in more complete and adequate models of reality. for example, over the past two decades, there has been an active use of the results of methodology (see arrow 2 in fig. 2) as a foundation of the 2 control theory, methodology, and psychology are not completely “parallel.” control theory does not study human beings, just as it does not study the motion of a rocket—this is the task of physics. methodology is the foundation of the theory of any activity, including control. psychology explores the human psyche, and control theory uses the methodology paradigm and the results of psychology to control human beings. in other words, the user’s position of control theory dictates: “provide a model!” 1 personality psychology man methodology control theory 2 3 control, activity, personality 115 copyright ©2020 assa. adv. in systems science and appl. (2020) theory of control in organizational systems (see [1, 18] and sections 3–4 below). there are long-standing and stable relations between methodology and the personality psychology (the subject matter of methodology––activity––is one of the basic categories of psychology); see dotted arrow 3 in fig. 2. however, the relation between the personality psychology and control theory (see arrow 1 in fig. 2) is still very weak. precisely this relation will be considered below: this paper is a “navigator” across the psychological aspects of the control problematics in systems that incorporate man. the main goal of this paper is not to contribute to the psychology of personality, but to establish an interface between the needs of control theory and the concepts and approaches of the existing psychological theories. making a small digression, let us discuss the differences and relations between a real cs and its model: how control theory “reflects” the reality; see fig. 4. (here the vertical dashed line indicates the boundary between the real world and the “world of models,” between the real cs and its model.) also, see fig. 2. assume that the problem is to design and create some real control system that consists of a controlling system and a controlled system. the corresponding activity has the structure shown in fig. 3. fig. 3. structure of control and organization [1] first (see fig. 4), analysis (i) is carried out: a model of the controlled system is constructed, and the controlled system and its model are studied (dynamics, sensitivity in parameters, initial and external conditions, etc.). also, the corresponding properties of the control system are examined. fig. 4. real systems and their models in control problems real cs model of cs subject of control controlled system models of subject of control model of controlled system v. assessment iv . « re g u la ti o n » ii. s y n th es is i. analysis iii. concretization 116 d. novikov copyright ©2020 assa. adv. in systems science and appl. (2020) then synthesis (ii) is carried out: a control law or algorithm that ensures the required properties of the control system and the behavior of the controlled system is designed. note that this stage is implemented in the “world of models.” in the process of concretization (iii), the synthesis results are implemented in a real control system that performs “regulation” (iv) of a real controlled system. assessment (v) is intended to identify the system, to compare the results of its activity with the goals, to make corrections if necessary (when the expected and observed behavior of the controlled system does not match one another), and to proceed to the next cycle. fig. 4 illustrates the significance of distinguishing between two worlds, i.e., the real world in which the systems (the controlling system, the controlled system, etc.) objectively exist, and the “world of models” in which the systems exist “subjectively.”3 note that the researchers operate the systems in the “world of models.” therefore, the interaction of control theory, modern psychology, and methodology is discussed in the “world of models”; also, see arrows 1 and 2 in fig. 2). in particular, the matter concerns the information about the “internal structure” of the controlled and controlling subjects, which are necessary for the formulation and solution of control problems. one of the classification bases for control systems is the type of control, determined by the degree of automation of control functions. depending on whether the elements of a control system include a man or not (who is always a subject!), there are three (out of four logically possible4) combinations—the types of control presented in table 1. a controlled system may have technical, natural, biological, social, and other components. automatic control corresponds to the case when both the controlling system and the controlled system are technical or cybernetic objects. automated control corresponds to the case when the controlling system is a subject, for example, man-machine systems in which the most important and responsible decisions are made by people. the controlled system is a technical or cybernetic object.5 table 1. types of control type of control subject of control controlled system automatic control (object–object) control device, controller controlled object automated control (subject–object) controlling subject controlled object organizational control (subject–subject) controlling subject controlled subject organizational control corresponds to the case when both the controlling system and the controlled system are subjects, i.e., when people control people. in this paper, automated control and organizational control will be considered. both this types of control involve man. if a man is included in a controlled system, then control of a human is nothing more than control of his behavior; see the definition of control above. in turn, behavior is an external reflection (manifestation and result) of his activity. therefore, the goal of control can be formulated as the desired result of the activity of a controlled subject. 3 system analysts are used to say, “any system is a model.” 4 the fourth type of control is the subject–object one, when a controller “operates” a human being. it also occurs, but will not be considered separately. some examples are simulators (control devices guiding the learner's activity), pacemakers, traffic lights (control devices regulating the behavior of drivers at crossroads), etc. 5 an automated control system generally includes both “automatic” and “human” components. for example, in industrial process control, information about the state of a controlled object can be acquired and processed automatically. this information is used by a human operator to makes an appropriate decision (perhaps, in combination with some “advising” information, which is automatically generated by a decision support system). then the decision is brought to the controlled object “automatically.” if a human being controls an object without automation, then such control is called manual (e.g., cycling). control, activity, personality 117 copyright ©2020 assa. adv. in systems science and appl. (2020) it’s almost impossible to control human behavior directly (the army is the ultimate example here). however, one can affect the activity of a man by indirectly influencing his behavior. activity is a nontrivial process with its own internal structure (an objective one as well as the one within the existing scientific principles). hence, the description and analysis of this structure, the identification of different subject matters of control (the components of activity that can and should be purposefully influenced), are necessary for formulating solving control problems for the systems involving people. moreover, activity is performed by a personality, who also has its own internal structure. the attributes of this structure significantly affect and often predetermine the process and results of activity. they have to be taken into account, since the components of personality structure can also be the subject matters of control. well, considering a personality in the role of a controlled system, for influencing its behavior one should analyze the interconnected structures of the personality and its activity. indeed, behavior is the “result of activity” see fig. 5 and fig. 14. in the case when a man performs control himself (being an element of a control system, or the entire control system, see fig. 1), he implements control activity [16]. his personality influences both the process and results of this activity. fig. 5. basic categories: personality, activity, control due to the aforesaid, it is necessary to consider the relations between the following categories: personality, activity, and control. in this paper, six aspects (see arrows in fig. 5) will be identified as follows: 1) the influence of personality attributes/traits on its activity (see section 7). 2) the influence of activity on personality. this phenomenon has been known for a long time and has been deeply studied in philosophy and psychology (the development of personality, first of all, its experience and volitional components) [19]. therefore, it is beyond the scope of the paper. 3) control of the components of personality structure (for their formation and development, see section 5). it is reasonable to distinguish between two cases: 3.1. self-development of personality (the subject controls these components himself); 3.2. the development of personality structure of one subject controlled by another subject, which is often implemented through education (the development of personality experience; see [17] and below). 4) the influence of personality attributes on the control activity performed by it or on the controlled activity. these phenomena are the subject matters of the psychology of management and organizational psychology, respectively, and are beyond the scope of the paper. 1 2 3 4 5 6 personality activity control 118 d. novikov copyright ©2020 assa. adv. in systems science and appl. (2020) 5) the specifics of control activity, which are the subject matter of control methodology [16] (also, partially considered within the methodology of complex activity [1]). they are beyond the scope of the paper as well. 6) control of activity components (see section 8). like in item 3, it is reasonable to distinguish between two cases: 6.1. self-control (the subject controls these components himself); 6.2. the activity of one subject controlled by another subject. the goals of this paper are to systematize: the structures of personality and activity; the influence of personality attributes on its activity; control of the components of the structures of personality and activity, as well as to harmonize the ideas about them among personality psychologists, methodologists, and control theoreticians. this paper is organized as follows. first, the psychological theories of personality (section 2) and the theories of activity (section 3) are discussed in brief. then, the methodology of complex activity (section 4) is sketched. such an approach makes it possible: to suggest the systemic structures of personality (section 5) and activity (section 6); to establish the correspondence between the components of personality structure and the structural components of the performed activity (section 7). the problematics of control are studied in the concluding part of the paper (section 8). 2. personality structure: history in soviet and russian psychology and philosophy, first the approach introduced by s.l. rubinstein and k.k. platonov should be mentioned. it will be adopted below to describe the components of personality structure and their relation with the components of activity structure. s.l. rubinstein [21] (the 1930s–1940s) proposed a personality structure that includes personality directivity, experience (knowledge, competences, skills) and individual typological characteristics (manifested in temperament, character, and capabilities). k.k. platonov [19] (the 1960s–1970s) suggested a four-level personality structure with the following substructures: personality directivity, experience, features of psychic (mental) processes, and biopsychic attributes. he reflected this structure in the system (!) of psychological concepts; see [19]. platonov’s system is the most structured one to date. the existing psychological approaches (in particular, the ones mentioned above) focus on various aspects (such as consciousness, attitude, etc.), are mainly conceptual in nature, have little appeal to experiment, and are weakly structured. in other words, the relations between the basic categories are not analyzed at the general systemic level, the completeness and redundancy of their set are not justified, and their operational use is awkward. for the sake of justice, note that these problems are typical not only for the soviet and russian psychology of personality, but also for any psychological research schools. in any monograph or handbook on the personality psychology (for example, see [4, 9, 10, 12, 22– 24]), the following four groups of personality theories are distinguished: 1) psychoanalytic/psychodynamic theories (s. freud, a. adler, e. erickson, c. jung, and others); 2) humanistic/phenomenological theories (a. maslow, c. rogers, and others); 3) theories of character traits (g. allport, h. eysenck, r. cattell, and others.); 4) social cognitive theories (g. kelly, a. bandura, and others). (in some of such books, the theories of learning, etc., are also included.) the approaches of american and european authors are more oriented towards experimental data; however, they are even less systematic and, in the final analysis, are mostly limited to the networks of correlations between personality traits measured in weak scales. control, activity, personality 119 copyright ©2020 assa. adv. in systems science and appl. (2020) the psychology of personality considers, among other things, individual characteristics, cognitive and motivational processes, volitional and emotional components, personality development, biopsychic grounds, social impact, and interaction. the psychology of personality has close ties with physiology, psychiatry, and sociology. 3. activity structure: history consider in brief the development of activity theories in psychology and methodology. s.l. rubinstein [21] (the 1930s–1940s) proposed the following structure of activity: “need → motive → activity.” (3.1) (hereinafter, the sequence of components reflects the logical and/or causal relations between them.) a.n. leontiev [12] (the 1950s–1960s) added the goals, tasks, and conditions of activity to the chain (3.1). in addition, he emphasized that the activity has a hierarchical structure “activity–action–operation.” k.k. platonov [19] (the 1960s–1970s) considered the following structure of activity: “goal → motive → method → result.” (3.2) a.m. novikov [15] (the 1990s) introduced the structure of activity: “need → motive → goal → methods and means → result.” (3.3) also, a.m. novikov’s idea was to consider the process of self-control/self-regulation, in which the subject corrects the procedural components of the activity by comparing the goals and results of activity, as well as by performing self-appraisal. in the 2000s, he added to the consideration the process of assessing/reflecting the results of activity, as well as the conditions, norms, and criteria of activity [15]. clearly, the “chains” (3.1)–(3.3) of the procedural components of activity were sequentially complicated, a next chain based on a previous one. following the development of methodology, more and more complex “activity models” were constructed, taking into account an increasing number of factors. (also, see (6.1) below.) control methodology and the theory of control in organizational systems (the 1990s– 2000s) involve a scheme presented in fig. 6 (for details, see [1, 15, 18]). fig. 6. activity structure [15] goal result need, motive en v ir o n m en t corrections action en v ir o n m en t a ss es sm en t norms principles environment conditions tasks technology (forms, methods, and means) c rit er ia self-regulation 120 d. novikov copyright ©2020 assa. adv. in systems science and appl. (2020) within organizational control (table 1), when both the controlling system and the controlled system are subjects, each of them performs the corresponding activity. their interaction is a control activity. (recall that control activity is an activity to organize (and regulate) another activity [15, 18].) a direct combination of fig. 1 and fig. 6 gives the structure of control activity shown in fig. 7. fig. 7. structure of control activity [16] despite the worldwide fame of his work, a.n. leontiev has only one notable follower among western psychologists, finnish psychologist and teacher j. engeström [7]. his first monograph [5] was published in 1987, and its second edition [6] appeared in 2012. almost all modern studies on activity theory refer to the non-hierarchical (!) structure of activity (chiefly, educational activity) described by engeström in [6, p. 78]; for example, see [20]. today, activity theory is widely used and developed, probably, only in the models of humancomputer interaction. for example, see a special issue of the scandinavian journal of information systems (2000, vol. 12). the other concepts of activity in the western psychology of personality (e.g., see the fundamental reference book [10]) are weakly structured in the sense discussed above. 4. methodology of complex activity presented in fig. 6, the structure of the procedural components of activity does not imply any decomposition of the goals, technology, and result of activity (and does not contain appropriate procedures). the description of the result or technologies in the form of nondetailed and unstructured objects significantly restricts the applicability of methodology for the study of practically interesting non-trivial examples of activity in complex social or organizational-technical systems. therefore, the methodology of complex activity was developed in the monograph [1] as follows. goal result need, motive corrections action a ss es sm en t requirements, norms principles conditions tasks technology (content and forms, methods and means) c ri te ri a self-regulation goal result need, motive corrections action a ss es sm en t requirements, norms principles conditions tasks technology (content and forms, methods and means) c ri te ri a self-regulation controlling subject controlled subject external disturbance control state of controlled system control, activity, personality 121 copyright ©2020 assa. adv. in systems science and appl. (2020) first, the concept of a structural element of activity (sea) was introduced therein (see fig. 8); in addition to activity (2), this element includes its subject (actor) (1) and subject matter (3). second, the concepts of elementary and complex activity were introduced therein: an elementary activity is an activity whose goals, technologies, and result have no non-trivial internal structure. fig. 8. structural element of activity [1] in particular, elementary activity is characterized by the three conditions as follows: 1) the technology does not change in the course of activity, and the activity itself has clear limits for the observer (researcher, subject of activity, consumer of its results). 2) the subject matter is single and changes in the course of activity according to the technology (which is actually the goal of activity); however, the subject matter does not change its place and role in the context (the requirements applied to the subject matter of activity remain the same). 3) the subject performing activity is single, clearly defined and limited, and also undergoes no transformations in the course of activity. in the case of elementary activity, the structure in fig. 6 suffices to describe the activity itself; there is no need to consider the subject matter and subject together with the activity itself. really, they play the role of a clear context: during the period of activity, only the subject matter evolves in accordance with the technology used by the subject. in contrast, a non-elementary activity was called complex in [1]. in other words, a complex activity is an activity with a non-trivial internal structure, with multiple and/or changing goals, subject, technology, and the role of the subject matter in its goal context. third, the temporal structure of any activity (elementary or complex) was proposed and considered in [1]; see fig. 9. actor technology actions result subject matter 1 2 3 needs goal tasks 122 d. novikov copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 9. phases and stages of life cycle of complex activity [1] fourth, the logical, cause-effect (causal), and process structures of complex activity (including a hierarchy of seas and elementary operations) were constructed in [1]. table 2 below describes the organizational specifics of activity during different periods of human development, from the stone age to the present day. the main conclusions that can be drawn from such a periodization are as follows. 1. starting from the appearance of man, human activity is characterized by multiple and changing goals and other attributes of the definition of complex activity. in other words, human activity has always been complex and the methodology of complex activity provides a uniform description for complex activity in all periods. unlike other living creatures acting jointly (a swarm of bees, a colony of ants, a pack of wolves, etc.), man organizes all types of complex activity (labor, learning, play, creativity, communication), i.e., performs structuring of the subject matter, in accordance with the goals of complex activity, not instinctively but consciously, as the forms of control. the second distinctive feature of human complex activity is the use of artificial means of activity. adv syst sci appl 2020; 03; 113-135 published online at https://ijassa.ipu.ru. table 2. organizational specifics of activity during different periods of human development social structures (k. marx) 2000 a.d. 0 500 a.d. 1000 a.d. 1500 a.d. 10000 b.c. 5000 b.c. … … primitive-communal system slave system feudalism capitalism … communism types of organizational culture (v.a. nikitin, a.m. novikov) … mass types of practical activity sources of energy muscle force of people and animals natural sources (water, wind) hydrocarbons (steam, electricity) nuclear energy, res hunting, fishing, collecting cattle-breeding, agriculture craftsmanship industrial production dominating types of production piece batch mass methods of normalization and translation of activity information production traditional myths and rituals corporate-handicraft sample and recipe for its recreation professional theoretical knowledge in the form of text projecttechnological projects and programs knowledge information models organizational forms of collective activity community state, army church workshop enterprise corporation extended or virtual enterprise time dominating links between actors performing joint activity organization technology kinship capital language faith property 124 d. novikov copyright ©2020 assa. adv. in systems science and appl. (2020) 2. besides the monotonously growing complexity of activity (the depth and width of its hierarchical logical structure), technologies have been the only evolving factor of complex activity. however, even for technologies, it is difficult to identify certain “historically specific” forms and methods of activity. the means of activity have been evolving! that is, in the course of human development, the means of complex activity and methods for performing “industry-specific” elementary operations (industry-specific technologies) have been created and further developed. at the system-wide level of generalization, the technology of complex activity has been remaining invariant: the target (logical) structure and the cause-effect structure, as well as the process model of complex activity as a universal algorithm for managing and/or implementing the life cycle of complex activity. 3. the maximum complexity of the projects implemented in different historical periods (which can be described, e.g., by the spatial dimensions of objects created and the number of their “elements,” the duration of projects and the number of their participants) has been demonstrating a moderate growth over time. 4. the gradually accelerating development of technologies has led to the mass creation of more and more complex artificial systems, which allows achieving results with fewer resources (time, energy, etc.). in fact, “the set of achievable results” over the entire history of mankind has not changed much (a few exceptions are: hydronautics, astronautics, etc.). thus, the typical architecture formalized by the methodology of complex activity is universal for any types of activity, throughout the entire past history of mankind. hence, it can be hypothesized that such universality will hold in the future (at least, until the set of subject matters of activity that is accessible to mankind changes). 5. systemic personality structure in the previous sections of the paper the history of personality theories and activity theories has been briefly traced. methodology provides an operational and systemic structure of activity, including the set of its elements with the functional, causal, and other relations between them. personality theories, however, are still lacking these important properties. this section is an attempt of maximal systematizing of main categories, basing on the personality structure introduced by k.k. platonov [19]. this will be done by analyzing the basic terminology, i.e., by establishing the cause-effect and particular-general relations between the relevant concepts. the analysis results are demonstrated in fig. 10. 1.1. incentive forms inclination aspiration 1.2. view forms 1. p er so na lit y di re ct iv ity 3. p sy ch ic pr oc es se s a nd p ro pe rt ies 3.1. cognition forms (processes and attributes) intellect, intelligence attention memory intellection 3.2. emotion-volition forms (processes and attributes) senses, feelings will 2. experience 4. biopsychic attributes temperament 6. character 5. capabilities giftedness imagination emotions talent competence habits skills knowledge worldview valuation value orientation life position ideal norm aspiration level principle belief attitude opinion judgment position image sensation perception desire, wish interest drive, attraction control, activity, personality 125 copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 10. systemic personality structure the resulting structure agrees with k.k. platonov’s hierarchy and has the following form6; see fig. 10. 1. personality directivity: 1.1. incentive forms: drive, attraction → desire, wish → interest → inclination → aspiration; (5.1) 1.2. view forms––worldview, valuation, value orientation, image,7 and life position, which includes three groups of components (fig. 11): opinion, judgment, position; ideal, norm, aspiration level; principle, belief, attitude. 2. experience––knowledge, competence, skills, habits. 3. psychic (mental) processes and attributes: 3.1. cognition forms: processes sensation → perception → image and capabilities –– intellect, intelligence (attention, intellection, memory, imagination). (5.2) 3.2. emotion-volition forms: will, senses, feelings, and emotions. 4. biopsychic attributes (temperament). 5. capabilities (giftedness, talent). 6. character. of course, the six groups of components of the personality structure (see the list above) are not isolated. for example, the appearance of inclination requires not only the presence of interest, but also the strong-willed aspiration to implement the latter in activity [19]; images, opinions, and judgements are naturally associated with the experience of an individual, etc. unfortunately, modern psychology has not suggested any systemic picture of such relations between different components of the personality structure so far, but their relevance is already obvious even to the representatives of adjacent sciences. among these six groups of components of the personality structure, additional systematization is needed for the view forms of personality directivity, or rather, the components of a life position. (indeed, their number is quite large, and the relations between them were not clearly established in the literature; in particular, any “structuring” like (5.1) or (5.2) has not been proposed to date.) to do this, consider the process by which a subject assesses the state of a certain system, as is illustrated in fig. 11. the state of this system is described in the state space by some indicators, measured in appropriate scales. the subject assesses the functioning of this system according to some (perhaps its own internal) criteria, also measured in appropriate scales. generally speaking, the assessment process consists in the transition from the state space of the system to the criteria space (fig. 11), i.e., in establishing a relationship between the values of the indicators of the system’s state and the values of the criteria (in a special case, the criteria can coincide with the indicators). an opinion, a position, and a judgement (in a certain sense, all three terms can be regarded as synonyms for subjective appraisals, possibly comparative and multicriteria) are “points” in the criteria space. according to this terminology, a norm is a system of criteria, i.e., a set of their “admissible” values; a principle is a norm of activity (including assessment); the aspiration level is the self-appraisal of a personality (i.e. a special case of self-assessment in relations with the environment); beliefs are a set of criteria, norms, ideals, and assessment results (i.e., opinions, positions, and judgements) in accordance with them; 6 note that genetic, neurocognitive, and many other related aspects remain almost beyond consideration. in recent time, they have been given much attention within instrumental studies at the junction of psychology and physiology, cognitive sciences, etc. 7 images can be treated as a view form (item 1.2 of the list) as well as a cognition form (item 3.1). 126 d. novikov copyright ©2020 assa. adv. in systems science and appl. (2020) an attitude is the readiness for a certain activity in a typical (assessed) situation according to beliefs and principles. as a matter of fact, the representation of the reflection process shown in fig. 11 is somewhat simplified. it would be more correct to consider the following chain (see also cognitive processes in fig. 10 and (5.2)): “objective state space → sensation → perception → image → → subjective state space → criteria space,” (5.3) thereby attributing reflection in fig. 11 to the last pair of this chain. fig. 11. view forms of personality directivity the subjective state space, existing in consciousness, reflects the subject’s awareness–– the information (the set of images) the latter has at the time of decision-making, at the time of the assessment process, etc. this information can be either narrower than the real world (due to insufficient observations, limited sensitive and cognitive capabilities of the subject, etc.) or wider (e.g., due to the results of the subject’s auto-reflection, or second-kind reflection [14], as well as due to fantasy and creativity). an important role in the content of the subject’s awareness and in the formation of this content can be played by other subjects interacting with the former, which is the subject matter of social psychology. concluding this section, let us analyze the components of the personality structure from the viewpoint of their “dynamism” (as the reciprocal of the characteristic time required for changes), i.e., the possibility of purposeful change by the subject itself (in the process of selfdevelopment) or by other subjects (in the process of control). components 1–6 in fig. 10 are ordered almost in the descending order of dynamism. indeed, the biopsychic attributes, capabilities, and temperament are the least susceptible to changes, although they can and must be formed and developed during training [19], but the time and effort expended for this are great. the emotion-volition forms and cognitive abilities (intelligence) also change rather slowly (in the process of exercises [17, 19]). concrete cognitive processes (sensation – perception – image) are much more dynamic and timely. experience is formed and then developed rather slowly in the process of education. personality directivity is formed in the process of upbringing and then developed in the process of activity. the components of the view forms of personality directivity have significantly different characteristic times. for example, the upper-level components (worldview, valuations, value orientations, and life position) change extremely slowly; principles, beliefs, and attitudes, slightly faster. the ideals, norms, and aspiration level are even more dynamic. and the processes of forming concrete subjective appraisals (opinions, judgements, and positions; see (reflection process) criteria space belief criterion 2 assessment criterion 1 indicator 3 indicator 1 indicator 2 environment consciousness state space 2 1 1 2 principle at tit ud e ideal opinion judgement position norm control, activity, personality 127 copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 11) are timely. the components of the incentive forms of personality directivity have almost the same dynamism. 6. systemic activity structure a direct combination of fig. 6, fig. 8, and fig. 10 gives the systemic activity structure presented in fig. 12. it includes the subject of activity, the activity itself (the procedural components are indicated by a dashed rectangle), and the subject matter of activity. the systemic activity structure in fig. 12 allows identifying three groups of internal components of activity: psychic (mental), procedural, and behavioral ones; see fig. 13. the psychic components of activity are the components of the personality structure, need, motive, reflection (more precisely, auto-reflection), assessment, and self-regulation. the procedural components of activity (cf. (3.1)–(3.3)) are: need → motive → goal → tasks → technology → action → result → reflection/assessment. (6.1) the behavioral components of activity are action, the state of the subject of activity, and result. the external components of activity are criteria, norms, principles, requirements, and conditions. fig. 12. systemic activity structure intention goal result need, motive en vi ro nm en t action requirements, norms principles conditions tasks technology (forms, methods, and means) self-regulation criteria subject matter reflection, assessment subject 1. directivity 2. experience 3. psychic processes state psychic (mental) components of activity behavioral components of activity procedural components of activity 4. biopsychic attributes 5. capabilities 6. character control of components of activity structure control of components of personality structure i ii iii 128 d. novikov copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 13. internal components of activity note that the systemic activity structure under consideration is “maximal”, and in particular cases, real activity can be described by a fragment of this general structure. for example, as the result of an impulse, the subject can perform some action not quite consciously (without formulating a clear goal, without decomposing it into tasks, without choosing an appropriate technology, etc.). another example is the so-called regular activity (see the classification of activity types in [1]) in which goals are fixed (and often set externally), the technology is mastered and does not change over time, and uncertainties do not occur (there is no need to develop the technology for new conditions of activity). regular activity is adequately described only by the behavioral components (“action → result,” etc.). the systemic activity structure presented in fig.12: includes/further develops the well-known approaches to the description of activity structure; see sections 3 and 4, as well as the publications [7, 12, 15, 19] on the psychology of activity and the monographs [1, 15] on methodology; integrates the well-known approaches of psychology to the description of personality structure; see sections 2 and 5, as well as [11, 19, 23]; identifies one external and three internal substructures of activity: the psychic (unobservable from the outside), procedural (“partially observable”), and behavioral (observable) components of activity; structurally coincides with the sea model and can be integrated [1]; is an input-output model with feedback,8 a classical model for control theory [13]; is complementary to decision theory, game theory, theory of collective behavior (the models of opinion dynamics and action dynamics under the influence of the environment [2, 3]), the models of learning (the development of experience) [17], and the theory of control in organizations [18]; see section 8. 7. personality and activity now let us establish the correspondence between the systemic structures of personality ( fig. 10) and activity (fig. 12). for this, consider a scheme in which the components of the personality structure influence the activity; see fig. 14 and fig. 15, where the 8 more precisely, with two feedback loops. one loop (through self-regulation, see arrow iii) corresponds to the timely (insignificant, “automatic”) correction of the activity components by the subject. the second loop (through the subject itself, see arrows i and ii) corresponds to relatively slower and more significant corrections; see also arrow 2 in fig. 5. activity psychic components: components of personality structure, need, motive, reflection, assessment, and self-regulation procedural components: need, motive, goal, tasks, technology, action, result, and reflection/assessment behavioral components: action, state of subject matter of activity, and result control, activity, personality 129 copyright ©2020 assa. adv. in systems science and appl. (2020) significant/determining influence is reflected by a continuous arrow, and the less obvious one by a dashed arrow. for the procedural components of activity, the following assumptions seem natural: for need and motive, the incentive forms of personality directivity are primary; for goal, the view forms; for tasks, cognitive processes and attributes; for technology, the experience of a personality; for action, its biopsychic attributes; for reflection/assessment, the view forms of personality directivity. fig. 14. influence of components of personality structure on procedural components of activity (see arrow i in fig. 12) the external components (in the first place, the constraints and conditions of activity) significantly depend on the environment. but personality characteristics are also important for the external components of activity. for example, the norms of the subject’s activity are influenced by social, corporate, and group norms (on the one hand) and its personal norms (on the other). for the external components of activity, it can be assumed that the view forms of personality directivity play the key role. various personality characteristics have the greatest influence on the norms of activity and the criteria for assessing its results; see fig. 15. of course, the arrows in fig. 14 and fig. 15 do not exhaust the influence of “personality” on “activity.” an in-depth study, systematization and justification of this influence are global and topical problems for future research. p e r s o n a l it y 1.1. incentive forms 1.2. view forms 1. p er so na lit y di re ct iv ity 3. p sy ch ic pr oc es se s 2. experience 4. biopsychic attributes 3.1. cognitive processes and attributes 3.2. emotion-volition processes and attributes 5. capabilities 6. character a c t iv it y need motive goal tasks technology action result reflection, assessment self-regulation 130 d. novikov copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 15. influence of components of personality structure on external components of activity (see arrow ii in fig. 12) 8. control as it has been underlined in section 1, the goal of control of a subject is to ensure the desired result of its activity. this statement equally applies to the case when a subject controls his activity himself (see fig. 12) and to the case when the goals are set by another subject (see fig. 7). the set of different factors (the components of the personality structure, the external components of activity, and the influence of the environment), as well as the relations between them (“the source of influence” → “the subject matter of influence”) and the components of activity (see causal relations in fig. 12 and fig. 14), can be used to construct models of activity control. (for details, see [16, 18] and chapter 7 of the monograph [1], devoted to the management of complex activity.) such models are based on the chains of relations between the components of the structures of activity and/or personality. the beginning of the chain is a component of the personality structure (or a component of the activity structure) to which control is applied, the subject matter of control; the end of the chain is always9 the result of activity. the most complete, but at the same time the least specific, is the set of all the relations between the entities in fig. 12 and fig. 14. any model implies abstraction; therefore, in each case, some of the elements of the chain can be omitted (eliminated from consideration). indeed, the model allows getting abstracted from the intermediate elements of the chain. but the researcher needs to keep all the elements in his field of vision, in order to “see the big picture”, understand clearly the assumptions introduced (what is abstracted, which factors are consciously neglected) and be sure of the observability/measurability of the factors used. the issues of measuring the personality traits of people were discussed, e.g., in the reference book [24, 25]. here are some examples. the theory of control in organizations rests on the assumption that the choice of action by the subject is a decision-making process described by a corresponding mathematical model (a decision rule, often a game-theoretic or collective behavior model; see below). 9 if the researcher does not solve any particular control problem, the “chain” of his interest may not be terminated by the result of activity. anyway, he needs to see the fragment of his interest within the general personality and/or activity structure. a c t iv it y conditions norms principles criteria requirements/ constraints p e r s o n a l it y 1.1. incentive forms 1.2. view forms 1. p er so na lit y di re ct iv ity 3. p sy ch ic pr oc es se s 2. experience 4. biopsychic attributes 3.1. cognitive processes and attributes 3.2. emotion-volition processes and attributes 5. capabilities 6. character control, activity, personality 131 copyright ©2020 assa. adv. in systems science and appl. (2020) within this theory, there are several types of control, classified by the subject matter of control; numerous models were given in the book [18]. 1. institutional control (control of the constraints (conditions, requirements, and principles) and norms of activity) is the most stringent types of control. here the controlling subject deliberately restricts the set of all admissible technologies, norms, principles, actions, and results of activity of the controlled subject. such a restriction can be implemented through explicit or implicit impacts, i.e., legal acts, orders, etc., or moral and ethical standards, corporate culture, etc. the resulting “model chain” has the following form: “constraints and norms of activity → decision rule → result of activity.” (8.1) 2. motivational control (control of preferences, i.e., the criteria of assessing the results of activity, both by the controlled and controlling subjects) is “softer” than institutional control. here the preferences of the controlled subject are changed in a purposeful way. such a change can be implemented by introducing a system of incentives for choosing proper actions and/or for achieving proper results of activity. the resulting “model chain” has the following form: “criteria for evaluating performance → decision rule/model → action → result of activity”. (8.2) the “real chain” on the example of motivational control has the following form: “criteria for assessing the results of activity → motive → goal → tasks → technology → action → result of activity.” in other words, the fragment “→ motive → goal → tasks → technology →” is immersed in the decision model. 3. informational control is the “softest” (indirect) type of control, as compared with institutional and motivational ones. here the subject’s awareness (beliefs and opinions) about essential parameters are properly changed [14]. first of all, the matter concerns the parameters of the environment. such essential parameters are often the opinions, actions, or results of activity of other subjects (e.g., the so-called social pressure, studied in social psychology). a subject has some degree of trust to other subjects and considers their reputation to change its opinion in accordance with a particular model of opinion dynamics. the two most widespread classes of mathematical models of opinion dynamics are the models of consensus (in which the subject “averages” its opinion with the opinions of those subjects with which it interacts; see [3]) and the so-called threshold models (in which the subject demonstrates conformity behavior: being subjected to social pressure, the subject makes its own decision or agrees with some opinion only when the proportion of other subjects behaving in this way exceeds its own conformity threshold; see [2]). the interested reader is also referred to [11]. the complete chain is as follows: “environment → sensation → perception (+ emotions) → image (+ experience) → → opinion → belief (+ experience + will) → attitude → motive → goal → tasks → → technology → action → result of activity.” (8.3) the corresponding “model chain” for (8.3), which incorporates some fragments of (8.3) within the opinion dynamics model and the decision rule, has the following form: “environment → belief (+ experience) → opinion dynamics model → → opinion → decision rule → action or result of activity.” (8.4) in a particular case, beliefs may coincide with opinions, and the resulting opinion of the subject can be considered the final result. note that the decision rule in (8.4) can be treated as an attitude model, i.e., a mapping of opinions into actions; also, see standard solutions in [1]. interestingly, in (8.1), (8.2), and (8.3), the action or result of activity is the only “variable,” which is observable (at least, partially detected by objective means). the 132 d. novikov copyright ©2020 assa. adv. in systems science and appl. (2020) overwhelming majority of modern models suggested by the theory of control in organizational systems consider this particular case, regardless of the exact action of the controlled subject (e.g., the information about the uncertain parameters reported by the latter to the controlling subject; the volumetric or temporal characteristics of its activity). the model chain (8.4) contains two “variables”: one “internal” (perhaps directly unobservable), which is the belief or opinion of the subject, and the other “external,” which is the action or result of its activity. on the one hand, chain (8.4) has been underinvestigated: the only exceptions are several strategic behavior (manipulation) models for the opinions of subjects [3], probabilistic models (bayesian models or controlled markov processes), and “logical” models, widely used in artificial intelligence and robotics within the bdiarchitecture of software agents (b–“belief,” d–“desire,” and i–“intention”). on the other hand, it is “richest” and most promising for future research. in order to illustrate the transition from the general activity scheme (see fig. 12) to the basic model of control in organizational systems, consider a somewhat simplified model of the subject’s decision-making about its actions. the details can be found in [18]. assume that a subject (an actor of some sea) chooses actions (strategies, states, etc.) from a set a of all admissible actions. hereinafter, an action will be denoted by y (y î a). let a parameter r ∈ r, further called the type, reflect all significant individual characteristics of the subject. in many applications, the subject’s type is interpreted as the efficiency of his activity, or the rate of learning, or the most desired result of activity [18]. the choice of an action y î a, together with the impact of an environment, lead to a result of activity, further denoted by z î a0; here a0 is the set of all admissible results of activity. a possible noncoincidence of the subject’s action and result of its activity is due to the impact of the environment, in particular, the actions of other subjects, etc. the functional relation between the subject’s action y î a and the result z î a0 of the activity may have a complex character, described by probability distributions, fuzzy information functions, etc. [18]. for the sake of simplicity, let the result of activity be a known function w(∙, ∙) of the action and the state of the environment " ∈ ω: z = w(y, "). this parameter reflects the impact of the environment. the function w(∙, ∙) is often called the technological function. it describes all variations in the state of the subject matter of activity due to the actions of the subject and the impact of the environment. (recall that the result of activity is the state of its subject matter achieved in the course of activity.) assume that the subject has preferences on the set of all results z î a0, being able to compare different results of activity with each other. the subject’s preferences will be described by its utility function (goal function, payoff function) f: a0 × r → â1: of two results of activity, the subject prefers the one corresponding to a greater value of its utility function. the subject’s goal is to maximize the utility. when choosing an appropriate action y î a, the subject considers its own preferences and the effect of the action on the result of activity z î a0. given the type and the state of the environment, the subject will choose an action maximizing its utility: . (8.5) expression (8.5) describes the decision rule model. subject’s choice depends on: the set of admissible actions a; the technological function w(∙, ∙); the utility function f(∙, ∙); the state of the environment " ∈ ω; the subject’s type r ∈ r. control consists in a purposeful influence on each of these factors or their combinations. (here the term “purposeful” means that the influence shall lead to the desired action of the ( ( ) ( ), , ) arg max ( ( ) )* y a y a,w , f r f w y, ,rq q î × × = ( ( ) ( ), , )*y a,w , f rq× × control, activity, personality 133 copyright ©2020 assa. adv. in systems science and appl. (2020) subject or the desired result of its activity.) depending on the subject matter of control, different types of control are distinguished. for example, the influence on the set of all admissible actions of the subject (the conditions, requirements, and norms of activity; see the general activity scheme in fig.12) and the technological function (the technology of activity) is institutional control; see (8.1) and [18]. the influence on the subject’s utility function (the criteria for assessing its activity in terms of the general activity scheme in fig.12) is motivational control; see (8.2) and [18]. the influence on the subject’s awareness of the state of the environment (and, possibly, its awareness of the values of other factors) is informational control; see [14, 18]. the influence leading to a change in the subject’s type can be interpreted as control of components of personality structure. within this model of the subject’s decision-making, the general activity scheme (fig. 12) takes the simplified form presented in fig. 16 (here, abstraction consists in omitting a number of entities, and some relations become trivial or degenerate). the scheme in fig. 16 is an “elementary brick,” i.e., a particular model of the structural element of activity (see section 4) of the subject making a decision on its action. such bricks can be used to describe uniformly even more sophisticated types of complex activity and their control, in particular, in multi-element, multilevel, dynamic systems operating under uncertainty [1, 2, 3, 13, 14, 18]. fig. 16. decision model, activity, and basic types of organizational control for example, consider the set n = {1, 2, …, n} of interacting subjects. hereinafter, the subjects will be indicated by subscripts and the time instants by superscripts. for the sake of simplicity, let the state of the environment θi for subject i be the vector y-i = (y1, …, yi result z î a0 action y î a conditions, requirements, norms (a) technology ( )w y,q criteria (f(∙, ∙)) subject matter subject r r en vi ro nm en t control of components of personality structure goal state of environment informational control motivational control institutional control 134 d. novikov copyright ©2020 assa. adv. in systems science and appl. (2020) 1, yi+1, …, yn) of the actions of all other subjects. the decision rule of the subject can be written as , % ∈ &. (8.6) expression (8.6) is the definition of a nash equilibrium in the normal form game. the (institutional, motivational, or informational) control problems for the action vector (8.6) are formulated and solved by analogy with (8.5), taking into account the corresponding specifics (see [1, 18]). if the subjects make their decisions { } at the time instants t = 0, 1, 2, … , then their decision rules can be described by the mapping , % ∈ &, (8.7) where is the action vector of the subjects at the time instant t; is the control variable; , % ∈ &. under definite condition, the dynamic system (8.7) may have a fixed point (8.6). the approaches to solve the analysis and synthesis problems for the controlled dynamic system (8.6) were considered in [1, 13, 18]. 9. conclusion let us return to the problem formulated in section 1 of this paper: find an appropriate description of a controlled system in terms of the personality psychology that can be used to state and solve control problems for the systems containing man. conventionally, the general scheme of interaction between psychologists and control theoreticians can be described as follows: 1) create and negotiate by both sides a single system of basic concepts characterizing the subject matter. 2) create and negotiate by both sides the representations of the basic structures (a combination of factors and their relations) of a personality and its activity as the subject matters of control. 3) formulate jointly the specific tasks of research in the form of models (“fragments” of the basic structures, like the “chains” described above, primarily (8.4)), with detailed specification of the following elements: the processes and phenomena to be studied, their qualitative and quantitative (if possible) description; the possibilities for “measuring” the key characteristics and identifying both the source data and the modeling results. 4) develop and study specific mathematical models (chiefly, by control theoreticians). 5) discuss jointly and give practical interpretations of the modeling results, perhaps with return to stages 3 and 4. this paper is a control theory-based view on stages 1 and 2. probably, the next steps are to coordinate the positions of representatives of various branches of science on stages 1 and 2, with further transition to more concrete research. indeed, in this paper, the structures of personality and activity have been systematized, and the interconnections of their components have been established. as a result, the control problems for systems containing man have been uniformly described. a promising area of joint researches by control theoreticians, personality psychologists and social psychologists is the accumulation of empirical material that describes the relationship (preferably quantitative) between the key components of the structures of personality and activity. the quantitatively measurable “fragments of chains” (8.1)–(8.4) can be used to construct and explore new applications-relevant control models for organizational, technical and social systems. ( ( ) ( ), , ) arg max ( ( ) ) i i * * * i i i i i i i i i i iy a y a ,w , f y r f w y , y ,r-î × × = t iy 1 1( , ) ( , )t t t t t i i i i iy y r g y ,r u-= = ( )t t t i iy y , yt i iu uî 1 t i n i i ig : a ... a r u a´ ´ ´ ´ ® control, activity, personality 135 copyright ©2020 assa. adv. in systems science and appl. (2020) references 1. belov, m. & novikov, d. (2020) methodology of complex activity foundations of understanding and modelling. heidelberg: springer. 2. breer, v., novikov, d. & rogatkin, a. (2017) mob control: models of threshold collective behavior. – heidelberg: springer. 3. chkhartishvili, a., gubanov, d. & novikov, d. (2019) social networks: models of information influence, control and confrontation. heidelberg: springer. 4. cloninger, s. (2012) theories of personality: understanding people. 6th ed. upper saddle river: pearson. 5. engeström, y. (1987) learning by expanding. helsinki: orienta konsultit. 6. engeström, y. (2014) learning by expanding: an activity-theoretical approach to developmental research. 2nd ed. cambridge: cambridge university press. 7. engeström, y. (2019) the future of activity theory: a rough draft / learning and expanding with activity theory. cambridge: cambridge university press. 8. feist, j. & feist, g. (2017) theories of personality. 9th ed. new york: mcgraw-hill education. 9. handbook of personality: theory and research. (2008) 3rd ed. new york: the guilford press 10. handbook of psychology. vol. v: personality and social psychology. (2003). new jersey: wiley. 11. hunter, j., danes, j. & cohen, s. (1984) mathematical models of attitude change. orlando: academic press. 12. leontiev, a. (1979) activity, consciousness, and personality. upper saddle river: prentice-hall. 13. novikov, d. (2016) cybernetics: from past to future. heidelberg: springer. 14. novikov, d. & chkhartishvili, a. (2014) reflexion and control: mathematical models. boca raton: crc press. 15. novikov, a. & novikov, d. (2013) research methodology: from philosophy of science to research design. amsterdam: crc press. 16. novikov, d. (2013) control methodology. new york: nova science publishers. 17. novikov, d. (2019) regularities of iterative learning. 2nd ed. moscow: institute of control sciences of the russian academy of sciences. 18. novikov, d. (2013) theory of control in organizations. new york: nova science publishers. 19. platonov, k. (1984) kratkii slovar sistemy psihologicheskih ponyatii [a short glossary for the system of psychological concepts]. moscow: vysshaya shkola [in russian] 20. proceedings of the first international workshop on activity theory based practical methods for it-design (2004). copenhagen. 21. rubinstein, s. (1981) probleme der allgemeinen psychologie [problems of general psygchology]. new york: springer-verlag. [in german] 22. schultz, d. & schultz, s. (2016) theories of personality. 11th ed. boston: cengage learning. 23. the cambridge handbook of personality psychology (2009) / ed. by p. corr and g. matthews. cambridge: cambridge university press. 24. the sage handbook of personality theory and assessment. vol. 1. personality theories and models (2008). los angeles: sage books. 25. the sage handbook of personality theory and assessment. vol. 2. personality management and testing (2008). los angeles: sage books. adv syst sci appl 2020; 02:98–118 published online at https://ijassa.ipu.ru. student mixture and its machine learning applications to pvt properties of reservoir fluids nikita volkov1,2*, elizaveta dakhova1,2, semen budennyy1,2, alla andrianova3 1moscow institute of physics and technology, moscow, russia 2center for engineering and technology of mipt, moscow, russia 3gazprom neft science and technology center, st.-petersburg, russia abstract: distribution mixture models are widely used in cluster analysis. particularly, a mixture of student t-distributions is mostly applied for robust data clustering. in this paper, we introduce em algorithm for a mixture of student distributions, where at the e-step, we apply variational bayesian inference for parameters estimation. based on the mixture of student distributions, we construct a machine learning method that allows to solve regression problems for any set of features, clustering, and anomaly detection within one model. each of these problems can be solved by the model even if there are missing values in the data. the proposed method was tested on real data describing the pvt properties of reservoir fluids. the results obtained by the model do not contradict the basic physical properties. in majority of conducted experiments our model gives more accurate results than well-known machine learning methods in terms of mape and rmspe metrics. keywords: student mixture, em algorithm, variational bayesian inference, clustering, regression, anomalies, missing values, pvt properties 1. introduction normal distributions often occur in various data analysis problems and they are fairly well studied. however, their disadvantage for evaluating parameters is their light distribution tails. in the presence of outliers, as it usually occurs in real problems, the parameter estimates are strongly biased towards outliers. to eliminate this disadvantage, the student distribution (or t-distribution) is often considered, because its properties are similar to those of the normal distribution, but it has heavy tails. thus, the student distribution has a certain degree of stability to emissions. the properties of the student distribution were first studied by william gossett. the author has published his first results on that under the pseudonym student. gosset noted that the distribution of the standardized (centered and scaled) normal sample average where the unknown variance is replaced with its estimation is different from the normal one [1]. there are many other theoretical properties of the student distribution. all the most important of them used in the paper are given in section 2. mixtures of normal distributions are often used to describe data. parameters of such a mixture are usually estimated using the em algorithm [6]. for a description of the em algorithm and some theoretical properties of a mixture of distributions, see section 3. if there are outliers in data, it is natural to consider a mixture of student distributions. some ideas ∗corresponding author: volkov.na@cet-mipt.ru student mixture and its machine learning applications to pvt properties 99 for the parameters estimation of the student mixture were described in [7], [8], [9] and [12]. in particular, [7] describes a conditional em algorithm. in this paper, section 4 provides a complete derivation of the parameter estimation method using a similar variation of the em algorithm, which optimizes the variational bayesian inference at the e-step (see [6]). this probabilistic model for data description has many practical applications, which allow it to be flexibly configured and conduct extensive data analytics. let us list the applications discussed in detail in section 5: 1. clustering with a predefined number of clusters. 2. detecting anomalies. 3. regression to predict any set of real features using any other set of features. 4. filling missing values. this large number of applications is due to the fact that the mixture model is generative, since it describes the joint distribution of all features. this distribution also allows to create artificial data. the model of a mixture of distributions can be recommended to solve problems with expected continuous dependence of features between each other, for example, physical problems. in sections 6 and 7, we apply our model to pvt properties of reservoir fluids, where it shows high quality relative to widely known machine learning models. it should be remarked that, the obtained experimental results do not contradict physical laws, unlike outcomes of many machine learning methods, in particular those based on decision trees. 2. distributions and their properties this section provides definitions and some basic properties of the normal and student distributions that are in the basis of the developed model. the properties of the gamma distribution are also given as they are to represent student random vector in a convenient form. all statements in this section can be found, for example, in the literature [1], [2], [3], [4], [5], [6]. 2.1. normal distribution the density of the multidimensional normal distribution centered at a point µ ∈ rd and a symmetric positive definite covariance matrix σ equals q(x|µ,σ) = 1 (2π)d/2 √ det σ exp ( −1 2 (x− µ)tς−1(x− µ) ) , where detς means the determinant of matrix σ. we will use this notation for the normal distribution density throughout. let matrix b be the root of the matrix σ that is satisfying the condition bbt = σ. then, if ξ has distribution n (0, id), where id is identity matrix of dimension d, then η = µ+bξ has distribution n (µ,σ). 2.2. gamma distribution the gamma distribution density γ(α, β) is γ(x|α, β) = αβ γ(β) xβ−1e−αxi{x > 0}. we will use this notation for the density of gamma distribution throughout. let ξ ∼ γ(α, β). it is not difficult to make sure that eξ = β/α [1], e ln ξ = ψ(β)− lnα, where ψ(x) = d ln γ(x) dx is digamma function. if β > 1 we also get eξ−1 = α β−1 . copyright © 2020 assa. adv syst sci appl (2020) 100 n.a. volkov, e.yu. dakhova, s.a. budennyy, a.m. andrianova 2.3. student distribution the student distribution has parameter ν indicating the number of degrees of freedom. the less the number of degrees of freedom, the heavier student distribution tails, and the more resistant it is to outliers. moreover, the student distribution converges in distribution to the normal distribution when the nu converges to infinity. we denote multidimensional student distribution with ν > 0 degrees of freedom, with mean vector µ ∈ rd and symmetric positive semi-definite and non-degenerate scale matrix σ by tν(µ,σ). the density of this distribution in x ∈ rd equals p(x|µ,σ, ν) = γ ( ν+d 2 ) γ(ν/2)νd/2πd/2|σ|1/2 [ 1 + 1 ν (x− µ)tς−1(x− µ) ]− ν+d 2 . the mathematical expectation and the covariance matrix of the distribution tν(µ,σ) equal ex = µ if ν > 1 and varx = ν ν−2 σ if ν > 2 respectively. claim 2.1: [5] let random vector ξ and random variable η be independent and have distributions n (0,σ) and γ(ν/2, ν/2) respectively. let µ ∈ rd be a fixed vector. than random vector x = µ+ ξ/ √ η has student distribution with ν degrees of freedom, mean vector µ and scale matrix σ. the student distribution density can be represented in an integral form p(x|µ,σ, ν) = +∞∫ 0 q(x|µ,σ/y)γ(y|ν/2, ν/2)dy. 2.4. marginal distributions let a, b be disjoint sets of indexes, and a t b = {1, ..., d}. without any loss of generality, we set a = {1, ..., da}, b = {da + 1, ..., d}, db = d− da. vectors and matrices are represented as follows x = ( xa xb ) , µ = ( µa µb ) , σ = ( σaa σab σt ab σbb ) , where xa ∈ rda , xb ∈ rdb , µa ∈ rda , µb ∈ rdb ,σaa ∈ rda×da ,σab ∈ rda×db ,σbb ∈ rdb×db . the following property is known [10]. claim 2.2: let random vector x have normal distributionn (µ,σ). then random vector xa has normal distribution n (µa,σaa). if random vector x has student distribution tν(µ,σ), then random vector xa has student distribution tν(µa,σa). 2.5. conditional distribution our model of probability distributions mixture allows to build a regression on arbitrary features using a conditional distribution. in this section, we recall relations for parameters of conditional distribution for a normal vector (claim 2.3) and for a student vector (theorem 2.1). let random vectorx have normal distributionn (µ,σ), where σ is positive semi-definite scale matrix. moreover let a, b, c are disjoint sets of indexes, and a t b t c = {1, ..., d}. without loss of generality, set a = {1, ..., da}, b = {da + 1, ..., da + db}, c = {da + db + 1, ..., d}. vectors and matrices are represented as follows x = ( xa xb xc ) , µ = ( µa µb µc ) , σ = ( σaa σab σac σt ab σbb σbc σt ac σt bc σcc ) . copyright © 2020 assa. adv syst sci appl (2020) student mixture and its machine learning applications to pvt properties 101 denote also λ = ( λaa λab λt ab λbb ) = ( σaa σab σt ab σbb )−1 . claim 2.3: vector xa conditioned on xb has distribution n (µ̃, σ̃), where µ̃ = µa − λ−1 aa λab(xb − µb), σ̃ = λ−1 aa . the proof of this statement is given in [10]. it follows that the conditional distributions of the components of a normal vector are also normal. let us now consider the case in which the vector x has a student distribution tν(µ,σ) [5]. theorem 2.1: vector xa conditioned on xb has distribution tν̃(µ̃, σ̃), for where ν̃ = ν + db, µ̃ = µa − λ−1 aa λab(xb − µb), σ̃ = ν + ϕ(xb) ν + db λ−1 aa , ϕ(x) = (x− µb)t ( λbb − λt abλ −1 aa λab ) (x− µb). remark. function ϕ(x) is non-negative since all angular minors of the matrix λbb − λt abλ −1 aa λab are positive. the latter fact follows from the block matrix determinant formula [11] and positive definiteness of the scale matrix. 3. mixture model in this section, we formalize the concept of a mixture of probability distributions and give its properties that appear helpful for solving machine learning problems mentioned in introduction. in addition, iterative methods for estimating parameters of a mixture of normal distributions and a mixture of student distributions are provided. 3.1. properties of a mixture model consider a mixture of probability distributions p = k∑ j=1 wjpj, where pj is a ”simple” probability distribution that defines the component of the mixture and wj ∈ [0, 1] are components weights, ∑n i=1 wj = 1. these components are often called clusters. a random vector x obeys the model of a mixture of distributions if it is represented as x = k∑ j=1 xji{t = j}, where xj is distributed as pj and random variable t is equal to the cluster number, i.e. p(t = j) = wj . moreover, variables xj are independent of t . next statement can be found in [12]. claim 3.1: let exj = µj,varxj = σj (their existence is assumed). then the expectation and copyright © 2020 assa. adv syst sci appl (2020) 102 n.a. volkov, e.yu. dakhova, s.a. budennyy, a.m. andrianova covariance matrix for a mixture of distributions are equal to ex = k∑ j=1 wjµj, varx = k∑ j=1 wjσj + k∑ j=1 wjµ j(µj)t − k∑ j,s=1 wjwsµ j(µs)t . claim 3.2: let xt = . . . ( xt a , x t b , x t c ) . then conditional probability of the cluster j conditioned on xb equals w̃j = p(t = j | xb) = wjp (b) j (xb)∑k j=1 wjp (b) j (xb) , where p(b) j is the density of the vector component xb conditioned on pj . let vector xa has distribution p̃ (a|b) j conditioned on xb and i{t = j}. then vector xa conditioned on xb has a distribution of mixture of distributions p̃(a|b) j with weights w̃j p̃(a|b) = k∑ j=1 w̃jp̃ (a|b) j . 3.2. normal mixture distribution the density in the model of a mixture of normal distributions equals p(x) = k∑ j=1 wjq(x|µj,σj). let x1, . . . , xn be a sample from such a mixture of distributions. the estimation of mixture parameters is performed by solving the problem of maximizing model likelihood using an iterative em algorithm [6]. this procedure consists in selecting some random initial approximation of the parameters and then alternating two steps. for a mixture of normal distributions they are conducted in the following way: e-step. compute the following auxiliary values rij = wjq(xi|µj,σj) k∑ s=1 wsq(xi|µs,σs) . rij is the probability that the random vector xi is obtained from the jth component of the mixture at the current approximation of the parameters wj, µj,σj . m-step. compute a new approximation of parameters wj = 1 n n∑ i=1 rij, µj = n∑ i=1 rijxi / n∑ i=1 rij, σj = n∑ i=1 rij(xi − µj)2 / n∑ i=1 rij. stopping criterion. iterations of the method are made up to the convergence of the variational lower bound on the logarithmic likelihood function l(w, µ,σ, r) = n∑ i=1 k∑ j=1 rij [lnwj + ln q(xi|µj,σj)]− n∑ i=1 k∑ j=1 rij ln rij. the procedure terminates as l changes in no more than a pre-set small number ε > 0 [6]. copyright © 2020 assa. adv syst sci appl (2020) student mixture and its machine learning applications to pvt properties 103 3.3. student mixture distribution the density of a mixture of multidimensional student distributions equals p(x) = k∑ j=1 wjp(x|µj,σj, ν), where p(x|µj,σj, ν) is the density of multidimensional student distribution with ν degrees of freedom centered at µj and the scale matrix σj . parameter ν is a hyperparameter of the model. let x = (x1, . . . , xn) be sample vectors from that mixture distribution. the proposed method for estimating the parameters of the mixture consists in selecting some initial approximation of the parameters and performing the next steps at each iteration. e-step. perform several iterations of the next two steps: i. compute the following auxiliary values rij = wjq (xi |µj,σjai/bi ) k∑ s=1 wsq (xi |µs,σsai/bi ) , as above, rij is the probability that the object xi is obtained from the jth component of the mixture at the current approximation of the parameterswj, µj,σj . ii. compute ai = ν + d 2 , bi = ν 2 + 1 2 k∑ j=1 rij (xi − µj)tς−1 j (xi − µj), ci = bi/ai, where d is the dimension of the feature space. m-step. compute a new approximation of parameters wj = n∑ i=1 rij / n,k∑ i,j=1 rij, µj = n∑ i=1 rijci xi / n∑ i=1 rijci, σj = 1 n n∑ i=1 ci (xi − µj)(xi − µj)t . stopping criterion. iterations of the method are performed up to convergence of l(w, µ,σ, r, a, b) = n∑ i=1 k∑ j=1 rij [ lnwj − d 2 ln 2π − 1 2 ln det σj− − bi 2ai [ ν + (xi − µj)tς−1 j (xi − µj) ] + ν 2 ln ν 2 − γ (ν 2 ) + ( ν + d 2 − 1 ) (ψ(bi)− ln ai) ] − − n∑ i=1 k∑ j=1 rij ln rij − n∑ i=1 [bi ln ai − ln γ(bi) + (bi − 1) (ψ(bi)− ln ai)− bi] . copyright © 2020 assa. adv syst sci appl (2020) 104 n.a. volkov, e.yu. dakhova, s.a. budennyy, a.m. andrianova using those iterations, the method approximates a local maximum of the logarithmic likelihood function, which makes it necessary to run the procedure several times from different initial points. for more information about method convergence see section 4. the hyperparameter ν can be chosen in a usual way by performing the algorithm for several values ν and choosing the one that maximizes l. 4. formulas derivation for estimation the parameters of student mixture let x = (x1, ..., xn) be a sample of vectors from a mixture of student distributions. the likelihood equals l(w, µ,σ) = n∏ i=1 k∑ j=1 wjp(xi|µj,σj, ν). the family of distributions does not belong to the exponential class of distributions. therefore, a maximization of likelihood is not straightforward. let us use the representation of a student random vector via a normal random vector and a gamma random vector. next, we introduce hidden (i.e. unknown) values. 1. for each xi, introduce the cluster number as ti = (ti1, . . . , tik) ∈ {0, 1}k, with∑k j=1 tij = 1. the value tij = 1 if the object xi is taken from the cluster j and tij = 0 otherwise. denote t = (t1, . . . , tn). 2. also for each xi, we introduce a random variable yi that has distribution γ(ν/2, ν/2) such that xi = k∑ j=1 ( µj + ξij /√ yi ) i{tij = 1}, where the random vector ξij has distribution n (0, σj) and is independent from yi. denote y = (y1, . . . , yn). vector xi has the normal distribution n (µj,σj/y) conditioned on tij = 1 and yi = y. the joint distribution of vectors (x,t, y ) has the density (the component t has discrete density) p(x, t, y|w, µ,σ, ν) = n∏ i=1 k∏ j=1 [wjq(xi|µj,σj/yi)γ(yi|ν/2, ν/2)]tij . compute ln p(x, t, y|w, µ,σ, ν) = n∑ i=1 k∑ j=1 tij [lnwj + ln q(xi|µj,σj/yi) + ln γ(yi|ν/2, ν/2)] = = n∑ i=1 k∑ j=1 tij [ lnwj − d 2 ln 2π − 1 2 ln det σj + d 2 ln yi − yi 2 (xij − µj)tς−1 j (xij − µj)+ + ν 2 ln ν 2 − ln γ(ν/2) + (ν 2 − 1 ) ln yi − ν 2 yi ] . the estimation of the mixture parameters is performed by solving the problem of maximizing the model likelihood function using em algorithm. at the e-step the distributions copyright © 2020 assa. adv syst sci appl (2020) student mixture and its machine learning applications to pvt properties 105 of t and y are estimated iteratively using variational bayesian inference. independence of posteriori distributions (conditioned on x) is assumed. these results are in the following iterative scheme. 1. selection a random initial values wj, µj,σj . vectors µj can be obtained from normal distribution n (0, id), vectors wj — from the uniform distribution on the simplex (dirichlet distribution). the matrices σj can be generated using the wishart distribution [13]. 2. e-step: (a) select a random initial value for y . (b) having the current distribution of y , approximate the distribution of t . (c) having the current distribution of t , approximate the distribution of y . (d) repeat steps b-c until convergence of l. 3. m-step. 4. repeat e and m steps until convergence of l. 4.1. e-step, internal step i at this step, the distribution of t conditioned on x is approximated using the relation ln r(t) ∝ eγ ln p(x, t, y |w, µ,σ, ν), where mathematical expectation eγ is computed under the condition that the current distribution of y was computed on the previous iteration of step ii. remark. everywhere below symbol ∝ means equality up to a multiplicative constant for probabilities and equality up to an additive constant for logarithms of probabilities. each ti has a discrete distribution with values in a set of binary vectors that have exactly one unit. for this distribution, density logarithm is ∑k j=1 tij ln rij , where rij = p(tij = 1) and ∑k j=1 rij = 1. ln r(t) ∝ eγ ln p(x, t, y |w, µ,σ, ν) ∝ ∝ n∑ i=1 k∑ j=1 tij [ lnwj − 1 2 ln det σj − eγyi 2 (xi − µj)tς−1 j (xi − µj) ] ∝ ∝ n∑ i=1 k∑ j=1 tij [ lnwj − 1 2 ln det (σj/eγyi)− 1 2 (xi − µj)t (σj/eγyi) −1 (xi − µj) ] ∝ ∝ n∑ i=1 k∑ j=1 tij [lnwj + ln q (xi |µj,σj/eγyi )] . the resulting expression implies that the optimal approximation of conditional distribution of t (provided x) is such that the values t1, . . . , tn are independent and rij ∝ wjq (xi |µj,σj/eγyi ). from the condition ∑k j=1 rij = 1 we get rij = wjq (xi |µj,σj/eγyi ) k∑ s=1 wsq (xi |µs,σs/eγyi ) . the expectation eγyi is taken of the current approximation of yi. copyright © 2020 assa. adv syst sci appl (2020) 106 n.a. volkov, e.yu. dakhova, s.a. budennyy, a.m. andrianova 4.2. e-step, internal step ii at this step, the distribution of y conditioned on x is computed: ln γ(y) ∝ er ln p(x,t, y|w, µ,σ, ν), where the current distribution of t is assumed. let us write this expression up to a constant that does not depend on y , given that∑k j=1 rij = 1 ln γ(y ) = er ln p(x,t, y|w, µ,σ, ν) ∝ = n∑ i=1 k∑ j=1 ertij [ d 2 ln yi − yi 2 (xi − µj)tσj(xi − µj) + (ν 2 − 1 ) ln yi − ν 2 yi ] ∝ ∝ n∑ i=1 k∑ j=1 rij [( ν + d 2 − 1 ) ln yi − ( ν 2 + 1 2 (xi − µj)tς−1 j (xi − µj) ) yi ] = ∝ n∑ i=1 [( ν + d 2 − 1 ) ln yi − ( ν 2 + 1 2 k∑ j=1 rij(xi − µj)tς−1 j (xi − µj) ) yi ] . thus, we obtain the gamma distribution with parameters ai = ν 2 + 1 2 k∑ j=1 rij (xi − µj)tς−1 j (xi − µj), bi = ν + d 2 . the values of mathematical expectations are updated using relations eγyi = bi ai and eγ lnyi = ψ(bi)− ln ai, which are stated in section 2. for brevity, denote ci = bi/ai. 4.3. m-step at this step, the values of the mixture parameters are updated by maximizing er,γ ln p(x,t, y |w, µ,σ, ν) , where the current distributions of t and y are assumed. we leave only those summands that depend on wj, µj,σj fx,ν(w, µ,σ) = er,γ ln p(x,t, y |w, µ,σ, ν) ∝ ∝ n∑ i=1 k∑ j=1 ertij [ lnwj − 1 2 ln det σj − 1 2 (xi − µj)tς−1 j (xi − µj)eγyi ] = = n∑ i=1 k∑ j=1 rij [ lnwj − 1 2 ln det σj − ci 2 (xi − µj)tς−1 j (xi − µj) ] . maximization by wj of fx,ν(w, µ,σ) is equivalent to finding a solution of the problem n∑ i=1 k∑ j=1 rij lnwj −→ max w , k∑ j=1 wj = 1, wj > 0. copyright © 2020 assa. adv syst sci appl (2020) student mixture and its machine learning applications to pvt properties 107 let us forget about the restrictions of the inequality type for a while, make a lagrange function, and find its maximum l = n∑ i=1 k∑ j=1 rij lnwj − λ ( k∑ j=1 wj − 1 ) , ∂l ∂wj = 1 wj n∑ i=1 rij − λ = 0, wj = 1 λ n∑ i=1 rij. from the condition ∑k j=1 wj = 1 we get wj = n∑ i=1 rij / n∑ i=1 k∑ s=1 ris. note that the conditions wj > 0 are met. the problem is solved due to the convexity. to maximize by µj , we equate the derivative of fx,ν(w, µ,σ) with respect to vector µj to zero ∂fx,ν(w, µ,σ) ∂µj = n∑ i=1 rijci σ−1 j (xi − µj) = 0, n∑ i=1 rijcixi = n∑ i=1 rijciµj, µj = n∑ i=1 rijci xi / n∑ i=1 rijci. to maximize by σj , we equate the derivative of fx,ν(w, µ,σ) with respect to matrix σj to zero. note that (xi − µj)tς−1 j (xi − µj) = tr ( (xi − µj)tς−1 j (xi − µj) ) = tr ( σ−1 j (xi − µj)(xi − µj)t ) . using this transformation, as well as matrix derivative formulas for square matrices [14] ∂ ∂x detx = detx ·x−t , ∂ ∂x tr ( x−1a ) = − ( x−1ax−1 )t , get ∂fx,ν(w, µ,σ) ∂σj = −1 2 n∑ i=1 k∑ j=1 rij [ ∂ ∂σj ln det σj + ci ∂ ∂σj (xi − µj)tς−1 j (xi − µj) ] = = −1 2 n∑ i=1 rij ( σ−1 j − ciς−1 j (xi − µj)(xi − µj)tς−1 j ) = 0. copyright © 2020 assa. adv syst sci appl (2020) 108 n.a. volkov, e.yu. dakhova, s.a. budennyy, a.m. andrianova multiplying both parts of the last equation by σj we get n∑ i=1 rijσj = n∑ i=1 rijci (xi − µj)(xi − µj)t , σj = n∑ i=1 rijci (xi − µj)(xi − µj)t / n∑ i=1 rij. 4.4. variational lower bound and convergence of the method iterations of the em algorithm continue until the convergence of the variational lower bound [6] l(w, µ,σ, r, a, b) = er,γ ln p(x,t, y |w, µ,σ, ν)− er ln r(t )− eγ ln γ(y ). let us write each summand separately er,γ ln p(x,t, y |w, µ,σ, ν) = = n∑ i=1 k∑ j=1 ertij [ lnwj − d 2 ln 2π + d 2 eγ lnyi − 1 2 ln det σj− −1 2 (xi − µj)tς−1 j (xi − µj)eγyi + ν 2 ln ν 2 − γ (ν 2 ) + (ν 2 − 1 ) eγ lnyi − ν 2 eγyi ] = = n∑ i=1 k∑ j=1 rij [ lnwj − d 2 ln 2π − 1 2 ln det σj− − bi 2ai [ ν + (xi − µj)tς−1 j (xi − µj) ] + ν 2 ln ν 2 − γ (ν 2 ) + ( ν + d 2 − 1 ) (ψ(bi)− ln ai) ] , er ln r(t ) = n∑ i=1 k∑ j=1 rij ln rij, eγ ln γ(y ) = n∑ i=1 eγ ln γ(yi) = n∑ i=1 [bi ln ai − ln γ(bi) + (bi − 1)eγ lnyi − aieγyi] = = n∑ i=1 [bi ln ai − ln γ(bi) + (bi − 1) (ψ(bi)− ln ai)− bi] . let us study the convergence of the method. the standard em algorithm is based on the formula (see [6]) logl(w, µ,σ) = l(w, µ,σ, q) +kl(q, pw,µ,σ(t, y|x)), where q is a distribution of the vector (t, y ), kl is kullback-leibler divergence. here, for the variational lower bound, we use the notation l(w, µ,σ, q) since, in general, q does not depend on parameters. in order to maximize the likelihood function, the variational lower bound is maximized at each step, at the e-step by q, at the m-step by the parameters w, µ,σ. copyright © 2020 assa. adv syst sci appl (2020) student mixture and its machine learning applications to pvt properties 109 thus, its values do not decrease. maximization at the e-step is equivalent to minimizing the divergence by q, which is equivalent to choosing q = pw,µ,σ(t, y|x), where the divergence is exactly zero. thus, the variational lower bound after the e-step is equal to the logarithm of the likelihood function logl(w, µ,σ).therefore the likelihood function also does not decrease and converges to the local maximum. for the modification of the em algorithm under consideration, the formula takes the form logl(w, µ,σ) = l(w, µ,σ, r, a, b) +kl(r × γ, pw,µ,σ(t, y|x)). variational output at the e-step minimizes divergence. however in the class of distributions under consideration, where the distributions of t and y are conditionally independent, the zero divergence may not be achieved. thus, the convergence of l to the local maximum cannot be guaranteed. in the practical problems we are considering, it is natural to assume that the distributions t and y are close to be conditionally independent, that should provide a good approximation of the local maximum using the resulting estimation. 5. applications of probabilistic model the model of a mixture distribution allows solving various machine learning problems listed in the introduction and obtaining consistent results. let us describe each of them in more detail, assuming a mixture model with density p(x) = k∑ j=1 wjp(x|θj), where θj is the parameter of the jth component distribution (for example, the mean vector and the covariance matrix). 5.1. clustering the components of the mixture can be considered as overlapping clusters. each object x ∈ rd can be assigned to one of the clusters with some probability. according to claim 3.2, the conditional probability that an object x corresponds to a cluster j is pj(x) = wjp(x|θj) k∑ s=1 wsp(x|θs) . if ŵj, θ̂j are estimations of the parameters wj, θj respectively, then we can estimate p̂j(x) = ŵjp(x|θ̂j) / k∑ s=1 ŵsp(x|θ̂s) . these probability estimation can be considered as the confidence level of the method when assigning an object x to a cluster j, which is sufficient to solve the following problem. performing ”hard” clustering, where an object x must be strictly attributed to one of the clusters, the cluster with the maximum probability p̂j(x) is selected j∗ = arg max j p̂j(x) = arg max j ŵjp(x|θ̂j). 5.2. anomalies object x ∈ rd is considered abnormal if density value p(x) is less than some threshold value q. the value q is chosen as density value p(x) so that the probability of getting an object with copyright © 2020 assa. adv syst sci appl (2020) 110 n.a. volkov, e.yu. dakhova, s.a. budennyy, a.m. andrianova a density value not exceeding q is exactly 0.05. in other words, the value q is the solution of the equation ∫ rd p(x)i{p(x) 6 q}dx = 0.05. the described procedure for determining anomalous objects is a special case of the procedure for checking statistical hypotheses. in that case, the hypothesis that object x is typical is tested against the alternative one about object abnormality. if the density at the point x is less than the threshold value q, the hypothesis of object typicity is rejected in favor of an alternative one at the significance level 0.05. this rule is a criterion for testing a hypothesis. a probability of getting objects with a density at least p(x) can be considered as the level of typicity of object x. this value is analogous to p-value and is computed using the integral∫ rd p(y)i{p(y) 6 p(x)}dy. 5.3. missing data both methods discussed above work only if all values of components x ∈ rd are known, i.e. there are no missing values. otherwise, the density of the object x can be estimated as the density integral over a subspace of omitted values. formally, let xk be a vector of known object values, and xu are all other object values that are omitted. then∫ rdu p(x)dxu, where du is a dimension of vector xu. for a normal or student distribution mixture, the density is equal to the mixture of marginal distributions obtained in claims 2.2. 5.4. conditional distribution and probabilistic regression on features let object x ∈ rd be recognized as anomalous. let us select elements of the vector x to trust in and evaluate others through them. without loss of generality we assume that xt = (xta , x t b ) and the values of xb are trusted. in addition, due to missing data, some values of xa may not be known. according to claim 3.2, vector xa under the condition of values xb has density p̃(x) = k∑ j=1 w̃jp(x|θ̃j), where θ̃j is the parameter of the mixture distribution jth component given xb. in the case of the normal mixture, the parameters θ̃j for each cluster are computed in accordance with relations from claim 2.3. in the case of the student mixture, the parameters are computed in accordance with relations from the theorem 2.1. replacing the parameters with their estimations, we get an estimation of the conditional distribution of vector xa, which is sufficiently informative for making various conclusions about the vector xa in practice. in particular, it can be used to compute • expectation value e(xa | xb) according to claim 3.1. this estimation solves the problem of regression of xb features to xa features. note that the regression problem can be solved for different features xa and xb using only one model. copyright © 2020 assa. adv syst sci appl (2020) student mixture and its machine learning applications to pvt properties 111 • variance estimation var(xa | xb) according to claim 3.2. • evaluation of conditional cluster distributions for all objects that havexb attributes fixed and equal to xb. • set of the highest density, i.e. set of values xa, for which the density of the conditional distribution is greater than for the other values. such a set is analogous to the confidence domain of the minimum volume. in the one-dimensional case, this set is an interval or set of intervals. 6. modeling pvt properties of reservoir fluids using a probabilistic model 6.1. data description there are several of methods for evaluating the representativeness of reservoir fluid samples [15], such as checking the tightness of sampling chambers, comparing the oil saturation pressure with the separation pressure at the separation temperature, etc.; the hoffman-kramp-hockot method, based on the correlation of equilibrium constants; determining the representativeness of samples by the criterion of contamination with process fluids used in drilling, perforation and development of wells. in conditions where only raw data is available the above methods cannot be applied. thus, it is reasonable to develop algorithms for detecting potentially incorrect values from raw data. for practical application of research on pvt-properties of fluids, a database containing the results of studies of more than 3,200 samples of reservoir fluids was analyzed. among the considered features, there are the following values: reservoir pressure, reservoir temperature, surface gas density, surface oil density, gas content, saturation pressure, reservoir oil density, oil volume coefficient, and reservoir oil viscosity. the problem of predicting pvt properties using machine learning methods was previously considered in a very limited version. for example, in [16], [17], [18] the prediction of saturation pressure through other properties using artificial neural networks (ann) is considered. in [18], [19] predictions of the oil volume coefficient are made in the same way. in [20] svm regression is used to predict the features mentioned above. this paper offers a fundamentally different approach to solving these problems. it is based on the introduction of a probabilistic model in the space of reservoir fluid properties, where it is assumed that the characteristic description of the sample is obtained independently of all other samples from a certain probability distribution. 6.2. normal mixture model first we use a mixture of normal distributions of four components to describe data. since the iterative procedure of the em algorithm converges to the point of the local maximum of the logarithmic likelihood function, the method was run several times from random initial values of parameters. the final score is obtained in the iteration with the highest value of the variational lower score. the result of parameter estimation is shown on figure 6.1. the diagonal shows the densities of features for the estimated mixture. each non-diagonal cell on figure corresponds to the projection of the feature space on all possible coordinate planes. lines of the density level of the resulting mixture of distributions are drawn above the diagonal. below the diagonal, ellipses of different colors indicate the relative location of clusters. notice that a gray semi-transparent cluster covers other two clusters, due to the presence of noise objects in the data. the normal distribution has light tails, so the estimations of its parameters are not stable to the presence of noise objects in the data. when constructing a model of a mixture of normal distributions on pvt data, the em algorithm tries to describe the main part of the data using three clusters, and the other less typical objects using the fourth one. copyright © 2020 assa. adv syst sci appl (2020) 112 n.a. volkov, e.yu. dakhova, s.a. budennyy, a.m. andrianova fig. 6.1. data visualization and results of application of the normal mixture model. the plots above the diagonal show the density levels of joint distribution of features projected on subspace of every two features. the diagonal plots show the density of every feature. the plots under the diagonal show the clusterization based on the application of the normal mixture model. every ellipse corresponds to one of four clusters/components of the mixture 6.3. student mixture model to eliminate the above disadvantages, the student distribution mixture model is applied. the number of degrees of freedom nu needs some expertise. on figure 6.2 the dependence of the variational lower bound on the iteration for the best result among several runs from different initial approximations is shown. the result of evaluating parameters for four clusters is shown on figure 6.3. the blue and red clusters obtained using the student multidimensional distribution mixture model correspond to the same clusters in the multi-dimensional normal distribution mixture model. the green cluster is split into two. note the clusters have no noise. this result is a consequence of the stability of the student distribution to emissions. copyright © 2020 assa. adv syst sci appl (2020) student mixture and its machine learning applications to pvt properties 113 fig. 6.2. the dependence of the variational lower bound on iteration number table 6.1. centers of clusters obtained using the student mixture model cluster 0 cluster 1 cluster 2 cluster 3 sample rate 28.3% 27.7% 29.0% 15.0% reservoir pressure, mpa 23.86 25.36 25.63 36.04 reservoir temperature, oc 75.33 80.47 83.91 96.12 surface gas density, kg / m3 1.01 1.07 1.17 1.11 surface oil density, kg / m3 859.74 855.61 836.75 823.55 gas content, m3 / t 49.20 69.58 138.74 425.62 saturation pressure, mpa 8.46 11.25 13.61 24.49 oil reservoir density, kg / m3 808.33 772.59 716.74 598.29 volume coefficient. oil, m3 / m3 1.12 1.20 1.36 2.04 reservoir oil viscosity, mpa * s 2.29 1.24 0.67 0.27 oil type heavy & medium medium light extra light cluster centers are given in table 6.1. during training, clusters are defined by the model up to permutation so the order of clusters is determined by experts. one can notice that for most clusters their centers are strictly ordered by most attributes. the type of oil corresponding to each cluster is also determined by experts. 7. model research the behavior of the model based on a mixture of four components of student distributions is tested on artificial data, the results of experiments are given below. in addition, the quality of model predictions is tested by test data that is not involved in the training set. 7.1. artificial experiments in our research, the results of the model application to artificial samples are analyzed. some sample features are fixed in three different versions of the experiments. their values are given in table 7.2. in each of three variants of experiments, the gas content values are varied between 0 and 800 m3/t. for each value, the probabilistic density of the sample (i.e., the probability of belonging to each of the three clusters), the expected values of saturation pressure, reservoir oil density, and oil volume coefficient are computed using a mixture of student distributions. the sample density depending on the gas content for the three variants under study is given on figure 7.4. the black dotted line shows the anomaly threshold. if the density is below the threshold, the sample is considered abnormal. the graph shows that in the first variant typical samples correspond to gas content values from 90 to 190 m3/t, in the second variant from 0 to 110 m3/t, in the third from 0 to 90 m3/t. copyright © 2020 assa. adv syst sci appl (2020) 114 n.a. volkov, e.yu. dakhova, s.a. budennyy, a.m. andrianova fig. 6.3. data visualization and results of application of the student mixture model. the plots above the diagonal show the density levels of joint distribution of features projected on subspace of every two features. the diagonal plots show the density of every feature. the plots under the diagonal show the clusterization based on the application of the student mixture model. every ellipse corresponds to one of four clusters/components of the mixture the figure 7.5 shows the probability estimations that the sample corresponds to each of four clusters for three variants of experiments. for example, considering the first variant of experiments, we can conclude the sample belongs to a blue or green cluster depending on the value of the gas content . figure 7.6 contains the graphs of saturation pressure predictions, reservoir density of oil, and volume coefficient of oil in the three above variants depending on the gas content. orange dots corresponds the samples used in train data. the shaded area denote the predictive interval. the less typical the sample, the greater the uncertainty of the model and the confident interval is wider. note the trajectories of the predicted values are smooth that follows from the model construction. it is also worth noting that the observed shift in predictions relative copyright © 2020 assa. adv syst sci appl (2020) student mixture and its machine learning applications to pvt properties 115 table 7.2. fixed values of features in model experiments feature variant 1 variant 2 variant 3 reservoir pressure, mpa 35 25 18 reservoir temperature, oc 100 75 50 surface gas density, kg / m3 1.2 0.95 0.68 surface oil density, kg / m3 800 845 880 reservoir oil viscosity, mpa * s 0.5 2 5 gas content, m3 / t 0 . . . 800 0 . . . 800 0 . . . 800 saturation pressure, mpa ? ? ? oil reservoir density, kg / m3 ? ? ? volume coefficient. oil, m3 / m3 ? ? ? fig. 7.4. sample density depending on the gas content for three variants of experiments to the total mass of points is due to the fact that each variant of experiments has fixed features not presented on the charts. if there weren’t fixed features the predictions would look like averages. fig. 7.5. probability estimation of the sample belonging to each of four clusters depending on the gas content 7.2. predictions quality the developed model calculates the expected values for unknown feature values based on the entered sample. the degree of deviation of the estimated values from the true values is estimated using quality metrics. let xi be a true value of the feature, and x̂i be a predicted value under the assumption that xi is unknown. then the relative error of the prediction xi is ei = (xi − x̂i)/xi. it measures the deviation of the predicted value from the true value in relation to the actual value of the true value. prediction errors are calculated for a test set containing about 650 samples that are not involved in building the model. when predicting each value of the model passed all the known values of the samples in addition to predicted. the prediction quality is calculated as the average absolute and standard error in percentages, copyright © 2020 assa. adv syst sci appl (2020) 116 n.a. volkov, e.yu. dakhova, s.a. budennyy, a.m. andrianova fig. 7.6. predictions of three features depending on gas content. shaded area denotes confidence interval which are defined as mape = 100% n n∑ i=1 |ei|, rmspe = 100% √√√√ 1 n n∑ i=1 e2 i . for metrics evaluation, 2.5% of the highest and 2.5% of the lowest values are excluded from the set of numbers ei due to outliers and incorrect values in the source data. for comparison, 9 regression models were built for each feature for the following machine learning methods: gradient boosting (xgboost†, lgbm‡, catboost§), as well as sklearn implementations¶ of random forest (rf), svm regression, neural network (ann). each such model is trained to predict one of the features, considering all the other features as a feature description. optimal hyperparameters for each model are selected using cross-validation on the training set. tables 7.3, 7.4 contain the values of the mape and rmspe metrics for the test dataset. one can notice in 5 out of 9 cases the model prediction of the student mixture is more accurate than all the other models, and in the other cases it is not far behind them. moreover, experts usually trust reservoir pressure and temperature, which are more accurately predicted by gradient boosting, and therefore their prediction is not very significant in this task. †https://xgboost.ai/ ‡https://lightgbm.readthedocs.io/ §https://catboost.ai/ ¶https://scikit-learn.org/ copyright © 2020 assa. adv syst sci appl (2020) student mixture and its machine learning applications to pvt properties 117 table 7.3. comparison of prediction quality based on the mape metric t-mix xgboost lgbm catboost rf svm ann reservoir pressure, mpa 8.13 6.70 6.56 6.53 8.31 6.77 9.63 reservoir temperature, oc 5.96 5.50 5.67 5.75 5.78 6.58 6.25 surface gas density, kg / m3 3.42 6.21 6.20 7.08 6.23 4.30 7.38 surface oil density, kg / m3 0.54 0.87 0.95 0.88 0.87 0.67 1.02 gas content, m3 / t 3.92 6.51 6.85 7.83 7.33 7.28 9.03 saturation pressure, mpa 11.69 9.15 9.25 9.34 10.33 9.23 18.07 oil reservoir density, kg / m3 0.71 1.35 1.27 2.04 1.58 1.77 1.85 volume coefficient. oil, m3 / m3 0.68 1.43 1.57 2.70 2.13 2.23 2.62 reservoir oil viscosity, mpa * s 22.32 30.60 26.86 27.30 57.60 123.73 9.66 table 7.4. comparison of prediction quality based on the rmspe metric t-mix xgboost lgbm catboost rf svm ann reservoir pressure, mpa 10.15 9.05 8.96 8.56 12.53 8.86 15.85 reservoir temperature, oc 7.50 7.15 7.26 7.15 7.61 8.23 7.93 surface gas density, kg / m3 4.75 7.80 7.79 8.72 7.95 5.44 9.38 surface oil density, kg / m3 0.88 1.11 1.21 1.07 1.12 1.01 1.58 gas content, m3 / t 5.64 8.57 8.57 9.92 11.12 10.15 11.21 saturation pressure, mpa 16.52 12.79 13.27 12.57 14.89 12.71 21.63 oil reservoir density, kg / m3 1.47 2.08 1.78 2.98 2.36 3.95 2.37 volume coefficient. oil, m3 / m3 1.49 2.12 2.36 3.74 3.34 4.03 3.29 reservoir oil viscosity, mpa * s 30.22 44.11 36.69 36.82 91.35 398.20 12.68 8. conclusions the probabilistic model of a mixture of multidimensional student distributions proposed in the paper for describing the properties of pvt samples has a wide range of practical applications, including checking the samples for abnormality, dividing the samples into four clusters, computing recommended values for missing values in the sample, and in the case of sample abnormality – for all features not selected as trusted. experiments have shown that the recommended values obtained do not contradict the physical properties of pvt samples, in particular, they have smoothness in terms of arguments. the division of samples into four clusters corresponding to components of the multidimensional student mixture is empirically justified by comparison with a mixture of multidimensional normal distributions. in the latter case, the quality is unsatisfactory due to the presence of noise objects that the model adjusts to a separate cluster. the student distribution has heavier tails, so it adjusts less to emissions. the probabilistic model has significant advantages over other models since it can be used to solve several problems at once and obtain consistent results. considering only the regression problem, the probabilistic model, in contrast to the traditional approach, allows getting predictions of set of features for other set of features and vise verse without repeated model training. furthermore, in the conducted experiments, the quality of the obtained predictions is also superior to other models. references 1. lagutin, m. b. 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(1997). old and new matrix algebra useful for statistics. mit media lab note. 15. brusilovskij a. i. (2002) fazovye prevrashcheniya pri razrabotke mestorozhdenij nefti i gaza [phase transformations in the development of oil and gas fields]. moscow, russia: graal, [in russian]. 16. alakbari f., elkatatny s., baarimah s.. (2016). prediction of bubble point pressure using artificial intelligence ai techniques. proc. of the spe middle east artificial lift conference and exhibition, 10.2118/184208-ms. 17. numbere, o. g., azuibuike, i. i., ikiensikimama, s. s. (2013). bubble point pressure prediction model for niger delta crude using artificial neural network approach. society of petroleum engineers. doi:10.2118/167586-ms 18. alcocer y., patricia r.. (2001). neural networks models for estimation of fluid properties. proc. of the spe latin american and caribbean petroleum engineering conference, 10.2523/69624-ms. 19. osman, e. a., abdel-wahhab, o. a., al-marhoun, m. a. (2001). prediction of oil pvt properties using neural networks. society of petroleum engineers. doi:10.2118/68233ms 20. el-sebakhy, e. a., sheltami, t., al-bokhitan, s. y., shaaban, y., raharja et. al. (2007). support vector machines framework for predicting the pvt properties of crude oil systems. society of petroleum engineers. doi:10.2118/105698-ms copyright © 2020 assa. adv syst sci appl (2020) introduction distributions and their properties normal distribution gamma distribution student distribution marginal distributions conditional distribution mixture model properties of a mixture model normal mixture distribution student mixture distribution formulas derivation for estimation the parameters of student mixture e-step, internal step i e-step, internal step ii m-step variational lower bound and convergence of the method applications of probabilistic model clustering anomalies missing data conditional distribution and probabilistic regression on features modeling pvt properties of reservoir fluids using a probabilistic model data description normal mixture model student mixture model model research artificial experiments predictions quality conclusions microsoft word 939 article text, copyedited.docx adv syst sci appl 2020; 03; 136-152 published online at https://ijassa.ipu.ru. original russian text © yu. v. mitrishkin, n. m. kartsev, a. e. konkov, m. i. patrov, 2019, published in problemy upravleniya / control sciences, 2019, no. 4, pp. 15-26. plasma control in tokamaks. part 3.2. simulation and realization of plasma control systems in iter and constructions of demo y. v. mitrishkin 1,2*, n. m. kartsev 2, a. e. konkov 1,2, m. i. patrov 3 1) m.v. lomonosov moscow state university, faculty of physics, moscow, russia 2) v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia 3) a.f. ioffe institute of russian academy of sciences, st. petersburg, russia e-mail: y_mitrishkin@hotmail.com e-mail: n.kartsev@yandex.ru e-mail: konkov@phys.msu.ru e-mail: michael.patrov@mail.ioffe.ru abstract: experimental testing of iter scenarios on the diii-d (us) and west (france) tokamaks, approaches for simulation and realization of iter plasma control systems, the preparation of iter plasma control systems for starting and exploitation are presented. the roadmaps that are known in europe for the development and creation of the first fusion power plant demo (the next step after iter) are reported. these maps show two directions of demo development: (i) based on conventional tokamaks with relatively large aspect ratio and (ii) based on spherical tokamaks of module type giving a chance to noticeably reduce the time needed to create the demo and to get cheap competitive electrical energy. the basic trends in the design of demo poloidal systems, as well as the initial version of the demo plasma vertical position control system, are given. keywords: tokamak, plasma, plasma magnetic control, iter, demo constructions introduction in the first part of the survey [1], tokamaks with plasmas in their magnetic fields together with diagnostics and actuators as controlled plants are considered. the second part [2] presents systems for the magnetic control of the position, current, and shape of the plasma, as well as resistive wall-modes in operating tokamaks. part 3.1 [3] shows the magnetic plasma control systems for iter (international thermonuclear experimental reactor). the systems include original engineering solutions for plasma position, current and shape control systems for the two versions of iter (iter-1 and iter-2) including those proposed and made in the institute of control sciences of ras. it was noted that in iter-1 the position and shape of the plasma were controlled by all pf-coils and robust н∞-controllers, and an additional non-linear control circuit was designed to reduce the peaks of control power during the suppression of minor disruptions. due to the reduction of the large installation radius from 8.1 m to 6.2 m, a special control circuit with a relatively fast actuator connected to the pf2-pf5 coils was used for vertical plasma stabilization in iter-2. but at the same time, the vertical control area of the plasma turned out to be a catastrophically small scale of 3-4 cm with a minor radius of iter2 equal to 2 m. in order to increase the controllable area of the iter-2 project, additional * corresponding author: y_mitrishkin@hotmail.com plasma control in tokamaks. part 3.2. simulation and realization... 137 copyright ©2020 assa. adv. in systems science and appl. (2020) horizontal field coils were introduced into the vacuum vessel of the tokamak, which made it possible to increase the controllable area by about one order of magnitude. modeling has shown that the best result in suppressing vertical instability of the plasma is achieved by combining the external and internal coils of the poloidal field. in section 1 of this part of the survey, the experimental development of scenarios for iter2 on two operating tokamaks, diii-d and west, is considered. all 4 planned iter-2 scenarios with a scale factor of 3.7 were tested on diii-d, and a tungsten diverter plate for iter-2 with a discharge duration of 1000 s will be studied on the west tokamak. in section 2 there is information about the algorithm of the development of the plasma control systems in iter-2, about the software-computing platform, about the real-time testbed for the implementation of the algorithms of plasma control, about the information-control system codac (control, data access and communication) for iter-2, about the scheme of the interfaces of the plasma control systems with codac, about the simulator of the plasma control systems, and about the developed software package imas: the iter integrated modelling & analysis suite. section 3 lists the measures taken to prepare the plasma control system in iter-2 for start-up and operation. the first roadmap shows the way to create a demo for traditional tokamaks with a relatively large aspect ratio of 3-4, and the second roadmap shows the prospect of faster creation of the demo on modular spherical tokamaks with an aspect ratio of 1.5-1.7 and cheaper electricity. this part concludes with a brief overview (section 5) of the planned demo designs and with information on the demo vertical plasma position control system for one of the constructions listed above. 1. experimental development of scenarios for iter-2 the iter international consortium attaches special importance to the development of scenarios for iter not only based on plasma-physical codes by numerical simulation [1, issues 4.2.1 and 4.2.2] but also on experiments on operating tokamaks even before starting iter and bringing it to the nominal operating modes. four iter operating scenarios were tested on the diii-d tokamak in a scaled form [4]. the unique properties of this work are that the plasma parameters reflect the essential properties of the iter scenarios and the expected performance characteristics, such as the plasma vertical section and aspect ratio in the diii-d discharges, correspond to the iter project with a reduction factor of 3.7 (fig. 1.1). key aspects of all four scenarios, such as bn and h98, have been successfully imitated on diii-d, providing an improved and unified physical basis for transport and stable modeling as well as for extrapolation to the iter operating modes. (the value is a normalized b value, where a is the minor radius, bt is the toroidal field induction, ip is the plasma current, is the ratio of the plasma gas kinetic pressure to the external magnetic field pressure, is the average plasma pressure, b is the average value of the full field, is the confinement factor, where is the confinement time, is the confinement time in l-mode, where p is the full input power, n is the plasma density, while ti and te are the ion and electronic temperatures). an example of such a discharge for the iter base case scenario is shown in fig. 1.1, b. in all four scenarios, the normalized quality of control coincided or was close to achieving the required one to realize the physical and technological goals of iter, and the diii-d discharge displays were coordinated with iter, reaching its goal of generating at least 400 mw of thermonuclear power and q ³ 10, where q = pout/pin is the ratio of the output power of thermonuclear fusion pout to the input power of pin. these studies are also related to many key physical issues concerning the iter project, including the l-h power threshold crossing, the level of edge localized modes, the scaling of parameters at the flat discharge phase, the effect of tearing /n t pab ib b= 2 0( / 2 )p bb µ= p τ τle еh = τe τle ( ) 33τ 2e i en t t d x p = +ò 138 y.v. mitrishkin, n.m. kartsev, a.e. konkov, m.i. patrov copyright ©2020 assa. adv. in systems science and appl. (2020) modes on confinement and disruptions, the limitation on β and the required capabilities of the plasma control system. an example of the direct impact of this work on the iter project is the modification of the physical requirements for the set of poloidal field coils at the 15 ma plasma current, based on observations that the internal inductance in the basic scenario changes to a level that lies outside the original iter specification. a b fig. 1.1. a) comparison of scaled (reduced by a factor of 3.7) iter plasma cross-section (black) and experimental diii-d plasma cross-section (red/grey). b) time evolution of key plasma parameters of a baseline scenario demonstration discharge, operating at a normalized plasma current equivalent to 15 ma on iter (131498). illustrated are (a) plasma current ip, (b) normalized beta βn and confinement factor h98 with iter target values, (c) fusion performance factor with target value for q = 10 operation on iter indicated, (d) line average electron density and divertor dα emission dα, indicating elm timing, (e) neutron injection beam power pnb the closest scenarios for iter in the experiment are supposed to be achieved on the west tokamak (w for tungsten environment in steady-state tokamak), which is an upgrade of the tora supra tokamak (france, cadarache), which has a round cross-section. in the west tokamak, inside the circular vacuum vessel, there are the coils of the poloidal magnetic field, allowing the achievement of a diverter configuration of the plasma (fig. 1.2). the main purpose of the west tokamak is to investigate the tungsten diverter plate for iter on long plasma discharges in iter scenarios, which will be created due to the superconducting coils of the west tokamak. the west tokamak parameters are as follows: plasma current ip = 1 ma, toroidal field bt = 3.7 t, major radius r = 2.5 m, minor radius a = 0.5 m, aspect ratio a = 5-6, elongation k = 1.3-1.8, triangularity d= 0.5-0.6, and the discharge time at a plasma current of 0.8 ma is tflattop = 1000 s. in the west tokamak, the first plasma was obtained in december 2016 with the new control system based on the asdex upgrade real-time discharge control system and adapted to the specific needs of the west tokamak. currently, the development phase of the west project is coming to an end, and the first operational version is constantly being used in experiments [6, 7]. 98 95/ng h qb= plasma control in tokamaks. part 3.2. simulation and realization... 139 copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 1.2. cross-section of the west tokamak: (a) design [7], (b) magnetic configuration of the plasma [8] 2. approaches in modeling and implementing plasma control systems in iter development of algorithms for plasma control systems. fig. 2.1 shows the scheme of the development of plasma control system algorithms which has been used at the v.a. trapeznikov institute of control sciences of the russian academy of sciences for the development of magnetic plasma control systems in iter. the main model of the controlled plant was a non-linear dina code [9], in which the plasma model in iter was implemented and plasma discharge scenarios were simulated. the necessary plasma parameters were loaded into the code with the arm. with the help of the code, linear plasma models were obtained by means of its linearization [10, 11] or identification. then linear models were used for synthesis of a plasma vertical instability suppression controller as well as a plasma current and shape controller. the obtained controllers were used for mathematical modeling of the control processes for the plasma position, current and shape in a closed-loop control system on linear plasma models and dina code. fig. 2.1. development of plasma control systems for iter: awp is an automated workplace software and computing platform. for the mathematical modeling of the systems of magnetic plasma control at the v.a. trapeznikov institute of control sciences of the russian 140 y.v. mitrishkin, n.m. kartsev, a.e. konkov, m.i. patrov copyright ©2020 assa. adv. in systems science and appl. (2020) academy of sciences, there was created a software and computing platform based on the dina plasma code and matlab/simulink environment [12]. the purpose of the platform is to organize a convenient software tool for numerical experiments on linear and nonlinear plasma models, to implement the possibility of transition from one model to another without changing the control system, and the subsequent analysis of the results of the modeling. the developed software and computing platform allows • loading into the matlab workspace all the variables necessary for the simulation (more than two hundred); • switching the operation of the control system from a linear model to a non-linear model and back by a special switch; • carrying out modeling in batch mode (sequentially with different parameter values), for example, to search for the best controller settings; • in the process of modeling, saving all registered signals in the variables of the matlab workspace; • saving the simulation results in a database, viewing them in the form of graphs, and comparing the results of various experiments by superimposing them on each other. the platform is a technical tool to improve the efficiency of scientific work and allows more productive research into various approaches to plasma control in the iter reactor tokamak, as well as comparing the results obtained to make a decision on the best plasma control methods. the platform in the matlab environment enables the implementation of developed control systems in the codac control complex of the iter reactor tokamak. the platform includes: • a set of matlab m-files, • simulink models, which are built on a block principle and include subsystems implementing plasma models in the tokamak, actuators, diagnostics, and controllers, • dina code files, • as well as files that store parameters from the iter database, discharge scenario, and other data. real time testbed. nowadays it is generally recognized that experimental real-time testbeds are necessary for the application of control methods and systems on real dynamic plants. these include computer testbeds capable of operating in a hardware in the loop simulation (hil) mode [13]. such a testbed consists of a plant model and controller, each of which is implemented on its own industrial computer (fig. 2.2, a). these computers form a feedback system, the signals in which are circulating in real time. data from the external computer of the automated workplace.} are loaded to the industrial computers in real time. such a testbed, in particular, is used on the diii-d tokamak [14]. a real-time testbed for the plasma position, current and shape control system in iter has been developed, created and tested at the v.a. trapeznikov institute [15]. fig. 2.2, b shows the real time operation of the plasma model in iter realized on the computer, which is controlled by another computer in real time under the simulinkrt operating system from the mathworks company. plasma control in tokamaks. part 3.2. simulation and realization... 141 copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 2.2. real-time testbed: (a) testbed structure; (b) real-time results of the plasma model in iter when controlling the model by the real-time controller at minor disruption (from right to left and from top to bottom): gaps, plasma current, vertical position of the plasma, control voltages the iter information-control system. all information and control systems in iter, including the plasma control systems, are integrated into a common system called iter codac (control, data access and communication) [16, 17]. the purpose of the common system is to integrate more than 30 subsystems, based on the open software platform epics (experimental physics and industrial control system) [18], which provides flexible interfaces for components and the principle of their description sdd (self-description data). the iter codac system is designed to process about one million signals in an united system. different codac subsystems, which provide the operation of the installation, contain specific local devices through which the logic is formed and the control and data acquisition is performed. such devices include programmable logic controllers plc (programmable logic controller), in the iter project the models of the siemens simatic s7 line are used [19], as well as "fast controllers" with input/output through the pcie bus, which provide fast processing а 142 y.v. mitrishkin, n.m. kartsev, a.e. konkov, m.i. patrov copyright ©2020 assa. adv. in systems science and appl. (2020) of signals at the control rate; in the project, the devices of national instruments company are accepted [20]. each subsystem also includes a psh (plant system host), providing the standard functions inherent in all elements of codac. the following functions for integration of subsystems are implemented in the system: • monitoring the status of the psh host and the state of the "fast controllers"; • a time synchronization service, which is a subnet for the transmission of accurate time signals, namely, tcn (time communication network) according to the standard ieee1588 precision time protocol (ptp), using devices ni pxi-6683h; • a synchronous data service that is a subnet for real-time data transmission, specifically, sdn (synchronous databus network) to ensure data exchange between all elements of the system; • data archiving: a high-performance data transmission subnet for dan (data archiving network) archiving and visualization; • a standard state machine for cos (common operating state) subsystems provides control over system elements by reading the states and requesting transitions between them. the subsystems are connected via a pon (plant operation network) control subnet to the codac servers that provide the monitoring, control, and processing of the upper level data, as well as to the human-machine interface of the locking and safety systems. fig. 2.3. scheme of interfaces between plasma control systems and codac [21] plasma control systems are included as components in iter codac, but can be implemented on specialized computing facilities, which allows the use of virtually any modern information processing technology, such as calculations on a gpu (graphics processing unit), on multi-core multiprocessor systems, and programmable logic integrated circuits. in contrast to diagnostic subsystems and actuators, plasma control systems use a wide range of codac system services: time synchronization, real-time data access, data archiving and plasma control in tokamaks. part 3.2. simulation and realization... 143 copyright ©2020 assa. adv. in systems science and appl. (2020) lockout and security systems. a scheme of the interfaces for the interaction of the plasma control systems with codac is shown in fig. 2.3 [21]. the huge number of requirements and tasks in the project have created the need to develop modeling tools. a joint team of specialists from general atomics, ipp garching and the create group developed the pcssp (plasma control system simulation platform) within the codac system, which provides: • development of plasma control systems, • creation and validation of discharge scenarios, • analysis and troubleshooting, • support for development and modification of the system as a whole. a functional diagram of the plasma control system simulator in pcssp is shown in fig. 2.4 [22, 23]. fig. 2.4. functional diagram of the plasma control system simulator [22, 23] the work on the plasma simulation software and the pcs (plasma control system) continues. one of the approaches to solving this task is the developing imas software package: the iter integrated modelling & analysis suite [24]. it will be available to all participants of the iter project as a key tool for the scientific operation of iter. the imas package will enable the collective development of integrated modeling tools through the exchange of data, code components and sequences of technological operations based on the different code components that are interconnected. the development of imas started in 2011 and the first prototype of the imas infrastructure has been implemented in the iter organization. this package is also used in the west tokamak. 144 y.v. mitrishkin, n.m. kartsev, a.e. konkov, m.i. patrov copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 2.5. verification of the corsica-imas composition (‘kepler’, solid lines) with respect to the stand-alone corsica code (‘corsica’, circles): evolution of the plasma boundary (left), safety factor q profiles, and plasma current density (right) during plasma current rise in iter. both schemes provided exactly the same results: the curves completely coincided the kepler [25] graphics tool has been implemented in imas, which allows accessing the corsica code embedded in imas. fig. 2.5 shows the result of the verification of the integrated corsica code and its stand-alone version. 3. preparing the plasma control system in iter for start-up and operation the iter tokamak reactor is an extremely complex controlled plant, so it is important to determine the necessary elements and algorithms for further operation of the pcs together with the rest of the plant systems [26-29]. the list of basic control functions required for development when obtaining the first plasma in iter has been compiled in [27]: • control of currents in the poloidal field (pf) and central solenoid (cs) coils; • magnetic zero feedback; • initial control of plasma current; • initial control of plasma position; • initial control of plasma boundary; • gas start-up control; • initial density control; • control of electron-cyclotron heating (ech); • prevention of scattered radiation at ech. the list of models (plasma, actuators, and sensors) corresponding to the final pcs development for the first plasma in iter is also presented in [27]: § pf and cs models; § power supply models; § magnetic sensor models; § passive component models (e.g. eddy current model on the vessel); § models of gas supply; plasma control in tokamaks. part 3.2. simulation and realization... 145 copyright ©2020 assa. adv. in systems science and appl. (2020) § model of the gas valve; § model of pressure in the vessel; § model of plasma fuel; § model of the vacuum gauge (penning sensor); § model of the density interferometer; § model of ech; § model of gas breakdown; § model of the plasma position. the paper [27] shows a structural scheme of the design, implementation, and operational testing of pcs, which consists of three steps. ü step 1 confirms that the implementation of the system meets the design goals and technical requirements, ü step 2 checks that the components of the system technically function correctly in accordance with the design goals, ü step 3 checks that the complete system functions correctly in accordance with the design goals in all operating modes. to facilitate the implementation, all the controllers and models are presented as modules of the pcssp system platform [21-23], which is capable of automatically generating implementation code by means of simulink®. in the same paper [27] the scheme of designing and evaluation of the control is presented: step 0 and the three basic commissioning steps in connection with the modeling of the controller, plant and on-line testing. step 2 will require pcs capability testing in the loop testing of real equipment (hil testing). commissioning of the pcs architecture is an important feature of this stage, which ends with the transfer of the system for operational verification. the iter development and research plan, including pcs, defines a detailed development program [27]. once the program is adopted, based on predictive modeling and risk analysis, the prepared stages can be executed in a workflow. each planned stage should be checked before and after its execution, the measurements should be verified, and the results should be compared with the ones that were predicted by the modeling. depending on the result, the planned stage will be configured within the approved range to perform the next test. the models employed in the simulation and the preparation of the work and, moreover, those needed to evaluate the pcs design, will be tested and can be refined. the latter may cause changes in the future work program, in the redesign of pcs or may affect the research plan of the next stage of the work in iter. as an example, fig. 3.1 shows the modeling of the initial phase of the discharge in iter at operation of the rt-efit equilibrium reconstruction code for the iter scenario with a plasma current of 15 ma at t = 0.5 s, 4 s, 6 s, and 28 s, together with the deviations of the internal and external gaps between the plasma boundary and the first wall [26]. there is another code for the reconstruction of plasma equilibrium in real time, namely, liuqe, which works well on tcv and includes a detailed model of the eddy currents. this code is also approved for iter and is being tested [26]. in addition to the magnetic position, current, and plasma shape control system, in iter there are also algorithms developed to improve the reliability of the plasma, so as to ensure the survivability of the entire plant. the limited number of actuators, in particular the additional plasma heating systems, necessitates the introduction of heterogeneous control functions during a plasma discharge, either sequentially or simultaneously with failures in the plasma or in the controlled plant systems. a partial failure in the actuator, e.g. in a gyrotron providing electronic cyclotron resonance heating, or in a gas valve, must be compensated for by automatic switching to another gyrotron or gas valve. to overcome failures and malfunctions, an advanced level of control of the actuators and processing of exceptional (emergency) situations in the plasma control system has been developed [30, 31]. 146 y.v. mitrishkin, n.m. kartsev, a.e. konkov, m.i. patrov copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 3.1. restoring plasma equilibrium using rt-efit code minor disruptions in iter will be counteracted by pcs. the very high level of thermal and magnetic energy in iter plasma requires an in-depth approach to protect the tokamak components from potential damage due to thermal loads, electromagnetic forces, and increasing electron flux in the event of major disruptions. the pcs is the first tokamak protection line that will try to avoid disruptions by confining the tokamak plasma within the specified limits and according to the specified conditions. the pсs will be equipped with advanced prediction algorithms of disruptions to ensure achieving the stability levels based on multiple real-time stability measurements and calculations. there are two approaches to predicting disruptions: one, based on physical measurements extrapolated into iter [32], and the other, on machine learning techniques to analyse large data sets [33]. if a major disruption is unavoidable, the pcs sends a request to the central interlock system (cis) to initiate a disruption mitigation system (dms) so that the dms injects large amounts of z impurities to reduce the energy of the plasma, control the rate of the current decrease, and attenuate any thermal and electromagnetic loads due to the disruption. the dms is designed with 12 independent granule (pellet) injectors to attenuate the thermal load and 15 independent injectors to attenuate the increase in the electron flow. in [26] the scheme of the interaction of pcs, cis, and dms is shown. the pcs decides on the sequence of injector initiations in the dms to ensure the required amount and mixture of gas for the required time of the emergency situation, the pcs determines or predicts the inevitable cooling, current decrease, loss of vertical control, the presence of a rising flow of electrons or critical system failure in the plant. 4. demo development roadmaps the development trend of conventional tokamaks with an aspect ratio of about 3-4 [1-3] has led to the design of iter and next-generation projects of demo (thermonuclear power plants on tokamak reactors) with relatively large dimensions: the major radius r0 of iter is 6.2 m, and r0 of demo is about 9 m. in doing so, spherical tokamaks with a relatively small aspect ratio of 1.4-1.6 were developed parallel to the conventional tokamaks. the plasma physics of these tokamaks differs from the physics of conventional tokamaks. hence, it is much cheaper and faster to use them as the basis for commercial thermonuclear power stations than it would plasma control in tokamaks. part 3.2. simulation and realization... 147 copyright ©2020 assa. adv. in systems science and appl. (2020) be to use conventional tokamaks. in particular, it has been shown that on the basis of one of the versions of the modular demo one can get an output of thermonuclear electricity costing less than 6 cents per 1 kwh [34]. fig. 4.1. roadmaps for the development and construction of the first commercial tokamak reactor power plants in this regard, two roadmaps for the development and construction of commercial power plants are currently available in europe (fig. 4.1) [35, 36]. it is estimated that commercial thermonuclear power plants of the conventional type may appear not earlier than 2075. at the same time the estimates for commercial thermonuclear power plants of spherical type have a more favorable forecast: a modular thermonuclear power plant can be created by 2030. creating such a power plant is much easier, because first of all only one module is created, and then a power plant of full capacity is assembled from a number of modules. a special company has been established to develop such a power plant in the uk: tokamak energy ltd (url: https://www.tokamakenergy.co.uk/). 5. demo structures and poloidal systems offered designs of poloidal systems in demo on conventional tokamaks essentially repeat similar designs in the iter-2 tokamak. one of the first conceptual designs of the demo tokamak was published in 2000 [37]. fig. 5.1 shows the designs of demo poloidal systems presented by groups of researchers from different countries. the papers [38, 39] demonstrate schemes of chinese and koreanamerican demo tokamaks: the hcsb-demo project and the k-demo project. it should be noted that the presented list is not exhaustive: only the most different variants are given. in fig. 5.1, the project of one spherical tokamak module [34] is presented, the other options are conventional and non-modular. in the module, the poloidal field coils are located inside the toroidal field coil, and they are concentrated in two groups of four coils at the top and bottom of the vacuum vessel. 148 y.v. mitrishkin, n.m. kartsev, a.e. konkov, m.i. patrov copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 5.1. vertical cross sections of demo tokamaks with various poloidal system options in [40] (fig. 5.1, c) it is proposed to use 9 pf-coils with two lower pf coils placed inside the toroidal coil, in [34] (fig. 5.1, a) and [44] (fig. 5.1, d) there are 8 pf-coils, and in [42] (fig. 5.1, b), [41] (fig. 5.1, e), and [38] (fig. 5.1, f) there are 6 pf-coils. in [34] (fig. 5.1, a) and [42] (fig. 5.1, b) it is proposed to apply the whole cs, in other projects there is a sectioned cs, and in [41] (fig. 5.1, f) and in [43] (fig. 5.1, e) the cs consists of 5 sections, where the height of the central section is twice as high as all the others, in [44] (fig. 5.1, d) the cs contains 6 sections, and in [40] (fig. 5.1, c) the cs consists of 8 sections of the same height. table 5.1. key features of demo projects project major radius, m minor radius, m aspect ratio maximum toroidal field, t plasma current, ma uk, spherical demo module, fig. 5.1, a 1.6 0.94 1.7 3.17 6.7 france, fig. 5.1, b 9 2.65 3.4 10.5 17 japan, fig. 5.1, c 8.2 2.6 3.2 17 14.6 japan, fig. 5.1, d 8.2 2.57 3.19 17 14.6 germany, fig. 5.1, e 9 2.25 4 7.1 20 european union (eu demo), fig. 5.1, f 9 2.9 3.1 13.3 20 china (hcsb-demo) 7.2 2.1 3.4 6.86 14.8 korea – usa (k-demo) 6.8 2.1 3.2 16 12 demo-s (russia) 7.8 1.5 5.2 8.75 12.8 plasma control in tokamaks. part 3.2. simulation and realization... 149 copyright ©2020 assa. adv. in systems science and appl. (2020) all presented projects are executed according to the conventional scheme [1], according to which the pf-coils are located outside the toroidal field coil, except for the spherical module [34] (see fig. 5.1, a). until recently, no comparisons of the different magnetic configurations of demo had been conducted, except for [45], who partly conducted simulations for different sizes of the vacuum vessel and configurations of the pf-coils. it follows from the presentation in that article that the development of demo is decentralized, unlike the iter project, in which the development is conducted by an international consortium of researchers. table 5.1 shows the main characteristics of the above demo projects. the project of the plasma position control system in demo, in fact, was presented only in [44], the results on current and plasma shape control systems for demo have not yet been given, the areas of the vertically controllable plasma and the areas of achievability of the separatrix have not been investigated, which can be explained by the need to check the corresponding algorithms in real work on iter. in [44], the assumptions made are as follows: the plasma is considered to be "rigid", taking into account the plasma current density profile; the components inside the vessel and the vacuum vessel itself are approximated as a three-dimensional and "thin" structure; the evolution of the eddy currents and coil currents is estimated using closed equations. the numerical simulation code for the plasma position control analysis consists of three modules: tosca plasma equilibrium code [46] for creation of a plasma current density profile and a magnetic surface by solving the grad-shafranov equation; eddycal eddy current code [47] to evaluate the passive effect of stabilization of the vacuum vessel and components inside it by the finite element method; plasma position control code for calculation of the time evolution of the vertical and radial plasma positions, eddy currents and coil currents of the feedback with a pid controller taking into account the results of calculating the equilibrium coefficients in the plasma and eddy currents [48]. fig. 5.1, d shows a cross-section including the plasma, protective modules (bm), conductive shell (shell), rear plates (bp), vacuum vessel (vv) and the system of poloidal coils (pf). the sequence of calculations in the structural diagram of the plasma position control system is as follows: after calculating, using the equilibrium reconstruction code, the plasma current density profile !𝐽#$, the eddy current time constant (𝜏') and the mutual inductances between the plasma and coils are calculated using the eddy current code eddycal. then, taking into account the data from the equilibrium and eddy current analysis, the evolution of the plasma motion (in vertical and radial directions), eddy currents and control coil currents are calculated. the developed numerical simulation code can work out exact models of the vacuum vessel and the components inside it [44]. 6. conclusion in addition to modeling and implementing plasma control systems in iter, this part of the survey has provided information on the development of the first demo thermonuclear power plant, including the design, poloidal systems, and plasma position control system. two demo development roadmaps have been presented for conventional tokamaks with relatively high aspect ratios and spherical tokamaks of modular type with a low aspect ratio. according to current estimates, the first commercial thermonuclear power plant with relatively cheap electricity can be developed much faster when based on the latter. iter experiments in conditions of thermonuclear reaction will allow formulating more precise requirements for the plasma control systems for demo on conventional tokamaks. it will then be possible to proceed with the development of the appropriate systems for demo conventional types with some certainty. for demo modular type on spherical tokamaks it is 150 y.v. mitrishkin, n.m. kartsev, a.e. konkov, m.i. patrov copyright ©2020 assa. adv. in systems science and appl. (2020) advisable to work out in detail plasma control systems on existing spherical tokamaks such as st40 (uk), globus-m2 (russia), nstx (usa). research is being carried out on the integration of the magnetic and kinetic plasma control systems, where an important direction is the control of the profile of the plasma parameters (current density, safety factor q, density, temperature). this direction, in particular, should lead to an understanding of which plasma parameter profiles must be provided so that no major disruptions occur. the methods and systems of kinetic plasma control will be reviewed in the fourth part. acknowledgements the work was supported by the russian foundation of basic research (grant no. 17-18-00293) and the russian science foundation (grant no. 17-19-01022). references 1. mitrishkin, yu. v., korenev, p. с., prokhorov, a. a., et al. 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(1988). simulation studies on plasma position control for the next generation tokamak machines with up-down asymmetry, fusion engineering design, 5, 343–356. microsoft word 1009 article text, copyedited.docx adv syst sci appl 2021; 02; 42-57 published online at https://ijassa.ipu.ru. deep learning techniques for detection of covid-19 using chest x-rays alampally naveen1*, boddu manoj1, golla akhila1, maunil bipinkumar nakarani1, jaini rathna sreekar1, pradyumn beriwal1, nipun gupta1, swathi jamjala narayanan1 1) school of computer science and engineering, vellore institute of technology, vellore, india e-mail: alampallynaveen123@gmail.com, manumanoj0010@gmail.com, akhilagolla2020@gmail.com, maunil2399@gmail.com, j.rathnasreekar94@gmail.com, pradyumnberiwal@gmail.com, nipungupta.2702@gmail.com, swathi.jns@gmail.com abstract: the covid-19 pandemic situation keeps on ruining and affecting the wellbeing and prosperity of the worldwide population and due to this situation, the doctors around the world are working restlessly, as the coronavirus is increasing exponentially and the situation for testing has become quite a problematic and with restricted testing units, it’s impossible for every patient to be tested with available facilities. effective screening of infected patients through chest x-ray images is a critical step in combating covid-19. with the help of deep learning techniques, it is possible to train various radiology images and detect covid-19. the dataset used in our research work is gathered from different sources and a specific new dataset is generated. the proposed methodology implemented is beneficial to the medical practitioner for the diagnosis of coronavirus infected patients where predictions can be done automated using deep learning. the deep learning algorithms that are used to predict the covid with the help of chest x-ray images are evaluated for their prediction based on performance metrics such as accuracy, precision, recall, and f1-score. in this work, the proposed model has used deep learning techniques for covid-19 prediction and the results have shown superior performance in prediction of covid19. keywords: covid, x-ray images, transfer learning, radiology, deep learning, diagnosis 1. introduction covid’19 has influenced many nations in a very little amount of time. the coronavirus has given a devastating blow to the entire world which is detrimental to the health condition of many people and continues to intimidate the world. this newly identified virus is extremely dangerous and pathogenetically different from sars-cov, mers-cov, avian influenza, influenza, and other various common respiratory viruses [1]. although the diagnosis process has become relatively rapid, the financial issues arising from the cost of diagnostic tests affect both the patients as well as the gdp of the country, particularly in countries with private health systems, or restricted access to health systems. so far, due to the lack of availability of public images of covid-19 patients, detailed studies reporting solutions for automatic detection of covid-19 from x-ray (or chest ct) images are not available [2]. medical predictions are not accurate a lot of times and their uncertainty always has been seriously underestimated [3]. in march 2020, there has been an enormous increase in covid cases which consequently, lead to a rapid increase in the x-ray dataset. this helps us * corresponding author: alampallynaveen123@gmail.com deep learning techniques for detection of covid-19 43 copyright ©2021 assa. adv. in systems science and appl. (2021) to analyze the medical images and recognize potential trends that help in the automatic diagnosis. the automated and early covid-19 diagnosis can be helpful for the countries to immediately refer the patient to quarantine, rapid intubation, or severe cases in specialist hospitals, and control the spread of the disease. the model can detect the covid from x-ray images or ct images using image classification with the help of various deep learning techniques [4]. due to a gradual increase in cases, the testing of covid has become the most difficult situation and it also takes more time. to overcome this situation, a model is designed, by which the coronavirus can easily be predicted with the most appropriate deep learning techniques which have the highest accuracy. the effectiveness of the deep learning technique proposed is pre-trained fully convolutional neural networks with regard to their expertise in the automatic diagnosis of covid-19 from thoracic x-rays. the use of machine learning techniques for automated medical diagnosis has recently gained prominence by being an adjunct method for clinicians [5]. researchers are increasingly applying the recent advancements of deep learning to the analysis of chest x-ray images to increase performance and relieve the burden of radiologists. deep learning, allows end-toend models to be built to achieve promised results using input data without the need for manual extraction of features. the rapid rise of the covid-19 epidemic has required the need for expertise in this field. the interest in designing automated detection systems based on machine learning techniques has increased. a weakly-supervised classification and localization system proposed for the computer-aided diagnosis of common thoracic diseases developed 121-layer dense convolutional neural networks [6]. for many image processing applications, such as image analysis, image classification, and image segmentation, deep learning techniques have demonstrated high efficiency. image classification is achieved through a descriptor extracting the import features from the images, and then these features can be used using classifiers in the classification task. in this research work, four major convolution neural networks namely resnet50, vgg16, inceptionv3, xception are used for evaluating the performance against coronaviruses detection. 2. literature survey covid is one of the major causes of death nowadays. ali et al. [7] had proposed a deep transfer learning technique-based approach for covid-19 detection using chest x-ray images. they have classified the images from the three datasets used for implementation into four classes namely covid-19, normal (healthy), viral pneumonia, and bacterial pneumonia. five pre-trained convolutional neural network models were used in this study – resnet50, resnet101, resnet152, inceptionv3 and inception-resnetv2. the performance analysis showed that the resnet50 convolutional neural network model showed the highest accuracy in all the three datasets. a deep convolutional neural network design for the detection of coronavirus cases in humans from chest x-ray (cxr) images is called covid-net with the help of an opensource, available to the general public covid-19 dataset. through this method, they gained deeper insights into critical factors that are related to the covid-19 cases, which helped doctors with an improvement in screening. the obtained model is a successful one as it has a success ratio of 95.9% in detecting the covid-19 virus and it can also identify non-covid accurately from a person's chest x-ray. but also, a slight problem would be that the detection decisions are taken only by covid-net [8]. they adapted detrac (decompose, transfer and compose) – a deep cnn architecture that depends on a class decomposition approach for the classification of covid-19 images. the experimental results showed that detrac had high accuracy and with high sensitivity and specificity respectively [9]. the first-ever data collection of covid-19 images are made accessible to researchers. they have collected cxr image data, which is about 542 chest x-ray images from about 44 naveen et al. copyright ©2021 assa. adv. in systems science and appl. (2021) 262 covid-19 patients from 26 different countries. frontal and lateral view imagery has been collected and all the metadata such as when did the patient experience first symptoms and what was his intensive care unit (icu) status, intubation status, or his survival status [10]. suat toraman, talha burak alakus, and ibrahim turkoglu proposed a novel artificial neural network for detecting covid-19 from the chest x-ray images and also to classify the dataset images into covid-19, and pneumonia x-ray images using capsule networks. the results displayed using capsule networks will not only lead to higher accuracy but at the same time can be effectively used for classification in a limited dataset too [11]. the following study attempts to give insights on the feasibility of using a deep learningbased decision tree classifier for detecting covid-19 from chest x-ray images. this proposed classifier consists of three binary decision trees. the first tree classifies the chest x-ray images as normal and abnormal. the second tree identifies the abnormal images with signs of tuberculosis and the third does the same thing for covid-19. the average accuracy achieved by the proposed classifier was found to be 95%. [12]. a deep learning model was used for detecting coronavirus, which is a sub-branch of artificial intelligence. the dataset used for this research consists mainly of three classes that are coronavirus, pneumonia, and normal x-ray imagery. fuzzy color techniques were used to restructure the data classes that would be used as a pre-processing step and the images that were structured with the original images were stacked. in the following step, deep learning models like mobilenetv2, squeeze net were used to train the stacked dataset and the social mimic optimization method was used to process the feature sets obtained by the models. subsequently, all the features were combined and categorized using svm that is support vector machines [13]. in this research paper, the authors used a dataset of x-ray images from patients that have diseases similar to coronavirus (covid-19) like pneumonia. incidents from daily news and cases were utilized for the automatic detection of the covid-19. the purpose of this study is to evaluate the performance of state-of-the-art convolutional neural network architectures that are suggested over the past years for medical image analysis and classification. particularly, the method called transfer learning was implemented. it is shown in the paper that even for small medical image datasets detection of various abnormalities is an achievable target with the help of transfer learning, and it often yields remarkable results. the outcome of the research proposed that deep learning with x-ray imaging may extract quite noteworthy biomarkers related to the coronavirus disease, with the highest accuracy, highest sensitivity, and highest specificity which is quite remarkable [14]. a deep learning-based lung ct diagnosis system [15] was developed that detects the patients with the presence of covid-19. the results of the experiment show that the trained model can identify the covid-19 patients from others accurately with high sensitivity and an excellent auc. besides, the obtained model is capable of distinguishing the coronavirus patients and pneumonia-infected patients with an astonishing high auc and recall. furthermore, the trained model is of great help to assist doctors in diagnosing coronavirus disease since it can also localize the main lesion features, especially the ground glass opacity (ggo). also, the diagnosis is super-fast. the diagnosis of a patient just takes about 30 seconds. so, the conclusion obtained from this research is that the established models can achieve a rapid and accurate classification of coronavirus, therefore allowing the identification of patients. the authors of this paper compared the deep learning-based feature extraction frameworks for the automatic classification of coronavirus. to achieve the most accurate feature, algorithms such as xception, vggnet, resnet, mobilenet, inceptionv3, inceptionresnetv2, densenet, and nasnet were used in the accurate classification of the virus. the features were extracted and then fed into several machine learning classifiers to distinguish the subjects as a case of coronavirus or not. the approach used to avoid taskspecific data pre-processing methods supports a better generalization ability for unseen data. the performance of the proposed method collaborates on a freely available dataset of chest deep learning techniques for detection of covid-19 45 copyright ©2021 assa. adv. in systems science and appl. (2021) x-ray and ct images of coronavirus patients. the best performance was achieved by the densenet121 feature extractor with a bagging tree classifier which was a classification accuracy [16]. in this report, the authors implemented a deep convolutional neural network architecture for the detection of cases of coronavirus in humans from chest x-ray (cxr) images, called covid-net, accessible to the general public dataset of covid-19 with the aid of an opensource. in addition to this, they researched how covid-net can use an explanatory capacity approach to make predictions. they gained greater insights into crucial factors linked to the covid-19 cases through this approach, which allowed doctors to enhance screening. the model obtained is a good one as it has a high success ratio in covid-19 virus detection and can also reliably distinguish non-covid from the chest x-ray of a human [17]. but also, a slight problem would be that only covid-net is taking the detection decisions. mahesh gour and sweta jain introduced a novel stacked convolutional neural network for the automatic analysis and diagnosis of the covid-19 from the chest x-ray images of the dataset. the authors obtain different sub-models from vgg19 and create a new 30 layered cnn model called covnet30 [18]. for the training and creation of our model, have used heterogeneous online sources for data collection. our dataset includes images under 2 categories covid19 chest x-ray images and normal chest x-ray images [19,20]. during the development of classification models, the initial stage included the merging of the following github repositories and kaggle resources which maybe have been updated even in our implementation. we would like to acknowledge them for their contributions to this research work and also making the data sources publicly available. 2.1. github repositories the links are provided below: • https://github.com/ieee8023/covid-chestxray-dataset • https://github.com/shervinmin/deepcovid • https://www.dropbox.com/s/09b5nutjxotmftm/data_upload_v2.zip?dl=0 • https://github.com/cssegisanddata/covid-19 2.2. kaggle repositories the links are provided below: • https://www.kaggle.com/tawsifurrahman/covid19-radiography-database • https://www.kaggle.com/bachrr/covid-chest-xray?select=images • https://www.qmenta.com/covid-19-kaggle-chest-x-ray-normal/ the reason behind creating the new dataset is that researchers till now have used a smaller number of covid chest x-ray images which is not enough to validate or justify that the model will be accurate in any study. so, a huge dataset has been used i.e. a greater number of images compared to other researchers’ datasets and is getting better accuracy for some algorithms in our model. a heterogeneous online data collection sources have been used for the training and development of our model. the initial stage involved combining the dataset from various credible sources during the creation of classification models and also making the data sources publicly accessible [21, 22]. 3. dataset collection the dataset is generated by collecting a set of chest x-ray images of covid -19 from heterogeneous online public sources like github, kaggle, and certain data sources mentioned in multiple research papers. in the new dataset created, there are 4539 number of covid-19 chest x-ray images and 5961 number of non-covid-19 chest x-ray images, used for training and testing using deep learning models. a total of 10500 images was collected, in that 8400 was used for the training phase of the model and the rest 2100 was 46 naveen et al. copyright ©2021 assa. adv. in systems science and appl. (2021) used for the testing phase of the model. the training of models also includes data argumentation (image data generator) for increasing the dataset size for both training and testing [23, 24]. fig. 3.1. x-ray images of covid infected patients. fig.3.2. x-ray images of normal patients. after data argumentation 8736 training images were generated and 2220 testing images. 4. methodology diagnostic imaging modalities, such as chest x-ray images are playing an important role in confirming the primary diagnosis from the polymerize chain reaction test for covid-19. convolutional neural networks are algorithms that are used to predict the class of images [25]. in our proposed work, deep learning techniques have been applied like vgg16, resnet50, inceptionv3, and xceptionfor covid-19 prediction. preprocessing of data has been implemented to establish the same size for all data and to normalize the images in the dataset. different techniques such as image resizing, image data generator, data labelling, and data normalization has been applied to the dataset. deep learning techniques for detection of covid-19 47 copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 4.1. block diagram of the model. . the dataset is prepared for the training and testing phase by applying the required preprocessing. then, the data is split into an 80:20 ratio for training and testing the dataset to predict the outcome of the chest x-ray image. the deep learning models are trained for 500 epochs with a batch each consisting of 32 images. the requirement of the model is to recognize the necessary features from the training phase and after training make the predictions and classify the images. in this work, the implementation of all the architectures included the addition of three custom layers in the pre-trained models for training the dataset. the first layer (flatten layer) was used to flatten out the input features. the next layer is the dropout layer, which buried the problem of overfitting. a dense layer with a softmax activation function is used as an output layer. 4.1 vgg16 vgg16 technique is a deep learning predictive model that is based on the ability of imitation effects shown by multiple small filters in a sequence with respect to large filters [26]. for interpreting the effect of depth on accuracy a simple cnn with a small convolution filter of size 3x3 with stride and padding of 1 along with a 2x2 max-pool with a stride of 2 for 16 layers was used. fig. 4.1.1. architecture of the vgg model. it shows improvement over alex net by substituting large kernel sized filters with multiple 3x3 kernel sized filters one after another. the algorithms were specifically trained 48 naveen et al. copyright ©2021 assa. adv. in systems science and appl. (2021) for image localization. the input to the first convolutional layer is an image after preprocessing the dataset. the image is passed over a stack of layers where small size filters of 3x3 have been used and pooling is also been applied after every convolutional layer. a 1x1conv filter had also been used for the conversion of input channels to linear. to help us extract the generic low-level descriptors or patterns from the chest x-ray image data, then froze the weights of the earlier layers of the pre-trained backbone. the first few layers learn very basic and generic characteristics that generalize to almost all types of images in the convolutional networks that are used in this research work. the features became increasingly more unique to the dataset on which the model was trained. the objective of fine-tuning is to adapt these specialized features rather than overwrite the generic learning, to work with the newly fed covid-19 dataset. vgg16 is an improvement over alexnet, which used to be a state of art model until the introduction of vgg16 that reduced the image size very drastically using convolution to avoid without needing sgd (stochastic gradient descent) necessarily and to increase the number of channels to make the image while reducing height and width of the image simultaneously. this improves the computational speed and accuracy of our model without going through a deep neural network. 4.2 resnet50 resnet50 is a deep learning pre-trained model, which is trained on the imagenet dataset. the backbone of this architecture involves introducing an identity shortcut connection that will enable us to skip multiple layers. it is the short name for the residual network and it introduces the concept of residual layers. fig. 4.2.1. architecture of the resnet50 model. resnet50 architecture is pretty similar to resnet18, the main difference being having more layers. resnet50 network each 2-layer block in the 34-layer net is replaced by a 3layer bottleneck block leading to a 50-layer model. the idea behind this is to skip layers and establish direct short connections to prevent saturation of accuracy [27]. the insertion of short connections will lead the network to turn into its residual counterpart. as the network gets deeper and more complex, it avoids the distortion that happens in the model during training. the shortcut connections perform identity mappings and the outputs obtained are added to the outputs of other stacked layers. one of the advantages of the architecture deep learning techniques for detection of covid-19 49 copyright ©2021 assa. adv. in systems science and appl. (2021) includes helping the network to create a path, which will help us simplify the gradient updates for the preceding layers. 4.3 inceptionv3 inceptionv3 architecture aims to assist in image analysis and object detection. it is a type of neural network model of convolution. it consists of several stages of convolution and full pooling. it increases the depth and width of the network and improves the involvement of computing resources inside the network while keeping computation operations unchanged [28]. fig. 4.3.1. architecture of the inceptionv3 model. in general, it has 48 layers and uses inception modules consisting of a concatenated layer with convolutions of 1 x 13 x 3 and 5 x 5. reducing the number of channels in the image is the basic concept of the 1x1 convolution. thus, it also decreases the number of parameters. the size of the matrix is usually not affected by 1x1 convolution; however, if the input matrix is multi-channel, the channel number of the 1x1 convolution method output matrix is equivalent to the number of 1x1 convolution filter channels added. this architecture's key benefit is that it enables one to reduce the number of parameters and increase the speed of training. another benefit of this architecture is that it enables the user to perform the localization according to their parameters and helps us to recognize with better accuracy. 4.4 xception xception is a 71-layer deep convolutional neural network. it involves depth wise separable convolutions instead of standard inception modules. it is an extension of the inception architecture. a pre-trained version of the network trained on more than 10 lakh images from the imagenet database can be loaded. in this model, the initiation of convolution modules is replaced by depth-wise separable convolutions. point-wise convolution layers obey the depth of wise convolution layers. for a large range of images, the network is capable of learningrich feature representations and is able to perform a little better than the inception network. the input image was first sent through the entry flow in this analysis, then eight times through the middle flow, and finally through the exit flow that includes a softmax layer. 50 naveen et al. copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 4.4.1. architecture of the xception model. 5. computational experimental setup in this research paper, the proposed architecture is to demonstrate the pipeline model that assembled would optimize the outcome and evaluate all model results, then compare model accuracy, training time, and model loss of the vgg16, resnet50, inceptionv3, and xception algorithms on the dataset while training. the classification model is evaluated in terms of precision, recall, and f1-score and is applied to the dataset after data set collection and using different data preprocessing techniques [29]. transfer learning modules from deep learning were implemented. several machine learning libraries such as tensorflow 2.3.2, sklearn, seaborn, numpy, matplotlib, and opencv are used to incorporate all transfer learning, and the training and testing procedures are carried out on the google colab platform. all the experiments presented in this paper have been performed on google colaboratory with the use of gpu, tesla k80 graphical processing unit with 12gb gddr5 vram hardware from the online cloud service that is available for free. the cnn models (resnet50, vgg16, inceptionv3, and xception) pre-trained on the imagenet dataset are used with random initialization weights, and adaptive moment estimation (adam) optimization of the cross-entropy function has been performed. there has been a random division of data into two separate datasets (train & test) of 80 % and 20% for training and testing, respectively. for all experiments, the batch size of 32, a learning rate of 0.00001, and a number of epochs of 500 were respectively used. these are some of the results after applying all the algorithms. , deep learning techniques for detection of covid-19 51 copyright ©2021 assa. adv. in systems science and appl. (2021) 5.1 vgg16 fig. 5.1.1. epoch vs accuracy in fig. 5.1.1. in this model accuracy, the x-axis indicates the epoch and the y-axis indicates the accuracy of training and testing of data which displays the graph of our dataset. fig. 5.1.2. epoch vs loss in fig. 5.1.2. in this model loss, the x-axis indicates the epoch and the y-axis indicates the accuracy of training and testing of data which displays the graph of our dataset. 5.2 resnet50 fig. 5.2.1. epoch vs accuracy in fig. 5.2.1, the graph displays the model accuracy of epoch with respect to accuracy for the resnet50 model. 52 naveen et al. copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 5.2.2. epoch vs loss in fig. 5.2.1, the graph displays the model loss of epoch with respect to for resnet50 model. 5.3 inceptionv3 fig. 5.3.1. epoch vs accuracy in fig. 5.3.1. in this model accuracy, the x-axis indicates the epoch and the y-axis indicates the accuracy of training and testing of data which displays the graph of our dataset for the inceptionv3 model. fig. 5.3.2. epoch vs loss in fig. 5.3.2. in this model loss, the x-axis indicates the epoch and the y-axis indicates the accuracy of training and testing of data which displays the graph of our dataset for the inceptionv3 model. deep learning techniques for detection of covid-19 53 copyright ©2021 assa. adv. in systems science and appl. (2021) 5.4 xception fig. 5.4.1. epoch vs accuracy in fig. 5.4.1, the graph displays the model accuracy of epoch with respect to accuracy for the xception model. fig. 5.4.2. epoch vs loss in fig. 5.4.2, the graph displays the model loss of epoch with respect to loss for the xception model. if true positive is categorized as the number of covid-19 chest x-ray images; false negative is the number of covid-19 chest x-ray images misclassifies as normal; false positive is the number of normal chest x-ray images misclassified as covid-19; the true negative is the normal number of chest x-ray images. in the convergence graph field, it is a standard measure to check how quickly the model is able to classify the data. accuracy, recall, f1score, accuracy is defined and the following equations will describe them. precision = !" !"#$" . (5.1) recall = !" !"#$& . (5.2) f1-score = '∗("*+,-.-/0∗1+,233) "*+,-.-/0#1+,233 . (5.3) accuracy: it is defining as the number of true positives and true negatives by the number of true positives, true negatives, false positives, and false negatives. accuracy = 𝑇𝑃+𝑇𝑁 !"#!&#$"#$& . (5.4) 54 naveen et al. copyright ©2021 assa. adv. in systems science and appl. (2021) table 5.1. classification report of all algorithms. algorithms precision recall f1-score accuracy vgg16 0.90 0.89 0.89 0.91 resnet50 0.83 0.82 0.82 0.83 inceptionv3 0.96 0.96 0.96 0.96 xception 0.91 0.91 0.91 0.92 fig. 5.5.2. classification report in the above table. 5.1, the precision, recall, f1-score and accuracy performances are listed after processing via various algorithms. the algorithms are trained and probed on the chest x-ray image dataset of the pre-trained models. in the above figures, the model accuracy and model loss of the trained models are given. it can be said that for the initiation, the highest model training accuracy is obtained for inceptionv3 and the training is carried out for 500 epochs. it can be analyzed that from the model loss figures that the loss values are reduced in all pre-trained models, and with the help of the confusion matrix, a comparison of all algorithms is listed out based on performance from the classification table. after comparing the precision, recall, f1-score, and accuracy from the above table, the inceptionv3 gives the best results in both the training and testing phase and shows better output than the other three models. 0,75 0,8 0,85 0,9 0,95 1 vgg16 resnet50 inceptionv3 xception classification report precision recall f1-score accuracy deep learning techniques for detection of covid-19 55 copyright ©2021 assa. adv. in systems science and appl. (2021) 6. conclusion early prediction of coronavirus is vital to prevent the spread of the disease among other people. so, the use of deep learning algorithms for the diagnosis of covid-19 has become a major concern during this pandemic considering the accuracies and f1 score of the present algorithms. keeping that in mind, we came up with better accuracies for the algorithm compared to the one which is present. we collected a huge dataset of x-ray images from different reliable sources and combined them to form one. large datasets could lead us to identify coronavirus infection more accurately and reliably when training and testing the model. in an effort to identify the covid19 affected patients more accurately and precisely we have implemented four different algorithms and compared each among themselves for better results. out of the four models vgg16, resnet50, inceptionv3, and xception. inceptionv3 showed better results among other algorithms and also with the previously proposed algorithms with an f1 score of 96% whereas the previously proposed inceptionv3 algorithm has a score of 94.8%. further, current research work can be enhanced by improving the efficiency of the model and also by reducing the training time with an improved version of the present code. finally, the current model can be deployed across various devices such as mobile phones such that patients can scan their x-rays and can get the prediction. references 1. abbas, a., abdelsamea, m. m., & gaber, m. m. 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(2020). covid-19 screening on chest x-ray images using deep learning based anomaly detection. arxiv preprint arxiv:2003.12338. microsoft word 1589-article text-7120-1-6-20240527 adv syst sci appl 2024; 03; 58-72 published online at https://ijassa.ipu.ru. recent advances in the development of polymeric floating biomaterials for helicobacter pylori therapy: current status and future directions manivasakam prakash1,2,3, venkateswaramurthy nallasamy1,2*, kalaiyarasi chinnasamy1,2 1) j.k.k. nataraja college of pharmacy, komarapalayam, namakkal-638183, india 2) the tamil nadu dr. m.g.r. medical university, chennai-600032, india 3) vellalar college of pharmacy, thindal, erode-638012, india abstract: helicobacter pylori (h. pylori) infection is a common global health problem linked to peptic ulcers, gastric cancer, and other digestive problems. current h. pylori eradication therapy must address issues such as antibiotic resistance, poor patient compliance, low stomach ph, and insufficient medication concentration at the infection site. with an emphasis on the present state and potential future directions in this developing field, this review attempts to give a broad overview of recent developments in the creation of such biomaterials for h. pylori therapy. this review examines the benefits of microballoon-based drug administration, utilizes the situation of h. pylori therapy, and covers several formulation and evaluation techniques for microballoon. additionally, it discusses current studies and clinical trials using microballoon drug delivery techniques to treat h. pylori. microballoon have the potential to improve drug delivery, reduce dose frequency, increase drug stability, and improve drug targeting. the assessment of challenges and opportunities in this burgeoning field provides insights into the evolving landscape of biomaterial-based h. pylori management, offering a roadmap for forthcoming research and clinical implementation. keywords: helicobacter pylori; microballoon; hallow microsphere; biomaterials; gastroretentive drug. 1. introduction helicobacter pylori (h. pylori) infection is a typical bacterial infection that affects the stomach. about 50% of people worldwide have h. pylori infection, which is more prevalent in developing nations, according to the world health organization [1]. infection with h. pylori can have serious effects. the infection is an important factor in peptic ulcer disease and has been associated with a higher risk of stomach cancer, which can be quite dangerous [2,3]. non-ulcer dyspepsia, a disorder characterized by persistent or recurrent abdominal pain or discomfort as well as other digestive symptoms, can also be brought on by h. pylori infection. h. pylori can have considerable negative repercussions on the economy and society in addition to the direct effects of peptic ulcer disease treatment. biomaterials, defined as materials engineered to interact with biological systems, provide specific benefits in the treatment of h. pylori. therapeutic agents can be transported by these materials, which offer precise delivery, controlled and prolonged release, and resistance to degradation by stomach acid. several study endeavors have examined the possibility of biomaterials augmenting the effectiveness of h. pylori treatment. for example, in one study, antibiotic-loaded polymeric nanoparticles showed improved stability and extended release, which led to increased antibacterial activity against h. pylori [4]. the development of biomaterials with particular characteristics, like floating or mucoadhesive attributes, is * corresponding author: vasakampharm@gmail.com recent advances in the development of polymeric floating biomaterials 59 copyright ©2024 assa. adv. in systems science and appl. (2024) required to overcome the particular difficulties presented by the acidic gastric environment and extend the duration of contact with the gastric mucosa. these characteristics may improve the therapeutic efficacy of medications administered locally. the application of floating microspheres containing antimicrobial agents showed enhanced antibacterial activity against h. pylori and a longer period of stomach residence [5]. 1.1. surveillance of h. pylori h. pylori colonizes the human and primate stomach mucosa. h. pylori thrives in an acidic environment that gastroenterologists traditionally believe to be sterile. h. pylori employs a two-pronged strategy to live in the harsh environment of the human stomach [6-8]. first, the bacteria possess several polar flagella, which allow them to travel through the mucus layer lining the stomach until they reach the cells at the bottom. second, they create a microenvironment where the acidity and alkalinity balance is close to neutral, preventing direct contact with the hydrochloric acid produced by the stomach. urease's enzymatic activity generates chemical compounds that neutralize the acidity in the mucus immediately surrounding the bacteria, thereby establishing a microzone that protects the bacteria. h. pylori is non-invasive, yet it colonizes the antral region and mucosal surfaces of the human stomach, releasing pathogenic proteins that induce cell injury and inflammation. these alterations make the stomach and duodenum more susceptible to damage by digestive fluids, such as stomach acid. 1.2. pathogenesis of h. pylori infection although the precise methods by which h. pylori causes disease are not fully known, it is believed that they entail several elements, including the development of toxins and enzymes that harm the stomach lining and the ability of the bacteria to elude the immune system. [6] the ability of h. pylori to produce the enzyme urease, which converts urea into ammonia and carbon dioxide, is one of its primary virulence factors [9]. the ammonia aids in balancing the stomach's acidic environment, enabling the bacteria to persist and colonize the stomach lining. in addition, h. pylori secretes a variety of toxins and enzymes that can harm the stomach lining and aid in the growth of gastritis and ulcers. the inflammation brought on by the h. pylori infection and its progression are influenced by the immune system. inflammation and tissue damage may result from h. pylori's ability to increase the production of cytokines and other immune mediators [10]. in addition, h. pylori produces chemicals that suppress the activity of immune cells as one of its many methods for evading the immune system and surviving in the stomach. 1.3. current status of h. pylori therapy a proton pump inhibitor (ppi) in addition to antibiotics is the first-line treatment for h. pylori infection. while the ppi aids in lowering the amount of acid produced in the stomach, which helps the antibiotics function more effectively, the antibiotics are used to kill the bacteria. depending on the patient and the exact strain of h. pylori implicated, different antibiotics and ppis may be utilized. it is important to remember that a ppi is typically used together with a combination of two or three different antibiotics to treat h. pylori infections. depending on how many antibiotics are taken, this is referred to as "triple therapy" or "quadruple therapy." the american college of gastroenterology and the european helicobacter and microbiota study group, among other medical organizations, have published guidelines outlining the recommended first-line treatment for h. pylori infection [11-16]. alternative treatment alternatives may be taken into account if the first-line therapy for treating h. pylori infection is ineffective or if the patient cannot handle the drugs. second-line therapy: this typically involves a combination of different antibiotics and a ppi, or a bismuth-based quadruple therapy. 60 n. venkateswaramurthy, c. kalaiyarasi, p manivasakam copyright ©2024 assa adv. in systems science and appl. (2024) sequential therapy: this involves taking a ppi and an antibiotic for the first week, followed by a ppi and two different antibiotics for the second week. concomitant therapy: this involves taking a ppi and two different antibiotics simultaneously for the entire treatment course. current treatment for h. pylori infection mainly uses antibiotic-based triple or quadruple therapy with acid-suppressing agents. however, rising antibiotic resistance challenges these treatments' efficacy. alternative non-pharmacological methods being explored include probiotics, oxygen-rich environments, antibacterial photodynamic therapy, nanomaterials, antimicrobial peptide therapy, phage therapy, and modified lysins. the increasing antimicrobial resistance necessitates new strategies and continuous assessment of treatment regimens based on local resistance rates, eradication rates, and cost [17, 18]. additionally, improved diagnostic methods, like rapid nucleic acid amplification tests, are crucial for guiding treatment based on resistance patterns. 1.4. reasons for h. pylori eradication failure h. pylori eradication failure can occur due to several reasons (fig. 1), including bacterial resistance to antibiotics, poor patient compliance, low stomach ph, and insufficient drug concentration. antibiotic resistance: h. pylori strains can become resistant to clarithromycin, metronidazole, and levofloxacin. conventional antibiotic therapy for h. pylori is less effective because of increasing drug resistance [19]. poor compliance: h. pylori eradication requires 7–14 days of antibiotics and acidsuppressants. most physicians and investigators believe that patient compliance and antibiotic resistance are the primary challenges to generating the best h. pylori treatment regimens. to ensure compliance, patients must be informed and educated before starting h. pylori eradication medication, which can be complex and have negative effects [20]. low stomach ph: h. pylori can adapt to acidic surroundings and become resistant to acid-suppressing treatments. gastric acid secretion varies widely among people. some people with normal gastrin levels have increased basal acid production. lower eradication rates are observed in patients with hypersecretion, who most likely have more parietal cells. people can have weak, moderate, or extensive metabolizers due to gene polymorphism. conventional ppi doses appear to be insufficient in patients who are extensive metabolizers to give enough ph inhibition to permit antibiotic activity in the stomach mucosa, leading to lower eradication rates despite some contradicting results [21]. insufficient antibiotic concentration at the h. pylori site: due to the synthesis of urease and the presence of flagella, h. pylori may survive in the gastrointestinal environment over a wide ph range, pass through the gastric mucous layer, and enter the gastric epithelium, where it can be found attached to and even inside of cells. numerous antibiotics appear to have significantly reduced in vivo efficacy compared to their good in vitro performance because of poor mucous layer transport and low ph inactivation. after eradication from the antrum and gastric body, h. pylori may survive in the gastric cardia. lack of antibiotic concentration at the h. pylori infection site is an important cause of treatment failure [22]. recent advances in the development of polymeric floating biomaterials 61 copyright ©2024 assa. adv. in systems science and appl. (2024) fig 1. reasons for h. pylori eradication failure 2. need of novel drug delivery because antimicrobial medications only stay in the stomach for a short time, they are unable to effectively concentrate on the gastric mucus layer or epithelial cell surfaces, which are home to h. pylori. tablets and capsules are frequently used in eradication therapy, but they do not remain in the stomach long enough to reach the minimum inhibitory concentration in the gastrointestinal mucus where h. pylori thrives. since h. pylori is still present in some people, new, more efficient medication regimens are required. researchers tried many methods to eliminate h. pylori from the stomach in the 1980s. the stomach niche, bacterial load, and coccoid morphology require the medicine to persist in high concentration at the site of action to be therapeutic. gastric retentive delivery methods may improve mucus penetration and medication concentration at the site of action. higher antibiotic concentrations from gastroretentive drug delivery systems can eradicate h. pylori [23]. 3. gastroretentive dosage forms gastroretentive dosage forms are drug delivery systems that are made to stay in the stomach for a long time and release the active pharmaceutical ingredient (api) gradually. as these systems can aid in increasing the bioavailability and duration of action of the api, they are frequently employed for drugs that are poorly absorbed in the small intestine or that have a short half-life. gastroretentive dosage forms come in a variety of formats, such as floating systems, swellable systems, and high-density systems [24-29]. floating systems: these systems are designed to remain buoyant in the stomach by incorporating a gas-generating agent or a hollow, air-filled structure. examples of floating systems include floating tablets and capsules. swellable systems: these systems are made to inflate or expand in the stomach after intake, which aids in keeping them there for longer. swellable systems can take the form of hydrogels and hydrocolloids, for instance. high-density systems: these systems have been developed to have a higher density compared to gastric fluids, which helps to keep them in the stomach. beads and microspheres are a few examples of high-density systems. to address the issue, floating drug delivery devices with site-specific drug delivery are suggested among the gastroretentive drug delivery systems. by using floating drug delivery devices, a topical narrow-spectrum antibiotic may be able to completely eradicate bacteria in the stomach. 62 n. venkateswaramurthy, c. kalaiyarasi, p manivasakam copyright ©2024 assa adv. in systems science and appl. (2024) 4. polymers used in floating drug delivery system several types of polymeric materials are suitable for floating drug delivery systems, providing the buoyancy necessary for prolonged gastric retention [30,31]. here are some commonly used types. hydroxypropyl methylcellulose (hpmc): hpmc is a hydrophilic polymer that swells upon contact with gastric fluids, forming a gel barrier [31]. this gel enhances buoyancy, allowing sustained release of drugs. hpmc is often used in the formulation of floating microspheres or tablets for controlled drug delivery. polymethacrylates: polymethacrylates, such as eudragit polymers [32], can be designed to have ph-dependent solubility. this property allows them to remain intact in the acidic stomach environment, providing sustained drug release. polymethacrylates are utilized in the development of floating dosage forms, particularly gastroretentive tablets. polyvinyl acetate (pva): pva is a water-soluble polymer that can be used in combination with other polymers to create floating drug delivery systems [31]. it provides mechanical strength and stability to formulations. pva is employed in the preparation of floating microspheres and matrices for controlled drug release. chitosan: chitosan is a biodegradable and mucoadhesive polymer derived from chitin. it exhibits good floating properties and can adhere to the gastric mucosa. chitosan is used in the formulation of floating drug delivery systems, particularly for its bioadhesive characteristics [31]. ethylcellulose: ethylcellulose is a hydrophobic polymer that provides resistance to gastric fluids [31]. it is commonly used in the coating of drug particles to achieve controlled release. ethylcellulose-coated formulations are employed to create floating drug delivery systems, ensuring prolonged gastric residence time. poly (lactic-co-glycolic acid) (plga): plga is a biodegradable and biocompatible polymer that can be tailored for sustained drug release. it is often used in the preparation of microspheres or nanoparticles. plga-based formulations are employed for their controlled release properties in floating drug delivery systems [31]. polyethylene oxide (peo): peo is a water-soluble and hydrophilic polymer that can contribute to the buoyancy of drug formulations. it is often used in combination with other polymers. peo is used in the development of floating dosage forms for controlled drug delivery [31]. sodium alginate: sodium alginate is a natural polysaccharide derived from seaweed. it forms a gel in the presence of calcium ions, contributing to controlled drug release and buoyancy. sodium alginate is often utilized in the formulation of floating microspheres and beads for oral drug delivery [31]. starch-based polymers: starch and its derivatives, such as hydroxypropyl starch, can be used to create floating drug delivery systems. these polymers can provide both buoyancy and controlled drug release. starch-based polymers are employed in the development of gastroretentive formulations for sustained drug delivery [31]. guar gum: guar gum is a natural polysaccharide with excellent swelling and gelling properties. it can contribute to the buoyancy and controlled release of drugs. guar gum is used in the formulation of floating tablets and capsules for oral drug delivery [31]. acrylic acid derivatives: acrylic acid derivatives, such as carbomer, can be used to create mucoadhesive floating drug delivery systems [32, 33]. these polymers adhere to the gastric mucosa, prolonging drug release. acrylic acid derivatives are employed in formulations designed for enhanced gastric retention and localized drug delivery. polymeric floating biomaterials and advanced drug delivery systems are vital in treating h. pylori infections. these technologies protect drugs from stomach acid, target lesions, control drug release, and disrupt biofilms, enhancing drug efficacy. hydrophilic swellable polymers in bilayer tablets enable sustained drug release for up to 24 hours, while floating tablets with specific polymers ensure gastric retention, providing prolonged exposure to h. recent advances in the development of polymeric floating biomaterials 63 copyright ©2024 assa. adv. in systems science and appl. (2024) pylori and effective eradication [34]. these systems leverage polymers' properties to overcome challenges like drug inactivation, intracellular bacteria, and biofilm formation, offering promising solutions for improved treatment outcomes. 5. microballoon drug delivery system a microballoon or hollow microsphere drug delivery system is a pharmaceutical formulation that uses non-effervescent floating, or small spherical particles, to deliver drugs to the site. because the relative density of these microballoons is less than 1, their sphere-shaped cavity, which is protected by a strong polymer shell, enables them to float on top of gastric acid. microballoons with drug-loaded outer polymer shells are created by combining enteric acrylic polymers with the emulsion solvent diffusion method. this allows for controlled drug release and continuous floating for more than 12 hours in vitro. the drug is released gradually as the microballoons float over the stomach components [32,33,35]. the performance and potential advantages of microballoon drug delivery systems will be affected by various characteristics, like the size and shape of the microballoons, the composition of the microballoons, the drug loading and release profile, and the manufacturing process. 6. advantages of microballoon-based drug delivery systems there are several advantages of microballoon-based drug delivery systems over other drug delivery methods [34,35], including: enhanced drug targeting: microballoon-based drug delivery systems are designed to target specific areas of the gut, which may improve the local concentration of the drug and enhance its effectiveness. improved drug delivery: microballoon drug delivery systems can reduce the frequency of dosing, which may improve compliance. to solve this problem, microballoon drug delivery systems were created, which prolong the duration that drugs remain in the stomach. floating drug delivery systems, mucoadhesive drug delivery systems, and controlled-release drug delivery systems are a few examples of these systems. gastroretentive drug delivery devices can increase the efficacy of h. pylori eradication therapy and lower the risk of treatment failure by enhancing eradication concentration in the stomach. to make the drug locally available and effective at eradicating infections, the stomach residence duration of the antibiotic must be extended. reduced dosing frequency: grdds can reduce the dosing frequency required for h. pylori eradication therapy. this can improve patient compliance and reduce the risk of treatment failure due to missed doses. improved safety: grdds can reduce the risk of systemic side effects by limiting drug absorption into the bloodstream. this can improve the safety profile of h. pylori eradication therapy. increased drug stability: microballoon-based drug delivery systems can also improve the stability of drugs, which may be beneficial for drugs that are sensitive to ph or degradation. this can help to increase the shelf life of the drugs and improve their effectiveness. 7. formulation of microballoons several methods can be used to prepare microballoons, which are small, spherical particles containing the active pharmaceutical ingredient [36-38]. these methods include: solvent evaporation: in this method, the drug is dissolved in a solvent, which is then sprayed into a drying chamber. the solvent is allowed to evaporate, leaving behind the drug in the form of small, spherical particles. 64 n. venkateswaramurthy, c. kalaiyarasi, p manivasakam copyright ©2024 assa adv. in systems science and appl. (2024) emulsion solvent diffusion: in this method, the drug is dissolved in a solvent and then emulsified in an aqueous phase. the mixture is then poured into a non-solvent, causing the drug to form small, spherical particles as the solvent diffuses out. coacervation: in this method, two immiscible polymers are dissolved in a solvent to form a polymer solution. the drug is then added to the solution, and the solvent is gradually removed, causing the polymers to coacervate around the drug to form microballoons. spray drying: in this method, the drug is suspended in a solution or a suspension, which is then atomized and sprayed into a drying chamber. the solvent or water is evaporated, leaving behind the drug in the shape of small, spherical particles. solvent diffusion: within this method, a solvent is used to dissolve the drug and a polymer, which are then mixed to form a solution. the solution is then poured into a nonsolvent, causing the drug and polymer to form small, spherical particles as the solvent diffuses out. phase separation: in this method, the drug is mixed with a polymer and a solvent, and the mixture is then poured into a non-solvent. the non-solvent causes the drug and polymer to separate into small, spherical particles as the solvent diffuses out. 8. evaluation of microballoons several parameters can be used to evaluate microballoons, which are small, spherical particles containing the active pharmaceutical ingredient [39-42]. these parameters include: particle size: the size of the microballoons can affect the drug release profile and the long-term sustainability of the particles. microscopy or laser diffraction are two methods that can be used to determine the particle size distribution. surface morphology: the stability of the particles and the drug release profile of the microballoons can both be impacted by their surface shape. scanning electron microscopy (sem) can be used to assess surface morphology. drug content: the drug content in the microballoons can be determined using analytical techniques such as spectrophotometry or chromatography. drug release: in vitro release tests, which can be carried out utilizing methods like dialysis or sink conditions, can be used to assess the drug release profile of the microballoons. stability: the stability of the microballoons can be evaluated by storing the particles under different conditions (e.g., different temperatures, and humidity levels) and analyzing their physical properties (e.g., particle size, drug content) over time. flow properties: the flow properties of the microballoons, such as their bulk density and angle of repose, can affect their handling and processing during manufacturing. these properties can be determined using techniques such as tap density measurements or the funnel method. surface charge: the surface charge of the microballoons can affect their stability and their interactions with other particles or surfaces. surface charge can be evaluated using techniques such as zeta potential measurements. in vitro drug release: the release profile of the drug from the microballoons can be assessed using in vitro drug release tests under various conditions (such as various ph levels and surfactant concentrations). in vivo performance: in vivo studies can be used to evaluate the performance of the microballoons in an animal model or clinical trials. these studies can provide information on the bioavailability, pharmacokinetics, and safety of the drug delivered using microballoons. compatibility with excipients: the materials used to make the microballoons must be compatible with the drug and any excipients (inactive substances) that may be utilized in the formulation. techniques like differential scanning calorimetry (dsc) or fourier transform infrared spectroscopy (ftir) can be used to assess compatibility. recent advances in the development of polymeric floating biomaterials 65 copyright ©2024 assa. adv. in systems science and appl. (2024) overall, the characterization of microballoons involves the evaluation of a range of physical and chemical properties, which can help to understand the performance of these particles and to optimize their formulation and manufacture. 9. current research in microballoon drug delivery systems for h. pylori designing targeted delivery systems: to increase the drug local concentration and efficacy, researchers are working on creating microballoon-based drug delivery systems that can target particular bodily regions, like the gastrointestinal tract. one way to extend the duration of drug residence in the stomach is to utilize floating dose forms, which have a lower density than gastric fluids and can float above the gastric juice in the stomach. the mechanism of buoyancy is used in the production of floating dosage forms for both effervescent and noneffervescent systems. there is no breaking down of the fundamental rule, which stipulates that things must float on gastric liquid that has a specific density of less than 1.004 g/cm of the gastric fluid in the stomach [43,44]. ramachandran s (2010) et al., designed and characterized the floating drug delivery system using eudragit l-100 polymer over the improvement of bioavailability of famotidine by prolonging the gastric residence time. long-lasting release (>8 h) and buoyancy (>10 h) were both characteristics of the hollow microspheres [45]. in comparison to the control and standard groups of rats, the in-vivo study revealed strong antiulcer capabilities of famotidineloaded microspheres. by employing the pyloric ligation method on rats, the hollow microsphere formulation showed strong antiulcer activity that was comparable to that of the common drug famotidine, showing its success as a potential formulation. awasthi r (2011) et al., created an h. pylori infection treatment using an amoxicillin gastroretentive floating system (gfs). the ratios of dichloromethane/ethanol/isopropyl alcohol, amoxicillin-eudragit s100, and cellulose acetate phthalate were selected [46]. according to the study, the developed gfs is an efficient design for amoxicillin for targeted drug release in the gastric region, which can help treat h. pylori infection. sanjay bansal (2014) et al., with a quality-by-design (qbd)-based methodology, multiple-unit gastroretentive microballoons containing itopride hydrochloride (ith) were created and improved [47]. excellent floating qualities and acceptable drug release control could be seen in the optimized formulation. experiments on the compatibility of drug excipients and sem experiments validated the formulation's appropriateness. studies using in-vivo x-ray imaging on rabbits verified the microballoons' buoyant properties for 8 hours in the upper gi tract. overall, this work was successful in creating gastroretentive floating microspheres for ith injection once a day. akash jain (2015) et al., developed and evaluated nizatidine-loaded microballoons for improved drug bioavailability and prolonged residence time in the gastrointestinal tract [48]. eudragit s-100 and hpmc were used to make microballoons, which were then optimized for several formulation factors. sem, drug entrapment, buoyancy, in-vitro drug release, and invivo floating efficiency experiments were used to characterize the optimised formulation. the drug-loaded microballoons demonstrated considerable ulcer prevention in comparison to free drug treatment in a mouse model of ethanol-induced ulcers. the improved formulation demonstrated sustained drug release for 12 hours and stayed afloat in the gastric contents for the same amount of time. these findings suggest that floating microballoons containing nizatidine extended a drug delivery system for the efficient treatment of stomach ulcers. sandeep choudhary (2016) et al., in a study designed for the treatment of peptic ulcer disease, a new gastroretentive floating system was created for the regulated delivery of rabeprazole sodium and amoxicillin tri-hydrate. using eudragit s-100 and hydroxypropyl methylcellulose along with various drugs, polymers, stirring speed, and time concentrations [49], low-density microballoons were created. the formulation factors had a big impact on 66 n. venkateswaramurthy, c. kalaiyarasi, p manivasakam copyright ©2024 assa adv. in systems science and appl. (2024) the drug release in vitro. in rabbits and a mouse model with ulcers, the gastro-retentive capacity of the microballoons was confirmed, with enhanced buoyancy and a decreased ulcer index compared to conventional drugs. according to the study's findings, these microballoons could enhance the treatment of ulcers by delivering drugs to the stomach in a targeted and sustained manner to get rid of h. pylori. ritesh kumar (2018) et al., developed and optimised famotidine-loaded microballoons for prolonged gastric residence, sustained release, and improved bioavailability [50]. utilizing hpmc k4m, microballoons were created, and their performance in terms of in vitro drug release, buoyancy, and drug entrapment was all assessed. the floating behavior and anti-ulcer activity of the optimized formulation were assessed in vivo. the mixture that was chosen demonstrated sustained release for more than 12 hours while remaining buoyant in the stomach. studies conducted in vivo revealed a higher ulcer prevention index when compared to the commercial brand. according to the findings, famotidine-loaded microballoons extended the drug's stay in the patient's system when treating h. pylori infection. future research to combat h. pylori includes exploring probiotics, nanoparticles, plantbased natural products, and developing new vaccines such as multi-epitope and vector-based types. studies focus on h. pylori-related gastric ulcer pathogenesis, oxidative stress mechanisms, competitive acid inhibitors, gut microbiota, probiotic adjuvant therapy, and neutrophil activator protein's immune protective effects [51]. addressing antibiotic resistance in h. pylori also requires investigating clarithromycin resistance and alternative therapies. these diverse approaches offer promising strategies for future treatment. 10. clinical trials involving microballoon drug delivery systems sato y (2004) et al., investigated the behavior of microballoon and nonfloating microspheres labeled with 99mtc in fasted and fed human subjects using gamma scintigraphy [33]. it also examined the pharmacokinetics of riboflavin released from these particles. according to the findings, microballoon dispersed in the upper stomach after feeding and stayed there for up to 300 minutes, whereas nonfloating progressively went to the lower stomach after 90 minutes. microballoons floated for roughly 60 minutes in the fasting condition before being quickly eliminated through the inner digestive migrating motor complex, whereas nonfloating was gone after 60 minutes. the pharmacokinetic parameters excretion half-life time (t1/2) and total urinary excretion, as well as the stomach residence time as evaluated by gamma scintigraphy, were shown to be strongly correlated in the study. the study's overall conclusion is that microballoon can increase medication bioavailability, resulting in pharmacological benefits that last longer. sato y (2003) et al., created floating controlled drug delivery systems using hollow microspheres known as microballoons. enteric acrylic polymers were used to create these microballoons, and they were dissolved in a solution of dichloromethane and ethanol [52]. the riboflavin release profiles of non-floating microspheres and the floating characteristics of the microballoons were compared. in comparison to riboflavin powder and non-floating microspheres, the microballoons showed delayed urine excretion of riboflavin, especially under fed conditions. additionally, both in the fed and fasted conditions, the microballoons showed greater total riboflavin urine excretion levels. the study concluded that the microballoons' ability to float in the stomach may contribute to the long-lasting pharmacological activity, improving drug absorption. hussein o ammar (2016) et al., created as a gastro-retentive microballoon for cinnarizine continuous release. utilizing cellulose acetate butyrate as the hosting polymer and a solvent mixture of 100% ethanol and dichloromethane, the microballoon were created utilizing the emulsion solvent diffusion technique [53]. the formulation was improved using recent advances in the development of polymeric floating biomaterials 67 copyright ©2024 assa. adv. in systems science and appl. (2024) a factorial experimental approach, producing an improved microballoon formulation with a particular drug loading capacity and release efficiency as well as a suitable size. the improved microballoon spherical form and porous structure were validated by sem. the modified microballoon formulation outperformed traditional tablets in terms of bioavailability metrics, according to in vivo tests on human volunteers. indicating prolonged stomach stay, this comprised quicker absorption, a larger area under the concentration-time curve, and a much longer mean residence duration. sato y (2004) et al., study involving three healthy volunteers, orally administered riboflavin-containing microballoon were analyzed to investigate the pharmacokinetics of riboflavin [41]. the findings demonstrated that microballoon consumption under both water and "fed" circumstances prolonged riboflavin urine excretion. the buoyancy of the microballoon in the stomach, which extended gastric residence time, was credited for the prolonged excretion. additionally, compared to microballoons with lesser buoyancy, those with increased buoyancy displayed longer excretion half-life (t1/2). an in vivo evaluation using an analysis of urine excretion showed that mixing microballoon with hydroxypropylmethylcellulose in various ratios improved riboflavin release qualities. strong correlations between buoyancy and t1/2, as well as between the release of riboflavin from microballoon and total urine excretion, were found in the study, underlining the potential advantages of microballoon intragastric floating capabilities for achieving the sustained pharmacological effect. combining outcomes from in vivo and in vitro studies enhances the understanding of novel drug delivery systems for h. pylori therapy [54, 55]. in vivo studies with animal models like mongolian gerbils and transgenic mice provide insights into h. pylori-related complications. in vitro studies on alternative treatments, such as lactic acid bacteria (lab) and their secretions, show promising anti-h. pylori effects by inhibiting bacterial growth, reducing virulence, and suppressing pro-inflammatory responses. additionally, research into advanced drug delivery systems like nano and microparticles highlights their potential to improve drug retention and release rates at the target site, offering solutions against antibiotic-resistant h. pylori strains [56]. integrating data from these studies can lead to a comprehensive review of innovative therapeutic approaches for h. pylori infections. conclusion recent advances in biomaterials have led to innovative polymeric membranes from marine biopolymers for tissue engineering [57, 58]. these membranes, made from collagen, chitosan, and fucoidan, offer structural stability, suitable swelling ability, and mechanical properties similar to native articular cartilage, making them ideal for tissue regeneration. sustainable polymers from renewable sources are also being emphasized, focusing on environmentally friendly processing and end-of-life considerations [59, 60]. an interdisciplinary approach has resulted in diverse polymeric biomaterials for wound care therapies, addressing the challenges of complex, chronic wounds. these advancements highlight the ongoing importance of research and innovation in developing biomaterials for biomedical applications. the development of novel biomaterials, especially polymeric floating biomaterials, presents a promising avenue for targeted and sustained drug delivery to the gastric mucosa is a novel approach to combat h. pylori. these particles can be made using a variety of materials, including polymers, proteins, and natural polysaccharides, and can be used to deliver a wide range of drugs. the use of microballoons in treating h. pylori infection, a common cause of peptic ulcer disease, has been investigated. the use of microballoon-based extended drug delivery systems in h. pylori therapy possesses the potential to enhance the effectiveness and safety of the treatment. microballoons may assist in addressing some of the issues with conventional methods of treating h. pylori, such as low bioavailability and the 68 n. venkateswaramurthy, c. kalaiyarasi, p manivasakam copyright ©2024 assa adv. in systems science and appl. (2024) emergence of antibiotic resistance, by regulating the release rate of the drug and lengthening its duration of action. microballoons may also be useful for delivering prodrugs, natural products, gene therapies, and vaccines over the treatment of h. pylori infection. in general, research into the application of microballoon-based extended drug delivery systems for h. pylori therapy is ongoing, and more research is required to fully grasp their potential and optimize their formulation and use. references 1. hooi, j. k. y., lai, w. y., ng, w. k., suen, m. m. y., underwood, f. e., et al. (2017). global prevalence of helicobacter pylori infection: systematic review and metaanalysis, gastroenterology, 153(2), 420–429. doi: 10.1053/j.gastro.2017.04.022 2. wroblewski, l. e., peek, r. m., & wilson, k. t. 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(2023). the potential of alternative therapies and vaccine candidates against helicobacter pylori, pharmaceuticals, 16(4), 552. doi: 10.3390/ph16040552 52. sato, y., kawashima, y., takeuchi, h. & yamamoto, h. (2003). in vivo evaluation of riboflavin-containing microballoons for floating controlled drug delivery system in healthy human volunteers, journal of controlled release, 93(1), 39–47. doi: 10.1016/s0168-3659(03)00370-5 53. ammar, h. o., ghorab, m., kamel, r. & salama, a. h. (2016). design and optimization of gastro-retentive microballoons for enhanced bioavailability of cinnarizine, drug delivery and translational research, 6(3), 210–224. doi: 10.1007/s13346-016-0280-4 54. le-tien, c., blemur, l. & baltzis, d. 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(2022). evaluating polymeric biomaterials to improve next generation wound dressing design, biomaterials research, 26(1). doi: 10.1186/s40824-022-00291-5 microsoft word 1409-article text-5913-1-4-20230511 adv syst sci appl 2024; 02; 219-231 published online at https://ijassa.ipu.ru. the effect of space flight factors on the correction disorders of the vaginal and cervical microflora in ground-based models daria komissarova1*, vyacheslav ilyin1, nonna usanova1, igor goldman2, elena sadchikova2, julia morozova1, vera muravieva3, gyuldana bayramova3, tatiana priputnevich3 1) ibmp ras, moscow, russia 2) igb ras, moscow, russia 3) fsbi national medical research center for obstetrics, gynecology and perinatology named after academician v.i. kulakov, ministry of healthcare of the russian federation, moscow, russia abstract: this work shows results of a comprehensive study of the vaginal and cervical microflora of women participating in a ground-based model experiment simulating some factors of space flight. experiments with dry immersion allow to simulate such factors of a space flight as physical inactivity and microgravity, as well as the redistribution of liquid media associated with them. 16 volunteers, participated in 5-days dry immersion experiment, used an oral prebiotic drug based on lactoferrin and intravaginal probiotic capsules based on lactobacillus acidophilus. it was found that the oral use of lactoferrin has a delayed positive effect on the state of the microflora of the vagina and cervical canal of volunteers. this effect is especially pronounced on the 35th day after the end of the immersion. while the use of probiotic capsules has a faster effect, noticeable after 5 days of use during immersion. the combination of lactoferrin and probiotic capsules did not give noticeable effect, which can be explained both by a small group of volunteers or lack of a synergistic effect of these drugs. one of the means that can give high efficiency in the prevention of the dysbiosis of the vaginal microflora are intravaginal capsules made of autostrains isolated individually from each volunteer and addition of lactoferrin into the capsules, the effectiveness of which when using locally was also proven in previously conducted studies in gynecological patients. keywords: lactoferrin, vaginal microflora, ground experiments, space medicine 1. introduction the modern development of technology makes space flights one of the necessary conditions for the development of mankind. a person's stay in space flight conditions leads to the need to correct health in an unfavorable environment, which includes microgravity and concomitant changes in the blood supply of organs, psycho-emotional chronic hypo-stress and, most importantly, a change in the ecological system of man-environment, including microflora. if for men these issues were mostly solved in almost half a century of the space age, then the mass participation of women in space flights is a relatively new phenomenon and requires specific solutions to problems, including those related to feminine hygiene. first of all, this concerns what men do not have (and, accordingly, has not been studied and corrected) in previous years a change in the vaginal microflora. since the direct study of the parameters of the vaginal microflora does not allow to obtain statistically significant sample volumes and, most importantly, to test methods for correcting violations, ground-based model experiments are a necessary condition and tool for solving this problem. this approach also makes it possible to assess the risks of developing adverse * corresponding author: d.komisarova@yandex.ru 220 d. komissarova et al. copyright ©2024 assa adv. in systems science and appl. (2024) disorders and the effectiveness of interventions before these violations lead to serious medical, social and economic (which is extremely important in space flight) consequences. [1,2] the microbiota of any human biotopes has individuality. [3,4] such factors as changes in diet, frequency and quality of hygiene procedures, as well as psycho-emotional stress can provoke a cascade of negative changes in the human body, and, in particular, decrease in colonization resistance of all biotopes. the vaginal microbiota plays an important role not only in the prevention of inflammatory diseases of the pelvic organs, but also affects the success of fertilization, regardless of whether fertilization occurred naturally or as a result of the in vitro fertilization procedure [5]. the main component of the normal vaginal microbiota of women of reproductive age is lactobacillus spp., which create an environment that prevents the reproduction of opportunistic and pathogenic bacteria, providing colonization resistance of the vaginal biotope. the normal amount of lactobacillus spp. is 106-108 cfu/ml, therefore, maintaining high quantity of lactobacilli in the vaginal microflora is important for maintaining the health of women participating in long-term space missions and is directly related to the ability to realize reproductive function. in one of the studies, it was found that in 192 studied patients the ones who had low content of vaginal lactobacilli had lower the percentage of pregnancy [6]. in another study, it was shown that the low percentage of pregnancy was associated with the presence of such opportunistic pathogenic microorganisms (opm) as gardnerella vaginalis and atopobium vaginae in the vaginal microflora. [7] conducting ground-based model experiments, such as dry immersion, makes it possible to study the microbiota under conditions simulating weightlessness. during dry immersion, volunteers experience the influence of such space flight factors as hypodynamia, redistribution of liquid media [8]. the essence of dry immersion is that a volunteer, dressed in underwear, lies on a platform on a waterproofing film, covered with a cotton sheet, in an ergonomically designed immersion bath filled with water. the immersion bath is designed in such a way that when the platform is lowered, the volunteer is completely immersed in water, except for the head. thus, during the experiment, a volunteer is in an unsupported suspended state (simulated weightlessness). the volunteer is allowed to leave the immersion bath only for ten minutes a day to take a shower and perform hygiene procedures. in previous studies of the microflora of women participating in a 3-day dry immersion, a decrease in the colonization resistance of the vaginal biotope was revealed, which was accompanied by an increase in the proportion of opm (especially the aerobic component) and a decrease in the number of protective lactobacillus spp. [9] to maintain the microbiota, prebiotic and probiotic drugs are currently actively used. prebiotic preparations are aimed, first of all, at creating a favorable environment for the reproduction of protective microorganisms already living in a particular biotope. probiotic preparations are “live strains of carefully selected microorganisms which, when administered in adequate amounts, confer a beneficial effect on the health of the host.” [10] both types of drugs have proven relatively effective in maintaining the state of the microbiota and have been studied most of all in the intestinal microbiota. [11] however, it is possible that pro or prebiotic drugs can cause allergic reactions. also in previous isolation experiments, it has been shown that probiotic drugs indeed support the colonization resistance of the intestinal biotope and suppress the development of opm, but only during the time that a volunteer takes the drug [12]. meanwhile, autoprobiotic drugs made from dominative protective strains of the microflora of a particular volunteer do not have these disadvantages. it should be noted that all biotopes of the human body are interconnected with each other and the violation of colonization resistance of one can lead to a deterioration of microbiota in another [13]. it has been found that oral intake of a preor probiotic that directly affects the intestinal microflora can also have a positive effect on the vaginal microflora. for example, this was shown in an experiment involving 8 female volunteers who took a fermented drink based on saccharomyces, bifidobacteria, lactobacilli and enterococci as a dietary supplement. [14] the effect of space flight factors on the correction disorders 221 copyright ©2024 assa adv. in systems science and appl. (2024) in this regard, it seems optimal to use a combination of an oral prebiotic preparation and a local autoprobiotic preparation, which will simultaneously strengthen the colonization barrier of a particular biotope and create an environment for favorable reproduction of autoflora. in present study, we used lactoferrin as an oral prebiotic preparation. lactoferrin is a polyfunctional protein of the transferrin family found in human secretory fluids: milk, saliva, and tears [15]. lactoferrin in the human body, being its component, is easily recognized and does not cause allergies. lactoferrin also has antibacterial activity. it binds iron to prevents a bacterial cell from assimilating this microelement, which is necessary for growth and reproduction. [16] lactoferrin receptors also have the ability to bind lipopolysaccharides of the bacterial cell wall, in addition, they initiate peroxidation of the proteins that make up the wall, which leads to a violation of the membrane permeability of the bacterial cell and its subsequent death. there are suggestions that the anti-infective activity of lactoferrin may also be based on stimulation of phagocytosis [17]. the purpose of this study is to evaluate the effect of the combined use of lactoferrin and intravaginal capsules based on lactobacillus acidophilus on the state of the microflora of the vagina and cervical canal of volunteers of the 5-day dry immersion experiment. 2. methods and materials the research programs in the isolation experiments "sirius-21" was approved by the bioethical commission of the ibmp (protocol no. 539 of march 17, 2020) and fully complies with the principles of the helsinki declaration of 1964. to study the state of the microflora of the vagina and cervical canal in a 5-day dry immersion, samples of vaginal and cervical canal microflora were taken from 16 volunteers three times: a) before the start of the experiment on 19-22 days of the menstrual cycle; b) on 5-7 day after the end of immersion (19 -22 days of the menstrual cycle); c) on 34-36 day after the end of immersion (19-22 days of the next cycle). probiotic capsules containing lactobacillus acidophilus (strains nk1, nk2, nk5, nk12) were chosen as a local drug for the correction and prevention of dysbiosis of the vaginal microflora. according to the results of in vitro studies, these strains have a pronounced positive effect on vaginal microflora and do not enhance the growth of fungi of the candida albicans species. [18,19]. a food supplement based on a biosimilar of human lactoferrin was used as a probiotic component. [20] all volunteers were divided into 4 groups.  group "placebo + placebo" (4 volunteers). during 30 days, starting from the first day of immersion, volunteers once per day took orally the placebo drug with 200 mg of maltodextrin and used glucose-based placebo intravaginal capsules (400 mg glucose in each capsule) once during five days of stay in immersion.  group "placebo + lactobacilli" (3 volunteers). during 30 days, starting from the first day of immersion, volunteers once per day took orally the placebo drug with 200 mg of maltodextrin and used intravaginal capsules based on lactobacillus acidophilus (titer of lactobacillus acidophilus -10⁸ cfu, excipients magnesium stearate 3 mg, lactose monohydrate a sufficient amount to obtain the mass of the contents of the capsule 400 mg), once during five days of stay in immersion.  group "lactoferrin + placebo" (6 volunteers). during 30 days, starting from the first day of immersion, volunteers once per day took orally the drug with 200 mg of lactoferrin and used glucose-based placebo intravaginal capsules (400 mg glucose in each capsule) once during five days of stay in immersion.  group "lactoferrin + lactobacilli" (3 volunteers). during 30 days starting from the first day of immersion volunteers once per day took orally the drug with 200 mg of lactoferrin and used intravaginal capsules based on lactobacillus acidophilus (titer of lactobacillus acidophilus -10⁸ 222 d. komissarova et al. copyright ©2024 assa adv. in systems science and appl. (2024) cfu, excipients magnesium stearate 3 mg, lactose monohydrate a sufficient amount to obtain the mass of the contents of the capsule 400 mg), once during five days of stay in immersion. microbiological analysis of the composition of the microbiota of the vagina and cervical canal of women in all experiments was carried out on the basis of fsbi “national medical research center for obstetrics, gynecology and perinatology named after academician v.i. kulakov” ministry of healthcare of the russian federation. all examiners underwent gram-stained microscopy of the vaginal discharge and cultural examination in accordance with the medical technology “integral assessment of the state of the vaginal microbiota. diagnosis of opportunistic vaginitis” [21]. the material was taken with a sterile dacron swab from the posterior vaginal fornix and, after treating the cervix with a sterile cotton swab, from the cervical canal. obtained sample was put into test tubes with amies transport medium (medical wire, england). the vaginal and cervical canal biomaterial was seeded on selective and non-selective agar solid nutrient media. facultative anaerobic microorganisms were isolated using columbia agar, brilliance gbs agar (oxoid, uk), mannitol salt agar (himedia, india), enterococcal agar, endo medium, and sabouraud agar (gitspm and b, obolensk, russia). microorganisms were incubated in a co2 incubator (jouan, france). lactobacilli were cultivated on lactobacillus medium (gitspm and b, obolensk, russia), strict anaerobes were cultivated on prereduced schedler agar (oxoid, uk) with the necessary additives in an anaerobic box (whitley dg 250 anaerobic workstation, uk) in three-component gas atmosphere mixtures (n2-80%; co2-10%; h2-10%). species identification of microorganisms was performed by maldi-tofms analysis using a microflex lt time-of-flight mass spectrometer with maldi biotyper software (bruker daltoniks; germany) version 4.0. the obtained data were analyzed using cluster analysis, chi-square test, nonparametric friedman and mann-whitney tests. [22.] statistical processing was carried out in the program statistica 12. 3. results and discussion 3.1. the results of comparing the state of the vaginal microflora "before", "after", "after + 30 days" in four groups of volunteers with different drugs combination according to the results of the analysis with non-parametric friedman criterion, a significant difference was revealed between the points “before”, “5-7 day” (5-7 days after the end of dry immersion) and “34-36 day” (34-36 days after the end of dry immersion) both for facultative anaerobic opm and obligate anaerobic opm in the “placebo + placebo” group. figure 1. dynamics of facultative anaerobic and obligate anaerobic opm in the 5-day dry immersion experiment in the “placebo + placebo” group of volunteers 1 10 100 1000 10000 100000 before 7-5 day 34-36 day ti tr e of m ic ro or ga ni sm s (c fu /m l) period of sample taking number of facultative anaerobic opm number of obligate anaerobic opm the effect of space flight factors on the correction disorders 223 copyright ©2024 assa adv. in systems science and appl. (2024) as can be seen from figure 1, the number of facultative anaerobic opm increased by the end of the experiment and remained consistently higher than before immersion, including 30 days after the end of exposure. the number of obligate anaerobic opm decreased after immersion with a subsequent increase on days 34-36 after the end of the experiment. in the “lactoferrin + placebo” group, a significant difference was found in obligate anaerobic opm (figure 2). figure 2. dynamics of obligate anaerobic opm in 5-day dry immersion in the “lactoferrin + placebo” group of volunteers as can be seen from figure 2, the number of obligate anaerobes in this group increased by the end of immersion and then decreased on days 34-36 after the experiment. it should be noted that the use of lactoferrin began on the 1st day of immersion and continued for 30 days, i.e. most of the time after the end of the immersion, the volunteers continued to take lactoferrin, which may have affected the state of the microflora and caused a decrease in the obligate anaerobic opm. in the “placebo + lactobacilli” and “lactoferrin + lactobacilli” groups, the nonparametric friedman test did not reveal any differences. the obtained data show that immersion has a negative impact on the vaginal microflora: the number of facultative anaerobic opm increases. oral intake of lactoferrin has a slight stabilizing effect on the vaginal microflora: the number of facultative anaerobic opm does not increase, although there is a slight increase in obligate anaerobic opm immediately after the end of immersion. after long-term use of lactoferrin, this group of microorganisms decreases with a consistently low titer of facultative anaerobic opm, which indicates a positive effect of this drug after prolonged use. the lack of differences in the groups “placebo + lactobacilli” and “lactoferrin + lactobacilli” on three points may be due to the small number of volunteers in these groups. 3.2. the results of comparing the state of the vaginal microflora in the placebo group and groups with different combinations of preand probiotic preparations in different periods of the experiment to compare the groups with each other, cluster analysis and mann-whitney test were used. the results of cluster analysis on days 5-7 after the end of immersion are shown in figure 3. 1 10 100 1000 10000 100000 before 5-7 day 34-36 day ti tr e of m ic ro or ga ni sm s (c fu /m l) period of sample taking number of obligate anaerobic opm 224 d. komissarova et al. copyright ©2024 assa adv. in systems science and appl. (2024) а) b) figure 3. results of cluster analysis for lactobacillus spp (a) and obligate anaerobic opm (b) for all four groups of volunteers with different drugs combination on 5-7 day after finishing of dry immersion. as can be seen from the figures with anaerobic opm used for clusterization, the groups “placebo + placebo”, “lactoferrin + placebo” and “placebo + lactobacilli” are the closest to each other. these groups form a single cluster, from which the group “lactoferrin + lactobacilli” is equidistant. for lactobacilli, the groups form somewhat different clusters: “placebo + placebo” and “lactoferrin + placebo” and separately the cluster of “placebo + lactobacilli” and “lactoferrin + lactobacilli”. these results suggest that, firstly, the “lactoferrin + lactobacilli” group differs from the other three groups, and secondly, the greatest effect on lactobacilli, apparently, is not the oral use of lactoferrin, but the local use of capsules. figure 4. results of cluster analysis of lactobacillus spp for all four groups of volunteers with different drugs combination at 34-36 day after finishing of dry immersion cluster analysis for obligate anaerobic and facultative anaerobic opm showed that groups “placebo + placebo”, “lactoferrin + placebo” and “placebo + lactobacilli” again turned out to be the closest, which coincides with the situation on days 5-7 after the end of the dry immersion. for lactobacilli, the situation on days 34–36 is somewhat different: the “lactoferrin + lactobacilli” group forms a cluster with the “placebo + lactobacilli” group, and then combines with the “placebo + placebo” group (figure 4). perhaps this is due to the prolonged action of lactoferrin, which the volunteers took for 30 days from the first day of immersion, while probiotic or placebo capsules were used just during the immersion, i.e. for 5 days only. the state of the microflora in different periods of time in different groups was also assessed using the nonparametric mann-whitney test. a significant difference was found between the “placebo + placebo” and “lactoferrin + placebo” groups on days 34-36 in number of different kinds of lactobacilli. the amount of the effect of space flight factors on the correction disorders 225 copyright ©2024 assa adv. in systems science and appl. (2024) lactobacillus jensenii, lactobacillus vaginalis and lactobacillus gasseri in the "lactoferrin + placebo" group was higher than in the “placebo + placebo” group, and the number of lactobacillus iners was, on the contrary, lower (figure 5). figure 5. the number of lactobacillus spp in the "placebo + placebo" and "lactoferrin + placebo" groups of volunteers on days 34-36 after the end of immersion. * statistically significant differences, p<0.05. these data correlate with data from cluster analysis, where the “placebo + placebo” group was quite distant from the “lactoferrin + placebo” group (figure 4b). it should be noted that the species lactobacillus iners has the lowest probiotic potential and is most often found in patients with infectious and inflammatory pathologies of the vagina. [23,24]. therefore, the decrease in the number of this kind of lactobacilli in the “lactoferrin + placebo” group indicates rather positive change in this group. based on the data, it can be concluded that long-term use of lactoferrin has a positive effect on the vaginal microflora: it increases the number of most species of lactobacilli by 30th day after the end of the dry immersion, at the same time inhibiting the growth of lactobacillus iners. significant differences were also found between groups “lactoferrin + placebo” and “placebo + lactobacilli” on obligate anaerobic opm immediately after the end of immersion. in the “placebo + lactobacilli” group, the number of obligate anaerobic opm was significantly lower than in the “lactoferrin + placebo” group. these data partly correlate with cluster analysis (figure 3b), where the cluster was formed by the “lactoferrin + placebo” and “placebo + placebo” groups. additionally, using the chi-square an eubiotic index (ei) was calculated and assessed. eubiotic index shows the ratio of positive changes in the microflora to negative ones. [14] significant differences between the three groups were found at the "before/5-7 day" point (figure 6). figure 6. eubiotic index for volunteers of three groups with different drugs combination for the "before/5-7 day" point. 1 10 100 1000 10000 100000 1000000 10000000 100000000 lactobacillus crispatus lactobacillus iners lactobacillus jensenii lactobacillus gasseri lactobacillus vaginalis ti tr e of m ic ro or ga ni sm s (c fu /m l) lactoacillus spp placebo+lactobacilli lactoferrin+placebo * 0.00 0.50 1.00 1.50 2.00 2.50 placebo+placebo lactoferrin+placebo placebo+lactobacilli e i groups of volunteers before / 5-7day * * * * * 226 d. komissarova et al. copyright ©2024 assa adv. in systems science and appl. (2024) as can be seen in figure 6, the lowest eubiotic index was noted in the “placebo + placebo” group, while the highest was in the “placebo + lactobacilli” group, i.e. in this group, the number of positive changes in the microflora was greater than the negative ones. also, it should be noted that the eubiotic index in “lactoferrin + placebo” group is higher than one, that is, despite the fact ut is not the highest, nevertheless, positive changes in the microbiota prevail over negative ones, i.e. oral intake of lactoferrin has a slight stabilizing effect on the vaginal microflora. 3.3. the results of the analysis of the cervical canal microbiota "before", "after", "after + 30 days" in four groups of volunteers the analysis using friedman's criterion revealed significant differences in obligate anaerobic opm in the microflora of the cervical canal in the “placebo + placebo” group (figure 7). figure 7. the number of obligate anaerobic opm in the "placebo + placebo" group of volunteers as can be seen from the figure, the number of obligate anaerobes decreased after the end of immersion and remained consistently low even on days 34–36 after the end of the experiment. these data fit with those obtained in a 3-day dry immersion, where a decrease in the anaerobic conditionally pathogenic component of the microbiota of the vagina and cervical canal was also observed. in other groups, no significant differences in facultative anaerobic and obligate anaerobic opm, as well as in lactobacilli, were found. 3.4. the results of comparison of the cervical canal microbiota of the volunteers in the placebo group and groups with different combinations of preand probiotic preparations in different periods of the experiment the results of cluster analysis on days 5-7 after the end of immersion are shown in figure 8. а) b) 1 10 100 1000 10000 before 5-7 day 34-36 dayti tr e of m ic ro or ga ni sm s (c fu /m l) period of sample taking the effect of space flight factors on the correction disorders 227 copyright ©2024 assa adv. in systems science and appl. (2024) figure 8. results of cluster analysis for opm (a) and lactobacillus spp. (b) for all four groups of volunteers with different drugs combination on 5-7 day after the end of the experiment as can be seen in the figure, for opportunistic microorganisms, the cluster “placebo + placebo” and “lactoferrin + placebo” is formed and “placebo + lactobacilli” group is close to this cluster. the group “lactoferrin + lactobacilli” is the furthest. the results of cluster analysis on days 34-36 after the end of the experiment are shown in figure 9. а) b) figure 9. results of cluster analysis for opm (a) and lactobacillus spp. (b) for all four groups of volunteers drugs combination on days 34-36 after the end of the experiment it can be seen that using opportunistic component of microflora for grouping “lactoferrin + lactobacilli” group is the most distant from the other three groups. for lactobacilli, clusters are formed, first of all, by “placebo + lactobacilli” and “lactoferrin + placebo” groups, then, quite close to them, is a “lactoferrin + lactobacilli” group, and the most distant group is “placebo + placebo”. according to the results of comparison of all groups immediately after the end of immersion, no significant differences were found, however, on days 34-36 after the end of the experiment, according to the kruskal-wallis criterion, significant differences were noted both in opm and in lactobacilli (figure 10). figure 10. number of opm and lactobacillus spp. in the placebo group and groups of volunteers with different combinations of probiotic preparations in participants of the 5-day "dry" immersion on days 34-36 after the end of immersion 1 10 100 1000 10000 100000 1000000 opm lactobacillus spp ti tr e of m ic ro or ga ni sm s (c fu /m l) group of microorganisms placebo+placebo lactoferrin+placebo placebo+lactobacilli lactoferrin+lactobacilli 228 d. komissarova et al. copyright ©2024 assa adv. in systems science and appl. (2024) amounts of lactobacilli in all four groups look quite close to each other, while significant differences were noted between groups “placebo + placebo” and “lactoferrin + placebo”. in “placebo + placebo” group, the number of lactobacilli is slightly higher than in “lactoferrin + placebo” group. these data fit with the data of cluster analysis, in which these two groups are the most distant from each other. the highest number of amount of opm was noted in the “lactoferrin + lactobacilli” group, and the lowest in the “placebo + lactobacilli” group, thus, the local probiotic preparation created the most optimal conditions for maintaining the normobiota: a low amount of opm with a consistently high number of lactobacilli. this is also reflected in the results of the cluster analysis presented above, where the group “lactoferrin + lactobacilli” was the most distant from all other groups. the eubiotic index for cervical canal is showed in figure 11. significant differences were found between the “placebo + placebo” and “lactoferrin + placebo” groups. significance of the results were proved by a nonparametric chi-square test. figure 11. eubiotic index (ei) calculated for the "placebo+placebo" and "lactoferrin+placebo" groups as can be seen in the figure 11, in the “placebo + placebo” group, the state of the microflora was initially close to the normobiota: a large number of lactobacilli was combined with a low amount of opm. in the “lactoferrin + placebo” group, the eubiotic index was initially below 1, while in the “placebo + placebo” group ei was almost three times higher. at the same time, after the end of the immersion, the state of the microflora of the “placebo + placebo” group worsened, and the state of the “lactoferrin + placebo” group improved ei became higher, i.e. the number of positive changes in the microflora was higher than the number of negative ones. by the 30th day after the end of the immersion, the index in the “lactoferrin + placebo” group continued to grow, significantly exceeding the initial value, while in the “placebo + placebo” group, the index returned to its original value. this indicates a positive effect of long-term use of lactoferrin on the microflora of the cervical canal. 4. conclusion 1. immersion has negative impact on the vaginal microflora: the number of facultative anaerobic opm increases, but the number of protective lactobacilli decreases. 2. oral use of lactoferrin during immersion has slight stabilizing effect on the vaginal microflora: despite the fact that the amount of obligate anaerobic opm on days 5-7 after the end of immersion slightly increases, the number of facultative anaerobic opm was stably low. 3. it was found that the amount of obligate anaerobic opm was decreasing at the time of 30-day taking of lactoferrin. another positive effect of lactoferrin was that small amount of facultative anaerobic opm and high amount of most species of lactobacilli was observed. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 normal-microbiota / before before / 5-7 day 5-7 day / 34-36 day e i period of sample taking placebo+placebo lactoferrin+placebo the effect of space flight factors on the correction disorders 229 copyright ©2024 assa adv. in systems science and appl. (2024) 4. according to the results of the cluster analysis of data on the vaginal microflora, as well as the kruskal-wallis analysis for the cervical canal, local use of the probiotic capsules had the greatest effect on lactobacilli in both biotopes. 5. when evaluating the general state of the vaginal microflora using the eubiotic index, its highest value was noted in the “placebo + lactobacilli” group, which confirms the effectiveness of intravaginal capsules against dysbiosis that is often observed in dry immersion. 6. the results of comparison of cervical canal microflora showed significant differences between the “placebo + placebo” and “lactoferrin + placebo” groups. at the same time, despite the fact that in the “placebo + placebo” group the microbiota was more balanced at the beginning of the experiment (the eubiotic index was higher than in the “lactoferrin + placebo” group), the eubiotic index in “lactoferrin + placebo” goup was increasing, starting from the background period, and did not have a sharp decline on the 5-7 days after immersion as in the “placebo + placebo” group. 7. summarizing obtained results, we can conclude that long-term use of lactoferrin has positive effect on the vaginal microflora, while the fastest and most effective countermeasure are probiotic vaginal capsules. the combination of lactoferrin and a local probiotic drug did not significantly increase their effectiveness. 8. the most promising, probably, is the use of probiotic capsules based on autologous strains of vaginal lactobacilli, as well as the addition of lactoferrin to these capsules, because effectiveness of lactoferrin when applied locally has been established in previous studies in gynecological patients [25]. this is currently the subject of further study. acknowledgements the work was made within the framework of the theme of fundamental scientific research of the russian academy of sciences with using the transgenbank unique scientific facility with financial support from the russian federation represented by ministry of education and science of russia within the framework of the project (agreement no. 075-15-2021-668 of 07/29/2021) references [1] khabriev, r. u., bezmelnitsyna, l. yu., isaeva, a. v., meshkov, d. o., cherkasov, s. n., et al. 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(2019). principy lechenija urogenital'nyh recidivirujushhih bakterial'nyh i gribkovyh infekcij u ginekologicheskih bol'nyh [principles of treatment of urogenital recurrent bacterial and fungal infections in gynecological patients], issues of gynecology, obstetrics and perinatology, 18(2), 57–60, [in russian]. adv syst sci appl 2023; 04:119–147 published online at https://ijassa.ipu.ru. identifiability of a family of dynamical systems: application to crops identification abdes bernoussi1*, edyta woźniak2, abdelaâziz belfekih1 1faculty of sciences and techniques tangier, university abdelmalek essaadi, tangier, morocco 2centrum badań kosmicznych pan, warsaw, poland abstract: in this paper we consider the problem of identifying a system among a family of given systems. thus, from measurements collected on an unidentified system but that is part of a family of known model systems, we seek to determine this unidentified system. this differs from identifying the parameters of a given system through experimental observations [15]. the determination (identification) in a given family not always being possible, we refer to the identifiable family as any family for which this identification is possible. we thus introduce the concept of identifiability of a family of systems through a given measurement function. for localized linear systems we give algebraic characterizations that use the notion of system observability. we then propose algorithms which, in case of identifiability of the family and by a process of elimination, identify the system to which the collected measurements correspond. we have given some examples to illustrate these algorithms. we have also added an exemplified extension to discrete localized systems. keywords: dynamical systems, identifiability, observability, remote sensing 1. introduction the modeling of a given phenomenon consists in describing it mathematically by the equations that describe its dynamics. moreover, in the model we have to take into account various input-output parameters that allow interaction with the system. for the output, the information are obtained via a measurement function. this function actually models the sensors installed on this system and provides the collected measurements. in general, the purpose of collecting measurements through these means of observation is to estimate, when possible, the state of the system which evolves over time, which subsequently allows corrections or adjustments of actions that one might have to exert on the system. so knowing the model of the dynamics of the system and the measurement function, the observability concept consists in the possibility to reconstruct the state of the system and even predict its future evolution and this in order to control it to achieve predetermined goals. the observability concept was introduced and characterized for localized systems by kalman [11, 12]. due to the interest of this concept, many works were devoted to it in many aspects for numerous applications [4, 19]. it was later generalized to distributed systems [2, 8, 18]. different degrees of observability were then introduced depending on the considered problem: ”total” observability ( [10]), partial observability ( [6]) and regional observability ( [1, 7]). the global state of the system can be determined by measurements in the case of total observability; the ”visible” part of this state can be determined in the case ∗corresponding author: abernoussi@uae.ac.ma 120 a. bernoussi, e. woźniak & a. belfekih of partial observability and the trace of the state of the distributed system over an observable region can be determined in the case of regional observability. the problem that we are considering in this paper differs both in the data of the problem and in the goal to be achieved. we assume that we have a family of models and a measurement function, then the problem is the following: can we determine, from the collected measurements, the model of our system knowing that it is in the given family? if we manage to determine it, we will be able to do everything that could be done on a system of which we have both the model and the measurements. as it is not always possible to identify the system (or, more precisely, the model) that can generate the measurements among any family of systems (see an example below), we introduce the mathematical concept of identifiability of a family of systems: this is the ability to differentiate the systems of this family from one another by a given measurement function, which then makes it possible to identify exactly which of them corresponds to the phenomenon under study. we define this concept for dynamic systems governed by equations of the ”cauchy problem” type in a fairly general way. we will then distinguish two levels of identifiability: the first level is the identifiability relative to a given function of measurement g, which we will refer to as the g identifiability (or the c identifiability for linear systems); while the second level is absolute identifiability which corresponds to the existence of a measurement function g which ensures g identifiability. it is immediate to see that the first level, which relates to a measurement function, can only be achieved if the second level is achieved, which is an intrinsic property of the family of systems independently of the measurement function. we noted that the observability was generally characterized by a rank condition of a given matrix. the computation of such rank depends on the particularity of the matrices and for large matrices many numerical methods are used [9, 14]. the study of the identifiability of a family of localized linear systems, both continuous (in time) and discrete, reveals an abstract system and uses its “observability”. the concept of identifiability, introduced in this paper, and observability one differ in their philosophies. while observability is the possibility for a measurement function that is associated with a system to reconstruct the state of that system from collected measurements, identifiability is the possibility to differentiate a group of systems from one another using a set of measurements, and to identify the system that generated those measurements. this study is motivated by several real issues. let’s quote two: • in agriculture, can we identify the crop growing on a given plot of land from measurements collected with satellite images? this is the problem of identifying a crop by comparing its signature with the signatures of a number of crops. • among a certain number of physical systems, only one of these systems emits (or has emitted) a signal; can we know which of these systems is sending this signal? this paper is organized as follow: in a first part we define the identifiability of a family of systems and we give a characterization for the linear localized systems by a rank condition, similar to the one given for the observability or the controllability of the linear systems. we then propose algorithms that test the identifiability of a family of systems, and other algorithms which determine, for an identifiable family, the evolving system that has generated the collected measurements. these algorithms are based on an elimination process. numerical simulations and an application are provided to illustrate our approach by applying the developed algorithms. we then propose algorithms that test the identifiability of a family of systems, and other algorithms that determine, for an identifiable family, the evolving system that has generated the collected measurements. these algorithms are based on a process of elimination. numerical simulations and an application are provided to illustrate our approach by applying the developed algorithms. copyright © 2023 assa. adv syst sci appl (2023) identifiability of a family of dynamical systems 121 2. identifiability of a family of systems: introduction of the concept in this section we introduce the concept of identifiability of a family of two, and more then many systems and this for systems that are not necessarily linear. next, we give some characteristics of the general case before dedicating ourselves to finite dimensional linear systems in the following sections. consider two systems (s1) and (s2) governed by the following state equations (t > 0) : (s1) { ż (t) = f1 (z (t)) z (0) = z10 ∈ z (s2) { ż (t) = f2 (z (t)) z (0) = z20 ∈ z where fi : z → z and z an hilbert space (states space). we assume that for each z10 ∈ z, the first equation admits a unique solution z1 (t) , t ⩾ 0, and that for each z20 ∈ z, the second equation admits a unique solution z2 (t) , t ⩾ 0. consider on the other hand an output function y (t) = g (z (t)) , t ∈ [0, t ] (2.1) where g : z → y and y an hilbert space (observations space). z (t) in (2.1) is one of the states of these systems (s1) or (s2) , but we do not know which one. actually, we have one of two statements: y (t) = g ( z1 (t) ) , t ∈ [0, t ] or y (t) = g ( z2 (t) ) , t ∈ [0, t ] to be able to identify the system that has generated the collected measurements, it is necessary that through the output y(t), the systems (s1) and (s2) give different signals. moreover, it is necessary that this difference appears for any state of (s1) and any state of (s2) because when they have the same state then their outputs coincide and in this case the origin of the measurements cannot be detected. these conditions allow us to define the identifiable families concept to which we give the following definition: definition 2.1: (i) we say that the family of the two systems f = {(s1) , (s2)} is identifiable on [0, t ] by the observation (2.1), or g−identifiable on [0, t ], if the obtained measurements of (s1) are different from the obtained measurements of (s2) and this for all possible states of (s1) and all possible states of (s2) , provided that at least one of the states is non-zero: ∀z1 (.) state of (s1) and ∀z2 (.) state of (s2) checking z1 (.) ̸= 0 or z2 (.) ̸= 0 on [0, t ] , we have g (z1 (.)) ̸= g (z2 (.)) on [0, t ] (2.2) (ii) we say that the family of the two systems f = {(s1) , (s2)} is identifiable on [0, t ] , if there exist a space y and an application g : z → y such as the family f i.e. g−identifiable on [0, t ] . remark 2.1: according to (2.2), the family f = {(s1) , (s2)} is not g−identifiable if there exist a state z1 (t) of the system (s1) and a state z2 (t) of the system (s2), non-zero, such that g (z1(t)) = g (z2(t)) , ∀t ∈ [0, t ]. in this case we can have the same measurement for one or more states of (s1) and one or more states of (s2) . we have an important particular case which consists in taking y = z and g = id; the observation in this case is y (t) = z (t) , t ∈ [0, t ] , which corresponds to direct and complete access to the state of the system. in this case we speak of id−identifiable family whose copyright © 2023 assa. adv syst sci appl (2023) 122 a. bernoussi, e. woźniak & a. belfekih definition is as follows ∀z1 (.) state of (s1) and ∀z2 (.) state of (s2) satisfying z1 (.) ̸= 0 or z2 (.) ̸= 0 on [0, t ] we have z1 (.) ̸= z2 (.) on [0, t ] (2.3) we then have the following proposition: proposition 2.1: the family f = {(s1) , (s2)} is identifiable on [0, t ] if, and only if it is id−identifiable on [0, t ] . proof • it is immediate to see that if f is id− identifiable on [0, t ] , then it is identifiable on [0, t ] since it is g−identifiable with y = z and g = id. • conversely if f is identifiable on [0, t ], then there exists y and g : z → y such that f be g−identifiable on [0, t ] . for a state z1 (.) of (s1) and a state z2 (.) of (s2) , with z1 (.) ̸= 0 or z2 (.) ̸= 0 on [0, t ] , we have g (z1 (.)) ̸= g (z2 (.)) on [0, t ] . we deduce that z1 (.) ̸= z2 (.) on [0, t ] because z1 (.) = z2 (.) on [0, t ] would give g (z1 (.)) = g (z2 (.)) on [0, t ] . this proves that f is id−identifiable on [0, t ]. remark 2.2: it is therefore sufficient, to know if f = {(s1) , (s2)} is identifiable on [0, t ] to check (2.3) (if it is id−identifiable on [0, t ]). however, since in practice the measurement function is given, it is necessary to test if the family is c−identifiable. this concept can be generalised to several systems. consider a family f = {(s1) , . . . , (sn)} of n systems and suppose that each system (si) is governed by the state equation (si) { ż (t) = fi (z (t)) , t > 0 z (0) = zi0 ∈ z with fi : z → z. we also assume that for each zi0 ∈ z, this equation admits a unique solution denoted by zi (.) . the output equation is always given by (2.1), we consider the following definitions which generalize the definition 2.1. definition 2.2: (i) we say that the family of systems f = {(s1) , . . . , (sn)} is g−identifiable on [0, t ] if every subfamily {(si) , (sj)}, i ̸= j, is g−identifiable on [0, t ] . (ii) we say that the family of systems f = {(s1) , . . . , (sn)} is identifiable on [0, t ] if every subfamily {(si) , (sj)} is identifiable on [0, t ] . as in the case of two systems, we will say that the family {(s1) , . . . , (sn)} is id−identifiable on [0, t ] if each subfamily {(si) , (sj)} is id−identifiable on [0, t ] . proposition 2.2: the family of systems f = {(s1) , . . . , (sn)} is identifiable on [0, t ] if and only if it is id−identifiable on [0, t ] . proof the family f is identifiable on [0, t ] if and only if each subfamily {(si) , (sj)}, i ̸= j, is identifiable on [0, t ] , or if and only if, thanks to the proposition 2.1, each {(si) , (sj)} , i ̸= j, is id−identifiable on [0, t ] , which is equivalent to what the family f i.e. id−identifiable on [0, t ] . copyright © 2023 assa. adv syst sci appl (2023) identifiability of a family of dynamical systems 123 3. identifiability of families of finite dimensional linear systems consider a family of two systems whose dynamics are governed by the two following linear state equations: (s1) { ż (t) = a1z (t) z (0) = z10 ∈ rn , (s2) { ż (t) = a2z (t) z (0) = z20 ∈ rn with a1 and a2 are two square matrices of order n. the state of (s1) , noted by z1 (.), is given by z1 (t) = exp (ta1) z 1 0 and the state of (s2) , noted by z2 (.), is given by z2 (t) = exp (ta2) z 2 0 . the system is augmented by the following linear measurement function (e) y (t) = cz (t) , 0 ⩽ t ⩽ t with c a matrix with q rows and n columns. z (t) is either z1 (t) or z2 (t); so we have y (t) = cz1 (t) , t ∈ [0, t ] , or y (t) = cz2 (t), t ∈ [0, t ] . remark 3.1: the matrices a1 and a2 are necessarily of the same order n since we use the same output (and therefore the same matrix c) to measure each of the two systems. note that the output (e) is of the form (2.1) with the choice g(u) = cu, u ∈ rn; hence, and in all that follows, the families g−identifiable will be called c−identifiable and families id−identifiable will be called in−identifiable. let us go back to the definition 2.1. the characterisation (2.2) takes the form ∀z1 (.) state of (s1) and ∀z2 (.) state of (s2) satisfying z1 (.) ̸= 0 or z2 (.) ̸= 0 on [0, t ] we have cz1 (.) ̸= cz2 (.) on [0, t ] by contraposition, and denoting zk0 = zk(0) the initial state of the system (sk), we obtain the characterisation of the c−identifiability of a family of two systems as given in the following proposition: proposition 3.1: (i) the family f = {(s1) , (s2)} is c−identifiable on [0, t ] if and only if the only states of the two systems which give the same measurements are the null states (generated by zero initial conditions): z1 (.) state of (s1) , z 2 (.) state of (s2) and cz1 (.) = cz2 (.) on [0, t ] } ⇒ z10 = z20 = 0 (3.4) (ii) the family f = {(s1) , (s2)} is identifiable on [0, t ] if, and only if, there exists an integer q ⩾ 1 and a matrix c ∈mq,n (r) such that f is c−identifiable on [0, t ] . (iii) f = {(s1) , (s2)} is identifiable on [0, t ] if, and only if, it is in−identifiable: z1 (.) state of (s1) z2 (.) state of (s2) and z1 (.) = z2 (.) on [0, t ] } ⇒ z10 = z20 = 0 this proposition which characterises c−identifiable (resp. identifiable) families of two linear systems can be easily generalised to families of several linear systems. example 3.1: (i) consider the two systems {(s1) , (s2)} given by the following state equations: (s1) ( ż1 (t) ż2 (t) ) = ( 1 0 0 2 )( z1 (t) z2 (t) ) , (s2) ( ż1 (t) ż2 (t) ) = ( 3 0 0 1 )( z1 (t) z2 (t) ) copyright © 2023 assa. adv syst sci appl (2023) 124 a. bernoussi, e. woźniak & a. belfekih with the measurement function y (t) = z1 (t) + z2 (t) , 0 ⩽ t ⩽ t. this corresponds to c = ( 1 1 ) . for z10 = ( 1 0 ) and z20 = ( 0 1 ) the respective states are z1 (t) = ( et 0 ) and z2 (t) = ( 0 et ) which give cz1 (t)− cz2 (t) = ( 1 1 ) ( et 0 ) − ( 1 1 ) ( 0 et ) = 0, ∀t ⩾ 0; the family {(s1) , (s2)} is therefore not c−identifiable. (ii) consider now the two systems {(s1) , (s ′ 2)} given by the following state equations: (s1) ( ż1 (t) ż2 (t) ) = ( 1 0 0 2 )( z1 (t) z2 (t) ) and (s ′ 2) ( ż1 (t) ż2 (t) ) = ( 3 0 0 4 )( z1 (t) z2 (t) ) with the same measurement function y (t) = z1 (t) + z2 (t) , 0 ⩽ t ⩽ t. for z10 = ( a b ) and z20 = ( c d ) the respective states are z1 (t) = ( aet be2t ) and z2 (t) = ( ce3t de4t ) which give cz1 (t)− cz2 (t) = aet + be2t − ce3t − de4t. the family of functions {et, e2t, e3t, e4t} is free, equality aet + be2t − ce3t − de4t = 0, t ∈ ]0, t [ , gives a = b = c = d = 0 or z10 = z20 = 0, which proves that the family {(s1) , (s2)} is c−identifiable. before characterising the c-identifiable families, we will recall in the following section the definition of the observable system as well as its algebraic characterisation. then we will, in this case, give the expression of the state from the collected measurements. this will be useful for us to characterize algebraically the c−identifiable families. 3.1. observable system and state reconstruction the observability concept of a localised system reflects the ability of an output to reconstruct the state of the system in a “faithful”manner. let us now the definition of observable system. consider a linear system whose dynamics is described by the following state equation (s) { ż (t) = az (t) , t > 0 z (0) = z0 ∈ z = rn where a is a square matrix of order n representing the dynamics of the system, and consider the output function given by the following equation: y (t) = cz (t) , t ∈ [0, t ] (3.5) where c is a given matrix with q rows and n columns. definition 3.1: the system (s) is said to be observable for the measurement function (3.5) on [0, t ] if two different states of the system give two different measurements: z (t) ̸= z̃ (t) on [0, t ] ⇒ cz (t) ̸= cz̃ (t) on [0, t ] we then say that (a,c) is observable on [0, t ] . thanks to the linear character of the above equations, we can see that this definition is equivalent to the following (cz (t) = 0 , t ∈ [0, t ]) ⇒ z0 = 0 (3.6) copyright © 2023 assa. adv syst sci appl (2023) identifiability of a family of dynamical systems 125 where z0 = z (0) is the initial state of the system. let us introduce the matrix o with nq rows and n columns o =  c ca ca2 ... can−1  called observability matrix of the system (s), and the symmetric square matrix m of order n: m = ∫ t 0 exp ( tat ) ctc exp (ta) dt (3.7) then we have the following result ( [1]): proposition 3.2: (i) the system (s) is observable on [0, t ] for the measurement function (3.5) if and only if rank (o) = n, or again if and only if the matrix m is invertible. (ii) if the system (s) is observable on [0, t ] for the observation (3.5) and if we have the measurements ymes (t) , t ∈ [0, t ], then the state of the system can be determined by z (t) = exp (ta) z0 , t ⩾ 0, (3.8) where z0 = m−1 ∫ t 0 exp ( sat ) ctymes (s) ds (3.9) the observability of a localised system is often characterised by the rank condition: rank (o) = n. 3.2. characterisation of identifiable families let us consider the family of two systems f = {(s1) , (s2)} as well as the observation (e) and let us look for a condition equivalent to condition (3.4) of the c−identifiability of f , where z1(t) is the state of (s1) and z2(t) is the state of (s2). we have cz1 (t)− cz2 (t) = c exp (ta1) z 1 0 − c exp (ta2) z 2 0 = [ c exp (ta1) −c exp (ta2) ] [ z10 z20 ] = [ c −c ] exp ( t [ a1 o o a2 ])[ z10 z20 ] = c exp (ta ) ξ0 where we have noted that a = [ a1 o o a2 ] , c = [ c −c ] , ξ0 = [ z10 z20 ] the condition given by (3.4) then becomes (c exp (ta ) ξ0 = 0 , t ∈ [0, t ]) ⇒ ξ0 = 0 which is exactly the characterisation of the observability of the couple (a ,c ) . the matrices a and c can be associated to an abstract linear system of order 2n and an observation of order q; then we have the following: copyright © 2023 assa. adv syst sci appl (2023) 126 a. bernoussi, e. woźniak & a. belfekih proposition 3.3: the family f = {(s1) , (s2)} is c−identifiable on [0, t ] if, and only if, the following 2n order system: (s0) { ξ̇ (t) = a ξ (t) , t > 0 ξ (0) = ξ0 ∈ rn ×rn (3.10) is observable on [0, t ] for the output η (t) = c ξ (t) , t ∈ [0, t ] (3.11) we deduce a characterisation of the c−identifiability of {(s1) , (s2)} via the observability of (s0) . this observability is verifiable by calculating the rank of the observability matrix of the latter system. indeed the system (s0) is observable if and only if its observability matrix o =  c c a c a 2 ... c a 2n−1  has rank 2n. and since for any integer k we have c a k = [ c (a1) k −c (a2) k ] , then o is expressed as a function of a1, a2 and c. noting it dc , this matrix becomes dc =  c −c ca1 −ca2 c (a1) 2 −c (a2) 2 ... ... c (a1) 2n−1 −c (a2) 2n−1  (3.12) thus, we obtain an algebraic characterisation of the c−identifiability of a family of two systems and thus of its identifiability. this gives us an algebraic characterisation of the c−identifiability of a family of two systems and hence of its identifiability: theorem 3.1: (i) the family f = {(s1) , (s2)} is c−identifiable on [0, t ] if and only if the matrix dc , called the c−identifiability matrix of f , has rank 2n: rank (dc) = 2n (ii) the family f = {(s1) , (s2)} is identifiable if and only if the matrix din , called matrix of identifiability matrix of the family f , is of rank 2n: rank (din) = 2n remark 3.2: 1. the (ii) in theorem 1 shows that the c−identifiability and identifiability do not depend on the observation time interval; it suffices that it is of non-zero length. 2. if a1 = a2, the problem of identifying does not arise because we have the same system. note that in this case, and as one might expect, the family f is not c−identifiable since the matrix of c−identifiability is in this case of rank ⩽ n (by adding in dc each column k to column n+ k we get n null columns that can be eliminated). the theorem 3.1 proves that: copyright © 2023 assa. adv syst sci appl (2023) identifiability of a family of dynamical systems 127 • if rank (din) < 2n then the family {(s1) , (s2)} is non identifiable and consequently non c−identifiable for none matrix c (i.e. none given output). • if rank (din) = 2n then the family {(s1) , (s2)} is identifiable and therefore there are c matrices for which this family is c−identifiable i.e. rank (dc) = 2n. example 3.2: (i) let us take the example 3.1 where a1 = ( 1 0 0 2 ) and a2 = ( 3 0 0 4 ) . the identifiability matrix is then din =  i2 −i2 a1 −a2 (a1) 2 − (a2) 2 (a1) 3 − (a2) 3  =  1 0 −1 0 0 1 0 −1 1 0 −3 0 0 2 0 −4 1 0 −9 0 0 4 0 −16 1 0 −27 0 0 8 0 −64  din is of rank 4 so the family {(s1) , (s2)} is identifiable. (ii) now take c = ( 1 0 ); the c−identifiability matrix is given by dc =  c −c ca1 −ca2 c (a1) 2 −c (a2) 2 c (a1) 3 −c (a2) 3  =  1 0 −1 0 1 0 −3 0 1 0 −9 0 1 0 −27 0  which is of rank 2. the family {(s1) , (s2)} is therefore not c−identifiable (although it is identifiable). remark 3.3: consider the matrix c̃ defined by blocks of type (q, n) , having 2n row blocks, 2n column blocks, the diagonal blocks are equal to c and non-diagonal blocks are zero (c̃ is therefore of type (2nq, 2n2)). so we have dc = c̃ din which gives us rank (dc) ⩽ rank (din) ⩽ 2n , ∀c inequality allowing to find that any c-identifiable family is identifiable. in the case where the matrices a1 and a2 commute, we have a characterisation of identifiability with a reduced dimension matrix. the proof uses the following lemma: lemma 3.1: let a be a square matrix of order n with real coefficients. so (exp (ta) z0 = ct on ]0, t [) ⇒ z0 = 0 (3.13) if and only if the following matrix has rank n a a2 ... an  (3.14) where ct denotes a constant vector. copyright © 2023 assa. adv syst sci appl (2023) 128 a. bernoussi, e. woźniak & a. belfekih proof since d dt exp (ta) z0 = a exp (ta) z0, ∀t ⩾ 0, then we have equivalence (exp (ta) z0 = ct , ∀t ∈ [0, t ]) ⇔ (a exp (ta) z0 = 0 , ∀t ∈ [0, t ]) and the implication (3.13) becomes (a exp (ta) z0 = 0 , ∀t ∈ [0, t ]) ⇒ z0 = 0 this implication is none other than the characterisation (3.6) of the observability for the following observed system{ η′ (t) = aη (t) , t ⩾ 0 η0 ∈ rn ; y (t) = aη (t) t ∈ [0, t ] whose observability matrix is exactly the matrix (3.14). the proposition 3.2 then makes it possible to obtain the equivalence. using this lemma, we have the following proposition: proposition 3.4: suppose that the matrices a1 and a2 commute. then family {(s1) , (s2)} is identifiable if and only if the following matrix has rank n m =  a2 − a1 (a2 − a1) 2 ... (a2 − a1) n  proof the characterisation of identifiability given by (3.4) becomes, when c = in ,( exp (ta1) z 1 0 = exp (ta2) z 2 0 , ∀t ∈ [0, t ] ) ⇒ z10 = z20 = 0 since a1 and a2 commute the equality exp (ta1) z 1 0 = exp (ta2) z 2 0 is equivalent to exp (t (a2 − a1)) z 2 0 = z10 , ∀t ∈ [0, t ] . taking a = a2 − a1 in the lemma 3.1, the family {(s1) , (s2)} is identifiable by the observation (e) on [0, t ] if and only the matrix m has rank n. example 3.3: consider the systems (s1) and (s2) whose dynamics are defined by the matrices a1 = ( 1 0 0 1 ) and a2 = ( 2 1 2 −1 ) in this case the identifiability matrix di2 admits 8 lines. note that the matrices a1 and a2 commute, and let us apply the previous proposition; the matrix m admits 4 rows and it is equal to m = [ a2 − a1 (a2 − a1) 2 ] =  1 1 2 −2 3 −1 −2 6  m is of rank 2, so the family {(s1) , (s2)} is identifiable. copyright © 2023 assa. adv syst sci appl (2023) identifiability of a family of dynamical systems 129 on the other hand, the observability of both systems (s1) and (s2) cannot guarantee the c−identifiability of the family {(s1) , (s2)}; however this observability is necessary as shown the following proposition: proposition 3.5: if the family {(s1) , (s2)} is c−identifiable then each system of this family is observable by the observation defined by c. the converse is not true. proof • let z1 (.) be a state of (s1) such that cz1 (t) = 0, t ∈ [0, t ] , and let z2 (.) be a state of (s2) such that cz2 (t) = 0, t ∈ [0, t ]; then cz1 (t)− cz2 (t) = 0, t ∈ [0, t ] . the family {(s1) , (s2)} being c−identifiable, we get z10 = 0 and z20 = 0. this proves that both (s1) are observable and (s2) is observable. • the converse is not true. simply consider both systems in r2 and the observation defined by a1 = ( 1 0 0 2 ) , a2 = ( 2 0 1 1 ) , c = ( 1 −1 ) we verify that {(s1) , (s2)} is not c−identifiable while every system is observable. as a consequence of proposition 3.5 we have: corollary 3.1: if one of the two systems is not observable for the observation given by c then the family {(s1) , (s2)} is not c−identifiable. remark 3.4: if the family {(s1) , (s2)} is identifiable then we can find at least one matrix c which makes it possible to differentiate these systems (and sometimes several matrices). by introducing a comparison criterion between these c matrices, we can take the one that verifies a certain optimality. we take this into account in our outlook using the above we can give a first algorithm which detects if a family of two systems is c−identifiable or not: algorithm 1 c−identifiability test of a family of 2 systems. 1. form the matrix dc ; 2. if rank (dc) = 2n: display “family c−identifiable.”; stop 3. else, display “family non c−identifiable.” 3.3. algorithm for identifying the searched system in a family once the possibility of differentiating between two specific systems is acquired, the problem arises of the effective determination of the system which has generated the collected measurements {ymes (t) , t ∈ [0, t ]} . the algorithm we present allows to identify this system. it relies on the c−identifiability of the family {(s1) , (s2)} and its unfolding is done in two steps. first, we estimate the states of the two systems from measurements collected over an arbitrary interval [0, t1]; as a second step, and with measurements collected over a second interval [t1, t2] that we will determine, comparisons are made which eliminate the system which cannot generate these measurements. principle. suppose the family {(s1) , (s2)} is c−identifiable over an interval of time [0, t ]. copyright © 2023 assa. adv syst sci appl (2023) 130 a. bernoussi, e. woźniak & a. belfekih a) we assume that measurements are taken ymes (t) on a time interval [0, t1] (t1 < t ) and supposing that these measurements are non-zero. if the system (s1) generated these measurements then its state, which is necessarily non-zero, is given by z1 (t) = exp (ta1) z 1 0 , t ⩾ 0 with z10 = (m1) −1 ∫ t1 0 exp ( sat 1 ) ctymes (s) ds where we have noted (k = 1, 2) mk = ∫ t1 0 exp ( sat k ) ctc exp (sak) ds (3.15) if on the other hand it is the system (s2) which generated these measurements then its state, which is necessarily non-zero, is given by z2 (t) = exp (ta2) z 2 0 , t ⩾ 0 with z20 = (m2) −1 ∫ t1 0 exp ( sat 2 ) ctymes (s) ds b) if we have the information that the system sought is one of the two systems then we have only two cases to study. in the general case, we have four cases (the first three of which lead to outputs): case 1. cz1 (t1) = ymes (t1) and cz2 (t1) ̸= ymes (t1), then the measurements come from (s1) and not from (s2); (s1) is therefore the sought system; case 2. cz1 (t1) ̸= ymes (t1) et cz2 (t1) = ymes (t1), then the measurements come from (s2) and not from (s1); (s2) is therefore the sought system; case 3. cz1 (t1) ̸= ymes (t1) and cz2 (t1) ̸= ymes (t1), then the measurements come neither from (s1) nor from (s2); case 4. cz1 (t1) = ymes (t1)cz2 (t1) = ymes (t1), so we cannot differentiate these systems on [0, t1]; we then apply the following: c) the family {(s1) , (s2)} is c−identifiable on [t1, t ] and one of the states is non-zero, we can find an instant t2 such that t2 ∈ ]t1, t [ and cz1 (t2) ̸= cz2 (t2) (3.16) indeed in the opposite case we will have cz1 (t) = cz2 (t) , ∀t ∈ ]t1, t [ , and c−identifiability from {(s1) , (s2)} gives z10 = z20 = 0 in contradiction with non-zero measurements. d) we resume step b) with t2 instead of t1; the output is then made by one of the first three cases only, the last case cannot be realised thanks to (3.16). in this algorithm, tolerance plays a key role. we set a tolerance ε > 0 and for two vectors u, v ∈ rn, we will try to fulfill the condition u = v by the test ∥u− v∥ < ε and to fulfill the condition u ̸= v by the test ∥u− v∥ ⩾ ε, where ∥u∥ denotes the norm of the vector u. this gives the following algorithm: copyright © 2023 assa. adv syst sci appl (2023) identifiability of a family of dynamical systems 131 algorithm 2 identification of the system sought in a family c−identifiable from 2 systems. 1. take measurements ymes (t) on [0, t1]; 2. compute initial states z10 and z20 , and the states z1 (t) and z2 (t); 3. if ∥∥cz1 (t1)− ymes (t1) ∥∥ ⩾ ε : reject the system (s1); 4. if ∥∥cz2 (t1)− ymes (t1) ∥∥ ⩾ ε : reject the system (s2); 5. if only one system is not rejected: it is the sought system; stop 6. if both systems are rejected: go to 13 7. determine t2 > t1 such that ∥∥cz1 (t2)− cz2 (t2) ∥∥ ⩾ ε; 8. take the measurements ymes (t) on [t1, t2]; 9. if ∥∥cz1 (t2)− ymes (t2) ∥∥ ⩾ ε : reject the system (s1); 10. if ∥∥cz2 (t2)− ymes (t2) ∥∥ ⩾ ε : reject the system (s2); 11. if only one system is not rejected: it is the sought system; stop 12. if no system is rejected: display “tolerance too large.”; stop 13. display “no system matches the measurements or tolerance too low.” determination of the moment t2 the existence of t2 is ensured by the c−identifiability of the family but this moment is unknown. to determine this we present two methods, both at random: method 1 (monte-carlo method) we choose t2 ∈ ]t1, t [ randomly (according to the normal law) and, as long as we have cz1 (t2) = cz2 (t2), we do a another choice of t2. method 2 • we choose a time step δ > 0; we start with t2 = t1 + δ and while we have cz1 (t2) = cz2 (t2) we increase t2 by δ: t2 ← t2 + δ. if we do not fulfill this condition, we have cz1 (kδ) = cz2 (kδ) , k = 0, 1, 2, . . . , t δ we can then avoid this situation by slightly changing δ. • we can also take δ large enough and t2 = t1 + δ, and as long as we have cz1 (t2) = cz2 (t2), we change δ by δ/2 and resume t2 = t1 + δ; the condition cz1 (t2) = cz2 (t2) will be then checked at a certain step, except in very special cases where the changing δ fixes the problem. 3.4. generalisation to families of several systems in applications, it is rare to have to identify one system among two systems. the most usual way is to seek to identify it among several given systems (in a database). therefore, we assume that a given family f = {(s1) , . . . , (sn)} of n linear systems is governed by (si) { ż (t) = aiz (t) , t > 0 z (0) = zi0 ∈ rn i = 1, . . . , n copyright © 2023 assa. adv syst sci appl (2023) 132 a. bernoussi, e. woźniak & a. belfekih where the matrices ai which model their dynamics are all of dimension n. we also assume that we have an observation (e) y (t) = cz (t) , t ∈ [0, t ] (3.17) where z (t) is the state of the system and c a matrix of type (q, n). we can generalise the theorem 3.1 which characterises c−identifiability and identifiability to families of several systems. let us introduce the matrices d ij c =  c −c cai −caj c (ai) 2 −c (aj) 2 ... ... c (ai) 2n−1 −c (aj) 2n−1  i, j = 1, . . . , n i ̸= j then we have the following characterisations: theorem 3.2: (i) the family of systems f = {(s1) , . . . , (sn)} is c−identifiable if and only if rank ( d ij c ) = 2n , i, j = 1, . . . , n , i ̸= j (ii) the family of systems f = {(s1) , . . . , (sn)} is identifiable if and only if rank ( d ij in ) = 2n , i, j = 1, . . . , n , i ̸= j from this theorem we deduce the following algorithm which allows to test the c−identifiability of family f : algorithm 3 c−identifiability test for a family of systems. 1. for i = 1, . . . , n and for j = 1, . . . , n, j ̸= i 2. determine the matrix d ij c ; 3. if rank ( d ij c ) < 2n : display “family {(si) , (sj)} non c−identifiable.”; stop 4. next j ; next i 5. display “family c−identifiable.” for a family {(s1) , . . . , (sn)} c−identifiable we generalise the previous algorithm to the case of n systems with a slight adaptation imposed by the case of several systems using the algorithm below. to identify the system sought we take in the first step the measurements on [0, t1] and we calculate the state of each system; then, using the measurements collected on a second interval [t1, t2] that is determined by one of the methods we make comparisons that eliminate systems that cannot generate these measurements. this last operation is repeated for more than one system, as long as the observation generated by its state coincides in t2 with the measurement obtained. in more details: a) we take measurements on [0, t1] (arbitrary), and we determine the state zi (t) of each system (si) by zi0 (t) = exp (ta1) z i 0 , t ⩾ 0, i = 1, . . . , n copyright © 2023 assa. adv syst sci appl (2023) identifiability of a family of dynamical systems 133 with zi0 = (mi) −1 ∫ t1 0 exp ( sat i ) ctymes (s) ds b) at each step τ we reject each system whose output associated with its state at the time tτ does not coincide with the measurement at this instant. if a single system is not rejected, it is the system sought. c) if all the systems are rejected it is that, or none of the systems does not correspond to the measurements, or the tolerance is too low. d) otherwise, we take two non-rejected systems, we determine an instant tτ+1 > tτ where their observations at this instant are distinct, we take measurements on [tτ , tτ+1] and we start again at step b) with this time tτ+1 instead of tτ . step by step the number of non-rejected systems is reduced, until one is reached and then and only then the sought system is found. if at some stage the number of non-rejected systems is ⩾ 2 and is not reduced, it is because the tolerance is too large: we change tolerance and resume at step b). remark 3.5: in some cases the only tests in t1 are enough to determine the sought system (see the example in section 3.5). this gives the algorithm 4. algorithm 4 identification of the sought system in a family c−identifiable. 1. take the measurements ymes (t) on [0, t1]; 2. compute the initial states zi0 and the states zi (t) , 1 ⩽ i ⩽ n ; 3. reject each system (sk) that checks ∥∥czk (t1)− ymes (t1) ∥∥ ⩾ ε; 4. if all systems are rejected: go to 14 5. if only one system is not rejected: this is the sought system; stop 6. else, choose 2 non-rejected systems (si) and (sj) , i ̸= j; 7. determine t2 > t1 such that ∥∥czi (t2)− czj (t2) ∥∥ ⩾ ε; 8. take measurements ymes (t) on [t1, t2]; 9. reject every system (sk) that matches ∥∥czk (t2)− ymes (t2) ∥∥ ⩾ ε; 10. if all systems are rejected: go to 14 11. if only one system is not rejected: this is the sought system; stop 12. if the number of unrejected systems has not decreased: display “tolerance too large.”; stop 13. do t1 ← t2 and resume at 6 14. display “none of the systems in the family matches the measurements or tolerance too low.” remark 3.6: the algorithm 3is applied once to find out if the systems registered in the database form a c−identifiable family. it is in the affirmative that we apply algorithm 4 and we reapply it each time we have an identification to make both with this family of systems and for any measurements collected. copyright © 2023 assa. adv syst sci appl (2023) 134 a. bernoussi, e. woźniak & a. belfekih 3.5. numerical example consider a family of three systems whose dynamics are modeled by the matrices a1 = ( 1 0 0 2 ) , a2 = ( 3 0 0 4 ) , a3 = ( 2 −1 1 2 ) , c = ( 1 1 ) (3.18) where c intervenes in the measurement function. to verify the c−identifiability of the family {(s1) , (s2) , (s3)} we calculate the matrices d ij c d12 c =  1 1 −1 −1 1 2 −3 −4 1 4 −9 −16 1 8 −27 −64  , d13 c =  1 1 −1 −1 1 2 −3 −1 1 4 −7 1 1 8 −13 9  d23 c =  1 1 −1 −1 3 4 −3 −1 9 16 −7 1 27 64 −13 9  (3.19) since they are all of rank 4 then the family {(s1) , (s2) , (s3)} is c−identifiable. we consider the tolerance ε = 0.01. • first example of measurements 1) we take measurement on [0, t1] with t1 = 1. we obtain ymes (t) = 2e4t − e3t , t ∈ [0, 1] . 2) we compute the matrices mi by (3.15), we obtain m1 = ( 1 2 e2 − 1 2 1 3 e3 − 1 3 1 3 e3 − 1 3 1 4 e4 − 1 4 ) , m2 = ( 1 6 e6 − 1 6 1 7 e7 − 1 7 1 7 e7 − 1 7 1 8 e8 − 1 8 ) m3 ≃ ( 25.70 0.22 0.22 1.09 ) (3.20) which make it possible to calculate the initial states z10 ≃ ( −23.70 19.06 ) , z20 = ( −1.0 2.0 ) , z30 ≃ ( 5.76 −9.08 ) and from this the states of the three systems z1 (t) ≃ ( −23.7et 19.06e2t ) , z2 (t) = ( −e3t 2e4t ) , z3 (t) ≃ ( 5.76 cos t+ 9.08 sin t 5.76 sin t− 9.08 cos t ) 3) (i) for (s1) : since ∣∣cz1 (t1)− ( 2e4t1 − e3t1 )∣∣ ≃ 12.69 ⩾ ε we reject (s1). (ii) for (s2): since ∣∣cz2 (t1)− ( 2e4t1 − e3t1 )∣∣ = 0.0 < ε, we keep (s2). (iii) for (s3) : since ∣∣cz3 (t1)− ( 2e4t1 − e3t1 )∣∣ ≃ 298.95 ⩾ ε, we reject (s3) . 4) only (s2) is not rejected: therefore (s2) is the sought system. • second example of measurements copyright © 2023 assa. adv syst sci appl (2023) identifiability of a family of dynamical systems 135 1) we take the measurement on [0, t1] with t1 = 1. we get ymes (t) = 4e2t cos (t) , t ∈ [0, 1] . 2) the matrices mi are already calculated by (3.20); we calculate the initial states: z10 ≃ ( 12.52 −5.88 ) , z20 ≃ ( 4.27 −1.31 ) , z30 = ( 2.0 2.0 ) and the states z1 (t) ≃ ( 12.52et −5.88e2t ) , z2 (t) ≃ ( 4.27e3t −1.31e4t ) , z3 (t) = e2t ( 2 cos t− 2 sin t 2 cos t+ 2 sin t ) 3) (i) for (s1) : since ∣∣cz1 (t1)− 4e2t1 cos (t1) ∣∣ ≃ 25.38 ⩾ ε, we reject (s1) . (ii) for (s2) : since ∣∣cz2 (t1)− 4e2t1 cos (t1) ∣∣ ≃ 32.47 ⩾ ε, we reject (s2) . (iii) for (s3) : since ∣∣cz3 (t1)− 4e2t1 cos (t1) ∣∣ = 0.0 < ε, we keep (s3) . 4) (s3) is the only system not rejected; so (s3) is the sought system. finally we obtain the following assignments: plot 1 2 system (s2) (s3) 4. case of discrete finite dimensional linear systems we note that the majority of environmental phenomena has a very slow evolution; this allows us to assume for one of these systems that are modelling these phenomena that, after an instant tk, its state only shows a significant change at an instant tk+1. the measurements can then be considered as taken at instants tk. all these considerations lead us to consider what becomes of everything discussed earlier when the systems have discrete models. 4.1. identifiability of a family of finite dimensional discrete linear systems suppose the two dynamical systems are governed by the following two recurring equations (s1) { zk+1 = a1zk k = 0, 1, . . . z0 ∈ z (s2) { zk+1 = a2zk k = 0, 1, . . . z0 ∈ z the state of (s1) is denoted z1k and the state of (s2) is denoted z2k. suppose also that the measurements are discrete (e) yk = czk , k = 0, 1, . . . ,m− 1 where zk is either z1k or z2k. if ymes k , k = 0, 1, . . . ,m− 1, are collected measurements; so we have either ymes k = cz1k , k = 0, 1, . . . ,m− 1 or else ymy k = cz2k , k = 0, 1, . . . ,m− 1. we take up the same problem of the identification posed in the previous sections. definition 4.1: (i) we say that the family {(s1) , (s2)} is c−identifiable for the first m measurements if[ z1k state of (s1) , z 2 k state of (s2) and ∃k ⩽ m− 1 such as z1k ̸= 0 or z2k ̸= 0 ] ⇒ ( ∃j ⩽ m− 1 such as cz1j ̸= cz2j ) (ii) we say that the family {(s1) , (s2)} is identifiable for the first m measurements if there exist an integer q ⩾ 1 and a matrix c of type (q, n) such that this family is c−identifiable for the first m measurements. copyright © 2023 assa. adv syst sci appl (2023) 136 a. bernoussi, e. woźniak & a. belfekih by taking the contrapositive, and making the initial states z10 and z20 appear, the definition of c−identifiability becomes[ z1k state of (s1) , z 2 k state of (s2) and cz1k = cz2k, k = 0, . . . ,m− 1 ] ⇒ z10 = z20 = 0 (4.21) we will adopt this implication to characterize the families c−identifiable formed by two discrete systems. in the general case, the sought system is among n systems of a given family. we assume that these systems have the equations (si) { zk+1 = aizk , k = 0, 1, 2, . . . z0 = zi0 ∈ rn i = 1, . . . , n where the matrices ai, which model their dynamics, are all of dimension n. we can generalize the definition of identifiability to families of several systems as follows: definition 4.2: (i) the family of systems {(s1) , . . . , (sn)} is said to be c−identifiable for the first m measurements if each subfamily {(si) , (sj)} , i ̸= j, is c−identifiable for the observation via the first m measurements. (ii) the family of systems {(s1) , . . . , (sn)} is said to be identifiable for the first m measurements if each subfamily of systems {(si) , (sj)} , i ̸= j, is identifiable for the first m measurements. 4.2. observable discrete system and state reconstruction as for continuous systems in time, we use the observability of an abstract discrete system for the characterisation of c−identifiability. consider a dynamical system governed by the following recurring equation of state: (s) { zk+1 = azk , k = 0, 1, 2, . . . z0 ∈ z = rn where a is a given square matrix of order n with real coefficients. the state of the system has the expression zk = akz0 , k = 0, 1, 2, . . .. this equation of state is augmented by a measurement function (output equation ): yk = czk , k = 0, 1 . . . ,m− 1 (4.22) where c is a given matrix with q rows and n columns. definition 4.3: the system (s) is said to be observable for observation (4.22) on the discrete interval 0, 1, . . . ,m− 1 if for two states any distinct zk and z̃k of this system correspond to two distinct measurements: (∃k ⩽ m− 1 such as zk ̸= z̃k) ⇒ (∃j ⩽ m− 1 such as czj ̸= cz̃j) we then say that the couple (a,c) is observable for the first m measurements (or on 0, 1, . . . ,m− 1). one can easily see that this definition is equivalent to the following characterisation: (czk = 0 , k = 0, 1, . . . ,m− 1) ⇒ z0 = 0 copyright © 2023 assa. adv syst sci appl (2023) identifiability of a family of dynamical systems 137 consider the linear map k : z0 ∈ rn → {yk}0⩽k⩽m−1 ∈ (rq)m. since yk = cakz0, 0 ⩽ k ⩽ m− 1, the matrix expression o of the application k is given by o =  c ca ca2 ... cam−1  the system (s) is then observable with m measurements if, and only if, k is injective, which is equivalent to rank (o) = n. the matrix m = oto is symmetric; it is expressed as functions of a and c by m = m∑ k=1 ( ak−1 )t ctcak−1 then we have: proposition 4.1: (i) the system (s) is observable for the observation (4.22) with the first m measurements if and only if rank (o) = n, or even if and only if the matrix m is invertible. (ii) if the system (s) is observable for the observation (4.22) with the first m measurements and if we have the measurements ymes k , k = 0, 1, . . . ,m− 1, then the state of the system is given by zk = akz0 ; k = 0, 1, 2, . . . where z0 = (m)−1 m−1∑ j=0 ( aj )t ctymes j corollary 4.1: the system (s) is not observable for the observation (4.22) on the discrete interval 0, 1, . . . ,m− 1 if mq < n this shows that for the system to be observable, a necessary but not sufficient condition must exist and that the number of measurements should be greater than or equal to the value n/q. 4.3. characterisation of the c−identifiability of a family similar to the 3.2 section, we have, in the case of a family with two systems, cz1k − cz2k = cak 1z 1 0 − cak 2z 2 0 = [ c (a1) k −c (a2) k ] [ z10 z20 ] = [ c −c ] [ a1 o o a2 ]k [ z10 z20 ] = c a kξ0 where a = [ a1 o o a2 ] , c = [ c −c ] , ξ0 = [ z10 z20 ] copyright © 2023 assa. adv syst sci appl (2023) 138 a. bernoussi, e. woźniak & a. belfekih the characterisation (4.21) of the c−identifiability of the family {(s1) , (s2)} then becomes( c a kξ0 = 0, k = 0, 1, . . . ,m− 1 ) ⇒ ξ0 = 0 which characterises the observability of the couple (a ,c ) over the time interval 0, 1, . . . ,m− 1. this couple models the dynamics of a certain abstract discrete system and a certain observation, which allows us to have the following characterisation: proposition 4.2: the family {(s1) , (s2)} is c−identifiable for the first m measurements if and only if the following system of order 2n (s0) { ξk+1 = a ξk k = 0, 1, . . . ξ0 ∈ z × z = rn ×rn is observable during the time interval 0, 1, . . . ,m− 1 by the observation ηk = c ξk , k = 0, 1, . . . ,m− 1 we know (proposition 4.1) that the observability of the system (s0) over 0, 1, . . . ,m− 1 is equivalent to its matrix observability o of rank 2n. o =  c c a ... c a m−1  with the remark that c a k = [ c (a1) k −c (a2) k ] the matrix o can be expressed in terms of a1, a2 and c. this matrix becomes dc =  c −c ca1 −ca2 c (a1) 2 −c (a2) 2 ... ... c (a1) m−1 −c (a2) m−1  we then get the following theorem: theorem 4.1: (i) the family {(s1) , (s2)} is c−identifiable for the first m measurements if, and only if, the so-called c−identifiability matrix has rank 2n : rank (dc) = 2n (ii) the family {(s1) , (s2)} is identifiable for the first m measurements if and only if, the so-called identifiability matrix has rank 2n: rank (din) = 2n we deduce from this theorem that for a given number m of measurements, if rank (din) < 2n then the family {(s1) , (s2)} is not identifiable and therefore it is not c−identifiable for any matrix c. if on the other hand rank (din) = 2n then this family is identifiable and therefore there are matrices c for which this family {(s1) , (s2)} is c−identifiable. copyright © 2023 assa. adv syst sci appl (2023) identifiability of a family of dynamical systems 139 corollary 4.2: the family {(s1) , (s2)} cannot be c−identifiable with the first m measurements if m < 2n q (4.23) proof dc admits mq rows so rank (dc) ⩽ mq. if mq < 2n then rank (dc) < 2n and in this case the family {(s1) , (s2)} is not c−identifiable. remark 4.1: 1. in the particular case of a scalar measurement (q = 1) the condition (4.23) becomes m < 2n; a minimum of 2n measurements is therefore necessary. if besides the system states are scalars (n = 1) a minimum of 2 measurements is required. 2. there exist non-identifiable families for a given number m of measurements. an example of such families is given by the family of the two systems modeled by the matrices a1 = ( 1 0 0 2 ) and a2 = ( 3 0 0 2 ) with m = 4 since its identifiability matrix din is of rank < 4. however, and contrary to the case of continuous models in time, it is possible that an unidentifiable family of discrete systems for a number of measurements m becomes identifiable for a greater number m′ ⩾ m. 3. as for continuous linear systems in time (remark 3.3) we can show that rank (dc) ⩽ rank (din) ⩽ 2n , ∀c inequality allowing to find that any family c−identifiable is identifiable. example 4.1: consider the two systems (s1) zk+1 = a1zk , (s2) zk+1 = a2zk where a1 = ( 1 0 0 2 ) , a2 = ( 3 0 0 4 ) and m = 4 • the identifiability matrix of the family {(s1) , (s2)} is given by din =  1 0 −1 0 0 1 0 −1 1 0 −3 0 0 2 0 −4 1 0 −9 0 0 4 0 −16 1 0 −27 0 0 8 0 −64  and its rank is 4. the family {(s1) , (s2)} is therefore identifiable for 4 measurements. • for observation (e) yk = czk , k = 0, 1, 2, 3 with c = ( 1 0 ) copyright © 2023 assa. adv syst sci appl (2023) 140 a. bernoussi, e. woźniak & a. belfekih the c−identifiability matrix of the family {(s1) , (s2)} is given by dc =  c −c ca1 −ca2 c (a1) 2 −c (a2) 2 c (a1) 3 −c (a2) 3  =  1 0 −1 0 1 0 −3 0 1 0 −9 0 1 0 −27 0  which is of rank 2. the family {(s1) , (s2)} is therefore not c−identifiable. • since the family {(s1) , (s2)} is identifiable we can find a matrix c ′ ̸= c such that this family is c ′−identifiable. it is the case of the following observation: (e ′) yk = c ′zk , k = 0, 1, 2, 3 with c ′ = ( 1 1 ) the c ′−identifiability matrix of this family is given by dc′ =  c ′ −c ′ c ′a1 −c ′a2 c ′ (a1) 2 −c ′ (a2) 2 c ′ (a1) 3 −c ′ (a2) 3  =  1 1 −1 −1 1 2 −3 −4 1 4 −9 −16 1 8 −27 −64  which is of rank 4. the family {(s1) , (s2)} is therefore c ′−identifiable. we have similar results for the case of continuous systems, which we give in the following proposition: proposition 4.3: (i) when a1 and a2 commute the family {(s1) , (s2)} is identifiable by the first m measurements if, and only if, the matrix with mn rows m =  a2 − a1 (a2 − a1) 2 ... (a2 − a1) m  is of rank n. (ii) for the first m measurements, if the family {(s1) , (s2)} is c−identifiable then each system of this family is observable. the opposite is false. we therefore deduce from this proposition, as well as from corollary 4.2, that if one of the systems is not observable by the first m measurements or if m < n = q then the family formed by these two systems cannot be c−identifiable for these measurements. for families with several systems the algebraic characterisation of the c−identifiability and identifiability of a family of systems {(s1) , . . . , (sn)} is done through the following matrices d ij c =  c −c cai −caj c (ai) 2 −c (aj) 2 ... ... c (ai) m−1 −c (aj) m−1  i, j = 1, . . . , n i ̸= j theorem 4.2: (i) the family of systems {(s1) , . . . , (sn)} is c−identifiable for first m measurements if and copyright © 2023 assa. adv syst sci appl (2023) identifiability of a family of dynamical systems 141 only if rank ( d ij c ) = 2n , i, j = 1, . . . , n , i ̸= j (ii) the family of systems {(s1) , . . . , (sn)} is identifiable for the first m measurements if and only if rank ( d ij in ) = 2n , i, j = 1, . . . , n , i ̸= j 4.4. algorithm for the identification of the sought system for a given family f = {(s1) , . . . , (sn)} of n discrete systems, we give below the first algorithm that determines, when it exists, the first integer for which f is c−identifiable. a second algorithm identifies in the f family the system that generated the measurements, assuming that this family is c−identifiable. consider an integer m0 ⩾ 2 (to be set by the user). we search for the first integer m ⩽ m0, if it exists, that ensures the c−identifiability of the family f . e [x] is always the integer part of x we have the following algorithm: algorithm 5 c−identifiability test of a family and determination of m. 1. if 2n q is an integer, take m← 2n q else take m← e [ 2n q ] + 1 2. for i = 1, . . . , n and for j = 1, . . . , n, j ̸= i 3. determine d ij c ; if rank ( d ij c ) < 2n go to 6 4. next j ; next i 5. display “family c−identifiable for m =”m; stop 6. m← m+ 1; si m ⩽ m0 resume in 2 7. else, display “family non c−identifiable for m ⩽”m0 suppose in the following that the family f is c−identifiable for an integer m, determined by the algorithm 5, and that we have collected m measurements ymes 0 , . . . , ymes m−1 assumed to be non-zero. for each system (si) of the family, we can estimate the state zik by zik = ak i z i 0 (4.24) where zi0 = (mi) −1 m−1∑ k=0 ( ak i )t ctymes k and mi = m−1∑ k=0 ( ak i )t ctcak i (4.25) let us then denote by (si0) the system (still unknown) which has given these measurements. so, other than system (si0), the state of each system (si), i ̸= i0, check czik ̸= ymes k for at least one integer k ⩽ m− 1; otherwise we would have( czi0k = ymes k , k = 0, 1, . . . ,m− 1 ) and ( czik = ymes k , k = 0, 1, . . . ,m− 1 ) which cannot be achieved because of the c−identifiability of the subfamily {(si0) , (si)} . so just test (czik = ymes k , k = 0, 1, . . . ,m− 1) for each system (si) and to eliminate it if it does not fulfill this test. at the end of this step we should, theoretically, obtain a single not eliminated system. practically three cases can arise: copyright © 2023 assa. adv syst sci appl (2023) 142 a. bernoussi, e. woźniak & a. belfekih case 1. several systems check for equalities czik = ymes k , i.e. ∥czik − ymes k ∥ < ε , k = 0, . . . ,m− 1. in this case the tolerance ε is too large to be able to identify the sought system; case 2. no system fulfills these equalities: ∥czik − ymes k ∥ ⩾ ε , k = 0, 1, . . . ,m− 1 , 1 ⩽ i ⩽ n. in this case, either the tolerance is too low, or no system in the family has generated these measurements; case 3. only one system checks for ties; this is then the sought system.we deduce from all the above the following algorithm: algorithm 6 identification of the sought system in a family c−identifiable by m measurements. 1. take the measurements ymes k ; k = 0, 1, . . . ,m− 1; 2. compute the states zik , i = 1, . . . , n ; 3. for each system (si) 4. for k = 0, 1, . . . ,m− 1 5. if ∥∥czik − ymes k ∥∥ ⩾ ε: reject (si) and go to 7 6. next k 7. next system (si) 8. if all systems are rejected: display “none of the systems generated these measurements, or tolerance was too low.”; stop 9. if the number of non-rejected systems is ⩾ 2: display “tolerance too large.”; stop 10. the only non-rejected system is the sought system. remark 4.2: the algorithms of the discrete case are slightly different from those of the continuous case (in time). this results from the very nature of these systems. 4.5. numerical example consider a family of three systems whose dynamics are modeled by the matrices (3.18). we determine the first integer m for which the family of these 3 systems is c−identifiable. for m < 4 the matrices d ij c have less of 4 rows and therefore their ranks are < 4. for m = 4 the d ij c are given by (3.19) which all have rank 4 = 2n; hence the c−identifiability of this family for measurements in k = 0, 1, 2, 3. in what follows we take ε = 0.01 as tolerance. • first example of measurements the observation at moments k = 0, 1, 2, 3 gave the measurements {ymes 0 , ymes 1 , ymes 2 , ymes 3 } = {1, 5, 23, 101} . 1) compute of the states: we first calculate the matrices mi by (4.25); we obtain m1 = ( 4 15 15 85 ) , m2 = ( 820 1885 1885 4369 ) , m3 = ( 228 −120 −120 84 ) (4.26) so the initial states, computed by (4.24), are z10 = ( −523 23 1694 115 ) , z20 = ( −1 2 ) , z30 = ( 65 22 −449 66 ) copyright © 2023 assa. adv syst sci appl (2023) identifiability of a family of dynamical systems 143 2) for (s1) : we test (i) |cz10 − ymes 0 | = 1036 115 ⩾ ε : we reject (s1) . 3) for (s2) : we test (i) |cz20 − ymes 0 | = 0.0 < ε; (ii) |cz21 − ymes 1 | = |ca2z 2 0 − 5| = 0.0 < ε; (iii) |cz22 − ymes 2 | = ∣∣c (a2) 2 z20 − 23 ∣∣ = 0.0 < ε; (iv) |cz23 − ymes 3 | = c (a2) 3 z20 = 0.0 < ε. on conserve (s2). 4) for (s3) : we test (i) |cz30 − ymes 0 | = 160 33 ⩾ ε : we reject (s3) . (s2) is the only system not rejected; so (s2) is the sought system. • second example of measurements the observation at moments k = 0, 1, 2, 3 gave the measurements {ymes 0 , ymes 1 , ymes 2 , ymes 3 } = {1, 5, 15, 35} . 1) the matrices mi are always given by (4.26); we calculate the initial states by (4.24): z10 = ( −101 23 564 115 ) , z20 = ( 74 039 29 355 −3028 5871 ) , z30 = ( 2 −1 ) 2) for (s1) we test : (i) |cz10 − ymes 0 | = 56 115 ≃ 0.48 ⩾ ε : we reject (s1) . 3) for (s2) we test: (i) |cz20 − ymes 0 | = 0.0 < ε; (ii) |cz21 − ymes 1 | = |ca2z 2 0 − 5| = 778 1545 ≃ 0.5 > ε : we reject (s2) . 4) for (s3) , we keep it because: (i) |cz30 − ymes 0 | = 0.0 < ε; (ii) |cz31 − ymes 1 | = |ca3z 3 0 − 5| = 0.0 < ε; (iii) |cz32 − ymes 2 | = ∣∣c (a3) 2 z30 − 15 ∣∣ = 0.0 < ε; (iv) |cz33 − ymes 3 | = ∣∣c (a3) 3 z30 − 35 ∣∣ = 0.0 < ε. (s3) is the only system not rejected; so (s3) is the sought system. finally we obtain the following assignments: plot 1 2 system (s2) (s3) 5. application to the crop identification using radar in this section we present the use of radar for crop identification as an application of the general approach given in the previous sections. we recall the radar principle. the radar, considered as a sensor, is a satellite-based synthetic aperture radar (sar). it transmits microwave pulses and receives the signals backscattered from the objects on the earth, which are recorded as an image data and data products. radar signal backscatter depends mainly on the physical properties of the observed objects: their roughness, geometry and dielectric constant. in case of vegetation it is mostly linked to the plant external structure, biomass and water content ( [13]). the possibility of observation of crop characteristics is also dependent on sensor properties, e.g.: copyright © 2023 assa. adv syst sci appl (2023) 144 a. bernoussi, e. woźniak & a. belfekih • the wavelength used for the observation and the associated depth of penetration, for instant: when using c-band (5,6 cm) we detect mainly plant’s leaves, but when using l-band (23 cm) mostly only branches are visible. • the polarization of the transmitted and received signal. using different polarization we can observe better different kind of objects. cross-polarization is better to use for observation of non-structured crops (cereals, maize), co-polarization is more suitable for detection of structured crops (vineyards, hop). to use the radar for crop identification, it is necessary to develop a classification algorithm/approach. first, it should be noted that even if crops evolve continuously in time we study them using discrete measurements as shown in figure 5.1. fig. 5.1. evolution of a crop (potato) (bcch the phenological development stages of plants) for identification of crops using radar we proceed in two steps as in the mathematical approach: 1. first step: dataset and crops dynamics. to assume that the dynamics of the crops, and consequently the discrete states, are known, a reference dataset is needed. this requires ”a first” measurement to construct the database. in the case of crops, ground truth data are needed to know the different types of crops, their dynamics and states in specific fields, as well as the measurements of radar backscatter must be acquired for these fields using satellite sensors (e.g.: sentinel-1). it must be noticed that the sar measurement can be realized using different polarization at the same time so several parameters can be derived to extract more information about crops, e.g.: polarimetric coherence matrices and h/α ( [5]), which show the degree of the depolarization of the signal in the contact with the object, its scatter type or entropy. such multidimensional observation increases the accuracy of the systems identification ( [16]). also the speckle noise must be removed. using these measurements we estimate the standard dynamic of each crop (reference models of crops) which we want to identify (figure 1). so the dataset consists to have the equivalent of figure 1 for each crop as in figure 2. 2. second step: identification. to identify a system on a given plot, where the crop type is unknown, we consider two steps: 2.1 we must ensure that our sensor can distinguish crops. this is done through the analysis of our reference database. in figures 5.3 we show accordingly a case where one crop can be identified whereas the other cannot. it must be noticed that different parameters can or cannot identify the systems. 2.2 if the sensor can identify the crops, a ”second measurement” is made. we collect measurements of all parameters for all the fields in the area where we want to identify copyright © 2023 assa. adv syst sci appl (2023) identifiability of a family of dynamical systems 145 fig. 5.2. reference models of different crops (dataset) fig. 5.3. examples of identifiable crops (a) and non-identifiable crops (b). unknown crop types (systems) and use the equivalent of the algorithm given in the mathematical approach. in the case of crop classification using radar we will look for all fields which can give the similar measurements given by the standard crops (reference models of crops). this step is carried out using one specific classifier (e.g. random forest (rf) ( [3]) which compares directly measurements to the reference models of the systems (crops) and identifies crops. in the mathematical approach, we have pointed out that measurements must be made over a time interval i = ]0, t ] in the continuous case and in the discrete cases, the measurements must be greater than a given threshold iterations. it is the case for crop identification using radar. as an example, in figure 5.4, we show that in order to identify the systems we have to consider a relatively greater time. in fact, we could include this (the time of the measurements) in our classification algorithm and choose this time according to this data. we can mention that different times of measurement have better or worse potential for the identification. the case study carried out in poland using sentinel-1 sar images ( [17]) showed that, e.g. sugar beet and winter rapeseed systems can be very well discriminated from other 23 crops (systems) – f1 score which evaluate an accuracy of the classification of a specific class is 0.98 and 0.99, respectively. so sentinel-1 can be considered as an identifying sensor for these systems. on the contrary, the cases of spring triticale and spring wheat systems, which were very poorly discriminated 0.53 and 0.6, respectively. in our mathematical approach, the first measurement consists in searching for standard states of systems (reference models of crops) given through the measurement of radar parameters. the second measurement consists in the identification among systems of both: mathematical and radar algorithms. copyright © 2023 assa. adv syst sci appl (2023) 146 a. bernoussi, e. woźniak & a. belfekih fig. 5.4. example of two systems identifiable or non-identifiable depending on the time of measurement. 6. conclusion in this paper, we considered the problem of identifying a family of dynamical systems. for the case of finite dimensional linear systems, we have given characterizations and proposed algorithms to determine the sought system, among n systems, from the collected measurements. we have developed numerical examples and an application to illustrate our approach. it would be interesting to extend this work, and in particular the algorithms, to nonlinear systems and to distributed parameter systems. it would also be very useful to automate the identification of the system sought for a possible application to the automation of the identification of cultures using radar. these issues are currently under investigation. acknowledgements this work have been supported by mesrsfc, cnrst and add, morocco under the research project al khawarizmi/2020/11 ”intelligent management tool for irrigation water and forest heritage”. references 1. belfekih, a., el jai, a. & bernoussi a. (2023). observabilité et opacité – systèmes spatio-temporels [observability and opacity of distributed parameter systems]. perpignan, france: presses universitaires perpignan, [in french]. 2. boukhobza, t., hamelin, f., martinez-martinez, s. & sauter, d. (2009). structural analysis of the partial state and input observability for structured linear systems: application to distributed systems, european journal of control, 15(5), 503–516. 3. breiman, l. (2001). random forests, machine learning, 45(1), 5–32. 4. castillo, e., conejo, a. j., pruneda, r. e. & solares, c. (2006). observability analysis in state estimation: a unified numerical approach, ieee transactions on power systems, 21(2), 877–886. 5. cloude, s. r. & pottier, e. (1997) an entropy based classification scheme for land applications of polarimetric sar, ieee transactions on geoscience and remote sensing 35(1), 68–78. 6. danine, m. e., bernoussi, a. s. & bel fekih, a. 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(1987). capteurs et actionneurs dans l’analyse des systèmes distribués [sensors and actuators in distributed systems]. international journal of control, 46(4), 1139-1153, [in french]. 11. kalman, r. e. (1962). canonical structure of linear dynamical systems, proceedings of the national academy of sciences, 48(4), 596–600. 12. kalman, r. e. (1963). mathematical description of linear dynamical systems, journal of the society for industrial and applied mathematics, series a: control, 1(2), 152–192. 13. noernberg, m. (1999). the use of biophysical indices and coefficient of variation derived from airborne synthetic aperture radar for monitoring the spread of aquatic vegetation in tropical reservoirs, international journal of remote sensing, 20(1), 67–82. 14. ubaru, s., saad, y. (2016, june). fast methods for estimating the numerical rank of large matrices, proc. of the 31st international conference on machine learning (beijing, china), 468–477. 15. walter, e., pronzato, l. & norton, j. (1997). identification of parametric models from experimental data, vol. 1. berlin, germany: springer. 16. woźniak, e., kofman, w., lewinski, s., wajer, p., rybicki, m., aleksandrowicz, s. & włodarkiewicz, a. (2018). multi-temporal polarimetry in land-cover classification, international journal of remote sensing, 39(22), 8182–8199. 17. woźniak, e., rybicki, m., kofman, w., aleksandrowicz, s., wojtkowski, c. et al. (2022). multi-temporal phenological indices derived from time series sentinel-1 images to country-wide crop classification, international journal of applied earth observation and geoinformation, 107, 102683. 18. wyrwal, j. (2019). simplified conditions of initial observability for infinite-dimensional second-order damped dynamical systems, journal of mathematical analysis and applications, 478(1), 33–57. 19. zhang, l., zhang, q. (2015). observability conservation by output feedback and observability gramian bounds, automatica, 60, 38–42. copyright © 2023 assa. adv syst sci appl (2023) introduction identifiability of a family of systems: introduction of the concept identifiability of families of finite dimensional linear systems observable system and state reconstruction characterisation of identifiable families algorithm for identifying the searched system in a family generalisation to families of several systems numerical example case of discrete finite dimensional linear systems identifiability of a family of finite dimensional discrete linear systems observable discrete system and state reconstruction characterisation of the c-identifiability of a family algorithm for the identification of the sought system numerical example application to the crop identification using radar conclusion adv syst sci appl 2019; 03; 93-117 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/603 interaction between incumbent while entrenched and modern while agile companies at a freezing moment of time jeffrey yi-lin forrest 1*, diane d. galbraith1, yong liu2, rafael pla-lopez3 1) school of business, slippery rock university, slippery rock, pa 16057, u.s.a. e-mail: jeffrey.forrest@sru.edu; diane.galbraith@sru.edu 2) school of business, jiangnan university, wuxi 214122, jiangsu, china e-mail: clly1985528@163.com 3) department of applied mathematics, universtat de valencia, valencia 46010, spain e-mail: rafael.pla@uv.es received may 30, 2018; revised august 4, 2019; published october 1, 2019 abstract: this paper aims at designing a game-theoretical model – without any unrealistic assumptions – to study the competition between a company that is operating under an old paradigm that continuously exploits its established competitive advantage, and a modern, agile company that anticipates and adapts to market changes. by looking at the 2 × 2 competition of the companies, established are several results on how the incumbent while entrenched company loses its competition to the modern company. then by including a customer into the interplay of the two companies, analyzed is the dynamics of competition between the two companies and why the oldfashioned company has to become modern by adopting the philosophy of transient competitive advantages. specifically, this paper shows when the old-fashioned company could invest in raising the entry barrier in order to protect its customer base, and when such investment would be useless. practically, this work provides useful guidelines for when firms that are incumbent in their markets should start to prepare to ride turbulent waves of a paradigm change, and what modern firms should constantly scan with respect to their environments for the next breakthroughs in order to design and to implement their next competitive advantages. keywords: competition, game, hybrid product, nash equilibrium, technology, market invasion 1. introduction with the increasing globalization of the present world of business, many well-established companies have either disappeared or become irrelevant due to various internal or external reasons or a combination of both, as competitive forces reshape the strategies companies employ (porter, 1979). when an organization is unable to forecast and/or adjust to changing trends or paradigm shifts, it will definitely exit the market soon as in the case of kodak, xerox, and motorola’s one-time dominance in the analog cellular telephone business (barker, 1993). in today’s world, clinging to established competitive advantages is no longer viable (forrest & tallapally, 2018). for a leadership to be futuristic and visionary, it generally requires confidence and narcissism (navis & volkan, 2016). leadership is essential for the creation of transient competitive advantage in order to stay abreast of the speed of business. the staying power of a singular competitive advantage has been shortened substantially by technology and automation (collins, 2001). an example of this is the rivalry between the cellular tech-giants, * corresponding author: jeffrey.forrest@sru.edu mailto:jeffrey.forrest@sru.edu mailto:diane.galbraith@sru.edu mailto:clly1985528@163.com mailto:rafael.pla@uv.es mailto:jeffrey.forrest@sru.edu 94 j. yl forrest , d. d. galbraith, y. liu, r. pla-lopez copyright ©2019 assa adv. in systems science and appl. (2019) at&t and verizon. even in the 1990’s and the turn of the century, as one of the authors actually experienced, when one company launched a new product, service, phone, data or pricing plan, another company would surpass the competition in a matter of weeks or months. in other words, companies that cannot successfully ride the waves of transient advantages become victims of quickly shifting business landscapes (mcgrath, 2013). that is because there are very few blue oceans, as explained by (kim & mauborgne, 2005), whereby there is little to no competition. this realization, for example, was deployed by cirque du soleil when they transformed the picture of a circus from animals to acrobatic, nimble human performers. considering the fact that in the modern world of business, some companies still hold on to the once-championed philosophy and practice – spend time to find a market niche, develop a particular advantage, and then reap in the profits by exploiting the occupied market share, while some other companies are preparing themselves by shaping their organizational cultures to face the challenge of transient competitive advantages, because for these companies to remain ahead of the fierce competition, they need to be super vigilant in their collective pursuit of the next trend or transient advantage (chan & chan, 2010). to this end, a natural question arises: how do companies of the former type actually lose their shirts in the competition with companies of the latter type? in other words, how can a company really tell when it is losing its market competition so that it needs to disengage itself from the current competitive advantage and adopt another competitive advantage in order to stay viable? other than theoretical value, investigations of this question are extremely important in practice, because, for example, in the field of management a commonly accepted exercise is to implement ‘best practices,’ identified out of the successes of most noticeable companies (mcgrath, 2013; porter, 1979), without much theoretical reasoning. in other words, beyond its academic value (see explanations below), this work brings forward a way of thinking for decision-making managers and entrepreneurs in their daily operations. aiming at addressing this question, this paper considers markets as systems (lin, 1987; 1999; senge, 1990), and organizations also systems that are comprised of smaller component parts, they must be agile and adaptable to respond quickly to continuous market and technological changes (johnson, 1998). in fact, morgan (2006, p. 44) also treats organizations as systems that are “comprised of human resources, people who are organisms, open to their environment… (they) are open systems with interrelated subsystems… these organic subsystems continuously interact with their environment in a mutual state of dependence.” in particular, this paper employs respectively the methods of 2 × 2 and 3 × 2 games to model the direct competition between two companies, named a and b, where company a is of the oldfashioned type of sustainable advantages, while company b the modern type of transient advantages. the appropriateness of summarizing a real-life scenario of business competition as that between only two companies is completely adequate in terms of the question this paper addresses. for example, when we look at a market occupied by n incumbent firms, for a natural number n, we can theoretically divide these incumbents into two theoretical firms: one focal firm of our interest, the other the aggregate of the other 𝑛 − 1 firms. if the focal firm desires to occupy a leading position in its competition against of the other firms, it needs to compete itself way ahead of the rest, which in practice is seen as one loosely identified entity. in particular, based on the 2 × 2 game theoretical modeling, among others, the following results are shown: (1) if expansion is expected for company b to generate profits, which are more than the cost for company a to take defensive actions, then company a would move first and take defensive actions against company b although the result is uncertain; and (2) if company b’s decision on whether or not to design and produce its hybrid product is based on the existence of an expanding market beyond that of company a, then company b will definitively introduce its hybrid product. interaction between incumbent and modern companies 95 copyright ©2019 assa adv. in systems science and appl. (2019) because it is consumers that ultimately determine the fortune and fate of companies, in the 3 × 2 game theoretical modeling, we include a customer as the third player who interacts with the two companies. based on this model, among others, we develop the following results: (1) if company b’s revenue is greater than its cost and its revenue from outside the market base of company a is smaller than its cost, and company a does not take any defensive actions to protect itself, then the probability for the customer to purchase company a’s product is inversely proportional to company b’s activities of designing and producing its hybrid product; and when the revenue company b earns from the market base of company a is equal to its fixed cost of designing and producing its hybrid product minus the revenue generated from beyond the customer base of company a, then the probability for the customer to purchase company b’s hybrid product is equal to 1. (2) if company b’s revenue is greater than its cost and its revenue from outside the market base of company a is smaller than its cost, and company a does take defensive actions to protect itself, then the higher the difference between the cost for company b to design and produce its hybrid product and the revenue generated beyond the customer base of company a, the lower the probability for company a’s customer to purchase company b’s hybrid. (3) if the profits of company b’s expansion come mainly from company a’s market share, then company a can successfully delay the deterioration of its territory as long as less than 2/3 of its maximum expected revenue is taken by company b. the rest of the paper is organized as follows. section 2 reviews the relevant literatures. section 3 details the game-theoretic models to be employed in the rest of this work. section 4 derives the results by looking at a two-player game and a three-player game, respectively. section 5 concludes this presentation. 2. literature review although what is studied in this paper is somewhat relevant to the vast literatures on market entry (see, for example, (zachary, et al., 2015) and references listed there) and r & d races to develop new products, which might be used to create new market spaces and reduce market scope from competitors (see, for example, (jindal, et al., 2016)), the main focus here is on the interaction between firms that practice the philosophy of sustainable competitive advantages and those that ride the wave of transient advantages. in the literature, a lot of attention has been given to the question of how to develop a competitive advantage. for example, purkayastha and sharma (2016) inductively analyze three firms that develop a competitive advantage by shaping their business model. koller (2016) introduces the concept of adaptive advantage and its implementation in an organization. bashir and verma (2017) highlight how business model innovation can serve as a competitive advantage. christensen, suárez and utterback (1998) confirm that firms that target new market segments with an architectural innovation tend to be successful. and, chan and chan (2010) reveal that in the fast-changing fashion market, being flexible and adaptive is a key to survival. in terms of adapting organizational cultures and strategies to the constantly changing environment, tushman and o'reilly iii (1996) conclude that managers must create an ambidextrous organization that is capable of simultaneously pursuing both incremental and discontinuous innovation. in terms of how to achieve and sustain market-driven orientation, day (1994) finds that the emerging capabilities approach to strategic management, coupled with total quality management, offers a rich array of ways to design change programs that will enhance a market orientation with increased capabilities in market sensing and customer linking. in terms of transient competitive advantage readiness, kaharuddin, handaru, sardan, and mohammed (2017) use hotels, cafes, and fashion retail industry in bandung, indonesia, to study the relevant measurement. in terms of competing in a world of transient advantages, http://www.inderscienceonline.com/author/mohammed%2c+hayder+alhadey+ahmad 96 j. yl forrest , d. d. galbraith, y. liu, r. pla-lopez copyright ©2019 assa adv. in systems science and appl. (2019) leavy (2016) considers what dynamic capabilities will be needed. in terms of how important innovation is, dobni and klassen (2015) look at whether or not the u.s.a., the world's largest economy, is still a leading nation in innovation. in terms of using patents as a competitive advantage, harrigan and diguardo (2016) empirically show that the duration of patent-based advantages appears to wane with time in the highly-dynamic u.s. communications-services industry during 1998 – 2012, when technological changes occurred rapidly. in terms of the long-standing game-theoretical studies of duopoly competition, the literature is quite large. for example, puu (1991) investigates the problem of chaos in duopoly pricing, agliari and puu (2002) look at a similar problem by using a cournot model with a bounded inverse demand function. zhang et al (2009) report the dynamics of bertrand model with bounded rational players. dubiel-teleszyński (2011) examines a bertrand duopoly with heterogeneous (bounded rational and adaptive) players engaged in manufacturing differentiated products. fan et al (2012) study the dynamics of a cournot duopoly with one player being a bounded rational player and the competitor following naive expectations. wang and ma (2013) employ the same combination of expectations to study the dynamics of cournot-bertrand game. ma and pu (2013) employ bounded rationality as the expectation of firms to investigate the dynamics of a cournot-bertrand duopoly model. ma and xie (2016) investigate the dynamic pricing game of the duopoly air conditioner market with disturbances in demand. zhu et al (2014) investigate the stability of nash equilibrium in a dynamic nonlinear mixed cournot model, where the competing firms are a semi-public firm that is interested in maximizing both social welfare and profit, and a private firm that is only interested in profit maximization. zhang and ma (2016) consider two distinct pricing policies in a dual-channel supply chain with a fairly caring retailer by examining the complex nonlinear dynamics. huang et al (2016) examine the influence of parameters on the stability of two manufacturers dealing with complimentary products by considering three games based on the channel power of the manufacturers. pecora and sodini (2018) look at a cournot duopoly in a continuous time framework by considering a system of differential equations, where the competitors are assumed to be heterogeneous in determining output decision. by employing nonlinear dynamics, sice et al (2000) assess and simulate interactions in duopoly competition for quality between two consumer durable products. these authors find that when the speed of adaptation to customer demand reaches a certain value, a hopf bifurcation occurs and the duopoly interaction converges into a limit cycle, and that further increasing the speed leads to quasi-periodicity and chaos. lotfi and sarkar (2016) study price competition in a duopoly with an arbitrary number of buyers with specific setting that captures a secondary spectrum access network, a non-neutral internet, or a micro-grid network in which unused spectrum bands, resources of isps, and excess power units constitute the respective commodities of sale. they identify a set of necessary and sufficient properties for the nash equilibrium. by considering market forces affecting switching costs, villas-boas (2015) investigates the effects of firms being forward-looking, consumers being forward-looking, degree of stability of consumer preferences, and market time horizon, respectively. by analyzing a game theoretic model of duopoly competition, xin and choudhary (2019) show, among other results, that it implementation can fail and such failure creates a possibility of cost-based differentiation and mitigates competition, while a higher probability of implementation failure can lead to lower investment risks and higher expected profits. through considering both competitive interactions in general, and the potential price effects of a merger between the two largest players in particular by using a unique route-level price data set of the recently deregulated german interurban bus industry, dürr et al (2016) find that route-level average prices depend not only on the number of competitors, but also on the composition of firms operating on a particular route. as motivated by real-life practices, huang et al (2019) interaction between incumbent and modern companies 97 copyright ©2019 assa adv. in systems science and appl. (2019) investigate in their duopoly competition setting a government’s optimal subsidies for energyefficient products in a market with two competing firms that differ in their production costs and compete on their product prices and product energy efficiencies. these authors find that very low subsidy might make the low-cost firm produce an uncertified product while the highcost firm a certified product, and that minimizing the average energy consumption helps to sustain product variety/duopoly outcome. in particular, the differentiated duopoly models, such as those mentioned above or in (singh & vives, 1984; zanchettin, 2006), could be potentially useful for our study here, considering the fact that this work addresses issues related to two firms selling differentiated products to a representative consumer. however, none of these prior models actually fits our needs due to the fact that these models violate some or all of the following problems. first, these models start with the optimization of the representative customer’s utility function. based on the knowledge of behavioral economics, this very practice is extremely questionable if we aim at producing practically applicable results (minton & kahle, 2013). in other words, customers are predictably irrational in ways that defy economic theory developed on optimizing utility functions. second, the representative customer’s utility function used in these models is of a particularly chosen type in order to produce the desired outcomes. for example, to produce theoretically ‘nice’ results, particular quadratic utility functions are used in order to generate linear demand functions. in fact, when studying real-life events and processes, linearity only stands for (very) local, (extremely) special cases with nonlinearity being the norm of interactions in the universe (lin & ouyang, 2010). third, the cost function is assumed, see, for example, (zanchettin, 2006), to be linear in the quantity of the product produced with constant marginal cost. this is simply untrue in real life especially in the economy of information age, for the reason why please go to proposition a.1 in appendix. and fourth, either the price competition or the quantity competition is assumed to be linear except a few recent works. as a matter of fact, as soon as such basic setup is taken to be linear, all the practically useful conclusions of realistic nonlinear interactions are assumed away. for details, see (liu, 2013; forrest & ouyang, 2010). considering all the listed and unlisted issues with the previously established models for duopoly competition, for the purpose of producing practically useful conclusions, we have no choice but develop a different and straightforward game-theoretical model based on a few very realistic and intuitive assumptions. and instead of optimizing the representative customer’s utility function, we maximize companies’ profits, which is most likely the practice of relevant companies when they try to decide their courses of action. so, comparing with the literature, such as those listed above, including particularly (christensen, suárez & utterback, 1998; porter, 1979; mcgrath, 2013), this work enriches the relevant knowledge at the height of theoretical abstraction with a much wider range of real-life applicability. and beyond addressing the question posed at the previous section, the general question this paper attempts to address is to show that other than inductive reasoning, deductive reasoning should be employed to produce scientifically sound theories and conclusions. here, inductive reasoning is the exclusively used logic of thinking in the literature in areas related to this work, where anecdotes and data mining, as well as models developed on particular and/or unrealistic assumptions, are employed to draw general conclusions. however, such conclusions are known in science to be generally not reliable. to this end, this paper establishes a theoretical model for how market competition should develop and evolve by using game theory without imposing any unrealistic condition and then provides scientifically sound managerial suggestions. as a side contribution, this work is the first to show that the literature of duopoly competition urgently needs to be enriched through developing new models that are beyond those established under unrealistic conditions – such as linear price and quantity interactions, linear demand and cost functions, etc. – and processes – such as optimizing customers’ utilities. 98 j. yl forrest , d. d. galbraith, y. liu, r. pla-lopez copyright ©2019 assa adv. in systems science and appl. (2019) another important note, appropriate at this junction, is the question of whether or not the approach and setup employed in this paper are relevant to today’s world of business. in particular, this paper looks at two competing companies that produce products or services that are similar but can be differentiated from each other. it is the differences of the products that allow the companies to compete for a similar customer base. examples of such setup are seemingly endless in the business world. for example, if a person needs a haircut, there are likely several salons in his/her area that could satisfy the need. they are all quite similar in terms of service and quality. so, each of the salons’ business goals is to entice the person to use its business instead of those of its competitors. now, the situation of competition can be treated both theoretically and practically as one salon competes against the aggregate of the rest, which falls within the setup of our model. in this example, hair salons can be seen as dummy entities that can be specified as health insurance companies, video streaming services, etc. 3. game-theroetical modeling this section describes the base game-theoretic models on which we derive our conclusions. 3.1. competition without involving customers consider two companies, named a and b, under two different business models. assume that a is a successful, old fashioned company that enjoys a full line of well received products based on its time-honored sustainable competitive advantage developed over a long period of time. for the sake of convenience of communication, let us assume that company a produces only one successful product. on the other hand, although company b also enjoys its traditional success with its unique product(s), as before, assume this company produces only one product, its leadership adopts the philosophy of transient competitive advantages. that is, company b is marketand technology-driven (day, 1994) and fully ready to embrace next fleeting advantage (kaharuddin, et al., 2017). so, the management of company b considers the question of whether or not to design and produce a new product that improves the functionality, user friendliness, and the features of the two products produced respectively by a and b. that is, the new product will be better and more advanced than any of the existing products of a and b. if it decides “yes, it will design and produce the imagined hybrid product,” then the production of the company’s old product will be terminated, and all its loyal customers will move on to enjoy the hybrid (better) product at the same price as that of the retired product. at the same time, a portion of company a’ customers will switch to buy this new product, which, in this case, would be considered cheaper and better than the existing product of a due to the enhanced functionality, user friendliness, and combined features. (note: our setup here is different from those studies touched on in the afore-mentioned literature of the previous section on how firms profit from short-term and long-term opportunities. in particular, the time variable does not enter our setup.) so, we have the two-player game in table 3.1, where 𝑆𝐴 stands for the total sales revenue of company a without company b’s invasion of a’s territory, 𝑆𝐵 the sales revenue of company b generated from the market share of company a by producing the imagined hybrid product, c the cost of company a for it to take actions, such as raising the barrier of entry, in order to deter the invasion of company b, 𝑅0 (respectively, 𝑅1 ) the cost of company b, including risk and opportunity costs, for designing and producing the hybrid product without (respectively, with) company a taking defensive actions. interaction between incumbent and modern companies 99 copyright ©2019 assa adv. in systems science and appl. (2019) table 3.1 the payoffs of the 2 × 2 pure strategy game company a company b design/produce don’t design/produce take defense 𝑆𝐴 – 𝐶 – 𝑆𝐵, 𝑆𝐵 – 𝑅1 𝑆𝐴 – 𝐶, 0 don’t take defense 𝑆𝐴 – 𝑆𝐵, 𝑆𝐵 – 𝑅0 𝑆𝐴, 0 in the setup of this game, the assumptions need to be read with generality in order to make the results of this work practically useful. in particular, what is assumed represents a momentary ‘camera’ snapshot of the interaction between companies a and b with the time variable frozen. in other words, if time is considered, with the given assumptions company a could still be monitoring what its competitor b is doing and trying to expand its existing advantage by identifying and investing in resources and capabilities that strengthen its current market position and potentially open up future opportunities. it could still be systematically using these strategic resources and capabilities to respond to the competitive pressures created by company b. that is, what is assumed does not mean that the design and production of the product of a are static and never changing over time; instead, it means that at the particular time moment of the ‘camera’ snapshot of the model company a’s identification of what resources and capabilities to invest in is hindered by its existing culture, philosophy and value system. for example, historically the rivalry between montgomery ward and sears had been in a game situation as described here at almost any chosen moment of time since the late 1800s. when sears introduced hybrid methods of sales and innovative methods of managerial operations, montgomery ward monitored and analyzed what sears was doing and planning on doing, and decided to develop itself and strengthen and improve its competitive advantages in a totally different direction. as a matter of fact, the very idea beyond what sears was doing came from one of the senior leaders of montgomery ward and was rejected as not an acceptable strategy by the company’s leadership (sobel, 1999). another good example is the rivalry between the eastman kodak and fuji films. the former surely identified and invested in resources and capabilities while monitored and took actions against what fuji films was doing in order to fence off the competition of the latter. however, its culture, philosophy and value system led the eastman kodak to a wrong direction (mcgrath, 2013). on the other hand, by the philosophy of transient competitive advantages, we mean that the leadership of company b, as well as the company’s culture and value system, is more in sync with the technological changes and desires of consumers than a so that it knows how to design its hybrid product to excite consumers. a good example that illustrates the point here is the evolving designs and productions of apple and lenovo laptop computers – one continuously improves its functionality and user friendliness, such as the appearance, while the other keeps its look the same. what is just discussed here simply means that when a company decides to design and produce a hybrid product, in real-life there are still many different ways to do it. in general, when a company is in sync with the technological changes and desires of consumers, its hybrid product will more likely excite the market than the hybrid products of other companies. once again, this end is very well demonstrated by the case of kodak vs fuji photo. a second point of notice regarding this very simply game model is that company a, although it is described as one firm, really stands for the aggregate of all incumbent firms of the marketplace that is in a state of mutual forbearance. in other words, incumbent firms mitigate rivalry by dividing markets in proportion to firm strength (bernheim & whinston, 1990). they cede dominance to their stronger competitors in those market segments where they are less efficient, while in exchange the latter do the same in segments where the former are more efficient (li & greenwood, 2004). the firms’ codependence gradually motivates them to de-escalate rivalry (yu & cannella, 2012). eventually, the rates of entry and exit in the market decrease (fuentelsaz & gómez, 2006), and interfirm hostility declines (haveman & 100 j. yl forrest , d. d. galbraith, y. liu, r. pla-lopez copyright ©2019 assa adv. in systems science and appl. (2019) nonnemaker, 2000). on the other hand, company b also represents the aggregate of all firms, either incumbent or not, which always think about designing and introducing newer and better products or services. a third point of notice is that it might seem ‘logical’ that if the additional profits of b come from the loss of a’s market share, then company a has an incentive to spend an amount equal to this potential loss to create entry barriers; so in equilibrium a would always spend an amount equal to 𝑆𝐵 to eliminate any incentive for b to introduce a new hybrid product. as a matter of fact, this seemingly logical feeling is incorrect. first, in practice the magnitude of 𝑆𝐵 cannot be estimated in advance or at the chosen moment of time the model is developed and is changing over time depending on how much the hybrid product of b excites the market. second, although 𝑆𝐵 could potentially turn out to be zero or very close to zero, the theorem established in (forrest, amatucci & markman, 2017) shows that company b still has to introduce its hybrid product to just maintain its current customer base. because of these reasons, the amount c company a spends on building its entry barriers is not a function in 𝑆𝐵, which is unknown at the moment of time when the model – a snapshot of the interaction between a and b – is fastened, and fluctuates over time. as a matter of fact, a theorem, established by forrest, buttermore and wajda (2017), shows that 𝑆𝐵 (less cost) could potentially be greater than 𝑆𝐴 (less cost). so, if the afore-mentioned ‘logic’ holds true, then company a has to simply close its door down because it has to spend all of its revenue and some additional capital on building the entry barrier. 3.2. competition involving customers in real life, no matter how companies a and b compete with each other, the ultimate fortune and fate of the companies are really determined by the consumers. so, in this second situation, let us include a fictitious customer in the interplay of the competition of the companies. in this case, company a takes an initiative to offer a product in the market, and company b correspondingly provides its improved hybrid product, while the customer takes the final action of either to purchase or not to purchase. the customer decides which product to buy and significantly influences the operations of these companies in the market. as a practitioner of the philosophy of transient competitive advantages, company b observes the initiatives of company a and then acts accordingly, while as a believer of sustainable competitive advantages, company a continues to identify and invest in its set of resources and capabilities that will not only maintain its current market position but also extrapolate it into the future by exploring how these resource and capabilities could be bettered in order to respond to competitive pressures of the market. in other words, company b follows the vibration of the market and the changing desires of the consumers by either rejuvenating its existing advantages or establishing new ones through absorbing those of its competitors, while company a continues its improvement and expansion of its proven advantages, believing that continuously improvement will strengthen and expand its market success. (at this junction, montgomery ward and kodak were good examples of company a described here.) so, we assume that although the customer is from the customer base of company a, she would try the new product designed and produced by company b if such a product is available. in other words, initiatives of company a are closely monitored and followed by company b, while the customer follows the actions of both companies and would try out the new product, as exactly what happens in real life. so, company a has to increase the consumer confidence on its product while deciding on whether or not to take defensive actions to deter company b’s possible invasion of its territory. that is, we have established the following game theory modeling: company a chooses either to take defensive actions (with cost c) or not to take any defensive action (with cost 0); company b elects between design/produce hybrid product (with fixed cost d) and not design/produce the imagined hybrid product (with cost 0). if company interaction between incumbent and modern companies 101 copyright ©2019 assa adv. in systems science and appl. (2019) a takes defensive actions to deter company b’s following, assume that company b will spend additional 𝛼𝐶 to design/produce its hybrid product, where 0 < 𝛼 < 1. this end is due to the fact that to follow up with company a’s defensive actions, company b has to push its organization’s knowledge envelope outward in order to incorporate unknown technologies in its effort of introducing the imagined hybrid product (harrigan & diguardo, 2016). ultimately the customer decides to purchase or no to purchase. if she purchases company a’s product, her utility is (−𝑢), because after purchasing the product she feels being exploited by a product that is not as innovative as what the market is expected to provide. in other words, instead of felling happy at least slightly from the consumption, she actually feels bad from the purchase. such situations occur in real life quite frequently and almost surely when the market is controlled by a monopolist. if she purchases company b’s hybrid product, her utility is 𝑢, because company b’s hybrid innovatively reflects the features of both company a’s product and the old product of company b and more. when the customer decides to purchase, if there is no hybrid product available in the market, company a earns revenue 𝑆𝐴 . in this case, although the customer does not feel good from the purchase, she does not really have a choice. however, if there is hybrid product in the market, then company b grabs revenue 𝑆𝐵 from company a by selling its hybrid to the customer. so, it follows that 𝑆𝐴 ≥ 𝑆𝐵 . when the customer decides not to purchase, the revenues of company a and b will be both 0. table 3.2 payoffs of customer and company b when company a takes defense customer company b design/produce (p) don’t design/produce (1 – p) buy () 𝑢, [(𝑆𝐵 + 𝑆𝐵−)– 𝐷 − 𝛼𝐶] −𝑢, 0 don’t buy (1 − ) 0, [(𝑆𝐵−) – 𝐷 − 𝛼𝐶] 0, 0 table 3.3 payoffs of customer and company b when company a does not take any defense customer company b design/produce (q) don’t design/produce (1 – q) buy (𝜀) 𝑢, [(𝑆𝐵 + 𝑆𝐵−)– 𝐷] −𝑢, 0 don’t buy (1 − 𝜀) 0, [(𝑆𝐵−) – 𝐷] 0, 0 so, the payoffs of company b and the customer are respectively given in table 3.2 for the case when company a takes defensive actions and in table 3.3 for the case when company a does not take any defensive action, where 𝑆𝐵− represents the revenue company b generates from the market of the hybrid product beyond the market share of company a and its expected revenue from its old product. as the conclusion of this section on game-theoretic modeling, we like to see to what practical scenarios this work can be potentially employed in order to produce tangible economic benefits. indeed, our modeling only focuses theoretically on the interaction between two companies a and b, characterized by that a’s success is based on its time-honored sustainable competitive advantage developed over a long period of time, while b is market-, technology-driven, and embraces next fleeting advantages. in practically applications, the results of this work can be beautifully applied to analyzing the strategic management and planning of any focal firm that competes against a host of other firms within a market, as long as the focal firm desires to introduce its hybrid product that is better than all the substitutable but differentiated products offered by the other firms. in particular, this focal firm will be seen as company b in our game-theoretic models, while the aggregate of all other firms as company a. 102 j. yl forrest , d. d. galbraith, y. liu, r. pla-lopez copyright ©2019 assa adv. in systems science and appl. (2019) 4. results in this section, we look at the competition of companies a and b in two cases as described in the previous section: (i) two-player competitions without involving customers; and (ii) threeplayer competition involving a fictitious customer. 4.1. two-player competitions based on the setup given in the previous section without involving any customer, we have the following implications. (1) for both companies, each unit of their product produced is successfully sold. (2) when the sales revenue of company b satisfies 𝑆𝐵 > 0, the sales revenue of company a is 𝑆𝐴 − 𝑆𝐵. that is, 𝑆𝐵 stands for the portion of sales revenue company b takes away from company a, assuming that that is the total increased sale of company b. (3) when company a takes actions to deter company b from invading its territory, the total cost for company b to design and produce the hybrid product goes higher. that is, 𝑅1 > 𝑅0. in the rest of this paper, we assume that all players establish their best responses by playing the nash equilibrium through pure self-analyses. in the following, let us analyze this game in different cases. case 1: 𝑆𝐵 ≤ 𝑅0, that is, the increased sales revenue of company b is less than the sum of the cost of designing and producing the hybrid product and the risk and opportunity cost. so, it is logical that company b does not pursue after the strategy of the hybrid product, a losing proposition. and, company a does not need to take any action to deter company b from invading its territory. indeed, in this case, (don’t take defense, don’t design/product) is the nash equilibrium, although in real life such a situation does not really exist. it is because at least some of the features of company a’s product can be combined with those of company b’s product without breaking the budget by adopting appropriate technologies. case 2: 𝑅0 < 𝑆𝐵 < 𝑅1 and 𝑆𝐵 < 𝐶. so, (don’t take defense, design/produce) is the nash equilibrium. in other words, if company a takes defensive actions against the potential entry of company b into its territory, a has to spend more than its loss in terms of revenue. consequently, due to the high costs, company a does not take any defensive action; and company b will enter the territory of company a. next, if 𝑅0 < 𝑆𝐵 < 𝑅1 and 𝑆𝐵 > 𝐶, then company a would decide which strategy to use to maximize its revenue, assuming this company chooses its strategy first before company b. in this case, a will do better by taking defense and by expelling b from the market, because the strategy profile (take defense, don’t design/produce) will give a the maximum revenue. that is, in the subgame for a to take defensive actions, (take defense, don’t design/produce) is the nash equilibrium. table 4.1 a 2 × 2 mixed strategy game company a company b design/produce don’t design/produce take defense (𝑝) 𝑆𝐴 – 𝐶 – 𝑆𝐵, 𝑆𝐵 – 𝑅1 𝑆𝐴 – 𝐶, 0 don’t take defense (1 − 𝑝) 𝑆𝐴 – 𝑆𝐵, 𝑆𝐵 – 𝑅0 𝑆𝐴, 0 to make the decision making process more dynamic, instead of using pure strategies, let us assume that company a assigns probability p for the event of itself taking defensive actions. it is because at the time moment when the model is established, company b does not really know whether or not company a will take defensive actions. so, we have the payoff matrix in table 4.1. analyzing this game leads to the following practically useful results: interaction between incumbent and modern companies 103 copyright ©2019 assa adv. in systems science and appl. (2019) proposition 4.1: assume that the revenue of company b from the territory of a is greater than its costs when a does not take defense, while less than the costs when a does take defense, and greater than the defense costs of a and that all other conditions are held constant. then the probability for company a to take defensive actions and the sales revenue of company b generated from the market share of company a by producing its hybrid product are directionally proportional to each other. what this result means practically is that the more company a expects to lose its revenue, the higher probability company a would take defensive actions against company b. and on the other hand, the higher probability company a would take defensive actions, the more revenue company b is expected to earn from the share of company a. proposition 4.2: assume that the revenue of company b from the territory of a is greater than its costs when a does not take defense, while less than the costs when a does take defense, and greater than the defense costs of a and that all other conditions are held constant. then the probability for company a to take defensive actions and the cost for company b to design and produce its hybrid product with company a taking defensive actions are inversely proportional to each other. proof of propositions 2.1 and 2.2. from table 4.1, for company b, its indifference condition is 𝑝(𝑆𝐵 – 𝑅1) + (1 − 𝑝)(𝑆𝐵 – 𝑅0) = 0, from which we have 𝑝 = 𝑆𝐵 – 𝑅0 𝑅1– 𝑅0 , where 𝑅1 > 𝑆𝐵 > 𝑅0 is assumed for the analysis here. and when 𝑅1 > 0 , 𝑅0 will be a nonexistent value. in this case, we assume 𝑅0 = 0. □ in practice, proposition 4.2 says that the higher the cost for company b to design and produce its hybrid product, the smaller probability for company a to take any defensive actions. and on the other hand, the smaller probability for company a to take any defensive actions, the higher cost will be for company b to successfully design and produce its hybrid product. the defensive actions company a could take might include ways to tighten the control of its sales network, additional advertisement campaigns, etc. so, the cost for company a to design and produce and especially sales of its product will increase. as a result, the chance for company b to move ahead to design and produce its imagined hybrid product will be lower. however, a lower degree for company a to take defensive actions against the potential invasion of company b, the lower level of costs will be for company b to design and produce its hybrid product. therefore, each initiative company a takes to minimize the chance of potential invasion will create a situation with increased costs for company b to crack the entry barrier and to improve its design of its hybrid product. hence, in practice, company a has to continuously improve its product in order to retain and even grow its market share. other the other hand, if company a does not implement strong barriers for entry, then various hybrid products could more easily appear in the marketplace and will sooner or later push company a out of the competition (this result follows from corollary 1 in (forrest & anderson, 2017), 104 j. yl forrest , d. d. galbraith, y. liu, r. pla-lopez copyright ©2019 assa adv. in systems science and appl. (2019) which says that: when the competition of a market grows with an increasing number of firms entering the market, the base of loyal customers for each incumbent firm will gradually diminish). more specifically, when the probability for company a to take defensive actions is 𝑝 = 𝑆𝐵 𝑅1⁄ , from the assumption that 𝑅1 > 𝑆𝐵 and 𝑆𝐵 > 𝐶, it means that a will take defense and b will not design and produce its imagined hybrid product. in other words, company b will have to consider its design and production of a different hybrid product with the output of another company. in such a case, the expected revenue of company a will be maximized at 𝑆𝐴 − 𝑝𝐶. next, let us consider the case that when deciding on designing and producing its hybrid product, other than the possibility of taking a portion of company a’s revenue, company b also knows the fact that its new product will appeal to other customers who were not within the individual customer bases of company a and b. for example, in 1979 although tom whiteley, the head of kodak’s emulsion research division, observed presentations of overhead projector slides generated on xerox alto, an early version of a personal computer, he somehow did not see the future of the new technology. instead, he was very much engulfed in how well-entrenched the kodak technology was. so, kodak continued to exploit and strengthen its long-standing advantages in film, while fuji photo moved into a future without film (mcgrath, 2013). here, fuji could be identified with our company b – other than potentially taking market shares of company a (kodak), it also competes within a new market where company a is absent. to study how the incumbent firms in this new market would raise entry barriers to keep new comers, especially company b in our case, from entering, assume that the market the imagined hybrid product will enter competitively is occupied by m incumbent firms, m = 1, 2, … these firms provide customers with mutually substitutable products; and each of the firms enjoys the backing of its loyal customers who only purchase the product of their firms provided that the price is no more than their reservation value, which is set to 1. assume that these firms use the strategy of raising the entry barriers for company b by competing over those customers who switch from the product of one firm to that of another firm with adjustable prices. to reflect the fact that these incumbent firms work together to prevent company b from entering, assume that these firms are well aware of the pricing strategies of each other, and that these firms have established their best responses by playing the nash equilibrium through pure selfanalyses. then, we have the following result, motivated by forrest, zhao and shao (2018). proposition 4.3: in the nash equilibrium, the following two statements are equivalent: 1) company b can profitably enter the pre-described market, as a competitor of the incumbent firms; and 2) the size of the market segment of switchers is greater than zero. proof. let 𝛼 stand for the portion of the market of the loyal customers of the m firms and 𝛽 = 1 − 𝛼 be the market segment of switchers, where 0 ≤ 𝛼, 𝛽 ≤ 1. for the sake of convenience of communication, assume that the constant marginal costs of the incumbent firms and company b are set to zero without loss of generality – so in the rest of this proof, revenues are the same as profits. and consider the aggregate of the incumbent firms as one firm, because these m firms are in a state of mutual forbearance (bernheim & whinston, 1990). so, the market share of this aggregate firm is 𝛼 so that 𝛽 = 1 − 𝛼 represents the market size of switchers who base their purchase decision on which price is lower. (⇒) suppose that company b enters profitably into the oligopoly market of m firms. then, the size of the market switchers or customer surplus must satisfy 𝛽 = 1 − 𝛼 > 0. interaction between incumbent and modern companies 105 copyright ©2019 assa adv. in systems science and appl. (2019) (⇐) assume that the customer surplus satisfies 𝛽 = 1 − 𝛼 > 0. let 𝛼0 be a real number satisfying 𝛽 = 1 − 𝛼 > 𝛼0 > 0 and 𝛼 = ℓ𝛼0 , where ℓ is a large natural number, indicating that the market has been largely taken by the incumbent firms. imagine that the aggregate firm is divided into ℓ many identical “firms”, named i, i = 1, 2, …, ℓ, each of which provides customers with identical products and enjoys the market share 𝛼0 = 𝛼/ℓ of loyal customers. these imaginary firms compete over the switchers with adjustable prices. because these imaginary firms are really equal partitions of the same aggregate firm, they have the same constant marginal cost, which is set to zero without loss of generality, the managements of these firms are fully aware of the pricing strategies used by all the firms (because the firms are managed by the same administrative unit), and they establish their best, identical responses by playing the nash equilibrium through their unified selfanalyses. these ℓ imaginary firms do not have any symmetric pure strategy nash equilibrium. because the reservation price of the loyal customers is set to be 1, all the prices (or pure strategies) are values in the interval [0,1]. (for the setup here, there is no need to consider asymmetric pure strategy nash equilibrium, because all these imaginary firms take identical actions). in fact, for any symmetric pure strategy portfolio (𝑥1, 𝑥2, … , 𝑥ℓ), where 𝑥𝑖 = 𝑥𝑗, for i, j = 1, 2, …, ℓ, a randomly chosen firm j (∈ {1, 2, … , ℓ}) can slightly lower its price from 𝑥𝑗 to 𝑥𝑗 ′ to produce additional profits for all the firms as long as 𝑥𝑗 ′𝛽 > (𝑥𝑗 − 𝑥𝑗 ′)𝛼, which is possible to do by adjusting 𝑥𝑗 ′ sufficiently close to 𝑥𝑗 . so, (𝑥1, 𝑥2, … , 𝑥ℓ) is not an nash equilibrium. even so, (forrest, buttermore, & wajda, 2017) shows that these ℓ firms do have a symmetric mixed-strategy nash equilibrium. for the rest of this proof, it suffices to show that company b will be expected to profit by entering this market through uniformly randomizing its price strategy over the interval [0,1]. let 𝐹(𝑃) be the price distribution of firm j, one of the imaginary firms of the aggregate firm. the aggregate firm or equivalently each of the ℓ imaginary forms sets its price after taking into account the price of company b and those of all other imaginary firms. hence, the profits for firm j from its loyal customers is 𝛼0𝑃 and those from its share of the switchers is 𝛽𝑃(1 − 𝑃) ∏ [1 − 𝐹(𝑃)]ℓ 𝑖≠𝑗 = 𝛽𝑃(1 − 𝑃)[1 − 𝐹(𝑃)]ℓ−1, where (1 − 𝑃) stands for the portion of the switchers not taken by company b, and [1 − 𝐹(𝑃)] the portion of the switchers not taken by one of the other imaginary and identical firms. hence, the profits π firm j generates when the firm sells its product at price p are 𝛼0𝑃 + 𝛽𝑃(1 − 𝑃)[1 − 𝐹(𝑃)]ℓ−1 and the objective function of firm j is 𝑚𝑎𝑥𝐹(𝑃)𝐸(π) = ∫ {𝛼0𝑃 + 𝛽𝑃(1 − 𝑃)[1 − 𝐹(𝑃)]ℓ−1}𝑑𝐹(𝑃) +∞ −∞ = ∫{𝛼0𝑃 + 𝛽𝑃(1 − 𝑃)[1 − 𝐹(𝑃)]ℓ−1}𝑑𝐹(𝑃) 1 0 where 𝐸(π) stands for firm j’s expected profits for all possible prices, and the objective for firm j is to maximize its expected profits by choosing its price distribution 𝐹(𝑃). the reason why the upper and lower limits of the integral are changed respectively from +∞ and −∞ to 1 and 0 is because when p < 0 or when p > 1, the profits are zero. the equilibrium indifference condition of firm j is      111 0 1 0     pfppp (4.1) 106 j. yl forrest , d. d. galbraith, y. liu, r. pla-lopez copyright ©2019 assa adv. in systems science and appl. (2019) so, for the ℓ imaginary firms, solving equation (4.1) leads to their symmetric equilibrium pricing strategy as follows:   1 1 01          p pf   (4.2) from 𝛽 > 𝛼0 , it follows that 𝛼0 𝛽⁄ < 1 . so, for any price p, satisfying 1 ≥ 𝑃 ≥ 𝛼0 𝛽⁄ , equation (4.2) is a well-defined probability distribution. this end implies that for the ℓ imaginary firms, or equivalently, the aggregate firm, the lowest allowed price is 𝛼0 𝛽⁄ . from 𝑙𝑖𝑚𝑃→1−𝐹(𝑃) = 1 − (𝛼0 𝛽⁄ )1 (ℓ −1)⁄ ≠ 𝐹(1) = 1 , it follows that the cumulative price distribution function 𝐹(𝑃) has a jump discontinuity at the reservation value 𝑃 = 1, where the amount of jump is (𝛼0 𝛽⁄ )1 (ℓ −1)⁄ . that is, 𝐹(𝑃) has a mass point of size (𝛼0 𝛽⁄ )1 (ℓ −1)⁄ at the reservation price 𝑃 = 1. so, the expected profits of company b are the following:     dppfppdpe       / / 0 0 0 )(1  (4.3)    1/ 0 1 / / 0 0 0 )(1                 dppfppdp (4.4) where the first term in the right-hand side of equation (4.3) stands for the expected profits of company b when it charges the lowest price in the marketplace and captures the entire segment of the switchers, and the second term is company b’s expected profits when it is in direct competition with the ℓ incumbent firms. it can be readily seen from equation (4.4) that the expected profits 𝐸(π) of company b is greater than zero. so, if the customer surplus satisfies 𝛽 = 1 − 𝛼 > 0 , company b can profitably enter the market to compete with the incumbent firms. □ the assumption of proposition 4.3 generally means that the technology involved and the relevant business operations of the new market have been standardized. from the assumption that company b’s leadership adopts the philosophy of transient competitive advantages and that it is r&d-driven and fully ready to embrace next fleeting advantage, it is reasonable to assume that company b has come up with a more efficient technology and/or operation system that can greatly reduce the overall business expenditure. for example, christensen, et al., (1998) confirm empirically that technological and market strategies of a new entrant are highly interrelated and that their joint effect plays an important role in a firm's probability of survival. and friedman (2005) maintains that technology is a critical factor and force in shaping and flattening the world and balancing the global field. so, each competitive advantage is fleeting, since technology continues to change the way how people do business as it becomes cheaper, more available, lighter, smaller, etc. what the proof of proposition 4.3 says is that it will be difficult and even impossible for the incumbent firms to stop company b from entering because to counter the pricing strategy of the incumbents company b randomizes its selling price between the unit cost and the reservation price the incumbents charge their loyal consumers. at the same time, this proof also implies that company b could potentially take the entire segment of switchers, which in turn means that if this segment of the market is larger than any of the royal-customer bases of the incumbents, then company b could make more profits than any of the incumbents. and from proposition 4.3, it follows that company b expects to expand its market share by taking first a portion of company a’s customers and second a portion of the greater market of the hybrid product. in other words, it is practically possible that the additional revenue interaction between incumbent and modern companies 107 copyright ©2019 assa adv. in systems science and appl. (2019) company b expects to generate from its hybrid product will be more than 𝑆𝐵, the sales revenue of b from the market share of company a, by producing the imagined hybrid product. a good example here is fuji photo vs kodak. let 𝑆𝐵− represent the revenue company b generates from the market beyond the market share of company a, as described in proposition 4.3. then, we naturally have the following case. case 3: 𝑆𝐵 + 𝑆𝐵− ≥ max {𝑅1, 𝐶}. that is, the expanded sales revenue of company b from producing the imagined hybrid product is at least the larger of company b’s cost (𝑅1) of designing and producing its hybrid product and the cost (𝐶) of company a for it to take defensive actions. if, additionally, 𝑆𝐵 > 𝐶 , then company a takes defense and the nash equilibrium is (take defense, design/produce). if also 𝑆𝐵 + 𝑆𝐵− = 𝑅1 , then the nash equilibrium is (take defense, don’t design/produce). on the other hand, if 𝑆𝐵 + 𝑆𝐵− > 𝑅1 and 𝑆𝐵 > 𝐶, then (don’t take defense, design/produce) is the pure strategy nash equilibrium with the corresponding payoffs (𝑆𝐴 – 𝑆𝐵 , 𝑆𝐵 – 𝑅0), see table 3.1 for details. in other words, the market share of company a erodes and is gradually taken by company b’s hybrid product while company a cannot do anything about it. that is, company a will soon disappear from existence when more such companies as b enter the territory of a. by summarizing what is established above, we have the following result: proposition 4.4: assume that company a exploits its sustainable competitive advantages, while company b looks forward to adopting transient competitive advantages (mcgrath, 2013), as described above. then the following hold true: 1. if expansion is expected for company b to generate profits, which are more than the cost for company a to take defensive actions, then company a would move first and take defensive actions against company b; 2. if company b’s decision of whether or not to design and produce its hybrid product is based on the existence of an expanding market beyond that of company a, then company b will introduce its hybrid product. the second result comes from proposition 4.3 and theorem 5 in (forrest, buttermore & wajda, 2017), where the expanding market means that the segment of switchers exists in this market, and is also expanding with the overall increasing market scale. 4.2. three-player competitions in this subsection, we focus on the competition between the companies while a representative customer is involved. proposition 4.5: assume that the additional revenue of company b from beyond its original territory is less than its cost of designing and producing its hybrid product. then the pure strategy nash equilibrium is reached when company a does not take defense, company b does not design/produce its hybrid product, and the customer does not make any purchase. in this nash equilibrium, none of the companies earns any revenue while the customer does not spend any money. proposition 4.6: assume that the additional revenue of company b from beyond its original territory is greater than its cost of designing and producing its hybrid product. then the pure strategy 108 j. yl forrest , d. d. galbraith, y. liu, r. pla-lopez copyright ©2019 assa adv. in systems science and appl. (2019) nash equilibrium is reached when company a does not take defense, company b designs and produces its hybrid product, and the customer makes purchase. in this nash equilibrium, company a collects no revenue, company b makes profits, while the customer enjoys her utility 𝑢. and, if, moreover, the additional revenue of b from a’s market territory is greater than the cost for a to take defense, then when a moves first, the nash equilibrium is (take defense, design/produce). proof of propositions 3.1 and 3.2. this result follows from the payoffs of company a that are corresponding to the scenarios that company a either takes defensive actions or does not take any defensive actions (either table 3.2 or table 3.3). for details see tables 3.3 and 3.4, respectively. □ table 4.2 the payoffs of company a when it takes defensive actions customer company b design/produce (p) don’t design/produce (1 – p) buy () −𝐶 𝑆𝐴 − 𝐶 don’t buy (1 − ) −𝐶 −𝐶 table 4.3 the payoffs of company a when it does not take any defensive actions customer company b design/produce (q) don’t design/produce (1 – q) buy (𝜀) 0 𝑆𝐴 don’t buy (1 − 𝜀) 0 0 what proposition 4.5 indicates is that when the additional revenue company b expects to generate from its hybrid product is less than the total cost of designing and producing the hybrid product, then it is foolish for company b to design and produce its hybrid product. so, consequently, company a does not need to take any defensive actions; and if the customer decides to purchase, she has no choice but purchase company a’s product. on the other hand, what proposition 4.6 implies is that when the additional revenue company b expects to generate from its hybrid product is more than the total cost of designing and producing the hybrid product, then company b will move ahead with its design and production of the hybrid product; in this case, because any defensive action is useless, company a does not even attempt to take any action. however, considering the assumption that company a is a diehard believer and practitioner of the old school of sustainable competitive advantages, such as montgomery ward and kodak mentioned above, it in really life would most likely not passively watch for its base of customers to deteriorate without trying to save it and/or even trying to expand it. because of this reason, let us next consider mixed strategies under the condition of 𝑆𝐵 + 𝑆𝐵− > 𝐷 + 𝛼𝐶, that is, the additional revenue of company b from beyond its original territory is greater than the cost of designing and producing its hybrid product. in this case, we also assume that 𝑆𝐵− < 𝐷 + 𝛼𝐶 (that is, the revenue company b earns outside the market base of company a is less than b’s cost), because otherwise company b will definitely design and produce its hybrid product. assume that when company a takes defensive actions, the probability for company b to design/produce hybrid product is p, (because before making the decision of what to do, company b has to firstly sense, secondly seize and transform itself, and then acquire the critical capabilities for successful organizational adaptation (day & schoemaker, 2016)), and the probability for the customer to purchase is 𝜏. on the other hand, when company a does not take any defensive action, the probability for company b to design/produce hybrid product is q, and the probability for the customer to purchase 𝜀 . solving for mixed strategy nash equilibrium provides 𝑝 = 𝑞 = 1/2, and interaction between incumbent and modern companies 109 copyright ©2019 assa adv. in systems science and appl. (2019) b b s scd     (4.5) b b s sd   (4.6) proposition 4.7: if company b’s additional revenue from beyond its original territory is greater than its cost of designing and producing its hybrid product and its additional revenue from outside the market base of company a is smaller than the cost, and company a does not take any defensive actions to protect itself, then 1. the probability for the customer to purchase company a’s product is inversely proportional to company b’s activities of designing and producing its hybrid product; and 2. when the revenue company b earns from the market base of company a is equal to its fixed cost of designing and producing its hybrid product minus the revenue generated from beyond the customer base of company a, then the probability for the customer to purchase company b’s hybrid product is equal to 1. proposition 4.8: if company b’s additional revenue from beyond its original territory is greater than its cost of designing and producing the hybrid product and its additional revenue from outside the market base of company a is smaller than the cost, and company a does take defensive actions to protect itself, then 1. the higher the difference between the cost for company b to design and produce its hybrid product and the additional revenue generated from beyond the customer base of company a, the higher the probability for company a’s customer to purchase company b’s hybrid; and 2. if, additionally, the 𝛼-value is fixed, then the higher investment company a makes in its effort to raise the barriers for company b to invade its territory, the lower probability its customer will buy company b’s hybrid. this result means that the defensive actions company a takes actually work and can help slow down the deterioration of company a’s customer base. proof of propositions 3.3 and 3.4. to solve for the mixed strategy nash equilibrium, we solve the following indifference equations: 𝑝𝑢 + (1 − 𝑝)(−𝑢) = 0 𝑞𝑢 + (1 − 𝑞)(−𝑢) = 0 𝜏[(𝑆𝐵 + 𝑆𝐵−)– 𝐷 − 𝛼𝐶] + (1 − 𝜏)[(𝑆𝐵−) − 𝐷 − 𝛼𝐶] = 0 and 𝜀[(𝑆𝐵 + 𝑆𝐵−)– 𝐷] + (1 − 𝜀)[(𝑆𝐵−) − 𝐷] = 0 so, we have 𝑝 = 𝑞 = 1/2 and equations (4.5) and (4.6). now, proposition 4.7 follows from equation (4.6), while proposition 4.8 follows from equation (4.5). 110 j. yl forrest , d. d. galbraith, y. liu, r. pla-lopez copyright ©2019 assa adv. in systems science and appl. (2019) proof of proposition 4.9: let 𝜋𝐴 represent the expected revenue of company a. then we have        ))(1)(1(()11, cpcspcpscsp abadefensea   (4.7) and     abadefensetdona sqssq   1', (4.8) so, by substituting the values of 𝑝 = 𝑞 = 0.5, 𝜏 and 𝜀 from equations (4.5) and (4.6) into equations (4.7) and (4.8), we have     cscd s scds b b ba defensea         2 1 , (4.9) and  ba b b defensetdona ss s sd     2 2 1 ', (4.10) because 1 2 1 1 2 ,          b a b adefensea s s s s c   (4.11) and 𝛼 is a constant between 0 and 1, it follows that when 𝑆𝐴 < ( 1 𝛼 + 1 2 ) 𝑆𝐵 , we have 𝜕𝜋𝐴,defense 𝜕𝐶 < 0. that is, when 𝑆𝐴 < ( 1 𝛼 + 1 2 ) 𝑆𝐵, the expected revenue 𝜋𝐴,defense of company a becomes a decreasing function in variable 𝐶. in other words, as soon as the revenue company b generates from company a’s customer base is greater than company a’s revenue when a does not experience any challenge from company b divided by ( 1 𝛼 + 1 2 ), then company a can no longer exploit its sustainable competitive advantages by continuously raising its investment in raising the entry barrier in order to prevent company b from invading its territory. on the other hand, equation (4.10) indicates that when 𝑆𝐵− > 𝐷, the expected revenue 𝜋𝐴,don′t defend of company a will be a loss. in other words, if company a decides not to defend itself against company b’s challenge and the expected revenue company b generates from the market except company a’s territory is greater than the fixed cost of designing and producing the hybrid product, then company a’s expected revenue will be a loss. □ 4.3. cost-benefit analysis under the assumption that company b’s additional revenue from beyond its original territory is greater than its cost of designing and producing the hybrid product and its revenue from outside the market base of company a is smaller than its cost (or 𝑆𝐵 + 𝑆𝐵− > 𝐷 + 𝛼𝐶 and 𝑆𝐵− < 𝐷 + 𝛼𝐶 ), the previous discussion indicates that to slow down the deterioration of company a’ customer base the most significant decision is to raise the investment on building a costly barrier for company b’s entry into company a’s territory and to increase the customer retention. as part of the most important investment, company a could constantly make the features and functionalities of its product design better, which makes the design and production of company b’s hybrid difficult to follow up. for the comparison of the deterrence cost 𝐶 of company a and its revenue, we have: interaction between incumbent and modern companies 111 copyright ©2019 assa adv. in systems science and appl. (2019) proposition 4.9. assume that company b’s additional revenue from beyond its original territory is greater than its cost of designing and producing the hybrid product and its additional revenue from outside the market base of company a is smaller than the cost. then the following hold true: 1. if company a decides to defend its territory by raising entry barriers for company b, then it can successfully delay the deterioration of its territory as long as less than ( 1 𝛼 + 1 2 ) of its maximum expected revenue is taken by company b; and 2. if company a decides not to take any defensive actions against company b’s possible invasion through designing and producing hybrid product, then it can expect to make positive revenues. however, this positive revenue decreases with increasing revenue company b generates from outside company a’s territory. what proposition 4.9 indicates is that in order to continuously stay in business, company a has no choice except also adopt the philosophy of transient competitive advantages. that is, as pointed out by dobni and klassen (2015), when the company matures and becomes slow in growth, innovation is a key driver in maintaining the company’s market advantage. another fact the proof of proposition 4.9 reveals is that with the constant improvement of technology and growing convenient availability of information, the fixed cost 𝐷 and the cost coefficient 𝛼 for company b to design and produce its hybrid product are getting lower with time, while the cost 𝐶 for company a to raise the entry barrier is getting higher. so, the expected revenue of company a will be crashed by company b quickly unless company a also starts to improve its product by designing and producing hybrid product for the purpose of expanding its market share, as what company b tries to do throughout this presentation. however, tushman and o'reilly iii (1996) confirm this end as a real challenge for the managers of company a, just as the cases of montgomery ward and kodak as discussed previously. 5. some final words mcgrath (2013) demonstrates vividly by using anecdotes and data that failure to respond and adapt to the ever-evolving market conditions by not focusing on multiple or transient competitive advantages may place a firm in an unenviable deficit with a commensurate serious loss in market share and/or dominance. the competitive landscape will continue to fluctuate and change. organizations can no longer remain at the pinnacle of their respective industries through sustaining main competitive differentiators or advantages. the acceptance that any advantage will be short-lived will be critical in a continuous learning and innovative company. that is, the once sustainable competitive advantages have become transient in the modern world of business, although they were indeed seemed to be long-lasting in the past. so, theoretically, there is a need to show why any company that plans on continuously exploiting its competitive advantages for the long term will disappear from the market place or soon become irrelevant when more and more of its competitors are adopting and practicing the philosophy of transient competitive advantages. this paper employs a very intuitive, straightforward game-theoretic model to clearly show how unresponsively the company that plans on continuously reaping in its profits by exploiting its sustainable competitive advantages is losing its future to those that adopt and practice transient competitive advantages. to make the conclusion definite, this paper looks at the scenarios of both pure strategies and mixed strategies. other than avoiding the weaknesses, such as deriving unreliable conclusions from maximizing the representative customer’s utility 112 j. yl forrest , d. d. galbraith, y. liu, r. pla-lopez copyright ©2019 assa adv. in systems science and appl. (2019) function, developing specific results from particularly chosen utility functions, and creating beliefs from imposed unrealistic linearities, of the previously developed models – say, for example, those by singh and vives (1984) and zanchettin (2006) – for duopoly competition, this work establishes several thought provoking results, which can be practically applied, based on the mathematical formulation of the problem by providing the feasible solutions that incorporate the differences in actions between two companies, one of which exploits its sustainable competitive advantage, while the other rides the wave of market conditions. the most important conclusion this study produces is that in order to avoid becoming history, any company should adopt the philosophy of transient competitive advantages. because the model developed in this paper does not involve the time variable, all the established conclusions only describe aspects of the momentary situation of the competition between companies a and b. so, what is both theoretically and practically significant is to see what would follow after company a realizes what it needs to do at the next time moment in the competition with company b. another important issue this paper does not address is that in practical applications, how will a host of incumbent firms interact with each other when more than one of them are market-, technology-driven, and embrace next fleeting advantages. these and other relevant issues will be theoretical interesting and practically significant research topics for future works. the managerial implications of this study include, but not limited to, the following: 1. if the potential loss to an emerging competition is more than the cost of taking defensive actions, then the incumbent firm should proactively approach the forthcoming challenge by taking defense although the chance of success is uncertain; 2. if a new product has the potential to penetrate several markets simultaneously, although the expected success in each of these markets is limited initially, the product has a great promise for market success; 3. although a business expansion into several markets simultaneously seems to be only break-even in terms of cost-benefit analysis, the company should jump ahead to expand; 4. if less than 2/3 of its maximum expected revenue could be potentially taken by a competitor who has trouble entering any other market, then the incumbent firm can successfully delay the deterioration of its territory by taking defensive actions. based on what the literature shows (e.g., mcgrath, 2013 and references found there), this study is the first of its kind that employs a rigorous approach to address, either through confirming or disapproving, issues related to many overriding conclusions – drawn mostly on anecdotes, or data mining, or models developed on limiting or unrealistic conditions (for example, kaharuddin, et al., 2017; jindal, et al., 2016; purkayastha & sharma, 2016; zachary, et al., 2015; zanchettin, 2006; tushman & o'reilly iii, 1996; singh & vives, 1984). because of such an approach employed in this paper, we are able to provide insights into the practically significant question: how do companies of the philosophy of sustainable competitive advantages actually lose their shirts in the competition with companies that ride the current wave of transient advantages (e.g., bashir & verma, 2017; koller, 2016; purkayastha & sharma, 2016; chan & chan, 2010)? at the same time, the model developed in this paper surely enriches the game-theoretical literature of duopoly competition (e.g., pecora & sodini, 2018; villas-boas, 2015; dubiel-teleszyński, 2011; zhang, et al., 2009; zanchettin, 2006; agliari & puu, 2002; sice, et al., 2000) through avoiding some of the major weaknesses of the models previously established for studying either cournot competition or bertrand competition (for related references, see those listed in the literature review section on why existing models of duopoly do not fit our need of this research project). interaction between incumbent and modern companies 113 copyright ©2019 assa adv. in systems science and appl. (2019) appendix proposition a.1: the cost function of a company that produces consumer goods cannot be generally assumed to be linear in the quantity of the product produced and sold with constant marginal cost, as given in (zanchettin, 2006). figure a.1. the cost function as a matter of fact, in the framework of fast technological progress, the marginal cost cannot be constant. instead, the cost function in reality is a piecewise defined function in number 𝑛(𝑝𝑠) of units of product produced and sold, because when the employed technology is fixed, by slightly improving the managerial operation of the program – such as changing the efficiency of the promotion, making the usage of resources and technology more effective, etc. – without or only slightly increasing the cost, the production can be increased majorly. that is, the total cost function should be a piecewise defined function, as shown in figure a.1. here, within each interval [𝑛𝑗 , 𝑛𝑗+1], 𝑗 = 1, 2, …, the cost is relatively flat while the productivity rises through improving efficiency in terms of the usage of resources, deployment of technology, managerial strategies, etc. the decreasing lengths of the intervals [𝑛𝑗 , 𝑛𝑗+1] represent the fact that with time technology needs to be updated more frequently, where at each location 𝑛𝑗 , 𝑖 = 1, 2, …, a large expenditure occurs due to investments in capital assets, human resources, etc. as an empirical illustration for the previous general argument, one can simply look at the upgrading of office computers. when new computers replace the existing ones, the productivity subsequently increases, while the cost for the increased productivity jumps drastically or nonlinearly due to the one-time expenditure on the new equipment. now, with time people will become more and more efficient regarding how they operate the new equipment. such gradual increasing efficiency can help improve productivity without additional expenditure. □ references 1. agliari, a. & puu, t. 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(2014) analysis of a nonlinear mixed cournot game with boundedly rational players, chaos, solitons and fractals, 59, 82 – 88. adv syst sci appl 2017; 3:49–57 published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/500 haar wavelets and subdivision algorithms on the plane tatyana zaitseva1∗ 1department of mechanics and mathematics, moscow state university, moscow, russia abstract: this paper presents a classification of all two-digits haar systems and tiles on the twodimension plane up to affine similarity. we obtain three cases, only two of them (rectangular and dragon tile) are well known. in all of them we compute the hölder regularity in l2 of corresponding haar functions. the technique of calculating the hölder regularity is well known for univariate wavelets and it was recently expended into multivariate wavelets. these values also give us the information about the rate of convergence of the corresponding subdivision algorithms and the rate of convergence of the cascade algorithm of the corresponding haar decomposition. keywords: self-similar tilings, hölder regularity, haar wavelets, subdivision schemes 1. introduction the subdivision schemes have been studied in the literature in great detail due to many applications in the approximation algorithms and in the curve and surface design [6, 7, 11, 13]. these algorithms possess several remarkable properties: linearity, shift-invariance, computational simplicity. a subdivision algorithm extrapolates a function by its values on a regular grid. in some sense they are development of de rham cutting corner algorithm. in many numerical problems, they are much more effective than spline algorithms, etc. [5,8]. in this paper, we address the problem of construction of multivariate subdivision algorithms based on haar functions in rd. the procedure of construction of haar systems is well elaborated [2, 10]. every haar basis is generated by a characteristic function of a tile, which is a self-similar attractor in rd, whose translates form a partition of the entire space. such an attractor is defined by an expanding integer matrix m and a system of integer points (”digits”) d0, . . . , dm−1, where m = | detm |. we classify all two-digits haar systems and tiles on the two-dimension plane. it turns out, that there are exactly three types of two-digit flate tiles up to affine similarity. then we compute the hölder regularity in l2 for all those haar functions. the exponents of hölder regularity are responsible for the rate of convergence of the corresponding subdivision algorithms. in the next section we introduce all notation and formulate basic properties of tiles and of haar functions in rd. then, in section 3, we formulate and prove our main result, the classification theorem. finally, in section 4, we compute the hölder regularity of haar functions and make conclusions on the rate of convergence of the corresponding subdivision algorithms. ∗corresponding author: zaitsevatanja@gmail.com http://ijassa.ipu.ru/ojs/ijassa/article/view/500 50 t. zaitseva 2. attractors, tilings, and haar functions let m ∈ zd×d be an integer dilation matrix all of whose eigenvalues are larger than one in the absolute value. let m = | detm |. the matrix m splits the integer lattice zd into m equivalence (quotient) classes defined by relation x ∼ y ⇔ y − x ∈mzd. choosing one representative di ∈ zd from each equivalence class, we obtain a set of digits d(m) = {di : i = 0, . . . ,m− 1}. we always assume that 0 ∈ d(m). we use the notation 0.d1d2 . . . = ∞∑ i=1 m−idi, di ∈ d(m). consider the following set g = { ∞∑ i=1 m−idi : di ∈ d(m) } . we can construct this set even if d(m) is not a correct set of digits and is just a set of m integer vectors including the zero vector (we will call it attractor). by [1, 2], for every expansive integer matrix m and for an arbitrary set of digits d(m), the set g is a compact set with a nonempty interior and possesses the properties: 1. the lebesgue measure µ(g) ∈ n; 2. g = ⋃ d∈d(m) m−1(g+ d), the sets m−1(g+ d) have intersections of zero measure; 3. the indicator function χ = χg(x) of g satisfies the refinement equation χ(x) = ∑ d∈d(m) χ(mx− d), x ∈ rd; 4. ∑ k∈zd χ(x+ k) ≡ µ(g), i.e. integer shifts of χ cover rd with µ(g) layers; 5. µ(g) = 1 if and only if the function system {χ(·+ k)}k∈zd is orthonormal. if µ(g) = 1, theng is called a tile. the integer shifts of a tile define a tiling. this is a partition of rd to integer translates of the tile. this partition is with disjoint interia. every tile defines a system of haar functions, ψ(s) jk (x) = m j 2ψ(s)(mj − k), j, k ∈ zd, where s = 1, . . .m− 1; ψ(s), s = 1, . . .m− 1, are generating function of the haar system. each of them is defined as ψ(s)(x) = √ m(χ(mg)− χ(mg+ ds)). this is a complete orthonormal system inl2(rd) [2]. thus, to classify haar wavelets on rd one needs to classify all attractors. we will do that for the case m = 2, d = 2 in the next section, and then compute the smoothness of those functions. let us remark, that the smoothness is equal to the rate of convergence of the corresponding subdivision algorithms, therefore computing the regularity give us tight bounds for the rate of convergence. 3. the classification theorem theorem 1: for m = 2, d = 2, there exist three types of attractors up to affine similarity, those types are: 1) m = ( 0 −2 1 0 ) , d = {( 0 0 ) , ( 1 0 )} 2) m = ( 1 −2 1 0 ) , d = {( 0 0 ) , ( 1 0 )} 3) m = ( 1 1 −1 1 ) , d = {( 0 0 ) , ( 1 0 )} copyright c© 2017 assa. adv syst sci appl (2017) haar wavelets and subdivision algorithms on the plane 51 thus, every two-digit attractor on the plain is affinely similar to one of those three. we illustrate these three tiles in fig. 1, 2, 3 (the bounding square in all figures has vertices (−1,−1), (−1, 1), (1, 1), (1,−1)). in fig. 4, 5, 6, we see the plain tilings by those attractors (the black attractor is initial). this theorem classifies not only all possible attractors but also all possible types of plain two-digit haar wavelets and the corresponding subdivision schemes. fig.1: rectangular tile fig.2: third type of tile fig.3: dragon tile fig.4: rectangular tile fig.5: third type of tile fig.6: dragon tile the proof of this theorem will be splitted to several lemmas. lemma 1: if m = 2, d = 2, attractors g1 and g2 with the same dilation matrix m and different sets of digits d1(m) and d2(m) are affinely similar. proof since m = 2, we can consider an attractor g0 with matrix m and set of digits d0(m) ={( 0 0 ) , ( 1 0 )} . it is sufficient to proof that g1 and g0 are affinely similar, then analogously g2 and g0 are affinely similar and then g1 and g2 are affinely similar. let m = ( u v w z ) , d1(m) = {( 0 0 ) , ( a b )} . let x = ( a x1 b x2 ) , x1 ∈ r, x2 ∈ r. we will proof that we can choose such x1 ∈ r and x2 ∈ r that xm = mx . it is equivalent to ( a x1 b x2 )( u v w z ) = ( u v w z )( a x1 b x2 ) .( au+ x1w av + x1z bu+ x2w bv + x2z ) = ( au+ bv x1u+ x2v aw + bz x1w + x2z ) copyright c© 2017 assa. adv syst sci appl (2017) 52 t. zaitseva { x1w = bv bu+ x2w = aw + bz av + x1z = x1u+ x2v (3.1) suppose that w = 0. then | detm | = uz = 2. case 1: detm = 2. since u ∈ z and z ∈ z, we have [ u = 2 z = 1 or [ u = −2 z = −1 or [ u = 1 z = 2 or[ u = −1 z = −2 . then trm = ±3, characteristic polynomial is λ2 ± 3λ+ 2 = 0 and it has a root ∓1, then it is not a dilation matrix, the contradiction concludes the proof for this case. case 2: detm = −2. since u ∈ z and z ∈ z, we have [ u = −2 z = 1 or [ u = 2 z = −1 or[ u = −1 z = 2 or [ u = 1 z = −2 . then trm = ±1, the characteristic polynomial is λ2 ± λ− 2 = 0 and it has root ±1, then it is not a dilation matrix. so w 6= 0 and x1 = bv w . from the second equality we have x2 = 1 w (aw + bz − bu). it remains to verify the third equality if x1, x2 are chosen as above. since av + x1z = av + bvz w = bvu w + awv+bzv−buv w = x1u+ x2v, we arrive at the third equality in (3.1). thus, we find a proper matrix x . since xm = mx , it follows that xm−i = m−ix ∀i ∈ z. by construction x ( 1 0 ) = ( a b ) . g1 = {∑ i∈i m−i ( a b ) , i ⊂ n } = {∑ i∈i m−ix ( 1 0 ) , i ⊂ n } = = {∑ i∈i xm−i ( 1 0 ) , i ⊂ n } = x {∑ i∈i m−i ( 1 0 ) , i ⊂ n } = xg0. lemma 2: let m1 and m2 be integer dilation 2× 2 matrices such that trm1 = trm2, detm1 = detm2 = ±2 and let d1 and d2 be arbitrary two-digit sets from z2; then the attractors g1 and g2 produced by pairs (m1, d1) and (m2, d2) are affinely similar. proof it is easy to prove that eigenvalues ofm1 are different (otherwise the discriminantd = t2 ± 8 of the characteristic polynomial (x2 − tx± 2) has to be zero, which is impossible for integer t). then we have a basis of eigenvectors for m1 and m2. since the assumptions of the lemma are satisfied, there is x such that xm1 = m2x . this system with four integer coefficients is degenerate because it has a non-trivial solution x . then it has a rational solution x0, and therefore an integer solution x∗. m1 = x−1 ∗ m2x∗. using lemma 1 we can choose any sets of digits. let d(m1) = {( 0 0 ) , ( 1 0 )} , ( a b ) = x∗ ( 1 0 ) , a, b ∈ z, d(m2) = {( 0 0 ) , ( a b )} . g1 = {∑ i∈i m−i 1 ( 1 0 ) , i ⊂ n } = {∑ i∈i x−1 ∗ m−i 2 x∗ ( 1 0 ) , i ⊂ n } = = x−1 ∗ {∑ i∈i m−i 2 ( a b ) , i ⊂ n } = x−1 ∗ g2. copyright c© 2017 assa. adv syst sci appl (2017) haar wavelets and subdivision algorithms on the plane 53 lemma 3: if m = 2, d = 2, m is a dilation integer matrix, there are six possible pairs of (trm , detm). proof let t = trm . the characteristic polynomial is x2 − tx± 2 = 0. let x1, x2 be eigenvalues of m . then |x1| > 1, |x2| > 1, |x1 · x2| = 2, and hence |x1| < 2, |x2| < 2. |t| ≤ |x1|+ |x2| < 4. t is an integer. case 1: detm = 2: t 6= ±3, because x2 ± 3x+ 2 has root∓1, but m is a dilation matrix, a contradiction. in this case t = 0,±1,±2, it is easy to construct examples for all these cases. case 2: detm = −2: since x1 · x2 = −2, let x1 < 0, x2 > 0, then −2 < x1 < −1, 2 > x2 > 1, then −1 < t = x1 + x2 < 1, t is an integer, then only t = 0 is possible, there are easy examples of dilation m with t = 0. now we are ready to prove theorem 1. proof as we can see from these lemmas, there are six types of tiles with different characteristic polynomials. but it is easy to see that cases 1) tiles with tr = 0, det = 2 and tr = 0, det = −2 are affinely similar, for example,( 0 −2 1 0 ) , ( 0 2 1 0 ) are both rectangles. 2) tiles with tr = 1, det = 2 and tr = −1, det = 2 are affinely similar, for example,( 1 −2 1 0 ) , ( −1 −2 1 0 ) . 3) tiles with tr = 2, det = 2 and tr = −2, det = 2 are affinely similar, for example,( 1 1 −1 1 ) , ( −1 1 −1 −1 ) . so we can see that there are only three types. we can obtain the fact, that they are affinely different from their different smoothness, it is calculated in the next section. 4. regularity of two-digit attractors in this section we compute the exponent of regularity of the three types of the haar wavelets, classified in theorem 1. the regularity is important for estimating the rate of convergence of the cascade algorithm of the haar decomposition and of the subdivision algorithms in approximation theory and surface design. we fix again an expanding matrix m and the set of digits d. they, as we know, define a unique attractor set k. the characteristic function of this set can also be defined in the framework of refinement equations. a refinement equation is the linear functional difference equation with a contraction of the argument by the matrix m . thus, general refinement equation has the form ϕ(x) = ∑ k∈zd ckϕ(mx− k), (4.2) where ck are prescribed coefficients. refinement equations have been studied in the literature in great detail due to countless applications in wavelets, approximation theory, etc. (see [1, 2, 5, 7]). we are interested in a special case of refinement equations, where all the coefficients ck are zeros and ones. this case will be referred to as haar case. refinement equations of the haar case generate scaling function for haar wavelets. in the univariate case, copyright c© 2017 assa. adv syst sci appl (2017) 54 t. zaitseva there is only one haar scaling function ϕ(x) = χ[0,1]. in the multivariate case, an arbitrary tile generate haar system by its characteristic function used as a scaling function. thus, we consider refinement equation (4.2) and suppose that ck = 1 ∀k ∈ d and ck = 0 if k /∈ d, then all solutions of refinement equation are ϕ = λχk [2]. next part is devoted to calculating the exact hölder exponent (regularity) αϕ = sup{α ≥ 0 : ‖ϕ(·+ h)− ϕ‖ ≤ c|h|α,∀h ∈ rd} of these solutions ϕ in all three cases. the hölder exponent is responsible for the rate of convergence of wavelet expansions as well as of subdivision approximation. it is one of the most important characteristics of wavelet systems. for univariate wavelets, it is well known that hölder exponent can be expressed with spectral characteristic of special matrices. recently, this technique was expended into multivariate wavelets. to attack the computation of hölder regularity, we first construct these matrices. they will be matrices build by coefficients of refinement equation. let d0 = {( 0 0 ) , ( 1 0 )} . we construct the attractor g using m and d0, it is a tile in these 3 cases. let ω ⊂ zd be a minimal subset of zd with the property k ⊂ ω +g = ⋃ k∈ω (k +g). we denote n = |ω|. we denote by td the transition n ×n matrices (td)ab = cma−b+d, a, b,∈ ω ∀d ∈ d0. in our case these matrices are boolean (consist in 0 and 1). there are two matrices, we will call them t0, t1. it is easy to see that we can have at most one 1 in column: otherwise, for some a1, a2, b ∈ ω, d ∈ d0, d1, d2 ∈ d ma1 − b+ d = d1,ma2 − b+ d = d2, then d2 − d1 = m(a2 − a1), which contradicts the fact that digits are from different classes of equivalence. we consider the following affine subset of the space rn : v = { ω = {ω1, ω2, . . . , ωn} ∈ rn : n∑ j=1 ωj = 1 } . we denote the linear part of subspace v by w = { ω = {ω1, ω2, . . . , ωn} ∈ rn : n∑ j=1 ωj = 0 } . as we have seen above, all of td are invariant respectively to v and w . let v(x) = (ϕ(x+ k1), . . . , ϕ(x+ kn), ki ∈ ω) ∈ rn . then the refinement equation is equal to v(x) = tdv(mx− d), ∀x ∈m−1(g+ d), d ∈ d0. let u = span{v(x1)− v(x2) | x1, x2 ∈ g}, n = dimu . since v(x) ∈ v ∀x ∈ g, it follows that u ⊂ w and n ≤ n − 1. since all of td are invariant respectively to v and w , we have they are invariant respectively to u . then the restrictions ad = td|u of the operators td are well-defined. we have only two operators in our case, we will call them a0, a1. from [3, theorem 7] it follows that αϕ = − log2 (ρ2(a0, a1)), where ρ2(a0, a1) = lim m→∞ (2−m · ∑ σ ‖aσ(1) . . . aσ(m)‖2) 1 2m , σ : {1, 2, . . . ,m} → {0, 1}. ρ2 is called l2 joint spectral radius. we will use the short notation l2 spectral radius. let us see an example. let m = ( 1 −2 1 0 ) , it is the second type from theorem 1. let d = {( 0 0 ) , ( 2 1 )} , it is a correct set of digits. copyright c© 2017 assa. adv syst sci appl (2017) haar wavelets and subdivision algorithms on the plane 55 in this case we obtain that ω = {( −1 −1 ) , ( −1 0 ) , ( 0 −1 ) , ( 0 0 ) , ( 0 1 ) , ( 1 −1 ) , ( 1 0 ) , ( 1 1 ) , ( 2 0 )} . as we can see, n = 9. further there are illustrations of the process (the bounding square has vertices (−1,−1), (−1, 1), (1, 1), (1,−1)): fig.1: attractor k fig.2: tilings of g fig.3: how to find ω let t0 be the matrix which corresponds to d = ( 0 0 ) , let t1 correspond to d = ( 1 0 ) . t0 =  0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0  t1 =  0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0  let v be the eigenvector of t0+t1 2 which corresponds to the value 1 (such an eigenvector exists because both t0 and t1 have a common left eigenvector (1, 1, . . . , 1)). then we have v ∈ v and t0v ∈ v , then it follows that u0 := v − t0v ∈ u . using the algorithm from [3, section 3.2], we regard u0 as the first basis vector in u . then for all vectors v1 in the current basis we try to complement to basis vectors t0v and t1v (evidently these vectors are in u ). if we complement vector t0v and vectors in the basis are still linear independent, then we keep this vector in the basis. the same is done with vector t1v. we keep doing this for all vectors v1 until our system stays the same. in our case, t0|u and t1|u are 5× 5 matrices (the coefficients are rounded to three decimal digits after the point): a0 = t0|u =  0.0 0.0 −1.044 −0.159 −0.23 0.0 0.0 0.693 0.106 0.153 1.088 0.0 0.214 −0.167 0.256 0.0 0.0 −0.291 −0.044 −0.064 0.0 0.618 −0.29 −0.449 −0.17  copyright c© 2017 assa. adv syst sci appl (2017) 56 t. zaitseva a1 = t1|u =  0.0 0.0 −0.174 −0.425 −0.843 0.817 0.0 −0.354 −0.888 0.56 0.0 0.0 0.282 0.142 0.938 0.0 1.0 0.463 0.34 −0.235 0.0 0.0 −0.36 0.558 −0.622  now we have a0, a1 and we can find the coefficient of smoothness. there are different ways to calculate ρ2(a0, a1). firstly, we can use formula ρ2 = λmax( 1 2 (a0 ⊗ a0 + a1 ⊗ a1)). secondly, we can consider operator ax = 1 2 (at0xa0 + at1xa1) in the space of symmetric matrices and then find its spectral radius: ρ2 = λmax(a ) (these formulas can be found in [4, 12, 14]. but both ways led us to the same result. in our case ρ2 ≈ 0.7607 and αϕ ≈ 0.3946. we can calculate hölder regularity of other types of tiles using the same method. let m = ( 1 1 −1 1 ) , it is the third type from theorem 1. we suppose that in this case d = {( 0 0 ) , ( 2 1 )} too. here n = 21 and a0,a1 are 5× 5 matrices. we obtain that ρ2 ≈ 0.8478 and αϕ ≈ 0.2382. in the first case from theorem 1, we suppose m = ( 0 −2 1 0 ) , d = {( 0 0 ) , ( 1 2 )} . here n = 18 and a0,a1 are 9× 9 matrices. we obtain that ρ2 ≈ 0.7071 and αϕ = 0.5. this answer is natural because the hölder l2 regularity of a charasteristic function of a polygon is always 0.5. remark 1: we can note, that in the case of dragon type hölder regularity is higher than in the case with tr = 1, this means that the convergence of subdivision schemes in dragon case is worse although this case is better known. 5. conclusion we have classified all two-digit tiles on the plain up to affine similarity. it turned out that there only three types, two of them (rectangular and dragon tiles) are well known, the third one is much less studied. for all the types, we computed the hölder regularity of the corresponding characteristic functions. the hölder exponents are different, which proved that these classes are also different (0.5 for rectangular, ≈ 0.2382 for dragon type, ≈ 0.3946 for third type). we find not only the smoothness of two-digit haar functions, but also the rate of convergence of the corresponding subdivision algorithms. thus, for two-digit plain haar system the problem is completely solved. the generalization for bigger dimensions (d ≥ 3) is possible for two-digit tiles, this is a subject of future research. on the other hand, the generalization to a bigger number of digits seems to be a hard problem, because already for three digits there are infinitely many types of flat tiles (but in case, when our digits are d0 = 0, d1, d2 and det(d1, d2) = ±1, there are 10 types of tiles). the method for computing the l2 hölder exponent (and correspondingly, the rate of convergence of the subdivision schemes) is universal and can be applied for every tile, with an arbitrary number of digits and arbitrary dimension. acknowledgements the author is grateful to her supervisor v. yu. protasov for posing the problem and permanent support. copyright c© 2017 assa. adv syst sci appl (2017) haar wavelets and subdivision algorithms on the plane 57 references 1. gröchenig, k. & haas, a. (1994) self-similar lattice tilings, j. fourier anal. appl., 2, 131–170. 2. gröchenig, k. & madych, w.r. (1992) multiresolution analysis, haar bases and selfsimilar tilings of rn, ieee trans. inform. theory, 38, 556–568. 3. charina, m. & protasov, v. yu. (2017) smoothness of anisotropic wavelets, frames and subdivision schemes. arxiv:1702.00269 [math.na]. 4. blondel, v. d. & nesterov, yu. (2005) computationally efficient approximations of the joint spectral radius. siam journal of matrix analysis, 27(1), 256–272. 5. cabrelli, c. a., heil, c. & molter, u. m. (2004) self-similarity and multiwavelets in higher dimensions, memoirs amer. math. soc., 170 (807) 6. chui, c. & de viles, j. (2011) wavelet subdivision methods: gems for rendering curves and surfaces. boca raton, fl: crc press. 7. dyn, n. & levin, d. (2002) subdivision schemes in geometric modeling, acta numer., 11, 73–144. 8. jia, r.-q. (1995) subdivision schemes in lp spaces, advances comp. math., 3, 455– 454. 9. lagarias, j. & wang, y. (1997) integral self-affine tiles in rn. ii. lattice tilings, j. fourier anal. appl., 3, 83–102. 10. lagarias, j. & wang, y. (1999) corrigendum and addendum to: haar bases for l2(rn) and algebraic number theory, j. number theory, 76, 330–336. 11. peter, j. & reif, u. (2008) subdivision surfaces, geometry and computing. berlin, germany: springer-verlag. 12. protasov, v. yu. (1997) the generalized spectral radius. a geometric approach, izvestiya math., 61, 995–1030. 13. warren, j. & weimer, h. (2002) subdivision methods for geometric design: a constructive approach. morgan-kaufmann. 14. zhou, d.-x. (1998) the p-norm joint spectral radius for even integers, methods appl. anal., 5 (1), 39–54. copyright c© 2017 assa. adv syst sci appl (2017) https://arxiv.org/abs/1702.00269 introduction attractors, tilings, and haar functions the classification theorem regularity of two-digit attractors conclusion microsoft word 1071 article text, copyedited.doc adv syst sci appl 2021; 03; 12-21 published online at https://ijassa.ipu.ru. advanced technologies for supporting operational decision-making in civil aviation valentin g. lebedev1*, elena l. kulida1 1) v.a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia e-mail: lebedev@ipu.ru abstract: in civil aviation, many tasks require the use of modern technologies, in particular artificial intelligence technologies. the features of the crew and dispatcher decision-support tasks that can be solved based on deep neural networks are considered. the features of the intelligent crew interface for improving interaction with the operational decision support system are considered. optimization of air traffic management by using genetic algorithms is illustrated by the example of two tasks. a special feature of the genetic algorithm for solving the problem of optimizing the sequence and landing times of arriving aircraft is a non-standard crossing operator. a special feature of the genetic algorithm for solving the problem of constructing conflict-free trajectories of a given length at low altitude in conditions of complex terrain is the method of forming the objective function. in conclusion, there is a list of problems that need to be solved for the wider use of artificial intelligence in civil aviation. keywords: decision support system, air traffic management, artificial intelligence, deep neural networks, genetic algorithms 1. introduction one of the main priorities of civil aviation is to ensure flight safety, which is largely determined by the correct and timely actions of pilots and dispatchers, and the serviceable technical condition of aircraft. the development of modern technologies, in particular artificial intelligence technologies, will support the adoption of adequate operational decisions by the crew and dispatcher, especially in critical situations. the complexity of tasks, a large amount of incoming information, short time for decisionmaking, and responsibility for decisions lead in some cases to a mismatch between human capabilities to the requirements of the process efficient operational management of humanmachine systems. in this regard, computer-based operational decision support systems are being increasingly used. the interface between the crew and the system is the most important component of the decision support systems. the goal of creating an intelligent crew interface is to reduce the emotional and sensory burden on the crew and create a human-machine interaction similar to human-to-human communication. promising technologies to support decision-making by the dispatcher are related to solving the following tasks ‒ • collection, processing, and intelligent analysis of information about the movement of aircraft and its application for air traffic management and organization systems; • increase in throughput by optimizing the structure of the air space, adapting as much as possible to the real flows of aircraft; • optimization of aircraft flight routes, including in the airport area. * corresponding author: lebedev@ipu.ru advanced technologies for supporting operational decision-making in civil aviation 13 copyright ©2021 assa. adv. in systems science and appl. (2021) the use of data mining methods is of significant interest for finding difficult-to-formalized patterns in the data on the state of aircraft aggregates and nodes. there are a large number of modern data mining methods for solving the problem of classification of the technical condition of the aircraft. the most well-known of them are neural networks, decision trees, the method of reasoning based on similar cases, the method of support vectors. the listed methods and their various modifications differ from each other according to the principle of forming the final result, but they have a common property – they use information about previously occurring cases (precedents) to search for and formalize patterns. one of the promising areas of application of intelligent technologies is the use in aviation training complexes. training of individual skills and abilities with the help of such aviation training complexes can be automatically improved based on objective testing data, knowledge verification, work protocols, and then optimized by correlating changes in the training methodology with changes in the learning outcome indicators. artificial intelligence will allow pilots to have even more realistic simulations, allowing them to observe not only "virtual reality", but also those objects that are difficult to see during a real flight. aviation training complexes with implemented artificial intelligence generates individual training tasks (training exercises), collect all training data, and records the behavior of the pilot during training. the data collected during training can also be used to improve autopilots. in the modern practice of training civil aviation pilots, the most widespread are complex (full flight simulator) and procedural (flight procedures training device) aviation training complexes with a single virtual environment. the use of artificial intelligence tools allows you to practice piloting in a wide range of external conditions. let's look at some of these areas of use of artificial intelligence methods in civil aviation in more detail. 2. support for operational decision-making by the crew the analysis of aviation accidents shows that the introduction of artificial intelligence methods can be especially useful for the crew in the following situations: • special situations: the approach to critical flight modes and output from them (the output modes at supercritical angles of attack, beyond the altitude and speed characteristics, going beyond the operational limits of alignment, etc.); • special situations relating to the technical status of the onboard systems, equipment, and assemblies (engine failures, hydraulic, fuel, and other systems, fire, etc.); • special situations related to external threats to the safety of flight and the use of airspace (meteorological, geomagnetic phenomena, terrain, risk of collision in the air); • special situations related to the psychophysiological state of the crew (increased psychophysiological load, fatigue, etc.); • special situations of the tactical and special use of the aircraft (unacceptable distance from the runway, violation of the state border, etc.). hybrid technologies, which combine different approaches, are an integral part of the implementation of decision support systems. expert systems were the first artificial intelligence approaches for creating decision support systems. for use in aviation, onboard operational-advisory systems were developed to solve specific tactical tasks, including military applications, the purpose of which was to determine a rational way to achieve an operationally assigned goal. during the development of the expert systems, it became clear that one of the most timeconsuming tasks is to obtain knowledge from experts and formalize it, that is, to fill in the knowledge base to describe a specific subject area that is the core of the expert systems. in this regard, the task of automatic acquisition of knowledge by the system has come to the fore – the task of machine learning, in the process of solving which deep neural networks and 14 v.g. lebedev, e.l. kulida copyright ©2021 assa. adv. in systems science and appl. (2021) methods of their training has been further developed. deep optimization methods are being actively developed based on deep neural networks. this group of methods is designed to solve a wide range of problems. 2.1. automatic object recognition; development of technical vision systems for solving navigation problems further automation of the decision support systems requires analysis of the state of the external environment based on aerial observation data. due to the large volume of aerial observation data, it is obvious that artificial intelligence methods and models need to be used to process this information. currently, training methods for solving problems of pattern recognition and big data analysis have been developed [1]. in the field of image processing and analysis, deep convolution neural networks are used in combination with deep machine learning. neural network algorithms based on deep convolution neural networks demonstrate good results and can be effectively used, including in solving problems of combining multispectral images, for example, based on a generative adversarial neural network. with the help of the deep neural networks on board the aircraft, image recognition tasks can be solved: monitoring of gas pipelines, power lines, etc. to the problems of image recognition solved by technical vision systems, the problem of recognizing phenomena occurring in space can be added. the task of autonomous navigation of the aircraft based on the images from the technical vision systems is also relevant. in the practical implementation of this task, various difficulties arise associated with image interference caused by weather conditions, errors in the alignment of the technical vision systems sensors, etc., which require the development of algorithms that are invariant to noise and geometric distortions. one of the ways to solve this problem is to use a convolutional neural network that allows you to detect objects even with partial noise. as research shows, the use of a convolutional neural network is quite effective for recognizing characteristic points of the terrain (reference points). the main difficulty of this approach is the selection of reference points and the process of training a neural network to solve such a problem. 2.2. intelligent interface of the crew the intelligent crew interface should provide: • a visual, integrated, comprehensive, multi-modal representation of information; • improved, synthesized, and combined vision, which is very important especially in landing conditions with poor visibility, as well as in low-altitude flight; • interact in natural language, adaptive to the pilot and the problem being solved, including the dialogue, including by transmitting and receiving commands and unstructured data; the use of artificial intelligence in providing a voice will allow achieving a more rapid and informative when sending messages; • face, voice, and gesture recognition for more accurate identification; • tracking the psychophysical state of the pilot and his actions with the generation of warning signals regarding the detected state, characterizing the level of real human abilities to solve problems and control the aircraft. in addition to the above-mentioned ways of interaction between the crew and the decision support systems, interaction is of interest when, as a result of an in-depth analysis of the state of the aircraft systems, the crew is automatically offered the information in the appropriate form that is most relevant at the current time. the development of the so-called neuro interface is also underway. it is a new promising direction in which a special headset that measures the signals of human brain activity, controls them, and can react in time, for example, when the pilot has begun to fall asleep. such an interface can be an integral part of the module for monitoring the psychophysical advanced technologies for supporting operational decision-making in civil aviation 15 copyright ©2021 assa. adv. in systems science and appl. (2021) state of a person. machine learning models can be trained to distinguish between normal and abnormal patterns about pilots ' eye movements, data from other psychophysiological sensors such as heart rate or blood pressure, and assist pilots in abnormal situations [2]. the top-level architecture of the intelligent interface of the crew is shown in figure 2.1. fig. 2.1. the top-level architecture of the intelligent crew interface 3. improving the efficiency of air traffic management promising artificial intelligence methods for improving the efficiency of air traffic management are, in particular, genetic algorithms. the main difference between a genetic algorithm and a random search is the active use of information accumulated during iterations. the main advantage of genetic algorithms is their "creativity", i.e. the ability to obtain unexpected effective solutions that are not generated using other methods. let's consider the application of genetic algorithms for solving the following two problems: 16 v.g. lebedev, e.l. kulida copyright ©2021 assa. adv. in systems science and appl. (2021) • improving the efficiency of using airport runways by optimizing the sequence and landing times of arriving aircraft; • construction of conflict-free routes of aircraft in the concept of air traffic "free flight" (free flight). 3.1. improving the efficiency of the use of airport runways by optimizing the sequence and landing times of arriving aircraft one of the most difficult tasks in the airport area is the task of optimizing the planning of the sequence and landing times of arriving aircraft, taking into account the attenuation of the resulting vortices and turbulence zones. the mathematical formulation of the problem of forming optimal aircraft queues for landing as a linear or quadratic optimization problem (depending on the selected objective function) is np-complete [3]. the task of optimizing the sequence and landing times of arriving aircraft is to optimize the global target function for a group of aircraft that are in the airport area to land, p is the number of aircraft waiting to land. between the aircraft, according to icao standards, it is necessary to provide a certain minimum interval to avoid falling into the wake vortex formed from the aircraft going ahead. this interval depends on the types of consecutive aircraft. known matrix s of size p*p, where – the minimum interval between landing aircraft after the aircraft , , ci is the type of i-th su. also, for each aircraft, a time window is defined , during which the aircraft with the number i can land by its flight characteristics, fuel availability, flight duration, etc. ei – the earliest possible landing time of the i-th aircraft; xi – the assigned landing time of the i-th aircraft, li – the latest possible landing time of the i-th aircraft. for each aircraft, the time ti is also known – the optimal time of arrival of the i-th aircraft under the condition of a free runway, . depending on the air traffic conditions, the optimized global objective function may depend on many factors, but as a rule, the determining values are the deviation of the real landing times xi from the optimal landing times ti. for example: • the piecewise-linear objective function is the minimization of the sum of absolute deviations from the optimum time of planting: , where (3.1) • the nonlinear objective function is the minimization of the sum of squares of deviations from the optimal time of planting: , where (3.2) in this task the subject of numerous scientific works, an overview of exact and approximate methods of solving this problem is presented in [3, 4], which deals with the genetic and memetic algorithms, algorithm diffused search and bionic algorithm, ant algorithm, etc. the main obstacle to using the proposed methods in practice in real-time is the unacceptably long counting time. the conducted research shows that heuristic and genetic algorithms are the most promising for solving the problem [5]. a special feature of the genetic algorithm for solving the problem of optimizing the sequence and landing time of arriving aircraft is a non-standard crossing operator. the landing sequence is encoded using the vector , which is a permutation of the aircraft numbers. the standard crossing operator cannot be applied in this case, because as a result of its application, the resulting vectors may not be permutations of aircraft numbers. figure 3.1 shows this with the example of a single-point crossing. advanced technologies for supporting operational decision-making in civil aviation 17 copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 3.1. standard single-point crossing operator it is proposed to perform the crossing operator of the corresponding vectors of the assigned landing times instead of the crossing operator of the vectors y. figure 3 shows this by the example of a single-point crossing. then, the corresponding vectors are reconstructed from the new vectors obtained. fig. 3.2. proposed single-point crossing operator in [6], a heuristic algorithm for solving the problem is proposed and the results of the study of the effectiveness of this algorithm are presented. the algorithm is based on using some initial solution, which is then consistently improved during the iterative process. as a result of the research, it was found that the result of applying the algorithm significantly depends on the quality of the selected initial solution. on the other hand, the peculiarity of genetic algorithms is that they do not guarantee an optimal solution, but allow you to quickly get an acceptable solution and then use it as the initial solution for a heuristic algorithm. this hybrid approach proved to be very effective. table 3.1 shows the results of 20 tests of applying algorithms for a sequence of 17 aircraft for a nonlinear objective function (3.2). the effectiveness of the algorithm is estimated by the value of the minimized objective function for the resulting solution. the values of the objective function are given for the initial solution, the solution obtained using a heuristic algorithm, the solution obtained using a genetic algorithm, the solution obtained using a hybrid algorithm, and for the optimal solution, which for a sequence of 17 aircraft can be obtained in the absence of restrictions on the counting time. table 3.1 shows that both heuristic and genetic algorithms provide a significant improvement of the initial solution, and the hybrid algorithm allows you to get optimal or fairly close to the optimal solution. for clarity, the results of comparing the algorithms are shown in figure 3.3. 3.2. construction of conflict-free routes of aircraft in the concept of air traffic "free flight» a serious problem with the perspective concept of air traffic management "free flight" in the airport area is associated with the limited capabilities of the human dispatcher. the way to overcome this problem is to further improve the automation of air traffic control, in particular, in the automatic synthesis of trajectories and the calculation of speeds along the trajectories to accurately perform the specified time intervals between aircraft during landing. in [7, 8], an algorithm for generating the shortest conflict-free approach trajectories in complex terrain conditions and an algorithm for controlling the speed along the constructed trajectories to maintain the necessary separation rates between aircraft is proposed. 18 v.g. lebedev, e.l. kulida copyright ©2021 assa. adv. in systems science and appl. (2021) table 3.1. the values of the objective function test № initial solution heuristic algorithm genetic algorithm hybrid algorithm optimal solution 1 704400 307564 310052 293100 293100 2 633057 227376 172175 171687 171687 3 279885 191400 185789 174344 166424 4 286196 141035 144304 141035 141035 5 912663 213026 213026 213026 213026 6 544427 165802 158362 158362 158362 7 1027015 301946 300370 300370 299743 8 261039 74370 74370 74370 74370 9 233146 100882 102166 102166 100882 10 2103394 457870 444770 444770 444770 11 232913 172292 168932 168932 162332 12 162565 150013 150013 150013 150013 13 211737 206756 206756 206756 206756 14 430268 119752 123816 119752 119752 15 323899 243954 230094 230094 230094 16 462540 266424 212784 212784 211584 17 220370 130446 130446 130446 130446 18 1060169 295720 251620 251620 251620 19 345061 153205 135352 134212 134212 20 957920 355969 291844 286465 286465 fig. 3.3. comparison of algorithm results advanced technologies for supporting operational decision-making in civil aviation 19 copyright ©2021 assa. adv. in systems science and appl. (2021) however, the proposed approach does not provide a solution in the case when maintaining the necessary separation standards between the aircraft using speed control on the shortest trajectory is impossible due to speed and acceleration restrictions. in this case, it is necessary to build a trajectory of the desired length, taking into account the difficult terrain, limited maneuverability of the aircraft, restrictions on the trajectory, taking into account the trajectories of other aircraft. the solution to this complex problem is possible with the help of a genetic algorithm, in which the necessary constraints are taken into account in the form of penalties in the optimized objective function [9]. we will explain this by the example of penalties for deviation of the trajectory length from a given target length d, for conflicts of the trajectory with the terrain, and sharp angles between successive segments of a piecewise linear trajectory. each solution in the genetic algorithm has the form: . this is a set of n integers, which are the numbers of points in the maneuvering area. the coordinates of the point e_i are denoted by . then: (3.3) deviation from the specified target length (3.4) based on the elevation matrix h, a set of horizontal and vertical segments is constructed, so that the trajectory that does not intersect these segments does not have conflicts with the terrain. each segment of a piecewise linear trajectory is checked for intersection with all segments from the set n1 denotes the number of segments of a piecewise linear trajectory that intersect with segments from the set the total number of acute angles between successive segments of a piecewise linear trajectory is denoted by n2: (3.5) then the objective function has the form (3.6) where f1 is the penalty for crossing the terrain, f2 is the penalty for an acute angle. the penalties are chosen large compared to the length of the trajectory, for example, , in this case, the genetic algorithm generates such chromosomes that the corresponding trajectories do not intersect with the terrain and do not have sharp corners. figure 3.4 shows the results of the algorithm for constructing a trajectory with a given length of 250,000 meters in the mountainous surroundings of yelizovo airport at an altitude of 1500 m. areas of terrain that need to be flown at the specified altitude due to the risk of collision with the terrain are shaded in the drawing. the constructed trajectory has a length of 250134 meters, has no sharp corners between consecutive segments, and passes through safe areas of the terrain. 1, 1, 1, 1 2 1 1, если 0, , где 0 в противном случае. n i i i i i i i i i d d d n n n + + = + ³ìï= = í ïî å 20 v.g. lebedev, e.l. kulida copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 3.4. comparison of algorithm results 4. conclusion in addition to the aforementioned areas of application, the use of ai techniques in civil aviation includes detection and prevention of collisions between aircraft and unmanned aerial systems; automation and autonomy of management; cybersecurity; regulation of traffic flow (simulation of demand for air travel, the facial recognition at airports, baggage and carry-on baggage), and others [10]. the prospects for using artificial intelligence capabilities in civil aviation are extremely diverse, but at present, they are still underutilized [11]. this is due to some problems that need to be solved for the wider use of artificial intelligence: 1. it is important to find the right balance between the use of ai technologies and human skills in ensuring flight safety in terms of piloting, control, and maintenance. 2. it is necessary to take into account the person in the system of support for operational decision-making. a person must be confident in the solutions offered by the ai system, trust them, while the methods of obtaining solutions using artificial intelligence are difficult to understand and explain. 3. it requires analysis of a large amount of source data obtained from different sources, and a large number of calculations to obtain the result. 4. it is necessary to take into account the problem of vulnerability of neural networks, which can lead to serious errors. advanced technologies for supporting operational decision-making in civil aviation 21 copyright ©2021 assa. adv. in systems science and appl. (2021) despite these difficulties, the results show that it is difficult to overestimate the possibilities of using artificial intelligence methods in civil aviation. in the future, the construction of an aircraft with one pilot is being considered, the second pilot is planned to be replaced by artificial intelligence, and fully autonomous passenger aircraft are being developed (boeing, airbus, etc.). references 1. vizilter, yu. v., gorbatcevich v. s., zheltov s. yu. (2019) structure-functional analysis and synthesis of deep convolutional neural networks, computer optics, 43(5), 886–900. doi: https://doi.org/10.18287/2412-6179-2019-43-5-886-900 u [in russian]. 2. lutnyk, l., rudi, d., martin, r., (2020) towards pilot-aware cockpits, conference: 1st international workshop on eye-tracking in aviation (etavi 2020), doi:10.3929/ethz-b-000407661. 3. veresnikov, g.s., egorov, n.a., kulida, e.l., lebedev, v.g. (2019). methods for solving of the aircraft landing problem. i. exact solution methods // automation and remote control, 80(7), 1317–1334. 4. veresnikov, g.s., egorov, n.a., kulida, e.l., lebedev, v.g. (2019). methods for solving of the aircraft landing problem. ii. approximate solution methods // automation and remote control, 80(8), 1502-1518. 5. kulida, e.l. (2020). analysis of algorithms for solving the aircraft landing problem. proc. of the 13th international conference "management of large-scale system development" (mlsd). moscow: ieee, https://ieeexplore.ieee.org/document/ 9247839. 6. kulida, e., egorov, n., lebedev, v. (2020). algorithm for scheduling a flight of a group of aircraft to implement an optimal landing plan. proc. of the 12th international conference "management of large-scale system development" (mlsd). мoscow: ieee, https://ieeexplore.ieee.org/document/8911115. 7. bazhenov, s.g., egorov, n.a., kulida, e.l., lebedev, v.g. (2016). control of aircraft trajectory and speed to avoid terrain and traffic conflicts during approach maneuvering. automation and remote control. 77(10), 1827-1837. 8. kulida, e.l., lebedev, v.g., egorov, n.a. (2018). the heuristic algorithm for planes queue optimization. proc. of the 11th international conference «management of large-scale system development» (mlsd). мoscow: ieee, https://ieeexplore.ieee.org/document/8551908. 9. kulida, e.l., lebedev, v.g. (2017). genetic algorithm for generating trajectories of specified length for the onboard flight path safety system. procedia computer science, 112, 1015-1022. 10. maheshwari, a., davendralingam, n., delaurentis, d.a. (2018). a comparative study of machine learning techniques for aviation applications. aviation technology, integration, and operations conference, doi: 10.2514/6.2018-3980. 11. kashyap, r. (2019). artificial intelligence systems in aviation, in book: cases on modern computer systems in aviation, doi: 10.4018/978-1-5225-7588-7.ch001. adv syst sci appl 2021; 01; 11-21 published online at https://ijassa.ipu.ru. purchase price control in high-tech product markets under competition of suppliers with multi-channel production capacities irina e. selezneva1* 1) v.a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia e-mail: ir.seleznewa2016@yandex.ru abstract: the strategic control of the purchase price for high-tech products and market competition is considered. for this purpose, a new bilateral oligopoly market model is proposed. the new model allows determining the purchase prices for high-tech products under the competition of suppliers with flexible production capacities. parametric calculations of the purchase prices, the number of competing suppliers and their production capacities are performed depending on the unit price offered by the monopsonist customer. the purchase prices offered by the customers to guarantee the competitiveness of the market are determined. the customers’ optimal pricing policy that will reduce the purchasing costs and/or ensure the existence of competing suppliers in the long run is identified. a possible price reduction for high-tech products when applying this policy is calculated. the ranges of some characteristic market parameters (the capital-output ratio, the rates of order execution and order flow, and the number of potential order sources) for which this policy is efficient are determined. keywords: strategic control of purchase prices, bilateral oligopoly market model, suppliers competition, production capacity, customer pricing policy 1. introduction as a rule, the customers and suppliers in high-tech product markets include one or several key players. in other words, these markets represent either a bilateral oligopoly, or a unilateral monopoly, or a unilateral monopsony; see a generally accepted classification in [7]. due to the strict budget constraints, it is necessary to reduce the costs of developing and manufacturing high-tech products. among other things, the high-cost prices of such products are due to the great market power of the suppliers. as a result, the prices for high-tech products are overstated, their quality is decreased, and the delivery times are violated. the monopolistic position of some manufacturers demotivates the adoption of advanced technologies, i.e., the institutional factors of product price increase amplify the effect of the technological factors. contrary to expectations, the great market power of the government as the main customer (for example, in the defense industry) does not guarantee low purchase prices. even if this market represents a monopsony (which occurs in the absence of exported products), there are usually only one or few potential suppliers. therefore, the buyer’s great market power is balanced by the great market power of the suppliers. the same applies to the markets of components. many markets of final defense industry products and their components are monopolized, primarily due to the degradation of many manufacturers during the almost 20-year crisis period. (in the soviet era, all manufacturers competed with each other for the right to develop and produce their products). in other words, the customers’ market power is at least balanced by the suppliers’ market power. the natural problems are therefore to keep the competing suppliers in the industry and simultaneously encourage new suppliers to enter the market. this paper attempts to solve the first problem * corresponding author: ir.seleznewa2016@yandex.ru mailto:ir.seleznewa2016@yandex.ru 12 i.e. selezneva copyright ©2021 assa. adv. in systems science and appl. (2021) (keeping the competing suppliers in the market in the long run) via an appropriate customer pricing policy. for example, an excessively “strict” pricing policy may withdraw the competing suppliers from the market. the paper also considers the possibility of reducing the purchase prices for high-tech products in the long run. the idea is to construct a high-tech product market model that reflects the relationship between the unit purchase price and the number of competing suppliers established in the market in the long run, depending on the unit price offered by the monopsonist customer at the initial time. a significant number of works on modeling markets of high-tech products were published in foreign economic literature. despite this fact, few results of practical value (or, at least, of methodological interest) were obtained therein. for example, in [3, 4, 11], strict assumptions on the classes of manufacturer’s cost functions and product demand functions were introduced. often even the number of players is limited (usually 1 or 2 on each side). in the papers [5, 6], the original goal was to optimize the customers’ pricing policy in the long-term and forecast the number of suppliers and their capacities. however, the cited authors investigated the application of their results (often qualitative, despite the presence of mathematical models) to antimonopoly policy, focusing on the welfare of consumers and producers (measured by their surpluses). many researchers directly indicate that the pricing mechanism is not the subject of study and describe it as the “black box.” in the paper [9], the markov models of bargaining were considered for bilateral oligopoly markets as a primary application. nevertheless, the main attention was paid to establishing links between firms in network organizations rather than to the processes of pricing, survival, or closing of enterprises in such markets. these processes will be studied below. 2. model for determining equilibrium purchase prices and number of competing suppliers in the paper [7], a model was proposed for estimating the purchase prices for high-tech products in the markets where competing suppliers have only one service channel. this paper considers a similar model, but competing suppliers have flexible production capacities. let us modify the stochastic bilateral oligopoly model [7] as follows. assume that n competing suppliers (contractors, manufacturers, etc.) and m potential customers (buyers, including the government, or system integrators of final products, etc.)) operate in a market. the suppliers and customers are supposed homogeneous. each supplier can simultaneously execute x orders for purchased products of different standard sizes (for example, using homogeneous production lines). after an appropriate reconfiguration, each production line can simultaneously execute only one order. assume that each customer implements y projects in parallel. within each of these projects, the need to conclude another contract for the supply of high-tech products occurs in pert years on average. denote by cntrt the average duration of contract execution. then n n x=  is the total number of production lines (service channels) in this market, and m m y=  is the total number of parallel projects implemented by the customers. the prices in the bilateral and unilateral monopoly markets are exogenous parameters, and the price in the unilateral monopsony market is the control variable of the model. let the number m of potential order sources be fixed. then the number n of the suppliers remaining in the market, as well as the number x of their service channels, can be treated as the model variables and optimized in the following way. each supplier optimizes its production capacity (the number of service channels) regardless of the other suppliers. therefore, each supplier can be represented as a multi-channel closed queuing system with the simultaneous execution of orders from several consumers; for details, see the paper [11]. in real networks, the centers of competence operate exactly in such a mode, optimizing their load and saving the available resources. the equilibrium market condition arising under the competition of suppliers in the high-tech product market will be forecasted in two stages; see formulas (2.1) and (2.2) below. purchase price control in high-tech product markets under competition of suppliers… 13 copyright ©2021 assa. adv. in systems science and appl. (2021) 1) for a fixed unit price of the monopsonists, we optimize the total number of service channels available to an individual supplier by maximizing its total profit: [ ( ; ) ] max,monopsony x п x p p x q avc q fc=   −  − → (2.1) where п is the total profit of an individual supplier; x is the number of its service channels; p is the weighted average unit purchase price of high-tech products; monopsonyp is the unit price in the unilateral monopsony market; q is the average annual output under one contract; avc is the average variable costs; fc is the fixed costs of maintaining one service channel. the solution of this problem yields the optimal production capacity optx of an individual supplier. 2) we estimate the possible presence of profitable suppliers in the market. for a fixed unit price of the monopsonists and the corresponding optimal production capacity optx , we find the maximum possible number of suppliers in the market under which the profit of an individual supplier (or service channel) remains positive: max : [ ( ; ) ] 0.opt monopsony optn п x p p n x q avc q fc→ =    −  −  (2.2) the solution of this problem yields the maximum possible number maxn of profitable suppliers in the market. following this approach, we construct the dependence of the weighted average unit purchase price on the monopsonist’s unit price. therefore, this model allows determining the equilibrium purchase prices established in the market in the long run, the number of competing suppliers, and the optimal capacity of each supplier, depending on the monopsonist’s unit price. note that this paper considers a static (steady-state) mode of operation of the queuing system. the steady-state mode is achieved at times exceeding those of all system processes (order receipt and execution). moreover, all conclusions will be valid if the forecasting horizon exceeds the order receipt and execution times. 3. parametric calculations of optimal purchase price offered by customer to competing suppliers with multi-channel production capacities using an example typical for some branches of the high-tech industry, we calculate the purchase prices for high-tech products and oligopoly surcharges to the cost price due to the market power of competing suppliers. let the unit product prices in the unilateral and bilateral monopoly markets be monopolyp = 200 monetary units (m.u.) and bilateralp = 150 m.u., respectively. example 1. assume that: the average annual output under one contract is q = 24 units; the unit average variable costs are avc = 95 m.u.; the unit fixed costs of maintaining one service channel are fc = 120 m.u. these parametres are typical for some branches of the high-tech industry. that is, afc = 5 m.u. and ac = 100 m.u., but only for an active service channel loaded by one order. roughly speaking, such a production cost structure is typical for the main subsectors of the foreign aircraft industry; see [1, 2, 10]. the average fixed costs constitute no more than 5% of the total production costs, but only for the enterprises executing orders (and not the idle ones). consider three average durations of contract execution, cntrt = 2, 5, and 20 years, and two average periods between the orders, pert = 1 and 5 years. we explore the cases of one and several potential sources of orders for high-tech products (m = 1 and m = 2,…, 10, respectively). for each case, the weighted average unit purchase prices for high-tech products calculated depending on the customer’s pricing policy are shown in figure 3.1. 14 i.e. selezneva copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 3.1. dependence of weighted average unit purchase price on customer policy (capital-output ratio 5%; order execution rate 0.2; order flow rate 1). suppose that the unit price offered by the monopsonists is close to the average variable costs. in this case, the competing service channels cannot survive in the long run: they can only succeed if they periodically receive a higher unit price in the bilateral or unilateral monopoly conditions. at the same time, when the unit purchase price in a competitive supplier environment (i.e., a parameter controlled by the customers with market power) exceeds some threshold allowing the competing service channels to cover the number of potential order sources profitably ( n m ), the average unit purchase price can sharply drop: in any state of the queuing system there will be free service channels, and the average unit purchase price will coincide with the unit price offered by the customers, .monopsonyp p moreover, the average unit purchase price will increase infinitely with the growth of the unit price offered by the customers. thus, the dependence of the average unit purchase price on the monopsonist’s unit price, ( )monopsonyp p p= , may achieve minimum at some price exceeding the minimum possible one in the short run ( monopsonyp avc ). even being monopsonists, the customers should assign a high enough unit purchase price for high-tech products to attract the competing suppliers and contractors. in this case, the customers will reduce the purchasing costs in the long run, avoiding the suppliers’ monopoly. figure 3.1 shows the gain from applying the pricing strategy proposed in this paper, i.e., a decrease in the average unit purchase prices with an increase in the monopsonist’s unit price. in all cases, the monopsonist can select such a unit price so that the total production capacity in the market will exceed the number of potential order sources, i.e., in any state of the queuing system, the market will be competitive. tables 3.1 and 3.2 present some fragments of the calculated data for the model. the green row in these tables corresponds to the monopsonist’s optimal unit price and the minimum weighted average unit purchase price. 95 115 135 155 175 195 215 95 98 101 104 107 110 113 116 119 122 125 128 131 134 137 140 143 146 149 w ei gh te d a ve ra ge u n it p u rc h as e p ri ce o n cu st o m er p o lic y, m .u . unit price offered by customer, m.u. m=1 m=2 m=3 m=4 m=5 m=6 m=7 m=8 m=9 m=10 purchase price control in high-tech product markets under competition of suppliers… 15 copyright ©2021 assa. adv. in systems science and appl. (2021) in this case, compared to the saving-based purchasing policy, the weighted average unit purchase price may decrease by 7–47% for cntrt = 20 years and pert = 5 years; by 11–47% for cntrt = 5 years and pert = 1 year; by 27–48% for cntrt = 20 years and pert = 1 year; up to 45% for cntrt = 2 years and pert = 1 year. for cntrt = 2 years, pert = 1 year, and m = 9, there is no characteristic decrease in the weighted average unit purchase price. the optimal unit purchase price in the market is defined as the minimum weighted average unit purchase price at which the total production capacity in the market exceeds the number of potential order sources (the market is competitive). for m varying from 1 to 10, the optimal unit purchase prices are as follows: for cntrt = 20 years and pert = 5 years, 108, 108, 104, 105, 103, 105, 103, 103, 102, and 103 m.u., respectively; for cntrt = 5 years and pert = 1 year, 108, 108, 104, 105, 103, 104, 102, 103, 104, and 103 m.u., respectively; for cntrt = 20 years and pert = 1 year, 106, 106, 106, 103, 102, 103, 103, 103, 104, and 103 m.u., respectively; for cntrt = 2 years and pert = 1 year, 111, 107, 106, 107, 105, 107, 105, 107, 106, and 107 m.u., respectively. table 3.1. fragments of calculated data based on the model (capital-output ratio 5%; order execution rate 0.2; order flow rate 1; the number of potential order sources 1) m=1 monopsonyp p optx maxn max optn n x=  95 150 1 1 1 102 150 1 1 1 107 150 1 1 1 108 108 1 2 2 109 109 1 2 2 114 114 1 3 3 120 120 1 4 4 126 126 1 5 5 132 132 1 6 6 138 138 1 7 7 144 144 1 8 8 149 149 1 8 8 table 3.2. fragments of calculated data based on the model (capital-output ratio 5%; order execution rate 0.2; order flow rate 1; the number of potential order sources 5) m=5 monopsonyp p optx maxn max optn n x=  95 177 3 1 3 101 177 3 1 3 103 103 3 2 6 104 104 3 2 6 106 106 3 3 9 110 110 3 4 12 114 114 3 5 15 117 117 3 6 18 139 139 3 12 36 142 142 3 13 39 149 149 3 14 42 16 i.e. selezneva copyright ©2021 assa. adv. in systems science and appl. (2021) example 2. within the data of example 1, let the annual fixed costs of maintaining one service channel be fc = 600 m.u. thus, the share of the fixed costs in the cost structure now constitutes almost 21% of the total costs; see figure 3.2. fig. 3.2. dependence of weighted average unit purchase price on customer policy (capital-output ratio 20%; order execution rate 0.2; order flow rate 1) figure 3.2 demonstrates well the gain from applying the pricing strategy proposed in this paper. in this case, compared to the saving-based purchasing policy, the weighted average unit purchase price may decrease by 34% for cntrt = 20 years and pert = 5 years or pert = 1 year; by 35% for cntrt = 5 years and pert = 1 year; by 33% for cntrt = 2 years and pert = 1 year. tables 3.3 and 3.4 present some fragments of the calculated data for the model. the green row in these tables corresponds to the monopsonist’s optimal unit price and the minimum weighted average unit purchase price. for cntrt = 20 years and pert = 5 years, or cntrt = 5 years and pert = 1 year, the monopsonist cannot select a unit purchase price so that the total production capacity in the market will exceed the number of potential order sources if m=1 and m=2. in these cases, the total production capacity may even be equal to the number of potential order sources. the same situation is observed for cntrt = 2 years and pert = 1 year if m=1, m=2, m=4, m=6, m=8, and m=10. for example, a single customer always has to interact with a single supplier. in other cases, only the equality between the total number of service channels and the number of potential order sources can be achieved. a characteristic decrease in the weighted average unit purchase price will not occur for cntrt = 20 years and pert = 5 years if m=1, m=2, and m=3; for cntrt = 5 years and pert = 1 year if m=1, 120 130 140 150 160 170 180 190 200 95 98 101 104 107 110 113 116 119 122 125 128 131 134 137 140 143 146 149 w ei gh te d a ve ra ge u n it p u rc h as e p ri ce o n cu st o m er p o lic y, m .u . unit price offered by customer, m.u. m=1 m=2 m=3 m=4 m=5 m=6 m=7 m=8 m=9 m=10 purchase price control in high-tech product markets under competition of suppliers… 17 copyright ©2021 assa. adv. in systems science and appl. (2021) m=2, m=3, and m=4; for cntrt = 2 years and pert = 1 year if m=1 and m=5; for cntrt = 20 years and pert = 1 year if m=1 and m=2. for m varying from 3 to 10, the optimal unit purchase prices are as follows: for cntrt = 20 years and pert = 5 years, 137, 142, 133, 142, 131, 135, 130, 133 m.u., respectively; for cntrt = 5 years and pert = 1 year, 136, 141, 132, 141, 130, 133, 129, and 132 m.u., respectively; for cntrt = 20 years and pert = 1 year, 148 m.u. (m = 1, 2, 3) and 135, 127, 131, 133, 135, 131, and 132 m.u. for m varying from 4 to 10, respectively; for cntrt = 2 years and pert = 1 year, 146, 141, 133, 144, and 146 m.u. for m = 3, 5, 6, 7, and 9, respectively. table 3.3. fragments of calculated data based on the model (capital-output ratio 20%; order execution rate 0.2; order flow rate 1; the number of potential order sources 7) m=7 monopsonyp p optx maxn max optn n x=  95 190 4 1 4 100 190 4 1 4 105 190 4 1 4 110 190 4 1 4 120 190 4 1 4 125 190 4 1 4 130 130 4 2 8 136 136 4 2 8 140 140 4 2 8 147 147 4 3 12 148 148 4 3 12 149 149 4 3 12 table 3.4. fragments of calculated data based on the model (capital-output ratio 20%; order execution rate 0.2; order flow rate 1; the number of potential order sources 9) m=9 monopsonyp p optx maxn max optn n x=  95 196 5 1 5 100 196 5 1 5 105 196 5 1 5 110 196 5 1 5 120 196 5 1 5 129 129 5 2 10 144 144 5 2 10 145 145 5 2 10 146 146 5 3 15 147 147 5 3 15 149 149 5 3 15 18 i.e. selezneva copyright ©2021 assa. adv. in systems science and appl. (2021) example 3. consider now another example in which the annual fixed costs of maintaining a single service channel are fc = 1000 m.u. compared to the previous scenario, the share of the fixed costs in the cost structure is even higher, reaching 31% of the total costs; see figure 3.3. fig. 3.3. dependence of weighted average unit purchase price on customer policy (capital-output ratio 30%; order execution rate 0.2; order flow rate 1) in this case, compared to the saving-based purchasing policy, the weighted average unit purchase price may decrease by 28% for cntrt = 20 years and pert = 5 years; by 23% for cntrt = 20 years and pert = 1 year if m=5; by 25% for cntrt = 5 years and pert = 1 year; by 21% for cntrt = 2 years and pert = 1 year. tables 3.5 and 3.6 present some fragments of the calculated data for the model. the green row in these tables corresponds to the minimum weighted average unit purchase price; in table 3.6, it also corresponds to the monopsonist’s optimal unit price. the monopsonist can select a unit purchase price so that the total production capacity in the market will exceed the number of potential order sources only in the case cntrt = 20 years, pert = 1 year and m = 5. the optimal unit purchase price in the market is 148 m.u. for example, for cntrt = 20 years and pert = 5 years, a single order source always has to interact with a single service channel. for m = 2, 3, 4, 6, and 8, it is possible to achieve the equality between the number of potential order sources and the total capacity in the market. for m = 5, it is possible to attract a maximum of 4 service channels to the industry. for m = 7, 9, and 10, only 4, 5, and 6 service channels, respectively, can survive in the market. for cntrt = 20 years, pert = 1 year, and m = 1, 2, 3, and 4, only the equality between the number of potential order sources and the total capacity in the market is possible. for m = 6, 7, 8, 9, and 10, only 4, 5, 5, 6, and 7 service channels, respectively, can survive in the market. 140 150 160 170 180 190 200 95 98 101 104 107 110 113 116 119 122 125 128 131 134 137 140 143 146 149 w ei gh te d a ve ra ge u n it p u rc h as e p ri ce o n cu st o m er p o lic y, m .u . unit price offered by customer, m.u. m=1 m=2 m=3 m=4 m=5 m=6 m=7 m=8 m=9 m=10 purchase price control in high-tech product markets under competition of suppliers… 19 copyright ©2021 assa. adv. in systems science and appl. (2021) for cntrt = 5 years and pert = 1 year, a single order source always has to interact with a single service channel. for m = 2, 3, 4, and 6, it is possible to achieve the equality between the number of potential order sources and the total capacity in the market. for m = 5, a maximum of 4 service channels are possible. for m = 7, 8, 9, and 10, only 4, 5, 5, and 6 service channels, respectively, can survive in the market. for cntrt = 2 years, pert = 1 year, and m = 1, the monopsonist customer cannot select a unit purchase price so that at least one supplier will profitably operate in the market. in the other cases, the total capacity is smaller than the total number of potential order sources. for example, for m = 2, 3, 9, and 10, only 1, 2, 4, and 5 service channels, respectively, can survive in the market. for m = 4, 5, 6, 7, and 8, a maximum of 3, 4, 4, 6, and 4 service channels, respectively, can be achieved in the market. a characteristic decrease in the weighted average unit purchase price will not occur with an increase of the monopsonist’s unit price for cntrt = 20 years and pert = 5 years if there are 1, 7, 9, and 10 potential order sources in the market; for cntrt = 5 years and pert = 1 year, if m = 1, 5, and 7–10; for cntrt = 2 years and pert = 1 year, if m = 1–3, 6, 9, and 10; for cntrt = 20 years and pert = 1 year, if m = 1–4 and 6–10. table 3.5. fragments of calculated data based on the model (capital-output ratio 30%; order execution rate 0.2; order flow rate 1; the number of potential order sources 6) m=6 monopsonyp p optx maxn max optn n x=  95 194 3 1 3 100 194 3 1 3 110 194 3 1 3 120 194 3 1 3 130 194 3 1 3 142 145 3 2 6 144 146 3 2 6 145 147 3 2 6 146 148 3 2 6 147 148 3 2 6 148 149 3 2 6 149 149 3 2 6 table 3.6. fragments of calculated data based on the model (capital-output ratio 30%; order execution rate 0.05; order flow rate 1; the number of potential order sources 5) m=5 monopsonyp p optx maxn max optn n x=  95 193 3 1 3 100 193 3 1 3 105 193 3 1 3 110 193 3 1 3 120 193 3 1 3 129 193 3 1 3 144 193 3 1 3 145 193 3 1 3 147 193 3 1 3 20 i.e. selezneva copyright ©2021 assa. adv. in systems science and appl. (2021) 148 148 3 2 6 149 149 3 2 6 4. conclusions this paper has proposed a model for determining the purchase prices for high-tech products in the markets of competing suppliers with flexible production capacity. as has been established, in some cases, the customer benefits by offering a higher price for high-tech products to ensure the existence of competing suppliers and (or) to reduce the purchase price in the long run. under a higher value of the capital-output ratio, the gain from applying the proposed pricing strategy decreases. compared to the saving-based purchasing policy, the achievable decrease in the weighted average unit purchase price is reduced. for example, under a low value of the capital-output ratio, it is possible to reduce the unit purchase price by 48%; under medium and high values of the capital-output ratio, by 35% and 28%, respectively. in addition, the optimal unit purchase prices for which the market will be competitive in any state increase with the growth of the capital-output ratio. for example, under a low value of the capital-output ratio, the optimal unit purchase price is 103–111 m.u.; under medium and high values of the capital-output ratio, 127–146 m.u. and 148 m.u., respectively. as a rule, the relative gain from applying the proposed pricing strategy decreases with an increase in the order execution rate or a decrease in the order flow rate. in these cases, as a rule, the optimal unit purchase prices increase and/or the opportunities to attract the competing suppliers to the industry decrease. under a low value of the capital-output ratio, the customer always selects a price so that the market in any state will be competitive: there is always a choice from the competing suppliers. under a medium value of the capital-output ratio, the customer selects the optimal price with a low value of the order execution rate and a high value of the order flow rate; in other cases, as a rule, with a middle or high number of potential order sources. with a high value of the capitaloutput ratio, the customer finds an optimal price only with a low value of the order execution rate, a high value of the order flow rate, and many potential order sources. a decrease in the weighted average unit purchase price with an increase in the monopsonist’s unit price occurs under a low value of the capital-output ratio. however, if the order execution rate is high, this is possible only under a small or medium number of potential order sources. under a medium value of the capital-output ratio, a decrease in the unit purchase price occurs for a high value of the order execution rate and a certain number of potential order sources; in other cases, only for a medium or high number of potential order sources. under a high value of the capital-output ratio, a decrease in the unit purchase price occurs in each case only for a certain number of potential order sources. references 1. aircraft engine and engine parts manufacturing: 1997, 2002, 2007 (1999, 2004, 2009), economic census. manufacturing. industry series. u.s. census bureau. 2. aircraft manufacturing (1999, 2004, 2009), economic census. manufacturing. industry series. u.s. census bureau. 3. collard-wexler a., gowrisankaran g., lee r.s. (2014). bargaining in bilateral oligopoly: an alternating offers representation of the “nash-in-nash” solution. [online]. available https://pdfs.semanticscholar.org/f39c/ce5b1ea82072f260144229a932577cc73d20.pdf 4. dickson a., hartley r. (2013). bilateral oligopoly and quantity competition, economic theory, 52, 979–1004. https://doi.org/10.1007/s00199-011-0676-9 purchase price control in high-tech product markets under competition of suppliers… 21 copyright ©2021 assa. adv. in systems science and appl. (2021) 5. funaki y., houba h., motchenkova e. (2019). market power in bilateral oligopoly markets with non-expandable infrastructures, international journal of game theory. https://doi.org/10.1007/s00182-019-00695-z 6. hendricks k., mcafee r.p. (2010). a theory of bilateral oligopoly, economic inquiry, 48(2), 391–414. https://doi.org/10.1111/j.1465-7295.2009.00241.x 7. klochkov v.v. (2012). ekonomika: uchebnoe posobie dlya vuzov. moscow: infra-m, [in russian]. 8. klochkov v.v., selezneva i.e. (2020). a stochastic model of bilateral oligopoly and price control for competition on high-tech product markets, automation and remote control, 81(2), 340–357. https://doi.org/10.1134/s0005117920020125 9. lee r.s., fong k. (2013). markov-perfect network formation. an applied framework for bilateral oligopoly and bargaining in buyer-seller networks. [online]. available https://pdfs.semanticscholar.org/7559/d599bcb0e3a1ff5b635c163fbf89ce22a469.pdf 10. other aircraft parts and auxiliary equipment manufacturing: 1997, 2002, 2007 (1999; 2004, 2009), economic census. manufacturing. industry series. u.s. census bureau. 11. rubinstein a. (1982). perfect equilibrium in a bargaining model, econometrica, 50(1), 97– 109. https://doi.org/10.2307/1912531 12. taha, h.a. (1997). operations research: an introduction. upper saddle river: prentice hall. https://link.springer.com/journal/182 1. introduction 2. model for determining equilibrium purchase prices and number of competing suppliers 3. parametric calculations of optimal purchase price offered by customer to competing suppliers with multi-channel production capacities 4. conclusions adv syst sci appl 2018; 04; 52-63 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/664 costs function optimization for prevention of firm’s industrial risks with regard to reinvestment of profit mikhail geraskin1, elena rostova2* 1) samara national research university, samara, russia e-mail: innovation@ssau.ru 2) samara national research university, samara, russia e-mail: el_rostova@mail.ru received february 18, 2018; revised november 9, 2018; published december 31, 2018 abstract: we consider the problem of the firm’s cost function optimization to prevent the industrial risks (i.e., the voluntary risk costs), by the criterion of the total costs with regard to a reinvestment of the firm’s profit. in the paper the voluntary risk costs function of the firm in the perfect competition by the total costs criterion is optimized. the rational strategy of the firm’s risk management consists in a gradual increase of the costs to prevent the risk situations with the production volume growth. the voluntary risk costs function is determined for the power function of the production costs and the exponential decreasing function of the industrial damage for the firm in the perfect competition. keywords: industrial risk, industrial damage, costs, optimization, risk management. 1. introduction the industrial risks are treated as the risks, arising in the commercial activity of the industrial firms. these risks were comprehensively considered in the economic theory. the problems of the industrial risks management cover various levels of the risks: first, the risks, connected with firm’s entry into the world market (i.e., risk of globalization) [13], second, the risks of commercial activity’s impact on the regional economic systems [20], [22], [26], and the risks, caused by the non-performance of some technological operations. in the context of the further analysis, the risks are considered in the firm’s level. in this aspect, the traditional reactive environmental management and the active enterprise’s risk management (erm) [30] was compared as the behaviors of the firm's management in the conditions of an uncertainty. the effects of the equipment maintenance frequency [6] and the organizational environment in industrial damage were analyzed: the risk analysis was conducted in the framework of the unified modeling language (uml) models [5], the analysis method of dysfunctional systems (mads), and then the risk management is simulated to ensure the safety of personnel from the possible industrial hazards. the effectiveness of the erm system was investigated for nine firms from various industries [2]. the risks of the goods and resources supply breaks chain were investigated as part of the technological process within the firms [29], and in the external supply chain [19]. researches of the decision-making models under conditions of uncertainty [21] proved an applicability of the system approach to the complex challenges modeling of the industrial equipment maintenance imperfection on the basis of the uml model and the method organized for a systemic analysis of risk (mosar) [12]. additionally, authors investigated the regional level risks [26]. for example, shelkov a. b. investigated the problems of the regional security [16], [24], [15] by means of the scenario approach. the risks in the firms were analyzed taking into account a human factor [1] by means of the business games [7], the mechanisms of penalties [9], and on the basis of the innovations [3], [11]. *corresponding author: el_rostova@mail.ru mailto:el_rostova@mail.ru costs function optimization for prevention of firm’s industrial risks 53 in general, it was shown [30], [4] that the production activity in any economy sector is connected with the uncertainty and the risk. in contrast to the aforementioned authors, we consider the problem of the firm’s total costs minimization. the solution of the problem enables to define the voluntary risk costs function. additionally, we determine the production volume, when the firm’s profit is maximum for derived voluntary risk costs function with regard to a reinvestment of the firm’s profit. further the industrial risk is treated [28], [25], [17] as a certain technogenic factor of the probabilistic nature, resulting in a loss of the firm’s assets. consequently, the industrial risk leads to the additional firm’s expenses, but does not lead to the revenue decrease. the risk measure is the mathematical expectation of the damage from risk factors (e.g., accident, incident, equipment stopping, production failure, etc.), measured in the monetary units of the firm’s costs; the degree of risk represents the probability of this event. the risk costs are divided into two components [8]: 1) the unexpected expenses (e.g., the penalties for an exceeding the extent of environmental damage over existing norms); 2) the planned expenses, caused by a necessity of the risk situation prevention or a decrease in the risk emergence probability. we divide the risk damage in the internal damage and the external damage in relation to the firm. the internal damage includes assets loss and, additionally, the expenses for a mitigation of the consequences and recovery works. the size of the possible internal damage is proportional to the firm's output, which growth increases the production capacity of the firm (i.e., production facilities, equipment, personnel). consequently, the potentially internal damage is affected an assets increases. therefore, the reduction of the possible internal damage can be reached by the additional expenses on the actions for the risk decrease, which we call further the voluntary risk costs (vrc). the structure of the vrc includes the costs of the following types: the training personnel, the advanced equipment installation, the sewage treatment plants installation, the control systems and alarms, etc. the firm's external damage (called further the obligatory risk costs) we accept equal to a sum of the penalties, imposed on the firm for an excess of the environmental damage level over the established norms. thus, the total costs of the industrial firm, among with the production costs, additionally include the vrc, the obligatory risk costs, and the internal damage. further, the model of the firm without regard to the obligatory risk costs is considered. the choice of the vrc size is predetermined by the following contradictory factors. on the one hand, a reduction of these costs leads to the firm’s total costs reduction that contributes to the profit’s growth and further expansion of the production; however, at the same time, an increasing in the internal damage and the obligatory risk costs, on the contrary, increases the total costs. on the other hand, an increase in the vrc leads to a decrease in the internal damage and the obligatory risk costs, however, the total costs of the firm increase. additionally, a reinvestment of the firm’s profit causes contradictory influence on the potential costs, revenue and assets in the future periods: the using of the current profit for the risk prevention measures leads to the further production decreasing, and, to an increasing in the further vrc and the obligatory risk costs. consequently, a reinvestment of the firm’s profit influences on the firm’s vrc function in the current period, because the optimal vrc function is derived, taking into account the both tendencies. thus, the relevant problem of the industrial risks management is the choice of firm’s vrc function, which is optimized by criterion of the total costs. additionally, the problem of the firm under the vrc optimal function consists in optimization of the output by criterion of the profit. the problems represent the model of the firm’s management taking into account the industrial risks. this problems is the subject of the paper. 54 m. geraskin, e. rostova copyright ©2018 assa adv. in systems science and appl. (2018) 2. methodological framework to search for the optimal vrc function, the firm’s total costs are decomposed into the production costs, the internal damage costs, and the vrc. in contrast to the traditional cost accounting methodology [10], according to which the last two components are included in the production costs, we consider the risk costs separately. the problem of the determining a nonnegative, real, limited from above function vrc is considered, taking into account the minimization of the firm’s total costs function: )},,0(,)(:)({ )),(,(minarg)( maxmaxmax )( *       cfffrfa fqcqf f af f (2.1) where q is the output, c(q) is the production costs function, f(q) is the vrc function, fmax is the vrc maximum possible amount. the symbol «*» indicates the optimal value. the firm’s total costs function cς(q, f) is considered in the two types. first, the total costs in the current period (t=0), indicated by )(0 c , is considered as a function of the following type: ,)(),,()()(),( max 00000   сcfqxqfqcfqc (2.2) where x(q,f) is the internal damage function, сmax is the maximum possible total costs of the period on the basis of the firm’s manufacturing capacity. second, the cumulative costs of the future periods is considered as a sum of the costs )( c up to the period τ, which is discounted to the current period: ,)(),,()()(),( , )1( ),( max 1         ccfqxqfqcfqc r c fqc ttttt t t t  (2.3) where r is the discount rate, t is the index of the period. we formulate the firm’s main objective as the problem of the output choice under the following conditions: first, the output is limited from above, second, it maximizes the profit, and, third, the vrc function is optimal according to (2.1). thus, the problem is as follows: },0,:{ )),(,(пmaxarg maxmax **     qqqrqa qfqq q aq q    (2.4) пτ(q, f)=rτ – cτ σ(q, f*), (2.5) rτ (q)=pqτ. (2.6) where пτ, rτ are the profit and the revenue in the period τ, respectively, p is the product price, qmax is the maximum possible manufacturing output taking into account the production capacity. the profit function (2.5) doesn’t include the increment from prevention of industrial risk. because, according to the accepted in the article concept, the risk costs reduce in the internal damage from the industrial risk, but it does not affect the revenue growth. we introduce the following assumptions that determine the limits of the model (2.1), (2.3) applicability on the basis of the generally accepted provisions of the economic theory. 1. the hypothesis of the perfect competition in the goods market [28]: for the firm the product price is an exogenous variable, that is, the firm does not affect the market price: .0)(  qpq 2. the hypothesis of the return from production increased scale, which corresponds to a relatively large firm [29]: .0)(  qcq costs function optimization for prevention of firm’s industrial risks 55 3. the hypothesis of the control parameters influence on the internal damage [10]: an increase in the volume of the production assets leads to an increasing in the damage; the internal damage is reduced with an increase in the vrc, the internal damage is limited from above due to the nature of technology and limited by the production volume: ,0],,0[),(,0),(,0),( maxmax  xxfqxfqxfqx fq where xmax is the maximum possible internal damage. 4. the hypothesis of the reinvested profit influence [18]: the profit in the current period is invested only in the fixed capital, that is, the reinvested profits affects the intensity (flow) of the capital, but does not affect the flow of other resources in the future periods. the costs are covered by the revenue of the corresponding period. therefore, the production function of the firm in the t-th period is: ,)],([  ttt lxfkq  where tt lk ),( are the capital flow and non-capital resources flow in the t-th period, α, β are the elasticity coefficients of resources. here the cobb-duglas’s function (i.e., the constant elasticity production function) is considered, but the further results are not proved for the general type of the constant elasticity function. the production costs function and the damage function, satisfying the hypotheses 2, 3, have the forms [10], [32] ,0],2,1(],,1(,)( maxmax  bbqqc tt  (2.7) .0)(],1,0(],,0(,)(),( maxmax   qt f tt qeqfqx   (2.8) the power-type production cost function (2.7) [10], [32] is introduced for a long-term period, because it includes only variable costs. the damage function (2.8) expresses the exponential distribution of damage [14], which corresponds to man-made accidents. therefore, the solution of the problems (2.1), (2.3) with the functions (2.7), (2.8) is not general, but it corresponds to the typical firm’s characteristics. under consideration the vrc and the industrial damage, in the t-th period, the capital flow is equal to the initial capital 0k , which is reduced by the amount of the not reinvested profit; the reduction in the profit in accordance with (2.1), (2.5) is equal to the amount of the vrc and the industrial damage:           1 1 1 1 0 )1()1( t i i ii t i i i t r xf r kk  , (2.9) where i is the potential profit of the company in the i-th period, which is calculated according to the model (2.5) without taking into account the risk costs, that is, when 0),(,0)(  iiiii fqxqf . in contrast to the classical model [10] of the firm’s capital growth, the model of the capital flow (2.9) describes the process of capital growth as the annual returns, and the process of the capital reduction due to the influence of the damage and the vrc, which enables to determine the risk costs. taking into account hypothesis 4, the production function in the t-th period is:    t t i i ii t i i i t l r xf r kq                   1 1 1 1 0 )1()1( . (2.10) here the financial capital is considered, that is, the capital is equal to the share capital and the retained earnings. let’s consider the optimal control problem: to search for the pair <f*(∙), q*>, which is optimized by the criteria (2.1), (2.3) on the respective admissible sets for the costs function and the damage function (2.6), (2.7), and taking into account the restrictions (2.10). 56 m. geraskin, e. rostova copyright ©2018 assa adv. in systems science and appl. (2018) 3. results in the first stage, we define the vrc function f*(∙) under condition (2.1) in the following propositions. proposition 1. for the continuously differentiable functions с(∙), f(∙), and the function x(∙) of the type (2.8) the function )](ln[ 1 )(* qqf    (3.1) is the solution of the problem (2.1) qaq at .1)( q proof. let’s solve the problem (2.1) by the lagrange multipliers method. this method is applicable as the criterion function 0 с (q, f) is continuously differentiable, the function x(q, f) – χ(q) e– ξf=0 is continuously differentiable with the partial derivatives, which are not equal to zero simultaneously (the equation x (q, f) – χ (q) e– ξ=0 defines a convex curve). we write the lagrange function: l(q, f, λ)=c (q)+f +x (q,f) + λ(x (q, f) – χ (q) e– ξ). (3.2) we search for the partial derivatives, which are equal to zero:           .0)( ,0)(1 ,01 f f f x eqxl eql l       we solve the resulting system:          ,)( ,)( ,1 f f eqx eq      and we derive the vrc function, subject to the total costs minimizing: ))(ln( 1 )(* qqf    . we check for the lagrange function (3.2) the fulfillment of the minimum sufficient conditions at f=f*. for this purpose we define a sign of * 2 | ffld  : 22222 )(2 dfeqdqldqdfldflld f qqqfff   . 0| 2 * 2  dfld ff  . this condition are fulfilled qaq . then function f*(q) is the solution of the problem (2.1) for ),(0min fqc faf   in the case of .1)( q █ proposition 2. for the continuously differentiable functions сt(∙), ft(∙), xt(∙) and qt form (2.10) the function )](ln[ 1 )( * qqf tt    (3.3) is the solution of the problem (2.1), if the costs functions are of the type (2.2) and (2.3) qaq at 1)( q and .02 2   proof. let’s solve the problem (2.1), (2.2), (2.3) by lagrange multipliers method. this method is applicable as the criterion function cς(q, f) is continuously differentiable, the function (2.10) is continuously differentiable with partial derivatives that are not equal to zero simultaneously. we write the lagrange function: costs function optimization for prevention of firm’s industrial risks 57 l(q, f, λ)= )(),(    tt lkqfqc  , (3.4) where q=(q1, q2, … qτ), f=(f1, f2, … fτ). we transform the function (3.4): ).( )1( )( ),,( 1       ttt t t f ttt lkq r eqfbq fql t        (3.4') we search for the partial derivatives of the function (3.4), which are equal to zero:                          .0 ,0 )1( )(1 ,0 )1( ' 1 1 1          ttt t t f t f t t f t q lkql r eq l r eqb l t t t t we solve the resulting system, and transform the second equation of the system:           11 )1( )( )1( 1 t t f t t t r eq r t . we can write the last equality as:    ,1 )1( )( )1( 1      t r eq r t f t t t or .,11)(     teq tf t here we derive the vrc function, subject to the total costs minimizing in t-th period: ))(ln( 1 )(** qqff ttt    . we will check for the lagrange function (3.4') fulfillment of the minimum sufficient conditions at f=f*. for this purpose we define a sign of * 2 | ffld  . . )1()1( '1 2) )1( '')1( |)2( 2 1 2 1 2 1 * 222 2 df r e dqdf r e dq r eqb dfdqldfldqlld t t f t t f t t f t fffqffqq ttt                         . .) 2 )1(( )1( 1 | 2 1 2 2 2 * 2 dqqb r ld t ttff                  .020| 2 * 2  ifld ff this condition are fulfilled qt aq  . then function ft*(q) is the solution of the problem (2.1) for ),(min fqc faf    in the case of 1)( qt . █ the logarithmic function of the firm’s output in the t-th period (3.1) minimizes the total costs under the exponentially decreasing function of the industrial damage. this function enables to determine the level of the vrc, which minimizes the total cost, depending on the production volume and the parameter ξ, characterizing the vrc efficiency. in the next stage, we determine the production volume, when the firm’s profit is maximum taking into account the function (3.1). proposition 3. for the continuously differentiable functions с(∙), f(∙),χ(∙) the equation 58 m. geraskin, e. rostova copyright ©2018 assa adv. in systems science and appl. (2018)        1qbp (3.5) is the solution of the problem (2.4), (2.5), (2.6), if 02   . (3.6) thus, when q0* satisfies (3.5) and (3.6), the profit function (2.5) has the maximum value. proof. let’s determine the production volume qτ* in the current period, in which the criterion function of the profit reaches its maximum value. for this purpose, we search for the partial derivative of function (2.5) on qt, and we equate it to zero taking into account that ft=ft* according to (3.1): 0    qqq crп  . 01          qbpп q .        1qbp . we check the fulfillment of the maximum sufficient condition of the function п(q, f) subject to (3.1): .)1()1( 2 2 2 22 2 2                    qbqbп qq 0 qqп at qaq   , if 02   .█ we consider two variants of the function χ(q): the power function and the exponential function. if χ(q)=qn, then from (3.1) the follows function may be written: q n qqf n ln ln )ln( 1 )(*      . (3.1') if χ(q)=eq, then from (3.1) the follows function may be written:     q eqf q  ln )ln( 1 )(* . (3.1'') the results of the simulation of these functions are presented in tables 1, 2. the calculations are performed in the program maple for the various parameters, whose values are indicated for each variant of the model. for the function χ(q)=q3, under all values of the parameters γ and ξ, satisfying (2.7) and (2.8), respectively, the problem (2.1) has no solution for sufficiently small values of the price p and the parameter b (model 1). that is, with low market price and low coefficient of the production costs function, the firm produces such low volume of the product that does not have revenue, sufficient for making the vrc. with the price growth, the revenue increases, and, as a result of the solution (3.4), we receive two possible values of q*, from which only q*<1 satisfies (3.5) (model 2). with higher values of the parameter b, the production costs increase, therefore, the problem (2.1) has no solution (models 3 to 5) with higher levels of the internal damage and the vrc. that is, the production costs are so high that the firm’s revenue does not cover the expenses for the control of the risks. for the function χ(q)= eq, under small values of the price p (model 6), the problem (2.1) has no solution for the reason, described above for model 1. the problem (2.1) has the solution, when the price increases that speaks about the increased revenue, which provides the vrc (model 7). an increasing in the parameters γ and b leads to an increase in the production costs that does not leave to the firm the revenue for making the vrc, therefore, the problem (2.1) has no solution under (2.8), (2.9). model 10 illustrates the ratio of the parameters, when the price is high enough to bring the revenue that is able to cover the expenses for the control of the risks. costs function optimization for prevention of firm’s industrial risks 59 let’s consider the simulation of optimal mechanisms (3.1), (3.5) with example of the real industrial company (samara bearing plant) at various values of p, b, γ, ξ, presented in table 1. table 1. results of simulation for χ(q)=q3. model number p b ξ γ q* 1 1 1 (0; 1] (1; 2] q*r 2 100 1 0,7 1,2 q*1=0,08 q*2=4018775720 3 100 [80; +∞) 0,7 1,2 q*r 4 1000 1000 0,7 1,2 q*1=0,018 q*2=0,3557 5 1000 (1257; +∞) 0,7 1,2 q*r table 2. results of simulation for χ(q)=eq. model number p b ξ γ q* 6 (0; 2) 1 (0; 1] 2 q*<0 7 10 1 1 1,3 8 8 10 1 1 (1,8; 2] 9 10 [6; +∞) 0,7 1,2 10 100 3 0,7 1,2 14,306·106 the graphic illustration of these examples is presented in the figure 1. fig. 1. function of profit at p =1000, ξ=0,7, γ=1,9. let’s consider modeling with the change of parameters q and в. 60 m. geraskin, e. rostova copyright ©2018 assa adv. in systems science and appl. (2018) fig. 2. results of modeling at p=1000, b=80, ξ=0,7, γ=1,9 (different view). the results of the graphic simulation show that at high values of parameter b revenue is not enough for a covering of the expenses. it means that with increase in the costs, the production volume has to increase. in this case the revenue exceeds the expenses. 4. discussions the problem of the firm’s risk management including the industrial damage from unexpected production situations is examined. the probable firm’s expenses in these situations are classified into the voluntary costs and the obligatory costs. it enables to identify the optimization problem of searching for the vrc as a separate component of the overall objectives for the costs management, which, in this case, is the component of the profit maximization problem. the vrc function optimum is determined by the total costs criterion for the power function and the exponential function of the production costs and for the decreasing function of the industrial damage. the optimal vrc function represents the logarithmic dependence on the monotonic function of the output volume. therefore, with the output growth, the rational strategy of firm’s risk management consists in smooth increase in the costs for prevention of risk situations. the influence of the costs function and the industrial damage function parameters on an existence of the solution of the profit maximization problem is analyzed. under the vrc optimal function, low firm’s goods price leads to low revenue, which together with high values of the cost function parameters does not enable to implement the vrc. under sufficiently high price, the existence of firm’s output is proved, which enable to make the vrc, with the profit maximizing, that is, the revenue is sufficient to conduct the risk reduction activities. thus, the boundaries of the market prices and parameters of the cost function are a set, in which the firm’s risk management is critical for the firm’s existence. 5. conclusion the results of the research enable to determine the vrc depending on the price and the production function parameters. the expediency of the vrc also depends on these parameters. thus, the firm can simulate the different development options and determine the vrc depending on the output volume, the product prices, the production functions parameters and the internal damage function. as a result of modeling, the high value of production costs together with low market price do not provide to make the vrc. with the price increase, the firm’s revenue increases, and, together with the large output volume, enable to search for an optimal solution of the problems of profit maximizing and expenses minimization. costs function optimization for prevention of firm’s industrial risks 61 references 1. abramova, n.a., kovriga, s.v. & makarenko, d.i. 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(1776) an inquiry into the nature and causes of the wealth of nations. london: a. strahan and t. cadell. 29. thun. j.-h., drüke, m. & hoenig, d. (2011) managing uncertainty – an empirical analysis of supply chain risk management in small and medium-sized enterprises, international journal of production research, 49, 5511 – 5525. 30. varcholová, t. (2003) active enterprise risk management, ekonomicky casopis, 51, 997 – 1010. 31. walters, a. a. (1963). production and cost functions: an econometric survey. econometrica. the econometric society, econometrica. http://www.tandfonline.com/author/thun%2c+j%c3%b6rn-henrik http://www.tandfonline.com/author/dr%c3%bcke%2c+martin http://www.tandfonline.com/author/hoenig%2c+daniel http://www.tandfonline.com/toc/tprs20/current https://www.scopus.com/authid/detail.uri?authorid=6505832674&amp;eid=2-s2.0-0346395910 https://www.scopus.com/sourceid/20126?origin=recordpage adv syst sci appl 2022; 01:35–50 published online at https://ijassa.ipu.ru. evolutionary game to model risk appetite of individual investors zohreh lashgari1*, madjid eshaghi1, alireza bahiraie1,2, abdul km azhar3 1department of mathematics, faculty of mathematics, semnan university, semnan, iran 2hec montréal, 3000, chemin de la côte-sainte catherine, montréal (québec), canada 3graduate school of management, universiti putra malaysia, serdang, malaysia abstract: in this paper, we present a novel mathematical model of the development and progression of investments based on evolutionary game theory. we consider four different types of investments in the market: bank account, bond, stocks, and risky derivatives. despite the relative sincerity of the model, it supplies a way to explore the interactions between the different investment types in the market. we assume the market is complete and four assets constitute the total capital market. we also assume that the risk-averse individual, the risk-neutral individual, and the risk-seeking individual enter the market, and that each of these three individuals buys at least two assets. we examine the interaction of the two assets and find evolutionary stable strategy in interactions and weights. we explore the heard effect on decision-making and investment. by providing examples in the capital market, we find that the results are consistent with the conditions in the capital market. the results show that investors who enter the market may change their behavior due to their herds effect and tend to other strategies that are more appropriate according to evolutionary game theory. keywords: game theory, portfolio risk, expected payoff, evolutionary stable strategy, heard effect 1. introduction in this paper, we examine the application of evolutionary game theory the risk and present a new mathematical model of investment development based on evolutionary game theory. a branch of mathematics is game theory such that analyzes interactions involving strategic financial decision-making. this interaction is called a game. the parties included in the game are the players, who can be persons or groups. the players are assumed to act rationally [1]. game theory has controlled many areas, from economics (historically its initial focus) to political science to biologies, such as business, wireless networks, computer science, psychology, and many others [2, 3]. evolutionary games examine behaviors in evolving populations. the goal is to examine changes in population over time. in human interactions, players who gain a good experience of their choice in one period pass it on to those around them so that they can make the same choice in the next round. humans interact with each other in many situations, and many of their situations and interactions depend on financial circumstances. in these interactions, it is important to choose the best strategy that will enable people to achieve the highest results. analysis and understanding of situations in which decision-makers are interacting is one of the goals of game theoretic. ∗corresponding author: z.lashgari@semnan.ac.ir 36 z. lashgari, m. eshaghi, a. bahiraie, a.k.m. azhar risk attitude is a concept of decision theory and allows individuals to determine their preferences from identified risks. game theory helps solve risk problems and risk management. game theory is combined with risk management theory to capture any opportunity that a business can turn the risk into opportunity. game theory for managers dealing with a huge amount of data can be an alternative perspective that they can view the process of solving hard problems. some studies have shown how game theory helps managers to solve problems, manage risks and make strategic decisions. financial risk is the possibility of losing money on an investment or business activity and is a type of risk that can lead to the loss of capital for investors. risk analysis supports game theory by providing various possibilities. methods and concepts of game theory can be effective in improving risk analysis. also, risk analysis and game theory are complementary [4]. investors have different and complex behaviors due to interaction with each other. if we have a solution for their behavior when entering the market and their behavior is predictable, then other people can choose the best strategy when entering the market and investing and have a more successful investment. given that investors have different preferences, in this article, we are looking for a solution to be able to predict the behavior of these investors using evolutionary game theory in the market and express the best strategy for them. in this article, we answer the following questions in the capital market: which strategy investors choose based on their behavior in the capital market? in the capital market, which strategy for investors is an evolutionary stable strategy? why do investors tend to pursue this evolutionary stable strategy? is it possible to predict the behavior of investors in terms of their type in the capital market? there are models for people’s risk appetite using game theory. the model presented in this article for investors’ risk appetite is a complete, logical, and close to a reality model. because if we compare this model with the nash model, we will see that each investor has chosen his strategy based on his opinion, but in the proposed model, each investor inherently has his own strategy and we know that in the real world people are inherently risk-seeking, neutral-risk, and risk-averse, so this model makes more sense. also, in nash model, players represent real players, but in our model, players (investors) represent players who are randomly selected from the population, and because the risk appetite of investors in a population is examined, so our model which expresses the choice of investors in the population is a model close to reality. therefore, our model shows this correct choice. nash model does not pay much attention to the mechanism of change over time, but our model examines changes in representative population strategy over time. also, in nash model, equilibrium is obtained, but in the proposed model, we obtain ess, which is the more general state of nash equilibrium. comparing this model with the nash model, we found that this model is an efficient model to examine the risk appetite of investors and provide the best strategy in the population of investors, and this model can be used to predict the behavior of investors and express their optimal strategy. 2. literature review evolutionary game theory is one of the important theories. numerous studies and researches in the field of evolutionary games have been done by scientists and researchers. we will review some of the most important of these researches. evolutionary game theory began with an article by price and maynard smith [5]. evolutionary game theory (in short egt) has burgeoned into a base of computational biology and mathematical [6–8]. the mathematical foundations of game theory experienced a revival when price and maynard smith revolved their attention to evolutionary game theory. maynard smith coined the term evolutionary stable strategy (in short ess), to mean a move (or play) that would assure the type of animal that wields it an evolutionary advantage over their rival, with the meaning that copyright © 2022 assa. adv syst sci appl (2022) evolutionary game to model risk appetite of individual investors 37 the evolutionary stable strategy can not ever be extinct [9]. using the basic tools in hand, continuous-trait game theory can be clearly extended to model evolution under situations of disruptive choice and nonequilibrium population dynamics, stochastic environments, speciation, coevolution, and others. multitude models applying these tools to evolutionary ecology and coevolution have been developed in the past two decades. trait evolutionary game theory and new applications are important in biological issues [10]. in the basic description of altruistic behaviors in darwinian evolution, evolutionary game theory has helped. it has in turn become of interest to sociologists, anthropologists, philosophers, and economists. evolutionary game theory is used in many areas, including sociology, economic, anthropology, social networks political science, and social sciences [11, 12]. in 1990 ”an evolutionary game theory model for risk-taking” was published by s ellner and a shmida s ellner and a shmida in 1990 in a paper entitled ”an evolutionary game theory model for risk-taking” published [13]. in 1998, the financial applications of game theory were proposed by franklin allen and stephen morris. in this study, it is stated that traditional financial theory based on the assumptions of symmetric information and complete and competitive markets has provided many insights [14]. using theory of game, risk analysis can be based on values of benefit or preferences which the objects can supply rather than subjective probability. furthermore, it can also be used in settings where statistical data is unavailable. this may expand the appropriateness and quality of the overall risk analysis process. an article was written by james raneke in 2009 on a game theory formula that discusses decision-making in terms of uncertainty and risk. the game environment is examined to decide in terms of uncertainty and risk for a general set of issues. the method is demonstrated using the assumptions governing future oil prices and environmental degradation to evaluate long-term investment options [15]. in 2011, rajabandari and snekkenes published an article entitled ”using game theory for private risk analysis: a basic insight”. in this paper, the authors state that with the advancement of information technology, there is a growing risk of privacy due to the widespread use of identity information. this article discusses some of the key issues surrounding the use of game theory in privacy risk analysis. using game theory, risk analysis can be based on preferences or profit margins [16]. the results and reasoning of game theory have overshadowed economic theory widely used in other behavioral and social sciences. also, it is widely used in economic decisions and other gaming situations that involve risk coalition game theory for security risk management. this paper examines the possibility of cooperation between the autonomous departments of an organization with dependent security assets or vulnerabilities to reduce overall security risks. a coalition game has been formed to model the possibilities of cooperation in these sectors based on their positive and negative interdependencies. the proposed game forms a framework that allows us to examine how an organization can maximize its overall profitability through collaboration between its various departments [17]. in 2012, farshchi wrote an article about finding the dominant strategy for risk in incomplete play discussed. he argued that game theory reasoning exists in economic theory and is widely used in other social and behavioral sciences, and that it is widely used in decision-making about economic behavior and other risk-playing conditions. the paper proposes a flawed multiplayer information system using a dominant risk strategy for decision-making. finally, it discusses the operation of the strategy [18]. in 2013, an article on financial markets and evolutionary beliefs was published by ellis jovini, clotilde knop, and yannick wyt [19]. 2016 financial risk of supply chain of small and medium enterprises based on evolutionary game theory by cheng zhang checked out. this article examines the game between commercial banks and corporate supply chains. and examines the stability of the balance between it zhang. in 2018 ”an evolutionary game model for behavioral gambit of loyalists: global copyright © 2022 assa. adv syst sci appl (2022) 38 z. lashgari, m. eshaghi, a. bahiraie, a.k.m. azhar awareness and risk-aversion” published by e alfinito, a barra, m beccaria, and a fachechi [20]. in 2018, ashkan hafezalkotob published an article entitled ”market risk and uncertain demand in the supply chain”. the paper argues that uncertainty in market demand poses a significant risk to retailers who supply the market, and that retailers’ risk-aversion behavior toward risk may have changed over time. considering a supply chain including the manufacturer and the retail population, the authors intend to examine how the retail population evolves toward risky behaviors. the purpose of this study is to evaluate the effects of producer wholesale selling price on retailers’ sustainable evolutionary strategy. a supply chain with a population of risk-averse and neutral risk retailers is examined. the wholesale pricing strategy is determined by a manufacturer who acts as a leader, while the retailers who make quantitative decisions act as the leader. to model this situation, kornot dual equilibrium approach and evolutionary game theory have been used [21]. in 2018, ”evolutionary game dynamics for financial risk” was studied. in the same year, zhi li, guanghao jin, and shen duan conducted a study on global supply chain performance based on egt. they wrote the evolutionary game between the main player in the supply chain and, using replicator dynamics, developed an ess for the game. the authors used evolutionary game theory to model the performance of supply chain members in a financial risk environment. the results of this study showed that in high financial risk, the cooperation strategy is a sustainable evolutionary strategy for both producers and suppliers [22]. game theory has a lot of applications in the business world. in 2019, a book entitled ”introduction to derivative securities, financial markets and risk management” was written by jarrow and chatterjea. this book explores the interest rates on derivatives, stocks, options, trade, financial engineering, futures markets, futures, and other issues [23]. in 2020, an article entitled ”heterogeneity in evolutionary games: an analysis of risk perception” was written by marco a. amaral and marco a. javarone. which examines the relationship between cooperation and heterogeneity. the approach of this modeling is based on the framework of evolutionary game theory [24]. ”the evolutionary game of stakeholders ’ coordination mechanism of new energy power construction ppp project: a china case” was released by l gao and zy zhao in 2020. in this study, government, investors, and the general public have established a relationship with multiple dynamic public-private partnership (ppp) play. the authors first developed a three-dimensional game model consisting of government, investors, and the people based on evolutionary game theory. in the second stage, they study the three-way evolution of strategic behaviors with a system dynamics model. finally, the effect of changes in key factors in behavioral strategies was studied and the results showed that the three parties eventually reach equilibrium and the benefits of the project are at their highest at this time and the three parties achieve a win-win situation. also, the security factor plays a vital role in choosing the overall strategy. public choice is affected not only by its own income and expenditure, but also by the amount of compensation promised by the government and the amount estimated by the investor [25]. in 2021, an evolutionary game theory model for the relationship between financial regulation and financial innovation was proposed by hui an, r yang, x ma, and s zhang. this paper builds an evolutionary game model for simulating financial regulation and financial innovation to analyze sustainable equilibrium strategies between financial institutions and regulators. the article also uses us macroeconomic financial data [26]. 3. evolutionary game theory (egt) egt be created as an application of the mathematical theory of games to biological content, arising from the awareness that frequency-dependent fitness establishes a strategic aspect to evolution. the attention among scientists and scholars in theory with detailed biological roots derives from three facts. first, the evolution process by evolutionary game theory require not be biological evolution. evolution can, in this situation, often be understood as cultural copyright © 2022 assa. adv syst sci appl (2022) evolutionary game to model risk appetite of individual investors 39 evolution, where this refers to changes in norms and beliefs over time. second, the rationality suppositions underlying evolutionary game are in lots categories more appropriate for the modelling of social systems than those suppositions underlying the classical game theory. third, egt, as a clearly dynamic theory, supplies an important factor missing from the classical theory. egt has been used to describe a large number of aspects of human behavior. evolutionary games have important unrealized potential for modeling substantive economic issues [11]. evolutionary game theory is used in different sciences, such as social dilemmas, economic, sociology, and anthropology [27, 28]. the beginning of evolutionary game theory can be dated to the definition of an evolutionarily stable strategy (ess) by price and maynard smith. in certain populations, all individuals be possible have the same strategy phenotype. such a strategy is called to be an ess if that strategy cannot be replaced, or invaded by any other strategy through natural selection. in the egt, the concept of evolutionary stable strategy is a population in which all those who play this strategy are resistant to the invasion of a group of mutant persistent. function u represents a player ’s payoff. in short, the following definition is given: α is an (ess) such that: for each β 6= α, there are some ε′ ∈ (0, 1), which may depend on β, such that: for all ε ∈ (0, ε′) u(α, εβ + (1− ε)α) > u(β, εβ + (1− ε)α). (3.1) that is, α is (ess) if, after mutation, non-mutants are more successful than mutants, in which exposition mutants cannot invade and will eventually get extinct. 4. portfolio optimization markowits apprcach in financial affairs, markowitz model-put forward by h.markowitz is a portfolio optimization model, this model helps in the option of the most efficient portfolio by studding various possible portfolios of the given securities. markowitz made the following hypothesizes: risk of a portfolio is established on the variability of returns from the said portfolio. an investor is risk-averse. an investor would like to increase consumption. the investor’s utility function is increasing and concave, due to his risk aversion and consumption preference. examination is based on the alone period model of investment. an investor either maximizes his return for the minimum risk or maximizes his portfolio return for a given level of risk. an investor is rational in nature. a portfolio that gives maximum return for a given risk, or minimum risk for given return is an efficient portfolio. therefore, portfolios are thus, portfolios are chosen as follows: an investor will prefer from the portfolios that have the same return, the portfolio with lower risk. and the investor will prefer the portfolios that have the same risk level from the portfolio with a higher rate of return. because he is a reasonable investor, he tends to be more profitable, and because he is a risk averse, he tends to take less risk. for choice of the best portfolio, the risk-return preferences are studied. an investor who is highly risk-averse will hold a portfolio on the lower left hand of the frontier, and an investor who is not too risk-averse will select a portfolio on the upper portion of the frontier [29, 30]. 5. prospect theory first allais, researchers noticed that expected utility theory (in short eut) doesn’t sufficiently explain decision-making under risk. the leading alternative to expected utility theory, prospect theory, supposing that individuals appraise gains differently from losses:a gain profit persons less than a loss of the same size copyright © 2022 assa. adv syst sci appl (2022) 40 z. lashgari, m. eshaghi, a. bahiraie, a.k.m. azhar benefits them. on the other hand, the weights that a person attributes to outcomes of risky decisions do not coincide with the probabilities of the respective outcomes. this probability weighting typically leads to over weighting of low probabilities and under weighting of high probabilities. one of economics theory is the prospect theory that improved by amos tversky and daniel kahneman. it challenges eut, improved by oskar morgenstern and von neumann. it is the founding theory of behavioral finance and of behavioral economics, and comprises one of the first economic theories created using experimental manners. founded on results from regulated studies, it explains how persons appraise in an asymmetric mode their gain and loss perspectives. for example, for some persons, the hurt from losing $500 could just be paid back by the pleasure of $1, 000. the prospect theory begins with the idea of loss aversion, an asymmetric form of risk aversion, from the observation that people react differently between potential losses and potential gains. consequently, people make decisions founded on the potential losses or gain relative to their specific condition rather than in absolute terms: facing a risky choice leads to profit, persons are risk-averse, appointing solving that lead to a lower expected utility but with a higher definiteness (function is concave). facing a risky choice leading to losses, persons are risk-seeking, appointing solutions that lead to a lower expected utility as long as it has the ability to nullify losses (function is convex). these examples are so in contrast with eut, which just considers options with the maximum utility. eut has influenced the analysis of decision-making under risk. generally, it is accepted as a normative model of rational choice, and greatly applied as an illustrative model of economic behavior [31–33]. consequently, it is supposed that all credible people would wish to obey the axioms of the theory, and that most people in factf do, most of the time [34, 35]. the prospect theory probability value function v(r) and weight function π(p) [36]. v(r) explicit formula is the value function of the prospect theory that is as follows [37]: v(r) = { (r − r0)α if r ≥ r0 −λ(r0 − r)β if r < r0 (5.2) where (ri) mean return of asset i, (r0) reference point,(λ) is positive value of risk and (−λ) is negative value of risk, α and β are risk aversion coeffcients with respect to gains and losses respectively. the prospect theory utility function can be written in terms of v and π as: ptu = s∑ s=1 π(ps)v(rs) = s∑ s=1 psv ( n∑ i=1 rsiwi ) . (5.3) the parameters are as follows: (ps) the probability of scenario s, (rsi) the return of asset i in scenario s, i = 1, . . . , n , s = 1, . . . , s, (wi) the weight of asset i in the portfolio, wi ≥ 0, (s) the number of scenarios, (ri) mean return of asset i, (r0) reference point, (n) the number of assets. the diagram prospect theory is shown in figure 5.1 [37]. copyright © 2022 assa. adv syst sci appl (2022) evolutionary game to model risk appetite of individual investors 41 fig. 5.1. prospect theory value function v(r) with λ = 2.25 and α = β = 0.88. 6. game theory and prospect theory extant research includes lots of effective applications of prospect theory, specially in insurance, finance, and betting markets. however, prospect theory studies are similar in that the risk that people face is exogenous. in lots of real-world conditions, contradiction, risk does not come from an exogenously given random device but rather is a result of the interaction of one person with others. in developing the prospect theory model, show (i) a solution strategy for incorporating prospect theory into a dynamic game theoretic model and its empirical applicability to field data, (ii) the usefulness of prospect theory in depicting people behavior, and (iii) managerial connotations that result from using prospect theory over the more commonly used expected utility theory. when we enter a game and play that game, payoffs are mostly given in monetary (and other similar) amounts. as morgenstern and von neumann observed, game theory has therefore to take into account the players preferences on uncertain and sure monetary outcomes of game. to incorporate prospect theory into a game theoretical framework, at first glance looks straightforward: the naive approach would be to transform the monetary outcomes via the value function and to transform actual probabilities for chance moves into experienced probabilities by implementing the probability weighting functions. this procedure mimics the technique one successfully applied when dealing with eut. focusing on the theory of perspective, which is devoted to human behavior in risky financial decisions, the idea of the expected utility theory has been developed by adding psychological components that consider human behavior in the decision-making process. if we want to show the minimum risk in choosing a stock portfolio with a fixed return, with a behavioral approach versus the traditional markowitz approach, we can impose a behavioral approach on the traditional approach. the reason for imposing behavioral characteristics on the theory of perspective is to minimize the risk with a certain level of income, in the matter of choosing a portfolio. according to the prospect theory portfolio selection problem looks as follows: maximizeptx = s∑ s=1 psv ( n∑ i=1 rsiwi ) . (6.4) where (ps) the probability of scenario s, (wi) represents the weight of asset i in the portfolio, (rsi) the return of asset i in scenario s. copyright © 2022 assa. adv syst sci appl (2022) 42 z. lashgari, m. eshaghi, a. bahiraie, a.k.m. azhar subject to the constraints r(x) = n∑ i=1 riwi ≥ d, (6.5) n∑ i=1 wi = 1. (6.6) wi ≥ 0, i = 1, . . . , n where (wi) represents the weight of asset i in the portfolio, wi ≥ 0, (d) represents the desirable level of return, (ri) mean return of asset i. 7. new model development in this chapter, we examine the issue of people’s risk appetite for investment and present this issue with the evolutionary game theory approach. also, we develop a new model to investigate the investment strategies in real market. our model includes a demonstration of real-world investment. investors have different and complex behaviors due to interactions with each other. this diversity and complexity of investor behavior leads to a model in evolutionary game theory. this mathematical model is used to examine interactions between investors. we assume that three types person enter the market. risk appetite reflects the willingness of investors to take risks. we express the preferences of investors using a level of risk that is acceptable to them. we denote the person whose behavioral preferences are risk-averse by p1, person whose behavioral preferences are riskneutral by p2, and person whose behavioral preferences are risk-seeking by p3. the risk-averse agent prefers to play the strategy that takes the least risk. a risk-neutral agent only attempts to get the highest expected utility. a risk-seeking agent chooses to play the strategy that gives the highest payoff without taking the risk into account. suppose the x-axis represents income and the y-axis represents utility. the graph of the utility function of these individuals is shown in figure 7.2. fig. 7.2. the utility of individuals who enter the market. we assume there are four different investment types in the market which the players may invest with respect of risk and return of each asset and his own preferences. these are four assets which are: investment on a bank account, bond, stock, and risky derivatives. copyright © 2022 assa. adv syst sci appl (2022) evolutionary game to model risk appetite of individual investors 43 we assume the market is a complete market and we assume that the assets on which the bank account, bond, stocks, and derivatives constitute the total capital market. the complete market occurs when all the assets are invested, and their behavior affects others. we consider that there are four types of assets in the market: (a): bank account (b): bond (c): stocks (d): risky derivatives the weight of the various types of assets are denoted by aa, ab, ac, and ad respectively. the weights are satisfy d∑ i=a ai = unity, (7.7) 0 ≤ ai ≤ 1. the following relationship applies to all assets.∑ (ai ×wi) = ∑ ci. (7.8) where c is the total capital of the individuals and can be defined. we will now set up costs and benefits for each type of asset. the notation and model parameters are summarized in table 7.1. table 7.1. model parameters symbol meaning aa weight of the bank account ab weight of the bond ac weight of the stocks ad weight of derivative µ benefits of asset i r risk of asset i cp the cost of call price dp the cost of discounted price eps the earning per share w0 initial fitness wx net benefit for a given type of the investment x ∈ {a, b, c, d} the concept of risk plays a key role in the financial market. risk is the main pillar of decisionmaking for investment. investors who start investing regardless of risk will make themselves in a loss-making situation. we consider this risk and show with r. earning per share (eps) is the earning obtained by each share stock. call price is the value at which a corporation can purchase and exercise preferred stock from its callable preferred derivating which shows the cost of call price with cp . discount price is reduced prices or something being sold at a price lower than that item is normally sold for which shows with dp . asset a indicates the bank accounts which is the symptom of the minimax risky asset. asset b indicates the bonds with the lowest investment risk and asset c is the stock with market fluctuations. asset d is all the other types of risky derivatives which indicates the highest risk related investment in the market. we have: ra < rb < rc < rd, (7.9) copyright © 2022 assa. adv syst sci appl (2022) 44 z. lashgari, m. eshaghi, a. bahiraie, a.k.m. azhar and µa < µb < µc < µd. (7.10) when the weight of the investment aa, ab, ac and ad, using the symbols in table 7.1, the net benefits to each type of the investment are waa = w0 −dp + µ(aa + ab)− r(aa + ab), wab = w0 − eps + µ(aa + ab + ac)− r(aa + ab + ac), wac = w0 − eps + µ(ab + ac + ad)− r(ab + ac + ad), wad = w0 − cp + µ(ac + ad)− r(ac + ad). (7.11) waa reads the individual in the investment bank has the initial fitness w0, pays the cost of discounted pric dp , receive benefits from investment in a and b, shown by the term µ(aa + ab) and pays the cost of risk, shown by the term r(aa + ab). wab reads the individual in the investment bond has the initial fitness w0, pays earning per share eps, receive benefits from investment in a, b, and c, shown by the term µ(aa + ab + ac) and pays the cost of risk, shown by the term r(aa + ab + ac). wac reads the individual in the investment stocks has the initial fitness w0, pays earning per share eps, receive benefits from investment in b, c, and d, shown by the term µ(ab + ac + ad) and pays the cost of risk, shown by the term r(ab + ac + ad). wad reads the individual in the investment stocks has the initial fitness w0, pays the cost of call price cp , receive benefits from investment in c and d, shown by the term µ(ac + ad) and pays the cost of risk, shown by the term r(ac + ad). the payoff matrix between the different assets is shown in table 7.2 . table 7.2. the payoff matrix a b c d a µ− r µ(a+ b)− r(a+ b)−dp 0 0 b µ(a+ b)− r(a+ b)−dp µ− r µ(b+ c)− r(b+ c) −dp − eps 0 c 0 µ(b+ c)− r(b+ c) −dp − eps µ− r µ(c+ d)− r(c+ d) −cp − eps d 0 0 µ(c+ d)− r(c+ d) −cp − eps µ− r for example, in table 7.2, the only payoff obtained is µ− r when the asset a is interacting with another asset a, when an asset a is interacting with the asset b, the payoff is µ(a+ b)− r(a+ b)−dp , when the asset a is interacting with another asset c the payoff is 0 because they are not interacting with each other, and similarly, when an asset a is interacting with the asset d, the payoff is 0. 8. empirical application assuming that the investors were three types: risk-averse individual, riskneutral individual, and risk-seeking individual. we assume that one kind of individual enters the market from these three types of individuals. and we assume that individuals purchase at least two assets. assuming a risk-averse individual enters the market. this person only evaluates a and b. this person is not interested in investing in c and d because the person is risk-averse. so, in this case, the bank account and the bond are interacting. in situations where bank account and bond are interacting, we are interested to know which is an evolutionary stable strategy. meaning that we find evolutionary stable strategy in this interaction. but the condition ess of such a situation is: copyright © 2022 assa. adv syst sci appl (2022) evolutionary game to model risk appetite of individual investors 45 waa −wab = eps−dp − µac + rac. (8.12) recall that the net benefit from interaction (fitness) for the asset bank account is denoted by waa and for the bond, by wab . we have: waa = wab . (8.13) then ac = eps−dp µ− r . (8.14) as the risk-averse enters the market, equilibrium is c and investor prefers to choose c. this result is due to the herd effect. that is, there is a herd effect in the financial market when a group of investors ignores their own information and instead only follows the decisions of other investors. so, in this situation, if every other risk-averse individual enters into the market, it will bring the same choice c, and therefore the whole population will choose this strategy. and this strategy is an ess. assuming a risk-neutral individual enters the market. this person only evaluates b and c. and is not interested in investing in a and d because the person is risk-averse. because assets a has low income and assets b are high risk, the person is unwilling to choose them. so, in this case, the bond and the stocks are interacting. we will now find the conditions evolutionary stable strategy in interaction between bond and stocks. but the condition ess of such a situation is: wab −wac = µaa − raa − µad − rad, (8.15) aa = µad − rad µ− r . (8.16) or ad = µaa − raa µ− r . (8.17) as the risk-neutral individual enters the market equilibrium are a or d and investor prefers to choose a or d. in this case, the investor behavior changes due to the herd effect. that’s mean, when a riskneutral enters the market, it is influenced by the behavior of others, the best strategy for this individual is the choice of a or d, so chooses a or d. then a and d are mixed evolutionary stable strategy. in the evolutionary game theory a situation where two or more types are best suited to one of the other types, mixed evolutionary stable strategy is called. assuming a risk-seeking individual enters the market. this person only evaluates c and d. because the person is risk-seeking, this person is looking for higher incomes then is not interested in a and b. so, in this case, the stocks and the risky derivatives are interacting. now find the conditions evolutionary stable strategy in the interaction between stocks and derivatives. but the condition ess of such a situation is: wac −wad = µab − rab + cp − eps, (8.18) copyright © 2022 assa. adv syst sci appl (2022) 46 z. lashgari, m. eshaghi, a. bahiraie, a.k.m. azhar ab = eps− cp µ− r . (8.19) as the risk-seeking individual enters the market, equilibrium is b and investor prefers to choose b. that is, in this group of investors ignore their information and change their behavior and instead follow only the decisions of other investors, so there is a herd effect in the financial market. many times in financial markets, as in our daily life, we parrot the decisions of predecessors, instead of analyzing available information and making our own decisions. this following the decision can lead to collective hysteria and investment calls may be influenced by these panicked conditions. each investor gets information (or signals) about how they should act. they also know the decisions of their predecessors, although they don’t know the signs they received. using this knowledge, each investor makes their own decision. but this unawareness of previous information can sometimes cause investors to disregard the signals they get in the here and now, and instead take in the same decisions as their predecessors. so, in summary a herd effect exists in the financial market when a group of investors ignores their own information and instead only follows the decisions of other investors. some studies (such as herd behavior in financial markets, or daring to be different) treat this effect as an intraday phenomenon, which intuitively makes sense. investors are much more likely to parrot others than to explicate the information they get when they have a few times to make investment decisions. 9. illustration results in this section, we review the research findings and compare the findings with the market results by providing an example. the data are derived from the european average 10-year interest rate on financial derivatives [38]. each investor is interested in the risk and return of his investments in the market economy. in this example, we consider four assets a, b, c, and d. asset a is considered in bank accounts and we consider it as a risk-free investment. for asset a we consider the benefit to be 2, the risk of the asset to be 0.5 and the discount price (dpc) to be 4. asset b is considered an investment on commodity. for asset b we consider the benefit to be 3.5, the risk of the asset to be 2 and discount rate to be (dr) to be 1.5. asset c is considered investment on stocks and equity. for asset c we consider the benefit to be 4, the risk of the asset to be 34 and minimum eps which is actually the lowest eps in the stock market (min eps d) to be 1.5. asset d is considered investment on forex (curvency exchang) and we considered as a high risk investment. for asset d we consider the benefit to be 6, the risk of the asset to be 4.5 and cost of call price (cp ) to be 0.5. consider table 9.3 for the assets. table 9.3. numerical examples asset benefit (%) risk weight cost a µa = 2 ra = 0.5 aa dp = 4 b µb = 3.5 rb = 2 ab 3.5 = 2 +drc c µc = 4 rc = 3 ac 3.5 = 2 + min eps d d µd = 6 rd = 4.5 ad cp = 0.5 we assume that a risk-neutral individual enters the market in this case, according herd effect, copyright © 2022 assa. adv syst sci appl (2022) evolutionary game to model risk appetite of individual investors 47 investor behavior changes and prefers to choose aa because aa = µdad − rdad µa − ra → aa = 6(ad)− 4.5(ad) 2− 0.5 . (9.20) when a risk-neutral individual enters the market ad = µaaa − raaa µd − rd → ad = 2(aa)− 0.5(aa) 6− 4.5 . (9.21) then we have: aa = ad. (9.22) in this case, this person has two choices aa or ad. that is, we have a mixed evolutionary stable strategy. it is a strategy in which two strategies are permanently present. finally, these two strategies become pervasive in the population with the type of risk-neutral behavior. on the other hand, when a risk-seeking individual enters the market according to herd effect, investor behavior changes and prefers to choose ab and the whole population will go to this strategy and follow it because ab = eps− cp µb − rb → ab = 3− 0.5 3.5− 2 . (9.23) finally, this strategy becomes pervasive in the population with the type of risk-seeking. and when a risk-averse individual enters the market according to herd effect, investor behavior changes and prefers to choose ab because ac = eps−dp µc − rc → ac = 3− 4 4− 3 . (9.24) finally, this strategy becomes pervasive in the population with the type of risk-averse. in this example, when three individuals enter the market with different preferences and buy from the four assets on the market, we have the following result in the weight of the assets: d∑ i=a ai = unity. (9.25) and in each interaction, investors tend to change their behavior because they are influenced by others experiences and prefer buying an asset that most people have bought. and as a result, this strategy is more appropriate than any other strategy, and the whole population will go to this strategy and follow it. this strategy is an ess. the results presented in this numerical example correspond to real-world results (according to the herd effect) and therefore, this model is useful for investing and predicting investment results. 10. conclusion in this paper, we showed and analyzed a game-theoretical novel model of investments. we assumed the market is a complete market and the assets bank account, bond, stocks, and risky derivatives constitute the total capital market. also, assumed that the risk-averse individual, risk-neutral individual and risk-seeking individual enter the market and each individual buys at least two assets. using evolutionary game theory, we showed how each investor behaves according to their preferences when this investor enters to the market and which strategy is copyright © 2022 assa. adv syst sci appl (2022) 48 z. lashgari, m. eshaghi, a. bahiraie, a.k.m. azhar an evolutionary stable strategy in the interaction between the two assets. we found that when risk-averse individual enters the market, two assets a and b are interacting, and ac is an ess. in this case, the investor behavior changes due to the herd effect and the whole population will go to this strategy. when risk-averse individual enters the market, it is influenced by the behavior of others and most people have invested in this asset. when riskneutral individual enters the market, two assets b and c are interacting, and aa or ad are mixed ess. so, the investor behavior changes and most people have invested in these two assets. when risk-seeking individual enters the market, two assets c and d are interacting, and ab is an ess. in this case, the investor behavior has changed due to the herd effect and prefers to choose ab and the whole population will go to ab. by giving an example, we find that the results are consistent with real-world results. therefore, using the new model presented in this article, we can predict their choice when entering the capital market according to the type of behavior of investors and explain the reason for this choice according to the herd effect. comparing the model presented in this article for investors’ risk appetite and nash model, we conclude that this model is a more complete, logical and close to a reality model. references 1. jameson, k. 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(2013) investment and investment finance in europe. annual economic conference and publication, luxembourg. copyright © 2022 assa. adv syst sci appl (2022) https://dx.doi.org/10.29252/jncog.1.2.88 https://dx.doi.org/10.1007/bf00122574 introduction literature review evolutionary game theory (egt) portfolio optimization markowits apprcach prospect theory game theory and prospect theory new model development empirical application illustration results conclusion adv syst sci appl 2020; 04:60–69 published online at https://ijassa.ipu.ru. multi-criteria estimation of input parameters in natural gas quality analysis ivan brokarev1⇤, sergei vaskovskii2 1 national university of oil and gas gubkin university, moscow, russia. e-mail: brokarev.i@gubkin.ru 2 v. a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia. e-mail: v63v@yandex.ru abstract: in this paper, we present a method for assessing input parameters in a statistical model used for the quality analysis of the natural gas. the analysis is done by measuring physical quantities of natural gas through the hierarchy analysis, compromise programming, correlation analysis and assessing the practical possibility of measuring selected input physical quantities by available means. the problems arising when selecting input parameters are also considered. the results are compared with the previously obtained results of the model input parameters correlation analysis in order to improve the developed system for natural gas quality analysis. the proposed method is applied to assess input parameters for certain samples of natural gas based mixtures. based on the results of the multicriteria assessment, the speed of sound, thermal conductivity and concentration of carbon dioxide were selected as input parameters for the developed natural gas quality analysis system. keywords: multi-criteria estimation, analytic hierarchy process, correlation analysis, natural gas quality analysis, neural network analysis 1. introduction to analyze the quality of natural gas is to determine the energy characteristics out of the measured physical parameters of the gas. at present, this problem is usually solved in the gas industry by traditional methods, but there are alternative approaches [1]. the essence of alternative methods and systems based on them is to determine the required indicators by a statistical model, where the measured physical parameters of natural gas are input parameters. these models, mostly neural networks, seem to be the most effective means to solve the problem in comparison with other predictive models [2]. however, if we use artificial intelligence methods, we encounter a number of problems that need to be solved for the developed method to succeed in analyzing the quality of natural gas. the main problems of the approach are high computational costs, a large number of input physical parameters to measure, no general algorithm to choose the architecture and parameters of the model, and the model’s decrease in accuracy, when the parameters deviate from standard values. some of the listed problems are associated with suboptimal choice of input parameters for the models. for instance, if the number of input variables is excessive, the time spent on the analysis procedure increases, up to the point when it is impossible to analyze in real time. in addition, some of the input parameters are uninformative, so the model becomes less accurate. ⇤corresponding author: brokarev.i@gubkin.ru multi-criteria estimation of input parameters in natural gas quality analysis 61 in practice, we choose the input parameters for a statistical model by heuristic methods and a complete enumeration of the available parameters. however, there are algorithms aimed to optimize the set of input parameters and to choose the parameters more efficient. this article discusses our methodology to select input parameters for a natural gas quality analysis system; the methodology is based on a combination of correlation analysis and a multi-criteria estimation method. 2. methodology for multi-criteria estimation of input parameters the authors of [2, 3] chose the measured physical parameters out of a relationship between the parameter under consideration and the component composition of the gas. in addition, the authors took into account if it is possible to measure this parameter with commercially available and relatively inexpensive measuring instruments. to determine the relationship between the parameters and the component composition of natural gas, the authors conducted a correlation analysis between the physical parameters and simulated samples of natural gas of various component composition. based on the calculated pearson correlation coefficients for the simulated sample, the authors figured out if they chose the input parameters correctly. if the parameter had a high correlation (the value of the correlation coefficient in the range 0.7 1 modulo) with certain gas components and at the same time low correlation (0 0.3 modulo) with other input parameters, then this parameter was added to a set of model input parameters. we propose to marry the described approach with multi-criteria estimation methods. the first step of the proposed methodology is to select criteria to estimate the input parameters for the model. the selected criteria are divided into three groups, the first of which is the availability of technology to measure the parameter (0 if absent, 1 if available), high correlation with the model output parameters (component composition), and low correlation with other input physical parameters of the gas. the second group includes three criteria: correlations with the main components of the equivalent pseudogas methane (ch4), propane (c3h8), and nitrogen (n2). the above criteria are not equivalent. the most important criterion is the availability of technology to measure the parameter. the second is the criterion for high correlation with the model output parameters. correlation criteria are expressed as pearson correlation coefficients r. the least important criterion is the one of low correlation between the input parameters; this is necessary mainly to eliminate the possible multicollinearity of the input parameters. based on the results of the previous studies and analysis [4-8], we choose the following physical parameters as the input parameters: speed of sound (c), thermal conductivity (�), concentration of carbon dioxide (co2), dynamic viscosity (⌘), and dielectric permittivity ("). to form an optimal set of input physical parameters of natural gas, we propose to use the analytic hierarchy process (ahp) [9]. this method allows you to find such an alternative, that is best consistent with how you understand the essence of the problem and the problem requirements. the main advantage of the analytic hierarchy process is high versatility the method can be used to solve a wide variety of problems, regardless of the field of application. the disadvantage of the analytic hierarchy process is a large amount of information needed to be obtained from experts. however, we compensate for this disadvantage if we calculate the correlation coefficients and ask an expert to assess only an availability of technology to measure the investigated parameter. it should be noted that we could represent our system hierarchically to describe how changing priorities at higher levels influences the priorities of lower level elements. hierarchies provide more detailed information about the structure and function of the system at the lower levels. to satisfy the constraints on level elements, it is best to reproduce the constrains at the next higher level. hierarchically structured natural systems are much more copyright c� 2020 assa. adv syst sci appl (2020) 62 i. brokarev, s. vaskovskii efficient. hierarchies are stable in the sense that small changes have little effect, and flexible in the sense that if we add elements to a well-structured hierarchy, we do not destroy the hierarchy’s characteristics. the first stage of the ahp is to build a hierarchical structure that includes the goal, criteria, alternatives, and other relevant factors. the structure to choose the optimal set is shown in fig. 1. fig. 2.1. the structure of how the optimal set of the model input parameters is formed as we analyze the hierarchies. the next step is to build a matrix of pairwise criteria comparisons. we compare each criterion relative to all others. to conduct paired comparisons, t. saaty [9] developed a scale of relative importance, presented in table 1. copyright c� 2020 assa. adv syst sci appl (2020) multi-criteria estimation of input parameters in natural gas quality analysis 63 table 2.1. scale of relative importance level of importance index level of importance essence of the level of importance 0 incomparable difficult to compare criteria 1 equal importance two criteria contribute equally 3 moderate superiority one criteria has moderate advantage over the other 5 strong advantage one criteria has strong advantage over the other 7 substantial advantage one criteria is so much superior over the other, that the advantage is substantial 9 very strong advantage one criteria is obviously superior over the other 2,4,6,8 intermediate solutions between two importance criteria applied in edge cases where it is difficult to determine what criteria wins according to the list of important criteria given in the introduction, we pairwisecompared the criteria on a qualitative scale and subsequently converted the criteria to points. the matrix of pairwise comparisons is compiled as we calculate the coefficients aij the ratio of the weight of criterion i to criterion j. the matrix of pairwise comparisons of criteria is shown in table 2. table 2.2. matrix of pair-wise criteria comparisons aij tekh. r(ch4) r(c3h8) r(n2) r(c) r(�) r(co2) r(⌘) r(") tekh. 1 2 2 2 3 3 3 3 3 r(ch4) 1/2 1 1 1 2 2 2 2 2 r(c3h8) 1/2 1 1 1 2 2 2 2 2 r(n2) 1/2 1 1 1 2 2 2 2 2 r(c) 1/3 1/2 1/2 1/2 1 1 1 1 1 r(�) 1/3 1/2 1/2 1/2 1 1 1 1 1 r(co2) 1/3 1/2 1/2 1/2 1 1 1 1 1 r(⌘) 1/3 1/2 1/2 1/2 1 1 1 1 1 r(") 1/3 1/2 1/2 1/2 1 1 1 1 1 then the matrix of pairwise criteria comparisons is normalized. for this, the new element of the matrix aij is calculated as the ratio of the original element aij to the sum of the elements of the corresponding column. the average values for each row (avgi) are found as well; the value is called the criterion weighted column for the goal. the normalized matrix of pairwise comparisons of criteria is shown in table 3. table 2.3. normalized matrix of pair-wise criteria comparisons aij tekh. r(ch4) r(c3h8) r(n2) r(c) r(�) r(co2) r(⌘) r(") avgi tekh. 0.240 0.267 0.267 0.267 0.214 0.214 0.214 0.214 0.214 0.235 r(ch4) 0.120 0.133 0.133 0.133 0.143 0.143 0.143 0.143 0.143 0.136 r(c3h8) 0.120 0.133 0.133 0.133 0.143 0.143 0.143 0.143 0.143 0.136 r(n2) 0.120 0.133 0.133 0.133 0.143 0.143 0.143 0.143 0.143 0.136 r(c) 0.080 0.067 0.067 0.067 0.071 0.071 0.071 0.071 0.071 0.070 r(�) 0.080 0.067 0.067 0.067 0.071 0.071 0.071 0.071 0.071 0.070 r(co2) 0.080 0.067 0.067 0.067 0.071 0.071 0.071 0.071 0.071 0.070 r(⌘) 0.080 0.067 0.067 0.067 0.071 0.071 0.071 0.071 0.071 0.070 r(") 0.080 0.067 0.067 0.067 0.071 0.071 0.071 0.071 0.071 0.070 copyright c� 2020 assa. adv syst sci appl (2020) 64 i. brokarev, s. vaskovskii ones we determined the weights of the criteria, we can proceed to the next step – to determine the weights of the alternatives. for this, it is proposed to use the method of compromise programming [10]. this is a multi-criteria optimization method, where a solution is determined to minimize the distance from the target point to a set of effective solutions. the first step in this method is to calculate the value of the criteria for each alternative. the value of the criterion for the presence of the technology to measure a parameter is equal to zero if the technology is absent and to unity if the technology is present. the rest of the criteria values were calculated as pearson’s correlation coefficients modulo for a sample of natural gas with the following concentration ranges for each component: 85 100% for methane, 0 5% for nitrogen, carbon dioxide and propane. table 4 shows the values of the criteria for each alternative. table 2.4. values of criteria for each alternative oki c � co2 ⌘ " tekh. 1 1 1 0 0 r(ch4) 0.899 0.848 0.628 0.785 0.117 r(c3h8) 0.477 0.557 0.053 0.339 0.906 r(n2) 0.787 0.764 0.083 0.750 0.129 r(c) 1 0.992 0.233 0.629 0.451 r(�) 0.992 1 0.141 0.535 0.554 r(co2) 0.233 0.141 1 0.541 0.445 r(⌘) 0.629 0.535 0.541 1 0.403 r(") 0.451 0.554 0.445 0.403 1 at the next step, the proximity degree values of the criteria are calculated. for this, the criteria are divided into two categories for minimum and maximum. in this problem, it is necessary both to maximize the availability of measurement technology and correlation with the output parameters, and to minimize the correlation between the input parameters. for the minimum criteria, the proximity degree of the each alternative according to the each criterion oki is calculated by the formula: oki = max(oki)�oki max(oki)�min(oki) , i = 1...n (2.1) where oki is an estimated value of alternative i over criteria k, min and max are min and max operations, n is the number of alternatives. for the max criteria, we compute the proximity degree of alternatives as: oki = oki �min(oki) max(oki)�min(oki) , i = 1...n (2.2) copyright c� 2020 assa. adv syst sci appl (2020) multi-criteria estimation of input parameters in natural gas quality analysis 65 the computation results for the proximity degree of alternatives over the criteria are shown in table 5. table 2.5. proximity degree of alternatives over the criteria oki c � co2 ⌘ " tekh. 1 1 1 0 0 r(ch4) 1 0.935 0.653 0.854 0 r(c3h8) 0.497 0.591 0 0.335 1 r(n2) 1 0.967 0 0.947 0.065 r(c) 0 0.010 1 0.484 0.716 r(�) 0.009 0 1 0.541 0.519 r(co2) 0.893 1 0 0.534 0.646 r(⌘) 0.621 0.779 0.769 0 1 r(") 0.920 0.747 0.930 1 0 after we determined the proximity degree of alternatives, it is necessary to determine how to assess each alternative according to the criteria. we need to take into account the criteria’ significance calculated above. for this, the set of criteria g for each pair of alternatives is divided into two subsets: g1 is a subset of criteria, where the considered i-th alternative is superior to the j-th; g2 is a subset of criteria, where the considered i-th alternative is inferior to the j-th one. for example, the first alternative (the speed of sound c) exceeds the fifth (dielectric permittivity ") according to the first, second, fourth, fifth, sixth, and eighth criteria, but is inferior in all others. the general formula to calculate �ij estimates is as follows: �ij = p k2g1 avgk ⇤ (oki � okj) p k2g2 avgk ⇤ (okj � oki) (2.3) here avgk are the values of criteria weights from table 2; oki and okj -are the degrees of proximity of i and j alternatives over k-th criteria from table 4. since �ii = 1, the assessment matrix for the alternative is the one in table 6. table 2.6. the assessment matrix for the alternatives �ij c � co2 ⌘ " c 1 1.277 6.391 5.207 4.046 � 0.783 1 5.667 4.230 4.391 co2 0.156 0.176 1 0.850 1.452 ⌘ 0.192 0.236 1.177 1 2.036 " 0.247 0.228 0.688 0.491 1 the final step of methodology proposed is to calculate the eigenvector of the assessment matrix for the alternatives and subsequently normalize the matrix to determine the required vector of weights for the alternatives. the resulting vector looks like this: [0.412; 0.351; 0.074; 0.095; 0.068]. based on the obtained vector, it can be concluded that the first (speed of sound) and second (thermal conductivity) alternatives are superior. next, it is necessary to evaluate the data obtained for consistency of the alternatives assessment matrix and to reassess the alternatives. for this, the following parameters were calculated: the maximum (main) eigenvalue �max, the concordance index (ci) and the concordance ratio (cr). we calculate the maximum eigenvalue. then we use the matrix for evaluating alternatives as follows: the sum of each column of the matrix is multiplied by the corresponding element of the weight vector of the alternatives, and then the resulting numbers are summed up. the closer the maximum eigenvalue is to the number of alternatives (the dimension of the alternative assessment matrix) n, the more consistent the result is. for the obtained assessment copyright c� 2020 assa. adv syst sci appl (2020) 66 i. brokarev, s. vaskovskii matrix for the alternatives and the vector of alternatives’ weights �max is 5.106 for five alternatives. the results are then compared with the randomly scored data for a scoring matrix of the same dimension. for this, the consistency index is calculated using the formula: ci = �max � n n� 1 (2.4) the consistency index for the received data is 0.026. the consistency index of a randomly generated inversely symmetric matrix with the corresponding reciprocal values of the elements is called the random index (ri), and the value of the random index increases as the order of the matrix increases. table 7 shows the order of the matrix and the average values of the random index, determined for random samples [9]. table 2.7. dependence of the random index on the order of the matrix. matrix order 1 2 3 4 5 6 7 8 9 10 average ri 0.00 0.00 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.49 for the analyzed matrix dimension, the random index is 1.12 according to table 6. the ratio of the consistency index to the mean random index for a matrix of the same order is called the consistency ratio. cr = ci ri (2.5) a value of this parameter that is less than or equal to 0.1 is considered acceptable. if the consistency ratio is 0, it means that the data is completely consistent. if the concordance ratio exceeds 0.1, then we need to revise the estimates and recalculate the concordance parameters. in the case study, this is not required, since the agreement ratio is 0.024. based on the calculated characteristics, it follows that the selected estimates of alternatives are consistent. 3. confirmation of the multi-criteria estimation results by neural network analysis to confirm the results of the multi-criteria estimation of the input parameters for the neural network model, we performed a neural network analysis. for our experiment, we simulated a neural network model on the calculated data. the developed recurrent neural network [11] was trained, tested, and simulated with different sets of input parameters to determine the required component composition of gas mixtures. for five sets of input parameters, ten sets of parameters are possible, three in each. all possible sets of input parameters for this case are shown in table 8. table 3.8. possible sets of input parameters for the considered case set number 1 2 3 4 5 6 7 8 9 10 input parameters c, �, co2 c, �, ⌘ c, �, " c, co2, ⌘ c, co2, " c, ⌘, " �, co2, ⌘ �, co2, " �, ⌘, " co2, ⌘, " for each option from the set, the accuracy characteristics to determine the required parameters were calculated: maximum absolute deviation (maxae), mean absolute deviation (mae), maximum relative deviation (maxape), mean relative deviation (mape) according to the formulas presented below. copyright c� 2020 assa. adv syst sci appl (2020) multi-criteria estimation of input parameters in natural gas quality analysis 67 table 3.9. simulation results of the developed neural network model for various sets of input parameters component characteristic set number 1 2 3 4 5 6 7 8 9 10 maxae, % 0.425 0.696 0.751 0.516 0.567 0.832 0.524 0.535 0.556 0.987 methane mae, % 0.008 0.011 0.012 0.009 0.010 0.015 0.009 0.009 0.010 0.018 maxape, % 0.451 0.714 0.789 0.562 0.579 0.875 0.542 0.551 0.561 0.991 mape, % 0.009 0.011 0.012 0.010 0.011 0.016 0.010 0.011 0.010 0.018 maxae, % 0.285 0.482 0.621 0.378 0.415 0.781 0.394 0.419 0.402 0.871 nitrogen mae, % 0.011 0.013 0.016 0.012 0.013 0.018 0.013 0.014 0.013 0.018 maxape, % 0.308 0.482 0.568 0.376 0.421 0.790 0.382 0.407 0.399 0.862 mape, % 0.012 0.017 0.018 0.012 0.013 0.017 0.012 0.012 0.012 0.018 maxae, % 0.256 0.521 0.535 0.279 0.293 0.541 0.311 0.321 0.361 0.671 propane mae, % 0.007 0.012 0.012 0.007 0.007 0.012 0.008 0.008 0.009 0.014 maxape, % 0.235 0.539 0.546 0.291 0.318 0.567 0.309 0.311 0.354 0.649 mape, % 0.006 0.011 0.011 0.007 0.007 0.012 0.007 0.007 0.008 0.013 mae = 1 n nx i=1 |youtput � ytarget| (3.6) mape = 100% n nx i=1 |youtput � ytarget ytarget | (3.7) maxae = max |youtput � ytarget| (3.8) maxape = 100%max |youtput � ytarget ytarget | (3.9) here youtput – the value obtained from the statistical model, ytarget – the initial values of the target parameters, n – sample size, max – operator to compute maximum value. the proposed calculation was performed in matlab 2019b [12] with a plug-in for nist refprop [13]. the data included 150000 gas mixtures with the ranges described above and the calculated physical input parameters under study. the simulation results of the developed neural network model are shown in table 9. the accuracy of carbon dioxide characteristics are not shown, since the concentration of this component is an input parameter and is considered known. from the results of the neural network analysis, it can be concluded that the best accuracy is in the first set; the set contains the alternatives prevailing from the multi-criteria estimation. at the same time, the worst accuracy is in the tenth set with the least weighty alternatives. the complete ranking of all sets of input parameters is shown in table 10. this procedure was carried out using expert estimations to clarify the calculated accuracy characteristics. based on the results obtained, it follows that the results of the multi-criteria estimation and neural network analysis coincide. table 3.10. ranged sets of input parameters set number 1 2 3 4 5 6 7 8 9 10 set rank i vii viii ii v ix iii iv vi x copyright c� 2020 assa. adv syst sci appl (2020) 68 i. brokarev, s. vaskovskii 4. conclusion this paper investigates how to choose input parameters for the statistical model to analyze the quality of natural gas. it is shown that the proposed technique based on multi-criteria estimation methods allows us to choose the quantitative and qualitative values of the criteria for each input physical parameter. also, the proposed methodology makes it possible to choose the parameters before training the statistical model. this significantly reduces the computational and time costs; this is extremely useful when we analyze hierarchies in real time. note, that our technique showed the speed of sound and thermal conductivity to dominate over the rest of the input parameters. these results partially coincide with previous studies, where only the correlation analysis of input parameters was used [2, 3]. it was also found that the concentration of carbon dioxide, inferior to most of the considered alternatives, might not be used in further studies as an input parameter of the statistical model, although it is needed to reduce the amount of unknown concentrations of the components of the analyzed gas. the results of our multi-criteria estimation of the input parameters are confirmed by the results of our neural network analysis carried out on various sets of input parameters. to investigate this area further, we need to apply our system to analyze the quality of natural gas [14] in laboratory and industrial conditions. we plan to use a recurrent neural network model as the main model to determine the required quality characteristics of natural gas by analogy with the neural network model of a multilayer perceptron, which was used earlier [15]. acknowledgements the authors would like to thank the international cooperation projects between brisk ii ta and erasmus + 2017-1-se01-ka107-034292 staff mobility for the opportunity to conduct this study. copyright c� 2020 assa. adv syst sci appl (2020) multi-criteria estimation of input parameters in natural gas quality analysis 69 references 1. dorr h., koturbash t., kutcherov v. (2019) review of impacts of gas qualities with regard to quality determination and energy metering of natural gas. measurement science and technology, 30(2), 1—20. 2. vaskovskii s.v., brokarev i.a. (2020) comparative analysis of statistical models for the task of natural gas composition analysis. journal of information technologies and computing systems, 1, 34—43. 3. koturbash t.t., brokarev i.a. (2018) metod opredeleniya svoystv i sostava prirodnogo gaza po izmereniyam ego fizicheskikh parametrov [the method of determination of the properties and the composition of natural gas by measuring of its physical parameters]. sensors and systems, 1, 43—50, [in russian]. 4. koturbash t., bicz a., bicz w. (2016) new instrument for measuring velocity of sound and quantitative characterization of binary gas mixtures composition. measurement automation monitoring, 1, 254—258. 5. löfqvist t., delsing j., sokas k. (2003) speed of sound measurements in gas — mixtures at varying composition using an ultrasonic gas flow meter with silicon based transducers. international conference on flow measurement, groningen, netherlands. 6. xensor integration website [online]. available: http://www.xensor.nl 7. dynament infrared gas sensors [online]. available: https://www.dynament.com 8. bright sensors blueeye [online]. available: https://www.brightsensors. com 9. saati t. l. (1989) prinyatie reshenij. metod analiza ierarhij. radio i svyaz’, 316 p. 10. ringuest j.l. (1992) compromise programming. in: multiobjective optimization: behavioral and computational considerations. springer, boston, ma, 51—59. 11. vaskovskii s.v., brokarev i.a. (2019) distributed data gathering system for natural gas composition analysis. proceedings of the 22nd international scientific conference on distributed computer and communication networks: control, computation, communications (dccn-2019, moscow), 381—388. 12. matlab 2018b software [online]. available: https://www.mathworks.com. 13. refprop software [online]. available: https://www.nist.gov/srd/refprop. 14. brokarev i.a., vaskovskii s.v. (2019) distributed data gathering system to analyze natural gas composition. advances in systems science and applications, 19(4), 14— 24. 15. brokarev i.a. iskusstvennyie neyronnyie seti dlya resheniya zadachi analiza komponentnogo sostava gazovyih smesey [artificial neural networks for solving the problem of analyzing the composition of gas mixtures]. upravlenie bolshimi sistemami [large-scale systems control], 80, 98—115. copyright c� 2020 assa. adv syst sci appl (2020) introduction methodology for multi-criteria estimation of input parameters confirmation of the multi-criteria estimation results by neural network analysis conclusion microsoft word source+texts,+doc_vershkov,+kuchukova+application+of+modular+neural+networks+for+image+recognition+in+foggy+computing+environments adv syst sci appl 2025; 1; 22-29 published online at https://ijassa.ipu.ru. application of modular neural networks for image recognition in foggy computing environments nikolay verskov*, natalia kuchukova north-caucasus federal university, stavropol, russia abstract: the paper considers various approaches to the decomposition of artificial neural networks for the purpose of their application on fog computing nodes. based on the requirements for the organization of fog computing, a method of dividing the input information into subspaces by means of wavelet transform and subsequent proportional division of all layers of the neural network is proposed. the proposed approach achieves a significant gain in the amount of information transferred between modules compared to the currently used layer-by-layer partitioning. in addition, the proposed method optimizes the load on fog computing nodes by partially utilizing the modules. keywords: artificial neural networks, image recognition, fog computing, wavelet transform, proportional division of layers of the neural network. 1. introduction today, the iot is becoming a significant part of society and the volume of connected devices is growing rapidly. existing approaches based on cloud computing technologies are unable to provide the required quality of service due to the increasing latency of data transmission [10]. therefore, more and more attention is being paid to the fog computing (fc) model. the fc model involves placing computing power at the edge of the network [11], i.e., close to the hardware in use [8]. this approach allows to significantly reduce the cost of data transmission, as well as to increase the privacy of user data. more and more tasks, such as virtual assistant, computer vision and object recognition, require the use of artificial neural networks (anns). highly accurate ann models have significant volumes and require large computational resources. with the cloud computing model, there is no problem in using anns, computing resources in the cloud are sufficient. however, the shift in focus to fog computing requires the placement of anns on fc nodes, whose computing power is much lower than cloud computing. in this regard, multiple fc nodes are required to be used simultaneously to support the high-fidelity ann model. application of ann is usually divided into 2 stages: training and intended use. the growth of requirements to model accuracy is directly related to the growth of the number and volume of ann layers, which, in turn, requires large expenditures of computational resources that fc nodes do not possess. to solve the emerging contradiction, different approaches are used today: 1. the training process is performed on the cloud server and the ann is deployed on fc nodes only directly for the intended application [9]. 2. utilizing the client-server computing paradigm [12], which requires infrastructure support. 3. ann model compression [12], which requires infrastructure support or special training steps. 4. using a distributed programming model such as mapreduce [3]. * corresponding author: vernick61@yandex.ru application of modular neural networks for image recognition… 23 copyright ©2025 assa. adv. in systems science and appl. (2025) 5. using mathematical methods of dividing the input information into modules with subsequent separate processing of each module at fc nodes [19]. approach 1 is widely applicable and can be used if the computational power of the fc node is sufficient for ann operation. this approach is not modular and allows to perform the most energy-intensive training operation on the server. approach 2 is also non-modular and moves energy-intensive operations from the fc domain to the cloud server. this approach does not solve the problem of service time delay and is actually a veiled form of cloud computing. model compression (approach 3), as a rule, requires additional costs for ann optimization, but allows to obtain a performance gain of 20-30% compared to approach 1. the use of distributed programming model (approach 4) is the most popular today. this approach allows to divide the ann into layers (by one or other attribute) and execute one or more layers on a separate fc node [2, 12]. a significant disadvantage of this approach is the heavy load on data transmission channels, since the amount of information transferred between layers is very large. at the same time, this approach allows training and intended use of ann without accessing the cloud server. approach 5 has a significant advantage over approach 4: with its help, the ann is divided into modules not along the layers, but across, i.e., all the layers of the module reduce their volume so that the computational power of the fc node is sufficient for its functioning [19]. due to the fact that almost all module layers are on the same node, the channel load is significantly reduced. another advantage of the proposed approach is the non-uniform distribution of features within modules, i.e., the use of a smaller number of modules only slightly reduces the recognition accuracy of ann and allows optimizing the load of fc nodes. one of the key characteristics of fc is scalability and dynamism [7], i.e. fc should be adaptive in nature. adaptability, in turn, implies support for the following key mechanisms: elasticity of computation, pooling of resource capabilities, adjustment to changes in data load and changes in network composition [10]. this paper is devoted to the use of the latter approach for building scalable and dynamic anns on fc nodes. 2. justification of using wavelet transform to create modular anns for fc the proposed approach is based on a well-known concept in the theory of information transmission power spectral density (psd). if x (t) is a periodic signal with period t, then the average power per period can be defined as 2 2 2 1 ( ) . t t p x t dt t   parseval's theorem for a real periodic signal x (t) has the form 22 2 2 1 ( ) , t nt n p x t dt c t      (2.1) where cn are the complex coefficients of the fourier series. the power spectral density p (f) of a periodic signal x (t), which is a real, even and non-negative function of frequency, gives the distribution of signal power over the frequency range, is defined as follows:  2 0( ) n n p f c f nf     (2.2) here f0 is the center (average) frequency. the input values of the ann can be considered as a discrete periodic signal, since all input values have the same digit capacity, and the period of the signal coincides with the digit capacity of the ann input. this discrete periodic signal at the ann input defines a 24 n. vershkov, n. kuchukova copyright ©2025 assa adv. in systems science and appl. (2025) finite set of classifiable images. differences in images belonging to the same cluster can be considered as additive noise on the ideal (averaged) image. it follows from riemann's theorem that the fourier coefficients cn of the function  ,rf l    tend to zero when n  [17], i.e., the partial sums   1 2, ,k k kp f k n n    , of expression (2) will also decrease as n increases. this means that if we divide the spectrum of the signal at the ann input into 2 equal parts, the psd in the left (low-frequency) part will be higher than in the right one. based on (2.1), it can be stated that the power of features in the signal representing the low-frequency part will be higher than in the high-frequency part. in [4] it is proved that in the presence of nonlinearity in ann, it is possible to construct connections and select coefficients between neurons in such a way that ann can compute any continuous function from its inputs with any accuracy. consequently, an ann trained with the low frequency part of the signal will perform image recognition with higher accuracy than an ann trained with the high frequency part of the signal. the fourier transform of a numerical sequence is a 2π-periodic function. therefore, when one speaks of low and high frequencies, one means close to zero or ±π, and the sampling step δt = 1. in this case, the kotelnikov (nyquist) frequency is equal to π [1, 16]. for ideal low-pass h (ω) and high-pass g (ω) filters, the transfer characteristics are described by expressions:   1, , , 2 2 , 0, , , 2 2 h                                 0, , 2 2 . 1, , , 2 2 g                               (2.3) from expression (2.3) it is not difficult to find the coefficients of ideal filters (by performing the inverse fourier transform):     1 , 02 0, 2 , 1 , 2 1 2 1 k n h n k n k k                1 1 , 02 0, 2 . 1 , 2 1 2 1 k n g n k n k k             (2.4) when the signal is decomposed into highand low-frequency components      ,g h n n nx x x the number of samples doubles, but using the kotelnikov-shenon theorem it is possible to discard half of the samples [16]. this process is called decimation. thus, to realize the process of dividing the input sequence into highand low-frequency parts it is enough to perform its convolution with coefficients of ideal filters followed by decimation. consider 2 haar filters with coefficients [1, 6, 16]: 0 1 0 1 1 1 1 1 , ; , . 2 2 2 2 h h g g     the convolution of the low-pass filter kernel  ih with the input sequence  jx will result in the following sequence 1 1 1 2 2n k n k n nk z a h x x x     (2.5) application of modular neural networks for image recognition… 25 copyright ©2025 assa. adv. in systems science and appl. (2025) and convolution of the high-pass filter kernel  kg with the input sequence  jx the sequence will be obtained 1 1 1 2 2n k n k n nk z d g x x x     (2.6) the sequence an, defined by expression (2.5), is obtained as the arithmetic mean of two neighboring values of  jx , reducing the oscillation frequency, while the sequence dn (2.6), being essentially the first derivative of the input values of  jx , reflects its oscillations. thus k k kx a d  , i.e., the input values  kx have been decomposed into a low-frequency part ak and a high-frequency part dk and can be recovered by a postal addition of the components. turning to the wavelet transform, we define in the space l2(r) the orthonormalized haar basis [16] as     1, 0,1 ( ) 0, 0,1 x x     . (2.7) in [16] it is shown that the spaces vj, generated by the system of functions , 2 (2 ),j j j n x n   (2.8) created on the basis of the parent function (2.7), form an infinite system of nested subspaces 0 1 2 ...v v v   . this system of subspaces can be used to go from an arbitrary function f (x) from l2(r) to its more or less exact approximation in the space jv z , the spaces vj are being scaled versions of the space v0. thus, the use of scaled versions of spaces for ann training will allow to create modular neural networks with cross-layer separation [19], since the dimensionality of spaces decreases proportionally to the level of nesting due to decimation. at the same time, the dimensionality of the ann module can be reduced in proportion to the level of nesting. in this case, the accuracy of space approximation vj determines the classification accuracy of the ann module from more accurate (low-frequency part) to less accurate (high-frequency part). the ann modules can be used separately, for example, for optimization (dimensionality reduction) of ann, or together. a single linear layer was used to combine the results of the modules in [19]. however, the haar wavelet transform assumes an inverse transform [6], but due to the nonlinearity of the ann, the inverse transform kernel cannot be used, so the inverse transform layers need to be trained. 3. exploring the potential of a modular cross-layer partitioned ann for use in fc the task of using ann at fc nodes can be divided into several subtasks: wavelet transformation of the input signal with decimation, division of the initial ann into modules (according to the number of input information module), training of the network and its application as intended. thanks to the mcculloch-pitts model [13], it has become possible to realize spectral decomposition operations in direct propagation anns and convolutional layers for almost any, including random signals. here, the most important feature is the correlation radius of the information field [15]. most conveniently, this operation can be performed using convolution layers, which allow to perform, for example, wavelet transform simultaneously with several kernels. this operation is considered in detail in [20]: 26 n. vershkov, n. kuchukova copyright ©2025 assa adv. in systems science and appl. (2025) ( ) ( ) ( ) ( ) , ( ) ( ) ( ) ( ) k k x n h n x k h n k x n g n x k g n k             if the signal at the input of the ann is a one-dimensional sequence, one convolution layer with kernels h (n) and g (n) is sufficient to perform the haar wavelet transform. in image processing, the input signal is a two-dimensional matrix, and 3 convolution layers will be required to perform the two-dimensional haar wavelet transform (fig. 1). fig. 1. two-dimensional wavelet transform as noted earlier, the division of ann into modules is most often performed layer-bylayer [18]. in this case, the volume of the ann does not change, and the lack of computing resources is compensated by using several computing devices connected in series with the placement of several layers of the network on each of them. the authors rely on the thesis that it is impossible to divide one layer into 2 or more computing devices [18]. as it was shown in [19], such separation is possible not at the level of the ann layer, but at the level of input information. it was proposed to divide the input information into several modules and to carry out joint or separate processing of modules on anns, proportionally reduced compared to the original network. the authors in this paper set out to compare the performance of the approaches proposed in [18] and [19] for fc nodes. for this purpose, 2 sets were used: the simplest one, mnist [14], for visual demonstration of comparative performance and a much larger set gtsrb [5] (german traffic sing recognition benchmark) for working with alexnet-like network. for the first experiment, an ann with two convolution layers and a classifier of two linear layers was used on a personal computer running the "windows 10 home" operating system with an intel© core™ i7-10510u processor and 16 gb of ram. each of the layers was placed in a separate ann, which were placed sequentially, i.e. the output of the previous ann is the input of the subsequent one. thus, the entire network was partitioned layer by layer and trained for later use at the fc nodes. it is clear that with such an architecture, it is impossible to remove at least one ann, because in this case a complete retraining is required. for comparison, a similar ann was divided across the layers into 4 modules and trained. at the same time, 3 convolution layers were added to the ann architecture to perform the two-dimensional haar wavelet transform to the first layer and 3 layers at the end of the network for the inverse transform. the training results are presented in table 1. table 1. training results of anns with longitudinal and cross-layer separation maximum training cycle time minimum training cycle time average training cycle time mean square deviation separation of anns along the layers 0.21533 0.14007 0.16213 0.01076 separation of anns across layers 0.21266 0.14084 0.16620 0.00917 application of modular neural networks for image recognition… 27 copyright ©2025 assa. adv. in systems science and appl. (2025) table 2 summarizes the amount of data transferred between modules. table 2. dimensionality of data transferred between modules ins input 1 module output 2 module output 3 module output 4 module output separation of anns along the layers 28х28х100 32х14х14х100 3136х100 1000х100 10х100 separation of anns across layers 28х28х100 4х196х100 4х10х100 10х100 table 1 shows that the time characteristics of both types of architectures are very close. taking into account that there are 6 more layers in the ann with cross-layer partitioning (forward and inverse two-dimensional wavelet transform), the difference in the average training time by 0.004 sec does not look critical. it became possible due to parallelization of operations between modules. here, it should be noted that the number of training cycles in ann with cross-layer partitioning turned out to be ≈1.5 times greater than in ann with layer-by-layer partitioning: 314 vs. 192 cycles. this is quite understandable, since in addition to the training of 4 modules, 3 layers of the inverse two-dimensional wavelet transform are subject to training. as noted earlier, introduction of nonlinearity into ann leads to impossibility of the inverse transform using the standard kernel. table 2 gives an idea of the amount of data transferred between modules. with layer-bylayer partitioning of the ann, the amount of data transferred is proportional to the size of the layer (the size of the batches is the same and equals 100). at the same time, the amount of data to be transmitted through communication channels is 8 times less at cross-layer separation of ann after the first layer, 78.4 times less after the second layer, and not required after the third layer. at the same time, the number of modules in cross-layer partitioned ann is much larger: 6 vs. 4, since 2 more modules are required to perform forward and inverse wavelet transform. but the biggest advantage in cross-layer partitioning of anns is that not all 4 modules can be used in network operation, but for example 2 or 3 (table 3). table 3: comparison of recognized quality and execution time using different number of modules 1 module 2 module 3 module 4 module recognition quality, % 96.08 97.18 97.72 98.11 average turnaround time, sec. 0.014 0.025 0.037 0.047 during the experiment, zero tensors were fed instead of the output data of the missing modules. at the same time, once trained ann does not require retraining if 2, 3 or 4 modules are used depending on the availability of fc computational resources. from the conducted experiment the advantages of cross-layer separation of ann become clear: firstly, the amount of transferred information between layers is tens of times less than at layer-by-layer separation, secondly, the possibility of using not all trained modules, which allows to reduce the load on fc nodes at some reduction of recognition accuracy. the disadvantages of the proposed method include the need for two additional modules: to perform the wavelet transform and to combine the data using linear or convolution layers. the following libraries for the python 3.7 (64 bit) compiler: numpy (v. 1.21.4), pywavelets (v. 1.3.0), pytorch (v. 1.12.1) were used for the experiment using the gtsrb dataset [5]. the experiment was conducted on a server running the linux 2operating system, cpu e5-2690v4 processors, 1 tb of ram. the purpose of the experiment was to evaluate the feasibility of the proposed cross-layer separation method for volumetric high-precision alexnet-like anns, as well as to assess the advantages of using linear and convolutional layer modules for combining the results of the linear and convolutional layer modules. the german traffic sign recognition test (gtsrb) includes 43 different types of road signs divided into 39,209 training and 12,630 test color images. the pictures depict a variety of lighting and environments. to reduce the volume of ann, the authors allowed themselves 28 n. vershkov, n. kuchukova copyright ©2025 assa adv. in systems science and appl. (2025) to convert color images of signs into grayscale. such approach allowed, on the one hand, to reduce the volume of ann due to the lack of necessity to process each color, on the other hand, increased the complexity of the problem due to the lack of 3 processing channels. table 4 shows comparative data on the recognition quality and performance of two alexnet-like networks, one of which used a linear layer as a unifying layer and the other used 3 convolutional layers simulating the inverse haar wavelet transform with the possibility of training kernels. table 4. comparison of the features of the unifying linear and convolutional layers 1 module 2 module 3 module 4 module recognition accuracy, line layer, % 78.57 96.24 99.38 99.49 recognition accuracy, convolution layer, % 99.23 99.29 99.31 99.34 recognition time of test images, line layer, с 14.8 21.84 28.47 35.65 recognition time of test images, convolution layer, с 15.04 21.93 28.32 35.35 table 4 shows that the temporal characteristics of both architectures are very close and it can be argued that there are no advantages. but the recognition quality of the ann with a convolutional layer showed a significant advantage: even 1 low-frequency module performs recognition with a quality almost as good as a full network of 4 modules. application of linear layer shows a significant decrease in recognition quality by more than 20%. and only 3 modules out of 4 show quality comparable to full ann. however, it should be noted that this approach does not always give such an advantage. a similar experiment on the mnist dataset showed almost the same drop in recognition quality when the number of modules decreases for anns with linear and convolutional combining layer. obviously, such an effect appears at significant redundancy of the network and/or input data. conclusion in this paper, methods of partitioning anns into modules were investigated. the proposed method of cross-layer partitioning of ann and further modular application showed a significant advantage over the currently used layer-by-layer partitioning, primarily in terms of the volume of data transferred between modules. in addition, the proposed method allows to use not all trained modules at the same time, but only a part of them. in this case, no additional training of the network is required when excluding a part of modules. the layerby-layer separation of anns does not allow using this approach. these advantages allow the proposed ann partitioning principle to be used in fog computing when the number and computational power of fc nodes are not known in advance. in this case, once trained ann can be placed on the number of fc node that is available without the need for additional training. in addition, when fc nodes are released, the used ann can be augmented to increase the recognition accuracy. thus, the scalability and dynamism of computation, which is one of the most important indicators of fc, are achieved. acknowledgements the research was supported by the russian science foundation grant no. 24-21-00149, https://rscf.ru/en/project/24-21-00149/. references 1. ahmed, n. & rao, k. r. (1975) orthogonal transfarms for digital signal processing. berlin, germany: springer-verlag. 2. bahtin, v. v. (2022) a monolithic neural network separation algorithm for implementing foggy computing in programmable logic devices, bulletin of pnrpu. application of modular neural networks for image recognition… 29 copyright ©2025 assa. adv. in systems science and appl. (2025) series: electrical engineering, information technology, control systems, 41, 123– 145, [in russian], doi: 10.15593/2224-9397/2022.1.06 3. dean, j. & ghemawat, s. (2008) mapreduce: simplified data processing on large clusters, communications of the acm, 51, 107–113. 4. gorban, a. n. (1998) generalized approximation theorem and exact representation of polynomials from several variables by superpositions of polynomials from one variable, russian math. 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(2022) optimization of artificial neural networks using wavelet transforms, program comput soft, 48, 376–384, https://doi.org/10.1134/s036176882206007x adv syst sci appl 2017; 3:9–21 published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/501 analysis of heterogeneity of transport flows igor o. molybog1,2∗, yury v. chehovich2 1skolkovo institute of science and technology, moscow, russia 2dorodnitsyn computing center, russian academy of sciences, moscow, russia abstract: the situation when two independent transport flows are passing through the same edge of a transport network with different speeds is quite common for modern cities. the precision of mobile gps devices, which are widely used for traffic monitoring, does not allow to determine the characteristics of transport flows in this situation because of their proximity. it causes difficulties in the analysis of the traffic situation, which may result in significant errors of transport routing systems. we propose a method to get the average speed of transport flows from common gps traffic data using machine learning techniques. to do that we introduce a formal statement for the flow separation problem, which makes it possible to divide the problem into two sequential parts: statistical and optimizational. we analyze the possible approaches to the solution of both, construct features space for the statistical part and determine the complexity of the optimizational one. the developed techniques were implemented and tested to be working on real traffic data. keywords: traffic data analysis, classification, sequence separation 1. introduction traffic data analysis plays important role in different areas of civil engineering such as infrastructure planning [1], transport flows control [2], and vehicles routing [3]. there is plenty of ways to collect the traffic data that can be used for different purposes. there are ways, based on counting flow through a section [4, 5], but in this work we focus on a way that based on following movements of individual particles in a system [6–10]. we consider gps data from drivers’ mobile devices, which are commonly used to estimate average speeds on different parts of a transport system. commonly used ways to analyze the traffic data include time series analysis [11], and classification [12]. modern routing systems consider average speed on a road to build optimal route for a vehicle. however, there is an effect called flow separation that may cause serious errors in average speed estimation. it is commonly observed before crossroads or exits from highways and consists in velocity heterogeneity across the flow. the problem that prevents us from considering the effect with standard engineering approaches is low precision of mobile gps devices [13, 14]. it makes it impossible to determine the lane a vehicle holds on a road. nevertheless, the accuracy becomes more and more important characteristic of routing systems and its improvement is of our interest. here we propose to apply machine learning techniques to get separate estimation of each transport flow in case of their separation. we use statistical approach to find the most likely value of the average speed of each flow and state optimizational problem to determine the distribution of the vehicles between flows. ∗corresponding author: igormolybog@gmail.com http://ijassa.ipu.ru/ojs/ijassa/article/view/501 10 i. o. molybog, y. v. chehovich while the flow separation effect arises in many applications [6, 15], the problem was neither stated, nor solved for the purposes we consider here. 2. informal description for simple explanation and intuition behind the problem take an example. consider a twolane road which is 8 meters wide and splits into two one-lane sleeves that go in opposite directions. assume that cars can’t change the lane along the way. let’s say the satellite can determine the position of a car with precision of 15 meters. this means that for any car that moves through the two-lane segment there is no way to detect immediately the lane it is moving through. when the car took its sleeve, we can conclude about the lane it was on and estimate the speed of the whole flow on that lane with the average speed of the car when it was on the two-lane segment. however, this information is out-of-date by the time when the car has left the two-lane segment and the speed of the flow could be changed during the time the car was changing to the sleeve. one possible way to deal with the problem is to track cars that are following the current one and are still on the move through the two-lane segment when the current car left. we can pick those have similar behavior in the past and assume that they should be moving along the same lane as the current car used to. those cars can help us to build a real-time estimation of the speed of the flow. the following section generalizes and formalizes the problem. fig. 2.1. illustration for heterogeneous transport flow 3. formalization consider a continuous (does not contain forks, exits or crossroads) segment s0 of a multilane one-way road. assume that the road has branching into w directions at some distance after the segment. denote them as {direction1, . . .directionw}. consider also s−1, s−2, . . . s−b — a continuous chain of straight segments of the road before s0. every vehicle on every segment has the following parameters: average speed v(vehicle), time of exit τ(vehicle) from the segment and the further direction on the next branching δ(vehicle) ∈ {direction1, . . .directionw}. at every moment t for every vehicle the parameters can be known or unknown. using this we can divide all of the vehicles into three groups: g1(t), g2(t), g3(t). their description you can see in the table 3.1. note that there is a time lag between the moment when a vehicle left the segment s0 and the moment when we found out its further direction. this time lag depends on the average speed of the flow, changes with time and plays crucial role in division into the groups. copyright c© 2017 assa. adv syst sci appl (2017) analysis of heterogeneity of transport flows 11 table 3.1. groups of vehicles g1(t) g2(t) g3(t) speeds on vs−b , . . . , vs0 known known unknown further direction known unknown unknown formally: ∃ lag(t) such that g1(t) = {vehicle|τ(vehicle) < t− lag(t)}, g2(t) = {vehicle|t− lag(t) ≤ τ(vehicle) < t}, g3(t) = {vehicle|t ≤ τ(vehicle)}. assume that the further direction ({direction1, . . .directionw}) is determined by the lane the vehicle is passing through. we calculate the average speed of a flow as exponential moving average with parameter λ of speeds of the vehicles in it: avsp(t) = λv(t) + (1− λ)avsp(t− 1), λ ∈ (0, 1) the parameter λ is assumed to be given. it is usually tuned for a system of average speed estimation beforehand. in respect to the introduced assumption, at the moment t average speeds on each lane avsp1(t− lag(t)), . . . ,avspw (t− lag(t)) can be calculated using information about vehicles in g1(t). at the same time, avsp1(t), . . . ,avspw (t) are impossible to calculate because of lack of information about g2(t) vehicles. this may cause serious errors in speed estimations and eventually in routing results. this paper deals with the problem of estimation of avsp1(t), . . . ,avspw (t). we are also interested in distribution of the vehicles between traffic flows. for practical purposes, it is especially important to have solutions of the problem for cases when the average speed differs significantly on lanes. 4. sequence separation let us consider a sequence of real numbers (speeds of vehicles as they are passing) {ul}nl=1. it can be broken into w subsequences (according to the transport flows): {ul}l∈l1 , {ul}l∈l2 , . . . {ul}l∈lw , where {lw}ww=1 is an unknown splitting of {1, . . . n}. every element l of a splitting is a subsequence of {1, . . . n} and contains elements in ascending order. let’s consider the statistics of exponential moving average as a function of element of the splitting: a(lw) = λ |lw|∑ i=1 (1− λ)i−1uli , li ∈ lw. the function represents the average speed of the flow that contains only vehicles with indexes from lw. copyright c© 2017 assa. adv syst sci appl (2017) 12 i. o. molybog, y. v. chehovich the problem is to find splitting {kw}ww=1 such that the values of a on the elements of {kw}ww=1 are as close as possible to the values of a on the true splitting {lw}ww=1, which is unknown. to formalize the sense of distance, introduce the loss function: f = w∑ w=1 αw(a(lw)− a(kw))2, αw > 0. 4.1. vector representation introduce 2w vectors of size n each: definition 4.1: jw = j(lw, λ) : jkw := { 0 k /∈ lw (1− λ)m−1 k = lm ∈ lw where jkw is the slightly modified indicator that a vehicle k is actually in the flow w. definition 4.2: iw = i(kw, λ) : ikw := { 0 k /∈ kw (1− λ)m−1 k = km ∈ kw where jkw is the slightly modified indicator that a vehicle k is assumed by the model to be in the flow w. in the vector jw, the components with numbers in {lm}|lw| m=1 are equal to (1− λ)m−1, all the other components are zeros. similarly for the vector iw. define the loss for the flow w: definition 4.3: fw = |lw|∑ m=1 (1− λ)m−1ulm − |kw|∑ m=1 (1− λ)m−1ukm form vector u out of the sequence {ul}nl=1. then fw = itwu− jtwu = (itw − jtw)u . note that the total loss now is f = λ2 w∑ w=1 αwf 2 w = λ2 w∑ w=1 αw(itw − jtw)u(itw − jtw)u 4.2. risk minimization we introduce a class probability distribution on 1, . . . , n, and assume that numbers belong to classes independently. a number belongs to class t if it lies in lw. matrix p of size n× t contains probabilities that index i belongs to the subsequence lw. also piw is the probability that the i-th component of jw is not zero: [jw]i 6= 0. definition 4.4: p(jw) — probability of the realization of jw. it is the probability that the floww really consists of vehicles with indexes that correspond to nonzero components of jw. introduce the expected loss on a flow of the prediction iw: definition 4.5: ew(i) = ejw{f 2 w|iw = i} = ∑ jw f 2 wp(jw) ∣∣ iw=i copyright c© 2017 assa. adv syst sci appl (2017) analysis of heterogeneity of transport flows 13 lemma 4.1: ew(iw) = itwuu t iw − 2( ∑ jw p(jw)jw)tuut iw + ut ( ∑ jw p(jw)jwj t w)u proof from definition 4.3 and the expressions followed: ew(i) = ∑ jw (it − jtw)u(it − jtw)up(jw) = ∑ jw ut (i− jw)(it − jtw)up(jw) = = ∑ jw ut (iit − jwit − ijtw + jwj t w)up(jw) = = ∑ jw (ut iitup(jw)− ut jwitup(jw)− ut ijtwup(jw) + ut jwj t wup(jw)) decompose the sum: • ∑ jw ut iitup(jw) = ituut i ∑ jw p(jw) = ituut i • ∑ jw ut jwi tup(jw) = ∑ jw p(jw)jtwuu t i = ( ∑ jw p(jw)jtw)uut i • ∑ jw ut ijtwup(jw) = ∑ jw ituut jwp(jw) = ituut ( ∑ jw p(jw)jw) = ( ∑ jw p(jw)jtw)uut i • ∑ jw ut jwj t wup(jw) = ut ( ∑ jw p(jw)jwj t w)u joining the sum, we get the statement of the lemma. lemma 4.2: k-th component of the vector π(pw, λ) = ∑ jw p(jw)jw can be written as: [ ∑ jw p(jw)jw]k = pkw k−1∑ s=0 (1− λ)s ∑ m⊂{1,...k−1}:|m |=s ( ∏ m∈m pmw ∏ m/∈m (1− pmw )) proof a component of expectation of a vector is equal to expectation of the component of the vector because of linearity of mathematical expectation. [ ∑ jw p(jw)jw]k = k−1∑ s=0 (1− λ)sp{[jw]k = (1− λ)s}. [jw]k = (1− λ)s ⇔ [jw]k 6= 0,∃m ⊂ {1, . . . k − 1}, |m | = s : ∀m ∈m : [jw]m 6= 0, ∀m /∈m : [jw]m = 0. p{[jw]k 6= 0} = pkw. as the event {[jw]k 6= 0} is independent, we can get the formula. similarly, for π(pw, λ) = ∑ jw p(jw)jwj t w : copyright c© 2017 assa. adv syst sci appl (2017) 14 i. o. molybog, y. v. chehovich lemma 4.3: [ ∑ jw p(jw)jwj t w ]lk =  pkwp l w l−1∑ s=0 (1− λ)s k−l+s∑ m=s+1 (1− λ)m ∑ a⊂{1,...l−1}:|a|=s ∑ b⊂{l+1,...k−1}:|b|=m−s−1( ∏ a∈a∪b paw ∏ a∈{1,...k−1}\(a∪b∪{l}) (1− paw) ) , k > l pkw k−1∑ s=0 (1− λ)2s ∑ m⊂{1,...k−1}:|m |=s ( ∏ m∈m pmw ∏ m/∈m (1− pmw )), k = l (4.1) proof a component of expectation of a matrix is equal to expectation of the component of the matrix. case k = l is very similar to previous lemma. consider k 6= l. the matrix is symmetric, so prove for k > l. [ ∑ jw p(jw)jwj t w ]lk = l−1∑ s=0 (1− λ)s k−l+s∑ m=s+1 (1− λ)mp{[jw]l = (1− λ)s; [jw]k = (1− λ)m}. [jw]l = (1− λ)s ⇔ [jw]l 6= 0, ∃a ⊂ {1, . . . l − 1}, |a| = s : ∀a ∈ a[jw]a 6= 0,∀a /∈ a[jw]a = 0 [jw]k = (1− λ)m ⇔ [jw]k 6= 0,∃b ⊂ {1, . . . k − 1}, |b| = m : ∀b ∈ b[jw]b 6= 0,∀b /∈ b[jw]b = 0 take into account that l ∈ b. as the event {[jw]k 6= 0} does not depend on {[jw]k′ 6= 0}∀ k′ 6= k, we can derive the formula from the statement of the lemma. both vector π and matrix π depend only on parameter λ and column pw of the matrix p . therefore, ew(i) is a quadratic function, and its minimization is a particular case of discrete quadratic programming problem. if p consists of 0 and 1, ew(i) has evident zero in i = π(pw, λ). in other words, if vector jw is not stochastic, optimal value of i is iw = jw. total risk function, which is the cumulative expected loss for all flows: e(i1, . . . , iw ) = ej1,...,jw {f |i1, . . . iw} = λ2 w∑ w=1 αwew(i) e(i1, . . . , iw ) = λ2 w∑ w=1 αw(itwuu t iw − 2π(pw, λ)tuut iw + utπ(pw, λ)u) 5. feasibility set describe the set of all possible vectors that iw can be chosen from. it is denoted i and consists of indicator-like vectors for all possible subsequences [k1, . . . , kd] of 1, . . . , n. formally: i = { i ∣∣∣∃d ≤ n,∃[k1, . . . , kd] subsequent of [1, 2, . . . , n] : ik = { 0 k /∈ [k1, . . . , kd] (1− λ)m−1 k = km } copyright c© 2017 assa. adv syst sci appl (2017) analysis of heterogeneity of transport flows 15 fig. 5.2. illustration for the lemma 5.1 lemma 5.1: conv(i) = k ∩ {x ∈ rn−1 : 0 ≤ x1 ≤ 1}, wherek is a multifaceted cone with 2n− 2 facets and ( 1 λ , 0, . . . , 0)t as the apex. proof • for each l 6= 1 consider set {x ∈i: xl = 0}. all of the points there lie in the n− 1dimensional hyperplane with ( 1 λ , 0, . . . , 0)t . there are n− 1 such hyperplanes • for each l 6= 1 consider set {x ∈i: xl 6= 1}. a vector from the set has the form ... (1− λ)r ...  if there are r non-zero coordinates among its coordinates number 1, . . . , l − 1. denote nl =  1 ... 1 1 λ 0 ... 0  , where 1 λ is on the position l. copyright c© 2017 assa. adv syst sci appl (2017) 16 i. o. molybog, y. v. chehovich now, show that ntl  ... (1− λ)r ...  = 1 λ = ntl  1 λ 0 ... 0 . to do that we can use λ(1−λ)r−1+(1−λ)r λ = 1 λ (1− λ)r−1: ntl  ... (1− λ)d ...  = r−1∑ k=0 (1− λ)k + 1 λ (1− λ)r = 1 λ • all of the vectors with x1 = 1 share a hyperplane, just like all of the vectors with x1 = 0. in this way, all of the vectors from i with same state of a coordinate (zero or non-zero), share a n− 1-dimensional hyperplane. 2(n− 1) of them have intersection in ( 1 λ , 0, . . . , 0)t . definition 5.1: n =  1 λ 1 · · · 1 0 1 λ · · · 1 ... ... . . . ... 0 0 · · · 1 λ  it is easy to check that from the proof above follows: conv(i) = {x ∈ rn ∣∣ntx ≤ 1 λ 1;x ≥ 0} for compact notation, make several formal definitions: definition 5.2: i = [i1, . . . , iw ] mutual constraints on vectors i1, . . . , iw consist in that only one element in every row of i is nonzero. it can be written in the form of sparsity constraints{ ‖i‖0 = n i1 > 0 definition 5.3: a = diag( √ α1, . . . , √ αw ) definition 5.4: π = [π(p1, λ), . . . , π(pw , λ)] lemma 5.2: e(i1, . . . , iw ) = e(i) = λ2(aitu)2 + λ2(aitu)(aπtu)t + const proof directly from the definition and lemma 4.1. 6. final problem formulation to estimate avsp1(t), . . . ,avspw (t) we propose to use information about vehicles from g2(t). let {vehiclem}nm=1 sequence of vehicles from g2(t) in descending order of exits from s0. then {um}nm=1 is a corresponding sequence of speeds of the vehicles on s0. copyright c© 2017 assa. adv syst sci appl (2017) analysis of heterogeneity of transport flows 17 in the applications, average speed of a flow is calculated as follows: avspw(t) = λ |lw|∑ i=1 (1− λ)i−1uli + λ(1− λ)|lw|avspl(t− lag(t)); these functions are very similar to a. to apply all of the results from thr previous section, slightly modify the problem. form û of size n+w , which is u stacked with avsp1(t− lag(t)), . . . ,avspw (t− lag(t)). expand p with w ×w identity matrix. new formulation for risk function: e(i1, . . . , iw ) = λ2(ait û)2 + λ2(ait û)(aπt û)t + const 6.1. optimization complexity as a result of the analysis, the problem of optimization stage is equivalent to: (ait û)2 + (ait û)(aπt û)t −→ mini ∈ i × · · · × i ‖i‖0 = n i1 > 0 where i is the set of extreme points of the convex polyhedron: i = extr ( {x ∈ rn+w ∣∣λntx ≤ 1;x ≥ 0} ) this problem belongs to the class of mixed-integer linear programming problems, which is proved to be np-complete. there are several approaches to solving the problem. some of them are considered in [16, 17]. because of asymptotic complexity of calculation of π and possible errors, which can be caused by estimating p , we will not conduct further investigation of the optimization problem and algorithms of its solution. 6.2. distribution estimation p is the only unknown parameter of the model. possible ways to estimate the probability for each vehicle depend on the data we have and include: • without respect to the vehicle, as an unconditional probability. • with respect to the type of a vehicle (a truck or a car) • with respect to the short-term history of a vehicle (speed history) • with respect to the long-term history of a vehicle (previous passages) • with respect to the planned route of a vehicle (for routed vehicles) 7. proposed solution as it was already mentioned, if p consists of 0 and 1, the optimizational part becomes trivial. another remarkable feature of such matrices is that it can be built by any classification method, which gives us a powerful tool of machine learning to perform the estimation. therefore, for practical purposes we will work with the class of matrices that consist of 0 and 1. in the following experimental part we will work with information that can be extracted from gps tracks data. more specifically, every vehicle at each moment t is characterized by a copyright c© 2017 assa. adv syst sci appl (2017) 18 i. o. molybog, y. v. chehovich vector, composed of its short-term speed history. we can use the metric space of these vectors to build a classifier on information about some vehicles from g1(t) as a training set and use it on a test set from g2(t). 8. experiment the data for the experimental part was provided by yandex. in the model that was used for data collection the roads system was approximated with a graph of straight segments. an id was assigned to every vehicle that was available for data collection. using this id we could follow the tracks through the transport system. collected data included length of every actual segment, speeds of vehicles on it and times of their exits from the segment. the data is very sparse because of low concentration of vehicles that were available for data collection. we use assumption that the concentration of vehicles that were available for observation is homogeneous and constant. also the data is noisy. besides significant variance of speeds of vehicles, it was hard to build proper tracks. for example, a track could be interrupted in one part of the transport system and continued in another part. in the experiments we used data collected during a weekday in the area of third ring road in moscow. we considered parts of the roads that are adjacent to crossroads or exits that meet three requirements: • the number of recorded passages per day exceeds 5000. • the number of vehicles in the classes doesn’t differ more than 1.5 times. • average speeds of vehicles in two classes differ by more than 10%. these conditions are considered as necessary for flow separation problem. all of the parts of the road have further separation into two, which means that in the further discussion w = 2. the classification problem was set for every straight segment of the road and was solved for each time of injection of information into the system. in terms of section 2, for every moment t and every s0 we composed train sample out of vectors of speeds on s0, s−1, . . . s−b of vehicles from g1(t). test samples were built similarly for vehicles from g2(t). 8.1. i.i.d. algorithms here we used an assumption about independency of the vehicles on the road. it is not always true, but we were dealing with sparse data, which means that most of the vehicles we studied did not follow each other immediately. for each moment t for each segment s0 we built an i.i.d. classifier. for training set we select vehicles that passed s0 in the h = 20 minutes interval. this time window is characteristic for transport processes of such scale [18]. in the table 8.2 we provided information about used algorithms with tuned and constant parameters. we used accuracy to measure the performance of the algorithms because it is proportional to a norm of p − p̂ as both of the matrices consist of 0 and 1: accuracy = n∑ i=1 i(estimated classi = real classi) n = = 1 n n∑ i=1 i{pi1 = p̂i1} = · · · = 1 n n∑ i=1 i{piw = p̂iw} = 1 nw n∑ i=1 w∑ w=1 i{piw = p̂iw} = copyright c© 2017 assa. adv syst sci appl (2017) analysis of heterogeneity of transport flows 19 table 8.2. hyperparameters of i.i.d. classification algorithms algorithm constant parameters adjusted parameters svm [19] rbf kernel with auto coefficienting, weights associated with classes, shrinking enabled penalty parameter randforest [20] gini criterion, bootstraped sample number of trees-estimators; maximal depth of a tree baggingknn [21] [22] base estimator is knn, bootstraped sample number of estimators; the number of samples to train each base estimator gradboosting [23] deviance loss (log reg) number trees-estimators; learning rate; maximal depth of a tree logreg [24] l2 norm of penalty, weights associated with classes inverse of regularization strength naivebayes [25] hypothesis: multivariate bernoulli distribution of binarized vectors. lidstone smoothing parameter neuralnet [26] two layers with 10 and 3 neurons, activation function is logistic regularization term parameter; the initial learning rate table 8.3. i.i.d. algorithms results svm randfor bagknn gradbst logreg nb nnet 1 0.722 0.879 0.873 0.886 0.717 0.727 0.880 2 0.750 0.857 0.866 0.865 0.768 0.860 0.864 3 0.744 0.751 0.765 0.765 0.731 0.548 0.796 4 0.907 0.920 0.929 0.925 0.912 0.715 0.925 5 0.769 0.799 0.799 0.801 0.761 0.722 0.800 6 0.711 0.772 0.777 0.778 0.703 0.692 0.769 7 0.725 0.742 0.749 0.753 0.750 0.690 0.751 = 1 nw n∑ i=1 w∑ w=1 (piw − p̂iw)2 = 1 nw ‖p − p̂‖2f the averaged throughout a weekday performance of the algorithms one can see in the table 8.3. the results presented are obtained on test samples for 7 different two-lanes problematic crossroads in moscow. it is easy to notice that results very much depend on the crossroad, which means that for some crossroads it is easier to solve the flow separation problem. authors tried to find the reasons why it is easier to solve the problem on some crossroads, but apparent connections with the length of the segment or difference between speeds of the flows were not found. probably, the data collected on some crossroads was simply more consistent then on the others. 9. results and discussion the results of the numerical experiment show that there is no general model that fits every crossroad. complex models like neural networks showed the best results. however, the hyperparameters of each model were tuned differently for crossroads but performed nearly copyright c© 2017 assa. adv syst sci appl (2017) 20 i. o. molybog, y. v. chehovich the same for each segment near a crossroad, which means that every case of transport flow separation has unique features that strongly depend on its position on the roadmap. there are other possible approaches to estimate matrix p with matrices that do not consist of 0 and 1. in those approaches, we have to solve the optimization problem stated in section 5. despite its asymptotic complexity, we still can solve it because of its limited dimension. numerical results show that n is up to 10 for real problems and depend on lag(t)). empirically we can say that the problem of traffic separation is characteristic for certain locations on the roadmap. as a result, there are other possible approaches for solving the problem more accurate. for example, cameras set on a problematic crossroad could give us exact solution to the problem without extra approximations and assumption. however, the theory in sections 3-5 is more general and can be applied to solve similar problems for networks, where it is possible to follow movements of single particles of the flow. 10. conclusion in the paper we propose a formal statement for traffic flow separation problem. we divide the problem into two: statistical and optimization. the statistical part can be solved numerically using algorithms for recovery probability distribution. to simplify the optimizational part, we relax the problem and use standard classification algorithms. we trained and tested several types of prediction models on the constructed features space, compare their performance for several problematic parts of moscow road system. the optimization part was theoretically analyzed, and stated in a form suitable for relaxations and solution with standard optimizational techniques. future works in this area might be to solve the optimization problem explicitly in the algorithm, without relaxations of the statistical part. the theoretical results obtained in the work can find applications beyond the civil engineering, in other areas related to study of flow dynamics. references 1. french, s. p, barchers, c. & zhang w. 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(2014) adam: a method for stochastic optimization. arxiv preprint no. 1412.6980, [online]. available: https://arxiv.org/abs/1412.6980 copyright c© 2017 assa. adv syst sci appl (2017) https://arxiv.org/abs/1412.6980 introduction informal description formalization sequence separation vector representation risk minimization feasibility set final problem formulation optimization complexity distribution estimation proposed solution experiment i.i.d. algorithms results and discussion conclusion adv syst sci appl 2021; 03:113–118 published online at https://ijassa.ipu.ru. on solvability of equations defined by continuous and smooth regular mappings sergey e. zhukovskiy1* 1v. a. trapeznikov institute of control sciences of ras, moscow, russia abstract: we consider equations defined by continuous mappings acting between finitedimensional real vector spaces. it is assumed that the mappings are differentiable in the first variable. a regularity condition for this type of equations is obtained. it is shown that the regularity assumption implies the existence of solutions to the considered equations. systems of two equations defined by continuous mappings acting between finite-dimensional real vector spaces are considered. it is assumed that the first mapping is differentiable in the first variable and the second mapping is differentiable in the second variable. a regularity condition for this type of systems is obtained. it is shown that the regularity assumption implies the existence of solutions to the considered system. the proofs of the main results of the paper are based on brouwer’s fixed point theorem and global implicit function theorem. keywords: nonlinear equations, regularity, covering, fixed point 1. introduction given positive integers n, k and a continuous mapping f : rn × rn → rk, consider the following equation f(x, x) = 0 (1.1) with unknown x ∈ rn. our goal is to obtain sufficient conditions for the existence of a solution to this equations. one of the standard approaches to this problem is based on the application of the covering mappings theory (see, for example, [1, 2]). the corresponding results guarantee that if f is covering in the first variable and is lipschitz continuous in the second variable with a sufficiently small lipschitz constant, then there exists a solution to equation (1.1). in the most general settings, these assertions provide sufficient solvability conditions for analogous equations defined by mappings acting between metric spaces. in this paper, we consider a specific case of finite-dimensional real linear spaces and differential mappings. we show that in this specific case, the assumption of lipschitz continuity is redundant. the proof of our main result is based on two assertions. one of them is the well-known brouwer’s fixed-point theorem (see, for example, chapter ii, §5.7 in [3]). the second is a global implicit function theorem (see theorem 2 in [4]). in the second section of this paper, we recall this implicit function theorem as well as the related concepts and assertions. in the third section, we present solvability conditions for equation (1.1) and provide a proof of this result. the last section is devoted to a development of the main result to systems of equations. ∗corresponding author: s-e-zhuk@yandex.ru 114 s.e. zhukovskiy 2. preliminaries let us recall the concept of covering constant of a linear operator. denote byln×k the space of linear operators a : rn → rk, denote by sln×k the set of all surjective operators a ∈ ln×k. denote by bn(r) the closed ball in the space rn centered at a point x ∈ rn with a radius r ≥ 0. here and below we assume that rn and rk are equipped with norms which we denote by | · |, and the space ln×k is equipped with the corresponding operator norm. for a linear operator a ∈ ln×k, put cova := sup{α ≥ 0 : bk(α) ⊂ abn(1)}. it is a straightforward task to ensure that cova > 0 if and only if a ∈ sln×k. let us recall the global implicit function theorem from [4]. given a topological space σ and a mapping f : rn × σ→ rk, assume that for every σ ∈ σ the mapping f(·, σ) : rn → rk is differentiable. for t ≥ 0, put α(t) := inf { cov ∂f ∂x (x, σ) : x ∈ bn(t), σ ∈ σ } . theorem 2.1: (see theorem 2 in [4]) assume that (a1) the mapping f(·, ·) is continuous on rn × σ, for every σ ∈ σ the mapping f(·, σ) : rn → rk is differentiable on rn, the mapping ∂f ∂x (·, ·) is continuous on rn × σ. if +∞∫ 0 α(t) dt = +∞ or sup σ∈σ |f(0, σ)| < +∞∫ 0 α(t) dt, then for every ε > 0 there exists a continuous mapping g : σ→ rn such that f(g(σ), σ) = 0 ∀σ ∈ σ, |g(σ)|∫ 0 α(t) dt ≤ (1 + ε)|f(0, σ)| ∀σ ∈ σ. below we also use the following corollary of theorem 2.1. corollary 2.1: let f satisfies the assumption (a1). if there exists r̄ > 0 such that sup σ∈σ |f(0, σ)| < α(r̄)r̄, then for every ε > 0 there exists a continuous mapping g : σ→ rn such that f(g(σ), σ) = 0 ∀σ ∈ σ, |g(σ)| ≤ (1 + ε)|f(0, σ)| α(r̄) ∀σ ∈ σ. note that in [4] these assertions were proved under more general assumptions. in particular, it was assumed that the domain of f in the variable x as well as the target space are banach spaces. however, the considered here weak form of implicit function theorem from [4] is enough for the subsequent constructions. copyright © 2021 assa. adv syst sci appl (2021) solvability of equations defined by continuous and smooth mappings 115 3. solvability condition for equations let us turn back to equation (1.1). assume that for every x2 ∈ rn the mapping f(·, x2) is differentiable. for t > 0 put a(t, r) := inf { cov ∂f ∂x (x1, x2) : x1 ∈ bn(t), x2 ∈ bn(r) } , b(r) := sup x2∈bn(r) |f(0, x2)|. theorem 3.1: assume that (a) the mapping f(·, ·) is continuous on rn × rn, for every x2 ∈ rn the mapping f(·, x2) : rn → rk is differentiable on rn, the mapping ∂f ∂x (·, ·) is continuous on rn × rn. if there exists r̄ > 0 such that b(r̄) < r̄∫ 0 a(t, r̄) dt, (3.2) then there exists a point x̄ ∈ bn(r̄) such that such that f(x̄, x̄) = 0. proof apply theorem 2.1 to the mapping f with σ = bn(r̄). we have α(t) = a(t, r) ∀ r > 0. therefore, assumption (3.2) implies that sup x2∈bn(r̄) |f(0, x2)| = b(r̄) < r∫ 0 a(t, r̄) dt = +∞∫ 0 α(t) dt. take an arbitrary ε > 0 such that (1 + ε)b(r̄) < r̄∫ 0 a(t, r̄) dt. it follows from theorem 2.1 that there exists a continuous mapping g : bn(r̄)→ rn such that f(g(x2), x2) = 0, |g(x2)|∫ 0 a(t, r̄) dt ≤ (1 + ε)|f(0, x2)| ∀x2 ∈ bn(r̄). (3.3) obviously the function a(·, r̄) is decreasing. thus, the inequality in (3.3) and the assumption (3.2) imply that |g(x2)| ≤ r for every x2 ∈ bn(r̄). so, g(x2) ∈ bn(r̄) ∀x2 ∈ bn(r̄). copyright © 2021 assa. adv syst sci appl (2021) 116 s.e. zhukovskiy therefore, by virtue of continuity of g brouwer’s fixed-point theorem implies that there exists a point x̄ ∈ bn(r) such that x̄ = g(x̄). we have f(x̄, x̄) = f(g(x̄), x̄) = 0. so, the point x̄ is the desired one. let us derive a stronger but simpler solvability condition for the equation (1.1). corollary 3.1: let the assumption (a) hold. if there exist ᾱ > 0 and β̄ ≥ 0 such that β̄ < ᾱ ≤ cov ∂f ∂x (x1, x2) ∀x1 ∈ rn, ∀x2 ∈ rn, |f(0, x2)| ≤ |f(0, 0)|+ β̄|x2| ∀x2 ∈ rn, then there exists a point x̄ ∈ rn such that f(x̄, x̄) = 0, |x̄| ≤ |f(0, 0)| ᾱ− β̄ . (3.4) proof we have a(t, r) ≥ ᾱ, b(r) ≤ |f(0, 0)|+ β̄r ∀ t > 0, ∀ r ≥ 0. take rj := |f(0, 0)| ᾱ− β̄ + 1 j , j = 1, 2, ... . by construction we have b(rj) ≤ |f(0, 0)|+ β̄ |f(0, 0)| ᾱ− β̄ + β̄ j < ᾱ ( |f(0, 0)| ᾱ− β̄ + 1 j ) = rj∫ 0 a(t, rj) dt. therefore, theorem 3.1 implies that there exists a point x̄j ∈ bn(rj) such that f(x̄j, x̄j) = 0 for every j = 1, 2, ... . by virtue of the compactness of bn(r1) there exists a subsequence {x̄ji} of the sequence {x̄j} which converges to a point x̄. obviously, the point x̄ satisfies the inequality in (3.4). passing to the limit in the equalities f(x̄ji , x̄ji) = 0 as i to∞ we obtain that f(x̄, x̄) = 0. 4. solvability condition for systems of equations consider now the following system { f1(x1, x2) = 0, f2(x1, x2) = 0. (4.5) here f1, f2 : rn × rn → rk are given mappings. let us derive solvability conditions for the system (4.5) analogous to those in section 3. copyright © 2021 assa. adv syst sci appl (2021) solvability of equations defined by continuous and smooth mappings 117 assume that f1 is differentiable in x1 and f2 is differentiable in x2. given numbers r̄1 > 0 and r̄2 > 0, denote a1 := inf { cov ∂f1 ∂x1 (x1, x2) : x1 ∈ bn(r̄1), x2 ∈ bn(r̄2) } , b1 := sup x2∈bn(r̄2) |f1(0, x2)|, a2 := inf { cov ∂f1 ∂x1 (x1, x2) : x1 ∈ bn(r̄1), x2 ∈ bn(r̄2) } , b2 := sup x1∈bn(r̄1) |f2(x1, 0)|. theorem 4.1: assume that mappings f1(·, ·) and f2(·, ·) are continuous on rn × rn, for every x1, x2 ∈ rn the mappings f1(·, x2), f2(x1, ·) : rn → rk are differentiable on rn, the mappings ∂f1 ∂x1 (·, ·) and ∂f2 ∂x2 (·, ·) are continuous on rn × rn. if b1 < a1r̄1, b2 < a2r̄2, (4.6) then there exists a solution (x̄1, x̄2) ∈ bn(r̄1)×bn(r̄2) to the system (4.5), i.e.{ f1(x̄1, x̄2) = 0, f2(x̄1, x̄2) = 0. proof take ε > 0 such that (1 + ε)b1 a1 ≤ r̄1, (1 + ε)b2 a2 ≤ r̄2. the existence of such number ε follows from the assumption (4.6). since b1 < a1r̄1, applying corollary 2.1 to f = f1 and σ = bn(r̄2) we obtain that there exists a continuous mapping g1 : bn(r̄2)→ rn such that f1(g1(x2), x2) = 0 ∀x2 ∈ bn(r̄2), |g1(x2)| ≤ (1 + ε)|f1(0, x2)| a1 ≤ (1 + ε)b1 a1 ≤ r̄1 ∀x2 ∈ bn(r̄2). since b2 < a2r̄2, applying corollary 2.1 to f = f2 and σ = bn(r̄1) we obtain that there exists a continuous mapping g2 : bn(r̄1)→ rn such that f2(g2(x1), x1) = 0 ∀x1 ∈ bn(r̄1), |g2(x1)| ≤ (1 + ε)|f2(x1, 0)| a1 ≤ (1 + ε)b2 a2 ≤ r̄2 ∀x1 ∈ bn(r̄1). consider the mapping g : bn(r̄1)→ bn(r̄1), g(x1) = g1(g2(x1)), x1 ∈ bn(r̄1). this mapping is well-defined, since the above relations imply g2(x1) ∈ bn(r̄1) and g1(g2(x1)) ∈ bn(r̄1) for all x1 ∈ bn(r̄1). moreover, g is continuous since it is a composition copyright © 2021 assa. adv syst sci appl (2021) 118 s.e. zhukovskiy of continuous mappings g1 and g2. therefore, it follows from brouwer’s fixed point theorem that there exists a point x̄1 ∈ bn(r̄1) such that x̄1 = g(x̄1). take x̄2 := g2(x̄1). let us show that (x̄1, x̄2) is a desired point. obviously x̄2 ∈ bn(r̄2). moreover, f1(x̄1, x̄2) = f1(g(x̄1), x̄2) = f1(g1(g2(x̄1)), g2(x̄1)) = 0, f2(x̄1, x̄2) = f2(x̄1, g2(x̄1)) = 0. so, (x̄1, x̄2) is a desired point. acknowledgements the research is supported by the grant of the president of russian federation (project no md-2658.2021.1.1) and by the rfbr grant (project no 19-01-00080). the results in section 4 were obtained under the financial support of the russian science foundation (project no 20-11-20131). references 1. arutyunov, a.v., avakov, e.r. & zhukovskiy, s.e. (2009) covering mappings and their applications to differential equations unsolved for the derivative, differ. equations, 45(5), 627–649. 2. arutyunov, a., de oliveira, v.a., pereira, f.l., zhukovskiy, e. & zhukovskiy, s. (2015) on the solvability of implicit differential inclusions, appl. anal., 94(1), 129–143. 3. granas, a. & dugundji, j. (2003) fixed point theory, n.y., usa: springer. 4. arutyunov, a.v. & zhukovskiy, s.e. (2021) on global solvability of nonlinear equations with parameters, doklady mathematics, 103(1), 57–60. copyright © 2021 assa. adv syst sci appl (2021) introduction preliminaries solvability condition for equations solvability condition for systems of equations microsoft word 781 leveraging category theory in model based enterprise adv syst sci appl 2020; 01; 50-65 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/781 leveraging category theory in model based enterprise serge p. kovalyov* v. a. trapeznikov institute of control sciences, russian academy of sciences moscow, russia e-mail: kovalyov@sibnet.ru abstract: a mathematical framework based on category theory is proposed to formally describe and explore procedures of modeling engineering products and processes that comprise operation of a model-based enterprise. the framework is intended to provide interoperability across a variety of engineering modeling languages and tools, supplying them with a common abstract foundation capable to represent, generate, and verify diverse design and production knowledge. the framework is leveraged via algebraic representation of product configurations as diagrams in categories with models as objects and descriptions of actions involved into products assembly as morphisms. relevance of the framework is justified by appealing to systems engineering standards such as iec 81346. category theoretical methods for solving direct assembly problems that consist in constructing a product model from a given configuration are presented. specifically, solutions are obtained via the universal construction called a colimit of a diagram. much attention is then paid to stating and solving inverse assembly problems that consist in recovery and subsequent optimization of the configuration from the product model and assembly actions. inverse problem solving is in demand for generative design, viz. an emerging fully automatic product development and manufacturing technology. example solutions to direct and inverse problems are described in categories that represent two major areas of model-based enterprise operation: solid body geometric modeling of mechanical products and discrete-event simulation of production processes. keywords: model-based enterprise, model-based systems engineering, assembly planning, generative design, category theory, colimit 1. introduction a model-based enterprise is a type of industrial organization in which all product life cycle processes are arranged around a digital 3d-model of the product annotated with detailed product manufacturing information (pmi) [1]. techniques and tools to develop and leverage such a model, with applicability ranging from requirements elicitation and analysis through design to manufacturing, maintenance, and disposal, are provided by model-based systems engineering (mbse). more than a decade has passed since the introduction of the model-based enterprise concept, during which it gradually matured to an industrial level of readiness for implementation, with supporting standards, neutral formats, techniques, and software tools for maintaining pmi. recently, in the emerging industry 4.0 context, new challenges arise, associated with upgrading a product model to the level of a digital twin, viz. a virtual copy of the product that reliably reproduces and imposes the structure, state, and behavior of the physical original in real time [2]. the degree of design and production automation is increased steadily, up to complete elimination of human intervention in the socalled generative design cycle [3]. at the same time, the core problem of specifying the product model in a neutral form that guarantees complete undistorted access to it for all life cycle participants and tools cannot be * corresponding author: kovalyov@sibnet.ru leveraging category theory in model based enterprise 51 copyright ©2020 assa. adv. in systems science and appl. (2020) considered solved [4]. there are many specific modeling languages, techniques, and tools that are poorly compatible with each other. a natural, albeit “difficult”, approach to make them interoperable consists in supplying them with a common abstract formal framework of reference that is capable to represent, generate, and verify diverse design and production knowledge. the highest level of formality, generality, and verification power is achievable by leveraging appropriate mathematics. in this paper, following a recent trend [5, 6, 7], category theory is employed as such a framework. models are treated as objects of suitable categories, in which morphisms describe actions involved into assembling complex products. category theoretical constructions that represent mbse techniques on abstract conceptual level are defined and studied. considerable amount of experience in such a study has been gathered previously in model-based software engineering [8] and is now being transferred to systems engineering in general. specifically, a product configuration is established to be represented as the diagram in a category of models, and the assembly procedure is formally described by a universal construction called the colimit of this diagram [9]. in the context of generative design, the inverse assembly problems are stated, solving of which amounts to reconstructing a diagram from a fragment of its colimit [10]. some approaches to “categorify” the engineering domain are presented in the literature. for example, the framework for ontology and database design is proposed, where objects of the categories represent information entities that specify various aspects (facets) of products; and morphisms describe functional relationships like “has” or “contains” that connect entities to meaningful databases [11]. in [12], a symmetric monoidal category is constructed whose associated graphical language is capable to represent signal-flow diagrams of control theory: control system state spaces, comprised by finite vectors of signals, are objects of the category, and signal processing devices are represented by morphisms. our approach differs from these and others in strong adhering to the model-based paradigm, where models and manipulations with them are taken for primary entities and thus represented by objects and morphisms, respectively. the paper is composed as follows. section 2 is devoted to overview of problems pertaining to assembling products in model-based setting. in sections 3 and 4, category theoretical means to solve direct and inverse assembly problems are presented, respectively. some conclusions and directions for further research are outlined in section 5. 2. direct and inverse problems of assembling products in mbse, products, parts, and equipment are represented by formalized models of various types: digital images of geometric shapes and bodies, numerical approximations of differential equations, labeled graphs, databases, etc. herewith, the internal structure of models is not so much necessary to be known at decomposing the product into assembly units and planning the production process. the range of model capabilities to connect with other models is more important in order to create a hierarchy of assembly unit models. in other words, the models are usually considered as “black boxes” with known behavior with respect to other models, and are combined into structural schematics of complex products and processes, as shown in fig. 1 [13]. hence, model-based production planning procedures and software tools operate over a virtual catalog of models and descriptions of possible actions involved into composing assembly units. this point of view is substantiated by systems engineering standards such as iec 81346 “industrial systems, installations and equipment and industrial products—structuring principles and reference designations”. this standard prescribes to represent the complex product structure as a directed graph of the hierarchy of units that are denoted by reference designations. the structure and related labeled graphs of various other types routinely occur while performing various mbse procedures [10]. a graph that consists of models of 52 s.p. kovalyov copyright ©2020 assa adv. in systems science and appl. (2020) constituents of a certain unit, which are related by the unit assembly actions, is called the unit configuration. fig. 1. assembly planning [13] as an example that is particularly relevant to mechanical design, consider a mechanical product consisting of rigid solid parts (see the left half of fig. 1). for such a product, models of parts and assembly units are geometric bodies that can be represented for computer processing by various means: constructive solid geometry (csg), finite array of small spatial cells (voxels), boundary representation (brep) [14]. representations of particular product parts or assembly units are obtained from such models by affine isometries and stretches. for instance, the model of a barbell (heavy sporting equipment) consists of several cylinders of different sizes. in the barbell structure graph, the cylinders are represented by different vertices, although all of them are generated by the same geometric model. so the solid body model catalog contains a cylinder, with which several different actions of inclusion into the barbell are associated. another example of modeling technique, widely used in production planning, is discreteevent simulation (see the right half of fig. 1). its widespread support is regarded among the most significant achievements of mbse. here, a model represents an operational scenario, viz. a fragment of the imagined history of the simulated object behavior described as a discrete sequence of events of various types. some events may trigger others or prevent others from occurring, inducing causal relations between some events within a scenario. descriptions of actions used to build scenarios of the complex systems behavior reflect the contribution of the components. specifically, the operational scenario of the model-based enterprise plant is composed from operational scenarios of plant equipment units that interact with each other in the course of the production process. so, in discrete-event simulation, a structure graph represents the process hierarchy. the direct product assembly problem is stated as follows. given a product configuration, the model of the product as a whole needs to be constructed along with actions involved into assembling it, to complete the hierarchical product structure graph. the principle of constructing the target product model is easy to see from the structure representation: the model should be located at precisely one hierarchical level above the level of largest leveraging category theory in model based enterprise 53 copyright ©2020 assa. adv. in systems science and appl. (2020) constituent units. in other words, the product model should include all constituents’ models respecting their relations within the configuration, and nothing more in a sense that it should be included into any model that include all constituents’ models respecting the configuration relations. this principle is easy to explain on a simple example. suppose that a product should be assembled from two given parts p and s, and a manufacturing engineer has decided to join them by means of a certain glue, viz. the intermediate part g that can be directly applied on both p and s. action of the glue is described by the following configuration: parts g and p comprise an intermediate unit pg produced as a result of applying the glue on p, and similarly, parts g and s comprise a unit sg. the model of the target product r assembled by gluing p and s with g is selected among candidate models that contain units pg and sg and respect the glue g in a sense that tracing the glue inclusion through either one of the two units amounts to the same action. the selection obeys the following structural criterion: r should be unambiguously identifiable within any candidate model t. this criterion is shown schematically in fig. 2 [15], where models included into the iec 81346 compliant product structure graph are colored in yellow and tagged by reference designations, and the glue is appended as a manufacturing information annotation. fig. 2. schematics of gluing [15] provided that such a model r exists, it clearly represents the product that is assembled by gluing two parts without any extra changes or augmentations. moreover, it is easy to see that such model is unique in a sense that any two such models, by definition, contain each other. on the contrary, if the model r doesn’t exist (cannot be found in the model catalog), then the wrong glue was chosen: g is incapable to connect p with s. notice that the gluing configuration is significantly simplified in case when influence of the glue on parts is negligible, so we can assume that pg = p and sg = s. in this case, the configuration takes a shape of a span, viz. a graph consisting of two edges with the same source: p  g  s. we will refer to this kind of gluing as non-intrusive. in real world, it routinely occurs in solid body modeling: applying thin layer of a glue leaves the shape of a part unchanged. for example, non-intrusive gluing is intended to fix parts together at assembly tasks m1-m4 shown in fig. 1. let’s return to the general setting. if models contain values of some parameters, and descriptions of assembly actions contain rules for transforming the values, then the structure graph allows to calculate values of the parameters for the target product, including geometry, topology, dimensions, tolerances, etc. emergent parameters that can’t be attributed to any particular unit can also be calculated. examples of calculations of this kind are known in the field of composite materials design [16]. the averaged (effective) physical properties of a composite, such as a young’s modulus and a poisson’s ratio, depend in a complex way on properties of components and methods of manufacturing the composite from them. using elasticity theory methods, these dependencies are specified as linearized matrix relations that label arrows of the composite material structure graph. then it becomes possible to predict 54 s.p. kovalyov copyright ©2020 assa adv. in systems science and appl. (2020) the properties of the material virtually by computation over a database of components shown in fig. 3, avoiding costly physical experiments. fig. 3. materials database [16] although direct calculation of the product model by its configuration is so much important, it plays only a secondary, instrumental role in preparing a large complex product for manufacturing. according to the standards like iec 81346 and manufacturing practices, the assembly process is routinely designed in a top-down fashion, from the product as a whole to progressively smaller units. therefore, a typical assembly planning problem has inverse rather than direct character: given the unit model, it is needed to determine (recover) the configuration from which the unit can be effectively assembled. the assortment of theoretically possible configurations is unimaginably huge, but in practice, it is severely constrained by plant capabilities, stakeholders’ requirements, suppliers’ offers, engineers’ preferences, and other factors. so, the primary assembly planning procedure consists in identification of the class of all configurations that produce the given unit and satisfy all constraints imposed by practical factors. once this class is identified, search for the final unit configuration is performed across it aiming to optimize conventional process objective functions, such as cost, duration, and complexity [17]. for this reason, we will call this class the assembly design space. developing algorithms and tools to explore the space on a solid mathematical basis powered by computer algebra and artificial intelligence would allow to automate the design for manufacturability (dfm) practices. the ultimate target is the end-toend generative design cycle of complex products in model-based enterprise environment. for example, the inverse problem of gluing is stated as follows. in accordance with the iec 81346 standard, the structure of the target product r is specified as a graph consisting of three models p, s, r and two actions p  r, s  r. an admissible gluing configuration is any combination of five models and four actions that comprise the “zigzag” shape highlighted by solid arrows in fig. 2. among all such configurations, those are of interest for which solving the direct problem produces specified actions p  r and s  r as two-step inclusions of the two outer models into the target product. in particular, outer models themselves are required to be precisely p and s, and the target product model is required to be r. 3. category theory in product assembly modeling as stated in the introduction, the natural source of mathematical means for representation, generation, and verification of product structure graphs is category theory, viz. a branch of higher algebra that “starts with the observation that many properties of mathematical systems can be unified and simplified by a presentation with diagrams of arrows” ([18, p. 1]). recall that a category is a class of abstract objects pairwise related by morphisms (arrows). the leveraging category theory in model based enterprise 55 copyright ©2020 assa. adv. in systems science and appl. (2020) precise definition takes just a few lines: a category c consists of a class of objects ob c and a class of morphisms mor c endowed with the following operations. firstly, two objects are associated with each morphism f: a domain (arrow source) dom f and a codomain (target) codom f, so that the pair of equalities dom f = a and codom f = b is depicted as an arrow f : a  b. secondly, for any pair of morphisms f, g that satisfies the condition codom f = dom g, the composition of morphisms g ◦ f : dom f  codom g is defined, and it is associative: for any three morphisms f, g, h such that codom f = dom g and codom g = dom h, the equality h ◦ (g ◦ f) = (h ◦ g) ◦ f holds. thirdly, any object a has an associated identity morphism 1a : a  a such that f ◦ 1a = 1b ◦ f = f for any morphism f : a  b. a classic example of a category is set that consists of all sets and all maps between sets: the composition law for maps is defined by the standard substitution, and the identity map of a set sends each element to itself. the axioms of a category are inherently symmetrical with regard to swapping dom with codom along with reversing the order of composing morphisms: the category obtained this way from c is called dual of c and denoted by cop. the axioms also tolerate restriction to subclasses, so the notion of subcategory is defined: a subcategory of c is a pair consisting of a subclass of ob c and a subclass of mor c that are closed with respect to operations inherited from c. a subcategory of c is called full if it contains any c-morphism whose domain and codomain are contained in it. together with category, the concept of functor is routinely introduced, which is a structure-preserving map between categories. specifically, a functor fun : c  d from category c to d is a pair of maps fun : ob c  ob d (an object function) and fun : mor c  mor d (a morphism function) that satisfies the following conditions for any c-morphisms f, g and c-object a: (i) fun(dom f) = dom fun(f), fun(codom f) = codom fun(f); (ii) fun(g ◦ f) = fun(g) ◦ fun(f) whenever the composition g ◦ f is defined; (iii) fun(1a) = 1fun(a). obviously, all categories and all functors comprise a (formal) category denoted by cat. to explore relationships between functors, the following construction is introduced: the natural transformation  of a functor fun : c  d to a functor fun' : c  d is a function that assigns to each c-object a a d-morphism a : fun(a)  fun'(a) in such a way that b ◦ fun(f) = fun'(f) ◦ a for any c-morphism f : a  b. categories are routinely constructed from graphs: any directed multigraph induces a category with vertices as objects and finite paths as morphisms. indeed, the domain and the codomain of a path are its source and target, respectively; composition of morphisms acts as concatenation of paths; the identity morphism of a vertex is an empty “path” from the vertex to itself. this construction leads to the fundamental notion of a diagram in c, viz. a functor of the kind  : x  c where x is a category generated by some graph called the shape of the diagram. a diagram is visually depicted as a graph that generates x, endowed with labels from c: each vertex a is labeled by the c-object (a) and each edge k is labeled by the c-morphism (k). all diagrams in c comprise a category denoted by dc (a covariant modification of a “super-comma” category introduced in [18, exercise v.2.5]), in which a morphism from a diagram  : x  c to  : y  c is a pair , fd consisting of a functor fd : x  y and a natural transformation  :    ◦ fd; the composition law for morphisms is 56 s.p. kovalyov copyright ©2020 assa adv. in systems science and appl. (2020) specified as follows: , fd ◦ , gd = gd(–) ◦ , fd ◦ gd. category theory provides a number of algebraic techniques for construction and analysis of diagrams. the effectiveness of employing category theory as a mathematical framework for modelbased enterprise stems from the observation that a catalog of models and actions introduced in section 2 is nothing but a category. indeed, the catalog has composite morphisms (sequential execution of actions) and identity morphisms (idle “doing nothing” action with any model). for example, in solid body modeling of mechanical products, a model is algebraically represented as a subset of ℝ3 which is bounded, regular (i.e., it coincides with the closure of its interior wrt the standard topology), and semi-analytical (i.e., it admits a representation as a finite boolean combination of sets of the kind {(x, y, z) | f1(x, y, z)  0, f2(x, y, z)  0, …} where fi : ℝ 3  ℝ is a real analytic function for any index i) [14]. in order to support specifications of assembly procedures such as gluing bodies together by surface fragments, all bounded regular semi-analytical subsets of ℝn, 0  n  3, are considered as valid solid body models. then, the quotient is formed: any two such sets that can be turned into one another by a composition of affine isometries and stretches are identified with one another. morphisms of such equivalence classes, which describe assembly actions of complex mechanical products, are generated by isometric embeddings and stretches. this way, the subcategory of set emerges, which we will denote by sbm (from solid body modeling). for many well-known mbse techniques, formal description of the model catalog produces a category of sets endowed with some structure, e.g., algebraic systems, topological spaces, graphs, etc. morphisms in such categories are maps of sets compatible with the structure. a canonical functor to set acts on such a category by “forgetting” the structure. an example is the following algebraic representation of discrete-event simulation. a scenario is represented by a set of events partially ordered by causal dependencies and labeled by event types. since neither events nor dependencies, nor labels could be “lost” when composing a complex system behavior scenario from components behavior scenarios, actions involved into assembling scenarios are described as maps that preserve the order and the labeling [19]. all scenarios and all such actions form a category called pomset and equipped with a functor to set that forgets the order and the labeling. let c be a category that represents some model catalog. a product structure and related graphs are diagrams in c, so the category dc represents the respective catalog of such graphs and actions involved into product structure development. we show how to rigorously solve various problems of assembling products by means of the category theoretical framework provided by dc. for instance, the solution to the direct problem is represented by the following universal construction called the colimit of the diagram. denote by 1 the singleton category that consists of one object 0 and one morphism 10. for any category x, there exists a unique functor from x to 1, denoted by !x : x  1, which sends any x-object to 0 (in other words, 1 is a terminal cat-object). there is a full embedding (i.e., an injective functor onto a full subcategory of its codomain) ┌–┐ : c ↪ dc that takes a c-object q to the singleton diagram ┌q┐ : 1  c : 0 ↦ q. a dc-morphism with a singleton diagram as the codomain is called a cocone. this name is justified by visual representation: a cocone , !x :   ┌q┐ over a diagram  : x  c can be depicted by adding an extra vertex q and associated edges a : (a)  q, a  ob x, to the graph of the base diagram . if the extra edges were directed reversely (dually), i.e., from q to vertices of , then category theorists would call this picture by the familiar word “cone”, hence they apply the prefix “co-” to indicate the duality. a colimit (named as a dual to a limit) of  is a cocone colim  :   ┌r┐ that is universal in the following sense: for any c-object t and any cocone  :   ┌t┐ there exists a unique c-morphism w : r  t that satisfies the equality  = ┌w┐ ◦ colim . it is easy to see leveraging category theory in model based enterprise 57 copyright ©2020 assa. adv. in systems science and appl. (2020) that this universality condition is precisely the structural criterion from section 2 which determines the model r of the target product built from the configuration . thus, given a product configuration represented as a diagram, rigorous analysis and verification of the product assembly process amounts to checking that the diagram has a colimit and calculating its vertex and associated edges. for example, a vertex of a colimit of a discrete diagram is known as a sum of objects that label the diagram’s vertices, and its edges are called injections. another useful kind of a colimit is that of a diagram consisting of a parallel pair of morphisms  ⇉ . the colimit edge that starts from the right vertex of the diagram is called a coequalizer of the pair; it generalizes the set-theoretical concept of a quotient. a morphism is called a regular epimorphism whenever there exists a parallel pair that it coequalizes. yet another fundamental type of a colimit is constructed as follows. let v be a spanshaped two-arrow graph      and  : v  c be a v-shaped diagram populated by a pair of c-morphisms f : p  g  s : g. the colimit of , called a pushout (or a universal square) of the pair, is specified by the vertex r and the pair of edges p : p  r  s : s that satisfy the following two conditions: the equality p ◦ f = s ◦ g holds (it establishes naturality of the pushout and determines the colimit edge directed from g to r), and, for any object t and a pair of morphisms u : p  t  s : v such that u ◦ f = v ◦ g, there exists a unique morphism w : r  t that satisfies the equalities w ◦ p = u and w ◦ s = v (this is the universality condition). clearly, a pushout provides a solution to the direct non-intrusive gluing problem stated in section 2. a pushout can be constructed via a sum and a coequalizer provided that they exist [18, exercise iii.3.2]. let p = q ◦ inp and s = q ◦ ins, where inp : p  p  s  s : ins are injections of the components into the sum and q is the coequalizer of the following parallel pair of morphisms: inp ◦ f, ins ◦ g : g ⇉ p  s. for the pushout constructed in such a way, the naturality condition (q ◦ inp) ◦ f = (q ◦ ins) ◦ g and the universality condition directly follow from the definition of a coequalizer. in the category set, the sum is the disjoint union of sets p and s, and the coequalizer is its quotient map modulo the equivalence relation generated by the set of pairs {(f(x), g(x)) | x  g}. so the edge map p takes each element of the set p to the element’s equivalence class, and the map s acts similarly on s. if maps f and g are injective, then they establish that g is the intersection of the sets p and s, and the pushout produces precisely the union of the sets. it often occurs at modeling a mechanical product assembly by means of the category sbm. illustrative examples are shown in fig. 1 as assembly tasks m1-m4. for instance, the task m3 consists in actions depicted by dashed arrows of the commutative square shown in fig. 4 (drawn upside-down in accordance with iec 81346). it is easy to verify directly that this square is a pushout in sbm. observe that the glue is represented by means of sbm as a 58 s.p. kovalyov copyright ©2020 assa adv. in systems science and appl. (2020) parts’ contact 2d surface on which thin layer of a red-colored chemical adhesive is applied to fix parts together. fig. 4. non-intrusive gluing as a pushout non-intrusive gluing is also widely used as a method of producing laminated composite materials. the database used in digital design and verification of the composite properties is outlined schematically in fig. 3 above. an interested reader with some background in materials science may construct example pushouts and evaluate properties of composites over them following [16], as an exercise. the construction of a pushout in set admits generalization to many familiar categories of sets with structure. for example, any v-shaped diagram in the category of discrete-event simulation pomset has a pushout. however, if it were constructed literally as in set, then the order on the vertex inherited from base components might turn into a preorder, so an extra quotient may apply to restore the partial order on the vertex in a standard way. the realworld example that includes a pushout in pomset is shown in fig. 5 below. a colimit of any diagram in any category can be constructed via a sequence of sums and pushouts provided that it exists (for example, the whole process shown in fig. 1 amounts to assembling the product from parts a, b, and c via a sequence of pushouts in sbm). furthermore, in practical applications simplifying the shape of a diagram helps to calculate colimits. we present the following result to pursue this direction. theorem 1: let  : x  c be a diagram, y be a full reflective subcategory of x, and isc : y ↪ x be an embedding. for each cocone  :  ◦ isc  ┌s┐, there exists a unique cocone ' :   ┌s┐ such that  = ' ◦ 1 ◦ isc, isc. moreover, the cocone  is a colimit of the diagram  ◦ isc if and only if the cocone ' is a colimit of the diagram . proof. recall that a subcategory y of x is called reflective if the embedding isc : y ↪ x has a left adjoint functor isc* : x  y called a reflector [18, § iv.3]. specifically, there exists a natural transformation  : 1x  isc ◦ isc*, called the reflection unit, that consists of universal arrows. the unit induces a dc-morphism (), isc* :    ◦ isc. by definition of the cocone, for each x-object i the edge f : (i)  s of the desired cocone ' satisfies the condition f = h ◦ (i) where h : (isc(isc*(i)))  s is an edge of the cocone , so ' =  ◦ (), isc*. assume that  is a colimit, and consider an arbitrary cocone  :   ┌t┐. let w : s  t be a canonical c-morphism that satisfies the condition  ◦ 1 ◦ isc, isc = ┌w┐ ◦ . hence, ┌w┐ ◦ ' =  ◦ 1 ◦ isc, isc ◦ (), isc* =  and, in addition, if some c-morphism w̃ : s  t satisfies the condition  = ┌w̃┐ ◦ ', then  ◦ 1 ◦ isc, isc = ┌w̃┐ ◦ ' ◦ 1 ◦ isc, isc = ┌w̃┐ ◦ , so w̃ = w. we conclude that ' is a colimit of the diagram . conversely, assume that ' is a colimit, and consider an arbitrary cocone  :  ◦ isc  ┌u┐. let u : s  u be a canonical c-morphism that satisfies the condition ' = leveraging category theory in model based enterprise 59 copyright ©2020 assa. adv. in systems science and appl. (2020) ┌u┐ ◦ ' where ' =  ◦ (), isc*. hence, ┌u┐ ◦  =  ◦ (), isc* ◦ 1 ◦ isc, isc =  and, in addition, if some c-morphism ũ : s  u satisfies the condition  = ┌ũ┐ ◦ , then ' = ┌ũ┐ ◦  ◦ (), isc* = ┌ũ┐ ◦ ', so ũ = u. we conclude that  is a colimit of the diagram  ◦ isc. □ as an application of theorem 1, consider the direct general gluing problem. its solution amounts to constructing a colimit of a diagram with the following shape that we will denote by m. there exists an embedding iv : v ↪ m that maps v onto a full subcategory of m with a set of object {b, c, b'}. this subcategory is reflective, with the reflector iv* taking a to b, a' to b', and other vertices to themselves, so that two outer arrows a  b and a'  b' are the reflection unit components associated with objects a and a', respectively. by theorem 1, the colimit of a diagram  : m  c is determined one-to-one by the pushout of the diagram  ◦ iv : v  c comprised from two central arrows of m. real-world examples of colimits of this kind can be constructed in the category sbm by representing “intrusive” means of joining mechanical parts, such as riveting. yet another example is inspired by the right half of fig. 1: the production process simulation can be glued from simulations of robot a, robot b, and the conveyor. according to the production graph, the conveyor acts as a “process glue” between two robots, delivering parts and assembly task results from one to another. one possible production scenario consisting of tasks m3 and m4 performed by robots b and a, respectively, is represented in fig. 5 by a colimit of an m-shaped diagram in the category pomset. 60 s.p. kovalyov copyright ©2020 assa adv. in systems science and appl. (2020) fig. 5. constructing a production process simulation via a colimit we conclude this section with the following remarks. a colimit of an arbitrary diagram  is unique up to an isomorphism provided that it exists: a cocone of the kind ┌i┐ ◦ colim  is a colimit of  if and only if i is a c-isomorphism. moreover, colimits allow to represent assembling products from configurations as a functor. still further, the construction of the category of diagrams dc induces a monad in cat whose associated category of algebras contains instances of the colimit functor and other functors that are relevant for mathematical framework of mbse. therefore, this monad is regarded as the mbse metamodel [10]. 4. inverse assembly problems and solutions to explore inverse product assembly problems, the following technical concept of a subcocone is needed. a cocone  is called a subcocone of a cocone  if there exists a dc-morphism , iy : dom   dom  such that the natural transformation  consists of identity c-morphisms, the functor iy is injective, and  ◦ , iy = . the inverse problem is stated mathematically as follows. given a cocone  (that represents the known fragment of the target product structure) and a class cd of diagrams (that represents the space of all practically feasible configurations), evaluate the subclass cds  cd consisting of all diagrams from cd with a colimit that contains  as a subcocone. the subclass cds is precisely the assembly design space across which (sub-, pareto-) optimal configurations are sought. category theory may improve the search as follows. let q be a linearly ordered set in which some search objective function takes values (most often, q is the set of all real or integer numbers). obviously, q can be regarded as a category, since any ordered set can be depicted as a directed graph. of particular interest is the situation where the objective function acts as an object function of a functor that takes values in the category q and arguments in a subcategory of dc (or of the dual category (dc)op) which has cds as class of objects and enough non-identity morphisms. in this case, optimization algorithms of gradient descent type can be employed, which navigate along morphisms of this subcategory, calculating the path by means of computer algebra. performant computer algebra packages for such category theory calculations are available [20]. for example, consider the inverse gluing problem stated in section 2. here, the class cd from the general configuration recovery problem statement consists of all m-shaped diagrams with a colimit in c. the cocone  can easily be reconstructed from the product structure presented in fig. 2 above: the cocone base is the diagram  : a ↦ p, a' ↦ s with the two-vertex discrete shape {a, a'}  m, and the cocone vertex is the target product r, so the cocone is specified as a pair of c-morphisms p : p  r  s : s. this problem is solved as follows. proposition 1: for any c-objects p, s, r and any pair of c-morphisms p : p  r  s : s, there exists a diagram  : m  c with a colimit , !m :   ┌r┐ that satisfies the equalities a = p and a' = s. this diagram is unique up to a dc-isomorphism if and only if the pair consists of isomorphisms and every epimorphism with the codomain r is an isomorphism. proof. construct a desired diagram  in two steps. first, choose an arbitrary factorization of the kind p = o ◦ q in order to label the outer arrow a  b of  by the morphism q, along with the similar factorization s = o' ◦ q'. second, having that, as established in section 3 by theorem 1, a colimit of an m-shaped diagram amounts to a pushout, find a diagram  : v  c that has a pushout specified by edges o and o', and use it to label two central arrows of  (i.e., to satisfy the equality  ◦ iv = ). leveraging category theory in model based enterprise 61 copyright ©2020 assa. adv. in systems science and appl. (2020) consider trivial factorizations with o = o' = 1r. a c-morphism e with the codomain r is an epimorphism if and only if the equality 1r ◦ e = 1r ◦ e determines a pushout [18, exercise iii.4.4]. hence, we shall take the following m-shaped diagram, which we will denote by e, as  provided that e is an epimorphism. in particular, with e = 1r we obtain somewhat paradoxical formal solution to the glue finding problem: if the complex product r that includes parts p and s is already specified, then let it serve as a glue fixing these parts together! if p is not an isomorphism, then another solution (non-isomorphic to the above) is obtained by changing the factorization of p to another trivial one with q = 1p, so that the part p plays the role of a glue as follows. other pairs of morphisms that have a pushout specified by edges p and 1r lead to other solutions. “symmetrical” solutions are obtained from factorizations with o = 1r and q' = 1s provided that s is not an isomorphism. other factorizations may produce yet other solutions. if both p and s are isomorphisms, then morphisms o and o' are epimorphisms with the codomain r [21, proposition 7.42]. if any such epimorphism is an iso, then the target diagram  is essentially unique: it can be comprised solely from isomorphisms. otherwise, at least two non-isomorphic diagrams of the kind e exist. □ for example, in the category set the uniqueness criterion of the glue finding problem solution is equivalent to the condition that the set r is empty. if this is not the case, then any larger set can be mapped onto r, so the target class cds contains at least infinitely many pairwise non-isomorphic m-shaped diagrams of the kind e. the stepwise glue finding procedure executed to prove proposition 1 admits generalization for recovering configurations with more complicated shapes than m. such a general procedure essentially amounts to “unwinding” the construction of a colimit from sums and pushouts. crucial steps of the procedure consist in recovering a pushout from its edges. this is the case for an inverse non-intrusive gluing problem. indeed, the class cd in this problem’s statement consists of all v-shaped diagrams with a colimit in c, while the cocone  has a discrete base consisting of two outer vertices of v and is specified by a pair of c-morphisms with the same codomain. this pushout recovery problem is quite difficult to solve. we start with solving it in the category set as follows. 62 s.p. kovalyov copyright ©2020 assa adv. in systems science and appl. (2020) proposition 2: let p, s, r be sets and p : p  r  s : s be a pair of maps. (1) a pair of maps that has a pushout specified by edges p and s exists if and only if the following condition holds for each r  r: (|p–1(r)| = 0  |s–1(r)| = 1)  (|s–1(r)| = 0  |p–1(r)| = 1). (2) let prp : p  p × s  s : prs be a cartesian product of sets equipped with projections and [p, s] : p  s  r be the canonical map that acts as p on p and as s on s. a pair of maps f : p  g  s : g has a pushout specified by edges p and s if and only if the map [p, s] is surjective and there exists a map h : g  p × s such that f = prp ◦ h, g = prs ◦ h, and the set h(g), considered as a binary relation on p  s, generates the equivalence relation that coincides with the kernel equivalence relation of the map [p, s]. (3) there exists a unique pair of maps that has a pushout specified by edges p and s if and only if the map [p, s] is bijective. proof. (1) assume that the maps p and s specify the pushout of some pair of maps f : p  g  s : g. from the construction of pushout in set described explicitly in section 3, it is easy to see that if the set p–1(r)  p is empty for some r  r, then the element r has been brought into r from the set s \ g(g): there exists a unique y  s such that s(y) = r. the case for empty set s–1(r)  s is considered similarly. conversely, assume that p and s satisfy the condition (1); in particular, p(p)  s(s) = r. construct a pullback (a limit, viz. a universal construction dual to a pushout) of the vop-shaped diagram in set populated by the pair [18, § iii.4]: let g0 = {(x, y) | p(x) = s(y)}  p × s, f0 : g0  p : (x, y) ↦ x, g0 : g0  s : (x, y) ↦ y. the equality p ◦ f0 = s ◦ g0 holds and determines the pullback; we claim that it also determines a pushout of the pair of pullback edges f0 : p  g0  s : g0. indeed, construct the pushout of this pair in a standard way, with p' : p  r'  s : s' being its edges. define the map i : r  r' as follows: for each r  r, if r  p(p), then let i(r) = p'(x) for an arbitrary x  p –1(r) (obviously, the value i(r) doesn’t depend upon the choice of x); otherwise, let i(r) = s'(y) where y is a unique element of the set s such that s(y) = r (such an element exists by assumption). it is easy to see that i is a bijection, p = i–1 ◦ p', and s = i–1 ◦ s', so the maps p and s are edges of a pushout of the pair consisting of f0 and g0. (2) for a map h : g  p × s, the requirement that the map [p, s] is surjective and its kernel equivalence relation is generated by h(g) is equivalent to the condition that [p, s] is a coequalizer of the following parallel pair of maps: inp ◦ prp ◦ h, ins ◦ prs ◦ h : g ⇉ p  s. if h satisfies this condition, then, by the construction of a pushout via a coequalizer, the maps p and s specify the pushout of the pair prp ◦ h : p  g  s : prs ◦ h. conversely, assume that p and s specify the pushout of some pair of maps f : p  g  s : g. let h = f, g : g  p × s : x ↦ (f(x), g(x)). leveraging category theory in model based enterprise 63 copyright ©2020 assa. adv. in systems science and appl. (2020) (3) if the map [p, s] is bijective, than its kernel equivalence relation is discrete (coincides with the identity). hence, a pair of maps, constructed from a map h : g  p × s composed with projections as in the proof of (2) above, may have a pushout with such edges p and s only if the set h(g) is empty, which implies that g itself is empty. indeed, the pair of empty maps p    s is the only one that has a pushout with such edges. conversely, assume that the map [p, s] is not bijective. if the condition (1) doesn’t hold, then no pushout exists with edges p and s. otherwise, the set g0 constructed above is nonempty, so any larger set can be mapped onto it. in this case, infinitely many pairwise nonisomorphic v-shaped diagrams exist that have a pushout specified by edges p and s: by (2), each of these diagrams is populated with a pair of maps of the kind f0 ◦ e : p  g  s : g0 ◦ e, where e : g  g0 is an epimorphism (surjection). □ if the maps p and s are injective, then the maps f0 and g0 constructed via the pullback in the proof of proposition 2 are injective as well [21, proposition 11.18]. hence, the method of the proof in many cases allows to recover gluing configurations in the category of solid body modeling sbm. for instance, recovering the glue for the assembly task m3 from fig. 1 by the pullback yields a pushout isomorphic to the one shown in fig. 4. proposition 2 can also be adapted for many familiar categories of sets with structure. in such a category, the solution existence condition stated for a pair p : p  r  s : s in proposition 2(1) is augmented with a suitable structure preservation requirement. one possible form of such a requirement is the following regularity condition: the canonical morphism [p, s] is a regular epimorphism. at least, by the construction of a pushout via a coequalizer, the regularity condition is necessary for morphisms p and s to be edges of some pushout in any category where the sum p  s exists. the glue can be constructed, as in set, via the pullback of the pair: the statement dual of [21, lemma 11.15(1)] implies that in any category, if a pair of morphisms with the same codomain admits both pushout recovery and a pullback, then the latter is the former, i.e., the pullback is at the same time a pushout, so it provides a solution to the pushout recovery problem. other glues can be constructed similarly to proposition 2(2) that, restated in terms of coequalizer as in its proof, holds in any category where both the product p × s and the sum p  s exist. analogues of proposition 2(3) can be obtained as well. the above considerations apply for the category of discrete-event simulation pomset. for a pair of maps p : p  r  s : s that preserve the order and the labeling, the pushout recovery problem has a solution if and only if the pair satisfies both the condition of proposition 2(1) and the regularity condition. the example glue is the set g0 constructed as in the proof of proposition 2 and endowed with the order and the labeling inherited from the product p × s. in enterprise simulation, such a glue represents a plant control system that ensures coherent operation of manufacturing equipment units p and s jointly executing the target process r. the construction of the glue demonstrates how category theory can be leveraged to automate control system programming. this kind of automation is in demand for the emerging approach to production planning and operation on the basis of a digital twin [22]. for example, recovering a glue of two upper maps from fig. 5 by the pullback yields a control program specification isomorphic to the conveyor simulation shown at the bottom. 5. conclusion category theory demonstrates significant potential in improving model-based enterprise performance. in particular, the theory provides mathematically rigorous solutions to direct and inverse problems associated with assembling products. a number of real-world 64 s.p. kovalyov copyright ©2020 assa adv. in systems science and appl. (2020) examples of such problems can be found in the cited literature and elsewhere. yet, theorems and proofs are of little utility to engineers: in order to hand category theoretical techniques over to industry, they shall be implemented in appropriate software tools. the tools, given requirements to the target product (e.g., a 3d-model) and to the manufacturing plant (e.g., a factory simulation), would represent and evaluate them by means of category theory as described above, and then automatically synthesize the production plan that is optimal wrt specified objective functors. provided that such synergy between computer algebra and optimization algorithms will be achieved, the tools will open new horizons in automatic production planning and generative design. development and deployment of the tools is a major area for further theoretical and applied research. acknowledgements this work was supported by the russian foundation for basic research (grant 19-01100799). references 1. frechette s. 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(1986). modeling concurrency with partial orders, int. j. parallel prog., 15(1), 33–71. https://doi.org/10.1007/bf01379149 20. gross j., chlipala a. & spivak d. i. (2014). experience implementing a performant category-theory library in coq. in klein g. & gamboa r. (eds.), 5th conf. (intl.) on interactive theorem proving proceedings (pp. 275–291). lecture notes in computer science ser., 8558, springer. https://doi.org/10.1007/978-3-319-08970-6 21. adámek j., herrlich h. & strecker g. e. (1990). abstract and concrete categories. new york, ny: john wiley. 22. liu q., zhang h., leng j. & chen x. (2019). digital twin-driven rapid individualised designing of automated flow-shop manufacturing system, intl. j. production research, 57:12, 3903–3919. https://doi.org/10.1080/00207543.2018.1471243 adv syst sci appl 2019; 03; 38-51 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/724 improved steganography techniques for different types of secret data rasha thabit* computer techniques engineering department, al-rasheed university college, hay alhussain, baghdad, iraq e-mail: rashathabit@yahoo.com received april 9, 2019; revised september 18, 2019; published october 1, 2019 abstract: steganography is an interesting aspect of data hiding art where the secret data such as (binary bits, logo, image, audio, and others) are embedded in a cover media without raising any suspicion about the existence of that data. over the years, different steganography techniques for digital images have been presented to be applied in various applications such as security and privacy control. some of these techniques are based on transforming the image from the spatial domain to the transform domain as a pre-processing step before embedding the secret data. this paper presents two improved steganography techniques in the transform domain based on slantlet transform (slt) and different data embedding methods. in the first proposed technique, a secret image can be embedded in a cover image by calculating the difference blocks of slt coefficients and modifying the coefficients according to the root mean square error values. in the second proposed technique, a sequence of binary bits can be embedded in a cover image by modifying the carrier slt subbands according to the secret data bits and the mean values of slt coefficients for image’s blocks. the experimental results proved the efficiency of the proposed techniques in comparison with the previous transform-based steganography techniques. keywords: data hiding, steganography, slantlet transform, security, sharing secret data. 1. introduction data hiding techniques are widely used in digital communication for different purposes such as security and privacy control [1]. the steganography [2-5] and watermarking [6,7] techniques are the two main parts of data hiding science. these techniques are based on hiding secret data in the cover media either by directly changing the contents of the cover media (i.e., embedding in spatial domain) or by transforming the cover media and embedding the secret data in the resultant coefficients after transformation (i.e., embedding in the transform domain). in the transform-based data hiding techniques, different transforms have been applied as a pre-processing step before embedding the secret data such as discrete cosine transform (dct) [8, 9], discrete wavelet transform (dwt) [5, 10, 11], and an orthogonal dwt called slantlet transform (slt) [4,12,13]. on the other hand, the implementation of the steganography algorithm is affected by the type of secret data that is intended to be embedded in the cover image. in [14], a steganography algorithm has been presented to hide a secret text in a cover image by embedding each character from the text in a window of pixels of size 5×5 using a rule of non-multiples of 5. in [15], a secret text file is compressed and embedded in the pixels of the red, green, and blue (rgb) channels of the color image. in [16], a text message is converted to a sequence of binary bits and embedded in the least significant bits (lsb) of the pixels in the cover image. in [4], another steganography technique has been introduced to embed binary bits in a cover image using slantlet transform (slt) [17] and two data *corresponding author: rashathabit@yahoo.com mailto:rashathabit@yahoo.com the effect of two major factors on the performance of steganography 39 copyright ©2019 assa. adv. in systems science and appl. (2019) embedding methods (i.e., lsb substitution method and the threshold embedding method). the performance of the steganography technique has been improved in comparison with the dwt-based scheme, however, the visual quality of the stego images still needs more improvement. other steganography techniques have been presented to hide a secret image in a cover image. in [18], a secret image is embedded in the lsbs of a cover image which could be a color or grayscale image. the exclusive-or (xor) of a block of pixels is used to determine the smoothness of the neighborhood in order to obtain the number of bits that can be embedded in the cover image. in [5], a dwt-based steganography technique has been presented to hide a secret image in a cover image. the technique converts the cover image and the secret image using dwt and the resultant coefficients are divided into 4-subbands named as approximation, horizontal, vertical, and the diagonal coefficients. each subband is divided into non-overlapping blocks then the approximation coefficients of the secret image are compared with the approximation coefficients of the cover image and the root mean squared error (rmse) is calculated to find the locations of the best similar blocks. the differences between these coefficients are calculated and saved as error blocks then the rmse is used to find the locations of the best matched horizontal coefficients blocks with the error blocks, thereafter, the error blocks are embedded in the horizontal coefficients blocks. the work in this research paper is directed towards the steganography techniques in the transform domain and its aims can be summarized as follows:  introducing an improved steganography technique to hide secret image in a cover image.  introducing another improved steganography technique to hide a sequence of binary bits in a cover image. to meet the first aim of this work, the transform type (i.e., dwt) is changed to (slt) in a previous steganography technique [5] while keeping the same data embedding method then the data embedding method is modified to improve the performance. to meet the second aim of this work, an improved steganography technique is introduced to hide a sequence of binary bits in a cover image by modifying the carrier slt subbands according to the secret data bits and the mean values of slt coefficients for the cover image’s blocks. the rest of the paper is organized as follows: section 2 explains the implemented algorithms, section 3 illustrates the experiments and their discussion, and section 4 contains the conclusion. 2. the implemented techniques the following subsections presents the steps of the proposed algorithms in this paper. the first algorithm (slt-em1) is implemented to hide a secret image in a cover image using slt and an embedding algorithm from [5]. then the same algorithm has been modified to improve the performance in terms of visual quality and the modified algorithm is named as slt-mem1. the second steganography algorithm has been implemented to embed a sequence of binary bits in a cover image and it is named as slt-em2. the abbreviations of the proposed algorithms can be explained as follows:  slt-em1: refers to slantlet transform and embedding method 1.  slt-mem1: refers to slantlet transform and modified embedding method 1.  slt-em2: refers to slantlet transform and embedding method 2. 2.1. slt-em1 the algorithms of slt-em1 for the secret image embedding and extraction processes are summarized in fig. 1 and fig. 2, respectively. the detailed steps are explained in the following subsections. 40 r. thabit copyright ©2019 assa adv. in systems science and appl. (2019) 2.1.1. slt-em1 data embedding procedure step 1: read the original cover image ( oi ) and the secret image ( si ). step 2: apply the 2d-slt to oi and divide the resulting coefficients into four subbands ( cdandcvchca ,,, ). fig. 1. slt-em1 data embedding procedure step 3: apply the 2d-slt to si and divide the resulting coefficients into four subbands ( sdandsvshsa ,,, ). step 4: divide saandchca ,, subbands into non-overlapping blocks of size 4×4. the subbands’ blocks can be represented as follows: }1,{ }1,{ }1,{ sk i i nkbssa nibhch nibaca    where iba is the ith block in ca , ibh is the ith block in ch , and kbs is the kth block in sa , n is the total number of 4×4 blocks in ca and in ch , sn is the total number of 4×4 blocks in sa . step 5: calculate the rmse to find the best matched block iba with kbs , save the locations i of these best matched blocks in a secret key 1k . step 6: calculate the difference (i.e., error block) keb between iba and kbs as follows: kik bsbaeb  …(1) step 7: for each block in keb , use the rmse to find the best matched block ibh ; save the locations i of these best matched blocks in a secret key 2k . then replace ibh with keb . step 8: continue to embed all the difference blocks in ch then replace the original ch subband with the new ch subband. step 9: to obtain the stego image g apply the inverse 2d-slt to ( cdandcvchnewca ,,, ). the effect of two major factors on the performance of steganography 41 copyright ©2019 assa. adv. in systems science and appl. (2019) the previous algorithm in [5] searches for the locations of the best matched blocks to obtain (k1 and k2) but some of the locations may be repeated more than once and this will cause losing some of the error blocks (because they will be embedded in the same location). to avoid this problem, the embedding algorithm in this paper has been modified to check the locations that has been obtained and if the same location appeared again then the location of the next block with the minimum rmse must be found. 2.1.2. slt-em1 data extraction procedure fig. 2. slt-em1 data extraction procedure step 1: apply the 2d-slt to the stego image g and divide the resulting coefficients into four subbands ( gcdandgcvgchgca ,,, ). step 2: divide gchandgca, into non-overlapping blocks of size 4×4. the blocks can be represented as follows: }1,{ }1,{ nibhgch nibagca i i   step 3: use secret key 1k to find the iba and use the secret key 2k to extract the error blocks keb from ibh . the secret image block kbs can be recovered as follows: kik ebbabs  …(2) step 4: repeat (step 3) to extract all the secret blocks. rearrange the blocks to obtain sa. step 5: set sdandsvsh ,, as zeros then apply inverse 2d-slt to ( sdandsvshsaextracted ,,, ) to obtain the secret image. to embed the secret image in cdorcv, subbands, the same procedures can be used but the ch subband must be replaced by the required subband. 42 r. thabit copyright ©2019 assa adv. in systems science and appl. (2019) in order to improve the visual quality of the stego image, the embedding algorithm is modified and the technique is named as slt-mem1. in the modified algorithm the error blocks are divided by a variable α and the procedure of the embedding process (section 2.1.1) is the same but in step 6 the following equation will be used: )( 1 kik bsbaeb   …(3) in the extraction process (section 2.1.2) the same procedure repeated but in step 3 the following equation will be used: kik ebbabs   …(4) 2.2. the proposed steganography technique (slt-em2) the previous study of slt-em1 and slt-mem2 proved that slt performs better in comparison with dwt, therefore it will be adopted in the proposed steganography technique (slt-em2) in which a new data embedding method is suggested in order to improve the performance. the proposed embedding process depends on modifying the carrier slt subband according to the secret data bit and the mean value of slt coefficients in a block. the algorithms of slt-em2 for the secret data embedding and extraction processes are summarized in the block diagrams shown in fig. 3 and fig. 4, respectively. the steps of the algorithms are detailed in the following subsections. 2.2.1. slt-em2 data embedding procedure step 1: apply 2d-slt to the cover image ( mi ) and divide the coefficients into four subbands ( cdandcvchca ,,, ). step 2: divide each subband ( cdandcvch ,, ) into non-overlapping blocks of size ( ss bb  ) as follows: }1,{ }1,{ }1,{ ncibcdcd ncibcvcv ncibchch i i i    where ibch is the ith block in ch, ibcv is the ith block in cv, ibcd is the ith block in cd, nc is the total number of ( ss bb  ) blocks in each subband. step 3: calculate the mean value of the coefficients in each block as follows:     s sb x b y yx s block c b m 1 1 ,2 1 …(5) }1,{ }1,{ }1,{ ncimcdcdmean ncimcvcvmean ncimchchmean i i i    the effect of two major factors on the performance of steganography 43 copyright ©2019 assa. adv. in systems science and appl. (2019) where blockm is the mean value for the block of size ss bb  , yxc , is the slt coefficient in the coordinates ( yx, ), imch is the mean value of the ith block in ch , imcv is the mean value of the ith block in cv , imch is the mean value of the ith block in cd . step 4: the rules of the proposed data embedding method can be explained as follows:  each bit from a binary sequence of data ( kw ) {where k=1, 2, …, length ( w )} can be embedded using two successive blocks. the main idea of the proposed data embedding process is that, when kw is ‘1’ then the mean value of the first block should be more than the mean value of the second block, when kw is ‘0’ the reverse is true. the embedding process in ch will be explained and the same idea can be applied for embedding data in cv and cd .  to embed a binary bit kw in two successive blocks the mean values of these blocks will be used to control the embedding process as follows: let 121 ,  ii mchxmchx if 211 xxandwk  , then exchange the blocks 1ii bchandbch else if 211 xxandwk  , then do nothing; if 120 xxandwk  , then exchange the blocks 1ii bchandbch else if 120 xxandwk  , then do nothing; this process must be repeated until embedding all data bits. step 5: rearrange the blocks to construct the modified subbands. step 6: apply inverse 2d-slt to obtain the stego image. fig. 3. slt-em2 data embedding procedure 44 r. thabit copyright ©2019 assa adv. in systems science and appl. (2019) 2.2.2. slt-em2 data extraction procedure step 1: apply the slt to the stego image (g ) and divide the coefficients into four subbands ( gcdandgcvgchgca ,,, ). step 2: divide each subband ( cdandcvch ,, ) into non-overlapping blocks of size ( ss bb  ) as explained in (section 2.2.1, step 2). step 3: calculate the mean value of the coefficients for each block in subband as explained in (section 2.2.1, step 3). step 4: the length of the data is required at the receiver side where the data bits can be extracted as follows: )(...,,2,1, ,0 ,1 1 1 wlengthkfor mchmchif mchmchif w ii ii k          fig. 4. slt-em2 data extraction procedure 3. the experimental results and discussion this section presents the experiments that have been conducted to evaluate the implemented algorithms: slt-em1, slt-mem1 and slt-em2. the first subsection contains the performance evaluation of the implemented algorithms (slt-em1 and slt-mem1) in comparison with the dwt-based technique from [5]. the second subsection contains the performance evaluation of the proposed technique (slt-em2) in comparison with the previous slt-based technique from [4]. in [4,5], the metric that have been used to evaluate the visual quality of the stego image is the peak signal-to-noise ratio (psnr), therefore, the same metric is adopted here to make the comparison valid. 3.1. experimental results of slt-em1 and slt-mem1 for comparison purpose, the same test images that have been used in [5] are utilized in the experiments of this section. the cover images (peppers, goldhill, cameraman, and barbara) are shown in fig. 5 where each image is of size (256×256) pixels. fig. 6 contains the secret images (airplane and bird) with size (128×128) pixels. the effect of two major factors on the performance of steganography 45 copyright ©2019 assa. adv. in systems science and appl. (2019) (a) (b) (c) (d) fig. 5. cover images: (a) peppers, (b) goldhill, (c) cameraman, and (d) barbara. (a) (b) fig. 6. secret images: (a) airplane, and (b) bird. the secret images are embedded in ch subband of the cover images using (slt-em1 and slt-mem1); α in slt-mem1 is set to 2 (as an example). fig. 7 presents the psnr results of the stego images for the two secret images (i.e., airplane and bird). the results proved that, replacing the dwt by slt improved the visual quality results of the stego images, however, further improvement has been obtained using slt-mem1 in which the embedding method has been modified. the psnr between the original secret image and the extracted secret image has been calculated to evaluate the quality of the extracted secret image and the results are shown in table 1. as illustrated in the results, the extracted secret images using slt-em1 and sltmem1 obtained better visual quality results in comparison with the dwt-based scheme [5]. the difference between slt-em1 and slt-mem1 is only in the embedding process, therefore, the same secret information is embedded and extracted and this is proved in the results of table 1. (a) (b) fig. 7. psnr(db) results for the stego images when the secret image: (a) airplane, and (b) bird. 46 r. thabit copyright ©2019 assa adv. in systems science and appl. (2019) table 1 the psnr (db) of the extracted secret images from ch subband cover images (256×256) secret images (128×128) airplane bird dwt [5] slt-em1 slt-mem1 dwt [5] slt-em1 slt-mem1 peppers 22.3544 25.2799 25.2796 28.8265 32.0501 32.0500 goldhill 22.3544 25.2793 25.2793 28.8265 32.0506 32.0502 cameraman 22.3544 25.2794 25.2795 28.8265 32.0505 32.0502 barbara 22.3544 25.2791 25.2791 28.8265 32.0500 32.0500 the effect of embedding the secret images in different subbands ( cdandcvch ,, ) of the cover images has been studied and the results of slt-em1 and slt-mem1 are shown in table 2 and table 3, respectively. as illustrated in table 2 and table 3, the obtained results for ch and cv subbands are convergent and both are better than the results of cd subband; on the other hand, the slt-mem1 technique obtained better results in comparison with slt-em1 technique. table 2 the psnr(db) results for the stego images at different subbands using slt-em1 cover images (256×256) secret images (128×128) airplane bird ch subband cv subband cd subband ch subband cv subband cd subband peppers 28.2451 28.4586 27.8485 29.6938 29.6936 29.1933 goldhill 28.3697 28.2927 27.7174 29.0134 29.2456 28.4965 cameraman 28.6842 28.8816 27.7867 29.8682 30.0028 29.2273 barbara 28.1230 28.2932 27.7850 29.4041 29.6504 29.0477 table 3 the psnr(db) results for the stego images at different subbands using slt-mem1 cover images (256×256) secret images (128×128) airplane bird ch subband cv subband cd subband ch subband cv subband cd subband peppers 32.6267 32.7314 32.3668 34.1096 33.9971 33.2923 goldhill 32.6697 32.5951 32.0575 33.0514 33.2603 32.4745 cameraman 33.1888 33.3216 32.5160 33.8336 33.8251 33.3362 barbara 32.5431 32.6211 32.2960 33.6547 33.7584 33.4449 3.2. experimental results of slt-em2 in order to prove the efficiency of the proposed data embedding method, the experiments have been conducted to compare the performance of the proposed slt-based steganography scheme with a previous slt-based steganography scheme from [4] which used two different embedding methods (i.e., lsb and threshold). for comparison purposes, the experiments have been conducted for the same test image (i.e., grayscale lena image) that has been used in [4]. in the proposed steganography scheme (slt-em2), different block sizes can be used and the data can be embedded in different subbands, therefore, the experiments have been conducted to evaluate the visual quality for the three subbands ( cdandcvch ,, ) at different block sizes ( ss bb  ). fig. 8 and fig. 9 present the experimental results and comparison with the lsb embedding method from [4] for the payload (36 bits) at block size (2×2) and block size (4×4), respectively. the results proved that the stego image using the proposed slt-em2 technique obtained better visual quality results in comparison with the lsb embedding method. the highest psnr values are obtained when the secret data is embedded in cd subband. on the other hand, the the effect of two major factors on the performance of steganography 47 copyright ©2019 assa. adv. in systems science and appl. (2019) results proved that in slt-em2, the lower the block size the better the visual quality results because less changes will be conducted in the stego image. fig. 8. visual quality comparison of slt-em2 with lsb method at block size (2×2). fig. 9. visual quality comparison of slt-em2 with lsb method at block size (4×4). to compare the performance of slt-em2 and the threshold embedding method from [4], the same previous experiments have been conducted but the payload is changed to (44 bits). fig. 10 and fig. 11 present the experimental results which proved that the proposed sltem2 technique obtained better visual quality results in comparison with the threshold embedding method. 48 r. thabit copyright ©2019 assa adv. in systems science and appl. (2019) fig. 10. visual quality comparison of slt-em2 with threshold method at block size (2×2). fig. 11. visual quality comparison of slt-em2 with threshold method at block size (4×4). the visual quality using slt-em2 has been tested at higher payloads and the comparison results are illustrated in table 4. the results proved that for higher payloads the stego image using slt-em2 has better visual quality in comparison with lsb and threshold embedding methods. table 4 visual quality comparison at higher payloads payload (bits) threshold embedding [4] lsb embedding [4] slt-em2 1000 ― ― 39.8666 1673 24.6087 25.8256 36.7899 3727 24.6042 25.8249 31.7497 4000 ― ― 31.1905 another experiment has been conducted to evaluate the error level in the extracted binary bits for the proposed algorithm slt-em2 by calculating the normalized correlation (nc) between the original bits and the extracted bits. table 5 presents the nc results for binary sequences of length (1000 and 1500 bits) which are embedded in the test images that have been used in the previous experiments (i.e., lena, peppers, goldhill, cameraman, and barbara) at block sizes (2×2 and 4×4). the results proved that there is a small amount of error in the extracted bits which is resulted from the rounding operation due to the embedding in the transform domain. the effect of two major factors on the performance of steganography 49 copyright ©2019 assa. adv. in systems science and appl. (2019) table 5 normalized correlation (nc) results test image binary sequence (1000 bits) binary sequence (1500 bits) block size (2×2) block size (4×4) block size (2×2) block size (4×4) lena 0.8960 0.9580 0.9150 0.9680 peppers 0.9520 0.9530 0.9520 0.9467 goldhill 0.9620 0.9860 0.9707 0.9575 cameraman 0.8940 0.9201 0.8933 0.9094 barbara 0.9500 0.9600 0.9402 0.9494 4. conclusion this paper presents two improved steganography techniques based on slt and different embedding methods. the first proposed technique embeds secret image in a cover image while the second proposed technique embeds binary data in a cover image. the work in this paper proved that the change of the transform type while using the same data embedding method can improve the visual quality of the stego images, on the other hand, the change of the data embedding method while applying the same transform type can also improve the performance of the steganography technique. the experimental results of the first technique proved that the slt-based steganography techniques perform better in comparison with the dwt-based techniques. in addition, the modification of the embedding algorithm causes further improvement in the results. the experimental results of the second technique proved the efficiency of the proposed technique in comparison with a previous slt-based steganography technique. from the work presented in this paper, one can conclude that the researcher who intend to implement a new steganography technique in the transform domain should investigate the performance of his / her technique using different transforms to choose the best transform that can match the embedding method to obtain the optimum results. references 1. hassan, n. a., & hijazi, r. 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(2013). a new approach to hide text in images using steganography, international journal of advanced research in computer science and software engineering, 3(4), 701-708. https://pdfs.semanticscholar.org/1243/57f41f1b3a87d7380763d2a78041786a35be.pdf 16. kaul, n., bajaj, n. (2013). audio in image steganography based on wavelet transform, international journal of computer applications, 79(3), 7-10. https://pdfs.semanticscholar.org/4d0f/50b782f10b01e5e963088d07d25037096263.pd f 17. selesnick, i. w. (1999). the slantlet transform, ieee transactions on signal processing, 47(2), 1304-1313. https://doi.org/10.1109/78.757218 https://doi.org/10.1016/s1665-6423(14)71612-8 https://doi.org/10.1007/978-90-481-3660-5_10 https://doi.org/10.1016/j.proenv.2011.09.180 https://doi.org/10.5281/zenodo.556248 https://www.ripublication.com/ijaer18/ijaerv13n8_106.pdf https://doi.org/10.1109/icsipa.2013.6708032 https://doi.org/10.1049/iet-ipr.2013.0862 https://arxiv.org/ftp/arxiv/papers/1307/1307.0642.pdf https://pdfs.semanticscholar.org/1243/57f41f1b3a87d7380763d2a78041786a35be.pdf https://pdfs.semanticscholar.org/4d0f/50b782f10b01e5e963088d07d25037096263.pdf https://pdfs.semanticscholar.org/4d0f/50b782f10b01e5e963088d07d25037096263.pdf https://doi.org/10.1109/78.757218 the effect of two major factors on the performance of steganography 51 copyright ©2019 assa. adv. in systems science and appl. (2019) 18. tamimi, a. a., abdalla, a. m., al-allaf, o. (2013). hiding an image inside another image using variable-rate steganography, international journal of advanced computer science and applications (ijacsa), 4(10), 18-21. https://pdfs.semanticscholar.org/878b/17246ca62e2a62c77fa47c5d6baa0fec3d6f.pdf https://pdfs.semanticscholar.org/878b/17246ca62e2a62c77fa47c5d6baa0fec3d6f.pdf adv syst sci appl 2020; 01:91–103 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/854 on properties of coincidence points of mappings between (q1, q2)-quasimetric spaces richik sengupta1, zukhra t. zhukovskaya2, sergey e. zhukovskiy2,3* 1peoples’ friendship university of russia. 2 v.a. trapeznikov institute of control sciences of ras. 3 moscow institute of physics and technology. abstract: in this paper, the properties of coincidence points of mappings acting between (q1, q2)quasimetric spaces are studied. for a pair of mappings, we obtain estimates for the distance from a point to the coincidence points set and intersection of the respective graphs of the mappings. in addition, the stability of coincidence points is studied. a generalization of lim’s lemma is obtained. keywords: (q1, q2)-quasimetric spaces, coincidence points, set-valued mappings 1. introduction and statement of the problem the paper is devoted to the investigation of coincidence points of pairs of set-valued mappings acting between (q1, q2)-quasimetric spaces. in order to proceed to the statement of the problem, let us recall the definitions of the concepts in use. let x be a nonempty set, numbers q0 ≥ 1, q1 ≥ 1, q2 ≥ 1 be given. a function ρx : x ×x → r+ is called a (q1, q2)-quasimetric if • ρx(x, y) = 0⇔ x = y ∀ x, y ∈ x (the identity axiom); • ρx(x, z) ≤ q1ρx(x, y) + q2ρx(y, z) ∀x, y, z ∈ x (the (q1, q2)-generalized triangle inequality). if ρx is a (q1, q2)-quasimetric, then the space (x, ρx) is called a (q1, q2)-quasimetric space. the concept of the (q1, q2)-quasimetric space was introduced in [1]. if q1 = q2 = 1 then this concept coincides with the concept of a quasimetric space. if we additionally assume that a quasimetric satisfies the symmetry axiom, i.e. ρx(x, y) ≡ ρx(y, x), it becomes a metric. the detailed description of topological properties of (q1, q2)-quasimetric spaces was provided in [2]. recall some basic definitions. a sequence {xi} ⊂ x is said to converge to a point x ∈ x if ρx(x, xi)→ 0 as i→∞. the point x is called a limit of {xi}. a subset of x is said to be closed if every limit of every convergent sequence from this set belongs to this set. a sequence {xi} ⊂ x is said to be a cauchy sequence if ∀ ε > 0 ∃n ∈ n : ρx(xj, xi) < ε ∀ i > j > n. let (x, ρx) and (y, ρy ) be (q1, q2)-quasimetric spaces, φ,ψ : x ⇒ y be set-valued mappings that map points x ∈ x to non-empty subsets of y. ∗corresponding author: s-e-zhuk@yandex.ru 92 r. sengupta, z.t. zhukovskaya, s.e. zhukovskiy a point ξ ∈ x is called a coincidence point of the set-valued mappings φ,ψ if φ(ξ) ∩ψ(ξ) 6= ∅. below we use the following notation: coin(ψ,φ) stands for the set of all coincidence points of mappings ψ and φ; gphψ stands for the graph of ψ, i.e. gphψ = {(x, y) ∈ x × y : y ∈ ψ(x)}; γ(ψ,φ) stands for the intersection of the graphs of ψ and φ, i.e. γ(ψ,φ) := {(x, y) ∈ x × y : y ∈ ψ(x) ∩ φ(x)}. it is obvious that coin(ψ,φ) 6= ∅ ⇔ γ(ψ,φ) 6= ∅. for non-empty sets u, v ⊂ x, denote dist(u, v ) = inf{ρx(x1, x2) : x1 ∈ u, x2 ∈ v }, h+ x(u, v ) = sup u∈u dist(u, v ), hx(u, v ) := max{h+ x(u, v ), h+ x(v, u)}. the function h+ x is called the hausdorff deviation; the function hx is called the hausdorff (q̂1, q̂2)-quasimetric (note that (q̂1, q̂2) may differ from (q1, q2)). for hausdorff deviation h+ x , the (q1, q2)-generalized triangle inequality holds (see [3, page 25]), i.e. h+ x(u,w ) ≤ q1h + x(u, v ) + q2h + x(v,w ) ∀u, v,w ⊂ x. moreover, the definitions above directly imply that hx(u, v ) ≥ h+ x(u, v ) ≥ dist(u, v ) ∀u, v ⊂ x. let us recall now some definitions related to set-valued mappings. definition 1.1: ( [3, definition 5.5]) given a number β ≥ 0, the set-valued mapping φ : x ⇒ y is called β-lipschitz if h(φ(x1),φ(x2)) ≤ βρx(x1, x2) ∀x1, x2 ∈ x. if (x, ρx) = (y, ρy ) and β < 1 then the β-lipschitz set-valued mapping φ is said to be a contraction. definition 1.2: ( [3, definition 4.4]) the set-valued mapping φ : x ⇒ y is said to be closed, if for all sequences {xi} ⊂ x, {yi} ⊂ y and points x ∈ x, y ∈ y such that xi → x, yi → y and (xi, yi) ∈ gph(φ) for all i, we have (x, y) ∈ gph(φ). definition 1.3: we will say that the graph of the set-valued mapping φ : x ⇒ y is complete, if for all cauchy sequences {xi} ⊂ x and {yi} ⊂ y such that {(xi, yi)} ⊂ gph(φ), there exists a point (x, y) ∈ gph(φ) such that xi → x and yi → y. denote by bx(x0, r) a closed ball in x centered at x ∈ x with the radius r > 0, i.e. bx(x0, r) = {x ∈ x : ρx(x0, x) ≤ r}. copyright © 2020 assa. adv syst sci appl (2020) properties of coincidence points in (q1, q2)-quasimetric spaces 93 definition 1.4: ( [3, definition 5.4]) given a number α > 0, the set-valued mapping ψ : x ⇒ y is called α-covering if ⋃ y∈ψ(x) by (y, αr) ⊆ ψ(bx(x, r)) ∀r ≥ 0,∀x ∈ x. let us now recall the coincidence point existence theorem from [3]. let numbers α > 0, β ∈ [0, α) and set-valued mappings ψ,φ : x ⇒ y be given. denote mψ,φ(x, r) := {y ∈ φ(x) : dist(ψ(x), y) < r}, x ∈ x, r > 0, s(θ, n) := 1− θn 1− θ , θ ∈ [0, 1), n = 0, 1, 2... , m0 := min { j ∈ n : q2 ( β α )j < 1 } . theorem 1.1: ( [3, theorem 5.7]) let numbers α > 0 and β ∈ [0, α) be given. assume that • the set-valued mapping ψ : x ⇒ y is α-covering and its graph is closed; • set-valued mapping φ : x ⇒ y is β-lipschitz; • at least one of the graphs gph(ψ) or gph(φ) is complete. then for all x0 ∈ x, r0 > dist(ψ(x0),φ(x0)), y1 ∈mψ,φ(x0, r0) there exists ξ ∈ x such that ψ(ξ) ∩ φ(ξ) 6= ∅, lim λ→ξ ρx(x0, λ) ≤ q2 1α m0−1s(q2 β α ,m0 − 1) + q1(q2β)m0−1 αm0 − q2βm0 r0. this assertion not only provides the sufficient conditions for the existence of a coincidence point but also an estimate of the distance from a point x0 ∈ x to a coincidence point ξ of the given mappings. a problem to obtain an estimate of distance from the point y1 to a point η ∈ ψ(ξ) ∩ φ(ξ) naturally arises. for the case when (x, ρx) and (y, ρy ) are metric spaces, this problem was solved in [4, 5]. the main goal of our paper is to obtain results analogous to those in [4,5] for set-valued mappings acting between (q1, q2)-quasimetric spaces. we also discuss a similar problem for fixed points of set-valued mappings and derive propositions on fixed points properties similar to those in [6]. the results of this paper may have applications in the investigation of various nonlinear equations. one of the possible applications of the results is the investigation of nonlinear equations in banach spaces equipped with an additional bimodule structure over a group ring based on the theory developed in [7]. note that the results on coincidence points and their analogs (see, for example, [8], [9]) are applied in the study of equations appearing in economic models (see [10]), differential inclusions (see [11], [12], [13]), and other problems. 2. main results. estimates of distance from a point to the intersection set of two graphs let q1 ≥ 1, q2 ≥ 1 be given numbers, (x, ρx), (y, ρy ) be (q1, q2)-quasimetric spaces. given α > 0, β ∈ [0, α), denote byfα,β the set of all ordered pairs of set-valued mappings (ψ,φ), ψ,φ : x ⇒ y, such that copyright © 2020 assa. adv syst sci appl (2020) 94 r. sengupta, z.t. zhukovskaya, s.e. zhukovskiy • the set-valued mapping ψ : x ⇒ y is α-covering; • set-valued mapping φ : x ⇒ y is β-lipschitz; • either gph(ψ) is complete and φ(x) is a closed set for every x ∈ x or ψ is closed and gphφ is complete. theorem 2.1: let numbers α > 0, β ∈ [0, α) and an arbitrary ordered pair of set-valued mappings (ψ,φ) ∈ fα,β be given. then for all x0 ∈ x, r0 > dist(ψ(x0),φ(x0)), y1 ∈mψ,φ(x0, r0) there exist ξ ∈ x and η ∈ y such that η ∈ ψ(ξ) ∩ φ(ξ), lim λ→ξ ρx(x0, λ) ≤ q2 1α m0−1s(q2 β α ,m0 − 1) + q1(q2β)m0−1 αm0 − q2βm0 r0, (2.1) lim κ→η ρy (y1, κ) ≤ β q2 1α m0−1s(q2 β α ,m0 − 1) + q1(q2β)m0−1 αm0 − q2βm0 r0. (2.2) before proving theorem 2.1, let us prove the following lemma. lemma 2.1: let the set-valued mapping ψ : x ⇒ y be α-covering, the set-valued mapping φ : x ⇒ y be β-lipschitz. then for arbitrary δ > 0, x0 ∈ x, y1 ∈mψ,φ(x0, αδ + dist(ψ(x0),φ(x0))) there exist sequences {xi} ⊂ x and {yi} ⊂ y such that ρx(x0, x1) ≤ δ + dist(ψ(x0),φ(x0)) α , (2.3) ρx(xi−1, xi) ≤ (β α + δ ) ρx(xi−2, xi−1) ∀i ≥ 2, (2.4) yi ∈ ψ(xi) ∩ φ(xi−1) ∀i ≥ 1, (2.5) ρy (yi−1, yi) ≤ (β + αδ)ρx(xi−2, xi−1) ∀i ≥ 2. (2.6) proof let us take an arbitrary x0 ∈ x and δ > 0. set r0 = αδ + dist(ψ(x0),φ(x0)). let us take an arbitrary point y1 ∈mψ,φ(x0, r0). since the mapping ψ is α-covering, there exists a point x1 ∈ bx(x0, r0) such that y1 ∈ ψ(x1). therefore, y1 ∈ ψ(x1) ∩ φ(x0). let us construct the sought sequences by induction. if x0 = x1 then set x2 := x1, y2 := y1. assume that x0 6= x1. set r1 := (β + αδ)ρx(x0, x1). since the mapping φ is β-lipschitz, we have hy (φ(x0),φ(x1)) < r1. therefore, there exists a point y2 ∈ φ(x1) such that ρy (y1, y2) < r1. since y1 ∈ ψ(x1), we have y2 ∈ ⋃ y∈ψ(x1) by (y, r1). copyright © 2020 assa. adv syst sci appl (2020) properties of coincidence points in (q1, q2)-quasimetric spaces 95 therefore, since the set-valued mapping ψ is α-covering, there exists a point x2 ∈ bx(x1, r1) such that y2 ∈ ψ(x2) and ρx(x1, x2) ≤ r1 α . the sought x2, y2 are constructed. let us now assume that for a certain j, the sought points xi, yi, i = 1, j, are constructed. let us construct xj+1, yj+1. if xj = xj−1 then set xj+1 := xj , yj+1 := yj . assume that xj 6= xj−1. set rj := (β + αδ)ρx(xj−1, xj). since the mapping φ is β-lipschitz, we have hy (φ(xj−1),φ(xj)) < rj. therefore, there exists a point yj+1 ∈ φ(xj) such that ρy (yj, yj+1) ≤ rj. since yj ∈ ψ(xj), we have yj+1 ∈ ⋃ y∈ψ(xj) by (y, rj). since the set-valued mapping ψ is α-covering, there exists a point xj+1 ∈ bx(xj, rj) such that yj+1 ∈ ψ(xj+1) and ρx(xj, xj+1) ≤ rj α . the sought xj+1, yj+1 are constructed. proof of theorem 2.1. without loss of generality, we assume α = 1. take a δ > 0 such that min { j ∈ n : q2 ( β α + δ )j < 1 } = m0, r0 > dist(ψ(x0),φ(x0)) + δ. let us consider the corresponding sequences {xi} and {yi}, that were constructed in lemma 2.1. let us show that {xi} is a cauchy sequence. for integers i, j ≥ 0, we have ρx(xi, xi+j) ≤ q1ρx(xi, xi+1) + q2ρx(xi+1, xi+j) ≤ ≤ q1r0(β + δ)i + q2 ( q1ρx(xi+1, xi+2) + q2ρx(xi+2, xi+j) ) ≤ ≤ q1r0(β + δ)i + q1q2r0(β + δ)i+1 + q2 2 ( q1ρx(xi+2, xi+3) + q2ρx(xi+3, xi+j) ) ≤ ≤ · · · ≤ q1r0(β + δ)i ( 1 + q2(β + δ) + · · ·+ qj−2 2 (β + δ)j−2 + qj−1 2 (β + δ)j−1q−1 1 ) = = q1r0(β + δ)is̃(j). here s̃(j) = s(q2(β + δ), j − 1) + qj−1 2 (β + δ)j−1q−1 1 , j ∈ n, s̃(0) = 0. thus, for any non-negative integer i and k, we have ρx(xi, xi+k) ≤ q1ρx(xi, xi+m0) + q2ρx(xi+m0 , xi+k) ≤ ≤ q1ρx(xi, xi+m0) + q2 ( q1ρx(xi+m0 , xi+2m0)+ +q2ρx(xi+2m0 , xi+k) ) ≤ q1ρx(xi, xi+m0) + q1q2ρx(xi+m0 , xi+2m0)+ +q2 2 ( q1ρx(xi+2m, xi+3m0) + q2ρx(xi+3m0 , xi+k) ) ≤ copyright © 2020 assa. adv syst sci appl (2020) 96 r. sengupta, z.t. zhukovskaya, s.e. zhukovskiy ≤ q1ρx(xi, xi+m0) + q1q2ρx(xi+m0 , xi+2m0) + q1q 2 2ρx(xi+2m0 , xi+3m0) + q2 3ρx(xi+3m0 , xi+k) ≤ ≤ · · · ≤ q1ρx(xi, xi+m0) + q1q2ρx(xi+m0 , xi+2m0)+ + · · ·+ q1q p−1 2 ρx(xi+(p−1)m0 , xi+pm0) + qp2ρx(xi+pm0 , xi+k) ≤ ≤ q2 1r0β is̃(m0) ( 1 + q2(β + δ)m0 + q2 2(β + δ)2m0 + · · ·+ qp−1 2 (β + δ)(p−1)m0 ) + +qp2q1r0(β + δ)i+pm0s̃(k − pm0) = = q2 1r0(β + δ)is̃(m0)s(q2(β + δ)m0 , p) + qp2(β + δ)i+pm0q1r0s̃(k − pm0), where p is the integer part of k/m0. since, q2(β + δ)m0 < 1, we have ρx(xi, xi+k) ≤ ≤ q2 1r0(β + δ)i ( s̃(m0)s(q2(β + δ)m0 , p) + qp2r0(β + δ)pm0q−1 1 s̃(k − pm0) ) ≤ ≤ q2 1r0(β + δ)i ( s̃(m0) 1− q2(β + δ)m0 + q−1 1 s̃(k − pm0) ) . (2.7) since, 0 ≤ k − pm0 ≤ m0, the value q−1 1 s(k − pm0) is uniformly bounded for all k. therefore, {xi} is a cauchy sequence. let us show that {yi} is also a cauchy sequence. according to the lemma, ρy (yi+1, yi+j+1) ≤ (β + δ)ρx(xi, xi+j) for every i and j. therefore, repeating the arguments above, we get ρx(yi+1, yi+k+1) ≤ (β + δ)[q2 1r0(β + δ)i ( s̃(m0)s(q2(β + δ)m0 , p)+ +qp2(β + δ)pm0q−1 1 s̃(k − pm0) ) ] ≤ ≤ q2 1r0(β + δ)i+1 ( s̃(m0) 1−q2(β+δ)m0 + q−1 1 s̃(k − pm0) ) . (2.8) therefore, {yi} is a cauchy sequence. consider now two cases. at first, assume that the gph(ψ) is complete and each value of φ is a closed set. then the cauchy sequences {xi}, {yi} converge to points ξ ∈ x, η ∈ ψ(ξ) respectively as (xi, yi) ∈ gph(ψ). we have hy (φ(xi),φ(ξ)) = hy (φ(ξ),φ(xi)) ≤ βρx(ξ, xi), and thus h+ y (yi+1,φ(ξ)) ≤ βρx(ξ, xi). therefore, h+ y (η,φ(ξ)) ≤ q1ρy (η, yi+1) + q2h + y (yi+1,φ(ξ)) ≤ q1ρy (η, yi+1) + q2βρx(ξ, xi). since {yi+1} tends to η and {xi} tends to ξ, we have h+ y (η,φ(ξ)) = 0. this equality and the closedness of φ(ξ) imply η ∈ φ(ξ). assume now that gph(φ) is complete and ψ is closed. then the cauchy sequences {xi}, {yi} converge to some points ξ ∈ x , η ∈ φ(ξ) respectively, since (xi, yi+1) ∈ gph(φ). the set gph(ψ) is closed, therefore η ∈ ψ(ξ). so, it is proved that η ∈ φ(ξ) ∩ψ(ξ). passing to the limit in (2.7) as k → +∞ and putting i = 0 we obtain lim ζ→ξ ρx(x0, ζ) ≤ q2 1α m0s ( q2 β+αδ α ,m0 − 1 ) + q1q m0−1 2 (β + αδ)m0−1 αm0 − q2(β + αδ)m0 ( δ + dist(ψ(x0),φ(x0)) α ) copyright © 2020 assa. adv syst sci appl (2020) properties of coincidence points in (q1, q2)-quasimetric spaces 97 hence, as the choice of δ is arbitrary, it implies (2.1). analogously passing to the limit as k →∞ in (2.8) and substituting i = 0 by virtue of the choice of δ we obtain (2.2). 2 let us now obtain an estimate of the distance from a point (x, y) ∈ x × y to the set γ(ψ,φ). put k(m0) := q2 1α m0−1s(q2 β α ,m0 − 1) + q1(q2β)m0−1 αm0 − q2βm0 . for vectors z = (x, y) ∈ x × y, a = (ax , ay ) ∈ r2 and a subset γ ⊂ x × y, we write d(z,γ) ≤ a if ∀ ε > 0 ∃(ξ, η) ∈ γ : lim λ→ξ ρx(x, λ) ≤ ax + ε, lim κ→η ρy (y, κ) ≤ ay + ε. theorem 2.2: let α > 0 and β ∈ [0, α) be given. if (ψ,φ) ∈ fα,β then the set γ(ψ,φ) is non-empty and, moreover, for arbitrary x ∈ x, y ∈ y, the inequality d ( (x, y),γ(ψ,φ) ) ≤ a(x, y, yφ) ∀ yφ ∈ φ(x), (2.9) holds. here a(x, y, yφ) := ( k(m0)dist(ψ(x), yφ), q1ρy (y, yφ) + q2βk(m0)dist(ψ(x), yφ) ) . proof theorem 2.1 implies that γ(ψ,φ) 6= ∅. let us take an arbitrary x ∈ x, y ∈ y, yφ ∈ φ(x), ε > 0. set r := dist(ψ(x), yφ). the definition of dist(·, ·) implies dist(ψ(x),φ(x)) ≤ r. therefore, yφ ∈mψ,φ(x, r + ε). it follows from theorem 2.1 that there exist ξ ∈ x and η ∈ y such that (ξ, η) ∈ γ(ψ,φ) and lim λ→ξ ρx(x, λ) ≤ k(m0)(r + ε), lim κ→η ρy (yφ, κ) ≤ βk(m0)(r + ε). since ρy (y, κ) ≤ q1ρy (y, yφ) + q2ρy (yφ, κ), we have lim κ→η ρy (y, κ) ≤ q1ρy (y, yφ) + q2 lim κ→η (yφ, κ) ≤ q1ρy (y, yφ) + q2βk(m0)(r + ε). the arbitrariness of ε > 0 implies that (2.9) holds. theorem 2.3: let α > 0 and β ∈ [0, α) be given. if (ψ,φ) ∈ fα,β then the set γ(ψ,φ) is non-empty and, moreover, for arbitrary x ∈ x, y ∈ y, the inequality d((x, y),γ(ψ,φ)) ≤ a(x, y) (2.10) holds. here a(x, y) := ( k(m0)(r1 + r2), q1r2 + q2βk(m0)(q1r1 + q2r2) ) , r1 := dist(ψ(x), y), r2 := dist(y,φ(x)). copyright © 2020 assa. adv syst sci appl (2020) 98 r. sengupta, z.t. zhukovskaya, s.e. zhukovskiy proof take arbitrary points x ∈ x, y ∈ y, yψ ∈ ψ(x), yφ ∈ φ(x) and a number ε > 0 such that ρy (yψ, y) < dist(ψ(x), y) + ε 2 , ρy (y, yφ) < dist(y,φ(x)) + ε 2 . set ε1 := ε(q1 + q2) 2 , r := q1r1 + q2r2. let us show that yφ ∈mψ,φ(x, r + ε1). indeed, by the assumption yφ ∈ φ(x). moreover, ρy (yψ, yφ) < q1r1 + q2r2 + ε(q1 + q2) 2 ≤ r + ε1 and dist(ψ(x),φ(x)) ≤ dist(ψ(x), yφ) ≤ ρy (yψ, yφ). hence, dist(ψ(x), y) < r + ε1. therefore, yφ ∈mψ,φ(x, r + ε1). theorem 2.1 implies that there exist ξ ∈ x and η ∈ y such that (ξ, η) ∈ γ(ψ,φ) and lim λ→ξ ρx(x, λ) ≤ k(m0)(r + ε1), lim κ→η ρy (yφ, κ) ≤ k(m0)β(r + ε1). (2.11) since ρy (y, κ) ≤ q1ρy (y, yφ) + q2(yφ, κ) the inequality (2.11) implies lim κ→η ρy (y, κ) ≤ q1ρy (y, yφ) + q2 lim κ→η (yφ, κ) ≤ q1r2 + q2k(m0)β(r + ε1). moreover, it follows from (2.11) that lim λ→ξ ρx(x, λ) ≤ k(m0)(r + ε1). the arbitrariness of ε > 0 implies that the inequality (2.10) holds. remark 2.1: in the case when (x, ρx) and (y, ρy ) are metric spaces, theorem 2.1 coincides with [4, theorem 1] and theorem 2.2 coincides with [4, theorem 3]. 3. corollaries. estimates of distance between intersections of graphs and sets of coincidence points let q1 ≥ 1, q2 ≥ 1 be given numbers, (x, ρx), (y, ρy ) be (q1, q2)-quasimetric spaces. let us define one more function which characterizes a distance between subsets of (q1, q2)-quasimetric spaces. for u, v ⊂ x set e+(u, v ) := sup v∈v dist(u, v). even though the definitions of e+ and h+ look quite similar, these functions are actually different as ρx is not necessarily symmetric. let us describe some properties of the function e+. copyright © 2020 assa. adv syst sci appl (2020) properties of coincidence points in (q1, q2)-quasimetric spaces 99 proposition 3.1: for arbitrary sets u, v, w ⊂ x, the following inequalities hold e+(u,w ) ≤ q1e +(u, v ) + q2e +(v,w ), (3.12) dist(u,w ) ≤ q1e +(u, v ) + q2e +(v,w ), (3.13) proof let us prove (3.12). take arbitrary sets u, v, w ⊂ x, a point w ∈ w and a number ε > 0. according to the definition of e+ there exists v ∈ v and such that ρx(v, w) ≤ e+(v,w ). moreover, there exists u ∈ u such that ρx(u, v) ≤ e+(u, v ). we have, dist(u,w) ≤ ρx(u,w) ≤ q1ρx(u, v) + q2ρx(v, w) ≤ q1e +(u, v ) + q2e +(v,w ). due to the arbitrariness of w ∈ w, the above inequality implies (3.12). inequality (3.13) follows from (3.12) as dist(u,w ) ≤ e+(u,w ). we will also use the following inequality which was proved in ( [3, property 5.1]): dist(u,w ) ≤ q1dist(u, v ) + q2h +(v,w ). (3.14) theorem 3.1: let α > 0 and β ∈ [0, α) be given. if (ψ,φ) ∈ fα,β then for arbitrary (ξ̃, η̃) ∈ γ(ψ̃, φ̃) and ε > 0 there exists (ξ, η) ∈ γ(ψ,φ) such that lim λ→ξ ρx(ξ̃, λ) ≤ k(m0) ( q1e +(ψ(ξ̃), ψ̃(ξ̃)) + q2h +(φ̃(ξ̃),φ(ξ̃)) ) +ε, lim κ→η ρy (η̃, κ) ≤ q1q2h +(φ̃(ξ̃),φ(ξ̃)) + q2βk(m0) ( q2 1e +(ψ(ξ̃), ψ̃(ξ̃)) + q2 2h +(φ̃(ξ̃),φ(ξ̃)) ) + ε. proof fix an arbitrary pair (ξ̃, η̃) ∈ γ(ψ̃, φ̃) and take an arbitrary ε > 0. applying theorem 2.3 to the mappings φ and ψ we obtain that there exists (ξ, η) ∈ γ(ψ,φ) such that lim λ→ξ ρx(ξ̃, λ) ≤ k(m0)(r1 + r2) + ε, (3.15) lim κ→η ρy (η̃, κ) ≤ q1r2 + q2βk(m0)(q1r1 + q2r2) + ε. (3.16) here r1 := dist(ψ(ξ̃), η̃), r2 := dist(η̃,φ(ξ̃)). it follows from (3.13) that dist(ψ(ξ̃), η̃) ≤ q1e +(ψ(ξ̃), ψ̃(ξ̃)) + q2e +(ψ̃(ξ̃), η̃). since η̃ ∈ ψ̃(ξ̃), we have r1 = dist(ψ(ξ̃), η̃) ≤ q1e +(ψ(ξ̃), ψ̃(ξ̃)). since η̃ ∈ φ̃(ξ̃), we have dist(η̃, φ̃(ξ̃)) = 0. so, it follows from (3.14) that r2 = dist(η̃,φ(ξ̃)) ≤ q2h +(φ̃(ξ̃),φ(ξ̃)). hence, the desired inequalities follow from (3.15) and (3.16). let us introduce one more notation. given arbitrary sets γ̃,γ ⊂ x × y and a vector a = (ax , ay ) ∈ r2, the notation h+(γ̃,γ) ≤ a copyright © 2020 assa. adv syst sci appl (2020) 100 r. sengupta, z.t. zhukovskaya, s.e. zhukovskiy means that for arbitrary (x, y) ∈ γ̃ we have d((x, y),γ) ≤ a. let arbitrary set-value mappings ψ,φ, φ̃, ψ̃ : x ⇒ y and numbers α > 0, β ∈ [0, α) be given. set ax(x) := k(m0) ( q1e +(ψ(x), ψ̃(x)) + q2h +(φ̃(x),φ(x)) ) , ay (x) := q1q2h +(φ̃(x),φ(x)) + q2βk(m0) ( q2 1e +(ψ(x), ψ̃(x)) + q2 2h +(φ̃(x),φ(x)) ) , a(x) := (ax(x), ay (x)), x ∈ x. given a set ξ ⊂ x, denote sup ξ∈ξ a(ξ) = {λ ∈ r2 : λ ≥ a(ξ) ∀ ξ ∈ ξ}. here the inequality is understood in the coordinate-wise sense. theorem 3.2: let α > 0 and β ∈ [0, α) be given. if (ψ,φ) ∈ fα,β then h+(γ(ψ̃, φ̃),γ(ψ,φ)) ≤ λ ∀λ ∈ sup ξ̃∈coin(ψ̃,φ̃) a(ξ̃). (3.17) if, in addition, ρx is lower semicontinuous with respect to the second argument, then h+(coin(ψ̃, φ̃),coin(ψ,φ)) ≤ sup ξ̃∈coin(ψ̃,φ̃) ax(ξ̃) (3.18) proof take arbitrary λ ∈ r2 such that λ ≥ a(ξ) for all ξ̃ ∈ coin(ψ̃, φ̃). take arbitrary (ξ̃, η̃) ∈ γ(ψ̃, φ̃). theorem 3.1 implies that d((ξ̃, η̃),γ(ψ,φ)) ≤ λ. hence, (3.17) is proved. let us prove (3.18). assume now that ρx is lower semicontinuous with respect to the second argument. theorem 2.3 implies that for every pair (ξ̃, η̃) ∈ γ(ψ̃, φ̃) and every ε > 0 there exists a point ξ ∈ coin(ψ,φ) such that ρx(ξ̃, ξ) ≤ k(m0)(dist(ψ(ξ̃), η̃) + dist(η̃,φ(ξ̃))) + ε, (3.19) it follows from (3.13) that dist(ψ(ξ̃), η̃) ≤ q1e +(ψ(ξ̃), ψ̃(ξ̃)) + q2e +(ψ̃(ξ̃), η̃). since η̃ ∈ ψ̃(ξ̃), we have dist(ψ(ξ̃), η̃) ≤ q1e +(ψ(ξ̃), ψ̃(ξ̃)). since η̃ ∈ φ̃(ξ̃), we have dist(η̃, φ̃(ξ̃)) = 0. thus, (3.14) implies dist(η̃,φ(ξ̃)) ≤ q2h +(φ̃(ξ̃),φ(ξ̃)). substituting these estimates into (3.19) we obtain ρx(ξ̃, ξ) ≤ ax(ξ̃) + ε. hence, h+(coin(ψ̃, φ̃),coin(ψ,φ)) = = sup ξ̃∈coin(ψ̃,φ̃) dist(ξ̃,coin(ψ,φ)) ≤ sup ξ̃∈coin(ψ̃,φ̃) ax(ξ̃). inequality (3.18) is proved. copyright © 2020 assa. adv syst sci appl (2020) properties of coincidence points in (q1, q2)-quasimetric spaces 101 corollary 3.1: let assumptions of theorem 3.1 hold and ρx be lower semi-continuous with respect to the second argument. then for arbitrary (ξ̃, η̃) ∈ γ(ψ̃, φ̃) and ε > 0 there exists (ξ, η) ∈ γ(ψ,φ) such that ρx(ξ̃, ξ) ≤ k(m0)(q1e +(ψ(ξ̃), ψ̃(ξ̃)) + q2h +(φ̃(ξ̃),φ(ξ̃))) + ε, (3.20) ρy (η̃, η) ≤ q1q2h +(φ̃(ξ̃),φ(ξ̃)) + q2βk(m0)(q2 1e +(ψ(ξ̃), ψ̃(ξ̃)) + q2 2h +(φ̃(ξ̃),φ(ξ̃))) + ε. recall lim’s lemma (see [14]). letx be a complete metric space, β ∈ [0, 1), φ, φ̃ : x ⇒ x be β-contractive set-valued mappings such that φ(x), φ̃(x) are closed for every x. h(fix(φ),fix(φ̃)) ≤ 1 1− β sup x∈x h(φ(x), φ̃(x)). here fix(φ) is the set of fixed points of the mapping φ. let us now derive a generalization of lim’s lemma for coincidence points of mappings between (q1, q2)-quasimetric spaces. corollary 3.2: let α > 0 and β ∈ [0, α) be given. assume that ρx is lower semicontinuous with respect to the second argument. if (ψ,φ), (ψ̃, φ̃) ∈ fα,β then h+(coin(ψ̃, φ̃),coin(ψ,φ)) ≤ k(m0) sup x∈x ( q1e +(ψ(x), ψ̃(x)) + q2h +(φ̃(x),φ(x)) ) (3.21) proof it follows from theorem 3.2 that h+(coin(ψ̃, φ̃),coin(ψ,φ)) ≤ sup ξ̃∈coin(ψ̃,φ̃) ax(ξ̃) ≤ sup x∈x ax(x) = = k(m0) sup x∈x ( q1e +(ψ(x), ψ̃(x)) + q2h +(φ̃(x),φ(x)) ) . hence, (3.21) holds. let us now derive a generalization of lim’s lemma for fixed points of self-mappings of (q1, q2)-quasimetric spaces. corollary 3.3: assume that (x, ρx) is a complete (q1, q2)-quasimetric space and ρx is lower semicontinuous with respect to the second argument. given a number β ∈ [0, 1), assume that mappings φ, φ̃ : x → x are β-contractions and closed. then h+(fix(φ̃),fix(φ)) ≤ q2 q2 1s(q2β, n0 − 1) + q1(q2β)n0−1 1− q2βn0 sup x∈x h+(φ̃(x),φ(x)). here n0 := min{j ∈ n : q2β j < 1}. proof set ψ(x) := {x}, ψ̃(x) := {x}, α := 1. then (ψ,φ), (ψ̃, φ̃) ∈ fα,β, m0 = n0, e+(ψ(x), ψ̃(x)) = 0, and k(m0) = q2 1s(q2β, n0 − 1) + q1(q2β)n0−1 1− q2βn0 . hence, applying corollary 3.2 we obtain the desired inequality. copyright © 2020 assa. adv syst sci appl (2020) 102 r. sengupta, z.t. zhukovskaya, s.e. zhukovskiy corollary 3.4: let (x, ρx) be a complete (q1, q2)-quasimetric space, ρx be lower semicontinuous in the second argument, (σ, ρς) be a (q1, q2)-quasimetric space, φ : x × σ→ x be given. given numbers β ∈ [0, 1) and l ≥ 0, assume that mapping φ(·, σ) is a β-contraction and closed for every σ ∈ σ, φ(x, ·) is l-lipschitz. then the set-valued mapping σ 7→ fix(φ(·, σ)) is lipschitz. proof take arbitrary σ, σ̃ ∈ σ. it follows from corollary 3.3 that h+ ( fix(φ(·, σ̃)),fix(φ(·, σ)) ) ≤ c sup x∈x h+(φ(x, σ̃),φ(x, σ)), where c = q2 q2 1s(q2β, n0 − 1) + q1(q2β)n0−1 1− q2βn0 . hence, h ( fix(φ(·, σ̃)),fix(φ(·, σ)) ) ≤ c sup x∈x h(φ(x, σ̃),φ(x, σ)).moreover, since φ(x, ·) is l-lipschitz, we have sup x∈x h(φ(x, σ̃),φ(x, σ)) ≤ lρς(σ̃, σ). thus, h ( fix(φ(·, σ̃)),fix(φ(·, σ)) ) ≤ lcρς(σ̃, σ), which completes the proof. let us now discuss the question of the stability of coincidence points. given a sequence of pairs of set-valued mappings (ψn,φn),ψn,φn : x ⇒ y,which tend in a certain sense to a pair of set-valued mappings (ψ,φ), ψn,φn : x ⇒ y, and a point ξ ∈ coin(ψ,φ). our goal is to derive conditions for the existence of points ξn ∈ coin(ψn,φn) such that ξn → ξ. corollary 3.5: assume that ρx is lower semicontinuous with respect to the second argument, {(ψn,φn)} ⊂ fα,β for every n, and there exists a point ξ ∈ coin(ψ,φ) such that e+(ψn(ξ),ψ(ξ))→ 0, h+(φ(ξ),φn(ξ))→ 0. then there exists a sequence {ξn}, such that ξn ∈ coin(ψn,φn) ∀n, ξn → ξ as n→∞. proof take an arbitrary point η ∈ ψ(ξ) ∩ φ(ξ). corollary 3.1 implies that for every n there exists a point ξn ∈ coin(ψn,φn) such that ρx(ξ, ξn) ≤ k(m0) ( q1e +(ψn(ξ),ψ(ξ)) + q2h +(φ(ξ),φn(ξ)) ) + 2−n. since e+(ψn(ξ),ψ(ξ))→ 0 and h+(φ(ξ),φn(ξ))→ 0, we have ρx(ξ, ξn)→ 0. therefore, ξn → ξ. acknowledgements the research is supported by the ministry of science and higher education of the russian federation (goszadaniye) and russian foundation for basic research (project no. 20-3170013). the results in section 3 were obtained with the support of the russian science foundation (project no. 17-11-01168). copyright © 2020 assa. adv syst sci appl (2020) properties of coincidence points in (q1, q2)-quasimetric spaces 103 references 1. arutyunov a.v. & greshnov a.v. 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(1985) on fixed-point stability for set-valued contractive mappings with applications to generalized differential equations, journal of mathematical analysis and applications, 110, 436–441. copyright © 2020 assa. adv syst sci appl (2020) introduction and statement of the problem main results. estimates of distance from a point to the intersection set of two graphs corollaries. estimates of distance between intersections of graphs and sets of coincidence points adv syst sci appl 2018; 03; 90-110 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/563 vikor madm based optimization method for vertical handover in heterogeneous networks silki baghla 1, savina bansal 2 1research scholar, ikg punjab technical university, kapurthala, india 2professor, giani zail singh campus college of engineering and technology, bathinda, india abstract: optimum network selection is one of the major issues for vertical handover in heterogeneous networks so as to provide requisite quality of service (qos) to the user. in this context, multiple attribute decision making algorithms provide promising solution. normalization process used can play a vital role for selecting the most appropriate network during handover process. in this work, vector normalized preferred performance based (v-vpp) normalization technique is proposed and applied to multiple attributes decision making (madm) based vlsekriterijuska optimizacija i komoromisno resenje (vikor) algorithm. performance of the proposed technique is analyzed extensively by varying qos parameters and weighting methods for different traffic classes in a heterogeneous network. the results obtained are compared with the available max, max-min and vector normalization. proposed method is also compared with the popular madm algorithms. performance of proposed method is quite optimistic in terms of lesser number of handovers, ranking abnormality, selection of appropriate network though with slight increase in handover latency in comparison to the available normalization methods. keywords: qos, madm, v-vpp, vikor, vertical handover, ranking abnormality 1. introduction in the era of beyond next generation wireless mobile networks, different cellular technologies and wireless standards such as umts, cdma2000, wlan, lte and wimax shall coexist. efforts are being made to integrate all these technologies in the form of heterogeneous network rather than developing a new technology. mobile terminals will be equipped with several interfaces to have ubiquitous and seamless access to these technologies under the principle of “always best connected” [1]. accordingly, different networks need to be integrated in an optimum manner with the ultimate objective to provide end-user with the requisite qos. to coordinate between different types of technologies effectively and seamlessly, with desired quality of service to the user, is a big challenge. to address the same, vertical handover (vho) is being suggested by the researchers. it is a process which decides handing over the mobile node to the most appropriate network among different types of available networks [2]. vertical handover is broadly executed in three phases(i) system discovery (ii) handover decision, and (iii) handover execution. in system discovery phase, a mobile terminal equipped with multiple interfaces has to discover the available networks and services offered by them. the handover decision phase aims at determining the optimal access network among those available. during handover execution, connections with the current network are needed to be rerouted to the selected network in a vikor madm based optimization method 91 copyright ©2018 assa adv. in systems science and appl. (2018) seamless manner. handover decision is the most critical phase, as selection of the optimal network includes dealing with large number of quality of service parameters offered by individual networks like available bandwidth, delay, jitter, cost, velocity and security. traditional decision making algorithms based on received signal strength (rss) cannot deal with the multi-criterion based network selection, as desired in vertical handover. several approaches based on genetic algorithms, fuzzy logic, utility function and multiple attribute decision making (madm) have been suggested in literature to provide seamless connectivity during vertical handover in heterogeneous networks [3-10]. vertical handover needs to select a network with largest coverage area for maintaining an on-going call or data session so as to reduce number of handovers. in addition, it should deliver even if the mobile station is moving at a high velocity, which tends to deteriorate the performance of the algorithm by increasing unnecessary handovers. further, handover latency and optimum network selection are also crucial parameters for the design of an efficient handover technique. wireless networks such as umts, wlan and wimax were typically developed for specific applications. umts networks are basically meant for voice communication and offer lesser amount of delay and jitter within limited bandwidth availability. on the other hand, wlans provides larger bandwidth to the end user thereby making them more suitable for data connections but suffers from higher delay. wimax networks that are developed to provide broadband wireless access to users, offer slightly higher delay than umts but provide highest bandwidth as compared to other networks. going by the nature of user application and requirements, the network selection during vertical handover becomes a multiple attribute decision making problem, for which the available madm techniques can offer a promising solution. the salient technical parameters of three networks are shown in table 3. madm method utilizes normalization process and weighting method at different stages of decision making. normalization methods are used for scaling diverse network attributes given in different units of measure in the range from 0 to 1. weighting methods are used to assign priority weights to various attributes as per the service demanded by user. normalization methods can be broadly classified into i) performance based normalization methods, and ii) distance based normalization methods. in the former method, performance ratings of all the network attributes is taken into account and do not include user’s preference in the normalization process. max, sum and vector methods [12] come under this category. distance based normalization utilizes the distance between the performance value of the network attribute and appropriate reference point. max-min normalization is a distance based normalization method. user’s preferences can be incorporated in distance based normalization process to improve the performance of madm method for optimum network selection. among the available weighing methods, analytic hierarchy process (ahp) and analytic network process (anp) are popular weighting methods used for assigning priority weights to different attributes of a network. the hierarchical structure of ahp weighting method assumes independence among all attributes at the same or different hierarchy level, and it does not consider the interdependence among attributes at various levels. anp has been developed for problems which involves inter-dependence (within same level) as well as outerdependence (between different levels) among attributes. fig.1 shows the hierarchical structure of ahp and anp. owing to the interdependence of various attributes in vertical handover, anp weighting method is more suitable for network selection. 92 s. baghla, s. bansal copyright ©2018 assa adv. in systems science and appl. (2018) fig. 1. structure of ahp and anp [13] based on different normalization techniques, many vho strategies have been suggested such as saw, mew, topsis, gra and electre by employing max, sum, vector and max-min normalizations. performance of these algorithms can be evaluated on the basis of number of handovers, latency, optimum network selection and ranking abnormality. ranking abnormality refers to change in the ranking of selected network if the worst network is removed from the candidate networks. handover latency refers to the time taken by the decision making algorithm to suggest an alternative network from those available for service continuation. network selection is considered as optimum if it also takes into account qos requirements of user along with the received signal strength offered by available networks. conventional madm based handover techniques suffer from large number of handovers and ranking abnormality [13]. in this work, to address these issues, we are proposing a vector normalized preferred performance based normalization (v-vpp) technique that incorporates user’s preference with the purpose of selecting most appropriate network, reducing number of handovers and ranking abnormality. rest of the paper is organized as follows: following section presents the work done by various researchers in this area so as to appreciate and appraise their contributions made in this upcoming field, section 3 provide motivation behind the work with section 4 presents procedural details of conventional vikor method. section5 differentiate the process of proposed v-vpp algorithm with respect to conventional vikor method. section 6 describes the simulation set up used and compares the performance analysis with present. section 7 provides results of simulation and section 8 conclude the work with final remarks. 2. related work multi criteria nature of vertical handover decision and ability of madm methods to deal with large number of conflicting criterion reciprocate each other. traditionally, madm techniques have been used in the field of manufacturing, construction, financial management, supply chain management, human resource management, and health care [14]. however, in recent times these techniques have been explored for optimum network selection in heterogeneous environment as well. many madm methods such as simple additive weighting (saw)[6], multiplicative vikor madm based optimization method 93 copyright ©2018 assa adv. in systems science and appl. (2018) exponent weighting (mew)[6], technique for order preference by similarity to ideal solution (topsis)[6], vikor[10,11], grey relation analysis (gra)[6], elimination and choice translating priority (electre)[7], ahp [8], and anp [4] have been proposed for ranking of networks participating in the vertical handover process. ahp and anp methods are typically used for weight assignments of network attributes. saw employs max normalization method while topsis is based upon vector normalization process. max-min normalization method is employed by vikor and gra algorithms. effect of various normalization methods applied with vikor algorithm was analyzed in [15]. improved performance in terms of number of handovers and ranking abnormality was observed with euclidean distance based vector normalization. weighting methods also play a significant role to improve the performance of decision making algorithms. effectiveness of various weighting methods as analyzed in [16] indicates that anp based weighing method results in reduced ranking abnormality. similar conclusions were drawn in the work available in [17], where vikor method was used as the baseline algorithm with anp method. a comparative analysis of seven madm based algorithms for network selection by martinez et al. in [18], it was indicated that saw, vikor and topsis are more suitable methods for voice application, as these resulted in the selection of umts and wimax networks which offer lower packet delay and jitter. similarly, for data connections as well, the saw, mew, topsis and vikor were able to select the most appropriate network like wlan and wimax, which offer higher bandwidths. however, the work did not consider number of handovers and ranking abnormalities. madm methods provides network selection index which is used to rank the available alternatives for the purpose of decision making. variation in the rank of any alternative with respect to change in the priority weight of an attribute is termed as sensitivity. sensitivity analysis of an algorithm adjudges the most sensitive network attributethe least change in the value of which can change the overall ranking of available alternatives [19]. u. kumaran et al. [20] performed sensitivity analysis of saw, mew and topsis and found these to be quite sensitive to the variation in attribute weights. traditional madm methods are applicable for the situations where attributes weights are definite or can be determined by user’s preferences. to deal with other situations, tao ding et.al [21] proposed cross-evaluation based madm method, which results in weight independent ranking of available attributes. the method is further extended for the situations where magnitude of attributes is not defined. an exponential fuzzy numbers based method was proposed by fu et al. [22] to deal with the situation when both the attribute value and attribute weight are exponential fuzzy numbers. their scheme calculated expected value, variance and altitude of exponential fuzzy numbers and determined the score function in accordance with user’s preference. the effectiveness of the scheme is verified on the basis of distance between exponential fuzzy numbers and positive negative ideal solution. chandavarkar et al. [23] suggested a simplified and improved multiple attributes alternate ranking method (si-maar) with improved results. madm based methods tends to address the issue of unnecessary handovers and ranking abnormality. however, these two parameters are not sufficient to describe the appropriateness of a network for a particular application or as per the service demanded by user. moreover, the impact of varying number of network attributes has also not been considered in the available works. the vertical handover decision should be capable of handling large number of qos parameters and user preferences for optimum network selection. the handover latency also needs consideration for fast switching over. traditional distance based normalization techniques do not include user’s preference into account. in this work, a new preferred performance based normalization technique is proposed to be used with the baseline vikor algorithm by varying normalization techniques for vertical handover in heterogeneous scenario. vikor method is used as it provides compromise ranking list for selection of alternatives in the presence of conflicting 94 s. baghla, s. bansal copyright ©2018 assa adv. in systems science and appl. (2018) criteria, thus making it suitable method for network selection in heterogeneous environment. basically, it is aimed to suggest an algorithm with i) linear ranking of available networks and iii) optimum network selection based on preferred performance 3. conventional vikor method a brief overview of vikor is given in this section as a ready reference for the work. the vikor method is based on distance based normalization process, which does not include user’s preference. the step by step procedure for the conventional vikor method [24] as used for network selection here is given below: i. construction of decision matrix: this step involves construction of a decision matrix, 𝐷 = [𝑎𝑖𝑗] 𝑚𝑥𝑛 , of attribute values offered by the available networks. here 𝑖 = 1,2, . . , 𝑚 represents the candidate networks (alternatives) participating in the decision and 𝑗 = 1,2, . . , 𝑛 represents network attribute. d= [ 𝑎𝑖𝑗 ⋯ 𝑎𝑖𝑛 ⋮ ⋱ ⋮ 𝑎𝑚𝑗 ⋯ 𝑎𝑚𝑛 ] (1) ii. selection of highest and lowest attribute value: the values of various network attributes available in the decision matrix are inspected, among all the alternatives, to get the best and worst values of an attribute for the aggregation process that is next following, as 𝑓𝑗 ∗ = 𝑀𝑎𝑥 𝑎𝑖𝑗 and 𝑓𝑗 − = 𝑀𝑖𝑛 𝑎𝑖𝑗 iii. aggregation: depending upon the type of an attribute, it can be termed as a benefit or cost attribute. for example, since it is desired that the bandwidth of a network should be maximum so it is termed as a benefit attribute, while delay comes under the cost attribute. for benefit attributes, positive and negative ideal solutions indicate the maximum and minimum value of that attribute among the alternative networks, and vice-versa for the cost attributes. depending upon the type of traffic class or application, every attribute of a network is assigned a weight (0 < 𝑤𝑗 < 1) indicative of its priority for that application. the weights of the attributes can be calculated by any of the available weighting schemes such as ahp or anp, as explained above. aggregation process provides a ranking index for every network(𝑆𝑖). it is aggregation of all attributes and their relative importance. basically, it is sum of weighted normalizations (maxmin) of all attributes in a particular network. 𝑆𝑖 = ∑ 𝑤𝑗(𝑓𝑗 ∗𝑛 𝑗 − 𝑎𝑖𝑗)/(𝑓𝑗 ∗ − 𝑓𝑗 −), (2) further, the maximum value of the weighted normalization of all attributes in a network is taken as another boundary measure to find distance rate from non acceptable solution and is termed as 𝑅𝑖. 𝑅𝑖 = 𝑀𝑎𝑥𝑗 [𝑤𝑗 (𝑓𝑗 ∗ − 𝑎𝑖𝑗)/ (𝑓𝑗 ∗ − 𝑓𝑗 −)], (3) thus, 𝑆𝑖 represents a measure of group utility where as 𝑅𝑖 represents individual regret for an alternative. iv. determination of network selection index 𝑄𝑖 : for each alternative, network selection index is calculated as 𝑄𝑖 = 𝑣(𝑆𝑖 − 𝑆∗)/ ( 𝑆− − 𝑆∗) + (1 − 𝑣)(𝑅𝑖 − 𝑅∗)/(𝑅− − 𝑅∗) (4) vikor madm based optimization method 95 copyright ©2018 assa adv. in systems science and appl. (2018) where 𝑆∗ = 𝑀𝑖𝑛 𝑆𝑖 , 𝑆− = 𝑀𝑎𝑥 𝑆𝑖, 𝑅∗ = 𝑀𝑖𝑛 𝑅𝑖 , 𝑅− = 𝑀𝑎𝑥 𝑅𝑖 . the solution obtained by 𝑀𝑖𝑛 (𝑆𝑖) represents measure of maximum group utility and 𝑀𝑖𝑛 (𝑅𝑖) is minimum individual regret among all alternatives. the parameter ʋ is the weighting reference with 0 ≤ 𝑣 ≤ 1, and represents the weight of the strategy of criteria used; majority rule with 𝑣 > 0.5, as consensus with 𝑣 ≈ 0.5 and as vito with 𝑣 < 0.5. in this work, 𝑣 = 0.5 has been used. v. ranking of the alternatives: sort the available networks in decreasing order of their values in three lists generated as 𝑄𝑖, 𝑆𝑖 𝑎𝑛𝑑 𝑅𝑖. it will provide three different sets of ranking list. the selected network can be either one or set of networks. the selected network is obtained by 𝑄 = 𝑀𝑖𝑛 (𝑄𝑖) if it satisfies following two conditions: a) if 𝑄𝑖 ′′ − 𝑄𝑖 ′ ≥ 𝐷𝑄 with 𝐷𝑄 = 1 (|𝑀| − 1)⁄ here 𝑖′′ 𝑎𝑛𝑑 𝑖′ are alternatives with second smallest and smallest value of 𝑄. this condition is termed as condition of acceptable advantage. b) if the selected alternative is also the best alternative for 𝑆𝑖 𝑎𝑛𝑑 𝑅𝑖. the condition is termed as acceptable stability in decision making. if only one of these conditions is satisfied, vikor method suggests set of compromising solutions given by:  if condition (a) is not satisfied, the solution will contains all the alternatives 𝑖′, 𝑖′′, … , 𝑖|𝑀| as long as 𝑄𝑖|𝑀| − 𝑄𝑖′ < 𝐷𝑄  if condition (b) is not fulfilled, then the set is compromise of 𝑄𝑖′ 𝑎𝑛𝑑 𝑄𝑖′′. vi. the selected network 𝐴𝑉𝐼𝐾 ∗ is 𝐴𝑉𝐼𝐾 ∗ = arg 𝑀𝑖𝑛 𝑄𝑖 (5) 4. proposed vector normalized preferred performance based normalization (v-vpp) method the traditional vikor method suffers from the limitation that at any stage of implementation if the two attributes become equal, the method will lead to undefined value of parameters at different steps and the procedure will be stopped at that point. this is due to the fact that while calculating different measures for alternatives, denominator term is using a subtraction term (eq 3). chia-ling chang [25] worked on this drawback of vikor method and suggested modifications which can be applied in that particular step where undefined value appears in calculation. but, these modifications do not avoid the undefined terms prior to their occurrence. in this work, the main focus is to modify the process of normalization in aggregation and ranking of network alternatives such as to avoid the subtraction term in the denominator. moreover, in conventional method, user’s preference was not included during normalization process. the proposed method also incorporates user’s preferences in the normalization phase as an additional advantage. the step by step procedure of the proposed v-vpp method is shown in fig. 1. all the available attributes are normalized in the range of 0 to 1 prior to application to step 3 in conventional vikor method. the normalized value of a network attribute in conventional method is replaced by proposed method as step 1: get user’s preference 𝑎𝑝𝑗 for all the attributes considered for decision making: these values can be greater than, equal to or less than the different attribute values of available alternatives. step 2: perform preferred performance based vector normalization: for each alternative, determine preferred performance based vector normalized values for all the attributes as given by 𝛾𝑖 = |𝑎𝑝𝑗 − 𝑎𝑖𝑗| √∑ 𝑎𝑖𝑗2𝑚 𝑖=1⁄ (6) 96 s. baghla, s. bansal copyright ©2018 assa adv. in systems science and appl. (2018) step 3: determine modified aggregation function 𝑆𝑣𝑖 : the aggregation function given in step 3 of conventional vikor method is modified as 𝑆𝑣𝑖 = ∑ 𝑤𝑗 ∗ 𝛾𝑖 𝑛 𝑗 (7) fig.2: block diagram of proposed v-vpp method where 𝑤𝑗 represents priority weight of individual attribute assigned by user. this aggregating function helps to determine acceptable solution among available alternatives for decision making. the weighted normalized values of all the attributes are given by 𝑆𝑣𝑡 = 𝑤𝑗 ∗ 𝛾𝑖 (8) step 4: determine maximum regret attribute value for each alternative: it is the maximum value of attributes of individual alternative after weighted normalization. distance based normalization of vikor method results in minimization of desired attributes and the attribute having maximum value after normalization is referred to as regret attribute for that alternative. these values as determined from weighted normalized values of all the attributes are given by 𝑅𝑣𝑖 = 𝑀𝑎𝑥𝑗 [𝑤𝑗 ∗ 𝛾𝑖], (9) step 5: find boundary values for acceptable and non acceptable solution: boundary values for acceptable and non acceptable solution are determined from 𝑆𝑣𝑖 𝑎𝑛𝑑 𝑅𝑣𝑖 and given by 𝑆∗ = 𝑀𝑖𝑛 𝑆𝑣𝑖 , 𝑆− = 𝑀𝑎𝑥 𝑆𝑣𝑖 , 𝑎𝑛𝑑 𝑅∗ = 𝑀𝑖𝑛 𝑅𝑣𝑖 , 𝑅− = 𝑀𝑎𝑥 𝑅𝑣𝑖 step 6: assign priority weights to sets of acceptable and non acceptable solutions and aggregate: the aggregating function or network selection index is similar to the conventional vikor method and is given by get the value of utility measure compute the value of qi find value of regret measure sorting the values of qi in descending order assign priority weights to attributes in accordance with traffic class as wj get user’s preference for network attribute values as apj normalization get attributes from available alternatives get absolute difference between preferred and available attribute values|𝑎𝑝𝑗 − 𝑎𝑖𝑗| get sum of squares of attributes to avoid null values in denominator √∑ 𝑎𝑖𝑗2𝑚 𝑖=1 ranking of alternatives vikor madm based optimization method 97 copyright ©2018 assa adv. in systems science and appl. (2018) 𝑄𝑣𝑖 = 𝑣(𝑆𝑣𝑖 − 𝑆∗)/ ( 𝑆− − 𝑆∗) + (1 − 𝑣)(𝑅𝑣𝑖 − 𝑅∗)/(𝑅− − 𝑅∗) (10) here 𝑣 is strategy weight and is decided by user. it represents the user’s approach to reach to the desired solution and can be determined either by consensus (𝑣 ≈ 0.5)or majority 𝑣 > 0.5 or veto (𝑣 < 0.5). if equal consideration is given to acceptable and non acceptable solutions then the approach is termed as by consensus. the final solution is obtained in a similar manner as that of conventional vikor method. fig.2 presents the various steps involved in proposed v-vpp method. the method is further explained with the help of a case example by considering four attributes available from three wireless networks umts, wlan and wimax. table 1 provides attribute values of delay, jitter, bandwidth and cost available from these three networks as constituents of decision matrix. values of these attributes are taken from [18]. 𝑤𝑗 represents priority weights assigned to different attributes in accordance with user’s preference for conversational traffic [26], as considered in this example. for conversational traffic, delay and jitter are of prime importance so umts will be the appropriate network due to lesser values of these attributes. table 1: constituents of decision matrix containing attribute values of different alternative alternatives/ attributes delay(ms) jitter(ms) bandwidth(mhz) cost (%) umts 30 12 1 0.6 wlan 80 15 11 0.1 wimax 50 10 54 0.5 priority weights for different attributes in conversational traffic class are taken as 𝑤𝑗 = [0.3162 0.3456 0.1265 0.2107] preferred values of different attributes as provided by the user are: 𝑎𝑝𝑗 = [40 11 3 0.3] preferred performance based vector normalized matrix is given by: 𝛾𝑗 ∗ = 𝑈𝑀𝑇𝑆 𝑊𝐿𝐴𝑁 𝑊𝐼𝑀𝐴𝑋 [ 𝐷𝑒𝑙𝑎𝑦 𝐽𝑖𝑡𝑡𝑒𝑟 𝐵. 𝑊 𝐶𝑜𝑠𝑡 0.1010 0.0462 0.0363 0.3810 0.4041 0.1847 0.1451 0.2540 0.1010 0.0462 0.9253 0.2540 ] weighted normalized matrix of all attributes is calculated as: 𝑆𝑣𝑡𝑖 = [ 0.0319 0.0160 0.0046 0.0803 0.1278 0.0638 0.0184 0.0535 0.0319 0.0160 0.1170 0.0535 ] aggregation matrix 𝑆𝑣𝑖 = [ 0.1328 0.2635 0.2185 ] regret attribute matrix of 3 alternatives is determined as 𝑅𝑣𝑖 = 𝑀𝑎𝑥 𝑆𝑣𝑡𝑖 = [ 0.0803 0.1278 0.1170 ] network selection index for each alternative is given by 𝑄𝑖 = 𝑈𝑀𝑇𝑆 𝑊𝐿𝐴𝑁 𝑊𝐼𝑀𝐴𝑋 [ 0 1 0.7149 ] 98 s. baghla, s. bansal copyright ©2018 assa adv. in systems science and appl. (2018) it shows that for given set of preferred values umts is the most suitable network among the available alternatives. table 2 provides ranking lists of the proposed v-vpp method along with popular topsis method for comparison. as umts is selected by three ranking lists of v-vpp method, it is compromised solution among available alternatives. moreover umts is ideal network for conversational traffic thus network selection is optimum in this regard also. topsis method selects wimax as suitable network followed by umts and wlan as indicated in table 2. as this method does not include user’s preference for attribute values, network selection is solely dependent upon priority weights assigned by user. table 2: ranking lists generated by v-vpp and topsis alternatives ranking list v-vpp topsis si ri qi c umts 0.1328 0.0803 0 0.4759 wlan 0.2635 0.1278 1 0.3996 wimax 0.2185 0.1170 0.7149 0.5857 inclusion of user’s preference in v-vpp during normalization results in different ranking list of available alternatives. thus to justify the effectiveness of proposed method, euclidean distance between and preferred set of attribute values and alternative selected by two methods is calculated and is given by 𝐸𝑈𝑉−𝑉𝑃𝑃 = 10.251 𝑎𝑛𝑑 𝐸𝑈𝑇𝑂𝑃𝑆𝐼𝑆 = 51.98 it can be concluded that network selection by v-vpp method is closed to user’s preference for the given attributes. moreover, difference between selection indexes of top ranked networks in vvpp is greater than topsis (0.7149 in case of v-vpp and 0.1098 in case of topsis). so, it will be easier to select the suitable alternative among available ones with the help of proposed method. from the case study presented above, it is observed that proposed v-vpp method offers following advantages over conventional topsis and vikor methods:  avoids undefined values at any step occurred by ‘0’ in difference term in denominator  includes user’s preference at early stage of decision making i.e. normalization  maintains adequate gap between two top ranked alternatives/ networks so that clear decision can be taken by user.  benefit attributes can be given majority in the decision by varying the value of strategy weight 𝑣. provision of compromised ranking if two alternatives have equal values for selection index. 5. simulation set-up and performance parameters for performance analysis, three networks wlan, wimax and umts are considered here to simulate heterogeneous environment with different attributes such as cost, delay, jitter, packet loss, security, throughput, velocity, rss, and traffic load. the values of these attributes vary randomly in the range given in table 3. the various traffic classes considered are conversational, streaming, interactive and background. ahp and anp methods have been used for the assignment of priority weights to the attributes. further, 600 vertical handover points for all simulation scenarios are considered to evaluate the performance of proposed method. these simulation scenarios have been implemented with matlab platform. vikor madm based optimization method 99 copyright ©2018 assa adv. in systems science and appl. (2018) table 3: attributes range for different networks [18, 27] various attributes considered are as follows:  received signal strength (rss): this attribute represents the signal strength received from the available networks. signal strength received from various networks considered in this work lies in the range from -160 to -55 db.  delay: it is measure of approximate delay provided by the access network. as given in table 3, wlan offers higher delay as compared to others.  jitter: jitter is a measure of the average delay variations within the access network.  packet loss: this attribute is a measure of the average packet loss rate within the access network over a considerable duration of time. it can be measured in (number of packets lost per million packets).  mobility: this attribute is considered to incorporate mobility characteristics of available networks. mobility refers to service continuation of mobile terminal with present point of attachment even at high velocity. umts network offers highest mobility to mobile terminals due to larger area covered by concerned base station. small coverage area of access point leads to low mobility characteristics of wlan. wimax offers medium mobility to mobile terminals.  bandwidth: it provides the specified bandwidth offered by available access networks and is expressed in mhz wimax offers higher bandwidth as compared with other wlan and umts.  throughput: it refers to actual rate of packet transmission offered by available networks and is expressed in mbps.  cost: this attribute is a measure of the operator’s transport cost for a particular access network. it is expressed as percentage cost per byte. in this work, cost of accessing umts is considered as 60% per byte as compared with other two networks.  security: security attribute represents security level of link layer of available network and expressed as percentage security.  load handling capability: it represents load handling capability of concerned network. performance of wlan deteriorates with increase in traffic load in the network where as load handling capacity of umts is greater among networks considered in this work. the performance evaluation scenarios are summarized in table 4. table 4: simulation scenarios criteria network rss (db) delay (ms) jitter (ms) packet loss (per 10^6) throughput (mbps) mobility bandwidth (mhz) load handling capability security (%) cost (%) umts -150 to -90 25-50 5-10 20-80 0.1-3 high 0.1-3 high 70 60 wlan -110 to -55 100-150 10-20 20-80 50-150 low 1-11 low 50 10 wimax -160 to -100 60-100 3-10 20-80 20-100 medium 1-60 medium 60 40 traffic classes  conversational  streaming  interactive  background no. of attributes  4 (cost, delay, jitter and packet loss 100 s. baghla, s. bansal copyright ©2018 assa adv. in systems science and appl. (2018) metrics considered for performance evaluation and comparison are:  number of handovers: it indicates number of times an algorithm changes the present point of attachment in heterogeneous environment. lesser number of handovers is desirable as excessive handovers result in undue network overheads.  ranking abnormality: it is the change in the ranking of available alternatives if worst alternative is removed from the list. conventional madm methods suffer from increased ranking abnormality reflecting inconsistent algorithm performance.  optimum network selection: network selection is considered to be optimum if it satisfies the quality of service requirements of user.  handover latency: it is the time taken by an algorithm to decide next point of attachment or service continuation. handover decision must be taken prior to decrease in relative signal strength received from new point of attachment. 6. performance analysis and comparison the proposed v-vpp algorithm as outlined in section 5 was implemented using matlab platform along with three other relevant algorithms based on different normalizations on vikor namely v-max (max normalization based vikor method), v-vec (vector normalization based vikor method) and v-prn (performance rating normalization based vikor method). the performance of these algorithms is analyzed over 600 vertical handover points in all scenarios. each handover point represents combination of all network attributes offering random value in the range given in table 3. further, comparative performance analysis of v-vpp with four conventional madm methods viz. saw, mew, topsis and gra is also presented, for additional 100 handover points, to strengthen the analysis and adjudge its effectiveness for network selection in heterogeneous environment. 6.1. number of handovers: the performance comparison of proposed v-vpp algorithm in terms of handovers with v-max, v-vec and v-prn is shown in table 5. these results are tabulated over the two (ahp and anp) weighting methods, for four traffic classes, and with 4, 6, and 9 number of network attributes, which were varied randomly in the range given in table 3. fig. 3 shows average number of handovers taken over all the four traffic classes for the four vertical handover techniques.  anp weighting method: it is observed that in conversational traffic no handover occurred in v-vpp, v-prn and v-vec techniques when lesser number of network attributes are considered, whereas, with increasing network attributes (equal to 9) v-max offered better results followed by v-vpp, v-prn and v-vec techniques. in streaming traffic class as well, the proposed v-vpp technique and v-prn gave optimized performance (zero handover) at lower attributes, while v-vec did not perform well and resulted in highest number of handovers. the performance of proposed technique is again better (no handover with 4 and 6 attributes) as compared with other techniques in background traffic class. for interactive traffic, v-vec resulted in minimum number of handovers. however, on an average, over all the traffic classes, the proposed v-vpp technique generated around 40%, 65%, and 48% lesser number of handovers in  6 (cost, delay, jitter, bandwidth, security and packet loss)  9 (rss, cost, delay, jitter, throughput, velocity, security, packet loss and traffic load) weighting methods  analytic hierarchy process (ahp)  analytic network process (anp) vikor madm based optimization method 101 copyright ©2018 assa adv. in systems science and appl. (2018) comparison to v-max, v-vec, and v-prn algorithms respectively for the anp weighing method (fig. 3). table 5: number of handovers in different traffic classes with varying attributes and weighting method traffic class attributes weighing method algorithm v-max v-vec v-prn v-vpp 4 anp 11 0 0 0 conversational ahp 9 0 0 0 6 anp 2 0 0 0 ahp 6 0 0 0 9 anp 9 17 15 13 ahp 7 12 15 12 streaming 4 anp 2 10 0 0 ahp 4 10 0 0 6 anp 2 6 0 0 ahp 2 6 0 0 9 anp 2 17 0 0 ahp 6 6 2 6 interactive 4 anp 0 0 10 8 ahp 8 0 12 2 6 anp 10 0 12 0 ahp 12 0 8 2 9 anp 4 12 0 4 ahp 4 13 2 12 background 4 anp 4 8 16 0 ahp 4 6 2 0 6 anp 0 4 1 0 ahp 2 4 1 0 9 anp 2 10 2 4 ahp 4 12 2 12  ahp weighting method: similarly the results obtained for ahp based weighting method are tabulated in table 5. it is seen that the proposed v-vpp vertical handover technique offered least number of handovers in all the traffic classes, for 4 and 6 number of network attributes, except in the interactive traffic class where v-vec performed marginally better than v-vpp. with 9 attributes, v-max performed better for all the traffic classes except in streaming traffic where v-vpp and v-prn resulted in optimized performance. on an average over all the traffic classes, the v-vpp generated 32%, 33%, and -5% lesser handovers than v-max, v-vec, and v-prn respectively 102 s. baghla, s. bansal copyright ©2018 assa adv. in systems science and appl. (2018) fig.3: average number of handovers over all the traffic classes 6.2. optimum network selection optimum network refers to the one that is best suited as per qos requirements of user. conversational traffic demands lesser delay and jitter, which can be honored by umts or wimax networks. in case of streaming traffic, jitter and throughput are given high priority among other network attributes. thus wlan and wimax networks will be appropriate networks for streaming traffic class. similarly, interactive traffic can be efficiently supported by lower delay and loss parameters. since umts and wimax networks offer lesser amount of delay and thus are able to support this kind of traffic. higher throughput and lesser delay are the necessities of background traffic, which can be provided by wlan or wimax networks. in this work, a network selection is considered optimum if at any point in heterogeneous environment, attribute values offered by the available network matches with the requirement of particular traffic class. table 6 provides number of times an optimum network is selected by various handover algorithms in all scenarios for each traffic class (2 weighting methods x3 network attribute classes x 25 handover points per scenario). fig. 4 shows the percentage optimum network selection for different algorithms for the four traffic classes. table 6: optimum network selection comparison for all attribute and weighing classes traffic class handover algorithm v-max v-vec v-prn v-vpp conversational 33 143 138 148 streaming 147 118 150 147 interactive 115 133 118 125 background 145 112 144 144 average (%) 73.3 84.3 91.7 94 it is seen that proposed v-vpp technique provides largest percentage of optimum network selection among all the vertical handover techniques. 17 17,25 11 11,5 12 21 14 7,25 0 5 10 15 20 25 v-vmax v-vec v-prn v-vpp # avg. handovers anp ahp vikor madm based optimization method 103 copyright ©2018 assa adv. in systems science and appl. (2018) fig.4: optimum network selection (%) comparison for different traffic classes 6.3. ranking abnormality how much consistent an algorithm is in its performance it is indicated by the ranking abnormality. higher ranking abnormality may result in inappropriate network selection by that algorithm. the performance of all the handover techniques with respect to ranking abnormality is shown in table 7. it is seen that the proposed v-vpp technique results in minimum ranking abnormality among all, and the goodness over all the four traffic classes is v-vpp>v-prn>vvec>v-max. however, it is seen that v-max and v-prn methods resulted in undefined values (16nan, 36nan) while making calculations for handover, indicating limitations of these methods. v-vec and v-vpp methods offer advantage in this regard, however. table 7: ranking abnormality comparison under different traffic classes traffic class v-max v-vec v-prn v-vpp conversational 0 1 16 5 streaming 56 13 11 1 interactive 35* 25 7* 10 background 40 24 12*’ 1 total 131 63 46 17 *16nan *15nan, *’36nan average ranking abnormality offered by all handover algorithms over the 600 scenarios ( 4 traffic classes x 25 handover points x 2 weighing methods x 3 sets of attributes ( 4, 6, and 9 number)) is shown in fig. 5. the superiority of v-vpp method in terms of considerably reduced ranking abnormality justifies its goodness. 0 10 20 30 40 50 60 70 80 90 100 v-max v-vec v-prn v-vpp o p ti m u m n et w o rk s el ec ti o n ( % ) handover algorithm conversational streaming interactive background 104 s. baghla, s. bansal copyright ©2018 assa adv. in systems science and appl. (2018) fig. 5: average ranking abnormality comparison over 600 scenarios 6.4. handover latency it is indicative of the time taken by the decision making algorithm for network selection. the results for handover latency in all the normalization techniques are shown in fig.6. as the number of network attributes increases, to accommodate the qos desired by a user, handover latency increases among all the algorithms. however, for the v-vpp algorithm, it is slightly higher than that of v-max and v-prn but much lower than that for v-vec. for 9 attributes, handover latency offered by v-vpp is 0.827 and 0.582 times more than that of v-max and v-prn due to two step normalization process. fig. 6: handover latency comparison for different number of network attributes 6.5. performance comparison with conventional madm methods the performance comparison of v-vpp algorithm with traditional multiple attribute decision making algorithms saw, mew, gra and topsis is carried out here on the basis of two major performance metrics (i) number of handovers and (ii) optimum network selection. 0% 5% 10% 15% 20% 25% v-max v-vec v-prn v-vpp a v er a g e ra n k in g a b n o rm a li ty handover algorithm 0 0,002 0,004 0,006 0,008 0,01 v-max v-vec v-prn v-vpp h a n d o v er la te n cy (s ec ) h a n d o v er a lg o ri th m 4 6 9 vikor madm based optimization method 105 copyright ©2018 assa adv. in systems science and appl. (2018) 6.5.1. number of handovers performance comparison of v-vpp with the well explored topsis algorithm is shown in table 8 for 9 network attributes with all the four traffic classes at 100 handover points in terms of number of handovers. the proposed method results in lesser number of handovers for all the traffic classes and with both weighing methods. results are seen to be much better with anp technique as compared to those with ahp weighting method for all the traffic classes. accordingly, the comparison is further extended with saw, mew, and gra, other popular madm methods, with anp weighting method and results are presented in fig.7. table 8: performance comparison of v-vpp with topsis method number of handovers (%) weighting algorithm ahp anp traffic class handover algorithm v-vpp topsis v-vpp topsis conversational 20 29 4 21 streaming 16 38 0 34 interactive 26 39 8 29 background 19 51 12 30 average 20.25 39.25 6 28.5 the goodness of the proposed v-vpp technique gets clearly demonstrated as it results in lesser number of handovers for network selection among all these algorithms at almost all traffic classes. average number of handovers over all traffic classes shown in fig. 8, also substantiates the observation as the proposed vertical handover technique results in about 78%, 89%, 73%, and 87% lesser number of handovers in comparison to topsis, gra, mew, and saw respectively over all the traffic classes. fig. 7: performance comparison of proposed v-vpp method with conventional madm methods 0 10 20 30 40 50 60 saw mew gra topsis v-vpp # h a n d o v er s handover algorithm conversational streaming interactive background 106 s. baghla, s. bansal copyright ©2018 assa adv. in systems science and appl. (2018) fig 8: performance comparison with traditional madm algorithms for number of handovers 6.5.2. optimum network selection in section 6.2, quality of service requirements of different traffic classes were discussed for the purpose of optimum network selection. as outlined in section 4, euclidean distance is calculated for critical evaluation of proposed method and to justify its effectiveness for network selection. user’s preferences can never be exactly matched with quality of service offered by available networks thus a compromise solution is provided by handover algorithm. euclidean distance will help to determine the effectiveness of handover algorithm for selecting most appropriate network for given set of user’s preferences. it is measured between attributes of selected network by any algorithm and predefined set of preferred attributes. a network selection will be considered to be optimum if euclidean distance between attributes of selected network and predefined set of preferred attributes is minimum. performance comparison of v-vpp with topsis assuming anp weighting method with 9 attributes for the conversational traffic class is shown in table 9 for the optimum network selection parameter. in total 100 handover points are considered for a given set of preferred values of attributes that vary randomly in the range as given in table 3. table 9: optimum network selection comparison selection of preferred values of attributes method # handovers average euclidean distance optimum network selection (%) user defined v-vpp 20 120 46 topsis 36 147.86 17 best values among available attributes v-vpp 10 80.4 57 topsis 36 98.4 27 the results are analyzed for i) user defined set of attribute values and ii) considering best values of network attributes from available networks. in user defined approach, set of preferred network attributes remains same for 100 handover points, whereas a new set of preferred values of network attributes is chosen for each handover point for the second option at all the 100 47 22,25 53,75 28,5 6 0 10 20 30 40 50 60 saw mew gra topsis v-vpp # avg. handovers h a n d o v er a lg o ri th m vikor madm based optimization method 107 copyright ©2018 assa adv. in systems science and appl. (2018) handover points. thus, number of handovers are determined for 200 vertical handover points (100 in each approach) to compare the performance of both methods. fig. 9: network selection in heterogeneous environment using v-vpp and topsis as outlined in section 6.2, umts and wimax are the most suitable network selections for conversational traffic. both v-vpp and topsis methods change their point of attachments between umts and wimax thereby justifying the requirement of conversational traffic class. in v-vpp, same network (umts or wimax) is selected most of the time so point of attachment remains same. on the other hand, with topsis network selection switched between umts and wimax frequently (table 9 and fig. 9). as, the number of handovers increases load on the network thus lesser number of handovers provided by v-vpp method is advantageous. moreover, to extend the level of performance evaluation of v-vpp and topsis methods, euclidean distance is calculated between selected network and preferred set of attributes for every handover point. average euclidean distances measured at 100 handover points for network selected by both methods is also shown in table 9 and it can be seen that v-vpp method provides lesser euclidean distance as compared with topsis. in addition, percentage optimum network selection indicates number of times the euclidean distance between selected network and user’s preferred set of attributes is minimum as compared with other networks. it can be concluded from table 9 that performance of proposed method is much better than topsis method in percentage optimum network selection as well. 7. conclusion madm based vertical handover techniques are being explored by researchers for appropriate network selection for the imminent heterogeneous wireless and mobile network environment. however, the effectiveness of these algorithms is highly dependent upon the underlying weighting and normalization techniques adopted. in this paper, we have proposed a vector normalized preferred performance rating based normalization technique to be used in conjunction with the madm based vikor method for developing a novel vertical handover algorithm named v-vpp. it incorporate user’s preference in the normalization process during the aggregation and ranking 108 s. baghla, s. bansal copyright ©2018 assa adv. in systems science and appl. (2018) of alternatives with the purpose of selecting most appropriate network, reducing number of handovers and minimizing ranking abnormality. performance of the proposed algorithm in comparison to vikor based other algorithms using different normalization techniques (v-max, v-vec and v-prn) distinctively shows an edge over others with widely varying attributes range for all the traffic classes on a heterogeneous network comprising of wi-max, umts and wlan. the simulation results for the v-vpp vertical handover are quite optimistic in terms of lower handovers, reduced ranking abnormality, and optimum network selection though at the cost of slight increase in handover latency. the proposed technique performs well as compared with conventional madm methods like saw, mew, gra and topsis under the given simulation scenarios. based on these results, v-vpp seems to be a potential vertical handover technique for usage in the upcoming beyond 3g or fourth generation of wireless mobile communication networks. references 1. gustafsson e and jonsson a. always best connected. ieee wireless communications magazine, 2003; 10(1): 49-55. 2. mcnair j and zhu f. vertical handovers in fourth-generation multi network environments. ieee wireless communications 2004; 11: 8-15. 3. zhang w. handover decision using fuzzy madm in heterogeneous networks. ieee conference on wireless communications and networking (wcnc’04), atlanta, usa ; 653-658. 4. saaty tl: theory and applications of the analytic network process. rws publications: 2005. 5. ormond o, murphy j and muntean gm. utility-based intelligent network selection in beyond 3g systems. in ieee international conference on communications, jun. 2006; 4: 1831-1836. 6. stevens-navarro e and wong vws. comparison between vertical handover decision algorithms for heterogeneous wireless networks. in 63rd ieee conference on vehicular technology (vtc’06-spring), melbourne, australia 2006; 947-951. 7. bari f and leung v. application of electre to network selection in a heterogeneous wireless network environment. in ieee conference on wireless communication and networking (wcnc’07), hong kong, china 2007; 3810-3815. 8. saaty t l. decision making with analytic hierarchy process. international journal of services sciences 2008; 1(1):83-98. 9. nkansah-gyekye, agbinya y, i j. a vertical handover decision algorithm for next generation wireless networks. in 3rd international conference on broadband communications, information technology and biomedical applications, nov. 2008; 358-364. 10. gallardo-medina j r, pineda-rico u and stevens-navarro e. vikor method for vertical handover decision in beyond 3g wireless networks. in 6th ieee international conference on electrical engineering, computing science and automatic control (cce’09), toluca, mexico 2009; 1-5. 11. opricovic s. multi-criteria optimization of civil engineering systems. faculty of civil engineering, belgrade, 1998. vikor madm based optimization method 109 copyright ©2018 assa adv. in systems science and appl. (2018) 12. stanujkic dragisa, magdalinovic nedeljiko and jovanovic rodoljub. a multiattribute decision making model based on distance from decision maker’s preferences. informatica 2013; 24(1): 103-118. 13. silki, bansal savina, lahby mohamed. a modified vikor method for network selection in heterogeneous network. international journal of mobile & adhoc network (ijman), feb. 2014; 4(1): 85-90. 14. mardani abbas et al.vikor technique: a systematic review of the state of the art literature on methodologies and applications. sustainability, 2016; 8, 37:1-38. 15. baghla silki, bansal savina. effect of normalization techniques in vikor method for network selection in heterogeneous networks. in ieee international conference on computational intelligence and computing research (iccic-2014), dec.2014, tamilnadu: india; 335-340. 16. lahby m, leghris c and adib a. a survey and comparison study on weighting algorithms for access network selection. in 9th annual conference on wireless on demand network systems and services (wons), italy, jan. 2012; 35-38. 17. silki, bansal savina. performance of vikor madm method for vertical handoffs in heterogeneous networks with various weighting methods. in 8th international conference on advanced computing & communication technologies (icacct™2014), nov. 2014: 29-34. 18. martinez-morales j. d., rico v p and steven e. performance comparison between madm algorithms for vertical handoff in 4g networks. in 7th international conference on electrical engineering computing science and automatic control (cce), 2010, pp. 309-314. 19. alinezhad alireza , amini abbas. sensitivity analysis of topsis technique: the results of change in the weight of one attribute on the final ranking of alternatives. journal of optimization in industrial engineering, 2011; pp 23-28. 20. kumaran u. and jeyakumar m. k.. comparative analysis of multi attribute decision making in vehicular ad-hoc networks, i j c t a , international science press, vol.9, issue 34, 2016; pp. 283-294. 21. ding tao, liang liang, yang min, and wu huaqing. multiple attribute decision making based on cross-evaluation with uncertain decision parameters. mathematical problems in engineering , hindawi publishing corporation, vol. 2016, 2016; pp.1-11. 22. fu sha and fan guo-bing. a multiple attribute decision-making method based on exponential fuzzy numbers. mathematical and computational applications, vol. 21,issue 19, may 2016; pp.2-9. 23. chandavarkar b.r., reddyguddeti ram mohana. simplified and improved multiple attributes alternate ranking method for vertical handover decision in heterogeneous wireless networks. computer communications, vol.83, issue c, june 2016; pp.81-97. 24. opricovic s., tzeng g.h. the compromise solution by mcdm methods: a comparative analysis of vikor and topsis (2004).european journal of operational research 156 (2), pp.445–455. 110 s. baghla, s. bansal copyright ©2018 assa adv. in systems science and appl. (2018) 25. chang chia-ling. a modified vikor method for multiple criteria analysis. environmental monitor assess (2010) 168;pp.339–344. 26. universal mobile telecommunications system (umts): quality of service (qos) concept and architecture. 3gpp ts 23.107 version 5.10.0 release 5. 27. kaleem faisal. vhits: vertical handoff initiation and target selection in a heterogeneous wireless network. florida international university 2012:pp.1-256. adv syst sci appl 2019; 04; 25-44 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/659 improving semi-supervised clustering algorithms with active query selection walid atwa*1, mahmoud emam2 1) computer science department, faculty of computers and information, menoufia university, 32511, egypt e-mail: walid.atwa@ci.menofia.edu.eg 2) mathematics and computer science department, faculty of science, menoufia university, 32511, egypt e-mail: ma7moud_emam@yahoo.com received october 30, 2018; revised april 3, 2019; published december 31, 2019 abstract: semi-supervised clustering algorithms use a small amount of supervised data in the form of pairwise constraints to improve the clustering performance. however, most current algorithms are passive in the sense that the pairwise constraints are provided beforehand and selected randomly. this may lead to the use of constraints that are redundant, unnecessary, or even harmful to the clustering results. in this paper, addressing the problem of constraint selection to improve the performance of semisupervised clustering algorithms. based on the concepts of maximum mean discrepancy, proposed method selects a batch of most informative instances that minimize the difference in distribution between the labeled and unlabeled data. then, querying these instances with the existing neighborhoods to determine which neighborhood they belong. the experimental results with state-of-the-art methods on different real-world dataset demonstrate the effectiveness and efficiency of the proposed method. keywords: semi-supervised clustering, active learning, pairwise constraints. 1. introduction recently, semi-supervised clustering (also known as constraint-based clustering) algorithms have become a topic of significant interest for many researchers. these algorithms aim to improve the clustering performance with the help of user-provided side information. there are several types of side information but pairwise constraints are the most used one. there are two types of pairwise constraints: must-link (ml) and cannot-link (cl) constraints. the must-link constraint indicates that the two objects must be grouped into the same cluster while the cannot-link constraint indicates that the two objects must be in different clusters. the existing constraint-based clustering algorithms assumed that they can improve the clustering performance with a suitable passively chosen set of constraints [1, 2]. however, if the constraints are selected improperly, they may also degrade the clustering performance [3, 4]. moreover, selecting constraints typically requires a user to manually inspect the data instances that can be time consuming and costly. for those reasons, the proposed method optimize the selection of the constraints to improve the performance of semi-supervised clustering algorithms. * corresponding author: walid.atwa@ci.menofia.edu.eg mailto:walid.atwa@ci.menofia.edu.eg mailto:ma7moud_emam@yahoo.com 26 w. atwa, m. emam copyright ©2019 assa adv. in systems science and appl. (2019) active learning is well motivated in many supervised learning scenarios where unlabeled instances are abundant and easy to retrieve but labeled instances are difficult, time-consuming and expensive to obtain. for example, it is easy to gather large amounts of unlabeled documents or images from the internet, whereas querying them requires manual effort from experienced human annotators. hence there is a need to select an optimal set of instances from the pool of unlabeled data for querying. randomly selection of unlabeled instances for querying is inefficient in many situations, since non-informative or redundant instances might be selected. active learning algorithms select the most informative unlabeled instances from enormous amount of unlabeled data for querying. specifically, the goal of active learning is to query data as little as possible to achieve a certain performance, thus saving considerable cost for generating good queries. in this paper, the active learning is applied in an iterative manner. in each iteration, set of queries are selected and queried with the existing neighborhoods to improve the clustering results. specifically, selecting a batch of informative query instances such that the distribution represented by the selected query set and the available labeled data is closest to the distribution represented by the unlabeled data. in other words, the proposed method select a set of samples s from the unlabeled data, denoted by du, such that the probability distributions represented by dl ∪ s and du \ s, where dl is the set of available labeled data, are similar to each other. then measuring the difference in the probability distribution between the two sets of data using the maximum mean discrepancy (mmd) [5, 6]. maximum mean discrepancy is a statistical test based on the fact that two distributions are different if and only if there exists at least one function in reproducing kernel hilbert space (rkhs) [6] having different expectations on the two distributions. once the batch of informative query instances are selected, the proposed method query them with the existing neighborhoods to determine which neighborhood they belong. a neighborhood contains a set of data instances that are known to belong to the same cluster (i.e., connected by must-link constraints) and different neighborhoods are known to belong to different clusters (i.e., connected by cannot-link constraints). well-formed neighborhoods can provide valuable information regarding what the underlying clusters look like. we empirically evaluate the proposed method with baseline and state-of-the-art methods on uci real datasets. the evaluation results demonstrate that the proposed method achieves consistent improvements over the baseline methods. the remainder of the paper is organized as follows. section 2 presents a brief review of the related work. section 3 introduces the proposed method. experimental results are presented in section 4. finally, conclude the paper and discuss future directions in section 5. 2. related work active learning has a long history in supervised learning algorithms [7, 8, 9, 10, 11]. recently, few studies reported the result of using active learning in constrained-based clustering problems. the first study was conducted by basu et al. [1] that proposed an active k-means clustering using the farthest-first strategy that has two-phases (explore and consolidate). the first phase (explore) uses the farthest-first scheme to form appropriate queries for getting the required pairwise disjoint neighborhoods. at the end of explore, at least one point has been obtained per cluster. the second phase (consolidate) iteratively expands the neighborhoods. where in each iteration it selects a random point outside any neighborhood and queries it against the existing neighborhoods until a must-link is found. an improvement version called min-max approach [12], which modifies the consolidate phase by selecting the most uncertain point to query, instead of selecting the point improving semi-supervised clustering algorithms with active query selection 27 copyright ©2019 assa adv. in systems science and appl. (2019) randomly. the idea is to select the data point whose largest similarity to the skeleton is the smallest. by this way, data points with largest uncertainty in cluster membership are chosen first to express the user queries. however, both previous methods do not work well in the case of a dataset with a large number of clusters or unbalanced datasets with small clusters. xu et al. [13] proposed an active constrained spectral clustering algorithm that examines the eigenvectors to identify the boundary points (of two classes) and sparse points; then it queries the oracle for constraints based on these points. it has shown limited applicability because it requires many queries to the oracle and assumes that errors in the clustering result only occur on the boundary points. wang et al. [14] presented another spectral active clustering technique that identifies informative pairs according to the entropy of the pair example. but, these approaches have limited their work to two classes, and the direct generalizations to multi-classes cases are not known. vu et al. [15, 16] proposed an active query selection method based on a constraint utility function called ability to separate between clusters. this method relies on two aspects: (1) a knearest neighbors graph is used to determine the best candidate queries in the sparse regions of the dataset between the clusters, and (2) a propagation procedure allows each user query to generate several constraints which limits the user intervention. the propagation procedure discovers new constraints from the information stored in already chosen constraints using the notion of strong paths. subsequently, the size of the candidate set is reduced by a refinement procedure that removes constraints between objects that are likely to be in the same cluster. specifically, the refinement procedure removes candidate constraints that are linked by a strong path. recently, xiong et al. [17] proposed an active learning method based on the classic uncertaintybased principle. they studied the selection of constraints by selecting the most informative instance to form queries accordingly. the responses to the queries are then used to improve the clustering results. however, this method selects only single instance that can become very slow for retraining with each single instance being queried. furthermore, if a parallel querying system is available, e.g., multiple annotators working in parallel, these methods would not be able to make the effective use of the resources. li et al. [18] proposed an active learning method that makes embeddings of labeled examples to those of unlabeled ones and back via deep neural networks. the active scheme makes association cycles that end up at the same class from that the association was started, which considers both the informativeness and representativeness of examples. the above mentioned methods form arrange of studies performed in active selection of constraints. each method considered a basic assumption on utility of constraints. their applicability on a specific problem is highly dependent on correlation between their assumption and the actual structure of data. 3. active constraint selection method semi-supervised clustering algorithms attempt to partition the unlabeled data into a set of clusters with the help of a small amount of pairwise constraints (must-link and cannot-link). in this section, addressing the problem of how to effectively choose pairwise constraints to produce accurate clustering results. the proposed method first selects a batch of most informative instances that minimize the difference in distribution between the labeled and unlabeled data. then, querying these instances with the existing neighborhoods to determine which neighborhood they belong. 28 w. atwa, m. emam copyright ©2019 assa adv. in systems science and appl. (2019) 3.1 batch instances selection traditional data mining and machine learning algorithms are based on the assumption that the training data (x, y) represents the true underlying distributions of x and y where x={x1, x2,…, xn} is the training data and their corresponding labels y={y1, y2,…, yn}. hence a model learned on this data works well for the test data (xtest, ytest) which is also drawn independently and identically distributed from the same distribution [5]. thus, a batch of query instances can be selected from unlabeled data such that the distribution represented by the queried and labeled data is similar to the probability distribution of the unlabeled data set. in other words, selecting a batch of instances s from the unlabeled data (denoted by du) such that the joint probability distribution represented by dl ∪ s and du \ s are similar to each other, where dl is set of available labeled data. this function is summarized in algorithm 1. to measure the difference between two distributions, maximum mean discrepancy (mmd) has been shown to be an effective measure of the difference in their marginal probability distributions [5, 6]. the principal underlying the maximum mean discrepancy is to find a function that assumes different expectations on two different distributions so that when evaluated empirically on samples drawn from the different distributions it would tell us whether the distributions are similar or not. let ℱ be a class of functions𝑓: 𝒳 → ℝ.. let p and q be probability distributions defined on a domain 𝒳, and let x = (x1, . . . , xm) and z = (z1, . . . , zn) be samples composed of independent and identically distributed observations drawn from p and q, respectively. the maximum mean discrepancy (mmd) [5, 6] and its empirical estimate are defined as: mmd[ℱ, 𝑝, 𝑞] ≔ sup 𝑓∈ℱ (𝐸𝑝[𝑓(𝑥)] − 𝐸𝑞[𝑓(𝑧)]) (1) mmd[ℱ, 𝑋, 𝑍] ≔ sup 𝑓∈ℱ ( 1 𝑚 ∑ 𝑓𝑚 𝑖=1 (𝑥𝑖) − 1 𝑛 ∑ 𝑓𝑛 𝑖=1 (𝑧𝑖)) (2) there is a class of functions for which mmd may easily be computed, while retaining the ability to detect all discrepancies between p and q without making any simplifying assumptions. let ℋ be a complete inner product space (i.e., a hilbert space) of functions𝑓: 𝒳 → ℝ, where 𝒳 is a nonempty compact set. then ℋ is termed a reproducing kernel hilbert space if for all x ∈ 𝒳, the linear point evaluation functional mapping f → f(x) exists and continuous. in this case, f(x) can be expressed as an inner product via 𝑓(𝑥) = 〈𝑓𝜙(𝑥)〉ℋ (3) where 𝜙 : 𝒳 → ℋ is known as the feature space map from 𝒳 to ℋ. when ℱ is the unit ball in a characteristic rkhs [6], mmd is defined as the difference between the means of two distributions after mapping onto the characteristic rkhs. an empirical estimate of mmd is then obtained as follows: mmd[𝜙, 𝑋, 𝑍] ≔ ‖ 1 𝑚 ∑ φ(𝑥𝑖) − 1 𝑛 ∑ φ(𝑧𝑖) 𝑛 𝑖=1 𝑚 𝑖=1 ‖ ℋ 2 (4) now, let us assume that we have u instances of unlabeled data du and l instances of labeled data dl and we would like to select a batch s of b instances such that the distribution of dl ∪ s is similar to the distribution of du \ s. thus, the mmd between the sets dl ∪ s and du \ s is defined by f(s), can be computed using the expression in equation (4), as follows: 𝑓(𝑆) = ‖ 1 𝑙+𝑏 ∑ φ(𝑥𝑗) − 1 𝑢−𝑏 ∑ φ(𝑥𝑖)𝑖∈𝐷𝑈∖𝑆𝑗∈𝐷𝐿∪𝑆 ‖ ℋ 2 (5) improving semi-supervised clustering algorithms with active query selection 29 copyright ©2019 assa adv. in systems science and appl. (2019) since we want to select a set s from unlabeled data set du to minimize the mismatch between dl ∪ s and du \ s. defining a binary vector 𝛼 of size u where each entry 𝛼𝑖 indicates whether the data xi ∈ du is selected or not. if a point is selected, the corresponding entry 𝛼𝑖 is 1 else 0. thus the minimization problem reduces to finding 𝛼 that minimizes the cost function f(s): min 𝛼:𝛼𝑖∈{0,1},𝛼𝑇1=𝑏 ‖ 1 𝑙+𝑏 (∑ φ(𝑥𝑗)𝑗∈𝐷𝐿 + ∑ αiφ(𝑥𝑖)𝑖∈𝐷𝑈 ) − 1 𝑢−𝑏 ∑ (1 − αi)φ(𝑥𝑖)𝑖∈𝐷𝑈 ‖ ℋ 2 (6) where 1 is a vector of the same dimension as 𝛼 with all entries 1 and symbol t is used to represent the matrix or vector transpose operation. evidently, the cost function in equation (6) is an alternative (equivalent) representation of the cost function f(s) in equation (5). the first term denotes the mean of the mapped features of the labeled and selected points. note that if a point xi is not selected in the current set then 𝛼𝑖 will be 0 and this term would not get added in the summation. the second term is mean of the mapped features of the unlabeled data set minus the selected query set. the first constraint ensures that each entry in 𝛼 is either 0 or 1 and the second constraint ensures that exactly b entries of 𝛼 are 1, meaning exactly b instances are selected from the unlabeled data set, where b is specified a priori by the user. algorithm 1. queryinstancesselection(dl, du, b); input: a set of labeled instances dl; set of unlabeled instances du; batch size b. output: a batch of query instances s. compute 𝛼 that minimize the distribution between the sets dl ∪ s and du \ s (eq. (6)) sort du in descending order of 𝛼 select top b instances of du as s update dl and du : dl← dl ∪ s, du← du \ s return s 3.2 a neighborhood-based active learning method once the batch of most informative query instances are selected, the proposed method query them against the existing neighborhoods to determine which neighborhood they belong to. in this section, introducing the concept of neighborhood, which is instrumental in the design of many existing methods for active learning of pairwise constraints [1, 12, 17]. then, explain how the proposed active learning method can expand the neighborhoods using the selected query instances. a neighborhood contains a set of data instances that are known to belong to the same cluster (i.e. connected by must-link constraints). different neighborhoods are connected by cannot-link constraints and thus are known to belong to different clusters. for example, figure 1 shows a set of must-link constraints (x1; x2), (x1; x3), (x4; x5), and (x4; x6), as well as a set of cannot-link constraints (x1; x4) and (x1; x5). two neighborhoods n1 and n2 can be generated as shown in figure 1. neighborhood n1 includes three instances x1, x2, and x3 as described by must-link constraints (x1; x2) and (x1; x3). similarly, x4, x5, and x6 should also be included in the same neighborhood. indicated by cannot-link constraints (x1; x4) and (x1; x5), x1 should not be in the neighborhood to which x4 and x5 belong. therefore, neighborhood n2 is discovered and it contains instances x4, x5, and x6. 30 w. atwa, m. emam copyright ©2019 assa adv. in systems science and appl. (2019) fig. 1: an example of neighborhoods a key advantage of using the neighborhood concepts is that by leveraging the knowledge of the neighborhoods, we can acquire a large number of pairwise constraints via a small number of queries. in particular, if we can identify the neighborhood of an instance x, we can infer its pairwise relationship with all other points that are currently confirmed to belong to any of the existing neighborhoods. this naturally encourages us to consider an active learning strategy that incrementally expands the neighborhoods by selecting the most informative instances and querying them against the known neighborhoods. we summarize the active learning method in algorithm 2. we begin by initializing the neighborhoods by selecting a random point to be the initial neighborhood (line 1). a selection criterion is then applied to select the batch query instances s as explained in the previous section (line 2). each selected instance 𝑠 ∈ 𝑆 is then queried against each existing neighborhood ni to identify where s belongs, during which the constraint set c is updated (lines 5-13). to determine the neighborhood of s with the smallest number of queries, we go through the neighborhoods in decreasing order based on p(s ∈ ni), i ∈ {1, · · · , k}, i.e. the probability of selected instance s belonging to the neighborhood ni, which is assumed to be the average similarity between s and the instances in ni . 𝑝(𝑠 ∈ 𝑁𝑖) = 1 |𝑁𝑖| ∑ 𝑀(𝑠,𝑥𝑗)𝑥𝑗∈𝑁𝑖 ∑ 1 |𝑁𝑝| ∑ 𝑀(𝑠,𝑥𝑗)𝑥𝑗∈𝑁𝑝 𝑘 𝑝=1 (7) where m(s, xj) denote the similarity between instance s and instance xj, |ni| indicates the number of instances in neighborhood ni, and k is the total number of existing neighborhoods. we should always start by querying s against the neighborhood that has the highest probability of containing s to minimize the total number of required queries. if a must-link is returned, we can stop with only one query. otherwise, one should ask the next query against the neighborhood that has the next highest probability of containing s. this process is repeated until a must-link constraint is returned or we have a cannot-link constraint against all neighborhoods. if no must-link is achieved, a new neighborhood will be created using the instance s (lines 14-16). finally, we apply the semi-supervised clustering algorithm using the selected active pairwise constraints to generate the final clusters (line 18). in this paper, we consider the semi-supervised clustering algorithm as a black-box and any existing algorithm can be used here. n1 n2 x1 x3 x2 x4 x5 x6 improving semi-supervised clustering algorithms with active query selection 31 copyright ©2019 assa adv. in systems science and appl. (2019) algorithm 2. the proposed active learning method input: a set of instances d (divided into labeled set dl and unlabeled set du); total number of queries q; batch size b. output: a set of clusters. 1. initialization: set n1 = {x}, where x is a random instance; c = ∅; q = 0; 2. while q<q 3. s = queryinstancesselection(dl, du, b); 4. for each instance s ∈ 𝑆 5. for each neighborhood ni ∈ 𝑁 in decreasing order of p(s∈ ni) 6. query instance s against any instance xi ∈ ni ; 7. q++; 8. update the constraint set c based on the results; 9. if a must-link achieved between s and xi then 10. add instance s to neighborhood ni; 11. break; 12. end if 13. end for 14. if no must-link is achieved then 15. create new neighborhood with the instance s; 16. end if 17. end for 18. apply semi-supervised clustering(d, c); 19. end while 4. experiments in this section, we evaluated the accuracy and efficiency of the proposed method on a diffrent realworld datasets. the results compared the proposed method with other constraint selection heuristics and evaluated in conjunction with two different constraint-based clustering algorithms to show the adaptability of the proposed method. the rest of this section is organized as follows. section 4.1 mentions the datasets used for evaluating the proposed method. section 4.2 describes some constraint selection heuristics that are compared with the proposed method and section 4.3 describes different constraint-based clustering algorithms used for constraints evaluation. section 4.4 explains the evaluation metrics used in this paper. the experimental results are described in section 4.5. 4.1 datasets experiments are conducted on 8 datasets from uci machine learning repository† (each with the following number of instances, attributes and clusters): protein (116/20/6) [15], heart (270/13/2) [17], ionosphere (351/34/2) [13, 14], breast (683/9/2) [15, 17], yeast (1484/8/10) [15], image segmentation (2310/19/7) [15, 17], digit-389 (3165/16/3) [17] and magic (19020/10/2) [10]. these datasets have been chosen because they facilitate the reproducibility of the experiments and because some of them have already been used in constraint-based clustering articles. also, provide † http://www.ics.uci.edu/~mlearn/mlrepository.html http://www.ics.uci.edu/~mlearn/mlrepository.html 32 w. atwa, m. emam copyright ©2019 assa adv. in systems science and appl. (2019) a good representation of different characteristics: number of instances ranges from 116 to 19020, dimensionalities from 8 to 34, and number of clusters from 2 to 10. 4.2 constraint selection heuristics in all experiments, we consider the following heuristics to select the constraints:  random: this policy corresponds to a completely random selection of the constraints. this method generates a set of ml and cl constraints based on the comparison of the labels of randomly chosen objects. if both labels are in the same cluster, a ml constraint is generated, and else, a cl constraint is generated.  min-max: this approach is neighborhood-based approach that works in two phases [12]. in the first phase, it builds c disjoint neighborhoods using farthest-first traversal, where c is the total number of clusters. in the second phase, it incrementally expands the neighborhoods by selecting a point to query using a distance-based min-max criterion.  asc: an active learning algorithm that relies on a k-nearest neighbors graph and a new constraint utility function to generate queries to the human expert. asc is based on two parameters (i.e. the number of nearest neighbors k and the threshold θ). these parameters k and θ are set to 6 and ⌊(𝑘/2) + 1⌋ respectively as recommended in their method [15].  npu: an active learning method based on the classic uncertainty-based principle that takes a neighborhood based approach, and incrementally expands the neighborhoods by selecting a single instance to query each time [17]. 4.3 constraint-based clustering algorithms in all experiments, we report the obtained results in conjunction with two different constraintbased clustering algorithms: the constrained k-means (mpckmeans) [19] and the agglomerative hierarchical clustering with constraints (ahcc) [20]. the choices of these algorithms are not critical and the proposed method can be used with any constraint-based clustering algorithm. they have been chosen because they are representative of the most popular clustering algorithms. when evaluating the performance of particular methods on a given dataset d, we apply it to select up to 150 pairwise queries, starting from no query at all. the queries are answered based on the ground-truth class label for the dataset. the constraint-based clustering algorithms (mpckmeans and ahcc) are then applied to the data with the selecting constraints. 4.4 evaluation metricsto evaluate the performance of the methods, we used normalized mutual information (nmi) and pairwise f-measure as the clustering validation metrics. nmi is an external validation metric, which is used to estimate the quality of clustering with respect to the given true labels of the datasets. nmi measures how closely the clustering algorithm could reconstruct the underlying label distribution in the data. if x is the random variable representing the cluster assignments of the instances and y is the random variable representing the class labels of the instances, then nmi is defined as follows: 𝑁𝑀𝐼 = 𝐼(𝑋;𝑌) (𝐻(𝑋)+𝐻(𝑌))/2 (8) where i (x; y) = h(y) − h(y|x) is the mutual information between the random variables x and y, h(y) is the shannon entropy of y , and h(y|x) is the conditional entropy of y given x [21]. the range of nmi values is 0–1. in general, the larger the nmi value, the better the clustering quality. improving semi-supervised clustering algorithms with active query selection 33 copyright ©2019 assa adv. in systems science and appl. (2019) pairwise f-measure is another evaluation metric to evaluate how well we can predict the pairwise relationship between each pair of instances in comparison to the relationship defined by the ground truth class labels. pairwise f-measure is defined as the harmonic mean of pairwise precision and recall, where the traditional information retrieval measures are adapted for evaluating clustering by considering pairs of points. for any pair of points, the decision to cluster this pair into same or different clusters is considered to be correct if it matches with the underlying class labeling available for the points. pairwise f-measure is defined as follows: 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 = 𝑛𝑐 𝑛𝑠 𝑅𝑒𝑐𝑎𝑙𝑙 = 𝑛𝑐 𝑛𝑓 𝐹 − 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 = 2×𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛×𝑅𝑒𝑐𝑎𝑙𝑙 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛+𝑅𝑒𝑐𝑎𝑙𝑙 (9) where nc is the number of point pairs that are correctly predicted as in the same cluster; ns is the number of point pairs that are predicted as in the same cluster; nf is the number of point pairs that are actually in the same cluster. 4.5 evaluation results 4.5.1 effectiveness and performance analysis in this section we present the evaluation results of the proposed method compared to min–max [12], asc [15], npu [17] heuristics and the random selection of the constraints. to evaluate the performance of the proposed method to adapt to distinct clustering algorithms, the heuristics are compared in conjunction with the clustering algorithms (mpckmeans and ahcc). figures 2 and 3 show the clustering performance on mpckmeans and ahcc respectively. to evaluate the performance in both mpckmeans and ahcc, we repeat this process for 50 independent runs and report the average performance using evaluation criteria described above. it can be observed from figures 2 and 3, that the proposed method generally outperforms other constraint selection heuristics in conjunction with the clustering algorithms mpckmeans and ahcc. this implies that the usefulness of constraints depends on how they are utilized by a clustering algorithm. furthermore, the proposed method keeps a smooth increase in the clustering performance while the other constraint selection heuristics drop in performance when the number of queries increases. it is interesting to note that the random selection of constraints degrade the clustering performance in some datasets when the number of queries increases (e.g. heart, breast, segment, and digit-389), while it is expected that the performance increases monotonically with the number of queries. this problem is a well-known issue in constraint-based clustering that has been addressed in [22, 23] and may be due to either the variability of the random approaches in some cases or due to a bad selection of some constraints that leads constraint-based clustering algorithm to poorer clustering results. this further demonstrates the importance of selecting the right set of constraints. in comparison, min-max method obtains better results with mpckmeans only for the heart dataset. this can be explained by the fact that for simpler datasets with a small number of clusters like heart dataset, the min-max method generally better manages to define the skeleton of cl constraints during its exploration step, which in turn promotes the initialization of the centers in 34 w. atwa, m. emam copyright ©2019 assa adv. in systems science and appl. (2019) mpckmeans. in the case of more complex datasets like segment and magic datasets, the minmax method needs more constraints to improve the clustering performance. more generally, it can be seen from figures 2 and 3, that the asc method obtains better results than the proposed active learning method with small number of queries (e.g. yeast, segment and digit-389 datasets). however, drop inefficiency with increasing the number of queries makes the proposed method superior to asc. the superiority of asc in a small number of queries comes from its propagation procedure that discovers new constraints from the information stored in previously chosen constraints and gives asc the capability to select a well propagated set of constraints in a small number of queries. on the other hand, the efficiency of asc is highly depending on its parameters k and θ. any improper assignments of k and θ may result in an inefficient set of constraints. in addition, some values of k and θ may result in incorrectly propagated constraints in the propagation step. it means that asc may provide incorrect-labeled constraints and mislead the clustering algorithms. in comparison with npu method that generally outperforms random, min-max, and asc method. npu is generally able to improve the clustering performance consistently as increasing the number of queries. however, its performance is dominated by the proposed method in most cases. improving semi-supervised clustering algorithms with active query selection 35 copyright ©2019 assa adv. in systems science and appl. (2019) protein heart ionosphere breast yeast segment digit-389 magic figure 2: comparison of the constraint selection heuristics with mpckmeans algorithm. 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0 25 50 75 100 125 150 n m i v al u e number of queries our method npu asc min-max random 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0 25 50 75 100 125 150 n m i v al u e number of queries our method npu asc min-max random 0,4 0,5 0,6 0,7 0,8 0,9 0 25 50 75 100 125 150 n m i v al u e number of queries our method npu asc min-max random 0,65 0,75 0,85 0,95 1,05 0 25 50 75 100 125 150 n m i v al u e number of queries our method npu asc min-max random 0,6 0,7 0,8 0,9 1 0 25 50 75 100 125 150 n m i v al u e number of queries our method npu asc min-max random 0,5 0,55 0,6 0,65 0,7 0,75 0,8 0 25 50 75 100 125 150 n m i v al u e number of queries our method npu asc min-max random 0,55 0,6 0,65 0,7 0,75 0,8 0,85 0 25 50 75 100 125 150 n m i v al u e number of queries our method npu asc min-max random 0,5 0,6 0,7 0,8 0,9 0 25 50 75 100 125 150 n m i v al u e number of queries our method npu asc min-max random 36 w. atwa, m. emam copyright ©2019 assa adv. in systems science and appl. (2019) protein heart ionosphere breast yeast segment digit-389 magic fig. 3: comparison of the constraint selection heuristics with ahcc algorithm. 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0 25 50 75 100 125 150 n m i v al u e number of queries our method npu asc min-max random 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 0 25 50 75 100 125 150 n m i v al u e number of queries our method npu asc min-max random 0,4 0,5 0,6 0,7 0,8 0,9 0 25 50 75 100 125 150 n m i v al u e number of queries our method npu asc min-max random 0,4 0,5 0,6 0,7 0,8 0,9 1 0 25 50 75 100 125 150 n m i v al u e number of queries our method npu asc min-max random 0,6 0,7 0,8 0,9 1 0 25 50 75 100 125 150 n m i v al u e number of queries our method npu asc min-max random 0,6 0,65 0,7 0,75 0,8 0 25 50 75 100 125 150 n m i v al u e number of queries our method npu asc min-max random 0,55 0,6 0,65 0,7 0,75 0,8 0,85 0,9 0 25 50 75 100 125 150 n m i v al u e number of queries our method npu asc min-max random 0,5 0,6 0,7 0,8 0,9 0 25 50 75 100 125 150 n m i v al u e number of queries our method npu asc min-max random improving semi-supervised clustering algorithms with active query selection 37 copyright ©2019 assa adv. in systems science and appl. (2019) we also use pairwise f-measure to evaluate the clustering quality; as it focuses on how accurately we can predict the pairwise relationship between any pair of instances. tables 1 2 show the pairwise f-measure results with different query size on the datasets with mpckmeans and ahcc algorithm respectively. the best performing method is then highlighted in boldface. once again, the proposed active methods show a clear advantage over the baseline methods. when using small number of queries, the performance of the methods is fairly close. however, as we increase the number of queries, the proposed method becomes better than all other methods. the experimental results demonstrate that the constraints selected by the proposed active learning process are generally more beneficial for constraint-based clustering algorithms than other baseline methods. table 1: comparison on pairwise f-measure (mean ± std) with mpckmeans algorithm. dataset constraint selection heuristics number of queries 25 50 75 100 125 150 protein proposed method npu asc min-max random 0.63±0.045 0.59±0.047 0.56±0.019 0.49±0.001 0.44±0.035 0.65±0.054 0.60±0.042 0.58±0.022 0.52±0.004 0.45±0.035 0.66±0.055 0.62±0.046 0.59±0.048 0.54±0.022 0.48±0.038 0.68±0.066 0.64±0.048 0.63±0.041 0.55±0.005 0.52±0.038 0.72±0.066 0.66±0.052 0.65±0.022 0.55±0.005 0.55±0.038 0.74±0.047 0.69±0.052 0.66±0.045 0.57±0.015 0.56±0.038 heart proposed method npu asc min-max random 0.68±0.002 0.65±0.011 0.57±0.024 0.55±0.004 0.48±0.025 0.71±0.006 0.66±0.044 0.58±0.055 0.57±0.012 0.49±0.030 0.73±0.005 0.67±00.72 0.59±0.044 0.57±0.012 0.52±0.030 0.75±0.005 0.68±0.007 0.62±0.006 0.57±0.014 0.55±0.033 0.75±0.008 0.70±0.036 0.63±0.008 0.58±0.015 0.55±0.036 0.76±0.009 0.71±0.074 0.64±0.009 0.59±0.015 0.56±0.036 ionosphere proposed method npu asc min-max random 0.71±0.054 0.65±0.001 0.67±0.024 0.64±0.033 0.60±0.053 0.77±0.027 0.70±0.028 0.69±0.056 0.68±0.035 0.65±0.005 0.79±0.077 0.75±0.024 0.74±0.032 0.72±0.024 0.66±0.006 0.84±0.030 0.83±0.043 0.81±0.064 0.75±0.034 0.61±0.009 0.86±0.034 0.86±0.025 0.83±0.057 0.77±0.084 0.57±0.005 0.87±0.077 0.86±0.035 0.84±0.065 0.80±0.075 0.55±0.012 breast proposed method npu asc min-max random 0.74±0.015 0.67±0.012 0.66±0.018 0.64±0.044 0.64±0.025 0.83±0.015 0.79±0.012 0.77±0.022 0.77±0.044 0.62±0.025 0.88±0.015 0.85±0.026 0.84±0.018 0.83±0.045 0.63±0.028 0.88±0.014 0.87±0.022 0.84±0.041 0.85±0.045 0.61±0.028 0.91±0.016 0.90±0.022 0.85±0.034 0.85±0.047 0.62±0.028 0.91±0.016 0.90±0.022 0.86±0.015 0.85±0.048 0.62±0.028 yeast proposed method npu asc min-max random 0.83±0.023 0.81±0.021 0.81±0.066 0.78±0.064 0.66±0.012 0.85±0.035 0.83±0.022 0.82±0.066 0.79±0.064 0.70±0.012 0.88±0.039 0.84±00.22 0.84±0.064 0.81±0.068 0.74±0.013 0.91±0.045 0.86±0.023 0.85±0.067 0.82±0.067 0.77±0.014 0.93±0.067 0.89±0.024 0.87±0.067 0.84±0.067 0.73±0.013 0.94±0.078 0.91±0.024 0.88±0.068 0.86±0.069 0.78±0.014 segment proposed method npu asc min-max random 0.61±0.024 0.57±0.031 0.64±0.081 0.55±0.014 0.52±0.024 0.65±0.024 0.59±0.032 0.66±0.004 0.55±0.015 0.54±0.025 0.68±0.025 0.61±0.035 0.66±0.022 0.54±0.017 0.54±0.025 0.69±0.026 0.62±0.035 0.67±0.024 0.54±0.017 0.52±0.024 0.70±0.027 0.63±0.038 0.68±0.037 0.53±0.020 0.51±0.024 0.71±0.027 0.64±0.041 0.68±0.038 0.53±0.022 0.51±0.027 digit-389 proposed method npu asc min-max random 0.73±0.025 0.68±0.018 0.66±0.022 0.65±0.014 0.70±0.055 0.75±0.025 0.70±0.018 0.69±0.022 0.68±0.014 0.73±0.055 0.77±0.025 0.75±0.019 0.71±0.024 0.70±0.015 0.75±0.058 0.84±0.025 0.81±0.021 0.77±0.031 0.73±0.015 0.70±0.058 0.86±0.026 0.83±0.022 0.79±0.035 0.74±0.017 0.72 ±0.045 0.88±0.026 0.84±0.023 0.81±0.035 0.75±0.018 0.73±0.046 magic proposed method npu asc min-max random 0.69±0.045 0.69±0.062 0.65±0.011 0.61±0.057 0.60±0.074 0.74±0.045 0.68±0.062 0.68±0.015 0.61±0.057 0.60±0.074 0.75±0.046 0.69±0.065 0.71±0.017 0.62±0.057 0.61±0.074 0.75±0.041 0.70±0.065 0.72±0.025 0.61±0.057 0.61±0.078 0.76±0.041 0.71±0.066 0.71±0.018 0.61±0.059 0.60±0.078 0.76±0.039 0.72±0.067 0.71±0.018 0.61±0.062 0.60±0.079 38 w. atwa, m. emam copyright ©2019 assa adv. in systems science and appl. (2019) table 2: comparison on pairwise f-measure (mean ± std) with ahcc algorithm. dataset constraint selection heuristics number of queries 25 50 75 100 125 150 protein proposed method npu asc min-max random 0.64±0.034 0.64±0.061 0.63±0.025 0.64±0.014 0.67±0.002 0.68±0.034 0.66±0.063 0.64±0.022 0.66±0.014 0.66±0.008 0.71±0.033 0.69±0.063 0.65±0.027 0.68±0.015 0.68±0.008 0.73±0.035 0.71±0.066 0.65±0.025 0.70±0.013 0.64±0.008 0.74±0.035 0.73±0.064 0.66±0.022 0.71±0.014 0.66±0.011 0.77±0.036 0.74±0.063 0.67±0.022 0.73±0.014 0.65±0.012 heart proposed method npu asc min-max random 0.70±0.043 0.67±0.012 0.68±0.026 0.67±0.035 0.54±0.072 0.74±0.044 0.72±0.012 0.71±0.022 0.73±0.045 0.51±0.062 0.77±0.042 0.75±0.014 0.74±0.033 0.75±0.054 0.55±0.015 0.79±0.043 0.78±0.013 0.78±0.064 0.77±0.003 0.53±0.014 0.83±0.043 0.83±0.013 0.82±0.021 0.82±0.042 0.55±0.016 0.86±0.043 0.85±0.012 0.84±0.032 0.83±0.008 0.52±0.035 ionosphere proposed method npu asc min-max random 0.75±0.023 0.74±0.031 0.72±0.024 0.68±0.031 0.65±0.003 0.78±0.023 0.77±0.038 0.75±0.026 0.69±0.031 0.65±0.005 0.81±0.024 0.80±0.034 0.77±0.022 0.74±0.032 0.68±0.006 0.84±0.024 0.83±0.033 0.81±0.024 0.75±0.034 0.64±0.007 0.87±0.025 0.87±0.035 0.84±0.027 0.77±0.034 0.65±0.009 0.90±0.030 0.89±0.035 0.86±0.025 0.78±0.035 0.67±0.010 breast proposed method npu asc min-max random 0.81±0.038 0.81±0.021 0.77±0.054 0.71±0.084 0.66±0.005 0.83±0.038 0.82±0.022 0.78±0.054 0.71±0.084 0.65±0.005 0.85±0.039 0.83±00.22 0.79±0.055 0.74±0.088 0.60±0.006 0.87±0.040 0.84±0.023 0.81±0.056 0.75±0.087 0.59±0.007 0.89±0.042 0.87±0.024 0.83±0.057 0.76±0.087 0.57±0.008 0.90±0.042 0.88±0.024 0.84±0.057 0.77±0.089 0.55±0.008 yeast proposed method npu asc min-max random 0.82±0.043 0.84±0.051 0.80±0.066 0.80±0.074 0.73±0.022 0.84±0.045 0.86±0.052 0.83±0.066 0.83±0.074 0.77±0.022 0.87±0.049 0.87±00.52 0.84±0.064 0.84±0.078 0.79±0.023 0.90±0.045 0.90±0.023 0.85±0.067 0.87±0.067 0.82±0.014 0.92±0.067 0.91±0.024 0.86±0.067 0.88±0.067 0.84±0.013 0.93±0.078 0.92±0.024 0.87±0.068 0.89±0.069 0.86±0.014 segment proposed method npu asc min-max random 0.60±0.035 0.58±0.017 0.61±0.034 0.57±0.045 0.52±0.012 0.61±0.037 0.59±0.017 0.62±0.028 0.58±0.045 0.54±0.012 0.63±0.059 0.60±0.018 0.62±0.030 0.58±0.045 0.53±0.013 0.65±0.024 0.60±0.032 0.62±0.004 0.59±0.025 0.51±0.025 0.67±0.037 0.62±0.053 0.62±0.021 0.60±0.055 0.50±0.014 0.69±0.026 0.63±0.035 0.63±0.054 0.60±0.067 0.48±0.024 digit-389 proposed method npu asc min-max random 0.80±0.055 0.81±0.001 0.78±0.026 0.76±0.044 0.70±0.012 0.83±0.055 0.83±0.002 0.81±0.026 0.79±0.044 0.70±0.012 0.86±0.056 0.85±0.002 0.83±0.025 0.81±0.048 0.74±0.013 0.88±0.055 0.86±0.003 0.85±0.027 0.82±0.047 0.77±0.014 0.91±0.057 0.89±0.004 0.86±0.027 0.84±0.047 0.78±0.013 0.92±0.055 0.91±0.004 0.88±0.028 0.86±0.049 0.82±0.014 magic proposed method npu asc min-max random 0.64±0.033 0.58±0.007 0.57±0.024 0.56±0.005 0.48±0.012 0.70±0.035 0.62±0.007 0.58±0.028 0.58±0.005 0.51±0.012 0.72±0.035 0.66±0.008 0.61±0.030 0.58±0.005 0.55±0.013 0.74±0.036 0.67±0.012 0.64±0.011 0.61±0.004 0.51±0.014 0.75±0.037 0.68±0.013 0.66±0.021 0.62±0.005 0.49±0.014 0.77±0.038 0.69±0.012 0.67±0.022 0.62±0.008 0.47±0.015 4.5.2 efficiency and scalability analysis one of the primary motivations of the proposed active learning method is to reduce the amount of computation in iterative active learning. figure 4 shows the average cpu time of the proposed selection method (using batch size value b = 10) with the baseline methods on a 3.5 ghz intel core i5 and 4 gb main memory. the horizontal axis indicates the total number of queries and the vertical axis shows the cpu running time (in seconds). from the results, we clearly see that the proposed method is significantly more efficient and scalable than other methods. as we observe, when the dataset size increases, the time cost of npu and asc increases dramatically, while the proposed method increases linearly. specifically, on the magic dataset with 150 queries, npu takes about 240 seconds and asc takes about 260 seconds, while the proposed method needs only about 24 second. hence, we can conclude that the proposed method is more efficient and scalable than other methods for large applications. finally, as indicated in figure 4, the time cost of the proposed method increases moderately as the number improving semi-supervised clustering algorithms with active query selection 39 copyright ©2019 assa adv. in systems science and appl. (2019) of queries increases. while, the time cost of other methods increases tremendously as the number of queries increases. generally, the results show that the proposed method is slower than random method that selects set of constraints randomly without performing additional cpu time for selecting constraints. protein heart ionosphere breast yeast segment 0 5 10 15 20 25 25 50 75 100 125 150 c p u t im e ( se co n d s) number of queries our method npu asc min-max random 0 5 10 15 20 25 25 50 75 100 125 150 c p u t im e ( se co n d s) number of queries our method npu asc min-max random 0 5 10 15 20 25 30 35 25 50 75 100 125 150 c p u t im e ( se co n d s) number of queries our method npu asc min-max random 0 10 20 30 40 50 60 25 50 75 100 125 150 c p u t im e ( se co n d s) number of queries our method npu asc min-max random 0 20 40 60 80 100 25 50 75 100 125 150 c p u t im e ( se co n d s) number of queries our method npu asc min-max random 0 20 40 60 80 100 120 25 50 75 100 125 150 c p u t im e ( se co n d s) number of queries our method npu asc min-max random 40 w. atwa, m. emam copyright ©2019 assa adv. in systems science and appl. (2019) digit-389 magic fig. 4: comparison of average run time over different number of queries with batch size value b = 10. 4.3.3 analysis of different batch size values in this section, we carried out an analysis of the proposed active learning with varying the value of the batch size b. figure 5 shows the performance versus the number of queries on the datasets with mpckmeans algorithm. from figure 5, selecting small b values results in similar (or better) performance compared to those obtained selecting only one instance. on the contrary, high b values decrease the performance without decreasing the computational time if compared to small b values. figure 6 shows the computational time taken for different b values on both yeast and magic datasets. from figure 6, it could be noticed that the largest learning time is obtained in the case where one instance is selected (i.e. b = 1). also, we can notice that the cpu time is increased as the value of batch size b is increased. therefore, it could be interesting to automatically identify the best value of the batch size b that can achieve the best clustering performance. (a) protein (b) heart (c) ionosphere (d) breast (e) yeast (f) segment 0 20 40 60 80 100 120 140 160 180 25 50 75 100 125 150 c p u t im e ( se co n d s) number of queries our method npu asc min-max random 0 50 100 150 200 250 300 25 50 75 100 125 150 c p u t im e ( se co n d s) number of queries our method npu asc min-max random 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0 25 50 75 100 125 150 n m i v al u e number of queries b=1 b=10 b=50 b=100 0 0,2 0,4 0,6 0,8 0 25 50 75 100 125 150 n m i v al u e number of queries b=1 b=10 b=50 b=100 0,5 0,55 0,6 0,65 0,7 0,75 0,8 0,85 0 25 50 75 100 125 150 n m i v al u e number of queries b=1 b=10 b=50 b=100 0,7 0,75 0,8 0,85 0,9 0,95 1 0 25 50 75 100 125 150 n m i v al u e number of queries b=1 b=10 b=50 b=100 0,75 0,8 0,85 0,9 0,95 0 25 50 75 100 125 150 n m i v al u e number of queries b=1 b=10 b=50 b=100 0,6 0,65 0,7 0,75 0,8 0 25 50 75 100 125 150 n m i v al u e number of queries b=1 b=10 b=50 b=100 improving semi-supervised clustering algorithms with active query selection 41 copyright ©2019 assa adv. in systems science and appl. (2019) (g) digit-389 (h) magic fig.5: clustering performance versus different batch size (a) protein (b) heart (c) ionosphere (d) breast (e) yeast (f) segment (g) digit-389 (h) magic fig. 6: cpu time versus different batch size 5. conclusion and future work identifying the most beneficial set of clustering constraints was considered in this paper. we present an iterative active learning method that selects a batch of query instances from the 0,65 0,7 0,75 0,8 0,85 0 25 50 75 100 125 150 n m i v al u e number of queries b=1 b=10 b=50 b=100 0,35 0,4 0,45 0,5 0,55 0,6 0,65 0,7 0 25 50 75 100 125 150 n m i v al u e number of queries b=1 b=10 b=50 0 10 20 30 40 50 0 25 50 75 100 125 150 c p u t im e ( se co n d s) number of queries b=1 b=10 b=50 b=100 0 5 10 15 20 25 0 25 50 75 100 125 150 c p u t im e ( se co n d s) number of queries b=1 b=10 b=50 b=100 0 5 10 15 20 25 30 35 40 0 25 50 75 100 125 150 c p u t im e ( se co n d s) number of queries b=1 b=10 b=50 b=100 0 20 40 60 80 0 25 50 75 100 125 150 c p u t im e ( se co n d s) number of queries b=1 b=10 b=50 b=100 0 20 40 60 80 100 120 0 25 50 75 100 125 150 c p u t im e ( se co n d s) number of queries b=1 b=10 b=50 b=100 0 20 40 60 80 100 120 140 160 0 25 50 75 100 125 150 c p u t im e ( se co n d s) number of queries b=1 b=10 b=50 b=100 0 20 40 60 80 100 120 140 0 25 50 75 100 125 150 c p u t im e ( se co n d s) number of queries b=1 b=10 b=50 b=100 0 20 40 60 80 100 120 140 160 180 0 25 50 75 100 125 150 c p u t im e ( se co n d s) number of queries b=1 b=10 b=50 b=100 42 w. atwa, m. emam copyright ©2019 assa adv. in systems science and appl. (2019) unlabeled data so that the marginal probability distribution represented by the labeled data after annotation, is similar to the marginal probability distribution represented by the unlabeled data. moreover, incrementally expands the neighborhoods by using the selected queries. experiments carried out on different real datasets show that the constraints selected by the proposed active learning process are generally more beneficial for constraint-based clustering algorithms than those provided by the npu and asc methods and achieving high clustering performance with minimizing the amount of computation for selecting the active constraints. however, the efficiency of the proposed method strongly depends on the value of the batch size b. in future work, we are interesting to automatically identify the best value of the batch size b that achieves the best clustering performance. also, the problem of clustering the big data with an incrementally growing constraint set. to address this problem, we interest to consider an incremental semi-supervised clustering method. references 1. basu, s., banerjee, a., & mooney, r. j. 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(2006). measuring constraint-set utility for partitional clustering algorithms. in european conference on principles of data mining and knowledge discovery, 115-126. microsoft word revised paper final adv syst sci appl 2024; 01; 129-141 published online at https://ijassa.ipu.ru. vision based classification of nocturnal road traffic using a custom deep convolution neural network sofiane abdelkrim khalladi1*, asmâa ouessai1,2, mokhtar keche1 1) signals and images laboratory, faculty of electrical engineering, department of electronics, university of sciences and technology of oran mohamed boudiaf usto-mb, b.p. 1505, el mnaouar-bir el djir-oran, algeria 2) faculty of technology, department of telecommunications, dr. tahar moulay university, saida, algeria abstract: intelligent transport systems (its) have emerged as an efficient solution for enhancing road utilization efficiency, ensuring convenient and safe transportation, and reducing energy consumption. this research addresses the challenging problem of accurately estimating road traffic congestion from videos recorded in low visibility and adverse weather conditions such as rainy, overcast, and sunny weather. to solve this problem, we propose a method based on the use a custom deep convolutional neural network (dcnn) that we have designed and trained to classify nighttime road traffic videos into three distinct categories. to train and test this dcnn, we used nighttime videos from the ucsd (university of california san diego) public dataset. for comparison, we have also tested a second method, which uses an existing cnn that we have trained on the same dataset to classify road traffic into the same three categories. the performances of the two methods were assessed and compared using the ucsd nighttime dataset. the first method achieves an impressive correct classification rate (ccr) of 98.91%, which is higher than the 96.18% ccr achieved by the second method. these rates surpass the 89.47% state-of-the-art ccr obtained with this dataset. this exceptional precision in estimating road traffic congestion in challenging low visibility conditions is achieved thanks to the integration of advanced deep learning techniques and cnns. the proposed method can be integrated as part of a traffic monitoring system, thus contributing to the advancement of intelligent transport systems and their potential to create cleaner, safer, and more efficient transportation networks. keywords: cnn, congestion, its, traffic classification, night traffic, macroscopic approach, ucsd. 1. introduction road traffic accidents are a worldwide calamity. in fact, over a million people die in road accidents worldwide each year, and an even greater number end up with some form of disability. low visibility is one of the causes of road accidents; indeed, the risk of a fatal accident during the night is seven times higher than during the day [1]. road traffic congestion is a significant cause of accidents and has social, economic, health, and environmental impacts, as well. several measures have been taken to combat this epidemic, which can occur during the day or night. among these measures, the employment of recent technology occupies a place of prominence. indeed, it enables the supervision of road traffic using sensors installed on the road. the data provided by these sensors is processed to extract relevant information for road users and road traffic management and control services, in order to help plan departures (vacation periods), avoid traffic bottlenecks, and thereby lessen their misdeeds. many cameras are now put on urban roadways and highways for security purposes. these cameras can be used to estimate the status of road traffic. currently, there is a high demand for the interpretation of nocturnal traffic situations. in this paper, we propose a traffic framework for the classification of traffic congestion from sofiane abdelkrim khalladi, asmâa ouessai, mokhtar keche 130 copyright ©2024 assa adv. in systems science and appl. (2024) nighttime surveillance videos. video analysis at night represents a challenge. night-time vehicle surveillance is even more complex since heavy traffic flows and incidents might occur on city streets or highways at night. the use of night vision sensors such as infrared-thermal cameras to identify automobiles is an efficient method. unfortunately, due to their high cost, infrared sensors are rarely used for traffic analysis. ccd cameras, unlike thermal cameras, are sensitive to visible light. however, some challenging variables, such as poor sight and loud surroundings, make some weather or illumination circumstances (such as day, night, sun, rain, snow, and occlusions) difficult to deal with. primarily, these circumstances are difficult because the traditional daytime surveillance framework, which is based on changes detection (such as background subtraction), is unable to work at night. this is due to two factors: first, the camera's generally poor light sensitivity and contrast, second, the numerous foreground/background ambiguities caused by the changing headlamp reflections. to overcome these limitations, we suggest a new macroscopic method for traffic nighttime classification that does not rely on vehicle monitoring (detection and tracking) and extraction of parameters, such as density and velocity. the proposed method is very efficient and outperforms the state-of-the-art methods in terms of accuracy; it can contribute to making transportation safer, cleaner, and more economical. the rest of the paper is organized as follows. section ii presents a literature review regarding the topic covered by the present paper. section iii provides a brief overview of convolution neural networks. in section iv the proposed custom dcnn and the cnn architectures, used for road traffic classification, are successively described. in section v, after an overview of the used dataset, the classification results obtained with the methods based on these two models are analyzed and compared with those obtained with previous works. lastly, section vi presents some conclusions. 2. related work there are two major approaches for road traffic congestion classification. the first approach, known as the microscopic [2] approach, is based on the extraction of the movement parameters of individual vehicles; it thus requires the detection and tracking of all the vehicles in the region of interest. the second approach, known as the macroscopic [3] approach, analyses the observed scene as a whole, without the need of extracting the movement parameters of individual vehicles. in this approach, the flow of automobiles is viewed as a liquid traveling through a tube, making it very responsive to changing road conditions, but lacking specific information about individual vehicles motions. this characteristic underpinned our decision to adopt the macroscopic approach in this research. important efforts have been made in the literature in the field of nighttime surveillance systems and image processing. c. h. hsia, y. kong, y. k. lin, and y. r. chien [4] led a research team who developed a technology that can track vehicle headlights at night using multiple features. the technology analyses the distribution of light sources in each captured image and separates the headlights based on their length and width ratios. it distinguishes between vehicles to avoid counting the same vehicle twice. based on their space and color characteristics, an algorithm that uses similarity analysis is used to determine whether two headlights belong to the same vehicle. pushkar s. and s. b. dhonde [5] devised a method for detecting and selecting an appropriate vehicle beam suitable for temporary blindness reduction. researchers used two methods to reduce headlight glare and temporary blindness. the image sequence is then analysed to determine the presence of a vehicle. a light intensity sensor is another option for dealing with temporary blindness. another work was presented in [6] by y. l. chen, b. f. wu, h. y. huang, and c. j. fan, who proposed a system for identifying and controlling vehicles at night. to distinguish vehicles from bright objects in image sequences, the system employs automated multilevel histogram thresholding. furthermore, illuminating objects in vehicle lighting clusters are grouped using a spatial analysis and a clustering method based on their likely motion across successive sofiane abdelkrim khalladi, asmâa ouessai, mokhtar keche 131 copyright ©2024 assa. adv. in systems science and appl. (2024) frames. in another research [7], the proposed stereo configuration incorporates an alignment that displaces the cameras along a vertical baseline to extract useful information pertaining to negative obstacle features for various illustrative terrain settings. the convex framework exploits these properties to evaluate depth jumps in the disparity space image and perform geometrical analysis of potential occlusion regions. the corresponding convex formulations are based on linear matrix inequalities to detect sudden disparity, angle profile, and intensity variations of potential negative obstacles and to estimate the internal depth of the detected obstacles. experimental results demonstrate that consecutive processes can be structured in a convex framework to efficiently identify negative obstacle attributes at a range of distances, for texture-varying environments, providing a vital extension to real-time vehicular system implementations with superior detection rates. sayed et al. [8] presented an efficient approach for nighttime contrast enhancement that alters standard histogram equalization to keep the original nighttime image's color information. each color channel is improved separately by multiplying the increased brightness ratio by the original luminance ratio. furthermore, cai et al. [9] merge a daylight image with a midnight image. the low-quality static elements of a nighttime image can be replaced by the high-quality counterparts in the daytime image, using the object extraction approach. liu and payeur [10] attempted to eliminate the static false positives introduced by cucchiara's technique [11] by performing background subtraction before detecting motion. this solution, however, cannot deal with the moving headlight's beam. it is also not possible to apply it to illuminated road scenes. the concept of rear-view monitoring was first proposed by yoneyama et al. [12] who, in order to improve detection accuracy, installed two cameras that detect both vehicle headlights and taillights, where in the past, most studies focused on daytime traffic monitoring [13]. however, the proposed solutions may not work properly at night. it should be reported that very few papers addressed the problem of nighttime traffic congestion classification. to the best of our knowledge, the last work that dealt with this problem dates from 2013, and since then no other work with improved performance was published. one can cite two existing works that estimate the road traffic status from videos recorded in the daytime and the nighttime. the first one [14] is based on scene modeling by dynamic textures and classification by support vectors machine (svm). the second one [15] uses motion vectors estimation to extract the mean traffic speed and density, which feed a svm classifier that categorizes the traffic congestion. five years later, kurniawan, j et al. [16] proposed a technique that uses a cnn for image classification on cctv camera feeds, to create an intelligent traffic congestion detection system. this solution requires less preprocessing for small grayscale images than traditional methods, which need high-quality images and manual feature calculation. a dataset of 1000 evenly distributed cctv monitoring photos was used to train the cnn model by binary classification. the results revealed that a basic cnn architecture is effective for real-time road traffic situation assessment, with a high average classification accuracy of 89.50%. in order to address the need for efficient traffic estimation in low light to improve transportation management and safety, this paper presents a framework for the classification of traffic congestion in nighttime films. the suggested approach employs a deep convolutional neural network (dcnn) model, which was designed to provide accurate and consistent assessment of traffic congestion even in difficult environmental circumstances. the study comprises several contributions, which are listed below:  we propose a macroscopic technique that, unlike the traditional microscopic technique, does not require vehicle identification and tracking.  excellent performance is achieved by our proposed dcnn model, when applied to the nighttime ucsd dataset.  our work focuses on road traffic categorization during night and puts an end to the dearth of works in this area since 2018.  we use video footage taken in a variety of weather conditions and difficult lighting circumstances, including at night, to systematically classify traffic congestion. sofiane abdelkrim khalladi, asmâa ouessai, mokhtar keche 132 copyright ©2024 assa adv. in systems science and appl. (2024)  we tested the same architecture for a recent research paper [16] which used a different cnn model to classify traffic, on the ucsd nighttime dataset. the aim was to compare and demonstrate the effectiveness of the best-performing model.  our method outperforms the three state of art methods in terms of classification accuracy. 3. convolutional neural network: an overview the success of traditional methods in solving computer vision problems is largely dependent on the feature extraction procedure. convolutional neural networks (cnn) [17], on the other hand, provide an alternative by automatically picking up domain-specific information. this paradigm change has led to a reassessment of several issues in the context of computer vision as a whole. currently, face recognition, object identification, scene labeling, and image categorization are all accomplished by means of cnns, with an improved performance, compared to traditional methods. 3.1. cnn architecture a convolutional neural network is composed of several types of layers that are described below: convolutional layers, pooling layers and fully connected layers. 3.1.1. convolution layer to extract information, the convolutional layer [18], leverages local correlations within the image. the process entails inserting a kernel in the image's upper-left corner. the pixel values within the kernel's coverage are multiplied by the corresponding kernel values, and the results are added to the bias. the kernel is then shifted by one pixel, and the process is continued recursively until all feasible locations in the image have been filtered. figure 1 depicts the convolution operation. fig. 1. convolution process operation 3.1.2. pooling layer the pooling layer [19] provides the model translation, rotation, and scaling invariance while sampling to extract features from the higher layer's feature map. a popular method called maximum pooling (maxpooling) utilizes the filter size to divide the input picture into rectangular parts and then outputs the maximum value for each zone, as shown in figure 2. convolutional and pooling layers often alternate in real-world applications. the integration of multidimensional characteristics is facilitated by connections made between neurons in the fully linked layer and higher neurons. the classifier then translates these features into onedimensional features that may be used for detection or classification tasks. sofiane abdelkrim khalladi, asmâa ouessai, mokhtar keche 133 copyright ©2024 assa. adv. in systems science and appl. (2024) fig. 2. max pooling process operation 3.1.3. fully connected layer convolutional neural networks (cnns) conclude by taking the output feature maps generated from the pooling or final convolution layer, and transforming them into a flattened onedimensional array. next, this array is joined to one or more dense layers, also referred to as fully connected layer [20], where trainable weights connect each input to every output. the features retrieved by convolution and down sampling are transformed by these fully connected layers into the final outputs of the network, such as class probabilities in classification tasks. the number of classes in the classification task is usually equal to the number of nodes in the final fully linked layer. each fully connected layer is followed by a nonlinear activation function. 3.2. activation functions the last fully connected layer in a neural network often employs a different activation function [21] than the preceding layers. the particular task at hand determines which activation function to use. one popular activation function used in multiclass classification is softmax, defined by the equation: 𝑆𝑜𝑓𝑡𝑚𝑎𝑥 = 𝑒𝑥𝑖 ∑ 𝑒𝑥𝑖𝑘 𝑖=1 , (1) where 𝑥 represents the i-th element of the input vector, which is typically a representation of the neuron activations in the last layer of the neural network before applying the softmax function. each 𝑥 corresponds to the activation of the neuron associated with the respective output class. the parameter 𝑘 in the formula represents the total number of elements in the input vector. in the context of classification, this corresponds to the total number of output classes, to provide a probability estimate for the output classes. the softmax function takes in a vector of real values as input and produces an output vector of real numbers. the resulting values, which represent estimates of the output classes probabilities, are constrained between 0 and 1, and the entire vector sums up to one. sofiane abdelkrim khalladi, asmâa ouessai, mokhtar keche 134 copyright ©2024 assa adv. in systems science and appl. (2024) 4. the tested methods 4.1. the proposed method the proposed method uses a deep convolution neural network (dcnn) to classify road traffic into three categories, namely, free, medium, and heavy. dcnns are regarded as the state-of-the-art image classification approach. in this study, a standard convnet (convolutional network) architecture with convolution layers and pooling layers was used. this architecture is shown in figure 3, where ‘conv 1’ refers to the first convolution layer, ‘fc’ denotes the fully connected layer, ‘s’ is the number of stride and ‘f’ is the number of filters. fig. 3. architecture of the proposed convolutional network the inputs of the network are images extracted from videos provided by the ucsd dataset. each video contains 40-50 frames at a 320x240 resolution. these frames are cropped and then downsized to get photos of size 64x64 pixels, to avoid memory allocation issues during model training. these photos are fed into the model at the first 64x3x3 convolution layer, which is followed by a max pooling layer of size 2x2. the network is then stretched six times. to prevent overfitting [22], a 0.25 probability dropout layer [23] is included before the final max pooling layer. softmax [24] is employed as an activation function throughout the model. table 1 shows the dcnn model architecture. table. 1. dcnn custom model architecture used layer kernel stride output shape input [64, 64, 3] convolution2d 3x3 [62,62, 64] maxpooling 2x2 1 [62,62, 64] convolution2d 3x3 [60,60, 64] maxpooling 2x2 1 [60, 60, 64] convolution2d 3x3 [58, 58, 64] maxpooling 2x2 1 [58, 58, 64] convolution2d 3x3 [56, 56, 64] maxpooling 2x2 1 [56, 56, 64] convolution2d 3x3 [54, 54, 64] maxpooling 2x2 1 [54, 54, 64] convolution2d 3x3 [52, 52, 64] maxpooling 2x2 1 [52, 52, 64] convolution2d 3x3 [50, 50, 64] dropout [50, 50, 64] maxpooling 2x2 1 [50, 50, 64] flatten 160000 dense 3 sofiane abdelkrim khalladi, asmâa ouessai, mokhtar keche 135 copyright ©2024 assa. adv. in systems science and appl. (2024) 4.2 the method based on an existing cnn in order to compare our proposed cnn architecture with a state of art method we have tested the cnn architecture proposed in [16] with same ucsd dataset and the same parameters configuration. as shown in figure 4, this architecture consists of: two convolutional layers, a max pooling layer, and a fully connected layer. fig. 4. flowchart of the cnn architecture proposed in [16]. the initial layer, denoted as c1, is a convolutional layer employing 3x3 filters and generating 32 feature maps. given the 100x100 input size, each feature map has a 98x98 dimension. subsequently, the second layer, denoted c2, is another convolutional layer utilizing 3x3 filters and producing 32 feature maps, each with a 93x96dimension. the third layer, referred to as p3, is a 2x2 max pooling layer employed for down-sampling; it reduces the size of each feature map to 48x48. the last hidden layer is a fully connected layer comprising 128 perceptron. each perceptron is fully connected to every unit within the feature maps from p3. the rectified linear units (relu) [25] activation function is applied in both the convolutional and fully connected layers. in the output layer, a single perceptron is utilized with a sigmoid activation function. 5. experimental evaluation of the two methods 5.1. dataset preparation the ucsd dataset contains 19 nighttime highway traffic videos recorded by a stationary camera, under different weather conditions: clear, rainy, and overcast. each video has a duration of about five seconds and contains between 45 and 50 frames. the dataset comprises diverse traffic categories: light, medium, and heavy. hand-labeled ground truth was provided; it assigns one of these three categories to each video sequence. in total, the 19 videos produced a set of 1112 frames for all traffic categories, which was used to train and test the network. some of these frames are shown in figure 5. sofiane abdelkrim khalladi, asmâa ouessai, mokhtar keche 136 copyright ©2024 assa adv. in systems science and appl. (2024) fig. 5. example frames from the nighttime ucsd dataset all experimental results were averaged over four trials. in each trial, the dataset was split differently and randomly into 2 sub-sets, containing, respectively, 75% and 25% of the 1112 frames. the first sub-set was used for training and cross-validation [26], and the second one was reserved for testing. to avoid overfitting, dropout regularization was combined with the data augmentation technique [27]. overfitting is a fundamental problem in supervised machine learning; it limits the ability to perfectly generalize models and fit observed data during training, as well as unseen data in the test set. overfitting occurs due to the presence of noise, the limited size of the training set, and the complexity of classifiers. a variety of algorithms have been proposed to reduce the effect of overfitting, like dropout and data augmentation where basic image transformations, such as rotation, flipping, and cropping, are applied. most of these techniques manipulate the images directly and are easy to implement. to guarantee data augmentation, images were rotated by 30 degrees, flipped horizontally and vertically, with horizontal flipping including flipping the image along the vertical axis (left to right) and vertical flipping involving flipping the image along the horizontal axis (top to bottom). rotation and flipping are two transformations that let the model generalize to objects in different orientations. zooming in entails magnifying a section of the image, enlarging the specified location. this can help the model become more resistant to scale changes and increase its ability to distinguish objects of varying sizes. these changes enable for the creation of a more diversified training dataset. this is advantageous because it subjects the model to a broader range of changes that it may meet in real-world circumstances, making it more resilient and less prone to overfitting. if m distinct augmentation procedures are applied to each of the n original images, each with a unique set of parameter parameters, then the total number of enhanced images can be calculated by the following formula: 𝑇 = 𝑁 𝑥 𝑀 (2) in our approach for each original image, we generate a total of 7 augmented images including the original image. with 834 original training images, the total number of augmented images would be: 834 original training images × 7 augmentations per image = 5838 augmented images 5.2. training process in the first method on a google colab gpu, the model training took ten minutes. we employed mini-batch gradient descent with adam [28] optimization with a learning rate of 0.001 to train our cnn model using python and the keras library [29], which is built on top of the theano library [30]. our implemented cnn model utilizes a batch of size 64 and 100 epochs, for the mini-batch gradient descent. this requires 13 iterations to complete an epoch. for each iteration, a batch of 64 images was submitted to the cnn model, and the weights were then changed through backpropagation. the categorical cross entropy [31] was used as the loss function to adjust the model weights during training. it is defined by: 𝐿 = −∑𝑡 log(𝑃 ) , (3) where ti is the truth label and 𝑃𝑖 is the softmax probability of the 𝑖𝑡ℎ class. sofiane abdelkrim khalladi, asmâa ouessai, mokhtar keche 137 copyright ©2024 assa. adv. in systems science and appl. (2024) the aim is to minimize the loss, i.e, the smaller the loss the better the model. a perfect model has a cross-entropy loss of 0. the loss and accuracy graphs presented in figure 6 show that the training was successful. indeed 100% of accuracy was reached during training. fig. 6. training model loss and accuracy versus epoch (first method). 5.3. training process in the second method the cnn model was trained using the mini-batch gradient descent algorithm with a batch size of 250, and the training process encompassed 100 epochs. as a result, it required 3 iterations to complete one epoch. a batch of 250 images was presented to the cnn model and then the weights were updated by backpropagation. for multiclass classification, categorical cross entropy is used for the objective function without data augmentation. figure 7 presents the results of the training and validation process. fig. 7. training model loss and accuracy versus epoch (second method). 5.4. obtained results our proposed custom dcnn model-based method is compared in table 2 with the method proposed in [15] and the second method based on the cnn used in in [16]. the method [15] is the only published method that was tested with the ucsd nighttime dataset. the method [14] is not included in the comparison, since only accuracy results with the daytime ucsd dataset were presented therein. from table 2, it can be seen that with an outstanding accuracy of 98.91%, our proposed outperforms the two other methods. in terms of speed, our method is faster than the method in [15] but slower than the method in [16]. sofiane abdelkrim khalladi, asmâa ouessai, mokhtar keche 138 copyright ©2024 assa adv. in systems science and appl. (2024) table. 2. comparison between the proposed method and the motion vector method. motion vectors method [15] cnn [16] proposed method accuracy 89.47% 96.18% 98.91% speed 127.23 sec 53 sec 95 sec the better accuracy achieved by our proposed custom model, compared to the model proposed in [16] may be explained by the fact that our model is deeper. more convolutional layers and maxpooling layers result in a complex architecture that allows capturing and extracting more complex hierarchical features in the data. on the other hand, deeper layers with a greater number of kernels enable the network to grasp higher-level abstractions. for comparison, our model uses 64 kernels per each layer instead, which is twice the number of layers used in the model of [16]. moreover, for a multi-class classification task, softmax activation in the last layer is more appropriate and performs better than sigmoid activation. the latter generates pseudo probabilities that do not necessarily sum up to 1. conversely, softmax guarantees that the probabilities sum is 1. for a further comparison between the two methods, we present in figures 8 and 9 the confusion matrices of the congestion classification obtained with these two methods. as shown from figure 8, there were only 3 missclassifications by the first method, which occurred between the light and the medium neighboring classes, denoted 0 and 1, respectively. comparatively, 11 missclassifications occurred by the second method, among which 8 between neighboring classes and 3 between distant classes (light and heavy). fig. 8. confusion matrix of congestion classification with the proposed method, using the ucsd dataset. sofiane abdelkrim khalladi, asmâa ouessai, mokhtar keche 139 copyright ©2024 assa. adv. in systems science and appl. (2024) fig. 9. confusion matrix of congestion classification with the method proposed in [16], using the ucsd dataset. 6. conclusion this research proposes a novel method for effectively classifying traffic congestion in surveillance films captured during nighttime. the proposed method employs a deep-learning neural network, which enables accurate identification of traffic congestion status under various weather conditions. the outcome of the performance assessment of this method using the ucsd nighttime dataset is an outstanding classification accuracy of 98.91%, which surpasses the previously known highest accuracy of 89.47% achieved by the method presented in [15] that also utilized the same dataset. furthermore, our approach outperforms, in terms of accuracy, the method based on the cnn proposed in [16], trained and tested by the same ucsd nighttime dataset. incorporation this method into a traffic monitoring system may help to assist drivers in avoiding traffic congestion and reaping the associated benefits. in our future work, we plan to further enhance our model's classification accuracy by refining the architecture of the convolutional neural network. acknowledgements the authors would like to thank dr. antoni. b. chan from the city university of hong kong for providing the ucsd night-time video traffic dataset. references 1. papageorgiou, m., diakaki, c., dinopoulou, v., kotsialos, a. & wang, y. 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(2018). generalized cross entropy loss for training deep neural networks with noisy labels, arxiv: 1805.07836, [online]. available: https://arxiv.org/abs/1805.07836 microsoft word 1216 article text, copyedited.doc adv syst sci appl 2022; 02; 62-72 published online at https://ijassa.ipu.ru. mathematical model of the infection spread in transport based on the theory of porous medium alexandra ocheretyanaya1*, alexander bratus1 1) russian university of transport (miit), moscow, russia e-mail: sandra.och@yandex.ru abstract: mathematical model of the spread of a virus in trains which considers transmission of infection from two types of sick passengers (clearly infected and latently infected) was built and named sei (susceptible-exposed-infected) model. ventilation and movements of the passengers inside the train are considered through the porous medium equation. an algorithm for filling empty seats in the train is developed. a route of ten hours duration with two stop stations was simulated. the results of numerical calculations allow to conclude that asymptomatic infected passengers are the most dangerous group in the spread of infection. keywords: mathematical model, sei model, train, spread of infection, porous medium equation 1. introduction the spread of coronavirus disease covid-19 caused serious problems for the economies and health systems of all the world. most of countries took the necessary precautions to slow down the spread of the infection, but these steps do not allow to live without limits. the results of mathematical modeling of disease dynamics can be used in emergency situations [21]. transport is a very dangerous channel which has huge influence on disease dynamics. bo xu et al. (2019) constructed the national highway network between 333 cities in china to show that road transport has significant impacts on pandemic influenza a (h1n1) in mainland china [19]. colizza et al. (2006) used worldwide air travel infrastructure to forecast global epidemics [6]. balcan et al. (2003) used airline traffic and small-scale commuting ows to analyze their role in the global epidemics [2]. therefore, after the outbreak of covid-19 many countries limited transport mobility. linka et al. (2020) suggested that an unconstrained mobility would have accelerated the spreading of the virus, especially in such countries as spain, and france [14]. analysis of the situation in italy confirms that, in addition with other factors (socio-economic, territorial and pollutant variables), the availability of the transport system had the main impact on covid-19 crisis [5]. cartenì et al. (2021) in further researches showed that the certified covid-19 cases certified are directly related to the number of public transport trips [4]. rail transport is a dangerous source of infection. zou et al. (2021) analyzed the influence of train-induced wind on the covid-19 pandemic and their results confirmed significance of rail transport to the transmission of the virus [22]. an outbreak of covid-19 started in wuhan, hubei. wuhan is located in the center of china and has a convenient transportation system and hubei is identified as a hub of traveling and transport in the country which played a big role in the spreading of covid-19 [15]. the transmission risk of covid-19 on high-speed trains in china in the period from 19 december 2019 through 6 march 2020 is quantified in [9]. this analysis helps to understand more clearly the risk of covid-19 * corresponding author: sandra.och@yandex.ru mathematical model of the infection spread in transport based on the theory 63 copyright ©2022 assa. adv. in systems science and appl. (2022) transmission among train passengers. viruses are transmitted through surfaces [18] and via airborne droplets, especially quickly indoors [12]. most of the studies connected with the modeling of covid-19 spread do not considered the influence of transport [10,16,17,20]. though there are various works about modeling of covid-19 spread via air travel (for instance, [7,13,14]), a few attention is paid to the disease transmission through rail travel passengers. the aim of this study is to investigate the infection spread connected with passengers transportation. to achieve it, a modified sei model is built. the infection spread within a train is described by the porous medium equation. a random process of filling free seats is considered through the developed algorithm and a pseudo-random number generator is used. 2. mathematical model modified sei model (susceptible-exposed-infected) with constant population n divided into three groups: susceptible (s), exposed (e) and infected (i) passengers. as for the rail travels the group of recovered is not so important because we study only the dynamics of the virus spreading, the infection itself is instantaneous. the consequences stretch over the time. isolation can be discontinued between 5 or 10 days after symptom onset. one of the longest train journeys is the trans–siberian railway, and one of the routes on it from moscow to vladivostok is about 7 days. these periods are very similar to each over, that is why it is assumed that during rail travel most of passengers spend less time than is required for full recovery. also, in the case of covid-19 the impact of asymptomatic patients is significant [8]. thus, modified sei model is the r group excluded and included the group of exposed (asymptomatic) passengers e, which also can infect susceptible passengers s as symptomatic infected individuals i. the compartmental model is formulated by the following set of ordinary differential equations: (2.1) (2.2) (2.3) (2.4) when symptomatic infected passengers i infect susceptible passengers s with the transmission rate , individuals from group s can become symptomatic infected with the rate or asymptomatic infected e with the rate at = . if asymptomatic infected passengers e infect susceptible passengers s with the transmission rate , the number of asymptomatic individuals increases at the rate . movements of susceptible, symptomatic infected and asymptomatic infected along the train mutually influence on each other. to consider it parabolic partial differential equations are used to enter variables responsible for motion. the train can be represented as a rod with a length l, which is insulated at the boundaries. characteristics of the spread of infection for groups inside the train (road) are functions of space and time ( ). they are presented as a porous medium equation: (2.5) 64 a. ocheretyanaya, a. bratus copyright ©2022 assa. adv. in systems science and appl. (2022) where is a constant which describes porosity. it is equal to the ratio of the volume occupied by pores to the total volume [3, pp. 7-9]: (2.6) the effect of porosity considers that the train has a general ventilation, through which the infection can be transmitted, and movements of the passengers inside the train and the restaurant car. the solutions or the porous medium equation converge to solutions of the heat equation when [1]. variables which describes each group of sei model and space (x) are normalized to one. the passengers from all the groups have influence on susceptible s and exposed e individuals. dynamics of symptomatic infected passengers i depends on the groups s and i. thus, considering the equations (2.1)(2.3) as the source terms and porosity (2.5) the model is formulated by the following set of equations: (2.7) (2.8) (2.9) (2.10) where is a positive definite matrix that characterized the intensity of infections. from station to station the train is isolated at the ends and the actual state of the system cannot be known. that is why we used second-type boundary condition, as known as neumann boundary condition, which specifies the value of the normal derivative of the function. this means that for an interval the boundary conditions for each group are: (2.11) (2.12) (2.13) each passenger has his own place and at every point of the train we can calculate to which group the passenger belongs to. we consider that asymptomatic e and symptomatic i passengers are formulated as functions, where parameters , specify the proportion of the sick passengers at each place of the train, the points determine the area of infection, and variance measures how sick passengers impact on susceptible neighbors. the initial conditions for the groups e and i are described by the following equations: (2.14) (2.15) parameters ( ) and ( ) refers to the groups e and i, respectively. mathematical model of the infection spread in transport based on the theory 65 copyright ©2022 assa. adv. in systems science and appl. (2022) the number of susceptible is determined from the (2.10): (2.16) determination of seats that sick passengers can buy is a random process. that is why, if the train is empty, values are determinated, using the pseudo-random number generator. equations (2.14) – (2.15) are calculated given number of times. the results obtained at each point are summarized. the arithmetic mean is taken. the algorithm to set the initial conditions at interchange points is very similar. it consists of the following steps: 1. divide the length of the train by the number of cars. 2. select the car that people leave using a pseudo-random number generator 3. choose how many percent leaves the car using a pseudo-random number generator. 4. divide this percentage evenly between each of the groups 5. make steps 2-4 until the train is free for a chosen percent 6. fill selected cars according to the statistics of each group depending on the city. 7. make steps 2-6 a given number of times to average the results. the transmission rate between susceptible passengers s with symptomatic infected passengers i is assumed to be =0,42. since passengers are indoors and the disease may not occur clearly immediately, probability to become asymptomatic infected e is higher than probability to become symptomatic infected i. therefore, the rate between susceptible passengers s with asymptomatic infected passengers e is equal to . the rate between susceptible passengers s with symptomatic infected passengers i is equal to . according to [11], the transmission rate is even higher than , and within this study of the model is assumed to be = 0,5. the matrix has positive definite eigenvalues, which ensures the non-negativity of the solutions, and is presented in the such form in this study: 3. results the total travel time is restricted by ten hours. there are two stop stations at which structure of passengers changed. the travel time between stations no greater than five hours. the train consists of sixteen cars, each with nine compartments. considering staff, the number of the passengers is . we set the porosity model (2.7) (2.9) with boundary and initial conditions (2.11) (2.16) are implemented with explicit method on python. the random module was used to generate pseudo-random numbers. fig. 3.1 shows the initial conditions for each group (s, e, i) at first station. panels (b)–(c) from fig.1 shows that the asymptomatic infected e and symptomatic infected i passengers are unevenly distributed throughout the train. the share of each group is 66 a. ocheretyanaya, a. bratus copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 3.1. the initial conditions for susceptible s (a) asymptomatic infected e (b) and symptomatic infected i (c) passengers at station one. the ordinate axis shows the share of each group ( ) at the point , the abscissa axis presents the location in the train ( ) mathematical model of the infection spread in transport based on the theory 67 copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 3.2 shows the dynamics over the time and space for each group through the trip from first to second station. it can be seen from the surfaces that the susceptible group (fig. 3.2 (a)) tends to fall, while the number of asymptomatic infected (fig. 3.2 (b)) passengers grows. the asymptomatic infected people almost do not change due to their inactivity. fig. 3.2. the surfaces for susceptible s (a) asymptomatic infected e (b) and symptomatic infected i (c) passengers during the trip. the abscissa axis presents the location in the train ( ), on the ordinate axis is time ( ), on the applicate axis is the share of each of each group ( ) consider the simulation results on arrival at station two of each group. fig. 3.3 shows the share at each point. the share of each group at last step is . during a five-hour trip the number of susceptible passengers decreased by 0,5 percent, number of asymptomatic infected passengers increased by 0,4 percent. it is clearly seen that location of symptomatic infected passengers (fig. 3.3 (с)) is very similar to initial conditions (fig.3.1 (с)). by comparing the dynamics over the time of asymptomatic infected passengers from departure (fig. 3.1 (b)) to arrival (fig. 3.3 (b)) it can be seen at the beginning most of the sick passengers are in the first half of the train. due to the effect of porosity, this difference is more noticeable at the end. the more asymptomatic infected passengers are in the area, the more possibility to get infected. 68 a. ocheretyanaya, a. bratus copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 3.3. the share for susceptible s (a) asymptomatic infected e (b) and symptomatic infected i (c) passengers on arrival at station two. the ordinate axis shows the share of each group ( ) at the point , the abscissa axis presents the location in the train ( ) fig. 3.4 shows the initial conditions for each group (s, e, i) at second station. based on the algorithm at the interchange points there are 7 percent who leaves the train and the free cars are filled by 98 percent susceptible and asymptomatic infected passengers mathematical model of the infection spread in transport based on the theory 69 copyright ©2022 assa. adv. in systems science and appl. (2022) respectively. it is assumed that the probability of taking seats by symptomatic infected passengers is very small with such filling. despite the small change in the initial conditions, there is a small difference between at arrival at second station (fig. 3.3) and departure from this station (fig. 3.4). the share of each group is . fig. 3.4. the initial conditions for susceptible s (a) asymptomatic infected e (b) and symptomatic infected i (c) passengers at station two. the ordinate axis shows the share of each group ( ) at the point , the abscissa axis presents the location in the train ( ) 70 a. ocheretyanaya, a. bratus copyright ©2022 assa. adv. in systems science and appl. (2022) the share of each group at last step is: the number of asymptomatic infected passengers e has already increased by 0,6 percent which is one and a half bigger than the results of simulation from first to second station. higher asymptomatic infected people lead to a faster spread od virus. the number symptomatic infected passengers, on the other hand, was almost unchanged. despite their danger during the trip, they are inactive and with a small change of number of passengers there is a small probability that free seats will be filled by people from this group. 5. conclusion modified sei model in the porous medium was built. the probability of infecting from clearly infected and latently infected groups, passenger movement and ventilation were considered. the numerical calculations showed that the effect of a porous medium affects the breadth of the spread of infection. an algorithm to determine the initial conditions in the case of filling a free train and at interchange stations was developed. this algorithm considers factor of random filling of seats of passengers of different groups. within the model problem, a small change in the composition of the train at the stop station led to increase of the virus spread. the numerical calculations made it possible to conclude that asymptomatic infected passengers have the greatest influence on the spread of infection than symptomatic infected. the increase in the number of asymptomatic infected passengers has led to a rapid increase spreading of disease. for a more detailed analysis, it is necessary to obtain reliable data on passenger traffic. acknowledgements this paper was written with the financial support of rfbr (grant no. 20-04-60157). references 1. aronson, d. g., vazquez, j. l., (1987). the porous medium equation as a finitespeed approximation to a hamilton-jacobi equation. annales de l'i.h.p. analyse non linéaire, 4 (3),203-230, http://www.numdam.org/item/aihpc_1987__4_3_203_0/ 2. balcan, d., colizza, v., gonçalves, b., hu, h., ramasco, j. j., et al. 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(2021). effect of train-induced wind on the transmission of covid-19: a new insight into potential infectious risks. international journal of environmental research and public health, 18(15), 8164, https://doi.org/10.3390/ijerph18158164 adv syst sci appl 2020; 03:73–90 published online at https://ijassa.ipu.ru. delay differential equation model of gene expression amit sharma1*, neeru adlakha1 1applied mathematics & humanities department, s.v. national institute of technology, surat-395007, gujarat, india. abstract: in this study, a delay differential equation model of gene expression for both retroviruses and normal cell is proposed to study the dynamics of functional gene products. the model is categorised into two sub-models to understand the characteristics of a cell by incorporating time delays in the processes of gene expression. the first model which is for retroviruses, involves time delay in replication, transcription, reverse transcription and translation processes taking place in the cell, while in the second model which is for normal cell, the time delay in transcription and translation processes are incorporated. a numerical solution is obtained using semi-temporal data set. the impact of time delays on temporal concentration profile of dna, mrna and proteins have been analysed which gives better insight into the normal cell as well as retroviruses. further, sensitivity analysis has been performed for both models to study the behaviour of gene expression in the cell. the results obtained from such models can be useful for biomedical applications. keywords: dna, rna, proteins, delay differential equations 1. introduction the cell has a very sophisticated control system which dynamically initiates, sustains and terminates the processes of replication, transcription, reverse transcription and translation in the cell. another vital characteristic of a control system of the cell is its ability to dynamically delay the processes of replication, transcription, reverse transcription and translation in order to regulate the concentrations of dna, mrna and proteins. apart from this, the delay may be caused in these processes of the cell due to some external influence, noise in the cell signal, any abnormality in the environment or due to any disease. the delay in these processes may either be useful to the cell or may have some harmful effects on the cell and the organism. thus, these delays in the processes may contribute positively to the control system of the cell and its gene expression or it may have a negative impact on the control system of the cell. experimental investigations have been performed by a number of research workers to study the expression [2, 3, 6, 8]. the experimental investigations are quite expensive and time-consuming, and therefore the scientists have also explored the theoretical approaches to study the gene expression [5, 15, 18, 20, 21, 22]. further, mathematical modeling plays a vital role for better understanding of the complex real-world problems in every area, for example, supply chain management, inventory theory etc [23]. in terms of differential equations, delay differential equations are more appropriate for modeling of the complex real-world problem, for example, prey-predator model [12, 17], chemostat models [31], circadian rhythms [24], epidemiology [7], the respiratory system [27], tumor growth [28] and neural networks [4]. thus, delay differential equation is one of the possible approaches to deal and unravel the complexities of real-world problems, and gene ∗corresponding author: amitsharmajrf@gmail.com 74 a. sharma, n. adlakha expression entities are also one of them. according to experimental data, there are some delays in the processes of replication, transcription, reverse transcription and translation to be completed, for example, the transcription process takes about 20 minutes to complete and a model is reported having both transcriptional and translational time delays [13]. some delay differential equation models are reported in the literature for the study of gene expression [1, 10, 11]. a model with discrete time delay, by using lindstedt’s method and the hopf bifurcation is reported to study the gene expression [26]. also, a temporal model, which demonstrates the intracellular signalling using delay differential equations is reported in the literature [25]. also, some research workers have developed a model with distributed time delay and with translational time delay using the fourth order runge-kutta method to study gene expression [9, 16]. further, in spite of experimental and theoretical investigations, the gene expression has still not been well understood. thus, in order to have a better understanding of gene expression and control system of the cell, it is necessary to develop models of gene expression involving delays in the processes of a cell. also, from the literature survey, it is evident that very few attempts are reported in the literature for modeling of gene expression using delay differential equation, and no attempts are reported in the literature for the development of michaelis-menten’s mechanism-based delay differential equations model to study gene expression. in this paper, two models based on delay differential equation are proposed to study the gene expression. the michaelis-menten’s mechanism is incorporated in these models. in the first model, the time delay in all four processes replication, transcription, reverse transcription and translation has been considered. in the second model, transcription and translation processes are considered with different time delay. the impact of time delays on the temporal concentration profile of dna, mrna and proteins have been analysed with the help of numerical results. the mathematical model is presented in the next section. 2. mathematical model in this section, two different delay differential equation models are proposed for the gene expression in retroviruses and a normal cell, respectively. in order to develop the model, the following assumptions are made: 1. there can be time delays in the processes of replication, transcription, reverse transcription and translation. 2. the variation in the temperature is constant throughout the model, so that, there is no effect of temperature on replication, transcription, reverse transcription and translation processes in the gene. 3. the rates of replication, transcription, reverse transcription and translation processes in the cell lie in the interval [0, 1] . 4. a single cell is considered throughout the model, and as the replication of dna occurs at the time of cell division in the baby cell from the mother cell, it is assumed that the replication occurs in the cell. 5. a single protein is synthesized in a single mrna transcript. in the present study, two mathematical models have been proposed using michaelismenten’s mechanism. in the first model, the processes of replication, transcription, reverse transcription and translation are incorporated as these processes take place in retroviruses. the first model called “model-i” is described by the system of delay differential equations (2.1)− (2.3): dw dt = k0w(t− τ1)− k1w(t− τ2) + k2x(t− τ3), (2.1) dx dt = k1w(t− τ2)− k2x(t− τ3)− k3x(t− τ4), (2.2) copyright © 2020 assa. adv syst sci appl (2020) delay differential equation model of gene expression 75 dp dt = k3x(t− τ4), t ≥ 0, τi > 0, i = 1, 2, 3, 4. (2.3) here k0, k1, k2 and k3 are constants. in the second model, it is assumed that the replication and reverse transcription processes are absent, and transcription and translation processes are taking place in the cell. in general, the replication process stops taking place in a mature cell and the reverse transcription is absent in a normal cell. in view of the above, the second model is proposed incorporating transcription and translation processes only. the second model called “model-ii” is described by the system of delay differential equations (2.4)− (2.6): dw dt = −k1w(t− τ2), (2.4) dx dt = k1w(t− τ2)− k3x(t− τ4), (2.5) dp dt = k3x(t− τ4), t ≥ 0, τi > 0, i = 2, 4. (2.6) we consider systems of differential equations (2.1)− (2.3) and (2.4)− (2.6) with the initial condition function φ : [−τ, 0]→ r3, where τ represents time delay. the time delays are taken as positive constants [30]. further, if all τi = 0, then there is no delay in the system and for without delay system, the initial condition is consider as w(t) = w0, x(t) = x0, p(t) = p0. here —————————————————————————————w(t) concentration of dna in the cell at time t (in second). x(t) concentration of mrna in the cell at time t (in second). p(t) concentration of protein in the cell at time t (in second). k0 rate of replication (microgram/second). k1 rate of transcription (microgram/second). k2 rate of reverse transcription (microgram/second). k3 rate of translation (microgram/second). φ initial history function. τ time delay. τ1 delay in replication process. τ2 delay in transcription process. τ3 delay in reverse transcription process. τ4 delay in translation process. —————————————————————————————for numerical solution, we use the built-in matlab programme in the optimization toolbox [19] for both model-i and model-ii, which is presented in the next section. 3. results the rates of replication, transcription, reverse transcription and translation processes will depend on the circumstances and capacity of the cell, and therefore k0, k1, k2 and k3 can be assigned different values. the data of the strain tjk16 is used to compute the results [6]. generally, the values of k0, k1, k2 and k3 lie between [0, 1] [14, 29]. the results obtained for the above system of delay differential equations of both model-i and model-ii, table 3.1, along with the comparison study between with time delay and without time delay of both models, are shown in fig. 3.1 and fig. 3.2. here, four cases are discussed regarding the time lag of replication, transcription, reverse transcription and translation processes, which are shown in table 3.2. in the first case, copyright © 2020 assa. adv syst sci appl (2020) 76 a. sharma, n. adlakha 0 20 40 60 80 100 120 140 160 180 200 −50 0 50 100 150 time c on ce nt ra tio n model−i, with delay (initial condition are 25, 0, 0) 0 20 40 60 80 100 120 140 160 180 200 0 10 20 30 40 50 time c on ce nt ra tio n model−i, without delay (initial condition are 25, 0, 0) 0 20 40 60 80 100 120 140 160 180 200 −50 0 50 100 150 time c on ce nt ra tio n model−i, with delay (initial condition are 25, 25, 25) 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 time c on ce nt ra tio n model−i without delay (initial condition are 25, 25, 25) dna mrna protein fig. 3.1. graphical representation of the solution of the concentrations of dna, mrna and protein with time delay in the replication, transcription, reverse transcription and translation processes (both left figures where the initial conditions of dna, mrna and protein are (25, 0, 0) and (25, 25, 25) respectively), and without time delay in the replication, transcription, reverse transcription and translation processes are shown (both right figures where the initial conditions of dna, mrna and protein are (25, 0, 0) and (25, 25, 25) respectively, of model-i). the values of parameters are spatio-temporal which exhibit the stability of the system. 0 20 40 60 80 100 120 140 160 180 200 −20 −10 0 10 20 30 40 50 time c on ce nt ra tio n model−ii, with delay (initial condititons are 25, 0, 0) 0 20 40 60 80 100 120 140 160 180 200 0 5 10 15 20 25 time c on ce nt ra tio n model−ii, without delay (initial condititons are 25, 0, 0) dna mrna protein 0 20 40 60 80 100 120 140 160 180 200 −20 0 20 40 60 80 100 120 time c on ce nt ra tio n model−ii, with delay (initial condititons are 25, 25, 25) 0 20 40 60 80 100 120 140 160 180 200 0 10 20 30 40 50 60 70 time c on ce nt ra tio n model−ii, without delay (initial condititons are 25, 25, 25) fig. 3.2. graphical representation of the solution of the concentrations of dna, mrna and protein with time delay in the replication, transcription, reverse transcription and translation processes (both left figures where the initial conditions of dna, mrna and protein are (25, 0, 0) and (25, 25, 25) respectively), and without time delay in the replication, transcription, reverse transcription and translation processes are shown (both right figures where the initial conditions of dna, mrna and protein are (25, 0, 0) and (25, 25, 25) respectively, of model-ii). the parameter values of transcription and translation rates k1 = 0.42, k3 = 2.59 respectively, are taken from the literature [29]. copyright © 2020 assa. adv syst sci appl (2020) delay differential equation model of gene expression 77 table 3.1. parameters related to functional gene products dna, mrna and protein for strain tjk16 of both models i and ii, and the time is t=200 sec. model-i model-ii replication lag τ1 (in second) 1.7 transcription lag τ2 (in second) 2.3 3.7 reverse transcription lag τ3 (in second) 5.2 translation lag τ4 (in second) 8.6 0.6 replication rate k0 (microgram/second) 0.1425 transcription rate k1 (microgram/second) 0.4785 0.42 reverse transcription rate k2 (microgram/second) 0.2568 translation rate k3 (microgram/second) 0.3691 2.59 time t (in second) 200 200 table 3.2. parameters related to the time lag in the replication, transcription, reverse transcription and translation processes of functional gene products dna, mrna and protein for strain tjk16 at time t=200 sec. model-i τ1 < τ2 < τ3 < τ4 τ1 > τ2 > τ3 > τ4 τ1 = τ2 > τ3 = τ4 τ1 > τ2 < τ3 < τ4 replication lag τ1 (in second) 1.7 7.0 4.2 1.7 transcription lag τ2 (in second) 2.3 6.2 4.2 1.2 r. transcri. lag τ3 (in second) 5.2 3.1 1.9 5.2 translation lag τ4 (in second) 8.6 1.9 1.9 8.6 time t (in second) 200 200 200 200 the time lag of replication, transcription, reverse transcription and translation processes are taken in increasing order as τ1 < τ2 < τ3 < τ4 respectively. in the second case, the time lag of replication, transcription, reverse transcription and translation processes are taken in decreasing order as τ1 > τ2 > τ3 > τ4 respectively. in the third case, the time lag of replication process is equal to the transcription process, and the time lag of reverse transcription process is equal to the translation process, which is taken as τ1 = τ2 > τ3 = τ4. finally, in the fourth case, the time lag of the replication process is greater than that of the transcription process and in turns lesser than both the reverse transcription and translation processes, which are taken as τ1 > τ2 < τ3 < τ4. the above mentioned conditions of time lag for model-i are taken in the replication, transcription, reverse transcription and translation processes, respectively and the results are shown in fig. 3.3. copyright © 2020 assa. adv syst sci appl (2020) 78 a. sharma, n. adlakha 0 50 100 150 200 −200 0 200 time c on ce nt ra tio n tau 1 < tau 2 < tau 3 < tau 4 0 50 100 150 200 −2000 0 2000 time c on ce nt ra tio n tau 1 > tau 2 > tau 3 > tau 4 0 50 100 150 200 −100 0 100 time c on ce nt ra tio n tau 1 > tau 2 > tau 3 > tau 4 0 50 100 150 200 −1 0 1 x 10 11 time c on ce nt ra tio n tau 1 < tau 2 < tau 3 < tau 4 0 50 100 150 200 −100 0 100 time c on ce nt ra tio n tau 1 = tau 2 > tau 3 = tau 4 0 50 100 150 200 −1 0 1 x 10 9 time c on ce nt ra tio n tau 1 = tau 2 < tau 3 = tau 4 0 50 100 150 200 −200 0 200 time c on ce nt ra tio n tau 1 > tau 2 < tau 3 < tau 4 dna mrna protein 0 50 100 150 200 −100 0 100 time c on ce nt ra tio n tau 1 > tau 2 > tau 3 < tau 4 fig. 3.3. change in the concentrations profiles of functional gene products with the change in time lags of model-i. the above graphical representation shows the change in the concentrations of dna, mrna and protein with different time lags taken in the replication, transcription, reverse transcription and translation processes, respectively. the left figures show the stability of the model while the right figures except for the fourth one (right lower figure) indicate that the system is not stable and the system bursts. the right figures are generated after reversing the order of time lags corresponding to each left figures, respectively. 4. sensitivity analysis the model can be demonstrated precisely using sensitivity analysis. based on sensitivity, it can be revealed that the particular model is stable, semi-stable or unstable. regarding this, we consider both models i and ii for sensitivity analysis. first, we analyze the sensitivity of both the models based on the rates of replication, transcription, reverse transcription and translation processes. for model-i, the results are given in table 4.3 and the graphical solutions are shown in fig. 4.4, fig. 4.5, fig. 4.6 and fig. 4.7, while for model-ii, the results are shown in table 4.4, and the graphical solutions are shown in fig. 4.8 and fig. 4.9. secondly, we analyze the sensitivity of both models based on lags or delays in replication, transcription, reverse transcription and translation processes. for model-i, the results are given in table 4.5, and shown in fig. 4.10, 4.11, 4.12 and 4.13, while for model-ii, the results are given in table 4.6, and shown in fig. 4.14 and 4.15. 5. discussion in the left parts of fig. 3.1 we observe the variation in the concentration of dna, mrna and protein with delay in the processes and without delay in the processes for the initial values of dna, mrna and protein as (25, 0, 0) for a newly born cell and (25, 25, 25) for mature cell, respectively. the oscillations are observed in dna, mrna and protein concentrations due to the delays in the processes in replication, transcription, reverse transcription and translation (both left parts of fig. 3.1), and in case of without delays, the smooth curves are seen in dna, mrna and proteins concentrations when the delay is absent in replication, transcription, reverse transcription and translation processes (both right parts of fig. 3.1). due to change in the initial values of dna, mrna and proteins concentrations from (25, 0, 0) to (25, 25, 25), the small oscillation is observed (both left parts of fig. 3.1). in case of processes copyright © 2020 assa. adv syst sci appl (2020) delay differential equation model of gene expression 79 table 4.3. sensitivity analysis of model-i with the parameter k0, k1, k2 and k3, respectively, and the initial condition of each dna, mrna and protein concentration as 25 micrograms. the value of delays τ1, τ2, τ3 and τ4 is 1.7, 2.3, 5.2 and 8.6, respectively. model-i replication rate k0 (microgram/second) 0.1425 +20% +50% -20% -50% replication rate k0 (microgram/second) 0.1710 0.2138 0.1140 0.0713 dna concentration (in microgram) -52.3469 2.4164e+03 -0.3831 -5.3790 mrna concentration (in microgram) 35.1480 -2.6285e+03 0.2160 -7.9647 protein concentration (in microgram) 144.1702 34.4349 87.6179 96.5009 transcription rate k1 (microgram/second) 0.4785 transcription rate k1 (microgram/second) 0.5742 0.7178 0.3828 0.2393 dna concentration (in microgram) 126.7853 5.6552e+05 1.8871e+03 -1.7556e+06 mrna concentration (in microgram) -93.0348 -4.6064e+05 -1.1285e+03 9.6616e+06 protein concentration (in microgram) 8.8427 -1.9215e+05 -1.4282e+03 -1.2712e+07 reverse transcription rate k2 (microgram/second) 0.2568 reverse transcription rate k2 (microgram/second) 0.3082 0.3852 0.2054 0.1284 dna concentration (in microgram) 1.4106e+03 -1.9018e+03 0.2606 488.0431 mrna concentration (in microgram) -1.3330e+03 1.6340e+06 -0.0292 -8.5203 protein concentration (in microgram) -32.0213 7.4441e+05 90.3924 -416.6525 translation rate k3 (microgram/second) 0.3691 translation rate k3 (microgram/second) 0.4430 0.5537 0.2953 0.1846 dna concentration (in microgram) -0.4949 -6.2232e+03 30.9181 -750.3591 mrna concentration (in microgram) 0.4716 -5.5662e+03 -31.0534 742.9808 protein concentration (in microgram) 88.9815 1.5255e+03 103.0534 295.4582 without delay, when the initial values of dna, mrna and proteins concentrations changes from (25, 0, 0) to (25, 25, 25), the oscillation in the concentration of mrna has vanished (both right parts of fig. 3.1). for the proposed model-ii, there are only transcription and translation processes with delays. in fig. 3.2, the same behaviour of the dna, mrna and copyright © 2020 assa. adv syst sci appl (2020) 80 a. sharma, n. adlakha 0 50 100 150 200 −100 −50 0 50 100 150 time c on ce nt ra tio n +20% 0 50 100 150 200 −50 0 50 100 150 time c on ce nt ra tio n −20% 0 50 100 150 200 −3000 −2000 −1000 0 1000 2000 3000 time c on ce nt ra tio n +50% 0 50 100 150 200 −50 0 50 100 150 time c on ce nt ra tio n −50% dna mrna protein fig. 4.4. sensitivity analysis of the parameter replication rate of model-i with an increase of 20% and 50%, and decrease of 20% and 50% in the base value. 0 50 100 150 200 −200 −100 0 100 200 time c on ce nt ra tio n +20% 0 50 100 150 200 −2000 −1000 0 1000 2000 3000 time c on ce nt ra tio n −20% 0 50 100 150 200 −6 −4 −2 0 2 4 6 x 10 5 time c on ce nt ra tio n +50% 0 50 100 150 200 −2 −1 0 1 2 x 10 7 time c on ce nt ra tio n −50% dna mrna protein fig. 4.5. sensitivity analysis of the parameter transcription rate of model-i with an increase of 20% and 50%, and decrease of 20% and 50% in the base value. protein concentration profiles are observed as shown in fig. 3.1 where oscillations occur due to delays in transcription and translation processes and smooth curves are observed due to no delay in transcription and translation processes. the values of concentrations profiles of dna, mrna and protein are different in both fig. 3.1 and fig. 3.2. overall, both figures depicted that model-i and model-ii are stable. further, four cases related to delays in the replication, transcription, reverse transcription and translation processes of model-i are shown in fig. 3.3. the concentration profiles of dna, mrna and protein vary with the change in delays of replication, transcription, reverse copyright © 2020 assa. adv syst sci appl (2020) delay differential equation model of gene expression 81 0 50 100 150 200 −2000 −1000 0 1000 2000 time c on ce nt ra tio n +20% 0 50 100 150 200 −50 0 50 100 150 time c on ce nt ra tio n −20% 0 50 100 150 200 −2 −1 0 1 2 x 10 6 time c on ce nt ra tio n +50% 0 50 100 150 200 −1500 −1000 −500 0 500 1000 1500 time c on ce nt ra tio n −50% dna mrna protein fig. 4.6. sensitivity analysis of the parameter reverse transcription rate of model-i with an increase of 20% and 50%, and decrease of 20% and 50% in the base value. 0 50 100 150 200 −50 0 50 100 150 time c on ce nt ra tio n +20% dna mrna protein 0 50 100 150 200 −50 0 50 100 150 time c on ce nt ra tio n −20% 0 50 100 150 200 −2 −1 0 1 2 x 10 4 time c on ce nt ra tio n +50% 0 50 100 150 200 −1000 −500 0 500 1000 time c on ce nt ra tio n −50% fig. 4.7. sensitivity analysis of the parameter translation rate of model-i with an increase of 20% and 50%, and decrease of 20% and 50% in the base value. transcription and translation processes, respectively. the left four figures along with right lower figure indicate the stability of the model while the figures on right hand side except for the fourth one(right lower figure) indicate that the model is unstable and the system bursts. the sensitivity analysis of model-i, fig. 4.4 shows the concentration profiles of dna, mrna and protein when the replication rate increases by 20% and 50%, and decreases by 20% and 50% in the base value. the oscillation decreases with the decrease in the replication rate (both right parts of fig. 4.4). it is observed from the figure that when the replication rate is lower, the model is stable (both right parts of the fig. 4.4) while the increase in replication copyright © 2020 assa. adv syst sci appl (2020) 82 a. sharma, n. adlakha table 4.4. sensitivity analysis of model-ii with the parameter k1 and k3, respectively, and the initial condition of each dna, mrna and protein concentration as 25 micrograms. the value of delays τ2 and τ4 is 3.7 and 0.6, respectively. model-ii transcription rate k1 (microgram/second) 0.42 +20% +50% -20% -50% transcription rate k1 (microgram/second) 0.50 0.63 0.34 0.21 dna concentration (in microgram) 1.8443e+03 1.3452e+08 -0.0028 5.3398e-11 mrna concentration (in microgram) 2.7789e+03 3.9139e+06 0.8346 0.8983 protein concentration (in microgram) -4.5482e+03 -1.3844e+08 74.1682 74.1017 translation rate k3 (microgram/second) 2.59 translation rate k3 (microgram/second) 3.11 3.89 2.07 1.30 dna concentration (in microgram) -16.4334 -16.4334 -16.4106 -16.4070 mrna concentration (in microgram) -1.3011e+19 3.2367e+42 -1.5741 -1.8137 protein concentration (in microgram) 1.3011e+19 -3.2367e+42 92.9847 93.2207 0 50 100 150 200 −1 −0.5 0 0.5 1 x 10 4 time c on ce nt ra tio n +20% 0 50 100 150 200 −20 0 20 40 60 80 100 time c on ce nt ra tio n −20% dna mrna protein 0 50 100 150 200 −1 −0.5 0 0.5 1 x 10 8 time c on ce nt ra tio n +50% 0 50 100 150 200 −20 0 20 40 60 80 100 time c on ce nt ra tio n −50% fig. 4.8. sensitivity analysis of the parameter transcription rate of model-ii with an increase of 20% and 50%, and decrease of 20% and 50% in the base value. rate causes bursts in the system and the model will be unstable (both left parts of the fig. copyright © 2020 assa. adv syst sci appl (2020) delay differential equation model of gene expression 83 0 50 100 150 200 −1 −0.5 0 0.5 1 x 10 19 time c on ce nt ra tio n +20% 0 50 100 150 200 −20 0 20 40 60 80 100 time c on ce nt ra tio n −20% 0 50 100 150 200 −3 −2 −1 0 1 2 3 x 10 42 time c on ce nt ra tio n +50% 0 50 100 150 200 −20 0 20 40 60 80 100 time c on ce nt ra tio n −50% dna mrna protein fig. 4.9. sensitivity analysis of the parameter translation rate of model-ii with an increase of 20% and 50%, and decrease of 20% and 50% in the base value. 0 50 100 150 200 −50 0 50 100 150 time c on ce nt ra tio n +20% 0 50 100 150 200 −50 0 50 100 150 time c on ce nt ra tio n −20% 0 50 100 150 200 −50 0 50 100 150 time c on ce nt ra tio n +50% 0 50 100 150 200 −1500 −1000 −500 0 500 1000 time c on ce nt ra tio n −50% dna mrna protein fig. 4.10. sensitivity analysis of the parameter replication with time delay for model-i with an increase of 20% and 50%, and decrease of 20% and 50% in the base value. 4.4). in fig. 4.5, it can be seen that the increase or decrease in transcription rate leads to system burst and makes the model unstable. thus in fig. 4.5, the model is highly sensitive as the variation in transcription rate leads to instability. in fig. 4.6, it is seen that for the small decrease in reverse transcription rate, the model remains stable (right upper part of the fig. 4.6) but for a substantial and larger amount of decrease in reverse transcription rate, the model becomes unstable (right lower part of fig. 4.6). also, the model becomes unstable on increasing the reverse transcription rate (both left parts of fig. 4.6). in fig. 4.7, it is observed that the model is stable for a small increase in translation rate (left upper part of fig. 4.7) copyright © 2020 assa. adv syst sci appl (2020) 84 a. sharma, n. adlakha table 4.5. sensitivity analysis of model-i with the parameter τ1, τ2, τ3 and τ4, respectively, and the initial condition of each dna, mrna and protein concentration is 25 micrograms. the value of rates k0, k1, k2 and k3 are 0.1425, 0.4785, 0.2568 and 0.3691, respectively. model-i replication lag τ1 (second) 1.7 +20% +50% -20% -50% replication lag τ1 (second) 2.04 2.55 1.36 0.85 dna concentration (in microgram) -0.3808 0.0017 35.9046 -1.0507e+03 mrna concentration (in microgram) 0.2621 2.1784e-04 -31.9812 815.3509 protein concentration (in microgram) 96.1922 99.6338 89.4417 348.8821 transcription lag τ2 (second) 2.3 transcription lag τ2 (second) 2.76 3.45 1.84 1.15 dna concentration (in microgram) -4.3923e+05 -6.3182e+07 0.00029 7.2881 mrna concentration (in microgram) 4.4777e+05 5.6421e+08 7.0222e-04 -24.9418 protein concentration (in microgram) 8.0491e+03 -1.5083e+09 96.8251 125.2183 reverse transcription lag τ3 (second) 5.2 reverse transcription lag τ3 (in second) 6.24 7.8 4.16 2.6 dna concentration (in microgram) -211.9863 6.6169e+05 0.3168 -2.2200 mrna concentration (in microgram) 209.0774 -7.9784e+05 -0.1869 -1.6293 protein concentration (in microgram) 127.0256 -4.6294e+04 89.3957 87.2496 translation lag τ4 (second) 8.6 translation lag τ4 (second) 10.32 12.9 6.88 4.3 dna concentration (in microgram) -10.3475 -1.1970e+04 -3.2942e+05 1.5779e+10 mrna concentration (in microgram) -8.1428 5.9625e+03 1.1022e+05 -3.3152e+10 protein concentration (in microgram) 109.7224 8.6237e+03 4.2001e+05 1.7112e+10 but for the larger increase in translation rate, the model becomes unstable (left lower part of fig. 4.7). also, with a decrease in translation rate, the increase in the size of oscillations is observed and it leads to instability (both right parts of fig. 4.7). for model-ii, in fig. 4.8, it can be seen that the size of oscillations increases with the increase in the transcription rate causing the system burst, and thus, the model will be unstable (both left parts of fig. 4.8). also, the size of oscillations decreases with a decrease copyright © 2020 assa. adv syst sci appl (2020) delay differential equation model of gene expression 85 0 50 100 150 200 −5 0 5 x 10 5 time c on ce nt ra tio n +20% 0 50 100 150 200 −50 0 50 100 150 time c on ce nt ra tio n −20% dna mrna protein 0 50 100 150 200 −3 −2 −1 0 1 2 3 x 10 9 time c on ce nt ra tio n +50% 0 50 100 150 200 −50 0 50 100 150 time c on ce nt ra tio n −50% fig. 4.11. sensitivity analysis of the parameter transcription with time delay for model-i with an increase of 20% and 50%, and decrease of 20% and 50% in the base value. 0 50 100 150 200 −300 −200 −100 0 100 200 300 time c on ce nt ra tio n +20% 0 50 100 150 200 −50 0 50 100 150 time c on ce nt ra tio n −20% dna mrna protein 0 50 100 150 200 −50 0 50 100 150 time c on ce nt ra tio n −50% 0 50 100 150 200 −1 −0.5 0 0.5 1 x 10 6 time c on ce nt ra tio n +50% fig. 4.12. sensitivity analysis of the parameter reverse transcription with time delay for model-i with an increase of 20% and 50%, and decrease of 20% and 50% in the base value. in transcription rate and the oscillations converge to make the system stable (both right parts of the fig. 4.8). in fig. 4.9, it can be seen that with the decrease in translation rate, the oscillations converge thereby making the model stable (both right parts of fig. 4.9). but the increase in translation rate leads to an increase in the size of oscillations, thereby making the model unstable (both right parts of the fig. 4.9). in fig. 4.10, it can be seen that the decrease in the time delay in replication process makes the model unstable (both right parts of fig. 4.10), while model attains stability with the increase in the time delay in replication process (both left parts of fig. 4.10). copyright © 2020 assa. adv syst sci appl (2020) 86 a. sharma, n. adlakha 0 50 100 150 200 −50 0 50 100 150 time c on ce nt ra tio n +20% 0 50 100 150 200 −10 −5 0 5 x 10 5 time c on ce nt ra tio n −20% 0 50 100 150 200 −1.5 −1 −0.5 0 0.5 1 1.5 x 10 4 time c on ce nt ra tio n +50% 0 50 100 150 200 −4 −3 −2 −1 0 1 2 x 10 10 time c on ce nt ra tio n −50% dna mrna protein fig. 4.13. sensitivity analysis of the parameter translation with time delay for model-i with an increase of 20% and 50%, and decrease of 20% and 50% in the base value. table 4.6. sensitivity analysis of model-ii with the parameter τ2 and τ4, respectively. here the initial values of each dna, mrna and protein concentration are 25 micrograms. the value of rates k1 and k3 is 0.42 and 2.59, respectively. model-ii transcription lag τ2 (second) 3.7 +20% +50% -20% -50% transcription lag τ2 (second) 4.44 5.55 2.96 1.85 dna concentration (in microgram) 5.7725e+03 7.6598e+05 -2.1140e-05 7.1306e-23 mrna concentration (in microgram) -604.6839 3.2312e+04 0.8043 0.8564 protein concentration (in microgram) -5.0929e+03 -7.9822e+05 74.1957 74.1436 translation lag τ4 (second) 0.6 translation lag τ4 (second) 0.72 0.9 0.48 0.3 dna concentration (in microgram) -16.4332 -16.4329 -16.4184 -16.4080 mrna concentration (in microgram) 7.9425e+15 -4.8743e+28 -1.2182 -0.9982 protein concentration (in microgram) -7.9425e+15 4.8743e+28 92.6366 92.4061 in fig. 4.11, it can be seen that the increase in the time delay in transcription process makes the model unstable (both left parts of fig. 4.11), while a small decrease in the time delay in transcription process, oscillations converge and lead to the stability of the system (right upper part of fig. 4.11), and further, for a larger decrease in the time delay copyright © 2020 assa. adv syst sci appl (2020) delay differential equation model of gene expression 87 0 50 100 150 200 −4000 −2000 0 2000 4000 6000 time c on ce nt ra tio n +20% 0 50 100 150 200 −20 0 20 40 60 80 100 time c on ce nt ra tio n −20% dna mrna protein 0 50 100 150 200 −5 0 5 x 10 5 time c on ce nt ra tio n +50% 0 50 100 150 200 −20 0 20 40 60 80 100 time c on ce nt ra tio n −50% fig. 4.14. sensitivity analysis of the parameter transcription with time delay for model-ii with an increase of 20% and 50%, and decrease of 20% and 50% in the base value. 0 50 100 150 200 −1 −0.5 0 0.5 1 x 10 16 time c on ce nt ra tio n +20% 0 50 100 150 200 −20 0 20 40 60 80 100 time c on ce nt ra tio n −20% dna mrna protein 0 50 100 150 200 −4 −2 0 2 4 x 10 28 time c on ce nt ra tio n +50% 0 50 100 150 200 −20 0 20 40 60 80 100 time c on ce nt ra tio n −50% fig. 4.15. sensitivity analysis of the parameter translation with time delay for model-ii with an increase of 20% and 50%, and decrease of 20% and 50% in the base value. in transcription process, model is semi-stable (right lower part of fig. 4.11). in fig. 4.12, for the decrease in the time delay in reverse transcription process, the oscillations converge and lead to stability of the system (both right parts of fig. 4.12), while for the increase in the time delay in reverse transcription process, the model becomes unstable (both left parts of fig. 4.12). in fig. 4.13, for the small increase in the time delay in translation process, the oscillations converge and the model becomes stable (left upper part of fig. 4.13), while a larger increase in the time delay in translation process, leads to instability as oscillations do not converge (left lower part of fig. 4.13). the decrease in the time delay in translation copyright © 2020 assa. adv syst sci appl (2020) 88 a. sharma, n. adlakha process makes the model unstable as oscillations do not converge (both right parts of fig. 4.13). for model-ii, in fig. 4.14, it can be seen that the model is stable as oscillations converge when time delays in transcription process decrease (both right parts of fig. 4.14), while the model is unstable when the time delays in transcription process increase (both left parts of fig. 4.14). in fig. 4.15 it can be seen that when the time delay in translation process increases then the model becomes unstable and due to oscillations in an increasing manner the system bursts (both left parts of fig. 4.15), while the decrease in the time delay in the translation process causes a decrease in size of oscillations and it makes the system stable (both right parts of fig. 4.15). 6. conclusion the proposed model, categorised into two models for retroviruses and normal cell, is employed to study the effect of time delays in the processes of replication, transcription, reverse transcription and translation of the gene expression. it is concluded from the results that the delay in these processes like replication, transcription, reverse transcription and translation causes more dynamic changes in concentrations of dna, mrna and proteins in comparison to the system without delay in these processes. the concentration profiles of dna, mrna and proteins in the cell for the case without delay are smooth. but the delay in the processes causes the disturbance in the system and therefore the control system of the cell exerts control on these processes to coordinate with each other to regulate the concentration profiles of dna, mrna and proteins and thus leads to oscillations. if the delay in these processes is larger, it causes larger disturbances which go beyond the control of the cell thereby making the system unstable. it is concluded that the system is highly sensitive to the rates of these processes replication, transcription, reverse transcription and translation as well as time delay in these processes. the impact of time delays on temporal concentration profile of dna, mrna and proteins have been analysed which gives better insight into the normal cell as well as retroviruses. thus, these two models give us interesting and useful information about the impact of rates of replication, transcription, reverse transcription and translation in presence or absence of time delay in these processes of gene expression which can be useful for various biomedical applications. acknowledgements the first author is grateful to council of scientific & industrial research (csir), new delhi, india, award no.09/1007(0002)/2009 for giving financial assistance as jrf/srf. also, the authors are thankful to department of biotechnology (dbt), new delhi, india for providing bioinformatics infrastructure facility at svnit, surat to carry out this work. references [1] bernard, s., čajavec, b., pujo-menjouet, l., mackey, m. c., and herzel, h. (2006). modelling transcriptional feedback loops: the role of gro/tle1 in hes1 oscillations. philosophical transactions of the royal society of london a: mathematical, physical and engineering sciences, 364(1842):1155–1170. [2] blake, w., kaern, m., cantor, c., and collins, j. (2003). noise in eukaryotic gene expression. nature, 422(6932):633–637. copyright © 2020 assa. adv syst sci appl (2020) delay differential equation model of gene expression 89 [3] bremer, h. and dennis, p. (1996). escherichia coli and salmonella: cellular and molecular biology, chapter modulation of chemical composition and other parameters of the cell by growth rate, pages 1553–1569. american society for microbiology, washington (dc). [4] campbell, s. a., edwards, r., and van den driessche, p. (2004). delayed coupling between two neural network loops. siam journal on applied mathematics, 65(1):316– 335. [5] chen, t., he, h. l., church, g. m., et al. 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(2009). on nonlinear dynamics of predator-prey models with discrete delay. mathematical modelling of natural phenomena, 4(02):140–188. [18] rué, p. and garcia-ojalvo, j. (2013). modeling gene expression in time and space. annual review of biophysics, 42:605–627. [19] shampine, l. f., thompson, s., and kierzenka, j. (2002). solving delay differential equations with dde23. manuscript, available at http://www. mathworks. com/dde tutorial. copyright © 2020 assa. adv syst sci appl (2020) 90 a. sharma, n. adlakha [20] sharma, a. and adlakha, n. (2014). markov chain model to study the gene expression. advances in applied science research, 5(2):387–393. [21] sharma, a. and adlakha, n. (2015). a computational model to study the concentrations of dna, mrna and proteins in a growing cell. journal of medical imaging and health informatics, 5(5):945–950. [22] sharma, a. and adlakha, n. (2018). fuzzy system model for gene expression. egyptian journal of medical human genetics, 19(4):301–306. [23] sharma, a., goel, r., and dua, n. k. (2012). optimal policy for eoq model with two level of trade credits in one replenishment cycle. american journal of operations research, 2(1):51–58. [24] smolen, p., baxter, d. a., and byrne, j. h. (2002). a reduced model clarifies the role of feedback loops and time delays in the drosophila circadian oscillator. biophysical journal, 83(5):2349–2359. [25] sturrock, m., terry, a. j., xirodimas, d. p., thompson, a. m., and chaplain, m. a. (2011). spatio-temporal modelling of the hes1 and p53-mdm2 intracellular signalling pathways. journal of theoretical biology, 273(1):15–31. [26] verdugo, a. and rand, r. (2008). hopf bifurcation in a dde model of gene expression. communications in nonlinear science and numerical simulation, 13(2):235–242. [27] vielle, b. and chauvet, g. (1998). delay equation analysis of human respiratory stability. mathematical biosciences, 152(2):105–122. [28] villasana, m. and radunskaya, a. 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(2018) application of algorithms of objectifying expert clustering of multiparameter objects in the analysis of big arrays of information vladimir guchuk v.a. trapeznikov institute of control sciences of the russian academy of sciences, 117997, profsoyuznaya street, 65, moscow, russia e-mail: polma@bk.ru abstract: the use of expert assessments is relevant for the development of new complex scientific and technical products, in medical diagnostics, in the study of unusual phenomena. to successfully apply these estimates, it is proposed to use a procedure that allows improving the quality of expert clustering of objects based on the analysis of measured parameters a procedure that objectifies expert estimates. this interactive procedure is based on elementary assumptions about the properties of objects. the paper describes the features of the implementation of the algorithm of this interactive procedure, proposed by the author. this procedure is essential during the creation of the technology of work to with complex objects that are difficult to fully or even partially formalize. a method for representing the proposed procedure for objectifying expert clustering in terms of the theory of fuzzy sets is described. the developed objectifying procedure was used to create algorithms for medical diagnostics based on pulsed signals of the radial artery. subjective clustering of the shape of pulse signals was used. keywords: expert assessments; clustering; objectifying; analysis algorithms. 1. introduction intellectual analysis of large amounts of information is a complex multi-step procedure [1]. at the first stages it is necessary to create a general idea about the researched objects, especially if these objects are a little studied or created on the basis of new the physical principles. in such cases, expert assessments are often used for subsequent formalization. the use of expert assessments is relevant for the development of new complex scientifictechnical products, in medical diagnostics, in the study of unusual phenomena. to successfully apply these estimates, it is proposed to use a procedure that allows improving the quality of expert clustering of objects based on the analysis of measured parameters a procedure that objectifies expert estimates. the paper describes the features of the implementation of the algorithm of this interactive procedure, proposed by the author in [2]. this procedure is necessary when creating a technology that works with complex objects that are difficult to fully or even partially formalize. in many cases, the aggregate of real objects [o1… on], described by the vectors ][ 1 nvv    of the measured parameters, has the property, which is called poly-attraction, when the objects are not evenly distributed… over the subspace of valid parameters [p1… pl]. in this case, the objects are as if attracted to one of several coordinate values of jx  , j=1… q. moreover, these values can form classes. this property is a powerful argument for carrying out a clustering, though the imperative clustering can also be carried out in case of even distribution. poly-attraction is manifested in the fact that most objects are identified with sufficient confidence as belonging to one of the types tj, j=1... m objects (m ≠ q). if the correlation between the assignment to a specific type tk and the corresponding parametric hit in the range of a particular coordinate value is application of algorithms of objectifying expert clustering 103 copyright ©2018 assa. adv. in systems science and appl. (2018) high, the compactness hypothesis [3] is fulfilled. it is necessary to consider that in practice while operating with the experimental material such context-sensitive tools and concepts as polynomial (multinomial) distribution, the hindering parameters, criterion student’s it is, etc. are expedient to apply only after the heuristic or expert analysis of a situation and additional processing of the obtained experimental data. as a specific example of the experimental data, in fig. 1 the diagrams characterizing distribution of the discrete values of several measured parameters for one class of objects in normalized interval are provided. differ from a polygon of distribution of a graphic in the scale of the ordinate axis in which the unit corresponds with the largest frequency of appearance of a parameter value. firstly, it is necessary to decide on the possible sources, generating features of the distribution of the value of the measured parameter. fig. 1. examples of value distribution of several measured parameters for example, local outbursts on graphs must be linked: with the design features of the system being analyzed; with the natural frequencies of component nodes and components; with the presence of extraneous noise and interference; with the presence of various biorhythms in the analysis of pulse and other signals in medical diagnostics. it is also necessary to take into account the inevitable presence of harmonics of the fundamental frequencies of signals that generate clones on the distribution, as well as the nature of the amplitude-frequency characteristic of the system under investigation, which contributes to the manifestation of the resonance phenomenon or vice versa to the suppression of the signal (parameter). it should be noted that in practice, often such methods as summation, finding the average, etc., which are used to specify the basic value of the parameter (such as mathematical expectation) do not work. for example, if third-party noises and induced signals have more power than the main signal, or the main value characterizing the class, there is a blurred range of values, etc. fig. 2. examples of value distribution of the measured parameters for two classes of objects fig. 1a can confirm the existence of harmonicas, as well as the existence of objects of more than one class. the presence of several classes can be clarified both by using other parameters, and by using expert estimates. surge in the right part of fig. 1b requires the additional analysis, for example, regarding the existence of a subclass in an represented class of objects. fig. 1c also generates similar questions, and fig. 1d is sufficiently clear. in fig. 2 the diagrams characterizing distribution of the discrete values of several measured parameters for two classes of objects are provided. fig. 2a shows the potential suitability of parameter for identification of classes. fig. 2b can confirm both unfitness of parameter for identification of classes and the illegality of division of objects into these classes. fig. 2c rejects classification opportunities of parameter, the parameter of fig. 2d can be used for this 104 v. guchuk copyright ©2018 assa. adv. in systems science and appl. (2018) purpose only for a certain subset of objects of two classes if finding parameter with more explicit distinctions for the identified classes isn’t possible. 2. use of expert estimates the subjective clustering is preceded by a formalization stage – formations of a set of the measured parameters [p1… pr] objects. in most cases it is naturally difficult to define the parameters which are most informative. another hindrance is that not everything can be meaningfully formalized. therefore, it is necessary to check as much as possible r of measured parameters of objects. the solution of the problem of parameter reliability is the choice of l-parameters whose distribution of values is most correlated with the deviation from the values forming the class jx  , j=1… m, m<q. it is performed after expert clustering, which can be performed both with measured parameters and without them, in particular, as a result of visual evaluation of objects. the stage of an expert clustering of multi-parameter objects shall come to an end with the generalized clustering. based on the judgement of different experts taking into account qualification and specialization of experts a significant part of objects shall be carried to one of the m classes, the number and character of which are created in the course of execution of this stage. additionally, experts can deduce subsets of reference objects, the most typical for each of the classes. hypothetically, it can increase efficiency of a row of procedures at an objectifying stage. however, in practice it isn't always expedient to demand such a detailed estimation from experts. it is often necessary to deal with clusters without the linear order in between them, and also with the very non-uniform structure of selection of the objects. though the large number of researches is devoted to finding a solution to such problems [4], in most cases reliability of the carried-out expert clustering, owing to a combination of the objective reasons and subjectivity of estimates is insufficient for practical use. this circumstance is one of the reasons for holding a procedure of increase in the reliability of expert estimation procedure of objectifying of an expert clustering. after formalization and subjective clustering, the analysis of the obtained data is carried out. in the simplest version on two sub-stages ranging of each of n vectors on a level of belonging to class appropriated to an object by expert estimates is made. of all the vectors of each class, the first sub-step randomly selects a part of the vectors. they are reference samples (rs). the remaining vectors are testing samples (ts). next, based on rs, we look for the l most effective parameters for identification. the identification of vectors from ts is carried out further. the sequence of the described procedures, starting with the formation of rs, cyclically repeats. the number of cycles depends on the rs and ts volume. during each cycle for each vector nv  the current (total) number sn of getting into the ts is counted and, if the identification score e of the vector class coincides with the expert evaluation of the *e class, the current (total) number of correct identifications increases: ( ) ( ) ( )*, 1 & 1 .n n n n n n nj njv ts s s ev e v s s  = + = → = + , ,0 & (( )( )) (( )( 1) .) i ii ia a     = =  as a result, the preliminary cluster coefficient of membership is calculated for the j-the class knj=snj/sn. also levels of belonging to others classes are calculated ( " nkk ). the second substep is similar to the first one, with the difference that the rs is created from those vectors that have high preliminary cluster coefficients. then, the cluster coefficients for the entire array of vectors are calculated, and the vectors are ranked according to these coefficients. the need for two sub-stages are due to the use in rs of the first sub-stage of vectors that could be incorrectly classified. it reminds the idea of a busting [5]. the application of algorithms of objectifying expert clustering 105 copyright ©2018 assa. adv. in systems science and appl. (2018) difference between the ordered sequences of the first and the second ranging depends on the quality of an expert clustering and can make 10% and more. as practice [2] showed, carrying out additional ranging after the second one doesn't give noticeable effect any more. this kind of adjustment approach is used in tasks with overlapping class distributions, when those vectors on which the classifier is mistaken are removed from the rs. in our case owing to influence of a human factor the composition of these false vectors increases at the expense of trivial errors of experts. as for the procedure of search of the most effective parameters [p1… pl] for the identification made every time when rs is created, execution of this procedure practically is the execution of the task of retraining [6] with the minimum level of formalization of models. in this case there are no explicit assumptions about the probability distributions used in the majority of operations on a clustering, recognition, etc. in the classical task of retraining after training at rs a retraining on test selection usually takes place. in the described procedure, it also occurs in the course of iterations of an algorithm of ranging. 3. procedure of objectifying next, objectifying is carried out. an iterative procedure for adjusting evaluation results applies to expert assessments. generally, an expert takes part in the decision to reclassify. that is, the procedure is interactive. this is justified by the non-absolute validity of the parameters used, by the complexity of the choice of minimum and maximum threshold values, and also in the initial subjective formation of the number of classes and the composition of classes. as practice shows, as a result of completely automatic objectifying, for instance the class, which has important application-oriented properties, can disappear. as a result of objectifying certainly erratic decisions of experts are eliminated, reliability of the results of a clustering increases and more adequate formalized methods of identification of the experimental data are created. in addition to the function of "heal the subset of classes", the objectifying procedure also has other useful properties. first of all, it allows to estimate the competence of a clustering for each of classes, and also in addition certifies measurable parameters on professional suitability to the identification of vectors. the latter property is extremely important, since the procedure of formation and selection of measurable parameters is subjective and a priori does not guarantee the very possibility of solving the identified problem. reclassification based on the system is carried out:  ( ) ( )( ) ( ) ( )( )  ( )  ( ) ( ) ( )( )  ( )  ( ) ( )( ) ( ) ( )( )  ( )  ( ) ( )( ) ** , * , , ** ** , , ** ** , * , , ** * * * , , , & , &  , ; ( , , & , ;    , & , &  , ; , & , n n i n j n j i ni j n n j n l j ni j n n i n j n l i mi m n n i n ji i v v k k j k k j k k v v i v v j k k l k k v v i v v k k j k k l k k v v i v v k k k j k k       →     →    →     →     →     →      →  ( )  ( )* , .m m m v v          →   here: n is the sequence number of the vector, {v  } is the set of vectors of the class (the class of the vector is defined according to expert judgment or previous reclassification), knj is the coefficient of belonging of n-the vector to the j-the class, k**, k*, k **, k* minimum and maximum threshold values for pertaining coefficients. the region of values below of k* (k**) corresponds to the absence of the expressed signs of a specific class and region of values above of k* (k**) – to the explicit existence of signs of a class. values with one "star" index are used in the analysis of a level of belonging to the class (at the moment), 106 v. guchuk copyright ©2018 assa. adv. in systems science and appl. (2018) otherwise the other threshold value is used. after carrying out several iterations on objectifying some sets can be divided into two or more subsets, others, on the contrary, can even increase the dimensionality. the first option suggests an idea of multi-label classification, known in slightly other sphere, in which the classified object can belong to several classes at the same time, and classes are not mutually exclusive (perhaps, even enclosed). tasks of this kind, for example, are characteristic for the analysis of images. the second option potentially corresponds to a case of execution of a hypothesis of compactness. in this case the identifiability level of (a share of truly identified objects at examination) as a result of objectifying can increase by 20–30%. the also possible to assimilation of some set (the degeneration of a set). application of the described procedure of objectifying can not only correct composition of a set, but expand it. another important characteristic for objectifying is the presence and minimization of identification errors of the second kind [6]. or the expert treats the object not to his class, or the algorithm assigns the object to one class, and further parametric analysis identifies it as belonging to another class. the coefficient " njk of belonging of the n-the vector to the j-the class, previously attributed by the expert or algorithm to other classes, will be called the penetration coefficient. in addition to the composition of the selection, the validity of the measured parameters, etc., the choice of an algorithm of identification (recognition) as well influences an error amount of identification. as the practice of working with several algorithms has shown, the algorithm of the method of potential functions (mpf) has proved to be the most unpretentious with respect to the choice of parameters used for training in rs and identification on ts. it showed the minimum errors of identification of the first kind. only on minimization of errors of the second kind the algorithm of an mpf has no advantages over the other used algorithms. another its feature is a need of a large number of computations that isn't an essential shortcoming, considering high-speed performance of the modern computers. the optimum length of a tuple of the parameters used for identification was rather general for different algorithms. the main result of the application of objectifying was manifested in the increase in the degree of identifiability (improvement can reach 10-20%). when counted the level of identifiability values of errors of identification of the first and second kind were accounted. the higher level be reached in some cases and an identification algorithm choice (a recognition algorithm) has no basic value. the effect of applying the procedure of objectifying is illustrated in fig. 3. the dashed line shows the dependence of the membership coefficient on the vector number (in the sequence ranged by results of expert estimation) [2]. the solid line reflects the results of objectifying. fig. 3. the dependence of the in cluster membership coefficient in the number of the vector in the ranked sequence up to (dashed line) and after (solid line) objectifying this example shows a case when the procedure of objectifying not only adjusts composition of a set, but also expands it. in fig. 4 dependence of the coefficient of the identifiability kid from number of the used parameters m to (dashed line) and after (solid line) of the objectifying is shown [2]. when counted the coefficient of identifiability values of errors of identification of the first and second kind were added. the figure displays the identifiability improvement reached by the procedure of objectifying. often, you can get more effect, and the choice of the algorithm of identification (recognition) does not play a application of algorithms of objectifying expert clustering 107 copyright ©2018 assa. adv. in systems science and appl. (2018) determining role. in general, defining influence on the efficiency of the procedure of objectifying is having an by several factors render: factors of internal (formal) character: 1.1. completeness of selection of vectors. 1.2. a validity of the measured parameters. factors of the external (introduced) plan: 2.1. potential possibility of expert estimation. 2.2. quality of expert estimates. fig. 4. dependence of coefficient of identifiability kid from number of the used parameters m to (dashed line) and after (solid line) of objectifying 4. formalization of the procedure of objectifying complexity of formalization of the procedure of objectifying of the expert clustering is defined, as a rule, by absence of mathematical models of real objects. for interpretation of the content of the objectifying procedure in terms of the theory of fuzzy sets [7] {v  } (the set of vectors of the class) we define as a fuzzy set fj, namely as a set of pairs fj = {(v, μj (v)) | vu}, where v is a vector belonging to the universe u. the set of all vectors, μj (v): u → [0,1] is the function (degree) of the vector v belonging to the fuzzy set fj (defined above as a cluster coefficient). as the threshold value of a level of attachment the value of so-called transition point of an indistinct set, namely 0.5 which can be used as an initial reference point, is normally used. the carrier of a fuzzy set fj is a subset of jf ~ vectors having explicit attributes of the class, that is, the degree of belonging of μj (v) which is very high. since in our case objectifying is a priori performed for sufficiently representative samples, the height of the fuzzy set sup jf μj (v) = 1, that is, the fuzzy set fj is normal. for the same reason, the fuzzy set is not unimodal (the degree of belonging to the class reaches the value "1" for more than one vector). filtering of vectors using the maximum threshold values k*=α (k**=α) for the membership coefficients generates an α-slice of the fuzzy set fj, that is, generates a clear set ajα, which is determined by the characteristic function χajα: ((μi < α)→(χai,α = 0)) & ((μi ≥ α)→(χai,α = 1)). for the alpha-slices of the fuzzy set fj, the mutual implication α1<α2aj, α1 aj, α2 is valid, reflecting the fact that filtering vectors using a larger threshold value, to generate sets of lower power than filtering with a smaller threshold value. as for such important concept 108 v. guchuk copyright ©2018 assa. adv. in systems science and appl. (2018) as convexity of a set, in most cases it isn't applicable to a real experimental data. it is necessary to point out that hypothetically in the evaluation plan the above-mentioned indistinct set of fi can be convex because of simplicity of creation of boundaries of sets in the space of the measured parameters. further the reclassification of the vectors defined by an expert estimation to fi and belonging to the fi area, but as a result of objectifying switched on in the set can violate the convexity conditions. for already objectified clustering, a deeper tool of the theory of fuzzy sets can also be used, in particular, based on maxi-min, algebraic and bounded operations, and using the t-norm and so on. if two fuzzy sets are combined for practical purposes, the combination of sets fi fj is obtained the smallest fuzzy set fi j containing simultaneously fi and fj, for which μi j (v) = max (μi (v), μl (v)). in this case, the largest degree of membership (cluster membership coefficient) is used for the first (for example, l-the) or second (j-the) class. the reclassification algorithm uses more complex procedures. in particular, a logical analysis of the absolute values of the membership levels and the correlation of the levels of belonging to different classes of the same vector is used. as a result of such logical analysis, it is possible to make an adjustment of the above-mentioned product of sets of fi fj, in particular. to formalize the objectifying of expert clustering, fuzzy estimates of the degree of belonging is more appropriate. this is due to the fact that at this stage it is impossible to obtain fairly accurate and final estimates. in the course of objectifying, the composition of the fi set, as well as a subset , ~ if can undergo significant changes that affect the parametric formation of the degrees of belonging. in addition, it is possible to introduce the notion of the level of blurring of estimates and the concept of the reliability of these estimates [2] or to use probable characteristics for the degree of belonging. as for the fuzzy classification, this concept in the conventional sense is difficult to apply to the parametric classification performed using recognition algorithms (classification algorithms). in a certain sense, the problem of fuzzy classification is decided by the expert in the subjective evaluation of the degree of similarity of the vector v with the standard of the class fi that he creates, that is, using pair comparisons and a previously undefined number of classes. the task of fuzzy ordering with this approach is not put at all we use the ranking of vectors by the calculated cluster membership coefficients. the index of blur of an indistinct set understood as a measure of internal uncertainty can be used also for the characteristic of compactness of a class in the parametric space and for the evaluation of identifiability of vectors of j-the of a class in the total mass of vectors. in general, the device of the theory of indistinct sets is intended firstly for the description and the analysis of a static situation when there is some fixed in the analyzed condition of the sets and an evaluation conglomeration. for full formalization of the procedure of objectifying it is necessary to enter dynamic constructions initially. for the first step it is possible to use introduction of such concept as not hardened set (a dynamic set) which in the course of the development changes composition, power, etc. in addition to the known attributes, we add here such a notion as the d-index (dynamic index) of a set. in the simplest case, the d-index is discrete, reflects a step or iteration in the dynamic process of adjusting the settings (in our case iteration in the objectifying procedure for clustering multi-parameter objects according to expert estimates). while fixing the value of the d-index, that is when reviewing the recorded status at a certain stage of objectifying, the situation enters the standard course for which the developed mathematical apparatus is available. there are some known precedents of the use of d-index, for example, in the methods of genetic optimization developing j. holland's ideas [3]. its presence is also tracked in the concept of the iterative sets [8] which are a special case of not hardened (dynamic) sets. for the description and the analysis of the procedure of objectifying in this case it is necessary to enter such concepts as a conditionally of a set, degeneration of a set, stability of presence of elements it is also possible to use such analytics as convergence of procedures, for example, aspiration of the power of a set in the course of its adjustment to a certain value, stability of a set concerning the nomenclature of elements etc. application of algorithms of objectifying expert clustering 109 copyright ©2018 assa. adv. in systems science and appl. (2018) 5. conclusion there are works on the objectifying of expert assessments [9]. procedures were developed to clarify the expert quality ratings of one group of objects, displayed in rank scales. it should be noted that for the objectifying of expert assessments, it is possible to use standard procedures for processing estimates [4]. for example: involving a sufficiently large group of experts; selection of the most competent experts formation of balanced assessments from individual evaluators. the developed objectifying procedure was used to create algorithms for medical diagnostics based on pulsed signals of the radial artery. subjective clustering of the shape of pulse signals was used [10]. the author expresses gratitude to e.s. nesterova for assistance in preparing materials for publication. references [1] dorofeyuk, y.a., dorofeyuk, a.a., & pokrovskaya, i.v. (2013). the expertanalytical forecasting model in the problem of railway track facilities control. proc. of the ifac conference on manufacturing modelling, management and control. st. peterburg, russia, 1826-1831, [in russian]. [2] guchuk, v.v. (2015). the technology of objectification of expert clustering of weakly formalizable objects, bulletin of the usatu, 66 (5), 149–154, [in russian]. [3] de jong, k.a. (2007). introduction to the second special issue on genetic algorithms, machine learning, 5 (4), 351–353. [4] gusev, v.b. (2011). autonomous mechanism of adaptation the data processing system to the service conditions. proc. of the iciea conf. beijing, china, 753–757. [5] friedman, j. (2001). greedy function approximation: a gradient boosting machine, annals of statistics, 29, 1189–1232. [6] omidvar, o. & dayhoff, j. (1998) neural networks and pattern recognition. new york: academic press. [7] zadeh, l. a. (1965). fuzzy sets, information and control, 8, 338–353. [8] maddy, p. (2007) second philosophy: a naturalistic method. oxford: oxford university press. [9] kuznetsov, m.p. & strijov, v.v. (2014). methods of expert estimations concordance for integral quality estimation, expert systems with applications, 41 (4), 1988–1996. [10] desova, a.a., guchuk, v.v. & dorofeyuk, a.a. (2014). a new approach to pulse signal rhythmic structure analysis, international journal of biomedical engineering and technology, 14, 148–158. microsoft word 861 article text, copyedited.docx adv syst sci appl 2020; 03; 50-72 published online at https://ijassa.ipu.ru. value potentials in innovation, resources, strategic networks and blocks jeffrey yi-lin forrest1*, sailau baizakov2, yohannes g. haile3, abel gyan1, yong liu4 1) school of business, slippery rock university, slippery rock, pa 16057, usa e-mail: jeffrey.forrest@sru.edu; abel.gyan@sru.edu 2) institute of mathematics and mathematical modeling, almaty, kazakhstan e-mail: baizakov37@mail.ru 3) black school of business, pennsylvania state university behrend, erie, usa e-mail: yxh313@psu.edu 4) school of business, jiangnan university, wuxi, china e-mail: clly1985528@163.com abstract: the present globalizing world of business has been witnessing the emergence of unprecedented kinds of markets. these new kinds of markets post significant challenges to the world of learning and decision-making managers, entrepreneurs and alike. to confront these challenges, this paper investigates the potentials of value creation and value capture at the firm level from different perspectives. by employing systems methodology and the logical reasoning that parallels the one commonly used in mathematics and natural science, this paper establishes 16 formal propositions on value creation and value capture from the respective perspectives of market competition, innovation, resource, inter-organizational network, and direct association of sellers and buyers. in particular, this work points out (1) when innovation provides additional potentials for value creation; (2) when resources’ latent values can be practically made visible through value creation; (3) how markets’ mutual forbearance and relatively sustainably increased profitability are positively related, (4) why a firm’s systemic hole position offers advantages and profitability to the firm, (5) how a firm’s profitability is positively affected by its membership in a strategic block of increasing market influence; and (6) when a firm creates and maintains convenient platforms it will readily create and capture values. this paper concludes with some practically reliable general recommendations and a few topics and directions for future research. keywords: market competition, product, profitability, supply-chain network, system, systemic hole 1. introduction with economic globalization, there have appeared new ways for businesses and individuals to create wealth [33]; and the once accustomed form and rule of competition have been changing. all these fast-paced changes and emergence of opportunities have been posting challenges to scholars, decision-making managers and entrepreneurs from around the world [47], and led to accelerated knowledge development and acquisition and sharing of information and knowhows [5]. however, accompanying the expanding literature on value creation and capture, most studies provide managerial suggestions of limited validity due to their various empirical constraints. so, theoretically, the following question arises naturally: based on the pioneering works of empirical studies, seen in the literature, on issues and matters of value, can one establish generally valid conclusions that do not suffer from dataand/or anecdote-specific constraints? that is, the gap this paper strives to fill is bridging conjectures developed on empirical studies to generally-true principles by introducing a new methodology. in particular, * corresponding author: jeffrey.forrest@sru.edu value potentials in innovation, resources, strategic networks and blocks 51 copyright ©2020 assa. adv. in systems science and appl. (2020) this gap of the literature appears due to at least two reasons, for details, see the literature review below and the relevant references quoted there: the key concepts of concern, such as innovation and resource, cannot be directly measured numerically, and each sample of data only reflects certain specific and limited aspects of the environment from which it is from. there is no doubt that this question is theoretically essential, because attempts of addressing it will potentially lead to breakthrough(s) in terms of methodology that can be used widely to develop useful conclusions in business studies. and this is also a practically significant question, because the business world is currently situated in the era of transient competitive advantages, where consumer preferences change quickly, while once sustainable competitive advantages become short-lived [66]. to this end, the current paper attempts to address this question at least partially by establishing a series of 16 formal propositions by employing the logical reasoning that is commonly used in mathematics and natural science and systems thinking [61] through respectively looking at innovations, resources, networks, and platforms that directly connect sellers and buyers. specifically, among others, this paper establishes the following general conclusions, among others: • when a market operates under the condition of free competition, protective property rights and complementary assets are positively correlated to the effectiveness of innovations’ value creation. in this case, exchanges that take place in the market help dormant values of resources emerge. • the more mobile resources are, the more capable value creation is. • the level of profitability of a firm can likely be sustained in a state of mutual forbearance. • when an inter-organizational network experiences its expanding market influence, it provides its members with a positive effect on their profitability. the rest of the paper is organized as follows. section 2 provides a literature review and introduces the basics of systems science. section 3 presents our main results on value creation and capture. in this section, we establish propositions on value creation in a market of free competition by means of innovation and resource, and study how such phenomena as market forbearance, supply-chain networks and strategic blocks can positively affect member firms’ profitability, and how firms grow by joining inter-organizational network(s) and by making use of platforms that directly connect sellers and buyers. this presentation is concluded in section 4 with practically useful recommendations for managers, entrepreneurs and retailers. 2. preparations this section provides the literature review and necessary background knowledge. 2.1. the literature this paper contributes to three literature areas related/focused on value: innovation, resource and business networks. in the following, this literature review will look at each of these three areas individually by focusing on only closely related studies and then respectively explain how this work enriches these relevant areas. first, let us look at the literature area of innovation. adam smith [86] (1776) is the first person on record to realize the importance of innovation in wealth creation. since then, many scholars have been involved in the investigation of the concept of innovation and consequent value creation and capture. even so, the main focus of research has been on what factors determine the innovativeness of a firm [3,13]. in terms of innovations and accompanying values brought forward by scientific and technological progress, baizakov et al [6] study how an economic entity evolves over time – the dynamics of an economic entity – by examining the relationship between incomes and expenses, as characterized by the coefficient of scientific and technological progress. different from this literature, this work presents a brand new approach of reasoning and modeling so 52 j.y-l. forrest, s. baizakov, y.g. haile, a. gyan, y. liu copyright ©2020 assa. adv. in systems science and appl. (2020) that a much clearer relationship is established between protective property rights (and complementary assets) and the effectiveness innovations have in terms of value creation. secondly, regarding resources, penrose [73] is the first scholar who recognizes the important role of organizational resources played in firms’ success. on top of many scholars’ works, the so-called resource-based view of the firm begins to emerge in the 1980s [54,79,91] for the reason that resources can logically explain synergistic and differential effects of various factors on firms’ performance [27]. for example, cacciolatti and lee [17] employ resources to analyze the capability-performance relationship, and explain how different types of marketing capabilities contribute to firm performance. ombaka et al. [72] reveal the fact that resources possess mixed influences on firm performance indicators, although resources have a statistically significant influence on the non-financial performance of insurance companies. enriching this literature, this paper shows that the capabilities of a firm also stand for resources for the firm, in what possible ways resources’ latent values can be mobilized, when resources can create value, and when they are inimitable and not readily substitutable. thirdly, other than looking at firms as atomistic actors that compete against each other, they should be treated as indispensable members of networks (or systems) of various businesses and exchange relationships [36-38]. such realization helps explain why firms’ conduct and performance need to be studied by examining relationships, in which the firms are part of. additionally, firms, when positioned in strategic networks, have access to information, resources, markets, and technologies, with advantages of learning and economies of scale and scope. such advantages bring these firms multiple steps closer to materializing their respective business objectives [41]. in terms of business networks, harrigan [44] and kogut [52] consider the formation of inter-firm partnerships, baum and dutton [12] study the behavior and performance of firms, forsgren [34] considers the uppsala internationalization process model, öberg [70] investigates innovation and several relevant issues, among others. building on this literature, this work shows • why within a supply-demand network systemic holes convene control and represent profitability for firms that occupy the positions of the holes; • why inter-organizational networks that possess expanding market influences can provide their member firms with increasing levels of profitability; and • why creating and maintaining platforms that directly connect sellers and buyers can be enormously beneficial for the underlying firms, among others. other than the afore-described contributions this paper makes, another important contribution is the introduction of systems methodology into the study of value creation and capture. this methodology is generally applicable to studies of organizations, business evolutions and interactions without suffering from limitations of data, anecdotes and relevant methods of analysis. that is, conclusions developed herein can be employed to produce general recommendations in practice instead of suggestions provided by most of the relevant studies in the literature. 2.2. some background information because the systems methodology employed in the current work is not widely employed in the literature of relevant studies, this subsection presents the basics for the purpose of making the rest of this presentation self-contained. by system it stands for an abstract concept that models business entities as organizations and structures so that components, treated as isolated in classical sciences, along with their associations with each other become organic wholes (or systems) [56]. so, systems are everywhere in life, because organizations (and structures) are the main features of the world. that is the reason why this concept of systems has appeared in discussions of business-related matters and issues [28,80,81]. indeed, both numbers and systems are rooted in the same world. however, they reflect two completely opposite aspects of physical and intellectual existences, where numbers come into value potentials in innovation, resources, strategic networks and blocks 53 copyright ©2020 assa. adv. in systems science and appl. (2020) being when internal structures either do not exist or are ignored, and systems focus on internal structures. speaking differently, numbers and systems are different in two major ways: • when one looks at a small scale, regional phenomenon, one can mostly make use of numbers. but, when one studies a large-scale existence or phenomenon, such as organizations or structures, he or she will need to employ the concept of systems (lin, 1999); and • only with the existence and occurrence of events, numbers appear. in the contrary, systems surface instantaneously with existences and occurrences [61]. these differences explain why systems logic and methodology represent more appropriate tools than those developed on numbers for the investigation of economic entities where internal structures do play a role. due to its general applicability, scholars from different disciplines have defined this concept of systems by using diverse languages since the time when it was initially proposed by von bertalanffy in 1924 in biology [90]. for our purpose, a system is identified with an ordered pair 𝑆 = (𝑀, 𝑅) of an object set 𝑀 and a relation set 𝑅 [60]. in particular, 𝑀 consists of all the isolated objects of system 𝑆 and 𝑅 a set of relations that connect the objects in 𝑀. for example, each business firm is composed of a set of employees, properties, equipment, etc. this set is collectively known as that of objects of the organizational system of the firm. more importantly, these objects are associated with each other through a set of particular relations. it is through particular relations that the whole is acknowledged as a functional firm. for theoretical completeness, a discrete system 𝑆 = (𝑀, 𝑅) is such a system that satisfies: r = ∅ or r = {∅} and m ≠ ∅, where ∅ stands for the empty set. in other words, a system 𝑆 is discrete when it consists of isolated objects only. a system 𝑆 = (𝑀, 𝑅) is said to be trivial, if m = ∅. theoretically, the concept of trivial system is equivalent to that of zero on the real number line. for a given system 𝑆 = (𝑀, 𝑅), another system 𝑆) = (𝑀), 𝑅)) is said to be a subsystem of 𝑆, if 𝑀) is a subset of 𝑀 and 𝑅) = 𝑅|𝑀), consisting of relations in 𝑅 but restricted on 𝑀). that is, for each relation 𝑟 in the subsystem 𝑆), there is a relation s in r of the original system 𝑆 such that 𝑟 consists of all relationship descriptions in 𝑠 of objects except those descriptions involving objects outside the subset 𝑀). for convenience, we write the subsystem 𝑆) as 𝑆|𝑀). let 𝑆= (𝑀-, 𝑅-) be a system for each 𝑖 ∈ 𝐼, where 𝐼 is an index set. if for any two systems 𝑆 and 𝑆1 , 𝑖 ≠ 𝑗 ∈ 𝐼 , their objects sets are disjoint, then the free sum of this collection of systems, denoted ⊕ {𝑆-: 𝑖 ∈ 𝐼} or 𝑆) ⊕ 𝑆8 ⊕…⊕ 𝑆:, if i = {1, 2, …, n} is finite, is simply defined as the system whose object set is equal to the totality of the objects in the given systems, and whose relation set that of all the relations in the individual systems. that is, ⊕ {𝑆-: 𝑖 ∈ 𝐼} = 𝑆) ⊕ 𝑆8 ⊕…⊕ 𝑆: = ;< 𝑀-∈= ,< 𝑅-∈= >. mathematically, we can show that for any set of systems, its free sum exists uniquely up to a similarity [60]. a connected system 𝑆 = (𝑀, 𝑅) is such a system that satisfies: for any 𝑚) and 𝑚8 ∈ 𝑀, there are a sequence 𝑛), 𝑛8, … , 𝑛c ∈ 𝑀, and a sequence 𝑟), 𝑟8, … , 𝑟cd) ∈ 𝑅, for some natural number 𝑘, satisfying that 𝑟) relates 𝑚) and 𝑛) , 𝑟8 relates 𝑛) and 𝑛8 , …, 𝑟c relates 𝑛cf) and 𝑛c, and 𝑟cd) relates 𝑛c and 𝑚c. that is, a system is connected if any two objects in the system are related through a finite number of relations of the system. otherwise, the system 𝑆 is known as a disconnected system, meaning that the system is really equal to the free sum of two or more subsystems. at this junction, let us emphasize that beyond adding additional explanatory power to the existing knowledge on value creation and capture, applying systems methodology enables the discovery of new conclusions and provides general and practically applicable 54 j.y-l. forrest, s. baizakov, y.g. haile, a. gyan, y. liu copyright ©2020 assa. adv. in systems science and appl. (2020) recommendations. unmistakably, such outcomes stand for an academic endeavor that is worthy of further pursuing, within which the true power of systems methodology lies. to help with the smooth flow of logical reasoning, assume that each business firm considered herein exists to satisfy a particular market niche with its operation financially maintained by a positive cash flow from the product market. such positive cash flows used to be backed by firms’ performance and profits [87]. but, it is no longer the case. for example, in recent times online retailers have focused on pumping up their future promises and potentials so that they continuously attract sufficient venture capitals by placing emphasis on increasing their market shares. as a matter of fact, some of the well-known retailers have been losing money one year after another since their inception [59]. although it is defined differently by competing schools of economics, by value in this paper it means the benefit offered by a product, service or good (such as informational goods). by value creation, it means a process a firm goes through to offer its output with value-added. by value capture, it stands for the value the firm, which implemented a process of value creation, actually captures from satisfying market demand(s). by consumers, it means the endusers of what is offered; and, by customers those firms that employ their inputs to produce their outputs. by virtual market, it means such a setting that the internet infrastructure helps facilitate business transactions and deals, where the internet helps create virtual communities and unconventional commercial arrangements [42]. it helps make it possible for sharing knowledge and know-hows among firms and information about products and services instantly available to consumers. the following game-theory based result [30] will be used in the rest of this paper: theorem 1: assume that an oligopoly market is perfectly competitive without any outside interference. then, in the nash equilibrium, at least one new enterprise enters the market competitively and profitably, if and only if there is a market segment of switchers who make purchase decisions based on which supplier’s price is lower. 3. main results this section focuses on various potentials of how values can be either created or captured. 3.1. potentials of value creation this subsection focuses on various potentials of how values can be created from two angles: innovation and systems of resources. it consists of two subsections accordingly. 3.1.1. innovation and value creation because the concept of product used in management and marketing is different of that in economics (for relevant references, see the following discussions), let us introduce the following: by product, it represents a tangible or intangible thing or process, made available by a business entity, which consumer(s) demand to have or use. for example, an iphone is a (tangible) product that consumers can physically use in their daily lives. an annuity offered by an insurance firm represents a (intangible) product, consisting of services and promises that consumers can count on when certain prescribed conditions are satisfied. such services as selfserving laundry facilities, indoor tanning salons, etc., are products that consumers can ‘receive’ through using the facilities and salons. a consulting firm’s investment recommendations are (informational) products (goods), produced by the firm through gathering, organizing, and analyzing facts. in real life, different kinds of products, such as a physical product and followup services, are usually bundled together to attract additional customers. to make our logical reasoning flow smoothly in the rest of this paper, let us define the concept of innovation as follows, parallel to that defined for the manufacturing sector [31]: by innovation, it stands for a set of activities, which contains at least one activity, that is related to the production and/or offer of a product or a bundle of products, which produces value potentials in innovation, resources, strategic networks and blocks 55 copyright ©2020 assa. adv. in systems science and appl. (2020) exceptionally added value for a firm when compared to other activities that are simultaneously taking place in the same economic sector. in terms of the literature, the concept of innovation has been conceptualized by many authors due to its theoretical and practical importance [13,18,22,48,56,75,77,83] and by the organization for economic cooperation and development [71]. however, the definition of this concept has been evolving with new particulars added over time in order to keep up with the pace of the constantly changing business world. considering the objective of this research, the situation of such forever changing concept of innovation needs to be elevated to the height of general abstraction so that this work can potentially contribute to the long-lasting knowledge of value creation and capture. that is, the definition of innovation adopted here needs to be as general, as time invariant as possible. in the following paragraphs, let us illustrate this end in more details. in this definition, the phrase ‘the production and/or offer of a product’ means implicitly the design of one or several original products that generates exceptionally added value for the firm. that is, innovation means any of the pertinent activities, such as the redesign of an existing business model, marketing efforts to adequately understand market signals, acquiring the necessary knowledge, converting know-hows to actual production, etc., that lead to the eventual introduction of the product(s). in the contrary, the meaning of ‘offer of a bundle of products’ is profoundly different. it means placing several products, including relevant services, together in one delivery to consumers. here, each of the bundled products may not be special or original on its own characteristics. the creativity that underlies a bundle of mundane products lies in the potential advancement of consequent simultaneous consumer utilities and multi-sided markets [95] with the former created for consumers who can use several products concurrently and the latter when different markets are served and pulled together. theoretically, the abstract concept of innovation and its emphasis on the comparatively added value implicitly make this notion include many applied aspects. for instance, the added value, as emphasized in the definition, suggests that • creativity can potentially develop over time; • market demands, either current or forecasted that are both comprehended within the firm and accepted externally, motivate the creativity, • intended values are actualized through introducing necessary processes to produce and offer products in order to create and capture the expected value; • other than inventions, this definition of innovation also implies translating inventions into products that satisfy the identified demand; • this definition of innovation leaves the door open for the potential of the newness of product(s) and process(es); and • it incorporates the role of technological changes in particular and processes and outcomes in general implicitly. this definition of innovation directly indicates that innovative creativity leads to new and exceptional value, as maintained by schumpeter in 1934 [83], and that innovation transpires from diverse contexts and in dissimilar means, such as design and offer of new products, new methods of production, procedures of additional efficiency, new markets, new sources of supply, and reformation of industries [83]. the inherent ability of collecting highly-priced rents is guaranteed by the originality and scarcity in various facets of innovation and tightly attracted entrepreneurs, although the values of these rents gradually diminish as the originality becomes routine industry exercises and the scarcity vanishes from the market. this end explains how the notion of creative destruction emerges [84]: the initial risks entrepreneurs undertake lead to the consequent originality and scarcity; and the insights that underlay entrepreneurial initiatives destroy themselves as knowledge accrues and spreads over time. summarizing what has been discussed so far, we conclude that each innovation can be seen as a collection of determined activities devised to satisfy an acknowledged demand of the marketplace, either large or small. the role played by technology in innovation is manifested 56 j.y-l. forrest, s. baizakov, y.g. haile, a. gyan, y. liu copyright ©2020 assa. adv. in systems science and appl. (2020) through the improvements it provides, including, but not limited to, making production more efficient, producing more targeted discoveries, reducing the overall expenditure, etc. with their individually different understandings of what the market demands, entrepreneurs provide their alternative innovations. their joint efforts transform markets and industries, leading to continuous economic development. proposition 1: within a market of free competition, the effectiveness of protective property rights and complementary assets are positively correlated to the potential innovations have in creating value. as a matter of fact, the previous discussion implies that free competition emboldens people’s spirit of entrepreneurship. on top of this backdrop, if effective protection of property rights is in place, and mobilization of complementary assets is assured, then product markets will motivate enthusiastic participants to innovate. here, the assumption of free competition means that the economy is not a shared one. for a related discussion based on anecdotes, see [88]. proposition 2: within a market of free competition, exchanges facilitated by various markets help make resources’ latent values practically materialized. this result follows directly from theorem 1. in particular, the assumed oligopoly market of free competition encourages consumers to make their preferences relentlessly and tastes more sophisticated. such forever evolution in consumer preferences and tastes inevitably enlarges the market segment of consumer surplus. that subsequently intensifies the market competition among the incumbent firms. this reasoning explains how dormant values in resources become visible and practically developed through creative combinations of entrepreneurs’ individually available resources. for related empirical studies, see [67]. and for the literature on what factors could potentially affect the innovativeness of a firm, see [13,47] and references listed there for relevant details. williamson [94] defines a transaction as such a trade that the ownership of a product is transferred across a technologically separable interface so that a cycle of production and delivery ends and another one starts. demsetz [23] defines transaction cost as the cost of making a trade within a market, where the term ‘trade’ stands for buying and selling, as commonly known, daily emotional interactions, informal gift exchanges, etc. for our purpose, this work does not develop conclusions from the definition of transaction costs given by cheung [20], although this definition is much broader than what is cited above. institutions, defined as rules accepted and followed by members in a society, are the essence in determining the level of transaction costs. and factors, such as trading frequency, specificity, informational asymmetry, uncertainty, limited rationality, opportunistic behaviors in small-numbers situations, etc., generally determine transaction costs [93]. that explains why companies, which are able to facilitate low transaction costs, can graciously increase their performance and improve growth [69]. let us refer the capability for a firm to reduce its transaction costs more than its competitors to as the firm’s transaction efficiency. then we naturally have the following result. proposition 3: each specific transaction efficiency represents a source of value that can be created for consumers and captured by the underlying firm. this conclusion follows readily from theorem 1: each transaction efficiency can directly lower the selling price of the underlying firm’s product. that surely helps the firm to maintain value potentials in innovation, resources, strategic networks and blocks 57 copyright ©2020 assa. adv. in systems science and appl. (2020) its existing loyal consumers while attracting the entire consumer surplus. realistically, this fact means that the firm has created value for its consumers and captured value for itself. for related empirical discussions on how firms are aware of the need to lower costs, the risks of coordination and transaction and how information technology can help reduce costs and risks, see [21]. and for empirical cases on how the internet or any other networked environment can help lower transaction costs, see [24]. by combining what is obtained earlier and proposition 3, it follows that a firm can create and capture value through the reduction of such determinants of transaction costs as informational asymmetry, uncertainty, complexity, limited rationality, and bargaining behavior in small-numbers situations. hence, firms, capable of economizing on transaction costs with innovation, are able to extract values from their business transactions. additionally, humanizing reputation, trust and transactional experience (e.g., frequent transactions) can help lower the cost of idiosyncratic exchanges among firms. that, of course, will in turn improve the transaction efficiency of a firm. 3.1.2. uniqueness of resources and value creation a resource stands for an asset that can be either tangible or intangible [43] a firm can mobilize to introduce and implement its strategies and to accomplish its objectives [9]. that is, a resource is such a thing, which can take a physical, or financial, or intellectual, or organizational form, that a firm can utilize to actualize its business objectives. capability generally refers to an information-based process that empowers a firm to organize its resources in its effort of reaching its objectives [64]. proposition 4: any firm-specific capabilities are also unique resources of the firm. by combining all the discussions above, we can identify each firm with its system of firmspecific resources, where the resources organically relate to each other for particular purposes. consequent to this identification between firms and their individually different systems of resources, the well-known resource-based view or theory (rbv) of the firm [54] follows naturally based on the following three basic, while also intuitive, axioms [32]: axiom 1 (resource heterogeneity): different firms possess individually different systems of resources, even when firms operate within the same industry. axiom 2 (resource immobility): these differences in systems of available resources persist over time, because of practical difficulties of trading resources across firms and of connecting available resources in particular ways. axiom 3 (different levels of efficiency): firms’ performance differences stem from differences in their systems of available resources and intrinsically different efficiency levels of available resources. a resource is said to be valuable to a firm [8], if the resource permits the firm to increase its profits through executing a particular strategy by considering external opportunities and threats beyond the case of not having the resource. through combining the rbv and the concept of value creation, we have the following result: 58 j.y-l. forrest, s. baizakov, y.g. haile, a. gyan, y. liu copyright ©2020 assa. adv. in systems science and appl. (2020) proposition 5: assume that a firm possesses the control of some valuable and rare resources. if the firm can mobilize these resources, then it is able to create values. this conclusion follows from the resources’ scarcity and the firm’s capability to mobilize these resources. it is because the scarcity and capability jointly enable the firm to create certain advantages that few other firms can compete with until the scarcity diminishes. proposition 6: assume the same as in proposition 5. if the management and stakeholders of the firm have non-conflicting interests, then the firm is in a good position to create value through utilizing its resources. in fact, the assumption that the management and stakeholders have non-conflicting interests implies that the firm’s valuable and rare resources can be effectively exploited by the management. so, propositions 4 and 5 jointly indicate that this firm is capable of creating value by employing these resources. proposition 7: when a firm is able to work adequately and jointly with such capabilities as marketing, research and development (r&d) and operation, the firm will be able to create value. in fact, for any firm, its marketing capability associates the firm with markets by identifying what the firm is, what the product market demands for, and what changes the firm needs to make in order to accomplish its goal [53]. that is, marketing helps a firm know and expect in terms of its current and future competitions [50]. in other words, a firm’s marketing affects the firm’s orientation of innovation and operation and presents the firm to the market and what the market demands to the firm. a firm’s r&d capability translates the vital information of the market, acquired through marketing, into the terms that are specific to the firm and into designs and production of relevant products for potentially satisfying consumers’ forever evolving demand. such capability establishes barriers for competitors to imitate the new products [49]. strong and reputable track records of a firm’s r&d help the firm enhance its image and create favorable consumer expectations. the firm’s operational capability practically actualizes the information and knowledge acquired by marketing efforts and r&d’s ideas of new products into deliverable offers. when the information and knowledge gained from marketing is novel, the r&d’s designs of new products are of state-of-the-art, and the corresponding operations are appropriate and accommodating, the entire chain of the firm’s activities will become practically mobile and imperfectly imitable. in such circumstances, the firm is in a perfect position to create values. that is, the conclusion in proposition 7 follows readily from propositions 4 and 5. for related anecdotal and empirical discussions, see [46,66]. in summary, what are established above implies, from the point of view of the rbv, that marshaling a system of specialized valuable and scarce resources can potentially lead to value creation. speaking differently, when a firm is identified with its individually unique system of resources, the firm can potentially create value out of its valuable and rare resources. for related empirical considerations of this conclusion, see, for example, [4,7,74,91]. if the concepts of resource and capability, as previously defined, are seen as timedependent or functions of time, then one can readily develop a general resource-based theory of firm evolution. in particular, a resource’s scarcity is clearly time-dependent; the real worth of $1 million investment is a function of time; an inimitable resource gradually becomes imitable as time elapses. from this discussion and proposition 4, it follows that each capability value potentials in innovation, resources, strategic networks and blocks 59 copyright ©2020 assa. adv. in systems science and appl. (2020) in real life is truly time-dependent or a function of time. therefore, the so-called dynamic capability [89] is really a special case of the concept of capability defined in this paper. with time introduced implicitly, one can readily use the resource-based theory to address such questions as • how are values claimed and demanded? • how can competitive advantages be sustainable [7,66]? and • how are the positions of valuable resources established over time? it is not a simple mental exercise to consider each resource (or capability) as a function of time. such consideration is indeed entrenched deeply in how firms’ operations consist mostly of managerial and organizational processes, be they aimed at forming coordination, accomplishing integration, materializing a revitalization, or necessitating a transformation [26,89], or learning [57]. such time-dependent resources/capabilities as strategic decisionmaking, knowledge acquirement, product development, how-to skill creation, social network formation, etc., facilitate firms to create and capture values practically through schumpeterian rents [89]. proposition 8: with the emergence of virtual markets, the sustainability of each created value is reduced, and new opportunities for value creation and capture appear. this conclusion holds true, because virtual markets operate on the widely available internet, which makes financial exchanges convenient and readily accomplishable. and the internet plays the role of a constantly available platform for entrepreneurs to uncover new kinds of resources, to explore potential complementarities among resources, and to exploit available resources. on top of this discussion, proposition 2 explains why virtual markets provide new opportunities for value creation and value capture. additionally, the internet offers timely sharing of information and knowledge either freely or at very low prices. that affordability and convenience of sharing accelerate the circulation speed of knowledge. so, various know-hows that were once difficult to acquire become almost instantly available on fingertips. that explicates why the sustainability of each created value is reduced. 3.2. potentials of value capture as the title suggests, this subsection studies potentials a firm can mobilize to capture value. it consists of two subsections. the first one focuses on those potentials brought forward by the state of market forbearance, supply-chain networks, and strategic blocks. and the second subsection looks at opportunities that exist along with inter-organizational networks and direct seller-buyer platforms. 3.2.1. mutual forbearance, supply-chain network & strategic blocks in the present global economy, when firms compete with each other autonomously in the marketplace for advantages in general and for profits in particular, firms are also parts of interorganizational networks. they collaborate with each other both socially and professionally [36]. by a strategic network, it stands for an established network of inter-organizational ties that are strategically formed to develop certain advantages for all partner firms [41]. for example, as an input-output system, each firm is naturally a part of a supply chain (or ecosystem network), consisting of upstream components and downstream complements [1]. here, suppliers of a firm are some of the upstream components, while customers, supporters and assistants who help to make the product of the firm available to consumers are the downstream complements. in this ecosystem network, supporters and assistants are literally not any part of 60 j.y-l. forrest, s. baizakov, y.g. haile, a. gyan, y. liu copyright ©2020 assa. adv. in systems science and appl. (2020) the firm’s direct supply chain. however, they have to invest and develop the necessary infrastructure for the firm to make its product practically offered to the ultimate consumer. the beauty of strategic networks is that they provide their member firms with access to information, markets, and technologies [41], opportunities of risk sharing, economies of scale and scope [85], as well as knowledge sharing and advanced learning [25]. the symbiotic activities of these networks help their member firms harvest the resultant benefits [15]. proposition 9: if a mutual forbearance exists among the incumbent firms in an oligopolistic industry, then the incumbents enjoy increased profitability that is relatively sustainable until the state of mutual forbearance breaks down. by mutual forbearance, it represents such a situational state that all incumbent firms lighten the intensity of their rivalry by dividing their market into segments proportional to the firms’ individual strengths [14]. so, each firm cedes its dominance to stronger ones in those segments where it is less competent [58]. consequently, these incumbent firms become inter-reliant on each other in their effort to dominate the market. such state of affairs motivates the firms to restrain their rivalries [96], leading to a decreased number of entries into and that of exits from the market [25], while inter-firm hostility lessens [45]. due to their strengths of dominance, market controls relative to upstream components and downstream complements, and deepened extents of collusion, tacit, etc., the incumbent firms exert an extensive influence on their industry’s profitability. in addition, the strongly fortified barriers hinder the entry of any new firm into the industry. hence, the profitability of these firms increases with the strengthening of the state of mutual forbearance and can be seen as relatively sustained. here, the word ‘relative’ stands for the situation that the profitability sustains until the incumbent firms, or at least some of them, can no longer keep up with consumers’ evolving preferences and tastes (theorem 1). for relevant empirical studies, see [36,82]. these scholars show that both dense and strong ties among incumbent firms within an industry or market can be conducive to oligopolistic coordination, tacit or otherwise. and, podolny [78] finds how top-tier banks employ their associations to maintain their good returns within the banking industry. for a connected system 𝑆 = (𝑀, 𝑅), if there is an object 𝑚 ∈ 𝑀 such that 𝑆g = h𝑀 − {𝑚}, 𝑅|(𝑀 − {𝑚})j = 𝑆)⨁𝑆8 ⨁…⨁𝑆c, where 𝑆= (𝑀-, 𝑅-), 𝑖 = 1, 2, …, k, is a connected system and 𝑀∩ 𝑀1 = ∅, for 𝑖, 𝑗 = 1, 2, …, k, satisfying 𝑖 ≠ 𝑗, for a whole number 𝑘 ≥ 2, then we say that the object 𝑚 is a (systemic) hole of the original system 𝑆. speaking less mathematically, an object in a connected system is a hole, if removing the object from the system makes the originally connected system into at least two separate systems, each of which is still connected. to help with an intuitive understanding of this concept, let 𝑆 = (𝑀, 𝑅) stand for the systemic structure of a complete supply chain, where the topmost upstream players are the providers of raw materials and the bottommost downstream players the consumers or the endusers. that is, the object set 𝑀 of this model consists of the set of all players, such as suppliers, producers, retailers, consumers, etc., in the specified supply chain, and the relation set 𝑅 describes how these players are associated with each other based on how their corresponding inputs and outputs are connected into each other. for example, we say that two players 𝑚) and 𝑚8 are associated, if the outputs of 𝑚) are the inputs of 𝑚8 either directly or indirectly. in this systemic model, a particular player 𝑚 ∈ 𝑀 is a hole in this system 𝑆 or supply chain, if when this player 𝑚 is removed from this system, this supply chain 𝑆 falls apart into two disjoint subsystems, neither of which constitutes a complete supply chain, as follows: 𝑆op = h𝑀op, 𝑅opj and 𝑆qrs: = (𝑀qrs:, 𝑅qrs:), value potentials in innovation, resources, strategic networks and blocks 61 copyright ©2020 assa. adv. in systems science and appl. (2020) where the object set 𝑀op of the upstream system 𝑆op contains all the components that are upstream to 𝑚 and the object set 𝑀qrs: of the downstream system 𝑆qrs: all the downstream complements to 𝑚. the reason why these two sub-supply chains 𝑆op and 𝑆qrs: is not complete is that the outputs of the former cannot eventually reach the demands of the latter, while the latter cannot produce its product(s) without the input from the former. in business terms, in the customersupplier network of an industry, if two trading partners can only complete their transactions through a specific focal industry, then this particular industry is a systemic hole in the entire supply-demand network. figuratively, fig. 1 demonstrates the concepts of systemic hole 𝑚, supply-chain system 𝑆 , and the disjoint subsystems 𝑆op and 𝑆qrs: . the arrowed curves represent the direction of the input-output flows of the supply chain 𝑆. fig. 1. how a systemic hole appears in a supply-chain ecosystem proposition 10: in each supply-demand network, a systemic hole assembles power and profitability because of its controlling location advantage. to see why this conclusion holds true, let us model a general supply-demand network as a connected system 𝑆 = (𝑀, 𝑅), where the object set 𝑀 consists of all players in the network and the relation set 𝑅 describes the supply-and-demand relationship of the players. hence, if an object 𝑚 ∈ 𝑀 is a systemic hole in 𝑆, then its locational advantage means that the object 𝑚 can cut the network 𝑆 into two disjoint and dysfunctional subsystems 𝑆op and 𝑆qrs: , as defined previously. and no one in the bottommost layer of the subsystem 𝑆op can find buyers for its products; no one in the topmost layer of the subsystem 𝑆qrs: can offer its outputs to downstream customers and consumers because of a shortage of necessary supplies. speaking differently, this particular object 𝑚 ∈ 𝑀 determines the survival of all other players in the network 𝑆. that power of life or death certainly points to profitability for object 𝑚. in terms of the literature of relevant empirical studies, burt [16] finds that industries that are in a position of systemic holes are capable of achieving good returns through appropriating over-weighted proportional large shares of resources. proposition 11: if a focal firm is a member of a supply-chain network that experiences an expanding base of consumers, then the membership in this network boosts its profitability by improving its market position in a broader network of resource flows. 62 j.y-l. forrest, s. baizakov, y.g. haile, a. gyan, y. liu copyright ©2020 assa. adv. in systems science and appl. (2020) this conclusion follows directly from theorem 1. in fact, the assumption that the said supply-chain network experiences an expanding base of consumers means that no new or existing competitor is taking market share away from the membership of the network (theorem 1) in the sense of market percentages. at the same time, the membership in the network is attracting additional business over time. jointly, these facts mean that the focal firm’s position in the marketplace helps boost, at least potentially, its profitability. on the other hand, if a supply-chain network suffers from a dwindling consumer base, then the focal firm’s location in its broader network of resources, and consequently its profitability are unfavorably disturbed. for example, when the investment in the military of a nation weakens, the fortune of the defense industry suffers. let 𝑆 = (𝑀, 𝑅) be a symbolic representation of the industry of concern, satisfying that each object 𝑚 ∈ 𝑀 stands for the systemic expression of a firm within the industry and every firm of the industry has a systemic expression and is an object in 𝑀. if 𝑆 = ⨁-t) : 𝑆= 𝑆)⨁𝑆8⨁…⨁𝑆: such that each system 𝑆= (𝑀-, 𝑅-) is connected with its object set disjoint with any other firm system, that is, 𝑀∩ 𝑀1 ≠ ∅, for any 𝑖, 𝑗 = 1,2, … , 𝑛, satisfying 𝑖 ≠ 𝑗, then each 𝑆is called a strategic block within the industry. here, we borrow the term from [68], where these scholars studied the global automobile industry. speaking in nontechnical terms, each block 𝑆stands for a group of firms that have formed a strategic alliance with each other but not with any other firms in the industry. proposition 12: each strategic block within an industry generally helps their member firms reach a certain level of profitability that varies from one block to another. if a strategic block experiences an expanding market influence, then its members enjoy a positive effect on their profitability. the first conclusion is a direct consequence of the fact that each strategic block in real life implements an individually unambiguous strategy with its particular accent. hence, different strategic blocks affect the market differently due to their differences in their employed strategies. that naturally means different scales of profitability for the blocks, causing firms with memberships in different blocks to enjoy different levels of performance. the second conclusion follows directly from proposition 11. for relevant empirical studies, see [11,76,98]. specifically, by looking at strategic networks within the venture capital industry, piskorski and nohria find major variations in profitability among firms due to their varied memberships in different strategic blocks. by considering interactions between banking firms, zaheer and zaheer assess the competition in the global industry of currency trading. through configuring alliances into networks, baum et al. find it possible for startups in the biotechnology industry to improve their performance if these firms purposefully tap into the capabilities and knowledge of their alliance partners by using data from the canadian biotechnology industry. 3.2.2. inter-organizational networks & direct seller-buyer platforms riding on the discussions above, the emphases of this section are on the values of a firm’s membership in an inter-organizational network(s), those in a strategic block, and what benefits a firm could enjoy from establishing a platform that bridges direct communication between sellers and buyers. proposition 13: if a firm is a member of an inter-organizational network or a strategic block, then the membership generally represents the firm’s resource that is inimitable and not readily substitutable by other firms and that the firm can potentially mobilize the resource. value potentials in innovation, resources, strategic networks and blocks 63 copyright ©2020 assa. adv. in systems science and appl. (2020) because each inter-organizational network helps its member firms maintain their profitability and boost their locational advantages in broader networks of resource flows, and because each strategic block provides its member firms relatively sustained levels of profitability, propositions 8 – 11 jointly illustrate why the conclusion in this proposition holds true. additionally, networks and strategic blocks practically serve as entry barriers for others to either enter an industry or move across strategic groups, either entering or leaving or both. so, both inter-organizational networks and strategic blocks represent opportunities, advantages and constraints for member firms. on the other hand, theorem 1 supports the conclusion in the proposition. specifically, that theorem states that if sufficient demand exists, the market will invite additional competitions and innovations. however, due to their varied backgrounds, available resources and abilities to learn, firms receive the same invitation in their respectively different ways. when firms develop their respectively different answers to the market call, these firms demand appropriate innovations and supplies correspondingly from their upstream suppliers and from their downstream complementors in order to successfully offer their products to the eventual consumers [2]. such cognitive variations in comprehending market signals, materialistic differences in available resources and in demands on upstream suppliers and downstream complementors make a firm’s membership and association with other firms within an interorganizational network and strategic block imperfectly imitable and imperfectly substitutable resources the firm can potentially mobilize. regarding the literature, gulati et al. [41] derive a related but dissimilar conclusion. in terms of applications, a firm’s membership in a network and strategic block provides the firm with access to significant external resources, such as relationships and information [39], knowledge, alertness and responsiveness [97], opportunities [92], capital, goods, services, technology, etc., [13]. all of these external resources possess the potential for a firm to enhance its competitive position in its industry. and because each inter-organizational network and every strategic block are distinctive and developed through history-specific paths, they are difficult, if not impossible, for others to imitate or substitute [40]. this end once again demonstrates why a firm’s membership in an inter-organizational network and association with a strategic block are resources that are inimitable or extremely difficult to imitate by other firms and are not readily substitutable with any other available resources. it also confirms why that firm can enjoy and potentially mobilize these resources. the so-called opportunism is defined as an observant policy and resultant advantage taking of circumstances, primarily driven by motives of self-interest(s) without considering underlying principles of any kind and consequences for others. the essence in the definition of this concept is the ‘advantage taking of circumstances’ without considering the underlying principles, which generally and assuredly exist with people, either individually or collectively [63]. hence, no matter what specific case and context are concerned with, the ‘advantage taking’ happens within and on top of adopted philosophical beliefs and value systems that dictate what behaviors are considered right, acceptable and which ones are wrong and immoral. that explains why opportunistic behaviors (or opportunism) can be quite easily detected through comparing what is happening against the norms or principles, even though there is no universal standard of morality within any market of free competition except laws and regulations regarding contracts and transaction settlements. our discussion here illustrates the concept of opportunism in detail while also demonstrating that there is no difficulty in defining what opportunism really means, as otherwise claimed by chen et al [19]. proposition 14: assume that an inter-organizational network contains member firms from different industries, and that a strategic block consists of firms within one industry. then, within such a network and strategic block firm-level trust is generally enhanced, firm-level asymmetry of information reduced, and opportunistic behaviors of firms made costly. 64 j.y-l. forrest, s. baizakov, y.g. haile, a. gyan, y. liu copyright ©2020 assa. adv. in systems science and appl. (2020) by firm-level trust, it means the assurance that no firm exploits the weaknesses of any other firm [10]. generally, business transactions take place within a historical succession of prior direct or indirect associations and a broad network of connections. therefore, interorganizational networks and strategic blocks play the role of important resources and necessary origins of referrals, where firms’ characters are the basis of each referral. although characters generally take time to develop and to be recognized, they can be destroyed quickly. therefore, each inter-organizational network and every strategic block require or expect their member firms to behave appropriately within the bound of their norms and principles. that shows why inter-organizational networks and strategic blocks generally enhance firm-level trust and make opportunistic behaviors costly. as for the reason why member firms’ asymmetry of information within these networks and blocks is reduced, it is because through close associations, such as strengthened business ties and frequent business interactions in terms of their respective capabilities, resources and vulnerabilities, the networks’ and blocks’ very nature empowers their member firms to know each other very well. regarding the relevant literature, gulati et al. [41] developed the first draft of this proposition, although stated it very differently. transaction cost is a decreasing function of firm-level trust and opportunism cost, and is an increasing function of informational asymmetry. in other words, enhanced firm-level trust, dropped informational asymmetry, and elevated opportunism cost all help lower appropriation worries. so, we naturally have the following conclusion. proposition 15: assume that an inter-organizational network contains member firms from different industries and a strategic block consists of firms within one industry. then the membership within this network and strategic block generally helps boost the member firms’ value creation and capture. by generalizing the concepts of strategic networks across industries and strategic blocks within an industry, one can develop such a network, or known as a platform, with members from the product market that directly connect competing sellers with buyers. assume that a particular firm plays the role of a systemic hole between sellers and buyers. graphically, fig. 2 demonstrates this concept of networks within the product market that directly connect sellers and buyers with firm 𝑚 being the systemic hole. so, we have the following conclusion naturally. proposition 16: assume that a focal firm offers a convenient platform for each buyer to reach multiple sellers simultaneously and each seller to reach multiple buyers concurrently. then, this firm brings value to sellers and buyers at the same time, and is able to capture its share of value handsomely until such a time when similar platforms are widely available. this conclusion follows directly from the result below developed by forrest and anderson [29]: in the consumer market, finding decided buyers represents a great challenge for sellers due to the availability of similar products and substitutes, and locating willing takers of bargaining bid prices denotes a major struggle for buyers. in particular, in the afore-described direct seller-buyer interactions on a convenient platform, all parties save hugely on their respective searches and information costs, bargaining costs, and policing and enforcement costs. so, when a firm develops a platform to connect a bid price to multiple sellers and each offer to multiple buyers, it practically creates values for both sellers and buyers. simultaneously, the firm is able to capture its share of value continuously until its service is no longer a systemic hole within the network of various platforms that directly connect sellers and buyers (proposition 10). that is the time when the total value potentially capturable by all platform providers has to be shared among many that offer similar platforms. value potentials in innovation, resources, strategic networks and blocks 65 copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 2. a network where sellers and buyers connect directly additionally, proposition 3 in a different light also supports this conclusion. it is because each platform that directly connects sellers and buyers in multiplicity increases the efficiency of transactions for the firm that operates the platform. in real life, priceline.com, for example, has developed such a platform, as described above, that directly connects sellers and buyers. it forwards consumers’ desired prices directly to multiple potential sellers. when a seller willingly accepts a bid price, the item, such as an airline ticket, car rental service, or hotel reservation, is sold over the internet. to protect how its business transactions are completed, priceline.com has patented its innovative business method, making itself different from other online travel agencies [5]. 4. some final words in the current era of transient competitive advantages, business enterprises have to be constantly in the lookout for market signals that either are new or can be comprehended in an unconventional way, technological advances that are either incremental or disruptive and business paradigm shifts in order to create value and to capture value [66]. with globally widening range of application of the internet, the form and rule of market competition have been changing with accelerating speed. platforms developed for buyers to place orders and for sellers to deliver products have become extremely effective at very low costs [5]. to cope with such fast pace of change both academically and practically, this paper investigates the potentials for a firm to create value and to capture value from various different angles, including innovation, resource, market forbearance, supply-chain network, strategic blocks, inter-organizational networks, and direct seller-buyer platforms. due to the novelty of the methodology employed in this research, established conclusions herein are expected to provide managers and entrepreneurs dependable bases to make their decisions. in particular, this paper studies the fast-changing world of business by using such logical reasoning that parallels the one commonly used in mathematics and natural science and systems methodology from the angles of innovation, resources, networks, and direct association of sellers and buyers. in the perspective of innovation, what is pointed out is when innovation provides additional potentials for value creation. in the perspective of resources, we show when resources’ latent values can be practically made visible through value creation. in the perspective of networks and strategic blocks, we exhibit how market’s mutual forbearance and relatively sustainably increased profitability are positively related, why a firm’s systemic hole position offers advantages and profitability to the firm, and how a firm’s profitability is positively affected by its membership in a strategic block of increasing market 66 j.y-l. forrest, s. baizakov, y.g. haile, a. gyan, y. liu copyright ©2020 assa. adv. in systems science and appl. (2020) influence. and, in the perspective of direct association of sellers and buyers, we show that when a firm creates and maintains convenient platforms that conveniently bridge market demands and supplies, it will readily create and capture values. to help the reader quickly glance over what are established in this paper, table 1 lists all 16 propositions in four blocks in terms of value creation and capture. table 1. four categories of all 16 established propositions value creation value capture 3.1.1. innovation & value creation 3.2.1. market association & value capture p1: within a market of free competition, the effectiveness of protective property rights and complementary assets are positively correlated to the potential innovations have in creating value. p2: within a market of free competition, exchanges facilitated by various markets help make resources’ latent values practically materialized. p3: each specific transaction efficiency represents a source of value that can be created for consumers and captured by the underlying firm. p9: if a mutual forbearance exists among the incumbent firms in an oligopolistic industry, then the incumbents enjoy increased profitability that is relatively sustainable until the state of mutual forbearance breaks down. p10: in each supply-demand network, a systemic hole assembles power and profitability because of its controlling location advantage. p11: if a focal firm is a member of a supply-chain network that experiences an expanding base of consumers, then the membership in this network boosts the firm’s profitability through improving its market position in a broader network of resource flows. p12: each strategic block within an industry generally helps their member firms reach a certain level of profitability that varies from one block to another. if a strategic block experiences an expanding market influence, then its members enjoy a positive effect on their profitability. 3.1.2. resource & value creation 3.2.2. networks/buy-sell platforms p4: any firm-specific capabilities are also unique resources of the firm. p5: assume that a firm possesses the control of some valuable and rare resources. if the firm can mobilize these resources, then it is able to create values. p6: assume the same as in proposition 5. if the management and stakeholders of the firm have non-conflicting interests, then the firm is in a good position to create value through utilizing its resources. p7: when a firm is able to work adequately and jointly with such capabilities as marketing, research and development (r&d) and operation, the firm will be able to create value. p8: with the emergence of virtual markets, the sustainability of each created value is reduced, and new opportunities for value creation and capture appear. p13: if a firm is a member of an inter-organizational network or a strategic block, then the membership generally represents the firm’s resource that is inimitable and not readily substitutable by other firms and that the firm can potentially mobilize the resource. p14: assume that an inter-organizational network contains member firms from different industries, and that a strategic block consists of firms within one industry. then, within such a network and strategic block firm-level trust is generally enhanced, firm-level asymmetry of information reduced, and opportunistic behaviors of firms made costly. p15: assume that an inter-organizational network contains member firms from different industries and a strategic block consists of firms within one industry. then the membership within this network and strategic block generally helps boost the member firms’ value creation and capture. p16: assume that a focal firm offers a convenient platform for each buyer to reach multiple sellers simultaneously and each seller to reach multiple buyers concurrently. then, this firm brings value to sellers and buyers at the same time, and is able to capture its share of value handsomely until such a time when similar platforms are widely available. regarding practical applications, conclusions developed in this paper can provide general recommendations instead of suggestions for decision-makers, such as entrepreneurs, value potentials in innovation, resources, strategic networks and blocks 67 copyright ©2020 assa. adv. in systems science and appl. (2020) managers, and retailers. specifically, this paper shows the critical need for entrepreneurs to spot innovatively market signals in their individually different ways so that they can produce their idiosyncratic products (theorem 1) while positioning themselves in systemic hole locations (proposition 10). to increase their likelihoods of success, this work explains why entrepreneurs need to join a supply chain or several that occupy an expanding market territory (proposition 11) and inter-organizational networks (propositions 12 – 15). second, this work shows that managers can potentially make their resources create additional values by encouraging information sharing and recognition of success, such as widely and openly recognizing employee contributions, offering financial incentives, promoting exchanges of ideas, and sharing resources (propositions 1 – 2, 4). managers need to adopt and implement policies and various procedures to acquire and mobilize their firmspecific resources (proposition 5), while tying their interests with stakeholders’ (proposition 6). recommended is that their marketing efforts, r&d achievements, and operational practicalities need to be seriously considered jointly in their decision making (proposition 7). with such joint thoughtfulness, managers will foster collaborations within their organizations while more reliably create value for consumers and capture value for their firms. and, to grow their relatively sustainable performance, managers need to consider the potential of reaching a state of mutual forbearance with competitors by either joining or organizing interorganizational networks (propositions 11 – 15). on the other hand, even after having established in a state of mutual forbearance with competitors (proposition 9), continuous innovations and introduction of new products are still necessary in order for a firm to stay ahead of the changing market environment (theorem 1). and possibly, managers should go after such crucial innovations that would place their firms at systemic holes within their respective supply-chain networks and eco-systems (proposition 10). some of the recommendations for retailers include: • they need to incessantly make their transactions more efficient (proposition 3); • they need to take advantage of whatever new convenience, such as the internet, instant messaging, etc., technology offers (proposition 8); and • they need to provide and recurrently advance their platforms that directly connect buyers and sellers conveniently and massively (proposition 16). before our conclusion of this paper, there is a need to say a few words about the methodology – logical reasoning and systems thinking – employed in this paper. first, the logical reasoning used here parallels that successfully employed in euclidean geometry and most parts of mathematics [51]. it starts with a few most basic and intuitive postulates (or axioms) in the development of generally true propositions. such logical reasoning has been universally utilized for knowledge generation and won successes for such well-established disciplines as physics, chemistry, etc., where derived conclusions are not constrained by specific data and anecdotes [55]. speaking differently, one can derive general managerial recommendations from our conclusions instead of suggestions of limited validity from data and/or anecdote-based theories. so, the main difference between conclusions established in this paper and the related ones developed empirically in the literature can be articulated as follows: the latter represent pioneering works that reveal potential facts, while providing empirical supports or illustration for the truthfulness of the former. as for systems thinking, it has been gradually, while successfully, used in various studies on business-related topics [28,65,80]. because of the new perspectives of systems methodology, brand new conclusions can be established. as for the limitations of this work, first, there are many available tools in systems science developed specifically for analyzing organizations, their evolutions and interactions [62]. when these available tools of systems research are employed one by one in studies of businessrelated issues and problems, one will establish finer conclusions that are expected to be reliable when applied in real life. second, all conclusions in this work are developed on the assumption of why a firm actually exists. but, in reality, firms exist for various reasons beyond that of 68 j.y-l. forrest, s. baizakov, y.g. haile, a. gyan, y. liu copyright ©2020 assa. adv. in systems science and appl. 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(1999). the structure of global competition: a network approach. paper presented at the smj special issue conference, northwestern university. adv syst sci appl 2024; 01:1–19 published online at https://ijassa.ipu.ru. copulas and quantiles in fork-join queueing systems anastasia v. gorbunova1*, alexey v. lebedev2 1v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia 2lomonosov moscow state university, moscow, russia abstract: the article considers a classic fork-join queueing system with a poisson input flow and exponential service times on homogeneous servers. when entering the system, tasks are divided into smaller components (subtasks), the number of which is equal to the number of subsystems. then the subtasks are sent for service to the corresponding subsystems, consisting of a storage device of unlimited capacity and one server. the described functioning mechanism allows, using fork-join systems, to simulate processes occurring in many different real physical systems where tasks are parallelized. the article studies the dependence between the sojourn times of subtasks in subsystems, which is at the same time the main reason for the complexity of analyzing such systems. an approach is proposed for determining the quantiles of the response time distribution, while most works in the field of analysis of fork-join systems are concentrated on obtaining approximations only for the mathematical expectation of the response time. in addition, the approximation of the copula of sojourn times of subtasks (parts of one task) by the gumbel copula and a new kendall’s correlation coefficient estimation are obtained. keywords: fork-join queueing system, response time, distribution quantiles, correlation coefficients, gumbel copula, diagonal section, simulation modeling. 1. introduction this paper examines a classic fork-join queueing system (qs) with a poisson input flow and exponential service times. fork-join qs is a mathematical model for many real-life systems in which tasks are parallelized. when entering the system, the task is divided (fork point) into a number of subtasks equal to the number of subsystems k ≥ 2. all subsystems are actually independent queueing systems with an infinite queue and a single server. each of the subtasks after entering one of the subsystems is serviced there, and then enters a conditional synchronization buffer (join point), where it awaits servicing of the remaining parts of the task. after servicing of all subtasks is completed, the entire task is instantly assembled and can leave the system. previously, such systems were studied by the authors in the works [10, 11, 14]. this work continues the article [11], where a study began on the dependence of the sojourn times of subtasks (parts of one task). the analysis of this system, but with a more complex architecture, can be found, for example, in one of the latest work [12]. the first publications aimed at studying the performance characteristics of mathematical models of fork-join qs and similar systems began to appear in the second half of the 20th century. then research activity decreased somewhat. however, currently there is a new round of interest in the analysis of fork-join such systems, especially in the field of modeling information systems and the processes occurring in them. one of the reasons ∗corresponding author: avgorbunova@list.ru 2 a.v. gorbunova, a.v. lebedev for this situation is the growing popularity of data-intensive applications. among the basic principles of the functioning of high-performance computing environments the parallel data processing, i.e. the simultaneous execution of several operations, commands or actions, is distinguished. for example, one of the main big data technologies from google is called mapreduce, the essence of which is to divide information flows into parts, process them in parallel and further combine the results obtained. this allows you to significantly increase the performance of relevant applications and at the same time reduce the time spent on processing big data, the volume of which continues to increase and, according to idc (international data corporation) forecasts, and can reach 175 zettabytes by 2025 [25, 30, 31, 34]. of course, the scope of application of fork-join models is not limited only to information and computing systems. issues of optimizing processes in production systems (for example, assembling orders in warehouse systems, manufacturing multi-component products) or improving the efficiency of organizing the process of patient stay in medical institutions(and etc.) continue to be in demand to this day [1–3, 9, 18, 22, 32]. another reason for the actualization of research on fork-join systems is the emergence of new methods and approaches to the analysis of complex queueing systems, in particular, an approach based on the use of machine learning methods and its various modifications [4–7, 19,27,28,35]. in this case, we are talking about the further development of this approach and the inclusion of graphical analysis and optimization methods in its composition. despite the apparent simplicity of operation, the study of fork-join qs is one of the most difficult problems to solve. the main reason for the complexity lies in the commonality of the moments of appearance of subtasks in the subsystems, which makes their sojourn times dependent random variables. this is the main difference between a fork-join system and simply parallel operating qss of the same type as the fork-join qs subsystems. therefore, accurate results were obtained only for the average response time in the case of two subsystems with a poisson input flow and exponential service times [24]. for other variants of fork-join qs architectures, which imply, for example, an increase in the number of subsystems or servers in them, limited storage capacity, or other types of distributions for input and service flows, only approximations of the mathematical expectation of response time were obtained in various ways. as for assessing other performance characteristics of a fork-join system, there is much less research in this direction. for example, the variance and standard deviation of response time was analyzed in the papers [8, 14]. in the work [11] you can find analytical expressions (exact or their estimates) for the correlation coefficients between the sojourn times of subtasks in subsystems however, in addition to the first or second moments of the response time random variable, the quantiles of its distribution are of interest. in [15], by using computer simulation, quantiles of response times were found in a system with k = 3 at various load values. in [29], using vector-matrix techniques and phase distributions, theoretical estimates of the tail and quantiles of high levels are obtained. in [26] estimates were obtained for high-level quantiles under conditions of high system load for several types of distributions and fork-join qs architectures. this paper proposes an approach for finding response time quantiles of various levels for a wider range of loads, which allows us to get a complete picture of the behavior of the random variable under study. despite the fact that the object of study of the article is the classic fork-join qs, the approach proposed in the article can be extended to other architectures of fork-join systems and beyond. our approach to constructing an estimation of response time quantiles is based on working with copulas and their diagonal sections. copulas are multivariate distribution functions on a unit cube with uniform marginal distributions. according to sklar’s theorem, any multidimensional distribution can be decomposed into marginal distributions and a copula. thus, copula exhaustively describes the dependence of random variables in its pure form. the copyright © 2024 assa. adv syst sci appl (2024) assa latex template 3 modern mathematical apparatus of copula theory has been actively developed and applied in recent decades, but it is still poorly represented in queueing theory. for example, in [33] a system with dependent service times for subtasks in two parallel queues was considered. a task is considered served if at least one subtask is serviced. marginal distributions of service times are assumed to be shifted exponential or hypoexponential, and their dependence is described by an artificially introduced copula. in our works we study copulas that arise naturally during the functioning of the system. in this regard, we note the article by the authors of [13], where copulas of maximum remaining service times in infinite-server fork-join systems were found. the article is organized as follows. section 2 presents a description of the system under consideration and provides some new results for the correlation coefficients between the sojourn times of subtasks in subsystems, section 3 gives the necessary elements of the theory of copulas, section 4 describes the approach to determining the approximation of the diagonal section of the copula and the quantiles of the response time distribution, in section 5 presents the results of approximating the copula of sojourn times of subtasks by the gumbel copula, section 6 compares it with previously known results, and the conclusion summarizes some results. 2. mathematical model of fork-join qs and correlation coefficients let us describe in more detail the process of functioning of the fork-join system. we will consider the special case of two subsystems (k = 2), however, note that the number of subsystems does not in any way affect the dependence in any pair of sojourn times of subtasks of one task (fig. 2.1). the system receives a poisson flow of tasks with rate λ > 0. at the moment the task is received into the system, it is instantly divided into 2 subtasks, each of which falls into the corresponding subsystem, which has a storage device of unlimited capacity and one server. all servers are homogeneous, the service time has an exponential distribution with parameter µ > 0. thus, the subsystems represent two identical qss of type . . . fork point join point ... ... ... λ λ λ λ µ µ µ fig. 2.1. fork-join model of a queue ing system with subsystems of type mλ|mµ|1. m |m |1. since a task is considered to be serviced only after the completion of servicing of both of its subtasks, the random sojourn time of the task in the qs (response time) r is the maximum of two random sojourn times of subtasks ξi, i = 1, 2, in each from two subsystems: r = max{ξ1, ξ2}. (2.1) random variables ξ1 and ξ2 are correlated due to the fact that all subtasks (parts of one task) arrive at the subsystems at the same time. in the work [11] you can familiarize yourself in detail with the analysis of this dependence, as well as with the derivation of analytical expressions for various correlation coefficients between any pair of sojourn times of subtasks in subsystems. since this article is a logical continuation of the research begun in [11], in copyright © 2024 assa. adv syst sci appl (2024) 4 a.v. gorbunova, a.v. lebedev order to understand the whole picture we present the exact analytical formulas obtained for the pearson and spearman correlation coefficients rp = ρ(4− ρ) 8 , (2.2) rs = 12 √ 2 √ 2− ρ 8− 3ρ − 3. (2.3) as well as an approximate expression for the kendall correlation coefficient rk ≈ ρ(0.25134 + 0.02517ρ), (2.4) where ρ = λ/µ is the system load factor. in what follows, for simplicity, we assume λ = 1, µ = 1/ρ. such parameters were used in the simulation [10, 14]. note that various correlation coefficients, in the general case, only partially reflect the dependence. just copulas fully reflect the dependence. 3. elements of copula theory a copulac is a multivariate distribution function on [0, 1]d, d ≥ 2, if all marginal distributions are uniform on [0, 1]. according to the famous sklar theorem, any multivariate distribution function in rd can be represented in the form f (x1, . . . xd) = c(f1(x1), . . . fd(xd)), where fi, 1 ≤ i ≤ d, are functions of private distributions. thus, every multidimensional distribution can be associated with its copula. if marginal distributions are continuous, then such a representation is unique. as a classic textbook on copulas we point out [23]. in what follows, we restrict ourselves to the case of two-dimensional copulas (d = 2). the diagonal section of a (two-dimensional) copula is the function δ(u) = c(u, u), u ∈ [0, 1]. it has the following (necessary and sufficient) properties: max{2u− 1, 0} ≤ δ(u) ≤ u; 0 ≤ δ(u2)− δ(u1) ≤ 2(u2 − u1), 0 ≤ u1 ≤ u2 ≤ 1. (3.5) the point of studying diagonal sections, for example, is as follows. if random variables x1 and x2 are given with identical marginal distributions f1 = f2 = f and joint distribution copula c, then their maximum is xmax = max{x1, x2} has a distribution function fmax(x) = p (x1 < x,x2 < x) = c(f (x), f (x)) = δ(f (x)), (3.6) so to calculate it we need to know only the diagonal section, and not the entire copula. it is easy to see that the conditions (3.5) are satisfied by the power function δ(u) = uα, 1 ≤ α ≤ 2, then the case α = 1 corresponds to a perfect positive dependence (comonotonicity), and the case α = 2 corresponds to independence of random variables. a classic example of an absolutely continuous (having density) copula with a power-law diagonal section is the gumbel copula c(u1, u2) = exp{−((− lnu1) θ + (− lnu2) θ)1/θ}, θ ≥ 1, u1, u2 ∈ [0, 1], copyright © 2024 assa. adv syst sci appl (2024) assa latex template 5 then δ(u) = u2 1/θ . more precisely, a gumbel copula belongs to the class of extreme value copulas, which always have power-law diagonal sections. you can read about such copulas in [23, § 3.3.4], [16], etc. authors’ works related to copulas can be listed as [13, 20, 21]. 4. approximations of the diagonal section and response time quantiles determining the quantiles of the response time distribution is no less important than finding the average response time, and sometimes even more important. understanding how a system behaves under high load and what maximum response time delays are possible is of great value. despite the fact that in this work we consider fork-join qs with m |m |1 subsystems, the results of estimating the upper quantiles can be used to initially predict the behavior of models that clearly do not have heavy-tailed distributions, for example, these can be various options for small warehouse or production systems, modeling the process of admission, stay and discharge of patients from medical institutions or the process of reviewing a client’s loan application in a financial institution and, possibly, even some private computing systems with the implementation of distributed or parallel computing, operated by employees of a small company. to approximate the quantiles of the distribution of the random variable response time r = max{ξ1, ξ2} we will use elements of copula theory. we will consider a two-dimensional copula c(u1, u2) of random vectors of sojourn times in subsystems (ξ1, ξ2). each component of the random vector has an exponential distribution with the distribution function f (x) = 1− e−(µ−λ)x, x ⩾ 0. then, in accordance with sklar’s theorem, a representation using the copula of the joint distribution (ξ1, ξ2) exists and is unique fξ1,ξ2(x1, x2) = p (ξ1 < x1, ξ2 < x2) = c(f (x1), f (x2)). due to (3.6) we obtain fr(x) = c(f (x), f (x)) = δ(f (x)), (4.7) where δ(u) = c(u, u) is the diagonal section of the copula, which gives us the equation for the quantile level p of the response time distribution fr(xp) = δ(f (xp)) = p, so xp = f−1 r (p) = f−1(δ−1(p)). (4.8) taking into account the inverse transformation method for generating random variables with a given distribution function, consider ui = 1− e−(µ−λ)ξi , i = 1, 2, these random variables will have a uniform distribution on the interval [0, 1], i.e. ui ∼ r[0, 1]. then v = max{u1, u2} = 1− e−(µ−λ)·max{ξ1,ξ2} = 1− e−(µ−λ)r. (4.9) figure 4.2 shows the set of points (u1, u2) at ρ = 0.9 for a fork-join system with two subsystems m |m |1 and for the case of two independent parallel operating qss m |m |1 with copyright © 2024 assa. adv syst sci appl (2024) 6 a.v. gorbunova, a.v. lebedev the same parameters. the number of pairs of dots is 200 thousand; increasing their number overloads the illustration. as you can see, in figure 4.2 b) the points are distributed uniformly inside the unit square, which is typical for the case of independent random variables. in 4.2 a) such uniformity in the distribution of points is no longer observed, even though the value of the pearson correlation coefficient between ξ1 and ξ2 is small (rp = 0.34875), and visual analysis is somewhat difficult, however, the relationship betweenu1 = f (ξ1) andu2 = f (ξ2) in this case is obviously traceable. a) b) fig. 4.2. illustration of the presence/absence of dependence between u1 and u2 at ρ = 0.9 in the case of a) fork-join qs with two subsystems m |m |1; b) two parallel functioning qs m |m |1. the diagonal section of the copula can be estimated as follows. we have δ(u) = c(u, u) = p (u1 < u,u2 < u) = p (max(u1, u2) < u) = p (v < u) = p, i. e. δ(up) = p (v < up) = p, where up is the quantile of the distribution of the random variable v . using realizations vi of the random variable v , obtained through simulation of the values of random stay times in the fork-join qs ri and further substituting them into the formula (4.9), we construct an estimate of the diagonal section δ(u), but in fact, probabilities p. in other words, we construct an empirical estimation of the diagonal section using quantiles of the v distribution. to do this, we order the values v obtained through simulation: v(1), v(2),..., v(n), where v(k) — this is the k-th order statistic, k = 1, ..., n , and from the points (v(k), k/(n + 1)) we determine the estimates (up, p) for probability values from the interval of interest to us p ∈ {0.2, 0.25, 0.30, ..., 0.90}, for a specific fixed value of the load factor ρ ∈ {0.10, 0.15, 0.20, ..., 0.90}. the choice of p values is due to the fact that, as a rule, quantiles of higher levels are of greater interest, so we begin to consider p values from 0.2. next, based on the available data, we will build a forecast of probabilities p depending on the quantiles up and the load factor ρ. p ≈ p̂ = δ̂(up, ρ). copyright © 2024 assa. adv syst sci appl (2024) assa latex template 7 fig. 4.3. dependence of ln p on lnup. fig. 4.4. dependence of (ln p/ lnup) on ρ. copyright © 2024 assa. adv syst sci appl (2024) 8 a.v. gorbunova, a.v. lebedev now, to determine the type of functional dependence, we will conduct a graphical analysis of the obtained data. first of all, it was noticed that the dependence of p on up is well described by a power function, which corresponds to a linear dependence for logarithms (see fig. 4.3). the dependence of the exponent α on ρ also turned out to be close to linear (see fig. 4.4). note that for ρ→ 0 the sojourn times of subtasks are asymptotically independent, hence α → 2. as can be seen from figure 4.4, the dependence graph resembles a bunch of close straight lines passing through the point (0, 2), so it is natural to assume (as a first approximation) that ln p lnup ≈ 2− c · ρ, and hence p = δ(up, ρ) ≈ u2−c·ρ p . (4.10) all that remains is to calculate the value of the coefficient c. similar to the situation with the kendall correlation coefficient [11], we will minimize the module of the relative approximation error relative to the simulation data using the nelder-mead method, as a result of which we obtain the value c ≈ 0.370608. (4.11) thus we have p = δ(up, ρ) ≈ u2−0.370608·ρ p . (4.12) figure 4.5 shows the results of simulation modeling of the probabilities or levels of p 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 a n a ly ti c a l fo r m u la simulation simulation formula fig. 4.5. comparison of analytical results of the formula (4.12) with simulation of the values p of quantiles up of the random variable v = f (r) for values ρ ∈ {0.10, 0.15, 0.20, ..., 0.90}. quantiles up of the random variable v = f (r) in comparison with the results of calculations using the analytical formula (4.12) in the range [0.20, 0.95] with increments of 0.05. each point shown on the graph is actually a set of 17 points based on the number of load factor values ρ ∈ {0.10, 0.15, 0.20, ..., 0.90}, which overlap each other and practically merge, which should be the case with a good level of approximation of the probabilities p. a slight stratification (deviation within 2%) is observed with increasing values of p. for clarity, table 4.1 shows the absolute values of the relative approximation errors for 272 calculated p values. now, taking into account (4.8), we can write δ−1(p) = f (xp), (4.13) copyright © 2024 assa. adv syst sci appl (2024) assa latex template 9 table 4.1. errors in approximations of probability values p calculated using the analytical formula (4.12) in comparison with the results of simulation modeling evaluated types of errors characteristic max ape, % min ape, % mape, % probability p from the formula (4.12) 1.679144 0.002731 0.438597 moreover, from (4.10) it follows that δ−1(p) ≈ p 1 2−c·ρ . we substitute the estimate δ−1(p) into (4.13) and get the relation p 1 2−c·ρ = 1− e−(µ−λ)xp from which it follows that the quantile level p of the distribution of the random variable fork-join response time qs r is determined by expression xp ≈ − ln(1− p 1 2−c·ρ ) µ− λ . (4.14) next, we evaluate the quality of approximation of the resulting expression on the following data set: ρ ∈ {0.10, 0.15, ..., 0.90}, p ∈ {0.20, 0.25, ..., 0.90}, i.e., we get a total of 272 values, for which we will estimate the approximation error using the formula 4.14. in the table 4.2 in the first row for the value c ≈ 0.370608 the relative errors of approximation are presented, and the modulus of the maximum approximation error (max ape) is about 3%, and the average value of the modulus of the relative error does not exceed 1%. however, if we again use the nelder-mead optimization method, but this time to minimize the quantile error xp from the formula (4.14), then the result will be the value c ≈ 0.348284. in this case, the result will improve somewhat, however, as further research shows, the value of the coefficient c ≈ 0.37 turns out to be more justified in other aspects. table 4.2. errors in approximations of the response time quantile for two variants of the values of the coefficient c in the formula (4.14) evaluated types of errors characteristic max ape, % min ape, % mape, % quantile xp, c ≈ 0.370608 3.123971 0.002328 0.734956 quantile xp, c ≈ 0.348284 2.819299 0.007276 0.699130 the figure 4.6 clearly demonstrates the quality of approximation of response time quantiles. as follows from the graphs, large errors arise for large values of the system load factor ρ, but do not exceed 3%, which is an acceptable result. for the sake of clarifying the approximation of quantiles, returning to fig. 4.4, we can note the dependence of the slope of the lines on p. this suggests that instead of the constant c in (4.14) we need to use expressions of the form c1 − c2p or c1 − c2p 2. the selection of constants by the nelder-mead method and a comparative analysis of the accuracy show that the second option is better, namely, the approximation xp ≈ − ln(1− p 1 2−(c1−c2p 2)ρ ) µ− λ , (4.15) where c1 ≈ 0.390327, c2 ≈ 0.237842, moreover, the error is only 0.62%, which is 4.6 times less than before. the obtained results in more detail are presented in the table 4.3. copyright © 2024 assa. adv syst sci appl (2024) 10 a.v. gorbunova, a.v. lebedev 0 5 10 15 20 25 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x p ρ p = 0.2 simulation p = 0.3 simulation p = 0.4 simulation p = 0.5 simulation p = 0.6 simulation p = 0.7 simulation p = 0.8 simulation p = 0.9 simulation fig. 4.6. comparison of analytical results of the formula (4.14) with simulation modeling of quantiles xp of the random variable of response time r. table 4.3. errors in response time quantile approximations calculated by using the analytical formula (4.15) compared to simulation results evaluated types of errors characteristic max ape, % min ape, % mape, % quantile xp 0.617316 0.004912 0.304632 unfortunately, this approach does not provide an explicit expression or convenient approximation for the diagonal section, so we will return to the formula (4.10) and move from there to copulas. 5. approximation of the copula of sojourn times of subtasks by the gumbel copula in the previous section, the estimate for the diagonal section of the copula δ(u) was obtained. in this section, we will present an analytical expression that evaluates the copula c(u1, u2) itself. this will require empirical data, after analyzing which it will be possible to conclude that the copula under study is close to one of the known families. the algorithm for constructing an empirical copula will be as follows: 1) the simulation modeling of a set of pairs (ξk1 , ξ k 2 ) of random variables of sojourn times in subsystems m |m |1 fork-join qs, where k is the serial number of the simulated pair of values, k = 1, ..., n , n is the sample size (total number of pairs of random variables); 2) the transformation of random variables with exponential distribution ξi ∼ exp(µ− λ) by the inverse function method into random variables with uniform distribution on the copyright © 2024 assa. adv syst sci appl (2024) assa latex template 11 interval [0, 1], ui ∼ r[0, 1] , i = 1, 2 (uk 1 , u k 2 ) = (f (ξk1 ), f (ξ k 2 )) = (1− e−(µ−λ)ξk1 , 1− e−(µ−λ)ξk2 ); 3) the division of the unit square into smaller squares (grid) with sides of length h = 1/m, where, for example, m = 20 and the determination of the number of points (uk 1 , u k 2 ) falling into each of squares whose vertices are the points (0, 0), (ih, 0), (0, jh), (ih, jh), i, j = 1, ...,m, and normalization of the resulting value, i. e. cij = c(ih, jh) ≈ ĉij = 1 n n∑ k=1 1{uk 1 < ih, uk 2 < jh}, where 1{·} is an indicator function for the event {·}. figure 5.7 shows a graph of an empirical copula or, what is the same, a joint distribution function of a random vector (u1, u2), constructed in accordance with the algorithm presented above. we will also construct the density of the copula using the following algorithm: 1. by using the results of steps 1 and 2 from the previous algorithm, we obtain a set of pairs of random variables (uk 1 , u k 2 ), k = 1, ..., n ; 2. we divide the unit square into smaller squares (grid) with sides of length h = 1/m, where, for example, m = 20 and determine the number of points (uk 1 , u k 2 ) falling into each of squares whose vertices are the points ((i− 1)h, (j − 1)h), (ih, (j − 1)h), ((i− 1)h, jh), (ih, jh), i, j = 1, ...,m, and normalize the obtained values, i. e. cij = c(ih, jh) ≈ ĉij = 1 nh2 n∑ k=1 1{(i− 1)h < uk 1 < ih, (j − 1)h < uk 2 < jh}. a) b) fig. 5.7. a) empirical copula ĉ(u1, u2), ρ = 0.9; b) empirical copula density ĉ(u1, u2), ρ = 0.9. based on the appearance of the obtained empirical functions in figures 5.7 a) and b), and also taking into account that the diagonal section of the copula under consideration was copyright © 2024 assa. adv syst sci appl (2024) 12 a.v. gorbunova, a.v. lebedev approximated in section 4 by an expression of the form δ(u) ≈ uα, α = 2− cρ, (5.16) we will approximate the desired copula c(u1, u2) gumbel copula, which has the form cg(u1, u2) = exp{−[(− lnu1) θ + (− lnu2) θ] 1 θ }, (5.17) where θ ∈ [1,+∞) is the copula parameter to be estimated. the density of a gumbel copula is determined by taking the second-order mixed partial derivative of the copula function (5.17) cg(u1, u2) = ∂2cg(u1, u2) ∂u1∂u2 = = cg(u1, u2)(− lnu1) θ(− lnu2) θ[θ + [(− lnu1) θ + (− lnu2) θ] 1 θ − 1] u1u2 lnu1 lnu2[(− lnu1)θ + (− lnu2)θ] 2θ−1 θ . (5.18) since for the gumbel copula the diagonal section has the following form δg(u) = cg(u, u) = u2 1/θ , taking into account (5.16) we obtain that θ ≈ ln 2 lnα = ln 2 ln(2− cρ) . (5.19) next, we will again use the nelder–mead optimization method to minimize the modulus of the relative error of approximation of the gumbel copula function (5.17), taking into account that the parameter θ is determined by the expression (5.19), when compared with “true” values of the gumbel copula function obtained by simulation for various load factors ρ ∈ {0.10, 0.15, 0.20, ..., 0.90}. as before, we will not consider low-level quantiles, i.e., let u1, u2 ∈ {0.20, 0.25, ..., 0.90}. as a result, we obtain the following value of the required coefficient c ≈ 0.369250, (5.20) therefore c(u1, u2) ≈ exp{−((− lnu1) ln 2 ln(2−0.36925ρ) + (− lnu2) ln 2 ln(2−0.36925ρ) ) ln(2−0.36925ρ) ln 2 }. (5.21) as can be seen, from (4.11) and (5.20) the coefficient values are very close and actually agree with each other, which confirms the quality of the obtained approximations. as for the approximation error of the formula (5.21), the table 5.4 presents the values of the maximum (max ape), minimum (min ape) and average relative approximation error (mape), the first of which does not exceed 5%, on a data set of 4913 triples (ρ, u1, u2). figure 5.8 a) also shows graphs of the empirical copula function and the copula defined by the expression (5.21) on a given range of values 0.2 ⩽ u1, u2 ⩽ 0.9. table 5.4. errors in approximation of the gumbel copula function c(u1, u2) by the formula (5.21) and the gumbel copula density by the formula (5.18) evaluated types of errors characteristic max ape, % min ape, % mape, % c(u1, u2) 4.966424 0.000000 0.411003 c(u1, u2) 11.164042 0.000805 1.634484 copyright © 2024 assa. adv syst sci appl (2024) assa latex template 13 a) b) fig. 5.8. comparison of empirical and analytical values at ρ = 0.9 for: a) copula function c(u1, u2) (formula (5.21)); b) copula distribution density c(u1, u2) (formula (5.18)). note also that if we estimate the parameter θ using the classical maximum likelihood method (the corresponding python function for the gumbel copula), then the resulting values, the number of which in this case will correspond to the number of values of the correlation coefficient ρ ∈ [0.1, 0.9] with a step of 0.05 (i. e. there will be only 17 of them) on the same 4913 triples of values (ρ, u1, u2) the gumbel copula approximation shows large errors. in this case, max ape ≈ 12.385819%, min ape ≈ 0.000000% and mape ≈ 1.037800%. as for the copula density (5.18), here the result of comparison with empirical data (simulation modeling data) is somewhat worse, but remains acceptable. table 5.4 presents the values of relative approximation errors for ρ ∈ {0.10, 0.15, 0.20, ..., 0.90}, u1, u2 ∈ {0.225, 0.250, ..., 0.875}, i. e. the total number of triplets (ρ, u1, u2) for which the calculation was carried out is 3332. some narrowing of the range of values u1, u2 is explained by the peculiarities of the calculation of the empirical density of the copula in this particular case. moreover, despite the fact that the maximum relative error is about 11%, in the total set of values considered, the number of relative errors exceeding the threshold of 10% is only 2. the number of approximation errors exceeding 5% is only approximately 1.95% of the total amount of data, which is confirmed by the mape indicator, which is approximately equal to 1.63%. note that the error in approximating the copula density increases with increasing values of u1 and u2, however, the resulting estimates will be upper estimates; in addition, this phenomenon can be explained by the insufficient number of tests in the region of upper quantiles and high values of the load factor. as mentioned above, increasing the accuracy of simulation estimates in the range of ρ values, as well as quantiles close to unity, requires a significant increase in the duration of the simulation [10]. 6. comparison of the gumbel copula approximation with previously known results next, let’s check how much the obtained result agrees with the exact formula for the mathematical expectation of the response time obtained in [24], since it must be true e[r] = +∞∫ 0 [1− fr(x)]dx = +∞∫ 0 [1− δ(f (x))]dx = +∞∫ 0 [1− δ(1− e−(µ−λ)x)]dx, (6.22) copyright © 2024 assa. adv syst sci appl (2024) 14 a.v. gorbunova, a.v. lebedev where the estimate of the diagonal section (based on the copula selected in section 5) has the form δ(u) ≈ u2−cρ, c ≈ 0.369250. (6.23) according to [24], the average response time of a fork-join qs with two subsystemsmλ|mµ|1 is e[r] = 12− ρ 8 · 1 µ− λ . taking into account the fact that in the case under consideration we assume for simplicity λ = 1 and, accordingly, µ = 1/ρ, we can rewrite this expression as follows: e[r] = (12− ρ)ρ 8(1− ρ) . (6.24) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 e[r] ρ nelson-tantawi gumbel's copula fig. 6.9. comparison of the results of calculating the average response time of the fork-join qs e[r] using the exact formula (6.24) and using the gumbel copula approximation (5.21) in accordance with the equality (6.27) from [17] it is known that a random variable x with a standard generalized exponential distribution with a function and distribution density of the form fη(x) = (1− e−x)α, fη(x) = α(1− e−x)α−1e−x, x ⩾ 0, α > 0, has the mathematical expectation e[x] = ψ(α + 1)− ψ(1) (6.25) and the variation v ar[x] = ψ′(1)− ψ′(α + 1), (6.26) where ψ(·) is the digamma function, which is defined as the logarithmic derivative of the gamma function [36]. in fact, based on the power-law diagonal section, we obtain an approximation for the response time distribution by a generalized exponential distribution of the general form fr(x) ≈ (1− e−x/β)α, x ⩾ 0, α, β > 0. this approximation was previously postulated in [26], but we derived it naturally, with empirical and theoretical justification. copyright © 2024 assa. adv syst sci appl (2024) assa latex template 15 due to the assumption (6.23), taking into account (6.25)), the integral (6.22) is transformed as follows e[r] = +∞∫ 0 [1− δ(1− e−( 1 ρ −1)x)]dx ≈ ≈ +∞∫ 0 [1− (1− e−( 1 ρ −1)x)2−cρ]dx = [ψ(3− cρ)− ψ(1)] ρ 1− ρ . (6.27) in the figure 6.9 you can compare the approximations of the integral from (6.27) with the true values of the average response time (6.24) according to [24] for values of ρ in{0.10, 0.15, ..., 0.90}. in addition, table 6.5 shows the results of calculations of the mathematical expectation of the response time, from which it follows that the modulus of the maximum relative error of approximation by the gumbel copula does not exceed ρ from the specified range 2.03%, which implies good consistency of the obtained approximations. note that if instead of c ≈ 0.369250 we take c ≈ 0.370608 from (4.11), then the result differs very little. table 6.5. error in approximation of the average response time by the gumbel copula in accordance with the formula (6.27) no. ρ e[r]nt e[r]g error, % 1 0.10 0.16527778 0.16503456 0.14716 2 0.15 0.26139706 0.26080338 0.22712 3 0.20 0.36875000 0.36760144 0.31148 4 0.25 0.48958333 0.48762334 0.40034 5 0.30 0.62678571 0.62369050 0.49382 6 0.35 0.78413462 0.77949216 0.59205 7 0.40 0.96666667 0.95994706 0.69513 8 0.45 1.18125000 1.17176220 0.80320 9 0.50 1.43750000 1.42432705 0.91638 10 0.55 1.74930556 1.73120372 1.03480 11 0.60 2.13750000 2.11273490 1.15860 12 0.65 2.63482143 2.60088709 1.28792 13 0.70 3.29583333 3.24893708 1.42290 14 0.75 4.21875000 4.15278226 1.56368 15 0.80 5.60000000 5.50421629 1.71042 16 0.85 7.89791667 7.75075620 1.86328 17 0.90 12.48750000 12.23495071 2.02242 since the average response time for k = 2 is known exactly, the example discussed is illustrative, but this approach may be useful for k > 2. let us now apply the copula method to estimate the kendall correlation coefficient, which, unlike the pearson and spearman coefficients, is not calculated exactly, and was previously estimated by the formula (2.4). according to [23, p. 164], for the gumbel copula (5.17) the kendall correlation coefficient is rk = 1− 1 θ , from which, taking into account the formula (5.19), we obtain the approximation rk ≈ 1− ln(2− cρ) ln 2 . (6.28) copyright © 2024 assa. adv syst sci appl (2024) 16 a.v. gorbunova, a.v. lebedev let us analyze the quality of the approximation taking into account the simulation results from [11]. unfortunately, at c = 0.37 (which corresponds to the values obtained when estimating the diagonal section and copula), the formula gives overestimated values with an error from 5.19 to 7.46%. however, if we optimize (6.28) over c using the nelder-mead method based on simulation results (as we did earlier), then at the optimal value c ≈ 0.349237 (which is close to the value of c ≈ 0.348284 obtained by estimating response time quantiles) we obtain an error of only 1.04%, which is close in quality to the empirical approximation by the quadratic function (2.4), previously obtained by the authors in [11]. the latter option therefore represents an alternative approximation of the kendall correlation coefficient. thus, copula approximations should not always be taken literally. they may suggest convenient analytical expressions for some characteristics, while the parameters of these expressions may require refinement through additional optimization based on actual data. let us also note the interesting fact that the formula (6.28) allows us to re-estimate the limiting value of the kendall correlation coefficient at ρ→ 1. in [11], based on the (2.4) approximation, an estimate of rk ≈ 0.276 was found, now we obtain a value that coincides with the previous one with an accuracy of three digits, which indicates a good correspondence. as it turned out during further research, the resulting approximation (copula and diagonal section), unfortunately, does not work for estimating the variation (and therefore the standard deviation). taking into account (6.26), the variation of the normalized response time should decrease with increasing load, just like the mathematical expectation, but in fact it increases, according to the simulation results [10, 14], and no c > 0 here fits. this once again shows that the same approximation can be good for some purposes and bad for others, so you should be careful with this in applications. 7. conclusion this work continues the author’s series devoted to the study of the characteristics of forkjoin systems with a poisson input flow and exponential service times. despite the simplicity of the systems and the long history of their research (since the 1980s), there is still a lot of uncertainty in this area. there are few accurate results here, and many estimates need improvement. there are issues that few or no one has dealt with. research mainly focuses on average response time, while variance, quantiles, etc. are also of interest. the key problem is the presence of a relationship between the sojourn times of subtasks (parts of one task), due to the commonality of the input flow into the subsystems. this dependence, although not very strong, has a significant impact on the characteristics, and it is far from being described by classical models (for example, multivariate normal distribution, linear regression, etc.). therefore, the authors in recent works have focused on studying this dependence. the case of two subsystems was considered on the basis that for any number of subsystems, for any pair of subsystems, the sojourn times of subtasks will have the same joint distribution. previous work found the exact values of the pearson and spearman correlation coefficients, as well as an estimate of the kendall correlation coefficient. in this paper, approximations of the joint distribution of subtask sojourn times were studied using copula theory. good agreement with the data for power-law diagonal sections and the gumbel copula is obtained. based on the estimates of diagonal sections, estimates of response time quantiles are derived over a wide range of levels and loads. a new estimate of the kendall correlation coefficient was also obtained. the developed approach, based on copula theory, can be attempted to be generalized to systems with a large number of subsystems or the case of more complex subsystems (for example, with heavy tails of service time distributions). copyright © 2024 assa. adv syst sci appl (2024) assa latex template 17 references 1. armony, m., israelit, s., mandelbaum, a., marmor, y. n., tseytlin, y., et al. 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(2024) digamma function – from wolfram mathworld, [online]. available: https://mathworld.wolfram.com/digammafunction.html copyright © 2024 assa. adv syst sci appl (2024) introduction mathematical model of fork-join qs and correlation coefficients elements of copula theory approximations of the diagonal section and response time quantiles approximation of the copula of sojourn times of subtasks by the gumbel copula comparison of the gumbel copula approximation with previously known results conclusion microsoft word 1384 adv syst sci appl 2023; 03; 104-118 published online at https://ijassa.ipu.ru. a comprehensive examination of the current state of the art in fractal array antennas anuj kumar sharma*, vipul sharma, sanjay singh electronics & communication engineering department, faculty of engineering & technology, gurukul kangri deemed to be university, haridwar, 247667, india abstract: modern astronomical and advanced wireless communication systems necessitate the utilization of array antennas that provide programmable multibeam capabilities, broadband coverage, high-end coverage range, high gain, reduced side-lobe levels with broader side-lobe level angles, enhanced signal-to-noise ratio, and compact dimensions. this has led to many array antenna theories, including fractal array antennas. in order to enhance comprehension of the operational mechanisms of fractal antennas, an introductory exposition on the underlying theoretical principles is provided. this paper provides an in-depth analysis of current developments in fractal array antenna design. to better understand how fractal antenna function, a primer on the theory behind them is presented. in addition, comparative research of the present state-of-the-art in antenna miniaturisation, gain, and bandwidth augmentation with fractal array are performed. keywords: metamaterials, fractal antenna, array antenna, high-gain antenna array, cantor linear array, circular polarization, side lobe level (sll) 1. introduction in antenna theory, microstrip antennas are highly active. a thin, grounded dielectric substrate holds a metallic patch for microstrip antennas. microstrip antennas are portable, compact, cheap, flexible, and easy to integrate with active electronics. their limitations include low power, bandwidth, polarization purity, and spurious feed radiation [1]. improved bandwidth and gain are difficult for microstrip antenna design and mimo communication [2]. in situations where the gain, bandwidth, and power handling capabilities of an individual microstrip antenna element are deemed inadequate, the utilization of an array configuration can be employed to enhance overall performance. in order for the array to operate as an efficient radiator, it is necessary to ensure that each patch within the array is adequately supplied with [3, 4]. the feed arrangement may consist of one or more lines. a network consisting of a solitary feed line is commonly referred to as a series feed network, whereas a network comprising several feed lines is commonly known as a corporate feed network. corporate feed networks provide the efficient and adaptable transfer of power among various antenna components. the manipulation of the beam's direction within the corporate feed network can be achieved by the introduction of a phase alteration [5]. the use of an array consisting of numerous smaller antennas has the potential to offer equivalent functionality to that of a solitary, cumbersome antenna. antenna arrays are favored due to their greater directivity and gain, which are essential attributes for numerous applications. the aforementioned categories encompass several domains such as communication systems, radar technology, satellite systems, electronic warfare mechanisms, and the field of radio astronomy. various geometries, such as linear, rectangular, circular, triangular, and spherical, among others, can be employed to construct an antenna array of identical elements. the electronic scanning of the radiation pattern of a phased array antenna * corresponding author: anujsharma@gkv.ac.in a comprehensive examination of the current state… 105 copyright ©2023 assa. adv. in systems science and appl. (2023) is a highly advantageous and innovative capability. the radiation pattern of the array can be electrically controlled and modified in terms of its shape by manipulating the amplitude and phase of the excitation of each radiating component. the resultant electric field generated by the array is obtained by calculating the vector sum of the electric fields created by each individual radiating component. when the constituent fields of the antenna arrays exhibit cooperative behavior, they generate a highly efficient directional pattern. the antenna array holds significant importance in contemporary communication and industrial systems that rely on radio frequency (rf), microwave, or millimeter wave frequencies. in order to address the challenges faced by modern and future communication systems, it is imperative to employ devices that possess the capability to dynamically change the electromagnetic field. antenna arrays are considered to be the optimal and highly versatile method for facilitating the functionality of these devices 1.1. the application of fractal technology in antenna design a structure characterized by a perpetual absence of termination is referred to as a fractal pattern. the aforementioned structures exhibit patterns that possess unlimited complexity and demonstrate self-similarity across multiple scales [6]. fractals has the capacity for selfreplication, rendering them applicable in diverse domains such as science and engineering. the term "fractal" originates from the latin term "fractus," denoting a surface characterized by a fragmented or non-uniform arrangement. several instances of naturally occurring fractals include the shoreline of the ocean, mountain formations, sea shells, snowflakes, leaves, and the intricate pattern of a peacock's plumage. all natural fractals can be categorized as random fractals due to the absence of a deterministic process for their generation and their existence on non-integral surfaces. these structures are also commonly known as "stochastic fractals." various statistical methodologies are employed in the field of fractal formation. the level of unpredictability exhibited by these fractals varies across different structures and manufacturing techniques. the phenomenon of brownian motion exhibited by microscopic particles in a fluid might be considered as a prominent example of random fractal activity. this can be noticed by looking at the diagram. geometric structures known as deterministic fractals are characterised by their ability to recur at a variety of sizes. in contrast to fractals that are generated at random, these ones have dimensions that have been accurately measured for their development. the construction of deterministic fractals always involves the use of iterated function systems (ifs) and complex number approaches. scaling, twisting the plan axes, and dislocating the source are the three steps involved in the generation of a fractal structure employing these approaches. the koch curve, sierpinski triangle, julia sets, and sierpinski square are only a few of the examples of well-known ifs and complex number fractals. within these deterministic fractal structures, the fundamental generator, also known as the seed, duplicates itself indefinitely [7–9]. 1.2. literature review when it comes to long-range communication systems, array antennas are a better option than aperture antennas. since the emission pattern can be fine-tuned with a larger number of antenna components, side lobes are reduced and scanning beams may be made more directed. these basic characteristics [11–14] are what make array antennas so useful in military settings. traditional array antennas' performance is inadequate for modern wireless communication systems. due to their new ideas about antenna properties like reconfigurable multi-beams, excellent array factor properties, reduced mutual coupling losses, ultra-wide band, and multiband characteristics, fractal array antennas have recently become a candidate for use in both military and civilian fields [15, 16]. 106 a. k. sharma, v. sharma, s. singh copyright ©2023 assa adv. in systems science and appl. (2023) y. kim, et al. constructed random fractal array antennas using fractal geometric technology to cut down on unwanted side lobes [17]. authors presented a technique to combines the best features of both random array antennas and periodic array antennas. for this purpose, the idea of self-similarity is integrated into the theory of random array antennas to control the sidelobe levels. strong and with a reasonable number of side lobes, fractal random array antennas are an excellent option for large thinning array antennas. carles puente-baliarda, et al. purposed a multiband operation and reduced side lobe levels fractal cantor array antenna [18]. in this article authors also discusses and analyses two approaches to building multi-frequency fractal array antennas. a fractal design of array antennas is introduced, and the relative current distribution function of such a fractal array is derived; in addition, the fractal distance between array components is studied. l. dwight, et al. [19] discussed the cantor array antenna of fractal nature, which had an expansion factor of three for multiband operation and lower side lobe levels than other antennas. this article examines two potential approaches to the fabrication of multifrequency fractal array antennas and evaluates, compares, and contrasts the benefits offered by each of these approaches. in this study, authors investigate the fractal distance between array components. additionally, authors introduce a fractal design of array antennas and derive the relative current distribution function of such a fractal array. d. h. werner, et al. discusses and designed a methodical strategy for the process of developing linear and planar fractal array antennas [20]. this geometric method, which is based on the concept of sub-array concentric circular rings, may be used to construct any kind of polygon shape imaginable. cantor linear, sierpinski triangular, square, and hexagonal fractal array antennas are some types of fractal array antennas that may be designed using this geometric design technique. other examples include. they demonstrated numerous essential characteristics of fractal array antennas by making use of the recursive structure of fractal geometric technology. some of these characteristics include multiband behavior, methods for getting lower sidelobe levels, and thinning. because of the recursive nature of this design process, developing algorithms for it is a very straightforward task. douglas h. werner, et al. [21] purposed a peano-gosper fractal array antennas. all of the antenna's elements are arranged in a straight line following a peano-gosper curve that allows them to avoid colliding with each other. in comparison to more conventional periodic planar array antennas, which typically have square or rectangular shapes and regular boundary borders, the frequency range of these fractal array antennas is relatively wider, which is just one of the many benefits that come along with using these antennas. these arrays do not display any grating lobes, and this is true even when the distance between antenna components is as close as physically feasible (at least one wavelength apart). radiation patterns of peano-gosper fractal array antennas have been described, along with a wellorganized iterative technique that can be used to quickly calculate radiation patterns up to an infinite number of stages of development. this technique can be used to determine the radiation patterns of peano-gosper fractal array antennas. the authors w. kushirun, et al. [22] offer a concentric circular design process for creating antennas with a fractal array shape, such as a pentagon, an octagon, or a honeycomb, with an expansion factor of 2. iterative improvements in these arrays' directional performance. the impact of the array's center antenna element has also been studied. the suggested fractal arrays' center element enhanced the behavior of the array's factors. additionally, a technique for designing conformal fractal array antennas (3d) using concentric spheres is presented. broadband antenna arrays employing the menger sponge and sierpinski gasket conformal fractal designs are created with this 3d design process. designing such arrays is a difficult task. pingjuan l. werner, et al. [23] have designed a class of antennas they name peanogosper fractal array antennas because of their self-similar structure. in order to prevent the various components of the antenna from coming into contact with one another, they are a comprehensive examination of the current state… 107 copyright ©2023 assa. adv. in systems science and appl. (2023) arranged in a row that follows a peano-gosper curve in a straight line. the frequency range of these fractal array antennas is rather extensive in comparison to that of more common periodic planar array antennas, such as those with square or rectangular geometries and regular boundary borders. fractal array antennas also have regular boundary bounds. even when the spacing between the antenna components is made as small as is practically possible, these arrays still do not exhibit any symptoms of grating lobes (at least one wavelength apart). we have defined the radiation patterns of peano-gosper fractal array antennas and coupled them with a streamlined iterative approach that can be used to rapidly compute radiation patterns up to an infinite number of stages of evolution. this approach will be discussed in more detail in the following section. anirban karmakar, et al. proposed microstrip fractal planar array antennas using a sierpinski carpet structure [24] as shown in fig. 1. as a result of their geometrical make-up, the mutual coupling between the antenna components dropped to below -20db, and low side lobes began to form. fig. 1. illustration of the procedure for converting a 3x3 rectangular microstrip planar array in to a first stage (5'=1) of sierpinski carpet based thinned array by removing the central element. the authors a. n. jabbar, et al. showed off three different forms of new fractal array antennas for use in space and high-tech wireless networks [25]. they outperform the widelyused uniform square array, as well as random conventional array antennas and the sierpinski fractal array antenna. antenna arrays inspired by twig, dragon, and flap fractal structures are presented. n. deepika rani, et al. compare the cantor fractal linear array antenna's benefits to those of more traditional arrays [26]. fractal structures like those used in cantor linear arrays allow them to outperform more traditional linear array antennas in terms of all of the array factor attributes. over time, narrower beams and fewer side lobes may be achieved by increasing the expansion factor and iteration count. v. s. rao, et al. developed cantor linear array as a solution to the problem of having an odd number of antenna components [27]. cantor arrays normally consist of three different components, however at any given moment only two of those elements will be functioning. in this case, the authors suggest making use of four different components, with two of them being in the "on" position and the other two being in the "off" position. because of this, there is an indirect change in the quality of the array factor as a consequence of the varying distance between the antenna components. the array that is detailed in this paper has 108 a. k. sharma, v. sharma, s. singh copyright ©2023 assa adv. in systems science and appl. (2023) obtained greater resolution and side lobes in comparison to both the cantor fractal array and the standard linear array antenna. r. t. hussein, et al. [28] explain the idea of fractal patterns, which are used in antenna arrays to provide multiband behavior. in designing multiband fractal linear array antennas, the cantor set is a key component shows in fig. 2. the characteristics of array factors are compared to those of the more common linear array. fig. 2. the first four iterations in the construction of the cantor set array. m. levy, et al. purposed a quick method for smart antenna applications that presents itself via the use of fractal array antenna theory, time and location tag algorithms, and beam shaping algorithms [29-30]. the new techniques that have been suggested continue to enhance the distribution of computational sources for the progression of rapid beams, and the method that has been suggested significantly reduces the amount of time that is required for scheming the array factor. it also considerably reduces the amount of reliance placed on the user's memory. levy mounissamy, et al. presented a conceptual study of fractal array antennas for use in optical communication systems [31-32]. both the diverging and converging optical fractal antennas, in addition to the non-linear one, are suggested and investigated. the arrays that have been described have great potential for a variety of forthcoming applications, including efficient quantum light sources, nano-scale spectroscopy, and high-speed data transmission systems. s. hebib, et al. purposed a cantor spiral thinning as a way for achieving both lower side lobe levels and narrower main beams simultaneously in a thinned array antenna [33] as shows in fig. 3. thinner by roughly 10%, this array yet exhibits the same array factor behavior as a standard cantor circular array antenna. fig. 3. (a) symmetric polyadic cantor set at stage of growth s = 1; 2; 3; and 4. (b) resulting concentric rings at stage s = 4. a novel thinning array antenna form to reduce unwanted side lobes was discussed by anirban karmakar, et al. [34]. for the reduction of the side lobes, this paper proposes a new a comprehensive examination of the current state… 109 copyright ©2023 assa. adv. in systems science and appl. (2023) technique using the sierpinski square design, which is called sequential elimination of components. for the purpose of showing the suggested thinning method, a 99 array with microstrip patch antenna components has been explored. carsten metz, et al. [35] presented effective thinning strategies using non-linear multiplicative processing of array antennas for high resolution digital beam-forming radar applications. these sub-array-based, nonlinear multiplicative thinning processing algorithms show off very promising properties, such as thinning by as much as 80% with very minor sacrifices in picture detail. carlo bencivenni, et al. have presented a novel thinning procedure for linear array antennas used in satellite systems [36]. this technique reduces the size of the array antenna by "switching off" the antenna components with the lowest weights in the "weight vector" that get the maximum gain. the method's efficacy is validated by comparing it to a precise combinatorial search strategy for finding the best solution for designing irregular array antennas of varying sizes. ovidio mario bucci, et al. suggested a novel tapering strategies to address the shortcomings of local and global optimization techniques currently employed in satellite communications [37-38]. it turns out that local optimization techniques aren't very useful for huge arrays, and global optimization requires intensive computer procedures. in this paper, oneand two-dimensional tapering strategies, respectively, are presented for linear and circular array antennas. the analytical methods take into account the geometry of the array antenna's central node and the array antenna's components. these approaches are helpful in computing the whole geometry of density tapered linear and concentric circular arrays quickly, iteratively, and deterministically. k. ulichny, et al. [39] proposed a thinned array antennas developed by randomly eliminating 22 array antennas (meaning, four neighboring elements) in large aperture arrays, with varying degrees of success. roughly 5-to-10% of the antenna elements may be thinned in this way without compromising the array factor qualities of the rest of the antenna. alessandro ramalli, et al. developed an antenna with a density tapered array configuration for use in ultrasonic imaging [40]. a method is presented in this article for designing the layout of medium, big, and very large circular array antennas with a minimal number of antenna components that conform to fermat's spiral seeds, and spiral density that tapers off toward the center. because of characteristics like aperiodic and deterministic geometry, a special effort is made to guarantee consistent performance throughout a broad range of wheel orientations. noor ainniesafina zainal, et al. [41] explored antenna for 5g mobile applications with high density tapering. density tapered arrays of two different forms are shown, one optimized for low side lobes and the other for broad band applications. instead of turning off certain antenna components, electrical tapering has been employed here by adjusting the space between them. p. m. n. keizer, et al. [42] have implemented thin circular arrays on a vast scale, with a wavelength range of 25 to 133, to create a synthesis with a low side lobe. synthesis of a lowlevel side lobe by use of amplitude-only taping applied to the "on" components of a large circularly thinned array antenna. efri sandi, et al. [43] have presented two new methods for simplifying the design and lowering the price of linear sparse array antennas. the first way is combinatorial, using integer cyclic difference sets to greatly decrease the initial set size. to mitigate the impact of side lobes, the second procedure employs amplitude tapering using a binomial array. this hybrid method considerably reduces the number of antenna components required, the time required for calculation, and the size of the array antennas. optimizing array antennas is a tried-and-true practice that has been around for at least four or five decades. however, because to the dynamic nature of algorithms, this process of applying optimization strategies to array antennas continues until now. numerous methods for synthesizing array antennas, 110 a. k. sharma, v. sharma, s. singh copyright ©2023 assa adv. in systems science and appl. (2023) including those that optimize the amplitude, phase, and distance between the antenna components, may be found in the literature. in this study, we focused on reducing the size of antenna components by adjusting their amplitude. this article presents the existing literature on amplitude optimization of antenna arrays, as well as fractal array antennas, and highlights the dearth of research into the practical use of optimization methods, notably thinning, in the context of fractal array antennas. r. l. haupt, et al. realize the advantages of tiny antenna arrays, was an early user of optimization methods such as genetic algorithms, particle swarm optimizations, and adaptive algorithms. to attain low side lobe levels, a thin array antenna is required, which is a challenging goal to meet. the production of non-uniform arrays via the use of traditional statistical methods is very inefficient. using conventional techniques of optimization, it is possible to synthesize large arrays of antennas. the person who came up with the genetic algorithm did so because he saw fundamental flaws in the methods that came before it. this approach determines which of an array antenna's components have to be disabled in order to get the fewest amount of side lobes that are feasible. in this paper, both linear and circular antenna arrays are given a more streamlined appearance [44–47]. rajesh bera, et al. [48–50] explained the process of thinning concentric hexagonal and cylindrical elliptical array antennas with evenly stimulated isotropic antenna components that are able to provide a directional beam with a low amount of relative side lobes. in order to reduce the size of the array antenna's individual antenna components, the particle swarm and craziness-based particle swarm optimization approaches were used. because of these optimization approaches, roughly "60%" of the antenna elements were able to become thinner while maintaining the same beam width and reducing the degree of side lobes. p. lombardo, et al. discussed the purpose of optimizing the design of a planar thinned array antennas with varying characteristics [51]. the method that has been presented may either function with individual antenna components or with more compact groupings of a variety of subarray kinds that are positioned on a flat surface. the strategy that has been suggested centers on lessening the intensity of the side lobes while also taking into account a fixed number of active antenna components or subarrays. the approach that is being proposed is suitable for a variety of different forms of the output array antenna that is chosen, which makes it possible to achieve the required directivity attributes on the corresponding antenna pattern. in the context of industrial production, the use of subarrays with a limited number of distinct form variants is suitable. this would result in lower costs for both the design and the mechanization of the process. the modularity that is produced as a consequence makes it possible to scale antenna designs to suit a variety of applications. u. singh, et al. proposed a biogeography-based optimization approach for the purpose of thinning bulky multiple concentric circular array antennas [52]. the objective is to construct an array antenna with evenly excited current amplitudes using isotropic antenna components in order to produce pencil beams with minimal amounts of side lobes. because of this optimization, over half of the antenna components have been reduced in thickness while maintaining or improving their array factor behavior. urvinder singh, et al. discussed the firefly optimization algorithm to the design of thinned concentric circular array antennas [53]. the goal is to build an array antenna consisting of evenly stimulated isotropic components. this kind of antenna is designed to generate a pencil beam pattern with minimal levels of side lobes. within the scope of this research project, two particular use cases for the suggested technique to thin down concentric circular array antennas were described. the main example uses an inter-element spacing that is uniform and is either fixed at 0.5 microns or one of its multiples, while the secondary case utilizes an inter-element spacing that is either optimal or one of its multiples. better array factor characteristics have been attained in both of these examples, and the results of this research make it abundantly evident that the firefly method is appropriate and easy to use for any kind of array antennas. a comprehensive examination of the current state… 111 copyright ©2023 assa. adv. in systems science and appl. (2023) douglas h. werner, et al. [54] proposed a synthesis of a fractal radiation pattern was used to describe a combined approach to the design of multiband array antennas. this innovative approach to the design of multi-band array antennas has a number of advantages, including a significant reduction in the amount of mutual coupling losses, the fact that only a minimal amount of element switching is necessary, and the ability to easily realize the design in the form of reconfigurable apertures. additionally, a thinning procedure that safeguards all of the benefits provided by the band switching architecture was established. this process is predicated on the selection of an appropriate window function. there were several other window functions that were taken into consideration, such as the blackman–harris, kaiser– bessel, rectangular, and blackman. a.n. bondareako, et al. proposed a new synthesis approach for the creation of various array antennas of sierpinski gasket nature [55]. this technique is based on the theory of atomic function. a significant advancement may be attributed to the use of this synthesis method to these arrays. authors d. h. werner, et al. addressed nature-based optimization methodologies for the construction of poly fractal antenna arrays, such as the genetic algorithm and the covariance matrix adaptation evolutionary approach [56]. these arrays make use of compact direct techniques, raised power series, and aperiodic tiling. the working frequency bandwidths of poly fractal array antennas may reach up to 30:1 and beyond, and they have no grating lobes and extremely low sidelobe levels. sierpinski fractal array antenna optimization utilizing the differential evolutionary method was taken into consideration by anirban karmakar, et al. [57]. in a normal situation, fractal array antennas like sierpinski carpet structures may suffer from an elevated side lobe level, in addition to suffering from complicated array factor calculation, which isolates it from the application of any evolutionary optimization strategy. a novel iterative feed matrix is proposed, and it is based on the sierpinski carpet array. its insertion simplifies the computation complexity of the sierpinski fractal array antenna at different iterations and expansion factors, and it makes them suitable for the application of any evolutionary optimization methods. chen w-l, et al. [59] proposed a wide-slot antenna with a fractal-shaped slot for bandwidth improvement. experimental results show that operating bandwidth can be greatly increased by etching the wide slot as fractal shapes, and that there is a correlation between bandwidth and iteration order (io). the experimental investigation of the fractal shapes' scale factor (if) continues. at operating frequencies around 4 ghz, the proposed fractal slot antenna achieves an impedance bandwidth of 2.4 ghz, defined by a 10 db reflection coefficient. this is roughly 3.5 times that of a conventional microstrip-line-fed printed wideslot antenna. in addition, it has a 1.59 ghz bandwidth with a 2-db gain. m. shirazi, et al. [70] introduces a slot-ring antenna/array that is capable of reconfiguration and supports dual polarization. the antenna/array can switch between the s and c bands, and it offers a large bandwidth in each operational condition. this configuration is depicted in fig.4. the reconfiguration of a 2×2 c-band antenna array to an s-band antenna unit cell, and vice versa, can be achieved by simultaneously activating eight pin diode switches. the integration of fractal shapes inside the antenna structure yields a substantial improvement in bandwidth. the antenna exhibits a fractional bandwidth (fbw) of 69.1% in the s-band working state and 58.3% in the c-band operating state. the measured maximum realized gain is 2.4 dbi in the s-band and 3.1 dbi in the c-band. in the case of the 2×2 c-band array, the inter-element spacing is designed to be smaller than 0.5λ0 over the frequency range. this specific configuration allows for beam scanning without the presence of grating lobes. in addition, a variant of this antenna/array is showcased, exhibiting a frontto-back ratio exceeding 12.0/16.0 db at frequencies of 3.5/8.0 ghz. the wideband antenna/array has the potential to function as a unit cell inside a larger array, with element spacing equal to half the wavelength at the center frequency of both frequency bands. 112 a. k. sharma, v. sharma, s. singh copyright ©2023 assa adv. in systems science and appl. (2023) fig. 4. configuration of reconfigurable dualpolarized wideband slot ring antenna/array m. shirazi, et al. [71] introduces a modular antenna array that operates in both the s-band and c-band frequencies, offering dual polarization capabilities. the reconfiguration of a 2×2 s-band antenna array to a 4×4 c-band antenna array can be achieved while maintaining the same physical aperture as shows in fig.5. the implementation of a short ground strip within the slotting antenna serves the purpose of mitigating the mutual coupling between antenna components and minimizing the presence of nulls in the radiation patterns within the c-band array. furthermore, the implementation of the vertical coax-to-microstrip transition facilitates the feeding of the antenna array from the rear of the ground plane, hence enabling a modular configuration. the achievement of dual polarization capacity is observed in each band. the antenna array exhibits a fractional bandwidth of 64.3% and 66.7% in the sand cband working modes, with corresponding gains of 8.4 and 14.3 dbi, respectively. the measurements conducted include the assessment of isolation between various antenna ports, analysis of radiation patterns at different frequency points, and determination of the thirdorder intercept point (iip3) of the antenna. the utilization of a triangular lattice is employed in the s-band array in order to decrease the physical separation between the individual antenna elements. this arrangement facilitates the ability to guide the beam towards larger angles. x. l. li, et al. [72] presents a wideband vehicular planar inverted f antenna (pifa) array that incorporates a fractal structure to achieve high isolation. the pifa antenna components are arranged in a face-to-face configuration, with the inclusion of a decoupling structure (ds) between the elements. this decoupling structure comprises four fractal unit slots and a narrow line slot (ls), which serves the purpose of minimizing the mutual coupling between the antenna elements. the suggested configuration is constructed, and the inhibitory impact of mutual coupling is confirmed by both simulation and measurement techniques. the obtained measurements indicate that the impedance bandwidth spans from 3.02 ghz to 3.96 ghz, encompassing the operational frequency range of 5g. a significant increase in isolation, ranging from 8 to 32 db, is achieved within the frequency range of 3.17–4.15 ghz when employing a decoupling structure. this improvement is observed when the spacing between two pifa antenna components is merely 0.17 times the free-space wavelength at the center frequency of 3.5 ghz. a comprehensive examination of the current state… 113 copyright ©2023 assa. adv. in systems science and appl. (2023) fig. 5. 2x2 dual polarized reconfigurable s-band array (4x4 c-band array) w. t. li, et al. [73] presents a bandwidth-enhanced, low cost, compact, inkjet-printed multilayer microstrip fractal patch antenna for integration into flexible and conformal devices. the antenna consists of two layers of patches, with the first layer inkjet-printed directly on a 0.125-mm (only 0.005 of the operating wavelength) kapton polyimide substrate. on top of it, a 0.12-mm-thick su-8 polymer is covered. to achieve the desired miniaturization and good impedance match, a minkowski fractal geometry patch is employed as the second layer inkjet-printed on top of the su-8 polymer. the proposed antenna has compact dimensions of only 22 × 31mm and covers 4.79–5.04-ghz frequency spectrum with < –10 db. moreover, a 2-bit 1×4 phased array antenna (paa) is constructed, and its scan ability is estimated to demonstrate its potential application in the true timedelay flexible paa systems. the prototype is fabricated and tested for impedance and radiation characteristics. the measured and the simulation results show the superiority of the proposed antenna. 2. results and discussion table 2.1 comparisons between previous related fractal antennas ref. no. bw (ghz) gain (db) directivity (db) size (mm) efficiency (%) array size type weak/strong points [18] 1.1-1.7 30 64 element koch not cover all bands, complex geometry [19] 40 754 elements cantor ring array complex geometry, [20] 60 64 elements sierpinski not cover all bands [21] 21.27 70 19x19 array koch large size [22] 22.5 60 3x3 array koch not cover all bands [23] 1024 elements flap fractal not cover all bands [24] 9x9 sierpinski gain is improved with sll (db) is -18 [55] 2.1-3.2 7 sierpinski gasket gain is not good 114 a. k. sharma, v. sharma, s. singh copyright ©2023 assa adv. in systems science and appl. (2023) [56] 3.2-4.2 10 sierpinski gasket reduce sidelobe levels [57] 10.25-10.75 5 sierpinski elevated side lobe level [59] (2.7–4) (4.3–5.2) 3~5.5 70×70×1.5 fractal slot large size, lp, does not cover all required bands, [60] (2.1–6) 2~5 48×40×1 80–90 sierpinski lp, complex design [61] (3–12) 31×28×1.6 80–90 koch does not cover 2.4 ghz band, lp, no gain values [62] (2.4–2.49) (1.9~7) 120×120×1.6 h-fractal large size, does not cover all required bands, lp, [63] (5.1–5.8) (2.4-2.48) (3.4–3.6) 1.1~3.1 100×100×5 50–72 minkowski large size, lp, low efficiency at lower band. [64] (5.1–5.8) (1.5–4) 2.2~2.4 80×40×1.58 60–79 kochsnowflake lp, does not cover the (5-6) ghz band, complex geometry [65] (1.3–20) −2~10 62×50.8×0.8 20–90 fern leaf lp, low gain and efficiency at lower band [66] (2.4–2.5) 35×35×2.5 48–62 koch lp, no values of gain, low efficiency values [67] (2.5–2.7) 263×164×2.3 mandelbrot large size, lp, does not cover all required bands, no values of gain and efficiency. [68] (5.4–14.2) 158×158×3.6 quasi-fractal large size, lp, does not cover all bands requirement, no values of gain and efficiency. [69] (3–25.2) 3~9.8 25×30×1 hexagonal triangular lp, no efficiency values, does not cover all required frequencies. [70] 2-8 2.4/3.1 23 61 2x2 slot-ring antenna gain is low, and sll (db) is -20 [71] 2-8 8.4/14. 3 25 70 2x2 and 4x4 modular antenna array gain is improved [73] 4.79–5.04 28 22 × 31 1x4 minkowski fractal geometry sll (db) is -10 3. conclusion several of the potential functionalities associated with fractal antenna arrays remain undeveloped, indicating that the technology is still in its nascent stage. the radiation characteristics exhibited by individual elements are characterized by a diffuse nature and possess a relatively low level of directivity, also known as gain. in certain instances, there is a requirement for antennas that possess highly directional characteristics in order to fulfill the demands of long-range communication, resulting in significant gains in signal strength. in order to achieve this objective, it is necessary to augment the electrical dimensions of the antenna. enlarging the dimensions of an individual component of an antenna sometimes confers it with more authoritative characteristics. one approach to augmenting the overall dimensions of an antenna without enlarging the individual elements involves employing an assemblage of radiating devices that are organized in a specific electrical and geometrical configuration. the newly developed antenna can be classified as an array due to its composition consisting of numerous discrete components. it can be inferred that each element inside the array possesses an equal value to every other element within the same array. while not obligatory, opting for this approach is frequently seen as the most effective utilization of one's time. the constituents of an array possess the capacity to assume various physical manifestations, such as wires, holes, and other forms. this topic serves as a source of inspiration for researchers who want to enhance the performance of implanted antennas through novel approaches. a fractal antenna array is a feasible choice for activities that require high frequency. the utilization of a millimeter wave communication system necessitates the presence of a highly directed and high-gain antenna array. the endeavor to achieve this objective poses a persistent challenge within this particular field. metamaterials a comprehensive examination of the current state… 115 copyright ©2023 assa. adv. in systems science and appl. 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(2021). historical development of spatial modulation and massive mimo communication system with implementation challenges: a review. international journal of sensors wireless communications and control, 11(2), 207-215. adv syst sci appl 2022; 02:73–84 published online at https://ijassa.ipu.ru. accelerating sequential quadratic programming for inequality-constrained optimization near critical lagrange multipliers alexey f. izmailov1*, ivan s. rodin1 1lomonosov moscow state university, vmk faculty, io department, moscow, russia abstract: we consider a sequential quadratic programming algorithm for optimization problems with equality and inequality constraints, equipped with the standard armijo linesearch procedure for a nonsmooth exact penalty function, intended for globalization of convergence. we are interested in the case when the standard assumptions for local superlinear convergence of the method may not hold. specifically, we allow for violation of standard constraint qualifications and second-order sufficient optimality conditions, in which case attraction to so-called critical lagrange multipliers is known to have a negative impact on convergence rate. in these circumstances, some known acceleration techniques can be expected to take effect only provided the true hessian and the full sqp step are asymptotically accepted, and these are the main issues addressed in this work. the presented constructions extend some previously known ones to the case when inequality constraints are involved. keywords: constrained optimization, karush–kuhn–tucker optimality system, critical lagrange multiplier, 2-regularity, newton-type methods, sequential quadratic programming, linesearch globalization of convergence, merit function, nonsmooth exact penalty function, true hessian, unit stepsize, extrapolation 1. introduction in this work we consider the equalityand inequality-constrained optimization problem minimize f(x) subject to h(x) = 0, g(x) ≤ 0, (1.1) where the objective function f : rn → r and the constraint mappings h : rn → rl and g : rn → rm are sufficiently smooth, and we are concerned with some crucial peculiarities of performance of newton-type methods near solutions or stationary points of problem (1.1) that are in a sense singular. let l : rn × rl × rm → r be the lagrangian of problem (1.1), i.e., l(x, λ) = f(x) + 〈λ, h(x)〉+ 〈µ, g(x)〉, where 〈·, ·〉 stands for the euclidian inner product. then the stationary points and associated lagrange multipliers of problem (1.1) are characterized by the karush–kuhn–tucker (kkt) optimality system ∂l ∂x (x, λ, µ) = 0, h(x) = 0, µ ≥ 0, g(x) ≤ 0, 〈µ, g(x)〉 = 0. (1.2) ∗corresponding author: izmaf@cs.msu.ru 74 a. f. izmailov, i. s. rodin for a feasible point x̄ of problem (1.1), the linear independence constraint qualification (licq) consists of saying that the gradients h′j(x̄), j ∈ {1, . . . , l}, g′i(x̄), i ∈ a(x̄), are linearly independent, where a(x̄) = {i ∈ {1, . . . , m} | gi(x̄) = 0} is the set of indices of inequality constraints active at x̄. if x̄ is a local solution of (1.1), satisfying licq, then there exists the unique pair (λ̄, µ̄) ∈ rl × rm such that the triple (x̄, λ̄, µ̄) solves (1.2). in this work, we are mostly interested in the case when licq does not hold, but x̄ is stationary for (1.1) with some (possibly nonunique) associated lagrange multiplier. in this case, the set of lagrange multipliers may naturally contain some special instances, called critical multipliers (to be defined formally in section 2 below), that are known to strongly attract dual sequences of various primal-dual optimization algorithms, and this phenomenon has a strong negative impact on convergence rate; see [14–16] and the summaries of this research in [18, section 7.1] and [19]. the effect of attraction to critical multipliers can be locally avoided by using some dual stabilization mechanisms, like the one of the stabilized sequential quadratic programming method; see the original proposals in [22, 24], as well as a more recent treatment in [17] and [18, section 7.2.2]. however, even such modified algorithms typically have large domains of convergence to critical multipliers when they exist [11], and this becomes even more of an issue when globalization of convergence is concerned. generally, there can be at least two different approaches to the specified criticality issue. one is indeed to try to avoid convergence to a critical multiplier, and some further tools for this were developed very recently in [5]. the essence of an alternative approach, adopted in this work, can be expressed as follows: since it is difficult to avoid convergence to critical multipliers, this convergence can be accelerated by employing the knowledge of its rather special character. this point will be explained in section 2, first for generic systems of nonlinear equations and for optimization problems with equality constraints only, using the results from [6–9] and [10], respectively. one of the outcomes of this analysis is a very simple extrapolation strategy for accelerating convergence of the basic sequential quadratic programming (sqp) method to critical lagrange multipliers, that can be easily incorporated into globalization schemes based on linesearch. the main goal of this paper is to extend this approach to optimization problems involving inequality constraints as well, and this is accomplished in section 3. section 4 presents some numerical results confirming the use of the approach, and section 5 contains some concluding remarks and discussion of directions for further research. some words about our notation and terminology. the euclidian (l2), l1, and l∞ norms will be denoted by ‖ · ‖, ‖ · ‖1, and ‖ · ‖∞, respectively. for a function ϕ : rn → r, by ϕ′(x; ξ) we denote the standard directional derivative of ϕ at x ∈ rn in a direction ξ ∈ rn. the null space and the range space of a linear operator λ : rn → rl are denoted by ker λ and im λ, respectively. for a given z ∈ rm and an index set i ⊂ {1, . . . , m}, the symbol zi stands for the subvector of z with components corresponding to i ∈ i . a set u ⊂ rp is called starlike with resect to ū ∈ rp if for every u ∈ u and t ∈ (0, 1], it holds that tu+ (1− t)ū ∈ u . any v ∈ rp is called an excluded direction for a set u starlike with respect to ū if ū+ tv 6∈ u for all t > 0. a set which is starlike with respect to a given point is called asymptotically dense if the complement of the corresponding set of excluded directions is open and dense. 2. nonlinear equations and equality-constrained optimization consider a system of nonlinear equations φ(u) = 0 (2.3) with a smooth mapping φ : rp → rp. a solution ū of (2.3) is called (non)singular if the jacobian φ′(ū) is a (non)singular matrix. for a given current iterate uk ∈ rp, the basic copyright © 2022 assa. adv syst sci appl (2022) accelerating sequential quadratic programming for inequality-constrained 75 newton method (nm) defines the next iterate as uk+1 = uk + vk, where vk is a solution of the linearized (at uk) equation φ(uk) + φ′(uk)v = 0. (2.4) the classical theory says that when u0 ∈ rp is taken close enough to a nonsingular solution of (2.3), then the nm defined this way generates the unique sequence of iterates {uk}, and this sequence converges to ū superlinearly; see, e.g., [18, section 2.1.1]. here we are mainly interested in the cases of when the solution in question can be singular, perhaps even nonisolated. to that end, we make use of the following second-order regularity concept assuming that φ is twice differentiable at ū: the mapping φ is said to be 2-regular at ū in a direction v ∈ rp if im φ′(ū) + φ′′(ū)[v, ker φ′(ū)] = rp. if φ′(ū) is nonsingular, then φ is 2-regular at ū in every direction, including v = 0. at the same time, 2-regularity may hold at singular (and even nonisolated) solutions in nonzero directions, including those from ker φ′(ū). the following theorem summarizes the results on local convergence of the basic nm, obtained in [8, theorem 6.1] and [9, theorem 2.1]. theorem 2.1: let φ : rp → rp be twice differentiable near ū ∈ rp, with its second derivative lipschitzcontinuous with respect to ū, that is, φ′′(u)− φ′′(ū) = o(‖u− ū‖) as u→ ū. let ū be a singular solution of equation (2.3), and assume that there exists v̄ ∈ ker φ′(ū) such that φ is 2-regular at ū in this direction. then there exists a set u ⊂ rp starlike with respect to ū, asymptotically dense at ū, and such that for every starting point u0 ∈ u , there exists the unique sequence {uk} ⊂ rp such that for all k, the step vk = uk+1 − uk is the solution of (2.4), and uk 6= ū, {uk} converges to ū, lim k→∞ ‖uk+1 − ū‖ ‖uk − ū‖ = 1 2 , (2.5) and the sequence {(uk − ū)/‖uk − ū‖} converges to some v ∈ ker φ′(ū). the convergence pattern specified by (2.5) suggests that acceleration of convergence can be achieved by doubling the basic nm step. this, however, should be done with due care, as simply doubling each step may force the resulting sequence to leave the domain of convergence, which is asymptotically dense but need not contain a full neighborhood of the solution in question. to that end, somehow more involved acceleration techniques have been developed in [7,9], such as extrapolation and overrelaxation. the simplest extrapolation technique consists of generating, along with the basic nm sequence {uk}, an auxiliary sequence {ûk} obtained by doubling the nm step: for each k compute ûk+1 = uk + 2vk. (2.6) according to [9, theorem 4.1], under the assumptions of theorem 2.1 above, this auxiliary sequence {ûk} converges linearly to ū, with the asymptotic ratio of 1/4 (instead of 1/2 for {uk}, given by (2.5)), from all points in the domain of convergence of nm sequences {uk}. we now turn our attention to globalization issues, and recall that convergence of the nm can be globalized in standard ways, e.g., by the armijo linesearch procedure for the residual ‖φ(·)‖ used as a merit function; see [18, section 5.1.1]. the following is a prototype algorithm of this kind. copyright © 2022 assa. adv syst sci appl (2022) 76 a. f. izmailov, i. s. rodin algorithm 2.1: fix the parameters σ ∈ (0, 1) and θ ∈ (0, 1). choose u0 ∈ rp, and set k = 0. 1. compute vk ∈ rp solving the linear system (2.4). if (2.4) cannot be solved, stop with failure. 2. set α = 1. if the inequality ‖φ(uk + αvk)‖ ≤ (1− σα)‖φ(uk)‖ (2.7) holds, set αk = α and go to step 3. otherwise, keep replacing α by θα until (2.7) is satisfied. 3. set uk+1 = uk + αkv k, increase k by 1, and go to step 1. according to [18, theorem 5.3], this algorithm converges globally in a sense that any accumulation point ū of any sequence {uk} generated by this algorithm satisfies (φ′(ū))>φ(ū) = 0. moreover, according to [18, theorem 5.4], step 2 of algorithm 2.1 accepts the unit stepsize when uk is close to a nonsingular solution, thus allowing the globalized algorithm to inherit the superlinear convergence rate of the basic nm. near singular solutions, the crucial issue of asymptotic acceptance of the unit stepsize is much more involved. the following result was obtained in [6, theorem 1]. theorem 2.2: under the assumptions of theorem 2.1, a set u in it can be chosen so that for every starting point u0 ∈ u , algorithm 2.1 with σ ∈ (0, 3/4) uniquely defines the sequence {uk}, and αk = 1 is accepted at step 2 of this algorithm for all k large enough. combining theorems 2.1 and 2.2, one can conclude that once a sequence generated by algorithm 2.1 enters the domain u specified in those theorems, this sequence converges linearly to ū with the asymptotic ratio of 1/2. moreover, for the “extrapolated” sequence {ûk} generated according to (2.6), the asymptotic ratio is 1/4. it is important to note that the main iterative sequences {uk} are not affected at all by computing the auxiliary sequences {ûk}, and hence, doing so does not affect the global convergence properties of the algorithm. it should also be mentioned that generating the auxiliary sequence {ûk} costs essentially nothing. getting back to optimization, we first consider the case when there are no inequality constraints in (1.1) (i.e., m = 0): minimize f(x) subject to h(x) = 0. (2.8) in this case, the kkt system (1.2) can be written as (2.3) with p = n+ l, φ : rn × rl → rn × rl, φ(u) = ( ∂l ∂x (x, λ), h(x) ) , (2.9) where u = (x, λ), and the missing argument µ of l is dropped. algorithm 2.1, as well as its version with extrapolation, and theorems 2.1 and 2.2 are certainly applicable to this special case of a nonlinear equation (2.3). the problem, however, is that in the optimization context, using the residual of a first-order optimality system as a merit function can hardly be justified: a merit function should be reflecting the intention to find a solution of the optimization problem rather than just any stationary point of it. one typical choice of an optimization-related merit function for problem (2.8) is the l1 exact penalty function ϕc : rn → r, ϕc(x) = f(x) + c‖h(x)‖1, copyright © 2022 assa. adv syst sci appl (2022) accelerating sequential quadratic programming for inequality-constrained 77 where c > 0 is a penalty parameter; see [1, section 17], [23, section 18.4], [18, section 6.2]. observe, however, that after this change of the merit function, theorem 2.2 is no more applicable, and the issue of asymptotically accepting the unit stepsize becomes even more involved, in particular, due to inherent nonsmoothness of ϕc, and to the presence of a penalty parameter c. another source of complications is that the primal part ξk of the nm step vk = (ξk, ηk) computed at the current iterate uk = (xk, λk) can be guaranteed to be a direction of descent for ϕc (with an appropriate choice of c) at uk only provided the hessian of the lagrangian hk = ∂2l ∂x2 (xk, λk) (2.10) is positive definite, which is not at all automatic even close to a solution/multiplier pair satisfying the second-order sufficient optimality condition. to that end, the algorithm for problem (2.8), presented next, has step 3, where the sufficient descent property of the generated primal direction is verified. algorithm 2.2: fix the parameters c̄ > 0, c̃ > 0, ρ > 0 and σ, θ ∈ (0, 1). choose u0 = (x0, λ0) ∈ rn × rl, and set c−1 = 0 and k = 0. 1. compute vk = (ξk, ηk) solving the linear system ∂l ∂x (xk, λk) +hkξ + (h′(xk))>η = 0, h(xk) + h′(xk)ξ = 0, (2.11) with hk given by (2.10). if (2.11) cannot be solved, stop with failure. 2. set ck = max { ck−1, 4(1− σ)‖λk + ηk‖∞ + ‖λk‖∞ 3− 4σ + c̄ } . (2.12) if maximum in (2.12) is attained at the second argument, replace ck by ck + c̃. 3. if the inequality ϕ′ck(xk; ξk) ≤ −ρ‖ξk‖2 (2.13) is violated, stop with failure. 4. set α = 1. if the inequality ϕck(xk + αξk) ≤ ϕck(xk) + σαϕ′ck(xk; ξk) (2.14) holds, set αk = α and go to step 5. otherwise, keep replacing α by θα until (2.14) is satisfied. 5. define uk+1 = (xk+1, λk+1) as uk+1 = uk + αkv k, increase k by 1, and go to step 1. observe that (2.11) with hk given by (2.10) is nothing else but the nm equation (2.4) with v = (ξ, η), for φ defined in (2.9). on the other hand, this method can be seen as the sqp algorithm, since (2.11) is the lagrange optimality system for the equality-constrained quadratic programming subproblem minimize 〈f ′(xk), ξ〉+ 1 2 〈hkξ, ξ〉 subject to h(xk) + h′(xk)ξ = 0; (2.15) for details see, e.g., [18, section 4.2]. asymptotic acceptance of the true hessian and unit stepsize by algorithm 2.2 has been investigated very recently in [10]. in order to present the main result of that work, some more terminology will be needed. according to [11, proposition 1] and [12, proposition 2], the copyright © 2022 assa. adv syst sci appl (2022) 78 a. f. izmailov, i. s. rodin assumptions of theorem 2.1 for φ defined in (2.9) may hold at ū = (x̄, λ̄) only if λ̄ is a critical lagrange multiplier associated with a stationary point x̄ of problem (2.8). criticality of λ̄ means that the linear subspace q(x̄, λ̄) = { ξ ∈ kerh′(x̄) ∣∣∣∣ ∂2l ∂x2 (x̄, λ̄)ξ ∈ im(h′(x̄))> } (2.16) is nontrivial; see [18, definition 1.41]. in particular, the assumptions of theorem 2.1 do hold under the following requirements: • the multiplier µ̄ is critical of order 1, which means that dimq(x̄, λ̄) = 1, or, in other terms, q(x̄, λ̄) = span{ξ̄} (2.17) with some ξ̄ ∈ rn \ {0}. • it holds that rankh′(x̄) = l − 1 (2.18) and h′′(x̄)[ξ̄, ξ̄] 6∈ imh′(x̄). (2.19) the following counterpart of theorem 2.2 for equality-constrained optimization problems appears in [10, theorem 3.1]. theorem 2.3: let f : rn → r and h : rn → rl be three times differentiable near x̄ ∈ rn, with their third derivatives lipschitz-continuous with respect to x̄. let x̄ be a stationary point of problem (2.8), with an associated lagrange multiplier λ̄ ∈ rl, and assume that (2.17) with q(x̄, λ̄) defined in (2.16), (2.18), and (2.19) hold with some ξ̄ ∈ rn \ {0}. assume also that h′(x̄) 6= 0, and set ū = (x̄, λ̄). then there exist ρ > 0 and a set u ⊂ rn × rl starlike with respect to ū, asymptotically dense at ū, and such that for every starting point u0 ∈ u , algorithm 2.2 with σ ∈ (0, 3/4) uniquely defines the sequence {uk}, this sequence converges to ū, and αk = 1 is accepted at step 4 of this algorithm for all k large enough. getting back to a general optimization problems involving inequality constraints, we recall that according to [18, definition 7.8], a lagrange multiplier (λ̄, µ̄) associated with a stationary point x̄ of problem (1.1) is called critical with respect to an index set a ⊂ a(x̄) if µ̄a(x̄)\a = 0 and (λ̄, µ̄a) is a critical lagrange multiplier associated with the stationary point x̄ of the equality-constrained optimization problem minimize f(x) subject to h(x) = 0, ga(x) = 0. (2.20) as discussed in [15] and [18, section 7.1.2], the effect of criticality on performance of primaldual algorithms shows up precisely this way, through stabilization of specific activity patterns. this is confirmed by the numerical experiments discussed in section 4 below. 3. optimization with inequality constraints unlike algorithms 2.1 and 2.2, the algorithm for problem (1.1) that we present next is not just a prototype algorithm in the following sense: it is supplied by a mechanism intended to avoid unnecessary terminations with failure, especially far from solutions, and it is exactly what we used in numerical experiments in section 4. more precisely, the issue of possible infeasibility of the iteration subproblem (3.23) (a counterpart of (2.15) involving linearized copyright © 2022 assa. adv syst sci appl (2022) accelerating sequential quadratic programming for inequality-constrained 79 inequality constraints) is not addressed in this algorithm, and it still stops with failure in such cases, as this is a general issue concerned with sqp methods, and there are known tools for tackling it; see, e.g., [18, section 6.2]. however, when the iteration subproblem is feasible but cannot be solved, or when the generated primal direction ξk does not pass the sufficient descent test (3.25) (a counterpart of (2.13)), the algorithm does not stop with failure, but rather sequentially adjusts the basic choice of hk = ∂2l ∂x2 (xk, λk, µk) (3.21) in such a way that eventually the needed ξk satisfying (3.25) is always found. for every value of the penalty parameter c > 0, we now need to re-define the penalty function ϕc : rn → r in order to incorporate the inequality constraints: ϕc(x) = f(x) + c(‖h(x)‖1 + ‖max{0, g(x)}‖1). (3.22) algorithm 3.1: fix the parameters c̄ > 0, c̃ > 0, ρ > 0 and σ, θ ∈ (0, 1). choose u0 = (x0, λ0, µ0) ∈ rn × rl, and set c−1 = 0 and k = 0. 1. define hk according to (3.21). 2. compute ξk as a stationary point of the quadratic programming problem minimize 〈f ′(xk), ξ〉+ 1 2 〈hkξ, ξ〉 subject to h(xk) + h′(xk)ξ = 0, g(xk) + g′(xk)ξ ≤ 0, (3.23) and (λk+1, µk+1) as an associated lagrange multiplier. if (3.23) is infeasible, stop with failure. if (3.23) is feasible, but a stationary point of (3.23) cannot be found, go to step 5. otherwise, set ηk = λk+1 − λk, ζk = µk+1 − µk, vk = (ξk, ηk, ζk). 3. set ck = max { ck−1, 4(1− σ)‖(λk+1, µk+1)‖∞ + ‖(λk, µk)‖∞ 3− 4σ + c̄ } . (3.24) if max in (3.24) is attained at the second argument, replace ck by ck + c̃. set ∆k = 〈f ′(xk), ξk〉 − ck(‖h(xk)‖1 + ‖max{0, g(xk)}‖1). 4. if the inequality ∆k ≤ −ρ‖ξk‖2 (3.25) is satisfied, go to step 6. 5. choose τk > 0, replace hk by hk + τki , and go to step 2. 6. set α = 1. if the inequality ϕck(xk + αξk) ≤ ϕck(xk) + σα∆k (3.26) holds with ϕck defined according to (3.22), set αk = α and go to step 7. otherwise, keep replacing α by θα until (3.26) is satisfied. 7. define uk+1 = (xk+1, λk+1, µk+1) as uk + αkv k, increase k by 1, and go to step 1. in step 7 of this algorithm, it is quite typical to use the stepsize parameter in primal updates only; see, e.g., [18, algorithm 6.7]. however, the theory developed in [10] for the equality-constrained case requires using the stepsize parameter in dual updates as well. this copyright © 2022 assa. adv syst sci appl (2022) 80 a. f. izmailov, i. s. rodin variant of linesearch sqp algorithms is also quite common; see, e.g., [23, algorithm 18.3]. theoretical global convergence properties of this kind of algorithms, as well as their rate of convergence properties under “standard” assumptions were recently discussed in [20]; see also references therein. here, however, we are concerned with the cases when those “standard” assumptions for fast local convergence may fail to hold, and in particular, with the cases of convergence to multipliers critical with respect to a relevant index set. motivated by its success in the equality-constrained case, our proposal being tested in this work is to generate an extrapolated auxiliary sequence {ûk} according to (2.6), with expectation that its convergence will be faster than that of the main sequence {uk}. 4. numerical experiments algorithm 3.1 will be abbreviated as sqp-mh, while its variant supplied with extrapolation, and hence, generating an auxiliary sequence {ûk} according to (2.6), will be abbreviated as sqp-mh-ep. note that (2.6) always employs the direction vk obtained by the basic choice of hk specified in (3.21). if for some k such vk cannot be defined, we put ûk+1 = uk+1. instead of modifying the hessian of the lagrangian, possible lack of positive definiteness of the true hessian can be tackled by defining hk as quasi-newton approximations of the hessian. a standard choice adopted here consists of using bfgs approximations complemented by powell’s correction at step 1 of algorithm 3.1 instead of (3.21); see [18, section 4.1] for details. furthermore, in the resulting algorithm abbreviated as sqp-bfgs, steps 4 and 5 of algorithm 3.1 are dropped, and the stepsize parameter is used for primal updates only, i.e., λk+1 = λk + ηk and µk+1 = µk + ζk. the parameter values used in our computations were as follows: c̄ = c̃ = 1, ρ = 10−9, σ = 0.01, θ = 0.5. furthermore, we adopted the following rule for controlling τk at step 5 of algorithm 3.1: for every given k, when step 5 is invoked for the first time, we set τk = 1, and then multiply it by 2 every next time algorithm 3.1 invokes step 5. runs of algorithms sqp-bfgs and sqp-mh were terminated with success if a newly generated iterate uk satisfied ‖φ(uk)‖ ≤ 10−6. (4.27) for sqp-mh-ep, for every k = 1, 2, . . . we first compute ûk, and terminate the run with success if ‖φ(ûk)‖ ≤ 10−6; (4.28) otherwise, we proceed with computing uk and verifying (4.27). when successful termination did not occur after 200 iterations, or the backtracking procedure at step 6 of algorithm 3.1 produced a trial value α such that α‖vk‖ ≤ 10−10, the process was terminated with failure. the experiments were performed in matlab environment, with path solver [3, 4] used for quadratic programming subproblems. problem instances for the experiments were taken from degen test collection [13] that contains optimization problems with constraints violating licq. we employed all problems from degen involving inequality-constraints (numerical results for purely equality-constrained problems can be found in [10]), except for those where the solutions of interest is not a stationary point (problems 20112, 20220, 20224, 30212 and 30302). this leaves 59 test problems. for each of those problems, all algorithms being tested were initialized at the same 100 starting points u0 = (x0, λ0, µ0) generated randomly in the l∞-ball of radius 100, centered at the primal solution of interest (reported in degen) for the primal part x0, and at (0, 0) for the dual part (λ0, µ0), with the additional nonnegativity restriction on µ0. copyright © 2022 assa. adv syst sci appl (2022) accelerating sequential quadratic programming for inequality-constrained 81 as measures of efficiency, we used the average number of quadratic programming subproblems solved and the average number of evaluations of the objective function, constraint mappings, and their derivatives, per one successful run. these results are presented in the form of performance profiles [2]. for degen test set, we employed a modified construction of performance profiles, intended for the case of multiple runs for each test problems; see [21] for details. 2 4 6 8 10 12 14 16 18 20 22 0 0.2 0.4 0.6 0.8 1 sqp-mh sqp-mh-ep bfgs-sqp (a) performance profile 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.2 0.4 0.6 0.8 1 sqp-mh sqp-mh-ep (b) subproblems per iteration fig. 4.1. subproblems solved the performance profile in figure 4.1a demonstrates that sqp-mh-ep is as robust as sqp-mh, somehow more robust than sqp-bfgs, and by far more efficient than both by the number of quadratic subproblems solved per a successful run. the effect of using extrapolation is achieved because the true hessian and the unit stepsize are typically asymptotically accepted, and the set of active constraints of subproblems asymptotically stabilizes. in addition, the plots in figure 4.1b show which portion of problems required solving no more than a given number of quadratic subproblems per iteration on the average. it can be seen that the behavior of sqp-mh and sqp-mh-ep in this respect is quite similar. in particular, for 40% of test problems, only one quadratic subproblem per iteration had to be solved on the average, while solving two or more subproblems was needed for less than 10% of problems. copyright © 2022 assa. adv syst sci appl (2022) 82 a. f. izmailov, i. s. rodin 2 4 6 8 10 12 14 16 18 20 22 0 0.2 0.4 0.6 0.8 1 sqp-mh sqp-mh-ep bfgs-sqp (a) evaluations of f 2 4 6 8 10 12 14 16 18 20 22 0 0.2 0.4 0.6 0.8 1 sqp-mh sqp-mh-ep bfgs-sqp (b) evaluations of h and g 2 4 6 8 10 12 14 16 18 20 22 0 0.2 0.4 0.6 0.8 1 sqp-mh sqp-mh-ep bfgs-sqp (c) evaluations of derivatives fig. 4.2. performance profiles by evaluations the performance profiles in terms of evaluations of f , of the constraint mappings, and of their derivatives are presented in figures 4.2a, 4.2b and 4.2c, respectively. according to copyright © 2022 assa. adv syst sci appl (2022) accelerating sequential quadratic programming for inequality-constrained 83 figure 4.2a, the relative efficiency of sqp-mh-ep by evaluations of f is lower than that by the count of subproblems solved, but the positive effect of using extrapolation is still evident. the picture in figure 4.2b by evaluations of h and g, and in figure 4.2c by evaluations of the derivatives is quite different though from that in figures 4.1a and 4.2a: sqp-mh-ep demonstrated the best results by these measures of efficiency for about 10% of problems only. this behavior is explained by the fact that even if the true hessian and the full step are typically asymptotically accepted, and the set of active constraints stabilizes, additional evaluations of h, g, and derivatives are required at extrapolated points for verifying the stopping test (4.28). 5. concluding remarks we have discussed the use of extrapolation techniques for acceleration of convergence of the sqp method equipped with linesearch, when convergence is to a critical lagrange multiplier of an optimization problem involving inequality constraints. a specific algorithmic implementation of these constructions has been developed and numerically tested. it would be interesting to develop a reasonably complete supporting theory, i.e., conditions ensuring asymptotic acceptance of the true hessian and the unit stepsize, as it is known to be possible for equality-constrained problems, and hopefully can be achieved by some kind of asymptotic reduction to the equality-constrained case. observe, however, that the key difficulty for such development apparently consists of establishing the relation between the values and directional derivatives of the penalty functions for (1.1) and for (2.20) in the sqp directions. acknowledgements this research was supported in part by the russian foundation for basic research grants 19-51-12003 nnio a and 20-01-00106 a, and by the volkswagen foundation. references 1. bonnans, j.f., gilbert, j.ch., lemaréchal, c. & sagastizábal, c. (2006) numerical optimization. theoretical and practical aspects. 2nd edition. berlin: springer-verlag. 2. dolan, e.d. & more, j.j. (2002) benchmarking optimization software with performance profiles. math. program. 91, 201–213. 3. ferris, m.c. & munson, t.s.. path. [online]. available http://pages.cs.wisc.edu/˜ferris/path.html. 4. ferris, m.c. & munson, t.s. (1999) interfaces to path 3.0: design, implementation and usage. comput. optim. appl. 12, 207–227. 5. fischer, a., izmailov, a.f. & scheck, w. (2020) adjusting dual iterates in the presence of critical lagrange multipliers. siam j. optim. 30, 1555–1581. 6. fischer, a., izmailov, a.f. & m.v. solodov. (2021) unit stepsize for the newton method close to critical solutions. math. program. 187, 697–721. 7. griewank, a. (1980) analysis and modification of mewton’s method at singularities. phd thesis. australian national university, canberra. 8. griewank, a. (1980) starlike domains of convergence for newton’s method at singularities. numer. math. 35, 95–111. 9. a. griewank. (1985) on solving nonlinear equations with simple singularities or nearly singular solutions. siam rev. 27, 537–563. copyright © 2022 assa. adv syst sci appl (2022) 84 a. f. izmailov, i. s. rodin 10. izmailov, a.f. (2021) accelerating convergence of a globalized sequential quadratic programming method to critical lagrange multipliers. comput. optim. appl. 80, 943– 978. 11. izmailov, a.f., kurennoy, a.s. & solodov, m.v. (2018) critical solutions of nonlinear equations: local attraction for newton-type methods. math. program. 167, 355–379. 12. izmailov, a.f., kurennoy, a.s. & solodov, m.v. (2018) critical solutions of nonlinear equations: stability issues. math. program. 168, 475–507. 13. izmailov, a.f. & solodov, m.v. (2009) degen. [online]. available http://w3.impa.br/˜optim/degen collection.zip. 14. izmailov, a.f. & solodov, m.v. (2009) on attraction of newton-type iterates to multipliers violating second-order sufficiency conditions. math. program. 117, 271– 304. 15. izmailov, a.f. & solodov, m.v. (2009) examples of dual behaviour of newton-type methods on optimization problems with degenerate constraints. comput. optim. appl. 42, 231–264. 16. izmailov, a.f. & solodov, m.v. (2011) on attraction of linearly constrained lagrangian methods and of stabilized and quasi-newton sqp methods to critical multipliers. math. program. 126, 231–257. 17. izmailov, a.f. & solodov, m.v. (2012) stabilized sqp revisited. math. program. 133, 93–120. 18. izmailov, a.f. & solodov, m.v. (2014) newton-type methods for optimization and variational problems. cham, switzerland: springer series in operations research and financial engineering, springer international publishing. 19. izmailov, a.f. & solodov, m.v. (2015) critical lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it. top 23, 1–26. 20. izmailov, a.f. & solodov, m.v. (2016) some new facts about sequential quadratic programming methods employing second derivatives. optim. meth. software 31, 1111–1131. 21. izmailov, a.f., solodov, m.v. & uskov, e.i. (2015) combining stabilized sqp with the augmented lagrangian algorithm comput. optim. appl. 62, 405–429. 22. li, d.-h. & qi, l. (2000) stabilized sqp method via linear equations. applied mathematics technical reptort amr00/5. university of new south wales, sydney. 23. nocedal, j. & wright, s.j. (2006) numerical optimization. 2nd edition. new york, berlin, heidelberg: springer-verlag. 24. wright, s.j. (1998) superlinear convergence of a stabilized sqp method to a degenerate solution. comput. optim. appl. 11, 253–275. copyright © 2022 assa. adv syst sci appl (2022) introduction nonlinear equations and equality-constrained optimization optimization with inequality constraints numerical experiments concluding remarks microsoft word the+methods+for+aviation+systemsˇ+safety+analysis+during+combinations+of+dangerous+heterogeneous+events+including+human+factor adv syst sci appl 2025; 1; 99-107 published online at https://ijassa.ipu.ru. the methods for aviation systems’ safety analysis during combinations of dangerous heterogeneous events including human factor leonid filimonyuk*, denis bobrov, egor zavialov, aleksandr baklashov v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia abstract: this study presents a probabilistic method for assessing the safety and reliability of aviation systems based on modeling transitions between functional and failed elementary states. the analysis includes five core components of the aircraft system: crew, engines, avionics, power supply, and structural elements. using failure and recovery flow intensities derived from realworld data for various aircraft types in 2022, the authors construct a system of differential equations to calculate dynamic safety indicators. the method applies state-transition graphs and formal mathematical models to describe the system’s behavior under different conditions. particular attention is given to the non-technical elementary state representing crew condition, with a refined classification of critical human factor events such as specific headache types. the study performs calculations in matlab and provides comparative safety assessments for multiple aircraft models, such as airbus (a-319 and a-320), bombardier crj200, boeing (737, 747 and 777), sukhoi superjet 100 (rrj-95) and an-26b-100. keywords: aviation safety, probabilistic modeling, system reliability, failure intensity, recovery intensity, state-transition graph, human factor, aircraft systems, mathematical modeling, flight risk assessment, headaches. 1. introduction nowadays, ensuring the safety and reliability of technical systems, particularly in critical areas, such as aviation, represents a complex and multifaceted task. this complexity stems from the necessity of considering numerous factors that may affect system performance. such factors include the class of the system, its functional purpose, the goals and tasks it must perform, as well as external operating conditions. safety and reliability of the system must always consider its specific characteristics and requirements. otherwise, the analysis becomes excessively complicated and redundant, thus complicating both the development and implementation of effective reliability and safety measures [5, 11]. for aviation systems [7, 11, 12], where errors or failures may lead to catastrophic consequences, the approach to evaluating safety and reliability requires particular thoroughness. it is crucial to clearly define the system's objectives and tasks, and then conduct calculations strictly based on these data. this approach allows us to narrow the scope of considered factors, enabling researchers to focus precisely on those that are genuinely influencing system safety and reliability under specific operating conditions. mathematical modeling represents one of the key tools for evaluating the safety and reliability of technical systems. typically, such modeling involves specialized algebra that formalizes the processes occurring within the system and assesses their impact on overall safety. mathematical models used for these calculations often include elements from various mathematical areas, including differential equations, graph theory, probability theory, and others. these models allow you to not only describe the system's current state but also * corresponding author: filimonyukleonid@mail.ru 100 l. filimonyuk, d. bobrov, e. zavialov, a. baklashov copyright ©2025 assa adv. in systems science and appl. (2025) predict its behavior under various conditions, which proves particularly essential for aviation systems operating within a dynamic and changeable environment. 2. problem ststement and mathematical models the problem is formulated as ensuring the maximum safety criterion and determining the probabilities of accidents for a set of failures’ intensities in the aviation system’. in the context of aviation systems, researchers previously developed two mathematical models that enable assessment of probabilistic safety characteristics. the first model relies on analyzing four elementary states of different aviation system components. these states may include normal operation, partial failure, complete failure, and recovery. a graph constructed from these states represents system behavior, with vertices corresponding to potential system states and arcs reflecting transitions between them. each system state corresponds to a differential equation that describes how the probability of the system occupying that state changes, considering incoming and outgoing transitions. solving this system of equations yields dynamic probabilistic safety characteristics for the aviation system. the second model builds upon the first by incorporating five elementary states, which enables the knowledge of a more detailed description of system behavior and accounts for additional factors that may influence its safety. for instance, this model may include states associated with external influences such as changing weather conditions or human factors [9]. as in the first model, each state corresponds to a differential equation that captures the probability dynamics based on transitions into and out of that state. solving the resulting system of equations provides a more accurate assessment of the system’s probabilistic safety characteristics. both models enable researchers to evaluate not only the current state of the system but also to predict its future behavior. this predictive capability proves especially critical for aviation systems, where anticipating potential failures and implementing preventive measures remains essential. however, each model presents specific limitations and applies only under certain conditions. therefore, selecting an appropriate model requires careful consideration of the system’s unique characteristics and the specific objectives of the analysis. further, in figure 2.1, we can see the graph of the second mathematical model for five elementary states of the system. the number of vertices in the graph is equal to 2n, in our case for 𝑛 = 5 elementary states there are 32 vertices. the methods for aviation systems’ safety analysis … 101 copyright ©2025 assa. adv. in systems science and appl. (2025) fig. 2.1. graph for five elementary states of the aviation system the following provides a brief overview of how these models operate. as shown in figure 1, the graph begins and ends with a single vertex. the initial vertex represents the fully functional state in which all of the elementary components operate correctly. this state always starts with a probability value of 1, as the system remains entirely operational. subsequent vertices represent combinations of 1, 2, 3, and 4 failed elementary states (in the case of a model based on five elementary components). the final vertex corresponds to the state in which all elementary components of the aviation system have failed. this state approaches a critical condition, or may already represent a non-recoverable state, where any attempt to prevent a catastrophic outcome becomes ineffective. the model also introduces 𝜆 and 𝜇 for 𝑖 = 1, 5, representing the intensities of the random variables responsible for failure and recovery, respectively. based on this graph, we may construct a system of differential equations (2.1), where each equation corresponds to a vertex and describes the interactions between the vertexes for the five elementary states of the aviation system [3]. 𝑑𝑃 (𝑡) 𝑑𝑡 = −(𝜆 + 𝜆 + 𝜆 + 𝜆 + 𝜆 ) ∗ 𝑃 (𝑡) + 𝜇 𝑃 (𝑡) + 𝜇 𝑃 (𝑡) + 𝜇 𝑃 (𝑡) + 𝜇 𝑃 + 𝜇 𝑃 (𝑡); 𝑑𝑃 (𝑡) 𝑑𝑡 = 𝜆 𝑃 (𝑡) − (𝜆 + 𝜆 + 𝜆 + 𝜆 + 𝜇 ) ∗ 𝑃 (𝑡) + 𝜇 𝑃 (𝑡) + 𝜇 𝑃 (𝑡) + 𝜇 𝑃 (𝑡) + 𝜇 𝑃 (𝑡); … 𝑑𝑃 (𝑡) 𝑑𝑡 = 𝜆 𝑃 (𝑡) + 𝜆 𝑃 (𝑡) + 𝜆 𝑃 (𝑡) + 𝜆 𝑃 (𝑡) + 𝜆 𝑃 (𝑡) − (𝜇 + 𝜇 + 𝜇 + 𝜇 + 𝜇 ) ∗ 𝑃 (𝑡). (2.1) here, 𝜆 , 𝑖 = 1. .5 , are the failure flow intensities, 𝜇 , 𝑖 = 1. .5 are the recovery flow intensities, and 𝑃 , 𝑖 = 0. .31 denotes the probability of the aviation system being in a specific combination of elementary states, as defined in fig. 2.1. 102 l. filimonyuk, d. bobrov, e. zavialov, a. baklashov copyright ©2025 assa adv. in systems science and appl. (2025) 3. practical scenarios let us explore how these models and systems could be applied in practical scenarios. the following elementary states represent key components of aviation systems: 1. 𝑅 is the crew condition; 2. 𝑅 is the engine condition; 3. 𝑅 is the avionics condition; 4. 𝑅 is the power supply condition; 5. 𝑅 is the condition on structural elements. the table below represents events associated with technical failures in class 1-3 aircraft reported in 2022 [2, 4, 6]. the data presented in the table supports the calculation of failure intensity rates 𝜆 , where 𝑖 = 1, 5. these rates then serve as the basis for computing the probabilistic safety characteristics 𝑃 , where 𝑛 = 0, 31. table 3.1. events related to technical failures in class 1–3 aircraft in 2022 aircraft type aircraft system a -3 19 a -3 20 c r j20 0 b -7 37 b -7 47 b -7 77 r r j95 a n -2 6b -1 00 t ot al engine 6 10 1 4 1 2 1 25 exhaust system (thrust reverser) 1 1 11 13 fuel system 6 6 hydraulic system 2 8 1 9 20 cabin pressurization 3 9 2 4 1 19 wing mechanization 1 3 2 6 12 landing gear 3 3 5 11 air conditioning system 5 5 instrumentation equipment 1 1 1 3 radio communication equipment 1 1 1 3 automatic flight control system (afcs) 1 2 3 electrical system 3 1 1 4 crew 1 3 1 3 3 1 12 total 20 38 6 15 3 6 47 2 137 the intensity 𝜆 of the failure flow for each elementary state 𝑅 (where 𝑖 = 1, 5) of the aviation system derives from the corresponding data in the table. the calculation uses the full duration of the year 2022 as the reference period. thus, the intensity can be calculated as 𝜆 = , where 𝑘 represents the number of failure-related events associated with each of the elementary system state, and the denominator denotes the conditional time unit (1 year). complex technical systems require the use of availability coefficients 𝐾 to enhance their safety and reliability [10]. given the use of both failure flow intensities 𝜆 and recovery flow intensities 𝜇 the availability coefficient follows the expression: the methods for aviation systems’ safety analysis … 103 copyright ©2025 assa. adv. in systems science and appl. (2025) 𝐾 = 𝜇 𝜆 + µ (3.1) only systems with high availability coefficient – typically close to one – enter operational service. in our case, the model assumes an availability coefficient of 𝐾 = 0.95. based on the above formula of the availability coefficient, the recovery intensity can be expressed as: 𝜇 = 𝐾 ∗ 𝜆 1 − 𝐾 (3.2) let's allocate the aircraft systems under each elementary state: crew condition 𝑅 :  visual acuity of the pilot (ability to perceive instrument data and the external environment);  auditory perception of the pilot (ability to hear signals, communication, and warnings);  physical capability (ability to operate controls and perform tasks);  attention and focus (ability to maintain task concentration and avoid distractions). engine condition 𝑅 :  engine;  exhaust system (including thrust reverser);  fuel system. avionics condition 𝑅 :  instrumentation equipment;  radio communication systems;  automatic flight control system (afcs). power supply condition 𝑹𝟒:  electrical system. structural components condition 𝑹𝟓:  cabin pressurization;  wing mechanization;  landing gear;  air conditioning system;  hydraulic system. for a more detailed system-level analysis, event trees are used to trace how individual or combined elementary events may trigger events that are more critical. these chains of events can ultimately lead to catastrophic outcomes, especially when the recovery flow intensity becomes insufficient and any crew intervention fails to resolve the situation. special attention must focus on the elementary condition 𝑅 of the crew of the aircraft system, as it represents the only non-technical state. traditional technical analysis and measurement techniques do not apply here. therefore, researchers must adopt alternative approaches to assess the safety and reliability of this component. as an illustrative example, consider various types of headaches as elementary events that may affect the crew's condition and, consequently, flight safety. these events fall under the classification of 𝑅 : 104 l. filimonyuk, d. bobrov, e. zavialov, a. baklashov copyright ©2025 assa adv. in systems science and appl. (2025) fig. 3.1. elementary events for the crew condition 𝑅 of the aviation system the study analyzes a risk tree constructed from the previously described elementary events [3]. we expanded the tree to a three-level structure. the third level includes the following key designations [8]:  pr 1.2.1 – migraine with typical aura;  pr 1.2.2 – migraine with brainstem aura;  pr 1.2.3 – hemiplegic migraine;  pr 1.2.4 – retinal migraine;  pr 2.1.1 – infrequent episodic tension-type headache with pericranial tenderness;  pr 2.1.2 – infrequent episodic tension-type headache without pericranial tenderness;  pr 2.2.1 – frequent episodic tension-type headache with pericranial tenderness;  pr 2.2.2 – frequent episodic tension-type headache without pericranial tenderness;  pr 2.3.1 – chronic tension-type headache with pericranial tenderness;  pr 2.3.2 – chronic tension-type headache without pericranial tenderness;  pr 2.4.1 – probable infrequent episodic tension-type headache;  pr 2.4.2 – probable frequent episodic tension-type headache;  pr 2.4.3 – probable chronic tension-type headache. to assess the probability of an aviation incident influenced by human factors, the study evaluates combinations of critical events within the event tree. these critical combinations represent the minimal set of interconnected events that may trigger an emergency situation. after identifying the necessary elementary states of the aviation system, systematizing them, assigning failure and recovery intensities, and collecting failure statistics for selected aircraft types in 2022, study proceeds to calculate 𝜆 and 𝜇 for several types of aviation systems: table 3.2. intensities of flow and recovery flows for certain aircraft types. 𝝀𝒊, 𝝁𝒊 aircraft type 𝝀𝟏, 𝝁𝟏 𝝀𝟐, 𝝁𝟐 𝝀𝟑, 𝝁𝟑 𝝀𝟒, 𝝁𝟒 𝝀𝟓, 𝝁𝟓 a-320 3, 57 10, 190 1, 19 1, 19 23, 437 crj-200 1, 19 2, 38 1, 19 0, 0 2, 38 b-747 0, 0 1, 19 0, 0 0, 0 2, 38 rrj-95 0, 0 18, 342 3, 57 0, 0 26, 494 after obtaining the failure (𝜆 ) and recovery (𝜇 ) flow intensities for specific aircraft types listed in table 1, the study proceeds to compute the probabilistic safety metrics using the methods for aviation systems’ safety analysis … 105 copyright ©2025 assa. adv. in systems science and appl. (2025) the matlab software environment. the differential equation system (2.1) can be implemented using the following program code: dp0 = (lyambda1 + lyambda2 + lyambda3 + lyambda4 + lyambda5) * p(0) + mu1 * p(1) + mu2 * p(2) + mu3 * p(3) + mu4 * p(4) + mu5 * p(5); dp1 = lyambda1 * p(0) (lyambda2 + lyambda3 + lyambda4 + lyambda5 + mu1) * p(1) + mu2 * p(6) + mu3 * p(7) + mu4 * p(8) + mu5 * p(9); … dp31 = lyambda5 * p(26) + lyambda4 * p(27) + lyambda3 * p(28) + lyambda2 * p(29) + lyambda1 * p(30) (mu5 + mu4 + mu3 + mu2 + mu1) * p(31); 4. results perform calculations for each aircraft type based on the obtained failure and recovery flow intensities, along with the previously presented system of differential equations. fig. 4.1. probabilistic safety characteristics for the a-320 fig. 4.2. probabilistic safety characteristics for the crj-200 106 l. filimonyuk, d. bobrov, e. zavialov, a. baklashov copyright ©2025 assa adv. in systems science and appl. (2025) fig. 4.3. probabilistic safety characteristics for the b-747 fig. 4.4. probabilistic safety characteristics for the rrj-95 figures 4.1-4.4 demonstrate that the initial vertex of the graph — representing the state in which all elementary components of the system remain fully operational — serves as the primary indicator for safety evaluation. this probabilistic characteristic remains high, while the values associated with other combinations of functional or failed elementary states 𝑅 appear significantly lower, often approaching zero or remaining around 0.05. among the analyzed aircraft types, the b-747 exhibits the highest safety level in 2022, with a probability of 𝑃 ≈ 0.9. in contrast, the a-320 shows the lowest safety estimate, with 𝑃 ≈ 0.78. 5. conclusion this study developed and applied a structured probabilistic method for evaluating the safety and reliability of aviation systems based on mathematical modeling of system states and the failure and recovery flow intensities. by combining real-world failure statistics, formal statetransition models, and computational tools such as matlab, the methodology provides a quantitative foundation for identifying and mitigating operational risks in aircraft systems. the analysis of probabilistic safety and reliability characteristics of aviation systems based on the assessment of elementary states and their corresponding failure and recovery flow intensities provides a quantitative evaluation of operational risks [1]. the use of statetransition models enables formalization of transitions between operational and failed system states. the methods for aviation systems’ safety analysis … 107 copyright ©2025 assa. adv. in systems science and appl. (2025) failure (𝜆) and recovery (𝜇 ) intensities play a central role in these models, as they determine the probability of the system occupying a particular state at any given moment. based on these characteristics, engineers and decision-makers implement technical and organizational measures to enhance system reliability and ensure the required level of flight safety. the probabilistic approach offers an objective foundation for designing, evaluating, and optimizing aviation systems throughout their entire life cycle. in future work, the approach can be expanded by integrating more complex human factor models, environmental variability, and data from more aircrafts to further increase accuracy and applicability. the proposed methodology demonstrates strong potential for supporting data-driven decisions in aviation safety engineering. acknowledgements the research is partly supported by the russian science foundation, project № 24-1900430. references 1. altus, s. (2012). quantitative problem solving methods in the airline industry. berlin, germany: springer, 295–315. 2. aviation safety network. [online]. available: https://asn.flightsafety.org 3. filimonyuk, l. yu. (2021). models and methods for a safe aviation transport systems’ functioning subject to the human factor, ifac-papersonline, 54(13), 627–630. 4. gunaev, a. g. (2023). analiz sostoyaniya bezopasnosti polyotov v organizatsiyakh kommercheskoy ga i aon, podvedomstvennykh sz mtu rosaviatsii za 2022 god [analysis of flight safety status in commercial and general aviation organizations under north-west interregional territorial directorate of rosaviatsiya for 2022]. [in russian]. 5. julyev, v. i. & ivanov, v. s. (1986). bezopasnost polyotov letatelnykh apparatov. teoriya i analiz [flight safety of aircraft. theory and analysis]. moscow, ussr: transport, [in russian]. 6. mezhgosudarstvennyy aviatsionnyy komitet [interstate aviation committee] – official website. [online]. available: https://mak-iac.org (accessed march 6, 2025). 7. novozhilov, g. v., neimark, m. s., tsesarskiy, l. g., et. al. (2015). problema kriticheskikh sochetaniy sobytiy v sisteme «ekipazh – vozdushnoe sudno – dispetcher» [the problem of critical event combinations in the “crew – aircraft – traffic controller” system], polet, 2, 10–16, [in russian]. 8. olesen, j. (2018). the international classification of headache disorders, cephalalgia, 38(1), 211. 9. reason, j. (1990). human error. cambridge. cambridge, uk: cambridge university press. 10. viktorova, v. s. & stepanyants, a. s. (2013). modeli i metody raschyota nadezhnosti tekhnicheskikh sistem [models and methods for reliability analysis of technical systems]. moscow, russia: lenand, 229 p., [in russian]. 11. vishnevsky, v. m., barabanova, e. a., vytovtov, k. a. & vytovtov, g. k. (2024). analysis of tethered unmanned high-altitude platform reliability, proc. of the 26th int. scientific conf. on distributed computer and communication networks: control, computation, communications (dccn-2023) (dordrecht, netherland), 105–116. 12. zakharova, a. a., kutakhov, v. p., meshcheryakov, r. v., podvesovskiy, a. g. & smolin, a. l. (2023). modelirovanie zadach transportirovki gruzov v bespilotnoy aviatsionnoy transportnoy sisteme [modeling cargo transportation tasks in an unmanned aerial transport system], aviakosmicheskoe priborostroenie, 3, 3–15, [in russian]. adv syst sci appl 2022; 03:36–48 published online at https://ijassa.ipu.ru. dynamics of a generalized fractional epidemic model of covid-19 with carrier effect khalid hattaf1,2, ahmed a. mohsen3,4 1centre régional des métiers de l’education et de la formation (crmef), casablanca, morocco 2 laboratory of analysis, modeling and simulation (lams), hassan ii university of casablanca, casablanca, morocco 3 department of mathematics, college of education for pure science (ibn al-haitham), university of baghdad, iraq 2 ministry of education, baghdad, iraq abstract: currently, coronavirus disease 2019 (covid-19) continues to cause several new cases and deaths, and further new variants of the virus responsible this disease appear. the present paper proposes a new mathematical model to understand the mechanisms of the spread of covid-19 and better describe its dynamics. the modes of infection spread of covid-19 via asymptomatic and symptomatic individuals are modeled by two general nonlinear incidence functions in order to include several types of incidence rates existing in the literature. when a disease outbreak within a community, individuals acquire information about this disease. therefore, the proposed model take into account the memory effect on the outbreaks of covid-19. this effect is modeled by a fractional order derivative in caputo sense. the mathematical analysis of the proposed model is rigorously investigated, including the computation of the basic reproduction number r0 and the stability of equilibria. keywords: covid-19, sars-cov-2, incidence rate, basic reproduction number, fractional differential equations, stability analysis. 1. introduction coronavirus disease 2019 (covid-19) is a infectious disease caused by a new coronavirus called severe acute respiratory syndrome coronavirus 2 (sars-cov-2). world health organization (who) first learned of this new virus on 31 december 2019, following a report of a cluster of cases of viral pneumonia detected in wuhan city, hubei province of china. due to rapid worldwide outbreak of covid-19, the who declared it a pandemic on 11 march 2020. over 2.7 million new cases were reported on 7 march 2021 with 116 166 652 cumulative cases, 60 323 new deaths and 2 582 528 cumulative deaths [18]. since the appearance of the first case in morocco on 2 march 2020 in the city of casablanca, the disease has caused 8 716 deaths on 12 march 2021 according to new statistics given by moroccan ministry of health (mmh) [13]. however, a total of 13 671 deaths was confirmed in iraq on the same day [19]. the outbreak covid-19 impacted nearly every side of society worldwide. therefore, the pandemic has required a substantial response by public health authorities at every level and considered aformidable global public health challenge. ∗corresponding author: corresponding author: aamuhseen@gmail.com dynamics of a generalized fractional epidemic model... 37 despite the prevention measures, such as hand washing, social distancing, quarantine of suspected cases, isolation of confirmed cases, the closure of schools and non-essential businesses and services, and also the start of vaccination in some countries, covid-19 continues to cause new cases and deaths, and new variants of sars-cov-2 appear. in fact, the transmission or outbreaks of the covid-19 consists of direct and indirect contacts between infective and susceptible individuals, it seems that the several mathematical models well described the covid-19. however, attention must be paid to the unique traits of covid-19. these traits are (a) the presence of individuals who exhibit infectious symptoms even during the incubation period, such as such as high body temperature, body aches and pains, queasiness, or diarrhea typically (2 14) days after infective to the virus, (b) there are asymptomatic individuals called a carrier, who can move freely and transmit infection, which makes controlling the situation complicated such that, approximately half of the covid-19 patients (40− 50%) in one study did not show any symptoms from above in (a). for example, there are several reports that have investigated the possibility of asymptomatic carriers of covid-19. a report of japanese travelers from wuhan, china approximates the proportion of carriers of disease but asymptomatic at 30 %. in addition, studies of every person in an isolated italian village revealed that 50-75 % of people were asymptomatic. a report of a nursing home in the usa found 30 % of patients were asymptomatic on the day of testing for covid-19, with 4 % remaining asymptomatic upon follow-up a week later. a report in hospitalized patients in beijing found that 5 % of patients testing positive for covid-19 had asymptomatic infections see [3–5,10,15]. recently, many mathematical models have been developed to describe the spread of infectious diseases, such as hepatitis b, hiv transmission and ebola virus disease. for more information, we refer the reader to the recent book published by hattaf and dutta about this subject in [7]. for covid-19, moussaoui and zerga [14] proposed a mathematical model that takes into account strategies against covid-19 such as wearing masks and respecting safety distances. a model for the dynamics of covid19 with quarantine strategy and media coverage effect was presented by mohsen et al. in [12], while a mathematical model of covid-19 pandemic involving the infective immigrants was also studied in [11]. a fractional order model with bilinear incidence rate was introduced in [2] to study transmission dynamics of covid-19 in japan 2020. modeling of the dynamics of covid-19 inside the human body was presented in [9]. the main objective of this study is to develop a new mathematical model in order to better describe the dynamics of the covid-19 in human population by taking into account the effects of memory and carrier, and also others aspects such as the nonlinearity of the incidence function, the death rate due to covid-19 and the recovery rates of the asymptomatic and symptomatic individuals. to do this, the structure of this work is outlined as follows. the next section is devoted to the formulation of model, the existence of equilibria and the computation of the basic reproduction number r0. in section 3, we use mathematical analysis to establish global stability results for the proposed model. some applications of our main results are presented in section 4. eventually, the conclusion is given in section 5. 2. model formulation and equilibria in this section, we first construct a mathematical model that takes into account the effects of memory and carrier. then we divide the total population into four classes s(t), c(t), i(t) and r(t) that represent susceptible, carrier (asymptomatic infected individuals), infected and recovered individuals at time t, respectively. the dynamics copyright© 2022 assa. adv syst sci appl (2022) 38 khalid hattaf, ahmed a. mohsen of the four classes is governed by the following nonlinear system of fdes: dαs(t) = a− νs − φ(s, c)c −ψ(s, i)i, dαc(t) = φ(s, c)c +ψ(s, i)i − (ν + d+ γ + r1)c, dαi(t) = γc − (ν + d+ r2)i, dαr(t) = r1c + r2i − νr, (2.1) where the susceptible individuals are recruited at a rate a and become infected either by effective contact with carrier at rate φ(s, c)c or by effective contact with infected individuals at rate ψ(s, i)i. so, the term φ(s, c)c + ψ(s, i)i denotes the total asymptomatic infection rate of susceptible individuals. the natural death rate in all classes is denoted by ν, while d is the death rate due to covid-19. the rate of transfer from the asymptomatic to symptomatic is denoted by γ. the parameters r1 and r2 are recovery rates of the asymptomatic and symptomatic individuals, respectively. finally, dα denotes the caputo fractional derivative with α ∈ (0, 1] that describes the memory effect. a schematic diagram of model (2.1) is shown in figure 1. fig. 2.1. the diagram of model (2.1). obviously, the first three equations of (2.1) do not depend on the variable r, model (2.1) can be rewrite by following format{ dαs(t) = a− νs − φ(s, c)c −ψ(s, i)i, dαc(t) = φ(s, c)c +ψ(s, i)i − d1c, dαi(t) = γc − d2i, (2.2) where d1 = ν + d+ γ + r1 and d2 = ν + d+ r2. furthermore and according to [8], we assume that the infection rate is the general incidences φ and ψ are continuously differentiable in the interior of ir2 + and satisfy the two following conditions: (h1) φ(0, c) = 0, ∂φ ∂s (s, c) > 0, ∂φ ∂c (s, c) ≤ 0 for all s, c ≥ 0. copyright© 2022 assa. adv syst sci appl (2022) dynamics of a generalized fractional epidemic model... 39 (h2) ψ(0, i) = 0, ∂ψ ∂s (s, i) > 0, ∂ψ ∂i (s, i) ≤ 0 for all s, i ≥ 0. next, we study the existence of equilibria. clearly, ef (s0, 0, 0) is a disease-free equilibrium of (2.2), where s0 = a ν . hence, the basic reproduction number of (2.2) can be calculated as follows r0 = d2φ(s0, 0) + γg(s0, 0) d1d2 , (2.3) which can be rewritten as r01+r02, where � r01 = φ(s0, 0) d1 is the basic reproduction number for asymptomatic mode of transmission. � r02 = γψ(s0, 0) d1d2 is the basic reproduction number for symptomatic mode of transmission. the second equilibrium point of (2.2) satisfies the following system of equations a− νs − φ(s, c)c −ψ(s, i)i = 0, (2.4) φ(s, c)c +ψ(s, i)i − d1c = 0, (2.5) γc − d2i = 0. (2.6) hence, c = a− νs d1 , i = γc d2 = γ(a− νs) d1d2 and d2φ(s, a− νs d1 ) + γψ(s, γ(a− νs) d1d2 ) = d1d2. since c = a− νs d1 ≥ 0, we have s ≤ a ν . then there is no epidemiological equilibrium when s > a ν . the function h can be define on the closed interval [0, a ν ] by h(s) = d2φ(s, a− νs d1 ) + γψ(s, γ(a− νs) d1d2 )− d1d2. we have h(0) = −d1d2 < 0, h( a ν ) = d1d2(r0 − 1) and h′(s) = d2 ( ∂φ ∂s − ν d1 ∂φ ∂c ) + γ ( ∂ψ ∂s − γν d1d2 ∂ψ ∂i ) > 0. therefore, the equation h(s) = 0 has a unique root s∗ ∈ (0, a ν ) when r0 > 1. this implies that our model has a unique endemic equilibrium e∗(s∗, c∗, i∗) under the condition r0 > 1. summarizing all the above cases in the following result. theorem 2.1: let r0 be defined by (2.3). copyright© 2022 assa. adv syst sci appl (2022) 40 khalid hattaf, ahmed a. mohsen (i) if r0 ≤ 1, then system (2.2) has only disease-free equilibrium of the form ef (s0, 0, 0), where s0 = a ν . (ii) if r0 > 1, then system (2.2) has a unique endemic equilibrium e∗(s∗, c∗, i∗), where s∗ ∈ (0, a ν ), c∗ = a− νs∗ d1 and i∗ = γ(a− νs∗) d1d2 . also, the disease-free equilibrium exists. 3. global dynamics in this section, we analyze the stability of the disease-free equilibrium ef and the endemic equilibrium e∗. theorem 3.1: the disease-free equilibrium ef is globally asymptotically stable if r0 ≤ 1 and unstable if r0 > 1. proof we define the following lyapunov functional f(t) = c + ψ(s0, 0) d2 i. then dαf(t) = dαc + ψ(s0, 0) d2 dαi = φ(s, c)c +ψ(s, i)i − d1c + ψ(s0, 0) d2 (γc − d2i). using a = νs0, we get dαf(t) = d1 ( d2φ(s, i) + γψ(s0, 0) d1d2 − 1 ) c + ( ψ(s, i)−ψ(s0, 0) ) i. from the first equation of (2.2), we have dαs ≤ a− νs, which leads to s(t) ≤ s(0)eα(−νtα) + a ν [1− eα(−νtα)] , where eα(z) = ∞∑ k=0 zα γ(αk + 1) denotes the mittag-leffler function of one parameter α. then lim sup t→∞ s(t) ≤ a ν , which involves that all omega limit points verify s(t) ≤ s0. thus, it suffices to consider solutions for which s(t) ≤ s0. according to the explicit expression of r0 given in (2.3) and (h1)-(h2), we obtain dαf(t) ≤ d1 ( r0 − 1 ) c. then dαf(t) ≤ 0 when r0 ≤ 1. furthermore, dαf(t) = 0 if and only if s = s0, c = 0 and i = 0. therefore, {ef} is the largest invariant set in {(s, c, i) | dαf(t) = 0}. it copyright© 2022 assa. adv syst sci appl (2022) dynamics of a generalized fractional epidemic model... 41 follows from the lasalle’s invariance principale [16] that ef is globally asymptotically stable when r0 ≤ 1. on the other hand, the characteristic equation of model (2.2) at ef is given by (ν + λ)(α + ν + λ)θ(λ) = 0, (3.7) where θ(λ) = λ2 + ( d1 + d1 − φ(s0, 0) ) λ+ d1d2(1−r0). we have lim λ→+∞ θ(λ) = +∞ and θ(0) = d1d2(1−r0) < 0 when r0 > 1. then we have the eigenvalue λ0 ∈ (0,+∞) such that θ(λ0) = 0, which implies that the equation (3.7) at ef has at least one positive root if r0 > 1. therefore, ef is saddle (unstable) whenever r0 > 1. in the following, we assume that r0 > 1 and the incidence functions φ and ψ satisfy, for all s, c, i > 0, the following assumption( 1− φ(s, c) φ(s, c∗) )( φ(s, c∗) φ(s, c) − c c∗ ) ≤ 0,( 1− φ(s∗, c∗)ψ(s, i) φ(s, c∗)ψ(s∗, i∗) )( φ(s, c∗)ψ(s∗, i∗) φ(s∗, c∗)ψ(s, i) − i i∗ ) ≤ 0. (h4) to establish the global dynamics of system (2.2) when r0 > 1, we need the following lemma. lemma 3.1: let k be a continuous function and u be a continuously differentiable function. for any constant c and α ∈ (0, 1], the caputo fractional derivative of the function k defined by k(t) = ∫ u(t) c k(x)dx, (3.8) satisfies the following property dαk(t) = k(u(t))dαu(t)− t−αv(0) γ(1− α) − α γ(1− α) ∫ t 0 (t− τ)−α−1v(τ)dτ, (3.9) where v(τ) = k(u(t)) ( u(t)− u(τ) ) + ∫ u(τ) u(t) k(x)dx. in particular, we have dαk(t) = k(u(t))dαu(t) when α = 1. moreover, (i) if k is a increasing function, then dαk(u(t)) ≤ k(u(t))dαu(t). (3.10) (ii) if k is a decreasing function, then dαk(u(t)) ≥ k(u(t))dαu(t). (3.11) proof by using the definition of the caputo fractional derivative, we have dαk(t)− k(u(t))dαu(t) = 1 γ(1− α) ∫ t 0 (t− τ)−αu′(τ) ( k(u(τ))− k(u(t)) ) dτ. copyright© 2022 assa. adv syst sci appl (2022) 42 khalid hattaf, ahmed a. mohsen clearly, v′(τ) = u′(τ) ( k(u(τ))− k(u(t)) ) and v(t) = 0. integrating by parts the last integral, we get dαk(t)− k(u(t))dαu(t) = 1 γ(1− α) [ (t− τ)−αv(τ) ]τ=t τ=0 − α γ(1− α) ∫ t 0 (t− τ)−α−1v(τ)dτ. from l’hopital’s rule, we have lim τ→t (t− τ)−αv(τ) = lim τ→t v(τ) (t− τ)α = lim τ→t u′(τ) ( k(u(τ))− k(u(t)) ) −α(t− τ)α−1 = lim τ→t − 1 α (t− τ)1−αu′(τ) ( k(u(τ))− k(u(t)) ) = 0. hence, dαk(t) = k(u(t))dαu(t)− t−αv(0) γ(1− α) − α γ(1− α) ∫ t 0 (t− τ)−α−1v(τ)dτ. now let us consider the following function ψc,k(τ) = k(c) ( c− τ) + ∫ τ c k(x)dx. it is obvious that ψ′ c,k(τ) = k(τ)− k(c). assume that k is a increasing function. in this case, the function ψc,k(τ) is decreasing on the interval (−∞, c] and increasing on [c,+∞) with ψc,k(c) = 0. then ψc,k(τ) has the global minimum at τ = c. thus, ψc,k(τ) ≥ 0, for all (c, τ) ∈ ir2. since v(τ) = ψu(t),k(u(τ)), we deduce that v(τ) ≥ 0 and then (i). by using the same technique, we can easily prove (ii). by applying of the fundamental theorem of analysis, we obtain dαk(t) = k(u(t))dαu(t), when α = 1. (3.12) this completes the proof. remark 3.1: lemma 3.1 extends the lemma presented in [1] that allows to find lyapunov candidate functions for demonstrating the stability of many fractional order systems, and also the elementary lemma given in [17] that estimates fractional derivatives of volterra-type lyapunov functions in the sense caputo for any monotone function. in fact, we have 1 2 dαu2(t) = dα ∫ u(t) c xdx ≤ u(t)dαu(t), (3.13) and dα [ u(t)− u∗ − u∗ ln u(t) u∗ ] = dαu(t)− u∗dα ∫ u(t) u∗ 1 x dx ≤ ( 1− u∗ u(t) ) dαu(t). (3.14) copyright© 2022 assa. adv syst sci appl (2022) dynamics of a generalized fractional epidemic model... 43 theorem 3.2: assume that r0 > 1 and (h4) holds. then the endemic equilibrium e∗ is globally asymptotically stable. proof we define a lyapunov functional as follows g(t) =s − s∗ − ∫ s s∗ φ(s∗, c∗) φ(x, c∗) dx + c∗χ ( c c∗ ) + ψ(s∗, i∗) d2 i∗χ ( i i∗ ) , (3.15) where χ(x̂) = x̂− 1− ln x̂ for x̂ > 0. since s 7→ 1 φ(x,c∗) is a decreasing function, it follows from (ii) of lemma 3.1 and (3.14) that dαg(t) ≤ ( 1− φ(s∗, c∗) φ(s, c∗) ) dαs(t) + ( 1− c∗ c ) dαc(t) + ψ(s∗, i∗) d2 ( 1− i∗ i ) dαi(t) = ( 1− φ(s∗, c∗) φ(s, c∗) )( a− νs − φ(s, c)c −ψ(s, i)i ) + ( 1− c∗ c )( φ(s, c)c +ψ(s, i)i − d1c ) + ψ(s∗, i∗) d2 ( 1− i∗ i )( γc − d2i ) . since a = νs∗ + φ(s∗, c∗)c∗ +ψ(s∗, i∗)i∗, φ(s∗, c∗)c∗ +ψ(s∗, i∗)i∗ = d1c∗ and γc∗ = d2i∗, we have dαg(t) ≤ νs∗ ( 1− s s∗ )( 1− φ(s∗, c∗) φ(s, c∗) ) +φ(s∗, c∗)c∗ ( 2− φ(s∗, c∗) φ(s, c∗) + φ(s, c)c φ(s, c∗)c∗ − c c∗ ) +ψ(s∗, i∗)i∗ ( 3− φ(s∗, c∗) φ(s, c∗) + φ(s∗, c∗)ψ(s, i)i φ(s, c∗)ψ(s∗, i∗)i∗ − i i∗ − ψ(s, i)ic∗ ψ(s∗, i∗)i∗c − i∗c ic∗ ) . copyright© 2022 assa. adv syst sci appl (2022) 44 khalid hattaf, ahmed a. mohsen hence, dαg(t) ≤ νs∗ ( 1− s s∗ )( 1− φ(s∗, c∗) φ(s, c∗) ) +φ(s∗, c∗)c∗ ( − 1− c c∗ + φ(s, c∗) φ(s, c) + φ(s, c)c φ(s, c∗)c∗ ) +ψ(s∗, i∗)i∗ ( − 1− i i∗ + φ(s, c∗)ψ(s∗, i∗) φ(s∗, c∗)ψ(s, i) + φ(s∗, c∗)ψ(s, i)i φ(s, c∗)ψ(s∗, i∗)i∗ ) +φ(s∗, c∗)c∗ ( 3− φ(s, c∗) φ(s, c) − φ(s∗, c∗) φ(s, c∗) − φ(s, c) φ(s∗, c∗) ) +ψ(s∗, i∗)i∗ ( 4− φ(s∗, c∗) φ(s, c∗) − φ(s, c∗)ψ(s∗, i∗) φ(s∗, c∗)ψ(s, i) − ψ(s, i)ic∗ ψ(s∗, i∗)i∗c − ci∗ c∗i ) . from above inequality, we get that 3− φ(s, c∗) φ(s, c) − φ(s∗, c∗) φ(s, c∗) − φ(s, c) φ(s∗, c∗) ≤ 0, and 4− φ(s∗, c∗) φ(s, c∗) − φ(s, c∗)ψ(s∗, i∗) φ(s∗, c∗)ψ(s, i) − ψ(s, i)ic∗ ψ(s∗, i∗)i∗c − ci∗ c∗i ≤ 0. the hypothesis (h1) leads to( 1− s s∗ )( 1− φ(s∗, c∗) φ(s, c∗) ) ≤ 0. by (h4), we get −1− c c∗ + φ(s, c∗) φ(s, c) + φ(s, c)c φ(s, c∗)c∗ = ( 1− φ(s, c) φ(s, c∗) )( φ(s, c∗) φ(s, c) − c c∗ ) ≤ 0 and −1− i i∗ + φ(s, c∗)ψ(s∗, i∗) φ(s∗, c∗)ψ(s, i) + φ(s∗, c∗)ψ(s, i)i φ(s, c∗)ψ(s∗, i∗)i∗ = ( 1− φ(s∗, c∗)ψ(s, i) φ(s, c∗)ψ(s∗, i∗) )( φ(s, c∗)ψ(s∗, i∗) φ(s∗, c∗)ψ(s, i) − i i∗ ) ≤ 0. therefore, dαg(t) ≤ 0 with equality holds if and only if s = s∗, c = c∗ and i = i∗. from lasalle’s invariance principle, we conclude that e∗ is globally asymptotically stable. 4. applications and numerical simulations in this section, we first apply our above analytical results to the following fractional covid-19 model  dαs(t) = a− νs − β1sc 1 + ϵ1c − β2si 1 + ϵ2i , dαc(t) = β1sc 1 + ϵ1c + β2si 1 + ϵ2i − d1c, dαi(t) = γc − d2i. (4.16) copyright© 2022 assa. adv syst sci appl (2022) dynamics of a generalized fractional epidemic model... 45 where β1 and β2 are the infection rates caused by carrier and infected individuals, respectively. the nonnegative constants ϵ1 and ϵ2 are the saturation rates. model (4.16) is a special case of system (2.2) with φ(s, c) = β1s 1 + ϵ1c and ψ(s, i) = β2s 1 + ϵ2i . it easy that see the basic reproduction number r0 is given by r0 = a(d2β1 + γβ2) νd1d2 . (4.17) evidently, the infected rates φ and ψ satisfy the conditions (h1)− (h3). according to theorems 3.1 and 3.2, we get the following result. corollary 4.1: (i) when r0 ≤ 1, the disease-free equilibrium ef of model (4.16) is globally asymptotically stable. (ii) when r0 > 1, the equilibrium ef gets unstable and the endemic equilibrium e∗ of model (4.16) is globally asymptotically stable. the second application of our results concerns a fractional model proposed in 2020 by das and samanta [2] to study the transmission dynamics of covid-19 in japan. this recent fractional model is a particular case of system (4.16), it suffices to take ϵ1 = ϵ2 = 0 and r1 = r2 = d = 0. in [2], the authors just discussed the local stability of the equilibria and calculated epidemic peak in japan scenario. they proved that the disease-free equilibrium is locally asymptotically stable if r0 < 1 and the endemic equilibrium is locally asymptotically stable under r0 > 1 and others three conditions (see, theorem 2 [2]). by applying corollary 4.1, we deduce that when r0 ≤ 1, the disease-free equilibrium of model [2] is globally asymptotically stable. however, this equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable under only r0 > 1. this improve the asymptotic stability results presented recently in [2]. for numerical simulations, we take the some parameters of model (4.16) accordingly to the world health organization reports as well as adopted by many references as stated in table 4.1 below. parameter definition value a recruitment rate 50 β1 transmission contact rate between s and c 1.2× 10−5 β2 transmission contact rate between s and i varied γ symptoms period 1/7 day−1 ϵ1 the measure of inhibition effect for carrier 0.04 ϵ2 the measure of inhibition effect for infected 0.01 ν natural death rate 0.01 day−1 r1 recovery rate from carrier 1/21 day−1 r2 recovery rate from infected 1/15 day−1 d death due to disease rate 0.1 day−1 table 4.1. parameter values of model (4.16). for β2 = 5.5× 10−5, we have r0 = 0.9532 ≤ 1. then model (4.16) has a diseasefree equilibrium ef (5000, 0, 0, 0). by corollary 4.1 (i), we know that ef is globally asymptotically stable. figure 2 demonstrates this result. copyright© 2022 assa. adv syst sci appl (2022) 46 khalid hattaf, ahmed a. mohsen 0 100 200 300 400 500 3500 4000 4500 5000 days s u s c e p ti b le ( s ) α=0.3 α=0.5 α=0.8 α=1 0 100 200 300 400 500 0 200 400 600 days c a rr ie r (c ) α=0.3 α=0.5 α=0.8 α=1 0 100 200 300 400 500 0 100 200 300 days in fe c te d ( i) α=0.3 α=0.5 α=0.8 α=1 3500 4000 4500 5000 0 500 0 200 400 sc i fig. 4.2. dynamics of the model (4.16) when r0 = 0.9532 ≤ 1. 0 50 100 150 200 3400 3600 3800 4000 days s u s c e p ti b le ( s ) α=0.3 α=0.5 α=0.8 α=1 0 50 100 150 200 0 200 400 600 days c a rr ie r (c ) α=0.3 α=0.5 α=0.8 α=1 0 50 100 150 200 0 100 200 300 days in fe c te d ( i) α=0.3 α=0.5 α=0.8 α=1 3400 3600 3800 4000 0 500 0 200 400 sc i fig. 4.3. dynamics of the model (4.16) when r0 = 1.9489 > 1. for β2 = 1.3× 10−4, we have r0 = 1.9489 > 1. figure 3 shows that the trajectories of model (4.16) converge to e∗(3670.8515, 35.1197, 43.4314) from different values of order fractional derivative α. this confirms the global stability result given by corollary 4.1 (ii). 5. conclusion in this work, we have presented a new fractional-order model of covid-19 that takes into modes of infection spread of covid-19 due to the direct contact between the susceptible individuals, with symptomatic and asymptomatic individuals. the incidence of infection is described by general nonlinear functions. it is proved that the proposed model has two steady states namely disease-free equilibrium and endemic equilibrium. the stability analysis of the model shows that the disease-free equilibrium is globally asymptotically stable if the basic reproduction numberr0 ≤ 1, which biologically means that the covid-19 is cleared. while the endemic equilibrium is globally asymptotically stable when r0 > 1. in this case, the covid-19 persists in the world and the infection copyright© 2022 assa. adv syst sci appl (2022) dynamics of a generalized fractional epidemic model... 47 becomes similar to many endemic viral diseases to date, such as influenza, aids and viral hepatitis. on the other hand, we have established a new generalized lemma to analyze the global stability of the endemic equilibrium. this lemma presents a new property for caputo fractional derivative when α ∈ (0, 1] and extends the results presented in [1, 17] for any monotone function. also, the new generalized lemma can be an important tool to study the global dynamics of many fractional order systems modeling infectious diseases such as covid-19. furthermore, the asymptotic stability results of the recent model of covid-19 in japan [2] are improved. several countries around the world have started vaccination against covid-19 in order to reduce the prevalence of this dangerous disease. we will extend our model to study the impact of vaccination on the propagation of covid-19. also, our model used a fractional derivative with singular kernel. it will be more interesting to extend the model by using a fractional derivative with non-singular kernel as in [6]. data availability the data used to support the findings of this study are available from the corresponding author upon request. the articles used to support the findings of this study are included within the article and are cited at relevant places within the text as references. conflicts of interest the authors declare that they have no conflicts of interest. acknowledgements the authors would like to thank the editor and anonymous reviewers for their constructive comments and suggestions, which have improved the quality of the manuscript. references 1. aguila-camacho, n., duarte-mermoud, m. a., & gallegos, j. a. (2014) a lyapunov functions for fractional order systems. commun. nonlinear. sci. numer. simulat. 19, 2951–2957. 2. das, m., samanta, g. p. (2020) a fractional order covid-19 epidemic transmission model: stability analysis and optimal control. [online]. available https://ssrn.com/abstract=3635938. 3. dehingia, k., mohsen, a.a., alharbi, s.a., alsemiry, r.d., & rezapour, s. (2022) dynamical behavior of a fractional order model for within-host sars-cov-2. mathematics, 10, 2344. 4. dobrovolny, h. m. (2020) modeling the role of asymptomatics in infection spread with application to sars-cov-2, plose one journal, 15(8), 1–14. 5. feng, y, xu, s., rong, z., xu, r., liu, x. et al. (2020) delivery of infection from asymptomatic carriers of covid-19 in a familial cluster. international journal of infectious diseases, 94, 133–138. 6. hattaf, k. (2020) a new generalized definition of fractional derivative with nonsingular kernel. computation, 8(2), 1–9. 7. hattaf, k., & dutta, h. (2020) mathematical modelling and analysis of infectious diseases, studies in systems, decision and control, springer. copyright© 2022 assa. adv syst sci appl (2022) 48 khalid hattaf, ahmed a. mohsen 8. hattaf, k., lashari, a. a., louartassi, y., & yousfi, n. (2013) a delayed sir epidemic model with general incidence rate. electronic journal of qualitative theory of differential equations, 3, 1–9. 9. hattaf, k., & yousfi, n. (2020) dynamics of sars-cov-2 infection model with two modes of transmission and immune response. mathematical biosciences and engineering, 17 (5), 5326–5340. 10. kim, b. n., kim, e., lee, s., & oh, c. (2020) mathematical model of covid-19 transmission dynamics in south korea: the impacts of travel restrictions, social distancing, and early detection. mdpi processes journal, 8. 11. mohsen, a. a., al-husseiny, h. f., hattaf, k., & boulfoul, b. (2021) a mathematical model for the dynamics of covid-19 pandemic involving the infective immigrants. iraqi journals of science 62(1), 295–307. 12. mohsen, a. a., al-husseiny, h. f., zhou, x., & hattaf, k. (2020) global stability of covid-19 model involving the quarantine strategy and media coverage effects. aims public health journal 7(3), 587–605. 13. moroccan ministry of health (mmh), the official coronavirus portal in morocco. [online]. available http://www.covidmaroc.ma/pages/accueilfr.aspx. 14. moussaoui, a., & zerga, e. h. (2020) transmission dynamics of covid-19 in algeria: the impact of physical distancing and face masks. aims public health, 7(4), 816–827. 15. peirlink, m., linka, k., costable, f. s., bhattacharya, j., bendavid, e. et al. (2020) visualizing the invisible: the effect of asymptomatic transmission on the outbreak dynamics of covid-19. comput. methods appl. mech. engrg. j., 372. 16. huo, j., zhao, h., & zhu, l. (2015) the effect of vaccines on backward bifurcation in a fractional order hiv model. nonlinear analysis: real world applications, 26, 289–305. 17. vargas-de-león, c. (2015) volterra-type lyapunov functions for fractional-order epidemic systems. commun. nonlinear. sci. numer. simulat., 24, 75–85. 18. world health organization (who) (2021, march). weekly epidemiological update. [online]. available https://www.who.int/docs/defaultsource/coronaviruse/situation-reports/20210309. 19. worldometer, covid-19 coronavirus pandemic. [online]. available https://www.worldometers.info/coronavirus/country/iraq/. copyright© 2022 assa. adv syst sci appl (2022) introduction model formulation and equilibria global dynamics applications and numerical simulations conclusion adv syst sci appl 2019; 01; 75-97 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/613 measuring the opinion of textile industries of punjab (india) towards the reliability of their connected power systems and its effects on the industrial operations. harjit singh mangat1*, dr harpuneet singh2 1) phd research scholar, department of mechanical engineering, i.k. gujral punjab technical university, kapurthala, punjab, india e-mail: mangat84@gmail.com 2) assistant professor, department of production engineering, guru nanak dev engineering college, ludhiana, punjab, india e-mail: prof.harpuneetsingh@gmail.com received june 27, 2018; revised march 19, 2019; published april 15, 2019 abstract: the purpose of this research was to measure the opinion of textile industries of punjab (india) towards the reliability of their connected power systems of the sole public utility company punjab state power corporation limited considering the period from 2010 to 2014. out of the 400 randomly selected, 148 registered textile industrial units from central, north and south punjab, respectively, were responded successfully by conducting the interviews using a questionnaire schedule and dropping questionnaires with the responders, over a period of two years from april 2015-april 2017 which revealed that majority of the industries from the whole punjab was discontented towards the occurrence of planned outages, unplanned outages and load shedding, work efficiency of staff, condition of infrastructure and tariff structure except the limited number of units from central punjab that have shown agreement towards the different aspects of power system reliability. an internal consistency and intra-rater reliability using cronbach’s alpha and krippendorff’s alpha, respectively, calculated, was found very decent. a kruskal wallis test was applied and mean rank of central punjab was found considerably higher followed by south and north punjab for the majority of the variables. dunn’s bonferroni post hoc exposed that a central punjab is significantly different from both north and south punjab but no statistical difference was found between the north and south punjab. a kruskal wallis test statistic (h) and mann whitney test statistics (u) was used to calculate the overall effect sizes and the effect sizes between the groups for all the dependent variables, respectively. out of the forty variables, thirty-three has shown the statistically significant difference from which twenty-eight, three and five were found in the zone of large, intermediate and small effect size, respectively. keywords: reliability, power systems, questionnaire schedule, cronbach’s alpha, krippendorff’s alpha, kruskal wallis test, bonferroni post hoc test, mann-whitney u test, effect size, punjab (india) 1. introduction electricity is one of the most important inventions that science has given to the humankind. the survival of the modern civilization is almost impossible in the absence of electric power. electricity has been playing a foremost role in the growth and development of the entire world since the day it was commercialized. the economy of the whole world is driven by the electric power as almost every system needs an electricity as an input for its functioning. india is a densely populated developing country with a population of around 1.34 billion (2017), globally ranked second after china [8]. such a big population is considered to be the prominent reason * corresponding author: mangat84@gmail.com mailto:prof.harpuneetsingh@gmail.com 76 h.s. mangat, h. singh copyright ©2019 assa adv. in systems science and appl. (2019) for the exponential increase in the demand for electric power supply. additionally, more and more industries and commercial businesses are being established with a very fast pace which is adding burden to the already existing power systems. an electricity distributor can’t promise for the continuous power supply because the power system can never be 100 per cent reliable. a very bulk complex infrastructure is involved at all the three levels of electricity i.e. generation, transmission and distribution which not only includes just the electrical components but also have mechanical and electronic components too. any failure of these components may occur due to natural calamities, ageing of the components or by any other cause can lead to an interruption of the electric power supply. any disruption in electric power not only causes inconvenience but also have firm noticeable impacts. in the residential sector, it can impact the psychology and can become the major cause of discomfort and boredom which ultimately reduces the performance of individuals in other tasks. in other sectors such as commercial, industrial and agricultural, any interruption in electric power is equivalent to the loss of economy. the occurrence of outages of planned and unplanned nature has been a major concern for the industries since ages. in 2009, zimbabwe lost us$1.8 billion which was 33% of their gdp [11]. another study from accra (ghana) revealed that small and medium enterprises were incapable of producing improved quantities and quality indicating poor sales that resulted in low levels of profitability because of the power outages [4]. in zimbabwe, about 10,000 jobs were gone and about 25% of businesses were closed due to power outages and load shedding, at the end of 2007. over a period of 2010-2013, an average of 24 monthly power outages was recorded which had an unwanted effect on the production efficiency of industries in africa [1]. a customer survey conducted in nepal for the years 2008-10 revealed that electricity supply outage costs experienced by the customers were found significantly higher than current electricity rates [10]. the study presented here is the first ever effort in the region to measure the opinion of textile industries of punjab for the period 2010-2014 towards the reliability aspects of their connected power systems of the sole public utility company, punjab state power corporation limited. all the definitions, terms and concepts that are relevant to the study are thoroughly discussed in the succeeding sections. the facets with supporting statistics pertaining to the utility company, power system reliability and textile industries, respectively, are discussed below: 1.1. description of utility company india lines sixth in terms of power generation. around 65% of the electricity used in india is produced by thermal power, 22% by hydroelectric power , 3% by nuclear power and remaining by 10% from other alternative sources such as solar, wind, biomass etc. [9]. punjab, one of the northernmost states of india has only two public utility companies: punjab state power corporation limited (powercom) also called p.s.p.c.l allocated with the key functions of generation and distribution, and (b) punjab state transmission corporation limited (transco) also called p.s.t.c.l assigned with the main role of transmission. the punjab state power corporation limited serves residential, commercial, industrial and agriculture sector in south zone, west zone, central zone, north zone, and border zone of punjab with a total installed generation capacity of 11054 mw as on 31 march 2015. in punjab, the electricity is primarily produced by thermal and hydropower. at the end of the financial year 2014-15, a total connected load of 31859 mw was evident from the official website of p.s.p.c.l. in terms of percentage share, the connected load of the general, industrial, agricultural and others sectors are 44.8, 26.6, 27.2 and 1.4 per cent, respectively [24]. 1.2. concepts of power system reliability from the power system reliability point of view, the downtime may occur at industrial premises because of the broadly three reasons: (a) planned outages, (b) unplanned outages and (c) load shedding. the ability of a component or system to perform required functions under stated conditions for a stated period of time is called power system reliability. power system measuring the opinion of textile industries of punjab 77 copyright ©2019 assa. adv. in systems science and appl. (2019) reliability has been categorized into system adequacy and operating reliability. “adequacy” is the capability of the electric system to supply the aggregate electric power and energy requirements of the electricity consumers continuously, considering planned and reasonably expected unplanned outages of system components. “operating reliability” is the capability of the electric system to withstand sudden disturbances such as electric short circuits or unanticipated loss of system components [19,22]. the total loss of electric power on, at least, one normally energized conductor to one or more customers connected to the distribution portion of the system is called an “outage”. the time period from the initiation of an interruption until service has been restored to the affected customers is called outage duration. a planned outage arises when electrical lines or equipment have to be temporarily taken out of service for repair or to allow network maintenance and alterations. a planned outage will be scheduled in advance and affected customers will be made alert of the outage before it arises. an unplanned outage is a consequence of the loss of a section of the network due to a fault, which can include: equipment failure, cable faults, damage by third parties, lightning strikes, high winds, pole fires, trees or branch interference, bird or animal interference etc. in these situations, there is no possibility of giving customers an early warning of an approaching outage [6]. load shedding can be required when there is an inequality between electricity demand (customers’ usage) and electricity supply (the ability of the electricity network to generate and transport the required amount of electricity to meet this demand). to relieve the burden on themselves and their consumers, providers may begin to share electricity. the flow of electricity is discontinued in one area for a predetermined amount of time. then, supplies are reconnected in that region and disconnected somewhere else. in several cases, people in the affected regions, especially the businesses that are dependent on power are notified in advance. for firms in the manufacturing sector, this implies that they cannot run their machinery and therefore cannot produce [25]. 1.3. description of textile industries at the national and regional level india positions second after china in the worldwide textile trade with 8 per cent of the total. it also lines second to china in the making of both cotton yarn and fabric and fifth in the creation of synthetic fibres and yarns [3]. it accounts for 14 per cent of total industry output and almost 5 per cent of gross domestic product (gdp) [12]. unlike other foremost textile-producing nations, india’s textile industry is covered typically of small-scale, non-integrated spinning, weaving, finishing, and apparel-making enterprises. this sole industry structure is principally a inheritance of government strategies that have encouraged labour-intensive, small-scale operations and discriminated against larger scale industries. punjab is amongst the major producers of cotton and blended yarn along with mill-made fabrics in india. ludhiana is often referred to as the ‘manchester of india [7]. punjab has arisen as a main center for textile-based industries including yarn, readymade garments and hosiery. the textile business accounts for 19% of the entire industrial production of punjab and pays about 38% of the whole exports from the state. amritsar, ludhiana and panipatthe three chief textile centres in north indiaare strongholds in their corresponding areas of knitting, hosiery and carpets and industrial yarns. the joint problems faced by these industries are like the high tariff, power outages, deficiency of modernization and inconsistent policies of the state governments [18]. as per the district wise reports of ministry of micro, small and medium enterprises, the joint problem confronted by all type of industries was the unreliable power supply and costly tariff rate [16]. comprising all types of industries, there were only 446 medium and large scale industries in punjab against the huge count of 111743 micro and small scale industries in the state [15]. based on the nationwide classification, the firms whose investment in plant and machinery does not increase 25 lakh rupees are called micro-scale enterprises, the firms with investment between 25 lakh and 05 crore rupees in plant 78 h.s. mangat, h. singh copyright ©2019 assa adv. in systems science and appl. (2019) and machinery are termed as small-scale enterprises and the firms having investment in plant and machinery more than 05 crore rupees are known as medium scale enterprises [17]. 2. methodology an extensive literature review was done before the commencement of the study and it has been found that a considerable research has been done on the impact of power outages on residential, commercial, industrial and agriculture sector throughout the world that involved the use of questionnaire containing a considerable number of questions related to cost patterns of industries. most of the previous research studies were conducted at macro level using customer survey approach in which the sample is highly distributed over a large geographical region making the process expensive and also they consider the industrial sector as a whole without focussing on subsectors or categories of industries. this is evident from the studies contributed by big organizations by world bank [2], berkley lab [13], nexant sari/energy [20] considering both developing and developed countries. numerous calculations are needed to assess the cost of different duration of outages in hypothetical scenarios that are very helpful to understand the broad impact of power outages on different sectors but other aspects are not covered in these studies. this fact is evident from the studies [23,27,21]. generally, the industries have very busy schedules and moreover majority of the time they are found not to disclose their confidential cost information which makes the response rate very poor. therefore, it was decided to conduct a micro level regional study to make the state government aware of the situation of electric power by measuring the opinion of the textile consumers towards the different aspects of power system reliability by presenting the results in a simple and effective manner. a very limited number of studies related to the power system reliability and its impact on industrial sector were conducted in india. in order to cover this gap, a decision was made to perform the study at the regional state level by considering only the industrial textile sector of punjab which is considered to be playing the major role in the economy of punjab. every industry depends on an interconnected network of generation, transmission and the distribution to run different processes of an industry. even a high reliable power system sometimes can be hit by the natural disasters which outcomes in unplanned outages that are unavoidable. a duration of unplanned power outage because of the occurrence of the fault in a power system is not generally the same for industrial units but most of the times this length becomes prolonged because of the inability of the industry to bring back the production system to normal operation i.e. the power system may recover immediately say after 5 minutes of unplanned outage but the outage of this unannounced nature results in damage of in-process materials in the industries which requires cleanup adding additional restart costs. depending on the type of industry, the recovery time may be as low as minutes or few hours but for some industries involving critical operations may last up to a considerable number of days. a database of 1756 registered textile industries of punjab was created to form a population list which contains the name of the company, location and their postal addresses. these companies were obtained from the following sources: ministry of corporate affairs and micro, small and medium scale enterprises (msmes). after removing the common industries and the industries falling in the districts with low concentration, a final sampling frame of 1163 registered textile consumers representing a population from central punjab (district-ludhiana), north punjab (districtsamritsar and jalandhar) and south punjab (districtspatiala and mohali) was obtained. a total of 400 industries were randomly selected from a sampling frame of 1163 industries out of which 257 industries were from central punjab (district-ludhiana), 86 from north punjab (districtsamritsar and jalandhar) and 55 from south punjab (districts-patiala and mohali). a questionnaire schedule was designed which contains the simple demographic questions and 7-point likert scale questions. a pilot testing of the questionnaire schedule was performed which involved brainstorming sessions measuring the opinion of textile industries of punjab 79 copyright ©2019 assa. adv. in systems science and appl. (2019) and important discussions with the management and shop floor staff of textile consumers resulted in modifications, changes and deletion of some questions. most of the responses (more than 90%) were achieved by conducting face to face interviews with shop floor and management staff by taking the appointments and few were accomplished through dropping the questionnaire schedule with the responder for a set period of time. sometimes multiple visits were made to the different responders in order get the pending questions of the questionnaire schedule filled in a manner desired. during the survey process, it was observed that large and medium scale industries were marginally very less. moreover, it is evident from the official website of msme as mentioned above. also, because of their busy schedules, they were not interested to take part in the questionnaire filling process. so, it was decided not to consider these industries in the sample. only the completed questionnaire schedules are considered in the study. over a period of 2 years from april 2015 to april 2017, a total of 169 textile industries were responded, however, a response of 148 industries was found complete and valid. out of total 148 successful responders, 84 industries were found from central punjab (district-ludhiana), 40 from north punjab (districtsamritsar and jalandhar) and 24 from south punjab (districts-patiala and mohali). 3. results and analysis a software package spss 24 was used for most of the data analysis however for calculating the krippendorff's alphaan ordinal reliability measure, spss syntax (macro) was used which otherwise was absent in spss menu [5]. further, as there was no provision of calculating effect sizes after kruskal wallis and mann-whitney u in spss, an online effect size calculator was used [14]. the formulas that are used for the calculation of effect size are thoroughly explained in the research article [26]. a tableau desktop 2018.1 software package was used to draw diverged 100 per cent stacked bar charts in order to visualize the data effectively. an internal consistency measure calculated using cronbach's alpha was found very decent for all the variables. moreover, an intra-rater reliability measure krippendorff's alpha for all the three independent groups was also calculated, separately, which was also found very good. the results of reliability are shown in table 3.1. a kruskal wallis test was applied to determine if there are statistically significant differences between three levels of an independent variable on a 7-point likert scale ordinal dependent variables, separately, which are shown in the tables for the different categories of variables. dunn’s bonferroni post hoc was applied to know further the statistically significant difference between the groups. a kruskal wallis test statistic (h) and mann-whitney test statistics (u) was used to calculate the overall effect sizes and the effect sizes between the groups for all the dependent variables, respectively. figure 3.1 shows the overall orientation of textile industries of punjab (india) towards the various aspects of power system reliability. in all the tables, the p-values are shown inside the round brackets together with the test statistics, with significant and insignificant values in the normal and italics bold font, respectively. in all the 100 per cent diverged stacked charts depicted as figures, only the likert scale categories with maximum percentage are shown as labels on either side of the charts. in the tables, η1-2 and η3-1 depict the effect sizes for the group combinations 1-2 and 2-3 respectively. also, le, ie and ie stand for large effect, intermediate effect and small effect sizes, respectively. the legend shown below is showing the colour coding of the seven ordinal categories of the dependent variables used in the 100 per cent diverged stacked charts. 80 h.s. mangat, h. singh copyright ©2019 assa adv. in systems science and appl. (2019) table 3.1. reliability measures of the survey responses opinion survey of textile industries towards power system reliability (2010 to 2014) time frame: april 2015-april 2017 cd = completely disagree (1) md = mostly disagree (2) sd = somewhat disagree (3) n = neutral (4) sa = somewhat agree (5) ma = mostly agree (6) ca = completely agree (7) independent variable (location) districts (punjab) responders (textile industries) reliability (krippendorff's alpha) reliability (cronbach’s alpha) 1. central punjab ludhiana 84 0.63 0.95 confidence level= 0.95 significance level= 0.05 2. north punjab amritsar & jalandhar 40 0.79 3. south punjab patiala & mohali 24 0.79 total 148 0.61 figure 3.1. overall percentage orientation of textile industries towards the different aspects of power system reliability. measuring the opinion of textile industries of punjab 81 copyright ©2019 assa. adv. in systems science and appl. (2019) a statistical analysis summary of the dependent variables related to planned outages is presented in table 3.2. a kruskal-wallis test was conducted on the dependent variables a1a5 and a14 that are shown in the table 3.2 to determine if there were differences in level of agreement scores between groups that differed in their geographical location: the "central punjab" (n = 84), "north punjab" (n = 40) and "south punjab" (n = 24) geographical location level groups. a level of agreement scores was statistically significantly different between the different levels of geographical location group based on the h (degree of freedom) or χ2 (degree of freedom) = test statistics value, p-value. the h or χ2 test statistics values for the variables a2, a3, a4, a5 and a14 are 87.363, 25.906, 37.206, 45.025 and 48.412, respectively. thereafter, effect sizes (ηh 2 ) for all the variables were calculated using the online effect size calculator and were found in the zone of large effect. the effect sizes (ηh 2 ) calculated for the variables a2, a3, a4, a5 and a14 are 0.589, 0.165, 0.243, 0.297 and 0.320, respectively. subsequently, pairwise comparisons were performed using dunn's (1964) procedure. a bonferroni correction for multiple comparisons was made with statistical significance accepted at the p < .0167 level. this post hoc analysis revealed statistically significant differences in the level of agreement scores between central punjab and north punjab (p = 0.000) and central punjab and south punjab (p = 0.000) geographical location groups, but not between the north punjab and south punjab geographical location groups. for all the variables, mean rank was found higher for central punjab followed by south punjab and north punjab except for the variable a14 in which the mean rank for north punjab is higher than south punjab. moreover, most of the effect sizes (ηu 2 ) for the statistical significant groups also came in the zone of large effect except for the group 1-3 (central punjab and south punjab) under the variables a3 and a4 which were found in the zone of intermediate effect. the effect sizes were calculated from the u test statistics of mann whitney u test. the effect sizes considering the group 1-2 (central punjab and north punjab) for the variables a2, a3, a4, a5 and a14 are 0.502, 0.153, 0.229, 0.252 and 0.158, respectively. similarly, the effect sizes considering the group 1-3 (central punjab and south punjab) for the variables a2, a3, a4, a5 and a14 are 0.348, 0.084, 0.114, 0.150 and 0.233, respectively. the values of effect sizes, η2 ≥ 0.0390, are considered to be falling in the zone of the desired effect. the zone with values of effect sizes from 0.060-0.110 is called the zone of intermediate effect and values greater than 0.140 shows the large effect as per [14]. a statistical analysis summary of the dependent variables related to unplanned outages is presented in table 3.3. a kruskal-wallis test was conducted on the dependent variables a6a10 and a15 that are shown in the table 3.3 to determine if there were differences in level of agreement scores between groups that differed in their geographical location: the "central punjab" (n = 84), "north punjab" (n = 40) and "south punjab" (n = 24) geographical location level groups. a level of agreement scores was statistically significantly different between the different levels of geographical location group based on the h (degree of freedom) or χ2 (degree of freedom) = test statistics value, p-value except for the variable a7 in which the score was not found significantly different. the h or χ2 test statistics values for the variables a6, a8, a9, a10 and a15 are 57.627, 86.318, 116.024, 114.728 and 121.615, respectively however for the non-significant variable a7, the value was found 2.947. thereafter, effect sizes (ηh 2 ) for all the variables, were calculated using the online effect size calculator and were found in the zone of large effect except for the variable a7 which were found in the zone of “no effect” effect. the effect sizes (ηh 2 ) calculated for the variables a6, a7, a8, a9, a10 and a15 are 0.384, 0.007, 0.582, 0.786, 0.777 and 0.825, respectively. subsequently, pairwise comparisons were performed using dunn's (1964) procedure. a bonferroni correction for multiple comparisons was made with statistical significance accepted at the p < 0.0167 level. this post hoc analysis revealed statistically significant differences in the level of agreement scores between central punjab and north punjab (p = 0.000) and central punjab and south punjab (p = 0.000) geographical location groups, but not between north punjab and south 82 h.s. mangat, h. singh copyright ©2019 assa adv. in systems science and appl. (2019) table 3.2. analysis summary of dependent variables related to planned outages dependent variables levels of independent variable survey response kruskal wallis dunn’s bonferroni post hoc mann-whitney u post hoc cd md sd n sa ma ca test statistic mean rank group test statistics (p-values) test statistic ηu 2 a2number of planned electric power outages your firm has experienced annually were negligible 1. central punjab 0 1 36 11 7 26 3 87.363 (0.000) 101.95 1-2 66.221 (0.000) 204 (0.00) 0.502 2. north punjab 2 28 10 0 0 0 0 35.73 2-3 -7.338 (1.000) 405 (0.204) 0.017 3. south punjab 0 15 9 0 0 0 0 43.06 3-1 58.884 (0.000) 178.5 (0.00) 0.348 total sample 2 44 55 11 7 26 3 effect size: ηh 2= 0.589 (large effect) η1-2= le & η3-1= le a3average duration of planned power outages your firm has experienced was negligibly small 1. central punjab 0 24 24 10 24 2 0 25.906 (0.000) 89.07 1-2 36.671 (0.000) 864 (0.00) 0.153 2. north punjab 0 24 16 0 0 0 0 52.40 2-3 -7.933 (1.000) 412 (0.274) 0.014 3. south punjab 0 11 13 0 0 0 0 60.33 3-1 28.738 (0.006) 600 (0.002) 0.084 total sample 0 59 53 10 24 2 0 effect size: ηh 2= 0.165 (large effect) η1-2= le & η3-1= ie a4advance notification was given well in advance in an event of planned outage 1. central punjab 0 18 30 2 26 8 0 37.206 (0.000) 92.00 1-2 44.775 (0.000) 684 (0.00) 0.229 2. north punjab 4 21 15 0 0 0 0 47.23 2-3 -11.483 (0.822) 385 (0.144) 0.027 3. south punjab 2 8 14 0 0 0 0 58.71 3-1 33.292 (0.001) 534 (0.00) 0.114 total sample 6 47 59 2 26 8 0 effect size: ηh 2= 0.243 (large effect) η1-2= le & η3-1= ie a5electric power was restored within the stipulated time frame in an event of planned outage 1. central punjab 0 8 38 2 7 26 3 45.025 (0.000) 93.42 1-2 46.942 (0.000) 635 (0.00) 0.252 2. north punjab 2 17 21 0 0 0 0 46.48 2-3 -8.525 (1.000) 404 (0.223) 0.017 3. south punjab 0 8 16 0 0 0 0 55.00 3-1 38.417 (0.000) 464 (0.00) 0.150 total sample 2 33 75 2 7 26 3 effect size: ηh 2= 0.297 (large effect) η1-2= le & η3-1= le a14utility people keep record of every planned outage your firm has experienced 1. central punjab 0 0 0 0 11 68 5 48.412 (0.000) 92.43 1-2 36.335 (0.000) 852 (0.00) 0.158 2. north punjab 0 0 0 0 24 16 0 56.10 2-3 13.704 (0.445) 388 (0.117) 0.025 3. south punjab 0 0 0 0 19 5 0 42.40 3-1 50.039 (0.000) 329.5 (0.00) 0.233 total sample 0 0 0 0 54 89 5 effect size: ηh 2= 0.320 (large effect) η1-2= le & η3-1= le measuring the opinion of textile industries of punjab 83 copyright ©2019 assa. adv. in systems science and appl. (2019) table 3.3. analysis summary of dependent variables related to unplanned outages dependent variables levels of independent variable survey response kruskal wallis h dunn’s bonferroni post hoc mann-whitney u post hoc cd md sd n sa ma ca test statistic mean rank group test statistics (p-values) test statistic ηu 2 a6number of unplanned electric power outages your firm has experienced annually was negligible 1. central punjab 0 35 44 5 0 0 0 57.627 (0.000) 95.35 1-2 49.376 (0.000) 564 (0.00) 0.287 2. north punjab 10 29 1 0 0 0 0 45.98 2-3 -3.087 (1.000) 454.5(0.646) 0.002 3. south punjab 5 18 1 0 0 0 0 49.06 3-1 46.289 (0.000) 372 (0.00) 0.205 total sample 15 82 46 5 0 0 0 effect size: ηh 2= 0.384 (large effect) η1-2= le & η3-1= le a7average duration of unplanned power outages your firm has experienced was negligibly small 1. central punjab 11 47 25 1 0 0 0 2.947 (0.229) there is no statistically significant difference between the groups. 2. north punjab 9 23 8 0 0 0 0 3. south punjab 1 18 5 0 0 0 0 total sample 21 88 38 1 0 0 0 effect size: ηh 2= 0.007 (no effect) a8responsiveness of utility people was rapid in an event of unplanned outage 1. central punjab 0 2 28 20 31 3 0 86.318 (0.000) 102.11 1-2 64.207 (0.000) 227 (0.00) 0.486 2. north punjab 2 27 10 1 0 0 0 37.90 2-3 -0.975 (1.000) 469 (0.857) 0.00 3. south punjab 2 14 8 0 0 0 0 38.88 3-1 63.232 (0.000) 142 (0.00) 0.379 total sample 4 43 46 21 31 3 0 effect size: ηh 2= 0.582 (large effect) η1-2= le & η3-1= le a9utility always communicated with the cause of outage in an event of unplanned outage 1. central punjab 0 0 0 1 30 51 2 116.024 (0.000) 106.48 1-2 75.476 (0.000) 1.5 (0.00) 0.649 2. north punjab 0 18 19 3 0 0 0 31.00 2-3 -4.083 (1.000) 418.5(0.333) 0.011 3. south punjab 0 7 16 1 0 0 0 35.08 3-1 71.393 (0.000) 0.5 (0.00) 0.513 total sample 0 25 35 5 30 51 2 effect size: ηh 2= 0.786 (large effect) η1-2= le & η3-1= le a10restoration time taken by the utility people in an event of unplanned outage was rapid 1. central punjab 0 0 0 0 40 37 7 114.728 (0.000) 106.50 1-2 77.013 (0.000) 0.00(0.00) 0.650 2. north punjab 2 20 17 1 0 0 0 29.49 2-3 -8.033 (1.000) 359.5(0.061) 0.044 3. south punjab 1 6 16 1 0 0 0 37.52 3-1 68.979 (0.000) 0.00(0.00) 0.514 total sample 3 26 33 2 40 37 7 effect size: ηh 2= 0.777 (large effect) η1-2= le & η3-1= le a15utility people keep record of every unplanned outage your firm has experienced 1. central punjab 0 0 0 0 59 25 0 121.615 (0.000) 106.50 1-2 74.700 (0.000) 0.00 (0.00) 0.650 2. north punjab 0 9 31 0 0 0 0 31.80 2-3 -1.867 (1.000) 452 (0.577) 0.002 3. south punjab 0 4 20 0 0 0 0 33.67 3-1 72.833 (0.000) 0.00 (0.00) 0.514 total sample 0 13 51 0 59 25 0 effect size: ηh 2= 0.825 (large effect) η1-2= le & η3-1= le 84 h.s. mangat, h. singh copyright ©2019 assa adv. in systems science and appl. (2019) punjab geographical location groups. for all the variables excluding the non-significant variable a7, mean rank was found higher for central punjab followed by south punjab and north punjab. moreover, the effect sizes (ηu 2 ) for the statistical significant groups 1-2 (central punjab and north punjab) and 1-3 (central punjab and south punjab) also came in the zone of large effect. the effect sizes were calculated from the u test statistics of mann whitney u test. except for the non-significant variable a7, the effect sizes considering the group 1-2 (central punjab and north punjab) for the variables a6, a8, a9, a10 and a15 are 0.287, 0.486, 0.649, 0.650 and 0.650, respectively. similarly, the effect sizes considering the group 1-3 (central punjab and south punjab) for the variables a6, a8, a9, a10 and a15 are 0.205, 0.379, 0.513, 0.514 and 0.514, respectively. a post hoc analysis was not performed for the nonsignificant variable a7. figure 3.2 and figure 3.3 depicts the groupwise percentage orientation of textile industries towards the aspects related to “planned outages” and “unplanned outages”, respectively. fig. 3.2. groupwise percentage orientation of textile industries towards the aspects related to planned outages fig. 3.3 groupwise percentage orientation of textile industries towards the aspects related to unplanned outages a statistical analysis summary of the dependent variables related to load shedding are presented in table 3.4. a kruskal-wallis test was conducted on the dependent variables a11 measuring the opinion of textile industries of punjab 85 copyright ©2019 assa. adv. in systems science and appl. (2019) a13, a16, a27 and a28 that are shown in the table 3.4 to determine if there were differences in level of agreement scores between groups that differed in their geographical location: the "central punjab" (n = 84), "north punjab" (n = 40) and "south punjab" (n = 24) geographical location level groups. a level of agreement scores was statistically significantly different between the different levels of geographical location group based on the h (degree of freedom) or χ2 (degree of freedom) = test statistics value, p-value except for the variable a11 in which the score was not found significantly different. the h or χ2 test statistics values for the variables a12, a13, a16, a27 and a28 are 45.533, 33.570, 46.417, 13.475 and 8.210, respectively however for the non-significant variable a11, the value was found 5.107. thereafter, effect sizes (ηh 2 ) calculated for the variables a11 and a28 were found in the zone of small effect. the effect sizes for the variable a27 were found in the zone of “intermediate effect” and for the variables a12, a13 and a16 the effect sizes were found in the zone of large effect. the effect sizes (ηh 2 ) calculated for the variables a11, a12, a13, a16, a27 and a28 are 0.021, 0.300, 0.218, 0.306, 0.079 and 0.043, respectively. subsequently, pairwise comparisons were performed using dunn's (1964) procedure. a bonferroni correction for multiple comparisons was made with statistical significance accepted at the p < 0.0167 level. this post hoc analysis revealed statistically significant differences in the level of agreement scores between central punjab and north punjab geographical location group for the variables a12, a13, a16, a27 and a28 at p values of 0.000, 0.000, 0.000, 0.001 and 0.004, respectively. further, for the geographical location group of central punjab and south punjab, the level of agreement scores were also found statistically significantly different for the variables a12, a13 and a16 at p values of 0.000, 0.000 and 0.000, respectively, however, for the variables a27 and a28, the scores were not found statistically different at p values of 0.049 and 0.623 which were greater than a significant value of 0.0167. furthermore, no statistical difference based on the level of agreement scores was observed between the north punjab and south punjab geographical location groups. for all the variables excluding the non-significant variable a11, mean rank was found higher for central punjab followed by south punjab and north punjab. moreover, the effect sizes (ηu 2 ) considering the 1-2 (central punjab and north punjab) groups for the variables a12, a13 and a16 were found in the zone of large effect, however, the effect sizes for the variables a27 and a28 were found in the zone of small and intermediate effect respectively. furthermore, the effect sizes (ηu 2 ) considering the 1-3 (central punjab and south punjab) groups for the variables a12 and a16 came in the zone of large effect, however, the effect size of the variable a13 was found in the zone of intermediate effect. the effect sizes considering the group 1-3 for the variables a27 and a28 were found in the zone of small effect and no effect, respectively. the effect sizes were calculated from the u test statistics of mann whitney u test. except for the non-significant variable a11, the effect sizes considering the group 1-2 (central punjab and north punjab) for the variables a12, a13, a16, a27 and a28 are 0.209, 0.182, 0.232, 0.049 and 0.062, respectively. similarly, the effect sizes considering the group 1-3 (central punjab and south punjab) for the variables a12, a13, a16, a27 and a28 are 0.167, 0.080, 0.170, 0.023 and 0.002, respectively. a post hoc analysis was not performed for the insignificant variable a11. a statistical analysis summary of the dependent variables related to utility, staff and infrastructure are presented in table 3.5. a kruskal-wallis test was conducted on the dependent variables a17-a23 and a25 that are shown in the table 3.5 to determine if there were differences in level of agreement scores between groups that differed in their geographical location: the "central punjab" (n = 84), "north punjab" (n = 40) and "south punjab" (n = 24) geographical location level groups. a level of agreement scores was statistically significantly different between the different levels of geographical location group based on the h (degree of freedom) or χ2 (degree of freedom) = test statistics value, p86 h.s. mangat, h. singh copyright ©2019 assa adv. in systems science and appl. (2019) table 3.4. analysis summary of dependent variables related to load shedding dependent variables levels of independent variable survey response kruskal wallis dunn’s bonferroni post hoc mann-whitney u post hoc cd md sd n sa ma ca test statistic mean rank group test statistics (p-values) test statistic ηu 2 a11load shedding (weekly off days) your firm has experienced annually were negligible 1. central punjab 5 62 17 0 0 0 0 5.107 (0.078) there is no statistically significant difference between the groups. 2. north punjab 5 32 3 0 0 0 0 3. south punjab 0 20 4 0 0 0 0 total sample 10 114 24 0 0 0 0 effect size: ηh 2= 0.021 (small effect) a12load shedding (peak load restriction hours/day) your firm has experienced were negligible 1. central punjab 0 23 58 3 0 0 0 45.533 (0.000) 92.70 1-2 41.727 (0.000) 726.5 (0.00) 0.209 2. north punjab 5 27 8 0 0 0 0 50.98 2-3 0.975 (1.000) 467.5 (0.837) 0.00 3. south punjab 4 15 5 0 0 0 0 50.00 3-1 42.702 (0.000) 432.5 (0.00) 0.167 total sample 9 65 71 3 0 0 0 effect size: ηh 2= 0.300 (large effect) η1-2= le & η3-1= le a13advance notification was given well in advance in an event of load shedding 1. central punjab 0 0 0 0 4 38 42 33.570 (0.000) 89.82 1-2 39.359 (0.000) 792 (0.00) 0.182 2. north punjab 0 0 0 0 8 31 1 50.46 2-3 -10.475 (0.839) 406.5 (0.168) 0.016 3. south punjab 0 0 0 0 3 18 3 60.94 3-1 28.884 (0.003) 609 (0.001) 0.08 total sample 0 0 0 0 15 87 46 effect size: ηh 2= 0.218 (large effect) η1-2= le & η3-1= ie a16utility people keep record of load shedding your firm has experienced 1. central punjab 0 0 0 0 15 56 13 46.417 (0.000) 93.36 1-2 44.595 (0.000) 675.5 (0.00) 0.232 2. north punjab 0 0 0 2 27 11 0 48.76 2-3 -2.633 (1.000) 455 (0.668) 0.002 3. south punjab 0 0 0 0 17 7 0 51.40 3-1 41.961 (0.000) 428.5 (0.00) 0.170 total sample 0 0 0 2 59 74 13 effect size: ηh 2= 0.306 (large effect) η1-2= le & η3-1= le a.27. labour was sent to home in an event of load shedding (weekly off days) 1. central punjab 0 0 0 0 2 57 25 13.475 (0.001) 82.50 1-2 20.000 (0.002) 1220 (0.001) 0.049 2. north punjab 0 0 0 0 0 40 0 62.50 2-3 -4.000 (1.000) 460 (0.551) 0.001 3. south punjab 0 0 0 0 2 19 3 66.50 3-1 16.000 (0.073) 796 (0.049) 0.023 total sample 0 0 0 0 4 116 28 effect size: ηh 2=0.079 (intermediate effect) η1-2= se a.28. labour was sent to home in an event of load shedding (peak load) 1. central punjab 0 7 41 0 1 28 7 8.210 (0.016) 81.42 1-2 22.454 (0.013) 1161.5 (0.004) 0.062 2. north punjab 2 17 5 0 3 13 0 58.96 2-3 -17.225 (0.309) 377 (0.130) 0.032 3. south punjab 0 7 3 0 3 11 0 76.19 3-1 5.229 (1.000) 945.5 (0.623) 0.002 total sample 2 31 49 0 7 52 7 effect size: ηh 2= 0.043 (small effect) η1-2= ie measuring the opinion of textile industries of punjab 87 copyright ©2019 assa. adv. in systems science and appl. (2019) table 3.5. analysis summary of dependent variables related to utility, staff and infrastructure dependent variables levels of independent variable survey response kruskal wallis bonferroni post hoc mann-whitney u post hoc cd md sd n sa ma ca test statistic mean rank group test statistics (p-values) test statistic ηu 2 a17utility people were skilled and reliable in their respective job areas 1. central punjab 0 0 0 0 29 46 9 116.927 (0.000) 106.50 1-2 77.375 (0.000) 0.00 (0.00) 0.65 2. north punjab 0 12 27 1 0 0 0 29.12 2-3 -9.000 (1.000) 345 (0.017) 0.055 3. south punjab 0 2 19 3 0 0 0 38.12 3-1 68.375 (0.000) 0.00 (0.00) 0.514 total sample 0 14 46 4 29 46 9 effect size: ηh 2= 0.793 (large effect) η1-2= le & η3-1= le a18utility people strongly followed preventive maintenance practice of electrical equipment 1. central punjab 0 3 37 10 30 4 0 70.676 (0.000) 99.17 1-2 57.748 (0.000) 377.5(0.00) 0.391 2. north punjab 4 21 14 1 0 0 0 41.42 2-3 -1.846 (1.000) 459.5 (0.750) 0.001 3. south punjab 1 14 8 1 0 0 0 43.27 3-1 55.902 (0.000) 238 (0.00) 0.300 total sample 5 38 59 12 30 4 0 effect size: ηh 2= 0.474 (large effect) η1-2= le & η3-1= le a19electric network was protected against lightning & storms 1. central punjab 0 10 33 10 26 5 0 72.327 (0.000) 99.40 1-2 60.180 (0.000) 347 (0.00) 0.409 2. north punjab 5 28 6 1 0 0 0 39.23 2-3 -6.900 (1.000) 402 (0.170) 0.018 3. south punjab 0 18 6 0 0 0 0 46.12 3-1 53.280 (0.000) 249 (0.00) 0.291 total sample 5 56 45 11 26 5 0 effect size: ηh 2= 0.485 (large effect) η1-2= le & η3-1= le a20reinforcement and upgradation of the network was effective to improve reliability 1. central punjab 0 0 0 1 50 31 2 118.257 (0.000) 106.50 1-2 74.838 (0.000) 0.00 (0.00) 0.65 2. north punjab 3 32 5 0 0 0 0 31.66 2-3 -2.233 (1.000) 446.5 (0.530) 0.003 3. south punjab 2 17 5 0 0 0 0 33.90 3-1 72.604 (0.000) 0.00 (0.00) 0.514 total sample 5 49 10 1 50 31 2 effect size: ηh 2= 0.802 (large effect) η1-2= le & η3-1= le a.21inspection and maintenance of the network were effective & in time by utility company 1. central punjab 0 17 31 0 17 19 0 75.138 (0.000) 99.18 1-2 55.960 (0.000) 388 (0.00) 0.385 2. north punjab 0 38 2 0 0 0 0 43.23 2-3 2.996 (1.000) 439 (0.294) 0.005 3. south punjab 3 19 2 0 0 0 0 40.23 3-1 58.955 (0.000) 226.5(0.00) 0.309 total sample 3 74 35 0 17 19 0 effect size: ηh 2= 0.504 (large effect) η1-2= le & η3-1= le a22ageing assets was always regularly monitored & replaced before their failure 1. central punjab 0 14 34 0 10 23 3 44.673 (0.000) 93.82 1-2 46.659 (0.000) 633 (0.00) 0.253 2. north punjab 1 24 15 0 0 0 0 47.16 2-3 -5.275 (1.000) 433.5 (0.453) 0.006 3. south punjab 0 13 11 0 0 0 0 52.44 3-1 41.384 (0.000) 432 (0.00) 0.168 total sample 1 51 60 0 10 23 3 effect size: ηh 2= 0.294 (large effect) η1-2= le & η3-1= le a23utility people adopted safe & standard techniques while performing inspection and maintenance work 1. central punjab 0 17 31 1 13 22 0 42.808 (0.000) 93.45 1-2 44.352 (0.000) 676 (0.00) 0.232 2. north punjab 0 26 14 0 0 0 0 49.10 2-3 -1.400 (1.000) 468 (0.841) 0.00 3. south punjab 0 15 9 0 0 0 0 50.50 3-1 42.952 (0.000) 420 (0.00) 0.175 total sample 0 58 54 1 13 22 0 effect size: ηh 2= 0.281 (large effect) η1-2= le & η3-1= le a25condition of electrical infrastructure from the feeder to your industry premises was sophisticated 1. central punjab 0 3 45 0 15 21 0 58.203 (0.000) 95.93 1-2 47.629 (0.000) 588 (0.00) 0.275 2. north punjab 0 18 22 0 0 0 0 48.30 2-3 5.133 (1.000) 436 (0.481) 0.006 3. south punjab 0 13 11 0 0 0 0 43.17 3-1 52.762 (0.000) 300 (0.00) 0.253 total sample 0 34 78 0 15 21 0 effect size: ηh 2= 0.388 (large effect) η1-2= le & η3-1= le 88 h.s. mangat, h. singh copyright ©2019 assa adv. in systems science and appl. (2019) value. the h or χ2 test statistics values for the variables a17, a18, a19, a20, a21, a22, a23 and a25 are 116.927, 70.676, 72.327, 118.257, 75.138, 44.673, 42.808 and 58.203, respectively. thereafter, effect sizes (ηh 2 ) for all the variables from a17-a23 and a25 were calculated using the online effect size calculator and were found in the zone of large effect. the effect sizes (ηh 2 ) calculated for the variables a17, a18, a19, a20, a21, a22, a23 and a25 are 0.793, 0.474, 0.485, 0.802, 0.504, 0.294, 0.281 and 0.388, respectively. subsequently, pairwise comparisons were performed using dunn's (1964) procedure. a bonferroni correction for multiple comparisons was made with statistical significance accepted at the p < 0.0167 level. this post hoc analysis revealed statistically significant differences in the level of agreement scores between central punjab and north punjab (p = 0.000) and central punjab and south punjab (p = 0.000) geographical location groups, but not between the north punjab and south punjab geographical location groups. for all the variables, mean rank was found higher for central punjab followed by south punjab and north punjab except for the variables a21 and a25 in which the mean ranks for north punjab are higher than south punjab. moreover, the effect sizes (ηu 2 ) considering the statistical significant groups for all the variables a17-a23 and a25 were found in the zone of large effect. the effect sizes were calculated from the u test statistics of mann whitney u test. the effect sizes considering the group 1-2 (central punjab and north punjab) for the variables a17, a18, a19, a20, a21, a22, a23 and a25 are 0.650, 0.391, 0.409, 0.650, 0.385, 0.253, 0.232 and 0.275, respectively. similarly, the effect sizes considering the group 1-3 (central punjab and south punjab) for the similarly, the effect sizes considering the group 1-3 (central punjab and south punjab) for the variables a17, a18, a19, a20, a21, a22, a23 and a25 are 0.514, 0.300, 0.291, 0.514, 0.309, 0.168, 0.175 and 0.253, respectively. figure 4 & figure 5 shows the groupwise percentage orientation of textile industries towards the aspects related to “load shedding” and “utility, staff & infrastructure”, respectively. fig. 3.4. groupwise percentage orientation of textile industries towards the aspects related to load shedding measuring the opinion of textile industries of punjab 89 copyright ©2019 assa. adv. in systems science and appl. (2019) fig. 3.5. groupwise percentage orientation of textile industries towards the aspects related to utility, staff and infrastructure a statistical analysis summary of the dependent variables related to generic nature is presented in table 3.6. a kruskal-wallis test was conducted on the dependent variables a1, a24, a26, a29, a30, a33 and a40 that are shown in the table 3.6 to determine if there were differences in level of agreement scores between groups that differed in their geographical location: the "central punjab" (n = 84), "north punjab" (n = 40) and "south punjab" (n = 24) geographical location level groups. a level of agreement scores was statistically significantly different between the different levels of geographical location group based on the h (degree of freedom) or χ2 (degree of freedom) = test statistics value, p-value except for the variable a40 in which the score was not found significantly different. the h or χ2 test statistics values for the variables a1, a24, a26, a29, a30 and a33 are 70.495, 116.727, 63.948, 11.364, 6.303 and 16.484, respectively however for the non-significant variable a40, the value was found 1.259. thereafter, effect sizes (ηh2 ) calculated for the variables a29 and a33 were found in the zone of intermediate effect. the effect sizes for the variables a30 and a40 were found in the zone of “small effect” and “no effect”, respectively and for the variables a1, a24 and a26 the effect sizes were found in the zone of large effect. the effect sizes (ηh2 ) calculated for the variables a1, a24, a26, a29, a30, a33 and a40 are 0.472, 0.791, 0.427, 0.065, 0.030, 0.100 and 0.005, respectively. subsequently, pairwise comparisons were performed using dunn's (1964) procedure. a bonferroni correction for multiple comparisons was made with statistical significance accepted at the p < 0.0167 level. this post hoc analysis revealed statistically significant differences in the level of agreement scores between central punjab and north punjab geographical location groups for the variables a1, a24, a26, a29 and a33 at p values of 0.000, 0.000, 0.000, 0.008 and 0.011, respectively, however, for the variable a30, the scores were not found statistically different at p-value of 0.018 which was greater than a significant value of 0.0167. further, for the geographical location group of central punjab and south punjab, the level of agreement scores were also found statistically significantly different for 90 h.s. mangat, h. singh copyright ©2019 assa adv. in systems science and appl. (2019) the variables a1, a24, a26, a29 and a33 at p values of 0.000, 0.000, 0.000, 0.004 and 0.000, respectively, however, for the variable a30, the scores were not found statistically different at p-value of 0.897 which was greater than a significant value of 0.0167. furthermore, no statistical difference based on the level of agreement scores was observed between the north punjab and south punjab geographical location groups. for the variables a1, a26, a29 and a30, mean rank was found higher for central punjab followed by south punjab and north punjab, however, for the variables a24 and a33, mean rank of north punjab was found to be greater than south punjab. moreover, the effect sizes (ηu2 ) considering the 1-2 (central punjab and north punjab) groups for the variables a1, a24 and a26 were found in the zone of large effect, however, the effect sizes for the variables a29, a30 and a33 were found in the zone of small effect. furthermore, the effect sizes (ηu2 ) considering the 1-3 (central punjab and south punjab) groups for the variables a1, a24 and a26 were found in the zone of large effect, however, the effect sizes for the variables a29, a30 and a33 were found in the zone of small effect, no effect and intermediate effect, respectively. the effect sizes were calculated from the u test statistics of mann whitney u test. except for the non-significant variable a40, the effect sizes considering the group 1-2 (central punjab and north punjab) for the variables a1, a24, a26, a29, a30 and a33 are 0.412, 0.650, 0.376, 0.034, 0.034 and 0.034, respectively. similarly, the effect sizes considering the group 1-3 (central punjab and south punjab) for the variables a1, a24, a26, a29, a30 and a33 are 0.258, 0.514, 0.225, 0.047, 0.00 and 0.077, respectively. a post hoc analysis was not performed for the non-significant variable a40. a statistical analysis summary of the dependent variables related to outage after affects are presented in table 3.7. a kruskal-wallis test was conducted on the dependent variables a31, a32 and a34-a39 that are shown in the table 3.7 to determine if there were differences in level of agreement scores between groups that differed in their geographical location: the "central punjab" (n = 84), "north punjab" (n = 40) and "south punjab" (n = 24) geographical location level groups. a level of agreement scores was statistically significantly different between the different levels of geographical location group based on the h (degree of freedom) or χ2 (degree of freedom) = test statistics value, p-value except for the variables a31, a32, a34 and a35 in which the score was not found significantly different. the h or χ2 test statistics values for the variables a36, a37, a38 and a39 are 29.711, 31.037, 31.124 and 34.960, respectively however for the non-significant variables a31, a32, a34 and a35, the values were found 1.359, 0.980, 5.479 and 0.285, respectively. thereafter, effect sizes (ηh 2 ) calculated for the variables a31 and a32 were found in the zone of no effect, the effect sizes for the variables a34 and a35 were found in the zone of “small effect” and “no effect”, respectively and for the variables a36-a39, the effect sizes were found in the zone of large effect. the effect sizes (ηh 2 ) calculated for the variables a31, a32, a34, a35, a36, a37, a38 and a49 are 0.004, 0.007, 0.024, 0.012, 0.191, 0.200, 0.201 and 0.227, respectively. subsequently, pairwise comparisons were performed using dunn's (1964) procedure. a bonferroni correction for multiple comparisons was made with statistical significance accepted at the p < 0.0167 level. this post hoc analysis revealed statistically significant differences in the level of agreement scores between central punjab and north punjab geographical location groups for the variables a36, a37, a38 and a39 at p values of 0.000. further, for the geographical location group of central punjab and south punjab, the level of agreement scores were also found statistically significantly different for the variables a36, a37, a38 and a39 at p values of 0.002, 0.000, 0.000 and 0.001, respectively. furthermore, no statistical difference based on the level of agreement scores was observed between the north punjab and south punjab geographical location groups. for the variables a36, a37, a38 and a39, mean rank was found higher for north punjab followed by south punjab and central punjab. moreover, the effect sizes (ηu 2 ) considering the 1-2 (central punjab and north punjab) groups for the variables a36, a37 and a39 were found in the zone of large effect, however, measuring the opinion of textile industries of punjab 91 copyright ©2019 assa. adv. in systems science and appl. (2019) table 3.6. analysis summary of dependent variables related to generic nature dependent variables levels of independent variable survey response kruskal wallis bonferroni post hoc mann-whitney u post hoc cd md sd n sa ma ca test statistic mean rank group test statistics (p-values) test statistic ηu 2 a1supply of electric power from the utility has been reliable 1. central punjab 0 4 34 9 10 27 0 70.495 (0.000) 98.94 1-2 60.140 (0.000) 342 (0.00) 0.412 2. north punjab 2 24 14 0 0 0 0 38.80 2-3 -9.658 (1.000) 390 (0.156) 0.024 3. south punjab 1 10 13 0 0 0 0 48.46 3-1 50.482 (0.000) 293 (0.00) 0.258 total sample 3 38 61 9 10 27 0 effect size: ηh 2= 0.472 (large effect) η1-2= le & η3-1= le a24overall behaviour of utility people towards your firm in an event of outages was courteous 1. central punjab 0 0 0 1 27 52 4 116.727 (0.000) 106.49 1-2 73.494 (0.000) 0.5 (0.00) 0.65 2. north punjab 0 14 25 1 0 0 0 33.00 2-3 1.312 (1.000) 460.5 (0.748) 0.001 3. south punjab 0 9 15 0 0 0 0 31.69 3-1 74.807 (0.000) 0.00 (0.00) 0.514 total sample 0 23 40 2 27 52 4 effect size: ηh 2= 0.791 (large effect) η1-2= le & η3-1= le a26tariff structure provided by the utility company was economical 1. central punjab 0 13 35 0 9 25 2 63.948 (0.000) 97.65 1-2 57.386 (0.000) 402 (0.00) 0.376 2. north punjab 4 29 7 0 0 0 0 40.26 2-3 -10.279 (0.982) 388.5 (0.119) 0.025 3. south punjab 1 15 8 0 0 0 0 50.54 3-1 47.107 (0.000) 341.5 (0.00) 0.225 total sample 5 57 50 0 9 25 2 effect size: ηh 2= 0.427 (large effect) η1-2= le & η3-1= le a29workers were found losing their interest in an event of sustained power outages 1. central punjab 0 0 0 0 1 66 17 11.364 (0.000) 66.27 1-2 -16.970 (0.032) 1294.5(.008) 0.034 2. north punjab 0 0 0 0 0 23 17 83.24 2-3 -5.513 (1.000) 444 (0.563) 0.004 3. south punjab 0 0 0 0 0 12 12 88.75 3-1 -22.482 (0.015) 702 (0.004) 0.047 total sample 0 0 0 0 1 101 46 effect size: ηh 2= 0.065 (intermediate effect) η1-2= se & η3-1= se a30utility company gave compensation if it took more time to restore power than stipulated time 1. central punjab 34 48 2 0 0 0 0 6.303 (0.043) 78.90 1-2 17.030 (0.052) 1295 (0.018) 0.034 2. north punjab 25 15 0 0 0 0 0 61.88 2-3 -18.250 (0.174) 360 (0.054) 0.043 3. south punjab 9 15 0 0 0 0 0 80.12 3-1 -1.220 (1.000) 993 (0.897) 0.00 total sample 68 78 2 0 0 0 0 effect size: ηh 2= 0.03 (small effect) η1-2= le a33your business has been heavily dependent on stable and secure power supply 1. central punjab 0 0 0 0 0 50 34 16.484 (0.000) 83.75 1-2 16.887 (0.028) 1294 (0.011) 0.034 2. north punjab 0 0 0 0 0 33 7 66.86 2-3 12.008 (0.505) 399.5 (0.064) 0.019 3. south punjab 0 0 0 0 1 22 1 54.85 3-1 28.896 (0.001) 617 (0.00) 0.077 total sample 0 0 0 0 1 105 42 effect size: ηh 2= 0.1 (intermediate effect) η1-2= se & η3-1= ie a40alternative power supplies are expensive comparing with that of electricity 1. central punjab 0 0 0 0 23 38 23 1.259 (0.533) there is no statistical significant difference between the groups. 2. north punjab 0 0 0 0 1 35 4 3. south punjab 0 0 0 0 1 18 5 total sample 0 0 0 0 25 91 32 effect size: ηh 2= 0.005 (no effect) 92 h.s. mangat, h. singh copyright ©2019 assa adv. in systems science and appl. (2019) table 3.7. analysis summary of dependant variables related to outage after affects dependent variables levels of independent variable survey response kruskal wallis bonferroni post hoc mann-whitney u post hoc cd md sd n sa ma ca test statistic mean rank group test statistics (p-values) test statistic ηu 2 a31 electric power outages have led to a decrease in your profit margin 1. central punjab 0 0 0 0 2 70 12 1.359 (0.507) there is no statistically significant difference between the groups. 2. north punjab 0 0 0 0 1 34 5 3. south punjab 0 0 0 0 3 13 8 total sample 0 0 0 0 6 117 25 effect size: ηh 2= 0.004 (no effect) a32electric power outages have led to closed down part of your branch operations 1. central punjab 0 0 0 0 8 66 10 0.980 (0.613) there is no statistically significant difference between the groups. 2. north punjab 0 0 0 0 6 29 5 3. south punjab 0 0 0 0 4 18 2 total sample 0 0 0 0 18 113 17 effect size: ηh 2= 0.007 (no effect) a34 electric power outages have led to increasing in the cost of preserving your products 1. central punjab 4 51 29 0 0 0 0 5.479 (0.065) there is no statistically significant difference between the groups. 2. north punjab 4 27 9 0 0 0 0 3. south punjab 2 19 3 0 0 0 0 total sample 10 97 41 0 0 0 0 effect size: ηh 2= 0.024 (small effect) a35electric power outages have led to increase in maintenance & replacement cost 1. central punjab 0 0 0 0 41 43 0 0.285 (0.867) there is no statistically significant difference between the groups. 2. north punjab 0 0 0 0 21 19 0 3. south punjab 0 0 0 0 13 11 0 total sample 0 0 0 0 75 73 0 effect size: ηh 2= 0.012 (small effect) a36electric power outages have led to decrease in sales 1. central punjab 0 0 0 0 23 52 9 29.711 (0.000) 60.45 1-2 -37.173 (0.000) 851 (0.00) 0.158 2. north punjab 0 0 0 0 0 22 18 97.62 2-3 12.500 (0.572) 384 (0.112) 0.028 3. south punjab 0 0 0 0 0 18 6 85.12 3-1 -24.673 (0.012) 657 (0.002) 0.062 total sample 0 0 0 0 23 92 33 effect size: ηh 2= 0.191 (large effect) η1-2= le & η3-1= ie a37 electric power outages have led to disposed of some of your assets 1. central punjab 0 0 0 1 58 25 0 31.037 (0.000) 59.63 1-2 -36.882 (0.000) 848.5 (0.00) 0.159 2. north punjab 0 0 0 0 9 29 2 96.51 2-3 6.658 (1.000) 431 (0.386) 0.007 3. south punjab 0 0 0 0 7 17 0 89.85 3-1 -30.223 (0.001) 590.5 (0.00) 0.088 total sample 0 0 0 1 74 71 2 effect size: ηh 2= 0.2 (large effect) η1-2= le & η3-1= le a38electric power outages have led to reduction in your product lines 1. central punjab 0 0 0 0 21 62 1 31.124 (0.000) 61.63 1-2 -30.157 (0.000) 996.5 (0.00) 0.108 2. north punjab 0 0 0 0 0 31 9 91.79 2-3 1.058 (1.000) 472 (0.877) 0.00 3. south punjab 0 0 0 0 0 19 5 90.73 3-1 -29.098 (0.000) 610.5 (0.00) 0.08 total sample 0 0 0 0 21 112 15 effect size: ηh 2= 0.201 (large effect) η1-2= ie & η3-1= ie a39electric power outages have led to decrease in purchasing of your products by existing customers 1. central punjab 0 0 0 0 0 48 36 34.960 (0.000) 59.14 1-2 -39.582 (0.000) 768 (0.00) 0.192 2. north punjab 0 0 0 0 2 36 2 98.72 2-3 10.850 (0.766) 423 (0.260) 0.01 3. south punjab 0 0 0 0 6 15 3 87.88 3-1 -28.732 (0.002) 630 (0.001) 0.072 total sample 0 0 0 0 56 87 5 effect size: ηh 2= 0.227 (large effect) η1-2= le & η3-1= ie measuring the opinion of textile industries of punjab 93 copyright ©2019 assa. adv. in systems science and appl. (2019) the effect sizes for the variable a38 was found in the zone of intermediate effect. furthermore, the effect sizes (ηu 2 ) considering the 1-3 (central punjab and south punjab) groups for all the significant variables a36-a39 were found in the zone of intermediate effect. the effect sizes were calculated from the u test statistics of mann whitney u test. except for the nonsignificant variables a31, a32, a34 and a35, the effect sizes considering the group 1-2 (central punjab and north punjab) for the variables a36, a37, a38 and a39 are 0.158, 0.159, 0.108 and 0.192, respectively. similarly, the effect sizes considering the group 1-3 (central punjab and south punjab) for the variables a36, a37, a38 and a39 are 0.062, 0.088, 0.080 and 0.072, respectively. a post hoc analysis was not performed for the non-significant variables a31, a32, a34 and a35. figure 3.6 and figure 3.7 depicts the groupwise percentage orientation of textile industries towards the aspects related to “generic nature” and “outage after affects”, respectively. fig. 3.6. groupwise percentage orientation of textile industries towards the aspects related to generic nature. fig. 3.6. groupwise percentage orientation of textile industries towards the aspects related to outage after affects 94 h.s. mangat, h. singh copyright ©2019 assa adv. in systems science and appl. (2019) 4. conclusion and discussion the results from the analysis reveal that the majority of the textile industries, especially from north and south punjab, were shown a high level of disagreement towards the reliability of their connected power systems of the public utility companypunjab state power corporation limited (p.s.p.c.l) for the period 2010 to 2014. it has been observed that the textile industries falling in the region of central punjab are somewhat agreed on various aspects of power system reliability but overall this effect is not dominant and opinions are partially oriented on either side. the outcomes of the conducted study for the different categories of dependent variables are highlighted below: planned outagesonly 24.3% of the sampled textile industries, all of them belong to the subgroup central punjab, were found agreed towards the occurrence of planned outages annually and rest of the industries from north and south punjab have shown the very high level of disagreement. similarly, merely 17.6% of the total industries, again all of them from central punjab, thought that average duration of the planned outages they experienced was negligible, against the rest who stated very high level of disagreement. almost 23% of the total industries were agreed with the advance notification practice of the utility company and 77% of the total from north and south punjab believed that they were not notified well in advance before the occurrence of the planned outages. following the same trend, more than 76% sampled industries from north and south punjab exposed that the utility company failed to restore the electric power in a promised time, however, rest of the 24% from central punjab were found satisfied with the restoration time. though the whole of the sampled industries from punjab was found agreed towards the record keeping habit of the utility company however high mean ranks and effect sizes revealed that central punjab was more fulfilled than the north and south punjab. above discussion revealed the condition of network and infrastructure in the state. unplanned outagesthough the whole of the sampled industries was found disagree towards the occurrence of unplanned outages but mean ranks and effect sizes revealed that central punjab opted the lower order category of agreement making its mean rank higher than north and south punjab. further, all the industries were found dissatisfied towards the average duration of unplanned outages they experienced. moreover, 23% of the industries, all of them from central punjab, believed that response of the utility was rapid in an event of an unplanned outage. more than 56% of the sampled industries, again all of them from central punjab was confident that the utility responds faster in an event of unplanned outages against the rest of the industries from north and south punjab. following the same trend, the group central punjab was also found high on agreement than other groups on the restoration time and record keeping practice as far as the unplanned outages are concerned. above discussion exposed the protection of power systems and infrastructure against the natural disasters. load sheddingas far as the load shedding (weekly off days) is concerned whole of the sampled industries were found oriented towards the disagreement which is indicating the inability of industries to use electric power because of the demand surpassing supply. based on the load shedding (peak load) experienced by the industries, all of them have shown orientation towards the disagreement, however, a group central punjab selected high order of agreement category making its mean rank higher than the other groups. further, all the sampled industries were found agreed on the advance notification and record keeping practice in an event of load shedding, however, central punjab have shown higher mean rank than the north and south punjab. when it was asked from the industries that labour should be sent to a home in an event of a load shedding (weekly off days), all of them responded towards the high level of agreement and only one combination i.e. central punjab and north punjab has shown the statistically significant difference. similarly, when thought was taken on that the labour should be sent to a home in an event of a load shedding (peak load), again only one group combination i.e. central punjab and north punjab have shown the statistically significant difference with central punjab showing the higher mean rank. measuring the opinion of textile industries of punjab 95 copyright ©2019 assa. adv. in systems science and appl. (2019) utility, staff and infrastructureabout 56% of the total sampled industries, all of them from central punjab revealed that utility people were skilled and reliable in their respective job areas, and reinforcement and upgradation of the network were effective to improve the reliability against the thinking of industries from north and south punjab. moreover, more than 20% of the total industries, again from central punjab believed that utility people strongly followed preventive maintenance practice of the electrical equipment and electric network was protected against lightning & storms as compared to north and south punjab. around 24% industries of the total, whole of them from central punjab thought that inspection and maintenance of the network were effective and in time by utility company, ageing assets was always regularly monitored and replaced before their failure, utility people adopted safe and standard techniques while performing inspection and maintenance work, condition of electrical infrastructure from the feeder to your industry premises was sophisticated, respectively, against the north and south punjab. this discussion has revealed that the overall the maintenance and upgradation practices of the networks by the utility are not effective in north and south punjab, however, central punjab have shown considerable agreement. genericfrom the whole punjab, just 25% of the sampled industries from central punjab have shown agreement towards the reliability of supply of electric power from the utility against north and south punjab which have shown the very high level of disagreement. further, almost whole of the industries from central punjab was agreed that the overall behaviour of the utility people towards their firms in an event of outages was courteous against rest of the industries from north and south punjab. more than 24% of the total industries, all of them from central punjab were found agreed on that the tariff structure provided by the utility company was economical against the north and south punjab which have shown the very high level of disagreement. further, the whole of the industries from punjab believed that workers lose their interest in an event of sustained power outages. though the statistically significant difference is evident from the analysis, the effect sizes are falling in the zones of intermediate and small effect showing the smaller overall difference and difference between the groups. furthermore, it has also been observed that almost whole of the industries was shown a high level of disagreement that utility company gave compensation if it took more time to restore power than stipulated time. also, all the sampled industries believed that businesses are heavily dependent on the stable and secure power supply. it has been observed from the results of a survey that almost whole of the sample have the same thinking that the cost of choosing alternative power supply is quite high. outage after affectswhen it was asked from the industries about the impact of power outages on their different industrial operations, all the industries throughout punjab was found almost having the same opinion of high negative affect. no statistically significant difference was found and almost whole of the sample believed that electric power outages led to a decrease in their profit margin, closing down part of their branch operations and increase in maintenance & replacement cost. it has also been observed that sampled industries from north and south punjab have the strong belief as compared to central punjab that the power outages led to decrease in their sales, disposed of some of their assets, reduction in the product lines, decrease in the purchasing of their products by the existing customers. overall, the conducted survey at the regional state level revealed the standing of the utility company. this kind of surveys considering the specific industrial sector at a local state level can be very helpful to draw the attention of the government towards the reliability of utility company without the use of any calculations. the results of the survey demand large scale modifications in order to improve the power systems of the utility company. the outcomes revealed that the textile industries had faced major power system reliability issues from 2010 to 2014 which not only affected their operations but also the goodwill among their customers. a high level of disagreement towards the power system reliability stresses the direct intervention of the state government in order to avoid the negative impact of power outages. 96 h.s. mangat, h. singh copyright ©2019 assa adv. in systems science and appl. (2019) acknowledgments author is highly thankful to i.k gujral, punjab technical university, kapurthala (punjab), india, for giving him the opputunity to conduct this research work. author expresses his sincere gratefulness to research guide dr harpuneet singh (assistant professor, department of production engineering), guru nanak dev engineering college, ludhiana (punjab), india, for his ongoing supervision, valuable suggestions, persistent inspiration and tireless efforts throughout the length of this research work. also, the author expresses his thanks to micro and small-scale industries in punjab for giving their precious time and required information to support the research to be completed in a effective way. references [1] abotsi, a. k. (2016). power outages and production efficiency of firms in africa. international journal of energy economics and policy, 6(1), 98-104. [2] arlet, j. 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(2009). an assessment of impact of zessco's power rationing on firm productivity and profitability: a case of selected luaska based manufacturing firms. http://dspace.cbu.ac.zm:8080/jspui/handle/123456789/214. retrieved june 14, 2018, from www.dspace.cbu.ac.zm: http://dspace.cbu.ac.zm:8080/jspui/handle/123456789/214 https://www.psychometrica.de/effect_size.html http://www.msmedildh.gov.in/state%20profile%20of%20punjab%2015-16.pdf http://www.msmedildh.gov.in/ http://www.msmedildh.gov.in/report_msme.html http://www.msme.gov.in/ https://msme.gov.in/faqs/q1-what-definition-msme http://www.indiantextilejournal.com/news.aspx?news=textile-centres-in-north-strength-vs-weakness&nid=0+hc9irqxmtiqa4mq3/szg http://www.indiantextilejournal.com/news.aspx?news=textile-centres-in-north-strength-vs-weakness&nid=0+hc9irqxmtiqa4mq3/szg https://www.nerc.com/pa/rapa/ra/reliability%20assessments%20dl/2009_ltra_v1_1_errata.pdf https://www.nerc.com/pa/rapa/ra/reliability%20assessments%20dl/2009_ltra_v1_1_errata.pdf https://www.nerc.com/docs/pc/definition-of-alr-approved-at-dec-07-oc-pc-mtgs.pdf https://www.nerc.com/docs/pc/definition-of-alr-approved-at-dec-07-oc-pc-mtgs.pdf http://www.pspcl.in/ http://www.pspcl.in/about-us/statistics/ http://dspace.cbu.ac.zm:8080/jspui/handle/123456789/214 http://www.dspace.cbu.ac.zm/ http://dspace.cbu.ac.zm:8080/jspui/handle/123456789/214 98 h.s. mangat, h. singh copyright ©2019 assa adv. in systems science and appl. (2019) [26] tomczak, m., & tomczak, e. (2014). the need to report effect size estimates revisited-an overview of some recommended measures of effect size. trends in sports science, 1(21), 19-25. [27] wijayatunga, p. d., & jayalath, m. (2004, january). assessment of economic impact of electricity supply interruptions in the sri lanka industrial sector. energy conservation and management, 45(2), 235-247. adv syst sci appl 2019; 01; 1-11 published online at https://ijassa.ipu.ru/index.php/ijassa/article/view/656 a simple improved algorithm to find all the lower boundary points in a multiple-node-pair multistate flow network majid forghani-elahabad1*, nelson kagan2 1) centro de matemática, computação e cognição – cmcc, universidade federal do abc – ufabc, santo andré, sp, brazil e-mail: majid@usp.br 2) department of electrical engineering, university of sao paulo, sao paulo, brazil e-mail: nelsonk@pea.usp.br received october 17, 2018; revised february 20, 2019; published april 10, 2019 abstract: there are several real-world systems which can be modeled as multistate flow networks (mfns). many researchers have worked on reliability evaluation of an mfn, and accordingly variety of approaches have been proposed in this regard. the problem of finding all the lower boundary points (lbps) in a multistate flow network (mfn) has been attracting significant attention in the recent decades as these vectors can be used for reliability evaluation of mfns. once all the lbps are determined, the reliability of the network can be calculated by using some techniques such as the inclusion–exclusion principle or the sum of disjoint products (sdp). several algorithms have been proposed to address this problem in the literature in terms of minimal paths or minimal cuts. as in the real systems the data (flow or commodity) are usually transmitting from several sources to a number of destinations, the problem of determining all the lbps in a multi-node-pair mfn is considered in this work. an improved algorithm is proposed to address the problem. the algorithm is illustrated through a benchmark network example. the complexity results are computed. the efficiency of the algorithm is demonstrated through a numerical example and the complexity results. keywords: lower boundary points; multi-node-pair multistate flow network; minimal paths; dmp problem; system reliability; sum of disjoint product. 1. introduction due to ever-increasing importance of transport networks such as communication or telecommunication networks, there is certainly need for designing and constructing the most possible reliable systems. a transport network can be generally modeled as a multistate flow network (mfn, also called stochastic-flow network) whose arcs (and possibly nodes) have randomly variable capacities due to failures, the need for maintenance, operations activities, etc. [1-22]. traditionally, several indices have been proposed and employed to measure the reliability of an mfn in the literature. the most attractive reliability index is the twoterminal reliability (ttr) or the source-destination reliability index [2-4, 14]. for a given demand value of d, the ttr in an mfn is the probability of transmitting at least d units of flow (data or commodity) from a source node to a destination node [5-9]. however, the realistic networks such as communication or telecommunication networks need to transmit concurrently several demands of signals (flow or data) from various source nodes to a number of destination nodes, and consequently the problem of reliability evaluation of multinode-pair mfn (mnp-mfn) has been emerged [5, 8, 14, 16]. in an mnp-mfn, there is a * corresponding author: email address: majid@usp.br, phone: +55(11)3091-5317, fax: +55(11)3091-5404. mailto:nelsonk@pea.usp.br 2 m. forghani-elahabad, n. kagan copyright ©2019 assa adv. in systems science and appl. (2019) set of node pairs with related demand values which should be satisfied. therefore, the reliability of such a system is defined as the probability of transmitting all the required demands from the sources to the corresponding destinations concurrently [5, 8, 14, 16]. although many algorithms have been proposed in the literature to address the problems, evaluating the system reliability is an np-hard problem [3], and thus the problem continues to be interesting to investigate. to evaluate the reliability of an mfn with single source and destination nodes, many algorithms have been proposed in terms of minimal cuts (upper boundary points – ubps or d-mcs) [5-8, 12, 15, 21] or minimal paths (lower boundary points – lbps or d-mps) [9-11, 13, 14, 16]. once all the ubps or lbps are determined, the reliability can be calculated by using some techniques such as the inclusion–exclusion principle [6] or the sum of disjoint products (sdp) [2]. in [13], the notion of d-mp candidate was introduced, and it was demonstrated that the minimal elements of all the d-mp candidates are the set of all the lbps. afterwards, several algorithms have been proposed in the literature to address the problem [9-11, 13, 14]. the algorithms generally consist of two main stages; finding all the d-mp candidates, and then checking each candidate for being a d-mp. the focus of this work is on determining all the d-mps (or lbps) in an mnp-mfn. in general, to compute the system reliability of an mnp-mfn, one can find all the system state vectors under which all the given demands can be satisfied simultaneously, and then calculate the probability of such vectors according to the given probability distributions of the arcs’ capacities. to search for such vectors, it is usually assumed that all the required minimal paths or minimal cuts are at hand in advance [5-11, 13, 14], and after determining these vectors, the system reliability can be computed using some techniques such as the inclusion-exclusion principle [6] or the sum of disjoint products [2]. lin and yuan [14] considered an mnp-mfn and proposed an algorithm to evaluate the system reliability in terms of minimal paths. the algorithm first finds all the flow patterns which satisfy the demands, then transforms these patterns to system-state vectors (ssvs) which are d-mp candidates, and finally checks each candidate for being a minimal one to find all the lbps. it is reminded that as there is a set d of the demand values in the multi-pair-node case, the ssvs are called d-mp rather than d-mp. the most time consuming part of the proposed algorithm in [14] is the checking process as it needs to compare all the candidates to each other. in [16], lin and yeng applied this algorithm to a real-world network, namely taiwan advanced research and education network, for evaluating the reliability of a computer network with multiple sources. chakraborty and goyal [5] proposed a three-step algorithm in terms of minimal cuts which uses sdp technique to evaluate the reliability of an mnp-mfn. the notable point on the algorithm is that it considers all the arcs with two states of failure or fully working, whereas usually in the real-world networks it is possible to have multistate arcs. here, considering the problem of determining all the lbps in an mnp-mfn, some new results are presented. based on the presented results, a simple improved algorithm is proposed to address the problem. the algorithm is illustrated through a benchmark network. the complexity results are computed, and accordingly the proposed algorithm is shown to be more efficient than the existing ones in the literature. the reminder of this work is organized as follows. section 2 states the required notations and explains some preliminaries on computing the system reliability. in section 3, an algorithm is proposed and illustrated through a benchmark example. then, the complexity results are computed, and the algorithm is compared with an existing one in the literature. section 4 concludes the work. 2. preliminaries here, first the required notations, nomenclature, and assumptions are stated, and then some preliminaries on computing the system reliability are described. lower boundary points in a multistate flow network 3 copyright ©2019 assa. adv. in systems science and appl. (2019) 2.1. notations, nomenclature, and assumptions here, the required notations and assumptions are stated. table 2.1. the required notations. g(n, a, m, q) an mnp-mfn with n = {1, 2, ..., n} being the set of nodes (n is the number of nodes), a= {ai | 1 ≤ i ≤ m} being the set of arcs (m is the number of arcs), m = (m1, m2, ..., mm) being a vector, where mi denotes the maximum capacity of arc ai, for i = 1, 2, ..., m, and q = {(i,j) | i is a source node with corresponding destination node j} being the set of all the node pairs. xi xi denotes the current capacity of the arc ai, for i = 1, 2, …, m. x x= (x1, x2, ..., xm) is a system-state vector representing the current capacities of all the arcs. di,j is a given demand of data (flow) to be transmitted from source node i to destination node j for every node pair (i, j) q. v, w are the number of sources and destinations in the network, respectively. oi ,1 w i i jj o d   is the total flow outgoing from source node i, for i = 1, …, v. uj ,1 v j i jj u d   is the total flow incoming to destination node j, for j = 1, …, w. s, t are the added artificial source and destination nodes to the network. bi is the added artificial arc from node s to the source node i with the fixed capacity of oi, for i = 1, …, v. cj is the added artificial arc from the destination node j to node t with the constant capacity of uj, for j = 1, …, w. g is the new network after adding artificial nodes and arcs into g. v(x) is the maximum flow on the network g from node s to node t. z(x) = {ai a | xi > 0} is the set of nonzero-capacity arcs, under x. ei = 0(ai) is a system-state vector in which the capacity level is 1 for ai and 0 for the other arcs. hi,j is the set of all the minimal paths from source node i to destination node j, for every (i,j)  q. h is the set of all the minimal paths in the network g, i.e., , ( , ) i j i j q h h   . h is the number of all the minimal paths in g, i.e., h = |h|. pj is the jth minimal path in g, for j = 1, …, h. cpj = min {mi | ai pj} is the capacity of pj, for j = 1, 2, …, h. d = {di,j | (i, j) q} is the set of all the demands in the network. σ is the number of all the d-mp candidates. a system state x = (x1, x2, ..., xm) is less than or equal to system state y = (y1, y2, ..., ym), written as x ≤ y, when xi ≤ yi, for i = 1, 2, ..., m, and also x < y, when x ≤ y with at least one j, j = 1, 2, ..., m, such that xj < yj. a vector x  is said to be a minimal vector when there is no vector y  such that y < x. a path is a sequence of adjacent arcs from a 4 m. forghani-elahabad, n. kagan copyright ©2019 assa adv. in systems science and appl. (2019) source node to a destination node, and a minimal path (mp) is a path with no cycle. the following assumptions are considered in this work: 1. each node is perfectly reliable. 2. each added artificial arc in g is perfectly reliable. 3. flow in g and g satisfies the flow conservation law [1]. 4. the capacity of each arc ai in a is modelled as a non-negative integer-valued random number less than or equal to mi. 5. the arcs’ capacities are statistically independent. 2.2. reliability in a multi-node-pair multistate flow network (mnp-mfn), for each node pair (i, j) in q, there is a demand dij of flow (data or commodity) to transmit from source node i to destination node j. hence, the system reliability is the probability of all the demands being satisfied, that is for each node pair (i, j) in q, at least dij units of flow (data) can be transmitted from node i to node j, simultaneously. to compute such a probability, one can first find the set of all the system-state vectors (ssvs) on which all the demands can be satisfied simultaneously, and then compute the probability of those ssvs. thus, considering  = {x | x is an ssv on which all the demands can be satisfied concurrently}, we see that rd = pr{x | x  }. now, assuming min = {x | x is a minimal vector in  } = {x1, x2, …, xδ} and ai = {x | x ≥ xi}, for i = 1, 2, …, δ, it is seen that 1 i i a     , and consequently the system reliability can be computed using methods such as inclusion-exclusion technique or sum of disjoint products (sdp) [2, 5]. for example, in sdp technique, it is set b1 = a1 and 1 1 rr r i i b a a     for each r = 2, …, δ, and then 1 pr( )i d i r b    (2.1) where pr( ) pr( ) i i x b b x    , and 1 pr( ) pr( )jj m x x    . hence, to evaluate the reliability of an mnp-mfn, it is enough to find all the set min which is the set of all the lower boundary points (lbps) in the network. the problem of finding all the lbps is called d-mp problem in the literature [8, 13, 14, 16]. 3. main block to find all the lbps in an mnp-mfn, we first define the notion of flow pattern at level d (called d-fp) as follows. it is reminded that d is the set of all the demand values. definition 3.1: a vector f = (f1, f2, …, fh) is named a flow pattern at level d (d-fp) when it satisfies the followings: , , : : ( ) , for all ( , ) , ( ) , 1,2,..., . i j l i l l i j l p h l i l a p i f d i j q ii f m i m             (3.1) it is noted that in a d-fp, the component fl shows the amount of flow on minimal path (mp), pl, and this is why it is called a flow pattern. in the system (3.1), the first equation guarantees that the summation of flows on all the mps from node i to node j is di,j, and so the given lower boundary points in a multistate flow network 5 copyright ©2019 assa. adv. in systems science and appl. (2019) demand can be satisfied. the second inequality guarantees that the summation of flows on each arc is less than the maximum capacity of the arc. for example, considering q = {(1, 2), (1, 3), (4, 3)} as the set of node pairs in fig. 1, the set of mps are h1,2 = {{a1}, {a5, a4}}, h1,3 = {{a1, a2}, {a5, a6}, {a1, a3, a6}, {a5, a4, a2}}, h4,3 = {{a6}, {a4, a2}}, and hence h = {{a1}, {a5, a4}, {a1, a2}, {a5, a6}, {a1, a3, a6}, {a5, a4, a2}, {a6}, {a4, a2}}. now, considering d1,2 = 1, d1,3 = 3, and d4,3 = 2 as the given demand values, and m = (3, 3, 2, 3, 2, 2) as the max-capacity vector, the set of all the d-fps is obtained as follows. (0,1,3,0,0,0,2,0), (1,0,2,0,0,1,2,0), (0,1,2,0,0,1,2,0), (1,0,1,0,0,2,2,0), (0,1,2,0,1,0,1,1), (1,0,2,1,0,0,1,1), (0,1,2,1,0,0,1,1), (1,0,1,0,1,1,1,1), (0,1,1,0,1,1,1,1), (1,0,1,1,0,1,1,1), (1,0,0,0,1,2,1,1), (0,1,1,0,2,0,0,2), (1,0,1,1,1,0,0,2), (0,1,1,1,1,0,0,2), (1,0,1,2,0,0,0,2), (1,0,0,0,2,1,0,2), (1,0,0,1,1,1,0,2). fig. 1. a benchmark example it is reminded that each d-fp shows the amount of flow on all the mps, and hence for each d-fp, say f = (f1, f2, …, fh), one can find the corresponding system-state vector (ssv), x = (x1, x2, …, xm) using the following equation. : , 1,2,..., . i l i l l a p x f i m    (3.2) the ssv, x which is calculated through eq. (3.2) above is called a d-mp candidate. a notable point is that as the number of mps is usually far greater than the number of arcs in a network, it is possible to obtain duplicate d-mp candidates from different d-fps, and so it is required to remove the duplicate candidates. for example, transforming the above-mentioned seventeen d-fp vectors in fig. 1, only five different ssvs x1 = (3, 3, 0, 1, 1, 2), x2 = (2, 3, 0, 2, 2, 2), x3 = (3, 3, 1, 2, 1, 2), x4 = (2, 3, 1, 3, 2, 2), x5 = (3, 3, 2, 3, 1, 2) are obtained and the other twelve ssvs are duplicates. the following result proven in [13, 14] shows that to find all the lbps, it is enough to first find all the d-mp candidates and then check each candidate for being an lbp. lemma 3.1: each lbp is a d-mp candidate. proposition 3.1: there is a possibility of generating duplicate d-mp candidates from different d-fps through eq. (3.2). in fact, it can be considered as a common drawback in the presented works in the literature to not notice for generating duplicate candidates from different flow pattern vectors [8]. the first trivial method is to compare d-mp candidates for detecting and removing the duplicates. however, as they are m-tuple vectors, the time complexity of comparison process is relatively high. hence, the proposed data structure in [7] is used here instead of comparing the vectors. next, it is briefly explained how the duplicate candidates can be removed using the proposed data structure in [7]. let m* = max {m1, m2, …, mm}, x = (x1, …, xm) be a d6 m. forghani-elahabad, n. kagan copyright ©2019 assa adv. in systems science and appl. (2019) mp candidate, k be the number of digits in m*, and ki be the number of digits in xi, for i = 1, 2, …, m. since xi ≤ mi ≤ m*, for i = 1, …, m, it is clear that ki ≤ k. now, to construct a k-digit number ni for component xi, if k = ki, then let ni = xi, and if ki < k, then let times 00...0 l i l k k n x   . thereby, the ml-digit number 1 2...x mn n n n can be constructed associated with x. corollary 3.1: the time complexity of removing the duplicate d-mp candidates by using the method above is o(σlogσ), where σ is the number of all the candidates. it should be noted that under each d-mp candidate x obtained by eq. (3.2), when the flows are transmitting simultaneously from all the sources to all the destinations, the maximum flow from each source node i to the corresponding destination node j equals the sum of flows on all the mps between them, that is exactly , ,: i j l l i jl p h f d   . hence, the following result is at hand. proposition 3.2: under d-mp candidate x, the maximum flow between each node pair (i,j) in q is exactly di,j when the flows are transmitting concurrently from all the sources to all the destinations. to find all the lbps from all the candidates, the authors in [13, 14, 16] compared all the candidates to detect the minimal ones. however, the number of candidates grows exponentially with the size of network and as they are m-tuple vectors, it is really time consuming to compare all the candidates. here, an improved technique is proposed to check each candidate to be an lbp. in fact, two artificial nodes and a few artificial arcs are added into the network to make use of max-flow algorithm for checking the d-mp candidates. for a given mnp-mfn, say g, first two artificial nodes s (a source) and t (a destination) are added to g. then, from node s to each source node i, an artificial arc bi with the fixed capacity of ,1 w i i jj o d   is added, for i = 1, 2, …, v. moreover, from each destination node j to node t, an artificial arc cj with the fixed capacity of ,1 v j i ji u d   is added, for j = 1, 2, …, w. the new network with artificial nodes and arcs is named g . for example, considering q = {(1, 2), (1,3), (4,3)} as the set of node pairs, and d = {1, 3, 2} as the set of demand values in the given network in fig. 1, the new network g with artificial nodes and arcs is constructed as given in fig. 2. the capacities of b1, b2, c1, and c2 are respectively o1 = 4, o2 = 2, u1 = 1, and u2 = 5. fig. 2. the benchmark with added artificial nodes and arcs. as the added artificial arcs are perfectly reliable with a constant capacity, they are not considered in the system state vectors (ssvs) of the network, and an ssv in g is still exhibited as x= (x1, x2, ..., xm) where m is the number of arcs in g. it is reminded that the lower boundary points in a multistate flow network 7 copyright ©2019 assa. adv. in systems science and appl. (2019) capacity of an artificial arc from node s to source node i is always assumed to be oi, for each i = 1, …, v, and the capacity of an artificial arc from destination node j to node t is always assumed to be uj, for j = 1, …, w. this way, it is possible to check all the node pairs’ demands by sending 1 1 v w i ji j d o u      units of flow from node s to node t. corollary 3.2: 1 1 v w i ji j o u     . proof. according to definition, we have ,1 w i i jj o d   , and consequently it is concluded that , ,1 1 1 1 1 1 v v w w v w i i j i j ji i j j i j o d d u               . □ lemma 3.2: if x is a d-mp candidate, then v(x) = d in g , namely the maximum flow from node s to node t equals d, where 1 1 v w i ji j d o u      . proof. first, the d units of flow from node s are transmitted through the artificial arcs bis to the source nodes (oi units on arc bi, for i = 1, 2, …, v), and then according to proposition 2, these d unites of flow are transmitted from the source nodes to the destination nodes so that uj units of flow is received at destination node j, for j = 1, 2, …, w. finally, according to considered capacities for the artificial arcs, the flows are transmitted to node t through the arcs cjs. this shows that v(x) = d which completes the proof. □ theorem 3.1: a d-mp candidate x is an lbp if and only if for each ai in z(x), we have v(x ei) < d in g . proof. according to lemma 1, each lbp is a d-mp candidate, and thus the first side has been already demonstrated. now, assume that x is a candidate which is not an lbp (or equivalently a minimal vector in  ). thus, there is a vector y in  such that x > y. without loss of generality, assume that xl > yl, for an l. since y is a vector in  , all the demands are satisfied under y, and so we have v(y) ≥ d in g . thus, we have v(x el) ≥ v(y) ≥ d, and this contradicts the definition of x being a candidate. hence, the proof is complete. □ considering σ as the number of all the d-mp candidates, since each candidate is an m-tuple vector, the time complexity of comparing them is o(mσ2). whereas considering the time complexity of max-flow algorithm as o(mn) [21], and that in the worst case the condition v(x ei) < d should be checked for all the m arcs, the time complexity of checking each candidate is o(m(m+v+w)(n+2)). therefore, the time complexity of checking all the candidates is o(mnσ(m+v+w)) ≈ o(m2nσ) as the number of all the sources and sinks are usually less than m. it is clearly seen that the number of all the d-mp candidates, σ, is really far greater than mn, and thus our technique proposed here is highly more efficient than the proposed ones in [14, 16]. lemma 3.3: the time complexity of using theorem 2 to check each d-mp candidate for being an lbp is o(m2nσ). now, we are at the position of stating our proposed algorithm for determining all the lbps in an mnp-mfn. 8 m. forghani-elahabad, n. kagan copyright ©2019 assa adv. in systems science and appl. (2019) 3.1. the proposed algorithm algorithm 1. (find all the lbps in an mnp-mfn) input: an mnp-mfn, g(n, a, m, q) along with the set of all the demand values, d. output: the set of all the lbps. step 1. let   (set of all the lbps) and   (set of all the associated numbers). then, compute ,1 w i i jj o d   , for i = 1, …, v, ,1 v j i ji u d   , for j = 1, …, w, and 1 v ii d o   . afterwards, add the artificial nodes s and t and the related artificial arcs with fixed capacities into g, as explained in preceding section for constructing g . step 2. find a d-fp, say f = (f1, f2, …, fh) by solving the system (3.1). if there is no solution, skip to step 7. step 3. considering d-fp, f, compute the ssv, xf = (x1, x2, ..., xm) through eq. (3.2). step 4. compute the associated number nx with xf. if it is duplicate number, return to step 2 to find the next d-fp, else add nx to  and determine z(xf) = {ai in a | xi > 0}. step 5. if z(xf) =  then xf is a lbp, add it to  and return to step 2 to find the next d-fp. step 6. select an arc in z(xf), say ai, and remove it from z(xf). if v(xf – ei) < d in g , then return to step 5, else x is not a lbp, return to step 2 to find the next d-fp. step 7. set  is the set of all the lbps. stop. the following example is provided to illustrate algorithm 1. example 3.1: there are two smart meters connected to two control centers in a smart grid. each smart meter may need to transmit some data on the network to one or both control centers. assume that the network is the one given in fig. 1 in which nodes 1 and 4 are the smart meters and nodes 2 and 3 are the control centers. as it is seen in fig. 1, there are some arcs (links) between the smart meters and control centers to transmit data on. suppose that the first smart meter (node 1) needs to send one unit of data to the first control center (node 2) and three units of data to the second control center (node 3) per hour. moreover, the second smart meter (node 4) needs to send only two units of data to the second control center (node 3) per hour. if the maximum capacities of the arcs in the network are given by m = (3, 3, 2, 3, 2, 2), the manager needs to know the reliability of the network to satisfy the demands. the probability distributions for the arcs’ capacities are given in table 2. solution: to compute the reliability of the system, algorithm 1 is employed as follows. step 1. let   ,   . then, we have o1 = 1+3 = 4, o2 = 2, u1 = 1, u2 = 2 + 3 =5, and d =6. afterwards, g is obtained as given in fig. 2. step 2. the d-fp, f = (0,1,3,0,0,0,2,0) is obtained by solving the system (3.1). step 3. the corresponding ssv, x = (3, 3, 0, 1, 1, 2) is obtained through eq. (3.2). step 4. the associated number with x is nx = 330112 which is not duplicate, and so we have  = {330112} and z(x) = {a1, a2, a4, a5, a6}. step 5. z(xf)  . step 6. arc a1 is selected and let z(x) = {a2, a4, a5, a6}. since v(2, 3, 0, 1, 1, 2) = 5<6, the transfer is made to step 5. step 5. z(x) = {}, and hence x is an lbp, letting  = {(3, 3, 0, 1, 1, 2)}, and transfer is made to step 2. lower boundary points in a multistate flow network 9 copyright ©2019 assa. adv. in systems science and appl. (2019) finally,  = {(3, 3, 0, 1, 1, 2), (2, 3, 0, 2, 2, 2)} is obtained. it is noted that among the seventeen d-fps, only five unduplicated d-mp candidates are obtained containing only two lbps. it is also noted that all the d-fps and d-mp candidates for this example have already stated in the previous sections. now, to compute the system reliability by using the sdp technique, let a1 = {x | x ≥ x1 = (3, 3, 0, 1, 1, 2)}, a2 = {x | x ≥ x2 = (2, 3, 0, 2, 2, 2)}, b1 = a1, and b2 = a2 – a1. thereby, the system reliability is equal to 1 2 1 2 3 4 5 6 1 2 3 4 5 6 9 7 pr( ) pr( ) pr( 3) pr( 3) pr( 0) pr( 1) pr( 1) pr( 2) pr( 2) pr( 3) pr( 0) pr( 2) pr( 2) pr( 2) 8 75 1 95 95 85 10 1 75 1 9 9 85 10 0.460275 0.0516375 0.5119125. dr b b x x x x x x x x x x x x                                              table 1. probability distributions for arcs’ capacities of fig. 1. arc capacity and probability 0 1 2 3 a1 0.05 0.05 0.10 0.80 a2 0.05 0.10 0.10 0.75 a3 0.05 0.05 0.90 0 a4 0.05 0.05 0.05 0.85 a5 0.05 0.05 0.90 0 a6 0.10 0.05 0.85 0 3.2. complexity results the time complexity of step 1 is o(v + w). assuming the number of all the d-fps as σ, the complexity of solving the system (3.1) in step 2 to find all the d-fps is of order of o(hσ) [7]. step 3 is of order of o(mh) for constructing each d-mp candidate, and thus is of order of o(mhσ) for computing all the candidates. according to corollary 1, the time complexity of step 4 is o(σlogσ). steps 5 and 7 are of order of o(1). according to lemma 3, the time complexity of step 7 is o(m2nσ). as a result, the time complexity of algorithm 1 is o(mhσ + σlogσ + m2nσ). by the way, in large networks, h grows exponentially with increasing m and n, and thus h m, n, and consequently the time complexity of algorithm 1 is o(mhσ) for large networks. theorem 3.2: the time complexity of algorithm 1 for large network is o(mhσ). 3.3. comparison results here, the proposed algorithm in [14] is compared with algorithm 1 proposed here. as the proposed algorithm in [14] needs to solve the system (3.1) to find all the d-fps, and also compares all the d-mp candidates to remove the duplicates and detect the minimal ones, the time complexity of this algorithm is at least o(mhσ + mσ2) = o(mσ2), where the first part is for solving the system (3.1) and the second part is for the comparison process. it should be kept in mind that each d-mp candidate is an m-tuple vector. even in small network such as 10 m. forghani-elahabad, n. kagan copyright ©2019 assa adv. in systems science and appl. (2019) the given benchmark in fig. 1 the number of d-fps, i.e., σ, is really greater than the number of mps, i.e., h. hence, the time complexity of the proposed algorithm in [14] is highly greater than the one for algorithm 1, the proposed algorithm here. it should be noted that in example 1, the proposed algorithm in [14] needs to compare seventeen 6-tuple vectors, that is 102 comparison job, to remove the duplicates and find the five d-mps, while algorithm 1 needs only to compare seventeen numbers, that is seventeen comparison job, to do the same work. it should be noted that as the number of d-fps is increased exponentially with increasing the size of the network, the practical efficiency of our proposed algorithm will be notable in large networks in comparison with the one proposed in [14]. 5. conclusion reliability evaluation of multistate flow networks (mfns) has been a very attractive problem in the recent decades among the researchers from several research areas. majority of the proposed algorithms have considered a single source-destination mfn whereas the realworld systems generally contain more than one node pair, and can be modeled as multiplenode-pair mfns (mnp-mfns). the reliability of an mnp-mfn can be computed in terms of minimal paths by determining all the lower boundary points (lbps) or in terms of minimal cuts by computing all the upper boundary points. here, for evaluating the reliability of an mnp-mfn, the problem of determining all the lbps in such a network was considered. some results were presented to improve the solution. based on the results, a simple improved algorithm was proposed to address the problem. the algorithm was illustrated through a benchmark example. the complexity results were computed. the algorithm was shown to be more efficient than an existing one based on the complexity results and the illustrative example. acknowledgements the authors thanks fapesp (grant 15/21924-1) for supporting this work. references 1. ahuja, r.k., magnanti, t.l. & orlin j.b. 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(2008). evaluation of the reliability of transport networks based on the stochastic flow of moving objects. reliability engineering & system safety 93, 838–44. https://doi.org/10.1080/24725854.2018.1504358 adv syst sci appl 2022; 04:1–10 published online at https://ijassa.ipu.ru. dynamics of evolutionary equations with 1+2 independent variables alexei g. kushner1,3*, elena n. kushner2, tao sinian1 1lomonosov moscow state university, moscow, russia 2moscow state technical university of civil aviation, moscow, russia 3moscow pedagogical state university, moscow, russia abstract: the paper is devoted to a method for constructing exact solutions of evolutionary differential equations with two space variables.the method uses the theory of symmetries of completely integrable distributions. as an example, we consider a linear parabolic equation that arises in filtration theory, thermodynamics, and mathematical biology. dynamics are calculated for this equation and classes of their exact solutions are constructed. keywords: integrability, completely integrable distributions, symmetry, evolutionary equations, filtration, thermodynamics 1. introduction consider the following partial differential equation: ∂u ∂t = f ( x, u, ∂|σ|u ∂xσ ) , (1.1) where σ = (σ1, . . . , σn) is a multi-index whose elements are non-negative integers, |σ| = σ1 + · · ·+ σn, x = (x1, . . . , xn), ∂|σ|u ∂xσ = ∂|σ|u ∂xσ1 1 , . . . ∂xσn n . here the variables x1, . . . , xn are called spatial, and the variable t is called temporal. for equations (1.1) with one spatial variable, the method of finite-dimensional dynamics was proposed in [1, 2]. this method was further developed in [3, 4]. it allows us to select finite-dimensional submanifolds of solutions from the infinite set of all solutions of evolutionary equations. these submanifolds are “numbered” by solutions of ordinary differential equations. however, this method does not allow direct generalization to equations with several spatial variables. in this case, using of ordinary differential equations are not enough anymore. in [5], a method was proposed in which systems of partial differential equations of finite type are used instead of ordinary differential equations. ∗corresponding author: kushner@physics.msu.ru 2 a. kushner, e. kushner, s. tao 2. symmetries of completely integrable distributions here we give the necessary information about a symmetry of distributions (see [6, 7]). let m be a smooth manifold and p a completely integrable distribution on m . a vector field x on m is called an infinitesimal symmetry of a distribution p if the local group of translations φt along trajectories of x preserves this distribution, i.e. (φt)∗ (p) = p . in what follows, for brevity, an infinitesimal symmetry will be called symmetry. the set of all symmetries of the distribution p forms the lie r-algebra symp with respect to the lie bracket. this lie algebra contains the ideal of characteristic symmetries charp that consists of vector fields lying in the distribution p . the quotient lie algebra shufp = symp/charp is called the lie algebra of shuffling symmetries of the distribution p . elements of this lie algebra “shuffle” the maximal integral manifolds of the distribution. 3. symmetries of finite type differential equations a system of differential equations is called a system of finite type if the space of its solutions is finite-dimensional [6]. examples of systems of finite type are systems of ordinary differential equations. a more important example is given by overdetermined systems of differential equations. for simplicity, we consider the case n = 2 and finite type equations of the second order. consider the following overdetermined system of three differential equations ∂2v ∂x2 = p ( x, y, v, ∂v ∂x , ∂v ∂y ) , ∂2v ∂x∂y = q ( x, y, v, ∂v ∂x , ∂v ∂y ) , ∂2v ∂y2 = r ( x, y, v, ∂v ∂x , ∂v ∂y ) . (3.2) in the space of 1-jets j1 = j1(r2) with canonical coordinates x1 = x, x2 = y, v, p1, p2, this system defines the two-dimensional distribution p : a ∋ j1 7→ p(a) = 2⋂ i=0 kerωi,a ⊂ taj 1, (3.3) where differential 1-forms ω0 = dv − p1dx1 − p2dx2, ω2 = dp1 − p (x1, x2, v, p1, p2)dx1 −q(x1, x2, v, p1, p2)dx2, ω2 = dp2 −q(x1, x2, v, p1, p2)dx1 −r(x1, x2, v, p1, p2)dx2. this distribution is completely integrable if the conditions of the frobenius theorem are satisfied: dωi ∧ ω0 ∧ ω1 ∧ ω2 = 0 (i = 0, 1, 2). (3.4) copyright © 2022 assa. adv syst sci appl (2022) dynamics of parabolic equations 3 if these conditions are satisfied, then system (3.2) is a system of finite type. in what follows, we assume that these conditions are satisfied. the distribution p can be given by two vector fields d1 = ∂ ∂x1 + p1 ∂ ∂v + p ∂ ∂p1 +q ∂ ∂p2 , d2 = ∂ ∂x2 + p2 ∂ ∂v +q ∂ ∂p1 +r ∂ ∂p2 . let a vector field s be a shuffling symmetry of the distribution p . it means that ls(ωi) ∧ ω0 ∧ ω1 ∧ ω2 = 0 (i = 0, 1, 2). here l is the symbol of lie’s derivative. theorem 3.1: each shuffling symmetry of distribution (3.3) is uniquely determined by a function φ on the space j1 and has the form sφ = φ ∂ ∂v +d1(φ) ∂ ∂p1 +d2(φ) ∂ ∂p2 . (3.5) the function φ is called generating function of the vector field sφ. theorem 3.2: the generating function φ satisfies the following system of differential equations: d2 1(φ)− sφ(p ) = 0, d2 2(φ)− sφ(r) = 0, d1d2(φ)− sφ(q) = 0, d2d1(φ)− sφ(q) = 0. (3.6) 4. dynamics let φt be a shift transformation along the trajectories of the vector field sφ from t = 0 to t and l be an integral manifold of the distribution p . then the manifold φt(l) is also integral manifold of this distribution. a completely integrable distribution p is called a dynamics of the equation ∂u ∂t = f ( x, y, u, ∂u ∂x , ∂u ∂y , ∂2u ∂x2 , ∂2u ∂x∂y , ∂2u ∂y2 ) (4.7) if φ = f ( x1, x2, v, p1, p2, p̄ , q̄, r̄ ) (4.8) is a generating function of a shuffling symmetry of the distribution p . here x1 = x, x2 = y, p̄ = p (x1, x2, v, p1, p2), q̄ = q(x1, x2, v, p1, p2), r̄ = r(x1, x2, v, p1, p2) (4.9) are restrictions of the functions to the distribution p . copyright © 2022 assa. adv syst sci appl (2022) 4 a. kushner, e. kushner, s. tao the following theorem allows us to obtain solutions of equation (4.7) from the known integral manifolds of the distribution p . theorem 4.1: let l be a maximal integral manifold of the distribution p . then the three-dimensional manifold φt(l) is the 1-graph of a solution of evolutionary equation (4.7). let us show how we can calculate the dynamics. 1. first, we construct function (4.8) with unknown p,q and r. 2. second, we construct vector field (3.5) and the differential 1-forms ω0, ω1, ω2. 3. third, we construct equations (3.4) and add to them equations (3.6). since equation (3.4) with i = 0 is trivial, we obtain a system of six equations for the unknown functions p,q,r. 4. solving it we find dynamics (3.2). since system (3.2) is involutive, it has a three-parameter family of solutions. in order to solve this system, we can use the lie–bianchi theorem (see [7]). according to this theorem, a completely integrable distribution p is integrable by quadratures if the lie algebra symp is solvable and, moreover, dimsymp = codimp . in our case, codimp = 3, and the vector field sφ is a symmetry of the distribution p . therefore, we need two more symmetries. for example, if the function f in equation (4.7) does not explicitly depend on variables x, y. then we obtain these two symmetries automatically. namely, these are vector fields with generating functions p1 and p2. due to theorem 3, to construct an explicit solution of equation (3.2) we should calculate the flow φt of the vector field sφ. suppose that an integral manifold of the distribution p is defined by the following system: h1(x1, x2, v, p1, p2) = 0, h2(x1, x2, v, p1, p2) = 0, h3(x1, x2, v, p1, p2) = 0. (4.10) acting to this system by the transformation φ−1 t , we get the system u1(t, x1, x2, v, p1, p2) = 0, u2(t, x1, x2, v, p1, p2) = 0, u3(t, x1, x2, v, p1, p2) = 0. (4.11) solving this system concerning the function v(t, x), we get an explicit solution u = v(t, x) of equation (3.2). if the flow φt cannot be found explicitly, then we can construct solutions numerically. 5. dynamics and solutions of linear parabolic equations the linear equation ∂u ∂t = n∑ i=1 ∂ ∂xi ( ki(x) ∂u ∂xi ) (5.12) is among the most commonly used equations in mathematical physics. such equations are often encountered in biology [8], thermodynamics, diffusion theory, and filtration theory. these equations describe, for example, linear conservation laws. copyright © 2022 assa. adv syst sci appl (2022) dynamics of parabolic equations 5 5.1. case n = 1 for n = 1 equation (5.12) takes the form (see [9]) ∂u ∂t = ∂ ∂x ( k(x) ∂u ∂xi ) . (5.13) theorem 5.1: suppose that the function k is not vanishing. then equation (5.13) has a finite dimension second-order dynamics (k(x)v′) ′ + αv − β ∫ dx k(x) − γ = 0, (5.14) where α, β, and γ are constants. if α = 0 then general solution of equation (5.13) is v(x) = ∫ ∫ ( β ∫ dx k(x) + γ ) dx+ δ k(x) dx. (5.15) here δ is an arbitrary constant. the corresponding vector field sφ = ( β ∫ dx k(x) + γ ) ∂ ∂v . applying translations along its trajectories to solution (5.15), we obtain a three-parameter family of solutions of equation (5.13): u(t, x) = ( β ∫ dx k(x) + γ ) t+ ∫ ∫ ( β ∫ dx k(x) + γ ) dx+ δ k(x) dx. note that this solution was obtained for any non-zero function k(x). 5.2. case n = 2 for n = 2 equation (5.12) takes the form ∂u ∂t = ∂ ∂x1 ( k1(x) ∂u ∂x1 ) + ∂ ∂x2 ( k2(x) ∂u ∂x2 ) . (5.16) let us look for dynamics (3.2) in the form ∂2v ∂x2 = p (x, y), ∂2v ∂x∂y = q(x, y), ∂2v ∂y2 = r(x, y). (5.17) copyright © 2022 assa. adv syst sci appl (2022) 6 a. kushner, e. kushner, s. tao the distribution p integrability condition has the form p (x1, x2) = ∫ ∂q ∂x1 dx2 + p(x1), r(x1, x2) = ∫ ∂q ∂x2 dx1 + q(x2), where p, q are arbitrary functions. here x1 = x, x2 = y. note that in this case [d1,d2] = 0. consider the case when the functions are quadratic: k1 = k120x 2 1 + k111x1x2 + k102x 2 2 + k110x1 + k101x2 + k100, k2 = k220x 2 1 + k211x1x2 + k202x 2 2 + k210x1 + k201x2 + k200, where kins are constants. the index i means the number of the function, and the indices n and s mean the powers of x1 and x2, respectively. 5.2.1. trivial dynamics p = q = r = 0 equation (5.16) admits trivial dynamics when the functions p,q, and r are zero. we can construct the corresponding explicit solution to equation (5.16). however, the form of this solution is very cumbersome, and we will consider the special case when k211 ̸= 0 and k2 120 − 2k202k120 + k2 202 + k111k211 = 1. the differential 1-forms are ω1 = dp1, ω2 = dp2. linear functions v = c1x1 + c2x2 + c0 (5.18) form a 3-parameter family of solutions to equation (3.2). the vector field sφ = (2k120x1p1 − k−1 211(k 2 120 − 2k202k120 + k2 202 − 1)x2p1 + p1k110 + 2p2k202x2 + p2k211x1 + p2k201) ∂ ∂v + (2k120p1 + k211p2) ∂ ∂p1 + (−k−1 211(k 2 120 − 1− 2k202k120 + k2 202)p1 + 2k202p2) ∂ ∂p2 is a shuffling symmetry. copyright © 2022 assa. adv syst sci appl (2022) dynamics of parabolic equations 7 fig. 5.1. vector field (k1, k2). acting to function (5.18) by the transformation, we obtain the following explicit solution to equation (5.16): u =− [(bk211 + a(k120 − 1− k202)(k120 + k202 + 1) (k120 + k202 + 1)((k120 x1 + k110 + x1 k202 − x1)k211 − (k120 + 1− k202)(x2 k120 + k201 − x2 + k202 x2)) × e(k120+k202−1)t − (k120 + k202 − 1) (bk211 + a(k120 + 1− k202))((k120 x1 + k110 + x1 k202 + x1)k211 − (k120 − 1− k202)(x2 k120 + k201 + x2 + k202 x2))e (k120+k202+1)t − 2 bk2 211k110 + (4 ak110 k202 + 4 bk120 k201)k211 + 2 ak201 (k120 + 1− k202)(k120 − 1− k202)] × (2k211(k120 + k202 + 1)(k120 + k202 − 1))−1 , where a = c1, b = c2 are arbitrary constant. for brevity, we assume c0 = 0. 5.2.2. dynamics p = q = r = 1 consider the following partial case of equation (5.16) with k1(x1, x2) = x1x2 − x2 2, k2(x1, x2) = −x1x2 + x2 1 (see fig. 5.1). then equation (5.16) admits the dynamics with p = q = r = 1. the corresponding solution of equation (3.2) is v = 1 2 (x2 1 + x2 2) + x1x2 + ax1 + bx2 + c, (5.19) copyright © 2022 assa. adv syst sci appl (2022) 8 a. kushner, e. kushner, s. tao where a, b and c are arbitrary constants. the generating function is φ = x2 1 − x2 2 + x2p1 − x1p2. therefore, sφ = (x2 1 − x2 2 + x2p1 − x1p2) ∂ ∂v + (x1 + x2 − p2) ∂ ∂p1 + (p1 − x1 − x2) ∂ ∂p2 . the translation along this vector field is φt :  x1 7→x1, x2 7→x2, v 7→v + (−x2 2 + (−2x1 + p2)x2 − x1(−p1 + x1)) cos t + (−x2 2 + x2p1 + x1(x1 − p2)) sin t + x2 2 + (2x1 − p2)x2 + x2 1 − x1p1, p1 7→(x1 + x2) cos 2 t+ (p1 − x2 − x1) cos t + ((x2 + x1) sin t+ x1 + x2 − p2) sin t, p2 7→x2 + x1 + (p2 − x1 − x2) cos t + (p1 − x2 − x1) sin t. the inverse transformation is φ−1 t :  x1 7→x1, x2 7→x2, v 7→v + (x2 1 + (2x2 − p1)x1 + x2(x2 − p2)) cos 2 t + (−x2 1 + (p1 − 2x2)x1 − x2(x2 − p2)) cos t + (x2 1 + (2x2 − p1)x1 + x2(x2 − p2)) sin 2 t + (−x2 1 + x1p2 + x2(−p1 + x2)) sin t, p1 7→(p1 − x2 − x1) cos t+ (p2 − x1 − x2) sin t + x1 + x2, p2 7→(p2 − x1 − x2) cos t+ (x1 + x2 − p1) sin t + x1 + x2. applying the transformations φ−1 t to the vector function [ v − 1 2 (x2 1 + x2 2) + x1x2 + ax1 + bx2 + c, p1 − x1 − x2 − a, p2 − x2 − x1 − b ] we get system (4.11): copyright © 2022 assa. adv syst sci appl (2022) dynamics of parabolic equations 9 fig. 5.2. the graph of solution (5.20) at t = 0.  (x2 1 + (2x2 − p1)x1 + x2(x2 − p2)) cos 2 t + (−x2 1 + (p1 − 2x2)x1 − x2(x2 − p2)) cos t + (x2 1 + (2x2 − p1)x1 + x2(x2 − p2)) sin 2 t + (−x2 1 + x1p2 + x2(−p1 + x2)) sin t + v − x2 1 + x2 2 2 − x1x2 − ax1 − bx2 − c = 0, − a+ (p1 − x2 − x1) cos t+ (p2 − x1 − x2) sin t = 0, − b+ (p2 − x1 − x2) cos t+ (x1 + x2 − p1) sin t = 0. solving this system, we get the following 3-parameter solutions family to evolutionary equation (5.16): u = 1 2 ( x2 1 + x2 2 ) + (ax1 + bx2) cos t+ (ax2 − bx1) sin t+ x1x2 + c, (5.20) where a, b and c are arbitrary constants. the evolution of a graph of function (5.20) in time, see fig. 5.2, 5.3. here a = b = 1, c = 0. references 1. kruglikov, b.s. & lychagina, o.v. (2005) finite dimensional dynamics for kolmogorov – petrovsky – piskunov equation. lobachevskii journal of mathematics, 19, 13–28. copyright © 2022 assa. adv syst sci appl (2022) 10 a. kushner, e. kushner, s. tao fig. 5.3. the graph of solution (5.20) at t = 5. 2. lychagin, v.v. & lychagina, o.v. (2007) finite dimensional dynamics for evolutionary equations. nonlinear dyn., 48, 29–48. 3. akhmetzyanov, a.v., kushner, a.g. & lychagin, v.v. (2017) attractors in models of porous media flow. doklady. mathematics, 472:6, 627–630. 4. kushner, a.g. & matviichuk, r.i. (2020) exact solutions of the burgers – huxley equation via dynamics. journal of geometry and physics, 151, 103615. 5. kushner, a.g. (2022) dynamics of evolutionary equations with multiple spatial variables. proc. of conf. “symmetries: theoretical and methodological aspects, 9(33), astrakhan, 15-16 sept. 2022, 37-43, [in russian]. 6. duzhin, s.v, lychagin, v.v. (1991) symmetries of distributions and quadrature of ordinary differential equations. acta appl. math. 24(1), 29–57. 7. kushner, a.g., lychagin, v.v. & rubtsov, v.n. (2007) contact geometry and nonlinear differential equations. cambridge: cambridge university press, xxii+496 pp. 8. pethame, b. (2015) parabolic equations in biology. springer, heidelberg, 203 p. 9. kushner, a.g., kushner, e.n. & tao, s. (2022) finite-dimensional dynamics of filtration equation. differential equations and mathematical modeling, 4, 62–63, [in russian]. copyright © 2022 assa. adv syst sci appl (2022) introduction symmetries of completely integrable distributions symmetries of finite type differential equations dynamics dynamics and solutions of linear parabolic equations case n=1 case n=2 trivial dynamics p=q=r=0 dynamics p=q=r=1 microsoft word 1177 article text, copyedited.doc adv syst sci appl 2022; 01; 130-147 published online at https://ijassa.ipu.ru. leakage power reduction in read and write enhanced macro memory circuit design using transistor stacking and reversible approach mucherla usha rani1*, siva sankara reddy n.2, rajendra naik b.3 1) ace engineering college, jntuh hyderabad, india e-mail: mucherlausharaniphd@gmail.com 2) vasavi college of engineering, osmania university hyderabad, india 3) osmania university hyderabad, india abstract: power dissipation is one of the on-demand problems of low power very large-scale integration (vlsi) circuit design and it is related with threshold voltage. generally, the subthreshold leakage current and the leakage power dissipation are increased while reducing the threshold voltage. the overall performance of the total power dissipation is completely depending on this leakage power dissipation. this leakage power causes the components that are functioned by battery for a long period washed-out rapidly. in this paper, a new read decoupled and write enhanced macro memory design is proposed using transistor stacking and reversible logic to reduce the leakage power. for the reduction of power consumption of static random access memory (sram) that are placed in system on chips (socs), the proposed model includes three features: a reversible cross coupled stacked inverter with transmission gate is adopted in sram cell to reduce the leakage power, a reversible differential power generator is adopted to improve the write ability of the design with less power and a read burst mode is used to reduce the read energy while reading consecutive addresses. furthermore, a reversible logic is applied in all the peripheral circuits, which gives additional level of confidence in terms of leakage power. the simulation results show that the leakage power of the proposed macro memory design is reduced to 2.22nw at 0.6v supply, which is 4.32nw less than that of the conventional macro design. furthermore, the write and read power consumption of the proposed macro design is reduced by 57% and 66% as compared to conventional one at a supply voltage of 0.6 v. keywords: macro memory design, sram, leakage power, reversible logic, toffoli gate, reversible differential power generator 1. introduction generally, the portable and battery-operated devices often embed system-on-chips (socs). static random-access memory (sram) occupies most of the chip area in soc [1-2]. furthermore, the progression of low power neural network accelerators in different applications has made a strong request for power-efficient sram design [3-4]. similarly, the battery-less socs, needs three different types of memories: a “hold mostly” memory to save critical information for the longest possible time when energy is scarce, a “read-mostly” memory to hold the program instructions that run on the soc, and a “read-write” memory to save the data gathered. for the socs to support all three types of memories and still operate on harvested energy, the power and energy consumption of these memories must be kept at a minimum. in the last few decades, scaling is utilized for accomplishing highly standard cmos devices. this scaling process is used to reduce the delay and area of the systems. however, the shrinking of device dimensions rises the leakage current and power as well. in * corresponding author: mucherlausharaniphd@gmail.com leakage power reduction in read and write enhanced macro memory 131 copyright ©2022 assa. adv. in systems science and appl. (2022) general, sram cells contains nanoscale components while minimizing the size of cmos technology. as a result of this, the vulnerability of sram to soft errors is increased because of their more integrated density and lower supply voltage [5-6]. sram requires to be more vigorous, cost-effective, and consume less power when operating well at low supply voltages. furthermore, the data storing ability of the reading and writing processes affects the sram structure [7-8]. therefore, it is necessary to minimize the leakage power dissipation and improve the stability of srams. the traditional sram designs are susceptible to regular read upsets due to their worsened read static noise margin (rsnm). the traditional sram design uses same port for both read and write operation. this degrades the stability of the design while the voltage at the ‘zero’ storage node surpasses the nmos transistor’s threshold voltage [9]. for improving the stability as well as rapidity of dual port memories, different write and read bit lines are required. the dependency of temperature is the main reason for leakage power dissipation on cmos circuits. recently, several leakage power reduction methods have been designed specifically for sram structures. they are sleep transistor approach, sleepy stacking method, power gating method and lector approach. these approaches stack leakage path with the help of off transistors [10-12]. the stacking transistor aims to develop power efficient sram structure with simpler design construction. this approach minimizes the leakage noise current at the time of reading process. the stacking method serially connects the transistors for avoiding the leakage current [13]. the sleepy transistor approach uses additional control circuity to generate sleep signals. it introduces additional area and power penalties in the sram design. the effectiveness of lector approach is lesser than those of the complete off transistor method [14]. power gating method is also named as low power design methodology. this approach adds additional header and footer transistors for isolating the supply and ground terminals when it is in inactive condition [15]. this approach uses several sleep transistors and hence it significantly increases the area usage. furthermore, power gating method generates more voltage oscillations in the power/ground rail while changing the mode from power gating mode to active mode. this voltage oscillations are named as power/ground bouncing noise and it generates undesirable changeovers in the nearby non-power gated regions. when the supply voltage (vdd) is decreased the dynamic or active power will be reduced drastically. hence, the power-efficiency in sram cell has been achieved by downscaling the supply voltage [16]. this downscaling of supply voltage helps to reduce both static and dynamic power efficiently. however, this down sampled supply voltage is limited due to the inconsistency among the read and write necessities and direct-read-access strategy. the read line is separated from the primary cell of the single-ended sram design to optimize the enabling power supply of the read and write processes self-sufficiently. also, the half-select disturbance suffers majority of the sram cell designs. hence, the sram design structure needs to be improved for the minimization of leakage power as well as stability. furthermore, most of the available leakage reduction techniques can only moderate the circuit leakage power in standby mode. so, more efficient active leakage power reduction techniques are still essential to keep the leakage power and this motivate us to develop an enhanced technique for macro memory circuit design. the design of transistor stacking reversible logic circuits is a trending concept to achieve zero power dissipation. series connected transistors or the stacking technique are utilized to minimize the subthreshold leakage currents. similarly, the reversible data handing methods decreases the heat dissipation through the calculations, which preserves the energy and data without dissipating it into heat. this paper mainly aims to develop an efficient leakage power reduction approach for memory circuit. the main contributions of the proposed macro memory design are summarized as follows: 132 mucherla usha r., siva sankar reddy n., rajendra naik b. copyright ©2022 assa. adv. in systems science and appl. (2022) • to combine the benefits of both transistors stacking and reversible logic circuits in the macro sram design for the reduction of leakage power. • to adopt the sram cell design using reversible cross coupled stacked inverter with transmission gate for reducing the leakage power in the discharging path of sram cell. • to enhance the write stability of sram cell by proposing a new reversible differential power generator write assist circuitry. this circuit is used to enhance the write static noise margin (wsnm) of the sram without compensating the power dissipation. • to introduce a new feature of read burst operation (rbo) in the sram macro design, which reduced the read power whenever the successive addresses must be read for further processes. this feature will be more useful for “frequently read” applications such as image and video processing. • to introduce an additional level of confidence in terms of stability, area and leakage/static power by developing a reversible logic based peripheral circuits in the macro sram design. rest of the paper is organized as follows: section 2 reviews recent related works on sram design. section 3 provides detailed description about the proposed macro memory design. section 4 validates the simulation results of the proposed method. finally, the paper is concluded in section 5. 2. related works some of the recent related works are summarized as follows: saranya et al [17] used power gating method to reduce the power consumption of sram design. this approach ignored the straight link between the supply and ground by producing a virtual link between them. pal et al [18] proposed a low power single ended sram cell that supported the concept of bit-interleaving for healthcare application. the primary cell of this sram design contained cross coupled inverters and feedback cutting mechanism to improve write stability. this structure exhibited a slightly lengthier write time due to the utilization of feedback cutting approach. kumar et al [19] proposed a triple-threshold method for optimizing the power in sram design that improved the system stability in lesser cmos technologies. these multiple threshold transistors played an important role in leakage power minimization task. but the sensitivity of leakage current developed from different transistors were high on process variants. kumar et al [20] performed a comparative study on the design constraints of different sram design such as such as sleepy keeper, stacked keeper with body bias, and leakage feedback approach for leakage power reduction. among that, the sram cell that used stack transistor in the read trail reduced the read power dissipation. satyaraj et al [21] compared the performance of different leakage power reduction methodologies such as sleepy stack approach, sleepy keeper approach, dual stack power gating and dual controlled stacking (dcs) on sram memory. gavaskar et al [22] used hybrid vlsi methods to reduce and improve the leakage and speed of the sram cell respectively. initially, they described the schematics of the conventional leakage power reduction methods such as galeor [23], lector [23], multi-threshold cmos (mtcmos) [24], drain gating [25], sleepy keeper [26], leakage control nmos transistor (lcnt), dynamic threshold mos (dtmos) and mtcmos ground gated approach [27]. to tackle the issues of these methods, two or more leakage power reduction methods were combined to form hybrid models, for example sleepy keeper and drain gating approach, drain gating approach and helper transistors, upper self-controllable voltage level (usvl) and trimode mtcmos ground gated approach, lower self-controllable voltage level (lsvl) and trimode mtcmos power gated method. among them, the hybrid trimode mtcmos power and ground gated method provided good performance in terms of power, delay and area. however, the size and cost of this sram cell is high as leakage power reduction in read and write enhanced macro memory 133 copyright ©2022 assa. adv. in systems science and appl. (2022) compared to dram cells. eslami et al [28] presented a new sram design that improved the stability of read, write, and hold processes without degrading the performance in terms of leakage power. the data has been floated temporarily at lower voltage to increase the capability of writing process. at the same time, the data storage node has been separated from the read bit line with the help of a transistor to maintain the read stability as same as that of the hold state. this structure connected an additional p-type access transistor for mitigating the write ‘1’ problems of single ended sram design. sharma et al [29] introduced a data-dependent-power-supply approach to reduce the leakage power and improve the read/write stability for a 11t sram cell. this cell consumed picowatt range static power and enhanced the write static noise margin (wsnm) considerably. furthermore, a read decoupling has been used with column-based read buffer for improving the read stability. the use of a data dependent power supply approach helped to avoid the requirement of an exclusive power supply, so that the leakage power of the sram cell has been reduced considerably. wang et al [30] introduced a novel leakage minimization approach by splitting the array cell into two voltage regions. these regions were connected serially between the supply and ground for the straight generation of subthreshold holding voltage from the nominal supply. here, the zigzag cell arrangement has been selected with optimal transistor for balancing the hold stability, leakage, and area density. lorenzo et al [31] used a power gating transistor and transmission gate in sram design to reduce the leakage power and improve the reliability of writing process. the capabilities of read and write processes have been improved by separating the read and write path. moreover, it eliminated the irrelevant bit-line discharging using a row dependant virtual ground signal. peng et al [32] proposed a new 12t sram design by incorporating reverse bias current eliminating feature and read decoupling strategy. as a result of this, the read and hold static noise margin have been increased. alternatively, sachdeva et al [33] introduced a 12t sram cell on the bases of schmitt trigger. here, the properties of an inverter design have been improves using schmitt trigger inverter. this analysis shows that the leakage power dissipation and data stability are considered as the two main concerns while developing sram memory cell. to tackle the issues of traditional designs of sram, several read stabilities and write enhanced methods were developed. however, all the advancement in the existing sram design exhibited very high dynamic power consumption. the existing stacking and sleep power gating methods needed several rest transistors to reduce the leakage power. also, none of the existing approaches considered the macro sram for reducing the leakage power. in addition, all the existing designs did not perform well when they considered for “frequently read” applications. hence, there is a requirement of an alternative approach to design sram circuits in deepsubmicron technologies with minimum leakage power. 3. proposed macro memory circuit design in this paper, the design of sram macro cell is modified by introducing a reversible stacked inverter designs and reversible logic in both differential power generation and peripheral circuits. the proposed sram macro design contains memory cell array with reversible row drivers (rrds) and reversible column drivers (rcds), a read/write controlling module, reversible row decoder and reversible column decoder with data controlling module (dcm). these units are joined together to design sram macro as illustrated in figure 1. here, the sram cell array is designed using 4x4 sram cell. the smaller module of sram macro design is sram cell that stores a single bit value (i.e., “0” or “1”). here, the control module is used to read the inputs of each array cell, determine the exact operating mode and generate suitable control signals for reading and writing operations. the proposed macro design also introduces a read burst operating (rbo) mode for reducing the read power by experiencing a 134 mucherla usha r., siva sankar reddy n., rajendra naik b. copyright ©2022 assa. adv. in systems science and appl. (2022) read operation in the entire row while rwl (read word line) is stated. it enables the read data to store in reversible latches. as a result of this, a single read operation is sufficient when there is a need of repeated reads on the same row. it reduces the total read power because the reading from the latches consume lesser power than that of the normal read operation. furthermore, the flow of data in the sram cell array is managed by the dcm unit. the reversible tg based row decoder is used to detect the row in the sram cell array at which the address is existing and feeds it to the reversible tg based column decoder for decoding the respective column. in this way, the accurate position of the address can be detected for performing the read or write operation. the rrds and rcds are utilized for setting the data on the wordline (wls) and bitlines (bls) respectively. in addition, the proposed design utilizes a naïve output buffer for reading the value on rbl (read bl), which reduces the problems of general-purpose sense amplifier while functioning at subthreshold voltages. fig. 1. block diagram of the proposed macro memory design 3.1. sram cell the design of sram cell contains two cross-coupled inverters. here, reversible cross coupled stacked inverter is introduced to minimize the leakage power and ground bouncing noise. the sram cell design must consume less power feasibly. also, the arbitrary circuit of sram design should not dissipate the power and therefore, they should be constructed using reversible gate. hence, an attempt is made to introduce reversible logic in the peripheral circuits of sram memory. first, a reversible cross coupled stacked inverter with transmission gate is adopted in sram cell to reduce the leakage power. the pmos transistors (p1, p2) and nmos transistors (n1, n2) forms a reversible stacked inverter design. furthermore, a transmission gates t1 is kept in the inverter design for avoiding the storage cell 4 x 4 sram cell array re ve rsi bl e t g b as ed ro w d ec od er rrd1 wl1 wl2 wl3 wl4 wwl1 rwl1 rrd2 rrd3 rrd4 wwl2 rwl2 wwl3 wwl4 rwl3 rwl4 rc d 1 w bl 1 w bl b1 rb l1 rc d 2 w bl 2 w bl b2 rb l2 rc d 3 w bl 2 w bl b2 rb l2 rc d 4 w bl 2 w bl b2 rb l2 reversible row driver (rrd) reversible column driver (rcd) read/write controlling unit data controlling module (dcm) with reversible tg based column decoder d 1 d 2 d 3 d 4 rb l1 rb l2 rb l3 rb l4 rw rbo clock enb rst row address <1:2> l-clock r-clock re we re we column address <1:2> input data output data leakage power reduction in read and write enhanced macro memory 135 copyright ©2022 assa. adv. in systems science and appl. (2022) generation of spurious feedback signals between two conflicting outputs. here, we/web is set to gnd/vdd for keeping the t1in on state, which preserve the cell stability and protect the saved data via feedback path. here, the stacking method is introduced by considering the statistic that when more than one transistor is turned off in the supply to ground path there is less leakage as compared to single transistor is in off state. if input “0” is given to the inverter design, the nmos transistors are directed to on condition while the pmos transistor is directed to off condition. hence, it decreases the leakage current passing through the stacked nmos transistor by increasing the source to substrate voltage at the upper nmos transistor and by increasing the drain to source voltage at the lowest nmos transistor. implementation of reversible stacked inverter technique with transmission gate leads to low leakage power. in addition, word line includes two access transistors (n5, n6) for performing peruse and compose activities. the read-decoupled methodology uses access buffers made up of 2 nmos transistors (n7, n8) to improve read stability. here, the read current path is decoupled from the storage node for removing the read-disturbances, evading read reverse bias current and improving the rsnm. furthermore, the writing process of the sram is enhanced by introducing reversible logic based differential power generation approach. figure 2 shows the proposed sram cell with assistant circuitry. in this figure, the write wl (wwl) activates the sram cell in row direction and the write bitline pairs (wbl and wblb) activates the sram cell in column direction. (a) (b) fig. 2. proposed sram design (a) cell circuitry (b) reversible differential power generation circuitry second, a reversible differential power generation circuitry is used for eliminating the half-selection issue. in general, the write operation is degraded as a result of the halfselection issue. in figure 2 (a), two distinct power supplies (vdd1 and vdd2) that are generated by the reversible logic based differential power generation approach are connected to the sram cell. here, reversible toffoli gate is used to obtain vddm1 that combines wwl and wblb for driving an inverter to turned on. here, the pmos and nmos transistor’s source terminals are linked to vdd. likewise, vddm2 is obtained by we web wwl rwl wbl wblb rbl p1 p2 n1 n2 p3 p4 n3 n4 n5 n6 n7 n8t1 q qb vdd1 vdd2 tg inv vdd1vdd2 vdd vdd vddvdd wwl wblb inv tg wwl wbl 136 mucherla usha r., siva sankar reddy n., rajendra naik b. copyright ©2022 assa. adv. in systems science and appl. (2022) combining cwl and wbl through toffoli gate for driving the power of second inverter to turned on. (a) (b) fig. 3. toffoli gate design (a) conventional design in cmos technology (b) based on scrl method with standard power supply one of the modest and familiar universal reversible gates is toffoli gate. here, the splitlevel charge recovery logic (scrl) and reversible inverter gates are used to implement toffoli gate for observing energy recycling. this reversible design is compatibly developed in a cmos technology. the scrl method usually saves power by recycling the output capacitance thorough the use of two clocks. these clocks are used for charging and discharging the output capacitance consecutively. this idea is utilized in the reversible tg design for saving the power. the tg gate design using scrl method is illustrated in figure 3. wbl and wblb are inputs and it provides and output at its third output line. here, the transmission gates are controlled by the third input of toffoli gate and its inverse (i.e., 0). in figure 3 (b), the standard scrl charge recycling theory is utilized in a different format since two different components such as a nand gate and not gate are used to invent a new toffoli gate design. the scrl nand gate contains two additional transmission gates t1 and t2. initially, t1 and t2 are open to enable vdd as the source for charging the complete circuit. the gate is usually transmitting the signal when the transmission gate is in open condition. at the closed condition, it will not transmit any signal. in the subsequent stage, t1 and t2 are switched off. in this condition, the residual energy saved in the wires between the t1 and t2 is imposed the earlier output produced by the nand into the scrl not for generating a conflicting logic value. this is the reason for lower power consumption of scrl method as compared to the conventional cmos toffoli gate (figure 3 (a)). table 1 reports the various signals employed during different operations of the proposed sram cell. wwl wblb 1 wwl wblb (wwl) and (wblb) ‘0’ ‘1’ vdd vdd ‘1’ ‘0’ wwl wblb wwl wblb (wwl) and (wblb) nand in ve rte r t1 t2 leakage power reduction in read and write enhanced macro memory 137 copyright ©2022 assa. adv. in systems science and appl. (2022) table 1. applied signals during various operations of sram cell signal write ‘1’ write ‘0’ read hold wbl 1 0 0 0 wblb 0 1 1 0 we 1 1 0 0 web 0 0 1 1 wwl 1 1 0 0 rwl 0 0 1 0 3.1.1. write operation in the proposed design, the differential vdds (vdd1 and vdd2) are dropping differentially based on the estimates of the needed write data. the dropping of vdd1 indicates that the data “0” is accessed on wbl (wblb = 1). at the same time, vdd2 is dropping in reverse direction for accessing write “1”. at the time of write operation, the write enable signal we/web is set to vdd/gnd for deactivating the t1, which upset the recognized feedback path among the inverter designs. in write “0” operation, the bitline wbl is turned to “0” (wblb=1) and wordline wwl is turned to “1”. thus, the nmos transistor placed at the left side of the reversible differential power generation circuitry is turned on (because wwl and wblb=1). as a result of this, the voltage is dropping on vdd1, while vdd2 maintains at vdd because the combined result of wwl and wbl is “0”. hence, the pulling up intensity of the pmos transistors (p1 and p2) are attenuated due to the drooping voltage of vdd1. this allows the access transistor n5 to pull the storage node q to “0”. likewise, the write “1” operation allows wbl to turned on and wblb to drop to “0”. hence, the voltage is dropping on vdd2 (wwl=wbl=1) and the pulling up intensity of pmos transistors (p3 and p4) are ensured that the node qb is discharging to zero by the use of n6. the proposed reversible logic based differential power generation approach provides outstanding and valuable influence on the write ability to facilitate the write operation by consuming less power. 3.1.2. hold and read operation during the hold mode, the access transistors (n5 and n6) are turned off due to the lower values on wwl, wbl and wblb. hence, the proposed two reversible stacked crosscoupled inverters latch the data. during the read mode, rbl is first pre-charging to the vdd and the higher value of rwl (rwl=1) directs the transistor n8 to turned on. it activates the read deciding transistor n7 when the node qb saves “1” and it discharges the rbl to gnd via n7 and n8. alternatively, rbl upholds its pre-charging power when the node qb saves “0”. during the hold and read mode, the signals we/web is set to gnd/vdd for ensuring the feedback path between the two reversible stacked cross-coupled inverters by maintaining the transistor t1 in on stage. 3.2. control and data controlling modules the control unit reads input to each memory cell, determines the exact operating mode, and generates suitable read, write, and control signals. the inputs of controlling unit are enb, rbo and read/write (rw) and its outputs are read enable (re), write enable (we), clock signals (l-clock for latch and r-clock for output buffer). when enb=1, the controlling module is prompted for reading/writing data into the cell structure. if rw=1, it performs read operation to output the data from the memory cell. hence, the controlling unit directs re to turned to “1” and maintain “0” on we signal. hence, re directs the rrds for setting rwls. after completing the read operation, the signal l-clock is driven to high for enabling the reversible latches in the dmu to store the data. at last, the controlling unit generates rclock signal for enabling the output register to output the data. here, the reversible latches are designed using peres gate and the peres gate design utilizes majority gates (mg). the 138 mucherla usha r., siva sankar reddy n., rajendra naik b. copyright ©2022 assa. adv. in systems science and appl. (2022) reversible t latch design using mg based peres gate is shown in figure 4 (b). when a=”0”, the data saved in the loop will not be changed and hence the input is ineffective. but the input data is directed towards the loop when the a signal is set to “1”. the reversible t latch design can be characterized using the following expression: (1) the above equation is developed using a single peres gate. table 2 provides the truth table of the reversible t latch. these latches are utilized for holding the read information to be utilized while rbo is activated. the detailed diagram of dcm is illustrated in figure 4 (a). (a) (b) qatq ä=+ tg enb rbl<4:1> reversible tlatch l-clock reversible m u x reversible tlatch t q t q reversible tg based column decoder input data <3:0> output data <4:1> d<4:1> to rcds r-clock rst column address <2:1> mg mg mg mg mg mg mg -1 inv-1 inv-1 inv -1 -1 inv 1 q+ a t q p r leakage power reduction in read and write enhanced macro memory 139 copyright ©2022 assa. adv. in systems science and appl. (2022) (c) fig. 4. data controlling module (a) schematic of dcm (b) reversible t-latch using peres gate with mg gate implementation (c) majority gate design table 2. truth table for the reversible t latch input output a t q 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0 when the signal rw=0, it performs write operation by enabling we signal. hence, we directs the rrds and rcds for setting wwl and wbl/wblb. in addition, the active input of rbo enables the burst control module placed within the read/write controlling module to save the accessing addresses. if two successive addresses of the same row have been read then the controlling module intimate that the entire data of the respective row is already saved in the latches. as a result of this, the signals re and and l-clock aren’t activated. the dcm block receives rbl, input data, l-clock, r-clock and column addresses to show the output data via register during read operation and gives the input data to the write drivers during write operation. 3.3. reversible row decoder and driver circuits in general, the sram cell array is formed by arranging the cells in rows and columns. the reversible row and column decoders are mainly used to choose the suitable row and column based on the address bits. the proposed macro memory design uses 2×4 reversible row decoder which receives 2-bit address to drive four reversible toffoli gates (tgs) and activates one of the wls. likewise, the reversible column decoder activates different bls. however, the design of reversible row/column decoders must be adapted to activate one of the rw l or wwl at any interval. to do this, all outputs of the row decoders are connected with 1-to-2 reversible toffoli gate based demultiplexer. the similar design has been used in the column decoder design. hence, the rwl and wwl needs different drivers as illustrated in figure 5. vdd x y x y z z x y x z z y p qatq ä=+ 140 mucherla usha r., siva sankar reddy n., rajendra naik b. copyright ©2022 assa. adv. in systems science and appl. (2022) (a) (b) fig. 5. block diagram of row decoder (a) reversible tg based row decoder (b) reversible 1:2 demux it should be noted that the cell structure must write valid data at its storage node for reading exact data (i.e., “0” or “1”). hence, a write driver is needed to enable exact writing of data. every writing process begins with pre-charge circuit for pre-charging the bit-lines (i.e., wbl and wblb). later, one of the bit-lines is forced to “0” using write driver. at the same time, other bit-line maintains its pre-charged value. if the line wwl is high, the written values are pointed on wbl and wblb lines. the pre-charge circuit is designed by connecting 3 pmos transistors on the bit lines (i.e., wbl and wblb). these bit lines have been connected to the cell. the write driver circuitry is designed by the use of 2 pass transistors (nmos) transistors and 2 two reversible inverter circuitries. the pass transistors are acting as a transfer component of the sram cell, that forces a logic “0” efficiently rather than “1”. the write process is executed by writing a “0” into either wbl or wblb. inv inv tg tg tg tg row address [1] row address [2] wl1 wl2 wl3 wl4 reversible 1:2 demux reversible 1:2 demux reversible 1:2 demux reversible 1:2 demux re/we wwl4 rwl4 wwl3 rwl3 wwl2 rwl2 wwl1 rwl1 inv tg tg re/we wl wwl rwl leakage power reduction in read and write enhanced macro memory 141 copyright ©2022 assa. adv. in systems science and appl. (2022) (a) (b) fig. 6. implementation block for (a) pre-charge circuit (b) write driver circuit 4. simulation results here, the effectiveness of the proposed macro memory design is verified through simulations. the proposed macro memory unit has been simulated in a standard 14 nm cmos process. the sram array consists of 4 rows and 4 columns. every column contains self-supporting differential power rails including vdd1 and vdd2. the area of the proposed cell is slightly larger when compared to recently proposed read and write enhanced sram cell. however, the total macro area of a sram is smaller due to its simple peripheral circuits. the timing diagrams of write, hold and read operation of the proposed sram cell is shown in figure 7. fig. 7. timing diagrams of hold, write ‘1’ and write ‘0’ operation of the proposed sram cell 4.1. static noise margin analysis the stability of the sram design can be analysed using the measure of static noise margin. here, the conventional static method (i.e., butterfly method) is introduced to measure the static noise margin (snm). this snm can be categorized as hold static noise margin (hsnm), read static noise margin (rsnm) and write noise margin (wsnm) based on the mode of operation of sram cell. in this section, the snm analysis of the proposed stacked sram design is compared with conventional sram design. the wsnm can be determined by embedding a smallest square between the read and write voltage transfer characteristics (vtc) as shown in figure 8 (a). in wsnm measurement, the bit lines (wbl inv inv wbl wblb we input data wbl wblb bit line precharge wbl wblb 142 mucherla usha r., siva sankar reddy n., rajendra naik b. copyright ©2022 assa. adv. in systems science and appl. (2022) and wblb) are pre-charging to vdd and set wwl to “1”. this biasing condition yields high voltage at node (q) while the read vtc curve can be obtained from node qb. the write vtc can be obtained by pre-charging bitlines wbl to vdd and by connecting wblb to gnd. in this condition, the write vtc is measured. the proposed sram design provides wsnm of 250 mv and it is two times higher as compared to conventional sram at the supply voltage of 0.6 v. this proves the dominance of the proposed design during write condition. the proposed reversible differential power generator exercises a noticeable beneficial impact on the write ability, easing the write processes. (a) (b) (c) fig. 8. static noise margin analysis (a) wsnm (b) hsnm (c) rsnm leakage power reduction in read and write enhanced macro memory 143 copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 8 (b) illustrates the hsnm of the proposed cell based on butterfly curves at the supply voltage of 0.6 v. this hsnm is computed by inserting a maximum nested square into the butterfly curve. the hsnm of the proposed circuit is increased to 180 mv that is much better than that of the conventional sram design. during hold condition, the proposed model does not generate a reverse bias current due to the usage of outward access transistors. here, the reverse bias current has been eliminated by connecting the access transistors n5 and n6 to ground (since, wbl=wblb=”0”). furthermore, the stability of the proposed design has also been maintained by keeping the transistor t1 in on state, which preserved the data saved at storage nodes via feedback path. in rsnm measurement, the rbl is precharging to vdd and rwl is set to “1”. this biasing condition enable the voltage at node q to turned vdd from “0”. then, the data is exerted from the tanner tool and drawn a voltage transfer curve of qb vs. q as illustrated in figure 8 (c). due to the usage of read decoupling method, the storage nodes are isolated from the bit lines that evades the participations of capacitive noise. as a result of this the read upsets have been eliminated by exhibiting a peak rsnm. 4.2. power analysis in this section the active power consumption of the proposed macro memory design is analysed in terms of leakage power. the leakage power is also termed as static power and the transistors that aren’t fully turned off increases the leakage power. here, the initial analysis is carried out by comparing the proposed reversible transistor stacked sram cell design with the existing cell designs to prove its efficiency in leakage power reduction. table 3 provides the summary of the leakage power of different sram cell. the leakage power of the proposed sram cell is much better than that of the existing cell designs including 12t sram, se-sram and sbl-11t cell design due to the presence of transistor stacking and transmission gate in discharging trail of the proposed cell design during standby mode. also, the proposed sram avoids the reverse bias current due to the usage of outward access transistors. the leakage power of the proposed stacked and reversible logic-based sram cell has also been compared with different leakage reduction methodologies to prove its effectiveness in table 4. the results of this table shows that the proposed sram cell with reversible transistor stacking inverter design with transmission gate consumes only 1.35x10-9 power. it minimizes the leakage power as compared to the sleep transistor, sleep and stack, sleep keeper, dual stack and dcs respectively. table 3. leakage power analysis of different sram cell table 4. leakage power analysis of sram cell with different reduction methods method leakage power (w) method leakage power (w) sleep transistor 5.30x10-2 dual stack 7.00x10-2 sleep and stack 3.40x10-2 dcs 1.80x10-2 sleep keeper 4.90x10-2 proposed method 1.35x10-9 second, the analysis is carried out by comparing the proposed macro sram design with the conventional design in terms of read power, write power and static power. note that, the sram cell leakage power (w) 0.3 0.4 0.5 0.6 12t sram [33] 0.10x10-6 0.15x10-6 0.20x10-6 0.25x10-6 se-sram [28] 1.20x10-9 2.20x10-9 3.30x10-9 5.40x10-9 sbl-11t [31] 5.90x10-6 6.40x10-6 7.10x10-6 8.30x10-6 proposed 1.18x10-9 1.24x10-9 1.16x10-9 1.35x10-9 144 mucherla usha r., siva sankar reddy n., rajendra naik b. copyright ©2022 assa. adv. in systems science and appl. (2022) conventional macro design is designed with 8t sram cells and irreversible logic based peripheral designs. due to the introduction of reversible differential power generator in the write assist circuitry, the write power consumption of the proposed macro sram is significantly reduced compared with that of the conventional one. figure 9(a) shows that the write power of the proposed macro memory design is decreased by 57% as compared to conventional macro design at the supply voltage of 0.6 v. it demonstrates the assurance of the proposed macro sram for the upcoming memory-dominating low-power applications. (a) (b) (c) fig. 9. power analysis of macro memory design (a) write power (b) static power (c) read power leakage power reduction in read and write enhanced macro memory 145 copyright ©2022 assa. adv. in systems science and appl. (2022) the static power consumption of the proposed macro sram design is compared with the conventional macro design in fig. 9(b). the leakage power of the proposed macro memory design is 66% lesser as compared to the conventional design at the supply voltage of 0.6 v. because, the proposed sram cell don’t produce reverse bias current during hold operation due to the introduction of transistor stacking, transmission gate and outward access transistor in the discharge path. also, most of the peripheral circuits of the macro sram design is designed using toffoli gate with scrl logic. the energy reversibility property of this logic further used to reduce the power consumption in the proposed macro design. similarly, the read power consumption of the proposed macro sram design is decreased as compared to the conventional macro design. the proposed design also employed a read burst operating mode (rbmo) feature to reduce the read power when consecutive addresses in the same row should be read. the use of this feature has been proven through the results in figure 9 (c). the proposed model reduced the read power by performing the read process once and storing the data in reversible laches to support the successive reads. the process of data access from the latch consumes considerably lesser power than that of normal read operation. as a result of this, the overall read energy has been reduced. 5. conclusion in this paper, a new macro memory design is proposed based on transistor stacking and reversible logic for solving the issues caused by the leakage power. the proposed sram cell provided the best wsnm as compared to the conventional technique through the adoption of a reversible differential power generation based write assist circuit. as a result of this, it has a lower write power. in addition, a read burst mode has been introduced for reducing the read energy while accessing consecutive addresses and it saved 48% of the active read energy. the proposed sram cell reduces the leakage current for showing an enhanced hsnm performance, which indicates the proposed design as a guaranteeing contestant for ultra-low power applications. the introduction of transistor stacking and reversible logics reduced the power consumption down to 1.35nw with 0.6v supply. the proposed macro memory design provided the lesser leakage power of 2.22nw for the compete cell array as compared to the conventional macro memory design. this research work on sram reliability have mainly focused on the memory cell array and memory’s peripheral circuitry such as row and column decoders. however, the degradation of the sense amplifier (sa) and the area footprints of the whole circuits were not analysed. this work only compares the power performances of the designs. the existing sense amplifiers may latch incorrect signal and degrades the output of the sram system due to offset voltage. in future, a new power-delay compensation circuitry and an ultra-fast offset compensated sense amplifier will be implemented to speed up the read operation, solve the problem of column (row) half-select and determine a good trade-off between delay, energy, area efficiency and sensing reliability. references 1. krishna, r and duraiswamy, p. 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(2020). a schmitt-trigger based low read power 12t sram cell. analog integrated circuits and signal processing, 105(2), 275–295. adv syst sci appl 2021; 04:87–99 published online at https://ijassa.ipu.ru. implementation of the deffuant model within the flame gpu framework armen l. beklaryan1*, levon a. beklaryan2, andranik s. akopov1,2 1national research university higher school of economics, moscow, russia 2central economics and mathematics institute, russian academy of sciences, moscow, russia abstract: the deffuant model is a model of opinion dynamics based on the factor of the degree of doubt of agents, called uncertainty. despite its simplicity, the deffuant model turned out to be technically extremely difficult to analyze, and its basic convergence properties, which are easy to observe numerically, are only empirical results. in the presented work, the agent-based deffuant model is implemented within the flame gpu framework, designed to parallelize simulations of agent-based models based on gpus. the identity of the results with the original single-thread model is demonstrated. this approach allows us to study various characteristics of the model, its development, modification of the configuration of the ensemble of agents, to conduct various analyses, in particular, cluster analysis. keywords: deffuant model, opinion dynamics, extremism, flame gpu 1. introduction following the detailed introduction in the article [1], we recall that the field of opinion dynamics studies various processes related to the formation, dissemination and evolution of public opinion about certain events and objects of public interest in social systems. the systematic study of the dynamics of opinions began with the two-stage communication flow model in [2], as well as with the averaging model of social power in [3]. subsequently, french’s model of social power was refined by harari [4] and completely rethought by degroot [5]. other notable works include the friedkin and johnsen model [6], which takes into account the initial distribution of agents’ opinions, a general influence network theory [7], social impact theory [8] and dynamic social impact theory [9]. a comprehensive review of these and other opinion dynamics models is given in the two papers [10, 11] and in the textbook [12]. in recent years, considerable attention has been paid to the so-called models of opinion dynamics with bounded confidence. in such models, one individual is ready to influence another only if the difference of opinion between them is below a certain threshold value. deffuant, neau, amblard, and weisbuch [13, 14] proposed their own model with bounded confidence, now called the deffuant–weisbuch model (dw) or simply the deffuant model. in this model, a pair of individuals is randomly selected at each discrete time step, and each individual updates his opinion if the opinion of another individual is within his confidence bound. a second well-known model with bounded confidence is the hegselmann–krause (hk) [15] model, in which all agents update their opinions synchronously, averaging the opinions of individuals within their confidence bounds. ∗corresponding author: abeklaryan@hse.ru 88 a.l. beklaryan, l.a. beklaryan, a.s. akopov as shown in the papers of lorenz [16, 17], the simulation results for the dw model are reduced to three dominant phenomena, such as consensus, polarization, and segmentation. however, the dw model itself does not lend to rigorous analysis due to intramodel strong nonlinear dependencies. the available theoretical (not empirical) results of the model analysis, as a rule, focus on the homogeneous case in which all the agents have the same confidence bound. the convergence of the homogeneous dw model was proved by lorenz in [18], and its convergence rate was established in [19]. some studies consider various modifications of the dw model. for example, in [20], the scaling limits of the dw model are considered when the number of agents increases to infinity; in [21], the model is generalized, assuming that each agent can choose several neighbors to exchange opinion at each time step. a separate area of study is the consideration of the dw model in various topologies [22–24]. nevertheless, the question of convergence of the heterogeneous dw model remains open. it is worth noting that the analysis of the hk model is also limited to the homogeneous case. the convergence of the heterogeneous hk model is partially studied in [1, 25, 26], but only for special cases. this article discusses the implementation of the quasi-homogeneous dw and hk models within the flame gpu framework. the observed convergences for different values of the control parameters are studied, which makes it possible to develop further directions for the study of the model. 2. model description 2.1. base model following [14], we give a description of the opinion dynamics model. a model with a population of n agents is considered. each agent i has two characteristics: opinion xi ∈ [−1, 1] and uncertainty ui ∈ [0, 2]. the interval [max(xi − ui,−1),min(xi + ui, 1)] is called the opinion segment of i-th agent. agents interact randomly, for the dw model there is one random pair at each step of the model, for the hk model, at each step, each agent interacts with all others. when agents interact, one of them can influence the other if their opinion segments overlap. if the opinion segments do not overlap, it is assumed that the agents are so different from each other in their opinions that they have no chance to influence each other. if the opinion segments of agents i and j overlap, then agent j depends on the opinion of agent i by an amount proportional to the difference between their opinions multiplied by the amount of intersection and divided by the uncertainty of agent i minus 1. the meaning of this formula is that undecided agents influence other agents less than agents who are confident in their opinion. note that the influence of agents on each other is not symmetrical if the agents have different uncertainties (see fig. 2.1). after updating the opinions and uncertainties of agents i and j, a new pair is randomly selected and the same process is repeated until the attractors of the dynamical system with invariant opinions and uncertainties of all agents are reached. more formally, let us consider the opinion segments si = [max(xi − ui,−1),min(xi + ui, 1)] and sj = [max(xj − uj,−1),min(xj + uj, 1)]. the overlap hij of opinions is defined as follows: hij = min(xi + ui, xj + uj, 1)−max(xi − ui, xj − uj,−1). the non-overlap, respectively, is equal to 2ui − hij . the agreement of opinions is equal to hij − (2ui − hij) = 2(hij − ui). thus, the relative agreement of opinions (not symmetric by agents) is equal to 2(hij − ui) 2ui = hij ui − 1. copyright © 2021 assa. adv syst sci appl (2021) implementation of the deffuant model within the flame gpu framework 89 xj xi hij 2ui hij x'j xi fig. 2.1. the influence of agent i on agent j. now let us consider the modification of the opinion of agent j and its uncertainty after interacting with agent i. if hij ≤ ui, that is, the agreement of opinions is negative, then agent j is not influenced by agent i. if hij > ui, then the modification of the opinion and uncertainty of agent j is proportional to the relative agreement of opinions, namely: x ′ j = xj + µ ( hij ui − 1 ) (xi − xj), u ′ j = uj + µ ( hij ui − 1 ) (ui − uj), where µ is the parameter of elasticity of opinion and uncertainty (intensity of interaction). the main features of the described model are the following properties: • in the process of interaction, agents influence not only each other’s opinions, but also uncertainties; • the influence is not symmetrical: when agents have different uncertainties (see fig. 2.2) “confident” agents (low uncertainty) are more influential. this corresponds to life experience, in which confident people, as a rule, are easier to convince uncertain people than vice versa; • continuous dependence of influence (increment of opinion) on the parameters xi, ui, xj , and uj . the abrupt change in increments in other opinion dynamics models is difficult to explain from a psychological point of view. as part of the further analysis of the model, we will consider two scenarios of agent interaction at the step: 1. at each step, one random pair of agents is selected (step of the first type). this corresponds to the original dw model. 2. at each step, each agent interacts with each others, and the interaction effect is averaged by the number of pairs at the model step (step of the second type). this corresponds to the modification within the hk model. 2.2. extremists in the model the next step in building the model is to introduce extremists into the model: we assume that agents at the extreme points of the distribution of opinions are more confident (their uncertainty is lower). this hypothesis can be justified by the fact that often people who hold extreme opinions tend to be more convinced of their views. on the contrary, people who have moderate initial opinions often express uncertainty. copyright © 2021 assa. adv syst sci appl (2021) 90 a.l. beklaryan, l.a. beklaryan, a.s. akopov fig. 2.2. relative agreement is plotted on the left graph as a function of xi for different values of ui. the right graph shows the increment of xj when modifying the opinion as a function of xi for different values of ui. to implement such a mechanism, we define two values of initial uncertainty: ue is the uncertainty of extremist agents (small one) and u is the uncertainty of moderate agents, presumably greater than ue. we also define pe as the global proportion (presumably small) of extremist agents in the general population (we believe that all extremists have the same uncertainty), and p+ and p− are the proportion of extremists with positive and negative extreme opinions, respectively. for ease of initialization of the model, we introduce a parameter of relative bias between the proportion of positive and negative extremist agents according to the following formula: δ = |p+ − p−| p+ + p− = |p+ − p−| pe . as part of the practical implementation, we first randomly assign an initial opinion to all agents of the population from a uniform distribution in the segment [−1, 1]. then we initialize np+ agents with the most positive initial opinions and np− with the most negative opinions with uncertainty ue, and the rest with uncertainty u. 2.3. parameters of the model to identify the effect of the initialization of the model on the type of convergence, the following parameters are introduced: • pe is the global proportion of extremist agents; • u is the initial uncertainty of moderate agents; • ue is the initial uncertainty of extremist agents; • δ is the relative bias between the proportions of positive and negative extremist agents; • µ is the intensity of interaction (the degree of trust in someone else’s opinion). due to the stochasticity of the model, each experiment was repeated 100 times after which the results were averaged. the total number of agents in the population for all experiments is 1000. to obtain a slightly more detailed picture of convergence, in comparison with the original model, instead of a purely homogeneous model (all moderate agents have the same initial uncertainty), we considered quasi-homogeneous model, in which the uncertainties of moderate agents are subject to a normal distribution with a single mean and extremely small variance. unless otherwise specified, ue = 0.1. copyright © 2021 assa. adv syst sci appl (2021) implementation of the deffuant model within the flame gpu framework 91 fig. 4.3. simulation results for low uncertainty and first type step. 3. flame gpu realization the software implementation of the model is performed using the flame gpu simulation platform [27], providing efficient parallelization of large-scale agent-oriented models. at the same time, calculations were performed on the forsite dsws pro supercomputer based on quadro rtx 6000 with a capacity of 16.3 tflops. a feature of the flame gpu platform is the use of xml language to describe the structure of the simulation model, global environment variables, functions implemented at the cpu level, in particular, providing dynamic change of opinion and uncertainty of agents, clustering, processing simulation results and interaction with the system database, etc., as well as functions implemented at the individual level of each agent and performed independently on gpus. 4. analysis of simulation results 4.1. without extremists the authors of the original article [14] note that the number of clusters of opinions in the model without extremists empirically is equal to w/(2u), where w is the width of the initial distribution of opinions (equal to 2, since opinions lie in the segment [−1, 1]), and u is the initial uncertainty of moderate agents. the results obtained by us are completely consistent with this conclusion. 4.1.1. many clusters with low uncertainty. with a low uncertainty of 0.2, we get 1/0.2 = 5 clusters almost equidistant from each other (u = 0.2). the results are shown in figures 4.34.4. 4.1.2. two clusters with average uncertainty. with an average uncertainty of 0.5, we get 1/0.5 = 2 clusters (u = 0.5). the results are shown in figures 4.5-4.6. 4.1.3. one cluster with high uncertainty. with a high uncertainty of 1, we get 1 cluster (u = 1). the results are shown in figures 4.7-4.8. 4.2. within extremists 4.2.1. central convergence. weak radicalization: if the moderate majority is confident in their opinion, and there are not too many extremists, then most of the moderate majority does copyright © 2021 assa. adv syst sci appl (2021) 92 a.l. beklaryan, l.a. beklaryan, a.s. akopov fig. 4.4. simulation results for low uncertainty and second type step. fig. 4.5. simulation results for average uncertainty and first type step. fig. 4.6. simulation results for average uncertainty and second type step. not become extremists, although some are radicalized (u = 0.3, µ = 0.5, pe = 0.2, δ = 0). the results are shown in figures 4.9-4.11. copyright © 2021 assa. adv syst sci appl (2021) implementation of the deffuant model within the flame gpu framework 93 fig. 4.7. simulation results for high uncertainty and first type step. fig. 4.8. simulation results for high uncertainty and second type step. fig. 4.9. simulation results for central convergence and first type step. 4.2.2. bipolarization. if the moderate majority is not sure of their opinion, then most of them become extremists. at the same time, the larger the proportion of extremists, the copyright © 2021 assa. adv syst sci appl (2021) 94 a.l. beklaryan, l.a. beklaryan, a.s. akopov fig. 4.10. joint distribution of opinion and uncertainty for central convergence. fig. 4.11. simulation results for central convergence and second type step. fig. 4.12. simulation results for bipolarization and first type step. more moderate agents become extremists (u = 1.2, µ = 0.5, pe = 0.2, δ = 0). the results are shown in figures 4.12-4.14. 4.2.3. single polarisation. if there are more extremists of one pole, then they can win over most of the moderate majority. at the same time, even if the proportion of extremists is not copyright © 2021 assa. adv syst sci appl (2021) implementation of the deffuant model within the flame gpu framework 95 fig. 4.13. joint distribution of opinion and uncertainty for bipolarization. fig. 4.14. simulation results for bipolarization and second type step. fig. 4.15. simulation results for single polarisation and first type step. very large, and the moderate majority is very uncertain, then the majority is able to completely radicalize (u = 1.2, µ = 0.5, pe = 0.2, δ = 0.7). the results are shown in figures 4.15-4.17. 4.2.4. slow weak radicalization. if trust in society is very low (µ parameter), then even extremely convincing radicals will not be able to persuade the entire moderate majority to copyright © 2021 assa. adv syst sci appl (2021) 96 a.l. beklaryan, l.a. beklaryan, a.s. akopov fig. 4.16. joint distribution of opinion and uncertainty for single polarisation. fig. 4.17. simulation results for single polarisation and second type step. fig. 4.18. simulation results for slow weak radicalization and first type step. themselves. at the same time, due to the low level of confidence, convergence will be slow (u = 0.5, µ = 0.1, pe = 0.2, δ = 0, ue = 0.05). the results are shown in figures 4.18-4.20. copyright © 2021 assa. adv syst sci appl (2021) implementation of the deffuant model within the flame gpu framework 97 fig. 4.19. joint distribution of opinion and uncertainty for slow weak radicalization. fig. 4.20. simulation results for slow weak radicalization and second type step. 5. conclusion the identity of the parallelized multiparametric implementation of the deffuant model, as an extension of the original version of the model, with the original result up to the ergodic components of the configuration space is established. such an extension makes it possible to study the dependence of the state of the configuration space on the control parameters, ergodic components of the configuration space, cluster partitions in the dynamics of the evolution of the system, to localize the points (regions) of the bifurcation of the system, as well as to investigate the dependence of the characteristics of the system on the topology of the ensemble of agents. a separate area of further research is the development of the model within the framework of the general concept of control, activity, personality [28]. acknowledgements this work was funded by the russian foundation for basic research (rfbr), project number 20-01-00002. references copyright © 2021 assa. adv syst sci appl (2021) 98 a.l. beklaryan, l.a. beklaryan, a.s. akopov 1. chen, g., su, w., mei, w., & bullo, f. 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(2020) control, activity, personality. advances in systems science and applications, 20, 113–135, https://dx.doi.org/10.25728/assa.2020.20.3.931. copyright © 2021 assa. adv syst sci appl (2021) https://dx.doi.org/10.1016/j.physa.2013.07.014 https://dx.doi.org/10.1016/j.physa.2019.123462 https://dx.doi.org/10.1016/j.physa.2010.08.025 https://dx.doi.org/10.1016/j.physa.2004.06.102 https://dx.doi.org/10.1137/11082751x https://dx.doi.org/10.1109/tac.2016.2644266 https://dx.doi.org/10.5281/zenodo.5428985 https://dx.doi.org/10.25728/assa.2020.20.3.931 introduction model description base model extremists in the model parameters of the model flame gpu realization analysis of simulation results without extremists many clusters with low uncertainty. two clusters with average uncertainty. one cluster with high uncertainty. within extremists central convergence. bipolarization. single polarisation. slow weak radicalization. conclusion adv syst sci appl 2022; 02:109–119 published online at https://ijassa.ipu.ru. the optimal control approach in problems of the income distribution into consumed and invested parts alexander p. chernyaev1* 1moscow institute of physics and technology (state university), dolgoprudnyi, russia abstract: in this paper, we investigate dynamic problems of the income distribution into consumed and invested parts using optimal control methods. here it is assumed that the basic economic identity holds true, i.e., the income is the sum of the consumption and the investment. the income is understood either in the pure form or the gross domestic product, or the national income. the consumption is also understood in different ways: in its pure form or the aggregate consumption. we start with the analysis of the harrod–domar macromodel with the capital intensity of the income growth depending on the continuous time. in particular, it is shown that the balance equation for accumulated household savings also satisfies the basic economic identity and the capital intensity of the income growth can depend on time. since households are the best savers and they demonstrate the best survival, we modify the considered macromodels replacing the given production functions with the problem of maximizing the integral discounted utility of the consumption. in this case, the utility function satisfies the arrow–pratt condition of the relative risk aversion. keywords: optimal control, income, consumption, investment introduction throughout the paper, we assume the basic economic identity: the income is equal to the sum of the consumption and the investment [1]. this identity has many various applications; see, e.g., [2]. for example, this identity holds true in the harrod–domar and the solow macromodels [3]– [15], and also in the household micromodel [?], [19]. hence, these models have some common properties. one of the main goals of the paper is to establish such properties. it should be noted that in these models the income is understood in different ways. the income is understood either in the pure form or the gross domestic product, or the national income. the consumption is also understood in different ways: in its pure form or the aggregate consumption. first, consider the harrod–domar macromodel with time-dependent capital intensity of the income growth. we also introduce a new variable, namely, the relative increase in the income of the cobb–douglas production function, and consider the balance equation for the capital. as a result, we establish the time dependence for the capital intensity of the income growth of the solow model. further, we show that the capital intensity of the income growth in the household model can also depend on time. since households are the best savers and they demonstrate the best survival, we shall modify the considered macromodels. namely, we replace the production functions given a priori with the problem of maximizing the integral discounted utility of the consumption, as ∗corresponding author: chernyaev49@yandex.ru 110 a.p. chernyaev households usually do. moreover, in all these models the utility function satisfies the arrow– pratt condition of the relative risk aversion; see [1, 2, 13, 14]. 1. the preliminary results 1.1. the main economic identity in this paper, we shall always assume that the main economic identity holds true: y (t) = c(t) + i(t), (1.1) where t means the continuous time, y (t) is the net income or the gross domestic product or the national income, c(t) is the consumption or the total consumption, i(t) is the general investment. 1.2. the dependence of the income on the consumption in the harrod–domar model with varying capital intensity of the income growth the main assumption in the harrod–domar model is that the investment linearly depends on the derivative of the income by t, that is, i(t) = by ′(t); see [1,15]. here the coefficient b is a positive constant. assuming that b depends on t, we have: y (t) = c(t) +b(t)y ′(t), (1.2) where b(t) is the capital intensity of the income growth. setting for (1.2) the initial condition y (t0) = y0 > 0, (1.3) we obtain the cauchy problem, whose solution is given by the following formula (see, e.g., [1, 8]): y (t) = exp {∫ t t0 ds b(s) }( y0 − ∫ t t0 c(τ) b(τ) exp { − ∫ τ t0 ds b(s) } dτ ) . (1.4) formula (1.4) determines the dependence of the income on the consumption. 1.3. the relative increase of the income for the cobb–douglas production function consider the cobb–douglas production function [17, 18]: y = f (k,l) = akαl1−α, a = const > 0, 0 < α < 1, (1.5) where k = k(t) is the capital, l = l(t) is the labor, i.e., the number of employees. for convenience, let us introduce a new variable σ = σ(t) = y ′(t) y (t) = α k ′(t) k(t) + (1− α) l′(t) l(t) , which is called the relative increase of the income. introduce the standard value ν = ν(t) = l′(t) l(t) , (1.6) which we call the rate of the growth of labor resources. then, taking into account (1.6), for the relative increase in income we have the expression σ = σ(t) = y ′(t) y (t) = α k ′(t) k(t) + (1− α)ν. (1.7) copyright © 2022 assa. adv syst sci appl (2022) the optimal control approach in problems of the income distribution 111 1.4. the balance equation for capital obviously, the balance equation for capital has the form k ′(t) = i(t)− µk(t), (1.8) where µ is the share of the retired capital. the standard assumption i(t) = ρy (t), (1.9) where ρ = ρ(t) is the rate of accumulation [1, 15, 17, 18], leads to the equality k ′(t) k(t) = ρy (t) k(t) − µ = ρa ( k(t) l(t) )α−1 − µ. (1.10) then, taking into account (1.7), we have σ(t) = αρak(t)α−1 + ν − α(µ+ ν), (1.11) where k(t) = k(t)/l(t) is the capital-labor ratio. 1.5. capital capacity of increase in the solow model sine equation (1.11) contains the capital-labor ratio, it leads to the solow macromodel [12]– [15]. using the equality i(t) = b(t)y ′(t) for determination the incremental capital intensity and taking into account (1.9), we get i(t) = b(t)y ′(t) = ρy (t), b(t) = ρ σ(t) . (1.12) the first equality has a deep economic meaning. when investing, an investor has to monitor not only the high capital intensity of the income growth. one must be sure that this incremental capital intensity is high not due to the small relative increase in income. substituting expression (1.11) into the right-hand formula (1.12), we obtain b(t) = ρ αρak(t)α−1 + ν − αλ , λ = µ+ ν. (1.13) in order to check that (1.13) is not constant and it really depends on time, one can use the exact solution of the solow model for the cobb–douglas production function. for the capital-labor ratio, this yields the equations k′(t) = ρf(k(t))− λk(t) = ρak(t)α − λk(t), λ = const. (1.14) setting the initial condition k(t0) = k0 > 0, (1.15) we get the cauchy problem, whose exact solution is given by the formula k(t) = e−λ(t−t0) ( ρa λ eλ(1−α)(t−t0) − ρa λ + k1−α 0 ) 1 1−α . (1.16) finally, from (1.13) and (1.16) we have b(t0) = ρ αρakα−1 0 + ν − αλ , b(+∞) = ρ ν . (1.17) the obtained formula shows that the capital intensity of the income growth depends on time and in the general case it is not constant. copyright © 2022 assa. adv syst sci appl (2022) 112 a.p. chernyaev 1.6. the balance equation for accumulated household savings now consider the household micromodel as an economic model governed by equation (1.1); see [?, 14]. then we have the equation x′(t) = p(t)x(t) + p (t)− c(t), (1.18) where x(t) is the accumulated savings, p(t) is the bank interest, p (t) is determined by salaries, pensions and extra earnings. as before, c(t) means the consumption. if we assume that y (t) = p(t)x(t) + p (t) is the income and i(t) = x′(t) is the investments, then we come to the basic economic identity (1.1). similarly to [?, 1, 14], we maximize the integral discounted utility of consumption j = ∫ t1 t0 u(c(t))e−δt dt, (1.19) where u is the utility function, δ is the coefficient of discounting the future utility. 1.7. the utility function and the risk avoidance according to the arrow–pratt theory [?, 1, 14], the utility of consumption is estimated by the function u(c), which describes the constant aversion to risk: − u′′(c)c u′(c) = a ≥ 0. (1.20) the relative measure of the arrow–pratt risk aversion is the consumption elasticity of the marginal utility taken with the opposite sign. let us denote by g(c) = u′(c) the marginal utility of consumption, and by ec(g) = g′(c) g(c)/c the elasticity of variation of the variable g with respect to the variable c. then, similarly to [?, 1], one can consider (1.20) as a differential equation for u(t). this yields u′(c) = g(c) = γc−a, where γ > 0 is the constant of integration. finally, we have u(c) =  γc1−a 1− a + χ, a 6= 1, γ lnc + χ, a = 1, where χ is an arbitrary constant. 2. the optimal control approach 2.1. optimal control in the modified harrod–domar model according to [17], households are the best savers. this motivates the maximization of the functional (1.19) in the framework of the modified harrod–domar model, which is governed by the equation y (t) = c(t) +by ′(t). consider the maximization of the functional (1.19), where the utility function satisfies the following conditions: − u′′(c)c u′(c) = a ≥ 0, g(c) = u′(c), ec(g) = −a, (2.21) copyright © 2022 assa. adv syst sci appl (2022) the optimal control approach in problems of the income distribution 113 under the following constrains: y (t) = c(t) +by ′(t), y (t0) = y0 > 0, y (t1) = y1 > 0. (2.22) this maximization problem can be solved using the standard variational method [?, 1]. a necessary condition of optimality is the euler–lagrange equation, which reads d dt ( g(c(t))b(t)e−δt ) + g(c(t))e−δt = 0. (2.23) integrating (2.23), we have c(t) = g−1 ( d b(t) exp { δt− ∫ t t0 dτ b(τ) }) , (2.24) where d is the constant of integration. finally, condition (2.21) gives a sufficient condition of optimality. theorem 2.1: under the above conditions, the total consumption c(t) is given by the formula c(t) = ( y0 − y1 exp{−β(t1)} ) b(t) 1 a exp{ 1 a β(t)− δ a t}∫ t1 t0 b(τ) 1 a −1 exp{( 1 a − 1)β(τ)− δ a τ} dτ , (2.25) where the auxiliary function β is defined as β(t) = ∫ t t0 ds b(s) . proof from (2.24) we have g(c(t))b(t) = d exp{δt− β(t)}, d = const > 0. (2.26) on the other hand, from (2.21) we get g(c) = γc−a, γ = const > 0. substituting this expression into (2.21), we have c(t) = ( γb(t) d ) 1 a exp{a−1(β(t)− δt)}. (2.27) now from (1.4) and (2.22) we obtain the equality∫ t1 t0 c(τ) b(τ) e−β(τ)dτ = y0 − y1e −β(t1), (2.28) and from (2.27) and (2.28) one can express the constant:( γ d ) 1 a = y0 − y1e −β(t1)∫ t1 t0 b(τ) 1 a −1 exp{( 1 a − 1)β(τ)− δ a τ} dτ , (2.29) finally, substituting the expression (2.29) into (2.27) and taking into account (2.22), we obtain (2.25). copyright © 2022 assa. adv syst sci appl (2022) 114 a.p. chernyaev 2.2. optimal control in the modified solow model similarly to the above, here we consider the maximization of the functional j = ∫ t1 t0 u(c(t))e−δt dt, where c(t) = c(t) l(t) , (2.30) and the utility function u satisfies the conditions similar to (2.21): − u′′(c)c u′(c) = a ≥ 0, g(c) = u′(c), ec(g) = −a, (2.31) under the following constrains (see (1.14)): k′(t) = ρ(t)f(k(t))− λk(t), k(t0) = k0 ≥ 0, k(t1) = k1 ≥ 0. (2.32) taking into account the formulas c(t) = c l = y − i l = y − ρy l = (1− ρ) y l = (1− ρ)f(k), k′(t) = cρ 1− ρ − λk, c(t) = 1− ρ ρ (k′ + λk), 0 < ρ < 1, the euler–lagrange equation (a necessary condition of optimality) reads − d dt ( g(c(t)) 1− ρ ρ e−δt ) + g(c(t)) 1− ρ ρ λe−δt = 0. (2.33) integrating (2.33) (see [1]), we have c(t) = g−1 ( dρ 1− ρ exp { δt+ ∫ t t0 λ(s)ds }) . (2.34) the condition (2.31) gives a sufficient condition of optimality. theorem 2.2: under the above conditions, the total consumption c(t) is given by the formula c(t) = ( k1e λ(t1) − k0 ) r(t) 1 a exp { − 1 a (λ(t) + δt) } ∫ t1 t0 r(τ) 1 a −1 exp { (1− 1 a )λ(τ)− δ a τ } dτ (2.35) where the auxiliary functions l and r are defined as λ(t) = ∫ t t0 λ(s)ds, r(t) = 1− ρ(t) ρ(t) . proof from (2.34) we have g(c(t))r(t)e−δt = deλ(t), d = const > 0. (2.36) copyright © 2022 assa. adv syst sci appl (2022) the optimal control approach in problems of the income distribution 115 on the other hand, from (2.31) we get g(c) = γc−a, γ = const > 0. substituting this expression into (2.36), we have c(t) = ( γ d r(t) ) 1 a exp{−a−1(λ(t) + δt)}. (2.37) substituting (2.37) into the equation k′(t) = cρ 1− ρ − λk, we get a linear differential equation for k(t). integrating it, we found k(t) = e−λ(t) ∫ t t0 c(τ) r(τ) eλ(τ)dτ + k0e −λ(t). (2.38) from (2.37) and (2.38) it follows k1e λ(t1) − k0 = ( γ d ) 1 a ∫ t1 t0 r(τ) 1 a −1 exp{( 1 a − 1)λ(τ)− δ a τ}dτ. (2.39) the obtained equality allows us to express the constant (γ/d) 1 a . therefore, from the equalities (2.37) and (2.39) we obtain (2.35). 2.3. optimum consumption management in household model in this section, we study a household model in more detail, see [17, 21]. as in the solow model, the study is based on the maximization of the functional (2.30), and the utility function u(c) satisfies the conditions (2.31). however, the constrains now have the form of the balance equation for accumulated household savings (see (1.19)), that is, x′(t) = p(t)x(t) + p (t)− c(t), x(t0) = x0 ≥ 0, x(t1) = x1 ≥ 0. (2.40) then the euler–lagrange equation (a necessary condition of optimality) reads d dt ( g(c(t))e−δt ) + g(c(t))p(t)e−δt = 0. (2.41) integrating (2.41), we have c(t) = g−1 ( d exp { δt− ∫ t t0 p(s)ds }) . (2.42) as before, the condition (2.31) gives a sufficient condition of optimality. moreover, for the cauchy problem obtained from the condition (2.40) by elimination the last equality (for x(t1)) one can write the solution: x(t) = ew (t) ( x0 + ∫ t t0 (p (τ)− c(τ))e−w (τ)dτ ) , w (t) = ∫ t t0 p(s)ds. (2.43) theorem 2.3: under the above conditions, the total consumption c(t) is given by the formula c(t) = ∫ t1 t0 p (τ)e−w (τ)dτ + x0 − x1e −w (t1)∫ t1 t0 exp { ( 1 a − 1)w (τ)− δ a τ } dτ exp { 1 a (w (t)− δt) } . (2.44) copyright © 2022 assa. adv syst sci appl (2022) 116 a.p. chernyaev proof from (2.34) we have g(c(t)) = d exp{δt−w (t)}, d = const > 0. (2.45) from (2.31) we get g(c) = γc−a, γ = const > 0. substituting this expression into (2.45), we obtain the equation c(t) = c0 exp{ 1 a (w (t)− δt)}, c0 = ( γ d ) 1 a . (2.46) after that, from (2.40), (2.43), (2.46) we have the equality x1e −w (t1) − x0 = ∫ t1 t0 ( p (τ)− c0 exp{ 1 a (w (τ)− δτ)} ) e−w (τ)dτ, (2.47) which yields c0 = ∫ t1 t0 p (τ)e−w (τ)dτ + x0 − x1e −w (t1)∫ t1 t0 exp{ 1 a (w (τ)− δτ)−w (τ)}dτ . (2.48) finally, from (2.46) and (2.48) we obtain (2.44). 2.4. restrictions on the accumulated savings in household model in this section, we obtain conditions for no debt in the in the household model considered above. in our notations, no debt is the inequality x(t) ≥ 0, t0 ≤ t ≤ t1. (2.49) theorem 2.4: a sufficient condition for (2.49) is given by the inequality φ(t)− φ(t1) ≥ 0, t0 ≤ t ≤ t1. (2.50) proof first, let us prove that the inequality x(t) ≥ x0 + ∫ t t0 p(τ)x(τ)dτ, t0 ≤ t ≤ t1, (2.51) is equivalent to ∫ t t0 (p (τ)− c(τ))dτ, t0 ≤ t ≤ t1. (2.52) indeed, integrating the both sides of the equation (2.40), we have the equality x(t)− x0 = ∫ t t0 p(τ)x(τ)dτ + ∫ t t0 (p (τ)− c(τ))dτ, which shows the equivalence of (2.51) and (2.52). it should be noted that the inequality (2.51) has an obvious economic meaning: the righthand side of (2.51) is the amount of money that would be on the deposit account at the moment t if the amount x0 was deposited on the account at the moment t0. copyright © 2022 assa. adv syst sci appl (2022) the optimal control approach in problems of the income distribution 117 second, let us present (2.52) in the form x(t) = ψ(t)ew (t)(φ(t)− φ(t1)) + x1 φ(t) φ(t1) ew (t)−w (t1), (2.53) with the functions ψ(t) = ∫ t t0 exp { ( 1 a − 1)w (τ)− δ a τ } dτ, φ(t) = ( x0 + ∫ t t0 p (τ)e−w (τ)dτ ) ψ−1(t). (2.54) since the function ψ is strictly positive and x1 ≥ 0, the equality (2.53) obviously shows that (2.50) implies (2.49). the function φ defined above is called the indicator function of the model. theorem 2.4 states that if the indicator function decreases, then the condition (2.49) holds true. 3. examples of indicator functions here we discuss several examples of monotonically decreasing indicator functions in the household model. let p (t) = p0e r(t−t0), p0 = const > 0, p(t) = p = const > 0. (3.55) then from (2.54) we have φ(t) = x0e p(t−t0) + p0e pt−rt0 ∫ t t0 e(r−p)τdτ ep(t−t0/a) ∫ t t0 exp{(( 1 a − 1)p− δ a )τ}dτ . (3.56) to simplify further calculations, we introduce new notations: θ = ( 1 a − 1)p− δ a , θ = r − p. first, consider the generic case: θ 6= 0 and θ 6= 0. then formula (3.56) yields the indicator function φ(t) = e δ a t0 ( x0 θ eθ(t−t0) − 1 + p0 θ(eθ(t−t0) − 1) θ(eθ(t−t0) − 1) ) . (3.57) now consider the partial cases: 1. assume that θ = θ and θ 6= 0. then we have the function φ(t) = e δ a t0 ( x0 θ eθ(t−t0) − 1 + p0 ) , which is monotonically decreasing. 2. assume that θ > 0 and θ = 0. then taking to the limit θ → 0 in (3.57), we get φ(t) = e δ a t0 ( x0 θ eθ(t−t0) − 1 + p0 θ(t− t0) eθ(t−t0) − 1 ) , copyright © 2022 assa. adv syst sci appl (2022) 118 a.p. chernyaev which is also a monotonically decreasing function. 3. assume that θ = θ = 0. then taking to the limit θ, θ → 0 in (3.57), we obtain the function φ(t) = e δ a t0 ( x0 t− t0 + p0 ) . this is a monotonically decreasing function as well. 4. conclusion the validity of the basic economic identity in all the models considered in the paper allows to determine general laws. the introduced new variable – the relative increase of the income – makes it possible to establish a correspondence between the harrod–domar models with variable incremental capital intensity and the solow model with variable rate of accumulation. since households are the best savers and they have the best survival rates, the savings regimes in the harrod–domar model and the solow model are examined. finally, we analyze a modification of the above macromodels replacing the given production functions by setting the problem of maximizing the integral discounted utility of the consumption. in this case, the utility function satisfies the arrow–pratt condition of the relative risk aversion. references 1. chernyaev a.p. (2020) comparison of two dynamic models of economic growth, adv. syst. sci. appl., 20 2, 71–81. 2. chernyaev a.p., meerson a.yu., sukhorukova i.v., & fomin g.p. (2020) features of mathematical formulation and solution of the problem of optimal division of funds in the construction business, iop conf. ser.: materials science & engineering, 012002. 3. harrod, r.f. (1939) an essay in dynamic theory, economic jornal, 49, 14–33. 4. domar, e. (1946) capital expansion, rate of growth and employment, econometrica, 14 (2), 137–147. 5. hamburg, d. (1981) early growth theory of the domar and harrod. moscow: progress. 6. meerson, a.y. & chernyaev, a.p. (2011) exact solution of the macroeconomic harrod– domar model with exogenous dynamics of the volume of consumption of arbitrary character, russian economic university bulletin, 1, 142–147. 7. meerson, a.y. & chernyaev, a.p. (2013) the exact solution of the cauchy problem for the differential equation of the harrod–domar macroeconomic model with variable coefficient of capital intensity of the income growth, the journal of mgup, 3, 252–255. 8. meerson, a.y. & chernyaev, a.p. (2014) the variational problem of optimization of consumption of the harrod–domar model of economic dynamics with variable coefficient of capital intensity of the income growth, proc. free economic society of russia, 186, 502–506. 9. arutyunov, a., kotyukov, a., pavlova, n. (2021) equilibrium in market models with known elasticities. advances in systems science and applications, 21(4), 130–144. 10. pavlova, n., zhukovskaya, z., zhukovskiy, s. (2020) equilibrium in continuous dynamic market models. proceedings of 2020 15th international conference on stability and oscillations of nonlinear control systems (pyatnitskiy’s conference), stab 2020, 9140586. 11. pavlova, n.g. (2020) applications of the theory of covering maps to the study of dynamic models of economic processes with continuous time. springer proceedings in mathematics and statistics, 318, 123–129. copyright © 2022 assa. adv syst sci appl (2022) the optimal control approach in problems of the income distribution 119 12. solow, r.m. (1956) contribution to the theory of economic growth, the quarterly journal of economics, 70 (1), 65–94. 13. solow, r.m. (1957) technical change and the aggregate production function, the review of economics and statistics, 39 (3), 312–320. 14. romer, d. (2014) the higher macroeconomics. higher school of economics, 2014. 15. samarov, k.l. & samarova, s.s. (2014) robert solow’ s model of economic growth in the course of differential equations, information technology bulletin, 2, 81–84. 16. chernyaev a.p., meerson a.yu., masenko a., & kucherenko d. (2020) setting and solution of the problem of optimum separation of material resources for the consumed and accumulated parts in microeconomics, j. physics: conference series, 32076. 17. guriev, s.m., & pospelov, i.g. (1994) model of general equilibrium of economy of transition period, mathematical modeling., 6 (2), 3–21. 18. zamkov, o.o., tolstopyatenko, a.v., & cheremnykh, yu.n. (1998) mathematical methods in economy. moscow: msu, dis publishing house. 19. malychin, v. (2001) mathematics in economics: tutorial. moscow: infra-m. 20. kolemayev, v.a. (2002) mathematical economics: textbook for higher education institutions. moscow: unity-dana. 21. dikusar, v.v., meerson, a.y., & chernyaev, a.p. (2004) optimal resource allocation problems using households as an example. moscow: dorodnitsyn comp. cent. ras. copyright © 2022 assa. adv syst sci appl (2022) the preliminary results the main economic identity the dependence of the income on the consumption in the harrod–domar model with varying capital intensity of the income growth the relative increase of the income for the cobb–douglas production function the balance equation for capital capital capacity of increase in the solow model the balance equation for accumulated household savings the utility function and the risk avoidance the optimal control approach optimal control in the modified harrod–domar model optimal control in the modified solow model optimum consumption management in household model restrictions on the accumulated savings in household model examples of indicator functions conclusion microsoft word 1258-source texts-5235-1-11-20220928.doc adv syst sci appl 2022; 03; 96-105 published online at https://ijassa.ipu.ru. determination of the rollover limitation of a vehicle when moving by 4-dimensional plots tuan a. nguyen automotive engineering department, thuyloi university, 175 tay son, dong da, hanoi, vietnam e-mail: anhngtu@tlu.edu.vn abstract: a vehicle rollover can occur when the driver steers at high speed. the vehicle is rolled over when the wheels are entirely separated from the road surface, i.e., the vertical force at the wheel will approach zero. this article aims to determine the rollover limitation of the vehicle under different conditions. the model of a complex dynamic is used to simulate vehicle motion. this model is a combination of three more petite models. it considers the impact of additional nonlinear variables. therefore, it is considered the first novelty of the article. the calculations and simulations are performed in the matlab environment with specific conditions. simulation results are given in the form of 4-dimensional plots; this is the second novel point of the article. this graph can fully show the dependence of the rollover limitation on other parameters. according to this result, if the vehicle's speed or height is increased, the vehicle's limited roll angle also increases. this reduces the vertical at the wheel. once this force value reaches zero, the vehicle's roll angle will reach its maximum. the vehicle has the ability to roll over at any time. in addition, the magnitude of the steering angle also increases the risk of rollover. the article's results can be used to establish a rollover function, which describes the relationship between the rollover limitation and other factors. this content will be further developed in the upcoming studies. keywords: rollover; simulation; 4-dimensional plot; vehicle dynamics. 1. introduction the automotive industry has been developing comprehensively for more than 130 years. today, vehicles are much larger in size and travel speed than those of the past. therefore, the vehicle's stability and safety are important issues that need full attention. the vehicle's instability when moving on the road can be described by several phenomena, of which the phenomenon of vehicle rollover is the most dangerous. rollover accidents often have significant consequences for passengers and cargo on board [8]. the rollover phenomenon occurs when the wheels are entirely lifted off the road, i.e., the vertical force at the wheels will approach zero [2,20,22]. it is different from the phenomenon that the wheel is separated from the road surface when the vehicle body vibrates (this problem is often studied in the control problems of the suspension system). as for the problem of rolling over, the wheel cannot return to its original state after being lifted off the road. the vehicle is rolled over due to the excessive roll angle of the vehicle's body. the roll angle will reach its maximum limit once the vertical force at the wheel is zero [1]. the rollover phenomenon usually occurs when the vehicle steers to avoid an obstacle or enters a roundabout [4]. then, centrifugal force can appear. this force is referred to as the center of gravity of the vehicle. therefore, it will cause a moment that causes the vehicle's body to tilt [2]. there are four common causes of rollovers. firstly, external factors such as lateral wind force can affect the vehicle when traveling in bad weather conditions [9,26]. however, the value of the lateral wind force is usually not significant. the cases of rolling over due to the determination of the rollover limitation of a vehicle… 97 copyright ©2022 assa. adv. in systems science and appl. (2022) lateral wind force are not many. the second cause of this phenomenon is the size of the vehicle. the larger the vehicle's height and the smaller the wheelbase, the more the vehicle's roll moment will increase. this was pointed out by nguyen et al. [15]. it is not easy to adjust the size of the vehicle because it will affect the performance of the vehicle. the third cause of this phenomenon is the speed of the vehicle. this is considered to be the leading cause of the transverse rollover phenomenon. if the velocity is more significant than remarkable, the centrifugal force will increase proportionally to the square of the velocity. therefore, the risk of a rollover will be higher. finally, the value of the steering angle also dramatically affects the problem of vehicle instability. if the steering angle or steering acceleration is greater, the rollover limitation will be reduced rapidly, making the vehicle more prone to rolling over. in [12], li and bei introduced the concept of a static roll index. according to xin et al., this index depends on the size of the vehicle [25]. however, the static roll index was not significant in the study. therefore, the dynamic roll index replaces the static roll index. the dynamic roll index, also known as ltr (load transfer ratio), is described in [24]. according to shin et al., this index is a function of the difference in vertical force between the wheels. the vehicle rolls over once the value of this indicator reaches 1 [10]. this index is used in many rollover studies [5,28]. when studying roll dynamics, there are several commonly used dynamical models. in [27], zhang et al. combined a half-dynamics model and a linear single-track dynamics model. this is a fairly common model. however, it is relatively simple. however, the accuracy is usually not high. with this model, the tire is treated as a linear deformation [6]. a spatial dynamics model should be used to comprehensively evaluate the vehicle's motion. this model takes into account the influence of all wheels [29]. it can be used for commercial vehicles or special vehicles [13]. the spatial dynamics model must be integrated with nonlinear double-track dynamics. nguyen in [19] did this combination. however, establishing this model is quite complicated. several solutions have been used to improve the vehicle's stability when traveling at high speed. according to nguyen, the stabilizer bar can reduce the roll angle of the vehicle body when steering [21]. in addition, the active suspension system also improves vehicle stability [11,16,18,23]. besides, redistributing power to the drive wheels can also reduce the rollover phenomenon [7]. the rollover phenomenon can be correctly predicted if the rollover limitation is predefined. this article focuses on investigating the vehicle's rollover limitation when steering. simulation methods are used to collect the data. a model of complex dynamics is used to simulate vehicle oscillations. this model combines a spatial dynamics model, a nonlinear double-track dynamics model, and a nonlinear tire model. this is the first new point in the article. the article's results are presented in the form of a 4-dimensional plot. it describes the dependence of the rollover limitation on the vehicle's velocity and the height of the center of gravity. this is considered the second novelty of the article. the layout of the article consists of four sections: introduction, material, results, and conclusions. the specific contents are presented in the sections below. 2. method a dynamics model describes a vehicle's oscillations. this article uses a complex dynamic model, a combination model of spatial dynamics (7 dofs), and a nonlinear double-track dynamics model (3 dofs). considering the model of the spatial dynamics in figure 1, the system of equations describing the vehicle's oscillation is given as follows: 98 tuan a. nguyen copyright ©0000 assa adv. in systems science and appl. (0000) fig. 1. model of the spatial dynamics.         2 1 1 1 1 , 1 1 1 1 1 j i j i s s ij s uij wi i ij s uij wi i i j m z k z z t a c z z t a                              (1)                1 1 2 12 1 1 , 1 1 1 1 1 1                                                         j i ij s uij wi ij s wij i i j ij s uij wi i y x s k z z t a j m h t c z z t a gsin v v cos m h (2)             1 1 2 12 1 1 , 1 1 1 1 1 1                                        j i ij s uij wi ii s ij i i j ij s uij wi i k z z t a j m h a c z z t a (3)           1 1 1 1 1 1 1 1                               j i uij uij tij rij uij ij s uij wi i j i ij s uij wi i m z k z z k z z t a c z z t a (4) the vx, vy, and  are determined by the nonlinear double-track dynamics model (figure 2) [14,17]. this model has 3 degrees of freedom, so it is necessary to establish three equations corresponding to the three directions of motion of the vehicle.     2 2 2 1 , 1 , 1 , 1 s uij x xij ij yij ij s uij y i j i j i j m m v f cos f sin f m m v                              (5) determination of the rollover limitation of a vehicle… 99 copyright ©2022 assa. adv. in systems science and appl. (2022)     2 2 2 2 , 1 , 1 , 1 s uij y xij ij yij ij s uij x i j i j i j m m v f sin f cos f m m v                              (6)         2 1 w , 1 1 1 j i xij ij yij ij i xij ij yij ij i i i zij i j j f cos f sin t f sin f cos a fc m                (7) fig. 2. model of the nonlinear double-track dynamics. the values of longitudinal force fx, lateral force fy, and aligning moment mz at the wheel are determined by the tire model. in this study, the pacejka tire model with experimental parameters was used.  x x x x xf d sin c artan b     (8)  y y y y y vyf d sin c artan b s    (9)  z z z z z vzm d sin c artan b s    (10) after the dynamics model has been established, the simulation process will be carried out. 3. results 3.1. simulation conditions three cases are used in the simulation, which is done in the matlab-simulink system (figure 3). the steering angle value will be modified in these instances. table 1 lists the vehicle's specifications. the speed, steering angle, and distance from cg (center of the gravity) to ra (roll axis) are all inputs to the simulation problem. the value of ra can be 100 tuan a. nguyen copyright ©0000 assa adv. in systems science and appl. (0000) referenced from the carsim software. using this range of values helps to describe better the roll angle's dependence on the height of the center of gravity. the maximum roll angle and the minimum vertical force at the wheel are among the problem's outputs. fig. 3. steering angle. table 1. details about the simulation parameters. symbol description value unit ms sprung mass 1740 kg muij unsprung mass 38 kg twi half of the track width front/rear axle 730/725 mm ai the distance between the center of gravity and the front or rear axle 1230/1620 mm j moment of inertia (longitudinal-axis) 725 kgm2 j moment of inertia (lateral-axis) 2750 kgm2 j moment of inertia (vertical-axis) 2730 kgm2 3.2. simulation results figures 4, 5, and 6 depict the simulation's outcomes. the change in maximum roll angle and the lowest vertical force at the wheel is shown in figure 4. according to this conclusion, the maximum roll angle rises as the distance between the cg and the ra increases. as a result, the vertical force at the wheel's minimal value is lowered. the maximum value of the roll angle increases as the vehicle's speed increases. the vertical force at the wheel is likewise reduced due to this. the vehicle may roll over if the vertical force at the wheel is lowered to zero. therefore, the maximum roll angle of the vehicle will tend to decrease gradually, although the speed and distance from cg to ra will still increase. determination of the rollover limitation of a vehicle… 101 copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 4. maximum roll angle (case 1). as the steering angle increases, the change in these values is also more considerable. the maximum roll angle of the vehicle will increase at the same speed and distance from cg to ra. at the same time, the minimum force at the wheel will also be smaller. this is shown in figure 5. because of the larger steering angle, the vehicle is at greater risk of rolling over sooner. if the velocity is greater or the height of the center of gravity is greater, the limited roll angle of the vehicle will decrease. fig. 5. maximum roll angle (case 2). 102 tuan a. nguyen copyright ©0000 assa adv. in systems science and appl. (0000) fig. 6. maximum roll angle (case 3). in the third case, this change is more pronounced (figure 6). as a vehicle's speed rises, so does the chance of it rolling over. besides, the distance from the object's gravitational center to the roll axis also increases the risk of rolling over. however, the effect of velocity is still more significant than size. the slope of the 4-dimensional graph demonstrates this. 4. conclusion the stability of the vehicle is a critical issue. it may affect the safety of passengers and cargo when the vehicle is on the road. there are many causes of instability, of which the rollover phenomenon is considered the most important. this phenomenon occurs when the wheel is lifted off the road, i.e., the vertical force at the wheel will approach zero. the consequences of rollover accidents are often very severe. this article studies the vehicle's rollover limitation when traveling on the road. a model of complex dynamics is used to simulate vehicle oscillations. this is a nonlinear model. it includes the influence of many factors. therefore, its accuracy will be higher than that of other conventional models. simulation results are shown through 4-dimensional graphs. the parameters of the graph that show the relationship include velocity, distance from the cg to ra, maximum roll angle, and minimum vertical force. according to this result, if the speed or distance is increased, the maximum roll angle of the vehicle will also increase. this causes the minimum vertical force at the wheel to decrease. once the value of the wheel's vertical force is reduced to zero, the risk of the vehicle rollover will appear, i.e., the maximum roll angle is reached. as the velocity or distance rises, the value of the restricted roll angle decreases. this indicates that the vehicle will be at risk of rollover sooner. if the value of the steering angle is increased, the change in the roll angle and the vertical force at the wheel will also be much larger. this is a warning signal for possible danger. the results obtained from this study are the basis for determining the rollover limitation of the vehicle. it is necessary to provide a function representing the relationship between these parameters. it can help predict the vehicle's rollover limitation under different determination of the rollover limitation of a vehicle… 103 copyright ©2022 assa. adv. in systems science and appl. (2022) conditions. however, this is not very easy. in the subsequent studies, this content can be conducted. nomenclature  : roll angle, rad  : yaw angle, rad  : pitch angle, rad  : steering angle, rad  : heading angle, rad fc : damping force, n fk : spring force, n fkt : tire force, n fx : longitudinal force, n fy : lateral force, n fz : vertical force, n mz : aligning moment, nm vx : longitudinal velocity, m/s vy : lateral velocity, m/s zs : sprung mass displacement, m zu : unsprung mass displacement, m references 1. anh, n. t. (2020). predict the rollover phenomenon of the vehicle when steering, international journal of mechanical & mechatronics engineering, 20(5), 31-40. 2. anh, n. t., binh, h. t. (2019). research on dynamic vehicle model equipped active stabilizer bar, advances in science, technology and engineering systems journal, 4(4), 271-275. https://doi.org/10.25046/aj040434 3. anh, n. t., binh, h. t. 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(2014). vehicle velocity and roll angle estimation with road and friction adaptation for four-wheel independent drive electric vehicle, mathematical problems in engineering. https://doi.org/10.1155/2014/801628 adv syst sci appl 2022; 01:176–191 published online at https://ijassa.ipu.ru. on order covering set-valued mappings and their applications to the investigation of implicit differential inclusions and dynamic models of economic processes evgenii o. burlakov1,2, elena a. panasenko3,4, irina d. serova1, evgeny s. zhukovskiy2,3* 1university of tyumen, tyumen, russia 2v.a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia 3derzhavin tambov state university, tambov, russia 4leonhard euler international mathematical institute, st. petersburg, russia abstract: the present work is devoted to investigation of operator inclusions in partially ordered spaces and application of the obtained results to differential inclusions. we consider the inclusion υ(x, x) 3 y with respect to the unknown x ∈ x, where υ : x ×x ⇒ y is a setvalued mapping, x and y are partially ordered spaces. it is assumed that the mapping υ is order covering with respect to the first argument and antitone with respect to the second argument. we prove that for any x0 ∈ x, if the set g(x0) contains an element y0 such that y � y0, then there exists a solution to the inclusion under consideration, which satisfies the inequality x � x0. this statement is applied to investigation of a cauchi problem for the differential inclusion f(t, x, ẋ, ẋ) 3 0 with a bound for the derivative of the unknown function ẋ(t) ∈ b(t) (here f : [a, b]× rn × rn × rn ⇒ rm, b : [a, b] ⇒ rn). we obtain conditions of solvability in the space of absolutely continuous functions, conditions of existence of a solution with the least derivative, and derive the solutions estimates. the latter results are applied to the analysis of the dynamic walrasian-evans-samuelson model of economic processes, which can be reduced to a system of implicit differential inclusions. we establish the existence of the equilibrium and obtain estimates of the equilibrium prices. keywords: operator inclusion, covering mapping of partially ordered spaces, implicit differential equation, supply-and-demand model 1. introduction the present work is devoted to the problem of solvability of operator inclusions in partially ordered spaces. the research uses the notion of order covering introduced for the “standard single-valued” mappings in [1, 2], and for set-valued mappings in [3, 4]. for a set-valued mapping g acting from a partially ordered space (x,�) to a partially ordered space (y,�), the property of covering means that if for any u ∈ x and y ∈ y , the set g(u) contains an element v such that y � v, then there exists a solution x of the inclusion g(x) 3 y, which satisfies the inequality x � u. in the cited papers, the existence of a coincidence point of a covering mapping and an isotone mapping (for the cases of singleand set-valued mappings) is demonstrated. in the present work, the stability of the solvability property of inclusions with respect to antitone perturbations is investigated. this problem is formalised in terms of the inclusion ∗corresponding author: zukovskys@mail.ru on order covering set-valued mappings and their applications 177 υ(x, x) 3 y with a set-valued mapping υ : x ×x ⇒ y that is order covering with respect to the first argument and antitone with respect to the second argument. an analogous problem for “standard single-valued” mappings was considered in the papers [5–8]. based on the statements on antitone perturbations proved in these works, comparison theorems (of the same type with the well-known chaplygin theorem [9] on differential inequality) for the unresolved with respect to the derivative (i.e. implicit) differential equations were obtained (see [5, 10]). the results on antitone perturbations of order covering mappings presented in this article are applied to the investigation of implicit differential inclusions. conditions of existence and estimates of solutions of a cauchi problem are obtained. these results are applied to the analysis of the dynamic walrasian-evans-samuelson supply-and-demand model (see [11–13]) that is reduced to a system of implicit differential inclusions. the existence of the equilibrium state is established and the estimates of the equilibrium prices are obtained. these statements extend the results on the models of economic processes of a.v. arutyunov, n.g. pavlova, s.e. zhukovskiy, a.a. shananin (see [14–17]) which employ single-valued functions of supply and demand. the fundamental mathematical tools for these works were represented by the theorems of the papers [18–21] on coincidence points of single-valued covering mapping of metric spaces. 2. a comparison theorem for operator inclusions in partially ordered spaces let partially ordered spaces (x,�), (y,�) be given. for any u, v ∈ x andu ⊂ x , we denote [u, v]x . = {x ∈ x : v � x � u}, ox(u) . = {x ∈ x : x � u}, ox(u) . = ⋃ ∀u∈u ox(u). consider a set-valued mappingg : x ⇒ y, i.e. a mapping that puts to any element x ∈ x into the correspondence a non-empty set g(x) ⊂ y. if for any x ∈ x , the set g(x) contains only one element, then the mapping g becomes a “standard single-valued” mapping (we conventionally denote such a mapping as g : x → y ). thus, set-valued mappings naturally generalise “standard” mappings. let us remind definitions of some properties of set-valued mappings used in the present work. definition 2.1: a mappingg : x ⇒ y is called antitone (isotone) on the set u ⊂ x, if for any x, u ∈ u such that x � u and for z ∈ g(u), there exists y ∈ g(x) satisfying the inequality y � z (y � z). if a mapping is antitone (isotone) on the whole space x , it is called antitone (isotone). the given definition of the antitone property for a mapping g : x → y means that ∀x, u ∈ u x � u ⇒ g(x) � g(u), matches the definition of the antitone property for single-valued mappings. the given isotone property in the case of single-valued mappings also matches the “classical” isotone property. definition 2.2: a mapping g : x ⇒ y order covers the set v ⊂ y if ∀u ∈ x oy ( g(u) ) ∩ v ⊂ g ( ox(u) ) . (2.1) the definition of order covering for set-valued mappings was introduced in [3, 4] for the case v = y. note that (2.1) is equivalent to the relation( ∀u ∈ x ∀v ∈ g(u) ∀y ∈ v y � v ) ⇒ ( ∃x ∈ x y ∈ g(x) and x � u ) . copyright © 2022 assa. adv syst sci appl (2022) 178 e.s. zhukovskiy, i.d. serova, e.a. panasenko, e.o. burlakov moreover, if v is a one-element set, i.e. v = {ŷ}, then the relation (2.1) is equivalent to the implication ∀u ∈ x ŷ ∈ oy ( g(u) ) ⇒ ŷ ∈ g ( ox(u) ) . thus, the property of order covering of the set {ŷ} means that for any u ∈ x such that the set g(u) contains an element v � ŷ, the inclusion ŷ ∈ g(x) (2.2) has a solution x ∈ x satisfying the inequality x � u. in the same manner, a single-valued mapping g : x → y order covers the set {ŷ} if ∀u ∈ x g(u) � ŷ ⇒ ∃x ∈ ox(u) g(x) = ŷ. we consider the problem of stability of the solvability property of the inclusion (2.2) under antitone perturbations of the order covering mapping. let a “perturbed” mapping υ : x ×x ⇒ y that order covers the set {ŷ} with respect to the first argument and antitone with respect to the second argument be given. define the mapping f : x ⇒ y by the relation ∀x ∈ x f (x) = υ(x, x) and consider the inclusion ŷ ∈ f (x) (2.3) (with respect to x ∈ x). let u ⊂ x. in order to formulate a theorem on existence of solutions to (2.3), we define the set s(υ, u, ŷ) of all chains s ⊂ u such that the following relations take place: ∀x ∈ s ∃y ∈ υ(x, x) y � ŷ, ∀x, u ∈ s x ≺ u ⇒ ∃ξ ∈ [x, u] ŷ ∈ υ(ξ, u). (2.4) theorem 2.1: let there exist u0 ∈ x and y0 ∈ y such that y0 ∈ υ(u0, u0), y0 � ŷ, (2.5) and the follwing conditions are satisfied: (a1) for any x ∈ ox(u0), the mapping υ(·, x) : x ⇒ y order covers the set {ŷ}; (a2) for any x ∈ ox(u0), the mapping υ(x, ·) : x ⇒ y is antitone on the set [x, u0]x ; (a3) any infinite chain s ∈ s ( υ,ox(u0), ŷ ) is bounded from below, and for some its lower bound ω ∈ x , one can find z ∈ υ(ω, ω) satisfying the inequality z � ŷ. then the inclusion (2.3) has a solution and the set of solutions possesses a minimal element belonging to the set ox(u0). proof we introduce the set u0 = {x ∈ ox(u0) : ∃y ∈ f (x) y � ŷ}. note that u0 6= ∅ as u0 ∈ u0. we define binary relations � and � on u0 as follows: ∀v, u ∈ u0 v � u ⇔ ( v ≺ u and ∃ξ ∈ [v, u] ŷ ∈ υ(ξ, u) ) , ∀v, u ∈ u0 v � u ⇔ ( v = u or v � u ) . copyright © 2022 assa. adv syst sci appl (2022) on order covering set-valued mappings and their applications 179 let us demonstrate that � and � are strict and non-strict order relations, respectively. as the relation ≺ is antisymmetric, the relation � is antisymmetric as well. prove that � is transitive. for v, w, u ∈ u0 such that v � w � u, it holds true that v ≺ w ≺ u and ∃ξ ∈ [w, u] ⊂ [v, u] ŷ ∈ υ(ξ, u). (2.6) thus, we have v � u,, which means that the relation � is transitive. obviously, the relation � is also antisymmetric and transitive, and, moreover, reflexive. we point out that the relation (2.6) also holds true for v, w, u ∈ u0 such that v ≺ w � u,, i.e. the relation v � u takes place in this case as well. let us consider the partially ordered set (u0,�). according to the hausdorff maximal principle (see e.g. [22, chapter 1]), this set possesses a maximal chain s that contains u0. we first prove the theorem statement in the case when the chain s contains the least element ω, i.e. ω � x for any x ∈ s. if ω is not a solution of inclusion (2.3), then for some z ∈ υ(ω, ω), it holds true that z � ŷ and, hence, by the virtue of the condition (a1), there exists an element υ ∈ x such that ŷ ∈ υ(υ, ω) and υ ≺ ω. according to (a2), there exists y ∈ υ(υ, υ) such that y � ŷ. so, υ ∈ u0 and υ � ω. we obtained that υ � x for any x ∈ s which contradicts with the maximality of the chain s. thus, ω is a solution to (2.3). consider now the case when the chain s does not possess the least element, i.e. for any x ∈ s, there exists u ∈ s, u� x. obviously, this chain is infinite. note that s is a chain with respect to the initial order � in x and belongs to s ( υ,ox(u0), ŷ ) . the assumption (a3) implies that this chain has a lower bound ω (with respect to the initial order �), for which there exists z ∈ υ(ω, ω) such that z � ŷ, i.e. ω ∈ u0 and ω /∈ s. for any x ∈ s, there is u ∈ s, u� x. as ω ≺ u, we get ω � x. the latter contradicts with the maximality of the chain s, so the situation where the maximal chain s does not possess the least element is not possible. let us demonstrate that the obtained element ω ∈ ox(u0) is minimal in the set of solutions of the inclusion (2.3). assume the contrary. then one can find z ≺ ω such that ŷ ∈ υ(z, z). hence, z ∈ u0 and z � ω. however, this inequality is not possible by the virtue of the maximality of the chain s in the space (u0,�). the theorem 2.1 implies the theorem on perturbations of an order covering single-valued mapping υ : x ×x → y. the corresponding properties of order covering and monotonicity of a single-valued mapping are defined above, and the relations (2.4) can be written as: ∀x ∈ s υ(x, x) � ŷ, ∀x, u ∈ s x ≺ u ⇒ ∃ξ ∈ [x, u] ŷ = υ(ξ, u). (2.7) the statement on perturbations of order covering single-valued mappings is analogous to the results obtained in the papers [5, 6]. these works employed a less restrictive assumptions on the chains s compared to (2.7), namely: ∀x, u ∈ s x ≺ u ⇒ ŷ � υ(ξ, u), so the condition corresponding to (a3) was more strict. 3. conditions of order covering for the nemytskii operator in the application of theorem 2.1 to concrete functional inclusions, the most of the difficulties connected with the verification of the order covering property of the corresponding set-valued operators, primarily the superposition operator that is also called the nemytskii operator. here we demonstrate that the nemytskii operator is order covering provided that the corresponding set-valued function possesses this property with respect to the corresponding argument. copyright © 2022 assa. adv syst sci appl (2022) 180 e.s. zhukovskiy, i.d. serova, e.a. panasenko, e.o. burlakov typically, the verification of the order covering property for functions does not involve significant difficulties. below we assume that the space rn is equipped with the natural order, i.e for x = (x1, ..., xn) ∈ rn and u = (u1, ..., un) ∈ rn, we have x ≤ u if xi ≤ ui for all i = 1, n. we denote by c(rn) and k(rn) the sets of all non-empty closed and all non-empty compact subsets of the space rn, respectively. we denote by cc(rn) and kc(rn) the sets of all non-empty connected closed and all non-empty connected compact subsets of the space rn, respectively. let a set-valued mapping b : [a, b] ⇒ rn having closed images b(t) ⊂ rn for any t ∈ [a, b] be given. we denote such a mapping as b : [a, b]→ c(rn). let this mapping be measurable. we define w (b) to be the set of all lebesgue measurable functions x : [a, b]→ rn such that x(t) ∈ b(t) for almost all t ∈ [a, b]. thus, the set w (b) is the set of all measurable selections of the set-valued mapping b. we endow the set w (b) with an order by assuming that for any two its elements x, u, the inequality x ≤ u holds true if x(t) ≤ u(t) for almost all t ∈ [a, b]. in the case b = rn, this set of all measurable functions x ∈ [a, b]→ rn we denote by w n. consider a set-valued mapping g : [a, b]× rn → c(rm). assume that it satisfies the caratheodori conditions (i.e. it is measurable with respect to the first argument and continuous in the hausdorff metric in the second argument). these assumptions allow to define the setvalued nemytskii operator ng : w n ⇒ wm, ∀x ∈ w n ng x = { y ∈ wm : y(t) ∈ g(t, x(t)) for almost all t ∈ [a, b] } . we denote ∆ = { (t, x) : t ∈ [a, b], x ∈ b(t) } and define g∆ : ∆→ c(rm) to be the restriction of the set-valued mapping g to the set ∆, and by ng∆ : w (b) ⇒ wm — the restriction of the operator ng to w (b). the following statement establishes connection between the property of order covering of the set-valued mapping g∆ with respect to the second argument and the order covering property of the corresponding nemytskii operator ng∆ . let a measurable function ŷ : [a, b]→ rm be given. theorem 3.1: if at almost all t ∈ [a, b], a set-valued mapping g∆(t, ·) : b(t)→ c(rm) order covers the set {ŷ(t)} ⊂ rm, then the operator ng∆ : w (b) ⇒ wm order covers the set {ŷ} ⊂ wm. proof let the mapping g∆(t, ·) order cover the set {ŷ(t)} ⊂ rm. assume that for some measurable function u ∈ w (b), there exists y ∈ ng∆ u such that y ≥ ŷ. thus, y(t) ∈ g∆ ( t, u(t) ) and y(t) ≥ ŷ(t) for almost all t ∈ [a, b]. we define the set-valued mappings o,u : [a, b] ⇒ rn by the relations: o(t) . = orn(u(t)), u(t) . = o(t) ⋂ b(t) = { x ∈ b(t) : x ≤ u(t) } , t ∈ [a, b]. for almost all t ∈ [a, b], the closedness of the sets orn(u(t)) and b(t) in rnimplies the closedness of the set u(t), i.e. u : [a, b]→ c(rn). the measurability of o,b : [a, b]→ c(rn) guarantees the measurability of the mapping u (see [23, § 1.5.1, § 1.5.8]). as g∆(t, ·) order covers the set {ŷ(t)} ⊂ rm, for almost all t ∈ [a, b], the inclusion ŷ(t) ∈ g∆ ( t, u(t) ) takes place for almost all t ∈ [a, b]. therefore, according to filippov’s lemma on implicit function (see [23, § 1.5.15]), there exists x̂ ∈ w n such that x̂(t) ∈ u(t) and ŷ(t) ∈ g∆ ( t, x̂(t) ) for almost all t ∈ [a, b]. thus, the relations ŷ ∈ ng∆ x̂ and x̂ ≤ u are fulfilled, i.e. the operator ng∆ order covers the set {ŷ} ⊂ wm. example 2.1: let ordinary single-valued” functions q0, q1 : [a, b]→ r be given. consider the function copyright © 2022 assa. adv syst sci appl (2022) on order covering set-valued mappings and their applications 181 g : [a, b]× r→ r defined by the formula g(t, x) . = q0(t) + 2q1(t)x− x2, t ∈ [a, b], x ∈ r. (3.8) obviously, g satisfies the caratheodori conditions. let r > 0, b(t) . = [q1(t)− r, q1(t) + r] and ∆ = { (t, x) : t ∈ [a, b], |x− q1(t)| ≤ r } , respectively. the values of g∆(t, ·) : b(t)→ r constitute the segment [ q0(t) + q2 1(t)− r2, q0(t) + q2 1(t) ] . if for the measurable function ŷ : [a, b]→ r, it holds true that ŷ(t) ≥ q0(t) + q2 1(t)− r2, (3.9) then g∆(t, ·) : b(t)→ r order covers the set {ŷ(t)} ⊂ r at almost all t ∈ [a, b] and, hence, by the virtue of theorem 3.1, the nemytskii operator ng∆ : w (b) ⇒ w order covers the set {ŷ} ⊂ w. in the particular case whenb(t) ≡ r, the operatorng : w ⇒ w is order covering (i.e. order covers the whole space w ). now we pick δ > 0 and define two set-valued mappings g+, g− : [a, b]× r→ k(r) by the formulas: g+(t, x) . = [ g(t, x), g(t, x)+δ ] , g−(t, x) . = [ g(t, x)−δ, g(t, x) ] , t ∈ [a, b], x ∈ r. (3.10) the images of the set-valued functions g+ ∆(t, ·) : b(t)→ k(r) and g−∆(t, ·) : b(t)→ k(r) are the sets [ q0(t) + q2 1(t)− r2, q0(t) + q2 1(t) + δ ] , and [ q0(t) + q2 1(t)− r2 − δ, q0(t) + q2 1(t) ] , respectively. using theorem 3.1 one can easily demonstrate that if the measurable function ŷ : [a, b]→ r satisfies the inequality (3.9) for almost all t ∈ [a, b], then the operator ng+ ∆ : w (b) ⇒ c(w ) order covers the set {ŷ(t)} ⊂ r. theorem 3.1 also implies that the operator ng− ∆ : w (b) ⇒ c(w ) order covers the set {ŷ(t)} ⊂ r provided that the inequality ŷ(t) ≥ q0(t) + q2 1(t)− r2 − δ, is valid almost everywhere on [a, b], 4. implicit differential inclusion in this section, based on theorems 2.1 and 3.1 we investigate differential inclusions not resolved with respect the derivative (which are also called implicit differential inclusions). let a measurable set-valued mapping b : [a, b]→ c(rn) be given. denote by l(b) the space of all integrable selections of the set-valued mapping b and denote by ac(b) the space of all absolutely continuous functions x : [a, b]→ rn such that ẋ ∈ l(b). in the case b(t) ≡ rn, we denote these spaces these spaces by ln and acn. we will require the fulfillment of the following analogue of the one-sided continuity property known for “ordinary single-valued” functions applicable to a set-valued mapping g : r→ k(rm), which was defined in [24]. such mapping is called right continuous at the point x0 ∈ r if for any ε > 0, there exists δ > 0 such that for all x ∈ (x0, x0 + δ), it holds true that hrm ( g(x0), g(x) ) < ε (hereinafter the symbol hrm denotes the hausdorff distance between the sets in the space rm). let a set-valued function f : [a, b]× rn × rn × rn → k(rm) and a vector γ ∈ rn be given. we assume that for any x, v, u ∈ rn, the function f(·, x, v, u) : [a, b]→ k(rm) is measurable; for almost all t ∈ [a, b] and all v, u ∈ rn, the function f(t, ·, v, u) : rn → k(rm) is right continuous in each of the arguments x1, . . . , xn; for almost all t ∈ [a, b] and any x, u ∈ rn, the function f(t, x, ·, u) : rn → k(rm) is right continuous in each of the copyright © 2022 assa. adv syst sci appl (2022) 182 e.s. zhukovskiy, i.d. serova, e.a. panasenko, e.o. burlakov arguments v1, . . . , vn; for almost all t ∈ [a, b] and any x, v ∈ rn, the function f(t, x, v, ·) : rn → k(rm) is continuous. we consider the differential inclusion f(t, x, ẋ, ẋ) 3 0, t ∈ [a, b], (4.11) with the following additional restriction on the derivative of the unknown function: ẋ(t) ∈ b(t), t ∈ [a, b]. (4.12) a solution of the system of inclusions (4.11), (4.12) is understood to be a function x ∈ ac(b) that satisfies the inclusion (4.11) for almost all t ∈ [a, b]. let us formulate a statement on solvability and estimates of solutions to the cauchi problem for the system (4.11), (4.12) with the initial condition x(a) = γ. (4.13) denote ω = { (t, x, v, u) : t ∈ [a, b], x ∈ rn, v ∈ b(t), u ∈ b(t) } and define fω : ω→ c(rm) to be the restriction of the set-valued mapping f to the set ω. theorem 4.1: let a function v0 ∈ ac(b) such that v0(a) ≥ γ and f ( t, v0(t), v̇0(t), v̇0(t) ) ∩ rn + 6= ∅ for almost all t ∈ [a, b] (4.14) be given. let the set of measurable selections of the set-valued mapping b(·) ∩ orn ( v̇0(·) ) : [a, b]→ c(rn) be integrally bounded from below (i.e. there exists a number c such that for any measurable function u ∈ w n, satisfying the relations u(t) ∈ b(t) and u(t) ≤ v̇0(t) for almost all t ∈ [a, b], it holds true that ∫ b a u(t)dt ≥ c) and the following conditions are fulfilled: (b1) for almost all t ∈ [a, b], any x ∈ rn and v ∈ b(t), the mapping fω(t, x, v, ·) : b(t)→ k(rm) order covers the set {0} ⊂ rm; (b2) for almost all t ∈ [a, b] and any u ∈ b(t), the mapping fω(t, ·, ·, u) : rn ×b(t)→ k(rm) is antitone. then there exists a solution x ∈ ac(b) to the problem (4.11), (4.12), (4.13) such that ẋ(t) ≤ v̇0(t) for almost all t ∈ [a, b]. proof the problem (4.11), (4.12), (4.13) with respect to the unknown function x ∈ ac(b) can be rewritten in the form of the following inclusion fω ( t, γ + ∫ t a u(s)ds, u(s), u(s) ) 3 0, t ∈ [a, b], (4.15) with respect to the unknown function u = ẋ ∈ l(b). let us demonstrate that the inclusion (4.15) can be represented in the form of the operator inclusion (2.3), that can be investigated using theorem 2.1. according to [24, theorem 2.1], the assumptions made for the set-valued function f provide its superpositional measurability, hence, for any x ∈ acn, the set-valued function f ( ·, x(·), ẋ(·), ẋ(·) ) : [a, b]→ k(rm) is measurable. superpositional measurability of f allows to define the mapping υ : l(b)× l(b) ⇒ wm, ∀u, v ∈ l(b) υ(u, v) . = fω ( t, γ + ∫ t a v(s)ds, v(t), u(t) ) , t ∈ [a, b], (4.16) copyright © 2022 assa. adv syst sci appl (2022) on order covering set-valued mappings and their applications 183 and the corresponding mapping f : l(b) ⇒ wm, f (u) . = υ(u, u). let us verify the conditions of theorem 2.1 (where we put ŷ = 0 ∈ wm) for these mappings. first of all, there exists a measurable function y ∈ wm satisfying the relations y(t) ∈ f ( t, v0(a) + ∫ t a v̇0(s)ds, v̇0(t), v̇0(t) ) = fω ( t, v0(a) + ∫ t a v̇0(s)ds, v̇0(t), v̇0(t) ) , y(t) ≥ 0 for almost all t ∈ [a, b]. these relations, according to (b2), imply the existence of a measurable function y0 ∈ wm such that y0(t) ∈ fω ( t, γ + ∫ t a v̇0(s)ds, v̇0(t), v̇0(t) ) , y0(t) ≥ y(t) ≥ 0 for almost all t ∈ [a, b]. thus, for the mapping υ defined by (4.16), we have υ(v̇0, v̇0) 3 y0, y0 ≥ 0, i.e. the relations (2.5) where u0 = v̇0 ∈ l(b) are fulfilled. secondly, for any v ∈ l(b), the mapping υ(·, v) : l(b) ⇒ wm is the nemytskii operator generated by the restriction g∆ to the set ∆ = { (t, u) : t ∈ [a, b], u ∈ b(t) } of the function g : [a, b]× rn → k(rm), g(t, u) . = fω ( t, γ + ∫ t a v(s)ds, v(t), u ) for almost all t ∈ [a, b] and any u ∈ rn. the assumption (b1) implies that for almost all t ∈ [a, b], the function g∆(t, ·) : b(t)→ k(rm) order covers the set {0} ∈ rm. according to theorem 3.1, for any v ∈ l(b), the mapping υ(·, v) order covers the set {0} ∈ wm. thus, the condition (a1) of theorem 2.1 is satisfied. next, due to the assumption (b2), for any u ∈ l(b), the mapping υ(u, ·) : l(b) ⇒ wm is antitone, so the condition (a2) of theorem 2.1 is fulfilled as well. in order to verify the assumption (a1) of theorem 2.1, we consider an arbitrary chain s ∈ s ( υ,ol(b)(v̇0), 0 ) and demonstrate that it possesses an infimum u . = inf s and, moreover, there exists a decreasing sequence {un} ⊂ s having the same lower bound inf{un} = inf s = u. for any u ∈ s, we define the number lu . = ∫ a 0 u(s)ds. the set ls . = {lu, u ∈ s} is a chain in r, which is bounded from below due to the integral boundedness of the set of measurable selections of the set-valued mapping b(·) ∩ orn ( v̇0(·) ) : [a, b]→ c(rn) from below. thus, there exists λ . = inf ls, and, hence, exists a decreasing sequence {un} ⊂ s such that limn→∞ lun = λ. in the space l, any decreasing norm bounded sequence has an infimum (see e.g. [25, p. 257]), so there exists u = inf{un}, and, moreover, for almost all t ∈ [a, b], it holds true that u(t) = inf{un(t)} = limn→∞ un(t). therefore, for almost all t ∈ [a, b], due to closedness of the set b(t) ⊂ rn, the inclusion u(t) ∈ b(t) takes place, so u ∈ l(b). let us now demonstrate that inf s = u. for any u ∈ s, it holds true that lu > λ, so for some number n, we get lu > lun, hence u > un > u. thus, u is a lower bound of s, and any function that is greater than u, is not a lower bound of the sequence {un} and moreover, not a lower bound of the chain s. so, u = inf s and u(t) = inf{un(t)} = limn→∞ un(t) for almost all t ∈ [a, b]. it remains now to prove the existence of a non-negative measurable selection of the setvalued mapping υ(u, u) : [a, b]→ k(rm) to verify the condition (a3). by the definition of the chain s, for any natural n, there exists a measurable function yn ∈ wm such that yn ∈ υ(un, un) and yn ≥ 0.as the mapping υ(un, ·) is antitone, there exists a measurable function ζn ∈ wm such that ζn ∈ υ(un, u) and ζn ≥ 0. the mapping υ(·, u) : l(b) ⇒ wm is the restriction to l(b) of the nemytskii operator generated by the function g : [a, b]× rn → k(rm), g(t, u) . = f ( t, γ + ∫ t a u(s)ds, u(t), u ) for almost all t ∈ [a, b] and any u ∈ rn, copyright © 2022 assa. adv syst sci appl (2022) 184 e.s. zhukovskiy, i.d. serova, e.a. panasenko, e.o. burlakov that satisfies the caratheodori conditions. therefore hrm ( g(t, un(t)), g(t, u(t)) ) → 0. (4.17) for almost all t ∈ [a, b], it holds true that g(t, un(t)) ∩ rm + 6= ∅ (as ζn(t) ∈ g(t, un(t))). assume that the set-valued mapping g(·, u(·)) does not possess non-negative selections. then for some set t ⊂ [a, b] of a positive measure, it holds true that g(t, u(t)) ∩ rm + = ∅. in this case, for t ∈ t there is ε > 0 such that the ε-neighborhood of the set g(t, u(t)) does not have intersections with the cone rm + . the latter implies that hrm ( g(t, un(t)), g(t, u(t)) ) > ε for any n, which contradicts with the convergence (4.17). thus, the condition (a3) is also fulfilled. theorem 2.1 implies the existence of a solution u ∈ l(b) of the inclusion (4.15) and, hence, the existence of a solution x ∈ ac(b) of the problem (4.11), (4.12), (4.13) (which is defined by the formula x = γ + ∫ t a u(s)ds). let us now illustrate applications of theorem 4.1 to investigation of concrete differential equations and inclusions. example 3.1: we denote by χ the heaviside function, i.e. χ(x) = 1 for x ≥ 0 and χ(x) = 0 for x < 0. let a measurable function q0 : [a, b]→ r, a measurable integrable function q1 : [a, b]→ r, and a positive number r be given. we consider the differential equation ẋ2 − 2q1(t)ẋ+ χ(x)− q0(t) = 0, t ∈ [a, b], (4.18) together with the additional condition ẋ ∈ b(t) . = [q1(t)− r, q1(t) + r], t ∈ [a, b]. (4.19) let us demonstrate that if the inequalities − 1 ≤ q0(t) + q2 1(t) ≤ r2, t ∈ [a, b], (4.20) are fulfilled, the cauchi problem for the system (4.18), (4.19) with the initial condition (4.13) for any γ has a solution x ∈ ac(b) such that ẋ(t) ≤ q1(t), t ∈ [a, b]. we treat the equation (4.18) as the inclusion (4.11) with the single-valued mapping f : [a, b]× r× r→ r defined by the relation f(t, x, u) = g(t, u)− χ(x), t ∈ [a, b], x, u ∈ r, where g is given in the example 2.1 by the formula (3.8). obviously, for all x, u, the function f(·, x, u) is measurable, for almost all t and any u, the function f(t, ·, u) decreases (i.e. the condition (b2) of theorem 4.1 is fulfilled) and right continuous, and for almost all t and any x, the function f(t, x, ·) is continuous. as it was demonstrated in example 2.1, the inequality q0(t) + q2 1(t) ≤ r2 implies the condition (b1) of theorem 4.1. if one takes v0(t) = γ + ∫ b a p(s)ds, t ∈ [a, b], the relations f ( t, v0(t), v̇0(t) ) = −v̇2 0(t) + 2q1(t)v̇0(t)− χ(vo(t)) + q0(t) ≥ q2 1(t)− 1 + q0(t) ≥ 0 take place for all t ∈ [a, b]. thus, the condition (4.14) is also satisfied, and, according to theorem 4.1, the problem (4.18), (4.19), (4.13) has a solution x ∈ ac(b) satisfying the estimate ẋ(t) ≤ q1(t), t ∈ [a, b]. consider now the following differential inclusion g+(t, ẋ)− χ(x) 3 0, t ∈ [a, b], (4.21) copyright © 2022 assa. adv syst sci appl (2022) on order covering set-valued mappings and their applications 185 where the set-valued mapping g+ : [a, b]× r→ k(r) is given by the first formula in (3.10) (see example 2.1). in the investigation of this inclusion, one can also make use of theorem 4.1. in order to do this, one should use the condition that guarantees that the mapping g+(t, ·) order covers the set {0} ⊂ r, which was derived in example 2.1. one thus concludes that if the inequalities − 1− δ ≤ q0(t) + q2 1(t) ≤ r2, t ∈ [a, b], (4.22) are satisfied, the problem (4.21), (4.19), (4.13) possesses a solution for any γ. finally, we consider the differential inclusion g−(t, ẋ)− χ(x) 3 0, t ∈ [a, b], (4.23) with the set-valued function g− : [a, b]× r→ k(r) given by the second formula in (3.10). for this inclusion, we analogously find that the validity of the inequalities − 1 ≤ q0(t) + q2 1(t) ≤ r2 + δ, t ∈ [a, b], (4.24) guarantees the solvability of the problem (4.23), (4.19), (4.13) for any γ. consider a particular case of the inclusion (4.11), which allows to obtain the existence of a lower solution to the corresponding cauchi problem. let m = n and the components of f be given by the functions fi : [a, b]× rn × rn × r→ kc(r), i = 1, n (having compact and connected values, i.e. the real line segments). let us assume that for any x, v ∈ rn, z ∈ r, the function fi(·, x, v, z) : [a, b]→ kc(r) is measurable; for almost all t ∈ [a, b], any v ∈ rn and z ∈ r, the function fi(t, ·, v, z) : rn → kc(r) is right continuous in each of the arguments x1, . . . , xn; for almost all t ∈ [a, b], any x ∈ rn and z ∈ r, the function fi(t, x, ·, z) : rn → kc(r) is right continuous in each of the arguments v1, . . . , vn; for almost all t ∈ [a, b] and any x, v ∈ rn, the function fi(t, x, v, ·) : r→ kc(r) is continuous. we investigate the following system: fi(t, x, ẋ, ẋi) 3 0, t ∈ [a, b], i = 1, n. (4.25) theorem 4.2: let functions u0, v0 ∈ acn such that u0(a) ≤ γ ≤ v0(a), −fi ( t, u0(t), u̇0(t), u̇0i(t) ) ∩ r+ 6= ∅ for almost all t ∈ [a, b], i = 1, n, fi ( t, v0(t), v̇0(t), v̇0i(t) ) ∩ r+ 6= ∅ for almost all t ∈ [a, b], i = 1, n, be given. define a set-valued mapping b : [a, b]→ cc(rn) as follows: b(t) . = [ v̇0(t), u̇0(t) ] rn , t ∈ [a, b]. (4.26) assume that for any i = 1, n, for almost all t ∈ [a, b] and all u ∈ b(t), the mapping fi(t, ·, ·, u) : rn × rn → kc(r) is antitone on the set rn ×b(t). then there exists a solution x ∈ ac(b) to the problem (4.25), (4.12), (4.13). moreover, the set of solutions to the problem (4.25), (4.12), (4.13) contains a solution with the least derivative. remark 4.1: by the virtue of the definition of the mapping b by (4.26), the fact that a solution x belongs to ac(b) means that this absolutely continuous function satisfies the inequalities u̇0(t) ≤ ẋ(t) ≤ v̇0(t) for almost all t ∈ [a, b]. proof we will denote the restrictions of mappings by the same symbols as the initial mappings copyright © 2022 assa. adv syst sci appl (2022) 186 e.s. zhukovskiy, i.d. serova, e.a. panasenko, e.o. burlakov provided that leads to no ambiguity. we denote the components of the set-valued mappingb : [a, b]→ cc(rn) as bi : [a, b]→ cc(r), i.e. bi(t) . = [ v̇0i(t), u̇0i(t) ] , t ∈ [a, b]. we define the set ti ⊂ [a, b] of such t ∈ [a, b] that for all x ∈ rn, v ∈ b(t), the mapping fi(t, x, v, ·) : bi(t)→ kc(r) is continuous, and−∞ < v̇0(t), u̇0(t) < +∞. the lebesgue measure of this set equals to b− a, and for all t ∈ ti, the set bi(t) is a segment of the real line. let us verify the condition (b1) of theorem 4.1. let for any i = 1, n, for some t ∈ ti, there exist x ∈ rn, v ∈ b(t) such that the mapping fi(t, x, v, ·) : bi(t)→ kc(r) does not order cover the set {0} ⊂ r. then there exists z0 ∈ r such that the set fi(t, x, v, z0) contains some positive number, and for any z ∈ [v̇0i(t), z0], zero does not belong to the set fi(t, x, v, z). let us demonstrate that there exists δ > 0 such that for any z ∈ [v̇0i(t), z0], it holds true that fi(t, x, v, z) ∩ (−δ, δ) = ∅. (4.27) otherwise, there exists a sequence {zk}∞k=1 ⊂ [v̇0i(t), z0] whose elements satisfy the inequality |zk| < 2−k. this sequence is compact, so it contains a subsequence converging to some z ∈ [v̇0i(t), z0]. due to continuity of the mapping fi(t, x, v, ·) at z, the inclusion 0 ∈ fi(t, x, v, z) holds true, which contradicts with the assumptions made. the relation (4.27), due to the connectedness of the values fi(t, x, v, z), implies that the segment u . = [v̇0i(t), z0] is a union of the following two sets: u+ . = { z ∈ u : fi(t, x, v, z) ⊂ [δ,+∞) } , u− . = { z ∈ u : fi(t, x, v, z) ⊂ (−∞, δ] } . both these sets should be closed as the mapping fi(t, x, v, ·) is continuous. however, this is not possible by the virtue of the connectedness of the segment u. thus, it is proved that for any t ∈ ti, for all x ∈ rn and v ∈ b(t), the mapping fi(t, x, v, ·) : bi(t)→ kc(r) order covers the set {0} ⊂ r. considering the fact that the lebesgue measure of the set ti equals to b− a, the condition (b1) is satisfied. the validity of the rest of the conditions of theorem 4.1 under the assumptions of the statement being proved is obvious. thus, there exists a solution x ∈ ac(b) of the problem (4.25), (4.12), (4.13). we now prove the existence of a solution with the least derivative. let us remind the reader (see the proof of theorem 4.1) that the derivatives of the solutions of the problem (4.25), (4.12), (4.13) are the solutions of the following system of inclusions fi ( t, γ + ∫ t a u(s)ds, u(t), ui(t) ) 3 0, t ∈ [a, b], i = 1, n. (4.28) thus, it remains to prove that the set of solutions of the system (4.28) possesses the least element. denote by r the set (in the space l(b)) of solutions of the system (4.28). according to theorem 2.1, the set r possesses a minimal element û ∈ l(b). assume that this element is not the least in r. then there exists a solution z ∈ r of (4.28) such that z � û. for each i = 1, n, we define the sets ei+ = { t ∈ [0, 1] : ûi(t) ≤ zi(t) } , ei− = { t ∈ [0, 1] : ûi(t) > zi(t) } . define a measurable function ẑ with the components equal to ẑi(t) = min { zi(t), ûi(t) } = { ûi(t) for t ∈ ei+ , zi(t) for t ∈ ei− , i = 1, n. thus, the inequality ẑ < û is fulfilled. hence, for almost all t ∈ ei+, the inclusion 0 ∈ fi ( t, γ + ∫ t a û(s)ds, û(t), ûi(t) ) , i = 1, n, copyright © 2022 assa. adv syst sci appl (2022) on order covering set-valued mappings and their applications 187 due to antitonicity of the mapping fi(t, ·, ·, u) on rn ×b(t), implies that for any i = 1, n, the set fi ( t, γ + ∫ t a ẑ(s)ds, ẑ(t), ẑi(t) ) = fi ( t, γ + ∫ t a ẑ(s)ds, ẑ(t), ûi(t) ) contains a non-negative number. analogously, for any i = 1, n, we get that for almost all t ∈ ei−, some non-negative number belongs to the set fi ( t, γ + ∫ t a ẑ(s)ds, ẑ(t), ẑi(t) ) . by theorem 2.1, there exists a solution ξ ∈ r of the system (4.28) such that ξ ≤ ẑ < û. the latter, however, contradicts with the minimality of û inr. finally, we note that the equation (4.18) considered in example 3.1 under the restriction (4.20) satisfies not only the assumptions of theorem 4.1, but also the assumptions of theorem 4.2 if one takes v0(t) = γ + ∫ b a q1(s)ds, u0(t) = γ + ∫ b a ( q1(s)− r ) ds, t ∈ [a, b]. therefore, the set of solutions of the problem (4.18), (4.19), (4.13) such that q1(t)− r ≤ ẋ(t) ≤ q1(t), t ∈ [a, b], (4.29) contains a solution with the least derivative. analogously, for the system of inclusions (4.21), (4.19) with the initial condition (4.13), which was considered in example 3.1, if the relations (4.22) take place, the set of solutions that satisfy the inequalities (4.29) contains a solution with the least derivative. the same statement is valid for the cauchi problem (4.23), (4.19), (4.13). namely, if the condition (4.24) takes place, the set of solutions satisfying the inequalities (4.29) possesses a solution with the least derivative. 5. existence and estimates of equilibrium prices in dynamical continuous supply-and-demand models in this section, theorems 4.1 and 4.2 are applied to investigation of equilibrium in dynamical supply-and-demand models. the model of economic processes we consider has its roots in the classical works of g.c. evans [11], p.a. samuelson [12], and r. allen [13]. a considerable advance in the investigation of models describing equilibrium processes in market models were the results of a.v. arutyunov, n.g. pavlova, s.e. zhukovskiy, a.a. shananin (see [14–17]) obtained using the results on covering mappings of metric spaces. in particular, in [17] the authors managed to investigate the models with the mappings of supply and demand that are dependent not only on the market prices, but also on the the prices change rates. in this section, the models under consideration contain set-valued mappings of supply and demand. such generalisation is natural as the values of these mappings are defined in the economic problems as the set of solutions of constraint minimisation problems (see e.g. [23, §4.2.2]). in the present research, in contrast to the cited works that base on the results on mappings of metric spaces, methods of analysis of mappings of partially ordered spaces are employed. let us formulate the problem. let us have n types of goods, whose prices at any time t ∈ [a, b] we denote by pi(t), i = 1, n. let these prices be known at the initial moment of time pi(a) = γi, i = 1, n. (5.30) copyright © 2022 assa. adv syst sci appl (2022) 188 e.s. zhukovskiy, i.d. serova, e.a. panasenko, e.o. burlakov we define the vector p(t) = ( p1(t), . . . , pn(t) ) ∈ rn + of prices. assume that there are some restrictions on the change rates of the prices, i.e. there is a measurable set-valued mapping b : [a, b]→ k(rn) and the inclusion ṗ(t) = ( ṗ1(t), . . . , ṗn(t) ) ∈ b(t) for almost all t ∈ [a, b] (5.31) takes place. next, let the set-valued mappings d : [a, b]× rn + × rn × rn → k(rn +) and s : [a, b]× rn + × rn → k(rn +) be given. we assume that for any x ∈ rn + and any v, u ∈ rn, the functions d(·, x, v, u), s(·, x, v) : [a, b]→ k(rn +) are measurable; for almost all t ∈ [a, b] and any v, u ∈ rn, the functions d(t, ·, v, u), s(t, ·, v) : rn + → k(rn +) are right continuous in each of the arguments x1, . . . , xn; for almost all t ∈ [a, b] and any x ∈ rn +, u ∈ rn, the functionsd(t, x, ·, u), s(t, x, ·) : rn → k(rn +) are right continuous in each of the arguments v1, . . . , vn; for almost all t ∈ [a, b] and any x ∈ rn +, v ∈ rn, the function d(t, x, v, ·) : rn → k(rn +) is continuous. we assume that the aggregate consumer demand at time t is defined by the formula d ( t, p(t), ṗ(t), ṗ(t) ) and aggregate manufacturer’s supply — by the formula s ( t, p(t), ṗ(t) ) . for the defined above set-valued mappings of supply and demand, we consider the problem of existence of an equilibrium — a solution p ∈ ac(b) (i.e. absolutely continuous function, whose derivative satisfies the restriction (5.31)) of the inclusion d ( t, p(t), ṗ(t), ṗ(t) ) − s ( t, p(t), ṗ(t) ) 3 0, (5.32) with the initial condition (5.30). here the difference d − s of sets in the space rn is the set of all vectors x− y, where x ∈ d, y ∈ s. we will denote the restrictions of mappings by the same symbols as the initial mappings indicating their domains and ranges. applying theorem 4.1 to the mapping f : [a, b]× rn + × rn × rn → k(rn +) defined by the relation f(t, x, v, u) . = d(t, x, v, u)− s(t, x, v) for almost all t ∈ [a, b] and any x ∈ rn +, v, u ∈ rn +, we get the following statement. corollary 5.1: let a non-negative function p ∈ ac(b) such that p(a) ≥ γ and( d ( t, p(t), ṗ(t), ṗ(t) ) − s ( t, p(t), ṗ(t) )) ∩ rn + 6= ∅ for almost all t ∈ [a, b], be given. let the set of measurable selections of the set-valued mapping b(·) ∩ orn ( p(·) ) : [a, b]→ c(rn) be integrally bounded from below (i.e. there exists a number c such that for any measurable function u ∈ w n, satisfying the relations u(t) ∈ b(t) and u(t) ≤ ṗ(t) for almost all t ∈ [a, b], it holds true that ∫ b a u(t)dt ≥ c) and the following conditions take place: (c1) for almost all t ∈ [a, b], any x ∈ rn + and v ∈ b(t), the mapping d(t, x, v, ·) : b(t)→ k(rn +) order covers the set {0} ⊂ rn; (c2) for almost all t ∈ [a, b] and any u ∈ b(t), the mapping d(t, ·, ·, u) : rn + ×b(t)→ k(rn +) is antitone, and the mapping s(t, ·, ·) : rn + → k(rn +) is isotone. then there exists a solution p ∈ ac(b) of the problem (5.32), (5.31), (5.30) such that ṗ(t) ≤ ṗ(t) for almost all t ∈ [a, b]. (5.33) copyright © 2022 assa. adv syst sci appl (2022) on order covering set-valued mappings and their applications 189 note that the assumptions (c2) have natural economic sense: a decrease of the prices and the growth rates of prices result in an increase of the demand and a decrease of supply. the main distinctions of corollary 5.1 from the results of [14–17] are: set-valued mappings of supply and demand, omission of the additional condition of essentially boundedness of the derivative ṗ of the prices function, and the estimate of the equilibrium price derivative of the form (5.33). corollary 5.1 has the type of the chaplygin comparison theorem for the differential inclusion modelling the economic processes under consideration. finally, let us formulate a statement allowing to obtain two-sided estimates of the equilibrium prices and guarantee the existence of the equilibrium price having the least rate of change. let the componentsdi, si of the mappingsd,s be the functionsdi : [a, b]× rn + × rn × r→ kc(r), si : [a, b]× rn + × rn → kc(r), i = 1, n (having compact connected values). we assume that for any x ∈ rn +, v ∈ rn, z ∈ r, the functionsdi(·, x, v, z) : [a, b]→ kc(r), si(·, x, v) : [a, b]→ kc(r) are measurable; for almost all t ∈ [a, b], any v ∈ rn and z ∈ r, the functions di(t, ·, v, z) : rn + → kc(r), si(t, ·, v) : rn + → kc(r) are right continuous in each of the arguments x1, . . . , xn; for almost all t ∈ [a, b], any x ∈ rn + and z ∈ r, the functions di(t, x, ·, z) : rn → kc(r), si(t, x, ·) : rn → kc(r) are right continuous in each of the arguments v1, . . . , vn; for almost all t ∈ [a, b] and any x ∈ rn +, v ∈ rn, the function di(t, x, v, ·) : r→ kc(r) is continuous. consider a particular case of the inclusion (5.32) — the system di(t, x, ẋ, ẋi)− si(t, x, ẋ) 3 0, t ∈ [a, b], i = 1, n. (5.34) theorem 4.2 implies the following statement. corollary 5.2: let functions u0, v0 ∈ acn such that u0(a) ≤ γ ≤ v0(a),( si ( t, u0(t), u̇0(t) ) −di ( t, u0(t), u̇0(t), u̇0i(t) )) ∩ r+ 6=∅ for almost all t ∈ [a, b], i = 1, n,( di ( t, v0(t), v̇0(t), v̇0i(t) ) − si ( t, v0(t), v̇0(t) )) ∩ r+ 6= ∅ for almost all t ∈ [a, b], i = 1, n, be given. define a set-valued mapping b : [a, b]→ cc(rn) by the relation (4.26). assume that for any i = 1, n, for almost all t ∈ [a, b] and all u ∈ b(t), the mapping di(t, ·, ·, u) : rn + × rn → kc(r) is antitone on the set rn ×b(t) and the mapping si(t, ·, ·) : rn + × rn → kc(r) is isotone on the set rn ×b(t). then there exists a solution p ∈ ac(b) of the problem (5.34), (5.31), (5.30). moreover, the set of solutions of the problem (5.34), (5.31), (5.30) contains a solution with the least derivative. acknowledgements the reported study was funded by rfbr, project number 20-04-60524. the results in section 4 were obtained with the support of ministry of science and higher education of the russian federation (agreement no. 075-15-2019-1619). the results in sections 2 and 3 were obtained with the support of the russian science foundation (grant no. 20-11-20131) in v.a. trapeznikov institute of control sciences of ras. references 1. arutyunov, a.v., zhukovskiy, e.s. & zhukovskiy, s.e. 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(2011) introduction to the theory of multi-valued mappings and differential inclusions. moscow, russia: librokom. copyright © 2022 assa. adv syst sci appl (2022) on order covering set-valued mappings and their applications 191 24. serova, i.d. (2021) superpositional measurability of a multivalued function under generalized caratheodory conditions, russian universities reports. mathematics, 26(135), 305–314. 25. krasnoselskii, m.a. & zabreiko, p.p. (1984) geometrical methods of nonlinear analysis. berlin, germany: springer. copyright © 2022 assa. adv syst sci appl (2022) introduction a comparison theorem for operator inclusions in partially ordered spaces conditions of order covering for the nemytskii operator implicit differential inclusion existence and estimates of equilibrium prices in dynamical continuous supply-and-demand models microsoft word b-1252-article _ fin2 adv syst sci appl 2022; 04; 126-143 published online at http://ijassa.ipu.ru. raw material price forecasting on commodity markets: application of expert and quantitative information zinaida k. avdeeva1*, elena a. grebenuyk1, svetlana v. kovriga1 1) trapeznikov v.a. institute of control sciences of ras, moscow, russia e-mail: zinaida.avdeeva@gmail.com e-mail: lngrebenuk12@yandex.ru e-mail: kovriga@ipu.ru abstract: the article considers the problem of forecasting prices on the commodity market for the year ahead, broken down by months. the uncertainty level of the forecast increases with the increase of its horizon, due to changes caused by events of the external environment on the forecasting horizon. to reduce this uncertainty, we have proposed a hybrid model for the formation and correction of a monthly price forecast for the year ahead. this model uses for forecasting expert-analytical information processed using the fcm model, and data from time series of commodity market prices and macro indicators. model of forecasting has based on the use of an ensemble of vecm and var models built at various time scales. for correcting the target indicator forecast on the forecasting horizon we have developed an algorithm based on signals of situation monitoring conducted on the fсм-model of commodity market situation. we demonstrate the efficiency of the proposed technique for monthly forecasting of prices for black scrap a year ahead for 2019. the accuracy of the obtained forecast we compared with naive and arima forecasts. keywords: commodity markets; nonstationary processes; forecasting; fuzzy cognitive map, situation and digital monitoring, time series. 1. introduction one of the most important activities of enterprises is to reduce their expenses without losing the value of the products created for the consumer. forecasting commodity prices for the year ahead, broken down by months, is a necessary element of enterprise procurement planning and inventory management and is crucial for reducing enterprise costs. the purpose of our study is to build a monthly forecast of price of a certain type of for raw material or product (target indicator) in the commodity market for the year ahead. commodity markets, as objects of forecasting, have the following features:  prices for various commodities follow common trends, such price trends are called co-movement [7,22];  the product value chain determines the dynamics of price interaction between goods [21];  the structure of the interrelations depends on the time scale in which we consider it and may change with changes in scale;  there is a dependence of prices on macro indicators, such as the real interest rate, exchange rates, oil prices, world demand for goods, etc. [8];  the processes of price changes are nonstationary, events occurring in the external environment, such as natural and economic disasters, political events, changes in participants' strategies in commodity markets, etc., can lead to the emergence of structural breaks in prices [13,33]. * corresponding author: zinaida.avdeeva@gmail.com raw material price forecasting on commodity markets 127 copyright ©2022 assa. adv. in systems science and appl. (2022) forecasting a time series several steps ahead for an extended period is a difficult problem, because the level of forecast uncertainty increases with the growth of the horizon [20]. to predict nonstationary processes, researchers use various methods: building e parametric models, singular spectral analysis, empirical mode decomposition, support vector machines, neural networks, etc. [5,10]. however, the high level of uncertainty in the dynamics of processes on commodity markets creates problems for the application of traditional forecasting models. to reduce this uncertainty some of authors [29,31] explored different multistep forecasting strategies and shown that the most efficient for multistep forecasts are strategies with several outputs. the application of these strategies implies the preservation of stochastic dependencies between the members of the predicted sequence. regular processes satisfy these conditions, because their dynamics can be described with sufficient accuracy by some dynamical system. however, for processes whose dynamics change is often associated with external events, the approach based on extracting and accounting in the forecast additional information about the events of the forecast period that affect the value of the forecasted indicator is more effective. according to the efficient market hypothesis [14], changes in the external environment are reflected in the quantitative data used in forecasting and thus taken into account in the forecasts. however, complex interactions of supply and demand, and extreme and unexpected events in the external environment are not predictable from commodity market data and are one of the sources of forecasting errors [11]. to take into account environmental events in the forecast, the authors of [32] proposed a methodology of multiscale forecasting, using search systems data in different time scales. to assess the impact of extreme events on the volatility of crude oil prices, the authors of [34] developed an estimation methodology based on the decomposition of the series by empirical modes at different time scales. the experiments have shown the promise of the method for analyzing possible changes in the price series under the influence of extreme events. modern approaches to forecasting point to the need to use expert information. when constructing medium-term (and long-term) forecasts, information about future events that change the dynamics of the target indicator in the interval between forecasts is not available in the data at the time of making the forecast. if the price dynamics on the forecasting horizon changes under the influence of external environment events, the forecasting model becomes unsuitable. for timely replacement and adjustment of the forecasting model, monitoring is necessary, the purposes of which are: detection of changes in time series (structural shifts), revealing of the reasons, which caused or may cause the occurrence of structural shifts, formation of scenarios of possible development of the situation, evaluation of the power and duration of the forthcoming changes. the time series monitoring algorithms detect the structural shifts of the forecast indicators and determine the type of these changes (change in the trend, volatility, the type of non-stationarity of the process). however, the results of digital monitoring do not allow solving such problems as:  identifying the causes of structural shifts;  forecasting further development of the situation which led to the occurrence of a structural shift;  forecasting the situation which may lead to a structural shift. situation monitoring based on processing and analysis of expert information is needed to support the solution of the above tasks. therefore, improving the quality of process forecasts in a changing external environment is impossible without the account of expert and analyst judgments [2,9,15,28]. the task of decision support systems for forecasting is to develop tools for extracting, integrating, and 128 z.k. avdeeva, e.a. grebenuyk, s.v. kovriga copyright ©2022 assa adv. in systems science and appl. (2022) processing expert knowledge and generating signals to incorporate it into forecasts obtained by processing quantitative data. in this study, we use methods for analyzing and modelling expert information based on fuzzy cognitive maps. fuzzy cognitive maps (fcm) refer to a class of models representing expert knowledge and judgments. they are used to analyze and model ill-structured situations. these are situations involving many elements of various nature, and the relationships between the elements can be both quantitative and qualitative [1,12,26]. the use of fcm in production sphere is associated with the support of managerial decision-making in situations of uncertainty or lack of quantitative data to solve the problems posed. typical fcm applications are the following examples:  analysis of factors supporting collaborative planning, forecasting and replenishment to ensure effective supply chain management [6];  risk analysis of erp (enterprise resource planning) maintenance in the system lifecycle [23];  risk analysis in a production system, for which the possibility of loss of profit is considered a key risk factor [33];  factors' analysis of influence on planning of production in manufacturing plants using the fcm of multi-criteria decision making [30];  selection and justification of technological solutions for creating final products with specified goal characteristics, where the technology solutions are products and technologiescomponents of the final product [4], etc. in accordance with the purpose outlined in the article, the contribution of this study is as follows: 1. we propose a hybrid model for the formation and correction of the monthly price forecast for the year ahead, combining the information obtained from the processing of expert judgments through fcm, and the data of time series of commodity market prices and macroeconomic indicators. the developed hybrid model performs the following functions:  processes expert judgments and analytical information and simulates scenarios of possible consequences of the impact of significant events of environment on the change of the target indicator by means of fcm – a formalized model of causal influences between the system-forming factors of the situation and the target indicator;  forms a monthly price forecast for the year ahead on the basis of the information obtained from the processing of expert judgments and time series data as a result of applying the ensemble of multidimensional time series models and taking into account the cause-andeffect effects between the factors of the situation and the target indicator. 2. we have developed an algorithm for correcting the target indicator forecast on the forecast horizon based on signals  situation monitoring implemented on the basis of analysis and modelling methods on fcm [3,4];  digital monitoring carried out by methods of sequential analysis of nonstationary processes [19]. 3. we tested the effectiveness of the hybrid model and algorithms for generating and correcting on monitoring signals forecast when generating a monthly forecast of prices for ferrous scrap for the year ahead. the constructed forecast showed a significant increase in forecasting accuracy compared to the plan-fact ("naive") forecast and the forecast on the arima model. raw material price forecasting on commodity markets 129 copyright ©2022 assa. adv. in systems science and appl. (2022) 2. problem statement, its features and general solution principle we consider the problem of forming and adjusting the monthly forecast of commodity prices for the year ahead. the forecast indicator (target) 1 2{ , ,..., ,...}ty y y y is a nonstationary time series, whose properties may change under the influence of:  factors affecting the domestic market of the forecasted indicator;  factors affecting changes in world prices for raw materials and products;  factors that characterize the global economy;  factors, characterizing the state policy of commodity markets regulation. we call these factors system-forming, because in their unity they form a system that reflects a holistic view of the situation in the context of the problem under consideration. the forecasted target at time t may depend on its previous values; previous values of other processes; significant events in external environment, uncontrolled and unpredictable factors described by a random sequence. we call significant events (inf) events that can lead to changes in the characteristics of the process: disruption of relationships, gradual or abrupt changes in trends, increase or decrease in volatility. required to build: a) an algorithm for monthly forecasting of the target indicator 1 2{ , ,..., ,...}ty y y y for the year ahead; b) a monitoring system that includes digital monitoring of price time series and situation monitoring to detect events and system-forming factors that change the dynamics of processes; c) an algorithm for correcting the forecast of the indicator 1 2{ , ,..., ,...}ty y y y on the forecasting horizon based on monitoring signals. d) assessing the quality of the forecast obtained, built according to algorithms (a)-(c). .2.1. general principle of the solution we divide the problem of constructing a monthly forecast of the values of the target indicator for the year ahead into two stages. stage 1 is a building a forecast for the year ahead (performed before the start of the forecast period); stage 2 is a correction of the forecast on the forecasting horizon based on situation and digital monitoring signals. to assess the state of the external environment and the strength of the influence of the factors that may change the values of the forecasted target indicator we build a formalized description of the current situation in the form of the fcm. to receive information about the effects of changes in the external environment that are not present in the time-series at this moment we use situation monitoring, according to the results of which the fcm generates signals about possible changes in the dynamics of the forecasting target indicator. to detect changes (structural shifts) in the processes and determine the type of these changes (trend change, volatility, type of process non-stationarity) we perform digital monitoring. figure 1 shows a general scheme of the forecasting process. stage 1 consists of three steps. step 1.1. based on the constructed description of the situation in the form of fcm we form groups of system-forming factors (sf-groups) that have a direct or indirect impact on the forecasted target indicator. the basis of fcm is a graph, the vertices of which represent the factors of the situation under study; the edges are the direct causal effects of some factors on others; any factor with all the factors of influence on them forms a node of fcm. the constructed fcm allows you to formalize the process of transferring expert information to the prediction algorithm (figure 1). 130 z.k. avdeeva, e.a. grebenuyk, s.v. kovriga copyright ©2022 assa adv. in systems science and appl. (2022) situation monitoring mode digital monitoring mode defining infosearch parameters forecast correction quantitative database 2.1 expert knowledge building ensembles of predictive models 1.2construction and correction of fcm. estimation of the significance sf-gropus 1.1 formation of the combined forecast 1.3 2.4 identification and estimation of system-forming factors associated with significant info-occasions 2.2 current observations fcm of situation qualitative database of commodity markets, etc; significant events data from heterogeneous sources of information detection and identification of the structural shift types 2.3 fig. 1. general scheme of the forecasting process each of the sf-groups describes one aspect of the situation under study (production factors related to the predicted indicator, market factors of the business environment related to supply and demand in commodity markets, macro-environment factors related to financial and economic indicators, factors of influence of state regulators, etc.). separate sf-groups correspond to time series of the data, changes in which arise under the influence of system-forming factors of the given sf-group. the formation of sf-groups is aimed at expanding the information space of the search for quantitative data to replenish the quantitative database and its structuring by distributing the series by the formed sfgroups. we calculate the weights of sf-groups according to the significance of their influence on the target indicator and pass them to step 1.3 to calculate the final forecast. step 1.2. to build ensembles of forecast models we use monthly data. each model includes a target indicator and time series of a certain sf-group. for each sf-group for which the corresponding time series are in the database, we compute a forecast on the ensemble of models. we selected only those time series that are granger-causes y [17]. since the series are non-stationary, integrated of the first order, we build a model describing the target indicator using the differences of the series. if there is cointegration between the series, we build vec models to trace the long-run dependencies. step 1.3. we form a combined forecast based on the forecasts obtained in step 1.2, and sf-group weights calculated in step 1.1. the proposed strategy allows us to obtain more accurate forecasts [16,22,32]. stage 2. to account for the influence on the built forecast of environmental events on the forecasting horizon, we adjust it according to the signals received from the results of situation and digital monitoring. the signals of actualization of the forecast correction process in the mode of current observations are of different nature: raw material price forecasting on commodity markets 131 copyright ©2022 assa. adv. in systems science and appl. (2022)  signals triggered by significant events in the external environment, which may coincide with digital monitoring signals or may serve as a warning of possible changes that will occur in the lagged observations;  signals about the presence of changes in the predicted indicator and the related price series (different for each type of change). 3. analysis of commodity markets based on historical data and expert knowledge and formation of a target indicator forecast 3.1. fcm construction and estimation of the sf-groups’ significance we conduct the structuring of the subject knowledge domain to form a holistic representation of the situation associated with the impact of the external environment on the dynamics of the predicted process. according to the results of the structuring, we obtain a conceptual model of the situation, which includes the main subject-oriented concepts and related subject knowledge, characterizing the production cycle and the value chain of the target-related products, as well as its businessand macro environment in the context of the forecasting problem set by us. taking into account, the heterogeneity of environmental factors affecting the predicted processes, we used expert knowledge (in the mode of direct interaction with experts in the analysis of the commodity market) and analytical information from heterogeneous sources as input data for structuring. based on the conceptual model of the situation, we identify, together of experts, the system-forming factors influencing the predicted process and establish the relationship between them and the target indicator, i.e., formalize the selected subject knowledge in the form of the fcm. the constructed fcm allows formalizing the process of transferring expert information into the forecasting algorithm (figure 1). in figure 2, we show a fragment of the fcm associated with the commodity market (grey arrows with ellipsis show the fragment boundaries in the full fcm). this map includes 59 system-forming factors, including the y factor associated with the target forecasted indicator y (purchase prices for black scrap metal (bsm)), which are related by causeeffect relationships. we represent factors as variables 1 2 1{ , ,..., , }n nx x x x x y       , described by certain sets of values (figure 2). let ( , , }fk x w f be an fcm of the situation ( )s in commodity markets related to a target indicator ,y where 1 2 1{ , ,..., , }n nx x x x x y       , y is a factor characterizing ;y [ ]ijw w is the n n -matrix of factors mutual influence, where the weight [ 1;1]ijw   determine the degree of expert confidence that activity of factor ix causes on the factor [ 1;1]jx   ; f is a function of factors' activation at any discrete time t : ( 1) ( ) ( ) ( ) i i i ji j i j i f a t a t w a t g t       , (1) where ( ), ( )i ja t a t are factor activity values ix and jx at the moment ,t ( ) ( ) ( 1)j j ja t a t a t    ; ii is a set of factor codes { }jx that directly influence factor { }ix ; ( ) (0)i ig t g for 0t  and ( ) 0ig t  for 0t  , (0) [ 1;1]ig   . 132 z.k. avdeeva, e.a. grebenuyk, s.v. kovriga copyright ©2022 assa adv. in systems science and appl. (2022) export duties the bsm offer for export purchase prices for bsm in the domestic market the offer of bsm for domestic consumption us dollar rate export prices of the bsm local collection and recycling of bsm bsm demand in the domestic marketthe bsm demand in the foreign market transportation of bsm from long distances bsm reserves demand for finished products of metallurgical companies the need for bsm (+): increase (decrease) in the factor causes increase (decrease) in the factor (-): increase (decrease) in the factor causes decrease (increase ) in the factor link: direct causal influence: weight of influence factor (concept): variable node(bundle): influence aggregation function ... ... ... ... ... ...... ... ... ... ... ... ... ... ... vat rate 14x 12x 13x 13x 1x 2x 3x 4x y  5x 6x 7x 8x 9x 10x 11x 13x 14x 12x 13 12 14( , )x f x x   fig. 2. the fragment of an fcm associated with the commodity market we define the state ( 1)a t  of the situation s at time 1t  as the result of the influence of all factors x on the interval [ 0, t ]: ( 1) ( )( (0) (0))a t q t a g   (2) where 2( ) ... ,t nq t e w w w     (0)a is an initial state of factors x , (0)g is a vector of external impacts on factors at 0t  , kw is a matrix of influences to k-th degree, 1,...,k t , ne is the identity matrix. the property 1lim ( )t nt q q e w      allows us to identify the steady states by factors. then q is the matrix of integral influences of fcm factors, the elements of which characterize all the direct and indirect influences between factors. q-matrix analysis is based on the method of [3], which allows  to determine the cumulative (direct and indirect) influence of factors on the target;  determine the nature (favorable or unfavorable) and weight of the cumulative influence of one or another factor (taking into account the current dynamics of this factor) on the favorable dynamics of the target indicator;  to estimate efficiency of influence of positive and negative factors on a target indicator and their ranking (taking into account their inclusion in different sf-groups);  to assess the activity of factors with a high rank of influence on the target indicator. based on this data, we calculate for each sf-group the integral positive q and negative q influence on the target indicator of all factors included in the sf-group. then the total influence of the sf-group is q q q   . according to estimates of type{ }iq , we calculate the normalized weights of the influence of sf-groups in the form raw material price forecasting on commodity markets 133 copyright ©2022 assa. adv. in systems science and appl. (2022) 1 2 , ... i i m q c q q q     (3) where iq is the total integral influence of sf-group, m is number of sf-groups. we transfer the weights of sf-groups to the numerical forecasting block 1.3 to account for the influence of systemic factors of the situation on the target. 3.2. construction of model assembles, and formation of a target indicator forecast from the structured time series database (step 1.1 on figure 1), we select the sets of series f , , 1, 2,..., ,i y i m  where y is the series of target indicator values, 1 2f ( , ,..., )i nix x x  are the series of sf-group with number , 1,2,...,i i m . we break down the available set of time series f  according to the structure of the available sfgroups: 1 2f = (f , f ,...,f ),m     . we obtain the sets of series f ,m y  , each of which describe the dynamics of the target indicator as a function of its past values and the system-forming factors of its corresponding sf-group. the process of building a forecast includes the steps below, which we do for each set of f ,i y  , and for each time scale (month, quarter, half-year, year). the models we apply vary for different time scales, because of changes in the relationships between successive observations and decreases in the size of the training sample as the scale increases. step 1. we apply structural shift detection algorithms to series y [19], to determine the presence of trends, their direction, and the moments of their changes. step 2. we check if the statistical properties of the series of the set fi  correspond to the properties of the targety . we include in the forecast model only those series from fi  , which have the order of integration and seasonality index equal to the corresponding characteristics of the series y . step 3. granger causality check [24] of the series from fi  relative to the target indicator y . for each group, we form variants of the var models that include the target indicator and the series from fi  .we exclude a series fi ix   from further analysis if it is not granger causal for the target indicator. step 4. building models. for series with a unit root, regression models linking the past values of the series y with its current values and with the values of the series from fi  , describe a spurious regression if there is no cointegration between the series included in the model [17]. for these series we build var-models, taking the necessary number of differences in advance, but such var-models reflect only short-term dependence. if there is cointegration between the series, we build a vec model to build a forecast that allows us to take into account the long-term dynamics of the series. the construction of var and vec models is associated with the estimation of a large volume of parameters: variables, lags of variables, constants and coefficients of trend, coefficients of error correction equations correction term (in the case of vec models). the volume of estimated parameters increases with the number of variables included in the model, and large samples are required to obtain consistent estimates of model parameters. we consider series in which the length of intervals without structural shifts rarely exceeds 45 years, which even for monthly data is about 50-60 points. with increasing data scale (transition to 2-month, quarterly, semi-annual data, etc.), the sample size decreases significantly, therefore, in order to obtain statistically significant results, it is necessary to reduce the number of regressors, included in the model. 134 z.k. avdeeva, e.a. grebenuyk, s.v. kovriga copyright ©2022 assa adv. in systems science and appl. (2022) according to the results of our experiments on real data, we chose models with two and three variables for monthly data, and models with two variables for quarterly data. analysis of the dynamics of annual and semi-annual data we performed according to the arima models built for the series y . in order to take into account, the possible effects on the values of the target indicator of all the series of the group that have an impact on it, we use an ensemble of models, because as the results of research [25] show, estimates by ensembles of models allow us to obtain more accurate forecasts. step 5. for each of the sf-groups we construct forecasts for ensembles of models at different time scales. forecast construction for individual sf-groups on a certain time scale includes:  evaluating the quality of the built models by schwartz 's criterion, 1 2, ,..., , jkr r r where ir is the value of the schwartz criterion of the model number ,i jk is the quantity of models in sf-group , 1, 2,...,j j m ; calculation of the average value of the schwartz criterion for each sf-group: 1 1 jk j ii j r r k    ;  calculation of forecast weights 1 2, ,..., jkw w w for each of the models in the sf-group by the formula: 1 , 1,2,..., , j j k jk ii p w k k p     where 1 ,i i p r  j is a number of sf-group;  formation of the forecast by the ensemble of models, built using time series from a separate sf-group, by the formula: 1 jk i j i ji y wy     (4) where i jy  is the vector of forecasts for the model with number i of sf-group ,j 1, 2,..., .j m the sizes of vectors are 12, 4, 2, 1, respectively, depending on the scale. for each sf-group, we build models using monthly, quarterly, semi-annual, and annual data. as a result, we get 12 monthly forecasts, four quarterly forecasts constructed using ensembles of models, two semi-annual and one annual forecasts constructed using a time series of target indicator. step 6. correction of forecasts of sf-groups with respect to time scales. we adjust the monthly forecasts jy  (equation (4)) for each sf-group 𝑗 using forecasts made at other time scales. we adjust the monthly forecasts for march and september by quarterly forecasts, the forecast for june with quarterly and semi-annual forecasts, and the forecast for december with quarterly, semi-annual, and annual forecasts. we calculate the adjusted forecast as the weighted average of the forecasts included in it, the weights of which are inversely proportional to the errors of the forecasting models calculated at the interval of model building. step 7. final forecast with expert information. we construct the final forecast as a weighted average of the sf-group weights calculated using equation (3) (section 3.1): 1 , m f i ji y c y    where jy  we calculate by equation (4). raw material price forecasting on commodity markets 135 copyright ©2022 assa. adv. in systems science and appl. (2022) 4. forecast correction in the forecast interval 4.1. application of situation monitoring to identify significant environmental events situation monitoring signals the effects of changes in the external environment that are not currently reflected in the time series data. situation monitoring implements the following functions  regular (or at the request of digital monitoring) tracking and preliminary filtering of information from diverse sources (specialized resources that publish weekly analytical reports, reviews and summaries of the dynamics of commodity markets; news portals and media that provide information about various events);  processing of filtered information about significance events (information occasions) by forming scenarios of the situation development to assess the significance of the consequences of these events on the predicted process (target indicator);  recalculation of sf-groups weights and/or generation of a signal associated with the identification of a change in the trend of the target indicator based on the results of scenario modelling. the basis for tracking and filtering information is the fcm of the situation, which we consider as a semantic model of observation in information sources on significant events and topics. this approach allows us to organize a directed search for information (step 2.1 in figure 1) on the selected system-forming factors of the fcm, taking into account their relationships with each other and the target indicator. the processing of filtered information is as follows. upon detection of an event inf (information occasion), which, according to a preliminary expert assessment, is significant for changing the target indicator and/or other system-forming factors of fcm, we model the "what if" scenario. to form the scenario infs , we determine which system-forming factors inf{ }x can be used to reflect info-occasion inf in infs . next, we, together with experts, evaluate the activity of change factors inf{ }x (their increasing or decreasing and how much) at the time of detection of an info-occasion. these values are the initial data for modelling (according to equation (1)) the net effect of factors inf{ }x on the target indicatory . we get the resulting state, which is a qualitative forecast for the infs scenario. this state determines at a qualitative level the change in the activity of the y (and other factors of the fcm) in the infs scenario. the obtained value of the factor y allows us to assess whether inf affects the change in the target indicatory . after evaluating 0y – value (activity) of the factor y at the initial moment ,t we compare it with the resulting *y – value in the infs scenario. if 0*y y  , then the target indicatory has changed. if 0 0 0( ) ( ) ( * )sign y sign y sign y y      then the trend of y does not change (the trend change indicator ( ) 0in y  ). otherwise 0 0 0( ) ( ) ( * )sign y sign y sign y y      , we fix a trend change y . moreover, if 0( ) 1sign y   and 0 0( ) ( * ) 1sign y sign y y      then the increase of y is replaced by a decrease (trend change indicator 1);in   in the case of 0( ) 1sign y   and 0 0( ) ( * ) 1sign y sign y y      , the decrease of y is replaced by an increase ( ( ) 1in y   ). for recalculation the weights { }ic of sf-groups based on the results of the infs scenario, we use the formula 136 z.k. avdeeva, e.a. grebenuyk, s.v. kovriga copyright ©2022 assa adv. in systems science and appl. (2022) inf inf inf inf 1 2 , ... i i p с c с с с     where inf iс is an estimate of the significance of the sf-group i influence on y , taking into account the activity of system-forming factors in the infs scenario, p is number of sf-groups in which there are active factors associated with the info-occasion inf. each inf iс estimate is calculated as the following sum inf inf , i j j j i q a   where inf jq is cumulative influence of the active factor from the sf-group i on y and inf ja is its degree of activity in the infs scenario, ii is a set of active factor codes for the sf-group in the infs scenario. the weights { }ic of sf-groups characterize the contribution of each group to the change in the target indicator, taking into account the influence significance of active factors in these groups ony and the activity degree of manifestation of these factors in the infs scenario. in other words, only those groups participate in the formation of weights, in which the system-forming factors associated with the info-occasion inf were activated. the obtained weights { }ic of sf-groups and the trend change indicator ( )in y are transferred to step 2.4 of forecast correction (figure 1). 4.2. correction of the forecast based on the results of digital monitoring if the prediction model matches the process, then we add new observations to the prediction formula when they arrive, to reduce uncertainty and increase the accuracy of the prediction. however, the forecast interval is large enough to assume that events may occur within it, leading to drastic changes in the process and requiring correction of the forecasting model. therefore, after making a forecast, it is necessary to monitor the situation and the forecasted process in the macro and business environment. we consider a generalized algorithm for monitoring the situation and the signals it generates in section 4.1. based on the results of monitoring performed during the forecasting period, we adjust the forecast on the forecasting horizon in the following cases:  when we receive new data at each forecast time horizon: month, quarter, six months;  when we get signals from the monitoring algorithms of situation using fcm;  when we receive the signals from time series monitoring algorithms. to successfully adjust the forecast based on the results of detected changes, it is necessary not only to detect the change, but also to determine its type and magnitude. to improve the quality of the numerical forecast, we apply algorithms for detecting changes in series level, volatility, trend direction and relationships between time series: granger causality, and cointegration [18,19]. rules for adjusting the forecast. if the drift has changed in the monitored target indicator, we adjust the deterministic trend coefficient, if the process volatility has changed or we detected deviations in the model coefficients, we recalculate the model. if the relationship between the predicted indicator and the regressors in any of the models has changed (granger causality has been violated), then we exclude this model from the forecast formula. if cointegration is broken for the rows included in the vecm model, we replace the vec model with a var model based on differences. if we find that the cointegration between the series in the vecm model is broken, we replace the vec model with a difference-based var model. raw material price forecasting on commodity markets 137 copyright ©2022 assa. adv. in systems science and appl. (2022) 5. results of experimental study the results of the implementation of the proposed technique we have demonstrated on the example of building a monthly forecast for 2019 prices for ferrous scrap in the commodity market. stage 1. because of the structuring and formalization of the subject area, we developed a fcm, the core of which are the key parameters of the scrap value chain: the price of primary and secondary raw materials, production cost and demand for the final product and system factors affecting changes in the price of scrap. we identified four groups of series corresponding to four groups of system factors: primary (sf1-group) and secondary (sf2group) raw materials for the creation of metal products, finished metal products (sf3-group), macroeconomic indicators (sf4-group). the 2019 forecast was constructed using monthly data (january 2016 december 2018), quarterly, semi-annual, and annual data (2010-2018). for each sf-group, we built a forecast by ensembles of models, taking into account the dynamics of the target indicator and related series at different time scales. we combined the constructed forecasts in the final forecast, taking into account the influence of sf-group factors on the dynamics of the current situation (section 3.2). figure 3 shows for comparison the actual prices for scrap metal for 2019, the monthly forecast for the year ahead obtained at stage 1, the "naive" forecast (for which the forecasts for january 2019, february 2019, ..., etc. are equal to the actual prices in january 2018, february 2019, ...., etc.). fig. 3. the result of the forecasting on stage 1 and naïve. stage 2. the algorithm adjusts the forecast during 2019 based on signals of situation and digital monitoring signals. as part of the situation monitoring information on commodity markets and other events of the external environment in 2019, we conducted a month-bymonth analysis. when detecting events (information occasions), which according to preliminary expert evaluation are potentially significant for the change in the forecasting indicator, we modeled infs scenarios "what happens if" to assess the significance of the consequences of these information occasions on the forecasting process now of their occurrence. in forming the 138 z.k. avdeeva, e.a. grebenuyk, s.v. kovriga copyright ©2022 assa adv. in systems science and appl. (2022) scenario, we correlated each information occasions with the system-forming factors of the fcm. a fragment of the situation monitoring log for 7 months is presented in table 1, which contains signals to be transmitted to the digital forecasting correction block. let us explain the formation of the trend change signal and the formation of weights of significance of groups by the examples of january 2019, march 2019 and august 2019 events. figure 4 demonstrates the initial data and the results of the scenario modeling for the formation of the signals by monitoring the events of january 2019. for example, in january 2019, we identified a potentially significant information factor a fire at a large steel plant in brazil and its shutdown, which caused a sharp increase in steel prices in the u.s. market. the baseline scenario factor activity for this information trigger is given on a linguistic scale (0.2 weak increase, -0.2 weak decrease, 0.5 average increase, etc., see the legend to figure 4). table 1. situation monitoring results: signals generated from the results of scenario modelling for transmission to the forecast correction block observation period 01’19 02’19 03’19 04’19 05’19 06’19 07’19 … 1 2 3 4 5 6 7 8 ( )in y +1  = 0,3 0 -1  = -0,1 0 +1  = 0,1 -1  = -0,1 +1  = 0,3 0( )sign y +1 +1 -1 +1 -1 im 0 ic 1 ic 2 ic 3 ic 4 ic 5 ic 6 ic 7 ic sf1 0,4 0,7 0,6 0,3 0,3 0,7 0,4 sf2 0,3 0,4 0,2 0,2 0,6 sf3 0,2 0,5 0,5 0,3 sf4 0,1 0,3 based on the results of the simulation, we formed a trend change signal for the scrap price ( )in y in january 2019, and adjusted the significance weights for the influence of sfgroups (see column 2 in table 1 and the histogram on the right in figure 4). in march 2019, the situation monitoring algorithm identified changes in demand for steel products in europe and the introduction of us duties on imported steel products. based on these data, the qualitative forecast for the fcm ( infs scenario) showed that in march the significance of the quantitative forecast for the group of factors reflecting demand for finished products (sf3-group) is the highest (see column 2 in table 1). in addition, in august 2019, the numerical monitoring algorithm revealed a lack of relationships between causal factors and target indicator in some forecast models and, as a consequence, a violation of cointegration in them. based on changes identified (march 2019, august 2019), the algorithm assigns the maximum weight (march 2019) to the group of factor models reflecting demand for finished goods, and replaces models with co-integration violations with other models from the same sf group (august 2019). thus, given the signals generated by situation monitoring (table 1), the numerical prediction algorithm recalculates for each sf-group the prediction for ensembles of models. figure 5 demonstrates the results of forecasting in two variants: the forecast at stage 1 (the broken line "stage 1 (algorithm 1)") and with its correction by signals from monitoring carried out on the forecast horizon (the broken line "stage 2 (algorithm 2)"). raw material price forecasting on commodity markets 139 copyright ©2022 assa. adv. in systems science and appl. (2022) fig 4. the examples of scenarios: results for dec’18 and jan’19 we compared the results of forecasting with the following algorithms: "naive" forecast, forecasting using arima autoregressive model, forecast generated at stage 1 without correction at forecast horizon (stage 1) and algorithm with correction at forecast horizon (stage 2). the following metrics were used to assess the quality of the prediction: mean absolute error in percent (mape) and root mean square error (rmse) to assess the accuracy of the prediction: | 1 ˆ100 n t t t h i t y y mape n y      , 2 | 1 1 ˆ( ) n t t t h i rmse y y n     ; mean percentage error (mpe) to assess the systematic error | 1 ˆ100 n t t t h i t y y mpe n y             , where ty is the value of the forecast indicator at time t (monthly), calculated with the horizon h (2019 year) at time t h (2018 year). comparison of forecasting errors was carried out according to the following algorithms: naive forecasting (naive), forecasting by arima model. the results are presented in table 2. the experimental results presented here demonstrate the advantage of our technique. at step 1, the forecast accuracy increases almost twice as much as the "naive" forecast and arima, and the bias, 10.5 and 3 times, respectively. applying correction on the forecast horizon reduces the absolute percentage error by 20% and the bias by a factor of 5 compared to the forecast constructed from data up to december 2018. the results shown in table 2 demonstrate the advantage of the proposed forecasting technique relative to naive and arima forecasts by the criteria of accuracy and bias of forecast. 140 z.k. avdeeva, e.a. grebenuyk, s.v. kovriga copyright ©2022 assa adv. in systems science and appl. (2022) fig. 5. the result of the forecasting on stage 1 and stage 2 table 2. average error of the annual forecast with a monthly split built by different algorithms algorithm mape 2019 rmse 2019($) mpe 2019 algorithm 2 5,29 17,09 -0,2 algorithm 1 6,26 17,9 -1,09 arima 12,5 38,57 -3,5 naive 11,2 35,14 -9,67 6. conclusion this article considered the problem of forecasting commodity prices for the year ahead, broken down by months. the purpose of this study was to form a monthly forecast of the price of a certain type of raw material (target indicator) on the commodity market for the year ahead. when constructing mediumand long-term forecasts, information about future events that change the dynamics of the forecasted value in the interval between forecasts is not available in the data at the time of the forecast. for such situations of uncertainty, we proposed a forecasting technique based on monitoring the situation and regularly including expert information in the generated digital forecast over the forecasting interval. this technique includes  structuring and processing of expert judgments and information from heterogeneous sources through the use of following instruments: a) the fcm, which reflects the structure of relationships (causal influences) between system-forming factors that characterize the main processes in the studied situation on commodity markets; b) scenario modelling on fcm of possible consequences the influence of significant events in the external environment on the change in the target indicator; raw material price forecasting on commodity markets 141 copyright ©2022 assa. adv. in systems science and appl. (2022)  formation of a monthly price forecast for the year ahead, based on information obtained as a result of processing expert judgments and time series data by constructing ensembles of vecm, var models in different time scales. this technique allows us to automate the process of systematic processing of expert and analytical information and its transmission in digitized form for inclusion in a forecast based on quantitative data. at the same time, we consider modelling on the fcm of situation as a driver (driving force) of the forecasting process, which ensures the directed and regular inclusion of expert information in the formation of the forecast and its correction in the forecasting interval. it is important to note that the structure of representing the mutual influence of factors in the fcm is quite flexible, which allows us to quickly make changes to it (adding factors, links) to reflect new aspects of the situation caused by external environment events. the results of the experimental study demonstrate the performance of the proposed technique with respect to the most important criterion for the practical applicability of the forecasting method the accuracy of the resulting forecasts. the experiment showed that the proposed method the forecast error reduced the forecast error in comparison with other methods due to the inclusion of the results of the analysis and modelling of the situation on the fcm in the forecasting process in the current observation mode. the experiment showed that the proposed method significantly reduces the forecast error compared to other methods due to the inclusion in the process of forecast the results of analysis and simulation of the situation on fcm. we see further development of the proposed forecasting methodology in the following directions:  to expand the composition of signal types about a possible change in the development of the situation according to situation and digital monitoring data and their formalization – for the interaction of blocks of qualitative and quantitative forecasting;  to develop a formalized model of the situation monitoring system, combining the fcm of situation, considered as the basis for identifying and filtering of information-search parameters, and modern approaches for setting up search and intellectual analysis of text content.  to develop a formalized model of the situation monitoring system, combining the fcm of the situation, considered as the basis for the identification and filtering of information-search parameters, and modern methods of organization of search and intellectual analysis of texts. the practical significance of the developed forecasting methodology is to increase the effectiveness of expert-analytical and predictive activities in situations of uncertainty and instability. references 1. andreou, a., neophytou, h. & stylianou, c. 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(2009). estimating the impact of extreme events on crude oil price: an emd-based event analysis method, energy economics, 31(5), 768–778, https://doi.org/10.1016/j.eneco.2009.04.003. adv syst sci appl 2025; 1:86–98 published online at https://ijassa.ipu.ru. multiple capture of coordinated evaders in the linear group pursuit problem with a simple matrix and phase restrictions nikolay petrov udmurt state university, izhevsk, russia abstract: in finite-dimensional euclidean space, an analysis is made of the problem of pursuit of two evaders by a group of pursuers, which is described by a system of the form żij = αzij + ui − v, ui, v ∈ v. it is assumed that the evaders use the same control and do not move out of a convex polyhedral set. the pursuers use counterstrategies based on information about the initial positions and the prehistory of the evaders’ control. the set of admissible controls v is a sphere of unit radius with its center at the origin, and the goal sets are the origin. the goal of the group of pursuers is the capture of at least one evader by a given number of pursuers. in terms of the initial positions and parameters of the game, a sufficient condition for capture is obtained. the method of resolving functions, which is used as a basis for analysis, provides sufficient conditions for solvability of the problem of pursuit in some guaranteed time. keywords: differential games, pursuer, evader, capture, multiple capture, conflict-controlled processes 1. introduction mathematical control theory and dynamical game theory provide a fundamental framework for investigating controlled processes of different nature. the schools of n.n. krasovskii and l.s. pontryagin played a key role in establishing these theories and developing a number of classical methods in this area. the methods for investigating dynamical games include those aimed at constructing optimal strategies [9] and methods that ensure a guaranteed result [19]. the main condition in this case is to achieve the goal in hand and to accomplish the task under specific conditions. to obtain the guaranteed result, use is often made of the first method of l.s. pontryagin [19] and the method of resolving functions [3,6,7]. these methods give sufficient conditions to complete the game in finite time from given initial positions. two-player differential game theory was developed further by studying the problem of pursuit of one evader by a group of pursuers and the problem of evasion of one evader from a group of pursuers [4, 5, 8, 10, 22]. b.n. pshenichny obtained [20] necessary and sufficient conditions for capture of the evader. n.l. grigorenko introduced the notion of multiple capture. for the problem with simple motions and equal opportunities he presented necessary and sufficient conditions [7] for multiple capture of the evader. in [14], sufficient conditions were obtained for multiple capture of the evader in l.s. pontryagin’s example with equal opportunities for all participants. the problem of multiple capture of an evader in the presence of defenders was discussed in [2]. ∗corresponding author: kma3@list.ru multiple capture of coordinated evaders... 87 a natural generalization of these group pursuit problems is the situation of conflict interaction of a group of pursuers and a group of evaders. the goal of the group of pursuers is to capture a given number of evaders, and the goal of the group of evaders is the opposite one [1, 12, 21]. the authors of [23] addressed the problem of pursuit of a group of evaders by a group of pursuers under the condition that all evaders use the same control. sufficient conditions for capture of at least one evader were obtained. in what follows, we will call the pursuit problem in which all evaders use the same control the problem of pursuing coordinated evaders. the study presented in [23] was developed, in particular, in [11, 16, 17], where sufficient, and in some cases necessary, conditions for capture of at least one evader were obtained under the condition that all participants have equal opportunities and all evaders use the same control. the problem of pursuing two coordinated evaders in which the goal of the group of pursuers was the capture of at least one evader by two pursuers was treated in [15, 18]. this paper addresses the linear problem of pursuit of two coordinated evaders by a group of pursuers in a differential game with a simple matrix and phase restrictions on the states of the evaders. the goal of the group of pursuers is the capture of at least one evader by a given number of pursuers. 2. formulation of the problem in the space rk (k ⩾ 2) we consider a differential game γ(n+ 2) involving n+ 2 players: n pursuers p1, . . . , pn and two evaders e1, e2. the law of motion of each of pursuers pi has the form ẋi = αxi + ui, xi(0) = x0 i , ui ∈ v. (2.1) the law of motion of each of evaders ej has the form ẏj = αyj + v, yj(0) = y0j , v ∈ v. (2.2) here i ∈ i = {1, . . . , n}, j ∈ {1, 2}, xi, yj , ui, v = { v | v ∈ rk, ∥v∥ ⩽ 1 } , α ∈ r1, α ⩽ 0. in addition, x0 i ̸= y0j for all i ∈ i, j = 1, 2. additionally, it is assumed that each evader ej, j ∈ {1, 2} does not move out of a convex polyhedral set ω = {y ∈ rk | (pl, y) ⩽ βl, l = 1, . . . , r}, where p1, . . . , pr are the unit vectors rk, β1, . . . , βr are the real numbers, and (u, v) is the scalar product. assume that ω = rk for r = 0. introduce new variables zij = xi − yj. then instead of the systems (2.1) and (2.2) we obtain the system żij = αzij + ui − v, zij(0) = z0ij = x0 i − y0j . (2.3) the measurable function v : [0,∞) → rk is called admissible if v(t) ∈ v, y1(t) ∈ ω, y2(t) ∈ ω for all t ⩾ 0. let us call the restriction of the function v to the interval [0, t] the prehistory vt(·) of the function v(·) at time t. the actions of the evaders can be interpreted as follows: there is a center that chooses the same control v(t) for evaders e1 and e2. definition 2.1: we will say that a quasi-strategy ui of pursuer pi is given if a map ui(t, z 0, vt(·)) is defined which associates a measurable function ui(t) = ui(t, z 0, vt(·)) with values in v to the initial position z0 = (z0ij), time t and an arbitrary prehistory vt(·) of admissible control v(·) of evaders ej . copyright © 2025 assa. adv syst sci appl (2025) 88 n. petrov definition 2.2: a q-fold capture (if q = 1, then it is a capture) occurs in the game γ(n+ 2) if there exist time t0 = t (z0), quasi-strategies u1, . . . ,un of pursuers p1, . . . , pn such that for any admissible measurable function v(·) there are numbers i1, . . . , iq ∈ i, l ∈ {1, 2}, and time instants τi1 , . . . , τiq ∈ [0, t0] such that zisl(τis) = 0 for all s = 1, . . . , q. remark 2.1: the capture condition implies that q pursuers perform a capture of one evader and that the time instants may or may not coincide. such a situation can arise if the system consists of q blocks, and in order to put it out of operation, one needs to damage all blocks. 3. auxiliary results definition 3.1 (see [13]): the vectors a1, a2, . . . , as form a positve basis in rk if for any x ∈ rk there exist nonnegative real numbers α1, α2, . . . , αs such that x = α1a1 + α2a2 + . . .+ αsas. we introduce the following notation. intx and cox are the interior and the convex hull of the set x ⊂ rk, respectively, and |j | is the number of elements of the finite set j , λ(h, v) = sup{λ ⩾ 0 | − λh ∈ v − v}, ωl(k) = {(i1, . . . , il) | i1, . . . , il ∈ k and are pairwise different}, where k is a finite set of natural numbers and l is a natural number. theorem 3.1 (see [13]): let a1, a2, . . . , am ∈ rk. the following statements are equivalent. 1. the vectors a1, a2, . . . , am form a positive basis in rk. 2. 0 ∈ intco {a1, . . . , am}. 3. for any p ∈ rk, p ̸= 0, there exists a number l ∈ {1, . . . ,m} for which (al, p) > 0. lemma 3.1: let a1, . . . , am, c1, p1 ∈ rk such that, for each j0 ⊂ j = {1, . . . ,m}, |j0| = q − 1, one has 0 ∈ intco{as, s ∈ j \ j0, c1, p1}. then for any b1, . . . , bq ∈ rk there is µ̂ > 0 such that for all µ > µ̂ the following inequality holds: δ(µ) = min v∈v max { max λ∈ω0 q(j1) min i∈λ λ(ωi, v), (p1, v) } > 0, where j1 = j ∪ {m+ 1, . . . ,m+ q}, ω0 q(j1) = ωq(j) ∪ {(m+ 1, . . . ,m+ q)}, ωi = { ai if i ∈ j, bi−m + µc1 if i = m+ 1, . . . ,m+ q. proof assume that the statement of the lemma is not valid. then there exist b01, . . . , b 0 q ∈ rk for which for any µ̂ > 0 there is µ > µ̂ such that δ(µ) = 0. it follows from the condition δ(µ) = 0 copyright © 2025 assa. adv syst sci appl (2025) multiple capture of coordinated evaders... 89 that there exists vµ ∈ v such that (p1, vµ) ⩽ 0 and max λ∈ω0 q(j1) min i∈λ λ(ωi, vµ) = 0. (3.4) we show that it follows from condition (3.4) that there exist a set j(µ) ⊂ j0, |j(µ)| = q − 1, and a number s(µ) ∈ {m+ 1, . . . ,m+ q} for which the following inequalities hold: (wi, vµ) ⩽ 0 for all i ∈ ( j0 \ j(µ) ) ∪{s(µ)}. (3.5) it follows from condition (3.4) that for any λ ∈ ω0 q(j1) there is a number iλ ∈ λ such that λ(ωiλ , vµ) = 0. by virtue of the properties of the function λ [7] we find that ∥vµ∥ = 1. therefore, the condition λ(ωiλ , vµ) = 0 is equivalent to the condition (ωiλ , vµ) ⩽ 0. let λ1 = {1, . . . , q}. then there is a number i1 ∈ λ1 such that (wi1 , vµ) ⩽ 0. take λ2 = λ1 \ {i1} ∪ {q + 1}. then there is a number i2 ∈ λ2 such that (ωi2 , vµ) ⩽ 0. continuing this process further, we find that for the set λm−q+1 = λm−q \ {im−q} ∪ {m} there is a number im−q+1 ∈ λm−q+1 for which (ωim−q+1 , vµ) ⩽ 0. in addition, for the set λ = {(m+ 1, . . . ,m+ q)} there is a number s(µ) such that (ωs(µ), vµ) ⩽ 0. consider a set j(µ) = j \ {i1, . . . , im−q+1}. then |j(µ)| = q − 1 and (ωs, vµ) ⩽ 0 for all s ∈ j \ j(µ). this proves (3.5). we now show that there exists a set j0 ⊂ j, |j0| = q − 1, such that 0 /∈ intco{as, s ∈ j \ j0, c1, p1}. (3.6) let µ̂ = 1. then there is µ1 > µ̂ for which there are a vector v1 ∈ v, ∥v1∥ = 1, a set j(µ1) ⊂ j, |j(µ1)| = q − 1, and an index s(µ1) ∈ {m+ 1, . . . ,m+ q} such that (ai, v1) ⩽ 0 for all i ∈ j \ j(µ1), (b0s(µ1)−m + µ1c1, v1) ⩽ 0, (p1, v1) ⩽ 0. take µ̂ = µ1 + 1. then there is µ2 > µ̂ for which there are a vector v2 ∈ v, ∥v2∥ = 1, a set j(µ2) ⊂ j, |j(µ2)| = q − 1, and an index s(µ2) ∈ {m+ 1, . . . ,m+ q} such that (ai, v2) ⩽ 0 for all i ∈ j \ j(µ2), (b0s(µ2)−m + µ2c1, v1) ⩽ 0, (p1, v1) ⩽ 0. continuing this process further, we find that there exist a sequence of real numbers {µl}∞l=1, lim l→+∞ µl = +∞, a sequence of vectors {vl}∞l=1, ∥vl∥ = 1, a sequence of sets {j(µl)}∞l=1, j(µl) ⊂ j , |j(µl)| = q − 1, and a sequence of natural numbers {s(µl)}∞l=1, s(µl) ∈ {m+ 1, . . . ,m+ q} for which the following inequalities hold: (ai, vl) ⩽ 0 for all i ∈ j \ j(µl), (b0s(µl)−m + µlc1, v1) ⩽ 0, (p1, v1) ⩽ 0. (3.7) it follows from (3.7) that there exist a subsequence {vlp}∞p=1, ∥vlp∥ = 1, a set j0 ⊂ j, |j0| = q − 1, and a natural number s ∈ {m+ 1, . . . ,m+ q} for which for all p = 1, 2, . . . the following inequalities hold: (ai, vlp) ⩽ 0 for all i ∈ j \ j0, (b0s−m + µlpc1, vlp) ⩽ 0, (p1, vlp) ⩽ 0. (3.8) from the sequence {vlp} one can choose a subsequence converging to v0, ∥v0∥ = 1. assume that the sequence {vlp} itself converges. passing in (3.8) to the limit as p → +∞, we find that copyright © 2025 assa. adv syst sci appl (2025) 90 n. petrov the following inequalities hold: (ai, v0) ⩽ 0 for all i ∈ j \ j0, (c1, v0) ⩽ 0, (p1, v0) ⩽ 0. by theorem 3.1, this implies that 0 /∈ intco{ai, i ∈ j \ j0, c1, p1}. this proves (3.6). we have obtained a contradiction with the condition of the lemma. the lemma is proved. lemma 3.2: let r = 1, α = 0, a1, . . . , am, c1, p1 ∈ rk such that for any j0 ⊂ j = {1, . . . ,m}, |j0| = q − 1, one has 0 ∈ intco{al, l ∈ j \ j0, c1, p1}. then for any b1, . . . , bq ∈ rk there is µ̂ > 0 such that for each µ > µ̂ there is a time instant t (µ) > 0 such that for any admissible control v(·) of evaders e1 and e2 there is a set λ∗ ∈ ω0 q(j1) such that for all l ∈ λ∗ the following inequalities hold:∫ t (µ) 0 λ(ωl, v(s)) ds ⩾ 1, where j1 = j ∪ {m+ 1, . . . ,m+ q}, ω0 q(j1) = ωq(j) ∪ {(m+ 1, . . . ,m+ q)}, ωi = { ai if i ∈ j, bi−m + µc1 if i = m+ 1, . . . ,m+ q. proof take arbitrary vectors b1, . . . , bq ∈ rk. by lemma 3.1, there exists µ̂ > 0 such that for all µ > µ̂ the inequality δ(µ) > 0 is satisfied, where δ(µ) = min v∈v max { max λ∈ω0 q(j1) min i∈λ λ(ωi, v), (p1, v) } . prove that the statement of the lemma is valid for all µ > µ̂. let v(·) be an arbitrary admissible control of the evaders. for each t > 0, µ > µ̂ we define the sets t1(t, µ) = {τ | τ ∈ [0, t], (p1, v(τ)) ⩾ δ(µ)}, t2(t, µ) = {τ | τ ∈ [0, t], (p1, v(τ)) < δ(µ)}. since v(·) is the admissible control of the evaders, the following inequalities hold for all t ⩾ 0: (p1, y1(t)) ⩽ β1, (p1, y2(t)) ⩽ β1. therefore, for all t ⩾ 0 the following inequality holds:∫ t 0 (p1, v(s)) ds ⩽ β0, where β0 = min{β1 − (p1, y 0 1), β1 − (p1, y 0 2)}. then one has β0 ⩾ ∫ t 0 (p1, v(s)) ds ⩾ δ(µ) ∫ t1(t,µ) ds− ∫ t2(t,µ) ds, t = ∫ t1(t,µ) ds+ ∫ t2(t,µ) ds. copyright © 2025 assa. adv syst sci appl (2025) multiple capture of coordinated evaders... 91 this implies that the following inequality holds:∫ t2(t,µ) ds ⩾ δ(µ)t− β0 1 + δ(µ) . next, we have max λ∈ω0 1(j1) min i∈λ ∫ t 0 λ(ωi, v(s)) ds ⩾ max λ∈ω0 1(j1) ∫ t 0 min i∈λ λ(ωi, v(s)) ds. for any nonnegative numbers γλ ( λ ∈ ω0 1(j1) ) the following inequality holds: max λ∈ω0 1(j1) γλ ⩾ 1 m ∑ λ∈ω0 1(j1) γλ, where m = m! q!(m− q)! + 1. therefore, max λ∈ω0 1(j1) ∫ t 0 min i∈λ λ(ωi, v(s)) ds. ⩾ 1 m ∫ t 0 ∑ λ∈ω0 q(j1) min i∈λ λ(ωi, v(s)) ds ⩾ ⩾ 1 m ∫ t 0 max λ∈ω0 1(j1) min i∈λ λ(ωi, v(s)) ds ⩾ 1 m ∫ t2(t,µ) max λ∈ω0 1(j1) min i∈λ λ(ωi, v(s)) ds ⩾ ⩾ δ(µ) m ∫ t2(t,µ) ds ⩾ δ(µ) m (δ(µ)t− β0 1 + δ(µ) ) . hence, max λ∈ω0 1(j1) min i∈λ ∫ t 0 λ(ωi, v(s)) ds ⩾ δ(µ) m (δ(µ)t− β0 1 + δ(µ) ) . the required inequality follows from the last inequality. this proves the lemma. lemma 3.3: let r = 1, α < 0, β1 = 0, a1, . . . , am, c1, p1 ∈ rk such that for any j0 ⊂ j = {1, . . . ,m}, |j0| = q − 1, one has 0 ∈ intco{al, l ∈ j \ j0, c1, p1}. then for any b1, . . . , bq ∈ rk there is µ̂ > 0 such that for each µ > µ̂ there is a time instant t (µ) > 0 such that for any admissible control of evaders e1 and e2 there is a set λ∗ ∈ ω0 q(j1) such that for all l ∈ λ∗ the following inequalities hold:∫ t (µ) 0 e−αsλ(ωl, v(s)) ds ⩾ 1, where j1 = j ∪ {m+ 1, . . . ,m+ q}, ω0 q(j1) = ωq(j) ∪ {(m+ 1, . . . ,m+ q)}, ωi = { ai if i ∈ j, bi−m + µc1 if i = m+ 1, . . . ,m+ q. copyright © 2025 assa. adv syst sci appl (2025) 92 n. petrov proof let v(·) be the admissible control of the evaders. then for all t ⩾ 0 the following inequalities hold: (p1, y1(t)) ⩽ 0, (p1, y2(t)) ⩽ 0. therefore, for all t ⩾ 0 the following inequality is valid: ∫ t 0 e−as(p1, v(s)) ds ⩽ β0, where β0 = min{−(p1, y 0 1),−(p2, y 0 2)}. further reasoning is similar to that used in the proof of lemma 3.2. 4. sufficient conditions for capture with α = 0 theorem 4.1: let r = 1 and suppose that there exists a set i0 ⊂ i, |i0| = n− q, such that for any j0 ⊂ i0, |j0| = q − 1, one has 0 ∈ intco{x0 l − y01, x 0 l − y02, l ∈ i0 \ j0, p1}. (4.9) then a q-fold capture occurs in the game γ(n+ 2). proof assume that i0 = {1, . . . , n− q}. denote c = y01 − y02. since the equation x0 l − y02 = x0 l − y01 + c holds for all l ∈ i , it follows from (4.9) that for any j0 ⊂ i0, |j0| = q − 1, we have 0 ∈ intco{z0l1, l ∈ i0 \ j0, c, p1}. it follows from lemma 3.1 that there exists µ > 0 such that δ(µ) = min v∈v max { max λ∈ω0 q(i) min i∈λ λ(ωi, v), (p1, v) } > 0, where ω0 q(i) = ωq(i0) ∪ {(n− q + 1, . . . , n)}, ωi = { z0i1 if i ∈ i0, z0i2 + µc if i = n− q + 1, . . . , n. it follows from lemma 3.2 that t0 = min{t ⩾ 0 | inf v(·) max λ∈ω0 q(i) min i∈λ ∫ t 0 λ(ωi, v(s)) ds ⩾ 1} is finite. let v(·) be the admissible control of the evaders. define the functions hi(t) = 1− ∫ t 0 λ(ωi, v(s)) ds. let pursuer pi, i ∈ i , construct a control as follows. if the inequality hi(t) ⩾ 0 holds at time t, then we assume ui(t) = v(t)− λ(ωi, v(t))ωi. if τ is the first time instant for which hi(τ) = 0, then we assume that λ(ωi, v(t)) = 0 for all t ⩾ τ. copyright © 2025 assa. adv syst sci appl (2025) multiple capture of coordinated evaders... 93 from the definition of controls and the system (2.3) it follows that for all t ⩾ 0 the following equations hold: zi1(t) = z0i1hi(t), i ∈ i0, zi2(t) = z0i2hi(t)− µc(1− hi(t)), i ∈ i \ i0. (4.10) it follows from lemma 3.2 that there exists λ∗ ∈ ω0 q(t ) for which hi(t0) = 0 for all i ∈ λ∗. two cases are possible. 1. λ∗ ∈ ωq(i0). in this case, zi1(t0) = 0 for all i ∈ λ∗. hence, a q-fold capture of evader e1 occurs. 2. λ∗ = {(n− q + 1, . . . , n)}. then zi2(t0) = −µc for all i ∈ i \ i0. we show that for any j0 ⊂ i , |j0| = q − 1, we have 0 ∈ intco{zi2(t0), i ∈ i \ j0, p1}. (4.11) assume that this is not the case. then there exists j0 ⊂ i , |j0| = q − 1, for which 0 /∈ intco{zi2(t0), i ∈ i \ j0, p1}. hence, there exists a vector v0 ∈ v, ∥v0∥ = 1 such that (p1, v0) ⩽ 0, (zi2(t0), v0) ⩽ 0 for all i ∈ i \ j0. it follows from the condition |j0| = q − 1 that there exists a number l ∈ i \ i0 such that l ∈ λ∗. then from the equation zl2(t0) = −µc we find that (c, v0) ⩾ 0. since the equation zi1(t0) = zi2(t0)− c holds for all i ∈ i0, the inequality (zi1(t0), v0) ⩽ 0 holds for all i ∈ i0 \ j0. therefore, it follows from (4.10) that (z0i1, v0) ⩽ 0 for all i ∈ i0 \ j0. in addition, from the equation (i ∈ i0) zi2(t0) = zi1(t0) + c = z0i1hi(t0) + z0i2 − z0i1 it follows that z0i2 = zi2(t0) + z0i1(1− hi(t0)) and hence (z0i2, v0) ⩽ 0 for all i ∈ i0 \ j0. thus, the following inequalities hold: (p1, v0) ⩽ 0, (z0i1, v0) ⩽ 0, (z0i2, v0) ⩽ 0 for all i ∈ i0 \ j0. let j2 ⊂ i0 \ j0 be a set such that for the set j1 = (i0 ∩ j0) ∪ j2 the equation |j1| = q − 1 is satisfied. then, by theorem 3.1, 0 /∈ intco{z0i1, z0i2, i ∈ i0 \ j1, p1}, which contradicts the condition of the theorem. thus, (4.11) is proved. taking t0 as the initial time instant and using theorem [14], we find that a q-fold capture of evader e2 occurs in the game γ(n+ 2). this proves the theorem. theorem 4.2: let d = rk and suppose that there exists a set i0 ⊂ i, |i0| = n− q, such that for any j0 ⊂ i0, |j0| = q − 1, we have intco{x0 l , l ∈ i0 \ j0} ∩ co{y01, y02} ≠ ∅. (4.12) then a q-fold capture occurs in the game γ(n+ 2). copyright © 2025 assa. adv syst sci appl (2025) 94 n. petrov proof we prove that it follows from condition (4.12) that 0 ∈ intco{x0 l − y01, x 0 l − y02, l ∈ i0 \ j0}. (4.13) assume that there exists j0 ⊂ i0, |j0| = q − 1, for which (4.12) is satisfied, but 0 /∈ intco{x0 l − y01, x 0 l − y02, l ∈ i0 \ j0}. hence, {0} and co{x0 l − y01, x 0 l − y02, l ∈ i0 \ j0} are separable. therefore, there exists v0 ∈ v, ∥v0∥ = 1, such that (x0 l − y01, v0) ⩽ 0, (x0 l − y02, v0) ⩽ 0 for all l ∈ i0 \ j0. therefore, (x0 l , v0) ⩽ γ ⩽ (y0j , v0) for all l ∈ i0 \ j0, j ∈ {1, 2}, where γ = min{(y01, v0), (y02, v0)}. hence, the sets co{x0 l , l ∈ i0 \ j0} and co{y01, y02} are separable. therefore, intco{x0 l , l ∈ i0 \ j0} ∩ co{y01, y02} = ∅, which contradicts (4.12). thus, (4.13) is proved. further reasoning is similar to that used in the proof of theorem 4.1. this proves the theorem. theorem 4.3: let there exist a vector p ∈ rk, p ̸= 0, a number γ ∈ r1 and a set i0 ⊂ i , |i0| = n− q, such that 1) d ⊂ {x ∈ rk | (p, x) ⩽ γ}; 2) for any set j0 ⊂ i0, |j0| = q − 1, we have 0 ∈ intco{x0 l − y01, x 0 l − y02, l ∈ i0 \ j0, p}. then a q-fold capture occurs in the game γ(n+ 2). the validity of this theorem follows from theorem 4.1. example 1. let q = 2, α = 0, k = 2, d = r2, x0 1 = (0, 0), x0 2 = (1, 1), x0 3 = (−1, 0), x0 4 = (−1, 1), x0 5 = (1,−1), x0 6 = (0, 1), y01 = (0,−2), y02 = (1, 2). taking i0 = {2, 3, 4, 5}, we find that the conditions of theorem 3 are satisfied and hence a two-fold capture occurs in the game γ(8). example 2. let q = 2, α = 0, k = 2, d = r2, p1 = (0,−1), β1 = 3, x0 1 = (−1, 1), x0 2 = (1, 1), x0 3 = (−1, 2), x0 4 = (1, 2), x0 5 = (0, 1), x0 6 = (0, 4), y01 = (0, 0), y02 = (0,−1). taking i0 = {2, 3, 4, 5}, we find that the conditions of theorem 4.2 are satisfied and hence a two-fold capture occurs in the game γ(8). taking i0 = {1, 2, 3, 4}, we find that the conditions of theorem 4.1 are satisfied and hence a two-fold capture occurs in the game γ(8). 5. sufficient conditions for capture with α < 0 theorem 5.1: let r = 1, α < 0, β1 = 0 and suppose that there exists a set i0 ⊂ i, |i0| = n− q, such that for any j0 ⊂ i0, |j0| = q − 1, one has 0 ∈ intco{x0 l − y01, x 0 l − y02, l ∈ i0 \ j0, p1}. (5.14) then a q-fold capture occurs in the game γ(n+ 2). copyright © 2025 assa. adv syst sci appl (2025) multiple capture of coordinated evaders... 95 proof assume that i0 = {1, . . . , n− q}. denote c = y01 − y02. since the equation x0 l − y02 = x0 l − y01 + c holds for all l ∈ i , it follows from (5.14) that for any j0 ⊂ i0, |j0| = q − 1, we have 0 ∈ intco{z0l1, l ∈ i0 \ j0, c, p1}. it follows from lemma 3.1 that there exists µ > 0 such that δ(µ) = min v∈v max { max λ∈ω0 q(i) min i∈λ λ(ωi, v), (p1, v) } > 0, where ω0 q(i) = ωq(i0) ∪ {(n− q + 1, . . . , n)}, ωi = { z0i1 if i ∈ i0, z0i2 + µc if i = n− q + 1, . . . , n. it follows from lemma 3.3 that t0 = min{t ⩾ 0 | inf v(·) max λ∈ω0 q(i) min i∈λ ∫ t 0 e−αsλ(ωi, v(s)) ds ⩾ 1} is finite. let v(·) be the admissible control of the evaders. define the functions hi(t) = 1− ∫ t 0 e−αsλ(ωi, v(s)) ds. let pursuer pi, i ∈ i , construct a control as follows. if the inequality hi(t) ⩾ 0 holds at time t, then we assume ui(t) = v(t)− λ(ωi, v(t))ωi. if τ is the first time instant for which hi(τ) = 0, then we assume that λ(ωi, v(t)) = 0 for all t ⩾ τ. it follows from (2.3) that the solution to the cauchy problem has the form zij(t) = eαt · ( z0ij + ∫ t 0 e−αs(ui(s)− v(s)) ds ) . (5.15) from the definition of the controls of the pursuers and (5.15) we find that for all t ⩾ 0 the following equations hold: zi1(t)e −αt = z0i1hi(t), i ∈ i0, zi2(t)e −αt = z0i2hi(t)− µc(1− hi(t)), i ∈ i \ i0. (5.16) it follows from lemma 2 that there exists λ∗ ∈ ω0 q(t ) for which hi(t0) = 0 for all i ∈ λ∗. two cases are possible. 1. λ∗ ∈ ωq(i0). in this case we find that zi1(t0) = 0 for all i ∈ λ∗. hence, a q-fold capture of evader e1 occurs. 2. λ∗ = {(n− q + 1, . . . , n)}. then zi2(t0) = −µeαt0c for all i ∈ i \ i0. we show that for any j0 ⊂ i , |j0| = q − 1, 0 ∈ intco{zi2(t0), i ∈ i \ j0, p1}. (5.17) copyright © 2025 assa. adv syst sci appl (2025) 96 n. petrov assume that this is not the case. then there exists j0 ⊂ i , |j0| = q − 1, for which 0 /∈ intco{zi2(t0), i ∈ i \ j0, p1}. consequently, there exists a vector v0 ∈ v, ∥v0∥ = 1, such that (p1, v0) ⩽ 0, (zi2(t0), v0) ⩽ 0 for all i ∈ i \ j0. since |j0| = q − 1, there exists a number l ∈ i \ i0 such that l ∈ λ∗. then it follows from the equation zl2(t0) = −µeαt0c that (c, v0) ⩾ 0. since the equation zi1(t0) = zi2(t0)− ceαt0 holds for all i ∈ i0, the inequality (zi1(t0), v0) ⩽ 0 holds for all i ∈ i0 \ j0. therefore, it follows from (5.16) that (z0i1, v0) ⩽ 0 for all i ∈ i0 \ j0. also, from the equation (i ∈ i0) zi2(t0) = zi1(t0) + eαt0c = z0i1hi(t0) + eαt0(z0i2 − z0i1) it follows that z0i2 = e−αt0zi2(t0) + z0i1(1− hi(t0)) and hence (z0i2, v0) ⩽ 0 for all i ∈ i0 \ j0. thus, the following inequalities hold: (p1, v0) ⩽ 0, (z0i1, v0) ⩽ 0, (z0i2, v0) ⩽ 0 for all i ∈ i0 \ j0. let j2 ⊂ i0 \ j0 be a set such that for the set j1 = (i0 ∩ j0) ∪ j2 the equation |j1| = q − 1 is satisfied. then, by theorem 3.1, 0 /∈ intco{z0i1, z0i2, i ∈ i0 \ j1, p1}, which contradicts the condition of the theorem. thus, (5.17) is proved. taking t0 as the initial time instant and using theorem [14], we find that a q-fold capture of evader e2 occurs in the game γ(n+ 2). this proves the theorem. theorem 5.2: let d = rk, α < 0 and suppose that there exists a set i0 ⊂ i, |i0| = n− q, such that for any j0 ⊂ i0, |j0| = q − 1, intco{x0 l , l ∈ i0 \ j0} ∩ co{y01, y02} ≠ ∅. (5.18) then a q-fold capture occurs in the game γ(n+ 2). this theorem is proved along the same lines as theorem 4.1. theorem 5.3: let there exist a vector p ∈ rk, ∥p∥ = 1, and a set i0 ⊂ i , |i0| = n− q, such that 1) d ⊂ {x ∈ rk | (p, x) ⩽ 0}; 2) for any set j0 ⊂ i0, |j0| = q − 1, one has 0 ∈ intco{x0 l − y01, x 0 l − y02, l ∈ i0 \ j0, p}. then a q-fold capture occurs in the game γ(n+ 2). the validity of this theorem follows from theorem 5.1. acknowledgements this research was funded by the ministry of science and higher education of the russian federation in the framework of state assignment, project fews-2024-0009. copyright © 2025 assa. adv syst sci appl (2025) multiple capture of coordinated evaders... 97 references 1. alias, i. a. & ibragimov, g. 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[in russian]. copyright © 2025 assa. adv syst sci appl (2025) introduction formulation of the problem auxiliary results sufficient conditions for capture with = 0 sufficient conditions for capture with < 0 microsoft word 1507 adv syst sci appl 2023; 04; 179-187 published online at https://ijassa.ipu.ru. machine learning and geometric mathematical models in kimberlite well classification problems nailia gabdrakhmanova*, pavel klimtsev peoples friendship university of russia (rudn university), moscow, russia abstract: nowadays, the application of mathematical models in geology becomes more and more relevant. the steady trend towards the global digitalization has led to the possibility of using the most modern computational methods in the construction of mathematical models. digitalization, further processing of digital data, their analysis and subsequent modeling contributes to the improvement of production efficiency. the purpose of this paper is the development of various methods of classification of kimberlite wells. the paper presents neural network, statistical and geometric mathematical models for solving the problem of kimberlite well classification. the problem was solved using geological and exploration data from wells drilled in the süldükar and ulakhan-kurung-yuryakh areas located in western yakutia. for the constructed models the estimations of the models' qualities were obtained, the comparative analysis of the models was carried out. the analysis of mathematical models showed that the most accurate models are neural network models and models using geometric methods. keywords: five factor analysis, neural network, kimberlite wells, classification, persistent diagram, curvature, topological data analysis. 1. introduction nowadays in geology there is a lot of quantitative data obtained during exploration subjected to digitization. this makes it possible to use geological and exploration data in mathematical modeling to predict certain processes and phenomena of interest to geologists. in particular, such a task arises in the exploration of diamond deposits where kimberlite is present. since its search is very costly, it is desirable to use mathematical models to predict the presence of kimberlite based on samples taken from the wells. a kimberlite pipe [1] is understood as a vertical or near-vertical geologic body formed when magma breaks through the earth's crust. a kimberlite pipe is usually filled with kimberlite [2], a series of magmatic ultramafic rocks of extrusive facies that form explosion tubes as well as dikes and sills. they often contain xenoliths of mantle rocks and sometimes contain diamonds of industrial concentrations. the purpose of the study is the development of mathematical models to solve the problem of classifying wells by the presence of kimberlite based on the data of geological exploration of wells. 1.1. data the material for the study was the data of geological surveys of kimberlite wells drilled in the süldükar and ulakhan-kurung-yuryakh areas located in western yakutia. the object of the study is two data sets of geologic features, where such important parameters as: their occurrence, their presence and their number in each drill hole are indicated. the data were collected from two areas located geographically in western yakutia: süldükar, where kimberlite was found, and ulakhan-kurung-yuryakh, where kimberlite is absent. during special documentation of core [3,4] – rock samples extracted from wells of two areas in western yakutia – with kimberlite (46 wells, süldükar) and without kimberlite (103 wells, ulakhan-kurung-yuryakh), tectonic and mineralogical features in the host strata of the lower * corresponding author: gabd-nelli@yandex.ru 180 n. gabdrakhmanova, p. klimtsev copyright ©2023 assa adv. in systems science and appl. (2023) paleozoic were recorded. the total area of both sites under consideration is 4 km2. the core is a cylindrical column (pillar) of rock strong enough to maintain monolithicity. the density of drilling networks is correlated, and for a competent comparison of areas, wells of near-tubular space were excluded on a 20x20m network. also, for correct mathematical analyses, the parameter of feature occurrence per drilled meter of rock was adopted. 1.2. method to solve the problem of object classification, several parallel models were built: a model using factor and cluster analysis, neural network models, models using methods of differential geometry and topological data analysis. all models showed a satisfactory solution to the problem. according to the analysis of solutions, all models have an acceptable calculation error. methods and models of factor analysis, as well as the method of principal components [6], are aimed at compression of information, i.e. reduction of the dimensionality of the feature space (this premise of the possibility of reducing the feature space in factor analysis is based on the mutual correlation of the initial features). within the development of the factor analysis model, the methods proceed from a common basic idea, in which the structure of relationships between p analyzed features (𝑋( ), 𝑋( ), … , 𝑋( )), can be explained in such a way that all these variables depend (either linearly or otherwise) on a smaller number of other, not directly measurable (latent) factor 𝑓( ), 𝑓( ), … , 𝑓( ) , (𝑟 < 𝑝), which are called common and which in most models are constructed so that they turn out to be mutually uncorrelated. cluster analysis was performed on the constructed factors using the k-means method. the action of the clustering algorithm is such that it seeks to minimize the total quadratic deviation of cluster points from the centers of these clusters: 𝑉 = ∑ ∑ (𝑧 − 𝜇 )∈ , where 𝑘 is the number of clusters, 𝑆 is the clusters obtained, 𝑖 = 1,2, … , 𝑘 , 𝜇 is the centers of masses of all vectors 𝑧 from the cluster. neural networks [7] are mathematical models that do well in the task of classification. classification is the process of finding a function that helps to divide a set of data into classes. the task of a classification algorithm is to find a mapping function to map an input vector x to a discrete output y. in neural network modeling, a neural network is first trained on a training dataset and then, the trained neural network can be used to classify new data. the quality of the solution depends on the choice of neural network architecture, the amount of data, and the degree of data noise. to evaluate the accuracy of the solution of the problem, the solution of the neural network on the test set is used. geometric mathematical models are based on the idea that topology and geometry provide a powerful approach to obtaining reliable qualitative and sometimes quantitative information about the structure of data. they are developed from the incorporation of geometric and topological methods, dealing with point clouds, i.e. finite sets of points equipped with a distance function. point clouds can be viewed as finite samples taken from a geometric object, possibly with noise. mathematical models adapt tools from different sections of geometry to study point clouds. the classical ricci curvature plays an important role in the geometric analysis of riemannian manifolds. the ricci curvature at a given point characterizes the average curvature of sectional curvatures in all directions. the notion of ricci curvature for metric spaces of general form was first introduced by bakri and emery [8], ollivier [9] in 2009 gave a definition of rough ricci curvature on markov chains, which can be used for metric spaces generated by graphs. and in 2011, lin, lu and yau [10] modified ollivier's definition for the ricci curvature of markov chains on metric spaces. this paper uses a method that combines network analysis techniques with classical correlation, graph theory and local clustering coefficient. this provides a novel graphical representation of the features that solves the problem. by analogy with curvature in riemannian geometry, we machine learning and geometric mathematical models… 181 copyright ©2023 assa adv. in systems science and appl. (2023) interpret ricci curvature as the amount of overlap between the neighborhoods of two neighboring vertices. for the solution we use the notion of local clustering coefficient, which shows the density of triangular relations. studies show that curvature is usually extremely low in random graphs. clusters of high curvature have a very non-random structure. depending on the problem to be solved, different methods of ricci curvature estimation are used. to solve the clustering problem, the watts-strogatz formula [11] is used in this paper: 𝑐𝑢𝑟𝑣(𝐴) = 𝑡 ( 𝑣(𝑣 − 1) 2 ) . here 𝑐𝑢𝑟𝑣(𝐴) is the vertex curvature, 𝑣 is the numbers of vertices and 𝑡 is the number of triangles that are formed by the edges of the graph containing vertex 𝐴. this function is a function of two variables. note that the value 𝑣(𝑣 − 1)/2 is the maximum number of triangles that can be formed by all vertices of the graph, hence 𝑐𝑢𝑟𝑣(𝐴) lies between 0 and 1. figure 1 shows examples of graphs and vertex curvature. (a) (b) fig 1. (a) the graph has vertices 𝑉 = 6, for vertex 5 : 𝑐𝑢𝑟𝑣(5)=0; (b) the graph has vertices v = 6, for vertex 3 : 𝑐𝑢𝑟𝑣(3) = 1 5. recently, there has been an increasing interest in topological data analysis (tda) [1217]. topological data analysis is a valuable tool for data analysis and visualization, which can be applied in various fields such as computer vision, bioinformatics and graph theory. the tools of topological analysis include barcodes, battie numbers and persistence diagrams. we propose a method to compare two data representations of two clusters in feature space. to investigate the point cloud in this problem, we used the vietoris-rips complex. the vietoris-rips complex is a symplectic complex whose simplexes are all possible sets of points of a given metric space in which pairwise distances between points do not exceed a given positive number. the constructed persistence diagram can be used to understand the features and relationships between features in the data. 2. results and discussion mathematical statement of the problem. we are given 10 attributes of 46 objects of class ‘0’and 10 attributes of 107 objects of class ‘1’. objects of class ‘0’ – wells where there is kimberlite; objects of class ‘1’ – wells where there is no kimberlite. it is necessary to build a mathematical model to classify objects by 10 features. the features investigated were: 0) tectonic fractures (r) 1) extensive pyritization (q) 2) fluidizite breccias (l) 3) gaping fractures (m) 4) recrystallization (t) 182 n. gabdrakhmanova, p. klimtsev copyright ©2023 assa adv. in systems science and appl. (2023) 5) drag folds (x) 6) microfossils (a) 7) subhorizontal stylolite seams (v) 8) ogling (d) 9) slip mirrors with subvertical grooves (f). we will use the number in the list later in the article as a feature code. 2.1. statistical models initially we have 10 geological features under consideration, but only 4 of them, namely tectonic fractures (r), extensive pyritization (q), fluidisite breccias (l) and drag folds (x), were used in the factor analysis because they occur in both areas. the factor analysis is based on the correlation matrix between variables. tables 2.1 and 2.2 present the correlation matrices calculated using ibm spss statistics 23 program. the table 2.3 and 2.4 below presents two different tests: the kaiser-meyer-olkin (kmo) measure of sampling adequacy and bartlett's test of sphericity and bartlett's test of sphericity for süldükar and ulahan respectively, found using statistics. table 2.1 correlation matrix of indicators (süldükar) variable r q l x r 1 0.02 -0.07 0.2 q 0.02 1.0 -0.05 -0.07 l -0.07 -0.05 1.0 0.11 x 0.2 -0.07 0.11 1.0 table 2.2 correlation matrix of indicators (ulahana) variable r q l x r 1.00 -0.06 -0.02 -0.13 q -0.06 1.00 -0.06 -0.09 l -0.02 -0.06 1.00 -0.01 x -0.13 -0.09 -0.01 1.00 the table 2.3 and 2.4 below presents two different tests: the kaiser-meyer-olkin (kmo) measure of sampling adequacy and bartlett's test of sphericity and bartlett's test of sphericity for süldükar and ulahan respectively, found using statistics. table 2.3 kmo and bartlett’s (süldükar) kmo 0,461 bartlett’s test of sphericity approx.chi-square 2.956 df 6 sig. 0.814 table 2.4 kmo and bartlett’s (ulahana) kmo 0.456 bartlett’s test of sphericity approx.chi-square 3.336 df 6 sig. 0.766 both tables do not show high values of correlations, therefore, the variables are not correlated with the same factors. we accept the null hypothesis in both cases according to bartlett's criterion of sphericity. the approximate values of 𝜒 statistics are 2.956 and 3.336 respectively with 6 machine learning and geometric mathematical models… 183 copyright ©2023 assa adv. in systems science and appl. (2023) degrees of freedom, they are insignificant at 0.05 level of significance. the values of kmo statistics (0.461 and 0.456) are also smaller (< 0.5); significance 0.814 and 0.766. thus, it is questionable to consider factor analysis as an acceptable method for analyzing correlation matrix data. tables 2.5, 2.6 show the eigenvalue tables and figures 2.1, 2.2 stony scree plots for süldükar and ulahan, respectively, plotted using statistica program. table 2.5 table of eigenvalues (süldükar) the initial eigenvalues component total % variance total % 1 1,213 30,328 30,328 2 1,101 27,526 57,853 3 0,951 23,786 81,640 4 0,734 13,360 100,000 fig. 2.1. graph of rocky scree (süldücar) table 2.6 table of eigenvalues (ulahana) the initial eigenvalues component total % variance total % 1 1,133 28,321 28,321 2 1,077 26,919 55,240 3 0,988 24,688 79,927 4 0,803 20,073 100,000 184 n. gabdrakhmanova, p. klimtsev copyright ©2023 assa adv. in systems science and appl. (2023) fig. 2.2. graph of rocky scree (ulahana) the result of this clustering method is as follows: süldükar 32 out of 46, ulahan 37 out of 103. in summary, the classification accuracy of the factor analysis method was approximately 46.3%. 2.2. neural network models tuples were used to build and implement the neural network model <x,y> : input vector x consisting of 10 elements (geologic features), one output y producing 0 or 1 (class number – süldükar and ulahan respectively). neural networks of multilayer perceptron (mlp) architecture were chosen to solve the problem. deep learning neural network was not considered because the data sample size is small. the architecture and parameters of the constructed mlp: two hidden layers of dimensionality 20 and 15 neurons respectively, one output producing 0 or 1 (class number süldükar and ulahan respectively), number of epochs 200, learning rate 0.2 (const), activation function logic function (sigmoid). 65% of the total data sample was used for training the neural network, while the remaining 35% (53 wells) was used for testing the network. the model was implemented using the open-source web application jupyter notebook in the python programming language. the mlp multilayer perceptron model performed better than the factor analysis method, namely, the total percentage of correctly classified objects, i.e. wells, in the two classes was approximately 85 % while the factor model performed approximately 46 %. 2.3. modelling using differential geometry methods to solve the problem, we used methods of graph curvature estimation using the watts–strogatz formula. curvature estimates were found for each cluster separately. first, we switched from continuous space to discrete space and constructed a complete graph. each vertex of the graph was assigned one of 10 features and connected all vertices in pairs by edges. each edge has a weight equal to the sample correlation coefficient of adjacent vertices. the constructed graph is called a correlation network. algorithm 1 step 1. construct the correlation network. step 2. remove edges with weight less than h, where h is the selected threshold value. step 3. for the resulting connected graph, compute the curvature estimate for all vertices using the watts-strogatz formula. an example of correlation network for süldükar cluster is presented in fig. 2.3. machine learning and geometric mathematical models… 185 copyright ©2023 assa adv. in systems science and appl. (2023) an example of correlation network for ulahan cluster is presented in fig. 2.4. the arithmetic mean of the vertex curvature is taken as the curvature of graph g: 𝑐𝑢𝑟𝑣(𝐺 ) = 1 𝑛 𝑐𝑢𝑟𝑣(𝑗) (1) where 𝑖 is the cluster number, 𝑗 is the vertex (feature) number, 𝑛𝑖 is the number of vertices of the graph 𝐺𝑖. fig. 2.3. the graph of cluster ‘0’ (süldükar) fig. 2.4. the graph of cluster '1' (ulahana) curvature estimates calculated by formula (1): 𝑐𝑢𝑟𝑣(𝐺0) = 0.1, 𝑐𝑢𝑟𝑣(𝐺1) = 0. we obtained that 𝑐𝑢𝑟𝑣(𝐺0) > 𝑐𝑢𝑟𝑣(𝐺1). this indicates that the feature space of cluster ‘1’ is flat, while the feature space of cluster ‘0’ is convex. the graphs in fig. 2.3, 2.4. give a visualization of the observation points of each cluster. for the study, it is important which vertices are included in the connected graph and the shape of the vertex connectivity. the research conducted using this method identified the features of cluster ‘0’ that have the most stable connection with each other. these features were used in the topological analysis of the data. algorithm 2 is a well classification algorithm. algorithm 2 1. identify the vertices (features) that form a connected graph for each cluster. 2. compute the centers of each cluster 3. for new features, calculate distances to the centers 4. assign the object to the cluster for which the distance to the center is smaller. classification of objects that were not used in the solution of the problem showed the correctness of the solution 88%. 2.4. topological data analysis let x be the space to which the observation points belong, and let x have a euclidean metric. vjetoris–rips complexes with different radii have been constructed for the considered metric space (x,d). the symplectic complexes are filtered by successively increasing the radius. the algorithm for their construction from the point cloud and the metric contains information about the filtering of the complex, that is, the increasing chain of embeddings of subcomplexes. the filtering thus contains, among other things, geometric information about the original point cloud, which is encoded in the form of a so-called persistence diagram computed on it [7]. we have 186 n. gabdrakhmanova, p. klimtsev copyright ©2023 assa adv. in systems science and appl. (2023) constructed persistence diagrams using the package ripser. figures 2.5, 2.6 shows the persistence diagram. in the persistence diagram, the abscissa axis marks the time of birth of the complex (birth), the ordinate axis time of death (death). the difference between the time of death and birth of the simplex is taken as the lifetime of the complex. fig. 2.5 shows the persistence diagram for cluster ‘0’ (süldükar). fig. 2.6 shows the persistence diagram for cluster ‘1’(ulahana). barcodes were constructed from the persistence diagrams. let us denote by 𝐿 the total length of barcodes for homologs 𝐻 . then the average value of the euler characteristic can be defined by the formula 𝜒 = 𝐿 − 𝐿 + 𝐿 . fig. 2.5. the persistence diagram for cluster ‘0’ (süldükar) fig. 2.6. the persistence diagram for cluster ‘1’ (ulahana) as a result of the calculations, we obtained that the euler characteristic is larger for cluster ‘0’, than for cluster ‘1’. the results of calculating barcode lengths are summarized in table 2.7. table 2.7. the euler characteristic cluster ‘0’ cluster ‘1’ l(h0) 7.4 3.8 l(h1) 2.8 0.4 l(h2) 2.6 0 𝜒 7.1 3.4 algorithm 2 is chosen as the classification algorithm. conclusion neural network models and mathematical models constructed using the tools of statistical analysis, differential geometry and topology are the main tools in the ongoing research. algorithms for solving the problem by different methods have been developed and comparative analysis has been carried out. at the first stage, the classification problem was solved using mathematical statistics and neural networks. the comparison of the accuracy of the problem solution showed the advantage of machine learning and geometric mathematical models… 187 copyright ©2023 assa adv. in systems science and appl. (2023) neural network models. however, neural network models do not always allow to interpret the results of the problem solution. in contrast to ns models, geometric models allowed not only to qualitatively solve the classification problem, but also gave visual and numerical description of the observation points in the feature space. as numerical characteristics of the observation point arrays of the two clusters, we used the estimates of the ricci curvatures of the correlation networks and the estimates of the euler features computed from barcodes. the solution to the problem of computing the ricci curvature estimates identified the most significant relationships in the feature space for wells with the presence of kimberlite. the study showed that the integration of digitalization, neural network modeling, and geometric mathematical models can qualitatively solve the classification problem on exploration data. references 1. ignatov, p. a. & novikov, k. v. (2019). field diagnostics of tectonic disturbances and fluid fracture formations in kimberlite-bearing sediments of the lower paleozoic: methodical manual. mirny, russia: alrosa. 2. milashev, v. a. (1984). explosion tubes. leningrad, ussr: nedra. 3. zorina, t. g. & slonimskaya, m. a. (2010). marketing research: textbook. minsk, belarus: bseu. 4. ilupin, i. p., vaganov, v. i., prokopchuk, b. i. (1990). kimberlites: handbook. moscow, ussr: nedra. 5. afonin, p. n., afonin, d. n. (2017). statistical analysis with the application of modern software tools: textbook. saints-petersburg, russia: 1c "intermedia". 6. aivazyan, s. (2010). methods of econometrics: textbook. moscow, russia: infra-m. 7. haikin, s. (2006) neural networks: a complete course, 2nd edition. moscow, russia: williams. 8. bakry, d. & emery, m. (1985). diffusions hypercontractives [hypercontractive diffusions], séminaire de probabilités de strasbourg, 19, 177–206, [in french]. 9. ollivier, y. (2009). ricci curvature of markov chains on metric spaces, j. funct. anal., 256(3), 810–864. 10. lin, y., lu, l. y. & yau, s. t. (2011). ricci curvature of graphs, tohoku mathematical journal, 63, 605–627. 11. watts, d. j. & strogatz, s. h. (1998). collective dynamics of 'small-world' networks, nature 393, 440–442. 12. edelsbunner, h., letscher, d., zomorodian, a. (2002). topological persistence and simplification, discrete comput. geom., 28, 511–533. 13. carlsson, g., zomorodian, a. computing persistent homology // in “proc. 20th ann. sympos.comput. geom, 2004, 347–356. 14. cohen-steiner, d., edelsbrunner, h. & harer, j. (2007). stability of persistence diagrams, discrete & computational geometry, 37(1), 103–120. 15. bubenik, p. (2015). statistical topological data analysis using persistence landscapes, the journal of machine learning research, 16(1), 77–102. 16. gabdrakhmanova, n. & pilgun, m. (2021). neural network technologies and topological analysis of social media data, advances in systems science and applications, 21(3), 101–112. 17. gabdrakhmanova, n., fedin, v. & matsuta.b. (2020). the modeling of forecasting new situations in the dynamics of the economic system on the example of several financial indicators, proc. of the 14th international symposium intelligent system (moscow, russia). microsoft word abrosimov_mazurov_assa_2021_edited adv syst sci appl 2022; 04; 162-175 published online at http://ijassa.ipu.ru. the collective behavior of robots based on the internet-of-things paradigm vyacheslav abrosimov1*, alexander mazurov2 1) ministry of defence of the russian federation, moscow, russia e-mail: avk787@yandex.ru 2) trapeznikov institute of control sciences, russian academy of sciences, moscow, russia e-mail: alexander.mazurov08@gmail.com abstract: this paper considers a group of autonomous robots performing collective tasks in an antagonistic environment. the environment can significantly hamper information exchange and decision-making in the group. well-known methods are intended for performing collective tasks by a large group of relatively simple objects and have obvious advantages (high reliability, potential complexity increase, etc.). however, these methods neglect the specifics of collective behavior in an antagonistic environment. moreover, they are inapplicable when a collective mission is fulfilled by a few objects expensive for production and maintenance. we propose a new approach to collective mission fulfillment based on the principle of collectivism and the internet-of-things paradigm. the novelty of the approach is that each robot in the group is valuable: with different resources and functionality, they assist each other. the robots execute different roles within the group, operating independently to fulfill a collective mission. the robots are equipped with observation sensors and motion detectors. the functions and resources of robots are represented as external services. a service-oriented mutual assistance architecture is designed for robots to resolve conflicts and problem situations caused by the environment: each robot can use the resources and functionality of other robots via online requests when needed. such spontaneous relations provide a fundamental opportunity for the collective management of unpredicted situations and the possibility of changing behavioral scenarios when fulfilling a collective mission, leading to self-organization. the effectiveness of this approach is illustrated by an example: a group of robots attacks a complex target secured by a defense system. keywords: robot, antagonistic environment, internet-of-things, collective mission, mutual assistance, conflict resolution, cooperative strategy. 1. introduction in many applications, heterogeneous robots have to move cooperatively in time and space to fulfill a collective mission. preset cooperative strategies are almost not implemented in practice due to different factors. here, we mention objective natural factors (night-time, fog, rain, etc.) and subjective factors, e.g., an active counteraction of other robots pursuing opposite goals. this counteraction of an environment may have critical consequences for the efficiency of collective mission fulfillment. theoretically, there are two fundamental approaches to construct collective behavioral strategies of robots. within the first approach, robots use some mechanism for adapting to a current situation. note that such adaptation must be collective, and biological adaptation mechanisms [7] are efficient. like biological specimen, robots in these mechanisms are comparatively independent and obey rather simple rules without cooperation. the development of biological algorithms requires much time, accumulated experience, and knowledge. the second approach is self-organization: under limited time to solve tasks and * corresponding author: avk787@yandex.ru the collective behavior of robots based on the internet-of-things paradigm 163 copyright ©2022 assa. adv. in systems science and appl. (2022) an active counteraction of an environment, the group control of robots should be designed in the class of self-organizing systems [20]. self-organization implies a joint assessment of situational awareness but an autonomous motion of robots. in this case, each robot performs its task as part of the collective mission. however, under existing threats to the collective mission (e.g., failure, termination, or restricted functionality), cooperation and assistance can be reasonable or vital for robots. this paper draws an analogy between the collective control principles for a group of autonomous robots and the internet-of-things (iot) paradigm. network design and communication of things are crucial for the iot. speaking simplistically, the sensors of systems communicate and control the sensors of the other systems. such a statement seems intuitively clear for things (systems) controlled without intelligent means. if control systems of objects are intelligent, then sensor commands from objects to other objects may contradict their individual behavioral strategies. collective control design for groups of such intelligent objects requires negotiations. as a rule, negotiations are organized using the principles of competition [16]. considering earlier publications, we hypothesize (and try to validate) that in such conditions, an efficient approach is to train control systems of autonomous robots based on the principles of collectivism. each robot should be ready to execute another (generally, subordinate) role, allocating its functions and resources to other robots whenever necessary, for fulfilling the collective mission in a current situation. 2. a survey of related literature the subject matter is at the junction of several fields, namely, machine-to-machine interaction (m2m), the internet of things (iot), and the group motion of robots. however, all three fields together were addressed in a few publications. according to the surveys on m2m methods [10, 13, 17, 18], a unified platform would hardly be developed. at the same time, numerous ореn sourсe realizations are expected to yield good engineering solutions for interacting robots with collective tasks. as emphasized in the paper [12], robots performing a common task must exchange information about the environment. in particular, jha and gupta proposed a communication architecture with an algorithm for searching the lost network members to route the information from a lost source robot to a destination robot. an appropriate robot for routing in the network is selected based on its load, estimated by different factors (ongoing processes, battery power, connectivity, or storage space). the idle robots have an effective role in the network to increase the efficiency of the communication system. the authors [11] extended the computation and information-sharing capabilities of networked robotics by proposing a cloud robotic architecture. the cloud robotics architecture leverages the combination of a virtual ad-hoc cloud formed by machine-to-machine (m2m) communications among participating robots. cloud robotics utilizes elastic computing models, in which resources are dynamically allocated from a shared resource pool in the cloud to support task offloading and information sharing in robotic applications. the paper [14] considered the conceptual similarities and differences of m2m and iot. as noted, m2m supplier competencies tend to focus on the “plumbing” aspects mentioned earlier, particularly embedded hardware and cellular telecommunications networks. many are starting to add cloud capability through internal development, acquisition, or partnering, but this represents new terrain for most m2m suppliers. iot solution suppliers, on the other hand, tend to emphasize software capabilities and particularly enterprise integration. these are important distinctions. duarte et al. [8] proposed swarm robotics systems to carry out marine environmental monitoring missions. in swarm robotics systems, each individual unit is relatively simple and inexpensive. the robots rely on decentralized control and local communication, allowing the swarm to scale to hundreds of units and cover large areas. the cited authors studied the 164 v. abrosimov, a. mazurov copyright ©2022 assa adv. in systems science and appl. (2022) application of a swarm of aquatic robots to environmental monitoring tasks. the first part of the study synthesized swarm control for a temperature monitoring mission and validated the results with a real swarm robotics system. then, the authors conducted a simulation-based evaluation of the robots’ performance over large areas and with large swarm sizes and demonstrated the swarm’s robustness to faults. according to the results obtained, swarm robotics systems are suited for environmental monitoring tasks by efficiently covering a target area, allowing redundancy in the data collection process, and tolerating individual robot faults. the paper [6] presented a survey of related work in the area of self-organization and discussed future research opportunities and challenges for self-organization in the iot. athreya and tague considered the process of discovering available peer devices to support and initiate communications during self-organization. they discussed how establishing peer connectivity could lead to end-to-end path establishment allowing for connectivity in the self-organized network. service recovery management is the process of recovering from local failures of devices and avoiding network service disruptions in the self-organized network. yadav, mccann, and pereira [19] introduced an extended and improved emergent broadcast slot (ebs) scheme, which facilitates collaboration for robust communication and is energy efficient. in the ebs, nodes communication units remain in sleeping mode and are awake to communicate. the ebs scheme is fully decentralized: nodes coordinate their wakeup window, partially overlapping within each duty-cycle, to avoid message collisions. in the paper [15] closely relating to the subject matter, ray introduced a new concept called the internet of robotic things (iort). this concept tackles the issues for supporting control and monitoring activities at deployment sites and industrial automations, where intelligent things can monitor peripheral events, induce sensor data acquired from various sources, and use ad-hoc, local, and distributed “machine intelligence” to determine an appropriate course of actions. the ultimate goal is to control or disseminate static or dynamic position-aware robotic things in the physical world through a seamless manner by providing a means for utilizing them as the iort. although progressive advancements can be seen in multi-robotic systems, and robots are constantly getting enriched by easier developmental functionalities, such vertical robotic service-centric silos are not enough for continuously and seamlessly supporting for which they are meant. thus, the collective management of unpredicted situations and the possibility of changing scenarios were not properly considered in the literature on the collective behavior of objects (particularly robots). if an object cannot perform an assigned task (due to exhausted resources, new conditions in the environment, etc.), it becomes unnecessary for the group. well-known methods are intended for performing collective tasks by a large group of relatively simple objects. they have obvious advantages such as high reliability (loss of one object does not affect the entire group’s performance), potential complexity increase (the complexity of tasks can be increased by adding new objects to the group), and others. however, these methods neglect the specifics of collective behavior in an antagonistic environment. moreover, they are inapplicable when a collective mission is fulfilled by a few objects expensive for production and maintenance. of particular interest are practical situations when an object enters the enemy’s counteraction zone (is exposed to a threat), and other objects in the group have the resources and functionality to eliminate this threat. therefore, we propose a new approach to collective mission fulfillment based on the principle of collectivism and the internet-of-things paradigm. the novelty of the approach is that each robot in the group is valuable: with different resources and functionality, they assist each other. the robots execute different roles within the group, operating independently to fulfill a collective mission. the robots are equipped with observation sensors and motion detectors. the functions and resources of robots are represented as external services. a service-oriented mutual assistance architecture is designed for robots to resolve conflicts and the collective behavior of robots based on the internet-of-things paradigm 165 copyright ©2022 assa. adv. in systems science and appl. (2022) problem situations caused by the environment. within the internet-of-things paradigm, each robot can use the resources and functionality of other robots via online requests when needed. such spontaneous relations provide a fundamental opportunity for the collective management of unpredicted situations and the possibility of changing behavioral scenarios when fulfilling a collective mission, leading to self-organization. the effectiveness of the proposed approach is illustrated by an example: a group of robots attacks a complex target secured by a defense system. 3. robot’s situational awareness each robot has to implement an individual control strategy and coordinate its actions with other group robots. therefore, each robot needs to know and predict its position and the situation in an environment (particularly the states of other robots). in other words, each robot must have situational awareness [9]. situational awareness forms the basis of the individual behavioral rules of a robot designed to fulfill a collective mission within a group. nowadays, researchers identify three main elements in the concept of situational awareness: information about the environmental situation in time and space, situation assessment and prediction (scenarios of further development) in the form of events, individual actions, and other participants’ actions. in practice, situational awareness is realized by specific models of data acquisition, accumulation, storage, and analysis, including access rules for robots. 3.1. information about the environment there exist two main sources of information about the environment. the first source is the observation systems of a robot. this group includes various technical means, e.g., optical, ultrasonic, infrared sensors, etc. however, the technical capabilities of such sensors to observe and measure environmental parameters are limited in principle. the second source is external monitoring systems, e.g., aerospace monitoring systems, remote information support systems deployed at a considerable distance to the operational space of robots, and others. as a rule, such monitoring systems enable robots to perform collective tasks. the sources mentioned can be used to design a full-scale geo-informational system to describe situational awareness. this paper considers the following simple case. we divide the operational space of a group of robots into a finite number of domains (segments). each segment can be characterized by different parameters: coordinates, size, the probability of problem occurrence, and others. (a problem is an event in the environment that affects the operation of a robot in a segment.) let us adopt an elementary description of a segment:  , , , , , , , ,e e r r r k kx y z h seg e pseg  (3.1) with the following notations: , ,r r rx y z are the coordinates of the center of the segment seg in a given coordinate system; h is the shape of the segment seg; δseg are the dimensions of the segment seg; e is an event observed or predicted in the segment seg; e kp is the probability (or possibility) of the event е for the k-robot in the segment seg; e k is the probability (or possibility) of threat to the k-robot’s task under the event е in the segment seg. assume that an event is detected by the robot’s sensors or external monitoring systems. the range of possible events and threats (including their characteristics e kp and e k ) should be defined by simulating group motion and learning in advance. 166 v. abrosimov, a. mazurov copyright ©2022 assa adv. in systems science and appl. (2022) 3.2. robot’s status now we introduce the concept of the robot’s status, stat. the intelligent robots under consideration act autonomously to fulfill a collective mission. each robot has an individual functionality (capabilities) and individual characteristics to realize the functionality. a robot may execute several roles. with a role assigned, a robot acts and observes the environment in its narrow sphere of responsibility due to the limited capabilities of its observation systems. for moving in the operational space, a robot uses a system of actuators with different sensors (position, tilt, displacement, etc.) implementing the corresponding functions (movement, maneuver, stop, and others. let each k-robot be represented by an ordered set  , , , , ( ), ( ), ( ), , , , , ,( ) k k k k k k k k kk role m seg t seg t t r t func g kstat kt        (3.2) where: k is the robot’s number in the group; krole is the robot’s role; km is the robot’s route in the environment; k is the robot’s value (significance) for the group; ( )seg t is the robot’s location (segment) at a time instant t; ( )seg t t is the predicted robot’s location (segment) at a time instant ( )t t ; ( )kr t is the robot’s residual resources at a time instant t; kfunc is the robot’s functionality (set of functions); k is a coefficient reflecting the robot’s skills and experience; k is the robot’s attribute of activity (operability); k is the robot’s willingness to assist other robots (the behavioral paradigm formed by learning); finally, kg is the robot’s capabilities to reduce the environment’s degree of counteraction. all robots transmit their statuses online to an information resource on the internet. any robot has online access to this information resource. 3.3. influence factors the environment is antagonistic and has different influence factors f. the first source of such factors, denoted by af , is uncontrolled objective conditions of the environment that hamper collective mission fulfillment (e.g., night-time, fog, or rain). the second source bf relates to a specially organized counteraction (e.g., the defense systems of a potential target under attack). the factors can be formally described by a set of attributes. in many cases, they are not deterministic. for example, reconnaissance means detect the opponent’s antagonistic systems and capabilities only approximately; the probability of a sustainable malicious interference for communication means is high. on the other hand, robots and their sensors may fail due to different reasons. the unpredictability of possible situations complicates the definition and fulfillment of tasks by robots. therefore, it is necessary to use probabilistic or fuzzy variables to describe the attributes of influence factors. 3.4. events in the environment an event occurs if a set of influence factors shows itself in specific conditions. an event e e is the consequence of the influence factors ( )af t and ( )bf t affecting the robot’s status integrally at a given time instant t. an event e occurs due to realizing a set of factors with definite characteristics: : ( ).a b a b f f e stat t    (3.3) the collective behavior of robots based on the internet-of-things paradigm 167 copyright ©2022 assa. adv. in systems science and appl. (2022) an event e is characterized by its probability ep , a segment eseg of the operational space where it occurs, and its threat e k to the task of the k-robot. in principle, the events can be divided into groups. according to practical evidence, such groups are finite for a given environment and a given class of collective missions. however, influence factors as the sources of an event are often unknown, while an event itself may critically affect robots. 3.5. situation in the environment a set of events occurring in the environment forms a certain situation s s for each robot. the current situation affects the fulfillment of a collective mission by robots. in an antagonistic environment, a realized situation causes a loss for the group of robots. for the irobot ( )i i k  , the loss consists in its missed contribution iw  to a collective objective function w: : ( ) .ke s s e stat t w    (3.4) the total loss for the group is the sum of the missed contributions of all robots from the set i k for which this situation becomes critical: .ii w w   (3.5) any situation requires collective decision-making by the group of robots. in the set of all admissible decisions d for the s-situation, an optimal decision s optd minimizes the total loss: min .s opt dd w (3.6) 3.6. precedents there is nothing new under the sun despite an intrinsic uncertainty of collective mission fulfillment in antagonistic environments. humankind has accumulated a rich experience of actions in different situations. this experience is yielded by mathematical, physical, and simulation modeling of different situations, carried out a priori. the correctness of modelbased solutions is verified a posteriori in practice. robots are trained to choose proper actions in different situations using learning procedures and the results of modeling. the robot’s behavioral paradigm forms its intelligence. therefore, we can specify a set of precedents (similar situations in the past) for any collective mission. what is important, we know the decisions made in such precedents, including their efficiency and consequences. hence, a robot can analyze its status and the statuses of other robots, detect current situations, and seek appropriate decisions in a neighborhood of well-known (validated) decisions to establish self-organization rules. 3.7. state of the environment the set of factors f, related events e, and situations s generated by these events represents the environment’s current state ( )env t at a given time instant t. therefore, the state ( )env t is formed according to the following scheme: realization of influence factors in the environment => occurrence of events affecting robots => formation of a situation in a given segment of the operational space and losses for robots => the environment with fixed factors, events, situations, and precedents. consider an illustrative example. a group of fire-suppression robots is eliminating the consequences of an earthquake. the earthquake is described by factors (the time and duration of earth shocks). a set of these factors, realized within 24 hours, generates events (destruction of buildings). a sequence of such events forms a situation (fire). there exist numerous precedents of fire suppression depending on the intensity and rate of propagation. robots make decisions considering the current situation and the experience of similar operations. 168 v. abrosimov, a. mazurov copyright ©2022 assa adv. in systems science and appl. (2022) the scheme described above (factors => events => situations => environment’s state => precedents) has no crisp mathematical relations. physical and other regularities or patterns are also absent. the underlying processes can be described by variations (increase or decrease) in the activity of the environment and robots. it is possible to identify several types of such variations. for example, the probability of destruction increases when a robot approaches a dangerous zone. if a robot can affect the environment, the latter’s negative impact can be reduced. when external observation systems monitor a territory, events occur in discrete time: an event can be detected at a time instant *( 0)t  although it was not the case at a previous time instant *( 0)t  . discontinuities of activity are also possible, e.g., when robots lose communication. in contrast, the phenomenon of “hyperactivity” can be observed: as robots restore communication, they have to act in another (essentially complicated) situation. what are possible ways to model situational awareness without crisp mathematical and physical relationships among the observed phenomena? the theory of neural networks provides one solution; see the paper [1]. hopefully, the neural model described therein can be implemented in practice. 4. robot as a thing in the iot paradigm we accept a conventional proposition of the internet of things: each thing provides a statefunction interface for interaction, making it accessible to external users and other things. nowadays, researchers are oriented towards iot-based technologies with a high market priority (intelligent house, intelligent public transport, and others). there exists scarce information on using the iot in complex, intelligent controlled objects (and their interaction), mostly for marketing purposes. let us endow a robot with the following properties intrinsic to the iot paradigm. 4.1. collectivism as the main principle of interaction each robot operates autonomously but can interact with other robots simultaneously and independently. unlike the existing approaches, we incorporate the principles of collectivism in the interaction of robots, as described in the book [3]. each robot in a group is learned to one of the three paradigms of collective behavior: altruistic behavior (a higher priority for group’s tasks), egoistic behavior (a higher priority for individual tasks), and pragmatic behavior (decision-making with a complex priority choice algorithm). the details were described in the paper [2]. in this statement, while performing individual tasks according to its behavioral paradigm, each robot possesses information about the existing problems of other robots in their tasks and expresses its willingness to solve these problems using its functionality and residual resources (if necessary). this collectivism of the autonomous behavior of robots within the group is a prerequisite to creating an innovative serviceoriented interaction architecture of robots, which guarantees a proper operation of the entire group regardless of the states of individual robots through the role and functional interchangeability of robots and their mutual assistance (cooperation). 4.2. functionality as service robot’s roles are defined by its functionality, behavioral paradigm, and the scale of learning for different actions in complex situations. modern autonomous robots have a multifunctional design. while fulfilling a collective mission, in addition to its main functional role, each robot may have (reveal) another potentially admissible role depending on its status, role, functionality, and residual resources. some examples of roles include leader (a robot initiating control commands), cooperator (a robot collaborating with other robots in collective mission fulfillment), communicator (a robot responsible for interaction and communication), executor (a robot implementing actions), resource (a partially or the collective behavior of robots based on the internet-of-things paradigm 169 copyright ©2022 assa. adv. in systems science and appl. (2022) completely informed robot placed in reserve; participates in collective mission fulfillment by request only), and other roles. under the principle of collectivism, any robot accepting another role offers its functionality and resources to other robots as an external service available to solve their tasks. on the other hand, any robot may request required functionality and resources from other robots and use them if necessary. with interaction processes organized in this way, each robot disposes of additional functionality, which appreciably enlarges its capabilities. moreover, the entire group considerably increases the adaptability to the negative impact of the antagonistic environment. 4.3. mutual assistance as a basis of behavioral strategies a robot may implement different levels of access to its functionality. this property allows creating networks of robots differing by structure and complexity: common networks, networks with complex structure and subordination, networks with hidden elements, and others. (a hidden element shows itself if necessary, e.g., a sacrifice.) a robot can be identified in a network as a part of another robot. it can be hidden or identified through another robot, forming new networks. such networks can implement different behavioral strategies for a group of robots in the antagonistic environment, depending on the events and current situation. here, major aspects are allocating tasks and goals before collective mission fulfillment, routing, and coordinating the individual behavioral strategies of robots during mission fulfillment. thus, we interpret a robot as a thing in the iot paradigm. each robot informs other robots about its status and acts as an external service for them, and the robot’s attributes describe its admissible roles, functionality, and residual resources to solve tasks within a collective mission. the robot’s functions and resources can be requested and used by other robots. on the other hand, the capabilities of a given robot can be enlarged (if necessary) using the functionality, roles, and resources of other robots. 4.4. provision of robot’s functionality to other robots in the operational space with an antagonistic environment, robots need different functionality (e.g., reconnaissance, the capability to reduce or even neutralize counteraction, concealed motion, etc.). in standard solutions, the robot’s functionality decreases over time due to consumed resources. however, the approach with assistance from other robots allows increasing and extending the robot’s functionality. each k-robot (3.2) can be assigned a set of functions (its functionality):  1 2, ,..., ,..., .k k k jk jkfunc f f f f k k   (4.1) a characteristic jkf in the set kfunc can be another set containing several numerical, probabilistic, and fuzzy parameters. for example, the characteristic 3kf can be the robot’s effective distance (in terms of its impact on the environment), the angular range and efficiency of this impact depending on the environment’s counteraction (a probabilistic or fuzzy variable), the admissible number of such impacts, etc. combining the functional characteristics of robots in rows and their parameters in columns, we construct matrices completely describing the robot’s functionality. different matrices of this type can be formed depending on the robot’s design, functionality, and learning paradigms. the paper [4] considered a group of controlled objects (agents) and developed a role allocation algorithm with a ranking procedure based on the closeness to role requirements in terms of functionality and characteristics. applying this algorithm, we find an appropriate robot for assistance. assume that due to an event e e , an m-robot ( )m k was impacted by an antagonistic environment when performing its individual tasks. as a consequence, some functional capabilities jmf of the robot were decreased to an inadmissible level, * .jm jmf f requesting or rendering assistance can be formally described by activating a special variable 170 v. abrosimov, a. mazurov copyright ©2022 assa adv. in systems science and appl. (2022) ma of the robot’s status. in response to the alarm signal ( 1),ma  negotiations are organized among the other k-robots ( ).k k in a technical sense, it is necessary to find a g-robot, g k , with maximum functional capabilities for assistance and enough resources to implement this functionality: * { }: max , .jg k m jkg f f k k j j     (4.2) the g-robot with the value * jgf will be assigned the highest priority during negotiations. according to practical evidence, the following parameters should be introduced for choosing an appropriate robot for assistance:  the weights f that describe the significance of each function jf for collective mission fulfillment;  the significance k of each k-robot for the entire group (e.g., leader must be eliminated from the list of potential assistants);  the practical experience k of each k-robot in the same (or similar) role in precedents;  the skill of each k-agent in a required function acquired by learning (e.g., the “sacrifice” function [0,1]k  ; see the paper [2] for details). 5. an approach to design local self-organization rules of robots 5.1. specific features of collective strategy formation consider a group of robots, and let their collective mission be the passage through the counteraction zones of an antagonistic environment with a minimum loss of a collective objective function. the counteraction means seek to hamper collective mission fulfillment as much as possible through a negative impact on the robots for disrupting their normal operation and even neutralizing them. the collective objective function is given by the sum of the contributions kw of all krobots, k k : .kk w w (5.1) we will interpret the environment’s impact on a robot as an event .e e due to this impact, the contribution of each k-robot is reduced by a value e kw , which describes the corresponding loss. while fulfilling the collective mission, the group of robots seeks to achieve maxe w over all possible events e, whereas the environment’s goal is to generate events e e to achieve max , .e e kw e e k k      a collective strategy of the group realizes the following. a) the collective mission is decomposed into tasks with certain roles for each robot; for details, see the algorithm presented in the paper [4]. b) information about the antagonistic environment and robots’ statuses is registered. considering the allocated roles, each robot forms algorithms for its individual tasks. (an example of a routing algorithm in the earth surface monitoring problem by a group of heterogeneous robots was given in the book [3].) c) while performing its tasks, each robot updates the situational awareness model by the following elements with a given period t : (a) the robot’s current status and (b) the current environment’s state observed by the robot’s sensors. d) in response to occurring events (failures of robots or their sensors, exhaustion of robot’s non-replenishable resources, detection of new counteraction means, etc.), the group organizes negotiations. the new roles and control strategies of robots are negotiated the collective behavior of robots based on the internet-of-things paradigm 171 copyright ©2022 assa. adv. in systems science and appl. (2022) depending on their statuses and learning paradigms to fulfill the collective mission in new situations. note that the existing precedents are used for this purpose. an important problem is to hide the actual roles of robots operating in the antagonistic environment. one interesting solution involves irrational behavioral strategies or robots; see the recent paper [5]. 5.2. specific features of individual strategy formation as demonstrated by several researchers, the emergent collective behavior of objects can be constructed using their elementary interactions. for the class of problems under consideration, such elementary interactions include the following: updating the situational awareness model with the robot’s current status; requesting the current situation at a given location (point on a path); requesting assistance; rendering assistance; and so on. due to the negative impact of the antagonistic environment, some robots may fail to perform their tasks. therefore, we design the robot’s individual strategy by realizing two principles as follows. first, each robot performs the individual tasks within its role defined when decomposing the collective mission. second, to the extent possible, each robot considers the problems of other robots fulfilling the collective mission as unpredictable situations occur. a typical strategy of each k-robot consists of two mutually complementing components, general and special. the general component includes: 1. permanently monitoring the environment by robot’s sensors and quantitatively assessing the influence factors (events and situations) and their attributes; 2. predicting the path for a time instant ( ),t t where t is a given period; 3. estimating the probability e kp of a critical event * ( )ke t t in the predicted motion zone and assessing the threats e k created by the environment; 4. updating of the situational awareness model by the observed information; 5. analyzing an occurred situation s s and incurred losses kw  and assessing the individual functional capabilities to solve the problem; 6. acting to solve the problem; 7. when necessary (no individual solution), requesting assistance from the group ( 1)ka  ; 8. participating in negotiations (see the scheme (4.1), (4.2)) to provide the functionality kfunc to other robots for performing their tasks. the special component of the robot’s behavioral strategy reflects its particular features and depends on the roles. 5.3. behavioral rules for a robot-thing assisting other robot-things: an example of design consider an illustrative example to design robot’s behavioral rules within the iot paradigm. robots a and b have the same functionality (see table 5.1) and perform individual tasks, moving towards targets to attack them. the targets are secured by the technical means of a defense system c. table 5.1 functionality of robots (k, m = a, b) identifier function values 1kf (role) robot’s role in group 1––leader; 2––attacker; 3––supporter; 4––communicator 2kf robot’s motion v––forward motion with velocity v; 0––stop; (–v)––backward motion with velocity v 172 v. abrosimov, a. mazurov copyright ©2022 assa adv. in systems science and appl. (2022) 3kf detection of threat to k-robot from defense system c k zp ––the probability of threat from z-defensive means :zc 0k zp  (no threat), 0 1,k zp  1k zp  (annihilation of k-robot a by the defensive means zc ) 4kf response or preventive impact of krobot on defense system c 0––no preventive impacts; z kq ––the number of preventive impacts on zc ; z kqp ––the reduced probability of threat from zc due to z kq impacts of k-robot; 1––eliminated threat from zc 5kmf assistance from k-robot to m-robot 0––no assistance required; 1––assistance required from any robot with appropriate functionality and sufficient resources; :k m k-robot assists m-robot 6kmf activation of the motion system of m-robot by k-robot 0k mv   ––the motion system of m-robot not activated by k-robot; 1k mv   ––the motion system of m-robot activated by k-robot 7kmf activation of the combat system of m-robot by k-robot to impact defensive means 0k mh   ––the combat system of m-robot not activated by k-robot; 1k mh   ––the combat system of m-robot activated by k-robot 8kf activation of the observation system of m-robot by k-robot to acquire information on its route 0k mb   ––the observation system of m-robot not activated by k-robot; 0k mb   ––the observation system of m-robot activated by k-robot 9kf the “sacrifice” paradigm of k-robot 1––“altruist”; 0––“egoist”; k ––“pragmatist” with the willingness for sacrifice k consider two common events: the predicted path of a k-robot intersects the responsibility zone of the defense system c (the event 1e ), and the operation of a k-robot is violated by the defense system c with a threat e k under an admissible threat *e k (the event 2e ). let us formulate the behavioral rules of robots a and b with functionalities afunc and bfunc , respectively; see table 5.2. table 5.2 strategies of robots a and b in the responsibility zone of defense system c step events in environ ment data in the situational awareness model local rules for robots a and b r o b o t func 1 2 3 4 5 6 7 8 9 r ol e v k zp z kq h a bv  a bh  a bb  k 1 1 0e  *e e a a  *e e b b  a afunc 2 av 0 0 0 0 0 0 0 b bfunc 2 bv 0 0 0 0 0 0 1 2 1 1e  *e e a a  a afunc 2 av 0 0 0 0 0 0 0 the collective behavior of robots based on the internet-of-things paradigm 173 copyright ©2022 assa. adv. in systems science and appl. (2022) *e e b b  b bfunc 2 bv 0 0 0 0 0 0 1 3 1 1e   2 1e  *e e a a  *e e b b  a afunc 2 0 a zp z aqp 0 0 0 0 0 b bfunc 2 bv 0 0 0 0 0 0 1 4 1 1e   2 1e  *e e a a  *e e b b  a afunc 2 0 1 0 1 0 b bfunc 3 0 0 z bqp 0 1bv 1 1 1 5 1 1e   2 0e  *e e a a  *e e b b  a afunc 2 av 1 b a 0 b bfunc 3 0 1 b a 1bv 0 1 1 now, we analyze the actions performed by robots as things whose control systems can communicate and control each other. at step 1 of the process, robots a and b are outside the potential threat zone. both robots move to the given targets. at step 2, the following event occurs: the predicted path of robot a intersects the responsibility zone of the defensive means .zc however, the probability of annihilation does not exceed the given thresholds for robots a and b. there is no impact from zc on robots a and b. the current situation on the predicted paths of the robots is not dangerous. robots a and b continue their motion to the targets. at step 3, a new event occurs in the environment: the defensive means zc apply impact on robot a with the probability a zp . the current situation on the predicted path of robot a becomes dangerous. learned in the egoistic paradigm, robot a stops applying z aq impacts on the defensive means zc with the total success probability z aqp . in turn, robot b executes the same role and continues to perform its individual tasks since robot a requests no assistance. at step 4, the defensive means zc apply impact on robot a without complete success: the latter’s functionality is partially reduced (e.g., the observation system is damaged). robot a requests assistance from robot b. learned in the sacrifice paradigm, robot b assesses its functionality and available resources and suspends its individual tasks for executing the supporter role: robot b authorizes using its observation and motion systems to robot a. responding to the control system of robot a, robot b changes its velocity 1bv and direction. the combat system of robot b applies impact on the defensive means zc with the probability of annihilation z bqp . at step 5, due to z b bq zp p , both robots win the duel, and robot a continues its motion to the target without threat (using the observation system of robot b for situational awareness). robot b assesses the residual resources to optimize its route to the targets. 5. conclusion therefore, applying the internet-of-things paradigm is a fundamentally new solution to design collective behavioral strategies of robots performing collective missions in antagonistic environments. implementing a group control strategy based on the principles of collectivism, we create and refine a networked service-oriented infrastructure. in a group of autonomous mobile robots, various relations arise and disappear among the observation and actuation systems of robots, depending on the events and situations in the environment. the generation processes of new relations are spontaneous, a priori unknown, and are dictated by the current situation and events occurring in the environment. the control systems of upper-level robots in the role hierarchy activate the control systems of the lower-level ones, reallocating their roles. if 174 v. abrosimov, a. mazurov copyright ©2022 assa adv. in systems science and appl. (2022) the robots have no hierarchy, control issues are settled during negotiations using the principle of collectivism. thus, the iot paradigm-based solution leads to self-organization. this aspect is particularly important due to the current engineering trend of constructing multifunctional robotic complexes using a group of simple robots with a small set of functions (instead of single complex platforms). another determinant factor to use the iot paradigm is the activation of control systems of robots by other robots according to the roles and structural hierarchy of the group. many issues are beyond the scope of this paper. what are the consequences of communication failures? how will a robot “agree” to execute an auxiliary role in addition to its individual tasks? what is the efficiency of the suggested approach under different scales and counteraction of an environment? how can we design a service-oriented architecture of interaction for multifunctional controlled objects? how many levels should the resulting network be composed of? what is their structure? what are the required geo-informational models of situational awareness? these issues will be considered in future work. references 1. abrosimov, v. 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(2016). a survey of self-organization mechanisms in multiagent systems. ieee transactions on systems, 47(3), 441–461. microsoft word 806-source texts-5863-1-11-20230417 adv syst sci appl 2023; 01; 115-128 published online at https://ijassa.ipu.ru. learning methods for distributed diagnosis: a failure classification methodology in discrete event systems hanane serradj1, myriam noureddine1** 1) department of computer science, université des sciences et de la technologie (usto), mohamed boudiaf bp. 1505 el m’naouer, oran, 31000, algeria abstract: in the dependability context, diagnosis is fundamental for identifying and classifying failures to reduce system malfunctions and accidents that can cause serious damage to property and people. currently, systems are generally complex and the distribution approach of diagnostic function makes it possible to better managing systems to be diagnosed. this article proposes a methodology for the distributed diagnosis of discrete event systems, which are dynamic systems widely spread. the proposed approach, based on a generic model, is formed of two sequential steps. the first one consists in the system modeling in order to understand real behavior of the studied system, through a distributed recovery of signals. the second step is to standardize and process data for the classification of failures. to achieve this challenge, two learning classification methods are adopted: the learning algorithm for multivariate data analysis (lamda) coupled with the k-nearest neighbor algorithm (k-nn). the methodology is applied on a particular discrete event system, a railway transport network following a real case of railway line. the obtained results showed improvement of the classification rate by recognition, also becoming better that the supervised classification rate. the adopted methodology offers a distributed diagnosis framework, which emerges as an efficacy and robust way for classify any failures of systems. keywords: dependability, failure, distributed diagnosis, learning methods, classification, lamda, k-nn algorithm. 1. introduction the need to ensure dependability [2] of systems and to monitor their actual behavior makes diagnosis an indispensable and fundamental operation. indeed, slightest dysfunction of the system can generate disasters and it's important to identify failures that cause. the diagnosis aim [33] is to detect probable cause of failure. in this context, many approaches [11,13,24,28] have been developed: these methods are based on a model of the normal and/or failing system behavior. diagnosis methods differ according to different criteria as process dynamics (discrete, continuous or hybrid), complexity, implementation (on-line and/or offline), nature of information (qualitative and/or quantitative), depth (structural, functional and/or temporal) and distribution (centralized, decentralized or distributed). the difference can also about localization, external and internal methods [26]. whatever orientation of methods, it is generally adopted that industrial diagnosis is a deterministic causality relation between the cause and the effect [33]. knowing that diagnosis becomes very complex when systems to be diagnosed are no longer elementary, integration of the distribution notion becomes an important issue. we are interested in distribution approach of the diagnostic function, especially on discrete event systems (des). these are dynamic systems allowing modeling and analysis of human-made systems like manufacturing, communication and transportation offering an interdisciplinary view [29]. given complexity of systems, this work focuses on a model which is an abstract and partial view of the system behavior. * corresponding author: myriam.noureddine@univ-usto.dz 116 h. serradj, m. noureddine copyright ©2023 assa adv. in systems science and appl. (2023) to exploit potentialities of a des, physically distributed, authors propose a methodology, on two steps and integrating a generic model. the first step consists in modeling the des using a method based on logical models. so, simulation of obtained model makes it possible to better understand real behavior of the studied system. the second step is to standardize and process obtained data for classification of failures according in terms of severity. to obtain best classification, the learning algorithm for multivariate data analysis (lamda) coupled with the k-nearest neighbor algorithm (k-nn) is used. remainder of the paper is organized as follows: section 2 presents distributed diagnosis and the two learning methods. in section 3, complete methodology is described by assembling the two paradigms that represent respectively the fuzzy lamda method and the k-nn algorithm. section 4 consists of methodology application on a particular des, a rail transport network; system specification and simulation of the distribution modules are given. the proposed classification algorithm is applied in section 5, thus validating the approach, where comparative and interpretation of the obtained results are discussed. 2. background the diagnosis function consists of detecting, locating and identify a fault in order to determinate its causes. identified dysfunctions require corrective actions to regain a normal functioning and to limit consequences of breakdowns that can be catastrophic. this becomes an important issue when systems to be diagnosed are large and complex. in this case, the criteria dedicated to the distribution and the process are to be considered. many works focus in this way, like an embedded diagnosis development [4,16], based on three types of architecture (centralized, decentralized and distributed), a centralized diagnostic approach for timed systems and dynamic systems [9]. a diagnosis structure based on decentralized models of a complex system represented by qualitative constraints is proposed in [7]. authors in [25] work on telecommunications networks which are based on the fact that the system model is very large. many models-based diagnostic approaches are derived from the des [8,14,27,28,31]. to be effective, the diagnosis function must rely on the system attitude before and after failures. therefore, classification of failures is fundamental and often used to view the system behavior [3,19,22]. following this context, the aim in this work is to classify failures of spatially distributed discrete events systems, helping the operator to limit the number of failures, to identify them and to obtain the best distribution of elements. considering that the learning techniques are adequate and widely applied in classification problems, the authors adopt this research framework through the two learning classification methods, lamda and k-nn method. lamda method has shown its effectiveness when it comes to classification with multi-source learning based on fuzzy logic. this gives the expert a real perception on the system behavior as well as the state change and it allows better model human reasoning based on uncertainty. k-nn method is used for indistinguishable elements, which are reclassified and a final ranking is obtained. 2.1. lamda method lamda method has been used in very diverse fields such as psychology [1], industrial processes [18], selection of sensors [23] and segmentation of biological images [10]. a software salsa (situation assessment uses lamda classification analysis) [17–19] allows its implementation. lamda approach is a learning technique based on fuzzy logic. the principle of lamda methodology is classification of objects, which representing information. objects are represented by descriptors with standardization of values. each object is affected to a class through the marginal adequacy degree (mad) function. this function defines a degree between attribute adequacy to a class and inadequacy of the attribute to this class. learning methods for distributed diagnosis… 117 copyright ©2023 assa. adv. in systems science and appl. (2023) for numeric descriptors, the mad evaluation can be made through many functions, in particular fuzzy extension of binomial distributions. in this work, the functions [17] lamda1 and lamda2 are used: lamda1: xpxppxmad  1)1()/( , (2.1) where p is the parameter value of the class c which describes the descriptor and x is the normalized value of the descriptor z. lamda2: xpxkppxmad  1)1()/( , (2.2) where 1p2 p1 p log )( k   . the normalized value x is given by equation (2.3): minmax min zz zz x    , (2.3) where zmax and zmin are respectively the maximum and minimum values of z. next, the global adequacy degree (gad) is calculated. for an element (described by 1 to d descriptors), the gad evaluation is obtained by aggregation of all mad of this element to a class through logic operators like intersection and union functions. so, the linear compensation is done according to the following equation (2.4): gad (mad1,..., madd) =  t (mad1,..., madd) + (1-s) (mad1,..., madd) , (2.4) where t is the t-norm connector (intersection), s is the t-conorm connector (union), α is the exigency index  [0,1]. an important feature of the lamda methodology resides in the taking into account of the information lack for indistinguishable elements and in this case, a non informative class (nic) is created. the exigency parameter measures affectation of an individual to a class. for α = 0, the classification is not exigent and the highest requirement is obtained for α = 1. so, increasing the value α implies increasing the number of elements assigned to the nic [18]. we bring the nic notion closer to the residual class (rc) notion. indeed, in classification domain, introduction of this particular class allows considering the unclassified elements: objects not belonged to any class are assigned to an additional residual class [32]. the lamda method can adapt to a situation evolving over time, so it can handle qualitative and quantitative variables simultaneously by using appropriate marginal adequacy functions. despite this property, there is no guarantee to obtain the best partition as results depend on choice of the connectives and the exigency index. to palliate of these disadvantages, the proposed solution in this article is introduction of the well-known k-nn method. 2.2. k-nn method the k-nn method is the simplest and widely used classifiers [30,32]; it is a powerful tool for any classification study on unknown specification about data distribution. the k-nn 118 h. serradj, m. noureddine copyright ©2023 assa adv. in systems science and appl. (2023) method uses two parameters: the number k and the similarity function to compare new case with the already classified cases. the k-nn method [30] follow three actions: calculate distances to obtain a learning base, sort to have the k smallest and choose the distance which belongs to the majority class. the similarity function which represents distance between the test sample and the training data samples, is paramount to the correct algorithm functioning. there are several measures of distances, and the most used is euclidean distance [30], given by equation (2.5):    n i yxyxd ii1 2)(),( , (2.5) where xi and yi represent coordinates of the points x and y. 3. proposed methodology 3.1. the generic model centralized approach is not satisfying for design of diagnostic method dedicated to systems having a distributed architecture. so, distribution notion of the diagnosis function becomes essential. in this context, we propose a methodology (fig. 3.1) based on a generic model, carried out in two modules. fig. 3.1. proposed methodology the first module involves aim at the distributed recovery of signal and transformation of this signal via intelligent devices, taking into account communication between the physically distributed stations. highlighting the need to have a model representing different relationships between generated events by the studied system, this generic module relies either on existing systems like scada (supervisory control and data acquisition) system [6, 15] or on systems dedicated to particular domains. distributed diagnosis distributed recovery recover classification algorithm stations physically distributed system to study: des signal recovered station 1 station 2 station n classification of failures learning methods for distributed diagnosis… 119 copyright ©2023 assa. adv. in systems science and appl. (2023) second module is the main issue of the article; it consists in providing a diagnosis aid based on recovered signal analysis by integrating an improved classification algorithm through two learning paradigms. the first one is based on fuzzy logic and more particularly lamda method and the other one uses k-nn method. the aim is giving the operator an orientation to know better the real situation of the system in order to make the right decision concerning a specific failure. 3.2. the classification algorithm as already mentioned, the proposed classification algorithm involves two principles; the first is the lamda method application, giving classified elements and indistinguishable elements affected to a specific class (nic). these last elements are reclassified using the principle of k nearest neighbors in order to correct the obtained classification. this article studies influence of the k-nn algorithm integration in the recognition part of the lamda method, in particular by the studying non classed elements. these individuals are in the residual class which has the same characteristics as the non informative class. obtained results are compared with the supervised part of the method using same parameters chosen by the operator to have the correct classification. the general algorithm of classification integrates the following steps for an element x, described by 1 to d descriptors. we included data preprocessing to have normalized values of descriptors. complete classification algorithm: input: a sequence of elements x / x = x'1, x'2, ..., x'd step 1: normalization of each element using equation (2.3) giving x / x = x1, x2, ..., xd such as xj is a normalized value. step 2: for each element x, calculate the mad(xj/ck) with regard to class ck using the functions lamda 1 or lamda 2 through the equations (2.1) and (2.2). step 3: calculate the gad(x/ck) for the element x using the equation (2.4). step 4: select the maximal value of gadi. step 5: assignment the element x to the class ci. if an element has low membership in all existing classes, this element is affected to the nic. these steps are issued from the lamda method. at the end of the 5th stage, a first ranking is obtained according to different classes and the non informative class. the next steps introduce the k-nn method and let us considering the nic as residual class (rc). step 6: choose a value for k. step 7: for each element x of rc, calculate distance (equation 2.5) between the individual of the rc and these neighbors. output: classification of all elements x. by application of this methodology, a final ranking is obtained with corrected classes where all elements are classified. 4. methodology application on a des: rail transport system as an illustration of the proposed generic model, we consider a particular discrete event system, a railway transport network. this transportation system is considered a safety critical system [30]. so, a management environment of failures is important for increasing dependability and ensuring safety of property and persons. 4.1. specification of the rail transport system the railway system comprises two elements closely linked to infrastructure and vehicle: the traffic loads, the type of trains and rolling stock on the track lead to an irregular degradation of the geometry. the speed accelerates this degradation and a reduction in the component life. 120 h. serradj, m. noureddine copyright ©2023 assa adv. in systems science and appl. (2023) a real case is studying through the line oran (west city of algeria) to algiers (the capital city); distance between these two cities is of 422km. a measurement car [21] is used for maintenance and renewal of railways. the vehicle exhibits dynamic behavior to be improved, in particular case when the vehicle passes over geometrical defects on the track or on switches. the measuring car is intended to detect and record geometrical parameters of the track, mainly by intervening urgently to ensure safety of the train movements through checking types of faults [5, 12]. the controls concern: the longitudinal leveling of 2 lines of rail. the cant values. the left, which represents the torsion of the track. the defect of dressage. the variation of the spacing. the major disadvantages of the measurement car are on the one hand inability of the car to interrupt monitoring and diagnosis, and on the other hand the high probability (75%) of having a false registration if the sensors do not work. furthermore, as it is a towed car without motor, this can cause a delay of the train with no information distribution. 4.2. distribution of the recovery phase between communicating modules a solution is proposed to delimit the disadvantages generated by the measuring car: the rail transport system has a control center and physically distributed stations, through remote terminal units (rtu). for the considered line (fig. 4.1), the command center is located at the algiers level; departure will start from oran (kilometer point 0) which sets post1, relizane (kilometer point 1) is workstation2 and ain defla (kilometer point 2) is post3. fig. 4.1. the railway line with rtus each station has cells except workstation1 (departure) and control center (arrival); this one has circuit breakers which can be either the data logger or protection. the data logger contains various measures such as train speed, size and time; the protection contains defects (spacing, left, long profile, vertical alignment, high and down points, devers, arrow). graphical data (fig. 4.2) between kilometer points 1 and 2 are recovered by rtus, giving simulation of the two defects, spacing (s) and left (l). this result is sent to the command center. control station and other stations are connected to a telecommunication network via a transport medium using standard protocols. a database allows storage of failures, as: ts: tele signalization (simple or double) which represents defects and the track state. tr: tele regulation that represents target speed for modification of parameters. tm: tele measure for controlled measurements to monitor current speed of the train. learning methods for distributed diagnosis… 121 copyright ©2023 assa. adv. in systems science and appl. (2023) fig. 4.2. graphical data for simulation of two defects 5. application of the proposed algorithm the following section illustrates in detail the methodology through application of the proposed algorithm on a set of individuals recovered from a distributed architecture; at first, lamda method is applied followed by k-nn method. the obtained results show the classification improvement in using same parameters thereby, the operator will carry out a multidisciplinary analysis on the collected data, so that he can make the right decision in real time regarding maintenance during the failure occurrence. he also manages a new failure that can belong to a class already predefined, by using the adopted classification and notes its influence to the given results in the case where learning is supervised. 5.1 classification obtained by lamda 5.1.1. specification of descriptors consider a sample of 23 individuals collected from intelligent equipments installed on railway transport network, in the oran-algiers line. we wish classifying these elements in supervised learning mode. descriptors are quantitative type, representing respectively the left and the spacing. in agreement with the expert and following defined measurement values, two classes are identified: the first represents right track and the second represents wrong track. table 5.1 shows initial values retrieved from rtus with the two defined descriptors for the 23 elements; each individual is stored with an affectation to the reference class it belongs. table 5.1. initial values of the descriptors individual descriptors reference class left (l) spacing (s) 1 9.900 1435.00 2 2 16.200 1400.00 2 3 -9.300 1430.00 2 4 14.400 1420.90 2 5 -12.300 1400.00 2 122 h. serradj, m. noureddine copyright ©2023 assa adv. in systems science and appl. (2023) 6 1.500 1435.500 1 7 -9.300 1200.00 2 8 12.600 1435.00 2 9 8.600 1435.00 1 10 5.00 1435.00 1 11 -10.800 1600.00 2 12 9.900 1435.00 2 13 2.300 1435.00 1 14 9.300 1700.00 2 15 9.600 1820.00 2 16 15.900 1400.000 2 17 9.100 1400.00 2 18 8.00 1435.00 1 19 3.300 1435.00 1 20 15.300 2000.00 2 21 19.200 2152.00 2 22 13.200 1435.00 2 23 5.400 1435.00 1 5.1.2. implementation of the lamda method the proposed algorithm is applied on data and the first steps concern lamda method. however, as initial values are evaluated on different scales, it is necessary to define a context which groups normalized values of descriptors. this data translation is supported in step 1 of the algorithm. the normalization (equation 2.3) allows values to always be in the interval [0,1] and following figure (fig. 5.1) presents description of context obtained with the new values normalized for the two descriptors. fig. 5.1. description of the context two types of classification (supervised and recognition) can be used in lamda method implementation. in this context, a classification rate (table 5.2) can be obtained by varying the three parameters: the first one is exigency index α, getting values in [0,1]. after simulation, we retain the extreme values as well as two intermediate values. the second parameter are the connectives and we used the two most known family (probabilistic and the minimum-maximum) for t-norm and t-conorm values. the last parameter are the two functions lamda1 and lamda2 given by following equations (2.1) and (2.2), specified previously. 0 0,2 0,4 0,6 0,8 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 n or m al iz ed v al ue s individuals left spacing learning methods for distributed diagnosis… 123 copyright ©2023 assa. adv. in systems science and appl. (2023) table 5.2. learning and recognition rate corresponds to two types of classification su pe rv is ed c la ss if ic at io n α values connectives functions rate of learning and recognition c la ss if ic at io n by r ec og ni tio n rate of learning and recognition 0 probab lamda1 69.56 % 56.52 % class nic {11,15,20} lamda 2 47.82 % 34.78 % class nic {11,15,20,21} minmax lamda1 52.17 % 52.17 % lamda 2 47.82 % 47.82 % 0.3 probab lamda1 73.91 % 52.17 % class nic {11,15,20,21} lamda 2 47.82 % 34.78 % class nic {11,15,20,21} minmax lamda1 69.56 % 47.82 % class cr{11} lamda 2 47.82 % 47.82 % class cr {11} 0.7 probab lamda1 73.91 % 52.17 % class nic {11,15,20,21} lamda 2 47.82 % 34.72 % class nic {11,15,20,21} minmax lamda1 73.91 % 60.86 % lamda 2 73.91 % 39.13 % class nic {11,15,20,21} 1 probab lamda1 73.91 % 56.52 % class nic {11,15,20,21} lamda 2 43.47 % 34.78 % class nic {11,15,20,21} minmax lamda1 65.21 % 43.47 % class nic {3,5,6,7,11, 13,14,15,19,20,21} lamda 2 65.21 % 34.78 % class nic {3,5,6,7,11, 13,14,15,19,20,21} from the table, difference between the two classification modes is appearance of the non-informative class (nic) in recognition classification. in this case, to improve the rate keeping same parameters, next steps of the proposed algorithm are applied. it consists in knn method implementation on elements belonging to residual class (rc), which has the same characteristics that the nic. so, individuals already classified in the nic are ranking again by adopting set of last steps of the classification algorithm. final results will be compared with results whose learning is supervised while preserving same parameters predefined by the expert. 5.2. application of the k-nn method 5.2.1. identification of residual class let us take the classification obtained by recognition and two cases can appear following two or three classes, with the rc identification (class 0). for example, look at following two cases (fig. 5.2, a and b) defined with specific values for the three parameters (exigency index α, connectives and functions). the first case (parameters: 0.3, probab, lamda1) gives the presence of three classes: class 1 is {1,3,5-7,9,10,12,13,16-19,23}, class 2 is {2,4,8,14,22} and {11,15,20,21} for rc. the second one (parameters: 1, minmax, lamda2) deals with two classes: {1,2,4,8-10,12, 1618,22,23} for class 2 and {3,5-7,11,13-15, 19-21} for rc. 124 h. serradj, m. noureddine copyright ©2023 assa adv. in systems science and appl. (2023) fig. 5.2. identification of residual class consider the highest value for exigency index, which is located in the case 2; in this case, recognition rate is 65.21% in supervised classification mode. with same parameters, in classification by recognition the rate is 34.78% with the appearance of rc giving unrecognized individuals. these elements are reclassifying by application of the two last steps of the proposed algorithm through classifier k-nn, in order to improve classification rate by recognition. 5.2.2. reclassification of the 6th individual for illustration, we apply the method on one element of rc, the 6th individual. values of the two descriptors l and s are respectively 1.500 and 1435.500; this element has three neighbors (k=3): the 4th, 8th and 9th individual. following the reference classification, the 9th belongs to class 1 and the two others elements belong to class 2. so, euclidean distance between this 6th individual and his neighbors is calculated (table 5.3), by application of equation (2.5) defined previously. table 5.3. calculation for the reclassification of the 6th individual neighbors class descriptors (l _ s) distance 4 2 (14.4_1420.9) 19.48 8 2 (12.6_1435) 11.11 9 1 (8.6_ 1435) 7.11 this element belongs to class which has the lower evaluated distance compared to other neighbors. hence, the 6th individual belongs to class 1 as its distance is the smallest from all calculated distances. 5.2.3. reclassification of all elements same approach is applied on full elements of rc; table 5.4 summarizes the classification decision for all other individuals after algorithm execution. table 5.4. reclassification of rc elements individual k neighbors class distance decision 3 2 2 2 39.37 individual 3 belongs to class 2 4 2 11.12 5 2 4 2 33.36 individual 5 belongs to class 2 8 2 49.95 7 2 4 2 222.16 individual 7 belongs to class 2 8 2 236.01 11 2 10 1 165.75 individual 11 belongs to class 2 12 2 165.06 13 3 10 1 2.7 individual 13 learning methods for distributed diagnosis… 125 copyright ©2023 assa. adv. in systems science and appl. (2023) 12 2 7.6 belongs to class 1 16 2 37.54 14 2 12 2 265 individual 14 belongs to class 2 16 2 300 15 2 12 2 385 individual 15 belongs to class 2 16 2 420 19 2 18 1 4.7 individual 19 belongs to class 1 22 2 9.9 20 2 18 1 565.04 individual 20 belongs to class 2 22 2 565.00 21 2 18 1 717.08 individual 21 belongs to class 2 22 2 717.02 5.3. results and discussion according the above table, final results are illustrated in next figure (fig. 5.3). fig. 5.3. evolution of the classification interpretation of different graphs shows classification evolution: all individuals belonging to the rc are classified. compared to reference classification, ranking of some elements did not move while others were refined on this sample of individuals, the rate of learning and recognition is calculated again, giving an improved rate (fig. 5.4) from 34.78% to 82.60% with a growth of 47.82%. using function lamda 1 and with same parameters (exigency value equal to 1 and the connective minmax), an improvement in the rate is also found (fig. 5.4), from 43.47% to 78.26% with an increase of 34.79%. note that supervised classification rate is 65.21% with same parameters and the rate classification by recognition that was lower is now better. 126 h. serradj, m. noureddine copyright ©2023 assa adv. in systems science and appl. (2023) fig. 5.4. improvement of the classification rate 6. conclusion diagnosis is fundamental to master failures of systems in order to protect human lives, particularly in specific systems such as rail networks. this article is focused on this framework, by proposing a methodology around a distributed approach of the diagnosis function for these complex discrete event systems. the approach includes two parts, a distributed recovery of signal and processing part with a classification algorithm. aim is to facilitate operator task in the decision-making phase about diagnosis. in this context, we proposed an algorithm for classification of failures, based first on the fuzzy lamda method, giving ranking results with unrecognized individuals, assigned to the non informative class. considering this particular class as the residual class, the k-nearest neighbor method is then introduced to reclassify unranked elements. the methodology has been applied to a real case study, which is a railway line. physically distributed stations are used to collect information and on recovered elements, the proposed algorithm is implemented. following the two types of classification, supervised and by recognition, ranking of individuals are provided with a classification rate. results of simulation show that the proposed approach gives an improvement of the classification rate by recognition. moreover, this rate becomes better that the supervised classification rate. future work will aim two main ways, design and application of the methodology. thereby, in order to perfect the approach, several perspectives can be considered in particular concerning the unrecognized elements by lamda method. in this work, euclidean distance for k-nn approach has been used and other metrics could be studied. more, additional methods from artificial intelligence might be taken up to reclassify these elements. furthermore, the presented approach can also be applied to other des and currently, the generic methodology is being studied on a system of electricity distribution. the real considered system has a control center and substations physically distributed; digital signal is recovered by a scada system. the algorithm implementation, based on the two learning methods, is in progress. so, by using the full methodology on different systems, the proposed distributed diagnostic framework, in a dependability scheme, emerges as an efficacy and robust way for classify any failures, while helping in decision making of an operator. learning methods for distributed diagnosis… 127 copyright ©2023 assa. adv. in systems science and appl. 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(1995). diagnostic des défaillances. théorie et pratique des systèmes industriels [failures diagnosis. theory and practice for industrial systems], paris, france: hermès editions, [in french]. adv syst sci appl 2020; 01:27–49 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/775 average predictive control for nonlinear discrete dynamical systems dmitriy dmitrishin1, emil iacob2∗, alex stokolos2 1odessa national polytechnic university, odessa, ukraine 2georgia southern university, statesboro ga 30460, usa abstract: we explore the problem of stabilization of unstable periodic orbits in discrete nonlinear dynamical systems. this work proposes the generalization of predictive control method for resolving the stabilization problem. our method embodies the development of control method proposed by b.t. polyak. the control we propose uses a linear (convex) combination of iterated functions. with the proposed method auxiliary, the problem of robust cycle stabilization for various cases of its multipliers localization is solved. an algorithm for finding a given length cycle when its multipliers are known is described as a particular case of our method application. also, we present numerical simulation results for some well-known mappings and the possibility of further generalization of this method. keywords: non-linear discrete systems, stabilization, predictive control. 1. introduction nonlinear dynamical systems are often characterized by extremely unstable movements in the phase space, defined as chaotic movements [5]. in practice, it is generally desirable to suppress or prevent such chaotic behavior due to its adverse effect on the physical systems normal operation. due to its theoretical significance and engineering applicability, much attention has been paid to the problem of chaos controlling in various fields and numerous studies [2, 3, 15, 22]. by chaos control we mean small external influences on the system or a small change in the system structure in order to transform the system chaotic behavior into a regular (or chaotic, still specific with other properties) one [13]. it is assumed that the dynamical system includes a chaotic attractor, which contains a countable set of unstable in lyapunov sense cycles with different periods. if, by using some control effect, a certain cycle is stabilized locally, the system path will remain in its neighborhood, i.e. regular movements will be observed in the system. hence, one of the ways for chaos controlling refers to the local stabilization of certain orbits from a chaotic attractor. the problem of stabilizing cycles is closely related to the problem of finding periodic points. the various control schemes [1, 4, 9] that were proposed for solving these problems ∗corresponding author: ieiacob@georgiasouthern.edu 28 d. dmitrishin, e. iacob, a. stokolos can be divided into two large groups: direct and indirect methods. the indirect methods either use the initial mapping t iterations or imply the construction a system of which order is t times greater than the initial system order (t being the desired cycle length). then one of the methods of finding a fixed point is applied. the most common among fixed point finding techniques is the newton-raphson relaxation method and its further modifications [12, 19, 22, 31]. the next step is to select periodic points from the entire set of fixed points. in direct methods, all the points in the cycle are found concurrently, i.e. the whole cycle is stabilized. in this case, the initial system is closed by control, based on the feedback principle [7, 20, 25, 30]. among such control schemes, the most simple in terms of physical implementation are the linear ones. however, they have significant limitations, as they can only be applied to a narrow domain of the space of parameters that are part of the initial nonlinear system [6, 28, 32]. to overcome the restricted stabilization condition ushio and yamamoto [29] introduced a prediction-based feedback control method for discrete chaotic systems with accurate mathematical model. b.t. polyak [23] (see also [24]) proposed a direct predictive control method that works well for one-dimensional as well as multidimensional maps. l. shalby [27] used predictive feedback control method for stabilization of continuous time systems. another possible control schemes classification into two groups: methods using the jacobi matrix and methods not based on this matrix. naturally, it is assumed that the jacobi matrix at the cycle points is not properly known, otherwise it would be possible to use the whole powerful apparatus of the linear control theory applied to systems linearized in the cycle neighborhood. the jacobi matrix is an indispensable attribute of newton-raphson-type methods. this matrix is also used in one of the modifications derived from polyak predictive control method. one of the main disadvantages of polyak scheme refers to the need for knowing the jacobi matrix for the cycle, or at least the need for sufficiently good estimates of the cycle multipliers. the research exposed herein is purposed to improve the polyak method by replacing it with mixed predicted values. a jacobi matrix representation for the t cycle at a controlled system is found through the jacobi matrix of the same cycle in the initial system. therefore, the correspondence between the cycle multipliers of the open loop system and those of closed loop system is established. below, it is assumed that the initial system cycle multipliers are not exactly known, we only know the range of their localization. then the solution of cycles robust stabilization problem for various localization of multipliers is given, and taking into account these general provisions the polyak method is considered. it is worth noting that in the general case of complex multipliers, we must know precisely enough their localization regions. at the end, the applications of proposed predictive control scheme to stabilize the cycles of some common systems in physics literature are considered. copyright c© 2020 assa. adv syst sci appl (2020) average predictive control 29 2. problem statement considered is a nonlinear discrete system xn+1 = f (xn) , xn ∈ rm, n = 1, 2, . . . , (2.1) where f(x) is a differentiable vector function of corresponding dimension. the differentiability is redundant and can be relaxed (f(x) needs to be differentiable only at the cycle points, c.f. [23]). it is assumed that the system (2.1) has an invariant convex set a, i.e. if ξ ∈ a, then f(ξ) ∈ a. we emphasize that we do not assume that the set a is a minimally convex set. it is also assumed that this system has one or several unstable t cycles {η1, . . . , ηt}, where all vectors are different and belong to the invariant set a, i.e. ηj+1 = f (ηj), j = 1, . . . , t − 1, η1 = f (ηt ). the considered unstable cycles multipliers are determined as eigenvalues of the product of jacobi matrices t∏ j=1 f ′ (ηt−j+1) of dimensions m×m at the cycle points. the matrix t∏ j=1 f ′ (ηt−j+1) is called the jacobi matrix of the cycle {η1, . . . , ηt} . typically, a priori the cycles of the system (2.1) are not known. consequently, the spectrum {µ1, . . . , µm} of the matrix t∏ j=1 f ′ (ηt−j+1) is unknown as well. the spectrum elements are called cycle multipliers. below, we assume that some estimates on the localization set m for the cycle multipliers are known. let us consider the control system xn+1 = f(xn) + un, (2.2) where un = (θ1 − 1)f(xn) + n∑ j=2 θjf ((j−1)t+1)(xn), f (1)(x) = f(x), f (k)(x) = f ( f (k−1)(x) ) , k = 2, . . . , (n − 1)t + 1. the numbers θ1, . . . , θn are real. it can be easily verified that at n∑ j=1 θj = 1 the system (2.2) also includes the cycle {η1, . . . , ηt}. we aim to choose such parameter n and coefficients {θ1, . . . θn} so that the system (2.2) cycle {η1, . . . , ηt} would be locally asymptotically stable. naturally, when constructing these coefficients, there will be used information on set m of multipliers localization. it is also desirable [16, 17] to fulfill an additional condition: the system (2.1) invariant convex set a must be also invariant for the system (2.2). this requirement will be fulfilled, for example, if 0 ≤ θj ≤ 1, j = 1, . . . , n . ) polyak method [23] utilizes the case θ1 = 1, θ2 = · · · = θn−2 = 0, θn−1 = −θn = ε. regarding the set m , it was assumed that m = d ∪ {µ∗} where d = {z : |z| < 1} is the central unit circle on the complex plane, and µ∗ is a known real number. in the case m = 1 the required coefficient formula has the form ε = 1∓ (|µ∗| /ρ)− 1 t (µ∗)n−2 (µ∗ − 1) where 0 < ρ < 1. in this article the control problem is solved for a wider class of multipliers localization set m . copyright c© 2020 assa. adv syst sci appl (2020) 30 d. dmitrishin, e. iacob, a. stokolos 3. constructing the jacobi matrix for a controlled system investigating stability of t cycles of the system (2.2) consists in constructing of jacobi matrix t∏ j=1 f ′ (ηt−j+1) of that cycle and studying the eigenvalues of this matrix. to derive the jacobi matrix, we use the ideas from [23]. let jj = f ′ (ηj), j = 1, . . . , t , then we write the jacobi matrix of the system (2.1) cycle {η1, . . . , ηt} as j = jt · . . . · j1. we introduce the following auxiliary matrices: a1 = i, a2 = j1, a3 = j2 · j1, . . . , at−1 = jt−1 · . . . · j1 (i − unity matrix of order m×m); b1 = jt · . . . · j1 = j, b2 = jt · . . . · j2, . . . , bt = jt ; then bkak = j , k = 1, . . . , t , akbk = (jk−1 · . . . · j1) · (jt · . . . · jk) and, consequently, (akbk) s = akj s−1bk, s = 1, 2, . . .. by chain rule:( f (s)(x) )′∣∣∣ x=ηi = ( f (s−1)(x) )′∣∣∣ x=ηi+1 · (f(x))′ ∣∣ x=ηi = ( f (s−1)(x) )′∣∣∣ x=ηi+1 · ji, we get ( f ((j−1)t )(x) )′∣∣∣ x=ηi = aij j−2bi, j = 2, . . . , n and therefore ( f ((j−1)t+1)(x) )′∣∣∣ x=ηi = jiaij j−2bi, j = 2, . . . , n. next we find that f ′ (ηi) = n∑ j=1 θj ( f ((j−1)t+1)(x) )′∣∣∣ x=ηi = θ1ji + n∑ j=2 θjjiaij j−2bi. for the jacobi matrix of the system (2.2) cycle {η1, . . . , ηt} we can write: f ′ (ηt ) · . . . · f ′ (η1) = jt ( θ1i + at ( n∑ j=2 θjj j−2 ) bt ) · jt−1 ( θ1i + at−1 ( n∑ j=2 θjj j−2 ) bt−1 ) · . . . · j1 ( θ1i + a1 ( n∑ j=2 θjj j−2 ) b1 ) copyright c© 2020 assa. adv syst sci appl (2020) average predictive control 31 taking into account that jk ( θ1i + ak ( n∑ j=2 θjj j−2 ) bk ) = jkak ( θ1i + ( n∑ j=2 θjj j−2 ) bkak ) a−1k = jkak ( n∑ j=1 θjj j−1 ) a−1k and jkak = ak+1 it follows that f ′ (ηt ) · . . . · f ′ (η1) = = jtat ( n∑ j=1 θjj j−1 ) a−1t · jt−1at−1 ( n∑ j=1 θjj j−1 ) a−1t−1 · . . . · · j1a1 ( n∑ j=1 θjj j−1 ) a−11 = j ( n∑ j=1 θjj j−1 )t (the reader is gently advised that the superscript t in the formula above and all subsequent formulas denotes power, not transpose.) for deriving the jacobian formula above it was assumed that the matrix j was not degenerated. this limitation can be easily circumvented using a well-known topological technique: considering the matrix j + δi instead of the degenerated matrix j and after all calculations taking the limit as δ → 0. thus, the following result is obtained. lemma 3.1: the jacobi matrix of the cycle {η1, . . . , ηt} in the system (2.2) can be represented as j ( n∑ j=1 θjj j−1 )t , (3.3) where j is the jacobi matrix of the cycle {η1, . . . , ηt} in the system (2.1). we now consider another control system, instead of system (2.2): xn+1 = f ( θ1xn + n∑ j=2 θjf ((j−1)t ) (xn) ) . (3.4) when n∑ j=1 θj = 1 then the system (3.4) preserves the cycle {η1, . . . , ηt} . in addition, according to formula (3.3), the jacobi matrix of the system (3.4) cycle is expressed in the terms of jacobi matrix of the system (2.1). the advantage of the control system (3.4) over the system (2.2) consists of a fewer calculation of the values for function f(x) (more precisely, the difference is n − 2). copyright c© 2020 assa. adv syst sci appl (2020) 32 d. dmitrishin, e. iacob, a. stokolos 4. main result all results presented in this section are formulated for system (2.2), however they hold without change for system (3.4). theorem 4.1: suppose f ∈ c1 and that the system (2.1) has an unstable t cycle with multipliers {µ1, . . . , µm}. then this cycle will be a locally asymptotically stable cycle of the system (2.2) if µj [r (µj)] t ∈ d, j = 1, . . . ,m, where r(µ) = n∑ j=1 θjµ j−1. proof according to lemma 3.1, the characteristic polynomial for a system of linear approximation in the cycle neighbourhood in the case of system (2.2) can be written as ϕ(λ) = det ( λi − j [r(j)]t ) . by reducing the matrix j to the jordan form, this characteristic polynomial can be represented as ϕ(λ) = m∏ j=1 ( λi − µj [r (µj)] t ) , from where the theorem conclusion follows. note that the condition r(1) = 1 is obligatory. if, additionally, θj ∈ [0, 1] for j = 1, . . . , n , then µj [r (µj)] t ∈ d when µj ∈ d, and hence ∣∣∣µj [r (µj)] t ∣∣∣ < |µj|1+t . this means that if some multiplier of the system (2.1) cycle lies in the unit circle, the corresponding multiplier of the system (2.2) will lie closer to zero. thus for the closed loop system, the stabilization quality is improving. various estimates for multipliers allow us to construct control systems that stabilize cycles. 4.1. case m = {µ1, . . . , µm} if the multipliers are exactly known, we can choose n = m+ 1 and the coefficients {θ1, . . . , θm+1} from the condition r(µ) = m+1∑ j=1 θjµ j−1 = 1 m∏ k=1 (1−µk) m∏ k=1 (µ− µk). then from theorem 4.1 we get the following conclusion. conclusion. suppose that f ∈ c1 and the system (2.1) has an unstable t cycle with multipliers {µ1, . . . , µm}, and the coefficients θ1, . . . , θm+1 are found as exposed above. then this cycle will be a locally asymptotically stable cycle of system (2.2). moreover, if the initial point belongs to the cycle basin of attraction, the convergence to the cycle is superlinear. the superlinearity of the convergence rate follows from the fact that all multipliers of system (2.2) {η1, . . . , ηt} cycle turn out to be zero. note that the authors are unaware about any other method that allow to stabilize a cycle by knowing the cycle multipliers only. unfortunately, in a typical situation the multipliers are either unknown. the best we can expect is to localize them approximately. what to do in that case is considered in the next section. copyright c© 2020 assa. adv syst sci appl (2020) average predictive control 33 4.2. case m = {z : re z ≤ 0} ∪ d theorem 4.2: suppose f ∈ c1 and that system (2.1) has an unstable t -cycle with multipliers {µ1, . . . , µm} satisfying the conditions: |µj − µ̂j| < δj, re {µj} ≤ 0, j = 1, . . . , n1, |µj| < 1, j = n1 + 1, . . . ,m. let the coefficients θj , j = 1, . . . , n , of the system (2.2) be determined from the condition n1+1∑ j=1 θjµ j−1 = 1 n1∏ k=1 (1− µ̂k) n1∏ k=1 (µ− µ̂k) (here n = n1 + 1). then, for sufficiently small values δj , j = 1, . . . , n1, the t -cycle will be a locally asymptotically stable cycle of system (2.2). proof since re {µj} < 0, j = 1, . . . , n1, then all coefficients θj > 0, j = 1, . . . , n1. that means that ∣∣∣µ [r (µ)]t ∣∣∣ < |µ|1+t when |µ| < 1, i.e. the eigenvalues of the jacobi matrix of system (2.2) cycle corresponding to multipliers µj , j = n1 + 1, . . . ,m, are smaller than the multipliers absolute values. let δj = 0, j = 1, . . . , n1, then the eigenvalues corresponding to multipliers µj , j = 1, . . . , n1, are equal to zero. when δj , j = 1, . . . , n1, are sufficiently small in magnitude, these eigenvalues will lie in the central unit circle, as follows from rouche theorem. thus, all eigenvalues will less than 1 in absolute value, which means local asymptotic stability. 4.3. case m = d ∪ {µ∗} , |µ∗| > 1 [23] in [23], the coefficients θ1, . . . , θn were chosen as θ1 = 1, θ2 = . . . = θn−2 = 0, θn−1 = −θn = ε, where ε = 1∓ (|µ∗| /ρ)− 1 t (µ∗)n−2 (µ∗ − 1) , 0 < ρ < 1. such a choice ensures that the multipliers belong to the open central unit interval corresponding to µ∗. however, the condition∣∣∣µ [r(µ)]t ∣∣∣ < 1 with |µ| < 1 is not necessarily satisfied. nevertheless, the value ε can be made arbitrarily small by choosing the number n large. and then, from rouche’s theorem, it follows that with a sufficiently large n the other multipliers will remain within the central unit circle. this ensures the local asymptotic stability of the system (2.2) cycle. when it is known that µ∗ < −1, the control scheme can be simplified, namely: xn+1 = θ1f (xn) + θ2f (t+1) (xn) , (4.5) where θ1 = |µ∗| 1 + |µ∗| , θ2 = 1 1 + |µ∗| . copyright c© 2020 assa. adv syst sci appl (2020) 34 d. dmitrishin, e. iacob, a. stokolos 4.4. case m = d ∪ {µ∗, µ∗} , |µ∗| > 1 the case of general localization of multipliers {µ1, . . . , µm} for the system (2.1) cycle was considered in [23] but only for t = 1. in that case, the coefficients θ1, . . . , θn were no longer scalars but matrices and, as before, were chosen as θ1 = i , θ2 = . . . = θn−2 = 0, θn−1 = −θn = ε, where i is identity matrix, 0 is zero matrix, ε = sλs−1, λ = diag {ε1, . . . , εm}, εj = 1 + eıϕ (ρ/ |µj|) (µj) n−2 (µj − 1) , if |µj| < 1, and εj = 0, if |µj| > 1, and 0 < ρ < 1, ϕ ∈ {0, π} if µj as a real number. the matrix s consists of the eigenvectors of the jacobi matrix j for equilibrium point. thus, to apply the stabilization method, it is necessary to know not only all the multipliers of the equilibrium, but also the jacobi matrix itself. that is impossible when the equilibrium is not known. now, let us we apply the scheme (2.2). let µ∗ = ρeıϕ. then r(µ) = (µ− µ∗) (µ− µ∗) (1− µ∗) (1− µ∗) = ρ2 ρ2 − 2ρ cosϕ+ 1 + −2ρ cosϕ ρ2 − 2ρ cosϕ+ 1 µ+ 1 ρ2 − 2ρ cosϕ+ 1 µ2. if the complex number µ∗ lies in the left half-plane, then the coefficients of polynomial r(µ) are positive, so ∣∣∣µ [r(µ)]t ∣∣∣ < |µ|1+t with |µ| < 1. therefore, each multiplier of system (2.2) cycle lying in the central unit circle turns out to be in absolute value less then the multiplier of the corresponding cycle of the system (2.1). also, the multipliers corresponding to µ∗ and µ∗ change to zero. if the multipliers µ∗ and µ∗ are not exactly known, but they can be well estimated, then for the coefficients θ1, θ2, θ3, although different from the calculated ones, the values of the polynomial with these coefficients at points µ∗ and µ∗ will not exceed 1 in absolute value, as follows from the rouche theorem. the desired control system is xn+1 = θ1f (xn) + θ2f (t+1) (xn) + θ3f (2t+1) (xn) , where θ1 = ρ2 ρ2 − 2ρ cosϕ+ 1 , θ2 = −2ρ cosϕ ρ2 − 2ρ cosϕ+ 1 , θ3 = 1 ρ2 − 2ρ cosϕ+ 1 . 4.5. case t = 1, m = b−µ∗, µ∗c, m = b−µ∗, 1c suppose m = b−µ∗, µ∗c. from theorem 4.1 it follows that in order to stabilize the equilibrium, it would be necessary to construct a polynomial µr(µ), so that r(1) = 1 and |µr(µ)| ≤ 1 for all |µ| < µ∗. theorem 4.3: let f ∈ c1 and the system (2.1) has unstable equilibrium with multipliers {µ1, . . . , µm} ⊂ [−µ∗, µ∗]. let the value n be odd and be chosen from the condition csc π 2n > µ∗, and the copyright c© 2020 assa. adv syst sci appl (2020) average predictive control 35 coefficients θ1, . . . , θn from the condition µr(µ) = µ n∑ j=1 θjµ j−1 = (−1) n−1 2 tn ( µ sin π 2n ) , where tn(x) is the first kind chebyshev polynomial of odd order n . then this equilibrium will be a locally asymptotically stable equilibrium of system (2.2) (modulo a finite number of cases when µj = cos πk/n sinπ/2n for some k = 1, . . . , n − 1). the proof follows from the properties of the first kind chebyshev polynomials: |µr(µ)| ≤ 1 at ∣∣∣µ sin π 2n ∣∣∣ ≤ 1, r(1) = 1.note that µ∗ →∞ (n →∞) with asymptotics 2 π n . now we will consider the case of m = b−µ∗, 1c. theorem 4.4: let f ∈ c1 and the system (2.1) has unstable equilibrium with multipliers {µ1, . . . , µm} ⊂ b−µ∗, 1c. let the n value be chosen from the condition cot2 π 4n > µ∗, and the coefficients θ1, . . . , θn from the conditions µr(µ) = µ n∑ j=1 θjµ j−1 = tn ( µ ( 1− cos π 2n ) + cos π 2n ) , where tn(x) is the first kind chebyshev polynomial of order n . then this equilibrium will be locally asymptotically stable equilibrium of the system (2.2) (modulo a finite number of cases). proof note that r(0) = tn(cos π/2n) = 0, r(1) = tn(1) = 1. in addition |tn(x)| ≤ 1, at |x| ≤ 1, whence |µr(µ)| ≤ 1 at ∣∣∣µ(1− cos π 2n ) + cos π 2n ∣∣∣ ≤ 1. the last inequality is equivalent to − cot2 π 4n ≤ µ ≤ 1, which proves the theorem. note that µ∗ →∞ (n →∞) with asymptotics 16 π2 n2. 4.6. the general case using the ideas from theorem 4.1 cases, we can propose the following t -cycle stabilization scheme, for which the coefficients θj are not necessarily constants: a) find the matrix f ′(x), b) find the vectors f (s)(x), s = 1, . . . , t − 1, c) find the matrix f ′ ( f (t−1)(x) ) · . . . · f ′(f(x)) · f ′(x) d) find the matrix characteristic polynomial m+1∑ j=1 θj(x)µj−1, copyright c© 2020 assa. adv syst sci appl (2020) 36 d. dmitrishin, e. iacob, a. stokolos e) normalize the characteristic polynomial 1 m+1∑ j=1 θj(x) m+1∑ j=1 θj(x)µj−1, f) build the control system xn+1 = f (xn) , where f (x) = 1 m+1∑ j=1 θj(x) m+1∑ j=1 θj(x)f ((j−1)t+1)(x) or f (x) = f  1 m+1∑ j=1 θj(x) ( θ1x+ m+1∑ j=2 θj(x)f ((j−1)t )(x) ) let us consider how this scheme looks like in the case of a linear problem. let f(x) = ax wherea is a non-degeneratem×mmatrix. then η = 0 is a single fixed point, in the absence of any higher order cycles. we choose θj from the condition 1 det (i − a) det (µi − a) = m+1∑ j=1 θjµ j−1. then the control system is xn+1 = 1 m+1∑ j=1 θj m+1∑ j=1 θja jxn. by the hamilton-cayley theorem it follows that this system right-hand side is an identical zero. in the general case, applying this method to stabilizing chaotic motion tending to mixing, one can expect that after a certain number of iterations the trajectory falls into the basin of attraction for the stabilized cycle. then the convergence to the cycle will be superlinear. note that if in all the considered cases |θj| is being used instead of θj , then it becomes possible to stabilize the system (2.1) cycles with multipliers lying inm = d ∪ {µ : re (µ) ≤ 0}. moreover, the convex invariant set of system (2.1) will remain such for system (2.2). in addition, the system (2.2) multipliers, corresponding to those multipliers of system (2.1) that lie in the unit circle, will become closer to zero. 5. examples let us illustrate the effectiveness of the generalized predictive control method for finding periodic orbits with several well-known examples of scalar and vector chaotic systems [26]. the scheme applied for the logistic and triangular mappings (examples from sections 5.1–5.2) was a general scheme{ xn+1 = θ(xn) 1+θ(xn) f (xn) + 1 1+θ(xn) f (t+1) (xn) , θ (xn) = −f ′ ( f (t−1) (xn) ) · . . . · f ′ (f (xn)) · f ′ (xn) . copyright c© 2020 assa. adv syst sci appl (2020) average predictive control 37 it was possible to find a large number of cycles for all considered periods t ; in general, different initial conditions are producing different cycles. numerical calculations show that with sufficiently dense initial values grid, all cycles of a given length can be found. however, in this case it is necessary to ensure that the point xn remains within the invariant set, otherwise, as a rule it goes to infinity. if we use |θ(x)| instead of θ(x), the point xn will always remain in the invariant set. however, in this case we can find cycles only with multipliers from the set m = d ∪ {µ : re (µ) ≤ 0}. in the two-dimensional case, the scheme used was xn+1 = f ( θ 1 + θ xn + 1 1 + θ f (t ) (xn) ) . (5.6) the value θ should be chosen according to the condition x ( θ 1 + θ + 1 1 + θ x )t ∈ d at x = µ∗j , where µ∗j are cycle multipliers (j = 1, 2), and in general, they are unknown. in the examples below, one of the multipliers never exceeds one in magnitude, while the second one is negative, greater than one in absolute value. the theorem 4.2 guarantees stability conditions even if the parameter θ satisfies x ( θ 1 + θ + 1 1 + θ x )t 6= 0. it is enough have θ in the neighborhood of multiplier. in general theorem 4.2 does not provide the estimate on the parameters, however when one multiplier is in the unit disc and the other is real and negative an elementary trial works quite effectively. thus θ > 0, and we only have to check the compliance with the condition for the second multiplier ∣∣∣∣∣µ∗2 ( θ 1 + θ + 1 1 + θ µ∗2 )t ∣∣∣∣∣ < 1. (5.7) let θ = |µ∗2|+ ∆ and assume that |µ∗2| < 2t . if required that ∣∣∣∣ θ 1 + θ + 1 1 + θ µ∗2 ∣∣∣∣ < 1 2 , which is equivalent to−1 3 (1 + |µ∗2|) < ∆ < 1 + |µ∗2| , or θ ∈ ( 2 3 |µ∗2| − 1 3 , 2 |µ∗2|+ 1 ) . now, if θ < 2 3 |µ∗2| − 1 3 then 2θ ≤ 2 |µ∗2|+ 1. therefore, choosing θ = 2k subsequently for k = 1, 2, ... we are sure that for some k we get θ ∈ ( 2 3 |µ∗2| − 1 3 , 2 |µ∗2|+ 1 ) , then the condition (5.7) will be satisfied. thus, the grid for sorting parameter θ should be chosen rather coarse. this justifies the procedure we used in our examples: running (5.6) with small values for θ and then doubling them until obtaining required cycles. to our surprise, the procedure turns out to be copyright c© 2020 assa. adv syst sci appl (2020) 38 d. dmitrishin, e. iacob, a. stokolos quite efficient. however, this idea has been successfully used recently in adaptive interiorpoint methods by lesaja and potra [18]. therefore, the scheme (5.6) allows finding cycles both with small multipliers and with large ones. in general, the large value of multipliers is not the main obstacle. more challenging is the problem of small basins of attraction for long cycles. therefore, it is convenient to either select a dense grid for initial values or use a sufficiently large number of iterations so that the point xn would fall into the desired basin of attraction. one can achieve any acceptable accuracy in determining the cyclic point. we have experimented with various number of cycles (28, 50, 101, etc.) using maple and python. all the results we include in the subsequent subsections are for t = 101 (sections 5.1–5.13), with a precision of 250 decimals. more results, including the python code to replicate the results, can be found in [10]. 5.1. logistic mapping the logistic mapping xn+1 = hxn (1− xn) , (5.8) is, perhaps, the most popular example. let us consider the case h = 3.99, t = 101. figure 5.1 illustrates one of its numerous t = 101-cycles. fig. 5.1. a 101-cycle of the logistic mapping (5.8). copyright c© 2020 assa. adv syst sci appl (2020) average predictive control 39 5.2. triangular mapping our next example is the triangular mapping: xn+1 = h (1− |2xn − 1|) , h = 0.99. (5.9) figure 5.2 shows a t = 101-cyclic point. fig. 5.2. a 101-cycle of triangular system (5.9). 5.3. burgers mapping for the burgers mapping: xn+1 = axn − y2n, yn+1 = byn + xnyn, a = 0.75, b = 1.75 (5.10) a 101-cyclic point is illustrated in figure 5.3. 5.4. tinkerbell mapping the tinkerbell mapping: xn+1 = x2n − y2n + axn + byn, yn+1 = 2xnyn + cxn + dyn, a = 0.9, b = −0.6, c = 2.0, d = 0.5 (5.11) has a 101-cyclic point illustrated in figure 5.4. copyright c© 2020 assa. adv syst sci appl (2020) 40 d. dmitrishin, e. iacob, a. stokolos fig. 5.3. a 101-cycle of the burgers system (5.10). fig. 5.4. a 101-cycle of the tinkerbell mapping (5.11). copyright c© 2020 assa. adv syst sci appl (2020) average predictive control 41 5.5. gingerbredman mapping the gingerbredman mapping: xn+1 = 1 + |xn| − yn, yn+1 = xn (5.12) has a 101-cyclic point represented in figure 5.5. fig. 5.5. a 101-cycle of the gingerbredman mapping (5.12). 5.6. prey-predator mapping for the prey-predator mapping: xn+1 = xn exp (a (1− xn)− byn) , yn+1 = xn (1− exp (−cyn)) , a = 3, b = 5, c = 5 (5.13) a corresponding 101-cyclic point is illustrated in figure 5.6. 5.7. delayed logistic mapping figure 5.7 shows a 101-cyclic point of the delayed logistic mapping: xn+1 = hxn (1− yn) , yn+1 = xn, h = 2.27 (5.14) copyright c© 2020 assa. adv syst sci appl (2020) 42 d. dmitrishin, e. iacob, a. stokolos fig. 5.6. a 101-cycle of the prey-predator system (5.13). fig. 5.7. a 101-cycle of the delayed logistic mapping system (5.14). copyright c© 2020 assa. adv syst sci appl (2020) average predictive control 43 5.8. hénon mapping the hénon mapping: xn+1 = 1 + ax2n + yn, yn+1 = bxn, a = −1.40000001, b = 0.30000002 (5.15) has a 101-cyclic point represented in figure 5.8. fig. 5.8. a 101-cycle of hénon system (5.15). 5.9. elhadj-sprott mapping the elhadj-sprott mapping: xn+1 = 1 + a sinxn + byn, yn+1 = xn, a = −4.0, b = 0.9 (5.16) has a 101-cyclic point illustrated in figure 5.9. 5.10. lozi mapping the lozi mapping: xn+1 = 1 + a |xn|+ byn, yn+1 = xn, a = −1.7, b = 0.5 (5.17) has a 101-cyclic point shown in figure 5.10. copyright c© 2020 assa. adv syst sci appl (2020) 44 d. dmitrishin, e. iacob, a. stokolos fig. 5.9. a 101-cycle of the elhadge-sprott system (5.16). fig. 5.10. a 101-cycle of the lozi system (5.17). copyright c© 2020 assa. adv syst sci appl (2020) average predictive control 45 5.11. ikeda mapping the ikeda mapping is given by the equations: xn+1 = 1 + u (xn cos τn − yn sin τn) , yn+1 = u (xn sin τn + yn cos τn) , (5.18) where u = 0.9, τn = 0.4− 6 1 + x2n + y2n . the mapping has a 101-cyclic point illustrated in figure 5.11. fig. 5.11. a 101-cycle of the ikeda system (5.18). 5.12. holmes cubic mapping the holmes cubic mapping: xn+1 = yn, yn+1 = axn + byn − y3n, a = −0.2, b = 2.77 (5.19) has a 101-cyclic point shown in figure 5.12. 5.13. multihorseshoe mapping for the multihorseshoe mapping [14]: xn+1 = xne a−0.8xn−0.2yn yn+1 = yn(0.2xn + 0.8yn)eb−0.2xn−0.8yn a = 3.0, b = 3.0 (5.20) copyright c© 2020 assa. adv syst sci appl (2020) 46 d. dmitrishin, e. iacob, a. stokolos fig. 5.12. a 101-cycle of the holmes cubic system (5.19). we show a 101-cyclic point in figure 5.13. fig. 5.13. a 101-cycle of the multihorseshoe system (6.21). copyright c© 2020 assa. adv syst sci appl (2020) average predictive control 47 6. conclusion this article deals with the problem of stabilization for nonlinear systems of two categories: those unstable and those with a priori unknown periodic orbits at discrete time. a well-known method of stabilizing controls, called the predictive control method, first proposed by b.t. polyak, have been thoroughly investigated in this work. we have found that this method has several disadvantages: it is necessary to know the cycle exact multiplier or its sufficiently accurate estimate even in the scalar case; in the vector case, one must know the whole cycle jacobi matrix; consequently, the proposed control does not have the required robustness with respect to the system parameters perturbations; the control gain coefficients have different signs, which can trigger the initial system multiplier’s shifting beyond the central unit circle (where it lies) when applying the control; therefore, the gain coefficients must be small, and, in order to evaluate them at every instance, we need to know the multiplier’s value. all these shortcomings imply the necessity to modify the predictive control method. we propose not only to use the first and last iterations of the original mapping, but also all previous ones, by considering their linear combination. this linear combination’s coefficients are sought as being of a special polynomial, characterized by certain properties. as a result, it was possible for us to extend the predictive control scope. in addition, if the coefficients are non-negative, then for the initial system cycle multipliers lying in the central unit circle the corresponding multipliers of the control system cycle become closer to zero. an algorithm is given as a special case of this method application, for finding a cycle of a given length when its multipliers are known. one of the possible directions for future research is related to investigating new control schemes that combine the use of control system previous states and the initial system predicted states, i.e. the predictive control shall be considered together with the semi-linear control [8] as follows:  xn = n1∑ j=1 ajxn−jt+t yn = n2∑ j=1 bjxn−jt+1 f (x) = n3∑ j=1 θjf ((j−1)t+1)(x) xn+1 = (1− γ)f (xn) + γyn (6.21) where n1∑ j=1 aj = 1, n2∑ j=1 bj = 1, n3∑ j=1 θj = 1. clearly, the t -cycles of systems (2.1) and (6.21) coincide. the conditions of the system (6.21) t -cycle local asymptotic stability can be formulated as µj [r (µj)] t ∈ ( c \ φ ( d ))∗ , j = 1, . . . ,m, φ(z) = (1− γ)t z(q(z))t (1− γp(z))t , q(z) = n1∑ j=1 ajz j−1, p(z) = n2∑ j=1 bjz j−1, copyright c© 2020 assa. adv syst sci appl (2020) 48 d. dmitrishin, e. iacob, a. stokolos where c is an extended complex plane, and the asterisk denotes the reciprocal operation: (z)∗ = 1 z . the semilinear control method (when n3 = 1 in (6.21)) has also certain disadvantages [8,9]. further studies shall aim to eliminating (reducing) inherent disadvantages of predictive control and semi-linear control, synthesizing these approaches together. references 1. andrievsky b.r., fradkov a.l., control of chaos: methods and applications. i. methods, avtomat. i telemekh., (2003), no. 5, 3-45. 2. bartuccelli m., constantin p., doering c.r., gibbon j.d., gisselfalt m., hard turbulence in a finite dimensional dynamical system, phys. lett. a 142 (6-7) (1987) 349-356. 3. chen g., dong x., from chaos to order: methodologies, perspectives and application. world scientific, singapore (1999) 4. davidchack r.l., lai y.-c., klebanoff a., bollt e.m., towards complete detection of unstable periodic orbits in chaotic systems, phys. lett. a 287 (2001), 99-104. 5. devaney r.l., an introduction to chaotic dynamical systems. new york: addisonwesley publ. co., second edition., (1993) 6. dmitrishin d., hagelstein p., khamitova a., and stokolos a., limitations of robust stability of a linear delayed feedback control, siam j. control optim. 56, (2018), 148-157. 7. dmitrishin d. and khamitova a., methods of harmonic analysis in nonlinear dynamics, comptes rendus mathematique, volume 351, issues 9-10, 367 370 (2013) 8. dmitrishin d., khamitova a. and stokolos a., fejér polynomials and chaos, springer proceedings in mathematics and statistics, 108 (2014), 49–75. 9. dmitrishin d., skrinnik i., lesaja g., stokolos a., a new method for finding cycles by semilinear control, physics letters a, 383 (2019), 1871 – 1878. 10. dmitrishin d., iacob e., stokolos a., average predictive control for nonlinear discrete dynamical systems, arxiv:1906.02925v2. 11. elaydi s., an introduction to difference equations, isbn 0-387-23059-9. springer, new york (2005), 539p. 12. galias z., rigorous investigations of ikeda map by means of interval arithmetic. nonlinearity, (2002) 15:1759-1779. 13. jackson e.a., perspectives of nonlinear dinamics. vol. i, ii, cambridge univ. press, cambridge, 1980, 1990 chaos ii, ed. hao bai-lin. world sci., (1990). 14. joshi, y.a. and blackmore, d., strange attractors for asymptotically zero maps. chaos, solitons & fractals volume 68, 2014, 123-138. 15. kittel a, parisi j, pyragas k., delayed feedback-control of chaos by self-adapted delaytime. phys lett a 1995, vol. 198 (5-6),433-436. 16. kuntsevich a.v., kuntsevich v.m., estimates of stable limit cycles of nonlinear discrete systems, journal of automation and information sciences, volume 44, issue 9, (2012), 1-10. copyright c© 2020 assa. adv syst sci appl (2020) average predictive control 49 17. kuntsevich a.v., kuntsevich v.m., invariant sets for families of linear and nonlinear discrete systems with bounded disturbances, automation and remote control, vol. 73, no. 1, (2012), 83-96. 18. lesaja g., potra f., adaptive full newton-step infeasible interior-point method for sufficient horizontal lcp, optimization methods and software, 21(1), (2018), doi: 10.1080/10556788.2018.1546857 19. miller j.r., yorke j.a. finding all periodic orbits of maps using newton methods: sizes of basins. physica d (2000);135:195-211. 20. morgul o., further stability results for a generalization of delayed feedback control, nonlinear dynamics, 1-8 (2012). 21. morgul o., on the stability of delayed feedback controllers, phys. lett. a 314 (2003) 278-285. 22. ott e., grebodgi c., yorke j.a., controlling chaos. phys. rev. lett. 64, 1196-1199 (1990). 23. polyak b. t., stabilizing chaos with predictive control, automation and remote control, vol. 66 (11), (2005), 1791-1804. 24. polyak b.t., gryazina e.n., iterations of perturbed tent maps with applications to chaos control, ifac proceedings, volume 39, issue 8, 2006, pp 125-129. 25. pyragas k., continuous control of chaos by self controlling feedback. phys. rev. lett. a 170, 421-428 (1992). 26. sprott, j. c., chaos and time-series analysis, oxford university press, oxford, vol. 69 (2003). 27. shalby l., predictive feedback control method for stabilization of continuous time systems. advances in systems science and applications. 17 (2017), pp 1-13. 28. ushio t., limitation of delayed feedback control in nonlinear discrete-time systems. ieee trans circuits syst i 1996; vol. 43 (9), 815-816. 29. ushio t., yamamoto, s., prediction-based control of chaos. phys. letts. a. 264(1):439446(1999). 30. vieira de s.m., lichtenberg a.j. controlling chaos using nonlinear feedback with delay. phys. rev. e 54, 1200-1207 (1996). 31. ypma t.j., historical development of the newton-raphson method. siam rev., (1995) 37:531-551. 32. zhu j., tian y.p., necessary and sufficient conditions for stabilizability of discrete-time systems via delayed feedback control. phys lett a 2005; vol. 343 (1-3), 95-107. copyright c© 2020 assa. adv syst sci appl (2020) introduction problem statement constructing the jacobi matrix for a controlled system main result case m={1,…,m} case m={z: rez0}d case m=d{*}, |*|>1 15 case m=d{*,*}, |*|>1 case t=1, m="4262304 -*,*"5263305 , m="4262304 -*,1"5263305 the general case examples logistic mapping triangular mapping burgers mapping tinkerbell mapping gingerbredman mapping prey-predator mapping delayed logistic mapping hénon mapping elhadj-sprott mapping lozi mapping ikeda mapping holmes cubic mapping multihorseshoe mapping conclusion adv syst sci appl 2023; 02:24–45 published online at https://ijassa.ipu.ru. object recognition by a minimally pre-trained system in the process of studying the environment dmitry maximov*, sekou diane trapeznikov institute of control science, russian academy of sciences, moscow, russia abstract: we refine a method for describing and evaluating a previously proposed process of studying an abstract environment by a system (robot). in the process, we do not model any biological cognition mechanisms and consider the system as an agent (or a group of agents) equipped with an information processor. the robot (agent) makes a move in the environment, consumes information supplied by the environment, and gives out the next move (thus, the process is considered as a game). the robot moves in an unknown environment and should detect new objects located in it and recognize them. in this case, the system should build comprehensive images of visible things and memorize them if necessary (and it should also choose the current goal set). the main problems here are object recognition and the assessment of information reward in the game. thus, the main novelty of the paper is a new method of evaluating the amount of visual information about the object as the reward. in such a system, we suggest using a minimally pre-trained neural network to be responsible for the recognition: at first, we train the network only for biederman geons (geometrical primitives). training sets of geons are generated programmatically and we demonstrate that such a trained network recognizes geons in real objects quite well. sets of geons connected with objects (schemes) are used as the rewards. we also expect to generate procedurally new objects from geon schemes obtained from the environment in the future and to store them in a database. keywords: robot intellect, cognition, goal lattice, conway game semantics, synthesis of training sets, image classification 1. introduction the universal artificial intelligence (uai) is a unifying framework and a general formal foundational theory for artificial intelligence investigations [10]. its primary goal is to give a mathematical answer to the question: what is the right thing to do in an unknown environment, and how can an intelligent system learn to behave in the environment? such a learning process should be active in this case.“with active learning, the [robotic] system may deviate from the implementation of the main application and perform actions aimed at collecting information about the environment.” [7]. investigations in the field are focused on systems which act rationally. the artificial intelligence is represented in the approach as an information processor that consumes and gives out information. the theory also tries to answer, in general, the question, “how can a system composed of relatively unintelligent parts (say, neurons or transistors) behave intelligently?” [6]. a formal description of the most intelligent agent behaviour, in the sense of some intelligence measure, is suggested in the uai framework [17], [18]. the framework specifies how an agent interacts with an environment. the model is based on probabilistic modelling of the environment, and determination of the next system move, based on previous experience. ∗corresponding author: dmmax@inbox.ru object recognition by a minimally pre-trained system... 25 also, it is based on a numerical estimation of the system position reward, and the expected reward maximization along the trajectory. however, the method to obtain this numerical estimation is absent. it has been demonstrated in [22], [23], [21], that the existence of a structure (a lattice structure or, else, a monoid structure) in the set of tasks of the system [22], [23] or its goals [21] is sufficient for the system to behave quite reasonably. the behaviour even looks like that of an ant in some things [22]. thus, we may suppose that system intelligence is the consequence of the system’s purpose or predestination to use in practice. the approach does not assume environment modelling, unlike [17]. in this paper, first, we refine the results of [24], [25]† and develop the approaches mentioned above. our approach is based on the idea that one can imagine the process of studying the environment as the execution of parallel processes of detecting and recognizing various objects (i.e., processes to achieve them) in the environment. a tensor multiplication in a linear logic corresponds to these parallel processes. the logic is modelled in some game category [28] (first mention in [1], [2]). thus, the process of revealing and achieving objects by the intelligent system in the environment can be described as a game. this technique allows us to give a natural description of concurrency in such an object recognition process and to gain the reward expressions from a construct associated with the logic. these expressions are consistent with our understanding of the meaning of the reward. position rewards in the game are represented by some sets which define the information about these objects. the process of obtaining information is just the process of knowing the environment. thus, the main problems that we will explore in this paper are the recognition of interesting objects in the environment and the method for estimating information about them to obtain the game rewards. usually, such a system has a pre-existing set of such items and a subsystem that recognizes them. however, what to do in an unknown environment? we suppose in this paper that the robot behaves like a baby: it moves toward objects which have attracted its attention, chooses the most attractive item, and looks at it from all sides. thus, such an artificial intelligence system should be able to highlight unknown objects in the environment which were not specified in its pre-existing object set. to do this, we suggest using biederman’s theory of image recognition by humans [4]. according to the theory, people recognize object types by schemes that consist of geometrical primitives. therefore, we can initially train the system’s neural network to recognize only these primitives. then, we can build their incidence matrixes for images of environmental objects‡. thus, unknown objects (combinations of primitives in the environment) with the same set of primitives and the same incidence matrix would belong to the same object type. after that, these objects may be saved in a database. this approach is based on the idea that the robotic system must go through the whole learning process as a real person, from childhood to adulthood to become an intellectual one. during the process, it must explore objects of the outside world, recognize and classify them, and evaluate their utility based on intent to use the system in practice. the first step is to teach the system to recognize objects’ types and assign them names (classify) in its internal database. when the robot moves, such an unknown object becomes more discernible, and the set of its primitives obtains new elements. thus, we get an increment of information about the object. when this thing is studied from all sides, and we no longer have an increase in information, the process of the object cognition stops. therefore, we may take these sets of geometrical primitives as the rewards in the game that describes the system movement in the environment. similarly, while the robot is moving toward a possibly known object, the †we give the improved notion of the payoff function in the game category used, adapt proofs of propositions 2.1 and 2.2 to this case, and clarify the whole construction. ‡we experimentally demonstrate in the paper that such a trained network recognizes the primitives in real objects quite well. all experiments with incidence matrixes are left for the future. copyright © 2023 assa. adv syst sci appl (2023) 26 d. maximov, s. diane uncertainty of its identification diminishes, and the set of possible identification variants may be taken as the game anti-reward (which decreases during the game in this case). our method of object recognition belongs to the class of local-based recognition methods [33]. however, these methods may consider any kind of local features that describe a local area of the object, instead of having appearance-based parts [34]. this technique requires a dataset composed of training images to select parts of the thing to be later recognized. before training, the method itself does not allow you to distinguish between multiple classes of objects. a similar technique is used in [11] with other methods of part selection and another learning algorithm. a more complicated approach is described in [12], [13], [14], [29], [36] which uses the joint model of the parts to facilitate the detection of the features. each part encodes local visual properties of the object, and the deformable configuration is characterized by a tree structure of connections between certain pairs of components. however, this method also requires a set of images to train part selection and to create an object class template. on the contrary, we know features in advance in our method, and can discriminate between different object classes immediately, without any training. also, in the first steps, we do not determine these classes precisely. the information obtained is increased from step to step up to the end. after that, we get an item in the database of such classes as a set which consist of all the object schemes at once. the item represents from all sides a comprehensive image of the corresponding object at the end of the process of learning it. subsequently, a graphical skeleton of these geon schemes could be generated, which matches the item, and a neural network can be trained to store such a 3d type as a new object as in [8] (see sec. 4). the idea of the latter image recognition approach [8] is to place items of a typical shape in a different orientation in a virtual environment and to get their images. these objects are generated from some models, and the item in our database can serve as such a model. the position and the appearance of each object in each perspective are known immediately without additional calculations. in our article, we initially have such images of basic geometric primitives only. in the process of the environment cognition, we could add more intricate object schemes in different orientations, built from primitives, as a new item in the memory. initially, this memory is the database; afterwards, these items can be stored in the network. the second difference of our method from the others is that we store shape information in a static incidence matrix and use neither a probabilistic method for this purpose nor the procedure of minimizing the energy of the constellation of parts. the paper is organized as follows. section 2 presents the backgrounds necessary for the understanding of the results. we adapt the theory of [28] to the paper purposes in subsection 2.2. section 3 represents an updated variant of a formal description of intelligent system behaviour from [24], [25]. section 4 represents the use of a neural network to obtain the information rewards in the game that describes the system behaviour. in sec. 5 we discuss the reward valuations. in sec.6, we conclude the paper. 2. backgrounds 2.1. lattices we suppose visible objects and, hence, the game rewards form different lattices. thus, we start with them. definition 2.1: the partially-ordered set p is a set with such a binary relation x ⩽ y on its elements, that for all x, y, z ∈ p the following relationships hold: • x ⩽ x (reflexivity); • if x ⩽ y and y ⩽ x, then x = y (anti-symmetry); copyright © 2023 assa. adv syst sci appl (2023) object recognition by a minimally pre-trained system... 27 fig. 2.1. a lattice example. an element j12 is the join of b1 and b2, j12e is the join of b1, b2, and e, and etc. elements {b1, b2, b3, e} are the set of generators. • if x ⩽ y and y ⩽ z, then x ⩽ z (transitivity). the definition means that in the partially-ordered set, not all elements are compared with each other. this property distinguishes these sets from linearly-ordered ones, i.e., from numeric sets which are ordered by a norm. thus, the elements of the partially-ordered set are the objects of a more general nature than numbers. in the partially-ordered set diagram, the greater the element (i.e., vertex, node), the higher it lies, and the elements that are compared with each other lie in the same path from a bottom element to a top one. an example of a partially-ordered set diagram is represented in fig. 2.1 which is also a lattice diagram. definition 2.2: the upper bound of a subset x ⊂ p in a partially-ordered set p is the element a ∈ p such that x ⩽ a for all x ∈ x . the supremum or join is the smallest subset x upper bound. the infimum or meet defines dually as the greatest element a ∈ p such that a ⩽ x for all x ∈ x . definition 2.3: a lattice is a partially-ordered set, in which every two elements have their meet, denoted by x ∧ y, and join, denoted by x ∨ y. in the lattice diagram, the elements join is the nearest upper element to both of them, and the meet is the nearest lower one to both. definition 2.4: generators are such elements that generate all other elements by joins and meets (they are b1, e, b2, b3 in fig. 1). definition 2.5: the lattice is referred to as a complete lattice if its arbitrary subsets have the join and the meet. thus, any complete lattice has the biggest element 1, and the smallest one 0 and every finite lattice is complete [5]. if we take such a lattice as a scale of truth values in multi-valued logic, then the largest element will correspond to complete truth (1), and the smallest to complete falsehood (0). intermediate elements will correspond to partial truth in the same way as in fuzzy logic, partial truth is estimated by elements of the segment [0, 1]. copyright © 2023 assa. adv syst sci appl (2023) 28 d. maximov, s. diane in logics with such a scale of truth values, the implication can be determined through the multiplication of lattice elements (residual logics), or internally, only from lattice operations. definition 2.6: a lattice that has internal implications is called a brower lattice. in such a lattice, the implication c = a ⇒ b is defined as the largest c : a ∧ b = a ∧ c. definition 2.7: the implication ¬a = a ⇒ 0 is called the pseudo-complement. distribution laws for union and intersection are satisfied in the brower lattices. the converse is true only for finite lattices. 2.2. game semantics definition 2.8: a conway game is defined as a rooted graph with vertices v as the game positions and edges e ⊂ v × v as the game moves. each edge has a polarity ±1 which depends on whether it is the proponent or the opponent move. definition 2.9: a trajectory or a play is some path from the graph root ∗. the path is alternated if the adjacent edges are of different polarities. definition 2.10: a strategy σ of a conway game is defined as a non-empty set of alternated plays (paths) of even length. they start from the opponent move, closed up to the prefix of even length, i.e., for all plays s and all moves m, n, s ·m · n ∈ σ implies s ∈ σ, and determined. determinism means that two different paths with a common prefix should coincide, i.e., for all plays s, and all moves m, n, and n’, s ·m · n ∈ σ, and s ·m · n′ ∈ σ implies n = n′. definition 2.11: a dual play x⊥ is obtained from the play x by reversing the polarity of moves. definition 2.12: the tensor product x ⊗ y of two conway games x and y is the product of the two underlying graphs, i.e., its positions x⊗ y are vertices vx⊗y = vx × vy with the game root ∗x⊗y = ∗x × ∗y , its moves are x⊗ y → { z ⊗ y;x → z in x x⊗ z; y → z in y . the polarity of a move in x ⊗ y is inherited from the polarity of the underlying move in x or y. generalized linear logic is modelled in the category conw of such games [28]. the category objects are conway games, and morphisms x → y are strategies in x⊥℘y . the definition of the categorical construction of the operation ℘, which is dual to tensor ⊗, is not discussed here for simplicity because this is not essential for our description. it is enough to mention that on game graphs, these two operations are the same, so in [28], they are not even distinguished. the morphism composition and identity morphism are apparent [28]. we do not need the construction of the linear logic here, except the notion of linear implication. definition 2.13: [28], [20] the linear implications x ⊸ y in the category are defined as x ⊸ y = x⊥℘y since the category is symmetric monoidal closed (thus, we may define linear logic and the implication in the category). thus, morphisms x → y are strategies in the linear implication x ⊸ y . a conway game x with a payoff is the game with an additional weight kx = {1, 1/2, 0} in each vertex [28]. the weight depends on whether the position is winning or not. in copyright © 2023 assa. adv syst sci appl (2023) object recognition by a minimally pre-trained system... 29 the tensor production and implication, these weights obey the usual rules of conjunction and implication for such truth values scale [28]. thus, the conway’s payoff game x ⊗ y is defined as the underlying conway game x ⊗ y , equipped with the payoff function kx⊗y (x⊗ y) = kx(x) ∧ ky (y). the conway’s payoff game x ⊸ y is defined as the underlying conway game x ⊸ y , equipped with the payoff function kx⊸y (x ⊸ y) = kx(x) ⇒ ky (y). a strategy σ on a conway’s payoff game x is winning when every play s : x 7→ y in the strategy ends in a winning position y, i.e., in a position of payoff 1/2 or 1. it is possible to prove that the categorical construction is conserved if the weights’ numbers are replaced with some sets which form a brouwer lattice. also, the operations of [28] must be replaced with lattice operations. thus, we may declare by definition kx⊗y (x⊗ y) = kx(x) ∧ ky (y) (2.1) with lattice ∧, kx⊥℘y (x ⊥℘y) = ¬kx(x) ∨ ky (y) (2.2) with lattice ∨, and kx⊸y (x ⊸ y) = kx(x) ⇒ ky (y) (2.3) where ⇒ is now the lattice implication. both of the last two definitions are suitable as the payoff function of the game x ⊸ y . the first definition is a particular case of the second one in boolean lattices, and it may be used in practice as more convenient. however, we must emphasize that these payoff definitions are purely voluntaristic. they do not follow from any mathematical construction. the unique reason to introduce them is interpreting the tensor production ⊗ as a multiplicative conjunction and the co-tensor production ℘ as multiplicative disjunction in linear logic. hence, we may invert the payoff definitions 2.1 and 2.2, since operations ⊗ and ℘ are mutually dual as ∧ and ∨. thus, we introduce: kx⊗y (x⊗ y) = kx(x) ∨ ky (y) (2.4) kx⊥℘y (x ⊥℘y) = ¬kx(x) ∧ ky (y). (2.5) we will use these definitions in sec. 3 since we interpret in the section the fulfilling of parallel processes in the tensor production as the processes’ join in the corresponding lattice. we will prove that the categorical construction is also conserved for these definitions. the larger the set is associated with a position, the more preferable it is. we suppose the existence of a universal set containing all the others. thus, all such estimation sets form a complete lattice. definition 2.14: we call a strategy σ on a conway’s payoff game x with position estimations in a lattice as winning if every play s : x 7→ y in the strategy ends in a position y of payoff in the lattice which is different from 0. let us note here, that linear implication x ⊸ z means the process (game) x consuming and the process (game) z obtaining [16]. therefore, the payoff kz(z) may not be 0 in the end position of the winning strategy σ of the game x ⊸ z, because we must not obtain a process (game) z ended in 0 in the winning case. however, this restriction is unnecessary in the case of definitions (2.4) and (2.5). we show now that such defined winning strategies do compose as in [28]. proposition 2.1: the strategy ρ ◦ σ : x ⊸ z is winning when the two strategies σ : x ⊸ y and ρ : y ⊸ z are winning. proof let us consider first the proof for (2.3). it is known that strategies do compose [28]. thus, it copyright © 2023 assa. adv syst sci appl (2023) 30 d. maximov, s. diane is sufficient to check the winning condition. we should observe that the composition of two winning positions is winning: kx(x) ⇒ ky (y) > 0, and ky (y) > 0, i.e., x ⊸ y is winning; ky (y) ⇒ kz(z) > 0, and kz(z) > 0, i.e., y ⊸ z is winning; implies kx(x) ⇒ kz(z) > 0, i.e., x ⊸ z is winning. however, it is evident, since kx(x) ⇒ kz(z) ⩾ kz(z) > 0. the proof for (2.2): it is evident that in the case of kx⊸y (x ⊸ y) ≡ kx⊥℘y (x ⊥℘y) = ¬kx(x) ∨ ky (y) and ky (y) > 0 winning strategies do compose. the proof for (2.5): it is evident that in the case of the winning strategy with kx⊸y (x ⊸ y) ≡ kx⊥℘y (x ⊥℘y) = ¬kx(x) ∧ ky (y), ¬kx(x) > 0 and ky (y) > 0, thus winning strategies do compose. definition 2.15: let us, setpayoff is a category whose objects are conway’s payoff games, in which position weights take values in a lattice, and morphisms x → y are winning strategies in x ⊸ y . proposition 2.2: the category setpayoff is symmetric monoidal closed. proof the proof for (2.1) and (2.3): the category of conway games is symmetric monoidal close [28]. therefore, it is sufficient to check if (kx(x) ∧ ky (y)) ⇒ kz(z) = kx(x) ⇒ (ky (y) ⇒ kz(z)) for all positions. but this formula is valid in heyting algebras (i.e., in brouwer lattices). the proof for (2.1) and (2.2): ¬(kx(x) ∧ ky (y)) ∨ kz(z) = ¬kx(x) ∨ (¬(ky (y) ∨ kz(z)) in brouwer lattices. the proof for (2.4) and (2.5): ¬(kx(x) ∨ ky (y)) ∧ kz(z) = ¬kx(x) ∧ (¬(ky (y) ∧ kz(z)) in brouwer lattices. thus, the symmetric monoidal closed categorical construction for conway’s payoff games from [28] is conserved for lattice payoffs, and morphisms x → y are strategies (in this case winning) in the linear implication x ⊸ y as in conway’s games’category. 2.3. biederman geons irving biederman has proposed in [4] a theory of image recognition by humans, in which an image is segmented into a set of geometric primitives, such as blocks, cylinders, wedges, and cones. the collection of the components, called geons, is rather limited (n ⩽ 36) and can be derived from easily detectable properties of edges: curvature, collinearity, parallelism, and convergence. the detection of these properties is invariant over viewing position and image quality, and thus allows objects to be perceived in different positions and in the case of noised image. the experiments of [4] showed low errors in object naming of such geon schemes of different objects. some of such simple geon schemes are represented in fig. 2.2. the approach is similar to speech perception: in both cases, we can code tens of thousands of objects by mapping the input onto a limited number of primitives and then using a representative system to build free combinations of these primitives. however, this method does not allow us to distinguish images inside one object type. to do this, we should use other methods. 3. system behaviour description we consider the cognition process as a game in which a system (a robot) investigates an environment, i.e., the system obtains information about the environment objects in the form of geon sets. copyright © 2023 assa. adv syst sci appl (2023) object recognition by a minimally pre-trained system... 31 fig. 2.2. examples of geon schemes of simple objects. fig. 3.3. an example of a winning strategy of a game and the visibility horizon. the bold line shows the resulting play (one of the winning strategy) with the rewards k(pi, bj) in the positions pi of the process of reaching objects bj and with polarities ±1. e and bj are objects of interest i, j = 1, 2, .... the environment moves — obtaining information by the robot — are depicted by circles, since they are not real moves. supposition 3.1: it is supposed that the robot investigates the environment visible up to some horizon in each direction (fig. 3.3) and builds images of the observed objects. supposition 3.2: it is supposed that objects’ degrees of attractiveness guide the robot to behave in a certain way: it investigates things in the environment which have attracted its attention, and builds their images. the theory of attention is discussed, e.g., in [15]. “attention is defined as a concentrated mental activity. in general, we can think of attention as a form of mental activity or energy that is distributed among alternative information sources” [15]. both general classes of theories that attempt to explain attention — bottleneck theories and capacity theories — explain how attention selects information sources. still, they tell us nothing about why exactly it makes the selection. we suppose the system has some pre-existing preferences to choose the sources. in the simplest cases, the choice can be based on the detection of areas of difference in hue, contrast, degree of distance from the observation point, etc. [7] in more complicated cases, copyright © 2023 assa. adv syst sci appl (2023) 32 d. maximov, s. diane the system can have some frames or gestalt images. thus, we suppose that the system mission or purpose for practical use determines the attention preferences. then, the system should distribute the objects according to the degree of attraction during the investigation. thus, some objects are more attractive, some of them less attractive, and some cannot be compared in terms of attractiveness. therefore, we get a partial order at the object set. it is supposed that the set has a bottom element, i.e., the element of zero interest and the top element, i.e., object of greatest interest, which is the most attractive element. the latter is the lattice join of all its elements (see supposition 3.3). other elements may be represented as some joins (combinations) or meets in the case when an attractive object is some part of another one. therefore, the system builds a complete lattice of the environment objects’ images, their combinations, and allotments (e.g., fig. 2.1 for some objects from fig. 3.3). such a lattice may already exist; thus, the attractiveness of new objects may be combined with the attractiveness of pre-existing ones. many simple biological systems, e.g., ants colonies, possibly have such a pre-built lattice of objects with which they can deal. perhaps the partial order in the lattice can be dynamically changed for more complicated systems. in any case, we assume that the lattice of the environment object images can dynamically change due to the fact that the system does not see everything around at every moment of time. the system looks into a sector and builds images of objects that lie inside it. supposition 3.3: it is supposed that the preferable behaviour of the system is to achieve all its goals — i.e., to investigate and recognize all interesting objects in the environment at a given time, i.e., to get maximum information. we identify the system goals (where “goals” mean interesting objects in the environment) and the processes of achieving these goals. thus, the most preferable behaviour variant corresponds to the top lattice element 1. and the bottom element 0 (zero) corresponds to complete inactivity and to the least significant behaviour variant. all the estimations may be considered as partially true truth values. thus, we can say that the more essential the system behaviour is, the truer it is. when it is not possible to determine the most crucial estimation for a behaviour due to the order being only partial, some additional methods may be used to select the optimal variant [24]. the process of moving a robot in the space of the environment, i.e., the cognitive process of studying the environment by a robotic system, can be represented as a game. during the game, the environment informs the system about (partially) visible objects at each step, and the system estimates the information and formes the position reward. in the next paragraph, we will see that the rewards are not numbers, but sets of geometric primitives which make up the objects. certainly, the environment has no agency of its own, and it is passive as a player, thus, this is all very reminiscent of a reinforcement learning approach [32]. however, using the construction of the category of conway games allows a natural description of the process of parallel exploration and achievement of various objects before choosing one of them. moreover, we are interested not so much in the recognition of objects as in their cognition. the main thing is to get as much information as possible about an individual object, and not teach the robot any specific behavior. also, this information as the rewards is represented by sets, not by numbers. therefore, usual methods of reinforcement learning do not fit. we regard the environment as the opponent and the system as the proponent to use the categorical construction of sec. 2.3. the proponent moves from one position in the environment to another, using the information to achieve his goals, i.e., environment objects. the more fortunate the position, i.e., the larger the reward, the more precise information the environment provides about the item in the position. thus, the completely winning position is the last game step of the proponent, in which it can get no more information. the robot is initially placed in the configuration space (environment) at the root * of the system game a with the system goal/object lattice ms partially ordered according to robot attention preferences. copyright © 2023 assa. adv syst sci appl (2023) object recognition by a minimally pre-trained system... 33 game a represents the possible robot’s moves in the environment. but the real trajectory or play is chosen from the requirement of the maximum total reward for a position along the projected path ending in a winning position (in this position the reward cannot be improved). the system movement in the environment is estimated according to this criterion with the reward k(pi, bj) in the position pi of the process of reaching the object bj . it may happen that the system initially does not intend to achieve any goal and moves according to the criterion of optimal movement in the absence of goals. thus, the system has a goal (task) a of free movement in the environment, which should be included in the lattice ms. hence, such free movement must have its own value. we do not discuss here the optimality criterion, since the robot or agent systems’ designers may suggest such ones according to their needs in a specific environment. however, we assume that the rewards for free movement in the corresponding game are also estimated by some sets. these sets are of the same nature as the sets which correspond to the rewards in the game of achieving goals. therefore, the rewards of the free movement are some sets of geometric primitives that are placed in the environment but are not treated as objects of interest. thus, we suppose the system sees different combinations of geometric primitives in the environment. the primitives (geons) which are located in one place make up an object scheme. these schemes may have joins and meets combining other such schemes (which may no longer be in one place). hence, such primitives’ sets form a lattice. the lattice is complete since it contains any joins and meets. this lattice is different from the lattice ms of the system goals/objects. the lattice ms contains only a part of all objects, and it is partially ordered according to attention preferences. the primitives’ lattice contains all visible things and their combinations, and it is ordered according to the relation of the set inclusion. we demand the geons’ lattice is brouwer (in particular, it must be distributive); thus, it includes all that is necessary for these joins and meets even when they are not evident. let us n objects b1...bn are discovered in the environment by the system with information about them k(pi, bj) in positions pi of the game a (fig. 3.3). the rewards k(pi, bj) take values in the geons’ lattice. only k goals from these n’s ones may be chosen due to limited system opportunities. game a corresponds to the process of achieving the free movement goal a. then, a winning strategy of the game a ⊸ b1 ⊗ ...⊗bk (i.e., a set of winning plays) defines a transition (morphism) from the game a to this new game a′ = b1 ⊗ ...⊗bk of moving and achieving goals b1...bk in the games b1...bk. it is supposed that the system can achieve several objects in parallel consistently reducing their number up to the moment when only one object rests to be chosen. thus, the game a′ corresponds to parallel processes of reaching those k goals from discovered n’s ones, which may be better achieved in the next sense. it is reasonable to choose the trajectory (one play from the winning strategy) from the requirement to maximize the reward along the path within the visibility horizon, in the following way (using (2.4), (2.5) in contrast to [24], [26])§: ka⊸b1⊗...⊗bk play (a ⊸ b1 ⊗ ...⊗ bk) ≡ ka⊥℘b1⊗...⊗bk play = = max plays [ ⋃ play ¬ka(a) ∧ (kb1(b1) ∨ ... ∨ kbk(bk))] (3.6) here, the reward ka⊸b1⊗...⊗bk play = ka⊥℘b1⊗...⊗bk play corresponds to that process of achieving k goals that has the greatest priority (i.e., the highest truth value) in the system goal/object §we may also use formulas (2.3) and (2.1) for rewards: k a⊸b1⊗...⊗bk play = max plays ⋃ play [ka ⇒ (kb1 ∧ ... ∧ kbk )], i.e., with meets in the rewards for games bi, i = 1...k. but, this form contradicts our reward sense (sec. 5). however, the implication calculation is no more difficult [27], hence, the form could also be used with some another reward definition. copyright © 2023 assa. adv syst sci appl (2023) 34 d. maximov, s. diane lattice (at the current time). thus, these k processes are the most important parallel ones from the viewpoint of the system goal lattice. the priority is maximal among all possible parallel processes of reaching n discovered objects¶. the maximum in (3.6) is taken among all possible plays, and it joins the rewards along these plays (i.e., trajectories) in games a⊥, b1, ... and bk. thus, information about all objects bi and their system images is demanded to maximize along the resultant path up to the moment when all the images together may not be able to improve. then, the total number of chosen objects should be decreased by the same method: the system should pick those l objects with l < k, the processes of parallel achievement of which have the highest estimation in the system goal lattice. and so on, up to the moment, when perhaps only one goal remains. however, some external method of the path calculation may suggest an objects’ traversal path to obtain the maximum cumulative reward from as many angles as possible (see discussion in sec. 5). the meaning of formula (3.6) is that, following the semantics of linear implication, the system moves from the execution of the process of free movement a to the processes of achieving goals bj . the gain is calculated within the strategy (i.e., a set of pathes) of game a⊥℘b (with simplified notation). at the same time, information about targets kbj(bj) is maximized, and information about the free movement of ka(a) is minimized (since pseudocomplements ¬ka(a) are maximized). such k and ¬k can be considered as arguments and counterarguments for the corresponding movement following the ideas of the jsm method of plausible inference [3]: the more information we have about the object, the stronger the arguments for the transition to achieving it. also, the stronger the arguments against free movement (i.e., the less information we have about the possibility of such a movement), the stronger the arguments for moving from free movement to achieving a goal. 4. object recognition thus, we come to the main problem: how to select (recognize) objects in the environment and how to evaluate the amount of information about them, i.e., the rewards? we offer a solution to the latter problem in sec. 5 “discussion of system movement and reward valuations”, and we discuss below the former one. we suggest to decompose an environment image to biederman’s geons (paragraph “biederman geons” 2.3 in sec. 2). then, during the cognition process, the system obtains such images from different camera angles, and gets a tuple of sets of geons localized in one place. these sets with their geon incidence matrixes constitute one type of environment objects. the incidence matrix for each geon pair indicates their facets which intersect (see paragraph “geon recognition” 4.4 below). if the system recognizes such a set in future, it will know the possible object type. to perform the programme, we need to automatically train the neural network that will be used for recognition, first only these geons (this is done in the paper), and then such sets of geons (which we suppose to perform later). thus, we consider first an automatic generation of training samples in a virtual environment [8], [9]. 4.1. generation of training samples the presence of a high-quality training set largely determines the efficiency of machine learning algorithms. it is worth mentioning that when preparing training samples, you should pay attention not only to the volume of data, but also to such things as the balance of classes ¶we consider here the join of several goals in the lattice ms as the process of their possibly parallel achieving (like in [22]), though, in [24] a linear logic structure on the goal lattice was used in this case. we did so because it is hard making sense of lattice elements multiplication in linear logic. copyright © 2023 assa. adv syst sci appl (2023) object recognition by a minimally pre-trained system... 35 fig. 4.4. the structure of the dataset synthesis software (dss). and the order of their sequence. the data must contain a comparable amount of instances for each class, and must be mixed. it is advisable to include such data in the training set that is as close as possible to the conditions of further use of the neural network. in this study, we propose a technology of synthesis of training sets based on the use of three-dimensional graphics (opengl library [19]) within the developed software-algorithmic complex (fig. 4.4). the dataset synthesis software (dss) [9] is based on the module for rendering images of virtual scenes. a virtual scene is a collection of three-dimensional objects of different categories, equipped with a description of their position, orientation and colour characteristics. the input of the scene rendering module is the information about the geometric and textural features of the scene, received from the virtual scene manager according to the data stored in the file system. alternatively, an instantly generated scene created with a virtual scene generator is used. unlike the loaded scene, the positions and number of objects in the generated scene are usually randomly selected, which creates the required variety for generating training examples for a neural network. the coordinates of the camera are also transmitted to the input of the virtual scene rendering module during its movement along a given trajectory. this feature allows simulating the video image from the onboard camera to model navigation tasks of mobile robots. the output image of the scene rendering module is transmitted to the modules for generating training examples. thus, e.g., the module for generating detection examples generates text descriptions containing the coordinates of target objects locations on the video frame. in turn, the module for analyzing the stereo image synthesizes a depth map corresponding to the observed visual scene. the segmentation module generates maps of pixel belonging to certain objects in the field of view of the camera. and finally, the camera position annotation module saves the camera transformation matrix at the current time in the training example. 4.2. training sets formation in problems of visual classification in accordance with tasks of visual image analysis, dss allows you to generate training samples for solving the problems of visual classification, localization, segmentation and copyright © 2023 assa. adv syst sci appl (2023) 36 d. maximov, s. diane image depth evaluation. in addition, the virtual environment provides access to the exact position of the camera at successive times, which makes it possible to synthesize training samples to solve the problem of visual odometry. there are two ways to create virtual scenes in the pre-rendering phase. we use the following approach for the tasks of coarse tuning of neural network classifiers, when the mutual position of different objects is not important and, on the contrary, requires the greatest possible variety of objects moving around the scene. in the approach, the number of n objects simultaneously observed in the scene is manually set or randomly selected. a list of random positions pi, i = 1...n for these objects is formed: p = {p1, ..., pn}. (4.7) the situations of mutual penetration of objects are eliminated on the basis of the method of potential fields: p′i = pi +min(dmax, n∑ j=1,j ̸=i ηij/ ∥ pj − pi ∥2), (4.8) where p′i = {x′, y′, z′} is an updated object position; dmax is a maximum shift; η is the repulsion coefficient for the pair of objects i and j, ∥ • ∥ is the euclidean norm. coefficient η is chosen for reasons of repulsion normalization. let r1 be the safety radius of the first object, r2 the safety radius of the second object. then the target distance between objects is d = r1 +r2. the condition for the empirical calculation η can be chosen as follows: ηij/ ∥ (pj − pi) ∥= 0.5dij. (4.9) for the tasks of configuring neural network classifiers to solve specific application problems, an approach based on loading pre-prepared virtual scenes is used. in such a case, the variety of training examples is achieved not by changing the position of objects in the scene, but by changing the angle of observation when the camera moves along the specified trajectory. at the first stage of solving a specific applied problem of setting up a visual classifier, software is used that allows to form descriptions of virtual scenes in the following form: w = {o1, ..., on}; (4.10) where oi is a programme structure that encapsulates the object’s position, orientation, type and specifics of its visual appearance. compatibility of virtual scenes’ storage format is ensured to provide their correct loading in the virtual environment. in the second stage, automatically generated scenes are loaded into the dss: the text descriptions of the scenes are interpreted and the corresponding software representations are formed for the objects listed in the scene file. next, reliable results of visual analysis are determined on the basis of a direct access to the properties of the loaded programme structures. annotations containing the desired results of the analysis of the type and position of objects are saved together with the images in a directory on the disk for further tuning neural networks. let us note that the first stage can be performed either using ready-made three-dimensional models as the elements of the set, or using procedurally synthesized models. the latter approach is preferable due to the unlimited possibilities for altering the parameters and, as a result, achieving variability in the appearance of these objects. 4.3. procedural model generation procedurally generated objects are specified as a set of parameters, which, on the one hand, is more compact compared to the explicit enumeration of constituting geometric primitives in the three-dimensional model file. on the other hand, the set allows you to diversify the appearance of the object to form high quality training sets. in general, the generated object is copyright © 2023 assa. adv syst sci appl (2023) object recognition by a minimally pre-trained system... 37 fig. 4.5. examples of automatically generated objects: a) wall; b) stairs; c) tree; d) a fragment of the landscape. specified by the formula o = {l, p1..., pk}, (4.11) where l is object class, pi, (i = 1...k) are parameters determined by an expert, which affect the geometric shape of the object through the relations embedded in the corresponding analytical model. examples of procedurally synthesized objects used in modelling both indoor and outdoor areas are presented in fig. 4.5. for example, to generate an object of type “tree” (fig. 4.5,c), the set (4.11) takes the form: o = {“tree”, h, k, nb, nl}, (4.12) where h is the tree trunk height, k is a thickness coefficient, nb is a number of branches at every level of the tree, nl is a number of levels of recursive branching. each branch is formed as a typical tetrahedral pyramid, the displacement and rotation of which are set randomly within the limits allowed by the parent object (trunk or branch). its length is selected in proportion to the distance from the positioning point of the branch to the final vertex of the parent object. this approach allows you to create a wide class of objects that look like real trees of different types. the same can be said about the other objects presented in fig. 4.5. the described possibility of procedural generation of objects can be used not only for the formation of training sets in the classification of visual images, but also when setting up vision systems that solve the problem of avoiding obstacles. the presence of comprehensive information about the geometry of the synthesized object enables optimal trajectory planning in the nearby space. comparison of the obtained plan of movement with partial visual information available for the moving camera makes it possible to form a dataset with samples copyright © 2023 assa. adv syst sci appl (2023) 38 d. maximov, s. diane that capture implicit relationships when choosing an adequate movement of the robot for various observed situations in the external environment. an additional increase in the number of training examples can be obtained using wellknown methods of augmentation: shifting, rotating, reflecting, scaling, noising, blurring images, as well as a relatively new method of neural network data augmentation [37]. 4.4. geon recognition now, we can flesh out the formula (4.11) for geons: geoni = {idi, {(facetj, linej, deformed, curved)}}, (4.13) where “line” is the linear parameter of the j’s facet, which takes values {long, short}, “curved” is its curvature parameter, which takes values {yes, no}, and the parameter “deformed” takes values {no, bloat, depressed}. the facet set is: {top, bottom, front, back, left, right} with the corresponding geometrical form for each value. all geons are modifications of a cylinder and a brick. accordingly, each brick is a combination of rectangles (and trapeziums up to triangles) that have a long and a short size. the cylinder also has such an aspect ratio. the “line” parameter indicates this characteristic. additionally, geons may be parameterized with scaling factors, e.g., sx = [0.5...2], sy = [0.5...2], sz = [0.5...2], and surface noise f = [0...1], which allow us to diversify their visual appearance during the generation of the training images. we have used such a diversification in this paper. then, we get the following values for the incidence matrix of a geon pair: mi,j = { {zonej}, geoni ⋂ geonj ̸=∅; 0, geoni ⋂ geonj=∅, (4.14) where the variable “zonej” refers to j’s geon and takes the next values: zonej ∈ {top, bottom, left, right, front, back}j. (4.15) the set of zones denotes those frame sides of j’s geon which intersect with the frame of i’s geon. we get corresponding frame sides of i’s geon in the mj,i element. thus, we can reconstruct the whole geon scheme of the object in a certain foreshortening by its incidence matrix. the incidence matrix and zone values can be obtained with fuzzy operations (especially, when the objects intersect by two or more zones) from the objects’ bounding box coordinates given by the used yolo (“you only look once”) neural network. naturally, this is rather a rough object classification, and we offer to use it as the first simple variant. at the first stage, these matrices can be storage in the system’s (robot’s) memory in order to compare them with the matrices of observed objects. subsequently, we would like to build a 3d object by a corresponding incident matrix set and to train a network for such objects as we do in this paper for geons. such a 3d object could be supplemented by a graph of geon connections to each other. in this paper, as a first step, we have used the dataset synthesis software (dss) [9] (subsec. 4.1) to get a geon data set||, and trained a yolo neural network on them with the resources [30], [31] with predefined training parameters there. at the network input, we have a raster rgb image, presented as an array of pixel brightness with a resolution of 416 * 416. at the output of the network, we obtain 169 cells, each of which contains 63 values: (4 coordinates of the frame +1 estimate of the presence of an object +16 geon classes) * 3 proportions of the frames = 63 values. total: 169 ∗ 63 = 10647 values. fig. 4.6 — fig. 4.11 show examples of decomposing real images on geons after the network training. ∥we used a set of only 16 main geons for our limited calculation capabilities. copyright © 2023 assa. adv syst sci appl (2023) object recognition by a minimally pre-trained system... 39 fig. 4.6. an example of a simple image decomposition in the case of automatically generated images in the training set. fig. 4.7. an example of a simple image decomposition in the case of manually marked up images in the training set. fig. 4.8. an example of a simple image decomposition in the case of automatically generated and manually marked up images in the training set. here, you can see that only an automatically generated image set is not enough for good image recognition under limited training (fig. 4.6, 4.9): we have used together three series of 150 images with 10, 20 and 30 geons respectively, with 10x augmentation (resource [35] copyright © 2023 assa. adv syst sci appl (2023) 40 d. maximov, s. diane fig. 4.9. an example of a complicated image decomposition in the case of automatically generated images in the training set. has been used). the manually marked up image set is also not good (fig. 4.7, 4.10)**, and only the combination of automatically generated images with a small part of the hand-marked ones gives an acceptable result (fig. 4.11)†† — geon recognition becomes much more exact and complete. we use here about 150 hand-marked images with the same augmentation. 5. discussion of system movement and reward valuations formula 3.6, which maximizes reward along the path, provides a mathematical justification for generally obvious behaviour. however, we can now clarify some points to use the formula in practice. obviously, the system should move in such a manner as to increase all the rewards ¬ka(a) and kbi(bj) at some discrete time moments for the maximum possible number of goals. thus, what amounts should we take as the objects’ rewards? clearly, the number of visible objects’ geons may not increase as the system gets closer to the object. this number may even decrease with the foreshortening change. however, the information amount increases since we recognize the object more confidently and from different sides. ∗∗an image is not presented in these image sets if it was not recognized in the corresponding case ††authors are grateful to k. rusakov for this advice copyright © 2023 assa. adv syst sci appl (2023) object recognition by a minimally pre-trained system... 41 fig. 4.10. an example of a complicated image decomposition in the case of manually marked up images in the training set. hence, we use the set ∪j{gl}j of geons in geon schemes of goals bj to be achieved as the anti-reward ¬ka(a) for the free movement. the larger the set, the higher the reward of the non-free movement and the lower the reward of the free movement. for kbi(bj) in the process of achieving goals, we use (possibly successively) two reward types : — in the first type, the system approaches an object and sees it better and better. it tries to identify the object with some ones in the system database, and the uncertainty of its identification diminishes. at this stage, we use the anti-reward ¬kbi(bj) as if it is a possibly known object: the anti-reward is the set of possible recognition variants. when the system sees all visible geons {gl}j of the thing bj precisely, without any hesitation, then the antireward is minimal, and the reward kbi(pi, bj) in position pi is maximal (the object has been identified). thus, the system should move in such a manner as to decrease the anti-reward set; — in the second one, the system has no association in its database with the object. in this case, we take the union ⋃ pk k⩽i {gl}j of geons’ sets (remembered up to the position pi) of the object bj visible from different camera angles, as the reward kbi(pi, bj). the geon sets are chosen to be localized in one place. this reward is maximal when the system has investigated the object from all sides and cannot get more information (i.e., cannot add new geons to the set corresponding to the item). such a reward definition may demand external methods to decide that the object is investigated from all sides. copyright © 2023 assa. adv syst sci appl (2023) 42 d. maximov, s. diane fig. 4.11. an example of a complicated image decomposition in the case of automatically generated and manually marked up images in the training set. suppose, there are left only two objects of interest (goals) b1 and b2, and the reward of these two games b1 and b2 cannot be enlarged in parallel:⋃ play ¬ka(a) ∧ (kb1(b1) ∨ kb2(b2)) = max. (5.16) the formula means that ka(a) cannot be diminished, as well as kb1(b1) and kb2(b2) cannot be increased, i.e., the set of known geons in b1 and b2 cannot be increased in parallel. hence, we should choose the game and the play with only one object achieving. we do this according to our supposition to pick those objects which have greater assessments in the goals/objects lattice ms (sec. 3). this lattice is partially ordered by attention preferences. thus, we reduce the number of goals that need to be achieved not from comparing their geon sets, but from comparing the degree of their attractiveness (attention preferences) for the robot. thus, formula 3.6 mathematically justifies the apparent general behaviour of the system, but not a specific space trajectory. let us also note, that completeness of information is not assumed, since the environment is unlimited and is only partially visible. the criterion of optimality is the maximum information received at the current time. if the visibility changes, then the number of visible objects may also change, thus, the priority lattice and, hence, the path may change. copyright © 2023 assa. adv syst sci appl (2023) object recognition by a minimally pre-trained system... 43 6. conclusion in this paper, we present the method of evaluating visual information about rewards along the path of a system (e.g., a robot), the method for recognizing objects around the system (robot), and the method for describing the process of studying the environment. we decompose the images of visible objects into sets of geometric primitives (geons) that the system’s neural network is pre-trained to recognize. complete combinations of these geons (geons’ schemes) corresponding to unknown objects can be stored in the system’s memory as new items. during the study of the object, the numbers of geons in such schemes increase, and they are accepted as the position rewards. thus, the system should move in such a way as to place in its database as many geons associated with the object as possible. later, we suppose to represent in detail a method for generating 3-d objects using geon schemes in order to train a neural network to recognize them without the database. as one can see in fig. 4.6 – 4.11, the calculation on the limited training database gives quite satisfactory geon recognition in images of real objects when the database includes not only automatically generated images, but also a number of hand-marked ones. this approach may be useful when creating intelligent robotic systems. in these systems, “brains” can be transferred from one robot to another one. however, the first such system must go through the entire learning process as a real person, from childhood to adulthood. during the process, it must explore objects of the outside world as they are, recognize and classify them and evaluate their usefulness based on the goals of the system. our approach offers one of the steps in this direction. the method has obvious limitations: we only store information about the shape of objects. therefore, the method is not suitable for recognizing lines and individual features of items. only objects that can be decomposed into geometric primitives can be recognized. thus, we consider such a process of studying the environment by an intelligent system as some movement in the environment. we used the intuition of baby-like behaviour when exploring an object in the environment to simulate the behavior of a robot-like system. the movement of the system was represented as a game in the specific game category in which the environment corresponds to the opponent. he provides the proponent (system) with some information about the objects of the environment, which can also be considered as the goals of the system. achievement of various goals is considered as parallel processes, which are presented as tensor products of the corresponding games, and form a comprehensive game. every game position has a certain reward (a geon set) that estimates the quantity of information provided by the environment. we demand the greatest total reward along the play of the system to choose the desired path. when the total reward of all parallel processes can no longer be improved, we can reduce the number of selected goals (processes for achieving the objects) to perhaps one at the end. an external method of the path calculation can modify this algorithm to create an objects’ traversal path with the maximum cumulative reward. of all the possible, we choose those processes for achieving goals that have the highest estimations in the goal/object lattice of the system. it is so because every goal (an object of interest) and the corresponding process of achieving it have a definite correspondent truth value in the lattice. the higher the value lies on the lattice diagram, the higher the priority of the process. the partial order in the lattice is generated by attention degrees of the system to its goals. thus, we consider two types of lattice estimations: the goal lattice value determines the choice of the goal achieving processes from all possible ones, and the position rewards of the game (geon sets) determine the optimal path of these chosen processes in the environment. let us note again that this path is the game play. it is not the real space trajectory. it only shows how to obtain the best position estimates, but there must be an external algorithm to build the trajectory in such a way as to get these assessments. such a model corresponds to the approach in which system intelligence is considered as the consequence of the mission or purpose of the system for practical use, i.e., the lattice of copyright © 2023 assa. adv syst sci appl (2023) 44 d. maximov, s. diane the goals of the system. in simple systems, e.g., ant colonies, such preference structures may be preexisting. in more complicated ones, these structures can be built and changed over the life of the system. references 1. 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(2017) data augmentation in emotion classification using generative adversarial networks, arxiv.org, 1–15, url: https://arxiv.org/pdf/1711.00648.pdf. copyright © 2023 assa. adv syst sci appl (2023) introduction backgrounds lattices game semantics biederman geons system behaviour description object recognition generation of training samples training sets formation in problems of visual classification procedural model generation geon recognition discussion of system movement and reward valuations conclusion adv syst sci appl 2018; 4:1–12 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/660 tuning hybrid distributed storage system digital twins by reinforcement learning vladislav belavin1∗, kenenbek arzymatov1, maksim karpov1, andrey nevolin2 andrey sapronov1,3, andrey ustyuzhanin1,4 1national research university higher school of economics, myasnitskaya 20, 101000, moscow, russia 2yadro, rochdelskaya 15/13, 123022, moscow, russia 3joint institute for nuclear research, joliot-curie 6, dubna, moscow region, 141980, russia 4moscow institute of physics and technology, institutskiy per. 9, dolgoprudny, moscow reg., 141700, russia received november 11, 2018; revised december 12, 2018; published december 31, 2018 abstract: in this paper we develop a concept of data-driven control of distributed storage systems digital twin. the design of the digital twin is supported by an optimal control strategy created by reinforcement learning technique. in the proposed approach we consider a combination of a trained neural network that tunes the parameters of a discrete event-driven storage system simulator to gain the best resemblance to the actual system. the proposed method has several benefits compared to conventional approaches providing trade-offs between the simplicity of customization, the ability to infer nontrivial patterns from the real system represented by the simulation model and interpretable behavior. we consider different optimization metrics and demonstrate the viability of the approach using a toy training system until physical system becomes available. keywords: storage area network, digital twin, simulation, machine learning, reinforcement learning 1. introduction digital twins [1, 2] are used to simulate a behavior of complex systems. the distributed storage systems are built of multiple components, many of which have measurable parameters altogether providing valuable diagnostic information. the task of storage system simulation converges to building its digital twin with a highly accurate description of components behavior and its interaction. such approach allows for prediction of the components failure and studying the failure outcomes. in this paper we consider the problem of building a credible digital twin of the distributed storage system on the assumption that we have a model-based simulator and data from the real system. it is also assumed that the simulator has a set of parameters that can be changed during the runtime thus affecting following simulation states. to solve this problem, we propose an approach that combines conventional computer simulation with machine learning. the first one is usually based on a mathematical model devised by an expert and the second one infers dependencies from the observed data. the control over these parameters is delegated to a neural network trained on real data using reinforcement learning (rl) techniques. in the case of distributed storage systems the following use case scenario can be applied. the real-time prediction of possible failures of the real system with the digital twin which is continuously provided with information from the real system: load and environment variables (temperature, pressure, etc.) collected from sensors. 2 v. belavin et al. one of the limitations of the simulators based on domain knowledge is the rigidity of the underlying mathematical model, resulting in a limited ability to reproduce unexpected patterns that might be observed in real data. such inability to imitate real system behavior implies the necessity for a revision of the mathematical model over and over again. this cycle of trial and error can take a lot of time and human resources. besides, simulators are usually ignorant of the appearance of new data, i.e. cannot adapt to newly observed patterns in the data due to, for example, the degeneracy of some components or/and changes in external conditions. on the other hand, simulation approaches in machine learning, for instance, generative adversarial networks [3], require only data to reproduce behavior of the real system. the indisputable advantage of the gan approach is its simplicity. however at least two significant problems are arising. firstly, without any underlying model, it is hard to interpret the neural network[4]. secondly, it might be inaccurate in operating regions that have not been presented in the dataset. we can achieve a synergy between two inherently different methods by loosening constraints on the simulator and delegating the control over parameters to the neural network. in this case, the simplification of the simulator model provides more freedom for the neural network, simultaneously regularizing it and making interpretation of the hybrid model much more manageable than for the single neural network. our approach has following key differences in comparison with existing solutions: 1. a novel idea of recovery of the functional dependency of control parameters, instead of point estimation; 2. a fusion of the discrete event simulator with the neural network; 3. the usage deep reinforcement learning for a such task; in this paper, we test the viability of such an approach. 2. related works the task of storage area network simulation has been approached in several publications. the majority of works are focused on the discrete event simulation methods. the researches codes project uses a highly parallel simulation to explore the design of exascale storage architectures and distributed data-intensive science facilities [5]. the general purpose network simulator ns-3 [6] allows a detailed modelling of a storage system from the view of interconnected nodes with different functionality. another general approach to networklike structure simulation is the omnet++ framework [7]. another work presents a sansim tool [8], a simulation compliant with the fiber channel technology often used in contemporary san architectures. more simulation methods descriptions and studies dedicated to san system modelling can be found in [9, 10, 11, 12]. from this overview we can see that the most popular approach to the storage area network modelling is the deterministic method based on discrete event sequences. below we going to discuss an extension to this method by coupling it with the dynamic parameter tuning by means of a neural network trained with real data sample. initially, the topic of simulation parameters optimization using reinforcement learning was discussed in [13]. in this work, the discrete-valued parameters are optimized to construct a matrix of optimal actions for each state. our approach features a solution for real-valued parameters with the usage of deep reinforcement learning which primarily became possible due to advances in gpu computing technology. our crucial difference from the mentioned research is the derivation of parameter dynamics during the whole simulation cycle instead of just optimal initial parameters. another way to find optimal parameters in the presence of real data is a validation of the simulator for each set of parameters output[14, 15]. in this setup for each set of parameters for simulated data and real data the hypothesis is constructed: copyright c© 2018 assa. adv syst sci appl (2018) tuning dss digital twins with rl 3 h0 : model output is the same as system output vs h1 : model output is not the same as system output the choice of an appropriate statistic for outputs comparison as well as the selection of a statistical test depends on a particular problem. however, this methodology can only give an estimation of a single parameter point. 3. simulator 3.1. simulated system tatlin (figure 3.1) is a pcie-centric distributed storage system designed by the yadro company [16]. it consists of hardware computational tools and a special software designed for storing and transmitting large amounts of data. due to the unified access protocol(hybrid unified storage), this system supports a wide range of disk interfaces: nvme/sas ssd and sas/nl-sas/sata hdd. the platform consists of three major building blocks: • the pcie fabric controller is a pci express bus that provides interconnection between storage controllers and storage media; • the storage controllers, the computer servers that provide access to storage media to end users; • the drive enclosures generally known as jbods (just a bunch of disks). to ensure data integrity in tatlin, the yadro uses customizable data protection technology based on reed-solomon codes with minimal redundancy instead of conventional raid. however, writing data on disks using this technology requires a lot of computational resources and can take considerable amount of time. to meet this problem and reduce cpu load the yadro proposed a concept of hardware deduplication acceleration with nonvolatile cache (nvram) built into the storage controllers. 3.2. description of the simulator the gotatlin framework (figure 3.2) is designed to simulate the tatlin distributed storage system. it has been implemented using the go programming language [17]. the storage system simulation is based on a discrete-event simulation model where the system’s evolution in time is presented by a discrete sequence of changes in its state. each change in the system state occurs at a distinct point of time due to some triggering event. 3.3. output data of the simulator we construct a vector of features from the output of the simulator. the features are summarized below. • storage controllers features: – traffic [gbs−1]; – load [%]; • storage media features: – used space [gb]; – average read/write speed [gbs−1]; • general tatlin features: copyright c© 2018 assa. adv syst sci appl (2018) 4 v. belavin et al. fig. 3.1. dss tatlin. fig. 3.2. gotatlin architecture view. – amount of data transferred in read/write mode [gb]; – number of read/write requests per unit of time [ms−1]; – read/write requests response time [ms]; – read/write requests processing time [ms]; 3.4. control parameters a full list of the control parameters of the simulator is the following: • storage media: – read – read speed [gbs−1]; – write – write speed [gbs−1]; • drive enclosures: – speed – computational power [gflops]; • storage controllers: – speed – computational power [gflops]; • pcie fabric controller: copyright c© 2018 assa. adv syst sci appl (2018) tuning dss digital twins with rl 5 – speed – computational power [gflops]; • physical network links: – bandwidth [gbs−1]; – latency [ms−1]. as one might imagine, the real system should have much more parameters, which represent the system’s behaviour (l1/l2 cache sizes, ram parameters, etc.), than the simulated one has. but we intentionally reduced the number of parameters implying that these variables represent most ‘effectively’ the real system. on the one hand, changing those parameters helps to relax rigidity of the simulator, and on the other hand, these effective parameters can’t be derived from the physical system and should be calibrated by the algorithm explained in the following chapter. in the rest of the paper, we control those effective parameters unless stated otherwise. 4. control system the core part of the control system is a neural network which analyses the output of the simulator at the time t and generates a new set of control parameters that will be used for simulation until next moment of time t+ ∆t. ∆t is an external parameter of the simulator fixed a priori. it should be noted that ∆t is a hyper-parameter of the simulation control system and can be chosen arbitrarily. we set ∆t to 3s a priori because in the real tatlin we are expecting the delay between successive log outputs to be approximately 1− 10s. general outlines of the simulation and training pipelines are presented at the figures 4.3a and 4.3b respectively. 4.1. reinforcement learning in reinforcement learning [18] the following terminology is commonly used: • environment (st) is an observed system (in our case, the system is a simulator); • state is a current status of the simulation represented by output data; • goal is a desired state of the simulation; • agent is a system that takes actions (at) in the environment and changes its state; • reward (rt) is a scalar value that shows how close is our system’s state to the goal state. in our case, goal state of the simulator is a state in which its output fully resembles the real storage system’s output. we will discuss the construction of the reward function in more details in section 4.2. the most common approach is to assume that the environment is modelled as a markov decision process. in this case our model can be described with 4-tuples (s,a, p (·, ·, r(·, ·))), where • s = {si}ni=0 is a set of all possible states of the simulator; • a = {ai}ki=0 is a set of all possible actions of the agent; • p (st+1|st, at) is a probability of transition from state st at time t to state st+1 at the next moment of time t+ 1 under action at; • r(st+1, st) = r(st, at) is a reward after transition from state st to state st+1; the most successful approach that use markovian assumption about environment is the so-called q-learning [19] and its successor in deep learning domain deep q network (dqn) [20]. in dqn method the neural network generates probability distribution over all possible actions in the current state of environment πθ(a|s) and the next action is sampled from it (at ∼ πθ(s)). copyright c© 2018 assa. adv syst sci appl (2018) 6 v. belavin et al. (a) proposed simulation control system pipeline. (b) the training pipeline. fig. 4.3. simulation control system training pipeline. the maximization of agent rewards in all possible states of the system can be represented as the maximization of the following objective: l(πθ) = es,a [r(s, a)] = ∫ s r(s) ∫ a πθ(s)dsda in the case of neural networks, optimization can be easily performed with monte-carlo evaluation of this mathematical expectation with following gradient descent and backward propagation of errors: ∇θl(πθ) = es,a [∇θπθ(s)r(s, a)] however, this naive approach has an extremely slow convergence rate. this happens due to a large number of control parameters in continuous real-valued space. to deal with this problem we decided to use ddpg (deep deterministic policy gradients) [21], an algorithm that was designed to tackle the continuous control problems. the authors of ddpg propose to predict not just a distribution over all possible actions that agent can make but a specific action. to endorse exploration of state-action space they suggest adding random noise sampled from ornstein–uhlenbeck process. in our realization the ornstein–uhlenbeck noise was sampled this way: nt+1 = nt + θ(n0 − nt) + σn(0, σ), where θ defines a momentum of regression toward the mean(n0),n(0, 1) is normal random variable, σ is a variance of the process. the action predicted by the neural network is consequently modified by addition of the noise: copyright c© 2018 assa. adv syst sci appl (2018) tuning dss digital twins with rl 7 ât = at + nt. this choice of the noise provides an adequate level of “curiosity” due to its random nature. furthermore, autocorrelation property of this process ensures smooth transitions of actions thus making it more suitable for control tasks and additionally reducing variance during training. 4.2. metrics we implemented various metrics to compare output data from different simulation runs: • cosine similarity of spectra or wavelet spectra; • mean square error; • kolmogorov distance (kd), wasserstein distance (w), cramér–von mises distance (cvm) between distributions of two sets of output parameters. our goal is to find a reward function that would be sensitive to differences in the control parameters, but smooth and robust to small perturbations. these requirements are important for the convergence of neural network training. we tested metrics on simulated data with different fixed parameters. it was done without complex analysis but with simple hand-selection based on plots of metric dependencies. we propose following loss function that combines several metrics. numeric coefficients were chosen so that all values in loss function would have about the same order of magnitude and similar contribution. rt = |sreft − scontrolledt | 107 + cvm(wv ref 1,...,t, v controlled 1,...,t ) 20 + cvm(wqref 1,...,t, q controlled 1,...,t ) 2 , where • st is a total occupied storage space at the time t. occupied storage space is a cumulative and robust characteristic of the system. one drawback of this metric is that it accumulates simulation errors. • v1,...,t is an empirical distribution function of transferred data amount in “write” mode from t′ = 0 to t′ = t; • q1,...,t is an empirical distribution function of write requests response time from t′ = 0 to t′ = t; these two metrics represent dynamical behavior of the system. then we can easily define reward function as a temporal difference between losses at consequent moments of time, i.e. rt = rt−1 −rt. 5. training 5.1. experiments for the moment, the data from the real system is not yet available for the optimization of simulation. due to this, the experiment is conducted in the following way: we take two configurations of the system (two sets of initial control parameters), the first configuration is assumed to be the target system that we want to be able to properly simulate the behavior of. the second configuration is used for initialization of the neural network. we initialize all layers randomly and for the last layer, which represents parameters of the simulator, we copyright c© 2018 assa. adv syst sci appl (2018) 8 v. belavin et al. fig. 5.4. load parameters sampling areas. loads for training set are sampled from red area, loads for validation set are sampled from blue area. black dots are sampled points. zero the weights and set biases of layer to parameters from the second configuration. on the assumption that in the real system the optimal parameters could be far away from the initial parameters we deliberately set parameters of the second configuration to be very different from the target system. also, we made one more simplification. our configurations differ only in the following sets of parameters: bandwidth of physical network links and the computational power of storage controllers. these parameters have the highest impact on the behavior on the system in comparison with other parameters like latency or computation power of drive enclosures. another reason for reduction of parameter space is an exponentially increasing complexity of optimization problems. overall, neural network generates 5 parameters: four parameters account for computational power of four storage controllers and the fifth one accounts for bandwidth(figure 5.5a). the system load is modelled as poisson process with mean time between incoming files equal τ̄ [s]. the file size is modelled with exponential distribution parameterized by average file size s̄[mb]. for tuning our algorithm we consider a finite set of possible load configurations: l = s × t = {si}ni=1 × {τj}kj=1. the loads for training ltrain and loads for validation ltest are sampled from uniform distributions on a 2d-grid (figure 5.4). it is important to note that ltrain and ltest distributions do not overlap and this allows us to test generalization of our model on unseen situations. we chose quite simple architecture (figures 5.5a and 5.5b), thus allowing us to avoid over-fitting and speed-up training. the pytorch [22] framework was used. at each epoch of training the neural network is fed with data from 40 new and 40 old (experience replay) runs of simulation. fed data is time differences between the outputs of simulator(section 3.3) at the moment t+ ∆t and t. optimization is done with adam with learning rate set to 10−2 and all other parameters left as default. evaluation on validation set of loads is done every 20 epochs. 5.2. results we judge our improvement in simulation by looking at difference in occupied storage space (|sreft − scontrolledt |[b]). as can be noted, this metric is the first part of rt definition(section 4.2). it was chosen for comparison mostly due to simplicity of interpretability. copyright c© 2018 assa. adv syst sci appl (2018) tuning dss digital twins with rl 9 input dim data 16 tanh speed of controller 1 speed of controller 2 speed of controller 3 speed of controller 4 bandwidth (a) actor architecture input dim data actor dim actor 4 tanh 1 (b) critic architecture fig. 5.5 the improvement of the metric on the training set of loads (figure 5.6a) and validation set of loads (figure 5.6b) have a similar pattern. taking this into account, two conclusions may be derived: • our agent (neural network) is able to tune the parameters of the simulation and make it similar to the target system; • the trained neural network is robust and capable of generalizing on unseen situations. (a) (b) fig. 5.6. averaged metric with confidence interval on training (left) and validation (right) set of loads. lower is better. at figure 5.7a and figure 5.7b one can see metric(rt) dependence from simulation time and epoch number. as expected, the distance between the hybrid and target systems is increasing while simulation time is growing, i.e. we are observing the disorder between two processes. this disorder diminishes during training of neural network making system more and more compliant. 6. conclusion we demonstrated the viability of our approach with a fusion of conventional simulation techniques and rl-trained neural network on the problem of optimization of discrete event copyright c© 2018 assa. adv syst sci appl (2018) 10 v. belavin et al. (a) (b) fig. 5.7. contour plot of logarithm of the metric as a function of simulation time (vertical axis) and epoch (horizontal axis) during training (left) and validation (right). simulator of the distributed storage system. in this setup, a neural network is trained by the reinforcement learning algorithm to find an optimal control strategy for parameters of the simulator. the hybrid simulator has a wide range of benefits in comparison with vanilla simulators: simplified customization of the simulator; ability to infer patterns from data. and in contrast with gans, it has interpretable behavior and ability to generalize on partially unseen scenarios. further research may cover such steps as: • increase complexity of the system under control by adding more control parameters; • assess the sensitivity of the agent to the choice of reward functions; • build and validate the approach on a realistic multiparametric data sample. 7. acknowledgements the research was carried out with the financial support of the ministry of science and higher education of russian federation within the framework of the federal target program “research and development in priority areas of the development of the scientific and technological complex of russia for 2014-2020”. unique identifier – rfmefi58117x0023, agreement 14.581.21.0023 on 03.10.2017. references 1. tao, f., cheng, j., qi, q., zhang, m., zhang, h., & sui, f. 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(1992) q-learning. machine learning, 8, 279–292, https://dx.doi.org/10.1007/bf00992698. 20. mnih, v., kavukcuoglu, k., silver, d., graves, a., antonoglou, i., wierstra, d., & riedmiller, m. a. (2013) playing atari with deep reinforcement learning. corr, abs/1312.5602. 21. silver, d., lever, g., heess, n., degris, t., wierstra, d., & riedmiller, m. (2014) deterministic policy gradient algorithms. in proceedings of the 31st international copyright c© 2018 assa. adv syst sci appl (2018) 12 v. belavin et al. conference on international conference on machine learning volume 32, i–387– i–395, icml’14, jmlr.org. 22. paszke, a., gross, s., chintala, s., chanan, g., yang, e., devito, z., lin, z., desmaison, a., antiga, l., & lerer, a. (2017) automatic differentiation in pytorch. copyright c© 2018 assa. adv syst sci appl (2018) adv syst sci appl 2021; 03:22–30 published online at https://ijassa.ipu.ru. on spectral decomposition of generalized bessel potentials m.l. goldman1, n.h. alkhalil1* 1 mathematical institute named s.m. nikolskii peoples’ friendship university of russia (rudn university) moscow 117198, miklukho-maklaya str. 6, russia abstract: we establish the localization condition for the γ-means spectral decomposition by the system of fundamental functions of the laplace operator in an arbitrary multi-dimensional domain. the result is obtained in terms of belonging of decomposing function to the spaces of the generalized bessel potential. for conditions of localization, we apply the exact estimates for the modulus of continuity of the potential. the generalized bessel potentials are constructed using convolutions of functions with kernels that generalize the classical bessel–macdonald kernels. in contrast to the classical case, non-power singularities of kernels are allowed in the vicinity of the origin. differential properties of potentials arc described by using the k-th order modulus of continuity in the uniform norm. keywords: spectral decomposition, the generalized bessel potential, the modulus of continuity of the potential, laplace operator 1. introduction the paper is organized as follows. section 1 contains basic definitions of the potential theory. the main properties of kernels are considered and basic spaces for potentials are described. sections 3.1, 3.2 contain some auxiliary results. in section 3.1 we give the condition of the embedding of the space hg e (rn) of generalized bessel potentials into the space c(rn) of bounded uniformly continuous functions, and obtain the estimates for modulus of continuity of potentials, see theorem 3.1. in section 3.2 we give the conditions for localization of γ-means of spectral decomposition in terms of properties of modulus of continuity for decomposing function, see theorem 3.2. in section 4.1 we prove theorem 4.1 giving the conditions for including of the space hg e (rn) into the scheme of spectral decomposition. finally, theorem 4.2 in section 4.2 gives the conditions for localization of spectral decomposition for generalized bessel potentials constructed over the basic weighted lorentz spaces. ∗corresponding author: khaleel.almahamad1985@gmail.com on spectral decomposition of generalized bessel potentials 23 2. basic definitions let v > 0 be measurable function on r+. the lorentz space λp(v) is the space of measurable functions on rn with finite (quasi) norms (see [1]) ‖f‖λp(v) =  (∫∞ 0 f ∗(t)pv(t) dt ) 1 p ; 0 < p <∞; ess sup t∈r+ { f ∗(t)v(t) } ; p =∞. (2.1) here f ∗ : r+ → [0,∞] is the decreasing rearrangement of function f : rn → r, i.e f ∗ is a nonnegative decreasing right-continuous function on r+ = (0,∞) which is equimeasurable with f : µn {x ∈ rn : |f(x)| > y} = µ1 {t ∈ r+ : |f ∗(t)| > y} , y ∈ r+, (2.2) where µn is the n-dimensional lebesgue measure. we assume that 0 < v (t) := ∫ t 0 v(τ)d(τ) <∞, t ∈ r+, and sup t∈r+ [ v (2t) v (t) ] <∞, (2.3) is the so called ∆2-condition. under these assumptions e(rn) = λp(v) is a (quasi) banach space which gives an important example of a rearrangement invariant space (shortly:ris), because of property: g∗ ≤ f ∗, f ∈ e(rn) ⇒ g ∈ e(rn), ‖g‖e ≤ ‖f‖e. (see c. bennett and r. sharpley [1]). moreover, e ′ = e ′(rn) is the associated ris for e(rn), i.e. e ′ is ris with the norm: ‖g‖e′ = sup {∫ rn |fg| dµn : f ∈ e, ‖f‖e ≤ 1 } . (2.4) for 1 < p <∞ the description of the associated space for e(rn) = λp(v) was obtained by e. sawyer [11]. namely, ‖g‖e′ = sup 0≤h↓ ∫∞ 0 g ∗ (τ)h(τ) dτ(∫∞ 0 h(τ)pv(τ) dτ ) 1 p ≈ ( ∞∫ 0 ( ξ∫ 0 g∗(τ)dτ )p′ v(ξ)dξ v (ξ)p′ ) 1 p′ , (2.5) here the symbol ≈ means that the ratio of left and right hand sides is bounded between positive constants depending only on p (and not on v or g). the potential space hg e ≡ hg e (rn) for e(rn) = λp(v) is defined as the set of convolutions of potential kernel g with all functions belonging to the basic ris e(rn): hg e (rn) = { u = g ∗ f : f ∈ e(rn) } . (2.6) we define ‖u‖hg e = inf { ‖f‖e : f ∈ e(rn), g ∗ f = u } . (2.7) we assume that the kernel g of a representation (2.7) is admissible, i.e g ∈ l1(rn) + e ′(rn). copyright © 2021 assa. adv syst sci appl (2021) 24 m.l. goldman, n.h. alkhalil here the convolution g ∗ f is defined as the integral (g ∗ f)(x) = ∫ rn g(x− y)f(y) d(y). for function φ: r+ → [0,∞) we define ϕ(τ) = φ (( τ vn ) 1 n ) , τ ∈ r+. (2.8) definition 2.1: let k, n ∈ n; r ∈ r+. we say that function φ belongs to the class ik,n(r) if it satisfies the following conditions: 1. 0 < φ ↓ on (0, r); ∃ c ∈ r+ such that r∫ 0 φ(ρ)ρn−1 dρ ≤ cφ(r)rn, r ∈ (0, r); ∞∫ r φ(ρ)ρn−1 dρ <∞. (2.9) 2. g(x) := φ(|x|) ∈ ck(rn\0), and for gk(x) := ∑ |α|=k |dαg(x)|, x ∈ rn\0, the estimate holds: for some c1 ∈ r+ |gk(x)| ≤ c1ψk(|x|), x ∈ rn\0; (2.10) where ψk ∈ c(r+), and for t = vnr n ϕk(τ) := ψk (( τ vn ) 1 n ) ≤ τ−k/nϕ(τ), τ ∈ (0, t ]; (2.11) ∞∫ t ϕk(τ) dτ <∞. (2.12) definition 2.2: for rise(rn) the spacehg e (rn) (2.6)-(2.7) with kernelg(x) = φ(|x|), where φ ∈ ik,n(r), is called the space of generalized bessel potentials. remark 2.1: note that the classical bessel–mcdonald kernels have the form gα(x) = c(α, n)ρ−βkβ(ρ), ρ = |x| ∈ r+, α ∈ (0, n), β = n− α 2 , where kβ is the mcdonald function, see [10]. the well-known properties of these kernels show that φ(ρ) = ρ−βkβ(ρ) ∈ ik,n(r) for fixed r ∈ r+, moreover φ(ρ) ∼= ρα−n, ρ ∈ (0, r); φ(ρ) ∼= ρ−β− 1 2 e−ρ, ρ > r, and our scheme includes classical bessel potentials. remark 2.2: note that functions φ ∈ ik,n(r) may have the property φ(ρ) = 0, ρ ∈ [2r,∞). (2.13) thus, the case of kernels with compact support is included in our scheme. copyright © 2021 assa. adv syst sci appl (2021) on spectral decomposition of generalized bessel potentials 25 definition 2.3: let c(rn) be the space of bounded and uniformly continuous functions with the norm ‖u‖c = sup x∈rn |u(x)|. for u ∈ c(rn) modulus of continuity of order k ∈ n is defined as ωkc(u; t) = sup { ‖∆k hu‖c : |h| ≤ t } , t ∈ r+, ∆k h(u;x) = k∑ m=0 (−1)k−mcm k u(x+mh), is the k-th difference of function u with the step h ∈ rn at the point x ∈ rn. for u ∈ l2(ω), where ω is domain in rn, we define ωk2,ω(u; t) = sup |h|≤t ‖∆k hu‖l2(ωkh); where ωkh = { x ∈ ω : [x, x+ kh] ⊂ ω } . definition 2.4: let ω ∈ c(0, 1], 0 ≤ ω(t) ↑; t−kω(t) ↓ on (0, 1]. the nikolskii-type space with generalized smoothness hω(·) 2 (ω) is defined as h ω(·) 2 (ω) = { u ∈ l2(ω) : ‖u‖ h ω(·) 2 (ω) <∞ } , (2.14) where ‖u‖ h ω(·) 2 (ω) = ‖u‖l2(ω) + sup 0<t≤1 [ ωk2,ω(u; t) ω(t) ] . (2.15) 3. auxiliary theorems 3.1. let rise(rn) = λp(v) be the lorentz space, see notations and assumptions (2.1)–(2.5), and hg e (rn) be the related space of generalized bessel potentials (2.6)–(2.7) withg(x) = φ(|x|), φ ∈ ik,n(r), see definitions (2.1), (2.2). for 0 < p <∞ we define function on (0, t ], t = vnr n, ap(t) = sup τ∈(0,tn) { 1 v (τ) 1 p ∫ τ 0 ϕ(ξ) dξ } , 0 < p ≤ 1; (3.16) ap(t) = {∫ tn 0 (∫ τ 0 ϕ(ξ) dξ )p′ v(τ) dτ v (τ)p′ + (∫ tn 0 ϕ(τ) dτ )p′ v (tn)− p′ p } 1 p′ , (3.17) for 1 < p <∞, p′ = p p− 1 . we require that ap(t) <∞. (3.18) copyright © 2021 assa. adv syst sci appl (2021) 26 m.l. goldman, n.h. alkhalil theorem 3.1: [6]. in the notations and assumptions of this section the following statements hold: 1. hg e (rn) ⊂ c(rn), and there exists constant c1 ∈ r+ such that ‖u‖c(rn) ≤ c1‖u‖hg e (rn), ∀u ∈ hg e (rn). (3.19) 2. let additionally t∫ t τ−k/nϕ(τ) dτ ≤ b0t 1−k/nϕ(t), t ∈ (0, t ], (3.20) with b0 ∈ r+ independent of t ∈ (0, t ]. then, there exists constant c2 ∈ r+ such that for any u ∈ hg e (rn) ωkc(u; t) ≤ c2ap(t)‖u‖hg e (rn), t ∈ (0, t ]. (3.21) 3.2. let f ⊂ rn be an arbitrary domain, and (−∆̂) a self adjoint non-negative extension of laplace operator in d; y(x, t) an ordered spectral representation of l2(f ) with respect to (−∆̂), and dρ(t) the related spectral measure; y(x, t) = {yi(x, t)}mi=1 be a system of fundamental functions, that is yi(·, t) ∈ c∞(f ), and ∆yi(x, t) + t2yi(x, t) = 0, x ∈ g. here m ≤ ∞ is the multiplicity of the representation. for f ∈ l2(f ) are defined fourier transforms f̂ := {f̂i(t)}mi=1; f̂i(t) = ∫ f f(x)yi(x, t) dx and spectral decomposition sµ(f ;x) = µ∫ 0 f̂(y)y(x, t) dρ(t), µ > 0, x where f̂y = m∑ i=1 f̂iyi. let s > 0 and ψ be a function on (0, 1] with properties 0 < ψ ↑ on (0, 1], and ψ(t) ∼= ψ(τ) if t ∼= τ . moreover, for s > 0 and s0 = s if s ≤ 1; s0 = 1 if s > 1 we require that 1) ψs0(t) = t∫ 0 τ s0−1ψ(τ) dτ <∞, t ∈ (0, 1], (3.22) 2) ψ ∈ c2(0, 1); |ψ′(τ)| ≤ cψ(τ)τ−1, |ψ′′(τ)| ≤ cψ(τ)τ−2, τ ∈ (0, 1], (3.23) 3) 1∫ 0 (1− τ)s−1ψ(τ) dτ = γ(s). (3.24) copyright © 2021 assa. adv syst sci appl (2021) on spectral decomposition of generalized bessel potentials 27 we define γ as s-th riemann–liouville integral γ(t) = 1 γ(s) t∫ 0 (t− τ)s−1ψ(τ) dτ, t ∈ (0, 1]. (3.25) let us introduce the γ-means of spectral decomposition as σγµ(f ;x) = µ∫ 0 f̂(t)y(x, t)γ ( 1− t2 µ2 ) dρ(t), µ > 0, (3.26) for f ∈ l2(f ). we see that γ(1) = 1 and if ψ(τ) ≡ γ(s+ 1), τ ∈ (0, 1], then γ(t) = ts, and γ-means reduce to the classical riesz means of orders, see [2, 3]. we define ω0(t) = t n−1 2 + s0 − s ψs0(t) , t ∈ (0, 1]. (3.27) let α, β ≥ 0 be such that n− 2 2 − s < α ≤ β < min { α + 3 2 , n 2 + 1 } . (3.28) let function ω satisfy the conditions ω(t)t−α ↑, ω(t)t−β0 ↓ on (0, 1]; β0 = min{β, k}, (3.29) and lim t→+0 ω(t) ω0(t) = 0. (3.30) let ω ⊂⊂ f , that is ω is bounded domain, and ω̄ ⊂ f , f ∈ hω(·) 2 (ω) ∩ l2(f ). (3.31) theorem 3.2: [4, 5]. in the notations and assumptions of section 3.2 (3.22)–(3.31), let d ⊂ ω and function f satisfies the condition f(x) ≡ 0, x ∈ d. then, for each compactk ⊂ d uniformly in x ∈ k the relation holds: lim µ→∞ σγµ(f ;x) = 0. (3.32) remark 3.1: theorem 3.2 gives sharp conditions for localization of γ-means of spectral decomposition. in typical situations we have s < n−1 2 , ψs0(t) ∼= ts0ψ(t), ω0(t) ∼= t n−1 2 −s ψ(t) , t ∈ (0, 1]. in particular, for riesz means, ψ(t) ≡ γ(s+ 1) and (3.30) gives the condition ω(t) = ¯̄0 ( t n−1 2 −s ) . copyright © 2021 assa. adv syst sci appl (2021) 28 m.l. goldman, n.h. alkhalil 4. localization for γ-means of spectral decomposition for generalised bessel potentials 4.1. theorem 4.1: let the notation and assumptions (3.16)–(3.18) and (3.20) be satisfied, v (+∞) =∞, and moreover bp := sup [ t 1 2v (t)− 1 p : t ∈ r+ ] <∞, 0 < p ≤ 2; (4.33) bp := (∫ ∞ 0 [ t 1 2v (t)− 1 p ]s v(t) dt v (t) ) 1 s <∞, 2 < p <∞, s = 2p p− 2 . (4.34) then, fore(rn) = λp(v) there is the embedding of the space of generalized bessel potentials hg e (rn) ⊂ l2(rn). (4.35) proof note that the conditions (4.33), (4.34) give the criterion of embedding e = λp(v) ⊂ l2(rn). (4.36) it follows from the assertions (see [7, 8]): let cp := sup f∈λp(v) [(∫ ∞ 0 f ∗(τ)2 dτ ) 1 2 (∫ ∞ 0 f ∗(τ)p v(τ) dτ )− 1 p ] . then, cp = bp, 0 < p ≤ 2; cp ∼= bp, 2 < p <∞. the embedding (4.36) implies the embedding hg e (rn) ⊂ hg l2 (rn). but, in the assumptions of section 3.1 the kernel g(x) = φ(|x|) ∈ l1(rn), and by the generalized minkovkii inequality for convolutions f ∈ l2(rn)⇒ u = g ∗ f ∈ l2(rn), ‖u‖l2(rn) ≤ ‖g‖l1(rn)‖f‖l2(rn). this gives embedding (4.35). 4.2. now, let us recall the notations and assumptions of section 3.2. let the conditions of theorem 4.1 be satisfied, and hg e (rn) be the space of generalized bessel potentials with e(rn) = λp(v). we assume that the notations and assumptions of section 3.2 hold, see (3.22)–(3.28). under assertions (3.16)–(3.18) and (3.20), we define ω(t) := ap(t) in (3.29), (3.30). theorem 4.2: under assumptions of this section, let d, ω, f be domains in rn, and d ⊂ ω ⊂⊂ f . if u ∈ hg e (rn); u(x) ≡ 0, x ∈ d, (4.37) then for each compact k ⊂ d uniformly in x ∈ k the relation holds: lim µ→∞ σγµ(u;x) = 0. (4.38) copyright © 2021 assa. adv syst sci appl (2021) on spectral decomposition of generalized bessel potentials 29 proof by theorem 4.1 we have the embedding (4.35). it means that u ∈ hg e (rn)⇒ u ∈ l2(rn)⇒ u ∈ l2(f ), and γ-means of spectral decomposition σγµ(u;x) are correctly defined by formula σγµ(u;x) = µ∫ 0 û(t)y(x, t)γ ( 1− t2 µ2 ) dρ(t), (4.39) where y(x, t) is the system of fundamental functions ûy = m∑ i=1 ûiyi; ûi(t) = ∫ f u(x)yi(x, t) dx, see notations in section 3.2. moreover, for u ∈ hg e (rn) we have ω2,ω(u; t) ≤ ωc(u; t)(mes ω) 1 2 . (4.40) indeed, ‖∆k hu‖l2(ωkh) = (∫ ωkh |∆k hu|2 dx ) 1 2 ≤ sup x∈ωkh |∆k hu(x)|(mes ωkh) 1 2 , and for ωk2,ω(u; t) = sup |h|≤t ‖∆k hu‖l2(ωkh) we have the estimates ωk2,ω(u; t) ≤ sup |h|≤t sup x∈rn |∆k hu(x)|(mes ω) 1 2 = = sup |h|≤t ‖∆k hu‖c(rn)(mes ω) 1 2 = ωkc(u; t)(mes ω) 1 2 , and (4.40) follows. thus, by theorem 3.1, see (3.21), we have ωk2,ω(u; t) ≤ c2(mes ω) 1 2‖u‖hg e (rn)ap(t), t ∈ (0, t ]. (4.41) now, we apply theorem 3.2. this estimates shows that u ∈ hg e (rn)⇒ u ∈ hω(·) 2 (ω), ω(t) = ap(t). (4.42) finally, u ∈ hω(·) 2 (ω) ∩ l2(f ), and we obtain the assertion (4.38) by applying of the theorem 3.2. remark 4.1: note that if kernel g of generalized bessel potentials is compactly supported on b2r = {x ∈ rn : |x| < 2r}, see the condition (2.13), we have for u ∈ hg e (rn) u(x) = (g ∗ f)(x) ≡ 0, x ∈ d, copyright © 2021 assa. adv syst sci appl (2021) 30 m.l. goldman, n.h. alkhalil function f ∈ e(rn) satisfies the condition f(x) ≡ 0, x ∈ d2r = { x ∈ rn : ρ(x,d) < 2r } , where ρ(x,d) = inf { |x− y| : y ∈ d } is the distance from x to d. remark 4.2: the posing of the problem and the results of sections 3.2 and 4.1 belong to m.l. goldman. the other results of paper obtained by n.h. alkhalil. remark 4.3: in this paper we consider the differential properties of generalized bessel potentials. their integral properties are connected with estimates of hardy-type operators on the cones of monotone functions considered in [12]. acknowledgements the research of m.l. goldman is supported by the ministry of science and higher education of the russian federation: agreement no. 075-03-2020-223/3 (fssf-2020-0018). references 1. bennett, c., & sharpley, r. (1988) interpolation of operators. boston: acad. press, 1988. (pure and appl. math.; v. 129). 2. il’in, v.a., & alimov, sh.a. (1971) conditions for the convergence of spectral decompositions that correspond to self-adjoint extensions of elliptic operators. i, differential equations 1971, 7(4), 670–710. 3. il’in, v.a., & alimov, sh.a. (1971) conditions for the convergence of spectral decompositions that correspond to self-adjoint extensions of elliptic operators. ii, differential equations 1971, 7(5), 851–882. 4. ayele, t.g., & goldman, m.l. (2014) spaces of generalized smoothness in summability problems for φ-means of spectral decomposition. eurasian mathem. journal 2014, 5(1), 61–81. 5. goldman, m.l., & tsegaye, g.a. (2015) spaces with generalized smoothness in summability problems for φ-means of spectral decompositions. springer international publishing, switzerland 2015, current trends in mathematics, 163–169. 6. alkhalil, n.kh. modulus of continuity for bessel type potentials over lorentz space. eurasian mathem. journal, (to appear). 7. burenkov, v.i., & goldman, m.l. (1995) calculation of the norm of a positive operator on the cone of monotone functions. proc. of the steklov inst. math., 1995, 210, 65–89. 8. goldman, m.l. (1998) hardy-type inequalities on the cone of quasimonotone functions. russian academy of sciences, far-eastern branch, research report 98/31, khabarovsk, 1998, 1–70. 9. goldman, m.l., & malysheva, a. (2013) estimates of the uniform modulus of continuity for bessel potentials. proc. of the steklov inst. math. 2013, 283, 1–12. 10. nikol’skii, s.m. approximation of functions of several variables and imbedding theorems. springer, berlin-heidelberg-new york, 1975. 11. sawyer, e. (1990) boundedness of classical operators on classical lorentz spaces. studia math., 96, 145–158. 12. almohammad, kh. (2021). the modular inequalities for hardy-type operators on monotone functions in orlicz space. advances in systems science and applications, 21(2), 133–141. copyright © 2021 assa. adv syst sci appl (2021) introduction basic definitions auxiliary theorems localization for -means of spectral decomposition for generalised bessel potentials microsoft word 1176 article text, copyedited.docx adv syst sci appl 2022; 01; 117-129 published online at https://ijassa.ipu.ru. robust reversible watermarking application for fingerprint image security rasha thabit1, bee ee khoo2* 1) department of computer techniques engineering, al-rasheed university college, baghdad, iraq e-mail: rashathabit@yahoo.com 2) school of electrical and electronic engineering, engineering campus, universiti sains malaysia, 14300, nibong tebal, penang, malaysia e-mail: beekhoo@usm.my abstract: fingerprint biometric systems can automatically identify individuals based on their fingerprint characteristics. different image watermarking schemes have been applied to fingerprint images for security and protection purposes. the previous schemes are either robust watermarking schemes or reversible watermarking schemes. in order to ensure the intactness of the minutiae, this work applies the robust reversible watermarking scheme for fingerprint images which can recover the original image after the watermark extraction process and provide robustness against different kinds of attacks. the proposed scheme embeds the identification number (id) of the person in his fingerprint image to provide security while saving and exchanging the image. the scheme has two stages of security; in the first stage, the id of the received image is extracted and compared with the saved id in the database, if they are identical the system proceeds to the second stage in which the matching score between the received fingerprint and the saved fingerprint is calculated to authenticate the received image. the experimental results proved the efficiency of the proposed scheme in terms of visual quality, robustness, reversibility, and intactness of the fingerprint’s minutiae. keywords: information security, fingerprint images protection, robust reversible watermarking, image authentication, biometric data security 1. introduction biometrics technology has been used to differentiate between authorized and unauthorized people in personal identification systems. for instance, the biometrics signals such as fingerprint, iris, face, and voice have been used as an input to the automated biometrics-based person authentication systems [1-4]. these biometrics signals can be acquired from the user, either locally or remotely. the biometrics-based verification system needs to verify that the biometric data came from the legitimate person at the time of enrollment, therefore, the security and integrity of the biometrics data became important issues [5,6]. different data hiding and watermarking techniques have been applied to protect biometrics signals and provide a secure transmission of their related data [7-12]. fingerprint-based authentication system is one of the most mature biometrics authentication systems. the main processes in these systems are acquisition, representation, feature extraction, and matching [13]. in order to present a suitable watermarking technique for fingerprint images, the watermarking process should not affect the features (i.e., minutiae) of the image and should not affect the fingerprints matching process. * corresponding author: beekhoo@usm.my 118 r. thabit, b.e. khoo copyright ©2022 assa. adv. in systems science and appl. (2022) over the years, different wavelet transforms have been applied in many different applications such as in [14-17] because of their efficiency. most of the watermarking and data hiding schemes for fingerprint images have concentrated on the robustness of the watermark to withstand different kinds of attacks [7-10,18-20]. in [10], a wavelet-based watermarking scheme for fingerprint images has been presented. the horizontal and vertical high frequency subbands have been used to carry the watermark. the scheme has robustness against gaussian noise, filtering, and jpeg2000 compression. in [9], a wavelet-based watermarking method has been applied to embed the fingerprint minutiae data in the fingerprint image in order to increase the level of security. the horizontal, vertical, and diagonal dwt coefficients have been selected to carry the watermark bits. the scheme provides robustness against jpeg compression, additive noise, and filtering. in [7,8], the fingerprint image has been divided into two regions called region of interest (roi) and region of non-interest (roni). only roi (ridges area) has been used to carry the watermark bits in order to provide robustness against cropping attack. in addition, the schemes have robustness against wavelet scalar quantization (wsq) compression, filtering, and additive white gaussian noise (awgn). in [18,19], image watermarking schemes have been presented to hide the minutiae data or text data in fingerprint images. the horizontal, vertical, and diagonal dwt coefficients have been selected to carry the watermark bits. the schemes have robustness against salt & pepper noise, gaussian noise, filtering, and jpeg compression. in [20], the method used dct and neural network-particle swarm optimization (nn-pso) to watermark host gray scale fingerprint images with their corresponding facial images. the scheme divides the image into blocks and the nn is used to find the features of the block. then, the output of the nn is used as input in the pso module to find the best dct coefficients’ locations in that block where the secret facial image data can be embedded. the scheme has robustness against different kinds of attacks. on the other hand, some fragile watermarking techniques have been presented for fingerprint images [21-24]. in [21], at the sender side, the sha-256 hash of the original fingerprint image and sensitive personal information are encrypted and embedded into the fingerprint image using lossless data hiding. at the receiver, the fingerprint image can be recovered without loss. in [22], a reversible watermarking based on difference expansion (de) [25] and rotational replacement of the lsb has been presented to hide text watermark in fingerprint image. the scheme can recover the original image after extracting the embedded text and it obtained better visual quality in comparison with the scheme in [25]. in [23], the watermarking process has been applied to the template of the fingerprint (the template is found after the feature extraction process). any tampering in the template will destroy the watermark and thus the absence of the watermark makes the template unauthentic. according to the reported results, the method did not affect the verification process of the fingerprint images (except for only three images among the test images that have been used in the experiments). in [24], a fragile image watermarking technique has been used to hide the watermark into the fingerprint images by changing the least significant bit value of randomly chosen pixels of the image. the results proved that the presented scheme did not affect the verification system except for three test images. the schemes in [7-10,18-20] obtained robustness against different kinds of attacks, however, the process of embedding the watermark in the fingerprint images causes distortions in these images and the original images cannot be recovered after the watermark extraction process (i.e., irreversible watermarking). in addition, there is no test for the effect of the watermarking process on the number of the fingerprint minutiae and the matching process. the schemes in [21,22] are reversible watermarking techniques where the original fingerprint image can be recovered without loss; the schemes in [23,24] are irreversible but they tested the effect of the watermarking on the fingerprints matching process. however, the watermarks of the schemes in [21-24] are fragile and they can be destroyed if the watermarked image undergone unintentional attacks. robust reversible watermarking application for fingerprint image security 119 copyright ©2022 assa. adv. in systems science and appl. (2022) recently, the researches in the reversible watermarking techniques highlighted the need for reversible watermarking techniques that can withstand the unintentional attacks such as channel noise and compression which are unavoidable attacks [17,26-31]. therefore, this paper suggested the use of robust reversible watermarking (rrw) method as a better candidate for fingerprint images watermarking because this kind of watermarking can recover the original image after the watermark extraction process and at the same time provide robustness against unintentional attacks. the proposed scheme embeds the identification number (id) of the person in his fingerprint image. embedding the id using rrw in the fingerprint image has the following advantages: (a) provides security where the personal information has been embedded in a cover image and thus it will not be a clear text for any user, (b) provides direct link between the image and its related information, (c) the embedded id is used for database authentication and the absence of the id means the image is not authentic and it may come from illegal alteration in the database, (d) the scheme is reversible watermarking, thus the original image can be recovered correctly after extracting the hidden data, and (e) the scheme is robust and the embedded id can withstand different attacks. the rest of this paper is structured as follows: section 2 explains the details of the proposed robust reversible watermarking (rrw) technique for fingerprint image watermarking. section 3 presents the experimental results for several fingerprint images. the conclusions are drawn in section 4. 2. proposed scheme as mentioned before, the rrw schemes have been suggested as better candidates for the practical applications because they can recover the original image after the watermark extraction process and can withstand attacks [27,32,33]. in this paper, we suggest using robust reversible watermarking for fingerprint images which is based on our previous work in [34]. the rrw scheme in [34] has been studied for different types of images, different block sizes, and different threshold values and the results proved that it has better performance in comparison with previous rrw schemes. in this paper, the proposed rrw scheme is applied for a specific type of images (i.e., fingerprint images). to find the best block size for dividing the fingerprint image, we will modify our previous algorithm [34] to automatically calculate the best block size according to the number of bits in the id number and the size of the fingerprint image. a general block diagram for the watermark embedding and extraction processes is shown in fig. 2.1. the procedures of the watermark embedding and extraction process are explained in the following subsections. fig. 2.1. proposed rrw for fingerprint images 120 r. thabit, b.e. khoo copyright ©2022 assa. adv. in systems science and appl. (2022) 2.1. watermark embedding algorithm the flow chart of the proposed algorithm for embedding the id number in the fingerprint image is shown in fig. 2.2 which can be summarized in the following steps (for more details about the original rrw algorithm the reader can refer to [34]): step 1: input the id number and the fingerprint image. the current format of the malaysian national registration identity card (nric) number which contains 12 digits has been used in the proposed scheme as an example. the id number is written in a text file. the software reads the id text file and the fingerprint image. step 2: convert the id number to binary sequence. each digit in the id number is converted from decimal number to 8 bits binary number. then, the binary numbers are reshaped to form a sequence of binary bits. (i.e., 12 digits × 8 bits gives a binary sequence of length 96 bits). step 3: selecting the best block size for dividing the fingerprint image. the best block size is selected according to the size of the fingerprint image and the length of the id binary sequence. according to our rrw scheme [34], better robustness and better visual quality can be obtained with the larger block size (bsize). at the same time, the larger bsize gives lower capacity. the capacity depends on the size of the fingerprint image and bsize. the algorithm must search for the largest bsize that can be used by calculating the available capacity at a specific bsize and comparing it with the length of the binary sequence. the best bsize is found using the following algorithm: a) read the size of the fingerprint image, denote the height and width by h and w, respectively. b) set the (bsize) to 64 (which means the block size is 64×64). c) calculate the capacity using c=⌊(h×w)/(bsize )2⌋, where ⌊ ⌋ refers to the floor function in which the result is the nearest integer number that is less than or equal to the element inside the function. d) compare c with the length of the binary sequence (lseq) as follows: e) if c ≥ lseq then stop f) else divide bsize by 2 and repeat steps c and d. g) the final bsize is chosen for dividing the fingerprint image. step 4: divide the fingerprint image into non-overlapping blocks (size bsize × bsize). step 5: transform each block using slantlet transform (slt) matrix as shown in the following equation: 𝑆 = 𝑆𝐿𝑇! 𝑠 𝑆𝐿𝑇!", (2.1) where (s) is the image block, (s) is the slantlet transform of the block, (sltn) is an n×n slantlet matrix, and t is the transpose of matrix. note that s, s, and sltn have the same size. step 6: divide the resultant coefficients. the resultant coefficients in the matrix (s) into four subbands and select the high frequency subbands (high-low (hl) and low-high (lh)) to carry the id bits. step 7: apply the modification rules according to the bit value as explained in [34]. this process will generate some side information. step 8: replace the original high frequency subbands by the modified subbands. step 9: apply inverse slt to obtain the watermarked blocks using the following equation: 𝑠 = 𝑆𝐿𝑇!" 𝑆 𝑆𝐿𝑇!. (2.2) step 10: rearrange the blocks to obtain the watermarked fingerprint image. step 11: apply post-processing as explained in [34] to prevent the overflow/underflow of the pixel values. robust reversible watermarking application for fingerprint image security 121 copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 2.2. proposed id embedding algorithm 2.2. watermark extraction algorithm the flow chart of the proposed algorithm for extracting the id number from the watermarked fingerprint image is shown in fig. 2.3 which can be summarized in the following steps (for more details about the original rrw algorithm the reader can refer to ([34]): step 1: input the side information and the watermarked fingerprint image. step 2: return the pixel values (that have been adjusted) to their places. step 3: divide the resultant fingerprint image into non-overlapping blocks. step 4: transform each block using slt matrix. step 5: select the high frequency subbands (high-low (hl) and low-high (lh)) to extract the id bits. step 6: apply the extraction rules as explained in [34]. step 7: recover the original transform domain blocks according to the side information and the extracted bits. step 8: apply inverse slt to obtain the original spatial domain blocks. step 9: rearrange the blocks to obtain the original fingerprint image. step 10: recover the id number from the extracted binary sequence. step 11: compare the recovered id with the saved id in database, if they are equal then continue to the next step, else show a message that notify the receiver about the inauthenticity of the received fingerprint image and end the algorithm. step 12: calculate the matching score, if the minutiae of the received fingerprint image is identical to the saved one then show a message that notify the receiver about the authenticity of the received fingerprint image. 122 r. thabit, b.e. khoo copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 2.3. proposed extraction algorithm 3. results and analysis to evaluate the proposed scheme, different fingerprint images have been collected from [fvc2000 and fvc2004 databases] [35]. each database contains four sub-databases (db1, db2, db3, and db4). in each sub-database, there are 80 fingerprint images. thus, the total number of fingerprint images that are used in the experiments is (640 fingerprint images). samples of the test images are shown in fig. 3.1. the experiments are conducted to evaluate the visual quality of the watermarked fingerprint images, the reversibility of the watermarking scheme, the robustness against attacks, and the effect of the proposed watermarking process on the features (i.e., minutiae) of the fingerprint images. fig. 3.1. sample test images from each database robust reversible watermarking application for fingerprint image security 123 copyright ©2022 assa. adv. in systems science and appl. (2022) 3.1. visual quality evaluation to evaluate the visual quality, the peak-signal-to-noise ratio (psnr) and the structural similarity index (ssim) are calculated for each image in each sub-database of fingerprint images. a sample id number has been embedded in the images. the average psnr values and the average ssim values have been calculated for each sub-database (for 80 images in each sub-database). the threshold value affects the visual quality of the watermarked image therefore the visual quality of the scheme has been tested at different threshold values. fig. 3.2 shows samples of the watermarked images that have been obtained at different threshold values. the psnr values and ssim values for each watermarked image are in table 3.1. table 3.2 and table 3.3 contain the average values for each sub-database. the results proved that the proposed scheme does not affect the visual quality of the image. as shown in the results, when the threshold value increased, the visual quality decreased. fig. 3.2. (a) original fingerprint image, watermarked images at different threshold values: (b) at t=1, (c) at t=2, (d) at t=3, (e) at t=4, (f) at t=5, (g) at t=6. table 3.1. the psnr and ssim values for the watermarked images in fig. 3.2 image name psnr (db) ssim wdb1_1_thr1 59.6101 0.9995 wdb1_1_thr2 55.2978 0.9986 wdb1_1_thr3 52.3700 0.9973 wdb1_1_thr4 50.2032 0.9957 wdb1_1_thr5 48.4958 0.9937 wdb1_1_thr6 46.9980 0.9912 table 3.2. the average psnr(db) values at different threshold values name of database size t=1 t=2 t=3 t=4 t=5 t=6 t=7 2000_db1 300×300 59.872 55.526 52.544 50.362 48.610 47.114 45.768 2000_db2 256×364 51.532 50.155 48.885 47.731 46.674 45.702 44.802 2000_db3 448×478 55.058 52.020 49.628 47.687 46.055 44.657 43.436 2000_db4 240×320 55.419 53.292 51.349 49.831 48.419 47.234 46.087 2004_db1 640×480 60.234 56.532 53.813 51.763 50.030 48.548 47.259 124 r. thabit, b.e. khoo copyright ©2022 assa. adv. in systems science and appl. (2022) 2004_db2 328×364 54.174 50.507 47.805 45.691 43.941 42.468 41.192 2004_db3 300×480 48.227 46.797 45.452 44.220 43.082 42.037 41.076 2004_db4 288×384 54.115 50.156 47.343 45.182 43.402 41.905 40.617 table 3.3. the average ssim values at different threshold values name of database size t=1 t=2 t=3 t=4 t=5 t=6 t=7 2000_db1 300×300 0.9995 0.9987 0.9974 0.9958 0.9938 0.9913 0.9883 2000_db2 256×364 0.9996 0.9993 0.9990 0.9985 0.9980 0.9974 0.9967 2000_db3 448×478 0.9996 0.9990 0.9982 0.9971 0.9957 0.9940 0.9921 2000_db4 240×320 0.9997 0.9992 0.9986 0.9979 0.9969 0.9958 0.9944 2004_db1 640×480 0.9997 0.9991 0.9981 0.9969 0.9954 0.9935 0.9912 2004_db2 328×364 0.9992 0.9978 0.9958 0.9932 0.9897 0.9857 0.9810 2004_db3 300×480 0.9991 0.9982 0.9968 0.9951 0.9929 0.9903 0.9873 2004_db4 288×384 0.9990 0.9969 0.9937 0.9896 0.9843 0.9780 0.9709 3.2. reversibility evaluation the image reversibility is evaluated by comparing the pixel values in the original image with the pixel values in the recovered image after the watermark extraction process. consider the original image (𝐼$) and the recovered image (𝐼%) are both of size (h×w), if the two images are equal then the image reversibility is ‘1’ else the image reversibility is ‘0’. the image reversibility is calculated using the following equation: 𝐼𝑚𝑎𝑔𝑒 𝑅𝑒𝑣𝑒𝑟𝑠𝑖𝑏𝑖𝑙𝑖𝑡𝑦 = 61 𝑖𝑓 𝐼$ (𝑖, 𝑗) = 𝐼%(𝑖, 𝑗), 0 𝑖𝑓 𝐼$(𝑖, 𝑗) ≠ 𝐼%(𝑖, 𝑗), (3.1) where 𝐼$(𝑖, 𝑗) is the pixel value at the coordinates (𝑖, 𝑗) in the original image, 𝐼%(𝑖, 𝑗) is the pixel value at the coordinates (𝑖, 𝑗) in the recovered image, 𝑖 = 1, 2, … , 𝐻, 𝑗 = 1, 2, … ,𝑊. table 3.4 shows the average image reversibility results for each database. as illustrated in the results, the proposed scheme obtained complete reversibility for all test images at different threshold values. table 3.4. the average ssim values at different threshold values name of database size t=1 t=2 t=3 t=4 t=5 t=6 t=7 2000_db1 300×300 1 1 1 1 1 1 1 2000_db2 256×364 1 1 1 1 1 1 1 2000_db3 448×478 1 1 1 1 1 1 1 2000_db4 240×320 1 1 1 1 1 1 1 2004_db1 640×480 1 1 1 1 1 1 1 2004_db2 328×364 1 1 1 1 1 1 1 2004_db3 300×480 1 1 1 1 1 1 1 2004_db4 288×384 1 1 1 1 1 1 1 3.3. robustness evaluation the robustness is evaluated by calculating the bit error rate (ber) in the extracted watermark after different kinds of attacks (i.e., additive gaussian noise (agn), jpeg/jpeg2000 compression, histogram equalization, wiener filtering, and median filtering). fig. 3.3 shows samples of the results from the fingerprint images database (2000_db1) where the average ber against the mentioned attacks is calculated at different threshold values. fig. 3.3(a) shows the robustness results against additive gaussian noise (agn) with zero-mean and the variance equals to (0.001, 0.002, …, 0.01) for different threshold values. fig. 3.3(b) shows the robustness results against jpeg compression with the quality factor equals to (40, 50, …, 100) robust reversible watermarking application for fingerprint image security 125 copyright ©2022 assa. adv. in systems science and appl. (2022) for different threshold values. fig. 3.3(c) shows the robustness results against jpeg2000 compression with the rate equals to (0.2, 0.4, …, 2) for different threshold values. fig. 3.3(d) shows the robustness results against histogram equalization, wiener filtering (3×3), and median filtering (3×3). as shown in the results, the higher the threshold value the better the robustness. fig. 3.3. robustness results at different threshold values against (a) agn, (b) jpeg compression, (c) jpeg2000 compression, and (d) histogram equalization, wiener filtering, and median filtering. 3.4. similarity measure the number of the extracted minutia from the original fingerprint images and the recovered fingerprint images after the watermark extraction process has been calculated. the matching process between the original and the watermarked images are based on calculating the similarity measure [36] of the extracted minutiae in both images. the similarity measure value is (0 to 1), the more similar the minutiae the higher the similarity measure; when the minutiae of both images are identical the similarity measure equals 1. the proposed algorithm is completely reversible therefore the number of the minutiae is identical for each test image. table 3.5 shows the average similarity measure results for all fingerprint images databases at different threshold values. table 3.5. the average similarity measure at different threshold values name of database size t=1 t=2 t=3 t=4 t=5 t=6 t=7 2000_db1 300×300 1 1 1 1 1 1 1 2000_db2 256×364 1 1 1 1 1 1 1 2000_db3 448×478 1 1 1 1 1 1 1 2000_db4 240×320 1 1 1 1 1 1 1 2004_db1 640×480 1 1 1 1 1 1 1 2004_db2 328×364 1 1 1 1 1 1 1 2004_db3 300×480 1 1 1 1 1 1 1 2004_db4 288×384 1 1 1 1 1 1 1 126 r. thabit, b.e. khoo copyright ©2022 assa. adv. in systems science and appl. (2022) 3.5. general comparison with previous fingerprint image watermarking schemes this section contains a general comparison between the proposed robust reversible watermarking scheme and the previous schemes in [7-10,18-20,22-24,37]. as mentioned in the introduction section, the previous fingerprint image watermarking schemes are either robust or reversible therefore the proposed scheme can be considered as a better candidate for fingerprint images. table 3.6 summarizes the differences between the proposed scheme and the previous schemes. the proposed scheme is better than the scheme in [7-10,18-20] in terms of reversibility where these schemes cannot recover the original image after the watermark extraction process. in comparison with the schemes in [22,37], the proposed scheme is better in terms of robustness where these schemes are fragile and the watermark cannot withstand attacks. the proposed scheme is better than the schemes in [23,24] in terms of robustness and reversibility. table 3.6. the average similarity measure at different threshold values scheme domain robustness reversibility tested minutiae [7]-[10], [18], [19] transform domain dwt have robustness against different attacks irreversible no test for the effect of watermarking process on the fingerprint features. [20] transform domain dct have robustness against different attacks irreversible no test for the effect of watermarking process on the fingerprint features. [37] comparative study for different domains (lsb, dct, and dwt) fragile the watermark cannot withstand any attack. reversible the scheme is reversible therefore the minutiae of the fingerprint images have not been changed. [22] transform domain dct fragile the watermark cannot withstand any attack. reversible the scheme is reversible therefore the minutiae of the fingerprint images have not been changed. [23], [24] spatial domain fragile the watermark cannot withstand any attack. irreversible the effect of the watermarking process has been tested and the verification process has been affected for only some images. proposed scheme transform domain slt have robustness against different attacks reversible the effect of the watermarking process has been tested and the verification process has not been affected for all test images. 4. conclusion the proposed scheme in this paper protects the fingerprint image by embedding the id number in the fingerprint image using rrw without destroying fingerprint features. the scheme based on applying the slantlet transform (slt) on fingerprint image’s blocks and modifying the slt coefficients to carry the watermark bits. the proposed scheme is completely reversible therefore the matching process has not been affected. the experiments proved that the proposed scheme obtained high visual quality, good robustness results against different attacks, and it performs better in comparison with the previous fingerprint image watermarking schemes, therefore, it can be considered as a good candidate for practical applications. robust reversible watermarking application for fingerprint image security 127 copyright ©2022 assa. adv. in systems science and appl. 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(2002). distortionless data hiding based on integer wavelet transform,” electron. lett., vol. 38, no. 25, pp. 1646– 1648, doi: 10.1049/el:20021131. adv syst sci appl 2025; 1:55–85 published online at https://ijassa.ipu.ru. stability analysis of a delayed epidemic model with general incidence and treatment functions amine bernoussi1*, khalid hattaf2,3, brahim el boukari4 1laboratory: équations aux dérivées partielles, algèbre et géométrie spectrales, faculty of science, ibn tofail university, bp 133, 14000 kenitra, morocco 2equipe de recherche en modélisation et enseignement des mathématiques (ermem), centre régional des métiers de l’education et de la formation (crmef), 20340 derb ghalef, casablanca, morocco 3laboratory of analysis, modeling and simulation (lams), faculty of sciences ben m’sick, hassan ii university of casablanca, p.o box 7955 sidi othman, casablanca, morocco 4laboratory of applied mathematics and scientific calculus (lmacs), higher school of technology, sultan moulay slimane university, 23000 béni mellal, morocco abstract: in this paper, we study an sir epidemic model with general nonlinear incidence function, general function of treatment and two discrete time delays, the first described the time delay due to the latent period of the disease and the second is the time delay due to the period that the infected individuals use to move into the recovered class. lyapunov’s method is used to show the global stability of the disease-free equilibrium if the basic reproduction number r0 ≤ 1, while if r0 > 1 and under some conditions of delays, the existence of hopf bifurcation appears for the endemic equilibrium. keywords: treatment, nonlinear incidence, time delay, global stability, hopf bifurcation. 1. introduction mathematical modeling has been used in several areas, it is considered as a decision support tool on any situation. in the epidemiological area, it has contributed to the epidemiological surveillance of the disease because it makes to predict the health consequences of actions as varied as vaccination, quarantine or the distribution of screening tests. among the models used in epidemiology, there are the sir models which are compartment models, where s stands for susceptible subpopulation, i is infected subpopulation, and r is recovered subpopulation. the incidence rate of a disease measures how fast the disease is spreading and it plays an important role in the research of mathematical epidemiology. in many previous epidemic models, the bilinear incidence rate βsi was frequently used [13, 21, 22, 26, 27, 29, 34, 40, 43, 44]. however, there are some advantages for adopting more general forms of incidence rates. for instance, capasso et al. [14–16] observed that the incidence rate may increase more slowly as i increases. so, they proposed a saturated incidence rate βsi 1+αi which was used in [2, 9, 33, 41], where α is the saturation factor that measures the inhibitory effect, βi measures the infection force of the disease and 1 1+αi measures the inhibition effect from the behavioral change of the susceptible individuals when their number increases or from the crowding effect of the infected individuals. this incidence rate seems more reasonable than the bilinear incidence rate βsi , because it includes the behavioral change and crowding ∗corresponding author: amine.bernoussi@yahoo.fr 56 a. bernoussi, k. hattaf, b. el boukari effect of the infected individuals and prevents the unboundedness of the contact rate by choosing suitable parameters. in the last years, many forms of incidence function have been considered by the researchers in mathematical epidemiology. for example, the first one is the saturated incidence βsi d+s+i [3], where β and d are the positive constants. the second one is the beddington-deangelis incidence βsi 1+α1s+α2i [6], where α1 and α2 are the positive constants. the effect of saturation factor (refers to α1 and α2) stems from epidemic control and the protection mesures. the third one is the standard incidence βsi n [18, 26]. in addition, a recent hattaf-yousfi incidence βsi α0+α1s+α2i+α3si that includes the three above functions was introduced in [24] and used in [36]. models with incidence functions of the form g(i)h(s) have been studied in [30, 37]. the most general incidence function f(s, i)i [25] which generalizes the previous incidence functions has been studied by the many authors [4,10–12]. it is well known that treatment is an important and effective method to prevent and control the spread of various infectious diseases. therefore, it very important to adopt a suitable treatment function. for instance, wang and ruan [40] introduced a constant treatment in sir model as follows: { r, i > 0, 0, i = 0, which simulated a limited capacity for treatment. further, wang [39] considered the following piecewise linear treatment function: { ri, 0 ≤ i ≤ i0, ri0, i > i0, where i0 is the infective level at which the health care system reaches capacity. based on this, zhang et al. [42] proposed the following saturated treatment function ri 1+ki , where r is the maximal medical resources supplied per unit time and k is the saturation factor that measures the effect of the infected being delayed treatment. a very general from of treatment function t (i) was considered by elazzouzi et al. [19]. epidemiological models can contain a delay that is either discrete [8, 17] or continuous [7,10,35] because the delays appear in differential equations to describe the time lag between the action on the system and the system’s response to this action, or because a some threshold must be reached before the system is activated. in [33], liu proposed an seir epidemic model with saturated incidence and saturated treatment function with two discrete delays: the time delay due to the latent period of the disease and the time delay due to the period that the infected individuals use to move into the recovered class. motivated by the above works, we propose a mathematical sir model that generalizes the above models and incorporates the general nonlinear incidence function f(s, i)i , general function treatment t (i) and two discrete time delays, the first τ1 described the time delay due to the latent period of the disease and the second τ2 is the time delay due to the period that the infected individuals use to move into the recovered class. this model is given by the following nonlinear system: ds(t) dt = a− µs(t) + γ1i(t)− f(s(t), i(t))i(t), di(t) dt = f(s(t− τ1), i(t− τ1))i(t− τ1)− η1i(t)− γ2i(t− τ2)− t (i(t− τ2)), dr(t) dt = t (i(t− τ2)) + γ2i(t− τ2)− µr(t), (1.1) where η1 = µ+ γ1 + α. here a is the constant recruitment rate into the population, s(t) represents the number of individuals who are susceptible to disease, that is, who are not yet infected at time t, i(t) represents the number of infected individuals who are infectious and are able to spread the disease by contact with susceptible individuals at time t, r(t) is the copyright © 2025 assa. adv syst sci appl (2025) stability analysis of a delayed epidemic model... 57 number of individuals who have been infected and temporarily recovered at time t, µ is the natural death rate of the population, τ1 is the units of time after infection expressing latent period, τ2, is the time delay due to the period that the infected individuals use to move into the recovered class, γ1 is the transfer rate from the infected class to the susceptible class, γ2 is the transfer rate from the infected class to the recovered class, α is the disease-induced death rate, t (i) is the general treatment function and f(s, i)i is the incidence function, i.e., the number of susceptible individuals infected through their contacts with the infectious individuals. model (1.1) generalizes several special cases existing in the literature. for example, abta et al. [1] studied the model (1.1) with f(s, i)i = βsi 1+α1s+α2i , γ1 = 0, τ2 = 0 and t (i) = 0. in [25], hattaf et al. studied the model (1.1) with γ1 = 0, τ2 = 0 and t (i) = 0. in [40], wang and ruan studied the model (1.1) with f(s, i)i = βsi, γ1 = 0, τ1 = τ2 = 0 and t (i) = r. in [39], wang studied the model (1.1) with f(s, i)i = βsi, γ1 = 0, τ1 = τ2 = 0 and t (i) = { ri, 0 ≤ i ≤ i0, ri0, i > i0. in [5], balamuralitharan et al. studied the model (1.1) with f(s, i)i = βsi, γ1 = 0, τ1 = τ2 = 0 and t (i) = ri . in this paper, the initial condition for the system (1.1) is: s(θ) = φ1(θ), i(θ) = φ2(θ), r(θ) = φ3(θ), θ ∈ [−τ, 0], (1.2) where τ = max{τ1, τ2} and φ = (φ1, φ2, φ3) t ∈ c such that φi(θ) ≥ 0 for −τ ≤ θ ≤ 0 and i = 1, 2, 3. the space c denotes the banach space c([−τ, 0],r3 +0) of continuous functions mapping the interval [−τ, 0] into r3 +0 with the supremum norm, where r+0 = {x ∈ r | x ≥ 0}. also, we assume that φi(0) > 0 for i = 1, 2, 3. on the other hand, the first two equation in system (1.1) do not depend on the third equation, and therefore this equation can be omitted without loss of generality. system (1.1) can be rewritten as{ ds(t) dt = a− µs(t) + γ1i(t)− f(s(t), i(t))i(t), di(t) dt = f(s(t− τ1), i(t− τ1))i(t− τ1)− η1i(t)− γ2i(t− τ2)− t (i(t− τ2)), (1.3) the incidence function f(s, i)i : r+ × r+ → r+ is a continuously differentiable and locally lipschitz function on r+ × r+ and satisfying the following hypotheses (see [11, 25]): (h0) f(0, i) = 0 for i ≥ 0; (h1) f(s, i) is a strictly monotone increasing function of s > 0, for any fixed i ≥ 0; (h2) f(s, i) is a monotone decreasing function of i ≥ 0, for any fixed s ≥ 0; (h3) ϕ(s, i) = f(s, i))i is a monotone increasing function of i ≥ 0, for any fixed s ≥ 0. moreover, the treatment function t : r+ → r+ is a locally lipschitz continuous differentiable function on r+ satisfying the following hypotheses (see [19]) (t0) t (0) = 0; (t1) t (i) i is a monotone increasing function of i > 0. the rest of the paper is organized as follows. in section 2, we carry out mathematical analysis about the basic reproduction number and the existence of equilibria of the model (1.1). in section 3, we study the global stability of the disease-free equilibrium. section 4 is devoted to the stability and hopf bifurcation of the endemic equilibrium. an application of our results and some numerical simulations are presented in section 5. at the end, we present some concluding remarks. copyright © 2025 assa. adv syst sci appl (2025) 58 a. bernoussi, k. hattaf, b. el boukari 2. mathematical analysis in this section, we prove the basic results which guarantee the positivity of solutions as well as the existence and uniqueness of the endemic equilibrium for system (1.3) under initial condition (1.2). lemma 2.1: the closed set: ω := { (s, i) ∈ (r+)2|s + i ≤ a µ } is positively invariant with respect to system (1.3). proof let n(t) = s(t) + i(t), and (s, i) ∈ (r+)2. then it follows from system (1.3) that dn(t) dt = a− µs(t)− µi(t)− αi(t)− γ2i(t− τ2)− f(s(t), i(t))i(t) + f(s(t− τ1), i(t− τ1))i(t− τ1)− t (i(t− τ2)) = a− µn(t)− ∫ t t−τ1 f(s(σ), i(σ))i(σ)dσ − t (i(t− τ2))− αi(t)− γ2i(t− τ2). by the hypotheses, we have t (i(t− τ2)) ≥ 0 and f(s(t), i(t))i(t) ≥ 0 for all t ≥ 0 and τ2 ≥ 0. then dn(t) dt ≤ a− µn(t), which implies that n(t) ≤ a µ when n(0) ≤ a µ . this completes the proof. hence, we discuss system (1.3) in the closed set ω. next, we discuss the existence of equilibria for system (1.3). the basic reproduction number of system (1.3) can be defined by r0 = f(a µ , 0) η1 + η2 , where η2 = γ2 + t ′ (0). note that the system (1.3) always has a disease-free equilibrium p0 = (a µ , 0). on the other hand, to prove the existence and uniqueness of an endemic equilibrium, we need the following lemma as in [19]. lemma 2.2: assume that the assumptions (t0) and (t1), are satisfied. then b2 − a2u− t (u) = 0, for a2 > 0 and b2 > 0, has a unique positive solution u1, and b2 − a2u− t (u) > 0, for all u ∈ [0, u1), and b2 − a2u− t (u) ≤ 0 for all u ∈ [u1, b2 a2 ]. proof let l be the function defined on r+ by l(u) := b2 − a2u− t (u), we have: l(0) = b2 > 0 and l( b2 a2 ) = −t ( b2 a2 ) < 0 and l ′ = −a2 − t ′ < 0. since l is continuous and is strictly monotone decreasing function, then the equation l(u) = 0, has a unique positive solution in the interval (0, b2 a2 ). this completes the proof. copyright © 2025 assa. adv syst sci appl (2025) stability analysis of a delayed epidemic model... 59 theorem 2.1: under the hypotheses (h1), (h2), (t0) and (t1) if r0 > 1, then system (1.3) admits a unique endemic equilibrium p ∗ = (s∗, i∗), with s∗ = a µ − (µ+ γ2 + α)i∗ + t (i∗) µ , and i∗ is the unique solution of the following equation: f( a µ − (µ+ γ2 + α)i + t (i) µ , i) = η1 + γ2 + t (i) i . proof for simplicity, we put k(i) = t (i) i . at an equilibrium point (s, i) of system (1.3), the following equations hold.{ a− µs + γ1i − f(s, i)i = 0, f(s, i)i − (η1 + γ2)i − t (i) = 0. (2.4) substituting the second equation into the first equation of (2.4), we obtain the following system: s = a µ − (µ+ γ2 + α)i + t (i) µ , f(s, i)i = (η1 + γ2)i + t (i). (2.5) if i = 0, we obtain the disease-free equilibrium point p0 = (a µ , 0). if i ̸= 0, then using (2.5) we get the following equation f( a µ − (µ+ γ2 + α)i + t (i) µ , i) = η1 + γ2 +k(i). we have s = a µ − [(µ+γ2+α)i+t (i)] µ , which implies that s ≤ a µ . by lemma 2.2, we have a µ − (µ+γ2+α)i+t (i) µ > 0 if and only if i ∈ [0, i1), where i1 is a unique positive solution of the equation a µ − [(µ+γ2+α)i+t (i)] µ = 0. hence, there is no positive equilibrium point if s > a µ or i ≥ i1. now, we consider the function g defined on the interval [0, i1] as follows g(i) := f( a µ − (µ+ γ2 + α)i + t (i) µ , i)− (η1 + γ2 +k(i)). we have lim i→0 g(i) = f( a µ , 0)− (η1 + η2) = (η1 + η2) (f(a µ , 0) η1 + η2 − 1 ) = (η1 + η2)(r0 − 1) > 0 for r0 > 1, and g(i1) = −(η1 + γ2 +k(i1)) < 0. copyright © 2025 assa. adv syst sci appl (2025) 60 a. bernoussi, k. hattaf, b. el boukari furthermore, g ′ (i) = −µ+ γ2 + α + t ′ µ ∂f ∂s + ∂f ∂i −k ′ . according to the hypotheses (h1), (h2) and (t1), we have g ′ (i) < 0. hence, there exists a unique endemic equilibrium p ∗ = (s∗, i∗) with i∗ ∈ (0, i1) and s∗ > 0 satisfies the equations s∗ = a µ − (µ+γ2+α)i∗+t (i∗) µ . this completes the proof. 3. global stability of the disease-free equilibrium in this section, we establish the global stability of the disease-free equilibrium p0 of system (1.3). we can defined rc = f(a µ , 0) η1 . note that r0 < rc. theorem 3.1: suppose the hypotheses (h1), (h2), (t0) and (t1) hold. • if r0 > 1, then the disease-free equilibrium p0 of system (1.3) is unstable. • if r0 ≤ 1 then we distinguish three cases: (i) the disease-free equilibrium p0 of system (1.3) is globally asymptotically stable whenever τ1 ≥ τ2. (ii) when rc ≤ 1, the disease-free equilibrium p0 of system (1.3) is globally asymptotically stable for all τ1 ≥ 0 and τ2 ≥ 0, then in particularly for τ2 ≥ τ1. (iii) when rc > 1, the equation (3.9) has a purely imaginary root. if more h ′ 0(ω ∗ 22,0) ̸= 0, then there exists a positive τ ∗22,0 where system (1.3) undergoes hopf bifurcation at p0 when τ2 = τ ∗22,0. however, the steady state p0 is locally asymptotically stable when τ2 ∈ [0, τ ∗22,0) and unstable when τ2 > τ ∗22,0, where τ ∗22,0 is given by τ ∗22,0 = 1 ω∗ 22,0 arccos ( f(a µ , 0) cos (ω∗ 22,0τ1)− η1 η2 ), and ω∗ 22,0 is the positive root of equation (3.9), and h0 are defined in equation (3.13). proof • if r0 > 1, then the characteristic equation at p0 is given by (λ+ µ)(λ− f( a µ , 0)e−λτ1 + η1 + η2e −λτ2) = 0. (3.6) obviously, λ = −µ is eigenvalue for (3.6), and hence, the stability of p0 is determined by the distribution of the roots of equation λ− f( a µ , 0)e−λτ1 + η1 + η2e −λτ2 = 0. (3.7) copyright © 2025 assa. adv syst sci appl (2025) stability analysis of a delayed epidemic model... 61 we put ψ(λ) = λ− f( a µ , 0)e−λτ1 + η1 + η2e −λτ2 . we have lim λ→+∞ ψ(λ) = +∞, and ψ(0) = −f( a µ , 0) + η1 + η2 = (η1 + η2)(1− f(a µ , 0) (η1 + η2) ) = (η1 + η2)(1−r0). since the function ψ(λ) is continuous on the interval [0,+∞), we conclude that the equation ψ(λ) = 0 has a positive real root and the disease-free equilibrium is unstable when r0 > 1. • if r0 ≤ 1, then there are three cases: first case τ1 > τ2 consider the following lyapunov functional v0(t) = ∫ s(t−τ1) a µ (1− f(a µ , 0) f(σ, 0) )dσ + i + η1 ∫ t t−τ2 i(ξ)dξ + ∫ t−τ2 t−τ1 f(a µ , 0) f(s(σ), 0) f(s(σ), i(σ))i(σ)dσ. we will show that dv0(t) dt ≤ 0 for all t ≥ 0. we have dv0(t) dt = (1− f(a µ , 0) f(s(t− τ1), 0) )(a− µs(t− τ1) + γ1i(t− τ1) − f(s(t− τ1), i(t− τ1))i(t− τ1)) + f(s(t− τ1), i(t− τ1))i(t− τ1)− η1i(t)− γ2i(t− τ2)− t (i(t− τ2)) + η1i(t)− η1i(t− τ2) + f(a µ , 0) f(s(t− τ2), 0) f(s(t− τ2), i(t− τ2))i(t− τ2)) − f(a µ , 0) f(s(t− τ1), 0) f(s(t− τ1), i(t− τ1))i(t− τ1)) = µ(1− f(a µ , 0) f(s(t− τ1), 0) )( a µ − s(t− τ1)) + γ1i(t− τ1)(1− f(a µ , 0) f(s(t− τ1), 0) ) + f( a (µ) , 0) f(s(t− τ2), 0) f(s(t− τ2), i(t− τ2))i(t− τ2)) − (η1 + γ2)i(t− τ2)− t (i(t− τ2)). copyright © 2025 assa. adv syst sci appl (2025) 62 a. bernoussi, k. hattaf, b. el boukari then we have dv0(t) dt = µ(1− f(a µ , 0) f(s(t− τ1), 0) )( a µ − s(t− τ1)) + γ1i(t− τ1)(1− f(a µ , 0) f(s(t− τ1), 0) ) + ( (η1 + γ2)i(t− τ2) + t (i(t− τ2)) ) × ( f(a µ , 0)f(s(t− τ2), i(t− τ2))i(t− τ2) f(s(t− τ2), 0)[(η1 + γ2)i(t− τ2) + t (i(t− τ2))] − 1 ) = µ(1− f(a µ , 0) f(s(t− τ1), 0) )( a µ − s(t− τ1)) + γ1i(t− τ1)(1− f(a µ , 0) f(s(t− τ1), 0) ) + ( (η1 + γ2)i(t− τ2) + t (i(t− τ2)) ) × ( r0 η1 + η2 η1 + γ2 +k(i(t− τ2)) f(s(t− τ2), i(t− τ2)) f(s(t− τ2), 0) − 1 ) . furthermore, it follows from the hypotheses (h2), (t0) and (t1) that f(s(t− τ2), i(t− τ2)) f(s(t− τ2), 0) ≤ 1, and η1 + η2 η1 + γ2 +k(i(t− τ2)) = η1 + γ2 + limi→0 t (i(t−τ2)) i(t−τ2) η1 + γ2 + t (i(t−τ2)) i(t−τ2) ≤ 1. hence, dv0(t) dt ≤ µ(1− f(a µ , 0) f(s(t− τ1), 0) )( a µ − s(t− τ1)) + γ1i(t− τ1)(1− f(a µ , 0) f(s(t− τ1), 0) ) + [(η1 + γ2)i(t− τ2) + t (i(t− τ2))](r0 − 1). by the hypothesis (h1), we obtain that (1− f(a µ , 0) f(s(t− τ1), 0) )( a µ − s(t− τ1)) ≤ 0, and i(t− τ1)(1− f(a µ , 0) f(s(t− τ1), 0) ) ≤ 0, where equality holds if and only if s = a µ . since r0 ≤ 1, ensures that dv0(t) dt ≤ 0, for all t ≥ 0, thus, the disease-free equilibrium p0 is stable and dv0(t) dt = 0 holds if and only if s = a µ and [(η1 + γ2)i(t− τ2) + t (i(t− τ2))] ( r0 η1+η2 η1+γ2+k(i(t−τ2)) f(s(t−τ2),i(t−τ2)) f(s(t−τ2),0) − 1 ) = 0. we discuss two cases: copyright © 2025 assa. adv syst sci appl (2025) stability analysis of a delayed epidemic model... 63 • if r0 < 1, then it follows from s = a µ and lemma 2.1 that i = 0. • if r0 = 1, then it follows from s = a µ and lemma 2.1 that i = 0. by the above discussions, we deduce that {p0} is the largest invariant set in{ (s, i)|dv0(t) dt = 0 } . from the lyapunov-lasalle theorem [31, 32], we conclude that p0 is globally asymptotically stable. second case τ1 = τ2 = τ consider the following lyapunov functional w0(t) = ∫ s(t−τ) a µ (1− f(a µ , 0) f(σ, 0) )dσ + i + η1 ∫ t t−τ i(ξ)dξ. we will show that dw0(t) dt ≤ 0 for all t ≥ 0. we have dw0(t) dt = (1− f(a µ , 0) f(s(t− τ), 0) )(a− µs(t− τ) + γ1i(t− τ) − f(s(t− τ), i(t− τ))i(t− τ)) + f(s(t− τ), i(t− τ))i(t− τ)− η1i(t)− t (i(t− τ)) + η1i(t) − (η1 + γ2)i(t− τ) = µ(1− f(a µ , 0) f(s(t− τ), 0) )( a µ − s(t− τ)) + γ1i(t− τ)(1− f(a µ , 0) f(s(t− τ), 0) ) + f(a µ , 0) f(s(t− τ), 0) f(s(t− τ), i(t− τ))i(t− τ)) − (η1 + γ2)i(t− τ)− t (i(t− τ)). then we have dw0(t) dt = µ(1− f(a µ , 0) f(s(t− τ), 0) )( a µ − s(t− τ)) + γ1i(t− τ)(1− f(a µ , 0) f(s(t− τ), 0) ) + ( (η1 + γ2)i(t− τ) + t (i(t− τ)) ) × ( f(a µ , 0)f(s(t− τ), i(t− τ)) f(s(t− τ), 0)(η1 + γ2 +k(i(t− τ))) − 1 ) = µ(1− f(a µ , 0) f(s(t− τ), 0) )( a µ − s(t− τ)) + γ1i(t− τ)(1− f(a µ , 0) f(s(t− τ), 0) ) + ( (η1 + γ2)i(t− τ) + t (i(t− τ)) ) × ( r0 η1 + η2 η1 + γ2 +k(i(t− τ)) f(s(t− τ), i(t− τ)) f(s(t− τ), 0) − 1 ) . furthermore, it follows from the hypotheses (h2), (t0) and (t1), we have f(s(t− τ), i(t− τ)) f(s(t− τ), 0) ≤ 1, copyright © 2025 assa. adv syst sci appl (2025) 64 a. bernoussi, k. hattaf, b. el boukari and η1 + η2 (η1 + γ2) +k(i(t− τ)) = η1 + γ2 + limi→0 t (i(t−τ)) i(t−τ) η1 + γ2 + t (i(t−τ)) i(t−τ) ≤ 1. then dw0(t) dt ≤ µ(1− f(a µ , 0) f(s(t− τ), 0) )( a µ − s(t− τ)) + γ1i(t− τ)(1− f(a µ , 0) f(s(t− τ), 0) ) + [(η1 + γ2)i(t− τ) + t (i(t− τ))](r0 − 1). by the hypothesis (h1), we get that (1− f(a µ , 0) f(s(t− τ), 0) )( a µ − s(t− τ)) ≤ 0, and i(t− τ)(1− f(a µ , 0) f(s(t− τ), 0) ) ≤ 0, where equality holds if and only if s = a µ . since r0 ≤ 1, we deduce that dw0(t) dt ≤ 0 for all t ≥ 0. thus, the disease-free equilibrium p0 is stable and dw0(t) dt = 0, holds if and only if s = a µ , and [(η1 + γ2)i(t− τ) + t (i(t− τ))] ( r0 η1+η2 η1+γ2+k(i(t−τ)) f(s(t−τ),i(t−τ)) f(s(t−τ),0) − 1 ) = 0. we discuss two cases: • if r0 < 1, then it follows from s = a µ and lemma 2.1 that i = 0. • if r0 = 1, then it follows from s = a µ and lemma 2.1 that i = 0. therefore, {p0} is the largest invariant set in { (s, i)|dw0(t) dt = 0 } . from the lyapunovlasalle theorem [31, 32], we conclude that p0 is globally asymptotically stable. third case τ2 > τ1 suppose that rc ≤ 1. consider the following lyapunov functional u0(t) = ∫ s(t) a µ (1− f(a µ , 0) f(σ, 0) )dσ + i + ∫ t t−τ1 f(s(ξ), i(ξ))i(ξ)dξ. copyright © 2025 assa. adv syst sci appl (2025) stability analysis of a delayed epidemic model... 65 we will show that du0(t) dt ≤ 0 for all t ≥ 0. we have du0(t) dt = (1− f(a µ , 0) f(s(t), 0) )(a− µs(t) + γ1i(t)− f(s(t), i(t))i(t)) + f(s(t− τ), i(t− τ))i(t− τ)− η1i(t)− γ2i(t− τ2)− t (i(t− τ2)) + f(s(t), i(t))i(t)− f(s(t− τ), i(t− τ))i(t− τ) = µ(1− f(a µ , 0) f(s(t), 0) )( a µ − s(t)) + γ1i(t)(1− f(a µ , 0) f(s(t), 0) ) + f(a µ , 0) f(s(t), 0) f(s(t), i(t))i(t))− η1i(t)− γ2i(t− τ2)− t (i(t− τ2)). then we have du0(t) dt = µ(1− f(a µ , 0) f(s(t), 0) )( a µ − s(t)) + γ1i(t)(1− f(a µ , 0) f(s(t), 0) ) + η1i(t) (f(a µ , 0)f(s(t), i(t)) f(s(t), 0)η1 − 1 ) − γ2i(t− τ2)− t (i(t− τ2)). hence, du0(t) dt = µ(1− f(a µ , 0) f(s(t), 0) )( a µ − s(t)) + γ1i(t)(1− f(a µ , 0) f(s(t), 0) ) + η1i(t) ( rc f(s(t), i(t)) f(s(t), 0) − 1 ) − γ2i(t− τ2)− t (i(t− τ2)). (3.8) furthermore, it follows from the hypotheses (h2) that f(s(t), i(t)) f(s(t), 0) ≤ 1, and by the hypotheses, we have i(t− τ2) ≥ 0 and t (i(t− τ2)) ≥ 0. then du0(t) dt ≤ µ(1− f(a µ , 0) f(s(t), 0) )( a µ − s(t)) + γ1i(t)(1− f(a µ , 0) f(s(t), 0) ) + η1i(t) ( rc − 1 ) . by the hypothesis (h1), we obtain that (1− f(a µ , 0) f(s(t), 0) )( a µ − s(t)) ≤ 0, copyright © 2025 assa. adv syst sci appl (2025) 66 a. bernoussi, k. hattaf, b. el boukari and i(t− τ)(1− f(a µ , 0) f(s(t), 0) ) ≤ 0, where equality holds if and only if s = a µ . since rc ≤ 1, ensures that du0(t) dt ≤ 0 for all t ≥ 0, thus, the disease-free equilibrium p0 is stable and by the equation (3.8) du0(t) dt = 0, holds if and only if s = a µ and i = 0. by the above discussions, we deduce that {p0} is the largest invariant set in{ (s, i)|du0(t) dt = 0 } . from the lyapunov-lasalle theorem [31, 32], we conclude that p0 is globally asymptotically stable for all τ1 ≥ 0 and τ2 ≥ 0. then p0 is globally asymptotically stable for all τ2 > τ1. now, we suppose that r0 < 1 and rc > 1 the characteristic equation at p0 is given by the equation (3.6), obviously, λ = −µ, is eigenvalue for (3.6), and hence, the stability of p0 is determined by the distribution of the roots of equation λ− f( a µ , 0)e−λτ1 + η1 + η2e −λτ2 = 0. (3.9) equation (3.9) has a purely imaginary root iω22, with ω22 > 0 if and only if f( a µ , 0) cos (ω22τ1)− η1 = η2 cos (ω22τ2), (3.10) ω22 + f( a µ , 0) sin (ω22τ1) = η2 sin (ω22τ2). (3.11) squaring and adding the squares together, we obtain ω2 22 + η21 + f( a µ , 0)2 − η22 − 2η1f( a µ , 0) cos (ω22τ1) + 2ω22f( a µ , 0) sin (ω22τ1) = 0. (3.12) we put h0(ω22) = ω2 22 + η21 + f( a µ , 0)2 − η22 − 2η1f( a µ , 0) cos (ω22τ1) + 2ω22f( a µ , 0) sin (ω22τ1). (3.13) on the other hand, we have −1 ≤ cos (ω22τ1) ≤ 1 and − 1 ≤ sin (ω22τ1) ≤ 1. then we have g01(ω22) ≤ h0(ω22) ≤ g02(ω22), where g01(ω22) = ω2 22 + η21 + f( a µ , 0)2 − η22 − 2η1f( a µ , 0)− 2ω22f( a µ , 0), and g02(ω22) = ω2 22 + η21 + f( a µ , 0)2 − η22 + 2η1f( a µ , 0) + 2ω22f( a µ , 0). copyright © 2025 assa. adv syst sci appl (2025) stability analysis of a delayed epidemic model... 67 we have lim ω22→+∞ g01(ω22) = lim ω22→+∞ g02(ω22) = +∞. then lim ω22→+∞ h0(ω22) = +∞, and h0(0) = η21 + f( a µ , 0)2 − η22 − 2η1f( a µ , 0) = (f( a µ , 0)− η1) 2 − η22 = ( f( a µ , 0)− η1 − η2 )( f( a µ , 0)− η1 + η2 ) = ( η1 + η2 )( r0 − 1 )( η1(rc − 1) + η2 ) ≤ 0. since h0(ω22) is continuous in [0,+∞), then the equation (3.9) has at least one positive root. we assume that equation (3.9) admits a finite family of solution ω∗ 22,o, with o = 1, ...,m and m ∈ n. by the equation (3.10), we have τ f2,o = 1 ω∗ 22,o ( arccos ( f(a µ , 0) cos (ω∗ 22,oτ1)− η1 η2 ) + 2πf ) , f = 0, 1, ...; o = 1, 2, ...,m. then ±iω∗ 22,o is a pair of purely imaginary root of equation (3.9), with τ2 = τ f2,o, f = 1, 2, ....; o = 1, 2, ...,m. clearly, lim f→∞ τ f2,o = ∞, o = 1, 2, ...,m. thus, we can define τ ∗22,0 = τ f02,o0 = min f=0,1...,o=1,2,...,m (τ f2,o), ω ∗ 22,0 = ω∗ 2,o0 . lets show that iω∗ 22,0 is simple, consider the branche of characteristic roots λ(τ2) = x1(τ2) + iy1(τ2), of equation (3.9) bifurcating from iω∗ 22,0 at τ2 = τ ∗22,0. by derivation (3.9) with respect to the delay τ2, we obtain dλ dτ2 ( 1 + τ1f( a µ , 0)e−λτ1 − τ2η2e −λτ2 ) = λη2e −λτ2 . (3.14) if we suppose, by contradiction, that iω∗ 22,0 is not simple, the right hand side (3.14) gives iω∗ 22,0η2 = 0, and leads a contradiction with the fact that ω∗ 22,0 > 0, and η2 > 0. next, we need to grantee the transversality condition of the hopf bifurcation theorem (see [23]). clearly, λ(τ2) = x1(τ2) + iy1(τ2), is a root of equation (3.9) if and only if x1 + η1 − e−x1τ1f( a µ , 0) cos (y1τ1) = −e−x1τ2η2 cos (y1τ2), (3.15) copyright © 2025 assa. adv syst sci appl (2025) 68 a. bernoussi, k. hattaf, b. el boukari y1 + e−x1τ1f( a µ , 0) sin (y1τ1) = e−x1τ2η2 sin (y1τ2). (3.16) let x1(τ ∗ 22,0) and y1(τ ∗ 22,0) satisfying x1(τ ∗ 22,0) = 0, and y1(τ ∗ 22,0) = ω∗ 22,0. by differentiating equations (3.15) and (3.16) with respect to τ2 and then set τ2 = τ ∗22,0. doing this, we get g1,0 dx1(τ ∗ 22,0) dτ2 +g2,0 dy1(τ ∗ 22,0) dτ2 = h1,0, −g2,0 dx1(τ ∗ 22,0) dτ2 +g1,0 dy1(τ ∗ 22,0) dτ2 = h2,0, (3.17) where g1,0 = 1 + τ1f( a µ , 0) cos (ω∗ 22,0τ1)− τ2η2 cos (ω ∗ 22,0τ2), g2,0 = τ1f( a µ , 0) sin (ω∗ 22,0τ1)− τ2η2 sin (ω ∗ 22,0τ2), h1,0 = ω∗ 22,0η2 sin (ω ∗ 22,0τ2), and h2,0 = ω∗ 22,0η2 cos (ω ∗ 22,0τ2). calculating dx1(τ∗22,0) dτ2 , we get dx1(τ ∗ 22,0) dτ2 = g1,0h1,0 −g2,0h2,0 g2 1,0 +g2 2,0 . therefore, according to the equation (3.10) and (3.11) we have dx1(τ ∗ 22,0) dτ2 = ω∗ 22,0h ′ (ω∗ 22,0) 2(g2 1,0 +g2 2,0) ̸= 0. this proves the theorem. 4. stability and hopf bifurcation of the endemic equilibrium in the next, we will study the local stability of the positive equilibrium p ∗ with respect to the time delay. we set x = s − s∗, and y = i − i∗. then the linearized system of equations around the equilibrium point p ∗ is given as follows: dx dt = (−µ− ∂f(s∗, i∗) ∂s i∗)x(t)− ( ∂f(s∗, i∗) ∂i i∗ + f(s∗, i∗)− γ1)y(t), dy dt = ∂f(s∗, i∗) ∂s i∗x(t− τ1) + ( ∂f(s∗, i∗) ∂i i∗ + f(s∗, i∗))y(t− τ1)− η1y(t) −(γ2 + t ′ (i∗))y(t− τ2). for simplicity, we put η3 = γ2 + t ′ (i∗), and η4 = µ+ i∗ ∂f(s ∗,i∗) ∂s . hence, the characteristic equation at p ∗ is given by ∆(λ, τ1, τ2) = ∣∣∣∣ λ+ η4 i∗ ∂f(s ∗,i∗) ∂i + f(s∗, i∗)− γ1 −i∗ ∂f(s ∗,i∗) ∂s e−λτ1 λ+ η1 − (i∗ ∂f(s ∗,i∗) ∂i + f(s∗, i∗))e−λτ1 + η3e −λτ2 ∣∣∣∣ = 0. copyright © 2025 assa. adv syst sci appl (2025) stability analysis of a delayed epidemic model... 69 by a simple computation, we get ∆(λ, τ1, τ2) = p (λ) +q(λ)e−λτ1 + j(λ)e−λτ2 = 0, (4.18) where p (λ) = λ2 + a1λ+b, q(λ) = cλ+d, j(λ) = eλ+ f, a1 = η1 + η4, b = η1η4, c = −( ∂f(s∗, i∗) ∂i i∗ + f(s∗, i∗)), d = −η4( ∂f(s∗, i∗) ∂i i∗ + f(s∗, i∗)) + ∂f(s∗, i∗) ∂s i∗( ∂f(s∗, i∗) ∂i i∗ + f(s∗, i∗))− γ1 ∂f(s∗, i∗) ∂s i∗ = −µ( ∂f(s∗, i∗) ∂i i∗ + f(s∗, i∗))− γ1 ∂f(s∗, i∗) ∂s i∗, e = η3, and f = η4η3. by the hypotheses (h2) and (t1), we easily deduce that (see [20, 28]) t ′ (i∗)− t (i∗) i∗ ≥ 0, (4.19) and f(s∗, i∗)− ∂ϕ(s∗, i∗) ∂i ≥ 0. (4.20) several cases arise. 4.1. case τ1 = τ2 = 0 theorem 4.1: if r0 > 1 and τ1 = τ2 = 0, then the endemic equilibrium p ∗ is locally asymptotically stable. proof when τ1 = τ2 = 0, the characteristic equation (4.18) reads as λ2 + (a1 + c + e)λ+ (b +d + f ) = 0 by using the second equation of system (1.3), we have f(s∗, i∗) = (η1 + γ2) + t (i∗) i∗ . (4.21) copyright © 2025 assa. adv syst sci appl (2025) 70 a. bernoussi, k. hattaf, b. el boukari then a1 + c + e = η1 + η4 − ( ∂f(s∗, i∗) ∂i i∗ + f(s∗, i∗)) + η3 = η4 − ∂f(s∗, i∗) ∂i i∗ + (t ′ (i∗)− t (i∗) i∗ ), and b +d + f = η1η4 − µ( ∂f(s∗, i∗) ∂i i∗ + f(s∗, i∗))− γ1 ∂f(s∗, i∗) ∂s i∗ + η4η3 = µ(t ′ (i∗)− t (i∗) i∗ ) + ∂f(s∗, i∗) ∂s i∗[(µ+ α) + η3]− µ ∂f(s∗, i∗) ∂i i∗. by the hypotheses (h1), (h2), (t1), and equation (4.19), we have a1 + c + e > 0 and b +d + f > 0. hence, according to the routh-hurwitz criterion, the endemic equilibrium p ∗ is locally asymptotically stable. this completes the proof. 4.2. case τ1 > 0 and τ2 = 0 theorem 4.2: if r0 > 1, τ1 > 0 and τ2 = 0, then the endemic equilibrium p ∗ is locally asymptotically stable. proof when τ1 > 0 and τ2 = 0 the characteristic equation (4.18) becomes ∆(λ, τ1) = λ2 + (a1 + e)λ+b + f + (λc +d)e−λτ1 = 0. (4.22) equation (4.22) has a purely imaginary root iω1, with ω1 > 0 if and only if ω2 1 − (b + f ) = d cos (ω1τ1) + ω1c sin (ω1τ1), and ω1(a1 + e) = d sin(ω1τ1)− ω1c cos (ω1τ1). squaring and adding the squares together, we obtain ω4 1 + cω2 1 + d = 0, (4.23) with c = (a1 + e)2 − 2(b + f )− c2 and d = (b + f )2 −d2. letting z1 = ω2 1, then equation (4.23) becomes the following equation z21 + cz1 + d = 0. on the other hand, we have d = (b + f )2 −d2 = (b + f −d)(b + f +d). using the case τ1 = τ2 = 0, we have (b + f +d) > 0 and by using the hypotheses (h1), (h3) and (t1), we have (b + f −d) > 0 then we have d > 0. now, we will prove c > 0. copyright © 2025 assa. adv syst sci appl (2025) stability analysis of a delayed epidemic model... 71 indeed c = (a1 + e)2 − 2(b + f )− c2, we have (a1 + e)2 = (η1 + η3 + η4) 2. by using the equation (4.21), find that (a1 + e)2 = ( f(s∗, i∗) + η4 + t ′ (i∗)− t (i∗) i∗ )2 = f(s∗, i∗)2 + η24 + 2f(s∗, i∗)η4 + (t ′ (i∗)− t (i∗) i∗ )2 + 2(t ′ (i∗)− t (i∗) i∗ )f(s∗, i∗) + 2(t ′ (i∗)− t (i∗) i∗ )η4, and −2(b + f ) = −2η4(η1 + η3) = −2f(s∗, i∗)η4 − 2η4(t ′ (i∗)− t (i∗) i∗ ), and −c2 = −( ∂f(s∗, i∗) ∂i i∗ + f(s∗, i∗))2 = −( ∂ϕ(s∗, i∗) ∂i )2. then we have c = ( f(s∗, i∗)− ∂ϕ(s(t), i(t)) ∂i )( f(s∗, i∗) + ∂ϕ(s(t), i(t)) ∂i ) + η24 + (t ′ (i∗)− t (i∗) i∗ )2 + 2(t ′ (i∗)− t (i∗) i∗ )f(s∗, i∗). by the hypothesis (h3) and equations (4.19) and (4.20), we have c > 0. then the equation (4.23) has non positive solution. consequently, using the case τ1 = τ2 = 0, the endemic equilibrium p ∗ is locally asymptotically stable . this completes the proof. 4.3. case τ1 = τ2 = τ consider the assumption: η1 − γ2 + µ∂ϕ(s∗,i∗) ∂i + γ1 ∂f(s∗,i∗) ∂s i∗ η4 < t ′ (i∗). (4.24) theorem 4.3: if r0 > 1 and (4.24) holds, then there exists a positive τ0 where system (1.3) undergoes hopf bifurcation at p ∗ when τ = τ0. however, the steady state p ∗ is locally asymptotically stable copyright © 2025 assa. adv syst sci appl (2025) 72 a. bernoussi, k. hattaf, b. el boukari when τ ∈ [0, τ0) and unstable when τ > τ0. here τ0 is given by τ0 = 1 ω0 arccos ( (d + f )(ω2 0 −b)− ω2 0a1(c + e) ω2 0(c + e)2 + (d + f )2 ), and ω0 = √ −x+ √ (x2 − 4y) 2 , where x and y are defined in equation (4.28). proof when τ1 = τ2 = τ , the characteristic equation (4.18) becones ∆(λ, τ) = λ2 + a1λ+b + [λ(c + e) + (d + f )]e−λτ = 0. (4.25) equation (4.25) has a purely imaginary root iω, with ω > 0 ∆(iω, τ) = 0, if and only if ω2 −b = ω(c + e) sin (ωτ) + (d + f ) cos (ωτ), (4.26) ωa1 = (d + f ) sin (ωτ)− ω(c + e) cos (ωτ). (4.27) squaring and adding the squares together, we obtain ω4 + xω2 + y = 0, (4.28) with x = a2 1 − 2b − (c + e)2, and y = b2 − (d + f )2. letting z = ω2, equation (4.28), becomes the following equation z2 + xz + y = 0. (4.29) on the other hand, we have y = b2 − (d + f )2 = ( b +d + f )( b −d − f ) . using the case τ1 = τ2 = 0, we have b + f +d > 0, and the assumption (4.24) implies that b −d − f < 0, then we have y < 0. consequently, the equation (4.29) has a unique solution positive z0 = −x+ √ (x2 − 4y) 2 . then the equation (4.28) has a unique positive solution ω0 = √ −x+ √ (x2 − 4y) 2 . copyright © 2025 assa. adv syst sci appl (2025) stability analysis of a delayed epidemic model... 73 on the other hand, the equations (4.26) and (4.27) imply that τ0 = 1 ω0 arccos ( (d + f )(ω2 0 −b)− ω2 0a1(c + e) ω2 0(c + e)2 + (d + f )2 ). lets show that iω0 is simple, consider the branche of characteristic roots λ(τ) = k(τ) + ij(τ), of equation (4.25) bifurcating from iω0 at τ = τ0. by derivation (4.25) with respect to the delay τ, we obtain dλ dτ ( 2λ+ a1 + e−λτ ([c + e]− [λ(c + e) + (d + f )]τ) ) = λ[λ(c + e) + (d + f )]e−λτ . (4.30) if we suppose, by contradiction, that iω0 is not simple, the right hand side (4.30) gives iω0(c + e) + (d + f ) = 0. on the other hand, the equation (4.24) implies that b −d − f < 0, and by the hypotheses (h1), (h3) and (t1), we have b > 0,−d > 0 and f > 0 then we have −d < f. this contradicts −d = f. next, we need to guarantee the transversality condition of the hopf bifurcation theorem (see [23]). clearly, λ(τ) = k(τ) + ij(τ) is a root of equation (4.25) if and only if k2 − j2 + a1k +b = −e−kτ ( [k(c + e) + (d + f )] cos (jτ) + j(c + e) sin (jτ) ) , (4.31) and 2kj + ja1 = −e−kτ ( j(c + e) cos (jτ)− [k(c + e) + (d + f )] sin (jτ) ) . (4.32) let k(τ0) and j(τ0), satisfying k(τ0) = 0, and j(τ0) = ω0. by differentiating equations (4.31) and (4.32) with respect to τ and then set τ = τ0. doing this, we get g1 dk(τ0) dτ +g2 dj(τ0) dτ = h1, −g2 dk(τ0) dτ +g1 dj(τ0) dτ = h2, where g1 = a1 + [(c + e)− τ0(d + f )] cos (ω0τ0)− τ0ω0(c + e) sin (ω0τ0), g2 = −2ω0 + τ0ω0(c + e) cos (ω0τ0) + [(c + e)− τ0(d + f )] sin (ω0τ0), h1 = ω0(d + f ) sin (ω0τ0)− ω2 0(c + e) cos (ω0τ0), and h2 = ω2 0(c + e) sin (ω0τ0) + ω0(d + f ) cos (ω0τ0). also, we have dk(τ0) dτ = g1h1 −g2h2 g2 1 +g2 2 , therefore, according to the equations (4.26) and (4.27) we have dk(τ0) dτ = ω2 0 √ x2 − 4y g2 1 +g2 2 . this completes the proof. copyright © 2025 assa. adv syst sci appl (2025) 74 a. bernoussi, k. hattaf, b. el boukari 4.4. case τ2 > 0 and τ1 = 0 theorem 4.4: if r0 > 1 and (4.24) holds, then there exists a positive τ2,0 where system (1.3) undergoes hopf bifurcation at p ∗ when τ2 = τ2,0. however, the steady state p ∗ is locally asymptotically stable when τ2 ∈ [0, τ2,0) and unstable when τ2 > τ2,0. here τ2,0 is given by τ2,0 = 1 ω2,0 arccos [ ω2 2,0(f − e[a1 + c])− (b +d)f ω2 2,0e 2 + f 2 ], and ω2,0 = √ −p+ √ (p2 − 4q) 2 , where p and q are defined in equation (4.36). proof when τ2 > 0 and τ1 = 0, the characteristic equation (4.18) becomes ∆(λ, τ2) = λ2 + (a1 + c)λ+b +d + (λe + f )e−λτ2 = 0. (4.33) equation (4.33), has a purely imaginary root iω2, with ω2 > 0, ∆(iω2, τ2) = 0, if and only if ω2 2 − (b +d) = ω2e sin (ω2τ2) + f cos (ω2τ2), (4.34) ω2(a1 + c) = f sin (ω2τ2)− ω2e cos (ω2τ2). (4.35) squaring and adding the squares together, we get ω4 2 + pω2 2 + q = 0, (4.36) with p = (a1 + c)2 − 2(b +d)− e2, and q = (b +d)2 − f 2. letting z2 = ω2 2, equation (4.36) becomes the following equation z22 + pz2 + q = 0. (4.37) on the other hand, we have q = (b +d)2 − f 2 = ( b +d + f )( b +d − f ) . using the case τ1 = τ2 = 0, we have b +d + f > 0, and the equation (4.24) implies that b −d < f, and by the hypotheses (h1), (h2), and (t1), we have b > 0, d < 0 and f > 0, then we have b +d < f, then, we have q < 0. consequently, the equation (4.37) has a unique solution positive z2,0 = −p+ √ (p2 − 4q) 2 . copyright © 2025 assa. adv syst sci appl (2025) stability analysis of a delayed epidemic model... 75 then the equation (4.36) has a unique positive solution ω2,0 = √ −p+ √ (p2 − 4q) 2 . on the other hand, the equations (4.34) and (4.35) imply that τ2,0 = 1 ω2,0 arccos [ ω2 2,0(f − e[a1 + c])− (b +d)f ω2 2,0e 2 + f 2 ]. lets show that iω2,0 is simple, consider the branche of characteristic roots λ(τ2) = u(τ2) + iv(τ2), of equation (4.33) bifurcating from iω2,0 at τ2 = τ2,0. by derivation (4.33) with respect to the delay τ2, we obtain dλ dτ2 ( 2λ+ a1 + c + e−λτ2(e − [λe + f ]τ2) ) = λ(λe + f )e−λτ2 . (4.38) if we suppose, by contradiction, that iω2,0 is not simple, the right hand side (4.38) gives iω2,0e + f = 0, and leads a contradiction with the fact that ω2,0 > 0, e > 0 and f > 0. next we need to guarantee the transversality condition of the hopf bifurcation theorem (see [23]). clearly, λ(τ2) = u(τ2) + iv(τ2), is a root of equation (4.33), if and only if u2 − v2 + (a1 + c)u+b +d = −e−uτ2 ( [ue + f ] cos (vτ2) + ve sin (vτ2) ) , (4.39) 2uv + v(a1 + c) = −e−uτ2 ( ve cos (vτ2)− (ue + f ) sin (vτ2) ) . (4.40) let u(τ2,0) and v(τ2,0), satisfying u(τ2,0) = 0, and v(τ2,0) = ω2,0. by differentiating equations (4.39) and (4.40) with respect to τ2 and then set τ2 = τ2,0. doing this, we get g3 du(τ2,0) dτ2 +g4 dv(τ2,0) dτ2 = h3, −g4 du(τ2,0) dτ2 +g3 dv(τ2,0) dτ2 = h4, where g3 = a1 + c + (e − τ2,0f ) cos (ω2,0τ2,0)− τ2,0ω2,0e sin (ω2,0τ2,0), g4 = −2ω2,0 + τ2,0ω2,0e cos (ω2,0τ2,0) + (e − τ2,0f ) sin (ω2,0τ2,0), h3 = ω2,0f (0) sin (ω2,0τ2,0)− ω2 2,0e cos (ω2,0τ2,0), and h4 = ω2 2,0e sin (ω2,0τ2,0) + ω2,0f (0) cos (ω2,0τ2,0). further, we have du(τ2,0) dτ2 = g3h3 −g4h4 g2 3 +g2 4 . therefore, according to the equations (4.34) and (4.35) we have du(τ2,0) dτ2 = ω2 2,0 √ p2 − 4q g2 3 +g2 4 . this completes the proof. copyright © 2025 assa. adv syst sci appl (2025) 76 a. bernoussi, k. hattaf, b. el boukari 4.5. case τ1 > 0 and τ2 > 0 theorem 4.5: if r0 > 1, g5h5 −g6h6 ̸= 0 and (4.24) holds, then there exists a positive τ ∗2,0 where system (1.3) undergoes hopf bifurcation at p ∗ when τ2 = τ ∗2,0. however, the steady state p ∗ is locally asymptotically stable when τ2 ∈ [0, τ ∗2,0) and unstable when τ2 > τ ∗2,0. here τ ∗2,0 is given by τ ∗2,0 = 1 ω∗ 2,0 arccos[ (f − ea1)(ω ∗ 2,0) 2 − fb + ω∗ 2,0(ed − cf ) sin (ω∗ 2,0τ1) (ω∗ 2,0e)2 + f 2 − (df + ec(ω∗ 2,0) 2) cos (ω∗ 2,0τ1) (ω∗ 2,0e)2 + f 2 ]. where ω∗ 2,0 is the positive root of the equation (4.43), and g5, g6, h5, h6 as defined in (4.48). proof equation (4.18) has a purely imaginary root iω∗ 2, with ω∗ 2 > 0, ∆(iω∗ 2, τ1, τ2) = 0, if and only if (ω∗ 2) 2 −b − ω∗ 2c sin (ω∗ 2τ1)−d cos (ω∗ 2τ1) = eω∗ 2 sin (ω ∗ 2τ2) + f cos (ω∗ 2τ2), (4.41) and a1ω ∗ 2 + ω∗ 2c cos (ω∗ 2τ1)−d sin (ω∗ 2τ1) = f sin (ω∗ 2τ2)− eω∗ 2 cos (ω ∗ 2τ2). (4.42) squaring and adding the squares together, we obtain (ω∗ 2) 4 + (ω∗ 2) 2(c2 + a2 1 − e2) +b2 +d2 − f 2 + 2 ( [−(ω∗ 2) 3c + ω∗ 2(bc − a1d)] sin (ω∗ 2τ1) + [(ω∗ 2) 2(a1c −d) +bd] cos (ω∗ 2τ1) ) = 0. (4.43) we define h(ω∗ 2) = (ω∗ 2) 4 + (ω∗ 2) 2(c2 + a2 1 − e2) +b2 +d2 − f 2 + 2 ( [−(ω∗ 2) 3c + ω∗ 2(bc − a1d)] sin (ω∗ 2τ1) + [(ω∗ 2) 2(a1c −d) +bd] cos (ω∗ 2τ1) ) . (4.44) on the other hand, we have −1 ≤ sin (ω∗ 2τ1) ≤ 1 and − 1 ≤ cos (ω∗ 2τ1) ≤ 1, and by the hypotheses (h1), (h3) and (t1), we have bd < 0, a1c < 0,−d > 0,−c > 0, bc < 0 and − a1d > 0. then g2(ω ∗ 2) ≤ h(ω∗ 2) ≤ g1(ω ∗ 2), where g1(ω ∗ 2) = (ω∗ 2) 4 − 2(ω∗ 2) 3c + (ω∗ 2) 2[c2 + a2 1 − e2 − 2a1c − 2d] − ω∗ 2(2bd + 2a1d) +b2 +d2 − f 2 − 2bd, copyright © 2025 assa. adv syst sci appl (2025) stability analysis of a delayed epidemic model... 77 and g2(ω ∗ 2) = (ω∗ 2) 4 + 2(ω∗ 2) 3c + (ω∗ 2) 2[c2 + a2 1 − e2 + 2a1c + 2d] + ω∗ 2(2bd + 2a1d) +b2 +d2 − f 2 + 2bd. we have lim ω∗ 2→+∞ g1(ω ∗ 2) = lim ω∗ 2→+∞ g2(ω ∗ 2) = +∞. then we have lim ω∗ 2→+∞ h(ω∗ 2) = +∞, and h(0) = b2 +d2 + 2bd − f 2 = (b +d)2 − f 2 = ( b +d + f )( b +d − f ) . by the case τ1 = τ2 = 0, we have (b +d + f ) > 0, and by the case (τ2 > 0 and τ1 = 0) if the equation (4.24) is satisfied then we have (b +d − f ) < 0. therefore, h(0) < 0. since h is continuous in [0,+∞), then the equation (4.43) has at least one positive root. we assume that equation (4.43) admits a finite family of solution ω∗ 2,i with i = 1, 2, ..., n. n ∈ n. by the equations (4.41) and (4.42), we have τ li = 1 ω∗ 2,i ( arccos [ (f − ea1)(ω ∗ 2,i) 2 − fb + ω∗ 2,i(ed − cf ) sin (ω∗ 2,iτ1) (ω∗ 2,0e)2 + f 2 − (df + ec(ω∗ 2,i) 2) cos (ω∗ 2,iτ1) (ω∗ 2,0e)2 + f 2 ] + 2πl ) , l = 0, 1, ....; i = 1, 2, ..., n. then ±iω∗ 2,i is a pair of purely imaginary roots of equations (4.18), with τ2 = τ li , l = 0, 1, ...; i = 1, 2, ..., n. clearly, lim l→∞ τ li = ∞, i = 1, 2, ..., n. thus, we can define τ ∗2,0 = τ l0i0 = min l=0,1...,i=1,2,...,n (τ li ), ω ∗ 2,0 = ω∗ 2,i0 . lets show that iω∗ 2,0 is simple, consider the branche of characteristic roots λ(τ2) = u1(τ2) + iv1(τ2), of equation (4.18) bifurcating from iω∗ 2,0 at τ2 = τ ∗2,0. by derivation (4.18) with respect to the delay τ2, we obtain dλ dτ2 ( 2λ+ a1 + [c − τ1(λc +d)]e−λτ1 + [e − τ2(λe + f )]e−λτ2 ) = λ[λe + f ]e−λτ2 . (4.45) if we suppose, by contradiction, that iω∗ 2,0, is not simple, the right hand side (4.45) gives iω∗ 2,0e + f = 0, copyright © 2025 assa. adv syst sci appl (2025) 78 a. bernoussi, k. hattaf, b. el boukari and leads a contradiction with the fact that ω∗ 2,0 > 0, e > 0 and f > 0. next we need to guarantee the transversality condition of the hopf bifurcation theorem (see [23]). clearly λ(τ2) = u1(τ2) + iv1(τ2), is a root of equation (4.18) if and only if u2 1 − v21 + a1u1 + e−u1τ1 ( cu1 cos (v1τ1) + cv1 sin (v1τ1) +d cos (v1τ1) ) = −e−u1τ2 ( eu1 cos (v1τ2) + ev1 sin (v1τ2) + f cos (v1τ2) ) , (4.46) and 2u1v1 + a1v1 + e−u1τ1 ( cv1 cos (v1τ1)− cu1 sin (v1τ1)−d sin (v1τ1) ) = −e−u1τ2 ( ev1 cos (v1τ2)− eu1 sin (v1τ2)− f sin (v1τ2) ) . (4.47) let u1(τ ∗ 2,0) and v1(τ ∗ 2,0), satisfying u1(τ ∗ 2,0) = 0, and v1(τ ∗ 2,0) = ω∗ 2,0. by differentiating equations (4.46) and (4.47) with respect to τ2, and then set τ2 = τ ∗2,0. doing this, we get g5 du1(τ ∗ 2,0) dτ2 +g6 dv1(τ ∗ 2,0) dτ2 = h5, −g6 du1(τ ∗ 2,0) dτ2 +g5 dv1(τ ∗ 2,0) dτ2 = h6, (4.48) where g5 = a1 − τ1cω∗ 2,0 sin (ω ∗ 2,0τ1) + (c − τ1d) cos (ω∗ 2,0τ1)− τ ∗2,0eω∗ 2,0 sin (ω ∗ 2,0τ ∗ 2,0) + (e − τ ∗2,0f ) cos (ω∗ 2,0τ ∗ 2,0), g6 = −2ω∗ 2,0 + (c − τ1d) sin (ω∗ 2,0τ1) + τ1cω∗ 2,0 cos (ω ∗ 2,0τ1) + (e − τ ∗2,0f ) sin (ω∗ 2,0τ ∗ 2,0) + τ ∗2,0eω∗ 2,0 cos (ω ∗ 2,0τ ∗ 2,0), h5 = fω∗ 2,0 sin (ω ∗ 2,0τ ∗ 2,0)− e(ω∗ 2,0) 2 cos (ω∗ 2,0τ ∗ 2,0), and h6 = e(ω∗ 2,0) 2 sin (ω∗ 2,0τ ∗ 2,0) + fω∗ 2,0 cos (ω ∗ 2,0τ ∗ 2,0). calculating du1(τ∗2,0) dτ2 , we get du1(τ ∗ 2,0) dτ2 = g5h5 −g6h6 g2 5 +g2 6 . this completes the proof. 5. numerical simulation in this section, we shall give some simulations to illustrate the previous results. for the following, let’s consider the saturated incidence rate function: f(s, i)i = βsi 1 + α1i , copyright © 2025 assa. adv syst sci appl (2025) stability analysis of a delayed epidemic model... 79 and the treatment function t (i) = ri. we take the parameters of the system (1.3) as follows: a = 8, β = 0.955, µ = 0.5, γ2 = 0.935, α = 0.03 γ1 = 0.3, r = 0.9, and α1 = 0.5. some parameters are for the spread of tuberculosis disease in turkey from 2005 to 2015 (see, [38]). the rest of the parameters are the hypothetical set of parameter values. 5.1. case τ1 = τ2 = 0 by applying theorem 4.1, the endemic equilibrium p ∗ is locally asymptotically stable when r0 > 1 and τ1 = τ2 = 0. figure 5.1 illustrates this result. fig. 5.1. dynamics of system (1.3) when τ1 = τ2 = 0 and r0 = 5.7336 > 1. 5.2. case τ1 > 0 and τ2 = 0 according to theorem 4.2, the endemic equilibrium p ∗ is locally asymptotically stable if r0 > 1, τ1 > 0 and τ2 = 0. figure 5.2 demonstrates this result. fig. 5.2. dynamics of system (1.3) when τ1 = 1.7678, τ2 = 0 and r0 = 5.7336 > 1. copyright © 2025 assa. adv syst sci appl (2025) 80 a. bernoussi, k. hattaf, b. el boukari 5.3. case τ1 = τ2 = τ from theorem 4.3, the following figures show that if the delay is below τ0 = 1.7678, the endemic equilibrium p ∗ is locally asymptotically stable (see, figure 5.3a). when the value of the delay τ increases we lose the stability of the endemic equilibrium (see, figure 5.3b) and p ∗ becomes unstable for τ > τ0 = 1.7678 (see, figure 5.3c), and vice versa when the delay is decreasing the model converges rapidly to p ∗. (a) dynamics of system (1.3) when τ1 = τ2 = 1.4178 and r0 = 5.7336 > 1. (b) for τ1 = τ2 = 1.7678, hopf bifurcation occurs and periodic solutions appear for model (1.3) with r0 = 5.7336 > 1. (c) for τ1 = τ2 = 2.1178, all solutions (s, i) of model (1.3) are unstable with r0 = 5.7336 > 1. fig. 5.3. case τ1 = τ2 = τ copyright © 2025 assa. adv syst sci appl (2025) stability analysis of a delayed epidemic model... 81 5.4. case τ1 = 0 and τ2 > 0 as in theorem 4.4, the following figures show that if the delay is below τ2,0 = 0.8357, the endemic equilibrium p ∗ is locally asymptotically stable (see, figure 5.4a). when the value of the delay τ2 increases we lose the stability of the endemic equilibrium (see, figure 5.4b) and p ∗ becomes unstable for τ2 > τ2,0 = 0.8357 (see, figure 5.4c), and vice versa when the delay is decreasing the model converges rapidly to p ∗. (a) dynamics of system (1.3) when τ1 = 0, τ2 = 0.7857 and r0 = 5.7336 > 1. (b) for τ1 = 0 and τ2 = 0.8357, hopf bifurcation occurs and periodic solutions appear for model (1.3) with r0 = 5.7336 > 1. (c) for τ1 = 0 and τ2 = 0.8651, all solutions (s, i) of model (1.3) are unstable with r0 = 5.7336 > 1. fig. 5.4. case τ1 = 0 and τ2 > 0 copyright © 2025 assa. adv syst sci appl (2025) 82 a. bernoussi, k. hattaf, b. el boukari 5.5. case τ1 > 0 and τ2 > 0 based on theorem 4.5, the following figures show that if the delay τ2 is below τ ∗2,0 = 0.9249 with τ1 fixed by taking τ1 = 3.6688, the endemic equilibrium p ∗ is locally asymptotically stable (see, figure 5.5a). when the value of the delay τ2 increases we lose the stability of the endemic equilibrium (see, figure 5.5b) and p ∗ becomes unstable for τ2 > τ ∗2,0 = 0.9249 (see ,figure 5.5c), and vice versa when the delay is decreasing the model converges rapidly to p ∗. (a) dynamics of system (1.3) when τ1 = 3.6688, τ2 = 0.8228 and r0 = 5.7336 > 1. (b) for τ1 = 3.6688 and τ2 = 0.9249, hopf bifurcation occurs and periodic solutions appear for model (1.3) with r0 = 5.7336 > 1. (c) for τ1 = 3.6688 and τ2 = 0.9976, all solutions (s, i) of model (1.3) are unstable with r0 = 5.7336 > 1. fig. 5.5. case τ1 > 0 and τ2 > 0 6. concluding remarks in this work, we have proposed and analyzed a delayed sir model with generalized incidence and treatment functions. our analysis proved that the two delays τ1 and τ2 have a very big influence on the stability of the equilibrium points. in fact, • when r0 ≤ 1, we have: if τ1 ≥ τ2, then the disease-free equilibrium is globally asymptotically stable for all rc, this shows that the disease disappears. this biologically mean that when the latent period is greater than the healing period and the basic reproduction number is copyright © 2025 assa. adv syst sci appl (2025) stability analysis of a delayed epidemic model... 83 less than or equal to one, the infected individuals cured before becoming infectious and the disease will disappear. if rc ≤ 1, then the disease-free equilibrium is globally asymptotically stable independently of delays. if rc > 1 and τ1 is held fixed, then there exists a positive constant τ ∗22,0 such that when τ2 ∈ [0, τ ∗22,0), the disease-free equilibrium is locally asymptotically stable. when τ2 = τ ∗22,0 a hopf bifurcation occurs and when τ2 > τ ∗22,0 the disease-free equilibrium is unstable. • when r0 > 1, the disease-free equilibrium is unstable and the proposed model admits an endemic equilibrium p ∗. in this case, we have: if τ2 = 0 and τ1 ≥ 0, then the endemic equilibrium p ∗ is locally asymptotically stable. if τ2 > 0, then there exists a positive constant τ̃ such that when τ2 ∈ [0, τ̃), the endemic equilibrium is locally asymptotically stable. when τ2 = τ̃ a hopf bifurcation occurs and the host populations and the disease coexists within an oscillatory mode, and when τ2 > τ̃ the endemic equilibrium is unstable and the disease persists. from the aforementioned, we conclude that the longer the recovery period is for the infected individuals, the higher the spreading risk becomes and vice versa. acknowledgments the authors would like to thank the editor and the anonymous referees for their valuable comments and suggestions that have led to improvement of the quality of this work. references 1. abta, a., kaddar, a. & alaoui, h. t. 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(2003). stability of periodic solutions for an sis model with pulse vaccination, mathematical and computer modelling. 38, 299-308. copyright © 2025 assa. adv syst sci appl (2025) introduction mathematical analysis global stability of the disease-free equilibrium stability and hopf bifurcation of the endemic equilibrium case 1 = 2 = 0 case 1 > 0 and 2 = 0 case 1 = 2 = case 2 > 0 and 1 = 0 case 1 > 0 and 2 > 0 numerical simulation case 1 = 2 = 0 case 1 > 0 and 2 = 0 case 1 = 2 = case 1 = 0 and 2 > 0 case 1 > 0 and 2 > 0 concluding remarks adv syst sci appl 2020; 02:32–43 published online at https://ijassa.ipu.ru. original russian text © v. v. zakharov, a. v. mugayskikh, 2018, published in upravlenie bol’shimi sistemami / large-scale systems control, 2018, no. 73, pp. 108-133. dynamic adaptation of genetic algorithm for solving routing problems on large scale systems viktor zakharov1, alexander mugayskikh1* 1saint petersburg state university, saint petersburg, russia abstract: this paper is devoted to the implementation of the dynamic adaptation procedure for the genetic algorithm used for solving large-scale travelling salesman problem. this procedure serves to obtain more profitable solutions by a fixed operating time. in order to evaluate effectiveness of new approach computational experiments were performed on well-known problem instances from tsplib library. as a result, generated solutions reduce the length of the routing plans in considered problem instances compare to classical genetic heuristics. by that, we show how to use the property of time inconsistency of heuristics to get better solutions. new criteria for estimating the efficiency of heuristics algorithms called experimental level of time consistency is introduced. keywords: time consistency, genetic algorithm, vehicle routing problem 1. introduction one of the most challenging goals in the field of transportation logistics is cost optimization, which allows to increase company’s benefit for the same amount of resources used. for transportation companies, reduced costs can be achieved by constructing effective route plans for the vehicles. a significant effect of cost reduction can be shown on large transport networks. for this reason, researches pay much attention to models and methods which generate less expensive solutions in the routing problems. probably, the most important objective of mathematical modelling in transportation logistics is the vehicle routing problem. a classical example of a vehicle routing problem is the travelling salesman problem, which will be examined in details. this article is motivated by the problem of time inconsistency (or dynamic instability) of heuristic algorithms solving vehicle routing problems and developing methods to avoid this: to increase the level of dynamic stability and efficiency of the algorithms with experiments hold on a genetic algorithm. the article is structured as follows. after introduction, it includes seven sections. the first section is devoted to the description of general model of the travelling salesman problem (tsp) and contains the review of methods solving it. the second section provides the basic concepts and terms of the genetic algorithm used in the text. section 3 includes a description of the basic scheme of the genetic algorithm and one of the most effective tools for constructing a solution — the greffenstett’s crossover. the fourth section discusses the problem of dynamic instability of heuristics in solving routing problems and a new criterion is suggested — the experimental level of time inconsistency. in section 5 we introduce the description of dynamic adaptation procedure which can improve basic solutions of heuristic algorithms. section 6 presents the results of numerical experiments to evaluate ∗corresponding author: alexander.mugaiskih@gmail.com dynamic adaptation of genetic algorithm for solving routing problems 33 the level of time inconsistency of the genetic algorithm and a comparative analysis of the results obtained by the genetic heuristics and its dynamic adapdation procedure. comparative analysis was hold on a set of problem instances from the tsplib library, widely known among specialists. in the seventh section we place brief conclusions and plans for further research. 2. travelling salesman problem: formulation and methods 2.1. problem formulation the travelling salesman problem consists in finding the shortest trip between a set of vertices, visiting each vertice exactly once, starting and finishing at a specified vertice. the tsp can be represented as a graph g = (v,e), where set of vertices v = {v1, v2, . . . , vn} stands for cities from the problem instance and set of edges e = {(vi, vj) : vi, vj ∈ v, i 6= j} represents paths between the cities. the weight of an edge (vi, vj) is the distance or travel cost between vertices vi and vj . travelling salesman problem can be defined as an problem of finding a hamiltonian cycle in graph g with the least weight. let i = {1, . . . , n} be the set of indices of vertices from the problem instance. the problem objective function f is total length of the route that includes all vertices from the problem instance. suppose xij is a binary variable which is equal to one if route passes the arc (i, j), and zero otherwise. length of the route between two vertices i and j is defined as euclidean distance cij . then the problem can be defined in the form of linear programming proposed by [9, 27, 29]: f = ∑ i∈i ∑ j 6=i,j∈j cijzij → min, subject to: 0 ≤ zij ≤ 1, zij ∈ z∑ j∈i,j 6=i zij = 1,∀i ∈ i, (2.1) ∑ i∈i,i 6=j zij = 1,∀j ∈ i, (2.2) ∑ i∈s,j∈s zij = |s| − 1 (s ⊂ v, |s| > 1) (2.3) equalities (2.1) – (2.2) enforce that every vertice is visited exactly once. the last constraints guarantee the absence of two or more disjoined parts in the final route. the travelling salesman problem takes a central role in combinatorial optimization. it has a wide range of practical application in various fields: transport logistics [1], robotics [6, 8], job shop scheduling. among its applications one can mention constructing routing plans for a fleet of vehicles of logistic company; performing video surveillance by wireless robots [33]; robot-assisted sensor network deployment and data collection [37]; generating assembly sequences in an autonomous multi-robot coordinated furniture system [26]; planning routes in controlled airspace [12]. despite the seeming simplicity of the mathematical model of the travelling salesman problem, it still receives close attention from specialists while considering the problem in new formulations and under various restrictions and constraints. copyright © 2020 assa. adv syst sci appl (2020) 34 v. v. zakharov, a. v. mugayskikh 2.2. algorithms and methods all of approaches solving travelling salesman problem can be assigned either to exact or heuristics algorithms. brute-force search, branch and bound algorithm, branch and cut methods and dynamic programming are among the exact algorithms. the branch-and-bound method and the branch-and-cut method represent the modification of exhaustive search. the main idea is to form the rooted tree, check the bounding criterion bounding and discard the current branch if it is not possible to produce a better solution than the best one found so far by the algorithm. some variants of this approach can solve the travelling salesman problem with several hundred vertices, however, they require high computation power of the computers used. the idea of dynamic programming, proposed by r. bellman [3], m. held, and r. karp [19], can be also applied to the tsp problem and consists in a many-step decision-making process, at each step of which it is necessary to determine the bellman function and find the best route for the left vertices. the number of operations required to calculate the final route increases exponentially with of size of problem instance. due to the np-complexity of the tsp problem, exact methods cannot always be effectively applied to large-scale problems. for this reason, heuristic methods are used that generate solutions that are close to optimal, but in an acceptable time compared to exact algorithms. an experimental analysis of heuristic algorithms for solving the tsp problem and its subclasses was carried out in [17]. heuristic algorithms can be divided into 2 classes according to the method of generating the final route: constructive and iterative heuristics. tour construction heuristics obtain only one solution without its further improvement [24]. for example: nearest neighbour algorithm [23], greedy algorithm [18], insertion heuristics [14] and christofides approximation algorithm presented in [7]. the algorithms above sequentially build a feasible solution by adding vertices to the final route until a complete route is formed. the part of the solution built to the current moment will remain unchanged until the end of the algorithm. the length of generated solutions is about 10–15% away from the optimal solution. more useful for getting shorter routes are tour improvement heuristics. there methods begin to work with an already prepared route generated by one of the constructive methods, iteratively improving the solution at each step. there are several ways to do this: 2-opt swap and 3-opt swap, or perform as a part of local search algorithms and the lin-kernigan heuristics [20, 21]. metaheuristics as a higher-level procedure also refer to tour improvement heuristics [4, 13]. these are quite general iterative procedures using randomization and self-learning elements, intensification and diversification of the search, adaptive control mechanisms, as well as constructive heuristics and local search methods. metaheuristics work either with single candidate or with whole population of feasible solutions. single solution approaches include variable neighborhood search, tabu search, simulated annealing [15] and others. population-based approaches are genetic algorithms, ant colony optimization [10], greedy randomized adaptive search procedure and some others [36]. 3. terminology and definitions further in the article, the following terms and definitions are used. heuristic algorithm is a problem-solving method, which solutions may not necessarily be optimal, but about which it is known that it gives a fairly good solution in most cases. genetic algorithm is a heuristic algorithm for solving optimization problems by randomly selecting, combining and varying the desired parameters using mechanisms similar to natural selection occurred in nature, generating different solutions at each launch due to the randomization mechanisms used in it. individual is a feasible solution to a problem instance in a genetic algorithm. copyright © 2020 assa. adv syst sci appl (2020) dynamic adaptation of genetic algorithm for solving routing problems 35 population is the final set of individuals in a problem instance, which the genetic algorithm will work with. chromosome is a set of genes that describe an individual. in the travelling salesman problem, genes in the chromosome correspond to the vertices of the test problem. chromosome contains the order in which the vertices are passed through in this solution. gene is an atomic element of a chromosome. crossover is the main stage of the genetic algorithm, in which two individuals of the same population take part. by combining and inheriting the characteristics of both parent individuals (parents), new individuals (children) are generated, and the quality of new individual is improved as well as the quality of whole offspring population. mutation is one of the stages of the genetic algorithm, in which one individual from the population takes part. mutation is carried out with the aim of restoring the individuals dropped out of the population and forming genes that were not present in the original population. selection is the final stage of the genetic algorithm aims to form a new population of individuals by selecting the most adapted individuals for their inclusion in the further process of evolution. 4. genetic algorithm with greffenstte’s crossover the genetic algorithm is a metaheuristic algorithm, which is based on operations of natural selection that occurs in nature. it was first introduced in 1975 by john holland as a subcategory of evolutionary algorithms [22]. without detailing the description of all stages of the genetic algorithm, we briefly give a pseudocode of the classical genetic algorithm, which consists of the following steps: 1. initialization of stating population, s individuals. 2. while stop criterion is not met, do: 3. several crossover operators (with 2 two random individuals) 4. several mutation operators (random individual) 5. evaluation of fitness function and selection applied to the travelling salesman problem, each individual in population stands for one feasible solution of a problem instance and represents the order in which the vertices are visited. the fittest individual in the last population is accepted as the final solution. the genetic algorithm is a genetal heuristic procedure that can be implemented in a software package for solving different flavors of routing problems, such as tsptw (travelling salesman problem with time windows), irp (inventory routing problem), vrp (vehicle routing problem), mdvrp (multi-depot vehicle routing problem), pdp (pickup and delivery problem) and others. in addition, genetic heuristics is capable of solving both the original problem and its sub-problem, which is necessary condition for applying dynamic adaptation procedures of this algorithm. as it is known, the main operator of the genetic algorithm, which will be discussed in details in section 5, is crossover. the crossover methods proposed in the literature are valid for the travelling salesman problem and give generally acceptable results, but they do not use information about the distance between the vertices of the tsp problem instances. john greffenstett solved this problem by proposing in his work [16] a new type of crossover that uses information about the distance between vertices. as applied to tsp intances with a symmetric distance matrix of 70 and 100 vertices of the tsplib test library [32], the use of the greffenstett’s heuristic at the stage of crossover helped to get shorter routes compared to partially mapped crossover, cycle crossover, edge recombination crossover, two-point crossover, uniform order-based crossover, shuffle crossover and sub-tour exchange crossover. for more information on these types of crossovers, see [25]. for a problem instance with 100 vertices, greffenstett’s crossover copyright © 2020 assa. adv syst sci appl (2020) 36 v. v. zakharov, a. v. mugayskikh showed the best result among other kinds of crossovers and differs from optimal solution by 11.9% [25] in work [31] authors also compared 8 types of crossover to solve the vrp problem, generalization of the tsp problem on the number of vehicles (2 and more). the following crossovers were described: order crossover, partially mapped crossover, edge recombination crossover, cycle crossover, alternating edges crossover, greffenstett’s crossover and its two modifications [31]. the best results among the mentioned algorithms on the vrp test problems of the christofides standard library [11] were shown by the greffenstett’s heuristic crossover and its two modifications. thus, the proposed kind of crossover outperforms its alternatives in generating shorter routes, and it was implemented in the genetic algorithm for solving problem instances for further research. the results are presented in the following sections of the article. the following scheme presents the classic greffenstett crossover operation: 1. randomly chosen vertice v is placed in the first gene of the children chromosome 2. while complete route is not formed, do: 3. compare the length of two edges connected with v, choose the shortest among them and denote the corresponding vertice as vmin 4. if vmin is already in children’s chromosome, then vmin will be the randomly chosen vertice in the set of non-used in the current route 5. place vertice vmin in the next gene of children’s individual v = vmin. 5. time inconsistency of heuristic algorithms in work [5], the authors note that the process of finding the optimal solution in genetic algorithms is guided exclusively by the obtained values of the objective function at the previous points in the solution space. and assumptions about such properties of the objective function such as convexity, differentiability, and bellman’s optimality property optimality are not used. formulated in [2] principle of optimality claims that at any period of time restriction of multi-period optimal routing plan found by exact algorithm for any remaining time horizon appears to be optimal in current sub-problem. it’s worth noting, restriction of solution obtained by tour improvement heuristic algorithm could be not optimal in a current subproblem for at least one of the periods. constructive heuristics almost always generate time consistent routes, however, they do not provide a high level of efficiency compared to iterative ones. using the principles of dynamic stability described in [30], we introduce the following notation. let us consider set of problem instances p for the travelling salesman problem. suppose that for each instance p ∈ p we can obtain a set of different solutions s(p) generated by heuristics. each solution s(p) corresponds to an order in which vertices of the problem instance are visited in a route. we consider the order of the vertices of one route and divide it into t parts so that the number of vertices that were visited in the first t parts of the route is calculated by the formula n(t, s(p)) = bn0t/t c, where n0 is the size of the problem instance p, and t — parameter defined before the experiment. thus, each part of the route (except the last) contains the same number of vertices. by the period t, where t = 0, 1, . . . , t − 1, we mean the time interval in the route that corresponds to the part t of the original solution s(p). define s+(t, p) as remaining consequence of nodes after period t and s−(t, p) stands for such part of the route that includes nodes visited during periods τ = 0, 1, . . . , t. thus each solution can be represented as s(p) = s−(t, p) ∪ s+(t, p). copyright © 2020 assa. adv syst sci appl (2020) dynamic adaptation of genetic algorithm for solving routing problems 37 for t = 1, . . . , t − 1 we consider the sub-problem p(s−(t, p)). this sub-problem differs from initial problem instance. firstly, nodes already visited before period t are excluded for visit in next periods. secondly, new depot is located at the last node of part s−(t, p) that belongs to the route s(p). then we obtain solution s(p(s−(t, p))) for the sub-problem p(s−(t, p)) using the same heuristics. definition 5.1: solution s(p) (routing plan), obtained by heuristic algorithm h , is time consistent according to algorithm h , if for each t = 1, . . . , t − 1 and each s(p(s−(t, p))) generated by h , the following inequality holds: f(s+(t, p)) 6 f(s(p(s−(t, p)))), (5.4) where f is distance value for corresponding route. definition 5.2: solution s(p) (routing plan), obtained by heuristic algorithm h , is time inconsistent according to algorithm h , if exists t ′ and s(p(s−(t ′ , p))) generated by h , and the following inequality holds: f(s+(t, p)) > f(s(p(s−(t ′ , p)))). (5.5) the property of time consistency of the solution in the tsp shows that moving according to the initial route plan, no shorter solution will be found than the current one in the subproblem generated by the method described above. and vice versa, time inconsistency of the solution means that at least in one of the periods a routing plan can be generated that will more profitable than the current one. we introduce the procedure for calculating the level of time consistency of the heuristic algorithm h for a certain class of vehicle routing problems. 1. consider set p of problem instances of certain class in routing problems. 2. generate set n of various solutions for every p ∈ p by heuristic algorithm h . obtained solutions (routing plans) will be different from each other due to the randomization used in heuristic operators: crossover, mutation, selection. 3. perform m runs for each solution s(p) ∈ s(p), where each run includes checking whether solution is time consistent or not. parameter m aims to average the obtained results of calculation, assume m = 5. starting with t = 1, formulate sub-problem p(s−(t, p)) and find its solution by algorithm h . check initial solution to be time consistent after first period. if inequality (5.4) holds, move to the second period. if not, mark the period at which property of time consistency is violated and start the next run. let b(s, t) be the number of runs, in which time consistency is violated for the solution s(p) for period t. if s is optimal, then regarding bellman’s principle of optimality we would have the following quality: t−1∑ t=1 b(s, t) = 0. (5.6) definition 5.3: experimental level of time consistency conh for heuristics h to be the value calculated as follows conh = 1− 1 m |p | ∑ p∈p 1 |n | ∑ s(p)∈n t−1∑ t b(s(p), t). (5.7) the experimental level of time consistency can vary for problem instances but one can note that 0 ≤ conh ≤ 1. high value of experimental level of time consistency for heuristics forms expectation that considered heuristics generates solutions which tend to stay optimal during the realization of its initial routing plan. copyright © 2020 assa. adv syst sci appl (2020) 38 v. v. zakharov, a. v. mugayskikh 6. dynamic adaptation of genetic algorithm the concept of dynamic stability has been widely studied in recent years for problems of game theory. using the main idea of the algorithm proposed in [40], we suggest dynamic adaptation procedure for the class of the travelling salesman problem and genetic algorithm solving it. at the initial stage, we generate a set of n different solutions of the problem instance by the genetic algorithm. from the set n we choose the best solution, with the minimum value of the objective function. for it, we will obtain m experiments on checking if property of time consistency holds. if this property is violated, starting from the period t, i.e. after the period t, the better solution will be found in the current subproblem p(s−(t, p)). to reduce the total length of the routing plan through all vertices, the current routing plan should be changed according to the new solution obtained in the considered subproblem. the general scheme of dynamic adaptation is as follows [28]. 1. obtain set n of different solutions generated by heuristics h for p ∈ p. 2. define s1(p) = arg min s(p)∈n f(s(p)). 3. from t = 1 to t − 1 do 4. formulate sub-problem p(s−t (t, p)), obtain set n of solution. 5. define s∗t+1(p) = arg min s(p(s−t (t,p)))∈n f(s(p(s−t (t, p)))). 6. check if the property of time consistency is valid for st(p). 7. if f(s+t (t, p)) > f(s∗t+1(p)) , then routing plan should be changed to st+1(p) = s−t (t, p) ∪ s∗t+1(p). 7. computational results 7.1. experimental level of time consistency of the genetic algorithm with greffenstett’s crossover for solving the tsp to conduct the experiment, five test instances from the standard tsplib library were considered and a genetic algorithm with a greffenstett’s heuristic crossover to solve them was applied. the following set of parameters was used in the computational experiment: |p | = 5,m = 5, t = 5, |n | = 20. for each of the test problems, 20 different solutions were generated. the number of runs for one solution s(p) equaled 5. the original route was divided into five periods. the period number t was marked, after which the solution s(p) lost the time consistency property. these values correspond to the columns b(s, t) in table 7.1. table 7.1. the experimental level of time consistency of the ga for the tsp problem instance number of runs b(s, t) number of time consistent runst = 1 t = 2 t = 3 t = 4 att48 100 31 16 26 8 19 eil51 100 64 14 11 3 8 berlin52 100 19 27 4 4 46 st70 100 16 22 4 1 57 eil101 100 15 28 3 2 52 sum 500 145 107 48 18 182 the average level of experimental level of time consistency of the genetic algorithm for solving the tsp equals 0, 364. conga = 0,364. copyright © 2020 assa. adv syst sci appl (2020) dynamic adaptation of genetic algorithm for solving routing problems 39 this value is quite low: only a third of the solutions generated at the initial stage keep the optimality property in the process of their implementation. this means that there are other routes that can be obtained dynamically using the same heuristic algorithm, the values of the objective function of which will be less than in current solutions. we calculate the level of time consistency for solutions obtained using the dynamic adaptation procedure. we will carry out the experiment according to the same scheme and with the parameters determined earlier. the experimental results are shown in table 7.2. table 7.2. the experimental level of time consistency of the dynamic adaptation for the ga for the tsp problem instance number of runs b(s, t) number of time consist. runst = 1 t = 2 t = 3 t = 4 att48 100 24 5 11 1 59 eil51 100 47 11 7 5 30 berlin52 100 2 12 5 3 78 st70 100 1 7 5 5 82 eil101 100 1 6 4 4 85 sum 500 75 41 32 18 334 the average value of the experimental level of time consistency of a dynamically adapted genetic algorithm for solving the tsp problem is 0.668, which is almost two times higher than for the genetic algorithm before adaptation. we consider the dependence of the number of time-consistent solutions obtained during the experiment in the first t periods on period t. the total number of runs performed is m |n ||p |. function (8) is introduced which states for the number of time-consistent solutions after the completion of each period. c(s, t) =m |n ||p | − t∑ k=0 b(s, k). (7.8) fig. 7.1. number of time consistent solution after period t for the genetic algorithm and its dynamic adaptation in figure 7.1, the continuous line corresponds to the values of the function c(s, t) during calculation the experimental level of the time consistency of the genetic algorithm with the greffenstett’s heuristic crossover, and the dashed line corresponds to the values of this function for the dynamic adaptation of this heuristics. copyright © 2020 assa. adv syst sci appl (2020) 40 v. v. zakharov, a. v. mugayskikh 7.2. comparison of route plans obtained by genetic heuristics and its dynamic adaptation procedute using the example of the tsp problem and the genetic algorithm, it was shown that the dynamic adaptation procedure allows to obtain solutions with a higher level of time consistency. we will check how the length of the route plans changes with this procedure. table 7.3 presents a comparison of the lengths of routes (solutions) generated by the genetic algorithm and its dynamic adaptation. the average values of the objective function are given for 500 runs of algorithms. the stopping criterion for each start of the genetic algorithm with the greffenstett’s heuristic crossover was reaching 100 generations of population. the dynamic adaptation procedure was carried out according to the scheme given in algorithm 3, with the following set of parameters: t = 5, |n | = 20. we introduce the following notation: the value lga(p) equals the length of the route obtained by the genetic algorithm for the test problem p, ldaga(p) is the length of the route obtained by using a dynamically adapted genetic algorithm. the last two columns of table 3 show the values of the objective function corresponding to the effective start. an effective run of the dynamic adaptation procedure for the problem instance p is a run for which the value of lga(p)− ldaga(p) is the maximum in the experiment. table 7.3. comparison of length of routing plans generated by genetics and its dynamic adaptation problem instance mean length value effective run genetic heuristics dynamtic adaptation genetic heuristics dynamtic adaptation att48 12270,72 11723,79 11657 10928 eil51 493,72 464,64 493 448 berlin52 8836,31 8257,36 8570 7893 st70 803,25 758,61 765 708 eil101 779,30 720,338 725 665 table 3 shows that for each of the problem instances considered, the average length of the generated solutions by the dynamic adaptation algorithm is less than the classical genetic algorithm with the greffenstett’s heuristic crossover. the percent of improvement we calculate by the equation below: k = lga(p)− ldaga(p) lga(p) 100%. (7.9) the average percentage improvement is 6, 01%. the maximum improvement value was obtained for the problem instance eil51 and amounts to 9, 12%. 7.3. time analysis for applying dynamic adaptation procedure the procedure of dynamic adaption of a genetic algorithm requires additional runs on each period and leads to increasing the time to get one routing plan. we will carry out the following experiment. let tdyn be the average duration (in seconds) of the dynamic adaptation algorithm to generate a route. let us limit the runtime of classical genetic heuristics to tdyn. for each problem instance, we will make 100 launches of the genetic algorithm and its dynamic adaptation and compare the results of the two methods with a fixed duration of their work equal to tdyn. as can be clearly seen from table 7.4, with a fixed operating time, the method of dynamic adaptation of genetic heuristics is able to generate better solutions compared to the genetic algorithm with the greffenstett’s heuristic crossover. this remark should be taken into account in large-scale problem instances, when the generation time of one route can take copyright © 2020 assa. adv syst sci appl (2020) dynamic adaptation of genetic algorithm for solving routing problems 41 table 7.4. comparison of length of routing plans obtained by fixed runtime tdyn problem instance mean length value tdyngenetic alrorithm dynamic adaptation att48 12273,51 11723,79 120,1 eil51 492,44 464,64 165,5 berlin52 8820, 25 8257,36 206,35 st70 807,37 758,61 1300,4 eil101 786,54 720,338 3403,7 tens of minutes. the experiment also showed, in the case of a fixed runtime for generating the routing plan, it is more profitable to use the dynamic adaptation procedure instead of the classical heuristic, distributing the time between the corresponding periods of the algorithm. 8. conclusion in this article, a genetic algorithm with a greffenstett’s heuristic crossover solving the tsp problem was described and implemented in java programming language. the level of time consistency of the solutions generated by this heuristic is estimated. the calculations for a set of problem problems from the tsplib library demonstrated that the genetic algorithm with the greffenstett’s heuristic crossover generates routes with a low level of time consistency. to generate solutions with a higher level of time solvency, a dynamic adaptation procedure for a genetic algorithm with a greffenstett’s heuristic crossover is proposed. as a result, the level of time consistency of the new solutions increased by two times. at the same time, the average value of the route length decreased by 6.01%. thus, the practical application of the proposed dynamic approach allows to generate shorter routes. it was also shown that with a fixed restriction on the operating time, the procedure for dynamically adapting the genetic algorithm with the greffenstett’s crossover leads to obtain shorter routes. it is worth noting that the proposed method of dynamic adaptation can be applied to other heuristic algorithms for solving the tsp problem. the usage of the this method to solve vehicle routing problems such as tsptw, irp, mdvrp, pdp is possible if the heuristic algorithm is able to solve both the problem itself and its subproblems [34, 35]. references 1. anbuudayasankar s. p., ganesh k., mohapatra s. 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(2015). stable cooperation in dynamic vehicle routing problems, automation and remote control, springer, 76(5), 935–943. copyright © 2020 assa. adv syst sci appl (2020) introduction travelling salesman problem: formulation and methods problem formulation algorithms and methods terminology and definitions genetic algorithm with greffenstte's crossover time inconsistency of heuristic algorithms dynamic adaptation of genetic algorithm computational results experimental level of time consistency of the genetic algorithm with greffenstett's crossover for solving the tsp comparison of route plans obtained by genetic heuristics and its dynamic adaptation procedute time analysis for applying dynamic adaptation procedure conclusion adv syst sci appl 2022; 04:31–50 published online at https://ijassa.ipu.ru. the influence of image morphology on neural network-based segmentation results efim v. lavrukhin1,2, kirill m. gerke 2 1 computational mathematics and cybernetics, lomonosov moscow state university, moscow, russia 2schmidt institute of physics of the earth of russian academy of sciences, moscow, russia abstract: in this article, we have provided initial results of evaluating the influence of training data morphology on convolutional neural networks segmentation quality. to do this, we selected four different porous media 3d images obtained with the help of x-ray microtomography (xct) and segmented them using a local thresholding ”converging active contours” algorithm. with the help of synthetic tomography algorithm, we created synthetic xct image based on these segmentations. as a result, we obtained true xct – ground-truth pairs for considered porous media samples. then with the help of stochastic reconstructions, we created a set of samples with varying degrees of morphological similarity to the original porous media samples. in order to check the effect of morphological properties on the quality of segmentation, we trained a u-net model with resnet50 encoder on pairs of synthetic xct – ground-truth data, and assessed the segmentation quality of synthetic xct images as obtained based on stochastically reconstructed images. based on the metrics, we concluded that the quality of segmentation is more influenced by the morphological differences of the original porous media samples than by difference from the generated stochastic reconstructions. we discussed possible ways to improve the future experiments design in order to finally resolve the issue of training for xct image segmentation with neural networks. keywords: porous medium, image processing, convolutional neural network, image segmentation, synthetic tomography, correlation functions 1. introduction it is not a secret that porous media structure, i.e., spatial distribution of forming materials (or phases), defines its major physical properties [2, 11, 58, 64]. in case structure information is known up to a relevant spatial resolution, a great number of such properties, including flow and transport characteristics, can be effectively obtained using so-called pore-scale modeling techniques that include direct voxel based [29,43,55], meshing-based [10] or computationally efficient pore-network modeling approaches [22,50,54]. computed or/and upscaled from the first principles, flow properties are of primary importance to parameterize darcian scale and to explain or design flow mechanics relevant for petroleum engineering [12, 30], hydrology [6, 27], food engineering [21] and numerous other applications. with recent progress in 3d imaging [19,68], stochastic reconstructions based on limited input data [3,17,35,39,63,71], or their combination [47,48], obtaining detailed structural information for a given porous media sample is not an issue. moreover, recently proposed multiscale image fusion techniques [24] provide a solution for complex/hierarchical porous media samples such as complex rocks [19, 25] and soils [40]. regardless of the methodology employed, some input experimentally ∗corresponding author: e.lavrukhin@digital-core.ru 32 efim v. lavrukhin, vasily postnicov, kirill m. gerke obtained structural information is necessary, and is mainly obtained by scanning electron microscopy (sem), focused ion beam-sem imaging, or x-ray computed tomography (xct) [25]. this information comes in the form of grayscale images where pixels/voxels have the intensity proportional to some local physical property, e.g., for xct it depends on local x-ray absorption [68]. yet, as we previously defined structure as spatial distribution of constituting phases, the crucial step necessary to perform pore-scale simulations is the transition from local attenuation coefficients to local material(s) distribution. in the simplest case that is needed, for instance, to solve stokes equation to simulate single phase flow and compute permeability [29,43,55], we need to distinguish two porous media components — pores and solids. such a transition from local physical properties to phases (or materials) is called image segmentation, and, in case of two phases (e.g., pores and solids), segmentation is also referred to as binarization. the biggest problem of any segmentation procedure is that it is never strictly rigorous and always contain some degree of uncertainty. apart from numerous experimental and numerical problems associated with image acquisition, the main reason for this issue to appear is due to the limited imaging resolution – partial volume effects [19, 20, 28, 68], i.e., precise segmentation would be possible only for images with nearly infinite resolution. in other words, each pixel/voxel usually contain a mixture of different phases, but segmentation requires it to be affiliated with a relevant phase, ideally with the material which dominates the volume under study. utilization of grayscale values to evaluate local abundance of a given material within each pixel/voxel is possible only for mono-component (consisting of a single material, which is rarely the case in relevant applications) or requires complex dual energy scanning with subsequent inverse modelling [70]. the primary target of the segmentation procedure in this context is to provide as accurate as possible approximation of the spatial distribution of phases under given image resolution conditions. numerous segmentation techniques are available for this task and can be roughly divided into four categories: 1. manual segmentation; 2. global thresholding; 3. local thresholding methods; 4. other approaches including machine learning and unsupervised learning algorithms. manual approach implies the choice of a single threshold grayscale value between two phases to be segmented by an operator and still a popular way to go. global segmentation approach is basically the same as manual one, but the threshold is chosen with the help of some automatic computational technique based on grayscale histogram (e.g., by minimizing intra-class variance in the method by otsu [53]). local thresholding methods identify two thresholds for each two phases to be segmented out. these thresholds represent trustworthy boundaries which define, for example, 100% certainty that all voxels below the lower threshold are pores, while all voxels above the upper threshold are pores. the pixels/voxels in between these two bounds are treated and classified according to some algorithms: statistically kriged in the popular indicator kriging method [52], grown from seeds (100% certain areas) using region growth [31] or converging active contours [61]. these thresholds are chosen either manually or automatically [60]. here we mentioned only some popular methods, more detailed list of techniques applied in porous media research can be found in the review of schlüter et al. [59]. machine learning methods fall into supervised and unsupervised bins, in the former case some input training information is provided to the algorithm, whilst in the latter case the algorithm makes all decisions by itself. despite the abundance of segmentation techniques and some auxiliary image processing methods such as filtering, none of the existing methodologies can pretend to provide universal and adequately accurate results. it is widely acknowledged that manual and global thresholding techniques are inferior to the local thresholding methods [34], which, copyright © 2022 assa. adv syst sci appl (2022) the influence of image morphology... 33 in turn, suffer from necessity to choose confidence bound threshold values. automatic choice of thresholds [59, 60] does not necessarily provide adequate results, as was observed in numerous studies [40]. an interesting study by baveye et al. [7] concluded that human operator performs much better than any automatic thresholding algorithm. whether it is true or not is still an open question, as the majority of segmentation techniques tested by baveye et al. [7] are not necessarily suitable for porous media applications. the results of the unsupervised segmentation techniques are not robust in the sense of the number of phases [33] and accuracy [15, 16, 42]. supervised segmentation methodologies are very limited in xct porous media applications [38, 46, 65], but their recent usage in numerous other fields such as medical imaging [9], neuroscience [18], and satellite image segmentation for urban structures [4] and flood identification [51] motivated this study. moreover, testing and verification of all existing methodologies in the area of porous media research is seriously hampered by the absence of the true data — precise segmentation results for different samples. artificial (synthetic) xct images produced by rescaling, noise and artifact addition (e.g., wang et al. [67] or schlüter et al. [59]) do not possess all necessary complexity and represent real xct only to some extent. at this point we can pose a relevant question — what the ideal segmentation methodology would look like? any expert working with 3d xct images or porous media would generally agree with the following definition: with the help of the ideal segmentation technique one should be able to take any xct image and obtain accurate segmentation by using only a stack of images as input data, i.e., without any operator input or (fitting) parameters. from our above-mentioned analysis of current segmentation technology, we can confidently conclude that none of existing methods provide a solid performance according to that definition. supervised machine learning algorithms seem to provide a basis for an ideal technique, but require ground-truth (precise segmentation results) data to train neural networks. we believe that the holy grail of the ideal training data lies with synthetic xct datasets. such datasets can be created from any structure consisting of any number of materials or phases [8, 14, 69]. now, we come to the problem of obtaining these original true structures or spatial phase distributions, especially considering that real porous media are infinite resolution structures, at least up to molecular scale. we argue that the original should be no more than an order of magnitude finer (better resolution) than the optimal resolution for the images we shall segment. for example, typical xct images for sandstones has resolution of 2.5-5 µm and in some instances it is enough to see all details relevant for structure characterization and porescale modelling. for carbonates and soils this may be different due to a wide range of pore sizes, some of them well below 1 µm which is an approximate limit for resolution with current desktop scanners such as bruker-skyscan-1172. in short, if the images we want to segment will have the resolution up to 1 µm, the true phase distributions of 100 nm will suffice to create accurate projections assuming the receiver matrix (ccd camera in real scanners) to end up with 1 µm pixel size. such ”magnified” images can be obtained easily using a number of techniques: discretization of a predefined grain packings for sandstones (e.g., [45]), stochastic constructions of a synthetic porous media [36,41], multiscale stochastic reconstruction/image fusion [24, 62], or super-resolution stochastic technique [49]. the proxies for magnification can be obtained from real porous media images by segmenting them using current state-ofthe-art techniques (online reference) enhanced by mineralogical composition overlay from 2d analysis [13, 32]. in the above analysis we arrive to a universal framework to create an ideal segmentation technology based on convolutional neural networks (cnn) or possibly other (better/emerging) type of machine learning methodology. the general scheme of this framework is shown in fig. 1.1. but in this case, some data on the spatial materials’ distribution in the porous media is required. then the question arises about the size and diversity of the sample that will be copyright © 2022 assa. adv syst sci appl (2022) 34 efim v. lavrukhin, vasily postnicov, kirill m. gerke fig. 1.1. universal workflow to train machine learning based segmentation models. used in training: is it necessary to use patterns of spatial distribution from different types of porous media, or how accurately is it necessary to reproduce the original (and unknown) morphological structure? the objective of this paper is to understand how accurate the morphological structure of the samples used to construct a synthetic xct should be – or in other words, how similar to the images we want to segment. we also want to elucidate how morphological structure errors of various sizes will affect the quality of the trained segmentation model. for this, we construct stochastic samples with different levels of likelihood relative to the original structures. for all these samples we create synthetic xct images and train segmentation cnn model on original sample-synthetic xct pairs. the segmentation quality of these samples analyzed depending on their likelihood, as well as on the morphological properties of the original structure. 2. methodology 2.1. convolutional neural networks (cnns) for image segmentation cnn for segmentation are tuned to segment entire input image into sub-regions of particular origin, e.g., in landscape images cnn can distinguish between grass, woods, sky, etc. therefore, the task is to identify which object a given pixel of the image belongs to. thus, in segmentation cnns classifiers are applied to each pixel independently. furthermore, to reduce noise, segmentation cnns use two procedures: 1. down-sampling (encoder), when original image is compressed in size; 2. up-sampling (decoder), when compressed image is restored to the size of the original image. then each pixel of the up-sampled image is a subject for it own classifier, which is independent from the classifiers of other pixels. a general structure of the segmentation cnn‘s is presented in fig. (2.2) copyright © 2022 assa. adv syst sci appl (2022) the influence of image morphology... 35 fig. 2.2. general structure of the encoder-decoder cnns architecture for segmentation. the most prominent networks in image segmentations cnns are u-net [56] and segnet [5]. in this paper, we are solving a task of dissociating between porous and non-porous space within xct images. furthermore, there can be only either porous or non-porous space in the image, i.e., no background pixels. therefore, we need to assign each pixel to be either of porous or non-porous type. segmentation cnns are natural solutions to this 2-class segmentation problem. in this study, we used a specific version of the u-net architecture: a u-net with a res-net 50 [37] encoder that was pretrained on the imagenet dataset [44]. we expect this combination provides a balance between the speed of the architecture and the quality of the segmentation result. 2.2. porous media acquisition and specs we gathered a library of xct images containing 4 natural porous media samples of different genesis. all 3d xct images were obtained using skyscan-bruker 1172 desktop scanner with different resolutions, which depended on the size of the cylindrical samples subdrilled (in case of rocks) from core material. all resolutions were carefully chosen to capture major pore sizes. some amount of porosity could be below resolution and, thus, not represented on the images. all these samples were chosen to be very different in pore structure. moreover, one of the samples contains cracks and fissures, that provide additional complexity to the segmentation procedure. after acquisition, we have cropped regions of interest with volumes of 10003 voxels from each full size scan. subcropping is useful is useful for a number of reasons: 1. to remove all border effects due to cylindrical sample shape; 2. to make subvolume cubic, such shape is convenient for further processing, as it basically represents a 3d matrix of the same dimension; 3. to reduce input size, the size of 10003 voxels provides a good compromise between image representability and computational efforts needed for its processing. next, the original porous media sample was segmented into two phases using the conventional local thresholding approach – converging active contours method [61]. obtained morphological binary structures were used as a basis for constructing synthetic xct image. the pore space of these structures was filled with air, the solid phase was considered to be made of kaolinite (al2si2o5oh4). while creating synthetic xct (see details in the next subsection), we compressed the resulting images (as well as the corresponding markup images) in 5 times to 2003 in order to obtain and study the partial volume effect – this made segmentation task on synthetic images closer to real xct conditions. copyright © 2022 assa. adv syst sci appl (2022) 36 efim v. lavrukhin, vasily postnicov, kirill m. gerke sample 1 sample 2 sample 3 sample 4 fig. 2.3. examples of 2d slices through 3d synthetic xct images. the example 2d slices from all synthetic xct images are shown in fig. 2.3. this figure also identifies sample references used throughout this study: sample 1, sample 2, sample 3, and sample 4. 2.3. synthetic xct algorithm the synthetic tomography algorithm consists of three stages: 1. construction a sample model based on physical and morphological information; 2. calculation of shadow projections; 3. reconstruction of xct image. the first step is required to calculate absorption for each phase independently (2.1) using the numerical direct radon transform: al m = σm ∫ l ρm(l)dl, (2.1) where: m phase number; l direction to calculate absorption; σm attenuation coefficient of phase m material; ρm linear density of phase m material. the total absorption (2.2) is calculated as the sum of the absorption of all materials (2.1): al = ∑ m al m. (2.2) directional transmittance (2.3) is calculated using total absorption (2.1) according to: t l = ea l . (2.3) further, to complete second step, the shadow projection for each detector pixel and each rotation angle of the sample (which are set by the direction l) can be calculated as the number of photons that reached the detector through the sample. the number of photons (2.4) is a random variable with a poisson distribution, and with an average proportional to the number of photons n0 released by source, and the transmittance in a given direction: n l ∼ pois(n0t l). (2.4) next step is to mathematically reconstruct the image from projections. this can be done using the numerical inverse radon transform. synthetic tomography was implemented copyright © 2022 assa. adv syst sci appl (2022) the influence of image morphology... 37 sample 1 sample 2 sample 3 sample 4 fig. 2.4. example 2d slices from obtained 3d stochastic reconstructions. in python with the help of astra toolbox [1], that includes forward and backward radon transform. at this point we have the ground-truth original phase distribution and its synthetic xct 3d grayscale image. moreover, during projections simulation and reconstruction procedures we can apply any software/hardware setups and parameters, including noise addition and specified scanning resolution. 2.4. stochastic reconstruction algorithm stochastic reconstruction is an algorithm that iteratively performs permutations of voxels within an image, trying to match its statistical properties to the original (or training) image. as input data, the stochastic reconstruction algorithm receives the calculated values of a set of two-point correlation functions (here: s2, l2, fss) in orthogonal and diagonal directions for a range of correlation lengths r. each correlation function measures a probability of some event for a given correlation length r. specifically, s2 measures the probability that both ends of the r line segment lie within the pore phase, l2 – the probability that the whole segment lies within the pore space, and fss – that both ends fall into a pore-solid interface. comprehensive information about correlation functions can be found in [64]. the reconstruction algorithm starts with mono-disperse sphere packing with similar porosity and close s2 correlation function. the porosity is known from s2(0) or l2(0) value. with each iteration, the algorithm swaps two voxels of an image to minimizes a special energy function (2.5), which is calculated from the difference between the current values of the image’s correlation functions f and the specified correlation functions values f̂ summed over all correlation lengths r: e(i) = ∑ f ∑ r ( f(r)− f̂(r) )2 −→ min . (2.5) the energy function is minimized using the simulation annealing algorithm. the algorithm stops after minimizing the error close to zero, or after a large number of consecutive unsuccessful attempts. during annealing we saved the structure dynamics after some particular number of permutations attempts – after 1, 4, 16 and 64 ×104 iterations. a more detailed description of this algorithm can be found in [23, 26]. stochastic replicas were also used as input for synthetic xct 3d images generation, see examples on fig.2.4. 2.5. image preparation for cnn segmentation cnn has a supervised learning architecture. this means that to train the network we need a pair of samples, i.e., the input examples and desired segmentation for each such image. here we describe the way the images to be fed into the network have been prepared for the segmentation experiments. copyright © 2022 assa. adv syst sci appl (2022) 38 efim v. lavrukhin, vasily postnicov, kirill m. gerke we refer to each 3d volume of synthetic xct as a stack. there are 200 images with resolution 200× 200 pixels within each such stack. we consider each image of resolution 200× 200 taken in parallel to the same plane as a single independent image, i.e., no 3d spatial information about image location in the stack is provided to cnn. fig. 2.5. preparation of the network inputs: convert stack image into image samples. next, we collect a pool of n = 32 samples of 128× 128 pixels from each image with minimal overlapping. this procedure is explained in detail on fig. 2.5. the pixel intensity within each sample is standardized by scaling with mean and standard deviation from imagenet (due to the use of an imagenet-pretrained encoder). this pool of samples constitutes the input samples to be fed into the segmentation cnn. 2.6. implementation the stochastic reconstructions algorithm is implemented on the julia programming language. the synthetic tomography algorithm is implemented on the python programming language using the astra framework for calculating linear projections and the numpy and pytorch libraries for matrix calculations parallelization. all neural networks models were trained on the python programming language using the pytorch library. we used a computer with ubuntu 18.04 operation system, 128gb ram and a nvidia geforce rtx 2090 graphics card as a working station for computational experiments. 3. experiments 3.1. model description for the purpose of this study, we have selected the u-net segmentation cnn architecture [56], as shown in fig.3.6. the u-net is a fully-convolutional encoder-decoder architecture with skip-connections, which are feedforward connections from encoder to decoder layers. u-net uses convolution and max-pooling layers for compression (down-sampling) in the encoder part. transposed convolutional layers and concatenation with skip-connected encoder feature map for decompression (up-sampling) in the decoder part. convolutional layers use only with 3x3 filters. in this study we used a u-net architecture with a imagenet-pretrained res-net 50 encoder. 3.2. training process as input to the network we use 128× 128 from the pool of samples with corresponding binarized 128× 128 desired output samples. these input samples are arranged into batches copyright © 2022 assa. adv syst sci appl (2022) the influence of image morphology... 39 3232 i 64 64 i/2 128 128 i/4 256 256 i/8 512 512 i/16 256 256 256 256 i/8 128 128 128 128 i/4 64 64 64 64 i/2 32 32 32 32 i fig. 3.6. u-net’s architecture. of size 32 for model training. therefore, the output of the model constitutes tensor with pore and non-pore probabilities within each pixel. we use combination of cross-entropy (3.6) and smoothed intersection jaccard index (3.7) as loss function: ce(x, y) = − ( y log p(x) + (1− y) log ( 1− p(x) )) , (3.6) jaccard(x, y) = − log ( p(x)y + ε p(x) + y − p(x)y + ε ) , (3.7) l(i,m) = 1 n n∑ i=1 ∑ x,y∈ii,mi 1 wh ( ce(x, y) + αjaccard(x, y) ) , (3.8) where: n total number of images; ii i-th image to segment; mi i-th true segmentation mask; w width of the output image; h height of the output image; x, y image/mask pixels with corresponding spatial positions; p(x) neural network prediction for pixel x; ε = 10−6; α = 1. we used adam optimizer [57] with following parameters: lr = 10−2 initial learning rate; wd = 10−2 weight decay; β1 = 0.95 coefficient used for computing running averages of gradient; β2 = 0.999 coefficient used for computing running averages of gradient square; lrd = 0.5 multiplicative learning rate decay on validation loss plateau; lrmin = 10−6 minimal learning rate. 3.3. experimental setup after the preparation of binarized and synthetic xct images as was described in the previous sections, we split each stack into 4–1–1 proportion – training, validation and testing datasets. we trained the model on the training samples (fig. 3.7) and tuned the hyperparameters to copyright © 2022 assa. adv syst sci appl (2022) 40 efim v. lavrukhin, vasily postnicov, kirill m. gerke fig. 3.7. u-net training process. minimize the loss function on the validation samples. as a result, we obtained the model trained on samples with real porous media morphological properties. we used this model for segmentation of the images from the test dataset, as well as for the segmentation of four sequences of synthetic xct images, that were created from stochastic reconstructions (fig. 3.8). finally, we measured the segmentation quality metric by the initial morphological properties and by the number of annealing iterations (iterations numbers 1, 4, 16, 64) against the morphological properties original porous media ground-truth images. 3.4. segmentation accuracy metrics to measure model performance in our experiments we employed intersection over union, or iou metric – the jaccard index. it describes the ratio between the size of the intersection of true and predicted labels sets, and the size of the union of these sets. this metric is widely used to assess quality in segmentation studies. we calculate it for the positive class (i.e., pore phase) for greater sensitivity. in order to get the class labels from the probabilities that are returned by the model, we used the probability threshold of 0.5. the iou metric was measured for each synthetic xct – ground-truth pair. 4. results we obtained an average iou = 0.81 for the pore phase, which is a relatively good result. it should also be noted that measurements are carried out against true ground-truth labels (which were used to produce synthetic xct images). most segmentation errors are located around the phase separation boundaries and can be explained by the influence of partial volume effect. copyright © 2022 assa. adv syst sci appl (2022) the influence of image morphology... 41 iteration 1 iteration 4 iteration 16 iteration 64 fig. 3.8. examples of 2d slices from 3d sequences of samples produced by stochastic reconstruction algorithm. 4.1. visual segmentation results in this section we show visual results of the cnn-based segmentation (fig. 4.9, 4.10, 4.11, 4.12). fig. 4.9. examples of 2d slices through 3d segmented images for sample 1. also we can see (fig. 4.13) the visualization of false-positive (shown in red) and falsenegative (shown in blue) segmentation errors. copyright © 2022 assa. adv syst sci appl (2022) 42 efim v. lavrukhin, vasily postnicov, kirill m. gerke fig. 4.10. examples of 2d slices through 3d segmented images for sample 2. fig. 4.11. examples of 2d slices through 3d segmented images for sample 3. fig. 4.12. examples of 2d slices through 3d segmented images for sample 4. as we can see, our model is uncertain only about interface pixels, the amount of falsenegative and false-positive errors is almost equal. 4.2. quantitative segmentation results to quantitatively describe the observed results, we built a bar charts to compare the following options: copyright © 2022 assa. adv syst sci appl (2022) the influence of image morphology... 43 sample 1 sample 2 sample 3 sample 4 fig. 4.13. visualization of false-positive and false-negative segmentation errors. 1. segmentation quality depending on the number of the annealing iterations for each sample separately (fig. 4.14); 2. segmentation quality of the original porous media stacks (fig. 4.15); 3. segmentation quality depending on the number of the annealing iterations in average over all stacks (fig. 4.16). sample 1 sample 2 sample 3 sample 4 fig. 4.14. metrics for each stack. fig. 4.15. separate iou metrics for original (real porous media) samples. copyright © 2022 assa. adv syst sci appl (2022) 44 efim v. lavrukhin, vasily postnicov, kirill m. gerke fig. 4.16. averaged iou metrics over all samples 5. discussion first of all, we tried to understand why the segmentation quality for sample 2 was much better than for the rest of samples. we believe that this is due to the fact that this sample differs from others in its morphological properties (in terms of correlation functions). samples 1, 3, 4 are similar to each other in correlation functions l2 (sample 3 slightly differ from samples 1, 4) and fss (all samples are close in all directions), all samples are pretty close to each other in s2, the sample 2 is not similar to the others in any correlation function, except s2 (see fig. 5.17). next, we analyzed why metrics dynamics (changes in segmentation accuracy with each annealing iteration) for stochastic reconstructions is multidirectional for different samples. in particular sample 2 is different to other samples. in this regard, we established that sample 2 has the simplest structure (fig. 2.3), and therefore showed the best segmentation metrics. moreover, stochastic ”deconstruction” (structural dynamics in reversed direction on fig.3.8) further simplifies its structure (fig. 3.8) by increasing the iou metric. the presence of the peak in metrics at 16 and 64 iterations raised some questions. after reviewing the corresponding stochastic reconstructions (fig. 3.8), we can conclude that they are visually much more similar to the original samples than the reconstructions at the final iterations. this means that the limited information content of the correlation functions does not fully convey the morphological properties of the original stacks. the above-mentioned considerations allow to suspect that the deconstruction of the initial samples’ structure has less effect on the segmentation metrics than the initial difference in the morphological properties between replicas and original porous media images. to test this hypothesis, it is necessary to carry out a similar experiment with a large number of different original samples. porosity of the sample can be of great importance on the results, therefore, it is necessary to neutralize its influence. this can be achieved by collecting a a library of original samples with the same initial porosity (stochastic reconstructions preserve the porosity of the original). 6. conclusion in this paper we presented the initial results of measuring the influence of samples morphology on cnn-based segmentation quality. to do this, we segmented four different copyright © 2022 assa. adv syst sci appl (2022) the influence of image morphology... 45 sx 2 sy 2 sz 2 lx 2 ly 2 lz 2 f x ss f y ss f z ss fig. 5.17. plot of the used correlation functions set along all directions. porous media samples using an unsupervised converging active contours algorithm. next, we obtained the xct – ground-truth pairs for these samples with the help of synthetic tomography algorithm. we acquired a set of samples with varying degrees of morphological properties similar to the original stacks with the help of stochastic reconstructions. we trained u-net model with resnet50 encoder on pairs for the original porous media samples. next, we assessed the segmentation quality of synthetic xct obtained from stochastically reconstructed samples, and checked the effect of morphological properties on the segmentation quality. based on the metrics, we concluded that the quality of segmentation is influenced more by the morphological differences between the original soil samples than by difference from copyright © 2022 assa. adv syst sci appl (2022) 46 efim v. lavrukhin, vasily postnicov, kirill m. gerke the generated stochastic reconstructions. we discussed possible ways to improve the future model experiment design in order to finally uncover the relationship between morphology variations ans segmentation accuracies. the results as presented here can help in simplifying the construction of a general segmentation pipelines. we showed that it is not necessary to segment the original images perfectly. to build a universal pipeline, it is necessary to continue this research direction with a much larger library of images with various morphological properties. acknowledgements this work was supported by the russian foundation for basic research grant 20-3190118\20 (the application of neural networks to solve the problem of optimal segmentation of tomographic images of geological rocks). references 1. van aarle, w. et al. 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(1998). reconstructing random media. ii. threedimensional media from two-dimensional cuts. phys. rev. e, 58, 224–233. https://doi.org/10.1103/physreve.58.224 copyright © 2022 assa. adv syst sci appl (2022) introduction methodology convolutional neural networks (cnns) for image segmentation porous media acquisition and specs synthetic xct algorithm stochastic reconstruction algorithm image preparation for cnn implementation experiments model description training process experimental setup segmentation accuracy metrics results visual segmentation results quantitative segmentation results discussion conclusion microsoft word 1124 article text, copyedited 2021.10.05.docx adv syst sci appl 2021; 03; 91-100 published online at https://ijassa.ipu.ru. face mask perception during the covid-19 pandemic: an observational study of russian online social network vkontakte alexander g. chkhartishvili1, dmitry a. gubanov1, ivan v. kozitsin1,2* 1) v.a. trapeznikov institute of control sciences russian academy of sciences, moscow, russia 2) moscow institute of physics and technology, dolgoprudny, russia abstract: the spread of covid-19 has forced governments to impose unprecedented restrictive measures on their people. lockdowns have been introduced in many countries, and most governments have decided to require citizens to wear masks. the current study characterizes users’ attitudes towards the face mask requirements introduced by the russian government as a response to the covid-19 pandemic. we study how they relate to other users’ characteristics such as age, gender, and political attitudes. subscriptions to information sources serve as a proxy for users’ political attitudes. our results indicate that men and elder individuals—demographic groups that are most vulnerable to covid-19—underestimate the benefits of wearing face masks, comparing to young people and women who demonstrate a higher rate of approval. we also discovered that users in opposition to the russian government highly approve of this anticovid-19 measure, a result that is quite counterintuitive since the oppositionists should, a priori, negatively perceive all measures taken by the government. further, we developed a probabilistic model that fixes the relationship between pure types (conservative/oppositionist; reject maskwearing/approve of mask-wearing) and derived the method of estimating its parameters. we obtained that an oppositionist approves of the face mask requirements with the probability of 0.95. for those who support the russian government, the odds of approval are merely 0.45. keywords: covid-19, online social networks, face masks wearing, risk perception, political attitudes 1. introduction the spread of covid-19 has forced governments to impose unprecedented restrictive measures on their people. lockdowns have been introduced in many countries, and most governments have decided to require citizens to wear masks. this measure has been proven to have a positive impact on the spread of covid-19 [18]. however, requirements alone are often not enough to effectively combat a pandemic. an important factor in this is how closely citizens comply with the restrictions imposed [1,9]. from this perspective, governments essentially must understand what citizens think about the restrictions. online social networks (osns) are places where individuals can express their opinions on certain issues. the form of expression varies: individuals may write posts or comment on them. these textual messages can indicate their views. furthermore, users may like or repost other users’ posts and comments. these types of actions can be considered (indirect) manifestations of users’ views. during the pandemic, a huge amount of users’ attention was devoted to different pandemic-related topics such as wearing masks, vaccination, and even the issue of whether the pandemic itself exists or has been fabricated [4,17]. in this cross-sectional study, we analyze how users of osn vkontakte (vk) react to posts related to covid-19. our main goal is to characterize users’ attitudes towards the face mask requirements introduced by the russian government. these requirements were released in the russian federation at different times depending on the region. in most russian regions, it was necessary to wear protective masks in public places by 21 may, 2020. * corresponding author: kozitsin.ivan@mail.ru 92 a.g. chkhartishvili, d.a. gubanov, i.v. kozitsin copyright ©2021 assa. adv. in systems science and appl. (2021) we also investigate how users’ attitudes toward the face mask requirements relate to their political preferences (we use information sources users are subscribed to as a proxy of their political attitudes) and their demographic characteristics (age and gender). 2. literature face mask–wearing has been proven to have a positive effect on limiting the spread of covid-19 [18]. a remarkable number of studies have aimed to understand how citizens perceive the face mask requirements (as well as other measures such as lockdowns or social distancing) and whether they comply with them [9-11]. [9] reported that women, elder individuals, and citizens of urban areas are more likely to wear masks. in turn, [11] suggested that gender has no significant effect on whether an individual wears a face mask, but it affects how they perceive it. a group of studies has documented that political affiliation affects individuals’ risk perception and, in particularly, how they perceive the effectiveness of the recommendations of the government’s leaders to stay at home [2,6]. they argued that us counties that had higher rates of people voting for donald trump are associated with lower levels of covid19 risk perception. further, state government leaders’ recommendations were more effective in democratic-leaning counties. these results are quite nontrivial since, in the face of danger, individuals must be as objective and rational as possible. as such, it is essential to understand the profound nature of this effect. for example, [2] argue that political preferences largely shape the information space around individuals (due to individuals’ tendency to receive information from sources congruent with their own views) and thus potentially limit their access to objective information (since information broadcasted by politically biased media may be disturbed). in osns, this regularity may be exacerbated due to personalization systems that can increase the limiting effect and foster the formation of echo chambers [14,22]. 3. methods 3.1 data collection we gathered data from vk, which operates rather similarly to facebook. vk accounts can be naturally divided into two sets: information producers and information consumers. the former encompasses public pages, groups, bloggers, and event pages. note that representatives of news outlets usually have the status of public pages. information consumers are ordinary users that essentially have fewer followers (users with more than 1000 followers become the official status of bloggers). we focus on posts published by public pages and how ordinary vk users react to them. we have chosen two public pages—meduza and rt—that are representative of the corresponding news outlets [20,26] these outlets are considered to hold opposing political sides: rt is traditionally associated with the russian official government, while meduza is considered to have much less loyalty to the kremlin [3,27]. recently, meduza has been given the status of “foreign agent” [19]. because these two public pages have such contending political leanings, the posts published by them should be of different political sides, and the same tendency should be found in the political views of their followers: meduza followers should be more oppositional whereas rt followers should be more conservative. it provides an opportunity to analyze individuals with potentially different political attitudes. we gathered all posts published by these pages within a predefined period (1 april–1 june, 2020) that contained at least one word from a predefined set of words from the topic of covid-19 (accounting for different word forms): “ковид,” “коронавирус,” “covid,” face mask perception during the covid-19 pandemic: an observational study 93 copyright ©2021 assa. adv. in systems science and appl. (2021) “coronavirus,” “карантин,” “удаленк,” “самоизоляц,” “пандеми,” and “эпидеми.” next, we downloaded all comments related to these posts, as well as likes left by users on these posts and comments. in this conference paper, we concentrate only on a small subset of the information cascades induced by these posts. more precisely, we included individuals who took part in these information cascades by commenting the posts with words that included at least one word from the following: “маск,” “масочн,” and “намордник” (these words relate to the topic of face mask requirements). by doing so, we collected a sample 𝐼! of 1559 users. we denote the set of corresponding comments as 𝐶. we classified each comment from the set 𝐶 according to the attitude it conveyed toward the face mask requirements. we highlighted the following cases: (1) negative attitude; (2) neutral attitude or attitude is not provided; and (3) positive attitude. note that a negative attitude does not necessarily mean that an individual does not wear a face mask; it describes only how they perceive face mask–wearing. using this classification, for each user 𝑖 from the sample 𝐼! we identified 𝑖’s attitude towards the face mask requirements manually by labeling them with the variable 𝑎"! ∈ {−1,0,1}. this variable is referred to as the comment-attitude (meaning that the attitude stems from the user’s comments). we further classified individuals who had liked at least one comment from the set 𝐶 with a negative or positive attitude. we denoted the set of such users with 𝐼# (|𝐼#| = 847). each user 𝑗 ∈ 𝐼# was associated with the variable 𝑎$# ∈ {−1,1} that stems from the attitudes of comments liked by 𝑗 (the like-attitude). we focused on users from the set 𝐼 = 𝐼! ∪ 𝐼#. the sets 𝐼! and 𝐼# had 113 intersections— users who both made at least one comment on the topic of face mask requirements (under the posts on the topic of covid-19) and liked at least one such comment with a positive or negative attitude. we carefully processed all these instances. we found 10 users who had liked a comment with an orientation opposite to what they had written themselves. excepting these individuals, each user 𝑖 from 𝐼 was assigned the variable 𝑎" ∈ {−1,0,1} representing their overall attitude towards the face mask requirements. the detailed scheme of how we defined the variable 𝑎" can be found in table 3.1. note that the situation where 𝑎" = 0 may correspond to qualitatively different cases: it may relate to a user with a neutral attitude, or it may stand for a user that has a positive or negative attitude, but this cannot be inferred from the data. table 3.1. users’ attitudes towards the face mask requirements manifested by comments (𝑎!") and likes (𝑎!#) comment-attitude 𝑎"! like-attitude 𝑎"# overall attitude 𝑎" number of users -1 none -1 106 -1 -1 -1 10 -1 1 not defined 7 0 none 0 1098 0 -1 -1 15 0 1 1 41 1 none 1 242 1 -1 not defined 3 1 1 1 37 none -1 -1 178 94 a.g. chkhartishvili, d.a. gubanov, i.v. kozitsin copyright ©2021 assa. adv. in systems science and appl. (2021) none 1 1 556 total number of users 2293 for individuals from 𝐼, we obtained their age and gender (0: female; 1: male), as well as information on their connections to other accounts on vk. we considered two types of connections: (1) friendship (reciprocated) ties with users and (2) following (unreciprocated) ties directed on bloggers and public pages. among the sample (consisting of 2293 individuals) we found only six friendship connections: four connected pairs and one open triad. it ensures that the observations are independent (since in social networks, neighboring vertices’ characteristics are usually correlated). 3.2 users’ political attitudes we estimated the political attitudes of users from 𝐼 by determining how oppositional users are with respect to the russian government. each user 𝑖 was associated with the continuous variable 𝑥" ∈ [0,1] denoting the extent to which user 𝑖 is inclined to support the opposition movement. 𝑥" = 1 indicates that 𝑖 is a strong oppositionist whereas 𝑥$ = 0 means that 𝑗 is loyal to the government. for simplicity, we define 𝑗 as a conservative user. we calculated 𝑥" using a slightly modified version of the algorithm presented by [15] and used in [12,13,16]. simply, 𝑥" is a projection from the set of information sources (bloggers and public pages) that user 𝑖 is subscribed to. it reflects the fact that users tend to follow information sources that have a political bias congruent with their own views [5]. as such, 𝑥" = 1 means that 𝑖 follows highly oppositional information sources whereas 𝑥$ = 0 indicates that 𝑗 is subscribed to conservative public pages and bloggers. 3.3 additional filters in our analysis, we impose additional constraints. we considered only users with a welldefined overall attitude towards the face mask requirements (i.e., we did not consider those with 𝑎" = 0). furthermore, we considered only accounts that had more than 10 subscriptions to public pages and bloggers. this filter was proposed by [15], who discovered that estimations of the political attitudes of such users were of the highest accuracy. following these adjustments, we were left with 582 observations. 4. results our analysis revealed that the sample of interest was comprised of slightly more male users (see fig. 4.1). only 254 users noted their age (about half) and nine of them had anomalously high values of age (greater than 90 years), which were likely fabricated. most users were estimated to have moderate political attitudes; the relatively small peaks at 0 and 1 correspond to radical conservatives and oppositionists, respectively. we also see that the number of people who supported the face mask requirements was more than twice that of those who had a negative attitude toward wearing masks. face mask perception during the covid-19 pandemic: an observational study 95 copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 4.1. histograms displaying users’ characteristics. next, we studied how individuals’ attitudes toward the face mask requirements were connected to their other characteristics. we specifically questioned how these attitudes vary along political lines. similarly to [2,6], we report that attitudes toward the face mask requirements varied strongly across political lines (see fig. 4.2). the main trend observed is that the more oppositional a user is, the higher the odds are that they approve of the face mask requirements. 96 a.g. chkhartishvili, d.a. gubanov, i.v. kozitsin copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 4.2. the proportions of individuals having different attitudes towards the face mask requirements. the groups themselves are constructed as follows: conservatives are users with 𝑥! < 0.33, oppositionists are characterized by 𝑥! > 0.66, moderates are those in between. we performed regression analysis in which we used 𝑎" as the dependent variable and gender, age, number of friends, and 𝑥" as independent variables. in this analysis, we ignored users aged over 90. after fitting the logit model on the remaining instances (see table 4.1), we found that the tendency of oppositional users to have a higher rate of approval for wearing face masks holds even after controlling for other characteristics. we also found that an increase in age or being male leads to lower odds of face mask approval, other things being equal. the number of friends had no significant effect on the dependent variable. table 4.2. logistic regression summary coefficient estimate standard error pr(>|z|) intercept 0.309 0.791 0.696 age -0.022 0.01 0.036* gender -0.751 0.325 0.021* number of friends 0.011 0.008 0.148 political attitude 𝑥! 3.431 1.237 0.006** signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 the effect of demography characteristics confirms the irrational nature of individuals’ perceptions since men and elder individuals are the demographic groups that are considered face mask perception during the covid-19 pandemic: an observational study 97 copyright ©2021 assa. adv. in systems science and appl. (2021) to be most vulnerable to covid-19. note that our results do not contradict those of [9], who argued that elder individuals are more prone to wear face masks: unlike that study, we analyze users’ attitudes, not their actions. the way political attitudes (and more precisely, information embedding) affect attitudes toward wearing face masks is quite surprising since the oppositionists should, a priori, negatively perceive all measures taken by the government. conversely, those who highly approve of the government should also approve orders released by the government. we leave this puzzle for future research. 5. pure-types model in this section, we develop a probabilistic model that fixes the relationship between pure types (conservative/oppositionist; reject mask-wearing/approve of mask-wearing) and derive the method of estimating its parameters. we further illustrate how this method works on the data at hand. recall that the quantity 𝑥" characterizes users on a continuous scale. however, decisions made by individuals imply a finite number of alternatives. the simplest example is an election with two parties [25]. from this perspective, the variable 𝑥" can be conceptualized as the probability that, in a particular situation, user 𝑖 will act as an oppositionist (conversely, 1 − 𝑥" stands for the chance of acting as a conservative). next, let us assume that an oppositionist agrees with the face mask requirements with a fixed probability 𝑝 and a conservative person with probability 𝑞. here we do not explain the nature of these probabilities: it may be caused by some sort of information distortion or something else. we suppose that 𝑝 and 𝑞 are determined solely by the individual’s type (conservative or oppositional) and are not sensitive to external effects such as, say, peer influence processes (but this assumption does not hold for the probability 𝑥" of being oppositional [7,12,13]). the data at hand provide an opportunity to obtain quantities 𝑝 and 𝑞 using maximum likelihood estimation. the probability of complying is defined by 𝑥"𝑝 + (1 − 𝑥")𝑞. in turn, the probability of rejecting is given by 𝑥"(1 − 𝑝) + (1 − 𝑥")(1 − 𝑞). as such, we can write the likelihood function: 𝐿(𝑝, 𝑞) =<[𝑥"𝑝 + (1 − 𝑥")𝑞] "∈&$ ∗<[𝑥"(1 − 𝑝) + (1 − 𝑥")(1 − 𝑞)] "∈&% , where 𝐼' are individuals from the filtered sample with 𝑎" = 1 and 𝐼( are those for whom 𝑎" = −1. to estimate the values of 𝑝 and 𝑞, one should solve the following optimization problem: min )*+,-*.() − ln 𝐿(𝑝, 𝑞), where 𝛿 is a small positive constant (we set it as 10(/) that aims to avoid situations when multipliers under the sign of the logarithm are null (see fig. 5.1). we solve this optimization problem using the function minimize introduced in scipy package with the sequential least squares programming (slsqp) method [28]. this approach requires providing an initial guess point. our analysis reveals that for almost all initial points (excepting those of (0,0) and (1,1)) the result is largely the same: �̂� = 0.95, 𝑞g = 0.45. the value of 𝛿 does not have a significant impact on the outcome of the optimization algorithm. estimated values of 𝑝 and 𝑞 indicate that, with a probability of 0.95, oppositionists will approve the facemask requirements whereas conservatives will do so with a probability of 0.45. that is, oppositionists approve face mask–wearing at an extremely high rate while conservatives act approximately at random. 98 a.g. chkhartishvili, d.a. gubanov, i.v. kozitsin copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 5.1. contour lines of the function (− ln 𝐿(𝑝, 𝑞))&/()*. the red “x” marks the minima obtained. 6. conclusion this cross-sectional study characterizes users’ attitudes towards the face mask requirements introduced by the russian government as a response to the covid-19 pandemic. we study how they relate to other users’ characteristics such as age, gender, and political attitudes. our research can be useful in elaborating the government’s anti-covid-19 policies as it demonstrates that different demographic and ideological groups may perceive the same measures in different ways. possible directions for future research include analysis of the coevolution of political and pandemic-related attitudes, which may provide more understanding of how these kinds of attitudes are connected to each other and what the role of demographic characteristics is in this process [23]. another interesting avenue for research could be the analysis of the data from the perspective of multidimensional opinion formation models [8,10,21,24]. 7. data availability all data used in this study can be obtained upon a reasonable request. acknowledgements this work was partially supported by the russian foundation for basic research, project no. 20-04-60296. face mask perception during the covid-19 pandemic: an observational study 99 copyright ©2021 assa. adv. in systems science and appl. 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(2020). a model of propaganda battle with individuals’ opinions on topics saliency. 2020 13th international conference ‘management of large-scale system development’ (mlsd), 1–4. https://doi.org/10.1109/mlsd49919.2020.9247796 26. rt international (2021, september 20). rt – breaking news, shows, podcasts. (n.d.). rt international. [online]. available https://www.rt.com 27. rt international (2021, september 20). rt: a gazillionaire in a world of media paupers? [online]. available https://www.rt.com/op-ed/377715-rt-gazillionaire-mediafunding/ 28. scipy v1.7.1 manual (2021, september 20). minimize(method=’slsqp’). available https://docs.scipy.org/doc/scipy/reference/optimize.minimize-slsqp.html#optimizeminimize-slsqp microsoft word 1590-article text-7030-1-4-20240419 adv syst sci appl 2024; 02; 103-115 published online at https://ijassa.ipu.ru. savage's solution to the problem of three-currency deposit diversification: program tools and modeling results vitaly molostvov* international center of decision choice and analysis of hse university, russia, moscow abstract: this paper presents the development of computing tools for finding the optimal structures of multi-currency deposits in terms of guaranteed risk under uncertain exchange rates. the approach utilizes savage's minimax regret concept to calculate risk and guaranteed risk functions explicitly, assuming only the limits of possible changes in uncertain parameters are known. the excel environment implements the algorithm for calculating the optimal solution that minimizes income loss due to incomplete information. computational experiments analyzed the dependence of the optimal guaranteed risk on problem parameters, such as interest rates of currencies and boundaries of uncertain factors. the results can be used to analyze financial management problems in conditions of incomplete information. keywords: optimization, incomplete information, minimax regret solution, deposit diversification, excel 1. introduction the search of the optimal structure of a multi-currency deposit is complicated by the uncertainty of future values of economic parameters, in particular, the unknown values of the exchange rates of the currencies used. a typical situation is the so-called substantial uncertainty, in which there are not only no exact values of the uncertain parameters, but also no statistical characteristics. the decision maker (dm) knows only the limits of the possible values of these parameters. this leads to the emergence of risk, understood as a deviation of the result obtained from the desired or expected result. one approach to measuring and minimizing risk is savage's concept of minimax regret. according to this , risk is interpreted as a loss from the ignorance of uncertain factors. the principle of the best-guaranteed outcome [2] is usually applied to the risk function to compute the smallest guaranteed risk. this is the approach used below in the problem of optimal deposit diversification across three currencies and risk is understood according to savage or regret risk [10] . the point of view used is that of the dm who allows risk and seeks to minimize it. the concept of savage optimality [10] and other approaches to problems with uncertain parameters were considered in publications [5–7]. the paper [14] investigated the solution of savage's minimax regret in the problem of deposit diversification for two types of currencies. the article [8] is devoted to finding a solution with optimal guaranteed risk in a similar problem with three currencies. in it, an explicit form of the optimal solution is obtained and a constructive algorithm to compute it is proposed. the current paper continues this research. the proposed algorithm is implemented in excel. an economic interpretation of the initial and derived parameters of the problem is given. debugging and experimental calculations are carried out. the case of equal guaranteed yield of all three currencies (at minimum future rates) is investigated analytically. numerical modeling of the dependence of the minimum guaranteed risk and the structure of the optimal deposit on the parameters of the problem, such as the upper bounds of the future values of exchange rates and interest rates, is carried out. * corresponding author: molostvov@list.ru v.s.molostvov 104 copyright ©2024 assa adv. in systems science and appl. (2024) 2. statement of the problem at the beginning of the deposit period, the dm allocates one ruble among three types of deposits, buying dollars in the amount of 𝑥 rubles and euros in the amount of 𝑥 rubles at the initial rates 𝐾 and 𝐾 respectively, leaving 1 − 𝑥 − 𝑥 rubles in a ruble deposit. at the end of the deposit period, the accrued interest (at the known interest rates 𝑑 and 𝑑 ) in dollars and euros are converted into rubles at exchange rates unknown at the beginning of the deposit period: 𝑦 ∈ [𝑎 , 𝑏 ], 𝑦 ∈ [𝑎 , 𝑏 ]. here 𝑎 and 𝑏 (𝑎 and 𝑏 ) are the lower and upper limits of possible changes in the final dollar (euro) exchange rates. in accordance with these rules, the final income in ruble terms can be presented in the form: 𝑓(𝑥, 𝑦) = (1 + 𝑟)(1 − 𝑥 − 𝑥 ) + 𝑥 1 + 𝑑 𝐾 𝑦 + 𝑥 1 + 𝑑 𝐾 𝑦 . (2.1) it is desirable to maximize this indicator by choosing a strategy 𝑥 = (𝑥 , 𝑥 ) ∈ 𝑋 = {𝑥 + 𝑥 ≤ 1, 𝑥 ≥ 0 (𝑖 = 𝑑, 𝑒)} and the dm should take into account the possibility of realizing any uncertainty 𝑦 = (𝑦 , 𝑦 ) ∈ 𝑌 = [𝑎 , 𝑏 ] × [𝑎 , 𝑏 ]. thus, the mathematical model of the diversification problem is represented by an ordered triplet 𝛤 =< 𝑋, 𝑌, 𝑓(𝑥, 𝑦) > where 𝑓(𝑥, 𝑦) is the income function of the depositor (dm), and the set x of dm's strategies x and the set y of uncertainties y have the form: 𝑋 = {𝑥 = (𝑥 , 𝑥 )|𝑥 + 𝑥 ≤ 1, 𝑥 ≥ 0 (𝑖 = 𝑑, 𝑒)}, (2.2) 𝑌 = {𝑦 = (𝑦 , 𝑦 )| 𝑦 ∈ [𝑎 , 𝑏 ] (𝑖 = 𝑑, 𝑒)}. (2.3) the problem 𝛤 is a single-criteria decision-making problem under uncertainty: maximize a linear function of 𝑥 with uncertain coefficients on a polyhedron (triangle) 𝑋 given the uncertain factors. the presence of uncertainty leads to the concept of risk as a possibility of the deviation of some results from their desired or expected values. in the framework of wald's maximin approach [13], this problem can be considered as an antagonistic game of dm against "nature". its solution is a "cautious" maximin strategy obtained according to the principle of the best-guaranteed result. note that in the considered problem it has a rather trivial form: it is necessary to calculate the profitability of each type of currency at the most unfavorable possible combination of final exchange rates 𝑦 = (𝑎 , 𝑎 ) and invest all funds in the currency for which this indicator is maximized. if the maximum is reached on several currencies, the whole ruble amount is distributed among them in an arbitrary way. this result is true for any number of currencies [14]. as we know, the wald criterion is very pessimistic and focuses on the worst case. it does not allow for risk and often gives extremely underestimated results. on the other hand, many dms accept risk and need tools to minimize it. one of these is the principle of minimax regret (savage's criterion). it also uses the notion of the best-guaranteed outcome, but applies not to the original quality indicator 𝑓(𝑥, 𝑦) but to the so-called risk (regret) function. 3. savage's principle of minimax regret let 𝑓(𝑥, 𝑦) be the income, 𝑋 be the set of dm strategies, and 𝑌 be the set of uncertainties. then 𝑓 (𝑧, 𝑦) is the best possible income if there is an uncertainty 𝑦. however, the dm does not know in advance, what the value 𝑦 of the uncertain parameter will be. the difference φ(𝑥, 𝑦) = max ∈ 𝑓 (𝑧, 𝑦) − 𝑓(𝑥, 𝑦) (3.1) is called the savage risk (regret) function. it represents the loss due to uncertainty which is the difference between the best outcome that could have been obtained under known y and the actual outcome of some strategy 𝑥. this risk depends on both strategy x and uncertainty y. to minimize risk, the dm can use the concept of the best-guaranteed outcome (wald's principle [13]). these considerations lead to the following definition (savage's minimax regret principle [10]). 105 savage's solution to three-currency deposit: program tools copyright ©2024 assa adv. in systems science and appl. (2024) definition 3.1: let us call the solution 𝑥 the risk-guaranteed solution (rgs) of the problem 𝛤, if 𝛷 = 𝑚𝑎𝑥 ∈ 𝛷(𝑥 , 𝑦) = 𝑚𝑖𝑛 ∈ 𝑚𝑎𝑥 ∈ 𝛷(𝑥, 𝑦), (3.2) where the risk (loss) function 𝛷(𝑥, 𝑦) is defined in (3.1). for the sake of brevity, we also refer to the decision 𝑥 and risk φ as the optimal decision and the optimal risk, respectively. according to definition 3.1, the construction of a rgs contains four steps. step 1: calculate 𝑓[𝑦] = max ∈ 𝑓 (𝑧, 𝑦) for each uncertainty 𝑦 ∈ 𝑌. step 2: calculate the risk function 𝑓[𝑦] − 𝑓(𝑥, 𝑦). step 3. compute the internal maximum in (3.2) which is the guaranteed risk max ∈ φ(𝑥, 𝑦) = max ∈ (𝑓([𝑦] − 𝑓(𝑥, 𝑦)) = φ[𝑥] ≥ φ(𝑥, 𝑦) for each strategy 𝑥 ∈ 𝑋. step 4. compute the external minimum in (3.2) which is the best-guaranteed risk φ = min ∈ φ[𝑥] = φ[𝑥 ]. 4. constructing a risk-guaranteed solution the explicit form of the risk function for the diversification problem of a three-currency deposit was obtained in [8]: φ(𝑥, 𝑦) = 𝑓[𝑦] − 𝑓(𝑥, 𝑦) = = ⎩ ⎪⎪ ⎨ ⎪⎪ ⎧ φ (𝑥, 𝑦) = (1 + 𝑟) − 𝑓(𝑥, 𝑦) = [(1 + 𝑟) − 𝜉 ]𝑥 + [(1 + 𝑟) − 𝜉 𝑦 ]𝑥 , 𝑦 ∈ 𝑌 , φ (𝑥, 𝑦) = 𝜉 𝑦 − 𝑓(𝑥, 𝑦) = (1 − 𝑥 )[𝜉 𝑦 − (1 + 𝑟)] + [1 + 𝑟 − 𝜉 𝑦 ]𝑥 , 𝑦 ∈ 𝑌 , φ (𝑥, 𝑦) = 𝜉 𝑦 − 𝑓(𝑥, 𝑦) = [1 + 𝑟 − 𝜉 𝑦 ]𝑥 + [𝜉 𝑦 − (1 + 𝑟)](1 − 𝑥 ), 𝑦 ∈ 𝑌 , (4.1) where, for the sake of brevity, we used the notations 𝜉 = , 𝜉 = . the sets 𝑌 , 𝑌 , 𝑌 are subsets of the set 𝑌 defined by additional linear inequations: 𝑌 = {𝑦 ∈ 𝑌, 1 + 𝑟 ≥ 𝜉 𝑦 , 1 + 𝑟 ≥ 𝜉 𝑦 }, (4.2) 𝑌 = {𝑦 ∈ 𝑌, 𝜉 𝑦 ≥ 1 + 𝑟, 𝜉 𝑦 ≥ 𝜉 𝑦 }, (4.3) 𝑌 = {𝑦 ∈ 𝑌, 𝜉 𝑦 ≥ 1 + 𝑟, 𝜉 𝑦 ≥ 𝜉 𝑦 }. (4.4) it is obvious that 𝑌 ∪ 𝑌 ∪ 𝑌 = 𝑌. functions φ (𝑥, 𝑦) (𝑖 = 1,2,3) are bilinear functions of the variables 𝑥 and 𝑦. therefore, the risk function φ(𝑥, 𝑦) for any fixed strategy 𝑥 is a piecewise linear function of uncertainty 𝑦. the sets 𝑌 , 𝑌 , 𝑌 are regions (polygons) of linearity on y of the risk function, i.e. the risk function φ(𝑥, 𝑦) coincides with the linear (by x) function φ (𝑥, 𝑦), 𝑦 ∈ 𝑌 . in this case, it is said that the risk function φ(𝑥, 𝑦) is defined at the point (𝑥, 𝑦) by the function φ (𝑥, 𝑦). for each strategy 𝑥 ∈ 𝑋 the guaranteed risk φ[𝑥] was calculated [8]: φ[𝑥] = max{𝛼 𝑥 +𝛼 𝑥 , 𝛽 (1 − 𝑥 ) + 𝛼 𝑥 , 𝛽 (1 − 𝑥 ) + 𝛼 𝑥 }, 𝑥 ∈ 𝑋, (4.5) where the following notations are used: 𝛼 = [(1 + 𝑟) − 𝜉 𝑎 ], 𝛼 = [(1 + 𝑟) − 𝜉 𝑎 ], (4.6) 𝛽 = [𝜉 𝑏 − (1 + 𝑟)], 𝛽 = [𝜉 𝑏 − (1 + 𝑟)]. (4.7) the secondary (derivative) coefficients 𝛼 , 𝛼 , 𝛽 , 𝛽 depend on the above-mentioned initial parameters. the substantive interpretation for them is given below (see section 6). the best (smallest) guaranteed risk φ = min ∈ φ[𝑥] = φ[𝑥 ] is computed in [8] by considering the problem min ∈ φ[𝑥] separately on the boundary and on the interior of the set of the depositor's strategies 𝑋 (triangle a(1;0)o(0;0)b(0;1)) followed by the selection of the best solution: v.s.molostvov 106 copyright ©2024 assa adv. in systems science and appl. (2024) min ∈ φ[𝑥] = min { min ∈[ , ] φ[𝑥], min ∈[ , ] φ[𝑥], min ∈[ , ] φ[𝑥], min ∈ φ[𝑥]}. #(4.8) the results obtained can be restated in economic terms in the following form. proposition 4.1: if the optimal guaranteed risk is achieved by a combination of dollar and ruble components, then the solution of the form 𝑥 = (𝑥 , 0), 0 < 𝑥 < 1) wher 𝑥 = , 𝑖𝑓 𝛽 ≥ 0, , 𝑖𝑓 𝛽 ≤ 0, is optimal. the last two formulas can be combined into a single expression: 𝑥 = 𝛽 − 𝛽 (𝑠𝑖𝑔𝑛𝛽 + 1)/2 𝛼 + 𝛽 . proposition 4.2: if the optimal guaranteed risk is achieved by a combination of euros and rubles, the solution of the form 𝑥 = (0, 𝑥 ), 0 < 𝑥 < 1 where 𝑥 = , 𝑖𝑓 𝛽 ≥ 0, , 𝑖𝑓 𝛽 ≤ 0, is optimal. here also both formulas can be combined into a single expression: 𝑥 = 𝛽 − 𝛽 (𝑠𝑖𝑔𝑛(𝛽 ) + 1)/2 𝛼 + 𝛽 . proposition 4.3: if the optimal guaranteed risk is achieved by a combination of dollars and euros, then some solution from the set: {𝑥 = (𝑥 , 1 − 𝑥 ), 𝑥 = (𝑥 , 1 − 𝑥 ), 𝑥 = (𝑥 , 1 − 𝑥 )} ∩ 𝑋 where 𝑥 = , 𝑥 = , 𝑥 = is optimal. we speak about full diversification if the guaranteed risk function φ[𝑥] reaches the minimum value φ∗ on the set 𝑋 at some interior point 𝑋 and for all points of the boundary 𝑓𝑟𝑋 φ[𝑥] > φ∗. this condition means that the optimal diversification necessarily uses all three currencies. in the following, we restrict ourselves to this condition, since other possible situations are covered by propositions 4.1–4.3. proposition 4.4: if the optimal guaranteed risk is achieved only when all three currencies are used, then the optimal solution is the following: 𝑥∗ = (𝑥∗ , 𝑥∗) = ( 𝛽 𝛼 + 𝛽 , 𝛽 𝛼 + 𝛽 ) provided that 0 < 𝑥 , 𝑥 < 1 and 𝑥 + 𝑥 < 1. any point of a local minimum of a convex function on a convex set is simultaneously a point of global (on this set) minimum. hence, if the point of internal minimum 𝑥∗ exists and 𝑥∗ = (𝑥∗ , 𝑥∗) ∈ 𝑖𝑛𝑡𝑋, it will be the rgs. whether it exists or not depends on the relationship between the parameters of the problem under consideration. very often, the rgs is on the boundary of the set of admissible solutions. this means that the optimal diversification contains only two (or even one) of the three currencies. the list of all possible candidates for the optimal solution specified in propositions 4.1–4.4 should be accomplished by three single-currency variants: (0;0) – only rubles, (1;0) – only dollars, 107 savage's solution to three-currency deposit: program tools copyright ©2024 assa adv. in systems science and appl. (2024) (0;1) – only euros. as a result, we obtain an exhaustive set of nine solutions, at least one of which will be optimal. 5. algorithm for constructing a risk-guaranteed solution combining the results of propositions 4.1–4.4 leads to the following scheme [8] for computing the risk-guaranteed diversification strategy and the smallest guaranteed risk. step 1. set numerical values of the interest rates 𝑟, 𝑑 , 𝑑 and the current exchange rates 𝐾 , 𝐾 ; set numerical values of the boundaries of possible changes in dollar and euro exchange rates 𝑎 , 𝑏 and 𝑎 , 𝑏 , respectively. any available information can be used, e.g. expert estimates and forecasts, dm's own assumptions, etc. step 2. calculate the secondary parameters 𝛼 , 𝛼 , 𝛽 , 𝛽 from (4.6-4.7). step 3. compute the nine candidate points for the optimal solution. the corresponding formulas for these points are presented in table 1 below. step 4: remove points that do not belong to the set 𝑋. step 5: calculate the values of the guaranteed risk for the remaining points, select the solution with the best (minimum) guaranteed risk. table 1. candidates for solutions and verification of their admissibility conditions [8] n candidates for solutions formulas checking the conditions 1 the point o=(0;0) (rubles) (0;0) – 2 the point a=(1;0) (dollars) (1;0) – 3 the point b=(0;1) (euro) (0;1) – 4 the point (𝑥 , 0) (rubles and dollars) 𝑥 = 𝛽 − 𝛽 (𝑠𝑖𝑔𝑛𝛽 + 1)/2 𝛼 + 𝛽 0 < 𝑥 < 1 5 the point (0, 𝑥 ) (rubles and euro) 𝑥 = 𝛽 − 𝛽 (𝑠𝑖𝑔𝑛𝛽 + 1)/2 𝛼 + 𝛽 0 < 𝑥 < 1 6 the point (1 − 𝑥 , 𝑥 ) (dollars and euro) 𝑥 = 𝛼 𝛼 + 𝛽 0 < 𝑥 < 1 7 the point (1 − 𝑥 , 𝑥 ) (dollars and euro) 𝑥 = 𝛽 𝛼 + 𝛽 0 < 𝑥 < 1 8 the point (1 − 𝑥 , 𝑥 ) (dollars and euro) 𝑥 = 𝛼 + 𝛽 𝛼 + 𝛼 + 𝛽 + 𝛽 0 < 𝑥 < 1 9 the internal point in x (rubles, dollars and euro) (𝑥∗ , 𝑥∗) = = ( 𝛽 𝛼 + 𝛽 , 𝛽 𝛼 + 𝛽 ) 0 < 𝑥 , 𝑥 < 1 𝑥 + 𝑥 < 1 6. economic and game interpretation let us explain the economic meaning of the derived parameters of the problem. recall that 𝑟, 𝑑 , 𝑑 are the interest rates on the corresponding deposits (ruble, dollar, euro); 𝐾 and 𝐾 are dollar and euro exchange rates against the ruble, respectively, at which the currency is purchased at the beginning of the year; 𝑎 , 𝑎 are the lower and 𝑏 , 𝑏 are the upper limits of the dollar and euro exchange rates, respectively, at the end of the deposit period. these parameters are assumed to be known to the dm at the beginning of the year. however, the values of uncertain parameters, i.e. the final values of the dollar and euro exchange rates are known only to the precision of the sets: 𝑦 ∈ [𝑎 , 𝑏 ], 𝑦 ∈ [𝑎 , 𝑏 ]. income, measured in rubles, for a one ruble allocation (1 − 𝑥 − 𝑥 , 𝑥 , 𝑥 ) is determined by the linear function (2.1) with uncertain coefficients. v.s.molostvov 108 copyright ©2024 assa adv. in systems science and appl. (2024) the secondary coefficients 𝛼 , 𝛼 , 𝛽 , 𝛽 depend on the above initial parameters and have the following meaningful interpretations. the expression 𝛼 = (1 + 𝑟) − 𝑎 defines the difference between the yield (1 + 𝑟) of one ruble in a ruble deposit and the yield when using it to buy dollars at the beginning of the period at the rate of 𝐾 ruble/dollar and selling the total amount (1 + 𝑑 ) at the minimum rate 𝑎 at the end of the period. the parameter 𝛼 = (1 + 𝑟) − 𝑎 defines the difference between the yields of the ruble and the euro if the uncertain euro exchange rate takes the minimum possible value. coefficient 𝛽 (𝛽 ) sets the difference between the yield of one ruble invested in a dollar (euro) deposit and the yield of a ruble in a ruble deposit at the highest final dollar (euro) exchange rate. as noted above, the problem of the best-guaranteed income can be regarded as an antagonistic game between dm and nature. this game has, at the given uncertainty structure, a rather simple solution even in the case of any finite number of currencies [14]. savage's minimax regret principle is related to the consideration of a more complex structure, the target risk function (3.1), to which wald's minimax principle is then applied. thus, the problem of computing optimal risk can also be interpreted as an antagonistic game between dm and nature-uncertainty with the risk function as the payoff function. however, unlike the aforementioned dm-nature game with income as a payoff function, this game does not typically have a saddle point [12] in pure strategies. indeed, under the assumptions made, the maximin and minimax in this game exist but are not equal. the maximin is equal to zero, which is the optimal value of the risk function under the uncertainty known to the dm in advance, i.e. actually without uncertainty. the minimax for a non-negative risk function is non-negative and equals zero only in degenerate cases, when there exists a strategy that is optimal under any uncertainty. hence the absence of a saddle point [12]. in this paper, the dm's guaranteed strategy was found by direct computing ("by definition") the minimax of the risk function. in its original formulation, the problem under consideration is close to markowitz's problem of optimal portfolio structure [3], first formulated in 1952 and since developed in numerous publications. despite such a long history, it is still the case that "many investment efficiency indicators are based on markowitz's portfolio theory" [1]. in markowitz’s theory, the returns of individual portfolio assets are random variables with respect to which mathematical expectations and covariance matrices are assumed to be known [4]. the expected return of the whole portfolio and its variance are considered as two indicators to be optimized – return and risk. from a formal point of view, this is a two-criterion optimization problem [9] with linear (return) and quadratic (risk as variance of return) criteria and simple linear constraints. various methods of finding pareto-optimal solutions and ways to construct efficient portfolios have been developed [11]. most of them use techniques of parameterization, for example, optimization by one criterion while restricting the values of another criterion, to construct the pareto frontier. the subject of this paper is close to the topic of optimizing a portfolio with three assets, two of which are risky. however, there are two important differences – in the available information and in risk assessment. for a number of reasons, it is not always possible to obtain satisfactory estimates of the statistical characteristics of asset returns and one has to be satisfied with interval forecasts. the use of dispersion as a measure of risk is computationally convenient, but has wellknown disadvantages. among them is the need for information on the correlation of assets. the use of an alternative risk measure – savage risk – develops additional directions of portfolio theory and practice. 7. analytical solution in the special case consider the special case when 𝛼 = 𝛼 = 0. this means that at minimum final values of dollar and euro exchange rates all three currencies give equal returns. then, from the point of view of the wald principle, any admissible diversification plan gives the same (and, therefore, optimal) 109 savage's solution to three-currency deposit: program tools copyright ©2024 assa adv. in systems science and appl. (2024) result [14]. this is explained by the fact that this principle takes into account only the worst possible value 𝑦 = (𝑦 , 𝑦 ) ∈ 𝑌 for each strategy 𝑥 = (𝑥 , 𝑥 ) ∈ 𝑋. thus, in our case, obviously, for all 𝑥 will be 𝑦 = (𝑎 , 𝑎 ). in contrast to wald's principle, the savage optimal solution is found taking into account both the lower and upper bounds of uncertainty 𝑏 , 𝑏 and the coefficients 𝛽 , 𝛽 depending on them. the guaranteed risk function (4.5) takes the form φ[𝑥] = max{0, 𝛽 (1 − 𝑥 ), 𝛽 (1 − 𝑥 )}, 𝑥 ∈ 𝑋 (7.1) from the condition 𝛼 = 𝛼 = 0 and the definition of the coefficients 𝛼 , 𝛼 , 𝛽 , 𝛽 it follows that 𝛽 > 0, 𝛽 > 0. in addition, (1 − 𝑥 ) ≥ 0, (1 − 𝑥 ) ≥ 0 for 𝑥 ∈ 𝑋. therefore, φ[𝑥] = max{ 𝛽 (1 − 𝑥 ), 𝛽 (1 − 𝑥 )}, 𝑥 ∈ 𝑋 (7.2) recall that the set of admissible solutions has the form: 𝑋 = {𝑥 = (𝑥 , 𝑥 )|𝑥 + 𝑥 ≤ 1, 𝑥 ≥ 0 (𝑖 = 𝑑, 𝑒)} (7.3) obviously, a solution 𝑥 = (𝑥 , 𝑥 ) such that 𝑥 + 𝑥 < 1, cannot be the minimum point of the function (7.2), since a small increase in both variables will result in a decrease in the function. therefore, the point of minimum of the function φ[𝑥] on the set 𝑋 lies on the segment [a,b] = {𝑥 + 𝑥 = 1, 𝑥 ≥ 0, 𝑥 ≥ 0}, where 𝑥 = (1 − 𝑥 , 𝑥 ). in the particular case 𝛼 = 𝛼 = 0 taking into account the conditions 𝛽 > 0, 𝛽 > 0 we obtain from 4.3 as a corollary: proposition 7.1: if 𝛼 = 𝛼 = 0 then the guaranteed risk function 𝛷[𝑥] reaches the minimum value on the set 𝑋 at some point in the set {(1; 0), (0; 1), 𝑥 = (𝛽 /(𝛽 + 𝛽 ), 𝛽 /(𝛽 + 𝛽 )}. in the derivation of the corollary, it is taken into account that some points from proposition 4.3 are duplicated here and that condition 𝑥 ∈ 𝑋 is fulfilled automatically in this case. for the final choice of the solution, we compute the value of the guaranteed risk function at the candidate points: φ[(1,0)] = max{ 𝛽 (1 − 1), 𝛽 (1 − 0)} = 𝛽 , φ[(0,1)] = 𝛽 , φ[(𝑥 , 𝑥 )] = max{ 𝛽 𝛽 /(𝛽 + 𝛽 ), 𝛽 𝛽 /(𝛽 + 𝛽 ) } = 𝛽 𝛽 /(𝛽 + 𝛽 ). the last value is smaller than the values at the other two points. hence, the rgs in this case is unique and has the form: 𝑥 = 𝛽 𝛽 + 𝛽 , 𝛽 𝛽 + 𝛽 (7.4) it gives the minimum guaranteed risk φ = φ[𝑥 ] = 𝛽 𝛽 /(𝛽 + 𝛽 ). in the case of equal ruble profitability of all three currencies (at minimum values of the final exchange rates), the optimal solution in terms of guaranteed risk is to distribute the ruble between the dollar and the euro in proportion to the upper possible values of their final exchange rates. if 𝛽 = 𝛽 = 𝛽, the optimal solution is 𝑥 = (0.5; 0.5). in the general case, the analytical search for the optimal solution of the diversification problem is difficult and requires the use of numerical methods and algorithms. 8. computational tools and some results the suggested method is implemented in the excel 2016 environment in two variants – user and research . the first is designed for a user with minimal skills of working in a ready-made excel spreadsheet. the work is reduced to the input of the above-mentioned initial data and the analysis of the results. the corresponding spreadsheet is consistent with table 1 and contains the calculation formulas for the points, of which at least one is the desired solution. table 2 reproduces part of the spreadsheet to input data. table 2. input of initial data. input cells are highlighted in green v.s.molostvov 110 copyright ©2024 assa adv. in systems science and appl. (2024) table 3 shows a part of the excel spreadsheet with calculations corresponding to the given initial parameters from table 2. column b contains the text of the formulas from table 1, columns c and d contain the corresponding calculations. the results are checked by logical operators for the fulfillment of the condition 𝑥 ∈ 𝑋 (descriptions of condition are in the column j). if the condition is not fulfilled, the text "not candidate" is written in the cell of column e, otherwise the result of the calculation of the guaranteed risk function at the corresponding point is given. then the minimum of these results is calculated. the corresponding cell (or cells in case of non-singularity) is highlighted by means of conditional formatting, and the optimal solution is located to the left of it. according to the values of the target function at the other acceptable candidate points, it is possible to estimate the increase in risk when reducing the number of currencies used. table 3. calculating the optimal solution the research variant contains a number of additional features. for example, instead of using the initial data, it is possible to directly specify the secondary parameters 𝛼 , 𝛼 , 𝛽 , 𝛽 which also fully determine the solution. however, the direct assignment of the derived parameters cannot be arbitrary, since their values must be consistent with the original parameters (see formulas (4.6–4.7)). for example, as noted above, if 𝛼 = 𝛼 = 0, then the parameters 𝛽 , 𝛽 cannot be negative. table 4 corresponds to the case considered above: 𝛼 = 𝛼 = 0 at 𝛽 = 0,2, 𝛽 = 0,3. 111 savage's solution to three-currency deposit: program tools copyright ©2024 assa adv. in systems science and appl. (2024) table 4. the case of equal minimal currency returns: 𝛼 = 𝛼 = 0; 𝛽 = 0,2, 𝛽 = 0,3 the software implementation of the algorithm provides means for multivariate calculations, in which one or two parameters are varied. for example, by fixing the parameters 𝛼 , 𝛼 , it is possible to build a table showing the dependence of the optimal result on the varied values 𝛽 , 𝛽 . alternatively, it is possible to study the optimal risk dependence on the lower bounds of rates 𝛼 , 𝛼 when the values 𝛽 , 𝛽 are fixed . on the other hand, we can construct a table of the dependence of the optimal risk on the values of dollar and euro interest rates. there are also other experiments, for example, studying the dependence of the optimal deposit structure on the interest rate of one of the currencies. table 5 shows the results of multivariate calculations of the optimal risk at all combinations of upper bounds of rates 𝑏 from 97 to 117 and 𝑏 from 108 to 128 respectively with step 1. other parameters are fixed at the values from table 2. table 5. dependence of optimal risk on upper limits of exchange rates 𝑏 , 𝑏 . in the upper line are dollar exchange rate values, in the left column – euro it is convenient to analyze the obtained data using excel graphical tools. thus, fig. 1 shows graphs of the first 10 rows of table 5. for the remaining eleven values of 𝑏 the graphs are similar. the graphs for the columns of the table also look similar. as one would expect, the optimal risk is an increasing nonlinear function for each parameter 𝑏 , 𝑏 . 0,075541667 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 108 0,000 0,002 0,012 0,021 0,029 0,036 0,042 0,048 0,053 0,057 0,061 0,065 0,068 0,071 0,074 0,077 0,080 0,082 0,084 0,086 0,088 109 0,003 0,003 0,012 0,021 0,029 0,036 0,043 0,048 0,053 0,058 0,062 0,066 0,070 0,073 0,076 0,078 0,081 0,083 0,086 0,088 0,090 110 0,012 0,012 0,013 0,023 0,031 0,039 0,046 0,052 0,057 0,062 0,067 0,071 0,074 0,078 0,081 0,084 0,087 0,089 0,092 0,094 0,096 111 0,020 0,020 0,022 0,024 0,033 0,041 0,049 0,055 0,061 0,066 0,071 0,075 0,079 0,083 0,086 0,089 0,092 0,095 0,097 0,100 0,102 112 0,027 0,027 0,030 0,032 0,035 0,044 0,052 0,058 0,065 0,070 0,075 0,080 0,084 0,088 0,091 0,095 0,098 0,101 0,103 0,106 0,108 113 0,034 0,034 0,037 0,039 0,042 0,046 0,055 0,062 0,068 0,074 0,079 0,084 0,089 0,093 0,097 0,100 0,103 0,106 0,109 0,112 0,114 114 0,039 0,040 0,043 0,046 0,049 0,052 0,057 0,065 0,072 0,078 0,084 0,089 0,093 0,098 0,102 0,105 0,109 0,112 0,115 0,118 0,120 115 0,044 0,045 0,048 0,052 0,056 0,059 0,063 0,068 0,076 0,082 0,088 0,093 0,098 0,103 0,107 0,111 0,114 0,118 0,121 0,124 0,127 116 0,049 0,049 0,053 0,057 0,061 0,065 0,069 0,073 0,079 0,086 0,092 0,098 0,103 0,108 0,112 0,116 0,120 0,124 0,127 0,130 0,133 117 0,053 0,054 0,058 0,062 0,067 0,071 0,075 0,080 0,084 0,090 0,096 0,102 0,108 0,113 0,117 0,122 0,126 0,129 0,133 0,136 0,139 118 0,057 0,058 0,062 0,067 0,071 0,076 0,081 0,085 0,090 0,094 0,101 0,107 0,113 0,118 0,123 0,127 0,131 0,135 0,139 0,142 0,145 119 0,061 0,061 0,066 0,071 0,076 0,081 0,086 0,091 0,095 0,100 0,105 0,111 0,117 0,123 0,128 0,132 0,137 0,141 0,144 0,147 0,149 120 0,064 0,064 0,070 0,075 0,080 0,085 0,090 0,095 0,101 0,106 0,111 0,116 0,122 0,128 0,133 0,138 0,142 0,146 0,148 0,150 0,152 121 0,067 0,067 0,073 0,078 0,084 0,089 0,094 0,100 0,105 0,111 0,116 0,121 0,127 0,133 0,138 0,143 0,146 0,149 0,151 0,153 0,156 122 0,070 0,070 0,076 0,082 0,087 0,093 0,098 0,104 0,110 0,115 0,121 0,126 0,132 0,138 0,143 0,147 0,149 0,152 0,154 0,157 0,159 123 0,072 0,073 0,079 0,085 0,090 0,096 0,102 0,108 0,114 0,120 0,125 0,131 0,137 0,143 0,146 0,149 0,152 0,155 0,157 0,160 0,162 124 0,074 0,075 0,081 0,087 0,093 0,099 0,105 0,111 0,117 0,123 0,129 0,136 0,142 0,146 0,149 0,152 0,155 0,157 0,160 0,163 0,165 125 0,077 0,078 0,084 0,090 0,096 0,102 0,109 0,115 0,121 0,127 0,133 0,140 0,145 0,148 0,151 0,154 0,157 0,160 0,163 0,166 0,168 126 0,079 0,080 0,086 0,092 0,099 0,105 0,112 0,118 0,124 0,131 0,137 0,143 0,147 0,151 0,154 0,157 0,160 0,163 0,166 0,168 0,171 127 0,081 0,082 0,088 0,095 0,101 0,108 0,114 0,121 0,127 0,134 0,140 0,146 0,149 0,153 0,156 0,159 0,162 0,165 0,168 0,171 0,174 128 0,083 0,084 0,090 0,097 0,104 0,110 0,117 0,124 0,130 0,137 0,144 0,148 0,151 0,155 0,158 0,162 0,165 0,168 0,171 0,174 0,176 v.s.molostvov 112 copyright ©2024 assa adv. in systems science and appl. (2024) fig. 1. graphs of the dependence of the optimal risk on the parameter 𝑏 at different values of the parameter 𝑏 . row 1: 𝑏 = 108, …, row 10: 𝑏 = 117 table 6 shows the optimal deposit structures when the interest rate for the ruble changes from 15% to 6% with steps of -0.5%. other parameters are fixed at the following values: dollar rate – 3%, euro rate – 3%, initial exchange rates – 90 rubles/dollar and 100 rubles/euro, dollar exchange rate limits – from 85 to 105, euro – from 95 to 115. the last line shows the corresponding values of the optimal guaranteed risk. as can be seen, it decreases monotonically with the growth of the ruble interest rate. fig. 2 gives a visual representation of the dynamics. table 6. dependence of the deposit structure on the ruble interest rate fig 2. optimal deposit structure 0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 0.160 9 7 9 8 9 9 1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 5 1 0 6 1 0 7 1 0 8 1 0 9 1 1 0 1 1 1 1 1 2 1 1 3 1 1 4 1 1 5 1 1 6 1 1 7 m in im um g ua ra nt ee d ris k upper value for the dollar exchange rate dependence of optimal risk min𝛷[𝑥] on bd at different values of be row1 row2 row3 row4 row5 row6 row7 row8 row9 row10 ruble rate 15,0% 14,5% 14,0% 13,5% 13,0% 12,5% 12,0% 11,5% 11,0% 10,5% 10,0% 9,5% 9,0% 8,5% 8,0% 7,5% 7,0% 6,5% 6,0% dollar 0,075 0,075 0,075 0,075 0,075 0,075 0,075 0,075 0,075 0,075 0,075 0,075 0,075 0,075 0,513 0,513 0,513 0,513 0,513 euro 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,487 0,487 0,487 0,487 0,487 ruble 0,925 0,925 0,925 0,925 0,925 0,925 0,925 0,925 0,925 0,925 0,925 0,925 0,925 0,925 0,000 0,000 0,000 0,000 0,000 risk 0,048 0,052 0,057 0,062 0,066 0,071 0,076 0,080 0,085 0,089 0,094 0,099 0,103 0,108 0,109 0,109 0,109 0,109 0,109 0.000 0.200 0.400 0.600 0.800 1.000 1.200 deposit structure dollar euro ruble 113 savage's solution to three-currency deposit: program tools copyright ©2024 assa adv. in systems science and appl. (2024) table 7 shows the results of calculations of the minimum guaranteed risk for dollar and euro interest rates in the range from 0,25% to 5% with steps of 0.25%. other parameters are fixed at the following values: initial exchange rates – 90 rubles/dollar and 100 rubles/euro, dollar exchange rate limits – from 85 to 105, euro – from 95 to 115; ruble interest rate is assumed to be 12%. table 7. optimal guaranteed risk at different dollar (top row) and euro (left column) interest rates fig. 3 shows graphs of the dependence of the optimal risk on the dollar interest rate 𝑑 at the first 10 fixed values of the interest rate 𝑑 for euro (the first 10 rows of table 7). for the remaining 11 values of 𝑑 the graphs are similar. the graphs for the columns of the table also look similar. fig 3. graphs of the dependence of the optimal risk on the interest rate 𝑑 for dollar at different values of the interest rate 𝑑 for the euro. row 1: d = 0,25%, …, row 10: d = 2,5% 0,0755 0,25% 0,50% 0,75% 1,00% 1,25% 1,50% 1,75% 2,00% 2,25% 2,50% 2,75% 3,00% 3,25% 3,50% 3,75% 4,00% 4,25% 4,50% 4,75% 5,00% 0,25% 0,046 0,048 0,050 0,052 0,054 0,055 0,057 0,059 0,060 0,062 0,063 0,064 0,066 0,067 0,068 0,069 0,070 0,071 0,072 0,073 0,50% 0,047 0,049 0,051 0,052 0,054 0,056 0,058 0,059 0,061 0,062 0,064 0,065 0,067 0,068 0,069 0,070 0,071 0,072 0,073 0,074 0,75% 0,047 0,049 0,051 0,053 0,055 0,057 0,059 0,060 0,062 0,063 0,065 0,066 0,068 0,069 0,070 0,071 0,072 0,073 0,074 0,075 1,00% 0,048 0,050 0,052 0,054 0,056 0,058 0,059 0,061 0,063 0,064 0,066 0,067 0,069 0,070 0,071 0,072 0,074 0,075 0,076 0,076 1,25% 0,048 0,051 0,053 0,055 0,057 0,059 0,060 0,062 0,064 0,065 0,067 0,068 0,070 0,071 0,072 0,074 0,075 0,076 0,077 0,078 1,50% 0,049 0,051 0,053 0,055 0,057 0,059 0,061 0,063 0,065 0,066 0,068 0,069 0,071 0,072 0,073 0,075 0,076 0,077 0,078 0,079 1,75% 0,050 0,052 0,054 0,056 0,058 0,060 0,062 0,064 0,066 0,067 0,069 0,070 0,072 0,073 0,075 0,076 0,077 0,078 0,079 0,080 2,00% 0,052 0,053 0,055 0,057 0,059 0,061 0,063 0,065 0,067 0,068 0,070 0,071 0,073 0,074 0,076 0,077 0,078 0,079 0,081 0,082 2,25% 0,054 0,055 0,056 0,058 0,060 0,062 0,064 0,066 0,067 0,069 0,071 0,072 0,074 0,075 0,077 0,078 0,079 0,081 0,082 0,083 2,50% 0,056 0,057 0,058 0,059 0,061 0,063 0,065 0,067 0,068 0,070 0,072 0,073 0,075 0,077 0,078 0,079 0,081 0,082 0,083 0,084 2,75% 0,058 0,059 0,060 0,061 0,062 0,063 0,065 0,067 0,069 0,071 0,073 0,075 0,076 0,078 0,079 0,081 0,082 0,083 0,084 0,085 3,00% 0,060 0,061 0,062 0,063 0,063 0,064 0,066 0,068 0,070 0,072 0,074 0,076 0,077 0,079 0,080 0,082 0,083 0,084 0,086 0,087 3,25% 0,062 0,063 0,063 0,064 0,065 0,066 0,067 0,069 0,071 0,073 0,075 0,077 0,078 0,080 0,081 0,083 0,084 0,086 0,087 0,088 3,50% 0,063 0,064 0,065 0,066 0,067 0,068 0,069 0,070 0,072 0,074 0,076 0,078 0,079 0,081 0,083 0,084 0,085 0,087 0,088 0,089 3,75% 0,065 0,066 0,067 0,068 0,069 0,070 0,071 0,072 0,073 0,075 0,077 0,079 0,080 0,082 0,084 0,085 0,087 0,088 0,089 0,091 4,00% 0,066 0,067 0,068 0,070 0,071 0,072 0,073 0,074 0,075 0,076 0,078 0,080 0,081 0,083 0,085 0,086 0,088 0,089 0,091 0,092 4,25% 0,068 0,069 0,070 0,071 0,072 0,073 0,074 0,076 0,077 0,078 0,079 0,081 0,082 0,084 0,086 0,087 0,089 0,091 0,092 0,093 4,50% 0,069 0,070 0,071 0,073 0,074 0,075 0,076 0,077 0,078 0,079 0,081 0,082 0,084 0,085 0,087 0,089 0,090 0,092 0,093 0,095 4,75% 0,070 0,072 0,073 0,074 0,075 0,076 0,078 0,079 0,080 0,081 0,082 0,083 0,085 0,086 0,088 0,090 0,091 0,093 0,094 0,096 5,00% 0,072 0,073 0,074 0,075 0,077 0,078 0,079 0,080 0,081 0,083 0,084 0,085 0,086 0,087 0,089 0,091 0,093 0,094 0,096 0,097 0.040 0.045 0.050 0.055 0.060 0.065 0.070 0.075 0.080 0.085 0.090 0 . 0 0 % 1 . 0 0 % 2 . 0 0 % 3 . 0 0 % 4 . 0 0 % 5 . 0 0 % 6 . 0 0 % m in im um g ua ra nt ee d ris k dollar interest rate dependence of the opt imal r isk on the dol lar interest rate at d i f ferent values of the euro interest rate row1 row2 row3 row4 row5 row6 row7 row8 row9 row10 v.s.molostvov 114 copyright ©2024 assa adv. in systems science and appl. (2024) the results give a partial idea of the capabilities of the proposed method and its software implementation, which gives an experienced excel user other possibilities to analyze the problem under consideration by means of, for example, a "what-if" analysis. conclusion the paper considers the problem of the optimal allocation of deposits to three of currencies (ruble, dollar, and euro) with uncertain future exchange rates of the dollar and euro. it is assumed that only the limits of possible changes in these uncertain parameters are known. the concept of savage's minimax regret is used. according to this, the original problem is reduced to a minimax problem with the benefit lost due to uncertainty as the target function (the risk function). the risk function and the guaranteed risk function are found in explicit form. after that, the problem is reduced to finding the minimum point of a piecewise linear function under linear constraints. explicit formulas are obtained for nine "representative" candidate points, at least one of which is the optimal solution. the final choice is made by direct comparison of the criterion values at these points. computational tools have been developed in the excel environment to find the structures of a three-currency deposit, optimal in terms of guaranteed risk, under conditions of unknown future exchange rates. the use of this computing environment has allowed the development of tools that combine ease of use for individuals with minimal spreadsheet skills and extensive data analysis tools for advanced users. the method is implemented in two variants – user and research. the first one is designed for those who have minimal skills of working in a ready-made excel spreadsheet. this work is reduced to the input of the initial data and substantive analysis of the automatically obtained results. the second version contains a number of additional features: direct specification of the derived parameters of the profitability of individual currencies, multiple calculations varying one or two parameters of the problem, and a graphical visualization of the results. debugging and experimental calculations were performed. in particular, the case of the equal guaranteed profitability of all three currencies (at minimum future exchange rates) was analytically investigated and used for debugging the program. numerical modeling of the dependence of the minimum guaranteed risk on the maximum possible future values of currency rates was carried out. a monotonic increase in risk for each of these variables was revealed. the tables of dependence of the optimal deposit structure on the ruble interest rate (with other parameters fixed), as well as the tables of dependence of the optimal risk on the dollar and euro interest rates were constructed. the results can be used in analyzing problems of financial management under conditions of incomplete information. acknowledgements this article is an output of a research project “models of data analysis and decision making in the socio-economic sphere” implemented as part of the basic research program at the national research university higher school of economics (hse university). the work was partially supported by the international center of decision choice and analysis of the national research university higher school of economics. the author is grateful to the reviewers for their thorough and in-depth study of the paper, constructive comments, and valuable suggestions, which improved the paper significantly. 115 savage's solution to three-currency deposit: program tools copyright ©2024 assa adv. in systems science and appl. (2024) references 1. berzon, n. i. & doroshin, d. i. (2012). osobennosti primeneniia pokazatelei` effektivnosti` finansovykh investitcii`. [peculiarities of application of financial investment efficiency indicators]. finansy i kredit, 14, 21–33, [in russian]. 2. germeyer, y.b. (1971). vvedenie v teoriiu issledovaniia operatcii`. [introduction to the theory of operations research]. moscow, ussr: nauka, [in russian]. 3. markowitz, h. m. (1952). portfolio selection, j. of finance, 7(1), 77–91. 4. markowitz, h. m. (1990). mean variance analysis in portfolio choice and capital markets. basil, switzerland: blackwell. 5. molostvov, v. s. (1983). multiple-criteria optimization under uncertainty: concepts of optimality and sufficient conditions. in: theory and practice of multiple criteria decision making, north-holland, 91–105. 6. molostvov, v. s. 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(2014). multicurrency deposit diversification – three possible approaches to risk accounting, int. j. oper. and quant. manag., 20, 1–14. adv syst sci appl 2020; 02:131–145 published online at https://ijassa.ipu.ru. original russian text © a. v. kuznetsov, 2017, published in upravlenie bol’shimi sistemami / large-scale systems control, 2017, no. 66, pp. 68-93. allocation of limited resources in a system with a stable hierarchy (on the example of the prospective military communications system) alexander v. kuznetsov1* 1 voronezh state university, voronezh, russia abstract: the article is devoted to the formalization of the resource allocation problem in the hierarchical social system, where resources are distributed according to the agent’s role in the system. we study the application of a proposed mathematical model to the organization of the special purpose communication network. also, we propose an algorithm of the automatic partitioning of the radio communication system according to the role of nodes. none of the nodes has complete knowledge about the entire communication system, and the initially unknown information about the purpose of nodes is established in the process of the beacons exchange. keywords: resource allocation problem, graphs dissimilarity measures, abstract data structures, radio network planning, communication protocols, cognitive networks 1. introduction the problem of the limited resources allocation and an optimal schedule construction in network structures, including hierarchical structures of the transport type, is well known. a large number of articles on this topic appears every year. for example, we can mention the work [1] on the planning of the rail transportation, the article [2] studies the problems of resource allocation in hierarchical systems, the article [3] considers the problem of balancing the input and output parameters of a section of a gas transmission system, etc. usually in such problems, it is necessary to distribute resources between the vertices of a graph, and the flow of resources through the edges of the graph is somehow restricted. the solution to such problems will be, for example, an optimal (in some sense) schedule or distribution of resources (which are not elements of the graph itself) over the vertices of the graph. this article, however, describes a fundamentally different type of resource allocation problems, often found in military communications planning problems (and, to a lesser extent, in civilian communications), more precisely, in the problem of frequency allocation and assignment. in the described problem, the resource is the edges of a certain graph corresponding to communication channels. this resource should be distributed between pairs of vertices in accordance with a given template. the template itself, which later will be called the requirements graph, is similar to the network of needs and opportunities from [4]. the vertices consumers of the resources are users of the communication system and/or their telecommunication devices. moreover, the structure of the relationship between agents of the certain types in the system is much more constant than the composition of specific agents. the solution for this problem is a channel allocation variant that meets certain requirements. we will approach ∗corresponding author: avkuz@bk.ru 132 a.v. kuznetsov solving such problems, focused on working not with specific agents of the system, but with an abstract structure of the agent system, “untied” from the agents themselves. it is also worth noting that despite a large amount of literature about the frequency planning (see, for example, the monograph [5]), it focuses on the physical compatibility of frequencies and on the optimal placement of radio stations. the distribution and redistribution of already known electromagnetically compatible frequencies across several thousand mobile radio stations (and this is a common situation for a military formation of the tactical level) are addressed rather little. it is usually customary to formulate the problem of the optimal assignment of frequency channels as the problem of the minimal colouring of the graph, in which, unlike the classical formulation of the problem, the adjacency of the vertices is determined by the restrictions on the use of not only one colour, but also some of their combinations (see, for example, [6, 7]). however, this formulation usually does not take into account either the hierarchy of agents, to which frequencies must be assigned nor possible changes in the composition of agents nor a case of the lack of frequencies. at the same time, preparing a list of electromagnetically compatible frequencies is a rather difficult computational task and to solve it anew each time with each change in the composition of a radio network, as is sometimes suggested in more specialized publications, is an extremely hard task. there are other possible approaches to the spectrum management, which are mainly used in cognitive radio networks and based, for example, on game theory [8], but also oriented to a much simpler communication system (actually consisting of peer-to-peer devices with the “point-to-point” connection) than what is considered in this article. 2. statement of the problem for further discussion, we need to give a mathematical description of the communication system and “the communication system settings” (i.e. the channel resource allocation), using the concepts proposed by the author in [9]. the resource allocation problem mentioned in the introduction, in its simplest form, is usually formulated as follows. 2.1. objects let us define the following entities: 1. the set of the device classes ccomm = {ccomm1, . . . , ccomms}. 2. the set of the devices comm = {(comm1; ccommi1), . . . , (commn; ccommin)}, where commi is the identifier of the i-th device, and ccommij is the class of the ij-th device, 1 ≤ i ≤ s. 3. the set of the agents ag = {ag1, . . . , agm}. 4. the ownership relation between ag and comm fhave : comm→ ag. if for comm ∈ comm holds fhave(comm) = ag, then we say that the agent ag has the device comm. 5. the set of the channel classes cf = {cf1, . . . , cfl}. copyright © 2020 assa. adv syst sci appl (2020) allocation of resources on the example of military communications system 133 in real-world applications, a class corresponds to a frequency or a bandwidth, or similar channel characteristics. 6. the devices’ capabilities function fcomm : ccomm× cf → z, determining the maximal number of channels of a given type that can be formed simultaneously by a device of a given type; 7. the set of the channels f = {(f1; cfi1), . . . , (fp; cfip)}, where fi is the identifier of the i-th channel and cfij is the class of the ij-th channel, 1 ≤ i ≤ p. 8. the loop-free labelled requirements multigraph (an example of such a graph is shown in figure 2.1) req = (ag,ereq, ϕreq), ereq ⊆ ag2 × cf, ϕreq : ag × ereq → z is an incidence function, showing which agents should be interconnected as well as the number of channels in each connection; the channel classes act as edge labels; fig. 2.1. requirements graph req 9. the channel class capacity function fe : cf → n, determining how many agents can maximally use a channel of a given class. 2.2. conditions it is necessary to distribute the channels from f by all the devices from comm to obtain the communication graph γ = (comm,e, ϕ), e ⊆ f, ϕ : comm× e → {0, 1} is the incidence function, so that copyright © 2020 assa. adv syst sci appl (2020) 134 a.v. kuznetsov fig. 2.2. communication graph γ 1. each edge of the graph req corresponds to one edge of the graph γ or to a continuous path in the graph γ (a relay case); 2. each vertex ag of the graph req would correspond to a vertex comm of the graph γ such that fhave(comm) = ag; 3. the number of edges e of the graph γ marked with the same channel (f, cf) ∈ f does not exceed the capacity of the channel of this class fe(cf); 4. the number of edges e of the graph γ marked with the channel (f, cf) and incident to the vertex comm does not exceed the capabilities of the device fcomm(ccomm, cf). an example of the graph γ corresponding to the graph req shown in fig. 2.1 is presented in fig. 2.2. to summarize, in the model under study the incidence matrix of the graph γ, informally sometimes called “the radio frequency distribution variant”, is the sought variable. known parameters of the model are channels f , devices comm, agents ag, relationships fhave, fe, fcomm, and the graph req. the process of distributing channels from the set f between agents and devices belonging to them is often informally called “the communication system setup”. for simplicity, we assume that all channels from f are electromagnetically compatible. otherwise, it will be necessary to add more edge length labels γ and restrictions on possible channel assignments for close to each other devices, after which we will use already known methods such as the coordination rings method. 2.3. model deficiencies although the author did not see articles describing this type of a task of resource allocation, it is quite typical in certain areas of activity and solved by the ordinary brute-force search. concerning military communications (in which the agents are personnel of armed forces), such algorithm was implemented by the author as part of the large team in the form of the software developed by the jsc “concern “sozvezdie” [10]. in the traditional approach to the discussed distribution problem, the following disadvantages are observed: 1. the graph req is literally drawn by the operator for a fairly long time. each time the composition of the sets comm, ag is changed, it is necessary to redraw it again, which with the cardinality of the sets comm, ag in hundreds of elements can drag on indefinitely. 2. moreover, in reality, the set of channels f is not constant: usually, the channels drop out of it over time in an unpredictable way due to the intentional or unintentional interference. with a significant change in the set of channels, the graph γ, of course, needs to be redesigned. relations between new agents will have approximately same structure as between their predecessors. copyright © 2020 assa. adv syst sci appl (2020) allocation of resources on the example of military communications system 135 3. usually, agents fromag have unequal importance, and for each agi ∈ ag the weight αi can be set. this weight determines what part of the channels available in f can be allocated to this agent in a situation lack of channels on all agents. 4. the graph γ is brought to radio stations using various types of storage devices. this methodology dates back to the times when there were relatively few telecommunication devices and it was not difficult to group them. 5. the graph γ can also be sent over the communication channel, but the question arises: after all, γ sets the configuration of the communication channel (frequency, at least) and how to send γ when there are no configured channels at all? 6. finally, in the event of the accidental failure of the planning centre capable of calculating the graphs γ, the entire system generally gradually fails. the purpose of this article is twofold: 1. give the extended formulation of the previously formulated problem taking into account the above disadvantages. 2. describe the communication system (whether civilian or military, although the author initially worked with military communication systems), in which channels are not assigned in a centralized way, but are selected by each device based on the characteristics of the devices and the position of this devices in the whole system, thereby continuing the work begun in [11]. note that in the prospective military communications systems [12], currently being developed abroad, the problems 1–6 are being actively solved with the help of the cognitive network technology, in which all nodes have spectrum sensing ability, as in the previously proposed methodology of cognitive radio, and the general knowledge plane of the network, which contains, inter alia, the information about the form, purpose and state network nodes. among other technologies, the dynamic spectrum control is used, during which nodes consistently occupy or free one or another frequency channel depending on device’s position in the node hierarchy, changes in the interference environment, or other circumstances. in this article, we consider a particular problem of dynamic spectrum control in a cognitive radio network: the initial allocation of frequencies in accordance with the roles of agents. earlier, the author of [11] considered the organization of a radio network configuration over a previously unorganized communication channel through the exchange of configuration beacons (cb) and response beacons (rb) between a communication control centre (ccc) and radio stations which are necessary to configure. to do this, all the radio stations were supplied with the same grid of technical channels, the central control centre sent the cb on one of channels selected from the pre-set grid, while the radio stations scanned the abovementioned channel grid and, when the cb was detected, sent the rbs. after receiving the rb, the ccc sent to the radio stations pre-generated configuration data, including the operating frequency and other parameters. we assumed that the configuration data was calculated based on the information about the composition and structure of a custom communication system. the securing of the transmitted data was performed, using an asymmetric cryptosystem like rsa. this eliminated the need for the preliminary distribution of keys (the author do not consciously touch on the possibility of using rsa in russia for this purpose since this is a more political than a technical issue). it should be noted that automatic clustering algorithms have been known for a long time (see, for example, [13]). however, they are mainly used for the sensor networks’ organization, where the main principle of the cluster construction is the energy efficiency, rather than the position of the device in a hierarchy of some kind. copyright © 2020 assa. adv syst sci appl (2020) 136 a.v. kuznetsov 3. distribution of resources, depending on the role of the agent and in conditions of insufficient resources the unequal importance of agents from ag was already mentioned earlier. the problems of the specific organization of agents, which may be determined by the graph of the military hierarchy or other social structure, go far beyond the scope of this article. suffice it to say that each agent ag ∈ ag uniquely corresponds to the agent’s feature vector i : ag ↔ (ag1, . . . , agq), and the agent’s “weight” α(ag1, . . . , agq) = α(i(ag)) = α ◦ i(ag) , α ◦ i(ag) ∈ [0, 1] is defined. in the actual implementation of the system, which will be further described in the article, we will assume that the agent’s feature vector is stored in each agent. we can say that each agent even before the distribution of resources “knows” its (and only its) features. in military communication systems, such features may be the level of the military hierarchy to which the agent belongs, the type of agent’s military formation, a serial number of a device, and so on. i note that in military systems such information becomes secret only in aggregate and the fact that it is contained in the agent, more precisely, in the telecommunication devices belonging to the agent, does not compromise the entire system. the set of all agents’ feature vectors is denoted by s. thus, the entities of subsection 2.1 are supplemented by the entities: 10. the agent importance function α : s → [0, 1]; 11. the bijective function i : ag → s,∑ ag∈ag α ◦ i(ag) = 1. in the case of the lack of resources (channels), condition 1 of subsection 2.2 will be replaced by 1. ∑ ag∈ag ∣∣∣∣α ◦ i(ag)− |f (ag)| |f | ∣∣∣∣→ min, where f (ag) is the set of edges γ incident to at least one device from the set comm(ag) ⊂ comm, comm(ag) = {comm ∈ comm|fhave(comm) = ag}. as a result, the problem of channel allocation under conditions of limited channel resources will turn into a combinatorial optimization problem. if the graph req is set manually, the following happens. the human-compiler of the graph has in his mind a predicates pi : s |ag| → {0, 1}, allowing him to group agents into groups that should be connected on the basis of certain signs: pi(i(ag1), . . . , i(agm))→ ϕreq(agji , cfki) = ni ∧ ji ∈ ji ⊂ {1, . . . ,m} ∧ ki ∈ ki ⊂ {1, . . . , l}, (3.1) so, if some statement pi about agents ags, s = 1,m holds, then some of them should be connected by 1 ≤ ni ≤ p channels of class cfki . copyright © 2020 assa. adv syst sci appl (2020) allocation of resources on the example of military communications system 137 of course, these signs can be formalized, and the entire graph req of subsection 2.1 is completely replaced by 8. the predicate family preq = {pi}. thus, it is possible to replace the complexity of the constant (for each distribution of resources) construction of the graph req with the complexity of the initial classification of objects and the identification of signs of their association. when implementing the communication system, which will be described in the article below, it must be assumed that the set preq is also “contained inside” agent. it seems that one could restrict oneself to the introduced concepts when formalizing the task, but in the tasks of forming military communications networks (and indeed in mobile networks) the composition of the set of ag is not constant agents drop out over time, reinforcements arrive, etc., although the structure of relationships between agents remains largely unchanged. finally, such the situation is possible when it is necessary to sequentially distribute channels for several units of the same organizational structure and equipment, but different in composition. 4. structural similarity 4.1. definitions to complete the description of the problem posed in the article, it remains only to determine which agents can be considered similar. we can apply the approach from [14]: two graphs are considered similar if they are isomorphic or have isomorphic subgraphs. to simplify, we will assume that the graph req describes the needs for telecommunication, for example, of some ideal military formation consisting of exactly the same smaller units. definition 4.1: let req = (ag,ereq, ϕreq), req′ = (ag′, ereq, ϕ ′ req). assume that edges of req, req′ are marked with labels from the one labels set. a one-to-one mapping of sets of graph vertices h : ag → ag′ is said to be an isomorphism of the graphs req, req′, if it translates adjacent vertices with the edge marked by β to adjacent vertices with the edge marked by β. on the set of all subgraphs of the graph req (we denote it by 2req) we can use the graph isomorphism to introduce an equivalence relation: two subgraphs are equivalent, if they are homomorphic in the sense of the above definition. thus, it is possible to construct the factor set (2req/ ∼). in the above example of an ideal military formation, the equivalence classes req containing more than one subgraph are subgraphs corresponding to the telecommunication requirements for platoons, companies, battalions, and other military units of the same type. we say that two different equivalent subgraphs are maximal equivalent subgraphs if no subgraphs containing them are equivalent to any subgraph except themselves. the vertices ag1, ag2, h(ag1) = ag2 of two maximal equivalent subgraphs can be identified with each other by introducing the equivalence relation on the set of agents, assuming that ag1 ∼ ag2 if h(ag1) = ag2. denote ag = (ag/ ∼). definition 4.2: let req = (ag,ereq, ϕreq). the structure of the graph req is the graph req′ = (ag, ereq, ϕ̃req), where each vertex ag ∈ ag is replaced by the corresponding equivalence class ag ∈ ag, ag ∈ ag. copyright © 2020 assa. adv syst sci appl (2020) 138 a.v. kuznetsov for the most complete separation of the structure from req, we can use the following algorithm for isomorphism construction: input: minimum diameter of the selected subgraph δ > 0 1 req′ := req; 2 i := 0; 3 while diam(req′) > δ do 4 extract the maximum subgraph reqi which is isomorphic with the isomorphism hi to at least one other subgraph req′ from req′; 5 req′ := reqi // the way to select a subgraph will be described in 4.2; 6 i := i+ 1; 7 end 8 m := i; 9 for all reqj ,hj found do 10 if ag is the vertex of reqj and is not the vertex of reqj+1 then 11 construct the general isomorphismhδ,hδ(ag) = hj(ag); 12 end 13 exit; 14 end algorithm 1: structuring obviously, the construction of the factor set of agent classes can also be done through predicates from preq, assuming that agents ag1, ag2 are equivalent if the values of predicates from preq coincide on them when all possible values of other agents from the set ag are substituted into these predicates. thus, it is possible to consider not all features (ag1, . . . , agq) of the agent ag, but only those that are the same for all agents of the class ag, ag ∈ ag, and distribute resources not between agents, but between classes of agents, taking into account only the total number of agents. once we have selected the structure for a certain graph preq and then distribute channels over the given structure, we can use this distribution option for other requirements graphs, which can be presented as a variant of implementation for the source structure in whole or in part for a sufficiently large subgraph. in this case, it is possible to achieve both scalability and portability of the solution of the resource allocation problem to other sets of agents with the same structure, and to eliminate the dependence of the solution of the problem on the current composition of agents. it remains to consider the case if the requirements graph does not consist of subgraphs which are exactly isomorphic to each other, but of somewhat similar subgraphs? in this case, we define the measure of dissimilarity of the subgraphs as follows: definition 4.3: let req′ = (ag′, ereq, ϕ ′ req), req′′ = (ag′′, ereq, ϕ ′′ req), ag′ ⊆ ag, ag′′ ⊆ ag. assume that edges of req′, req′′ are marked with labels from the one set of labels. lethε be one-to-one mapping of graph vertices, hε : ag′ \bag′ → ag′′ \bag′′, translating adjacent vertices fromag′ \bag′ with the edge marked by β to adjacent vertices from ag′′ \bag′′ with the edge marked by β. then the corresponding tohε measure µ of graph dissimilarity is defined as µ(req′,req′′) = |bag′|+ |bag′′| |ag′|+ |ag′′| − |ag′ \bag′| . copyright © 2020 assa. adv syst sci appl (2020) allocation of resources on the example of military communications system 139 the so-defined measure of dissimilarity for isomorphic graphs req′, req′′ with an isomorphism h0 : ag′ → ag′′ is equal to zero. this measure, in essence, coincides with the biotopic distance, which, in turn, is a special case of the steinhaus distance [15]. definition 4.4: the mapping hε from definition 4.3 is called an ε-similarity of the graphs req′ and req′′ if µ(req′,req′′) < ε. if ε > 0, then one can define maximal ε-similar subgraphs on the analogy of maximal isomorphic subgraphs. that is, two ε-similar subgraphs are maximal if no subgraphs of the graph req containing them are ε-similar with any subgraph except themselves. the mapping hε generates the equivalence relation ag1 ∼ ag2 ≡ ag1 = hε(ag2), and all the above considerations about constructing the structure of the graph req. summing up, we can say that the problem of distributing channels between agents according to a given template is replaced by the problem of distributing channels between equivalence classes of agents according to a given template, followed by replacing the agent classes with arbitrary pairwise different representatives of these classes that are currently available in the system. in other words, the graph req of subsection 2.1 is completely replaced by 8. the structure of the requirements graph req′ = (ag, ereq, ϕ̃req) or its equivalent predicate set. 4.2. structuring the requirements graph as a clustering task the graph structuring described in the previous subsection is inherently close to the clustering problem known in machine learning [16]. to work with such a problem, we define the function ρ : s2 → r in the space of feature vectors of agents, ρ(sag1, sag2) > 0, ρ(sag1, sag2) = ρ(sag2, sag1), sag1, sag2 ∈ s, sag1 6= sag2, ρ(sag, sag) = 0, sag ∈ s, which shows how close the types of the two agents are. if the cluster structure of the graph req would be known in advance (in the case of a military communications system, this means that it is immediately completely known which regiments, battalions, and other units consist of the military formation for which we want to distribute frequency), then it would only be necessary to check the cluster of approximately the same size (differing by no more than e0 ≥ 0) for the isomorphism or for the ε-similarity. although there are several methods for constructing graph isomorphisms, obviously, not all copyright © 2020 assa. adv syst sci appl (2020) 140 a.v. kuznetsov of them are suitable for us. to select the desired mapping, consider the algorithm below. 1 if ag′, ag′′ ⊂ ag are two clusters, 0 ≤ |ag′′| − |ag′| < e0 then 2 identify with the vertex ag′ ∈ ag′ the vertex ag′′ ∈ arg min ag∈ag′′ ρ(i(ag′), i(ag)). (4.2) 3 end 4 introduce an additional condition in order to prohibit the choice of too dissimilar agents as similar “for lack of a better ones” ρ(i(ag′), i(ag′′)) < e1. (4.3) 5 get the family of mappings h1 = {mi : ag′ → ag′′}. from h1 we select only the injective mappingshε : d′ → ag′′, d′ ⊆ ag′, resulting in a family of mappings h2. // note that by virtue of (4.3) the map hε may not be defined on all ag′. 6 denote by b(hε) the number of edges such that if ag1, ag2 ∈ ag′ are connected by the edge with the label β, thenhε(ag1),hε(ag2) ∈ ag′′ are connected by the edge with the label β. 7 from h2, we choose a map with the largest by cardinality domain d′ and with the largest value b(hε). thus, when choosinghε, it is necessary to maximize the value of |d′|+b(hε). algorithm 2: construction of the isomorphism or the ε-similarity if the cluster structure req is not known in advance, and the agents are represented only by their feature vectors, since we know that the agents are organized hierarchically, a hierarchical clustering algorithm with the dendrogram construction can be applied. after that, it remains only to look at the dendrogram and apply algorithm 2 to clusters that are at the same level of the dendrogram. table 4.1. values of the function i for the agents from fig. 2.1 agent feature vector ag1 (1, 1, 1, 0) ag2 (1, 1, 2, 0) ag3 (1, 1, 3, 0) ag4 (1, 2, 4, 0) ag5 (1, 2, 5, 0) ag6 (1, 2, 6, 0) ag7 (1, 1, 7, 1) ag8 (1, 2, 8, 1) ag9 (2, 1, 9, 1) for illustration, consider the most simplified example of the graph req shown in fig. 2.1. let the agents represented on the graph have the feature vectors shown in the table 4.1, i.e. s ⊂ z4. here, if i(ag) = sag = (ag1, ag2, ag3, ag4), then ag1 is the agent’s level in the hierarchy, ag2 is the number of some community of agents within the hierarchy level (for example, “1 motorized rifle platoon of 1 motorized infantry battalion”), ag3 is the agent’s identifier, ag4 is the sign of whether the agent is the leader (for example, a platoon commander). as a measure of the dissimilarity of the objects ag1 and ag2, i(ag1) = sag1 = (ag11, ag 2 1, ag 3 1, ag 4 1), i(ag2) = sag2 = (ag12, ag 2 2, ag 3 2, ag 4 2) we use copyright © 2020 assa. adv syst sci appl (2020) allocation of resources on the example of military communications system 141 the function ρ(sag1, sag2) = 3∑ i=1 [agi1 − agi2] + 1 2 [ag41 − ag42], (4.4) where [x− y] = 0 if x− y = 0 and [x− y] = 1 in all other cases. fig. 2.1 with a dashed line shows that, for example, when clustering from agents with a dissimilarity measure ρ, which is strictly less than 2, we get the clusters ag′(2) and ag′′(2) (and possibly others). when forming clusters from agents with a dissimilarity measure ρ, which is less than 5, we obtain only one ag′(5) cluster. if we choose as the corresponding pair of agents from ag′(2) and ag′′(2) for which ρ is minimal (step 1 of algorithm 2), then we get the isomorphism h : ag′(2)→ ag′′(2) specified in table 4.2. in general, several options for constructing an isomorphism are possible, but due to the construction of the function ρ (4.4), the “leading” agent in one cluster will always go to the “leading” agent in the other, which is fundamentally important in applications. table 4.2. construction of the isomorphismh : ag′(2)→ ag′′(2) agent in ag′(2) ag1 ag2 ag3 ag7 agent in ag′′(2) ag6 ag5 ag4 ag8 the value of ρ 2 2 2 2 as a result of applying of the structuring algorithm with the minimum diameter of the extracted subgraph 2, we obtain the graph shown in fig. 4.3, ag1 = {ag1, ag6}, ag2 = {ag2, ag5}, ag3 = {ag3, ag4}, ag4 = {ag7, ag8}, ag5 = {ag9}. thus, we have the graph with fig. 4.3. structure req′ of the requirements graph from fig. 2.1 five vertices instead of the requirements graph with nine vertices. if we take a smaller minimum diameter of the selected subgraph, then it would be possible to construct a structure with three vertices. when constructing the communication graph γ, this can significantly accelerate the frequency distribution process, since it will be necessary to look through almost half as many vertices of the requirements graph. note that in the analysed case, the rules for constructing the graph req from the set preq mentioned in section 3 are formulated as follows: 1. if ag1 6= ag2 and ρ(i(ag1), i(ag2)) < 2, then the vertices ag1 and ag2 must be connected by an edge marked with cf2. copyright © 2020 assa. adv syst sci appl (2020) 142 a.v. kuznetsov 2. if (i(ag1)) 1 = 1 and (i(ag1)) 4 = 1, and (i(ag2)) 1 = 2, then the vertices ag1 and ag2 must be connected by an edge labelled cf1. 3. if (i(ag1)) 1 = 1 and (i(ag1)) 4 = 1, and (i(ag2)) 1 = 2, then the vertices ag1 and ag2 must be connected by an edge labelled cf2. if f = {(f1; cf2), (f2; cf2), (f3; cf2), (f4; cf1)} and fhave(commi; ccomm2) = agi, 1 ≤ i ≤ 9, fhave(comm10; ccomm1) = ag9, fhave(comm11; ccomm1) = ag7, fhave(comm12; ccomm1) = ag8, and also fcomm(ccomm1, cf1) = fcomm(ccomm2, cf2) = 16, fcomm(ccomm2, cf1) = fcomm(ccomm1, cf2) = 0, then we can easily obtain the communication graph shown in fig. 2.2 from the graph req shown in fig. 2.1, or from the structure req′ requirements graph (fig. 4.3) in accordance with the rules of section 2.2. 5. example of the communication system based on the structural approach we have presented a formal description of the problem of channel distribution over the agent system in accordance with a given template. this description corrects almost all the shortcomings of the conventional approach, given at the beginning of the work. there is only one question lefts: is it possible to build a communication system that generally eliminates a single distribution centre for a channel resource, but which can occupy communication channels in accordance with the mission of each agent, and not chaotically? as the answer to this question, we propose the telecommunication system depicted in fig. 5.4, which is a further development of the system described in [11]. let each agent ag ∈ ag “knows” its fig. 5.4. scheme of the hierarchical self-organizing network feature vector (descriptor) i(ag) = (ag1, . . . , agn) and the predicate family preq. further, each agent can have three roles: 1) the main communication control centre (mccc); 2) the communication control centre (ccc); 3) the network node – communication centre (cc). agents can change their roles as necessary. the cognitive radio device described in [17, 18] may be proposed as the prototype of the real telecommunication device with such functionality. all agents have one common technological grid of channels f ′ ⊂ f . the functioning of the communication system in brief looks like described in algorithm 3. the work of the algorithm is additionally illustrated by the uml diagram in fig. 5.5. as mentioned earlier, if necessary, the encryption and the authentication of the beacons can be carried out using an asymmetric rsa type cryptosystem. copyright © 2020 assa. adv syst sci appl (2020) allocation of resources on the example of military communications system 143 1 agm ∈ ag with the role “mccc” selects the best in certain sense channel ft ∈ f ′ and starts transmitting its descriptor dm = i(agm) on it as a part of a configuration beacon (cb), which also contains the response channel fr ∈ f ′. agents with other roles sequentially cycle through the channels from f ′ and wait for some descriptor to be received. at the same time, “mccc” is waiting for answers on the channel fr ∈ f ′ 2 if the agent ag1 with the role of “ccc” or “cc” and the family of predicates preq found cb with the descriptor d2 ∈ s of the agent ag2, such that from the predicates in accordance with (3.1) ag1 and ag2 should be connected in the process of cycling through the channels, then ag1 sends a response beacon (rb) via channel fr containing the descriptor d1 = i(ag1) and the transmitter identifier comm ∈ comm. 3 when “mccc” receives a sufficient amount of rbs, it constructs a fragment of the graph req from the descriptors of the agents and the corresponding transmitter identifiers and solves, according to the rules preq inside the mccc, the problem of distribution of the channels for this fragment. the resulting fragment of the communication graph γ is sent to all registered senders of the rb as a configuration data packet (cd). thus, “mccc” divides the frequencies set f into the subsets fi, ∪ifi = f between the agents agi with the role of “ccc”. 4 the agent agi with the role of “ccc”, after receiving the cd, configures its devices in accordance with cd and starts transmitting cb at a certain frequency f it ∈ fi, similar to the same as “mccc”. 5 assume that the agent ag1 has the role of “cc” and the family of predicates preq. also, assume that the agent ag2 with the descriptor d2 ∈ s must be connected with a1 in accordance with (3.1). if ag1 in the process of searching the channel founds cb from “ccc” with the descriptor d2 ∈ s, then ag1 sends the response beacon (rb) via channel fr, containing the descriptor d1 = i(ag1) and the transmitter identifier comm ∈ comm. 6 when “ccc” receives a sufficient amount of rbs, it builds a fragment of the graph req from the descriptors of the agents and the corresponding identifiers of the transmitter and decides according to the rules inside it preq for it distribution channel resource problem. the resulting fragment of the communication graph γ is sent to all registered senders of the rb as the configuration data packet (cd). thus, “ccc” distributes the frequencies from f between agents agi with the role “cc”. algorithm 3: operation of the communication system 6. conclusions and perspectives the article presents a mathematical description of a special kind of the resource allocation problem in accordance with a given template. an algorithm for the structuring the set of resource consumers is also proposed. it should be noted that this problem is not necessarily related to radio communication systems and frequency distribution. “channels” may correspond, for example, information channels of different bandwidth with specified qos between users of different ranges in the wide area network. also, there can be options for laying railway tracks between settlements with “given channels” for a given matrix of connectivity of settlements and with a restriction on the total length of rails. the choice of the problem statement presented in the article using the conceptual apparatus of graph theory is connected with the fact that the author used precisely such constructions in creating a simulation model of the movement and the communication of hierarchically organized agents, including for visualizing changes in the state of the communication system. the above model is based on a cellular automaton simulating the copyright © 2020 assa. adv syst sci appl (2020) 144 a.v. kuznetsov fig. 5.5. beacons exchange movement of groups of agents over rough terrain [19]. moreover, the vertices of the graph γ at each moment of time are mapped to the cells of the automaton in which the agents are located. in the process of movement, the agents lose each other out of visibility due to the characteristics of the terrain being travelled, as a result of which there is a need to rebuild the communication graph γ. the author thinks the requirements graph as the imaginary device that generates an endless stream of communication graphs according to certain rules. so, the structure of the communication system can be represented by entities of the codata type. further development may be in the direction of improving measures of dissimilarity for communication requirements graphs. for example, we can represent the communication requirements graph in the form of the state transition model and replace the isomorphism of subgraphs in the definition of the dissimilarity of subgraphs with relations of mutual simulation or bisimulation. references 1. lazarev, a., musatova, e., gafarov, e., & etc. (2012) teorija raspisanij. zadachi zheleznodorozhnogo planirovanija. nauchnoe izdanie. m.: ipu ran. 2. afraimovich, l. g. & prilutskii, m. k. (2006) multiindex resource distributions for hierarchical systems. autom. remote control, 67, 1007–1016, https://dx.doi.org/10.1134/s0005117906060130. 3. prilutsky, m. k. & dikarev, k. i. (2012) resource allocation in hierarchical systems containing active components [in russian]. vestnik nngu, 181–189. 4. vittikh, v. a. & skobelev, p. (2003) multiagent interaction models for constructing the needs-and-means networks in open systems. automation and remote control, 64, 162– 169. 5. byhovskij, m., vasil’ev, a., & lashkevich, a. (2012) osnovy upravlenija ispol’zovaniem radiochastotnogo spektra: chastotnoe planirovanie setej teleradioveshhanija i podvizhnoj svjazi. avtomatizacija upravlenija ispol’zovaniem radiochastotnogo spektra, vol. 3. moskva: krasand. 6. geniatulin, k. & nosov, v. 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(2017) a model of the joint motion of agents with a threelevel hierarchy based on a cellular automaton. computational mathematics and mathematical physics, 57, 340–349, https://dx.doi.org/10.1134/s0965542517020099. copyright © 2020 assa. adv syst sci appl (2020) introduction statement of the problem objects conditions model deficiencies distribution of resources, depending on the role of the agent and in conditions of insufficient resources structural similarity definitions structuring the requirements graph as a clustering task example of the communication system based on the structural approach conclusions and perspectives microsoft word 782-article text-2783-1-6-20200403.doc adv syst sci appl 2020; 02; 1-19 published online at https://ijassa.ipu.ru. effect of fiber orientation and residual stresses on the structural performance of injection molded short-fiberreinforced components giorgio ramorino1, silvia cecchel1, giovanna cornacchia1* 1) dimi, department of industrial and mechanical engineering, university of brescia, via branze 38, 25123 brescia, italy. e-mail: giovanna.cornacchia@unibs.it. abstract: the increasingly need of lightweight structures in different sectors is driving new researches about the substitution of metal with polymeric matrix composite. objective of this work is to optimize the strength design of short-fiber injection molding manifold block, originally made with brass, with the aid of a three-dimensional flow and structural analysis simulations. these finite element analyses (fea) were based on orthotropic, linear and elastic models including process-induced residual stresses. through an appropriate interface, the results of injection molding analysis of 35 wt.% fiber reinforced polyphthalamides (ppa) manifold block are transferred to the structural analysis software. autodesk moldlfow software was used to predict the fiber orientation and the in-cavity residual stresses considering the flow kinetics and moulding parameters. the abaqus interface for moldflow was used to translate this data into a form that can be used for the structural analysis. finally, this paper reports some experimental tests carried out on the injection moulded component in order to evaluate the internal burst pressure values. the results not only highlighted the importance of carrying out structural simulations, which consider both the orientation of the glass fibre and the residual stresses given by molding, but also showed the usefulness and accuracy of an integrated cad-fea approach. the application on a real case demonstrated good agreement with numerical analysis predictions. keywords: moldflow, abaqus, residual stress, injection moulding, short-fiber-reinforced components, metal replacement 1. introduction nowadays the research of technical solutions to answer at the lightweighting issue connected to emissions reduction, safety and economic targets is constantly increasing [1-3]. advances in researches concerning the materials choice are required in several field. in this context, it is possible to observe not only substitution of i.e. steel with light alloys [4-8] but ever more frequently, where possible, metal replacement with polymer matrix composites [910]. today, the metal replacement is present in different industrial sectors, i.e., automotive, medical and electronics, where benefits like weight reduction, corrosion resistance and improved weldability can be achieved with low costs [11-16]. in this persistent search of lightness and cost reduction, the use of thermoplastics composite materials is continuously increasing in place of metal parts. the most common method for plastic part manufacturing is injection molding, which is suitable for the mass production of complex short fiber plastic parts with precise dimensions [17]. the properties of plastic materials can vary over a wider range than all other engineering materials and they can be enhanced with the addition of various agents. fiber reinforced thermoplastics are a category of composite plastics used in technical applications at increasing loads or elevated temperatures, because they have a higher stiffness, strength, and a reduced tendency to creep compared with non-reinforced polymers. for instance, short-fiber-reinforced thermoplastic products obtained by injection molding are currently successfully used in metal replacement applications [18-22]. it is worthwhile to * corresponding author: giovanna.cornacchia@unibs.it 2 g. ramorino, s. cecchel, g. cornacchia copyright ©2020 assa adv. in systems science and appl. (2020) note that the structural performances of these products are greatly affected by the fibers’ orientation state [23-25]. during the filling stage of the injection molding process, the polymer molecules and reinforcing fibers (for fiber-filled polymers) are oriented in a direction influenced by the shear flow. this leads to anisotropy in the mechanical properties of the material. fiber orientation also causes differential shrinkage in the longitudinal and transverse directions, which can lead to warpage (bending and twisting of a molded part after it has been ejected from the mold and cooled down to ambient temperature). further, this orientation improves mechanical properties in the fiber direction while reduced the strength in the transverse direction [26-27]. therefore, the prediction of fiber orientation using computer-aided engineering (cae) tools is of great importance for the safe design of shortfiber reinforced thermoplastic parts [28-29]. the calculated fiber direction can lead to local mechanical properties, thus allowing precise structural analysis of a moulded component. if a fiber-reinforced material is used to produce the part, the formation of the weld-lines is one of the most significant problems for structural applications due to the potential failure in correspondence of these discontinuities [30-33]. weld lines occur when two or more flow fronts do not blend completely when they meet in the mould. this happens, for example, when the melts splits as it flows around an obstacle such as a core or pin in the mold cavity or when parts are gated at several points. these necessary features represent discontinuities during the filling of the mold and induce the formation of weld lines with a further potential increase of the damage since these are stressed areas. the quality-reduction factors in the weld line area are highly dependent on the type and content of filling and reinforcing material. the existence of weld-lines will generally reduce the local material strength by greater than fifty percent in glass-reinforced resins due to unfavorable molecular orientation, insufficient bonding and/or formation of a v-notch [34-36]. therefore, a part containing one or more weld lines that are subjected to a localized structural loading will not perform as well as a component containing no weldlines. weld-lines strength can be improved by modifying the geometry of the parts or changing the structure of the mould or adjusting the process parameters such as melt temperature, injection velocity, holding pressure, holding time, and mold temperature. injection moulding usually induces residual processing stresses (rps), stored in the components after ejection from the mould and cooling to ambient temperature, which are originated from two main sources: “flow-induced stresses” as well as “thermally and pressure induced stresses” [37-39]. the rps exist in the moulded part without application of any external load and may affect the plastic component similarly to externally applied load. thus, especially for structural components, rps have to be predicted and minimized through the determination of optimum processing conditions. this is particularly important for component with complex geometry like manifold block (figure 1) where the part is characterized by areas with high thickness. as a result, if the rps are strong enough to overcome the structural integrity of the part, the component may warp as a consequence of different sources: upon ejection from the mould, after its use for an extended period of time or due to the exposure to relatively high temperatures. some consequences are poor dimensional stability and potential cracks when external service load is applied due to a strength reduction. it is worthwhile to note that, weld-lines and fiber orientation together with the rps are the major factors that determine the mechanical strength and stiffness of a moulded component. design engineers typically solve the problem by applying a generous safety factor to their design, making them thicker than is actually required. concerning the orientation direction and the degree of orientation of the fibers, it is evident that in areas where fibers are strongly aligned, the material will have higher strength characteristics and higher modulus in that direction, but will be relatively weak with much lower modulus in the perpendicular direction (across the fibers). because the injection moulding process for short-fiberreinforced components can cause such great variation in strength and stiffness through the part, the effect of the injection process should be considered in the design of such parts. effect of fiber orientation and residual stresses on the structural performance of injection molded short-fiber-reinforced components 3 copyright ©2020 assa. adv. in systems science and appl. (2020) predicting the position of the weld-lines, as well as the amount of rps, is useful in the initial stage of product development, in order to avoid modifications on the part and mold design after their manufacturing. mould filling simulations during the preliminary design step could help to increase the final quality of the component, thanks to the opportunity of improving design and process parameters according to the indications given by simulations. further advantages are a reduction of the time to market with consequent lower costs. in particular, the use of this new design approach is increasing during the last years and consists in flow simulations and structural analysis employed at the same step of the product development to optimize the strength design. a comparison between this advanced approach and the traditional one, without mould filling simulations, is sketched in figure1. figure 1. advanced approach: coupling of flow simulations and structural analysis employed to optimize the strength design of component in the literature, to the best knowledge of the authors, there are few works that try to couple a three-dimensional flow simulations and structural analysis but not based on both orthotropic, linear, elastic simulations and includes rps [40-42]. a number of researchers have studied the flow induced anisotropy in short-fiber-reinforced composites employing experimental techniques and numerical simulations. however, in these research works, the residual processing stresses (rps) (stresses that remaining inside the moulded product under the condition of no external loads) are typically neglected in structural analysis where external loads are applied to the components or considered in structural analysis only to determine the optimum processing conditions, residual stress distributions, and final shrinkage and shape of the components. kulkarni et al. [43] conducted flow and structural analysis on a reinforced square plate composed of composites, by successfully coupling two commercial codes, moldflow and ansys. their results showed that unlike conventional materials, where simple isotropic models can be employed, the simulation with orthotropic configurations are to be employed to gain a better agreement with the experimental results. in particular, load vs. deflection isotropic results are 70% off as compared to experimental results while the othotropic model exhibits a better correlation with more than 92% accuracy. current autodesk moldflow 3d products solve the standard folgar-tucker equation for prediction of the fiber orientation distributions. folgar and tucker developed a continuum model for a non-dilute suspension based on jeffery’s model that is used for dilute fiber suspensions, but they included a term that accounts for fiber-fiber interactions which tend to reduce highly aligned orientation states predicted by jeffery's model for some flow conditions [44]. it was demonstrated that simulation results agree remarkably well with measured microstructural data [45]. a typical example concerns the injection molding of structural components used in hot-water fluid engineering application, where tubes as well as holes have to be integrated in the parts. 4 g. ramorino, s. cecchel, g. cornacchia copyright ©2020 assa adv. in systems science and appl. (2020) following these premises, in this experimental work a feasibility study was conducted on a manifold block, a device that regulates fluid flow between pumps and actuators and other components in a hydraulic system. aim of this analysis is to evaluate the possibility of substitute the brass, material usually employed for this component, with 35 wt.% fiber reinforced polyphthalamides (ppa). in this context, the present paper reports an actual case study where the results of flow simulations and structural analysis were combined to optimize the strength design of a shortfiber-reinforced component. the software autodesk moldflow was used to estimate the incavity residual stress and the fibers orientation in a moulded component while the structural software abaqus was used to evaluate how residual stresses and fibers’ orientation state affect the final shape and performance of the product (see figure 1 – advanced approach). the results of injection molding analysis of 35 wt.% fiber reinforced polyphthalamides (ppas) manifold block used in hot-water fluid engineering application (autodesk moldflow) are transmitted to the structural analysis program (abaqus). the abaqus interface for moldflow is used to translate these data into a form that can be used by abaqus for the structural analysis. after that, experimental tests are carried out on the injection moulded components in order to evaluate the internal pressure that the component can handle before rupturing called “burst pressure”. the burst strength of the molded manifold block samples is then compared with those predicted by the numerical analysis. the simulations are based on orthotropic, linear, elastic simulations that includes the process residual stress and could play a key role for well-founded predictions in order to optimized part strength, part geometry, mold design and processing conditions. 2. materials and methods 2.1. description of the component structure and materials figure 2 shows the selected manifold block, component used for hot-water fluid engineering application. the design of either the original brass ((a) and the new metal replaced application composed of 35 wt.% fiber reinforced polyphthalamides (ppa) (b) can be observed. the maximum dimension of the original component is 80x100 mm. figure 2. manifold block used for hot-water fluid engineering application brass (a) replacement by 35 wt.% fiber reinforced polyphthalamides (ppas) (b) the substitution of brass with 35 wt.% fiber reinforced ppa led to a final weight of 190 g, with a weight reduction of about 60% in comparison with the 480 g of the traditional one. the cost analysis reported in figure 2 shows a saving of about 65% respect to the original component. effect of fiber orientation and residual stresses on the structural performance of injection molded short-fiber-reinforced components 5 copyright ©2020 assa. adv. in systems science and appl. (2020) 2.2. mould filling analysis the material used for the injection moulding simulations was polyphthalamides (ppa) having 35% short glass fiber (gf) content (see properties listed in table 1). table 1. properties of zytel htnfg52g35hslbk011 from dupont engineering polymers (ppas+35%gf) stress at break (iso 527) – cond. 180 mpa tensile modulus (iso 527) cond. 12 gpa elongation at break (iso 527) cond. 2.7% hdt (iso 75-1/-2) 1.8 mpa 285°c glass transition temp. (iso 11357-3) 90°c melting temperature (iso 113571/-3) 310°c water absorption (iso 62) 24h. 0.4 % short-fiber-reinforced ppa material was chosen due to excellent preservation of strength and stiffness at elevated temperatures, great resistance to hydrolysis, high dimensional stability and low warpage [46-47].the mould was designed with a single cavity. the parts were filled with a cold submarine gate and the cavity pressure was controlled via a direct pressure transducer placed in the mould cavity located to the end of the flow path. the injection location, the size of sprue, runners and gate as well as the lay-out of the cooling channels were optimized based on the filling analysis using cae technology in order to minimize the warpage of the part, decrease cycle times, increase part quality and achieve dimensional stability of the component. more details will be reported in section 3.2. solid model of the boiler manifold block with the fed system is designed by solidworks software for the mould filling analysis. three-dimensional flow simulations based on a solid, tetrahedral finite-element volume mesh were performed with autodesk moldflow software. figure 3a shows the boiler manifold 3d model designed with solidworks, while figure 3b reports the three dimensional tetrahedral mesh created by moldflow mpi. in particular, the finite element mesh employed was tetrahedral – c3d4, having 910’000 elements. the analysis comprised the entire molding process from injection, through mold packing, to component solidification. this kind of simulations also permitted to predict material anisotropy, fibre orientation, in-cavity residual stresses and component warpage /shrinkage. the cooling circuits design, employed to model filling, packing and cooling stages of the injection moulding are reported in figure 3c. several iterations were carried out to optimize the injection location and the processing parameters in order to achieve the most balanced flow into the cavity along with uniform pressure drop and flow front temperature. in order to choose the optimum process settings, different conditions affecting injection molding are considered, i.e. injection time, packing holding time and pressure, mold temperature and melt temperature. the detail of optimum processing parameters is summarized in table 2. 6 g. ramorino, s. cecchel, g. cornacchia copyright ©2020 assa adv. in systems science and appl. (2020) figure 3. boiler manifold block: a) model imported in autodesk moldflow (mpi) from 3d cad system (solidworks); b) 3d-tetrahedral mesh model created by moldflow mpi; c) cooling circuits design table 2. processing parameters and mesh type for the mould filling analysis (autodesk moldflow mpi) mould temperature 90°c melt temperature 330°c fill time 1.4 s packing holding time 15 s packing holding pressure 65 mpa cooling time 34 s mesh type tetrahedral – c3d4 n° of elements 910.000 special attention was given to reducing weld-lines and increase their resistance by keeping the melt at high temperature. indeed, based on previous experience from laboratory tests it is worthwhile to note that the presence of weld lines reduces the ultimate tensile strength of gf ppa produced by injection moulding at a range between 65 to 75 mpa. in addition, once known the fiber orientation in the parts the mechanical properties were calculated. in particular, moldflow analysis is based on a well-established and theoretically well founded micromechanical model proposed by tandon-weng [48]. the thermomechanical property calculation for fiber-filled composites are based on the orthotropic assumption, that fiber-filled material properties are different in three orthogonal principal directions. the presence of fibers often results in a fairly linear elastic response before failure. in such cases, an orthotropic elastic material model serves as a reasonable model, which can be used to model anisotropic situations. effect of fiber orientation and residual stresses on the structural performance of injection molded short-fiber-reinforced components 7 copyright ©2020 assa. adv. in systems science and appl. (2020) moldflow pack analysis also calculated the in-cavity residual stresses developed during the mold filling process along with the material properties required for the orthotropic structural analysis that represent the stresses in the part before it is ejected. the mold physically prevents the material from shrinking while the part is in the cavity, so these results are nearly not affected by other external factors and are a good representation of the real condition. 2.3. geometry optimization: from metal to plastic part in order to obtain metal replacement and have a polymeric part resistant to the real stresses it has been necessary to re-design the component. some modification has been performed, the initial and the final geometry are shown in the figure 4. figure 4_a2, b2, c2 show the geometrical re-design. ribs (dark grey) have been added, in different areas, in order to obtain a plastic part enough resistant to the operating loads. some other areas have been modified, as shown in the circle in figure 4_c2. the geometry in the right circle in figure 4_c2 has been designed to allow a subsequent introduction of a threaded insert. thicknesses have been increased in many areas of the component in order to obtain a part resistant enough to undergo the operating loads. in figure 4_c1 are shown the main quotas of the thickness of the original metal part, in figure 4_c2 are shown the same quotas re-designed for the plastic part. both, figure 4_c1 and figure 4_c2 show a section of the part to make an easier comprehension of the modification done. besides, in the circle of the figure 4_c2 is shown the emptying performed in order to obtain a part suitable for the injection molding process. during the re-design of the part multiple steps of structural linear elastic analysis have been made in order to understand if the direction taken was good enough to obtain the desired performance. the anisotropic analysis (which uses moldflow data results as input) has been made only as the final check step on the one that was considered the optimized geometry. indeed, even if this analysis is more suited for these component and material it is more time and power-consuming. the boundary conditions used were the same as the one shown in figure 6, with the only exception of the value of the pressure which was set to 12 bar (value of a possible water hammer in hot-water fluid application). in figure 5, three different steps of the design-analysis-design modification cycles can be seen as example. the results show the von mises stresses on the part (considered made in ppa+gf) measured with fem done on starting geometry (as was in metal figure 5_a); one of the intermediary geometry (close to the final one, thicknesses have been increased as shown in figure 4_c2); the optimized geometry (some other ribs have been added – figure 5-c). the main difference between geometry in figure 5_b and figure 5_c are the ribs highlighted in figure 4_a2. these ribs, as can be seen comparing figure 5_b and 5_c, do not help significantly to decrease the stress, but they are useful during the molding of the part to vent the weld line that forms in the that area. 8 g. ramorino, s. cecchel, g. cornacchia copyright ©2020 assa adv. in systems science and appl. (2020) figure 4. geometrical re-design from metal to optimized plastic component effect of fiber orientation and residual stresses on the structural performance of injection molded short-fiber-reinforced components 9 copyright ©2020 assa. adv. in systems science and appl. (2020) figure 5. abaqus analysis study with isothotropic material properties: distribution of mises stress; (a) starting geometry – as was in metal; (b) intermediary geometry (close to the final one, thicknesses have been increased as shown in figure x_c2; (c) optimized final geometry 2.4. orthotropic structural analysis after cooling, when the injection moulded part is ejected from the mould, it undergoes warpage caused by the in-cavity residual stress and by additional actions of temperature gradient to room temperature. in particular, once removed the mold constraint a certain amount of the residual stress is relieved, the part shrinks and may warp. in order to analyse these deformations rps data are needed. during the present work, the rps data obtained by moldlfow are used for warpage analysis with abaqus software. from these analyses, shrinkage and rps on the boiler manifold block are predicted. based on the local fiber orientation states, calculated through mould filling analysis, the local thermoelastic properties can be calculated and used as input for the structural fea. during the present work abaqus software was employed. for the reasons mentioned above, structural simulations (in which an external load is applied) with the orthotropic, liner, elastic material should take into account also the rps for an accurate result. figure 6 shows boundary conditions highlighting the areas were the fixed constrains and the pressure are located. the internal pressure was set at 45 bar. 10 g. ramorino, s. cecchel, g. cornacchia copyright ©2020 assa adv. in systems science and appl. (2020) figure 6. abaqus boundary conditions for the structural analysis 2.5. injection moulding after the definition of the design and the process parameters, the injection moulded boiler manifold blocks (component shown in the figure 2b) were obtained by using a 200 t injection moulding machine (bmb-mc200) equipped with a 55 mm diameter screw. for the components molding, polyphthalamides (ppa) having 35% short glass fiber (gf) content (zytel htn from dupont), having mechanical properties reported in table 2, was employed. the resin was pre-dried in a convection oven for at least 8 h at 100 °c to remove any moisture from the pellets before processing. the dimension of the boiler manifold block and location of the submarine gate are shown in figure 3. 2.6. sem analysis a leo evo 40 scanning electron microscope (sem) was used to analyse the surface of the component in the area closed to the weld-line, previously prepared by means of standard gold sputtering procedure. 2.7. burst tests a burst test is a destructive test performed to determine the overload capacity of component operating under pressured conditions. this pre-production step is necessary to meet safety and quality standards. the burst test internally pressurizes a manifold block with a fluid until the component completely fails (bursts pressure). burst testing is often used to quantify the strength of the part. this test is also useful for measuring the ability to withstand rapid internal pressurization by yielding an ultimate failure pressure throughout the process and eventually to determine if an aging condition have reduced the strength of the component. effect of fiber orientation and residual stresses on the structural performance of injection molded short-fiber-reinforced components 11 copyright ©2020 assa. adv. in systems science and appl. (2020) 3. results and discussion 3.1. mould filling analysis the analysis of the mould filling was useful in order to evaluate the soundness of the process. the filling pattern shows how the part is filled and help to understand how weld lines and air trap will form. in particular, the weld lines location was determined looking at the flow fronts movements based on the analysis of the mould filling. an example of mould filling is reported in figure 7; here it can be seen the formation of two weld lines (wl) at time t4, wl1 and wl2, which are expected to be placed around the hole. further, the results showed that the boiler manifold block is completely filled with a high level of confidence without any freezing of the material. the part is filled in less than 1.5 seconds (t6). the weld lines are significant especially for short-fiber injection moulding component performances because the local mechanical properties in the weld line area differ significantly from those in the rest (bulk) areas of the molded parts. the extent of mechanical properties changes depending on the ability of the two melt flows to join together homogeneously through the part thickness. the average fibre orientations (i principal direction – flow direction) in the boiler manifold block predicted by moldflow is shown in figure 8. blue (0.000) indicates more randomly oriented fibers, while red (1.000) indicates highly oriented fibers, compared to the direction of material flow. the results of the orthotropic, linear, elastic structural simulations that includes rps are discussed in the next section. figure 7. (a) weld-lines wl1 and wl2 predicted by moldflow simulation analysis b) filling time plot and weld line (wl) prediction based on optimum runner system design 12 g. ramorino, s. cecchel, g. cornacchia copyright ©2020 assa adv. in systems science and appl. (2020) figure 8. fiber orientation plot on the first principal direction 3.2. structural analysis the deformed shape of the boiler manifold block evaluated in z (caused by the in-cavity residual stress distribution) is shown in figure 9. the results were also compared with the real component deformation after the component production, as it can be seen in the picture. the analytical results presented a good agreement with the real production. for the sake of clarity, the simulation is reported with a magnification factor of 10. figure 10 shows the predicted rps. figure 9. warping displacement (mm) for the boiler manifold block in z direction – comparison of model vs. experimental results: (a) prediction from computational analysis (moldflow/abaqus); (b) actual deformation on the moulded component effect of fiber orientation and residual stresses on the structural performance of injection molded short-fiber-reinforced components 13 copyright ©2020 assa. adv. in systems science and appl. (2020) figure 10. abaqus prediction of the residual processing stress (rps) distribution after warping (deformation scale = 10) it can be clearly seen that, also after the warping deflection and the consequent stress release, the magnitude of von mises stress stored in the component should not be neglected. this result was obtained with the optimum processing conditions reported in table 2; i.e. optimum mould temperature, melt temperature, packing pressure and time and injection time to minimize shrinkage and residual stresses. figure 11a reports the predicted von mises stress obtained using the orthotropic material properties by neglecting the rps, while figure11b represents the same simulation including rps. figure 11. abaqus analysis study with orthotropic material properties: distribution of mises stress (a) rps neglected, (b) rps included. same initial boundary conditions for both cases (internal pressure = 45 bar). the analysis revealed that the stress on the weld-line areas wl1 and wl2 was around 29 mpa and 36 mpa respectively in this condition, the part should widely resist if an internal pressure of 45 bar is applied. this is due to the strength of the material along the weld-lines for 35% glass fiber ppa which is evaluated to be in the range between 65 to 75 mpa. however, examining figure 11 b, it is possible to note that when the residual processing stress (rps) distribution (figure 11 ) is taken into account in the structural simulation, the von mises stress on the wl1 and 14 g. ramorino, s. cecchel, g. cornacchia copyright ©2020 assa adv. in systems science and appl. (2020) wl2 strongly increases to about 58 mpa and 72 mpa, respectively. these values are above the von mises strength of the weld-lines for this material, highlighting the relevance of this additional test. 3.3. sem analysis sem images reported in figure 12 show clearly some defects that could led to a premature fracture during the burst test. in particular, close to the sharp edge (fig 12 b), the surface is very irregular (see fig 12 c and d) and in this area occur the fracture in correspondence to the weld line (fig 12 a). the presence of these discontinuities, combined with the presence of the weld line, explains the drastic resistance properties decrease of the component in such areas, due to the development of certain displacement discontinuity surfaces, origin of the fracture. it is worthwhile to note that in this area, it’s impossible to insert any fillet radius because of the incidence of mold plugs which form the two cavities within the component. figure 12. sem observations: a) area near the weld line; b) discontinuous surface close to the sharp edge; c) and d) high magnitude se image of the irregular surface 3.4. burst tests the burst test device and burst strength peak value recorded in the present work are reported in figures 13 and 14. effect of fiber orientation and residual stresses on the structural performance of injection molded short-fiber-reinforced components 15 copyright ©2020 assa. adv. in systems science and appl. (2020) figure 13. (a) burst test device; (b) crack propagation along the weld-lines wl1 and wl2 at the end of the test figure 14. burts tests diagram the burst test value was found at about 45 ± 3 bar and the crack propagation can be observed in figure 11b along the weld-lines wl1 and wl2, as expected. these results are found in good agreement with those predicted only by the orthotropic, liner, elastic numerical analysis in which the rps were taken into account. 4. conclusion this paper describes the feasibility study of a new manifold block, traditionally made with brass and metal replaced with 35 wt.% fiber reinforced polyphthalamides (ppa). in particular, this work deals on structural simulations that take into account the effect of fiber orientation and residual stresses in injection moulded components. it is possible to summarize the main results in terms of process, structural simulations and experimental tests as follows: • the main innovation of this research regards the importance of carry out structural simulations which consider not only the orientation of the glass fibre but also the residual stresses given by molding. to this end the present paper demonstrates the 16 g. ramorino, s. cecchel, g. cornacchia copyright ©2020 assa adv. in systems science and appl. (2020) usefulness and accuracy of an integrated cad-fea approach composed of coupled 3d flow and structural simulations applied at the optimization of injection moulding. in particular, the simulations are based on orthotropic, linear, elastic simulations that include the residual process stress (rps). • the integration of rps data leads to a much more reliable prediction of the actual product response that allows to obtain optimized part strength, part geometry, mold design and processing conditions. in the examined case, the von mises stress on the wl1 and wl2 strongly increases respectively from 29 mpa and 36 mpa, in simulations without rps, up to about 58 mpa and 72 mpa, when the rps distribution is taken into account. these values highlighted the relevance of this additional test. • experimental tests, such as burst test and sem analysis, were conducted on real components. the results were in good agreement with those predicted by the orthotropic, linear and elastic numerical analysis which considered the rps, confirming the effective benefits obtained with this design methodology. • the substitution of brass with 35 wt.% glass fiber filled ppa led to a final weight of 190 g, with a weight reduction of about 60% in comparison with the 480 g of the traditional one. the cost analysis reported in figure 2 shows a saving of about 65% respect to the original component. this integrated cad-fea approach related to experimental work would be applied also in future applications for the further improvement of weight reduction and metal replacement. acknowledgements the authors are grateful to fonditalia spa for the realization of workpieces. references 1. s. cecchel, d. chindamo, m. collotta, g. cornacchia, a. panvini, g. tomasoni, m. gadola (2018). lightweighting in light commercial vehicles: cradle-to-grave life cycle assessment of a safety relevant component, the international journal of life cycle assessment, 1-12. 2. s. kulkarni, d.j. edwards, e.a. parn, c. chapman, c.o. aigbavboa, r. cornish (2018). evaluation of vehicle lightweighting to reduce greenhouse gas emissions with focus on magnesium substitution, journal of engineering, design and technology, 869-888. doi 10.1108/jedt-03-2018-0042. 3. s. cecchel, d. chindamo, e. turrini, c. carnevale, g. cornacchia, m. gadola, a. panvini, m. volta, d. ferrario, r. golimbioschi (2018). impact of reduced mass of light commercial vehicles on fuel consumption, co2 emissions, air quality, and socioeconomic costs, science of the total environment, 613(14), 409-417. 4. m. tisza, i. czinege (2018). comparative study of the application of steels and aluminium in lightweight production of automotive parts, international journal of lightweight materials and manufacture, 1, 229-238. 5. m. faccoli, d. dioni, s. cecchel, g. cornacchia, a. panvini (2017). an experimental study to optimize the heat treatment of gravity cast sr-modified b356 aluminum alloy, transactions of nonferrous metals society of china, 27(8), 1698-1706. effect of fiber orientation and residual stresses on the structural performance of injection molded short-fiber-reinforced components 17 copyright ©2020 assa. adv. in systems science and appl. 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(1984). the effect of aspect ratio of inclusions on the elastic properties of unidirectionally aligned composites, polym. comp., 5, 327. microsoft word 1128 article text, copyedited.doc adv syst sci appl 2021; 04; 16-30 published online at https://ijassa.ipu.ru. the model of the aggregated consumer with detailed description of consumer financial balance using the original method of relaxation of complementary slackness conditions alexei ujegov* national research university higher school of economics, moscow, russia e-mail: auzhegov@hse.ru abstract: this paper is complementary to a series of researches on aggregated consumer behavior and complementary slackness conditions application in linear nonautonomous optimal control type problems with mixed conditions. the model of an aggregated consumer is proposed. consumer maximizes utility function in consumption and labor and chooses the values of its planned variables: loans and deposits in national and foreign currency, amount of real estate, and currency. the key feature of the proposed approach is a detailed description of all financial instruments that are available to the consumer. keywords: households, financial balance, complementary slackness conditions, macroeconomics, consumer financial instruments 1. introduction aggregated consumer (or household) behavior can be modeled with various optimization problems starting from the relatively simple ramsey growth model and ending with heavily modified dsge models (see [3]-[6], [9], [10], [13]). however, the mainstream approach follows [11]. consequently, stable foundations were built, which are rarely discussed. as a result, there are many similarities in various papers in consumer description. one of the disadvantages is that aggregated consumer (household) rarely has in its financial balance various financial instruments (see [1]-[3], [5], [6], [10]). moreover, labor variable in utility function is not endogenous (if included) (see [1]-[3]). the main purpose of this research is to show how it is possible to describe an aggregated consumer in the macroeconomic model with the inclusion of a detailed description of consumer financial instruments. as various dsge models show (see [3]-[5], [9], [10], [13]), the consumer usually operates with several bonds and very rare (or never) operates with a set of instruments: loans, deposits, foreign currency, real estate or others. in this paper, it is proved that such a description is possible due to the implementation of complementary slackness conditions (see [7], [8], [14]). nowadays, there is a discussion about different types of households: ricardian and nonricardian (see [3]-[6], [13]). it complements the research vector on detailed consumer modeling. papers [12], [15] discuss utility function with endogenous labor. multiplicative relation between crra consumption and labor functions shows high accuracy in the standalone aggregated consumer model. however, current empirical testing proved that additive relation is better if there is a need to include an aggregated consumer as part of the multi-block model. also, additive relation between consumption and labor is used frequently (see [5], [6], [9], [10], [13]). * corresponding author: auzhegov@hse.ru the model of an aggregated consumer with detailed description of consumer… 17 copyright ©2021 assa. adv. in systems science and appl. (2021) 2. aggregated consumer model the model describes the economic behavior of an aggregated consumer (household) that consumes goods and services, provides labor, manages its financial instruments: takes loans in national and foreign currency, makes deposits in national and foreign currency, buys foreign currency, performs operations with real estate and receives transfers. by the time the consumer takes loans in rubles at an interest rate and makes interest payments . the equation (2.1) describes the dynamics of loans on consumer balance: , (2.1) where is a flow variable for consumer loan operations and is the consumer's average return frequency of loans. similarly, the consumer takes loans in foreign currency at an interest rate and makes interest payments in foreign currency , where is the exchange rate. the equation (2.2) describes the dynamics of loans on consumer balance: . (2.2) in each moment consumer saves at an interest rate and receives interest payments . the equation (2.3) describes the dynamics of deposits on consumer balance: , (2.3) where is a flow variable for deposit operations and is the average frequency of making deposits by the consumer. similarly, by the time the consumer saves at an interest rate and receives interest payments . the equation (2.4) describes the dynamics of deposits in foreign currency on consumer balance: . (2.4) the consumer performs operations with real estate at an interest rate and receives income . the equation (2.5) describes the dynamics of real estate amount on consumer balance: , (2.5) where is a flow variable that describes money received due to operations with real estate and is an average frequency of operations. additionally, the consumer can operate with foreign currency at an exchange rate . the equation (2.6) can describe the dynamics of foreign currency amount on consumer balance: t ( )l t ( )lr t ( ) ( )lr t l t ( ) ( ) ( ) ( )l d l t k t t l t dt b= ( ) 0k t ³ ( )l tb ( )vl t ( )vlr t ( ) ( ) ( )vl vlw t r t vl t ( )vlw t ( ) ( ) ( ) ( )vl d vl t vk t t vl t dt b= t ( )s t ( )sr t ( ) ( )sr t s t ( ) ( ) ( ) ( )s d s t v t t s t dt b= ( ) 0v t ³ ( )s tb t ( )vs t ( )vsr t ( ) ( )vsr t vs t ( ) ( ) ( ) ( )vs d vs t vv t t vs t dt b= ( )b t ( )br t ( ) ( )br t b t ( ) ( ) ( ) ( )b d b t ob t t b t dt b= ( )ob t ( )b tb ( )vq t ( )vqw t 18 a. ujegov copyright ©2021 assa. adv. in systems science and appl. (2021) , (2.6) where is the flow variable for foreign currency operations and is an average frequency of operations with foreign currency. in addition to described relations, there are several exogenous boundaries. the consumer receives transfers from the government , where the consumer knows . also, the consumer has other expenditures , which help to consider unseen or missed expenditures data. an aggregated household consumes by the price and earns , where is wage per capita, labor, and income tax rate. considering as cash that consumer has, it is possible to write the consumer financial balance, where the equation (2.7) describes cash dynamics: . (2.7) additionally, there is a liquidity constraint: (2.8) where are constants. this model's consumption deflator , wage , income tax rate , interest rates , exchange rate , transfers , and other expenditures are exogenous. considering all the above, the consumer chooses the values of its planned variables (controls): . the goal of the consumer is to maximize the total discounted utility from the consumption and labor as functional: , where is potential consumption when all economically active population is employed and receives wage per capita . is economically active population. it is a common technique to standardize labor using an exogenous variable. moreover, basic macroeconomics models, e.g., the ramsey model, consider labor as exogenous. here labor is an employed population, and it is endogenous. thus, aggregated consumer decision relies on the employed to economically active population ratio. utility function represents the discounted difference between crra functions of consumption and standardized labor: ( )( ) ( ) ( ) ( ) ( ) ( )vq vq vq dw t vq t w t ovq t t vq t dt b= ( )ovq t ( )vq tb 0( ) ( )tr t tr t£ 0 ( )tr t 0( ) ( )oc t oc t³ ( )c t ( )cp t (1 ( )) ( ) ( )r rt t r tt w( )r tw ( )r t ( )r tt ( )m t ( ) ( ) ( ) (1 ( )) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) c r r s vs l vl vq b d m t p t c t t t r t dt d d d dr t s t s t r t vs t vs t r t l t l t r t vl t vl t dt dt dt dt d dw t vq t r t b t b t tr t oc t dt dt t w= + + + + + + + + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), s s vs vs vs l l vl vl vl b b vq vq vq m t t t s t t t vs t w t t t l t t t vl t w t t t b t t t vq t w t n b n b n b n b n b n b ³ + + + + + + , , , , ,s vs l vl b vqn n n n n n ( )cp t ( )r tw ( )r tt ( ), ( ), ( ), ( ), ( )s vs l vl br t r t r t r t r t ( )vqw t 0 ( )tr t 0 ( )oc t ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( )c t r t s t vs t l t vl t vq t b t ( )c t ( )r t 0 ( ) ( ), ( ) ( ) t u ut c t r tu dt c t r t æ ö ç ÷ è ø ò ( ) ( )( ) ( ) r u u c t r tc t p t w = ( )r tw ( )ur t the model of an aggregated consumer with detailed description of consumer… 19 copyright ©2021 assa. adv. in systems science and appl. (2021) . (2.9) a terminal condition is required to solve this type of problem with constraints: , (2.10) where . (2.11) and are vectors that consist of constants for each of the variables at a time and respectively. the problem above is a linear nonautonomous optimal control type problem with mixed constraints. enormous equations are omitted to keep narration clear. the solution process takes two steps: 1. langrage functional is written where each constraint has an associated dual variable: nonnegative – to inequality and arbitrary – to equality. 2. a system of sufficient conditions for optimality is constructed that consists of four types of equations: initial differential equations, the initial conditions for which are assumed to be given; conjugate differential equations for variables dual to the constraints of the problem; transversality conditions that define terminal conditions for conjugate differential equations; complementary slackness conditions for inequalities. the example of a problem solution is described in [7], [8], [14]. several techniques mentioned in studies are used here to perform the transition from continuous to discrete time and to implement complementary slackness conditions. problem solution is a system of equations, where the variable is a part of equations written below. is a dual variable to consumer financial balance. will often appear in sufficient optimality conditions, making it logical to redesignate as : the rate of falling of dual variable or consumer profitability ratio. equations (2.12) (2.27) describe problem solution. 1 1( ) ( ) ( ) ( ) ( ) 1 1 u u t c t r t c t r t u t e b a b a -d æ öæ ö æ öç ÷ ç ÷ç ÷è ø è øç ÷= -ç ÷ ç ÷ è ø 0( ) ( )atercons t atercons t£ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )tercons t b t l t m t s t vl t vq t vs t= + + + + + + a a tercons 0t t ( ) ( ) ( ) d t dtt t x r x = ( )tx ( ) ( ) d t dt t x x ( )tr 20 a. ujegov copyright ©2021 assa. adv. in systems science and appl. (2021) consumer profitability ratio depends on exogenous variables and parameters , which are discussed later: , (2.12) . briefly, is a result of the substitution of all model variables in consumer financial balance (2.7) after sufficient optimality conditions equation system is solved. [7], [8], [14] describe the detailed solution. consumption at a time depends on consumption in the previous period and exogenous variables and parameters: . (2.13) labor is similar to consumption and follows from a problem as an endogenous variable: .(2.14) the following variables are described with the help of the complementary slackness conditions technique. loans in national currency: . (2.15) loans in foreign currency: . (2.16) ( )tr , ,i i ia b c ( )( ) ( ) h tt d t r = ( ) ( ) ( ) 1 1 1 2 2 5 5 5 5 6 ( ) ( 1) ( ) ( ) ( ) ( ) ( ) ( )( 1) ( ) ( 1) ( 1) ( ) ( 1) ( ) ( 1) ( ) ( ) ( ) ( ) ( )( 1) ( ) ( )( 1) ( ) ( 1) ( 1) ( vs w vs w vq vs vs vq b b b vs w vs w vs vl vs vl vl l h t m t a w t g t b w t g t w t w t b t vq t a r t b t b t a w t g t b w t g t w t a r t w t b t vl t a r t n b n b n b = + + + + + + + + + + + + + + +( ) ( ) ( ) 6 3 3 3 3 4 4 ) ( 1) ( ) ( 1) ( ) ( ) ( ) ( ) ( )( 1) ( ) ( )( 1) ( ) ( 1) ( 1) ( ) ( 1) ( ) ( 1) l l vs w vs w vs vs vs vs vs s s s b t l t a w t g t b w t g t w t a r t w t b t vs t a r t b t s t n b n b n b + + + + + + + + + + + + 1 1 1 2 2 2 5 5 5 6 6 6 3 3 3 4 4 4 ( ) ( )( ) ( 1) ( ( ) ) ( 1) ( ( ) ( ) ( ) ( )) ( 1) ( ( ) ) ( 1) ( ( ) ( ) ( ) ( )) ( 1) ( ( ) ) ( 1) vs vq b b vs vl vl vs vs l l vs vs vs vs vs s s d t w t a a b vq t t a a b b t w t t a a w t b w t vl t t a a b l t a w t t a w t b w t vs t t a a b s t n b n b n b n b n b n = + + + + + + + + + + + + + + + ( )tr t ( )( )1 ( ) ( 1)( ) ( 1) ( )( ) ( 1) ( 1) ( 1) ( 1) u uc c c u c t c tp t p t tc t c t c t p t c t b r b b b b -æ ö-d = + + -ç ÷-è ø ( ) ( )( ) ( ) ( )( ) ( ) ( ) 1 ( 1) ( ) ( 1( ) ( 1) ( 1) ( 1) 1 ( 1) 1 ( 1) ( ) ( 1) ( 1) 1 ( ) ( 1) ( ) ( ) ( 1) 1 ( 1) ( 1) 1 ( ) r r rr r r r r r u u r u r t t tt t t t t t r t r t r t r t r t t t t r t t t t w wt t t a t a t w a r x t a a t x -æ ö+ d -d + +ç ÷-ç ÷= -ç ÷ç ÷+ +ç ÷-è ø ( ) ( ) ( ) ( 1) ( 1)ll t k t t l t l tb= + ( ) ( ) ( ) ( 1) ( 1)vlvl t vk t t vl t vl tb= + the model of an aggregated consumer with detailed description of consumer… 21 copyright ©2021 assa. adv. in systems science and appl. (2021) deposits in national currency: . (2.17) deposits in foreign currency: . (2.18) real estate that consumer owns: . (2.19) foreign currency: . (2.20) the following equations describe flow variables. income received from operations with foreign currency: (2.21) income received from operations with real estate: (2.22) consumer deposit operations in foreign currency: (2.23) consumer deposit operations in national currency: (2.24) ( ) ( ) ( ) ( 1) ( 1)ss t v t t s t s tb= + ( ) ( ) ( ) ( 1) ( 1)vsvs t vv t t vs t vs tb= + ( ) ( ) ( ) ( 1) ( 1)bb t ob t t b t b tb= + ( ) ( ) ( ) ( 1) ( 1)vqvq t ovq t t vq t vq tb= + ( )( ) ( ) ( )( ) 0 0 1 1 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( 1) ( ) ( ) ( 1). ( 1) c r r vq vq vq vq vq vq p t c t tr t oc t t r t t c ovq t w t w t w t a b a a b t b t vq t w t w t n r b + = + æ öç ÷+ + + + ç ÷-è ø ( )( ) ( )( ) 0 0 2 2 2 2 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( 1). c r r b b b b ob t p t c t tr t oc t t r t t c t a a b t b t r t a b t w t b n r b = + + + + + + ( )( ) ( ) ( )( ) 0 0 3 3 3 3 3 3 3 3 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( 1) ( ) ( ) ( ) ( ) ( 1). ( 1) c r r vs vs vs vs vs vs vs vs p t c t tr t oc t t r t t c vv t w t w t w t a b a t a b t b t r t a vs t w t w t b n r b + = + -æ ö + + + + + -ç ÷-è ø ( )( ) ( )( ) 0 0 4 4 4 4 4 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( 1). c r r s s s s v t p t c t tr t oc t t r t t c t a a b t b t r t a s t w t b n r b = + + + + + + 22 a. ujegov copyright ©2021 assa. adv. in systems science and appl. (2021) consumer loan operations in foreign currency: (2.25) consumer loan operations in national currency: (2.26) consumer liquidity: (2.27) 3. empirical testing the model was tested on quarterly data from q1-2008 to q1-2021. the restriction for the chosen period is data availability. the primary data sources are the federal state statistics service (fsss) and the central bank of russia (cbr). the table below describes the connection between statistics and model variables : table 3.1. model variables matching with data variable statistics source final consumption expenditures, 2016 prices, billions of rubles fsss author's calculation using variables final consumption cost deflator fsss labor, million people fsss economically active population, mil people fsss wage rate per employee, thousand rubles/person fsss the personal income tax rate fsss transfers fsss nominal usd / rub exchange rate at the end of the period cbr author's calculation using variable loans to individuals, billions of rubles cbr ( )( ) ( ) ( )( ) 0 0 5 5 5 5 5 5 5 5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( 1) ( ) ( ) ( ) ( ) ( 1). ( 1) c r r vl vl vl vl vl vl vl vl p t c t tr t oc t t r t t c vk t w t w t w t a b a t a b t b t r t a vl t w t w t b n r b + = + +æ ö + + + + -ç ÷-è ø ( )( )( ) ( )( ) 0 0 1 2 3 4 5 6 6 6 6 6 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 ( ) ( ) ( ) ( ) ( 1). c r r l l l l k t p t c t tr t oc t t r t t c c c c c t a a b t b t r t a l t w t b n r b = + + + + + + ( ) ( ) ( 1) ( ) ( 1) ( ) ( ) ( 1) ( ) ( ) ( 1) ( 1) ( ). s s vs vs vs vl vl vl b b vq vq m t t s t t vs t w t t vl t w t t b t vq t w t n b n b n b n b n = + + + + + ( )c t ( )uc t ( ) ( )( ) ( ) r u u c t r tc t p t w = ( ), ( ), ( )r u ct r t p tw ( )cp t ( )r t ( )ur t ( )r tw ( )r tt ( )tr t ( )w t ( )wg t ( ) ( 1) w t w t ( )w t ( )l t the model of an aggregated consumer with detailed description of consumer… 23 copyright ©2021 assa. adv. in systems science and appl. (2021) loans to individuals, usd bn cbr individual deposits, billions of rubles cbr individual deposits, usd bn cbr current accounts of individuals, billions of rubles cbr loans granted to individuals, billions of rubles cbr loans granted to individuals, billions of dollars cbr deposits received from individuals, billions of rubles cbr deposits received from individuals, billions of dollars cbr the interest rate on loans to individuals in rubles cbr the interest rate on loans to individuals in us dollars cbr the interest rate on deposits of individuals in rubles cbr the interest rate on deposits of individuals us dollars cbr frequency of repayment of loans to individuals (rubles) cbr frequency of repayment of loans to individuals (dollars) cbr frequency of return of deposits of individuals (rubles) cbr frequency of return of deposits of individuals (dollars) cbr currency purchase costs, usd billions cbr consumer foreign currency, usd billions cbr fixed parameter (data na) constant 0.1 real estate purchase expenses fsss real estate amount, billions of rubles fsss (balance of income and expenses, obdx) fixed parameter (data na) constant 0.001 real estate profitability rate fsss minus all sums on the consumer balance author's calculation figures below (data – blue line, model estimation – orange line) describe the result of equations (2.13) (2.27), which provide model estimation results. ( )vl t ( )s t ( )vs t ( )m t ( )k t ( )vk t ( )v t ( )vv t ( )lr t ( )vlr t ( )sr t ( )vsr t ( )l tb ( )vl tb ( )s tb ( )vs tb ( )ovq t ( )vq t ( )vq tb ( )ob t ( )b t ( )b tb ( )br t ( )oc t ( )m t 24 a. ujegov copyright ©2021 assa. adv. in systems science and appl. (2021) the equation (2.14) describes endogenous labor in mil people: fig. 3.1. labor , mil people the equation (2.13) results in consumer consumption: fig. 3.2. consumption , billions of rubles ( )r t ( )c t the model of an aggregated consumer with detailed description of consumer… 25 copyright ©2021 assa. adv. in systems science and appl. (2021) the equation (2.15) describes consumer loans in national currency: fig. 3.3. loans in national currency , billions of rubles the equation (2.16) stays for consumer loans in foreign currency: fig. 3.4. loans in foreign currency , usd billion ( )l t ( )vl t 26 a. ujegov copyright ©2021 assa. adv. in systems science and appl. (2021) the equation (2.17) describes consumer deposits in national currency: fig. 3.5. deposits in national currency , billions of rubles the equation (2.18) results in consumer deposits in foreign currency: fig. 3.6. deposits in foreign currency , usd billions ( )s t ( )vs t 27 a. ujegov copyright ©2021 assa. adv. in systems science and appl. (2021) the equation (2.20) describes consumer foreign currency: fig. 3.7. foreign currency , usd billions the equation (2.19) describes consumer real estate amount: fig. 3.8. real estate amount, billions of rubles ( )vq t 28 a. ujegov copyright ©2021 assa. adv. in systems science and appl. (2021) the equation (2.7) describes consumer financial balance: fig. 3.9. consumer current assets, billions of rubles the figures above demonstrate high accuracy for consumption , labor , loans , and deposits . however, real estate and currency description accuracy are low, resulting in lower accuracy for consumer current accounts description. nevertheless, the figures represent overall trends. the reason for observed performance is that it is clear how to describe loans and deposits (see [7]). the currency and real estate variables description is studied briefly in the given type of problems. it is one of the research topics for future studies. table 3.2 shows the accuracy metric mape for two periods: including and excluding pandemic (q1-2020 – q1-2021). overall model accuracy is significantly better when the pandemic period is excluded. it is necessary to point out that model performance facing unpredictable events as pandemic crisis remains good enough. table 3.2. model accuracy measures (mape) model estimated variable mape (q1-2008 to q1-2021) mape (q1-2008 to q4-2019) labor 1.25% 1.13% consumption 3.46% 1.18% amount of currency 77.93% 89.4% amount of real estate 11.22% 12.16% loans in national currency 10.14% 11.34% loans in foreign currency 17.47% 11.61% deposits in national currency 5.26% 4.16% deposits in foreign currency 10.62% 8.61% consumer current accounts 39.69% 18.39% it is necessary to address the lagrange multiplier method to clarify parameters in equations (2.12), (2.21)-(2.26). the lagrange multiplier method results in a system of inequalities for dual variables and corresponding control variables where one variable becomes equal to zero and the other one to non-zero during the quarterly interval. the system stays in one of many possible regime states. variables change regimes frequently, which can be interpreted as when consumers take loans and buy a foreign currency. in other moments he makes deposits or everything altogether. the consumer consistently changes the ( )c t ( )r t ( ), ( )l t vl t ( ), ( )s t vs t ( )b t ( )q t ( )m t ( )r t ( )c t ( )vq t ( )b t ( )l t ( )vl t ( )s t ( )vs t ( )m t , ,i i ia b c the model of an aggregated consumer with detailed description of consumer… 29 copyright ©2021 assa. adv. in systems science and appl. (2021) regimes by performing various operations. data is discrete, and the system state is unobservable in statistics during the quarter, but this approach helps restore this information. consumer makes his decision about the regime with the help of exchange and interest rates included in (consumer profitability rate). parameters and describe how long the system stays in one regime during quarter intervals based on profitability (rates) of available financial instruments to the consumer. reflects all other money flows not connected with financial instruments (consumption, wage, transfers, and others) and helps identify regime states when the consumer does not operate with financial instruments, but performs other activities, e.g., consuming goods and services. to sum up, parameters are elasticity frequency ratios to income variables. all above summarizes the idea of relaxation of complementary slackness conditions. as it was stated, not all variables are described with high accuracy since it is still a research question of how to describe foreign currency amounts available to the consumer. that is why the received system contains a noticeable disturbance variable that influences overall model performance. 4. conclusion the method of relaxation of complementary slackness conditions allows including various financial instruments in the model of an aggregated consumer. it helps to describe consumer financial balances in detail corresponding to available data. on the example of the russian economy, it is proved that this method can provide high modeling accuracy even in crisis periods, but a study of foreign currency amounts available to the consumer is still needed. acknowledgements the reported study was funded by rfbr, project number 19-31-90106. references 1. algozhina a. (2012). monetary and fiscal policy interactions in an emerging open economy: a non-ricardian dsge approach, the center for economic research and graduate education economics institute, cerge-ei working papers wp476. 2. aloui, r. & guillard, m. (2020). the fiscal theory of the price level in a nonricardian economy, annals of economics and statistics, genes, 137, 65–82. 3. aursland, t. a., frankovic, i., kanik, b., & saxegaard, m. (2020). state-dependent fiscal multipliers in nora a dsge model for fiscal policy analysis in norway, economic modelling, elsevier, 93(c), 321–353. 4. ginn, w. & pourroy, m. (2019). optimal monetary policy in the presence of food price subsidies, economic modelling, elsevier, 81, 551–575. 5. khotulev, i. & styrin k. (2019). optimal monetary and macroprudential policies for financial stability in a commodity-exporting economy, bank of russia working paper series, 52. 6. marto, r. (2014). assessing the impacts of non-ricardian households in an estimated new keynesian dsge model, swiss journal of economics and statistics (sjes), swiss society of economics and statistics (sses), 150(iv), 353–398. 7. pilnik, n., radionov, s. & yazykov, a. (2019). the model of the russian banking system with indicators nominated in rubles and in foreign currency, optimization and applications 9th international conference, optima 2018, petrovac, montenegro, october 1–5, 2018, revised selected papers / ed. by m. jaćimović, m. khachay, y. kochetov, v. malkova, yu g. evtushenko, m. posypkin. springer. ( )tr ia ib ic , ,i i ia b c ( )vq t 30 a. ujegov copyright ©2021 assa. adv. in systems science and appl. (2021) 8. radionov, s., pilnik, n. & pospelov, i. g. (2019) the relaxation of complementary slackness conditions as a regularization method for optimal control problems, advances in systems science and applications, 19(2), 44–62. 9. rigon, m. & zanetti f. 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(2005). non-ricardian households and fiscal policy in an estimated dsge model of the euro area, computing in economics and finance, 102, society for computational economics. 14. vasilyev, s., pilnik, n., & radionov, s. (2019). the relaxation of complementary slackness conditions in dynamic general equilibrium models, mathematical models and computer simulations (usa), 11(4), 611–621. 15. vasilyev, s., stankevich, i. & ujegov, a. (2018). a model of the real sector of russian economy with several goods and agents-traders, hse economic journal, 22(3), 362–386, [in russian]. adv syst sci appl 2023; 02:184–194 published online at https://ijassa.ipu.ru. nonuniqueness of equilibrium in closed market model alexander kotyukov1*, natal’ya pavlova1,2,3,4 1v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia 2 moscow institute of physics and technology, moscow, russia 3 rudn university, moscow, russia 4 derzhavin tambov state university, tambov, russia abstract: in this paper, we consider a closed market model. in this model the supply and the demand functions are restored by their price elasticities. we obtain sufficient conditions on nonuniqueness of equilibrium in this model. for several special cases of closed market models we get a criteria of equilibrium uniqueness. the obtained results are illustrated with the example of the market with two goods. keywords: supply, demand, equilibrium, coincidence point, covering map 1. introduction equilibrium is one of the main concepts of modern economics. it is a state of the market in which all the produced goods are sold and every market participant does not want to change their position. consider an economic system which consists of producers and consumers. the producers manufacture goods which are sold to the consumers. we assume that the consumers work for the producers and spend all their salaries to purchase the goods and do not make savings. the total amount of the goods produced is called a supply. the total amount of the goods sold is called a demand. if the supply exceeds the demand, the producers suffer profit loss due to the goods unsold. this makes them cut salaries of the consumers. if the consumers have less money, they purchase less goods which, in turn, decrease the profit of the producers even more. it is easy to see that this process leads to a catastrophic economical situation. on the other hand, if the demand exceeds the supply, some consumers cannot purchase the goods they need since these goods are not available for purchase. this leads to an unfavorable situation in the region, e.g. a massive hunger, a pandemic, lack of building resources etc. therefore, we are interested in maintaining the market state in which the total amount of the goods produced is equal to the total amount of the goods needed, i.e., if the supply equals to the demand. such market state is called an equilibrium. the prices on the goods on such market are called equilibrium prices. equilibrium as an economical concept appeared in the second half of 18th century in the works of stewart [14] and smith [13]. the first mathematical notion of the concept was introduced by walras [15] in 1874. at that time there were no developed mathematical theory ∗corresponding author: amkotyukov@mail.ru nonuniqueness of equilibrium in closed market model 185 to obtain the substantial results on equilibrium existence. such results were first obtained by arrow and debreu [1] in 1956. in this work, we obtain sufficient conditions on equilibrium nonuniqueness in a closed market model. here the demand and the supply functions are restored by their price elasticities. the elasticity shows how one economic variable changes with a change of another. an equilibrium in this model is considered as the coincidence point of the supply and the demand functions. to obtain the mentioned conditions we use the results in the theory of covering maps and coincidence points [2]– [5]. let us formalize the problem. 2. problem statement denote rn + = {p = (p1, ..., pn) ∈ rn : pi > 0 ∀i = 1, n}. let the market consist of n goods. the prices on these goods are described by a vector p ∈ rn + which satisfies the inequalities: c1i ≤ pi ≤ c2i ∀i = 1, n. here c1 = (c11, ..., c1n), c2 = (c21, ..., c2n) ∈ rn + are natural constraints on the prices p (denote p = [c11; c21]× ...× [c1n; c2n]). let the demand function d : rn + → rn +, d(p) = (d1(p), ..., dn(p)), and the supply function s : rn + → rn +, s(p) = (s1(p), ..., sn(p)), be given. we assume that for some known prices p∗ ∈ p we know the demand d∗ ∈ rn +, d∗ = (d∗ 1, ..., d ∗ n), and the supply s∗ ∈ rn +, s ∗ = (s∗ 1 , ..., s ∗ n), i.e., d∗ = d(p∗), s∗ = s(p∗). next, suppose that we have a matrix e = (eij)i,j=1,n where eij ∈ r is the jth good price elasticity of the ith good demand. this quantity shows how the demand on the ith good changes with the change of the price on the jth good. similarly, we have a matrix ẽ = (ẽij)i,j=1,n where ẽij ∈ r is the jth good price elasticity of the ith good supply. this quantity shows how the demand on the ith good changes with the change of the price on the jth good. definition 2.1. a closed market model is a following set of parameters: σ = (c1, c2, p ∗, d∗, s∗, e , ẽ). this set uniquely defines the demand function d by: di(p1, ..., pn) = d∗ i n∏ j=1 (p∗j) −eijp eij j , i = 1, n; (2.1) and the supply function s by: si(p1, ..., pn) = s∗ i n∏ j=1 (p∗j) −ẽijp ẽij j , i = 1, n. (2.2) formulas (2.1), (2.2) are the solutions to the following systems of equations: eij = ∂di ∂pj pj di , i, j = 1, n; (2.3) ẽij = ∂si ∂pj pj si , i, j = 1, n (2.4) copyright © 2023 assa. adv syst sci appl (2023) 186 a. kotyukov, n. pavlova with initial conditions di(p ∗) = d∗ i , si(p ∗) = s∗ i , i = 1, n. (2.5) formulas (2.3), (2.4) are the definitions of ith good price elasticity of jth good demand and supply respectively. definition 2.2. a vector p0 ∈ p is called an equilibrium prices vector (an equilibrium) in the model σ if d(p0) = s(p0). (2.6) 3. on the uniqueness and nonuniqueness of equilibrium in a closed market model sufficient conditions on equilibrium existence were obtained in [3]. substitute (2.1), (2.2) into (2.6) and obtain the following system of linear equations: d∗ i n∏ j=1 (p∗j) −eijp eij j = s∗ i n∏ j=1 (p∗j) −ẽijp ẽij j , i = 1, n. by taking a logarithm of the left-hand and right-hand sides of the last equation we get: n∑ j=1 ( eij − ẽij ) ln pj = lns∗ i − lnd∗ i + n∑ j=1 (eij − ẽij) ln p ∗ j , i = 1, n. (3.7) note that this system is linear by ln pj, j = 1, n. theorem 3.1. let the parameters of the model σ satisfy the condition eij = ẽij ∀i, j = 1, n. (3.8) then ∀p ∈ p is an equilibrium in the model σ iff s∗ i = d∗ i ∀i = 1, n. proof indeed, let the parameters of the model σ satisfy (3.8). hence, system (3.7) is equivalent to the following system: lns∗ i − lnd∗ i = 0, i = 1, n. (3.9) it is obvious that any vector p ∈ p is an equilibrium in the model σ iff s∗ i = d∗ i ∀i = 1, n. corollary 3.1. let the parameters of the model σ satisfy (3.8). then the condition: ∃i = 1, n : s∗ i ̸= d∗ i is a criteria of equilibrium absence in the model σ. remark 3.1. the condition (3.8) means that the consumers and the producers react to the price changes in the same way. example 3.1. let us describe several examples of the market models satisfying the conditions of lemma 3.1. 1. consider a market model of electric energy inside the given country. there is only one good – electric energy, and only one producer – a number of state energetic companies which possess the natural monopoly on this good. in the short term the price elasticity of demand equals zero [9]. this is conditioned by the fact that the companies cannot rearrange their technological processes and change the volume of production rapidly. in this model the price elasticity of demand also equals zero in the short term [7]. copyright © 2023 assa. adv syst sci appl (2023) nonuniqueness of equilibrium in closed market model 187 2. the models of raw material market have zero price elastitcities of both supply and demand [6]. for consumers the cost of raw material has small influence to the goods’ prices. for producers raw material are inelastic for several reasons, such as the load capacity, production factors interchangeability or raw material availability [11]. now we introduce the following notation: a = (aij)i,j=1,n, aij = eij − ẽij; ω = (ω1, ..., ωn), ωi = lns∗ i − lnd∗ i + n∑ j=1 (eij − ẽij) ln p ∗ j . theorem 3.2. let the parameters of the model σ satisfy the condition deta ̸= 0. then there exists a unique equilibrium prices vector p0i = exp(a−1ω)i , i = 1, n (3.10) iff the parameters of the model satisfy the following condition: max i=1,n 2 ln c2i − ln c1i ∣∣∣∣(a−1ω)i − ln c1i + ln c2i 2 ∣∣∣∣ ≤ 1 (3.11) where (a−1ω)i is the ith component of the vector a−1ω. proof in conditions of lemma the system (3.7) is consistent by rouché-capelli theorem [12, theorem 2.14] since deta ̸= 0. moreover, by theorem [12, theorem 2.15] its solution is unique. firstly we prove the necessity. let p0 defined by (3.10) be an equilibrium in the model σ. then p0 satisfies (3.7). moreover, p0 ∈ p , i.e.: ln c1i ≤ (a−1ω)i, (a −1ω)i ≤ ln c2i, i = 1, n, (3.12) where (a−1ω)i is the ith component of the vector a−1ω. subtract (ln c1i + ln c2i)/2 from the inequalities (3.12): ln c1i − ln c1i + ln c2i 2 ≤ (a−1ω)i − ln c1i + ln c2i 2 , (a−1ω)i − ln c1i + ln c2i 2 ≤ ln c2i − ln c1i + ln c2i 2 , i = 1, n. thus, − ln c2i − ln c1i 2 ≤ (a−1ω)i − ln c1i + ln c2i 2 ≤ ln c2i − ln c1i 2 , i = 1, n. hence, we have: ∣∣∣∣(a−1ω)i − ln c1i + ln c2i 2 ∣∣∣∣ ≤ ln c2i − ln c1i 2 , i = 1, n. by dividing this inequality by (ln c2i − ln c1i)/2 > 0 we obtain: 2 ln c2i − ln c1i ∣∣∣∣(a−1ω)i − ln c1i + ln c2i 2 ∣∣∣∣ ≤ 1, i = 1, n. copyright © 2023 assa. adv syst sci appl (2023) 188 a. kotyukov, n. pavlova therefore, if p0 defined by (3.10) is an equilibrium in the model σ, the following condition holds: max i=1,n 2 ln c2i − ln c1i ∣∣∣∣(a−1ω)i − ln c1i + ln c2i 2 ∣∣∣∣ ≤ 1. now we prove the sufficiency. let the condition (3.11) be satisfied. since deta ̸= 0, the matrix a is invertible and, hence, the solution to the system (3.7) is unique and has the form: p = exp (a−1ω). from the condition (3.11) we obtain that p ∈ p . therefore, there exists a unique equilibrium p0 in the model σ. corollary 3.2. let the parameters of the model σ satisfy lemma 3.2 and the condition s∗ i = d∗ i ∀i = 1, n. then p0 = p∗. now we consider the case ranga = rang(a|ω) = k < n. theorem 3.3. in the model σ there exist an infinite number of equilibrium prices vectors iff the parameters of the model σ satisfy the following conditions: 1) ranga = rang(a|ω) = k < n; 2) max i=1,n 2 ln c2i − ln c1i ∣∣∣∣∣ n−k∑ j=1 cjxji − ln c1i + ln c2i 2 ∣∣∣∣∣ < 1, (3.13) where xji ∈ r is the ith component of the vector xj ∈ x , i = 1, n and x = {x1, ..., xn−k} is the fundamental system of solutions to the system (3.7). proof first let an infinite number of equilibrium prices vectors exist in the model σ. since any equilibrium p ∈ p is a solution to the system (3.7), we have ranga = rang(a|ω) = k (since the system (3.7) is consistent) and k < n (since system (3.7) has more than one solution). let x = {x1, ..., xn−k}, xi ∈ rn (3.14) be the fundamental system of solutions for the system (3.7). then any equilibrium p (which is the solution to the system (3.7)) can be written in the following form: ln pi = n−k∑ j=1 cjxj (3.15) where cj ∈ r, j = 1, n− k, are some constants. from the inclusion p ∈ p we get ln c1i ≤ n−k∑ j=1 cjxj, n−k∑ j=1 cjxj ≤ c2i, i = 1, n. by repeating the steps conducted in the proof of lemma 3.2 we obtain that if the vector p is an equilibrium in the model σ, then max i=1,n 2 ln c2i − ln c1i ∣∣∣∣∣ n−k∑ j=1 cjxji − ln c1i + ln c2i 2 ∣∣∣∣∣ ≤ 1 copyright © 2023 assa. adv syst sci appl (2023) nonuniqueness of equilibrium in closed market model 189 where xji is the ith component of the vector xj , i = 1, n. from this we easily obtain what we want. now let conditions 1) and 2) of the theorem be satisfied. then by rouché-capelli theorem this system is consistent, but its solution is not unique since ranga ̸= n. using (3.14) we get that any solution p ∈ rn + to the system (3.7) can be written in the form (3.15). moreover, from condition 2) inverting the conclusions made in the first part of the proof we obtain that p ∈ p . hence, any solution to the system (3.7) satisfying condition 2) is an equilibrium in the model σ. to demonstrate these results consider the following example. example 3.2. let n = 2, p∗ = (1; 1), c1 = (1; e), c2 = (1/(4e); 1). in table 3.1 we can see the existence and uniqueness and non-uniqueness for different parameters d∗, s∗, ω. table 3.1. existence, uniqueness and non-uniqueness of the solution in the case n = 2, p∗ = (1; 1), c1 = (1; e), c2 = (1/(4e); 1). s∗, d∗, ω∗ (1; 1) (1; 1) (0; 0) (2; 2) (1; 1) (ln 2; ln 2) (1; 1) (2; 2) (− ln 2;− ln 2) (2; 2) (1; 1 2 ) (ln 2; 2 ln 2) a ( 0 0 0 0 ) theorem 3.1 ∀p∈p corollary 3.1 ∅ corollary 3.1 ∅ corollary 3.1 ∅( 1 0 0 1 ) corollary 3.2 (1;1) theorem 3.2 (2;2) theorem 3.2 (1/2;1/2) theorem 3.2 (2;4)( 0 1 1 0 ) corollary 3.2 (1;1) theorem 3.2 (2;2) theorem 3.2 (1/2;1/2) theorem 3.2 (4;2)( 1/2 1/2 1/2 1/2 ) theorem 3.3 (p1, 1 p1 ), p1∈[1;e] theorem 3.3 (p1; 4 p1 ),p1∈[1;e] theorem 3.3 (p1; 1 4p1 ),p1∈[1;e] corollary 3.1 ∅( 1 0 1/2 1/2 ) corollary 3.2 (1;1) theorem 3.2 (2;2) theorem 3.2 (1/2;1/2) theorem 3.2 (2;8) now we consider the case n = 2. the system (3.7) is equivalent to the following system: a11 ln p1 + a12 ln p2 = ω1, a21 ln p1 + a22 ln p2 = ω2, (3.16) which is considered under the conditions ln c11 ≤ ln p1 ≤ ln c21, ln c12 ≤ ln p2 ≤ ln c22. (3.17) theorem 3.4. let n = 2. then: 1. if deta ̸= 0 and ln c11 ≤ a22ω1 − a12ω2 a11a22 − a12a21 ≤ ln c21, ln c12 ≤ a11ω2 − a21ω1 a11a22 − a12a21 ≤ ln c22, then there exists a unique equilibrium( a22ω1 − a12ω2 a11a22 − a12a21 , a11ω2 − a21ω1 a11a22 − a12a21 ) in the model σ; copyright © 2023 assa. adv syst sci appl (2023) 190 a. kotyukov, n. pavlova 2. if ranga = rang(a|ω) = 1 and ai2 = 0 ∀i = {1, 2}, then there exists an infinite number of equilibrium prices vectors in the model σ which belong to the set: ln p1 = ω1 a11 = ω2 a21 , c12 ≤ p2 ≤ c22. 3. if ranga = rang(a|ω) = 1 and ai1 = 0 ∀i = {1, 2}, then there exists an infinite number of equilibrium prices vectors in the model σ which belong to the set: ln p2 = ω1 a12 = ω2 a22 , c11 ≤ p1 ≤ c21. 4. if ranga = rang(a|ω) = 1, a11a12 > 0 and ω1 = a11 ln c11 + a12 ln c12, then there exists a unique equilibrium (c11, c12) in the model σ; 5. if ranga = rang(a|ω) = 1, a11, a12 > 0 and ω1 = a11 ln c21 + a12 ln c22, then there exists a unique equilibrium (c21, c22) in the model σ; 6. if ranga = rang(a|ω) = 1, a11a12 < 0 and ω1 = a11 ln c21 + a12 ln c12, then there exists a unique equilibrium (c21, c12) in the model σ; 7. if ranga = rang(a|ω) = 1, a11a12 < 0 and ω1 = a11 ln c11 + a12 ln c12, then there exists a unique equilibrium (c11, c12) in the model σ; 8. if ranga = rang(a|ω) = 1 and either: • a11a12 > 0 and a11 ln c11 + a12 ln c12 < ω1 < a11 ln c21 + a12 ln c22; or • a11a12 < 0 and a11 ln c21 + a12 ln c12 < ω1 < a11 ln c11 + a12 ln c22; then there exists an infinite number of equilibrium prices vectors in the model σ defined by the formula: ln p2 = ω1 − a11 ln p1 a12 , ln c11 ≤ p1 ≤ ln c21. proof consider all the cases consequently. copyright © 2023 assa. adv syst sci appl (2023) nonuniqueness of equilibrium in closed market model 191 1. let deta ̸= 0. then by rouché-capelli theorem the system (3.16) is consistent and by theorem [12, theorem 2.15] its solution is unique. let us find this solution using, e.g., cramer’s rule: a = ∣∣∣∣a11 a12 a21 a22 ∣∣∣∣ = a11a22 − a12a21, a1 = ∣∣∣∣ω1 a12 ω2 a22 ∣∣∣∣ = a22ω1 − a12ω2, a2 = ∣∣∣∣a11 ω1 a21 ω2 ∣∣∣∣ = a11ω2 − a21ω1. then the solution to this system has the form: ln p1 = a1/a = a22ω1 − a12ω2 a11a22 − a12a21 , ln p2 = a2/a = a11ω2 − a21ω1 a11a22 − a12a21 . this solution must satisfy (3.17). therefore, if ln c11 ≤ a22ω1 − a12ω2 a11a22 − a12a21 ≤ ln c21, ln c12 ≤ a11ω2 − a21ω1 a11a22 − a12a21 ≤ ln c22, then there exists a unique equilibrium in the model σ. otherwise an equilibrium in this model does not exist. now let ranga = rang(a|ω) = 1. 2. in the case ai2 = 0, ∀i ∈ {1, 2} the system (3.16) is equivalent to the following system: a11 ln p1 = ω1, from which it follow that all equilibrium prices vectors has the form: ln p1 = ω1 a11 = ω2 a21 , c12 ≤ p2 ≤ c22. 3. in the case ai1 = 0, ∀i ∈ {1, 2} all equilibrium prices vectors belong to the set: {(p1, p2) ∈ p | ln p2 = ω1 a12 = ω2 a22 , c11 ≤ p1 ≤ c21}. hereinafter we suppose that aij ̸= 0, i, j = 1, 2. then all the solutions to the system (3.16) belong to the line: ln p2 = ω1 − a11 ln p1 a12 . (3.18) now we find the conditions under which an equilibrium is unique even in the case when the system (3.16) has an infinite number of solutions. note that the angle of the line (3.18) equals −a11/a12. we consider several cases. copyright © 2023 assa. adv syst sci appl (2023) 192 a. kotyukov, n. pavlova 4. let a11a12 > 0. then an equilibrium in the model σ is unique in two cases: (a) if: ln c12 = −a11 ln c11 a12 + ω1 a12 ; (3.19) (b) if: ln c22 = −a11 ln c21 a12 + ω1 a12 ; (3.20) in the case 4a from (3.19) we obtain that if ω1 = a11 ln c11 + a12 ln c12, then in the model σ there exists a unique equilibrium p0 = (c11, c12). 5. in the case 4b from (3.20) we obtain that if: ω1 = a11 ln c21 + a12 ln c22, then in the model σ there exists a unique equilibrium p0 = (c21, c22). in the case when a11 ln c11 + a12 ln c12 < ω1 < a11 ln c21 + a12 ln c22, (3.21) in the model σ there exist an infinite number of equilibrium prices vectors belonging to the set: ln p2 = ω1 − a11 ln p1 a12 , ln c11 ≤ p1 ≤ ln c21. 6. let a11a12 < 0. this case is considered similarly to the previous case. here an equilibrium in the model σ is unique in two cases: (a) if: ln c22 = −a11 ln c11 a12 + ω1 a12 ; (3.22) (b) if: ln c21 = −a11 ln c12 a12 + ω1 a12 ; (3.23) in the case 6a from (3.22) we obtain that if: ω1 = a11 ln c11 + a12 ln c22, then in the model σ there exists a unique equilibrium p0 = (c11, c22). 7. in the case 6b from (3.23) we obtain that if: ω1 = a11 ln c12 + a12 ln c21, then in the model σ there exists a unique equilibrium p0 = (c12, c21). in the case when: a11 ln c12 + a12 ln c21 < ω1 < a11 ln c11 + a12 ln c22, (3.24) in the model σ there exist an infinite number of equilibrium prices vectors belonging to the set: ln p2 = ω1 − a11 ln p1 a12 , ln c11 ≤ p1 ≤ ln c21. copyright © 2023 assa. adv syst sci appl (2023) nonuniqueness of equilibrium in closed market model 193 8. the last statement is obtained from inequalities (3.21) and (3.24) with the corresponding conditions on the parameters a11, a12. example 3.3. consider a market of electric energy in the russian federation in 2022. in the table below we can see the data collected from federal state statistics service [8]: table 3.2. the production and consumption of electric energy in the russian federation in 2021–2022 2021 2022 production, trillion kw*hr 1.100 1.167 consumption, trillion kw*hr 0.965 1.110 prices, rubles per kw*hr 4.23 4.78 in terms of the model σ we have: p∗ = 4.78, s∗ = 1.167, d∗ = 1.11. we put c1 = 2.69, c2 = 4.81 as the lowest and the highest mean prices on electric energy in the russian federation in 10 years, respectively. using the economic definition of elasticity we obtain that: e = 1.15, ẽ = 0.46. now we check the conditions of theorem 3.2. here a = 0.69, ω = 1.1295. from (3.11) we obtain that 2 ln 8.5− ln 2.63 ∣∣∣∣1.12950.69 − ln 8.5 + ln 2.63 2 ∣∣∣∣ = 0.1422 < 1. therefore, the conditions of theorem 3.2 are satisfied. hence, there exists a unique equilibrium in the model σ. acknowledgements theorem 3.2 is due to n. pavlova who was supported by the russian science foundation (project no. 23-11-20020, https://rscf.ru/project/23-11-20020/). theorems 3.3 and 3.4 are due to a. kotyukov who was supported by the russian science foundation (project no. 20-11-20131, https://rscf.ru/project/20-11-20131/). references 1. arrow, k., & debreu, g. (1954). existence of an equilibrium for a competitive economy, econometrica, 22(3). 2. arutyunov, a. (2014). coincidence points of two maps, funct. anal. its. appl., 48, 72– 75. 3. arutyunov, a., kotyukov, a. & pavlova, n. (2021). equilibrium in market models with known elasticities, advances in systems science and applications, 24(4), 130–144. 4. arutyunov, a., & zhukovskiy. s. (2010). existence and properties of inverse mappings, proc. steklov inst. math., 271, 12–22. 5. arutyunov, a., zhukovskiy, s., & pavlova, n. (2013). equilibrium price as a coincidence point of two mappings, comput. math. math. phys., 53(2), 158–169. 6. bond, m.e., (1987). an econometric study of primary commodity exports from developing country regions to the world, staff papers (international monetary fund), 34(2), 191–227. copyright © 2023 assa. adv syst sci appl (2023) 194 a. kotyukov, n. pavlova 7. burke, p.j., & abayasekara, a. (2018). the price elasticity of electricity demand in the united states: a three-dimensional analysis, the energy journal, 39(2), 123–146. 8. federal state statistics service. (2023, may). proizvodstvo osnovnih vidov produkcii v natural’nom virajenii [the production of basic commodities in natural equivalent (annual data)]. [in russian]. [online]. available https://gks.ru/free doc/new site/ business/prom/natura/god.htm 9. knaut, a. & paulus, s. (2016) hourly price elasticity pattern of electricity demand in the german day-ahead market, ewi working paper, 16/07. 10. kotyukov, a., & pavlova, n. (2021). equilibrium in market models, proc of. 14th international conference management of large-scale system development (mlsd) (moscow, russia), 1–5. 11. radetzki, m. (2010, august). primary commodities: historical perspectives and prospects. [online]. available https://www.imf.org/external/np/seminars/eng/2010/ afrfin/pdf/radetzki2.pdf 12. shafarevich, i. & remizov, a. (2012). linear algebra and geometry. luxemburg: springer science & business media. 13. smith, a. (1776). an inquiry into the nature and cause of the wealth of nations. 2 vols. london, uk: w. strahan and t. cadell. 14. stewart, j. (1767). an inquiry into the principles of political economy, vol. 1. london, uk: a. millar and t. cadell, strand. 15. walras, l. (1874). elements d’economie politique pure [elements of pure political economy]. lausanne, france: revue de théologie et de philosophie et compterendu des principales publications scientifiques, [in french]. copyright © 2023 assa. adv syst sci appl (2023) https://gks.ru/free_doc/new_site/business/prom/natura/god.htm https://gks.ru/free_doc/new_site/business/prom/natura/god.htm https://www.imf.org/external/np/seminars/eng/2010/afrfin/pdf/radetzki2.pdf https://www.imf.org/external/np/seminars/eng/2010/afrfin/pdf/radetzki2.pdf introduction problem statement on the uniqueness and nonuniqueness of equilibrium in a closed market model adv syst sci appl 2022; 03:135–146 published online at https://ijassa.ipu.ru. three-currency deposit diversification: savage’s principle approach vitaly s. molostvov1* 1higher school of economics (national research university), moscow, russia abstract: the problem of optimal multi-currency deposit diversification with uncertain future exchange rates is studied as the problem of the minimization of the lost profit. it is assumed that only the ranges of these uncertain parameters are known. the savage minimax regret conception is used to minimize the lost profit (risk by savage) caused by uncertainty. the risk function and the function of the guaranteed risk are calculated in an explicit form. after that, the problem is reduced to finding the point of the minimum of a piecewise linear function under simple linear constraints. explicit formulas for nine ”representative” point-candidates for the optimal solution are found. the final choice is made by direct comparison of the values of the savage criterion at these points. keywords: deposit diversification, uncertainty, risk by savage, guaranteed on risk solution introduction the paper deals with finding the guaranteed risk solution (the savage minimax regret solution) in the deposit diversification problem with three types of currencies. an explicit form of the optimal solution is obtained, and simple and clear recommendations for decision makers are presented. in general, decision makers (dm) can be classified into three categories: risk averse, risk takers, and neutral. risk averse avoid any risk and try to maximize the guaranteed income. risk takers consider the risk only and seek to minimize it. neutral dms try to consider simultaneously income and risk. very often, the main cause of risk is the incompleteness of information when making decisions. if stochastic characteristics of uncertain parameters are known, the optimization problem is usually reduced to a deterministic one with average values of optimized indexes. a more complex case of information incompleteness is that the dm knows neither the values of uncertain parameters nor or their stochastic characteristics, but only the ranges. this very case will be investigated below in the framework of a problem of optimality with respect to guaranteed risk deposit diversification. there exist many definitions of risk. in the current study, we understand risk in terms of savage [4] (risk by savage or regret). it may be interpreted as the loss of the income (regret) due to the lack of knowledge of the uncertain parameters. we proceed from the risk taker’s viewpoint and use the best guaranteed result approach to get an explicit form of the optimal solution (the savage minimax regret solution) in a three-currency diversification problem. ∗corresponding author: molostvov@list.ru 136 v.s. molostvov other possible points of view (risk averse and neutral) were studied earlier in [9]. particularly, the best guaranteed income solution for the problem under consideration with n types of currencies was obtained in an explicit form. a neutral dm, who takes into account both the outcomes and the risks, seeking to increase the value of the outcome and to reduce the value of the risk, has to take into account at the same time that any possible values of the uncertain parameters can occur. the correspondent bicriteria problem under the uncertainty was studied in [9], where the explicit form of the optimal solution in the diversification problem with two currencies was obtained and simple rules for dm were formulated. various aspects of multiple-criteria optimization under uncertainty were investigated both in the static and dynamic cases in [1] – [3], [6] – [8] the savage minimax regret solution in a two-currency problem was obtained in [9]. in this paper we investigate the same problem for three currency case. 1. formulation of the problem we assume that the optimal structure of the deposit diversification for any amount of money (in rubles) is completely determined by the optimal allocation of one ruble. at the beginning of the given time interval (here, a year) the dm distributes one ruble among three deposits, for definiteness, in rubles, dollars and euros. dm aims to obtain the highest possible value (in terms of rubles) at the end of the deposit period. so, let kd and ke be the exchange rates of dollar and euro against ruble at the beginning of the year, and 1− xd − xe, xd, xe be the sizes of ruble, dollar and euro deposits respectively (in ruble terms). interest rates of all types of the deposits r, dd, de are assumed to be known. however, dm does not know the exact exchange rates of dollars yd and euro ye at the end of the deposit period and there are no available statistical characteristics concerning their possible values. only the ranges of these uncertain parameters are known: yd ∈ [ad, bd], ye ∈ [ae, be] . the consolidated result (income) at the end of the year after conversation to rubles depends both on a plan of diversification x = (1− xd − xe, xd, xe) and the exchange rates at the end of the period – uncertainties y = (yd, ye) ∈ y = [ad, bd]× [ae, be] . (1.1) this result is the sum of the future values of the different components of the deposits after back conversion into rubles. therefore, it can be presented in the following form: f(x, y) = (1 + r) (1− xd − xe) + ξdxdyd + ξexeye, (1.2) where ξd = 1 + dd kd , ξe = 1 + de ke . (1.3) at a meaningful level, the task of dm is the design of the optimal strategy x = (xd, xe) ∈ x = {xd + xe ≤ 1, xd ≥ 0, xe ≥ 0} (1.4) in order to achieve the greatest result f (x, y). however, dm should take into account the possibility of realizing any values of uncertainty y ∈ y . thus, the mathematical model of the problem of diversification is represented by the ordered triple γ = ⟨x, y, f(x, y)⟩, where f(x, y) is the utility function of a depositor (dm) defined in (1.2),x is the set of dm’s strategies defined in (1.4), and y is the set of uncertainties defined in (1.1). copyright © 2022 assa. adv syst sci appl (2022) three-currency deposit diversification 137 the problem γ is a single-criterion decision-making problem under uncertainty: to maximize a linear function on x with uncertain coefficients on the polyhedron x taking into account the range y of the uncertain factors. the presence of uncertainty leads to the concept of risk as the possibility of deviation of realistic results from the desired or expected values. 2. the savage minimax regret principle let f(x, y) be the objective function (the income), x be the set of dm strategies, and y be the set of uncertainties. then maxz∈x f(z, y) is the best result (income), if uncertainty y occurs. nevertheless, dm does not know which value of the uncertain parameter y will realize. the difference φ(x, y) = max z∈x f(z, y)− f(x, y) (2.5) is called the savage risk function (regret). it presents the loss due to non-acquaintance – the difference between the best result, obtained with known uncertainty, and the real result with any strategy x. this risk depends both on the strategy x and the uncertainty y. trying to minimize the risk, dm may use the concept of the best guaranteed result (wald’s principle; see [5]). this leads to the following definition (savage’s principle of the minimax regret; [4]). definition 2.1: a strategy xr ∈ x is called a guaranteed risk solution (grs) of the problem γ, if φr = max y∈y φ(xr, y) = min x∈x max y∈y φ(x, y), (2.6) where the savage risk function φ(x, y) is defined in (2.5). note some properties of the grs: • due to (2.5), the risk function is nonnegative (the best risk is zero risk): φ(x, y) ≥ 0 ∀x ∈ x, ∀y ∈ y. • if the function f(x, y) is continuous and x, y are compacts, then grs exists. • for every strategy x ∈ x , the inner maximum operation in (2.6) determines a guarantee on the risk φ [x] = max y∈y φ (x, y) ≥ φ(x, y) ∀y ∈ y. the outer minimum operation in (2.6) min x∈x φ[x] = φ[xr] = φr chooses the best (the least) guarantee, for φr ≤ φ[x], ∀x ∈ x, and φr = φ[xr] ≥ φ(x, y), y ∈ y. therefore, dm seeks to reduce his risk by choosing the strategy x ∈ x , assuming dm should take into account the possibility of realization every uncertainty y ∈ y . remark 2.1: the construction of grs consists of four steps: 1. construction of the function f [y] = max z∈x f(z, y), ∀y ∈ y. 2. construction of the risk function φ(x, y) = f [y]− f(x, y). copyright © 2022 assa. adv syst sci appl (2022) 138 v.s. molostvov 3. computation of the inner maximum in (2.6) that determines the guarantee on risk: max y ∈y φ(x, y) = max y∈y (f [y]− f(x, y)) = φ[x] ≥ φ(x, y), ∀x ∈ x. 4. calculation of the outer minimum in (2.6) yields the best guaranteed risk: φr = min x∈x φ[x] = φ[xr]. finally, the obtained strategy xr is the grs of the problem γ. remark 2.2: previous considerations are valid for arbitrary f ,x , and y . if the function f is defined in (1.2) and the sets x, y are from (1.4), (1.1), then the diversification plan (1− xrd − xre, x r d, x r e) is the grs in γ. 3. explicit form of grs assume that the function f is defined in (1.2) and the sets x, y are from (1.4), (1.1). in this section, we construct the grs following the algorithm described above (remark 2.1). step 1. the function f(x, y) is linear with respect to the variable x = (xd, xe). therefore, for every fixed y ∈ y it attains its maximum on the polyhedronx at one of the vertices (0, 0), (1, 0) or (0, 1). taking into account (1.2), we obtain: f [y] = max z∈x f(z, y) = max{(1 + r), ξdyd, ξeye}, (3.7) where ξd, ξe are defined in (1.3). step 2. consider the partition y = y1 ∪ y2 ∪ y3 (some of the subsets yi can be empty) defined by the following conditions: y1 = {y ∈ y : 1 + r ≥ ξdyd, 1 + r ≥ ξeye}, y2 = {y ∈ y : ξdyd ≥ 1 + r, ξdyd ≥ ξeye}, y3 = {y ∈ y : ξeye ≥ 1 + r, ξeye ≥ ξdyd}. (3.8) then the risk function has the form φ(x, y) = f [y]− f(x, y) =  φ1(x, y), y ∈ y1, φ2(x, y), y ∈ y2, φ3(x, y), y ∈ y3, (3.9) where φ1(x, y) = (1 + r)− f(x, y) = (1 + r − ξdyd)xd + (1 + r − ξeye)xe, φ2(x, y) = ξdyd − f(x, y) = (1− xd)(ξdyd − (1 + r)) + (1 + r − ξeye)xe, φ3(x, y) = ξeye − f(x, y) = (1 + r − ξdyd)xd + (ξeye − (1 + r))(1− xe). (3.10) the functions φi are bilinear by the variables x and y. that is why the risk function φ(x, y) with any fixed strategy x is a piece-wise linear function on the uncertainty y. the sets yi are the domains (polygons) of linearity of the risk function with respect to y. in other words, the risk function φ(x, y) coincides with the linear (with respect to x) function φi(x, y) for any fixed y ∈ yi. in this case, we shall say that the risk function φ(x, y) is specified at the point (x, y) of the function φi(x, y). copyright © 2022 assa. adv syst sci appl (2022) three-currency deposit diversification 139 step 3. for every strategy x ∈ x the guaranteed risk φ[x] is computed. due to natural conditions r > 0, di > 0, bi > ai > 0, ki > 0 for i = d, e, we obtain: φ [x] = max y∈y φ (x, y) = max i=1,2,3 φi [x], (3.11) where φ1[x] = φ1(x, ad, ae) = (1 + r − ξdad)xd + (1 + r − ξeae)xe, φ2[x] = φ2(x, bd, ae) = (1− xd)(ξdbd − (1 + r)) + (1 + r − ξeae)xe, φ3[x] = φ3(x, ad, be) = (1 + r − ξdad)xd + (ξebe − (1 + r))(1− xe). (3.12) for the sake of brevity, we introduce the following notations: αd = 1 + r − ξdad, αe = 1 + r − ξeae, βd = ξdbd − (1 + r), βe = ξebe − (1 + r). (3.13) then the formula (3.11) reads φ[x] = max{φ1[x], φ2[x], φ3[x]} = max{αdxd + αexe, βd(1− xd) + αexe, αdxd + βe(1− xe)}. (3.14) step 4. now we seek the best guaranteed risk φr = min x∈x φ[x] = φ[xr] by considering the problem minx∈x φ[x] separately in the interior and on the border of the set (triangle) x and choosing the final best result from these partial results. namely, min x∈x φ[x] = min { min x∈[o,a] φ[x], min x∈[o,b] φ[x], min x∈[a,b] φ[x], min x∈intx φ[x] } . (3.15) case 1. the cathetus oa. the strategy x ∈ [o,a] = {xd ∈ [0, 1], xe = 0}, that is, x = (xd, 0), and we deal with the function φ(xd) = φ[xd, 0] = max{αdxd, βd(1− xd), αdxd + βe} = max{φ1(xd), φ2(xd), φ3(xd)}, where φ1(xd) = αdxd, φ2(xd) = βd(1− xd), φ3(xd) = αdxd + βe. note that φ3(xd)− φ1(xd) = βe − const. subcase 1 a. if βe ≥ 0, then φ3(xd) ≥ φ1(xd) for every xd, and the function φ1 can be excluded from consideration. the graph of φ on the segment [0, 1] consists of one or two line segments. the function φ attains its minimum at the point a = (1, 0) or o = (0, 0) or at the intersection xoa d of the graphs φ2 and φ3 if xoa d ∈ [0, 1]. the point xoa d can be determined from the equation βd(1− xd) = αdxd + βe, which yields xoa d = βd − βe αd + βd . (3.16) the condition xoa d ∈ [0, 1] is equivalent to the following relations between the parameters: ad ≤ (1 + de)kd (1 + dd)ke ≤ bd. copyright © 2022 assa. adv syst sci appl (2022) 140 v.s. molostvov subcase 1 b. if βe ≤ 0, then φ3(xd) ≤ φ1(xd) for every xd, and the function φ3 can be excluded from consideration. the graph of φ on [0, 1] consists of one or two line segments. the function φ attains its minimum at the pointa = (1, 0) oro = (0, 0) or at the intersection xoa d of the graphs φ1 and φ2 if xoa d ∈ [0, 1]. the point xoa d can be determined from the equation αdxd = −βdxd + βd, which yields xoa d = βd αd + βd . (3.17) the condition xoa d ∈ [0, 1] is equivalent to the following relations between the parameters: ad ≤ 1 + r 1 + dd kd ≤ bd. the both formulas (3.16) and (3.17) may be combined in the single expression xoa d = βd − βe(sign βe + 1)/2 αd + βd . (3.18) further investigations of maximum of the function φ[x] on the interval [oa] is based on the monotonicity of the functions φ1, φ2, φ3. another approach is to calculate and compare the values of the function φ[x]. this comparison may be done after the calculation of all candidates for grs of the problem γ. thus we have proved the following assertion. proposition 3.1: the guaranteed risk function φ[x] attains its minimum on the interval [oa] at a point of the set [0, 1] ∩ {o = (0, 0), a = (1, 0), xoa = (xoa d , 0)}, (3.19) where xoa d is defined in (3.18). in other words, if the grs xr = (xd, xe) lies in the part [oa] of the boundary of the set x (xe = 0), then the set (3.19) contains the grs. case 2. the cathetus ob. the strategy x ∈ [0, b] = {xd = 0, xe ∈ [0, 1]}, that is, x = (0, xe), and we deal with the function ψ(xe) = φ[0, xe] = max{αexe, αexe + βd, βe(1− xe)}. in this case, all reasonings are similar to the previous one. we present the corresponding results: subcase 2 a. if βd ≥ 0, then the function ψ attains its minimum at the point b = (0, 1) or o = (0, 0) or the point xob e = βe − βd αe + βe , (3.20) if the condition xob e ∈ [0, 1] holds true. the letter is equivalent to the following relations between the parameters: ae ≤ (1 + dd)ke (1 + de)kd ≤ be. subcase 2 b. if βd ≤ 0, then ψ attains its minimum at the point b = (0, 1) or o = (0, 0) or xob e = βe αe + βe , (3.21) copyright © 2022 assa. adv syst sci appl (2022) three-currency deposit diversification 141 if the condition xob e ∈ [0, 1] holds true. the letter is equivalent to the following relations between the parameters: ae ≤ 1 + r 1 + de ke ≤ be. the both formulas (3.20) and (3.21) may be combined in the single expression xob e = βe − βd(sign βd + 1)/2 αe + βe . (3.22) recall that the final comparison will be done after the analysis of the remaining calculation for grs. thus we have proved the following assertion. proposition 3.2: the guaranteed risk function φ[x] attains its minimum on the interval [ob] at a point of the set [0, 1] ∩ {o = (0, 0), b = (0, 1), xob = (0, xob e )}, (3.23) where xob e is defined in (3.22). in other words, if the grs xr = (xd, xe) lies in the part [ob] of the boundary of the set x (xe = 0), then the set (3.23) contains the grs. case 3. the hypotenuse of the triangle aob: ab = {xd + xe = 1, xd ≥ 0, xe ≥ 0}, the strategy x = (1− xe, xe) and the guaranteed risk function (3.14) takes the form χ (xe) = φ [1− xe, xe] = max {χ1 (xe) , χ2 (xe) , χ3 (xe)} , xe ∈ [0, 1], (3.24) where χ1(xe) = φ1[1− xe, xe] = (αe − αd)xe + αd, χ2(xe) = φ2[1− xe, xe] = (βd + αe)xe, χ3(xe) = φ3[1− xe, xe] = (αd + βe)(1− xe). (3.25) let li be the graph of the linear function χi, i = 1, 2, 3. the function χ (xe) is a convex piecewise linear function on the variable xe ∈ [0, 1]. the graph of χ (xe) is the lower envelope of the triple l1, l2, l3 and it consists of one, two or three segments. a function of such kind attains its minimum at one of the ends of the segment [0, 1] or at the point of interception of some pair of l1, l2, l3. therefore, three new candidates for possible points of maximum of the function χ (xe) can be found from the following equations: χ1(xe) = χ2(xe), χ1(xe) = χ3(xe), χ2(xe) = χ3(xe). solutions of these equations have the following form: x12e = αd αd + βd , x13e = βe αe + βe , x23e = αd + βe αe + αd + βe + βd . (3.26) the solutions outside the open interval (0, 1) should be eliminated from consideration. the remaining points from the set (3.26) together with two inner points – candidates from the case 1 and the case 2 – and three vertices of the triangle x constitute the full set of possible points of the minimum of the function φ[x] on the boundary of the set x . thus we have proved the following assertion. proposition 3.3: the guaranteed risk function φ[x] attains its minimum on the interval (a,b) at a point of the set (0, 1) ∩ { x12 = (x12e , 1− x12e ), x13 = (x13e , 1− x13e ), x23 = (x23e , 1− x23e ) } . (3.27) in other words, if the grs xr = (xd, xe) lies in the part (a,b) of the boundary of the set x (xe = 0), then the set (3.27) contains the grs. copyright © 2022 assa. adv syst sci appl (2022) 142 v.s. molostvov case 4. the interior of the triangle aob: intx = {xd + xe < 1, xd > 0, xe > 0}. the strategy x = (xd, xe) and the guaranteed risk function (3.14) takes the form φ[x] = max{φ1[x], φ2[x], φ3[x]} = max{αdxd + αexe, βd(1− xd) + αexe, βe(1− xe) + αdxd}, x ∈ intx. (3.28) now we introduce the following condition tm (total mixing). the guaranteed risk function φ [x] attains its minimal value φ∗ on the set x at an inner point of x , and φ[x] > φ∗ for all points x ∈ intx . the condition tm means that the optimal diversification plan necessarily uses all three currencies. later on, in this section, we restrict our consideration by the condition tm, since other possible situations are covered by the propositions 3.1 – 3.3. the guaranteed risk φ [x] is a convex continuous function, it may attain its minimum at inner point of the convex set x . this minimum is the global minimum on the set x . let p1, p2, p3 be two-dimensional planes defined by the equations z = fi (x), x ∈ r2, i = 1, 2, 3. the graph of the guaranteed risk function φ [x] is the lower envelope of a family of the planes p1, p2, p3, or, more precisely, its part located above the trianglex . the epigraph of the function φ [x], x ∈ x , is an inverted obelisk-shaped three-dimensional body. calculation of the minimum of the function φ [x] on x is equivalent to finding the lowest point of that body. such a geometric interpretation allows us to simplify the minimization of the nonsmooth function φ [x] with linear restrictions. since the minimum of φ[x] on the boundary of x is already investigated, it remains to consider the minimum of φ[x] on intx . lemma 3.1: suppose that φi[x] is not identically constant for at least one i ∈ {1, 2, 3} and there exists a point x0 ∈ intx such that φi[x 0] > φj[x 0], φi[x 0] > φk[x 0], ∀ j ̸= i, k ̸= i. (3.29) then x0 is not a point of minimum of φ[x] on the set intx . proof due to the equality φ[x] = max{φ1[x], φ2[x], φ3[x]}, the condition of the lemma means that the value of φ [x] is determined at the point x0 only by the function φi [x]. since all functions φi[x] are continuous, the inequalities (3.29) hold true in a neighborhood of the point x0. in this neighborhood the risk function coincides with the linear function φ [x] = φi (x), which is not constant. therefore, it cannot reach the extremum (even local) at a point x0 ∈ intx . lemma 3.1 does not cover degenerate cases that all functions φi[x] are constant. however, it is obvious that in these cases the function φ[x] is also constant, whence all admissible solutions x ∈ x are optimal. let the function φ1[x] = αdxd + αexe is identically constant, that is, αd = αe = 0. then the risk function is defined by the formula φ[x] = max{0, φ2[x], φ3[x]} = max{0, βd(1− xd), βe(1− xe)}, where 0 < (1− xd) < 1, 0 < (1− xe) < 1 for any point x = (xd, xe) ∈ intx . let us show that then the condition tm is not satisfied, and the minimum is attained on the boundary of the set x . indeed, consider all possible combination of the signs of βd, βe: copyright © 2022 assa. adv syst sci appl (2022) three-currency deposit diversification 143 1. let βd ≤ 0 and βe ≤ 0, then φ2[x] ≤ 0 and φ3[x] ≤ 0. therefore, φ[x] = φ∗ = 0 for every x ∈ x . all admissible diversification plans are optimal. the condition tm is not satisfied. 2. let βd ≤ 0 and βe > 0, then φ2[x] ≤ 0 and φ3[x] > 0. therefore, the function φ[x] = φ3[x] = βe(1− xe) for every x ∈ intx . the condition tm is not satisfied, since the linear function φ3[x] does not reach minimum on the open set intx . 3. let βd > 0 and βe ≤ 0, then φ2[x] > 0 and φ3[x] ≤ 0. therefore, the function φ[x] = φ2[x] = βd(1− xd) for every x ∈ intx . as before, the condition tm is not satisfied. 4. let βd > 0 and βe > 0, then φ2[x] > 0 and φ3[x] > 0. therefore, the function φ[x] = max{βd(1− xd), βe(1− xe)} for every x ∈ intx . note that the functions φ2 [x], φ3 [x] are not constant. then, as well as in lemma 3.1, the function φ[x] can attain its minimum in intx only on the line βd(1− xd) = βe(1− xe). however, the linear function φ[x] = βd(1− xd) = βe(1− xe) does not attains its minimum on an open interval. the condition tm is not satisfied. from now, we may assume that the function φ1[x] is not constant. thus, the minimum of the function φ[x] on the set intx (if it exists) can be reached only at the intersection of some pair of the planes pi, i = 1, 2, 3. therefore, in what follows we consider only those interior points of the set x where the risk function is determined by more than one function φi: φ1[x] = φ2[x] or φ1[x] = φ3[x] or φ2[x] = φ3[x]. we denote the corresponding sets by l1, l2, l3, respectively. due to the linearity of φi, all li are open intervals (possibly, empty). consider the function φ[x] on the interval l1, where φ1[x] = φ2[x]. then only two situations are possible. subcase a. φ3[x] < φ1[x] = φ2[x], x ∈ l1 (the function φ3 is omissible). if the linear function φ[x] = φ1[x] is constant on an open interval l1, then it attains its global minimum at the border of the set x as well as on x . this minimum will be found when examining the boundary of the set x . if φ[x] = φ1[x] is not constant on l1, then it has no points of minimum on the open interval l1. subcase b. φ1[x ∗] = φ2[x ∗] = φ3[x ∗] at some point x∗ ∈ l1. this point corresponds to an interception of three plains p1, p2, p3, where pi is the graph of the function z = φi[x], i = 1, 2, 3. the uniqueness of such points under natural assumptions about the parameters is shown below. therefore, the internal point of the minimum of the guaranteed risk function satisfies the following system of linear equations and inequalities: φ1[x] = φ2[x], φ3[x] = φ2[x], xd + xe < 1, xd > 0, xe > 0. (3.30) taking into account the above expressions for the functions φi[x], this yields the system of linear equations αdxd + αexe = βd(1− xd) + αexe, αdxd + βe(1− xe) = βd(1− xd) + αexe, (3.31) whose solution x∗ = (x∗d, x ∗ e) = ( βd αd + βd , βe αe + βe ) (3.32) copyright © 2022 assa. adv syst sci appl (2022) 144 v.s. molostvov is a candidate to be the grs, if the inequalities in (3.30) hold true. we would like to emphasize that x∗ is the point of inner minimum, if the function φ [x] attains its minimum on intx . remark 3.1: due to the natural assumptions bd > ad, be > ae and the notation (3.13) we have αd + βd > 0, αe + βe > 0. therefore, the solution of the linear system (3.31) exists, it is unique and defined by the formula (3.32). thus we have proved the following assertion. proposition 3.4: suppose that function φ [x] attains its minimum on the open set intx . then the point x∗ given by the formula (3.32) is the point of minimum, if x∗d, x ∗ e ∈ (0, 1) and xd + xe < 1. remark 3.2: recall that any point of local minimum of a convex function on a convex set is at the same time the point of global minimum on the given set. hence, if the point of inner minimum (3.32) exists and it belongs to intx , it gives the gr solution. whether it exists or not depends on the interrelations between the parameters of the problem. the gr-solution very often belongs to the boundary of the set of diversification plans. it means that the optimal plan of diversification contains only two (or even one) of three currencies. let us rewrite the inequalities for (x∗d, x ∗ e) from (3.30) in terms of the initial parameters of the problem under consideration. natural conditions bi > ai, i = d, e, imply the following inequalities (see (3.13)): αi + βi = ξi(bi − ai) > 0, i = d, e, where ξi are defined in (1.3). hence the condition (x∗d > 0) ∧ (x∗e > 0) is equivalent to (βd > 0) ∧ (βe > 0), (3.33) which, in turn, is equivalent to ξibi > 1 + r, i = d, e. (3.34) assume that the condition (3.33) holds true. then the condition (x∗d < 1) ∧ (x∗e < 1) is equivalent to (βd < 1) ∧ (βe < 1), which, in turn, is equivalent to ξiai < 1 + r, i = d, e. (3.35) finally, we remark that the inequality x∗d + x∗e < 1 is equivalent to βd/(αd + βd) + βe/(αe + βe) < 1, that is, βdβe < αdαe, which is equivalent to (ξdbd − (1 + r))(ξebe − (1 + r)) < (ξdad − (1 + r))(ξeae − (1 + r)). (3.36) thus, the union of the inequalities (3.34), (3.35) and (3.36) is a necessary and sufficient condition for x∗ ∈ intx . remark 3.3: we emphasize that x∗ is only the candidate, but not necessarily a point of minimum. copyright © 2022 assa. adv syst sci appl (2022) three-currency deposit diversification 145 4. final algorithm to construct the guaranteed risk solution combining the above results (propositions 3.1 – 3.4), we obtain the following algorithm to calculate the guaranteed risk diversification strategy and the minimal guaranteed risk: 1. write down the numerical values of the interest rates r, dd, de and the current exchange rates kd, ke. set the possible ranges (per year) for dollar and euro: [ad, bd] and [ae, be], respectively. 2. calculate the secondary parameters: αd, αe, βd, βe using the formula (3.13). 3. calculate nine points – the candidates to be the optimal solution. it is convenient to represent these points in the form presents in the table below. 4. remove the points that do not belong to the set x of the admissible plans. 5. calculate the values of the guaranteed risk at the remaining points using the formula (3.15). choose the solution with the best (minimal) guaranteed risk. candidate formula condition 1 the vertex o xd = 0, xe = 0 – 2 the vertex a xd = 1, xe = 0 – 3 the vertex b xd = 0, xe = 1 – 4 the inner point (xoa d , 0) xoa d = βd−βe(signβe+1)/2 αd+βd 0 < xoa d < 1 of the cathetus (oa) 5 the inner point (0, xob e ) xob e = βe−βd(signβd+1)/2 αe+βe 0 < xob e < 1 of the cathetus (ob) 6 the inner point (1− x12e , x 12 e ) x12e = αd αd+βd 0 < x12e < 1 of the hypotenuse (ab) 7 the inner point (1− x13e , x 13 e ) x13e = βe αe+βe 0 < x13e < 1 of the hypotenuse (ab) 8 the inner point (1− x23e , x 23 e ) x23e = αd+βe αe+αd+βe+βd 0 < x23e < 1 of the hypotenuse (ab) 9 the inner point (x∗d, x ∗ e) x∗d = βd αd+βd 0 < x∗d, x ∗ e < 1 of the set x x∗e = βe αe+βe x∗d + x∗e < 1 5. conclusion we considered the problem of optimal multi-currency deposit diversification with uncertain future exchange rates as a problem of minimization the lost profits. it was assumed that only the ranges of the uncertain parameters are known, and there are no statistical characteristics. the savage minimax regret conception is used to minimize the lost profits due to uncertainty. it should be noted that the savage criterion is more sophisticated than the more often used wald criterion. therefore, the corresponding mathematical constructions are more complicated. nevertheless, in the problem under consideration, it is possible to find the risk copyright © 2022 assa. adv syst sci appl (2022) 146 v.s. molostvov function and the function of the guaranteed risk in an explicit form. after that, the problem is reduced to finding the point of the minimum of a piecewise linear function under simple linear constraints. to solve it, the problem is decomposed into two stages: finding the minimum on the boundary and on the interior of the given set. explicit formulas for nine representative points being candidates for the optimal solution are found. the final choice is made by the direct comparison of the values of the savage criterion at these points. under certain ratios of the parameters, the optimal solution is not unique. without dwelling in more detail, we only note that in the case of non-uniqueness, it is possible to consider an additional criterion (for example, the guaranteed income). this leads to multicriteria optimization problems with uncertainty. some problems of this class were considered earlier in [9]. it seems promising to apply the approaches developed in these works to the analysis of financial management problems with incomplete information. acknowledgements this article is an output of a research project implemented as part of the basic research program at the national research university higher school of economics (hse university). the work was partially supported by the international centre of decision choice and analysis of the national research university higher school of economics. the author is grateful to the reviewers for their thorough and in-depth study of the paper, constructive comments, and valuable suggestions, which gave possibility to improve significantly the presentation of the paper. references 1. molostvov, v.s. (1983). multiple-criteria optimization under uncertainty: concepts of optimality and sufficient conditions. in: theory and practice of multiple criteria decision making, north-holland, 91–105. 2. molostvov, v.s. (2004). savage’s principle for non-cooperative games under uncertainty. in: problems of control and power engineering, tbilisi, georgia, 38–39. 3. molostvov, v.s. (2011). multiple criteria optimization for stochastic systems with uncertain parameters. model assist. stat. appl., 3, 231–237. 4. savage, l.y. (1954). the foundation of statistics. new york: willey. 5. wald, a. (1939). contribution to the theory of statistical estimation and testing hypothesis, annuals math. stat., 10, 299–326. 6. zhukovskiy, v.i. & molostvov, v.s. (1980). on the pareto-optimality in cooperative differential games with multi-valued objective functionals. in: mathematical methods in systems theory, moscow: istit. syst. stud., 96–108 [in russian]. 7. zhukovskiy, v.i. & molostvov, v.s. (1988). multiple-criteria decision making in conditions of uncertainty. moscow: int. res. inst. manag. sci. [in russian]. 8. zhukovskiy, v.i. & molostvov, v.s. (1990). multiple-criteria optimization of systems in conditions of incomplete information. moscow: int. res. inst. manag. sci. [in russian]. 9. zhukovskiy, v.i., molostvov, v.s. & topchishvili, a.l. (2014). multicurrency deposit diversification – three possible approaches to risk accounting. int. j. oper. quant. manag., 20, 1–14. copyright © 2022 assa. adv syst sci appl (2022) formulation of the problem the savage minimax regret principle explicit form of grs final algorithm to construct the guaranteed risk solution conclusion adv syst sci appl 2020; 04; 1-10 published online at https://ijassa.ipu.ru. an algorithm to search for all minimal cuts in a flow network majid forghani-elahabad1*, nezam mahdavi-amiri2 1) centro de matemática, computação e cognição – cmcc, universidade federal do abc – ufabc, santo andré, sp, brazil e-mail: m.forghani@ufabc.edu.br 2) faculty of mathematical sciences, sharif university of technology, tehran, iran e-mail: nezamm@sharif.edu abstract: reliability evaluation approaches exploit a variety of tools for system modeling and calculation of reliability indices. most proposed approaches in the literature are in terms of minimal cuts (mcs) or minimal paths. hence, finding all the mcs in a network is an attractive problem in network flow and reliability theory. here, investigating some works related to the mc problem, we first point out a number of defects in certain proposed algorithms, and present certain new techniques to rectify the flaws in two algorithms. moreover, we present some techniques to improve the algorithms. using our results, we propose an improved algorithm for finding all the mcs in a flow network. a benchmark network example is used to illustrate the performance of the algorithm. the proposed algorithm is shown to be correct. finally, after establishing the complexity results, we demonstrate the proposed algorithm to be more efficient than some existing algorithms. keywords: minimal cut, network flow, algorithm, reliability 1. introduction most proposed algorithms for evaluating the reliability of a flow network make use of minimal cuts (mcs) [3, 4, 9, 10, 16-18, 19, 20] or minimal paths [5, 6, 7, 8, 11, 12, 14, 15, 22]. in the algorithms based on mcs, we usually need to know all the mcs in advance. this has promoted the mc problem that is, finding all the mcs in a network, as one of the most attractive problems in network flow and reliability theory. a cut is a set of edges whose removal from the network results in disconnection between the source (s) and sink (t) nodes. an mc is a cut with none of its proper subsets being a cut. the mc problem is an np-hard problem [20]. several algorithms have been proposed to obtain all the mcs in both undirected flow networks [1, 2, 13, 25, 26] and directed flow networks [1, 21]. to determine all the mcs, fard and lee [2] considered both link and node failures and proposed an algorithm for finding all the mcs of such networks by the use of networks having perfect nodes. the algorithm adopts the concept of common-cause failure and does not re-enumerate mcs for the additional supposition of node failures. yeh [25] defined an mc using a node set (called mcv), and then proposed a simple algorithm to find all the mcvs between the two special nodes s and t. unfortunately, the algorithm of [25] turned to be faulty and thus might not find all the mcvs of a network [13]. gomes and fernandes [13] tried to identify some but not all the existing defects of [25]. in fact, the modified algorithm of [13] has still some flaws which we will point out later on. yeh et al. [26] proposed an algorithm for the problem of finding all the mcs in a modified network. they showed the algorithm to be more efficient than the existing algorithms. a number of available related algorithms in the * corresponding author: m.forghani@ufabc.edu.br, phone: +55(11)4996-8330. 2 m. forghani-elahabad, n. mahdavi-amiri copyright ©2020 assa. adv. in systems science and appl. (2020) literature were stated in [1]. assuming nodes with k-out-of-n property, tan [21] extended the traditional directed s-t networks and presented an approach for finding all the mcs for all the nodes by using the definition of mc for the nodes and starting with the source node s and ending with the sink node t. here, the techniques of [13] for modifying the proposed algorithm of yeh [25] are improved. we note that the algorithms of [13] and [25] first determine all the mcvs. then, the algorithms transform each mcv to an mc by determining all the existing arcs between the two associating node sets of the mcv by using the adjacency matrix of the network. this may be a very time consuming step for large networks. here, some new techniques are presented to improve upon the existing algorithms. before that, we first state all the flaws of the algorithms of [13] and [25]. in the remainder of our work, in section 2 we provide the required definitions and state the flaws in the algorithms of [13, 25]. in section 3, some more efficient techniques are presented to modify the algorithm of [25]. then, an approach with less time complexity is presented to transform each mcv to an mc. afterwards, an improved algorithm is proposed to determine all the mcs in a network flow by using the presented results. the correctness and complexity results are provided in section 3. section 4 gives our concluding remarks. 2. finding all the minimal cuts the required notations are first described, and then the algorithm of [25] is exposed to explain its flaws in finding all the mcs. we also state some flaws of the modified algorithm of [13]. for brevity, here we only state the proposed algorithm of [25] and do not show the details of the algorithm of [13]. 2.1. problem description we use the same notations, nomenclature, and assumptions of [13, 25]. let g(v, e) be a connected network with the set of nodes v ={s, 1, 2, …, n-2, t} and the set of edges e, where s and t are the source and sink nodes, respectively, and euv ∈ e denotes an edge between nodes u and v. for an arbitrary set of nodes ,u v let ,u v u= − e(u) = {euv∈e | u, v ∈ u} be the associated edges with the set of nodes u, g(u, e(u)) be the sub network of g(v, e) including only the set of nodes u and its associated edges, and mc(u) = {euv ∈ e | u ∈ u and }v u be the corresponding cut. an mcv candidate is a subset of nodes whose removal will cause a disconnection of nodes s and t. an mcv candidate in g(v, e), say u, is an mcv when mc(u) is a minimal cut. let σ be the number of mcvs (or mcs), and for each i=1, 2, …, σ, ci=mc(ui) be the associated mc with mcv, ui. moreover, for each mcv, u, and each node ,v u let e(v,u) = {evu ∈ e | u ∈ u}. 2.2. flaws in the existing algorithms here, all the flaws in the algorithm of [25] are explained. for convenience, we rewrite the proposed algorithm of [25] as ‘algorithm 1’ below. algorithm 1 the algorithm of [25] for finding all the mcvs in a network g(v, e). step 0. let i = k = 0, s = u0 ={s}, t = v−{s}, n0 ={t}, and p=ϕ. step 1. if there is a node u ∈ t–ni adjacent to s, then s ∪ {u} is an mcv candidate and go to step 2, else go to step 4. step 2. if g(t−{u}, e(t−{u})) is a connected network then s ∪{u} is an mcv and go to step 3, else any node set containing s ∪{u} is not an mcv. step 3. let i = i+1, k = k+1, uk = s = s ∪{u}, p = p ∪{uk }, t = t−{u}, ni = ni-1, and go to step 1. an algorithm to search for all minimal cuts in a flow network 3 copyright ©2020 assa. adv. in systems science and appl. (2020) step 4. if i = 1 then stop, else remove the last node, say v, in s, let i = i−1, ni = ni ∪ {v}, t = t ∪ {v}, and go to step 1. we first use a similar example as the one given in [25] to show how “algorithm 1” fails in some cases, and also its obtained solution depends on the order of node selection in step 1. example 1. consider fig. 1 as a network, and find all its mcvs by using ‘algorithm 1’. solution: step 0. let i = k = 0, s = u0 = {s}, t = v−{s}, n0 ={t}, and p=ϕ. step 1. since t − n0 = {1, 2, 3, 4}, node 2 is selected and transfer is made to step 2. step 2. since g({1, 3, 4, t}, e({1, 3, 4, t})) is connected, {s,2} is found as an mcv, and transfer is made to step 3. step 3. let i = 1, k = 1, u1= s ={s,2}, p = {u1}, t = {1, 3, 4, t}, n1 = {t}, and transfer is made to step 1. step 1. since t – n1 = {1, 3, 4}, node 4 is selected and transfer is made to step 2. step 2. since g({1, 3, t}, e({1, 3, t})) is not connected, the algorithm deduces that any set containing the set {s, 2, 4} is not an mcv. fig. 1. a benchmark network next, if transfer is made to step 1, the algorithm may select node 4 recursively, and the algorithm does not terminate. if transfer is made to step 3, ‘algorithm 1’saves the set {s, 2, 4} as an mcv which is an incorrect result. if transfer is made to step 4, since i = 1, the algorithm stops without finding the other mcvs. hence, in any case, the algorithm leads to an incorrect result, and consequently it cannot find all the mcvs. it should be noted that if in the first place one selects node 1 instead of node 2, the algorithms end up with a very different final result. in fact, the obtained solutions by “algorithm 1” depend on the order of node selection in step 1. example 1 illustrated some flaws in ‘algorithm 1’. in the sequel, we mention all the defects in this algorithm in detail. (1) during steps 1-4, ‘algorithm 1’ never determines the mcv={s}, and so we should consider it as an initial input in step 0. to address this flaw, we can replace “p= ϕ” with “p = {u0}” in step 0. (2) as seen in example 1, there is an ambiguity in the second part of step 2 in ‘algorithm 1’, where g(t−{u}, e(t−{u})) is not connected. in fact, in such a case, it is not clear how the algorithm should proceed. (3) in step 2 of ‘algorithm 1’, when g(t−{u}, e(t−{u})) is not connected, the algorithm leads to a wrong result, namely, “any node set containing s ∪ {u} is not an mcv”. to see this more clearly, we note that in example 1, since g({1, 3, t}, e({1, 3, t})) is not connected, the algorithm concludes that any set containing the set {s, 2, 4} is not 4 m. forghani-elahabad, n. mahdavi-amiri copyright ©2020 assa. adv. in systems science and appl. (2020) an mcv, whereas it is apparent that the sets {s, 1, 2, 4} and {s, 1, 2, 3, 4} contain the set {s, 1, 3} and are indeed mcvs. (4) the stopping criterion for the algorithm, i.e., i = 1, is not appropriate and may lead to the loss of some mcvs. in fact, with this stopping condition, the algorithm may stop before determining all the mcvs. although goems and fernandes [13] stated the flaws (1), (2), and (4), but the algorithm of [13] still contained the flaws (3) and (4); see lines 15 (step 2) and 27 (step 4) in the proposed algorithm of [13]. to rectify the stated defect (2), the authors of [13] used a new set d in their proposed algorithm and added lines 4, 16, and 17 to the algorithm. even though this way they removed the flaw, but there is room for improvement. in fact, the proposed algorithm of [13], in such a case (flaw (2)), removes the node u form the set d (see line 16 of the algorithm of [13]) and does not change the sets t and ni. in this case, the algorithm may again add the node u to the set d going through lines 5-7, 11-13, 19-24, and 4. hence, the algorithm of [13] may check a node several times in a time consuming process, deteriorating the efficiency of the algorithm. furthermore, a common inefficiency of both algorithms of [13] and [25] is that both algorithms first determine all the mcvs and then transform each mcv into the associated mc. however, as we state in the next section, if one determines an mc, whenever the associated mcv is obtained, then the computing time would decrease. 3. new results here, we first make some proposals to modify the flaws in the proposed algorithm of [25], and then present some results to improve the efficiency of the algorithm. an improved algorithm is presented and shown to be correct. the complexity results are also provided. 3.1. modifications and improvements to address flaw (1), as we mentioned in section 2.2, we can simply replace “p= ϕ” with “p = {u0}” in step 0. to address flaw (2), we use a new set b and make some more changes in steps 0-2 as given in the proposed algorithm below, algorithm 2. to address the flaw (3), we just need to remove the wrong assertion, “any node set containing s ∪ {u} is not an mcv”, from step 2. to address flaw (4), the stopping criterion for the algorithm is replaced by i = 0. in addition to these modifications, we also make some improvements. we note that the final aim of the algorithm is to determine all the mcs in a flow network. for this, the algorithms of [13] and [25] first find all the mcvs and then transform each mcv into an mc by determining all the arcs between the two associated node sets. let u be an mcv. it is vivid that the associated mc with u, i.e., mc(u), given by ( ) ( , ). u u mc u e u u  = (3.1) for each ,u u the time complexity of obtaining ( , )e u u is o(n). since n is an upper bound for the number of nodes in u, the time complexity of determining each mc is o(n2), and consequently the time complexity for determining all the mcs is o(n2σ). however, if one determines each mc, whenever a new mcv is obtained, then the time complexity can be lessened to o(nσ). we note that a new mcv is obtained by adding to or removing some nodes from the current mcv in the algorithm, and hence considering this, we can determine the associated mc by adding and removing some edges to the current mc. let u be an mcv with the associated mc, mc(u), and { }w u v= be another mcv. in this case, we have ( ) ( ) ( , { }) ( , ).mc w mc u e v u v e v u= − − similarly, when we remove a node, say u, from an algorithm to search for all minimal cuts in a flow network 5 copyright ©2020 assa. adv. in systems science and appl. (2020) mcv, u, we should add the edges of the set ( , )e u u to and remove the edges of the set ( , { })e u u u− from the associated mc to update the mc. for this, corresponding to each node, the time complexity for updating the mc is o(n). as a result, the time complexity for determining all the mcs is o(nσ). in the next section, using the stated modifications, an improved algorithm is proposed to find all the mcs in a flow network. 3.2. proposed algorithm here, we propose an improved algorithm to solve the mc problem, and then provide its correctness and complexity results. note that since in our algorithm, algorithm 2 below, all the mcs are obtained straightforwardly, there is no need to save the mcvs. algorithm 2 an improved algorithm for finding all the mcs in a network g(v, e). step 0. let i = k = 0, s = {s}, t = v−{s}, a = c0 = d0 = e(s, t), n0 = {t}, b = ϕ, and p = {c0}. step 1. if there is a node v ∈ t – {b ∪ ni} adjacent to s then go to step 2, else go to step 4. step 2. if g(t−{v}, e(t−{v})) is a connected network then let b = ϕ and go to step 3, else let b = b ∪ {v} and go to step 1. step 3. let i=i+1, k=k+1, t = t − {v}, ( , ) ( , ),i ka d c a e v t e v s= = = − s = s ∪ {v}, p = p ∪ {ck}, ni = ni-1, and go to step 1. step 4. if i = 0 then stop, else remove the last node, v, in s, let i=i−1, a = di, ni = ni ∪ {v}, t = t ∪ {v}, and go to step 1. for a more convenient description of algorithm 2, a flowchart of the algorithm is given in fig. 2. next, we use a benchmark network named modified arpanet as given in fig. 1 to illustrate algorithm 2, and also show how our introduced changes turn the algorithm to work correctly with reasonable efficiency. fig. 2. a flowchart of algorithm 2 solution: step 0. let i = k = 0, s ={s}, t = {1, 2, 3, 4, t}, a = c0 = e(s, t) = {es1, es2} , n0 = {t}, b = ϕ, and p = {c0}. step 1. since t − {b ∪n0} = {1, 2, 3, 4}, node v = 1 is selected and transfer is made to step 2. step 2. since g({2, 3, 4, t}, e({2, 3, 4, t})) is connected, b = ϕ and transfer is made to step 3. 6 m. forghani-elahabad, n. mahdavi-amiri copyright ©2020 assa. adv. in systems science and appl. (2020) step 3. let i = 1, k = 1, t = {2, 3, 4, t}, a = d1 = c1 = 1 2 12 14 1{ , } { , } { }s s se e e e e− = 2 12 14{ , , },se e e s = {s, 1}, p = {c0, c1}, n1 = n0 = {t}, and transfer is made to step 1. step 1. since t − {b ∪ n1} = {2, 3, 4}, node v = 2 is selected and transfer is made to step 2. step 2. since g({3, 4, t}, e({3, 4, t})) is connected, b = ϕ and transfer is made to step 3. step 3. let i = 2, k = 2, t = {3, 4, t}, a = d2 = c2 = {e14, e24, e23}, s = {s, 1, 2}, p = {c0, c1, c2}, n2 = {t} and transfer is made to step 1. step 1. since t − {b ∪ n2} = {3, 4}, node v = 3 is selected and transfer is made to step 2. step 2. since g({4, t}, e({4, t})) is connected b = ϕ and transfer is made to step 3. step 3. let i = 3, k = 3, t = {4, t}, a = d3 = c3 = {e14, e24, e34, e3t}, s = {s, 1, 2, 3}, p = {c0, c1, c2, c3}, n3 = {t} and transfer is made to step 1. step 1. since t −{b ∪ n3} = {4}, node v = 4 is selected and transfer is made to step 2. step 2. since g({t}, e({t})) is connected b = ϕ and transfer is made to step 3. step 3. let i = 4, k = 4, t = {t}, a = d4 = c4 ={e4t, e3t}, s = {s, 1, 2, 3, 4}, p={c0, c1, c2, c3, c4}, n4 ={t} and transfer is made to step 1. step 1. since t − {b ∪ n4} = ϕ, transfer is made to step 4. step 4. since i = 4 ≠ 0, let v = 4, s = s − {4} = {s, 1, 2, 3}, i = i − 1 = 3, a = d3 = {e14, e24, e34, e3t}, n3 = {4, t}, t = {4, t}, and transfer is made to step 1. step 1. since t − {b ∪ n3} = ϕ, transfer is made to step 4. step 4. since i = 3 ≠ 0, let v = 3, s = s − {3} = {s, 1, 2}, i = i − 1 = 2, a = d2 = {e14, e24, e23}, n2 = n2 ∪ {3} = {3, t}, t = {3, 4, t}, and transfer is made to step 1. step 1. since t − {b ∪ n2} = {4}, node v = 4 is selected and transfer is made to step 2. step 2. since g({3, t}, e({3, t})) is connected b = ϕ and transfer is made to step 3. step 3. let i = 3, k = 5, t = {3, t}, a = d3 = c5 ={e23, e34, e4t}, s = {s, 1, 2, 4}, p = {c0, c1, c2, c3, c4, c5}, n3 = {3, t}, and transfer is made to step 1. step 1. since t − {b ∪ n3} = ϕ, transfer is made to step 4. step 4. since i = 3 ≠ 0, let v = 4, s = s − {4} = {s, 1, 2}, i = i − 1 = 2, a = d2 = {e14, e24, e23}, n2 = n2 ∪ {4} = {3, 4, t}, t = {3, 4, t}, and transfer is made to step 1. step 1. since t − {b ∪ n2} = ϕ, transfer is made to step 4. step 4. since i = 2 ≠ 0, let v = 2, s = s − {2} = {s, 1}, i = i − 1 = 1, a = d1 = {es2, e12, e14}, n1 = n1 ∪ {2} = {2, t}, t = {2, 3, 4, t}, and transfer is made to step 1. step 1. since t −{b ∪ n1} = {3, 4}, node v = 3 is selected and transfer is made to step 2. step 2. since g({2, 4, t}, e({2, 4, t})) is disconnected, b = {3}, and transfer is made to step 1. step 1. since t −{b ∪ n1} = {4}, node v = 4 is selected and transfer is made to step 2. step 2. since g({2, 3, t}, e({2, 3, t})) is connected b = ϕ, and transfer is made to step 3. step 3. let i = 2, k = 6, t = {2, 3, t}, a = d2 = c6 = {es2, e12, e24, e34, e4t}, s = {s, 1, 4}, p = {c0, c1, c2, c3, c4, c5, c6}, n2 = {2, t}, and transfer is made to step 1. step 1. since t −{b ∪ n2} = {3}, node v = 3 is selected and transfer is made to step 2. step 2. since g({2, t}, e({2, t})) is disconnected, b = {3}, and transfer is made to step 1. step 1. since t − {b ∪ n2} = ϕ, transfer is made to step 4. step 4. since i = 2 ≠ 0, let v = 4, s = s − {4} = {s, 1}, i = i − 1 = 1, a = d1 = {es2, e12, e14}, n1 = n1 ∪ {4} = {2, 4, t}, t = {2, 3, 4, t}, and transfer is made to step 1. step 1. since t − {b ∪ n1} = ϕ, transfer is made to step 4. step 4. since i = 1 ≠ 0, let v = 1, s = s − {1} = {s}, i = i − 1 = 0, a = d0 = {es1, es2}, n0 = n0 ∪ {1} = {1, t}, t = {1, 2, 3, 4, t}, and transfer is made to step 1. step 1. since t −{b ∪ n0} = {2, 4}, node v = 2 is selected and transfer is made to step 2. step 2. since g({1, 3, 4, t}, e({1, 3, 4, t})) is connected b = ϕ and transfer is made to step 3. step 3. let i = 1, k = 7, t = {1, 3, 4, t}, a = d1 = c7 = {es2, e12, e24, e23}, s = {s, 2}, p = {c0, c1, c2, c3, c4, c5, c6, c7}, n1 = {1, t} and transfer is made to step 1. step 1. since t −{b ∪ n1} = {3, 4}, node v = 3 is selected and transfer is made to step 2. step 2. since g({1, 4, t}, e({1, 4, t})) is connected b = ϕ and transfer is made to step 3. an algorithm to search for all minimal cuts in a flow network 7 copyright ©2020 assa. adv. in systems science and appl. (2020) step 3. let i = 2, k = 8, t = {1, 4, t}, a = d2 = c8 = {es1, e12, e24, e34, e3t}, s = {s, 2, 3}, p = {c0, c1, c2, c3, c4, c5, c6, c7, c8}, n2 = {1, t}, and transfer is made to step 1. step 1. since t −{b ∪ n2} = {4}, node v = 4 is selected and transfer is made to step 2. step 2. since g({1, t}, e({1, t})) is disconnected, b = {4} and transfer is made to step 1. step 1. since t − {b ∪ n2} = ϕ, transfer is made to step 4. step 4. since i = 2 ≠ 0, let v = 3, s = s − {3} = {s, 2}, i = i − 1 = 1, a = d1 = {es1, e12, e24, e23}, n1 = n1 ∪ {3} = {1, 3, t}, t = {1, 3, 4, t}, and transfer is made to step 1. step 1. since t − {b ∪ n1} = ϕ, transfer is made to step 4. step 4. since i = 1 ≠ 0, let v = 2, s = s − {2} = {s}, i = i − 1 = 0, a = d0 = {es1, es2}, n0 = n0 ∪ {2} = {1, 2, t}, t = {1, 2, 3, 4, t}, and transfer is made to step 1. step 1. since t − {b ∪ n0} = {3}, there is no node adjacent to s and transfer is made to step 4. step 4. since i = 0, the algorithm terminates. we see that algorithm 2 found all the 8 mcs in fig. 1. 3.3. the correctness and complexity results the following theorem demonstrates the correctness of algorithm 2. theorem 1: algorithm 2 determines all the mcs in a connected flow network. before presenting the proof, we note that in the proof below, ci is an mc with two corresponding node sets ui and iu such that i iu v u= − and ci = mc(ui) = { | , }.iuv ie u u v u  moreover, ( ) { | }.i i i uv iv v c u u e c= =   it is clear that ,i iv u and one can name this as the extreme nodes in ui. proof. it is vividly seen that algorithm 2 finds 0 { }svc e e=  as the first mc in step 0. now, to the contrary, assume that there are some mcs missed by the algorithm. without loss of generality, let ci be a missed mc with its corresponding node set ui having a minimal number of nodes. it is clear that vi is not empty, and so consider a .i ij v u  according to definition of the set vi, we know that there is at least one ik u so that ejk ∈ ci, and consequently, g (ui − {j}, e (ui − {j})) and ( { }, ( { }))i ig u j e u j are connected. hence, { }i iu u j = − is an mcv. because ui is the corresponding node set with a missed mc by the algorithm having a minimal number of nodes, the mc associated with the node set ,iu  i.e., ,ic  is found by the algorithm. it is observed that algorithm 2 finds mcs in step 3, and then goes to step 1 to find the next possible one. therefore, if algorithm 2 finds ic  in step 3, then it will determine ci in the subsequent iterations by selecting j as a node adjacent to .is u = this contradicts the earlier assumption that ci is a missed mc, and thus completes the proof. ■ now, we compute the time complexity of algorithm 2. the time complexity of step 0 is o(n). the time complexity for the construction of each mcv through the steps 1-4 is o(n+m), and as explained in section 3.1, the time complexity for updating an mc is o(n). thus, considering σ as the number of all the mcvs (or mcs) in the network, the time complexity of algorithm 2 to determine all the mcs is o(n+(m+n)σ), and since in a 8 m. forghani-elahabad, n. mahdavi-amiri copyright ©2020 assa. adv. in systems science and appl. (2020) connected flow network we have ( ) ( )o n o m [1], the time complexity of algorithm 2 is o(mσ). thus, we have the following result. theorem 2: the time complexity of algorithm 2 for determining all the mcs is o(mσ), where m and σ are respectively the number of edges and the number of mcs in the network. we note that theorem 2 shows that algorithm 2 is as efficient as the recently algorithm of [26] and is more efficient than the algorithms of [13, 25]. 5. concluding remarks there are a number of algorithms for finding all the minimal cuts in different types of networks such as directed or undirected networks, network with node failures, link failures, or both node and link failures, networks with k-out-of-n nodes, cyclic or acyclic networks, etc. here, two available algorithms in the literature were investigated and their flaws were stated. certain efficient techniques were proposed to modify the flaws and some new techniques were also developed. using the presented results, an improved algorithm was proposed to solve the mc problem and its correctness was established. the established complexity results showed the algorithm to be more efficient than some available algorithms. acknowledgements the second author thanks sharif university of technology for supporting this work. references 1. ahuja, r.k., magnanti, t.l. and orlin j.b. 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(1983). the complexity of counting cuts and of computing the probability that a graph is connected. siam journal of computing, 12: 777–788. 21. tan, z. (2003). minimal cut sets of s-t networks with k-out-of-n nodes. reliability engineering and system safety, 82: 49–54. 22. wu, w.w., ning, a. and ning, x.x. (2008). evaluation of the reliability of transport networks based on the stochastic flow of moving objects. reliability engineering and system safety, 93: 838–44. 23. xu, x.z., niu, y.f. and li, q. (2019). efficient enumeration of d-minimal paths in reliability evaluation of multistate networks. complexity, doi: https://doi.org/10.1155/2019/4561845. 24. xu, x.z., niu, y.f. and li, q. (2018). performance assessment of a freight network with stochastic capacities. complexity, article id 9142542, doi: https://doi.org/10.1155/2018/9142542. http://www.inescc.pt/documentos/11-2010.pdf http://www.inescc.pt/documentos/11-2010.pdf https://doi.org/10.1155/2019/4561845 https://doi.org/10.1155/2018/9142542 10 m. forghani-elahabad, n. mahdavi-amiri copyright ©2020 assa. adv. in systems science and appl. (2020) 25. yeh, w.c. (2006). a simple algorithm to search for all mcs in networks. european journal of operational research, 174: 1694–1705. 26. yeh, w.c., ho, h.c., chen, y.c., and yeh, y.m. (2012). a new algorithm for finding all minimal cuts in modified network. international journal of innovative computing, information and control, 8(1): 419–430. 1. introduction 2. finding all the minimal cuts 2.1. problem description 2.2. flaws in the existing algorithms 3. new results 3.1. modifications and improvements 3.2. proposed algorithm 3.3. the correctness and complexity results 5. concluding remarks acknowledgements microsoft word 1096 article text, copyedited 2.doc adv syst sci appl 2022; 01; 51-64 published online at https://ijassa.ipu.ru. evaluating the effectiveness of the regional industrial policy implementation ol'ga p. smirnova*, alena o. ponomareva institute of economics of the ural branch of russian academy of sciences, ekaterinburg, russia email: olygsmirnova95@gmail.com abstract: the industrial development is of great importance for balanced economic growth. emerging new trends such as circular transformation, digitalization, etc. predetermine the need to formulate new methodological approaches to evaluating the effectiveness of industrial policy. regional imbalances shift the research focus towards assessing the emerging gaps. to do so, we present a multi-criteria methodology for evaluating the effectiveness of the regional industrial policy. using statistical and structural analysis and 2015–2019 data for 19 regions of russia, the paper examines the comparative effectiveness of the ongoing industrial policy and identifies the strengths and weaknesses of the industrial regions. among the key development problems revealed in the study are low pay in the manufacturing industry; high depreciation of fixed assets of industrial enterprises; insufficient investment in technological innovation; falling labor productivity and profitability of the manufacturing industry; and insufficient level of environmental safety of industrial production. the research confirms the hypothesis that industrial policy’s effectiveness is influenced by legal regulations at regional level. the findings demonstrate that the top performers in the regional industrial policy implementation have introduced relevant regulations over 15 years ago. hence, the earlier the institutional norms for supporting industrial regions are adopted, the more efficient the industrial development is going to be in the future. the proposed methodology involves a comprehensive assessment of the industry growth, which allows performing comparative analysis of the regions’ development dynamics. keywords: industrial policy, industrial region, institutional regulation, structural imbalances, regional differentiation, socio-economic development 1. introduction an integrated industrial policy is becoming a significant tool for the balanced economic development of russia [16, 29, 37]. to attain this goal, pride of place goes to progressive changes in the structure of industry and its reconstruction on a new technological basis rather than increasing volumes of industrial production. structural imbalances and uneven regional socio-economic development can be easily traced in transitioning economies such as russia. currently, the development of the manufacturing industry in russia is influenced by an array of factors, including heightened global competition, low performance (profitability) of enterprises, poor labor productivity, staff shortage, high depreciation of fixed assets, administrative barriers to entering the world market, reduced investment, insufficient innovative activity, etc. [34-35]. deindustrialization is believed to be the major threat to the russian economy which resulted in heavy dependence on imports of key industries [15]. the process was initiated in the time of active post-soviet reforms and is largely associated with the destruction of the existing business ties between enterprises. another cause is the impairment of the material and technical facilities of the real sector of economy. for a long * corresponding author: olygsmirnova95@gmail.com 52 o.p. smirnova, a.o. ponomareva copyright ©2022 assa. adv. in systems science and appl. (2022) time, the main priority was the development of the non-material sphere and the consumer sector, which led to instability in the fundamental sector of the economy, i.e. manufacturing [24]. in russia, industrial policy is enacted at federal and regional levels. government authorities use a system of targets and indicators for assessing industrial policy’s effectiveness. these indicators are included in government programs for the development of the entire manufacturing sector, as well as in strategies and programs for the development of individual industries. most of them are focused on the increase in gross indicators and the scope of support. the complexity of the manufacturing industry structure hampers the assessment of the government support measures since even determining their effect for a certain area is an arduous task. the latest trends such as state-of-the-art technologies, circular economy, digitalization, etc. make it necessary to develop brand-new methodological approaches to evaluating the effectiveness of industrial policy. in today’s situation, industrial policy should be assessed according to a wider range of indicators and requires a more individual approach. thus, the present study aims to design a methodology for evaluating the effectiveness of the regional industrial policy. using the case of russia, we attempt to trace the dynamics of the industrial development over the past few years and identify stable patterns. we hypothesize that legal regulations at regional level influence the effectiveness of industrial policy. this allows formulating the following scientific tasks: to systematize the factors affecting the change in the structure of the regional industrial complex and to compare the level of industrial development and the time span during which legislative support for the regional industrial policy has been in effect. 2. literature review the problems of theoretical substantiation of industrial policy are widely debated in academic literature [3, 5, 12, 19, 27]. a plethora of interpretations of the industrial policy concept emphasizes the complexity of this phenomenon and its ambiguous and dynamic nature [14, 17, 28, 37]. a number of researchers highlight that active participation of the state as a bearer of the national interests should be the central factor in the formation and implementation of industrial policy since it is oriented towards resolving strategic challenges [2]. an alternative view is that selective support for the private industrial sector can decrease the competitiveness and efficiency of its functioning. establishing the objectives of industrial policy is the principal aspect for assessing its effectiveness and performance. experts from unido [40] describe new industrialization as a knowledge-intensive, digital and high-tech process focused on the reproduction of population and improving their quality of life rather than earning profit [13]. we believe these statements should become the main principles in assessing the effectiveness of industrial policy. thus, along with the economic dimension, special attention should be paid to such factors as social justice, human well-being and environmental protection [30]. the formal approach of the russian federal authorities implies evaluating the industrial policy’s effectiveness according to the industrial production index, the index of production by the type of economic activity “manufacturing”, the index of labor productivity, the amount of investment attracted to the region, the number of investment projects being implemented (aimed at developing industry), the number of high-performance jobs created, etc. [23]. the level of centralization is of special interest when analyzing the effectiveness of the government support measures for industrial production. excessive centralization of power can adversely affect self-sufficiency and initiatives of russia’s constituent territories [9, 33]. a decentralized approach to the implementation of industrial policy makes it possible to more accurately respond to problematic aspects and bottlenecks in a region’s economy. therefore, the purposes and objectives of the federal industrial policy can be evaluating the effectiveness of the regional industrial policy implementation 53 copyright ©2022 assa. adv. in systems science and appl. (2022) expanded and supplemented by the specifics of regional development while being in strong compliance with the federal concept. numerous researchers use the indicative approach as the basis for industrial policy evaluation [24]. in most cases, they distinguish between the following groups of indicators: resource, staff, technological, organizational, financial and infrastructural. the effectiveness of industrial policy is examined using the indicators such as the number of cutting-edge production technologies created; expenditures on technological innovation [6, 38]; results of intellectual activity; the level of staff competence including those with academic degree [7, 25]. goryacheva [10] analyzes the effectiveness of state support measures for the industrial complex from the perspective of achieving the tactical objectives, i.e. an increase in production volumes and a break-even activity. litvinova’s [21] integral approach uses the indicators of socio-economic development of a region and innovation activity. litau [20] proposes a scoring method of evaluation in innovative production. vorobyova [41] assesses the effectiveness of industrial policy according to the principle of diversification of production activities in a region through indicators of concentration and localization. a number of studies [22] often aim to conduct a simple evaluation of the government support for particular enterprises and territories. let us look at the world practice in assessing the industrial policy effectiveness. in sweden, it is analyzed from the standpoint of the development of scientific, technical and innovation spheres. the country’s policy harmonizes the targets set by the state and private business thus forming a responsible attitude [11]. in germany, an effective policy refers to a structural one that is of corrective or formative nature. in this vein, the effectiveness of industrial policy is determined by its contribution to structural transformations and leveling the consequences of excessively rapid structural shifts [4, 36]. priorities of the uk industrial policy are closely linked to r&d. there is active cooperation on the projects of research centers and industrial production. to accelerate the introduction of new technologies, special programs are being implemented, such as “knowledge transfer partnerships” and “knowledge transfer networks”. in recent decades, the eu economy has witnessed a gradual decline in industrial production, which, however, still serves as the main driver of the economy [39]. the abovementioned approaches to assessing the effectiveness of the regional industrial policy form the basis of the methodology developed in the paper. in the introduction and the literature review we highlight that in the context of the innovation economy it is insufficient to assess the effectiveness of the industrial complex development exclusively by the volume and financial indicators. this is a necessity for both developed and developing nations including russia [30, 35]. it is noteworthy that the indicator of growth in production volumes taken separately is unable to fully and adequately assess the effectiveness of the industry’s development, since its growth may be due to economic and non-economic factors. the case of product passports used in the eu countries is highly relevant here. products that do not comply with the environmental standards are not allowed for import. similar quality requirements are applicable to industrial products, and, therefore, they can be taken into account when assessing the development of the regional industrial complex. hence, we propose an integrated evaluation methodology accounting for new trends and requirements for competitiveness. under these conditions, the effectiveness of the regional industrial policy implementation needs to be comprehensively analyzed. 3. materials and methods in a narrow sense, industrial policy is referred to as a set of measures aimed at ensuring the competitive development of the industry, primarily in the manufacturing sector. in a broad sense, industrial policy is a structural policy that can be implemented in any economic sector. in the current study, industrial policy refers to a set of measures aimed at changing 54 o.p. smirnova, a.o. ponomareva copyright ©2022 assa. adv. in systems science and appl. (2022) the structure of industry (economy) in order to enhance the competitiveness of enterprises through active innovation and investment. the proposed methodology for evaluating the effectiveness of the regional industrial policy implements an integrated approach (fig. 1). within the methodology, the “effectiveness” category is viewed as the extent to which the selected indicators of the industrial development are met. the average value for the whole country is used as the criterion level. thus, the criterion of the regional industrial policy’s effectiveness is determined by the deviation from the average level in russia. juxtaposing the selected indicators will allow establishing whether there is a positive or negative dynamics of industrial development in a region. fig. 1. methodology for evaluating the effectiveness of the regional industrial policy. source: [34]. note: ki is comparable coefficient of an i-indicator; хi is the value of the indicator in the region; хip is average value of the indicator for the country; cvi is variation of the i-indicator; σ is the standard deviation; is average value; wi is weighting coefficient; ri is composite effectiveness index. at the initial stage of assessing the effectiveness of industrial policy, it is necessary to determine key indicators. particular indicators are selected according to modern trends in the socio-economic and industrial development. we have identified indicators that correspond to the definition of industrial policy. in total, 21 indicators were used [35]. they were evaluating the effectiveness of the regional industrial policy implementation 55 copyright ©2022 assa. adv. in systems science and appl. (2022) organized into 9 groups, each of which characterizes specific areas of industrial policy (table 1). the presented indicators are calculated based on the unified methodology of rosstat [31], which allows analyzing them in terms of territorial affiliation and evolution dynamics. the time span of the research is limited to the period of 2015–2019 that provides comprehensive data for the last 5 years. table 1. list of indicators characterizing the effectiveness of the regional industrial policy indicator of the regional industrial development group [i01] average monthly nominal accrued wages of employees of organizations salary and headcount [i02] share of manufacturing industry workers in the total number of employees in the region [i03] share of the manufacturing industry in grp structure of economy [i04] share of manufacturing enterprises in the total number of enterprises in the region [i05] percentage of fixed assets of manufacturing enterprises in the overall structure of fixed assets in the region fixed assets [i06] ratio between fixed assets introduced to industry and the total volume of fixed assets introduced [i07] depreciation of fixed assets of manufacturing enterprises at the end of the year [i08] growth rate of advanced production technologies created in the region innovation [i09] growth rate of advanced production technologies used in the region [i10] share of organizations engaged in innovation [i11] share of expenses incurred in technological innovations in the total volume of investment in fixed assets [i12] volume of innovative goods, works, services as a percentage of the total volume of goods shipped, works and services performed [i13] industrial production indices in the manufacturing industry compared to the previous year production growth [i14] index for investment volume in fixed assets investment [i15] labor productivity index in the manufacturing industry production efficiency [i16] return on assets of manufacturing enterprises [i17] share of captured and neutralized air pollutants in the total amount of waste pollutants environment [i18] ict use in organizations (portable computers, %) information and communications technology (ict) [i19] ict use in organizations (servers, %) [i20] ict use in organizations (computer networks, %) [i21] ict use in organizations (cloud services, %) in the study, we scrutinize the regions with the developed manufacturing industry (over 27% of grp). the industrial development was characterized using a number of multicriteria indicators. therefore, only those indicators were applied describing the state of the industrial development but incapable of affecting it directly. in the methodology, the criterion for evaluating the effectiveness of the industrial policy implementation is the average value of the indicators. comparison with the average value is a common approach and serves as a standard for determining the differentiation of regional development through ranking. the industrial policy effectiveness was analyzed for 19 out of 85 constituent entities of the russian federation, which are traditionally classified as industrially developed regions (the share of industrial production in grp is over 27%). among russia’s industrial regions are the oblasts of vladimir, kaluga, lipetsk, ryazan, tula, yaroslavl, vologda, leningrad, novgorod, kirov, nizhny novgorod, ulyanovsk, sverdlovsk, chelyabinsk and omsk, the republic of bashkortostan, the mari el republic, perm krai and krasnoyarsk krai. due to strong differentiation of regions, it is impossible to compile a single list of indicators suitable for all the constituent entities. therefore, selection of indicators is one of the methodology’s stages. the indicators are selected individually according to the goals and objectives of the socio-economic and industrial development of a particular territory. some 56 o.p. smirnova, a.o. ponomareva copyright ©2022 assa. adv. in systems science and appl. (2022) of the indicators characterize the situation in the entire region, which is due to the existing limitations on the availability of statistical data. a cross-regional comparison of the effectiveness of industrial policy is carried out through indicators norm setting and their aggregation [32]. as mentioned above, the criterion level is set by the indicators’ mean values. the integrated indicator of effectiveness was established using weighting coefficients. in the study, a formal method for calculating the weights based on the variation index was used. the methodology used is conditional, and the assessment of regional industrial policy measures requires a much deeper factor analysis. the immediate provision of statistical data is a significant obstacle. in the course of the analysis, no consequences of the covid-19 pandemic were considered [1, 8]. the decline in industrial production recorded during the pandemic (according to operational statistics for entire russia) can change the estimated position of the regions according to the methodology. despite the research limitations indicated, the proposed methodology allows one to comprehensively assess the development dynamics of industry, as well as conduct cross-regional comparative analysis. 4. results and discussion the first two stages of the methodology for assessing the regional industrial policy effectiveness were carried out earlier. based on the comparable coefficients calculated for the selected indicators (see appendix), one can identify the positive and negative aspects of the regional industrial complex development (stage 3 of the proposed methodology, fig. 1). these coefficients are categorized into three groups: the average level in russia (+/–5%), and those that are above and below it. for example, the weakest spots of vladimir oblast are labor productivity in the industry, low salary, and a significant amount of pollution emissions. among the region’s strongest points are the structural factors: a substantial share of manufacturing enterprises in the overall structure, and the share of the manufacturing industry in grp. the industrial complex generates a considerable number of jobs in vladimir oblast; this indicator is 1.7 times higher than the national average. the activity of the region’s industrial enterprises in the field of innovation is also above the average level throughout russia. at the same time, other regions such as kaluga oblast experience the opposite trend: the output volume of innovative products and services and the growth rate of advanced production technologies are below the national average. similarly, one can examine the effectiveness of the regional industrial policy for each of the 19 regions. to calculate the weighting coefficients for the selected indicators of the industrial policy effectiveness, we have applied a variation (stage 4 of the developed methodology). calculations were performed for each individual indicator for all 85 russian regions for 2019 (table 2). table 2. weighting coefficients of the significance of the industrial policy effectiveness indicators indicator variation (сv) weight (wi) [i15] labor productivity index in the manufacturing industry 3.697 0.283 [i16] return on assets of manufacturing enterprises 1.543 0.118 [i11] share of expenses incurred in technological innovations in the total volume of investment in fixed assets 1.136 0.087 [i12] volume of innovative goods, works, services as a percentage of the total volume of goods shipped, works and services performed 0.976 0.075 [i06] ratio between fixed assets introduced to industry and the total volume of fixed assets introduced 0.821 0.063 [i05] percentage of fixed assets of manufacturing enterprises in the overall structure of fixed assets in the region 0.654 0.050 [i03] share of the manufacturing industry in grp 0.630 0.048 [i10] share of organizations engaged in innovation 0.502 0.038 evaluating the effectiveness of the regional industrial policy implementation 57 copyright ©2022 assa. adv. in systems science and appl. (2022) indicator variation (сv) weight (wi) [i17] share of captured and neutralized air pollutants in the total amount of waste pollutants 0.496 0.038 [i01] average monthly nominal accrued wages of employees of organizations 0.422 0.032 [i02] share of manufacturing industry workers in the total number of employees in the region 0.403 0.031 [i04] share of manufacturing enterprises in the total number of enterprises in the region 0.283 0.022 [i08] growth rate of advanced production technologies created in the region 0.257 0.020 [i07] depreciation of fixed assets of manufacturing enterprises at the end of the year 0.196 0.015 [i21] ict use in organizations (cloud services, %) 0.188 0.014 [i09] growth rate of advanced production technologies used in the region 0.170 0.013 [i14] index for investment volume in fixed assets 0.166 0.013 [i13] industrial production indices in the manufacturing industry compared to the previous year 0.160 0.012 [i19] ict use in organizations (servers, %) 0.158 0.012 [i20] ict use in organizations (computer networks, %) 0.128 0.010 [i18] ict use in organizations (portable computers, %) 0.058 0.004 as shown in table 2, the three most significant industrial policy indicators are labor productivity (28% of the composite index), return on assets of manufacturing enterprises, i.e. financial performance of the industrial production (11%), and the share of expenses incurred in technological innovations in the total volume of investment in fixed assets (8%). once weighting coefficients (w) are calculated, all the obtained values of comparable coefficients (k) are aggregated into a composite indicator of the regional industrial policy effectiveness (stage 5 of the methodology). it is possible to juxtapose the dynamics of the obtained values and to compare it with the national average taken as the criterion level (table 3). assessing the relationships between the regions and their cooperation in various areas should be treated as a separate avenue for analysis. table 3. composite indices of the regional industrial policy effectiveness in 2015–2019 industrial region 2015 2016 2017 2018 2019 lipetsk oblast 1.53 1.51 1.79 1.73 1.50 tula oblast 1.47 1.45 1.44 1.62 1.49 vologda oblast 1.42 1.35 1.73 1.57 1.49 perm krai 1.43 1.29 1.30 1.36 1.41 nizhny novgorod oblast 1.45 1.42 1.58 1.55 1.39 kaluga oblast 1.28 1.32 1.54 1.52 1.39 leningrad oblast 1.34 1.33 1.37 1.22 1.37 omsk oblast 1.46 1.28 1.48 1.31 1.35 krasnoyarsk krai 1.12 1.12 1.10 0.99 1.31 sverdlovsk oblast 1.07 1.21 1.27 1.29 1.27 vladimir oblast 1.26 1.24 1.16 1.16 1.23 republic of bashkortostan 1.22 1.09 1.43 1.33 1.13 ryazan oblast 1.15 1.08 1.16 1.04 1.05 yaroslavl oblast 1.07 0.98 1.13 1.01 1.04 chelyabinsk oblast 1.07 1.14 1.25 1.21 1.01 novgorod oblast 1.13 1.16 1.10 1.06 0.99 ulyanovsk oblast 1.02 1.02 1.00 0.89 0.98 kirov oblast 0.93 0.88 0.94 0.93 0.92 mari el republic 1.06 0.96 1.24 0.77 0.63 in recent years, lipetsk, vologda and tula oblasts top the list of the regions with the most effective industrial policy. in lipetsk oblast, this is due to high profitability of the manufacturing industry, which in 2018–2019 amounted to 17.8–14.3%, respectively. this region is also characterized by intensive use of fixed assets, although their depreciation is 58 o.p. smirnova, a.o. ponomareva copyright ©2022 assa. adv. in systems science and appl. (2022) about 10% higher than the national average. sverdlovsk oblast closes the list of top 10 regions with the most effective industrial policy. the territory is lagging behind in the indicators such as average wage and the growth rate of advanced production technologies created in the region. despite a large number of research and educational institutions operating in sverdlovsk oblast, over the last 5 years it has witnessed a downward trend in advanced production technologies created (from 90 technological inventions in 2015 to 69 in 2019). in comparison, over the same period this indicator in chelyabinsk oblast has risen from 92 to 135 inventions per year. having analyzed the effectiveness of the regional industrial policy, we identified a number of development problems. these are low wage in the manufacturing industry; high depreciation of fixed assets of manufacturing enterprises; insufficient investment in technological innovation; falling labor productivity in the manufacturing industry; and a low level of environmental safety of industrial enterprises. these reference points can be used when adjusting the current measures for stimulating industrial development of the regions and imposing new ones. it is not allowed using a uniform list of indicators to assess the effectiveness of industrial policy at both the regional and federal levels. when organizing a monitoring system, it should be recognized that regions differ in opportunities for industrial development, and the use of uniform indicators is unacceptable. a cyclic scheme is needed that encompasses observation, analysis and identification of trends in the development of the industrial complex, as well as social, economic and environmental factors. the final step is to adjust the existing tools and mechanisms of industrial policy. obviously, a region’s institutional environment is of crucial importance for the effective implementation of industrial policy. in order to ensure consistency between the regional and federal industrial policy, each of the constituent territories should adopt a corresponding law. regional laws and regulations on industrial policy form the institutional basis by establishing purposes, objectives and development priorities. despite the diversity of these laws and different levels to which they are elaborated, the very fact of their existence affects (sometimes indirectly) the key indicators of the socio-economic and industrial development of the region [18, 37]. let us match the results of the industrial policy effectiveness analysis and the time span during which legislative support for the regional industrial policy has been in effect. table 4. regional laws and regulations on industrial policy region law no. and date of adoption last amended vladimir oblast no. 93-оz of july 10, 2015 december 5, 2019 kaluga oblast no. 33-oz of october 28, 1999 (repealed) no. 301-oz of february 21, 2018 october 31, 2018 lipetsk oblast no. 144-oz of june 4, 2001 (repealed) no. 508-oz of april 1, 2016 ryazan oblast no. 90-oz of august 10, 2006 (repealed) no. 61-oz of october 11, 2016 tula oblast no. 437-ztо of april 5, 2004 (repealed) no. 2402-zto of december 24, 2015 yaroslavl oblast no. 30-z of may 5, 2015 february 25, 2019 vologda oblast no. 3945-oz of april 28, 2016 march 13, 2020 leningrad oblast no. 93-oz of december 25, 2017 novgorod oblast no. 712-oz of march 30, 2010 (repealed) no. 922-oz of march 3, 2016 january 29, 2019 republic of bashkortostan no. 38-z of december 6, 1999 (repealed) no. 294-z of december 1, 2015 february 4, 2020 mari el republic no. 3-z of february 29, 2016 august 3, 2020 perm krai no. 440-pk of march 3, 2015 november 6, 2019 kirov oblast no. 125-zo of december 31, 2002 (repealed) no. 648-zo of may 5, 2016 july 27, 2020 evaluating the effectiveness of the regional industrial policy implementation 59 copyright ©2022 assa. adv. in systems science and appl. (2022) region law no. and date of adoption last amended nizhny novgorod oblast no. 190-z of december 21, 2011 (repealed) no. 206-z of december 25, 2015 december 24, 2019 ulyanovsk oblast no. 073-zо of november 9, 2004 (repealed) no. 218-zо of december 25, 2014 december 23, 2019 sverdlovsk oblast no. 136-oz of november 23, 2015 march 3, 2020 chelyabinsk oblast no. 197-zo of november 27, 2003 (repealed) no. 201-zo of june 18, 2015 august 5, 2020 krasnoyarsk krai no. 10-4346 of march 31, 2016 omsk oblast no. 1917-oz of december 8, 2016 december 3, 2019 source: compiled using data from [26]. many russian regions adopted local laws on industrial policy following the introduction of the federal law of 2014 “on the industrial policy in the russian federation”. it is worth noting that the fundamental law concentrates not on supporting russian industry, but stimulating its development. many laws on industrial development adopted at the regional level are of a framework nature and do not contain regional specifics. however, this is not always true. the present study showed that regions leading in industrial development were among the first to develop regional laws on industrial policy even before the federal law was introduced in 2014. these constituent entities are actively improving the adopted regional laws on industrial policy and making amendments that allow the use of more modern and relevant tools to support industry, such as investment contract. shifts in the global industrial market and new trends in technology require the industrial policy tools to be adjusted. investment contract is one of the major tools of industrial policy implementation in russia. being institutionalized in the regional laws, investment contract encourages industrial development. for instance, in the republic of bashkortostan, in accordance with the latest amendments (of february 4, 2020), regional authorities undertake to ensure the stable business environment for investors for the period of the contract and to apply incentive measures in industrial production. a similar amendment was made to kirov oblast’s law on industrial policy. the latest amendments dated july 27, 2020 guarantee the investor the invariability of the terms of the contract. we can conclude that the adjustment of regional laws on industrial policy contributes to the implementation of large-scale projects aimed at introducing or designing competitive innovative technologies. figure 2 displays that the top 3 regions with the most effective industrial policy (tula, lipetsk and kaluga oblasts) have had their local laws regarding industrial policy for more than 15 years. we can conclude that legal regulation concerning regional industrial policy accelerates the development of the industrial complex. 60 o.p. smirnova, a.o. ponomareva copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 2. correlation between the time span that the regional law on industrial policy has been in effect and the composite indicator of its effectiveness the obtained results demonstrate that a formal framework law on industrial policy introduced in a region cannot positively influence the effectiveness of industrial policy. however, in order to implement it, legislative support is needed that is part of the institutional framework. 5. conclusion when assessing industrial policy at regional level, it is crucial to allow for the peculiarities of the socio-economic development of a particular territory. the special features of the regional development can cause the expansion of the federal concept (key indicators), but cannot contradict it. we have worked out a multi-criteria methodology for evaluating the effectiveness of the regional industrial policy and identified the strengths and weaknesses in the regions’ industrial development. the performed analysis indicates that the current measures are effective and a growing number of industrial regions are improving their positions in comparison with the average level in russia. such factors as the level of labor productivity and the profitability of industrial enterprises exert the greatest influence on the effectiveness of the regional industrial policy. however, investment support for technological innovation is also of a significant importance. the given research shows that lipetsk, vologda and tula oblasts top the list of the regions with the most effective industrial policy in 2019. sverdlovsk oblast is lagging behind in such indicators as average wage and the growth rate of cutting-edge production technologies created in the region. despite a large number of research and educational institutions operating in the oblast, over the last 5 years it has witnessed a downward trend in advanced production technologies created. this fact proves the need for stimulating scientific research for industrial production at regional level and more active development of production modernization tools. in the course of the study, we have confirmed the hypothesis that the effectiveness of the local industrial policy is influenced by the region’s legal regulations. the earlier the institutional evaluating the effectiveness of the regional industrial policy implementation 61 copyright ©2022 assa. adv. in systems science and appl. 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(2012). assessment of far eastern federal district industry within the framework of state industrial policy, proceedings of irkutsk state technical university, 10, 270-275. appendix table. comparable coefficients (k) of industrial policy effectiveness indicators for the regions under study in 2019 in di ca to r v la di m ir ob la st k al ug a ob la st li pe tsk o bl as t ry az an o bl as t tu la o bl as t y ar os la vl o bl as t v ol og da o bl as t le ni ng ra d ob la st n ov go ro d ob la st re pu bl ic o f ba sh ko rto sta n m ar i e l r ep ub lic pe rm k ra i k iro v ob la st n iz hn y n ov go ro d ob la st u ly an ov sk o bl as t sv er dl ov sk o bl as t ch el ya bi ns k ob la st k ra sn oy ar sk k ra i o m sk o bl as t [i01] 0.69 0.87 0.72 0.72 0.80 0.75 0.82 0.97 0.67 0.76 0.63 0.82 0.63 0.74 0.64 0.86 0.78 1.04 0.74 [i02] 1.71 1.69 1.27 1.39 1.59 1.46 1.41 1.23 1.51 1.10 1.51 1.44 1.45 1.44 1.52 1.43 1.59 0.99 1.03 [i03] 1.91 2.36 2.48 1.62 2.51 1.61 1.52 1.74 1.84 1.95 1.77 1.69 1.64 1.77 1.49 1.81 2.00 1.77 2.02 [i04] 1.54 1.51 1.14 1.30 1.28 1.22 0.99 1.24 1.29 1.03 1.65 1.01 1.35 1.20 1.30 1.15 1.34 0.93 0.99 [i05] 1.71 2.91 3.03 1.42 2.80 1.85 2.14 2.58 1.68 1.52 0.95 1.68 1.63 2.14 2.08 2.15 2.49 1.71 2.14 [i06] 3.04 2.15 2.22 0.70 5.21 1.95 3.78 4.02 1.61 1.34 0.87 1.80 1.80 1.31 2.42 2.33 2.07 1.43 2.85 [i07] -1.01 -0.91 -1.05 -1.16 -0.72 -1.19 -0.99 -0.88 -0.95 -1.08 -1.02 -1.11 -0.80 -1.14 -0.94 -0.93 -1.09 -0.89 1.07 [i08] 0.97 0.34 1.35 1.14 1.93 0.83 1.59 1.00 1.69 1.14 1.35 1.09 0.97 1.69 0.12 0.78 0.90 0.80 1.61 [i09] 1.09 1.01 1.07 1.08 1.46 1.07 1.09 1.15 1.07 0.84 0.93 1.07 1.01 0.98 0.77 1.12 1.00 1.06 0.99 [i10] 1.15 1.26 1.22 1.30 1.29 1.16 1.27 0.89 1.08 1.13 1.24 0.96 1.60 1.51 1.60 1.27 1.15 0.76 0.82 [i11] 1.61 1.45 0.97 1.26 1.26 1.79 0.08 0.65 0.30 0.92 0.21 2.34 0.72 2.74 0.53 1.31 0.79 1.75 4.22 [i12] 1.23 0.34 1.32 1.83 1.55 1.13 0.53 0.38 0.21 1.23 2.00 2.26 1.85 2.58 2.08 1.19 1.02 0.89 0.25 [i13] 1.08 1.02 0.94 1.04 1.04 1.02 1.00 1.02 1.03 1.01 1.06 0.94 0.99 1.02 1.01 1.00 1.00 1.00 0.97 [i14] 1.11 1.08 1.10 1.01 1.07 1.00 1.16 0.75 0.67 1.12 0.91 1.09 1.10 1.02 0.81 1.08 1.08 0.94 1.28 [i15] 0.68 1.57 1.43 0.71 1.00 0.67 1.44 1.74 0.73 1.09 0.57 1.08 0.44 0.96 0.57 1.06 0.35 0.39 0.90 [i16] 1.59 1.19 2.07 1.22 0.83 0.74 3.03 0.71 1.64 1.48 -1.07 1.68 0.45 1.35 0.49 1.26 1.23 4.13 0.75 64 o.p. smirnova, a.o. ponomareva copyright ©2022 assa. adv. in systems science and appl. (2022) in di ca to r v la di m ir ob la st k al ug a ob la st li pe tsk o bl as t ry az an o bl as t tu la o bl as t y ar os la vl o bl as t v ol og da o bl as t le ni ng ra d ob la st n ov go ro d ob la st re pu bl ic o f ba sh ko rto sta n m ar i e l r ep ub lic pe rm k ra i k iro v ob la st n iz hn y n ov go ro d ob la st u ly an ov sk o bl as t sv er dl ov sk o bl as t ch el ya bi ns k ob la st k ra sn oy ar sk k ra i o m sk o bl as t [i17] 0.21 0.83 1.05 1.12 1.16 0.19 0.84 1.13 0.99 0.54 0.29 1.07 0.81 0.51 0.58 1.18 1.08 0.91 1.22 [i18] 1.03 1.02 1.00 1.01 0.98 1.03 1.04 1.02 1.01 1.01 1.03 1.00 1.03 1.02 0.98 1.04 1.02 1.00 0.97 [i19] 1.04 1.05 0.94 0.91 1.04 1.15 1.01 1.11 1.06 0.98 0.93 1.07 0.91 1.02 0.83 1.14 1.09 0.97 0.90 [i20] 1.08 0.99 0.96 0.97 1.06 1.16 1.01 1.07 1.05 1.00 1.03 1.05 1.08 1.01 0.93 1.09 1.08 1.03 0.91 [i21] 1.00 1.14 1.03 1.02 1.01 1.06 0.92 0.97 1.52 0.95 0.86 1.38 0.84 1.05 0.74 1.08 0.99 0.94 0.79 note: above the average value average value (+/–5%) below the average value microsoft word 1480-article text-7014-1-18-20240409 adv syst sci appl 2024; 02; 187-191 published online at https://ijassa.ipu.ru. lyapunov functions for periodic selector-linear difference inclusions mikhail morozov v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia abstract: the paper considers periodic selector-linear difference inclusions. a class of timeperiodic quasi-quadratic lyapunov functions is distinguished, as well as parametric classes of piecewise-quadratic and piecewise-linear lyapunov functions. these functions establish necessary and sufficient conditions for asymptotic stability. an example leading to periodic selector-linear differential and difference inclusions is given. the results can find applications in the stability analysis of control systems with periodic parameters, in particular, servomechanisms whose elements operate on alternating current, control systems with amplitude-frequency modulation. keywords: periodic selector-linear difference inclusion, asymptotic stability, lyapunov functions. 1. introduction the problem of the stability of periodic difference inclusions arises in the study of discrete control systems with periodic parameters and uncertainty, in particular, tracking systems that operate on alternating current and systems with amplitude-frequency modulation. in some cases, such as the absolute stability problem, the study of linear nonstationary systems whose right part matrix satisfies interval constraints can be used selector-linear difference inclusions. in [5] periodic difference inclusions are considered. the definitions of asymptotic, uniform asymptotic and uniform exponential stability are given and the equivalence of these properties for selector-linear difference inclusions is proved. on the basis of the variational approach, a necessary and sufficient condition for uniform asymptotic stability in the form of some limit relation is obtained. in this paper we obtain asymptotic stability criteria for periodic selectorlinear difference inclusions based on the lyapunov function method. the remainder of this paper is structured as follows. in section 2, we consider periodic selector-linear difference inclusions and set the problem of obtaining asymptotic stability criteria for such inclusions on the basis of the lyapunov function method. in section 3 we distinguish a class of periodic in time lyapunov functions of quasi-quadratic form and parametric classes of piecewise quadratic and piecewise linear lyapunov functions. theorems 1-3 are formulated and proven. they establish necessary and sufficient conditions for asymptotic stability of the considered inclusions. in section 4, an example of a system leading to periodic differential and difference inclusions is given. in section 5, we offer concluding remarks. 2. statement of the problem consider the dynamic systems described by periodic selector-linear difference inclusion 𝑥(𝑠 + 1) ∈ 𝐹(𝑠, 𝑥), 𝑠 = 0, 1, … , 𝑥 ∈ 𝑅 , (2.1) where the set-valued map 𝐹: 𝑅 → 𝑅 has the form 𝐹(𝑠, 𝑥) = {𝑦: 𝑦 = 𝐵(𝑠)𝑥, 𝐵(𝑠) ∈ ω(𝑠)}, 188 m. morozov copyright ©2024 assa adv. in systems science and appl. (2024) here 𝛺(𝑠), 𝛺(𝑠 + 𝑁) = 𝛺(𝑠) (𝑠 = 0,1, . .. , 𝑁is a natural number) is a convex, compact set of real (𝑛 × 𝑛) – matrices 𝐵. such set-valued maps are called selector-linear, since the righthand side is a union of linear maps. the sequence of vectors {𝑥(𝑠)}, satisfying for all 𝑠 = 0,1, … inclusion (2.1), is the solution of the inclusion (2.1). let 𝑥(𝑠, 𝑠 , 𝑥 ) be the solution of inclusion (2.1) with initial conditions (𝑠 , 𝑥 ). due to the periodicity of the multivalued function 𝐹(𝑠, 𝑥) in 𝑠 without generality restriction we can assume that 0 ≤ 𝑠 ≤ 𝑁. the equivalence of the properties of asymptotic stability, uniform asymptotic stability, and uniform exponential stability for inclusion (2.1) was proved in [5]. hereinafter we will refer to the asymptotic stability of inclusion (2.1). the problem is to identify the parametric classes of lyapunov functions establishing necessary and sufficient conditions for asymptotic stability of inclusion (2.1) and to construct the stability criteria for inclusion (2.1) using a discrete analogue of the direct lyapunov method. 3. results theorem 3.1: the following conditions are equivalent: 1. inclusion (2.1) is asymptotically stable. 2. there exists the lyapunov function 𝑣(𝑠, 𝑥) of the quasiquadratic form 𝑣(𝑠, 𝑥) = 𝑥 ′𝐿(𝑠, 𝑥)𝑥, 𝐿(𝑠, 𝑥) = (𝑙 (𝑠, 𝑥)) , , 𝐿(𝑠 + 𝑁, 𝑥) = 𝐿(𝑠, 𝑥), 𝐿′(𝑠, 𝑥) = 𝐿(𝑠, 𝑥) = 𝐿(𝑠, 𝜇𝑥), 𝑥 ≠ 0, 𝜇 ≠ 0, 𝑣(𝑠, 0) ≡ 0 (3.1) which is 𝑁-periodic in 𝑠, homogeneous (of second order), strictly convex in 𝑥, and it satisfies the following inequality: 𝑚𝑎𝑥 ∈ ( , ) 𝑣(𝑠, 𝑦) ≤ 𝜃𝑣(𝑠, 𝑥), 𝑥 ∈ 𝑅 , 𝑠 = 0,1, … (3.2) for some 𝜃 (0 < 𝜃 < 1). in (3.1) the prime means the transposition operation. the sufficiency of the conditions of theorem 3.1 is established (using the lemma given in [4]) by the reasoning used in the proof of exponential stability of discrete systems in [2]. the proof of the necessity follows the same scheme as the proof of the corresponding conditions of the theorem in [4], where exponential estimates for solutions of inclusion (2.1) are used. theorem 3.1 is an extension of the classical theorem for the discrete analogue of direct lyapunov method in [2] to selector-linear periodic difference inclusions (2.1). the reasoning used in the proof of the lemma in [4] proves that, under theorem 1, there exist constants such 𝜆 ≥ 𝜆 > 0 that for all 𝑠 ≥ 0 and 𝑥 ∈ 𝑅 , the inequalities 𝜆 ‖𝑥‖ ≤ 𝑣(𝑠, 𝑥) ≤ 𝜆 .‖𝑥‖ . (3.3) (3.3) implies positive definiteness of the function 𝑣(𝑠, 𝑥). the problem of constructing the lyapunov function is simplified if this function is selected from a certain parametric class of functions that depend on a finite number of parameters. the class of quasi-quadratic lyapunov functions (3.1) is not parametric. the existence of a parametric class of lyapunov functions, also defining necessary and sufficient conditions for the asymptotic stability of inclusion (3.1), can be formulated as a theorem. theorem 3.2: inclusion (2.1) is asymptotically stable iff, for some integer 𝑀 ≥ 𝑛, there exists a periodic in 𝑠 (of period 𝑁), piecewise-quadratic lyapunov function luapunov functions for periodic selector-linear difference inclusions 189 copyright ©2024 assa. adv. in systems science and appl. (2024) 𝑣 (𝑠, 𝑥) = 𝑚𝑎𝑥 𝑙 (𝑠), 𝑥⟩ , (𝑙 (𝑠 + 𝑁) = 𝑙 (𝑠), (3.4) for which the inequality (3.2) is satisfied for all 𝑠 ≥ 0 and 𝑥 ∈ 𝑅 , and the 𝑛-dimensional periodic vectors (𝑙 (𝑠) (𝑙 (𝑠 + 𝑁) = 𝑙 (𝑠)), 𝑗 = 1, 𝑀 satisfy the condition 𝑟𝑎𝑛𝑘𝐿(𝑠) = 𝑛 ≤ 𝑀, 𝐿(𝑠) = 𝑙 (𝑠), … , 𝑙 (𝑠) , 𝑠 = 0,1, … (3.5) (i.e., periodic (𝑛 × 𝑀) matrix 𝐿(𝑠)(𝐿(𝑠 + 𝑁) = 𝐿(𝑠)) has a maximum rank for all 𝑠 ≥ 0). we denote by ⟨⋅, ⋅⟩ a scalar product of vectors. the set of lyapunov functions (3.4) forms a parametric class. the parameters defining this class are the components of periodic vectors 𝑙 (𝑠) (𝑗 = 1, 𝑀, 𝑠 ≥ 0 ) and the integer 𝑀 ≥ 𝑛. if the rank condition (3.5) is satisfied, the function 𝑣 (𝑠, 𝑥) is positively defined in 𝑅 and its level surfaces at any fixed one 𝑠 ≥ 0 are centrally symmetric convex polyhedrons. the vectors 𝑙 (𝑠), 𝑗 = 1, 𝑀 defining the norms to their faces. proof. sufficiency. the sufficiency of the conditions of theorem 3.2 is established according to the standard scheme of the proof of exponential stability in [2] using inequality (3.2) and estimates 𝜆 ‖𝑥‖ ≤ 𝑣 (𝑠, 𝑥) ≤ 𝜆 ‖𝑥‖ , 𝜆 ≥ 𝜆 > 0 for function (3.4) under condition (3.5). necessity. it follows from theorem 3.1 that for inclusion (2.1) there exists a periodic in 𝑠 lyapunov function 𝑣(𝑠, 𝑥) of the quasiquadratic form satisfying the conditions of theorem 3.1. consider centrally symmetric convex bodies 𝑃 (𝑠) = {𝑥: 𝑣(𝑠, 𝑥) ≤ 1}, 𝑃 (𝑠) = {𝑥: 𝑣(𝑠, 𝑥) ≤ 𝑟}, 𝑠 = 0,1, . . . ,0 < 𝑟 < 1. let 𝑖𝑛𝑡 𝐴 be the set of interior points of the set 𝐴. since 0 < 𝑟 < 1, then 𝑃 (𝑠) ⊂ 𝑖𝑛𝑡 𝑃 (𝑠). it follows from theorem 20.4 in [6] that there exists a centrally symmetric convex polyhedron 𝐷 (𝑠), that the relations 𝑃 (𝑠) ⊂ 𝑖𝑛𝑡 𝐷 (𝑠) ⊂ 𝐷 (𝑠) ⊂ 𝑖𝑛𝑡 𝑃 (𝑠) ⊂ 𝑃 (𝑠). (3.6) let 2𝑀 be the number of faces of the polyhedron 𝐷 (𝑠), then there are 𝑀 pairs of centrally symmetric faces. let 𝑟 > 0 be the distance from a point 𝑥 = 0 to the faces of the𝑗 pair, and let ±𝑛 ( 𝑛 = 1), 𝑗 = 1, 𝑀 be the unit external normals to the faces of this pair. in order the surface of the polyhedron 𝐷 (𝑠) to be the surface of the level 𝛺 (𝑠) = {𝑥: 𝑣 (𝑙(𝑠), 𝑥) = 1} for function 𝑣 (𝑙(𝑠), 𝑥) (3.4), choose 𝑙 (𝑠) = 𝑟 𝑛 , 𝑗 = 1, 𝑀, 𝑠 = 0,1, . .. then the vectors 𝑙 (𝑠), 𝑗 = 1, 𝑀, 𝑠 = 0,1, . .. satisfy rank condition (3.5), because otherwise the surfaces of the level function 𝑣 (𝑙(𝑠), 𝑥) will not be bounded. let 𝛿(𝐾, 𝑧) = 𝑚𝑎𝑥 ∈ (𝑧, 𝑥), 𝑧 ∈ 𝑅 be the reference function of compact 𝐾 ⊂ 𝑅 (see [6]). since 𝛿(𝑃 (𝑠), 𝑧) = √𝑟𝛿(𝑃 (𝑠), 𝑧), it follows from (3.6) and property 9 in [1] √𝑟𝛿(𝑃 (𝑠), 𝑧) < 𝛿(𝐷 (𝑠),z) < 𝛿(𝑃 (𝑠),z), z ≠ 0. (3.7) let us show that there exists a number 𝑟 (𝑟 < 𝑟 < 1), such that the inequality √𝑟𝛿(𝑃 (𝑠), 𝑧) < 𝑟 𝛿(𝐷 (𝑠),z), z ≠ 0. (3.8) suppose that inequality (3.8) is not satisfied, i.e., for any 𝑟 (𝑟 < 𝑟 < 1) √𝑟𝛿(𝑃 (𝑠), 𝑧) ≥ 𝑟 𝛿(𝐷 (𝑠),z) or 𝑟 𝑟 𝛿(𝑃 (𝑠), 𝑧) ≥ 𝛿(𝐷 (𝑠),z), z ≠ 0, that contradicts inequality (3.7). function √𝑟 𝛿(𝐷 (𝑠),z) = 𝛿(𝐷 ,z) is the reference function of the polyhedron 𝐷 (𝑠) = {𝑥: 𝑣 (𝑙(𝑠), 𝑥) ≤ 𝑟 }, which is similar to polyhedron 𝐷 (𝑠) = {𝑥: 𝑣 (𝑙(𝑠), 𝑥) ≤ 1}. therefore (3.8) is equivalent to the relation 𝑃 (𝑠) ⊂ 𝑖𝑛𝑡 𝐷 (𝑠) ⊂ 𝐷 (𝑠). (3.9) 190 m. morozov copyright ©2024 assa adv. in systems science and appl. (2024) it follows from inequality (3.2) for the function 𝑣(𝑠, 𝑥) that 𝐹(𝑃 (𝑠)) ⊂ 𝑃 (𝑠), where 𝐹(𝑃 (𝑠)) = ∪ ∈ ( ) 𝐹(𝑠, 𝑥). (3.6) implies that 𝐷 (𝑠) ⊂ 𝑃 (𝑠). therefore 𝐹 𝐷 (𝑠) ⊂ 𝐹 𝑃 (𝑠) ⊂ 𝑃 (𝑠). (3.10) it follows from (3.9) and (3.10) 𝐹(𝐷 (𝑠)) ⊂ 𝑖𝑛𝑡 𝐷 (𝑠) and therefore the condition 𝑚𝑎𝑥 ∈ ( , ) 𝑣 (𝑙(𝑠), 𝑦) < 𝑟 , 𝑥 ∈ 𝐷 (𝑠), 𝑠 = 0,1, … (3.11) since 𝑣 (𝑙(𝑠), 𝜇𝑥) = 𝜇 𝑣 (𝑙(𝑠), 𝑥) and 𝐹(𝑠, 𝜇𝑥) = 𝜇𝐹(𝑠, 𝑥) for all 𝑠 ≥ 0, 𝑥 ∈ 𝑅 and 𝜇 ∈ 𝑅 , then, for the above-constructed function 𝑣 (𝑙(𝑠), 𝑥) from (3.11), the inequality 𝑚𝑎𝑥 ∈ ( , ) 𝑣 (𝑙(𝑠), 𝑦) ≤ 𝑟 𝑣 (𝑙(𝑠), 𝑥), 0 < 𝑟 < 1, 𝑥 ∈ 𝑅 , 𝑠 = 0,1, … (3.12) of the form (3.2). this completes the proof of theorem 3.2.  consider piecewise linear lyapunov functions 𝑉 (𝑠, 𝑥) = 𝑚𝑎𝑥 𝑙 (𝑠), 𝑥⟩ . (3.13) a corollary of theorem 3.2 is theorem 3.3: inclusion (2.1) is asymptotically stable iff, for some integer 𝑀 ≥ 𝑛 there exists a periodic on 𝑠 (period 𝑁 ), piecewise-linear lyapunov function 𝑉 (𝑠, 𝑥) (3.13), satisfying condition (3.5) and inequality (3.2) for all 𝑥 ∈ 𝑅 , 𝑠 ≥ 0. proof. since 𝑣 (𝑠, 𝑥) = 𝑚𝑎𝑥 ⟨𝑙 (𝑠), 𝑥⟩ = 𝑚𝑎𝑥 𝑙 (𝑠), 𝑥⟩ = 𝑉 (𝑠, 𝑥), then each function 𝑣 (𝑠, 𝑥) can be matched with function 𝑉 (𝑠, 𝑥) = 𝑣 (s,x). in the same way each function 𝑉 (𝑠, 𝑥) (3.13) can be associated with function 𝑣 (𝑠, 𝑥) = 𝑉 (𝑠, 𝑥). from this and theorem 3.2 follows the validity of theorem 3.3.  4. example consider the torsional vibrations of the crankshafts of a single-cylinder engine with a flywheel with allowance for the inertia of connecting rods and pistons. let us assume that the mass of the flywheel is sufficiently large, so that the rotation of the shaft can be considered uniform. let 𝜔 angular velocity of the flywheel, 𝑐 torsional stiffness coefficient of the shaft. in [3] it is shown that vibrations are described by second order differential equation 𝑑 𝑞 𝑑𝑡 + 𝑐 𝜔 𝑝(𝑡)𝑞 = 0, (4.14) where 𝑞 – generalized coordinate associated with the angle of rotation of the crank, and periodic function 𝑝(𝑡) of the period 2𝜋can be determined by the kinetic energy of the crank together with the associated moving masses (connecting rod and piston). equation (4.1) is a special case of an equation of more general form 𝑑 𝑧 𝑑𝑡 + 𝑏𝑓(𝑡)𝑧 = 0, (4.2) luapunov functions for periodic selector-linear difference inclusions 191 copyright ©2024 assa. adv. in systems science and appl. (2024) where 𝑓(𝑡) is a periodic function of time (with period 𝑇 > 0), and 𝑏 ∈ 𝐼, 𝐼 = [𝑏 , 𝑏 ] is a certain parameter. let us introduce notation 𝑥 = 𝑧, 𝑥 = = , 𝑥 = 𝑥 𝑥 , 𝐴(𝑡, 𝑎, 𝑏) = 0 1 −𝑏𝑓(𝑡) 0 . equation (4.2) will take the form of second order system 𝑑𝑥 𝑑𝑡 = 𝐴(𝑡, 𝑏)𝑥, 𝐴(𝑡 + 𝑇) ≡ 𝐴(𝑡), 𝑏 ∈ 𝐼, 𝑡 ≥ 0, 𝑇 > 0, 𝑥 ∈ 𝑅 . (4.3) consider discrete analogue of system (4.3) 𝑥(𝑠 + 1) = 𝐴(𝑠, 𝑏)𝑥(𝑠), 𝐴(𝑠 + 𝑀) ≡ 𝐴(𝑠), 𝑏 ∈ 𝐼, 𝑀 ∈ 𝑁, 𝑥 ∈ 𝑅 , (4.4) where 𝑠 = 0,1, . .. is discrete time. system (4.4) is equivalent to periodic selector-linear difference inclusion (2.1), where the multivalued function 𝐹(𝑠, 𝑥) (𝐹(𝑠 + 𝑀, 𝑥) ≡ 𝐹(𝑠, 𝑥)) is defined at each point (𝑠, 𝑥), 𝑥 ∈ 𝑅 by the relation 𝐹(𝑠, 𝑥) = {𝑦: 𝑦 = 𝐴(𝑠, 𝑏)𝑥, 𝑏 ∈ 𝐼}. 5. conclusion for periodic selector-linear difference inclusion (2.1) asymptotic stability criteria were obtained. they use quasi-quadratic, piecewise quadratic and piecewise linear lyapunov functions. the example of technical problem leading to consideration of periodic differential and difference inclusions is given. the obtained results can be used in the study of the stability of control systems with periodic parameters. in particular, such systems are the tracking systems, which elements operate on alternating current and the control systems with pulse-amplitude modulation. the extracted in theorems 3.2, 3.3 piecewise-quadratic and piecewise-linear lyapunov functions of the form (3.4), (3.13) establish necessary and sufficient conditions of asymptotic stability for inclusion (2.1). these functions can be used when developing numerical methods of stability analysis for systems equivalent to difference inclusion (2.1). references 1. blagodatskikh, v. i. & filippov, a. f. (1986). differential inclusions and optimal control, proc. steklov inst. math., 169, 199–259. 2. halanay, a. & wexler, d. (1968). theoria calitativa a sistemelor cu impulsuri [qualitative theory of pulse systems]. bucharest, romania: editura academiei republicii socialiste romania, [in romanian]. 3. malkin, i. g. (1952). theory of stability of motion. washington, dc: us atomic energy comission. 4. morozov, m. v. (2021). a criterion for the asymptotic stability of a periodic selectorlinear differential inclusion, automation & remote control, 82(1), 63–72, https://doi.org/10.1134/s0005117921010045 5. morozov, m. v. (2022). on the stability of periodic difference inclusions, advances in systems science and applications, 22(1), 167–175. 6. rockafellar, r. t. (1970). convex analysis. princeton, nj: princeton university press. adv syst sci appl 2023; 01:129–151 published online at https://ijassa.ipu.ru. verification of bounded rationality models of price dispersion on an online marketplace data nikolay bazenkov1*, elena glamozdina2, maria kuznetsova2, marina sandomirskaia2 1v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russian federation 2hse university, moscow, russian federation abstract: we explore how well the models of bounded rationality explain price dispersion observed in internet price-comparison sites. this paper considers three hypotheses about the strategic origin of price dispersion in homogeneous product online sales. two of them are εequilibrium and quantal-response equilibrium (qre) in a pure bertrand setting, they involve the bounded rationality of sellers. the third hypothesis introduces the fraction of loyal consumers into the model and assumes the sellers are rational. these behavioral models earlier were proposed in the literature as possible explanations of high price dispersion in online markets. they were supported by experimental lab data, but not tested on real prices. we test the hypotheses on real data for 30 models of household appliances collected from the largest russian online marketplace market.yandex.ru, which organizes e-competition in the closest to the bertrand setting way. we found no support for ε-equilibrium hypothesis and only limited support for loyal consumer hypothesis. qre showed the best performance on the data. for most of the products it predicts central tendency, i.e. the mean and the median, remarkably well. the shape of the observed price distributions is explained accurately enough in comparison with the two other hypotheses and random behavior. keywords: price dispersion, internet markets, marketplace, quantal-response equilibrium, εequilibrium, loyal consumers. structural estimations 1. introduction large price dispersion in internet sales of homogeneous goods is a steady but still unexplained phenomenon. while theoretical analysts originally expected perfect competition in these markets, data-based studies report significant price variations which persist over time [2, 16, 22]. we focus here on the dispersion in posted price, which is higher than that in transacted prices, i.e. the prices at which the real transactions occur. dispersion in transacted price is a related but a distinct problem which requires access to transactions data which are usually unavailable. so, it is hard to estimate its degree [15, 34]. below we will discuss several widely used approaches which explain the origin of price dispersion from different points of view. the common for most studies is that sophisticated models generally exist separately from data analysis, which checks tendencies and significant factors without structural estimation of the model. to fill this gap, we take three theoretical models included bounded rationality of market agents and fit them to real price data from a price-comparison site. we show that, in spite of the plausible assumptions behind all three models and their comparable success in prediction of experimental pricing behavior, one of them has much more explanation power than two others. ∗corresponding author: n.bazenkov@yandex.ru 130 n. bazenkov, e. glamozdina, m. kuznetsova, m. sandomirskaia the classical market theory argues that heterogeneous consumer characteristics, in particular, different search costs or information availability, makes a great influence on price setting and price dispersion. this line of research is inspired by the theory of consumer’s search behavior [32] and variation in strategies caused by varying costs and access to the complete list of offers [29, 31, 33]. the accurate estimation of the search factor requires consumer-level data, which are generally not available to researchers. using only price distribution data [20] admits a lot of assumptions on the structure of search costs, such that the analysis of search costs structure for similar products may be inconsistent. moreover, pricecomparison sites are organized in order to reduce consumers’ search costs by allowing buyers to order the offers from the cheapest to the most expensive, also with the restrictions on the rating of sellers or on the particular subset of e-shops. they intensify competitive pricing pressure between firms [30]. under this pressure, firms apply mixed strategies in order to prevent effective learning for consumers [1, 23]. the strategic competitive behavior of e-shops is an equally important source of observed price diversity, so the research of sellers price-setting policies is essential. the common belief [4, 5, 28] is that the large price dispersion must be an equilibrium phenomenon, but the mechanism is unclear. different theoretical models may produce different equilibrium mixed strategies, so the problem of their selection on data arises. some papers analyze markets with the mechanism of simultaneous competing auctions [18, 19]. together with the consideration of switching problem, as it is costly for consumer to move from one auction at a particular seller to another, authors highlight the influence of additional guarantees on online price dispersion. however, complicated mechanisms poorly fit markets where firms are allowed only to set the price for their good. in this paper, the models rely on static bertrand framework which appears a reasonable approximation of a price-comparison platform where firms announce the price for an item. this is caused by the high price-sensitivity of online demand [11], and the fact that most consumers consider the low price as the main factor of purchase, according to numerous consumers’ surveys [12, 36, 37]. the static setting is a more questionable modelling assumption, nevertheless, it is widely used in literature, see the survey [4], and the papers comparing multi-channel sales [7, 13, 35], and online advertising [3]. technically, building a fully dynamic model is complicated by the randomly varying number of active firms at each moment. pure bertrand competition itself can’t generate price dispersion, but adding bounded rationality on the firm or on the consumer sides [14] can generate dispersed prices in equilibrium. how these equilibrium distributions correspond to real-world prices is an open question which is addressed in our research. we investigate three hypotheses about the behavior of online sellers. the models underlying these hypotheses were theoretically developed in [2], and [27]. here they are tested on data from the price-comparison site market.yandex.ru. the dataset includes all prices for 30 popular models of household appliances available from july to october 2015 [6]. the prices at every moment show a large dispersion according to the three measures: the coefficient of variation, the relative price range, and the gap between the two minimal prices. we investigate whether the features of the price distributions observed in the data may be explained by the considered theoretical models. two hypotheses based on assumptions of bounded rationality of firms are ε-equilibrium and quantal-response equilibrium (qre) [2]. as indicated in the original paper [2], similar patterns of price dispersion arise both in internet data and lab experiments, where “unobserved heterogeneities” are excluded by design. however, testing with experimental data does not allow them to conclude that the market behavior is also explained by these bounded rationality models. both models better work for duopolies, than for triopoly and quadropoly with a large percent of statistical tests rejection, but still better than random or completely rational behavior. thus, the experiments do not allow to select the best fitted copyright © 2023 assa. adv syst sci appl (2023) verification of bounded rationality models of price dispersion 131 model. preliminary suggestions on their applicability to internet markets also looks like the task for this study. the alternative model concerns the limited rationality of consumers and introduces the fraction of loyal consumers into the classical bertrand setting [27]. these consumers pick a random seller, not the one who posted the minimal price. in this case, high price dispersion is observed under standard nash behavior. though strong experimental support was obtained to the main hypothesis, the question of how this prediction approximates real market behavior remained open and some systematic discrepancies were detected. this model lies close to [10] where some consumers stick to a randomly chosen seller while others search sequentially with an optimal reservation price. while we tested only the model of [27], the intuition is that similar results would be observed for [10]. though in [27] experiments were designed such that participants knew the share of loyal buyers, in the real data analysis we treat this share as a hidden parameter that is to be estimated. so our estimation strategy is similar with the two hypotheses above, and it is natural to consider all three models in comparison with each other. in this paper the three hypotheses are reformulated in a form suitable for fitting to the dataset. applying a similar strategy to [2] we estimated the hidden parameters for each model. then the goodness of fit is evaluated by three statistical tests. we examined whether each model could produce the mean, the median and the empirical probability distribution for the observed data. this study demonstrates that the qre hypothesis survived the tests. its predictions about the mean and the median are valid, and the prediction about the shape of price distribution is accurate for 50% of the observed time moments. that is much better than the uniform distribution and the two others. the model with loyal consumers performed significantly worse than qre, but better than pure randomness. there are a non-negligible fraction of days when the observed pricing pattern is similar to those predicted by the model. the εequilibrium failed dramatically to explain the observed data despite the reasonable underlying assumption. to the best of our knowledge this is the first attempt to fit the bounded rationality models to online pricing data. the paper reveals the structure of daily price distributions in homogeneous product online markets. because of the observed diversity in daily price distributions, the unique theoretical concept doesn’t allow to cover all of them. the most frequent situation with the single peak close to the minimal price can be explained by qre, which is associated with a natural learning process under the changing market conditions. the less frequently observed two-peaked distribution is partly explained by the model with heterogeneous consumers, but this line needs further development in order to select among several close models with potentially non-equal division of loyals [3]. the paper is organized as follows. section 2 provides a brief description of the dataset from market.yandex.ru. section 3 reproduces the theoretical concepts for further estimations and presents the analytical formulae for price distributions. section 4 describes the techniques applied, while section 5 contains the results of the structural estimations of the parameters for each hypothesis. in the last section the quality of theoretical models and their limitations are discussed. 2. online prices: empirical observations 2.1. data for the dataset, prices from the online marketplace market.yandex.ru were collected during several months from july, 24 to october, 20, 2015 [6]. market.yandex.ru is the most popular copyright © 2023 assa. adv syst sci appl (2023) 132 n. bazenkov, e. glamozdina, m. kuznetsova, m. sandomirskaia platform for online shopping in russia. more than 40% of population and more than 50% of internet users in moscow make online purchases†. the site is organized in the intuitive way (see figure 2.1) and free of charge for potential buyers. it is useful both for choosing a specific product by learning their detailed characteristics and for selecting a particular seller of this good. the option of ordering the prices from lowest to highest is available in one click. in addition, it is immediately easy to account for the rating of the e-shop and some delivery conditions. we believe that the rules of a price aggregator makes the structure of the market close to the perfect competition. fig. 2.1. view of market.yandex.ru description of the availablle sellers ordered by price for a selected good we fixed a list of 30 the most popular goods in 5 subcategories of household appliances, i.e. fridges, cooker hoods, warm ovens, dishwashers, cooktops, and washing-machines. the whole set of actual prices was downloaded 4 times per day for e-shops in the moscow region. household appliances are in the top-5 categories with the largest share of online sales. for this category the fraction of consumers who prefers to purchase on market.yandex.ru exceeds 30%. not every e-shop specializing in household appliances sells every product from our sample, and moreover, even if it sells the specific product, the e-shop may not sell it at every moment. this gives unbalanced panel data with 30 goods, 258 sellers and 324 moments containing 463502 unique price offers. this dataset is representative in several aspects. first, one may be sure that all the offers presented on yandex.market.ru are available for the announced price because the platform †here and further the survey characteristics of industry are based on [12], [36]. in 2021 these shares have obviously become greater. copyright © 2023 assa. adv syst sci appl (2023) verification of bounded rationality models of price dispersion 133 checks it regularly and sanctions the violators. second, the metropolitan online market is highly localized: 90% of respondents prefer to buy locally, in moscow, [36]. third, geolocation inside the city is not important since for the purchase of major household appliances most consumers use delivery services. hence this data though covering a limited subset of products along a relatively short time period, may serve a representative example of how competition is organized in modern online markets. 2.2. price dispersion the prices in the dataset demonstrate large dispersion both in the coefficient of variation and in the relative price range at each moment. the aggregate description of our data is presented in table 2.1. the following measures of price dispersion are used. the coefficient of variation is the standard deviation divided by the average. the relative price range is the difference between the maximal and minimal prices divided by the minimal price. the gap is the difference between the two minimal prices divided by the minimal price. according to these measures, the level of price dispersion we observe for russian data in 2015 is similar to those reported for usa data in 2001 [2]. direct comparison to [16] is difficult since they don’t report any measures of price dispersion. in [22] the coefficient of variation (relative standard deviation) is 10% for offline retail data from knrs dataset. this is very close to our average cv which equals 10.6%. we may conclude that general characteristics of our data are similar to those observed in other countries and other samples of goods. table 2.1. average characteristics of products and price dispersion for the whole period product coeff relative price average lowest # of price gap price price var range % % % rub rub cata ceres 600 negra 22.6 143.6 4.0 19443 12431 36 electrolux ehh 6340 fok 15.1 73 2.7 24524 19949 50 gorenje bo 73 cli 13.7 70.1 1.4 27435 22947 47 gorenje gw 65 cl i 13.5 69.3 2.1 17900 14836 28 electrolux ezb 53400 ax 13.5 51.2 5.2 19944 15750 18 hansa osc 511 wh 13.4 79.4 8.4 2813 2124 47 hotpoint-ariston 7hpc 640 13 80.5 2.2 11876 9833 67 krona kamilla 600 12.7 76.7 13.2 7546 5348 30 atlant xm 4008-022 12.5 101.1 0.7 14281 12698 59 hansa zim 436 eh 12.2 52.7 1.3 20252 17542 31 hansa boei62000015 11.7 85.7 0.6 16744 14442 68 electrolux ews 1052 ndu 11.6 57.5 2.5 19539 16906 31 hansa bhi68300 11.6 55.1 1.4 16494 14035 45 bosch dhl 545 s 11.3 65.7 3.3 9220 7706 36 elikor integra 60n-400 11.1 64.6 2.1 3778 3125 46 bosch pic645f17e 10.8 58.4 0.3 28000 23686 78 indesit bia 18 10.3 60.4 1.4 21264 17741 71 bosch hbg43t450 10 60.3 2.7 30429 25717 51 samsung wf8590nmw9 9.1 78.2 0.7 19111 17217 75 bosch activewater spv40e10ru 9.1 47.6 0.8 24489 21671 56 siemens sr 66t090 8.6 58.3 3.2 43908 37238 68 hotpoint-ariston ftr 850 8.5 52.2 1.4 21085 18539 65 bosch wlg 20061 oe 8.1 54.2 3.0 20417 17506 66 ariston lstb 4b00 ru 7.9 43.2 2.2 17519 15710 34 bosch kgs 39xw20 r 7.8 44.8 1.4 34584 30170 52 indesit wiun 81 (csi) f053525 7.6 26.7 4.4 13340 12102 9 candy cdcf 6-07 7 36.4 2.6 13619 12284 25 lg f-1096sd3 6.2 39.6 0.5 21768 19605 67 lg gab409svqa 5 28.8 1.6 32571 29610 44 liebherr sbsesf 7212 4.3 25.1 4.8 124142 111186 29 copyright © 2023 assa. adv syst sci appl (2023) 134 n. bazenkov, e. glamozdina, m. kuznetsova, m. sandomirskaia 2.3. sellers characteristics for each seller we calculated its average normalized price deviation, i.e. how much it deviates from the average price across all goods and all time moments. figure 2.2 shows the distribution of average normalized price deviations of the sellers. the distribution is unimodal and slightly skewed to the right. a common assumption is that the observed price dispersion may be to some extent explained by sellers heterogeneity in the quality of service. below we test this assumption for the data. fig. 2.2. distribution of sellers normalized price deviations yandex.market platform contains sellers ratings which are numbers from 0 to 5 based on the buyers responses to their experience with the given seller. so, the rating is a long-term reputation which aggregates all information such as delivery conditions, customer support, site reliability etc. the intuition that buyers prefer sellers with good reputation and therefore higher rated sellers will post higher prices doesn’t hold for our data. we found no correlation between rating and the average price deviation of the sellers. figure 2.3a show the relationship between the rating and the average price deviation. we also used the number of goods a seller offers as a proxy for its size and market share. the intuition is that large shops with diverse assortment are more recognizable, attract more loyal customers and thus may post higher prices. here we actually observed the opposite tendency. larger sellers less deviate from the average price than the “small” ones. the correlation between the number of goods available at a seller and its price deviation is −0.24 with p = 8.2× 10−5. figure 2.3b shows the relationship between the number of goods available at the seller and the average price deviation. another factor which is believed to influence price dispersion is the number of active sellers present in the market. our data contains dynamic prices and the number of sellers varies greatly across the period. figure 2.4 shows the distribution of nt, the number of active sellers observed for the moment t. in the sample at least two firms are present on the market at every moment. table 2.2 illustrates how the average price and the dispersion measures correlate with nt. the most certain tendency is the average price decreasing as nt rises for 21 of 30 products. 2.4. shape of the normalized price distributions here we describe the general features of the prices in yandex.market platform. a single price observation is ptik which is the price posted by shop i for product k at the time moment t ∈ {1, ..., t}. data for the given product is a set of price vectors p1, . . . , pt where each vector copyright © 2023 assa. adv syst sci appl (2023) verification of bounded rationality models of price dispersion 135 1-2 2-3 3-4 4-5 sellers rating -0.2 0 0.2 0.4 r e la ti v e p ri c e d e v ia ti o n 1-5 6-10 11-15 16-20 21-25 26-30 number of goods -0.2 0 0.2 0.4 r e la ti v e p ri c e d e v ia ti o n (a) (b) fig. 2.3. price deviation relationship with sellers rating and assortment. the widths of the boxes are proportional to the number of sellers in a given category fig. 2.4. distribution of the instant number of sellers. the data are aggregated across all days and all products. table 2.2. correlation of price dispersion measures with the number of sellers # of products positive corr. negative corr. not significant average price 5 21 4 gap 3 17 10 range 20 7 3 coeff of var 12 15 3 pt contains prices observed at moment t. we normalize each price vector to its mean pt: p̂tik = ptik pt . (2.1) the normalized prices across all days can be aggregated into a single price distribution for each product. while the shapes of these distributions vary from product to product, they share some key features. figure 2.5 shows the distribution for one of the products, a cooker hood bosch dhl 545 s. the distribution is unimodal, similar to normal, but skewed to the right and leptokurtic. this shape is observed with some variations for all of the examined products. the right tail and the central peak may be more or less prominent, but the general pattern is the same. in [21] similar distributions were observed though they are more symmetric than ours. copyright © 2023 assa. adv syst sci appl (2023) 136 n. bazenkov, e. glamozdina, m. kuznetsova, m. sandomirskaia we can see that the shape of aggregated price distribution is close to normal. however, fig. 2.5. the distribution of normalized prices across all days. the solid curve shows the normal distribution of the same mean and variance. the distributions for shorter time periods are expected to be be different from the aggregated one. moreover, they can belong to different classes of distributions, i.e. be one-peaked or twopeaked, for some particular days for the same product. testing several reasonable theoretical models in order to select one that better fits the data is addressed in the rest of the paper. 3. daily price distribution: bounded rationality models we examine the behavioral models developed in [2] and [27]. this section briefly introduces the theoretical framework, the intuition behind it and the price distributions under every hypothesis for fitting the models to the data. more formalities are in appendix a. 3.1. price competition with homogeneous product let us start with the pure homogeneous bertrand setting. the models are based on the classical bertrand price competition where n firms sell the same homogeneous product with costs c. the consumers are rational and always buy from the firms who announced the minimal price. this is in line with the design of the online marketplace where consumers are able get the complete list of prices for a chosen product and to sort these prices with a single click. moreover, we can expect that those consumers who consciously use pricecomparison site do this in order to find the best price. nash equilibrium in this setting yields the bertrand paradox with zero profits for all firms. this clearly contradicts the observations of real online markets. seeking a plausible theoretical background of online sellers behavior, the authors in [2] enriched the setting by two models of bounded rationality. ε-equilibrium the ε-equilibrium implicates the minimal level of extra payoff which is expected to be reached after a deviation. this also can be interpreted as the decision maker being insensitive to small price differences or having limited cognitive abilities. another logic beyond this concept involves the possible costly price changes and as a result of the lack of motivation for small extra gains from frequent price adjustments. copyright © 2023 assa. adv syst sci appl (2023) verification of bounded rationality models of price dispersion 137 for the profit function with non-elastic demand normalized to 1, the ε-equilibrium price distribution for ε ∈ (0, [n/(n− 1)]n−12−n(pm − c)) is given by f ε i (p) =  0, p < c+ θ; 1− ( θ p−c ) 1 n−1 , p ∈ [c+ θ, pm); 1, p ≥ pm, (3.2) where n is the number of active firms, pm is the monopoly price, c are costs which are the same for all firms, and θ = [ εn−1 ( n n− 1 )n−1 (pm − c) ]1/n (3.3) prices generated by this model are concentrated around the two points: θ and pm. figure 3.6 the bimodal shape with very low density between the two peaks. the equation for cdf (3.2) implies that the cdf is discontinuous in pm, so the density in pm is represented by dirac delta function. this is shown as the vertical line surmounted by an arrow in the figure. as n grows, θ approaches zero, and for n ≥ 8 almost all prices fall near the monopoly price pm or near the minimal price c. 0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 0.5 0.6 epsilon-equilibrium pdf n=4, =17.6 n=8, = 1.1 n=16, =0.004 fig. 3.6. price distribution for ε-equilibrium. ε is set as 0.5 of the upper bound for the given n. ε-hypothesis: the firms selling identical products on the internet play their ε-equilibrium price strategies in the bertrand model with homogeneous products. quantal-response equilibrium quantal-response equilibrium was proposed in [26]. the idea is that a player randomizes over all available actions with a tendency to choose actions proportionally to their expected utility. in [9] the concept is interpreted as an agent maximizing the “subconscious” utility function which is not completely known to her. it is proved that fictious play learning dynamics converges to qre [9]. in the context of online retail, this means that a seller can only approximately anticipate what profit it will gain from a particular decision, but after some learning process the policy will favor more profitable actions in a proportion predicted by qre. in qre the probability of setting a particular price depends on the expectations of how this price will influence the profit level under some fixed price distributions of competitors. the motivation for this concept stems from preference shocks [25], or decision errors [24]. copyright © 2023 assa. adv syst sci appl (2023) 138 n. bazenkov, e. glamozdina, m. kuznetsova, m. sandomirskaia for the decision rule power function with parameter λ ∈ [ 0, 1 n−1 ) , the symmetric qre is provided by the following price distribution: fq i (p) =  0, p < c; 1− ( 1− ( p−c pm−c )(1+λ) ) 1 1+λ−nλ , p ∈ [c, pm); 1, p ≥ pm. (3.4) the parameter λ expresses the degree of irrationality in a specific sense. the two limit cases correspond to nash equilibrium (λ → 1 n−1 ) and random behavior (λ → 0). the intermediate values of λ generate the set of price distribution functions with a larger mass of firms setting low prices in comparison to ε-equilibrium. according to this model the price distribution should be unimodal and right skewed, as shown in figure 3.7. the distribution spans the whole price range [c, pm]. the peak is close to the minimal price. as the players behavior tend from strict rationality to randomness, the distribution becomes flatter, tending to uniform. in contrast to ε-equilibrium, the probability function is smooth and shows more flexibility. the number of sellers n slightly affect, but the effect is much weaker than for ε-equilibrium. 0 20 40 60 80 100 price 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 d e n s it y qre distribution (n-1)=0.25 (n-1)=0.50 (n-1)=0.75 fig. 3.7. price distribution for quantal-response equilibrium. the number of sellers n = 8, lambda(n− 1) is the “degree of rationality”: 0 is complete randomness, 1 is strict nash rationality. q-hypothesis: the firms selling identical products on the internet set their prices according to the qre price distribution in the bertrand model with homogeneous products. 3.2. price competition with a share of loyal consumers an alternative model for explaining the behavior of online competitors is based on consumer heterogeneity in access to the complete list of offers. in particular, this line relates to modelling multi-channel competition via both online sites and bricks-and-mortar stores, in opposite to considering online and offline competitions as independent settings [8]. the paper [27] presents the clearinghouse model which modifies bertrand price competition by introducing a share of captive consumers. assume that the fraction α of all consumers (the total mass is normalized to 1) use a price aggregator and choose the firm with the minimal price. the rest of the consumers are not informed about the whole range of firms: they are loyal to a certain firm from the list and buy from it with the reservation price normalized to 1. here we do not specialize the reasons for this loyalty, which can be the result of prohibitively high search costs or advertising efforts of the firm [3]. as above, costs are equal to c for all firms. the utility of firm i is combined from two sources: non-competitive copyright © 2023 assa. adv syst sci appl (2023) verification of bounded rationality models of price dispersion 139 revenue coming from loyal consumers and a possible gain for the winner of the bertrand competition. the (nash) equilibrium price distribution is given by fα i (p) =  0, p < p; 1− [ 1−α αn pm−p p−c ] 1 n−1 , p ∈ [p, pm); 1, p ≥ pm, (3.5) where p = pm(1−α)+αnc 1−α+αn is the minimal price for an active firm in the market. while the assumptions under this model are different from those for ε-equilibrium, the shape of the distribution is similar. the probability density is u-shaped with the first peak at p, approaching c as α grows, and the second peak at pm. figure 3.8 shows that as the fraction of rational consumers α grows, more prices are concentrated around the left bound, though the right peak does not diminish completely even for very high α. 0 20 40 60 80 100 price 0 0.1 0.2 0.3 0.4 d e n s it y loyal consumers model =0.50 =0.75 =0.95 fig. 3.8. price distribution for loyal consumers equilibrium. the number of sellers n = 8, α is the fraction of rational consumers α-hypothesis: the firms selling identical products on the internet compete in price for the share of informed consumers within the bertrand model with homogeneous product and simultaneously get extra revenue from the consumers loyal to them. 4. estimation technique here we examine the shape of the price distributions observed at any particular moment of time. we address the following questions: 1. does any of the three hypotheses from section 3 provide a valid explanation for the shape of the observed price distributions; 2. what probability distribution is the best parametric model for the observed price data. to answer the first question we apply the technique similar to [2]. for each of the three hypotheses, we statistically tested how well it predicts the mean, the median, and the histogram of the observed price distributions. in the analyzed behavioral models the number of sellers n greatly influences the shape of the price distribution, so the analysis was performed separately for the different numbers of sellers which is exogenous under the complete access to information about the list of competitors. the data were grouped such that each group contains only price vectors of the copyright © 2023 assa. adv syst sci appl (2023) 140 n. bazenkov, e. glamozdina, m. kuznetsova, m. sandomirskaia given length n and the total number of prices in each group was at least n = 25. then the parameters were estimated separately for each group. each behavioral model contains three parameters. the monopoly price pm, the cost c, and the shape parameter: λ, θ, and α. in the previous work [2] the monopoly price pm and the cost c were fixed by the experimental design. here all three parameters are estimated by minimizing the sum of squared errors between the empirical and the theoretical cumulative distribution functions, following the same approach as [2]. after the parameters had been estimated for the given price group, we evaluate the observed and theoretical distributions by three tests. t-test was used to compare the observed mean and the mean predicted by each theoretical model. the sign test was used to compare the observed and the predicted medians. the goodness of fit was evaluated by the chi-square test. five bins were used, chosen such that the expected frequencies were approximately equal for each bin. the significance level was chosen as p = 0.05 for each test. we calculated the fraction of all time moments when the hypothesis was not rejected. this fraction estimates how well the given behavioral model explains the observed data post factum. the next section covers the results of these estimations. 5. results the complete results are summarized in tables 6.4, 6.5, and 6.6, provided in appendix. the percent of successful tests are shown in table 5.3. the “random” column corresponds to uniformly distributed prices. a detailed discussion for each hypothesis is presented below. table 5.3. the fraction of time when the hypotheses are not rejected, the significance level is p = 0.05 test q−hypothesis ε−hypothesis α−hypothesis random mean 98% 20% 33% 14% median 92% 12% 43% 15% distribution 49% 0.1% 3% 0.4% 5.1. success of q-hypothesis the results of estimation of q-hypothesis are the most optimistic. the mean and median tests fail to reject qre in almost 100% of the tests. this is caused by the fact that most of the observed price distributions are unimodal with positive skewness and this shape is in general similar to that induced by the qre hypothesis. one of the successful fits is presented in figure 5.9. the goodness-of-fit chi-square test rejects the null hypothesis for a half of the observed moments. the best result is for the cooktop hotpoint-ariston 7hpc 640, where qre hypothesis passed the test in 91% of the time moments. for a half of the products qre passed the test in more than 55% of time. there is a single product, the fridge liebherr sbsesf 7212, with 0% of qre passed, this may be due to its high price in comparison with other products in our sample, and thus, a specific consumer demand structure not covered by the model assumptions. figure 5.10 shows the success rate varying from 90% to 0%. the success rate does not depend on the number of sellers. the fraction of successful estimations convinces us that qre hypothesis performs much better than the uniform prices and the other two hypotheses. we assume that this result is due to the great flexibility of quantal response equilibrium which covers the broad range of distribution shapes. therefore, our study shows that qre can justify very different behavioral patterns, though this model is still falsifiable since it was rejected in 50% of the time. the model may serve as a useful technique to weaken the assumption of strict rationality. further copyright © 2023 assa. adv syst sci appl (2023) verification of bounded rationality models of price dispersion 141 fig. 5.9. empirical and the theoretical distributions for the kitchen hood bosch dhl 545 s. number of sellers is n = 37. the prices are from 2015, august 17 at 1am. only qre passed χ2 test (λ(n− 1) = 0.41, pχ2 = 0.18). 0 5 10 15 20 25 30 product number 0 0.2 0.4 0.6 0.8 1 degree of rationality 2 pass rate (n-1) avg. (n-1) lab avg. (n-1) fig. 5.10. fraction of qre passed χ2 test for all products (bars) and the related degree of rationality λ(n− 1) (solid line), average is 0.56 (dashed line) and the average value for lab experiments in [2] is 0.83 this can be applied in combination with some sophisticated market model involving multichannel competition or consumer search. 5.2. testing of εand α-hypotheses the estimations of εand α-hypotheses are much less successful than qre. the prediction quality of ε-hypothesis is comparable to the uniform distribution. even the mean and the median tests provide weak predictions, while the chi-squared test fails dramatically. the examples of successful fits are shown in figure 5.11 and figure 5.12. a possible reason for the poor performance is that both εand α-hypotheses predict high concentration of prices near the monopoly price for n ≥ 8. but this pattern is rarely observed in our data, most of the time the prices lie close to the minimal price, not maximal. table 2.2 indicates that, for most of the products, the average price correlates negatively with the number of sellers. this tendency also directly contradicts the predictions of both models. it is to be noted that the model with loyal consumers, α-hypothesis, performs significantly better than random and ε-hypotheses. the reason is that this model generates the smooth distribution curve which is more flexible than those of ε-equilibrium. nevertheless, the fraction of successful estimations is considerably lower than that for qre. it may be the copyright © 2023 assa. adv syst sci appl (2023) 142 n. bazenkov, e. glamozdina, m. kuznetsova, m. sandomirskaia fig. 5.11. empirical and the theoretical distributions for the washing machine indesit wiun 81 (csi) f053525. number of sellers is n = 5. prices are grouped across 5 time points starting from september 14. only εequilibrium passed χ2 test: θ = 183.8, pχ2 = 0.08. fig. 5.12. empirical and the theoretical distributions for the warm oven electrolux ezb 53400 ax. number of sellers n = 20, prices are grouped from august 19, 1am and 7am. only α-hypothesis passed χ2 test: α = 0.97, pχ2 = 0.13 case that for some particular situations the behavioral patterns predicted by α-hypothesis do prevail. the theoretical distributions generated by these two hypotheses have an important distinction: the right peak in ε-hypothesis is an “anomaly”, i.e. the price density is discontinuous here and the density just in the left neighbourhood of the monopoly price tends to zero. for α-hypothesis we observe the smooth, u-shaped distribution with positive densities on the both sides of the price range. this corresponds to the determination of each firm to serve either “shoppers” (for firms with the price close to the marginal cost), or “loyals” (local monopolists) at every moment. copyright © 2023 assa. adv syst sci appl (2023) verification of bounded rationality models of price dispersion 143 6. discussion the objective of this study was to verify the models from [2], and [27] on real price data. the results show that, from the three models to explain internet price dispersion, one performed significantly better than the other two, and one performed not better than random behavior. the principle difficulty concerns the number of competitors. in the bertrand model with ε-equilibrium and in the model with loyal consumers the higher the number of sellers, the higher the prices must be concentrated at the monopoly price, while our data shows the opposite tendency. in the experiments, reported in [2], and [27] there were no more than four competitors, which is in line with the model. however, in internet sales, the number of active firms is generally greater, such that these models are not applicable. even in the case of a relatively small number of sellers, the predictions under ε-equilibrium are not valid, and the predictions of α−hypotheses are rarely successful. these results were not obvious, because the assumptions underlying both models have a reasonable interpretation in the context of online markets. qre performs much better in comparison with the two others. it is shown that the central tendency is predicted very accurately by qre. the fraction of successful distribution fits is 50%, which is surprisingly high. the average value of the predicted degree of rationality λ(n− 1) is 0.56, closer to 0 than the average 0.83, reported in the experiments [2]. according to the model this means that real firms are less rational than the participants of the lab session. this interpretation contradicts the intuition that online markets are the highly competitive environment where sellers employ highly optimized strategies aided by modern software tools. one of the possible interpretation of this “irrationality” is the following. online markets are highly stochastic, and in contrast to lab experiments the firms do not know exactly what price is optimal at this particular moment. hence they are trying to pick the optimal price through some trial-and-error iterative process, which never converges since the environment evolves permanently in an unpredictable direction. the motivation behind quantal-response equilibrium proposed in [9] reflects this vision. the agents cannot compute the optimal price precisely, but they have more or less correct intuitive assumptions about what actions are more profitable. so, the result that agents in real market rely on randomization more than in lab experiments is not due to their less rationality, but due to the random nature of the environment. the relative success of qre is also explained by the fact that qre admits a much broader set of possible price distributions. an important note is that the general formulation of qre is proved to lack falsifiability until additional restrictions are introduced [17]. here we examined only symmetrical qre, and this formulation appeared to be sufficiently strict since only half of all price distributions doesn’t contradict this hypothesis according to χ2. to summarize, our observation supports the important role of bounded rationality for the adequate modeling of online competition. one more optimistic observation is that the pure bertrand model occurs to be enough for obtaining successful estimations with relatively high explanatory power. the question of what model is suitable still remains open, but qre has proved itself as a flexible model which can be fit successfully to the real-life data. the model with loyal consumers appears to catch at least some features of the observed behavior. however, this model requires the further elaboration. the combination of both approaches may be a promising direction of future research. another possible direction is to extend the homogeneous bertrand competition in a way which is more suitable for online markets, i.e. to introduce the idea of online product differentiation. copyright © 2023 assa. adv syst sci appl (2023) 144 n. bazenkov, e. glamozdina, m. kuznetsova, m. sandomirskaia acknowledgments maria kuznetsova and marina sandomirskaia acknowledge the suppport of the hse university basic research program. nikolay bazenkov acknowledges the financial support by russian foundation for basic research under project #20-07-00190a. the authors gratefully thank the yandex.market customer support api of commercial services for the data access. the authors are indebted to jacques-francois thisse, sergei izmalkov, alexander tarasov, fuad aleskerov, and alexis belianin for fruitful discussions and useful comments. we also thank ernan haruvy for the detailed recommendations which allow us to improve the performance of the results significantly. appendix a. theoretical framework bertrand competition let firm i ∈ {1, . . . , n} proposes price pi simultaneously and independently of all other firms. for simplicity, let pi ∈ [c, pm], pm be the monopoly price, and π(pm) be the monopoly payoff. the expected utility of firm i under the price vector (p1, . . . , pi, . . . , pn) is given by πi(p1, . . . , pi, . . . , pn) = { π(pi) k , pi = min{p1, . . . , pn}, k is the number of firms charging pi; 0, otherwise. (6.6) for estimation purposes the monopoly profit function with non-elastic demand normalized to 1 is used: π(p) = p− c. (6.7) the mixed strategy expansion is considered with a price cumulative distribution vector {f1, . . . , fn}. the expected payoff of firm i is eπi(f ) = ∫ [c, pm]n πi(p)df. ε-equilibrium definition 1. fix ε > 0. a strategy vector f ε is an ε-equilibrium if for any unilateral deviation f ′ i of player i eπi(f ′ i , f ε −i)− eπi(f ε) ≤ ε, i = 1, . . . , n. in ([2]), the solution for the bertrand model with arbitrary function π was obtained. for the profit function (6.7), the ε-equilibrium price distribution is given by formulae (3.2). quantal-response equilibrium let ti be a map from eπi(pi, f−i) into the probability of setting the price less than or equal to pi. definition 2. a strategy vector fq is a quantal-response equilibrium if for any pi fq i (pi) = ti(eπi(pi, f q −i)), i = 1, . . . , n. in ([2]) the decision rule function t was set as the power function with parameter λ ∈ [ 0, 1 n−1 ) [] ti(eπi(p, f−i)) = ∫ p c eπi(q, f−i) λ∫ pm c eπi(t, f−i)λdt dq. (6.8) copyright © 2023 assa. adv syst sci appl (2023) verification of bounded rationality models of price dispersion 145 for ti defined by (6.8), the symmetric qre is provided by the following price distribution: fq i (p) = 1− ( g(pm)− g(p) g(pm)− g(c) )1/(1+λ−nλ) , i = 1, . . . , n, p ∈ [c, pm], with g(p) ≡ ∫ π(p)λdp+k. accounting for (6.7), the final form of the qre price distribution is (3.4). bertrand competition with a share of loyal consumers the utility of firm i has a view πi(p1, . . . , pn) =  α(pi−c) k + (1−α)(pi−c) n , pi = min{p1, . . . , pn}, k is the number of firms charging pi; (1−α)(pi−c) n , otherwise. (6.9) by analogy with the logic in [27], it is easy to derive formula (3.5). the corresponding lower bound for the fraction α is determined by α = pm − p pm − nc+ p(n− 1) . (6.10) the search-theoretic model proposed in [10] employs a different set of assumptions, but provides similar u-shaped price distribution. so the results obtained for ε-equilibrium and α-hypotheses may be generalized to this model. copyright © 2023 assa. adv syst sci appl (2023) 146 n. bazenkov, e. glamozdina, m. kuznetsova, m. sandomirskaia appendix b. estimation results table 6.4. tests of equality of means, medians, and distributions: the shares of moments for every product for which qand ε-hypotheses are not rejected, the significance level is 0.05 qre ε−equilibrium product mean median χ2 mean median χ2 bosch dhl 545 s 1.00 0.99 0.60 0.12 0.10 0.00 hotpoint-ariston ftr 850 1.00 1.00 0.73 0.00 0.00 0.00 indesit wiun 81 (csi) f053525 0.85 0.93 0.17 0.43 0.39 0.04 elikor integra 60n-400 1.00 1.00 0.48 0.26 0.13 0.00 hansa osc 511 wh 1.00 1.00 0.76 0.28 0.19 0.00 krona kamilla 600 0.94 0.35 0.07 0.55 0.25 0.00 samsung wf8590nmw9 1.00 1.00 0.57 0.13 0.00 0.00 atlant xm 4008-022 1.00 1.00 0.47 0.00 0.00 0.00 liebherr sbsesf 7212 0.84 0.06 0.00 0.13 0.22 0.00 candy cdcf 6-07 1.00 0.97 0.60 0.20 0.20 0.00 siemens sr 66t090 1.00 0.88 0.19 0.00 0.00 0.00 bosch activewater spv40e10ru 1.00 0.91 0.35 0.03 0.02 0.00 hansa boei62000015 1.00 1.00 0.65 0.00 0.00 0.00 bosch hbg43t450 1.00 1.00 0.39 0.82 0.02 0.00 indesit bia 18 1.00 1.00 0.67 0.03 0.00 0.00 hotpoint-ariston 7hpc 640 1.00 1.00 0.91 0.03 0.00 0.00 cata ceres 600 negra 1.00 1.00 0.67 0.23 0.31 0.00 electrolux ehh 6340 fok 1.00 1.00 0.58 0.30 0.14 0.00 bosch kgs 39xw20 r 1.00 0.84 0.34 0.26 0.01 0.00 lg gab409svqa 1.00 1.00 0.65 0.12 0.12 0.00 bosch wlg 20061 oe 1.00 0.95 0.49 0.02 0.05 0.00 hansa zim 436 eh 0.99 1.00 0.48 0.28 0.20 0.00 bosch pic645f17e 1.00 0.92 0.29 0.35 0.00 0.00 hansa bhi68300 1.00 1.00 0.73 0.18 0.17 0.00 lg f-1096sd3 1.00 0.98 0.64 0.03 0.00 0.00 gorenje bo 73 cli 0.95 0.96 0.28 0.13 0.07 0.00 gorenje gw 65 cl i 1.00 0.97 0.39 0.13 0.23 0.00 electrolux ews 1052 ndu 0.96 1.00 0.63 0.00 0.04 0.00 electrolux ezb 53400 ax 0.99 0.96 0.08 0.98 0.55 0.00 ariston lstb 4b00 ru 1.00 1.00 0.73 0.12 0.16 0.00 copyright © 2023 assa. adv syst sci appl (2023) verification of bounded rationality models of price dispersion 147 table 6.5. tests of equality of means, medians, and distributions: the shares of moments for every product for which α-hypothesis and random behavior are not rejected, the significance level is 0.05 αrandom product mean median χ2 mean median χ2 bosch dhl 545 s 0.34 0.48 0.01 0.00 0.03 0.00 hotpoint-ariston ftr 850 0.10 0.38 0.00 0.00 0.00 0.00 indesit wiun 81 (csi) f053525 0.88 0.91 0.12 0.44 0.54 0.02 elikor integra 60n-400 0.83 0.38 0.01 0.06 0.12 0.00 hansa osc 511 wh 0.52 0.69 0.03 0.38 0.43 0.01 krona kamilla 600 0.45 0.32 0.07 0.37 0.17 0.00 samsung wf8590nmw9 0.08 0.43 0.00 0.00 0.01 0.00 atlant xm 4008-022 0.18 0.28 0.00 0.00 0.00 0.00 liebherr sbsesf 7212 0.50 0.33 0.03 0.27 0.03 0.00 candy cdcf 6-07 0.53 0.64 0.06 0.16 0.14 0.01 siemens sr 66t090 0.12 0.12 0.00 0.01 0.00 0.00 bosch activewater spv40e10ru 0.04 0.21 0.00 0.00 0.00 0.00 hansa boei62000015 0.08 0.25 0.00 0.00 0.00 0.00 bosch hbg43t450 0.04 0.15 0.00 0.04 0.00 0.00 indesit bia 18 0.21 0.53 0.00 0.02 0.02 0.00 hotpoint-ariston 7hpc 640 0.22 0.47 0.00 0.10 0.09 0.00 cata ceres 600 negra 0.31 0.29 0.00 0.29 0.62 0.00 electrolux ehh 6340 fok 0.35 0.41 0.01 0.24 0.21 0.00 bosch kgs 39xw20 r 0.13 0.21 0.00 0.09 0.11 0.00 lg gab409svqa 0.18 0.21 0.00 0.29 0.25 0.02 bosch wlg 20061 oe 0.12 0.44 0.00 0.01 0.00 0.00 hansa zim 436 eh 0.36 0.56 0.03 0.14 0.12 0.00 bosch pic645f17e 0.36 0.54 0.00 0.00 0.06 0.00 hansa bhi68300 0.30 0.40 0.00 0.09 0.23 0.01 lg f-1096sd3 0.17 0.54 0.00 0.08 0.08 0.00 gorenje bo 73 cli 0.24 0.31 0.04 0.00 0.01 0.00 gorenje gw 65 cl i 0.50 0.47 0.03 0.03 0.09 0.00 electrolux ews 1052 ndu 0.50 0.60 0.17 0.13 0.10 0.00 electrolux ezb 53400 ax 0.76 0.91 0.15 0.80 0.80 0.01 ariston lstb 4b00 ru 0.40 0.43 0.05 0.09 0.09 0.03 copyright © 2023 assa. adv syst sci appl (2023) 148 n. bazenkov, e. glamozdina, m. kuznetsova, m. sandomirskaia table 6.6. parameters of qre and α-hypothesis product λ(n− 1) α all days success fit days all days success fit days bosch dhl 545 s 0.64 0.61 0.99 1.00 hotpoint-ariston ftr 850 0.67 0.70 1.00 indesit wiun 81 (csi) f053525 0.62 0.86 0.84 0.97 elikor integra 60n-400 0.15 0.17 1.00 1.00 hansa osc 511 wh 0.34 0.33 0.99 0.99 krona kamilla 600 0.44 0.71 0.88 0.96 samsung wf8590nmw9 0.74 0.66 1.00 atlant xm 4008-022 0.84 0.78 1.00 liebherr sbsesf 7212 0.34 0.87 0.94 candy cdcf 6-07 0.64 0.64 0.99 0.95 siemens sr 66t090 0.67 0.73 1.00 bosch activewater spv40e10ru 0.67 0.71 1.00 hansa boei62000015 0.80 0.81 1.00 bosch hbg43t450 0.52 0.59 1.00 indesit bia 18 0.58 0.60 1.00 hotpoint-ariston 7hpc 640 0.67 0.67 1.00 cata ceres 600 negra 0.26 0.29 1.00 electrolux ehh 6340 fok 0.74 0.77 1.00 1.00 bosch kgs 39xw20 r 0.56 0.66 1.00 lg gab409svqa 0.32 0.28 1.00 bosch wlg 20061 oe 0.62 0.59 1.00 hansa zim 436 eh 0.76 0.76 1.00 1.00 bosch pic645f17e 0.30 0.37 1.00 hansa bhi68300 0.66 0.66 1.00 lg f-1096sd3 0.44 0.37 1.00 gorenje bo 73 cli 0.71 0.65 1.00 1.00 gorenje gw 65 cl i 0.67 0.62 1.00 1.00 electrolux ews 1052 ndu 0.59 0.63 0.98 0.97 electrolux ezb 53400 ax 0.35 0.58 0.98 0.97 ariston lstb 4b00 ru 0.62 0.60 1.00 0.99 copyright © 2023 assa. adv syst sci appl (2023) verification of bounded rationality models of price dispersion 149 references [1] quentin andré, nicholas reinholtz, and bart de langhe. “can consumers learn price dispersion? evidence for dispersion spillover across categories”. in: journal of consumer research 48.5 (2022), 756––774. 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[36] yandex and gfk. study of online-consumers in russia, august – september 2014 (in russian). tech. rep. 2014. url: https : / / cache mskstoredata04 . cdn . yandex . net / download.yandex.ru/company/figures/gfk 2014.pdf. [37] yandex and gfk. study of online-consumers in russia, september – november 2018 (in russian). tech. rep. 2018. url: https://cache-mskm905.cdn.yandex.net/download. yandex.ru/company/figures/2018/gfk/yandex market gfk 2018.pdf. copyright © 2023 assa. adv syst sci appl (2023) https://doi.org/https://doi.org/10.1108/10610420810856512 https://cache-mskstoredata04.cdn.yandex.net/download.yandex.ru/company/figures/gfk_2014.pdf https://cache-mskstoredata04.cdn.yandex.net/download.yandex.ru/company/figures/gfk_2014.pdf https://cache-mskm905.cdn.yandex.net/download.yandex.ru/company/figures/2018/gfk/yandex_market_gfk_2018.pdf https://cache-mskm905.cdn.yandex.net/download.yandex.ru/company/figures/2018/gfk/yandex_market_gfk_2018.pdf introduction online prices: empirical observations data price dispersion sellers characteristics shape of the normalized price distributions daily price distribution: bounded rationality models price competition with homogeneous product price competition with a share of loyal consumers estimation technique results success of q-hypothesis testing of and -hypotheses discussion references adv syst sci appl 2018; 02; 26-52 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/598 plasma control in tokamaks. part 1. controlled thermonuclear fusion problem. tokamaks. components of control systems. yuri mitrishkin1,2*, pavel korenev1,2, artem prokhorov1,2, nikolay kartsev2, mikhail patrov3 1) m. v. lomonosov moscow state university, faculty of physics, moscow, russia 2) v. a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia 3) ioffe institute, russian academy of sciences, st. petersburg, russia abstract: various concepts of tokamaks as leaders in solving the controlled thermonuclear fusion problem are presented. the evolution of tokamaks from round in vertical cross-section tokamaks with a large aspect ratio up to tokamaks with a small aspect ratio including spherical ones is characterized. the classification of modern tokamaks according to their poloidal systems with the location of the poloidal field coils inside and outside of the toroidal field coil is given, taking into account the presence of coils inside a vacuum vessel to stabilize the plasma position. the methods of plasma diagnostics by magnetic measurements outside the plasma, actuators for both plasma magnetic and kinetic control, associated with plasma additional heating, plasma magnetic and kinetic models, instabilities and disruptions of a plasma discharge are described. keywords: controlled thermonuclear fusion, thermonuclear power station, tokamak, plasma, tokamak classification, diagnostics, additional plasma heating, actuators, plasma equilibrium reconstruction, plasma magnetic and kinetic models, plasma instabilities, disruptions, spherical tokamaks. 1. introduction the aim of the survey is to present a plasma in a tokamak as a complex unstable time-varying multivariable nonlinear plant with distributed parameters and uncertainties to be controlled and to explore magnetic and kinetic plasma control systems and their development trends. the plasma in a tokamak [1–4] has toroidal axial-symmetric magnetic configuration which is superposition of fields created by the currents in toroidal field coils, plasma distributed current, and currents in poloidal field coils. an abbreviation “tokamak” (from russian toroidal chamber with magnetic coils) was proposed in the i.v. kurchatov institute of atomic energy (moscow, russia). a variable magnetic field produced by the current in a central solenoid (inductor) induces a vortex electrical field, which breaks down gas, typically hydrogen, in a vacuum vessel and creates plasma, which is fully ionized gas with distributed current. thus, tokamak is a current transformer. a plasma column tends to gain its major radius because of the integral plasma current at axial-symmetric points is oppositely directed which leads to self-pushing. additionally, the major radius is gained by the gas kinetic pressure in plasma. hence, a feedback control system is needed to stabilize a horizontal plasma position by a vertical magnetic field. the plasma in modern tokamaks is vertically elongated that allows to increase plasma pressure for the same toroidal field. unfortunately, it leads to plasma vertical instability. hence, * corresponding author: yvm@mail.ru original russian text © yu. v. mitrishkin, p. s. korenev, a. a. prokhorov, n. m. kartsev, m. i. patrov, 2018, published in problemy upravleniya / control sciences, 2018, no. 1, pp. 2–20. plasma control in tokamaks. part 1. 27 copyright ©2018 assa. adv. in systems science and appl. (2018) a feedback control system is necessary to stabilize an unstable plasma vertical position by a horizontal magnetic field. a multivariable plasma shape control system is needed to keep a high-temperature plasma near the tokamak first wall. the plasma current can be controlled either by a controller in a separate loop or by the multivariable controller for plasma shape and current simultaneously [5–7]. while plasma pressure increases unstable resistive wall modes (rwm) may also occur. hence, additional actuating coils and feedback control system are needed to suppress rwm [8]. based on the facts listed above one can conclude that modern tokamak devices must be developed in parallel with plasma magnetic control systems. neutral atom injection and electromagnetic waves are used for additional plasma heating, which provides ability to plasma kinetic control, i.e. control of plasma current profile [9, 10], stability margin [11], pressure, and temperature [12] in order to optimize plasma discharge parameters. magnetic and kinetic control integration [13] will be the basis for reliable plasma control systems of future thermonuclear reactors. the survey consists of four parts. different construction concepts of modern tokamaks and different magnetic and kinetic plasma control systems, which are used in real experiments or in numerical simulations of present tokamaks are considered. part 1 is devoted to the general problem of the controllable thermonuclear synthesis, main distinctive tokamak features, and plasma control systems. construction and diagnostics system of the spherical globus-m tokamak (ioffe institute, st. petersburg, russia) are described. results of globus-m experiments were used in lomonosov msu and trapeznikov ics to develop original plasma position, current, and shape control systems. plasma magnetic control systems are presented in parts 2 and 3. part 4 is dedicated to plasma kinetic control systems and their integration with magnetic control systems to achieve optimal tokamak operation regimes in stationary and transition modes to get maximum stability margins and acceptable performance. 2. controlled thermonuclear fusion at present, the primary source of energy for the humankind is burning organic fossil fuels namely coal, oil, and gas. although natural reserves of fossil fuels will not be completely exhausted for several hundred years, experts predict an energy deficit in less than 50 years at current consumption rates [14]. renewable energy sources such as solar energy, hydroelectric power, geothermal energy, and wind energy are attractive from an environmental point of view, but cannot provide sufficient energy output to become a full-fledged substitute for the fossil fuels. nuclear power stations, which harness energy released in the fission reactions of atoms of heavy elements, can produce sufficient amounts of energy, but, unfortunately, the byproducts of fission are very radioactive and long-lived. additionally, nuclear power plants require careful monitoring of all operational parameters, since an accidental release of radioactive substances would have the disastrous impact on the environment and could exceed the impact from thermoelectric power stations. one of the most promising sources of the future energy is thermonuclear fusion reactions, which are merging of the nuclei of the light elements like hydrogen, for example. although the controlled thermonuclear fusion is an extremely complex technological problem, the thermonuclear power has significant advantages over existing energy sources. in particular, fuel reserves for thermonuclear reactions on the earth will last for many thousands of years, since the necessary hydrogen isotopes can be obtained from water and widely available lithium. hydrogen nuclear fusion reactions do not lead to air pollution or formation of greenhouse gases, since the product of the fusion reaction is helium. in contrast with nuclear fission, nuclear fusion does not involve uncontrolled chain reactions, which means that there 28 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov copyright ©2018 assa adv. in systems science and appl. (2018) is no risk of a nuclear accident. the point is that at any time one can close the valves of fuel feeding into the tokamak vacuum vessel and by this action to stop the fusion process. we also note that a thermonuclear reactor does not produce long-lived radioactive waste. the main radioactive by-products of the reactor are neutron-activated materials (i.e., materials which become radioactive because of a neutron bombardment of the first wall of tokamaks), which can be minimized by material selection for the reactor vessel. most of the radioactive materials produced in the reactor can be safely and easily disposed of after use and utilized harmless in several decades, unlike most fission by-products that require special storage and processing for thousands of years. there are many possible reactions of nuclear fusion, which differ by the required conditions and the amount of energy released. however, the most promising is the fusion reaction of hydrogen isotopes: deuterium and tritium 2 3 4 1 1 1 2 0d+ t + n +17.6mev. (2.1) when the nuclei of deuterium and tritium fuse, an alpha-particle and a neutron are formed. the alpha-particles may remain in the plasma and contribute energy to maintain the thermonuclear reaction. the neutrons carry 80% of the released energy and are absorbed by reactor’s first wall, transforming energy into heat, which can be converted into electricity. the deuterium-tritium fuel has an extremely high density of energy: 1 g. of the mixture is equivalent to 10 000 liters of oil! a small portion of the energy generated by the reactor can be used to extract deuterium from seawater, while tritium can be produced by neutron bombardment of lithium. since the nuclei of atoms are positively charged, according to coulomb's law they repel each other. at ordinary temperatures, no fusion reactions occur, since the atoms do not have enough energy to overcome the coulomb barrier. the temperature required for the thermonuclear reaction is about 109 k. at these temperatures, the fuel is in the state of completely ionized gas such as plasma. for a positive energy output, the plasma in the reactor must satisfy the lawson criterion [2] 21 3τ 3 10 m ×kev,n t s  where τ is the energy confinement time, during which the plasma retains enough energy for the reaction to occur, n is the plasma particle density, and t is the plasma temperature. 3. tokamaks 3.1. tokamaks evolution the first tokamak was constructed in 1954 in the kurchatov institute of atomic energy, and after that the researches of plasma confinement in tokamaks began in many laboratories in the world. since 1954 221 tokamaks have been built and 41 of them are working nowadays [15]. plasma control in tokamaks. part 1. 29 table 3.1 shows comparative characteristics of the iter tokamak-reactor and the thermonuclear power plant demo [18] based on the tokamak principle. demo will be the last step before the construction of commercial thermonuclear power plants. in contrast to fig. 3.1. view of iter.1 the evolution of tokamaks began from devices with a circular cross section which were surrounded by a copper shell and led to modern vertically elongated ones [2]. in the first tokamaks the plasma speed was damped by induced foucault currents in the tokamak copper shell that was acting as the controller of a direct action. in modern devices, the magnetic and kinetic active feedback control systems are used. magnetic plasma control systems [2, 6, 7, 16] evolved from simple scalar plasma horizontal position control systems to multivariable plasma position, current, and shape control systems. for the last 30 years, power generated by tokamaks increased in 108 times [14]. such a significant progress was achieved due to the qualitative jump in the development of the plasma physics and in the level of understanding of processes occurring during plasma discharges. fig. 3.2. deuterium-tritium thermonuclear power plant scheme.2 the fundamental task, which arises when developing modern tokamaks, is the search of the methods and approaches to provide optimal and reliable thermonuclear fusion. most of researches of plasma confinement and heating on operating tokamaks are directed to support the iter project (international thermonuclear experimental reactor) [17] (fig. 3.1), which has to open the way to the demo project (demonstration power plant). the demo is the first future noncommercial thermonuclear power plant (fig. 3.2) [18]. the iter tokamak will be the first thermonuclear experimental reactor, which will produce ten times more energy than consumes. the international community with participants from european union, japan, usa, russia, china, south korea, and india is developing and constructing the iter. 1 iter organization, http://www.iter.org/. copyright ©2018 assa adv. in systems science and appl. (2018) 2 max-planck-institute-fur-plasmaphysik: http://www.ipp.mpg.de/986351/fusion_e.pdf. 30 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov copyright ©2018 assa adv. in systems science and appl. (2018) iter, which is the experimental device for high temperature plasma researches, demo will be more close to future industrial reactors, more technological for the aims of reliable exploitation and replacement of components. it will have optimized cooling system for higher temperatures at energy generation, less diagnostics, and minimal set of subsystems, which is necessary for working. the time of plasma discharge in demo will be more than two hours with self-sustained thermonuclear fusion reaction. the results obtained on iter will permit to narrow the range of requirements down and at the same time to optimize them on demo. table 3.1. characteristics of the iter experimental reactor and the thermonuclear power plant demo [18] parameter iter a = 3.1 demo 1 a = 3.1 demo 2 a = 2.6 aspect ratio а = r/a 6.2/2.0 9.1/2.9 7.5/2.9 elongation k / triangular δ 1.7/0.33 1.6/0.33 1.8/0.33 surface area, m2 / plasma volume, m3 683/831 1428/2502 1253/2217 plasma current, ma 17 20 22 major radius toroidal field, t 5.3 5.7 5.6 discharge time, sec 400 > 7200 > 7200 output thermonuclear power, mw / generated power, mw 500/0 2037/500 3255/953 r and а are major and minor radiuses, k and δ are elongation and triangularity (see fig. 5.2). 3.2. principles of tokamak operation the electromagnetic fields produced by a current-carrying coils in a tokamak effect on plasma charged particles and are used for plasma confinement and control in the limited vessel volume. the poloidal and toroidal magnetic fields generated by currents in poloidal and toroidal coils and the magnetic field generated by the plasma current compose the resulting helical magnetic field (fig. 3.3). plasma charged particles cannot leave such a magnetic configuration while moving on larmor radii along helical magnetic lines with cyclotron frequencies inside the torus. the toroidal field of tokamaks usually a tenfold greater than the poloidal field and is around several units of tesla. the toroidal field coil has a large number of turns and a current about several ka traveling through the coil creates the same field as a current about several ma traveling through one turn. depending on the resistance of the coil (usually copper or superconducting), the voltage from several units to hundred units is needed for steady state mode, but for transition processes the requirements are significantly higher. there are tokamaks, for example jet (uk), with an iron core inside the central solenoid to increase magnetic flux inducting plasma current. however, most of the tokamaks do not have iron cores, for example diii-d (usa). plasma control in tokamaks. part 1. 31 copyright ©2018 assa. adv. in systems science and appl. (2018) fig. 3.3. vertically elongated tokamak without an iron core: 1 is the poloidal field inner and outer coils; 2 is the vacuum vessel; 3 is the toroidal field coil; 4 is plasma and helical magnetic lines.3 tokamak magnetic fields make outer pressure, which balances plasma inner kinetic pressure. thus, a small magnetic field disturbance can lead to a local expansion of the plasma, which will exponentially grow if not suppressed. magnetohydrodynamic (mhd) theory [19] can describe most of such plasma instabilities. the varying field of the central solenoid induces the electrical field, which creates the plasma current analogously to a field produced by the current in the primary transformer winding induces a current in the secondary winding. however, the solenoid current must continuously grow to maintain the plasma current, this fact constrains the plasma discharge time. that is why other non-ohmic current sources are needed. the most promising are a bootstrap current inducing by a plasma charged particles density gradient, and an additional heating current. 3.3. modern tokamaks poloidal system classification table 3.2 shows the main characteristics of the present vertically elongated tokamaks with the most advanced plasma control systems (see survey parts 2–4 in the next journal issues), the iter experimental reactor under construction, and the demo project for the thermonuclear power plant. table 3.2. characteristics of present tokamaks, iter, and demo device country starting year major radius, m minor radius, m toroidal field, t plasma current, ma diii-d [20] usa 1986 1.66 0.67 2.2 3.0 nstx [21] usa 1999 1.85 0.65 0.3 1.4 jt-60u [22] japan 1991 3.40 1.00 2.7 5.5 tcv [23] switzerland 1992 0.88 0.25–0.7 1.4 1.2 jet [24] uk 1992 3.00 1.25–2.1 4.0 6.0 4 3 project center iter (russia): http://www.iterrf.ru/upload/docs/booklet_new.pdf. 32 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov copyright ©2018 assa adv. in systems science and appl. (2018) the poloidal systems of present vertically elongated tokamaks may be differentiated on three main groups.  «warm» poloidal field coils are inside the toroidal coil, the plasma vertical position control coil is outside the vacuum vessel. such tokamaks are nstx, diii-d (usa), jt-60u (japan), tcv (switzerland).  «warm» poloidal field coils are outside the toroidal coil, the plasma vertical position control coil is outside the vacuum vessel. such tokamaks are jet (uk), asdex upgrade (germany), globus-m (russia).  superconducting poloidal field coils are outside the toroidal coil, the plasma vertical position control coil is inside the vacuum vessel. such tokamaks are east (china), iter (france), kstar (south korea). in tokamaks kstar [27] and jt-60sa [30] the plasma vertical position control coil and the plasma horizontal position control coil are inside the vacuum vessel, which considerably enhances the efficiency of control. 3.4. plasma auxiliary heating in the tokamaks when the lawson criterion [2] is satisfied, the plasma generates enough heat to maintain its temperature, but to achieve that, joule heating by plasma current is not enough, there is a need for auxiliary plasma heating. auxiliary heating of the plasma can also be used to control the profiles of its current, temperature, and pressure in order to achieve energetically favorable operating conditions and suppress mhd instabilities. the plasma current ip generates joule heat 2~ η pp i where η is the plasma resistivity. however, with increasing the plasma temperature the resistivity of the plasma decreases according to the law 3/2~ t  , therefore ohmic heating ceases to be effective at high temperatures and auxiliary heating methods are needed. one of the main methods of auxiliary heating is neutral beam injection heating. the electromagnetic field does not prevent the entry of neutral particles into the plasma, where they transfer their kinetic energy to plasma in collisions with its particles. for example, deuterium beams with a power of 20 mw are used in tokamaks diii-d [20] and asdex upgrade [25]. another method of the plasma heating is based on the use of radio waves that resonate with plasma particles and transmit their energy to them. there are heating by the radio waves with the ion cyclotron frequency (tens of megahertz) and with an electron cyclotron frequency (hundreds of gigahertz). for example, on the diii-d tokamak, ion cyclotron heating (30-120 mhz, 6 mw) and electron cyclotron heating (110 ghz, 6 mw) are used [31], the asdex upgrade tokamak includes the ion (30-120 ghz, 6 mw) and electron (140 ghz, 4 mw) cyclotron heating [32, 33]. the tcv tokamak uses the electron cyclotron heating system, consisting of 9 gyrotrons (82 and 118 ghz, 4.5 mw) [12, 23]. the east tokamak uses ion cyclotron heating (27 mhz, 6 mw) [34]. asdex upgrade [25] germany 1991 1.65 0.5–0.8 3.9 1.4 east [26] china 2006 1.75 0.4–0.8 5.0 0.5 kstar [27] south korea 2008 1.8 0.5 3.5 2.0 iter [28, 29] france 2025 6.2 2.0 5.3 17 demo 1 [18] – 2040–2045 9.1 2.9 5.7 20.0 plasma control in tokamaks. part 1. 33 copyright ©2018 assa. adv. in systems science and appl. (2018) 4. plasma magnetic diagnostics measuring systems are the most important components of the plasma control systems of any tokamak. they allow to determine the physical quantities describing the behavior and structure of the heated plasma inside the tokamak vessel and to perform plasma diagnostics. measuring devices that require physical contact with plasma even for a short time period are extremely limited, since a high-temperature plasma can quickly disable them. infrared sensors are also used in experiments, but they do not provide measurements with sufficient time resolution for the most purposes of plasma control [35]. the simplest and most widely used method to detect the boundary and integral parameters of the plasma is the use of magnetic sensors measuring the changes in the magnetic field and flux outside the plasma [4, 6, 35], but within the region of the magnetic field produced by the plasma. all sensors related to magnetic diagnostics work on the same basic principle, namely: the induced electromotive force 0u in any coil is equal to the time rate of change of the magnetic flux passing through this coil, taken with a minus sign (in accordance with the faraday’s law): 0 φ , d d u n n bds dt dt      (4.1) where n is the number of coil turns, φ is the magnetic flux passing through the coil, b is the magnetic induction field. to describe electromagnetic phenomena in tokamaks, the function of the poloidal flux ( ) 1 2π ψ( , ) s r r t bds  as a magnetic flux per one radian passing through a circle s centered on the vertical axis of the tokamak is usually introduced (fig. 4.1). this function depends on the coordinate r on the poloidal plane and on the time t . the magnetic flux sensor consists of a single turn of wire, also called a loop. the flux loop is usually located outside the vacuum vessel in the toroidal direction. the poloidal magnetic flux in a tokamak is measured by integrating the voltage 0u , which is proportional to the electromotive force induced in the flux loop (fig. 4.1). the integrated loop voltage with a corresponding coefficient is a poloidal magnetic flux passing through its circuit: 0 0 0 1 ( )ψ( , ) ψ( , ). 2π t t u dr t r t    (4.2) for axisymmetric plasma, this measurement gives the value of the poloidal flux at a certain point of the poloidal plane. the saddle loops are constructed by connecting two sectors of poloidal loops measuring poloidal fluxes at points 1r and 2r . their integrated voltage is proportional to the difference between poloidal fluxes: 0 1 2 1 0 2 00ψ( , ) ψ( , ) ψ( , ) ψ( , 1 ) 2 .( ) π t t u dr t r t r t r t    (4.3) the magnetic field is measured using the magnetic probes (fig. 4.1), which react to the change in the magnetic flux passing through them. a magnetic probe is a multi-turn coil connected to a voltage sensor. assuming that there is the homogeneity of the magnetic field inside the coil, the induced voltage 0u is related to the local magnetic field by the relation 0 0 0 1 ( , ) ( ) ( , ) , t t b r t n u d b r t n ns     (4.4) where n is the normal vector to a cross section of the coil, n is the number of turns, s is the cross-sectional area of the coil. this expression is approximate since the magnetic flux is not constant at the edges of the probe. 34 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov copyright ©2018 assa adv. in systems science and appl. (2018) fig. 4.1. magnetic measurements in tokamak: 1 is the magnetic loop; 2 is the saddle loop; 3 is the magnetic probe; 4 is the rogowski coil. in order to measure the plasma current in tokamaks, rogowski coils are used. fig. 4.1 shows rogowski loop, which measures the sum isum of the plasma toroidal current рi and the current in the vacuum vessel that passes through the region bounded by the coil, based on integrating the magnetic field in a closed loop around the measured current. according to ampere’s circuital law, the electric current passing through the loop is determined through the integrated magnetic field around this closed loop 0 1 i bdl   , where 0 is the vacuum permeability. if individual turns of the rogowski loop are small in comparison with the overall size of the coil, then the magnetic field b remains almost unchanged along this turn and the flux measured per unit length of coil is given by d nsbdl  , where n is the number of turns per unit length of the coil, s is the cross-sectional area of the coil. then the total magnetic flux penetrating rogowski's loop is equal to the value ns b dl   and the measured plasma current is determined by the equation: 0 0 0 0 0 ( ) 1 ( ) ( ) ( ). t sum sum t t i t u d i t ns ns          (4.5) the magnetic diagnostics is the main function of the measuring complex of any tokamak. consideration of other types of diagnosis goes beyond the scope of this review. we note only their great diversity, which arises, first of all, because of the features of the high-temperature plasma emitting in a wide spectrum of wavelengths: from the range of visible wavelengths to the hard x-rays, and also because of the need to control not only the integral parameters of the plasma, but also local values of physical quantities inside the plasma. 5. plasma models 5.1. plasma evolution modelling the mhd plasma equilibrium in a tokamak is usually described by the poloidal flux function ψ proportional at the point p to the magnetic induction flux through the circle s perpendicular to the z axis and passing through the point p (fig. 5.1). knowing the poloidal flux distribution, the plasma boundary can be found as the largest closed constant level line of the poloidal flux inside the tokamak vessel, and in accordance with the gauss theorem div 0b  the poloidal magnetic field is expressed as ψpb    . plasma control in tokamaks. part 1. 35 copyright ©2018 assa. adv. in systems science and appl. (2018) fig. 5.1. cylindrical , ,r z and toroidal coordinates , ,   in a tokamak from the ampere’s law 0curl b j  , where j is the plasma current density vector, μ0 is the vacuum permeability, the force balance equation j b p  and the axial symmetry of the tokamak the grad-shafranov equation describing the plasma equilibrium is derived [2, 4, 6]. in cylindrical coordinates it has a form: 2 2 02 1 , dp df r r f r r r z d d              (5.1) where p is the plasma pressure, and function ( )f  is related to the magnetic toroidal field as ( )f rb  . in the limit of the infinite plasma conductivity, the contour lines of the poloidal flux are "frozen" into the plasma, and the value of the poloidal flow on the plasma boundary does not change. however, the plasma resistance is not small enough to be neglected during the tokamak discharge, and the mhd equilibrium of the plasma undergoes evolution in accordance with ohm's law. the projection of the ohm’s differential law onto magnetic lines in the plasma is called the magnetic diffusion equation. in the limit of the large aspect ratio and the circular cross section of the plasma, it has the form [36]: 0 1 ,nirj t             (5.2) where  is the conductivity of the plasma along the magnetic lines, and nij is the density of non-ohmic currents, including bootstrap current and currents created by the auxiliary heating systems. in order to calculate conductivity and non-ohmical currents for the magnetic diffusion equation, pressure and density profiles of the electrons and plasma ions are needed. they are found by solving the transport equations for plasma particles and energy in the plasma [37]:   , 3 5 , 2 2 n nu s t p q p u p t                       (5.3) where the index α ,i e indicates the type of particles (ions or electrons), αn is the particle density, αs is the particle sources, αq is the heat flux, αu is the particles velocity, αp is the power of the energy sources, including ohmic and auxiliary heating, including the energy exchange between electrons and ions, as well as the radiation losses. these quantities are taken from theoretical or (since there is currently no complete theory of transport processes in a tokamak) from empirical models, depending on the mode of tokamak operation. 36 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov copyright ©2018 assa adv. in systems science and appl. (2018) 5.2. nonlinear plasma evolution codes 5.2.1. free boundary codes to solve the plasma evolution equations, a number of plasma-physical codes have been developed. they consist of two groups. the first group contains codes that model plasma with a free boundary. for example, the non-linear dynamic plasmophysical tokamak plasma model was developed in the state reserch center of rf "troitsk institute for innovation & fusion research (triniti)" (troitsk). the model is based on a two-dimensional equilibrium of a plasma with a free boundary in the external magnetic field, the transport and magnetic diffusion equations, averaged over magnetic surfaces, and a system of active coils of the poloidal field, and a passive stabilization structure. the model is numerically implemented in the dina code [37]. in different countries and for different tokamaks, other nonlinear codes with a free plasma boundary were created, differing from the dina code by various features (a nonzero plasma mass, different numerical methods, etc.): tsc (tokamak simulation code, usa) [38], jetto (italy) [39], corsica (usa) [40], uedge (usa) [41], pet (russia) [42], scope (russia) [4], edge2d (england) [43], maxfea (italy) [44], b2-irene (germany) [45], parasol (japan) [46], and others. 5.2.2. fixed boundary codes the second group contains codes that model plasma with a fixed boundary specified by the user. these include many codes created to model the transport processes in the plasma. they are astra (russia) [47], cronos (france) [48], transp and ptransp [49, 50], onetwo [51], and baldur [52] (all usa). transport codes can be used in conjunction with free boundary plasma codes, forming a hybrid transport-magnetic model. for example, the dina-ch code has been combined with the cronos code [53], and the tsc code [38] has been combined with the code ptransp [50]. 5.3. linearization of nonlinear plasma models to solve the problems of controlling the plasma shape, current, and position in a tokamak it is generally required to keep the plasma near the desired equilibrium position. the smallness of the deviations from the equilibrium, which is provided by feedback control systems, allows to describe the plasma by linear models. there are different approaches to designing control systems for linear models. the basic equation for these problems is the nonlinear differential kirchhoff's equation describing the interconnected magnetic system of conductors with currents, including the plasma [54, 55]:   , , , , , , 0 m m n m с с n m m ri nu n i r u r n m                  (5.4) where  is the vector of poloidal fluxes averaged over the cross sections of the conducting elements and penetrating them in vertical planes, ci is the vector of currents in the active poloidal field coils and in the passive elements, r is the impedance matrix of the conducting elements, u is the vector of external voltages applied to the active poloidal field coils, u is the input of the controlled plant, m m is the unit matrix of the size m m ,   0 n m m  is the zero matrix of the size  n m m  . nonzero values in the matrix n correspond to the voltages applied to the active poloidal field coils, while the zero values correspond to zero voltages applied to the passive elements. the linearized kirchhoff’s equation for small deviations from the equilibrium position has the form: plasma control in tokamaks. part 1. 37 copyright ©2018 assa. adv. in systems science and appl. (2018) .с p p с с p p i i r r i n u i i r                    (5.5) where ip is the total plasma current, pr is the vector of plasma magnetic axis coordinates, ξ is the vector of parameters describing the current profile such as the ratio of the gas kinetic pressure to the pressure of the poloidal magnetic field βp and the internal plasma inductance il : t p il     . the form of the poloidal plasma cross section is conveniently described as follows:  , , , ,c p ph h i i r  (5.6) where iс is the vector of currents in the active poloidal field coils and in the passive elements, ip is the total plasma current, pr is the position of the magnetic axis and ξ is the perturbation vector. using the linearization of this equation, we obtain: .с p p с p p h h h h y h i i r i i r                   (5.7) deviation of the plasma current δ pi can be described by the linearized kirchhoff’s equation for the plasma or expressed in terms of deviations δi and δξ taking into account the physical condition that magnetic field line freezing-in into the plasma: 1 const p s j ds i     , where j is the plasma current density and hence: 0.с p p с p p i i r i i r                 (5.8) similarly, the displacement of the plasma position δ pr can be described by the motion equation pmr f , or, neglecting the mass of the plasma, is linearly expressed based on the forces f acting on the plasma balance equation: 0.p с p p с p f f f f r i i r i i                 (5.9) the resulting system of equations of the linear plasma model in the state space representation has the form: , , i a i b u e y c i f            (5.10) where а, в, c, e and f are jacobi matrices. the state vector δi includes current deviations in the active and passive tokamak elements and it could also include the deviation of the total plasma current δ pi , the displacement of the plasma position δ pr and its velocity δ pr . if we exclude the derivative δξ from the first equation of the given system by changing the variables x i e    , then the equations of the linear model, taking into account the notation δu u , take the standard form of the linear dynamical system representation. since the vector of the external perturbation is presented in both equations, it allows simulating "the minor disruptions" in the plasma of tokamaks: , ( ) . x ax bu ae y cx ce f         (5.11) we emphasize that in the model there is no static connection between the control vector u and the output vector y in terms of the abcd representation of the linear dynamical system: the matrix d is zero. through the matrices ae and ce + f, the perturbation δξ turns into the 38 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov copyright ©2018 assa adv. in systems science and appl. (2018) state and output equations of the plant model. the control action u is included only into the state differential equation and is not included in the output equation. physically, it is explained by the fact that the current in the plasma and its magnetic configuration cannot change instantaneously under the influence of control signals, since it takes some time for the electromotive force to penetrate into the plasma, all this creates the dynamics of the propagation of the control action into the plasma. mathematically, the description of this process reduces to solving a two-dimensional partial differential equation for the voltage being induced in plasma [4]. 5.4. plasma equilibrium reconstruction plasma evolution codes can calculate distributions of the current density and poloidal flux for any point of time during the modelled discharge. however, in the experiments these distributions are unknown, so plasma shape and the profiles of plasma parameters are to be identified from the signals of the tokamak diagnostic system. this inverse problem is called the plasma equilibrium reconstruction problem [35]. the plasma equilibrium reconstruction problem is to find the distribution of the poloidal flux ψ , the plasma region s and the plasma toroidal current density distribution j in it, satisfying the grad-shafranov equation: 2 0 2 0 , ( , ) ,1 0, ( , ) , 1 ( ) ( ) ( ) rj r z s r r r r z r z s j rp f f r                    (5.12) with boundary conditions 0 0 r r     and consistent with the indications of the available magnetic diagnostics, which usually include the total plasma current p s jds i , measurements of the poloidal flux in the set of n points outside the plasma ( ) , 1,...,i ir i n    , and the poloidal magnetic field in the set of m points outside the plasma ( ) ( ) , 1,...,j j pjr r b j m    . additional restrictions on the current density distribution, based on the measurements of kinetic parameters and safety factor profile q are possible. one of the most common approaches to solving the equilibrium reconstruction problem is to use the method based on the picard iterations with linear parametrization of the plasma current density [56-58]. in this method, at each iteration the functions 'p and 'ff from the grad-shafranov equation, are approximated as linear combinations of basis functions depending on the poloidal flux values of the previous iteration: 1 1 1( ), ( ).n k k n n n l n k l p a f f b          (5.13) here, coefficients ka and lb are obtained at each iteration through minimization of the error functional, which may be, for example, quadratic functional of the errors between the calculated and measured values of the total plasma current, the poloidal flux, and the poloidal magnetic field. since the plasma equilibrium reconstruction problem is ill-posed in the sense of hadamard, the regularization techniques are used, for example, tikhonov regularization or svd truncation [59]. this procedure is repeated at each iteration until the convergence of the solution is achieved. the described algorithm is computationally expensive, and simplifications are required for its application in real-time tokamak experiments. possible simplification is truncation to the single iteration step and using the distributions calculated at the previous time point as initial values. another possible modification is the representation of the plasma in the form of a set of filaments [60], which are concentric ideal conductors with currents that approximate the plasma control in tokamaks. part 1. 39 copyright ©2018 assa. adv. in systems science and appl. (2018) plasma current distribution of the plasma. in this case, the plasma current density is approximated as a linear combination of delta functions. filaments may be fixed in space, or may be moving, with the coordinates determined by the minimization of the error functional. a b c d fig. 5.2. magnetic configurations: a – magnetic surfaces in a tokamak; b – configuration with upper х-point in globus-m tokamak; c – definition of plasma elongation; d – definition of plasma triangularity the filament method does not allow reconstruction of the profiles of plasma parameters, but it can be used when it is sufficient to determine the plasma position and shape. algorithms that reconstruct the plasma shape but do not restore poloidal flux distribution are also suitable for such tasks. an example of such an algorithm is the xloc code [61], which approximates the poloidal flux in the vacuum region by the polynomial functions satisfying *ψ 0  but does not reconstruct the poloidal flux in the plasma region. the xloc algorithm determines plasma boundary, as a level line of the poloidal flux, containing the x-point characterized by ψ 0  . another possible solution for such problems is to use algorithms based on the regression analysis of a large amount of experimental data [62]. it is mathematically rigorously shown [4] that magnetic measurements alone are insufficient to determine the internal parameters of the plasma, such as the current and pressure profiles. being ill posed the problem of the equilibrium reconstruction, in general, may have several solutions that differ substantially in the internal plasma parameters and does not continuously depend on the input data. the method based on the use of ε-nets and implemented in the sdss code (substantially different solutions searcher, russia) [63-67] allows to overcome these difficulties. an equilibrium reconstructed by any other method, for example, one of the above, is used as an initial approximation in this method. then an ε-net of the right-hand sides of the gradshafranov equation is constructed in a neighborhood of this equilibrium, which allows one to approximate any equilibrium in the neighborhood with an error not exceeding  . the equilibria corresponding to the elements of the ε-net can be analyzed to find all essentially different solutions of the reconstruction problem and their errors estimations. 40 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov copyright ©2018 assa adv. in systems science and appl. (2018) in fig. 5.2, a magnetic surfaces (surfaces of the constant poloidal flux) in a tokamak plasma are shown, fig. 5.2, b shows equilibrium of the globus-m tokamak plasma, reconstructed with fcdi code [58]. fig. 5.2, c, d show the geometric parameters of the plasma separatrix. the major radius of the plasma is the arithmetic mean between the maximal and the minimal radial coordinate of the separatrix max min( ) 2r r r  , while the minor radius is their half-difference max min( ) 2a r r  , the plasma elongation is the ratio of the vertical and radial dimensions of the plasma max min max min( ) ( )k z z r r   , the upper triangularity of the plasma is the horizontal distance between the upper point of the separatrix and the major radius, divided by the minor radius  maxδ ( )up r z r a  , the lower triangularity is defined similarly  minδ ( )down r z r a  . 6. instabilities and disruptions the grad-shafranov equation describes the equilibrium plasma configuration in a tokamak, but it does not reflect equilibrium stability. the mhd stability studies consider evolution of the plasma equilibrium under various perturbations  , for example, ( )( ) i t т nr e     (see fig. 5.1). perturbations of this form are called modes and characterized by the mode numbers m, n and frequency  . to ensure the equilibrium stability it is necessary for all possible modes to be damped or have characteristic times much longer than the discharge duration. stability analysis is an extremely complex problem, which cannot be solved analytically in many cases. however, it is necessary to solve this problem, since plasma instabilities can lead to plasma disruptions, during which magnetic confinement is lost, and the plasma comes into contact with the walls of the tokamak, can lead to accidents and may damage the device. for example, during disruptions in the jet tokamak there were large mechanical forces acting on the wall, reaching values of several hundred kilo-newtons [2]. since plasma disruptions cannot be allowed to occur in an operating thermonuclear reactor, research of their causes is of crucial importance. in order to learn how to suppress plasma disruptions their models are being constructed from experimental data and plasma interactions with the tokamak walls are studied [69]. in these processes the halo-currents flowing between the plasma and tokamak walls seem to play an important role [2, 69]. plasma instabilities impose limitations on the operating parameters of the tokamak. it is because of mhd instabilities that strong magnetic fields are necessary for the tokamak operation. in order to suppress the instabilities the ratio β of the kinetic plasma pressure p to the magnetic field pressure 2 0/ 2b  is required to have values of the order of ~ 10–2 or less. after analysis of the large amount of experimental data of various tokamaks an empirical limit for the value of β in stable equilibrium was calculated [70]: max 2 0 max ma 0.028 , / 2 p i b ab m t              (6.1) where a is the minor plasma radius. this condition also limits the allowable plasma pressure in the tokamak. an important parameter describing plasma stability is the safety factor q [2], defined for a magnetic surface l as 1 ( ) 2 pl b q l dl rb     where the integration is performed over the poloidal section of the surface, bφ and bp are toroidal and poloidal magnetic fields. in the limit of the large aspect ratio (r/a) and the circular plasma cross-section, the profile ( )q  is plasma control in tokamaks. part 1. 41 copyright ©2018 assa. adv. in systems science and appl. (2018) expressed as 2 0 2 ( ) ( ) b q ri      where (ρ)i is the plasma current profile. thus, the restrictions on the profile q are lead to the restrictions on the current profile. the stability analysis of the mode 1m  leads to the crucial kruskal-shafranov stability limit [70, 71]     ( ) 1. ab a q a rb a     (6.2) the limit leads to the necessity of exceeding the ratio of the toroidal  b a and poloidal  b a components of the magnetic field over the aspect ratio r/a. this is equivalent to limiting the possible total plasma current for a given value of toroidal magnetic field: 2 0 2 .p a b i r    (6.3) the stability conditions can depend not only on the total plasma current, but also on the shape of the current profile. for example, a plasma with a flat profile and with an abrupt drop at the plasma edge is unstable with respect to all 1m  modes, therefore a small current gradient is desirable at the plasma boundary. the presence of a surface with a safety factor 2q  near the separatrix leads to excitation of the tearing modes associated with the finite conductivity of the plasma, which lead, in turn, to the tearing and the reconnection of the magnetic lines resulting in the formation of "magnetic islands." this phenomenon in turn enhances plasma transport processes and leads to the plasma disruption. therefore, in practice, the safety factor restriction is strengthened to ( ) 3q a  . as the plasma is heated, its conductivity and the current density at the center of the plasma also increase, which leads to a decrease of the safety factor (0)q . when a value (0) 1q  is reached, the unstable mode m = 1 is excited, which leads to a sudden drop of the density and temperature at the center of the plasma after which the plasma heating starts again. such temperature and density oscillations are called sawtooth oscillations. the physics of sawtooth oscilations is not completely understood, but it is known that the amplitude and period of the oscillations are influenced by the presence of fast ions and the plasma pressure, as well as the steepness of the q profile. sawtooth oscillations do not lead to the plasma disruptions and are considered as part of the normal tokamak operation, but they can negatively affect the stability of other modes. for this reason, some tokamaks have sawtooth oscillations control systems, which use neutral particles injectors and radio-frequencies heating to regulate amplitude of the oscillations. the plasma density limit is also related to the safety factor restrictions. as the density increases, the transport processes associated with the collision frequency are accelerated, which leads to the increasing of energy loss at the plasma boundary due to radiation and decreasing of the boundary layer temperature. as the temperature decreases, the resistance of the boundary layer increases, and the layer ceases to conduct current, diminishing the effective plasma volume, and reducing the safety factor q at the effective boundary, which causes subsequent plasma disruption. to maintain stability the plasma density n should not exceed the density limit established by greenwald [72]: 20 3 2 2 ma [10 m ] . m pi n a         (6.4) for the problems with the fixed plasma density n this condition may be interpreted as a restriction on the minimal total plasma current ip. overall mhd perturbations have a significant effect on the plasma discharges and are one of the primary causes of plasma confinement deterioration and abrupt ending of a discharge. for this reason, the search for the discharge scenarios free from the large-scale mhd 42 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov copyright ©2018 assa adv. in systems science and appl. (2018) instabilities is an urgent problem in modern tokamaks. in addition to creating relatively stable magnetic configurations of a plasma column, instability suppression active methods are also employed. for example, auxiliary heating at electron cyclotron resonance frequencies can be used to control the pressure profile, preventing the development of unstable modes. in the tokamak research, much attention is being paid to the search for plasma configurations with increased pressure, temperature, density, and energy confinement time, allowing to approach the lawson’s criterion for plasma ignition. however, such configurations are often subject to new instabilities. for example, usually on modern tokamaks the vertically elongated plasma is studied. in comparison with the circular configuration, the elongated plasma makes it possible to achieve a much higher pressure for the same magnetic field strength. to elongate the plasma vertically, a horizontal magnetic field, directed in opposite directions in upper and lower parts of the plasma is created (fig. 6.1). however, in this configuration, small plasma displacements along the vertical are enhanced by the ampere’s forces rj b , and the plasma is vertically unstable. therefore, a feedback control system that regulates currents in horizontal field coils and stabilizes the position of the plasma is necessary for the operation of modern tokamaks. fig. 6.1. illustration of the vertical instability of the elongated plasma another promising tokamak plasma configuration is the so-called h-mode (highconfinement), which is created by intense auxiliary heating and is characterized by a significantly increased energy confinement time. however, this mode is subject to the elminstabilities (edge localized modes). these modes, as a rule, have ~10, ~ 30n m , lead to periodic eruptions of plasma energy, cause confinement deterioration, and may damage the tokamak vessel. heating of the plasma edge region and axial asymmetric magnetic fields are applied to control amplitude and frequency of the elm outbursts. 7. actuators and their models of plasma magnetic control systems the plasma magnetic sensors and diagnostics are connected via controllers to power actuators (supplies) to control plasma position, current, and shape. these actuators connected to the magnetic field coils in the feedback control system are as follows: multiphase thyristor rectifier (fig. 7.1, a) [73–74], power transistor voltage inverter (fig. 7.1, b) [75–77], and thyristor current inverter (fig. 7.1, c) [78]. thyristor rectifiers are relatively slow so usually they are used for the toroidal field coil supply as well as for plasma current and shape control [79, 80]. in some cases, thyristor rectifiers are applied for plasma position control together with other thyristor rectifies of poloidal field coils, for example on asdex upgrade [33]. the thyristor rectifier model used for linear controller synthesis is approximated with sufficient accuracy by the transfer function of the first order with transport delay  –     ( ) ( )/    1 tsw s k e ts  , where k is the gain and the time constant t depends on particular rectifier properties [79]. plasma control in tokamaks. part 1. 43 copyright ©2018 assa. adv. in systems science and appl. (2018) a b c fig. 7.1. power parts of actuators: a ‒ multiphase thyristor rectifier; b ‒ transistor voltage invertor; c ‒ thyristor current inverter the voltage and current inverters have higher speed of response in comparison with thyristor rectifiers and are used to supply poloidal field coils or separate coils of vertical and horizon field to stabilize plasma horizon and unstable vertical position respectively [78, 80]. the models of the actuators are complex and nonlinear. however, for the controller synthesis purpose a simple gain coefficient approximation is often enough [80, 82, 83] because the actuator nonlinear dynamics is faster than the controlled processes. the voltage inverters can work in pulse-width modulation mode [76, 77] or relay mode providing sliding regime or selfoscillation processes [78, 83] in a closed-loop control system. 8. spherical tokamak globus-m 8.1. globus-m tokamak structure the spherical tokamak (st) globus-m is a new generation st that has been built for studying the physical processes in a plasma of the spherical configuration and generating engineering recommendations for sts of a ma diapason [84]. the main st globus-m parameters are as follows: plasma current 0.3ma, toroidal magnetic field bt ≤ 0.4t; major and minor radiuses are r = 0.36 m and a = 0.24 m, respectively, that correspond to the aspect ratio r/a = 1.5. the plasma elongation k upper limit is 2.2, while triangularity  is 0.4. the electro-magnetic tokamak system has been constructed according to the conventional scheme, when all the coils are located outside the vacuum vessel, and the poloidal magnetic field coils are kept outside the toroidal ones (fig. 8.1). 44 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov copyright ©2018 assa adv. in systems science and appl. (2018) 6 5 7 4 8 3 2 11 12 1 10 9 fig. 8.1. globus-m st: 1 – support structure; 2 – toroidal coils separators; 3 – poloidal coils binding; 4 – central solenoid; 5 – bandage ring; 6 – wedge clamp; 7 – toroidal field coils; 8 – “slow” poloidal field coils; 9 – “fast” control coils (horizontal field); 10 – “fast” control coils (vertical field); 11 – vacuum vessel; 12 – compensation coils sixteen single turn d-shaped coils, constructed from the electro-technical copper, are series-connected and form the toroidal field coil. the coil parts that go through the inner cylinder of the vacuum vessel are constructed as 16 isolated bronze segments. a coil used to generate a curl electric field, i.e. a central solenoid is located at the central kernel. this solenoid is created from the copper-silver alloy and has a form of a two-layer coil for which the total height is 1.3 m. nine pairs of the poloidal coils can be divided into three functional groups, which are scattered field compensation coils, “slow” and “fast” coils to trace the position and shape of plasma. a poloidal coil system allows to create both, single or double-nulled limiter and divertor configurations. a small distance (around 2-3 cm) between the outer magnetic surface of the plasma column and the first wall of the vacuum vessel is very attractive. this is a key feature that differs the globus-m st from other sts. application of the tokamak structure when the plasma fits the whole volume of the vacuum vessel has a range of benefits. the tokamak poloidal magnetic field coils can be set very close to the plasma itself without the necessity of putting them into the vacuum vessel. this, in turn, reduces a volume of created external magnetic field along with a reduction in a power used to create it. moreover, the coil structure itself becomes simpler, as there are fewer requirements to provide a heat removal compared to the coils in the vacuum. in addition, there are a number of benefits used in a diagnostic system design. for example, the magnetic sensors can be located close to the external magnetic surface of the plasma, the wider angle of view for the plasma observation may be realized via diagnostic sockets. there are some disadvantages associated with the globus-m construction specifically a higher load on the first wall of the tokamak due to a small area of the vessel compared to the plasma volume; also, much stronger requirements to the plasma control system (small transition process time, power supply speed of response in the control contour, and accuracy). the coils of the tokamak are fed from six-phase thyristor rectifiers and fast thyristor current inverters with frequency up to 3 khz connected to the high-voltage ac line of 110 kv through power transformers. the power supplies are enveloped by feedback loops, which allow to plasma control in tokamaks. part 1. 45 copyright ©2018 assa. adv. in systems science and appl. (2018) maintain plasma position, shape, and current at their references. the cooling system of the coils provides the tokamak operation at the frequency of six discharges per an hour with duration of a discharge about 0.3 s and the maximal achieved plasma current of 0.36 ma. the working plasma current range is 180–250 ka. the plasma current is maintained by the flux change of the poloidal coils with average volt-second area of 0.33 vs. the central solenoid contribution to the average magnetic flux is 90%. the globus-m vacuum vessel is an all-welded stainless steel construction and has the volume of about 1.1 m3. there are 38 diagnostic sockets having an overall area of 0.8 m2, which provide good access to plasma for diagnostics and additional heating. the vacuum vessel is welded from an inner cylinder with a wall thickness of 2 mm, two hemispheres with a wall thickness of 3 mm and a thick outer ring with a thickness of 14 mm. the total resistance of the vessel is 120 µohm in the toroidal direction. about 50% of the current flows through the inner cylinder, 35% through the hemispheres and only 15% through the outer ring. such a small current through the external ring is explained by the presence of sockets. in fact, the current flows in two rings with a height of about 4 cm. the construction of the sockets allows to inject additional heating power into the plasma up to 4 mw (fundamental and high harmonics of ion-cyclotron frequency [85], injection of a neutral atom beam [86]). the main part of the plasma-facing surface, which is more exposed to plasma flows, is protected by the tiles made of a special type of graphite rg-ti91. the details of construction of the tokamak are also described in [87]. 8.2. diagnostic systems of the globus-m tokamak the tokamak diagnostics complex used in the experiments consists from a large number of monitor diagnostic systems working permanently. these are current and voltage sensors in a tokamak magnetic system, the series of loops and probes for the magnetic measurements, uhf-interferometer, optic (collimated and survey sensors of luminosity lines of hydrogen and deuterium, sensors of light impurities radiation, survey spectrometers) and x-ray detectors (soft and hard radiation), bolometers, langmuir probes, etc. [88, 89]. depending on the experimental aims, systems that are more complicated can be connected. some of them require an operator action; others transfer a large amount of extra data into the database of the installation. they are thomson scattering diagnostic [90], charge exchange recombination systems, fast optical band video capturing. the scheme in fig. 8.2 shows the location of the main diagnostic units and general tokamak systems. reconstruction of the plasma magnetic configuration is based on the magnetic coils measurements [91] (21 in total on globus-m) located on the tokamak discharge vessel surface to measure the poloidal flux. the efit code [56] performs this reconstruction in the off-line mode. the uhf-interferometer measures the plasma density in a monitoring mode with the following settings: wavelength is 0.8 mm; vertical chords are located on 24, 42, and 50 cm from the tokamak axis. the apparatus allows reasonably measuring a linear density in a range till 0.6×1020 m–2 that corresponds to the averaged plasma density around 1.0×1020 m–3. thomson scattering (ts) [90] is used to measure the plasma electron temperature and density profiles. measurements are taken along the minor radius from the inner side of the plasma column. a total number of spatial points is 10. the ts makes it possible to achieve up to 20 measurements per discharge for the pre-defined time points. a minimal interval between the neighboring measurements is not more than 500 µs. the ion temperature is determined using the cxrs (charge exchange recombination spectroscopy) with the help of the akord12 analyzer [89]. its observation line is directed along the tokamak major radius. this analyzer can measure the neutral hydrogen and deuterium flows at the same time by 6 energetic channels for each isotope. a minimal time resolution is 1 ms. the observation line of the second similar analyzer is toroidally oriented. 46 y.v. mitrishkin, p.s korenev, a.a. prokhorov, n.m. kartsev, m.i. patrov copyright ©2018 assa adv. in systems science and appl. (2018) fig. 8.2. the scheme of location of general globus-m tokamak systems and diagnostics: view from the top; poloidal magnetic system is not showed the key feature of the globus-m st in comparison with the other spherical machines is an extremely high chord-averaged density in a plasma column. it varies from 1.4 to 1.8 ma /m2 in different operating modes. thus, the ohmic mode becomes more effective because of the specific heating power is proportional to the squared plasma current density. the high current density as well as a high ratio bt/r that reaches 1.8 t/m permit obtaining discharges with high averaged plasma density limit around 1020 m–3 even in the ohmic mode discharges. 9. conclusion the survey outlines different features of modern vertically elongated tokamaks as plants under control through examples of some present tokamaks and iter, which is currently under construction. the report from developers of the first thermonuclear power plant demo [18] shows that the project is realizable and time limit of construction is determined. described state of the art of the iter and demo projects defines the roadmap to final solution of controllable thermonuclear fusion problem in the foreseeable future. the main differences between present tokamaks are as follows: toroidal field coil placement inside or outside the poloidal field coils, putting the plasma position control coils inside or outside the vacuum vessel, superconductivity property of magnet coils, and so on. superconductive coils are proposed to use in the demo project, but its poloidal system structure is still in the phase of research. considered problems of the plasma diagnostics and design of tokamak actuators show an additional complexity of plasma control systems development, for synthesis and analysis of these systems linear and nonlinear plasma models presented are necessary. nowadays, main tokamak plasma instabilities are suppressed. the most dangerous phenomenon for future thermonuclear power plants are so called “major disruptions” [2, 69, 92] which must not occur in tokamaks-reactors. this phenomenon is being studied and the ways not to admit this effect are under research. these ways are right assignment of profiles plasma control in tokamaks. part 1. 47 copyright ©2018 assa. adv. in systems science and appl. (2018) of plasma kinetic parameters by means of plasma kinetic control systems and their optimal interaction with the plasma magnetic control systems. acknowledgements this work was supported by the russian science foundation (rsf), grant no 17-19-01022 (sections 1-7) and the russian foundation for basic research (rfbr), grant no 17-08-00293 (section 8). references 1. artsimovich, l. a. (1972). tokamak devices. nuclear fusion, 12(2), 215-252, https://doi.org/10.1088/0029-5515/12/2/012 2. wesson, j. (2004). tokamaks: 3rd ed. oxford: clarendon press. 3. dnestrovskii, yu. n. & kostomarov, d. p. (1993). matematicheskoe modelirovanie plazmy: 2nd ed. [mathematical modeling of plasma]. moscow, fismatlit, [in russian]. 4. zaitsev, f.s. (2014) mathematical modeling of toroidal plasma evolution. moscow, maks press. 5. samoilenko, yu. i., gubarev, v. f. & krivonos, yu. g. 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(2016). v.d. shafranov and tokamaks. journal of plasma physics, 82, 515820102, https://doi.org/10.1017/s0022377816000027 https://doi.org/10.1088/0029-5515/53/9/093013 https://doi.org/10.1016/j.fusengdes.2016.04.026 https://inis.iaea.org/search/search.aspx?orig_q=rn:39119971 https://inis.iaea.org/search/search.aspx?orig_q=rn:39119971 https://doi.org/10.1080/10519990108224485 https://doi.org/10.1080/10519990108224492 https://doi.org/10.1088/0029-5515/55/10/104016 https://doi.org/10.1017/s0022377816000027 adv syst sci appl 2021; 02:133–141 published online at https://ijassa.ipu.ru. the modular inequalities for hardy-type operators on monotone functions in orlicz space kh. almohammad1* 1 mathematical institute named s.m. nikolskii peoples’ friendship university of russia (rudn university) moscow 117198, miklukho-maklaya str. 6, russia abstract: this paper is devoted to the study of modular inequality for general hardy–copson type operator restricted on the cone of monotone functions from weighted orlicz space with general weight. keywords: modular inequalities, norm inequalities, orlicz space, cone of decreasing functions, positively homogeneous operators 1. introduction in this paper, we consider modular inequalities for hardy-type operators on the cone ω of positive decreasing functions from weighted orlicz spaces. we use a general theorem (proved in [10]) on the reduction of modular inequalities for positively homogeneous operators on the cone ω, which enables passing to modular inequalities for modified operators on the cone of all positive functions from orlicz space. it is based on the duality theorem describing the associated norm for the cone ω. we follow, mostly, the notation used in the book [2, sec. 8, chap. 4] of bennett and sharpley. in the paper, we concretize modular inequalities for the case in which the positive operator is a hardy-type operator. it is shown that, in that case, the modified operator is a generalized hardy operator in the jim quile sun notation [1]. this allows us to use approaches developed in [8–10], as well as results obtained by jim quile sun [1] to establish the explicit criteria for the validity of modular inequalities. 2. auxiliary definitions definition 2.1: (i) a banach function space, shortly bfs, e = e(rn) is a banach space of lebesgue measurable functions f : rn → c with monotone norm, i.e. such that |f | ≤ g, g ∈ e implies f ∈ e, ‖f‖e ≤ ‖g‖e. (2.1) (ii) a bfs e is called a rearrangement-invariant space, shortly: ris, if its norm is monotone with respect to rearrangements, f ∗ ≤ g∗, g ∈ e implies f ∈ e, ‖f‖e ≤ ‖g‖e. (2.2) ∗corresponding author: khaleel.almahamad1985@gmail.com 134 kh. almohammad here f ∗ is the decreasing rearrangement of the function, i.e. a positive decreasing right continuous function on r+ = (0,∞), which is equimeasurable with f : µn { x ∈ rn : |f(x)| > y } = µ1 { t ∈ r+ : |f ∗(t)| > y } , y ∈ r+. (2.3) where µn is n-dimensional lebesgue measure. definition 2.2: the potential space hg e (rn) on the n-dimensional euclidean space rn is defined by hg e (rn) = { u = g ∗ f : f ∈ e(rn) } , (2.4) where e(rn) — is a rearrangement-invariant space (shortly: ris), and ‖u‖hg e = inf { ‖f‖e : f ∈ e(rn), g ∗ f = u } . (2.5) here g is an admissible kernel, that is g ∈ l1(rn) + e ′(rn), the convolution g ∗ f is defined as the integral (g ∗ f)(x) = (2π)− n 2 ∫ rn g(x− y)f(y) dy. (2.6) here e ′ = e ′(rn) is the associated ris, i.e. ris with the norm: ‖g‖e′ = sup {∫ rn |fg| dµ : f ∈ e, ‖f‖e ≤ 1 } . (2.7) examples. e = lp, 1 ≤ p ≤ ∞⇒ e ′ = lp′ ; 1 p + 1 p′ = 1. l′1 = l∞; l′∞ = l1. for the ris e(rn), e ′(rn), we consider the spaces ẽ = ẽ(r+), ẽ ′ = ẽ ′(r+) — their luxemburg representations [2], i.e. ris for which the following equalities are satisfied ‖f‖e = ‖f ∗‖ẽ, ‖g‖e′ = ‖g∗‖ẽ′ . we denote: f ∗∗(t) = 1 t t∫ 0 f ∗(τ) dτ ; t ∈ r+. (2.8) we introduce the class of monotone functions in(r), r > 0 as follows. the function θ : (0, r)→ r+ belongs to the class in(r), if θ satisfies the following conditions: decreasing and continuous at (0, r); there is a constant c ∈ r+, such that r∫ 0 θ(ρ)ρn−1 dρ ≤ cθ(r)rn, r ∈ (0, r). (2.9) now, we introduce ϕ(τ) = θ (( τ vn ) 1 n ) ∈ i1(t ), t = vnr n. copyright © 2021 assa. adv syst sci appl (2021) the modular inequalities for hardy-type operators 135 where vn is the volume of the unit ball in rn. fϕ(t; τ) = min { ϕ(t), ϕ(τ) } = { ϕ(t), t > τ, ϕ(τ), τ > t. (2.10) definition 2.3: let θ ∈ in(∞). the potentials u ∈ hg e (rn) are called generalized riesz potentials, if g(x) ∼= θ(|x|), x ∈ rn, (∼= means two-sided estimate). definition 2.4: let θ ∈ in(r). the potentials u ∈ hg e (rn) are called generalized bessel potentials, if g(x) = g0 r(x) +g1 r(x); br = { x ∈ rn : |x| < r } , r ∈ r+, g1 r(x) = g(x)χbc r (x), g0 r(x) = g(x)χbr (x), g0 r(x) ∼= θ(|x|), x ∈ br, g1 r(x) ∈ (l1 ∩ e ′)(rn), ∫ rn gdx 6= 0. definition 2.5: function φ: [0,+∞)→ [0,+∞) is called n -function if φ(t) = t∫ 0 φ(τ) dτ ; where φ is continuous, 0 < φ ↑; φ(0) = 0, φ(∞) =∞. let φ−1 be the right continuous inverse function of φ, and define ψ(t) = t∫ 0 φ−1(τ) dτ. ψ is called the complementary function of φ. definition 2.6: a) ann -function φ is said to satisfy the ∆2-condition (we write φ ∈ ∆2) if there is a constant b > 0, such that φ(2t) ≤ bφ(t), ∀t > 0. (2.11) b) we write φ1 ≺≺ φ2 if there is a constant l0 > 0, such that inequality∑ i φ2 ◦ φ−1 1 (ai) ≤ l0φ2 ◦ φ−1 1 ( ∑ i ai), (2.12) holds for every sequence {ai} with ai ≥ 0. c) let v be a positive, measurable weight function and φ be an n -function. the orlicz space lφ,v consists of all measurable function f (modulo equivalence almost everywhere) with ‖f‖φ,v = inf { λ > 0, ∫ ∞ 0 φ ( λ−1|f(x)| ) v (x) dx ≤ 1 } <∞. (2.13) we call ‖ · ‖φ,v the luxemburg norm. the orlicz norm of a function f is given by ‖f‖′ψ,v = sup {∫ ∞ 0 |fg|v dx : ∫ ∞ 0 ψ(|g|)v dx ≤ 1 } . (2.14) copyright © 2021 assa. adv syst sci appl (2021) 136 kh. almohammad remark 2.1: lφ,v is a banach space and the luxemburg and orlicz norms are equivalent . in fact, ‖f‖φ,v ≤ ‖f‖′ψ,v ≤ 2‖f‖φ,v. we assumem(r+) is the set of lebesgue-measurable almost everywhere finite functions,m+ is the cone of almost everywhere positive functions from m = m(r+); m+ = { f ∈m(r+) : f > 0 } . consider the cone of positive decreasing functions from the orlicz space: ω = { f ∈ lφ,v : 0 ≤ f ↓} (2.15) for g ∈m+, we introduce the following associated norm on the cone ω: ‖g‖′ω = sup {∫ ∞ 0 fg dt : f ∈ ω; ‖f‖φ,v ≤ 1 } . (2.16) we formulate the result that generalize some previous results of papers [3], [5–7]. proposition 2.1: ([4]). let φ, ψ be the complementary n -functions, the n -function φ satisfies ∆2-condition, let v ∈m+, and let 0 < v (t) := t∫ 0 v dτ <∞, t ∈ r+, v (+∞) = +∞. (2.17) the following two-sided estimate holds: ‖g‖′ω ∼= ‖r0(g)‖ψ,v = inf { λ > 0 : ∫ ∞ 0 ψ ( λ−1|r0(g; t)| ) v(t) dt ≤ 1 } , (2.18) where r0(g; t) := v (t)−1 t∫ 0 g(τ) dτ, t ∈ r+. (2.19) here and below we use the notation a ∼= b ⇔ ∃ c ∈ [1,∞) : c−1 ≤ a/b ≤ c. (2.20) in the following considerations, we will use the formula for the conjugate operator: r∗0(f ; τ) = ∞∫ τ f(t) v (t) dt, τ ∈ r+. (2.21) let us now state the main result of this section allowing us to reduce modular inequalities for operators on the cone ω to modular inequalities for modified operators on the cone m+. proposition 2.2: ( [10]). let t and t ∗ be positively homogeneous operators that map m+ to m+ and are copyright © 2021 assa. adv syst sci appl (2021) the modular inequalities for hardy-type operators 137 adjoint, i.e., ∫ r+ gtf dτ = ∫ r+ ft ∗g dτ, f, g ∈m+. (2.22) let φ1, φ2 be n -functions, u, v, w ∈m+, and let condition (2.17) holds. let the operator r0 be given by formula (2.19). then the following inequalities are equivalent: ∃ c1 ∈ r+ : φ−1 2 {∫ r+ φ2(wtf)u dt } ≤ φ−1 1 {∫ r+ φ1(c1f)v dt } , f ∈ ω; (2.23) ∃ c3 ∈ r+ : φ−1 2 {∫ r+ φ2 ( wtr∗0(vf) ) u dt } ≤ φ−1 1 {∫ r+ φ1(c3f)v dt } , f ∈m+. (2.24) definition 2.7: the generalized hardy operators are operators of the form kf(x) = x∫ 0 k(x, t)f(t) dt, k∗g(t) = +∞∫ t k(x, t)g(x) dx, (2.25) where a) k : { (x, t) ∈ r2 : 0 < t < x < +∞ } → [0,+∞); b) k(x, t) ≥ 0 is nondecreasing in x, nonincreasing in t; c) k(x, y) ≤ d ( k(x, t) + k(t, y) ) , for some constant d, (2.26) whenever 0 ≤ y ≤ t < x < +∞ proposition 2.3: ( [1]). let φ1, φ2 be n -function and φ1 ≺≺ φ2, and k be a generalized hardy operator (2.25). let a, b, v and ω be positive weight functions. then there exists a constant a > 0 such that φ−1 2 ( +∞∫ 0 φ2(akf)ω dx ) ≤ φ−1 1 ( +∞∫ 0 φ1(afb dx)v ) for all positive, measurable functions f if and only there exists a constant c such that φ−1 2 ( +∞∫ r φ2 ( a(x) c ∥∥∥∥k(r; ·)χ(0,r)(·) εvb ∥∥∥∥ ψ1(εv) ) ω(x) dx ) ≤ φ−1 1 (1 ε ) and φ−1 2 ( +∞∫ r φ2 ( a(x) c ∥∥∥∥χ(0,r)(·) εvb ∥∥∥∥ ψ1(εv) k(x; r) ) ω(x) dx ) ≤ φ−1 1 (1 ε ) holds for ε, r > 0. copyright © 2021 assa. adv syst sci appl (2021) 138 kh. almohammad proposition 2.4: ( [1]). let φ1, φ2 be n -function and φ1 ≺≺ φ2, and k∗ be a generalized hardy operator (2.25). let a, b, v and ω be positive weight functions. then there exists a constant a > 0 such that φ−1 2 ( +∞∫ 0 φ2(ak∗f)ω dt ) ≤ φ−1 1 ( +∞∫ 0 φ1(abf)v dt ) holds for all positive, measurable functions f if and only there exists a constant c such that φ−1 2 ( r∫ 0 φ2 ( a(t) c ∥∥∥∥k(·; r)χ(r,+∞)(·) εvb ∥∥∥∥ ψ1(εv) ) ω(t) dt ) ≤ φ−1 1 (1 ε ) and φ−1 2 ( r∫ 0 φ2 ( a(t) c ∥∥∥∥χ(r,+∞)(·) εvb ∥∥∥∥ ψ1(εv) k(r; t) ) ω(t) dt ) ≤ φ−1 1 (1 ε ) holds for ε, r > 0. 3. applications for hardy-type operators let us now state the main result of this section allowing us to reduce modular inequalities for operators on the cone ω to modular inequalities for modified operators on the cone m+. i(f ; t) = +∞∫ 0 fϕ(t; τ)f(τ) dτ, τ ∈ r+, (3.27) where fϕ(t; τ) = min { ϕ(t), ϕ(τ) } . theorem 3.1: let φ1, φ2 be n -function and φ1 ≺≺ φ2, w, u, v be positive weight functions, i be hardytype operators (3.27). let the condition be satisfied aϕ = sup t∈r+ 1 tϕ(t) (∫ t 0 ϕ dτ ) <∞. (3.28) then there exists a constant c > 0 such that inequality φ−1 2 {∫ r+ φ2 ( w(t)if ) u(t) dt } ≤ φ−1 1 {∫ r+ φ1(cf)v dt } , f ∈ ω, (3.29) holds for all positive, nonincreasing functions f if and only if there is a constant b such that the following inequalities hold for all ε, r > 0: φ−1 2 { ∞∫ 0 φ2 ( w(t) b · fϕ(t, r) ϕ(r) ∥∥∥∥fϕ(·, r)(·) εv ∥∥∥∥ ψ1(εv) ) u(t) dt } ≤ φ−1 1 ( 1 ε ) . (3.30) copyright © 2021 assa. adv syst sci appl (2021) the modular inequalities for hardy-type operators 139 proof the purpose of the first step is to reduce estimate(2.24) to the estimate for the hardy-type operator given in [11]. for the hardy-type operator (3.27), by using (2.21), we obtain i ( r∗0(vf ; t) ) = ∞∫ 0 fϕ(t; τ)r∗0(vf ; τ) dτ = ∞∫ 0 fϕ(t; τ) ( ∞∫ τ f(ξ)v(ξ) v (ξ) dξ ) dτ, τ ∈ r+ . by changing the order of integration, we have i ( r∗0(vf ; t) ) = ∞∫ 0 f(ξ)v(ξ) v (ξ) ( ξ∫ 0 fϕ(t; τ) dτ ) dξ. (3.31) let us show that ξ∫ 0 fϕ(t; τ) dτ ∼= fϕ(t; ξ)ξ; t, ξ ∈ r+ = (0,+∞). (3.32) from the decrease of ϕ and from the condition aϕ <∞ it follows that ϕ(t)t ≤ t∫ 0 ϕ dτ ≤ aϕϕ(t)t, t ∈ r+. (3.33) 1) for ξ ≤ 1 we have fϕ(t; τ) = ϕ(t), τ ∈ (0, ξ), so that ξ∫ 0 fϕ(t; τ) dτ = ϕ(t) ξ∫ 0 dτ = ϕ(t)ξ = fϕ(t; ξ)ξ. (3.34) 2) for ξ > t we have ξ∫ 0 fϕ(t; τ) dτ = t∫ 0 ϕ(t) dτ + ξ∫ t ϕ(τ) dτ = = ϕ(t)t+ ξ∫ t ϕ(τ) dτ ∼= (3.34) t∫ 0 ϕ(τ) dτ + ξ∫ t ϕ(τ) dτ = ξ∫ 0 ϕ(τ) dτ ∼= (3.34) ϕ(ξ)ξ, that is, for ξ > t ξ∫ 0 fϕ(t; τ) dτ ∼= ϕ(ξ)ξ = fϕ(t, ξ)ξ. (3.35) from (3.34), (3.35) it follows (3.32). substitute (3.32) in (3.31): i ( r∗0(vf ; t) ) = ∞∫ 0 f(ξ)v(ξ) v (ξ) fϕ(t; ξ)ξ dξ, (3.36) copyright © 2021 assa. adv syst sci appl (2021) 140 kh. almohammad so i ( r∗0(vf ; t) ) = ∞∫ 0 fϕ(t; ξ)g(ξ) dξ, (3.37) where g(ξ) = f(ξ)v(ξ)ξ v (ξ) . (3.38) we obtain the equivalence of (2.24) and (3.39), where (3.39) is of the form ∃ c3 ∈ r+ : φ−1 2 {∫ r+ φ2 ( w(t) ∫ ∞ 0 fϕ(t; ξ)g(ξ) dξ ) u(t) dt } ≤ φ−1 1 {∫ r+ φ1(c3σg)v dt } , g ∈m+ (3.39) and σ(t) = v (t)v−1(t)t−1. as a result, introducing the operator i0(g; t) = ∞∫ 0 fϕ(t; ξ)g(ξ) dξ, t ∈ r+ (3.40) where the kernel fϕ(t; τ) = min { ϕ(t), ϕ(τ) } = { ϕ(t), t > τ, ϕ(τ), τ > t (operator i0 in the notations of [11]) we obtain the equivalence of the modular inequalities (2.24) and (3.39). 2. we now pass to the proof of the equivalence of inequality (3.39) and the set of conditions (3.29). to this end, we use a known result due to jim quile sun which was given in our paper [11] combined with the generalizations given in [4]. denote φ−1 2 {∫ r 0 φ2 ( w(t) b · tfϕ(t, r) ϕ(r) ∥∥∥∥fϕ(·, r) εv ∥∥∥∥ ψ1(εv) ) u(t) dt } ≤ φ−1 1 (1 ε ) . (3.41) thus, we have shown that (2.24)⇔ (3.39)⇔ (3.29). theorem 3.1 is proved. acknowledgements the author thanks professor m.l. goldman for helpful discussions. references 1. jim quile sun (1995) hardy type inequalities on weighted orlicz spaces. ph.d. thesis, the univ. of western ontario, london, canada, 1995. 2. bennett, c., & sharpley, r. (1988) interpolation of operators. boston: acad. press, 1988. (pure and appl. math.; v. 129). 3. bloom, s., & kerman, r. (1994) weighted orlicz space integral inequalities for the hardy–littlewood maximal operator. studia math. 1994, 110(2), 149–167. copyright © 2021 assa. adv syst sci appl (2021) the modular inequalities for hardy-type operators 141 4. goldman, m., & kerman, r. (1998) on the principle of duality in orlicz-lorentz spaces, function spaces. differential operators. problems of mathematical education. proc. intern. conf. dedicated to 75-th birthday of prof. kudrjavtsev 1998, 1, 179–183. 5. drabek, p., heinig, h.p., & kufner, a. (1997) weighted modular inequalities for monotone functions. j. inequal. and applications 1997, 1, 183–197. 6. sawyer, e. (1990) boundedness of classical operators on classical lorentz spaces. studia math., 1990, 96, 145–158. 7. heinig, h.p., & kufner, a. (1996) hardy operators on monotone functions and sequences in orlicz spaces. j. london math. soc. 1996, 53(2), 256–270. 8. goldman, m. (2016) estimates for restrictions of monotone operators on the cone of decreasing functions in orlicz spaces. mathematical notes 2016, 100(1), 24–37. 9. goldman, m. (2016) estimates for the norms of monotone operators on weighted orlicz-lorentz classes. doklady mathematics 2016, 94(3), 627–631. 10. bakhtigareeva, e., & goldman, m. (2017) inequalities for hardy-type operators on the cone of decreasing functions in a weighted orlicz space. doklady mathematics 2017, 96(3), 1–5. 11. almohammad, kh. on modular inequalities for generalized hardy operators on weighted orlicz spaces. eurasian mathem. journal, (to appear). copyright © 2021 assa. adv syst sci appl (2021) introduction auxiliary definitions applications for hardy-type operators adv syst sci appl 2023; 03:1–15 published online at https://ijassa.ipu.ru. frequency domain identification of the quadcopter attitude dynamics vadim alexandrov*, ilya rezkov, dmitrii shatov, yuri morozov v.a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia abstract: the paper presents the identification of a linearized model of the quadcopter attitude dynamics. the attitude is described by the euler angles of roll, pitch and yaw. closed loop identification is performed, when the quadcopter control system operates providing flight stability. the experimental flight data, when the sequence of the sine waves is fed as required value for each angle separately, are used. the transfer functions from the controls, which provide the torques related to the body frame axes, to the euler angles are obtained via the finite frequency identification procedure. moreover, components of the rotational dynamics are considered in detail. the unknown parameters of the transfer functions are found by optimization procedure using the experimentally obtained values of the frequency response for the set of test frequencies. the difference in the parameters values of the fixed linearized model structure, identified for two operating points, shows that the tilt angle dynamics is sufficiently nonlinear. keywords: system identification, frequency domain identification, quadcopter attitude, euler angles, transfer function 1. introduction the complex flight dynamics of a quadcopter is a widely studied benchmark for control design methods. although there are several well-functioning control systems for commercial quadcopters and there are open source projects for experimental multirotor setups, research is ongoing on the application of classical and original methods of control design for the quadcopter flight control system. identification is an essential step for model based control design. the quadcopter model identification was studied by various methods [14]. in [7, 10, 13] coefficients of the linearized quadcopter models were identified using the software cifer. the test signal called frequency sweep is applied during the test flight of the quadcopter, then the corresponding frequency response is calculated and model coefficients are found. in [13] the full linearized quadcopter model in form of the several transfer functions was identified, in particular, for lateral and longitudinal motions there were obtained the third order systems with pure differentiation plus time delay and the poles were stable, one was real and the rest were complex-conjugate. the authors of [7, 10] identified quadcopter models in state-space form basing on the results of transfer function identification. in [7] the third order functions similar to [13] were used but augmented with a real zero. the article [8] develops a method of linear model identification in state-space form for a multirotor vehicle based on the equation-error maximum likelihood estimator which reduces to least-square approach applied to experiment data in the time domain. the real-time identification method for a quadcopter model was proposed in [1]. here, the recursive fourier transform coupled with least-square estimation was used to find the state-space model coefficients estimates. the proposed method uses multisine excitation signal added to the full motor command. ∗corresponding author: va.alexandrov@yandex.ru 2 v. alexandrov, i. rezkov, d. shatov, yu. morozov the finite frequency identification approach [2] is used in our works to find the transfer function of the control plant model. the problem for altitude control loop identification has been studied in [5]. the transfer function from the horizontal projection of a full thrust to the horizontal velocity of the quadcopter was experimentally identified, and aerodynamic drag was detected in [4]. it should be noted, that the horizontal part of thrust is determined by the combination of pitch and roll angles. identification of the tilt angle dynamics is considered in [6], where, in addition to frequency domain identification, the time domain nonlinear graybox parameters estimation is used. this article develops the results of [6], considering also the rotational dynamics for the yaw angle. the problem of transfer functions identification for separated loops of roll, pitch and yaw angles that describe the quadcopter attitude is stated in section 2 more detailed. section 3 briefly describes the finite frequency identification method and results of the experimental flight data processing. the structures of the transfer functions for roll, pitch and yaw loops are introduced and evaluated via optimization procedure in section 4. moreover, the test flight data for another operating point of the pitch angle is analyzed. conclusion is in section 5. the experimental quadcopter [4–6] is the frame s500 with 10-inch propellers, the onboard flight controller pixhawk 4 and the standard ardupilot software suite. 2. problem statement the quadcopter rotational motion description based on newton-euler formalism [14] is ϕ̇ θ̇ ψ̇  = [ 1 sinϕ tan θ cosϕ tan θ 0 cosϕ − sinϕ 0 sinϕ/ cos θ cosϕ/ cos θ ][ p q r ] , (2.1) [ ṗ q̇ ṙ ] =  jy − jz jx qr jz − jx jy pr jx − jy jz pq +  1 jx τϕ 1 jy τθ 1 jz τψ  , (2.2) where ϕ, θ, ψ are roll, pitch, yaw euler angles, p, q, r are angular velocities in body frame coordinates x, y, z respectively, jx, jy, jz are inertia values assuming that the quadcopter is symmetric about the body axes, τϕ, τθ, τψ are torques around the respective axes, which are shown on fig. 2.1. the torques are determined by the difference in the rotational speeds ω1...4 of each propeller. formulas for the case of symmetric x-type quadcopter frame are τϕ = kτ (ω2 2 + ω3 2 − ω1 2 − ω4 2), τθ = kτ (ω2 2 + ω4 2 − ω1 2 − ω3 2), τψ = kψ(ω1 2 + ω2 2 − ω3 2 − ω4 2), (2.3) where kτ , kψ are proportionality coefficients. thus, the control signals u1...4 for each motor are formed as u1 = uf − 0.5uϕ − 0.5uθ + uψ, u2 = uf + 0.5uϕ + 0.5uθ + uψ, u3 = uf + 0.5uϕ − 0.5uθ − uψ, u4 = uf − 0.5uϕ + 0.5uθ − uψ, copyright © 2023 assa. adv syst sci appl (2023) frequency domain identification of the quadcopter attitude dynamics 3 fig. 2.1. body frame axes where uf is the control value for thrust providing the altitude hold, uϕ, uθ, uψ are controls for torques related to body frame axes x, y and z. it is assumed that u1...4 ∈ [0, 1], uϕ, uθ, uψ ∈ [−0.5, 0.5]. some dynamics of motors is present from ui to ωi. usually installed electronic speed controllers (esc) for each motor provide a constant gain from control ui to the rotational speed ωi. a simplified description for the rotor speed dynamics is a first order transfer function [9]: pω(s) = kω tωs+ 1 (2.4) with the time constant tω and the gainkω. a time delay tdelay should be added to this transfer function to account the esc discrete operation. if a separate control for a one channel is considered, when, for example, uϕ = const ̸= 0 and uθ = uψ = 0, then the rotational speed can be divided into parts ωf and ωϕ dependent from the controls uf and uϕ ωi = ωf ± ωϕ and then ωi 2 = ωf 2 ± 2ωfωϕ + ωϕ 2. therefore, squares are canceled in (2.3), and the torque is linear with respect to the additional rotational speed generated from the angular control uϕ, when ωf is assumed as a constant: τϕ = 4kτωfωϕ. moreover, when uθ = uψ = 0 for roll control consideration separately, then one can assumed that angular velocities q = r = 0 in (2.2). thus, a basic model for roll angular velocity p from the angular control uϕ can be described by transfer function pp(s) = kϕ s(tωs+ 1) e−stdelay , (2.5) kϕ = 4kτωfkω jx , where values of gain kϕ, time constant tω and time delay tdelay should be determined. the transfer function for roll angle ϕ can be obtained from (2.5) by adding one more integrator if copyright © 2023 assa. adv syst sci appl (2023) 4 v. alexandrov, i. rezkov, d. shatov, yu. morozov pitch and yaw angles values are negligible in (2.1): pϕ(s) = kϕ s2(tωs+ 1) e−stdelay . (2.6) the basic models for pitch and yaw angles loops are similar to (2.6) on the strength of (2.2). nevertheless, the experimentally identified models in a number of papers have a different structure. transfer functions for roll and yaw angular velocities obtained in [13] are pp(s) = k1s (t1s+ 1)(s2 + 2ξωps+ ωp2) e−stdelay , (2.7) pr(s) = k2 t2s+ 1 e−stdelay , (2.8) and the transfer function for the pitch angular velocity is similar to the roll one. another model structure for roll and pitch angular velocities is considered in [12], where the statespace model for the roll loop is[ v̇ ṗ ϕ̇ ] = [ a1 0 g a2 0 0 0 1 0 ][ v p ϕ ] + [ b1 b2 0 ] uϕ(t− tdelay), (2.9) where a1, a2, b1, b2 are identified parameters, g is the gravitational acceleration, v is the translational velocity projection onto the body frame y axis. the transfer function from control uϕ to angular velocity p for the model (2.9) is pp(s) = k̃s(s+ b̃1) s3 + ã2s2 + ã0 e−stdelay , where k̃, b̃1, ã2, ã0, are some parameters, calculated from parameters of the system (2.9). the pitch loop model in the paper [12] is similar to (2.9) and the yaw loop model is the same as (2.8). the first order rotor speed dynamics is added to pitch and roll model of the form (2.9) in [7]. moreover, the yaw loop model is complemented by a lead time constant tlead, i.e. the transfer function for yaw angular velocity is pr(s) = k2(tleads+ 1) (t2s+ 1)(tωs+ 1) e−stdelay . (2.10) one more example of the pitch and roll loop model is described in [10]. for brevity, the roll model only is separated:[ v̇ ṗ ϕ̇ ] = [ a11 a12 g a21 a22 0 0 1 0 ][ v p ϕ ] + [ 0 b2 0 ] uϕ, (2.11) where a11, a12, a21, a22, b2 are identified parameters. the corresponding transfer function is pp(s) = k̃s(s+ b̃1) s3 + ã2s2 + ã1s+ ã0 , (2.12) where k̃, b̃1, ã2, ã1, ã0 are coefficients calculated from the parameters of (2.11). copyright © 2023 assa. adv syst sci appl (2023) frequency domain identification of the quadcopter attitude dynamics 5 thus, the model structure (2.6) is not confirmed by the identification results. however, there is no generally accepted model, parameters of which need to be identified. therefore, the considered problem is to find the transfer functions from the controls uϕ, uθ, uψ to roll, pitch and yaw angles in the form p (s) = b̂ms m + . . .+ b̂0 sn + ân−1sn−1 + . . .+ â0 (2.13) with orders m,n under the condition m < n and parameters b̂0, . . . , b̂m, â0, . . . , ân−1. 3. finite frequency identification the finite-frequency identification approach is based on analysis of the experimental data when the sine wave test signals on several different frequencies from wide enough range are fed to the system input. the quadcopter cannot be tested in open loop mode, since horizontal stabilization is provided by default attitude controllers. in the case of closed loop identification [3,5] the sine wave test signals are added to the controller setpoint. the required values of the pitch, roll and yaw angles θ∗, ϕ∗, ψ∗ for attitude controllers in identification test for the pitch angle loop should be the follows ones: θ∗ = ηi sin(ωi(t− t0)), ϕ∗ = 0, ψ∗ = 0, (3.14) where ηi and ωi are amplitude and frequency of the test signal (i = 0, .., nω, where nω is a number of the test frequencies from a given test set that are fed consequently to the system input), t is time and t0 is the start time of the experiment on the current frequency. then, uθ, uϕ and uψ are outputs of corresponding controllers to stabilize the roll and yaw angles values and provide the sinusoidal variation of the pitch angle. identification tests for the roll and yaw angles loops are similar. the ardupilot onboard control system is used for the test flight, where the author’s new software procedures are added to form the test signals (3.14) and collect the experimental data. the finite-frequency identification procedure consists of two steps: 1) obtaining estimates of frequency response for several test frequencies ωi and 2) finding estimates of coefficients of the identified transfer function using the frequency response estimates. the frequency response estimates are obtained according to p (jωi) = αi + jβi = αyi + jβyi αui + jβui , i = 1, . . . , nω, (3.15) where αi and βi are called the frequency parameters, which are found using output αyi, jβyi and input αui, jβui values calculated from the experimental data by fourier filter formulae [2, 5]: αyi = 2 ηiti ∫ t0+ti t0 θ(t) sin(ωi(t− t0))dt, βyi = 2 ηiti ∫ t0+ti t0 θ(t) cos(ωi(t− t0))dt, αui = 2 ηiti ∫ t0+ti t0 uθ(t) sin(ωi(t− t0))dt, βui = 2 ηiti ∫ t0+ti t0 uθ(t) cos(ωi(t− t0))dt, i = 1, . . . , nω, (3.16) copyright © 2023 assa. adv syst sci appl (2023) 6 v. alexandrov, i. rezkov, d. shatov, yu. morozov where ti = 2πq/ωi is the filtration time (experiment duration), where q ∈ n is the number of periods of the test frequency. some examples of the experimental data are shown on fig. 3.2. it is seen that the system output value follows the setpoint sine wave very accurately for low frequency but with lag and reduced amplitude for high frequency, which is in line with expectations for the closed loop system. in other hand, the control signal absorbs the noises and nonlinearities. the frequency parameters estimates obtained from the experimental flight data, where inputs are the controls uϕ,θ,ψ ∈ [−0.5, 0.5] and outputs are angles ϕ, θ, ψ in radians, are in table 3.1. fig. 3.2. sine wave test for the pitch closed loop with frequencies 16 and 2 rad/s table 3.1. the frequency parameters estimates roll pitch yaw i frequency, ωi, rad/s αi βi αi βi αi βi 1 30.28 -0.044 -0.008 2 20.11 -0.397 0.344 -0.385 0.287 -0.094 -0.055 3 16.01 -0.736 0.552 -0.724 0.438 -0.139 -0.0864 4 12.03 -1.532 0.722 -1.409 0.628 -0.195 -0.122 5 8.00 -3.642 1.517 -3.571 0.972 -0.316 -0.243 6 5.34 -8.642 3.662 -6.815 2.142 -0.630 -0.422 7 4.00 -9.436 9.005 -9.040 5.771 -1.039 -0.552 8 3.50 -8.479 12.253 -7.689 6.591 -1.317 -0.731 9 3.00 -4.108 11.115 -6.914 7.417 -1.657 -0.838 10 2.00 0.559 8.897 -2.384 7.335 -3.217 -2.439 11 1.00 0.843 3.979 0.007 3.643 these estimates are used to write a system of linear equations, where they are equated for each test frequency to (2.13) with unknown parameters, but with chosen order values n,m : b̂m(jωi) m + . . .+ b̂0 (jωi)n + ân−1(jωi)n−1 + . . .+ â0 = αi + jβi, (3.17) where ωi, αi, βi are values from experiments and b̂0, . . . , b̂m, â0, . . . , ân−1 are unknowns. when condition 2nω > n+m+ 1 is satisfied, the overdetermined system of 2nω linear algebraic equations with n+m+ 1 unknowns is obtained by separating the real and imaginary parts in (3.17). then, the solution of this system provides estimates of the transfer function coefficients by the least squares criterion. the values n,m are chosen by decreasing until the identified transfer function contains no obviously cancellable stable poles and zeros. copyright © 2023 assa. adv syst sci appl (2023) frequency domain identification of the quadcopter attitude dynamics 7 the identification results for the data from table 3.1 are pϕ(s) = 58.65(s+ 0.304) (0.050s+ 1)(0.193s+ 1)(s2 − 3.44s+ 13.8) , (3.18) pθ(s) = 49.39(s− 0.156) (0.046s+ 1)(0.181s+ 1)(s2 − 3.79s+ 12.9) , (3.19) pψ(s) = 1294(s+ 10.93) s(0.834s+ 1)(s2 + 28.9s+ 1036) . (3.20) comparison of these functions with the data of table 3.1 is shown on fig. 3.3 – 3.5. fig. 3.3. nyquist diagram of identified transfer function for roll fig. 3.4. nyquist diagram of identified transfer function for pitch copyright © 2023 assa. adv syst sci appl (2023) 8 v. alexandrov, i. rezkov, d. shatov, yu. morozov fig. 3.5. nyquist diagram of identified transfer function for yaw the identified transfer functions differ from those given in section 2. the function structures for roll and pitch are expectedly the same. this structure is similar to the form of (2.12), where the differentiating element is added in the numerator, since (2.12) is the transfer function for angular velocity. the smallest time constant in the denominators of (3.18) and (3.19) can be considered a first-order rotor speed model as proposed in [7] added to (2.12). more detailed study of the experimental data is presented in the next section. 4. identification in details 4.1. rotors dynamics electronic speed controllers (esc) type blheli32 are installed on the experimental quadcopter, so the rotors angular velocities ω1...4 are measured. then, the rotor speed dynamics can be identified from the collected experimental data. the frequency parameters obtained via (3.15)–(3.16) during the yaw tests from the controls u1...4 ∈ [0, 1] to the rotors angular velocities ω1...4 in rad/s are in the table 4.2 for each rotor. table 4.2. the frequency parameters estimates for the rotors dynamics rotor 1 rotor 2 rotor 3 rotor 4 i frequency, ωi, rad/s αi βi αi βi αi βi αi βi 1 30.28 251 -570 285 -443 273 -508 221 -589 2 20.11 475 -568 538 -521 518 -560 390 -591 3 16.01 627 -551 665 -517 575 -557 620 -516 4 12.03 751 -503 772 -457 754 -490 737 -478 5 8.00 899 -378 908 -341 870 -334 907 -386 6 5.34 991 -281 936 -254 995 -248 991 -257 7 4.00 997 -211 990 -202 996 -203 980 -199 8 3.50 1012 -189 995 -147 1003 -178 1008 -175 9 3.00 1013 -145 1015 -125 -1002 -158 1009 -160 the structure of the main dynamics for rotors can be set as the first order (2.4) with delay. the delay is due to the discreteness of the flight controller and esc. thus, the parameters copyright © 2023 assa. adv syst sci appl (2023) frequency domain identification of the quadcopter attitude dynamics 9 kω, tω, tdelay of the transfer function should be identified: pω(s) = kω tωs+ 1 e−stdelay . (4.21) the estimates of (4.21) parameters can be found via optimization procedure with variables vector x = [kω, tω, tdelay]. for the convenience of notation, hereinafter the vector of variables for the optimization procedure will be denoted as a row vector. the objective function is min x nω∑ i=1 4∑ k=1 |pω(x, jωi)− (αki + jβki)| |αki + jβki| , where pω(x, jωi) is the value for s = jωi of the transfer function (4.21) with parameters from the variables vector x, k is the rotor number and αki, βki are corresponding estimates of the frequency parameters from the table 4.2, i.e., an average function for four rotors is found. the identification result is pω(s) = 1033 0.048s+ 1 e−0.0042s. (4.22) the delay value correlates with the fast loop of the flight controller, which operates with step of 0.0025 s. the rotors time constant is expectedly small, but it cannot be considered negligible when the controller sampling time is essentially less. its nyquist diagram and experimental points from the table 4.2 are shown on fig. 4.6. thus, one part of all considered loops is identified. fig. 4.6. nyquist diagram of identified transfer function for rotors 4.2. roll and pitch model further, the pitch model will be considered mainly, since the roll model is similar. translation velocity is included into the differential equation describing the angular velocity dynamics in (2.9) and (2.11) [10, 12], which is different from the base model copyright © 2023 assa. adv syst sci appl (2023) 10 v. alexandrov, i. rezkov, d. shatov, yu. morozov (2.2). for a detailed consideration, it is proposed here to use a system of three differential equations for the angular velocity dynamics with the state variables: angle, angular velocity and translation velocity. the nonlinear model for translation velocity is obtained in [4]. then, the pitch model is: v̇ = g tan θ − cx|v|v, q̇ = −a1θ − a2q − a3v + a4τθ, θ̇ = q, (4.23) where v is horizontal translation velocity under assumptions that the vertical velocity is zero, i.e. the vertical projection of the thrust mg compensates to the quadcopter mass m, the pitch loop is considered separately, i.e. roll and yaw angles are zero and v corresponds to the pitch angle, and no wind, since the airspeed is implied in (4.23) but the ground speed is measured in the flight controller; g is acceleration of gravity; cx is the aerodynamic parameter, which is not a constant, but can be considered as some value for the operating point; a1, a2, a3, a4 are other parameters of the model. the torque τθ is determined by the rotors dynamics (4.22) with some gain. the aerodynamic parameter cx is the product of 1) the aerodynamic coefficient cx, which in turn depends on the velocity, 2) air density, i.e. the value ofcx may vary slightly depending on the weather conditions and decreases noticeably in the mountains, and 3) surface area, which depends on the tilt angle, which in the case of a quadcopter can be a very complex dependence, since the rotating propellers are not a rigid body. the experimental flight data contains the horizontal translation velocity values. an analysis of these data shows that for low velocity up to 10 m/s the nonlinear first formula in (4.23) is not confirmed, and the dynamics of the translation velocity depending on the angle can be described by the transfer function pv(s) = g s+ a5 , (4.24) where g = 9.8065 and a5 is the parameter, which value is identified from the experimental flight data: a5 = 0.16. then, the transfer function from the control uθ to the pitch angle formed from the system (4.23) taking into account (4.21) and (4.24) is pθ(s) = ã4(s+ a5) (tωs+ 1)((s+ a5)(s2 + a2s+ a1) + ã3) e−stdelay , (4.25) where ã4 = a4 · k̃τ and ã3 = ga3 are new unknowns as products of the corresponding parameters of (4.23) and the gains of torque and translation velocity; tω and tdelay are evaluated in (4.22), a5 is already identified too. thus, the estimates of unknowns x = [a1, a2, ã3, ã4] of the model (4.25) should be found. the globalsearch optimization procedure from the matlab global optimization toolbox can be used to find these values from the experimentally obtained frequency parameters given in table 3.1. this procedure is based on the multistart approach [11] with nonlinear programming solver fmincon, where interior-point algorithm is chosen. this procedure usually provides a reliable result, while the most effective optimization procedure for the problem being solved is not discussed here. the objective function is min x nω∑ i=1 |p (x, jωi)− (αi + jβi)| |αi + jβi| . (4.26) copyright © 2023 assa. adv syst sci appl (2023) frequency domain identification of the quadcopter attitude dynamics 11 the estimates are found for roll and pitch: xϕ = [−1.28, 1.86, 73.07, 286.6], (4.27) xθ = [−14.61, 2.97, 83.06, 295.5]. (4.28) the identified transfer functions are pϕ(s) = 56.67 · (s+ 0.16) (0.048s+ 1)(0.198s+ 1)(s2 − 3.04s+ 14.4) e−0.0042s, (4.29) pθ(s) = 42.91 · (s+ 0.16) (0.048s+ 1)(0.145s+ 1)(s2 − 3.75s+ 11.7) e−0.0042s. (4.30) these functions match enough to the experimental data that is shown on fig. 4.7, 4.8 and differ little from (3.18), (3.19). fig. 4.7. nyquist diagram of identified transfer function for roll the problem of the validity of the identified models under other flight conditions remains. the following flight test was carried out, where the required value of the pitch angle is formed as θ∗ = ηi sin(ωi(t− t0)) + θ0, where θ0 is an operating point. the values for the test are θ0 = 5°, ηi = 3°. the obtained frequency parameters estimates are in table 4.3. the estimates of the model (4.25) parameters other than (4.28) are found for the pitch loop: xθ = [−34.01, 3.41, 26.38, 241.9]. (4.31) the identified transfer function is pθ(s) = 30.05 · (s+ 0.16) (0.048s+ 1)(0.124s+ 1)(s− 3.79)(s− 0.685) e−0.0042s. (4.32) the correspondence of the identified transfer function to the experimental data is shown on fig. 4.9 thus, the identified transfer functions (4.29), (4.30) represent the roll and pitch dynamics only for small values of the tilt angle and the translation velocity. the parameters a1 and a3 of (4.23) change significantly when the tilt angle is not near zero. copyright © 2023 assa. adv syst sci appl (2023) 12 v. alexandrov, i. rezkov, d. shatov, yu. morozov fig. 4.8. nyquist diagram of identified transfer function for pitch table 4.3. the frequency parameters estimates for the pitch loop (5°operating point) i frequency, ωi, rad/s αi βi 1 20.11 -0.328 0.223 2 16.01 -0.562 0.304 3 12.03 -1.073 0.404 4 8.00 -1.971 0.390 5 5.34 -3.68 0.296 6 4.00 -5.003 0.545 7 3.50 -5.885 0.225 8 3.00 -5.735 0.672 9 2.00 -6.186 1.727 10 1.00 -4.554 3.292 fig. 4.9. nyquist diagram of identified transfer function for pitch (5°operating point) copyright © 2023 assa. adv syst sci appl (2023) frequency domain identification of the quadcopter attitude dynamics 13 4.3. yaw model it should be noted that the structure of the identified transfer functions (4.29), (4.30) for roll and pitch angles does not contain an integration element, in contrast to the preliminary structure (2.6), while the integrator is present in (3.20) for the yaw loop. thus, the transfer function for angular velocity r can be considered, and then, the transfer function for yaw angle is obtained by adding the integrator. this approach is further approved by the fact that in the case of yaw loop the measured data of the angular velocity during the sine wave test have less fluctuations than the angle data that makes it possible to obtain more reliable values of the frequency parameters. it is obvious that the rotors dynamics (4.22) should be used for the model from control uψ to torque τψ. the next idea follows from the presence of numerator in identified transfer function (3.20). this suggests the existence of some state variable, such that the dynamics of the angular velocity can be described as a system ν̇ = −a4ν +kνr, ṙ = −a1ν − a2r + a3τψ, (4.33) where ν is some state variable, a1..4, kν are model parameters. then, the transfer function from the control uψ to the yaw angular velocity r has the form pr(s) = ã3(s+ a4) (tωs+ 1)((s+ a4)(s+ a2) + ã1) e−stdelay , (4.34) where ã3 = a3 · k̃τ and ã1 = a1 ·kν . the estimates of unknowns xψ = [ã1, a2, ã3, a4] are found via the globalsearch optimization procedure from the experimentally obtained frequency parameters for the yaw angular velocity: xψ = [−1875, 352.6, 992.0, 5.62]. (4.35) the identified transfer function is pr(s) = 992 · (s+ 5.62) (0.048s+ 1)(s+ 358)(s+ 0.296) e−0.0042s. (4.36) the identified function (4.36) match enough to the experimental data that is shown on fig. 4.10. the transfer function for yaw angle can be presented: pψ(s) = 992 · (s+ 5.62) s(0.048s+ 1)(s+ 358)(s+ 0.296) e−0.0042s. (4.37) this function differs from (3.20), since the frequency parameters for yaw angle are less accurate than for yaw angular velocity one, and the result obtained with criterion (4.26) and structure (4.34) is more reasonable. 5. conclusion frequency domain identification allows to identify and analysis complex non-obvious dynamics. the transfer functions (4.29), (4.30), (4.37) for roll, pitch and yaw angles of the quadcopter attitude are obtained in this article. the advantage of separated identification for simple loops is shown, such as the rotors dynamics, to reduce the complicity of the nonobvious part of the system. solution of the optimization problem can provide better results than the least squares due to the correct cost function and also allows to find the delay copyright © 2023 assa. adv syst sci appl (2023) 14 v. alexandrov, i. rezkov, d. shatov, yu. morozov fig. 4.10. nyquist diagram of identified transfer function for yaw angular velocity estimation. the unknown parameters for optimization procedure can be more sensible, than the coefficients of the transfer function. identification of the pitch loop for operating points of 0° and 5° proves that the system is nonlinear, since the obtained transfer functions differ significantly. further research is required to find limits and dependencies of the system parameters. acknowledgements research partially supported by rsf, project no. 23-29-00588, https://rscf.ru/en/project/23-29-00588/. references 1. alabsi, m.i. & fields, t.d. (2019) real-time closed-loop system identification of a quadcopter. j. aircraft, 56(1), 1–12. 2. alexandrov, a.g. (1999) finite-frequency identification and model validation of stable plant. ifac proceedings volumes, 32(2), 4064–4069. 3. alexandrov, a. & palenov, m. (2011) self-tuning pid-i controller. ifac proceedings volumes, 44(1), 3635–3640. 4. alexandrov, v., rezkov, i. & shatov, d. (2020) finite-frequency identification of the quadcopter translation dynamics. proc. of the 24th international conference on system theory, control and computing (icstcc), sinaia, romania, 471–476. 5. alexandrov, v., rezkov, i. & shatov, d. (2021) linearized model identification for quadcopter vertical translation dynamics. proc. of the 25th international conference on system theory, control and computing (icstcc), iasi, romania, 278–283. 6. alexandrov, v., rezkov, i., shatov, d. & morozov, y. (2023). identification of the quadcopter rotational dynamics for the tilt angle. in: ronzhin, a., sadigov, a., meshcheryakov, r. (eds) interactive collaborative robotics. icr 2023. lecture notes in computer science (pp. 334–344), vol 14214. springer, cham. copyright © 2023 assa. adv syst sci appl (2023) frequency domain identification of the quadcopter attitude dynamics 15 7. cho, s.h., bhandari, s., sanders, f.c., tischler, m. & cheung, k.k. (2019) system identification and controller optimization of coaxial quadrotor uav in hover. proc. of the aiaa scitech 2019 forum, san diego, ca, 1–14. 8. cunningham, m.a. & hubbard jr., j.e. (2020) open-loop linear model identification of a multi-rotor vehicle with active feedback control. j. aircraft, 57(3), 1–18. 9. murphy, m.d. (2016) a modular virtual laboratory for quadrotor control simulation. ifac-papersonline, 49(6), 93–98. 10. niermeyer, p., raffler, t. & holzapfel, f. (2015) open-loop quadrotor flight dynamics identification in frequency domain via closed-loop flight testing. proc. of the aiaa guidance, navigation, and control conference, kissimmee, fl, 1–14. 11. ugray, z., lasdon, l., plummer, j., glover, f., kelly, j., et. al. (2007) scatter search and local nlp solvers: a multistart framework for global optimization. informs journal on computing, 19(3), 328–340. 12. wei, w., cohen, k., & tischler, m.b. (2015). system identification and controller optimization of a quadrotor uav. proc. of the ahs international’s 71st annual forum and technology display, virginia beach, virginia. 13. wei, w., tischler, m.b., schwartz, n. & cohen, k. (2014) frequency-domain system identification and simulation of a quadrotor controller. proc. of the aiaa modeling and simulation technologies conference, national harbor, md, 1–9. 14. zhang, x., li, x., wang, k. & lu, y. (2014) a survey of modelling and identification of quadrotor robot. abstract and applied analysis, 2014, 320526. copyright © 2023 assa. adv syst sci appl (2023) introduction problem statement finite frequency identification identification in details rotors dynamics roll and pitch model yaw model conclusion adv syst sci appl 2022; 01:92–116 published online at https://ijassa.ipu.ru. secure image cloud storage using homomorphic password authentication with ecc based cryptosystem devipriya m., m. brindha* department of computer science and engineering, national institute of technology, tiruchirappalli, tamilnadu, india abstract: cloud based storage gives us ease of storing large data. cloud is a storage that can store multimedia images and other data, but what happens to the data after the storage is invisible to the user. to protect the image in the cloud and transmission to the cloud, image encryption is used. confidentiality is the way to protect from unauthorized access while transmitting and storing medical images, military images, online banking and telecommunication. recently, chaos has become an emerging trend towards the encryption of images. new techniques are adopted in image encryption to increase the randomness and to decrease the correlation between the pixels. in the present work, an authentication based framework and image cryptosystem has been proposed for secure storage in the cloud. authentication is provided using sha hash algorithm and elgamal encryption. image cryptosystem uses chaos based scrambling using bit-matrix generated from logistic chaotic map sequence. a novel key generation for chaos is done using fourier transform for the image cryptosystem. bit sequence and sorting based pixel permutation are performed using the chaotic sequence. totalistic cellular automata technique has been applied using the corner neighborhood values to diffuse the image pixels. pixels of the image is finally diffused using the random sequence formed using elliptic curve points/ coordinates to improve the security. this encryption technique is evaluated using statistical and differential analysis. ecc gives security using the discrete logarithm problem. from the analyses, it is shown that it possesses large keyspace, sensitivity towards a small change in pixel value and also key value. it also resists pattern attacks as well as differential attacks. keywords: trusted agent, elgamal encryption, fourier transform, bit matrix based scrambling, cellular automata diffusion, elliptic curve encryption 1. introduction cloud based storage is used as it is accessible anywhere using the internet facility and also does not crash like any hard disk. managing the data in the cloud and also sharing the data with people is much comfortable. this data includes multimedia data such as images which should be encrypted before storing it in the cloud. encryption before submitting the data to the cloud prevents leakage of information both in the transmission network as well as while stored in the cloud. many businesses either large scale or small scale use to store the documents, images and other data in the cloud. it is easy to access the data from anywhere without a need for maintaining a storage platform. the image data and other records of the medical field can be stored and retrieved in the cloud. similarly, different fields such as military, media, legal and even common people use the cloud to store the image and other documents. to protect the image files, there are a lot of applications available. stingle photo is an application that encrypts the data and stores it in the cloud environment. digilocker is a service initiated by the government of india for every citizen to store documents in the cloud securely. to provide confidentiality, data integrity and authorization, the images are ∗corresponding author: brindham@nitt.edu secure image cloud storage using homomorphic password authentication 93 encrypted before storing them in the cloud. even though the cloud is safe, it can analyze the data for advertisement purposes which can be prevented using encryption. authentication is used to provide credentials only to the authorized user. sha algorithm and elgamal encryption are used in the proposed authentication technique to serve the intended user by the trusted agent. in the proposed image encryption, in addition to the sorting based permutation, bit sequence based pixel permutation is performed to reduce the relationship between the adjacent pixels. a novel fourier transform based key generation technique is adopted to generate new keys each time to reduce the possibility of differential attacks. many researchers in the literature concentrate on the permutation part rather than the diffusion. hence, in this proposed technique, the totalistic cellular automata is adopted to give strength towards the effectiveness of the diffusion. the novel combination adopted in the diffusion is totalistic cellular automata with ecc based prime field which provides high level security that can be seen from the analyses obtained. correlation coefficient analysis shows lesser possibility to statistical attack as well as the npcr and uaci results show that the possibility of differential attack is less. from the study of the existing literature, it has been found that the solutions in the literature do not involve homomorphic authentication which is considered as very safe method as it does not decrypt the credentials to authenticate the user. in homomorphic encryption, the comparison for authentication is done in the encrypted value itself. in the existing literature, cryptosystems do not use plain image related key as it is used to overcome differential attacks. the following are the contributions of the proposed system: 1. homomorphic encryption based authentication algorithm to authorize the user. 2. initial keys for the proposed cryptosystem are formed using fourier transform, a novel technique to overcome the differential attack depending on the plain image. 3. a novel totalistic cellular automata based diffusion is employed to avert the statistical attack. 4. ecc encryption converts the chaotic sequence into coordinates and these coordinates are reduced to random sequence for diffusion to overcome both the correlation based attacks and to improve the substitution efficacy. the organization of the paper is as follows. section 2 is the study of available authentication and image encryption techniques in the literature. section 3 is the preliminary section that defines and elaborates the techniques adopted in the proposed system. the proposed authentication framework which describes the authentication technique designed using the elgamal encryption is presented in section 4. section 5 describes the proposed ecf image encryption scheme. section 6 presents the results obtained from the proposed authentication and image encryption. the summary of the techniques used in the proposed authentication systems and image encryption has been provided in section 7. 2. related works there are different kinds of authentication methods like passwords, biometric and smart cards etc., which are used to recognise the authenticated user. in [44], session initiation with user anonymity using ecc is developed. [3] uses three factor key agreement and authentication using ecc. fuzzy extractor function is developed to provide authentication of the mobile user for mobile cloud computing in [35]. authentication with user anonymity is developed using biometrics and ecc in [15]. authentication based on the chaotic map for wsn is proposed in [17]. different authentication schemes like [18], [16] and [13] are proposed for iot and cloud based environments. the computations mentioned in the literature are not using homomorphic technique which is highly required when authenticated data is outsourced to copyright © 2022 assa. adv syst sci appl (2022) 94 devipriya manivannan, m. brindha non trusted environments. the main advantage of homomorphic encryption is that there is no need to decrypt the credentials to authenticate the user. lorenz introduced the first chaotic attractor in the year 1963 [27] from then the research in the chaotic system has started and flourished. the chaotic system is disordered in its state and shows high sensitivity towards the initial state [34]. this stimulates chaotic systems to gain attention by the researchers in image encryption [37, 46]. confusion and diffusion structure using chaotic systems is followed by the researchers which were suggested by fridrich [7]. normally, permuting the pixels is done and it is followed by substitution or diffusion which changes the value of the pixel. elliptic curve cryptosystem (ecc) provides security using the discrete logarithm problem. the advantage of the elliptic curve system is, it uses less key space with greater security but the problem is, if it is directly applied to the image, the statistical attack might be possible. by changing it into a random irregular form and then applying ecc will prevent various kinds of attacks. many researchers used the chaos technique and ecc technique to encrypt the images from which few of them are described in this section. in [10], image is converted into serial bits using the steps such as transpose, reshape, to-frame and unbuffered blocks. in addition, different chaotic maps are used to create three different random sequences and these three sequences are xored to get a new pseudo random bit generator sequence to encrypt the plain image. even though three random sequences are involved, single diffusion is vulnerable to attacks. in [5], the image is split into three channels and it is converted to ycbcr color space. this combined ycbcr is converted into 8× 8 blocks and these subblocks are scrambled. these subblocks are inverted, rotated and applied a positive negative transformation to get the encrypted image but the drawback is, block level scrambling alone cannot provide a good permutation result. also, converting the color image into gray scale and the reverse may lead to lossy conversion. in [36], a new 3d chaotic system is realized and chaotic permutation by row wise as well as column wise is done. diffusion using xor with the permuted image is performed using the chaotic sequences. here the uniformity in the encrypted pixel distribution and the efficiency of the substitution are less. in [8], bit pair level rotation is performed by converting each pixel into four pairs of bits to confuse the image. paired bits based rotation is performed on 8× 8 bit pair blocks. these 64 bit block is converted into 4 regions. each region is rotated using the control instructions. the type of rotation, clockwise or anti-clockwise is determined by the parity. the chaotic sequence is derived from a modified pulsed coupled spiking neurons circuit map. also, xor and add operations are used to diffuse the image. the computational complexity for the confusion is high for bit pair level rotation. in [23], using 2d henon-sine map sequences, two sequences are obtained in which one sequence is added to the pixel of the image row wise and another sequence is added to the pixel of the image in column wise. in continuation, the sine map chaotic sequence is obtained and added to the image pixel in row wise. swapping of rows is used to interchange the place of the pixels. previous row pixel values are xored with the present row values. similarly, column wise swapping, addition and xor are done. the limitation of this work is swapping of rows alone cannot provide better permutation and also encryption has the possibility for attacks as the randomness and irregularity in encryption is less. in [28], crossover permutation is used to confuse the pixel values using the sequence obtained from the logistic-tent map and tent-sine map and the elliptic curve and elgamal are used to get cipher points from the confused pixels. these points are then converted to a pixel like value using addition, xor and modulo. crossover permutation is good to confuse the pixels and it provides a good result when it is adopted in higher order bits. however, the disadvantage is, correlation between the pixels in the encryption is high in this scheme. in [19], blake2 hash algorithm takes the plain image as input and produces two 256 bit keys. these 256 bit keys, key1 and key2 are given as input to blake2 hash algorithm to calculate two pseudo random streamskeyenc1 andkeyenc2. plain image is converted to 8× 8 blocks which undergo a single horizontal transform for 8 rows and 8 vertical different transforms for 8 columns and these blocks are quantized copyright © 2022 assa. adv syst sci appl (2022) secure image cloud storage using homomorphic password authentication 95 using a jpeg quantization table. keyenc1 pseudo random sequence and fisheryates shuffle algorithm are used to permute these blocks and dc coefficient is changed for each block. for ac coefficient encryption, keyenc2 pseudo random sequence is used for key1 embedding and the entropy based encoding of ac and dc coefficients is used to produce the encrypted values. in this algorithm, the randomness in the encryption is less, hence there is a possibility of statistical attacks. in [20], the image is water marked using the discrete wavelet transform function. josephus traversing map is generated to scramble the watermarked image. logistic map, xor and mod operation are used to encrypt the scrambled image. single diffusion has the risk towards known/ chosen plain text attacks. in [25], image is converted into blocks and converted to quantum image representation. these blocks are scrambled using quantum arnold transform and encrypted using the sine chaotic sequence and logistic map with xor operation. uniformity in the pixel distribution is less in this encryption. in [32], discrete wavelet transform is applied to sparsify the image to get the sparse coefficients. it is converted to a 2d array with the 2d-lasm chaotic sequence and is used for the permutation. compressive sensing based encryption is done using a 3d arnold cat map and a structural random map is created using arnold map sequence. finally, the encrypted value is quantized in the range [0, 255]. these encrypted values are embedded in a carrier image by dividing the carrier image and the intermediate image (im) into blocks and mapping the im to the carrier image to produce the cipher image using a reversible color transformation but the uniformity in the distribution of pixels is less in this encryption. in [38], plain image is confused using lorenz chaotic sequence and the permuted image is encrypted using forward and reverse diffusion. with this cipher image, the hash code of the plain image is sent to the receiver to check the data integrity. correlation between the pixels is high in this encryption. in [43], plain image is converted into blocks and a tent sine map is used to calculate the measurement matrix to compress the image blocks. arnold cat map is calculated to scramble the image blocks. 2d fractional fourier transform and chen hyperchaotic map are applied to diffuse the image. the initial values of the chaotic sequence are derived from the hash code of the plain image. randomness and irregularity in the encryption are less, hence there is a possibility for the statistical attack in this encryption. in [33], 2 pseudo sets and a set of keys are produced to shift column and row of the image circularly and to diffuse the scrambled image. the pseudo set and key for permutation are produced using chaotic permutation and multi circular shrinking is applied with gradual removal of the input set. the efficiency of the substitution is less, hence there is a possibility for the differential attack in this system. in [9], a 3d image is formed out of n different images. the initial value for chaotic maps is selected using the sha-256 hash code. an intertwining logistic map is utilized to choose the images which is also used to select the particular row and column in selected images for scrambling. rows and columns that are selected in the images using the intertwining logistic map are swapped. finally, this scrambled 3d image is encrypted using an improved piecewise linear chaotic map. swapping the whole rows cannot provide better permutation results in this work. in [14], two shared keys and nonces are produced for encryption. one key and nonce are calculated to construct an s-box whereas another key and nonce are used to construct the chaotic sequence. each pixel value maps to an s-box value. this s-box value along with previously encrypted pixel and chaotic value are used for diffusion. to diffuse a pixel, two s-box mappings are used, that may lead to a collision if different s-boxes are mapped to the same pixel. in [29], a new random sequence is formed using ecc and xored with the image pixel values. the permutation is done using the crossover technique which improves the efficiency of permutation only when it is performed in the high order bits. in [12], s box is created using ecc and chaotic sequence which is used to diffuse the pixel values. the efficiency of the substitution is less, hence there is a possibility for the differential attack. in the proposed encryption scheme, the bit sequence based pixel permutation and sorting based permutation are performed to resist the statistical attacks. the totalistic cellular automata based diffusion and ecc based diffusion are employed to resist the differential copyright © 2022 assa. adv syst sci appl (2022) 96 devipriya manivannan, m. brindha attacks. moreover, many of the existing solutions perform simple xor based diffusion but in the proposed work complexity of the diffusion has been increased to improve the security. adaptive plain image related key generation strategy is used in the proposed encryption technique to overcome the differential attack which is a highlight when compared with the existing solutions. 3. preliminary this section describes the techniques used by the proposed framework such as an authentication scheme using elgamal and an encryption scheme using elliptic curve with fourier transform (ecf). 3.1. elgamal encryption a prime p is chosen, a primitive root g (mod p) is chosen as generator. a random number a is chosen as exponent which is the secret key. public key is (p, g, a) and a is given by (3.1). a = ga (mod p) (3.1) given, y1 and y2 are the two plain values, the multiplicative homomorphism of elgamal encryption is given by (3.2). e(y1 · y2) = e(y1) · e(y2); (3.2) where e(y1) and e(y2) are the encrypted values of y1 and y2. 3.2. fourier transform the fourier transform is the main tool for image processing which is used to convert an image into sine and cosine components. the result of the transformation converts the image to the frequency domain. the equation for fourier transform is realized in eq. (3.3). z(k) = l∑ j=0 zj [ cos ( 2π l kj ) − i sin ( 2π l kj )] , (3.3) where k ∈ [0, l− 1]. 3.3. totalistic cellular automata the neighborhood values of the cell are used as a whole to apply the cellular automata rules. say, for instance, the neighborhood cell values are added or xored or multiplied or averaged to get a totalistic value and this value is used to get the next generation value of a cell. 3.4. finite field arithmetic finite field arithmetic is used in the ec cryptosystem in order to get a value bound to a finite set of numbers. overflow in addition or multiplication makes the data insufficient to revoke at the decryption side. 3.5. elliptic curve cryptography over prime field the discrete logarithm problem is mainly used by the elliptic curve cryptosystem to ensure the security. an elliptic curve on a prime field fr is realised by eq. (3.4). q2 = p3 + cp+ d (mod r), (3.4) copyright © 2022 assa. adv syst sci appl (2022) secure image cloud storage using homomorphic password authentication 97 where c, d ∈ fr, r 6= 2, 3 and satisfy the condition 4c3 + 27d2 6= 0 (mod r). the points satisfy the curve equation and point at infinity o includes the order of the curve #e(fr). group law for the curve is 1. identity: a+o = o + a = q for all a ∈ e(k). 2. negatives: if a = (p1, q1) ∈ e(k), then (p1, q1) + (p1,−q1) = o. the point (p1,−q1) is denoted by −a and it is called the negative of a. −a is a point in e(k). −o is equal to o. 3. point addition: let a = (p1, q1) and b = (p2, q2) ∈ e(k), where a 6= ±b. then a+b = (p3, q3). taking into account eq. (3.5), we have p3 = ( q2 − q1 p2 − p1 )2 − p1 − p2 (mod r), q3 = ( q2 − q1 p2 − p1 )2 (p1 − p3)− q1 (mod r). (3.5) 4. point doubling: let a = (p1, q1) ∈ e(k), where a 6= −a. then 2a = a+ a = (p3, q3). taking into account eq. (3.6), we have p3 = ( 3p21 + c 2q1 )2 − 2p1 (mod r), q3 = ( 3p21 + c 2q1 )2 (p1 − p3)− q1 (mod r). (3.6) 3.6. chaos based encryption chaotic systems are preferred in image encryption for their randomness, periodicity, sensitivity to initial values and ergodicity. it is not only dynamic but also has the property to generate the values depending on the initial value for effective decryption. in this paper, logistic map is used to derive random sequences for permutation and diffusion. 3.6.1. logistic map the logistic map is a chaotic function to calculate pseudo random sequence using equation (3.7). an+1 = αan(1− an), (3.7) where 0 < an < 1 and α = [0, 4] are used to get chaotic behaviour. 4. proposed authentication framework the classical method of storing the image in the cloud does not encrypt the image and hence there can be possibility of attacks to steal the information from the cloud. fig. 4.1 shows the classical method of storing the image in the cloud. in order to overcome this limitation, a novel authentication based cryptosystem has been developed. the authentication protocol is designed using sha 256 hash and elgamal encryption. the proposed cryptosystem is based on chaos, totalistic cellular automata and ecc. the framework consists of three levels such as user, trusted agent and cloud server. the user who wants to store images in the cloud will register with the trusted agent with user id and password. the trusted agent stores the hash copyright © 2022 assa. adv syst sci appl (2022) 98 devipriya manivannan, m. brindha fig. 4.1. classical framework value of user id and the encrypted hash value of the password. whenever the user has images to be stored in the cloud, the user will login into the trusted agent and this trusted agent checks the hash value of the user id for equality. the encrypted hash password stored in the database is divided by the encrypted hash of the password entered by the user which leads to a value of 1 since the encryption is homomorphic. subsequently, the trusted agent authenticates the user, encrypt the images and pass them to the cloud. on the other way, the trusted agent rejects the user whose user id or password does not match. fig. 4.2 shows the proposed authentication framework. fig. 4.2. proposed authentication framework 4.1. authentication system entities the proposed authentication system consists of user (u), trusted agent (t) and cloud server (cs). 1. user: this entity is the one which owns the images and wants to store the images in the cloud server. 2. trusted agent: this entity is the server that authenticates the user and encrypts the images. for authentication, elgamal homomorphic encryption has been used and for image encryption, the proposed ecf cryptosystem has been used. 3. cloud server: the cloud server stores the images encrypted by the trusted agent. the images in the cloud are retrieved by the trusted agent whenever there is a request raised by the user to access the images. 4.2. proposed framework design this section presents the sequence of the key generation process, user registration in t, authentication and encryption process done in t. copyright © 2022 assa. adv syst sci appl (2022) secure image cloud storage using homomorphic password authentication 99 table 4.1. hashed userid userid hashed userid user 0f96a6ba6b08e076efa346db1d1c59ff user@123 db9a02f4e902587f2b5bf34b4c774d84 table 4.2. encrypted password values password hashed encrypted value authen 131b3309235535ea00002e5338cb2 dcd103a00c938452fe51dd62a23 1fee2e100bc727852cc12d4707da 2c3b1af51682330933090c903b263 2c60c0a010c25f3 authen@123 170830ae103a0ddf2d4735ea0f2 e16823ed01cca2d8a14ad2ed93 0f12e96025b05f39510fb41f680b4 1252a096cc27423a5d3bef163f37 391af5170841f42679 4.2.1. key generation in key generation, only one private key a is chosen. the generator raised to a is multiplied with the message to get the encrypted result. as the system is homomorphic, multiplicative homomorphism is utilized to test the equality between the passwords. 4.2.2. user registration the user registration is a process of creating user id and password with t. the created user id is hashed using sha algorithm and the created password is hashed and encrypted using the sha and elgamal encryption. fig. 4.3 depicts the user registration. table 4.1 and table 4.2 show the hashed user id and hashed encrypted password. fig. 4.3. user registration 4.2.3. authentication and image encryption the authentication is performed by a trusted agent. 1. user enters the user id and password. copyright © 2022 assa. adv syst sci appl (2022) 100 devipriya manivannan, m. brindha 2. the hash value of the user id is compared with the authentication table. 3. if the user id in the authentication table matches with the entered user id, the hash value for password entered is created and it is encrypted using elgamal encryption. 4. the password in the authentication table is divided by the encrypted password. the authentication is said to be successful if the result is 1. otherwise, it fails. 5. if the authentication is successful, then the images of u are encrypted using the proposed ecf cryptosystem by t and stored in the cs. 6. whenever the image is needed, t is queried by u and t, in turn query the cloud server to retrieve the image and send it to u after decryption. figures 4.4 and 4.5 show the user authentication and storage of encrypted image to cs using t and retrieval of decrypted image from cs using t, respectively. fig. 4.4. user authentication 5. proposed encryption scheme the proposed elliptic curve with fourier transform based key (ecf) system starts with the generation of key value for the chaotic system. the initial key value is generated using fourier transform based equation. it takes the unique pixels from each color plane r, g and b. these pixel values are multiplied with the sine and cosine functions to get the initial keys which are used to generate the logistic map sequences. these sequences are used to scramble the pixels of each plane. the chaotic sequence is sorted and its index values are used to scramble the pixel values. the same chaotic sequence is diffused with the scrambled pixels using totalistic cellular automata technique. these image pixel values are diffused using the random sequence formed by the coordinates of an elliptic curve, given a prime number r. in this ecc encryption, r value and curve points (c and d in eq. (3.4) ) are also considered as the secret values in addition to the private key. fig. 5.6 shows the diagram of the encryption process. 5.1. proposed approach this section elaborates the encryption and decryption process of ecf. copyright © 2022 assa. adv syst sci appl (2022) secure image cloud storage using homomorphic password authentication 101 fig. 4.5. store and retrieval fig. 5.6. encryption process 5.1.1. encryption process 1. key generation for chaotic map fourier transform function is used to calculate the initial values for the chaotic map. equation (5.8) is used to calculate initial value for chaos. k = l∑ j=0 zj [ cos ( 2π l j2 ) sin ( 2π l j2 )] , (5.8) where α and x(1) for logistic map are derived fromk. zj is the pixel andl is the number of unique pixels in color plane r. similarly, k for g and b planes are calculated. 2. chaotic sequence the chaotic sequence is generated using the initial values derived from fourier function. copyright © 2022 assa. adv syst sci appl (2022) 102 devipriya manivannan, m. brindha the random chaotic sequence is generated using eq. (5.9). a(n+ 1) = αa(n)(1− a(n)) ∗ 104 mod 1 (5.9) 3. permutation using bits sequence and sorting the chaotic sequence is converted to bits and these bits act as a control parameter to jumble the image. the image pixel values are substituted sequentially in the places of ones. the remaining pixel values are substituted with zeros. the chaotic sequence is sorted and the index values are used to permute the pixels. figures 5.7 and 5.8 show the permutation using bit sequence and sorting, respectively. fig. 5.7. permutation using bit sequence fig. 5.8. permutation using sorting 4. diffusion using totalistic cellular automata chaotic sequence is converted to a matrix and the corner neighborhood values are added with the corresponding (i, j)th value. this sum value is added to the pixel value and it is copyright © 2022 assa. adv syst sci appl (2022) secure image cloud storage using homomorphic password authentication 103 reduced to 0 to 255 values. fig. 5.9 represents the function of totalistic cellular automata. fig. 5.9. totalistic cellular automata 5. ecc based diffusion in this diffusion, pixels are diffused using values obtained from ecc coordinates. (a) let r be 643, a = 1, b = 19; (b) let private key value of user a be ka = 148. (c) g = (14, 89) is the generator. (d) public key of user b, pb = kb.g = (276, 203). (e) ce is coordinate points calculated using eq. (5.10). ce = c.g+ pb.ka (5.10) (f) c is the chaotic sequence and the elliptic curve coordinate points formed using c is shown in table 5.3. table 5.3. elliptic curve coordinates using c 1 2 3 4 .... 512 1 (159,204) (193,620) (174,346) (555,272) .... .... 2 (479,179) (425,237) (469,600) (370,525) .... .... 3 (437,628) (466,22) (76,487) (309,156) .... .... 4 ... ... ... ... .... .... ... ... ... ... ... .... .... 512 ... ... ... ... .... .... (g) these coordinate points, x and y are added and reduced to [0, 255] in order to get a new random sequence for diffusion. finally, this random sequence is diffused with the pixel values. results of encryption are shown in fig. 5.10. fig. 5.10(a) shows the plain image, fig. 5.10(b) shows the permuted image and fig. 5.10(c) shows the encrypted image. copyright © 2022 assa. adv syst sci appl (2022) 104 devipriya manivannan, m. brindha fig. 5.10. results of encryption process 5.1.2. decryption process 1. public key of b and private key of a and generator g is used to generate the random sequence using the chaotic sequence with ecc. this random sequence is diffused with the image. 2. it is followed by the diffusion using totalistic cellular automata. 3. the image is permuted to its original position using chaotic bit sequence and sorted index to get the original image i . 4. results of decryption are shown in fig. 5.11. fig. 5.11(a) shows the encrypted images and fig. 5.11(b) shows the decrypted images, respectively. fig. 5.11. results of decryption process copyright © 2022 assa. adv syst sci appl (2022) secure image cloud storage using homomorphic password authentication 105 6. results and discussion the hardness of the authentication system and simulation results of the color image with the performance of the encryption algorithm are evaluated and presented in this section. the simulation has been done using matlab r2020a on 64 bit os windows 7 professional, intel core i5-3380m processor with operating frequency 2.90ghz and 8gb ram. 6.1. analysis of authentication system hash is a one way function and its hardness towards the computation of the same hash value i.e., collision with the available hash value makes the system safer. the encryption function is hard as it uses the discrete logarithm problem to encrypt the hash value of the password. the following subsections analyse the proposed cryptosystem using statistical and differential analyses to show the strength of the encryption. 6.2. analysis of key space in this algorithm, α and a1 values are taken as keys for chaotic diffusion. three different α and a1 values are used as there are three different planes in the color image. for ecc encryption, it uses 16 bit key. if the floating point accuracy of the system is 10−15 then the key space is 1093 > 2100. thus, the proposed encryption technique is strong enough to resist brute force attack. 6.3. key sensitivity analysis a small difference in the key should give a significant change in the encrypted image. the difference between the pixels of the encrypted image with the original key and the encrypted image with a very negligible change in the key gives a new scrambled image if they are completely different. if any negligible change in the key does not allow the decryption algorithm to decrypt the message, then the algorithm is said to be effective. key1 and key2 are changed by adding 0.0000001 and 0.0000002 to the keys, respectively. in fig. 6.12, fig. 6.12(a) presents plain image, fig. 6.12(b) is the image encrypted with original key, fig. 6.12(c) and fig. 6.12(d) are images encrypted with slight change in the keys, fig. 6.12(e) and fig. 6.12(f) are the images obtained as a result of the difference between fig. 6.12(b) & fig. fig. 6.12(c) and fig. 6.12(b) & 6.12(d), respectively. in fig. 6.13, fig. 6.13(a) is the image decrypted with the original key whereas, fig. 6.13(b) and fig. 6.13(c) are the images decrypted with a slight change in the key. 6.4. histogram analysis the histogram is one of the analyses to test the resistivity towards statistical attack. it shows how pixel values in an image are distributed. if the graph shows the flat histogram then it is assumed that the algorithm can resist statistical attack. fig. 6.14 shows the histogram of the plain image and the cipher image of lena. fig. 6.14(a), fig. 6.14(c) and fig. 6.14(e) are the histogram of the r, g and b planes of plain image whereas, fig. 6.14(b), fig. 6.14(d) and fig. 6.14(f) are the histogram of the r, g and b planes of cipher image. the flat histogram of the cipher image in fig. 6.14 conveys that the attack using the histogram is not possible. 6.5. correlation coefficient analysis the correlation coefficient (corr) is used to determine the degree of connectivity between the pixels. if the corr value is nearer to 1, it shows the high similarity between the pixel values. if the corr value is nearer to 0, it shows less similarity between the pixel values. it is copyright © 2022 assa. adv syst sci appl (2022) 106 devipriya manivannan, m. brindha fig. 6.12. key sensitivity results for encryption fig. 6.13. key sensitivity results for decryption fig. 6.14. histogram of plain and cipher image calculated using eq. (6.11). corr = d1((p−d1(p))(q −d1(q)))√ u1(p)(u1(q)) , (6.11) copyright © 2022 assa. adv syst sci appl (2022) secure image cloud storage using homomorphic password authentication 107 where d1(p) and u1(p) are the p gray level expectation and variance. corr of the proposed ecf scheme is nearer to 0 and proves that the cipher has less similarity between adjacent pixel values. fig. 6.15 shows the correlation distribution of lena plain image and cipher image. fig. 6.15(a), fig. 6.15(c) and fig. 6.15(e) are horizontal, vertical and diagonal correlation distribution of the plain image, respectively. fig. 6.15(b), fig. 6.15(d) and fig. 6.15(f) are the horizontal, vertical and diagonal correlation distribution of cipher image, respectively. table 6.4 conveys the correlation coefficient values of cipher image. table 6.5 conveys the comparison of average correlation coefficient values of the proposed solution with the existing solutions. lena image is taken for comparison. the obtained correlation coefficient values and correlation distribution in fig. 6.15 conveys that the correlation between the adjacent pixel values is less. fig. 6.15. correlation distribution of plain and cipher image table 6.4. correlation co-efficient values for cipher images image horizontal vertical diagonal lena1 0.0014 0.0094 0.0042 peppers1 0.0097 0.0119 0.0051 house1 0.0053 0.0043 0.0035 boat1 0.0067 0.0042 0.0050 baboon1 0.0111 0.0059 0.0072 copyright © 2022 assa. adv syst sci appl (2022) 108 devipriya manivannan, m. brindha table 6.5. comparison of correlation co-efficient of proposed with existing solutions encryption scheme horizontal vertical diagonal ecf 0.0014 0.0094 0.0042 [24] 0.0119 0.0087 0.0045 [40] 0.0082 0.0128 0.0012 [31] 0.0265 0.0044 0.0625 6.6. entropy analysis the quality of the confused image is determined by the entropy. if the entropy value is high, it means that the image is irregular. it is calculated using eq. (6.12). e(i) = 2n−1∑ i=1 p(ii) log2 p(ii), (6.12) where i is the image, n is the length of the pixel in binary, p(ii) denotes the probability of occurrence of ii. table 6.6 shows the entropy values of the cipher image for all three planes. table 6.7 compares the proposed entropy result with existing solutions and from the comparison, it is inferred that the proposed solution outperforms the existing solutions. table 6.6. entropy analysis of proposed solution image plane entropy lena1 r g b 7.9994 7.9992 7.9993 peppers1 r g b 7.9993 7.9993 7.9994 house1 r g b 7.9993 7.9993 7.9993 boat1 r g b 7.9993 7.9993 7.9993 baboon1 r g b 7.9993 7.9994 7.9993 6.7. chi square analysis this is also one of the tests to analyse the distribution of pixel values in an image which can be calculated using eq. (6.13). χ = 255∑ i=0 (ci − c0)2 c0 , c0 = (m ×n)/256, (6.13) where ci and c0 are the real and expected frequencies of pixel value i. chi square value for α = 0.05 is 293.24783. table 6.8 shows the chi square value of the cipher images and all the values are less than 280. if the values are less than 293, the possibility of statistical attack is less. copyright © 2022 assa. adv syst sci appl (2022) secure image cloud storage using homomorphic password authentication 109 table 6.7. comparison of entropy value of proposed with existing solutions algorithm plane entropy ecf r g b 7.9994 7.9992 7.9993 [24] r g b 7.9892 7.9902 7.9896 [40] r g b 7.9892 7.9896 7.9896 [30] r g b 7.9972 7.9972 7.9976 [31] r g b 7.9972 7.9973 7.9972 [22] r g b 7.9874 7.9872 7.9866 table 6.8. chi square analysis of ecf image chi square value lena1 271.3691 peppers1 272.7402 baboon1 255.9316 house1 251.8164 boat1 250.4531 6.8. maximum deviation this is a measure to convey the quality of the encryption which can be calculated using eq. (6.14). for larger values of maximum deviation, the encryption is efficient. the deviation between the plain and ciphertext should be higher to ensure that the image is random and irregular. me = t0 + t255 2 + 254∑ l=1 tl (6.14) 1. histogram of plain image hip and cipher image hic is calculated. 2. the absolute difference of the cipher and plain histogram is the value t which is calculated using (6.15). t = |hip − hic| (6.15) 3. the difference value t with specified intensity is tl. 6.9. irregular deviation this is a statistical measure of the uniform distribution of pixel values. if the irregular deviation is smaller, the uniformity in pixel distribution is high. it is calculated using the following steps, copyright © 2022 assa. adv syst sci appl (2022) 110 devipriya manivannan, m. brindha 1. the image pn and its cipher cn is subtracted and the absolute value dn is calculated using (6.16). dn = |pn − cn| (6.16) 2. the difference image dn is used to find the histogram hl using (6.17). hl = histogram(dn) (6.17) 3. havg is calculated from each hl and the total intensity value 256 using (6.18). havg = 1 256 255∑ l=0 hl (6.18) 4. isl is calculated using the absolute difference between the individual histogram and the the average value as given in (6.19). isl = |hl − havg| (6.19) 5. the irregular deviation is calculated by isavg using eq. (6.20). isavg = 255∑ l=0 isl (6.20) the maximum and irregular deviation are tabulated in table 6.9. the maximum deviation of the ecf cipher gives larger values and shows that the quality of encryption is high. irregular deviation of the ecf cipher is lesser than maximum deviation which implies an equal distribution of pixel values. table 6.9. maximum deviation and irregular deviation image maximum deviation irregular deviation r g b lena1 352876 129498 160072 190006 peppers1 308224 174436 120152 122900 baboon1 356369 166722 176616 148552 house1 516702 156870 145676 145726 boat1 292957 186676 112830 110884 6.10. deviation from uniform histogram deviation from uniform histogram is a measure to determine the uniformity in the distribution of pixel values. it is measured using eq. (6.22) and the deviation values are tabulated in table 6.10. 1. the ideal value calculated using the dimensions r and s of the image is hj which is expressed in (6.21). hj = r × s 256 (6.21) 2. using hj and the individual histogram value of the cipher hdu, he is calculated. he = ∑255 u=0 |hdu −hj| r × s (6.22) copyright © 2022 assa. adv syst sci appl (2022) secure image cloud storage using homomorphic password authentication 111 table 6.10. deviation from uniform histogram image deviation from uniform histogram r g b lena1 0.0246 0.0253 0.0247 peppers1 0.0250 0.0245 0.0230 baboon1 0.0249 0.0224 0.0250 house1 0.0265 0.0253 0.0261 boat1 0.0251 0.0239 0.0239 6.11. differential attack analysis this is used to test the quality of diffusion by comparing two cipher images where one cipher produced using the original plain image, another cipher produced using single bit value change in the original plain image. this single bit value change should produce a vast difference in the cipher image. number of pixel change rate (npcr) and unified average changing intensity (uaci) are the measures used to determine the efficiency of the diffusion which are calculated using eq. (6.23) and eq. (6.24), respectively. npcr = 1 m1×n1 ∑m1−1 i1=0 ∑n1−1 j1=0 e1(i1, j1)× 100, e1(i1, j1) = { 1, ifda(i1, j1) 6= db(i1, j1) 0, otherwise , (6.23) uaci = 1 m1 ×n1 ∑ i1,j1 |da(i1, j1)−db(i1, j1)| 256 × 100, (6.24) where da and db are cipher images with only single bit change. m1 and n1 are the image dimensions. npcr and uaci values are tabulated in table 6.11. table 6.12 shows the comparison of npcr and uaci values of proposed solution with existing solutions. for comparison, pepper image is used and the results obtained convey that the proposed solution has high substitution efficiency than existing solutions. table 6.11. npcr and uaci values image plane npcr uaci lena1 r g b 99.6349 99.5815 99.6014 33.4528 33.4553 33.4137 peppers1 r g b 99.6395 99.6098 99.6040 33.4835 33.5218 33.4457 house1 r g b 99.6193 99.5991 99.6113 33.4858 33.4465 33.4762 baboon1 r g b 99.6189 99.6208 99.6151 33.4233 33.4453 33.4773 boat1 r g b 99.6002 99.6063 99.6220 33.4625 33.4307 33.4085 copyright © 2022 assa. adv syst sci appl (2022) 112 devipriya manivannan, m. brindha table 6.12. comparison of average npcr and uaci value of ecf with existing solution algorithm npcr uaci ecf 99.6177 33.4836 [24] 99.61 32.20 [1] 99.62 28.62 [6] 99.29 33.18 [21] 99.63 31.87 [4] 99.60 33.45 [45] 99.58 33.44 6.12. overall comparison using metrics the image encryption scheme is compared with the available different encryption schemes in the literature. the correlation coefficient, entropy, npcr and uaci are the metrics used for comparison in which the proposed scheme outperforms the encryption scheme already in existence. few works perform well in the efficiency of substitution and lack in the correlation coefficient. some of the works lack randomness and uniform distribution of pixel values. the proposed scheme has consistency in the performance and there is no skewed performance like performing well in one metric and very poor in another metric. table 6.13 shows the comparison of the proposed scheme with the existing solution using the above mentioned metrics. table 6.13. overall comparison of proposed solution with existing solutions using metrics algorithm correlation coefficient entropy npcr uaci h v d ecf 0.0014 0.0094 0.0042 7.9993 99.6177 33.4836 [24] -0.0119 -0.0087 -0.0045 7.9896 99.61 32.20 [37] 0.0054 0.0064 0.0046 99.60 33.79 [10] 99.43 21.97 [36] -0.0423 0.0202 0.0212 7.9974 99.55 33.61 [39] -0.0331 0.0057 0.0169 7.9972 99.61 33.40 [11] -0.0070 0.0151 0.0003 7.8963 99.60 33.36 [26] 0.0074 -0.0094 -0.0054 7.9992 99.61 33.46 [42] 0.0925 0.0430 0.0533 99.65 33.50 [41] -0.0226 0.0041 0.0368 7.2072 99.61 33.53 [2] -0.0061 0.0130 0.0017 7.9992 99.60 33.49 6.13. overall comparison using features the features compared are key derivation, strong diffusion, one time key and decryption complexity. even though few encryption schemes have strong diffusion property and decryption complexity, it has drawback such as it does not use an adaptive key for each plain image. encryption scheme [2] has all features like the adaptive key, strong diffusion and decryption complexity but its computational overhead is high as it uses both ca and dna. the proposed encryption has all the features such as plain image related key, strong diffusion, high decryption complexity and less computational overhead. table 6.14 compares the proposed scheme and the existing schemes using the features used for encryption. copyright © 2022 assa. adv syst sci appl (2022) secure image cloud storage using homomorphic password authentication 113 table 6.14. overall comparison of proposed solution with existing solutions using features algorithm plain image related key strong diffusion one time key decryption complexity ecf yes yes no initial key for chaos and a private keys for ecc [24] no yes no initial key for chaos [37] no yes no initial key for chaos [10] no yes no non chaotic encryption [36] yes no no initial key for chaos [39] no no no initial key for chaos [11] no no no initial key for chaos [26] no yes no initial key for chaos using ecc [42] yes no no initial key for chaos [41] yes no no initial key for chaos [2] yes yes no initial key for chaos 7. conclusion the authentication system is proposed using sha hash and elgamal encryption scheme using the homomorphic property. the proposed ecf cryptosystem uses the logistic map to permute and diffuse the image. three iterations of logistic map are used for each plane to improve the security. initial values are derived from the image using fourier transform based equation. the permutation is performed using bit values obtained from the chaos sequence. additionally, sorting based chaotic permutation is also performed. finally, the image is diffused using totalistic cellular automata and encrypted using elliptic curve encryption. the performance is analyzed using cipher image. statistical and differential attacks can be reduced using the conversion of pixels with the random values obtained from ecc coordinates. overall, the proposed 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(2015). a novel image encryption algorithm based on chaos and line map. neurocomputing, 169:150–157. copyright © 2022 assa. adv syst sci appl (2022) introduction related works preliminary elgamal encryption fourier transform totalistic cellular automata finite field arithmetic elliptic curve cryptography over prime field chaos based encryption logistic map proposed authentication framework authentication system entities proposed framework design key generation user registration authentication and image encryption proposed encryption scheme proposed approach encryption process decryption process results and discussion analysis of authentication system analysis of key space key sensitivity analysis histogram analysis correlation coefficient analysis entropy analysis chi square analysis maximum deviation irregular deviation deviation from uniform histogram differential attack analysis overall comparison using metrics overall comparison using features conclusion microsoft word 1127 article text, copyedited.docm adv syst sci appl 2021; 03; 101-112 published online at https://ijassa.ipu.ru. neural network technologies and topological analysis of social media data nailia gabdrakhmanova1*, maria pilgun2 1) peoples friendship university of russia (rudn university), moscow, russia e-mail: gabdrakhmanova_nt@rudn.ru 2) institute of linguistics russian academy of sciences, moscow, russia e-mail: mpilgun@iling-ran.ru abstract: currently, due to the successful use of neural network technologies for analyzing data of various formats, the range of problems that can be solved using mathematical modeling methods has significantly expanded. the paper deals with the topical task of analyzing speech perception by social media data and assessing the level of conflictogenity, social approval / stress of the residents regarding urban planning projects. the purpose of the paper is to develop and show on a practical example the effectiveness of the integration of neural network and mathematical models for solving such tasks. mathematical models are built using the methods of mathematical statistics and topological data analysis. keywords: neural network, persistence diagrams, time series, speech perception, social media data 1. introduction analysis of speech perception by social media data is a topical task, since it makes it possible to reveal the attitude of users to certain events, conduct predictive analytics and identify conflict situations in the early stages of crisis situations, which is a demanded task in various fields. there are studies of possible adaptive sampling mechanisms for haptic data compression aimed at applications like tele-operations and tele-surgery [3]. machine understanding is being actively studied [6]. notably, fuzzy and natural language based approaches and solutions for a more human consistent dealing with decision support, time series analysis, forecasting, clustering, etc. are discussed [2]. much attention is paid to the study of the features of perception of robots [4, 7]. the purpose of the paper is to develop and show on a practical example the effectiveness of the integration of neural network and mathematical models for the analysis of user content and assessment of the level of conflictogenity, social approval/stress of the residents regarding urban planning projects or district conflicts based on the analysis of perception concerning the construction of the fiztekh subway station (moscow, russia). in our work, we show the potential of automatization of the conflictogenity level assessment according to social media data. this work is a continuation of the previous works of the authors [7]. to build mathematical models, methods of mathematical statistics and differential equations are used in the works of the authors [7]. in this paper, we supplement the developed methods and algorithms with methods of topological data analysis. such an extension of the algorithm was required due to the specifics of the data in the time series of the problem being solved. * corresponding author: gabdrakhmanova_nt@rudn.ru 102 n. gabdrakhmanova, m. pilgun copyright ©2021 assa. adv. in systems science and appl. (2021) topological data analysis (tda) is a fairly new direction in the field of data analysis. tda makes it possible to find the structure in time series data. persistence diagrams introduced by h. edelsbunner [8] are the most important tool in computational topology that allow, for example, to obtain qualitative information about the “topological dynamics” of a time series. an important feature of persistence diagrams is their robustness to “noise”. methods of computational topology were further developed in the works of g. carlsson, a. zomorodian [9,10]. topological data analysis can be combined with methods in machine learning (including deep learnin) as well as statistical methods [11]. 1.1. data the material for the study was the data of social media, microblogs, blogs, instant messengers, videos, forums and reviews concerning the implementation of the project of the terminal station for the lyublinsko-dmitrovskaya line (construction of the fiztech subway station) in moscow (russia). the data was collected between january 1, 2019 and december 22, 2019 (see table 1.1). table 1.1 data characteristics number of messages: 7063 max number of messages per day: 614 number of authors: 933 activity (posts per author): 7,57 number of sources: 213 1.2. method the study involved several parallel models. with the help of the semantic neural network model, a text analysis was performed; the topic structure and the semantic network were analyzed, as well as the sentiment and the level of aggression; digital conflictogenic zones were identified; and the indices of social stress and well-being were calculated. in parallel with the neural network model, a statistical analysis of experimental data was performed and dynamic models of processes were built. the following models were used for data analysis: a model based on the neural network paradigm of using neural-like elements with time summation of signals [5], methods of mathematical statistics and topological data analysis [9]. analysis of solutions for all models showed the consistency of solutions. 1.2.1. topological data analysis persistence diagrams are an important tool in tda. in this section, we will give basic information about persistence diagrams, following [9]. let x be a triangulated topological space, f a continuous function on x. for 𝑎 ∈ 𝑅, we denote the set of points of the sub-level: 𝑋! = 𝑓"#(−∞, 𝑎]. figure 1 shows a graph of the construction of sub-levels for two levels a=0.05 (green); a=0.1 (purple). neural network technologies and topological analysis of social media data 103 copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 1.1 sub-levels for levels a=0.05 (green) ; a=0.1 (purple). the number 𝑎 ∈ 𝑅 is called a homological critical value if for some k the homomorphism induced by the embedding of 𝑓∗: 𝐻%(𝑋!"&) → 𝐻%(𝑋!'&) is not an isomorphism for any sufficiently small 𝜀 > 0 ( homology groups are considered with coefficients in ζ(). a continuous function f is called tame if it has only a finite number of homological critical values, the homology groups 𝐻%(𝑋!) are finitely generated for all k and all 𝑎 ∈ 𝑅, and if the segment [x, y] does not contain homological critical values of the tame function, then the embedding-induced homomorphism of 𝑓) *: 𝐻%(𝑋)) → 𝐻%5𝑋*6 is an isomorphism. suppose 𝑓: 𝑋 → 𝑅 is a tame function. let 𝑎# < 𝑎( < ⋯ < 𝑎+ be its homological critical values. a persistence diagram 𝐷(𝑓) ⊂ 𝑅(of function f is the set of points 5𝑎, , 𝑎-6 , 𝑖, 𝑗 = 1, 𝑛????? i, j = 1, . . . , n combined with the set of points of the diagonal △= {(𝑥, 𝑥)|𝑥 ∈ 𝑅}. a remarkable feature of the persistence diagram d(f) is its stability with respect to perturbations of the function f. persistence diagrams can be used to calculate the lengths of the barcodes of connectivity components. here the term barcode stands for the component lifetime. let us denote the summarized lengths of barcodes of two homology groups 𝐻. and 𝐻# as 𝐿. and 𝐿# correspondingly. then the mean of the euler characteristic can be determined [9] as 𝜒 = 𝐿. − 𝐿# (1.1) to identify conflictogenity by social media data, it is proposed to use the estimates of euler’s characteristics. it is proposed to choose a certain step on the time axis and find estimates of the euler characteristics at each interval. based on the dynamics of the estimates of euler’s characteristics, it is proposed to develop an algorithm for classifying the situation. in section 2.3.3, we show the possibility of classifying the situation according to the dynamics of estimates of the euler’s characteristics of time series. 1.2.2. time series analysis to study the dynamics of processes according to observational data, methods of time series analysis are used [12]. a fairly general mathematical model for the time series x(t) is a model of the form: 𝑥(𝑡) = 𝑢(𝑡) + 𝑣(𝑡) , 104 n. gabdrakhmanova, m. pilgun copyright ©2021 assa. adv. in systems science and appl. (2021) where u(t) is a deterministic sequence or systematic component, v(t) is a random component. the purpose of mathematical modeling of time series is to study the dynamics and predict the values of x(t) several steps ahead. in this work, regression models were used to estimate u(t) and autoregressive models to estimate v(t). the results of building models are presented in paragraph 2.3.2. 2. results and discussion 2.1. general description of the content key topics of the content related to the construction of the fiztech subway station that had been generated by users, were found mostly in the social network vkontakte. it should also be noted that users’ interest in the topic increased in 2020 compared to 2019 (figures 2.1) fig. 2.1. digital platforms where actors generated content the peak of growth in the authors’ activity dynamics (value of 107), the total number of messages (614) and unique messages (436) falls on may 25, 2020 (figures 2.2; 2.3; 2.4) and is associated with information that the entire lublinsko-dmitrovskaya line of the moscow subway will become underground. analysis of the views dynamics shows that, in addition to this news, the greatest attention of users was also attracted by messages from august 20, 2020 (value of 2 524 187) regarding the mayor’s approval of the names of seven stations under construction on the big beltline of the subway and from november 30, 2020 (value of 2 860 008) about how the new subway stations will look like (figures 2.5). 0 20 40 60 80 100 120 01 .0 1. 20 19 01 .0 3. 20 19 01 .0 5. 20 19 01 .0 7. 20 19 01 .0 9. 20 19 01 .1 1. 20 19 01 .0 1. 20 20 01 .0 3. 20 20 01 .0 5. 20 20 01 .0 7. 20 20 01 .0 9. 20 20 01 .1 1. 20 20 0 100 200 300 400 500 600 700 01 .0 1. 20 19 01 .0 3. 20 19 01 .0 5. 20 19 01 .0 7. 20 19 01 .0 9. 20 19 01 .1 1. 20 19 01 .0 1. 20 20 01 .0 3. 20 20 01 .0 5. 20 20 01 .0 7. 20 20 01 .0 9. 20 20 01 .1 1. 20 20 neural network technologies and topological analysis of social media data 105 copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 2.2. dynamics of users’ activity (sec) fig. 2.3. dynamics of the total number of messages fig. 2.4. dynamics of unique messages fig. 2.5. views dynamics 2.2. semantic neural network model 2.2.1. key topics there is a small list of key topics related to complaints and dissatisfaction of the residents, which indicates a calm situation around this urban construction project: negative cluster: • surface construction of the subway from the lianozovo subway station to the severny settlement. • incorrect holding of public hearings on the project to extend the construction of the lyublinsko-dmitrovskaya line. • risk of destruction of buildings located in the vicinity of the construction sites. • environmental deterioration due to the project implementation. • deterioration of the traffic situation in the district during construction. • violation of the architectural appearance of the district and crudity of the design. • the residents are afraid that after the appearance of new subway stations, their yards will be filled with unauthorized cars. • the statements of the mayor’s office representatives about construction works in the capital during the pandemic were negatively perceived by users. positive cluster: • the strongest positive reaction was caused by the news of the abandonment of the elevated subway line. • messages about high rates of the construction. • improving the transport accessibility of districts after the commissioning of a new subway station. 2.2.2. sentiment analysis the overwhelming majority of the content with various types of sentiment was posted on the social network vkontakte. messages with negative sentiment were also generated on facebook and instagram, and messages with positive sentiment on mos.ru and m24.ru; while those with neutral sentiment were on mos.ru and facebook (figures 2.6; 2.7; 2.8.) 0 100 200 300 400 500 01 .0 1. 20 19 01 .0 3. 20 19 01 .0 5. 20 19 01 .0 7. 20 19 01 .0 9. 20 19 01 .1 1. 20 19 01 .0 1. 20 20 01 .0 3. 20 20 01 .0 5. 20 20 01 .0 7. 20 20 01 .0 9. 20 20 01 .1 1. 20 20 0 500000 1000000 1500000 2000000 2500000 3000000 3500000 01 .0 1. 20 19 01 .0 3. 20 19 01 .0 5. 20 19 01 .0 7. 20 19 01 .0 9. 20 19 01 .1 1. 20 19 01 .0 1. 20 20 01 .0 3. 20 20 01 .0 5. 20 20 01 .0 7. 20 20 01 .0 9. 20 20 01 .1 1. 20 20 106 n. gabdrakhmanova, m. pilgun copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 2.6. top-10 sources of negative sentiment fig. 2.7. top-10 sources of positive sentiment fig. 2.8. top-10 sources of neutral sentiment most digital footprints and audience reach are characterized by neutral sentiment, which indicates a calm situation around these objects and users’ calm perception (figures 2.9; 2.10.). fig. 2.9. sentiment of the digital footprints fig. 2.10. sentiment of the audience reach 46 30 8 3 3 3 2 2 2 2 0 10 20 30 40 50 vk.com instagram.com icmos.ru m24.ru youtube.com 452 81 41 34 32 32 25 25 19 18 0 100 200 300 400 500 vk.com m24.ru advis.ru mperspektiva.ru mskagency.ru 2764 391 312 285 158 133 100 91 81 81 0 1000 2000 3000 vk.com facebook.com twitter.com advis.ru telegram.org neural network technologies and topological analysis of social media data 107 copyright ©2021 assa. adv. in systems science and appl. (2021) 2.2.3. formation of the social tension rating regarding the construction of the fiztech subway station the dynamics of references by sentiment indicates a calm situation around the object, since there is a significant predominance of neutral reactions. negative messages make up a negligible portion of the content (figures 2.11.). it is natural that the analysis of social stress regarding the construction of the mosinzhproekt facility, the fiztech subway station, according to the database (collected between january 1, 2019 december 22, 2019), showed the presence of social stress with an extremely low index of 1.95 and a high index of social well-being of 18.92 (figures 2.12.). fig. 2.11. dynamics of references by sentiment fig. 2.12. social tension and well-being indices 2.3. mathematical models 2.3.1. statement of the problem as a result of the implementation of the steps described in 2.1 and 2.2, sequences were obtained that represent the number of negative neg(t), positive p(t) and neutral ntr(t) messages at each time moment t. for the problem being solved, it is important to know the proportion of negative, positive and neutral messages in each moment of time and to study the dynamics of these data. for this purpose, at the preprocessing stage, the entire data array is converted into several proportions. let s(t) be the number of posts at time t with a reference to the topic of the subway construction. we introduce the following notations: p(t) is the proportion of messages at time t with a positive attitude towards the subway construction; neg(t) is the proportion of messages at time t with a negative attitude towards the subway construction; 108 n. gabdrakhmanova, m. pilgun copyright ©2021 assa. adv. in systems science and appl. (2021) ntr(t) is the proportion of messages at time t with a neutral attitude to the subway construction. the data conversion formulas are as follows: p(t)= p(t)/s(t); neg(t)=neg(t)/s(t); ntr(t)=ntr(t)/s(t). the constructed time series are notes as: {p(t)},{neg(t)},{ntr(t)}, t=1,..,n. it is necessary, according to the data of time series {p(t), {neg(t)}, {ntr(t)} for the period t=1,..,n 1) estimate the trend of each time series; 2) classify the situation (the actors’ attitude towards the construction): a) there is a conflict or b) there is no conflict. 2.3.2 time series trend models regression analysis models were used to estimate the parameters of the systematic components. for a time series with a positive attitude towards the construction, the following model was built: pm (t) = 0.006+0.01t (2.1) where pm (t) is a model value for the proportion of positive messages at time t. a positive coefficient in (2.1) before t indicates an increase in the proportion of positive messages. for a time series with a negative attitude towards the construction, the following model was built: negm (t) = 0.028 0.001t (2.2) where negm (t) is a model value for the proportion of negative messages at time t. a negative sign before the coefficient t in (2.2) indicates a decrease in the proportion of negative messages over time. for a time series with a neutral attitude to the construction, the following model was built: ntrm (t) = 0.98 0.01t (2.3) where ntrm (t) is a model value for the proportion of neutral messages at time t. a negative sign before the coefficient t in (2.3) indicates a decrease in the proportion of neutral messages over time. the adequacy of models (1), (2), (3) was assessed using standard methods. in particular, for model (3), the multiple coefficient of determination r= 0.7, the hypothesis of the model inadequacy according to fisher was rejected at a significance level of p=0.02. these calculations indicate a fairly good quality of model (3). the results of checking the adequacy of models (2) and (3) showed the satisfactory quality of the models. based on the prediction using the constructed models, one can claim that there is no conflict. 2.3.3 topological analysis of time series data for the calculations, the tda rstudio package was used, which has some tools for topological data analysis. before calculations, the entire time interval was divided into several intervals with the same step. the effectiveness of this approach and algorithm was proposed in the work of the author of the paper [13]. at the next step, for each time section and for each time series, persistence diagrams and barcodes were built, and estimates of euler’s characteristics were calculated. below are the results for one time interval. on fig. 2.13 we presented the main results of the topological data analysis for the time series {p(t)}. fig. 2.13 a) shows the dynamics of the time series {p(t)}. in this plot, the abscissa is time, and the ordinate is the value of p(t). fig. 2.13 b) shows the persistence diagram of the time series {p(t)}. the component time of birth is plotted along the abscissa axis, and neural network technologies and topological analysis of social media data 109 copyright ©2021 assa. adv. in systems science and appl. (2021) the component time of death is plotted along the ordinate axis. on the diagram, points correspond to zero-dimensional simplexes, triangles to one-dimensional simplexes. the point on the diagram has the values of the component birth and death as coordinates. fig. 2.13 c) shows the barcodes for each simplex. the black line corresponds to zero-dimensional simplexes, the red line corresponds to one-dimensional simplexes. the length of the barcode corresponds to the lifespan of the simplex. based on the calculated values, the estimate of the euler’s characteristics for the series {p(t)} according to formula (1) is as follows: 𝜒 = −0.56 . fig. 2.13 a) plot of the time series {p(t)}; fig. 2.13 b) persistence diagram of {neg(t)}; fig. 2.13 c) barcode diagram of р(t) on fig. 2.14, we presented the main results of the topological data analysis for the time series {neg(t)}. based on the calculated values, the estimate of the euler’s characteristics for the series {neg(t)} according to formula (1) is as follows: 𝜒 = 1.77 . 110 n. gabdrakhmanova, m. pilgun copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 2.14 a) plot of the time series {neg(t)}; fig. 2.14 b) persistence diagram of {neg(t)}; fig. 2.14 c) barcode diagram of {neg(t)}. on fig. 2.15, we presented the main results of the topological data analysis for the time series {ntr(t)}. based on the calculated values, the estimate of the euler’s characteristics for the series {ntr(t)} according to formula (1) is as follows: 𝜒 = −0.73 . neural network technologies and topological analysis of social media data 111 copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 2.15 a) plot of the time series {ntr(t)}; fig. 2.15 b) persistence diagram of {ntr(t)}; fig. 2.15 c) barcode diagram of {ntr(t)}. from the above studies, we see that the highest value (1.77) of the euler’s characteristics corresponds to a number of message rates with a negative attitude towards the project. from the dynamics of this series, we observe rare outliers. the time series has a negative trend. these studies suggest the absence of a negative attitude towards the project. for a number of message rates with a positive attitude towards the project, the value of the euler’s characteristics is less (-0.56), the time series has a positive trend. consequently, the positive attitude towards the project grows. for the series with a neutral attitude towards the project, the value of the euler’s characteristics is even less (-0.73), and the time series has a negative trend. based on these studies, we believe that the euler’s characteristics in combination with regression and autoregression models make it possible to classify time series to solve the problem set. 0 0,5 1 1,5 0 5 10 15 20 25 30 ntr 112 n. gabdrakhmanova, m. pilgun copyright ©2021 assa. adv. in systems science and appl. (2021) 3. conclusion analysis of the data showed the absence of aggression, negative attitude towards the construction of the fiztech subway station. most of the actors express a neutral and positive attitude towards the project implementation. after solving the main problem that concerned the residents – that is, changing the subway type from elevated to underground, – the project is positively perceived by the residents as a planned improvement of the transport situation. also, the residents were promised the construction of intercept parking lots near the new stations, since the residents fear that after the appearance of new subway stations, their yards will be filled with unauthorized cars. taking into account the fact that the residents of the district have been waiting for the construction of the subway station for decades, the project justifiably receives social approval. the calculation results obtained using the constructed mathematical models confirmed the absence of a conflict. the algorithms developed using tda methods show the potential of using these methods to solve similar problems. references 1. batyrshin, i., sheremetov, l., zadeh, l.a. (eds.) (2007). perception-based data mining and decision making in economics and finance. berlin, germany: springer-verlag berlin heidelberg. 2. chaudhuri, s., bhardwaj, a. (2018). kinesthetic perception. a machine learning approach. singapore: springer singapore. 3. ferreira, j.f., dias, j.m. (2014). probabilistic approaches to robotic perception. cham, switzerland: springer international publishing. 4. kharlamov a.a., pilgun m. (eds.) neuroinformatics and semantic representations. theory and applications. newcastle upon tyne: cambridge scholars publishing. 5. les, z., les, m. (2020). machine understanding. machine perception and machine perception mu. cham, switzerland: springer international publishing 6. liu, h., sun, f. (2018). robotic tactile perception and understanding. a sparse coding method. singapore: springer singapore. 7. gabdrakhmanova, n., pilgun m. (2020) development of unified approaches to building neural network and mathematical models based on digital data. advances in systems science and applications, 20(4), 113-124. https://doi.org/10.25728/assa.2020.20.4. 8. h. edelsbunner, d. letscher, a. zomorodian (2002) topological persistence and simplification, discrete comput. geom., 28, 511–533. mr 1949898 9. carlsson, g., zomorodian, a. (2004) computing persistent homology, proc. 20th ann. sympos.comput. geom, 347–356. 10. d. cohen-steiner, h. edelsbrunner, j. harer (2007) stability of persistence diagrams, discrete &computational geometry, 37:1, 103–120. mr 2279866 11. bubenik, p. (2015). statistical topological data analysis using persistence landscapes. the journal of machine learning research, 16(1), 77–102. 12. borovikov, v.p., ivchenko, g.i. (2006) forecasting in the statistica system in the windows environment. fundamentals of theory and intensive practice on a computer. finance and statistics. moscow. 13. gabdrakhmanova n. (2018) forecasting time series using topological data analysis, itise 2018 international conference on time series and forecasting proceedings of papers 19-21 september 2018 granada (spain), 1367-1374. adv syst sci appl 2022; 02:21–33 published online at https://ijassa.ipu.ru. information spreading and evolution of non-homogeneous networks natalia m. markovich1*, maksim s. ryzhov1 1v.a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia abstract: information spreading among nodes of directed random networks by means of the linear preferential attachment (pa) schemes and the well-known spread algorithm is considered. the novelty of the paper is that schemes of the linear preferential attachment proposed in wan et al. (2020) for the network evolution are also used here for the information spreading. the spread algorithm proposed for undirected random graphs is adapted to directed graphs. moreover, we deal with non-homogeneous directed networks consisting of nodes whose inand out-degrees have different power law distributions that is realistic for practice and we find communities in a network that spread the information faster. we compare the minimum number of evolution steps k∗ required for the preferential attachment schemes and the well-known algorithm spread to spread a message among a fixed number of nodes. the evolution of the network in time starts from a seed set of nodes. we study the impact of the seed network and parameters of the preferential attachment on k∗ for simulated graphs. real temporal graphs are also investigated in the same way. the pa may be a better spreader than the spread algorithm. this is valid for the sets of the pa parameters with dominating proportions of created new edges from existing nodes to newly appending ones or between the existing nodes only. it is shown both for simulated and real graphs that the communities with the smallest tail indices of the out-degrees and pageranks may spread the message faster than other communities. keywords: non-homogeneous directed network, information spreading time, linear preferential attachment, leading community, tail index 1. introduction the evolution of networks is arising in numerous applications: citation networks [1], the web-page popularity by pagerank during evolutionary changes [2], or the evolution of the network [3]. information spreading, as a message delivery model in the whole network (the full spreading) [4] or in a part of the network (the partial spreading) [5], has an application for the parallel grid calculations in the computation network. in [6] the selection of a leading community in a network and the comparison of the spreading time by the latter and other communities are observed by an example of homogeneous geometric graphs. following [4] we consider nodes with asynchronous clocks. by the proposed model nodes of the network may spread their messages among other nodes by clicks of a global clock. the latter clicks are modelled as poisson process and the inter-arrival times between clicks are therefore exponentially distributed. thus, the spreading time required to spread messages may be calculated as a sum of a random number of exponentially distributed random variables. generally, the clicks can be generated by another random process with another distribution of the inter-click times. in this paper we use the poisson model and interpret the spreading ∗corresponding author: nat.markovich@gmail.com, markovic@ipu.rssi.ru 22 n. m. markovich, m. s. ryzhov rate in terms of a minimum number of clicks (or steps) required to spread the information among a part of nodes of the network. in order to evolve the network starting from a seed set of nodes containing at least one node, we use a linear preferential attachment (pa) schemes proposed in [3]. we consider the pa as a spreading model additionally to its evolution function and compare it with the algorithm spread proposed for spreading purposes in [4]. we aim to consider non-homogeneous networks containing subgraphs with stationary distributed inand out-degrees of nodes. to this end, we partition the network into communities and use these communities as such subgraphs. a community is a set of highly connected nodes which are weak connected with nodes outside the community. then we aim to find leading communities that may spread the information faster than the rest of the network. on each evolution step, a configuration of the evolving network may change due to newly appended directed edges that may change the distribution of the inand out-degrees of nodes. apart of the in-degrees, we consider pageranks as influence indices of the nodes. summarizing, the following novelties are implemented. firstly, the information spreading is studied in a directed evolving graph where a probability to create new edges is provided by the linear pa schemes. the spread algorithm proposed in [4, 5] for undirected stationary graphs is modified for directed graphs. regarding the pa schemes, the probability to create newly appended edges may depend on the inor (and) out-degrees and specific pa parameters. secondly, a basic graph for the spreading and evolution is non-homogeneous as it is made of subgraphs with different distributions of node characteristics. this may lead to a different nodes’ ability to spread information for different sub-graphs of the network. finally, the impact of the heaviness of the distribution tail on the spreading rate is demonstrated on a range of real graphs. the paper is organized as follows. in section 2 related works and a necessary methodology are provided. in section 3 we investigate simulated graphs regarding the spreading capacity of stationary parts of the network. in section 4 the impact of a seed network on the spreading rate is studied for a real graph. a set of real graphs are compared in section 5. the exposition is finalized with some conclusions. 2. preliminaries 2.1. information spreading the spread algorithm proposed in [4] models a possible information spreading in undirected graphs. let g = (v,e) be the undirected graph with a set of nodes v and edges e. considering an asynchronous time model (there are not quite synchronized clocks in all nodes), a node may initiate a communication by ticks of a global clock which are modelled as a poisson process of rate n = |v |, [4, 5]. let k ≥ 0 denote the index of a tick, on which at most one node can receive messages by communicating with another node. on a tick one of n nodes of the graph is chosen uniformly. then this node i chooses a node j uniformly among it neighbors with probability pij = 1/dmax, where dmax = maxi∈v di, di is the degree of node i. in [4, 5] nodes share with each other all information they have. here, we consider the case when only a single message is spreading in the network. in algorithm 1 we slightly change the spread with this respect. to apply the spread algorithm in directed graphs it is plausible to assume that the node i may share its message with node j, if there is a directed edge (i→ j) from i to j. thus, we use pij = 1/oi, where oi is the out-degree of the node i. copyright © 2022 assa. adv syst sci appl (2022) information spreading and evolution of non-homogeneous networks 23 algorithm 1 the spread • at the tick k a node i is uniformly chosen in the graph, and it initiates a communication. • the node i chooses a node j uniformly among its neighbors without the message with probability pij = 1/oi and sends its single message to the node j. 2.2. preferential attachment now we consider the information spreading in directed graphs. let us recall the linear pa model that is proposed and studied in [3, 7]. it starts with an initial directed graph g(k0) with at least one node and k0 ∈ n edges. for the non-negative parameters α, β, γ such that α + β + γ = 1 holds, and specific parameters ∆in,∆out, the model constructs a growing sequence of directed random graphs g(k) = (v (k), e(k)) depending on the step k. a graph g(k) is generated from g(k − 1) by adding a directed edge. furthermore, n(k), ik(w) and ok(w) denote the number of nodes, inand outdegrees of node w in the graph g(k) with k edges. the following three scenarios of the edge creation, which are called α−, β− and γ− schemes, respectively, are proposed in [3, 7]. to select schemes, one has to generate an i.i.d. sequence of trinomial random variables with values 1, 2 and 3 and the corresponding probabilities α, β and γ, see algorithm 2. algorithm 2 the linear preferential attachment schemes • by the α− scheme we append a new node wnew and create an edge (wnew → w) with probability α, where the existing node w ∈ v (k − 1) is chosen with probability p (choose w ∈ v (k − 1)) = ik−1(w) + ∆in k − 1 + ∆inn(k − 1) . • by the β − scheme the existing nodes w1, w2 ∈ v (k − 1) are chosen independently with probability p (choose w1, w2 ∈ v (k − 1)) = ok−1(w1) + ∆out k − 1 + ∆outn(k − 1) · ik−1(w2) + ∆in k − 1 + ∆inn(k − 1) and a new edge (w1 → w2) is appended with probability β. • by the γ − scheme the existing node w ∈ v (k − 1) is chosen with probability p (choose w ∈ v (k − 1)) = ok−1(w) + ∆out k − 1 + ∆outn(k − 1) , and an edge (w → wnew) is appended with probability γ. this algorithm implies that n(k) = n(k − 1) for β-scheme and n(k) = n(k − 1) + 1 for the others. these scenarios realize a ’rich-get-richer’ mechanism, when a node with a large number of in-/outedges can likely increase them with a high probability. as mentioned in [3], such model can create multiple edges between pairs of nodes and self loops. the parameters ∆in and ∆out may be estimated by the semi-parametric extreme value method (ev) based on the maximum-likelihood method or by the snapshot (sn) method proposed in [8]. the latter is summarized in algorithm 3. the snapshot graph g(n) = (n(n), e(n)) represents a point in time when the data about the graph are available. copyright © 2022 assa. adv syst sci appl (2022) 24 n. m. markovich, m. s. ryzhov algorithm 3 snapshot method 1. let us estimate β̂sn = 1−n(n)/v, where v = |e(n)|. we obtain ∆̂0 in and ∆̂0 out by solving the equations ∞∑ l=1 n in >l v l l + ∆̂0 in (1 + ∆̂0 in(1− β̂sn)) = n in 0 v + β̂sn 1− n in 0 n ∆̂0 in 1+(1−β̂sn )∆̂0 in , ∞∑ l=1 n out >l v l l + ∆̂0 out (1 + ∆̂0 out(1− β̂sn)) = nout 0 v + β̂sn 1− nout 0 n ∆̂0 out 1+(1−β̂sn )∆̂0 out , where n in l and n out l are the number of nodes with inand out-degree equal to l, and thus, n in >l = ∑ l′>l n in l′ , n out >l = ∑ l′>l n out l′ are the number of nodes with inand out-degree larger than l. 2. estimate α̂0 and γ̂0 by α̂0 = n in 0 v + β̂sn 1− n in 0 v ∆̂0 in 1+(1−β̂sn )∆̂0 in − β̂sn , γ̂0 = nout 0 v + β̂sn 1− nout 0 v ∆̂0 out 1+(1−β̂sn )∆̂0 out − β̂sn . 3. re-normalize the probabilities (α̂sn , β̂sn , γ̂sn) = ( α̂ 0(1−β̂sn ) α̂0+γ̂0 , β̂sn , γ̂ 0(1−β̂sn ) α̂0+γ̂0 ). obtain ∆̂sn in and ∆̂sn out by solving equations ∞∑ l=0 n in >l/v l + ∆̂sn in − 1− α̂sn − β̂sn ∆̂sn in − (α̂sn + β̂sn)(1− β̂sn) 1 + (1− β̂sn)∆̂sn in = 0, ∞∑ l=0 n out >l /v l + ∆̂sn out − 1− γ̂sn − β̂sn ∆̂sn out − (γ̂sn + β̂sn)(1− β̂sn) 1 + (1− β̂sn)∆̂sn out = 0. 2.3. tail index let {xn}n≥1 be a stationary sequence of independent identically distributed (i.i.d) random variables (r.v.s) with distribution function (df) f (x). the parameter αti is called the tail index (ti). it may be estimated with the hill’s estimator [9] α̂h(k) = ( 1 k k∑ i=1 log( x(n−i+1) x(n−k) ) )−1 , (2.1) where x(1) ≤ x(2) ≤ ... ≤ x(n) are the order statistics corresponding to the sample and the parameter k is a number of the largest order statistics. α̂h(k) is derived assuming that the distribution tail is regularly varying with the tail index αti , i.e. f (x) = p{x1 > copyright © 2022 assa. adv syst sci appl (2022) information spreading and evolution of non-homogeneous networks 25 x} = x−αti`(x), where `(x) is a slowly varying function. the optimal value of k may be obtained by a smoothing method of the asymptotic mean squared error of the hill’s estimator (samsee) [9]. the confidence interval around the optimal ti value is obtained by the bootstrap procedure with the 95% quantile. 2.4. test of stationarity the inhomogeneity is one of the problems of the statistical analysis of graphs. this means that the heaviness of the power law tail distributions of node characteristics, like inand outdegrees and pageranks may be different within the graph or its communities. to our best knowledge, there are no stationarity tests for graphs since there is no numeration of nodes in the graphs. such a numeration may be determined by random walks used in graphs as sample tools. at the same time, there are stationarity tests for random sequences. one of the simplest ways to solve the problem is to use a number of random walks within the graph and to test the stationarity of the obtained sequences of the gathered nodes. we use the stationarity test statistic v/s = vn/ŝ 2 n,q, vn = 1 n2  n∑ k=1 (s∗k) 2 − 1 n ( n∑ k=1 s∗k )2  , ŝ2 n,q = q−1 q∑ i,j=1 γ̂i−j, (2.2) s∗k = ∑k j=1(xj −xn), γ̂j = n−1 ∑n−j i=1 (xi −xn)(xi+j −xn), 0 ≤ j < n, where the bandwidth q = qn satisfies q →∞, q/n→ 0 proposed in [10] for stochastic processes with the stationary distributed short range dependent noise. the null hypothesis of stationarity is rejected, if v/s > cρ, where cρ is a quantile of the asymptotic distribution function of the kolmogorov’s statistic fk(π √ x). cρ ∈ {0.190, 0.153} holds for significant level ρ ∈ {5, 10}%, respectively. 2.5. a node’s pagerank by google’s definition [11] the pagerank is determined as the rank of node (web page) i by ri = ∑ j→i c oj rj + (1− c)qi, i = 1, n, (2.3) where the sum is taken over all pages with incoming links to node i (j → i implies that node j is linked to node i, i.e. (j, i) ∈ e). the number of such pages constitutes the in-degree of the node i. oj denotes the out-degree of the node j, i.e. the number of its outgoing links. c ∈ (0, 1) is a damping factor, i.e. a probability to browse a web-page connected with the current one that is set by google as c = 0.85. qi ≥ 0 is a personalization probability of node i. the pagerank is a numeric measure of inter-relations between nodes that reflects the local network structure. to estimate the pagerank we use the iterative formula r̂ (n,0) i = 1, r̂ (n,k) i = ∑ j→i c dj r̂ (n,k−1) j + (1− c), k ∈ n, (2.4) proposed in [12] for a given uniform personalization vector qi = 1/n, 1 ≤ i ≤ n = |v |. the iteration (2.4) is proceeding until the difference between two consecutive iterations |r(n,k) i −r(n,k−1) i | will be small enough which is sufficient for a moderate number of iterations k. in [13], it is proved that pageranks of nodes in a directed graph received by the pa are power law distributed. the tail of the latter distribution is heavier than the tail of the limiting in-degree distribution. copyright © 2022 assa. adv syst sci appl (2022) 26 n. m. markovich, m. s. ryzhov 3. simulation graphs let us describe the information spreading mechanism by means of the pa. let the initial directed graph g(k0) with n0 nodes and k0 edges be a seed set of nodes having a message to be spread. at global poissonian clock ticks, we do a step of the pa (section 2.2) with predefined parameters α, β, γ,∆in,∆out. the latter parameters may be estimated by inand out-degrees data sets as proposed in section 3.3 in [3]. a newly appended edge may increase the number of nodes receiving the message. the message can be delivered from the node i to the node j, if the directed edge (i→ j) is created from the node i to the node j which has not this message. this is possible only if the edge is created by the β− or γ− schemes. the α−scheme provides edges with the opposite direction, namely, from the newly appearing node without the message to the existing node. indeed, the message can be transmitted from the node having this message, only. at the kth clock tick we receive a graph g(k) = (v (k), e(k)) with |e(k)| = k + k0 edges and n(k) ≤ n0 + k nodes. the pa schemes in [3] may generate multiple edges and self loops for some nodes. as a result, messages may get a stuck in these nodes and the corresponding spreading time increases. we compare an ability to spread the information by the pa schemes and by the spread algorithm given in section 2.1. the graph is first simulated by the pa and the spread is operating in the prepared graph. we use the following parameters for the pa. the α, β and γ are taken within the interval [0.04, 0.96] with the step 0.04, such that α + β + γ = 1, and ∆in = ∆out = 1 holds. 100 simulated graphs are provided for the each set of the parameters. the graphs evolve starting from the initial triangle, that is a three nodes connected to each other, i.e. k0 = 3, n0 = 3 hold. at least one of the nodes has a message to spread. selecting this triple of nodes within one of the communities of the network, we aim to find, what community may spread the message faster. let us assume that the number of steps k is limited as k ≤ k ′. we define the number of clock ticks required to disseminate the message from an initial node to n nodes with probability not less than 1− δ as k∗(n, δ) = inf{0 < k ≤ k ′ : pr(|s(k)| = n) > 1− δ}, δ ∈ (0, 1). here, s(k) denotes the number of nodes which received the message at step k. let us take k ′ equal to 5000. then s(k∗) < n means that k ′ steps of the evolution are likely not enough to disseminate the message to n nodes. the delivery delay is not less than ∑k′ i=1 τi since the inter-arrival times {τi} between the evolution steps are exponential distributed and the message cannot be spread if the corresponding edge is created by the α− scheme. results of the comparison of the pa and the spread algorithms for n = 100 nodes are presented in fig. 3.1. fig. 3.1 (left) shows the average 〈k∗〉 against the average 〈q(k∗)〉 over 100 simulated graphs, where q(k) = |s(k)| n(k) is a proportion of nodes that received the message in the graph g(k). the q(k) that is close to zero means a small part of nodes received the message at the tick k. q(k) = 1 means that all nodes in the graph have received the message. in the top line of fig. 3.1, we obtain for the pa that the spreading rate depends on the β value for the same γ. the β value near 0 corresponds to the information spreading mostly to newly appearing nodes by the γ−schema. the β value near 1 means the information is delivered mostly between existed nodes by the β−schema. these situations may need the same tick number for the spreading. for the spread, values of the β and γ affect the number of ticks. the β value near 1 corresponds to the large connections between nodes in some sets, which create the information ”stucking” in nodes. however, for the larger γ 〈k∗〉 becomes less dependent on the β. in fig. 3.1 (middle) we show the proportion of the {s(k∗) < n} events for the α, β, γ values. for the pa, these events are extremely rare except the case of the value γ ≤ 0.04. this copyright © 2022 assa. adv syst sci appl (2022) information spreading and evolution of non-homogeneous networks 27 fig. 3.1. the average clock ticks 〈k∗〉 against the average 〈q(k∗)〉 (left); the parameter β against the probability of s(k∗) < n event (middle); 〈k∗〉 against β. 100 simulations with the same parameters are taken for the pa schemes (top line) and the spread algorithm (bottom line). in the left figures, the point sizes are proportional to the value of β ∈ [0.04, 0.96], in middle and right figures to the value of γ ∈ [0.04, 0.96]. is the opposite to the spread, where the increase of the β value causes the increase of the {s(k∗) < n} event probability. in fig. 3.1 (right) we show 〈k∗〉 for the α, β, γ values. for the pa and the spread, we receive the same result that the decrease of the β value allows to spread the message quicker. this corresponds to a graph with a less number of edges between the nodes, in which the information may spread quicker. as a result, we notice that the choice of the set α, β, γ can make the spreading by the pa more effective than by the spread. fig. 3.2 shows the impact of the parameters β and γ on log(k∗pa/k ∗ spread) as an averaging over 100 resamples. let us recall that α is calculated as α = 1− β − γ. we define the parameter sets corresponding tok∗pa = k∗spread,k∗pa > k∗spread andk∗pa < k∗spread. it can be seen, that the pa can spread a message better if γ > 0.51 or β > 0.6 holds. the options when γ + β > 1 were not considered because α + β + γ = 1 holds. 4. investigation of real graphs here, we study the impact of a seed network having the message on k∗(n, δ) by real data. we investigate the graph obtained by the berkeley-stanford dataset with 685230 nodes and 7600595 edges [14], which represents web pages from berkely.edu and stanford.edu domains that are connected in a union network by directed edges as hyperlinks between them. within this network we select a small part and partition it into communities. each of the latter communities may be used as the seed network. starting from the seed network we apply the pa schemes to evolve the graph. using the obtained evolved graph we apply the spread algorithm and compare its capacity with the pa with regard to the spreading rate. the underlying directed graph g = (v,e) is partitioned into communities {ci}5 i=1,⋃5 i=1ci = v , ci ⋂ cj = ∅, i 6= j, by means of the directed louvain’s algorithm in such a way that the graph modularity is maximized [15]. the modularity is a dependence measure of the graph that is determined by the identified communities. the modularity is large when communities are made of highly connected nodes. we calculate the pagerank of each node of the underlying graph by the iterative formula (2.4). the result of the graph partitioning into communities is shown in fig. 4.3. since some copyright © 2022 assa. adv syst sci appl (2022) 28 n. m. markovich, m. s. ryzhov fig. 3.2. the dependence of log(k∗ pa/k ∗ spread) with the minimum number of steps k∗ pa and k∗ spread for the pa and spread algorithms, respectively, required to propagate a single message to n = 100 new nodes regarding the pa-parameters of the graph evolution. the red line indicates the case k∗ pa = k∗ spread, to the left of the line is k∗ pa > k∗ spread, to the right of the line is k∗ pa < k∗ spread. communities are highly connected, we may combine such communities together and consider c1 ∪ c3 ∪ c5 and c2 ∪ c4. however, the pageranks of the latter combined communities may be non-stationary distributed. fig. 4.3. communities {ci}5i=1 built by the berkeley-stanford data, where the point sizes are proportional to the node pageranks. we aim to study the impact of the heaviness of tails of the node characteristics, namely, the inand out-degrees and pageranks on the spreading rate. the inand out-degrees are known to be power law distributed [1]. the pageranks are proved to be regularly varying distributed [16]. hence, the tis of node characteristics of each community and their mergers may be estimated by the hill’s estimator (2.1). indeed, this requires their stationarity that, copyright © 2022 assa. adv syst sci appl (2022) information spreading and evolution of non-homogeneous networks 29 |c| v/sin v/sout v/spr c1 266 0.0579 0.1040 0.0764 c2 266 0.0580 0.1058 0.0828 c3 135 0.0773 0.0827 0.0774 c4 86 0.0605 0.0632 0.0652 c5 85 0.0439 0.0757 0.0457 table 4.1. the v/s statistics (2.2) averaged over 100 random sequences and calculated by the inand outdegrees and pageranks of the communities {ci}5i=1 and denoted by v/sin, v/sout and v/spr, respectively. fig. 4.4. empirical distribution functions fn(x) of pageranks of communities {ci}5i=1 and their mergers c2 ∪ c4 and c1 ∪ c3 ∪ c5. in practice, cannot be precisely fulfilled due to the inhomogeneous data. the finding of homogeneous communities is an unsolved problem that is out of scope of this paper. in table 4.1, the v/s statistics (2.2) and the sample sizes of the communities denoted as |c| are shown. we use 100 sequences of pageranks which correspond to possible numerations of nodes in the graph built by random walks. the latter are formed by uniform selection of each next node among nearest neighbors of a node. random walks cover all nodes in the communites. the values in table 4.1 do not contradict the null hypothesis of stationarity, but one cannot insist on the stationarity. |c| α̂in α̂out α̂pr c1 266 0.895 (0.616, 1.175) 1.141 (1.035, 1.247) 0.474 (0.403, 0.544) c2 266 0.899 (0.629, 1.169) 1.141 (1.035, 1.247) 0.485 (0.444, 0.526) c3 135 1.117 (1.039, 1.195) 5.434 (5.351, 5.518) 0.667 (0.549, 0.786) c4 86 0.387 (0.2, 0.576) 3.712 (3.43, 4.009) 0.706 (0.3, 1.112) c5 85 0.425 (0.27, 0.58) 3.719 (3.427, 4.011) 1.798 (1.497, 2.099) c1 ∪ c3 ∪ c5 486 0.885 (0.669, 1.101) 3.236 (3.16, 3.312) 0.592 (0.523, 0.661) c2 ∪ c4 352 0.76 (0.474, 1.045) 1.48 (1.445, 1.515) 0.54 (0.478, 0.602) table 4.2. the ti estimates of the node inand out-degrees and pageranks of the communities {ci}5i=1, c1 ∪ c3 ∪ c5 and c2 ∪ c4 denoted as α̂in, α̂out and α̂pr, respectively, with their bootstrap confidence intervals in brackets. in table 4.2, the tis with their bootstrap confidence intervals are shown. the tis are estimated by the hill’s estimator (2.1) where the number of the largest order statistics k is calculated by the samsee method [9]. empirical distribution functions fn(x) of pageranks for communities and their mergers are shown in fig. 4.4. they are in agreement with table 4.2. one can see that the heaviness of tail of the merges is determined by the dominating community in the merger, namely, copyright © 2022 assa. adv syst sci appl (2022) 30 n. m. markovich, m. s. ryzhov the community with the smallest ti. really, the pagerank distribution and the ti α̂pr of the communityc1 are close to those ones ofc1 ∪ c3 ∪ c5. the same is valid for the communities c2 and the c2 ∪ c4. moreover, c1 and c2 have close values of the tis despite they are weak connected and the smallest tis of the out-degrees and pageranks. one may expect that the nodes of these communities can be the best spreaders. using the communities {ci}5 i=1 as seed networks for further evolution, we provide 50 simulations of the evolved graphs by the pa schemes. we check how the spread algorithm and the pa schemes spread one message from each node in {ci}5 i=1 to the first arbitrary 100 nodes. we examine values of k∗ for the parameters (α, β, γ) ∈ {(0.4, 0.2, 0.4), (0.3, 0.1, 0.6)} and ∆in = ∆out = 1 of the pa schemes. the number of the pa steps is upper bounded by k ′ = 35 · 104. the resulted triples (mink∗, 〈k∗〉,maxk∗) with the minimum, average and maximum ofk∗ over 50 simulations of the application of the spread algorithm and the pa schemes to each node of communities {ci}5 i=1 are presented in table 4.3. (α, β, γ) algorithm community (0.4, 0.2, 0.4) (0.3, 0.1, 0.6) c1 (18866, 128739.0, 232350) (31675, 158485.4, 322566) c2 (46166, 127821.3, 263166) (27075, 148624.7, 312100) spread c3 (44242, 128236.4, 214050) (37766, 149498.0, 305433) c4 (62450, 120647 .3 , 233325) (44366, 140530 .3 , 250233 ) c5 (57900, 125888.5, 189200 ) (82966, 159207.1, 332700) c1 (2808, 47019.0, 184560) (4804, 54782.3, 166520) c2 (3048, 46940.4, 190466) (3907, 55853.6, 161633) pa c3 (3061, 32599 .5 , 103911 ) (6461, 41223 .9 , 92542 ) c4 (3960, 39293.6, 116357) (6149, 44848.1, 141400) c5 (2885, 42192.8, 117966) (7470, 48430.5, 130366) spread c1 ∪ c3 ∪ c5 (18866, 128100.8, 232350 ) (31675, 156110.2, 332700) c2 ∪ c4 (46166, 126068 .6 , 263166) (27075, 146630 .1 , 312100 ) pa c1 ∪ c3 ∪ c5 (2808, 42169 .5 , 184560) (4804, 49905 .2 , 166520) c2 ∪ c4 (4804, 54782.3, 166520) (3907, 53164.7, 161633 ) table 4.3. the triples (mink∗, 〈k∗〉,maxk∗) corresponding to graphs generated by the pa schemes with two sets of parameters where the minimum values of mink∗ are marked by bold, the minimum values of 〈k∗〉 and maxk∗ by italic. conclusions from table 4.3 are the following. the communities c1 or c2 have the smallest values of minimum k∗ for each set of parameters and each spreading algorithm. both of them have the lowest tis αout and αpr, see table 4.2. since the ti of the out-degrees is smaller than 2, the out-degrees have an infinite variance. the nodes with such large numbers of outgoing links are in fact the best spreaders. the minimum averages and maximum values of k∗ relate to c3 and c4 with lighter distribution tails due to the relatively larger tis. 5. comparison of spreading models for real graphs here, we investigate the effectiveness of the spread and the pa spreading models (see, sections 2.1 and 2.2) for real graphs. the temporal graphs (i.e. networks where edges have timestamps†) provided in [17] were used as an example. among them there are graphs of messages and comments from websites (sx-mathoverflow, sx-askubuntu, collegemsg), †for instance, a directed edge (u, v, t) means that person u sent an e-mail to person v at time t. copyright © 2022 assa. adv syst sci appl (2022) information spreading and evolution of non-homogeneous networks 31 graphs of bitcoin transactions (soc-sign-bitcoin-otc, soc-sign-bitcoin-alpha) and graphs of email communication (email-eu, email-eu-dept1 and others). their description can be found in table 5.4. name number of number of description nodes temporal edges sx-mathoverflow 24818 506550 comments, questions, (m-overlow) and answers on math overflow sx-askubuntu 159316 964437 comments, questions, (askub) and answers on ask ubuntu email-eu 986 332334 e-mails between users (eu) at a research institution email-eu-dept1 (eu-dept1) 309 61046 email-eu-dept2 (eu-dept2) 162 46772 email-eu-dept3 (eu-dept3) 89 12216 email-eu-dept4 (eu-dept4) 142 48141 collegemsg 1899 20296 messages on a facebook (colmsg) -like platform at uc-irvine soc-sign-bitcoin-otc 5881 35592 bitcoin otc web of (bit-otc) trust network soc-sign-bitcoin-alpha 3783 24186 bitcoin alpha web of (bit-alpha) trust network table 5.4. temporal networks from [17] with their description. the size of some graphs makes it computationally difficult to model the information spreading by the pa and spread directly. therefore, we conducted our evolutionary simulations starting from a single node to spread one message to 100 new nodes. for each real graph the parameters of the pa model are evaluated by means of the snapshot method (see, algorithm 3) and denoted as (α̂sn , β̂sn , γ̂sn , ∆̂sn in , ∆̂ sn out ). the effectiveness of both the spread and pa algorithms is compared for different values of the latter parameters. the results are presented in fig. 5.5, left, where ∆̂sn in and ∆̂sn out are found to be close to 1 for all graphs. moreover, we estimate the tis of the inand out-degrees of nodes of all real graphs which are presented in fig. 5.5, right. fig. 5.5. the minimum log(k∗ pa/k ∗ spread) for each graph achieved for the snapshot estimates β̂sn and γ̂sn of the pa parameters, where the red line indicates the case k∗ pa = k∗ spread, the area to the left of the line corresponds to k∗ pa > k∗ spread, and to the right of the line to k∗ pa < k∗ spread, the condition α̂sn + β̂sn + γ̂sn = 1 is fulfilled for the area to the left of the dotted line (left); the tis of the in-degrees α̂in and out-degrees α̂out for the real networks (right). for most of the investigated graphs the pa delivers messages faster, with the exception of the bitcoin graphs (soc-sign-bitcoin-otc and soc-sign-bitcoin-alpha). the tis of the in-degrees copyright © 2022 assa. adv syst sci appl (2022) 32 n. m. markovich, m. s. ryzhov of the latter graphs are less than 1, α̂in < 1. this implies that their in-degrees have an infinite variance according to the properties of the power law distribution. the pageranks may have an even smaller ti due to [13]. such likely large in-degrees appear in the bitcoin graphs due to the dominating effect of the α− scheme. that is a consequence of the small values of β and γ for these graphs, see fig. 5.5, left. since the α− scheme creates a new edge directed from the newly appending node to a node existed before, it cannot increase the number of nodes receiving the message at the evolution step. this shows the impact of the heaviness of tail of the node influence indices on the spreading rate. 6. conclusions the novelties of this paper are that the linear pa schemes are applied for the information spreading purpose, the spread algorithm is reconsidered for directed graphs and both the pa and the spread are applied to non-homogeneous graphs. a message from one node is spreading to a fixed number of nodes in the network. the information spreading is investigated both for simulated (section 3) and real (section 4) non-homogeneous graphs. the nodes in the latter graphs may have different distributions of their inand out-degrees. we compare the pa and the spread algorithm on directed graphs, which may contain cycles and multiple edges generated by the pa with different sets of parameters. considering the simulated graphs which are assumed to be homogeneous, one may conclude that the pa may be the better spreader than the spread algorithm for such sets of its parameters (α, β, γ) where α is sufficiently small. the rest of the parameters (∆in,∆out) were taken equal to 1. regarding the real non-homogeneous graphs consisting of the interconnected communities of nodes with different tail indices of the inand out-degrees and pageranks, we found that some nodes in the community with the smallest tail index of the out-degrees and pageranks may spread the message faster than other nodes. we have classified the real temporal graphs in section 5 by the number of steps required to disseminate a message from one node to a fixed number of nodes. it was found that the pa is a better spreader than the spread tool for most of the considered graphs for which the pa α-scheme is not significant. the latter conclusion is plausible since the α-scheme creates new edges directed from the new nodes to the existing ones that cannot be useful for information spreading. acknowledgements the reported study was funded by the russian science foundation rsf, project number 22-21-00177 (recipient n.m. markovich, conceptualization, methodology development, formal analysis, writing – original draft preparation; recipient m.s. ryzhov, software, data validation). references 1. newman, m. e. j. 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(2014). snap datasets: stanford large network dataset collection, https://snap.stanford.edu/data. copyright © 2022 assa. adv syst sci appl (2022) https://snap.stanford.edu/data introduction preliminaries information spreading preferential attachment tail index test of stationarity a node's pagerank simulation graphs investigation of real graphs comparison of spreading models for real graphs conclusions microsoft word 1264-article text-5509-1-18-20221229 adv syst sci appl 2022; 04; 116-125 published online at http://ijassa.ipu.ru. building a digital twin of agricultural production based on mathematical modeling of sustainability and development processes rifkat sirazetdinov1, rustem sabitov 2, gulnara smirnova3, shamil sabitov4, alexander eponeshnikov5 1), 2), 3) kazan national research technical university named after a.n. tupolev – kai, kazan, russia e-mail: rif-kat@inbox.ru, r.a.sabitov@mail.ru, seyl@mail.ru 4) kazan federal university, kazan, russia e-mail: sh.r.sabitov@gmail.com 5) innopolis university, innopolis, russia e-mail: sashah275@gmail.com abstract: an approach to building a model of a digital twin of agricultural production is proposed. the modeling is based on the principle of sustainable economic development, harmoniously combining two interrelated processes: sustainability and development. development requires expanding external relations, markets, and developing specialization. however, in cases of crisis, interruption in the supply of raw materials, closure of sales markets and other unfavorable circumstances, there should always remain a part of the economy, closed on itself, using its own resources and working for its consumers. the paper considers a structural and mathematical model of sustainable development of the agricultural production process, based on the so-called model of a typical production unit. it is a system of algebraic equalities and inequalities that estimate the power of individual components of the process and are based on balance ratios. within the framework of the proposed model, it is possible to consider the sustainable development of the economy of individual agricultural regions, exploring both components of the region's economy and achieving their harmonious interaction. this model also allows us to consider the sustainable development of individual settlements, rural settlements based on local farms and private households, as well as individual agricultural production. in each case, it is necessary to ensure a harmonious combination of stability, self-sufficiency, at least food independence of farms and the possibility of their development. the proposed models, tuned to a specific online-managed production, can form the basis for the digitalization of agricultural production and serve as a tool to ensure its efficiency and continuous sustainable development. keywords: dynamic production models, industry 4.0, forecasting, agricultural production, digital twin, identification, sustainable development 1. introduction the growing problems associated with the unpredictable emergence of global crisis processes and situations, inconsistencies between sectors of the domestic economy and emerging digital transformations in most areas of activity, actualize the search for effective solutions for the strategic management of the digital transformation process in any organization, including large and medium agribusiness companies. it should be noted that in the modern world, sustainable development of agricultural enterprises is based on the use of digital technologies. however, in the digital economy, the elements of the mechanism of the digital transformation process in building a digital twin of agricultural production… 117 copyright ©2022 assa adv. in systems science and appl. (2022) agricultural organizations remain not fully defined, only a set of factors that determine the objective need for digitalization are constant, namely: ensuring the country's food security by significantly increasing yields and labor productivity, reducing material costs, protecting environment and so on. today, with the transition of the economy to a management system, which is based on the use of a large array of data generated by digital technologies, the issue of revising the targets, principles, mechanisms and tools of modernization in the agricultural industry becomes relevant, since the current practice of digital transformation does not contribute to a technological breakthrough in the studied industry [1]. the rapid and widespread introduction of digitalization, as a natural regularity of the functioning of economic systems in the modern world, presupposes a conceptual and methodological substantiation of the digitalization implementation process, taking into account: the peculiarities of the transition to a digital economy in the agricultural industry and the establishment of the readiness level of large and medium-sized companies in the agribusiness sector management to digital transformation [5,6]. as you know, one of the key areas of digitalization is the use of the so-called digital twin [8], which is based on modeling the production process in real time [2,5]. this paper proposes approaches to the application of mathematical models of agricultural production to build a digital twin of the production process that reflects online the main processes of sustainable development of production and allows you to visually represent the state of the production process and promptly make strategic management decisions. 2. agricultural production in the model of sustainable development of the region the problem of sustainable development [7] of regions is associated with the provision of two parallel processes taking place in the region this is the process of development and the process of ensuring sustainability [9]. development is closely related to the expansion of production, sales markets, cooperation with external suppliers and buyers. however, these processes lead to risks associated with factors external to the region, and therefore poorly controlled by the region. stability is determined by the degree of independence of the economy from external factors [10]. therefore, the less the economy is connected with external suppliers and consumers and the more it is "tied" to the domestic market, the more stable it is. however, in this case, the development of the economy is limited by the volume of the domestic market and stops at some point. hence, it follows that for the sustainable development of the region's economy, it is necessary to harmoniously combine both of these processes. for its development, the region must expand external relations, look for its niche in foreign markets, and specialize in them. however, in the event of a crisis, cessation of supplies from outside of raw materials, components, closure of sales markets and other unfavorable circumstances, a part of the economy should remain in the region, closed on itself, using its own resources and working for its consumers within the region. only in this case can we talk about sustainable development of the region. the sustainable development of a region is closely linked to its economic security. as you know, food security is one of the main directions of ensuring the national security of the country. therefore, agricultural production plays an important role in the sustainable development of the region. at the same time, the sustainable development of the agricultural sector can be considered as an independent problem within the framework of the economic security of the region. agricultural production naturally contains both components of the sustainable development process. first of all, this is the production of products for domestic consumption the consumption of agricultural products by the population of the region itself this process makes a certain contribution to the economic stability of the region. and besides this, the agricultural industry produces products for the external consumer these are grain, 118 r. sirazetdinov, r. sabitov, g. smirnova, s. sabitov, a. eponeshnikov copyright ©2022 assa adv. in systems science and appl. (2022) dairy products, and other types of products supplied to external consumers. and the development of the industry in the direction of markets external to the region, its specialization, works to enhance the economic power of the region as a whole. within the framework of the sustainable development model proposed in this work, it should be noted that the agricultural industry has a sufficiently developed part of ensuring sustainability, since for the most part it uses its own resources land, water bodies and others and works for internal consumption, which gives a certain independence from extra-regional factors. at the same time, there is also a component associated with external markets and allowing active development. 3. mathematical model of agricultural production as a typical production facility the modeling of agricultural production is based on a methodology that uses the concepts of the capacity of complex systems, the dynamics of an economic object, a typical production facility (tpf), as a structural and mathematical model of business processes [11-13,15,16]. when talking about a process, then, as a rule, there is an input through which something enters the process, and, after processing, the result of the process is obtained at the output. however, in addition to this, fixed assets are involved in the process, i.e. the equipment with which the process is carried out, as well as the personnel involved in the implementation of the process [18]. an important role is played by the information and intellectual component, which, in the case of the production process, is technology, technical regulations, licenses, intangible assets, etc. the structural part of the tpf model (fig. 1) is close to the models of business processes of the sadt methodology, including the idef0 standard [14]. however, the tpf, in addition to the structural component, also contains a mathematical model. further, we will consider the tpf in relation to agricultural production, which we will call an agricultural production facility (apf). agricultural production facilities can include both individual agricultural production, farms, and agriculture of a district or region, and even an agro-industrial complex as a whole. fig. 1. typical production facility. let some agricultural production facility produces 𝑛 types of products, including intermediate products, the volumes of which for a given time interval will be denoted by the column vector 𝑋 = (𝑥 , … , 𝑥 ) . some of these products are used for domestic consumption. to describe the processes of consumption, we will apply the model of the inter-industrial balance of v. leontiev (iib) [3]. let us introduce a square 𝑛 𝑥 𝑛 − 𝑚𝑎𝑡𝑟𝑖𝑥 𝐴 , which determines the norms of expenditures on domestic consumption, which is called the matrix of coefficients of direct production costs. let us introduce an m-dimensional column vector 𝑌 = (𝑦 , . . . , 𝑦 ), the components of which represent the volumes of products or services provided by the apf to external consumers. here, in contrast to the iib model, the nomenclature of the external product 𝑌 and internal 𝑋 are different, but the production of the external product is based on the internal product, and the amount of the internal product required to produce the output product is equal building a digital twin of agricultural production… 119 copyright ©2022 assa adv. in systems science and appl. (2022) to ayy, where 𝐴 is the 𝑛 𝑥 𝑚 matrix of consumption rates domestic product per unit of output product. in the special case when the vectors 𝑋 and 𝑌 coincide, 𝐴 is the identity matrix. the volume of domestic production 𝑋 is related to the volume of output 𝑌 by an expression similar to the iib equation: (𝐸 – 𝐴 )𝑋 = 𝐴 𝑌, (3.1) where 𝐸 is the n-dimensional identity matrix. expression (3.1) within the framework of the tpf structure is a mathematical model of the block of the apf production process. consider the rest of the tpf blocks (fig. 1). the “output” block in the tpf structure characterizes the result of the process and determines the requirements for all other blocks of the production process. the output determines the characteristics of the products. it contains both actual information about the release and the required characteristics of the manufactured product. these are the production volumes, market prices and lead times, etc. they entail requirements for the rest of the units of the production unit. then, having set some required values of the output 𝑌 = 𝑌 , it is possible, on the basis of expression (3.1), to determine the requirements for the volume of output of the domestic product 𝑋 . block "entrance" is the process of organizing the supply of raw materials, accessories, components coming from outside. this includes, for example, the purchasing processes that take place in the apf. the stability of this block significantly affects the stability of the whole apf. let's introduce the 𝑟-dimensional vector column 𝑉 = (𝑣 , … , 𝑣 ) of the volumes of products coming from outside. the need for these products is related to the vector 𝑋 using the 𝑟 𝑥 𝑛-matrix of the cost rates 𝐴 : 𝑉 = 𝐴 𝑋 (3.2) the components of the input vector, as a rule, have restrictions and are described by a system of inequalities of the form: 0 < 𝑉 < 𝑉∗, (3.3) where 𝑉∗ is the vector of the maximum possible volumes of input products for the considered time interval. an expression of the form (3) defines a certain set of possible volumes of input products, which in what follows we will call the input power of the system. the block "fixed assets" characterizes the production capacity of atf, which are determined by the availability of this or that equipment. let the enterprise have 𝑞 types of equipment. the productivity of equipment and their number directly determine the possibilities of producing one or another component of the vector 𝑋. let us introduce into consideration the vector 𝐾 = (𝐾 , . . . , 𝐾 ) , the components of which correspond to the number of units of operating equipment of each type. it is assumed that each type of equipment can produce several types of products. the capabilities of each piece of equipment, i.e. the power of a piece of equipment is described by the following inequality: 0 < 𝑎 𝑥 + ⋯ + 𝑎 𝑥 < 1, 𝑖 = 1, … , 𝑞, (3.4) here the coefficients 𝑎 are related to the productivity of the 𝑖 − 𝑡ℎ equipment for the 𝑗 − 𝑡ℎ type of product. then the capacity of the equipment, as the set of all possible options for the production of products 𝑋, is determined by the following system of equalities and inequalities: 0 < 𝑎 𝑥 + ⋯ + 𝑎 𝑥 < 𝐾 , 𝑖 = 1, … , 𝑞, (3.5) 𝑥 = 𝑥 + ⋯ + 𝑥 , 𝑗 = 1, … , 𝑛, 120 r. sirazetdinov, r. sabitov, g. smirnova, s. sabitov, a. eponeshnikov copyright ©2022 assa adv. in systems science and appl. (2022) where the first index of the variable 𝑥 corresponds to the number of the product component, and the second index to the number of the type of equipment on which it is produced. the "staff" block contains mathematical models of the capacity of the atf personnel. staff power, i.e. capacity of atf in personnel, represents a set of all the possibilities of production, limited by the presence and number of personnel of one or another qualification. let 𝑙 be the number of types of qualifications of personnel involved in the production process. let us introduce into consideration the vector 𝐿 = (𝐿 , … , 𝐿 ) , the components of which correspond to the number of employees of a particular qualification. then, introducing into consideration the coefficients 𝑝 , which characterize the productivity of a specialist of the 𝑖 − 𝑡ℎ qualification for the production of the 𝑗 − 𝑡𝑦𝑝𝑒 of product, by analogy with the capacity of the equipment, we obtain the expression for the capacity of the personnel of the atf: 0 < 𝑝 𝑥 + ⋯ + 𝑝 𝑥 < 𝐿 , 𝑖 = 1, … , 𝑙, 𝑥 = 𝑥 + ⋯ + 𝑥 , 𝑗 = 1, … , 𝑛, (3.6) where the first index of the variable 𝑥 corresponds to the number of the product component, and the second index to the number of the type of personnel qualifications. the block "information-technologies" provides mutual coordination of all processes of the enterprise on the basis of main production technologies, methods and management structure. this block determines the production technology, the availability of documentation and rights to one or another type of activity, permission to perform certain types of work, etc. therefore, it determines the structure of the matrices, the values of the coefficients included in the formulas considered above. the influence of this block on production activity is carried out through productivity, determined by the availability of a particular technology, the perfection of the management system, as well as the presence or absence of patents and licenses for certain technologies and activities. thus, expressions (3.1) (3.6) represent a mathematical model of the open source production process. in the proposed mathematical model, the types of available equipment, types of qualifications of specialists, the coefficients of their capacities, as well as the corresponding matrix of cost coefficients 𝐴 , 𝐴 , 𝐴 , should be specified. by specifying the vectors of the number of equipment 𝐾 and personnel 𝐿, as well as the constraints on the input 𝑉∗, we obtain the system power  , which is, ultimately, the set of all possible output vectors 𝑌 in the presence of appropriate equipment, personnel, and external supply capabilities. by setting a certain vector 𝑌 = 𝑌 , corresponding to the needs of the external market, it is possible to determine whether there is enough capacity to produce the volume of output 𝑌 in the presence of equipment 𝐾 and personnel 𝐿 and constraints on external resources 𝑉∗,. if the capacity is not enough, then the model allows you to determine the required number of equipment and personnel that must be added to ensure the production of 𝑌 . 4. an example of building a sustainable development model for an agricultural production system consider, as an illustrative example, a certain livestock farm that breeds cattle and produces meat and milk. let's say there are 𝐾 on a farm. let us denote the volume of annual production of meat 𝑋 and milk 𝑋 𝑋 calves are born on the farm per year, of which a part in the amount of 𝛥𝐾 is used to repair the herd, the rest is either sold or fattened in the amount of 𝑋 for meat production. than: 𝑋 < 𝑋 − 𝛥𝐾 (4.1) building a digital twin of agricultural production… 121 copyright ©2022 assa adv. in systems science and appl. (2022) in the following, for simplicity, we will use the average data: 1) the average annual milk yield per cow is 3.5 tons, i.e. annual milk production, in tons per year, is given by: 𝑋 = 3,5𝐾 , 2) a cow gives birth to one calf per year, i.e. 𝑋 = 𝐾 , 3) the replacement number of calves, taking into account the case, is determined as 20% of the herd size, i.e. 𝑋 = 0,2𝐾 = 0,2𝑋 , 4) annually, 20% of cows that are used for meat production also drop out of the herd. from one mature cow, 300 kg of meat comes out, and in total, due to the replacement, meat is produced in the amount of: 𝑋 = 0,06𝐾 , 𝑡𝑜𝑛𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟. 5) the yield of meat from a fattened bull is 200 kg, then the annual production of meat is determined by the expression: 𝑋 = 0,2𝑋 , 𝑡𝑜𝑛𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟. in total, the annual production of meat (in tons) is determined by the expression: 𝑋 = 𝑋 + 𝑋 = 0,06𝐾 + 0,2𝑋 (4.2) consider labor costs. here we will not describe in detail the structure of labor costs, but only assume that there are conditionally variable and fixed costs. conditionally variable costs proportional to the number of cows, calves, bulls on the farm. conditionally fixed costs do not depend on the composition of the herd, are associated with work that does not depend, within certain limits, on the number of cattle on the farm. for example, it is necessary to support a tractor driver, a mechanic, a veterinarian, a watchman, etc. labor costs, as the total number of staff units required on the farm, are determined by the expression: 𝐿 = 𝑇 𝐾 + 𝑇 𝑋 + 𝑇 𝑋 + 𝐿 , (4.3) where we assume 𝑇 = 0,07, 𝑇 = 0,05, 𝑇 = 0,05 people (staff units) per head of cattle, 𝐿 = 10 people, which cowresponds to the labor costs of some small average farm. in the future, it is assumed that part of the wages of workers is given out with meat and milk produced on the same farm, and in an amount determined by the consumption norms for all members of the employee's family. every worker has a family. the average number of people in a family is 𝑘 and for russia now it is 3-4 people, let's assume 𝑘 = 4. then the number of people who have to eat the farm products is 𝑘 𝐿. the rest that is produced on the farm is an external product that goes on sale. in accorcdance with the norms, each person consumes 𝑎 = 0,075 tons of meat (75 kg) and 𝑎 = 0,34 tons of milk (340 kg) per year. then the annual consumption by the population of the farm of its own products is determined by the expressions: 𝑋 = 𝑎 𝐿 tons meat per year and 𝑋 = 𝑎 𝐿 tons milk per year. hence, the system of equations describing the production of milk and meat on the farm has the following form: [3,5– 𝑎 (𝑇 + 𝑇 )]𝐾 – 𝑎 𝑇 𝑋 = 𝑎 𝐿 + 𝑌 , 122 r. sirazetdinov, r. sabitov, g. smirnova, s. sabitov, a. eponeshnikov copyright ©2022 assa adv. in systems science and appl. (2022) [0,06– 𝑎 (𝑇 + 𝑇 )]𝐾 – (0,2– 𝑎 𝑇 )𝑋 = 𝑎 𝐿 + 𝑌 (4.4) where 𝑌 is the annual milk production program, 𝑌 annual meat production program. this model allows solving both direct and inverse problems. the direct problem includes determining the amount of meat and milk produced by the farm for a given number of dairy cows and fattening bulls. some calculation results are shown in table 1. for example, a herd of 100 cows, with the maximum number of fattening bulls, i.e. with 𝐾 =100 heads and 𝑋 =80 heads, yields 20.0 tons of meat and 341.2 tons of milk per year. note that in the first line of table 1, with the number of dairy cows 𝐾 = 3 and the number of bulls 𝑋 = 3, the meat yield 𝑌 = 0. this means that such a number of animals makes it possible to provide meat only for farm workers, and meat for sale does not remain, although milk is produced for sale. table 1. calculation results № 𝐾 , heads 𝑋 , heads 𝑌 tons/ year 𝑌 tons/year 1 3 3 6,9 0 2 4 3 10,4 0,04 3 10 8 31 1,3 4 20 16 65,5 3,4 5 30 24 100 5.5 6 50 40 169 9.7 7 100 80 341 20 8 200 160 686 41 9 400 320 1375 62 the inverse problem is to determine the required herd size, the number of dairy cows and fattening bulls that would provide the required amount of milk 𝑌 and meat 𝑌 put on the market. some results of the calculations are shown in table 2. since the results of the calculations are fractional, herd sizes are shown in brackets, rounded to whole numbers. table 2. calculation results № 𝑌 tons/ year 𝑌 tons/ year 𝐾 , heads 𝑋 , heads. 1 10 0 3,85 (4) 2,8 (3) 2 50 0 15,4 (4) 0 3 100 5 30 21,5 (22) 4 400 20 117 75,3 (75) 5 400 25 117 100,7 (101) 6 400 40 117,5 (118) 177,1 (177) figures 2 and 3 show the results of calculations of the required number of dairy cows 𝐾 and bulls for fattening 𝑋 (ordinate axis), to ensure a different amount of milk yield 𝑌 (abscissa axis), provided that meat is not produced for sale, 𝑌 = 0. figure 2 considers the option when the farm has 𝐿 = 10 permanent workers. figure 3 considers the option when the farm has 𝐿 = 20 permanent workers. building a digital twin of agricultural production… 123 copyright ©2022 assa adv. in systems science and appl. (2022) fig. 2. calculation results: 𝐿 = 10 permanent workers. fig. 3. calculation results: 𝐿 = 20 permanent workers the number of cows is shown in blue, and the number of bulls is shown in red. it can be seen from the graphs that, despite the fact that the sale of meat is not provided, it must be produced to provide for the population of the farm. at the same time, in the first case, when 𝐿 = 10 people, if less than 7 tons of milk is produced per year, then the number of bulls is required more than there are cows. and in the second case, when 𝐿 = 20 people, the number of bulls is required more than there are cows, with the demand for milk already less than 13 tons per year. thus, in this calculation example, expression (4.4) is a static mathematical model of the production process of a meat and dairy farm, when part of the output is distributed among the workers themselves. it does not specifically consider the financial side of the farm, which is associated with market prices for products and raw materials. it illustrates the simulation of a closed part of the production process, when part of the production is used for internal needs. then, in the absence of demand for products, the farm continues to work, providing for the needs of its own workers and, thereby, ensuring its sustainability. as you know, most of the costs in meat and dairy production is animal food. therefore, this model can also include the production of animal food, and, accordingly, the payment of the labor of animal food production with products produced on the farm. thus, we will further expand and strengthen the sustainable component of the farm as a production system. 5. conclusions this model allows you to simulate the gradual transformation of agricultural production facility in the following directions. 124 r. sirazetdinov, r. sabitov, g. smirnova, s. sabitov, a. eponeshnikov copyright ©2022 assa adv. in systems science and appl. (2022) first, as shown above, the development and expansion of production is modeled, the structure of production is determined, which is necessary to ensure the production of new volumes, or new types of products. at the same time, this can be done both at the expense of previous technologies, increasing the amount of equipment of an existing type, or it is possible to acquire and launch new types of equipment, new technologies, which will affect the size and values of the corresponding matrices and model coefficients. secondly, it is possible to simulate the localization of production, which consists in reducing the dimension of the vector 𝑉 of production received from outside, and expanding the vector 𝑋 of domestic production. this again leads to the acquisition of new equipment, the adoption of new personnel, i.e. changing the structure and dimensions of the model. in fact, this adds stability to production and creates new jobs. thirdly, by changing the matrix 𝐴 , one can vary the ratio of how much production to keep for domestic consumption and how much to send to the external market. thus, to manage the degree of stability of the enterprise and its development. in general, within the framework of the proposed model of sustainable development, one can consider not only agriculture, but the entire agro-industrial complex of the region as a whole. on the other hand, within the framework of this model, one can also consider the sustainable development of individual regions, examining both components of the region's economy and achieving their harmonious interaction. the proposed model also makes it possible to consider the sustainable development of individual settlements, for example, rural settlements relying on local farms and private households, as well as individual agricultural production. in each case, it is necessary to ensure a harmonious combination of stability, selfsufficiency, at least food independence of farms and the possibility of their development. references 1. babkin a.v. et al. 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(2017). modeling production facility as a dynamic integrated interacting objects system. procedia computer science, 112, 965–970 adv syst sci appl 2023; 03:91–107 published online at https://ijassa.ipu.ru. global stability analysis of malaria model with prophylactic treatment saheed ajao1∗, isaac olopade2, sunday adewale3, adelani adesanya1 1department of mathematics and computer science, elizade university, ondo state, nigeria 2department of mathematics and statistics, federal university wukari, taraba state, nigeria 3department of pure and applied mathematics, ladoke akintola university of technology, oyo state, nigeria abstract: in this paper, we present a mathematical model of the interaction between the human population and the vector (mosquito) population to study the stability of a malaria model in the presence of prophylactic treatment. this study is aimed at investigating the effect of the prophylactic treatment and the long-term dynamics of the solutions of the model. the graph-theoretic method was used to obtain the basic reproduction number (r0). we obtained the disease-free equilibrium for the model which is locally and globally asymptotically stable when the basic reproduction number is less than unity. moreover, we showed that there exists a unique endemic equilibrium whenever r0 > 1, and the lyapunov function was used to establish that the endemic equilibrium is globally asymptotically stable whenever r0 > 1. the simulations show the impact of prophylactic treatment on the population of infectious individuals and the findings of this study suggest that prophylactic treatment is an effective approach to reduce the transmission of malaria as prophylactic treatment reduces the population of infectious individuals. further numerical simulations carried out conformed with the analytic results. keywords: malaria, prophylactic treatment, basic reproduction number, graph-theoretic method, global stability, lyapunov function 1. introduction malaria is one of the most serious public health problems in the world. in many developing countries, it is one of the main causes of death and disease [6]. globally, 247 million cases of malaria were recorded in 2021 and malaria claimed the lives of 619,000 people [46]. in 2020, malaria cases stood at 241 million cases globally, and an estimate of 627,000 deaths was also recorded with the who africa region having the larger share of the global malaria burden [44]. malaria is a fatal disease and one of the main causes of death for children under five years of age. it is caused by a parasite called plasmodium which is transmitted through the bite of an infected female anopheles mosquito. the parasite requires two hosts for the completion of its life cycle: namely, a female anopheles mosquito and a human [29]. the presence of malaria depends on the climatic factors of an area, such as rainfall, humidity, and temperature, and this can be linked to the prevalence of malaria in tropical and subtropical areas where the conditions allow mosquitoes to survive, multiply, and complete their life cycle [7]. when parasites enter the human body through the bites of infected mosquitoes, the symptoms manifest themselves after about 10 to 15 days, ∗corresponding author: saheed.ajao@elizadeuniversity.edu.ng 92 s. ajao, i. olopade, s. adewale, a. adesanya and children under the age of five, pregnant women, non-immune travellers, hiv/aids patients and people with low immune systems are more likely to develop the disease [45]. if one is exposed to the disease, it can cause morbidity and fatality if not treated in time. in recent times, the modelling of the transmission of infectious diseases is now influencing the theory and practice of disease management and control. mathematical modelling now plays an important role in policy decision-making regarding the epidemiology of disease in many countries [21]. mathematical models have become useful tools for analysing and controlling the transmission of infectious diseases, and they are used to compare, plan, implement, evaluate and optimize different detection, prevention, and control programs [16]. several models have been formulated to explain and show the interaction between the host and vector, its transmission and control (see [1, 3, 4, 8–10,17,19,26,37,38,40,42]). massad et al. [30] studied the malaria infection risk for travellers visiting the amazonian region of brazil using a mathematical modelling approach. the study reveals that the risk is high, heterogeneous and depends on the length of stay, arrival time to a locality, entomological potential, and force of infection for the population of the local vector. mohammed-awel and gumel [31] worked on the mathematics of an epidemiology-genetic model to assess the effect of insecticide resistance on the dynamics of malaria transmission. the epidemiology model of the disease is coupled with the genetics of the vector population and incorporates various fitness costs associated with the insecticide resistance for the malaria model. the results of their simulations show that the insecticides treated nets and indoors residual spraying (itns & irs) control strategies reduce the disease effectively in both moderate and high malaria transmission communities if the itns coverage level is high enough, but fail to manage insecticide resistance effectively in the community. woldegerima et al. [43] studied the mathematical analysis of transmission-blocking drugs’ impact on the population dynamics of malaria. rehman et al. [35] proposed and analysed a mathematical model of malaria with multiple compartments with an associated learning mechanism between the vector-to-host and vice-versa incorporating memory, reinfection, and relapse. traoré et al. [39] presented a mathematical model of malaria transmission with four different metamorphic stages of mosquitoes and the model also features seasonality. collins and duffy [11] studied the mathematical model for the dynamics of malaria and its control in nigeria. epidemiological features such as drug resistance, mosquito nets use and malaria treatment were introduced into the model. the results show that the disease will remain in the country if good control measures are not geared toward dominant resistant strains, mosquito net use, and treatment. kuddus and rahman [22] developed a human-mosquito model for the dynamical transmission of malaria. the sensitivity analysis was carried out and the contact rates for the human and the mosquito were found to have the greatest influence on the prevalence of malaria. moreover, the impacts of other features of the model like progression rate, disease-related death rate, recovery rate, and the rate of losing immunity were also examined. olaniyi et al. [34] worked on the mathematical analysis of a social hierarchy-structured model for the transmission dynamics of malaria. the human population of the model was classified based on two social classes namely: low and high, and the optimal control analysis was also carried out. beretta et al. [5] proposed a mathematical model for the transmission of malaria which features two-age and asymptomatic classes, and it was found that decreasing the biting rate and increasing the mortality rate of the mosquito can cause the endemic situation to fade off. one of the important aspects of modelling in epidemiology is to examine the global property of the equilibrium, and the lyapunov function, which serves as a basic and potent tool, has been widely used to study and establish the global stability of the equilibrium. korobeinikov and wake [20] presented the lyapunov functions for the sir, copyright © 2023 assa. adv syst sci appl (2023) global stability analysis of malaria model 93 sirs, and sis models to establish the global stability of the endemic states of the model. vargas-de-leon [41] studied the global stability of infectious disease models with relapse by constructing lyapunov functions in providing the conditions for the stability of the models with relapse. li et al. [25] presented an approach that is algebraic in nature to prove the global stability of a class of epidemic models and showed how the coefficients ai can be chosen for the classical lyapunov function ∑n i=1 ai(xi − x∗ i − xi ln xi/x∗ i ) so that the derivative of the lyapunov function is negative definite or semidefinite. khan et al. [18] examined the global stability of vector-host disease with variable population sizes, and numerical simulations were carried out to justify the theoretical results. gebremestal [15] studied the global stability of the transmission of a malaria model with logistic growth, and the simulations examined how changes in the sustainable level of the vectors affect the population of humans. the study of endemic global stability is not only of importance to mathematics but also helps in forecasting the long-term dynamics of the disease to design effective prevention and intervention strategies to combat the disease [36]. most of the studies in the literature failed to consider the prophylactic treatment in malaria models. hence, the novelty of this work is the focus on the exploration, using the lyapunov function, of the global stability properties of a malaria model incorporating prophylactic treatment. furthermore, it is assumed that a fraction of newly infected individuals with low immunity progress to the exposed class while the remaining fraction moves to the infectious class. the paper is organized as follows: section 2 has the method of model design/formulation and model analysis, sections 3 and 4 contain the results and discussion of results, respectively, while the conclusion follows in section 5. 2. methods 2.1. model design in this subsection, we describe how the model is designed and formulated. the malaria model has two populations, namely the human population and the vector (mosquito) population. the overall human population (denoted by nh) is subdivided into susceptible individuals (sh), exposed individuals (em), infectious individuals (im), and those that recovered (rm). also, the total mosquito population (denoted by nv) consists of three subpopulations, namely susceptible mosquitoes (sv), exposed mosquitoes (ev) and infectious mosquitoes (iv). the susceptible humans are assumed to be recruited into the population at the rate π and get infected with malaria after being bitten by infected mosquitoes at the rate λm , which is given by λm = βmaiv, where βm is the probability of transmission from mosquitos to humans provided that there is significant contact between the mosquito and the human, and a is the number of mosquito bites that one person has per unit time. the susceptible population has those who lost their immunity at the rate ϕ in the recovered class, and the population decreases due to natural death which does occur in all human subpopulations at the rate µh. it is assumed that a fraction ϵ of newly infected individuals with low immunity moves to the exposed class of humans em , while the remaining fraction moves to the infectious compartment im , the exposed humans progress into the infectious class at the rate κ, and the prophylactic treatment is given at the rate τ1 to the exposed humans making them move to the recovered class. those that are treated in the infectious class at the rate τ2 progress to the recovered class and death due to the disease causes a reduction in the population of the infectious compartment. the recovered population diminishes as a result of loss of immunity at the rate ϕ and natural death at the rate µh which occurs in all human subpopulations. copyright © 2023 assa. adv syst sci appl (2023) 94 s. ajao, i. olopade, s. adewale, a. adesanya the susceptible mosquitoes sv come into the population at a constant rate (recruitment rate πv), and they become infected after effective contact with people infected with malaria at the rate λv, where λv = βvb(em + ηim) and η ≥ 1 is the amplification parameter which accounts for the relative infectiousness of infectious individuals when compared to the exposed individuals, b is the number of humans bitten by mosquito per unit time and βv is the transmission probability from humans to mosquitoes. the newly infected mosquitoes move to the exposed class of mosquitoes, and there is a progression from the exposed class to the infectious class of mosquitoes at the rate σ. the death of mosquitoes does occur in all compartments of mosquitoes at the rate µv. the schematic diagram of the model showing the interaction among the classes is depicted in figure 2.1 and table 2.1 shows the definitions of the parameters. then, the system of equations of the model is as follows: fig. 2.1. flow chart of the malaria model copyright © 2023 assa. adv syst sci appl (2023) global stability analysis of malaria model 95 dsh dt = πh − λmsh − µhsh + ϕrm , dem dt = ϵλmsh − (κ + τ1 + µh)em , dim dt = (1 − ϵ)λmsh + κem − (τ2 + δ + µh)im , drm dt = τ1em + τ2im − (ϕ + µh)rm , dsv dt = πv − λvsv − µvsv, dev dt = λvsv − (σ + µv)ev, div dt = σev − µviv, (2.1) where λm = βmaiv, and λv = βvb(em + ηim). for convenience, we can rewrite the equations above as below: dsh dt = πh − λmsh − µhsh + ϕrm , dem dt = ϵλmsh − t1em , dim dt = (1 − ϵ)λmsh + κem − t2im , drm dt = τ1em + τ2im − t3rm , dsv dt = πv − λvsv − µvsv, dev dt = λvsv − t4ev, div dt = σev − µviv, (2.2) where t1 = κ + τ1 + µh, t2 = τ2 + δ + µh, t3 = ϕ + µh, and t4 = σ + µv. 2.2. model analysis the analysis of the model is carried out here, and the threshold for the eradication and persistence of malaria is determined and explored. 2.2.1. the invariant region theorem 2.1: the closed set d = { (sh, em , im , rm , sv, ev, iv) ∈ r7 + : nh ≤ πh µh , nv ≤ πv µv } , is positively invariant with non-negative initial values in r7 +. copyright © 2023 assa. adv syst sci appl (2023) 96 s. ajao, i. olopade, s. adewale, a. adesanya table 2.1. description of parameters used in the model (2.1) parameter description value source πh recruitment rate of humans 0.3913 [3] βm probability of transmission from mosquitoes to humans 0.001 [24] µh natural death rate of humans 0.00003913 [3] a number of mosquito bites per unit time 0.29 [17] ϕ rate of loss of immunity 0.25 [3] ϵ fraction of newly infected people with low immunity 0.6 assumed τ1 prophylactic treatment for the exposed class 0.002 assumed κ progression rate of humans from exposed to infectious class 0.0714 [32] b number of humans bitten by mosquitos per unit time 0.2980 [4] η amplification factor 2.5 assumed δ death due to the disease 0.00009 [8] τ2 treatment rate for the infectious 0.0092 [33] µv natural death rate of mosquitoes 0.0476 [3] πv recruitment rate of mosquitoes 0.7 assumed σ progression rate of exposed mosquitoes 0.0555 [24] βv probability of transmission from humans to mosquitos 0.0001 [24] proof consider the feasible region d = dh × dv ⊂ r7 + with dh = { (sh, em , im , rm) ∈ r4 + : nh ≤ πh µh } , and dv = { (sv, ev, iv) ∈ r3 + : nv ≤ πv µv } , and by adding the human and mosquito compartments of (2.1) separately with δ = 0 we have dnh dt = πh − µhnh and dnv dt = πv − µvnv. it follows that dnh dt ≤ πh − µhnh and dnv dt ≤ πv − µvnv. then nh(t) ≤ nh(0)e−µht + πh µh (1 − e−µht) and nv(t) ≤ nv(0)e−µvt + πv µv (1 − e−µvt). copyright © 2023 assa. adv syst sci appl (2023) global stability analysis of malaria model 97 if nh(0) ≤ πh µh , then nh(t) ≤ πh µh . also, nv(0) ≤ πv µv , then nv(t) ≤ πv µv . hence, all solutions of the model with initial values in d stay there for t > 0. this means that d is positively invariant and in this region, and the model is considered to be epidemiologically meaningful and mathematically well-posed. 2.2.2. disease-free equilibrium the disease-free equilibrium of equation (2.2) is obtained by setting all the rates of change to zero. hence, the disease-free equilibrium is given by e1 = (sh, em , im , rm , sv, ev, iv) = ( πh µh , 0, 0, 0, πv µv , 0, 0 ) . (2.3) 2.2.3. basic reproduction number we use the graph-theoretic method as given by de-camino-beck et al. [13] to obtain the basic reproduction number of the model. the digraph reduction procedure of the digraph fλ−1 − v for the model at disease-free is shown in figure 2.2. fig. 2.2. digraph reduction procedure of the model from figure 2d, − 1 + βvbπvσβmaπh(η − ϵη)λ−2 µ2 vµht2t4 + ϵβmaπhβvbπvσ(κη + t2)λ−2 µ2 vµht1t2t4 = 0, ∴ r0 = λ = √√√√βvbπvσβmaπh(t1η − t1ϵη + ϵκη + ϵt2) µ2 vµht1t2t4 . (2.4) the threshold r0 is the basic reproduction number, which is the average number of new infection cases emanating from a single infection source when introduced in a population consisting of only susceptibles [14]. this threshold determines whether malaria will die out or persist. in the next subsection, we try to show that a small influx of infected copyright © 2023 assa. adv syst sci appl (2023) 98 s. ajao, i. olopade, s. adewale, a. adesanya mosquitoes/infectious individuals into the community/environment will not generate a substantial malaria outbreak when r0 < 1. 2.2.4. local stability of the disease-free equilibrium theorem 2.2: the disease-free equilibrium of the system (2.2) given by (2.3) is locally asymptotically stable if r0 < 1 and unstable when r0 > 1. proof the jacobian matrix j(e1) of the system (2.2) evaluated at disease-free equilibrium (2.3) is as given below: j(e1) =  −µh 0 0 ϕ 0 0 −βmaπh µh 0 −t1 0 0 0 0 ϵβmaπh µh 0 κ −t2 0 0 0 (1 − ϵ)βmaπh µh 0 τ1 τ2 −t3 0 0 0 0 −βvbπv µv −βvbηπv µv 0 −µv 0 0 0 βvbπv µv βvbηπv µv 0 0 −t4 0 0 0 0 0 0 ϕ −µv  . (2.5) the first three eigenvalues of (2.5) are −µh, −t3, and −µv. the remaining submatrix is given by j(e1) =  −t1 0 0 ϵβmaπh µh κ −t2 0 (1 − ϵ)βmaπh µh βvbπv µv βvbηπv µv −t4 0 0 0 σ −µv  . (2.6) the characteristic polynomial of the matrix (2.6) is given by x4λ 4 + x3λ 3 + x2λ 2 + x1λ + x0 = 0. (2.7) where x4 = 1, x3 = µv + t4 + t2 + t1, x2 = t1t2 + t1t4 + t1µv + t2t4 + t2µv + t4µv, x1 = t2t4µv + t1t4µv + t1t2µv + t1t2t4 [ 1 − r2 0µv (η(1 − ϵ) + ϵ) [ϵ(ηκ + t2) + t1η(1 − ϵ)] ] , and x0 = t1t2t4µv(1 − r2 0). the characteristic equation (2.7) has negative roots if it satisfies routh-hurwitz criteria, such that xi > 0 for i = 0, 1, 2, 3, 4 and x1(x2x3 − x1) > x0x 2 3 . clearly, x4, x3 and x2 > 0, x1 > 0 if ϵ(ηκ + t2) + t1η(1 − ϵ) > µv(η(1 − ϵ) + η) and x0 > 0 for r0 < 1. hence, the disease-free equilibrium is locally asymptotically stable whenever r0 < 1. copyright © 2023 assa. adv syst sci appl (2023) global stability analysis of malaria model 99 this theorem 2.2 implies that there is possibility of malaria eradication (when r0 < 1) if the initial subpopulation sizes are in the basin of attraction of the disease-free equilibrium e1. to ensure that malaria eradication is not dependent on the initial subpopulations sizes, it is vital to show that the disease-free equilibrium of malaria is globally asymptotically stable. 2.2.5. global stability of disease-free equilibrium theorem 2.3: the disease-free equilibrium given by (2.3) is globally asymptotically stable when r0 < 1. proof we construct the lyapunov function to prove the global stability of the disease-free as thus: l = (κη + t2)em + t1ηim + t2t1r0 βvbsv ev + t4t2t1r0 σβvbsv iv. (2.8) differentiating equation (2.8) gives l̇ = (κη + t2) ˙em + t1η ˙im + t2t1r0 βvbsv ėv + t4t2t1r0 σβvbsv i̇v. (2.9) l̇ = (κη + t2)[ϵβmaivsv − t1em ] + t1η[(1 − ϵ)βmaivsh + κem − t2im ] + t2t1r0 βvbsv [βvb(em + ηim)sv − t4ev] + t4t2t1r0 σβvbsv [σev − µviv]. l̇ = κηϵβmaivπh µh + t2ϵβmaivπh µh − t1t2em + t1η(1 − ϵ)βmaivπh µh − t1ηt2im + t2t1r0em + t2t1ηr0im − t4t2t1r0ivµ2 v σβvbπv . l̇ = [√ t1t2t4βmaπhµ2 v σβvbπvµh iv + t1t2em + t1t2ηim ] r0 − 1. (2.10) therefore, l̇ < 0 if r0 < 1 with l̇ = 0 if and only if em = im = iv = 0. also, the largest invariant set in {(sh, em , im , rm , sv, ev, iv) ∈ d : l̇ = 0} is the singleton e1. by the lasalle invariance principle [23], every solution that starts in d approaches e1 as t → ∞. therefore, the disease-free equilibrium is globally asymptotically stable whenever r0 < 1. the implication of theorem 2.3 epidemiologically is that malaria can be eradicated regardless of the initial sizes of the subpopulations of the model whenever the basic reproduction number is less than unity. 2.2.6. existence of endemic equilibrium the condition for which the endemic equilibrium exists is investigated here. let e∗ 2 = (s∗ h, e∗ m , i∗ m , r∗ m , s∗ v , e∗ v , i∗ v ), represents an arbitrary endemic equilibrium of the system of equations (2.2). also, let λm and λv at endemic steady state be denoted by λ∗ m and λ∗ v and given by λ∗ m = βmai∗ v , (2.11) λ∗ v = βvb(e∗ m + ηi∗ m). (2.12) copyright © 2023 assa. adv syst sci appl (2023) 100 s. ajao, i. olopade, s. adewale, a. adesanya when we solve the equations of model (2.2) at steady state in terms of the forces of infection λ∗ m and λ∗ v , we will have s∗ h = πh µh + λ∗ m(1 − ϕp3) , e∗ m = p1λ ∗ ms∗ h, i∗ m = p2λ ∗ ms∗ h, r∗ m = p3λ ∗ ms∗ h, (2.13) s∗ v = πv λ∗ v + µv , e∗ v = p4λ ∗ vs∗ v , i∗ v = p5λ ∗ vs∗ v . (2.14) where p1 = ϵ t1 , p2 = (1 − ϵ)t1 + κϵ t1t2 , p3 = t2τ1ϵ + τ2((1 − ϵ)t1 + κϵ) t1t2t3 , p4 = 1 t4 , p5 = σ µvt4 . and t1, t2, t3, t4, and t5 are defined in (2.2) above. by substitution of (2.13) and (2.14) into (2.11) and (2.12), we have λ∗ m = βmap5λ ∗ vs∗ v , (2.15) λ∗ v = βvb(p1λ ∗ ms∗ h + ηp2λ ∗ ms∗ h). (2.16) further substitution leads to cλ∗ m − d = 0, where d = µvµh(r0 2 − 1) and c = βvbπh(p1 + ηp2) + µv(1 − ϕp3). then λ∗ m = d c . (2.17) from (2.17), it shows clearly that c > 0 and d > 0 if r0 > 1. then, λm is positive (greater than zero) whenever r0 > 1. hence, the following result is established: lemma 2.1: there exists a unique endemic equilibrium for the model if the basic reproduction number r0 > 1. 2.2.7. global stability of endemic equilibrium the global stability of the endemic equilibrium of the model is considered here for the case when ϕ = 0, and we take ϵ = 1. theorem 2.4: the unique endemic equilibrium of the model with ϕ = 0 and ϵ = 1 is globally asymptotically stable whenever r0|ϕ=0,ϵ=1 > 1. proof the volterra type of lyapunov function stated in 2.18 is used to prove the global stability of the endemic equilibrium. l = ( sh − s∗ hln sh s∗ h ) + ( em − e∗ m ln em e∗ m ) + t1 κ ( im − i∗ m ln im i∗ m ) + ( sv − s∗ v ln sv s∗ v ) + ( ev − e∗ v ln ev e∗ v ) + t4 σ ( iv − i∗ v ln iv i∗ v ) . (2.18) copyright © 2023 assa. adv syst sci appl (2023) global stability analysis of malaria model 101 taking the derivative of (2.18), we have l̇ = ṡh − s∗ h sh ṡh + ˙em − e∗ m em ˙em + t1 κ ( ˙im − i∗ m im ˙im ) + ṡv − s∗ v sv ṡv + ėv − e∗ v ev ėv + t4 σ ( ˙im − i∗ m im ˙im ) . (2.19) l̇ = 2λms∗ h + 2µhs∗ h − µhsh − λms∗2 h sh − µhs∗2 h sh − λmshe∗ m em + t1e ∗ m − t1e ∗ mim im − t1emi∗ m im + t1em + 2λvs∗ v + 2µvs∗ v − µvsv − λvs∗2 v sv − µvs∗2 v sv − λvsve∗ m ev + t4e ∗ v − t4e ∗ miv i∗ v − t4evi∗ v iv + t4e ∗ v . l̇ = t1e ∗ m ( 2 − s∗ h sh ) + µhs∗ h ( 2 − s∗ h sh ) − µhsh − t1e ∗2 m sh s∗ hem + t1e ∗ m − t1e ∗ mim i∗ m − t1emi∗ m im + t1e ∗ m + t4s ∗ v ( 2 − s∗ v sv ) + µvs∗ v ( 2 − s∗ v sv ) − µvsv − t4e ∗2 v sv svev + t4e ∗ v − t4e ∗ viv i∗ v − t4evi∗ v iv + t4e ∗ v . (2.20) by simplification, (2.20) becomes l̇ = µhs∗ h ( 2 − s∗ h sh − sh s∗ h ) + t1e ∗ m ( 4 − s∗ h sh − e∗ ms∗ h ems∗ h − emi∗ m e∗ mim − im i∗ m ) + µvs∗ v ( 2 − s∗ v sv − sv s∗ v ) + t4e ∗ v ( 4 − s∗ v sv − e∗ vsv evs∗ v − iv i∗ v − evi∗ v e∗ viv ) . (2.21) since the arithmetic mean is greater than the geometric mean, then 2 − s∗ h sh − sh s∗ h ≤ 0, 4 − s∗ h sh − e∗ ms∗ h ems∗ h − emi∗ m e∗ mim − im i∗ m ≤ 0, 2 − s∗ v sv − sv s∗ v ≤ 0, 4 − s∗ v sv − e∗ vsv evs∗ v − iv i∗ v − evi∗ v e∗ viv ≤ 0. with all the parameters of the model being positive, therefore, l̇ ≤ 0 for r0|ϕ=0,ϵ=1 > 1. hence, by the lasalle invariance principle [23], every solution to the model equations approaches the endemic equilibrium e2 as t → ∞ whenever r0|ϕ=0,ϵ=1 > 1. the epidemiological implication of theorem 2.4 is that malaria will persist in the community irrespective of the initial sizes of the populations of the model whenever r0 > 1. 3. results the results of the numerical simulations of the model are obtained using the ode45 solver of matlab and the values of the parameters used for the simulations are given in copyright © 2023 assa. adv syst sci appl (2023) 102 s. ajao, i. olopade, s. adewale, a. adesanya table 2.1. we investigate the influence of prophylaxis on the dynamics of malaria and the long-term dynamics of the model. the results of the simulations are depicted in figures 3.3 3.6. 0 5 10 15 20 25 30 5 10 15 20 25 30 35 time (days) in fe ct io u s in d iv id u al s no prophylaxis with prophylaxis fig. 3.3. graph of the effect of the prophylactic treatment on the infectious population 0 5 10 15 20 25 30 14 16 18 20 22 24 26 28 30 32 time (days) in fe ct io u s in d iv id u al s τ 1 =0.02 τ 1 =0.06 τ 1 =0.10 τ 1 =0.14 fig. 3.4. graph of the infectious population at different values of prophylactic treatment rate (τ1) copyright © 2023 assa. adv syst sci appl (2023) global stability analysis of malaria model 103 0 500 1000 1500 2000 0 20 40 60 80 100 120 140 160 180 time (days) in fe ct io u s in d iv id u a ls fig. 3.5. plot of the infectious individuals at different initial conditions when r0 = 0.65 and the parameters used are given in table 2.1 0 500 1000 1500 2000 0 50 100 150 200 250 time (days) in fe ct io u s in d iv id u al s fig. 3.6. plot of the infectious individuals at different initial conditions when r0 = 12.4, with ϵ = 1, ϕ = δ = 0, βm = 0.004, βv = 0.0004, τ1 = 0.001, τ2 = 0.004, µv = 0.0175 and other parameters used are given in table 2.1 4. discussion of the results a mathematical model of malaria with prophylactic treatment is presented, and its global asymptotic stability properties are studied. the threshold which determines whether malaria will fade out or persist is established. the qualitative analysis of the stability carried out showed that the disease-free equilibrium is locally asymptotically copyright © 2023 assa. adv syst sci appl (2023) 104 s. ajao, i. olopade, s. adewale, a. adesanya stable whenever r0 < 1. also, there exists an endemic equilibrium whenever the basic reproduction number goes beyond unity. the lyapunov functions were used to establish that the disease-free equilibrium and endemic equilibrium are globally asymptotically stable whenever r0 < 1 and r0 > 1, respectively. the graph of figure 3.3 illustrates the population of infectious individuals when prophylactic treatment is present and when there is no prophylactic treatment. the population of infectious individuals is at minimum when there is prophylaxis as compared to when there is none. hence, the presence of prophylactic treatment lowers the number of infectious individuals as it prevents exposed individuals to progress to the infectious class/state. the increasing effect of prophylactic treatment rate on the infectious population is depicted in figure 3.4. the population of infectious individuals diminishes as the prophylactic treatment rate increases and thereby reducing the burden of the disease. therefore, prophylactic treatment plays an important role in reducing the population of the infectious individuals. the practical situation of the study is the recent implementation of seasonal malaria chemoprophylaxis in nigeria by the malaria consortium [27,28] to reduce the disease burden. some states in nigeria were selected for the pilot phase and every child in the community under five years old is given prophylactic treatment every month. a study by de cola et al. [12] reported that the treatment programme reduces the prevalence of malaria in the country, which is in line with the result of this current study. the work of ambe et al. [2] also supports the result we obtained. we carried out numerical simulations on global stability to validate the analytic results. the plot in fig. 3.5 shows the long-term dynamics of the population of infectious individuals at different initial conditions when r0 < 1. the infectious population converges to disease-free equilibrium and vanishes, which means that the disease dies out of the community irrespective of the different initial sizes of the infectious class. also, fig. 3.6 shows a scenario of the population of the infectious class with different initial conditions when the basic reproduction number r0 > 1. in this case, the disease does clear out from the population, but persists, and this is in line with the analytic result. 5. conclusion we formulated a mathematical model for the dynamics of malaria with prophylaxis to study the global stability properties of the model. the basic reproduction number was obtained using the graph-theoretic method. the disease-free equilibrium is proved to be locally and globally asymptotically stable whenever the basic reproduction number is less than unity. we showed that there exists a unique endemic equilibrium whenever the basic reproduction number surpasses unity, and by the lyapunov function, the endemic equilibrium is globally asymptotically stable whenever r0 > 1. the presence of prophylaxis plays a vital role as it minimizes the population of infectious individuals and mitigates the burden of malaria. hence, there is a need to intensify efforts in the provision of good and appropriate prophylaxis to reduce the burden of the disease. further numerical simulations carried out justified the analytic results. references 1. abboubakar, h., buonomo, b. & chitnis, n. 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[online]. available http://www.who.int/news-room/fact-sheets/detail/malaria. copyright © 2023 assa. adv syst sci appl (2023) https://doi.org/10.1016/j.nonrwa.2019.103081 https://doi.org/10.1016/j.nonrwa.2019.103081 https://doi.org/10.1016/j.amc.2021.126005 https://doi.org/10.1016/j.amc.2021.126005 http://www.who.int/news-room/fact-sheets/detail/malaria http://www.who.int/news-room/fact-sheets/detail/malaria http://www.who.int/news-room/fact-sheets/detail/malaria introduction methods model design model analysis the invariant region disease-free equilibrium basic reproduction number local stability of the disease-free equilibrium global stability of disease-free equilibrium existence of endemic equilibrium global stability of endemic equilibrium results discussion of the results conclusion adv syst sci appl 2018; 03; 154-164 published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/568 preliminary design with the epistemic uncertainty of parameters georgy veresnikov, ludmila pankova, valeriya pronina v.a. trapeznikov institute of control sciences of russian academy of scienses, moscow, russia e-mail: veresnikov@mail.ru abstract: in design, especially in preliminary design, the assumption of parameter accuracy is not justified, since the parameters here are inaccurate (uncertain), due to insufficient knowledge or lack of statistics, as well as the fact that design parameters are further implemented in the production with some tolerance. the application of deterministic optimization methods under conditions of parametric uncertainty can lead to unacceptable solutions even with slight variation in the parameters. currently, to account for uncertainty of the parameters there are commonly used stochastic methods designed to account for aleatory uncertainty with a priori known distribution functions of random parameters. however, in the preliminary design, most of the parameters are not random variables with known distribution functions. the necessary information on the parameters is obtained from the experts. in this paper, we develop methods and algorithms for preliminary design in conditions of epistemic uncertainty arising from lack of knowledge and observation results, replenished by expert assessments. in the paper the problem of optimal design in the presence of input and design parameters with epistemic uncertainty is considered. the choice of liu's uncertainty theory for solving the problems of preliminary design is justified. the model of uncertain design parameter and optimization model with uncertain design and input parameters are proposed. the task of optimal design of the propulsion system parameters of supersonic maneuverable airplane is solved using the proposed models. the results are compared with the solution using monte carlo method. the solution time using the proposed model is two orders of magnitude less. keywords: preliminary design, uncertain quantity, epistemic uncertainty, optimization under conditions of parametric uncertainty, deterministic duplicate, expected mean, model of uncertain design parameters, optimal design. 1. introduction preliminary design of complex technical systems is usually understood as an iterative multidisciplinary optimization process of searching for design parameters (solutions) with given objective functions and constraints that determine the admissible set of solutions. in the deterministic formulation of optimal design, the parameters are considered accurate. the influence of parameter uncertainty on the objective functions and on the admissibility of the solutions is not taken into account. in the design, especially in preliminary design of new technical objects, the assumption of parameter accuracy and the search for a global maximum in the deterministic formulation is not justified, since the parameters here are inaccurate (uncertain), due to insufficient knowledge, lack of statistics, as well as the fact that design parameters are further implemented in the production with some tolerance. in addition, multidisciplinary optimization involves independent optimization of subsystems, i.e. subsystems should be optimized without full information on the output (design) parameters of other subsystems, using expert estimates. therefore, the search for design parameters in deterministic form can also cause inefficient solutions. the application of deterministic optimization methods under conditions of parametric uncertainty can lead to unacceptable solutions even with a slight variation in the parameters preliminary design with the epistemic uncertainty of parameters 155 copyright ©2018 assa. adv. in systems science and appl. (2018) [1]. the need to present and take into account uncertainty in engineering design is now universally recognized [2]. any method of decision-making in design under uncertainty must provide three basic tools:  uncertainty representation model;  method of calculations with uncertain variables (method of propagation of uncertainty on a function of uncertain variables);  model of optimization taking into account uncertainty of parameters. in the paper a brief overview of these tools for decision-making in the conditions of uncertainty ("theories of uncertainty") is given. unlike other uncertain theories liu's uncertainty theory provides a tool for reducing optimization models with epistemic input parameters to models of mathematical programming (for a certain class of functions). this provides the possibility of using classical numerical or meta-heuristic optimization methods without the use of simulation and approximation schemes, which lead to unacceptable computational time, especially with significant number of parameters, and introduce an additional error. in liu's theory input parameters are considered as uncertain. the work extends liu's theory of uncertainty, namely, the model of uncertain epistemic design parameters and the optimization model with epistemic input and design parameters are proposed. the problem of preliminary design of supersonic maneuverable airplane parameters in the conditions of parametric epistemic uncertainty using the liu's uncertainty theory [3] is solved. section 2 provides a classification of uncertainty types and a brief overview of existing models of uncertainty. section 3 gives an overview of the optimization models and methods under conditions of parametric epistemic uncertainty. the choice of liu's uncertainty theory for solving the problems of preliminary design of a supersonic maneuverable aircraft is justified. in section 4 a model of epistemic design parameters is proposed and the expected mean formula for the objective function with epistemic design and input parameters is derived. in section 5 the problem of calculating the parameters of the propulsion system for supersonic maneuverable aircraft using the proposed model is solved. 2. classification of uncertainty and model of epistemic uncertainty there are two types of uncertainty, depending on the available information: aleatory and epistemic. aleatoric (objective) uncertainty is associated with a random variable for which there is statistical data to help make reliable conclusions about the distribution of a random variable. in this case, the uncertain quantity is modeled by the probability distribution function. the epistemic (subjective) uncertainty of the first kind is connected with the fact that there is no or not enough information on random variables in order to make reliable conclusions about the distribution of a random variable. in the literature, such uncertainty is called an imprecise probability. in this case, it is common practice to obtain information from experts. an imprecise probability is sometimes called second-order uncertainty: uncertainty about an uncertain variable, i.e. the uncertainty associated with either the subjectivity of expert information or the methods of processing incomplete data is added to the uncertainty of the random variable. to date, more than 20 "uncertainty theories" have been proposed for modeling imprecise probabilities [4], including evidence theory [5], possibility theory [6], interval analysis [7], interval probability (p-box) [8], the liu's uncertainty theory [3]. epistemic uncertainty of the second kind is associated with a nonrandom variable, the value of which is currently unknown. the expert gives information about the predictive 156 g. s. veresnikov, l.a. pankova, v.a. pronina copyright ©2018 assa. adv. in systems science and appl. (2018) evaluation of this value. this kind of epistemic uncertainty is often called imprecise. the aleatory and epistemic uncertainties of the first kind can be classified as probabilistic form of uncertainty, while the second kind of epistemic uncertainty is non-probabilistic form of uncertainty. possibility theory [6], fuzzy sets theory [9], interval analysis [7], uncertainty theory [3] are used for model non-probabilistic form of uncertainty. to represent the epistemic uncertainty, there are many models. however, there is no well-defined procedure for selecting an epistemic uncertainty representation model. a variety of models is discussed in [10]. in practice, the choice of the model is determined by the available information and the ability to obtain expert information about the uncertain parameters of the problem as well as the complexity of the objective functions, i.e. the complexity of computations with uncertain variable. 3. optimization models and methods under conditions of parametric epistemic uncertainty. an objective function with uncertain parameters is an uncertain variable. in order to be able to compare uncertain objective functions and put the optimization task, it is necessary to introduce deterministic duplicates of objective functions, i.e. the numerical characteristics of these uncertain variables that will allow establishing the order relation on the set of uncertain objective functions. the choice of numerical characteristics as duplicates is the prerogative of the decision-maker (dm). the expected average value is most often used as the deterministic duplicate of the objective function. perhaps, the decision maker will prefer to obtain a result corresponding to the smallest spread of the objective function. then different spread indicators can be duplicates. dm also use a combination of both duplicates. dm may prefer to have a guaranteed result with a fairly high degree of belief. then the guaranteed value of the function is used as a duplicate. in this case, the decision maker sets the level of belief that the values of the function will be less than the guaranteed value when minimizing the objective function and more than the guaranteed value when maximizing the objective function. in a rigid setting, constraints must be satisfied for any uncertain parameter values. in a more conservative formulation (soft constraints) the expert sets levels for the belief degree in the performance of constraints in a multicriteria formulation for several objective functions fi, i=1, …, m, a model with uncertain parameters can use different set of deterministic duplicates for each objective function [11]: min(max (d[f1],…, d[fm]), где d[fi] – set of deterministic duplicates for fi. different aggregations of deterministic duplicates can be used as criteria. however, choosing a single metric for several objective functions is a very difficult task, which does not always have a solution. therefore the aggregation of objective functions is rarely used in real multi-criteria optimization tasks. on the other hand, when the number of objective functions is greater than 2, there is a problem of visualization of multidimensional data. in this case, it is important to have tools for graphical representation of the pareto front to facilitate the selection of solutions [12]. the problems of analysis and optimization with uncertain parameters usually have a large computational complexity, especially in applications to complex nonlinear systems. the application of epistemic parameter models using evidence theory, probability theory, fuzzy sets theory and interval representation often requires an excessively large number of samples and, as a rule, the found solution is not an extremum of the objective function. the calculation methods here rely on interval computations, which in general make optimization problems difficult to solve, in the worst case, it is not better than a brute force [13]. to preliminary design with the epistemic uncertainty of parameters 157 copyright ©2018 assa. adv. in systems science and appl. (2018) alleviate this problem, surrogate models were proposed [14]. however, the effectiveness of these approaches strongly depends on the accuracy of the surrogate model, and in addition the costs of building a surrogate model can be high. calculation methods for these types of models, using monte carlo simulation methods, are easy to implement, but for engineering applications their computational cost becomes unacceptable. for engineering design in conditions of epistemic uncertainty, it is important to use mathematical formalism, which will be effective for computation and decision making. in this paper, an approach based on the liu’s uncertainty theory was chosen to solve the problem of preliminary design. this theory for a certain class of functions ensures the reduction of optimization models with input epistemic parameters to models of mathematical programming. this allows to use classical numerical or meta-heuristic optimization methods without applying simulation and approximation schemes that lead to unacceptable computational time. in the future, uncertain variables will be called variables with epistemic uncertainty, modeled within the framework of liu’s uncertainty theory. let us describe a model of an uncertain variable in the uncertainty theory. the epistemic uncertainty of the event in uncertainty theory is the degree of the expert belief that the event will happen, i.e. measure of the uncertainty of this event m {*}. the measure of the uncertainty m is subadditive and the measure of the product of events is equal to the minimal measure of these events. uncertain variable ξ is given (on the basis of expert estimates) by the distribution ф(х) = m(ξ < x). m(ξ < x) is the degree of the expert belief that ξ is not greater than some value of x (measure of uncertainty). unlike the measure of uncertainty, the probability measure is additive, and the measure for the product of events is equal to the product of the measures of these events. in particular, it is asserted in [3] that experts tend to overestimate the probability of cases leading to negative consequences, therefore the application of probability theory with epistemic uncertainty of parameters can lead to significant errors in the analysis. in [3] optimization models under conditions of parametric uncertainty are considered for uncertain input parameters. let's give an analytical expression for the expected mean, which is used as a deterministic duplicate of an uncertain objective function. let x be the vector of the design exact parameters,  be the vector of independent input uncertain parameters with inverse distribution functions фi -1, i = 1, …, n, ),( xf be the objective function, ,0),( xg j j = 1, 2,…, p be constraints. if ),( xf is a continuous strictly increasing function with respect to ξ1, ξ2,…,ξm and strictly decreasing with respect to ξm+1, ξm+2,…, ξn, then:  dxfxfe nmm     1 0 11 1 11 1 ))1(),...,1(),(),...,(,()],([ 4. model of uncertain design parameters and optimization model with uncertain design and input parameters the need to take into account the uncertainty of the design parameters is caused by:  impossibility of providing accuracy of design parameter values in production conditions;  possible changes in the further iterative design process. design parameters can take any value within a specified range. if we consider them to be uncertain, then for each such parameter an uncertainty distribution function must be specified. the definition domain of distribution function changes during optimization and does not exceed the range in which the solution is searched. to ensure the possibility of solving optimization problems, it is necessary to consider the uncertain design parameter in the form of uncertain variable, including deterministic and epistemic variable. the deterministic 158 g. s. veresnikov, l.a. pankova, v.a. pronina copyright ©2018 assa. adv. in systems science and appl. (2018) variable provides the possibility of optimizing the objective function. the epistemic variable formed on the basis of expert information, takes into account the impressive of the design parameter. so, we consider that the uncertain design parameter x’ is modeled by the sum x’= x + δ, where x is deterministic design variable and δ is uncertain variable representing the impressive of this variable. the area for determining the uncertainty distribution function of δ depending on the preferences of dm can be given by absolute or relative limits. let ],[ ba xx be an interval for x’, xa≥0, xb≥0. in the first case, the uncertainty distribution function of δ is defined on the interval [-δ a, δ b], δ a≥0, δ b≥0, δ b+δ a < xbxa and x takes values from the range [xa+δ a, xb-δ b]. in the second case, the uncertainty distribution function of δ is defined on the interval [-kax, kbx], 0≤ka≤1, 0≤kb≤1, kbx+kax < xb-xa and x takes values from the range ] 1 , 1 [ b b a a k x k x  . in future we assume that the interval for δ is symmetric with respect to the mean value equal to 0, ka=kb, δ a=δ b. this option is most natural for expert. we derive an analytic expression for the expected mean, which is used as a deterministic duplicate of uncertain objective function. let ' ix be independent uncertain design parameters, xi be deterministic design parameters, δi be independent uncertain input parameters with inverse distribution functions 1 i , i = 1, …, p, and ξj, j = 1, …, n, be independent uncertain input parameters with inverse distribution functions 1 j . if ),( 'xf is a continuous strictly increasing function with respect to ' 1x , …, ' qx and strictly decreasing with respect to ' 1qx  , …, ' px , also a continuous strictly increasing function with respect to ξ1, …,ξm and strictly decreasing with respect to ξm+1, …, ξn, then the expected mean for ),( 'xf will have the form:    d))(),...,(),(),...,( ),(x),...,(x),(x,...,)(x(f)],x(f[e nmm p1qq ' pqq          11 11 11 1 11 1 111 1 0 1 1 11 (1) 5. calculating the parameters of the propulsion system for supersonic maneuverable aircraft we consider the task of preliminary design (pd) of a supersonic maneuverable aircraft, namely the calculation of aircraft propulsion system parameters ensuring the satisfaction of the requirements for the range of supersonic cruise flight (scf) and priority tactical and technical requirements (ttr) in the subsonic region under conditions of parametric uncertainty is considered [15, 16]. traditionally, pd is decomposed into several different disciplinary stages, where the parameters corresponding to the discipline are determined with the other parameters fixed, which ensure the satisfaction of ttr. then, the iterative process of multidisciplinary optimization takes into account the links between several stages and different ttr. since ttr are often contradictory, the choice of interrelated values of the aircraft design parameters, which ensure a compromise between the requirements of different disciplines is carried out. the values of the design parameters are formed as the results of solving various multicriteria optimization problems ttr to supersonic maneuverable aircraft contain requirements for subsonic and supersonic characteristics. at the preliminary design stage, the layout and propulsion parameters that satisfy the priority ttr on subsonic are first determined. then the calculation of the parameters of the propulsion ensuring the performance of ttr on supersonic is performed with the fixed parameters preliminary design with the epistemic uncertainty of parameters 159 copyright ©2018 assa. adv. in systems science and appl. (2018) ensuring the performance of ttr on subsonic. the main requirement for supersonic modes is the requirement for the range of supersonic cruise flight (scf), which depends to large extent on the parameters of the propulsion system in the scf mode. maximizing the range of the scf for a given mach number is equivalent to maximizing к/ce, where k is the aerodynamic quality of the aircraft, ce is the specific fuel consumption factor. the indicator of the level of wave resistance kw is a complex parameter reflecting the subsonic and supersonic aerodynamic characteristics of aircraft. an aircraft with a high value of kw is easier to construct and to ensure the performance of subsonic ttr. however, as kw increases, the maximum range of supersonic flight decreases. thus, the problem of finding compromise solutions for choosing aircraft propulsion system parameters arises that provide the largest possible range and the largest possible value of the level of the wave resistance. supersonic cruising flight is considered with mach number m = 1.5 at flight altitude h = 11 km. this task of calculating parameters is presented as a two-criteria problem of finding pareto solutions with criteria к/ce и kw [15, 16]. , ))(( )( dryopt p edryeopt weteqvfwoptosw e ppccp qsckpqkl с k                      1 2 2 p e drye dryopt opt eqvfwet opt0r w c c pp p qcs рp k ,)( optdryreheatdryopt stcpppp  stc = 0.4, where l is the wingspan, q is the velocity pressure magnitude at a given altitude and speed, kosw is the oswald coefficient, dryp is the relative thrust of the engine in the "maximum" mode, reheatp is the relative thrust of the engine in the "full afterburner" mode, p0r is the prospective thrust of the engine in the mode of "full afterburner" (h = 0, m = 0), stcopt is supersonic throttling factor, corresponding optр , eqvfc is the coefficient of equivalent friction, swet is surface area of the aircraft surface, dryec is specific hourly fuel consumption at the engine operating mode "maximum", p ec is slope of specific supersonic throttle characteristic. it is believed that the parameters of the geometry swet, l and the prospective thrust p0r , which ensure the performance of the main (subsonic) ttr, were previously calculated. in conditions with uncertain design and input parameters, the two-criteria maximization problem with duplicates of the expected average will have the form: max (е[к/ce], е[kw]). the results of calculations for three optimization models are presented in the paper. in the first model the input parameters eqvfc , kosw are considered uncertain. in the second model the design parameters reheatp , dryec , p ec are considered uncertain. in the third model, the input parameters eqvfc , kosw and the design parameters reheatp , dryec , p ec are considered uncertain. for uncertain parameters linear uncertainty distribution functions are given. the linear distribution function of uncertainty, reflecting the absence of expert preferences in the range of values of an uncertain parameter, can be regarded as corresponding to the uniform distribution of a random parameter. this makes it possible to compare results obtained using 160 g. s. veresnikov, l.a. pankova, v.a. pronina copyright ©2018 assa. adv. in systems science and appl. (2018) the theory of uncertainty and statistical multivariate modeling, without additional transformations. geometry parameters and characteristics of the engine correspond to the aircraft type f/a-22. multicriteria genetic algorithm was used to calculate optimal parameters. the following parameters of the multicriteria genetic algorithm from the matlab optimization toolbox were used. the population size is 10,000 chromosomes, the number of generations is 400. the remaining parameters were left by default. fig. 5.1 shows approximation of pareto fronts for the model with uncertain input parameters for different ranges of uncertain input parameters. using the jensen’s inequality [3], it is easy to show that depending on the convexity/concavity of the objective function relative to the input parameters, the pareto front for the model with uncertain input parameters can move up/down relative to the pareto front for deterministic model with nominal values of the input parameters. the blue pareto front corresponds to the calculation with the exact nominal values of the parameters. fig. 5.1. approximation of the pareto fronts for different ranges of uncertain input parameters consider the second model. the first objective function eс k is strictly decreasing with respect to reheatp , dryec , p ec . in accordance with formula (1) ][ eс k e has the form: preliminary design with the epistemic uncertainty of parameters 161 copyright ©2018 assa. adv. in systems science and appl. (2018) , ))))(21(()21(( )( ][ 1 0    d ppckckccp qsckpqkl с k e dryopt p e p edryedryeopt weteqvfwoptosw e     where optdryreheatreheatdryopt stcppkppp ))21((   . the second objective function wk is strictly decreasing with respect to dryec and strictly increasing with respect to reheatp , p ec . in accordance with formula (1) e[ wk ] has the form: ,1 )12( )21( 2 2 ][ 1 0    d ckc kcc pp p qcs рp ke p e p e dryedrye dryopt opt eqvfwet opt0r w                     where ,))12(( optdryreheatreheatdryopt stcppkppp   stc = 0.4. optimization of deterministic parameters reheatp , dryec , p ec is performed in intervals ] 1 ; 1 [ k p k p b reheat a reheat  , ] 1 ; 1 [ k c k c b drye a drye  , ] 1 ; 1 [ k c k c bp e ap e  , where ];[ b reheat a reheat pp , ];[ b drye a drye cc , ];[ bp e ap e cc are intervals of change of uncertain design parameters. fig. 5.2 shows the pareto fronts approximation for the model with uncertain design parameters for different ranges of their variation. fig. 5.2. approximation of the pareto fronts for different ranges of uncertain design parameters 162 g. s. veresnikov, l.a. pankova, v.a. pronina copyright ©2018 assa. adv. in systems science and appl. (2018) from fig. 5.2 it can be seen that pareto-fronts in the calculation with different ranges of uncertain design parameters are shifted in the direction of the worst values of the objective functions relative to the pareto-front for a deterministic model with nominal values of design parameters, i.e. solutions for the model with uncertain design parameters will always be more conservative. this task was solved using the monte carlo method to compare. the random variables reheatp , dryec , p ec were modeled in the intervals [-k reheatp ; k reheatp ], [-k dryec ; k dryec ], [-k p ec ; k p ec ] with a uniform distribution. fig. 5.3 shows the pareto-fronts approximation for calculating deterministic equivalents using the monte carlo method. fig. 5.3. approximation of pareto fronts obtained by the monte carlo method pareto fronts obtained from the second model with uncertain design parameters and obtained by the monte carlo method practically coincide. the monte carlo calculation time can be several orders of magnitude greater. this is because in the model with uncertain design parameters at each iteration of the genetic algorithm the expected mean of the functions k/ce, kw are calculated once. in monte carlo method a large number of computations of the functions k/ce, kw are required with the subsequent determination of the mean values e[k/ce], e[kw] (depending on the required accuracy). in solving this task by the monte carlo method there are 2000 calculations of k/ce, kw at each iteration. fig. 5.4 shows approximation of pareto fronts for the model with uncertain input and design parameters for different ranges of their changes. preliminary design with the epistemic uncertainty of parameters 163 copyright ©2018 assa. adv. in systems science and appl. (2018) fig. 5.4. approximation of pareto fronts for various ranges of uncertain design parameters with a range of uncertain input parameters 15%. by calculations of the model with uncertain design and input parameters it is impossible to estimate the influence of each type uncertainty parameters (input and design) on the optimization results, so it is advisable to carry out calculations on all three models. 6. conclusion the paper substantiates the choice of liu’ uncertainty theory for solving the problem of preliminary design for epistemic indeterminate parameters from the point of view of computational costs. a model of uncertain design parameters with epistemic uncertainty and an optimization model with uncertain design and input parameters are proposed. using the proposed model, the task of calculating the propulsion system parameters for the supersonic maneuvering aircraft is solved. a comparison is made with the results of deterministic optimization. the results are presented for three optimization models under conditions of parametric uncertainty: with uncertain input parameters, with uncertain design parameters, with uncertain design and input parameters. the results are compared with the monte carlo method. solution time using the proposed model is two orders of magnitude smaller. further research is planned to focus on the development of optimization methods for non-monotonic functions, as well as robust optimization models. 164 g. s. veresnikov, l.a. pankova, v.a. pronina copyright ©2018 assa. adv. in systems science and appl. (2018) references [1] nemirovski a. (2012) lectures on robust convex optimization. h. milton stewart school of industrial and systems engineering georgia institute of technology. atlanta georgia. [online]. available http://www2.isye.gatech.edu/~nemirovs /ro_ln.pdf [2] oberkampf w.l., deland s.m., rutherford b.m., diegert k.v., alvin k.f. (2002) error and uncertainty in modeling and simulation. reliab. eng. syst. saf., 75(3), 333– 357. [3] liu b. (2007) uncertainty theory. 2-rd edition. berlin: springer-verlag. [4] zadeh l. (2008) computation with imprecise probabilities. proceedings of ipmu’08. málaga, spain. [5] shafer g. a. (1976) mathematical theory of evidence. princeton, nj: princeton university press. [6] dubois d, prade h. (1988) possibility theory: an approach to computerized processing of uncertainty. new york: plenum. [7] jaulin l, kieffer m, didrit o, walter e. (2001) applied interval analysis. new york, ny: springer. [8] ferson, scott, vladik kreinovich, lev ginzburg, david s. myers, kari sentz (2003) constructing probability boxes and dempster-shafer structures. sand2002-4015. sandia national laboratories. [9] zimmerman h-j. (2001) fuzzy set theory and applications. 4th rev. ed. boston: kluwer academic publishers. [10] klir g.j. (2006) uncertainty and information: foundations of generalized information theory, wiley-interscience, hoboken. [11] veresnikov g.s., pronina v.a., pankova l.a. (2017) uncertain programming in preliminary design of technical systems with uncertain parameters. proceedings of the 12th international symposium "intelligent systems". atlanta: elsevier, 103, 36-43. [12] lotov, a.v., and k. miettinen. visualizing the pareto frontier. in j.branke, k.deb, k.miettinen, r.slowinski (eds.), multiobjective optimization. interactive and evolutionary approaches, lecture notes in computer science, 5252, springer, berlin-heidelberg, 213-244. [13] sharyy s. p. (2018) konechnomernyy interval'nyy analiz institut vychislitel'nykh tekhnologiy [finite-measured interval analysis] institute of computational technologies, novosibirsk: publishing house xyz, [in russian]. [14] helton j.c., johnson j.d., oberkampf w.l. (2004) an exploration of alternative approaches to the representation of uncertainty in model predictions. reliability engineering and system safety, 85, 39–71. [15] veresnikov g.s., pronina v.a., pankova l.a., ogorodnikov o.v. (2017) solution of preliminary design problems under conditions of parametric uncertainty. control sciences, 4, 65-73, [in russian]. [16] bashkirov i.g., irodov r.d. (1997) calculation of jet aircraft parameters under design requirements. world aviation congress, anaheim, ca, 975598, 1–3. http://www2.isye.gatech.edu/~nemirovs%20/ro_ln.pdf http://www2.isye.gatech.edu/~nemirovs%20/ro_ln.pdf microsoft word assa_1237-article text-4842-1-4-20220506 adv syst sci appl 2022; 04; 202-211 published online at http://ijassa.ipu.ru. the nature of the combined effect of low doses of radiation and chemical pollutants in doses close to the maximum permissible, on the state of somatic health of the population petr istratov1,2 1) moscow department of rospotrebnadzor; moscow, russia 2) medical university reaviz, moscow, russia e-mail: istratow@mail.ru abstract: technogenic development of the society besides improvement of social and living conditions of people leads to increase of emissions into the environment of industrial wastes, including radioactive wastes. the purpose of this work was to study the influence of low doses of radiation combined with chemical pollutants in doses close to the maximum permissible, on the health of people in comparison with similar health indicators of the population living in an area with similar chemical pollution (with radiological well-being) and in the area with "clean ecology". the study was conducted on the territory of one region for 3 years: from 2016 to 2018 in three districts: the first district district with the presence of radiation and chemical pollution, the second district (comparison) district with natural radiation background and chemical pollution and the third district (comparison) district of "environmental wellbeing". the state of health of the population, the specified settlements was studied on the basis of analysis of data from the accounting form № 12 (departmental form of state statistical observation of the ministry of health of the russian federation on form 12 "information on the number of diseases registered in patients residing in the service area of the medical organization" in the moscow region, for 2016-2018 years). it was found that both radiation and chemical pollution of the environment in low doses leads to an excessive risk of somatic diseases and mnd. when their effects on the human body are summarized, the resulting negative effect on human health is intensified. in the studied area of the rwdf, natural and climatic features and the nature of anthropogenic influence led to contamination, first of all, of the air and soil, which affected the structure of cancer mortality, risks of development of mns and the nature of pathology predominantly spread among the population upper and lower respiratory tract diseases, allergological diseases, which opens new directions for preventive and health-improvement work among the population. keywords: low doses of radiation, chemical pollutants, somatic pathology risks. 1. introduction despite being one of the driving forces for improving social and living conditions, technogenic development has led to an increased environmental pollution, including the production of radioactive waste. chemical pollutants and radionuclides contaminate soil, air, and water and then can enter the human body through inhalation and ingestion of contaminated food and water. regular intake of pollutants may result in chronic intoxication and trigger various somatic disorders, such as cancer [1,2,3,4]. the hypothesis we suggested p. istratov 203 copyright ©2022 assa. adv. in systems science and appl. (2022) stated that combination of low dozes of radiation and chemical pollution enhance each other and results in more complications compared with separate exposition of radiation and chemicals. the objective of the study: to clarify whether the combined effect of low doses of radiation and chemical pollutants (in doses less or close to the maximum allowable) on human health outcomes in compared with similar health indicators of the population living in an area with chemical contamination only (with radiological well-being) and in an area without any contamination ("clean ecology"). 2. objectives the objectives of this research were: 1. to assess the in a comparative aspect the radiation background and level of chemical contamination of three territorial formations: settlement 1 (in sergiev posad district), settlement 2 (in voskresensk district) and settlement 3 (in lotoshinsk district) of moscow region and their total impact on the health of the population; 2. in a comparative aspect to analyze the state of health of the population, the specified settlements to clarify the degree of influence of low doses of radiation and chemical pollutants in doses close to the maximum allowable, on the state of somatic health of residents. 3. methods the study was conducted on the territory of one region for 3 years: from 2016 to 2018 in three settlements of different districts of the region: the first district 4 kilometers from the radioactive waste dump (rw) and presence of territorial closeness of industrial enterprises settlement 1, the second district with only natural radioactive background and territorial closeness of industrial enterprises settlement 2, and the third district (according to the environmental well-being assessment one of the most environmentally safe areas settlement 3). when selecting residential settlements for the study, we relied on the statistical comparability of natural and climatic factors and the population structure in the studied settlements, which could affect the value of the parameters under study. the number of residents in these settlements is presented in table 1. table 1. population dynamics in the sites analyzed site/year number of residents (abs.) 2016 2017 2018 site 1 7157 7135 7114 site 2 7987 7988 8012 site 3 16906 16788 16567 the health status of the population of these settlements was studied based on the analysis of data from the accounting form № 12 (departmental form of state statistical observation of the ministry of health of the russian federation on form 12 "information on the number of diseases registered in patients residing in the service area of a medical organization" for the moscow region, for 2016-2018), obtained by excerpting. all residents permanently residing in the area were included in the study. the diagnosis was made on the basis of classification proposed in icd-10. the dynamics of pathology prevalence over the period from 2016 to 2018 were assessed.0 the prevalence of somatic pathology among the entire population of the settlement as a whole was evaluated. age periodization adopted by who among adults and children population was not considered due to the fact that the studied factors equally affected the entire population, in this regard, the health status of the population as a whole 204 the nature of the combined effect of low doses of radiation copyright ©0000 assa adv. in systems science and appl. (0000) was assessed. this study did not aim to compare the intensity of exposure to different age groups [6,7,8,9]. the questionnaire "do you have any problems with breathing?" was used to detect latent pulmonary pathology. interpretation of the results: if an interviewee has a total of 18 points or less, the probability of copd in the form of chronic bronchitis or pulmonary emphysema (or both) is high. the air condition in the residential area was assessed on the basis of reports on maximum permissible concentrations of substances (mpc) in the air. mpc levels were determined by levels approved by resolution of the chief state sanitary doctor of the russian federation on approval of hygienic standards gn 2.1.6.3492-17 "maximum allowable concentrations (mpc) of pollutants in the air of urban and rural settlements" from december 22, 2017 n 165 with amendments as of may 31, 2018, in force for the study period. radioactive contamination at all sites was measured and results were interpreted in accordance with the following regulatory documents: 1. sanpin 2.6.1.2523-09 ‘radiation safety standards (nrb-99/2009).’ 2. sp 2.6.1.2612-10 ‘basic sanitary rules for radiation safety (osporb -99/2010).’ 3. sanpin 2.6.1.2800 ‘requirements for radiation safety in case the population is exposed to natural ionizing radiation.’ measurements were performed according to the methods described in the regulatory document mu 2.6.1.2398-08 ‘radiation control and sanitary-epidemiological assessment of land plots intended for the construction of houses, buildings, and structures for public and industrial purposes to ensure radiation safety.’ information on measuring instruments is shown in table 2. table 2. measuring instruments used to assess radioactive contamination instrument serial number verification certificate measuring range measurement error dosimeter "dks at1123" (russia) 52711 no. 897 dated 11.02.2020 between 50 nsv/h to 10 nsv/h ≤ ± 15% statistical results were considered reliable at p<0.05. dependence between the signs was assessed using the paired correlation coefficient (r), its error (mr), and the significance level of differences. dependence was considered strong if r> 0.7, and average if the modulus of the pair correlation value lay within 0.3-0.7. if a correlation value less than 0.3 was found, it was considered to be weak. calculations were performed in microsoft excel 2007 spreadsheets and with the help of the licensed program statistica 6.1 for windows. 4. results analysis of the environmental situation of the main area and comparison areas also showed that the most polluted environment was the air environment, in connection with which we evaluated the number of nosological forms, whose relationship with environmental factors has now been established by numerous studies (table 3). table 3. average content of chemical pollutants in the air of residential areas of comparison settlements (compared with mpc (%) no. of the point chemical poppution allowed concentration in the air according to regulations (mg/m3) settlement 1: actual concentration in the air (mg/m3) settlement 2: actual concentration in the air (mg/m3)1 settlement 3: actual concentration in the air (mg/m3) hazard classification according to regulations 1 hexavalent chromium (converted to 0,0015 100 100 76 1 p. istratov 205 copyright ©2022 assa. adv. in systems science and appl. (2022) chromium trioxide) vanadium (v) oxide beryllium and its compounds in beryllium equivalent cobalt 0,002 0,00001 0,001 98 102 98 99 100 97 62 75 63 2 bromine chlorine carbon disulfide manganese and its compounds (in terms of manganese dioxide) nickel oxide (in terms of nickel) 0,040 0,030 0,005 0,001 0,001 104 103 114 100 100 105 99 112 98 101 57 62 76 54 51 2 3 nitrogen oxide sulfur dioxide anhydrite (sulfur dioxide) 0,06 0,05 94 105 98 100 77 80 3 4 carbon oxide 3 112 112 84 4 5 action summation coefficient 1,03 1,02 0,68 when several substances with a summation effect are present in the ambient air together, the sum of their concentrations should not exceed one when calculating by the formula: where p1, p2, p3....p actual concentrations of substances in the atmospheric air; mрc1, mрc2, mрc3 ... mрc maximum permissible concentrations of the same substances, n number of analyzed substances. in the case of settlements 1 and 2 the value of the action summation factor is 1.03 and 1.02, respectively. it means that during the long-term stay in the territory of their registration people can form the prerequisites for formation of somatic pathology associated with the mechanisms of impact of these compounds on human physiology, regardless of the fact that their concentrations in the air are practically close to the values of mpc. analysis of radiological situation showed that in settlement 1 the level of gammaradiation was at the upper limit of the norm (did not exceed 0,28-0,30 μsv/h), and in compared settlements did not differ from background values for the studied region (table 4). table 4. measurement results of the ambient dose equivalent rate of gamma radiation at the points with the maximum readings of the search instrument no. of the point place of measurement h, μsv/h ∆, μsv/h note 206 the nature of the combined effect of low doses of radiation copyright ©0000 assa adv. in systems science and appl. (0000) on the territory of the settlement 1 site 1 0,30 ±0,02 upper limit of normal 2 site 2 0,10 ±0,02 normal background 3 site 3 0,07 ±0,004 normal background the study of the nature of somatic pathology showed that in settlements 1 and 2 the most frequent pathology is bronchopulmonary pathology. the structure of bronchopulmonary pathology in the studied settlements is presented in tables 5.1, 5.2. table 5.1. characteristics of the prevalence of upper and lower respiratory tract diseases in the population (site 1, site 2) parameter site 1 site 2 abs. (cases) prevalence among the adult population, % abs. (cases) prevalence among the adult population, % δ with iudino parameters, % 2016 j30 1302 18.2 1296 16.2 -2.0 j30.1 1276 17.8 1231 15.4 -2.4 l50 154 2.2 104 1.3 -0.9 l27.2 134 1.87 112 1.4 -0.47 t78.1 138 1.9 101 1.3 -0.5 т 88.7 459 6.4 434 5.4 -1.0 2017 j30 1305 18.3 1299 16.3 -2.0 j30.1 1279 17.9 1239 15.5 -2.4 l50 154 2.2 109 1.4 -0.8 l27.2 135 1.9 118 1.5 -0.4 t78.1 142 2.0 115 1.4 -0.6 т 88.7 458 6.4 434 5.4 -1.0 2018 j30 1306 18.4 1332 16.6 -1.8 j30.1 1280 18.0 1249 15.6 -2.4 l50 155 2.2 109 1.4 -0.8 l27.2 137 1.9 119 1.5 -0.4 t78.1 148 2.1 117 1.5 -0.6 т 88.7 461 6.5 438 5.6 -0.9 note: data is adjusted to population dynamics table 5.2. characteristics of the prevalence of upper and lower respiratory tract disease in the population (site 3) parameter site 3 abs. (cases) prevalence among the adult population, % δ with iudino parameters, % 2016 j30 946 5.6 -12.6 j30.1 532 3.1 -14.7 l50 68 0.4 -1.8 l27.2 105 0.6 -1.27 t78.1 96 0.6 -1.3 т 88.7 98 0.6 -5.8 2017 j30 944 5.6 -12.7 j30.1 523 3.1 -14.8 l50 60 0.4 -1.8 l27.2 100 0.6 -1.3 t78.1 96 0.6 -1.4 т 88.7 95 0.6 -5.8 2018 j30 930 5.6 -12.8 p. istratov 207 copyright ©2022 assa. adv. in systems science and appl. (2022) j30.1 511 3.1 -14.9 l50 49 0.3 -1.9 l27.2 89 0.5 -1.4 t78.1 91 0.5 -1.6 т 88.7 92 0.6 -5.9 note: data is adjusted to population dynamics however, as the first international epidemiological study of the spread of chronic respiratory diseases in russia has shown, the official statistics do not reflect the true picture. the study was conducted by the global alliance for chronic respiratory diseases (gard) under the guidance of the research institute of pulmonology on the initiative of who. thus, for example, according to the official data only 1% of the russians were diagnosed with copd, while in the course of the international epidemiological study more than 15% of the russians included in the examination had copd symptoms and the experts consider that the real sickness rate is even higher. in order to increase the objectivity of the data on the prevalence of bronchopulmonary pathology in the study areas, we offered a random sampling of 300 people from each settlement to answer the questionnaire "do you have problems with breathing?", which was used in the above-mentioned study. in order to objectify the picture, we tried to include in the survey statistically comparable contingents that did not have, according to their medical records, pulmonary pathology. taking into account the fact that according to the who classification periodization of ages covers rather wide time periods, we divided all periods by 10 years (table 6). table 6. characteristics of people who took part in the survey age/sex 18-29 30-39 40-49 50-59 60-69 70-79 total, % site 1 male 29 27 27 26 25 24 53 female 21 23 23 24 25 26 47 total 50 50 50 50 50 50 300 site 2 male 28 28 27 28 25 23 53 female 22 22 23 22 25 27 47 total 50 50 50 50 50 50 300 site 3 male 30 27 29 27 24 26 55 female 20 23 21 23 26 24 45 total 50 50 50 50 50 50 300 when processing the results of the questionnaire, in accordance with the requirements of the questionnaire, it was considered that the interviewee had 18 points or less, the probability of developing copd in the form of chronic bronchitis or pulmonary emphysema (or both) was high. the results obtained during the study are shown in the figure below (figure 1). 208 the nature of the combined effect of low doses of radiation copyright ©0000 assa adv. in systems science and appl. (0000) figure 1. latent bronchopulmonary pathology identified in the questionnaire as can be seen from the presented diagram, the greatest difference in the amount of latent pathology is between settlement 3, where environmental factors are most favorable, and settlement 1 (by +42.9 %), where there is pronounced toxic-chemical and radiation pollution. in order to find out whether there is a connection between the qualitative composition of the air/soil of the zone of residential development within the radius of the impact of raw by the leading environmental factors, including sensitizing factors, and various nosological forms of respiratory disease, we conducted a correlation analysis with calculation of the values of pair linear correlation and the square of the correlation relation. the most significant indicators are presented in table 7. table 7. chemical pollutants that demonstrated significant squared correlation coefficients (n2xy) for respiratory pathology among the population and levels of pollutants in the environment (soil, air) no. chemical pollutant hazard class effect on the organism (distribution by type at inhaling) * η2xy 1 caesium strontium polonium group 2 group 2 group 2 r r i/r 0.91 0.91 0.91 2 chromium chlorine sulfur (carbon disulfide) manganese cobalt class 1 class 2 class 2 class 2 class 2 s r i i d 0.78 0.87 0.78 0.78 0.91 3 nickel class 1 i 0.68 4 bromine beryllium vanadium class 1 class 1 class 2 r s i/r 0.78 0.78 0.78 5 manganese dioxide class 2 r 0.68 type ‘s’–chemical pollutants slowly soluble in human lungs; type ‘i’–chemical pollutants soluble at an intermediate rate; type ‘r’–rapidly soluble chemical pollutants the second group of disorders closely related to environmental pollution is allergic diseases. we observed a significant difference between sites in the prevalence of allergic rhinitis (j30), pollinosis (j30.1), urticaria (l50), unspecified adverse effect of drug or medicament (т88.7), dermatitis due to ingested food (l27.2), and other adverse food reactions (t78.1). the prevalence of allergic diseases is shown in table 6. p. istratov 209 copyright ©2022 assa. adv. in systems science and appl. (2022) figure 2 demonstrates relative prevalence of allergic diseases in sites 2 and 3 compared to site 1 (its prevalence is considered as 100%). it allows us to evaluate the effect of increased levels of radiation and chemical pollution on human health. figure 2. ratio of allergic pathology among residents of the studied residential settlements (as of 2018) analysis of the mortality structure from malignant neoplasms (mn) in the included settlements showed some differences from the general statistical pattern (figure 4). thus, in settlement 1 there was an increase in the number of persons with cancer of the respiratory organs, as well as organic lesions of lymphatic and hematopoietic tissue compared to the average statistical data for the moscow region by 7.0% and 1.5% respectively, the number of deaths from colorectal cancer and stomach cancer was also increased by 2.9%. in settlement 2, the structure of mortality from cancers was close to that of the study region, except for mortality from respiratory cancers (figure 3). at the same time, it was noted that mch mortality in settlement 1 increased by 1.9% in 2018 compared to 2016, in settlement 2 by 1.6%, and in settlement 3 increased by only 0.2%. 42,3 24,2 9,1 7,2 17,2 region 1 cancer of gaster and colorectal cancer bronheal, tracheal and lung cancer cancer of female reproductive system 40,1 19,1 7,5 5,1 28,2 region 2 cancer of gaster and colorectal cancer bronheal, tracheal and lung cancer cancer of female reproductive system figure 4. structure of mortality from stds from 2016 to 2018: a settlement 1, b settlement 2 210 the nature of the combined effect of low doses of radiation copyright ©0000 assa adv. in systems science and appl. (0000) 5. discussion of the obtained results as can be seen from the data obtained, the combined effect of low doses of radiation and chemical pollutants in doses close to the maximum allowable, can have a negative impact on the somatic health of the population. what is this related to? most studies to date have been based on a study of the effects of high doses of radiation on the human body, due to the need to consider the presence of nuclear weapons in the world, and the high likelihood of technogenic accidents. until recently, when mentioning low doses of radiation, only its stimulating effect hormesis was considered. however, recent studies have shown that low-dose irradiation also induces numerous structural modifications in cells, resulting in changes in their functional activity, and thus may lead to the development of pathology. and small doses of radiation can kill the cell just as well as large ones, if the question is not about a single exposure, but a long-term exposure. what is the reason? weak irradiation acts indirectly, triggering the mechanisms of genomic regulation. it can provoke apoptosis programmed cell death. high doses of radiation simply cause death by directly damaging the dna molecule, while low doses do so through gene expression and the appearance of proteins that will trigger the mechanism of programmed cell death. the data obtained do not contradict the fundamental science, since any living organism can give different responses and they will not necessarily obey the linear "dose-effect" relation. this is due to the fact that a reparation (repair) reaction does not always occur to small doses. this is due to the fact that human evolution took place in conditions of natural background radiation, which resulted in the body not recognizing low doses of radiation as a hazard, not mobilizing, not trying to adapt, "not defending". the second important factor of negative impact on the human body is chemical factors. this is due to the development of both industrial and domestic and agricultural pollutants. the simultaneous impact of factors of chemical and radiological nature, even in low concentrations, as it was in settlement 1, or only chemical pollutants (settlement 2), leads to a significant increase in somatic diseases among the population. at the same time, the important fact remains that regardless of the nature of the factors acting on the person, the synergism is maximized when they are affected simultaneously. at the same time, experimental studies have shown that a decrease in the radiation dose (low doses) requires a proportional decrease in the concentration of the other agents in order to maintain maximum synergistic effect. this is precisely the condition that arises when low doses of radiation and concentrations of chemical agents close to the mac are combined. the result is a decrease in the health indicators of the population even if the sanitary-hygienic standards are observed. 6. conclusion 1. at summation even small doses of radiation and chemical pollution of environment (not exceeding maximum permissible norms) their resultant negative influence on human health increases. 2. the synergistic effect of low doses of radiation and chemical contamination of the environment (not exceeding maximum permissible norms) can lead to the development of excessive risk of somatic diseases and mnd, which opens new directions for preventive and health-improving work among the population. acknowledgements the research was supported by the moscow department of rospotrebnadzor and medical university reaviz, moscow, russia. p. istratov 211 copyright ©2022 assa. adv. in systems science and appl. (2022) references 1. astafyeva o.v. (2019). current aspects of standardization of negative impact on the environment. biosphere compatibility: person, region, technology, 2, 3–12. 2. chervoniy, m.a. & strelnikova, a.i. (2019). the role of protection against ionizing radiation in maintaining human health. international scientific and technical conference of young scientists of v.g.shukhov belgorod state technological university. belgorod, russia, 2346–2350. 3. kurochkina, a.i. & oganesyan, d.a. (2018) impact of ionizing radiation on humans. bulletin of modern research, 5:3(20), 63–65. 4. karpin, v.v., starinsky, g.v. & petrova, g.v. (2018, january 15) malignant tumors in russia in 2018 (incidence and mortality) [online]. available https://glavonco.ru/cancer_register/ 5. markin, yu.g. (2017) international and russian experience of responsibility for the elimination of accumulated environmental damage and approaches to risk management. ecological bulletin of russia, 7, 54–58. 6. kreuzer m., auvinen a., cardis e., durante m., harms-ringdahl m., et. al. (2011) multidisciplinary european low dose initiative (melodi): strategic research agenda for low dose radiation risk research. radiation and environmental biophysics, 57(1), 5– 15. 7. okhrimenko, s.e., korenkov, i.p., akopova, n.a. & ivanov s.i. (2016) hygienic assessment of contamination of the city territory with radioactive pollutants of technogenic origin. anri, 3, 64–71. 8. ostrovskiy, n.v. (2016) local self-government and environmental protection (review of legislative acts). moscow, russia: union of russian cities. 9. voronich, s.s. monitoring of atmospheric pollution in urban areas. moscow, russia: nauka. adv syst sci appl 2024; 02:54–65 published online at https://ijassa.ipu.ru. the mean-field approximation for the scardo model in the case of 3-element opinion space: fixed points and exact solutions vladislav n. gezha1,2, ivan v. kozitsin1,2* 1 moscow institute of physics and technology, dolgoprudny, russia 2 v. a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia abstract: we analyze the scardo model in the case of the 3-element opinion space under specific constraints on the transition table parameters that allows to link the problem at stake to the case of the 2-element opinion space that has been thoroughly studied previously. we characterize the properties of fixed points and support our findings by numerical experiments. further, we manage to find out those settings that ensure the system almost surely reaches a specific domain in the phase space, after which its behavior can be predicted analytically. keywords: opinion dynamics models, scardo-model, mean-field approximation 1. introduction the recent proliferation of social media platforms as well as emergent speculations around the issues of opinion polarization [18], destructive content [14], echo-chambers [5], and filter bubbles [1] have motivated scientists to focus their attention on the analysis of these social effects by applying the framework of opinion formation models [4, 12, 15–17]. these models, initially designated to describe how individuals’ opinions change following peer interactions [2], are now intensively upgraded to account for opinion formation patterns in the online domain [3]. the current paper is dedicated to the analysis of a relatively recent opinion formation model introduced in ref. [8]. this minimal model (hereafter – the scardo-model) was to suggest a flexible, easy-to-validate framework that could approximate a huge variety of micro-level mechanisms of opinion formation at both quantitative and qualitative levels. further, an extension of the model introduced in ref. [7] allows to account for the fact that individuals’ social power may depend on their socio-demographic characteristics [13]. as was reported in ref. [8], the typical behavior of the model displays out-of-equilibrium patterns and has a hard time getting an analytical description. nonetheless, under specific restrictions common in the field of socio-physics (the number of agents in the system is huge and they communicate via a complete graph), a mean-field approximation in the form of a system of ordinary differential equations can be derived. these equations explain opinion dynamics in terms of the populations of opinions, at the macroscopic level [15]. previously, the mean-field approximation for the scardo-model in the case of the 2element opinion space has been thoroughly investigated [6]. these settings correspond to the situation when there are two opinion factions. however, in many respects this description of real-world processes is too simplified. indeed, in the case of political attitudes, individuals’ ∗corresponding author: kozitsin.ivan@mail.ru the mean-field approximation for the scardo model... 55 opinions are not simply divided into two groups, but display rather substantial diversity, with various levels of political radicalism. at least, we should account for people with neutral views, as they usually substitute the majority of the population. in this paper, we focus on studying the properties of the mean-field approximation system in the situation when agents’ opinions belong to the 3-element opinion space. clearly, this case is much more complicated. however, under specific constraints on the model parameters, we will investigate fixed points and, in some cases, derive precise solutions for the system. 2. opinion dynamics model in this section, we briefly present the scardo-model that was introduced in ref. [8] and later investigated in ref. [6] for the case of the two-element opinion space. in this model with discrete time t = 0, 1, 2, . . . , n, agents update their opinions following consecutive pairwise interactions that unfold on a social network g. agents’ opinions (denoted by o) belong to a discrete opinion space with m elements: z = {z1, . . . , zm}. at each iteration, an agent i is chosen at random and then one of i’s peers (j) is selected also at random. then, agent j (influence source) influences on i (influence object). as a result, the influence object’s opinion updates in accordance with a specific opinion distribution that is a function of the opinions of interacting agents. the distribution that outlines how agent i updates their opinion is represented as follows: (ps,l,1, . . . , ps,l,m), with the element ps,l,k standing for the probability that i’s opinion will become zk following the interaction. the first two indices of ps,l,k stand for the opinions of i and j before the communication event respectively. in turn, the third index links to the index of probable opinion. as a result, the quantity ps,l,k is just a conditional probability: ps,l,k = pr {oi(t+ 1) = zk | oi(t) = zs, oj(t) = zs}. (2.1) coupled together, these probabilities form a 3-d object p = [ps,l,k] m s,l,k=1 , which is referred to as the transition table [8]. this table can be safely expressed via its slices over the first index: p = [p1, . . . , pm], where p1 = [ p1,1,1 . . . p1,1,m . . . . . . . . . p1,m,1 . . . p1,m,m ] , . . . , pm = [ pm,1,1 . . . pm,1,m . . . . . . . . . pm,m,1 . . . pm,m,m ] (2.2) within these shorthands, p1, . . . , pm are m×m matrices that encode opinion change strategies of individuals espousing opinions z1, . . . , zm correspondingly. all these matrices are row-stochastic: ps,l,1 + . . .+ ps,l,m = 1 copyright © 2024 assa. adv syst sci appl (2024) 56 v.n. gezha, i.v. kozitsin for each s and l. an important feature of the scardo-model is that it can describe both ordered and categorical opinions. the model mechanics does not depend on if the opinion alphabet is ordered or not. the presence/absence of order is important only at the stage of the transition table interpretation/estimation. once the transition table is defined, all the necessary information on the opinion dynamics is stored therein. 3. mean-field approximation for the scardo-model the following mean-field approximation was obtained in ref. [8] for the scardo-model. let yi(t) stand for the fraction of individuals espousing opinion zs at a time moment t: ys(t) = #{i | oi(t) = zs} n . with the assumptions n → ∞ and g is a complete graph, the mean-field approximation can be derived in a form of a system of ordinary differential equations in the scaled time τ : dy1 dτ = m∑ s,l=1 ysylps,l,1 − y1, . . . dy3 dτ = m∑ s,l=1 ysylps,l,m − ym, (3.3) where τ = t/n, δτ = 1/n. the fixed points of system (3.3) are given by the fixed point system: m∑ s,l=1 ysylps,l,1 − y1 = 0, . . . m∑ s,l=1 ysylps,l,m − ym = 0. (3.4) the initial point of system (3.3) is defined by y1(0) = q1, . . . , ym(0) = qm, (3.5) where q1 . . . , qm ≥ 0, q1 + . . .+ qm = 1. system (3.3) and the corresponding cauchy problem (3.3), (3.5) feature the following properties (see refs. [8] and [10] for proofs). corollary 3.1: the function u = y1 + . . .+ ym is the first integral of (3.3). theorem 3.1: the cauchy problem (3.3), (3.5) has a unique solution y(τ), which can be extended on the whole τ -axis. the components of y(τ) are nonnegative and sum up to one for each τ ∈ r. copyright © 2024 assa. adv syst sci appl (2024) the mean-field approximation for the scardo model... 57 4. main assumption hereinafter, we will study system (3.3) in the case of the 3-element opinion alphabet (m = 3). this situation is much more complex than the case m = 2 and we have failed to obtain analytical solutions. however, we report that it is possible to get some analytical results for a specific family of transition tables: p1 = [ 1− α α 0 1− β β 0 a31 a32 a33 ] , p2 = [ γ 1− γ 0 δ 1− δ 0 b31 b32 b33 ] , p3 = [ c11 c12 c13 c21 c22 c23 c31 c32 c33 ] , (4.6) where α, β, γ, δ ∈ [0, 1], and other parameters are arbitrary. from this point, we assume that the transition table belongs to the class (4.6). it is worth noting that transition tables (4.6) are not somewhat disconnected from the real world. in fact, the restraint (4.6) just prescribes that opinion shift z1 → z3 can only happen if the influence comes from opinion z3, an assumption that is quite natural. further, from (4.6) it follows that opinion shift z2 → z3 cannot occur if the influence subject has opinion z1 or z2, indicating thus that agents having opinion z2 are not sensitive to the negative and anti-conformity forms of influence. 5. auxiliary system we are going to need an auxiliary social system which is defined in the 2-element opinion alphabet by the following transition table: q1 = [ 1− α α 1− β β ] , q2 = [ γ 1− γ δ 1− δ ] , (5.7) where α, β, γ, δ are the same as in (4.6). one can easily notice that this transition table is just a chunk of transition table (4.6). below, this social system and the corresponding transition table (5.7) will be referred to as auxiliary ones. in order to avoid confusion, the initial 3-element opinion space system will be denoted as the main. the properties of auxiliary system have been thoroughly studied in ref. [6]. in particular, the exact solution of (3.3) has been found, and fixed points as well as their stability properties have been systematically characterized. for now, we will harness these findings for our purposes. 6. fixed points on the line y3 = 0 let us turn to the analysis of system (3.3) in the case of the main system. we will start with the characterization of fixed points that are located on the line y3 = 0. because of (4.6), the third equation of (3.3) turns out to have the following form: dy3 dτ = y3 × [ (y1p1,3,3 + y2p2,3,3 + y1p3,1,3 + y2p3,2,3 + y3p3,3,3)− 1 ] . (6.8) with y1 + y2 + y3 = 1 we can rewrite (6.8) as follows: dy3 dτ = y3 × [ ay1 +by3 + c ] , (6.9) where copyright © 2024 assa. adv syst sci appl (2024) 58 v.n. gezha, i.v. kozitsin a = p1,3,3 − p2,3,3 + p3,1,3 − p3,2,3, b = p3,3,3 − p2,3,3 − p3,2,3, c = p2,3,3 + p3,2,3 − 1. let us now assume that y3 = 0. in this case, dy3 dτ = 0 and the first equation of (3.3) (for the derivative of y1) turns out to: dy1 dτ = (1− α)y21 + (1− β)y1y2 + γy2y1 + δy22 − y1. let us denote y = y1 and make use of y1 + y2 = 1: dy dτ = (1− α)y2 + (1− β)y(1− y) + γ(1− y)y + δ(1− y)2 − y = (1− α)y2 + (1− β)y − (1− β)y2 + γy − γy2 + δ − 2δy + δy2 − y = = (δ + β − α− γ)y2 + (γ − β − 2δ)y + δ. now it is time to recall that this expression is exactly similar to the one obtained for auxiliary system (see ref. [6], p. 107, formula (4.7)). further, from y3 = 0 it follows that y1 + y2 = 1. with both of these facts, we obtain that the fixed points of (3.3) that are located on the line y3 = 0 can be found by solving the fixed point equation (3.4) for the auxiliary system. its solutions [y∗1 y∗2] t will define the fixed points of the main system as follows: [y∗1 y∗2 0]t . 7. simulation experiments let us now pinpoint the above findings with simulation experiments. we will consider two basic scenarios. 7.1. scenario 1 in this scenario, we will specify the transition table as follows: p1 = [ 0.8 0.2 0 0.5 0.5 0 0.5 0.5 0 ] , p2 = [ 0.8 0.2 0 0.5 0.5 0 0.5 0.5 0 ] , p3 = [ 0.5 0.5 0 0.5 0.5 0 0 0 1 ] on this occasion, the transition table for auxiliary system is given by: q1 = [ 0.8 0.2 0.5 0.5 ] , q2 = [ 0.8 0.2 0.5 0.5 ] . (7.10) it is known from ref. [6] that in this case, the only fixed point for auxiliary system is y∗1 = 5 7 , y∗2 = 2 7 . from our previous derivations, it follows that [ 5 7 2 7 0 ]t should be a fixed point of the main system. copyright © 2024 assa. adv syst sci appl (2024) the mean-field approximation for the scardo model... 59 7.2. scenario 2 in this scenario, we consider the following transition table: p1 = [ 0.8 0.2 0 0.5 0.5 0 0.24 0.24 0.52 ] , p2 = [ 0.8 0.2 0 0.5 0.5 0 0.24 0.24 0.52 ] , p3 = [ 0.24 0.24 0.52 0.24 0.24 0.52 0 0 1 ] . the transition table of the corresponding auxiliary system is given by (7.10) again. in this vein, one should expect [ 5 7 2 7 0 ]t to be a fixed point of the main system. 7.3. stability analysis for scenarios 1 and 2 from ref. [6], we know that y3 = 0 ensures dy1 dτ > 0 for every y1 = y∗1 − ε and dy1 dτ < 0 for every y1 = y∗1 + ε, where y∗1 is the first component of auxiliary system’s fixed point [y∗1 y∗2] and ε > 0. as such, the properties of the fixed point [y∗1 y∗2 0] of the main system are only defined by the sign of the derivative dy3 dτ . in the case of scenario 1, we end up with the following system: dy1 dτ = −0.3 · y1 · y3 − 0.5 · y3 · y3 − 0.7 · y1 + 0.5, dy3 dτ = y3 · y3 − y3. (7.11) the second equation in (7.11) has the negative coefficient before the linear term, which means that the equilibrium point [ 5 7 2 7 0 ]t is a stable node. in scenario 2, the system (3.3) turns out to be: dy1 dτ = −0.3 · y1 · y3 + 0.02 · y3 · y3 − 0.7 · y1 − 0.52 · y3 + 0.5, dy3 dτ = −0.04 · y3 · y3 + 0.04 · y3. (7.12) the second equation in (7.12) has the positive coefficient before the linear term. because of this, the equilibrium point [ 5 7 2 7 0 ]t is a saddle. that is, the difference between scenario 1 and scenario 2 lies in the stability of fixed point [ 5 7 2 7 0 ]t . it is worth noting that in the case of scenario 2, the phase portrait of the system contains one more fixed point [0 0 1]t , which is stable. 7.4. results of simulations in fig. 7.1, we present the results of the simulations conducted. we see that in the case of scenario 1, the system steadily converges to the fixed point [ 5 7 2 7 0 ]t , just as predicted by our theoretical derivations. however, in scenario 2, the system does not converge to copyright © 2024 assa. adv syst sci appl (2024) 60 v.n. gezha, i.v. kozitsin [ 5 7 2 7 0 ]t , which is unstable on this occasion. instead, the system first reaches a specific line located near the red and blue curves (these two lines are defined by the fixed point equation (3.4) for auxiliary system and their intersections outline fixed points) and then tends to drift along this line in the direction of the stable fixed point [0 0 1]t . let us look at this trajectory in more detail. one can notice that each point on this line is almost an equilibrium as it lies near the red and blue curves at once. nevertheless, the phase velocity on this trajectory is directed towards [0 0 1]t . to get a deeper understanding of this effect, let us consider a specific family of transition tables that satisfy the following constraints: p3,3,3 − p2,3,3 − p3,2,3 = 0, p1,3,3 − p2,3,3 + p3,1,3 − p3,2,3 = 0, (7.13) p2,3,3 + p3,2,3 − 1 = 0. in fact, the transition table from scenario 1 almost fits (7.13): whereas for the second equation, we get the perfect matching, the first and third ones have a residual of 0.04. with a transition table that meets criteria (7.13), one can ensure that dy3 dτ = 0 holds for all y1 and y2, and the infinite number of fixed points exists. these fixed points form a line which is extremely close to the trajectory of the system in scenario 2. each of these fixed points can be easily computed by reducing the problem at hand to auxiliary system. to this end, one should pose 0 ≤ y3 = c < 1 and find a fixed point [y∗1 y∗2] from auxiliary system subject to y1 + y2 = 1− c. 8. further properties the results presented above center around the line y3 = 0, which is an extremely narrow domain of the phase space. however, after imposing additional restrictions on the transition table components, we can guarantee that with probability 1, the system will always reach this line in a finite time interval. further, after reaching the line y3 = 0, the system will not leave it, and its behavior to be described by the solution of the corresponding auxiliary system. let us now look at equation (6.9) in more detail. because y3 ≥ 0, we can rewrite inequality dy3 dτ < 0 as y3 × (p3,3,3 − p2,3,3 − p3,2,3) + +y1 × (p1,3,3 − p2,3,3 + p3,1,3 − p3,2,3) + (p2,3,3 + p3,2,3 − 1) < 0, (8.14) y3 > 0. the first inequality in (8.14) is ensured if the following constraints hold (recall that y1 ≥ 0, y3 ≥ 0, y1 + y3 ≤ 1): p3,3,3 − p2,3,3 − p3,2,3 ≤ 0, p1,3,3 − p2,3,3 + p3,1,3 − p3,2,3 ≤ 0, (8.15) p2,3,3 + p3,2,3 − 1 < 0, note that inequalities (8.15) are imposed upon unspecified components of (4.6) and thus do not contradict it. copyright © 2024 assa. adv syst sci appl (2024) the mean-field approximation for the scardo model... 61 fig. 7.1. on panels (a) and (b), 5 simulation experiments for scenarios 1 and 2 are depicted. panels (c) and (d) demonstrate the phase portraits for these scenarios. the gray dots on panels (c) and (d) plot the fixed point[ 5 7 2 7 0 ]t . the same point is defined on panels (a) and (b) by the dashed lines. the blue and red lines are defined by the equations of (3.4). their intersections mark fixed points. the green curves stand for the trajectories obtained in simulations (one trajectory per scenario). if (8.15) is true, then equation dy3 dτ = 0 has the only root (y3 = 0), and dy3 dτ < 0, excepting for the only fixed point [y∗1 y∗2 0]t (we will characterize this fixed point later). for such systems, the following results can be obtained. theorem 8.1: let us consider the cauchy problem (3.3), (3.5) for m = 3. let assume that restrictions (4.6) and (8.15) hold. then, the system will almost surely reach the line y3 = 0 in a finite amount of time. theorem 8.2: let assume that the conditions of theorem 8.1 hold. let assume that the system has just (t = t ) reached the line y3 = 0 at the point [q̂1 q̂2 0]t . let us redefine the time as follows: τ̂ = τ − t, and initialize auxiliary system with the initial point [q̂1 q̂2] t . then the solution of the main system y(τ) is given by y(τ) = [y1(τ) y2(τ) 0] t , where y1(τ) = ŷ1(τ̂), y2(τ) = ŷ2(τ̂), copyright © 2024 assa. adv syst sci appl (2024) 62 v.n. gezha, i.v. kozitsin and [ŷ1(τ̂) ŷ2(τ̂)] t is the solution of auxiliary system. proof first of all, let us show that with probability 1, the line y3 = 0 can be reached from any initial point in finite time. let us denote p1,j,k as aj,k, p2,j,k as bj,k, and p3,j,k as cj,k. the probability that the number of agents with opinion z3 will increase during the next time step can be safely presented as follows: pr(oi = z1, oj = z1) · 0 + pr(oi = z1, oj = z2) · 0 + pr(oi = z1, oj = z3) · a3,3 + +pr(oi = z2, oj = z1) · 0 + pr(oi = z2, oj = z2) · 0 + pr(oi = z1, oj = z3) · b3,3. in turn, the probability that the number of agents with opinion z3 will decrease during the next step is given by the following expression: pr(oi = z3, oj = z1) · c1,1 + pr(oi = z3, oj = z1) · c1,2 + pr(oi = z3, oj = z2) · c2,1 + +pr(oi = z3, oj = z2) · c2,2 + pr(oi = z3, oj = z3) · c3,1 + pr(oi = z3, oj = z3) · c3,2. the expected change in the number of agents with opinion z3 during one step ∆y3 is equal to: e(∆y3) = +1 · [pr(oi = z1, oj = z1) · 0 + pr(oi = z1, oj = z2) · 0 + +pr(oi = z1, oj = z3) · a3,3 + pr(oi = z2, oj = z1) · 0 + pr(oi = z2, oj = z2) · 0 + +pr(oi = z1, oj = z3) · b3,3] + (−1) · [pr(oi = z3, oj = z1) · c1,1 + pr(oi = z3, oj = z1) · c1,2 + +pr(oi = z3, oj = z2) · c2,1 + pr(oi = z3, oj = z2) · c2,2 + pr(oi = z3, oj = z3) · c3,1 + +pr(oi = z3, oj = z3) · c3,2] = = y3 (y1 + y2 + y3)2 (y1 · a3,3 + y2 · b3,3 − y1(1− c1,3)− y2(1− c2,3)− y3 · (1− c3,3)). let us rewrite this as follows: e(∆y3) = y3 (y1 + y2 + y3)2 (y1 · (p3,1,3 + p1,3,3 − 1) + +(1− y1 − y3) · (p2,3,3 + p3,2,3 − 1)) + y3 · (p3,3,3 − 1) = = y3 (y1 + y2 + y3)2 (y3 · (p3,3,3 − p2,3,3 − p3,2,3) + +y1 · (p1,3,3 − p2,3,3 + p3,1,3 − p3,2,3) + (p2,3,3 + p3,2,3 − 1)). now from (8.15) we conclude that e(∆y3) ≤ 0 for every y1, y2, and y3. moreover, e(∆y3) is only equal to zero when y3 = 0. having (y1 + y2 + y3) = n is a finite natural number and y3 · (p3,3,3 − p2,3,3 − p3,2,3) + y1 · (p1,3,3 − p2,3,3 + p3,1,3 − p3,2,3) + (p2,3,3 + p3,2,3 − 1) ≤ d, where d = (p3,3,3 − p2,3,3 − p3,2,3) + (p1,3,3 − p2,3,3 + p3,1,3 − p3,2,3) + (p2,3,3 + p3,2,3 − 1)) copyright © 2024 assa. adv syst sci appl (2024) the mean-field approximation for the scardo model... 63 is a finite negative number for every fixed transition table, so we end up with |e(∆y3)| ≥ |d| n2 , which means that for every initial condition, with probability 1, the line y3 = 0 will be reached in a finite time t. after reaching the line y3 = 0, the system ends changing the third phase component: the probability of increasing the number of opinion z3’s followers is zero due to p1,1,3 = p1,2,3 = p2,1,3 = p2,2,3 = 0, and dy3 dτ = 0. as such, we can safely throw away the third dimension and analyze the behavior of the system along the first two phase components y1 and y2. in other words, we can switch ourselves to consider auxiliary system. remark 8.1: theorem 8.1’s conditions ensure that the state [0 0 1]t cannot be a point of no return for the agent system (not to be confused with the mean-field equation system), because from (8.15) it holds that p3,3,3 < p2,3,3 + p3,2,3 < 1. in this vein, even if the agent system reaches the state [0 0 1]t , there is always a nonzero probability of leaving it in the next time moment. 9. discussion in this paper, we studied the mean-field approximation for the scardo-model in the case of the 3-element opinion alphabet. these settings are more realistic and meaningful and, at the same time, much more complicated than the 2-element opinion alphabet, for which in ref. [6], the precise analytical description has been derived. unfortunately, for the 3element opinion space, we failed to get analytical solutions for the mean-field system without imposing additional constraints on the model parameters. the main idea of our approach was to reduce the 3-element opinion alphabet system to the 2-dimensional phase space and then harness the results obtained in ref. [6]. to this end, we focused on a specific family of transition tables that affords such a treatment. for such transition tables, we managed to jump to the 2-element opinion alphabet. after that, using a combination of analytical and simulation methods, we investigated the properties of fixed points located on the line y3 = 0. after that, after implementing additional restrictions on the transition table, we demonstrated that the system will reach the line y3 = 0 in a finite amount of time and then its behavior can be easily predicted. the reason is that after reaching the line y3 = 0, the system will remain on it. in this vein, we can make use of the analytical results for the case m = 2 from ref. [6]. as we already discussed in section 4, the constraints 4.6 that were imposed on the transition table are not too rigid and still allow handling many scenarios of opinion formation. it is worth noting that instead of (4.6), we could use different specifications of the transition table that are just the symmetric displacements of (4.6): p1 = [ 1− α α 0 1− β β 0 a31 a32 a33 ] , p2 = [ c11 c12 c13 c21 c22 c23 c31 c32 c33 ] , p3 = [ c11 c12 c13 0 δ 1− δ 0 γ 1− γ ] or copyright © 2024 assa. adv syst sci appl (2024) 64 v.n. gezha, i.v. kozitsin p1 = [ 1− α α 0 1− β β 0 a31 a32 a33 ] , p2 = [ b11 b12 b13 b21 b22 b23 b31 b32 b33 ] , p3 = p3 = [ c11 c12 c13 0 δ 1− δ 0 γ 1− γ ] . for these transition tables, all the derivations would be virtually the same. next, in section 8, we introduced a new set of restrictions on the transition table. in short, these restrictions take away the competitive edge from the third opinion camp and ensure that this opinion will not have supporters in the long run. we should say that, like (4.6), these restrictions are not somewhat exotic. for example, the transition table estimated on the empirical data from ref. [9]: p1 = [ 0.96 0.04 0 0.942 0.057 0.001 0.907 0.091 0.002 ] , p2 = [ 0.039 0.952 0.008 0.021 0.969 0.01 0.02 0.944 0.036 ] , p3 = [ 0.001 0.082 0.917 0.001 0.07 0.929 0.001 0.054 0.945 ] , meets inequalities (8.15). we believe that these results would be useful in performing research on opinion dynamics with the scardo-model, as they provide analytical insights into model behavior for some configurations of the transition table. acknowledgements the research is supported by a grant of the russian science foundation (project no. 2271-00075). references 1. combs, a., tierney, g., guay, b., merhout, f., bail, c. a., hillygus, d. s. & volfovsky, a. (2023) reducing political polarization in the united states with a mobile chat platform, nature human behaviour, 7(9), 1454–1461. https://doi.org/10.1038/s41562023-01655-0 2. degroot, m. h. 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(2023) users choose to engage with more partisan news than they are exposed to on google search, nature, 618(7964), 342–348. https://doi.org/10.1038/s41586-023-06078-5 copyright © 2024 assa. adv syst sci appl (2024) introduction opinion dynamics model mean-field approximation for the scardo-model main assumption auxiliary system fixed points on the line y3 = 0 simulation experiments scenario 1 scenario 2 stability analysis for scenarios 1 and 2 results of simulations further properties discussion microsoft word neural network models and cognitive computing from social media data.docx adv syst sci appl 2022; 02; 98-108 published online at https://ijassa.ipu.ru. neural network models and cognitive computing from social media data: perception of situation nailia gabdrakhmanova1*, maria pilgun2 1) peoples’ friendship university of russia (rudn university), s.m. nikol'skii mathematical institute, moscow, russia e-mail: gabd-nelli@yandex.ru 2) institute of linguistics, ras, moscow, russia e-mail: pilgunm@yandex.ru abstract: the paper presents the development of various types of models using cognitive computing based on speech data of social media actors to reveal the presence/absence of social tension in the areas where urban development projects are being implemented, as illustrated by the construction of the nizhegorodskaya transport interchange hub in moscow (russia). the empirical base of the study was data from social networks, microblogs, blogs, instant messengers, video hosting sites, forums, internet media, subject-related portals and reviews on the project implementation. the research was carried out using a transdisciplinary approach, including semantic analysis, neural network technologies and mathematical modeling methods. the study showed the consistency of the results obtained during the application of various types of models. semantic analysis of content using neural network technologies showed a neutral perception of the project by residents, the absence of social stress in the construction areas. the results of the analysis performed with autoregressive models confirmed the results obtained. keywords: social media, neural network technologies, speech perception, natural language processing, time series, differential equations 1. introduction cognitive research has recently attracted more and more attention from researchers. cognitive computing generalizes various process approaches, such as artificial neural networks, fuzzy systems and evolutionary computing, for modeling human cognitive abilities (thinking, learning, reasoning, etc.) using computer models [1]. cognitive computing is used in various professional fields: in medical information processing, video analytics, cognitive robotics, predictive analytics, machine learning, applications and many others [2, 3]. applications of cognitive management and cognitive computing in the fields of risk management, cognitive fraud detection, and in business decision making have shown their effectiveness. for example, by analyzing patterns in big data, small data, and “dark data,” cognitive technologies can detect human behavior and suggest options for personalizing of products and services [4]. new algorithms and methods in a variety of fields are presented, together with solution-based approaches. the topics addressed include various theoretical aspects and applications of computer science, artificial intelligence, cybernetics, automation control theory, and software engineering [5]. in the recent era, smart methods with human touch called as human cognition is adopted by many researchers in the field of information technology with the cognitive computing [6]. the high demand for cognitive computing in various areas of natural language processing is quite predictable, for example, in text analytics, for building conversational * corresponding author: gabd-nelli@yandex.ru neural network models and cognitive computing from social media data 99 copyright ©2022 assa. adv. in systems science and appl. (2022) interfaces, context based quantum language model with application to question answering, named entity recognition and others [3, 7]. aim of the study: development of various types of models using cognitive calculations based on speech data of social media actors to reveal the presence/absence of social tension in the areas where urban development projects are being implemented, as illustrated by the example of the construction of the nizhegorodskaya transport interchange hub (tih) in moscow (russia). 1.1. data the empirical base of the study was data from social networks, microblogs, blogs, instant messengers, video hosting, forums, internet media, subject-related portals and reviews on the implementation of the nizhegorodskaya tih project (fig. 1). date of collection: january 1, 2020 march 31, 2021 (table 1, fig. 2, 3, 4, 5). table 1. quantitative parameters of the data parameter value number of messages: 3 999 maximum number of messages per day: 533 number of active actors: 397 activity (posts per actor): 10,07 number of sources: 111 fig. 1. nizhegorodskaya tih fig. 2. dynamics of messages fig. 3. dynamics of unique messages 0 100 200 300 400 500 600 01 .0 1. 20 20 01 .0 2. 20 20 01 .0 3. 20 20 01 .0 4. 20 20 01 .0 5. 20 20 01 .0 6. 20 20 01 .0 7. 20 20 01 .0 8. 20 20 01 .0 9. 20 20 01 .1 0. 20 20 01 .1 1. 20 20 01 .1 2. 20 20 01 .0 1. 20 21 01 .0 2. 20 21 01 .0 3. 20 21 0 20 40 60 80 100 120 01 .0 1. 20 20 01 .0 2. 20 20 01 .0 3. 20 20 01 .0 4. 20 20 01 .0 5. 20 20 01 .0 6. 20 20 01 .0 7. 20 20 01 .0 8. 20 20 01 .0 9. 20 20 01 .1 0. 20 20 01 .1 1. 20 20 01 .1 2. 20 20 01 .0 1. 20 21 01 .0 2. 20 21 01 .0 3. 20 21 100 n. gabdrakhmanova, m. pilgun copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 4. actors’ activity dynamics fig. 5. dynamics of views the implementation of the nizhegorodskaya tih project began a long time ago. the first stage of the nizhegorodskaya transport interchange hub was started on march 27, 2020 simultaneously with the extension of the nekrasovskaya subway line and the opening of the nizhegorodskaya subway station; however, the tih has not yet started to function as an integral complex. now the tih implementation project is in the final stages of commissioning. according to the heads of the stroykompleks, a distribution hall for tih and subway users will open in the near future. in the media space, information about the project implementation is presented in a fairly complete manner. 1.2. method the data was collected using brand analytics algorithms. a transdisciplinary approach was used to interpret the data. neural network technology textanalyst 2.3 allowed to identify and analyze the topic structure of the database, identify the semantic network and analyze the core of the semantic network, conduct summarization and analysis of word associations. sentiment analysis was performed using the sentiment determination module, eureka engine. content analysis was performed using the automap service. for visual analytics, the tableu platform was used. an important result of data filtering is the creation of data subsets for building mathematical models in order to analyze the situation in detail. when formalizing the problem, the object of study was considered as a dynamic system. to analyze the behavior of a dynamic system, time series models and differential equations are used. time series models are built using regression models [9,10] and convolutional neural networks [11]. when choosing the general form of the differential equation [12], the problem under consideration is presented as a rivalry problem. when approximating the model coefficients, the nelder-mead method was used. the constructed models were studied for structural stability [12]. 2. results and discussions 2.1. general description of the content analysis of the data showed that the most significant, key content topics related to the nizhegorodskaya tih project were revealed in messages, the dynamics of which had only one peak of activity, that is, on 31.03.2020, with an insignificant quantitative value of 533 and, obviously, were a reaction to the peak of growth in the number of unique messages and the total number of authors observed on 27.03.2020 (fig. 2, 3, 4). the peak of the growth in the number of unique messages falls on 27.03.2020 with a quantitative value of 111 (fig. 3). the date 27.03.2020 is also the peak of the growth in the total number of authors with a quantitative value of 43. (fig. 4). such activity was caused by the announced opening of the subway station 0 5 10 15 20 25 30 35 40 45 50 01 .0 1. 20 20 01 .0 2. 20 20 01 .0 3. 20 20 01 .0 4. 20 20 01 .0 5. 20 20 01 .0 6. 20 20 01 .0 7. 20 20 01 .0 8. 20 20 01 .0 9. 20 20 01 .1 0. 20 20 01 .1 1. 20 20 01 .1 2. 20 20 01 .0 1. 20 21 01 .0 2. 20 21 01 .0 3. 20 21 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 01 .0 1. 20 20 01 .0 2. 20 20 01 .0 3. 20 20 01 .0 4. 20 20 01 .0 5. 20 20 01 .0 6. 20 20 01 .0 7. 20 20 01 .0 8. 20 20 01 .0 9. 20 20 01 .1 0. 20 20 01 .1 1. 20 20 01 .1 2. 20 20 01 .0 1. 20 21 01 .0 2. 20 21 01 .0 3. 20 21 neural network models and cognitive computing from social media data 101 copyright ©2022 assa. adv. in systems science and appl. (2022) and tih nizhegorodskaya (27.03.2020). the peak in the number of views on january 16, 2021 (58 011) was caused by the information that 11 new subway stations will appear in moscow in 2021. the largest number of views on 11.02.2021 (78 310) followed the message about anna merkulova, the head of the main engineering center of russia (coverage of 11 185), and on 11.02.2021, s. sobyanin’s message about the completion of the vostochny station construction in moscow was published. the most significant content topics related to the construction of the nizhegorodskaya tih were generated by actors from various countries. most of the actors with determined geolocation belonged to russia, which is natural, but users from ukraine, spain, indonesia and japan also took part. key topics related to the project were generated on various digital platforms. the audience coverage shows that microblogs took the first place; the second and third places in the rating were taken by social networks and blogs, respectively (fig. 6). fig. 6. ranking of types of sources by audience coverage meanwhile, analysis of comments, likes, reposts and duplicates showed that social networks were in the lead in the ranking of digital platforms. actors also left comments and reposts in microblogs and blogs. likes are predominantly presented on social networks; they are also found in messengers and microblogs. actors made duplicates of information about the project in social networks, as well as in the internet media and microblogs (fig. 7). fig. 7. ranking of types of sources by digital footprints thus, passive users received information about the nizhegorodskaya tih project from microblogs mainly, and active users preferred social networks, where they actively generated digital footprints that demonstrated a reaction to the project implementation. analysis of sources according to the number of messages shows that key content topics were covered mainly on official resources, and the ranking of digital platforms where the content with the highest ratings was generated is led by instagram, facebook, twitter and vkontakte (fig. 8). 102 n. gabdrakhmanova, m. pilgun copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 8. ranking of digital platforms based on user preferences 2.2. content sentiment the analysis of the data made it possible to determine the sentiment of the overwhelming number of messages as neutral (2 404 940). negative reactions were insignificant (21,234) (fig. 9). fig. 9. content sentiment by coverage analysis of the content sentiment by type of source and by coverage shows that the neutral content was generated mainly in microblogs, as well as in social networks and blogs. most of the positive content was posted on social networks, as well as messengers and blogs. the negative content was presented in blogs; a small amount was also presented in instant messengers and social networks (fig. 10). fig. 10. content sentiment by type of source and coverage the study of the distribution of actors’ digital footprints with various sentiment types by types of sources allowed creating rankings, where the top sources are as follows: in terms of neural network models and cognitive computing from social media data 103 copyright ©2022 assa. adv. in systems science and appl. (2022) positive sentiment: vk.com, mos.ru, mperspektiva.ru (fig. 11), in terms of neutral sentiment: vk.com, instagram.com, mos.ru (fig. 12), and in terms of negative sentiment: gucodd.ru, vk.com, livejournal.com (fig. 13). fig. 11. ranking of sources by positive sentiment fig. 12. ranking of sources by neutral sentiment fig. 13. ranking of sources by negative sentiment fig. 14. dynamics of messages with different types of sentiment in addition, the dynamics of messages with different types of sentiment and the ranking of sources by sentiment demonstrates an increase in negative messages and a decrease in neutral and positive references to the implementation of the nizhegorodskaya tih project over the analyzed period (fig. 14). 2.3. key topics to analyze the most important key topics of the content related to the project, the database was divided into three clusters according to the sentiment types (table 2). table 2. key topics of the clusters with different types of sentiment negative cluster positive cluster neutral cluster • station renaming. • information about the financial scheme of the mr group, which is linked to the future multifunctional center within the ryazan terminal 1 tih (next to the nizhegorodskaya mcc station). • design of nizhegorodskaya station; color scheme of the lobby. • plan of the areas adjacent to the nizhegorodskaya tih, public transport stops. • traffic restrictions due to road works. • alleged poor quality of construction, lack of confidence in inspections initiated by the moscow authorities. • network actors believe that it was counterproductive to connect the • information about the opening of the subway station and nizhegorodskaya tih. • the completion schedule for the construction of the eastern section of the subway big line by the end of 2022, as announced by the press service of the mayor and the moscow government. • announcement of the construction schedule for the elektrozavodskaya station of the big circle line (bcl). • information about the tih features (at grade parking, pedestrian bridge over the yauza river with a length of 85 meters), which will improve the transport situation, create comfortable aesthetic conditions for residents. • s. sobyanin’s tweet about the opening of the fifth flagship center @mydocs_mos in the southeast of moscow in the gorod shopping center. the mfc of the nizhny novgorod region has also moved here. the shopping center is located on ryazansky prospekt, next to the tih, which unites the stations of the bcl, the nekrasovskaya subway line, the mcc and the railway. • announcement that 11 new subwat stations will appear in moscow in 2021. • information about the construction of the largest tih in europe near the nizhegorodskaya subway station and of a cross-platform transfer hub. 104 n. gabdrakhmanova, m. pilgun copyright ©2022 assa. adv. in systems science and appl. (2022) kommunarskaya and biryulyovskaya subway lines; the initial decision to connect the kommunarskaya and nekrasovka lines seems more logical to them. • the cross-platform nature of the future tih will lead to increased complexity; passengers will get confused. • the ill-conceived location of the station. it is located far from the stops and near the cemetery, which does not look very good; the station had to be located according to the old routing projects. • premature opening of stations, three stations were opened too early and they could be opened already as part of the bcl • information that the nizhegorodskaya bcl station will become part of the ryazanskaya tih. • plans to build a multi-tiered landscape park on ryazansky prospekt, which will become a buffer zone between residential areas and the ryazanskaya transport hub. • information that the new section of the south-eastern chord from ryazansky prospekt to the third transport ring will improve the transport situation in the southeastern administrative district; in particular, comfortable approaches to the novokhokhlovskaya and nizhegorodskaya tihs will appear. • information about the details of the subway construction • information on the construction schedule for the subway big ring. • news about the rating of cheap new buildings near new tihs. • announcement of the awarding of anna merkulova. • announcement of plans for the construction of the ryazanskaya tih, which will be erected at the intersection of ryazansky prospekt with nizhegorodskaya street, on the basis of the nizhegorodskaya subway station. • information about plans for the tih and subway construction in the capital. • information on the schedule for the opening of the passengers lounge in nizhegorodskaya tih. • messages about moscow mayor’s personal control over the construction 2.4. revealing the presence/absence of social stress in the areas of the nizhegorodskaya tih construction the result of the database analysis showed the absence of social stress (0.89) and the average social well-being index (8.75) (fig. 16). the analysis of the sentiment of actors’ digital footprints confirmed the absence of social stress in the areas of the project implementation and the city of moscow in general. only 57 likes and 3 reposts indicated negative reactions (fig. 23). meanwhile, 79 062 likes, 2 749 comments and 744 reposts speak for neutral reactions, and positive reactions are expressed in 7 421 likes, 157 comments, 89 reposts (fig. 15). fig. 15. digital footprint sentiment fig. 16. social well-being index and social stress index 0,89 8,75 0 2 4 6 8 10 social stress index social well-being index neural network models and cognitive computing from social media data 105 copyright ©2022 assa. adv. in systems science and appl. (2022) 2.5. results of mathematical modeling the data obtained as a result of the filtration is time series of the studied dynamic process [13,14]. for time series, we introduce the following notation: poz(t) is the number of messages at time t with a positive attitude towards the tih construction; neg(t) is the number of messages at time t with a negative attitude towards the tih construction; ntr(t) is the number of messages at time t with a neutral attitude towards the tih construction. 2.5.1. arma models models for time series data with a positive attitude after identifying and correcting anomalous points in the time series {poz(t)}, a trend can be identified based on the time series data: poz$ (t)= 0.5+0.005t (1) the positive coefficient in front of t in (1) indicates that the number of positive messages increases over time. the adequacy of the construction of the model (1) has been verified by various methods. in particular, the hypothesis of the inadequacy of model (1) according to fisher’s criterion was rejected at a significance level of p = 0.001620, which indicates a good quality of the constructed model. to construct a model of the random component, we find the remainders of series e!"#(t)=poz(t)-poz$ (t). for the remainders of series e!"#(t)=poz(t)-poz$ (t) , after testing the hypothesis for stationarity, an autoregressive model arima (2,0,2) was constructed. e0!"#(t)=-0.1e!"#(t-1)-0.4e!"#(t-2)-0.24ε(t)+0.2ε(t-1), where ε(t)∈n(0; 4.4) models for time series data with a negative attitude models for time series {neg(t)} were constructed using an algorithm similar to. after correcting anomalous points, a trend can be identified according to the time series {neg(t)} data: neg$ (t) = 0.02 + 0.0001t (2) the positive coefficient in front of t in (2) indicates an increase in the number of negative messages, but the rate of growth in the number of negative messages is much less than the growth in the number of positive messages. the model was checked for adequacy. in particular, the hypothesis of the inadequacy of model (2) according to fisher’s criterion was rejected at a significance level of p = 0.14, which indicates a satisfactory quality of the constructed model. to construct a model of the random component, we find the remainders of series e$%&(t)=neg(t)-neg$ (t). for the remainders of the series, an arima (5,0,5) model was constructed. 𝐸<'()(𝑡) = −0.4𝐸'()(𝑡 − 4) − 0.4𝐸'()(𝑡 − 5) + 𝜀(𝑡) − 0.6𝜀(𝑡 − 5), where ε(t)∈n(0; 5.4). models for time series ntr (t) data models for time series {ntr(t)} were also constructed using an algorithm similar to. after identifying and correcting anomalous points, a trend can be identified according to the time series {ntr(t)} data: ntrc(t) = 7.86-0.01t (3) 106 n. gabdrakhmanova, m. pilgun copyright ©2022 assa. adv. in systems science and appl. (2022) the negative coefficient in front of t in (3) indicates a decrease in the number of messages with a neutral attitude to the construction. the model was checked for adequacy. in particular, the hypothesis of the inadequacy of model (3) according to fisher’s criterion was rejected at a significance level of p = 0.04, which indicates the reliable quality of the constructed model. to construct a model of the random component of time series e0$*+(t)=ntr(t)-ntr$ (t), estimates of the autocovariance and partial autocovariance functions are found. based on the results of the found estimates of these functions, the structure of the autocovariance model is chosen. for the remainders of the series, an arima (1,0,0) model was constructed. e0$*+(t)=0.4e$*+(t-1)+ε(t), where ε(t)∈n(0; 1). the models constructed allow making time series predictions several steps ahead. 2.5.2. ode models to analyze the behavior of a dynamic system, it is important to have continuous mathematical models. when building models using ode, data transformations are used. within the framework of the problem under consideration, we assumed that the percentage of all three groups of the part of the society that is not involved in social networks is in the same percentage as the one that is involved. due to the fact that we are primarily interested in the dynamics of the number of actors with a positive and negative attitude, and given that the activity of posts varies depending on the day of the week, holidays, etc. the following normalization is adopted. two new variables have been introduced: x(t)=poz(t)/ntr(t), y(t)=neg(t)/ntr(t). in table 3, a fragment of the initial data x(t), y(t). table 3. fragment of initial data x(t), y(t) t x y 1 0,139394 0 2 0,240876 0 3 0,22549 0 4 0,21875 0 5 0,032258 0,096774 6 0,382979 0,021277 general statement of the problem. time series {x(t)}, {y(t)}, where t=1,.., n, are given, which characterize the level of positive and negative attitude towards the project, respectively. the sampling step is constant. it is necessary, according to these data, to build dynamic models for forecasting and analyzing the situation. at the first step, the data were smoothed using convolutional neural networks, then estimates of the parameters of differential equations were found using the algorithm of the nelder-mead method. as a result of the implementation of the algorithms, several variants of models were obtained in the form of systems of differential equations. one of the equations looks like this: f 𝑑𝑥 𝑑𝑡 = 0.91𝑥 − 0.15𝑦 𝑑𝑦 𝑑𝑡 = 0.1𝑥 + 0.3𝑦 the constructed model was studied for structural stability. eigenvalues of the determinant of the system λ1=0.84 , λ2=0.36. therefore, the rest point (0, 0) is an unstable node. this indicates that as t increases, points arbitrarily close to the origin are removed from ε, the neighborhood of the origin. because of this, in order to track the dynamics of the development of the situation, it is necessary to use parallel models, in particular, models for forecasting time series. it can also be noted that estimates of the coefficients of the system of differential equations indicate that the series {y(t)} grows faster than {x(t)}. neural network models and cognitive computing from social media data 107 copyright ©2022 assa. adv. in systems science and appl. (2022) 2.5.3. mathematical model with lstm traditionally developed for 2d image data, lstms can be used to model univariate and multivariate time series forecasting problems. the lstm network is one of the modifications of recurrent neural networks most commonly used in deep learning. the work was carried out in the google colaboratory environment (a google research product that is used to write code), the models were implemented and trained using machine learning libraries such as: tensorflow, keras, sklearn. additionally, the mathematical libraries numpy, pandas and matplotlib were used for calculations and plotting. a recurrent neural network (rnn) was built, namely a modification of the lstm. the neural network architecture looks like this: adamax optimizer learning rate 0.001 lstm modification activation function relu. error function mse (root mean square error) metric for quality assessment mae (mean absolute error) number of learning epochs 100 batch size 16 in our problem, the data is considered in the form of a multivariate time series. data with a window of 4 were fed to the input of the neural network. the choice of the size of the number of input time steps has an important impact on how much data array will be used for training. an estimate of the size of the number of input time steps is obtained using autocorrelation estimates. the work was carried out in the google colaboratory environment, the models were implemented and trained using machine learning libraries such as: tensorflow, keras, sklearn. additionally, the mathematical libraries numpy, pandas and matplotlib were used for calculations and plotting. when training the cnn, the adamax optimizer was used, and relu was used as an activation function. the following learning outcomes have been obtained: epoch 98/100 164/164 0s loss: 0.0140 405ms/epoch 2ms/step epoch 99/100 164/164 0s loss: 0.0141 404ms/epoch 2ms/step epoch 100/100 164/164 0s loss: 0.0139 396ms/epoch 2ms/step train score: 0.18 rmse test score: 0.30 rmse 2.5.4. model building results the constructed arma, ode, lstm models were compared on the test set (table 4). in the test error column, the percentage of points where the calculation error is greater than the allowed threshold value. according to the performed calculations, lstm has the best result in terms of accuracy and computational speed. the arma and ode models are more suitable for analyzing the dynamics of the situation. table 4. model comparison test error arma 10% ode 8% lstm 5% 108 n. gabdrakhmanova, m. pilgun copyright ©2022 assa. adv. in systems science and appl. (2022) 3. conclusion the study showed the consistency of the results obtained during the application of various types of models using cognitive calculations based on speech data of social media actors to reveal the presence/absence of social tension in the area of the nizhegorodskaya tih construction. semantic analysis of the content involving neural network technologies showed a neutral attitude of the residents towards the project implementation, the absence of social stress in the construction areas. the content ranking is topped by official messages announcing the plans of the moscow authorities to improve the transport system in the area and the construction progress. the users’ digital footprints confirm the neutral perception and assessments of the residents’ needs to change the traffic situation after the completion of the project. however, slight concerns can be caused by the dynamics of messages with various sentiment types, which demonstrates an increase in negative messages and a decrease in neutral and positive references to the implementation of the nizhegorodskaya tih project. meanwhile, the insignificance of quantitative indicators does not give grounds for negative assumptions. the constructed mathematical models of time series according to the intensity of messages with a neutral, positive and negative attitude to the construction make it possible: 1) to assess the dynamics of these series quantitatively; 2) to predict the values of time series by several steps ahead. the results of the study are important for making management decisions in identifying and preventing urban planning conflicts. references 1. portmann, e., d’onofrio, s. 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(2021) neural network technologies and topological analysis of social media data. advances in systems science and applications, 21(3), 101– 112. https://doi.org/10.25728/assa.2021.21.3.1127 microsoft word 1190 article text, copyedited.doc adv syst sci appl 2022; 01; 167-175 published online at https://ijassa.ipu.ru. on the stability of periodic difference inclusions mikhail morozov* v.a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia e-mail: granmiguel@mail.ru abstract: the paper considers asymptotically stable periodic difference inclusions. the uniform character of the convergence of solutions to zero is established. for selector-linear difference inclusions the equivalence of the uniform asymptotic stability and the uniform exponential stability is proved, and a necessary and sufficient condition for the uniform asymptotic stability in the form of a certain limit relation is obtained. examples of systems leading to periodic difference inclusions are given. these results can find applications in the stability analysis of control systems with periodic parameters, in particular, servomechanisms whose elements operate on alternating current, control systems with amplitude-frequency modulation, systems used to solve problems associated with the study of large electric power systems in the presence of forced oscillations. keywords: periodic difference inclusion, periodic selector-linear difference inclusion, uniform asymptotic stability, uniform exponential stability 1. introduction the problem of the stability of difference and discrete inclusions arises in various areas of mathematics: in control theory, in linear algebra, in the study of convergence of iterative processes, in problems related to discrete wavelet transforms and markov chains. in some cases, selector-linear inclusions can be used. for example, this is the problem of absolute stability, the study of linear non-stationary systems, the matrix of the right side of which satisfies interval constraints, the study of the stability of control systems that contain elements with incomplete information. nonlinear discrete control systems were considered in [7,5]. their equivalence to autonomous selector-linear difference inclusions is proved. various stability conditions were obtained using the method of lyapunov functions. for autonomous selector-linear difference inclusions, the lyapunov indicator was introduced and some properties of solutions were established in [1], asymptotic stability conditions in the form of constraints on the right-hand side of inclusions in [2] and those in the form of the existence of smooth and finite-step lyapunov functions in [3,4] were obtained. the problems of absolute and robust stability of discrete control systems with periodically varying parameters were solved in [6,8]. in particular, it was established that the considered control systems with periodic parameters are equivalent, in the sense of the coincidence of the sets of solutions, to time-periodic selector-linear difference inclusions. in this paper, periodic difference inclusions are considered. it is known that the properties of the uniform asymptotic stability and the uniform exponential stability of zero solution of autonomous selector-linear difference inclusions are equivalent. for periodic selector-linear difference inclusions, the question of the equivalence of these properties has not previously been considered. the remainder of this paper is structured as follows. in section 2, we consider periodic difference inclusions of general form and periodic selector-linear difference inclusions. we * corresponding author: granmiguel@mail.ru 168 m. morozov copyright ©2022 assa. adv. in systems science and appl. (2022) give preliminary remarks and definitions. in section 3, it is proved that under some restriction of initial conditions, the convergence of solutions of a periodic difference inclusion to zero has the uniform character. we also establish the equivalence of the uniform asymptotic stability and the uniform exponential stability of the zero solution for periodic selector-linear difference inclusions. using the variational method for periodic selector-linear difference inclusions, a necessary and sufficient condition for uniform asymptotic stability is obtained in the form of a limit relation. in section 4, two examples of systems leading to periodic difference inclusions are given. in section 5, we offer concluding remarks. 2. statement of the problem consider the dynamic systems described by periodic difference inclusion (2.1) (2.1) where , is the set of natural numbers. everywhere below we assume that in some domain for all the set is nonempty, bounded, closed, convex, and the multivalued function is upper semicontinuous. we will call a solution of inclusion (2.1) a sequence of vectors satisfying for all inclusion (2.1). we assume that the sequence is the equilibrium position of the inclusion (2.1). due to the multivaluedness of the function a point , generally speaking, specifies not one but a set of solutions. due to the periodicity of the multivalued function in when studying the properties of solutions of inclusion (2.1), without loss of generality, we can assume that definition 2.1: inclusion (2.1) is stable if for any there exists such that, as soon as the initial conditions satisfy the condition , the solution with initial condition satisfies the inequality for all definition 2.2: inclusion (2.1) is asymptotically stable if the conditions of definition 2.1 are satisfied and the limit relation holds true, that is, for any there exists such that for all the solution of inclusion (2.1) satisfies the inequality . definition 2.3: inclusion (2.1) is uniformly asymptotically stable if the conditions of definitions 2.1 and 2.2 are satisfied and numbers and do not depend on solution , and consider a periodic selector-linear difference inclusion ),,()1( xsfsx î+ ),,(),( xmsfxsf += , ... ,1 ,0=s nrxî ,nm î n }:{ , ,0{ 00 rxxggxmsg rr £== gxs î),( nrxsf ì),( nn rrf ®+1: )(sx )},({ sx nsî nsxsx î= s 0,)( :)}({ 0)( ºsx ),( xsf ),( 00 xs ),( xsf s ),,( 00 xssx .0 0 ms ££ 0>e 0),),,,(( 000 >ed sxssx 00 )( xsx = ),),,,(( 0000 ed sxssxx < ),,( 00 xssx 0x e<),,( 00 xssx .00 ³³ ss 0),,(lim 00 = +¥® xssx s 0>h nxsxssxs î),,),,,(( 0000 h ),,),,,(( 00000 hxsxssxsss +³ ),,( 00 xssx h<),,( 00 xssx d s ),,( 00 xssx 00 ³s .0 nrx î on the stability of periodic difference inclusions 169 copyright ©2022 assa. adv. in systems science and appl. (2022) , , (2.2) where is the equilibrium position of the difference inclusion (2.2), is a convex compact set of real matrices. let be a solution of inclusion (2.2) defined by a matrix . definition 2.4: inclusion (2.2) is uniformly exponentially stable if there exist numbers and such that, for any solution of inclusion (2.2), the inequality holds true for any matrix , any and . the problem is to establish for inclusion (2.1) uniform character of the limit relation in definition 2.2. in the case of inclusion (2.2), the problem is to determine a condition for uniform asymptotic stability and to prove the equivalence of the properties of uniform asymptotic stability and uniform exponential stability. 3. results theorem 3.1: if inclusion (2.1) is asymptotically stable, then there exists such that all solutions of inclusion (2.1) satisfy the condition uniformly with respect to for any and the proof of theorem 3.1 is carried out mutatis mutandis by the scheme of theorem 1 in [9]. let be a solution of inclusion (2.2) with the initial conditions corresponding to a matrix to obtain a condition for uniform asymptotic stability of inclusion (2.2), we introduce into consideration the functions and (3.1) by the weierstrass theorem, taking into account the form of inclusion (2.2), the function is defined for any and since the functional is a continuous function of the variables and . using the function we can formulate the following criterion for asymptotic stability. ),,()1( xsfsx î+ { })()( ,)(:),( ssbxsbyyxsf wî== )()( sms w=+w , ... ,1 ,0=s ,nm î 0)( ºsx )(sw )( nn´ ),,( 00 xssxb )()( ssb wî 0>a 1³b ),,( 00 xssxb ))(exp(),,( 0000 ssxxssxb --£ ab )()( ssb wî 00 ³³ ss nrx î0 00 >d ),,( 00 xssx 0),,(lim 00 = +¥® xssx s ),( 00 xs ms ££ 00 .: 000 d£xx ),,( 00 xssxb ),,( 00 xs ).()( ssb wî 2 00)()(00 ),,(max),,( xssxxstw bssb wî = .0 ),,,(max),( 00010 0 ³³= = ssxsswss x r ),( 0ssr 0x nsî 2 00 ),,( xssxb 0x )(sb ),( 0ssr 170 m. morozov copyright ©2022 assa. adv. in systems science and appl. (2022) theorem 3.2: inclusion (2.2) is uniformly asymptotically stable if and only if the limit relation (3.2) holds true uniformly with respect to . proof. necessity. since inclusion (2.2) is uniformly asymptotically stable, then for any there exists such that for all from the last inequality and (3.1) follows that for all and , which proves the necessity of condition (3.2). sufficiency. due to the uniform boundedness of the elements of the matrix the solution of inclusion (2.2) with will be uniformly bounded for all therefore, for any there exists , such that for all it follows from the last inequality and relation (3.2) that there exists, that is independent of such that for all . for any , , , . (3.3) from (3.3), linearity and periodicity of inclusion (2.2) for any , any , and follows . we set . then it follows from (3.3) that solutions of inclusion (2.2) satisfy the inequality for all and , if only . it follows from (3.2) that for any and there exists such that is independent of and , such that for all the inequality holds true and hence the inequality due to the linearity and periodicity of inclusion (2.2). therefore, for all thus, the solution of inclusion (2.2) satisfies all the conditions of definition 2.2. theorem 3.2 is proved. � 0),(lim 0 = +¥® ss s r 00 ³s 0>h ,)( ns îh h<),,( 00 xssxb ,00 ³s ,0 sss +³ ),()( ssb wî .1: 00 =xx 2 0 ),( hr <ss sss +³ 0 00 ³s )(sb ),,( 00 xssxb 10 =x . ..., ,1 , 000 *++= sssss 0* >s 1)( * ³sg )(),( * 0 sss gr £ . ..., ,1 , 000 *++= sssss 10 ³r 0s 00 ),( rr <ss 00 ³³ ss 00 ³s 1: 00 =xx 0ss ³ )()( ssb wî 00 2 00 ),(),,( rr <£ ssxssxb 0>e 00 ³s ,1: 00 =xx 0ss ³ )()( ssb wî ere <)/,,( 000 xssxb 0/)( reed = e<),,( 00 xssxb 00 ³³ ss )()( ssb wî )(0 ed<x 0>h ,0>r ,),( nrs îh )()( ssb wî 00 ³s ,0{ 0 ³s ),,(0 rsss h+³ ,10 =x )}()( ssb wî h<),,( 00 rxssxb h<),,( 00 xssxb ,0{ 0 ³s ),,(0 rsss h+³ ,10 =x )}.()( ssb wî ,/),(),,( 22 0 2 00 rssxssxb hr <£ on the stability of periodic difference inclusions 171 copyright ©2022 assa. adv. in systems science and appl. (2022) consider the system of difference equations (3.4) where the transition matrix of system (3.4) is the matrix connecting the solution and i.e., satisfying the equalities and , where is the unit matrix of order from the equality it follows corollary 3.1: inclusion (2.2) is uniformly asymptotically stable if and only if the transition matrix of system (3.4) satisfies the condition , uniformly in and the equivalence of the uniform asymptotic stability and the uniform exponential stability for inclusion (2.2) is established by the following theorem. theorem 3.3: inclusion (2.2) is uniformly asymptotically stable if and only if inclusion (2.2) is uniformly exponentially stable. proof. the sufficiency follows directly from definitions 2.3 and 2.4, and only necessity needs to be proved. to prove the necessity we use the statement of theorem 3.2. let us show that the uniform exponential stability of inclusion (2.2) follows from condition (3.2). for solutions of inclusion (2.2) we have (3.5) for all where the function is defined in (3.1). note that solutions of inclusion (2.2) satisfy the equality for all let where ( denotes the integer part of ). using (3.5), we obtain ,)()1( xsbsx =+ ),()( ),()( smsssb w=+wwî , ... ,1 ,0=s ,nrxî .nm î ),( 0ssbp ),,( 00 xssxb ,0x 0000 ),(),,( xssxssx bb p= ,),( 00 nb ess =p 0ss ³ ne .n ,),(max),( 2 0)()(0 ssss bssb p= wî r ,0ss ³ ),( 0ssbp 0),(lim 0 =p +¥® ssbs 00 ³s ).()( ssb wî =200 ),,( xssxb ( ) £ 2 000 2 0 /,, xxssxx b ,),( 2 00 xssr ,0{ 0 ³s ,0: 00 ¹xx )},()( ssb wî ),( 0ssr ),,()),,(,,( 0020012 xssxxssxssx bbb = .012 sss ³³ ,0 mksss ++= ,0]/)[( 0 ³-= sssk sm <£0 ][z z =++ 2 000 ),,( xsmkssxb £+-++-+++ 2 00000 )),,)1((,)1(,( xsmsksxmsksmkssx bb ×+-+++£ ))1(,( 00 msksmkssr £+-+ 2 000 )),,)1(( xsmsksxb 172 m. morozov copyright ©2022 assa. adv. in systems science and appl. (2022) … (3.6) since inclusion (2.2) is uniformly asymptotically stable, relation (3.2) holds. therefore, for any there exists such that for each of the factors in (3.6). therefore, for any (3.7) taking into account that from (3.5) we obtain the estimate (3.8) where from (3.7) and (3.8) it follows since we have for solutions of inclusion (2.2), where and the numbers and do not depend on , and . consequently, under theorem 3.3, inclusion (2.2) is uniformly exponentially stable. theorem 3.3 is proved. � it follows from theorem 3.2 that for uniformly asymptotically stable inclusion (2.2) there exists a number independent of , such that using this property in the proof of theorem 3.3 allows us to establish the uniform exponential stability of inclusion (2.2). therefore, the following statement is true. corollary 3.2: for inclusion (2.2) to be uniformly asymptotically stable, it is necessary and sufficient that there exists the number such that 4. examples example 4.1: the problem of absolute stability was solved in [6] for nonlinear nonstationary discrete control systems with a periodic linear part described by the equations , (4.1) ),( sss -£ r ×-)2,( ssssr ×+--× ),)1(( 0 mssksr .),,( 2 000 xsmsxb + 0>h ),(hs 1)exp(),)1(( <-=<--chjsssjsr ).0( >c kjjsssjs ,1 ),,)1(( =---r ...2,1,0=k £200 ),,( xssxb .),,()exp( 2 000 xsmsxk b +-c ,1),( 0 =sss £+ 2 000 ),,( xsmsxb £+ 2 000 ),( xsmsr ,200 xb .1),(max 0000 ³+= ££ smsr sm b £200 ),,( xssxb ,...2,1,0 ),exp(2 00 =kkx cb ],/)[( 0 sssk -= £),,( 00 xssxb )),(exp( 00 ssx --ab ),2/(,10 se cabb c =³= 0>a 1³b 00 ³s 0x )()( ssb wî ,nsî ),( 00 xs .1),( 0 <ssr ,nsî .1),( 0 <ssr å = +=+ m j jj j ssxssbsxsasx 1 ))),(,(()()()()1( sj on the stability of periodic difference inclusions 173 copyright ©2022 assa. adv. in systems science and appl. (2022) where is an dimensional vector, characterizing the deviation of the system from the zero solution , is a square matrix of order , are dimensional vectors, is discrete time, the brackets denote the inner products. it is assumed that the matrix and the vectors are bounded and satisfy the periodicity conditions for each and it is also assumed that the nonlinear functions specifying the characteristics of nonlinear nonstationary elements are defined for all and satisfy the standard sector-type constraints. system (4.1) is equivalent to periodic selector-linear difference inclusion (2.2), where the multivalued function is defined in each point ( ) by the relation . (4.2) here and in the following example, equivalence is understood in the sense of coincidence of the sets of solutions of the considered system and the corresponding inclusion with the same initial conditions. example 4.2: consider the equation with real parameters и (4.3) where is an integrable piecewise continuous periodic function, an equation of this kind was first considered by the american mathematician and astronomer george hill for the case when is an even periodic function with period , in connection with the study of the average motion of the lunar perigee. later, the equation was studied in a large number of publications, mainly of an applied nature. a special case of hill's equation is the equation that was obtained by the french mathematician and astronomer emile mathieu to describe vibrations of an elliptical membrane. let’s introduce the notation and rewrite equation (4.3) ,,1,0),0(,)(),(),( 1 mjsxscxscxs n i ji j i j j =º=ñá= å = js nn ii rxxx î= = ,)( 1 -n 0ºx )(sa n mjscsb ,1),(),( = -n ,...1,0=s ñá.,. )(),( scsb jj ),()( samsa =+ ),()( sbmsb jj =+ ),()( scmsc jj =+ nm î ,...1,0=s .,1 mj = ,,1),),,(( mjsxsjj =sj 0, ³î srx n )),(),(( ),( xsfxmsfxsf º+ ),( xs ,0³s nrxî { ,,)()()(:),( 1 ñá+== å = xscsbxsayyxsf m j j j j µ }mjkk jjj ,1],[ 2,1 =îµ a b ,0))(( 2 2 2 =++ ztbfa dt zd ,0 ,0 ),()( , , 21 >³º+îî tttfttfibia )(tf ].,[ ],,0[ 21211 bbiai == )(tf p , 0 )(-1 0 ),,( , , , 2 2 11 21 ÷÷ ø ö çç è æ =÷÷ ø ö çç è æ ==== tbfa bata x x x dt dz dt dxxzx 174 m. morozov copyright ©2022 assa. adv. in systems science and appl. (2022) (4.4) consider the discrete analogue of system (4.4) (4.5) where is discrete time. system (4.5) is equivalent to periodic selector-linear difference inclusion (2.2), where the multivalued function is defined in each point by the relation 5. conclusion periodic difference inclusions are considered. it is proved that under some restriction on the initial conditions, the convergence of solutions to zero is uniform. for periodic selectorlinear difference inclusions the equivalence of the properties of uniform asymptotic stability and uniform exponential stability of the zero solution is established. using the variational method the necessary and sufficient condition for the uniform asymptotic stability in the form of the limit relation has been obtained. examples of systems leading to the consideration of periodic difference inclusions are presented. the results obtained can be used in the analysis of stability of control systems with periodic parameters, in particular, servomechanisms whose elements operate on alternating current, control systems with amplitude-frequency modulation, systems used to solve problems associated with the study of large electric power systems in the presence of forced oscillations. further investigation of the inclusions considered in this paper may be related to the search for parametric classes of lyapunov functions that establish necessary and sufficient conditions for asymptotic stability. references 1. barabanov n.e. (1988). lyapunov indicator of discrete inclusions. i–iii, automation & remote control, (49),152-157, 283-287, 558-565. 2. diamond, p.&opoitsev, v.i. (2001). stability of linear difference and differential inclusions, automation & remote control, 62(5), 605-703. 3. geiselhart, r. (2016) lyapunov functions for discontinuous difference inclusions. [online]. ifac-papersonline. 49(18). 223-228. available https://www.sciencedirect.com/science/article/pii/s2405896316317463 4 kellett, c.m. & teel, a.r. (2004) smooth lyapunov functions and robustness of stability for difference inclusions, systems & control letters, 52, 395-405. 5. molchanov, a. p. (1987). funktsii lyapunova dlia nelineynyh diskretnyh system upravlenia [lyapunov functions for nonlinear discrete-time control systems]. avtom. & telemekh., 6, 26-35, [in russian]. ),()( ,),,( tattaxbata dt dx º+= . ,0 ,0 , , 2 21 rxttibia î>³îî ),()( ),(),,()1( samsasxbasasx º+=+ , , , , 2 21 rxnmibia îîîî ,...1,0=s )),(),(( ),( xsfxmsfxsf º+ 2 ),,( rxxs î }. , ,),,(:{),( 21 ibiaxbasayyxsf îî== on the stability of periodic difference inclusions 175 copyright ©2022 assa. adv. in systems science and appl. (2022) 6. molchanov, a. p. & morozov, m.v. (1992) lyapunov functions for nonlinear nonstationary discrete-time control systems with periodic linear part, automation & remote control, 53(10), 1505–1513. 7. molchanov, a. p. & pyatnitskii, e.s. (1989). criteria of asymptotic stability of differential and difference inclusions encountered in control theory, systems & control letters, 13, 59-64. 8. morozov, m.v. (2014). kriterii robastnoy absolutnoy ustoychivosti diskretnyh system upravleniya s periodicheskimi ogranicheniyami [criteria of robust absolute stability for discrete control systems with periodic constraints]. proceedings of institute for systems analysis, 64, 2, 13-18. [in russian]. 9. morozov, m. v. (2000). properties of solutions of periodic differential inclusions. diff. equat. 36, 677–682. https://doi.org/10.1007/bf02754225 adv syst sci appl 2022; 01:148–166 published online at https://ijassa.ipu.ru. sensitivity and optimal control analysis of japanese encephalitis disease: a mathematical model naba kumar goswami1* 1department of mathematics, pet research foundation, pes college of engineering, karnataka, india abstract: japanese encephalitis is caused by flavivirus, which can affect both humans and animals. a japanese encephalitis disease model is proposed and analyzed to investigate the transmission dynamics of japanese encephalitis with saturated incident rate and saturated treatment function. assume that the infected pig populations are growing logistically in the environment. the basic reproduction number (r0) of the model is obtained using the nextgeneration matrix method. the sensitivity analysis is performed to identify the key parameters that affect the basic reproduction number, which can be regulated to control the transmission dynamics of the disease. the model is extended to optimal control model incorporating three timedependent inputs for the control of transmission route by using pontryagin’s maximum principle. this study is significant as japanese encephalitis disease poses serious challenges for public health in past few years. numerical simulation is performed to support our analytical findings. keywords: japanese encephalitis, basic reproduction number, sensitivity, optimal control, simulation 1. introduction the japanese encephalitis is primarily a mosquito-borne rural disease, which is transmitted to human through the bite of an infected culex species mosquitoes. japanese encephalitis virus (jev) is a flavivitus of the family of flaviviridae like the viruses of the yellow fever, west nile, and dengue. the virus is transmitted in an enzootic cycle among mosquitoes and vertebrate amplifying hosts, primarily domestic pigs and/or water birds (wild heron) [1, 2]. most impotently there is no such type of evidence for human to human transmission. the first case of jev disease was recorded in 1871 in japan [3]. the clinical cases of japanese encephalitis globally estimated every year nearly 68,000, with approximately 13,600 to 20,400 deaths. the japanese encephalitis primarily affects children less than 15 years of age. the fatality rate varies between 20%− 40%, but it may reach 58% and over. the incubation period for japanese encephalitis is between 4–14 days [1, 3]. the primary symptoms of the disease includes fever, chills, headache, and vomiting. other symptoms like neurological symptoms, drowsiness, dizziness, confusion, abdominal pain and diarrhea might develop over the next few days [5]. there is no specific medicine for treatment, but patients should be hospitalized for supporting care. during the infected period of japanese encephalitis, people should drink plenty of fluids and rest sufficiently. the vaccine is available to control japanese encephalitis virus disease but it does not protect 100 %. japanese encephalitis virus has been identified most common in rural areas in southeast asia, eastern asia, southeastern asia, and pascific island. the disease is investigated globally in the following territories : australia, bangladesh, bruni, burma, cambodia, china, guam, india, indonesia, ∗corresponding author: nabakrgoswami@gmail.com sensitivity and optimal control analysis of japanese 149 japan, laos, malaysia, nepal, north korea, pakistan, phillipines, russia, saipan, singapore, south korea, sri lanka, taiwan, thailand, timor-leste, vietnam. the main vector of japanese encephalitis is c. annulirostris. in indian condition many other secondary vectors have reported including anopheles subpictus, a. peditaeniatys, c.epidesmus, c.gelidus, c.whitmorei, c. pseudovishnui, m. uniform and mansonia indiana. pigs are considered to be principal amplifying host in india. the transmission is possible between may and september. the first case of japanese encephalitis disease was diagnosed in 1955 in tamil nadu in india [4]. consequently, several major out-breaks have been listed from different parts of the country, mostly in rural areas of assam, bihir, harayana,karnataka, tamil nadu, uttar pradesh. in india, 1,500 to 4000 jev cases are reported every year [3]. according to national health mission in assam most of the upper assam districts, including dibrugarh, golaghat, jorhat, lakhimpur and lower assam district of kamrup have been badly affected every year by the japanese encephalitis mosquito-borne disease. several mathematical models have been studied and analyzed the dynamics of japanese encephalitis transmission [3-13] as yet. in [3], authors discussed the current knowledge of the epidemiology and the parthenogenesis of this deadly disease have been summarized. in [4], authors deals with application of various control strategies to japanese encephalitis among human, pig and mosquito. in [5], authors proposed a mathematical model for the spread of japanese encephalitis with emphasis on the environmental effects on the aquatic phase of mosquitoes. in [6], authors proposed a mathematical model with impact of vaccination on of je with standard incidence rate of mosquitoes, pigs and humans. in [7], authors has been studied stability and bifurcation analysis of japanese encephalitis model with/without effect of some control parameters. in [8] authors developed a mathematical model and analysis the spread of japanese encephalitis with environmental effects. in [9] authors build an sirs epidemic model of japanese encephalitis and discussed analytically. in [10], authors have formulated a deterministic model with saturated treatment function and found that north east states of india need better treatment and awareness about the disease. in [11], author contracted a mathematical model with saturated incidence function and analysis optimal control theory. in [12], author discussed different causes of backward bifurcations in some epidemiological models. in [13] authors formulated a mathematical model and regulate important parameters in the spread of the disease through the sensitivity analysis. in this paper, we have constructed a japanese encephalitis mathematical model by considering standard incidence type interaction with saturated incident and saturated treatment. as india is a developing countries, so treatment may not available for all if the number of infectives are very large. so in the present study we have incorporated the saturated treatment as discussed in [10]. the remaining part of the paper is organized as follows: section 2 describes the model and established biological feasible region; section 3 presents basic reproduction number; section 4 deals with data; section 5 deals with sensitivity analysis of basic reproduction number r0; section 6 describes optimal control problem; section 7 demonstrates the numerical simulation of optimal control and finally section 8 concludes the paper. 2. the model formulation the model consists six different compartments such as susceptible humans (sh), infected humans (ih) and recovered humans (rh), susceptible japanese encephalitis mosquitoes (sj), infected japanese encephalitis mosquitoes (ij), and infected pigs (ip). many epidemic models has been used bilinear incident rate to prevent and control the spread of the infectious disease. here we introduced saturated incident rate in the proposed model as saturated incident rate is more suitable than bilinear incident rate. based on following assumption the system of differential equations formulated. 150 n.k. goswami • β1shij 1 + kih is the saturated incident rate, which tends to a saturated level when ih gets large, β1ij measures the infection force when disease is entering a fully susceptible population [14]. • β1shij is known as bilinear incidence rate and 1 1 + kih measures the inhibition effect from the behavioral change of susceptible individuals when their number increases or from the crowding effect of the infective individuals [9, 14, 15]. • 1 1 + kih is the incidence rate which is more reasonable than the β1shij (bilinear incidence rate), [10]. • assume that through treatment the infected individuals recover at a saturated function h(ih) = γhih 1 + αih , where γh is recovery rate. when α = 0, the saturated treatment function becomes linear [14, 16–18]. • the model is a non-linear standard incidence type interaction between human and mosquitoes. • the individuates are recruited in the region at a constant rate λh and they join the susceptible class. • the susceptible individuals become infected when the mosquito bites and join the infected human class at the rate of β1. • infected individuals get treatment and recover from encephalitis with recovery rate γh and join recover class. • the susceptible japanese encephalitis mosquitoes becomes infected and join infected mosquitoes class at the rate β2. • susceptible mosquitoes bite the infected pigs and becomes infected at the rate β3. • the infected pig population are growing logistically. keeping in view the above facts, the mathematical model is framed as follows: dsh dt = λh − β1sh ( ij 1 + kih ) − µhsh + δhrh dih dt = β1sh ( ij 1 + kih ) − γhih 1 + αih − (µh + µ1)ih drh dt = γhih 1 + αih − (δh + µh)rh dsj dt = λj − β2sj ih nh − β3sjip − µjsj dij dt = β2sj ih nh + β3sjip − µjij dip dt = rip ( 1− ip k ) − µpip (2.1) 2.1. positivity and boundedness here, we shall show that the positivity and boundedness of the population. from the system (1), we have dsh dt ∣∣∣ sh=0 = λh + δhrh > 0, dih dt ∣∣∣ ih=0 = β1shij ≥ 0, sensitivity and optimal control analysis of japanese 151 table 2.1. description of biological parameters parameter description value λh rate of recruitment of human population 20 λj rate of recruitment of vector(mosquito) 40 β1 transmission probability interaction between sh and ij 0.00256 β2 transmission probability interaction between sj and ih .031 β3 transmission probability interaction between sj and ip 0.013 α a positive constant 0.02 γh recovery rate of human population 0.005 δh rate at which recovered human become susceptible due to loos of immunity 0.05 µh natural mortality rate of human population 0.0421 µ1 mortality rate of human due to infection 0.0326 µj natural mortality rate of vector 0.112 µp natural mortality rate of pigs 0.51 r the growth rate coefficient of infected pigs 2 k carrying capacity of mosquito population 10 drh dt ∣∣∣ rh=0 = γhih 1 + αih ≥ 0, dsj dt ∣∣∣ sj=0 = λj > 0, dij dt ∣∣∣ ij=0 = β2sj ih nh + β3sjip ≥ 0, dip dt ∣∣∣ ip=0 ≥ 0 here, all the rates are non-negative, so if we start in the interior of the non-negative bounding r6, we shall always remain in this cone keeping mind of the fact that direction of the vector field is inward on all the bounding planes. we note the change rate of the total population nh = sh + ih +rh and nj = sj + ij are given by the following differential equations : dnh dt = λh − µhnh − µ1ih, dnj dt = λj − µjnj. this gives lim sup t→∞ nh ≤ λh µh , lim sup t→∞ nj ≤ λj µj . therefore, all sh(t), ih(t), rh(t) are bounded by λh µh and the solutions sj(t), ij(t) are bounded by λj µj . hence, the biological feasible region of the proposed system (1) is given by the following positively invariant region: ω = (sh, ih, rh, sj, ij, ip) ∈ r6 + : (sh + ih +rh) ≤ λh µh , (sj + ij) ≤ λj µj . 152 n.k. goswami 3. basic reproduction number the disease-free equilibrium for the system (2) as follows e0=(n0 h , i 0 h, r 0 h, n 0 j , i 0 j , i 0 p ) =( λh µh , 0, 0, λj µj , 0, 0 ) . the reproduction number r0 gives the average number of infected individuals generated by the one in a fully susceptible population and for our model, it is given by the above expression of r0. mathematically, r0 is a threshold parameter for the stability of a diseasefree equilibrium and is related to the highest and final size of an epidemic. to find the basic reproduction number r0, we consider ( dih dt , drh dt , dij dt , dip dt ) and using the next-generation matrix method as described in [19, 20]. the matrix f and v as follows: f =  β1(nh − ih −rh) ij 1 + kih 0 β2(nj − ij) ih nh + β3(nj − ij)ip 0  , v =  γhih 1 + αih + (µh + µ1)ih − γhih 1 + αih + (δh + µh)rh µjij −rip ( 1− ip k ) + µpip  now the matrix f ( jacobian of f at disease-free equlibrium e0) and v (jacobian of v at disease-free equilibrium e0) are obtain as f =  0 0 β1n 0 h 0 0 0 0 0 β2 n0 j n0 h 0 0 β3n 0 j 0 0 0 0  and v =  γh + µh + µ1 0 0 0 −γh µh + µp 0 0 0 0 µj 0 0 0 0 −r + µp  the largest eigenvalue of fv −1 is called the basic reproduction number r0 and is obtained as follows: fv −1 =  0 0 β1n 0 h µj 0 0 0 0 0 β2 n0 j n0 h(γh + µh + µ1) 0 0 β3n 0 j µp − r 0 0 0 0  , sensitivity and optimal control analysis of japanese 153 2005 2010 2015 2020 0 200 400 600 800 po pu la tio ns years cases deaths fig. 4.1. trend of japanese encephalitis cases in assam, india during 2004-2019 the spectral radius of the matrix fv −1 is called the basic reproduction number r0 and is obtained as follows: r0 = √ β1β2λj µ2 j(γh + µh + µ1) the quantity r0 is knownas basic reproduction number, the expected number of secondary cases produced in completely susceptible population, by a typical infective individual, 4. data scenario in india, approximately 597,542,000 people are live in japanese encephalitis region, and every year 1,500 to 4,000 japanese encephalitis cases are reported. here, from 2004 to 2019 annual reported cases of japanese encephalitis are using for graphical representation. for this study, we consider two states of india which are assam and uttar pradesh as these are japanese encephalitis infected states of india. the total reported cases from 2004 to 2019 in assam and uttar pradesh are 7,325 and 33,572, which are demonstrated in figure 1 and 2 respectively. in uttar pradesh highest number of cases were reported in 2006 and lowest number of cases were reported in 2019. in assam highest number of cases were reported in 2014 and lowest number of cases were reported in 2005. 154 n.k. goswami 2005 2010 2015 2020 0 2000 4000 6000 po pu la tio ns years cases death fig. 4.2. trend of japanese encephalitis cases in uttar pradesh, india during 2004-2019 5. sensitivity analysis ofr0 to visualize the effect of the parameters involved in the expression of basic reproduction number r0, we calculated the normalized forward sensitivity indices of r0 to those parameters [21]. we keep the parameters at the same values as in table 1. the normalized forward sensitivity index of a variable to a parameter is determined by the ratio of the relative change in the variable to the relative change in the parameter [13]. the sensitivity indices of r0 with respect to the parameters of interest are depicted in figure 3, 4 and 5. the figure shows that when the parameters β1, β2, and λj increase, keeping the other parameters constant, the value ofr0 increases as these parameters have positive indices. instead, increase in the values of parameters γh, µh, µ1, and µj leads to decrease in the values of r0 as they have negative indices. it is noted that the sensitivity index of r0 is 0.5 for the parameters β1, β2,λj . it means that 0.5% increase in the values of any of these parameters, keeping other parameters fixed, will result in 0.5% increment in the value of r0. precisely, lower values of r0 are of crucial importance as they increase the chances of disease eradication. therefore,it is imperative to prevent an increase in the parameters having positive sensitivity indices whereas increasing the values of parameters having negative indices is instead preferred.thus, any prevention measure aiming at reducing the former parameters and increasing the latter must be taken into serious consideration. if such variable is differentiable with respect to the parameter, then the sensitivity index is defined using partial derivatives, [13]. the normalized forward sensitivity index of r0, which is differentiable with respect to a given parameter p, is defined by y r0 p = p r0 ∂r0 ∂p sensitivity and optimal control analysis of japanese 155 1 2 h h 1 j j parameters -1 -0.5 0 0.5 s en ci tiv ity in di ce s fig. 5.3. normalized forward sensitivity indices of r0 with respect to model parameters. parameter values : β1 = 0.00256, β2 = 0.031, γh = 0.245, µh = 0.0421, µ1 = 0.0326, µj = 0.112,λj = 40 the above formula can be used to compute the analytical expression for the sensitivity of r0 to each parameter that it includes. plotted contour plot of some key parameters on r0. if we increase or decrease the value of β1 and β2, and then the value of r0 will increase or decrease, respectively, which is shown in figure 4(a). also, we can conclude that increasing the value of λj will increase the value of the r0, but if we increase the value of µj , it will decrease the value of the r0, which is the death rate of the mosquito, and this is the one of the best control policy to reduce the disease infection from the population, manifest in figure 4(b). similarly other significant changes ofr0 are shown in figures 4(c) and 5(a,b,c) based respective parameters. so, the correct estimation of these parameters is very important to predict transmission of this disease. 6. optimal control problem in this section, three different types of control intervention viz., u1(t), u2(t), u3(t) are incorporated into the model system (1) and extended to optimal control problem. there interventions are implementing either pharmaceutical(treatment) or nonpharmaceutical(effect of information). the main goal this research is to investigate the best control strategies with minimum cost of implementation as well as financial loss generated. however, the vaccination and effective medicines are the most useful strategies to prevent contagious transmission. the details of each intervention are described as follows: • control variable u1(t): the force of japanese encephalitis infections is reduce by (1u1(t)), where u1(t) measures effect due to the use of electronic devices, insecticidetreated bed nets, and mosquito repulsive lotions are used to reduce transmission between mosquitoes and human. if one can reduce between mosquitoes to human transmission rate, then it will be helpful in controlling the mosquito-borne disease. hence, we consider u1(t) as a control intervention to reduce human-mosquito interaction. • control variable u2(t): the control variable u2(t) is the pig vaccination lowered the mosquito infection rate. pig vaccination is associated with a reduction in human cases. in this an effective intervention but requires considerable amount of money. thus, our main objective is to minimize the disease effect in pig population with minimum cost. • control variable u3(t): the control variable u3(t) is the use of effective medicines for the treatment measure of infectious humans, proper vaccination and insecticide to the japanese encephalitis infection causing species [22]. η is the modification parameter 156 n.k. goswami 0 0.2 0.4 0.6 0.8 1 1 0 0.5 1 2 10 20 30 40 (a) 0 0.2 0.4 0.6 0.8 1 j 0 0.2 0.4 0.6 0.8 1 j 0.01 0.02 0.03 0.04 (b) 0 0.2 0.4 0.6 0.8 1 h 0 0.2 0.4 0.6 0.8 1 1 6 7 8 9 10 (c) fig. 5.4. contour plots of the basic reproduction numberr0 with respect to (a) β1 and β2 (b) λj and µj , (c) γh and µ1 sensitivity and optimal control analysis of japanese 157 0 0.2 0.4 0.6 0.8 1 1 0 0.2 0.4 0.6 0.8 1 h 5 10 15 20 25 (a) 0 0.2 0.4 0.6 0.8 1 1 0 0.5 1 h 5 10 15 (b) 0 0.2 0.4 0.6 0.8 1 2 0 0.2 0.4 0.6 0.8 1 j 0.1 0.2 0.3 0.4 0.5 0.6 0.7 (c) fig. 5.5. contour plots of the basic reproduction numberr0 with respect to (a) β1 and µh, (b) β1 and γh, (c) β2 and λj 158 n.k. goswami in control u3(t). in this an effective intervention but requires considerable amount of money. thus, our main objective is to minimize the disease effect in human population with minimum cost. based on different health care restriction , it is essential to impose some bounds on controls as 0 ≤ u1(t), u2(t), u3(t) ≤ 1. if u1(t), u2(t), and u3(t) are equal to zero, then there is no effort being placed in these controls at time t and if they are equal to one then the maximum effort is applied. keeping the view of the above assumptions, the optimal control model is formulated as follows: dsh dt = λh − (1− u1(t))β1sh ( ij 1 + kih ) − µhsh + δhrh dih dt = (1− u1(t))β1sh ( ij 1 + kih ) − ηu3(t)γhih 1 + αih − (µh + µ1)ih drh dt = ηu3(t)γhih 1 + αih − (δh + µh)rh dsj dt = λj − (1− u1(t))β2sj ih nh − (1− u2(t))β3sjip − µjsj dij dt = (1− u1(t))β2sj ih nh + (1− u2(t))β3sjip − µjij dip dt = rip ( 1− ip k ) − µpip (6.2) consider u1(t) = u1, u2(t) = u2, and u3(t) = u3 for further analysis. 6.1. cost construction and characterization of optima controls this particular section devotes into two parts which include determination of the total cost generated due to applied controls as well as diseases itself. whereas the second part determines the analytical forms of the controls. 6.1.1. total cost determination here, the total cost is determined for the applied control interventions and disease as well, which need to be minimized. • cost due to disease: the cumulative cost incurred due to the disease burden is modeled as follows: ∫ t 0 (x1ih +x2(sj + ij) +x3ip)dt this cost consists of various components such as cost due to loss of manpower, opportunity loss and other related wealth loss. • cost incurred in insecticide-treated bed nets: the total cost involved in reducing the transmission between human to mosquito is given as∫ t 0 y1u 2 1dt. • cost incurred intreatment and vaccination for pig population: the total cost involved in reducing pig infection is defined as∫ t 0 y2u 2 2dt. sensitivity and optimal control analysis of japanese 159 • cost incurred in treatment for infected human population: the cumulative cost in process of treating infected human is defined by∫ t 0 y3u 2 3dt. the cost functional corresponding to total cost incurred, for fixed time t , which need to be minimized is given by j(u1, u2, u3) = ∫ t 0 [ x1ih +x2(sj + ij) +x3ip + 1 2 (y1u 2 1 + y2u 2 2 + y3u 2 3) ] dt (6.3) subject to the model system (2). the parameterx1 ≥ 0,x2 ≥ 0, y1 ≥ 0, y2 ≥ 0, and y3 ≥ 0, are the weight and balancing constants, which measure the respective cost involvement over the interval [0, t ]. the term y1u1 2 denotes the cost associated with insecticide-treated bed nets are a form of personal protection that has been shown to reduce mosquito borne disease i.e. zika, dengue, malaria etc. and death due to mosquito borne disease in endemic regions. but at the same time, the execution of itns requires lot of money which is given by the term y1u 1 1 [11]. also, the total cost incurred in process primarily includes the cost involved in awareness and educational campaigns, cost of required manpower etc. the term y2u2 2 signifies the cost associated with pig vaccination to reduce the mosquito infection rate and the term y3u3 2 represents the cost associated with treatment for infected human population which primarily medical facilities, hospitals expenditure, manpower, human vaccination etc [21]. in order to find an optimal control, u1∗,u2∗, and u3∗ such that j(u1 ∗, u2 ∗, u3 ∗) = min (u1,u2,u3)∈u j(u1, u2, u3), (6.4) where u is the control set and is defined as u = {(u1, u2, u3) : 0 ≤ u1, u2, u3 ≤ 1, t ∈ [0, t ]} here, all the controls are bounded and measurable. 6.1.2. existence and characterization of optima controls here, we shall first establish the existence of such control functions that minimises the cost functional j . the lagrangian l of this problem is defined as: l(ih, sj, ij, ip, u1, u2, u3) = x1ih +x2(sj + ij) +x3ip + 1 2 y1u1 2 + 1 2 y2u 2 2 + 1 2 y3u 2 3 now, we shall use pontryagin’s maximum principle [?] for necessary conditions for optimal controls system (2). for that by choosing a = (sh, ih, rh, sj, ij, ip), u = (u1, u2, u3) and λ = (λ1, λ2, λ3, λ4, λ5, λ6), the associated hamiltonianh can be written as h(a,u , λ) = l(ih, sj, ij, ip, u1, u2, u3) + λ1 dsh dt + λ2 dih dt + λ3 drh dt + λ4 dsj dt + λ5 dij dt + λ6 dip dt (6.5) since u∗1,u ∗ 2, u∗3 are solutions to the control problem (2), there exists the adjoint variables λ1, λ2, λ3, λ4, λ5, λ6 satisfying the following conditions. dx dt = ∂h(t, x, u∗1, u ∗ 2, u ∗ 3, λ1, λ2, λ3.λ4, λ5, λ6) ∂λ 0 = ∂h(t, x, u∗1, u ∗ 2, u ∗ 3, λ1, λ2, λ3.λ4, λ5, λ6) ∂u dλ dt = −∂h(t, x, u∗1, u ∗ 2, u ∗ 3, λ1, λ2, λ3.λ4, λ5, λ6) ∂x (6.6) 160 n.k. goswami theorem 6.1: for the objective functional (3) and the control set (6) subject to control system (2) there exists an optimal control u∗ = (u1 ∗, u2 ∗, u3 ∗) ∈ u such that j(u1 ∗, u2 ∗, u3 ∗) = min u j(u1, u2, u3). proof to establish this result, we follow the theorem 4.1 mentioned in [23] for the existence of optimal controls. as, we have discussed above that all the state variables (population) are bounded for each bounded controls coming from the control set u . furthermore, lipschitz condition with respect to state variables is satisfied by the right hand part of the model system (2). the control variable set u is also convex and closed by the definition and the model system (2) is linear in control variables. the integrand of the function l = x1ih +x2(sj + ij) +x3ip + 1 2 y1u1 2 + 1 2 y2u 2 2 + 1 2 y3u 2 3 is convex on the control set u due to quadratic nature of control variables. moreover, l = x1ih +x2(sj + ij) +x3ip + 1 2 y1u1 2 + 1 2 y2u 2 2 + 1 2 y3u 2 3 ≥ 1 2 y1u1 2 + 1 2 y2u 2 2 + 1 2 y3u 2 3. now consider c1 = min(x1, x2, x3) > 0 and g(u1, u2, u3) = c1(u1 2 + u2 2 + u3 2). thus, l ≥ g(u1, u2, u3) holds true and g is continuous. also, g satisfies the condition |(u1, u2, u3)|−1g(u1, u2, u3)→ ∞ whenever |(u1, u2, u3)| → ∞. thus, all the conditions for the existence of controls are fulfilled (for more details one can follow [23, 24]. hence the result. theorem 6.2: let u∗1, u ∗ 2, u ∗ 3 be optimal control functions and s∗h, i ∗ h, r ∗ h,s∗j , i ∗ j ,i∗p are the corresponding state variable of the optimal control problem (2) (3). then there exists adjoint variables λ = (λi) t ∈ r6, i = 1, 2, 3, 4, 5, 6, which satisfies the following equations : dλ1 dt = µhλ1 + (1− u1)β1( ij 1 + kih )(λ1 − λ2)− (1− u1)β2 sjih n2 h (λ4 − λ5) dλ2 dt = −x1 + (1− u1) β1shijk (1 + kih)2 (λ2 − λ3) + ηu3γh (1 + αi2h) (λ2 − λ3) + (µh + µ1)λ2 dλ3 dt = δh(λ3 − λ1) + µhλ3 dλ4 dt = −x2 + µjλ4 + (1− u1)β2 ih nh (λ4 − λ5) + (1− u2)β3ip(λ4 − λ5) dλ5 dt = −x2 + (1− u1)β1sh( 1 1 + kih )(λ1 − λ2) + µjλ5 dλ6 dt = −x3 + (µp + r − 2rip k )λ6 (6.7) with transversality conditions λ1(t ) = λ2(t ) = λ3(t ) = λ4(t ) = λ5(t ) = λ6(t ) = 0 (6.8) proof let u∗1, u ∗ 2, u ∗ 3 be the optimal control functions and s∗h, i ∗ h, r ∗ h,s∗j , i ∗ j , i ∗ p are the corresponding state variables. then, pontryagin’s maximum principle ensures the existence of the following adjoint variable λi(i = 1, 2, 3, 4, 5, 6) ∈ r6, which satisfies the following canonical equations: dλ1 dt = − ∂h ∂sh , dλ2 dt = −∂h ∂ih , dλ3 dt = − ∂h ∂rh , dλ4 dt = −∂h ∂sj , dλ5 dt = −∂h ∂ij , dλ6 dt = −∂h ∂ip sensitivity and optimal control analysis of japanese 161 with transversality conditions (8), where h is the hamiltonian defined as above. thus, the adjoint system (7) can be obtained, in the following result, we shall state the analytical forms of the optimal controls. theorem 6.3: the optimal controls (u1 ∗, u2 ∗, u3 ∗) which minimizes j over the region u given by u1 ∗ = min{1,max(0, ũ1)} u2 ∗ = min{1,max(0, ũ2)} u3 ∗ = min{1,max(0, ũ3)} where, ũ1 = β1 ( sh + ij 1 + kih ) (λ2 − λ1) + β1shijk (1 + kih)2 (λ3 − λ2) + β2 ih nh (λ5 − λ4) y1 ũ2 = β3ip(λ5 − λ4) y2 (6.9) ũ3 = ηγh (1 + αi2h) (λ3 − λ2) y3 proof using the optimal condition (6), we get (9), as follows ∂h ∂u1 = y1u1 + β1 ( sh + ij 1 + kih ) (λ1 − λ2) + β1shijk (1 + kih)2 (λ2 − λ3) + β2 ih nh (λ4 − λ5) = 0 this implies, u1 = β1 ( sh + ij 1 + kih ) (λ2 − λ1) + β1shijk (1 + kih)2 (λ3 − λ2) + β2 ih nh (λ5 − λ4) y1 := ũ1, ∂h ∂u2 = y2u2 + β3ip(λ4 − λ5) = 0 this implies, u2 = β3ip(λ5 − λ4) y2 := ũ2, and, ∂h ∂u3 = y3u3 + ηγh (1 + αi2h) (λ2 − λ3) = 0 this implies, u3 = ηγh (1 + αi2h) (λ3 − λ2) y3 := ũ3. moreover, lower and upper bounds of these control are 0 and 1 respectively. thus, if ũ1 > 1, ũ2 > 1, ũ3 > 1, then u1 = u2 = u3 = 1. 162 n.k. goswami also, ũ1 < 0, ũ2 < 0 , ũ3 < 0, then u1 = u2 = u3 = 0. otherwise, we have u1 = ũ1, u2 = ũ2, and u3 = ũ3 hence, for these controls u∗1, u ∗ 2, u ∗ 3 we get optimum value of the function j . 7. simulation of optimal control in this section, by using matlab software, the optimal control model is simulated. the parameter values are remain same as esteemed in the table 1. the weight constants for the optimal control problem are taken as x1 = 1, x2 = 1, x3 = 1, y1 = 45, y4 = 65, y5 = 75. we solve the optimality system (2) by iterative method with the help of forward and backward difference approximations [23]. we consider the time interval as [0, 200]. profiles for optimal control u1,u2 and u3 are shown, respectively, in figures 6(a), 6(b), and 6(c). finally, to see the effects of optimal controls, the infected human and infected vectors are plotted against time with and without optimal control in figures 7(b). it is easy to notice that optimal control is more effective in reducing the number of infectives. we consider different types of strategies to see the impact of optimal control in the total number of human infectives. for this purpose, we shall define the following control strategies: • strategy a: employing control intervention (u1), only. • strategy b: employing control intervention (u2) only. • strategy c: employing control intervention (u2) only. • strategy d: employing control intervention (u1, u2), (u1, u3), (u2, u3) • strategy e: employing control intervention (u1, u2, u3) 7.1. strategy a: employing electronic devices, insecticide-treated bed nets, and mosquito repulsive lotions (u1), only. here, only control measure u1(t) is used to optimize the objective function j , while control intervention u2(t) = u3(t) = 0, were not employed. the influence of u1(t) is demonstrated in figure 6(a), to minimize the objective function, the optimal control u1(t) is maintained at the maximum level. a single preventive measure can influence the spread of japanese encephalitis in the population. 7.2. strategy b: increase pig vaccination to reduce the mosquito infection (u2) only. here, only control measure u2(t) is used to optimize the objective function j , while control intervention u1(t) = u3(t) = 0, were not employed. in figure 6(b), we present the plots of population and the due to effects of pig vaccination are demonstrated to minimizing the cost and reducing the number of japanese encephalitis infections in the population. 7.3. strategy c: use of effective medicines as treatment measure of infectious humans (u3) only. here, only control measure u3(t) is used to optimize the objective function j , while control intervention u2(t) = u3(t) = 0, were not employed. the influence of u3(t) is demonstrated in figure 6(c), to minimize the disease effect in human population with minimum cost. 7.4. strategy d: effective use of control intervention (u1, u2), (u1, u3), (u2, u3). here, double control interventions (u1(t), u2(t)) when u3(t) = 0, (u1(t), u3(t)) when u2(t) = 0, and (u2(t), u3(t)) when u1(t) = 0, are used to optimize the objective function j. sensitivity and optimal control analysis of japanese 163 the control intervention are demonstrated in figure 6(d), 6(e) and 6(f). it is observed that combination of two control intervention is most effective then using signal control intervention. significant number of japanese encephalitis infective cases reduce. 7.5. strategy e: employing all three control interventions (u1, u2, u3). here all three control interventions (u1(t), u2(t), u3(t)) are used to optimize the objective function j . from figure 7(a), it is easy to say that by combining all three optimal controls u1(t), u2(t), and u3(t), the total number of infectious individuals decreases significantly. the three optimal control application is the best control strategy to minimize the number of infectives, and will definitely reduce the spread of jev. 8. conclusion in this study, a non-linear mathematical model is formulated and analyzed for japanese encephalitis disease with saturated incident ( β1shij 1 + kih ) and saturated treatment function( γhih 1 + αih ) . the basic reproduction number is computed by next generation matrix method. here, it is clearly identify that the parameter β1 is the most sensitive parameter which transmitted japanese encephalitis disease in human. the sensitivity analysis of the basic reproduction and japanese encephalitis infected cases are executed in order to determine the relative importance of the model parameters to the disease prevalence. the results of the numerical simulation is executed to support our mathematical findings and demonstrated graphically. also, we perform here trend of japanese encephalitis cases of assam and uttar pradesh through line-graph. the infection rate is high in assam and uttar pradesh comparatively other part of the country(india). three time-department optimal control is incorporated in model system to eliminate the virus from the tropical region using pontryagin’s maximum principal. adapting all three optimal control parameters are the best control strategy to minimize the number of infectives, which will reduce the rate transmission of the disease. it is noticed easily that use optimal control in the mode is more effective than without optimal control in reducing the number of infectives in the considered period. references 1. world health organization, japanese encephalitis fact sheets. https://www.who.int/newsroom/factsheets/ details/japanese-encephalitis. accessed 9 may 2019 2. world health organization, vaccine-preventable disease. https://www.who.int/immunization/ monitoring-surveillance/burden/vpd/whosurveillancevaccinepreventable-10-je-r2. accessed 9 april 2015 3. tiwari, s., singh, r. k., tiwari, r., and dhole,t.n.(2012). japanese encephalitis: a review of the india perspective, the brazilian journal of infectious diseases.16(6):564573. 4. de, a., maity, k., jana, s., and maiti, m.,(2016). application of various control strategies to japanese encephalitis: a mathematical study with human, pig and mosquito, math. biosci. 282, 46–60. 5. ndairou, f., area, i., torres, d. f. m., (2020). mathematical modeling of japanese encephalitis under aquatic environmental effects. mathematics. 8 (11), 1880 . 6. baniya, v., and keval, r.,(2020). the influence of vaccination on of je with standard incidence rate of mosquitoes, pigs and humans.journal of applied mathematics and computing. 64, 519-550 . 164 n.k. goswami 0 20 40 60 80 100 120 140 160 180 200 time(in days) 0 0.2 0.4 0.6 0.8 1 c o n t r o l p r o f il e ( u 1 ) u 1 , when u 2 =u 3 =0 (a) 0 20 40 60 80 100 120 140 160 180 200 time(in days) 0 0.1 0.2 0.3 0.4 0.5 0.6 c o n t r o l p r o f il e ( u 2 ) u 2 , when u 1 =u 3 =0 (b) 0 20 40 60 80 100 120 140 160 180 200 time(in days) 0 0.2 0.4 0.6 0.8 1 c o n t r o l p r o f il e ( u 2 ) u 3 , when u 1 =u 2 =0 (c) 0 20 40 60 80 100 120 140 160 180 200 time(in days) 0 0.2 0.4 0.6 0.8 1 c o n t r o l p r o f il e ( u 1 ,u 2 ) u 1 , when u 3 =0 u 2 , when u 3 =0 (d) 0 20 40 60 80 100 120 140 160 180 200 time(in days) 0 0.2 0.4 0.6 0.8 1 c o n t r o l p r o f il e ( u 1 ,u 3 ) u 1 , when u 2 =0 u 3 , when u 2 =0 (e) 0 20 40 60 80 100 120 140 160 180 200 time(in days) 0 0.2 0.4 0.6 0.8 1 c o n t r o l p r o f il e ( u 2 ,u 3 ) u 2 , when u 1 =0 u 3 , when u 1 =0 (f) fig. 7.6. (a) control profile of u1, when u2 = u3 = 0, (b) control profile of u2, when u1 = u3 = 0, (c) control profile of u3, when u2 = u3 = 0, (d) control profile of u1, u2, when u3 = 0,(e) control profile of u1, u3, when u2 = 0, (f) control profile of u2, u3, when u1 = 0 sensitivity and optimal control analysis of japanese 165 0 20 40 60 80 100 120 140 160 180 200 time(in days) 0 0.2 0.4 0.6 0.8 1 c o n t r o l p r o f il e ( u 1 ,u 2 ,u 3 ) u 1 u 2 u 3 (a) 0 20 40 60 80 100 120 140 160 180 200 time(in day) 0 50 100 150 in fe c te d p o p u la ti o n s without optimal control with optimal control (b) fig. 7.7. (a) control profile of u1, u2, and u3, (b) variation of infected human populations against time with and without optimal control. 7. panja, p., mondal, s.k., and chattopadhyay, j.,(2016).stability and bifurcation analysis of japanese encephalitis model with/without effect of some control parameters. comput. appl. math 32(2), 155-175 . 8. naresh, r. and panday, s.,(2009). modelling and analysis of the spread of japanese encephalitis with environmental effects, appl. math. 4(1), 155–175 . 9. mukhopadhyay, b.b., and tapaswe, p.k.,(1994). an sirs epidemic model of japanese encephalitis. math. sci. 17(2), 347–356. 10. srivastav, a. k., and ghosh, m.,(2021). modeling the transmission dynamics of malaria with saturated treatment : a case study of india. journal of applied mathematics and computing.https://doi.org/10.1007/s12190-020-01469-7. 11. goswami, n.k.,(2021).modeling analysis of zika virus with saturated incidence using optimal control theory. international journal of dynamical systems and differential equations, 11(3-4), 287-301. 12. gumel, a.b., (2012), causes of backward bifurcations in some epidemiological models, j.math. anal. appl. 395, 355-365. 13. chitnis, n., hyman, j. m., and cushing, j.m. . determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, bull.math.biol. 70(5), 12721296. 14. zhang, j., jai, j., and song, x., (2o14). analysis of an seir epidemic model with saturated incidence and saturated treatment function. the scientific world journal. vol.2014, id 910421, 11 pages. https: //doi.org/10.1155/2014/910421. 15. goswami, n.k., and shanmukha, b.,(2020) stability and optimal control analysis of zika virus with saturated incidence rate. malaya journal of mathematik. 8(2).331-342. 16. dubey, b., patra, a., srivastava, p.k. and dubey, u. s.,(2013) modeling and analysis os an seir model with different types of nonlinear treatment rates, j. biol. syst. 21(3), 1350023. https://doi.org/10.1142/s021833901350023. 17. zhang, z., and suo, y., (2010). qualitative analysis of a sir epidemic model with saturated treatment rate. j. appl. math. comput. 34,177–194 . 18. li, x. z., li, w. s., ghosh, m., (2009). stability and bifurcation of an sis epidemic model with treatment. chaos solit. fract. 42 (5), 2822–2832. 19. castillo-chavez, c., feng, z.,and huang, w., (2002). on the computation of r0 and its role on global stability, mathematical approaches for for emerging and reemerging infectious diseases.springer-verlag. 9780387953540, 229-250 . 166 n.k. goswami 20. driessche, p.v., and watmough, j., (2002). reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. mathematical biosciences. 180, 29-48. 21. goswami, n. k., and shanmukha, b., (2020).modeling and analysis of symptomatic and asymptomatic infections of zika virus disease with non-monotonic incidence rate. appl. math. inf. sci. 14 (4), 655-671. 22. muniaraj, m. and rajamannar, v. (2019). impact of sa 14-14-2 vaccination on the occurrence of japanese encephalitis in india. hum vaccin immunother. 15(4),834-840. doi:10.1080/21645515.2018.1564435. 23. lenhart, s., and workman, j.t., (2007)optimal control applied to biological models, crc press 24. pontryagin, l. s., boltyanskii, v.g., gamkrelidze, r.v. and mishchenko, e.f. (1962) the mathematical theory of optimal processes, inter science publishers, geneva. introduction the model formulation positivity and boundedness basic reproduction number data scenario sensitivity analysis of r0 optimal control problem cost construction and characterization of optima controls total cost determination existence and characterization of optima controls simulation of optimal control strategy a: employing electronic devices, insecticide-treated bed nets, and mosquito repulsive lotions (u1), only. strategy b: increase pig vaccination to reduce the mosquito infection (u2) only. strategy c: use of effective medicines as treatment measure of infectious humans (u3) only. strategy d: effective use of control intervention (u1, u2),(u1, u3),(u2, u3). strategy e: employing all three control interventions (u1,u2 ,u3). conclusion microsoft word 1224-article text-5527-1-18-20230104 adv syst sci appl 2022; 04; 144-161 published online at http://ijassa.ipu.ru. analysis and dynamics of tuberculosis outbreak: a mathematical modelling approach festus abiodun oguntolu1, olumuyiwa james peter2,3*, kayode oshinubi4, tawakalt abosede ayoola5, asimiyu olalekan oladapo5, mayowa m. ojo6,7 1)department of mathematics, federal university of technology minna, minna, nigeria 2)department of mathematical and computer sciences, university of medical sciences, ondo city, ondo state, nigeria 3)department of epidemiology and biostatistics, school of public health, university of medical sciences, ondo city, ondo state, nigeria. 4)agies research unit, universite grenoble alpes, alpes, france 5)department of mathematics, osun state university oshogbo, nigeria 6)department of mathematical sciences, university of south africa, florida, south africa 7)thermo fisher scientific, microbiology division, lenexa, kansas, usa e-mail: peterjames4real@gmail.com abstract: tuberculosis (tb) is an infectious disease caused by mycobacterium disease which causes major ill health in humans. control strategies like vaccines, early detention, treatment and isolation are required to minimize or eradicate this deadly pandemic disease. this article presents a novel mathematical modelling approach to tuberculosis disease using vaccinated-susceptiblelatent-mild-chronic-isolated-treated model. we examined if the epidemiology model is well posed and then obtained two equilibria points (disease free and endemic equilibrium). we also showed that tb disease free equilibrium is locally and globally asymptotically stable if 𝑅 < 1. we solved the model analytically using homotopy perturbation method (hpm) and the graphical representations and interpretations of various effects of the model parameters in order to measure the impact for effective disease control are presented. the findings show that infected populations will be reduced when the isolation and treatment rates and their effectiveness are high. keywords: tuberculosis; homotopy perturbation method; infectious disease; basic reproduction number; vaccination 1. introduction tuberculosis (tb) is the third greatest killer worldwide caused by an infectious agent [12]. according to world health organization (who), one-third of the world’s population is currently infected by the tb bacillus bacteria. being a disease of poverty, the vast majority of tb deaths are in developing countries with more than half occurring in asia. furthermore, over 95% of these deaths occurred in lowand middle-income countries where the cost of diagnosis and treatment is high, and not readily accessible. tuberculosis is a chronic bacteria infectious disease caused by mycobacterium tuberculosis which poses a major health, social and economic burden globally, especially in lowand middle-income countries [5]. the surge in hiv-tb co-infection and the growing emergence of multidrug-resistant tb (mdr-tb) and extensively drug-resistant tb (xdr-tb) strains has further fueled tb epidemic. tb usually affects the lungs (pulmonary tb) but it can also affect other sites as well (extra-pulmonary tb). tuberculosis is transmitted by tiny analysis and dynamics of tuberculosis outbreak 145 copyright ©2022 assa. adv. in systems science and appl. (2022) airborne droplets which are expelled into the air when a person with active pulmonary tb coughs or talks [21,22]. diagnosis of latent tb infections (ltbi) and prompt treatment of active cases remains an important component of effective tb control as shown in previous studies. on the other hand, undetected tb infection and delay in the treatment of active tb cases leads to more severe disease conditions in the infected person which could result in wider disease spread in the community [3, 7, 10, 9]. some researchers have proposed some mathematical models to solve problems arising from tb models and we shall discuss some of them as follows: [8] presents a susceptibleexposed-infected-recovered (seir) tuberculosis model which incorporated treatment of infectious individuals and chemoprophylaxis (treatment for the latently infected). the model assumed that the latently infected individuals develop the active disease as a result of endogenous re-activation, exogenous re-infection and disease relapse. [4] presents the mathematical model of a tuberculosis transmission dynamics incorporating first and second line treatment. [7] proposed a seven-compartment model which included diagnosed and undiagnosed infectious population and their result shows that high vaccination rate is required to eradicate tb. in [16], the authors develop a mathematical model for control of tuberculosis epidemiology by incorporating some control strategies, the findings of the study show that using multiple controls is the best way to control the spread of tb. several studies have been conducted utilizing a mathematical model method in order to identify ways to control diseases in the population [1,13, 18-20]. [17] presented a deterministic seir model to described the transmission dynamics of tb in ashanti region of ghana and the results showed that the region has herd immunity against tb infection. in this study we consider some control strategies such as, treatment, isolation and vaccination into the mathematical formulation of tb outbreak with assumptions that people in each compartment have equal natural death rate and infection does not confer immunity to the treated and recovered individuals. the rest of the article is divided as follows: method which includes model formulation and mathematical analysis of the model are described in “section 2. next, section consists of the semi-analytic solution of the model formulated. numerical simulation and graphical representation of results is given in 4, while the discussion of the results is presented in section 5. finally, in section 6, we have provided conclusions of this article. 2. method 2.1. model formulation in this section, the tb transmission model is formulated. using a compartmental approach, the total host population can be partitioned into seven compartments according to their epidemiological status. the groups are the vaccinated 𝑉(𝑡) , susceptible 𝑆(𝑡) ,latent 𝐸(𝑡) , mild tb 𝐼 (𝑡) , chronic tb 𝐼 (𝑡) , isolated infectious 𝐽(𝑡) and treated 𝑇(𝑡) individual, where 𝑡 is the time variable. it is assumed that once the treatment of active tb cases is interrupted, there is no more treatment. vaccination reduces the risk of infection by a factor 𝜃 ∈ (0,1) and the efficacy of the vaccine is 𝜔. let a constant 𝜋 stands for the number of newborn babies into the population, then 𝜃𝜋 are the individuals in the vaccinated class while (1 − 𝜃)𝜋 are the susceptible individual. the susceptible class also increases with a waning rate of vaccine at 𝜔, due to fact that vaccine does not confer a total immunity. we assume that𝜇is per capital natural death rate and 𝑑 (𝑖 = 1,2,3) is the disease induced death rate in classes𝐼 (𝑡), 𝐼 (𝑡) and𝐽(𝑡) respectively. it is natural to assume that 𝑑 ≥ 𝑑 ≥ 𝑑 due to the treatment of active tb cases reducing the disease induced death rate are the transmission coefficients from class 𝑆(𝑡) , 𝐸(𝑡) and 𝑇(𝑡) respectively. we assume that 𝜆 > 𝜆 > 𝜆 because the treatment of active tb cases reduces the infectivity of active tb cases. take 𝜌(0 < 𝜌 < 1) as the fraction of the latent persons who have fast tb progression. the 146 f.a. oguntolu, o.j. peter, k. oshinubi, t.a. ayoola, a.o. oladapo, m.m. ojo copyright ©2022 assa adv. in systems science and appl. (2022) proportion 𝜙 of individual in the exposed class will progress to the chronic class via endogenous us reactivation. 𝜎is the reactivation rate from the latent persons to infected class. 𝛾is the reactivation rate of the individual in the mild tb 𝐼 (𝑡) to the chronic tb 𝐼 (𝑡) . the parameters are the recovery rates of the individual in the classes 𝐼 (𝑡) , 𝐼 (𝑡) and 𝐽(𝑡) respectively. and the parameter is the rate of isolation of the individuals in chronic class 𝐼 (𝑡) . in this article, the tb dynamic model describing the compartment is based on the following assumptions: that a proportion of the population of newborn is immunized against tb infection through vaccination. that the immunity conferred on individuals by treatment expires after some time at given rate. that people in each compartment have equal natural death rate of 𝜇 that there are no immigrants and emigrants. the only way of entry into the population is through new-born babies and the only way of exit is through death from natural causes or death from tb related causes. that the infection does not confer immunity to the treated and recovered individuals and so they go back to the susceptible class at a given rates. that all newborns are previously uninfected by tb and therefore join either the immunized compartment or the susceptible compartment depending on whether they are vaccinated or not. we combine the basic assumptions, model parameters, variables and the tb infection processes to formulate a schematic diagram for tb infection as shown in figure 1. the model equations are given as follow = 𝜋𝜃 − (𝜇 + 𝜔)𝑉 , (1) = 𝜋(1 − 𝜃) − 𝜆 𝑆 − 𝜇𝑆 + 𝜔𝑉, (2) = 𝜆 𝑆 + 𝜆 𝑇 − 𝜆 𝐸 − (𝜇 + 𝜎)𝐸, (3) = (1 − 𝜌)𝜎𝐸 − (𝛾 + 𝜇 + 𝑟 + 𝑑 )𝐼 + (1 − 𝜙)𝜆 𝐸, (4) = 𝜌𝜎𝐸 + 𝛾𝐼 − (𝑟 + 𝑟 + 𝜇 + 𝑑 )𝐼 + 𝜙𝜆 𝐸, (5) = 𝑟 𝐼 − (𝜇 + 𝑑 + 𝑟 )𝐽 , (6) = 𝑟 𝐽 + 𝑟 𝐼 + 𝑟 𝐼 − (𝜇 + 𝜆 )𝑇 , (7) where 𝜆 = 𝛽(𝐼 + 𝜀 𝐼 + 𝜀 𝐽),𝜆 = 𝛼(𝐼 + 𝜀 𝐼 + 𝜀 𝐽) and 𝜆 = 𝛾(𝐼 + 𝜀 𝐼 + 𝜀 𝐽). analysis and dynamics of tuberculosis outbreak 147 copyright ©2022 assa. adv. in systems science and appl. (2022) figure 1: schematic representation of the model table 1. notation and definition of other parameters symbol description 𝑟 treatment rate of individuals in mild class 𝑟 the treatment rate for those is 𝐼 class 𝑟 the progression rate from classes𝐼 to j 𝑟 the treatment rate for those in isolated class 𝛽 transmission rate among the susceptible 𝑑 death rate for mild tb individual 𝑑 death rate for chronic tb individuals 𝑑 death rate for isolated class 𝛼 effective transmission rate from latent class to infected class 𝜀 relative infectiousness of humans with mild tb compared to humans in the chronic class 𝜀 relative infectiousness of humans with tb in the isolated class compared to humans in the chronic class 𝜀 relative infectiousness of humans with mild tb due to endogenous reactivation compared to humans in the chronic class. 𝜀 relative infectiousness of humans on isolation after endogenous reactivation compared to humans in the chronic class 𝜀 relative infectiousness of humans with mild tb due to exogenous re-infection compared to humans in the chronic class 𝜀 relative infectiousness of humans on isolation after exogenous re-infection compared to humans in the chronic class 𝜌 fraction of the latent persons who have fast tb progression 𝜎 reactivation rate from the latent individuals 𝜙 proportion of individual in the exposed class 𝛾 reactivation rate of the individual in the mild tb class 𝜇 natural death rate 2.2. the positive invariant region the entire population size 𝑁 can be determined from by adding equations (1)–(7). hence, 𝑁(𝑡) = 𝑆(𝑡) + 𝑉(𝑡) + 𝐸(𝑡) + 𝐼 (𝑡) + 𝐼 (𝑡) + 𝐽(𝑡) + 𝑇(𝑡) then, = 𝜋 − 𝜇𝑁(𝑡) − 𝑑 𝑁 − 𝑑 𝑁 − 𝑑 𝑁 (8) in the absence of the disease (𝑑 = 𝑑 = 𝑑 = 0) then (8) gives 148 f.a. oguntolu, o.j. peter, k. oshinubi, t.a. ayoola, a.o. oladapo, m.m. ojo copyright ©2022 assa adv. in systems science and appl. (2022) = 𝜋 − 𝜇𝑁 (9) theorem 1: the system (1) to (7) has solution which are contained in the feasible region ω for all 𝑡 > 0. proof. let 𝛺 = (𝑆, 𝑉, 𝐸 , 𝐼 𝐼 , 𝐽, 𝑇) ∈ 𝑅 be any solution of the system (1)–(7) with non-negative initial condition. using theorem of differential inequality, equation (9) gives ≤ 𝜋 − 𝜇𝑁, then 0 ≤ 𝑁 ≤ , hence 𝜋 − 𝜇𝑁 ≥ 𝑘𝑒 , where 𝑘 is constant. thus, the feasible set of the model is given by 𝛺 = (𝑆, 𝑉, 𝐼 , 𝐼 , 𝐽, 𝑇) ∈ 𝑅 : 𝑆, 𝑉, 𝐼 , 𝐼 , 𝐽, 𝑇 ≥ 0, 𝑁 ≤ , which is positive invariant (i.e., solution remain positive for all time 𝑡) and the model is epidemiologically meaningful and mathematically well pose. □ 2.3. positivity of solutions since equations (1)–(7) represent the population in each compartment and all model parameters are all positive, then it lies in a region ω defined by 𝛺 = (𝑆, 𝑉, 𝐼 , 𝐼 , 𝐽, 𝑇) ∈ 𝑅 : 𝑆, 𝑉, 𝐼 , 𝐼 , 𝐽, 𝑇 ≥ 0, 𝑁 ≤ . theorem 2: let the initial value for the model equation be given as 𝑆(0), 𝑉(0) , 𝐸(0), 𝐼 (0) , 𝐼 (0), 𝐽(0), 𝑇(0) ∈ 𝑛 then the solution set {𝑆(𝑡), 𝑉(𝑡) , 𝐸(0), 𝐼 (𝑡) , 𝐼 (𝑡), 𝐽(𝑡), 𝑇(𝑡)} of the system (1)–(7) is positive for all 𝑡 > 0. proof. from equation (2) = 𝜋(1 − 𝜃) + 𝜔𝑉 − (𝜆 + 𝜇)𝑆 then, ≥ −(𝜆 + 𝜇)𝑆 on integrating it gives ∫ ≥ − ∫(𝜆 + 𝜇) 𝑑𝑡. hence, 𝑆(𝑡) ≥ 𝑆(0)𝑒 ( ) . applying the same approach to other equations in the model equations, we have: 𝑉(𝑡) ≥ 𝑉(0)𝑒 ( ) , 𝐸(𝑡) ≥ 𝐸(0)𝑒 ( ) , 𝐼 (𝑡) ≥ 𝐼 (0)𝑒 ( ) , 𝐼 (𝑡) ≥ 𝐼 (0)𝑒 ( ) ,𝐽(𝑡) ≥ 𝐽(0)𝑒 ( ) , 𝑇(𝑡) ≥ 𝑇(0)𝑒 ( ) . therefore, the solution to the model equations is positive for all 𝑡 > 0.□ 2.4. equilibrium points of the model the equilibrium state is the point in which there is zero disturbance on the system under consideration. that is, the rate of change of the model variables with time is zero. thus, at equilibrium, = = = = = = = 0 (10) let 𝐸∗ = (𝑉, 𝑆, 𝐸, 𝐼 , 𝐼 , 𝐽, 𝑇) = (𝑉∗, 𝑆∗, 𝐸∗, 𝐼 ∗, 𝐼 ∗, 𝐽∗, 𝑇∗) be arbitrarily equilibrium point substituting equations (10) into equations (1)–(7) gives: 𝜋𝜃 − (𝜇 + 𝜔)𝑉 = 0, (11) 𝜋(1 − 𝜃) − 𝜆 𝑆 − 𝜇𝑆 + 𝜔𝑉 = 0, (12) analysis and dynamics of tuberculosis outbreak 149 copyright ©2022 assa. adv. in systems science and appl. (2022) 𝜆 𝑆 + 𝜆 𝑇 − 𝜆 𝐸 − (𝜇 + 𝜎)𝐸 = 0, (13) (1 − 𝜌)𝜎𝐸 − (𝛾 + 𝜇 + 𝑟 + 𝑑 )𝐼 + (1 − 𝜙)𝜆 𝐸 = 0, (14) 𝜌𝜎𝐸 + 𝛾𝐼 − (𝑟 + 𝑟 + 𝜇 + 𝑑 )𝐼 + 𝜙𝜆 𝐸 = 0, (15) 𝑟 𝐼 − (𝜇 + 𝑑 + 𝑟 )𝐽 = 0, (16) 𝑟 𝐽 + 𝑟 𝐼 + 𝑟 𝐼 − (𝜇 + 𝜆 )𝑇 = 0. (17) from equation (11) we have 𝑉 = ( ) . (18) substitute equation (18) into equation (12) we have 𝜋(1 − 𝜃) + = 𝑆. (19) from equation (16) we have 𝐽 = 𝐾 𝐼 where 𝐾 = . from (15) we have [(1 − 𝜌)𝜎 + (1 − 𝜙)𝜆 ]𝐸 = [𝑟 + 𝑟 + 𝜇 + 𝑑 ]𝐼 ⇒ 𝐸 = 𝐾 𝐼 where 𝐾 = ( ) ( ) , from equation (16) we have (𝑟 + 𝑟 + 𝜇 + 𝑑 )𝐼 = 𝜌𝜎𝐸 + 𝜙𝜆 𝐸 + 𝛾𝐼 , then 𝐼 = 𝐾 𝐼 where 𝐾 = ( ) further substitution gives 𝐽 = 𝐾 𝐾 𝐼 and from equation (17) we have: 𝑇 = . then 𝑇 = 𝐾 𝐼 where 𝐾 = ( ) , 𝜆 = 𝛽(𝐼 + 𝜀 𝐼 + 𝜀 𝐽) 𝜆 = 𝛼(𝐼 + 𝜀 𝐼 + 𝜀 𝐽) 𝜆 = 𝛾 (𝐼 + 𝜀 𝐼 + 𝜀 𝐽) and 𝐽 = 𝐾 𝐾 𝐼 𝐼 = 𝐾 𝐼 by combining both equations: 𝜆 = 𝛽𝐼 (𝐾 + 𝜀 𝐾 𝐾 + 𝜀 ) 𝜆 = 𝛼𝐼 (𝐾 + 𝜀 𝐾 𝐾 + 𝜀 ) 𝜆 = 𝛾 𝐼 (𝐾 + 𝜀 𝐾 𝐾 + 𝜀 )⎭ ⎪ ⎬ ⎪ ⎫ further substitution gives [𝛽𝐾 𝑆 + 𝛾 𝐾 − 𝛼𝐾 𝐾 𝐼 − (𝜇 + 𝜎)𝐾 ]𝐼 = 0. therefore, 𝐼 = 0 or 𝛽𝐾 𝑆 + 𝛾 𝐾 − 𝛼𝐾 𝐾 𝐼 − (𝜇 + 𝜎)𝐾 = 0 where 𝐾 = 𝐾 + 𝜀 + 𝜀 𝐾 𝐾 𝐾 = 𝐾 + 𝜀 + 𝜀 𝐾 𝐾 𝐾 = 𝐾 + 𝜀 + 𝜀 𝐾 𝐾 and 𝜆 = 𝛽𝐾 𝐼 𝜆 = 𝛼𝐾 𝐼 𝜆 = 𝛾 𝐾 𝐼 ⎭ ⎪ ⎬ ⎪ ⎫ . (20) 2.5. disease free equilibrium (d.f.e) the disease-free equilibrium state is the point at which there exist no infection in the given population. at disease free equilibrium, we let (𝑉, 𝑆, 𝐸, 𝐼 , 𝐼 , 𝐽, 𝑇) = 𝐸∗ = (𝑉∗, 𝑆∗, 𝐸∗, 𝐼∗ , 𝐼∗, 𝐽∗, 𝑇∗). lemma 1: 150 f.a. oguntolu, o.j. peter, k. oshinubi, t.a. ayoola, a.o. oladapo, m.m. ojo copyright ©2022 assa adv. in systems science and appl. (2022) the d.f.e of the model exists and is given by 𝐸 = (𝑉∗, 𝑆∗, 𝐸∗, 𝐼∗ , 𝐼∗, 𝐽∗, 𝑇∗) = 𝜋𝜃 (𝜇 + 𝜔) , 𝜋(𝜇 + 𝜔 − 𝜇𝜃) 𝜇(𝜇 + 𝜔) , 0,0,0,0,0 . proof. suppose 𝐼 = 0. then equation (20) becomes 𝜆 = 0 𝜆 = 0 𝜆 = 0 and also 𝐽∗ = 0 𝐸∗ = 0 𝐼∗ = 0 𝑇∗ = 0 . from (19) 𝑆∗ = 𝜋(𝜇 + 𝜔 − 𝜇𝜃) 𝜇(𝜇 + 𝜔) . thus, the lemma is proved and ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ 𝑉∗ 𝑆∗ 𝐸∗ 𝐼 ∗ 𝐼 ∗ 𝐽∗ 𝑇∗ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ = ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ ( ) ( ) ( ) 0 0 0 0 0 ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ . (21) the above equation is the disease-free equilibrium. 2.7. effective reproduction number in biomathematics, the basic reproduction number (r0) is the average number of infected contacts per infected individual. it is one of the fundamental concepts to determine the future of an epidemics in a population. when 𝑅 < 1 the infection will die out in the long run, but if 𝑅 > 1. the infection will be able to spread in a population. in this model, the spectral radius of the equation is given as largest eigenvalue given as 𝑅 = 𝜌𝑓𝑣 following the procedure in [14, 15], the next generation matrix operator is used to estimate the effective reproduction number such that the jacobian matrices for the new infection terms and the remaining transfer terms are obtained below 𝐹 = 𝜆 𝑆 + 𝜆 𝑇 (1 − 𝜙)𝜆 𝐸 (𝜙𝜆 )𝐸 0 𝑉 = ⎝ ⎛ (𝜇 + 𝜎)𝐸 (𝛾 + 𝑟 + 𝑑 + 𝜇)𝐼 − (1 − 𝜌)𝜎𝐸 (𝑟 + 𝑟 + 𝑑 + 𝜇)𝐼 − 𝛾 𝐼 − 𝜌𝜎𝐸 (𝜇 + 𝑑 + 𝑟 )𝐽 − 𝑟 𝐼 ⎠ ⎞ 𝐹(𝐷𝐹𝐸) = ⎝ ⎜ ⎛ 0 𝛽𝜀 𝜋(𝜇 + 𝜔 − 𝜇𝜃) 𝜇(𝜇 + 𝜔) 𝛽𝜋(𝜇 + 𝜔 − 𝜇𝜃) 𝜇(𝜇 + 𝜔) 𝛽𝜀 𝜋(𝜇 + 𝜔 − 𝜇𝜃) 𝜇(𝜇 + 𝜔) 0 0 0 0 0 0 0 0 0 0 0 0 ⎠ ⎟ ⎞ analysis and dynamics of tuberculosis outbreak 151 copyright ©2022 assa. adv. in systems science and appl. (2022) 𝑉 = ⎝ ⎛ (𝜇 + 𝜎) 0 0 0 −(1 − 𝜌)𝜎 (𝛾 + 𝜇 + 𝑟 + 𝑑 ) 0 0 −𝜌𝜎 −𝛾 (𝑟 + 𝑟 + 𝑑 + 𝜇) 0 0 0 −𝑟 (𝜇 + 𝑟 + 𝑑 )⎠ ⎞ let 𝑉 = 𝑄 0 0 0 −𝐶 𝑄 0 0 −𝐶 −𝛾 𝑄 0 0 0 −𝑟 𝑄 where 𝑄 (𝜇 + 𝜎), 𝑄 = 𝛾 + 𝜇 + 𝑟 + 𝑑 , 𝑄 = 𝑟 + 𝑟 + 𝑑 + 𝜇, 𝑄 = 𝑑 + 𝑟 + 𝜇, therefore 𝑉 = ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ 1 (𝜇 + 𝜎) 0 0 0 (1 − 𝜌)𝜎 𝑄 𝑄 1 𝑄 0 0 (1 − 𝜌)𝛾𝜎 + 𝜌𝜎𝑄 𝑄 𝑄 𝑄 𝛾 𝑄 𝑄 1 𝑄 0 𝑟 [(1 − 𝜌)𝛾𝜎 + 𝜌𝜎𝑄 ] 𝑄 𝑄 𝑄 𝑄 𝑟 𝛾 𝑄 𝑄 𝑄 𝑟 𝑄 𝑄 1 𝑄 ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ 𝐹𝑉 = ⎝ ⎜ ⎛ 𝛽𝜀 𝜎𝑥(1 − 𝜌) 𝑄 𝑄 + 𝛽𝑥𝜎[(1 − 𝜌)𝛾 + 𝜌𝑄 ] 𝑄 𝑄 𝑄 + 𝛽𝜀 𝑥𝑟 𝜎[(1 − 𝜌)𝛾 + 𝜌𝑄 ] 𝑄 𝑄 𝑄 𝑄 𝛽𝜀 𝑥 𝑄 + 𝛽𝑥𝛾 𝑄 𝑄 + 𝛽𝜀 𝑥𝑟 𝛾 𝑄 𝑄 𝑄 𝛽𝑥 𝑄 + 𝛽𝜀 𝑥𝑟 𝑄 𝑄 𝛽𝜀 𝑥 𝑄 0 0 0 0 0 0 0 0 0 0 0 0 ⎠ ⎟ ⎞ , (22) where 𝑥 = ( ) ( ) . from (22), we calculate the eigen-values to determine the basic reproduction number, 𝑅 by taking the spectral radius (dominant eigenvalue) of the matrix 𝐹𝑉 , this is computed by |𝐽 − 𝜆𝐼| = 0, hence the matrix becomes |𝐽 − 𝜆𝐼| = 𝑇 − 𝜆 𝑇 𝑇 𝑇 0 −𝜆 0 0 0 0 −𝜆 0 0 0 0 −𝜆 = 0 where 𝑇 = 𝛽𝜀 𝜎𝑥(1 − 𝜌) 𝑄 𝑄 + 𝛽𝑥𝜎[(1 − 𝜌)𝛾 + 𝜌𝑄 ] 𝑄 𝑄 𝑄 + 𝛽𝑥𝜀 𝑟 𝜎[(1 − 𝜌)𝛾 + 𝜌𝑄 ] 𝑄 𝑄 𝑄 𝑄 𝑇 = 𝛽𝜀 𝑥 𝑄 + 𝛽𝑥𝛾 𝑄 𝑄 + 𝛽𝜀 𝑥𝑟 𝛾 𝑄 𝑄 𝑄 , 𝑇 = 𝛽𝑥 𝑄 + 𝛽𝜀 𝑥𝑟 𝑄 𝑄 , 𝑇 = 𝛽𝜀 𝑥 𝑄 and 152 f.a. oguntolu, o.j. peter, k. oshinubi, t.a. ayoola, a.o. oladapo, m.m. ojo copyright ©2022 assa adv. in systems science and appl. (2022) 𝑥 = 𝜋(𝜇 + 𝜔 − 𝜇𝜃) 𝜇(𝜇 + 𝜔) . this implies that 𝜆 = 𝑇 = 𝛽𝜀 𝜎𝑥(1 − 𝜌) 𝑄 𝑄 + 𝛽𝑥𝜎[(1 − 𝜌)𝛾 + 𝜌𝑄 ] 𝑄 𝑄 𝑄 + 𝛽𝜀 𝑥𝑟 𝜎[(1 − 𝜌)𝛾 + 𝜌𝑄 ] 𝑄 𝑄 𝑄 𝑄 𝜆 = 0, 𝜆 = 0, 𝜆 = 0. therefore 𝑅 = 𝛽𝜎𝑥[(1 − 𝜌)(𝜀 𝑄 + 𝛾 ) + 𝜌𝑄 ] 𝑄 𝑄 𝑄 + 𝛽𝜀 𝑥𝑟 𝜎[(1 − 𝜌)𝛾 + 𝜌𝑄 ] 𝑄 𝑄 𝑄 𝑄 = 𝛽𝜎𝜋(𝜇 + 𝜔 − 𝜇𝜃)([(1 − 𝜌)(𝜀 𝑄 + 𝛾 ) + 𝜌𝑄 ]𝑄 + 𝜀 𝑟 [(1 − 𝜌)𝛾 + 𝜌𝑄 ]) 𝜇(𝜇 + 𝜔)(𝜇 + 𝜎)𝑄 𝑄 𝑄 where 𝑄 = 𝛾 + 𝜇 + 𝑟 + 𝑑 , 𝑄 = 𝑟 + 𝑟 + 𝑑 + 𝜇, 𝑄 = 𝑑 + 𝑟 + 𝜇. 2.8. local stability of disease-free equilibrium theorem 3: the disease equilibrium of the model equations (1)–(7) is locally asymptotically stable (las) if 𝑅 < 1. proof. using jacobian stability techniques, the jacobian matrix at d.f.e is given by: 𝐽(𝐸 ) = ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ −(𝜇 + 𝜔) 0 0 0 0 0 0 𝜔 −𝜇 0 0 0 0 0 0 0 −(𝜇 + 𝜎) 𝛽𝜀 𝜋(𝜇 + 𝜔 − 𝜇𝜃) 𝜇(𝜇 + 𝜔) 𝛽𝜀 𝜋(𝜇 + 𝜔 − 𝜇𝜃) 𝜇(𝜇 + 𝜔) 𝛽𝜀 𝜋(𝜇 + 𝜔 − 𝜇𝜃) 𝜇(𝜇 + 𝜔) 0 0 0 (1 − 𝜌)𝜎 −(𝛾 + 𝜇 + 𝑑 + 𝑟 ) 0 0 0 0 0 𝜌𝜎 𝛾 −(𝑟 + 𝑟 + 𝜇 + 𝑑 ) 0 0 0 0 0 0 𝑟 −(𝜇 + 𝑑 + 𝑟 ) 0 0 0 0 𝑟 𝑟 𝑟 −𝜇⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ . let 𝑄 = (𝜇 + 𝜎), 𝑄 = (𝛾 + 𝜇 + 𝑑 + 𝑟 ), 𝑄 = −(𝑟 + 𝑟 + 𝜇 + 𝑑 ), 𝑄 = (𝜇 + 𝑑 + 𝑟 ), 𝑄 = (𝜇 + 𝜔), 𝐵 = ( ) ( ) , 𝐵 = ( ) ( ) , 𝐵 = ( ) ( ) , 𝐶 = (1 − 𝜌)𝜎 ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎫ , (23) analysis and dynamics of tuberculosis outbreak 153 copyright ©2022 assa. adv. in systems science and appl. (2022) 𝐽(𝐸 ) = ⎝ ⎜ ⎜ ⎜ ⎛ −𝑄 0 0 0 0 0 0 𝜔 −𝜇 0 0 0 0 0 0 0 −𝑄 𝐵 ∗ 𝐵 𝐵 0 0 0 𝐶 𝑄 0 0 0 0 0 𝜌𝜎 𝛾 −𝑄 0 0 0 0 0 0 𝑟 −𝑄 0 0 0 0 𝑟 𝑟 𝑟 −𝜇⎠ ⎟ ⎟ ⎟ ⎞ . then, |𝐽 − 𝜆𝐼| = ⎝ ⎜ ⎜ ⎜ ⎛ −𝑄 − 𝜆 0 0 0 0 0 0 0 −𝜇𝑄 − 𝜆 0 0 0 0 0 0 0 −𝑄 − 𝜆 𝐵 𝐵 𝐵 0 0 0 0 𝑄 − 𝜆 𝐶 𝐵 𝐶 𝐵 0 0 0 0 0 𝑄 − 𝜆 𝑄 0 0 0 0 0 0 −𝑄 − 𝜆 0 0 0 0 0 0 0 −𝑄 𝑄 − 𝜆⎠ ⎟ ⎟ ⎟ ⎞ and 𝑄 = 𝐶 𝐵 − 𝑄 𝑄 , 𝑄 = (𝐶 𝐵 − 𝑄 𝑄 )(𝜌𝜎𝐵 − 𝑄 𝑄 ) − (𝛾𝑄 − 𝜌𝜎𝐵 )𝐶 𝐵 , 𝑄 = (𝐶 𝐵 − 𝑄 𝑄 )𝜌𝜎𝐵 − (𝛾𝑄 − 𝜌𝜎𝐵 )𝐶 𝐵 , 𝑄 = −([(𝐶 𝐵 − 𝑄 𝑄 )(𝜌𝜎𝐵 − 𝑄 𝑄 ) − (𝛾𝑄 − 𝜌𝜎𝐵 )𝐶 𝐵 ]𝑄 ), +𝑟 [(𝐶 𝐵 − 𝑄 𝑄 )𝜌𝜎𝐵 − (𝛾𝑄 − 𝜌𝜎𝐵 )𝐶 𝐵 ], 𝑄 = 𝜇(𝐶 𝐵 − 𝑄 𝑄 )[(𝐶 𝐵 − 𝑄 𝑄 )(𝜌𝜎𝐵 − 𝑄 𝑄 ) − (𝛾𝑄 − 𝜌𝜎𝐵 )𝐶 𝐵 ] where − 𝑄 − 𝜆 = 0, −𝜇𝑄 − 𝜆 = 0, −𝑄 − 𝜆 = 0, 𝑄 − 𝜆 = 0, 𝑄 − 𝜆 = 0, −𝑄 𝑄 − 𝜆 = 0⎭ ⎪ ⎬ ⎪ ⎫ . (24) from equation (24) 𝜆 = −𝑄 , 𝜆 = − [(𝐶 𝐵 − 𝑄 𝑄 )(𝜌𝜎𝐵 − 𝑄 𝑄 ) − (𝛾𝑄 − 𝜌𝜎𝐵 )𝐶 𝐵 ]𝑄 +𝑟 [(𝐶 𝐵 − 𝑄 𝑄 )𝜌𝜎𝐵 − (𝛾𝑄 − 𝜌𝜎𝐵 )𝐶 𝐵 ] . (25) substitute equation (23) into equation (25), we have = −𝛽𝛿𝜋(𝜇 + 𝜔 − 𝜇𝜃) (1 − 𝜌)(𝜀 𝑄 + 𝛾)𝑄 + 𝜌𝑄 𝑄 + 𝑟 𝜀 (1 − 𝜌)𝛾 + 𝜌𝑄 𝜇(𝜇 + 𝜔)𝑄 𝑄 𝑄 𝑄 . therefore, 𝑅 = 𝛽𝛿𝜋(𝜇 + 𝜔 − 𝜇𝜃) ( )( ) ( ) ( ) . (26) 154 f.a. oguntolu, o.j. peter, k. oshinubi, t.a. ayoola, a.o. oladapo, m.m. ojo copyright ©2022 assa adv. in systems science and appl. (2022) the dfe is locally asymptotically stable since all the eigenvalues of (26) are negatives for 𝑅 < 1, hence the proof is established. □ 3. numerical method 3.1. homotopy perturbation method (hpm) the fundamental of homotopy perturbation method (hpm) was first proposed by [11]. the homotopy perturbation method, which provides analytical approximate solution, is applied to various linear and non-linear equations. [2, 18] used homotopy perturbation method to solve a susceptible-infected-recovered (sir) model of infectious diseases. the homotopy perturbation method is a series expansion method used in the solution of nonlinear partial differential equations [18]. to show the simple concepts of this method, we consider the following non-linear differential equation given as equations according to [2]. 𝐴 (𝑈) − 𝑓(𝑟) = 0, 𝑟 ∈ 𝛺 (27) subject to the boundary condition 𝐵 𝑈, = 0, 𝑟 ∈ 𝛤 (28) where a3 is a general differential operator, b3 a boundary operator, 𝑓(𝑟) is a known analytical function and γ is the boundary of the domain 𝛺. the operator a3 can be divided into two parts l and n, where l is the linear part, and n is the nonlinear part. equation (27) can be written as: 𝐿(𝑈) + 𝑁(𝑈) − 𝑓(𝑟) = 0, 𝑟 ∈ 𝛺. the homotopy perturbation structure is shown as follows 𝐻(𝑉, ℎ) = (1 − ℎ)[𝐿(𝑉) − 𝐿(𝑈 )] + ℎ[𝐴(𝑉) − 𝑓(𝑟)] = 0 (29) where 𝑉(𝑟, 𝑃): 𝛺 ∈ [0,1] → 𝑅. (30) in equation (29) 𝑃 ∈ [0,1] is an embedding parameter and 𝑈 is the approximation that satisfies the boundary condition. it can be assumed that the solution of the equation (30) can be written as power series in h given as equations (31) to (32): 𝑉 = 𝑉 + ℎ𝑉 + ℎ 𝑉 +. .. (31) and the best approximation for the solution is: 𝑈 = 𝑙𝑖𝑚 𝑣 = 𝑣 + ℎ𝑣 + ℎ 𝑣 + ⋯ ℎ → 1. (32) the series (31) is convergent for most cases. however, the convergent rate depends on the nonlinear operator a (v) 3.2. solution of the model equations using hpm from differential equations 1 to 7 + (𝜇 + 𝜆 )𝑆 − 𝜔𝑉 − 𝜋(1 − 𝜃) = 0, (33) + (𝜇 + 𝜔)𝑉 − 𝜃𝜋 = 0, (34) + (𝜎 + 𝜆 + 𝜇)𝐸 − 𝜆 𝑇 − 𝜆 𝑆 = 0, (35) analysis and dynamics of tuberculosis outbreak 155 copyright ©2022 assa. adv. in systems science and appl. (2022) + (𝛾 + 𝜇 + 𝑟 + 𝑑 )𝐼 − (1 − 𝜌)𝜎𝐸 − (1 − 𝜙)𝜆 𝐸 = 0, (36) + (𝜇 + 𝑑 + 𝑟 + 𝑟 )𝐼 − 𝜌𝜎𝐸 − 𝛾𝐼 − 𝜙𝜆 𝐸 = 0, (37) + (𝜇 + 𝑑 + 𝑟 )𝐽 − 𝑟 𝐼 = 0, (38) + (𝜇 + 𝜆 )𝑇 − 𝑟 𝐽 − 𝑟 𝐼 − 𝑟 𝐼 = 0. (39) with the initial condition given as 𝑆(0) = 𝑆 , 𝑉(0) = 𝑉 , 𝐸(0) = 𝐸 , 𝐼 (0) = 𝐼 , 𝐼 (0) = 𝐼 , 𝐽(0) = 𝐽 , 𝑇(0) = 𝑇 . (40) let 𝑆 = 𝑠 + ℎ𝑠 + ℎ 𝑠 +. . ., (41) 𝑉 = 𝑢 + ℎ𝑢 + ℎ 𝑢 +. . ., (42) 𝐸 = 𝑣 + ℎ𝑣 + ℎ 𝑣 +. . ., (43) 𝐼 = 𝑤 + ℎ𝑤 + ℎ 𝑤 + ⋯, (44) 𝐼 = 𝑥 + ℎ𝑥 + ℎ 𝑥 +. . ., (45) 𝐽 = 𝑦 + ℎ𝑦 + ℎ 𝑦 +. . ., (46) 𝑇 = 𝑧 + ℎ𝑧 + ℎ 𝑧 +. … (47) applying hpm into equation (33) (1 − ℎ) + ℎ + (𝜇 + 𝜆 )𝑆 − 𝜔𝑉 − 𝜋(1 − 𝜃) = 0. (48) substitute equation (41) and (42) into (48) (𝑠 + ℎ𝑠 + ℎ 𝑠 +. . . ) + ℎ (𝜇 + 𝜆 )(𝑠 + ℎ𝑠 + ℎ 𝑠 +. . . ) − 𝜔 (𝑢 + ℎ𝑢 + ℎ 𝑢 +. . . ) − 𝜋(1 − 𝜃) = 0. collecting the coefficient of power of h, we have ℎ : 𝑠 = 0, (49) ℎ : 𝑠 + (𝜇 + 𝜆 )𝑠 − 𝜔𝑢 − 𝜋(1 − 𝜃) = 0, (50) ℎ : 𝑠 + (𝜇 + 𝜆 )𝑠 − 𝜔𝑢 = 0. (51) from equation (49) 𝑠 = 0 , integrating both side 𝑠 = 𝐷 and applying initial condition 𝑠 (0) = 𝑆 = 𝑠 , 𝐷 = 𝑆 and 𝑠 = 𝑆 . from equation (51) 𝑠 = 𝜋(1 − 𝜃) + 𝜔𝑣 − (𝜇 + 𝜆 )𝑠 . integrate both sides and applying the initial condition we get: 𝑠 (𝑡) = (𝜋(1 − 𝜃) + 𝜔𝑣 − (𝜇 + 𝜆 )𝑠 )𝑡. (52) substitute equations (41) and (42) into (52) 𝑠 (𝑡) = (𝜋(1 − 𝜃) + 𝜔𝐸 − (𝜇 + 𝜆 )𝑆 )𝑡. (53) applying hpm to equation (34) (1 − ℎ) + ℎ + (𝜇 + 𝜔)𝑉 − 𝜋𝜃 = 0. (54) substitute equation (42) into (54) 156 f.a. oguntolu, o.j. peter, k. oshinubi, t.a. ayoola, a.o. oladapo, m.m. ojo copyright ©2022 assa adv. in systems science and appl. (2022) (𝑢 + ℎ𝑢 + ℎ 𝑢 +. . . ) + ℎ[(𝜇 + 𝜔)(𝑢 + ℎ𝑢 + ℎ 𝑢 +. . . ) − 𝜋𝜃] = 0. collecting the coefficient of power of ℎ, we have ℎ : 𝑢 = 0, (55) ℎ : 𝑢 + (𝜇 + 𝜔)𝑢 − 𝜋𝜃 = 0, (56) ℎ : 𝑢 + (𝜇 + 𝜔)𝑢 = 0. (57) from equation (56) 𝑢 = 𝜋𝜃 − (𝜇 + 𝜔)𝑢 . integrate both sides and applying the initial condition we have: 𝑢 (𝑡) = (𝜋𝜃 − (𝜇 + 𝜔)𝑢 )𝑡. (58) substitute equation (42) into (60) 𝑢 (𝑡) = (𝜋𝜃 − (𝜇 + 𝜔)𝑉 )𝑡. (59) from equation (51) 𝑠 = 𝜔𝑢 − (𝜇 + 𝜆 )𝑠 . (60) substitute equation (59) and (53) into (60) 𝑠 = 𝜔(𝜋𝜃 − (𝜇 + 𝜔)𝑉 )𝑡 − (𝜇 + 𝜆 )(𝜋(1 − 𝜃) + 𝜔𝐸 − (𝜇 + 𝜆 )𝑆 )𝑡, 𝑠 = 𝜔(𝜋𝜃 − (𝜇 + 𝜔)𝑉 ) − (𝜇 + 𝜆 )(𝜋(1 − 𝜃) + 𝜔𝐸 − (𝜇 + 𝜆 )𝑆 ) 𝑡. integrating both sides and applying the initial condition 𝑠 (𝑡) = 𝜔(𝜋𝜃 − (𝜇 + 𝜔)𝑉 ) − (𝜇 + 𝜆 )(𝜋(1 − 𝜃) + 𝜔𝐸 − (𝜇 + 𝜆 )𝑆 ) . (61) substitute the initial condition and equation (53) and (61) into (41) 𝑆(𝑡) = 𝑠 + ℎ𝑠 + ℎ 𝑠 +. . ., 𝑆(𝑡) = lim { → } (𝑠 + ℎ𝑠 + ℎ 𝑠 + ⋯ ), 𝑆(𝑡) = 𝑠 + 𝑠 + 𝑠 +. . ., hence, 𝑆(𝑡) = 𝑆 + (𝜋(1 − 𝜃) + 𝜔𝐸 − (𝜇 + 𝜆 )𝑆 )𝑡 + 𝜔(𝜋𝜃 − (𝜇 + 𝜔)𝑉 ) − (𝜇 + 𝜆 )(𝜋(1 − 𝜃) + 𝜔𝐸 − (𝜇 + 𝜆 )𝑆 ) 𝑡 2 . following the same process for other equations, we have: 𝑉(𝑡) = 𝑉 + (𝜋𝜃 − (𝜇 + 𝜔)𝑉 )𝑡 + (𝜇 + 𝜔)(𝜋𝜃 − (𝜇 + 𝜔)𝑉 ) 𝑡 2 , 𝐸(𝑡) = 𝐸 + 𝜆 𝑇 + 𝜆 𝑆 − (𝜎 + 𝜆 + 𝜇)𝐸 𝑡 + 𝜆 (𝑟 𝐽 + 𝑟 𝐼 + 𝑟 𝐼 − (𝜇 + 𝜆 )𝑇 ) + 𝜆 (𝜋(1 − 𝜃) + 𝜔𝐸 − (𝜇 + 𝜆 )𝑆 ) −(𝜆 + 𝜇 + 𝜎)(𝜆 𝑇 + 𝜆 𝑆 − (𝜎 + 𝜆 + 𝜇)𝐸 ) 𝑡 2 , 𝐼 (𝑡) = 𝐼 + (1 − 𝜌)𝜎𝐸 + (1 − 𝜙)𝜆 𝐸 − (𝛾 + 𝜇 + 𝑟 + 𝑑 )𝐼 𝑡 + (1 − 𝜌)𝜎 + (1 − 𝜃) (𝜆 𝑇 + 𝜆 𝑆 − (𝜎 + 𝜆 + 𝜇)𝐸 ) − (𝛾 + 𝑟 + 𝑑 + 𝜇) (1 − 𝜌)𝜎 + (1 − 𝜙)𝜆 𝐸 − (𝛾 + 𝜇 + 𝑟 + 𝑑 )𝐼 𝑡 2 , analysis and dynamics of tuberculosis outbreak 157 copyright ©2022 assa. adv. in systems science and appl. (2022) 𝐼 (𝑡) = 𝐼 + (𝜙𝜆 𝐸 + 𝜌𝜎𝐸 + 𝛾𝐼 − (𝜇 + 𝑑 + 𝑟 + 𝑟 )𝑥𝐼 )𝑡 + (𝜌𝜎 + 𝜙𝜆 )(𝜆 𝑇 + 𝜆 𝑆 − (𝜎 + 𝜆 + 𝜇)𝐸 ) + 𝛾 (1 − 𝜌)𝜎𝐸 + (1 − 𝜙)𝜆 𝐸 − (𝛾 + 𝜇 + 𝑟 + 𝑑 )𝐼 − (𝜇 + 𝑑 + 𝑟 + 𝑟 )(𝜙𝜆 𝐸 + 𝜌𝜎𝐸 + 𝛾𝐼 − (𝜇 + 𝑑 + 𝑟 + 𝑟 )𝑥𝐼 ) 𝑡 2 , 𝐽(𝑡) = 𝐽 + 𝑟 𝐼 − (𝜇 + 𝑑 + 𝑟 ) 𝐽 𝑡 + 𝑟 (𝜙𝜆 𝐸 + 𝜌𝜎𝐸 + 𝛾𝐼 − (𝜇 + 𝑑 + 𝑟 + 𝑟 )𝑥𝐼 ) −(𝜇 + 𝑑 + 𝑟 )(𝑟 𝐼 − (𝜇 + 𝑑 + 𝑟 )𝐽 ) 𝑡 2 , 𝑇(𝑡) = 𝑇 + (𝑟 𝐽 + 𝑟 𝐼 + 𝑟 𝐼 − (𝜇 + 𝜆 )𝑇 )𝑡 + ⎝ ⎜ ⎜ ⎜ ⎜ ⎛ 𝑟 (𝑟 𝐼 − (𝜇 + 𝑑 + 𝑟 )𝐽 ) + 𝑟 𝜙𝜆 𝐸 + 𝜌𝜎𝐸 + 𝛾𝐼 − (𝜇 + 𝑑 + 𝑟 + 𝑟 )𝑥𝐼 + 𝑟 (1 − 𝜌)𝜎𝐸 + (1 − 𝜙)𝜆 𝐸 − (𝛾 + 𝜇 + 𝑟 + 𝑑 )𝐼 −(𝜇 + 𝜆 ) 𝑟 𝐽 + 𝑟 𝐼 + 𝑟 𝐼 −(𝜇 + 𝜆 )𝑇 ⎠ ⎟ ⎟ ⎟ ⎟ ⎞ 𝑡 2 . 4. results 4.1. numerical parameters in this section, we give the values and source of the parameters used for simulating the model. we used the following initial values for 𝑆(𝑡) = 160,840,589, 𝐸(𝑡) = 1,700,000, 𝐼 (𝑡) = 90000, 𝐼 (𝑡) = 10,400,000, 𝐽(𝑡) = 1,000,000, 𝑇(𝑡) = 1,109,000 and 𝑉(𝑡) = 8,000,000. 𝑁 = 𝑆(𝑡) + 𝐸(𝑡) + 𝐼 (𝑡) + 𝐼 (𝑡) + 𝐽(𝑡) + 𝑇(𝑡) + 𝑉(𝑡) = 206,139,589. table 2. the parameters value used for the model parameters value source 𝜋 2,895,131 estimated 𝜇 0.018 estimated 𝑑 0.0365 assumed 𝑑 0.68 [21] 𝑑 0.1 [22] 𝑟 0.02 assumed 𝑟 0.02 assumed 𝑟 0.00375 [21] 𝜃 0.020 [22] 𝜌 0.075 [22] 𝜙 0.3 assumed 𝜆 0.2 assumed 𝜆 0.2 assumed 𝜎 0.01 assumed 4.2. graphical representation of solutions of the model equation the graphical representations are from the analytical solutions of the model equations. they are plotted using maple software. 158 f.a. oguntolu, o.j. peter, k. oshinubi, t.a. ayoola, a.o. oladapo, m.m. ojo copyright ©2022 assa adv. in systems science and appl. (2022) (a) (b) (c) (d) (e) (f) (g) fig. 2. (a) effect of waning rate of vaccine on the chronic class. (b) effect of waning rate of vaccine on vaccinated class. (c) effect of isolation rate on the isolated compartment. (d) treated individual against time for different values of recovery rate for chronic class. (e) effect of recovery rate for those in isolated class. (f) impact of effective interaction between susceptible and infected classes. (g) mild tb individual against time for different values of transmission rate. analysis and dynamics of tuberculosis outbreak 159 copyright ©2022 assa. adv. in systems science and appl. (2022) 5. discussion figure 2a is the graph of chronic infected individuals against time for different values of waning rate of vaccine. we carried out simulations by varying waning rate of vaccine as 0.2, 0.4 and 0.6. it could be observed that different level of waning rate does not have effect on the infected population. for different level of waning rate, the tb infection continues to persist in the given population. figure 2b is the graph of vaccinated individuals against time for different values of waning rate of vaccine. it was observed that the vaccinated population decreases with increase in the vaccination rate. therefore, high waning rate of vaccine reduces the vaccinated population and thus puts them at the risk of contracting the disease. figure 2c is the graph of isolated infectious individuals against time at different values of progression rate from chronic tb class to isolated infected class. it was observed that the number of isolated individuals increases as progression rate from chronic tb class to isolated infected class increases. figure 2d is the graph of treated tb individuals against time. it was observed that the number of treated tb individual increases as the recovery rate among the chronic tb individual increases. this implies that increase in the progression rate will lead to increase in number of individuals with chronic tb disease. figure 2e is the graph of recovered individual against time. the lower the treated rate the lower the number of recovered individuals. the lowest percentage almost decrease to zero. this shows that as the recovered are treated, they move to chronic tb population. figure 2f is the graph of exposed individual against time for different values of contact rate. we can observe that infected individuals increase as contact rate increases. the figure illustrates the great influence of effective contact rate on the exposed population. figure 2g is the graph of mild tb individual against time. it was observed that the number of mild tb individual increases as the transmission rate from the exposed to the chronic individuals increases. 6. conclusion in this study, a mathematical model of tuberculosis transmission dynamics incorporating treatment, isolation and vaccination using the system of first order ordinary differential equations was developed and analyzed. it was discovered that the model has two equilibria. the equilibrium states were obtained and analyzed for their stability relatively to the effective reproduction number. the result shows that, the disease-free equilibrium was stable. we are able to show that the tuberculosis infectious free equilibrium is locally and globally asymptotically stable if 𝑅 < 1 . the analytical solution was obtained using homotopy perturbation method and effective reproduction number was computed in order to measure the relative impact for individual or combined intervention for effective disease control. the graphs illustrate the impact of a combined effect of contact rate, waning rate of vaccine and rate of isolation. one can observe that this combines effects reduce the size of infected compartments. thus, the simultaneous increase of effectiveness of vaccination rate, isolation rate and treatment rate are effective control measures against tb infection. the model shows that the spread of tuberculosis infection depends largely on the contact rate, hence the ministry of health and other health workers should emphasize on the improvement in early detection of tuberculosis infection cases, so that transmission can be minimized. infected individuals should be isolated and treated immediately and individuals infected with tuberculosis should be given antiretroviral drugs immediately. in future work, we intent to incorporate optimal control strategy into the model for greater insight into the dynamics. 160 f.a. oguntolu, o.j. peter, k. oshinubi, t.a. ayoola, a.o. oladapo, m.m. ojo copyright ©2022 assa adv. in systems science and appl. (2022) references 1. abioye, a.i., peter, o.j., ogunseye, h.a., oguntolu, f.a., oshinubi, k. et al. (2021). mathematical model of covid-19 in nigeria with optimal control. results in physics, 28, 104598. 2. abubakar, s., akinwande, n.i., jimoh, o.r., oguntolu, f.a. & ogwumu, o.d. (2013). approximate solution of sir infectious disease model using homotopy perturbation method (hpm). pacific journal of science and technology, 14(2), 163–169. 3 al-darraji, h.a.a., altice, f.l. & kamarulzaman, a. (2016). undiagnosed pulmonary tuberculosis among prisoners in malaysia: an overlooked risk for tuberculosis in the community. tropical medicine and international health, 4(4), 401–871. 4. andrawus, j., eguda, f., usman, i., maiwa, s., dibal, i. et al. (2020). a mathematical model of a tuberculosis transmission dynamics incorporating first and second line treatment. journal applied science environment management, 24(5), 917–922. 5. asefa, a. & teshome, w. 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(2020). modelling and optimal control analysis of lassa fever disease. informatics in medicine unlocked, 20, 100419. 20 peter, o.j., qureshi, s., yusuf, a., al-shomrani, m., & idowu, a.a. (2021). a new mathematical model of covid-19 using real data from pakistan. results in physics, 24, 104098. 21 wang, m., fitzgerald, j.m. & richardson, k. (2011). is the delay in diagnosis of pulmonary tuberculosis related to exposure to fluoroquinolones or any antibiotic? union: int. journal of tuberculosis lung diseases, 15(8), 1062–1068. 22 world health organization (2019, august 25). tuberculosis fact sheet n 104. [online] available http://www.who.int/ mediacentre/factsheets/fs104/en/ index.html microsoft word article+text,+3)+integer+dea+and+flexible+factors adv syst sci appl 2025; 02; 1-10 published online at https://ijassa.ipu.ru. data envelopment analysis in the presence of fuzzy integer and flexible factors zahra moazenzadeh1, saber saati1*, reza farzipoor saen2, reza kazemi matin3, sevan sohraee1 1 dept. of mathematics, north tehran branch, islamic azad university, tehran, iran 2 faculty of business, sohar university, sohar, oman 3 dept. of mathematics, islamic azad university, karaj, iran abstract: in the traditional data envelopment analysis (dea) approaches, the inputs and outputs are usually regarded as exact and real values. decision making units’ (dmus’) relative efficiency is assessed and it is known that the factors are input and/or output. however, there are some conditions in which the dmus' efficiency should be calculated while the data is integer-valued and ambiguous. so, different integer dea models were proposed to determine the units’ performance when integer-valued data and fuzzy factors are present. furthermore, there are occasions which dmus’ efficiency score should be determined wherever integer-valued data and flexible factors are present. hence, some integer dea methods were suggested to calculate dmus’ performance and specify the role of flexible measures where some of the data are integer-valued and some of them are flexible factors. however, there are some situations that include integer data, fuzzy integer-valued measures and flexible factors. hence, the current paper sheds on the kind of model that evaluate the entities’ relative efficiency wherever integer-valued data, flexible factors and fuzzy integer-valued measures are present, and determines the role of factors with the uncertain input or output. keywords: dea; relative efficiency; slacks-based model; flexible factor; fuzzy integer-valued measure; integer-valued data. 1. introduction data envelopment analysis (dea) is a non-parametric methodology which assesses the performance of a set of comparable decision making units (dmus). the conventional dea models use the real-valued data and specify the roles of the data. nevertheless, we face the cases in real application that inputs and/or outputs are integer-valued and the role of data is unknown. the factors that can be either inputs or outputs are called “flexible factors”. cook and zhu [1] considered the flexible factors to assess the dmus’ relative efficiency and specified the role of flexible measures. amirteimoori, et al. [2] proposed a slacks-based measure with the flexible factors to evaluate the units’ performance. tohidi and matroud [3] introduced a method to define the role of flexible factors. lately, toloo, et al. [4] suggested a non-radial directional distance model to categorize the flexible measures. kordrostami, et al. [5] provided the methods to calculate the units’ efficiency scores where integer-valued data and flexible factors are present. lozano and villa [6] introduced models for examining integervalued data in dea. afterwards, kuosmanen and kazemi matin [7] provided a new axiomatic foundation for dea models which are integer-valued. jie, et al. [8] improved the kuosmanen and matin’s model [7] and showed the model truly solves problems. besides, due, et al. [9] provided methods that examine the slacks to estimate the relative efficiency and superefficiency scores of dmus where integer-valued data are present. furthermore, the traditional dea methods usually treat the data as continuous and exact measures. nevertheless, there are situations that entities’ performance should be estimated *corresponding author: s_saatim@iau-tnb.ac.ir 2 moazenzadeh z., saati s., saen r. f., matin r. k., sohraee s. copyright ©2025 assa adv. in systems science and appl. (2025) wherever integer-valued and imprecise data are present. models with fuzzy measures can be detected in dea context. fuzzy set was used in [10] for the first time. then, this theory was used to different problems; see [11,12,13] for more information. kordrostami, et al. [14] provided a number of models to assess the efficiency scores of dmus and to identify the roles of fuzzy flexible measures. saati and imani [15] suggested a procedure to categorize shared factors using fuzzy concept and they determined the role of shared factors. kordrostami, et al. [16] introduced the approaches to estimate dmus’ efficiency score wherever integer-valued data and fuzzy factors are present. however, yet there is no study considering integer-valued data, fuzzy integer-valued measures and flexible factors in the texts related to dea. that’s why, the current paper introduces the models to estimate dmus’ efficiency, and specify flexible factors’ role, where integer-valued measures, fuzzy integer-valued data and flexible factors are present. in section 2, the notations used in this paper are suggested and the main ideas of dea with integer data, fuzzy integer-valued model and flexible slacks-based model (fsbm) are noted. in section 3, a new model for calculation of efficiency with integer-valued measures, fuzzy integer-valued data and flexible factors is presented. in section 4, the example is presented. in section 5, conclusion is drawn. 2. preliminaries 2.1. notations suppose we deal with n dmus. symbols are introduced as follows: 𝑗 = 1, . . . , 𝑛: the set of dmus 𝑖 = 1, . . . , 𝑚: the set of inputs 𝑟 = 1, . . . , 𝑠: the set of outputs 𝑘 = 1,2, . . . , 𝐾: the set of flexible factors dmuj: the j-th unit, 𝑗 = 1, . . . , 𝑛 dmuo: the unit under consideration 𝑥 : the i-th input resource of j-th unit 𝑥 : the input resource i of dmuo 𝑦 : the output product r of dmuj 𝑦 : the output product r of dmuo 𝑧 : the k-th flexible factor of dmuj 𝑧 : the k-th flexible factor of dmuo 𝑠 : i-th input slacks for 𝑖 = 1, . . . , 𝑚 𝑠 : r-th output slacks for 𝑟 = 1, . . . , 𝑠 𝑔 ( ): flexible factor slacks as the input for 𝑘 = 1, . . . , 𝐾 𝑔 ( ): flexible factor slacks as the output for 𝑘 = 1, . . . , 𝐾 𝑑 ( ) , 𝑑 ( ): binary variables 𝜆 : intensity vectors of dmuj 𝐼 : the subset of inputs which are integer-valued 𝑂 : the subset of outputs which are integer-valued 𝐾 : the subset of flexible factors which are integer-valued 𝑥 : the i-th triangular fuzzy input of dmuj 𝑦 : the r-th triangular fuzzy output of dmuj 𝑥 : the i-th triangular fuzzy input of dmuo 𝑦 : the r-th triangular fuzzy output of dmuo data envelopment analysis in the presence… 3 copyright ©2025 assa. adv. in systems science and appl. (2025) 2.2. integer-valued dea suppose we deal with n dmus, dmuj (𝑗 = 1, . . . , 𝑛), with m input resources 𝑥 (𝑖 = 1, . . . , 𝑚) and s output products 𝑦 (𝑟 = 1, . . . , 𝑠). in the traditional dea methods, all data are regarded as real-valued measures. thus, the performance of units is measured while the reference points of units obtain values which are real. however, in many real worlds, some inputs and/or outputs can only be integer-valued measures. assume 𝑥 (𝑖 = 1, . . . , 𝑚) and 𝑦 (𝑟 = 1, . . . , 𝑠) are integer-valued data of dmuj (𝑗 = 1, . . . , 𝑛), thus some dea methods were developed and improved to get integer-valued projections for integer measures. the suggested model is used to estimate the entities’ performance where integer-valued data, fuzzy integer-valued factors and flexible measures are present. in the next subdivision, a fuzzy integer-valued number is determined. section 2.3 is similar to that of section 2.2 in kordrostami, et al. [16]. 2.3. main concepts of fuzzy integer-valued numbers let r be the set of real numbers and z be the set of integer numbers. definition 2.3.a. suppose 𝑢 : 𝑅 → [0,1] is a fuzzy set. it is called a fuzzy integer if its support is a closed integer interval (denoted as ⟨𝑢(0), �̄�(0)⟩) and satisfies the following: 1. 𝑢 is normal; i.e., there exists 𝑥 ∈ ⟨𝑢(0), �̄�(0)⟩ such that 𝑢(𝑥 ′) = 1, 2. 𝑢(𝑥 ) ≤ 𝑢(𝑥 ) for any 𝑥 , 𝑥 ∈ ⟨𝑢(0), 𝑥 ⟩ with 𝑥 ≤ 𝑥 , 3. 𝑢(𝑥 ) ≥ 𝑢(𝑥 ) for any 𝑥 , 𝑥 ∈ ⟨𝑥 , �̄�(0)⟩ with 𝑥 ≤ 𝑥 . note that an interval, which is closed and integer, is showed by ⟨𝑠 , 𝑠 ⟩ = {𝑥 ∈ 𝑍 : 𝑠 ≤ 𝑥 ≤ 𝑠 } for any 𝑠 , 𝑠 ∈ 𝑍 and 𝑠 ≤ 𝑠 . definition 2.3.b. suppose 𝑠 , 𝑠 , 𝑡 and 𝑡 ∈ 𝐼 with 𝑠 ≤ 𝑠 ≤ 𝑡 ≤ 𝑡 , and 𝑚, �̄� ∈ 𝑍. if the fuzzy set 𝑢 : 𝑅 → [0,1] is determined as: 𝑢(𝑥) = ⎩ ⎪⎪ ⎨ ⎪⎪ ⎧ 1, 𝑖𝑓 𝑥 ∈ ⟨𝑠 , 𝑡 ⟩; 𝑥 − 𝑠 𝑠 − 𝑠 , 𝑖𝑓 𝑥 ∈ 𝑚, 𝑠 ; 𝑡 − 𝑥 𝑡 − 𝑡 , 𝑖𝑓 𝑥 ∈ ⟨𝑡 , 𝑚⟩; 0, 𝑖𝑓 𝑥 ∈ 𝑚, 𝑚 . where 𝑠 ≤ 𝑚 ≤ 𝑠 and 𝑡 ≤ �̄� ≤ 𝑡 ; then 𝑢 is a trapezoidal fuzzy integer. a triangular fuzzy integer-valued number can obtain, if 𝑠 = 𝑡 . see kordrostami, et al. [16]. 2.4. dea models with fuzzy integer-valued measures in this part, the methods are suggested to assess the dmus’ performance wherever fuzzy integer-valued factors are present. assume there exist n units, that produce s outputs by consuming m inputs. the j-th unit showed by dmuj (𝑗 = 1, . . . , 𝑛), whose 𝑥 (𝑖 = 1, . . . , 𝑚) and 𝑦 (𝑟 = 1, . . . , 𝑠) are i-th input and r-th output, respectively. see kordrostami, et al. [16]. the following model, referred to as the ccr model, was introduced by charnes et al. [17] for estimating the entities’ relative efficiency with data that are precise and real numbers. 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝜃 𝑠𝑢𝑐 𝑡 𝑎𝑡 𝑦 ≤ 𝑦 𝜆 , 𝑟 = 1, . . . , 𝑠, (2.1) 4 moazenzadeh z., saati s., saen r. f., matin r. k., sohraee s. copyright ©2025 assa adv. in systems science and appl. (2025) 𝜃𝑥 ≥ 𝑥 𝜆 , 𝑖 = 1, . . . , 𝑚, 𝜆 ≥ 0, 𝑗 = 1, . . . , 𝑛; 𝜃 shows the efficiency score. 𝜆 (𝑗 = 1, . . . , 𝑛) indicate intensity vectors. in this model,𝑥 and 𝑦 are symbols of the inputs and outputs of dmuo, respectively. 𝑥 = (𝑥 , 𝑥 , 𝑥 ), 𝑦 = (𝑦 , 𝑦 , 𝑦 ) are inputs and outputs that are triangular fuzzy numbers. note that 𝑥 and 𝑦 are inputs and outputs of dmuo. the graded mean integration representation approach is used to calculate the dea models with fuzzy data, as follows: definition 2.4.a suppose 𝐴 = (𝑎, 𝑏, 𝑐) is a triangular fuzzy number, the graded mean integration representation𝐴 can be determined as (𝑎 + 4𝑏 + 𝑐)/6. in fact, abovementioned models are used because of the easiness and rational calculation. therefore, by considering definition 2.4.a, the ccr model with fuzzy factors can be changed with the model (2.2) as follows: 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝜃 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 1 6 (4𝑦 + 𝑦 + 𝑦 ) ≤ 1 6 4𝑦 + 𝑦 + 𝑦 𝜆 , 𝑟 = 1, … , 𝑠 ; 𝜃 6 (4𝑥 + 𝑥 + 𝑥 ) ≥ 1 6 4𝑥 + 𝑥 + 𝑥 𝜆 , 𝑖 = 1, … , 𝑚; 𝜆 ≥ 0, 𝑗 = 1, … , 𝑛. (2.3) definition 2.4.a will be true wherever integer-valued variables in the fuzzy linear programming are present. see [18,19,20] for more information. nevertheless, model (2.2) is not appropriate to assess dmus’ efficiency scores where fuzzy factors and integer-valued measures are present. indeed, as non-integer values may be the reference point of a dmu with integer. the aim of preparing model (2.2) is to compare its outcomes with the models with fuzzy factors and integer measures. 2.5. sbm model with flexible factor (fsbm) amirteimoori, et al. [2] provided the following model in terms of computing the dmus’ efficiency where the flexible measures are present: 𝜋∗ = 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 1 − (𝑚 + 𝑘) ∑ 𝑠 𝑥 + ∑ 𝑔 ( ) 𝑧 1 + (𝑠 + 𝐾) ∑ 𝑞 𝑦 + ∑ 𝑔 ( ) 𝑧 ; 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑥 = 𝜆 𝑥 + 𝑠 , 𝑖 = 1, … , 𝑚; 𝑦 = 𝜆 𝑦 − 𝑞 , 𝑟 = 1, … , 𝑠; 𝑧 = 𝜆 𝑧 + 𝑔 ( ) − 𝑔 ( ) , 𝑘 = 1, … , 𝐾; 𝑔 ( ) . 𝑔 ( ) = 0, 𝑘 = 1, … , 𝐾; 𝜆 , 𝑔 ( ) , 𝑔 ( ) , 𝑞 , 𝑠 ≥ 0, ∀𝑖, 𝑗, 𝑘, 𝑟. (2.3) data envelopment analysis in the presence… 5 copyright ©2025 assa. adv. in systems science and appl. (2025) aforementioned model evaluates the maximum of units’ efficiency scores and determines the role of the flexible measures. then, they changed model (2.3) into milp model (2.4) using charnes and cooper’s transformation [21] and some variables substitutions: 𝜋∗ = 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝜌 − (𝑚 + 𝑘) 𝑠 𝑥 + 𝑔 ( ) 𝑧 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝜌 + (𝑠 + 𝐾) 𝑞 𝑦 + 𝑔 ( ) 𝑧 = 1, 𝜌𝑥 = 𝜆 𝑥 + 𝑠 , 𝑖 = 1, … , 𝑚; 𝜌𝑦 = 𝜆 𝑦 − 𝑞 , 𝑟 = 1, … , 𝑠; 𝜌𝑧 = 𝜆 𝑧 + 𝑔 ( ) − 𝑔 ( ) , 𝑘 = 1, … , 𝐾; 0 ≤ 𝑔 ( ) ≤ 𝑀𝑑 ( ) , 0 ≤ 𝑔 ( ) ≤ 𝑀𝑑 ( ) , 𝑑 ( ) + 𝑑 ( ) = 1, 𝑘 = 1, … , 𝐾; 𝜆 , 𝑔 ( ) , 𝑔 ( ) , 𝑞 , 𝑠 ≥ 0, 𝑑 ( ) , 𝑑 ( ) ∈ {0,1}, ∀𝑖, 𝑗, 𝑘, 𝑟, (2.4) that 1 + (𝑠 + 𝐾) ∑ + ∑ ( ) ∧ {−1} = 𝜌 and 𝑀is a large positive number. unit o is efficient if and only if 𝜋∗ = 1. if 𝑔( ) = 𝑔 ( ) = 0, the factor which is flexible, can be input or output. 2.6. dea model with integer data the following model is a slacks-based nonlinear model with the integer data to analyze the dmus’ performance where integer-valued measures are present. in this part, a brief explanation of this model is provided. 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝜌 = 1 − (𝑚) ∑ 𝑠 𝑥∈ 1 + (𝑠) ∑ 𝑠 𝑦∈ 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝜆 𝑥 ≤ 𝑥 ∀𝑖, 𝑥 = 𝑥 − 𝑠 ∀𝑖, 𝜆 𝑦 ≥ 𝑦 ∀𝑟, 𝑦 = 𝑦 + 𝑠 ∀𝑟, 𝑦 ∈ 𝑍 ∀𝑟 ∈ 𝑂 . (2.5) in aforementioned formula, 𝑠 , 𝑠 show the non-radial slacks. 𝑥 ∈ 𝑍 , 𝑦 ∈ 𝑍 are the integer-valued projection points for inputs 𝐼 and outputs 𝑂 respectively. 𝜆 (𝑗 = 1, . . . , 𝑛) are the intensity vectors. 2.7. fsbm method with integer-valued data to assess the dmus’ relative efficiency where integer data and flexible measures are present, kordrostami et al. [5] suggested the following model: 6 moazenzadeh z., saati s., saen r. f., matin r. k., sohraee s. copyright ©2025 assa adv. in systems science and appl. (2025) 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑠 𝑥 ∉ + �̃� 𝑥 ∈ + 𝑔′( ) 𝑧 ∈ + �̄� ( ) 𝑧 ∉ + 𝑠 𝑦 ∉ + �̃� 𝑦 ∈ + 𝑔′( ) 𝑧 ∈ + �̄� ( ) 𝑧 ∉ 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑥 − 𝑠 = 𝜆 𝑥 , 𝑖 ∉ 𝐼 𝑦 + 𝑠 = 𝜆 𝑦 , 𝑟 ∉ 𝑂 , 𝑥 ≥ 𝜆 𝑥 , 𝑥 − �̃� = 𝑥 𝑖 ∈ 𝐼 , 𝑦 ≤ 𝜆 𝑦 , 𝑦 + �̃� = 𝑦 , 𝑟 ∈ 𝑂 , 𝑧 − �̄� ( ) + �̄� ( ) = 𝜆 𝑧 , 𝑘 ∉ 𝐾 , 0 ≤ �̄� ( ) ≤ 𝑀𝑑 ( ) , 0 ≤ �̄� ( ) ≤ 𝑀𝑑 ( ) , �̃� − 𝑔 ( ) + 𝑔 ( ) = 𝜆 𝑧 , 𝑧 − 𝑔′( ) + 𝑔′( ) = �̃� , 𝑘 ∈ 𝐾 , 0 ≤ 𝑔′( ) ≤ 𝑀𝑑 ( ) , 0 ≤ 𝑔′( ) ≤ 𝑀𝑑 ( ) , 0 ≤ �̑� ( ) ≤ 𝑀𝑑 ( ) , 0 ≤ 𝑔 ( ) ≤ 𝑀𝑑 ( ) , 𝑑 ( ) + 𝑑 ( ) = 1, 𝑥 , 𝑦 , �̃� ∈ 𝑧 , 𝑑 ( ) , 𝑑 ( ) ∈ {0,1}, 𝑖 ∈ 𝐼 , 𝑟 ∈ 𝑂 , 𝑘 ∈ 𝐾 , 𝜆 ≥ 0 𝑠 ≥ 0, 𝑖 ∉ 𝐼 , 𝑠 ≥ 0, 𝑟 ∉ 𝑂 , �̃� ≥ 0, 𝑖 ∈ 𝐼 , �̃� ≥ 0, 𝑟 ∈ 𝑂 , 𝑔′( ) , 𝑔′( ) ≥ 0, 𝑔 ( ) , 𝑔 ( ) ≥ 0, 𝑘 ∈ 𝐾 , �̄� ( ) + �̄� ( ) ≥ 0, 𝑘 ∉ 𝐾 , (2.6) 3. dea models with integer-valued data, flexible factors and fuzzy integer-valued measures suppose n dmus (i.e. dmuj (𝑗 = 1, . . . , 𝑛)) is present that produces s outputs (i.e. 𝑦 (𝑟 = 1, . . . , 𝑠)) by consuming m inputs (i.e. 𝑥 (𝑖 = 1, . . . , 𝑚)). the following model is provided to assess the dmus’ performance wherever integer-valued data, the flexible measures and fuzzy integer-valued factors are present and specify the role of flexible factors. data envelopment analysis in the presence… 7 copyright ©2025 assa. adv. in systems science and appl. (2025) 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 1 − (𝑚 + 𝐾) ∑ 𝑠 𝑥 + ∑ 𝑔 ( ) �̃� 1 + (𝑠 + 𝐾) ∑ 𝑠 𝑦 + ∑ 𝑔 ( ) �̃� 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝜆 𝑥 ≤ 𝑥 , 𝑥 = 𝑥 − 𝑠 ∀𝑖, 𝜆 𝑦 ≥ 𝑦 , 𝑦 = 𝑦 + 𝑠 ∀𝑟, 𝜆 �̃� = 𝑧 − 𝑔 ( ) + 𝑔 ( ) , 𝑘 ∈ 𝐾 , 𝑧 = �̃� − 𝑔 ( ) + 𝑔 ( ) , 𝑘 ∈ 𝐾 , 0 ≤ 𝑔 ( ) ≤ 𝑀𝑑 ( ) , 0 ≤ 𝑔 ( ) ≤ 𝑀𝑑 ( ) , 0 ≤ 𝑔 ( ) ≤ 𝑀𝑑 ( ) , 0 ≤ 𝑔 ( ) ≤ 𝑀𝑑 ( ) 𝑑 ( ) + 𝑑 ( ) = 1, 𝑥 , 𝑦 , 𝑧 ∈ 𝑧, 𝑖 ∈ 𝐼 , 𝑟 ∈ 𝑂 , 𝑘 ∈ 𝐾 , 𝑑 ( ) , 𝑑 ( ) ∈ {0,1} 𝑠 ≥ 0, 𝑠 ≥ 0, 𝑔 ( ) , 𝑔 ( ) , 𝑔 ( ) , 𝑔 ( ) ≥ 0, 𝜆 ≥ 0, 𝑥 ≥ 0, 𝑦 ≥ 0, ∀𝑖, 𝑗, 𝑟, 𝑘 (3.1) 𝜆 (𝑗 = 1, . . . , 𝑛) are intensity vectors. when imprecise data as triangular fuzzy numbers are present (i.e. 𝑥 = (𝑥 , 𝑥 , 𝑥 ), 𝑦 = (𝑦 , 𝑦 , 𝑦 ) in the model (3.1) that 𝑥 ≥ 0 and 𝑦 ≥ 0), a fuzzy model should be used to assess the units’ efficiency score. in above model 𝑥 and 𝑦 are inputs and outputs of dmuo . the data of dmuo are fuzzy integervalued measures. consider that in this case inputs and outputs of dmuo are showed by 𝑥 and 𝑦 , respectively. the graded mean integration representation method is used to calculate the dea models with fuzzy data and handle fuzzy factors. based on definition 2.4.a, the model with integervalued data, fuzzy integer-valued measures and flexible factors can be written in (3.2). in this formula, 𝜆 indicates intensity weights. 𝑀is a large positive number. 𝑥 , 𝑦 , 𝑧 are variables that are positive and integer-valued. they represent integer-valued projection points for data that are integer-valued. moreover, if 𝑔( ) > 0 then 𝑧 is considered as an input, and it will be an output if 𝑔( ) > 0. on the other hands if 𝑑( ) = 0, the flexible factor is called output and otherwise (i.e. 𝑑( ) = 0), it is considered as an input. in fact, fuzzy sets 𝑥 , 𝑦 and �̃� are replaced with (4𝑥 + 𝑥 + 𝑥 )/6, (4𝑦 + 𝑦 + 𝑦 )/6 and (4𝑧 + 𝑧 + 𝑧 )/6, respectively. the pessimistic and optimistic targets are used to show the fuzzy produced outputs and the fuzzy consumed inputs based on [20]; that is ,331221 ijijij xwxwxw  𝑤 𝑦 + 𝑤 𝑦 + 𝑤 𝑦 and 𝑤 𝑧 + 𝑤 𝑧 + 𝑤 𝑧 where 𝑤 + 𝑤 + 𝑤 = 1. as explained in [20], 𝑥 and 𝑦 are too optimistic and 𝑥 and 𝑦 are too pessimistic. therefore, we use the weights 𝑤 = 1/6, 𝑤 = 4/6and 𝑤 = 1/6 that can be substituted subjectively. thus, these boundary values provide us boundary solutions. notice that(𝑎, 𝑏, 𝑐) is a triangular integer fuzzy number, while (𝑎 + 4𝑏 + 𝑐)/6is gained as non-integer value, we will round it to the closest integer value. indeed, we consider ⌊(𝑎 + 4𝑏 + 𝑐)/6⌋ in terms of the effect of rounding (𝑎 + 4𝑏 + 𝑐)/6is almost insignificant. 8 moazenzadeh z., saati s., saen r. f., matin r. k., sohraee s. copyright ©2025 assa adv. in systems science and appl. (2025) 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 1 − 6 𝑚 + 𝐾 ∑ 𝑠 4𝑥 + 𝑥 + 𝑥 + ∑ 𝑔 ( ) 4𝑧 + 𝑧 + 𝑧 1 + 6 𝑠 + 𝐾 ∑ 𝑠 4𝑦 + 𝑦 + 𝑦 + ∑ 𝑔 ( ) 4𝑧 + 𝑧 + 𝑧 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 1 6 𝜆 (4𝑥 + 𝑥 + 𝑥 ) ≤ 𝑥 ∀𝑖, 𝑥 = 1 6 (4𝑥 + 𝑥 + 𝑥 ) − 𝑠 ∀𝑖, 1 6 𝜆 (4𝑦 + 𝑦 + 𝑦 ) ≥ 𝑦 ∀𝑟, 𝑦 = 1 6 (4𝑦 + 𝑦 + 𝑦 ) + 𝑠 ∀𝑟, 1 6 𝜆 (4𝑧 + 𝑧 + 𝑧 ) = 𝑧 − 𝑔 ( ) + 𝑔 ( ) , ∀𝑘, 𝑧 = 1 6 (4𝑧 + 𝑧 + 𝑧 ) − 𝑔 ( ) + 𝑔 ( ) , ∀𝑘, 0 ≤ 𝑔 ( ) ≤ 𝑀𝑑 ( ) , 0 ≤ 𝑔 ( ) ≤ 𝑀𝑑 ( ) , 0 ≤ 𝑔 ( ) ≤ 𝑀𝑑 ( ) , 0 ≤ 𝑔 ( ) ≤ 𝑀𝑑 ( ) , 𝑑 ( ) + 𝑑 ( ) = 1, 𝑑 ( ) , 𝑑 ( ) ∈ {0,1}, 𝑥 , 𝑦 , 𝑧 ∈ 𝑧, 𝑖 ∈ 𝐼 , 𝑟 ∈ 𝑂 , 𝑘 ∈ 𝐾 , 𝑠 ≥ 0, 𝑠 ≥ 0, 𝑔 ( ) , 𝑔 ( ) , 𝑔 ( ) , 𝑔 ( ) ≥ 0, 𝑥 ≥ 0, 𝑦 ≥ 0, 𝜆 ≥ 0, ∀𝑖, 𝑟, 𝑘, 𝑗. (3.2) table 1. data of an example dmu 𝑥 𝑥 𝑦 𝑦 flexible measure 1 280 182 (100,121,160) (160,182,195) (15,12,30) 2 370 280 (180,210,232) (145,156,160) (10,10,10) 3 230 124 (102,120,136) (150,175,190) (12,10,9) 4 430 210 (150,170,190) (50,60,70) (9,6,18) 5 325 122 (102,130,160) (280,286,293) (6,8,12) 6 315 240 (190,213,234) (72,85,94) (16,30,13) 7 253 170 (130,151,167) (260,275,286) (15,10,4) 8 305 185 (201,225,246) (76,87,93) (10,5,8) 9 245 129 (130,146,160) (230,242,251) (3,5,10) 4. numerical example suppose 9 dmu exist. inputs are numbers which are integer-valued and outputs are measures that are fuzzy integer-valued. the flexible measures are fuzzy integer-valued factors which are showen by triangular fuzzy numbers. the columns of table 1 indicate inputs and outputs. column 2 shows the first input and column 3 shows second input while columns 4 and 5 display outputs and column 6 indicates the flexible measures. our purpose is to assess the units’ efficiency score in the presence of the abovementioned measures. for defining the role of flexible measures and estimating the dmus’ efficiency, model (3.2) is used. the targets that are obtained from proposed method, can be seen in table 2. the efficiency score of model (3.2) data envelopment analysis in the presence… 9 copyright ©2025 assa. adv. in systems science and appl. (2025) are showen in column 2 from table 2. moreover, the scores are specified in columns 3,4,5 and 6. it is clear that the scores of integer data are assessed as non-integer scores. consider that firstly we have used one method to defuzzify the fuzzy numbers. secondly we have assessed the dmus’ performance. therefore, the reference points are evaluated as continuous and crisp values. based on the columns 5 and 6 of table 2, dmu 1, 3, 7 are regarded as inputs and the others are considered as outputs. consider that all integer-valued scores in model (3.2) have integer-valued projections. besides, unit 4 is the most inefficient dmu in model (3.2). however, it is reminded in lozano and villa [6] that rounding the real reference point may not be suitable. thus, apparently, the model proposed (model (3.2)) is reasonable for the conditions that all inputs and outputs are integer-valued numbers, fuzzy integer-valued factors and flexible measures. 5. conclusions in the conventional dea models, all data are usually regarded as exact and real numbers. classifying data is an important subject to analyze the performance. there are the situations in real application that the units’ efficiency score with integer-valued data, fuzzy integervalued measures and flexible factors should be evaluated. some studies have examined the units’ efficiency scores wherever integer-valued data, fuzzy integer-valued measures are present. furthermore, some researches studied the dmus’ efficiency where integer-valued data and flexible factors are present. this paper was suggested a slacks-based nonlinear programming problem to estimate the entities’ efficiency score and specify the roles of flexible factors in the presence of integer data, flexible factors and fuzzy integer-valued measures. the graded mean integration representation method was used on fuzzy data in terms of defuzzify them. an example was stated to describe and illustrate approaches. table 2. results of the proposed models dmu efficiency 𝑔 ∗( ) 𝑔 ∗( ) 𝑑 ∗( ) 𝑑 ∗( ) 1 0.644 4.5 0 1 0 2 0.524 0 5 0 1 3 0.738 4.167 0 1 0 4 0.279 0 4.5 0 1 5 0.852 0 4.667 0 1 6 0.395 0 4.167 0 1 7 0.862 2.833 0 1 0 8 0.463 0 4.667 0 1 9 0.846 0 4.5 0 1 references [1] cook, w. d. & zhu, j. (2007) classifying inputs and outputs in data envelopment analysis, eur. j. oper. res., 180, 692–699. [2] amirteimoori, a., emrouznejad, a. & khoshandam, l. (2013) classifying flexible measures in data envelopment snalysis: a slack-based measure, measurement, 46, 4100– 4107. [3] tohidi, g. & matroud, f. (2017) a new non-oriented model for classifying flexible measures in dea, j. oper. res. society, 68, 1019–1029. [4] toloo, m., allahyar, m. & hanclova, j. (2018) a non-radial directional distance method on classifying inputs and outputs in dea: application to banking industry, expert systems with applications, 92, 495–506. [5] kordrostami, s., amirteimoori, a. & noveiri, m. j. s. (2019) inputs and outputs classification in integer-valued data envelopment analysis, measurement, 139, 317–325. 10 moazenzadeh z., saati s., saen r. f., matin r. k., sohraee s. copyright ©2025 assa adv. in systems science and appl. (2025) [6] lozano, s. & villa, g. (2006) data envelopment analysis of integer-valued inputs and outputs, comput. oper. res., 33, 3004–3014. [7] kuosmanen, t. & matin, r. k. (2009) theory of integer-valued data envelopment analysis, eur. j. oper. res., 192, 658–667. [8] jie, t., yan, q. & xu, w. (2015) a technical note on “a note on integer-valued radial model in dea”, computers & industrial engineering, 87, 308–310. [9] due, j., chen, c. m., chen, y., cook, w. d. & zhu, j. (2012) additive superefficiency in integer-valued data envelopment analysis, eur. j. oper. res., 218, 186–192. [10] zadeh, l. a. (1965) fuzzy sets, inf. control., 8, 338–353. [11] bellman, r. e. & zadeh, l. a. (1970) decision-making in a fuzzy environment, manag. sci., 17, 141–164. [12] wang, y. m. & chin, k. s. (2011) fuzzy data envelopment analysis: a fuzzy expected value approach, expert syst. appl., 38, 11678–11685. [13] hatami-marbini, a., agrell, p. j., tavana, m. & khoshnevis, p. (2017) a flexible cross-efficiency fuzzy data envelopment analysis model for sustainable sourcing, j. clean. prod., 142, 2761–2779. [14] kordrostami, s., farajpour, g. & noveiri, m. j. s. (2014) evaluating the efficiency and classifying the fuzzy data: a dea based approach, int. j. industrial mathematics, 6(4), 321–327. [15] saati, s. & imani, n. (2015) classifying flexible factors using fuzzy concept, journal of new researches in mathematics, 1(2), 35–46. [16] kordrostami, s., amirteimoori, a. & noveiri, m. j. s. (2018) fuzzy integer-valued data envelopment analysis, rairo. oper. res., 52, 1429–1444. [17] charnes, a., cooper, w. w. & rhodes, e. (1978) measuring the efficiency of decision-making units, eur. j. oper. res., 2, 429–444. [18] herrera, f. & verdegay, j. l. (1995) three models of fuzzy integer linear programming, eur. j. oper. res., 83, 581–593. [19] lai, y. j. & hwang, c. l. (1992) fuzzy mathematical programming. new york, ny: springer-verlag. [20] wang, l. x. (1997) a course in fuzzy systems and control. london, uk: prenticehall. [21] charnes, a. & cooper, w. w. (1962) programming with linear fractional functionals, naval research logistics quarterly, 9, 181–186. adv syst sci appl 2022; 04:103–115 published online at https://ijassa.ipu.ru. binary opinion space in the scardo model vladislav n. gezha2, ivan v. kozitsin1,2* 1 v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia 2 moscow institute of physics and technology (state university), dolgoprudny, russia abstract: the paper presents the analysis of the scardo model in the case of the binary opinion space. the model itself and the conditions under which the analysis is performed are described and discussed in details. analytical solutions are found for the mean field approximation. the fixed points as well as their stability properties are characterized. furthermore, we precisely describe the hyperplane in the parameter space that defines which opinion will gather more supporters. extensive computational experiments are performed to demonstrate the applicability of our theoretical results. experiments suggest that the mean-field approximation can fairly accurately predict the behavior of the model provided there is a nonzero probability of anticonformity-type opinion changes. at the end of the paper, we outline the possible direction for future studies. keywords: opinion dynamics models, mean-field approximation, differential equations 1. introduction to mitigate the spread of fake news, stop opinion polarization, and convince citizens to wear face masks and vaccinate during the covid-19 pandemic [19], it is essential to understand key mechanisms that underlie the processes of opinion formation [4, 7, 11]. however, as was pointed out in ref. [18],”... understanding human behavior remains a grand challenge across disciplines.” in recent decades, the way information flows between individuals has dramatically changed from one-directional messages that come from the mass media and political elites upon citizens (one-step information paths) to complex communications whereby even ordinary people can play a crucial role and affect the macroscopic properties of the social system. this issue has become possible due to proliferation of online social networks that provide a low-cost way of communication without social, geographical, and economic borders. in online networks, complex information paths can be established, with one or more users serving as intermediaries between mass media and other people. a promising way to analyze these social dynamics is to use agent-based opinion formation models (aka social influence models) [22]. stipulating how agents (artificial entities that model real people or mass media) communicate with each other and how their opinions evolve following these interactions, scholars can analyze the outcomes that this or that opinion formation mechanism will lead to [2]. in this paper, we advance the recently introduced opinion formation model [13]. in the following we will refer to it as the stochastic conditional arranged discrete opinions model, abbreviated as the scardo model. in the scardo model, the agents update their opinions within a probabilistic sequential process whereby two randomly chosen neighboring ∗corresponding author: kozitsin.ivan@mail.ru 104 v.n. gezha, i.v. kozitsin agents (the influence recipient and the influence source) communicate at each iteration. in these local consecutive interactions, the agents’ new opinions are determined stochastically, according to a probability distribution, which is a function of the recipient’s and the source’s opinions (opinions are encoded by discrete variables). assuming that there are m possible opinion values, one can exhaustively describe opinion dynamics processes in the model via m3 probabilities standing for all possible pairwise interactions and outcomes. together, these quantities form what the author of [13] calls the transition matrix. however, strictly speaking, this mathematical 3-d construction is not a matrix. for this reason, in the current paper, we adopt a different terminology and call it by “transition table”. by tuning the values of the transition table, one can use the scardo model to approximate a broad range of opinion formation models, including those with discrete opinions [3, 9, 16] and those with continuous ones [5, 8, 17] (to achieve the latter purpose, one should assume that m is large enough). on this occasion, the scardo model can be used to compare opinion formation models against each other in terms of the transition table. further, the scardo model can be relatively easily calibrated on real data upon which it puts forward only a few requirements. thus, the scardo model can act as a link between theoretical opinion formation model and empirical studies of social dynamics processes [13]. in ref. [13], for the scardo model, a mean-field approximation was derived. this approximation describes successfully the dynamics of the populations of m opinion camps with the system of ordinary differential equations. the system was thoroughly investigated under empirically-calibrated settings (with the transition table calibrated on the data from refs. [12, 14]) for different configurations of the opinion spectrum m = 2,m = 3, and m = 10. in this paper, we apply a different strategy and focus on a low-dimensional case of this model (m = 2), for which analytical derivations are feasible and a general description of the mean-field approximation system of equations can be obtained for any values of the models’ parameters. for this low-dimensional case, we find exact solutions of the mean-field equations, characterize fixed points, and investigate their stability properties. we exemplify our analytical analysis with computational simulations. the rest of the paper is organized as follows. section 2 provides the description of the scardo model. in section 3, we present the mean-field approximation derived for this model. section 4 presents the main results of the paper. subsection 4.2 obtains and analyzes the solutions of the mean-field equations in the case of the binary opinion space while subsection 4.3 complemente these results with illustrative examples that compare theoretical predictions against simulation data. section 5 ends the paper with concluding remarks. 2. opinion dynamics model in this section, we present the scardo model that was first introduced in ref. [13]. in the model, n agents communicate via a (by default, static) connected (undirected and unweighted) social network g and discuss a socially important topic [21]. the time in the model is discrete. at each iteration, one chooses a random agent i (by default, all agents have the equal probability 1/n to be chosen). then, one selects at random one of their friends j in the network g (again, according to the uniform probability distribution). then, the agent j (influence source) influences the agent i (recipient) during a one-directional influence procedure. as a result, the recipient’s opinion changes, depending on the current opinions of the agents i and j. agent opinions (denoted by o) belong to a discrete opinion space with m elements: x = {x1, . . . , xm}. the elements x1, . . . , xm represent an opinion alphabet of the model and may encode a broad spectrum of opinion systems, such as competing alternatives (as in the voter model copyright © 2022 assa. adv syst sci appl (2022) binary opinion space in the scardo model 105 [16]) or arranged attitudes that may approximate continuous opinion spaces for large m—see ref. [1]: x1 ≺ ... ≺ xm. after receiving influence from j’s opinion oj(t) = xl, the agent i’s opinion oi(t) = xs may change or may remain the same. the outcome of the communication is determined stochastically, according to the following probability distribution: {ps,l,1, . . . , ps,l,m}, with the element ps,l,k standing for the probability of selecting the opinion xk. one can think of this quantity as the conditional probability that is defined as follows: ps,l,k = pr {oi(t+ 1) = xk | oi(t) = xs, oj(t) = xl}. (2.1) stuck together, these conditional probabilities are the elements of a 3-d object p = [ps,l,k] m s,l,k=1 , which is called the transition table [13]. note that this table can be safely expressed as a list of matrices: p1 = [ p1,1,1 . . . p1,1,m . . . . . . . . . p1,m,1 . . . p1,m,m ] , . . . , pm = [ pm,1,1 . . . pm,1,m . . . . . . . . . pm,m,1 . . . pm,m,m ] (2.2) within these shorthands, p1, . . . , pm are 2-d matrices with m rows and m columns that encode opinion change strategies of individuals espousing opinions x1, . . . , xm correspondingly. note that all these m matrices are row-stochastic because ps,l,1 + . . .+ ps,l,m = 1 for each s and l. 3. mean-field approximation for the model presented in the previous section, the following mean-field approximation was obtained in ref. [13]. first, let us denote the population of agents espousing the opinion xs at a time moment t by ys(t) : ys(t) = {j | oj(t) = xs}. correspondingly, the fraction of such individuals among the whole population is symbolized by ys(t) = ys(t)/n. using the scaled time τ, which is defined as τ = t/n, δτ = 1/n, one can ends up with the following system of differential equations (see ref. [13] for details):  dy1(τ) dτ = ∑m s,l=1 ys(τ)yl(τ)ps,l,1 − y1(τ), . . . dyi(τ) dτ = ∑m s,l=1 ys(τ)yl(τ)ps,l,i − yi(τ), . . . dym(τ) dτ = ∑m s,l=1 ys(τ)yl(τ)ps,l,m − ym(τ), (3.3) that is, from the micro-level description of the model we come to the macro-level [20]. one can equip system (3.3) with the initial condition copyright © 2022 assa. adv syst sci appl (2022) 106 v.n. gezha, i.v. kozitsin y1(0) = q1, . . . , ym(0) = qm (3.4) whereby q1 + . . .+ qm = 1 and all initial values are no less than zero. the cauchy problem (3.3), (3.4) features the following property (see [13]). corollary 3.1: the function u = y1(τ) + . . .+ ym(τ) is the first integral of the autonomous system (3.3). remember that the quantities y1, . . . , ym stand for the populations of all the opinion camps x1, . . . , xm and thus these variables should be nonnegative and sum up to one. the following theorem ensures that solutions of (3.3) meet this property (see [15] for proofs). theorem 3.1: assume that q1 ≥ 0, . . . , qm ≥ 0 and q1 + . . .+ qm = 1. then the cauchy problem (3.3), (3.4) has a unique solution y⃗(τ), which can be extended on the whole τ -axis (in fact, we are interested only in τ ≥ 0). the components of y⃗(τ) are nonnegative and sum up to one for each τ. theorem 3.1 indicates that the mean-field approximation is a reliable mathematical construction which gives the outputs consistent with the physical definitions of the models’ parameters. 4. binary opinion space 4.1. problem formulation let us now concentrate on the case of the binary opinion space m = 2. in this situation, the table matrix can be safely presented by only two row-stochastic matrices: p1 = [ p1,1,1 p1,1,2 p1,2,1 p1,2,2 ] , p2 = [ p2,1,1 p2,1,2 p2,2,1 p2,2,2 ] . for now, the mean-field predictions (3.3) will have the following form: { dy1 dτ = p1,1,1y 2 1 + (p1,2,1 + p2,1,1)y1y2 + p2,2,1y 2 2 − y1, dy2 dτ = p1,1,2y 2 1 + (p1,2,2 + p2,1,2)y1y2 + p2,2,2y 2 2 − y2 or { dy1 dτ = −p1,1,2y 2 1 + (p2,1,1 − p1,2,2)y1y2 + p2,2,1y 2 2, dy2 dτ = p1,1,2y 2 1 − (p2,1,1 − p1,2,2)y1y2 − p2,2,1y 2 2 (4.5) for the sake of simplicity, let us introduce notations α = p1,1,2, β = p1,2,2, γ = p2,1,1, and δ = p2,2,1. within this notation strategy, α and β represent the probabilities of anticonformitytype opinion changes for both the opinion camps (note that anticonformity refers to the situation when individuals change opinions after interacting with those holding similar positions [16]). in turn, β and γ represent the probabilties of the conformity-type opinion shuffles. these four variables represent the transition table exhaustively: p1 = [ 1− α α 1− β β ] , p2 = [ γ 1− γ δ 1− δ ] . (4.6) one of the equations in (4.5) is redundant. after making use of y2 = 1− y1 and applying encoding strategy (4.6), one can end up with the following differential equation: copyright © 2022 assa. adv syst sci appl (2022) binary opinion space in the scardo model 107 dy1 dτ = (δ − α− θ)y21 + (θ − 2δ)y1 + δ, where θ = γ − β. for simplicity, we denote y = y1 and focus on the analysis of the following differential equation: dy dτ = (δ − α− θ)y2 + (θ − 2δ)y + δ, (4.7) 4.2. equation (4.7): fixed points, solutions, and stability let us now concentrate on solving equation (4.7). for this equation, the fixed points can be found from the following quadratic trinomial: (δ − α− θ)y2 + (θ − 2δ)y + δ = 0. the discriminant of this equation is d = θ2 + 4δα ≥ 0, and one can easily obtain the following expression for fixed points: y∗−,+ = 2δ − θ ± √ θ2 + 4δα 2(δ − α− θ) . (4.8) depending on how θ, δ, and α, the denominator in (4.8) may be positive, negative, or may be equal to zero. let us consider these three situations separately. case 1. let θ = δ − α. in this case, equation (4.7) simplifies to y′ = −(δ + α) · y + δ. (4.9) thus, we end up with the following equation on the fixed points : −(δ + α) · y∗ + δ = 0. (4.10) taking into account the non-negativity of the parameters δ and α we obtain that for α = δ = 0 equation (4.10) has infinitely many solutions: any y∗ that lies in the interval [0; 1] is a fixed point. for other values of α and δ, we have one fixed point given by the expression y∗ = δ δ + α . if α = δ = 0, equation (4.9) turns out to be y′ = 0, i.e. the solution of the cauchy problem is y = y(0). if (α, δ) ̸= (0, 0), then we obtain dy −(δ + α) · y + δ = dτ. after integrating this equality, we obtain the solution of the cauchy problem y = δ − c · exp−(α + δ)τ δ + α , where c is determined from the initial condition y(0) = δ−c δ+α , whence c = δ − (δ + α) · y(0). copyright © 2022 assa. adv syst sci appl (2022) 108 v.n. gezha, i.v. kozitsin case 2. let θ ∈ [−1; δ − α). in this case, equation (4.7) has the form y′ = (δ − α− θ)(y − y∗−)(y − y∗+), (4.11) where y− and y+ are defined in (4.8). for fixed points, we have the following equation: (δ − α− θ)(y∗ − y∗−)(y ∗ − y∗+) = 0. (4.12) let’s see in which cases the solutions of (4.12) satisfy the condition of staying within [0; 1]. the denominator of (4.8) is positive, and we can rewrite the condition 0 ≤ y∗−,+ ≤ 1 as 0 ≤ 2δ − θ ± √ θ2 + 4δα ≤ 2(δ − α− θ) (4.13) consider the inequality on the left in (4.13). for the sign ”+” it is true because the root is positive and the inequality 2δ − θ > δ + α ≥ 0, where θ is estimated using the initial condition, is true. we proceed to the inequality 2δ − θ ≥ √ θ2 + 4δα. (4.14) for the sign ”−” . since its left side is always positive, we square both sides and come to 4δ · (δ − θ − α) ≥ 0. applying the initial condition to θ we obtain that the inequality is true. consider the right inequality in (4.13): for the sign ”−” we pass to the inequality 2α + θ ≤ √ θ2 + 4δα. (4.15) note that the right side is positive, and therefore, squaring both sides, we arrive at a more strict inequality: 4α · (α + θ − δ) ≤ 0. nevertheless, it is still satisfied, because α ≥ 0, and θ ≤ δ − α from the initial condition. and since the more strict condition is satisfied, the original condition is also true. for the sign ” + ” we get 2α + θ + √ θ2 + 4δα ≤ 0, which is only possible for θ ≤ −2α. let’s take this fact into account and move 2α + θ to the right. after that, we square the inequality. the resulting inequality is satisfied only if α = 0, θ ≤ 0. thus, there is a unique fixed point in the zone θ ∈ [−1; δ − α) given by the following expression: y∗ = 2δ − θ − √ θ2 + 4δα 2(δ − α− θ) . (4.16) copyright © 2022 assa. adv syst sci appl (2022) binary opinion space in the scardo model 109 further, if the condition α = 0, θ ≤ 0 is satisfied, then the second fixed point arises – y∗ = 1. let us now focus on solving equation (4.11). we rewrite it as dy (δ − α− θ)(y − y∗−)(y − y∗+) = dτ. we need to mention the case when y∗− = y∗+ which happens when θ2 + 4αδ = 0. due to parameter conditions, it occurs only if θ = 0, α = 0, δ > 0. in that case, the equation has the following form y′ = δ · (y − 1)2 with a solution y = 1− 1 δτ + c , whence c = 1 1−y(0) . it is worth pointing out that this notation is correct only in case of y(0) < 1. in case of y(0) = 1 the solution simply stays for y(0) = 1. after that, we end up with − 1√ θ2 + 4δα · (y − y∗−) + 1√ θ2 + 4δα · (y − y∗+) = dτ after integrating both parts and simplifying them, we get ln | y − y∗+ y − y∗− | = √ θ2 + 4δα · (τ + c), whence, after conversion, we get y = y∗+ − c · exp ( √ θ2 + 4δα · τ) · y∗− 1− c · exp ( √ θ2 + 4δα · τ) , (4.17) in turn, c can be easily found from the initial condition: y(0) = y∗+−c·y∗− 1−c , whence c = y∗+−y(0) y∗−−y(0) . it is worth pointing out that this notation is correct only in case of y(0) ̸= y∗−. in case of y(0) = y∗− the solution simply stays for y(0) = y∗−. case 3. it is simple to demonstrate that the only fixed point in the zone θ ∈ (δ − α; 1] is y∗ = 2δ − θ − √ θ2 + 4δα 2(δ − α− θ) . in the case δ = 0, θ ≥ 0, the second fixed point y∗ = 0 appears. it is also easy to show that the solution of the differential equation (4.7) in the zone θ ∈ (δ − α; 1] is y = y∗+ − c · exp ( √ θ2 + 4δα · t) · y∗− 1− c · exp ( √ θ2 + 4δα · τ) , (4.18) copyright © 2022 assa. adv syst sci appl (2022) 110 v.n. gezha, i.v. kozitsin where c is determined from the initial conditions as c = y∗+−y(0) y∗−−y(0) . it is worth pointing out that this notation is correct only in case of y(0) ̸= y∗−. in case of y(0) = y∗− the solution simply stays for y(0) = y∗−. we also need to mention the case when y∗− = y∗+ which happens when θ2 + 4αδ = 0. due to parameters conditions it occurs only if θ = 0, α > 0, δ = 0. in that case, the equation has the following form y′ = −α · y2 with a solution y = 1 ατ + c whence c = 1 y(0) from the initial solution. it is worth pointing out that this notation is correct only in case of y(0) > 0. in case of y(0) = 0 the solution simply stays for y(0) = 0. table 4.1 summarizes our findings. note that stability properties of the fixed points can be easily obtained given we know how the solution of the cauchy problem looks like in each particular situation. as one can note from table 4.1, if there is only one fixed point, then it is always a global attractor. if there are two of them, then one fixed point is always unstable and the other one is a global attractor excluding the case the initial point is exactly the unstable fixed point. if θ = δ = α = 0, then something unusual appears. in this case, we end up with the following configuration of the transition table (note that β = γ because θ = β − γ): p1 = [ 1 0 1− β β ] , p2 = [ β 1− β 0 1 ] . in other words, we have symmetric opinion dynamics with no anticonformity. the meanfield prediction says that in such settings, the opinion camps have no advantage over each other, so the populations of the camps should not change. from the fixed points depicted in table 4.1, one can easily compute the outcome of the competition between opinions x1 and x2. we will focus on three scenarios: (i) opinion x1 obtains a total victory (y∗ = 1—recall that within our notations, the variable y depicts the proportion of the opinion x1’s backers); (ii) a draw when both the opinions have the same number of supporters (y∗ = 1/2); (iii) opinion x2 obtains a total victory (y∗ = 0). in table 4.2, we depict how the outcomes of the competition depend on the transition table (we do not consider the trivial case θ = δ = α = 0). among other things, from table 4.2 one can conclude that a draw between competing opinions is ensured by the equality of θ = α− δ or γ − β = α− δ. (4.19) intuitively, equation (4.19) is quite clear because the quantity γ − β describes the advantage of the opinion x1 over the opinion x2 with respect to how these opinions attract the individuals via the conformity mechanism whereas α− δ in (4.19) demonstrates the relative outcome of x1-backers caused by anticonformity. therefore, if the right side of (4.19) outweighs the left one, then we should expect that x2 wins. correspondingly, if γ − β > α− δ, then x1 will have more supporters at the equilibrium for nearly all initial points. copyright © 2022 assa. adv syst sci appl (2022) binary opinion space in the scardo model 111 table 4.1. equilibrium points and exact solutions parameters fixed points exact solution (θ = δ − α) & (α = δ = 0) – or, simply, θ = δ = α = 0 y∗ is any number (in [0; 1]) – lyapunov stable y = y(0) (θ = δ − α) & ¬ (α = δ = 0) y∗ = δ δ+α – asymptotic stable (global attractor) y = δ−c·exp (−(α+δ)τ) δ+α , where c = δ − (δ + α) · y(0) (θ < δ − α) & (α = 0, θ = 0) y∗ = 1 – asymptotic stable (global attractor) if y(0) ̸= 1, then y = 1− 1 δτ+c , where c = 1 1−y(0) . if y(0) = 1, then y = 1 (θ < δ − α) & (α = 0, θ < 0) y∗ = 1 – unstable, y∗ = δ (δ−θ) – asymptotic stable (global attractor excepting the initial point y(0) = 1) if y(0) ̸= y∗ −, then y = y∗ +−c·exp (−θ·τ)·y∗ − 1−c·exp (−θ·τ) , where c = y∗ +−y(0) y∗ −−y(0) . if y(0) = y∗ − : y = y∗ − (θ < δ − α) & ¬ (α = 0, θ ≤ 0) y∗ = y∗ − – asymptotic stable (global attractor) if y(0) ̸= y∗ −, then y = y∗ +−c·exp ( √ θ2+4δα·τ)·y∗ − 1−c·exp ( √ θ2+4δα·τ) , where c = y∗ +−y(0) y∗ −−y(0) . if y(0) = y∗ − : y = y∗ − (θ > δ − α) & (δ = 0, θ = 0) y∗ = 0 – asymptotic stable (global attractor) if y(0) ̸= 0, then y = 1 ατ+c , where c = 1 y(0) . if y(0) = 0 : y = 0 (θ > δ − α) & (δ = 0, θ > 0) y∗ = 0 – unstable, y∗ = θ (α+θ) – asymptotic stable (global attractor excepting the initial point y(0) = 0) if y(0) ̸= y∗ − : y = y∗ +−c·exp (θ·τ)·y∗ − 1−c·exp θ·τ) , where c = y∗ +−y(0) y∗ −−y(0) . if y(0) = y∗ −, then y = y∗ − (θ > δ − α) & ¬ (δ = 0, θ ≥ 0) y∗ = y∗ − – asymptotic stable (global attractor) if y(0) ̸= y∗ −, then y = y∗ +−c·exp ( √ θ2+4δα·τ)·y∗ − 1−c·exp ( √ θ2+4δα·τ) , where c = y∗ +−y(0) y∗ −−y(0) . if y(0) = y∗ − : y = y∗ − 4.3. experimental data compared with theoretical results we complement our theoretical results with computational experiments with the scardo model. in this section, we present some ideal-typical simulation runs, with a special focus on those cases where theoretical results and experimental data can be very different from each other (see figure 4.1). copyright © 2022 assa. adv syst sci appl (2022) 112 v.n. gezha, i.v. kozitsin table 4.2. the outcome of the opinion competition as a function of transition table organization parameters total victory of x1 (y∗ = 1) draw (y∗ = 1/2) total victory of x2 (θ = δ − α) & ¬ (α = δ = 0) α = 0 θ = 0 δ = 0 (θ < δ − α) & (α = 0, θ = 0) any transition table none none (θ < δ − α) & (α = 0, θ < 0) y(0) = 1 (θ = −δ) & (y(0) ̸= 1) (δ = 0) & (y(0) ̸= 1) (θ < δ − α) & ¬ (α = 0, θ ≤ 0) α = 0 θ = α− δ δ = 0 (θ > δ − α) & (δ = 0, θ = 0) none none any transition table (θ > δ − α) & (δ = 0, θ > 0) (α = 0) & (y(0) ̸= 0 ) (θ = α) & (y(0) ̸= 0 ) y(0) = 0 (θ > δ − α) & ¬ (δ = 0, θ ≥ 0) α = 0 θ = α− δ δ = 0 first, we should say that the equilibrium points obtained within the framework of meanfield approximation are not necessarily of the same status from the perspective of the model’s dynamics. to be more specific, only states y = 0 and y = 1 can be pure equilibriums, and this occurs if and only if δ = 0 or α = 0 correspondingly. otherwise, if δ ̸= 0 and α ̸= 0, the system cannot reach an equilibrium. we report that if the transition table has no anticonformity (α > 0 and δ > 0), then the model behavior is nearly in agreement with the theoretical predictions (see panels c, d, and f) – the populations of opinion camps fluctuate around theoretical solutions, with the oscillation magnitude disappearing if n → ∞. note that in this case, the system cannot reach a pure equilibrium state because there is always a nonzero probability that the populations of the opposing opinion camps will change in the next few moments. further, we report that in this case the theoretical predictions are still valid even if considering structured populations that are characterized by non-complete social graphs. note that the mean-field predictions were derived under assumption that the underlying network is a complete graph whereby each two agents have the same probability of interaction. however, our numerical simulations indicate that this assumption is redundant if considering transition tables with non-zero anticonformity effects. if, however, at least one of the anticonformity parameters is equal to zero, then the model features deviations from the mean-field predictions (see panels a, b, and e). in some situations, these deviations can be understood as oscillations around the theoretically predicted solutions again (panel a). however, the same transition table configurations can result in qualitively different behaviors – see panels a and b. instead of panel a, on panel b we see that the system substantially deviates from the theoretical solution, with no sign of fluctuations around it. finally, on panel e, for t ≤ 250, 000 the system oscillates around copyright © 2022 assa. adv syst sci appl (2022) binary opinion space in the scardo model 113 the fixed point y∗ = 5/6, which, according to the mean-field predictions, should be a global attractor for all the trajectories that start from y ̸= 1. but because in the underlying simulation run there were only n = 200 agents, the magnitude of the fluctuations was quite huge and there is a high chance that the trajectory finds itself in the fixed point y∗ = 1, from which there is no coming back (because the probability of anticonformity for individuals espousing opinion x1 is equal to zero in this case). as a result, the system ends up in the unstable fixed point, which is a pure equilibrium. in a nutshell, if α > 0 and δ > 0, then the model behavior is well-predictable, with both the opinions having nonzero populations of supporters. in this case, the balance between populations is defined by the hyperplane (4.19) – if this equality is true, then both opinion camps will have the same number of supporters (n/2 agents). if the left side dominates, then the position x1 will have an advantage over x2. the reverse is also true. fig. 4.1. here, we demonstrate six simulation runs carried out on the fully-connected social graph for n = 2000 agents (excepting for the experiment on panel e, where there were only n = 200) for different transition tables (shown at the top of the panels). the model’s dynamics are presented with orange curves. mean-field solutions are symbolized with dashed lines. on each plot, we depict what the fixed points look like for a given transition table. copyright © 2022 assa. adv syst sci appl (2022) 114 v.n. gezha, i.v. kozitsin 5. conclusion this paper advances the recently published opinion formation model [13] by studying its behavior in the case of the binary opinion space—the most simple situation whereby only two opinions exist and compete in a society. in an attempt to get a comprehensive analytical description of the model, we focused on the mean-field approximation that describes the model’s dynamics in terms of the populations of two competing opinion camps via the autonomous system of ordinary differential equations under the assumptions that the number of agents is huge and the agents communicate on a complete graph. for this system, we found exact solutions, characterized fixed points, and investigated their stability properties across all possible configurations of the model parameters. furthermore, we obtained a hyperplane in the space of the model parameters that define which opinion will win the competition. we supported our theoretical results by computational experiments. these experiments revealed that for most points in the parameter space (excepting the set of measure zero, which represents absence of anticonformity effects in the model), the mean-field predictions manage to forecast the real model behavior. this finding is quite important because it means that for a wide range of transition tables (actually, this range does not include only specific transition table configurations, which are unlikely encountered in real life [1]), we have no need in searching for suitable network topologies and performing potentially expensive computational experiments (especially in the case of large systems)—the outcome of opinion dynamics (at the macroscopic level) does not depend on this factor and can be effectively predicted analytically, via the mean-field differential equations. further analysis may be devoted to incorporating ranking algorithms into the model [6] or to improving the mean-field predictions by accounting for local interactions [10]. acknowledgements the 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(2017). a tutorial on modeling and analysis of dynamic social networks. part i. annual reviews in control, 43, 65–79. copyright © 2022 assa. adv syst sci appl (2022) introduction opinion dynamics model mean-field approximation binary opinion space problem formulation equation (4.7): fixed points, solutions, and stability experimental data compared with theoretical results conclusion adv syst sci appl 2023; 02:46–69 published online at https://ijassa.ipu.ru. stress-strength reliability of a weibull-standard normal distribution based on type-ii progressive censored samples ramin kazemi department of statistics, imam khomeini international university, qazvin, iran abstract: in this paper, under the type-ii progressive censored scheme, we obtain the point and interval estimates of stress-strength parameter (r), when stress and strength are two independent weibull-standard normal variables. we study the problem in three cases. first, assuming that stress and strength have the different scale parameters and the common shape parameter, we obtain maximum likelihood estimation, approximation maximum likelihood estimation and two bayesian approximation estimates due to the lack of explicit forms. also, we construct the asymptotic and highest posterior density intervals for r. second, assuming that common shape parameter is known, we derive the maximum likelihood estimation and bayes estimate and uniformly minimum variance unbiased estimate of r. third, assuming that all parameters are unknown and different, we achieve the statistical inference of r, namely maximum likelihood estimation, approximation maximum likelihood estimation and bayesian inference of r. furthermore, we use the monte carlo simulations to compare of the performance of different methods. keywords: stress-strength model, type-ii progressive censored sample, weibull-standard normal distribution, bayesian inference, monte carlo simulation. 1. introduction one method of introducing distributions is to generalize previous distributions. in fact, the new distribution includes the previous distributions for different values of the parameters. another method to introduce new distributions is to put a specific function of distribution function of a random variable into the distribution function of another random variable. the behavior of the distribution and hazard functions of these new random variables determine the importance of studing them. various techinqes have been introduced, especially in recent years, to produce continuous distributions with the second method. if the support of these distributions is positive, they can be used in reliability analysis. the weibull distribution is one of the most widely used distributions in the reliability and survival studies. some works on the stress-strength model of weibull distribution and its related distribution under censored data can be found in [1, 10, 11] and some references therein. consider a continuous distribution g and the weibull cumulative density function (c.d.f.) fx(x) = 1− e−axc (x > 0) with positive parameters a and c. based on this density, by replacing x with g(x)/(1−g(x)), we can define the c.d.f. family by fx(x|a, c, θ) = 1− exp ( − a ( g(x; θ) 1−g(x; θ) )c) , x ∈ d ⊂ r, a, c > 0, where g(x; θ) is a baseline c.d.f., which depends on a parameter vector θ. henceforth, let g be a continuous baseline distribution. for each g distribution, the weibull-g distribution with ∗corresponding author: r.kazemi@sci.ikiu.ac.ir stress-strength reliability of a weibull-standard normal distribution 47 two extra parameters a and c is defined by the above c.d.f. [3]. for the normal distribution n(µ, σ2), g(x;µ, σ) 1−g(x;µ, σ) = φ ( x−µ σ ) 1− φ ( x−µ σ ) , where φ(·) is the c.d.f. of the standard normal distribution n(0, 1). then, the weibull-normal distribution (wnd) is defined by f (x) = 1− exp −a  φ ( x−µ σ ) 1− φ ( x−µ σ ) c , x, a, c, σ > 0, µ ∈ r. thus, wnd with scale parameter a and shape parameter c, respectively, which denoted by wn(a, c, µ, σ2), has the probability density function (p.d.f.), and failure rate function as follows: f(x) = ac ϕ ( x−µ σ ) σ φ ( x−µ σ )c−1 ( 1− φ ( x−µ σ ))c+1 exp −a  φ ( x−µ σ ) 1− φ ( x−µ σ ) c , h(x) = ac ϕ ( x−µ σ ) σ φ ( x−µ σ )c−1 ( 1− φ ( x−µ σ ))c+1 , x, a, c, σ > 0, µ ∈ r respectively. in the stress-strength modelling, r = p (x < y ) is a measure of component reliability when it is subjected to random stress x and has strength y . note that if x and y are two independent random variables from wn(a, c, µ, σ2) and wn(b, c, µ, σ2) distributions, respectively, then with the change of variable u =  φ ( x−µ σ ) 1− φ ( x−µ σ ) c , we have r = ∫ +∞ −∞ fx(x)(1− fy (x))dx = ∫ ∞ 0 ae−(a+b)udu = a a+ b . because the our purpose is to analysis of the strees-strength reliability we focus on the standard normal distribution (sn) case. the reason for this is r is a function of a and b. thus, for the wsnd, fwsn(x) = 1− exp { − a ( φ(x) 1− φ(x) )c} , fwsn(x) = ac ϕ(x) φ(x)c−1( 1− φ(x) )c+1 exp { − a ( φ(x) 1− φ(x) )c} , hwsn(x) = ac ϕ(x) φ(x)c−1( 1− φ(x) )c+1 , x, a, c > 0. copyright © 2023 assa. adv syst sci appl (2023) 48 r. kazemi it has been used very effectively for analyzing lifetime data, particularly when the data are censored. among various censoring schemes, the type ii progressive censoring scheme has become very popular one in the last decade. it can be described as follows: consider n units are placed under a study and only n(< n) units are completely observed until failure. at the time of the first failure (the first stage), r1 of the n − 1 surviving units are randomly withdrawn (censored intentionally) from the experiment. at the time of the second failure (the second stage), r2 of the n − 2− r1 surviving units are withdrawn and so on. finally, at the time of the nth failure (the nth stage), all the remaining rn = n − n− r1 − ...− rn−1 surviving units are withdrawn. we will refer to this as progressive type-ii right censoring with scheme (r1, r2, ..., rn). it is clear that this scheme includes the conventional type-ii right censoring scheme (when r1 = r2 = ... = rn−1 = 0 and rn = n − n) and complete sampling scheme (when n = n and r1 = r2 = ... = rn = 0). for further details on progressively censoring and relevant references, the reader may refer to the book by balakrishnan and aggarwala [2]. although, in complete sample case, many authors have been studied the stress-strength models, much attention has not been paid to censored sample case. whereas in really applicable situations, for many reasons such as financial plane or limited time, the researchers confront censored data. in this paper, based on type-ii progressive censoring, the reliability parameter r = p (x < y ) is estimated, when x and y are two independent random variables from the wsn distribution (wsnd). the rest of this paper is arranged as follows. in section 2, under the type-ii progressive censoring, assuming x ∼ wsn(a, c) and y ∼ wsn(b, c), we obtain the point and interval estimates of r = p (x < y ), from the frequentist and bayesian viewpoints. because the maximum likelihood estimations (mles) of unknown parameters and r cannot be earned in the closed forms, we obtain the approximation maximum likelihood estimations (amles) of parameters and r which have the explicit forms. also, we develop the bayes estimates of r, using lindley’s approximation and mcmc method due to the lack of explicit forms. moreover, different confidence intervals such as asymptotic and hpd intervals of r are provided. in section 3, assuming the common shape parameter is known, the mle and bayes estimate and uniformly minimum variance unbiased estimate (umvue) of r are earned. because the assumption which we study in section 2 is quite strong, we consider the statistical inference of r in general case. so, in section 4, under the type-ii progressive censoring scheme, assuming wsn(a, c) and wsn(b, d), we provide the mle, amle and bayes estimate of r. in section 5, we give the simulation results and conclude the paper in section 6. 2. inference on r with unknown common c 2.1. maximum likelihood estimation of r assume that x and y are two independent random variables from wsn(a, c) ≡ wn(a, c, 0, 1) and wsn(b, c) ≡ wn(b, c, 0, 1) distributions, respectively. in this section, under the type-ii progressive censoring, we derive the mle of r. because r is a function of the unknown parameters, first we obtain the mles of a, b and c. suppose x = (x1:n , x2:n , ..., xn:n) is a progressively type-ii censored sample from wsn(a, c) with censored scheme r = (r1, r2, ..., rn) and y = (y1:m , y2:m , ..., ym:m) is a progressively typeii censored sample from wsn(b, c) with censored scheme s = (s1, s2, ..., sm). for notation simplicity, we will write (x1, x2, ..., xn) for (x1:n , x2:n , ..., xn:n) and (y1, y2, ..., ym) for (y1:m , y2:m , ..., ym:m). therefore, the likelihood function of the unknown parameters a, b and copyright © 2023 assa. adv syst sci appl (2023) stress-strength reliability of a weibull-standard normal distribution 49 c, can be written as l(a, b, c) = [ k1 n∏ i=1 fwsn(xi)[1− fwsn(xi)] ri ] × [ k2 m∏ j=1 fwsn(yj)[1− fwsn(yj)] sj ] , where k1 = n(n − 1− r1)(n − 2− r1 − r2)...(n − n+ 1− r1 − ...− rn−1) and k2 = m(m − 1− s1)(m − 2− s1 − s2)...(m −m+ 1− s1 − ...− sm−1). based on the observed data, the likelihood function can be obtained as: l(data|a, b, c) = k1k2(ac) n  n∏ i=1 ϕ(xi) φ(xi) c−1( 1− φ(xi) )c+1  exp { − a n∑ i=1 (ri + 1) ( φ(xi) 1− φ(xi) )c} × (bc)m  m∏ j=1 ϕ(yj) φ(yj) c−1( 1− φ(yj) )c+1  exp { − b m∑ j=1 (sj + 1) ( φ(yj) 1− φ(yj) )c} . therefore, the log-likelihood function is: ℓ(a, b, c) = constant + n log(a) +m log(b) + (n+m) log(c) + n∑ i=1 log(φ(xi)) + (c− 1) n∑ i=1 log(φ(xi))− (c+ 1) n∑ i=1 log(1− φ(xi)) + m∑ j=1 log(φ(yj)) + (c− 1) m∑ j=1 log(φ(yj))− (c+ 1) m∑ j=1 log(1− φ(yj)) − a n∑ i=1 (ri + 1) ( φ(xi) 1− φ(xi) )c − b m∑ j=1 (sj + 1) ( φ(yj) 1− φ(yj) )c . (2.1) set w(x, t, c, k) := (t+ 1) ( φ(x) 1− φ(x) )c logk ( φ(x) 1− φ(x) ) , x > 0, k ∈ n ∪ {0}. so, to earn the mles of a, b and c, namely, â, b̂ and ĉ, respectively, we should solve the following equations: ∂ℓ ∂a = n a − n∑ i=1 w(xi, ri, c, 0) = 0, (2.2) ∂ℓ ∂b = m b − m∑ j=1 w(yj, sj, c, 0) = 0, (2.3) ∂ℓ ∂c = m+ n c + n∑ i=1 log ( φ(xi) 1− φ(xi) ) + m∑ j=1 log ( φ(yj) 1− φ(yj) ) − a n∑ i=1 w(xi, ri, c, 1)− b m∑ j=1 w(yj, sj, c, 1) = 0. (2.4) copyright © 2023 assa. adv syst sci appl (2023) 50 r. kazemi from the equations (2.2) and (2.3), we have â(c) = n ( n∑ i=1 (ri + 1) ( φ(xi) 1− φ(xi) )c)−1 = n ( n∑ i=1 w(xi, ri, c, 0) )−1 , b̂(c) = m ( m∑ j=1 (sj + 1) ( φ(yj) 1− φ(yj) )c)−1 = m ( m∑ j=1 (w(yj, sj, c, 0) )−1 . also, to derive ĉ, we apply one numerical method such as newton-raphson on the equation (2.4). after obtaining the mles of a, b and c, by using the invariance property, the mle of r can be derived as r̂mle = â â+ b̂ . (2.5) 2.2. approximation maximum likelihood estimation of r from the section 2.1, we see that the mles of unknown parameters and r cannot be earned in the closed forms. so in this section, we obtain the amles of the parameters which have the explicit forms. let z ′ and z ′′ be weibull and extreme value distributions, in symbols z ′ ∼ w (a, θ) and z ′′ ∼ ev (ξ, σ), if they have the following cumulative distribution functions respectively as: fz′(z) = 1− e− za θ , z > 0, a, θ > 0, fz′′(z) = 1− e−e z−ξ σ , z ∈ r, ξ ∈ r, σ > 0. the following simple theorem is critical to obtain the amles of the parameters. theorem 2.1: (i) if z ∼ wsn(a, c), then z ′ = φ(z) 1− φ(z) ∼ w (c, 1/a). (ii) if z ′ ∼ w (c, 1/a) and z ′′ = log(z ′), then z ′′ ∼ ev (ξ, σ), where ξ = −1 c log(a) and σ = 1 c . proof the proof is obvious. □ suppose that {x1, . . . , xn} and {y1, . . . , ym} be two type-ii progressive censoring samples with the above censoring schemes and x ′ i = φ(xi) 1− φ(xi) , ui = log(x ′ i), y ′ j = φ(yj) 1− φ(yj) , vj = log(y ′ j ). applying theorem 2.1, we have ui ∼ ev (ξ1, σ) and vj ∼ ev (ξ2, σ), where ξ1 = −1 c log(a), ξ2 = −1 c log(b), σ = 1 c . copyright © 2023 assa. adv syst sci appl (2023) stress-strength reliability of a weibull-standard normal distribution 51 after earning ξ̃1, ξ̃2 and σ̃ , the values of ã, b̃ and c̃ can be evaluated by (see appendix a): ã = e− ξ̃1 σ̃ , b̃ = e− ξ̃2 σ̃ , c̃ = 1 σ̃ . so, the amle of r, namely r̃, is r̃ = ã ã+ b̃ . (2.6) 2.3. asymptotic confidence interval in this section, we earn the asymptotic confidence interval of r by the asymptotic distribution of r̂, which obtain from the asymptotic distribution of λ̂ = (â, b̂, ĉ). we denote the observed fisher information matrix by i(λ) = [iij] = [ − ∂2ℓ ∂λi∂λj ] , i, j = 1, 2, 3. by differentiating twice from (2.1) with respect to a, b and c, the inlines of i(λ) matrix can be obtained as: i11 = n a2 , i22 = m b2 , i12 = i21 = 0, i13 = i31 = n∑ i=1 w(xi, ri, c, 1), i23 = i32 = m∑ j=1 w(yj, sj, c, 1), i33 = m+ n c2 + a n∑ i=1 w(xi, ri, c, 2) + b m∑ j=1 w(yj, sj, c, 2). theorem 2.2: let â, b̂ and ĉ be the mles of a, b and c. then [(â− a) (̂b− b) (ĉ− c)]t d−→ n3(0, i −1(a, b, c)), where i(a, b, c) and i−1(a, b, c) are symmetric matrices and i(a, b, c) = ( i11 0 i13 i22 i23 i33 ) , i−1(a, b, c) = 1 |i(a, b, c)| ( b11 b12 b13 b22 b23 b33 ) , in which |i(a, b, c)| = i11i22i33 − i11i 2 23 − i213i22, b11 = i22i33 − i223, b12 = i13i23, b13 = −i13i22, b22 = i11i33 − i213, b23 = −i11i23, b33 = i11i22. proof from the asymptotic normality of the mle, the theorem resulted. □ theorem 2.3: let r̂mle be the mle of r. then, (r̂mle −r) d−→ n(0, b), where b = 1 |i(a, b, c)| [(∂r ∂a )2 b11 + (∂r ∂b )2 b22 + 2 (∂r ∂a )(∂r ∂b ) b12 ] . (2.7) copyright © 2023 assa. adv syst sci appl (2023) 52 r. kazemi proof using theorem 2.2 and applying delta method, the asymptotic distribution of r̂ = â â+b̂ can be obtained as follows: (r̂mle −r) d−→ n(0, b), where b = bt i−1(a, b, c)b, with b = [∂r ∂a , ∂r ∂b , ∂r ∂c ]t = [∂r ∂a , ∂r ∂b , 0]t , in which ∂r ∂a = b (a+ b)2 , ∂r ∂b = − a (a+ b)2 . (2.8) also, i−1(a, b, c) is defined in theorem 2.2. therefore, b can be represented as (2.7) and the theorem is resulted. □ using theorem 2.3, the asymptotic confidence interval of r can be derived. it is notable that b should be estimated by the mles of a, b and c. so, a 100(1− γ)% asymptotic confidence interval of r can be constructed as, (r̂mle − z1− γ 2 √ b̂, r̂mle + z1− γ 2 √ b̂), where zγ is 100γ-th percentile of n(0, 1). 2.4. bayes estimation in this section, under the squared error loss function, we infer the bayesian estimation and corresponding credible interval of the stress-strength parameter, when a ∼ γ(a1, b1), b ∼ γ(a2, b2) and c ∼ γ(a3, b3) are independent random variables. based on the observed censoring samples, the joint posterior density function of a, b and c are given by: π(a, b, c|data) = l(data|a, b, c)π1(a)π2(b)π3(c)∫∞ 0 ∫∞ 0 ∫∞ 0 l(data|a, b, c)π1(a)π2(b)π3(c)dadbdc , (2.9) where π1(a) ∝ aa1−1e−b1a, a > 0, a1, b1 > 0, π2(b) ∝ ba2−1e−b2b, b > 0, a2, b2 > 0, π3(c) ∝ ca3−1e−b3c, c > 0, a3, b3 > 0. as we see from (2.9), the bayes estimates cannot be derived in the closed form. so, we approximate them by applying two methods: • lindley’s approximation, • mcmc method. 2.4.1. lindley’s approximation one of the most numerical methods to approximate the bayes estimate has been introduced by lindley in [8]. this method has explained in appendix b. based on this approximation, the bayes estimate of r is: r̂lin = r + [u1d1 + u2d2 + d4 + d5] + 1 2 [a(u1σ11 + u2σ12) +b(u1σ21 + u2σ22) + c(u1σ31 + u2σ32)]. (2.10) as we see, constructing the hpd credible interval is not possible, using the lindley’s approximation. so, we apply the markov chain monte carlo (mcmc) method to approximate the bayes estimate and construct the corresponding hpd credible intervals. copyright © 2023 assa. adv syst sci appl (2023) stress-strength reliability of a weibull-standard normal distribution 53 2.4.2. mcmc method by (2.9), we get the posterior p.d.fs of a, b and c as: a|c, data ∼ γ ( n+ a1, n∑ i=1 w(xi, ri, c, 0) + b1 ) , b|c, data ∼ γ ( n+ a2, m∑ j=1 w(yj, sj, c, 0) + b2 ) , π(c|a, b, data) ∝ cn+m+a3−1  n∏ i=1 φ(xi) c−1( 1− φ(xi) )c+1   m∏ j=1 φ(yj) c−1( 1− φ(yj) )c+1  × exp { − a n∑ i=1 w(xi, ri, c, 0)− b m∑ j=1 w(yj, sj, c, 0)− b3c } it is obvious that the posteriors p.d.f. of c are not the well known distributions. so, we utilize the metropolis-hastings method with normal proposal distribution to generate random samples from it. therefore, the gibbs sampling algorithm can be proposed as follows: 1. start with the begin value (a(0), b(0), c(0)). 2. set t = 1. 3. generate c(t) from π(c|a(t−1), b(t−1), data), using metropolis-hastings method. 4. generate a(t) from γ ( n+ a1, ∑n i=1 w(xi, ri, c, 0) + b1 ) . 5. generate b(t) from γ ( n+ a2, ∑m j=1 w(yj, sj, c, 0) + b2 ) . 6. calculate rt = at at+bt . 7. set t = t+ 1. 8. repeat steps 3-7, for t times. applying this algorithm, the bayes estimate of r, under the squared error loss function is given by r̂mc = 1 t t∑ t=1 rt. (2.11) moreover, a 100(1− γ)% hpd credible interval of r can be constructed by utilizing the method of chen and shao [4]. 3. inference on r with known common c 3.1. maximum likelihood estimation of r suppose x = (x1:n , x2:n , ..., xn:n) is a progressively type-ii censored sample from wsn(a, c) with censored scheme r = (r1, r2, ..., rn) and y = (y1:m , y2:m , ..., ym:m) is a progressively type-ii censored sample from wsn(b, c) with censored scheme s = (s1, s2, ..., sm). based on section 2.1, when the common shape parameter c is known, the mle of r can be earned easily by r̂mle = ( 1 + m ∑n i=1w(xi, ri, c, 0) n ∑m j=1 w(yj, sj, c, 0) )−1 . (3.12) copyright © 2023 assa. adv syst sci appl (2023) 54 r. kazemi in a similar manner as section 2.3, (r̂mle −r) d−→ n(0, c), where c = (∂r ∂a )2 1 i11 + (∂r ∂b )2 1 i22 , and ∂r ∂a and ∂r ∂b are given in (2.8). so, a 100(1− γ)% asymptotic confidence interval for r can be constructed as, (r̂mle − z1− γ 2 √ ĉ, r̂mle + z1− γ 2 √ ĉ), (3.13) where zγ is 100γ-th percentile of n(0, 1). 3.2. bayes estimation in this section, we infer the bayesian estimation and corresponding credible interval of the stress-strength parameter, when a ∼ γ(a1, b1) and b ∼ γ(a2, b2) are independent random variables. based on the observed censoring samples, the joint posterior density function of a and b are given by: π(a, b|c, data) = (v + b1) n+a1(u + b2) m+a2 γ(n+ a1)γ(m+ a2) an+a1−1bm+a2−1e−a(v+b1)−b(u+b2), (3.14) where v = ∑n i=1w(xi, ri, c, 0) and u = ∑m j=1w(yj, sj, c, 0). under the squared error loss function, to obtain r bayes estimate, we solve the following integral r̂b = ∫ ∞ 0 ∫ ∞ 0 a a+ b × π(a, b|c, data)dadb. now, we use the idea of kizilaslan and nadar [5] and obtain the r bayes estimate as r̂b =  (1− z)n+a1(n+ a1) w f 2 1 (w, n+ a1 + 1;w + 1, z), |z| < 1, (n+ a1) w(1− z)m+a2 f 2 1 (w,m+ a2;w + 1, z(1− z)−1), z < −1, (3.15) where w = n+m+ a1 + a2, z = 1− v + b1 u + b2 and f 2 1 (a, b; c, z) = 1 b(b, c− b) ∫ 1 0 tb−1(1− t)c−b−1(1− tz)−adt, |z| < 1, is the hypergeometric series, which is quickly evaluated and readily available in standard software such as matlab. moreover, we construct a 100(1− γ)% bayesian interval for the stress-strength parameter by (l,u), where l and u are the lower and upper bounds, respectively which satisfy∫ l 0 fr(r)dr = γ 2 , ∫ u 0 fr(r)dr = 1− γ 2 , (3.16) where fr(r) is the probability density function of r which obtained from (3.14) as fr(r) = 1 b(n+ a1,m+ a2) (1− z)n+a1rn+a1−1(1−r)m+a2−1(1−rz)−w, 0 < r < 1. copyright © 2023 assa. adv syst sci appl (2023) stress-strength reliability of a weibull-standard normal distribution 55 3.3. uniformly minimum variance unbiased estimate of r suppose x = (x1:n , x2:n , ..., xn:n) is a progressively type-ii censored sample from wsn(a, c) with censored scheme r = (r1, r2, ..., rn) and y = (y1:m , y2:m , ..., ym:m) is a progressively type-ii censored sample from wsn(b, c) with censored scheme s = (s1, s2, ..., sm). when the common shape parameter c is known, based on the observed data, the likelihood function is: l(data, c|a, b) ∝ (ac)n  n∏ i=1 ϕ(xi) φ(xi) c−1( 1− φ(xi) )c+1  exp{−au} × (bc)m  m∏ j=1 ϕ(yj) φ(yj) c−1( 1− φ(yj) )c+1  exp{−bv }. from the above equation, we conclude that v and u are complete sufficient statistics for a and b, respectively. we can verify that x∗ i = ( φ(xi) 1−φ(xi) )c , i = 1, . . . , n is one type-ii progressive censoring samples from an exponential distribution with mean a−1. now, using the transformations z1 = nx∗ 1 , z2 = (n − r1 − 1)(x∗ 2 −x∗ 1 ), ... zn = (n − n−1∑ i=1 ri − n+ 1)(x∗ n −x∗ n−1). from balakrishnan and aggarwala [2], we conclude that z1, . . . , zn are independent and identically distributed as an exponential distribution with mean a−1. so, v = n∑ i=1 zi ∼ γ(n, a). lemma 3.1: let x∗ i = ( φ(xi) 1−φ(xi) )c , i = 1, . . . , n and y ∗ j = ( φ(yj) 1−φ(yj) )c , j = 1, . . . ,m. then the conditional p.d.fs of x∗ 1 given v and y ∗ 1 given u are, respectively, as follows: fx∗ 1 |v=v(x) = n(n− 1) (v −nx)n−2 vn−1 , 0 < x < v n , fy ∗ 1 |u=u(y) = m(m− 1) (u−my)m−2 um−1 , 0 < y < u m . proof the proof of the lemma is similar to the proof of lemma 2 in [7]. copyright © 2023 assa. adv syst sci appl (2023) 56 r. kazemi theorem 3.1: for the complete sufficient statistics v and u , the umvue of r, say ˆ̂ r, is as follows: ˆ̂ r =  1− n−1∑ k=0 (−1)k(u v )k (n−1 k ) (m−1+k k ) , u < v, m−1∑ k=0 (−1)k( v u )k (m−1 k ) (n−1+k k ) , u > v. (3.17) proof it is observable that x∗ 1 and y ∗ 1 are exponential variables with mean (na)−1 and (mb)−1, respectively. by this, we can easily show that ϕ(x∗ 1 , y ∗ 1 ) = { 1, my ∗ 1 > nx∗ 1 , 0, my ∗ 1 < nx∗ 1 , is an unbiased estimate for r. so, ˆ̂ r = e(ϕ(x∗ 1 , y ∗ 1 )|v = v, u = u) = ∫∫ a fx∗ 1 |v=v(x)fy ∗ 1 |u=u(y)dxdy, where a = {(x, y) : 0 < x < v/n, 0 < y < u/m,my > nx}. moreover, fx∗ 1 |v=v(x) and fy ∗ 1 |u=u(y) are given in lemma 3.1. now, for u < v, we have ˆ̂ r = nm(n− 1)(m− 1) vn−1um−1 u m∫ 0 my n∫ 0 (v −nx)n−2(u−my)m−2dxdy = 1− m(m− 1) vn−1um−1 u m∫ 0 (v −my)n−1(u−my)m−2dy = 1− (m− 1) 1∫ 0 (1− t)m−2 ( 1− u v t )n−1 dt = 1− (m− 1) 1∫ 0 (1− t)m−2 n−1∑ k=0 ( n− 1 k ) (−1)k ( u v )k tkdt = 1− n−1∑ k=0 (−1)k ( u v )k ( n−1 k )( m−1+k k ) by a similar method, for u > v, the result can be verified. copyright © 2023 assa. adv syst sci appl (2023) stress-strength reliability of a weibull-standard normal distribution 57 4. estimation of r in general case 4.1. maximum likelihood estimation of r assume that x and y are two independent random variables from wsn(a, c) and wsn(b, d) distributions, respectively. we have r = a ∫ ∞ 0 exp{−au− bu d c }du. so, the likelihood function, based on the observed data can be obtained as: l(data|a, b, c, d) = k1k2(ac) n  n∏ i=1 ϕ(xi)φ(xi) c−1( 1− φ(xi) )c+1  exp { − a n∑ i=1 (ri + 1) ( φ(xi) 1− φ(xi) )c} × (bd)m  m∏ j=1 ϕ(yj) φ(yj) d−1( 1− φ(yj) )d+1  exp { − b m∑ j=1 (sj + 1) ( φ(yj) 1− φ(yj) )d} . therefore, the log-likelihood function is as: ℓ(a, b, c, d) = constant + n log(a) +m log(b) + n log(c) +m log(d) + n∑ i=1 log(φ(xi)) + (c− 1) n∑ i=1 log(φ(xi))− (c+ 1) n∑ i=1 log(1− φ(xi)) + m∑ j=1 log(φ(yj)) + (d− 1) m∑ j=1 log(φ(yj))− (d+ 1) m∑ j=1 log(1− φ(yj)) − a n∑ i=1 (ri + 1) ( φ(xi) 1− φ(xi) )c − b m∑ j=1 (sj + 1) ( φ(yj) 1− φ(yj) )d . (4.18) so, to earn the mles of a, b, c and d, namely, â, b̂, ĉ and d̂, respectively, we should solve the following equations: ∂ℓ ∂a = n a − n∑ i=1 w(xi, ri, c, 0) = 0, (4.19) ∂ℓ ∂b = m b − m∑ j=1 w(yj, sj, d, 0) = 0, (4.20) ∂ℓ ∂c = m c + n∑ i=1 log ( φ(xi) 1− φ(xi) ) − a n∑ i=1 w(xi, ri, c, 1) = 0, (4.21) ∂ℓ ∂d = m d + m∑ j=1 log ( φ(yj) 1− φ(yj) ) − b m∑ j=1 w(yj, sj, d, 1) = 0. (4.22) after obtaining the mles of a, b, c and d, by using the invariance property, the mle of r can be derived as r̂mle = â ∫ ∞ 0 exp{−âu− b̂u d̂ ĉ }du. (4.23) copyright © 2023 assa. adv syst sci appl (2023) 58 r. kazemi 4.2. approximation maximum likelihood estimation of r suppose that {x1, . . . , xn} and {y1, . . . , ym} be two type-ii progressive censoring samples from wsn(a, c) and wsn(b, d) distributions and x ′ i = φ(xi) 1− φ(xi) , ui = log(x ′ i), y ′ j = φ(yj) 1− φ(yj) , vj = log(y ′ j ). applying theorem 2.1, we have ui ∼ ev (ξ1, σ1) and vj ∼ ev (ξ2, σ2), where ξ1 = −1 c log(a), ξ2 = −1 d log(b), σ1 = 1 c , and σ2 = 1 d . in a similar manner as section 2.2, we derive the amles of ξ1, ξ2, σ1 and σ2, say ξ̃1, ξ̃2, σ̃1 and σ̃2, respectively, by ξ̃1 = a1 − σ̃1b1, ξ̃2 = a2 − σ̃2b2, σ̃1 = −d1 + √ d2 1 + 4c1e1 2c1 , σ̃2 = −d2 + √ d2 2 + 4c2e2 2c2 , where a1, a2, b1, b2, c1, c2, d1, d2, e1, e2 are given in appendix a. after earning ξ̃1, ξ̃2, σ̃1 and σ̃2, the values of ã, b̃ and c̃ can be evaluated by c̃ = 1 σ̃1 , d̃ = 1 σ̃2 , ã = exp (−ξ̃1 σ̃1 ) , b̃ = exp (−ξ̃2 σ̃2 ) . then, r̃ = ã ∫ ∞ 0 exp{−ãu− b̃u d̃ c̃ }du. (4.24) 4.3. bayes estimation in this section, under the squared error loss function, we infer the bayesian estimation and corresponding credible interval of the stress-strength parameter, when the unknown parameters a ∼ γ(a1, b1), b ∼ γ(a2, b2), c ∼ γ(a3, b3) and d ∼ γ(a4, b4) are independent random variables. in a similar manner as section 2.4, as the bayesian estimation of r has not a closed form, we approximate it by mcmc method. after simplify the joint posterior copyright © 2023 assa. adv syst sci appl (2023) stress-strength reliability of a weibull-standard normal distribution 59 density function of the unknown parameters, we get the posterior p.d.fs of a, b, c and d as: a|c, data ∼ γ ( n+ a1, n∑ i=1 w(xi, ri, c, 0) + b1 ) , b|d, data ∼ γ ( n+ a2, m∑ j=1 w(yj, sj, d, 0) + b2 ) , π(c|a, data) ∝ cn+a3−1  n∏ i=1 φ(xi) c−1( 1− φ(xi) )c+1 × exp { − a n∑ i=1 w(xi, ri, c, 0)− b3c } , π(d|b, data) ∝ dm+a4−1  m∏ j=1 φ(yj) d−1( 1− φ(yj) )d+1 × exp { − b m∑ j=1 w(yj, sj, d, 0)− b4d } . it is recognized that the posterior p.d.fs of c and d are not well known distributions. so, we utilize the metropolis-hastings method with normal proposal distribution to generate random samples from them. therefore, the gibbs sampling algorithm can be proposed as follows: 1. start with the begin value (a(0), b(0), c(0), d(0)). 2. set t = 1. 3. generate c(t) from π(c|a(t−1), data), using metropolis-hastings method. 4. generate d(t) from π(d|b(t−1), data), using metropolis-hastings method. 5. generate a(t) from γ ( n+ a1, ∑n i=1 w(xi, ri, c(t−1), 0) + b1 ) . 6. generate b(t) from γ ( n+ a2, ∑m j=1 w(yj, sj, d(t−1), 0) + b2 ) . 7. calculate rt = a(t) ∫ ∞ 0 exp { − a(t)u− b(t)u d(t) c(t) } du. 8. set t = t+ 1. 9. repeat steps 3-8, for t times. using this algorithm, under the squared error loss function, the r bayes estimate is given by r̂mc = 1 t t∑ t=1 rt. (4.25) moreover, a 100(1− γ)% hpd credible interval of r can be constructed by utilizing the method of chen and shao [4]. 5. simulation study using monte carlo simulations, we compare the behavior of different methods, in this section. to compare of point estimates, we compute the mean squared errors (mses). also, to comparing of interval estimates, we compute the average lengths and coverage percentages. different schemes, parameters and hyper parameters are employed to obtain the simulation results. we report all results, based on 3000 replications and the nominal level is 0.95. the copyright © 2023 assa. adv syst sci appl (2023) 60 r. kazemi censoring schemes which we used are as follows: scheme 1: r1 = . . . = rn = n − n n , scheme 2: r1 = . . . = rn 2 = 0, rn 2 +1 = . . . = rn = 2(n − n) n , scheme 3: r1 = n − n 2 , r2 = . . . = rn−1 = 0, rn = n − n 2 . first, when the common scale parameter c is unknown, without loss of generality, we put a = 3, b = 2, c = 4. also, bayesian inference are given based on three priors as: prior 1: aj = 0, bj = 0, j = 1, 2, 3, prior 2: aj = 1, bj = 0.1, j = 1, 2, 3, and prior 3: aj = 2, bj = 0.2, j = 1, 2, 3. we derive the mle using (2.5), amle using (2.6), bayes estimates of r via lindley’s approximation and mcmc method using (2.10) and (2.11), respectively. further, we obtain the asymptotic confidence and hpd credible intervals of r. the results are given in tables 5.1-5.2. second, when the common scale parameter c is known, without loss of generality, we put a = 2, b = 3, c = 4. also, bayesian inference are given based on three priors as: prior 4: aj = 0, bj = 0, j = 1, 2, prior 5: aj = 1, bj = 0.1, j = 1, 2, and prior 6: aj = 2, bj = 0.2, j = 1, 2. we derive the mle using (3.12), bayes estimates using (3.15) and umvue of r using (3.17). further, we obtain the asymptotic and bayesian intervals of r using (3.13) and (3.16), respectively. the results are given in table 5.3. third, when all parameters are unknown and different, without loss of generality, we put a = 1.5, b = 3, c = 2 , d = 4. also, bayesian inference are given based on three priors as: prior 7: aj = 0, bj = 0, j = 1, 2, 3, 4, prior 8: aj = 1, bj = 0.1, j = 1, 2, 3, 4, and prior 9: aj = 2, bj = 0.2, j = 1, 2, 3, 4. we derive the mle, amle and bayes estimates via mcmc method using (4.23), (4.24) and (4.25), respectively. the results are given in table 5.4. to monitor the convergence of mcmc method, all three cases, we considered the trace plots for different censoring schemes and parameters. in all cases, it is observed that the mcmc method is converged. some of this plots are shown in figures 5.1-5.2. from tables 5.1-5.2, we observe that bayes estimates and amle have the best and worst performance, based on mses, respectively. also, in bayesian inference, the informative priors perform better than non-informative ones, in point and interval estimates. furthermore, the lindley’s approximation performs worse that the mcmc method. from table 5.3, we observe that bayes estimates and umvues have the best and worst performance based on mses, respectively. also, in bayesian inference, the informative priors perform better than non-informative ones, in point and interval estimates. from table 5.4, we observe that the bayes estimates perform better that the mles based on mses. also, in bayesian inference, the informative priors perform better than noninformative ones, in point and interval estimates. as a fact, from tables 5.1-5.4, for fixed n , with increasing n, the mses of all estimates decrease, the average confidence lengths decrease and the associated coverage percentages increase, in all cases. this can be due to the fact, with increasing n, some additional information is gathered. 6. conclusion in this paper, we obtain different estimates of stress-strength parameter, under the hybrid progressive censored scheme, when stress and strength are two independent kumaraswamy random variables. the problem is solved in three cases. first, when x ∼ wsn(a, c) and y ∼ wsn(b, c), we derive ml, aml and two approximated bayes estimates using lindley’s approximation and mcmc method, due to the lack of explicit forms. also, we copyright © 2023 assa. adv syst sci appl (2023) stress-strength reliability of a weibull-standard normal distribution 61 ta bl e 5. 1. b ia se s an d m se s fo re st im at es of r w he n c is un kn ow n. pr io r1 pr io r2 pr io r3 (n , n ) c .s a m l e m l e m c m c l in dl ey m c m c l in dl ey m c m c l in dl ey e st . m se e st . m se e st . m se e st . m se e st . m se e st . m se e st . m se e st . m se (3 0, 10 ) (1 ,1 ) 0. 61 82 0. 01 82 0. 56 07 0. 01 70 0. 60 71 0. 00 97 0. 62 12 0. 01 34 0. 58 84 0. 00 83 0. 60 11 0. 01 10 0. 58 57 0. 00 68 0. 57 52 0. 00 88 (2 ,2 ) 0. 60 73 0. 01 72 0. 60 32 0. 01 47 0. 55 61 0. 00 93 0. 60 47 0. 01 56 0. 59 12 0. 00 85 0. 56 59 0. 00 87 0. 56 42 0. 00 67 0. 60 22 0. 00 86 (3 ,3 ) 0. 60 56 0. 01 82 0. 59 43 0. 01 71 0. 55 22 0. 00 89 0. 59 23 0. 01 44 0. 61 48 0. 00 79 0. 60 49 0. 00 93 0. 61 49 0. 00 68 0. 60 27 0. 00 87 (1 ,2 ) 0. 58 15 0. 01 96 0. 61 09 0. 01 52 0. 61 23 0. 00 96 0. 57 67 0. 01 49 0. 60 89 0. 00 86 0. 59 37 0. 00 93 0. 61 29 0. 00 75 0. 58 62 0. 00 83 (1 ,3 ) 0. 57 16 0. 01 79 0. 60 85 0. 01 38 0. 60 22 0. 00 86 0. 57 71 0. 01 57 0. 61 02 0. 00 78 0. 58 95 0. 00 93 0. 58 84 0. 00 65 0. 59 79 0. 00 87 (2 ,3 ) 0. 60 49 0. 01 98 0. 58 58 0. 01 39 0. 58 61 0. 00 91 0. 60 33 0. 01 29 0. 59 69 0. 00 73 0. 58 77 0. 00 97 0. 59 84 0. 00 65 0. 59 85 0. 00 88 (5 0, 20 ) (1 ,1 ) 0. 58 66 0. 01 77 0. 58 91 0. 01 68 0. 59 81 0. 00 98 0. 61 37 0. 01 42 0. 60 38 0. 00 83 0. 59 37 0. 00 96 0. 61 01 0. 00 66 0. 61 09 0. 00 87 (2 ,2 ) 0. 60 59 0. 01 86 0. 58 94 0. 01 50 0. 57 91 0. 00 97 0. 61 02 0. 01 45 0. 60 20 0. 00 82 0. 58 98 0. 00 96 0. 59 87 0. 00 66 0. 59 52 0. 00 92 (3 ,3 ) 0. 58 07 0. 01 72 0. 59 00 0. 01 51 0. 59 84 0. 00 99 0. 60 14 0. 01 23 0. 59 87 0. 00 81 0. 60 91 0. 01 10 2 0. 61 62 0. 00 65 0. 58 65 0. 00 86 (1 ,2 ) 0. 59 15 0. 01 79 0. 61 71 0. 01 69 0. 61 38 0. 00 92 0. 58 41 0. 01 34 0. 58 12 0. 00 87 0. 59 07 0. 00 96 0. 59 23 0. 00 66 0. 59 87 0. 00 78 (1 ,3 ) 0. 59 81 0. 01 94 0. 58 10 0. 01 32 0. 61 62 0. 00 97 0. 60 70 0. 01 41 0. 60 71 0. 00 88 0. 60 86 0. 00 97 0. 59 83 0. 00 70 0. 59 23 0. 00 86 (2 ,3 ) 0. 61 18 0. 01 79 0. 61 87 0. 01 42 0. 60 52 0. 00 98 0. 59 76 0. 01 33 0. 60 12 0. 00 79 0. 61 03 0. 01 15 0. 59 67 0. 00 68 0. 59 35 0. 00 87 (5 0, 30 ) (1 ,1 ) 0. 61 01 0. 01 64 0. 60 82 0. 01 41 0. 61 55 0. 00 87 0. 60 82 0. 01 07 0. 60 57 0. 00 64 0. 59 84 0. 00 81 0. 59 13 0. 00 42 0. 60 14 0. 00 69 (2 ,2 ) 0. 61 62 0. 01 75 0. 58 97 0. 01 37 0. 61 77 0. 00 73 0. 61 79 0. 01 08 0. 61 00 0. 00 65 0. 59 77 0. 00 89 0. 60 33 0. 00 43 0. 58 96 0. 00 68 (3 ,3 ) 0. 61 37 0. 01 72 0. 61 95 0. 01 47 0. 59 39 0. 00 77 0. 60 93 0. 01 06 0. 59 78 0. 00 62 0. 59 74 0. 00 89 0. 59 77 0. 00 44 0. 60 11 0. 00 65 (1 ,2 ) 0. 59 38 0. 01 71 0. 61 78 0. 01 48 0. 59 82 0. 00 78 0. 61 65 0. 01 06 0. 59 74 0. 00 68 0. 59 85 0. 00 87 0. 58 58 0. 00 41 0. 59 68 7 0. 00 58 (1 ,3 ) 0. 58 79 0. 01 66 0. 58 62 0. 01 41 0. 61 74 0. 00 71 0. 59 75 0. 01 08 0. 59 87 0. 00 65 0. 60 16 0. 00 85 0. 59 31 0. 00 43 0. 60 71 0. 00 67 (2 ,3 ) 0. 61 54 0. 01 70 0. 58 82 0. 01 41 0. 61 22 0. 00 77 0. 60 98 0. 01 25 0. 59 49 0. 00 62 0. 60 22 0. 00 81 0. 60 93 0. 00 49 0. 61 86 0. 00 57 (7 0, 30 ) (1 ,1 ) 0. 61 07 0. 01 63 0. 59 81 0. 01 45 0. 59 15 0. 00 75 0. 59 52 0. 01 21 0. 59 81 0. 00 65 0. 59 43 0. 00 79 0. 59 86 0. 00 39 0. 60 79 0. 00 61 (2 ,2 ) 0. 60 81 0. 01 62 0. 61 61 0. 01 45 0. 61 61 0. 00 77 0. 59 40 0. 01 08 0. 60 42 0. 00 65 0. 60 31 0. 00 81 0. 60 04 0. 00 51 0. 60 91 0. 00 58 (3 ,3 ) 0. 58 51 0. 01 67 0. 61 43 0. 01 46 0. 61 80 0. 00 77 0. 61 08 0. 01 09 0. 59 39 0. 00 66 0. 60 48 0. 00 79 0. 60 75 0. 00 42 0. 60 79 0. 00 63 (1 ,2 ) 0. 59 32 0. 01 61 0. 61 08 0. 01 37 0. 60 77 0. 00 71 0. 59 67 0. 01 06 0. 60 17 0. 00 64 0. 59 18 0. 00 87 0. 59 05 0. 00 41 0. 59 65 0. 00 58 (1 ,3 ) 0. 61 70 0. 01 76 0. 61 40 0. 01 45 0. 61 26 0. 00 74 0. 61 07 0. 01 06 0. 60 46 0. 00 61 0. 60 84 0. 00 83 0. 58 04 0. 00 44 0. 59 87 0. 00 65 (2 ,3 ) 0. 61 48 0. 01 70 0. 59 31 0. 01 39 0. 59 11 0. 00 74 0. 59 58 0. 01 08 0. 60 15 0. 00 63 0. 59 77 0. 00 88 0. 60 62 0. 00 42 0. 60 37 0. 00 63 copyright © 2023 assa. adv syst sci appl (2023) 62 r. kazemi table 5.2. average confidence/credible lengths and coverage percentages for estimates of r when c is unknown. (n,n) c.s amle mle prior 1 prior 2 prior 3 length c.p length c.p length c.p length c.p length c.p (30,10) (1,1) 0.5499 0.912 0.5391 0.925 0.3680 0.935 0.4253 0.942 0.2906 0.946 (2,2) 0.5400 0.911 0.5294 0.920 0.3687 0.935 0.4329 0.943 0.3830 0.945 (3,3) 0.5414 0.913 0.5311 0.927 0.4577 0.936 0.4172 0.942 0.3847 0.944 (1,2) 0.5534 0.912 0.5204 0.921 0.4539 0.935 0.4178 0.943 0.3856 0.945 (1,3) 0.5421 0.911 0.5302 0.924 0.3617 0.936 0.4351 0.942 0.3820 0.946 (2,3) 0.5500 0.911 0.5279 0.924 0.3626 0.935 0.4237 0.942 0.2902 0.945 (50,20) (1,1) 0.5415 0.913 0.5289 0.920 0.4568 0.935 0.4299 0.942 0.3800 0.944 (2,2) 0.5588 0.911 0.5296 0.923 0.4581 0.935 0.4155 0.943 0.2950 0.943 (3,3) 0.5532 0.910 0.5374 0.921 0.3626 0.937 0.4228 0.941 0.2971 0.945 (1,2) 0.5583 0.910 0.5206 0.927 0.3679 0.937 0.4317 0.942 0.3886 0.945 (1,3) 0.5569 0.910 0.5305 0.920 0.3639 0.936 0.4311 0.944 0.2939 0.944 (2,3) 0.5551 0.911 0.5374 0.923 0.3690 0.935 0.4297 0.943 0.2993 0.946 (50,30) (1,1) 0.4001 0.928 0.3740 0.939 0.3014 0.943 0.2622 0.947 0.3323 0.955 (2,2) 0.5194 0.928 0.3719 0.936 0.3022 0.944 0.2612 0.946 0.3205 0.952 (3,3) 0.4012 0.926 0.4507 0.937 0.3086 0.941 0.2589 0.947 0.3221 0.957 (1,2) 0.4085 0.927 0.3699 0.935 0.4156 0.944 0.2756 0.949 0.3305 0.950 (1,3) 0.5229 0.924 0.4569 0.936 0.4155 0.941 0.2749 0.946 0.3296 0.957 (2,3) 0.5147 0.925 0.3759 0.936 0.4150 0.943 0.2627 0.947 0.3359 0.960 (70,30) (1,1) 0.4079 0.924 0.3614 0.935 0.2956 0.940 0.2653 0.946 0.3344 0.959 (2,2) 0.5109 0.925 0.3625 0.938 0.3066 0.942 0.2503 0.947 0.3214 0.956 (3,3) 0.4067 0.924 0.4525 0.936 0.4124 0.940 0.2633 0.948 0.3257 0.954 (1,2) 0.4017 0.928 0.3696 0.935 0.2913 0.944 0.2552 0.946 0.3346 0.950 (1,3) 0.5126 0.926 0.3799 0.936 0.2981 0.943 0.2630 0.946 0.3336 0.956 (2,3) 0.5255 0.925 0.3626 0.938 0.4105 0.944 0.2773 0.945 0.3224 0.953 0 1000 2000 3000 4000 5000 0.4 0.45 0.5 0.55 0.6 0.65 prior 1 prior 2 prior 3 0 1000 2000 3000 4000 5000 0.4 0.5 0.6 0.7 0.8 prior 1 prior 2 prior 3 0 1000 2000 3000 4000 5000 0.55 0.6 0.65 0.7 0.75 0.8 prior 1 prior 2 prior 3 fig. 5.1. trace plots with c.s (2, 2) with (n,n) = (30, 10) (top left), (1, 2) with (n,n) = (70, 30) (top right) and (1, 3) with (n,n) = (50, 30)(down), in unknown common c. copyright © 2023 assa. adv syst sci appl (2023) stress-strength reliability of a weibull-standard normal distribution 63 ta bl e 5. 3. b ia se s, m se s, a ve ra ge co nfi de nc e/ cr ed ib le le ng th s an d co ve ra ge pe rc en ta ge s fo re st im at es of r w he n c is kn ow n. pr io r4 pr io r5 pr io r6 (n , n ) c .s m l e a sy m p. c .i e xa ct c .i e xa ct c .i e xa ct c .i u m v u e e st . m se le ng th c .p e st . m se le ng th c .p e st . m se le ng th c .p e st . m se le ng th c .p e st . m se (3 0, 10 ) (1 ,1 ) 0. 51 46 0. 01 53 0. 53 22 0. 92 9 0. 60 24 0. 01 35 0. 46 37 0. 93 7 0. 60 16 0. 00 90 0. 43 36 0. 94 3 0. 60 06 0. 00 67 0. 40 25 0. 94 6 0. 49 64 0. 01 59 (2 ,2 ) 0. 51 03 0. 01 56 0. 53 68 0. 92 9 0. 51 06 0. 01 33 0. 46 99 0. 93 6 0. 51 75 0. 00 81 0. 44 55 0. 94 4 0. 48 56 0. 00 65 0. 40 10 0. 94 7 0. 49 94 0. 01 58 (3 ,3 ) 0. 60 53 0. 01 58 0. 52 75 0. 92 8 0. 49 92 0. 01 36 0. 57 66 0. 93 6 0. 49 83 0. 00 85 0. 44 10 0. 94 3 0. 51 42 0. 00 64 0. 20 41 0. 94 7 0. 51 52 0. 01 55 (1 ,2 ) 0. 60 54 0. 01 53 0. 52 45 0. 93 0 0. 60 84 0. 01 32 0. 46 11 0. 93 7 0. 60 71 0. 00 93 0. 44 64 0. 94 3 0. 51 14 0. 00 65 0. 40 07 0. 94 6 0. 60 50 0. 01 59 (1 ,3 ) 0. 49 48 0. 01 55 0. 52 39 0. 92 9 0. 51 54 0. 01 35 0. 46 85 0. 93 6 0. 60 91 0. 00 99 0. 43 81 0. 94 4 0. 51 38 0. 00 67 0. 39 85 0. 94 6 0. 49 60 0. 01 54 (2 ,3 ) 0. 60 75 0. 01 50 0. 52 27 0. 93 0 0. 51 32 0. 01 30 0. 57 61 0. 93 6 0. 48 84 0. 00 86 0. 43 30 0. 94 3 0. 49 24 0. 00 62 0. 39 05 0. 94 6 0. 51 41 0. 01 52 (5 0, 20 ) (1 ,1 ) 0. 60 37 0. 01 52 0. 53 25 0. 92 9 0. 60 57 0. 01 31 0. 46 34 0. 93 7 0. 49 75 0. 00 85 0. 44 62 0. 94 3 0. 49 77 0. 00 60 0. 39 59 0. 94 6 0. 49 80 0. 01 58 (2 ,2 ) 0. 49 37 0. 01 54 0. 53 76 0. 92 8 0. 48 52 0. 01 38 0. 46 10 0. 93 7 0. 48 91 0. 00 97 0. 44 75 0. 94 3 0. 49 18 0. 00 66 0. 40 84 0. 94 7 0. 49 67 0. 01 52 (3 ,3 ) 0. 60 04 0. 01 58 0. 52 15 0. 93 0 0. 60 47 0. 01 35 0. 57 55 0. 93 6 0. 48 60 0. 00 91 0. 44 89 0. 94 3 0. 48 85 0. 00 62 0. 39 00 0. 94 6 0. 49 10 0. 01 57 (1 ,2 ) 0. 51 69 0. 01 54 0. 53 42 0. 93 1 0. 48 84 0. 01 32 0. 46 98 0. 93 7 0. 51 43 0. 00 88 0. 44 98 0. 94 4 0. 60 80 0. 00 69 0. 39 84 0. 94 7 0. 60 65 0. 01 55 (1 ,3 ) 0. 48 76 0. 01 57 0. 53 24 0. 92 8 0. 49 73 0. 01 33 0. 57 43 0. 93 7 0. 49 34 0. 00 86 0. 43 60 0. 94 4 0. 49 18 0. 00 62 0. 40 38 0. 94 7 0. 60 90 0. 01 59 (2 ,3 ) 0. 49 49 0. 01 56 0. 53 30 0. 92 8 0. 49 63 0. 01 34 0. 46 61 0. 93 6 0. 60 60 0. 00 99 0. 43 40 0. 94 4 0. 51 70 0. 00 60 0. 39 06 0. 94 7 0. 49 82 0. 01 52 (5 0, 30 ) (1 ,1 ) 0. 51 65 0. 01 35 0. 48 39 0. 93 4 0. 60 05 0. 01 10 0. 43 64 0. 94 1 0. 60 70 0. 00 70 0. 41 79 0. 94 6 0. 51 07 0. 00 48 0. 37 77 0. 95 2 0. 60 19 0. 01 50 (2 ,2 ) 0. 49 62 0. 01 36 0. 48 83 0. 93 4 0. 49 04 0. 01 09 0. 43 94 0. 94 1 0. 60 30 0. 00 67 0. 40 06 0. 94 7 0. 51 33 0. 00 47 0. 36 95 0. 95 4 0. 51 10 0. 01 50 (3 ,3 ) 0. 60 68 0. 01 36 0. 49 49 0. 93 4 0. 49 32 0. 01 07 0. 44 14 0. 94 2 0. 60 93 0. 00 66 0. 40 94 0. 94 7 0. 51 41 0. 00 55 0. 36 70 0. 95 0 0. 51 07 0. 01 53 (1 ,2 ) 0. 60 20 0. 01 37 0. 48 17 0. 93 3 0. 60 75 0. 01 03 0. 44 01 0. 94 3 0. 60 80 0. 00 71 0. 41 63 0. 94 8 0. 49 97 0. 00 52 0. 37 01 0. 95 9 0. 49 76 0. 01 52 (1 ,3 ) 0. 51 75 0. 01 35 0. 49 16 0. 93 4 0. 60 34 0. 01 02 0. 43 16 0. 94 1 0. 49 04 0. 00 65 0. 41 92 0. 94 7 0. 48 82 0. 00 47 0. 36 88 0. 95 0 0. 51 54 0. 01 55 (2 ,3 ) 0. 51 07 0. 01 30 0. 48 49 0. 93 3 0. 49 91 0. 01 11 0. 44 62 0. 94 2 0. 51 02 0. 00 78 0. 41 69 0. 94 8 0. 51 46 0. 00 49 0. 37 10 0. 95 3 0. 60 93 0. 01 50 (7 0, 30 ) (1 ,1 ) 0. 60 92 0. 01 30 0. 48 44 0. 93 3 0. 60 18 0. 01 09 0. 44 39 0. 94 2 0. 60 09 0. 00 66 0. 40 73 0. 94 6 0. 51 19 0. 00 50 0. 36 19 0. 95 5 0. 49 93 0. 01 39 (2 ,2 ) 0. 51 49 0. 01 37 0. 48 00 0. 93 4 0. 49 93 0. 01 11 0. 44 15 0. 94 2 0. 51 57 0. 00 69 0. 41 57 0. 94 8 0. 48 61 0. 00 46 0. 37 98 0. 95 6 0. 49 35 0. 01 52 (3 ,3 ) 0. 49 59 0. 01 34 0. 49 49 0. 93 3 0. 51 19 0. 01 09 0. 44 76 0. 94 1 0. 49 98 0. 00 67 0. 13 0. 94 7 0. 60 60 0. 00 53 0. 37 12 0. 95 3 0. 60 76 0. 01 39 (1 ,2 ) 0. 51 39 0. 01 38 0. 48 00 0. 93 4 0. 60 81 0. 01 08 0. 44 92 0. 94 2 0. 51 57 0. 00 73 0. 40 41 0. 94 7 0. 51 28 0. 00 52 0. 36 23 0. 95 5 0. 60 59 0. 01 50 (1 ,3 ) 0. 60 22 0. 01 33 0. 49 21 0. 93 3 0. 51 35 0. 01 09 0. 44 52 0. 94 2 0. 48 75 0. 00 74 0. 40 66 0. 94 7 0. 60 97 0. 00 47 0. 37 53 0. 95 0 0. 51 00 0. 01 54 (2 ,3 ) 0. 49 69 0. 01 32 0. 49 03 0. 93 3 0. 49 22 0. 01 10 0. 44 65 0. 94 2 0. 48 54 0. 00 65 0. 41 03 0. 94 8 0. 51 56 0. 00 43 0. 37 04 0. 95 8 0. 49 99 0. 01 37 copyright © 2023 assa. adv syst sci appl (2023) 64 r. kazemi ta bl e 5. 4. b ia se s, m se s, a ve ra ge cr ed ib le le ng th s an d co ve ra ge pe rc en ta ge s fo re st im at es of r in ge ne ra lc as e. pr io r7 pr io r8 pr io r9 (n , n ) c .s a m l e m l e m c m c c .i m c m c c .i m c m c c .i e st . m se e st . m se e st . m se le ng th c .p e st . m se le ng th c .p e st . m se le ng th c .p (3 0, 10 ) (1 ,1 ) 0. 58 62 0. 01 58 0. 61 13 0. 01 33 0. 59 54 0. 01 16 0. 48 40 0. 93 7 0. 58 69 0. 01 05 0. 45 82 0. 94 2 0. 58 21 0. 00 81 0. 40 23 0. 94 3 (2 ,2 ) 0. 58 58 0. 01 53 0. 59 14 0. 01 35 0. 61 46 0. 01 19 0. 48 46 0. 93 9 0. 61 95 0. 01 04 0. 45 24 0. 94 2 0. 61 07 0. 00 85 0. 41 65 0. 94 9 (3 ,3 ) 0. 60 81 0. 01 57 0. 60 92 0. 01 30 0. 61 67 0. 01 06 0. 57 05 0. 93 7 0. 58 54 0. 01 02 0. 45 94 0. 94 3 0. 60 97 0. 00 89 0. 39 87 0. 94 5 (1 ,2 ) 0. 60 86 0. 01 59 0. 59 61 0. 01 26 0. 59 15 0. 01 19 0. 57 32 0. 93 7 0. 61 19 0. 01 01 0. 44 59 0. 94 3 0. 58 86 0. 00 71 0. 40 65 0. 94 5 (1 ,3 ) 0. 60 80 0. 01 51 0. 61 83 0. 01 33 0. 58 06 0. 01 15 0. 57 26 0. 93 8 0. 61 28 0. 01 00 0. 44 04 0. 94 1 0. 60 03 0. 00 77 0. 39 45 0. 94 6 (2 ,3 ) 0. 58 98 0. 01 56 0. 60 26 0. 01 38 0. 61 70 0. 01 07 0. 57 08 0. 93 8 0. 60 08 0. 00 97 0. 45 27 0. 94 2 0. 58 37 0. 00 77 0. 39 75 0. 94 7 (5 0, 20 ) (1 ,1 ) 0. 59 15 0. 01 48 0. 61 17 0. 01 36 0. 60 14 0. 01 12 0. 48 80 0. 93 6 0. 58 95 0. 00 98 0. 45 29 0. 94 2 0. 61 35 0. 00 88 0. 40 44 0. 94 7 (2 ,2 ) 0. 59 90 0. 01 51 0. 59 49 0. 01 38 0. 61 79 0. 01 15 0. 57 66 0. 93 9 0. 59 13 0. 00 97 0. 45 14 0. 94 3 0. 61 78 0. 00 87 0. 41 45 0. 94 4 (3 ,3 ) 0. 61 97 0. 01 53 0. 61 94 0. 01 32 0. 60 04 0. 01 18 0. 47 20 0. 93 6 0. 60 42 0. 00 96 0. 45 94 0. 94 2 0. 58 89 0. 00 75 0. 41 76 0. 94 5 (1 ,2 ) 0. 60 14 0. 01 55 0. 59 73 0. 01 31 0. 59 09 0. 01 20 0. 47 07 0. 93 7 0. 61 20 0. 01 00 0. 44 41 0. 94 4 0. 59 56 0. 00 72 0. 39 65 0. 94 5 (1 ,3 ) 0. 60 50 0. 01 52 0. 60 26 0. 01 33 0. 60 16 0. 01 10 0. 47 30 0. 93 8 0. 61 07 0. 01 00 0. 45 09 0. 94 2 0. 60 14 0. 00 72 0. 40 75 0. 94 5 (2 ,3 ) 0. 59 03 0. 01 58 0. 60 61 0. 01 37 0. 59 24 0. 01 15 0. 48 46 0. 93 6 0. 59 01 0. 01 03 0. 46 16 0. 94 4 0. 58 54 0. 00 85 0. 39 65 0. 94 3 (5 0, 30 ) (1 ,1 ) 0. 61 08 0. 01 41 0. 59 94 0. 01 17 0. 61 54 0. 00 98 0. 44 05 0. 94 1 0. 58 87 0. 00 65 0. 40 92 0. 94 6 0. 61 41 0. 00 53 0. 35 29 0. 95 6 (2 ,2 ) 0. 58 63 0. 01 31 0. 58 71 0. 01 02 0. 61 13 0. 00 91 0. 43 83 0. 94 1 0. 60 05 0. 00 67 0. 39 44 0. 94 6 0. 61 86 0. 00 44 0. 35 56 0. 95 5 (3 ,3 ) 0. 61 00 0. 01 35 0. 61 14 0. 01 14 0. 58 96 0. 00 97 0. 44 45 0. 94 2 0. 59 66 0. 00 67 0. 40 90 0. 94 3 0. 59 94 0. 00 48 0. 35 37 0. 96 0 (1 ,2 ) 0. 60 28 0. 01 39 0. 59 23 0. 01 15 0. 61 59 0. 00 95 0. 43 69 0. 94 2 0. 59 12 0. 00 70 0. 41 05 0. 94 6 0. 58 65 0. 00 61 0. 35 75 0. 95 5 (1 ,3 ) 0. 59 00 0. 01 35 0. 60 50 0. 01 09 0. 58 37 0. 00 98 0. 43 81 0. 94 1 0. 59 53 0. 00 75 0. 41 62 0. 94 7 0. 61 96 0. 00 54 0. 35 40 0. 95 3 (2 ,3 ) 0. 58 23 0. 01 34 0. 60 88 0. 01 07 0. 61 38 0. 00 86 0. 43 55 0. 94 0 0. 59 21 0. 00 69 0. 40 54 0. 94 4 0. 59 31 0. 00 41 0. 36 14 0. 95 5 (7 0, 30 ) (1 ,1 ) 0. 61 45 0. 01 40 0. 59 16 0. 01 01 0. 60 87 0. 00 90 0. 43 02 0. 94 1 0. 59 65 0. 00 61 0. 40 69 0. 94 3 0. 59 01 0. 00 47 0. 35 61 0. 95 5 (2 ,2 ) 0. 60 97 0. 01 40 0. 61 50 0. 01 14 0. 60 74 0. 00 95 0. 43 08 0. 94 3 0. 60 71 0. 00 66 0. 41 04 0. 94 5 0. 60 18 0. 00 42 0. 36 23 0. 95 7 (3 ,3 ) 0. 60 05 0. 01 35 0. 61 26 0. 01 17 0. 61 50 0. 00 91 0. 43 74 0. 94 2 0. 58 22 0. 00 64 0. 41 64 0. 94 5 0. 58 33 0. 00 53 0. 36 61 0. 96 0 (1 ,2 ) 0. 58 06 0. 01 32 0. 60 85 0. 01 08 0. 59 46 0. 00 98 0. 43 59 0. 94 3 0. 58 81 0. 00 72 0. 41 54 0. 94 5 0. 59 32 0. 00 43 0. 35 24 0. 95 5 (1 ,3 ) 0. 61 52 0. 01 35 0. 58 51 0. 01 03 0. 58 78 0. 00 97 0. 43 35 0. 94 1 0. 59 31 0. 00 72 0. 40 80 0. 94 6 0. 60 80 0. 00 43 0. 35 22 0. 95 4 (2 ,3 ) 0. 59 08 0. 01 32 0. 61 17 0. 01 18 0. 59 15 0. 00 83 0. 44 70 0. 94 1 0. 58 68 0. 00 71 0. 39 64 0. 94 6 0. 61 30 0. 00 51 0. 36 22 0. 95 6 copyright © 2023 assa. adv syst sci appl (2023) stress-strength reliability of a weibull-standard normal distribution 65 0 1000 2000 3000 4000 5000 0.2 0.25 0.3 0.35 0.4 0.45 0.5 prior 7 prior 8 prior 9 0 1000 2000 3000 4000 5000 0.35 0.4 0.45 0.5 0.55 prior 7 prior 8 prior 9 0 1000 2000 3000 4000 5000 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 prior 7 prior 8 prior 9 fig. 5.2. trace plots with c.s (1, 1) with (n,n) = (50, 20) (top left), (2, 3) with (n,n) = (50, 30) (top right), (3, 3) with (n,n) = (70, 30) (down) in general case. consider the existence and uniqueness of the mle and construct the asymptotic and hpd intervals for r. second, when the common second shape parameter, c, is known, we obtain the mle and exact bayes estimate of r. third, in general case, when x ∼ wsn(a, c1) and y ∼ wsn(b, c2), we provide ml, aml and bayesian inferences of r. from the simulation results, which obtained by the monte carlo method, in point estimates, we observed that the bayes estimates have the minimum value of mses. also, in bayesian inference, the informative priors perform better than non-informative ones. furthermore, the mcmc method performs better than lindley’s approximation. in interval estimates, we observed that the hpd credible intervals have the better performance than the asymptotic confidence intervals. also, in bayesian inference, the hpd credible intervals based on informative priors have the smallest average lengths and largest coverage percentages. references 1. asgharzadeh, a. valiollahi, r., & raqab, m.z. (2011). stress-strength reliability of weibull distribution based on progressively censored samples, sort, 35(2), 103–124. 2. balakrishnan, n., & aggarwala, r. (2000). progressive censoring: theory, methods and applications. boston, ma: birkhauser. copyright © 2023 assa. adv syst sci appl (2023) 66 r. kazemi 3. bourguignon, m., silva, r. b., & cordeiro, g.m. (2014). the weibull-g family of probability distributions, journal of data science., 12, 53–68. 4. chen, m. h., & shao, q.m. (1999). monte carlo estimation of bayesian credible and hpd intervals, journal of computational and graphical statistics, 8, 69–92. 5. kizilaslan, f., & nadar, m. (2016). estimation of reliability in a multicomponent stressstrength model based on a bivariate kumaraswamy distribution, statistical papers, 59, 307—340. 6. kizilaslan, f., & nadar, m. (2016). estimation and prediction of the kumaraswamy distribution based on record values and inter-record times, journal of statistical computation and simulation, 86, 2471–2493. 7. kohansal, a. (2019). on estimation of reliability in a multicomponent stress-strength model for a kumaraswamy distribution based on progressively censored sample, statistical papers, 60, 2185–2224. 8. lindley, d.v. (1980). approximate bayesian methods, trabajos de estadistica, 3, 281– 288. 9. nadar, m., papadopoulos, a., & kizilaslan, f. (2013). statistical analysis for kumaraswamy’s distribution based on record data, staistical papers, 54, 355–369. 10. shoaee, sh., & khorram, e. (2015). stress-strength reliability of a two-parameter bathtub-shaped lifetime distribution based on progressively censored samples, communications in statistics-theory and methods, 44, 5306–5328. 11. singh, b., & goel, r. (2018). reliability estimation of modified weibull distribution with type-ii hybrid censored data, iranian journal of science and technology, transactions a: science, 42(3), 1395–1407. appendix a the standard extreme value distribution has the p.d.f. and c.d.f. as g(v) = ev−ev , g(v) = 1− e−ev . therefore, based on the observed data {u1, . . . , un} and {v1, . . . , vm}, ignoring the constant value, the log-likelihood function is as follows: ℓ∗(ξ1, ξ2, σ) ∝ −n log(σ) + n∑ i=1 ti − n∑ i=1 (ri + 1)eti −m log(σ) + m∑ j=1 zj − m∑ j=1 (sj + 1)ezj , (6.26) where ti = ui − ξ1 σ , zj = vj − ξ2 σ . now by taking derivatives with respect to ξ1, ξ2 and σ from (6.26), we obtain the following equations: ∂ℓ∗ ∂ξ1 = − 1 σ [ n− n∑ i=1 (ri + 1)eti ] = 0, ∂ℓ∗ ∂ξ2 = − 1 σ [ m− m∑ j=1 (sj + 1)ezj ] = 0, ∂ℓ∗ ∂σ = − 1 σ [ n+m+ n∑ i=1 ti − n∑ i=1 (ri + 1)tie ti + m∑ j=1 zj − m∑ j=1 (sj + 1)zje zj ] = 0. copyright © 2023 assa. adv syst sci appl (2023) stress-strength reliability of a weibull-standard normal distribution 67 to obtain the amles, let qi = 1− n∏ j=n−i+1 j + n∑ k=n−j+1 rk j + 1 + n∑ k=n−j+1 rk , i = 1, . . . , n, q̄j = 1− m∏ i=m−j+1 i+ m∑ k=m−i+1 sk i+ 1 + m∑ k=m−i+1 sk , j = 1, . . . ,m. by expanding the functions eti and ezj in taylor series around the points νi = log ( − log(1− qi) ) , ν̄j = log ( − log(1− q̄j) ) , respectively and keeping the first order derivatives, we have eti = ai + biti and ezj = āj + b̄jzj , where ai = eνi(1− νi), bi = eνi , āj = eν̄j(1− ν̄j) and b̄j = eν̄j . with the similar manner to [1], we derive the amles of ξ1, ξ2 and σ, say ξ̃1, ξ̃2, and σ̃, respectively, by ξ̃1 = a1 − σ̃b1, ξ̃2 = a2 − σ̃b2, σ̃ = −(d1 +d2) + √ (d1 +d2)2 + 4(c1 + c2)(e1 + e2) 2(c1 + c2) , where a1, a2, b1, b2, c1, c2, d1, d2, e1, e2 are given as follows: a1 = n∑ i=1 (ri + 1)biui n∑ i=1 (ri + 1)bi , b1 = n∑ i=1 ai − n∑ i=1 ri(1− ai) n∑ i=1 (ri + 1)bi , c1 = n, a2 = m∑ j=1 (sj + 1)b̄jvj m∑ j=1 (sj + 1)b̄j , b2 = m∑ j=1 āj − m∑ j=1 sj(1− āj) m∑ j=1 (sj + 1)b̄j , c2 = m, d1 = n∑ i=1 aiui − a1b1 ( n∑ i=1 (ri + 1)bi ) − n∑ i=1 riui(1− ai), d2 = j2∑ j=1 ājvj − a2b2 ( m∑ j=1 (sj + 1)b̄j ) − m∑ j=1 sjvj(1− āj), e1 = n∑ i=1 (ri + 1)bi(ui − a1) 2, e2 = m∑ j=1 (sj + 1)b̄j(vj − a2) 2. copyright © 2023 assa. adv syst sci appl (2023) 68 r. kazemi appendix b let u(λ) be a function of parameter value. the bayes estimate of u(λ), under the squared error loss function is e ( u(λ)|data ) = ∫ u(λ)eq(λ)dλ∫ eq(λ)dλ , where q(λ) = ℓ(λ) + ρ(λ), ℓ(λ) and ρ(λ) are the logarithm of likelihood function and prior density of λ, respectively. lindley has been approximated e(u(λ)|data) as e ( u(λ)|data ) = u+ 1 2 ∑ i ∑ j (uij + 2uiρj)σij + 1 2 ∑ i ∑ j ∑ k ∑ p ℓijkσijσkpup ∣∣∣∣ λ=λ̂ , where λ = (λ1, . . . , λm), i, j, k, p = 1, . . . ,m, λ̂ is the mle of λ, u = u(λ), ui = ∂u/∂λi, uij = ∂2u/∂λi∂λj , ℓijk = ∂3ℓ/∂λi∂λj∂λk, ρj = ∂ρ/∂λj , and σij = (i, j)th element in the inverse of matrix [−ℓij] all calculated at the mle of parameters. when we confront the case of three parameter λ = (λ1, λ2, λ3), lindley’s approximation conducts to e ( u(λ)|data ) = u+ (u1d1 + u2d2 + u3d3 + d4 + d5) + 1 2 [a(u1σ11 + u2σ12 + u3σ13) +b(u1σ21 + u2σ22 + u3σ23) + c(u1σ31 + u2σ32 + u3σ33)], calculated at λ̂ = (λ̂1, λ̂2, λ̂3), where di = ρ1σi1 + ρ2σi2 + ρ3σi3, i = 1, 2, 3, d4 = u12σ12 + u13σ13 + u23σ23, d5 = 1 2 (u11σ11 + u22σ22 + u33σ33), a = ℓ111σ11 + 2ℓ121σ12 + 2ℓ131σ13 + 2ℓ231σ23 + ℓ221σ22 + ℓ331σ33, b = ℓ112σ11 + 2ℓ122σ12 + 2ℓ132σ13 + 2ℓ232σ23 + ℓ222σ22 + ℓ332σ33, c = ℓ113σ11 + 2ℓ123σ12 + 2ℓ133σ13 + 2ℓ233σ23 + ℓ223σ22 + ℓ333σ33. in our case, for (λ1, λ2, λ3) ≡ (a, b, c) and u = r = a a+b , we have ρ1 = a1 − 1 a − b1, ρ2 = a2 − 1 b − b2, ρ3 = a3 − 1 c − b3, σij , i, j = 1, 2, 3 are obtained by using ℓij , i, j = 1, 2, 3 and ℓ111 = 2n a3 , ℓ222 = 2m b3 , ℓ133 = ℓ331 = ℓ313 = − n∑ i=1 w(xi, ri, c, 2), ℓ233 = ℓ332 = ℓ323 = − m∑ j=1 w(yj, sj, c, 2), ℓ333 = 2(m+ n) c3 − a n∑ i=1 w(xi, ri, c, 3)− b m∑ j=1 w(yj, sj, c, 3), copyright © 2023 assa. adv syst sci appl (2023) stress-strength reliability of a weibull-standard normal distribution 69 and other ℓijk = 0. moreover, u3 = ui3 = 0, i = 1, 2, 3, and u1, u2 are given in (2.8). also, u11 = −2b (a+b)3 , u12 = u21 = a−b (a+b)3 , u22 = 2a (a+b)3 . so, d4 = u12σ12, d5 = 1 2 (u11σ11 + u22σ22), a = ℓ111σ11 + 2ℓ131σ13 + ℓ331σ33, b = 2ℓ232σ23 + ℓ222σ22 + ℓ332σ33, c = ℓ113σ11 + 2ℓ133σ13 + 2ℓ233σ23 + ℓ223σ22 + ℓ333σ33. it is notable that all parameters are evaluated at (â, b̂, ĉ). copyright © 2023 assa. adv syst sci appl (2023) introduction inference on r with unknown common c maximum likelihood estimation of r approximation maximum likelihood estimation of r asymptotic confidence interval bayes estimation lindley's approximation mcmc method inference on r with known common c maximum likelihood estimation of r bayes estimation uniformly minimum variance unbiased estimate of r estimation of r in general case maximum likelihood estimation of r approximation maximum likelihood estimation of r bayes estimation simulation study conclusion microsoft word 1477-article text-6882-1-18-20240311 adv syst sci appl 2024; 01; 58-68 published online at https://ijassa.ipu.ru. optimizing mechanical strength in w-30cu/xtic alloys via solid-state sintering: a comprehensive characterization lahcene mebarki1*, abdelyamine boukhobza2, kamel fedaoui3, lazhar baroura4, ali abderrazak tadjeddine2 1) mechanics research center (crm), constantine, algeria 2) lseter laboratory, nour bachir university center, el-bayadh, algeria 3) lereesi, laboratory, hns-re2sd, batna, algeria 4) ista, university of constantine 1, algeria abstract: the objective of this study is to investigate the influence of titanium carbide (tic) on both the mechanical properties and microstructural characteristics of tungsten-copper (w-cu) alloys. the manufacturing process adopted for this exploration employs solid-phase sintering, conducted at a temperature of 1,200°c under a uniaxial pressure of 30 mpa. this process is employed to synthesize w-30cu/xtic alloy powders, where 'x' denotes the weight percentages of 5, 10, and 15 for tic. the alloys underwent a comprehensive analysis employing various analytical techniques to assess their mechanical and microstructural properties. these techniques include scanning electron microscopy (sem), x-ray diffractometry (xrd), density evaluations, hardness tests, and flexural strength testing. notably, the incorporation of titanium carbide (tic) resulted in substantial improvements in both hardness and flexural strength, with the alloy containing 15% tic exhibiting a microhardness value of 1420±5 hv0.1 and a flexural strength of 1745 n/mm². additionally, the sintering process led to a significant increase in density, rising from 8.6g/cm³ to 13.8g/cm³. these findings underscore the positive impact of tic and the sintering process on the alloy's mechanical and structural characteristics. keywords: wcu-tic alloys, sintering, density, microhardness. 1. introduction tungsten-based (w) alloys are renowned for their dual-phase structure, providing a favorable blend of hardness and ductility. among these, w-cu alloys are extensively used in electronic enclosures and heat sinks. however, the transformation of phases during sintering weakens the alloy's hardness [27]. this study aims to address this by exploring the addition of tic to wcu alloys. the miscibility and uncreativeness between copper and tungsten pose challenges in their manufacturing, leading to the adoption of powder metallurgy methods [3]. the incorporation of tic restricts the dissolution and precipitation of tungsten particles within the liquid copper phase, contributing to the prevention of phase growth during sintering [2]. heavy tungsten-based (w) alloys exhibit a biphasic structure comprising a high-density body-centered cubic (bcc) phase and a face-centered cubic (fcc) phase, which offers a favorable combination of hardness and ductility [6]. heavy tungsten-based alloys find applications in various industries, including aerospace, defense, medical, and nuclear fields, where their unique combination of properties is advantageous. these alloys are often used in situations where a balance of strength, radiation shielding, corrosion resistance, and hardness is crucial for optimal performance. renowned for their exceptional mechanical properties, including high strength, radiation absorption capacity, resistance to wear and corrosion, as well * corresponding author: lahcenemebarki@yahoo.fr optimizing mechanical strengthin w-30cu/xtic alloys… 59 copyright ©2024 assa. adv. in systems science and appl. (2024) as elevated hardness and density, these alloys find widespread utility across various industrial domains [10, 14]. within this spectrum, w-cu alloys have garnered substantial attention, notably in electronic enclosures and heat sinks for integrated circuits, owing to their compatibility with copper. nonetheless, the susceptibility to the formation of a brittle cuw phase during sintering due to the disparate solubility of copper in tungsten poses a challenge, potentially compromising the hardness of w-cu alloys [23, 5]. the adjustment of tungsten (w) and copper (cu) ratios is a critical factor in leveraging the exceptional characteristics of both elements. however, challenges arise in the manufacturing of w-cu alloys due to the immiscibility (limited ability to form a homogeneous mixture) and chemical inertness between copper and tungsten. these hurdles complicate the process of achieving a uniform distribution of the two elements in the alloy, impacting the alloy's overall properties and performance. overcoming these challenges requires careful consideration and innovative approaches in alloy design and manufacturing processes to optimize the benefits of both tungsten and copper in the final material [4, 26]. powder metallurgy methods such as infiltration melting, liquid phase sintering (lps), and hot-pressing sintering (hps) have been the recourse, augmented by the introduction of minute quantities of ceramic phases or rare earth oxide particles, enabling controlled recrystallization and enhanced dispersion [25, 7]. the process of augmenting tungsten-based alloys to enhance mechanical properties entails intentionally introducing specific elements such as nickel (ni), cobalt (co), iron (fe), copper (cu), and titanium carbide (tic). this augmentation is carried out with careful and precise control of transformations that are dependent on the presence of tungsten. the objective is to deliberately prevent polyphasic grain growth, where multiple phases or structures develop within the alloy, particularly during critical manufacturing processes such as sintering and solid-state fusion. the meticulous regulation of these transformations is crucial for achieving the desired microstructure and, consequently, improving the overall mechanical performance of the tungsten-based alloy [8, 13]. significantly, the incorporation of tic particles notably restrains tungsten particle dissolution and precipitation within the liquid copper phase, curtailing phase growth during sintering [17, 28]. the exploration of tic incorporation at varied concentrations aims to achieve a microstructure with finely distributed γ-cu bonding phases, leading to pronounced enhancements in mechanical properties [19, 11], demonstrating remarkable thermal stability at elevated temperatures and mitigating grain growth within the tungsten component [16]. this study is dedicated to a comprehensive exploration of tic's impact on the mechanical properties and microstructure of w-cu-tic alloys, intending to unravel the intricacies of densification, sintering behavior, and strengthening within the domain of tic-enriched w-cu alloys [9, 24]. 2. materials and mathematical formulation the experimentation involved the utilization of tungsten (w), copper (cu), and titanium carbide (tic) powders with a particle size ranging from 38 to 50 micrometers and approximately 99% purity, following established protocol [12]. a meticulous preparation process commenced with high-energy ball milling of sample blends at a rotation speed of 950 rpm for an 8-hour duration. subsequently, these blends underwent cold uniaxial compaction at a pressure of 30 mpa [18, 20]. the resultant powder mixture, encompassing 20 g of the composite, was then compressed at a pressure of 30 mpa, yielding cylindrical pellets measuring ø15 mm in diameter and 10 mm in thickness. moreover, (3555) mm specimens were fashioned specifically for flexural testing [15, 1]. density measurements were conducted both preand post-sintering using a digital densimeter voyager pro (vp6102c). the samples underwent sintering in a hydrogen sintering furnace equipped with a resistance filament, employing a constant heating rate of 5 °c/min. sintering transpired over a two-hour period at a temperature of 1200°c [21]. 60 l. mebarki, a. boukhobza, k. fedaoui, l. baroura, a. a. tadjeddine copyright ©2024 assa adv. in systems science and appl. (2024) the microstructural alterations were scrutinized using a scanning electron microscope (sem), the jeol jsm-6063 model, while computer-assisted microhardness measurements were executed employing an affri-dm2d hardness tester. furthermore, flexural testing was performed utilizing the we-10k universal hydraulic testing machine. the determination of various phases present in the samples was facilitated by employing a philips diffractometer. these comprehensive experimental procedures provided critical insights into the microstructural changes, mechanical properties, and phase compositions of the synthesized wcu-tic alloys [22]. 3. results and discussion sem analysis revealed evolving morphological features of wcutic alloys with varying titanium carbide (tic) proportions. post-sintering sem images displayed uniformity, corroborated by density measurements. phases included a tungsten-copper matrix integrated with tic particles, showing diverse carbon bonding arrangements with tungsten facets. xrd analysis identified predominant tungsten phases, copper, and tic, showcasing a phase evolution with increased tic content, emphasized by intensified tic peaks. mechanical tests unveiled tic's influence: hardness improved linearly with tic content, reaching 1273 hv0.1 at 15%, while flexural strength peaked at 1745 n/mm². density rose with sintering temperature and tic content, hitting 13.8 g/cm³ at 15%. excessive tic aggregation adversely affected performance. these results underscore tic's significance in structurally and mechanically shaping wcutic alloys. 3.1. scanning electron microscope sem analysis the sem analysis presented in figure 3.1. delineates the morphological progression of wcutic alloys characterized by varying titanium carbide (tic) proportions. notably, the absence of pores attests to the homogeneity achieved during the sintering process, a finding corroborated by density measurements. the recorded densities for the respective tic proportions of 5%, 10%, and 15% stand at 9.60 g/cm³, 11.40 g/cm³, and 13.80 g/cm³. the sintered tungsten and copper alloy, w30cu, exhibits a discernible solid matrix, while the identification of tic presents a more intricate analysis. figures 3.1a, 3.1b, and 3.1c, acquired via sem imaging of sintered samples containing 5%, 10%, and 15% tic, unveil distinct features within the microstructures. the white coloring denotes copper particles serving as binders, gray areas represent various tungsten phases, and dark gray tic particles are observed in limited quantities. the presence of tic particles fluctuates with the tic content due to carbon's varied bonding with distinct tungsten facets. (a) (b) (c) fig. 3.1. sem image by backscattered electrons of samples with different tic proportions, (a) x=5%, (b) x=10%, (c) x=15% of tic figure 3.1a illustrates the surface micrograph of a sample with 5% titanium carbide (tic), revealing black spots—identified by yellow arrows—attributed to tungsten cabride (wc) contamination during sintering. the 10% tic sample displays varied distribution of tic particles within the microstructure, yet achieving uniform dispersion is challenging, notably at optimizing mechanical strengthin w-30cu/xtic alloys… 61 copyright ©2024 assa. adv. in systems science and appl. (2024) higher tic contents, with 10% and 15% by weight, due to tungsten particle aggregation. agglomeration results in elongated conglomerates and persists even after sintering, indicating the stability of wc particles. the distribution of tic particles by weight between 5% and 10% accentuates solid phase growth effects, evidencing the diffusion of carbide particles into the tungsten matrix and forming larger wc particles. notably, decarburization of tic particles is more pronounced in samples with 15% tic. xrd patterns affirm tic diffusion into the tungsten matrix, forming solid solutions, corroborating earlier discussions. the growth and aggregation of tic particles are observed notably in samples with 5% and 10% tic, with larger conglomerates formed post-sintering for the 15% tic sample. 3.2. x-ray diffraction (xrd) analysis the xrd analysis, a standard technique for identifying crystalline phases within an alloy, divulges critical insights into the chemical composition of the w-cu-tic alloy reinforced with varying tic proportions (5%, 10%, and 15% by weight). figures 3.2a, 3.2b, and 3.2c elucidate the xrd diffraction patterns, delineating distinct characteristics for each composition. the positioning of peaks primarily suggests the dominance of tungsten (w) within the alloy. in addition to w, the distinct presence of copper (cu) and titanium carbide (tic) is evident across all xrd patterns. figure 3.2a emphasizes the presence of w and w2c phases in the sintered w powders. notably, the xrd analysis in figure 3.2b reveals peaks corresponding to (ti, w) alongside base phases such as w, w2c, and a solid solution of 30cu. remarkably, the intensity of cu-associated peaks diminishes as the weight percentage of tic increases (figure 3.2c). this decline in cu intensity suggests a controlled sintering environment, limiting the formation of undesired oxides. fig. 3.2. phase analysis by xrd of tic the xrd patterns further emphasize the augmentation of tic-associated peaks with increasing proportions of reinforcing particles. this emphasizes the presence of cu particles within the w matrix. additionally, the analysis highlights a consistent amount of cu across all samples, evidenced by peaks of similar intensity. however, the intensity of tic-related peaks notably escalates with the rise in tic content from 5% to 15% by weight. this observation signifies the proportional increase in tic particles within the alloy matrix, underlining the impact of tic content variation on the xrd patterns and the consequent phase composition changes within the alloy. the densification resulting from sintering reduces pores in the microstructure at 1200°c, as noted in figure 3.3 (highlighted by red arrows). this reduction in porosity positively influences the superior mechanical properties observed. 62 l. mebarki, a. boukhobza, k. fedaoui, l. baroura, a. a. tadjeddine copyright ©2024 assa adv. in systems science and appl. (2024) (a) (b) (c) fig. 3.3. scanning electron microscopy om image of porosity, (a) x=5%, (b) x=10%, (c) x=15% of tic 3.3. effect of tic on hardness the microhardness analysis of the wcutic alloy in figure 3.4 displays a discernible and consistent enhancement in microhardness values, spanning from 871 to 1273 hv0.1, concomitant with the augmentation of tic content. the rise in microhardness reaches its peak, approximately 1273 hv0.1, at a concentration of 15% by weight of tic. comparatively, microhardness values of 871 hv0.1 were recorded for 5% by weight of tic, and 1024 hv0.1 for the sample with 10% by weight of tic. (a) (b) fig. 3.4. .the evolution, (a) microhardness, (b) flexural strength this increase in microhardness can be attributed to the well-dispersed tic reinforcement particles within the tough phase. fig. 3.5. .density evolution before and after sintering the precision in conducting microhardness tests ensured the positioning of the indenter precisely within areas where both w and tic particles are present (figure 3.6a, 3.6b, and 3.6c). optimizing mechanical strengthin w-30cu/xtic alloys… 63 copyright ©2024 assa. adv. in systems science and appl. (2024) the integration of hard ceramic particles within the alloys enriched dislocations, a pivotal factor contributing to the observed peak microhardness in samples featuring a higher concentration of reinforcement. the reinforcement's contribution to enhancing microhardness lies in the enrichment of dislocations within the alloys. the integration of hard ceramic particles like tic curtails the energy of grain boundaries, impedes their mobility, and obstructs dislocation movement. this impediment preserves the size of tungsten particles and stabilizes tic grain size during sintering, ultimately resulting in escalated microhardness. (a) (b) (c) fig. 3.6. microhardness impression and values, (a) x=5%, (b) x=10%, (c) x=15% of tic additionally, the improved microhardness, owing to reinforcement addition, can also be attributed to the higher hardness of cu and tic particles in comparison to the ductile tungsten alloy matrix. the hardness superiority of these reinforcement particles significantly contributes to the overall enhancement in microhardness observed in the composite alloy. table 3.1 presents the measured values for hardness (hv), diagonal x, diagonal y, and force (gf) for the respective cases (a), (b), and (c), aligned with varying tic content percentages. table 3.1 indentation measurement result case hardness (hv) diagonal x (microns) diagonal y (microns) force (gf) case (a), x=05% 871 47.9 44.3 1000 case (b), x=10% 1024 41.6 43.6 1000 case (c), x=15% 1273 39.2 37.2 1000 3.4. effect of tic on flexural strength figure 4b delineates the flexural strength of samples featuring various weight proportions of tic particles, indicating a discernible rise in flexural strength as tic content within the 64 l. mebarki, a. boukhobza, k. fedaoui, l. baroura, a. a. tadjeddine copyright ©2024 assa adv. in systems science and appl. (2024) tungsten-copper alloy matrix increases. the graph demonstrates that alloys integrating 15% tic particles display notably elevated flexural strength compared to samples with 5% and 10% tic. initially, samples with 5% tic showcased a flexural strength of 1200 n, significantly elevating to 1745 n/mm² with 15% tic, attributing this enhancement directly to the addition of tic. comparatively, the w-cu-15% tic alloy experienced a remarkable 26% increment in flexural strength. the observed strength improvement can be attributed to various factors. primarily, the interfaces between copper and tic create substantial multidirectional thermal stresses, fortifying the alloy's structure. additionally, the refinement of the crystalline structure contributes significantly to enhanced alloy strength. the robust bond between the w-cu alloy matrix and tic particles, along with their uniform distribution, emerge as pivotal elements fostering increased flexural strength in composites (figure 6a, b, and c). moreover, the integration of tic particles restricts dislocation mobility through dispersion strengthening mechanisms. as the number of tic particles increases, the reduced distance between them necessitates more force to move dislocations through these particles. the resultant increase in composite strength is directly associated with this modification. these findings align with the established advantages of using powder metallurgy processes for manufacturing alloys, facilitating uniform grain distribution and robust bonding between phases, particularly enhancing composite sample strength through tic addition to the wcu matrix. the observed strength increase can also be attributed to microstructural modifications induced by the reinforcement's addition, resulting in a high density of dislocations and significant disparities between the reinforcements and the tungsten matrix. 3.5. effect of density the investigation of density in our study included two pivotal measurements: green density and post-sintering density, denoting the sample density before and after the sintering process. the determination involved measuring the mass and dimensions of the samples employing a caliper. the initial green density exhibited a progression from 7.2 for 5%, 7.8 for 10%, to 8.6 g/cm³ for 15% tic. the influence of sintering temperature on density is markedly apparent. postsintering, the density highlighted a proportional increase, starting at a minimum of 9.6 for 5%, escalating to 11.4 for 10%, and reaching 13.8 g/cm³ for 15% tic, as portrayed in figure 4c. the green density exhibited a gradual rise commensurate with the increase in tic content up to 15%, reaching 8.6 when the sintering temperature was adequately high. this rise confirms the cumulative impact of the cu binder and the complete pore filling between particles. moreover, the post-sintering density swiftly ascended to 13.8, nearing the theoretical density, signifying the substantial influence of both sintering temperature and tic content on the overall density. this observation underscores the critical interplay between tic content and sintering conditions in shaping the density profile, indicating their pivotal roles in the density enhancement process. 3.6. effect of adding tic on mechanical properties the investigation delved into the impact of integrating tic at various concentrations on the microstructure and mechanical attributes. addition of fine tic particles, exhibiting an average size, led to their cohesion around w particles, thereby enhancing interfacial bonding strength, wear resistance, and microhardness. the tic grains' distributed and uniform shape, as depicted in figure 6 (in the yellow rectangle), wielded a positive influence, culminating in the attainment of essential mechanical properties crucial in industrial applications, particularly in addressing wear and corrosion issues across diverse uses. nonetheless, excessive tic aggregation markedly compromised the overall performance. among the tested wcutic samples, those harboring 15% tic by weight optimizing mechanical strengthin w-30cu/xtic alloys… 65 copyright ©2024 assa. adv. in systems science and appl. (2024) showcased superior performance metrics, boasting an average microhardness of 1273±5 hv0.1, a flexural strength of 1745 n·mm², and a post-sintering density of 13.8 g/cm³, with uniform microstructures (figure 6a, b, and c). this study unequivocally demonstrates the applicability of the solid-state sintering method in alloy preparation, offering a promising path for further explorations in the domain of sintered alloy fabrication. the outcomes highlight the critical balance required in tic integration, emphasizing the necessity to avoid excessive tic aggregation to preserve and enhance the desirable mechanical attributes within the composite, while also underscoring the pivotal role of tic distribution and shape in optimizing the overall performance of wcutic alloys. (a) (b) (c) fig. 3.7. sem image by backscattered electrons of phase, (a) x=5%, (b) x=10%, (c) x=15% of tic 4. conclusion in summary, the successful fabrication of a tic-reinforced wcu alloy through solid-state sintering using metal powders has demonstrated promising wear resistance properties. while this study specifically focused on the wcu/tic ratios, it is crucial to acknowledge the potential for other compositions. the recommended manufacturing process involving grinding, compaction, and sintering at 1200°c proved integral in achieving reduced pore sizes through densification and particle size optimization. the density of wcu/tic alloys reached 13.8%, and significant porosity reduction was observed in samples containing 15% tic after a 2-hour sintering period. notably, the sample incorporating 15% tic displayed minimal porosity and highlighted an optimal hardness of approximately 1273 hv0.1. conversely, samples featuring 5% tic exhibited relatively lower hardness values, below 900 hv0.1. this alloy exhibited improved mechanical attributes alongside enhanced flexural strength, rendering it suitable for various application areas, aligning with the study's initial expectations. the performance of tic-w-cu alloy in real-world applications can be influenced by various external factors, including temperature and environmental conditions. tic-w-cu alloy 66 l. mebarki, a. boukhobza, k. fedaoui, l. baroura, a. a. tadjeddine copyright ©2024 assa adv. in systems science and appl. (2024) may exhibit good performance at high temperatures, as tungsten carbide (wc) is known for its high melting point and hardness. however, the copper (cu) component has a lower melting point, so extreme temperatures might affect the overall stability of the alloy. tungsten and copper have different coefficients of thermal expansion. this can lead to internal stresses within the alloy when exposed to temperature variations. these stresses may affect the structural integrity of the alloy over time. the presence of aggressive chemicals in the environment can influence the alloy's performance. copper is susceptible to corrosion, and the corrosive nature of the environment may affect the overall integrity of the alloy. tic-wcu alloys are often used in applications requiring high wear resistance. the alloy's performance can be influenced by the type and intensity of mechanical loading. extreme impact or abrasion can lead to wear and degradation over time. the manufacturing process, especially the sintering conditions, can affect the microstructure and properties of the alloy. proper control of these parameters is crucial for achieving desired performance. nevertheless, practical assessments under real operational conditions are crucial to provide concrete insights into potential technical challenges that may arise in industrial settings. these evaluations would further validate the alloy's performance and durability, offering invaluable insights for its industrial implementation. references 1. boukhobza, a., fedaoui, k., mebarki, l., arar, k. & baroura, l. 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(2022). ultrafine-grained tungsten heavy alloy prepared by high-pressure spark plasma sintering, materials, 15(17), 6168. https://doi.org/10.3390/ma15176168 adv syst sci appl 2024; 02:192–202 published online at https://ijassa.ipu.ru. a generalized bi-objective scheduling algorithm for batch-of-tasks on heterogeneous computing systems mohammad qasim1, mohammad sajid1, maria lapina2, mikhail babenko2* 1aligarh muslim university, aligarh, india 2north-caucasus federal university, stavropol, russia abstract: the high energy consumption of data centers and its contribution towards greenhouse gases demand energy-efficient management of resources. energy consumption of computing resources encourages the development of bi-objective scheduling algorithms optimizing the makespan of jobs and energy consumption of computing resources. in general, the problem of job scheduling and bi-objective optimization falls in the np-complete combinatorial optimization problem category. to address the bi-objective scheduling problem, a generalized biobjective scheduling algorithm (z*) for batch-of-tasks (bot) applications on the heterogeneous computing system (hcs) has been proposed. the bot represents the set of independent tasks from multiple applications, and the hcs represents the computational environment consisting of processors with different frequencies. to schedule tasks, the z* algorithm takes decisions using the optimization function of energy consumption and completion time of tasks based on the given weights. the weight could be fractional or integer, so the z* algorithm represents a set of different algorithms. the proposed algorithm is beneficial for cloud data centers/serviceoriented computing to execute customer jobs based on the demand, whether the customer needs high throughput or low cost of execution. keywords: batch-of-tasks, heterogeneous computing system, scheduling, makespan, energy 1. introduction on one side, the performance of data centers has been improved by increasing the parallel servers, but on the other side, the overall energy consumption of data centers housing thousands of servers needs to be minimized due to negative environmental effects [1, 2]. for example, a data center houses the ten fastest systems (tianhe-2) to provide services to users. since a tianhe-2 server requires 17.808 mw of power, making a single hour’s execution on tianhe-2 cost roughly $1780.80 on the assumption that the average price of electricity is $100/mw-hour. if the cloud data service is available for 8 hours per day, 20 days per month, the cost of operating a cloud data center apart from the cooling cost will be $34 million per year. to provide power to the data center, the powerhouse needs to consume 71230 kg coal and 712320l water while producing 177545.8 kg co2 and 5342.4 kg so2 [1–5]. furthermore, high energy use raises temperature since each electronic component’s temperature has a direct inverse relationship with its energy use, which lowers system reliability. according to nrdc [2], in 2013, it was predicted that 12 million computing servers across 3 million data centers in the us used 91 billion kilowatt-hours of electricity. by 2020, this energy usage is projected to increase by a direct 53%, reaching 140 billion ∗corresponding author: mgbabenko@ncfu.ru a generalized bi-objective scheduling algorithm... 193 kilowatt-hours. due to the high financial costs, decreased reliability of computer systems, and adverse impacts on the environment, the excessive energy usage of data centers has become a concern, which encourages the development of energy-efficient hardware and software management [6]. heterogeneous computing systems (hcss) are the base of data centers due to the replacement and up-gradation of computing nodes. compared to homogeneous computing, hcs can be thought of as the coordinated employment of a variety of processing units with varying capabilities to give higher performance, better energy efficiency, and more cost-effectiveness [7]. to deploy energy efficiency, energy-aware scheduling models are required to develop that execute jobs with minimum possible energy consumption as well as makespan. scheduling generally involves distributing tasks among processors with the purpose of optimizing one or more quality of service criteria. to find the task-to-processor mapping that minimizes both makepan and energy usage, scheduling methods are necessary. since data centers execute jobs in three categories, i.e., lower cost of execution with no constraints on time, lower execution time with no constraint on cost and the combination of both. the cost of execution is measured in terms of direct energy consumption and the time span holding the resources of data centers. therefore, the data centers require a bi-objective scheduling algorithm that has the capability to offer different weights to both execution time and energy consumption/cost. in this work, a generalized bi-objective scheduling algorithm (z*) for batch-of-tasks (bot) on a hcs has been proposed. the bot is a very common job representing the set of independent tasks from multiple applications, and hcs represents the computational environment consisting of processors with different fixed frequencies. the proposed z* algorithm schedules tasks based on a weighted aggregation of energy consumption and completion time of tasks. the remainder of the paper is organized as follows. the next section explains the related scheduling algorithm that schedules bot on hcs. the design principles, which include the bot model, the hcs model, and the energy consumption model, are presented in section 3. the proposed generalized z* algorithm and its optimization function are described in section 4. section 5 shows that z* is the same as some of the existing algorithms, and the simulation results present the behavior for different cases. the paper ends in section 6 with concluding remarks and possible future directions. 2. related work the standard methods for reducing the makespan of bot are maxmin [10, 11], minmin [10, 11], maxmin+ [12], minmin+ [12], duplication algorithms [14], analytical models [15] and dependent task algorithms [15]. the expected time to compute (etc) simulation model provides a standard by which to compare the performance of these algorithms on a distributed computing platform. several of the algorithms are described in the following list. the maxmin algorithm operates in two stages as well. in the first stage, it determines the shortest time to complete each unscheduled task, considering all processors. in the second stage, the task that will take the longest to finish overall is selected and assigned to the appropriate processor. this process is repeated until all tasks have been scheduled, at which point the newly scheduled item is removed from the list of unscheduled tasks [10, 11]. the minmin algorithm comprises two stages. it determines the minimal completion time for every unscheduled task in the first stage while considering all processors. the job that takes the shortest time to complete overall is selected and allocated to the suitable processor in the second stage. once a task has been scheduled, it is deleted from the list of unscheduled tasks, and the process is then repeated until every item has been scheduled [10, 11]. another form of a minmin algorithm is known as minmin+ with less time complexity. instead of being task-oriented like in the minmin, and minmin+ algorithm operates from a processor-oriented perspective. after initialization, the minmin+ algorithm determines the completion times of each task with regard to each processor. after that, algorithm runs in copyright © 2024 assa. adv syst sci appl (2024) 194 m. qasim, m. sajid, m. lapina, m. babenko two phases. the algorithm initially identifies the task that will take each processor the fewest amount of time to complete. the task that will take the least amount of time to complete will be scheduled and taken out of the set. since the current task’s scheduling decision only influences the single processor’s readiness time, this approach only calculates the time requirements of all unscheduled tasks with regard to the same single processor. this avoids calculating the completion times of tasks with respect to all other processors. the two-stage process continues until all tasks have been scheduled [12]. there have been proposed a plethora of bi-objective algorithms with the aim of optimizing energy consumption as well as makespan. some of the algorithms that are energy-aware are global scheduling with shared slack reclamation (gssr) [17], fixed-order list scheduling with shared slack reclamation (flssr) [17], shortest task first for computer with minimum energy (stf-cme) algorithm [18], longest task first for computer with minimum energy (ltf-cme) algorithm [18], maxmax min [19], minmean min [19], minmin [20], minsuff [20], minmin-minmin [20], kpb-sufferage [20], g2 [21], ests algorithm [22], min-energy algorithm and esheft algorithm [24]. to minimize energy consumption while meeting deadlines for all the tasks, zhang et al. presented several energy-efficient algorithms, such as the shortest task first for computer with minimum energy (stf-cme) algorithm and the longest task first for computer with minimum energy (ltf-cme) algorithm. the initial stage of each algorithm schedules tasks to the machine, and the final stage establishes the ideal or nearly ideal speeds for each task. according to simulation research, hybrid algorithms outperform and provide the optimum job schedules for both discrete and continuous-speed computers [18]. for independent tasks, diaz et al. suggested three low-cost energy-aware algorithms: maxmax min, minmax min and minmean min. the two phases of the low-cost heuristic’s operation. the method gives each task a priority in the first phase, sorts the tasks into decreasing priority order, and then uses the ordered task’s list to choose the best processor. in the next phase, calculate the scoring function (sf) for the chosen task relative to each processor and scheduling the work for the processor with the lowest sf value. up till all the tasks are scheduled, the process is repeated [19]. twenty bi-objective energy-conscious scheduling methods were proposed and assessed by nesmachnow et al. [20] utilizing the algorithms maxmin, minmin, suffrage, and k-percent best (kpb). based on a heterogeneous multi-core environment, the proposed algorithms are assessed—minmin functions in two stages. in the first stage, it chooses task and processor combinations that, when all processors are considered, have the shortest completion times for each unscheduled task. the work is allocated to the relevant processor in the second stage using pair from the stage 1 pairs that minimize energy consumption. the most recent task scheduled is then removed from the list of unscheduled tasks; this process is repeated for each task that is still on the list until all tasks have been scheduled [20]. sajid et al. propose two bi-objective energy-aware scheduling algorithms using the weighted aggregation method to schedule bot applications on heterogeneous processors with the objective of optimizing makespan and energy consumption. the proposed algorithms employ processor-oriented to schedule the tasks and hence, offer lower time complexity in comparison to minmin, maxmin, minsuff and minmin algorithms [10, 11, 20]. the effectiveness of the suggested algorithms has been compared with maxmin, minmin, minmin and minsuff using the etc simulation model. the simulation study reveals that the proposed algorithm (g2) performs better than peer algorithms in terms of flowtime. energy consumption and makespan [21]. in this work, we propose a generalized algorithm z* that offers the selection of weights is flexible to energy consumption and/or completion times. it is a useful algorithm to compare the different versions of the algorithms. copyright © 2024 assa. adv syst sci appl (2024) a generalized bi-objective scheduling algorithm... 195 3. the problem formulation in this section, the problem is formulated, including bot model, a hcs, design principles and energy consumption [21]. 3.1. batch-of-tasks and heterogeneous computing system the bot b consists of n tasks and is represented by b = t1, t2 . . . . . . ti . . . tn . every task ti ∈ b is independent and atomic in nature, i.e., the tasks have no precedence constraints and cannot be partitioned/ distributed/pre-empted during execution. the expected number of cycles (cy clei) required to execute a task ti ∈ b is also given. the bot applications have been used by many streams, such as high energy physics, geophysics, molecular quantum mechanics, weather forecasting, climate, cryptographic analysis, and tomographic reconstruction. the hcs is explained in the next paragraph and figure 3.1. shows the mechanism of hcs in cloud environments. fig. 3.1. heterogeneous computing in cloud environments. the hcs h represents the execution environment consisting of m heterogeneous processors and is characterized by h = p1, p2. . . . . . pj. . . pm . all processors are heterogeneous, i.e., the frequency fj and voltage vj of the processor pj ∈ h is different from the processor pk ̸= pj . a high-speed interconnection network connects all processors, e.g., myrinet, to execute real-time tasks. the processors are not dynamic voltage-frequency scaling-enabled, and each processor pj ∈ h runs on fixed voltage (vj) with corresponding fixed frequency (fj). 3.2. design principles the scheduler maintains information about a scheduler node, which assigns the bot b of n independent tasks to the hcs h of m heterogeneous processors and manages the batch of independent jobs on a periodic basis. in the sequence of their execution, all tasks assigned to such a processor pj is en-queued in a local queue qj . we employ the following notations and design ideas in this work. etc [n][m]: the proposed algorithm has been assessed using the simulation model known as expected-time-to-compute (etc). the estimated execution time for every task on each processor [10,11,20] is stored in a matrix called etc[n][m] that is ordered n-by-m. copyright © 2024 assa. adv syst sci appl (2024) 196 m. qasim, m. sajid, m. lapina, m. babenko it is possible to calculate the task ti expected runtime on the processor pj using frequency fj as follows: etc (i, j) = cy clei fj , (3.1) alloc [n][m]: the distribution of n tasks across m processors for execution is represented by an allocation matrix (alloc) of order n by m. whether or not the task ti has been assigned to the processor pj determines whether the value of alloc (i, j) is zero or one. the allocation matrix should satisfy the following restrictions after scheduling n tasks. m∑ j=1 alloc(i, j) = 1 ∀1 ≤ i ≤ n (3.2) n∑ i=1 alloc(i, j) = k, ∀0 ≤ k < n, ∀1 ≤ j ≤ m (3.3) prt j: it indicates the amount of time that the processor pj has been ready. the prt j for the processor pj is the earliest time processor pj has the ability to finish all of the tasks that were previously scheduled for completion. if there is no task allocated to the processor pj , the prt j = 0. rt ij: it indicates the task ti ready time on the processor pj . rtij is the processor-ready time for the processor pj on which task ti is scheduled i.e. rt ij = prt j (3.4) ct ij: it shows when a task ti on the processor pj was completed. it may be expressed mathematically as the summation ready time of the processor pj and the expected time it will take to complete the task ti. ct ij = prt j + etc (i, j) (3.5) after scheduling the task ti on processor pj , the processor ready time is also updated as prt j = ct ij (3.6) makespan of bot b: it is referred to as the maximum amount of time that elapsed between the first scheduled task’s start time and the last scheduled task’s completion time of the batch of n tasks, and it may be calculated as mkb,h = max {ctij|alloc(i, j) = 1, 1 ≤ i ≤ n, } (3.7) 3.3. heterogeneous computing systems energy model to execute the bot b on hcs h, the processors consume energy deepening on the schedule and order of tasks. the cmos-fabricated processor-based system’s energy comprises of static energy (estatic) and dynamic energy (edynamic) with the significant energy consumption factor being edynamic. for a specific frequency fj and voltage vj , the dynamic power (pdynamic) used by a processor pj is given by [24]. pdynamic (pj) = a ∗ cj ∗ vj2 ∗ fj (3.8) where the activity factor is represented by the constant α, cj represented physical capacitance, processor’s pj clock frequency (speed) denoted by fj and vj denotes the voltage copyright © 2024 assa. adv syst sci appl (2024) a generalized bi-objective scheduling algorithm... 197 applied to the processor corresponds to the frequency fj . the physical capacitance (cj) and activity factor (α) is determined during the manufacturing and design phases. the supply voltage vj and the clock frequency fj are configurable either statically (when compiling) or dynamically (when running). due to the heterogeneity of the processors utilized, the energy consumption of a task ti is dependent on the frequency of operation, voltage, and the amount of time necessary to complete the task. consequently, a task’s ti energy consumption on a processor pj may be expressed as ecij = a ∗ cj ∗ vj 2 ∗ fj ∗ etc (i, j) (3.9) the energy consumed by bot b on hcs h is given by eb, h = n∑ i=1 m∑ j=1 alloc (i, j) ∗ ecij (3.10) 4. the generalized bi-objective algorithm (z*) in this section, the proposed generalized bi-objective algorithm (z*) is explained. the z* algorithm is a static scheduling algorithm to schedule a bot of n independent tasks on hcs h of m heterogeneous processors. the objective of the algorithm is to optimize either makespan or energy or both depending on the weights assigned to objectives. the z* algorithm represents the set of 25 algorithms for integer weights and a set of infinite algorithms based on fraction weights. the algorithms a1 to a25 are given in table 4.1. the explanation of the z* algorithm is given as follows. 4.1. optimization function specification the bot b of n independent tasks results in negligible idle slots while scheduling on an hcs h of m processors [21]. the optimization function (of) employed by the z* algorithm does not consider the idle; however, it considers both the energy consumption and completion time of tasks as follows. completion time of task – as defined in equation (3.5). energy consumption of task – as per equation (3.9). the weighted sum approach, which is commonly used in the literature, has been applied for the optimization function (of), combining both objectives into one objective by multiplication of weights for each objective [8]. the drawback of this optimization technique is also widely known: only solutions found in the convex region of the pareto front (a group of solutions to the problem that are not dominated by any other solutions), and only one solution from the pareto front is discovered in each run. however, the solution obtained using the weighted sum method is always pareto-optimal if the weights are positive for all objectives. the definition of the of based on the normalized values of the goals measured in the same unit (constant). then, given task ti on the processor pj , the normalized energy (ne(ti, pj)) and the normalized completion time (nc(ti, pj)) can be calculated as ne (ti, pj) = ec (i, j) max(ec (i, k)) ∀pk ∈ p (4.11) nc (ti, pj) = ct ij max(ct ik) ∀pk ∈ p (4.12) it is obvious that 0 < ne(ti, pj) ≤ 1, 0 < nc(ti, pj) ≤ 1. the optimization function (of (ti, pj)) can then be written using (11) and (12) as copyright © 2024 assa. adv syst sci appl (2024) 198 m. qasim, m. sajid, m. lapina, m. babenko of (ti, pj) = β1 ∗nc (ti, pj) + β2 ∗ne (ti, pj) (4.13) the optimization function is computed for the task ti considering all processors followed by the scheduling decisions. the performance and priority of the z* algorithm depend on the constants (β1, β2) of an optimization function that integrates both objectives and based on the value, attempts to optimize both. therefore, the values of constants (β1, β2) are critical and must be chosen based carefully depending on the requirements. the constants beta1 and beta2 must be between –1 to +1, i.e., β1 ∈ [−1, 1], β2 ∈ [−1, 1] and must fulfil the following two constraints |β1|+ |β2| = 1 (4.14) |β1|+ |β2| ≠ 0 (4.15) here, constraint (4.14) states that the weights of the objectives cannot be zero, and the sum of weights assigned to objectives must be one. for positive values of β1 and β2, the of (ti, pj) minimizes the objectives, whereas it maximizes the objectives for negative values of β1 and β2. 4.2. the procedure of the z* algorithm algorithm 1 provides the pseudo code for the z* algorithm. algorithm 1: pseudo code of z* input: hcs h of m processors, bot b of n tasks output: schedule eb,h ,mkb,h , s 1 generate etc using equation (3.1) 2 assign values to β1 and β2 for optimization function 3 while b ̸= ϕ do 4 st = ϕ // the pair of task and processors that have the lowest cost functions 5 for ∀ti ∈ h do 6 for ∀pi ∈ h do 7 calculate ne(ti, pj) by using eq (4.11) 8 calculate nc(ti, pj) by using eq (4.12) 9 calculate of (ti, pj) by using eq (4.13) for specidied values of β1 and β2 10 end 11 find pair (ti, pj) with min value of (ti, pj) 12 add pair (ti, pj) to st i.e. st = st ∪ (ti, pj) 13 end 14 find pair (t′, pk) from st with min. value of of (t′, pk) 15 set value in alloc matrix corresponding to (t′, pk) 16 allocate task t′ on processor pk 17 eliminate task t′ from b i.e. b = b − t′ 18 end 19 find schedule s using alloc matrix 20 calculate mkb,h using (3.7) 21 calculate eb,h using (3.10) 22 return schedule eb,h ,mkb,h , s first of all, the z* algorithm assigns values to their objectives, i.e., β1 and β2 and then proceed to schedule the tasks using a while loop from steps 3 to 18. the z* algorithm computes the optimization function (of (ti, pj)) for each unscheduled task against all processors using equation (4.13) and determines the pair (ti, pj) of task ti and processor copyright © 2024 assa. adv syst sci appl (2024) a generalized bi-objective scheduling algorithm... 199 pj where processor pj offers a minimum value of optimization function. next, the algorithm finds the pair (t′, pk) that provides the lowest value for the optimization function (of (t′, pk)). the task t′ has now been assigned to pk and it has been withdrawn from bot b. this process is then repeated for the remaining tasks until they have all been assigned to a processor. 4.3. z* for integer weights of β1 and β2 for different values of β1 or β2 with the constraints in equations (4.14) and (4.15), z* represents a set of 25 algorithms a1 to a25 listed in table 4.1; some of them are already wellknown and established algorithms, viz. minmin [10, 11], maxmin [10, 11], minmin [20], minmin [20], minmin [20] and maxmin [20]. table 4.1 depicts 25 algorithms from a1 to a25. the a1, a2, a3, a11, a12 and a13 behave similarly to minmin, maxmin, minmin, minmin, maxmin and minmin, respectively. table 4.1. different forms of z* algorithm sr. no. phase i phase ii β1 β2 β1 β2 algorithm 1 1 0 1 0 a1 or minmin [10, 11] 2 1 0 -1 0 a2 or maxmin [10, 11] 3 1 0 0 1 a3 or minmin [20] 4 1 0 0 -1 a4 5 1 0 (-1, 1) (-1, 1) a5 6 -1 0 1 0 a6 7 -1 0 -1 0 a7 8 -1 0 0 1 a8 9 -1 0 0 -1 a9 10 -1 0 (-1, 1) (-1, 1) a10 11 0 1 1 0 a11 or minmin [20] 12 0 1 -1 0 a12 or maxmin [20] 13 0 1 0 1 a13 or minmin [20] 14 0 1 0 -1 a14 15 0 1 (-1, 1 ) (-1, 1) a15 16 0 -1 1 0 a16 17 0 -1 -1 0 a17 18 0 -1 0 1 a18 19 0 -1 0 -1 a19 20 0 -1 (-1, 1) (-1, 1) a20 21 (-1, 1) (-1, 1) 1 0 a21 22 (-1, 1) (-1, 1) -1 0 a22 23 (-1, 1) (-1, 1) 0 1 a23 24 (-1, 1) (-1, 1) 0 -1 a24 25 (-1, 1) (-1, 1) (-1, 1) (-1, 1) a25 4.4. z* for fractional weights of β1 and β2 it is very obvious that the weights of β1 and β2 between –1 to +1, there exist infinite weights for β1 and β2. therefore, the z* algorithm represents a set of infinite algorithms for different fractional weights. 4.5. z* as minmin the proof that z* behaves like minmin is given. the proof for maxmin, minmin, minmin, maxmin and minmin are similar. the outcome of the z* depends on the values of β1 and β2. for instance, it behaves similarly to the minmin algorithm (a1) if the values assigned to β1 and β2 are 1 and 0, respectively, for both phases. since the normalized completion time metric (nc(ti, pj)) for task ti on processor pj is the ratio of the completion time of a task ti on processors pj (ctij)) and a maximum of the completion times of the task ti against each processor pk (1 ≤ k ≤ m or pk ∈ p ). if β1 = 1 and β2 = 0, then the optimization function copyright © 2024 assa. adv syst sci appl (2024) 200 m. qasim, m. sajid, m. lapina, m. babenko (of (ti, pj)) in phase i minimizes the makespan of the tasks and generates pair of tasks and processors (ti, pj) such that pj offers minimum completion time to task ti. it is similar to phase i of the minmin algorithm. again, if β1 = 1 and β2 = 0, then the optimization function (of (ti, pj)) phase ii minimizes the makespan of the tasks. phase ii of the z* algorithm considers the pairs generated in phase i. it determines the pair of tasks and processors (tk, pl) such that pl offers minimum completion time to task tk among all the pairs. it is similar to phase ii of the minmin algorithm. 4.6. use of z* in data centers there are many ways to employ the z* algorithm based on the requirements. the z* algorithm can be used to: 1. minimize only completion time for higher throughput of data centers to generate more monetary benefits. 2. minimize only the energy consumption for lower negative environmental effects and cooling costs. 3. minimize only completion time for foreground jobs and minimize energy consumption for background jobs. 4. as a simulation model to compare the performance for different weights. 5. conclusion the energy consumption of data centers is a considerable issue from an environmental point of view due to ghg gas emissions and from the service provider’s view due to the extra burden of cooling costs. therefore, energy-efficient of resources at data centers is demanding. in this work, we proposed a generalized energy and makespan algorithm for batch-of-task (bot) on a heterogeneous computing system (hcs) which has the capability to minimize either the makespan, energy or both. the proposed z* algorithm schedules tasks based on a newly defined optimization function, and it represents the set of different algorithms based on the weights of objective functions. the z* algorithm can be employed by data centers in different ways, either for fast execution or low energy consumption, depending on the job types. this work can be further extended to a heterogeneous system of dvfs-enabled processors having different speed and voltage levels. this work can also be realized for a single dependent job, a batch of dependent jobs, open shop scheduling problems, as well as job shop scheduling problems. further, the proposed algorithm z* can be further extended for jobs having stochastic execution times. acknowledgements this work was supported by the russian science foundation 19-71-10033, 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(2016). energy-aware stochastic scheduling model with precedence constraints on dvfs-enabled processors. turkish journal of electrical engineering and computer sciences, 24(5), 4117–4128. copyright © 2024 assa. adv syst sci appl (2024) introduction related work the problem formulation batch-of-tasks and heterogeneous computing system design principles heterogeneous computing systems energy model the generalized bi-objective algorithm (z*) optimization function specification the procedure of the z* algorithm z* for integer weights of 1 and 2 z* for fractional weights of 1 and 2 z* as minmin use of z* in data centers conclusion adv syst sci appl 2024; 02:166–186 published online at https://ijassa.ipu.ru. data storage with increased survivability and reliability based on the residue number system nikolay kucherov1*, mikhail babenko1, egor shiriaev1, nguyen viet hung2 1north-caucasus federal university, stavropol, russia 2le quy don technical university, hanoi, vietnam abstract: the paper considers the principles of building reliable data storage and processing systems in cloud-fog environment. the factors affecting the reliability and survivability of systems are an-alyzed and the main types of failures are given. the principles of introducing information re-dundancy to improve reliability, as well as corrective capabilities and error detection algorithms of rns codes are considered. a scheme of data processing and storage in fog with increased re-liability is proposed. monte carlo modeling is carried out, and the results and conclusions are given, the probability distribution of failure occurrence is constructed, showing that failure probabilities obey the gamma probability distribution. keywords: fog-cloud computing, residue number system, error correction, reliability, fault tolerance 1. introduction as the volume of stored data continues to grow, an increasing number of users are transitioning to cloud storage solutions. in today’s context, the demands for data storage reliability within cloud environments are consistently escalating. reliability in cloud storage refers to its capacity to uphold data management and retention, all while upholding predefined quality benchmarks within operational conditions over time. this concept primarily encompasses the influence of intersystem factors, particularly the occurrence of unforeseen technological breakdowns, on the performance of cloud storage systems [1]. fog computing, often referred to as fog networks or edge computing, represents a decentralized computing model that draws data processing and storage closer to the devices and individuals responsible for generating and utilizing the data. it entails processing data at the network’s periphery, in close proximity to the data’s origin, rather than relying on a central server like a data center or cloud infrastructure. this approach notably di-minishes the volume of data necessitating transmission through the network, effectively curbing latency, bandwidth demands, and associated expenses. it also empowers re-al-time processing, expediting decision-making processes and enhancing overall respon-siveness [2, 3]. fog computing is particularly useful in the internet of things (iot) and other applications where devices are distributed and have limited processing power and connectivity. they allow these devices to process data locally and make decisions autonomously, without the need to constantly interact with the cloud. this can also improve security and privacy, as sensitive data can be processed and stored locally rather than being transmit-ted over a network. fog computing also provides a bridge between the cloud and the net-work edge, allowing data ∗corresponding author: nkucherov@ncfu.ru data storage with increased survivability and reliability... 167 to be processed and analyzed at different levels of granularity depending on the application requirements. they can be used in a variety of applications such as industrial control systems, smart city, transportation and healthcare [4]. one of the key advantages of edge computing lies in its capability to swiftly gather and analyze data in actual time, facilitating accelerated decision-making and heightened responsiveness. moreover, edge computing can serve to shift computational and storage burdens away from cloud environments, subsequently mitigating the expenses and de-lays typically linked with cloud-based solutions. nevertheless, this approach also intro-duces novel hurdles, including issues related to security, unpredictability, efficient data handling, and seamless deployment. fog computing infrastructure can include various devices such as routers, gateways, and embedded systems. these devices are capable of running various applications and providing services in a decentralized manner, in interaction with each other and with the cloud [5]. the survivability of a cloud-fog system refers to the robustness of its control and transceiver system against the action of causes from outside aimed at disabling the storage, as well as resistance to cascading failures. the causes of reduced survivability are divided into two types: spontaneous, deliber-ate, and suboptimal in terms of the choice of technical solutions in the construction of the system. the notions of reliability and survivability share commonalities while also present-ing notable distinctions. what binds them is the underlying principle of ”sustain-ability,” which encompasses a wide range of factors, encompassing diverse failures that can transpire. the sustainability index emerges as a composite metric that incorporates aspects of reliability, survivability, and fault tolerance indices. discrepancies between the concepts of reliability and survivability stem from distinct ways they manifest and their varying natures within the context of cloud and fog systems. factors leading to disruptions in their regular functioning vary significantly, including the nature and scope of failures, their duration, and the strategies employed for rectification and augmenting fault tolerance. consequently, the foundational data, calculation methodologies, precision, and fundamental nature of reliability and survivability metrics diverge substantially. the former group benefits from substantial statistical data, encompasses essential influencing factors, and boasts robust theoretical foundations. they prove useful for reasonably precise projections, calculations, design work, and simulations. on the other hand, the cluster of survivability metrics tends to convey a qualitative perspective on the behavior of the examined cloud-fog system when subjected to external impacts or the sequential spread of failures. now, let’s delve into the primary factors that impact system reliability: • mistakes associated with incorrect technical choices when designing and building a cloud storage system; • hard disk failures; • technical factors, factors that depend on the structure of the object and its operating modes, the use of redundancy, the organization of control and recovery after failure, the characteristics of component elements, the protection of elements from adverse factors, the quality of technological processes in the manufacturing process, the degree of adaptability for operation; • program factors depend on the quality of microprocessor software development; • operational factors depend on the external environment surrounding the object during operation; • climatic factors, temperature, humidity, solar radiation, wind loads, etc.; • the impact of maintenance on reliability is determined by the fact that facilities are often automated rather than automatic systems. a human being is a kind of link that fits into the structure of the system. relationship between man and technology. with the development of technology, the complication of its structure, the expansion of its functions, the question of the relationship between man and technology is becoming more and more acute. copyright © 2024 assa. adv syst sci appl (2024) 168 n. kucherov, m. babenko, e. shiriaev, n.v. hung the paper is further organized as follows. section 2 analyzes the factors affecting the reliability and survivability of control and communication systems. in section 3, the types of redundancy and redundancy introduction in rns are discussed. corrective capacity estimation and error correction algorithms in rns are given. section 4 deals with structural redundancy in cloud storage systems. section 5 considers the reliability of nonrepairable redundant objects and redundancy methods. section 6 is devoted to the principles of fault tolerant cloud systems. section 7 gives the structure of fault tolerant storage system in cloud fog environment. section 8 deals with the structure of a fault tolerant data processing system in fog environment. section 9 is devoted to the modeling of the proposed system for reliable data storage and processing in fog environment and the results obtained are described. 2. analysis of factors affecting reliability and survivability of control and communication systems the reliability of both cloud and fog storage systems becomes evident through instances of failures. the notion of failure is intricately connected to that of operability. operability represents the condition of a system in which it can execute designated functions in alignment with specified parameters and the quality demands of the services rendered. failure constitutes an arbitrary occurrence characterized by the disruption of the system’s operability. failures can be categorized as follows: • sudden – system parameters change instantly; • gradual – gradual change of system parameters due to wear and tear or aging; • work-in-progress – presence of defective elements, assembly and installation errors. enhancing the reliability of cloud and fog systems can be achieved through the utilization of mathematical tools, standardization practices, load distribution techniques, safeguarding against external influences, and the strategic selection of data storage schemes [6]. if the above-mentioned methods did not give the desired result, then redundancy should be used. at the stage of development of models and schemes of data storage the main reliability characteristics of the future system are laid down. it is necessary to reasonably choose the mathematical apparatus, the accuracy of the system, redundancy and, consequently, the optimal form of data representation (data packets) during their transmission and processing. the choice of the number system essentially and determines the reliability model of cloud storage. when building cloud storage, alternative non-positional and ad hoc number systems, which are oriented to solve a wider range of problems, are often not considered. in addition to applying mathematical approaches to improve reliability and survivability, structural approaches are also used, which consist in improving reliability by introducing additional redundancy and applying redundancy. the use of information methods in the form of corrective codes can improve the reliability and survivability of data storage systems. let’s examine the elements influencing the survivability of cloud systems. a notable distinction between evaluating survivability and other comparable concerns, like assessing reliability, is the general inability to apply probability concepts for the occurrence of specific scenarios. to enhance survivability, it becomes feasible to adopt a numerical system where the failure of a specific component does not result in the complete cessation of the entire system’s operation. it should be borne in mind that the number system significantly affects the algorithms of operation, exchange protocols, technical characteristics, reliability and survivability of the data storage system. for example, the decimal number system is irrational for use in computer technology, but is optimal for human perception of data. hence, the significance of selecting the operational numbering system for cloud storage systems is heightened. this choice primarily shapes the system’s reliability model, redundancy methodology, counteracting cascading failures, and the mitigation of accumulating errors. an intrinsic qualitative measure of survivability is the system’s copyright © 2024 assa. adv syst sci appl (2024) data storage with increased survivability and reliability... 169 resilience, denoted by e survived, subsequent to a predefined series of influences. for cloud and fog systems, the main reliability metrics are: 1. probability of uptime operation in the time interval from 0 to t0 p(t0) = p(0; t0) = p{ε ≥ to] = 1− f1(t0), where f1(t) = p[ε1 ≤ t] – distribution of failures in time to first failure; 2. probability of uptime operation of the object in the time interval from t to t+ t0 p(t, t+ t0) = p{ε1 ≥ t+ t0|ε1 > t] = p(0, t+ t0) p(0, t) = p(t+ t0) p(t) ; 3. failure distribution density f(t) = d dt f (t) = − d dt p(t); 4. intensity of object failures at the moment of time µ(t) = 1 1− f (t) · d dt f (t); 5. average time to failure ti = m(ε1) = ∫ ∞ 0 xdq(x) = ∫ ∞ 0 p(x)dx. presently, the primary pragmatic approach to enhance the dependability and survivability of storage systems involves the implementation of redundancy and error correction codes. redundancy entails augmenting the reliability of a cloud system by incorporating supplementary clouds beyond the minimum necessary for the system’s regular functioning. the main types of reservations: • time redundancy – utilizing excess time to accomplish tasks; • functional redundancy – the use of additional elements with which functions can be performed autonomously; • structural redundancy – a method of increasing reliability by introducing additional redundant elements included in the overall system structure; • information redundancy – improving reliability by storing multiple copies of data. optimal stability, reliability, and survivability emerge when a system seamlessly integrates and interweaves these methods. this state is exemplified by the residue number system (rns), which encompasses these attributes. when constructing cloud storage systems utilizing rns, a blend of redundant and operational clouds will be present (for each rns base), facilitating the harmonious incorporation of the various redundancy strategies outlined earlier. 3. information redundancy in cloud storage systems information redundancy is one of the main types of redundancy that is used to ensure the reliability and availability of information. it is the process of backing up data in order to restore it in case of data loss or corruption [7, 8]. the main task of information redundancy is to guarantee the safety of data in case of equipment failure, operator errors, natural disasters or other unforeseen circumstances. copyright © 2024 assa. adv syst sci appl (2024) 170 n. kucherov, m. babenko, e. shiriaev, n.v. hung there are various methods of information redundancy, including. full backup. with this method, a complete copy of all data is made, which requires a large amount of disk space and time to create a backup. however, data recovery after a failure is fast and easy. incremental backup. this method creates a copy of only the changed data since the last full or incremental backup. this method reduces the size of backups and backup creation time, but restoring data may take longer and require more complex procedures. differential backup. this method copies only the changed data since the last full backup. the difference from the incremental method is that the differential backup contains only changes since the last full backup, not since the last incremental backup. this method takes up more space than the incremental method, but the data recovery procedure is usually simpler and faster. snapshots. this method of data backup, which stores the state of the file system at a particular point in time. snapshots do not require additional copies of data, but they do require operating system support and are stored within the file system. replication is a method of creating copies of data on other servers or disk arrays in order to ensure availability and reliability of information in case of failure of the primary server. replication allows you to reduce the time it takes to recover data because if the primary server fails, you can switch to a backup that is already up and running. every information backup approach comes with its own strengths and weaknesses, and the selection of a method hinges on the distinct needs and constraints of the system. for instance, a complete backup might prove optimal for smaller systems handling modest data volumes, whereas in larger systems managing substantial data loads, incremental or differential backups could offer a more fitting solution. also an important aspect of information backup is the choice of storage location for backups, as they must be protected from potential threats such as hacking or natural disasters. in this case, it can be effective to use cloud data storage, which provides a high level of security and availability. diverse strategies are employed to enhance the dependability of data storage within cloud environments. the use of approaches that allow to perform error control contributes to greater utilization of cloud storage systems. to the code, which allows to control errors that occur in the process of information processing, we present the requirement of arithmetic. by arithmeticity of the code allowing to perform error control we will understand such a property of the code when we will have ”correct” information in the result when performing any actions with ”correct” information, i.e. we require transparency of performing actions with information. it is also necessary for the code to be able to fix error packets. this property is necessary to ensure system survivability. 3.1. principles of introducing information redundancy in the rns the residue number system is a numerical framework in which numbers are depicted as a collection of non-negative subtractions x1, x2, . . . , xn by mutually prime moduli p1, p2, . . . , pn [9, 10]. let the number x be represented by residues x1, x2, . . . , xn on the bases p1, p2, . . . , pn. let us assume that k residues are sufficient to uniquely represent the number x , k < n, p1 < p2 < · · · < pn, with working bases p1, p2, . . . , pk. the range of single-valued representation by selected modules is equal to the product of these modules pn = ∏n i=1 pi. corrective properties of the modular code appear when redundancy is introduced. let the number x be represented by residues x1, x2, . . . , xn on the bases p1, p2, . . . , pn. the number x in general case can vary in the range of |·|+pn . let us impose a restriction on the range of copyright © 2024 assa. adv syst sci appl (2024) data storage with increased survivability and reliability... 171 possible variation, for this purpose we will assume that k residues are sufficient for a one-toone representation of a number x , and k < n. let’s assume that p1 < p2 < · · · < pk < · · · < pn, whereby the working bases will be considered as p1 < p2 < · · · < pk. the range of single-digit representation of numbers on these bases is equal to pn =∏k i=1 pi. consequently, in the depiction of a number x using residues x1, x2, ..., xn, any subset of r residues can be omitted without compromising the clarity of the representation of the number x . this attribute empowers rns to manage errors effectively and establishes it as a non-linear code, recognized as an r-code. let us show by example the ability of r-code in rns to control errors. let some number 0 ≤ x ≤ 100. for representation in residues, we choose bases p1 = 3, p2 = 5, p3 = 7. the residues x1, x2, x3 on the corresponding bases can be used to represent a number in the range from 0 to 104. thus we have an almost lossless representation x = (x1, x2, x3). let x = 49 = (1, 4, 0). let’s add one more base p4 = 11. to the three existing bases. then our number x = 49 be represented by four residues x = (1, 4, 0, 5). suppose a fault occurred during data transmission and as a result an error occurred on the basis of p2 = 5. in the received data, we will leave only three parts of the data corresponding to the grounds p1 = 2, p2 = 7, p3 = 11, (1, 0, 5). let’s restore the data from the three parts and get that they correspond to the transmitted number 49. this erase operation can be performed with any of the bases and the data will be restored correctly as a result. a linear corrective l-code can also be constructed in rns. to construct the l-code it is enough to remove the restrictions on pairwise mutual simplicity of bases. removing the conditions of pairwise mutual simplicity leads to the fact that redundancy in the representation of a number x will appear not only due to the reduction of the number of working bases, but also due to the fact that the range of representation of a number x by k residues will be less than the power of the range of number representations. 3.2. estimation of corrective capacity of codes in rns in order to evaluate the ability of rns to detect, localize and correct errors we introduce the notion of weight wpn(|x|+pn ) of a number x . we will define the weight wpn(|x|+pn ) of a number x as the count of non-zero residues. this delineation of a number’s weight aligns with the description of code weight in the hemming metric. in wpn(|x|+pn ), the lower index of pn at w indicates that the number x is depicted using a quantity of residues such that x ∈ | · |+pn . evidently, the count of residues through which the number x , is represented is equal to ω in this case. writing the argument |x|+pn means that x ∈ | · |+pn . wpn (|x|) = n∑ i=1 σi (xi) , where σi = { 0, if xi = 0 1, if xi ̸= 0 utilizing this delineation of a number’s weight, wpn(|x|+pn ) it is possible to compute it both directly on x , and on the set of residuals x1, x2, . . . , xn. the concept of weight attributed to a number x depicted using residues can be employed to establish the concept of distance between two points in a space, where each point corresponds to the numbers x1 and x2 [11]. we denote the distance dx1,x2 between points in space x1 and x2 as the weight of the disparity between x1 and x2. dx1,x2 = wpn(|x1 −x2|+pn ) copyright © 2024 assa. adv syst sci appl (2024) 172 n. kucherov, m. babenko, e. shiriaev, n.v. hung to ascertain the error-correcting capabilities of the code, we calculate the mean weight w pn(| · |+pn ) of non-zero complexes. if the value of number x is modified within the range | · |+pn , the division period of zeros becomes pi, and the count of zeros in base pi will be equivalent to ωi = pk pi − 1, (i = 1, k) subsequently, the quantity of non-zero elements within the base becomes identical to ωi = pk − 1− ωi = pk − pk pi the sum of weights s of residues xi(i = 1, 2, .., k) when x changes from zero to pk − 1 is determined from the expression s = k∑ i=1 ( pk − pk pi ) = pk ( k − k∑ i=1 1 pi ) average weight w pn(| · |+pn ) can be defined as w pn(| · |+pn ) = s pk − 1 = pk pk − 1 · ( k − k∑ i=1 1 pi ) the more adept the code is in error correction, the greater the distinctions among numbers within the code representation, resulting in an increased code distance. consequently, the distance will vary between distinct numbers. when we compute the mean weight of the numbers composing the null space, the minimum code distance will always remain below the upper limit of the minimum code distance. the minimal weight of elements within the null code space is equivalent to dmin = min{wpn(| · |+pn )] = ω − k + 1 ≈ r + 1 when dmin ≈ r + 1 we can ensure the detection of any error. simultaneously, the utilization of rns empowers us to identify a greater quantity of errors. as an illustration, when utilizing rns with a single supplementary base exceeding the values of any worker, we can detect 100% of single errors and 95% of double errors. additionally, it’s feasible to identify all errors of a multiplicity t if t < dmin. this signifies that rns facilitates the detection of errors with a specific multiplicity, and the ascertainable error multiplicities are determined by the minimum code distance dmin. 3.3. algorithms for error detection in rns suppose a number x = (x1, x2, . . . , xn) has undergone a distortion and instead of a number x we have x̃ = (x̃1, x̃2, . . . , x̃n), such that x̃ ∈ | · |+pn . let us compute the difference e beetwin x̃ and x , which will determine the magnitude of the error e = |x̃− x|+pn since errors in distinct digits are mutually exclusive, it becomes viable to illustrate e in a manner akin to representing a number x , i.e., by residues e = (e1, e2, . . . , en) which will be defined ei = |x̃i − xi|+pn . copyright © 2024 assa. adv syst sci appl (2024) data storage with increased survivability and reliability... 173 from the accepted residuals x̃1, x̃2, . . . , x̃n we calculate the number x̃ by solving systems of comparisons x̃i ≡ x̃i(modpi), (i = 1, 2, . . . , k) to address this system of comparisons, we will employ the technique of orthogonalized vectors in the following format bi = (0, 0, . . . , 1, . . . , 0) = mipk pi where mi ∈ | · |+pi such that bi ≡ 1(modpi). by utilizing orthogonalized vectors, the number x can be expressed in the configuration x = k∑ i=1 xibi − rpk x(|x|+pk )pk where rpk (|x|+pk ) – core function [11–13], such that if the argument |x|+pk takes such a value that x is computed using orthonormalized vectors x ∈ | · |+pk . after computing the value of x̃ determine the values of the residues x̃ = (x̃1, x̃2, . . . , x̃n) of this number by the pk+1, . . . , pn. syndrome components are defined as cj = |xk+j − x̃k+j|+pk+j , j = 1, 2, . . . , n− k we will show that it is possible to determine the fact of error occurrence from syndrome components. let us represent the number x as x̃ = |x + e|+pk suppose that only residuals on bases p1, p2, . . . , pk, which we consider informational, are affected by errors. with this in mind. cj = ∣∣∣∣∣∣ ∣∣∣∣∣∑ ϕ∈jk eϕbϕ ∣∣∣∣∣ + pk ∣∣∣∣∣∣ + pk+j where eϕ – error value in ϕ-digit, jk – set formed from the numbers of information bases, the residuals of which are distorted. given that the value of ek is non-zero and the supplementary bases adhere to the criterion of mutual coprimality, any instance of at least one syndrome component being non-zero signifies the existence of an error. similarly, it can be demonstrated that error management is attainable when errors arise within the control bases, either individually or collectively. 3.4. error detection algorithm using positional features error control for any code is reduced to finding out whether the message belongs to the set of numbers that make up the zero space of the code. when performing an error finding operation for a number x , it is desirable that this range be close to the range p . the error detection process in rns is reduced to comparing the number x̃ with the value of the range pk [13]. the computed positional characteristic πpn(|x|+pk ) determines the position of x̃ inside the range | · |+pk . πpn(|x|+pk ) = { 0, if x < pk 1, if x ≥ pk copyright © 2024 assa. adv syst sci appl (2024) 174 n. kucherov, m. babenko, e. shiriaev, n.v. hung by the character of change, positional characteristics are divided into linear (monotonic) and nonlinear ones. an instance of a non-linear positional attribute is represented by the number rank function rpn(|x|+pk ). the function rpn(|x|+pk ) is periodic with period equal to pk. rpn(|x|+pk ) = ∣∣∣∣∣ k+1∑ i=1 xiβi ∣∣∣∣∣ + pk+1 , βi = ∣∣∣∣bi pk ∣∣∣∣+ pk+1 the function rpn(|x|+pk ) depends on k + 1 variables, and the possible domain of definition of the argument x is in the range | · |+pn , pn = pk+1pk. 4. structural redundancy in cloud storage systems structural redundancy in cloud storage systems is one method of ensuring the reliability and availability of data in the cloud. structural redundancy involves using multiple physical devices to store data and replicating data on these devices. cloud storage systems typically utilize distributed storage, which is a collection of physical storage devices working together. these devices can be located in different locations and have different characteristics such as capacity, speed, reliability and bandwidth [14]. to ensure the reliability and availability of data in cloud systems, various backup technologies such as replication, sharding, etc. are used. structural redundancy involves creating multiple copies of data on different physical devices using special algorithms and protocols that ensure data consistency and integrity. if one of the devices fails, data can be restored from backups stored on other devices. at the same time, the cloud continues to function and provide access to data without downtime or data loss. the basic idea behind structural redundancy in cloud storage systems is to create a distributed network of servers that work together to ensure data availability and reliability. each server in such a network can store only part of the data, which reduces the load on a single server and increases the speed of data access. classification of various methods of structural redundancy is performed according to the following features: • according to the redundant element inclusion scheme; • by homogeneity of redundancy; • by method of reserve activation; • on functional recovery of failed elements; • by reserve loading. the fundamental attribute of structural redundancy is its multiplicity, which signifies the proportion of redundant elements to the total number of elements (both redundant and primary). this proportion is expressed as a non-simplified fraction. the redundancy multiple m is determined by the ratio: m = p o where p – number of reserve elements; o – number of main elements. if m = 1, redundancy is executed with integral multiplicity. in practical terms, this signifies the presence of a single primary element and a corresponding redundant element, resulting in a total count of two elements for the object. when m = 3, redundancy is employed with integral multiplicity. this indicates the existence of one primary element along with three supplementary elements, resulting in a total of four elements. copyright © 2024 assa. adv syst sci appl (2024) data storage with increased survivability and reliability... 175 when m = 4 2 , it signifies a fractional multiplicity redundancy scenario, wherein there are 4 reserve elements, 2 primary elements, and a total of 6 elements. it’s not feasible to simplify the fraction, as reducing m = 4 2 to m = 2 would lead to redundancy with integral multiplicity, entailing 2 reserve elements and a total of 3 elements. as already mentioned, structural redundancy in cloud storage systems involves the use of distributed systems in which data is stored on multiple devices (servers) in different geographical areas. each server has its own copy of the data, and if one of the servers fails, the data can be restored from another server. an important aspect of structural redundancy is the organization of the network infrastructure that provides connectivity between servers and allows data to be transferred between them. typically, high-speed links such as optical networks are used as network connections to reduce latency and increase data transfer speeds. in addition, an important aspect of structural redundancy is the organization of data integrity control system, which allows detecting and correcting data errors. various methods can be used for this purpose, such as checksums, hamming codes and others. an illustration of structural redundancy within cloud storage is the redundant array of independent disks (raid), which amalgamates numerous physical disks into a unified logical storage unit to ensure elevated data reliability and accessibility. various raid levels offer distinct degrees of data safeguarding and recuperation should disk failure or other system anomalies arise. 5. reliability of non-recoverable redundant facilities reliability quantification of redundant facilities is determined by the type of redundancy. 5.1. total hot spare capacity with integer multiplicity in general, hot spare capacity, the main and each redundant object have the same number of elements connected in series in the reliability structural diagram. the redundant objects are included in parallel with the main object and themselves [15]. in each object with series connection of elements the probability of uptime operation is determined on the basis of expressions: p0(t) = n∏ i=1 p0i(t), p1(t) = n∏ i=1 p1i(t), . . . , pm(t) = n∏ i=1 pmi(t) the main and redundant facilities, consisting of elements connected in series, constitute the structural reliability scheme of the facility with parallel connection. then the probability of uptime operation pc(t) of the object: pc(t) = 1− [ 1− n∏ i=1 pi(t) ]m+1 where pi(t) – probability of uptime operation of i-th element. if the failure rate of an element is known and it is a constant value, then: pc(t) = 1− (1− e−λ0t)m+1 where λ0 = ∑n i=1 λi – failure rate of the main facility or any of the backup facilities. mean time between failures (mtbf) of the redundant object: t = 1 λ0 m∑ j=0 1 j + 1 copyright © 2024 assa. adv syst sci appl (2024) 176 n. kucherov, m. babenko, e. shiriaev, n.v. hung failure intensity of the redundant object: λc = (m+ 1)λ0p0(t)q m 0 (t) 1−qm+1 0 (t) general hot standby for non-recoverable objects gives a good effect at small values of the product λ0t. this type of redundancy is most appropriate for redundancy of sufficiently reliable single-use facilities with short continuous operation time. the redundancy multiplicity m required to achieve a given reliability of the redundant object at a certain point in time at a given reliability of the main object can be found from the expression: m = ln[1− p3(t)] ln[1− p0(t)] − 1 = lnq3(t) lnq0(t) − 1 where p3(t) – specified reliability of the redundant object at a certain point in time; p0(t) – reliability of the main facility without redundancy. 5.2. separate hot spare capacity with integer multiplicity in case of separate hot standby, the main and each redundant object can have the same or different number of elements. the redundant objects are switched on in parallel to themselves and in parallel to the elements of the main object. each i-th element of the main object is redundant by a redundant element. in case of separate redundancy, the object consists of n elements connected in series, each of which has m elements connected in parallel, in the sense of reliability. the probability of uptime of the object pc(t) under separate hot standby is equal to: pc(t) = n∏ i=1 {1− [1− pi(t)] mi+1} = n∏ i=1 [1−qmi+1 i (t)] where mi is the redundancy multiplicity of each i-th element of the main object. if the elements have the same reliability and the same redundancy multiplicity, the probability of uptime operation of the object: pc(t) = [ 1− (1− e−λt)m+1 ]n = [1−qm+1 i (t)]n mtbf of the redundant object: t = (n− 1)! λ(m+ 1) m∑ j=0 1 j+1 m+1 ( j+1 m+1 + 1 ) · · · ( j+1 m+1 + n− 1 ) failure intensity of the redundant object λc = n · (m+ 1) · λ · e−λt · (1− e−λt)m 1− (1− e−λt)m+1 = n(m+ 1) · λ · p (t)qm(t) 1−qm+1(t) where λ is the failure rate of one element. other things being equal, separate redundancy gives a significant reliability improvement compared to general redundancy. separate redundancy is used to increase the reliability of facilities with a large number of elements and longtime of use. copyright © 2024 assa. adv syst sci appl (2024) data storage with increased survivability and reliability... 177 5.3. hot spare capacity with fractional multiplicity a majoritarian object can be considered as a variant of an object with parallel connection of elements, the failure of which will occur if k elements out of n elements connected in parallel are operable. the presented scheme is operable when any two, three, four or all five of its five elements are operable. the combinatorial method can be used to calculate the reliability of majoritarian objects. then the probability of uptime of the object: pk = ck np k(1− p)n−k where ck n = n! k!(n−k)! mtbf of the redundant object with equal reliability elements is equal: tc = 1 λ k∑ j=0 1 n+ j the analysis of the last expression allows us to conclude that the average uptime of a redundant object is less than that of an unreserved object. therefore, redundancy with fractional multiplicity is inexpedient to use for objects with long continuous operation time. 5.4. cold backup with integer multiplicity in the case of general cold backup, the main facility operates, the backup facilities are disconnected by means of special switching devices. if the main object fails, it is disconnected and one of the redundant objects is connected instead. thus, the redundant object will fail when the main and all reserve objects fail. it is assumed that the switching devices are absolutely reliable. then the probability of uptime is equal to the probability of failure. pc(t) = m∑ j=0 p (aj) = e−λ0t m∑ j=0 (λ0t) j j! where λ0 = ∑n i=1 λi. failure rate: λc(t) = λ0(λ0t) m m! ∑m j (λ0t)j j! mtbf: tc = m+ 1 λ0 = (m+ 1)t0 5.5. separate cold backup with integer multiplicity probability of uptime of the main object when the elements are connected in series p0(t) = n∏ i=0 pi(t) where pi(t) – probability of uptime of the object elements redundant by the method of replacement. the probability of uptime of an object if all elements are equally reliable: pc(t) = eλ0t [ m∑ j=0 (λt)j j! ]n copyright © 2024 assa. adv syst sci appl (2024) 178 n. kucherov, m. babenko, e. shiriaev, n.v. hung failure rate: λc(t) = λ0(λt) m m! ∑m j=0 (λt)m′ j! where λ – failure rate of one element, λ0 = nλ. mtbf: tc = m+ 1 λ0 = (m+ 1)t0 separate redundancy by replacement, all other things being equal, gives the greatest reliability gain compared to other types of redundancy. and this gain is greater the more elements the redundant system has. 5.6. sliding reservation sliding redundancy is used to increase the reliability of several identical (or interchangeable) elements of the facility by one or more reserve elements. in this case, the failure of the object will occur if the number of failed main elements exceeds the number of redundant elements. this type of redundancy is used if all elements of the object perform the same functions. the main object has n elements, and m elements are in cold standby. in this case m < n. if any main element fails, any of the reserve elements is connected instead of it. in this case, the redundancy multiplicity is m/n. the probability of uptime of such an object is equal to: pc(t) = m∑ j=0 p (aj) = e−λ0t m∑ j=0 (λ0t) j j! where λ0 = ∑n i=1 λi. failure rate: λc(t) = λ0(λ0t) m m! ∑m j=0 (λ0t)j j! mtbf: tc = m+ 1 λ0 = (m+ 1)t0 this means that the reliability of the facility is equal to that of a facility with total redundancy with replacement, but, at the same time, it has n times fewer redundant elements. 6. principles of building a fault tolerant cloud system when building cloud-based high-reliability systems, there are two approaches to improve reliability: preventing failure and creating fault-tolerant and survivable systems [16]. in the first approach, one tries to eliminate all possible sources of failure by constructing a system from highly reliable elements. however, in this case, as the intensity of element failures approaches threshold values due to physical parameters or technologies, the cost of the system increases dramatically. in the second approach, the probability of failure of cloud system components is not reduced, but the system is subjected to additional requirements: it must be error free when individual parts or storages fail. these additional requirements and hence fault tolerant operation of the system is ensured by applying different approaches: • distributing load and data across multiple nodes so that if one node fails, the others can continue to operate without downtime; copyright © 2024 assa. adv syst sci appl (2024) data storage with increased survivability and reliability... 179 • backup data and distribute it across multiple nodes so that if one node fails, the data can be quickly restored to other nodes; • use of geographically distributed data centers, to reduce the risks associated with natural or man-made disasters in one region; • use of modern virtualization and containerization technologies, which allow to increase flexibility and scalability of the system, as well as to quickly transfer work from one node to another; • ensuring the security of stored or processed data and the system as a whole by means of multi-level protection, encryption, authentication and user authorization; • continuous testing and analysis of system operation to identify bottlenecks, optimize performance and improve reliability; • fault tolerance systems can be categorized into four types; • systems with multiple redundancy, having n identical modules and determining the solution result on the basis of majorization; • systems with structural redundancy, in which part of the modules are used for the solution, the remaining modules are in reserve and are activated as active modules fail; • hybrid redundant systems consisting of multiple redundant modules for failure replacement; • systems with gradual degradation (survivable), in which all serviceable modules are used to perform the assigned task, i.e., all modules are active and in case of failure the system is reconfigured and the task is continued . the performance of the first three types of systems does not change as individual modules fail. in the fourth type of system the performance decreases as degradation (failure of modules) occurs [17, 18]. the generalized algorithm of functioning of any fault-tolerant system is reduced to the following standard procedures: detecting a failure in the system, diagnosing the failed module, eliminating the influence of the failed device, reconfiguring the system. systems with constant performance are described by the following characteristics: time to system failure tc, system reliability r(t) = pc[tc ≥ t] and mtbf defined as the mathematical expectation of tc. in degraded systems, performance degrades over time, so characteristics linking performance to reliability characteristics are introduced. • reliability of operation [r∗(t, t )] – the probability that the system will correctly perform the assigned task in time t ; • average number of calculations between failures nav; • tτ , i.e., the duration of the interval at which the reliability of operation for a task with operation time t reaches a given value; • computation availability factor [dc(t)] i.e., the average value of system performance at time t; • performance threshold (tc), i.e., the time interval at which the value dc(t) reaches a given value. 7. structure of fault tolerant data storage and processing system in cloud fog environment typically, a fault-tolerant storage system in a cloud environment is constructed with multiple tiers, each assigned distinct functions aimed at ensuring data reliability and accessibility [18–20]. the physical infrastructure layer is the first layer in the structure of a resilient cloud storage system. it includes physical servers, storage, networking equipment, and other components that provide data storage and availability. copyright © 2024 assa. adv syst sci appl (2024) 180 n. kucherov, m. babenko, e. shiriaev, n.v. hung to provide fault tolerance at this level, various technologies such as server clustering, data replication, load balancing mechanisms, etc. are used. for example, server clustering allows you to combine several physical servers into one cluster, which increases system availability and performance, and data replication allows you to create copies of data on several servers, which allows you to ensure data availability in case of failure of one server. at the physical infrastructure level, virtualization technologies can also be used to make more efficient use of server resources and provide greater resiliency, for example, by enabling applications and data to be quickly migrated between physical servers in the event of a hardware failure. the virtualization layer in a cloud environment is responsible for managing and virtualizing the physical infrastructure. this layer abstracts away physical servers and data storage, allowing for more flexible resource management and efficiency. virtualization provides the ability to create virtual machines and containers that can run on a single physical server using the resources of that server. virtual machines can be quickly moved between servers in the event of a hardware failure or for load balancing. virtualization also enables you to create snapshots of virtual machines and datastores, making it easier to back up data and restore it in the event of failures. in addition, virtualization provides the ability to create virtual networks and network devices, allowing you to manage your network infrastructure more flexibly. thus, the virtualization layer in a cloud environment plays an important role in providing system resiliency by enabling rapid migration of applications and data between physical servers, data backup, and resource access control. the application layer is the layer in the cloud infrastructure that hosts the applications that utilize the data stored in the cloud. this layer includes software that is used to process, store, and transfer data, including databases, data processing applications, and web services. at the application layer, cloud infrastructure utilizes data backup mechanisms, monitoring and management mechanisms, and automatic disaster recovery mechanisms. in addition, the application layer typically utilizes mechanisms that provide scalability to support the growing number of users and data being processed. the data access layer is an abstraction layer that provides access to data from applications that utilize the cloud environment. this layer typically implements a mechanism to access data through apis or protocols such as http or https. to provide security and fault tolerance at this layer, data encryption, access control, and user authentication mechanisms are used. for example, authorization mechanisms such as oauth or openid connect can be used for access control, and encryption protocols such as ssl or tls can be used for data encryption. in addition, at the data access layer, a data caching mechanism can be used to increase the speed of data access. data caching allows a copy of the data to be stored in server memory or on other devices, which speeds up data access and reduces the load on the data warehouse. however, when using data caching, it is necessary to ensure that the cache is consistent with the data store to avoid possible data integrity issues. the rns allows to significantly improve the parameters of cloud storage compared to the storage based on traditional arithmetic, as well as to obtain new, more progressive design and structural solutions. application of the rns provides independent and parallel processing of each part of data, which determines the specifics of cloud storage organization. redundant coding in rns provides survivability of the system even in catastrophic situations, when the flow of faults and failures is very large, then the system will produce functioning. the main feature of rns based cloud structure is the potential ability to function in case of failures, losses, inaccessibility of data parts. this allows to preserve the operability at the expense of performance and performance degradation. the specified property provides high survivability of the system and is one of the main factors on which the development of ways to build fault-tolerant data storage systems that can degrade gradually is one of the main ways to build complexes with high viability, fault tolerance and reliability is based. copyright © 2024 assa. adv syst sci appl (2024) data storage with increased survivability and reliability... 181 let us consider issues related to fault tolerance of cloud storage system based on one of the most important properties of a residual class system, namely, the possibility of exchange operations between accuracy, speed and reliability. the cloud storage structure shown in fig. 7.1 consists of independent channels and cloud storage. in order to realize the adaptation methods, it is necessary to strive for the ultimate homogeneity and independence of all equipment. fig. 7.1. structure of the cloud fog system. 8. structure of a fault tolerant data processing system in a fog environment fog computing is an architectural concept that extends cloud computing capabilities by moving computing resources closer to the edge of the network, to internet of things (iot) devices and other end-user devices. the basic idea of fog computing is to process data and perform computations at a closer distance to the data source. this reduces latency, improves responsiveness, and enables more efficient utilization of network resources. there are certain reliability issues of data processing on fog nodes. here are some of them: lack of network connectivity: fog nodes may be connected to a network with limited bandwidth or may temporarily lose connectivity with cloud servers. this may lead to problems with data transfer or synchronization with cloud services. vulnerability to failure: fog nodes may be prone to failures or malfunctions due to hardware issues, software bugs, or external influences. this may result in data loss or temporary unavailability for data processing. data security: data processing on fog nodes may pose data security risks. such nodes may be more vulnerable to cyberattacks and may not have the same levels of protection as centralized cloud servers. appropriate security measures are required to protect data transmitted and processed on fog nodes. copyright © 2024 assa. adv syst sci appl (2024) 182 n. kucherov, m. babenko, e. shiriaev, n.v. hung scaling: managing and scaling a large number of fog nodes in a distributed environment can be challenging. as a fog computing system grows and new nodes are added, efficient management mechanisms are needed to ensure reliability and consistency across the entire system. resource optimization: allocating and optimizing computational resources on fog nodes can be a complex task. application requirements, node load, and available resources must be considered to achieve optimal reliability and performance. data synchronization: in a fog computing environment, there may be problems with data synchronization between different nodes. while transferring data between nodes and synchronizing computation, it is necessary to ensure data consistency and avoid possible conflicts. limited resources: fog nodes may have limited computing resources, memory, power consumption and network bandwidth. this can impact system reliability and performance, especially when processing large amounts of data or performing complex computations. change management: in a fog computing environment, changes to the network infrastructure, node configuration, or available resources may occur. change and update management can be complex and require effective mechanisms to minimize potential system disruption and ensure reliability. fig. 8.2. data storage and processing system in a fog environment. however, modern technologies and development techniques can cope with these challenges. for example, resource reservation, disaster recovery mechanisms, data replication, and cryptographic techniques can improve the reliability of data processing on fog nodes. 9. modeling of the proposed system for reliable data storage and processing in fog environment to validate our theoretical conclusions about the reliability of the proposed system, we have performed monte carlo simulation of the developed system. in a foggy system, any large number of devices can be disconnected from the access point because it has physically moved and no signal reaches it. also, it may become inaccessible due to battery drain and device disconnection. the proposed system allows to build a system with a small redundancy, designed for unforeseen failures. for this purpose, let’s build a mathematical model. the developed model of data storage and processing in a fog environment consists of k working devices and r control devices, to increase reliability with a slight increase in redundancy. in case of failure or unavailability of working devices, they can be replaced by control devices and vice versa. the requirement for the computational scheme to provide guaranteed protection against the issuance of unreliable results determines the need to maintain in an operable state kmin working and at least one control device. thus, the copyright © 2024 assa. adv syst sci appl (2024) data storage with increased survivability and reliability... 183 reliable structure of the computational model will correspond to the known method of sliding redundancy with the loaded state of redundant elements. when carrying out calculations on fog devices each of the above-mentioned possible system failures can occur both separately and jointly. we have introduced a single-tier architecture for data processing and storage on fog devices, offering simplified frameworks for crafting data processing strategies. the assembly of data storage and processing systems can be accomplished through a range of module counts. in this system, there are k operational bases and r control bases, summing up to n = k + r. establishing the data processing system is achievable through complete replication of the operational range (k = r) or through partial replication (k > r). we will determine the overall reliability of the information processing and storage design by employing rns for distributed data storage and processing. additionally, we will assess the available space across diverse system configurations. we will calculate the reliability of the system by the following construction parameters: 1. the system parameters were randomly selected as follows: k = 3 . . . 26, where r varies from 1 to 19. 2. the probability of complete failure of the data processing and storage system was calculated by the formula p0 = n∑ i=n−k+1 ci np i dq n−1 where q = 1− pd, pd – a device of size 3015 was randomly selected from the set of failures. the monte carlo method was used to calculate for 10,000 repetitions for each of the random failures from the set. the simulation results are shown in fig. 9.3, 9.4, 9.5. fig. 9.3. failure distribution of the system under parameters r = 3, k = 3 the failure probability distribution was then plotted and compared with the gamma probability distribution, and the results are shown in figure 4. 10. conclusion the principles of building reliable data storage and processing systems in cloud-fog environment were considered in the article. the factors affecting the reliability and survivability of systems have been analyzed and the main types of failures have been copyright © 2024 assa. adv syst sci appl (2024) 184 n. kucherov, m. babenko, e. shiriaev, n.v. hung fig. 9.4. failure distribution of the system under parameters r = 5 . . . 7, k = 1 . . . 5 fig. 9.5. failure distribution of the system under parametersr = 19 . . . 26, k = 15 . . . 19 fig. 9.6. probability distribution, for system configurations r = 3, k = 3 given: sudden, gradual, operational. the principles of introducing information redundancy to increase reliability are considered, as well as the corrective abilities and error detection algorithms of rns codes were considered. the scheme of data processing and storage in fog copyright © 2024 assa. adv syst sci appl (2024) data storage with increased survivability and reliability... 185 fig. 9.7. failure distribution of the system under parametersr = 5 . . . 7, k = 1 . . . 5 with increased reliability is proposed. monte carlo modeling was carried out, which showed that the proposed system allows you to process and store data in a foggy environment at a high level of reliability. a probability distribution of failure occurrence was constructed, which showed that failure probabilities obey the gamma probability distribution. funding: the research was supported by the russian science foundation grant no. 2271-10046, https://rscf.ru/en/project/22-71-10046/. references 1. al kez, d., foley, a. m., laverty, d., del rio, d. f., & sovacool, b. (2022) exploring the sustainability challenges facing digitalization and internet data centers. journal of cleaner production, 371, 133633. 2. yi, s., li, c., & li, q. (2015) a survey of fog computing: concepts, applications and issues. in proceedings of the 2015 workshop on mobile big data, 37–42. 3. priyadarshini, r., barik, r. k., & dubey, h. (2018) deepfog: fog computing-based deep neural architecture for prediction of stress types, diabetes and hypertension attacks. computation, 6, 62. 4. babenko, m., tchernykh, a., pulido-gaytan, b., avetisyan, a., nesmachnow, s., wang, x., & granelli, f. (2022) towards the sign function best approximation for secure outsourced computations and control. mathematics, 10, 2006. 5. apat, h. k., nayak, r., & sahoo, b. (2023) a comprehensive review on internet of things application placement in fog computing environment. internet of things, p. 100866. 6. dai, y.-s., yang, b., dongarra, j., & zhang, g. 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(2016) a survey of fault tolerance architecture in cloud computing. journal of network and computer applications, 61, 81–92. 15. araujo, j., oliveira, d., matos, r., alves, g., & maciel, p. (2023) availability and reliability modeling of mobile cloud architectures. ieee transactions on industrial informatics. 16. al-dulaimy, a., ashjaei, m., behnam, m., nolte, t., & papadopoulos, a. v. (2022) fault tolerance in cloud manufacturing: an overview. in international conference on mobile computing, applications, and services, 89–101, springer. 17. sheeba, a. & uma maheswari, b. (2023) an efficient fault tolerance scheme based enhanced firefly optimization for virtual machine placement in cloud computing. concurrency and computation: practice and experience, 35, e7610. 18. mohammadi, v., rahmani, a. m., darwesh, a., & sahafi, a. (2023) fault tolerance in fog-based social internet of things. knowledge-based systems, 265, 110376. 19. semmoud, a., hakem, m., benmammar, b., & charr, j.-c. 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(2022) fault tolerance mechanism for software application through fog computing as middleware. international journal of computing and digital systems, 11, 45–54. copyright © 2024 assa. adv syst sci appl (2024) introduction analysis of factors affecting reliability and survivability of control and communication systems information redundancy in cloud storage systems principles of introducing information redundancy in the rns estimation of corrective capacity of codes in rns algorithms for error detection in rns error detection algorithm using positional features structural redundancy in cloud storage systems reliability of non-recoverable redundant facilities total hot spare capacity with integer multiplicity separate hot spare capacity with integer multiplicity hot spare capacity with fractional multiplicity cold backup with integer multiplicity separate cold backup with integer multiplicity sliding reservation principles of building a fault tolerant cloud system structure of fault tolerant data storage and processing system in cloud fog environment structure of a fault tolerant data processing system in a fog environment modeling of the proposed system for reliable data storage and processing in fog environment conclusion adv syst sci appl 2022; 04:65–78 published online at https://ijassa.ipu.ru. the problem of safe evacuation of large floodplains population during flooding alexander a. voronin1, ann a. vasilchenko1, ann yu. klikunova1, oksana yu. vatyukova1, alexander v. khoperskov1* 1volgograd state university, volgograd, russia abstract: the natural conditions of the floodplains of regulated large rivers are attractive for the urbanization of such regions. however, severe floods on flat terrain pose a threat of extensive flooding of such areas. therefore, the effective organization of the population evacuation is an urgent problem for such lowland floodplains. the spatial and temporal distribution of the flooding threat of large floodplains makes it possible to organize the population evacuation using a limited number of vehicles in the presence of a high-quality forecast of the hydrological regime. we study the problem of organizing the safe evacuation of all floodplain residents using fairly accurate forecasts of the flooding time points of settlements and evacuation routes based on highperformance hydrodynamic numerical experiments. the use of empirical algorithms makes it possible to minimize the number of evacuation vehicles on the set of safe schedules. the result is an empirical function of the dependence of the minimum number of vehicles that ensure the safe evacuation of the given territory population on the values of the hydrograph and the beginning moments of the evacuation. as an example, we construct this function for the northern part of the volga-akhtuba floodplain. keywords: evacuation, flooding, schedule, optimization problem, simulations, volga-akhtuba floodplain 1. introduction the construction of hydroelectric power plants on large rivers makes it possible to regulate the hydrological regime and develop floodplains, accelerating the process of urbanization of such regions. however, the presence of a hydroelectric dam does not exclude the threat of catastrophic events during the spring floods and/or heavy rainfall when the safe values of the hydrograph (water discharge q) are exceeded. the large flat relief of the floodplain determines the specific features of flood hazards. even a relatively small (10–15 percent) excess of the average annual flood level can cause catastrophic events, when a significant part of the territory is flooded (up to 50 percent) with the formation of dangerous areas of hundreds of square kilometers [4, 23]. on the other hand, the flooding of urbanized large floodplains occurs relatively slowly, which makes it possible to solve the task of organizing effective evacuation of the population in conditions of limited resources. the problem of organizing the population evacuation during floods has a high degree of objective and subjective uncertainty, and therefore it belongs to the field of decisionmaking research. these uncertainties are the reason for a wide variety of models of the evacuation situation, criteria for the effectiveness of the evacuation organization, modeling methods and technologies. the objective functions of this problem are, for example, the ∗corresponding author: khoperskov@volsu.ru 66 a.a. voronin, a.a. vasilchenko, a.yu. klikunova, et al. time and risk [1, 11, 18], the number of evacuated people [13], the various types of damage [6, 25, 27], the cost of evacuation and assistance to the population [12], the optimal shelter locations [10], the personnel resettlement points [24], the comprehensive safety criteria [5]. this requires the use of a wide range of models and methods, such as flood dynamics [11], road network and traffic flow [5, 11], game theory [1, 10], scenario [6, 11, 27], multiagent [1, 27], econometric [6], dynamic [1, 5], stochastic [1] models, integer and linear programming [12, 24], heuristic methods optimization and decision making [1, 10, 12, 24], models of hydrotechnical projects [25]. consideration of unorganized evacuation may require traffic control [20]. a wide variety of approaches to evacuation modeling does not change the main problem of its organization, which is to ensure safety for specific terrain conditions and hydrological regime dynamics. therefore, we emphasize that the criterion for the effectiveness of the organization of evacuation directly affects its safety. for example, minimizing evacuation time can lead to traffic accidents when evacuation routes are overloaded using private vehicles for self-evacuation. the maximization of the number of organized evacuation of the population by ground transport with a lack of resources leads to high danger for a small part of those remaining in the flood zone in difficult conditions. the most universal criterion for the effectiveness of the evacuation organization is in our opinion the aggregate risk of loss of health and life of the population in the flooded area. the mathematical model of risk management in the evacuation organization has the following form f (e,s) −→ min s , (1.1) where f is the human life risk, e = e(t) = ⟨kfl(t, q), p⃗ , l⃗(g), t(0),m⟩ is the tuple of the evacuation situation, kfl(t, q) is the dynamic digital map of the flooded area, t [hours] is the time after the flood start, q = q(t) is the river hydrograph (water discharge or volume of water through the dam per unit of time), p⃗ is the vector of the number of inhabitants of settlements (n is the number of such settlements), g is the graph of the road network, l⃗ is the lengths vector of evacuation routes (the shortest paths from settlements to evacuation points in the graph g), t(0) is the start time of evacuation in the floodplain, m is the number of evacuation vehicles, s is the schedule of evacuation transport (the number of buses on each route at each moment of time). all components of the evacuation situation e have uncertainty, since part of the population leaves the danger zone ourselves, some refuse to evacuate, and so on. the accuracy of determining the flooding time of settlements and sections of the road network is determined by the errors of the hydrograph, the digital elevation model (dem), and the adequacy of the computational hydrodynamic model. estimating the number of vehicles involved is also inaccurate. since floods can have a multifactorial negative impact on human health, the mathematical definitions of the objective function f can be different [16]. simulation modeling and solving optimization problems can reduce a significant part of these uncertainties. the solution (1.1) may exceed the valid risk level for largeq, t(0) and smallm . therefore, the urgent problem is to find situations e and schedules s that do not allow this excess. we will assume that the valid risk corresponds to the timely evacuation of the inhabitants of the flooded part of the territory without specifying the form of the function f . such a safe evacuation is provided by the schedule s(sec). the optimization problem solution m(kfl(t, q), p⃗ , t (0),s(sec)) −→ min s(sec) (1.2) defines the boundary of the domain valid risk. task (1.2) is a variant of the resource investment problem or a mirror copy of the extension of the classical rcpsp (resource constrained project scheduling problem) with time windows (deadline) [2, 14, 15, 17, 26]. copyright © 2022 assa. adv syst sci appl (2022) the problem of safe evacuation 67 the purpose of this article is to create a algorithm for the approximate solution of the problem (1.2) and its implementation using the results of simulation modeling of flood dynamics in the northern part of the volga-akhtuba floodplain (vaf) as an example application. 2. research methods and technologies the proposed method for the approximate solution of the problem (1.2) is based on reducing the uncertainty of the parameters values of the evacuation situation using a series of computational experiments with our model of the development of an emergency situation. this allows us to reduce the problem of parametric optimization of a renewable resource (1.2), which is solved approximately by heuristic methods with an estimate of the effectiveness of the result and the use of upper and lower parameter estimates. our emergency development model consists of the numerical hydrodynamic flood model combined with the digital elevation model (dem), which calculates the time of water arrival at specific floodplain points. this model allows us to reduce the evacuation situation e(kfl(t, q),g,m) to the form e(t, q, p,m), since the hydrodynamic model instead of the flooding map and the graph of the road network g gives us the dependence τ⃗(q) (τ⃗ is the vector of the beginning times of flooding of settlements and sections of evacuation routes on the graph g). the hydrodynamic model allows one to pass from p⃗ to the vector p⃗ (sh), in which the components for non-floodable habitats are equal to zero. 2.1. development model of emergency situation the calculation of flood wave propagation is based on numerical hydrodynamic model in the shallow water approximation [4, 8, 9]: ∂h ∂t +∇ (hu⃗) = q(x, y, t) , (2.3) ∂hu⃗ ∂t +∇ ((h u⃗)⊗ u⃗) = −gh(∇h)− gh(∇ b) +hf⃗ , (2.4) where h(x, y, t) is the depth of the liquid, u⃗ is the velocity vector of the fluid, averaged over the vertical coordinate (z), q is the function of sources (q > 0) or drains (q < 0) of water, ∇ is the nabla operator in the plane (x, y), g is the gravitational acceleration, b(x, y) is the bottom topography, f⃗ is the specific external total force, including the contribution of bottom and viscous turbulent friction, wind stress, coriolis force, etc. the topography is given by dem on a fixed grid with a step ∆ℓ: xi+1 = xi +∆ℓ, xj+1 = xj +∆ℓ, i = 0, 1, . . . , nx, j = 0, 1, . . . , ny. the number of cells in two directions (nx andny) is determined by the sizes of the simulated arealx = ∆ℓ ·nx andly = ∆ℓ ·ny. we neglect the processes of infiltration and interaction with groundwater in the model (2.3), (2.4) [8], since the duration of the emergency situation at the stage of evacuation is only a few hours. the effect of sediment transport together with water movement can be more significant and depends both on the properties of the underlying surface and, to a greater extent, on the features of the terrain. then the system of equations (2.3), (2.4) must include an additional non-stationary equation for the function b(x, y, t) [9, 19]. the developed evacuation method is applicable to any complex large floodplains. an example of such a natural object is the volga-akhtuba floodplain (vaf) in the lower volga river (figure 2.1). we restrict ourselves to the northern part of the vaf (see inset on the right), which is the most urbanized area of the floodplain. the calculation of the hydrological regime dynamics requires a high-quality digital elevation model (dem) [3, 4, 9], and the non-stationarity of the flooding process imposes copyright © 2022 assa. adv syst sci appl (2022) 68 a.a. voronin, a.a. vasilchenko, a.yu. klikunova, et al. fig. 2.1. the location of the northern part of the volga-akhtuba floodplain is highlighted in the inset (bestmaps aggregator image) special requirements on the accuracy of modeling. the spatial resolution of dem is limited not only by topographic data, but also by the availability of computing resources. for example, halving the cell size increases the number of cells by a factor of 4 for a fixed area. there is an additional restriction of about two times on the integration step (∆t→ ∆t/2) due to the stability requirement for the explicit numerical scheme [9]. however, parallelization increases the load on gpu, which increases the efficiency of computing. the total increase in computation time turns out to be less than 8 times when ∆x→ ∆x/2 is passed. the cell size ∆x is determined by the specific conditions of the study area. we will consider below the algorithms for organizing evacuation using the example of the volgaakhtuba floodplain (see figure 2.1), a characteristic feature of which is a developed hierarchical system of large and small channels (figure 2.2). vaf is a very large floodplain and we use ∆x = 15 m as the base dem, which is built from a combination of various topographic data [7]. for example, the acceptable accuracy of flood modeling is provided by ∆x = 15 m for lower manhattan in new york city, which has an area of only about 15 sq km [3]. small water systems require a more detailed grid with ∆x ∼ 2− 5 m. the basis of the hydrodynamic flooding model for a specific area is the digital hydrological landscape model (dhlm), which includes the dem and other characteristics of the earth’s surface and soil. first of all, it is the roughness coefficient (or the so-called manning’s resistance coefficient) nm(x, y), which determines the intensity of the interaction between the water flow and the bottom as part of the force f⃗ in (2.4). the study of the hydrological regime over long time intervals requires additional spatial characteristics, such as the hydraulic conductivityk, the bottom porosity ψ, the groundwater bottom b(gw) [4,8,9]. all these parameters are specified in the cells of our grid (xi, yj) and form the dhlm as a set of spatial matrices ⟨b, nm , k, ψ, b (gw)⟩. the distribution heterogeneity of nm(x, y) is due to changes in the roughness of the earth’s surface (sand, gravel, arable land, etc.), various types of vegetation (meadow, shrub, sparse or dense forest, etc.) (table 2.1). hydrological resistance to surface water flow is one of the most important factors that changes the dynamics of water movement. figure 2.3 shows the difference in flooding of the vaf for the two calculations. the first takes into account the real inhomogeneity of the surface properties (nm(x, y)), the second is based on a constant roughness coefficient. the emergency situation development model is multilayer geoinformation model of the certain territory, including dhlm, layers of the road network (the topological structure is determined by the graph g), the settlements with data on the number of inhabitants, as well as a large number of layers of the dynamics of catastrophic floods that are built based on copyright © 2022 assa. adv syst sci appl (2022) the problem of safe evacuation 69 fig. 2.2. the result of numerical hydrodynamic simulation on the grid with cell of 15 meters. the hydrological situation in the vaf corresponds approximately to the low-water level, when large channels contain water without flooding the flat part of the floodplain table 2.1. roughness coefficient for different types of underlying surface in our model type of underlying surface roughness coefficient (nm ) riverbed 0.02 sand 0.02 bush 0.05 sparse forest 0.07 dense forest 0.09 cultivated land (arable land) 0.05 meadow without shrubs 0.04 hydrodynamic model (2.3), (2.4). direct hydrodynamic modeling gives a table of flooding times for settlements and evacuation routes for their population for a given catastrophic hydrograph q(t) with a maximum level q for the rectangular hydrograph model. 2.2. algorithm for finding safe evacuation schedule the evacuation schedule is determined by the matrix s = ∥mtj∥, t = 1, ..., t , j = 1, ..., n, t = max j (τj − t(0)), where mtj is the number of vehicles to evacuate from point j at time step t, τj is the components of τ⃗ vector. we call schedule s(sec) a valid (safe) schedule if it copyright © 2022 assa. adv syst sci appl (2022) 70 a.a. voronin, a.a. vasilchenko, a.yu. klikunova, et al. fig. 2.3. simulation results of the flooding of the northern part of the vaf at q = 45000m3/sec at the time t = 15 hr. areas that are flooded at the non-uniform roughness coefficient (nm (x, y)), but remain flooded at the constant nm = 0.045, are highlighted in red. yellow color shows areas flooded at nm = const, but not flooded at nm (x, y). blue areas are flooded in both models satisfies the condition m̃j ≥ 1 + [ p (sh) j a ] (j = 1, ..., n), m̃j = pmax j∑ pj=1 m̃pij , mtj = m (0) tj , m (pj) tj = m (pj−1) tj − m̃pj ,j, m̃pj ,j = min ( m (pj−1) pj ,j , ...,m (pj−1) pj+θj ,j ) , pj = 1, ..., pmax j , pmax j = τj − t(0) − θj + 1, t = pj, ..., pj − t(0) + τj − 1, j = 1, ..., n , (2.5) where a is the evacuation bus capacity, p (sh) j is the number of inhabitants in the settlement j subject to organized evacuation, θj is the stage duration of population evacuation from the settlement j. thus, the problem (1.2) reduces to minimizing the number of vehicles m on the set of safe schedules (2.5): m = max t (mt) → min s(sec) , mt = n∑ j=1 mtj, t = 1, ..., t , (2.6) copyright © 2022 assa. adv syst sci appl (2022) the problem of safe evacuation 71 where the time t is measured in hours. the heuristic algorithm for solving the problem (2.5) – (2.6) is a multi-step process of redistributing vehicles between evacuation routes. each redeployment results in a new valid evacuation schedule. the initial valid schedule s(sec) 0 is one that realizes the evacuation of the inhabitants number p (sh) j along each route j in a time not exceeding τj − t(0). this schedule requires m0 buses, and each of them travels along a single route. such a schedule is compiled by an elementary calculation. the first part of the algorithm implements the multiple distribution of the reserve of vehicles formed at the end of the evacuation on some routes to other routes. fig. 2.4. digital model of the road network of the vaf with the settlements locations and road numbers the appearance of a reserve on one or several routes at step k (time θk, k = 1, 2, ..., n− 1) leads to its assignment to one of the evacuation routes or to distribution over parts on several evacuation routes. a valid scheduling with a decrease in the number of machines in the initial schedule is generated for each possible reserve assignment. if the maximum of this reduction is reached on a single route, then the corresponding assignment is accepted. if the maximum is reached on several routes (in the amount of jk), then a new valid evacuation schedule for each of them is calculated together with the moment of appearance of the first reserve θkj and its value mkj (j = 1, 2, ..., jk). the assignment is made to the route with the maximum value mkj/θkj . if the maximum of this value is reached on several routes, then the reserve is assigned to the route with the minimum value of θkj . the equality of these values for several routes allows you to assign to any of them. the distribution of the reserve of buses on several routes is carried out in case of its redundancy for one route. after assigning a copyright © 2022 assa. adv syst sci appl (2022) 72 a.a. voronin, a.a. vasilchenko, a.yu. klikunova, et al. reserve, the algorithm step is considered completed and the schedule s(sec) k with mk ≤mk−1 is compiled. the algorithm ends when k = n− 1. the second part of the algorithm is the sequence of constructing valid schedules s(sec) w (w = n, n+ 1, . . .) while decreasing mn−1 by one at each step of the sequence. each valid schedule is constructed by sequential distribution of vehicles between evacuation routes at each time t = t, t − 1, . . . , 1 in the direction from the end of the evacuation period to its beginning. higher priority is given to routes with the longest allowable evacuation period. thus, the algorithm maximizes the reserve of vehicles for their use on routes with the minimum allowable evacuation periods. the algorithm stops at some step w if it is impossible to construct an valid schedule with the available number of buses mw . the final schedule is the schedule s̃(sec) = s(sec) w+1 at the minimum of vehicles number m̃ = mw+1. the efficiency of the two-stage algorithm is estimated by the vehicles load factor η = (m̃ t (act))−1 t (act)∑ t=1 mt, where t (act) ≤ t is the actual evacuation time in the generated schedule. 3. model of population evacuation in the northern part of the volga-akhtuba floodplain the vaf is located between the volga and akhtuba rivers (see figure 2.1). the water flow of the volga river is regulated by the passage of water through the dam of the volga hydroelectric power plant (vhpp). the development of the road network in the northern part of the vaf in recent decades has been a powerful driver of population growth, which now exceeds 35,000 people. figure 2.4 shows the road network map of the floodplain with the numbers of its local sections (edges of the g graph) and the settlements location. the final evacuation points are the bridges over the volga river and the akhtuba river marked with triangles. table 3.2 shows the evacuation routes (ers) of the l⃗-vector components. the calculation of the parameters of the evacuation situation e(t, q, p⃗ (sh),m) in the floodplain is based on a series of computational experiments on the study of the flooding dynamics of settlements and evacuation routes of the vaf forq = {30, 35, 40, 45} thousands m3/sec with flood map every hour. the modeling results show that the flooding of settlements can occur at maximum hydrographs greater than 29 000 m3/sec, and the territory flooding occurs during the first 4 days. figures 3.5, 3.6 a show floods maps and time dependences of the inhabitants’ number of flooded settlements, respectively, for several values of emergency situation parameters. table 3.2 contains the times of settlements flooding and local sections of evacuation routes for different hydrograph maxima. bold type highlights situations where the escape route is flooded before the settlement. figure 3.6 b shows the dynamics of the distribution of the settlements number into groups with different durations of safe evacuation periods for several values ofq. dependences of the flooded settlements number and their evacuation routes on the maximum value of hydrograph are shown in figure 3.7. we classified all settlements into three groups: 1) the settlement is not flooded; 2) the settlement is flooded later than the evacuation route; 3) the settlement is flooded earlier than the evacuation route. approximate solutions of the problem (2.5), (2.6) are constructed for two series of schedules for the vaf territory with an estimate of the efficiency factor η. the first series of safe schedules ssec 0 (t(0), q) with average efficiency η(ssec 0 ) = 44% describes safe evacuation from potentially floodable settlements during the entire safe period without reassignment of transport between settlements. copyright © 2022 assa. adv syst sci appl (2022) the problem of safe evacuation 73 fig. 3.5. flood maps of the vaf and hazard levels of settlements at q = 35 000 m3/sec, t = 32 hr and t = 72 hr (a, b); at q = 45 000 m3/sec, t = 32 hr and t = 48 hr (c, d). the heuristic algorithm described above underlies the constructed series of schedules s̃(sec)(t(0), q) with average efficiency η(s̃(sec)) = 72% and the corresponding approximate function of the minimum transport support for the safe evacuation of the vaf population m̃(t(0), q) (figure 3.8). the function m0(t (0), q) corresponds to the initial schedules s(sec) 0 . 4. discussions and conclusion a feature of flood development in floodplains is the high frequency of earlier flooding of evacuation routes than settlements. for example, the number of such cases is 5 times higher than the number of opposite cases in the territory of the volga-akhtuba floodplain (figure 3.7). this causes the high sensitivity of the transport support for safe evacuation (m̃ ) to the start time of evacuation t(0). solution of the considered problem (2.5), (2.6) at 30000 m3/sec ≤ q ≤ 45000 m3/sec exists only at t(0) ≤ 4 hours. evacuation by vehicles at t(0) > 4 hours can only be considered copyright © 2022 assa. adv syst sci appl (2022) 74 a.a. voronin, a.a. vasilchenko, a.yu. klikunova, et al. table 3.2. parameters of settlements evacuation: n is the settlement number, pi is the component of vector p⃗ , ner is the escape route (er) number, li is the components of vector l⃗ (km), “–” is the no flood sign, tp is the beginning time of the settlement flooding (hours), tr is the time the escape route flooded after the flood started (hours), nfp is the local section flooded number. the numbers of settlements for which evacuation routes are flooded before the settlement are in bold. q = 30 000 m3/sec q = 40 000 m3/secn pi ner list of evacuation roads li tp tr nfp tp tr nfp 1 1330 1 66,75,26 16 52 48 66 2 100 2 75,26 6 50 75 3 2000 3 26 5 4 1500 4 10 2 33 20 5 200 5 55,74,77,54,68,1,0 21 54 11 55 26 9 55 ... 15 18000 15 14,0 10 17 13 35 14 16 200 16 44,73,43,0 13 17 5 44 13 5 44 17 170 17 45,27,0 15 6 45 6 45 18 480 18 69,48,46,36,7,8,9,10 22 45 18 69 23 14 69 19 300 19 27,0 12 31 27 52 19 27 20 180 20 46,36,7,8,9,10 13 50 46 30 28 46 21 80 21 42,10 7 42 42 29 27 42 22 100 21 42,10 7 42 42 32 27 42 23 100 12 51,49,47,46,36,7,8,9,10 21 32 47 20 51 24 300 22 41,9,10 6 42 39 41 25 345 23 22,10 3 ... 50 185 2 75,26 6 50 75 51 100 27 38,8,9,10 15 67 63 65 43 42 65 52 100 44 78,65,75,26 16 63 65 52 42 65 53 200 45 29,3,2,1,0 14 54 460 46 25 10 63 38 25 31 29 25 55 10 47 35,6,5,4,3,2,1,0 19 31 52 35 56 20 48 33,5,4,3,2,1,0 19 50 46 33 30 28 33 fig. 3.6. a — dependences of the inhabitants number of flooded settlements of the vaf on time for different hydrographs with maximum values: q(1) = 30 000m3/sec, q(2) = 35 000m3/sec, q(3) = 40 000m3/sec, q(4) = 45 000m3/sec. b — diagram of the distribution of the settlements number into groups with different durations of safe evacuation periods for different maximum values of the hydrograph. copyright © 2022 assa. adv syst sci appl (2022) the problem of safe evacuation 75 fig. 3.7. dependences of the number of settlements in three groups (see text) on the maximum value of the hydrograph for the next situations: i — nonflooded settlements (n (i)), ii — the evacuation route is flooded before the settlement (n (ii)), iii — the settlement is flooded before its evacuation route (n (iii)). the results are shown for the four maximum hydrograph values q (see caption in figure 3.6) fig. 3.8. functions m0(t (0), q) and m̃(t(0), q)). conditionally safe, since residents of some settlements (n 16 and n 17, see table 3.2) can no longer be evacuated along the road network. thus, the constructed functions in figure 3.8 for t(0) > 4 hours provide only a conditionally safe evacuation of the vaf population. analysis of figure 3.8 for t(0) ≤ 4 hours shows that the ratio m̃(t(0), q) to m0(t (0), q) varies from 0.59 to 0.88. an important feature of flood development in a large flat floodplain is the spatial and temporal distribution of threats. the possibility of sufficiently accurate forecasting of the hydrological situation based on numerical simulation allows reducing the problem of hydrological risk minimization to a more deterministic and simple problem of finding a valid (safe) evacuation schedule with the available stock of vehicles for evacuation. the uncertainty of this reserve is overcome by solving the problem of minimizing the number of vehicles on the set of safe evacuation schedules. our results on the example of the volga-akhtuba floodplain show that the construction of the function of the minimum resource support for safe evacuation provides effective management of its organization for various hydrographs. the practical implementation of safe schedule for organized evacuation requires the advance compilation of a directory of safe schedules for each territory in a wide range of hydrographs for various moments of the evacuation beginning. the variety of tasks for effective evacuation of the population during floods is primarily due to the variability of the components of the evacuation situation e. the main factor in the evacuation situation is a catastrophic hydrograph. the nature of the flooding events is determined by the inflow rate and the total volume of water, the duration of high water, etc. these parameters depend both on the type of floods (snowmelt, rains, accidents at hydraulic structures) and on the nature of the river system (plain or mountain area, lowland rivers with mountain tributaries). it is necessary to take into account the factors that determine the disaster development: the relief height of the flooded area, the territorial compactness or low population density, the capacity of the road network and the quality of the roadway, the temporary and transport resources. various combinations of these factors lead to varying degrees of severity of the evacuation situation. copyright © 2022 assa. adv syst sci appl (2022) 76 a.a. voronin, a.a. vasilchenko, a.yu. klikunova, et al. the considered problem of finding the minimum resource support for the safe evacuation of the large floodplains population during flooding (zero life risk) is a special case of the problem of minimizing life risk under uncertainty. the effectiveness of its solution is provided by objective and subjective groups of factors. the first group includes the specificity of the conditions for the development of the disaster: a relatively small excess of the safe threshold of the hydrograph and the territorial distribution of settlements. subjective factors are the possibility of a fairly accurate forecast of the flooding moments of settlements and sections of evacuation routes, as well as solving the problem of minimizing the number of vehicles on a set of safe schedules. the absence of at least one of the listed factors greatly reduces the efficiency of solving our problem or makes the solution impossible. indeed, the compactness of the residents distribution in a big city or a catastrophic hydrograph lead to the rapid flooding of places where people live. this and the lack of forecast of the flooding of settlements do not allow for the safe evacuation of the entire population. in these cases, the evacuation effectiveness is measured by its speed and the number of residents evacuated. acknowledgements the study was supported by the ministry of science and higher education of the russian federation (government task no. 0633-2020-0003). references 1. davidson, r. a., nozick, l. k., wachtendorf, t., blanton, b., colle, b., et. al. 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(2008) solving the resource availability cost problem in project scheduling by path relinking and genetic algorithm, applied mathematics and computation, 196(2), 879-888. 18. rezchikov, a., ivaschenko, v., samartsev, a., kushnikov, v., filimonyuk, l., & bogomolov, a. (2019) multiagent model of people evacuation from premises while emergency, advances in systems science and applications, 19(1), 98-115, https://doi.org/10.25728/assa.2019.19.1.558 19. sukhinov, a.i., chistyakov, a.e., protsenko, e.a., sidoryakina, v.v., & protsenko, s.v. 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(2021) control model for hydrologic safety of flooded territories, proceedings 2021 3rd international conference on control systems, mathematical modeling, automation and energy efficiency, summa 2021, 344-348, https://doi.org/10.1109/summa53307.2021.9632213 23. voronin, a. a., khoperskov, a. v., isaeva, i. i., & klikunova, a. y. (2020) control model of the floodplain territories structure, advances in systems science and applications, 20(3), 153-165, https://doi.org/10.25728/assa.2020.20.3.964 24. wen, y., & zhang, n. (2019) evacuation and settlement model of personnel in major flood disasters and its application, iop conference series: earth and environmental science. – iop publishing, 304(4), 042016, https://doi:10.1088/17551315/304/4/042016. 25. wyzga, b., kundzewicz, z. w., konieczny, r., piniewski, m., zawiejska, j., et. al. (2018) comprehensive approach to the reduction of river flood risk: case study of the upper vistula basin, science of the total environment, 631, 1251-1267. 26. yamashita, d. s., armentano, v. a., & laguna, m. (2007) robust optimization models for project scheduling with resource availability cost, journal of scheduling, 10(1), 67-76, https://doi.org/10.1007/s10951-006-0326-4 27. yang, l. e., scheffran, j., süsser, d., dawson, r., & chen, y. d. (2018) assessment of flood losses with household responses: agent-based simulation in an urban catchment area, environmental modeling & assessment, 23(4), 369-388. copyright © 2022 assa. adv syst sci appl (2022) introduction research methods and technologies development model of emergency situation algorithm for finding safe evacuation schedule model of population evacuation in the northern part of the volga-akhtuba floodplain discussions and conclusion microsoft word 1280 article text, copyedited.docx adv syst sci appl 2022; 03; 147-167 published online at http://ijassa.ipu.ru. a revisit to the prevalent producer theory with emphasis placed on firms’ individualism jeffrey yi-lin forrest* slippery rock university, slippery rock, pa, usa https://orcid.org/0000-0002-5124-9012 e-mail: jeffrey.forrest@sru.edu abstract: as time elapses, an increasing number of theorists and practitioners expressed the urgent need for the community of economists to rebuild economic theories so that derived conclusions would be more readily applicable to real life. to answer this theoretically and practically important call, this paper attempts to reformulate some of the main conclusions of the producer theory so that firms are allowed to have their individually different criteria of optimality and methods of optimization, as the real-life business world dictates. to achieve this goal, on the basis of natural endowments of firms, this paper establishes a series of 7 generally true propositions, while it simultaneously examines how some of the presently well-known results hold true only conditionally. in the process of achieving this end, we generalize hotelling’s and shepard’s lemmas to much more relaxed scenarios than before. because natural endowments are generally different from one firm to another, what is considered better is defined differently so that firms do not collectively produce a better society as a whole, even though each of them maximizes its selfinterest. in other words, one of the main conclusions this paper derives formally is that the invisible hand, as proposed so convincingly by adam smith, is indeed not only invisible but also nonexistent in real life, unless a supernatural being is out there to tell what is better for everyone. in the conclusion, several open questions are listed for future research. keywords: decision making; invisible hand; modular function; natural endowments; optimization; preference order; rationality 1. introduction in studies of economic decision making, a commonly employed approach is to first introduce an objective function, such as a utility function, a production function, a profit function, etc., and then based on some kind cost-and-benefit analysis of the decision maker, this objective function is optimized (e.g., [15,16,18]). however, such an approach does not capture real-life scenarios [3,52] since not everybody is a maximizer or minimizer (e.g., [24, 26]), although it has been repeatedly confirmed with falsified empirical evidence, as so criticized by behavioral economists (e.g., [28,41]). hence, the following question arises naturally at the most fundamental level underneath all investigations of economic decision making, if one focuses only on the micro-level of individual firms: does a firm really go through such a general procedure when it decides on what to do in terms of making a production decision? theoretically speaking, the importance of this question is well witnessed by the vast amount of related literature in the name of rationality, where the aforementioned, commonly employed approach in studies of economic decision making is widely known as the assumption of rationality. although such rationality has been criticized only in recent decades by behavioral economists, some degrees of an inherent uncertainty this assumption implicitly embodies has been broadly felt and explored by a good number of leading scholars [23], including, among numerous others, gary becker [4], frank lovett [36], fritz machlup [37], * corresponding author: jeffrey.forrest@sru.edu 148 j.y-l. forrest copyright ©0000 assa adv. in systems science and appl. (0000) ariel rubinstein [46], paul samuelson [47], herbert simon [50], ll thurstone [53], max weber [55] and glen weyl [57]. in summary, after using this approach for so many decades, scholars are still debating on what the assumption of rationality really means [23]. this end indirectly explains the reason why a compelling need for a meaningful reconstruction of economic theory has been called for by recent events, in particular, the 2008 financial crisis. for example, considering the inability for existing economic theories to describe, to predict and to explain in a timely manner the recent financial turmoil, paul krugman commented as follows in new york times (2009-09-02), the economic profession went astray because economists, as a group, mistook beauty, clad in impressive-looking mathematics, for truth … as memories of the depression faded, economists fell back in love with the old, idealized vision of an economy in which rational individuals interact in perfect markets … unfortunately, this romanticized and sanitized vision of the economy led most economists to ignore … things that can go wrong. they turned a blind eye to the limitations of human rationality that often leads to bubbles and burst; to the problem of institutions that run amok; to the imperfection of markets … that can cause the economy … to undergo sudden, unpredictable crashes; and to the dangers created when regulators don’t believe in regulation. at the same time, paul de grauwe wrote the following in financial times (2009-07-21): mainstream (economic) models take the view that economic agents are superbly inform and understand the deep complexities of the world ... they have “rational expectations” … they all understand the same “truth”, they all act the same way. thus modelling the behavior of just one agent (the “representative” consumer and the “representative” producer) is all one has to do to fully describe the intricacies of the world. rarely has such a ludicrous idea been taken so seriously by so many academics. practically speaking, the significance of the previously posted question is witnessed by failures of predicting the occurrence of imminent economic crises, such as the great recession that started in 2008. in particular, economists surely failed to foretell that the crisis was brewing and failed to predict the surprisingly damaging aftermath when the noted home-price bubble burst [31]. in the contrast, many scholars provided inconsistent 20-20 hint-sights regarding the causes underneath the recession (e.g., [25,29]). additionally, such failures widely exist in applications of economic theories, such as what caused the industrial revolution [14,20,56] and whether or not economic policies actual work in real life [2,11]. that is, no matter whether one is talking about theoretical development or practical applications of economic theories and related business studies, there is a need to reestablish the relevant theories so that the derived conclusions are closer to real life than before. aiming at addressing the aforementioned question of fundamental importance, this paper represents a first step towards this goal by basing our reasoning and analysis on the four natural endowments of a firm: self-awareness, imagination, conscience and free will. by doing so unconventionally, this research is able to pay a revisit to some of the most well-known results in the prevalent producer theory, one of the economic theories that desperately needs to be rewritten, and show how some known conclusions are only true conditionally, while others do not hold true at all. the rest of this paper is organized as follows. section 2 prepares for the smooth flow of analysis and discussion of the following sections. section 3 develops all the main results of this work by generalizing the well-known hotelling’s lemma and shepard’s lemma and by examining how and why no societal-wide preference order can realistically exists. this presentation concludes in section 4 with a few important open problems for future research. a revisit to the prevalent producer theory 149 copyright ©2022 assa. adv. in systems science and appl. (2022) 2. motivation, firms’ decision making and operation this section prepares the presentation of the paper from two different angles. first, it quotes relevant conclusions about a firm’s natural endowments and the role they play in the firm’s decision making. and second, this section lays out the conventions and the necessary symbolic terms needed for later discussions. 2.1. roles natural endowments play in firms’ decision making to help develop the desired methodology, we notice the fact that most part of modern science and mathematics is intrinsically based on the concept of numbers, although more abstract objects have been gradually introduced over time by using different names [30]. at the same time, the systemic structures behind numbers are ignored [34]. for instance, the concept of number 1 is abstracted from such observations as 1 apple, 1 table, 1 chair, etc. when number 1 is singled out from these situations, the systemic structures of the apple, table, chair, etc., are ignored. however, in terms of studies of business, the established theories are mainly about how a business entity evolves and how it interacts with others. that is, theories of business emphasize on systemic structures that are conventionally ignored by the well-adopted methods in economic studies [12]. to overcome this deficit in focus, in our effort to make the producer theory a better fit to real life, let us treat each firm as a living being so that its decisions are made based on its individually unique system of values and beliefs. and such system exists on top of and is determined by the natural endowments of the firm. specifically, parallel to the four natural endowments of an individual – self-awareness, imagination, conscience and free will [35], forrest, hafezalkotob et al. [10] develop a set of corresponding natural endowments for a firm. specifically, by a firm’s self-awareness, it means the firm’s awareness that it exists as a business entity that is separate from other entities, such as people, firms and things, with its business secrets, such as adopted customer value propositions, operational strategies, protected product designs, etc. by a firm’s imagination, it describes the firm’s ability to learn and to acquire new knowledge, to innovatively imagine what might be the right offer, such as a newly designed product, or an improved product or new (or improved) service, to satisfy the deciphered information of market demand, and to develop the necessary process of materially introducing the imagined offer. by a firm’s conscience, it represents the ability for the firm to evaluate which business effort among a group of alternatives is more beneficial than other efforts. by a firm’s free will, it means the capability for the firm to keep, how to keep and to what degree to keep the promises written in its contracts with various business partners. as is well-known from real-world experiences [39], although each firm naturally possesses these natural endowments, how well a firm can mobilize these endowments is dependent on the specific composition and constraints of the firm. that is, the degree of how the natural endowments can be employed is really different from one firm to another. that explains why some firms do well in certain aspects of business while not as well in other aspects. real-life examples that can be used to confirm this end are plentiful. for instance, in any economic sector, there are a few companies that dominate the market while others do not seem prominent or matter at all. by using systemic thinking and logical reasoning, forrest, shao et al. [13] establish the following results, when these scholars attempt to find the true meaning of the assumption of rationality (e.g., [15,16,23]): (i) the goal of each firm’s effort is to materialize, at least partially or remotely, the firm’s clearly stated mission. (ii) at the macro-firm level, the assumption of rationality stands for finding an optimal choice among all available alternatives with the criteria of optimality determined by the focal firm’s management based on the firm’s natural endowments. (iii) each firm has its unique and dissimilar system of values and beliefs. (iv) when a firm faces a decision-making situation, it optimizes the potential subject to the given constraints by using its particular set of criteria of optimality, as formulated consistently with its system of values and beliefs. 150 j.y-l. forrest copyright ©0000 assa adv. in systems science and appl. (0000) example 1: to help confirm the importance of these systemic conclusions, let us revisit a decisionmaking situation from hu [22] and lin [34, p. 136]. assume that the production routine of a firm can be abstracted into the directed and weighted network in figure 1. the firm needs to minimize the path from node x, where the production starts, to node z, where the production ends. case 1: the firm orders the real-number weights in the same way as how real numbers are conventionally ordered. in this case, x  x1  y  z1  z is the path the firm looks for. this path has the weight of 1. and other paths from node x to node z respectively have weights 2, 3, and 4. case 2: the firm orders the real-number weights by referring to the mod4 function so that for any two real numbers 𝑥 and 𝑦, 𝑥 < 𝑦 if and only if 𝑥(mod4) < 𝑦(mod4). within such a system of decision-making, x  x2  y  z2  z is the path the firm looks for. the path’s weight is equal to 3 + 0 + 0 + 1 = 4 (mod 4) = 0, where 𝑥 (mod4) = the reminder of 𝑥 ÷ 4. in comparison, the weights of other paths respectively have weights 1, 2, or 3. figure 1. how minimization is defined differently in different value-belief systems before moving on, let us explain how the modular function, such as the mod4 function above, is actually applied in real life and how different systems of values and beliefs dictate different orderings of real numbers. firstly, modular functions appear frequently in life, because in principle mod𝑟 function represents periodicity 𝑟, for any positive real number 𝑟 [10]. commonly seen examples include 12-hour clocks, 7-day weeks, months of various numbers of days, where the first scenario represents a mod12 function, the second scenario the mod7 function, and the third scenario comprises a mixture of mod28, mod29, mod30 and mod31 functions. and, more generally, the projects a firm participates in also indirectly deal with modular operations. specifically, every time when a new project starts, the involved firm begins its new round of counting of, for example, costs and profits, and measurement of, for example, how success a new business procedure will be. secondly, according to the conventional ordering of real numbers, one has $30 k > $3 million. that is the order commonly used in the literature of neoclassical economics. however, if it is revealed that $30 k is the wage from a lawful employment, while $3 million is the individual share of a group effort of robbing a bank, then a lot of people with certain kinds of values and beliefs will order $30 k > $3 million. in terms of business firms, one can readily construct scenarios where values and beliefs make a difference in the ordering of real numbers, for example, if such a concept as corporate social responsibilities, see, e.g., [9] for relevant details, is involved. now, we are ready to summarize what example 1 tells us different criteria of priority, such as different orderings of real numbers, empower different methods of optimization, a revisit to the prevalent producer theory 151 copyright ©2022 assa. adv. in systems science and appl. (2022) leading to different optimal decisions. and all these differences stem from the varied systems of values and beliefs of individual firms. that is, although different firms might look at an identical objective function, the specifically employed criteria of priority or methods of optimization can be different from one firm to another. this end implies that example 1 analytically confirms what mises [40, p. 244] says – “the value judgements a man pronounces about another man’s satisfaction do not assert anything about this other man’s satisfaction. they only assert what condition of this other man better satisfies the man who pronounces the judgement.” specific to our current context, what is implied by case 1 of the example above, which is the commonly studied situation, is that the economist asserts the focal firm’s condition that better satisfies the economist, while case 2 may actually be the state of affairs of the firm. that is, there is discrepancy between what the economist expects and what the firm desires to achieve [52]. this discrepancy surely represents one source of uncertainties and risks the economist experiences or takes when he draws conclusions and makes claims†, if the firm goes after what its values and beliefs direct. based on the discussions above, assume that each firm has a particular way, defined by its system of values and beliefs, to order real numbers that fall within the domain 𝐷 of its decision-making activities. let the ordering relation be denoted by ≤ for less than or equal to, ≥ for greater than or equal to, and = for equality. 2.2. set-theoretical model of a firm’s operations this subsection and the rest of this paper consider a randomly chosen firm that will be referred to as the firm. when it purchases a quantity 𝑥 of a commodity at unit price (or simply price) 𝑝 , the firm creates for its account a debit in the amount of 𝑥𝑝 . in the contrary, when it sells a commodity in quantity 𝑦 at unit price or price 𝑝 , it creates in its account a credit of 𝑦𝑝 . the quantities of all commodities the firm purchases from others, referred to as inputs to the firm, and offers to the market, referred to as outputs of the firm, can be written as a vector. to separate inputs from outputs in such a vector, the former quantities are represented by negative numbers because they create debits for the firm, while the latter quantities by positive numbers as they represent revenues of the firm. assume that in the marketplace, all commodities are available for exchange and there are ℓ commodities in total that are ordered with such labels as 1, 2, …, ℓ, respectively. to make the following analysis possible, assume that the quantity of each commodity, either received as an input or offered to the market as an output, is a real number, as commonly done in economic analysis (e.g., [43]) if 𝑝 = (𝑝 , 𝑝 , … , 𝑝ℓ) stands for the price system of the quantity vector 𝑐 = (𝑐 , 𝑐 , … , 𝑐ℓ) of all commodities, then the firm’s account will have the overall cash flow 𝑝 ∙ 𝑐 = 𝑝 𝑐 ℓ , (2.1) where ∙ stands for the dot product of vectors 𝑝 and 𝑐. without loss of generality, assume that each price system 𝑝 = (𝑝 , 𝑝 , … , 𝑝ℓ) contains only positive components. that is, no commodity can be acquired without paying a price. and, for the commodity vector 𝑐, some or most of its components should be zero, representing that these commodities are neither inputs nor outputs of the firm. as for commodities, considered include specific times when they are available for delivery, although they are exchanged at the present time, and locations where exchanges of ownership take place. as in real life, for such purposes, the time axis is divided into intervals of equal length, labeled by using natural numbers 1, 2, …, chronologically with the first interval starts at the origin 0, seen as the present moment. assume that all time moments within an interval are not distinguishable. similarly, the land is divided into finite many regions, in each of which † because this paper does not deal with the set of possible consumptions of a consumer, the concepts of risk and knightian uncertainty defined for choice of possible consumptions [5] do not play any role here. 152 j.y-l. forrest copyright ©0000 assa adv. in systems science and appl. (0000) deliveries take place. so, when a commodity has different times of availability and/or different locations of delivery, it is seen as separate and different commodities with specified time and location. correspondingly, the price of a commodity is the amount a consumer needs to pay now for each unit of that commodity while interests and discounts over time are omitted in order to simplify our analysis. additionally, we ignore the value of money at different locations. that is, no issue with exchanges of money is involved. for the firm, its plan of action is to specify the quantity of each commodity it either consumes or offers. that is, its plan 𝑦 = (𝑦 , 𝑦 , … , 𝑦ℓ) of action is to choose an element from the ℓth dimensional euclidean space ℝℓ, representing the quantities of commodities it either consumes or offers, where ℝ is the set of all real numbers and the subscripts 1, 2, …, ℓ denote the individual commodities that are available for exchanges. so, the price of action 𝑦 is given by 𝑝 ∙ 𝑦 = 𝑝 𝑦 ℓ , (2.2) where 𝑝 = (𝑝 , 𝑝 , … , 𝑝ℓ) is the price system of all commodities and 𝑦 stands for the quantity of commodity ℎ (= 1, 2, …, ℓ). if the ℎth component of 𝑦 is 0, it means that the firm neither consumers nor produces commodity ℎ. within the boundary of its constraints, the firm chooses such a plan of action that best fits its specific system of values and beliefs, where maximizing profit, as conventionally studied in the literature, stands for only one such scenario that a particular system of values and beliefs might be demanding. let 𝑌 be the set of all feasible production plans of the firm. that is, each 𝑦 ∈ 𝑌 is a production possibility of the firm that is technically materializable and meets the moral codes of the firm’s system of values and beliefs. as the last part of this preparation section, note that two specific binary relations, ≤ and ≤ , exist. the first one is defined on 𝑌 such that for any 𝑥, 𝑦 ∈ 𝑌, 𝑥 ≤ 𝑦 if and only if 𝑥 ≤ 𝑦 , for each ℎ = 1,2, … , ℓ. and the second is the firm-specific ordering ≤ of real numbers. as defined in mas-collel et al. [38], if ≤ satisfies the conditions of completeness, transitivity and reflexivity, then the firm is seen as rational. evidently, this binary relation ≤ is not complete. on the other hand, because ≤ represents firm 𝐹 ’s specific criteria of priority defined for the real numbers in the domain 𝐷 of decision-making activities, when no confusion appears, assume that ≤ satisfies: (i) completeness (for any 𝑥, 𝑦 ∈ 𝐷, 𝑥 ≤ 𝑦 or 𝑦 ≤ 𝑧 holds true); (ii) transitivity (for 𝑥, 𝑦, 𝑧 ∈ 𝐷, if 𝑥 ≤ 𝑦 and 𝑦 ≤ 𝑧, then 𝑥 ≤ 𝑧); (iii) reflexivity (for any 𝑥 ∈ 𝐷 , 𝑥 ≤ 𝑥 ); and (iv) anti-symmetry (for different 𝑥, 𝑦 ∈ 𝐷 , 𝑥 ≤ 𝑦 and 𝑦 ≤ 𝑥 cannot hold true at the same time). in short, conditions (i) – (iv) are not equivalent to assuming that the firm considered in this paper is rational for the research economist who asserts conditions that lead to his expected optimal possibility, as so phrased in the language of mises [40]. 3. main results this section consists of three parts, where subsection 3.1 looks at how the well-known hotelling’s lemma can be generalized; subsection 3.2 similarly explores possible generalization of shepard’s lemma; and subsection 3.3 examines how and why no societalwide preference order can realistically exists. 3.1. profits and optimal production correspondence for the firm, its profit function 𝜋 : ℝℓ → ℝ is defined as follows 𝜋 (𝑝) = max ∈ 𝑝 ∙ 𝑦, (3.1) a revisit to the prevalent producer theory 153 copyright ©2022 assa. adv. in systems science and appl. (2022) where 𝑝 ∈ ℝℓ is a price system of commodities, if the maximum value exists according to the firm’s system of values and beliefs. that is, 𝜋 is not necessarily defined for some 𝑝 ∈ ℝℓ. in the rest of this paper, when the expression 𝜋 (𝑝) appears, what is implicitly assumed is the existence of the maximum above unless stated otherwise. in equation (3.1), firm 𝐹’s specific system of values and beliefs determines the meaning of maximum and how the maximization is carried out, if a potential result exists‡. this symbolism perfectly reflects the most common and most relevant concept of a rational action defined in psychology as an action that is in line with the values and beliefs of the individual concerned [27]. related to this problem of maximization, neoclassical economics has embraced such a convention that the objective of each firm is to maximize its profit (e.g., [15,16,18,58]). however, this convention is not generally true in real life (e.g., [24,26]). for example, an organization of important chief executives from the united states, chaired by jp morgan chase ceo jamie dimon, recently ceased to support the doctrine that businesses must maximize profits for shareholders above all else (https://opportunity.businessroundtable.org/ ourcommitment/, accessed on january 30, 2021). “americans deserve an economy that allows each person to succeed through hard work and creativity and to lead to a life of meaning and dignity” and “we commit to deliver value to all of them, for the future success of our companies, our communities, and our country,” (https://s3.amazonaws.com/brt.org/brtstatementonthepurposeofacorporationoctober2020.pdf, accessed on january 30, 2021), as so declared by the organization. once again, the reason why many firms don’t put profit maximization as the number one priority can be explained by using their natural endowments. it is because the conscience of the managers directs them to contribute more to their respective causes. this end surely supports the notion that how a firm behaves is dictated by its system of values and beliefs. proposition 1: the firm’s profit function 𝜋 is partially defined on ℝℓ ; and if the firm’s system of values and beliefs is consistent with the conventional ordering of real numbers, then 𝜋 is homogeneous of degree one on 𝑑𝑜𝑚𝑎𝑖𝑛(𝜋 ). and in general, 𝜋 is not homogeneous of degree one. proof. the first conclusion follows from the fact that there is no guarantee that 𝜋 (𝑝) = max ∈ 𝑝 ∙ 𝑦 exists for each price system 𝑝 ∈ ℝℓ . to see the second conclusion, for any scalar 𝜆 > 0 and any 𝑝 ∈ domain(𝜋 ), 𝜋 (𝜆𝑝) = 𝜆max ∈ 𝑝 ∙ 𝑦 = 𝜆𝜋 (𝑝). as for the third conclusion, it suffices for us to construct a counterexample to confirm the statement. to achieve this end, let us modify example 1 slightly. to do this, let us borrow the modular function (or mod function for short) that is generalized from the domain of integers to the set of all real numbers by forrest, hafezalkotob et al. [10]. in particular, let 𝑎 ∈ ℝ be a positive number. define a linear order relation < ( ) on ℝ as follows: for any 𝑥 and 𝑦 ∈ ℝ, 𝑥 < ( ) 𝑦 if and only if 𝑥 𝑚𝑜𝑑(𝑎) < 𝑦 𝑚𝑜𝑑(𝑎), (3.2) where the ordering < is the conventional one defined on ℝ, 𝑥 𝑚𝑜𝑑(𝑎) is the remainder of 𝑥 ÷ 𝑎 and 𝑦 𝑚𝑜𝑑(𝑎) the remainder of 𝑦 ÷ 𝑎, such that 0 ≤ 𝑥 𝑚𝑜𝑑(𝑎) < 𝑎 and 0 ≤ 𝑦 𝑚𝑜𝑑(𝑎) < 𝑎 . when all the involved numbers 𝑎 , 𝑥 and 𝑦 are integers, this order relation < ( ) degenerate into the one widely studied in number theory [6]. in other words, for any 𝑟 ∈ ℝ, ‡ one needs to pay attention to the fact that the concepts of optimality and optimization, employed here and throughout the rest of this paper, are firm-specific instead of those defined by the economist who looks at the issue in hands and believe the outcome is what the firm desires. 154 j.y-l. forrest copyright ©0000 assa adv. in systems science and appl. (0000) satisfying 0 ≤ 𝑟 < 𝑎, the following set of real numbers are classified into one equivalence class, denoted by 𝑟 without causing confusion: 𝑟 ≡ {𝑥 ∈ ℝ: ∃𝑞 ∈ ℤ(𝑥 = 𝑎𝑞 + 𝑟)}, (3.3) where ℤ stands for the set of all integers, that is, ℤ = {… , −3, −2, −1, 0, +1, +2, +3, … }. example 2: assume that a specific production of the firm involves one unit of each of the commodity inputs x, x1, x2, y, z1, z2, z, where x1 and x2 can substitute for each other and so do z1 and z2, figure 2. the arrows stand for the sequence these commodities are fed into the production line one after another, while the weights the relevant profits created by the production sequence from one node to the next. now, the manager of the production wants to maximize the total profit. figure 2. a specific production line with respective profits case 1: the firm orders the real-number weights in the same way as how real numbers are conventionally ordered. in this case, the four possible paths and their respective total weights are given as follows: x  x1  y  z1  z with weight 5; x  x1  y  z2  z with weight 6; (3.4) x  x2  y  z1  z with weight 7; x  x2  y  z2  z with weight 8. hence, the maximum total profit is equal to 8. to show that for any scalar 𝜆 > 0 , 𝜋 (𝜆𝑝) = 𝜆max ∈ 𝑝 ∙ 𝑦 = 𝜆𝜋 (𝑝) , assume that all the commodities involved here are ordered as follows: 𝐼 = {𝑋, 𝑋 , 𝑋 , 𝑌, 𝑍 , 𝑍 , 𝑍}. let the respective paths be 𝐼 , = {𝑋, 𝑋 , 𝑌, 𝑍 , 𝑍}, 𝐼 , = {𝑋, 𝑋 , 𝑌, 𝑍 , 𝑍}, (3.5) 𝐼 , = {𝑋, 𝑋 , 𝑌, 𝑍 , 𝑍}, 𝐼 , = {𝑋, 𝑋 , 𝑌, 𝑍 , 𝑍}. then the corresponding input and associated output vectors are given respectively by 𝑦 = 𝑦 ∈ , , 𝑦 = 𝑦 ∈ , , 𝑦 = 𝑦 ∈ , , and 𝑦 = 𝑦 ∈ , , (3.6) and 𝑍 , 𝑗 = 1, … , 4. now, each price system 𝑝 can be written as follows: 𝑝 = 𝑝 ∈ , 𝑍 . (3.7) and for any production 𝑦, there is 𝑘 (= 1, …, 4) such that when all zero components are eliminated, we have 𝑦 = 𝑦 and 𝑦 = 𝑍 . (3.8) therefore, for any price system 𝑝 and any production 𝑦, 𝑝 ∙ 𝑦 = 𝑝 ∙ 𝑦 + 𝑝 ∙ 𝑦 , (3.9) a revisit to the prevalent producer theory 155 copyright ©2022 assa. adv. in systems science and appl. (2022) where 𝑝 stands for the sub-vector of 𝑝 of the prices of the sub-vector 𝑦 of input commodities in 𝑦 , and 𝑝 the sub-vector of 𝑝 of those of the sub-vector 𝑦 of output commodities in 𝑦. therefore, we have that for any scalar 𝜆 > 0, 𝜋 (𝜆𝑝) = max ∈ 𝜆𝑝 ∙ 𝑦 + 𝜆𝑝 ∙ 𝑦 = 𝜆max ∈ 𝑝 ∙ 𝑦 = 𝜆𝜋 (𝑝) = 8𝜆. (3.10) case 2: the firm orders the real-number weights by referring to the mod4 function so that for any two real numbers 𝑥 and 𝑦, 𝑥 < 𝑦 if and only if 𝑥(mod4) < 𝑦(mod4). in this case, the respective total profits of the four possible paths, as listed above, are given as follows: 5 (mod4) = 1, 6 (mod4) = 2, 7 (mod4) = 3 and 8 (mod4) = 0. therefore, the maximum total profit is equal to 3. now, let 𝜆 = 3.2 be the scalar that is multiplied to each of the individual local values. the corresponding total profits for the four paths are respective equal to 5 × 3.2 (mod4) = 0, 6 × 3.2 (mod4) = 3.2, 7 × 3.2 (mod4) = 2.4, and 8 × 3.2 (mod4) = 1.6. that is, we have (3.2 times 7 (mod4)) ≠ (3.2 × 6 (mod4)). this end concludes the example and the proof of proposition 1. qed the optimal production correspondence of the firm [33] is the partial, set-valued function 𝜂 : ℝℓ → 𝑌 defined as follows: for any price system 𝑝 ∈ ℝℓ , 𝜂 (𝑝) = 𝑦 ∈ 𝑌: 𝑝 ∙ 𝑦 = max ∈ 𝑝 ∙ 𝑦 , (3.11) assuming that there is a production 𝑦 ∈ 𝑌 such that 𝑝 ∙ 𝑦 = max ∈ 𝑝 ∙ 𝑦 . speaking differently, 𝜂 maps each price system 𝑝 to the subset 𝜂 (𝑝) ⊂ 𝑌 of all profit-maximizing productions, if this subset is not empty. in the rest of this section, assume that the order relation of real numbers, implied by the firm’s system of values and beliefs, is the same as the conventional one. proposition 2 (generalized hotelling’s lemma): for a given price system 𝑝 = (𝑝 , 𝑝 , … , 𝑝ℓ) ∈ ℝℓ, if the firm’s profit 𝜋(𝑝) = 𝑚𝑎𝑥 ∈ 𝑝 ∙ 𝑦 exists, 𝜂(𝑝) is then a singleton in a neighborhood of 𝑝, if and only if 𝜋(∙) is differentiable at 𝑝 with respect to each 𝑝 and 𝜕𝜋(𝑝) 𝜕𝑝 = 𝑦 , 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑦 = (𝑦 , 𝑦 , … , 𝑦ℓ) ∈ 𝜂(𝑝), ℎ = 1,2, … , ℓ. (3.12) proof. (⇒) because 𝜂(𝑝) is a singleton in a neighborhood of 𝑝, the envelope theorem applies 𝜕𝜋(𝑝) 𝜕𝑝⁄ = 𝑦 , for the element 𝑦 = (𝑦 , 𝑦 , … , 𝑦ℓ) ∈ 𝜂(𝑝), ℎ = 1,2, … , ℓ. (⇐) for any 𝑦 = (𝑦 , 𝑦 , … , 𝑦ℓ ) and 𝑦 = (𝑦 , 𝑦 , … , 𝑦ℓ ) ∈ 𝜂(𝑝), we have 𝜋(𝑝) = 𝑝 ∙ 𝑦 = 𝑝 ∙ 𝑦 . so, equation (3.12) implies that 𝑦 = 𝜕𝜋(𝑝) 𝜕𝑝 = 𝑦 , ℎ = 1,2, … , ℓ. (3.13) therefore, 𝑦 = 𝑦 . that is, 𝜂(𝑝) is a singleton. qed define the following characteristic function 𝐹: 𝑌 → ℝ: 𝐹(𝑦) = 0, if 𝑦 is on the frontier of 𝑌 < 0, if 𝑦 is in the interior of 𝑌 > 0, if 𝑦 is outside of 𝑌 . (3.14) with the help of this function, the following maximization problem max 𝑝 ∙ 𝑦, s. t. 𝑦 ∈ 𝑌 can be rewritten as max 𝑝 ∙ 𝑦, s. t. 𝐹(𝑦) ≤ 0. the lagrangian of this problem is 𝐿 = 𝑝 ∙ 𝑦 − 𝜆𝐹(𝑦) (3.15) 156 j.y-l. forrest copyright ©0000 assa adv. in systems science and appl. (0000) which implies the first-order conditions: 𝑝 = 𝜆𝐹 (𝑦∗), 𝐹(𝑦∗) ≤ 0, for 𝑦∗ ∈ 𝜂(𝑝), ℎ = 1,2, … , ℓ. (3.16) proposition 3: for any price system 𝑝 = (𝑝 , 𝑝 , … , 𝑝ℓ) ∈ ℝℓ such that 𝑝 > 0 , for ℎ = 1,2, … ℓ , if 𝑦(𝑝) ∈ 𝜂(𝑝) is continuously differentiable, then the matrix 𝐷 𝑦(𝑝) = 𝐷 𝜋(𝑝) = 0. proof. first, let us compute 𝐷 𝑦(𝑝) as follows: 𝐷 𝑦(𝑝) = 𝜕𝑦(𝑝) 𝜕𝑝 = 𝜕𝑦 (𝑝) 𝜕𝑝 , 𝜕𝑦 (𝑝) 𝜕𝑝 , … , 𝜕𝑦ℓ(𝑝) 𝜕𝑝 = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 𝜕𝑦 𝜕𝑝 𝜕𝑦 𝜕𝑝 … 𝜕𝑦ℓ 𝜕𝑝 𝜕𝑦 𝜕𝑝 𝜕𝑦 𝜕𝑝 … 𝜕𝑦ℓ 𝜕𝑝 … … … … 𝜕𝑦 𝜕𝑝ℓ 𝜕𝑦 𝜕𝑝ℓ … 𝜕𝑦ℓ 𝜕𝑝ℓ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ ℓ×ℓ (3.17) = 𝜕 𝜋(𝑝) 𝜕𝑝 𝜕𝑝 ℓ×ℓ from proposition 2 = 𝐷 𝜋(𝑝) the general (𝑡, ℎ) cell of 𝐷 𝑦(𝑝), as given in equation (3.17), is equal to 𝜕𝑦 𝜕𝑝 = 𝜕𝑦 𝜕𝑝 𝑝 1 𝑝 since 𝑝 > 0, for ℎ = 1,2, … ℓ = 𝜕𝑦 𝜕𝑝 𝐹 (𝑦) 𝜆 𝑝 from equation (3.16) = 𝜕𝐹(𝑦) 𝜕𝑝 𝜆 𝑝 (3.18) = 0 ∙ 𝜆 𝑝 = 0, from envelope theorem where a different way to explain 𝜕𝐹(𝑦) 𝜕𝑝⁄ = 0 is that 𝑦 ∈ 𝜂(𝑝), no matter what value 𝑝 is equal to, 𝐹(𝑦) = 0. therefore, the derivative of 𝐹(𝑦) with respect to any variable will be zero. therefore, 𝐷 𝑦(𝑝) = 𝐷 𝜋(𝑝) = 0. qed proposition 3 stands for a major improvement of a theorem in the producer theory [33,38], where the matrix 𝐷 𝑦(𝑝) = 𝐷 𝜋(𝑝) is shown to be symmetric, positive semi-definite under the assumptions, which are not imposed here, that the firm produces a single product and that 𝜂(𝑝) is a singleton. 3.2. the cost minimization problem & conditional factor demands let 𝑞 = 𝑞 , 𝑞 , … , 𝑞 denote the required quantities of production outputs, satisfying that 𝑞 > 0 , for any 𝑗 = 1, 2, … , 𝑠 , and ℎ < ℎ < ⋯ < ℎ . then, the firm’s cost minimization problem can be written as follows, assuming that the firm is a price taker: for a given price system 𝑝 ∈ ℝℓ , a revisit to the prevalent producer theory 157 copyright ©2022 assa. adv. in systems science and appl. (2022) min ∈ 𝑝 ∙ 𝑦 s. t. 𝑓 𝑦 ≥ 𝑞 , (3.19) where 𝑓 𝑦 = 𝑦 satisfies that {ℎ , ℎ , … , ℎ } is generally a subset of the set of all commodity subscripts that appear in the components of 𝑦 . without loss of generality, we assume that these two sets are the same, because producing additional products beyond what are listed in 𝑞 requires at least an increased amount of labor input. let 𝑧 = 𝑧(𝑝, 𝑞) be a solution of the minimization problem in equation (3.19). this solution is known as a conditional factor demand [33], because of its dependence on the required production outputs 𝑞. let 𝑍 = 𝑧: ∃𝑦 ∈ 𝑌 𝑧 = 𝑦 and 𝑓(𝑧) ≥ 𝑞 and the optimal value of the objective function in equation (3.19), be 𝑐 (𝑝, 𝑞) = min ∈ 𝑝 ∙ 𝑧, for 𝑝 ∈ ℝℓ , (3.20) which gives the minimum cost at which the required outputs 𝑞 can be produced. because the implementation of any non-zero production plan has to use certain amounts of some inputs, such as labor, work space, etc., and produce certain outputs, such as waste, if nothing useful, we have 𝑍 ⊆ 𝑦 : 𝑦 ∈ 𝑌 ⊆ ℝ ℓ . (3.21) and, for given 𝑝 ∈ ℝℓ and 𝑞 ∈ ℝ , define the following set-valued function 𝜉 (𝑝, 𝑞) = 𝑧 ∈ 𝑍: 𝑝 ∙ 𝑧 = min ∈ 𝑝 ∙ 𝑧 , (3.22) known as the set of conditional factor demands (that is conditional on the desired level of outputs). proposition 4: for any production 𝑦 ∈ 𝑌 , if 𝑦 ∈ ℝ , for some 𝑡 = 1,2, … , ℓ − 1, and 𝑦 ∉ 𝑍 ∩ ℝ implies that there is a hyperplane 𝐿 in ℝ that separates 𝑦 and 𝑍 ∩ ℝ in terms of the firm’s order relation ≤ of real numbers so that 𝑦 ∉ 𝐿, then 𝑍 = 𝑧 ∈ ℝ : ∀𝑝 ∈ ℝℓ 𝑝 ∙ 𝑧 ≥ 𝑐 𝑝 , 𝑞 ℓ . (3.23) proof. from equation (3.21), it follows that 𝑍 = 𝑍 ∩ ℝ ℓ . (3.24) so, to show equation (3.23), it suffices to demonstrate that for each 𝑡 = 1,2, … , ℓ − 1, 𝑍 ∩ ℝ = 𝑧 ∈ ℝ : ∀𝑝 ∈ ℝℓ 𝑝 ∙ 𝑧 ≥ 𝑐 𝑝 , 𝑞 , (3.25) due to the fact that terms in the union in equation (3.23) are pairwise disjoint. let the set defined by equation (3.25) be 𝑍. it suffices to demonstrates that 𝑍 ∩ ℝ ⊆ 𝑍 and 𝑍 ⊆ 𝑍 ∩ ℝ . the former follows directly from the definition of 𝑐 (𝑝, 𝑞) in equation (3.20). next, let us examine that 𝑍 ⊆ 𝑍 ∩ ℝ . to this end, let us pick an arbitrary 𝑦 ∈ 𝑌 such that 𝑦 ∉ 𝑍 ∩ ℝ . then, the if-condition guarantees the existence of a hyperplane 𝐿 in ℝ that separates 𝑦 and 𝑍 ∩ ℝ . assume that the equation of the plane is 𝑝 ∙ 𝑥 = 𝛽, for 𝑥 ∈ ℝ , some non-zero 𝑝 ∈ ℝℓ and a scalar 𝛽 ∈ 158 j.y-l. forrest copyright ©0000 assa adv. in systems science and appl. (0000) ℝ , satisfying that for any 𝑧′ ∈ 𝑍 ∩ ℝ , 𝑝 ∙ 𝑧 ≥ 𝛽 and 𝛽 > 𝑝 ∙ 𝑦 . hence, taking minimum or infimum produces 𝑝 ∙ 𝑦 < 𝛽 ≤ min ∈ ∩ℝ 𝑝 ∙ 𝑧 or inf ∈ ∩ℝ 𝑝 ∙ 𝑧 . (3.26) once again, due to differences in dimensionality between ℝ and ℝ , for 𝑡 , 𝑡 = 1,2, … , ℓ − 1, 𝑡 ≠ 𝑡 , it can be seen that min ∈ ∩ℝ 𝑝 ∙ 𝑧 = 𝑐 𝑝 , 𝑞 , because 𝑝 ∈ ℝ the corresponding 𝑧 in the operation 𝑝 ∙ 𝑧 has to be from ℝ . therefore, equation (3.26) is the same as 𝑝 ∙ 𝑦 < 𝛽 ≤ min ∈ 𝑝 ∙ 𝑧 or inf ∈ 𝑝 ∙ 𝑧 . (3.27) that is, what is shown is that 𝑦 ∉ 𝑍 ∩ ℝ → 𝑦 ∉ 𝑍, which means 𝑍 ⊆ 𝑍 ∩ ℝ . therefore, equation (3.25), and then equation (3.23) follows from equation (3.24). qed proposition 5 (generalized shepard’s lemma): for a given price system 𝑝 ∈ ℝℓ , assume that the firm’s minimization of production cost is based on the conventional lagrangian approach. then, 𝜉 (𝑝, 𝑞) = {𝑧(𝑝, 𝑞)} is a singleton in a neighborhood of 𝑝 , if and only if 𝑐(∙, 𝑞) is differentiable at 𝑝 with respect to each 𝑝 such that 𝜕𝑐(𝑝, 𝑞) 𝜕𝑝 = 𝑧 (𝑝, 𝑞), for 𝑗 = 1,2, … , 𝑡, (3.28) assuming that 𝑝 = 𝑝 , 𝑝 , … , 𝑝 ∈ ℝ , for some 𝑡 = 1,2, … , ℓ − 1. proof. (⇒ ) because 𝜉 (𝑝, 𝑞) = {𝑧(𝑝, 𝑞)} is a singleton in a neighborhood of 𝑝 , the assumption on how the firm minimizes its cost means that the envelope theorem applies so that we have 𝜕𝑐(𝑝, 𝑞) 𝜕𝑝 = 𝜕 𝜕𝑝 𝑝 ∙ 𝑧(𝑝, 𝑞) = 𝜕𝑝 𝜕𝑝 ∙ 𝑧(𝑝, 𝑞) = 𝑧 (𝑝, 𝑞), (3.29) for 𝑗 = 1,2, … , 𝑡. (⇐) for any 𝑧 , 𝑧 ∈ 𝜉 (𝑝, 𝑞), by filling unmatching components with zeros if needed, we can assume that 𝑧 = 𝑧 , 𝑧 , … , 𝑧 and 𝑧 = 𝑧 , 𝑧 , … , 𝑧 . hence, we have 𝑐(𝑝, 𝑞) = 𝑝 ∙ 𝑧 = 𝑝 ∙ 𝑧 . so, equation (3.28) implies that 𝑧 = 𝜕𝑐(𝑝, 𝑞) 𝜕𝑝 = 𝑧 , 𝑗 = 1,2, … , 𝑡. (3.30) therefore, 𝑧 = 𝑧 . that is, 𝜉 (𝑝, 𝑞) is a singleton. qed for the rest of this subsection, assume that the firm’s order relation ≤ of real numbers is the same as the conventional one ≤. proposition 6: for a given price system 𝑝 ∈ ℝℓ , if 𝜉 (𝑝, 𝑞) = {𝑧(𝑝, 𝑞)} is a singleton in a neighborhood of 𝑝 and 𝑧(𝑝, 𝑞) is continuously differentiable with respect to 𝑝 = 𝑝 , 𝑝 , … , 𝑝 ∈ ℝ , then the matrix 𝐷 𝑧(𝑝, 𝑞) = 𝐷 𝑐(𝑝, 𝑞) = 0 × . a revisit to the prevalent producer theory 159 copyright ©2022 assa. adv. in systems science and appl. (2022) proof. first, let us compute 𝐷 𝑧(𝑝, 𝑞) as follows: 𝐷 𝑧(𝑝, 𝑞) = 𝜕𝑧(𝑝, 𝑞) 𝜕𝑝 = 𝜕𝑧 (𝑝) 𝜕𝑝 , 𝜕𝑧 (𝑝) 𝜕𝑝 , … , 𝜕𝑧 (𝑝) 𝜕𝑝 = 𝜕 𝑐(𝑝, 𝑞) 𝜕𝑝𝜕𝑝 , 𝜕 𝑐(𝑝, 𝑞) 𝜕𝑝𝜕𝑝 , … , 𝜕 𝑐(𝑝, 𝑞) 𝜕𝑝𝜕𝑝 from proposition 5 = 𝜕 𝑐(𝑝, 𝑞) 𝜕𝑝 𝜕𝑝 × (3.31) = 𝐷 𝑐(𝑝, 𝑞) that is 𝐷 𝑧(𝑝, 𝑞) = 𝐷 𝑐(𝑝, 𝑞). to show 𝐷 𝑧(𝑝, 𝑞) = 0, define 𝐹(𝑧) = 0, if 𝑧 is on the frontier of 𝑍 < 0, if 𝑧 is in the interior of 𝑍 > 0, if 𝑧 is outside of 𝑍 . (3.32) then, the following minimization problem min 𝑝 ∙ 𝑧, s. t. 𝑧 ∈ 𝑍 can be rewritten as min 𝑝 ∙ 𝑧, s. t. 𝐹(𝑧) ≤ 0. the lagrangian of this problem is 𝐿 = 𝑝 ∙ 𝑧 − 𝜆𝐹(𝑧) (3.33) which implies the first-order conditions: 𝑝 = 𝜆𝐹 (𝑧∗), 𝐹(𝑥∗) ≤ 0, for 𝑗 = 1,2, … , 𝑡, (3.34) where 𝑧∗ ∈ 𝜉 (𝑝, 𝑞) . from equation (3.31), it follows that for 𝑧(𝑝, 𝑞) = 𝑧 , 𝑧 , … , 𝑧 ∈ 𝜉 (𝑝, 𝑞), 𝐷 𝑧(𝑝, 𝑞) = 𝜕𝑧 𝜕𝑝 × = 𝜕𝑧 𝜕𝑝 ∙ 𝑝 𝑝 × = 𝜆 𝑝 ∙ 𝜕𝑧 𝜕𝑝 ∙ 𝐹 (𝑧) × from equation (3.34) = 𝜆 𝑝 𝜕𝐹(𝑧) 𝜕𝑝 × = 𝜆 𝑝 ∙ 0 × = 0 × from equation (3.33) or the envelope theorem (3.35) where 𝜕𝐹(𝑧) 𝜕𝑝 = 0 also comes from the fact that 𝑧 ∈ 𝜉 (𝑝, 𝑞) so that 𝐹(𝑧) = 0 no matter how 𝑝 changes. therefore, the derivative of 𝐹(𝑧) with respect to 𝑝 is zero. therefore, 𝐷 𝑧(𝑝, 𝑞) = 0 × . qed this result carries a well-known conclusion in the producer theory much forward. in particular, the known result claims [33,38] that the matrix 𝐷 𝑧(𝑝, 𝑞) = 𝐷 𝑐(𝑝, 𝑞) is symmetric, negative semidefinite and 𝐷 𝑧(𝑝, 𝑞)𝑝 = 0. 3.3. the non-existence of invisible hand in any large-scale economy 160 j.y-l. forrest copyright ©0000 assa adv. in systems science and appl. (0000) let 𝑝 = (𝑝 , 𝑝 , … , 𝑝ℓ) ∈ ℝℓ be a price system. then for any production 𝑦 ∈ 𝑌 , the profit 𝜋 of producer 𝑗 is 𝑝 ∙ 𝑦 and the total profit 𝜋 of all producers is 𝜋 = ∑ 𝜋 = ∑ 𝑝 ∙ 𝑦 = 𝑝 ∙ ∑ 𝑦 = 𝑝 ∙ 𝑦, where assumed is 𝑦 = ∑ 𝑦 , which is known as a total production. in this setup, each producer is treated as a price taker, while trying to maximize the realization of its mission by choosing an optimal production 𝑦 , where both the definition of maximum and method of maximization are defined by the producer 𝑗 ’s system of values and beliefs. considering that each price and every commodity are time and location specific, this setup requires each producer to choose its production so that its inputs and outputs are optimally distributed over both time and space. such an optimal production is referred to as an equilibrium production of producer 𝑗 with respect to the price system 𝑝 [7]. let ≤ be the order relation of real numbers producer 𝑗 employs in its determination of optimal decisions and ≤ the conventional order relation between real numbers. for the economy of 𝑛 producers, the collective order relation ≤ of real numbers is defined as follows, assuming that the society is democratic: for any 𝑎 and 𝑏 ∈ ℝ, 𝑎 ≤ 𝑏 if and only if 𝑎 ≤ 𝑏, for 𝑗 = 1,2, … , 𝑛. (3.36) proposition 7: assume that all involved order relations of real numbers are consistent with the conventional one. then, for a given price system 𝑝 = (𝑝 , 𝑝 , … , 𝑝ℓ) ∈ ℝℓ , and a total production 𝑦 = 𝑦 + 𝑦 + ⋯ + 𝑦 ∈ 𝑌 + 𝑌 + ⋯ + 𝑌 , where 𝑦 ∈ 𝑌 , for 𝑗 = 1, 2, …, 𝑛, 𝑝 ∙ 𝑦 = 𝑚𝑎𝑥 ∈ 𝑝 ∙ 𝑦 if and only if 𝑝 ∙ 𝑦 = 𝑚𝑎𝑥 ∈ 𝑝 ∙ 𝑦 , 𝑓𝑜𝑟 𝑗 = 1, 2, … , 𝑛. proof. (⇒) assume that 𝑝 ∙ 𝑦 = 𝑝 ∙ 𝑦 + 𝑝 ∙ 𝑦 + ⋯ + 𝑝 ∙ 𝑦 = max ∈ 𝑝 ∙ 𝑦 , where 𝑦 ∈ 𝑌 , for 𝑗 = 1, 2, …, 𝑛 , but there is a producer 𝑗 , 1 ≤ 𝑗 ≤ 𝑛 , such that 𝑝 ∙ 𝑦 < max ∈ 𝑝 ∙ 𝑦 . hence, we have max ∈ 𝑝 ∙ 𝑦 = 𝑝 ∙ 𝑦 + 𝑝 ∙ 𝑦 + ⋯ + 𝑝 ∙ 𝑦 < 𝑝 ∙ 𝑦 + ⋯ + 𝑝 ∙ 𝑦 + max ∈ 𝑝 ∙ 𝑦 + 𝑝 ∙ 𝑦 + ⋯ + 𝑝 ∙ 𝑦 . (3.37) that contradicts the meaning of 𝑝 ∙ 𝑦 = max ∈ 𝑝 ∙ 𝑦 . (⇐) assume that 𝑝 ∙ 𝑦 = max ∈ 𝑝 ∙ 𝑦 , for 𝑗 = 1, 2, …, 𝑛, but 𝑝 ∙ 𝑦 = 𝑝 ∙ 𝑦 + 𝑝 ∙ 𝑦 + ⋯ + 𝑝 ∙ 𝑦 < max ∈ 𝑝 ∙ 𝑦 . hence, there is 𝑦∗ = 𝑦∗ + 𝑦∗ + ⋯ + 𝑦∗ ∈ 𝑌, where 𝑦∗ ∈ 𝑌 , for 𝑗 = 1, 2, …, 𝑛, such that 𝑝 ∙ 𝑦 + 𝑝 ∙ 𝑦 + ⋯ + 𝑝 ∙ 𝑦 < 𝑝 ∙ 𝑦∗ + 𝑝 ∙ 𝑦∗ + ⋯ + 𝑝 ∙ 𝑦∗. so, there is at least one such 𝑗, 1 ≤ 𝑗 ≤ 𝑛, that 𝑝 ∙ 𝑦 = max ∈ 𝑝 ∙ 𝑦 < 𝑝 ∙ 𝑦∗. (3.38) that is a contradiction. therefore, the assumption 𝑝 ∙ 𝑦 < max ∈ 𝑝 ∙ 𝑦 is false. qed what this proposition says is that if all involved order relations of real numbers are consistent with the conventional one, then one has max ∈ 𝑝 ∙ 𝑦 = max ∈ 𝑝 ∙ 𝑦 . (3.39) this equation (3.39) provides a condition under which adam smith’s invisible hand exists, for more details, see the discussion right after example 3 below. however, the following a revisit to the prevalent producer theory 161 copyright ©2022 assa. adv. in systems science and appl. (2022) example shows that this equation is not generally true when some of the individual order relations of real numbers are allowed to be different from the conventional one. example 3: assume that an economy has two producers, named 1 and 2. both of them order real numbers with the order relation ≤ ( ). without loss of generality, assume that one unit of each involved commodity is either needed for production or produced as market offer. the particular production of producer 1 is shown in figure 3; it involves one unit of each of the commodity inputs a, a?, b, c?, c, where a? can be either a1 and a2, but not both, and similarly, c? can be either c1 and c2, but not both. that is, commodities a1 and a2 can substitute for each other and the same holds true for commodities c1 and c2. in figure 3, the arrows stand for the sequence the corresponding commodities are fed into the production line one after another, while the weights the relevant profits created by the production sequence from one node to the next. figure 3. productions of producer 1 let us label the individual productions of producer 1 as follows: 𝐼 : a  a1  b  c1  c; 𝐼 : a  a1  b  c2  c; 𝐼 : a  a2  b  c1  c; 𝐼 : a  a2  b  c2  c. (3.40) so, the set of production possibilities of producer 1 is 𝑌 = {𝐼 , 𝐼 , 𝐼 , 𝐼 } and corresponding profits are 1.345, 1.36, 6.085 mod(4) = 2.085, and 6.1 mod(4) = 2.1, respectively. that is, we have max ∈ 𝑝 ∙ 𝑦 = 2.1. (3.41) for producer 2, its production is shown in figure 4; it involves one unit of each of the commodity inputs u, u?, v, w?, w, where u? can be either u1 and u2, but not both, and similarly, w? can be either w1 and w2, but not both. that is, commodities u1 and u2 can substitute for each other and the same holds true for commodities w1 and w2. in figure 4, the arrows stand for the sequence the corresponding commodities are fed into the production line one after another, while the weights the relevant profits created by the production sequence from one node to the next. let us label the individual productions of producer 2 as follows: 𝐽 : u  u1  v  w1  w; 𝐽 : u  u2  v  w1  w; 𝐽 : u  u1  v  w2  w; 𝐽 : u  u2  v  w2  w. (3.42) 162 j.y-l. forrest copyright ©0000 assa adv. in systems science and appl. (0000) so, the set of production possibilities of producer 2 is 𝑌 = {𝐽 , 𝐽 , 𝐽 , 𝐽 } and corresponding profits are 6.1 mod(4) = 2.1, 4.135 mod(4) = 0.135, 3.31, and 1.315, respectively. that is, we have max ∈ 𝑝 ∙ 𝑦 = 3.31. (3.43) therefore, from equations (3.41) and (3.43), we have max ∈ 𝑝 ∙ 𝑦 + max ∈ 𝑝 ∙ 𝑦 = ( ) 2.1 + 3.31 𝑚𝑜𝑑(4) = ( ) 1.41. (3.44) figure 4. productions of producer 2 to compute max ∈ 𝑝 ∙ 𝑦 , we first have 𝑌 = {𝑦 + 𝑦 : 𝑦 ∈ 𝑌 , 𝑦 ∈ 𝑌 }, where the order relation of real numbers of the two-producer economy is equal to ≤ =≤ ( ) . that is, elements in 𝑌 are given in the form of 𝐼 + 𝐽 , for 𝑖, 𝑗, 𝑘, 𝑙 = 1,2. the computational results of 𝑝 ∙ 𝑦 = 𝑝 ∙ 𝑦 + 𝑝 ∙ 𝑦 mod(4) are shown in table 3.1. table 3.1. computing max ∈ 𝑝 ∙ 𝑦 p2 p1 2.1 0.135 3.31 1.345 1.345 3.445 1.48 0.655 2.69 1.36 3.46 1.495 0.67 2.705 2.085 0.185 2.22 1.395 3.43 2.1 0.2 2.235 1.41 3.445 note: p1 = producer 1; p2 = producer 2 from table 3.1, we have max ∈ 𝑝 ∙ 𝑦 = 3.46. combining this end with equation (3.44) leads to the conclusion that max ∈ 𝑝 ∙ 𝑦 > max ∈ 𝑝 ∙ 𝑦 . (3.45) where the order relation ≤ is given by ≤ ( ). in summary, it has been shown that equation (3.39) does not hold true in general in terms of systems of values and beliefs. qed a more general situation for equation (3.36) and in turn equation (3.39) not to hold is that in general the economy does not have any order relation ≤ of real numbers. for instance, assume that a small economy contains only two producers, named 1 and 2, such that producer 1’s order relation of real numbers is ≤ ( ), while that of producer 2 is ≤ ( ), then 2 and 3.1 cannot be ordered in the economy, because these producers have inconsistent order relations: 3.1 ≤ ( ) 2 𝑎𝑛𝑑 2 ≤ ( ) 3.1, so that equation (3.36) cannot be applied to define any ≤ that is consistent with each of the order relations of the individual producers. a revisit to the prevalent producer theory 163 copyright ©2022 assa. adv. in systems science and appl. (2022) this end is very important in terms of the “invisible hand,” as introduced by adam smith in 1759 in his work the theory of moral sentiments in part iv and chapter 1. in this work, the concept describes the unintended greater social benefits and public good brought about by individuals acting in their own self-interests [48]. here, the key is the word “greater”. what does it mean? according to the discussion above, the order relation ≤ of the economy might not exist in general, as long as there are producers whose order relations of real numbers are inconsistent with each other. that is, in general, within any economy in real life there is not a universally accepted way to tell what is meant by “greater social benefits and public good”, because real numbers in such a large-scale system are not ordered consistently in the individual systems of values and beliefs of the incumbent producers. in other words, what is obtained above analytically confirms what joseph e. stiglitz [1] believes the invisible hand is often not there, as he claimed based on greenwald and stiglitz [17]. 4. some final words there have been very loud and meaningful calls from both theorists and practitioners to reconstruct the prevalent business and economic theories. the reason underlying the calls is that applications of these theories don’t generally provide timely guidance for practical purposes, for related discussions, see, for example, [12,28,41], and the comments by paul krugman and paul de grauwe, given in the start of this presentation. to positively answer such calls and to address the question posted in the introduction section, this paper represents one step towards this end by examining how the widely accepted producer theory [33,38] can be made more realistic by basing logical reasoning on the four natural endowments of a firm [10,13]: self-awareness, imagination, conscience and free will. other than continuing the tradition, the reason why this work chooses set theory as the tool of analysis and reasoning is because this tool can readily bring systems methodology [34] into rigorous studies of economic theories. such a convenience is very significant, because studies of business and economies tend to involve organizations, their interactions and evolutions, while the focus of systems science is on the creation of tools for such studies. for relevant studies that confirm this point, see, for example, [12,45]. as for why we choose natural endowments as the basis of our reasoning and analysis is because, systemically speaking, they represent the most fundamental elements underlying human intellectual activities in general and decision-making in particular [35]. and we like to reconstruct relevant economic and business theories in a similar fashion as theories of mathematics, where every conclusion is derived from a small set of basic postulates of clear validity. one example of such a mathematical theory is the euclidean plane geometry, which is constructed on the basis of five axioms. by doing so, we expect that many endless debates that occurred in the history of economic studies can be possibly ended fruitfully with more or less definite verdicts. because of the novelty of choosing natural endowments as our starting points, this paper is able to examine a series of 7 most fundamental propositions of the producer theory. at the same time, it convincingly demonstrates the existence of firm-specific order relations of real numbers, reflecting differences in the decision criteria of priority from one firm to another, and consequently the existence of firm-specific methods of optimization. for related discussions, see, for example, hammerton [19], van fleet [54] and yang and andersson [59]. this end represents a major contribution this work makes to the literature beyond checking which known results of the producer theory hold true generally or only under specific conditions. in terms of the literature, several schools of economic thoughts also attempted similarly to either reconstruct or revisit the prevalent economic theory in general or traditional production theory in particular from different angles or starting points. for example, richard nelson and sidney g. winter contributed to the development of the whole evolutionary school by reviving 164 j.y-l. forrest copyright ©0000 assa adv. in systems science and appl. (0000) relevant discussions in the 1980s. they highlighted that the endogenous and permanent change in the environment, “largely driven by innovation, is a central characteristic of modern capital economies” [42, p. 3]. based on piero sraffa’s [51] interpretation of david ricardo, the neorecardian school emerged. scholars in this school argue, among other topic fronts, that natural growth is primarily demand-driven because growth in the labor force as well as in labor productivity both respond to the pressure of demand, both domestic and foreign [8,32]. the post-keynesian economics attempts to rebuild economic theory in the light of keynes' ideas and insights with its theoretical foundation placed on the principle of effective demand, that demand matters in the long as well as the short run so that a competitive market economy has no natural or automatic tendency towards full employment [21]. recently, shaikh [49] demonstrated by using statistics and mathematical modeling that he could derive most of the key propositions of economics without employing such basic assumptions as rationality, optimization, perfect competition, perfect information, representative agents or so-called rational expectations. in comparison, it can be seen that no matter which school one is looking at, its highlighted starting points should be theoretically confirmed instead of being claimed or believed to be. for instance, what the whole evolutionary school believes to be central to modern capital economies should be established by using a confirmative means. to this end, the approach adopted in this paper can fulfil this need. in particular, by using natural endowments, it has been shown that the never-ending need for innovation in modern capital economy is relentlessly driven by the forever changing consumer needs for better lives; that in turn increasingly intensifies competitions among the suppliers in the marketplace [12, chapter 3, 13, proposition 9]. in other words, whatever conclusions derived in the so-called evolutionary economics can be seen as rooted in the concept of natural endowments. as for shaikh’s [49] work, similar scenarios appear throughout, because many of his data-based claims are subject to different interpretations. for instance, in his critical review, patomäki [44] was able to draw a conclusion inconsistent to the one shaikh claimed. in other words, the approach adopted in this research can be expected to provide a new foundation for economic and business studies, because it problematizes the commonly used methodology, while reconstructs the ontology of related theories at a deeper level. as for topics for possible future research, many aspects of this paper call for follow-ups. for example, this paper explores two different order relations of real numbers – the conventional one and the one defined by a modular function. however, managers and entrepreneurs in real life surely employ other means to prioritize the alternatives available for their decision making. so, to make this work more compatible for real-life applications, it will be desirable for scholars to explore how the results developed in this paper remain reliable for different criteria of priority or different orderings of real numbers. specifically, propositions 2, 3, 5 and 6, most likely do not hold true for systems of values and beliefs that define order relations of real numbers through the modular function. so, with such an order relation, how will these results look like? or, in what forms will these results appear? other two important open questions for future studies are about (i) how generally one can specify the ordering of real numbers or criteria of priority when a particular system of values and beliefs is given; and (ii) how to systemically and strategically investigate the interactions of heterogeneous firms with their individually different systems of values and beliefs. references 1. altman, d. 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(2018). spatial aspects of entrepreneurship and innovation. the annals of regional science, 61, 457-462. https://doi.org/10.1007/s00168018-0888-z. adv syst sci appl 2022; 03:49–69 published online at https://ijassa.ipu.ru. model of calcium dynamics regulating ip3 and atp production in a fibroblast cell ankit kothiya*, neeru adlakha department of mathematics and humanities, svnit, surat-395 007, gujarat, india abstract: fibroblast cells are involved in the mechanism of growth and repair of tissue in human organs. calcium (ca2+) signaling is essentially required to maintain the microstructure and physiological function of the fibroblast cell. the study of ca2+ signaling in fibroblast cells is crucial to understanding the mechanisms and disorders of fibroblast cells. the inositol 1,4,5trisphosphate (ip3) participates in the release and extension of calcium from the endoplasmic reticulum (er). adenosine triphosphate (atp ) also regulate various biological activity like proliferation, migration, stimulation of cell growth, etc. any disturbance in ca2+ can disturb the levels of ip3 and atp in the cell. the ca2+ dynamics and its relationship with ip3 and atp production are not well understood. there is a need to understand the relationship among various parameters of ca2+ dynamics and its role in ip3 and atp production. the ca2+ dynamics in fibroblast have been investigated by several researchers, but no attention is reported for analyzing ca2+ dynamic production of ip3 and atp . a mathematical model is proposed for analyzing the role of ca2+ signaling in the production of ip3 and atp in fibroblast cells. the model incorporates the effect of diffusion, buffer, leak, sarco endoplasmic reticulum calcium atpase (serca) pump, etc. the model is formulated in the form of an initial-boundary problem. the solution is obtained using the finite element approach. the effects of excessive or low values of various parameters on ca2+ dynamics, atp production, ip3 production, and ip3 degradation have been analyzed for possible disorders in functions of a fibroblast cell. the alterations in these parameters cause alterations in atp production, ip3 production and ip3 degradation and ca2+ concentration profiles. this alteration in parameters can be responsible for the function and dysfunction of fibroblast cells, leading to cardiac disease, fibrosis, delayed wound healing, etc. the results lead to conclusion that the changes in source influx, buffers, and diffusion coefficient can cause an increase or decrease in atp and ip3 production and ip3 degradation leading to disorders of fibroblast cells like cardiac fibroblast cell proliferation and migration, which have a role in wound healing, inflammation, and cancer . obtained results provide insights into the ca2+-dependent production of ip3 and atp in a fibroblast cell. these insights can be useful for developing diagnostics and treating disorders of fibroblast cells. keywords: ip3 receptor, serca pump, atp , finite element method, calcium, fibroblast cell 1. introduction every system of an organism is made up of various types of cells. the functions of an organism are achieved by the cellular mechanism in the cell. one of the primary functions is to communicate with each part of the body and accomplish by intra and inter-cellular calcium signaling [3]. a fibroblast is a kind of organic cell that originates locally from mesenchymal cells and is a permanent resident of connective tissue. these mesenchymal cells produce extracellular matrices such as collagens, fibronectin, tenascin and proteoglycans. it creates the auxiliary system for creature tissues and assumes a basic job in the healing of ∗corresponding author: ankitkothiya1996@gmail.com 50 wounds [32]. fibroblast cells are virtually present in every tissue of our body. they are responsible for the three-dimensional architecture and mechanical strength of the tissue. it maintains the structural framework in the tissue of animals by the synthesis of extracellular matrix molecules [33]. the change in fibroblast number and properties are associated with chronic lung diseases. the alteration in injury repair is responsible for asthma and pulmonary fibrosis [23]. fibroblast cell plays a vital role in chronic lung disease. transforming growth factor-beta (tgf -β) regulates the fibroblast function. fibroblast response to cytokines, proinflammatory mediators and a variety of growth factors. the role of ca2+ signaling in these responses is still not well explored [22]. intracellular free ca2+ is a ubiquitous secondary messenger, and it regulates various physiological processes like metabolism, differentiation, proliferation, signal transduction, secretion and contraction [15]. there are various types of buffers found in the fibroblast cell. also, referred as proteins, buffer binds with ca2+ to reduce the intracellular ca2+ in fibroblast cells. there are many types of signaling processes in fibroblast cells that elevate intracellular ca2+ and initiate activities of the cell. two major mechanisms release ca2+ from intracellular stores. these are the ryanodine receptor (ryr) and ip3 receptors activations in a cell. the ryr receptor is reported to be absent in humans and rate cardiac fibroblast [5]. however, tgf-β acts on ryanodine sensitive channel in human pulmonary fibroblast leading to ca2+ wave activity [23]. the ip3 activation is the major ca2+ release mechanism in fibroblast cells. when the legend bind to plasmalemma receptors, they cause the release of ip3, which in term causes the release of sequestered ca2+ and influx of extracellular ca2+. ip3 receptor and serca pump act as the source of ca2+ from the er which is used by the cell to initiate and sustain the signaling process required for smooth cellular functions. adenosine triphosphate (atp ) is an important signaling molecule that regulates various biological activities like increasing or reducing proliferation in various cells. cardiac fibroblast cells are involved in maintaining structural, biochemistry, mechanical, and electrical characteristics of the heart [4]. atp regulates the proliferation and migration of cardiac fibroblast cells. the cell migration was detected in a wound-healing assay. cell proliferation increases with the increase in ca2+ concentration. a significant effect of atp at concentrations 0.1µm is observed in increasing the cell proliferation in cardiac fibroblast cells. but the extracellular atp decreases cell proliferation in human cancer, endometrial stromal cells, respectively and neonatal fibroblast cells, the proliferation is reduced by extracellular atp [19]. it stimulates cell growth in fibroblast cells. approximately 1µm of atp is present in the cell, and 1nm − 1µm is present outside the cell. atp release and its metabolism dynamics regulate the intra and extracellular levels, which have a role in wound healing, contraction, relaxation, cell migration, differentiation, inflammation and cancer. purinergic receptors (p2y ) present on the cell membrane detect the atp , thereby realizing ip3 present in the membrane, and it is moved into the cytoplasm. the ip3 then triggers the release of ca2+ from the endoplasmic reticulum into the cytosol of the cell [19]. therefore, it is of great interest to understand the functional aspects of ca2+ signaling in a fibroblast cell [33]. a number of researchers have performed the study of ca2+ distribution in various types of cells like neurons [13,43], astrocytes [12,41], myocytes [31,36,37], oocytes [24,28], acinar [20, 21], fibroblast [16], hepatocyte [11] cells, etc. naik and pardasani [25, 29] have studied a ca2+ signaling in oocytes involving voltage-gated channel, buffer and ryr receptor. harootunian et al. [7] explained the phenomena of oscillations in ca2+ concentrations occurring due to the positive feedback between ip3 and ca2+ in a fibroblast cell. they proposed four classes of models of generating mechanisms. van zoelen et al. [47] clarified the major discrepancies between transformed and normal cells anchorage-independent growth (aig) and loss of density-dependent growth inhibition (ddgi). in this investigation, they copyright © 2022 assa. adv syst sci appl (2022) 51 used normal rate kidney (nrk) fibroblast cell, which has been used widely as a model system for examining the role played by growth factors in phenotype transformation. wagner and keizer [48] analyzed the influence of buffer about ca2+ release from internal stores. they also reported the effects of ip3, rapid stationary and mobile ca2+ buffers. jafri and keizer [9] analyzed the influence of ca2+ diffusion in the cytosol, and ca2+ handling by the er on ip3 affected ca2+ wave propagation. smith et al. [38] obtained an analytical solution to estimate the ca2+ profile near the open ca2+ channel for the steadystate condition of the transport equation proposed by wagner and keizer. berridge et al. [2] reported that ca2+ signaling in the cell triggers new life at fertilization control and controlling various development processes. once the cells are differentiated, the ca2+ functions to control processes like memory, secretion, proliferation, contraction and metabolism. they discussed the source of ca2+, elementary ca2+ signals, the role of the endoplasmic reticulum or sarcoplasmic reticulum, ip3 receptors and voltage-gated ca2+ channel (v gcc) in ca2+ signaling. godefridus and harks [6] performed investigations on excitable fibroblasts, including ion channels, action potential and ip3 signaling in nrk fibroblasts. torres et al. [44] constructed a model for action potential generated in normal rate kidney fibroblasts. kuster et al. [18] created a model for stabilizing the role of ca2+/ip3 store-dependent plasma membrane and ca2+/ip3 channels responsible for potential firing and intracellular ca2+/ip3 fluctuations. chen et al. [4] investigated the mechanism of atp regulates the proliferation of human cardiac fibroblast. sun et al. [40] gave a dynamic model that provides a constitutive way for controlling the ca2+ dynamics and the electrical field-induced intracellular ca2+ response. schmitz et al. [34] investigated the whole-cell ca2+ dynamics using models of single trans-membrane ion-conducting proteins. they successfully described several experimental stages and captured important characteristics of ca2+ dynamics upon t-cell receptor (tcr) stimulation. hao and rovin et al. [8] represented a mathematical progress model from tubulointerstitial inflammation into fibrosis, which can be used for monitoring the treatment for interstitial fibrosis in lupus nephritis autoimmune disease (lnad). also, they give a mathematical model of idiopathic pulmonary fibrosis. lembong et al. [19] performed experiments in fibroblast cells under a variety of tissue conditions to study atp . kumar and pardasani [17] developed a one-dimensional mathematical model for t-lymphocytes representing the intracellular ca2+ distribution using a finite element approach. they also obtained the effect of source influx, buffers, and ryr receptor for exceptional cases and serca pump for the temporal case. kotwani and adlakha [16] constructed an excess buffer approximated model of the plasma membrane and endoplasmic reticulum ca2+ influx and efflux in fibroblast cell. jagtap and adlakha [10] constructed a framework to observe the impact of fluxes and buffer on ca2+ dynamics in a hepatocyte cell. naik, pardasani [26] studied the relationship of intracellular ca2+ with ryr receptor, na+/ca2+ exchanger, buffers, and serca pump, in oocytes cell. jagtap and adlakha [10, 11] studied the distribution of intracellular calcium in the presence of buffers by using the finite elements method (fem ) in hepatocyte cells and concluded that calcium concentration decreases sharply for fast buffers, especially for endogenous and bapta buffer in compression with exogenous ethylene glycol tetraacetic acid (egta) buffer. also, they studied the ip3 and calcium association in hepatocyte cells involving serca pump, er leak, buffer, ip3 diffusion and advection-diffusion of calcium. they concluded ca2+ concentration increases due to the decrements in serca pump rate constant, buffer concentration, and increment in advection velocity. joshi and jha [14] studied the fractional reaction-diffusion model for the one-dimensional case involving ryr, v gcc and calbindin-d28k, and they obtained analytical explanations for the physiology of parkinson’s brain. martin vaeth et al. [46] studied the effect of ca2+ release-activated calcium (crac) channels and stromal interacting molecule (stim ) in t cell-mediated copyright © 2022 assa. adv syst sci appl (2022) 52 immunity calcium signaling. joshi and jha [15], using the fractional reaction-diffusion model, showed the complex interaction between calbindin-d28k and ca2+ in the presence of v gcc in neuron cells. they used fourier and laplace transformation in the fractional form to deduce the solution for the spatio-temporal fractional reaction-diffusion equation. bazhutina et al. [1] propose a mathematical model of human cardiomyocytes to study the mechanical and electrical response with cardiac fibroblast cells to its electronic interaction. no work is noticed for studying the influence of ca2+ regulation on ip3 and atp production in a fibroblast cell. the attempt has been made in this direction in the paper. 2. mathematical model the partial differential equation representing the one-dimensional ca2+ dynamics in the presence of excess buffer and ca2+ flux from the er of the fibroblast cell is expressed below: [18, 44, 49] ∂[ca2+] ∂t = dca ∂2[ca2+] ∂x2 − k+ j [b]∞([ca2+]− [ca2+]∞) + aer vcyt jer. (2.1) here, [b]∞, [ca2+]∞ are the steady-state buffer and calcium concentration respectively. the diffusion coefficient of calcium is denoted by dca. the position variable and time-variable are denoted by ’x’ and ’t,’ respectively. the association rate of a buffer is denoted by k+ j . the area of the endoplasmic reticulum and volume of the cytosol of the fibroblast cell is denoted by aer and vcyt respectively. the various influx and outflux terms of equation (2.1) can be written as [16] jer = (jip3r + jlker − jserca). (2.2) jserca = jmax serca [ca2+cyt] 2 [ca2+cyt] 2 +k2 serca , (2.3) jlker = klker([ca2+er]− [ca2+cyt]), (2.4) jip3r = kip3r([ca2+er]− [ca2+cyt]). (2.5) to handle the nonlinearity in the equation (2.3), the following assumption [16] has been made. let kserca >> [ca2+cyt], (2.6) where kserca denotes the calcium dissociation constant. then we have, [ca2+cyt] [ca2+cyt] +kserca << [ca2+cyt] kserca . (2.7) also, [ca2+cyt] 2 [ca2+cyt] 2 +k2 serca << [ca2+cyt] k2 serca << [ca2+cyt] kserca . (2.8) substituting the values of equations (2.2) and (2.8) in (2.1), we get ∂[ca2+] ∂t = dca ∂2[ca2+] ∂x2 − k+ j [b]∞([ca2+]− [ca2+]∞)+ aer vcyt ( kip3r([ca2+er]− [ca2+cyt]) +klker([ca2+er]− [ca2+cyt])− jmac serca kserca [ca2+cyt] ) . (2.9) copyright © 2022 assa. adv syst sci appl (2022) 53 calcium channels are presented on the er membrane in fibroblast cells. in fibroblast cell intracellular calcium concentration is initially assumed at it’s resting condition and it’s value is 0.1µm . thus the initial condition is [27, 45], [ca2+](x, 0) = 0.1µm. (2.10) the boundary conditions for the cytosolic calcium at x = 0 and x = 10 are given by the following equations [16, 35] lim x→0 ( −dca ∂[ca2+] ∂x ) = σca, (2.11) where σca represents calcium flux on the left boundary of the fibroblast cell. lim x→10 [ca2+] = [ca2+]∞ = 0.1µm. (2.12) the calcium-dependent atp release are defined as [39] atp ([ca2+]) = [( x0 x0 − 1 ) − 2 ( [ca2+] [ca2+]max )][( 1 x0 − 1 ) − ( [ca2+] [ca2+]max )2 ]−1 (2.13) and the calcium-dependent ip3 production is defined as [49] ip3([ca2+]) = vproduction ( [ca2+]2 [ca2+]2 +k2 pump ) . (2.14) here, x0 is the constants for the atp feedback and [ca2+]max considered as a maximum value of calcium concentration of the cell. therefore, the set of equations (2.9), (2.10), (2.11) and (2.12) form an initial boundary value problem. the equation (2.9) is written as 1 dca ∂[ca2+] ∂t = ( ∂2[ca2+] ∂x2 ) − a1[ca2+] + b1, (2.15) where a1 = k+ j [b]∞ dca + 1 dcafc ( kip3r + jmac serca kserca + vlker ) , (2.16) b1 = k+ j [b]∞ dca [ca2+]∞ + 1 dcafc ( kip3r[ca2+er] +klker[ca2+er] ) . (2.17) if we assume that the value of kserca is very much less than the cytosolic calcium concentration. then, the equation (2.15) is expressed as ∂2[ca2+] ∂x2 − a2[ca2+] + b2 − 1 dca ∂[ca2+] ∂t = 0, (2.18) where a2 = k+ j [b]∞ dca + 1 dcafc (kip3r + vlker) , (2.19) b2 = k+ j [b]∞ dca [ca2+]∞ + 1 dcafc ( kip3r[ca2+er] +klker[ca2+er]− kserca 1 +m2 ) (2.20) copyright © 2022 assa. adv syst sci appl (2022) 54 with fc= aer vcyt and 0 < m < 1. from the equation (2.15) and equation (2.18), we get ∂2[ca2+] ∂x2 − a[ca2+] + b− 1 dca ∂[ca2+] ∂t = 0, (2.21) where a = a1, b = b1 for kserca >> [ca2+cyt] and a = a2, b = b2 for kserca << [ca2+cyt]. the equation (2.21) is solved by using the finite elements method [30]. here ′v′ represent cytosolic calcium concentration. the equation (2.21) in discretized variational form is shown below: j (el) = 1 2 xj (el)∫ xi (el) [( ∂v(el) ∂x )2 + a(v(el))2 − 2bv(el) + 1 dc ( ∂v(el) ∂t )2 ] dx− µ(el) ( σca dca v(el) ∣∣∣ x=0 ) . (2.22) here, el = 1, 2....10. the constant term µ(el) = 1.0 for el = 1.0 and µ(el) = 0.0 for remaining elements. 3. solution the linear interpolation function is employed for field variable in each element as given below: v(el) = c1 + c2x, (3.23) where linear elements is denoted by ’el’ and c1, c2 are constants. the equation (3.23) is written as v(el) = p tc(el), (3.24) where p t = [1 x], c(el) = [ c1 c2 ] , v(el)(xi) = c1 + c2xi = vi, (3.25) v(el)(xj) = c1 + c2xj = vj. (3.26) using equations (3.24) (3.26), we get: v̄(el) = p (el)c(el), (3.27) where v̄(el) = [ vi vj ] , p (el) = [ 1 xi 1 xj ] , c(el) = (p (el))−1v̄(el). from equation (3.27), we have c(el) = r(el)v̄(el). (3.28) thus, r(el) = (p (el))−1 = 1 xj − xi [ xj −xi −1 1 ] . copyright © 2022 assa. adv syst sci appl (2022) 55 from equation (3.24) and (3.28) we get v(el) = p tr(el)v̄(el) (3.29) and v(el) ′ = p t x r (el)v̄(e), (3.30) where p t x = [0 1] . substituting the equations (3.29) and (3.30) in equation (2.22) we get j (el) = 1 2 ∫ xj (el) xi (el) ( (p t x r (el)v̄(el))2 + a(p tr(el)v̄(el))2 − 2bp tr(el)v̄(el) + 1 dc ∂(p tr(el)v̄(el))2 ∂t ) dx− µ(el) ( σca dca p tr(el)v̄(el)|x=0 ) . (3.31) ∂j (el) ∂v̄(el) = ∫ xj (el) xi (el) (r(el)tpxp t x r (el)v̄(el)) + a(r(el)tpp tr(el)v̄(el))− b(r(el)tp ) + 1 dc ∂(p tr(el)v̄(el)) ∂t dx− µ(el) ( σca dca p tr(el)v̄(el)|x=0 ) . (3.32) we can write equation (3.32) as ∂j (el) ∂v̄(el) = ∂j (el) 1 ∂v̄(el) + ∂j (el) 2 ∂v̄(el) − ∂j (el) 3 ∂v̄(el) + ∂j (el) 4 ∂v̄(el) − ∂j (el) 5 ∂v̄(el) , (3.33) where ∂j (el) 1 ∂v̄(el) = ∫ xj (el) xi (el) (r(el)tpxp t x r (el)v̄(el))dx, (3.34) ∂j (el) 2 ∂v̄(el) = a ∫ xj (el) xi (el) (r(el)tpp tr(el)v̄(el))dx, (3.35) ∂j (el) 3 ∂v̄(el) = b ∫ xj (el) xi (el) (r(el)tp )dx, (3.36) ∂j (el) 4 ∂v̄(el) = 1 dc ∫ xj (el) xi (el) ( ∂(p tr(el)v̄(el)) ∂t ) dx, ∂j (el) 5 ∂v̄(el) = µel ( σca dca r(el)tp |x=0 ) . now the integral j (el) is minimized with respect to each nodal points as shown below: ∂i(el) ∂v̄(el) = 0, (3.37) ∂j (el) ∂v̄(el) = ∂j (el) 1 ∂v̄(el) + ∂j (el) 2 ∂v̄(el) − ∂j (el) 3 ∂v̄(el) + ∂j (el) 4 ∂v̄(el) − ∂j (el) 5 ∂v̄(el) = 0, (3.38) copyright © 2022 assa. adv syst sci appl (2022) 56 ∂j ∂v̄ = σm̄ (el)∂j (el) dv̄(el) m̄ (el)t = 0, (3.39) where m (el) =  0 0 . . 1 0 0 1 . . 0 0  , u=  v1 v2 . . . v11 . assembling the integral equations (3.39) we get j = σj (el). (3.40) the equations (3.39) and (3.40) lead to the following system of linear differential equations [p ]11×11 [ ∂ū ∂t ] 11×1 + [q]11×11[ū]11×1 = [r]11×1. (3.41) here, [v̄] = [v1, v2, v3, ..., .v11], p is a system matrix, q is the system vector and r is a force vector. the crank nicholson method is use for solving the system (3.41). matlab has been used to simulate the problem and obtain numerical results. 4. results and discussion the biophysical values and numerical constants used in computing the numerical results are shown in table 4.2. fig. 4.1 displays the spatial and temporal ca2+, atp , and ip3 concentration profiles for various values of time and positions. from fig. 4.1(a), it is clear that spatial ca2+ concentration continuously decreases at each nodal point in the cytosol of the fibroblast cell with the increased distance from the source. this is due to the presence of source flux at x = 0.0µm and the diffusion of ca2+ towards x = 10µm. in fig. 4.1(b), it is notice that the ca2+ concentration is 0.1µm at t = 0msec and increases with increases in time. the rise in calcium concentration at the node at x = 0µm is more than other points and becomes lower as we move from x = 0µm to x = 10µm. it achieves a steady-state in time t = 150msec. fig. 4.1(c, d) displays the calcium-dependent atp production in space and time. it is evident from fig. 4.1(c) at the initial node, the concentration of atp is higher and decreases gradually towards the ending node. from fig. 4.1(d), it is clear that the range of atp varies from a lower concentration of 0.4µm to a higher concentration of 1µm , which depends on calcium concentration, in this case, when the initial concentration of calcium is 0.1µm then the atp is 0.4µm . therefore calcium and atp are directly proportional and converge to a steady-state in time t = 150msec. fig. 4.1(e, f) shows the calcium-dependent ip3 production for space and time. the concentration of ip3 is higher at the mouth of source influx. it decreases gradually on moving far from the source, and later it converges to the 0.005 µm. in fig. 4.1(f) it is notice that ip3 concentration is 0.005µm at t = 0msec and increases with an increase in time. the rise in ip3 concentration is more at x=0 µm and becomes lower as we move from x = 0µm to x = 10µm. it achieves a steady-state in time at 150msec. therefore, it is clear that the ip3 production and ca2+ concentration show similar behavior. fig. 4.2 shows the profile of spatial and temporal ca2+, atp , and ip3 concentration for the different values of source amplitude along with the concentration of buffer 5µm and the value of diffusion coefficient 250µm2/s. in fig. 4.2, it is notice that the concentration of ca2+, atp , and ip3 is more for higher values of source amplitude. in fig. 4.2(a, c, e) copyright © 2022 assa. adv syst sci appl (2022) 57 distance (µm) 0 2 4 6 8 10 [c a 2 + ] c o n c e n tr a ti o n (µ m ) 0 0.2 0.4 0.6 (a). [ca2+] profile w.r.t space t =00ms t =10.0ms t =20.0ms t =30.0ms t =50.0ms t =100ms t =500ms time (msec) 0 100 200 300 400 500 [c a 2 + ] c o n c e n tr a ti o n (µ m ) 0 0.2 0.4 0.6 (b). [ca2+] profile w.r.t time x=1.0µm x=2.0µm x=3.0µm x=4.0µm x=5.0µm x=6.0µm x=11µm distance (µm) 0 2 4 6 8 10 a t p c o n c e n tr a ti o n (µ m ) 0.2 0.4 0.6 0.8 1 (c). atp profile w.r.t space t =00ms t =10.0ms t =20.0ms t =30.0ms t =50.0ms t =100ms t =500ms time (msec) 0 100 200 300 400 500 a t p c o n c e n tr a ti o n (µ m ) 0.2 0.4 0.6 0.8 1 (d). atp profile w.r.t time x=1.0µm x=2.0µm x=3.0µm x=4.0µm x=5.0µm x=6.0µm x=11µm distance (µm) 0 2 4 6 8 10 ip 3 c o n c e n tr a ti o n (µ m ) 0 0.02 0.04 0.06 (e). ip 3 profile w.r.t space t =00ms t =10.0ms t =20.0ms t =30.0ms t =50.0ms t =100ms t =500ms time (msec) 0 100 200 300 400 500 ip 3 c o n c e n tr a ti o n (µ m ) 0 0.02 0.04 0.06 (f). ip 3 profile w.r.t time x=1.0µm x=2.0µm x=3.0µm x=4.0µm x=5.0µm x=6.0µm x=11µm fig. 4.1. calcium, atp and ip3 concentration profile in a fibroblast cell at various time and positions for [b] = 5µm , dc = 250µm2/s and σ = 20pa. the calcium, atp , and ip3 concentration is higher at the source x = 0µm and decreases along the spatial direction and converges to 0.1µm , 0.25µm and 0.005µm at x = 10.0µm respectively. from fig. 4.2(b, d, f) it is clear that the calcium, atp , and ip3 concentration achieves a steady-state in time t = 180msec. therefore calcium, atp , ip3 are dependent on the source amplitude. fig. 4.3 displays the spatial and temporal calcium, atp , and ip3 concentration for the various values of association rate of the buffer in the fibroblast cell. from fig. 4.3(a, c, e), it is noticed that as buffer binding rate increases calcium, atp and ip3 concentration decreases. from fig. 4.3(b, d, f) it is clear that the calcium, atp , and ip3 dynamics achieves a steadystate in time t = 175msec. therefore calcium, atp , ip3 are reliant on the association rate of the buffer in the fibroblast cell. copyright © 2022 assa. adv syst sci appl (2022) 58 distance (µm) 0 2 4 6 8 10[c a 2 + ] c o n c e n tr a ti o n ( µ m ) 0 0.2 0.4 0.6 0.8 (a). [ca2+] profile at t = 0.5 sec sigma=10 pa sigma=20 pa sigma=30 pa time (msec) 0 100 200 300 400 500[c a 2 + ] c o n c e n tr a ti o n ( µ m ) 0 0.2 0.4 0.6 0.8 (b). [ca2+] profile at x = 0 µm sigma=10 pa sigma=20 pa sigma=30 pa distance (µm) 0 2 4 6 8 10 a t p c o n c e n tr a ti o n ( µ m ) 0.2 0.4 0.6 0.8 1 (c). atp production at t = 0.5 sec sigma=10 pa sigma=20 pa sigma=30 pa time (msec) 0 100 200 300 400 500 a t p c o n c e n tr a ti o n ( µ m ) 0.2 0.4 0.6 0.8 1 (d). atp production at x = 0 µm sigma=10 pa sigma=20 pa sigma=30 pa distance (µm) 0 2 4 6 8 10 ip 3 c o n c e n tr a ti o n ( µ m ) 0 0.02 0.04 0.06 (e). ip 3 production at t = 0.5 sec sigma=10 pa sigma=20 pa sigma=30 pa time (msec) 0 100 200 300 400 500 ip 3 c o n c e n tr a ti o n ( µ m ) 0 0.02 0.04 0.06 (f). ip 3 production at x = 0 µm sigma=10 pa sigma=20 pa sigma=30 pa fig. 4.2. ca2+-dependent atp and ip3 concentration profile in fibroblast cell for different source influx for [b] = 5µm and dc = 250µm2/s. fig. 4.4 shows the differences in spatial and temporal calcium, atp and ip3 concentration for the different values of diffusion coefficient i.e 220µm2/s, 270µm2/s, 320µm2/s. from fig. 4.4(a, c, e) it is notice that for the higher value of diffusion coefficient calcium, atp , and ip3 ion move fast from apical to the basal region of the cell. therefore, the concentration of calcium, atp , and ip3 decreases with an increase in the value of diffusion coefficient near the source and calcium concentration uniformly converges to 0.1µm . atp concentration and ip3 concentration converges to 0.22µm and 0.005µm respectively. the fig. 4.4(a, b) it is observed that calcium concentration is 0.58µm , 0.51µm , and 0.47µm for the diffusion coefficient 220µm2/s, 270µm2/s, 320µm2/s. therefore diffusion and calcium concentration is inversely proportional to each other. in fig. 4.4(c, d, e, f) concentration of atp and ip3 are inversely proportional to the value of diffusion coefficient and achieve a steady-state in time t = 125msec. copyright © 2022 assa. adv syst sci appl (2022) 59 distance (µm) 0 2 4 6 8 10[c a 2 + ] c o n c e n tr a ti o n ( µ m ) 0 0.2 0.4 0.6 0.8 (a). [ca2+] profile at t = 0.5 sec k j =0.32(µmsec) -1 k j =1.5(µmsec) -1 k j =13(µmsec) -1 time (msec) 0 100 200 300 400 500[c a 2 + ] c o n c e n tr a ti o n ( µ m ) 0 0.2 0.4 0.6 0.8 (b). [ca2+] profile at x = 0 µm k j =0.32(µmsec) -1 k j =1.5(µmsec) -1 k j =13(µmsec) -1 distance (µm) 0 2 4 6 8 10 a t p c o n c e n tr a ti o n ( µ m ) 0.2 0.4 0.6 0.8 1 (c). atp production at t = 0.5 sec k j =0.32(µmsec) -1 k j =1.5(µmsec) -1 k j =13(µmsec) -1 time (msec) 0 100 200 300 400 500 a t p c o n c e n tr a ti o n ( µ m ) 0.2 0.4 0.6 0.8 1 (d). atp production at x = 0 µm k j =0.32(µmsec) -1 k j =1.5(µmsec) -1 k j =13(µmsec) -1 distance (µm) 0 2 4 6 8 10 ip 3 c o n c e n tr a ti o n ( µ m ) 0 0.02 0.04 0.06 (e). ip 3 production at t = 0.5 sec k j =0.32(µmsec) -1 k j =1.5(µmsec) -1 k j =13(µmsec) -1 time (msec) 0 100 200 300 400 500 ip 3 c o n c e n tr a ti o n ( µ m ) 0 0.02 0.04 0.06 (f). ip 3 production at x = 0 µm k j =0.32(µmsec) -1 k j =1.5(µmsec) -1 k j =13(µmsec) -1 fig. 4.3. ca2+-dependent atp and ip3 concentration profile in fibroblast cell for different calcium binding association rate for [b] = 5µm , dc = 250µm2/s and σ = 20pa. fig. 4.5 displays the spatial and temporal distribution of ca2+, atp , and ip3 concentration for the different values of ethylene glycol tetraacetic acid (egta) buffers. from fig. 4.5, it is clear that the concentration of free ca2+, atp , and ip3 decreases at each nodal point of a fibroblast cell with the increase in the concentration of egta buffers 5µm , 10µm and 15µm . this egta binds with free ca2+ makes a calcium-bound buffer in the cytosol and reduces the amount of free ca2+. thus, the increase in the concentration of buffer causes a decrease in the free ca2+ concentration. the rate of decrease in ca2+ concentration increases with the rise in egta buffer concentration. the ca2+ concentration uniformly decreases towards the basal part of the fibroblast cell and attains a background equilibrium concentration of 0.1µm . it is clear from the results of fig. 4.1 and fig. 4.5 the behavior of calcium profiles are followed by the atp and ip3 profiles. ca2+, atp , and ip3 concentrations are inversely proportional to buffer binding affinity. copyright © 2022 assa. adv syst sci appl (2022) 60 distance (µm) 0 2 4 6 8 10[c a 2 + ] c o n c e n tr a ti o n ( µ m ) 0 0.2 0.4 0.6 (a). [ca2+] profile at t = 0.5 sec d ca =220µm 2 /s d ca =270µm 2 /s d ca =320µm 2 /s time (msec) 0 100 200 300 400 500[c a 2 + ] c o n c e n tr a ti o n ( µ m ) 0 0.2 0.4 0.6 (b). [ca2+] profile at x = 0 µm d ca =220µm 2 /s d ca =270µm 2 /s d ca =320µm 2 /s distance (µm) 0 2 4 6 8 10 a t p c o n c e n tr a ti o n ( µ m ) 0.2 0.4 0.6 0.8 1 (c). atp production at t = 0.5 sec d ca =220µm 2 /s d ca =270µm 2 /s d ca =320µm 2 /s time (msec) 0 100 200 300 400 500 a t p c o n c e n tr a ti o n ( µ m ) 0.2 0.4 0.6 0.8 1 (d). atp production at x = 0 µm d ca =220µm 2 /s d ca =270µm 2 /s d ca =320µm 2 /s distance (µm) 0 2 4 6 8 10 ip 3 c o n c e n tr a ti o n ( µ m ) 0 0.02 0.04 0.06 (e). ip 3 production at t = 0.5 sec d ca =220µm 2 /s d ca =270µm 2 /s d ca =320µm 2 /s time (msec) 0 100 200 300 400 500 ip 3 c o n c e n tr a ti o n ( µ m ) 0 0.02 0.04 0.06 (f). ip 3 production at x = 0 µm d ca =220µm 2 /s d ca =270µm 2 /s d ca =320µm 2 /s fig. 4.4. calcium, atp and ip3 concentration profile in a fibroblast cell for the different values of diffusion coefficient at [b] = 5µm and σ = 20pa. the ratio of calcium and atp in fig. 4.6(a) is highest at the calcium source and lowest along the spatial direction and became constant at the other end. they were comparing the fig. 4.6(a, b) with the calcium and atp profiles of fig. 4.1, it shows that the sharpness of the curves in calcium and atp profile in fig. 4.1 has got reduced in this fig. 4.6. this shows that the behavior of calcium and atp is very close. in fig. 4.6(b), the ratio of calcium and atp is lowest at time t = 0msec and increases with time and achieves a steady-state in 110msec. the steady-state time in fig. 4.6(b) is less in fig. 4.1(b, d) for the calcium and atp . this shows that the temporal profile of calcium and atp behavior is very close to each other. in fig. 4.6(c), the spatial profile of the ratio of calcium and ip3 is different from that in fig. 4.1 for calcium and ip3 profiles. for t = 25msec the special profile of calcium and ip3 ratio is increasing concave whereas for t = 500msec this ratio profile is increasing convex from x = 0µm to x = 10µm. copyright © 2022 assa. adv syst sci appl (2022) 61 distance (µm) 0 2 4 6 8 10[c a 2 + ] c o n c e n tr a ti o n ( µ m ) 0 0.2 0.4 0.6 (a). [ca2+] profile at t = 0.5 sec b=05µm b=10µm b=15µm time (msec) 0 100 200 300 400 500[c a 2 + ] c o n c e n tr a ti o n ( µ m ) 0 0.2 0.4 0.6 (b). [ca2+] profile at x = 0 µm b=05µm b=10µm b=15µm distance (µm) 0 2 4 6 8 10 a t p c o n c e n tr a ti o n ( µ m ) 0.2 0.4 0.6 0.8 1 (c). atp production at t = 0.5 sec b=05µm b=10µm b=15µm time (msec) 0 100 200 300 400 500 a t p c o n c e n tr a ti o n ( µ m ) 0.2 0.4 0.6 0.8 1 (d). atp production at x = 0 µm b=05µm b=10µm b=15µm distance (µm) 0 2 4 6 8 10 ip 3 c o n c e n tr a ti o n ( µ m ) 0 0.02 0.04 0.06 (e). ip 3 production at t = 0.5 sec b=05µm b=10µm b=15µm time (msec) 0 100 200 300 400 500 ip 3 c o n c e n tr a ti o n ( µ m ) 0 0.02 0.04 0.06 (f). ip 3 production at x = 0 µm b=05µm b=10µm b=15µm fig. 4.5. calcium, atp and ip3 concentration profile in a fibroblast cell for the different values of buffer for dc = 250µm2/s and σ = 20pa. the calcium profile in fig. 4.1 is decreasing convex. the ip3 profiles in fig. 4.1 are also decreasing convex for t = 25msec and 50msec. but for t = 500msec the ip3 profile was decreasing concave. the drastic change in the ratio of calcium and ip3 profiles indicates a significant difference in the behavior of calcium and ip3 profiles. in fig. 4.6(d), the temporal profile ratio of calcium and ip3 is almost similar for x = 3µm and x = 6µm in behavior. still, there is a change in behavior at x = 1µm compared to calcium and ip3 profiles in fig. 4.1 this shows that the behavior of curves of calcium and ip3 is very close at all spatial points for x > 1µm in fig. 4.1 but is different near the source. these fluctuations are due to a mismatch in calcium and ip3 profile near the source. the steady-state time of the ratio of calcium and ip3 profiles is less than the independent calcium and ip3 profiles. fig. 4.7 shows the influence of calcium on the spatial and temporal flux of ip3 degradation profile in a fibroblast cell. the maximum concentration of degradation is observed at copyright © 2022 assa. adv syst sci appl (2022) 62 distance (µm) 0 2 4 6 8 10 r a ti o (c a 2 + /a t p ) 0.2 0.3 0.4 0.5 0.6 (a). ratio(ca2+/atp) profile w.r.t space t =25 msec t =50 msec t =500 msec time (msec) 0 100 200 300 400 500 r a ti o (c a 2 + /a t p ) 0.2 0.3 0.4 0.5 0.6 (b). ratio(ca2+/atp) profile w.r.t time x=1 µm x=3 µm x=6 µm distance (µm) 0 2 4 6 8 10 r a ti o (c a 2 + /i p 3 ) 10 15 20 25 (c). ratio(ca2+/ip 3 ) profile w.r.t space t =25 msec t =50 msec t =500 msec time (msec) 0 100 200 300 400 500 r a ti o (c a 2 + /i p 3 ) 10 15 20 25 (d). ratio(ca2+/ip 3 ) profile w.r.t time x=1 µm x=3 µm x=6 µm distance (µm) 0 2 4 6 8 10 r a ti o (a t p /i p 3 ) 0.01 0.02 0.03 0.04 0.05 (e). ratio(atp/ip 3 ) profile w.r.t space t =25 msec t =50 msec t =500 msec time (msec) 0 100 200 300 400 500 r a ti o (a t p /i p 3 ) 0.01 0.02 0.03 0.04 0.05 (f). ratio(atp/ip 3 ) profile w.r.t time x=1 µm x=3 µm x=6 µm fig. 4.6. ratio of ca2+-dependent atp and ip3 with the calcium for different points of time and position for [b] = 5µm , dc = 250µm2/s and σ = 20pa. x = 0.0µm and it reduces from x = 0.0µm to x = 10.0µm. in fig. 4.7, it is clear that the high value of calcium leads to the higher degradation of ip3. further, it shows that ip3 concentration becomes low due to higher degradation of ip3. fig. 4.8 shows the ratio of calcium, atp , and ip3 concentrations for various values of source amplitude. it is noticed in fig. 4.8(a) that the peak value of calcium ratio for sigma 10pa and 20pa is 1.68 at x = 0µm. the peak value of atp and ip3 ratios are notice at x = 1µm and x = 3µm respectively. at the initial node, calcium concentration is 68 percent higher than the atp and lower than the ip3 percentage ratio, which is 81 percent. the minimum ratio is attained at the last node. in fig. 4.8(b) the ratio profile of calcium, atp and ip3 for sigma 10pa and 20pa is minimum at time t = 0msec and increases with time from t = 0msec to t = 15msec for atp and ip3. the calcium ratio gradually increases with copyright © 2022 assa. adv syst sci appl (2022) 63 distance (µm) 0 1 2 3 4 5 6 7 8 9 10 ip 3 c o n c e n tr a ti o n (µ m ) 0.09 0.095 0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135 0.14 (a). ip 3 degradation profile with respect to (w.r.t) space t =00 msec t =10 msec t =20 msec t =30 msec t =50 msec t =100 msec t =500 msec time (msec) 0 50 100 150 200 250 300 350 400 450 500 ip 3 c o n c e n tr a ti o n (µ m ) 0.09 0.095 0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135 0.14 (b). ip 3 degradation profile with respect to (w.r.t) time x=1 µm x=2 µm x=3 µm x=4 µm x=5 µm x=6 µm x=11 µm fig. 4.7. [ca2+]-dependent ip3 degradation for different points of time and position for [b] = 5µm , dc = 250µm2/s and σ = 20pa. distance (µm) 0 1 2 3 4 5 6 7 8 9 10 r a ti o c o n c e n tr a ti o n 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 (a). [ca2+], ip 3 and atp profile at t = 0.5 sec [ca 2+ ] σ=20pa/σ=10pa [atp] σ=20pa/σ=10pa [ip 3 ] σ=20pa/σ=10pa time (msec) 0 50 100 150 200 250 300 350 400 450 500 r a ti o c o n c e n tr a ti o n 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 (b). [ca2+], ip 3 and atp profile at x = 0 µm [ca 2+ ] σ=20pa/σ=10pa [atp] σ=20pa/σ=10pa [ip 3 ] σ=20pa/σ=10pa fig. 4.8. ratio of calcium, atp and ip3 for different source influx for [b] = 5µm and dc = 250µm2/s. time and achieves a steady-state in 90msec, atp , and ip3 ratios for sigma 10pa and sigma 20pa also achieve steady state in 90msec. fig. 4.9 shows the ratio graph of calcium, atp , and ip3 concentrations for different values of the buffer. it is observed in fig. 4.9(a) that the peak value of calcium, atp and ip3 ratios are at 4µm and 5µm respectively. initial node calcium concentration is 26%, which is higher for buffer 5µm and 10µm than the atp and ip3 ratio percentage. the minimum ratio is attained at the last node. in fig. 4.9(b), the ratio profile of calcium, atp , and ip3 is copyright © 2022 assa. adv syst sci appl (2022) 64 distance (µm) 0 1 2 3 4 5 6 7 8 9 10 r a ti o c o n c e n tr a ti o n 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 (a). [ca2+], ip 3 and atp profile at t = 0.5 sec [ca 2+ ] [b]=5µm/[b]=10µm [atp] [b]=5µm/[b]=10µm [ip 3 ] [b]=5µm/[b]=10µm time (msec) 0 50 100 150 200 250 300 350 400 450 500 r a ti o c o n c e n tr a ti o n 1 1.05 1.1 1.15 1.2 1.25 1.3 (b). [ca2+], ip 3 and atp profile at t = 0 µm [ca 2+ ] [b]=5µm/[b]=10µm [atp] [b]=5µm/[b]=10µm [ip 3 ] [b]=5µm/[b]=10µm fig. 4.9. ratio of calcium, atp and ip3 for different buffers for dc = 250µm2/s and σ = 20pa. table 4.1. steady state time period for the ca2+, ip3 and atp fig. no. temporal profile time required to achieve steady state 1-b calcium w.r.t time 150msec 1-d atp w.r.t time 150msec 1-f ip3 w.r.t time 150msec 2-b calcium at x = 0 180msec 2-d atp at x = 0 180msec 2-f ip3 at x = 0 180msec 3-b calcium at x = 0 175msec 3-d atp at x = 0 175msec 3-f ip3 at x = 0 175msec 4-b calcium 125msec 4-d atp 115msec 4-f ip3 100msec 5-b calcium 90msec 5-d atp 95msec 5-f ip3 100msec 6-b ratio calcium/atp 110msec 6-d ratio calcium/ ip3 100msec 6-f ratio atp / ip3 105msec 7 ip3 degradation 150msec 8 ratio of calcium, atp , ip3 90msec 9 ratio of calcium, atp , ip3 100msec lowest at time t = 0msec and increases with time and achieves a steady-state in 100msec for buffer 5µm and 10µm . in figures 4.1 to 4.9 the profiles of calcium, ip3 and atp are observed to be nonlinear which implies that the nonlinear phenomena is occurring due to changes in conditions like alteration in values of buffer, source influx, etc. to balance the effects of these alterations in parameters for regulating the concentrations of calcium, ip3 and atp at appropriate levels to maintain the structure and function of the cell. copyright © 2022 assa. adv syst sci appl (2022) 65 table 4.2. list of physiological parameters [44] symbol parameters values [ca2+]∞ background calcium concentration 0.1 µm jmax serca maximal pump rate of serca 8×10−5 µmol (s× dm2) kserca serca calcium dissociation constant 2.0µm klker leak constant 2.0s−1 [ca2+]er er calcium concentration 10.89 µm aer er area 0.3 × 10−7dm2 ver volume of er 0.1 × 10−12dm3 vcyt volume of cytosol 1 × 10−12dm3 k+ j association rate 1.5(mms)−1 [b]∞ total buffer concentration 20 µm σ source amplitude 15pa dca diffusion coefficient 250µm2/s table 4.3. error analysis of [ca2+] concentration profile at x = 0µm time n=10 elements n=20 elements error error % t = 100msec 0.7535093547 0.889250659 0.13574 0.15264 % t = 200msec 0.7638349682 0.994152451 0.23031 0.23165% t = 300msec 0.7640457957 1.015927426 0.25188 0.24794 % t = 500msec 0.7640501883 1.021388902 0.25739 0.25195 % t = 1000msec 0.7640501901 1.021634989 0.25758 0.25212% table 4.4. error analysis of atp production profile at x = 1µm time n=10 elements n=20 elements error error % t = 100msec 0.9892130695 0.918883410 0.07032 0.07653 % t = 200msec 0.9893086343 0.870996603 0.11831 0.13583% t = 300msec 0.9893087359 0.860896109 0.12841 0.14916 % t = 500msec 0.9893087360 0.858363403 0.13094 0.15255 % t = 1000msec 0.9893087361 0.858249304 0.13106 0.15270% table 4.5. error analysis of ip3 production profile at x = 2µm time n=10 elements n=20 elements error error % t = 100msec 0.0371489218 0.058113011 0.020964 0.36075 % t = 200msec 0.0371770587 0.0612872185 0.024112 0.39341% t = 300msec 0.0371770887 0.0618411302 0.024664 0.39883 % t = 500msec 0.0371770888 0.0619751619 0.024798 0.40013 % t = 1000msec 0.0371770888 0.0619811563 0.024804 0.40019% for the time period t = 100, 200, 300, 500 and 1000 msec, the accuracy of the model for calcium are 99.84736 %, 99.76835 %, 99.975206 %, 99.74805 % and 99.74788 %, respectively, and for atp production, the accuracy is 99.92347 %, 99.86417 %, 99.85084 %, 99.84745 % and 99.8473 %, respectively. for ip3 production, the accuracy is 99.63925 %, 99.60659 %, 99.60117 %, 99.59987 %, and 99.59981 %, respectively. thus the minimum accuracy and maximum error % for this model are 99.75 % and 0.40 %, respectively. the grid sensitivity is negligible; this implies that the solution is independent of the grid. to perform stability analysis, calculated the spectral radius, and the system is said to be stable if the spectral radius is less than or equal to unity [50]. in this case, the spectral radius was 0.9619, which is less than unity. thus the finite element solution in the present condition is stable. copyright © 2022 assa. adv syst sci appl (2022) 66 5. conclusion a one-dimensional model is proposed to investigate the impact of buffer, channel flux, and diffusion coefficient on spatial and temporal calcium profile in fibroblast cells and study their impact on atp and ip3 production in the cell. it is concluded that spatial and temporal calcium distribution variation is proportional to the source influx and inversely proportional to diffusion coefficient and buffers. the variation in spatial and temporal profiles of atp and ip3 production is also directly proportional to source influx and inversely proportional to buffer and diffusion coefficient. the ip3 degradation is also proportional to calcium distribution in the cell. the finite element simulation is quite effective in generating these insights, which may help frame measures for diagnosis and treatment. the outcomes of this study are in agreement with biological facts. the non-availability of the experimental results for the conditions of the present problem is constant, preventing validation impossible. the dysregulation effect of source influx, buffers, and diffusion coefficients calcium profiles are transferred to the atp , ip3 production, and ip3 degradation profiles. thus the changes in source influx, buffers, and diffusion coefficient can cause an increase or decrease in atp and ip3 production and ip3 degradation. the ip3 is involved in the function of the serca pump. therefore causes alteration in calcium profile due to the serca pump leading to dysregulation in calcium dynamics. the variation in atp can cause changes in differentiation, contraction, relaxation, migration, wound healing, etc. here the dysregulation of atp can lead to severe disorders such as inflammation, heart disease, and cancer. the results from our model here give us an insight into the parameters which may help regulate serious disorders such as cancer, inflammation, and heart disease. copyright © 2022 assa. adv syst sci appl (2022) 67 references 1. bazhutina, a., balakina-vikulova, n. a., kursanov, a., solovyova, o., panfilov, a. et. al. 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(2003). a finite element approach for solution of burgers’ equation. applied mathematics and computation, 139(2-3), 417–428. copyright © 2022 assa. adv syst sci appl (2022) introduction mathematical model solution results and discussion conclusion adv syst sci appl 2024; 03:31–42 published online at https://ijassa.ipu.ru. the problem of dynamic regulation of train formation on a marshalling yard elena musatova1*, alexander lazarev1, alexander osminin2, alexey baushev3 1v. a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia 2volga state transport university, samara, russia 3emperor alexander i st. petersburg state transport university, saint petersburg, russia abstract: the problem of dynamic freight trains formation at a railway station is considered. this problem is part of a two-step strategy to create a dynamic train formation plan. it is assumed that technical routes of all cars and the predicted arrival time of new groups of cars at the station are known. the problem is to determine the set of cars and the departure time of each train formed at this station, taking into account the availability of locomotives and restrictions on the length of trains. the goal is to minimise the total waiting time for cars at the station. mathematical models for different versions of the problem are presented as an integer and a mixed integer linear programming problems. results of computational experiments allow to estimate the influence of the amount of information about arriving trains to the station on the value of the objective function. keywords: train formation, integer linear programming, daily planning. 1. introduction the problem of developing a train formation plan (tfp) plays a key role in improving the quality of freight rail transportation. in different countries there are various approaches to the organization of freight rail transport. there are two main widely used operational strategies for freight rail transportation [1]. in the first case, the priority is the scheduled plan. this means that the departure of trains is carried out exactly according to a predetermined schedule, regardless of the actual number of cars that need to be transported on a certain day. such plans require an accurate forecast of the number of cars. in the usa this type of plan is used in rail transportation. in the second case, priority is given to the tonnage plan, i.e. the departure of a train is carried out immediately after the station has the necessary number of cars to form a train in a given direction. russia, china and many other countries use such freight transportation strategy. in accordance with these two approaches, the formulation of optimization problems corresponding to them also differs. in general, the problem of calculating the train formation plan refers to the problems of network routing [2]. from the point of view of the mathematical model structure, two areas of research can be distinguished: the construction of linear and nonlinear models. at the same time, most modern research is devoted to linear models of the tfp (see, for example, [1, 3, 4]), that is, problems of integer linear programming and mixed integer linear programming. examples of nonlinear models can be, for example, the problems considered in [5, 6]. ∗corresponding author: nekolyap@mail.ru 32 e. musatova, a. lazarev, a. osminin, a. baushev as a rule, the problem of train formation is a large-scale problem of designing a flow network and routing with millions, and even billions of decision variables. the size and mathematical complexity of this problem do not allow us to solve it accurately with the help of any commercial software [7]. in this regard, most of the studies in the field of train formation and the distribution of car traffic are devoted to the development of fast heuristic algorithms (see, for example, [3,4]). in addition, the decomposition of the general problem into a number of successively solved problems is used. the following subproblems of railway planning are distinguished [8]: 1) the problem of combining cars into blocks [9]. each block is characterized by a departure station and a destination station, while the cars included in it may have other stations of formation and destination; 2) the problem of constructing routes and train schedules [2]; 3) the problem of assigning blocks to trains [10]. however, the direct use of foreign experience in solving the tfp problem for russian railways is impossible due to differences in the organization of the freight transportation process. the existing methodology for creating the tfp was developed to solve the problems of a planned economy and does not meet the conditions of the modern market economy. as a result, the actual planning is reduced to a daily adjustment of the tfp, based on the experience of process engineers and decision makers. the fast operation of planning algorithms is extremely important in this problem due to the fact that the moments and volumes of incoming trains to a marshalling yard differ from year to year and from day to day. the formulation of the tfp problem in russian railways is determined by a specific procedure for constructing the formation plan. so there are two plans for the trains formation: a basic plan and an adaptive one. the basic tfp is built a year in advance and determines for each station destinations of trains formed at it. the problem of the adaptive formation plan is to determine technical routes for cars, i.e. sequences of stations through which the cars must go to destination stations, as well as determining which car is assigned to which train. our work deals with the adaptive tfp problem. building an adaptive formation plan is a high-dimensional np-hard problem. moreover, various local changes and inconsistencies that arise in practice do not allow solving the problem for the entire railway network at the same time. an approach is proposed that focuses on the possibility of implementing the developed algorithms for constructing an approximate solution of the tfp problem in real railway practice. the adaptive tfp problem is divided into two subproblems: the problem of technical routing and the problem of dynamic regulation. this article is devoted to the problem of dynamic regulation, where it is necessary for a single station to determine a content (set of cars) and a time of departure of each train formed at this station. our goal is to build mathematical models of dynamic formation of trains at the station and trace the influence of the length of the planning horizon on the solution to the problem. the article is organised as follows. section 2 includes a description of the problem, notation and basic terms. in section 3 a mathematical statement of the problem is presented. in section 4 we present results of computational experiment using solver cplex for problems of small dimensions. results of solving high-dimensional problems are given in section 5. in section 5 we consider a special case of the problem that is easier to solve. section 7 contains conclusions and suggestions. 2. problem description a train formation station (or marshalling yard) is considered. let k = {1, . . . , n} be a set of destinations of trains departing from the station. this means that it is possible to form n types of trains with different destination stations. as a planning horizon, we choose the time copyright © 2024 assa. adv syst sci appl (2024) problem of dynamic regulation of train formation 33 interval [t0, tmax], the length of which depends on the available information about arrival of freight trains at the station. in practice, planning can be carried out a day ahead. t = {t1, . . . , tm} specifies moments of trains arrival to the station in the considered planning horizon. at the initial moment of time, n storages corresponding to n destinations already contain groups of cars, called blocks. in addition, there are r0 locomotives at the station. these are the locomotives that delivered the blocks to the station before the initial point in time, but were not yet used for new train departures. at time moments ti, i ∈ i = {1, . . . ,m}, trains consisting of blocks arrive at the station. arrived blocks of cars, for which this station is not the final one, are distributed among the storages in accordance with their destinations, thereby increasing the number of cars in the corresponding potential trains. each arrived locomotive can be used to form a new train. let g be a set of all blocks considered on the given planning horizon, including the blocks located in the storages. each block g ∈ g has the following characteristics: • τg is the time moment of arrival at the station (if a block g arrived at the station before the initial moment, then we set τg = t0 = 0); • kg is the destination of g; • lg is the number of cars in the group. each block is an indivisible group of cars, i.e. all cars of the block must leave the station in one train. in section 5 we will additionally consider another version of the statement where the division of blocks is allowed. let gk be the set of blocks of destination k and lmin, lmax be respectively the minimum and the maximum numbers of cars included in one train formed at the given station, i.e. the departure of a train can only take place if the number of cars in it satisfies the upper and lower limits. at each time moment from set t it is necessary to make a decision whether to sent trains to some destinations. also we have to determine which blocks will be included in these trains. we assume that the time required to form a train is a constant equal to β. the goal function is the total number of car-hours spent at the station for the period under consideration. 3. mathematical statement let’s number all the locomotives that are at the station at the zero moment of time plus all the locomotives that will arrive in the considered time period, from 1 to r0 +m in the order of their arrival at the station. let j = {1, . . . , r0 +m} be the set of all locomotives. denote by φj the moment of arrival of the locomotive j at the station. if locomotive j arrived at the station before the beginning of the time period under consideration, then we set φj = 0. let’s introduce the following decision variables: • binary variable yjik, j ∈ j , i ∈ i , k ∈ k, equals to 1 if locomotive j leaves the station as part of a new train at time ti + β and has a destination k, and 0 otherwise. • binary variable xgij , g ∈ g, i ∈ i , j ∈ j , is equal to 1 if block g leaves the station at time ti + β with locomotive j, and 0 otherwise. • binary variable zg, g ∈ g, takes the value 1 if block g does not leave the station in the given planning horizon, and 0 otherwise. we list the main constraints. each block leaves the station once, or does not leave it at all in the current planning period: ∑ i∈i ∑ j∈j xgij + zg = 1 ∀g ∈ g. (3.1) each locomotive can be used at most once:∑ i∈i ∑ k∈k yjik ≤ 1 ∀j ∈ j. (3.2) copyright © 2024 assa. adv syst sci appl (2024) 34 e. musatova, a. lazarev, a. osminin, a. baushev each block leaves the station no earlier than it arrives there: xgij = 0 ∀g ∈ g, ∀j ∈ j, ∀i ∈ i : τg > ti (3.3) and no earlier than its locomotive arrives there: xgij = 0 ∀g ∈ g, ∀j ∈ j, ∀i ∈ i : φj > ti. (3.4) each locomotive can depart as part of a new train not earlier than its arrival at the station: yjik = 0 ∀j ∈ j, ∀i ∈ i : φj > ti, ∀k ∈ k. (3.5) there is a restriction on the number of cars in the trains being formed: lminyjik ≤ ∑ g∈g:kg=k lgxgij ≤ lmaxyjik ∀k ∈ k, ∀i ∈ i, ∀j ∈ j. (3.6) finally, we add the condition of binary variables: xgij, yjik, zg ∈ {0, 1} ∀g ∈ g, ∀i ∈ i, ∀j ∈ j, ∀k ∈ k. (3.7) the problem of minimizing the total car-hours at the station in the given planning horizon is considered. the total number of car-hours for blocks that will be sent in the planning period is determined as ∑ g∈g lg ∑ i∈i ∑ j∈j (ti + β)xgij − ∑ g∈g τglg ∑ i∈i ∑ j∈j xgij. for cars remaining at the station by the end of the period under consideration, the total waiting time is equal to tmax ∑ g∈g lgzg − ∑ g∈g τglgzg. the sum of these quantities can be written as∑ g∈g lg ∑ i∈i ∑ j∈j (ti + β)xgij + tmax ∑ g∈g lgzg − ∑ g∈g τglg( ∑ i∈i ∑ j∈j xgij + zg). due to the constraint (3.1), we get that the objective function of the problem can be written as ∑ g∈g lg ∑ i∈i ∑ j∈j (ti + β)xgij + tmax ∑ g∈g lgzg − ∑ g∈g τglg → min . (3.8) we denote problem (3.1)-(3.8) by (p1). it can be seen that the number of variables is equal to |j ||i||k|+ |j ||i||g|+ |g|, where | · | is the number of elements in the corresponding set. the number of constraints is a value of the same order. theorem 3.1: problem (p1) is strongly np-hard. proof we consider the multiple subset sum problem with identical capacities (mssp-i) which is strongly np-hard [11]. in this problem we are given a set of values wj , j ∈ {1, . . . , g}, and we are looking for m subsets of this values with the maximal total sum provided that the sum in each subset does not exceed a capacity c. problem mssp-i is a special case of problem (p1) where the number of train destinations n = 1, all m = m trains arrive at zero time moment, |g| = g, lj = wj , j ∈ {1, . . . , g}, copyright © 2024 assa. adv syst sci appl (2024) problem of dynamic regulation of train formation 35 lmin = 0, lmax = c and the storage already contains groups of cars. as all locomotives are available at zero time moment, in this special case we just have to redistribute the cars by m trains, taking into account the trains in the storage, to maximise the number of cars sent from the station (or minimise the number of cars remaining at the station). this reduction of the problem mssp-i to the special case of problem (3.1)-(3.8) proves the theorem. cars in storage: 30 10 arriving trains: 7:00 9:00 10:00 55 20 20 40 10 35 20 20 fig. 3.1. initial data in practice, a train is formed as soon as the required number lmin >= lmin of cars has accumulated at the station for some destination and a locomotive is available for this departure. we will call this strategy the current approach to planning (cap). let’s consider example 3.1 showing that cap may be non-optimal. example 3.1: let trains of two destinations be formed at the station. suppose that lmin = lmin = 65, lmax = 75, β = 0, the planning horizon is 24 hours. during the period under review, 3 trains arrive at the station at 7:00, 9:00 and 10:00, respectively. fig. 3.1 shows data on the size of blocks in each arriving train and on blocks located in the storages at the initial time moment. blocks of different destinations have different colours. using cap strategy we form a train at 7:00 (see fig. 3.2). but it turns out to be more profitable to wait for the arrival of the second train, then send two trains at once (see fig. 3.3). indeed, in the case of cap the first train contains fewer cars and block of 10 cars remains unshipped until the end of the planning period. the total car-hours at the station for cap is equal to 3035, while the optimal solution gives us 3015 car-hours. 7:00 9:00 10:00 55 10 30 20 20 35 40 undelivered cars: 20 10 20 fig. 3.2. cap strategy the purpose of this article is to evaluate the effect of implementing a planning method that uses more information about trains arriving at the station than is done in practice. in fact, copyright © 2024 assa. adv syst sci appl (2024) 36 e. musatova, a. lazarev, a. osminin, a. baushev 7:00 9:00 10:00 55 10 1030 20 20 35 40 undelivered cars: 20 20 fig. 3.3. optimal strategy if we use information about only one moment of train arrival, we apply cap: if it is possible to form trains at this time moment, then a train with the maximum possible number of cars is formed. applying this strategy at every moment of t , we get some feasible solution to the problem (p1). the task arises of choosing a suitable planning horizon that would improve the goal function compared to the cap, but would not require too much time to solve the problem. 4. solving problems of small dimensions in this section we analyse and compare the behaviour of cap and an exact method of solving problem (p1) using small-dimensional test instances. in this experiment, we vary values of parameters m and n to determine how the gain from using the exact method changes. during computational experiments, cap was compared with the optimization solver ibm ilog cplex 22.1.0.0. [12]. in cap, to make a decision at each point in time about which train should be sent, we also used cplex to solve the problem (p1) with |t | = 1. the planning horizon is 24 hours, lmin = lmin = 61, lmax = 75, β = 0. the datasets generated and analysed during the current study are available from the corresponding author on reasonable request. the running time of cap and the cplex solver was limited to 10 min., because, in practice, the decision-making system should work quickly. in table 4.1 the results of solving the randomly generated problems are given. the following notations are used in the table: ”m” is the number of incoming trains, ”n” is the number of train destinations generated at the given station, ”cap” is the value of the objective function obtained when applying cap, ”cplex” is the value of the objective function obtained as a result of cplex operation for 10 minutes, ”gapcplex” is the gap between the best integer solution and the best bound in cplex after 10 minutes of work (by default, the gap equal to 0.01% is set as a stop criterion in cplex), ”dif” is the percentage of improvement in the value of the objective function when using cplex for problem (p1). the calculations were performed on a personal computer (intel core i7-7700k, 4.2 ghz, 32.0 gb). as can be seen from the table, using the exact solution method gives a win in all cases, with the exception of instances of very small dimensions. it is also worth noting that the winning percentage varies depending on number of train destinations generated at the given station. in problems with m < 30, the smaller the value of n, the greater the winning percentage. this can be explained by the fact that the fewer the number of train destinations, the more likely it is that new cars of given destination will arrive in the near future, and therefore it is more profitable to wait for these cars. it can also be noted that starting from m = 25 cplex has not found a guaranteed exact solution in many cases. as the dimension of the problem increases, the gap will increase. so when running the test example at m = 140, n = 70 after 10 minutes of cplex operation, copyright © 2024 assa. adv syst sci appl (2024) problem of dynamic regulation of train formation 37 table 4.1. results of comparing cap and the exact method on small-dimensional instances. m n cap cplex gapcplex dif 5 2 5903 5865 0,01% 0,66% 5 3 6140 6140 0,01% 0,00% 5 4 6288 6288 0,01% 0,00% 10 2 11146 10978 0,01% 1,50% 10 5 11584 11565 0,01% 0,16% 10 8 14421 14400 0,01% 0,15% 15 5 14514 14167 0,01% 2,39% 15 7 17614 17394 0,01% 1,25% 15 10 22869 22738 0,01% 0,57% 20 5 17861 17693 0,01% 0,94% 20 10 20341 20242 0,01% 0,49% 20 15 31260 31201 0,01% 0,19% 25 5 20882 19753 1,40% 5,41% 25 10 26423 25442 2,36% 3,71% 25 15 30402 29590 0,45% 2,67% 25 20 35303 34870 0,01% 1,23% 30 5 26646 26375 3,31% 1,02% 30 10 34299 33836 0,43% 1,35% 30 15 35486 35243 0,01% 0,69% 30 20 40439 40431 0,01% 0,02% 40 5 34350 34185 2,37% 0,48% 40 10 40586 40566 2,89% 0,05% 40 20 47524 47467 1,42% 0,12% 40 30 53888 53780 0,11% 0,20% we got gap = 95.9% and the value of the objective function which is significantly worse than when using cap. that is why the next section discusses reducing the planning horizon, i.e. considering fewer arriving trains for solving high-dimensional problems. 5. solving high-dimensional problems as we noted above, the exact solution method is not applicable for problems with a large number of arriving trains on the planning horizon. however, in practice, some marshalling yards of russian railways (for example, sverdlovsk) operate with up to 8000 cars, i.e. with more than 130 arriving freight trains per day and form new trains in almost 70 destinations. one of the approaches to planning at stations of this size is to use less information about incoming trains, that is, to reduce the planning horizon of the problem. in our experiment, we examined randomly generated high-dimensional problems. to solve them, we first used cap. next, we used the following approach. let ν be the number of train arrival moments that we take into account when planning. the problem (p1) was solved for t = {t1, . . . , tν}. if in the received schedule some blocks and locomotives were sent, they were removed from the input data, and then the problem was solved with t = {tν+1, . . . , t2ν} and so on. since the size of ν strongly affects the complexity and time of solving the problem, we have considered those options for ν that allow us to solve the general problem (p1) in less than 10 minutes. as a result, the values 2, 3, 4, 5, 6 were considered for ν. note that ν = 1 coincides with cap. the test results are shown in the table 5.2. the cells of the table show the values of the objective function at different values of ν. the best values of the objective function for each example are highlighted in gray. dif here shows the maximum improvement in the value of the objective function relative to the value obtained by cap. copyright © 2024 assa. adv syst sci appl (2024) 38 e. musatova, a. lazarev, a. osminin, a. baushev we can see from the table that it cannot be argued that increasing the planning horizon necessarily improves the final value of the target function per day. for example, the algorithm with ν = 5 shows a worse result than with ν = 4. the largest number of minimum values of the objective function is obtained at ν = 6. however, in many other cases, the value of the function at ν = 6 is worse than when using cap. note also that the spread of the values of the objective function when using 6 variants does not exceed 1%, except the instance with m = 50, n = 10. thus, it can be argued that in high-dimensional problems the exact method of solving the problem (p1), which takes into account all trains arriving per day, is not applicable, since it requires too much time. besides, considering additional 1, ...,5 trains arriving at the station does not give a gain in the value of the objective function by more than 1%, while an increase in the number of trains under consideration does not necessarily lead to a decrease in carhours. the question remains how far the solution obtained by cap is from the optimal one. a partial answer to this question will be given in the next section, where a special case of the problem will be considered. table 5.2. solving high-dimensional instances. m n cap ν = 2 ν = 3 ν = 4 ν = 5 ν = 6 dif 50 10 52215 52478 50686 50865 51193 50060 4,13% 60 10 60947 61236 60932 60817 60928 61146 0,21% 70 10 63682 63787 63911 63609 63858 64071 0,11% 80 10 74505 74526 74265 74503 74442 73932 0,77% 90 10 79333 79063 78925 79434 79369 78792 0,68% 100 10 87419 87482 88023 87700 87684 88006 0,00% 110 10 97006 96985 97080 96575 96715 96572 0,45% 120 10 108222 108289 108266 108153 108344 108508 0,06% 130 10 121505 121650 121368 121266 121665 121521 0,20% 140 10 126714 127093 126214 126998 126683 126627 0,39% 50 20 51584 51609 51600 52009 51699 52041 0,00% 60 20 63448 63377 63351 63222 63548 63298 0,36% 70 20 74696 74687 74612 74013 74351 74656 0,91% 80 20 80400 80427 80372 80643 80381 80357 0,05% 90 20 85539 85464 85570 85704 85805 85591 0,09% 100 20 99386 99264 99420 99389 99356 99431 0,12% 110 20 110324 110354 110265 110235 110215 110418 0,10% 120 20 113795 113817 113510 113595 113677 113382 0,36% 130 20 120042 119951 120382 120157 120180 120192 0,08% 140 20 126933 127063 127258 127576 127277 126875 0,05% 50 25 55104 55087 54850 55138 55094 55096 0,46% 60 30 73218 73232 73170 73202 73188 73217 0,07% 70 35 83956 83907 83775 83895 83793 83788 0,22% 80 40 93114 93011 93026 92813 93295 92868 0,32% 90 45 105207 105079 105125 105038 105086 104977 0,22% 100 50 120615 120638 120565 120419 120436 120465 0,16% 110 55 130184 129782 129778 129948 129917 129558 0,48% 120 60 152286 152290 152265 152207 152250 152005 0,18% 130 65 153508 153507 153688 153961 153359 153190 0,21% 140 70 169040 169138 169043 169215 168975 168997 0,04% copyright © 2024 assa. adv syst sci appl (2024) problem of dynamic regulation of train formation 39 6. special case of the problem we will also consider a simplified formulation of the problem, where blocks can be separated, i.e. cars of the same block can be sent by different trains. obviously, the possibility of splitting blocks increases the bureaucratic difficulties of tracking cars, but reduces the load of stations and the number of car-hours. denote by αik, i ∈ i ∪ {0}, k ∈ k, the number of cars that arrive at the station at time moment ti for further shipment in the direction k, i.e. αik = ∑ g∈gk: τg=ti lg, i ∈ i ∪ {0}, k ∈ k. we introduce the following decision variables: • integer variable ỹik, i ∈ i , k ∈ k, equals to the number of trains formed at time moment ti + β for destination k. • integer variable x̃ik, i ∈ i , k ∈ k, is equal to the number of cars departing from the station at time moment ti + β to destination k. the constraints of the problem take the following form. at any given time moment, we cannot send more cars than are left at the station for a given destination: x̃ik ≤ i∑ j=0 αjk − i−1∑ j=1 x̃jk ∀i ∈ i, ∀k ∈ k. (6.9) we cannot use more locomotives than we have at the station:∑ k∈k ỹik ≤ r0 + i− ∑ k∈k i−1∑ j=1 ỹjk ∀i ∈ i. (6.10) the restriction on the number of cars in the trains can be written as follows: lminỹik ≤ x̃ik ≤ lmaxỹik ∀i ∈ i, ∀k ∈ k. (6.11) the integer condition is x̃ik, ỹik ∈ n0 ∀i ∈ i, ∀k ∈ k, (6.12) where n0 is the set of non-negative integers. the objective of the problem is to minimise the following function:∑ i∈i ∑ k∈k x̃ik(ti + β) + ( m∑ i=0 ∑ k∈k αik − ∑ i∈i ∑ k∈k x̃ik)tmax → min . (6.13) we denote problem (6.9)-(6.13) by (p2). unlike problem (p1) with binary variables, problem (p2) has integer variables, but the number of variables is equal to 2|i||k|, that is significantly fewer than in (p1). it should be noted that the optimal value of the objective function of the problem (p2) is a lower bound for the problem (p1). in addition, the integer condition (6.12) can be replaced with a more lenient condition x̃ik ≥ 0, ỹik ∈ n0 ∀i ∈ i, ∀k ∈ k. (6.14) indeed, denote by (p3) the problem (6.9)-(6.11), (6.14), (6.13), that is problem (p2) without requirements for the integers of variables x̃ik, i ∈ i , k ∈ k. we can use (p2) instead of(p3) due to the following theorem: copyright © 2024 assa. adv syst sci appl (2024) 40 e. musatova, a. lazarev, a. osminin, a. baushev theorem 6.1: in any optimal solution of problem (p3) variables x̃ik, i ∈ i , k ∈ k, take integer values. proof let’s assume the opposite. consider an optimal solution x̃∗ ik, ỹ ∗ ik, i ∈ i, k ∈ k. choose an arbitrary k̄ ∈ k for which there exists i ∈ i such that x̃∗ ik̄ is fractional. denote by ī(k̄) ⊂ i the set of all such indexes i. if this set consist of one index, i.e. ī(k̄) = {̄i}, then we change x̃∗ īk̄ by [x̃∗ īk̄ ] + 1, where [x] is the integer part of the number x. it is easy to see that constraints (6.9) and (6.11) are fulfilled for the new value of x̃∗ īk̄ . at the same time, the value of the goal function is strictly better than the previous one. now suppose that w = |ī| > 1. let’s arrange all indexes of |ī| in increasing order: ī = {i1, i2, . . . , iw}, i1 < i2 < . . . < iw. let ε1, . . . , εw be fractional parts of the corresponding optimal values of the variables x̃∗ i1k̄ , . . . , x̃∗ iwk̄ . if ε2 + . . .+ εm ≤ 1− ε1 we can change x̃∗ i1k̄ by [x̃∗ i1k̄ ] + 1 and x̃∗ ij k̄ — by [x̃∗ ij k̄ ] for j ∈ {2, . . . , jw} without violating constraints (6.9), (6.11) and with strict improvement of the value of the goal function. in the case of ε2 + . . .+ εm > 1− ε1 we denote by j̄ the maximal value of j for which ε2 + . . .+ εj ≤ 1− ε1. then we can change x̃∗ i1k̄ by [x̃∗ i1k̄ ] + 1 and x̃∗ ij k̄ — by [x̃∗ ij k̄ ] for j ∈ {2, . . . , j̄} by improving the value of the goal function. applying the same approach for the remaining indexes ij̄+1, . . . , iw the required number of times we will get solution with integer values of x̃∗ ik̄ for all i ∈ i . repeating this procedure for all k ∈ k for which there are fractional values of x̃∗ ik, we will get a feasible solution with integer x̃∗ ik, i ∈ i, k ∈ k, and better goal function value, which proves the theorem. it is worth noting that the cap technique is also simplified in the case when it is possible to divide blocks. at each step, there is no need to solve a combinatorial optimization problem of the knapsack type to make up the train with the largest number of cars. it is enough to choose the destination for which the maximum number of cars is located at the station. we used the same test data for testing the model (p3) as for (p1), but it was assumed that the blocks could be divided. the results are presented in table 6.3. the same notation was used as in table 4.1, except for a new column called ”gap1,2”, in which we have a percentage decrease in the value of the objective function obtained using cplex for problem (p3) compared to the best found solution to the problem (p1) (see table 5.2). values gap1,2 and gapcplex together allow us to estimate the error of solving the problem (p1). so, for the considered test problems, it can be argued that the error of the solutions found for (p1) is guaranteed not to be very large. at the same time, it can be seen from the table that the exact method of solving the problem (p3) allows us to find a solution with acceptable accuracy for high-dimensional problems in 10 minutes. 7. conclusion in the paper, we proposed and investigated two models of dynamic train formation at a marshalling yard with divisible and indivisible train blocks, respectively. for the first model, computational experiments have shown that in problems of small dimension (a small number of incoming trains per day and a small number of train destinations), it is possible to find an exact solution to the problem of minimizing car-hours at the station per day in an acceptable time. this is impossible for high-dimensional problems. however at the same time, the trains formation without taking into account the trains coming later in such problems gives an acceptable result relative to the found lower bounds. moreover, increasing information about copyright © 2024 assa. adv syst sci appl (2024) problem of dynamic regulation of train formation 41 table 6.3. solving the special case. m n cap cplex gapcplex dif gap1,2 50 10 50272 49046 0,01% 2,44% 2,03% 60 10 60490 59586 0,01% 1,49% 2,02% 70 10 63422 62210 0,01% 1,91% 2,20% 80 10 73466 72858 0,49% 0,83% 1,45% 90 10 78303 77259 0,29% 1,33% 1,95% 100 10 86858 85753 0,27% 1,27% 1,91% 110 10 96408 95057 0,47% 1,40% 1,57% 120 10 107656 106307 0,61% 1,25% 1,71% 130 10 120075 118958 0,55% 0,93% 1,90% 140 10 125843 124201 0,37% 1,30% 1,59% 50 20 50619 49591 0,01% 2,03% 3,86% 60 20 61268 60517 0,01% 1,23% 4,28% 70 20 73438 72211 0,01% 1,67% 2,44% 80 20 79151 78185 0,15% 1,22% 2,70% 90 20 83164 81306 0,32% 2,23% 4,86% 100 20 97922 96880 0,36% 1,06% 2,40% 110 20 107770 106505 0,53% 1,17% 3,37% 120 20 112236 110087 0,66% 1,91% 2,91% 130 20 118159 116098 0,71% 1,74% 3,21% 140 20 124708 122654 1,22% 1,65% 3,33% 50 25 52875 52308 0,01% 1,07% 4,63% 60 30 72030 71487 0,01% 0,75% 2,30% 70 35 83002 82538 0,01% 0,56% 1,48% 80 40 90429 89742 0,01% 0,76% 3,31% 90 45 102158 101367 0,01% 0,77% 3,44% 100 50 118017 117136 0,01% 0,75% 2,73% 110 55 125432 123572 0,01% 1,48% 4,62% 120 60 146170 143839 0,17% 1,59% 5,37% 130 65 148858 147924 0,06% 0,63% 3,44% 140 70 164399 163193 0,03% 0,73% 3,42% incoming cars of 2-5 trains in advance does not give a tangible benefit. if we talk about the model for the case of divisible cars blocks, then this problem can be solved with high accuracy for any dimensions that have a practical application. references 1. yaghini, m., mirghavami, m., & zare andaryan, a. (2021) a mathematical formulation and an lp-based neighborhood search matheuristic solution method for the integrated train blocking and shipment path problem. networks, 78, 523–542. 2. cordeau, j.-f., toth, p., & vigo, d. (1998) a survey of optimization models for train routing and scheduling. transportation science, 32, 380–404. 3. yaghini, m., momeni, m., & sarmadi, m. (2014) solving train formation problem using simulated annealing algorithm in a simplex framework. journal of advanced transportation, 48, 402–416. 4. zhang, x., kang, l., liu, w., & yao, y. (2016) solving train formation plan problem in freight transportation corridors. in 2016 international conference on engineering science and management, 102–104, atlantis press. copyright © 2024 assa. adv syst sci appl (2024) 42 e. musatova, a. lazarev, a. osminin, a. baushev 5. lin, b., zhao, y., lin, r., & liu, c. (2021) integrating traffic routing optimization and train formation plan using simulated annealing algorithm. applied mathematical modelling, 93, 811–830. 6. zhao, y. & lin, b. (2019) the multi-shipment train formation optimization problem along the ordered rail stations based on collection delay. ieee access, 7, 75935–75948. 7. ahuja, r. k., jha, k. c., & liu, j. (2007) solving real-life railroad blocking problems. interfaces, 37, 404–419. 8. xiao, j. & lin, b. (2016) comprehensive optimization of the one-block and two-block train formation plan. journal of rail transport planning & management, 6, 218–236. 9. barnhart, c., jin, h., & vance, p. h. (2000) railroad blocking: a network design application. operations research, 48, 603–614. 10. xiao, j., pachl, j., lin, b., & wang, j. (2018) solving the block-to-train assignment problem using the heuristic approach based on the genetic algorithm and tabu search. transportation research part b: methodological, 108, 148–171. 11. kellerer, h., pferschy, u., & pisinger, d. (2004) knapsack problems. springer-verlag, berlin, heidelberg. 12. (2024, december 06), ibm ilog cplex optimization studio. available https://www.ibm.com/products/ilog-cplex-optimization-studio. copyright © 2024 assa. adv syst sci appl (2024) introduction problem description mathematical statement solving problems of small dimensions solving high-dimensional problems special case of the problem conclusion microsoft word anton_sinelnikov,+assa+word+template_sin (1) adv syst sci appl 2025; 1; 120-131 published online at https://ijassa.ipu.ru. analysis of ring laser gyroscope cavity thermal deformations anton sinelnikov* patrice lumumba rudn university, moscow, russia abstract: ring laser gyroscopes are crucial for high-precision orientation, stabilization, and autonomous inertial navigation systems. however, their operational accuracy is significantly affected by temperature-induced cavity deformations, leading to potential performance degradation and data loss. this study addresses this challenge by developing a robust algorithm in the matlab environment designed to simulate the temperature deformations of a ring laser cavity. leveraging experimental data, the methodology enables the precise modeling of cavity deformation behavior at various stages of the ring laser gyroscope assembly. the developed mathematical model effectively demonstrates how the addition of structural elements influences the cavity's thermal response and overall deformation. this research provides a novel approach to quantitatively estimate the contribution of individual structural components to the total thermal deformation. the findings offer critical insights for optimizing ring laser gyroscope design, thereby enhancing their accuracy, reliability, and resilience in diverse operational conditions. keywords: inertial navigation systems, inertial measurement unit, ring laser gyroscope, cavity, control system, thermal deformations, matlab, mathematical model, simulation 1. introduction due to their high accuracy, reliability and resistance to external influences, ring laser gyroscopes (rlg) are actively used in high-precision systems for orientation, stabilization and navigation [1]. today, they are employed in aircraft and spacecraft for calculating the angular rotation of a broad range [2]. in the case of inertial navigation guidance, the aerospace vehicles use onboard sensors to determine their motion and acceleration with the help of rlg or other gyroscopes and accelerometers [3]. a gyroscope is used to measure angular rotation, whereas an accelerometer is used to measure linear motion. the two are combined into a single unit along with a control mechanism, and the unit is called inertial measurement unit (imu), or inertial navigation systems (ins). rlg, is an important optronic rate sensor and, in essence, the heart of any ins [4]. based on application, rlg market can be split into platform stabilizations as well as aeronautic, missile, submarine and satellite types of navigation [5]. based on the key end users, rlg commercial market can be divided into business jets, civil aircraft, helicopters and uavs, general aviation, satellites, spacecraft, and rockets. according to the forecasts, the rlg market volume should exceed $1 billion by 2030 [4,5]. today optical gyroscopes are one of the dominant technologies in the imus and inss. they are divided into active rlgs and passive fiber optical gyroscopes (fog) [6]. they uses the sagnac effect to calculate the rotation rate by measuring the phase shift of two counter propagating light beams in a passive interferometer and frequency difference in an active laser interferometer [7]. according to the sagnac equation, frequency difference is obtained by fundamental formula (1.1): * corresponding author: mr.sinelnikov.a@mail.ru analysis of ring laser gyroscope cavity thermal deformations 121 copyright ©2025 assa. adv. in systems science and appl. (2025) δf=4sω/λl (1.1) where δf is the frequency difference between two counter-propagating light waves in the ring laser; s and l are the square and length of the ring laser cavity; ω is the angular rate; λ is the wavelength of the laser. so, the ideal input-output characteristic of a rlg is described by a simple expression (1.2): δf=kω (1.2) where k=4s/λl is called a scale factor. for a more compact form of this expression a scale factor k is often included into ω, thereby making the direct relation between the rlg output signal and the measured angular rate more obvious. let us consider the analytical review of the yole development company to assess the market of inertial navigation systems over the past 10 years [8,9]. the main competitive optical gyroscopes are hemispherical resonator gyroscopes (hrgs), micro-machined gyros (mems) and dynamically tuned gyros (dtgs). figure 1.1 presents ring diagrams of the evolution of high-end inertial technology market from 2011 to 2019 in monetary value. as can be seen from the left diagram in figure 1.1, rlg occupied 40 percent of the inertial systems market in 2011. (a) (b) fig. 1.1. global inertial technology market evolution from 2011 (a) to 2019 (b) year it was about $510 million. the entire market was estimated at $1,3 billion. in 2019, the share of rlg increased to 52% as can be seen from the right diagram. this corresponds to $1,7 billion and a threefold increase in a cash volume. the share of fog decreased by 3 percent, the share of mems increased by 5%, the share of hrgs increased by 5%, and the share of other gyros types amounted to 1 percent. so, considering the presented data, rlg technology is dominant in the inertial technology market. figure 1.2 shows distribution diagrams of gyroscope types by accuracy class in 2011 and 2019. it can be seen that mems gyros consistently dominate in commercial and industrial navigation. optical gyros are dominating in tactical navigation. the share of rlg in midlterm navigation researches 61%. the use of rlg in high-end and strategic navigation is 78%. give the conservatism of inertial navigation market, the dominance of rlg will continue in the next 10-20 years. 122 a. sinelnikov copyright ©2025 assa adv. in systems science and appl. (2025) (a) (b) fig 1.2. gyro technology breakdown per bias stability categories from 2011 (a) to 2019 (b) year there are top-3 types of the most common rlg high-end navigation accuracy class suitable for use in spacecraft among a variety of optical gyroscopes. they are shown in figure 1.3. (a) (b) (c) fig. 1.3. most common types of rlg commonly used ring laser cavity structure is triangular geometry, as shown in figure 1.3.a. this type of geometry is popular with rlgs from honeywell [10]. figure 1.3.b shows square ring laser geometry, which is one form of geometry in use [11]. these two types of rlg using forced dithering to overcome lock-in effect [12]. figure 1.3.c shows non-polar square-type ring laser geometry that allows the using magneto-optical frequency shift based on zeeman effect to overcome lock-in effect [13]. due to the absence of dither, the zeeman rlg has increased resistance to external mechanical influence [14,15]. this allows using the term in many areas of aerospace industry. however, this may be due to sudden changes in external temperature, which may have a negative impact on the zeeman rlg operation [16]. the most important condition for achieving all types of rlg stable operation and high precision characteristics, including super-high stability of the scale factor k, is to ensure the constancy of the laser cavity so that the ring laser generates on a single mode [17,18]. the accuracy of timing and stabilizing the laser cavity to keep the generation frequency in the center of the gain curve should be less than a hundredth of die generation wavelength, i.e, units of nanometers (λ~ 1-10nm). figure 1.4 illustrates active and passive compensation methods been used to ensure stable operation of the rlg under temperature influences. analysis of ring laser gyroscope cavity thermal deformations 123 copyright ©2025 assa. adv. in systems science and appl. (2025) (a) (b) fig. 1.4. active and passive methods for the rlg cavity thermal stabilization the adjustment of the rlg to the working mode and the time of continuous operation in it is provided by the cavity control system (ccs), which is an active thermal compensation system [18, 19]. at the same time, a wide range of operating temperatures and heating of the zeeman rlg during its operation impose significant restrictions on the dynamic range of the ccs, therefore, in addition to the active method, a passive method of thermal compensation is used [20]. to provide passive thermal compensation, the gyroscope cavity materials should have ultra-low thermal expansion coefficient (tec) [21]. this materials include glass ceramics of the brands zerodur, clearceram, cervit and so-115m sitall [22-24]. in particular, the housings of the ring laser cavity and mirror substrates in russia are made of grade so-115m sitall. the use of this material in the manufacture of rlg structural elements is determined by a number of indisputable advantages, which include an ultra-low tec, optical transparency and increased retention of the active medium inside the cavity [24]. the main disadvantage of this material is the non-linear nature of the tec in the range of operating temperatures for measuring systems based on ring laser sensors, which limits the possibility of cavity perimeter active adjustment [25]. in addition to the cavity itself, there are other structural elements in the rlg. these include fasteners, piezoelectric actuators (executive elements of an active ccs), magnetic screens, etc. these structural elements are made of various materials, which have different tec than glassceramic material, which directly affects the temperature stability of the ring laser cavity [26]. to varying degrees, this problem is inherent in all types of modern optical gyroscopes, and mathematical modeling methods are actively used to solve it and temperature correction [2729]. the purpose of this work is to develop an algorithm in the matlab environment, with which it is possible to simulate the zeeman rlg cavity temperature deformations. to achieve this goal, the following tasks are solved in the work: • obtaining initial data for modeling; • development and implementation of the algorithm in the matlab environment; • development of a temperature drift model for path length of a ring laser cavity. 2. theoretical and experimental part a sensitive rlg is a he-ne laser with a non-planar optical cavity (fig. 1.3.c) that generates laser beams with circular polarization at a wavelength of λ= 632,8 nm. this makes it possible to use a magneto-optical frequency bias based on the zeeman effect to remove the rlg from the capture region and form a reference signal for the active ccs. the spectrum of natural 124 a. sinelnikov copyright ©2025 assa adv. in systems science and appl. (2025) frequencies of longitudinal modes of a non-planar four-mirror cavity zeeman rlg is half the beams wavelength λ/2, which corresponds to 316,4 nm [30]. two movable mirrors of the rlg resonator are equipped with piezoelectric actuators, which ensure the operation of the active ccs [31]. a part of the beam is put out through a semitransparent mirror to the mixing unit to form output signals and obtain information about rotation. the cavity is installed on the base, fixed with special fasteners and placed in magnetic screens. the ccs inputs receive a cavity detuning signal ap from a special photodetector installed in the rlg, and a reference signal synchronous with the current of the alternating frequency bias. cavity stabilization is carried out by integrating the signal rectified by a synchronous detector from the photodetector, followed by converting the output voltage of the integrator into a voltage proportional to it on the actuating piezoelectric actuators that move the movable mirrors in the direction of decreasing the error signal. from this it follows that the signal amplitude ap is proportional to the cavity detuning in units of the generation frequency, which in turn is proportional to the temperature deformation of the ring laser cavity. setting the path length to the center of the rlg amplification circuit in the open ccs mode corresponds to the minimum value of the cavity signal amplitude, and the detuning corresponds to the maximum. thus, in the studied rlg, from the amplitude of the signal ap, it is possible to estimate, with an accuracy of one fourth of the generation wavelength λ/4 the cavity temperature deformation and its direction from the change in the phase of the signal ap. to build a model of the rlg cavity thermal deformations, studies of the zlk-16 type device were carried out under long-term temperature effects according to the methodology presented in and obtained the necessary initial data [32]. figure 2.1 demonstrates a general view of the experimental setup for obtaining initial data for computer modeling. fig. 2.1. experimental study and initial data it shows the dependence of the amplitude of the cavity detuning signal with an non-working ccs. at a linearly changing temperature t in the range from –55°с to 75°с at a rate of 1°/min inside the heat and cold chamber, this characteristic reflects the sequential rearrangement of the rlg frequency spectrum relative to the center of the amplification circuit, caused by the cavity temperature deformation δl, nm. it is known from paper that the typical value of the path length drift δl for zeeman rlg type zlk-16 in the operating temperature range is 3λ-4λ (from 1900 to 2500 nm) and may depend on many factors: tec of glass-ceramic so-115m and other structural elements, the length of the cavity, the method of pumping the active medium, the magnitude of the working analysis of ring laser gyroscope cavity thermal deformations 125 copyright ©2025 assa. adv. in systems science and appl. (2025) current of the discharge, and other factors [33,34]. thus, having the time dependences of the signal ap on temperature t as initial data, it becomes possible to create a model of the cavity temperature drift using a software method. 3. mathlab simulation to build the model, a special algorithm was developed, which was implemented in the matlab environment. figure 3.1 shows a block diagram of the modeling of ring laser cavity temperature deformations algorithm. fig. 3.1. block diagram of the algorithm data from the text file work.txt obtained during the experimental study is written to the array "a" (3.1): a=dlmread('work.txt') (3.1) in the array "t" the 3rd column of the array "a" is written, containing the readings of the temperature sensor. the 2nd column of the array "a" is written to the "e" array, containing the amplitude values of the detuning signal (3.2): t=a(:,3) e=a(:,2) (3.2) the first empty intermediate arrays "e1" and "t1" are filled with values from the inflection point search algorithm. next, the search for points of the array "e", the values of which are 126 a. sinelnikov copyright ©2025 assa adv. in systems science and appl. (2025) greater or less than the values of their previous and subsequent points, which are entered into the intermediate array, also giving these points the values of the array "t" (3.3): e1=[] t1=[] indx=1 for i = 2:size(e)-1 if ( ( e(i-1)< e(i))&& ( e(i+1)< e(i)))||( ( e(i-1) > e(i) ) && ( e(i+1)> e(i) ) ); e1(indx) = e(i); t1(indx) =t(i); indx = indx + 1; end (3.3) after that, the values of the "e1" array are replaced by their corresponding serial numbers and converted to nanometers (3.4): e1=1:size(e1’); e1=633*0.25*e1 (3.4) figure 3.2 shows an intermediate graph of the dependence of the signal ap on temperature t, containing the key points of "peaks" and "dips" necessary to build the final model of the temperature deformation of the rlg cavity. on the intermediate graph (fig. 3.1) the number of the inflection point "ind" is selected, namely the number 9. this is the point at which the phase of the signal ap changes. fig. 3.2. intermediate chart corresponding to the step of determining the extremes of the cavity signal amplitude from the values of the points of the "e1" array, the value of the magnitude of the amplitude change at the point of change in the direction of deformation is subtracted, which allows you to move the graph to zero coordinates at the inflection point (3.5): ind=9 (3.5) analysis of ring laser gyroscope cavity thermal deformations 127 copyright ©2025 assa. adv. in systems science and appl. (2025) ind=9 e1=e1-e1(ind) next, the values "e1" and "t1" from the cycle are entered into the empty intermediate arrays "ex" and "te", which reflects the values of the change up to the inflection point relative to the abscissa axis (3.6): ex=[] te=[] i1=1 for k=1:size(e1') if k<ind ex(i1)=-e1(k)' te(i1)=t1(k)' i1=i1+1 end if k>=ind ex(i1)=e1(k)' te(i1)=t1(k)' i1=i1+1 end (3.6) based on the values from the "ex" and "te" arrays, a final graph of the dependence of the absolute change in the perimeter δl on temperature t is constructed. figure 3.3 shows the temperature deformation model of the rlg resonator obtained in the matlab environment, which reflects the dependence of the absolute temperature change in the path length. fig. 3.3. model of the ring laser cavity temperature drift having an idea about the nature of the path length temperature drift, it becomes possible to estimate the contribution of all structural elements of the rlg, which has tec different from the glass-ceramic so-115m, to the resulting temperature deformation of its cavity δl. 128 a. sinelnikov copyright ©2025 assa adv. in systems science and appl. (2025) 4. discussion of the results based on the initial data and thanks to the algorithm implemented in the matlab environment, it becomes possible to simulate and obtain the form of the dependence of the cavity temperature deformations at different stages of the rlg assembly, which will be described by expression (4.1): δl(t)=a·t3+ b·t2+c·t+ d (4.1) where, δl is the value of thermal deformation, nm; t is the temperature value, °c; a, b, c, d are the polynomial coefficients determined by the design parameters of the rlg. in the present model (4.1), the coefficient a characterizes the nonlinear tec of the glass-ceramic so-115m in temperature range from –55°с to 75°с. the coefficient b takes into account the contribution of piezo actuators and c determines the influence to other rlg structural elements. figure 4.1 (approximation of the ring laser cavity temperature drift) presents a graph of an approximating function of the type and its formula obtained using microsoft excel. fig 4.1. approximation of the ring laser cavity temperature drift it has a confidence level r=0,9932, which indicates the reliability of the obtained functional dependence. this opens up the possibility of using formula (4.1) to calculate the rlg cavity drift in the operating temperature range of the spacecraft, as well as to use it as a governing law for the ccs active stabilization (4.2). uccs(t) =up·δl(t(t)) (4.2) where, uccs is the ccs control voltage, v; t is the operating time, sec, °c; up·is the piezo actuators transmission coefficient, v/nm. 5. conclusion the presented mathematical model demonstrates the behavior of the temperature deformation of the rlg cavity at different stages of its generation. when adding new structural elements, a shift of the inflection point and a change in the resulting cavity deformation are observed. analysis of ring laser gyroscope cavity thermal deformations 129 copyright ©2025 assa. adv. in systems science and appl. (2025) this makes it possible to estimate the contribution of each structural element to the resulting thermal deformation of the ring laser cavity. the next step in this work will be to check the operability of the approximating function as the governing law of the active ccs. the proposed methodology not only addresses the critical challenge of rlg thermal compensation but also contributes to the advancement of fully autonomous navigation systems [35] for next-generation spacecraft – from earth observation satellites to interplanetary probes. further efforts will focus on implementing the model in industrial-grade gyroscopes for launch vehicles and crewed missions. references 1. hering, e., schönfelder, g., basler, s., biehl, k-e., burkhardt, t. et. al. (2022). geometric quantities. in hering e., schönfelder g. (eds.). sensors in science and technology (pp.147–372). springer, wiesbaden, https://doi.org/10.1007/978-3-65834920-2_3 2. logashina, i. v., chumachenko, e. n., bober, s. a., aksenov, s. a. (2009). thermal stress state of a laser-gyroscope housing for use in space. russian engineering research, 29, 751–755, https://doi.org/10.3103/s1068798x09080012 3. dell’olio, f., natale, t., wang, y. c., & hung, y. j. (2023). miniaturization of interferometric optical gyroscopes: a review. ieee sensors journal, 23(24), 29948– 29968, doi: 10.1109/jsen.2023.3327217 4. sinelnikov, a.o., tikhmenev, n.v., ushanov, a.a., medvedev, v.m. 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(2022) high-precision navigation independently of global navigation satellite systems data. gyroscopy navig., 13, 1–6, 2022, doi: 10.1134/s2075108722010059 adv syst sci appl 2023; 02:164–177 published online at https://ijassa.ipu.ru. pairwise similarity estimation for discrete optimization problems darya lemtyuzhnikova1,2*, pavel chebotarev3,4, mikhail goubko5, nikita shushko1, mikhail somov1 1institute of control sciences of ras, moscow, russia 2moscow aviation institute, moscow, russia 3technion – israel institute of technology, haifa, israel 4kharkevich institute for information transmission problems of ras, moscow, russia 5flöha, altran deutschland s.a.s. & co. kg abstract: in this paper, we propose a new method, called the pairwise similarity method, for assessing the similarity of instances of optimization problems. this method is a generalization of the metric approach proposed earlier for scheduling problems. it relaxes the requirements to the structure of the problem constraints. the method involves a dissimilarity function for the comparison of instances. it identifies “simple” instances that can be solved in polynomial time and uses them to get good approximations for other instances. we apply the pairwise similarity method to two discrete optimization problems: the majority domination problem and the maximum cut problem. keywords: pairwise similarity method, pairwise dissimilarity function, discrete optimization, np-hard problems, special case, majority domination problem, maximum cut problem 1. introduction the problem of np-hardness is related to one of the key problems of theoretical computer science of the relationship between complexity classes p and np. to date, there is no way to quantify the complexity of np-hard problems. analyzing the behavior of heuristic algorithms as applied to various problem instances leads to the identification of patterns in the data. studying such patterns allows one to gain insight into the complexity structure of np-hard problems. in this paper, we propose a general method for estimating the computational complexity of np-hard problems based on comparing the problem instances using a dissimilarity function. the proposed method is based on the metric approach, which was originally formulated for scheduling problems [2, 4, 6, 7, 8]. the metric approach was tested for problem instances of the same dimension without taking into account the difference in the structure of the constraints. first, instances were found that enables obtaining an exact solution in a reasonable time. to compare a general instance with such an instance, metrics were proposed that evaluate the difference between the objective functions of those two instances. an exact solution to the first instance induces an approximate solution for the second instance and the characteristics of the latter solution ∗corresponding author: darabbt@gmail.com pairwise similarity estimation for discrete optimization problems 165 have been assessed using the above metrics. more specifically, the metric was defined as the upper bound for the absolute error of the approximate solution. computational experiments for solving some problems of scheduling theory by approximate algorithms in polynomial time have shown that with increasing dimension, the upper bound for the relative error of the objective function tends to zero, while the proportion of problems that can be solved exactly in polynomial time tends to 100%. this means that the proposed algorithms can be effectively used for most of the instances of the problems under consideration. this made it possible to obtain exact or fairly close to exact solutions. moreover, with an increase in the dimension of the problem, the proportion of such instances increases [6, 7, 8]. furthermore, [2] studied the possibility of finding approximate schedules using the metric approach for an np-complete scheduling problem on two parallel identical machines with priority delays for jobs or for jobs of length 1 or 2 with minimization of execution time. an application of these methods for finding the optimal solution for p2 | pred, pj = 1 |cmax using the coffman and sethi algorithms has been implemented. in [4], the metric approach was developed for planning a two-station single-track railway. new metrics were proposed and computational tests were carried out. the pairwise similarity method generalizes the metric approach. it allows for different structures of instances and involves pairwise dissimilarity functions for which satisfaction of the axioms of metric is not required. this paper proposes a general method for assessing heuristics called the pairwise similarity method. it is applied to two discrete optimization problems: the problem of majority domination on graphs with a limited degree of vertices and the maximum cut problem. the focus is on finding instances with the required characteristics and using them to obtain efficient solutions for other instances. relations are established between the change in the value of the objective function and the difference between the graphs of the problems under consideration. the structure of the paper is as follows. section 2 introduces the pairwise similarity method and describes the algorithms that make it up. in section 3.1, the pairwise similarity method is applied for finding the strict majority domination number in a graph with cycles in the framework of a two-level voting model. in particular, it is proved that graphs with a limited cyclomatic number form a “semisimple” special case, a pairwise dissimilarity function is introduced for them, and an approximate algorithm of polynomial complexity is constructed that has a guaranteed bound for the absolute error of the objective function. in section 3.2, the pairwise similarity method is applied to the maximum cut problem in the case where it is difficult to analytically obtain an absolute upper bound for the error and the estimates are based on numerical simulation. in conclusion, a research program for further study of the proposed method is outlined. 2. pairwise similarity method we first introduce the formal definitions needed to construct the pairwise similarity method. definition 2.1. for an np-hard problem p and a pattern α, the special case pα is the subclass of p comprising all instances a ∈ pα that satisfy α. definition 2.2. a simple special case pα simple is a special case for which there is a polynomial (or pseudopolynomial) algorithm that finds the exact solution for all instances a ∈ pα simple. definition 2.3. a semisimple special case pα ssimple is a special case for which there is a polynomial (or pseudopolynomial) algorithm that finds an approximate solution with a guaranteed error for all instances a ∈ pα ssimple. definition 2.4. a numerical special case pα num is a special case for which there exist a polynomial (or pseudopolynomial) algorithm and an error estimate such that this estimate for the algorithm is not proved analytically, but is confirmed by numerical experiments on pα num. copyright © 2023 assa. adv syst sci appl (2023) 166 d. lemtyuzhnikova, p. chebotarev, m. goubko, m. somov,n. shushko definition 2.5. a hard special case pα hard is a special case for which there is no polynomial (or pseudopolynomial) algorithm that finds exact solutions for all instances a ∈ pα hard. in what follows, by special cases, unless otherwise indicated, we mean simple, semisimple, or numerically simple special cases. we now define the pairwise dissimilarity function. definition 2.6. a pairwise dissimilarity function is a function that measures the difference between instances according to some criterion. given a simple special case pα simple and the corresponding polynomial or pseudopolynomial algorithm, for any b ∈ pα simple and a ̸∈ pα simple, the absolute (or relative) error of the objective function for a when the polynomial (pseudopolynomial) exact solution for b is applied to a will be used as the value of the dissimilarity function, denoted by τ , on the pair (a,b). the steps of the pairwise similarity method are described below. note that different values, such as relative error, rate of convergence of the method, convergence scores of several methods, number of operations, etc., can be used to construct dissimilarity functions in the pairwise similarity method. let us move on to the main definitions assuming that the problem under consideration is a linear programming problem with one objective function. in the pairwise similarity method, a solution xb to the instance b obtained using a polynomial (or pseudopolynomial) algorithm is applied to an instance a, for which it may violate the internal constraints. depending on how the solution xb is applied, we consider three strategies. 1. evaluation of instance a: using the obtained solution xb to evaluate the solution xa, regardless of the feasibility of the resulting solution. 2. relaxation of instance a: using the obtained solution xb to evaluate the solution xa, taking into account which restrictions of instance a are violated when xb is applied to a. to implement this strategy, it is necessary to check the resulting solution for feasibility. that is, if instance b is a relaxation of instance a, then a procedure for checking the satisfaction of constraints is needed in order to determine which particular restrictions of instance a are violated by solution xb. 3. constructing a valid solution for instance a using the obtained solution xb as an approximate solution to instance a. to do this, it is necessary to check the feasibility of xb and, if some constraints are violated, find the new solution x∗ b closest to xb among the solutions that satisfy all the constraints of a. thus, to obtain xa = x∗ b, it is necessary to construct a function transforming xb into x∗ b. thus, the pairwise similarity method consists of the following steps. 1. find one or more special cases of the problem. for many discrete optimization problems, some special cases are known. below we present guidelines for finding new special cases. 2. construct a pairwise dissimilarity function that allows one to estimate the distance from any instance to each of the special cases. an algorithm for building such a function is proposed below. 3. given an instance a, find the closest special case using the chosen pairwise dissimilarity function. 4. find the solution to the closest to a instance that belongs to a special case. 5. estimate the guaranteed error of the obtained solution using the chosen pairwise dissimilarity function. 6. choose a strategy for applying the solution of item 4 to instance a. – for the “evaluation of instance a” strategy, we use xb without adjustment. copyright © 2023 assa. adv syst sci appl (2023) pairwise similarity estimation for discrete optimization problems 167 – for the strategy “relaxation of instance a”, we use xb, taking into account information about whether the constraints of the original instance are violated, and if so, which ones. – for the strategy “constructing a valid solution for instance a”, we check the obtained solution for feasibility, and if some restrictions are violated, then use the function transforming xb into xa = x∗ b. finding special cases is a non-trivial task, the solution of which is based on studying the special properties of the problem under consideration. for example, for the classes of problems where the graph included in a constraint is a tree, there are often fast exact algorithms based on classical tree traversal algorithms. expanding the set of special cases allows one to more accurately determine the absolute error of the original instance. we now give some guidelines that help find special cases. the most obvious way is to expand the known special cases by slightly changing the original pattern. section 3.1 gives an instance of such an extension by adding additional edges to the trees to form cycles. the second way is based on the following statement. if for problem p there exists an approximate algorithm ã, then among the instances of p one can single out a subset of instances p ∗ for which this algorithm provides exact solutions. to implement this guideline, it is necessary to solve a number of randomly selected instances of problem p with different parameters using the exact algorithm a and the approximate algorithm ã. next, a subset of instances p ∗ ⊂ p for which exact solutions (or solutions with a guaranteed error) were obtained by the approximate algorithm ã must be selected. note that the subset of instances p ∗ is, by definition, a numerically simple special case. then it is necessary to determine whether there is a subset of instances p ∗∗ ⊆ p ∗ for which some pattern α can be formulated. if the pattern α is found, then the resulting set of instances p ∗∗ is, by definition, a simple or semisimple special case. the third way involves an exact combinatorial algorithm a, for which the rate of convergence is investigated. if for problem p it is possible to select a subset of instances p ∗ ⊂ p for which algorithm a converges fast enough, then one may try to construct a heuristic algorithm ã based on a, which provides an exact solution or an approximate solution with a guaranteed error for all instances in p ∗. note that the concept of fast convergence of an algorithm in this context is used informally, since the formalization depends on a specific problem. to implement this guideline, it is necessary to randomly select a number of instances of p with different parameters and solve them using a. next, we should single out a subset of instances p ∗ ⊂ p for which a obtains exact solutions fairly quickly. then it is necessary to build a heuristic algorithm based on the steps of a, which are carried out in the course of solving the instances in p ∗. as a result, we get p ∗ as a numerically simple special case. further, similarly to the second way, we look for a pattern α that describes some subset of instances p ∗∗ ⊆ p ∗. if such a pattern has been found, then p ∗∗ is a simple special case. we now turn to the construction of a pairwise dissimilarity function. this includes the following steps. 1. find a special case p ∗. 2. fix all parameters of the p ∗ problem, except for one parameter q. 3. solve some set of instances of the problem p ∗ for various q. 4. determine the relationship βq between q and the exact value of the objective function. 5. construct a pairwise dissimilarity function ρq that approximately represents βq. 6. repeat steps (2)–(5) for the other problem parameters. 7. as the resulting pairwise dissimilarity function, consider the sum ρ = ∑n q=1 ρ q of partial functions for all parameters of problem p . copyright © 2023 assa. adv syst sci appl (2023) 168 d. lemtyuzhnikova, p. chebotarev, m. goubko, m. somov,n. shushko an example of the application of this procedure is the variation of the cyclomatic number of a graph in section 3.1, where the pairwise similarity method is used to build an analytical estimate with a limited absolute error. note that, due to the complexity of the problem, the pairwise dissimilarity function cannot always be found analytically. in this case, it is proposed to evaluate the pairwise dissimilarity function numerically. for instance, in section 3.2 the pairwise dissimilarity function is evaluated using computational experiments. it is proposed to majorize from above and tabulate the numerical values of the pairwise dissimilarity function. 3. applications of the pairwise similarity method 3.1. majority domination problem in this section, the pairwise similarity method is illustrated by the its application to the problem of finding the strict majority domination number for a graph with cycles within the two-tier voting model. the pairwise dissimilarity function is built using the procedure described in section 2. the special case is constructed by extending the well-known special case, where the constraint graph is a tree. in social and technical systems, two-tier majority procedures are often used to make decisions. in such procedures, at the first stage, majority voting is carried out in local groups of agents, at the second stage, the results of group voting are aggregated, also using majority, into the final decision [3]. one of the central questions in the analysis of twotier procedures is: “for what minimum proportion of agents that support a proposal can it ultimately be accepted by this procedure?” for a given graph of information connections between the agents, the minimum difference between the number of agents that support an accepted proposal and the number of agents that do not support it is called the strict majority domination number of the graph. we will use the following notation. g = (v,e), |v | = n, is a graph, each vertex of which has a loop. vertex v ∈ v has the set of neighbors nv = {w ∈ v : {v, w} ∈ e}, which includes v. we will write nv(g) when several graphs share the same vertex set. an opinion function f : v → {−1, 1} associates an opinion with each vertex. the opinion function is extended to subsets w ⊆ v as the sum of the opinions of the vertices that belong to w : f(w ) = ∑ v∈w f(v). each vertex v ∈ v votes “for” if f(nv) > 0 or “against” otherwise. a pair (g, f) ≡ gf will be called a configuration. let v + = {v ∈ v : f(nv) > 0} be the set of vertices voting “for”. definition 3.1. given a configuration, a proposal put to a vote is accepted if |v +| > |v |/2 and rejected otherwise; acceptance or rejection are the possible results of the vote. an opinion function for which a proposal is accepted is called a strict majority function for g. definition 3.2. the strict majority domination number of a graph g is γ(g) = minf∈f+f(v ), where f+ is the set of strict majority functions for g. a strict majority function f will be called optimal whenever f(v ) = γ(g). finding the strict majority domination number of a given graph g and the corresponding optimal opinion function is a discrete optimization problem whose solution space is given by 2n possible opinion functions. for the case of an arbitrary graph g, this is an np-complete problem [1], however, there are special cases for which a polynomial algorithm that solves it exists. in particular, if g is a tree, then an optimal opinion function can be constructed in polynomial time using a dynamic programming algorithm [9]. the problems of finding the strict majority domination number on trees act as a simple special case in the application of the pairwise similarity method described below. in accordance with the general scheme of the method given in section 2, we will show that the case of graphs with an upper bounded cyclomatic number is a semisimple special case for copyright © 2023 assa. adv syst sci appl (2023) pairwise similarity estimation for discrete optimization problems 169 the problem of finding the strict majority domination number of a graph, that is, it allows the construction of an approximate solution with a guaranteed absolute error by reduction to the case of a tree. thus, a limited number of independent cycles is the pattern α around which the application of the pairwise similarity method can be developed. in theorem 3.1 below, we prove that for any spanning tree t of graph g having k cycles, γ(g) ≥ γ(t )− 4k holds. it also follows from the proof that knowing the optimal opinion function f of a spanning tree t, one can construct a strict majority function f̃ for the graph g such that f̃(v ) ≤ γ(t ) + 2k, which follows that the error of the approximate solution f̃(v )− γ(g) does not exceed 6k. this theorem allows us to reduce the search for an approximate solution for g to the search for an optimal opinion function for a tree t , which can be done in polynomial time [1], while the guaranteed error of the solution depends only on the number of cycles in the graph, not on its order or size. we first introduce a technical notion and prove an auxiliary result. definition 3.3. let configurations gf and g′f̃ share vertex set v . we say that the voting conditions of v ∈ v in g′f̃ are no worse than in gf if v has at least as many neighbors in g′f̃ with opinion +1 than in gf and no more neighbors in g′f̃ with opinion −1 than in gf . lemma 3.1: let g be a graph with e(g) ̸= ∅ and h be the graph obtained from g by removing some edge uv. then −4 ≤ γ(g)− γ(h) ≤ 2. (3.1) proof suppose f is an optimal opinion function for h; therefore, f(v ) = γ(h). let us bound γ(g) from above and below. to do this, consider the cases where f is not a strict majority function for g along with modifications of f that make the resulting function f̃ a strict majority function for g. then f̃(v ) is an upper bound for γ(g). first consider the cases where vertices u and/or v vote “for” in hf , while without this the proposal would be rejected. in such cases, after adding edge uv, the voting conditions for u and/or v may worsen, which may lead to the violation of the strict majority condition for f in gf . if these vertices vote “against” in hf , then changing the voting conditions for them with the addition of edge uv does not cancel the acceptance of the proposal and preserves the strict majority condition for f in gf . 1. let f(u) = f(v) = 1. in this case, adding a neighbor with opinion +1 will not worsen the voting conditions for either vertex. therefore, f remains a strict majority function on g, but may cease to be optimal. hence γ(g) ≤ f̃(v ) = f(v ) = γ(h). 2. let f(u) = f(v) = −1. then the addition of edge uv worsens the voting conditions for both u and v. • if u and v vote “for” in hf and f(nu(h)) = 1 or f(nv(h)) = 1, then after adding edge uv, the vote of one of the vertices changes to ”against”. as a result, f may cease to be a strict majority function. to obtain a function f̃ having this property, it is sufficient to change the opinion of u or v (let it be u) from −1 to +1. then f̃(nu(g)) = f(nu(h)), while f̃(nv(g)) = f(nv(h)) + 1. thus, after adding edge uv and setting f̃(u) = f(u) + 2, the votes are preserved and so f̃ is a strict majority function for g. hence γ(g) ≤ f̃(v ) = f(v ) + 2 = γ(h) + 2. • if u or v votes “for” and the other vertex votes “against” in hf , then to define a strict majority function f̃ for g, it suffices to set f̃(u) = +1 or f̃(v) = +1. in this case also γ(g) ≤ f̃(v ) = f(v ) + 2 = γ(h) + 2. copyright © 2023 assa. adv syst sci appl (2023) 170 d. lemtyuzhnikova, p. chebotarev, m. goubko, m. somov,n. shushko 3. let f(u) = −1, f(v) = 1. for u, adding edge uv does not worsen the voting conditions. if v votes “against” in hf , then f remains a strict majority function after adding edge uv, since the number of votes “for” does not decrease. if v votes “for”, then obtaining a strict majority function f̃ for g reduces to setting f̃(u) = +1, which implies f̃(v ) = f(v ) + 2. therefore, γ(g) ≤ f̃(v ) = γ(h) + 2. now let f be an optimal strict majority function for g, which implies f(v ) = γ(g). let us bound γ(h) from above by transforming f into a strict majority function f̃ for h . 1. let f(u) = f(v) = −1. removing edge uv does not worsen the voting conditions for u or v. hence f remains a strict majority function for h and γ(h) ≤ f(v ) = γ(g). 2. let f(u) = f(v) = 1. then removing edge uv worsens the voting conditions for both u and v. • if u or v votes “for” in gf (let it be u), while the other vertex votes “against”, then f may cease to be a strict majority function for h . this may happen if f(nu(g)) = 1, in which case f(nu(h)) = 0. to obtain a strict majority function f̃ for h , it suffices to change the opinion of one of the neighbors of u from −1 to 1. this can always be done, since f(nu(h)) = 0 and f(u) = +1 imply the existence of a neighbor of u with opinion −1. therefore, f̃(v ) = f(v ) + 2. • if both u and v vote “for” in gf , then f may cease to be a strict majority function for h if f(nu(g)) = 1 or f(nv(g)) = 1 (or both). then a strict majority function f̃ for h can be obtained by changing the opinion of one neighbor of u or/and one neighbor of v from −1 to +1, as in the previous case. then f̃(v ) ≤ f(v ) + 4. thus, γ(h) ≤ f̃(v ) ≤ γ(g) + 4. 3. let f(u) = 1, f(v) = −1. removing edge uv improves the voting conditions for u and worsens them for v. if v votes “for” in gf , then this may lead to its vote “against” in hf . as a result, f may cease to be a strict majority function for h . to obtain a strict majority function f̃ for h , it suffices to set f̃(v) = +1, which yields f̃(nv(h)) = f(nv(g)) + 1 and f̃(nu(h)) = f(nu(g)) + 1. then γ(h) ≤ f̃(v ) = f(v ) + 2 = γ(g) + 2, since f(v ) = γ(g). collecting the obtained inequalities, we get (3.1). g h fig. 3.1. graph g and graph h obtained from g by removing one edge such that γ(g)− γ(h) = −4, since γ(g) = −7 and γ(h) = −3. inequality (3.1) is sharp. examples of graph g and graph h obtained from g by removing one that satisfy that γ(g)− γ(h) = −4 and γ(g)− γ(h) = 2 are shown in fig. 3.1 and fig. 3.1, respectively. copyright © 2023 assa. adv syst sci appl (2023) pairwise similarity estimation for discrete optimization problems 171 g h fig. 3.2. graph g and graph h obtained from g by removing one edge such that γ(g)− γ(h) = 2, since γ(g) = 1 and γ(h) = −1. theorem 3.1: let a connected graph g have cyclomatic number k, and let t be its arbitrary spanning tree. then γ(t )− 4k ≤ γ(g) ≤ γ(t ) + 2k. proof by the definition of cyclomatic number, tree t is obtained from g by removing k edges. deleting these edges one by one generates a sequence of graphs g :=h0, h1, ..., hk=:t . the proof of the theorem reduces to applying lemma 3.1 to pairs of graphs hi−1, hi, i = 1, ..., k and summing the resulting inequalities (3.1) for them. theorem 3.1 defines a pairwise dissimilarity function, according to which each spanning tree of a graph g with cyclomatic number k is at distance k from g. a spanning tree of g can be found via breadth-first search or depth-first search algorithms, which have linear time complexity. on the other hand, spanning trees refer to a simple special case of the strict majority domination number problem, which allows one to find the strict majority domination number for any of them in polynomial time o(n2) and use the tree’s optimal strict majority function to construct a strict majority function for the graph g, which gives an approximate solution to the original problem. a strict majority function for a graph g with cyclomatic number k can be found as follows: 1. find a spanning tree t of g. determine an optimal strict majority function for t using the polynomial algorithm [9]. 2. sequentially add k edges to t until g is obtained: • add an edge that belongs to e(g)∖e(t ). with the current opinion function, check if |v +| > |v |/2. if this is true, proceed by adding the next edge. • otherwise, change the opinion −1 of a vertex incident to the new edge to +1. in accordance with the general scheme of the pairwise similarity method, as an approximate strict majority domination number of g, the best solution constructed on the basis of the optimal strict majority functions of the spanning trees of g should be taken. in practice, one can limit oneself to constructing several randomly selected spanning trees. 3.2. maximum cut problem in this section, the pairwise dissimilarity function based on the relative error is estimated numerically. for this, an estimate of the error is constructed, based on the reduction of problem instances to a simple special case. we use bipartite graphs as a simple special case for the maximum cut problem. first, we introduce the main definitions related to this problem. copyright © 2023 assa. adv syst sci appl (2023) 172 d. lemtyuzhnikova, p. chebotarev, m. goubko, m. somov,n. shushko definition 3.4. a bipartite graph is a graph whose set of vertices can be divided into two parts in such a way that each edge of the graph connects a vertex of one part with a vertex of the other part. definition 3.5. a cut is a partition of the vertices of a graph into two subsets. any cut determines a cut-set, the set of edges that have one endpoint in each subset of the partition. these edges are said to cross the cut. definition 3.6. in the case of an unweighted graph, the size or weight of a cut is the number of edges crossing the cut; in the case of a weighted graph, the weight of a cut is the sum of the weights of the crossing edges. definition 3.7. a maximum cut of a graph is a cut whose size is at least the size of any other cut (an example is shown in fig. 3.3). the maximum cut problem for unweighted undirected graphs can be formulated as follows. given a connected graph g, find a subset s of its vertices such that the number of edges connecting s with v (g)∖s is maximum. fig. 3.3. maximum cut of a graph this problem is np-hard [5]. consider the maximum cut problems on bipartite graphs (definition 3.4) as a special case. for them, the solution is trivial: the maximum cut includes all edges. it can be easily checked in linear time in the number of edges whether a graph is bipartite, therefore, by definition 2.2, bipartite graphs determine a simple special case for the maximum cut problem. the following simple depth-first search algorithm finds an approximate solution to the maximum cut problem; this solution is exact for bipartite graphs. the error of this algorithm gives rise to a dissimilarity function, which is an upper-bound estimate of the error. the input of algorithm 1 is an arbitrary undirected connected graph g = (v,e). by removing edges, the algorithm selects a bipartite graph gb = (v,eb). it works recursively and paints the vertices in two colors, red and blue for definiteness. first, we color an arbitrary vertex blue and the adjacent vertices a different color (in this case, red). further, the algorithm recursively repeats this procedure on all adjacent vertices that were painted at the current iteration. if the current and an adjacent vertices have the same color, then instead of repainting the latter one, we remove the edge between them. as a result, the algorithm returns a bipartite graph with blue and red vertices, the colors of adjacent vertices of which differ. the complexity of the algorithm is o(|v |2), since the algorithm passes through all the vertices of the graph, and for each vertex, its adjacent vertices are checked. according to the general concept of dissimilarity function (section 2), the dissimilarity based on the relative error between the maximum cut problems ag with graph g and ah copyright © 2023 assa. adv syst sci appl (2023) pairwise similarity estimation for discrete optimization problems 173 algorithm 1 max-cut heuristic algorithm function max-cut heuristics(v) a – adjacency matrix of the connected graph color = [0 for v in v ] # 0 – not colored, −1 – first color, 1 – second color v := random(v) # choose an arbitrary vertex color[v] = 1 # paint first vertex color adjacent vertices(v) return ∑ u,v a[u,v] 2 end function procedure color adjacent vertices(v) for u in v do if a[u, v] = 1 and color[u] = color[v] then a[u, v] := 0 #remove the edge else if a[u, v] = 1 and color[u] = 0 then color[u] := −color[v] #paint the vertex in the opposite color color adjacent vertices(u) end if end for end procedure with a bipartite graph h such that e(h) ⊆ e(g) can be defined as τ(ag, ah) = f(ag)− f(ah) f(ag) , where f(ag) is the optimal value of the objective function for ag. in what follows, for simplicity, we will use f(g) as an abbreviation for f(ag). for the above algorithm, it is not easy to obtain the upper bound for the dissimilarity between ag and ah , where h is the bipartite graph provided by the algorithm, analytically. however, this bound can be approximated by numerical experiments. in such experiments, graphs gi(n, p) are randomly generated with two parameters: the number of vertices n and the density p (the ratio of the number of edges in the graph to the maximum possible number of edges n(n − 1)/2). that is, the probability that there is an edge between any two vertices is p. all edge weights are equal to 1. each point in the figures below is the average over 100 random instances. to estimate the upper bound of the pairwise dissimilarity function, the average relative error and the maximum relative error are used: δav(n, p) = 1 k k∑ i=1 f(gi(n, p))− f̂(gi(n, p)) f(gi(n, p)) ; δmax(n, p) = max i∈{1,...,k} f(gi(n, p))− f̂(gi(n, p)) f(gi(n, p)) , where k is the number of instances with given parameters, f̂(gi(n, p)) is the value of the objective function of the algorithm, and f(gi(n, p)) is the optimal value of the objective function. fig. 3.4 shows the dependence of the average relative error δav(n, p) of the objective function on the number of vertices in a graph with a density p = 0.6. it can be seen that with an increase in the number of vertices, the relative error decreases. copyright © 2023 assa. adv syst sci appl (2023) 174 d. lemtyuzhnikova, p. chebotarev, m. goubko, m. somov,n. shushko 50 100 150 200 250 300 350 400 n 0.050 0.055 0.060 0.065 0.070 0.075 0.080 0.085 av er ag e re la tiv e er ro r fig. 3.4. dependence of the relative error of the objective function on the number of vertices. fig. 3.5 shows the dependence of the average relative error δav(n, p) of the objective function on the density of a graph with a constant number of vertices n = 150. it can be seen that with increasing density the relative error decreases. fig. 3.6 shows the dependence of the average relative error δav(n, p) of the objective function on the density of a graph and on the number of vertices. it can be seen that the density of the graph has a much stronger effect on the relative error than the number of vertices. fig. 3.7 shows the dependence of the maximum relative error δmax(n, p) of the objective function on the density of a graph. the shape of this curve is very similar to that of the average relative error one. fig. 3.8 shows the dependence of the maximum relative error δmax(n, p) of the objective function on the number of vertices. this curve is also similar to the curve of the average relative error, but it is less robust. fig. 3.9 shows the dependence of the maximum relative error δmax(n, p) of the objective function on the density and number of vertices. it is not monotonic in the region of small numbers of vertices, since for such numbers, the solution is very unstable. since we choose a random vertex to start the algorithm, the algorithm works in a nondeterministic way. obviously, it is necessary to remove as few edges as possible from the initial graph, since the number of remaining edges determines the value of the objective function. therefore, in the general case, appropriate heuristics can be used to select the first vertex and study the influence of this choice on the size of the resulting bipartite graph. copyright © 2023 assa. adv syst sci appl (2023) pairwise similarity estimation for discrete optimization problems 175 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 p 0.04 0.06 0.08 0.10 0.12 0.14 av er ag e re la tiv e er ro r fig. 3.5. the dependence of the relative error of the objective function on the density. according to the general scheme of pairwise similarity method, it is suggested to run the heuristic algorithm 1 several times and choose the graph with the least number of removed edges. in this case δav(n, p) can be used as an estimate for the upper bound for the dissimilarity, while δmax(n, p) is more appropriate if the algorithm was run once. 4. conclusion in this paper, a new pairwise similarity method is proposed for measuring the quantitative complexity of np-hard problems. this method, which generalizes the metric approach, allows one to compare problem instances of different dimensions and with different constraint structures. this, in turn, makes it possible to use algorithms developed for some special cases of problems as heuristic algorithms for a wider class of problems. using the pairwise dissimilarity function, one can estimate the error of the obtained solutions. since the pairwise similarity method includes two non-trivial tasks, namely, searching for new special cases and defining the pairwise dissimilarity function, the paper provides recommendations for solving these problems. the proposed method is illustrated for two optimization problems. for the problem of finding the strict majority domination number of a graph with cycles in the framework of a two-level voting model, the pairwise similarity method is applied in the case where it is possible to analytically estimate the absolute error as specified in section 2. for the maximum copyright © 2023 assa. adv syst sci appl (2023) 176 d. lemtyuzhnikova, p. chebotarev, m. goubko, m. somov,n. shushko density 0.2 0.4 0.6 0.8 num be r o f v ert ixe s 50 100 150 200 250 300 350 400 av er ag e re la tiv e er ro r 0.04 0.06 0.08 0.10 0.12 0.14 fig. 3.6. the dependence of the relative error of the objective function on the density and number of vertices. cut problem, the pairwise similarity method is applied in the case where it is not possible to estimate the error analytically and the estimates are based on numerical experiments. in the future, it is planned to apply the pairwise similarity method to other well-known optimization problems on graphs and to study a number of variations of similarity criteria. it is also planned to develop a system of complexity maps for the most important similarity criteria. acknowledgements the research was supported by rsf (project no. 22-71-10131). pavel chebotarev is supported by the european union (erc, generalization, 101039692). the results of sections 1, 2, subsection 3.2 and theorem 3.1 were obtained within the rsf grant. views and opinions expressed are those of the authors only and do not necessarily reflect those of the european union or the european research council executive agency. neither the european union nor the granting authority can be held responsible for them. references [1] i. broere, j. h. hattingh, m. a. henning, and a. a. mcrae. majority domination in graphs. discrete mathematics, 138(1–3):125–135, 1995. copyright © 2023 assa. adv syst sci appl (2023) pairwise similarity estimation for discrete optimization problems 177 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 p 0.04 0.06 0.08 0.10 0.12 0.14 m ax im um re la tiv e er ro r n = 150 fig. 3.7. the dependence of the maximum relative error of the objective function on the density. [2] e. bukueva, i. kudinov, and d. lemtyuzhnikova. analysis of the feasibility to use metric approach for np-hard makespan minimization problem. ifac-papersonline, 55(10):2898–2901, 2022. [3] p. chebotarev and d. peleg. the power of small coalitions under two-tier majority on regular graphs. preprint [math.oc] 2212.03394, arxiv, 2022. https://doi.org/10.48550/arxiv.2212.03394. [4] t. e. cheng, a. lazarev, and d. lemtyuzhnikova. a metric approach for the two-station single-track railway scheduling problem. ifac-papersonline, 55(10):2875–2880, 2022. [5] r. m. karp. reducibility among combinatorial problems. springer, 2010. [6] a. lazarev, d. lemtyuzhnikova, n. pravdivets, and f. werner. polynomially solvable subcases for the approximate solution of multi-machine scheduling problems. in advances in optimization and applications: 11th international conference, optima 2020, moscow, russia, september 28–october 2, 2020, revised selected papers 11, pages 211–223. springer, 2020. [7] a. a. lazarev, d. v. lemtyuzhnikova, and n. a. pravdivets. metric approach for finding approximate solutions of scheduling problems. computational mathematics and mathematical physics, 61:1169–1180, 2021. [8] a. a. lazarev, d. v. lemtyuzhnikova, and f. werner. a metric approach for scheduling problems with minimizing the maximum penalty. applied mathematical modelling, 89:1163–1176, 2021. [9] h.-g. yeh and g. j. chang. algorithmic aspects of majority domination. taiwanese journal of mathematics, pages 343–350, 1997. copyright © 2023 assa. adv syst sci appl (2023) https://doi.org/10.48550/arxiv.2212.03394 178 d. lemtyuzhnikova, p. chebotarev, m. goubko, m. somov,n. shushko 50 100 150 200 250 300 350 400 n 0.055 0.060 0.065 0.070 0.075 0.080 0.085 0.090 m ax im um re la tiv e er ro r p = 0.6 fig. 3.8. the dependence of the maximum relative error of the objective function on the number of vertices. copyright © 2023 assa. adv syst sci appl (2023) pairwise similarity estimation for discrete optimization problems 179 density 0.2 0.4 0.6 0.8 num be r o f v ert ixe s 50 100 150 200 250 300 350 400 m ax im um re la tiv e er ro r 0.04 0.06 0.08 0.10 0.12 0.14 0.16 fig. 3.9. the dependence of the maximum relative error of the objective function on the density and number of vertices. copyright © 2023 assa. adv syst sci appl (2023) introduction pairwise similarity method applications of the pairwise similarity method majority domination problem maximum cut problem conclusion microsoft word 1184 article text, copyedited.doc adv syst sci appl 2021; 04; 115-129 published online at https://ijassa.ipu.ru. original russian text © d.a. novikov, 2021, published in upravlenie bolshimi sistemami / large-scale systems control, 2021, vol. 94, pp. 5–32. optimal schedule to test independent hypotheses dmitry novikov* v. a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia e-mail: novikov@ipu.ru tel.: +74953347569 abstract: the first stage of any creative activity consists in generating a set of hypotheses and testing them. generally, the time, required for testing a hypothesis is random and depends on its complexity (the prior probability of testing per unit time) and on acquired experience, determined by the set of hypotheses, successfully tested before. the problem is to choose an optimal schedule of testing, i.e. minimizing the sum of expected testing times, which are essentially nonlinear past-sequence-dependent and take into account learning and deterioration effects. for this aim, the general model of creative activity is formulated and the corresponding problem of optimal scheduling is stated; the classification of subproblems is introduced. analysis of related works demonstrates the absence of methods to find computationally “simple” solution of the problem in hand. the used method of analytical proof of certain monotonic schedule optimality consists in reordering of two adjacent hypothesis, violating monotonicity. main result is a set (for different subproblems) of sufficient conditions, under which the monotonic “simple-to-complex” schedule is optimal: the hypotheses are arranged in ascending order of their complexity. keywords: schedule theory, past-sequence-dependent scheduling problem, time-dependent processing times, learning and deterioration effects, creative activity, hypotheses testing 1. introduction in the introduction the structure of creative activity is analyzed (subsection 1.1), then a model of the first phase of creative activity is formulated (subsection 1.2) and the problem of optimal scheduling is stated (subsection 1.3). parallelly the motivation is exposed, as well as the main known results (related works) are discussed. 1.1. structure of creative activity activity is a dynamic interaction of a human with the reality in which this human represents a subject (actor) purposefully influencing an object (subject matter) [0]. activity is a form of human actions aimed at cognizing and transforming the surrounding world, humans themselves, and the conditions of their existence. creative activity is an activity that produces an a priori uncertain demand for the results of an a priori unknown activity with a technology created during the new activity. three phases of the life cycle of creative activity (subject matter) were identified in the paper [2]: 1) the discovery of a new subject matter and the accumulation of basic knowledge (the generation of hypotheses and their testing); 2) mastering the subject matter; * corresponding author: novikov@ipu.ru 116 d. novikov copyright ©2021 assa. adv. in systems science and appl. (2021) 3) mass productive use. as was shown therein, creativity is concentrated in goal-setting. for scientific or art activities, it is concentrated in the generation and testing of hypotheses [3, 4]. a reasonable approach to the second phase involves the mathematical models of experience [5]. the third phase can be described using structural and algorithmic models [0] and optimization models [6]. for reflecting the first phase, mathematical models for choosing an optimal schedule to test hypotheses will be suggested below. 1.2. a model of the first phase of creative activity consider a subject mastering k types of activity: a subject acquires k knowledge elements or tests k hypotheses on a single subject matter, choosing an appropriate schedule to do it himself. let any schedule be admissible. having chosen a schedule, the subject begins to test the hypotheses one by one (for the sake of convenience, in ascending order of their numbers.) the hypothesis testing process is not interrupted and runs as follows. by the end of a next discrete-time instant (step), the hypothesis can either have been already tested or not. the process continues until the hypothesis is tested. hypothesis k is characterized by the initial level of mastering (“initial knowledge”) and the prior probability of testing per unit time (one step) this probability conditionally characterizes the complexity of testing. (for example, the complexity of testing can be inversely proportional to the probability of testing and vice versa.) assume that testing of hypothesis k begins at step 0. according to [5], the learning level of hypothesis k (the probability that this hypothesis is tested) has the following form: , . (1) the learning level can be described either by the expected test time of hypothesis k (see below) or the time tk(ε, l(0), wk) when the probability that hypothesis k is still untested will not exceed ε > 0: , . (2) clearly, is a strictly monotonically decreasing and concave function of and a strictly monotonically decreasing and convex function of wk. we denote by the initial knowledge vector and by the vector of the prior probabilities of testing. consider the strictly sequential testing of hypotheses: testing of a next hypothesis begins immediately after the completed testing of the previous one. regardless of the schedule of testing, the total time to master the subject matter sequentially is given by tseq(l(0), w, ε) = . (3) the aggregate learning level has the following form: , (4) where , (see (2)), and . let the initial values of all learning levels be 0. under this assumption, we compare the time (3) with the time of mastering the subject matter by randomly choosing the hypothesis tested at each step; for details, see the models of mastering experience in [5]. in this case, the learning level takes the form (0)kl (0,1).kw î ( )( )1 1 (0) 1( ) t k k klt wl = 1k ,k= ( ) ( ) ln( ) ln ( , (0), 0 ) ) l 1 n ( 1 k k k k kt l l w w e e = 1k ,k= ( , (0), )k k kt l we (0)kl 1(0) = ( (0),..., (0))kl ll 1( ,..., )kw w=w ( ) ( )1 0l 1 ( )n( ) l ln 1 n k k k k l w e = å 0 1 1 1 min{1 ; (1 (1 }( ) ) ( ) (1 ) )seq k t k k k l w i t tt k e e = = ³ å 0 k j j k t t < =å ( ) ( ) ln( ) ln 1 (0) ln 1 j j j l t w e = , 1,j k k= optimal schedule to test independent hypotheses 117 copyright ©2021 assa. adv. in systems science and appl. (2021) . (5) as is known [5], under the same prior probabilities of testing and arbitrary distributions p = , the expected learning level (5) achieves maximum for the uniform distribution. therefore, assume that the “uniform random” testing strategy is used, and the prior probabilities of testing are the same and equal to w. then , and the time when the probability that hypothesis k is still untested will not exceed ε > 0 is . under the assumptions introduced above, the sequential test time (3) of all hypotheses is . obviously, : the sequential testing strategy faster covers the subject matter than the uniform random testing strategy. 1.3. an optimal schedule to test independent hypotheses: problem statement and known results we formulate the problem of an optimal schedule to test hypotheses. let the subject tests the hypotheses in ascending order of their numbers: from 1 to k. for each hypothesis , we define the set of its “predecessors” and the function of sets , where is a continuous function. suppose that the probability of testing depends on the complexity of the hypotheses already tested by the subject. two cases will be considered below: the multiplicative and additive dependencies on the sum of the prior probabilities of testing for the already tested hypotheses. the problem is to find an appropriate schedule to test all hypotheses in the minimum total time. this problem belongs to the class of scheduling theory problems [7, 8] with a single machine, several jobs/tasks, and the deterioration and learning effects: the time to execute a job or set up the machine depends on the previous trajectory (history). in scheduling problems, such setup times are called past-sequence-dependent (p-s-d) or generally timedependent [9]. the papers [10, 11] considered models with the deterioration effect (in terms of this work, the negative effect of the history, see figure 1) in which the job execution time increases linearly with the time of beginning. sufficient conditions were established under which an optimal schedule (by some criterion, e.g., the minimum time for completing all jobs, the minimum weighted delays, etc.) arranges the jobs in descending or ascending order of their characteristics (the minimum initial execution times, the linear dependence coefficients, etc.). in addition to linear constraints, scheduling theory deals with other, quite specific (!) constraints and dependencies (power [12], exponential [13, 14]). we will also consider a particular case of the multiplicative (7), (13) or additive (10), (14) effect of the history on the test times of hypotheses. the publications [15, 16, 17] were first to consider the learning effect in scheduling theory (in terms of this work, the positive effect of the history, see figure 1). within the models suggested therein, the job execution time decreases linearly with the number of preceding jobs. as was shown, the corresponding problem is generally an np-complete problem. particular polynomially solvable cases were presented. a v-shaped schedule turns out to be optimal in several models: the jobs are arranged first in descending order and then in the ( ) 1 ) 1 1( k t k k k k rndl p wt p = = -å { 1 }kp , k ,k= 1 11) 1( tk k k rnd wl t k k= æ ö= -ç ÷ è ø å ln( ) ln 1 rndt w k e = æ ö-ç ÷ è ø ( ) ln( ) ln 1seq kt w e = seq rndt t£ 1k ,k= {1,..., 1}kn k= 1 1 ( ) ( ) k k k k j j n wµ y = = å 1 1: (0 ]k k , w y +â ® 118 d. novikov copyright ©2021 assa. adv. in systems science and appl. (2021) ascending one [18]. clearly, monotonic schedules are special cases of v-shaped ones [19]. the model [20] includes both effects (deterioration and learning) simultaneously. some papers consider more complicated setup, but under certain assumptions: coupled tasks [21, 22], scenario-dependent processing times [23], a set of linearly [24] or quadratic deteriorating jobs, position-and-resource-dependent processing times [25], etc. a more general approach is treating the problem of an optimal schedule to test hypotheses as a complex modification of the traveling salesman problem. the traveling salesman problem [26] with complex precedence constraints and the dependence of times on the sequence of the graph vertices and other graph properties is called the megalopolis visit problem. for this problem, dynamic programming-based solution schemes are constructed and tested in computational experiments [27]. megalopolis visit problems are more general than scheduling problems with deterioration and learning. but as the latter, they don’t give analytical solution to the problem of optimal hypotheses testing schedule even for the simplest cases. the rest content of the paper is organized as follows. brief section 2 introduces a classification of the problems of an optimal schedule to test hypotheses. the main results are presented in section 3 sufficient conditions are established under which the monotonic “simple-to-complex” schedule is optimal. the discussion section contains some possible prospects for applying discrete optimization methods to creative activity modeling. 2. problems of an optimal schedule to test hypotheses: a classification we return to the problem of an optimal schedule to test hypotheses. let us introduce a binary classification system for particular cases of this problem (figure 1). sixteen options are possible: the set of already tested hypotheses affects the initial level of mastering or the probability of testing for a current hypothesis. this effect is positive, increasing the initial level of mastering or the probability of testing (the subject gains experience), or negative, decreasing the characteristics mentioned (the subject “gets tired” or overloads own cognitive capabilities). the dependence is multiplicative or additive (see the discussion above). the optimality criterion is the expected test time t* of a hypothesis or the time tε when the probability that the hypothesis is still untested will not exceed a given threshold ε. optimal schedule to test independent hypotheses 119 copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 1. problems of an optimal schedule to test hypotheses: a binary classification figure 1 shows the numbers of the propositions below corresponding to different particular cases (the corresponding rectangles are shaded). note that propositions 1–6 are proved using the same logic. each of these propositions claims that under certain assumptions, the optimal schedule is a monotonic sequence of the hypotheses by the prior probabilities of testing (from the largest to smallest probability, or vice versa). the proof considers an arbitrary schedule of hypotheses under the assumption that a pair of “adjacent” hypotheses breaks the monotonicity. then the test times in the original and reverse schedule are compared; it is shown that interchanging these two hypotheses will reduce the total test time of the pair. since the test times of the hypotheses with lower numbers do not change, like the initial conditions to test the hypotheses with higher numbers, the interchange will reduce the total test time of all hypotheses. this fact proves the optimality of their corresponding monotonic schedule. the main results are presented in the next section. 3. main results: optimal schedules in the multiplicative case, the learning level (unlike (1)) depends on the schedule of testing as follows: , . (6) then ts(l(0), w, ε) = . (7) the corresponding optimization problem is to minimize the time (7) of mastering the subject matter by choosing an appropriate schedule to test the hypotheses. generally speaking, this is ( )( )1 1 (0) (( ) 1 ) tk k kk kt l wl nµ= 1k ,k= ( ) ( )1 ln( ) ln 1 (0) ln 1 ( ) k k k k k k l w n e µ= -å 120 d. novikov copyright ©2021 assa. adv. in systems science and appl. (2021) a combinatorial problem: there exist k! admissible schedules. fortunately, within definite assumptions, a simple analytical solution can be found in some cases. let . we introduce the following assumption. a.1. the initial values of all learning levels are the same: . for all hypotheses, the previous experience has the same effect on the probabilities of testing, described by = , , where is a smooth and strictly monotonically increasing function. consider two auxiliary lemmas. lemma 1: if is a continuous, strictly monotonically increasing and concave function, then for any positive numbers x, y, and z such that y < z, we have . lemma 2: if is a smooth, strictly monotonically increasing and convex function, then: a) for any positive numbers x, y, and z such that q(x) < y < z, we have . b) for any positive numbers x, y, and z such that y < z < q(x), we have , where q(x) is the solution of the differential equation . (8) the proofs of all statements are given in the appendix. the assertions of lemma 2 are immediate from the properties of smooth and convex functions. proposition 1: under assumption a.1, let be a smooth, strictly monotonically increasing and convex function, and let j be the number of a hypothesis such that . then interchanging hypotheses j and j + 1 will reduce the total time (7) of mastering the subject matter sequentially. quite expectedly, lemma 1 can be used to establish a result similar to proposition 1 for concave monotonic functions . now consider the additive case in which the learning level (unlike (1)) depends on the schedule of testing as follows: , . (9) then ts(l(0), w, ε) = . (10) 1 1 j j l l wh = =å 0(0) 1jl l e= < ( )jy × ( )y × 1j ,k= 1 1: (0, min ] k kw y +â ® 1 1:y + +â ®â ( ) ( )x z x y z y y y+ + < 1 1:y + +â ®â ( ) ( )x z x y z y y y+ + > ( ) ( )x z x y z y y y+ + < ( ) ( ) x q x q d s q ds y y + + = ( )y × 1( )j j jq w wh +< < ( )y × ( )( )1 1 (0) (( ) 1 ) tkk k k kl t l w nµ= 1k ,k= ( ) ( )1 ln( ) ln 1 (0) ln 1 ( ) k k k k k k l w n e µ= -å optimal schedule to test independent hypotheses 121 copyright ©2021 assa. adv. in systems science and appl. (2021) we introduce another assumption. a.2. the initial values of all learning levels are the same: , where l0 can be interpreted as a characteristic of the subject matter. for all hypotheses, the previous experience has the same effect on the probabilities of testing, described by = , , where is a continuous and strictly monotonically increasing function satisfying (11) for any positive numbers x, y, and z such that y < z. some examples of the functions obeying assumption а.2 are: the linear function ; the quadratic function ; any strictly monotonically increasing and strictly convex function such that the derivative . proposition 2: under assumption a.2, let j be the number of a hypothesis such that . then interchanging hypotheses j and j + 1 will reduce the total time (10) of mastering the subject matter sequentially. corollary 1: consider the model (9), (10) of mastering the subject matter sequentially under assumption a.2. in terms of the minimum total time of mastering, the optimal schedule arranges the hypotheses in descending order of the prior probabilities of testing (in ascending order of their “complexities”). in the models described above, an optimality criterion is the time to reach the learning level 1 – ε. an alternative approach involves the expected test time of hypothesis k: τk(lk(0), wk) = (1 – lk(0)) / wk, . (12) clearly, τk(lk(0), wk) is a linear decreasing function of and a strictly monotonically decreasing and convex function of wk. in the multiplicative case, the total expected time of mastering the subject matter sequentially is given by τseq(l(0), w) = . (13) we introduce another assumption similar to assumption a.1: а.3. for all hypotheses, the previous experience has the same effect on the probabilities of testing, described by = , , where is a smooth and strictly monotonically increasing function. the following analog of propositions 1 and 2 holds. 0(0) 1jl l e= < ( )jy × ( )y × 1j ,k= 1: [0 1 min ]kk , wy +â ® ( ) ( )x z x y z yy y+ + ³ ( ) (0,1]ss ,y a a = î 2( )s sy = ( )y × min ( ) ( )| 1 kk y w d y y dy y y = > 1j jw w +< 1k ,k= (0)kl 1 1 (0) ( ) k k k k k k l w nµ= -å ( )jy × ( )y × 1j ,k= 1 1: (0, min ] k kw y +â ® 122 d. novikov copyright ©2021 assa. adv. in systems science and appl. (2021) proposition 3: under assumption a.3, let j be the number of a hypothesis such that and . then interchanging hypotheses j and j + 1 will reduce the total time (13) of mastering the subject matter sequentially. corollary 2: consider the model (6), (13) of mastering the subject matter sequentially under assumption a.3. in terms of the minimum total time of mastering, the optimal schedule arranges the hypotheses in descending order of the prior probabilities of testing (in ascending order of their “complexities”) if it coincides with the ordering of the prior probabilities of testing normalized by 1 minus the initial learning level. in the additive case, the total expected time of mastering the subject matter sequentially is given by τseq(l(0), w) = . (14) the following analog of proposition 3 holds. proposition 4: under assumption a.2, let j be the number of a hypothesis such that . then interchanging hypotheses j and j + 1 will reduce the total time (14) of mastering the subject matter sequentially. corollary 3: consider the model (6), (14) of mastering the subject matter sequentially under assumption a.2. in terms of the minimum total time of mastering, the optimal schedule arranges the hypotheses in descending order of the prior probabilities of testing (in ascending order of their “complexities”). now we study the case of a continuous and strictly monotonically decreasing function . the following counterpart of proposition 4 holds. proposition 5: let the initial values of all learning levels be the same: . for all hypotheses, let the previous experience have the same effect on the probabilities of testing, described by = , , where is a continuous and strictly monotonically decreasing function. also, let j be the number of a hypothesis such that . then interchanging hypotheses j and j + 1 will reduce the total time (13) of mastering the subject matter sequentially. corollary 4: consider the multiplicative case with a continuous and strictly monotonically decreasing function . in terms of the minimum total time of mastering the subject matter sequentially, the optimal schedule arranges the hypotheses in ascending order of the prior probabilities of testing (in descending order of their “complexities”). we finally examine the case in which the initial learning level depends on the schedule to test the hypotheses: 1 11 (0) 1 (0) j j j j w w l l + + < 1j jw w +< 1 1 (0) ( ) k k k k k k l w nµ= +å 1j jw w +< ( )y × 0(0)jl l= ( )jy × ( )y × 1j ,k= 1: (0,1 min ]kk wy +â ® 1j jw w +> ( )y × optimal schedule to test independent hypotheses 123 copyright ©2021 assa. adv. in systems science and appl. (2021) τk( , wk) = (1 – ) / wk, , (15) where is a continuous and strictly monotonically increasing function. the total expected time of mastering the subject matter sequentially is given by τseq(w) = . (16) we introduce the following assumption. а.4. for all hypotheses, let the previous experience have the same effect on the initial value of the learning level, described by = , , where is a continuous and strictly monotonically increasing function. in the case under consideration, the following analog of propositions 1–4 holds. proposition 6: under assumption a.4, let j be the number of a hypothesis such that . then interchanging hypotheses j and j + 1 will reduce the total time (16) of mastering the subject matter sequentially. corollary 5: consider the model (6), (15) of mastering the subject matter sequentially under assumption a.4. in terms of the minimum total time of mastering, the optimal schedule arranges the hypotheses in descending order of the prior probabilities of testing (in ascending order of their “complexities”). 4. conclusion: discrete optimization problems in creative activity modeling thus, in terms of the expected total time to test the independent hypotheses or the total time to reach a given learning level, the optimal schedule in several cases is a simple rule: the hypotheses are arranged in descending order of the prior probabilities of testing (the monotonic “simple-to-complex” schedule). now we present general sufficient conditions for the optimality of the “simple-to-complex” schedule. let be the test time of hypothesis , where the function is continuous and decreasing in both variables, and for any x, y, and z such that t(x, y) + t(x + y, z) > t(x, z) + t(x + z, y). as is easily checked in the case , interchanging hypotheses j and j + 1 will reduce the total time of their testing. attempts to weaken these sufficient conditions and give them in a practically interpretable form seem a promising line of further research. another area of interest is studying optimal schedules to test interdependent hypotheses. graph theory and discrete optimization offer a rich apparatus to formulate and solve optimization problems for hypothesis testing and creative activity modeling. here are some illustrative examples. let the subject area be described by a graph containing k vertices. graph vertices correspond to hypotheses to be tested. hypothesis k is characterized by a pair of nonnegative numbers (ck, dk): the first number reflects the cost to test it, whereas the second one reflects its contribution to the experience of mastering the subject area. assume that the experience is additive: under the sequential testing of all hypotheses in ascending order of their numbers, ( )kz h ( )kz h 1k ,k= 1: [0,1)z +â ® 1 1 ( )k k k kw z h = -å ( )jz × ( )z × 1j ,k= 1: [0 1),z +â ® 1j jw w +< ( )k k kt t ,wh= 1k ,k= ( , )t × × 1j jw w +< 124 d. novikov copyright ©2021 assa. adv. in systems science and appl. (2021) when the subject starts testing hypothesis k, the experience is , and the total cost of the already tested hypotheses is . we define the complexity of each hypothesis as . (the complexity depends on the schedule to test all hypotheses.) note that with such a definition, the complexity can be negative. we define the complexity of each schedule to test all hypotheses (each hamilton circuit ρ = (i1, i2, …, ik, …, ik) in the graph) as the maximum value among the complexities to test different hypotheses in this schedule: . the problem is to find a schedule to test all hypotheses (a hamilton circuit) of minimum complexity: . under the assumptions introduced above, this combinatorial problem has a simple analytical solution (see general results in [0]): – take the hypotheses satisfying , arrange them in ascending order of the cost сk, and include them in the schedule (hamilton circuit). – add the other hypotheses ( ) to this sequence in descending order of the contribution dk. in other words, we should first test the hypotheses whose cost is less than the contribution to the experience, arranging them from simple to complex. then we should proceed to the hypotheses whose contribution to the experience is smaller than the cost, arranging them in descending order of the contribution. of course, the assumptions about the “additive” complexity of the experience are strong enough. on the other hand, these assumptions allow obtaining a simple analytical solution with a practical interpretation. generally speaking, discrete optimization and graph theory provide a wide field for different problems of optimal hypothesis testing (author is grateful to prof. vladimir burkov for some ideas and discussion); for example, the following formulations are possible based on the well-known results [7, 0]: • the knapsack problem with the synergistic effect: adding two items to the knapsack (including two topics in the research plan, or deciding to test two hypotheses) reduces their total weight (the total time to deal with them). it is required to develop a research plan that maximizes the accumulated experience (the amount of scientific knowledge) under cost constraints (labor-intensiveness). • the editor problem: for example, two researchers test a given set of hypotheses (a theoretician and a programmer). the theoretician develops a testing method, whereas the programmer develops testing software. after that, the theoretician conducts computational experiments based on this software. it is required to determine a schedule to test hypotheses that minimizes the total test time. • the assignment problem: there are n hypotheses (scientific problems) and m researchers, where . the competencies of all researchers to solve these problems (to test the corresponding hypotheses) are known. it is required to assign one researcher to each hypothesis so that the total (or maximum) test time for all hypotheses achieves minimum. (the test time depends on the researcher’s competence.) 1 1 k k j j d d = =å 1 1 k k j j с с = =å k k k kc c dg = + 1 ( ) max{ } kik ,k g r g = = ( ) min r g r ® k kd c³ k kd c< m n³ optimal schedule to test independent hypotheses 125 copyright ©2021 assa. adv. in systems science and appl. (2021) appendix proof of lemma 1. assume on the contrary that . since the function is concave, . hence, , which finally implies due to the strict monotonicity of the function . this contradiction completes the proof. proof of proposition 1. first, note that interchanging hypotheses j and j + 1 will not modify the sets nl for all l from 1 to j – 1 and from j + 2 to k. therefore, it will not change the test times of all hypotheses except j and j + 1. this fact is essential for the other propositions. we compare the total test times of hypotheses j and j + 1: and . the first sum is equal to tj, j+1 = . the second sum is equal to tj+1, j = . assume on the contrary that tj+1, j ≥ tj, j+1. in view of , we obtain ≤ . we rewrite this inequality as ≤ . since , the left-hand side is strictly positive. hence, the right-hand side has the same property: > 0. as the denominator is strictly positive, the numerator is such as well: > 1. therefore, , which contradicts item a) of lemma 2. proof of proposition 2. we compare the total test times of hypotheses j and j + 1: and . ( ) ( )x y x z y z y y+ + £ ( )y × ( ) ( ) ( ( ) ( )) /x y x x z x y z y y y y y y+ + + ³ ( ) ( ) ( ) ( )x x z x x z y z z z y y y y+ + + £ z y£ ( )y × 1 1( , ( )) + ( , ( ))j j j j j j jt w t w we y h e y h+ + + 1 1 1( , ( )) + ( , ( ))j j j j j j jt w t w we y h e y h+ + ++ ( ) ( )0 1 1 1ln( ) 1 ln 1 ( ) ln 1 ( )j j j j jl w w w e y h y h+ é ù +ê ú +ê úë û ( ) ( )0 1 1 1 1ln( ) 1 ln 1 ( ) ln 1 ( )j j j j jl w w w e y h y h+ + é ù +ê ú +ê úë û 0ln( ) 0 1 l e < ( ) ( )1 1 1 1 ln 1 ( ) ln 1 ( )j j j j jw w wy h y h+ + + + ( ) ( )1 1 1 ln 1 ( ) ln 1 ( )j j j j jw w wy h y h+ + + ( ) ( )1 1 1 ln 1 ( ) ln 1 ( )j j j jw wy h y h+ ( ) ( )1 1 1 1 ln 1 ( ) ln 1 ( )j j j j j jw w w wy h y h+ + + + 1j jw w +< ( ) ( ) 1 1 1 1 1 ( ) ln 1 ( ) ln 1 ( ) ln 1 ( ) j j j j j j j j j j j j w w w w w w w w y h y h y h y h + + + + æ ö+ ç ÷ç ÷+è ø + + 1 1 1 ( ) 1 ( ) j j j j j j w w w w y h y h + + + + 1 1 ( ) ( )j j j j j j w w w w y h y h+ + + + < 1 1( , ( )) + ( , ( ))j j j j j j jt w t w we y h e y h+ + + 1 1 1( , ( )) + ( , ( ))j j j j j j jt w t w we y h e y h+ + ++ 126 d. novikov copyright ©2021 assa. adv. in systems science and appl. (2021) the first sum is equal to tj, j+1 = . the second sum is equal to tj+1, j = . assume on the contrary that tj+1, j ≥ tj, j+1. in view of , we obtain ≤ . we rewrite this inequality as ≤ . since , the left-hand side is strictly positive. hence, the right-hand side has the same property: > 0. as the denominator is strictly positive, the numerator is such as well: > 1. therefore, , which contradicts condition (11) of assumption a.2. proof of proposition 3. let . from the condition of this proposition it follows that . we compare the total expected test times of hypotheses j and j + 1: and . the first sum is equal to tj, j+1 = + . the second sum is equal to tj+1, j = + . assume on the contrary that tj+1, j ≥ tj, j+1. then tj+1, j tj, j+1 ≥ 0, and ( ) ( )0 1 1 1ln( ) 1 ln 1 ( ) ln 1 ( )j j j j jl w w w e y h y h+ é ù +ê ú +ê úë û ( ) ( )0 1 1 1 1ln( ) 1 ln 1 ( ) ln 1 ( )j j j j jl w w w e y h y h+ + é ù +ê ú +ê úë û 0ln( ) 0 1 l e < ( ) ( )1 1 1 1 ln 1 ( ) ln 1 ( )j j j j jw w wy h y h+ + + + ( ) ( )1 1 1 ln 1 ( ) ln 1 ( )j j j j jw w wy h y h+ + + ( ) ( )1 1 1 ln 1 ( ) ln 1 ( )j j j jw wy h y h+ ( ) ( )1 1 1 1 ln 1 ( ) ln 1 ( )j j j j j jw w w wy h y h+ + + + 1j jw w +< ( ) ( ) 1 1 1 1 1 ( ) ln 1 ( ) ln 1 ( ) ln 1 ( ) j j j j j j j j j j j j w w w w w w w w y h y h y h y h + + + + æ ö+ ç ÷ç ÷+è ø + + 1 1 1 ( ) 1 ( ) j j j j j j w w w w y h y h + + + + 1 1( ) ( )j j j j j jw w w wy h y h+ ++ + < 0 1 (0) j' j j w w l = > 1 ' ' j jw w +< 1 1( ( )) + ( ( ))j j j j j j jw w wt y h t y h+ + + 1 1 1( ( )) + ( ( ))j j j j j j jw w wt y h t y h+ + ++ 1 ( )' j jw y h 1 1 ( )' j j jw wy h+ + 1 1 ( )' j jw y h+ 1 1 ( )' j j jw wy h ++ optimal schedule to test independent hypotheses 127 copyright ©2021 assa. adv. in systems science and appl. (2021) + ≥ 0. due to assumption a.3 and , we can estimate the denominator of the second term as: + > 0, > 0, > 0. since , the numerator of the first factor is strictly negative whereas the denominator is strictly positive. by assumption а.3, the numerator and denominator of the second factor are strictly positive. this contradiction completes the proof. proof of proposition 4. we compare the total expected test times of hypotheses j and j + 1: and . the first sum is equal to tj, j+1 = + . the second sum is equal to tj+1, j = + . assume on the contrary that tj+1, j ≥ tj, j+1. then tj+1, j tj, j+1 ≥ 0, and + ≥ 0. trivial transformations, particularly cancelation by the strictly positive value , yield ≥ 0. the denominators of both fractions are strictly positive as the products of strictly positive values. since , the numerator of the first fraction is strictly negative. hence, this fraction has the same property as well. the numerator of the second fraction is nonnegative due to condition (11) of assumption a.2. therefore, the entire difference of these strictly negative and nonnegative values is strictly negative. this contradiction completes the proof. proof of proposition 5. we compare the total expected test times of hypotheses j and j + 1: and . the first sum is equal to tj, j+1 = + , the second sum is equal to tj+1, j = + . 1 1 ( )' j jw y h+ 1 1 ( )' j j jw wy h ++ 1 ( )' j jw y h 1 1 ( )' j j jw wy h+ + 1j jw w +< 1 1 1 1 ( )' ' j j jw w y h+ æ ö -ç ÷ç ÷ è ø 1 1 1 1 ( )' ' j j j jw w wy h+ æ ö -ç ÷ç ÷ +è ø 1 1 1 1 1 ( ) ( )' ' j j j j jw w wy h y h+ æ öæ ö -ç ÷ç ÷ç ÷ç ÷+è øè ø 1 1 ( ) ( ) ( ) ( ) ' ' j j j j j ' ' j j j j j w w w w w w y h y h y h y h + + æ öæ ö+ ç ÷ç ÷ç ÷ç ÷+è øè ø 1j jw w +< 1 1( ( )) + ( ( ))j j j j j j jw w wt y h t y h+ + + 1 1 1( ( )) + ( ( ))j j j j j j jw w wt y h t y h+ + ++ 01 ( )j j l w y h + 0 1 1 ( )j j j l w wy h+ + + 0 1 1 ( )j j l w y h+ + 0 1 1 ( )j j j l w wy h + + + 0 1 1 ( )j j l w y h+ + 01 ( )j j l w y h + 0 1 1 ( )j j j l w wy h + + + 0 1 1 ( )j j j l w wy h+ + + 01 l1 1( ( ))( ( )) j j j j j j w w w wy h y h + + + + 1 1 1 1 ( ) ( ) ( ( ))( ( )) j j j j j j j j j j j j w w w w w w w w y h y h y h y h + + + + + + + + + + + 1j jw w +< 1 1( ( )) + ( ( ))j j j j j j jw w wt y h t y h+ + + 1 1 1( ( )) + ( ( ))j j j j j j jw w wt y h t y h+ + ++ 01 ( )j j l w y h 0 1 1 ( )j j j l w wy h+ + 0 1 1 ( )j j l w y h+ 0 1 1 ( )j j j l w wy h + + 128 d. novikov copyright ©2021 assa. adv. in systems science and appl. (2021) assume on the contrary that tj+1, j ≥ tj, j+1. then tj+1, j tj, j+1 ≥ 0, and + ≥ 0. since and the function is decreasing, we cancel by the strictly positive value and estimate the denominator of the second term as: > 0, > 0, > 0. since , the numerator and denominator of the first factor are strictly positive. since and the function is strictly decreasing, the numerator of the second factor is strictly negative whereas the denominator of the second factor is strictly positive. this contradiction completes the proof. proof of proposition 6. we compare the total expected test times of hypotheses j and j + 1: and . the first sum is equal to tj, j+1 = + , the second sum is equal to tj+1, j = + . assume on the contrary that tj+1, j ≥ tj, j+1. then tj+1, j tj, j+1 ≥ 0, and + ≥ 0. + ≥ 0. since and the function is strictly monotonic, we have the estimate + < < 0. this contradiction completes the proof. references 1. belov, m.; novikov, d. 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(2013) mechanism design and management: mathematical methods for smart organizations; nova science publishers: new york. adv syst sci appl 2022; 04:176–192 published online at https://ijassa.ipu.ru. optimal control techniques for the transmission risk of nipah virus disease with awareness naba kumar goswami1*, fidele hategekimana2 1departmant of mathematics, pet research foundation, pes college of engineering, karnataka, india 2university of central africa, ngoma campus, karongi, rwanda abstract: nipah virus is one of the life-threatening infectious diseases in south-east asian regions. in this study, we developed a compartmental model of nipah virus transmission using an ordinary differential equation. we find the disease-free equilibrium and compute basic reproduction number (r0). the sensitivity analysis of the parameters of the basic reproduction number of the model is studied and identifies the most sensitive parameters which can control the transmission dynamics of the nipah virus. the model is extended to the optimal control problem and is analyzed by using pontryagin’s maximum principle. further, we analyze the cost-effective and three different time-dependent control strategies to minimize the number of infectives in during that period of time. finally, compare the results of the optimal control models using numerical simulation. keywords: nipah virus, basic reproduction number, sensitivity analysis, optimal control problem, pontryagin’s maximum principle. 1. introduction nipah virus disease is a zoonotic infection caused by nipah virus, a paramyxovirus belonging to the genus henipavirus of the family paramyxoviridae. nipah virus is one of the deadliest infectious diseases in south-east asian regions since end of the century. it is transmitted to humans from infected bats, infected pigs, or contaminated fruits. also it can be transmitted directly from human to human contact [1, 6]. the transmission of disease can be from bats to human, human to human and contaminated fruits to human. the symptoms of the nipah virus are fever, headache, dizziness, muscle pain, vomiting, drowsiness, and sore throat [5]. there is no medicine and vaccine available for either human or animals. the primary treatment for humans is supportive care. according who the case fatality rate is estimated at 40 % to 75 %. this rate can very by outbreak depending on local capabilities for epidemiological surveillance and clinical management. in 1998, the first of nipah virus case was recorded during malaysian outbreak. in 1999, nipah virus was detected from an infected human of sungai nipah village in malaysia [4]. in 1999, first largest nipah virus outbreak was identified in malaysia and singapore among pig farmers. in malaysia and singapore outbreaks 276 cases were reported [1, 21] between 1998 to 1999. in that period most of the infected people had contact with sick pigs [16], and death rate infected people were 39% [7]. in india and bangladesh, the evidence of nipah virus transmitted through raw date palm sap contaminated with infectious bat excretions. in 2001, bangladesh nipah outbreaks a total of 321 infection cases were reported [1, 16], and 161 of them deaths, accounting for a 70 percentage fatality rate. in india, the first nipah outbreak was reported ∗corresponding author: nabakrgoswami@gmail.com optimal control techniques of nipah virus disease 177 in siliguri town in 2001, followed by the second outbreak in nadia district in west bengal state in 2007. in 2018, a nipah outbreak in india and bangaladesh human to human [9, 17], transmission is regularly reported according to cdc, but in malaysia and singapore there is no evidence of human to human transmission. in 2018, a nipah virus outbreak was reported in kozhikode district and in 2019, another outbreak reported in kochi district in kerala. during this outbreak 17 nipah infected people were dead. in 2021 also few nipah virus suspected cases reported in kozhikode district of kerala. a 12-years old boy were infected and later the boy was death at a private hospital. the high risk of nipah virus infected countries other than above mention include australia, bhutan, brunei, cambodia, china, indonesia, laos, madagascar, myanmar, nepal, philippines, papua new guinea, taiwan, thailand, and vietnam. many nipah virus mathematical models are proposed and analysis in the literature to study the various aspects of transmission dynamics of disease. the study of biswas [1] shows that the control strategy of deadly niv in bangladesh. chua et. al. [6] studied isolation of nipah virus from malaysian island flying-fox. chua et. al. [5] studied emergent of deadly paramyxovitus of nipah virus. chua [4] discussed about nipah virus outbreak in malaysia. mondal et. al. [7], studied controlling the spread of nipah virus in bangladesh through a mathematical model. parashar et. al. [16], shows that the risk factors for human infection with a new zoonotic paramyxovirus and nipah virus during the malaysia outbreak in 1998-1999. goh et. al. [9], shows that clinical features of nipah virus encephalitis among pig farmers in malaysia. goswami [3], studied stability analysis of a mathematical model and discussed sensitivity analysis of basic reproduction number. paton et.al. [17], shows that the outbreak of niv infection among abattoir works in singapore. uppal [21], studied about emergence of malaysia nipha virus outbreak. this paper is organized as follows: section 2, formulates the model, analysis of the model includes positive invariant, boundedness. section 3, compute the basic reproduction number using next-generation matrix method. section 4, discuss sensitivity analysis of basic reproduction number and identify most sensitive parameter. section 5, deals with the optimal control techniques using pontryagin’s maximum principal. section 6, presents the numerical results and discussion. finally we concludes our the paper in section 7. 2. mathematical model a non-linear mathematical model is formulated for nipah virus by considering the total population variable and the standard incident type interaction for transmission of the disease. we incorporate the awareness parameter in the model as media has a positive effect on controlling the transmission of any type of infectious disease. an extensive study suggests that any awareness program related to the disease can alert susceptible individuals and will not contact with infective. also, due to the awareness program some infectives can take medical care and recover from the disease. this proposed study is an extension work of our previous mathematical model in [12], the mainly focuses of this study is on finding the optimal control strategies to reduce the nipah virus transmission. the model is divided into seven mutually exclusive compartments viz., susceptible humans (sh), exposed humans (eh), infected humans (ih), recovered humans (rh), susceptible fruits (sf ), infected fruits (if ), and infected bats (ib). the model is a non-linear standard action type and considering logistic growth bib ( 1− ib k ) on the infected bat’s population. here, the incident rate β1ih nh is the average number contacts with infectives per unit time of one susceptible [20]. although,( β1ih nh ) sh is the number of new cases per unit time among the susceptible. nipah virus copyright © 2022 assa. adv syst sci appl (2022) 178 n.k. goswami, f. hategekimana table 2.1. description of biological parameters parameter description value λh rate of recruitment of human population 15 λf rate of recruitment of fruits 20 β1 usual transmission probability between sh and ih 0.0541 β2 transmission probability between sh and if 0.0521 β3 transmission probability between ih and sf 0.0631 δh contact rate between eh and ih 0.53 γh recovery rate of human population 0.009 µh natural mortality rate of human population 0.19 µ1 natural mortality rate of human population due to infection 0.09 µf natural mortality rate of fruits 0.095 µb natural mortality rate of bats 0.0123 b growth rate coefficient of infected bats 0.005 k carrying capacity of infected bats 100 500 ah recovery rate of exposed humans due to effect of awareness 0.95 has been studied by many authors, most of the models are mass action type model but here we have formulated a standard incident type model. we have consider standard incidence type interaction between between human and fruits, which is more reasonable than mass action type model. this type of consideration is more suitable for many disease like nipah virus, zika virus, dengue fever etc. it is well known that nipah virus disease is one of the deadliest endemic, so long term study of the dynamics of disease is most significant. the biological interpretations of parameters and values of the parameters are shown in table 1. the mathematical model of nipah described as follows: s ′ h = λh − β1sh ih nh − β2sh if nh − µhsh e ′ h = β1sh ih nh + β2sh if nh − (δh + ah + µh)eh i ′h = δheh − (γh + µh + µ1)ih r′ h = aheh + γhih − µhrh (2.1) s ′ f = λf − β3sf ib nh − µfsf i ′f = β3sf ib nh − µfif i ′b = bib ( 1− ib k ) − µbib 2.1. positivity and boundedness of the system here, we shall show that the positivity and boundedness of the population. from the system (2.1), we have s ′ h ∣∣∣ sh=0 = λh > 0, e ′ h ∣∣∣ eh=0 = β1sh ih nh + β2sh if nh ≥ 0, i ′h ∣∣∣ ih=0 = δheh ≥ 0, (r′ h)rh=0 = aheh + γhih ≥ 0, copyright © 2022 assa. adv syst sci appl (2022) optimal control techniques of nipah virus disease 179 s ′ f ∣∣∣ sf=0 = λf > 0, i ′f ∣∣∣ if=0 = β3sf ib nh ≥ 0, i ′b ∣∣∣ ib=0 = 0 here, all the rates are non-negative, so if we start in the interior of the non-negative bounding r6, we shall always remain in this cone keeping mind of the fact that direction of the vector field is inward on all the bounding planes. we note the change rate of the total population nh = sh + eh + ih +rh and nf = sf + if are given by the following differential equations. n ′ h = λh − µhnh − µ1ih n ′ f = λf − µfnf this gives lim sup t→∞ nh ≤ λh µh , lim sup t→∞ nf ≤ λf µf . therefore, all sh(t), eh(t), ih(t), rh(t) are bounded by λh µh and the solutions sf (t), if (t) are bounded by λf µf . hence, the biological feasible region of the proposed system (2.1) is given by the following positively invariant region: ω = (sh, eh, ih, rh, sf , if , ib) ∈ r7 + : (sh + eh + ih +rh) ≤ λh µh , (sf + if ) ≤ λf µf 3. basic reproduction number r0 the disease-free equilibrium of the proposed model as e0=(n0 h , e 0 h, i 0 h, r 0 h, n 0 f , i 0 f , i 0 b ) =( λh µh , 0, 0, 0, λf µf , 0, 0 ) . to find the basic reproduction number r0, using the next-generation matrix method as described in [2, 8, 12, 13]. the matrix f and v as follows: f =  β1(nh − eh − ih −rh) ( ih + if nh ) 0 β3(nf − if ) ib nh 0  and v =  (δh + ah + µh)eh −δheh + (γh + µh + µ1)ih µfif −bib ( 1− ib k ) − µbib  jacobian of f and v at e0 as follows f=  0 β1 β1 0 0 0 0 0 0 0 0 β3 µhλf µfλh 0 0 0 0  and v =  δh + µh + ah 0 0 0 −δh γh + µh + µ1 0 0 0 0 µf 0 0 0 0 −b+ µb  the largest eigenvalue of fv −1 is called the basic reproduction number r0 and is obtained as follows: r0 = β1δhµfλhc3 − β3µhλfc1c2 2c1c2c3µfλh + √( β1δhµfλhc3 − β3µhλfc1c2 2c1c2c3µfλh )2 + β1β3δhµhλf µfλhc1c2k4 (3.2) where, c1 = δh + µh + ah; c2 = γh + µh + µ1, c3 = µb − b, copyright © 2022 assa. adv syst sci appl (2022) 180 n.k. goswami, f. hategekimana 4. sensitivity analysis of reproduction number in this section, we perform sensitivity analysis for the parameters involved in reproduction number r0, which reflects that increase or decrease in these parameter causes increase or decrease in r0 [3, 19]. the sensitivity of r0 to different parameters is shown in figs. 4.1 to 4.7. it is used to discover the parameters that have a high impact on r0 and should be targeted by intervention strategies. sensitivity indices allows to measure the relative change in a variable when parameter changes. for that we use the forward sensitivity index of a variable, with respect to a given parameter, which is defined as the ratio of the relative change in the variable to the relative change in the parameter. if such variable is differentiable with respect to the parameter, then the sensitivity index is defined using partial derivatives. the normalized forward sensitivity index of r0, which is differentiable with respect to a given parameter p , is defined by y r0 p = p r0 ∂r0 ∂p (4.3) the above formula can be used to compute the analytical expression for the sensitivity of r0 to each parameter that it includes. accordingly, the sensitivity indexes of the model (2.1) are illustrate in fig. 4.1. consequently, the value of r0 increases with increase in the values of all positive indices parameters β1, β3, δh, µf , µb, and λf with r0. also, the parameters ah, γh, µh, µ1, b, and λh have negative index with r0. it is clearly observed that the effect of the parameter λh is the maximum and hence it is the most sensitive parameter of r0. it means that small change (increase or decrease) in the parameters λh will significant change in the value of r0. other sensitive parameter r0 are β1, δh, µh as small change in these parameters can cause large change in the value of r0. so correct estimation of these parameters is very important to predict transmission of this disease. it is obvious that phenomenon of a lower value of r0 will boost to prevent the disease prevalence. thus, to control the disease from the population, we have to control the increase of parameters having positive indices with r0, whereas parameters with negative indices should be maintained. 1 3 h a h h h 1 f b b h f parameters -1.5 -1 -0.5 0 0.5 1 s en si tiv ity in di ce s fig. 4.1. forward sensitivity of r0 for the parameters as β1 = 0.0541; β3 = 0.0631; δh = 0.53; ah = 0.95; γh = 0.009; µh = 0.19; µf = 0.095; µ1 = 0.09; µb = 0.0123; b = 0.005; λh = 15; λf = 20. 5. the optimal control model in this section, three different types of control intervention viz., u1(t), u2(t), andu3(t) are incorporated into the model system (2.1) and extended to optimal control problem. copyright © 2022 assa. adv syst sci appl (2022) optimal control techniques of nipah virus disease 181 0 1 1 1r 0 0.8 3 0.5 0.6 1 2 0.4 0.20 0 fig. 4.2. influence of β1 and β3 on r0. 0 1 1 1r 0 0.8 h 0.5 0.6 1 2 0.4 0.20 0 fig. 4.3. influence of β1 and µh on r0. 0 1 1 1r 0 0.8 a h 0.5 0.6 1 2 0.4 0.20 0 fig. 4.4. influence of β1 and a on r0. there interventions are implementing either pharmaceutical(treatment) or nonpharmaceutical(effect of information). the main goal of this research is to investigate the best control strategies with minimum cost of implementation as well as financial loss generated. since there is no proper vaccination and appropriate medicines for nipah virus infections, so in the model we introduce three control strategies, viz, creating awareness u1(t), estimation of novel technologies u2(t), and the treatment u3(t). if u1(t), u2(t), and copyright © 2022 assa. adv syst sci appl (2022) 182 n.k. goswami, f. hategekimana 0 1 1 1r 0 0.8 f 0.5 0.6 1 2 0.4 0.20 0 fig. 4.5. influence of β1 and µf on r0. 0 1 1 1r 0 0.8 h 0.5 0.6 1 2 0.4 0.20 0 fig. 4.6. influence of β1 and δh on r0. 0 1 1 1r 0 0.8 h 0.5 0.6 h 2 0.4 0.20 0 fig. 4.7. influence of λh and γh on r0. u3(t) are equal to zero, then there is no effort being placed in these controls at time t and if they are equal to one then the maximum effort is applied. the details of each intervention are described as follows: copyright © 2022 assa. adv syst sci appl (2022) optimal control techniques of nipah virus disease 183 • control variable u1(t): the force of nipah virus infections is reduce by (1-u1(t)), where the effect control measures u1(t) increase awareness in the population which results in the reduction of the transmission rate β1 and the control measures the effort required for giving health cares for the infected people to reduce the infected individuals. creating awareness activity among the community about the risky areas before outbreak of the disease is more effective. personal care should be taken in order to avoid direct contact with the person infected by nipah virus. also, using masks, gloves and glasses will help in prevention and control effectively potential outbreaks. • control variable u2(t): the control variable u2(t) is estimation of novel technologies or methods to minimize spread of the virus within bat or pig population. controlling bat is associated with a reduction in human cases. this is an effective intervention but requires considerable amount of money. thus, our main objective is to minimize the disease effect in pig or bat population with minimum cost. • control variable u3(t): the control variable u3(t) is the use of effective medicines for the treatment measure of infectious humans and insecticide to the nipah virus infection causing species. η is the modification parameter in control u3(t). due to the expensive nature of the antibody drugs, in this effective intervention requires large amount of money. thus, the main objective of this intervention is to minimize the disease effect in human population with minimum cost. dsh dt = λh − (1− u1(t))β1sh ih nh − (1− u2(t))β2sh if nh − µhsh deh dt = (1− u1(t))β1sh ih nh + (1− u2(t))β2sh if nh − (δh + ah + µh)eh dih dt = δheh − (γh + ηu3(t) + µh + µ1)ih drh dt = aheh + (ηu3(t) + γh)ih − µhrh (5.4) dsf dt = λf − (1− u2(t))β3sf ib nh − µfsf dif dt = (1− u2(t))β3sf ib nh − µfif dib dt = bib ( 1− ib k ) − µbib consider u1(t) = u1, u2(t) = u2, and u3(t) = u3 for further analysis. 5.1. cost construction and characterization of optimal controls this particular section devotes into two parts which include determination of the total cost generated due to applied controls as well as diseases itself. whereas the second part determines the analytical forms of the controls. 5.1.1. total cost determination here, the total cost is determined for the applied control interventions and disease as well, which need to be minimized. • cost due to disease: the cumulative cost incurred due to the disease burden is modeled as follows: ∫ t 0 a1(eh + ih) + a2(sf + if )dt copyright © 2022 assa. adv syst sci appl (2022) 184 n.k. goswami, f. hategekimana this cost consists of various components such as cost due to loss of manpower, opportunity loss and significant economic losses for farmers. • cost incurred due to health care awareness : the total cost involved in reducing the transmission between human to human is given as∫ t 0 a3u 2 1dt. • cost incurred due to novel technologies: the total cost involved in reducing bat or pig infection is defined as ∫ t 0 a5u 2 2dt. • cost incurred in treatment for infected human population: the cumulative cost in process of treating infected human is defined by∫ t 0 a5u 2 3dt. the cost functional corresponding to total cost incurred, for fixed time t, which need to be minimized is given by j(u1, u2, u3) = ∫ t 0 [ a1(eh + ih) + a2(sf + if ) + a3ib + 1 2 (a4u 2 1 + a5u 2 2 + a6u 2 3) ] dt (5.5) subject to the model system (5.4). the parameter a1 ≥ 0, a2 ≥ 0, a3 ≥ 0, a4 ≥ 0, a5 ≥ 0, a6 ≥ 0 are the weight and balancing constants, which measure the respective cost involvement over the interval [0, t]. in order to find an optimal control, u1 ∗, u2 ∗, and u3 ∗ such that j(u1 ∗, u2 ∗, u3 ∗) = min (u1,u2,u3)∈u j(u1, u2, u3), (5.6) where u is the control set and is defined as u = {(u1, u2, u3) : 0 ≤ u1, u2, u3 ≤ 1, t ∈ [0, t]} here, all the controls are bounded and measurable. 5.1.2. existence and characterization of optimal controls here, we shall first establish the existence of such control functions that minimises the cost functional j . the lagrangian l of this problem is defined as: l(eh, ih, sf , if , ib, u1, u2, u3) = a1(eh + ih) + a2(sf + if ) + a3ib + 1 2 a4u1 2 + 1 2 a5u 2 2 + 1 2 a6u 2 3 now, we shall use pontryagin’s maximum principle [18] for necessary conditions for optimal controls system (5.4). for that by choosing x = (sh, ehih, rh, sf , if , ib), u = (u1, u2, u3) and λ = (λ1, λ2, λ3, λ4, λ5, λ6), the associated hamiltonian h can be written as h(x,u , λ) = l(eh, ih, sf , if , ib, u1, u2, u3) + λ1 dsh dt + λ2 deh dt + λ3 dih dt + λ4 drh dt +λ5 dsf dt + λ6 dif dt + λ7 dib dt (5.7) copyright © 2022 assa. adv syst sci appl (2022) optimal control techniques of nipah virus disease 185 since u∗ 1,u ∗ 2, u ∗ 3 are solutions to the control problem (5.4), there exists the adjoint variables λ1, λ2, λ3, λ4, λ5, λ6 satisfying the following conditions. dx dt = ∂h(t, x, u∗ 1, u ∗ 2, u ∗ 3, λ1, λ2, λ3.λ4, λ5, λ6, λ7) ∂λ 0 = ∂h(t, x, u∗ 1, u ∗ 2, u ∗ 3, λ1, λ2, λ3.λ4, λ5, λ6, λ6, λ7) ∂u dλ dt = −∂h(t, x, u∗ 1, u ∗ 2, u ∗ 3, λ1, λ2, λ3.λ4, λ5, λ7) ∂x (5.8) theorem 5.1: for the objective functional (5.5) and the control set (5.8) subject to control system (5.4) there exists an optimal control u∗ = (u1 ∗, u2 ∗, u3 ∗) ∈ u such that j(u1 ∗, u2 ∗, u3 ∗) = min u j(u1, u2, u3). proof to prove this theorem, we use [14]. as, we have discussed above that all the state variables (population) are bounded for each bounded controls coming from the control set u . furthermore, lipschitz condition with respect to state variables is satisfied by the right hand side functions of the model system (5.4). the control variable set u is also convex and closed by the definition and the model system (5.4) is linear in control variables u1,u2 and u3 with coefficients depending on state variables. the integrand of the function l = a1(eh + ih) + a2(sf + if ) + a3ib + 1 2 a4u1 2 + 1 2 a5u 2 2 + 1 2 a6u 2 3 is convex on the control set u due to quadratic nature of control variables u1,u2 and u3 respectively. moreover, l = a1(eh + ih) + a2(sf + if ) + a3ib + 1 2 a4u1 2 + 1 2 a5u 2 2 + 1 2 a6u 2 3 ≥ 1 2 a4u1 2 + 1 2 a5u 2 2 + 1 2 a6u 2 3. now consider c1 = min(a1, a2, a3) > 0 and g(u1, u2, u3) = c1(u1 2 + u2 2 + u3 2). thus, l ≥ g(u1, u2, u3) holds true and g is continuous. also, g satisfies the condition |(u1, u2, u3)|−1g(u1, u2, u3) → ∞ whenever |(u1, u2, u3)| → ∞. thus, all the conditions for the existence of controls are fulfilled. hence from result [10, 14, 15, 18], we conclude that there is a control pair u∗ 1, u ∗ 2, u ∗ 3 such that j(u1 ∗, u2 ∗, u3 ∗) = min u j(u1, u2, u3). theorem 5.2: for optimal controls measures u∗ 1, u ∗ 2, u ∗ 3 and the state solutions s∗ h, e ∗ h, i ∗ h, r∗ h, s ∗ f , i ∗ f , i∗b of the state system (5.4), there exists adjoint variables λ = (λi) t ∈ r7, i = 1, 2, 3, 4, 5, 6, 7 such that dλi dt = −∂h ∂ui (5.9) with transversality conditions λ1(t) = λ2(t) = λ3(t) = λ4(t) = λ5(t) = λ6(t) = λ7(t) = 0 (5.10) further, the optimal controls (u1 ∗, u2 ∗), u3 ∗ which minimizes j over the region u given by u1 ∗ = min{1,max(0, ũ1)} copyright © 2022 assa. adv syst sci appl (2022) 186 n.k. goswami, f. hategekimana u2 ∗ = min{1,max(0, ũ2)} u3 ∗ = min{1,max(0, ũ3)} where, ũ1 = β1ih (sh + eh + ih +rh)a4 ũ2 = ( 4β3 sfib n2 h + ib nh + sf nh ) (λ6 − λ5) a5 (5.11) ũ3 = (γh + η)(λ3 − λ4) a6 proof let u∗ 1, u ∗ 2, u ∗ 3 be the optimal control functions and s∗ h, e ∗ h, i ∗ h, r ∗ h, s∗ f , i ∗ f , i ∗ b are the corresponding state variables. then, pontryagin’s maximum principle ensures the existence of the following adjoint variable λi(i = 1, 2, 3, 4, 5, 6, 7) ∈ r6, which satisfies the following canonical equations: dλ1 dt = − ∂h ∂sh , dλ2 dt = − ∂h ∂eh , dλ3 dt = −∂h ∂ih , dλ4 dt = − ∂h ∂rh , dλ5 dt = − ∂h ∂sf , dλ6 dt = −∂h ∂if , dλ7 dt = −∂h ∂ib with transversality conditions (5.10) and the hamiltonian (5.7) . to establish the adjoint system (5.9) and the transversality conditions in (5.10), we use hamiltonian h in (5.7). by using third condition of (5.6), we get (5.11), as follows −dλ1 dt = (1− u1)β1 (eh + ih +rh)ih n2 h (λ1 − λ2) + (1− u2)β2 (eh + ih +rh)if n2 h (λ1 − λ2) +(1− u2)β3 sfib n2 h (λ6 − λ5) + µhλ1 dλ2 dt = −a1 + (1− u1)β1 shih n2 h (λ2 − λ1) + (1− u2)β2 shif n2 h (λ2 − λ1) +(1− u2)β3 sfib n2 h (λ6 − λ5) + (ah + µh)λ2 dλ3 dt = −a1 + (1− u1)β1 sh(sh + eh +rh) n2 h (λ2 − λ1) + (1− u2)β2 shif n2 h (λ2 − λ1) +(1− u2)β3 sfib n2 h (λ6 − λ5) + (γh + ηu3)(λ3 − λ4) + (µh + µ1)λ3 dλ4 dt = (1− u1)β1 shih n2 h (λ2 − λ1) + (1− u2)β2 shif n2 h (λ2 − λ1) copyright © 2022 assa. adv syst sci appl (2022) optimal control techniques of nipah virus disease 187 +(1− u2)β3 sfib n2 h (λ6 − λ5) + µhλ4 dλ5 dt = −a2 + (1− u2)β3 ib nh (λ6 − λ5) + µfλ5 dλ6 dt = −a2 + (1− u2)β2 sh nh (λ1 − λ2) + µfλ6 dλ7 dt = −a3 + (1− u2)β3 sf nh (λ5 − λ6) + µbλ7 − (b− 2bib k )λ7 using the second condition of (5.6), we get (5.11), as follows ∂h ∂u1 = a4u1 + β1ih(λ1 − λ2) sh + eh + ih +rh = 0 this implies, u1 = β1ih(λ2 − λ1) (sh + eh + ih +rh)a4 ∂h ∂u2 = a5u2 + ( 4β3 sfib n2 h + ib nh + sf nh ) (λ5 − λ6) = 0 this implies, u2 = ( 4β3 sfib n2 h + ib nh + sf nh ) (λ6 − λ5) a5 and, ∂h ∂u3 = a6u3 + (γh + η)(λ3 − λ4) = 0 this implies, u3 = (γh + η)(λ3 − λ4) a6 moreover, lower and upper bounds of these control are 0 and 1 respectively. thus, if ũ1 > 1, ũ2 > 1, ũ3 > 1, then u1 = u2 = u3 = 1. also, ũ1 < 0, ũ2 < 0 , ũ3 < 0, then u1 = u2 = u3 = 0. otherwise, we have u1 = ũ1, u2 = ũ2, and u3 = ũ3 hence, for these controls u∗ 1, u ∗ 2, u ∗ 3 we get optimum value of the function j . copyright © 2022 assa. adv syst sci appl (2022) 188 n.k. goswami, f. hategekimana 6. numerical results and discussion with the help of matlab, the optimal control model is simulated. we simulate our optimal control model by using the parameters shown in table 1. most of the values of table 1 are taking from [10]. the weight constants for the optimal control problem are taken as a1 = 1, a2 = 1, a3 = 1, a4 = 55, a5 = 75, a6 = 85. we solve the optimality system (5.4) by iterative method with the help of forward and backward difference approximations [14]. we consider the time interval as [0, 150]. first we solve the state equations by the forward difference approximation method then we use the backward difference approximation method to solve the adjoint equations. it is observed that from fig. 6.15, the optimal control model gives a better result as compacted to the model without the optimal control model as it reduces the number of infectives significantly in a desired interval of time. following are the different types of control strategies to see the impact of optimal control in the total number of human infectives. strategy i: when only one control is used at a time (i) employing only control measure u1 to optimize the objective function j , while control intervention u2 = u3 = 0, were not employed. the influence of u1 is shown in the fig. 6.8, to minimize the objective function, the optimal control u1 is maintained at the maximum level. a single preventive measure can influence the spread of nipah virus in the population. from the figures, it is clear that the control intervention u1 is more effective compared to other types of controls. (ii) employing only control measure u2 is used to optimize the objective function j , while control intervention u1 = u3 = 0, were not employed. the influence of u2 is demonstrated in fig. 6.9,to minimizing cost as well as infection of nipah virus. (iii) employing only control measure u3 is used to optimize the objective function j , while control intervention u2 = u3 = 0, were not employed. the influence of u3 is demonstrated in fig. 6.10, to minimize the disease effect in human population with minimum cost. strategy ii: when two controls are used employing double control interventions (u1, u2) when u3 = 0, (u1, u3) when u2 = 0, and (u2, u3) when u1 = 0, to optimize the objective function j. the control intervention are demonstrated in figs. 6.11, 6.12, and 6.13. it is observed that combination of two control intervention is most effective then using signal control intervention. significant number of nipah virus infective cases reduce. strategy iii: when all three controls are used employing all three control interventions (u1, u2, u3) to optimize the objective function j . this control intervention is demonstrated in fig. 6.14. it is easy to say that the total number of infectious individuals decreases significantly if we combine all three optimal controls u1, u2, and u3. the three optimal control application is the best control strategy to minimize the number of infectives, and will definitely reduce the spread of nipah virus. copyright © 2022 assa. adv syst sci appl (2022) optimal control techniques of nipah virus disease 189 0 50 100 150 time(in days) 0 0.2 0.4 0.6 0.8 1 c o n tr o l p ro fi le ( u 1) u 1 , when u 2 =u 3 =0 fig. 6.8. control profile of u1, when u2 = u3 = 0. 0 50 100 150 time(in days) 0 0.2 0.4 0.6 0.8 1 c o n tr o l p ro fi le ( u 2) u 2 , when u 1 =u 3 =0 fig. 6.9. control profile of u2, when u1 = u3 = 0. 0 50 100 150 time(in days) 0 0.2 0.4 0.6 0.8 1 c o n tr o l p ro fi le ( u 3) u 3 , when u 1 =u 2 =0 fig. 6.10. control profile of u3, when u2 = u3 = 0. 7. conclusion the transmission dynamics of nipah virus mathematical model is proposed and analyzed. disease-free equilibrium and the basic reproduction number (r0) is computed. the sensitivity of different parameters of (r0) is presented and it is clear that from the forward sensitivity indices figure that λh, β1, δh, and µh are very sensitive parameter. its shows that copyright © 2022 assa. adv syst sci appl (2022) 190 n.k. goswami, f. hategekimana 0 50 100 150 time(in days) 0 0.2 0.4 0.6 0.8 1 c o n tr o l p ro fi le ( u 1,u 2) u 1 , wnen u 3 =0 u 2 , wnen u 3 =0 fig. 6.11. control profile of u1, u2, when u3 = 0. 0 50 100 150 time(in days) 0 0.2 0.4 0.6 0.8 1 c o n tr o l p ro fi le ( u 1,u 3) u 1 , when u 2 =0 u 3 , when u 2 =0 fig. 6.12. control profile of u1, u3, when u2 = 0. 0 50 100 150 time(in days) 0 0.2 0.4 0.6 0.8 1 c o n tr o l p ro fi le ( u 2,u 3) u 2 , when u 1 =0 u 3 , when u 1 =0 fig. 6.13. control profile of u2, u3, when u1 = 0. small change in these parameters can cause huge change in the value of basic reproduction number r0. the parameter ah corresponds effects of awareness and it is the fact that due to awareness programs by media, the equilibrium level of the infected population decreases significantly. the proposed mathematical model is extended to optimal control problem by incorporating three time-dependent optimal control parameters to reducing transmission and cost duo to nipah virus outbreak by using pontryagin’s maximum principal. the copyright © 2022 assa. adv syst sci appl (2022) optimal control techniques of nipah virus disease 191 0 50 100 150 time(in days) 0 0.2 0.4 0.6 0.8 1 c o n tr o l p ro fi le ( u 1,u 2,u 3) u 1 u 2 u 3 fig. 6.14. control profile of u1, u2, and u3. 0 50 100 150 time (days) 2 4 6 8 10 in fe ct ed p op ul at io ns without control with control fig. 6.15. variation of infected human populations against time with and without optimal control. optimal control model provides a more reliable result as compacted to the model without the optimal control model. this control strategy reduces the number of infectives significantly in a desired interval of time. numerical simulation is performed to support our analytical findings. the fact of the control strategies are demonstrated with the help of matlab and it is observed that combining all three optimal controls give significant effect. the simulation result indicates the effectiveness of optimal control strategies in reducing the number of infectives. acknowledgments we appreciate the handling editor’s diligent perusal of our article, as well as the anonymous reviewers’ numerous informative remarks and recommendations. references 1. biswas, m. h. a. (2012). model and control strategy of the deadly nipah virus (niv) infections in bangladesh, res rev biosci, 6(1), 370–377. 2. castillo-chavez, c., feng, z., & huang, w. (2002). on the computation of r0 and its role on global stability, mathematical approaches for for emerging and reemerging infectious diseases. springer-verlag, 229–250. 3. chitnis, n., hyman, j. m., & cushing, j. m. (2008). determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, copyright © 2022 assa. adv syst sci appl (2022) 192 n.k. goswami, f. hategekimana bull.math.biol, 70(5), 12721296. 4. chua, k .b. (2003). nipah virus outbreak in malaysia, j clin virol, 26, 265–275. 5. chua, k .b., bellini, w. j., & rota, p. a. (2000). nipah virus: a recently emergent deadly paramyxovirus. science, 288,1432–1435. 6. chua, k. b., koh, c. l., & hooi, p. s. (2002). isolation of nipah virus from malaysian island flying-foxes, microbes infect, 4,145–151. 7. mondal, m. k., hanif, m., & biswas, h. a.(2017). a mathematical analysis for controlling the spread of nipah virus infectio, international journal of modelling and simulation, http://dx.doi.org/10.1080/02286203.2017.1320820 8. driessche, p.v., & watmough, j.(2002). reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. mathematical biosciences, 180, 29–48. 9. goh, k. j., tan, c. t., & chew, n. k. (2000). clinical features of nipah virus encephalitis among pig farmers in malaysia. n engl j med, 342, 1229–1235. 10. goswami, n. k.(2021). modeling analysis of zika virus with saturated incidence using optimal control theory, international journal of dynamical systems and differential equations, 11(3-4), 287–301. 11. goswami, n. k.(2020). modeling and sensitivity analysis of dynamics of nipah virus disease, world journal of modelling and simulation , 16(2), 107–119. 12. goswami, n. k., & shanmukha, b. (2020). modeling and analysis of symptomatic and asymptomatic infections of zika virus disease with non-monotonic incidence rate,appl. math. inf. sci ,14(4), 655–671. 13. goswami, n. k., srivastav, a. k., ghosh, m.,& shanmukha, b. (2018). mathematical modeling of zika virus disease with nonlinear incidence and optimal control. iop conf. series: journal of physics: conf, series 1000, 012114 doi :10.1088/17426596/1000/1/012114. 14. lenhart, s., & workman,j. t.(2007).optimal control applied to biological models, crc press. 15. olaniyi, s., okosun, k. o., adesanya, s. o., adesanya, & r. s. lebelo,, (2020) modelling malaria dynamics with partial immunity and protected travellers: optimal control and cost-effectiveness analysis, journal of biological dynamics, 14, 90–115. 16. parashar, u. d., sunn, l. m., & ong, f.(2000). case-control study of risk factors for human infection with a new zoonotic paramyxovirus, nipah virus, during a 1998–1999 outbreak of severe encephalitis in malaysia. j infect dis, 181,1755–1759. 17. paton, n. t., leo, y. s., & zaki, s. r.(1999). outbreak of nipah-virus infection among abattoir workers in singapore, the lancet, 354, 1253–1256. 18. pontryagin, l. s., boltyanskii, v. g., gamkrelidze, r. v., & mishchenko, e. f. (1962).the mathematical theory of optimal processes, inter science publishers, geneva. 19. srivastav, a. k., goswami, n. k., ghosh, m., & xue-zhi, li. (2018). modeling and optimal control analysis of zika virus with media impact, int. j. dyanm. control, 6, 1673–1689 https://doi.org/10.1007/s40435-018-0416-0. 20. stefan, m., & yingcun, x. (2009). mathematical understanding of infectious diseade dynamics,world scientific publishing co. pte. ltd. 21. uppal, p. k. (2000). emergence of nipah virus in malaysia, ann n y acad sci. 916, 354–357. copyright © 2022 assa. adv syst sci appl (2022) introduction mathematical model positivity and boundedness of the system basic reproduction number r0 sensitivity analysis of reproduction number the optimal control model cost construction and characterization of optimal controls total cost determination existence and characterization of optimal controls numerical results and discussion conclusion microsoft word il'in adv syst sci appl 2023; 01; 152-156 published online at https://ijassa.ipu.ru. evaluation of microalgae prebiotic activity in long-term humans isolation in confined habitat vyacheslav k. ilyin, denis v. korshunov, yulia a. morozova, nonna a. usanova, daria v. komissarova*, alexandra g. korosteleva, margarita a. levinskyh ibmp ras, moscow, russia abstract: it is known that long-term isolation in sealed compartment the first 10-14 days is called a period of "acute adaptation". it is manifested by an increase of number of opportunistic microflora species together with quantitative decrease of protective microflora species. this is fraught with the development of autoand cross-infections. in this regard, it is necessary to develop countermeasures against maladjustment processes in these conditions. ibmp conducted research on possibility of using an autotrophic element of biological life support systems, including as an additive in diets. a comparative assessment of the microflora of 6 people, participants of a long-term isolation experiment, was carried out. as countermeasures pills based on spirulina were used. two pills was taken by all participants times per day. for comparison, archival data from experiments in which subjects did not use any countermeasures were used. the results indicate the stabilization of the microbiocenosis of the intestines and upper respiratory tract when using the spirulina pills. the concentration of enterococci was maintained at a high level, most of the participants did not have proteus, staphylococci or yests. however there were minor fluctuations in the content of protective groups in the intestinal microflora (bifidobacteria and lactobacilli), which determines the need to include probiotics as countemesures. thus, studies have demonstrated the positive results of the use of spirulina as a countermesure. therefore, it is expedient to consider the issue of equipping the sealed compartments planned for use for the implementation of deep space exploration programs with water-mixture bioreactors. keywords: microbial fuel cell, isolation, microflora, spirulina. 1. introduction the human microflora is an integral and plastic system, considered by some researchers as a separate “organ” that functions as part of a particular biotope and is influenced by various factors [2,9]. maintaining the qualitative and quantitative balance of the microbiota seems to be especially important taking into account planned long-term space missions beyond the earth's orbit. previous studies of the microflora in sealed compartments revealed that due to the constant microbial exchange between crew members, combined with stress and intense psycho-emotional state associated with the performance of a space mission, there is a deterioration in the microflora of the upper respiratory tract and intestines. under such conditions, there is an activation and increase in the number of opportunistic microorganisms, and at the same time decrease in protective species for a given biotope, which leads to decrease in colonization resistance and increase in the risk of developing dysbiotic conditions, and in case of too strong an imbalance in the microflora – to inflammatory processes of one or more other localization [3,8]. the most dangerous from the point of view of the development of such situations is a socalled period of acute adaptation, which is observed from 1 to 10–14 days of the isolation experiment [1]. the experience of the volunteers' previous participation in isolation * corresponding author: d.komisarova@yandex.ru evaluation of microalgae prebiotic activity… 153 copyright ©2023 assa. adv. in systems science and appl. (2023) experiments slightly "smoothes" the severity of changes in this period, but still, it seems appropriate to develop effective countermeasures to reduce risks for crew members. one of such couontermeasures is probiotic and autoprobiotic pills [10], however, other means can also be used as a food additive, the effectiveness of which can also be high, especially when combined with the above mentioned drugs. one such option is spirulina. spirulina contains a large amount of protein, some vitamins and minerals, as well as phenolic acids. the toxicological safety of spirulina has already been proven, in addition, this algae is easy to cultivate. a number of studies have shown that spirulina has a good anti-inflammatory effect by inhibiting the release of histamine from mast cells [6,11]. also, spirulina is able to stimulate an increase in the production of iga immunoglobulin, an increase in which was noted in saliva in a study by ishii et al. [5]. this indicates the influence of algae on the formation of immunity of the oral mucosa. the purpose of this study was to analyze the data of the intestinal microflora and the microflora of the upper respiratory tract of the subjects who participated in a 3-month isolation experiment and used a spirulina-based preparation as a dietary supplement. research objectives: 1. assess the state of the microflora of the upper respiratory tract and intestines from 1st to 60th days of isolation; 2. determine the timing of the most pronounced changes; 3. evaluate the effectiveness of taking a biological food supplement based on spirulina to maintain the stability of the intestinal microflora and upper respiratory tract. 2. methods and materials this study is included in the scientific program of the isolation experiment, which simulates the classical adverse factors of space flight and the sealed compartment experiment, such as sensory deprivation, monotony, limited living volume and controlled habitat. the experiment also simulates the factors of autonomous interplanetary flight, including limiting the resources of the expedition and modeling the professional activities of the space crew. according to the scenario of the experiment, there is a delay in communication with the mission control center up to 5 minutes one way, as well as working out emergency situations. this study analyzed data on the microflora of the upper respiratory tract and intestines in 5 volunteer crew members. the study was reviewed and approved by the bioethical commission of the state scientific center of the russian federation ibmp ras. the subjects used a preparation with spirulina, which included microalgae spirulina (spirulina platensis) and an auxiliary component: aerosil (anti-caking agent). the drug was taken two pills three times a day with meals for one month (30 days). spirulina was taken from the 1st to the 30th day of the experiment. daily dose 6 pills. samples of microflora from the upper respiratory tract were taken 2–3 days before the start of the experiment (background) with a sterile swab, followed by inoculation on selective nutrient media, then by the crew doctor from each tester on the 30th and 60th day of isolation. fecal samples were also taken in the background period 2–3 days before the start of the isolation experiment, and then on the 14th, 30th, and 60th days of the experiment. a number of tenfold dilutions in sterile saline from 101 to 109 were prepared from fecal samples, and 100 μl of the inoculum were sown in petri dishes with agar nutrient media: blood agar, endo agar or macconkey agar, mannitol-salt agar, sabouraud medium, mrs, bactofoc medium, citrate agar, enterococcal agar, bifidoagar. samples from the upper respiratory tract were diluted with saline and plated on the following nutrient media: blood agar, chocolate agar, macconkey agar, mannitol-salt agar, sabouraud's medium. manufacturer of all media himedia, india. 154 v.k. ilyin, et al. copyright ©2023 assa adv. in systems science and appl. (2023) growing colonies were counted and identified. statistical processing of the results was carried out using the statistica 7.0 program; discriminant and variance analysis for small groups was used. [7] to assess changes in the microflora of the gastrointestinal tract, the eubiotic index was used, which shows the ratio of the number of positive changes in the microflora to the number of negative changes. [4] 3. results and discussion when analyzing the composition of the microflora of the upper respiratory tract, a significant difference was found between the state of the microbiota on the 30th day after the monthly use of the drug with spirulina and in the background period, as well as on the 60th day (30 days after stopping the use of the supplement). discriminant analysis showed a significant difference between groups in staphylococcus spp. taking into account the value of the contribution of various groups of microorganisms to canonical variables (root 1 and root 2), we can conclude that when moving along the abscissa axis (from left to right), the number staphylococcus and fungi (together in the "nose" and "mouth" biotopes) decreases, and when moving along the y-axis (from bottom to top), the number of neisseria spp increases (figure 1). thus, analyzing the position of the groups on the graph, we can conclude that after 30 days of using the drug with spirulina, there is a decrease in the opportunistic component of the microbiota of the upper respiratory tract. figure 1. projections of cenoses on two canonical variables (root 1 and root 2). each point on the graph corresponds to one participant. g_1:3006 30th day of isolation, g_2:6006 60th day of isolation. when analyzing the intestinal microflora, discriminant analysis revealed significant differences in 4 groups of microorganisms: enterobacteriaceae spp, enterococcus spp, lactobacillus spp and clostridium spp. evaluation of microalgae prebiotic activity… 155 copyright ©2023 assa. adv. in systems science and appl. (2023) figure 2. the number of microorganisms of the intestinal microflora of the participants who took pills with spirulina from days 1 to 30 of the isolation experiment (lg (cfu)). attention is drawn to the decrease and stabilization at a relatively low level of opportunistic microorganisms, such as enterobacteriaceae spp and clostriduim spp, while the number of clostridiim spp. by 14 days (by the end of the acute period of adaptation) increases, and then decreases by 30 days, and after discontinuation of the drug sharply increases again. the number of lactobacilli (a protective component of the intestinal microbiota) slightly decreases by the end of the acute period of adaptation, and then increases to the level of normocenosis (figure 2). in addition to assessing the number of different groups of microorganisms, the eubiotic index was also calculated using data on genera, for which statistically significant differences were found (figure 3). the figure shows that the index increases by day 30 (that is, the number of protective microorganisms increases and stabilizes, while the number of conditionally pathogenic microorganisms decreases), and then decreases slightly, while remaining slightly above one, which indicates the predominance of positive changes in the microflora over negative ones. figure 3. dynamics of the eubiotic index of intestinal microflora in subjects who took a supplement with spirulina from days 1 to 60 of the isolation experiment. ei – eubiotic index. separately, it is noteworthy that some subjects do not have any staphylococci, or proteus, or yeast, which indicates the stability of the colonization resistance of the intestinal microflora. nevertheless, a slight decrease in lactobacilli on the 14th day of the experiment as well as the absence of a significant difference in bifidobacteria, which are also a protective species, determines the need to include additional probiotic or autoprobiotic preparations based on these groups of microorganisms in the diet. thus, it seems appropriate to consider the issue of completing long-term isolation experiments with algal bioreactors for cultivating spirulina under conditions of long-term 156 v.k. ilyin, et al. copyright ©2023 assa adv. in systems science and appl. (2023) autonomous stay of isolated groups in a hermetic facility. it also appears that this bioreactor may be compatible with other biotechnological processes for regenerative life support systems, such as microbial fuel cells, in which the cathode chamber would be used for oxygen production purposes, where this latter would be optimized by the operation of the algae reactor. acknowledgements the work was supported by the basic theme of the russian academy of sciences no. 64.2 "study of the function of the gastrointestinal tract during adaptation of the human body to an artificial habitat and methods for correcting dysbacteriosis using autoprobiotics", as well as using the transgenbank unique scientific facility with financial support from the russian federation represented by ministry of education and science of russia within the framework of the project (agreement no. 075-15-2021-668 of 07/29/2021). references [1] antipov, v. v. 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(march, 2022). rf patent 2460778c12n1/20, a61k35/74, a23c9/127. a method for obtaining an autoprobiotic based on enterococcus faecium, a representative of the indigenous intestinal microflora of the host. [online]. available: http://www.findpatent.ru/patents/246/2460778.html. [11] yang, h. n., lee, e. h., & kim, h. m. (1997). spirulina platensis inhibits anaphylactic reaction, life sci., 61(13), 1237–1244, doi: 10.1016/s0024-3205(97)00668-1, pmid: 9324065. adv syst sci appl 2021; 01; 1-10 published online at https://ijassa.ipu.ru. the optimal control of the vessel’s automatic dynamic positioning system under deviation vadim kramar 1, vasilii alchakov2, alexander osadchenko3* 1) sevastopol state university, sevastopol, russia e-mail: kramarv@mail.ru 2) sevastopol state university, sevastopol, russia e-mail: alchakov@sevsu.ru 3) sevastopol state university, sevastopol, russia e-mail: aeosadchenko@rambler.ru abstract: an optimal algorithm for linear control of the unmanned vessel dynamic positioning system with provision for vessel characteristics, equipment, and the process has been developed. a block diagram for the automatic dynamic positioning system has been suggested which allows presenting a necessary set of devices for its technical implementation. keywords: automatic dynamic positioning system, deviation, control system, vessel, the optimal control 1. introduction automatic dynamic positioning systems or position-keeping systems are the main part of unmanned vessels that are developed in modern practice. vessel movement while station keeping is due not only to the type or nature of the external influence (wind, current, sea disturbance, the propulsive force of the steering) but also the type of the performed action since it can change dynamic properties of the vessel [1–3, 10]. when the vessel is under the action of current, wind, sea disturbance – manual control, if high accuracy of keeping the station is required, becomes almost impossible. there is consequently a need for automatic dynamic positioning system development. the main tasks of dynamic positioning system (stabilization of horizontal shifts of the ship with the required accuracy; vessel control by the heading angle so that its current value φ align with its optimal value φ* with the required accuracy, which ensures minimal energy consumption while steering etc.) [4], inaccuracy in the calculations of perturbation strengths from wind forcing and current predetermines system design. principles for system designing, its structure, performance, properties are mainly defined by the control object – the vessel, its mathematical model, and control problems. existing methods of dynamic positioning systems design are based on the application of mimo pid controllers [6, 9], neural networks [12], and fuzzy logic controllers [11], as well as approaches that rely on the application of adaptive [5] and hybrid [7] controllers. ways of angle speed control and vessel heading under “strong” maneuvers using the method of final conditions are suggested in the article. many control methods are based on the principle of optimal control and kalman filtering theory [1]. questions of designing a block diagram of the automatic dynamic positioning system and optimal linear control of unmanned vessels under deviation are considered in this article. * corresponding author: aeosadchenko@rambler.ru 2 v. kramar, v. alchakov, a. osadchenko copyright ©2021 assa. adv. in systems science and appl. (2021) 2. a block diagram of automatic position-keeping system position-keeping is possible with the use of two active control means (acm), each of them can have two control responses 2,1,,, =ip ii  , where ip – force intensity value, i – acm turn angles. the vessel control must be performed to the coordinates yx, and the heading angle  . this indicates the need for introducing intermediate control responses, which are vessel axisdirection forces yyxy ff , and the yawing moment zym that must be compensated by the acm, and the ones in the system must be converted into real control response 2211 ,,,  pp . thus, a device that can perform a one-to-one transformation of these controls is needed. coordination of both digital and analog elements must be performed by digital-to-analog (dac) and analog-to-digital (adc) converters. the system must also contain a deviation control unit (dcu) and a vessel horizontal shifts measurement system along with a coordinates calculator. a general block diagram of the automatic vessel position-keeping system is shown in fig. 1. the object of control (o) is the vessel. variables  ,, characterize motions of the vessel and are disturbers for the sonar sensors system (ss). external disturbance influence on the vessel in the form of wind, current, sea disturbance is characterized by the variables  – direction of current relative to a fore-and-aft axis of the vessel,  – wind direction, tb  , – wind speed and current speed respectively, f – uncontrolled disturbance, which characterizes the sea forces and moments. control actions on the vessel are thrust forces of the acm 21, pp and their steering angles 21, . sonar sensors system is intended for measuring vessel horizontal shifts, presented in the form of variables 4321 ,,,  which are parameters for calculation of the coordinates yx, by coordinates calculator (cc). variables  ,, that define the motion of the vessel are used as additional variables for calculations of the coordinates yx, . they can be measured and are disturbances. the deviation control unit generates intermediate control actions 000 ,, zyyyxy mff in the function of deviations of horizontal shifts yx, and deviations of the heading angle from the optimal value * . the choice of these control is a separate task. thrust block optimizer unit (ot) performs an optimal conversion of intermediate controls zyyyxy mff ,, , presented in digital form, into controls 20201010 ,,,  pp that are also presented in digital form. the optimal control of the vessel’s automatic dynamic positioning system... 3 copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 2.1. the general architecture of the system of the automatic hold of an unmanned vessel automatic gear (ag) tracks the forces of thrusts 2010 ,pp and their turn angles 2010 , , presented in analog form, into real thrusts and turn angles acm 2211 ,,,  pp . the connection between ot and ag is accomplished by dac. optimization unit by heading angle from measured magnitude  ,,,, tb calculates the optimal value of heading angle * to the direction of wind effect and current. it determines the value of force projection ybtxbt ff , and moment zbtm from the wind and current. a block diagram of the automatic position-keeping system assumes the following parameters to be measurable: wind angle and current angle to the center plane of the vessel, wind speed and current speed, horizontal shifts, vessel relative azimuth. to design an automatic position-keeping system, it is necessary to know many design and hydrodynamic parameters. 3. system model and problem formulation the mathematical model of the vessel can be presented in different forms. the spatial motion of the vessel considering all the coordinates correlations and vessel motion is described by a complex nonlinear nonstationary system of differential equations. the use of such a model for solving vessel motion optimization problems in the modes of position-keeping currently presents insuperable theoretical difficulties. the position-keeping regime is characterized by many favorable features: 1) the coordinates of vessel position and their velocity change in proximity to zero values; 2) all types of vessel motions must be insignificant; 3) the vessel heading angle in a time of vessel control process can be considered constant, equal to some required value. neglecting the influence of the oscillatory motion and performing linearization of ship motion control, neglecting also the influence of the equipment on the dynamics of vessel motion, neglecting the influence of hydrodynamic coefficient 26 [4, 10], we get a simplified vessel motion model in the horizontal plane in the aligned with the vessel coordinate system in the form of: 4 v. kramar, v. alchakov, a. osadchenko copyright ©2021 assa. adv. in systems science and appl. (2021)      =+ =+ =+ zz y x mj fym fxm       )( )( )( 66 22 11 , (1) where m – vessel mass, zj – body centroidal moment of inertia as regard to the z-axis, 662211 ,,  – hydrodynamical coefficient. it is also possible to present the vessel motion model in the fixed coordinate system. in addition to the system (1) let us introduce constraint equations     +−= +=   cossin sincos   y x , (2) where  , – coordinates of a fixed coordinate system connected with the vessel center of gravity. let us find second derivatives of coordinates:     −−+−= +−+=   sincoscossin cossinsincos   y x . (3) substituted (2), (3) to (1), we get:      =+ =+−+ =+++ zz y x mj fmm fmm       )( sin)(cos)( sin)(cos)( 66 * 22 * 22 * 11 * 11 , (4) where: ,cossin)sin( ,sincos)cos( *** ***   ++ −+ and variables  ,, – the deviations from the set values * 00 ,0  === . system (4) can be represented as:          = + + + = +  − + = z xy yx m c f сссс f сссс с f ccccc cc f сссс с 3 21212121 1 21212 21 2121 2 1 1 )(       , (5) where 1 * 11 cos)( cm =+  , 1 * 11 sin)( cm =+  , 2 * 22 cos)( cm =+  , 2 * 22 sin)( cm =+  , 366 cj z =+ . model (1) in the connection with the vessel frame of the axis, reflects only the inertial properties of the object. coordinate motions are independent. this vessel motion model reflects the main properties of the object in the position-keeping mode under the effect of external disturbance forces, presented in the form of force projection and the moment as regards the z-axis. apart from the disturbance forces, the control force will affect the acm. the optimal control of the vessel’s automatic dynamic positioning system... 5 copyright ©2021 assa. adv. in systems science and appl. (2021) if we present these forces in the form of projection of forces and moment, then a simplified vessel motion model in the horizontal plane will have the form:      +=+ +=+ +=+ zzyz yyy xxy mmj ffym ffxm       )( )( )( 66 22 11 . (6) forces and moment projection values, created by acm, depend on the number of acm, their placement on the vessel, thrusts values created by them, and their turn angles as regards the center plane of the vessel. for two acm:      +++= += += 222111222111 2211 2211 sinsincoscos sinsin coscos    pxpxpypym ppf ppf zy yy xy , (7) where 21, pp – acm thrusts values; 2121 ,,, yyxx – coordinates of acm placement on the vessel; 21, – acm turn angles. forces projection yx ff , and moment zm represent the algebraic sum of projections of forces and moments from wind, current, and sea disturbance. 4. synthesis of the optimal linear control for the synthesis problem, we shall accept an object model in the form (6), assuming that: xmm =+ 11 , ymm =+ 22 , zz mj =+ 66 , 0 xyxy ff = , 0 yyyy ff = , 0 zyzy mm = . it means that we shall design a control contour, which works on the principle of removal of random deflections of variables from the required values. a mathematical model of the control object can be given by:       += += += zzyz yyyy xxyx mmm ffym ffxm 0 0 0    . (8) we shall accept limitations on vessel horizontal shifts: 222 ryx + , (9) where r – circle radius, which determines legitimate vessel horizontal shifts. for an object (8) with limitations (9) let us consider a problem of synthesis of the optimal linear control under random stationary perturbations. as optimal control under random perturbations criterion, when the full-time of the control is big enough, we shall accept the root-mean-square criterion, which depends on the control coordinates  +++++= → t zyyyxy t dtmmfmfmmymxm t j 0 02 6 02 5 02 4 22 3 22 2 22 1 )( 1 lim 222  , (10) where )6,,2,1( =imi – weight coefficients, which determine the contribution of each sum and in the quality criterion value; t – the full time of the control. let us convert inequality (9) into equality constraint by extending the number of variables. for this purpose, let us introduce an additional real variable ),(1 −v , nonlinear conversion 1v e , and write over (9) in the form of equivalent constraint 6 v. kramar, v. alchakov, a. osadchenko copyright ©2021 assa. adv. in systems science and appl. (2021) 02221 =++− yxre v . (11) to minimize the performance functional (10) under constraints (8), (11) let us make use of the lagrange multiplier method and the necessary conditions in the form of the euler-poisson equation. lagrange function will be written in the form of ),( )()() ( 222 4 0 3 0 2 0 1 02 6 02 5 02 4 22 3 22 2 22 1 1 222 yxre mmmffymff xmmmfmfmmymxml v zzyzyyyyxxy xzyyyxy ++−+ +−−+−−+−− −++++++=      (12) where )4,3,2,1( =jj – lagrange indefinite multipliers, some functions of time. necessary conditions for the minimum criterion (10) in the form of the euler-poisson equation will be given by                    =−− =− =− =− =−− =−− =−− =+ =++ =++ 0)()10 02)9 02)8 02)7 0)6 0)5 0)4 02)3 022)2 022)1 222 4 3 02 6 2 02 5 1 02 4 0 0 0 3 2 3 24 2 2 14 2 1 yxr mm fm fm mmm ffym ffxm mm myym mxxm zy yy xy zzyz yyyy xxyx z y x               . (13) it should be noted that the last equation in (13) appeared due to constraints (12) under differentiation of the last term in (13) by 4 and 1v combination of the obtained correlations into one equation. this equation is satisfied under any yx, if 04 = or under any finite 4 , if 0222 =+− yxr . the last condition corresponds with the presence of the vessel in the boundary of the admissible deviation domain. from a physical standpoint, it cannot be realized for a long time, as the vessel under the reduced output of the acm can only cross this boundary. thus, there is no need to switch over controls and 04 = must be set. it means, that constraints (8) and equivalent to them (11) can be disregarded under the synthesis of optimal controls. it is enough to have optimal controls when the vessel faces admissible boundaries. the synthesis of optimal linear controls satisfies these conditions. at that, the set of equations (13) falls into three independent subsystems: 1) equations number (1), (4), (7); 2) equations number (2), (5), (8); 3) equations number (3), (6), (9), when 04 = . each subsystem gives equations for extremals which correspond to the coordinates and controls xxx fpmmxmpmm 22 4 2 1 42 4 2 )( =+ , (14) xxyx fmfmpmm 2 1 02 1 42 4 2 )( −=+ , (15) yyy fpmmympmm 22 5 2 2 42 3 2 )( =+ , (16) yyyy fmfmpmm 2 2 02 2 42 5 2 )( −=+ , (17) the optimal control of the vessel’s automatic dynamic positioning system... 7 copyright ©2021 assa. adv. in systems science and appl. (2021) zzz mpmmmpmm 22 6 2 3 42 5 2 )( =+  , (18) zzyz mmmmpmm 2 3 02 3 42 6 2 )( −=+ , (19) where dt d p = – operator of differentiation. controls (15), (17), (19) cause unstable coordinates motion. for control synthesis, which ensures stable extremals motions, let us make use of the results [8]. for perturbances, set by correlation functions of the form )3,2,1(,)( == − iek i , we get control laws in the form of xpy mm y pmf x xxy )]( )( [ 12 1 2 4 1220   −= , (20) ypy mm y pmf y yyy )]( )( [ 32 2 2 5 2420   −= , (21)    )]( )( [ 52 3 2 3620 py mm y pmm z zzy −= , (22) where xx mm m p mm m ppy 4 1 4 12 1 2 )( ++= ; xx mm m mm m y 4 1 1 4 12 112 2 )( +−=  ; yy mm m p mm m ppy 5 2 5 22 3 2 )( ++= ; yy mm m mm m y 5 2 2 5 22 224 2 )( +−=  ; zz mm m p mm m ppy 6 3 6 32 5 2 )( ++= ; zz mm m mm m y 6 3 3 6 32 336 2 )( +−=  . for perturbances, set by correlations functions of the form of 3,2,1),sin(cos)( =+= − iedk i i i ii i      we will have x pbam py pmf xxy ] )( )( [ 11 2 4 120 + −= , (23) where 2 1 1 11 cca   += ; 1 2 1  c b = ; coefficients 21,cc are determined from the equation 21 112 2 11 )( )( jcc jy jmx += − −   , y pbam py pmf yyy ] )( )( [ 22 2 5 320 + −= , (24) where 22 ,ba are determined from the expressions 4 2 2 32 cca   += ; 2 4 2  c b = ; 43 224 2 22 )( )( jcc jy jmy += − −   , ] )( )( [ 33 2 6 520 pbam py pmm zzy + −= , (25) 8 v. kramar, v. alchakov, a. osadchenko copyright ©2021 assa. adv. in systems science and appl. (2021) where 33,ba are determined from the expressions 6 3 3 53 cca   += ; 3 6 3  c b = ; 65 334 2 33 )( )( jcc jy jmz += − −   . for a complete definition of optimal control laws (20) – (25) it is necessary to find weight coefficients )6,,2,1( =imi . deviation control block which realizes optimal linear control laws (20), (21), (22) or (23), (24), (25), is comparatively complex, as it requires second differential coefficient coordinates measuring. from system engineering, for inertial objects control experience, it is known that control of such objects is quite efficient when the deviation signal and its first-order derivative are used. hence, control can be set in proportional-differential form. the optimal stop values will be determined by the relations [4]:          −−+− −+−+−− = −−+− −+−+−− == 2112 2 12 2 21 2 1 2 2112 2 12 2 21 2 2 1 )( )()( )( )()( xxmfxfxm mfxfxmxxfmfx p xxmfxfxm mfxfxmxxffxm p zyyz zyyzxzy zyyz zyyzxyz . (26) the rotation angles are determined by the relations [4]:        − − = − − = )( )(2 )( )(2 21 1 2 21 2 1 xxf mfx arctg xxf fxm arctg x zy x yz   ; (27) where 2,1, =ixi are the coordinates of the stops. let us cite the results of the study of the dynamics of the automatic position-keeping system. the model under study performed the calculation of optimal values of controls, presented in the form of projections of forces and moment on vessel coordinate axes with the subsequent calculation of the stop values 21, pp (see fig. 4.1) and their rotation angles 21, (see fig. 4.2). the studies were conducted for the vessel of the length of ml 104= and mass of тm 7,734= , which has the following characteristics: 21 11 22 stm −= ; 21 22 410 stm −= ; 215 66 102,2 stm −= ; 24109,48 sтмj z = . fig. 4.1, 4.2 shows the results of the calculations of thrust change of the vessel acm, provided that the following conditions for vessel deviation in the set time interval 600t s are met: 15,0x m, 05,0y m, 2,0 rad. the optimal control of the vessel’s automatic dynamic positioning system... 9 copyright ©2021 assa. adv. in systems science and appl. (2021) fig 4.1. thrusts change p1(t), p2(t) fig 4.2. angles change α1(t), α2(t) 5. conclusions a block diagram of the automatic vessel position-keeping system and the development of optimal linear control laws of unmanned coastal vessels under deviation was suggested. a designed block diagram and algorithms of its implementation allow presenting a necessary complex of devices for its technical implementation. an automatic vessel position-keeping system represents a digital-to-analog complex. a digital part of the system performs control laws generation, problems solution of optimization of variables conversion, makes calculations of vessel horizontal shifts based on the measurement system data. an analog part represents an automated control actions task drive and the control object. an optimal linear control law suggested in the study allows for effective position-keeping. 10 v. kramar, v. alchakov, a. osadchenko copyright ©2021 assa. adv. in systems science and appl. (2021) acknowledgments studies were supported by the ministry of education and science of the russian federation (project rfmefi57817x0259). references 1. balchen, j. g., n. a. jenssen, e. mathisen, and saelid, s. (1980) a dynamic positioning system based on kalman filtering and optimal control, modeling, identification and control, 1(3), 135–163. 2. fossen, t. i. (2011) handbook of marine craft hydrodynamics and motion control. john wiley & sons. 3. fossen, thor i. (1994) guidance and control of ocean marine vehicles. john wiley and sons ltd. new york. 4. kramar, v. (2018) the construction principle and system architecture of the automatic hold system of an unmanned vessel, proceedings of the international multi-conference on industrial engineering and modern technologies, fareastcon, vladivostok, russian federation. doi: 10.1109/fareastcon.2018.8602791. 5. lennon, w. k., and passino, k. m. (1997) strategies for genetic adaptive control, proceedings of the conference on decision and control, san diego, california, usa. doi: 10.1016/s0952-1976(98)00051-7. 6. martin, p. and katebi, r. (2005) multivariable pid tuning of dynamic ship positioning control systems, journal of marine engineering and technology, 1(a7):11_24. doi: 10.1080/20464177.2005.11020190. 7. nguyen, t.d., quek, s.t., sorensen, a. j. (2007) design of hybrid controller for dynamic positioning from calm to extreme sea condition, automatica (journal of ifac), 43(5):768_785, 2007. doi: 10.1016/j.automatica.2006.11.017. 8. petrov, y. (1973) optimization of controllable systems that are affected by wind and sea waves, sudostroenie [in russian]. 9. sorensen, a.j., sagatun, s.i. and fossen, t.i. (1996) design of a dynamic positioning system using model-based control, control eng. practice, 4(3), 359–368. doi: 10.1016/09670661(96)00013-5. 10. vadim, k. (2015) investigation of changes and development of a mathematical model for a drilling vessel, annals of daaam and proceedings of the international daaam symposium volume 2015-january, 2015, 49–55. doi: 10.2507/26th.daaam.proceedings.007. 11. cao, y., lee, t. (2003) maneuvering of surface vessel using a fuzzy logic controller, journal of ship research, 47(2), 101–130. 12. lee, z. h., cao, y. and lin, y. (2001) application of an on-line training predictor/controller to dynamic positioning of floating structures, tamkang journal of science and engineering, 4(3), 141–154. 1. introduction 2. a block diagram of automatic position-keeping system 3. system model and problem formulation 4. synthesis of the optimal linear control 5. conclusions acknowledgments microsoft word revised+paper+(study+of+the+effects+of+axially+defective+blades+of+wind+turbine+on+vibration+patterns).docx adv syst sci appl 2025; 3; 11-24 published online at https://ijassa.ipu.ru. study of the effects of axially defective blades of wind turbine on vibration patterns abdulhamid hamdan al-hinai*, karu clement varaprasad, vinod kumar sohar university, sohar, sultanate of oman abstract: axial defects in wind turbine blades represent a critical factor influencing both their vibration characteristics and overall operational performance. among these, the axial cracks are particularly concerning. they alter the structural dynamics of the blade, compromise stiffness and reduce the ability of the system to withstand fluctuating aerodynamic and mechanical loads. this study presents a systematic vibration analysis of wind turbine blades with three different levels of axial crack defects, specifically of lengths 50 mm, 100 mm, and 350 mm subjected to shaft rotational speeds ranging from 50 rpm to 200 rpm. the experimental investigations are conducted using the spectra quest wind turbine simulator. it replicates the conditions of a real wind turbine by incorporating actual wind data from sohar. the results reveal a strong correlation between defect size, shaft speed and vibration response with regression analysis yielding a high coefficient of determination (r² = 0.98). further, taguchi design of experiments is applied to identify the most effective parameter combinations for minimizing vibration risks. the proposed condition monitoring framework demonstrates an effective approach for detecting and mitigating vibration levels in defective blades. thereby contributing to enhanced reliability, extended service life and reduced maintenance costs in wind turbine systems. keywords: blade defect, crack size, rotational speed, taguchi analysis, vibration patterns, wind turbine. 1. introduction the quick development of wind energy as a clean and renewable control source has led to expanded attention to upgrading the reliability and efficiency of wind turbine systems [1]. regardless of significant advancements in wind turbine innovation, the presence of blade defects remains a challenge, affecting the execution and life expectancy of wind turbines [2]. blade vibration can be categorized into three types: stall flutter, classical flutter, and axial vibration. these vibrations can cause severe damage to the blade structure. the wind turbine blades are essential components that convert the kinetic energy of the wind into a mechanical form of energy. as these blades are subjected to significant mechanical loads and environmental forces throughout their working lives, they are critical to the longevity and productivity of wind energy systems. [3]. however, the presence of defects inside these blades, especially along the axial direction, can compromise structural integrity and performance, leading to operational downtime, upkeep costs, and security risks [4][5]. axial defects, such as axial cracks, can lead to catastrophic failures if not detected and treated on time. vibration-based investigation of wind turbines utilizes advanced monitoring methods to assess blade integrity and performance. data complexity and computational requirements are some challenges in vibration-based analysis. vibration analysis procedures, such as time domain and regression analysis, are employed to detect and analyze these defects, allowing for timely maintenance and repair interventions to prevent costly downtime and ensure the safe operation of wind turbines [6]. this is particularly useful for detecting sudden changes in vibration patterns and blade failures. regression analysis of vibration data involves modelling the relationship between various vibration parameters and the condition of wind turbine blades. * corresponding author: abdulhamidalhinai@gmail.com 12 abdulhamid hamdan al-hinai, karu clement varaprasad, vinod kumar copyright ©2025 assa adv. in systems science and appl. (2025) taguchi analysis is a statistical method used in this paper for optimizing the wind turbine performance as it is widely used in engineering research [7]. the main idea of this method is to design experiments efficiently so that it shows how different input factors affect the output [8]. this is done while minimizing variation (noise) and maximizing performance [9]. this analytical approach aims to correlate vibration measurements with the health status of the blades under different operating conditions. this can identify patterns and trends in vibration data indicative of potential defects or deterioration. these vibration amplitudes and frequencies may signal abnormalities or wear in turbine blades. condition monitoring procedures allow for proactive evaluation of turbine component health, enabling early defect detection and targeted maintenance interventions [10]. wind turbines are becoming increasingly larger in capacity [11]. this proactive maintenance technique empowers convenient intervention, reducing downtime, maintenance costs, and safety risks while optimizing energy generation and increasing the operational life expectancy of wind turbines. this research paper presents an analytical vibration analysis of axially defective wind turbine blades through time domain and regression analyses, as a condition monitoring approach. the novelty of this research study is to simulate sohar's actual wind speed data using a vibration lab wind turbine simulator to evaluate the vibration patterns of a wind turbine due to axially defective blades. 2. literature review currently, many research studies on blade vibration suppression focus on passive and active control methods. passive control adjusts the mass and stiffness distribution of the blade structure, changes the blade’s natural frequency, redesigns the blade structure, and replaces blade materials. lighter and harder carbon fibers were used to improve blade performance. the results showed that the new materials can reduce vibration speed by about 5% to ensure a certain bend-twist coupling [12]. active control suppresses blade vibration by installing dampers on the original blade and using advanced control strategies without adjusting the structure, designing new airfoils, or replacing blade materials. a new tuned damper was proposed to be installed inside the blade to improve the critical condition of blade vibration. the results showed that the tuned damper can increase the critical speed of blade flutter by 100% [13]. however, this method was mainly for the blade torsional flutter problem, rather than the axial vibration. an active optimal control scheme based on a piezoelectric actuator and linear rectangular gaussian theory was proposed to suppress wave vibration [14]. researchers aimed to decrease the vibration amplitude by reducing the fluctuations of the axial force moment [15][16][17]. additionally, as the blade vibrates, it is necessary to detect the vibration and modify the load control algorithm by identifying the blade dynamics [18]. blade vibration, which generally involves axial vibration, is related to the inherent dynamic characteristics and blade aerodynamic characteristics. however, there are still some deficiencies in current research on blade vibration due to various defect directions. with the increase in blade length, wind shear and tower shadow effects on blade vibration will become more prominent. therefore, it is worth considering how to prevent blade vibration caused by different effects. this paper investigates the analysis of blade vibration caused by axial defects and the effective optimization methods to suppress blade vibration. 3. methodology the research work involves wind speed data collection followed by practical design and setup for test measurement and signal processing. design of tests and model development are then employed. results are discussed through time-domain and analytical vibration analysis, alongside graphical analysis. regression and taguchi analyses are conducted to further understand vibration patterns. finally, prediction, optimization, and sensitivity analyses are study of the effects of axially defective blades… 13 copyright ©2025 assa. adv. in systems science and appl. (2025) performed to enhance comprehension of the effects of axially defective blades on wind turbine vibration. fig. 3.1 shows the methodology flow chart. fig. 3.1. methodology flow chart 3.1. data сollection the spectra quest wind turbine simulator is a lab facility installed in the vibration lab at sohar university, oman. the sohar town's monthly average wind speed is shown in table 3.1, which was measured at a height of 50 m using sohar university facilities. using the wind speed to rotational speed conversion for the spectra quest wind turbine simulator, the wind speed data was converted to a rotational speed at the low-speed shaft. then, this rotational speed was converted to a high-speed shaft rotational speed, at which the accelerometer was mounted, using the gearbox ratio of 4.571. a conversion error of 5% is considered in the spectra quest wind turbine simulator. the high-speed shaft rotational speed in table 3.1 ranges from almost 100 rpm to 150 rpm. to represent it clearly, a hypothetical range of rotational speed from 50 rpm to 200 rpm is proposed, considering the 50 rpm limit on both sides. considering an interval speed range of 50 rpm, the set of input rotational speeds is namely; 50 rpm, 100 rpm, 150 rpm and 200 rpm for the current research work. this research work includes practical design and setup, data measurement and signal processing methods, and vibration analysis techniques. the practical work was done by installing the axially defective wind turbine blades, one after the other, in the spectra quest wind turbine simulator and obtaining the resultant vibration patterns under the input set of rotational speeds. graphical representation and regression analysis were employed to correlate with the condition monitoring approach. table 3.1. monthly averaged wind speed and corresponding rotational speeds months average wind speed (m/s) low shaft speed (rpm) high shaft speed (rpm) jan 4.50 24.75 113.15 feb 5.12 28.16 128.74 mar 4.87 26.79 122.45 apr 4.80 26.40 120.69 may 5.47 30.09 137.54 jun 5.47 30.09 137.54 jul 5.35 29.43 134.52 aug 5.15 28.33 129.49 sep 4.92 27.06 123.71 oct 4.24 23.32 106.61 nov 3.86 21.23 97.06 dec 4.42 24.31 111.14 14 abdulhamid hamdan al-hinai, karu clement varaprasad, vinod kumar copyright ©2025 assa adv. in systems science and appl. (2025) 3.2. practical design and setup the spectra quest wind turbine simulator is a wind turbine lab facility with three blades. one of these blades was replaced by an axially defective blade for each test. based on the crack size, there are three different blades with different crack sizes, namely; low, medium and high. these axial defects were intentionally made for analysis purposes. after replacing the healthy blade with a defective blade, the system was tested for vibration analysis. the same was repeated using medium and high defective blades at each time. this simulator provides a labwise controlled environment for replicating real-world conditions, enabling precise testing and data collection. the varying rotational speeds were measured using a tachometer installed in the spectra quest wind turbine simulator, while the vibration level was measured by an accelerometer installed at the high-speed shaft of the spectra quest wind turbine simulator. each accelerometer measures the vibrational excitation in one direction [10]. the sensor’s readings are collected by a data acquisition system that is connected to a computer. vibra quest signal analysis software is used to analyze the signals the data acquisition system reads and generate output vibration reports. 3.3. design of experiments design of experiments (doe) is a systematic approach utilized to optimize processes and products by exploring the effects of various factors and their interactions. table 3.2 outlines the input parameters and their corresponding levels needed for conducting an effective design of experiments. these parameters include defect level with low, medium, and high categories, defect size with values of 50mm ≤ x1 ≤ 350mm, and shaft rotational speed with options of 50rpm ≤ x2 ≤ 200rpm. the impact of these inputs on vibration patterns can be evaluated by manipulating them at different levels. this facilitates the identification of optimal settings to enhance efficiency and reduce vibration levels. table 3.2. input parameters and their levels parameters input levels defect level low medium high defect size (mm) x1 50 100 350 shaft rotational speed (rpm) x2 50 100 150 200 3.4. model development to optimize machine learning model development for wind turbine performance, a multifaceted approach integrating wind to rotational speed conversion and crack size vibration effects is essential. leveraging monthly averaged wind speeds and corresponding rotational speeds in table 3.1 provides a foundational dataset for training and validation. additionally, the input parameters and their levels in table 3.2 offer critical variables for model refinement. these data can be analyzed to predict optimal shaft speeds based on wind conditions and detect potential defects by harnessing advanced algorithms, including regression analysis. this holistic optimization strategy not only enhances turbine efficiency but also minimizes maintenance costs and downtime, contributing to sustainable energy production. 4. results and discussion the vibra quest software exported the vibration results in ms excel format, after reading the data from the vibration sensor (accelerometer). at the same time, the software shows the timedomain vibration waveforms with the times of a maximum of two time-peaks set by the software. for each test, the generated file presents about 32768 vibration peaks (positive and negative waveform peaks) that were used for vibration analysis. in this results section, the time domain waveforms are discussed for each case of the defective blades (low, medium and high) study of the effects of axially defective blades… 15 copyright ©2025 assa. adv. in systems science and appl. (2025) run at each rotational speed (50 rpm, 100 rpm, 150 rpm and 200 rpm) followed by the analytical analysis of the vibration patterns. after that, the graphical and regression analyses are discussed. finally, the vibration model optimization is discussed for prediction and sensitivity analysis. 4.1. time-domain vibration analysis the practical tests of the vibration analysis presented the vibration patterns obtained by the accelerometer at different rotational speeds using the vibra quest wind turbine simulation system. at each case of defect size, four tests were performed at each rotational speed (50 rpm, 100 rpm, 150 rpm and 200 rpm) respectively. 4.1.1. case 1: low defective blade the spectra quest wind turbine simulation system operated first with two healthy blades and one low defective blade. figs. 4.1 to 4.4 show the vibration waveform readings at each rotational speed respectively, generated by vibra quest software. as the rotational speed increases, the amplitude of the vibration level increases. the figs. show also the pre-set twotime times when higher values of amplitudes occur. fig. 4.1. vibration waveform with a low defective blade at 50 rpm fig. 4.2. vibration waveform with a low defective blade at 100 rpm fig. 4.3. vibration waveform with a low defective blade at 150 rpm fig. 4.4. vibration waveform with a low defective blade at 200 rpm 4.1.2. case 2: medium defective blade in this case, the low defective blade is replaced by a medium defective blade. the spectra quest wind turbine simulation system is operated with two healthy blades and one medium defective blade. figs. 4.5 to 4.8 show the vibration waveform readings at each rotational speed respectively. same as in case 1, the amplitude of the vibration level increases with the increase in rotational speed. the times when higher values of amplitudes occur are shown also in these figs. 16 abdulhamid hamdan al-hinai, karu clement varaprasad, vinod kumar copyright ©2025 assa adv. in systems science and appl. (2025) fig. 4.5. vibration waveform with a medium defective blade at 50 rpm fig. 4.6. vibration waveform with a medium defective blade at 100 rpm fig. 4.7. vibration waveform with a medium defective blade at 150 rpm fig. 4.8. vibration waveform with a medium defective blade at 200 rpm 4.1.3. case 3: high defective blade in case 3, the spectra quest wind turbine simulation system operated with two healthy blades and one high defective blade. figs. 4.9 to 4.12 show the vibration waveform readings at each rotational speed respectively. the same as in cases 1 and 2, the amplitude of the vibration level increases as the rotational speed increases. these graphs also display the times at which the amplitudes reach larger values. fig. 4.9. vibration waveform with a high defective blade at 50 rpm fig. 4.10. vibration waveform with a high defective blade at 100 rpm study of the effects of axially defective blades… 17 copyright ©2025 assa. adv. in systems science and appl. (2025) fig. 4.11. vibration waveform with a high defective blade at 150 rpm fig. 4.12. vibration waveform with a high defective blade at 200 rpm 4.2. analytical vibration analysis for this research work, the vibra quest software generated the vibration patterns in the voltage unit. for each test, about 32768 vibration peaks (positive and negative) were generated by the software. an analytical analysis was conducted including the minimum (min) and maximum (max) peak values, the averages of (n) negative peaks (avg. (min)) and (m) positive peaks (avg. (max)). additionally, determining the vibration range, spanning from the average of positive peaks to the average of negative peaks shows the amplitude dispersion within the recorded data. table 4.1 tabulates these results from the generated reports. for each test, the following equations were used: 𝐴𝑣𝑔. (𝑀𝑖𝑛) = ∑ (𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑣𝑖𝑏𝑟𝑎𝑡𝑖𝑜𝑛 𝑝𝑒𝑎𝑘𝑠) 𝑛 (4.1) 𝐴𝑣𝑔. (𝑀𝑎𝑥) = ∑ (𝑝𝑜𝑠𝑖𝑡𝑣𝑒 𝑣𝑖𝑏𝑟𝑎𝑡𝑖𝑜𝑛 𝑝𝑒𝑎𝑘𝑠) 𝑚 (4.2) 𝑉𝑖𝑏𝑟𝑎𝑡𝑖𝑜𝑛 𝑅𝑎𝑛𝑔𝑒 = 𝐴𝑣𝑔. (𝑀𝑎𝑥) − 𝐴𝑣𝑔. (𝑀𝑖𝑛) (4.3) table 4.1. vibration waveform results test input parameters vibration waveform analysis defect level defect size (mm) rot. speed (rpm) min (v) max (v) avg. (min) (v) avg. (max) (v) vibration range (v) 1 low 50 50 -0.028766 0.025389 -0.004481 0.004612 0.009093 2 low 50 100 -0.080538 0.075132 -0.015803 0.015027 0.030830 3 low 50 150 -0.150698 0.122752 -0.017426 0.017276 0.034702 4 low 50 200 -0.141121 0.169997 -0.019577 0.019999 0.039577 5 medium 100 50 -0.025456 0.024302 -0.004495 0.004679 0.009174 6 medium 100 100 -0.077624 0.069251 -0.015792 0.015070 0.030862 7 medium 100 150 -0.207226 0.144662 -0.019152 0.018824 0.037976 8 medium 100 200 -0.144500 0.148000 -0.021969 0.022760 0.044729 9 high 350 50 -0.023882 0.027492 -0.004298 0.004885 0.009183 10 high 350 100 -0.076275 0.065869 -0.016437 0.014879 0.031316 11 high 350 150 -0.148958 0.135611 -0.020603 0.020607 0.041210 12 high 350 200 -0.200616 0.177882 -0.022574 0.023095 0.045668 as shown in table 4.1, the tests investigate how the vibration waveform features are affected by different rotation speeds and defect sizes. the generated waveforms are examined in terms of their minimum peaks, maximum peaks, average of minimum peaks, average of maximum peaks, and the range from maximum to minimum average peaks for every combination of parameters. the range of average peaks widens as the rotation speed increases at each defect level. 18 abdulhamid hamdan al-hinai, karu clement varaprasad, vinod kumar copyright ©2025 assa adv. in systems science and appl. (2025) at a low defect level, as the rotation speed increases from 50 to 200 rpm, there is a gradual expansion in the range of average vibration. for instance, as shown in fig. 4.1, where the defect size is 50 mm, the range of average voltages increases from 0.009093 v at 50 rpm to 0.039577 v at 200 rpm, as shown in fig. 4.4. this expansion suggests that higher rotation speeds exacerbate vibration fluctuations, likely due to increased stress and deformation within the system. similarly, at medium and high defect levels, the range of average voltages exhibits a notable increase as both defect size and rotation speed rise, as shown in figs. 4.5 to 4.8 for a medium defective blade and figs. 4.9 to 4.12 for a high defective blade. 4.3. graphical analysis of vibration the collected vibration data is graphically analyzed in fig. 4.13 to represent the peak-peak levels. the graphs show the relationship between vibration levels in terms of defect size and rotational speed. as the rotational speed increases, the vibration level increases for each defect size. the higher the defect size, the higher the vibration level at the same rotational speed. fig. 4.13. peak-peak vibration the presented graphs in fig. 4.13 reflect the dynamic behavior of the wind turbine system. till 100 rpm rotational speed, the graphs are almost linearly parallel and then they vary due to either non-linear response of the system, resonance effects, non-linearities in the defects or dynamic response of the system. table 4.2 presents a comparison of vibration levels for each defect level with the average vibration level across each rotational speed. the higher the crack size, the higher the vibration level. however, as rotational speed increases, the impact of crack severity becomes more pronounced, with higher speeds exacerbating vibration levels, particularly evident in cases of medium and high crack severities. table 4.2. vibration levels comparison rotational speed (rpm) average vibration (v) low crack vibration (v) medium crack vibration (v) high crack vibration (v) 50 0.009150 0.009093 0.009174 0.009183 100 0.031002 0.030830 0.030862 0.031316 150 0.037963 0.034702 0.037976 0.041210 200 0.043325 0.039577 0.044729 0.045668 4.4. regression analysis of vibration regression analysis is performed to quantify the relationship between defect size and shaft rotational speeds with vibration levels. the quadratic function represents the best regression expression for the research model compared with linear regression. table 4.3 shows the input factors and output response set for the quadratic regression model. table 4.3. input and output parameters for a quadratic regression model crack size (mm) crack size^2 (mm^2) rot. speed (rpm) rot. speed^2 (rpm^2) crack size × rot. speed (mm × rpm) vibration range (v) 50 2500 50 2500 2500 0.009093 50 2500 100 10000 5000 0.030830 study of the effects of axially defective blades… 19 copyright ©2025 assa. adv. in systems science and appl. (2025) 50 2500 150 22500 7500 0.034702 50 2500 200 40000 10000 0.039577 100 10000 50 2500 5000 0.009174 100 10000 100 10000 10000 0.030862 100 10000 150 22500 15000 0.037976 100 10000 200 40000 20000 0.044729 350 122500 50 2500 17500 0.009183 350 122500 100 10000 35000 0.031316 350 122500 150 22500 52500 0.041210 350 122500 200 40000 70000 0.045668 the quadratic regression analysis was performed using minitab software. table 4.4 gives the regression statistics of the regression model. table 4.4. regression statistics multiple r r square adjusted r square standard error observations 0.990879 0. 981841 0. 966708 0. 002504 12 as shown in table 4.4, the multiple r of 0.990879 indicates a strong positive and direct relationship between the independent variables (defect size and rotational speeds) and the dependent variable (vibration level). the r square of 0.981841 shows that approximately 98.18% of the variance in the vibration level is clarified by the defect size and rotational speeds in the regression model. the adjusted r square is calculated as 0.966708, slightly lower than the r square, showing that the model may have some degree of overfitting. the standard error of 0.25% indicates the average amount of error in the predicted values of the vibration level. the analysis of variance (anova) in this research work is conducted to evaluate the implications of the regression model and its components. this statistical technique helps determine if the observed variations in vibration levels can be attributed to the different defect sizes tested. table 4.5 shows the anova results and table 4.6 shows the regression summary. the anova table offers a clear understanding of the effects of vibration caused by axially defective blades at different rotation speeds, ranging from 50 to 200 rpm. the f-statistic of 64.88 indicates a strong correlation between rotation speed and vibration caused by defective blades. additionally, the significance f value of 3.85×10-5 suggests that this correlation is statistically significant, reinforcing the idea that changes in rotation speed affect vibration caused by axially defective blades. table 4.5. analysis of variance (anova) df ss ms f significance f regression 5 0.002034 0.000407 64.88184 3.85×10-5 residual 6 3.76×10-5 6.27×10-6 total 11 0.002071 table 4.6. regression summary coefficients standard error t stat p-value intercept -0.02011 0.005763 -3.48954 0.012991 crack size 5.73×10-5 6.5×10-5 0.881162 0.412133 crack size^2 -1.6×10-7 1.58×10-7 -1.0019 0.355069 rot. speed 0.000612 7.51×10-5 8.139658 0.000185 rot. speed^2 -1.6×10-6 2.89×10-7 -5.70372 0.001256 crack size × rot. speed 1.23×10-7 1×10-7 1.224465 0.266668 the regression equation derived from this analysis offers a mathematical model for predicting vibration levels (peak-peak) based on defect size and shaft rotational speed: 20 abdulhamid hamdan al-hinai, karu clement varaprasad, vinod kumar copyright ©2025 assa adv. in systems science and appl. (2025) vibration level = −0.02011 + 5.73 × 10 𝑥 − 1.6 × 10 𝑥 +0.000612𝑥 − 1.6 × 10 𝑥 + 1.23 × 10 𝑥 𝑥 (4.4) where x1 is the defect size and x2 is the shaft rotational speed. considering this equation, the predicted vibration level could be calculated. fig. 4.13 can be integrated with the predicted vibration (peak-peak) level. fig. 4.14 shows the predicted vibration levels graph for each defect size at different shaft rotational speeds as well as the actual vibration levels graph. fig. 4.14. peak-peak vibration (actual and predicted) fig. 4.14 presents an insightful visualization of a comparative analysis between predicted and actual vibration levels across varying defect sizes and shaft rotational speeds. the predicted vibration levels graphs intricately overlay the predicted outcomes based on the regression analysis, while the actual vibration levels graph reflects empirical data obtained from tested observations. the collocation of these two datasets shows a clear degree of alignment between theoretical predictions and practical manifestations. figs. 4.15 and 4.16 show the defect size line fit plot and the rotational speed line fit plot, respectively. figs. 4.15 and 4.16 collectively show a multi-layered investigation into the relationship between defect size estimated at varying rotational speeds and vibration levels as an investigation of anticipated and real vibration information. they express a relationship in a scatter plot of actual and predicted data points. fig. 4.15 shows how diverse defect sizes impact vibration levels, whereas fig. 4.16 shows how changes in the rotational speed of the wind turbine shaft correspond to changes in the vibration level of the turbine. fig. 4.15. defect size line fit plot fig. 4.16. rotational speed line fit plot 4.5. taguchi analysis in this taguchi analysis, the focus is on understanding the relationship between vibration range and crack size concerning rotational speed. table 4.7 shows the response table for signal to noise ratios (considering smaller vibration is better). this table 4.7 shows the signal to noise ratios (snr) for two factors: crack size and rotational speed. the smaller the ratio, the better the quality. it indicates less variation or noise in the output. the factors are tested at different levels, denoted as 1, 2, 3, and 4. delta represents the difference between the best and worst snr for each factor. rank indicates the ranking of each factor based on signal to noise ratios. fig. 4.17 shows the main effects plot for snr. study of the effects of axially defective blades… 21 copyright ©2025 assa. adv. in systems science and appl. (2025) the other consideration of taguchi analysis is the response table for means, shown in table 4.8. this table 4.8 shows the means for crack size and rotational speed. the means indicate the average values of the factors at different levels. delta represents the difference between the best and worst means for each factor. rank indicates the ranking of each factor based on the mean values. fig. 4.18 shows the main effects plot for means. table 4.7. response table for signal to noise ratios table 4.8. response table for means level crack size rot. speed 1 32.07 40.77 2 31.59 30.17 3 31.33 28.43 4 27.28 delta 0.74 13.49 rank 2 1 level crack size rot. speed 1 0.028550 0.009150 2 0.030685 0.031002 3 0.031844 0.037963 4 0.043325 delta 0.003294 0.034175 rank 2 1 fig. 4.17. main effects plot for sn ratios fig. 4.18. main effects plot for means taguchi analysis results can be interpreted as follows: 4.5.1. crack size: in terms of signal to noise ratio (snr), the best performance is achieved at level 2, where the snr is 31.59, indicating less variation in the output. in terms of means, the best performance is also at level 2, with a mean value of 0.030685. therefore, for crack size, level 2 performs the best in both snr and mean values. 4.5.2. rotational speed: in terms of snr, the best performance is at level 3, with an snr of 28.43. in terms of means, the best performance is at level 1, with a mean value of 0.009150. therefore, for rotational speed, there is a discrepancy between the optimal level based on snr and mean values. level 3 is optimal based on snr, while level 1 is optimal based on mean values. 4.5. prediction, optimization and sensitivity analysis the regression analysis determines the relationship between vibration responses and independent factors (crack size and rotational speed) by using defined regression algorithms and relationship estimates [19]. from the regression analysis shown in table 4.6, the p-value for the shaft rotational speed is 0.000185 (<0.05) which is fine, but the p-value for the crack size is 0.412133 (>0.05). for optimization, the p-value of the crack size is high and should be excluded in regression analysis. the longer bar represents the input that contributes more to the model. considering the prediction and optimization of the multiple regression model, the following terms are in the fitted equation that models the relationship between vibration range and the rotation speed variables (x2: rotational speed and x2^2). the model fits the data well. consequently, this equation can be used to predict the vibration range for specific values of the x variables and find the settings for the x variables that correspond to a desired value or range of values for the vibration range. the final model equation is: 35010050 42 40 38 36 34 32 30 28 26 20015010050 crack size (mm) m ea n of s n ra tio s rotational speed (rpm) main effects plot for sn ratios data means signal-to-noise: smaller is better 35010050 0.045 0.040 0.035 0.030 0.025 0.020 0.015 0.010 20015010050 crack size (mm) m ea n of m ea ns rotational speed (rpm) main effects plot for means data means 22 abdulhamid hamdan al-hinai, karu clement varaprasad, vinod kumar copyright ©2025 assa adv. in systems science and appl. (2025) 𝑉𝑖𝑏𝑟𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑛𝑔𝑒 = −0.01762 + 0.000631𝑥 + 0.000002𝑥 (4.5) the model-building sequences in fig. 4.19 display the order in which terms were added or removed in the regression model. fig. 4.19. model building sequences the final regression model equation shows that the relationship between the variables in the model is statistically significant (p<0.1) with 96.65% of the variation in vibration range can be explained by the regression model 5. conclusion the results of this study provide an important understanding of how vibration patterns are affected by axially defective wind turbine blades, with significant implications for improving maintenance and operating reliability. through time-domain analysis, the study revealed a noticeable increase in vibration amplitude corresponding to the increased rotational speeds and defect severity levels. this trend underscores the essential role of rotational speed as a primary determinant of vibration level, with defects exacerbating the vibrational response. utilizing analytical techniques such as regression analysis, a robust correlation between defect size, rotational speed, and vibration levels was established, with the regression model demonstrating a commendable explanatory power (r square = 0.981841). additionally, taguchi analysis discerned optimal parameter settings for mitigating vibration-induced risks, guiding how to reduce operational disturbances and improve wind turbine performance. furthermore, predictive and optimization models derived from regression analysis present actionable insights for targeted involvement strategies aimed at the limitation of vibration-related hazards. this research underscores the potential for proactive management of wind turbine integrity by integrating advanced analytical methodologies, thereby fostering sustained operational efficiency and reliability within renewable energy systems. the research findings contribute to advancing the understanding of wind turbine dynamics and asses performance and longevity while ensuring safety and sustainability in wind energy production, thereby supporting the continued growth of renewable energy technologies. acknowledgement the authors are thankful to the faculty of engineering at sohar university for their guidance and cooperation. competing interests the authors declare that they have no competing interests relevant to this research paper. availability of data and materials materials used in this study are available upon reasonable request from the corresponding author. step change step p final p 2 1 add x2^2 add x2 0.001 0.000 0.001 0.000 1007550250 r-squared(adjusted) % displays the order in which terms were added or removed. study of the effects of axially defective blades… 23 copyright ©2025 assa. adv. in systems science and appl. (2025) references [1] ogaili, a. a. f., jaber, a. a., & hamzah, m. n. (2023). wind turbine blades fault diagnosis based on vibration dataset analysis, data in brief, 49, 109414, doi: 10.1016/j.dib.2023.109414 [2] zhou, f., yang, j., pang, j., & wang, b. (2023). research on control methods and technology for reduction of large-scale wind turbine blade vibration, energy reports, 9, 912– 923, doi: 10.1016/j.egyr.2022.12.042 [3] pacheco, j., pimenta, f., guimarães, s., castro, g., cunha, á., et al. (2024). experimental evaluation of fatigue in wind turbine blades with wake effects, engineering structures, 300, 117140, doi: 10.1016/j.engstruct.2023.117140 [4] teng, w.; ding, x.; tang, s.; xu, j.; shi, b., et al. (2021). vibration analysis for fault detection of wind turbine drivetrains – a comprehensive investigation, sensors 2021, 21, 1686, doi: 10.3390/s21051686 [5] liu, x., lü, c., shi, l., godbole, a. r., & chen, y. (2015). influence of the vibration of large-scale wind turbine blade on the aerodynamic load, energy procedia, 75, 873–879. doi: 10.1016/j.egypro.2015.07.196 [6] xiao, f., tian, c., wait, i. w., yang, z., still, b., et al. (2020). condition monitoring and vibration analysis of wind turbine, advances in mechanical engineering, 12(3), 168781402091378, doi: 10.1177/1687814020913782 [7] kaushik, n. v., & shankar, n. n. (2022). statistical analysis using taguchi method for designing a robust wind turbine, journal of advanced research in fluid mechanics and thermal sciences, 100(3), 92–105, doi: 10.37934/arfmts.100.3.92105 [8] pane, m., & saragih, r. (2024). optimal design of gorlov turbine using computational fluid dynamics and taguchi method, vanos journal of mechanical engineering education, 9(1), 25, doi: 10.30870/vanos.v9i1.24496 [9] boukenoui, r., bradai, r., & kheldoun, a. 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(2016). individual pitch controller based on fuzzy logic control for wind turbine load mitigation, iet renewable power generation, 10(5), 687–693, doi: 10.1049/iet-rpg.2015.0320 24 abdulhamid hamdan al-hinai, karu clement varaprasad, vinod kumar copyright ©2025 assa adv. in systems science and appl. (2025) [17] zhang, y., cheng, m., & chen, z. (2015). load mitigation of unbalanced wind turbines using pi‐r individual pitch control, iet renewable power generation, 9(3), 262–271, doi: 10.1049/iet-rpg.2014.0242 [18] dai, y., xie, f., li, b., wang, c., & shi, k. (2023). effect of blade tips ice on vibration performance of wind turbines, energy reports, 9, 622–629, doi: 10.1016/j.egyr.2022.12.092 [19] popović, b. (2019). the optimal solution to the problems by regression analysis, journal of engineering management and competitiveness, 9(1), 25–37, doi: 10.5937/jemc1901025p adv syst sci appl 2023; 02:178–183 published online at https://ijassa.ipu.ru. a combinatorial view on derivations in bimodules andronick a. arutyunov* v. a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia moscow institute of physics and technology (state university), dolgoprudniy, russia abstract: this paper is devoted to derivations in bimodules over group rings using previously proposed methods which are related to character spaces over groupoids. we consider bimodules which are closures of a group algebra with respect to norms of subordinate supremum norma. the main theorem proves that all derivations in such bimodulues are quasiinner. we also consider some examples, in particular the case of (σ, τ)-derivations. keywords: derivations, bimodule, outer derivations 1. introduction derivations in various associative algebras have been actively studied since the middle of the last century. in particular, the following question, known as the johnson problem (or ”derivation problem”), is widely known: hypothesis ( [1], question 5.6.b): is it true that all derivations in l1(g) are inner? here and hereafter g is a finitely generated generally noncommutative group. a partial answer to this question was given by b. johnson himself in [2], and the most complete answer was found by v. losert in [3]. a more detailed description of the history of this problem is given in [1, 4]. in purely algebraic form, consider the group ring c[g], that is, the space of all linear combinations of the form ∑ g∈g x(g)g, where x(·) is a finite function – a function with a finite support. the derivation in this case is a linear operator d : c[g] → c[g] satisfying the leibniz rule d(uv) = d(u)v + ud(v), ∀u, v ∈ c[g]. the johnson’s problem is whether there are derivation other than inner, i.e., having the form da : x→ [a, x], a ∈ c[g]. in this formulation of the problem, the algebra of outer derivations will be nontrivial (see central derivations from [5] as well as more general results from [4]). in the present paper we focus on the study of banach spaces equipped with a bimodule structure over a group ring c[g] of the following form. let us fix a norm ∥ · ∥ in c[g] and let a be the closure of the group ring by this norm. a is not an algebra, but it is naturally understood as a free bimodule over the ring c[g]. ∗corresponding author: andronick.arutyunov@gmail.com a combinatorial view on derivations in bimodules 179 the ring c[g] will be defined as a space with the supremum norm ∥ · ∥|s, i.e. for ω = ∑ g∈g x(g)g we put ∥ω∥s := sup g∈g |x(g)|. the boundedness of operators will be defined as follows. let us define the norm of the operator d as ∥d∥ = sup 0̸=ω∈c[g] ∥d(ω)∥ ∥ω∥s . (1.1) definition 1.1: by the derivation over c[g] with values in bimodule a we will call a linear bounded operator d : c[g] → a such that d(uv) = d(u)v + ud(v), ∀u, v ∈ c[g]. (1.2) the space of such operators we will denote by der(a). for a wide class of norms (including natural class of ℓp-norms) the following statement will be proved. theorem 1.1: if the norm ∥ · ∥ is subordinate to the supremum norm, then all derivations in the bimodule a are quasi-inner. let us explain about which norms we are talking about. definition 1.2: we will say that the norm ∥ · ∥ is subordinated to the norm ∥ · ∥s iff ∥ω∥ <∞ =⇒ ∥ω∥s <∞ a rigorous definition of quasi-inner derivations is given below (see definition 2.1). informally speaking, quasi-inner derivations are such operators that can be represented as a formal infinite sum of inner derivations, i.e. the commutator x→ [a, x], for the element a not lying in a in general. 2. preliminaries the following definitions and results are based on [4, 6]. let γ be a groupoid of connected action in which objects coincide with elements of the group obj(γ) = g, and morphisms are pairs of elements of the group, i.e. hom (γ) = g×g. moreover, the morphism ϕ := (u, v) has the source s(ϕ) = v−1u and the target t(ϕ) = uv−1. we will call endomorphisms (i.e. morphisms whose source and target coincide) ϕ as loops for clarity. then we define a character as a complex-valued function χ : hom (γ) → c such that χ(ψ ◦ ϕ) = χ(ψ) + χ(ϕ), (2.3) for all pairs of linkable morphisms ϕ, ψ. the groupoid γ will be represented as an uncoupled union of subgroupoids γ[u], where [u] is a class of conjugate elements. [u] = {tut−1|t ∈ g}. objects of the subgroupoid γ[u] coincide with the class [u]. copyright © 2023 assa. adv syst sci appl (2023) 180 a. aurutyunov lemma 2.1: for every derivation d ∈ der(as) there exists a character χ ∈ x(γ) such that the following formula holds d(g) = g (∑ t∈g χ(gt, g)t ) ,∀g ∈ g. (2.4) proof the proof literally repeats the proof of theorem 1 of [5] except for the property of local finiteness, which was a consequence of the finiteness of nonzero terms in elements of group algebras. let us make it clear that character values are coefficients of the expansion of a linear operator by standard basis of the ring c[g]. due to the implementation of the leibniz rule, the calculation shows that the property (2.3) is fulfilled. the space of quasi-inner derivations was introduced earlier in [6]. it will play an important role below. definition 2.1: we call the derivation d ∈ der(a(g)) quasi-inner if the character χ in the formula (2.4) is zero on all loops. it is easy to see that the inner derivations are quasi-inner. indeed, let a ∈ g be a basis element, then the character χa corresponding to the inner derivation da : x→ [x, a] has the form (see section 2.2 in [5] for details) χa(ϕ) = 1, ϕ ∈ hom (a, b), b ̸= a, −1 ϕ ∈ hom (b, a), b ̸= a, 0 otherwise. the character χa is zero on all loops. the inner derivation is a sum of (possibly infinite) characters χa, and hence is itself trivial on loops. the role of quasi-inner derivations for group rings is that they form an ideal containing the ideal of inner derivations (theorem 4.1, [6]), with examples of quasi-inner derivations that are not inner in the group ring (section 3.3 of [5] – the case of heisenberg group). the coincidence of spaces of inner and quasi-inner derivations in a bimodule a is equivalent to the fact that inner derivations are dense in the space of quasi-inner derivations. 3. derivations in bimodules recall that in the group ring c[g] we have a ”supremum norm”. for x = ∑ g∈g x(g)g ∈ c[g], where x(·) is a finite function, the norm is ∥x∥s := sup g∈g |x(g)|. (3.5) the banach bimodule that is the closure of c[g] by the supremum norm is denoted by as(g). the bimodule as(g) can be understood as the space of elements of the form∑ g∈g x(g)g, where the function x(g) is bounded. this a sort of free bimodule over c[g]. lemma 3.1: all derivations with values in the banach bimodule as are quasi-inner. copyright © 2023 assa. adv syst sci appl (2023) a combinatorial view on derivations in bimodules 181 proof let χ be the character corresponding in the sense of lemma 2.1 to a nontrivial derivation d. let us show that if the character χ takes a nonzero value on some loop, then the operator d is unbounded. consider the loop ϕ ∈ hom (g, g). as it can be seen from the definition, for some t ∈ z(g), the notation ϕ = (gt, t) is valid. let χ(ϕ) ̸= 0. without loss of generality, we may assume that χ(ϕ) = 1. note that the element t cannot be of finite order in the group g, otherwise we would get that χ((g, e)) = ord(t), which is impossible since (g, e) is a neutral morphism. given that t is of infinite order and using the property (2.3), we get the following χ((gtn, tn)) = χ(ϕn) = n. with the formula (2.4) we get that ∥d(tn)∥s ≥ n. so ∥d(tn)∥s → ∞ at n→ ∞. at the same time ∥gtn∥s = 1. in this case the operator d – converts a bounded sequence into an unbounded one, and hence is not bounded itself. proposition 3.1: if the character χ defines the derivation d ∈ der(as) then for any two morphisms ϕ, ψ ∈ hom (a, b) we have that χ(ϕ) = χ(ψ). proof by the lemma 3.1 for each derivation d ∈ der(as), the corresponding character χ, in the sense of the lemma 2.1, is zero on loops. for two different objects a ̸= b, and two morphisms ϕ, ψ ∈ hom (a, b) (if they exist, of course) there exists a loop ζ ∈ hom (a, a) such that ϕ ◦ ζ = ψ. so, since χ(ζ) = 0, by the formula (2.3) we have that χ(ϕ) = χ(ψ). let us proceed to the proof of the 1.1 theorem, i.e. we show that if norm ∥ · ∥ is subject to the supremum norm ∥ · ∥s, then all derivations with values in the bimodule a are quasi-inner. proof of theorem 1.1 consider a bounded operator d over a. from boundedness we obtain that for some constant a holds an inequaluty ∥d(g)∥ < a for all basis elements g ∈ g. from the norm subordination we have that ∥d(g)∥s < ca. so the character χ corresponding to the derivation d is trivial on loops by the lemma 3.1, and so the derivation d is quasi-inner. 4. examples and applications consider the space ℓp(g), i.e., all elements of the form ω = ∑ g∈g x(g)g, bounded by the ℓp-norm ∥ω∥p := p √∑ g∈g |x(g)|p. (4.6) it is clear that such norm is subordinate to the supremum for p ≥ 1: ∥ω∥p ≤ sup g∈g |x(g)|. example 4.1: all derivations in der(ℓp(g)), p ≥ 1 are quasi-inner. in [8] was shown that for p = 1 all derivations are inner. the case p > 1 is new. the natural question remains as to whether inner and quasi-inner derivations coincide. this question look not obviouse. but note that qasi-inner derivations can be understood as formal sums of inner one. therefore, in terms of analysis difference between inner and quasi-inner derivations is minor. copyright © 2023 assa. adv syst sci appl (2023) 182 a. aurutyunov the case of (σ, τ)-derivations it is clear that in the proof of theorem 1.1 the ability to represent derivations via groupoid characters plays the key role. this construction can be applied to other structures as well, with a corresponding change in the groupoid structure. thus, in [9], (σ, τ)-derivations, i.e. operators satisfying a ”twisted” leibniz rule, were investigated. conditions and applications of (σ, τ)-derivations can be found in [10, 11]. let us give a slightly more general definition to fit our case. definition 4.1: a linear bounded operator d : c[g] → a such that for some endomorphisms σ, τ : c[g] → c[g] d(ab) = d(a)σ(b) + τ(a)d(v), a, b ∈ c[g], (4.7) let’s call a (σ, τ)−derivation. as shown in [9] for the ”twisted” groupoid (see [9], section 3) (σ, τ)−derivation can be presented by its characters ( [9], theorem 1). applying the same reasoning as in the proof of the theorem 1.1 to the case of (σ, τ)−derivations we obtain the ”twisted” analog of our theorem. example 4.2: all (σ, τ)−derivations with values in bimodule a generated by a norm subordinate supremum – are (σ, τ )-quasi-inner. here (σ, τ )-quasi-inner derivations are defined similarly as (σ, τ )-derivations given by characters identically equal to zero on loops. central derivations the proved theorem and the proposed approach also suggest examples of norms in which outer derivations appear. recall ( [5], section 2.3) that the central derivation dtz is an operator which is given by a center element of the group z ∈ z(g) and a non-trivial homomorphism τ : g→ (c,+) into the additive group of complex numbers on generators g ∈ g ⊂ c[g] by the formula dtz : g → τ(g)gz. (4.8) central derivations form a subalgebra ( [5], theorem 2). it is easy to see that the corresponding character χt z in the sense of the lemma 2.1 is non-trivial on loops (see [5], proposition 5). more precisely, its support is a subgroupoid with one object (just a fixed central element). from which we obtain that the central derivation cannot be quasi-inner. a simple calculation shows that the central derivations are not bounded in the supremum norm. it follows from the theorem 1.1 but this also can easily be checked by direct calculation. however, it is possible to choose a norm which is not subordinate to the supremum. this case was discussed in more detail in the joint paper with a. najanzin [7]. let us briefly describe one example of a norm in which the central differentials are bounded. we define the norm ∥ · ∥∗α for ω = ∑ g∈g x(g)g as follows ∥ω∥∗α := ∑ g∈g |x(g)|e−α|g|. (4.9) we denote the closure of c[g] by this norm as a∗ α. example 4.3 (theorem 2.1, [7]): for all sufficiently large α > 0, central derivations with values in the bimodule a∗ α are bounded. copyright © 2023 assa. adv syst sci appl (2023) a combinatorial view on derivations in bimodules 183 in the case of a group ring (i.e. the case of finite linear combinations without topological space structure) the ideals of inner and quasi-inner derivations do not coincide (see [5], the case of heisenberg group). moreover, the difference between inner and quasi-inner derivations is determined by the combinatorial properties of the group. as the proof of the theorem 1.1 shows in the case of normalized bimodules, this dependence vanishes. so the following assumption seems plausible: whether all derivations in ℓp(g) are inner. acknowledgements the research is supported by the ministry of science and higher education of the russian federation (goszadanie n. 075-00337-20-03, project 0714-2020-0005). references 1. dales, h. g. (2000). banach algebras and automatic continuity, oxford, uk: oxford university press. 2. johnson, b. e. (2001). the derivation problem for group algebras of connected locally compact groups, j. london math. soc., 63(2), 441–452. 3. losert, v. (2008). the derivation problem for group algebras, ann. of math., 168(1), 221–246. 4. arutyunov, a. a., & mishchenko, a. s. (2019). a smooth version of johnson’s problem on derivations of group algebras, mat. sb., 210(6), 3–29, doi: https://doi.org/10.4213/sm9119. 5. arutyunov, a. a. (2020). derivation algebra in noncommutative group algebras, proc. steklov inst. math., 308, 22–34, doi: https://doi.org/10.1134/s0081543820010022 6. arutyunov, a. a., & alekseev, a. v. (2019). cohomology of n-categories and derivations in group algebras, topology and its applications, 275, 107002 doi: https://doi.org/10.1016/j.topol.2019.107002. 7. arutyunov, a. a., & naianzin, a. v. (2022). nontrivial outer derivations in bimodules over group rings, eurasian mathemeatical journal, 13(4), 8–17, doi: https://doi.org/10.32523/2077-9879-2022-13-4-08-17. 8. johnson, b. e., & ringrose, j. r. (1969). derivations of operator algebras and discrete group algebras, bull. london math. soc., 1, 70–74. 9. alekseev, a., arutyunov, a., & silvestrov, s. (2020). on (σ, τ)-derivations of group algebra as category characters. arxiv: 2008.00390, [online]. available: https://arxiv.org/abs/2008.00390 10. hartwig, j. t., larsson, d., & silvestrov, s. d. (2006). deformations of lie algebras using σ derivations, j. algebra, 295(2), 314–361. 11. elchinger, o., lundengard, k., makhlouf, a., & silvestrov, s. d. (2016). brackets with (τ, σ)-derivations and (p, q)-deformations of witt and virasoro algebras, forum math., 28(4), 657–673. copyright © 2023 assa. adv syst sci appl (2023) introduction preliminaries derivations in bimodules examples and applications microsoft word source+texts,+revision_assa+osmcpid adv syst sci appl 2024; 03; 43-57 published online at https://ijassa.ipu.ru. enhancing comfort and stability of a vehicle suspension system when oscillating by an optimal hybrid control algorithm tuan anh nguyen* thuyloi university, hanoi, vietnam abstract: this article proposes a control algorithm for an automotive suspension system to improve ride comfort. the control method is based on the combination of sliding mode control (smc) and proportional-integral-derivative (pid), with parameters optimally selected by an inloop algorithm called osmcpid. the system's performance is evaluated by simulation, which goes through three specific cases. according to the article's findings, in the first two cases, the average value of vehicle body displacement when using the proposed algorithm is only 7.71% and 6.96%, respectively, compared to mechanical suspension. in the third case, vehicle body displacement approaches zero once the road excitation signal ends. meanwhile, the remaining cases have a more significant gap. in addition, the phase difference of the control signal is slight, while the chattering phenomenon almost does not occur when using the hybrid algorithm. in general, the system can adapt to changing external conditions. keywords: active suspension system, hybrid algorithm, chattering phenomenon, vehicle vibration. 1. introduction roughness on the road might cause the automobile to oscillate while it is in motion. this will cause passengers to feel very uncomfortable. additionally, the quality of the merchandise is also impacted. there are several criteria used to assess vehicle oscillations. commonly employed metrics include the change in the sprung mass displacement, acceleration, and the oscillation phase differential. the maximum values of the above criteria should be considered for a single impulse or discontinuous oscillations. their mean data will be evaluated regarding continuous or periodic oscillations. a suspension system is used to control the automobile's oscillations. each type of suspension system will have a variety of components. a passive suspension system uses a metal spring and conventional damper. the overall stiffness of these elements has not changed. therefore, the vehicle's smoothness cannot be guaranteed. the spring and damper stiffness must be altered to eliminate the vehicle's oscillation problems. in [1], marenkov and baurova presented the air suspension mechanism. this suspension has variable-stiffness air springs. internal pneumatic pressure variations alter the spring's stiffness [2, 3]. in addition, the suspension system employing a magnetorheological damper helps increase the vehicle's movement stability. the magnetorheological damper utilizes electromagnetic force to control the movement of liquid inside [4]. the oscillation quenching procedure can thus be adjusted easily [5]. active suspension technologies should be employed widely to improve oscillation [6, 7]. for each position of the active suspension, there is an extra hydraulic actuator. this actuator's operation depends on the movement of inside hydraulic valves [8]. these valves work once control signals from the controller is received [9]. in general, an active suspension system's performance can be more fantastic than other suspension systems. * corresponding author: anhngtu@tlu.edu.vn 44 nguyen t. a. copyright ©2024 assa adv. in systems science and appl. (2024) for fundamental control issues, the linearity of the object is frequently assumed. the pid algorithm may be implemented if the controlled object has only one input and one output, also known as the single input and single output (siso) system. the pid controller has three distinct phases. there will be a matching coefficient for each step [10]. the selection of optimal parameters for this controller is essential. in [11], han et al. presented a method for adjusting the pid controller's coefficients based on fuzzy principles. in another study, demir et al. used fuzzy method to tune these factors [12]. the fuzzy algorithm described in [12] had two inputs and one output. the defuzzification procedure was created from the perspective of the designer. the membership functions of fuzzy algorithms were commonly trapezoidal or triangular [13]. in certain instances, a gaussian functional form might be utilized [14]. the fuzzy technique could make factors of the pid controller change continuously. this was an effective adaptive response [15]. furthermore, particle swarm optimization approaches might optimize pid controller configurations [16-19]. when controlling both displacement and acceleration, two pid controllers could be coordinated [20]. however, this might result in undesirable interactions [21]. in [22], nguyen et al. controlled active suspension with the quarter-dynamics model using the linear quadratic regulator (lqr) method. this algorithm was often applied to multi-input and multi-output (mimo) systems. this approach was designed to minimize the cost function [23]. in order to efficiently solve the riccati algebraic equation, every oscillation state model of the automobile must be expressed in the form of a state matrix [24]. when this controller had incorporated the "cancellation term" further, it was referred to as i-lqr. haddar presented this concept in [25]. the smc technique was frequently applied to individuals with random or nonlinear stimuli [26]. this approach employed lyapunov theory to create asymptotically stable controllers [27]. according to nguyen [28], every controlled object would travel across the sliding surface until a steady state was reached. the sliding surface was built according to the model's high-order derivative signal [29]. therefore, the actuator must be linearized into an approximation of a linear differential equation. this procedure was established in [30] by nguyen et al. the order of the signal's derivative would depend on the system's order. the control model got more complicated if a system had many more state variables [31]. using the smc algorithm frequently resulted in the “chattering” effect. there might be a disturbance in the control signals [32-34]. combining the smc algorithm with the fuzzy or pid algorithm could help solve this problem. then, the system's controller would be referred to as a “hybrid control” [35, 36]. in addition, several more complex control algorithms [3739] were employed to regulate the performance of an active suspension system. overall, these strategies were also exceptionally practical. an optimal algorithm (osmcpid) is proposed in this work to control the suspension system to limit the influence of chattering and suppress the system error. two control signals are used to synthesize the ultimate controller's control signal. the controller's parameters are optimally selected by the in-loop algorithm. the integration method will depend on the researchers' design perspective. this article is divided into four sections: introduction, models, simulation and result, and conclusion. 2. models in this work, a model of the quarter-dynamic is utilized in order to simplify the modelling of vehicle oscillation (figure 2.1). this dynamic model incorporates both two mass, and the suspension system containing the conventional spring and damping connects them. the hydraulic actuator is installed with a suspension system to enhance ride comfort. consequently, a mechanical suspension system (passive) develops into an electronic suspension system (active). enhancing comfort and stability of a vehicle suspension system 45 copyright ©2024 assa. adv. in systems science and appl. (2024) figure 2.1. automobile dynamic model. an automobile's dynamic model has two degrees of freedom for each direction of movement, including zs and zu. using mechanic theory, equations (2.1) and (2.2) describe vehicle oscillations.    s s u s u s am z k z z c z z f       (2.1)      u u t r u u s u s am z k z z k z z c z z f         (2.2) where ms and mu are sprung mass and unsprung mass, respectively; zs and zu are sprung mass displacement and unsprung displacement, respectively; k is the suspension spring coefficient; c is the suspension damper coefficient; and kt is the tire spring coefficient. in order to decrease the oscillation of an automobile, a hydraulic actuator should provide an impact force. this force impacts the sprung and unsprung masses. the actuator force (fa) is created by the pressure difference between the two compartments within the actuator. this force's magnitude is proportional to the piston's area, sp, and the pressure differential, p, according to (2.3). a pf s p  (2.3) the actuator work thanks to the voltage signal provided by the controller. the internal valves will travel when the voltage signal is delivered to the hydraulic actuator. this movement changes the pressure of the fluid within the cylinder. a complicated multivariable function was used to describe a dependence between the control signal, u(t), the displacement of valves, xsv, and the pressure difference. even the suspension system displacement, xs, is a factor. this dependence is demonstrated by equations (2.4) and (2.5).   sv sv sv 1 x = k u t x dt τ  (2.4)   3 sv s sv 2 1 p sδp = x p sgn x δp dt δpdt s x dt        (2.5) using the original hydraulic actuator equations makes the design of the controller extremely challenging. according to [30], the impact force, fa, produced by an actuator may be linearized as (2.6), where i are coefficients.    1 2 3a a s uf u t f z z        (2.6) according to [28], sprung mass' acceleration may be estimated as (2.7), where  is a coefficient. 46 nguyen t. a. copyright ©2024 assa adv. in systems science and appl. (2024)  t s r u s k z z z m   (2.7) let state variables: 1 sx z 2 sx z  3 ux z 4 ux z  5 ax f taking derivatives of state variables, we get (2.8) to (2.12). 1 2x x (2.8)  2 1 2 3 4 5 1 s x kx cx kx cx x m       (2.9) 3 4x x (2.10)   4 1 2 3 4 5 1 t u x kx cx k k x cx x m       (2.11)  5 1 3 2 3 4 2 5x u t x x x       (2.12) let e1(t) be the error of two signals, include the output signal and the setpoint signal of the sliding mode controller.      1 1se t y t y t  (2.13) take the derivative of both sides of (2.13), we obtain (2.14).        1 2s se t y t y t y t x       (2.14) the second derivative of the equation (2.13) is shown in (2.15).          1 3 t s s r s k e t y t y t y t x z m         (2.15) take the derivative of both sides of (2.15), we get (2.16).                3 3 3(3) 1 4 t s s r s k e t y t y t y t x z m       (2.16) by continuing to derive both sides of (2.16), we can obtain (2.17).                  4 4(4) 1 4 1 2 3 4 5 s t t s t r s u s e t y t y t k k y t kx cx k k x cx x z m m m             (2.17) let's use the symbols as (2.18–2.23): 1 1 1 s u kc m m         (2.18) 2 2 2 3 s u c c k m m            (2.19) 3 t s u k kk c m m          (2.20) 2 2 4 3t s u c c k k m m              (2.21) 5 2 s u c c m m            (2.22) 1 6 t s u k m m    (2.23) the effect of systematic noise r(t) is ignored. the fifth derivative signal of the output is written according to (2.24). enhancing comfort and stability of a vehicle suspension system 47 copyright ©2024 assa. adv. in systems science and appl. (2024)      5 1 1 2 2 3 3 4 4 5 5 6y t x x x x x i t           (2.24) figure 2.2. sliding surface. following the lyapunov theory, a controlled object will return to a state of equilibrium by sliding down the surface (figure 2.2). therefore, the sliding surface must be selected according to the system's stability requirement. a sliding surface (2.25) is the function dependent on the system’s error, e1(t):   1 ( ) 1 1 0 n n n i j i j e t       (2.25) once a sliding surface has been established, it is simple to calculate output signals of the sliding mode controller using an equation (2.26).             1 1 1 1 ( ) ( 1 ) 1 6 1 1 1 1 0 0 0 sgn n n n n n n n i n l s k k j m k i j m l u t y t x t e t j e t                               (2.26) a lyapunov control function is chosen according to (2.27). taking the derivative of (2.27), we obtain (2.28).   21 0 0 2 v x x    (2.27)  v x   (2.28) take the derivative of the sliding surface (2.25). then, substituting the result just found and the control signal u1(t) mentioned in (2.26) into (2.28), we obtain (2.29).    sgn 0v x j    (2.29) let e2(t) be the error of a pid controller, as (2.30).      2 2se t y t y t  (2.30) a control signal, u2(t), is calculated using the following formula (2.31):        2 2 2 2 0 t p i d de t u t k e t k e d k dt     (2.31) using the in-loop algorithm, the pid and smc controller’s parameters are optimally chosen to increase the system's responsiveness to changing external inputs. the schemata of the control algorithm are shown in figure 2.3. this algorithm is established found some of the following points of view:  peak value and mean value of a vehicle body displacement are minimum 48 nguyen t. a. copyright ©2024 assa adv. in systems science and appl. (2024)  peak value and mean value of a vehicle body acceleration are minimum  the phase difference of oscillation is minimum  chattering phenomenon must not exist the integrated controller's common control signal is determined by the control signals of two distinct controllers. the process of calculation and simulation will be implemented as soon as the controller has been fully designed. figure 2.3. control system diagram. 3. simulation and result 3.1. simulation the numerical simulation method was used for this research. in this article, matlabsimulink software is used. three cases are investigated corresponding to the three types of input stimulus signals (figure 3.1). input stimulus signals are the bumpy road surface. the output of a simulation problem can include displacement values and acceleration values of an automotive body. in each case, four situations are considered, including:  an automobile utilizing the active suspension system controlled by a novel hybrid method (osmcpid) – the first situation.  an automobile utilizing the active suspension system controlled by the smc method – the second situation.  an automobile utilizing the active suspension system controlled by the pid method – the third situation.  an automobile utilizing the passive suspension system (none) – the fourth situation. enhancing comfort and stability of a vehicle suspension system 49 copyright ©2024 assa. adv. in systems science and appl. (2024) figure 3.1. the roughness of the road surface. the specifications of a reference automobile are shown in table 3.1. table 3.1. the reference vehicle. description symbol unit value sprung mass ms kg 470 unsprung mass mu kg 42 tire coefficient kt nm-1 177000 spring coefficient k nm-1 39500 damping coefficient c nsm-1 3150 3.2. results and discussion simulation results are discussed in three specific cases. the stimulus amplitude in all three cases is r(t) = 80 mm. however, the frequency and acceleration of the oscillation are different. the first case: sine-shaped stimulus is used for the first case. this is a low-frequency excitation. the trajectory of the excitation signal varies cyclically with time. therefore, the oscillation of the vehicle also has a cyclical form. the change in vehicle body displacement can be observed in figure 3.2. for vehicles using only the mechanical suspension, the peak displacement value can be up to 101.00 mm. this value is reached in the first half of the first phase of the 50 nguyen t. a. copyright ©2024 assa adv. in systems science and appl. (2024) oscillation. from the second half of the first phase onwards, a maximum amplitude of the oscillation is only 91.25 mm. the mean oscillation value determined based on the root mean square (rms) criterion is 65.36 mm. the vehicle body displacement is able to be declined once an active suspension is used instead. once a pid controller is utilized to direct an active suspension, the displacement value is reduced to 37.30 mm and 26.50 mm, respectively, the peak and mean oscillation value. smc controller can help reduce oscillation values more than the pid controller, only 13.84 mm and 9.78 mm. in particular, if the osmcpid algorithm is utilized for a suspension system, the maximum value and mean value of a vehicle body displacement are significantly reduced to only 7.12 mm and 5.04 mm. besides assessing the magnitude of the oscillation, the phase deviation of the oscillation should also be considered. the slightest phase deviation belongs to the first situation (osmcpid). meanwhile, the deviation between the second (smc) and third (pid) situations is more significant than the fourth. therefore, the osmcpid algorithm can provide better performance with high accuracy. figure 3.2. sprung mass displacement (the first case). a vehicle's acceleration is the value used to evaluate the smoothness of a vehicle when it oscillates. if the value of acceleration is high, there is a possibility that smoothness will be reduced. the change of acceleration throughout the simulation period is illustrated in figure 3.3. the acceleration of the vehicle body reaches its peak value at the first phase, respectively 0.69 m/s2, 0.91 m/s2, 1.68 m/s2, and 1.90 m/s2, corresponding to the four situations examined. in the following phases, the vehicle body acceleration oscillates with a stable amplitude. the mean value of the vehicle body acceleration determined according to the rms criterion reached 0.07 m/s2, 0.11 m/s2, 0.28 m/s2, and 0.68 m/s2, respectively. compared with the fourth situation, the average acceleration value, once the osmcpid algorithm is used for the suspension system, is only 10.29%. in some situations, during phase transition, the value of the acceleration may change suddenly. however, this change is quite small. therefore, this has no impact whatsoever on how smoothly the vehicle is. for controllers established based on a smc algorithm, a “chattering” problem will appear. this problem causes signals of the oscillation to be continuously noisy. therefore, the average value of the oscillation can be increased. this only happens for the single smc method for an electronic suspension system. meanwhile, an osmcpid algorithm can eliminate this enhancing comfort and stability of a vehicle suspension system 51 copyright ©2024 assa. adv. in systems science and appl. (2024) phenomenon if the controller's parameters are selected optimally. this is considered an important advantage of this new algorithm. figure 3.3. sprung mass acceleration (the first case). the second case: in the following case, the pavement bump is random. this form of excitation has a very high frequency, so the smoothness of the vehicle may be affected more than in the previous case. the change of vehicle body displacement is continuous and does not follow any rules, as demonstrated in figure 3.4. peak values of a vehicle body displacement corresponding to four situations is 6.62 mm, 12.24 mm, 31.53 mm, and 100.43 mm, respectively. the difference between the first and fourth situations can be up to 15.17 times. this is a huge difference. in this case, the rms criteria can also be used to evaluate oscillations over a continuous time. the average value of the oscillation is 2.38 mm, 4.42 mm, 12.04 mm, and 34.19 mm. if mean displacement values of the last situation are taken as a fixed point, the remaining values are 6.96%, 12.93%, and 35.21%, respectively. just using the advanced suspension system, automobile's oscillation can be better improved. this efficiency can be further enhanced once the osmcpid algorithm is utilized in order to control an active suspension system. acceleration values of an automotive body in the second case are more significant than in the first case. a graph in figure 3.5 explains this. in figure 3.5, the change of acceleration is continuous and random. peak and average acceleration values once the vehicle only has a mechanical suspension system are 8.60 m/s2 and 2.81 m/s2. this value can be reduced slightly to only 7.44 m/s2 and 2.80 m/s2 if an automobile utilizes an active hydraulic suspension directed by a pid controller. the difference in values between these two situations is relatively small. because this is a nonlinear oscillation, the pid algorithm’s efficiency for the suspension system is not high. as mentioned in the first section, the smc algorithm is more suitable for nonlinear systems. peak value and average value of the automotive body acceleration are decreased to 6.11 m/s2 and 2.01 m/s2 if the smc controller is utilized to direct the suspension system. once the smc and pid algorithms are combined, it becomes a hybrid algorithm. the hybrid algorithm with optimally selected parameters is called osmcpid. this is the new algorithm that is used in this article. the control system's performance will be much better if a controller is built based on this new method. the maximum and average values of the oscillation have decreased sharply, to only 5.69 m/s2 52 nguyen t. a. copyright ©2024 assa adv. in systems science and appl. (2024) and 1.89 m/s2. in addition, the "chattering" phenomenon does not exist when using the new hybrid algorithm. meanwhile, this phenomenon still occurs if the controller is only established based on a single smc algorithm. this helps to demonstrate that the quality of the osmcpid algorithm is higher than that of a single smc or pid algorithm. figure 3.4. sprung mass displacement (the second case). figure 3.5. sprung mass acceleration (the second case). the third case: in the last case, bump on the road takes the shape of a double step. the amplitude of the stimulus will change in three steps. therefore, the sprung mass displacement will change by three steps. this change is shown in figure 3.6. according to simulation results, the automotive displacement will decrease in order of none, pid, smc, and osmcpid. this order is valid for both steps of the oscillation. since the time t = 6 s, the excitation signal enhancing comfort and stability of a vehicle suspension system 53 copyright ©2024 assa. adv. in systems science and appl. (2024) ends, and the displacement of the vehicle body will tend to return to zero. however, this value will fluctuate around zero for a short period before finally becoming zero. the oscillation amplitude is minimal once the osmcpid algorithm controls the active suspension. in the other three situations, the amplitude of the oscillation is larger. therefore, the novel hybrid algorithm will help the vehicle quickly return to a steady state after the stimulation from the road surface has ended. figure 3.6. sprung mass displacement (the third case). in this situation, the acceleration of the vehicle's body also varies with each step of the excitation signal. the highest and lowest values of the automotive acceleration still belong to two situations: automobile utilizing a mechanical suspension system and automobile utilizing a hydraulic suspension directed by the osmcpid method, reaching 0.64 m/s2 and 0.23 m/s2. the chattering phenomenon still occurs once the smc controller is utilized to control the suspension system. meanwhile, this phenomenon does not exist if the novel hybrid algorithm is used (figure 3.7). this is a harmful phenomenon, and it can harm vehicle oscillation. thanks to the new algorithm, the automobile's comfort and stability have been increased. the values obtained from the simulation and calculation process are illustrated in table 3.2. 54 nguyen t. a. copyright ©2024 assa adv. in systems science and appl. (2024) figure 3.7. sprung mass acceleration (the third case). table 3.2. simulation results. average displacement (mm) maximum displacement (mm) average acceleration (m/s2) maximum acceleration (m/s2) the first case osmcpid 5.04 7.12 0.07 0.69 smc 9.78 13.84 0.11 0.91 pid 26.50 37.30 0.28 1.68 none 65.36 101.00 0.68 1.90 the second case osmcpid 2.38 6.62 1.89 5.69 smc 4.42 12.24 2.01 6.11 pid 12.04 31.53 2.80 7.44 none 34.19 100.43 2.81 8.60 the third case osmcpid 4.21 7.01 0.03 0.23 smc 8.22 13.60 0.04 0.30 pid 22.89 37.50 0.10 0.56 none 49.47 87.16 0.19 0.64 5. conclusion roughness on the road is the main causing of vehicle oscillation. these oscillations are able to affect negatively the comfort and smoothness of an automobile when travelling on different conditions. suspension systems are fitted to the automobile to regulate and suppress these oscillations. once the mechanical suspension is combined with the hydraulic actuator, it becomes an active suspension system (electronic suspension). an electronic suspension brings better quality than a mechanical suspension system. in this article, the author has proposed a novel hybrid control algorithm called osmcpid. this is an integrated algorithm. the final control signal is synthesized from two component algorithms, smc and pid. the controller factors are optimally selected based on the vehicle's smoothness and stability considerations. simulation is performed to evaluate the controller’s performance. three cases were examined; in each case, four situations were enhancing comfort and stability of a vehicle suspension system 55 copyright ©2024 assa. adv. in systems science and appl. (2024) mentioned. the input of the simulation problem is the bumpy signals of the road, and the output is the change in the automotive body's displacement and acceleration. according to simulation results, both displacement and acceleration values are reduced when an active hydraulic suspension is used. the peak and mean values of two values above reach the minimum once the osmcpid method is utilized to direct a suspension system. besides, the "chattering" phenomenon did not occur when using this new hybrid algorithm. vehicle smoothness and stability have increased dramatically. the efficiency of the novel hybrid method, which is established in this article, has been determined through simulation. however, there is still a problem: the acceleration signal overshoots during phase transition. this problem can be improved by adding the fuzzy algorithm to this original algorithm. in addition, some experimental processes also should be performed in the future to evaluate this control algorithm's performance accurately. references [1] marenkov, i. g., & baurova, n. i. 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(2020). design of fuzzy super twisting sliding mode control scheme for unknown full vehicle active suspension systems using an artificial bee colony optimization algorithm, asian journal of control, 23(4), 1966– 1981. doi: 10.1002/asjc.2352 adv syst sci appl 2025; 02:92–104 published online at https://ijassa.ipu.ru. vector fields with non-isolated singular points viewed from the inside and outside natalia pavlova1,2 alexey remizov1* 1moscow institute of physics and technology (state university), dolgoprudnyi, russia 2rudn university, moscow, russia abstract: we present a brief survey of recent results about vector fields with non-isolated singular points and their applications. one of the most interesting and promising application is connected with quasi-linear differential equations of the second order, including the equation of geodesics in signature varying (pseudo-riemannian) metrics. keywords: vector fields, singular points, center manifold, normal forms, resonances 1. introduction we start with a general construction, which naturally leads us to vector field with non-isplated singular points (more precisely, singular points fill a manifols of codimension two in the phase space). let m be a real smooth (c∞) manifold of dimension n+ 2. here and further we use the following standard notations: • c∞(m) is the ring of smooth functions on m • γ(tm) is the module of smooth vector fields on m • γ(t ∗m) is the module of smooth covector fields (1-forms) on m consider a distribution on m defined via n+ 1 differential 1-forms: ω1 = 0, . . . , ωn+1 = 0, ωi ∈ γ(t ∗m). (1.1) at points of m where the 1-forms ω1, . . . , ωn+1 are linearly independent, system (1.1) defines a one-dimensional distribution, that is, a direction field, which can be a class of collinear vector fields, such points we shall call regular. at points of m where the 1-forms ω1, . . . , ωn+1 are nor linearly independent, the distribution has dimension greater then 1, such points we shall call singular. in (local) coordinates u = (u1, . . . , un+2) on m system (1.1) yields the pfaffian system ωi = n+2∑ j=1 ωij(u) duj = 0, i = 1, . . . , n+ 1, (1.2) with (n+ 1)× (n+ 2) matrix ω = (ωij). ∗corresponding author: alexey-remizov@yandex.ru vector fields with non-isolated singular points... 93 singular points fill a stratified manifold σ ⊂ m that consists of the strata σi = {u ∈ m : rg ω(u) = (n+ 1)− i}, i = 1, . . . , n+ 1. the dimension of σi decreases rapidly with increasing i, namely: codimσi = i(i+ 1); see, for example, [2]. the maximal stratum σ1 of codimension 2 is stable with respect to small perturbation of ωi, while the strata of higher codimension are, generally speaking, not. the goal of the paper is to study integral curve of the distribution (1.1) entering its singular points of the maximal stratum σ1. we also consider some application of the obtained results for studying singularities of differential equations of special types, which are interesting due to various applications. 2. the main results the distribution (1.2) can be determined (at least locally) by a smooth vector field v̄ ∈ γ(tm) that is zero at points of σ. this field is unique up to multiplying by a scalar factor: v̄ = n+2∑ j=1 vj(u) ∂ ∂uj , vj = (−1)j+1∆j, j = 1, . . . , n+ 2, (2.3) where ∆j is the minor of ω obtained by elimination of the j-th column. example 2.1: consider the lowest dimension case: n = 1. then the distribution (1.2) in 3-dimensional space is the intersection of two fields of planes: ωi = ωi1du1 + ωi2du2 + ωi3du3 = 0, i = 1, 2. here ω is a 2× 3 matrix ω =  ω11 ω12 ω13 ω21 ω22 ω23  and v̄ is the vector product of its columns: v1 = ∣∣∣∣∣∣ω12 ω13 ω22 ω23 ∣∣∣∣∣∣ , v2 = − ∣∣∣∣∣∣ω11 ω13 ω21 ω23 ∣∣∣∣∣∣ , v3 = ∣∣∣∣∣∣ω11 ω12 ω21 ω22 ∣∣∣∣∣∣ . the maximal stratum σ1 is defined by the condition rg ω = 1, that is, the matrix ω has at least one non-zero element. without loss of generality, assume that ωi3 ̸= 0. then the equality ωi1v1 + ωi2v2 + ωi3v3 = 0 yields the expression v3 = −ωi1 ωi3 v1 − ωi2 ωi3 v2, (2.4) which is valid locally, in a neighborhood of a point where ωi3 ̸= 0. expression (2.4) shows that the components of the vector field v̄ are connected by functional relations: in a neighborhood of a point of σ1 they belong to the ideal in the ring c∞(m) generated by two of them. the letter property established for n = 1, is valid for arbitrary n: copyright © 2025 assa. adv syst sci appl (2025) 94 n. g. pavlova, a. o. remizov lemma 2.1: in a neighborhood of a point of σ1, all components of the field (2.3) belong to the ideal (in c∞(m)) generated by two of them. therefore, one can choose coordinates x, y, z = (z1, . . . , zn) such that the germ of the field (2.3) has the form ẋ = v, ẏ = w, żi = aiv + biw, i = 1, . . . , n, (2.5) where v, w, ai, bi ∈ c∞(m). proof the proof of the first statement of the lemma is similar to those for the case n = 1 considered above. the second statement obviously follows from the first one. lemma 2.1 establish some important properties of vector fields defined by pfaffian systems (1.2). the main distinctive feature is that singular points of such fields are not isolated, but filled a manifold of codimension two. for instance, singular points of the field (2.5) are given by two equations v(x, y, z) = 0, w(x, y, z) = 0. (2.6) therefore, the spectrum of the linear part of field (2.5) at every singular points t ∈ σ1 has the form spec (t ) = (λ1(t ), λ2(t ), 0, . . . , 0), #0 = n, where the eigenvalues λ1,2(t ) continuously depend on t ∈ σ1. further we shall use the notation λ1,2 = λ1,2(t0). further we shall consider the germ (2.3) at a generic singular point t0 ∈ σ1, where reλ1,2 ̸= 0. (2.7) in this case, the set of singular points σ1 is the (unique) center manifold w c of the field (2.5) passing through the point t0. this allows us to apply the reduction principle [1, 6], which yields the following result: theorem 2.1: the germ of the field (2.5) at a generic singular point t0 satisfying the condition (2.7) is topologically equivalent to the field ξ̇ = a1ξ, η̇ = a2η, ζ̇i = 0, i = 1, . . . , n, (2.8) where aj = sgn(reλj). proof this is a trivial corollary of the reduction principle, which states that the germ of the field (2.5) at t0 is topologically equivalent to the product of the field ξ̇ = a1ξ, η̇ = a2η and the restriction of (2.5) to its center manifold w c. since w c consists of singular points of the field (2.5), this yields normal form (2.8). the next step in the study of fields (2.5) is their smooth local classification. for this, we need to consider two types of resonances (integer relations) between the non-zero eigenvalues: s1λ1 + s2λ2 = 0, si ∈ z+, i = 1, 2, (2.9) s1λ1 + s2λ2 = λj, si ∈ z+, i, j = 1, 2. (2.10) from them we have to exclude trivial resonances, which always exist: resonance (2.9) with s1 = s2 = 0 and resonance (2.10) with s1 = 1, s2 = 0, j = 1 or s1 = 0, s2 = 1, j = 2. the number |s| = s1 + s2 is called the order of resonance (2.9) or (2.10). copyright © 2025 assa. adv syst sci appl (2025) vector fields with non-isolated singular points... 95 remark 2.1: 1. the absence of resonances (2.10) implies the absence of the resonances (2.9): 2. in the absence of resonances (2.9), resonances (2.10) may have only the simplest form λ1 = mλ2 or λ2 = mλ1 (2.11) with integer m ≥ 1. theorem 2.2: if between the eigenvalues λ1,2(t ) there are no non-trivial resonances (2.9) of any order |s| ≥ 1 for all t ∈ w c sufficiently close to t0, then the germ of the field (2.5) at t0 is c∞smoothly equivalent to ξ̇ = x(ξ, η, ζ), η̇ = y (ξ, η, ζ), ζ̇j = 0, j = 1, . . . , n, (2.12) where x and y are smooth functions vanishing on the center manifold. if, in addition, |λ1| > |λ2| at t0, then x = λ1(ζ)ξ + φ(ζ)ηm, y = λ2(ζ)η, (2.13) where φ(ζ) ̸≡ 0 only if λ1 = mλ2 with some integer m > 1. the condition that between λ1,2(t ) there are no non-trivial resonances (2.9) of any order |s| ≥ 1 for all t ∈ w c appears when we consider infinitely smooth classification of vector fields with non-isolated singular points. first it was formulated in the paper [17] and then it is often named after him. remark 2.2: 1. if the pair λ1,2 = λ1,2(t0) belongs to the poincaré domain, that is, λ1,2 are real and of the same sign or complex conjugate with the condition (2.7), then the roussarie condition holds true, whence theorem 2.2 is valid. 2. if the pair λ1,2 = λ1,2(t0) belongs to the siegel domain, that is, λ1,2 are real and of different signs, the roussarie condition holds true if and only if λ1(t ) : λ2(t ) ≡ const /∈ q, ∀t ∈ w c. if we consider ck-smooth equivalence with k < ∞, the absence of resonances of all orders |s| at all points t ∈ w c can be replaced with the absence of resonances of the orders |s| ≤ n(k) at t0, where n(k) = 2 [ (2k + 1) max |reλ1,2| min |reλ1,2| ] + 2, (2.14) the square brackets denote the integer part of a number. the estimation (2.14) is taken from [18]. theorem 2.3: if between the eigenvalues λ1,2 = λ1,2(t0) there are no non-trivial resonances (2.9) of any order 1 ≤ |s| ≤ n(k), then the germ of the field (2.5) at t0 is ck-smoothly equivalent to ξ̇ = x(ξ, η, ζ), η̇ = y (ξ, η, ζ), ζ̇j = 0, j = 1, . . . , n, where x and y are smooth functions vanishing on the center manifold. if, in addition, |λ1| > |λ2| at t0, then x = λ1(ζ)ξ + φ(ζ)ηm, y = λ2(ζ)η, where φ(ζ) ̸≡ 0 only if λ1 = mλ2 with positive integer m ≤ n(k). copyright © 2025 assa. adv syst sci appl (2025) 96 n. g. pavlova, a. o. remizov for the proofs of theorems 2.2, 2.3, see [5, 13]. remark 2.3: geometrically, theorems 2.2, 2.3 state that the vector field (2.5) has 2-dimensional invariant foliation such that the restriction of the field to its leaves has the spectrum λ1,2(t ), that is, it is a node or a saddle od a focus. see, for example, fig. 2.1. from the analytical viewpoint, theorems 2.2, 2.3 are generalization of the poincare – dulac normal form for vector fields that have zero eigenvalues. fig. 2.1. illustration of theorems 2.1 – 2.3: local phase portraits of the vector field (2.5). here n = 1 and w c coincides with the z-axis. 3. roussarie vector fields now we consider vector fields of the form (2.5) satisfying the additional condition: spec (t ) ≡ (λ1(t ), λ2(t ), 0, . . . , 0), ∀t ∈ w c, where λ1,2(t ) are non-zero real numbers such that qλ1(t ) + pλ2(t ) = 0, p, q ∈ z+, gcd (p, q) = 1, ∀t ∈ w c. (3.15) such vector fields are named after robert roussarie, who studied the partial case p = q = 1 (3.16) in his [17]. this study was motivated by the degeneracy of closed differential 2-forms. a generic closed 2-form ω on the 4-dimensional real space degenerates on a smooth 3dimensional manifold σ. at a generic point of σ, the 2-dimensional kernel of the form ω is transversal to σ, and the germ of ω can be reduced to p1dp1 ∧ dq1 + dp2 ∧ dq2. however, the kernel of ω is tangent to σ at some points, which generically fill a curve s ⊂ σ, the normal form of ω at points of s is more complicated. the kernel of ω cuts out a direction field on σ, which can be given (uniquely up to a scalar factor) by a vector field of the form (2.5), whose singular points fill the curve s. the condition that ω is closed implies that the trace of the linear part of the vector field at every its singular point is zero, that is, the non-zero eigenvalues λ1,2 satisfy the resonance (3.15) with p = q = 1. copyright © 2025 assa. adv syst sci appl (2025) vector fields with non-isolated singular points... 97 actually, in many problems we are interested not in vector fields themselves, but in the corresponding direction fields. therefore, when bringing a vector field to its normal forms, one can multiply it by a non-vanishing scalar function in addition to changes of the phase variables. normal forms thus obtained are called orbital. orbital normal forms allow us to get rid of a redundant module, and therefore, to simplify the classification. theorem 3.1: the germ of every roussarie vector field is c∞-smoothly orbitally equivalent to ẋ = px(1 + φ1(r, z)), ẏ = qy(−1 + φ2(r, z)), żi = rψi(r, z), i = 1, . . . , n, (3.17) where r = xqyp is the resonant monomial of the resonance (3.15), φ1,2,ψi ∈ c∞(m) and φ1(0, 0) = φ2(0, 0) = 0. theorem 3.1 establishes a generalization of the poincare – dulac normal form for roussarie vector fields: the functions φ1,2 and ψi contain all resonant terms. it is wellknown that the poincare – dulac normal form allows further simplification. to obtain such simplification for (3.17), we shall use the notion of the quotient vector field. the field (3.17) generates the field in the (r, z)-space: ṙ = ˙(xqyp) = qxq−1ypẋ+ pxqyp−1ẏ = qxq−1yppx(1 + φ1) + pxqyp−1qy(−1 + φ2) = pqr(φ1 + φ2), żi = rψi(r, z), i = 1, . . . , n. reducing the common factor r, we get the quotient vector field for (3.17): ṙ = pqφ(r, z), żi = ψi(r, z), i = 1, . . . , n, (3.18) where φ(r, z) = φ1(r, z) + φ2(r, z). remark 3.1: from the resonant relation (3.15) it follows that φ(0, z) ≡ 0 for all z, i.e., the restriction of φ(r, z) to the center manifold is identically zero. generically, for almost all points t0 ∈ w c the field (3.17) satisfies the following condition: ∃i ∈ {1, . . . , n} : ψi(0, 0) ̸= 0. (3.19) equivalently, 0 is not a singular point of its quotient field (3.18). this allows us to establish the existence of n independent first integrals of the field (3.18), which are obviously first integrals of the field (3.17). for details, see [5, 13]. using these first integrals, one can prove the following theorem: theorem 3.2: if the condition (3.19) for (3.17) holds true, the germ of the roussarie vector field is c∞smoothly orbitally equivalent to ẋ = px, ẏ = −qy, ż1 = xqyp, żi = 0, i = 2, . . . , n. (3.20) moreover, it is ck−1-smoothly orbitally equivalent to ẋ = px, ẏ = −qy, żi = 0, i = 1, . . . , n, (3.21) where k = max{p, q}, but in general it is not ck-smoothly equivalent. copyright © 2025 assa. adv syst sci appl (2025) 98 n. g. pavlova, a. o. remizov fig. 3.2. two examples of c0 saddle surfaces of the vector field ẋ = x, ẏ = −y, ż = xy: z = − 1 2xy ln |y/x| (left) and z = −xy ln |y| (right). proof the first statement of the theorem is proved in the case p = q = 1 by roussarie in 1975. other statements are proved in [10]. example 3.1: consider the vector field ẋ = x, ẏ = −y, ż = xy, (x, y, z) ∈ r3. any saddle surface of this field has the form z = −1 2 f (x, y), where f (x, y) = f(xy) + xy ln ∣∣∣y x ∣∣∣, if xy ̸= 0, and f (x, y) = 0, if xy = 0. here f is an arbitrary continuous function. the obtained formula shows that f is continuous (for continuous f ), but it is never c1. see examples in fig. 3.2. 4. applications: quasi-linear odes of the second order consider the differential equation ∆(x, y) dp dx = m(x, y, p), p = dy/dx, (4.22) where ∆(x, y), m(x, y, p) are smooth functions, m is analytic in p. generically, the set of singular points of equation (4.22) is a regular curve γ = {(x, y) : ∆(x, y) = 0}. if the point q0 = (x0, y0) /∈ γ, then for every direction p0 equation (4.22) has a unique solution satisfying the initial condition y(x0) = y0, p(x0) = p0. the situation is more complicated if q0 = (x0, y0) ∈ γ. for example, there may be solutions whose oscillations accumulate near a singular point. copyright © 2025 assa. adv syst sci appl (2025) vector fields with non-isolated singular points... 99 example 4.1: the equation x4dp/dx = 2x3p− (2x2 + 1)y has a family of solutions y(x) = x2(α cosx−1 + β sinx−1), α, β = const, except for α = β = 0, all solutions are oscillating at x = 0: ∃ lim x→0 y(x) = 0, ̸ ∃ lim x→x0 y′(x) (but y′(0) = 0). example 4.2: the equation x2dp/dx = xp− 2y has a family of solutions y = x(α cos ln |x|+ β sin ln |x|), α, β = const, except for α = β = 0, all solutions are oscillating at x = 0: ∃ lim x→0 y(x) = 0, ̸ ∃ lim x→x0 y′(x) (and ̸ ∃ y′(0)). the above examples motivate the following formal definition: definition 4.1: oscillating solutions entering q0 are solutions such that ∃ lim x→x0 y(x) = y0, ̸ ∃ lim x→x0 p(x). here x → x0 may be either two-sided or one-sided limit. the above examples show that oscillating solutions exist. however, the following theorem states that oscillating solutions do not exist generically. theorem 4.1: let q0 ∈ γ and m(q0, p) is an analytic function not identically zero. then equation (4.22) has no oscillating solutions entering the point q0. moreover, solutions can enter q0 at so-called admissible directions p that correspond to real roots of m(q0, p) only. the proof of theorem 4.1 is based on the analysis of the distribution defined by equation (4.22) in the (x, y, p)-space: ω1 = ∆dp−mdx = 0, ω2 = dy − pdx = 0, which generates the vector field ẋ = ∆(x, y), ẏ = p∆(x, y), ṗ = m(x, y, p). (4.23) all components of the field (4.23) belong to the ideal generated by ∆ and m . given q0 ∈ γ, the admissible directions p correspond to singular points (q0, p) of the vector field (4.23) given by two equations: ∆(q0) = 0, m(q0, p) = 0. for more details about oscillating solutions see [11]. further, we shall analyse vector filed (4.23) in order to study non-oscillating solutions of equation (4.22) entering its singular points. first, remark that vector filed (4.23) has the form (2.5) with n = 1, where the variables x, p in (4.23) correspond to x, y in (2.5), the variable y in (4.23) corresponds to z1 in (2.5). let q0 ∈ γ and m(q0, p∗) = 0, i.e., p∗ is an admissible direction at q0. the spectrum of the linear part of the field (4.23) at singular point (q0, p∗) is spec (q0, p∗) = (λ1, λ2, 0), copyright © 2025 assa. adv syst sci appl (2025) 100 n. g. pavlova, a. o. remizov where the eigenvalues λ1 = (∆x + p∆y)(q0, p∗), λ2 = mp(q0, p∗). assume that λ1,2 ̸= 0. then we define the value λ = λ2 : λ1, which determines the number and the behavior of solutions of equation (4.22) that enter q0 with the direction p∗. namely, the following result is obtained in [15]. theorem 4.2: if λ < 0, then equation (4.22) has only one solutions passing though the point q0 with the tangential direction p∗. if λ > 0, then equation (4.22) has an infinite number of solutions entering the point q0 with the direction p∗. in appropriate local coordinates, these solutions have one of two following forms: y = f (x, c|x|λ), if λ /∈ n, y = f (x, xλ(c+ ε ln |x|)), ε ∈ {0, 1}, if λ ∈ n, where f is a smooth function, c = const. fig. 4.3. illustration for theorem 4.2: integral curves of the field (4.23) and their projections to the (x, y)-plane – solutions of equation (4.22). from left to right: λ < 0 (a), 0 < λ < 1 (b), λ > 1 (c). the curve γ is depicted as a dashed curve on the (x, y)-plane. 4.1. quasi-linear odes of the second order cubic in p consider an important class of equations (4.22), where m(x, y, p) is a cubic polynomial in p: m = µ0 + µ1p+ µ2p 2 + µ3p 3, µi = µi(x, y). an attention to such equations is motivated by their role in physics and geometry, for instance, the description of various geometric structures (geodesic flows in affine or projective connection, etc.). equations of this class were studies by sophus lie, a. tresse, j. liouville, e. cartan, etc. see also the recent papers [7, 19, 20]. for a generic cubic polynomial m(q0, p) and almost all points q0 ∈ γ there are 4 possible cases (the abbreviation ad below means admissible direction): copyright © 2025 assa. adv syst sci appl (2025) vector fields with non-isolated singular points... 101 • 1 ad p0 with λ(q0, p0) > 0; • 3 ads p0, p1, p2 with λ(q0, p0) > 0 and λ(q0, pi) < 0, i = 1, 2; • 1 ad p0 with λ(q0, p0) < 0; • 3 ads p0, p1, p2 with λ(q0, p0) < 0 and λ(q0, pi) > 0, i = 1, 2. example 4.3: consider the equation x dp dx = αp(p2 − 1), α ̸= 0, (4.24) whose singular points fill the curve γ = {x = 0}. for every point q0 ∈ γ the cubic polynomial m(p) = αp(p2 − 1) has three different real roots: p0 = 0 and p1,2 = ±1, where λ(q0, 0) = −α and λ(q0,±1) = 2α. solution of equation (4.24) entering the origin are presented in fig. 4.4. fig. 4.4. on the left: the case α > 0, on the right: the case α < 0. the bold lines are solutions that are unique with given ad. example 4.4: consider the equation x dp dx = αp(p2 + 1), α ̸= 0, (4.25) whose singular points fill the curve γ = {x = 0}. for every point q0 ∈ γ the cubic polynomial m(p) = αp(p2 + 1) has one real root p0 = 0 and λ(q0, 0) = α. fig. 4.5. on the left: the case α > 0, on the right: the case α < 0. the bold lines are solutions that are unique with given ad. copyright © 2025 assa. adv syst sci appl (2025) 102 n. g. pavlova, a. o. remizov 4.2. equation of geodesics in signature varying metrics consider a pseudo-riemannian metric ds2 = a(x, y) dx2 + 2b(x, y) dxdy + c(x, y) dy2, where a, b, c ∈ c∞(m) and the discriminant function ∆ = ac− b2 vanishes and changes its sign on the regular curve γ ⊂ m . the latter condition implies that the coefficients a, b, c do not vanish simultaneously, therefore, the square polynomial a(q0) + 2b(q0)p+ c(q0)p 2 at every point q0 ∈ gamma has the double root p0 = −a b = −b c , which defines the isotropic direction of the metric at the point q0. the equation of (unparametrized) geodesics in this metric has the form ∆(x, y) dp dx = 1 2 m(x, y, p), p = dy/dx, where m is a cubic polynomial m = µ0 + µ1p+ µ2p 2 + µ3p 3 with the coefficients µ3 = c(2by − cx)− bcy, µ2 = b(2by − 3cx) + 2ayc− acy, µ1 = b(3ay − 2bx) + axc− 2acx, µ0 = a(ay − 2bx) + axb. this is a special case of quasi-linear equation of the second order. in this case, the isotropic direction p0 is always admissible, that is, m(q0, p0) = 0. at almost all points of the curve γ, the cubic polynomial m has either one or three real roots, and there are only two possible combinations: • one ad p0 with λ(q0, p0) = 1 2 , • three ads p0, p1, p2 with λ(q0, p0) = 1 2 and λ(q0, pi) = −1, i = 1, 2. a deep explanation of this surprising fact can be found in [5]. finally, we remark one more interesting property established recently. let q0 ∈ γ and the polynomial m at q0 has three different real roots p0, p1, p2, that is, three different admissible directions at q0. let pγ be the tangent direction to the curve γ at q0. then the relative position of the directions p0, p1, p2, pγ is determined by their cross-ratio: dv(p0, p1, pγ, p2) = 2. the proof can be found in [12]. in fig. 4.6, we present three families of geodesics issuing from a point q0 ∈ γ. here the curve γ coincides with the horizontal axis (dashed line) and the isotropic admissible direction p0 at all points q0 ∈ γ is vertical. the families on the left and in the center correspond to points q0 with unique admissible direction p0. the family on the right corresponds to a point q0 with three admissible direction p0, p1, p2; geodesics with non-isotropic tangential directions p1, p2 are depicted as bold lines. copyright © 2025 assa. adv syst sci appl (2025) vector fields with non-isolated singular points... 103 fig. 4.6. three families of geodesics issuing from a point q0 ∈ γ. 5. conclusion we presented a survey of recent results about vector fields with non-isolated singular points and some their applications. one of the most interesting and promising application is connected with quasi-linear differential equations of the second order, including the equation of geodesics in signature varying (pseudo-riemannian) metrics. this subject motivates research in various directions, see, for example, the papers [9,14,16]. applications of different types can be also found in [3, 4], [7, 19], and [8]. references 1. arnol’d, v. i. & ilyashenko, yu. s. (1988) ordinary differential equations, dynamical systems i. encycl. math. sci. 1, 1–148. 2. arnol’d, v. i., gusein-zade, s. m., & varchenko, a. n. (1988) singularities of differentiable maps, vol. ii. monogr. math. 83. birkhauser, boston, ma. 3. bonnard, b., glaser, s. j., & sugny, d. (2012) a review of geometric optimal control for quantum systems in nuclear magnetic resonance, adv. math. phys., article id 857493, 29 p. 4. bonnard, b., chyba, m., & marriott, j. (2013) singular trajectories and the contrast imaging problem in nuclear magnetic resonance, siam j. control optim., 51(2), 1325– 1349. 5. ghezzi, r. & remizov, a. o. (2012) on a class of vector fields with discontinuities of divide-by-zero type and its applications to geodesics in singular metrics, j. dyn. control syst., 18(1), 135–158. 6. hirsch, m. w., pugh, c. c., & shub, m. (1977) invariant manifolds, lect. notes math. 583, berlin: springer. 7. liang, j. (2009) a singular initial value problem and self-similar solutions of a nonlinear dissipative wave equation, j. differ. equations, 246, 819–844. 8. panov, a. v. (2024) on bifurcation of non-isolated singular points arising in a problem of two-phase fluid motion in a pipe, physica d, 470, part a, article id 134408, 12 p. 9. pavlova, n. g. & remizov, a. o. (2018) a brief survey on singularities of geodesic flows in smooth signature changing metrics on 2-surfaces, springer proc. math. stat., 222, 135–155. 10. pavlova, n. g. & remizov, a. o. (2021) smooth local normal forms of hyperbolic roussarie vector fields, moscow math. j., 21(2), 413–426. 11. pavlova, n. g. & remizov, a. o. (2022) oscillating and proper solutions of singular quasi-linear differential equations, adv. syst. sci. appl., 22(4), 51–64. 12. pavlova, n. g. & remizov, a. o. (2025) the cross-ratio at singular points of geodesic flows in signature changing metrics, lobachevskii j. math., to appear. copyright © 2025 assa. adv syst sci appl (2025) 104 n. g. pavlova, a. o. remizov 13. remizov, a. o. (2008) multidimensional poincaré construction and singularities of lifted fields for implicit differential equations, j. math. sci., 151(6), 3561–3602. 14. remizov, a. o. (2015) on the local and global properties of geodesics in pseudoriemannian metrics, diff. geom. appl., 39, 36–58. 15. remizov, a. o. (2023) singularities of quasi-linear differential equations, dal’nevost. mat. zh., 23(1), 85–105. 16. remizov, a. o. & tari, f. (2016) singularities of the geodesic flow on surfaces with pseudo-riemannian metrics, geom. dedicata, 185(1), 131–153. 17. roussarie, r. (1975) modèles locaux de champs et de formes, asterisque, 30, 1–181. 18. samovol, v. s. (1983) equivalence of systems of differential equations in a neighborhood of a singular point, trans. mosc. math. soc., 2, 217–237. 19. seiler, w. m. & seiss, m. (2021) singular initial value problems for scalar quasi-linear ordinary differential equations, j. differ. equations, 281, 258–288. 20. tunitsky, d. v. (2021) on some global properties of multivalued simple waves, adv. syst. sci. appl., 20(4), 125–131. copyright © 2025 assa. adv syst sci appl (2025) introduction the main results roussarie vector fields applications: quasi-linear odes of the second order quasi-linear odes of the second order cubic in p equation of geodesics in signature varying metrics conclusion microsoft word 1442-article text-6568-1-18-20231226 adv syst sci appl 2023; 04; 41-59 published online at https://ijassa.ipu.ru. balancing accuracy, fairness and privacy in machine learning through adversarial learning alexander eponeshnikov1, rustem sabitov2, gulnara smirnova2*, shamil sabitov1 1 kazan federal university, kazan, russia 2 kazan national research technical university named after a.n. tupolev – kai, kazan, russia abstract: this paper investigates balancing accuracy, fairness and privacy in machine learning through adversarial learning. differential privacy (dp) provides strong guarantees for protecting individual privacy in datasets. however, dp can impact model accuracy and fairness of decisions. this paper explores the effect of integrating dp into the adversarial learning framework called laftr (learning adversarially fair and transferable representations) on fairness and accuracy metrics. experiments were conducted using the adult income dataset to classify individuals into high vs low income groups based on features like age, education etc. gender was considered a sensitive attribute. models were trained with different levels of dp noise (controlled by the epsilon hyperparameter) added to different modules like the encoder, classifier and adversary. results show that adding dp consistently improves fairness metrics like demographic parity and equalized odds by 3-5% compared to an unfair classifier, albeit at a cost of 1-3% reduction in accuracy. stronger adversary models further improve fairness but require careful tuning to avoid instability during training. overall, with proper configuration, dp models can achieve high fairness with minimal sacrifice of accuracy compared to an unfair classifier. the study provides insights into balancing competing objectives of privacy, fairness and accuracy in machine learning models. keywords: machine learning, differential privacy, adversarial learning, fairness, accuracy, privacy-preserving models 1. introduction in the modern world, a huge number of different industries use machine learning to automate processes such as credit ratings, spam filtering. machine learning plays an important role in preventing financial losses in the banking industry. perhaps the most urgent task of forecasting is the assessment of credit risk (the risk of default on debt). such risks can lead to losses of billions of dollars annually. a good result of machine learning mainly depends on the huge amount of data in which biases can be found in favor of certain attributes that are not fair in reality [8]. for example, by learning from unfair data, a classification model of an automated recruitment system is more likely to hire candidates from certain racial or gender groups or to favor candidates of a certain age. algorithmic bias is a growing subject of much discussion and debate in the use of ai. this is a complex topic due to the potential complexity of the mathematical definition of what it means to be “fair” in decision making. fairness depends on the situation and is not only a reflection of values, ethics and legal norms. however, there are clear ways to approach ai fairness issues. quite often, individual and sensitive data (for example, financial transactions or tax payments) are taken to solve machine learning problems. because of this, algorithms must * corresponding author: seyl@mail.ru balancing accuracy, fairness and privacy in machine learning… 41 copyright ©2023 assa. adv. in systems science and appl. (2023) guarantee privacy. one of the methods is differential privacy. differential privacy (dp) is a mathematical definition of the loss of confidential data of individuals when their personal information is used to create a product. differential privacy allows to find a balance between privacy and accuracy using a positive value . if is small, then we keep more privacy, but we degrade accuracy. if is big, then privacy suffers for the sake of accuracy. meaning of varies from 0 to infinity. to train models with dp, use dp-sgd. the core idea is that training a model can be done through access to its parameter gradients, i.e., the gradients of the loss with respect to each parameter of your model. if this access preserves differential privacy of the training data, so does the resulting model, per the post-processing property of differential privacy. summarizing all of the above, machine learning models must guarantee data privacy while avoiding discrimination and ensuring fair decision-making. however, the use of these methods may have an impact on the accuracy of the model. the research questions below in this work aim to examine the impact of privacy on fairness and accuracy, as well as to compare fair metrics and accuracy in different approaches of privacy protection in different machine learning models.  what is the effect of privacy during encoding process on fairness in machine learning models?  how does privacy during encoding process impact the accuracy of machine learning models?  how do different privacy strategies impact fairness metrics in machine learning models?  how do the different privacy strategies impact on accuracy in machine learning models?  how do the different approaches in models impact on ability to balance fairness and accuracy while privacy in machine learning models?  how do different datasets configurations impact on fairness and accuracy results? 2. background in this section, we discuss the fundamental privacy and fairness concepts used throughout the work. 2.1. differential privacy differential privacy is a set of methods that provide the most accurate queries to a statistical database while minimizing the possibility of identifying individual records in it. let be a positive real number and be a probabilistic algorithm that takes a set of data as input (represents the actions of a trusted party that has the data). algorithm is differentially private if for all datasets and the following expression is executed [11]. where and are differing datasets by at most one element, and denotes the probability that is the output of . according to this definition, differential privacy is a condition of the data publishing mechanism (that is, determined by the trusted party that releases information about the data set), not the data set itself. intuitively, this means that for any two similar datasets, the differential private algorithm will behave approximately the same on both datasets. 2.2. privacy preservation (dp-sgd) stochastic gradient descent (sgd) is an iterative method for optimizing differentiable objective functions. it updates the weights and biases by calculating the gradient of the loss function for small data packets [3]. dp-sgd [26] is a modification of the stochastic gradient 42 a. eponeshnikov, r. sabitov, g. smirnova, sh. sabitov copyright ©2023 assa adv. in systems science and appl. (2023) descent algorithm that provides provable privacy guarantees. it differs from sgd in that it limits the sensitivity of each gradient and works in tandem with the moments accountant algorithm to amplify and track the loss of privacy when the weight is updated. the moments accountant accumulates and tracks privacy spending while training deep neural networks. moments accountant greatly improves on earlier sgd privacy analysis and provides meaningful privacy guarantees for deep learning trained on real-size datasets. to ensure that the sgd is differentially private (i.e. dp-sgd), two modifications must be made to the original sgd algorithm. first, the sensitivity of each gradient must be limited. this is done by trimming the gradient in normal . second, random noise is applied to the previously clipped gradient by multiplying its sum by the learning rate and then using it to update the model parameters. 2.3. fairness fair modeling is an area of artificial intelligence that ensures that machine simulation results are not affected by protected attributes such as gender, race, religion, sexual orientation, etc. in this work we will use specific metrics that can be used to evaluate the fairness of a model. the most commonly used measures of fairness are statistical (demographic) parity and equalized odds [19, 27]. the definition of these metrics is given below. given a dataset , where is the feature vector, the label, and the binary protected attribute (e.g. gender, race, etc.). the goal is to learn a new representation , such that will satisfy a certain fairness criteria such as:  statistical parity: a predictor trained on must satisfy:  equalized odds: a predictor trained on must satisfy: 3. related work the growing field of algorithmic fairness and differential privacy has led to the development of numerous methods for mitigating bias and ensuring data privacy. in recent research, the authors in [21] introduce a framework for evaluating and analyzing bias mitigation techniques by using a synthetic dataset and controlling different components of the data generation process. the study also analyzes the performance of several model architectures (such as mlp [22], cnn [13], laftr [17, 9], cfair [6], ffvae [15]) on the adult, cimnist datasets, to understand the sources and levels of bias. as an extension to this article, this work is focused on combining the principles of algorithmic fairness and differential privacy in the training of models under adversary. in particular, this article considers the laftr model, which aims to provide a fair data while maintaining good predictive accuracy. the methodology is based on the introduction of differential privacy to model training. furthermore, the examination of combining algorithmic fairness and differential privacy in training models in the presence of an adversary is extended in other studies. in [18] authors aim to show that it is possible to maintain good predictive accuracy while still providing a strong guarantee for the privacy of individuals' data. their paper is structured around the concept of differential privacy, which is a mathematical framework for protecting the privacy of individuals' data by adding noise to the data. in [16] authors introduce a differentially private (dp) neural representation framework that safeguards user privacy during network computations, using a dp noise layer and robust training algorithm while maintaining performance. the framework offers formal privacy guarantees through sensitivity-based noise injection. some other papers [28, 26, 2, 1] also learn how privacy affects accuracy in different models. the paper [28] proposes a new type of generative adversarial network (gan) called differentially private generative adversarial network balancing accuracy, fairness and privacy in machine learning… 43 copyright ©2023 assa. adv. in systems science and appl. (2023) (dpgan) that addresses the issue of privacy protection in training data. the dpgan adds carefully designed noise to gradients during the learning process to provide differential privacy, with a proof of privacy guarantee and empirical evidence to support its analysis. another paper [26] compares the fairness implications of two differentially private deep learning algorithms, dp-sgd and pate, and finds that pate has higher utility on underrepresented groups in imbalanced datasets. the impact of differential privacy on the accuracy of machine learning models is also examined [2], and it is found that the reduction in accuracy brought about by dp disproportionately affects underrepresented subgroups and subgroups with more complex data. the authors of next paper [1] present new algorithmic techniques and a refined analysis of privacy costs to train deep neural networks while ensuring privacy. the solution is evaluated on standard image classification tasks (mnist and cifar-10) and demonstrates the feasibility of the approach against a strong adversary who has full knowledge of the training mechanism and access to the model's parameters. there are some papers, which explore the effect of privacy introduction on fairness. the first paper [24] explores the impact of differential privacy algorithms on the fairness of machine learning systems. it focuses on two specific differential privacy methods and analyses the reasons for the disparities that arise among different groups of individuals in these methods. the paper proposes guidelines to mitigate the unfair impacts and contributes to the growing research at the interface between differential privacy and fairness. the second paper [25] considers importance of ensuring fairness in machine learning systems, specifically in the context of decisions that affect individuals, such as criminal assessment and hiring. the text highlights the trade-off between model accuracy and fairness and the importance of considering sensitive attributes in learning tasks to ensure non-discrimination. in this paper, the focus is on the integration of algorithmic fairness and differential privacy in the training process of machine learning models. differential privacy, which is a mathematical framework for the protection of the privacy of individuals' data by the addition of noise to the data, is utilized as the main method for privacy preservation. the laftr model, designed to provide a fair data representation while maintaining good predictive accuracy, is evaluated. the performance of the laftr model is assessed and the fairness and accuracy indicators are evaluated in a classification problem, which is a common application in machine learning. machine learning bias refers to a systematic error that occurs in the results due to incorrect assumptions. the objective of algorithmic fairness is to reduce this bias. this is achieved through three main categories of algorithms: pre-processing, in-processing, and post-processing. pre-processing techniques aim to reweight training samples [15], edit features and labels [4], and resample datasets [5]. in contrast, postprocessing methods aim to calibrate predictions [12, 7] by adjusting the learned predictor to remove discrimination based on the joint statistics of the predictor, target, and protected attribute. in-processing techniques [14, 18, 11, 29, 17, 6, 30] focus on removing sensitive information, such as racial or gender group, age, financial transactions or tax payments that may lead to discrimination or bias in decision making, from the data. there are various architectures of models for realizing fair representation in-processing, such as cfair [29], laftr [17], ffvae [6], among others. one of the architectures of the adversarial biasmitigation model consists of an encoder, adversary module, and classifier. the encoder takes in the data and considers the sensitive attribute while encoding the data into the latent space. the adversarial training setup, using a gradient-reversal layer and an attacker network, to train a classifier that accurately predicts main task labels while being oblivious to protected attributes [10]. finally, the data from the preserved latent space is passed on to the classifier, which has an integrated architecture. this architecture is further described in [17]. in this work, the focus is on using the laftr model to generate data for a third-party classifier that is both fair and private. the laftr model is designed to provide a fair representation. differential privacy, which adds noise to the data to protect individuals' privacy, is utilized as the main method for privacy preservation. the integration of 44 a. eponeshnikov, r. sabitov, g. smirnova, sh. sabitov copyright ©2023 assa adv. in systems science and appl. (2023) algorithmic fairness and differential privacy is a central focus in the training process of the laftr model, ensuring that the data generated is suitable for use by a third-party classifier. the paper suggests using a third-party classifier, such as logistic regression, to evaluate the fairness of the data generated by the laftr model. the logistic regression model will be trained on the data from the latent space, which has undergone in-processing methods, allowing us to compare the performance of models trained on both raw data and processed data. mathematical definition of the fairness variate depends on tasks. in this paper we will use two popular metrics: statistical (demographic) parity and equalized odds [20]. 4. methodology in this article, we investigate the dependencies of the impact of privacy injection on accuracy and fairness. the laftr-dp and laftr-eod [17] models with different privacy values in different modules of the model will be compared. in differential privacy represents the privacy budget that determines the amount of noise added to the gradients during training, where a smaller corresponds to stronger privacy guarantees but potentially higher noise levels, impacting the trade-off between privacy and utility in the learning process. these networks apply adversary training and optimization using dp-sgd to achieve fairness and guarantees of data confidentiality. 4.1. dataset adult dataset: the adult dataset is a widely used standard machine learning dataset for studying and demonstrating many common or specially designed machine learning algorithms for unbalanced classification. in total, the data set (train and test) contains 48842 samples. the dataset contains 16 columns, including a target field “income” and 14 attributes that describe a person's demographics and other features. the income is divided into two classes: <=50k and >50k, which serves as the target variable for the classification task. the dataset includes various personal information about individuals, such as their age, education level, gender, occupation, and so on. given the characteristics of adult recruitment, the classification task is to determine whether a person earns more than 50k or less than 50k. gender was selected as a protected attribute. the size of the test data was chosen to be equal to 30% of all data. 4.2. model the idea is to study a representation that satisfies a certain property of fairness while remaining differentially private. fig. 1 shows a network architecture that seeks to study a representation of data capable of reconstructing input data , classifying target labels , and protecting a sensitive attribute (fig. 1). the loss function of the network is defined as a linear combination of three loss terms, the reconstruction loss , the adversary loss , and prediction loss [9]: where , and are hyperparameters controlling the weighting of the competing objectives. the decoder reconstructs input data from the latent space. the classifier predicts the class label from latent space . the adversarial loss is to enforce the representation (latent space) to satisfy a certain fairness notion. any of the first two requirements can be omitted by setting the hyperparameters to zero. to improve fair metrics, which defined in background section, loss functions can be defined as [9]: , , balancing accuracy, fairness and privacy in machine learning… 45 copyright ©2023 assa. adv. in systems science and appl. (2023) , where adversary, classifier, encoder, decoder. also, the objective of the network is not only to make the representation fair, but also to be privacy preserving. in this regard, we propose to train the network with privacy preserving techniques such as dp-sgd. fig. 1. laftr model with dp-sgd in (a) encoder/ classifier, (b) encoder/classifier/adversary to make latent space confidential we need to introduce dp-sgd method in encoder. also, in this work we will inject privacy in classifier and adversary modules (fig. 1). in addition, in this work, two types of laftr models (laftr-dp and laftr-eod) are considered. the only difference between laftr-dp and laftr-eod models is that in laftr-eod approach we pass to adversary not only , but we pass class label in addition. 4.3. training laftr seeks to study the encoder that gives reliable representations, i.e. the output of the encoder can be used by third parties with confidence that their naively trained classifiers be fairly fair, private and accurate. algorithm below describes the training process of laftr with privacy preservation. the detailed pseudocode is described in algorithm 1. algorithm 1: differentially private sgd on learning under adversary input: dataset ; batch size ; learning rate ; iterations; noise ; clipping bound ; hyperparameters controlling the weighting ( , and ); component for privacy preservation ( (for and ) or (for ) or (for )) as ; loss function initialize network parameters (encoder, classifier, adversary respectively) true (boolean indicating whether to update parameters ) for k do if then select next batch; foreach in batch do compute gradients for each module: clip gradients: 46 a. eponeshnikov, r. sabitov, g. smirnova, sh. sabitov copyright ©2023 assa adv. in systems science and appl. (2023) end add noise to selected component: sum other gradients: descent: if then else end end in short, it is a combination of adversarial learned fair representations [9] and dp-sgd [1]. we learn encoder and classifier with gradient clipping and adding noise if necessary. then we freeze the learned encoder and classifier and learn the adversary part with gradient clipping and adding noise if necessary. the overall loss function is a combination of these loss functions with hyperparameters controlling the weighting. the training process consists of iterating over a number of epochs, each time selecting a random batch of data from the training set. the gradients for each component are computed and clipped to ensure that they do not exceed a certain bound. then, a noise is added to one of the components to achieve differential privacy. finally, the parameters of the corresponding component are updated by performing stochastic gradient descent with the computed gradients. this process is repeated until the desired number of epochs is reached. the training algorithm of laftr with privacy preservation is a crucial component of this approach, allowing for the training of complex models that can achieve both fairness and privacy. 4.4. evaluation metrics as we said in section 4.3, we try to train encoder which will give reliable representation. training encoder several times and taking the outputs from the encoder, we will pass it into a simple binary classifier (logistic regression), average them and calculate standard deviation. after that we will use fairness metrics which was mentioned in section 0 to evaluate these bias-mitigation strategies. also, we will measure accuracy of all models for comparison with bias-mitigation strategies and unfair strategies. to evaluate accuracy and fair metrics we will use logistic regression. logistic regression will learn on generated by encoder new data several times and return mean and standard deviation of accuracy, difference of demographic parity and difference of equalized odds . balancing accuracy, fairness and privacy in machine learning… 47 copyright ©2023 assa. adv. in systems science and appl. (2023) during the training of different model configurations, the fair metrics and accuracy will change. to understand advantages of provided models we need to compare accuracy-fairness trade-off. for evaluation trade-off we will use next function: where accuracy, fair metric. next we will call acc/fair. 4.5. model and hyperparameters at the outset of the experiment, the initial step was to define a set of hyperparameters that remained constant throughout all subsequent experiments. hyperparameters are parameters that cannot be learned directly from the training data and need to be set manually before the training process commences. table 1. model fixed hyperparameters model hyperparameters values encoder mlp depth as in depth*[width] 2 layers classifier mlp depth as in depth*[width] 2 layers encoder mlp width 32 neurons classifier mlp width 32 neurons latent (z) space dimension 8 neurons – reconstruction loss weight 0 – prediction loss weight 1 – adversary loss weight 1 activation function in hidden layers in autoencoder leakyrelu activation function in hidden layers in classifier leakyrelu activation function in hidden layers in adversary leakyrelu activation function after last hidden layer of encoder leakyrelu activation function after last hidden layer of classifier sigmoid activation function after last hidden layer of adversary sigmoid in table 1, we can observe the various model hyperparameters and their corresponding values that were used in our experiments. we trained our model without taking into account the reconstruction loss, as our primary goal was to obtain a good representation in the latent space. therefore, we did not require a good reconstruction of input data. additionally, we used the same architecture for all modules of the network, except for the activation function after the last hidden layer of the encoder. this difference in the activation function was necessary since the encoder generates a new representation of input data whereas the classifier and adversary try to predict label or attribute , which have binary nature. the choice of parameters in the model was based on the findings from prior research, particularly [21]. the model architecture was configured to have a two-layer multi-layer perceptron (mlp [22]) for both the encoder and classifier, with a width of 32 neurons in each layer, following the recommendation from [21]. the adversary module was set up in two different configurations. the latent space dimension was set to 8 neurons. having completed the previous step, the subsequent action is to establish the values of the training parameters. these parameters are essential in controlling the learning process and determining the behavior of the model during the training phase. the selection of suitable values for these parameters is a crucial process as it can significantly influence the performance of the model. after determining the appropriate values for the training parameters, they were fixed and used throughout the training process to optimize the model's performance on the given task. notably, these parameters were carefully chosen to produce results that could be compared to those reported in a previous study [17]. the process of defining these constant training parameters is complex, and various factors must be considered. the size of the dataset, the complexity of the model, and the available computational resources – factors which were taking into account during empirical 48 a. eponeshnikov, r. sabitov, g. smirnova, sh. sabitov copyright ©2023 assa adv. in systems science and appl. (2023) choosing hyperparameters. the most important hyperparameters which affect to results learning rate, and number of epochs, optimizer. finding the best combination of these parameters helps achieve optimal results. once the appropriate values for the training parameters have been determined, they are fixed and used throughout the training process to optimize the model's performance on the given task. regular monitoring and tuning of these parameters may also be required during the training process to ensure that the model continues to improve its performance. in some cases generally founded hyperparameters cannot provide satisfiable results. this iterative process involves observing the model's performance during training, making adjustments to the parameters, and retraining the model to ensure that it is optimized for the task at hand. table 2 training fixed parameters training parameters values max gradient norm in encoder 10 max gradient norm in classifier 10 max gradient norm in adversary 10 encoder/classifier optimizer nadam adversary optimizer nadam encoder/classifier learning rate scheduler polynomiallr(2) adversary learning rate scheduler polynomiallr(2) the table 2 provided outlines the specific training parameters and their corresponding values that were used during the training process. the values were carefully selected based on various factors, such as the size of the dataset and the complexity of the model. the maximum gradient norm in the encoder, classifier, and adversary was set to a value of 10. this parameter controls the maximum magnitude of the gradients during the training process, preventing the model from diverging or becoming unstable. the optimization algorithm used for both the encoder and classifier was nadam, while the adversary used the same algorithm with modifications to ensure differential privacy. nadam was selected over other popular optimization algorithms due to its superior performance in handling noisy or sparse gradients. nadam is a variant of the adam optimization algorithm that combines the benefits of adaptive learning rates and momentumbased optimization techniques [23]. it has been shown to outperform other popular optimization algorithms such as adagrad, rmsprop, and adam. in detail, dp-nadam works similarly to dp-sgd, with some modifications that introduce the nadam algorithm itself. these modifications include the calculation of the adaptive learning rates and momentum parameters, which are adjusted based on the first and second moments of the gradient, as well as the inclusion of noise to ensure differential privacy. the choice of nadam over sgd was made based on its superior performance on the given task, as well as its ability to handle noisy or sparse gradients that are common in differential privacy settings. furthermore, the use of dp-nadam provides additional benefits such as faster convergence rates. the learning rate scheduler used for both the encoder and classifier was polynomiallr(2), which gradually reduces the learning rate quadratically over the course of the training process to prevent the model from overfitting to the training data. similarly, the adversary also used the same scheduler. it is worth noting that the choice of parameters was not solely based on the findings from [21], as modifications were made to ensure the model could achieve similar results as reported in both [21] and [17]. while [21] provided valuable insights into the choice of architecture, latent space dimension, and the adversary module configurations, [17] provided guidance on the selection of the optimization algorithm and its modifications to ensure differential privacy. the modifications made to the optimization algorithm were necessary to achieve differential privacy while maintaining the model's performance. it should be noted that even with the use of dp-nadam, the learning rate still had to be increased dramatically (~0.1-0.15) to achieve the desired level of performance. despite the need to increase the learning rate dramatically, the model was able to achieve the desired level of performance balancing accuracy, fairness and privacy in machine learning… 49 copyright ©2023 assa. adv. in systems science and appl. (2023) without sacrificing its stability or accuracy. overall, the parameters were carefully selected based on a mixture of insights from both [21] and [17], with modifications to ensure that the model could achieve the desired level of performance while maintaining differential privacy. additionally, based on a large number of parameter tuning attempts, it was discovered that using a larger batch size improves model stability. therefore, the maximum feasible batch size was used for training, which could fit in vram. for the adult dataset, the entire dataset was used, for the german dataset, the entire dataset was used, and for celeba, 20,000 samples were used. empirically, it has been observed that the model may cease to be adequate under certain conditions. therefore, after training the model, a validation of its adequacy was conducted. the classifier within the neural model should yield accuracy values above 50%, while fair metrics should not exceed 2%. in the event that any of these requirements were not met, the model was retrained with different initial weights. 5. experiments in this section we will compare different approaches of privacy preservation and/or biasmitigation strategies. all models with privacy preservation and/or bias-mitigation was trained several times to calculate mean and standard deviation of accuracy and fair metrics. this work considers models with various configurations as described in table 3. since each configuration was trained multiple times, in addition to the histogram, a t-test table with a pvalue of 5 was used for comparison. table 3 changing parameters parameter values dataset adult model architecture laftr-dp, laftr-eod adversary module 2 layers * 32 neurons; 4 layers * 64 neurons privacy in no privacy; encoder/classifier; encoder/classifier/adversary 1, 3, 10, 30 to ensure that the results were robust a diverse set of 37 models were trained for each dataset. these models varied in terms of their architecture, hyperparameters, and level of privacy. this approach allowed for a comprehensive evaluation of the performance of the models under different configurations and helped to identify the most effective configurations for each dataset. a t-test will be utilized to compare the performance of models. t-test a statistical hypothesis test that is used to determine if there is a significant difference between the means of two groups. in the context of metrics, t-tests can be used to determine if there is a statistically significant difference in the performance of two models, or if a certain modification or intervention has had a significant effect on the performance of a model. by using t-tests, we can make informed decisions about the effectiveness of different strategies for improving the performance or fairness of machine learning models. there are several conventions in the model names. “unfair|no privacy” logistic regression learned on raw data without any modifications. “adversary = classifier” means that modules of laftr have the same structure, “adversary > classifier” means that adversary part stronger (2 times more layers and 2 times more neurons in each layer). 5.1. adult dataset the adult dataset is a well-known benchmark dataset in machine learning that contains demographic and employment-related information of individuals to predict whether their income is above or below a certain threshold. in this case each configuration of models was trained 10 times. number of epochs for each training was set to 250. in all next datasets, based on t-test, protected models demonstrated similar performance across different levels of 50 a. eponeshnikov, r. sabitov, g. smirnova, sh. sabitov copyright ©2023 assa adv. in systems science and appl. (2023) privacy, regardless of the level of noise in the gradients during training and the neural network modules in which this noise was injected. therefore, we will consolidate the results with different epsilon values and different privacy injection configurations. next, tables will be presented for each metric, comparing the minimum, maximum, and average values among all implementations. the results will be grouped by model type and adversary strength, taking into account the conditions described above. 5.2. difference of demographic parity there are results of measuring first fair metric difference of demographic parity – . based on the fig. 2, the comparison models with different privacy levels, epsilon values, and adversary-classifier strengths we can say for sure that the "unfair|no privacy" model is the most unfair. fig. 2. dependence of on for adult dataset as observed in table 4, for laftr-dp, both the minimum and average values of are very low, regardless of the strength of the adversary. however, for the private models, balancing accuracy, fairness and privacy in machine learning… 51 copyright ©2023 assa. adv. in systems science and appl. (2023) the minimum values are nearly zero, while for the unprotected models, they range from 6% to 9%. additionally, in the case of protected models, the average value is independent of the adversary's strength, with a difference of only 3%. the maximum value does not exceed that of the unfair approach in any case, but the difference ranges from 0.6% to 7%. regarding laftr-eod, the behavior of the results is comparable to laftr-dp, but with slight variations and more stability in the values. the maximum difference between laftr-eod and the unfair model ranges from 3.5% to 2.5%. on average, the minimum values are slightly lower for this architecture. in the absence of protection, the difference can reach up to 5% in the case of a weak adversary. additionally, the strength of the adversary has a stronger impact on the average value for the protected models compared to the unprotected ones. table 4 comparison. adult dataset. δdp model privacy adversary min max mean laftr-dp no privacy adversary = classifier 0.064 0.136 0.097 adversary > classifier 0.020 0.097 0.060 privacy in.. adversary = classifier 0.000 0.163 0.059 adversary > classifier 0.001 0.140 0.057 laftr-eod no privacy adversary = classifier 0.017 0.133 0.080 adversary > classifier 0.046 0.143 0.081 privacy in.. adversary = classifier 0.005 0.145 0.061 adversary > classifier 0.000 0.152 0.057 unfair no privacy 0.169 0.169 0.169 5.3. difference of equalized odds next step – results of measuring second fair metric difference of equalized odds . it is very difficult to draw any conclusions based on the fig. 3. based on table 5, in the case of , the situation is slightly different in comparisaon with . the strengthening of the adversary continues to have a positive impact on fairness and often allows for an additional 1-2% improvement on average. furthermore, all protected models can achieve better results than their unprotected counterparts. however, in the case of , techniques such as adversary strengthening or privacy injection can significantly deteriorate the outcomes. for instance, in the laftr-dp model, the addition of a strong adversary can lead to results that are 5% worse than the unfair model, and introducing noise can increase the difference up to 22% in the worst case. nevertheless, the average values are still lower than those of the unfair approach, but the introduction of noise brings the average values closer to those of the unfair solution, with a difference of 1-2%. in contrast, for nonprivate models, the difference ranges from 3.6% to 2.6%. the behavior of the results in laftr-eod models is similar to laftr-dp, but they exhibit less stability. the minimum values are lower in almost all cases, while the maximum values are consistently high, even in the case of a weak adversary without protection (unfair approach performs better by 7%). in other cases, the difference can reach up to 19%. due to the higher instability and a larger number of unfair results, the fairness metric's average value for laftr-eod is approximately 2.5% worse than that of laftr-dp. 52 a. eponeshnikov, r. sabitov, g. smirnova, sh. sabitov copyright ©2023 assa adv. in systems science and appl. (2023) fig. 3. dependence of on for adult dataset table 5 comparison. adult dataset. δeod model privacy adversary min max mean laftr-dp no privacy adversary = classifier 0.055 0.107 0.084 adversary > classifier 0.015 0.165 0.076 privacy in.. adversary = classifier 0.008 0.335 0.103 adversary > classifier 0.010 0.317 0.092 laftr-eod no privacy adversary = classifier 0.027 0.183 0.114 adversary > classifier 0.042 0.205 0.107 privacy in.. adversary = classifier 0.006 0.309 0.113 adversary > classifier 0.007 0.276 0.090 unfair no privacy 0.117 0.117 0.117 5.4. accuracy to fully understand the behavior of the model, it is necessary to investigate patterns of accuracy. balancing accuracy, fairness and privacy in machine learning… 53 copyright ©2023 assa. adv. in systems science and appl. (2023) as in demographical parity metric fig. 4 demonstrates that the "unfair" model, which has no privacy protection and an unfair adversary much better predict labels and consistently outperformed by all other models in terms of accuracy. fig. 4. dependence of accuracy on for adult dataset as shown in table 6, the average accuracy values are independent of the adversary's strength and the model type. the only factor that leads to a decrease in accuracy of around 23% is the introduction of privacy. the difference between the minimum and maximum accuracy values varies between 4% and 7%, and these values do not exhibit any clear correlation. in general, non-private models demonstrate a 5% deviation from the unfair solution, likely due to the reduced dimensionality of the latent z-space and the potential loss of information during data encoding. protected models show an 8% deviation, which can be attributed to the addition of noise in the training process, affecting the latent z-vector. to assess the trade-off between accuracy and fairness and understand the benefits of prioritizing fairness over accuracy, it is crucial to examine the compromise between these two metrics. by analyzing the relationship between accuracy and fairness measures, we can determine the extent to which sacrificing accuracy has resulted in improved fairness 54 a. eponeshnikov, r. sabitov, g. smirnova, sh. sabitov copyright ©2023 assa adv. in systems science and appl. (2023) outcomes. this analysis allows us to evaluate the value of prioritizing fairness and make informed decisions regarding the trade-off between these two important considerations. table 6 general accuracy comparison. adult dataset. accuracy model privacy adversary min max mean laftr-dp no privacy adversary = classifier 0.785 0.829 0.810 adversary > classifier 0.767 0.810 0.792 privacy in.. adversary = classifier 0.749 0.803 0.777 adversary > classifier 0.749 0.805 0.777 laftr-eod no privacy adversary = classifier 0.758 0.830 0.803 adversary > classifier 0.784 0.823 0.804 privacy in.. adversary = classifier 0.756 0.808 0.779 adversary > classifier 0.754 0.822 0.778 unfair no privacy 0.853 0.853 0.853 5.5. accuracy/fairness trade-off we compared fairness and accuracy of different approaches and explored that the bigger fairness the lower accuracy and vice versa. to understand advantages of provided models we need to compare accuracy-fairness trade-off using custom metric which was provided in section 4.4. the information presented in fig. 5 shows that “no privacy” approaches slightly better than “unfair” and protected approaches. but for a complete picture of understanding the compromise, it is also necessary to compare the accuracy-difference of equalized odds trade-off. it is evident from fig. 6 that “unfair” approach has better trade-off across all other models. as evident from table 7, in all cases of trade-offs between , the average values are better than those of the unfair solution (with a difference of 0.4% to 1.7%), indicating consistent improvements in terms of this metric. it is important to note that the introduction of noise increases the variability between the minimum and maximum values (from approximately 3% for non-private models to around 10% for private models), but it also leads to an increase of approximately 2-3% in the maximum value. there are no differences observed between architectures, but strengthening the adversary may yield improvements of 1-2% in some cases. when considering , the average values generally fall behind the unfair solution by 2-6%. however, in the maximum column, all values are equal to or greater than the unfair solution, indicating the need for more nuanced adjustments. on average, laftr-dp demonstrates values superior to laftr-eod by 1%. it is worth noting the high stability of laftr-dp without privacy, with a variability of 4% and 7% between the minimum and maximum values for different adversaries, while the others range from 12% to 18%. in general, for this dataset, it can be stated that with proper configuration, outstanding fairness results can be achieved while sacrificing less in terms of accuracy compared to gains in any fairness metric. furthermore, based on all the results, it becomes evident that strengthening the adversary has a positive impact on the trade-off, but only when carefully fine-tuned; otherwise, it may lead to poor outcomes. additionally, it is not possible to confidently assert that laftr-dp outperforms laftr-eod, as it depends on the specific task, where laftr-eod may perform better in certain cases. finally, it can be concluded that by incorporating privacy at various levels, it is feasible to attain either a highly fair model or avoid significant accuracy losses while improving fairness. balancing accuracy, fairness and privacy in machine learning… 55 copyright ©2023 assa. adv. in systems science and appl. (2023) fig. 5 dependence of acc/fair on for adult dataset 56 a. eponeshnikov, r. sabitov, g. smirnova, sh. sabitov copyright ©2023 assa adv. in systems science and appl. (2023) fig. 6 dependence of acc/fair on for adult dataset table 7 general accuracy-fairness trade-off comparison. adult dataset. acc/fair δdp model privacy adversary min max mean laftr-dp no privacy adversary = classifier 0.716 0.755 0.739 adversary > classifier 0.723 0.761 0.747 privacy in.. adversary = classifier 0.683 0.789 0.734 adversary > classifier 0.694 0.783 0.735 laftr-eod no privacy adversary = classifier 0.716 0.766 0.744 adversary > classifier 0.720 0.766 0.744 privacy in.. adversary = classifier 0.678 0.779 0.734 adversary > classifier 0.698 0.777 0.737 unfair no privacy 0.730 0.730 0.730 acc/fair δeod model privacy adversary min max mean laftr-dp no privacy adversary = classifier 0.723 0.764 0.747 adversary > classifier 0.694 0.776 0.737 privacy in.. adversary = classifier 0.595 0.773 0.707 adversary > classifier 0.604 0.763 0.714 laftr-eod no privacy adversary = classifier 0.687 0.769 0.722 balancing accuracy, fairness and privacy in machine learning… 57 copyright ©2023 assa. adv. in systems science and appl. (2023) adversary > classifier 0.668 0.785 0.728 privacy in.. adversary = classifier 0.594 0.776 0.702 adversary > classifier 0.624 0.774 0.716 unfair no privacy 0.764 0.764 0.764 6. conclusion this research study aimed to develop a benchmark to evaluate differentially private fair representations across various model configurations and datasets and research results. the following key findings were obtained during the study:  firstly, the impact of privacy integration on fairness in the encoding process was examined. it was demonstrated that with successful weight initialization, any model could generate nearly fair representations. additionally, statistical tests confirmed the equality of privacy-enabled models in all cases. the level of noise, as well as the number of parts it is injected into, did not affect the fairness outcome. the introduction of privacy improved fairness by up to 5%.  secondly, it was observed that privacy integration had a negative impact on model accuracy, with a typical decline of 1-3% compared to unprotected counterparts.  furthermore, the analysis of results revealed a pattern where strengthening the adversary could have a positive effect on the outcomes. however, careful fine-tuning and monitoring of the training process were necessary due to increased instability.  moreover, it was shown that any private fair models could achieve up to 5% advantage in the compromise between accuracy and fairness compared to the "unfair" model. based on the study's findings, it can be concluded that it is possible to train a fair and private model with acceptable accuracy and a high level of protection. however, achieving such results requires fine-tuning or fortunate circumstances, as privacy integration reduces stability in an already unstable model. increasing fairness may require strengthening the adversary, which further adds to the instability. references 1. abadi, m., chu, a., goodfellow, i., mcmahan, h. b., mironov, i., et al. 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(2020). conditional learning of fair representations. arxiv:1910.07162, [online]. available: https://arxiv.org/abs/1910.07162 30. smirnova g. s., sabitov r. a., korobkova, e. a., sabitov, sh. r. (2017). modeling production facility as a dynamic integrated interacting objects system, procedia computer science, 112, 965–970, https://doi.org/10.1016/j.procs.2017.08.136 adv syst sci appl 2023; 02:70–78 published online at https://ijassa.ipu.ru. a new approach on the modelling and analysis stability of a class of fractional-order quasi-polynomial systems mohamed reda lemnaouar1, mohamed khalfaoui1, rabie zine2*, younes louartassi1,3, 1mohammed v university in rabat, est salé, lastimi, salé, morocco 3school of science and engineering, al akhawayn university in ifrane, morocco 3 mohammed v university in rabat, faculty of sciences, laboratory of mathematics, computing and applications, rabat, morocco abstract: stabilization and observation for nonlinear fractional derivative systems remain open problems in automatic due to the fractional nature and nonlinearity of these systems. the present paper studies global stability by the return output for fractional systems. first, we give some definitions of fractional calculus and the quasi-polynomial (qp) and lotka-volterra (lv) systems. then, we analyze their stabilities as well as linear (lmi) and bilinear (bmi) matrix inequalities. in order to solve the controller design problem. the goal of this paper is to investigate the global and local stability of a dynamic fractional order system using the quasi-polynomial and lv representation. then, we use the lmi to study the stabilization of this fractional system. keywords: quasi-polynomial systems, fractional derivative, lmi, bmi, stability 1. introduction in the last decades, the analysis problem of control and stability of dynamic systems has attracted the attention of researchers [10, 20]. these studies are applied in several fields in reality such as chemistry, technology and physical systems including electrical networks, economic systems and electrical systems [2]. stabilization and observation for nonlinear fractional derivative systems remain open problems in automatic due to the fractional nature and nonlinearity of these systems. indeed, stabilization is one of the major concerns of both researchers and engineers. there are many methods of stabilizing linear systems; they are generally based on pole placement techniques or the minimization of a quadratic criterion and lead to feedback, the implementation of which requires the use of observers when the condition is partially measured. lyapunov’s first method makes it possible to use these results for the control of nonlinear systems from their linearizations and the use observers. many dynamical systems are represented by a non-integer order dynamic model, usually based on the notion of differentiation or integration of the non-integer order. studying the stability of fractional order systems [12–17] is more difficult than for their counterparts (integer order systems). on the one hand, fractional systems are regarded as memory systems, in particular for the consideration of initial conditions, and on the other hand they ∗corresponding author: r.zine@aui.ma a new approach on the modelling and analysis stability. . . 71 have a much more complex dynamic. lotka-volterra and quasi-polynomial are models proven to be well adapted to canonical forms generally applicable to models of ordinary non-linear ode differential equations, since a large class of these models can be written in these types [5]. furthermore, the approach of global and local stability analysis of generalized lv models are abundant [3, 4, 11]. lv systems are generally used in some scientific fields, such as population biology. the form and properties of the generalized and classical lv algebraic forms were treated by hernandez-bermejo and fairen in [19]. the goal of this paper is to investigate the global and local stability of a dynamic fractional order system using the quasi-polynomial and lv representation. then, we use the lmi to study the stabilization of this fractional system. 2. background material and preliminary results throughout this paper, mp,m(r) denotes the set of real matrices with p rows and m columns, mm(r) the set of square real matrices of size m, diag(ω1, . . . , ωm) represent the diagonal matrix which contains the elements ω1, . . . , ωm and bjk is the element of the j-th row and the k-th column of a matrix b. in this section, we use the new easy definition of the fractional derivative (see [7]). this new definition is a natural extension of the usual derivative, and it verifies the first four properties of the fractional derivative in the sense of riemann-liouville and in the sense of caputo (see [7]). also, this definition coincides with the known fractional derivatives on the polynomials. definition 2.1: let h : [0,∞) −→ r, u > 0 and α ∈ (0, 1). we define the conformable fractional derivative of h of order α by : t α(h)(u) = lim ϵ→0 h(u+ ϵu1−α)− h(u) ϵ . (2.1) furthermore, if h is α-differentiable (the conformable fractional derivative t α(h)(u) exist) in some (0, u1), u1 > 0 and lim u→0+ t α(u) exists, then clearly t α(u)(0) = lim u→0+ t αh(u). theorem 2.1: let α ∈ [0, 1[ and the functions y1, . . . , yp are α-differentiable at a point u > 0. then (1) t α(k1y1 + k2y2) = k1t α(y1) + k2t α(y2) for all k1, k2 ∈ r. (2) t α(u) = 0, for all u ∈ r. (3) t α(y1y2) = y1t α(y2) + y2t α(y1). (4) t α ( y1 y2 ) = y2t α(y1)−y1t α(y2) y22 . (5) if y is differentiable, then t α(y) = u1−α dy(u) du . (6) t α(ym) = mym−1tα(y) for all m ∈ n∗. (7) t α( p∏ k=1 yk) = p∑ k=1 t α(yk) p∏ i=1 i ̸=k yi for all m ∈ n∗. proof properties (1) through (5) have already been approved by khalil et al. in [7]. copyright © 2023 assa. adv syst sci appl (2023) 72 m. r. lemnaouar, m. khalfaoui, r. zine and y. louartassi to prove property (6). let u > 0 and m ∈ n∗, we have t α(ym)(u) = lim ϵ→0 ym(u+ϵu1−α)−ym(u) ϵ , = lim ϵ→0 ( y(u+ϵu1−α)−y(u) ϵ )lim ϵ→0 m−1∑ k=0 yk(u+ ϵu1−α)ym−k−1(u)  , = t α(y)(u) m−1∑ k=0 yk(u)ym−k−1(u), = t α(y)(u) m−1∑ k=0 ym−1(u), = mym−1(u)t α(y)(u). now, for the last property, we use recurrence reasoning. for all u > 0, we suppose that yp+1 is α-differentiable and t α( p∏ k=1 yk) = p∑ k=1 t α(yk) p∏ i=1 i̸=k yi. we have t α( p+1∏ k=1 yk) = t α(( p∏ k=1 yk)yn+1), = yn+1t α( p∏ k=1 yk) + ( p∏ k=1 yk)t α(yn+1), = ( p∑ k=1 t α(yk) p∏ i=1 i̸=k yi)yn+1 + ( p∏ k=1 yk)t α(yn+1), = p∑ k=1 t α(yk) p+1∏ i=1 i ̸=k yi + ( p∏ k=1 yk)t α(yn+1), = p+1∑ k=1 t α(yk) p+1∏ i=1 i ̸=k yi according to the reasoning by recurrence we proved the relation (7). 3. lotka-volterra and quasi-polynomial models we will define some notions concerning the qp and lv fractional order systems as well as their stability analysis and that of lmi and bmi. qp models are systems of fdes in this form : t α(yi) = yi li + m∑ j=1 mij p∏ k=1 y bjk k  , i = 1, . . . , p. (3.2) where t α is defined in definition 2.1, y = [y1, . . . , yp] t ∈ rp +, l = [l1, . . . , lp] t ∈ rp, m ∈mp,m(r) and b ∈mm,p(r). without the loss of generality, we can assume that rank(b) = p and m ≥ p (see [5]). copyright © 2023 assa. adv syst sci appl (2023) a new approach on the modelling and analysis stability. . . 73 considering zj = p∏ k=1 y bjk k and using the 6th and 7th properties of theorem 2.1. if rank(b) = p, then the fractional differential system fdes (3.2) becomes in the lv form : t α(zj) = zj nj + m∑ i=1 ajizi  , j = 1, . . . ,m (3.3) where a = bm and n = bl. 3.1. input-affine qp system models we consider the input-affine nonlinear system fractional order model following : t α(y) = ϕ(y) + r∑ l=1 φl(y)ϑl, θ = ψ(y). (3.4) where y, ϑ and θ are the state vector, the input vector and the output vector, respectively. the function ϕ, φl for l = 1, . . . , r and ψ are in qp -form. then the input-affine qp -system with p-inputs is : t α(yi) = yi l0 + m∑ j=1 m0ij p∏ k=1 y bjk k + r∑ l=1 yi lli + m∑ j=1 mlij p∏ k=1 y bjk k ϑl, i = 1, . . . , p. (3.5) where m0,ml ∈mp,m(r), b ∈mm,p(r) and l0, ll ∈mp,1(r) for l = 1, . . . , r. the related input-affine lv model in this form : t α(zj) = zj n0j + m∑ k=1 a0jkzk + r∑ l=1 zj nlj + m∑ k=1 aljkzk ϑl, j = 1, . . . ,m (3.6) where n0 = bl0 ∈mm,1(r), nl = bll ∈mm,1(r), a0 = bm0 ∈mm(r) and al = bml ∈mm(r) for l = 1, . . . , r. 3.2. transforming non-qp fdes models into qp -form the non-linear fdes is writing in this form : t α(yi) = ∑ is1,...,isn,js ais1,...,isn,jsy is1 1 . . . yisnn g(ȳ)js , yi(0) = yi0, i = 1, . . . , p (3.7) where g(ȳ) is not written in this form : p∏ k=1 y cjk k , j = 1, . . . ,m with c ∈mm,p(r). in addition, t αg ∂yi = ∑ es1,...,esn,es bes1,...,esn,esy es1 1 . . . yesnn g(ȳ)es . (3.8) copyright © 2023 assa. adv syst sci appl (2023) 74 m. r. lemnaouar, m. khalfaoui, r. zine and y. louartassi we introduce a new auxiliary variable η = f q p∏ k=1 ypkk , q ̸= 1. (3.9) then, equations (3.7) can write in qp -form : t α(yi) = yi  ∑ is1,...,isn,js ais1,...,is1,jsη js/q p∏ k=1 y isk−δsk−jspk/q k  , i = 1, . . . , n (3.10) where δsk = 1 if s = k and 0 otherwise. additionally, a new qp fdes appears for the new variable η : t αη = η p∑ s=1 pst α(ys)y −1 s + ∑ isβ esβ,es ,js aisα,js besβ,esqη (es+js−1)/q p∏ k=1 y isk+esk+(1−es−js)pk/q k  . (3.11) the choose of p and q can be in (3.9) in different approach in order to facilitate the calculations. the simplest choice is ps = 0, s = 1, . . . , n and q = 1. so, if the initial values of the variables introduced according to (3.9) is fixed, at that moment the dynamics of the integrated system will be equivalent to the original non-qp system shown in (3.9). hence, it is evident that the original system (3.9) is stable whenever the systems (3.10)-(3.11) are stable. 3.3. qp models of process systems nonlinear process system models with localized parameters are divided into two forms in terms of their representation in the form of qp . the systems operate ϕ from the input affine spatial state model (3.4) and they are not always in the form of qp . consequently, the integration of these models in the described qp -form crucial practice. the characteristics of the input function φi of the input-affine space-state model (3.4). frequently, φi is a simple homogeneous linear function of the corresponding state variable yi : φi(y) = const.yi implies al = 0 in (3.2) and ml = 0 in (3.6). if φi = const, this form is not accepted when an equation of state in qp -form comes from the incorporation of variables. 3.4. application new, we consider the following non-qp model (see [9]) : t α(x) = r1x ( 1− x k ) − σ1x + σ2y − ux2 − βxs α′+x − q1e1x − n1xy − γxi, t α(y ) = (r2 − σ2)y + σ1x − vy 2 − n2xy, t α(s) = σβxs α′+x − δsi − µs − q2e2s, t α(i) = δsi + σγxi − q3e3i − ηi. (3.12) where x , y , i and s represent biomass densities of the unreserved area, reserved area, infected and susceptible predator, respectively. for the harvesting in the unreserved area, e1, e2 and e3 denote the effort applied the susceptible and infected predator populations, copyright © 2023 assa. adv syst sci appl (2023) a new approach on the modelling and analysis stability. . . 75 respectively. r1, r2 the fish population growth rates in the reserved and unserved areas, respectively. q1 and q2 represents the coefficient of catchability in the unreserved area and the predator species. σ1 and σ2 represent the migration rates between the two zones. µ and η represent the rate of death of susceptible and infected predators. n1 and n2 are the coefficients of competition. γ the strength of intra-specific between x and i . δ the coefficient of disease transmission. β the rate of search of the prey toward susceptible predators. α′ constant of saturation while susceptible predators s attack the prey x . σ is the rate of conversion of susceptible predator i due to prey x . ux2 and vy 2 represents the reduction terms, in the unreserved area and reserved area respectively. to transform the model (3.12) into the qp form, we introduce a new variable z = 1 α′+x , we get the following qp-model : t α(x) = x ( r1 − σ1 − q1e1 + σ2v2 − (u+ r1 k )x − βv1 − n1y − γi ) , t α(y ) = y (r2 − σ2 + σ1v3 − vy − n2x) , t α(s) = s (σβv4 − δi − µ− q2e2) , t α(i) = i (δs + σγx − q3e3 − η) , t α(z) = z ( −v4(r1 − σ1 − q1e1) + ( r1 k + u)v5 + n1v6 + βv7 − σ2v8 + γv9 ) . (3.13) the 14 quasi-monomials of the qp system model given are : x , y , z, i , s, v1 = sz, v2 = x−1y , v3 = xy −1, v4 = xz, v5 = x2z, v6 = xy z, v7 = xz2s, v8 = y z and v9 = xzi . then, the qp-model (3.13) is transfomed in the lotka-volterra form : t α(x) = x ( r1 − σ1 − q1e1 + σ2v2 − (u+ r1 k )x − βv1 − n1y − γi ) , t α(y ) = y (r2 − σ2 + σ1v3 − vy − n2x) , t α(s) = s (σβv4 − δi − µ− q2e2) , t α(i) = i (δs + σγx − q3e3 − η) , t α(z) = z ( −v4(r1 − σ1 − q1e1) + ( r1k + u)v5 + n1v6 + βv7 − σ2v8 + γv9 ) , t α(v1) = v1 ( (σβ + r1 − σ1 − q1e1)v4 + ( r1k + u)v5 + n1v6 + βv7 − σ2v8 + γv9 + σγx − η − q3e3 ) , t α(v2) = v2 ( r2 − σ2 − (r1 − σ1 − q1e1) + ( r1k + u− n2)x + (n1 − v)y + βv1 − σ2v2 − γi ) , t α(v3) = −v3 ( r2 − σ2 − (r1 − σ1 − q1e1) + ( r1k + u− n2)x + (n1 − v)y + βv1 − σ2v2 − γi ) , t α(v4) = v4 ( t α(x) x + t α(z) z ) , t α(v5) = v5 ( t α(x) x + 2 t α(z) z ) , t α(v6) = v6 ( t α(x) x + t α(z) z + t α(y ) y ) , t α(v7) = v7 ( t α(x) x + 2 t α(z) z + t α(s) s ) , t α(v8) = v8 ( t α(z) z + t α(y ) y ) , t α(v9) = v9 ( t α(x) x + t α(z) z + t α(i) i ) . (3.14) 4. stability of qp systems fractional order according on the basic concepts of lv and qp systems gave in section 3, we will analysis of the stability of qp systems fractional order and a proposed numerical algorithm to solve the case of non-strict lmi. the analysis of stability is carried out around an equilibrium point y∗ = (y∗1, y ∗ 2, . . . , y ∗ p). the solution of the equilibrium point is determined from (3.2) as follows : 0 = yi li + m∑ j=1 mij p∏ k=1 y bjk k  , i = 1, . . . p, and m ≥ p (4.15) copyright © 2023 assa. adv syst sci appl (2023) 76 m. r. lemnaouar, m. khalfaoui, r. zine and y. louartassi suppose that y∗ is a positive equilibrium point of (3.2), so z∗ = (z∗1 , z ∗ 2 , . . . , z ∗ m) is a positive equilibrium in the case lv systems (3.6). the lyapunov function associated with lv-system in the following: f (z) = m∑ i=1 ωi ( zi − z∗i − z∗i ln( zi z∗i ) ) ωi > 0, i = 1, . . . ,m (4.16) where z∗ is correspondent point of equilibrium y∗ of system (3.2). the fractional derivate of lyapunov function is defined by : t αf (z) = f ′(z).t α(z) = 1 2 (z − z∗)(wa+ atw)(z − z∗), (4.17) where w = diag(ω1, . . . , ωm) and a is the invariant characterizing the lv-form (3.6). consequently, the decreasing nature of lyapunov’s function is equivalent to a feasibility problem of the following lmi : wa+ atw ≤ 0 w > 0, (4.18) the possibilities of finding a function of lyapunov that shows the global asymptotic stability of a qp system can be raised with the help of temporal reparametrization (see [3]). if the model (3.6) are ordered so that the first p lines of b are linearly independent, then ωi > 0 for i = 1, . . . , p and cj = 0 for j = p+ 1, . . . ,m ensure the global stability. the global stability of (3.6) is studied and demonstrated in [3,4] with the lyapunov function (4.16), which gives the boundedness of the solutions and the global stability of (3.2). it can be noted that the global stability is limited to the positive orthant only for lv and qp models, which is compatible with the nature of the variables. the global stability of the equilibrium points (3.2) as a function of lyapunov (4.16) are not related on the value of the vector l as will as the equilibrium points are in the positive orthant. this makes it possible to reach the equilibrium point of the closed-loop system (cls) when designing the stabilization controller [18]. by using temporal reparametrization [18], we increase the possibilities to find a function of lyapunov which demonstrates the global asymptotic stability of a qp system. 5. the controller design problem now, we will study the global feedback stabilization of (3.4) where the state feedback control is on the qp form of the cls. also, we will study its global stability using lmi if the return parameters are known and fixed. otherwise, a feedback design problem will be defined that globally stabilizes the cls. the global stabilization of the feedback design problem for qp systems is modeled on (5.19). let qp entries: φl = s∑ i=1 kilqi, l = 1, . . . , r (5.19) where the quasi-monomial functions qi = qi(y1, . . . , yp) for i = 1, . . . , s of the state variables of (3.7) and kil are constants gain. moreover, the cls will also be a qp system with the copyright © 2023 assa. adv syst sci appl (2023) a new approach on the modelling and analysis stability. . . 77 following matrices : m̄ = m0 + r∑ l=1 s∑ i=1 kilmil, b̄ l̄ = l0 + r∑ l=1 s∑ i=1 killil. (5.20) the qi in the cls, as well as the matrix b̄ can vary considerably depending on the choice of the feedback structure. hence, in the closed-loop lv, the matrix of the coefficients ā is expressed in this form : ā = b̄.m̄ = a0 + r∑ l=1 s∑ i=1 kilail. (5.21) thereafter, the analysis of the global stability of the cls with an unspecified gain feedback kil gives the next bmi : ātw +wā = at 0w +wa0 + r∑ l=1 s∑ i=1 kil(a t ilw +wail) ≤ 0. (5.22) where the feedback gain parameters is kil and the positive coefficients of the lyapunov function is cj , j = 1, . . . ,m. then, there exists globally stabilizing feedback whenever the bmi above is attainable. references 1. cao, y. y., lam, j., & sun, y. x. (1998). static output feedback stabilization: an ilmi approach, automatica, 34(12), 1641–1645. 2. duan, g. r. (2010). analysis and design of descriptor linear systems (vol. 23). springer science & business media. 3. figueiredo, a., gleria, i. m., & rocha filho, t. m. (2000). boundedness of solutions and lyapunov functions in quasi-polynomial systems, physics letters a, 268(4–6), 335– 341. 4. hernández-bermejo, b. (2002). stability conditions and liapunov functions for quasipolynomial systems. applied mathematics letters, 15(1), 25–28. 5. hernández-bermejo, b., fairén, v., & brenig, l. (1998). algebraic recasting of nonlinear systems of odes into universal formats, journal of physics a: mathematical and general, 31(10), 2415. 6. kaszkurewicz, e., & bhaya, a. (2012). matrix diagonal stability in systems and computation. springer science & business media. 7. khalil, r., al horani, m., yousef, a., & sababheh, m. (2014). a new definition of fractional derivative, journal of computational and applied mathematics, 264, 65–70. 8. kocvara, m., & stingl, m. (2005). tomlab/penbmi solver (matlab toolbox), [online]. available: https://tomopt.com/tomlab/products/penbmi/. 9. lemnaouar, m. r., khalfaoui, m., louartassi, y., & tolaimate, i. (2020). fractional order prey-predator model with infected predators in the presence of competition and toxicity, mathematical modelling of natural phenomena, 15, 38. 10. li, h., wang, s., lü, j., you, x., & yu, x. (2016). stability analysis of the shunt regulator with nonlinear controller in pcu based on describing function method. ieee transactions on industrial electronics, 64(3), 2044–2053. 11. magyar, a., szederkényi, g., & hangos, k. m. (2008). globally stabilizing feedback control of process systems in generalized lotka–volterra form, journal of process control, 18(1), 80–91. copyright © 2023 assa. adv syst sci appl (2023) 78 m. r. lemnaouar, m. khalfaoui, r. zine and y. louartassi 12. michelitsch, t. m., maugin, g. a., nicolleau, f. c., nowakowski, a. f., & derogar, s. (2009). dispersion relations and wave operators in self-similar quasicontinuous linear chains, physical review e, 80(1), 011135. 13. oldham, k., & spanier, j. (1974). the fractional calculus theory and applications of differentiation and integration to arbitrary order. elsevier. 14. podlubny, i. (1999). an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, math. sci. eng, 198, 340. 15. riesz, m. (1949). l’intégrale de riemann-liouville et le problème de cauchy, acta mathematica, 81, 1–222. 16. ross, b., samko, s. g., & love, e. r. (1994). functions that have no first order derivative might have fractional derivatives of all orders less than one, real analysis exchange, 20(1), 140–157. 17. samko, s. g., kilbas, a. a., & marichev, o. i. (1993). fractional integrals and derivatives (vol. 1). yverdon-les-bains, switzerland: gordon and breach science publishers. 18. szederkényi, g., hangos, k. m., & magyar, a. (2005). on the time-reparametrization of quasi-polynomial systems, physics letters a, 334(4), 288–294. 19. szederkényi, g., & hangos, k. m. (2004). global stability and quadratic hamiltonian structure in lotka–volterra and quasi-polynomial systems, physics letters a, 324(5–6), 437–445. 20. tan, s., wang, y., & lü, j. (2016). analysis and control of networked game dynamics via a microscopic deterministic approach, ieee transactions on automatic control, 61(12), 4118–4124. copyright © 2023 assa. adv syst sci appl (2023) introduction background material and preliminary results lotka-volterra and quasi-polynomial models input-affine qp system models transforming non-qp fdes models into qp-form qp models of process systems application stability of qp systems fractional order the controller design problem microsoft word 1142 article text, copyedited.doc adv syst sci appl 2022; 01; 80-91 published online at https://ijassa.ipu.ru. two-phase extraction of dye eosine b imperial red using non ionic surfactant triton x-100 iso-octyl phenoxy polyethoxy ethanol and dehydol lt7 nacera ghitri1*, hassiba bouabdesselam1, mostefa kameche2, zahera talbi2 1) département de physique/chimie, laboratoire de recherche en technologie de l'environnement, ecole nationale polytechnique d'oran-maurice audin, oran, algérie e-mail: ghitrinacera@hotmail.fr e-mail: has.bouab@yahoo.fr 2) laboratoire de physico-chimie des matériaux, catalyse et environnement, université des sciences et de la technologie d’oran-mohammed-boudiaf, oran, algérie e-mail: kameche@hotmail.com, mostefa.kameche@univ-usto.dz e-mail: talbizahera@yahoo.fr abstract: the objective of this work is the extraction by coacervate of eosin b. its extraction has been achieved in aqueous solution by using the extractant triton x-100 (iso-octyl phenoxy polyethoxy ethanol) and dehydol lt7 which are easily and rapidly biodegradable. the experimental results of the extraction, as a function of the concentration of the surfactant and the temperature, are expressed by the following three quantities: percentage of dye extracted (e), residual concentration of dye in the phase dilute (xs, w) and volume fraction of the coacervate (φc) at equilibrium. the results obtained for each parameter are represented on three-dimensional diagrams by using the empirical smoothing method. the calculated values are consistent with the measurements. for oxoc15e7, the extraction yields achieved under optimal conditions with tx100 surfactant are 99.18% and 98% successively. the results revealed that the reduction of the solute concentration in the dilute phase is about 48 and 88.5 times with dehydol lt7 and tx-100, respectively. besides, the addition of sodium sulfate is also investigated. keywords: cloud point, non ionic surfactants, eosin b dye, cloud point extraction 1. introduction synthetic dyes are widely used in printing, textile industries food, cosmetic, clinical, metal coloring, photography (sensitization), biology (staining of microscopic preparations), color indicators and medical care therapy (antiseptic) [32]. the research work carried out on these dyes, has shown that these chemical compounds have carcinogenic effects for humans and animals [11, 24, 34]. both, the coloured untreated or partially treated discharges in the ecosystem cause a dramatic source of pollution, eutrophication, inhibition of photosynthesis in aquatic plants by reducing the penetration of light [38]. eosine b is an acid, anionic used as a dye in cytology, for laboratory microscopy, to stain cell cytoplasm (plant or animal), collagen, muscle fibers, lymphocytes and bacteria. it is also used to stain raw wood of a burgundy red before application of varnish, disinfectant and drying in dermatology, hydrogeological tracer and photopolymerization [8].it is an irritant, harmful if swallowed and causes serious eye damage [19]. the degradation of such dyes often requires a physico-chemical process and also a biological treatment chain ensuring the elimination of the various pollutants in successive * corresponding author: ghitrinacera@hotmil.fr two-phase extraction of dye eosine b imperial red using non ionic surfactant 81 copyright ©2022 assa. adv. in systems science and appl. (2022) stages [46]. the physico-chemical treatments commonly utilized include adsorption [5], coagulation [44], flocculation, precipitation, ozonation [9], membrane processes [45] ultrafiltration and nanofiltration [1], filtration with coagulation [18], ozonation with coagulation. although, they are successfully used for the treatment of the industrial effluents, these separation methods have proven to be ineffective given the standards required for these discharges. as a matter of fact, the cloud point extraction treatment is proposed in this work as an efficient and reliable alternative [2]. the present work concerns the extraction of the problem point (pce) of the dye, and it has been observed that this method simultaneously includes the problem point and the phenomenon of solubilization of the medium of nonionic surfactants. besides, most polyethoxylated nonionic surfactants in aqueous solution form two phases above the cloud point (t,): a rich-surfactant phase (coacervate) and a dilute phase in which the concentration of the surfactant is close to its critical micelle concentration [38]. therefore, due to the micellar solubilizing property of surfactants, the hydrophobic, amphiphilic or even ionic solutes can be extracted into the coacervate after increasing the temperature above tc. epcs using micelles or mixed micelles. it is a promising method for setting up a new large-scale process in which the aqueous phase is used [2]. the small volume of the rich phase in biodegradable surfactants obtained by the cloud point method makes it possible to design an extraction strategy requiring low cost, extraction efficiency and good efficiency more than those using organic solvents. this fact is particularly interesting, as the concept of "green chemistry" is obyed. this novel process called liquid-coacervate extraction or cloud point extraction (cpe), was first time applied to the separation of metal ions in the presence of a chelating agent [41]. it has been largely utilized to remove organic [16, 43]. really, surfactants suitable for the cloud point extraction process should be poorly soluble in water (to limit the loss of surfactants in the dilute phase), have a low cloud point (to reduce the need for heating), solubilize pollutants, form low toxicity and of course biodegradable. (the results of the cloud point extraction of eosin b with tx-100 are consistent with the conclusions drawn from the experimental results found for eosin. indeed, the cloud point extraction of the dye eosin y can be successfully used to remove color from wastewater using tx-100 as a nonionic surfactant. the extraction efficiency increases with temperature, tx-100 and nacl concentration. it has been observed that for a dye concentration up to 10 ppm, 100% eosin y dye removal is possible using 0.2 m tx-100 at 95°c. for a higher dye concentration up to 200 ppm, 87.5-100% eosin y removal can be achieved using 0.1 m tx100 and 0.2 m salt concentration (nacl) at 80°c [37]). 2. materials and methods 2.1. reagents the non ionic surfactant used in this work, the triton x-100 (iso-octyl phenoxy polyethoxy ethanol), belonging to the family of polyethoxylated alkylphenols (formula [iso-c8h17-c6h4 (och2ch2) 10oh]), was supplied by sigma-aldrich, its critical micellar mass concentration (cmc) is 0.162g / l, its surface tension at cmc ϒcmc is 37.7 dyne / cm, a surface saturation concentration is 3.30 .1010 mol / cm2, head area polar s is 49.6 a ° 2 and its cloud point at 1% by weight in water was at 70°c. loxo-c15e7 belonging to the family of commercial polyethoxylated alcohols which is dehydol lt7 (oxo-c15e7) of formula [oxo-c15h31(och2ch2) 7oh] its critical micellar mass concentration, cmc is 0.01144g / l, its surface tension at cmc is ϒcmc is 31.60 dyne / cm a surface saturation concentration is 1.89. 1010 mol / cm2, area of the polar head s is 87.72 a ° 2. this sample was provided by henkel (ain temouchent, algeria). eosine b obtained from an algerian pharmacy, in the form of a 2% aqueous solution of disodium eosine b (double-distilled water 100 cc eosine disodium 2 g) contained in a plastic 82 n. ghitri, h. bouabdesselam, m. kameche, z. talbi copyright ©2022 assa. adv. in systems science and appl. (2022) bottle, its volume is 60 ml. for all the experiments we prepared an eosin b solution with a concentration of 8.01.10-4 mol/l equivalent to 500 mg / l. surfactants were used without further purification. 2.2. apparatus and procedure 2.2.1. apparatus the determination of the cloud point was carried out using a mettler fp 900 apparatus: temperature of the sample placed in a cell was measured using a precise sensor placed in a small oven. at the bottom of the measuring cell, there is a luminous source and an optic driver which illuminates the sample. the crossed sample light was converted by the photoelectric cell into an electric signal proportional to the transmitted light intensity. the transmission of light was measured continuously, while the cell temperature increased linearly according to the chosen heating rate. the cloud point designates the temperature of the single limpid phase which becomes cloudy, inducing a transmission decrease. the extraction efficiency of eosine b can be calculated by using (2.1): e = (civi – cdvd ) / civi (2.1) where ci is initial concentration of eosine b, cd is concentration of eosine b in dilute phase, vi is initial volume of solution (10 ml), and vd is volume of dilute phase. the phase separation was carried out in a memert-oven with good temperature control. the dye concentration in the diluted phase was measured at the wavelength 516 nm using the dr6000 uv-visible spectrophotometer. 2.2.2. procedure we prepared an aqueous solution of eosine b of concentration 500 mg/l equivalent to 8.10-4 mol/l, to which we added crude surfactant at different concentrations ranging from 1% up to 6% (by weight) and 4% (weight) of sodium sulphate salt na2so4. the mixture was then introduced into 10 ml test tubes, depending on the clouding temperature of the surfactant, the test tubes were then heated in a thermostatic oven at a desired temperature for two hours to allow a complete decantation of the two phases (diluted phase and coacervate). we waited until their compositions were stabilized reaching equilibrium, and then we recorded the values of the volumes of the two phases. thence, we took a sample of the diluted phase using a syringe, for spectrophotometric analysis. 3. results and discussions 3.1. phase diagram 3.1.1. cloud point measurement the cloud point (cp) of the water / surfactant / solute system refers to the temperature of a single clear phase, which becomes cloudy. below this temperature a single phase exists, often called "pseudo phase". really, beyond this point, cloudy solutions separate into two phases: one phase containing most of the surfactant called "coacervate" and another phase in which the concentration of the surfactant is near the critical micellar concentration [39] however, the solute (dye) in our case prevented the detection of the cp of such a system. it was therefore necessary to plot the phase diagrams of the binary water / surfactant systems in order to determine approximately the cp of the extraction domain for each surfactant. two-phase extraction of dye eosine b imperial red using non ionic surfactant 83 copyright ©2022 assa. adv. in systems science and appl. (2022) 3.1.2. cloud point curve the nonionic surfactants studied is the polyethoxylated alkylphenol (triton x100) and the ethoxylated alcohol, dehydol lt7 (oxo-c15e7). we first established the binary water / surfactant diagrams as shown in fig. 3.1. fig. 3.1. effect of sodium sulphate on the could point temperature of oxo-c15e7 and tx-100 in a homologous series of polyethoxylated nonionic surfactants, the cp increases with the number (of ethoxylation (eo)) of ethylene oxides in the hydrophilic chain of the surfactant, due to the increase in the solubility of the surfactant in water. on the other hand, it decreases when the number of carbon in the hydrophobic chain increases. this is due to the reduction in the solubility of surfactants in water when the number of carbon in the hydrophobic chain increases. the cp of oxo-c15e7 surfactant 1% (wt.) was approximately 57°c lower than that of tx-100 at 1% (wt.) which was approximately 70°c (fig. 3.1). this is due to the fact that tx-100 has one carbon atom less in the hydrophobic chain than oxo-c15e7 and counts three additional ethylene oxides in the hydrophilic chain [20]. the solubilisates can interact with the polar head of the surfactant or with its hydrophobic length after solubilization in micelles. owing to their chemical nature, the organic compounds can vary the cp of the surfactants [20, 21]. in our case, as it can be seen in the extraction test, the phase separation of the water / ta / dye system was possible at that of the water / ta system because 0.5g / l of eosin has insignificant cp value. 3.2. effect of na2so4 on cloud point it is well known that the presence of electrolytes causes a decrease in the solubility of an organic compound in water [40]. in fig. 3.1, we can clearly see that the electrolyte (na2so4) lowers the cloud temperature of the nonionic surfactants used. this is due to the dehydration of the ethylene oxide links which weakens the hydrogen bond between the water molecule and the polar head of the surfactant causing by the solvation of the ions [40] electrolytes have the structuring properties of water, making it less available to hydrate micellar aggregates. this phenomenon, known as the salting-out effect, therefore imposes a weakening of the hydrogen bonds between the ethylene oxide units of the nonionic surfactants and the water of hydration, as it has been previously noticed by [24]. the release phenomenon is mainly due to the anion of the electrolyte. in effect, the anion so42has twice as much effect on the cp as cl[16, 17]. in contrast, the results demonstrated 0 10 20 30 40 50 60 70 80 0 5 10 15 20 te m pe ra tu re (° c ) surfactant wt.% tx-100 (0 wt.% na2so4 ) oxoc15e7 (0 wt.% na2so4 ) tx-100 (4 wt.% na2so4 ) oxo-c15e7 (4 wt.% na2so4 ) 84 n. ghitri, h. bouabdesselam, m. kameche, z. talbi copyright ©2022 assa. adv. in systems science and appl. (2022) by [40] suggest that the effect of the cation does not appear to be significant compared to that of the anion. the cation has thus less influence although some of them can form complexes with the oxygen atoms of the ethylene oxide members of the surfactant. fig.3.1 shows the cp reduction of the water / tx-100 and water / oxo-c15e7 systems by the addition of amount of the electrolyte na2so4. despite the drop in the temperature caused by the electrolyte, there was no change in the shape of the demixing curve. it is consequently suggested that the solvated electrolyte slightly weakens the hydrogen bond between the water molecule and the polar head of the surfactant due to the change in the continuous medium. fig.3.1 displays how much 4% (by weight) of na2so4 could lower the cp of 5% (by weight) of tx-100 solution from 70°c down to 44°c and 5% (by weight) of oxo c15e7 solution, the cp decreased from 57°c to 38°c after the addition of 4% (by weight) of na2so4 [21]. the cloudy temperature (tc) is sensitive to external agents such as electrolytes, alcohols, organic additives and ionic ta [3]. in our case, the extraction of the dye was achieved in the presence of 4% (by weight) of na2so4 and the experimental study region for the extraction extends to a wider temperature range between 38 ° c and 51 ° c for dehydol lt7 (oxo-c15e7) and between 45°c and 54°c for triton x-100 (tx-100). the surfactant concentration range was from 2 to 6% (mass.) for dehydol lt7 and 1 to 5% (mass.) for triton x-100. 4. extraction of eosin b surfactants are used in many industrial and domestic activities, for the formulation of detergents, shampoo, maintenance product, machining product, metal cleaning agent, mineral flotation agent, and textile auxiliaries. in the present section, the nonionic surfactants are advantageous in the removal of the dye. in effect, the results of extracting eosin b from its aqueous solution at 500 mg/l with the surfactants tx-100 and oxoc15e7, are expressed by the answers are: mass percentage of solute extracted (e) concentration in mg/l of solute remaining in the diluted phase after extraction (xs,w) volume fraction of coacervate phase (φc) to be balanced [29, 30, 32]. the results are expressed by the smoothing method which consists in determining the mean of the experimental values, the polynomial constants which fit better the constants of a polynomial which best adjust the variation of the studied property. the verification of the relevance of the identification is carried out by comparison in plotting the calculated values against the experimental results. besides, the quadratic empirical correlation is chosen to give the slope and the (r2)-coefficient close to unity. such a correlation allows the construction of response surfaces: y = a0 + a1xt + a2t + a12xtt + a11xt2 + a22t2 (4.1) where “y” is the physico-chemical response of the studied property; t is the temperature; xt is the mass percentage of the surfactant and ai (i=0, 1, 2, 12, 11, 22) are the fitting parameters of the mathematical model used thoughout. 4.1. cloud point extraction using polyethoxylated alcohol, oxo-c15e7 (dehydol lt7) the quadratic equations for the three quantities (e; xs,w and φc) whose reliability, have been verified are as follows: e (oxoc15e7, eosine b) = -351,05-32,34xt+21,78t+0,93xtt-1,21xt2-0,27t2 (4.2) xs,w (oxo-c15e7, eosine b) = 1734,36-24,15xt-73,76t+0,41xtt-1,37xt2+0,81t2 (4.3) φc (oxo-c15e7, eosine b) = 0,673+0,394xt-0,027t-0,003xtt-0,055x2t+0,0003t2 (4.4) two-phase extraction of dye eosine b imperial red using non ionic surfactant 85 copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 4.1. three-dimensional iso-response curve e = f (xt,t) calculated by the quadratic model (4.2) for the system h2o / oxo-c15e7 / eosine b fig. 4.2. three-dimensional iso-response curve xs,w = f (xt,t) calculated by the quadratic model (4.3) for the system h2o / oxo-c15e7 / eosine b fig. 4.3. three-dimensional iso-response curve φc = f (xt,t) calculated by the quadratic model (4.4) for the system h2o / oxo-c15e7 / eosine b 86 n. ghitri, h. bouabdesselam, m. kameche, z. talbi copyright ©2022 assa. adv. in systems science and appl. (2022) the percentage of extracted dye (e) increases progressively, when both the initial mass percentage of the surfactant xt and temperature rise. the micelles of nonionic surfactant in which the dye has been solubilized, becoming more hydrophobic and concentrated in the coaceravat, when the temperature has risen above the cp causing an increase in the extraction. the extraction (e) decreases slightly with a strong increase in temperature above the cp and at low concentration of surfactant. then the rise in temperature beyond the cp had a negative effect on the extraction power. thus, the percentage of extraction of the eosine solute by dehydol lt7 (oxo-c15e7) is high in the region with a higher concentration of ta and at a temperature close to the cp. indeed, at 5% by weight of oxoc15e7 and 41°c, e reaches 98%. fig. 4.2, illustrates that the lowest residual concentration of the dye is obtained at high concentration of surfactant (xt) and temperature close to the cp and xs,w decreases slightly with temperature. in practice at 5% in weight and 41 °c, the concentration of eosin in the effluent has been reduced from 500 mg l down to 7.67 mg/l (i.e. 65 times). a minimum volume fraction of coacervate (φc) must be obtained when the temperature increases for the treatment of the maximum effluent. the values calculated from the quadratic model (4.4), the φc value is low at high temperature and at low concentration of the surfactant (xt). however, a low φc is unfavorable for the efficiency of the extraction (fig.4.3); while in high concentration of surfactant, φc increases due to the progressive enrichment of the coacervate in micelles. then the optimization of the process must make a compromise between the three studied parameters e, xs,w and φc. table 4.1. experimental results of the extraction parameters e, xs,w and φc (xt,t) e(%) xs,w(mg/l) jc eosine b; oxo-c15e7 (2,38) 85.37 125.93 0.42 (3,38) 94.88 55.51 0.54 (2,46) 93.38 35.59 0.10 (3,46) 96.04 24.70 0.20 (2,51) 88.15 62.93 0.06 (3,51) 92.23 43.08 0.10 (5,51) 98.20 10.40 0.16 eosine b; tx-100 (2,45) 24.23 378.54 0.00 (3,45) 68.54 206.77 0.24 (2,48) 84.66 78.22 0.12 (3,48) 96.35 22.23 0.18 (2,51) 96.65 21.45 0.08 (3,51) 98.22 11.39 0.12 (5,51) 99.19 5.65 0.20 4.2. cloud point extraction using polyethoxylated alkylphenol, triton x-100 so, by using a polyethoxylated alkylphenol surfactant, triton x-100 for the properties e, xs, w and φc, the extraction of eosin with tx-100, the quadratic model gives the following equations e, xs,w, φc: e(tx-100, eosine b) = -31,61+108.73xt 2,26t 1.39xtt 4.61x2 + 0.066t2 (4.5) xs,w(tx-100, eosine b) = 9263,25 499,03x 325,24t + 7.43xtt + 17.24x2 + 2.88t2 (4.6) φc(tx-100, eosine b) = -7.48 + 0.38xt + 0,302t 0,006xtt + 3.10-18xt2 0,0030t2 (4.7) two-phase extraction of dye eosine b imperial red using non ionic surfactant 87 copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 4.4. three-dimensional iso-response curve e = f (xt,t) calculated by the quadratic model (4.5) for the system h2o / tx-100 / eosine b fig. 4.5. three-dimensional iso-response curve xs,w = f (xt,t) calculated by the quadratic model (4.6) for the system h2o / tx-100 / eosine b fig. 4.6. three-dimensional iso-response curve φc = f (xt,t) calculated by the quadratic model (4.7) for the system h2o / tx-100 / eosine b 88 n. ghitri, h. bouabdesselam, m. kameche, z. talbi copyright ©2022 assa. adv. in systems science and appl. (2022) the extracted amount of dye with tx-100 was higher than that achieved with oxo c15e7 (table 4.1 exp.). therefore, the values of xs,w obtained with tx-100 were low compared to that given by oxo-c15e7 (table exp). indeed at 5% oxo-c15e7 and 51°c, the eosine b concentration in the effluent drops from 500 mg/l to 10.40 mg/l, i.e. 48.07 times less under the same conditions; the eosine b concentration in the effluent decreases drastically from 500 mg/l to 5.65 mg/l i.e. 88.49 times less using the tx-100. 5. conclusion under optimal experimental conditions of (xt, t), the percentages of extraction of the eosine b with the tx-100 surfactant are 99.19% and 98% for the oxo-c15e7 at higher concentration of the surfactant (i.e. 5 % mass). on the other hand, in order to increase the concentration factor of the solute, a minimum of the volume fraction of the coacervate (φc) should be obtained when xt decreases. in addition, a minimum volume fraction of coacervate (φc 0.1 for oxo-c15e7 and 0.08 for tx-100) was obtained at higher temperature. however, this latter has a slight effect on the extraction of the dye; the concentration of eosine b in the effluent was greatly reduced with oxo-c15e7 and even more with tx-100. these two surfactants have almost the same number of ethoxylates (eo * 7). the results of the extraction show that the metylene blue (mb) is solubilized thanks to hydrophobic interactions with the micelles of the surfactants in the hydrophobic chain. really, the solubilization of mb in the micelles seems to interact with the hydrophobic chain of tx-114 which is higher than that of oxo-c15e7. this allows to identify the solubilization site of the solute, namely the core of the micelles. similar results were obtained from the cp of orange ii with tx-100 and oxo-c15e7. in consequence, the results obtained in this study confirm the solubilization of the eosine b solution in the core of the micelles (close to the hydrophobic chain) of tx-100. the solubility of orange g is 50 g/l in water, whereas eosine b is 53.22%. under the same operating conditions, the solubility of eosine b is 300 g/l (i.e. 98%), showing that the solubility of the pollutant in water cannot be a criterion indicative of the extraction yield. on the other hand, the tx-100 is unfavorable from an energy point of view since it has a higher cp than that of oxo-c15e7. hence it is possible for extracting with high energy input and less biodegradable surfactant than oxo-c15e7. the comparison of the extraction yields by coacervate using the surfactant oxoc15e7 with a concentration of approximately 4% of the dyes treated are as follows: nr (85.29%); mb (34.91%); orange g (51 %); orange ii (98%) and eosin b (98%). it can be concluded that orange ii and eosin b display strong interaction with oxoc15e7. references 1. benítez, f.j., acero, j.l., leal, a.i., gonzález, m. 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(2014). cationic dye removal from aqueous solutions using ionic liquid and nonionic surfactanttwo-phase extraction of dye eosine b imperial red using non ionic surfactant 91 copyright ©2022 assa. adv. in systems science and appl. (2022) ionic liquid systems: a comparative study based upon experimental design. chemical engineering communications, 201(1), 41–52. 44. tan, b. t., teng, t. t., & omar, a. k. (2000). removal of dyes and industrial dye wastes by magnesium chloride. water research, 34(2), 597–601. 45. tan, l.& sudan, r.g. (1992). removing color from a groundwater source. j. am. water works assoc., 84(1), 79–87. 46. walsh, g. e., bahner, l. h., & horning, w. b. (1980). toxicity of textile mill effluents to freshwater and estuarine algae, crustaceans and fishes. environmental pollution series a, ecological and biological, 21(3), 169–179. 47. willmott, n., guthrie, j., nelson, g. (1998). the biotechnology approach to colour removal from textile effluent. journal of the society of dyers and colourists, 114(2), 384. 48. zollinger heinrich (1987). color chemistry [texte imprimé]: syntheses, properties, and applications of organic dyes and pigments. weinheim, federal republic of germany new york, ny, usa: vch distribution, usa and canada, vch publications. adv syst sci appl 2023; 03:48–65 published online at https://ijassa.ipu.ru. on a random topological characteristic for inclusions with nonlinear fredholm operators: applications to some classes of feedback control systems valeri obukhovskii*, sergey kornev, ekaterina getmanova voronezh state pedagogical university, voronezh, russia abstract: we define and study an oriented random coincidence index for a pair consisting of a nonlinear zero index fredholm operator f and a nonconvex valued random multivalued map g which is fundamentally restrictible with respect to f. it is shown how this characteristic can be used for the justification of the existence of random coincidence points. we present an application of developed results to the existence of a random solution for a control system whose dynamics is governed by an implicit integro-differential equation and the feedback is realized by a random differential inclusion. keywords: feedback control system, random differential inclusion, random coincidence index, random coincidence point, nonlinear fredholm operator, random multivalued map 1. introduction topological methods provide strong and useful tools to deal with problems in the control theory and optimization (among many others, let us mention monographs [3], [7], [8], [9], [15], [16], [19], [22] and references therein). an important place in these applications is occupied by the effective use of various topological characteristics. in the present paper we define a random coincidence index for a pair consisting of a nonlinear fredholm operator and a random nonconvex-valued multimap and use it for the study of a control system whose dynamics is presented by an implicit integro-differential equation and the feedback is realized by a random differential inclusion. it is worth noting that the problem of a coincidence of nonlinear fredholm operators and their perturbations of various types is of a great mathematical interest and finds contentable applications in many fields of contemporary mathematics (see, for example, [5], [27], [29], [32], [33] and references therein). it should be mentioned that the authors of the works [20], [25], [26], [30], [31] introduced topological coincidence index for various classes of multivalued perturbations of nonlinear fredholm operators and used it to obtain some applications. in the recent years the topological degree theory was successfully extended to the case of random maps and new random fixed point results and their applications were obtained (see, for example, [1], [11], [24], [28]). in particular, in papers [24], [28] the random coincidence degree theory was developed for some classes of multivalued perturbations of linear fredholm operators. the present paper is organized in the following way. in the next section we collect necessary preliminaries. the third section is devoted to the construction of an oriented random coincidence index, the description of its properties and its usage to random ∗corresponding author: valerio-ob2000@mail.ru on a random topological characteristic 49 coincidence points. in the forth section we present an application to the problem of existence of a solution for a feedback control system. 2. preliminaries 2.1. nonlinear fredholm operators by the symbols e,e ′ we will denote real banach spaces. everywhere, by y we will denote an open bounded set u ⊂ e (case (i)) or u∗ ⊂ e × [0, 1] (case (ii)). we recall some notions (see, e.g. [5], [33]). definition 2.1: a c1-map f : y → e ′ is called a fredholm operator of index k ≥ 0 (f ∈ φkc 1 (y )) if for every y ∈ y the frechet derivative f ′ (y) is a linear fredholm operator of index k, that is, dimker f ′ (y) <∞, dimco ker f ′ (y) <∞ and dimker f ′ (y)− dimco ker f ′ (y) = k . definition 2.2: a map f : y → e ′ is proper if f−1 (k) is compact for every compact set k ⊂ e ′. we recall now the notion of oriented fredholm structure on y . an atlas {(yi,ψi)} on y is said to be fredholm if, for each intersecting charts (yi,ψi) and (yj,ψj) and every y ∈ yi ∩ yj it is( ψj ◦ψ−1 i )′ (ψi (y)) ∈ cg ( ẽ ) , where ẽ is the corresponding model space, and cg ( ẽ ) denotes the collection of all linear invertible operators in ẽ of the form i +k, where i is the identity map and k is a compact linear operator. the set cg ( ẽ ) is divided into two connected components. the component containing the identity map will be denoted by cg+ ( ẽ ) . two fredholm atlases are said to be equivalent if their union is still a fredholm atlas. the class of equivalent atlases is called a fredholm structure. a fredholm structure on u is associated to a φ0c 1-map f : u → e ′ if it admits an atlas {(yi,ψi)} with model space e ′ for which( f ◦ψ−1 i )′ (ψi (y)) ∈ lc (e ′) at each point y ∈ u , where lc (e ′) denotes the collection of all linear operators in e ′ of the form: identity plus a compact map. let us note that each φ0c 1-map f : u → e ′ generates a fredholm structure on u associated to f . a fredholm atlas {(yi,ψi)} on y is said to be oriented if for each intersecting charts (yi,ψi) and (yj,ψj) and every y ∈ yi ∩ yj it is true that( ψj ◦ψ−1 i )′ (ψi (y)) ∈ cg+ (e) . two oriented fredholm atlases are called orientally equivalent if their union is an oriented fredholm atlas on y . the equivalence class with respect to this relation is said to be the oriented fredholm structure on y . copyright © 2023 assa. adv syst sci appl (2023) 50 v. obukhovskii, s. kornev, e. getmanova 2.2. multivalued maps we describe now some notions of the theory of multivalued maps that will be used in the sequel (details can be found, e.g. in [2], [4], [11], [13], [15], [22]). let x, z be metric spaces, k(z) [c(z)] denote the collection of all nonempty compact [resp., closed] subsets of z. a multimap f : x → c (z) is said to be upper semicontinuous (u.s.c.) [lower semicontinuous (l.s.c.)] if for every open [closed] set v ⊂ z, the set f−1 + (v ) = {x ∈ x : f (x) ⊂ v } is open [resp. closed] in x. if a multimap f : x → c (z) is both u.s.c and l.s.c. it is called continuous. a multimap f : x → c (z) is said to be closed if its graph γf = {(x, z) ∈ x × z : z ∈ f(x)} is a closed subset of x × z. to present the class of multimaps which will be considered, we recall some notions. definition 2.3 (see, e.g. [21], [4], [11], [12]): a nonempty compact subset a of a metric space z is said to be aspheric (or uv ∞, or ∞proximally connected) if for every ε > 0 there exists δ, 0 < δ < ε, such that for each n = 0, 1, 2, . . . every continuous map g : sn → oδ (a) can be extended to a continuous map g̃ : bn+1 → oε (a), where sn = {x ∈ rn+1 : ∥x∥ = 1} and bn+1 = {x ∈ rn+1 : ∥x∥ ≤ 1}. definition 2.4 (see [11]): a u.s.c. multimap f : x → k (z) is said to be a j-multimap (f ∈ j (x,z)) if every value f (x), x ∈ x is an aspheric set. we recall (see, for example, [6]) that a metric space m is said to be an absolute retract (an ar-space) (respectively, an absolute neighborhood retract (an anr-space)), if for each homeomorphism h, which maps it to a closed subset of the metric space z the set h(m) is a retract ofz (respectively, of some its open neighborhood inz). notice that the class ofanrspaces is sufficiently wide: in particular, a compact subset of a finite-dimensional space is an anr-space if and only if it is locally contractible. in turn, this means that compact polyhedra and compact finite-dimensional manifolds are anr-spaces. the union of a finite number of closed convex subsets of a normed space is also an anr-space. definition 2.5 ([14]): a nonempty compact space a is said to be an rδ-set, if it can be presented as the intersection of a decreasing sequence of compact ar-spaces. proposition 2.1 (see [11]): let z be an anr-space. in each of the following cases a u.s.c. multimap f : x → k (z) is a j-multimap: for each x ∈ x the value f (x) is a) a convex set; b) a contractible set; c) an rδ-set; d) an ar-space. in particular, every continuous map σ : x → z is a j-multimap. definition 2.6: by cj (x,x ′) we will denote the collection of all multimaps g : x → k (x ′) of the form g = φ ◦ f, where f ∈ j (x,z) for some metric space z, φ : z → x ′ is a continuous map. the composition φ ◦ f will be called the representation (or decomposition, cfr. [11]) of g. we will denote g = (φ ◦ f) ∈ cj (x,x ′). copyright © 2023 assa. adv syst sci appl (2023) on a random topological characteristic 51 it is worth noting that a multimap can admit different representations (see [11]). 2.3. an oriented coincidence index for compact triplets we will start with the following notion. definition 2.7: a map f : y → e ′, a multimap g = (φ ◦ f) ∈ cj ( y ,e ′) and the space y form a compact triplet ( f,g, y ) c if the following conditions are satisfied: h1) f is a continuous proper map, f|y ∈ φkc 1 (y ) with k = 0 in case (i), k = 1 in case (ii), and the fredholm structure on y generated by f is oriented; h2) g is compact, i.e., g ( y ) is a relatively compact subset of e ′; h3) coin (f,g) ∩ ∂y = ∅, where coin (f,g) = {y ∈ y : f(y) ∈ g(y)} is the coincidence point set of f and g. let us mention that from hypotheses (h1), (h2) it follows that the coincidence point set q = coin (f,g) is compact. as earlier, let u be an open bounded subset of a banach space e. for a compact triplet( f,g, u ) c the oriented coincidence index ind ( f,g, u ) c , the integer-valued topological characteristic with the following basic properties can be defined (see [26]). proposition 2.2 (the coincidence point property): if ind ( f,g, u ) c ̸= 0, then ∅ ≠ coin (f,g) ⊂ u . to formulate the topological invariance property of the coincidence index, we will give the following definition. definition 2.8: two compact triplets( f0, g0 = (φ0 ◦ f0) , u0 ) c and ( f1, g1 = (φ1 ◦ f1) , u1 ) c are said to be homotopic( f0, g0 = (φ0 ◦ f0) , u0 ) c ∼ ( f1, g1 = (φ1 ◦ f1) , u1 ) c if there exists a compact triplet ( f∗, g∗, u∗ ) c , where u∗ ⊂ e × [0, 1] is an open set, such that: a) ui = u∗ ∩ (e × {i}), i = 0, 1; b) f∗|ui = fi, i = 0, 1; c) g∗ has the form g∗ (x, λ) = φ∗ (f∗ (x, λ) , λ) where f∗ ∈ j ( u∗, z ) , φ∗ : z × [0, 1] → e ′ is a continuous map, and f∗ |u i = fi , φ∗ |z×{i}= φi , i = 0, 1 . copyright © 2023 assa. adv syst sci appl (2023) 52 v. obukhovskii, s. kornev, e. getmanova proposition 2.3 (the homotopy invariance property): if ( f0, g0, u0 ) c ∼ ( f1, g1, u1 ) c , then ∣∣ind ( f0, g0, u0 ) c ∣∣ = ∣∣ind ( f1, g1, u1 ) c ∣∣ . remark 2.1: if the fredholm operator f is constant under the homotopy, i.e., u∗ has the form u∗ = u × [0, 1] where u ⊂ e is an open set and f∗ (x, λ) = f (x) for all λ ∈ [0, 1], where f ∈ φ0c 1 (u), then ind ( f,g0, u ) c = ind ( f,g1, u ) c . proposition 2.4 (additive dependence on the domain property): let u0 and u1 be disjoint open subsets of an open bounded set u ⊂ e and ( f,g, u ) c be a compact triplet such that coin (f,g) ∩ ( u \ (u0 ∪ u1) ) = ∅ . then ind ( f,g, u ) c = ind ( f,g, u0 ) c + ind ( f,g, u1 ) c 2.4. random multivalued maps and random coincidence points let (ω,σ) be a measurable space and x, z separable metric spaces. let b(x) be the σalgebra of all borel subsets of x and σ⊗ b(x) denote the minimal σ-algebra containing the sets a×b, where a ∈ σ, b ∈ b(x). definition 2.9: a multimap φ: ω×x → c(z) is called random if it is measurable with respect to the σalgebra σ⊗ b(x), i.e., φ−1 + (v ) ∈ σ⊗ b(x) for each open v ⊂ z. as example of a random multimap we can consider a carathéodory type multimap φ: ω×x → c(z), i.e., it is supposed that φ is such that: 1) φ(·, x) : ω → c(z) is measurable w.r.t. σ for each x ∈ x; 2)φ(ω, ·) : x → c(z) is continuous for each ω ∈ ω (see, e.g., [13], proposition 7.9). definition 2.10: for a multimap φ: ω×x → c(z) and a map ψ : x → z, a measurable map ξ : ω → x is called a random coincidence point of ψ and φ if it satisfies the following inclusion ψ(ξ(ω)) ∈ φ(ω, ξ(ω)) for each ω ∈ ω. the following assertion on a random coincidence point holds true. proposition 2.5: let (ω,σ) be a complete measurable space (see, e.g., [13], definition 1.29); a space x is complete; φ: ω×x → c(z) a random multimap; a map ψ : x → z be measurable w.r.t. b(x). if for each ω ∈ ω the coincidence point set coinω(ψ,φ) = {x ∈ x : ψ(x) ∈ φ(ω, x)} is nonempty, then ψ and φ have a random coincidence point. copyright © 2023 assa. adv syst sci appl (2023) on a random topological characteristic 53 proof the measurable map ψ : x → z can be naturally extended to the random map ψ̃ : ω×x → z if we set ψ̃(ω, x) = ψ(x) for all ω ∈ ω. in fact, for each open v ⊂ z we will have then ψ̃−1(v ) = ω× ψ−1(v ) ∈ σ× b(x). notice that the function µ : ω×x → r, µ(ω, x) = dist(ψ(x),φ(ω, x)) = dist(ψ̃(ω, x),φ(ω, x)) is random (see [10]). define the multimap f : ω ⊸ x as f(ω) = coinω(ψ,φ). then for the graph γf of the multimap f we have γf = µ−1(0) ∈ σ× b(x) and applying to f the aumann selection theorem for a multifunction with a measurable graph (see, e.g., [13], theorem 2.2.14) we conclude that f admits a measurable selection ξ : ω → x which is the desirable random coincidence point. 3. a random coincidence index for fundamentally restrictible quadriples 3.1. a completely fundamentally restrictible triplet at first we recall some notions (see, e.g. [15]). let again y = u ⊂ e, or u∗ ⊂ e × [0, 1] be open bounded sets, f : y → e ′ a map; g : y → k (e ′) a multimap. definition 3.1: a convex, closed subset t ⊂ e ′ is said to be fundamental for a triplet ( f,g, y ) if: (i) g (f−1 (t )) ⊆ t ; (ii) for any point y ∈ y , the inclusion f (y) ∈ co (g (y) ∪ t ) implies f (y) ∈ t. the entire space e ′ and cog ( y ) are natural examples of fundamental sets for ( f,g, y ) . it is easy to verify the following properties of a fundamental set. proposition 3.1: the following holds: a) the set coin (f,g) is included in f−1 (t ) for each fundamental set t of ( f,g, y ) ; b) let t be a fundamental set of ( f,g, y ) , and p ⊂ t , then the set t̃ = co (g (f−1 (t )) ∪ p ) is also fundamental; c) let {tα} be a system of fundamental sets of ( f,g, y ) . the set t = ∩αtα is also fundamental. definition 3.2: we will say that a triple ( f,g, y ) is completely fundamentally restrictible if it possesses a nonempty, compact fundamental set t. the fact that a triple ( f,g, y ) is completely fundamentally restrictible will be denoted as ( f,g, y ) f . it is clear that each compact triple ( f,g, y ) c is completely fundamentally restrictible since we can take cog ( y ) as a compact fundamental set. to obtain more valuable copyright © 2023 assa. adv syst sci appl (2023) 54 v. obukhovskii, s. kornev, e. getmanova example of a completely fundamentally restrictible triplet, we will need the following notions. denote by p (e ′) the collection of all nonempty subsets of a banach spacee ′ . let (a,≥) be a partially ordered set. definition 3.3: a map β : p (e ′) → a is called a measure of noncompactness (mnc) in e ′ if β (cod) = β (d) for every d ∈ p (e ′) . a mnc β is called: (i) monotone, if d0, d1 ∈ p (e ′) , d0 ⊆ d1 implies β (d0) ≤ β (d1) ; (ii) nonsingular, if β ({a} ∪d) = β (d) for every a ∈ e ′ , d ∈ p (e ′); (iii) real, if a = r+ = [0,+∞] with the natural ordering, and β (d) < +∞ for every bounded set d ∈ p (e ′) . among the known examples of mnc satisfying all the above properties we can consider the hausdorff mnc χ (d) = inf {ε > 0 : d has a finite ε-net} . and the kuratowski mnc α (d) = inf {d > 0 : d has a finite partition with sets of diameter less than d} . definition 3.4: maps f , g and the space y form a β-condensing triplet, ( f,g, y ) β if they satisfy conditions (h1) and (h3) in definition 2.7 and h2β) a multimap g = φ ◦ f ∈ cj ( y ,e ′) is β-condensing w.r.t. f , i.e., β (g (d)) ≱ β (f (d)) for every d ⊆ y such that g (d) is not relatively compact. proposition 3.2: each β-condensing triplet ( f,g, u ) β , where β is a monotone, nonsingular mnc is completely fundamentally restrictible. proof consider the collection {tα} of all fundamental sets of ( f,g, u ) β containing an arbitrary point a ∈ e ′ . this collection is nonempty since it contains e ′ . then, taking t = ∩αtα ̸= ∅ we obviously have t = co ( g ( f−1 (t ) ) ∪ {a} ) and hence β ( f ( f−1 (t ) )) ≤ β (t ) = β ( g ( f−1 (t ) )) , so g (f−1 (t )) is relatively compact and t is compact. copyright © 2023 assa. adv syst sci appl (2023) on a random topological characteristic 55 3.2. a random coincidence index let (ω,σ) be a complete measurable space. definition 3.5: a map f : y → e ′, a multimapg : ω× y → k(e ′) and the spaces ω and y form a random fundamentally restrictible quadriple ( f,g,ω, y ) f if the following conditions are satisfied: g1) f is a continuous proper map, f|y ∈ φkc 1 (y ) with k = 0 in case (i), k = 1 in case (ii), and the fredholm structure on y generated by f is oriented; g2) g is a random multimap; g3) for each ω ∈ ω the multimap gω = g(ω, ·) = (φω ◦ fω) ∈ cj ( y ,e ′) ; g4) for each ω ∈ ω the triplet (f,gω, y ) is completely fundamentally restrictible and coin (f,gω) ∩ ∂y = ∅. now our purpose is to define an oriented random coincidence index for a random fundamentally restrictible quadriple ( f,g,ω, u ) f . to this end, define for each ω ∈ ω a coincidence index of a completely fundamentally restrictible triplet ( f,gω, u ) f . let t ω be a nonempty compact fundamental set of a triplet ( f,gω, u ) f and ρ : e ′ → t ω be any retraction. consider the multimap g̃ω = ρ ◦ φω ◦ fω ∈ cj ( u,e ′). from proposition 3.1(a) it follows that coin ( f, g̃ω ) = coin (f,gω) . (3.1) hence, f, g̃ω, and u form a compact triplet ( f, g̃ω, u ) c . we will say that ( f, g̃ω, u ) c is a compact approximation of the triplet ( f,gω, u ) f . definition 3.6: the oriented coincidence index of a completely fundamentally restrictrictible triplet( f,gω, u ) f is defined by the equality ind ( f,gω, u ) f := ind ( f, g̃ω, u ) c , where ( f, g̃ω, u ) c is a compact approximation of ( f,gω, u ) f . to prove the consistency of the above definition, consider two nonempty, compact fundamental sets t0 and t1 of the triplet ( f,gω = φω ◦ fω, u ) f with retractions ρ0 : e ′ → t0 and ρ1 : e ′ → t1 respectively. if t0 ∩ t1=∅, then by proposition 3.1 (a), (c), coin ( f, g̃ω0 ) = coin ( f, g̃ω1 ) = coin ( f, g̃ω ) = ∅, where g̃ωi = ρi ◦ φω ◦ fω, i = 0, 1. hence, by proposition 2.2, ind ( f, g̃ω0, u ) c = ind ( f, g̃ω1, u ) c = 0. otherwise, we can assume, w.l.o.g., that t0 ⊆ t1. in this case, consider the map φω : zω × [0, 1] → e ′ , given by φω (z, λ) = ρ1 ◦ (λφω (z) + (1− λ) ρ0 ◦ φ− ω (z)) and the multimap gω ∈ cj ( u × [0, 1] , e ′), gω (x, λ) = φω (fω (x) , λ). copyright © 2023 assa. adv syst sci appl (2023) 56 v. obukhovskii, s. kornev, e. getmanova the compact triplet ( f,gω, u × [0, 1] ) c realizes the homotopy( f, g̃ω0, u ) c ∼ ( f, g̃ω1, u ) c . indeed, the only fact that we need to verify is that coin ( f,gω ) ∩ (∂u × [0, 1]) = ∅, where f (x, λ) ≡ f (x) is the natural extension. to the contrary, suppose that there exists (x, λ) ∈ ∂u × [0, 1] such that f (x) = ρ1 ◦ (λφω (z) + (1− λ) ρ0 ◦ φω (z)) for some z ∈ fω (x). but in this case, x ∈ f−1 (t1) and hence φω (z) ∈ t1. since also ρ0 ◦ φω (z) ∈ t1 we have λφω (z) + (1− λ) ρ0 ◦ φω (z) ∈ t1 and so f (x) = λφω (z) + (1− λ) ρ0 ◦ φω (z) ∈ co (gω (x) ∪ t0) and we obtain that f (x) ∈ t0 and x ∈ f−1 (t0), implying φω (z) ∈ t0 and ρ0 ◦ φω (z) = φω (z). we conclude that f (x) = φω (z) ∈ gω (x) giving the contradiction. definition 3.7: for a given random fundamentally restrictible quadriple( f,g,ω, u ) f the oriented random coincidence index is defined as the following collection of integers: ind ( f,g,ω, u ) f = {ind ( f,gω, u ) f : ω ∈ ω}. by definition we set ind ( f,g,ω, u ) f ̸= 0 under the condition that ind ( f,gω, u ) f ̸= 0 for all ω ∈ ω. from proposition 2.5 we obtain the following random coincidence point property. theorem 3.1: if ind ( f,g,ω, u ) f ̸= 0 then the quadriple ( f,g,ω, u ) f has a random coincidence point, i.e., there exists a measurable map ξ : ω → u such that f(ξ(ω)) ∈ g(ω.ξ(ω)), ∀ω ∈ ω. definition 3.8: two random fundamentally restrictible quadriples( f0, g0 = (φ0 ◦ f0) ,ω, u0 ) f and ( f1, g1 = (φ1 ◦ f1) ,ω, u1 ) f are said to be homotopic( f0, g0 = (φ0 ◦ f0) ,ω, u0 ) f ∼ ( f1, g1 = (φ1 ◦ f1) ,ω, u1 ) f if there exists a random completely fundamentally restrictible quadriple ( f∗, g∗,ω, u∗ ) f , where u∗ ⊂ e × [0, 1] is an open set, such that: a) ui = u∗ ∩ (e × {i}), i = 0, 1 ; b) f∗ |ui = fi, i = 0, 1 ; copyright © 2023 assa. adv syst sci appl (2023) on a random topological characteristic 57 c) for each ω ∈ ω, the multimap g∗ω = g∗(ω, ·, ·) has the form g∗ω (x, λ) = φ∗ω (f∗ω (x, λ) , λ) where f∗ω ∈ j ( u∗, zω ) , φ∗ω : zω × [0, 1] → e ′ is a continuous map, and f∗ω |u i = fiω , φ∗ω |zω×{i}= φiω , i = 0, 1 . we get the following homotopy invariance property. theorem 3.2: if ( f0, g0,ω, u0 ) f ∼ ( f1, g1,ω, u1 ) f , then ∣∣∣ind (f0, g0,ω, u0 ) f ∣∣∣ = ∣∣∣ind (f1, g1,ω, u1 ) f ∣∣∣ in the sense that ∣∣∣ind((f0, g0ω, u0 ) f ∣∣∣ = ∣∣∣ind (f1, g1ω, u1 ) f ∣∣∣ for all ω ∈ ω. proof for a given ω ∈ ω, let t ω ∗ be a nonempty compact fundamental set of the triplet( f∗, g∗ω = (φ∗ω ◦ f∗ω) , u∗ ) f . it is easy to see that t ω ∗ is fundamental also for the triplets( fi, giω, u i ) f , i = 0, 1. let ρ∗ω : e ′ → t ω ∗ be any retraction, and ( f∗, g̃∗ω = ρ∗ω ◦ φ∗ω ◦ f∗ω, u∗ ) c the corresponding compact approximation of ( f∗, g∗ω, u∗ ) f . then ( f∗, g̃∗ω, u∗ ) c realizes a compact homotopy connecting the triplets ( fi, ρ∗ω ◦ φiω ◦ fiω, u i ) c , i = 0, 1 which are compact approximations of ( fi, giω, u i ) f , i = 0, 1 respectively. by proposition 2.3 we have∣∣ind ( f0, ρ∗ω ◦ φ0ω ◦ f0ω, u0 ) c ∣∣ = ∣∣ind ( f1, ρ∗ω ◦ φ1ω ◦ f1ω, u1 ) c ∣∣ giving the desired. remark 3.1: in connection with remark 2.1 if the operator f is constant under the homotopy, i.e., u∗ has the form u∗ = u × [0, 1], where u ⊂ e is an open set and f∗(x, λ) = f(x) for all λ ∈ [0, 1] with f ∈ φ0c 1(u) then ind ( f,g0,ω, u ) f = ind ( f,g1,ω, u ) f let us formulate the additive dependence on domain property for random fundamentally restrictible quadriples which follows from proposition 2.4. theorem 3.3: let u0 and u1 be disjoint open subsets of an open bounded set u ⊂ e and ( f,g,ω, u ) f be a fundamentally restrictible quadriple such that coin (f,gω) ∩ ( u \ (u0 ∪ u1) ) = ∅ copyright © 2023 assa. adv syst sci appl (2023) 58 v. obukhovskii, s. kornev, e. getmanova for each ω ∈ ω. then ind ( f,g,ω, u ) f = ind ( f,g,ω, u0 ) f + ind ( f,g,ω, u1 ) f in the sense that ind ( f,gω, u ) f = ind ( f,gω, u0 ) f + ind ( f,gω, u1 ) f for each ω ∈ ω. analogs of the above notions and assertions for the condensing case can be formulated in the natural way. let β be a monotone, nonsingular mnc in e ′. definition 3.9: a map f : y → e ′, a multimapg : ω× y → k(e ′) and the spaces ω and y form a random β-condensing quadriple ( f,g,ω, y ) β if they satisfy conditions (g1)− (g3) of definition 3.5 and the following condition g4β) for each ω ∈ ω the triplet (f,gω, y ) is β-condensing and coin (f,gω) ∩ ∂y = ∅. for a given ω ∈ ω, the oriented coincidence index of a β-condensing triplet (f,gω, u)β can be defined via its compact approximation as in definition 3.6 and the oriented random coincidence index of a random β-condensing quadriple (f,g,ω, u)β also is defined in accordance with definition 3.7: ind ( f,g,ω, u ) β = {ind ( f,gω, u ) β : ω ∈ ω}. the notion of homotopy for random β-condensing quadriples follows definition 3.8 with the substitution of ”completely fundamentally restrictible” with ”β-condensing” and the corresponding homotopy invariance property also holds true. remark 3.2: let us mention that in case of constant f and u : u∗ = u × [0, 1] f∗ (x, λ) ≡ f (x) , ∀λ ∈ [0, 1] , the condition of β-condensivity for a triplet ( f,g∗ω,u × [0, 1] ) β may be weakened: for the existence of a nonempty, compact fundamental set t ω it is sufficient to demand that β (g∗ω (d × [0, 1])) ≱ β (f (d)) for every d ⊆ u such that g∗ω (d × [0, 1]) is not relatively compact. in fact, it is enough to notice that in this case f−1 ∗ (t ) = f−1 (t )× [0, 1] and to follow the line of reasoning of proposition 3.2. taking into consideration the corresponding property of compact triplets, we can precise the above property of homotopy invariance. if ( f,g∗ω,u × [0, 1] ) β is a β-condensing triplet, where g∗ω has the form (c) of definition 2.8, then ind ( f,g0ω, u ) β = ind ( f,g1ω, u ) β where giω = g∗ω (·, {i}), i = 0, 1. as an example of the application of a random coincidence point property, let us consider the following assertion. copyright © 2023 assa. adv syst sci appl (2023) on a random topological characteristic 59 theorem 3.4: let f ∈ φ0c 1 (e,e ′) be odd; g : ω× e → k(e ′) a random multimap such that for each ω ∈ ω, gω ∈ cj (e,e ′) and β-condensing w.r.t. f on bounded subsets of e, i.e. β (gω (d)) ≱ β (f (d)) for every bounded set d ⊂ e such that g (d) is not relatively compact. if the set of solutions of two-parameter family of operator inclusions f (x) ∈ λgω (x) (3.2) is a priori bounded, then there exists a random coincidence point of f and g. proof from the condition it follows that there exists a ball b ⊂ e centered at the origin whose boundary ∂b does not contain solutions of (3.2). let φω ◦ fω be a representation of gω. if g∗ω : b × [0, 1] → k (e ′) has the form g∗ω (x, λ) = φ∗ω (fω (x) , λ) , (x, λ) ∈ b × [0, 1] , φ∗ω (z, λ) = λφω (z) then f,g∗ω and b × [0, 1] form a β-condensing triplet ( f,g∗ω,b × [0, 1] ) β . in fact, suppose that β (g∗ω (d)) ≥ β (f (d)) for some d ⊂ b. since g∗ω (d × [0, 1]) = co (gω (d) ∪ {0}) we have that β (gω (d)) ≥ β (f (d)) implying that gω (d) and hence g∗ω (d × [0, 1]) is relatively compact. so the triplet ( f,g∗ω,b × [0, 1] ) β induces a homotopy connecting the triplets ( f,gω,b ) β and ( f, 0,b ) β . since the triplet ( f, 0,b ) β is finite dimensional, from the odd condition on f and the odd field property of the brouwer degree, it follows that ( f, 0,b ) β is an odd number. then, from the equality ind ( f,gω,b ) β = ind ( f, 0,b ) β it follows that ind ( f,gω,b ) β ̸= 0 for all ω ∈ ω and hence ind ( f,g,ω,b ) β ̸= 0 and we can apply the coincidence point property. 4. an application: existence of a solution for a random feedback control system let (ω,σ, µ) be a σ-finite measure space. consider a random feedback control system governed by the following relations: a (t, x (t) , x′ (t)) = b ( t, x (t) , x′ (t) , ∫ t 0 yω (s) ds ) , t ∈ [0, a]; (4.3) yω (τ) ∈ c (ω, τ, x (τ)) , a.e. τ ∈ [0, a], ∀ω ∈ ω; (4.4) x (0) = x0, (4.5) where ω ∈ ω, a : [0, a]× rn × rn → rn, b : [0, a]× rn × rn × rm → rn are continuous maps; c : ω× [0, a]× rn ⊸ rm is a multimap, and x0 ∈ rn. by a random solution of problem (4.3)-(4.5) we mean a pair (xω, yω) consisting of a trajectory function xω(·) such that ω ∈ ω → xω ∈ c1 ([0, a] ;rn) is a measurable map and the function xω satisfies for each ω ∈ ω relations (4.3)-(4.5), whereas the control function yω(·) is such that ω ∈ ω → yω ∈ l1 ([0, a] ;rm) is a measurable map and the function yω satisfies for each ω ∈ ω relations (4.3)-(4.4). copyright © 2023 assa. adv syst sci appl (2023) 60 v. obukhovskii, s. kornev, e. getmanova our aim is to show that, under appropriate conditions, the problem of finding of a random solution to systrm (4.3)-(4.5) can be reduced to the study of a condensing quadriple of the above mentioned form (see section 3). consider the following condition: (a) for each (t, u, v) ∈ [0, a]× rn × rn there exist continuous partial derivatives a′ u (t, u, v), a ′ v (t, u, v) and moreover, deta′ v (t, u, v) ̸= 0. the following assertion holds true (see [26], proposition 5.1). proposition 4.1: under condition (a) a map f : c1 ([0, a] ;rn) → c ([0, a] ;rn)× rn defined as f (x) (t) = (a (t, x (t) , x′ (t)) , x (0)) is a fredholm map of index zero, whose restriction to each closed bounded set d ⊂ c1 ([0, a] ;rn) is proper. now we will describe the assumptions on the map b and the multimap c. denoting by the symbol kv (rm) the collection of all nonempty compact convex subsets of rm, we suppose that the multimap c : ω× [0, a]× rn → kv (rm) satisfies the following conditions: (c1) c is a random multimap, i.e., it is measurable w.r.t. the σ-algebra σ⊗ l⊗ b(rn), where l is the σ-algebra of lebesgue subsets of [0, a]; (c2) for all (ω, t) ∈ ω× [0, a] the multimap c (ω, t, ·) : rn → kv (rm) is upper semicontinuous; (c3) for each r > 0 there exists a function γr : ω× [0, a] → r+ such that: (i) γr(ω, ·) ∈ l1(0, a) for each ω ∈ ω; (ii) the function γr(·, t) : ω → r+ is measurable for a.e. t ∈ [0, a] and the following estimate holds for all ω ∈ ω and u ∈ rn, ∥u∥ ≤ r : ∥c (ω, t, u)∥ := sup {∥c∥ : c ∈ c (ω, t, u)} ≤ γr (ω, t) a.e. t ∈ [0, a]. notice that from condition (c1) it follows that for each (ω, u) ∈ ω× rn the multifunction c(ω, ·, u) : [0, a] → kv(rm) is lebesgue measurable. it is known (see, e.g. [8], [11], [13], [15], [22]) that under given conditions for each ω ∈ ω the superposition multioperator p(ω, ·) : c([0, a];rn) ⊸ l1([0, a];rm) : p(ω, x) = {ψ ∈ l1([0, a];rm) : ψ(t) ∈ c(ω, t, x(t)) a.e. t ∈ [0, a] is well defined. moreover, if we will consider the composition π: ω× c([0, a];rn) ⊸ c([0, a];rm) π(ω, x) = j ◦ p(ω, x), where j : l1([0, a];rm) → c([0, a];rm) is the integral operator j(ψ)(t) = ∫ t 0 ψ(s)ds, then π(ω, ·) : c([0, a];rn) ⊸ c([0, a];rm) is a closed multioperator for each ω ∈ ω. further, since the embedding c1([0, a];rn) ↪→ c([0, a];rn) is completely continuous, the restriction of π(ω, ·) toc1([0, a];rn) is locally compact and hence upper semicontinuous (see, e.g. [22], theorem 1.2.32). it is easy to see also that the multimap π has compact convex values and hence this restriction is a j-multimap. at last, from [23], theorem 3 it follows that π is a random multimap. now, we will assume that the maps a and b satisfy the following lipschitz type condition: copyright © 2023 assa. adv syst sci appl (2023) on a random topological characteristic 61 (ab) there exists a constant q, 0 ≤ q < 1 such that |b (t, u, v, w)−b (t, u, v, w)| ≤ q |a (t, u, v)− a (t, u, v)| for all t ∈ [0, a] , u, v, v ∈ rn, w ∈ rm. consider the continuous map ξ : c1 ([0, a] ;rn)× c ([0, a] ;rm) → c ([0, a] ;rn) defined as ξ (x, z) (t) = b (t, x (t) , x′ (t) , z (t)) and the multimap f̃ : ω× c1 ([0, a] ;rn) → k (c1 ([0, a] ;rn)× c ([0, a] ;rm)), f̃ (ω, x) = {x} × π(ω, x). consider the composition g̃ = ξ ◦ f̃ : ω× c1 ([0, a] ;rn) → k (c ([0, a] ;rn)). it is easy to see that the multimap f̃ and hence g̃ are random. further, from [22], theorem 1.3.17 it follows that for each ω ∈ ω the multimap f̃ω = f̃(ω, ·) is u.s.c. and hence it is a jmultimap, and therefore the composition g̃ω = ξ ◦ f̃ω : c1 ([0, a] ;rn) → k (c ([0, a] ;rn)) is a cj-multimap. it is clear that the set g̃ω (x) consists of all functions of the form b (t, x (t) , x′ (t) , πω (t)) where πω ∈ π(ω, x) . define now the multimap g : ω× c1 ([0, a] ;rn) → k (c ([0, a] ;rn)× rn) by g (ω, x) = g̃ (ω, x)× {x0} . it is easy to see that g is a random multimap and gω = g(ω, ·) is a cj-multimap for each ω ∈ ω. now, in accordance with proposition 2.5 the existence of a trajectory for system (4.3)(4.5) is equivalent to the existence of a coincidence point xω ∈ c1 ([0, a] ;rn) for the pair (f,gω) for each ω ∈ ω. if u ⊂ c1 ([0, a] ;rn) is an open bounded set, then to show, for a given ω ∈ ω that( f,gω, u ) form a condensing triplet w.r.t. the kuratowski mnc, it is sufficient to prove the following statement. proposition 4.2: the triplet ( f̃ , g̃ω, u ) , where f̃ : c1 ([0, a] ;rn) → c ([0, a] ;rn) , is defined as f̃ (x) (t) = a (t, x (t) , x′ (t)) is α-condensing w.r.t. the kuratowski mnc α in the space c ([0, a] ;rn) for each ω ∈ ω. proof take any subset d ⊂ u , and let α ( f̃ (d) ) = d. from the definition of kuratowski mnc it follows that taking an arbitrary ε > 0 we may find a partition of the setf̃ (d) into subsets f̃ (di) , i = 1, · · · , s such that diam ( f̃ (di) ) ≤ d+ ε. since the embedding c1 ([0, a] ;rn) ↪→ c ([0, a] ;rn) is completely continuous, the image dc of d under this embedding is relatively compact. it is known (see, e.g. [13], [15], [22]) that a u.s.c. compactvalued multimap sends compact sets to compact sets, then we can conclude that the set πω (d) is relatively compact. it means that taking a fixed δ > 0 and any di we may divide the sets dic and πω (d) into a finite number of subsets dijc , j = 1, · · · , pi, and balls bik (zik), k = 1, · · · , ri, centered at zik ∈ c ([0, a] ;rm) respectively, such that for each t ∈ [0, a] ; copyright © 2023 assa. adv syst sci appl (2023) 62 v. obukhovskii, s. kornev, e. getmanova u1 (·) , u2 (·) ∈ dijc , v ∈ rn; w1 (·) , w2 (·) ∈ bik (zik) we have that |a (t, u1 (t) , v)− a (t, u2 (t) , v)| < δ (4.6) |b (t, u1 (t) , v, w1 (t))−b (t, u2 (t) , v, w2 (t))| < δ. (4.7) now, the set g̃ω (d) is covered by a finite numbers of sets γijk, i = 1, · · · , s; j = 1, · · · , pi; k = 1, · · · , ri of the form γijk = {b (·, x (·) , x′ (·) , y (·)) : x ∈ dijc , y ∈ bik (zik)} . let us estimate the diameters of these sets. taking arbitrary x1, x2 ∈ dijc and y1, y2 ∈ bik (zik) and applying (4.6), (4.7) and condition (ab), for any t ∈ [0, a] we have |b (t, x1 (t) , x ′ 1 (t) , y1 (t))−b (t, x2 (t) , x ′ 2 (t) , y2 (t))| < |b (t, x1 (t) , x ′ 1 (t) , zik (t))−b (t, x2 (t) , x ′ 2 (t) , zik (t))|+ 2δ ≤ |b (t, x1 (t) , x ′ 1 (t) , zik (t))−b (t, x1 (t) , x ′ 2 (t) , zik (t))|+ + |b (t, x1 (t) , x ′ 2 (t) , zik (t))−b (t, x2 (t) , x ′ 2 (t) , zik (t))|+ 2δ ≤ q |a (t, x1 (t) , x ′ 1 (t))− a (t, x1 (t) , x ′ 2 (t))|+ 3δ ≤ q |a (t, x1 (t) , x ′ 1 (t))− a (t, x2 (t) , x ′ 2 (t))|+ + q |a (t, x2 (t) , x ′ 2 (t))− a (t, x1 (t) , x ′ 2 (t))|+ 3δ < q (d+ ε) + qδ + 3δ. now, if q = 0 it means, by the arbitrariness of the choice of δ > 0 that α ( g̃ (d) ) = 0 and then the triplet ( f̃ , g̃, u ) and therefore ( f,g, u ) is compact. otherwise, let us take ε > 0 and δ > 0 so small that qε+ (q + 3) δ < (1− q) d. then, q (d+ ε) + qδ + 3δ = µd where 0 < µ < 1 and, hence diamγijk ≤ µd, implying that α ( g̃ (d) ) ≤ µα ( f̃ (d) ) . now, if we suppose that the trajectory function xω(·) for system (4.3)-(4.5) is found already, the existence of the corresponding control yω(·) follows from the next reasonings. for a given trajectory xω, let a measurable function h : ω → c([0, a];rn) be given as h(ω)(t) = a(t, xω(t), x ′ ω(t)). further, consider the random map g : ω× l1([0, a];rm), g(ω, y) = b(t, xω(t), x ′ ω(t), ∫ t 0 y(s)ds) and a multimap v : ω ⊸ l1([0, a];rm), v (ω) = p(ω, xω). applying again theorem 3 from [23] we conclude that the multimap v is measurable. since xω is a trajectory, we get that h(ω) ∈ g(ω, v (ω)), ∀ω ∈ ω. copyright © 2023 assa. adv syst sci appl (2023) on a random topological characteristic 63 from a version of the filippov implicit function lemma (see proposition 2.2.25 of [13]) we conclude that there exists a measurable function v : ω → l1([0, a];rm) such that h(ω) = g(ω, v(ω)), ∀ω ∈ ω. it is clear that we can set yω(·) = v(ω) as the desired control function. the above arguments yield that the random coincidence index theory, developed in the previous sections, can be applied to the study of the solvability of problem (4.3)-(4.5). for example, the direct application of theorem 3.4 yields the following assertion. proposition 4.3: under above conditions, suppose that the map a is odd: a (t,−u,−v) = a (t, u, v) for all t ∈ [0, a]; u, v ∈ rn and the set of functions x ∈ c1 ([0, a] ;rn) satisfying the two-parameter family of relations a (t, x (t) , x′ (t)) = λb ( t, x (t) , x′ (t) , ∫ t 0 yω (s) ds ) , t ∈ [0, a], λ ∈ [0, 1]; (4.8) yω (τ) ∈ c (ω, τ, x (τ)) , a.e. s ∈ [0, a], ∀ω ∈ ω; (4.9) x (0) = x0, (4.10) is a priori bounded. then, there exists a random solution of problem (4.3)–(4.5). acknowledgements the results of v. obukhovskii and e. getmanova were supported by the russian science foundation (project number 22-71-10008). the research of s. kornev was supported by the ministry of education of the russian federation within the framework of the state task in the field of science (topic no fzgf-0640-2020-0009). references 1. andres, j., & górniewicz, l. 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(2019). topologicheskie metody v teorii nelineinyh fredgolmovyh otobrazhenii i ih prilozheniya [topological methods in the theory of nonlinear fredholm maps and their applications], moscow, russia: nauka, [in russian]. copyright © 2023 assa. adv syst sci appl (2023) introduction preliminaries nonlinear fredholm operators multivalued maps an oriented coincidence index for compact triplets random multivalued maps and random coincidence points a random coincidence index for fundamentally restrictible quadriples a completely fundamentally restrictible triplet a random coincidence index an application: existence of a solution for a random feedback control system adv syst sci appl 2021; 01; 46-59 published online at https://ijassa.ipu.ru. decoupling model-based fuzzy logic control of room temperature and humidity snejana yordanova department of automation, technical university of sofia, sofia, bulgaria e-mail: sty@tu-sofia.bg abstract: the control of room temperature and humidity is important for ensuring of the necessary indoor human comfort for optimal work capacity and effective rest. the plant nonlinearity and the variables coupling require intelligent control techniques in order to satisfy the high performance demands. the present paper suggests a procedure for the design of a simple for industrial implementation fuzzy logic controller on the principle of parallel distributed compensation (pdc) that consists of linear local decoupling two-variable controllers. it is based on a takagi-sugeno-kang (tsk) plant model, derived from experimentally obtained plant step responses using expert knowledge and parameter optimisation via genetic algorithms. the design is applied for the control of the temperature and the relative humidity of a laboratory airconditioning system. the pdc system outperforms an existing mamdani two-variable control system with adaptive properties in shorter settling time, higher robustness and reduced overshoot, estimated from simulations. keywords: decoupling nonlinear control, laboratory air-conditioning system, parallel distributed compensation, room temperature and humidity control, tsk plant modelling, simulations 1. introduction temperature and humidity are ones of the most often controlled variables in numerous industrial processes and air-conditioning systems in rooms, cars, green houses, public buildings, etc. their proper control ensures optimal process conditions and human indoor comfort for good work capacity and rest at a low cost [3, 7, 10]. the plant is nonlinear with coupling between the variables and subjected to various disturbances such as the changes of the number of people and their activity in the room, the outside air temperature and humidity, the wind, etc. it is difficult to be modelled which makes the control task complex for solving by classical approaches which are based on a simple and accurate mathematical plant model. besides, the classical controllers often fail to satisfy the increasing performance demands on the control system. controllers based on fuzzy logic (fl) can successfully ensure stability and robustness of the designed control system using only expert knowledge [4, 10, 11, 12, 13, 22]. various fuzzy logic controllers (flc) for air temperature and humidity have been developed in [1, 2, 4, 5, 7, 9, 10, 11, 12, 13, 18, 19, 21, 22, 24, 25] aiming at diverse performance improvements. energy efficient flc are designed in [1, 4]. in [1, 2, 9] evolutionary and genetic algorithms (ga) for flc tuning are applied to reduce the number of the fuzzy rules and hence the flc complexity, and optimize the flc parameters and membership functions (mf). the variables coupling effect is considered in the design of mimo flc in [18, 24, 25]. type 2 flcs are suggested in [21] to improve system robustness. more sophisticated adaptive flc are developed in [5, 19, 22, 23]. a simple neuro-fuzzy sugeno type structure is trained from expert data in [14] to adaptively control the temperature and the relative humidity via the compressor speed in an air-conditioning system. most of the fl control approaches suggested are tested only in simulations and often are difficult to apply for real time control due to the sophisticated design or/and algorithm. decoupling model-based fuzzy logic control of room temperature and humidity 47 copyright ©2021 assa. adv. in systems science and appl. (2021) the model-based flc on the principle of parallel distributed compensation (pdc) marks a significant progress in the development of flc. the pdc has a simple structure of a few rules and a stability and robustness based design supported by the well mastered advanced linear control technique [6, 20, 22, 23, 15]. this greatly facilitates the pdc wide industrial implementation and ensures improved performance of the pdc control system. the derivation of the necessary for the pdc design tsk plant model from plant input-output data is explained in [22, 23]. the data can be collected from experimental step responses in different operation points of the plant or during the real time operation control of the plant in a closed loop. if a nonlinear plant model is available the data can be obtained via simulations for various plant input signals. both simulations and experiments have to be correctly designed. the tsk plant model is built on ga optimised or expert defined few linear operation zones. it unites a number of fuzzy rules, one for each zone, that recognize the position of the current operation point with respect to each zone, and the linear plant models which accurately represent the plant in the zones. the pdc is built on the same zone recognition conditions of the tsk plant model fuzzy rules. the conclusions, however, represent linear controllers each designed for a stable and robust control of the corresponding local linear plant for the operation zone. the nonlinear pdc performs soft switching between the control actions of all linear controllers taken with weights that depend on the degree of belonging of the current operation point to each of the fuzzy defined linearization zones. the pdc principle allows designing of a nonlinear controller by the use of local linear controllers of any type, mimo included, in a simple unified manner by the help of the wellknown and proven effective linear systems techniques. based on a tsk plant model such plant peculiarities as nonlinearity, inertia, coupling, etc. can be better accounted for in the pdc design thus ensuring a stable, robust and more accurate control. therefore the pdc often outperforms the model-free mamdani controllers. the simple algorithm of operating in parallel local standard linear controllers is a prerequisite for an easy implementation under the real time restrictions for fast computations, traceability and reliability using the widely spread general purpose industrial programmable logic controllers (plc) or microcontrollers. a two-variable pdc-flc can be a perspective candidate for improvement of the performance of the control of temperature and relative humidity in an air-conditioning system. so, the aim of the present investigation is to develop a procedure for the design of a pdcflc with standard local linear controllers for decoupling two-variable control of the relative humidity and the temperature of a laboratory air-conditioning system and to demonstrate via simulation its implementation for the control and the performance improvement in comparison to a mamdani two-variable controller. the further paper organization is the following. in section 2 the theoretical background including previous relevant research and the problem formulation are presented. section 3 is devoted to the derivation of a simpler plant tsk model from experimental plant step responses. the design of a pdc with decoupling local two-variable controllers is described in section 4. section 5 deals with the simulation investigations of the developed pdc system for the control of the relative humidity and the temperature of a laboratory air-conditioning system. there the performance of the system is assessed and its advantages discussed on the basis of a comparison with the performance of a designed in a previous research mamdani flc system. section 6 contains the conclusion and plans for future research. 2. theoretical background and problem formulation in the present investigation a modified transfer functions based pdc, developed in [22], is considered. it is a tsk structure built on the basis of a modified transfer functions based tsk plant model. the pdc and the tsk plant model consist of a nonlinear static sugeno model and a dynamic linear part of operating in parallel elements described by transfer 48 s. yordanova copyright ©2021 assa. adv. in systems science and appl. (2021) functions or matrices in the multivariable case. the sugeno model contains expert defined input orthogonal mf that describe the overlapping operation zones of the plant, e.g. under, around and above the norm, where the plant model can be considered linear. input to it is the measured plant output y(t) which is used to represent the current operation point and to determine the degree k of its matching to all defined linearization zones. the number of the sugeno model outputs is equal to the number of the linearization zones. due to the specific rule base used each value k of the mf appears at a k-th output. for instance, the rule base for three linearization zones, k=13, is the following: r1: if y is zone 1 then output1 1=1 and output2 1=0 and output3 1=0 r2: if y is zone 2 then output1 2=0 and output2 2=1 and output3 2=0 (2.1) r3: if y is zone 3 then output1 3=0 and output2 3=0 and output3 3=1. the measured plant output yn at moment tn matches the three defined zones with degrees µ1, µ2 and µ3, respectively. then the sugeno model outputs for orthogonal mf (µ1+µ2+µ3=1) after weighted average defuzzyfication become: output1 о=(µ1•output1 1+µ2•output12+µ3•output1 3)=µ1, output2 о=(µ1•output2 1+µ2•output2 2+µ3•output2 3)=µ2, output3 о=(µ1•output3 1+µ2•output3 2+µ3•output3 3)=µ3. for n linearization zones and arbitrary mf each sugeno model output is computed using weighted average as ∑μμ= n 1=i i/k o koutput . in fig. 2.1 a modified transfer functions based pdc is depicted for three linearisation zones. it consists of linear controllers ri(s) – one for each linearisation zone and a sugeno model. the nonlinear controller output is computed as: .∑μ/∑μ=∑output= n 1=k k n 1=k k•k n 1=k k• o k o pdc uuu thus the sugeno model defines the premises of the fuzzy rules in the classic pdc [6, 20]. it is completed in a simple fu with a few mf and simple defuzzyfication which makes it easy to programme in a plc even if the plc supports no fl. the dynamic part is an analogue to the conclusions of the fuzzy rules in the classical pdc. it comprises standard linear controllers that exist as plc components. they are tuned using the linear control systems methods for the local linear plant model in the tsk plant model according to desired stability and performance criteria. the corresponding tsk plant model differs only in the dynamic part, where the common input is the control action u and for each zone a linear model of the local plant with proper transfer function pi(s) replaces ri(s). the transfer functions of the local plant models and controllers are replaced by transfer matrices for mimo plants and controllers. the development of a high performance pdc-flc for the air temperature (t) and relative humidity (rh) in a room should comply with the requirements for ensuring r2(s) r3(s) e sugeno model x 2 updc x y=20 1 2 3 1 3 x r1(s) u1 y u2 u3 fig. 2.1. a modified pdc-flc for three linearization zones of linear controllers and a sugeno model with expert designed orthogonal mf decoupling model-based fuzzy logic control of room temperature and humidity 49 copyright ©2021 assa. adv. in systems science and appl. (2021) thermal comfort for the people, defined by the comfort zone on the psychrometric chart in fig. 2.2 [3]. this chart shows that temperature and relative humidity have to be controlled as coupled variables at references within the comfort zone. the present research is based on the developed in [22-25] laboratory air-conditioning system. it is depicted in fig. 2.3 together with the designed in [22, 25] mamdani twovariable flc for the control of rh (y1) and t (y2), used later for comparison. the laboratory air-conditioning system consists of three connected via fans chambers – for air heating, for air humidifying and a cabin for the room with humidity and temperature measuring transmitters. the plant-controller interface includes pulse-width modulators (pwmi) for the control actions ui, digital outputs (doi) of the data acquisition (daq) board, solid state relays ssri to the humidifier and the heater, transmitters for rh and t, analog-to-digital converters (adci) of daq and converters of binary code to real physical quantities rh% and toc. the mamdani flc consists of two identical fuzzy units (fui) which are cross-connected via the two inputs to each fu [ei n, n jy ], where ei n are the normalized main channel errors, ei=yri-yi, i=1, 2, yri – the corresponding references and n jy , ji, j=1, 2, are the derivatives of the cross channel plant outputs, computed by first order differentiators wdj(s). the postprocessing is proportional-plus-integral (pi) with transfer function cpii(s). the flc parameters are tuned by a designed fuzzy supervisory controller in simulations mode for different references yri and disturbances using a model of the closed loop system based on a derived tsk plant model. the final flc tuning parameters are fixed to their mean values from the supervisor off-line auto-tuning. thus the designed and tuned to adapt two-variable flc is applied for real time control of rh and t in matlabtm environment [8, 15, 16]. the main tasks of the current research are the following. 1. derivation of a simpler tsk plant model from experimental plant step responses than the obtained in [22-24] where for each of the four channels a separate sugeno model with ga optimised gaussian mf is designed. the sugeno models in [22-25] are used as parts of the tsk plant model only in simulations. in this research they are included in the structure of the pdc and therefore have to be reduced in number and with fixed and simple in shape mf in order to facilitate the further plc implementation. 2. design and tuning of a pdc with local linear two-variable decoupling pi controllers. 3. simulation investigations of the pdc closed loop system based on the new tsk plant model and system performance assessment. 4. comparison of the step responses and the performance estimates of the pdc system with the flc system from [25] and assessment of improvements. rh, % t, o c x,g/kg comfort zone fig. 2.2. ashrae comfort zone representation 50 s. yordanova copyright ©2021 assa. adv. in systems science and appl. (2021) adci doi cabin yi software generalised plant pwmi ui converters to t o c and rh% yi=yii+yij a two-variable mamdani flc for y1 relative humidity and y2 temperature fui kеi pii wdj(s) yri ei yj fig. 2.3. laboratory air conditioning system and flc for control of air relative humidity and temperature 3. tsk plant modelling the accepted tsk plant model structure is depicted in fig. 3.1. two sugeno models are designed – one for each controlled variable. they describe three linearization zones via 10 60 y1 y2 y2 µ 1 2 µ 2 2 µ 3 2 15 40 sugeno 1 y1 µ1 1 µ 2 1 µ 3 1 sugeno 2 µ1 1 µ2 1 µ3 1 µ3 2 µ1 2 µ2 2 fig. 3.1. block diagram of tsk plant model structure decoupling model-based fuzzy logic control of room temperature and humidity 51 copyright ©2021 assa. adv. in systems science and appl. (2021) orthogonal mf triangular for the “norm” in the middle of the range of the corresponding plant output – y1[10,60]%, y2[15,40]oc, and two trapezoidal mf for “under the norm” and “above the norm” on both sides of the triangular mf. the dynamic part of each zone is represented by a transfer matrix: 𝐏k(𝑠) = [ 𝑃11 k (𝑠) 𝑃12 k (𝑠) 𝑃21 k (𝑠) 𝑃22 k (𝑠) ] , 𝑘 = 1 ÷ 3 . (3.1) the output of the i-th channel in the k-th zone is yi k(s)=yii k(s)+yij k(s)=рii k(s)ui k+pij k(s)uj k, where 𝑦ij k(𝑠) = 𝑃ij k(𝑠)𝑢j(𝑠), (i, j=1, 2). for a main channel i=j and for a cross channel ij. the superscript denotes the number of the zone. the parameters of the tsk plant model are the parameters of the transfer functions (tf) in the dynamic part of each zone qtsk=[qtf], i.e. the parameters of the local linear plant models and the initial conditions for the output variables. they are computed in an optimisation procedure using ga with the fitness function: 𝐅 = ∫ ∑ { [𝑦itsk(𝑡,𝐪tsk)−𝑦iexp(𝑡)] 𝑦iexp(𝑡) } 2 𝑑𝑡 → 𝐦𝐢𝐧⏟ 𝐪tsk m i=1 , (3.2) where m is the number of the plant output variables (for a two-variable plant m=2), yitsk(t, 𝐪tsk) and yexpi(t) are the corresponding outputs of the tsk plant model and of the real plant, obtained for the same inputs uexpi(t) via simulation using an available nonlinear plant model or an experimental study of the plant. the experimental and simulation investigations are designed in aspect of ensuring of rich in frequencies and magnitudes input signals to the plant that cover the whole range of operation of the plant. this is a prerequisite for high accuracy in modelling of the plant nonlinearity and other peculiarities. the experimental and simulation data are first processed to ensure fast convergence of the optimisation procedure by noise filtration, reduction of the size of the sample (elimination of noninformative data, dilution of data by discretization of time with higher step size preserving the data pattern) and reduction of the data correlation, normalization or standardization, etc. ga for tsk plant model parameter optimisation are selected for the parallel search of the parameter space, the avoidance of trapping in a local extremum and of computation of gradient. for the safety of the industrial plant operation ga are applied here off-line in combination with simulation in matlabtm environment. the modified tsk plant model is suitable for simulink simulation and fast computation of the fitness function. according to ga the model parameters called genes are ordered in an array to make a chromosome or an individual for specific parameters values, which is viewed upon as one possible solution of the optimization problem. randomly generated chromosomes create the population of the first generation. the individuals are rated according to the value of their computed fitness function. every next generation is filled by the better rated offspring produced by mating of chromosomes parents, crossover of genes between parents and random mutation of genes. the optimisation ends after an accepted number of generations or after reaching of a desired minimum of the fitness function. the tsk plant model with the optimal parameters is validated for independent input-output experimental or simulations data or in parallel operation to the original plant with “hanging end”. in case validation fails another optimisation with new random initialization of the first population, fitness function, ga parameters or parameter bounds, etc. or a new model is recommended. the tsk plant model of the relative humidity and the temperature in the laboratory airconditioning system in fig. 2.3 is derived from the experimental study of the two-variable plant in the whole range of its operation. the obtained step responses of the generalized plant with respect to the two outputs – rh and t are shown in fig. 3.2. the plant inputs applied are u= [u1, u2]=[2, 0; 4, 0; 6, 0; 2, 6; 4, 6; 6, 6], v and change stepwise at step times tstep=[0 800 1600 2400 3400 4400], s, where u1 and u2 are the control signals to the pwm1 and pwm2 52 s. yordanova copyright ©2021 assa. adv. in systems science and appl. (2021) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 20 30 40 50 60 70 experiment y11 y22 y12 y21 time, s temperature, o c rh, % tsk model fig. 3.2. step responses of rh and t table 3.1. optimal tsk plant model parameters zones for y1=y11+y12=rh, rh(0)=35%, to1=57s for y2=y21+y22=t, t(0)=22.8oc, to2=244s main channel 11 (y11) cross channel 12 (y12) cross channel 21 (y21) main channel 22 (y22) k11 k,% t11 k, s k12 k,% t12 k, s k21 k,% t21 k, s k22 k,% t22 k, s zone1, k=1 1.4 273 -1.1 895 -0.6 1954 2.6 228 zone2, k=2 2.8 284 -3.6 818 -0.6 2052 3.5 859 zone3, k=3 4.7 150 -6.5 456 -1.2 1810 3.2 854 respectively, which via ssr1 and ssr2 connect the humidifier and the electrical heater to the nets voltage during the corresponding pulses and disconnect them during the pauses. three linearization zones for each output are assumed described by two sugeno models, shown in fig. 3.1, with expert defined mf of standard shape with economical piecewise linear representation in plc. the local plants are two-variable. the type of step responses in fig. 3.2 suggests second order time lag transfer functions in (3.1) which are also convenient for the further model parameter optimisation: pij k(s)=kij k.[(tij k.s+1).(toi.s+1)]-1. (3.3) the role of the common for all zones time lag element (toi.s+1)-1 is to reflect the plant inertia in each output using only one tuning parameter – the time constant toi. the ga tuning parameters are qtsk=[qtsk 1 qtsk 2 qtsk 3 to1 to2 rh(0) t(0)], where qtsk k=[k11 k t11 k k12 k t12 k k21 k t21 k k22 k t22 k] and rh(0), t(0) are the initial conditions for humidity and temperature respectively. the optimal tsk plant model parameters, computed by ga minimization of the fitness function (3.2), are systemized in table 3.1. the tsk plant model is validated in the following way. the designed in previous research mamdani two-variable flc, shown in fig. 2.3, is first applied for the real time control of rh and t in the laboratory air-conditioning system. then it is used for the control of the tsk plant model via simulation for the same references. the two types of step responses – experimental and from simulation are presented in fig. 3.3. from their comparison it can be concluded that they are close and hence the tsk plant model correctly represents the real plant. in fig. 3.3 the step responses of two simulated systems are shown. decoupling model-based fuzzy logic control of room temperature and humidity 53 copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 3.3. step responses of flc closed loop system from real time control of relative humidity and temperature and from flc and pdc closed loop systems simulations using a tsk plant model in the first system the flc parameters are tuned from robust performance criterion while in the second the computed parameters are improved with correction of the gains after a fuzzy supervisor based off-line auto-tuning [22-25] which leads to a better system performance expressed in a reduced settling time and variables coupling effect. the second order time lag transfer functions (3.3) are approximated to the more convenient for the pdc design ziegler-nichols (zn) plant models pij k(s)=kij k[(tij ks+1).(tois+1)]-1kij ke-ijks(tij knews+1)-1 (3.4) the zn approximation is based on the representation of the time delay by the first term of the taylor’ series expansion e-ijks(ij k.s+1)-1. so, the time lag element with the smaller time constant in (3.3) can be accepted as the approximation of the time delay. in that case ij k =min(tij k, toi) and tij knew=max(tij k,toi). the parameters of all zn models are ij k=toi, tij knew=tij k, k=13, i, j=1, 2 with the exception for i=j=2 where 22 1=t22 1, t22 1new=to2. 4. design of pdc with local linear two-variable controllers the pdc for the control of a two-variable plant is shown in fig. 4.1. it is based on the derived tsk plant model. the sugeno models are the same as in the tsk plant model. the local linear controllers in each of the three linearization zones are two-variable, represented by the following transfer matrices: 𝐑k(𝑠) = [ 𝐶11 k (𝑠) 𝐶12 k (𝑠) 𝐶21 k (𝑠) 𝐶22 k (𝑠) ] , 𝑘 = 1 ÷ 3 . (4.1) the design of the pdc concludes in the design of the local linear controllers. the block diagram of a linear two-variable controller is presented in fig. 4.2. it consists of two standard pid based main controllers cii(s) and two cross controllers cij(s) the aim of which is to 0 1000 2000 3000 4000 5000 6000 7000 8000 20 22 24 26 28 30 32 34 36 38 40 mamdani flc system simulation temperature, c o humidity, % dotted line mamdani flc simulation without supervision references pdc system simulation mamdani flc real time with corrected gains two-variable pdc and flc time, s 54 s. yordanova copyright ©2021 assa. adv. in systems science and appl. (2021) u22 c22 2 (s) c22 1 (s) c22 3 (s) u2 sugeno 1 output k 1 y1 u11 c11 2 (s) c11 1 (s) c11 3 (s) x x x u1 u12 c12 2 (s) c12 1 (s) c12 3 (s) e2 e1 sugeno 2 output k 2 y2 u21 c21 2 (s) c21 1 (s) c21 3 (s) x x x u11 1 u11 2 u11 3 u21 1 u21 2 u21 3 x x x x x x fig. 4.1. modified pdc-tsk with local linear two-variable controllers for three linearization zones c11(s) c12(s) c21(s) c22(s) e1 u11 u12 u21 u22 e2 u1 u2 fig. 4.2. block diagram of linear two-variable controller compensate the cross connections between the two plant output variables. the controller outputs are ui=uii+uij (i, j=1,2; i j), where uij=cij(s).ej. the design of the linear two-variable controllers is based on the linear control theory methods [17, 22]. the design criterion is to achieve decoupling in the closed loop system and a desired performance of the two decoupled systems. the requirement for decoupling is the closed loop system transfer matrix with respect to reference yr (s)=[i+p(s).r(s)]-1p(s).r(s) to be turned into a diagonal matrix (s)=diag[ii(s)] by the help of cross controllers with proper transfer functions. the decoupling ensures that each plant output variable depends only on its reference, i.e. yi(s)=ii(s).yri+ij(s).yrj, ij(s)=0. the transfer functions of the decoupling cross controllers cij(s), computed from the conditions ij(s)=0, are: decoupling model-based fuzzy logic control of room temperature and humidity 55 copyright ©2021 assa. adv. in systems science and appl. (2021) 𝐶ij(𝑠) = − 𝑃ij(𝑠)𝐶jj(𝑠) 𝑃ii(𝑠) . (4.2) the decoupled closed loop systems are described by the following transfer functions: ii(s)=piieq(s).cii(s).[1+ piieq(s).cii(s)]-1, (4.3) where piieq(s) are equivalent plants 𝑃iieq(𝑠) = 𝑃ii(𝑠)[1 − 𝑃12(𝑠)𝑃21(𝑠) 𝑃11(𝑠)𝑃22(𝑠) ]. (4.4) relationships (4.2)-(4.4) determine the following procedure for tuning of the parameters of the two-variable controller. step 1. the equivalent plants (4.4) are computed and their zn models are derived by approximation of the obtained step responses. the zn plant model is necessary when engineering techniques are applied for the tuning of the main pid based controllers since most empirical tuning approaches are developed for zn plant model [17]. step 2. the main controllers are tuned to ensure desired performance indicators in the decoupled closed loop systems, e.g. for ensuring of a good compromise between a small overshoot  and a short settling time ts the parameters kpii and ti ii of a main pi controller 𝐶ii(𝑠) =kр[1+1/(ti ii.s)] are computed from [17]: kрii=a. 𝑇iieq/(𝐾iieq. iieq); тi ii=b. 𝑇iieq, a=0.32.5, b=0.63. (4.5) step 3. the cross controllers are computed from (4.2) where the plant time delay is represented by its taylor’s approximation as a time lag in order to obtain rational transfer functions 𝐶ij(𝑠)=(ansn+…+ao).(bmsm+…+bo)-1, mn. then the simulated step responses are approximated by low order dynamic elements – lead-lags, first order differentiators, pi or pid controllers in order to simplify the pdc completion. this procedure is repeated for each linearization zone of the pdc. the tuning of the pdc for rh and t considers local two-variable pi controllers. the zn model parameters of the approximated equivalent plants in the three linearisation zones are shown in table 4.1. the main controllers are tuned according to (4.5) and the cross controllers (4.2) are approximated with pi algorithms. the computed parameters of all pi controllers are presented in table 4.2. table 4.1. zn model parameters of the approximated equivalent plants zones main channel 11 main channel 22 k11eq k t11eq k 11eq k % s s k22eq k t22eq k 22eq k % s s zone1, k=1 1.15 300 18 2.15 400 40 zone2, k=2 2.2 200 40 2.75 750 150 zone3, k=3 2.26 100 10 1.57 550 150 table 4.2. parameters of the pdc local pi controllers zones main channel 11 cross channel 12 cross channel 21 main channel 22 kp11 k ti11 k % s kp12 k ti12 k % s kp21 k ti21 k oc s kp22 k ti22 k oc s zone1, k=1 1.03 164 0.08 65 0.03 60 0.36 147 zone2, k=2 0.53 171 0.4 220 0.038 160 0.9 515 zone3, k=3 0.17 90 0.65 150 0.03 80 1 513 56 s. yordanova copyright ©2021 assa. adv. in systems science and appl. (2021) 5. pdc system simulation investigation and performance assessment the designed pdc for the control of the relative humidity and the temperature of a laboratory air-conditioning system is tested in simulations based on the developed simple tsk plant model. the simulation investigations include the following experiments: 1. simulation of the step responses of humidity and temperature to reference changes applied to the pdc closed loop system in cases the local two-variable controllers rk(s) (4.1) are designed with and without (cij k(s)=0) decoupling controllers. this investigation aims at justification of the necessity of decoupling cross controllers in the local two-variable controllers for improving the performance of the closed loop system, e.g. reduction of the dynamic error due to plant output variables coupling. the parameters of the local twovariable controllers without decoupling controllers for which rk(s)=diag[cii(s)] are computed from (4.5) on the basis of the corresponding local main channels plant models, i.e. 𝑃iieq(𝑠) = 𝑃ii(𝑠), and are shown in table 5.1. both final defuzzyfied control actions are scaled by 2. the simulated step responses are shown in fig. 5.1 from where it can be established that the system with the decoupling cross controllers demonstrates reduced overshoot i=(ymaxi/yri).100, %, ymax i=ymax i-yri, settling time tsi and coupling effect, estimated by ceij=|ymax i|/|yrj|.100, %, for ij. besides, the step responses for each plant output for equal reference step changes differ a little despite the changes of the parameters of the nonlinear plant in the different operation points. this is a proof that the system is robust. so, the decoupling cross controllers contribute to the improvement of the system performance. 2. simulation of the step responses of humidity and temperature to reference changes applied to the flc and the decoupling pdc closed loop systems in order to be compared. the simulated step responses of the pdc closed loop system with the decoupling controllers in the local two-variable linear controllers from fig. 5.1 are added to the graphs in fig. 3.3. table 5.1. tuning of main controllers of the local diagonal two-variable controllers zones kp11,% ti11, s kp22, oc ti22, s zone 1 1.03 246 0.12 220 zone 2 0.53 256 0.3 773 zone 3 0.17 135 0.33 769 0 1000 2000 3000 4000 5000 6000 7000 20 22 24 26 28 30 32 34 36 r k (s) on main controllers (cij k (s)=0) temperature, o c humidity, % references r k (s) with decoupling controllers cij k (s) coupling effect time, s pdc with local two-variable controllers r k (s) fig. 5.1. simulated closed loop systems step responses for two-variable pdc with and without decoupling decoupling model-based fuzzy logic control of room temperature and humidity 57 copyright ©2021 assa. adv. in systems science and appl. (2021) the pdc system outperforms the flc system in reduced overshoot, settling time and coupling effect and better robustness – almost identical responses to equal reference changes from different operation points. considering the closeness between the simulated and real time step responses of the flc system it can be deduced that: the experimental step responses of the real time pdc system will not deviate significantly from its simulated responses; the performances of the pdc and flc real time systems will be related in the same manner the corresponding simulated systems performances are related, i.e. the pdc real time control system is expected to have faster step responses with smaller overshoot, insignificant decoupling and better robustness. the investigated systems assessed performance indicators overshoot i, %, settling time tsi, s, and coupling effect ceij, %, are presented in table 5.2. the step responses of temperature are characterised by higher ts. the coupling effect is higher on the relative humidity. both the correction of the flc gains with the effect of adaptation and the decoupling cross controllers of the pdc reduce at least by 2 the coupling effect in the corresponding systems without gain correction and decoupling cross-controllers respectively. in simulations the decoupling pdc outperforms the flc with the corrected gains in reduced coupling, settling time and overshoot as seen from the values in colour in table 5.2. the simulated system with flc with corrected gains shows on average worse perfomance than the flc real time system since  and ts are 2-2.5 times greater. this gives grounds to deduce that the decoupled pdc system in real time control can have much better performance than in simulations. besides, in real time control it can be expected to have smaller overshoot, coupling and settling time than the flc system with corrected gains. table 5.2. assessed performance indicators of all investigated systems systems yr1% yr2 oc y1, % y2, oc , % ts, s ce, % , % ts, s ce, % flc corrected gains real time 29 34 34 34 25 25 27 29 0 5 250 250 0 0 5 3 1300 1400 10 8 flc corrected gains simulations 29 34 34 34 25 25 27 29 0 30 700 700 12 15 25 12 2500 2500 4 4 flc simulations 29 34 34 34 25 25 27 29 0 32 1500 1400 20 20 25 10 2600 2400 5 10 decoupling pdc simulations 29 34 34 34 25 25 27 29 20 20 700 700 10 3 20 4 1200 1500 3 3 pdc simulations 29 34 34 34 25 25 27 29 24 26 700 700 25 32 0 0 1100 2600 4 4 58 s. yordanova copyright ©2021 assa. adv. in systems science and appl. (2021) 6. conclusion and future work the novelty and the main contributions of the research conclude to the following. a fuzzy logic controller on the principle of parallel distributed compensation for decoupling control of a two-variable nonlinear plant the coupled room relative humidity and temperature in a laboratory air-conditioning system, is developed in order to improve the pdc closed loop system accuracy and robustness with respect to a mamdani flc system. the pdc design is based on a novel two-variable tsk plant model with simple structure and local plant transfer matrices models. the tsk plant model is derived and validated from an objective procedure using experimental data in a genetic algorithms optimisation. the pdc has the same structure as the tsk plant model but with local standard linear two-variable decoupling controllers which unlike the two-variable mamdani flc simplifies the pdc completion in general purpose industrial programmable logic controllers for real time control. besides, the pdc design and tuning is based on the linear control theory approaches. simulations and real time experiments on a laboratory hvac system are performed to show that the developed pdc system outperforms the existing more complicated adaptive mamdani two-variable control system in improved dynamic accuracy and robustness. the future research will be focused on the testing of the designed pdc in real time control of the relative humidity and the temperature in a laboratory air-conditioning system using the facilities of matlabtm real time [16] and of an industrial plc. references 1. alcala, r., casillas, j., & gonzalez, a. 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(2011). methods for design of fuzzy logic controllers for robust process control. sofia: king [in bulgarian]. 23. yordanova, s. (2016). design of fuzzy supervisor-based adaptive process control systems. in k. nakamatsu, & r. kountchev (eds), new approaches in intelligent control: techniques, methodologies and applications (pp. 1–42), book series “intelligent systems reference library” 107, switzerland: springer int. publishing. 24. yordanova, s. (2017). energy efficient fuzzy logic control of indoor air-conditioning in real time. wseas trans. on circuits and systems, 16, 163–170. 25. yordanova, s., merazchiev, d., & jain, l. (2015). a two variable fuzzy control design with application to an air-conditioning system. ieee trans. on fuzzy systems, 23(2), 474–481. 1. introduction 2. theoretical background and problem formulation 3. tsk plant modelling 4. design of pdc with local linear two-variable controllers 5. pdc system simulation investigation and performance assessment 6. conclusion and future work microsoft word 1006-article text-6542-1-18-20231220 adv syst sci appl 2023; 04; 156-178 published online at https://ijassa.ipu.ru. a novel two-stage hybrid multi-objective differential evolution with opposition based-learning noureddine boukhari1*, mohamed amine nemmich2, fatima debbat3, nicolas monmarché4, mohamed slimane4 1) department of mathematics, evolutionary engeneering & distributed information systems laboratory (eedis), djillali liabes university of sidi bel-abbes, sidi belabbes, algeria 2) department of computer science, mathematics laboratory, djillali liabes university of sidi bel-abbes, sidi bel-abbes, algeria 3) department of computer science, university mustapha stambouli of mascara, mascara, algeria 4) université de tours, laboratoire d’informatique fondamentale et appliquée de tours (lifat), tours, france abstract: evolutionary algorithms have been shown to be powerful for solving multi-objective optimization problems, where non-dominated sorting is a widely adopted selection method. in this paper, a new two stage hybridized multi-objective evolutionary algorithm, are evolved. firstly, multi-objective differential evolution based on ranking mutation is applied using nondominated sorting and crowding distance. secondly, jumping probability is used in second stage in order to meet the objective of balancing the precision of the solution and the rate of convergence while maintaining the diversity of the population by opposition-based learning technique. through the validation of the new approach using a suite of carefully selected test benchmarks problems with complex pareto sets and solve a large portfolio complex problem , it is observed that our hybridization achieves overall better performance in terms of convergence and diversity compared to other algorithms of literature. keywords: evolutionary multi-objective optimization, differential evolution, pareto dominance, opposition-based learning, hybrid algorithm, portfolio optimization problem 1. introduction among the recent advances in optimization algorithms, evolutionary algorithms (eas) have proven successful in overcoming difficulties with traditional optimization techniques either in their standard version, or hybridized [1]. eas work with a set of solutions, called population. this feature is particularly suitable for solving multi-objective problems, as it enables approximating the efficient frontier in a single run. a multi-objective evolutionary algorithm (moea) is a modified version of the traditional genetic algorithm (ga) also known as the simple ga, designed to solve multi-objective optimization problems (mops). moeas have been recognized to be suitable for solving multi-objective optimization problems because of their ability to find good solutions for competing objective functions simultaneously and to search for multiple non-dominated solutions simultaneously in the population of candidate solutions. this capability enables them to find several trade-off solutions for all objectives in a single run of the algorithm, instead of having to perform a series of separate runs as in the case of the traditional techniques such as weighted sum, goal programming and weighted min-max methods. ideally, a moea returns a pareto-optimal * corresponding author: boukhari.noureddine@gmail.com a novel two-stage hybrid multi-objective differential evolution… 157 copyright ©2023 assa. adv. in systems science and appl. (2023) set, the solutions not dominated by any other solution in the search space. in addition, moeas are less susceptible to the shape or continuity of the pareto-optimal front, whereas these two issues pose a problem for the mentioned traditional techniques. in the past two decades , multi-objectives optimization has attracted increasing interests in the evolutionary computation community, and a large number of multi-objective optimization algorithms have been developed on the basis of different population based meta-heuristics, such as genetic algorithm [2], differential evolution algorithm [3], firefly algorithm [4], particle swarm optimization algorithm [5], sms-emoa[6]. although various approaches have been adopted for selection [7], most moeas adopt the pareto-based approach, i.e., the qualities of the candidate solutions are compared using pareto dominance. among various dominance comparison mechanisms, non-dominated sorting has been shown to be very effective for finding pareto-optimal solutions. much work has also been done to efficiently store non dominated solutions found during search in an archive. non-dominated sorting is a procedure where solutions in the population are assigned to different fronts based on their dominance relationships. differential evolution (de) is a simple and efficient population-based ea that has been reported in several studies for its high robustness, fast convergence speed, and good solution quality, making it a very popular ea in the evolutionary computing community. due to these strengths observed in de, the use of this ea may provide an answer to some, but not all, of the above limitations. in order to overcome these limitations, several improvement works were carried out on de in order to allow it to meet the objective of balancing the precision of the solution and the rate of convergence while maintaining the diversity of the population for different types of moop. in order to simultaneously achieve fast convergence speed and efficient global search capacity, researchers explored the use of hybridization of different eas for moops as well as the formulation of memory algorithms that integrate local search into eas, and these efforts can also be found in de [8][9]. researchers have also introduced the use of adaptive or self-adaptive de variants [10][11] which eliminate the need to undergo the tedious process of trial and error setting the control parameters in de to an optimum setting for the problems tested. differential evolution as one of the most evolutionary algorithms, has been widely applied to solve multi-objective optimization problems due to its simple implementation and fast convergence. in order to apply de to multi-objective optimization, there are at least two fundamental issues to be addressed. the first issue is how to define the best individual, given that there does not exist any individual which can perform the best on all objectives of an mop. the second issue is how to balance convergence and diversity of the population. since the target of multi-objective optimization is to obtain a set of trade-off solutions, diversity maintenance is particularly important. a de based multi-objective algorithm is very likely to be trapped in local optimum (or one of the many optima) of an mop due to its fast convergence. there-fore, striking a balance between convergence and diversity is crucial to the performance of multi-objective de algorithm. 2. differential evolution for mop several researchers have applied de to solve multi-objective problems [12], with superior and impressive performance to other multi-objective algorithms in benchmarks and widely practical applications. reported and verified [13][14], the general demo procedure is shown in algorithm 1. like de, the algorithm starts with a population p randomly created solutions. for each generation, the following steps are repeated. a candidate is constructed from its parent (and other solutions of p) using the de/rand/1/bin strategy described previously. after that, the candidate is assessed and compared to his parent. at this point, demo differs from de (see step 2.1 (c) in algorithms 1). in demo, the candidate replaces the parent only if he dominates him. if the parent dominates the candidate, the candidate is 158 n. boukhari, m. a. nemmich, f. debbat, n. monmarché, m. slimane copyright ©0000 assa adv. in systems science and appl. (0000) discarded. otherwise (when the candidate and the parent are incomparable), the candidate is added to the population. after repeating this step p times, we obtain a population of size between p and 2p. if the population has grown in size, it should be truncated to size p using one of the environmental selection approaches. at the end of the generation, the solutions of p are listed at random (as in de). the end result of demo consists of non-dominated solutions of p. general mode algorithm 1 [15] input: parameters of the algorithm. output: non-dominated solutions of p. 1. evaluate the initial population p of p random solutions. 2. while the stop criterion has not been met, proceed as follows: 2.1. for each solution 𝑥 (𝑖 = 1, … , 𝑝) of p repeat: (a) create candidate c from parent 𝑥 (b) calculate the candidate's goals. (c) if the candidate dominates the parent, the candidate replaces the parent. if the parent dominates the candidate, the candidate is discarded. otherwise, the candidate is added to the population. 2.2. if the population has more p solutions, apply an environment selection procedure to obtain the best solutions p. 2.3. randomly list the solutions in p. note that newly created candidates who enter the population (either by replacement or by addition) instantly participate in the creation of subsequent candidates. this makes it possible to achieve rapid convergence towards the optimal pareto front. the extension of de to mops has also given promising results [16]. in what follows, we will review the multi-objective de (mode) under five aspects, namely adaptive strategies, selection operator, diversity control, constraint management and practical applications. for adaptive mutation type and mode parameterization strategies, [17] have proposed a self-adaptive mode (mosade) capable of adaptively selecting the appropriate mutation strategies. later, a new version of the mosade hybrid with learning strategies by objectives (owmosade) was developed by [18], who can adaptively select mutation strategies and crossing parameters appropriate for each objective separately. [3] presented a self-adapting mode in which direction information from solutions archived below the current population is used in the mutation process. [19] used information from the evolution of solutions along each search direction to tailor mode control parameters. [20] proposed a variant of moea/d with de operators (moea/dfrrmab) where rates of mutation operators were adaptively determined by a multi-armed bandit scheme. for the selection operator in mode, traditional modes [18], and pareto differential evolution [21] used the one-to-one selection scheme. one, that is, the test vector was allowed into the next generation if and only if it could dominate its target solution. different from this scheme, some modes preferred to store the new test vectors and then select good solutions from the union of test vectors and the original population to build the next population. the application of such a scheme is found in the mode named pdea proposed by [22], demo proposed by (robič & filipič, 2005), ademo proposed by [23], modea proposed by [24], and the demon proposed by [25]. regarding the diversity control in mode, many studies have preferred to adopt the pareto ranking and the overpopulation distance of nsga-ii [26] to truncate the elitist archive. instead of crowding distance, [27] used a measure of diversity entropy of crowding in the self-adaptive mode to preserve the diversity of elite archives. [28] developed a measure of diversity based on harmonic distance to maintain the diversity of external archives and dominance has been integrated into mode to ensure the diversity of external archives. in addition to metrics for measuring diversity, the multiple populations strategy has also been adopted in mode to maintain diversity. a cooperative mode (cmode) with multiple populations has been presented by [29], in which each population treated only one a novel two-stage hybrid multi-objective differential evolution… 159 copyright ©2023 assa. adv. in systems science and appl. (2023) goal. another version of the multi-population mode has been proposed in [30], in which each subpopulation adopted its own mutation strategy and competed for evolutionary resources. for the mop with constraints, [31] incorporated local research based on approximate set theory into mode. [32] developed a new mode by designing a new constraint management method to guide the search for individuals using a number of good infeasible solutions. for the practical application of the mode, [33] presented a mode for the optimization of the operation of the pyrolysis of naphtha. [34] presented a mode incorporating a taboo list for chemical engineering applications. [35] studied the optimization of the functioning of the p-xylene oxidation reaction process and developed a mode with a self-adaptive strategy for the generation of test vectors and the control of settings. [36] proposed a mode with an adaptive parameter control strategy and non-dominated ranking for pom of electromagnetic problems. [37] proposed an adaptive mode for the problem of defining the operating point with four objectives focusing on the cost of fuel consumption and reduction of emissions. the work [38] proposed a de variant to solve multi-objective optimization problems called myodemr (multiple-objective de with mutation restriction). the algorithm uses the concept of pareto dominance coupled with the inverted generational distance metric to select the population for the next generation from a combination of the parent and offspring populations. the algorithm also uses a strategy of restriction of the difference vector in the de mutation to improve the convergence characteristics on multimodal fitness landscapes. recently, [39] proposed a multi-goal optimization algorithm that periodically rearranges goals based on their conflict status and selects a subset of conflicting goals for further processing. the authors used demo as the underlying metaheuristic algorithm, and implemented the technique of selecting a subset of conflicting goals using an order based on the correlation of goals. the resulting method is called α-demo, where α is a parameter determining the number of conflicting objectives to select. de has also been used as a core optimizer in recently developed decomposition-based moeas for multi-objective optimization such as [40]. 3. opposition based learning opposition based learning (obl) is a new concept of machine learning, inspired by the opposing relationship between entities. in 2005, for the first time, the concept of opposition was introduced, which has attracted a lot of research efforts over the past decade. a variety of soft computer algorithms such as optimization methods, reinforcement learning, artificial neural networks, and fuzzy systems have already used the concept of obl to improve their performance. the concept of computational opposition [41] was inspired by the concept of opposition in the real world and opposing numbers were simply defined as follows: definition 3.1: (opposite number). [41] let 𝒙 ∈ [𝒂, 𝒃] be a real number. its opposite, x  , is defined as follows: 𝑥 = 𝑎 + 𝑏 − 𝑥 (3.1) definition 3.1: (opposite point in space d). [41] let 𝑥 (𝑥 , . . . , 𝑥 ) be a point in dimensional space 𝐷 and 𝑥 ∈ [𝑎 , 𝑏 ], 𝑖 = 1, 2, . . . , 𝐷. the opposite of 𝑥 is defined by x  (𝑥 ,..., 𝑥 ) as follows: 𝑥 = 𝑎 + 𝑏 − 𝑥 (3.2) in fact, they indicate that to find the unknown optimal solution, the simultaneous search for a random direction and its opposite gives a higher chance of finding the promising regions. it is reasonable that if the current estimates (assume) are far from the unknown optimal solution, the computation of their opposites leads in the opposite direction to the unknown optimal solution. note that the base opposite point is calculated the same as a reflected point when it is calculated through the center point ((𝑥 + 𝑥 + ⋯ + 𝑥 )/2). 160 n. boukhari, m. a. nemmich, f. debbat, n. monmarché, m. slimane copyright ©0000 assa adv. in systems science and appl. (0000) obl was introduced as an attempt to increase the rate of learning in eas. since evolution is a slow process while revolution is a fast process, the simulation of revolution in eas might speed up their convergence. it has also been implemented in many other optimization algorithms as mentioned in section 01. several research works have been conducted in utilizing obl concept and extending the new schemes of obl to enhance ea algorithms. also, mathematical theorems were derived to demonstrate that opposite candidate solutions have higher probability to be closer to an unknown optimal solution than the randomly generated candidate solutions. in [42], some theorems are mathematically proved to conclude the advantage of using the obl concept. also, they conducted some experiments by utilizing the obl concept in the framework of ea algorithms to enhance their performance. in fact, according to probability theory, 50% of the time, a guess is further from the solution than its opposite guess. therefore, it will generate the opposite individual of the current individual, evaluate the fitness of both individuals, and select the better one as a new individual which can consequently improve the quality of the search population and accelerate convergence. the motivation for this work was twofold. first, we wanted to design an efficient algorithm for multi-objective optimization, which would use de for decision space exploration in a simple way. although de-based algorithms for multi-objective optimization have already been proposed in the past, they have either ignored the basic feature of de of comparing each new solution. to its parent, or applied it too strictly for multi-objective optimization. in addition, the existing approaches only use the environmental selection method of nsga-ii, while our objective was to allow combinations of exploration of the decision space to boost the algorithm to select better solutions with a good diversity compared to the algorithms proposed in the literature. the second objective is to apply an efficient hybridization of the de-based multi-objective optimization algorithm with other training technique in order to reduce the complexity of the compute and that the results are precise and simple. our hybrid approach could help by exploring space. 4. proposed hybrid algorithm mode-obl this section describes the multi-objective differential evolution based on the opposition mode-obl developed in this study on the basis of the mode-rmo algorithm [43] (see algorithm 2). in mode-obl, the candidate potential solutions are processed at initialization and the exploration/exploitation capacities during the various optimization processes are mainly concerned. obl can be used in two steps of mode-obl. first, at the initialization stage in order to arrive at candidate solutions for the most appropriate fit under conditions where there is no a priori knowledge about the initial individuals; second, during the implementation of mode-obl in order to force the current population to embark on new candidate solutions more suitable than current ones. these two steps are called, respectively, opposition-based population initialization and opposition-based generation leap. in this way, the proposed algorithm can converge faster while maintaining good diversity. figure 4.2 shows the overall operational architecture of the proposed algorithm. the majority of works have used obl in the context of mono-objective optimization and recently very few attempts are invested to hybridize obl with multi-objective evolutionary algorithms [44] [45]. hence our approach is inspired to incorporate a two-phase (serial) learning technique into multiobjective differential evolution based ranking mutation. 4.1. multi-objective differential evolution with ranking-based mutation operator in this section, the proposed mode-rmo algorithm is presented in detail. in mode-rmo, the ranking-based mutation operator is introduced in mode to improve its performance by accelerating the speed of convergence. therefore, in the next part, we will first describe the a novel two-stage hybrid multi-objective differential evolution… 161 copyright ©2023 assa. adv. in systems science and appl. (2023) ranking-based mutation operator for multi-objective optimization; then mode-rmo is minutely elucidated. in nature, good people always hold good information, and they are more likely to be used to guide other people. based on these considerations, [46] proposed a ranking-based mutation operator for de in single-objective optimization, in which parents are selected proportionally based on their ranking in the current population. the higher a parent ranks, the more they will be selected. the authors incorporated the proposed rank-based mutation operator in some de algorithms, and experimental results indicated that the rank-based mutation operator is able to improve the performance of de algorithms in the optimization of one goal. the ranking-based mutation operator was introduced in mode for mop. however, the challenge of this extension is that the population can be directly sorted from best to worst in single objective optimization, while there are many solutions that are not dominated among themselves in multi-objective optimization. whereas the mode algorithm should generate approximate pareto solutions with both good convergence and good propagation, therefore, non-dominated fast sort and saturation distance are incorporated into the rank-based mutation operator to treat mops. 4.1.1. fast sorting not dominated and crowding distance the unexpressed rapid sorting and crowding distance is proposed by [47] in the nsga-ii algorithm. in the quick sort procedure without markup, for each solution, two entities are calculated: 1) 𝑛 : the dominance account, i.e. the number of solutions which dominate the solution 𝑝; 2) 𝑆 : a set of solutions that the solution dominates. all solutions of the first non-dominated front will have their dominance count at zero. now, for each solution 𝑝 with 𝑛 = 0, we visit each member (𝑞) of its set 𝑆 and reduce its domination count by 1. if for a member the domination count becomes zero, we put it in a separate list 𝑄, these members belong to the second non-dominated front. now the above procedure continues with each limb and the third front is identified. this process continues until all fronts are identified. the crowding distance is used to get an estimate of the density of solutions surrounding a particular individual 𝑖 in the population; it calculates the average distance of its two neighboring solutions on along each of the goals. the computation of the gathering distance requires sorting the population according to each objective function value in an increasing order of magnitude. then, for each objective function, the boundary solutions (solutions with the smallest and largest function values) are assigned an infinite distance value. all other intermediate solutions are assigned a distance value equal to the absolute normalized difference of the function values of two adjacent solutions. this calculation is continued with other objective functions. the overall staging distance value is calculated as the sum of the individual distance values corresponding to each goal. each objective function is normalized before calculating the crowding distance. 4.1.2. ranking assignment and probability of selection after obtaining the number of non-naming fronts 𝑖 and the gathering distance 𝑖 for each solution 𝑖, we can define a partial order for the whole population. solution 𝑖 is better than solution 𝑗, if one of the two conditions is met: (𝑖) 𝑖 < 𝑗 ; (𝑖𝑖) 𝑖 = 𝑗 𝑎𝑛𝑑 𝑖 > 𝑗 now the population can be sorted in ascending order based on the partial order defined above. an individual's ranking is assigned as follows: 𝑅 = 𝑁 − 1, 𝑖 = 1,2, … , 𝑁 (4.1) where 𝑁 is the size of the population. 162 n. boukhari, m. a. nemmich, f. debbat, n. monmarché, m. slimane copyright ©0000 assa adv. in systems science and appl. (0000) according to the equation. (4.1), the best individual in the current population will obtain the highest rank. after assigning the ranking for each individual, the selection probability 𝑃 of the thi individual is calculated as follows: 𝑃 = 𝑅 𝑁 , 𝑖 = 1,2, … , 𝑁 (4.2) after calculating the probability of selecting each individual by the equation. (4.2), the mutation operator based on the ranking of de/rand/1 for multi-objective optimization can be presented in step 5 on algorithm 2, where the individual with a higher ranking will have a greater probability of being selected as the base vector or the terminal vector in the mutation operator; hence, it is beneficial for the propagation of good information in the population to the offspring. in the ranking-based mutation operator for multi-objective optimization, only the base vector and the terminal vector are selected based on their selection probabilities, while the start vector is selected randomly. indeed, if the two vectors of the difference vector are both chosen from among better vectors, the search step size of the difference vector can decrease rapidly and lead to premature convergence [48]. 4.1.3. selection operator in mode, the crossover is performed in the same way as in single objective optimization. however, the selection needs to be rethought, as the test vector and the target vector are often non-dominated over each other. in mode-rmo, we use the following selection operator, which has three steps: (1) if the test vector dominates the target vector, use the test vector to replace the target vector. (2) if the target vector dominates the test vector, the test vector is discarded. (3) otherwise, the test vector is added to the population. thus, at the end of a generation, the total population size is between np and 2np. this population is truncated for the next step of the algorithm. the truncation process involves sorting and rating individuals on the same front with the crowding distance. the truncation procedure retains only the best np vectors in the population. algorithm 2 general procedure of mode-rmo step 1: define the number of generations 𝒈 = 𝟎, randomly initialize the population 𝑷𝒈 = {𝒙𝟏 𝒈 , 𝒙𝟐 𝒈 , … , 𝒙𝑵𝒑 𝒈 } with 𝒙𝒊 𝒈 = {𝒙𝒊𝟏 𝒈 , 𝒙𝒊𝟐 𝒈 , … , 𝒙𝒊𝑫 𝒈 } (𝒊 = 𝟏, … , 𝑵𝒑) uniformly distributed in the search space define the mutation scale factor f, the crossbreeding constant cr and the maximum number of maxgen generations. step 2: evaluate the fitness value of each target vector 𝒙𝒊 𝒈. step 3: for each individual vector, perform the following steps from step 4 to step 7. step 4: determine the vector indices selected using the method described in 4.1.2 step 5: use the ranking-based mutation operator to generate a vector 𝒎𝒊 𝒈 𝟏 corresponding to the target vector 𝒙𝒊 𝒈 according to equation (5). step 6: use the crossover operation to generate a test vector 𝒗𝒊 𝒈 𝟏 for each target vector 𝒙𝒊 𝒈 according to equation (6). step 7: evaluate the test vector and use the following selection operation: (i) if 𝒗𝒊 𝒈 𝟏 dominates 𝒙𝒊 𝒈 , 𝒙𝒊 𝒈 𝟏 = 𝒗𝒊 𝒈 𝟏. (ii) if 𝒙𝒊 𝒈 dominates 𝒗𝒊 𝐠 𝟏, 𝒗𝒊 𝐠 𝟏 is rejected. (iii) if 𝒙𝒊 𝒈 and 𝒗𝒊 𝐠 𝟏 are not dominated with each other, add 𝒗𝒊 𝐠 𝟏 to the population. step 8: after step 7, the size of the population varies from np to 2np. sorting population based on non-dominated sorting and crowding distance, and best individual 𝑵𝒑 survive the next generation. step 9: set g = g + 1, go back to step 3 until the 𝑴𝒂𝒙𝒈𝒆𝒏 be reached. a novel two-stage hybrid multi-objective differential evolution… 163 copyright ©2023 assa. adv. in systems science and appl. (2023) 4.2. initialization based on opposition learning random initialization, in the absence of a priori knowledge, reduces the chances of sampling better regions in population-based algorithms. however, the use of obl can obtain more suitable starting candidates even in the absence of a priori knowledge and increase the probability of detecting better regions. obl is an effective mechanism in the field of optimization because of the promising potential to improve the performance of various optimization algorithms, including kh [49], de [50] and pso [51]. the optimization based on the opposition: let 𝑃 (𝑥 , 𝑥 , . . . , 𝑥 ), a point in a space of dimensions 𝐷 with 𝑥 ∈ [𝑎 , 𝑏 ] (𝑖 = 1, 2, . . . , 𝐷), or a candidate solution. suppose 𝑓 (𝑥) is a fitness function used to measure candidate optimality. according to the definition of the opposite point, 𝑃 (𝑥 , 𝑥 , . . . , 𝑥 ) is the opposite of 𝑃 (𝑥 , 𝑥 , . . . , 𝑥 ). if 𝑓 (𝑃) is better than 𝑓 (𝑃), then point 𝑃 can be replaced by 𝑃; otherwise, we continue with 𝑃. therefore, the point and its opposite point are evaluated simultaneously to continue with the adjuster [52]. inspired by the idea of [49], we conduct here a new method for initializing mode with obl, which is different from the survival selection in previous obl-based algorithms (including obl strategies choose the best 𝑁 individuals among the original 𝑁 individuals and the opposite 𝑁 individuals in initialization). the main steps to explain this procedure are given as shown in figure 4.1: step 1: divide the population 𝑃 (𝑁 ) into two parts, the first half of the population 𝑃 is generated by a random distribution. step 2: the remaining half of the population 𝑃 is initialized in terms of obls, as shown in section 4.2: 𝑃 = 𝑎 + 𝑏 − 𝑃 , (4.3) where (𝑖 = 1, 2, . . . , 𝐷) step 3: the set 𝑃 ∪ 𝑃 is restructured as the initial population 𝑁 . fig 4.1. obl-based initialization strategy in mode the obl strategy used in our proposed algorithm is different from the traditional obl based algorithm. regarding initialization, the traditional obl strategy first initializes the population at random, then calculates the opposite population. while the obl operation in our proposed algorithm first divides the population into two parts, then randomly generates and calculates the inverse respectively, which can obtain more suitable starting candidates when there is no a priori knowledge of the solution. also, after the above steps in generation and opposite calculation, the two subpopulations are made up of one population, which can 164 n. boukhari, m. a. nemmich, f. debbat, n. monmarché, m. slimane copyright ©0000 assa adv. in systems science and appl. (0000) make the population size unchanged in the optimization process and help the algorithm to operate efficiently. as for the use of obl in the evolutionary phase, traditional obl-based algorithms can apply the obl in the evolutionary phase with a jump rate or a jump probability. however, the proposed mode obl directly calculates the opposite without the jump rate, which can increase the likelihood of effectively detecting better regions and reduce complex setting, especially for the real-world problem. this strategy is used to accelerate convergence when there is no prior knowledge of solutions, thus achieving better solutions more quickly. 4.3. jumping generation based on opposition in order to improve overall convergence and avoid sub-optimal solutions, the obl technique is reapplied to the current population. at this point, if the jump condition is satisfied: 𝑗 𝑟𝑎𝑛𝑑( ) ≤ − + 2 , (4.4) where 𝐺𝑒𝑛 and 𝐺 are the current and maximum generations respectively [45]. the corresponding opposition population is calculated by forcing current one to pass to a new solution. after that, the fittest 𝑁 individuals are selected from the combined population of the current population and the opposition as the current population for the next generation. unlike the process of the opposition-based initialization phase, the generation jump computes the opposition population dynamically. instead of using the predefined interval limits of the variables [𝐿𝐵 , 𝑈𝐵 ], the generation jump calculates the opposite of each variable based on the minimum values 𝑀𝑖𝑛 and maximum 𝑀𝑎𝑥 for this variable in the current population. 𝑋 , , = 𝑀𝑖𝑛 + 𝑀𝑎𝑥 − 𝑥 , , 𝑤ℎ𝑒𝑟𝑒 𝑗 = 1, … , 𝐷; 𝑖 = 1, … , 𝑁 (4.5) by remaining within the static limits of the interval of variables, we jump outside the already reduced search space and lose knowledge of the current reduced space (converged population). therefore, we calculate opposite points using the current range of variables in the population which is, as we search, smaller and smaller than the corresponding initial range. a novel two-stage hybrid multi-objective differential evolution… 165 copyright ©2023 assa. adv. in systems science and appl. (2023) fig 4.2. flowchart of the proposed hybrid algorithm mode-obl 5. experimental results and comparisons five state-of-the-art algorithms, namely nsga-ii, mopso, nsde, moea/d-de and mode-rmo were chosen for the performance comparison with the proposed mode-obl. nsga-ii [53] is a popular algorithm in evolutionary multi-objective optimization because it has the ability to achieve promising solutions for most moops. this algorithm uses pareto rank and crowding distance as the operators of fitness assignment, binary tournament selection, uniform crossover, bit-flip mutation, and parent-offspring archiving. the nondominated sorting differential evolution (nsde) [54] is an extension of the basic differential evolution solving multi-objective optimization. it adopts the non-dominated sorting, ranking, and elitism techniques found in nsga-ii, but the main difference between them is that the nsde uses the differential evolution mutation operator instead of the sbx operator. for moea/d-de [55], it is an evolutionary algorithm that decomposes any given mop into a number of single objective subproblems. each subproblem is simultaneously optimized during the evolutionary research process. for the decomposition of moop, tchebycheff's approach is used in this algorithm and the difference between moea/d-sbx and moea/d-de lies in their genetic operators whereby the sbx crossover operator is used with a polynomial mutation. for moea/d-sbx while moea/d-de uses the de/rand/1 crossover with polynomial mutation. finally, for mopso [56], it is extended from the particle swarm optimization algorithm (pso), and it essentially combines the strong characteristics of pso. for fair comparisons, all parameters of the compared algorithms are set to recommended values as in their original articles. 166 n. boukhari, m. a. nemmich, f. debbat, n. monmarché, m. slimane copyright ©0000 assa adv. in systems science and appl. (0000) in the following subsections, the test issues and quality indicators used in our comparative experiments are first presented. then, the experimental parameters adopted in this study are provided. in addition, thirty independent tests are performed for each test problem in order to avoid the stochastic phenomenon, and the wilcoxon rank sum test is adopted with significance level of 0.05 to compare the results obtained by mode-obl and the five algorithms compared to determine if the best performing algorithm differs from competitors results in a statistically significant way. where the symbols "+", "-" and "≈" indicate that the result is significantly better, significantly worse and statistically similar to that obtained by the mode-obl, respectively. 5.1. test problems a total of 12 benchmark test problems were chosen to test the optimization performance of the proposed hybrid algorithm mode-obl in terms of convergence towards the true pareto front as well as the ability to maintain a set various solution. the test issues used included zdt and dtlz issues. for test problems, they can have two, three or five objective functions and have an evolving number of decision variables. these problems have been chosen because they cover different characteristics of multi-objective optimization, namely the convex pareto front, the non-convex pareto front, the discrete pareto front, multimodality, and non-uniformity of the distribution of solutions. the presence of these characteristics may pose challenges to a multi-objective optimization algorithm. table 5.1. multi-objective testing issues. s (scalability), m (the number of objective functions), k (scalar parameter), n (the number of decision variables), sp (separable), ns (non-separable). instance m n interval geometry sp/ns u/m zdt1 2 30(s) [0,1] convex sp u zdt2 2 30(s) [0,1] concave sp u zdt3 2 30(s) [0,1] deconnected sp m zdt4 2 30(s) [0,1] × [−5,5] convex sp m zdt6 2 30(s) [0,1] concave sp m dtlz1 3(s) 𝒎 + 𝑲 − 1(𝑆) [0,1] lineair sp m dtlz2 3(s) 𝒎 + 𝑲 − 1(𝑆) [0,1] concave sp u dtlz3 3(s) 𝒎 + 𝑲 − 1(𝑆) [0,1] concave sp m dtlz4 3(s) 𝒎 + 𝑲 − 1(𝑆) [0,1] concave sp u dtlz5 3(s) 𝒎 + 𝑲 − 1(𝑆) [0,1] degenerated ns u dtlz6 3(s) 𝒎 + 𝑲 − 1(𝑆) [0,1] degenerated ns u dtlz7 3(s) 𝒎 + 𝑲 − 1(𝑆) [0,1] deconnected sp m 5.2. performance indicators in our experimental study, in order to make a fair comparison of the different optimization algorithms, performance measures relevant and applicable to the optimization objectives of convergence and distribution must be used. two widely used measures are chosen to evaluate the performance of each algorithm, which are called the generational distance [57], and the inverted generational distance (igd) [58]. gd and igd can effectively measure the convergence and diversity of the obtained solutions, respectively. convergence describes the degree of approximation of the result obtained by the algorithm to the true pareto front (pf). the stronger the convergence of the algorithm, the closer the set of solutions is to the true optimal solution and the more precise the result. the distribution describes the distribution characteristics of the result obtained in the objective space. on the one hand, the results should be distributed as much as possible over the entire pf, and on the other hand, the results should be distributed as evenly as possible. the stronger the distribution of the algorithm, the better the overall exploration capacity of the algorithm. 5.2.1. generational distance (gd) generational distance (gd) is a unary performance indicator that is defined as: a novel two-stage hybrid multi-objective differential evolution… 167 copyright ©2023 assa. adv. in systems science and appl. (2023) 𝐺𝐷 = 𝑑(𝑃∗, 𝑃) 𝑁 , (5.1) where 𝑁 is the number of solutions in 𝑃𝐹∗, 𝑝 ∈ 𝑃𝐹, 𝑝∗ ∈ 𝑃𝐹∗ and 𝑑(𝑝∗, 𝑝) the minimum euclidean distance in the objective space between 𝑝∗ and 𝑝 for each member 𝑖. gd illustrates the convergence capacity of the algorithm by measuring the convergence between the optimal pareto front and the evolved pareto front. thus, a lower value of gd shows that the evolved pareto front is closer to the optimal pareto front. this indicator is a representative metric that provides a quantitative measure of the multi-objective optimization convergence goal. 5.1.2. inverted generational distance (igd): igd is a unary indicator by which the distance of each solution in the optimal pareto front to the obtained pareto front is calculated. let *p be a set of solutions uniformly distributed in objective space along the pareto front. p is an approximation of the pf, which is obtained by the algorithm. igd is described as: 𝐼𝐺𝐷(𝑃, 𝑃∗) = 𝑑𝑖𝑠𝑡(𝑃∗, 𝑃) | ∗| |𝑃∗| , (5.2) where dist(𝑃∗, 𝑃) is the euclidean distance between a point 𝑥∗ ∈ 𝑃∗ and its nearest neighbor in 𝑃, and |𝑃∗| is the cardinality of 𝑃∗. we can see from the definition of igd that for a large |𝑃∗|, it can cover approximately the entire pareto front, which is another aspect of the metric in terms of diversity. 5.3. results for the other algorithms compared, their parameter settings used in this study followed those used in their original studies. the experimental settings and overall parameters are summarized in table 5.2. the comparison was made to examine their optimization performance in the test problems described above. all algorithms were implemented in matlab and run on an intel® core ™ i3 2.53 ghz computer with 6 gb memory capacity. table 5.2. parameter values used in this comparison parameter settings parameter values population size for all algorithms 100 for problems with 2 objectives 300 for problems with 3 objectives 500 for problems with 5 objectives stop criterion 50,000 evaluations function with 2 objectives 150,000 evaluations function with 3 goals 250,000 evaluations fonction with 5 goals number of decision variables for zdt problems 300 for zdt1, zdt2 and zdt3, and 100 for zdt4 and zdt6 number of decision variables for dtlz problems 12 for dtlz1 and dtlz3, and 120 for all the other problems dtlz number of independent executions 30 for each algorithm mutation rate 1/n (where 𝑛 is the number of decision variables) crossover rate for nsga-ii and nsde 0.8 mutation scale factor for nsde 0.5 neighborhood size for moea/d algorithms 20 inertia weight for the mopso algorithm 0.5 (𝐶 , 𝐶 ) coéfficients for mopso algorithm 𝐶 = 1, 𝐶 = 2 comparative studies were conducted to evaluate the performance of the six algorithms as part of a comprehensive suite of benchmark test functions. the simulation results in terms of measurement of the mean and standard deviation values of the generational distance (gd) and the inverted generational distance (igd) over 30 simulation cycles are presented in tables 5.3, 5.4 and 5.5. the parentheses next to the test problems indicate the number of goals (m) and decision variables (d) for the problems. 168 n. boukhari, m. a. nemmich, f. debbat, n. monmarché, m. slimane copyright ©0000 assa adv. in systems science and appl. (0000) zdt test problems [59] are a set of simple bi-objective optimization problems that are scalable in the number of decision variables and have different characteristics in the pareto optimal front such as the convexity, concavity, discontinuity, local optimality and nonuniformity. since most evolutionary algorithms are able to solve zdt problems without difficulty, the number of decision variables was set at ten times its original parameters in this study. this would then pose greater challenges to the algorithms due to a larger search space. the results indicate that mode-obl has the best overall performance. a notable achievement of mode-obl is its ability to achieve convergence for zdt4 when all other algorithms cannot for this problem. the zdt4 problem is an extremely multimodal problem with the presence of many local optima, and so it is likely that the other algorithms have had difficulty being trapped in the local optima. the good overall performance obtained by mode-obl can be attributed to the complementary effects of de based on opposition. table 5.3. results obtained by the algorithms for zdt problems test problem algorithm igd gd zdt1(2,300) nsga-ii mopso nsde moea/d-de mode-rmo mode-obl 0.118 (7.5e-03) 0.072 (8.9e-03) 0.716 (1.9e-01) 0.339 (4.4e-02) 0.048 (1.1e-03) 0.047 (6.8e-04) 0.066 (3.1e-03) 0.488 (1.5e-01) 0.286 (4.9e-02) 0.189 (4.5e-02) 0.054 (3.3e-03) 0.052 (3.1e-04) zdt2(2,300) nsga-ii mopso nsde moea/d-de mode-rmo mode-obl 0.079 (1.5e-02) 0.073 (1.8e-03) 0.060 (2.2e-03) ≈ 0.099 (2.9e-03) 0.232 (6.8e-02) 0.059 (1.0e-02) 0.366 (1.9e-02) 0.652 (7.1e-02) 0.400 (2.2e-02) 2.889 (1.1e-01) 1.594 (2.1e-01) 0.339 (1.8e-02) zdt3(2,300) nsga-ii mopso nsde moea/d-de mode-rmo mode-obl 0.447 (2.3e-02) + 0.587 (7.1e-02) 0.810 (1.2e-01) 0.669 (2.6e-01) 0.454 (8.2e-02) 0.429 (5.4e-02) 0.507 (7.5e-01) 0.714 (4.9e-01) 0.870 (1.2e-01) 0.402 (3.8e-01) 0.859 (1.4e-01) 0.283 (3.8e-03) zdt4(2,100) nsga-ii mopso nsde moea/d-de mode-rmo mode-obl 0.053 (4.2e-03) 0.076 (4.6e-04) 0.067 (1.2e-03) 0.399 (2.0e-02) 0.067 (7.3e-03) 0.048 (7.0e-04) 0.011 (1.3e-03) 0.347 (2.0e-02) 0.021 (2.6e-03) 0.742 (1.0e-05) 0.024 (5.0e-03) 0.006 (3.8e-05) zdt6(2,100) nsga-ii mopso nsde moea/d-de mode-rmo mode-obl 0.034 (1.1e-03) ≈ 0.047 (4.1e-04) 0.090 (1.9e-02) 0.321 (5.5e-02) 0.034 (6.3e-04) + 0.042 (6.4e-04) 0.101 (1.2e-02) 0.143 (6.4e-03) 0.087 (2.9e-03) 1.156 (3.3e-01) 0.068 (1.5e-03) + 0.108 (9.8e-03) the suite of dtlz problems created by [2] can be extended to any number of objectives and decision variables. therefore, for this study, the dtlz problems consisted of three and five objective functions. the number of decision variables in dtlz1 and dtlz3 was fixed at 12 because they are highly multimodal problems and therefore more difficult. for the other dtlz problems, they are generally easier to solve, so the number of decision variables has been set at 120 instead. for the case of dtlz problems with three objective functions, mode-obl achieves competitive performances compared to the other algorithms of this study. based on the simulation results, mode-obl achieves the lowest igd and gd values or approaches the best values for most dtlz problems. however, for the case of dtlz5 and dtlz6, mode-obl did not perform as well compared to other algorithms. for dtlz5, we observe that the nsga-ii algorithm which incorporates the use of the sbx operator as well as mopso gives better results compared to algorithms with the de operator. this suggests that using the de operator may not be as powerful in solving degenerate pareto front problems. a novel two-stage hybrid multi-objective differential evolution… 169 copyright ©2023 assa. adv. in systems science and appl. (2023) in dtlz problems with five objective functions, the results demonstrate that modeobl achieves the best overall performance for dtlz1, dtlz3 and dtlz7 when opposed to all the algorithms compared. for dtlz2, mode-obl reaches the lowest igd value but not for the gd metric. for dtlz4, dtlz5 and dtlz6, we observe that the decompositionbased algorithm generally performs better in terms of better igd and gd values than the others for these three problems. this demonstrates the better ability of decomposition-based algorithms to solve multi-objective problems compared to domination-based algorithms. this is attributed to the fact that algorithms based on decomposition allow a better selection of promising solutions by using aggregated fitness values. for the other dominance-based algorithms in this study, dominance behavior between solutions must be determined before deciding which solutions are superior. however, as the number of objective functions increases, dominance behavior will be weakened. therefore, it will be more difficult for domination-based algorithms to select the best solutions in a higher objective space. we observe that mode-obl generally shows better performance for dtlz1, dtlz3 and dtlz7, and competitive performance for dtlz2, with three and five objective functions. dtlz1 and dtlz3 are highly multimodal problems, and the success of modeobl in dealing with these problems is probably attributed to the strong exploration capabilities inherent in its de operator which allows the algorithm to escape the optimal local. as with dtlz2, the opposition-based generation jump phase in mode-obl helps produce adequate selection pressure towards the large spherical pareto front in the large objective domain. the good performance shown by mode-obl for dtlz7 could also be attributed to the strong exploratory nature of its de operator complemented by the rankingbased mutation operator (rmo) as this helps the algorithm to discover distributed subpopulations. in all disconnected pareto-optimal regions. moreover, the environmental selection method used is also effective in keeping the solutions found in pareto-optimal regions disconnected. these factors may explain why mode-obl is able to handle the dtlz7 problem well. table 5.4. results obtained by algorithms for dtlz problems (3 objectives) test problem algorithm igd gd dtlz1(3,12) nsga-ii mopso nsde moea/d-de mode-rmo mode-obl 0.200 (2.3e-02) 0.198 (3.0e-02) 0.254 (7.1e-02) 0.764 (9.1e-02) 0.439 (2.8e-02) 0.151 (4.9e-03) 2.802 (9.8e-01) 2.711 (9.2e-02) + 4.069 (3.0e-01) 3.147 (1.3e-01) 3.716 (1.6e+00) 2.431 (5.6e-02) dtlz2(3,120) nsga-ii mopso nsde moea/d-de mode-rmo mode-obl 0.201 (1.5e-01) 0.178 (8.4e-04) 0.2719 (5.1e-01) 0.153 (1.1e-03) + 0.254 (4.3e-02) 0.154 (8.8e-03) 0.012 (1.9e-03) 0.006 (1.4e-05) + 2.130 (5.7e+00) 0.049 (1.4e-02) 0.742 (5.8e-05) 0.111 (2.4e-02) dtlz3(3,12) nsga-ii mopso nsde moea/d-de mode-rmo mode-obl 0.766 (4.0e-02) 1.288 (4.1e-01) 0.948 (2.1e-02) 0.519 (1.0e-01) + 0.877 (3.5e-01) 0.565 (1.2e-01) 0.3013 (2.1e-01) 0.2931 (3.7e-02) + 1.0623 (6.5e-01) 0.4865 (2.4e-02) 0.8574 (9.2e-01) 0.3021 (5.8e-01) dtlz4(3,120) nsga-ii mopso nsde moea/d-de mode-rmo mode-obl 0.146 (1.5e-03) + 0.394 (2.9e-02) 0.463 (8.5e-02) 0.180 (3.7e-02) 0.185 (8.2e-05) 0.158 (6.9e-03) 0.387 (5.0e-04) 0.368 (1.1e-02) ≈ 2.648 (4.1e-01) 0.391 (2.1e-02) 0.431 (5.2e-02) 0.368 (8.2e-03) dtlz5(3,120) nsga-ii mopso nsde moea/d-de mode-rmo mode-obl 0.100 (1.5e-03) + 0.177 (5.7e-03) 0.537 (6.8e-03) 0.099 (4.6e-06) + 0.231 (1.6e-02) 0.124 (2.7e-03) 0.253 (5.3e-03) 0.046 (3.2e-03) + 0.134 (6.6e-01) 0.026 (1.5e-02) + 0.668 (8.1e-01) 0.211 (4.7e-03) 170 n. boukhari, m. a. nemmich, f. debbat, n. monmarché, m. slimane copyright ©0000 assa adv. in systems science and appl. (0000) dtlz6(3,120) nsga-ii mopso nsde moea/d-de mode-rmo mode-obl 0.413 (7.1e-02) 9.086 (4.8e+00) 0.201 (4.1e-02) 0.112 (3.1e-03) 0.113 (2.5e-03) 0.099 (2.9e-03) 1.874 (6.3e-01) 1.272 (3.6e-02) 2.063 (6.1e-02) 0.246 (1.3e-03) + 1.346 (5.5e-02) 1.080 (7.3e-03) dtlz7(3,120) nsga-ii mopso nsde moea/d-de mode-rmo mode-obl 2.547 (2.1e-01) + 11.18 (9.6e+00) 18.07 (2.5e+02) 6.061 (6.1e-01) 7.645 (1.1e+00) 2.614 (7.7e-02) 0.813 (3.8e-01) 0.791 (4.2e-02) 0.955 (2.1e-01) 0.744 (6.4e-01) 0.669 (8.6e-02) 0.668 (2.2e-03) table 5.5. results obtained by algorithms for dtlz problems (5 objectives) test problem algorithm igd gd dtlz1(5,12) nsga-ii mopso nsde moea/d-de mode-rmo mode-obl 1.355 (2.6e-02) 8.413 (3.7e+00) 1.686 (1.1e-01) 1.344 (5.0e-06) 1.567 (5.7e-02) 1.227 (3.8e-03) 1.814 (1.6e-02) 3.297 (1.0e+00) 2.278 (3.2e-01) 1.745 (1.4e-02) 1.793 (2.2e-02) 1.726 (2.5e-02) + dtlz2(5,120) nsga-ii mopso nsde moea/d-de mode-rmo mode-obl 0.410 (5.9e-02) 1.397 (5.0e-05) 3.561 (3.2e-02) 0.257 (1.1e-03) 0.658 (7.4e-02) 0.224 (4.7e-03) 0.194 (7.2e-01) 0.438 (3.2e-02) 4.152 (4.2e-01) 0.156 (1.2e-03) + 0.306 (4.6e-02) 0.218 (2.1e-02) dtlz3(5,12) nsga-ii mopso nsde moea/d-de mode-rmo mode-obl 1.548 (5.9e-02) 3.157 (2.5e+00) 4.699 (2.1e+00) 0.881 (1.1e-01) 1.978 (4.1e-01) 0.682 (2.1e-01) 2.852 (4.4e-03) 4.043 (8.7e-02) 2.973 (2.2e+00) 1.663 (2.1e-02) + 2.528 (5.8e-02) 1.906 (4.5e-03) dtlz4(5,120) nsga-ii mopso nsde moea/d-de mode-rmo mode-obl 2.245 (8.9e-03) 4.398 (2.2e-02) 6.544 (3.9e-02) 2.224 (1.4e-03) + 2.365 (1.1e-02) 2.330 (1.6e-02) 0.956 (1.4e-02) 1.030 (8.0e-02) 3.680 (8.7e-01) 0.899 (2.0e-02) + 3.258 (3.9e-01) 1.269 (4.4e-02) dtlz5(5,120) nsga-ii mopso nsde moea/d-de mode-rmo mode-obl 1.267 (1.4e-02) 3.441 (4.7e-02) 2.811 (4.3e-02) 1.169 (5.0e-03) + 1.345 (2.8e-02) 1.219 (1.0e-02) 0.475 (1.2e-01) 8.762 (5e+05) 4.75 (1.4e-01) 0.097 (5.7e-02) + 1.682 (5.7e+01) 1.104 (2.0e-03) dtlz6(5,120) nsga-ii mopso nsde moea/d-de mode-rmo mode-obl 2.2e+09 (6e+09) 5.9e+10 (3e+10) 2.946 (2.7e+01) 1.061 (5.5e-02) 1.402 (8.5e-02) 0.927 (2.8e-03) 15.735 (1.1e+00) 17.512 (3.7e+01) 17.23 (1.3e+02) 4.98 (2.1e+00) + 12.887 (3.2e+00) 9.216 (6.2e+00) dtlz7(5,120) nsga-ii mopso nsde moea/d-de mode-rmo mode-obl 1.498 (1.7e-03) 1.463 (4.3e-01) 12.02 (2.7e+01) 1.320 (2.3e-02) 1.946 (4.3e-01) 1.053 (4.0e-03) 2.565 (4.6e-01) 7.127 (3.8e+02) 7.982 (2.2e-03) 3.287 (6.8e-00) 3.889 (1.4e+01) 1.996 (5.2e-01) 6. application on portfolio optimization problem computational finance is an emerging application field of metaheuristic algorithms. these optimization methods are becoming the solving approach alternative when dealing with realistic versions of several decision-making problems in finance. the portfolio selection problem can be defined as the optimal allocation of wealth among a finite number of assets that follows careful processing of all available information about a novel two-stage hybrid multi-objective differential evolution… 171 copyright ©2023 assa. adv. in systems science and appl. (2023) both investors and markets. markowitz’s mean-variance model is by far the most popular procedure in asset allocation [60]. there are a few key concepts in portfolio optimization. first, reward and risk are measured by expected return and variance of a portfolio. expected return is calculated based on historical performance of an asset, and variance is a measure of the dispersion of returns. second, investors are exposed to two types of risk: unsystematic risk and systematic risk. unsystematic risk is an asset’s intrinsic risk which can be diversified away by owning a large number of assets. these risks do not present enough information about the overall risk of the entire portfolio. systematic risk, or the portfolio risk, is the risk generally associated with the market which cannot be eliminated. third, the covariance between different asset returns gives the variability or risk of a portfolio. therefore, a well-diversified portfolio contains assets that have little or negative correlations [61]. eas work with a set of solutions, called population. this feature is particularly suitable for solving multi-objective problems, as it enables approximating the efficient frontier in a single run. as a result, multi-objective evolutionary algorithms (moeas) have received growing attention to financial applications [62]. in fact, portfolio optimization was one of the first successful applications of moeas in economics and finance. increasing complexity of practical applications has led researchers to develop heuristic procedures for solving their portfolio optimization problems. these techniques require less domain information to be considered than the standard gradient-based mathematical programming methods do. moreover, they guarantee satisfactory approximations to solutions in a fair computational time even when they deal with non-convexity, discontinuity, and integer decision variables. the approaches that have been proposed in the soft-computing literature can be categorized into the following two groups. on one hand, single objective methods optimize a weighted sum of the portfolio objectives. on the other hand, multiobjective evolutionary algorithms (moeas) attempt to tackle the allocation problem directly in its multi-objective form by simultaneously optimizing risk and reward. in the first case, the complete set of risk-return profiles is obtained by varying a parameter that represents the risk aversion of the investor [63] [64]. in the second case, the complete efficient frontier is represented in a single run [65] [66]. both categories pay great attention to encoding types and constraint–handling techniques [67]. while single objective optimization methods consider either a minimal risk for a given return or a maximum risk for a given expected return or an objective function that weights the two goals and thus have to be run several times with the respective weights [68], multiobjective optimization methods find a set of pareto solutions, while balancing two or more objective functions simultaneously. 6.1. problem formulation the key to achieving investors’ objectives is to provide an optimal portfolio strategy which shows investors how much to invest in each asset in a given portfolio. therefore, the decision variable of portfolio optimization problems is the asset weight vector  tnxxxx ,..., 21 with ix as the weight of asset i in the portfolio. the expected return for each asset in the portfolio is expressed in the vector form  tnpppp ,..., 21 with ip as the mean return of asset i . the portfolio expected return is the weighted average of individual asset return (eq 6.2) variance and covariance of individual asset are characterized by a variancecovariance matrix            nnn n v      1 111 , where ii , is the variance of asset i and ji , is the covariance between asset i and asset j . the portfolio variance is defined in eq (6.1). 172 n. boukhari, m. a. nemmich, f. debbat, n. monmarché, m. slimane copyright ©0000 assa adv. in systems science and appl. (0000) the mathematical representation of portfolio optimization was introduced by markowitz in the fifties and he was rewarded with a nobel prize in economics in 1990 [69]. the markowitz model assumes the investors would like to maximize the return under certain risk level or minimize the risk with certain return level and this model makes use of the mean and variance of normalized historical asset price to measure the expected portfolio return and risk [70].the model can be expressed as a bi-objective problem and formulated as following: 𝑀𝑖𝑛: risk 𝜎 = 𝑤 𝑤 𝜎 (6.1) 𝑀𝑎𝑥: return r = 𝑤 𝑟 (6.2) subject to 𝑤 = 1 ; 𝑤 ≥ 0, (6.3) where 𝑛 is the number of assets in the portfolio and the dimension of the problem, 𝑤 is the weight of 𝑖 asset. 𝜎 stands for the portfolio risk and 𝜎 is the covariance between asset 𝑖 and asset 𝑗. if 𝑖 = 𝑗, 𝜎 is just the variance of that particular asset. 𝑟 is the portfolio return, while 𝑟 is the individual return of asset 𝑖. 6.2. experimental settings and dataset all the experiments are conducted using standard markowitz (mean-variance) model. 100 days’ closing prices of 100 and 500 stocks were downloaded and used as the historical stock data in the simulation and used as the test data used in the simulation. stocks’ monthly returns and close prices are picked from [71]. a monthly return calculation is formulated as: 𝑀 = ln ; where, 𝑃(𝑡) is the closing price, 𝑃(𝑡 − 1) is the closing price in the day before and 𝑀(𝑡) is the monthly return. experiments are conducted on 100 and 500 stocks. thus, the chromosome size is 100 and 500, respectively. and the number of function evaluation is 100000 and 300000 respectively for all the three algorithms and all algorithms are run 25 times with random initialization. except mode-obl, two other multi-objective evolutionary algorithms are also tested on this portfolio optimization problem for comparison due to their superiority respected to gd and igd metrics in different benchmarks problems tested previously. for real-coded nsgaii, a population size of 100 is used, crossover probability of 0.9 and mutation probability of 1/n , where n is the number of decision variables, distribution indexes for crossover and mutation operators as presented in [72]. moea/d-de uses a population size of 100, f is set to 0.5 and cr is set to 0.1. to check the robustness of the results, 25 simulations for each algorithm and for each test problem are used. the algorithms are implemented in matlab r2019b and the experiments are carried out on a 2.6 ghz intel core i5 7300u laptop with 8 gb ram. 6.3. experimental results a set of non-dominated solutions generated by moeas can be measured with different performance metrics available in the literature. in case, we don't have information regarding true pareto front, then coefficient of variation measure 𝐶𝑉 defined as follow (eq 6.4) is used: 𝐶𝑉 = 𝑚 𝜎 = 𝑚𝑒𝑎𝑛 (𝐽(𝑥), 𝑥 ∈ 𝑋 | 𝑉𝑎𝑟{𝐽(𝑥), 𝑥 ∈ 𝑋} (6.4) which allows to determine how much volatility, or risk, is assumed in comparison to the amount of expected return from investments and spacing metric (sp) suggested by [73] is a novel two-stage hybrid multi-objective differential evolution… 173 copyright ©2023 assa. adv. in systems science and appl. (2023) calculated with a relative distance measure between consecutive solutions in the obtained non-dominated set, as follows: 𝑆𝑃 = 1 |𝑆| − 1 �̅� − 𝑑 | | , (6.5) where 𝑑 = 𝑚𝑖𝑛( , )∈ , 𝐹(𝑠 ) − 𝐹(𝑠 ) is the 𝑙 distance between a point 𝑠 ∈ 𝑆 and the closest point of the pareto front approximation produced by the same algorithm, and �̅� the mean of the 𝑑 . from the results obtained in table 6.1, it is observed that mode-obl obtains better pareto-optimal fronts with better convergence and diversity comparing with nsga-ii and moea/d-de on 100-stocks problem. the differences among various algorithms are small compared with the results obtained in terms of distribution metric with slight superiority of mode-obl. note that moea/d-de showed good performance when optimizing the 500stock portfolio indicating that multi-objective algorithm-based decomposition approach has a stronger ability in solving large-scale optimization problem. a better insight into the nature of the found solutions can be obtained by analyzing fig 6.1 and fig 6.2. the plots presented in these figures show the efficient frontiers. as can been seen from these plots, the best results (risk/return) are generated by moea/d-de in 500stocks and quasi-consistent with the results obtained by mode-obl in 100-stocks. although nsgaii performs the worst, it can still generate distributed and satisfactory fronts. the experiments conducted on portfolio optimization problems with 100 and 500 stocks show that the multi-objective differential algorithm based ranking mutation combined with opposition-based learning presented an excellent performance comparing with other two multi-objective evolutionary algorithms and was a potential solution for this kind of problems. table 6.1. the comparison results of algorithms mode-obl, moea/d-de, nsga-ii dataset metrics statistics mode-obl moea/d-de nsga-ii 100 stocks risk best avg 0.0281e-04(1.0019) 0.0742e-04(1.0093) 0.0697e-04(1.0017) 0.0665e-04(1.0087) 0.1351e-04(1.0016) 0.6894e-04(1.0083) return best avg 1.1056(2.0036e-04) 1.0093 (0.0742e-04) 1.1036(2.004e-04) 1.0087(0.0665e-04) 1.1036(2.004e-04) 1.0083(0.0689e-04) coefficient of variation (cv) best worst avg 6.7795e-03 1.4993e-02 1.1725e-02 2.5005e-01 2.9526e-01 2.7482e-01 5.2053e-01 5.6592e-01 5.4216e-01 spacing metric (sp) best worst avg 6.9662e-06 1.4376e-05 1.0603e-05 8.9935e-06 1.9869e-05 1.2816e-05 6.6883e-05 1.4525e-04 9.8367e-05 500 stocks risk best avg 0.2107e-04(1.0023) 0.4890e-04(1.0053) 0.1890e04(1.0025) 0.5156e-04(1.0057) 0.3772e-04(1.0024) 0.6135e-04 (1.0043) return best avg 1.0067(1.0956e-04) 1.0053(0.4890e-04) 1.0071(1.0253e-04) 1.0057(0.5156e-04) 1.0051(1.2715 e-04) 1.0043(0.6135e-04) coefficient of variation (cv) best worst avg 1.3851e-02 3.5104e-01 2.8757e-01 1.5036e-02 3.4263e-01 2.6981e-01 5.6942e-01 8.9172e-01 6.7819e-01 spacing metric (sp) best worst avg 2.1776e-04 2.5627e-04 2.3606e-04 1.4915e-04 5.1583e-04 2.7524e-04 1.3038e-02 4.1017e-02 2.9133e-02 174 n. boukhari, m. a. nemmich, f. debbat, n. monmarché, m. slimane copyright ©0000 assa adv. in systems science and appl. (0000) fig 6.1 pareto frontier generated of 100 stocks problem fig. 6.2. pareto frontier generated of 500 stocks problem this work applied multi-objective differential algorithm based ranking mutation combined with opposition-based learning to solve the large-scale portfolio optimization problems. the experiments conducted on portfolio optimization problems with 100 and 500 stocks show that the proposed algorithm presented a better performance comparing with other two multi-objective evolutionary algorithms and was a potential solution for this kind of problems. 7. conclusion this paper proposed a hybridized multi-objective differential evolution variant based on opposition-based learning. the proposed algorithm mode-obl inherited two operators from the original mode: selection count dominance and crowding distance. ranking based mutation from mode-rmo is used to accelerate the convergence rate, then a new generated population passed to the second stage in order to maintain diversity of solution by the opposition-based learning technique. experiments on mathematical benchmark functions and financial portfolio optimization problem were performed to make an extensive comparison of mode-obl with a series of multi-objective algorithms and mode-obl was considered to be competitive with other methods. however, there is still a lot of room for improvement in the improvement aspect of mode, and this paper aims to explore feasible ways to improve the basic mode algorithm to handle a wide range of mop. future research will be directed to applying others selection techniques and verifying the mode-obl on more real-world problems such as telecommunication network design. a novel two-stage hybrid multi-objective differential evolution… 175 copyright ©2023 assa. adv. in systems science and appl. 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(2000). comparison of multiobjective evolutionary algorithms: empirical results, evolutionary computation, 8(2), 173–195. adv syst sci appl 2022; 03:70–83 published online at https://ijassa.ipu.ru. the analysis of big data centers performance anastasia v. gorbunova1*, vladimir m. vishnevsky1 1v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia abstract: the article proposes the approach for predicting the performance of big data processing center services. to simulate the process of parallel computing, the fork-join queuing system (qs) with a truncated pareto distribution of service time and three variants of distributions for the incoming flow is used. the number of devices in each of the fork-join subsystems of the qs can be more than one. the performance scores are the average system response time and its tail delay (99th percentile of the response time distribution). the approach is based on a combination of simulation modeling and neural networks, as one of the methods of machine learning. for at least 93% of the data used to test the proposed method in the course of a numerical experiment the approximation error of both studied characteristics does not exceed 5%, while the maximum approximation error is not more than 10%. keywords: parallel computing, data-intensive applications, big data, fork-join queueing system, average response time, artificial neural networks, machine learning methods 1. introduction according to an idc (international data corporation) report [19], digital data continues to grow relentlessly. thus, according to forecasts, by 2025 the global datasphere will increase to 175 zettabytes [19]. this is more than five times the amount of digital data available as of 2018. in the field of scientific research, this trend has led to the formation of the so-called “fourth paradigm”, along with experimental, theoretical and computational ones. its feature is the use of big data technologies. at the same time, science that uses big data is also characterized by interdisciplinarity, i.e., it combines modeling, theory and experiment. working with big data for both scientific and engineering calculations has become available mainly due to the development of cloud technologies, as well as the emergence of many scientific and technical applications that use the services of cloud providers for computing and data processing [17]. at the same time, the issue of improving application performance comes to be in the foreground. one of the main ways to improve the performance of various kinds of data center services is to parallelize calculations, which can occur both at the hardware and software levels [17]. the distribution of workloads can be based on the parallelization of the data or tasks themselves. by now, many frameworks have been developed with a parallel approach to speed up data-intensive applications. these include, for example, the google mapreduce [7] framework, hadoop mapreduce [2], apache spark [3] and many others [11]. in addition to addressing development-level performance and support environments for data-intensive applications, there is a natural challenge in practice to optimally allocate ∗corresponding author: avgorbunova@list.ru the analysis of big data centers performance 71 data center resources while meeting quality-of-service agreements. due to the lack of good tools to objectively assess their performance characteristics, the main strategy at present is to overprovision resources, half of which are idle most of the time [1, 5]. this leads to a significant increase in the cost of maintaining equipment (servers). most data center services for large-scale computing are based on fork-join structures, they are the main component of the data processing of parallel and/or distributed computing systems. a fork-join structure is understood as a system, upon entering which the task is divided into parts – independent subtasks, each of which is sent for service to the corresponding node. the task is considered completed after processing the last of its components. thus, the parallelization of data or tasks is modeled. in this case, the processing time of the entire task will be determined by the maximum of the processing times of its subtasks. from the point of view of performance and efficient use of resources, it is of interest to predict such characteristics of big data analysis systems as its average response time, including in the area of high loads. the equally important characteristic that needs to be evaluated is the tail of latency, by which we mean the 99th percentile of the system response time distribution [5]. this article proposes the new approach for estimation of the average response time and its tail delay for data centers. the approach using neural networks allows one to quickly and with acceptable accuracy obtain estimates of the characteristics of interest. the advantage of the approach compared to previously known ones is its universality, since there are no restrictions on the architecture of fork-join systems. as a result, it becomes possible to construct a more realistic parallel computing model based on practical data. in addition, you can achieve good quality estimates with a basic knowledge of neural networks or some other machine learning methods. at the same time, the speed of the trained neural network is comparable to performing calculations using a simple analytical formula, as opposed to the existing complex computational algorithms. the article is organized as follows: the section 2 describes the structure of the classical fork-join qs, as well as the main approaches to assessing its characteristics, the section 3 describes in detail the approach proposed by the authors of using artificial neural networks (ann), the section 4 describes the mathematical model of the parallel structure of the big data processing center, the section 5 presents the results of numerous numerical experiments, the section 6 discusses the features of the proposed approach and the prospects for further research. in conclusion, some results are summarized. 2. related work most data centers are based on fork-join structures. by a fork-join structure (system), we mean a queuing system, each node of which is an independent qs. the fork-join type system functions as follows (fig. 2.1): 1) at the moment of its arrival a request is instantly split into k (k ⩾ 2) subtasks, each of which, in its turn, is queued for future service; 2) after the end of the service, the subtask enters the synchronization buffer and remains there until all related subtasks are processed, i.e. only after that the request is considered to be served and can leave the system. a classical fork-join qs assumes that each subsystem has one server. we will expand this system to the most general case, when each qs can contain n ≥ 1 servers, i.e., each subsystem will be a qs of type g|g|n. thus, it will be possible to analyze the characteristics of the system in the conditions of allocation of additional resources. the results of the study will avoid redundancy in their selection. one of the first works devoted to the analysis of fork-join qs with subsystems of the form m |m |1 is the article [14]. here, exact expressions were obtained for estimating the copyright © 2022 assa. adv syst sci appl (2022) 72 a.v. gorbunova, v.m. vishnevsky . . . fork point join point ... ... ... fig. 2.1. fork-join system model. mean response time of the system in the case of two subsystems m |m |1, k = 2. in most of the following works on the subject of fork-join qs, various methods were used to obtain only approximations of the mean response time for k > 2. for a detailed overview of publications up to 2014, see [20]. among the main approximate methods for studying fork-join qs with k subsystems of the form m |m |1 or m |g|1 are: matrix-geometric approach, interpolation based on data obtained in extreme cases of high and low input loads, an approach using elements of the theory of order statistics [6], an empirical approach (construction of analytical formulas based on data obtained by simulation), etc. note that there are no exact solutions for the mean response time even in the case of exponential incoming and servicing flows for k > 2. the complexity of the analysis is explained by the dependence of the residence times of subtasks in subsystems due to their common moments of arrival. therefore, the analytical approach is based, as a rule, on the assumption of the absence of this dependence. as a result, the application of the estimates obtained is limited either by the number of subsystems k, or by the specific type of distribution of the servicing flow, or by insufficient approximation accuracy. this narrows the scope of the obtained analytical expressions. in particular, there are problems with modern systems due to heavy-tailed distributions. as for the case with subsystems of a more general form, i.e., g|g|1, there are very few studies on this topic and they have appeared mainly in recent times. fork-join cmo is a natural model for many real-life systems in various areas where parallel processing of tasks is carried out in order to improve performance. in this case, we are talking not only about telecommunication systems, but also about production areas (assembly of orders, logistics, etc.). therefore, despite a slight decrease in the activity of research in this direction, their relevance is still great. among the recent works devoted to this subject, it is worth noting the following. [18] proposes an approach for approximating the distribution of response time in the case of homogeneous k fork-join nodes of a qs of the form map |ph|1. numerical experiments have shown that the method is quite accurate mainly for tail delays. it is based on analytical results only for k = 1 and k = 2. the accuracy of the algorithm developed by the authors of the article depends on the choice of the constant c. the larger the value of this constant, the higher the accuracy of the resulting estimates. however, this significantly increases the complexity of the algorithm itself and, accordingly, the running time of its software implementation, since it contains operations with high-dimensional matrices. the quality of the tail delay approximation also depends on the choice of specific distributions, the load level, and the value of k. as k increases, the error increases, but, for example, for large values of k, the error does not exceed 15% in the case of the erlang distribution of service time. and for the hyperexponential service time distribution, the approximation error does not exceed 5%. copyright © 2022 assa. adv syst sci appl (2022) the analysis of big data centers performance 73 the article [22] explores two modes of operation of the fork-join qs with subsystems of the form m |g|1, in which requests can be split into subtasks, the number of which is less than or equal to the total number of servers in the qs. propositions about the asymptotic independence of the sojourn times of subtasks (when the number of servers tends to infinity) are proved, and an upper bound on the mean response time for this system is obtained. in [8] for the fork-join qs with two subsystems m |g|1 and non-homogeneous servers, the remaining service time is analyzed, which is necessary for the correct shutdown of the system in the event of a disconnection of the incoming flow. the asymptotic independence of the maximum residual service times for a high system load is proved. the article [15] proposes an approach for estimating the average response time and its tail delay fork-join qs with g|g|n subsystems as a model of some data center. in this case, each qs node is proposed to be considered either as a white box model, i.e., under conditions where the type of service time distribution is known. in the second variant (when there are no assumptions about the structure of the node, namely, about the type of service time distribution on it), it is assumed that we are dealing with a node model as a black box. in this case, it is proposed to estimate the mean values and variance of the response time of an individual node empirically. in other words, it is proposed to represent each node as a qs of the form g|g|1 by measuring the first two response times of the node of the system under study, which is technically feasible. in accordance with numerical experiments, the approximation error of the estimated response time characteristics in the region of high loads (above 0.8 for the average response time and above 0.9 for the tail delay) does not exceed 20% and 15%, respectively. in the articles [10] and [9] for the study of fork-join qs with subsystems of type m |m |1 in the first case and m |g|1 in the second, the approach based on neural networks is proposed. the results of numerical experiments showed a good quality of approximation, which prompted the authors of the article to conduct a new study in relation to estimating the performance of data processing centers with parallel structures, the nodes of which are modeled by general systems of the g|g|n type. 3. analysis method of fork-join qs using neural networks neural networks have become widespread due to the possibility of their application to solving poorly formalized problems. in addition, they are one of the main tools for big data analysis. it is currently difficult to name an area in which neural networks or other machine learning methods would not find their application. the mathematical model that is used to describe the operation of a big data processing center is a queuing system. machine learning methods and neural networks, in particular, have been applied to solving complex problems of queuing theory relatively recently, although this prospect was predictable. complex problems in the field of queuing theory are understood as problems that cannot be solved by using well-known analytical methods, or the solution is so complicated that actually obtaining numerical results using the developed algorithms is difficult to implement even taking into account the capabilities of modern computers. an overview of publications on the application of machine learning methods to solving problems in the field of queuing theory can be found in [21]. the idea of using neural networks is based on the possibility of using them to solve the problem of forecasting or, in fact, the problem of approximating a function of several variables. it is clear that the desired estimates of the mean response time and its tail delay depend, for example, on the values of such parameters as the system load, the number of subsystems k and servers in them, the intensities for the incoming flow and for the servicing time on the servers. therefore, the problem of finding estimates of the response time can be considered as a problem of approximating a function depending on the listed parameters. to interpolate a function, a set of values of the input parameters and the corresponding true values of the approximated quantities are required. there are several ways to get the copyright © 2022 assa. adv syst sci appl (2022) 74 a.v. gorbunova, v.m. vishnevsky true values. for example, with the help of simulation modeling or with the help of an exact analytical solution, which is not available in this case. as an alternative to the exact analytical solution, one can use, for example, the time-consuming algorithm proposed in [18] to obtain tail delays in the region where a high-precision solution exists. the need to combine neural networks with simulation or algorithmic solution arises from the significant time costs that are required when using only a simulation or only an algorithm. therefore, we limit the amount of input data for which it is necessary to use simulation modeling or a computational algorithm. then, on the resulting set, we train the neural network, which will allow us to predict the characteristics for any intermediate values of input parameters without restrictions on their number in the minimum time comparable to the time required for calculations using a simple analytical formula. for clarity, we describe the main stages of the proposed method: 1. obtaining by means of simulation modeling or a labor-intensive computational algorithm the values of the characteristics of the analyzed system of interest for a finite set of values from the given numerical intervals for the input parameters on which the performance of the system depends; 2. training an intellectual model on the data obtained using simulation by one of the machine learning methods in order to solve the forecasting problem; 3. almost instantaneous assessment of the desired performance characteristics for any other intermediate values of input parameters on the same numerical intervals using a trained intelligent model. among the main advantages of the described technique, universality can be distinguished, since a simulation modeling is sometimes the the only possible way to analyze complex systems. however, it is a resource-intensive tool. thanks to the use of neural networks, this disadvantage can be eliminated. 4. mathematical model of the parallel structure of the big data processing center let us describe in more detail the architecture of the fork-join qs, which we will use to model the operation of a data center. the system consists of k subsystems of type g|g|n (fig.4.2). a request entering the system is split into k subtasks, each of which is queued in the corresponding subsystem. servicing in subsystems occurs in the order of receipt of subtasks (first in first out, fifo discipline). each subsystem contains the same number of servers n, n ⩾ 1. selecting a multiline system will allow you to analyze the performance improvement of nodes by provisioning additional resources, as additional server replicas are modeled in this way. based on the results of the [15] study, we will restrict ourselves to the case with 3 server replicas, i.e. we will check the accuracy of the proposed approach for 1 ⩽ n ⩽ 3. as a service time distribution we will consider a truncated pareto distribution with a distribution function and density of the form [4]: f (x) = 1− (l/x)α 1− (l/h)α , f(x) = αlαx−α−1 1− (l/h)α , 0 ⩽ l ⩽ x ⩽ h, α > 0. the truncated pareto distribution is a three-parameter distribution. the parameter α is a shape parameter, and the parameters l and h are in fact the minimum and maximum values of a random variable with a given distribution. in [15], based on empirical data for the google system from [12], the following values for parameters of service time distribution are proposed: α = 2.0119, l = 2.14, h = 276.6, (4.1) copyright © 2022 assa. adv syst sci appl (2022) the analysis of big data centers performance 75 ... . . . fork point join point n 1 ... 1 ... n 1 ... n ... ... fig. 4.2. fork-join model of qs with subsystems of type g|g|n. i.e. the minimum service time is 2.14 ms, and the maximum is 276.6 ms. the k-th moment of the truncated pareto distribution is given by: mk = lα 1− (l/h)α α(lk−α −hk−α) α− k , α ̸= k. accordingly, we have that the average service time is approximately equal to 4.22 ms, the variance is 22.34, and the coefficient of variation, which is the ratio of the root of the variance to the average value of a random variable, will be approximately cv ≈ 1.22. to distribute the incoming flow, consider several options. since some studies allow the assumption of the poisson nature of the incoming flow (cv = 1) for data processing centers [13, 16], we will not exclude the case of an exponential distribution for the time interval between adjacent incoming requests. we also consider two more types of distribution for the incoming flow with a coefficient of variation different from unity. in particular, for the erlang distribution it is known that its coefficient of variation is always less than one. therefore, consider the erlang distribution with a density of the form f1(x) = β2xe−βx, x ⩾ 0, β > 0, (4.2) and with cv = 1/ √ 2 ≈ 0.7. we also consider a heavy-tailed distribution (cv > 1), namely the gamma distribution with density f2(x) = γk γ(k) xk−1e−γx, x ⩾ 0, γ > 0, k > 0 (4.3) for which the coefficient of variation cv = 1/ √ k for k = 0.25 will be equal to 2. 5. numerical experiment let’s check the performance and quality of the approximation of the proposed approach using neural networks. the first version of the data center architecture is a fork-join qs with a poisson input flow with an intensity λ inversely proportional to the average time between adjacent arrivals of requests a = 1/λ. the service time has a truncated pareto distribution with parameters from (4.1) with an average value of b = 4.22 ms. in fact, each subsystem is a qs of the form m |g|n. copyright © 2022 assa. adv syst sci appl (2022) 76 a.v. gorbunova, v.m. vishnevsky to train the neural network, we will use the input data from the table 5.1, where the load factor ρ = b na = λb n takes values on the segment [0.1, 0.9] with a step of 0.1, the number of subsystems k varies from 2 to 24, and the number of servers n from 1 to 3. the output of the neural network will be the mean response time and its 99th percentile. the values of the output parameters were obtained by using simulations carried out in the python software environment. table 5.1. input data for constructing estimates of the mean response time and its tail delay. no. 1 2 ... 9 10 ... 27 28 29 ... 621 ρ 0.1 0.2 ... 0.9 0.1 ... 0.9 0.1 0.2 ... 0.9 n 1 1 ... 1 2 ... 3 1 1 ... 3 k 2 2 ... 2 2 ... 2 3 3 ... 24 let’s split the data from the table 5.1 into two sets corresponding to the level of low and high loads, i.e. for ρ ∈ [0.1, 0.5] and ρ ∈ [0.6, 0.9] . and we will train the neural network on each of these two sets. in this case, the data set is divided in the ratio of 80% and 20% into training and test samples. the latter will be used for validation during training. by the standards of neural networks, it cannot be said that the number of data sets for irradiation is large. nevertheless, we will check the accuracy of the proposed approach under these conditions. a two-layer perceptron with two hidden layers of 10 neurons each with a logistic activation function ϕ(x) = 1/(1 + e−x) was chosen as the neural network structure. to improve the approximation accuracy, we will build two neural networks: the first one is for predicting the mean response time e[rk ], the second one is for the tail delay rk(99). training will be carried out by the method of back propagation of errors or the adam method in the python software environment, where simulation modeling was carried out. for an objective check of the forecast quality of trained neural networks, we will use intermediate data absolutely unknown to it from the 5.2 table. for a neural network trained in the area of low loads, consider the values for the prediction ρ ∈ [0.15, 0.55], and for a neural network trained in the area of higher loads ρ ∈ [0.65, 0.85]. table 5.2. intermediate input data for constructing estimates of the mean response time and its tail delay. no. 1 2 ... 8 9 ... 24 25 26 ... 552 ρ 0.15 0.25 ... 0.85 0.15 ... 0.85 0.15 0.25 ... 0.85 n 1 1 ... 1 2 ... 3 1 1 ... 3 k 2 2 ... 2 2 ... 2 3 3 ... 24 the figures 5.3 show the deviation of the estimates of the studied characteristics from their true values obtained using simulation modeling. for more detail, let’s analyze the relative approximation error pe = ∣∣∣∣ x̂j − xj xj ∣∣∣∣ · 100% for each set of intermediate input data, as well as its average value mape = 1 n n∑ j=1 ∣∣∣∣ x̂j − xj xj ∣∣∣∣ · 100%, where x̂j is an estimate of the characteristic under study (mean value or tail delay of the response time), obtained by using analytical formulas, and xj is the real value of one of the copyright © 2022 assa. adv syst sci appl (2022) the analysis of big data centers performance 77 estimated characteristics, obtained as a result of fork-join qs simulation , j = 1, n , n is the number of data sets in the sample for the estimation of the approximation error. 5 15 25 35 45 55 65 5 15 25 35 45 55 65 a n n simulation a) 5 25 45 65 85 105 125 145 165 5 25 45 65 85 105 125 145 165 a n n simulation b) fig. 5.3. the mean response time of a system with poisson input, a truncated pareto distribution of service time and load factor a) ρ ∈ [0.15, 0.55]; b) ρ ∈ [0.65, 0.85] let us determine the maximum, minimum and mean value of the relative approximation error (table 5.3). we will also plot relative error histograms in the order in which k, ρ, and n values are presented in the 5.2 table. 98.85% of the mean response time approximation table 5.3. approximation errors of estimates of the mean response time and its tail delay for a system with poisson input, obtained using a neural network on data sets from the table 5.2 estimated error types characteristic maximum pe, % minimum pe, % mape, % e[rk ], ρ ∈ [0.15, 0.55] 5.61689 0.02737 1.22500 e[rk ], ρ ∈ [0.65, 0.85] 9.93912 0.05324 3.28332 rk(99) 8.11459 0.00244 1.62527 0 1 2 3 4 5 6 a) 0 1 2 3 4 5 6 7 8 9 10 b) fig. 5.4. approximation errors for the mean response time of a system with poisson input, a truncated pareto distribution of service time and load factor a) ρ ∈ [0.15, 0.55]; b) ρ ∈ [0.65, 0.85]. error for ρ ∈ [0.15, 0.55] does not exceed 3% (fig.5.4a). for ρ ∈ [0.65, 0.85], the result of predicting the average response time by the neural network turned out to be somewhat worse. in this case, the number of approximation errors, which does not exceed 7%, is approximately 92.27% of their total number (fig.5.4b). in this case, the maximum approximation error does not exceed 10%. for the 99th percentile of the response time distribution (fig.5.5a), the situation is such that 97.54% of the approximation errors do not exceed 5% (fig.5.5b). now let’s analyze the case with the erlang distribution of service time from (4.2). to do this, consider a smaller training input set from the 5.4 table. the erlang distribution parameter is given by β = 2nρ/b. after training the neural network, we build the corresponding copyright © 2022 assa. adv syst sci appl (2022) 78 a.v. gorbunova, v.m. vishnevsky 35 85 135 185 235 285 335 385 435 35 85 135 185 235 285 335 385 435 a n n simulation a) 0 1 2 3 4 5 6 7 8 9 b) fig. 5.5. for a system with poisson input and a truncated pareto distribution of service time a) the 99th percentile of the response time distribution, b) approximation errors of the 99th percentile. table 5.4. input data for constructing estimates of the mean response time and its tail delay in the case of the erlang or gamma distribution for the incoming flow of requests. no. 1 2 ... 27 28 29 ... 405 ρ 0.1 0.2 ... 0.9 0.1 0.2 ... 0.9 n 1 1 ... 3 1 1 ... 3 k 2 2 ... 2 3 3 ... 16 estimates for the intermediate data from the 5.5 table. so, for the mean response time (fig.5.6a) 98.51% of errors does not exceed 5% (fig.5.7a), and for its tail delay (fig.5.6b), this percentage is already 99.40%, i.e., for only two predictive values, the relative error ranges from 5% to 7% (fig.5.7b). the maxima, minima, and average values of the approximation errors are presented in the 5.6 table. the maximum error in the approximation of e[rk ] and rk(99) does not exceed 7.8% and 6.3%, respectively, which is acceptable, especially considering that almost 99% of the errors are within 5%, which evidenced by the low values of their average values. table 5.5. intermediate input data for constructing estimates of the mean response time and its tail delay in the case of the erlang or gamma distribution for the incoming flow of requests. no. 1 2 ... 24 25 26 ... 336 ρ 0.15 0.25 ... 0.85 0.15 0.25 ... 0.85 n 1 1 ... 3 1 1 ... 3 k 2 2 ... 2 3 3 ... 16 table 5.6. approximation errors of estimates of the mean response time and its tail delay for a system with an erlang distribution of the incoming flow, obtained using a neural network on data sets from the table 5.5. estimated error types characteristic maximum pe, % minimum pe, % mape, % e[rk ] 7.82757 0.00518 1.75565 rk(99) 6.27911 0.01538 1.55339 if the time intervals between adjacent incoming requests have a gamma distribution (4.3) with parameters k = 0.25 (cv = 2) and γ = knρ/b, then after training the neural network on the input data from the table 5.4 we get the following result for intermediate data from the table 5.5. for the mean response time (fig.5.8a) 96.13% of the relative errors of their total number does not exceed 6% (fig.5.9a). for the tail delay (fig.5.8b), the percentage of errors that do copyright © 2022 assa. adv syst sci appl (2022) the analysis of big data centers performance 79 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 a n n simulation a) 35 75 115 155 195 235 275 315 355 35 75 115 155 195 235 275 315 355 a n n simulation b) fig. 5.6. for a system with an erlang distribution of the incoming flow and a truncated pareto distribution of service time, a) the mean response time, b) the 99th percentile of the response time distribution. 0 1 2 3 4 5 6 7 8 a) 0 1 2 3 4 5 6 7 b) fig. 5.7. system approximation errors with erlang distribution of the incoming flow and a truncated pareto distribution of the service time for a) the mean response time, b) the 99th percentile of the response time distribution. not exceed the threshold of 6% is 94.64% (fig.5.9b). in this case, the maximum relative approximation error does not exceed 9.8% in both cases (table 5.7). 5 45 85 125 165 205 5 45 85 125 165 205 a n n simulation a) 30 130 230 330 430 530 630 30 130 230 330 430 530 630 a n n simulation b) fig. 5.8. for a system with a gamma distribution of the incoming flow and a truncated pareto distribution of the service time a) the mean response time, b) the 99th percentile of the response time distribution. if we analyze the level of load reduction in case of allocation of additional servers, then the situation is as follows. regardless of the type of distribution for the incoming flow out of the three considered, if each node has two servers (n = 2) instead of one, the mean response time and tail delay are approximately halved, which was expected. if the number of servers is n = 3, then the average response time and its tail delay are reduced by an average of 65% compared to n = 1. the calculation took place for the high load level ρ ∈ [0.6, 0.9], since it copyright © 2022 assa. adv syst sci appl (2022) 80 a.v. gorbunova, v.m. vishnevsky 0 1 2 3 4 5 6 7 8 9 10 a) 0 1 2 3 4 5 6 7 8 9 10 b) fig. 5.9. system approximation errors with a gamma distribution of the incoming flow and a truncated pareto distribution of the service time for a) the mean response time, b) the 99th percentile of the response time distribution. table 5.7. approximation errors of estimates of the mean response time and its tail delay for a system with a gamma distribution of the incoming flow, obtained using a neural network on data sets from the table 5.5. estimated error types characteristic maximum pe, % minimum pe, % mape, % e[rk ] 9.79003 0.00344 2.34142 rk(99) 9.42462 0.01472 2.31488 is in this case that the allocation of additional resources to improve the quality of service is an urgent issue. for clarity, the figure 5.10 shows graphs of the mean response time at a fixed k depending on the load level for various values of n = 1, 2, 3. 0 40 80 120 160 200 240 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 a v er a g e re sp o n se t im e , m s load 1 2 3 a) 0 40 80 120 160 200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 a v er a g e re sp o n se t im e , m s load 1 2 3 b) 0 40 80 120 160 200 240 280 320 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 a v er a g e re sp o n se t im e , m s load 1 2 3 c) fig. 5.10. mean response time, if the time intervals between adjacent incoming requests have a) an exponential distribution, k = 24, b) an erlang distribution, k = 16, c) a gamma distribution, k = 16; 1 — n = 1, 2 — n = 2, 3 — n = 3. summarizing, we can draw the following conclusions. for 93% of the input data in the case of an approximation of the mean response time and for 96% of the data for the tail delay, the approximation error does not exceed 5%. since we considered the absolute values of the relative error, then the compensation of the approximation error in the amount of 5% can, in the case of positive estimates, actually lead to an overallocation of resources in the amount of 10%. however, compared to the 50% overprovisioning of today’s data centers, the 40% savings is a significant benefit. 6. discussion now let’s analyze the results in more detail. the use of neural networks as one of the methods of machine learning for predicting the mean response time and its tail delay makes it possible to obtain qualitative estimates without any restrictions on the structure of the system copyright © 2022 assa. adv syst sci appl (2022) the analysis of big data centers performance 81 being modeled. this is one of the main advantages of the proposed approach compared, for example, with analytical methods. based on the analysis of the publications available to date, a partial review of which was presented in this paper, we can draw the following conclusion. the use of analytical methods, if possible, is usually effective only for certain types of incoming or serving flows or only for some areas of the parameters of these distributions and, accordingly, the level of load (high or low), and also limits the structure of the system (number of servers or buffer capacity in subsystems). on the other hand, the use of neural networks requires preliminary data preparation for its training. in this case, simulation modeling was used. by itself, simulation modeling makes it possible to obtain practically exact values of the desired characteristics. however, it requires serious computational resources and time costs, so its use in its pure form is not rational. machine learning significantly narrows the range of parameters for which simulation is required. due to this, the time spent on its implementation is significantly reduced. at the same time, after training the neural network, it becomes possible to obtain estimates of acceptable quality for any number of intermediate values of the input parameters without any time spent on this. training of neural networks aimed at a satisfactory forecast quality does not require too deep immersion in this area due to the fact that many algorithms for their training are qualitatively implemented in various computing environments, there are ready-made libraries, etc. nevertheless, if it is necessary to obtain more accurate forecasts, for example, in comparison with those presented in this paper, it is possible to involve relevant specialists. among the many listed advantages of the proposed approach, it also has some weaknesses in the context of making a decision to allocate additional resources to compensate for forecast errors. despite the low approximation error, which on average does not exceed 5% for 90% of the data, and in some cases even 3%, there are still a small number of errors that exceed, although not significantly, these thresholds. and it is impossible to predict in advance for which specific values (ranges of values) of the input parameters this will happen. this fact is not affected by the type of distribution, nor by its coefficient of variation, nor by the level of loading, etc., as is the case with approximate analytical approaches. the main reasons for this phenomenon, apparently, are the features of ann training, the algorithm chosen for this, the size of the training data set, the architecture of the neural network (the number of hidden layers, the number of neurons, activation functions, etc.). therefore, the continuation of research in this direction, aimed at compensating for this shortcoming, seems to be an urgent task. 7. conclusion the paper proposed the approach to assess the performance of big data processing center services. using a combination of simulation modeling with neural networks, estimates were obtained for the mean response time and its tail delay (99th percentile of the response time distribution). the advantage of the approach compared to previously known ones is its universality, since there are no restrictions on the architecture of fork-join systems. the speed of the trained neural network is comparable to performing calculations using a simple analytical formula, as opposed to the existing complex computational algorithms. at the same time, the approximation quality is quite high and does not depend on the architecture of the mathematical model used. acknowledgements the reported study was obtained within rfbr grant (project no. 19-29-06043). copyright © 2022 assa. adv syst sci appl (2022) 82 a.v. gorbunova, v.m. vishnevsky references 1. alesawi, s., nguyen, m., che, h. & singhal, a. (2015). tail latency prediction for datacenter applications in consolidated environments, in proc. ieee international conference on computing, networking and communications (icnc), honolulu, hi, usa, 265–269. 2. apache: apache hadoop. (2022, april). [online]. available: https://hadoop.apache.org 3. apache spark. (2022, april). [online]. available: https://spark.apache.org 4. harchol-balter, m. (2013). modeling and design of computer systems: queueing theory in action. cambridge, u.k.: cambridge univ. press. 5. blake, g. & saidi, a. g. (2015). where does the time go? characterizing tail latency in memcached. in proc. ieee international symposium on performance analysis of systems and software (ispass), 21–31. 6. david, h. a. & nagaraja h. n. (2003) order statistics. hoboken, nj: john wiley & sons. 7. dean, j. & ghemawat, s. 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(2018). forktail: a black-box fork-join tail latency prediction model for user-facing datacenter workloads, in proc. 27th int. symp. high-perform. parallel distrib. comput., 206–217. 17. de oliveira, d. c., liu, j. & pacitti, e. (2019). data-intensive workflow management: for clouds and data-intensive and scalable computing environments. synthesis lectures on data management, 14, 1–179. 18. qiu, z., prez, j. f. & harrison, p. (2015). beyond the mean in fork-join queues: efficient approximation for response-time tails, performance evaluation, 91, 99–116. 19. reinsel, d., gantz, j. & rydning, j. (2018). idc report: the digitization of the world from edge to core. idc white paper no.us44413318. framingham, ma: international data corporation. 20. thomasian, a. (2014). analysis of fork/join and related queueing systems, acm computing surveys (csur), 47(2), 17:1–17:71. copyright © 2022 assa. adv syst sci appl (2022) the analysis of big data centers performance 83 21. vishnevsky, v. m. & gorbunova, a. v. (2022). application of machine learning methods to solving problems of queuing theory, communications in computer and information science, in print. 22. wang, w., harchol-balter, m., jiang, h., scheller-wolf, a. & srikant, r. (2019). delay asymptotics and bounds for multitask parallel jobs, queueing systems, 91, 207–239. copyright © 2022 assa. adv syst sci appl (2022) introduction related work analysis method of fork-join qs using neural networks mathematical model of the parallel structure of the big data processing center numerical experiment discussion conclusion adv syst sci appl 2022; 01:65–79 published online at https://ijassa.ipu.ru. a model for analysis of gene expression in the cell involving protein degradation amit sharma*, neeru adlakha applied mathematics & humanities department, s.v. national institute of technology, surat-395007, gujarat, india abstract: the gene expression is responsible for the initiation sustenance and termination of various processes taking place in a cell in order to maintain the structure and the function of the cell. in the present paper, a model is proposed to analyze the gene expression in terms of temporal variation of dna, mrna and protein concentration along with the stability of molecular processes taking place in the cell. the model incorporates the parameters like replication, transcription, reverse transcription, translation and degradation of the protein. the model is obtained in terms of a system of differential equations along with initial conditions based on the physical and physiological conditions of the problem. the analytical solution is obtained for the model. the results have been obtained for a dataset of e. coli tjk16 strain and e. coli b/r strain, and employed to study the relationships of the parameters involved with the concentration of dna, mrna and protein. the asymptotic stability analysis at the equilibrium point has been performed for the molecular processes in the cell. the information generated from the model can be useful for developing the protocols for healthcare. keywords: differential equations, dna, mrna, protein, gene expression, tjk16, b/r 1. introduction mathematical modeling is indispensable now-a-days which plays a vital role in better understanding the complexity of any system related to the real world. the real-world problem may arise in different fields as business, agriculture, biology, etc. many models have been developed to unravel the complexity of the system of real-world problems [5], [16], [19]. a cell is considered as the primordial structural, functional and biological unit in all living organisms that can perform all the functions of life and can be classified as eukaryotic cell and prokaryotic cell. to understand the cell as a system, it is important to understand the molecular mechanisms of the cell. there is a nucleus in the cell which controls the structure and function of the cell. the cell nucleus contains the dna and mrna which controls protein synthesis [2]. during the synthesis of functional gene product protein, the genetic information in the cell flows from dna to mrna and mrna to protein. for the growth of the cells, the process has different rates of gene expression in an individual cell due to the different rates of transcription and translation as per the requirements of the cell for a particular protein [1], [9]. in the eukaryotic and prokaryotic cells, the gene expression is stochastic in nature and the stochastic models can be moderated at the transcriptional and translational levels [18]. also, it is shown experimentally that an increase in the stochasticity in the transcription process leads to an increase in cell variability for protein synthesis [3]. in the view of the central dogma, the needed information for producing protein is ∗corresponding author: amitsharmajrf@gmail.com 66 a. sharma, n. adlakha contained in dna sequence is then decoded into mrna and thereafter the information in mrna molecules is transformed to produce the sequence of amino acids [7][8]. further, the complexity of the bacterial cell along with the macromolecular composition of salmonella typhimurium involved in the protein synthesis was unravelled and it was discovered that at a given temperature, the cellular contents of dna, rna and protein depends on the growth rate of bacterial cell [12]. the growth of a cell depends on the growth of its cellular contents and a complete study about cell growth and the cellular contents has been reported in the literature [4]. in the gene expression profile of temporal dynamics and spatial patterning, many mathematical models have been reported in the literature which shows how modeling plays an important role to study the molecular mechanism of cells and their function [10] [11]. a differential equation model to study the complexity of gene expression is reported in the literature by using two methods; minimum weight solutions to linear equations (mwsle) and fourier transform for stable system (ftss) with the set of temporal data [5]. further, a comprehensive study is elaborated on the time delay model for gene expression. recently, a system of differential equations on mrna translocation in the ribosome is developed and employed to test the consistency and inconsistency of the model with different data set [19]. in a probabilistic manner, a markov chain model of gene expression has been reported in the literature considering dna, mrna and protein as the states of the system [17]. a differential equation-based model for gene expression considering michaelis-menten’s mechanism is reported in the literature to unravel the complexity of gene expression. in this study, the gene expression based on michaelis-menten’s mechanism involving the processes replication, transcription, reverse transcription and translation has been reported. they assume that there is no degradation of proteins and obtained the analytical solution and performed the stability analysis of the model [13]. in terms of fuzzy, a fuzzy system model is reported considering functional gene products [14]. the gene expression based on michaelis-menten’s mechanism involving the processes replication, transcription, reverse transcription and translation has been studied. the authors assume that there is no degradation of proteins and obtained the analytical solution and performed the stability analysis of the model. recently, a delay differential equation-based model on gene expression was reported in the literature which is more realistic to understand the functional gene products [15]. in this paper, an attempt has been made to model gene expression involving protein degradation. here, the different rates of replication, transcription, reverse transcription, translation and protein degradation of both strains tjk16 and b/r, in a single cell, are incorporated in the model. the e. coli tjk16 is a bacterial strain and due to mutationally altered control of replication initiation, all genes have a lower concentration than their wild type b/r parent. these genes produce the bulk of stable rna and mrna which are more active in strain tjk16 in the comparison of strain b/r. therefore, this strain is useful for studying the variation in the concentrations of dna, mrna and protein in the cell. secondly, the e. coli strain b/r, parent of strain tjk16, is also used for computing the result [6]. initially, the dna concentration of strains tjk16 and b/r is 25 micrograms and 10 micrograms, respectively. the exact solution is obtained for the system of differential equations. the model is proposed for both retroviruses and normal cells. the asymptotic stability of the system at the equilibrium point is also analyzed and it is demonstrated by the vector plot. 2. nomenclature the abbreviations used in the paper are as follows: dna: deoxyribonucleic acid rna: ribonucleic acid tjk16: bacterial strain with genotype thya deob b/r: bacterial strain e.coli: escherichia coli copyright © 2022 assa. adv syst sci appl (2022) a model for analysis of gene expression in the cell involving protein 67 w(t): concentration of dna in the cell at time t (second) x(t): concentration of mrna in the cell at time t (second) p(t): concentration of protein in the cell at time t (second) k0: rate of replication k1: rate of transcription k2: rate of the reverse transcription k3: rate of translation k4: rate of protein degradation 3. mathematical model and method in the present model, it is assumed that replication, transcription, reverse transcription, translation and degradation of protein are taking place in the cell. the graphical representation of the model is shown in fig.3.1. fig. 3.1. graphical representation of modelling of gene expressions. the following assumptions have been made: 1. the time delay in the processes of gene expression is considered to be negligible. 2. the constant temperature is maintained so that there is no effect of temperature on replication, transcription, reverse transcription, translation and protein degradation process in the gene. 3. the constant rates of the processes of gene expression are taken which lie in [0,1]. 4. a single cell is considered where replication occurs in the cell. the michaelis-menten’s mechanism is used to develop a differential equation-based model of gene expression. the motive to use the proposed mechanism is the applicability copyright © 2022 assa. adv syst sci appl (2022) 68 a. sharma, n. adlakha of enzyme-catalyzed (generally proteins) biochemical reactions and these enzymes (biomolecules: dna, mrna and protein) take place in the cell which are synthesized by ribosomes. in 1913, michaelis-menten proposed that the binding of the substrate (the molecule acted upon) and the enzyme is reversible and derived a kinetic model to study the kinetic behaviour of the biochemical reaction systems. the advantages of michaelismenten’s mechanism are simplicity and broad applicability in the field of biochemistry. the mathematical description of enzyme behaviour is an amazing tool which is useful in the analysis of enzyme activity. the mathematical model for these processes, which is based on michaelis-menten’s mechanism, is given below [2][19]: w′(t) = k0w(t)− k1w(t) + k2x(t) (3.1) x′(t) = k1w(t)− k2x(t)− k3x(t) (3.2) p′(t) = k3x(t)− k4p(t) (3.3) where w(t), x(t) and p(t) represent the concentration of dna, mrna and protein respectively. also, k0, k1, k2, k3 and k4 are rates of replication, transcription, reverse transcription, translation and degradation of protein, respectively. a sufficient condition of the stability of zero equilibrium for the system (3.1)− (3.3) is that all its eigenvalues λ1 = −k4, λ2 = −a+ √ ∆ 2 , λ3 = −a− √ ∆ 2 (3.4) have negative real parts, where a = (−k0 + k1 + k2 + k3), the discriminant ∆ = a2 − 4(−k0k2 − k0k3 + k1k3), and −(k0 − k1)(k2 + k3)− k1k2 6= 0. further, in view of the stability of the proposed system, three cases are considered as case (a): ∆ > 0, case (b): ∆ = 0, case (c): ∆ < 0. in this case, we have found different positive values of ∆ in the form of 0.5756, 0.9995, 0.7122, 1.0237, 2.9502 and 1.1767 for six different copy numbers of k0, k1, k2 and k3 where the data has been taken from table (5.1). further, we focus on the data of table (5.2) where the observed value of ∆ is 0 for all six different copy numbers of k0, k1, k2 and k3. thus, it is evident that the value of ∆ is positive and cannot be negative in the closed interval [0, 1] for all values of k0, k1, k2 and k3. further, it is also evident that the rates of replication, transcription, reverse transcription and translation can not be negative as per the cell physiology. so, we have discarded this case. thus, there are five conditions for stability of the system as 1. if λ1 < 0 < λ2 < λ3, then the system is unstable, 2. if λ1 < λ3 < 0 < λ2, then the system is unstable, 3. if λ3 < λ1 < λ2 < 0, then the system is stable, 4. if λ1 < λ2 = λ3 < 0, then the system is stable, 5. if λ2 = λ3 < λ1 < 0, then the system is stable. further, the stability of the system of linear differential equations is demonstrated. the three-dimensional vector plot of stability of the given system is shown in fig. 3.2 and fig. 3.3, respectively. the points of stability obtained above are for the data set available in the literature for a particular set of conditions. the rate of these processes like replication, transcription, reverse transcription, translation and protein degradation vary in response to environmental factors and the requirement of the organism. 4. solution initially, it is assumed that there is some concentration of dna in the cell and no concentration of both mrna and protein is present in the cell. thus, the initial conditions copyright © 2022 assa. adv syst sci appl (2022) a model for analysis of gene expression in the cell involving protein 69 fig. 3.2. three-dimensional vector plot of the system of linear differential equations converging to the origin. −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 mrna stable − node dna p r o te in fig. 3.3. three-dimensional vector plot of the system of linear differential equations where arrows showing the direction (blue colour) and streamlines (red colour) converging to the equilibrium point (0, 0, 0) which displays the stability of the system as a stable-node. the eigenvalues of the system are (−0.7655,−0.1016,−0.8603). for differential equations model of dna, mrna and protein are given below: w(0) = w0 > 0, x(0) = 0, p(0) = 0. using these initial conditions, the solution is obtained for both the cases (a) and (b) as follows: 4.1. case (a): when ∆ > 0, the concentration of dna, mrna and protein in the cell is expressed as w(t) = w0 λ3 − λ2 [ eλ3t(λ3 + k2 + k3)− eλ2t(λ2 + k2 + k3) ] , (4.5) x(t) = w0k1 λ3 − λ2 (eλ3t − eλ2t), (4.6) copyright © 2022 assa. adv syst sci appl (2022) 70 a. sharma, n. adlakha −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 d n a v s m r n a (i) asymptotically stable − node −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 m r n a v s d n a (ii) unstable − saddle node −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 −1 −0.5 0 0.5 m r n a v s p ro te in (iii) asymptotically stable − node −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 p ro te in v s m r n a (iv) unstable − saddle node −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 −1 −0.5 0 0.5 p ro te in v s d n a (v) asymptotically stable − node −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 d n a v s p ro te in (vi) unstable − saddle node fig. 3.4. the figures in (i): dna vs mrna, (iii): mrna vs protein and (v): protein vs dna are showing the stability behaviour as asymptotically stable-node, where arrows show the direction (blue colour) and streamlines (red colour) are converging towards the equilibrium point (0, 0, 0), and their eigenvalues are (−0.1016,−0.8603), (−0.7655,−0.6259) and (−0.7655,−0.3360) respectively. the figures in (ii): mrna vs dna, (iv): protein vs mrna and (vi): dna vs protein are showing stability behaviour as an unstablesaddle node where arrows showing the direction (blue colour) and streamlines (red colour) are diverging from the equilibrium point (0, 0, 0), and the eigenvalues for these figures are (0.8394,−0.1041), (0.9009,−0.5318), and (0.5072,−0.5072) respectively. the initial concentrations of dna and mrna, in figures (iii), (iv), (v) and (vi), are assumed to be zero. p(t) = w0k1k3 λ2λ3(λ3 − λ2) eλ1t [ e(λ3−λ1)t − 1 λ3 − λ1 − e(λ2−λ1)t − 1 λ2 − λ1 ] . (4.7) 4.2. case (b): when ∆ = 0, the concentration of dna, mrna and protein in the cell is expressed as w(t) = w0e λ2t [1 + (λ2 + k2 + k3)t] , (4.8) x(t) = w0k1te λ2t, (4.9) p(t) = w0k1k3 λ2 − λ1 eλ1t [ e(λ2−λ1)t ( t− 1 λ2 − λ1 ) + 1 λ2 − λ1 ] . (4.10) 5. results the rate of replication, transcription, reverse transcription, translation and degradation will depend on the circumstances and capacity of the cell depending on different conditions and therefore k0, k1, k2, k3 and k4 can be assigned different values. the data of the strains tjk16 copyright © 2022 assa. adv syst sci appl (2022) a model for analysis of gene expression in the cell involving protein 71 and b/r is used to compute the results [6]. generally, the values of k0, k1, k2, k3 and k4 lie in [0, 1] [1][9]. as a special case, we assign values, given in table (5.1), to all the abovementioned rates such that it satisfies the condition of stability. 5.1. case (a) firstly, for case (a), ∆ > 0, the nature of the concentration profiles of dna, mrna and protein is shown in the form of fig. 5.5, 5.6, 5.7 and table (5.1). the fig. 5.8 (both left and table 5.1. concentration profiles of six copies of dna (w(t)) mrna (x(t))and protein (p(t)) for case (a) ∆ > 0 of the strain tjk16 and b/r at time t = 180, along with the rates, k0, k1, k2, k3 and k4. copy 1 copy 2 copy 3 copy 4 copy 5 copy 6 w(t) tjk16 1.96e-07 6.62e-16 7.89e-22 5.95e-06 4.12e-06 9.33e-08 (µg) b/r 7.86e-08 2.65e-16 3.15e-22 2.38e-06 1.65e-06 3.73e-08 x(t) tjk16 1.79e-07 6.67e-16 1.61e-21 1.10e-05 2.40e-06 1.40e-07 (µg) b/r 7.17e-08 2.67e-16 6.44e-22 4.39e-06 9.58e-07 5.61e-08 p(t) tjk16 9.97e-08 8.44e-16 2.65e-13 4.52e-06 4.95e-06 8.76e-08 (µg) b/r 3.99e-08 3.38e-16 1.06e-13 1.81e-06 1.98e-06 3.50e-08 replication rate k0 0.1425 0.3232 0.1084 0.0848 0.2981 0.2477 transcription rate k1 0.4785 0.7696 0.8301 0.7630 0.7722 0.8631 r. transcri. rate k2 0.2568 0.2341 0.2142 0.3242 0.6690 0.3397 translation rate k3 0.3691 0.7404 0.4756 0.1689 0.7452 0.3382 protein deg. rate k4 0.7655 0.7952 0.1869 0.4898 0.4456 0.6463 time (sec.) 180 180 180 180 180 180 0 20 40 60 80 100 120 140 160 180 0 5 10 15 20 25 time d n a tjk16 copy 1 copy 2 copy 3 copy 4 copy 5 copy 6 0 20 40 60 80 100 120 140 160 180 0 5 10 15 20 25 time d n a b/r copy 1 copy 2 copy 3 copy 4 copy 5 copy 6 fig. 5.5. the graph between dna concentration (µg) and time for the strain tjk16 and b/r of the case (a) ∆ > 0. right figures) shows the protein concentration profiles for the strain tjk16 and b/r of the case (a) with protein degradation and without protein degradation, respectively. copyright © 2022 assa. adv syst sci appl (2022) 72 a. sharma, n. adlakha 10 0 10 1 10 2 0 5 10 15 time m r n a tjk16 copy 1 copy 2 copy 3 copy 4 copy 5 copy 6 10 0 10 1 10 2 0 5 10 15 time m r n a b/r copy 1 copy 2 copy 3 copy 4 copy 5 copy 6 fig. 5.6. the graph between mrna concentration (µg) and time for the strain tjk16 and b/r of the case (a) ∆ > 0. 0 20 40 60 80 100 120 140 160 180 0 2 4 6 8 10 12 14 time p ro te in tjk16 copy 1 copy 2 copy 3 copy 4 copy 5 copy 6 0 20 40 60 80 100 120 140 160 180 0 2 4 6 8 10 12 14 time p ro te in b/r copy 1 copy 2 copy 3 copy 4 copy 5 copy 6 fig. 5.7. the graph between protein concentration (µg) and time for the strain tjk16 and b/r of the case (a) ∆ > 0. 5.2. case (b) secondly, for the case (b), ∆ = 0, the parameters justify this condition when the rates of replication and reverse transcription are zero and the rate of transcription and rate of translation are equal to each other. the nature of the concentration profiles of dna, mrna and protein are shown in the form of fig. 5.9, 5.10, 5.11 and in table 5.2. the fig. 5.12 (both left and right figures) shows the protein concentration profiles for the strain tjk16 and b/r of case (b) with protein degradation and without protein degradation, respectively. further, the comparison of the general behaviour of concentration profiles of dna, mrna and protein for case (a) and copy number 1 respectively, is shown in fig. 5.13. also, for case (b) and copy number 1, the comparison of the general behaviour of concentration profiles of dna, mrna and protein respectively, can be predicted as we adjust the value of copyright © 2022 assa. adv syst sci appl (2022) a model for analysis of gene expression in the cell involving protein 73 0 20 40 60 80 100 120 140 160 180 0 2 4 6 8 10 12 14 time p ro te in tjk16 0 20 40 60 80 100 120 140 160 180 0 50 100 150 200 250 time p ro te in tjk16 0 20 40 60 80 100 120 140 160 180 0 2 4 6 8 10 12 14 time p ro te in b/r copy 1 copy 2 copy 3 copy 4 copy 5 copy 6 0 20 40 60 80 100 120 140 160 180 0 50 100 150 200 250 time p ro te in b/r fig. 5.8. the graph between protein concentration (µg) and time for the case (a) ∆ > 0 with protein degradation (both figures left) and without protein degradation (both figures right) of both strains tjk16 and b/r. this shows the comparison of protein concentrations for the strain tjk16 and b/r of the case (a). table 5.2. concentration profiles of six copies of dna (w(t)) mrna (x(t))and protein (p(t)) for the case (a) ∆ = 0 of the strain tjk16 and b/r at time t = 180, along with the rates k0 = k2 = 0 and k1 = k3 respectively, and k4 is greater than all the rates k0, k1, k2, k3. copy 1 copy 2 copy 3 copy 4 copy 5 copy 6 w(t) tjk16 4.13 0.68 0.11 0.02 3.08e-03 5.10e-04 (µg) b/r 1.65 0.27 0.05 0.01 1.23e-03 2.04e-04 x(t) tjk16 7.44 2.46 0.61 0.13 2.78e-02 5.51e-03 (µg) b/r 2.98 0.98 0.24 0.05 1.11e-02 2.20e-03 p(t) tjk16 0.78 0.51 0.19 0.06 1.45e-02 3.45e-03 (µg) b/r 0.31 0.21 0.08 0.02 5.79e-03 1.38e-03 replication rate k0 0.00 0.00 0.00 0.00 0.00 0.00 transcription rate k1 0.01 0.02 0.03 0.04 0.05 0.06 r. transcri. rate k2 0.00 0.00 0.00 0.00 0.00 0.00 translation rate k3 0.01 0.02 0.03 0.04 0.05 0.06 protein deg. rate k4 0.10 0.11 0.12 0.13 0.14 0.15 time (sec.) 180 180 180 180 180 180 parameter k3 but the change in the parameter k3 does not follow the condition of the case (b) ∆ = 0. copyright © 2022 assa. adv syst sci appl (2022) 74 a. sharma, n. adlakha 0 20 40 60 80 100 120 140 160 180 5 10 15 20 25 time d n a tjk16 copy 1 copy 2 copy 3 copy 4 copy 5 copy 6 0 20 40 60 80 100 120 140 160 180 0 5 10 15 20 25 time d n a b/r copy 1 copy 2 copy 3 copy 4 copy 5 copy 6 fig. 5.9. the graph between dna concentration (µg) and time for the strain tjk16 and b/r of the case (b) ∆ = 0. 0 20 40 60 80 100 120 140 160 180 0 2 4 6 8 10 12 time m r n a tjk16 copy 1 copy 2 copy 3 copy 4 copy 5 copy 6 0 20 40 60 80 100 120 140 160 180 0 2 4 6 8 10 12 time m r n a b/r copy 1 copy 2 copy 3 copy 4 copy 5 copy 6 fig. 5.10. the graph between mrna concentration (µg) and time for the strain tjk16 and b/r of the case (b) ∆ = 0. 6. discussion figs. 5.5, 5.6 and 5.7 show the variation of the concentrations of dna, mrna, and protein with degradation for the strains tjk16 and b/r, respectively. in fig. 5.5, for strain tjk16 (left figure), it is observed that the concentration of all six copy numbers of dna decreases sharply with respect to time from t = 0 up to t = 29, t = 15, t = 10, t = 30, t = 35 and t = 25, respectively and thereafter, decreases gradually up to t = 180. further, for strain b/r (right figure) the concentration of all six copy numbers of dna decreases sharply with respect to time from t = 0 up to t = 21, t = 11, t = 7, t = 19, t = 25 and t = 17, respectively and thereafter, decreases gradually up to t = 180. in fig. 5.6, for both strains, tjk16 (left figure) and b/r (right figure), the concentrations of copy number 1 of mrna increases copyright © 2022 assa. adv syst sci appl (2022) a model for analysis of gene expression in the cell involving protein 75 0 20 40 60 80 100 120 140 160 180 0 0.5 1 1.5 2 2.5 3 3.5 time p ro te in tjk16 copy 1 copy 2 copy 3 copy 4 copy 5 copy 6 0 20 40 60 80 100 120 140 160 180 0 0.5 1 1.5 2 2.5 3 3.5 time p ro te in b/r copy 1 copy 2 copy 3 copy 4 copy 5 copy 6 fig. 5.11. the graph between protein concentration (µg) and time for the strain tjk16 and b/r of the case (b) ∆ = 0. 0 20 40 60 80 100 120 140 160 180 0 1 2 3 4 time p ro te in tjk16 0 20 40 60 80 100 120 140 160 180 0 5 10 15 20 25 time p ro te in tjk16 0 20 40 60 80 100 120 140 160 180 0 1 2 3 4 time p ro te in b/r 0 20 40 60 80 100 120 140 160 180 0 5 10 15 20 25 time p ro te in b/r copy 1 copy 2 copy 3 copy 4 copy 5 copy 6 fig. 5.12. the graph between protein concentration (µg) and time for the case (b) ∆ = 0 with protein degradation (both figures left) and without protein degradation (both figures right) of both strains tjk16 and b/r. this shows the comparison of protein concentrations for the strain tjk16 and b/r of case (b). sharply from t = 0 up to t = 3, then decreases sharply from t = 3 up to t = 7 and thereafter decreases gradually up to t = 180 for both strains. the concentration of copy number 4 of both strains increases sharply from t = 0 up to t = 3, decreases sharply from t = 3 up to t = 10 for tjk16 and up to t = 6 for b/r strain, respectively, and thereafter decreases gradually up to t = 180 for both strains. the concentration of the remaining copy numbers 2, 3, 5 and 6 of both strains increases sharply from t = 0 to t = 2 respectively, decrease sharply from t = 2 tot = 9, t = 9, t = 5 and t = 11 for tjk16 (left figure), and for b/r strain (right figure), decreases sharply from t = 2 to t = 9, t = 8, t = 18 and t = 20, respectively and then decreases gradually up to t = 180 for both strains. in fig. 5.7, for both strains tjk16 and copyright © 2022 assa. adv syst sci appl (2022) 76 a. sharma, n. adlakha 0 20 40 60 80 100 120 140 160 180 0 5 10 15 20 25 time c on ce nt ra tio n tjk16 dna mrna protein 0 20 40 60 80 100 120 140 160 180 0 5 10 15 20 25 time c on ce nt ra tio n b/r dna mrna protein fig. 5.13. graphical representation of comparison of the concentrations of dna, mrna and protein of copy number 1 for the strain tjk16 (left figure) and b/r (right figure) of the case (a) ∆ > 0. b/r, the protein concentration of copy number 2 increases from t = 0 up to t = 3, thereafter decreases sharply up to t = 16 for tjk16 strain and up to t = 11 for b/r strain, respectively and then gradually decreases up to t = 180 for both strains. the concentration of protein of copy number 4 increases with time from t = 0 to t = 6 for both strains, falls sharply up to t = 26 seconds for tjk16 strain and up to t = 15 for b/r strain, respectively and thereafter gradually decreases up to t = 180 for both strains. the concentration of protein in remaining copy numbers 1, 3, 5 and 6 increases from t = 0 to t = 5 for both strains, respectively, decrease sharply up to t = 22, t = 25, 37 and t = 25 for tjk16 strain, respectively and up to t = 13, t = 20, 27 and t = 16 for b/r strain, respectively, and then decreases gradually up to t = 180, respectively for both strains. also, in fig. 5.8, the comparison of the concentration profile of protein with degradation and without degradation is shown for both strains tjk16 and b/r. the concentration of protein without degradation (right upper figure) of all copy numbers 1, 2, 3, 4, 5 and 6 increases sharply from t = 0 up to t = 13, t = 13, t = 9, t = 21, t = 45 and t = 24, respectively and thereafter the concentration of protein for all copy numbers becomes constant. in fig. 5.8 (right upper figure), there is no fall in the protein concentration due to the absence of protein degradation. further, for strain b/r, the concentration of protein without degradation (right lower figure) of all copy numbers 1, 2, 3, 4, 5 and 6 increases sharply from t = 0 up to t = 6, t = 10, t = 4, t = 6, t = 30 and t = 12, respectively and thereafter the concentration of protein for all copy numbers becomes constant. the concentration profile of protein with degradation for both strains tjk16 and b/r is shown as the left upper figure and left lower figure of fig. 5.8, respectively and their concentration behaviour has been shown in fig. 5.7. in this way, the concentration of all six copy numbers first increases sharply and thereafter decreases sharply and then decrease gradually up to t = 180, respectively. this is due to the presence of protein degradation which reduces the protein concentration. figs. 5.9, 5.10 and 5.11 show the variation of the concentrations of dna, mrna, and protein with degradation for the strains tjk16 and b/r, respectively. in fig. 5.9, the dna concentration of copy number 1 decreases sharply from t = 0 up to t = 180 for strain tjk16 (left figure). a similar behaviour in the concentration of the strain b/r (right figure) is found for copy number 1. the dna concentration of remaining copy numbers decrease sharply from t = 0 up to t = 127, t = 108, t = 82, t = 66 and t = 54, respectively for tjk16 and up to t = 160, t = 107, t = 80, t = 64 and t = 53 for strain b/r, respectively and thereafter decreases gradually up to t = 180, respectively for both strains. in fig. 5.10, for both copyright © 2022 assa. adv syst sci appl (2022) a model for analysis of gene expression in the cell involving protein 77 strains tjk16 (left figure) and b/r (right figure), the mrna concentration of copy number 1 increases sharply from t = 0 up to t = 100 and up to t = 101, respectively and then decreases gradually up to t = 180 for both strains. the mrna concentration of remaining copy numbers of both strains increases sharply from t = 0 up to t = 50, t = 33, t = 25, t = 20 and t = 17, respectively and up to t = 51, t = 34, t = 26, t = 21 and t = 18, respectively and thereafter decrease sharply up to t = 167, t = 160, t = 120, t = 96 and t = 80, respectively and up to t = 128, t = 120, t = 90, t = 72 and t = 60, respectively and then decreases gradually up to t = 180 for both strains. in fig. 5.11, for both strains tjk16 (left figure) and b/r (right figure), the protein concentration of copy number 1 increases sharply from t = 0 up to t = 45 and up to t = 21, respectively and then increasing gradually up to t = 112 and up to t = 85, respectively and thereafter decreases gradually up to t = 180 for both strains. the concentration of copy number 2 of both strains increases sharply from t = 0 up to t = 61 and up to t = 42, respectively and decreases sharply up to t = 180, respectively for both strains. the protein concentration of remaining copy numbers of both strains increase sharply from t = 0 up to t = 44, t = 35, t = 29 and t = 25, respectively and up to t = 44, t = 35, t = 30 and t = 26, respectively and thereafter decrease sharply up to t = 106, t = 89, t = 77 and t = 68, respectively and up to t = 104, t = 100, t = 76 and t = 65, respectively and then gradually decrease up to t = 180, respectively for both strains. also, in fig. 5.12, the comparison of the concentration profile of protein with degradation and without degradation is shown for both strains tjk16 and b/r. the concentration of protein in the absence of protein degradation (right upper figure) of copy numbers 1 and 2 increases from t = 0 up to t = 180, respectively. the concentration of remaining copy numbers increases sharply from t = 0 up to t = 122, t = 105, t = 84 and t = 70, respectively and thereafter increases gradually up to t = 180, respectively. there is no decrease in the concentration profiles of all six copy numbers of protein due to the absence of protein degradation. further, for strain b/r, the concentration of protein without degradation (right lower figure) of copy numbers 1 and 2 increases from t = 0 up to t = 180, respectively. the concentration of remaining copy numbers increases sharply from t = 0 up to t = 101, t = 76, t = 61 and t = 50, respectively and thereafter increases gradually up to t = 180, respectively. the concentration profile of protein with degradation for both strains tjk16 and b/r is shown as the left upper figure and left lower figure of fig. 5.13, respectively and their concentration behaviour has been discussed in fig. 5.11. the concentration profiles of protein first increase sharply and thereafter decrease sharply and then decrease gradually. this happens due to the presence of protein degradation. also, in fig. 5.12, the variation in the concentration profile of protein of both strains is observed due to the variation in the rates of replication, transcription, reverse transcription and protein degradation. further, fig. 5.13 shows the graphical representation of comparison of the concentrations of dna, mrna and protein of both strains and also shows the variation in the concentration profiles of dna, mrna and protein with respect to time in presence of protein degradation for case (a). dna concentration decreases with time, mrna concentration increases gradually with time and thereafter decreases gradually with time and the protein concentration increases gradually with time and thereafter decreases gradually with time. the same behaviour is obtained for all copy numbers in figs. 5.5, 5.6 and 5.7 with the same variations in the concentration level of dna, mrna and protein for both strains tjk16 and b/r. the information generated from these cases gives us an idea about the range of rate of processes for which the system or the cell will have normal functions and beyond this range indicate the impairment of the function of the cell due to some abnormal conditions like diseases or due to limited capacity of the cellular processes responding to changes in environmental factors. the results obtained here are in agreement with the biological facts. copyright © 2022 assa. adv syst sci appl (2022) 78 a. sharma, n. adlakha 7. conclusion a model on gene expression, in terms of variations of dna, mrna and protein concentration in the cell, is proposed and successfully employed for the strains tjk16 and b/r to analyze the dynamic behavior of dna, mrna and protein concentration in the presence and absence of protein degradation. the model has been successfully demonstrated for the normal cell as well as retroviruses. in absence of protein degradation, the protein concentration goes on increasing and the system will become unstable and burst, but the results are contrary to this under certain conditions. thus, it is concluded from the results that the cell exerts a beautiful control on the process of replication, transcription, reverse transcription and translation in absence of protein degradation in order to meet the condition of stability and avoid the condition of system burst. in the presence of protein degradation, the behaviour of the system will depend on the rates of protein degradation and translation which in turn depends on transcription and availability of dna by replication. it is concluded that in the cell initially when there is no protein concentration there is no question of degradation and therefore the replication, transcription, reverse transcription and translation will lead to rise in protein concentration in the first few seconds and when the protein becomes available in the cell the degradation starts thereby reducing the protein concentration which has grown in the cell in the initial few seconds. this degradation process decreases the protein concentration sharply and after some time the translation and degradation are in equilibrium thereby making the protein concentration constant in a steady state. further, it is observed from the results that the dna concentration decreases sharply for the initial few seconds due to transcription into mrna taking place in the cell. this process brings a sharp rise in mrna concentration for the initial few seconds and then due to translation into protein this mrna concentration decreases sharply which approaches constant value gradually after some time. the difference in the temporal concentration profiles for different copy numbers is observed due to different rates of replication, transcription, reverse transcription, translation and protein degradation for each copy number. the proposed model gives us better insights of the coordinated mechanism of dna, mrna and protein concentration in the cell leading to gene expression. also, the equilibrium points of the system are carried out and the condition of stability and instability of the system has been identified with the help of this model, which gives us a better insight into the range of rate of processes for which the cell will have normal function and the limit beyond which, the cell will have impaired functions due to any abnormality like diseases or abnormal environmental conditions. such models can be developed to study the gene expression behavior for different strains under different environmental conditions and for various diseases. the information generated from such models can be useful to biomedical scientists for developing protocols for the diagnosis and treatment of diseases. acknowledgements this work was supported by the council of scientific & industrial research (csir), new delhi, award no.-09/1007(0002)/2009. the first author is grateful to csir for financial assistance and sardar vallabhbhai national institute of technology (svnit) authority for providing computer lab facility. also, the authors are thankful to the department of biotechnology (dbt), new delhi, india for providing bioinformatics 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(2014). an explanation of biphasic characters of mrna translocation in the ribosome. biosystems, 118, 1–7. copyright © 2022 assa. adv syst sci appl (2022) introduction nomenclature mathematical model and method solution case (a): case (b): results case (a) case (b) discussion conclusion microsoft word assa_tran+thi+luong_spn.docx adv syst sci appl 2025; 3; 74-95 published online at https://ijassa.ipu.ru. a word-oriented substitution–permutation network cipher with security evaluation based on active s-box bounds and a dynamic advanced encryption standard-based variant nguyen van long1, tran thi luong1*, nguyen bui cuong2, truong minh phuong2 1) academy of cryptography techniques, hanoi, vietnam 2) institute of cryptographic science and technology, hanoi, vietnam abstract: the substitution–permutation network (spn) serves as a foundational structure in the design of modern block cipher algorithms due to its effective realization of two essential properties: confusion and diffusion. currently, the security of spns is actively studied, with the s-box structure and the diffusion layer being key components analyzed to enhance resistance against attacks. in this paper, we propose a generalized spn-based cipher model inspired by the advanced encryption standard (aes) structure. we then present a novel theoretical approach to evaluating the security of this spn cipher, based on a lower bound of the number of active s-boxes. this forms the basis for identifying the roles and cryptographic properties required of the component transformations in this type of cipher. next, we propose a dynamic block cipher algorithm based on the aes cipher, which not only ensures the required level of security but also inherits the implementation advantages of the original aes. the dynamic aes block cipher demonstrates higher security compared to the original aes, passes randomness evaluation standards, and is efficiently implementable. these results are significant in guiding the design of secure and flexible block cipher algorithms, while also providing a theoretical foundation for the selection and evaluation of secure cryptographic components in modern cryptosystems. keywords: dynamic block cipher, active s-box, spn block cipher, aes. 1. introduction block ciphers serve as fundamental building blocks in numerous modern network security protocols. typically, these ciphers are designed using an iterative round-based structure, where each round comprises two primary layers [1]: a confusion layer and a diffusion layer. among various designs, the spn architecture is widely adopted for symmetric encryption due to its effective combination of nonlinear substitution (via s-boxes) and permutation (p-layer) to achieve both confusion and diffusion. in most spn-based block ciphers [2–5], the diffusion layer is implemented using a linear transformation, while the confusion layer relies on small nonlinear substitution boxes, commonly 4 or 8 bits in size. given their small size, these sboxes offer limited nonlinearity, which highlights the importance of the diffusion layer in propagating the s-box nonlinearity across the entire cipher block. as emphasized in shannon’s seminal work [1], strong cryptographic security depends on the joint and inseparable use of both confusion and diffusion mechanisms. among the class of algorithms based on the above principle, aes [2] is arguably the most prominent representative of spn block ciphers and has attracted the most extensive research in this field. from a security perspective, the designers of aes demonstrated that the algorithm achieves at least 25 active s-boxes after four rounds of operation [2]. this issue was later * corresponding author: luongtranhong@gmail.com a word-oriented substitution–permutation network cipher 75 copyright ©2025 assa. adv. in systems science and appl. (2025) revisited and extended in [6] to the case involving an 8×8 mds matrix. however, the analysis in [6] is based on certain specific differential trails. as a result, the lower bounds on the number of active s-boxes for 3, 7, 11, ... rounds proposed in that study are not entirely accurate (see table 1 and table 3 in [6]). thanks to its widespread adoption and long-established security, aes has become the focus of numerous studies aimed at enhancing and developing more optimized variants. one prominent research direction involves the construction of dynamic aes versions, in which core components and transformations within the round function are designed to be keydependent or time-varying. from a design philosophy standpoint, the concept of dynamic block cipher security was introduced by l. knudsen in [7], and this viewpoint was later supported by the original designers of aes, also in [7]. for dynamic versions of the aes block cipher, numerous research studies have been published, each proposing different dynamicization methods. some approaches focus on making the aes s-boxes key-dependent [8–14], while others emphasize generating keydependent mixcolumn transformations [15–18]. notably, there are studies that explore the dynamicization of both the s-box and mixcolumn components of aes [19], and even the dynamicization of all three transformations—s-box, mixcolumn, and shiftrow—has garnered attention [20–22]. another ongoing research direction involves making the xor operation dynamic within aes [23–27]. additionally, the authors in [28] proposed a dynamic aes approach where the number of encryption rounds varies for each plaintext block. in the context of dynamic s-box techniques for aes, al-dweik and colleagues [9] proposed a way to create key-dependent s-boxes that exhibit desirable algebraic characteristics such as nonlinearity, bic, and sac. however, other crucial cryptographic attributes of these s-boxes were not addressed in their study. another approach to generating key-based s-boxes for aes was introduced in [10], which involves rearranging the s-box structure by employing a simulated key expansion algorithm. additionally, the authors in [11] presented a technique to produce modified s-boxes by permuting the original aes s-box. these adaptable s-boxes depend on a secret key and incorporate affine constants and an unconventional polynomial. with each additional bit of the key, a newly rearranged s-box is generated, thereby enhancing the cipher’s complexity. in [12], an s-box generation algorithm based on the playfair cipher was examined. the authors evaluated and compared criteria such as balance and avalanche standards between the modified block cipher with dynamic s-boxes and the original aes. however, similar to [9], other important cryptographic properties were not addressed. in [13], the authors proposed a key-dependent dynamic s-box generation algorithm for aes, utilizing a pseudo-random number generator (prng) based on three linear feedback shift registers (lfsrs). in [14], a completely different approach was explored, where dynamic s-boxes are generated based on epoch time or unix time during each encryption cycle. however, the method proposed in [14] lacks a clear explanation of the s-box generation mechanism when using epoch or unix time as input, which makes it difficult for readers to understand. notably, the cryptographic properties of the resulting dynamic s-boxes were not mentioned at all. returning to the problem of dynamic aes block cipher design at the mixcolumns transformation layer, murtaza et al. [15] proposed a key-dependent mixcolumns transformation based on scalar multiplication, where the scalar multiplication is performed on the rows of the mds matrix. similar approaches were also applied in [16]. additionally, the authors in [16] explored dynamic mixcolumns transformations using exponentiation. along the same lines, the authors in [17] utilized self-reciprocal recursive mds matrices to introduce dynamism into the mixcolumns layer. meanwhile, in [18], a collection of n×n binary matrices was presented that can be used to generate dynamic matrices resembling both the aes matrix and recursive mds matrices. regarding the dynamic modification of multiple transformation components within the aes round function, a notable example is presented in [19]. the authors introduced dynamic s-boxes and new mixcolumns matrices that retain favorable cryptographic properties while 76 n.v. long, t.t. luong, n.b. cuong, t.m. phuong copyright ©2025 assa adv. in systems science and appl. (2025) developing a dynamic version of aes. beyond the subbytes and mixcolumns components, the shiftrows component was also made dynamic in the study by the authors in [20], where the dynamic approach is similarly key-dependent. the approach of constructing keydependent, randomly generated dynamic transformation components was proposed in [21], where all three transformations—subbytes, shiftrows, and mixcolumns—within the aes round function are made dynamic. however, no security evaluations were provided in this study. continuing with the dynamic modification of these three cryptographic components, [22] introduced a new, efficient, key-dependent aes algorithm. for the addroundkey transformation in the aes round function, the authors in [23, 24] introduced improved techniques using key-dependent xor tables generated through 3d chaotic maps. these xor tables are based on initial secret parameters, resulting in a dynamic aes version where the addroundkey layer follows the rules of the new xor tables. this approach was further investigated and developed by t. t. luong et al. in [25-27], employing different xor table generation techniques. recently, in 2024, adamu et al. published a study on dynamic aes [28]. in their approach, the number of encryption rounds for each data block is made dynamic and key-dependent. while this represents a novel method, we assess that it may not be fully reasonable from a security standpoint, since the number of rounds—such as 10 for aes-128—is a threshold chosen based on known cryptanalysis attacks. randomizing the number of rounds without ensuring the necessary security bounds could compromise the entire system. our observations are as follows: according to the philosophy of dynamic cryptosystems proposed by knudsen in [7], introducing dynamism is meaningful from a security perspective. however, to the best of our understanding, dynamic methods must comply with minimal security principles. for example, cryptographic components such as the mds property of the matrix in aes’s mixcolumns transformation must be preserved, the cryptographic properties of the generated dynamic s-boxes should not be inferior to the original version, and the new shiftrow operation must still ensure effective diffusion of active bytes, among others. secondly, dynamic methods need to consider the potential significant impact on the implementation of the proposed solution. the research works we reviewed above seem to have overlooked these issues. in particular, those dynamic approaches fail to leverage aes’s table lookup implementation for optimizing speed. some studies mention recalculating these tables after each dynamic change. however, for applications requiring frequent key changes, this recalculation can be even more complex than the encryption operations themselves for a single data block. this is likely a limitation affecting the practical applicability of dynamic block ciphers. our contributions. in this study, we first propose a generalized aes-like spn cipher model. then, we introduce a novel approach to evaluate the security of these ciphers based on estimating a lower bound on the number of active s-boxes. accordingly, we present a generalized aes-like spn cipher model and provide a theoretical proof for the results obtained. from these results, we identify the crucial cryptographic roles and properties that each component of the cipher’s round function must possess. building on this foundation, we propose an improved dynamic aes version with dynamic components that satisfy the necessary security requirements. notably, our dynamic method does not require recomputing lookup tables in the optimized table-based implementation. this is a significant distinction compared to previous works, as it effectively addresses the practical implementation challenges of dynamic block ciphers. the dynamic aes cipher is carefully evaluated, demonstrating higher security than the original aes, meeting randomness standards, and enabling efficient implementation. based on this, the remainder of the paper is organized as follows. preliminary knowledge and notation are presented in section 2. section 3 introduces a generalized spn block cipher model along with theoretical results on the lower bound of active s-boxes for four rounds of this cipher. a dynamic aes block cipher algorithm is proposed in section 4. section 5 a word-oriented substitution–permutation network cipher 77 copyright ©2025 assa. adv. in systems science and appl. (2025) provides a security analysis of the dynamic aes cipher. section 6 evaluates the randomness properties of the dynamic aes cipher. section 7 analyzes the implementation efficiency of the dynamic aes cipher on a software platform. the conclusion is in section 8. 2. preliminaries some notations used in this paper include: 𝔽 : the binary field consisting of two elements, 0 and 1. 𝔽 : the galois field containing 2 elements, with addition denoted by xor (⊕) and multiplication denoted by (⊗). ℤ : the set of positive integers. 𝑥 ∥ 𝑦: the concatenation of bit string 𝑦 to bit string 𝑥. {𝑥} : a bit string 𝑥 of length 𝑑 bits. 𝑤𝑡(𝑥): the hamming weight of the binary vector 𝑥. given a vector 𝑥 represented as 𝑥 = 𝑥 ∥ 𝑥 ∥ ⋯ ∥ 𝑥 with 𝑥 ∈ {0,1} , the quantity 𝑤𝑡 (𝑥) is called the bundle weight of the vector 𝑥 and is defined as follows: 𝑤𝑡 (𝑥) = #{𝑥 |𝑥 ≠ 0}. let 𝐴 be a linear transformation: 𝔽 ⟶ 𝔽 , we define the branch number of 𝐴, denoted by 𝐵𝑟 (𝐴), as follows: 𝐵𝑟 (𝐴) = 𝑚𝑖𝑛 ∈𝔽 , (𝑤𝑡 (𝑥) + 𝑤𝑡 (𝐴(𝑥))) here, ⌈𝑥⌉ denotes the smallest integer 𝑞 such that 𝑞 ≥ 𝑥. 3. propose a generalized spn cipher model and evaluate a lower bound on the number of active s-boxes. in this section, we first introduce a generalized word-oriented spn cipher model. we then present some theoretical results on the lower bound of the number of active s-boxes for this type of cipher. the security of this block cipher structure against linear and differential cryptanalysis is typically based on the lower bound of the number of active s-boxes. 3.1. propose a generalized spn cipher model inspired by the aes structure in this section, we present a word-oriented spn block cipher model based on three transformations that play roles similar to the subbytes, shiftrows, and mixcolumns operations in aes. first, we define an 𝑚-diffuse linear transformation as follows: definition 1. let 𝜋 be a word-oriented linear transformation from {0,1} × to {0,1} × , where 𝑚, 𝑡 ∈ 𝑍 , and 𝑡 = 𝑚 × 𝑙 × 𝑛. then, 𝜋 is called an 𝒎-diffuse transformation if each input and output state (belonging to {0,1} × ) is partitioned into 𝑚 disjoint consecutive subsets of {0,1} , satisfying the following property: in each output subset, there exist 𝑙 distinct 𝑤-bit words originating from all the input subsets of the transformation. for each string 𝑥 ∈ {0,1} × with 𝑚, 𝑡 ∈ 𝑍 , and 𝑡 = 𝑚 × 𝑙 × 𝑛 , besides the representation as 𝑛-bit words (in which case we can compute the bundle weight 𝑤𝑡 (𝑥)), 𝑥 can also be represented as 𝑡-bit words, where 𝑡 = 𝑚 × 𝑙 × 𝑛. let 𝑥 = 𝑥 ∥ ⋯ ∥ 𝑥 , with 𝑥 ∈ {0,1} , 1 ≤ 𝑖 ≤ 𝑚. in this representation, we can also compute the bundle weight 𝑤𝑡 (𝑥). in this form, each 𝑥 is called a substate, and the transformation 𝜋 can be illustrated in figure 3.1. 78 n.v. long, t.t. luong, n.b. cuong, t.m. phuong copyright ©2025 assa adv. in systems science and appl. (2025) fig. 3.1. illustration of the transformation π, where the black arrows represent the transfer of l′ n-bit words from each input state to the output substates we present some properties related to the 𝑚-diffuse transformation as follows: lemma 3.1. let 𝜋 be an 𝑚 -diffuse transformation from the set {0,1} × to {0,1} × , where 𝑚, 𝑘 ∈ 𝑍 , and 𝑡 = 𝑚 × 𝑙 × 𝑛. then: 1. 𝜋 is also m-diffuse. 2. 𝑤𝑡 𝜋(𝑥 ) × 𝑙 ≥ 𝑤𝑡 𝑥 , for all 𝑗 = 1, … , 𝑚, where 𝑥 = 𝑥 , ∥ 𝑥 , ∥ ⋯ ∥ 𝑥 , , and 𝑥 , ∈ {0,1} . proof. it is straightforward to verify this from the definition of an 𝑚-diffuse permutation. consider the weight of 𝑥 as follows:  when 𝑤𝑡(𝑥 ) = 0, we have 𝑤𝑡 (𝑥 ) = 𝑚𝑖𝑛 ,…, 𝑤𝑡 (𝑥 , ) = 0.  when 𝑤𝑡(𝑥 ) > 0 , there exists some 𝑗 such that 𝑥 , ≠ 0 ∈ {0,1} . suppose 𝑤𝑡 𝑥 , = 𝑑 , where 1 ≤ 𝑑 ≤ 𝑡 . then, 𝑛 -bit blocks in 𝑥( , ) have nonzero bundle weight. since the permutation 𝜋 has the 𝑚 -diffuse property, these 𝑛 -bit blocks in 𝑥( , ) will be diffused into 𝑡-bit substate blocks in 𝜋(𝑥 ). thus, 𝑤𝑡 (𝜋(𝑥 )) ≥ 𝑑 𝑙′ = 𝑤𝑡 (𝑥 , ) 𝑙′ therefore, 𝑤𝑡 (𝜋(𝑥 )) × 𝑙 ≥ 𝑤𝑡 (𝑥 , ) for all 𝑗 = 1, … , 𝑚. ■ next, we consider the local diffusion transformation 𝜃: {0,1} × → {0,1} × , constructed from smaller-dimensional diffusion transformations 𝜃 : {0,1} → {0,1} for 1 ≤ 𝑖 ≤ 𝑚 , defined as follows: 𝜃(𝑥) = 𝜃 (𝑥 ) ∥ 𝜃 (𝑥 ) ∥ ⋯ ∥ 𝜃 (𝑥 ) (1) where 𝑥 = 𝑥 ∥ 𝑥 ∥ ⋯ ∥ 𝑥 , 𝑥 ∈ {0,1} . in this case, we obtain the following result about a transformation composed of 𝜋 and 𝜃 to achieve an optimally diffusive transformation with a higher dimension. theorem 3.1. let 𝜋, 𝜋 : {0,1} × → {0,1} × be two 𝑚 -diffuse permutations, and let 𝜃: {0,1} × → {0,1} × be a transformation defined by (1) based on 𝑚 n-bit-oriented transformations 𝜃 : {0,1} → {0,1} satisfying 𝐵𝑟(𝜃 ) = 𝑡 . then, the transformation 𝜎 = 𝜋 ∘ 𝜃 ∘ 𝜋 satisfies: 𝑚𝑖𝑛 ∈{ , } × , 𝟎{𝑤𝑡 (𝑥) + 𝑤𝑡 (𝜎(𝑥))} × 𝑙′ ≥ 𝑚𝑖𝑛 ∈{ ,.., } {𝑡 } proof. consider the input to the transformation 𝜎 of the form 𝑥 = (𝑥 , . . , 𝑥 × ), where 𝑥 ∈ {0,1} for 𝑗 = 1, … , 𝑡 × 𝑚 . let 𝑥 , 𝑥 , 𝑦 ∈ {0,1} × satisfy 𝑥 = 𝜋(𝑥 ), 𝑥 = 𝜃(𝑥 ) , and 𝑥 = 𝜋 (𝑥 ). then, we have: a word-oriented substitution–permutation network cipher 79 copyright ©2025 assa. adv. in systems science and appl. (2025) 𝑚𝑖𝑛 ∈{ , } × , 𝟎 𝑤𝑡 (𝑥 ) + 𝑤𝑡 𝜎(𝑥 ) = 𝑚𝑖𝑛 ∈{ , } × , 𝟎{𝑤𝑡 (𝑥 ) + 𝑤𝑡 (𝑥 )} = 𝑚𝑖𝑛 ∈{ , } × , 𝟎 𝑤𝑡 (𝜋 (𝑥 )) + 𝑤𝑡 𝜋 𝜃(𝑥 ) . since 𝑥 ≠ 𝟎 (where 0 is the all-zero vector), there exists a subset in the partition of 𝑥 into 𝑚 disjoint consecutive subsets of the form 𝑥 , = (𝑥( )× , … , 𝑥 × ) satisfying 𝑤𝑡 𝑥 , = 𝑑 with 1 ≤ 𝑑 ≤ 𝑙 × 𝑚, 1 ≤ 𝑗 ≤ 𝑚. then, since 𝐵𝑟 𝜃 = 𝑡, we have: 𝑤𝑡 𝑥 , + 𝑤𝑡 𝜃 𝑥 , ≥ 𝑡 . from lemma 3.1 and the fact that 𝜋 and 𝜋 are 𝑚-diffuse transformations, we have: 𝑤𝑡 (𝜋 (𝑥 )) × 𝑙′ ≥ 𝑤𝑡 𝑥 , và 𝑤𝑡 𝜋(𝜃 𝑥 , ) × 𝑙′ ≥ 𝑤𝑡 𝜃 𝑥 , it follows that: 𝑤𝑡 (𝜋 (𝑥 )) + 𝑤𝑡 𝜋 𝜃(𝑥 ) × 𝑙′ ≥ 𝑡 , ∀𝑥 ∈ {0,1} × \{0}, 𝑗 = 1, … , 𝑚. 𝑚𝑖𝑛 ∈{ , } × , 𝟎{𝑤𝑡 (𝑥) + 𝑤𝑡 (𝜎(𝑥))} × 𝑙′ ≥ 𝑚𝑖𝑛 ∈{ ,.., } {𝑡 }. ■ fig. 3.2. description of the round function of the proposed generalized spn cipher from the two diffusion transformations above, we propose a specific iterative block cipher with an spn structure and an 𝑙-bit word-oriented round function, where the block size is 𝑛 bits, with 𝑙 = 𝑚 × 𝑙 × 𝑤. the input and output states of the round function of the block cipher are strings represented as states consisting of 𝑚 × 𝑙 blocks of 𝑛 bits, denoted by 𝑥 = (𝑥 , 𝑥 , … , 𝑥 × ) , where each 𝑥 is an 𝑛 -bit block. these are transformed through the following four basic operations: the round key addition: 𝐾 with the input state 𝑥. however, this addition does not affect the number of active s-boxes for different fault patterns, so we can omit it from consideration. the nonlinear transformation 𝛾: uses a 𝑤-bit s-box 𝑠, which is applied in parallel across the state as follows: 𝛾(𝑥) = 𝛾(𝑥 , 𝑥 , … , 𝑥 × ) = 𝑠(𝑥 ) ∥ 𝑠(𝑥 ) ∥ ⋯ ∥ 𝑠(𝑥 × ) ∥ 𝑠(𝑥 × ) the permutation 𝜋: provides full diffusion, permuting the 𝑛-bit words within each state as follows: 𝜋(𝑥 , … , 𝑥 × ) = 𝑥 ( ) ∥ 𝑥 ( ) ∥ ⋯ ∥ 𝑥 ( × ), where 𝜌 is a permutation of the set {1, … , 𝑙′ × 𝑚 } such that 𝜋 s an 𝑚-diffuse from {0,1} × → {0,1} × , with 𝑡 = 𝑙′ × 𝑚 × 𝑛. 80 n.v. long, t.t. luong, n.b. cuong, t.m. phuong copyright ©2025 assa adv. in systems science and appl. (2025) the mixing step 𝜃: is applied in parallel to substates of size (𝑚 ⋅ 𝑙 ) × (𝑚 ⋅ 𝑙 ) over the field gf(2 ) , based on a matrix 𝑀 with branch number br(𝑀) . the state is divided into 𝑚 substates, each of which is transformed through this mixing step. we refer to the spn cipher with this proposed structure as 𝛾𝜋𝜃_𝑆𝑃𝑁. this structure is illustrated in figure 3.2. based on the proposed generalized spn cipher (𝛾𝜋𝜃_𝑆𝑃𝑁), we describe several existing spn block ciphers in terms of the transformations and parameters of the proposed generalized spn structure. table 3.1 lists several word-oriented block ciphers that use this structure with a block size of 128 bits and their corresponding transformations. for the case of 128-bit byte-oriented block ciphers, table 3.1 includes all existing spn ciphers whose transformations satisfy the properties and conditions we analyzed above. note that for 𝑚 = 8, there is no permutation 𝜋 that satisfies the 𝑚-diffuse property. table 3.1. some specific instances of the proposed generalized spn structure cipher 𝒏 𝒎 𝒍′ 𝜸 𝝅 𝜽 source aes 8 4 1 subbytes shiftrows mixcolumns (4 × 4 mds matrix over 𝔽 ) [2] kalyna128 8 2 4 𝜋 𝜏 𝜓 (8 × 8 mds matrix over 𝔽 ) [4] kuznyechik 8 1 16 𝜋′ identity mapping 𝐿 (16 × 16 mds matrix over 𝔽 ) [5] 3.2. lower bound evaluation of the number of active s-boxes in the proposed generalized spn cipher model in this section, we present the lower bound on the number of active s-boxes over four rounds of the proposed generalized spn cipher 𝛾𝜋𝜃_𝑆𝑃𝑁. theorem 3.2. four consecutive rounds of a block cipher using a round function structured in the form of 𝛾𝜋𝜃_𝑆𝑃𝑁 will have a number of active s-boxes no less than ( ) × 𝐵𝑟(𝑀). proof. for the convenience of presentation, we denote the input difference of a transformation— such as the transformation 𝜃 —at round 𝑖 as δ𝑖𝑛 , = (δ𝑖𝑛 , , … , δ𝑖𝑛 , ) , where δ𝑖𝑛 , ∈ {0,1}, 1 ≤ 𝑗 ≤ 𝑙. then, the total number of active s-boxes over rounds 1, 2, 3, and 4 is: 𝑛 + 𝑛 + 𝑛 + 𝑛 = 𝑤𝑡 (δ𝑖𝑛 , ) + 𝑤𝑡 (δ𝑖𝑛 , ) + 𝑤𝑡 (δ𝑖𝑛 , ) + +𝑤𝑡 (δ𝑖𝑛 , ). since the 𝑛-bit s-boxes only shuffle within 𝑛-bits, and the transformations 𝜋 and 𝜋 are word-oriented 𝑚-diffuse transformations over 𝑤-bit words, it is easy to prove that: 𝑤𝑡 (δ𝑖𝑛 , ) = 𝑤𝑡 (δ𝑜𝑢𝑡 , ), ∀𝑟 ∈ {1, … ,4} 𝑤𝑡 (𝑥) = 𝑤𝑡 𝜋 𝜃 𝜋 (𝑥) , ∀𝑥 ∈ {0,1} × . to facilitate the calculation of the number of active s-boxes, we add two transformations, 𝜋 and 𝜋 , to the output of the 𝜃 transformation in the second round. then, by applying the result of theorem 3.1 to the inputs of the three transformations—𝜋 (of the second round), 𝜃 (of the second round), and the added 𝜋 transformation—we obtain: 𝑤𝑡 (δ𝑖𝑛 , ) + 𝑤𝑡 𝜋 𝜃 𝜋(δ𝑖𝑛 , ) × 𝑙′ ≥ 𝐵𝑟(𝑀) since: 𝜋 𝜃 𝜋(δ𝑖𝑛 , ) = 𝜋 δ𝑖𝑛 , , therefore, we obtain: a word-oriented substitution–permutation network cipher 81 copyright ©2025 assa. adv. in systems science and appl. (2025) 𝑤𝑡 (δ𝑖𝑛 , ) + 𝑤𝑡 𝜋 δ𝑖𝑛 , × 𝑙′ ≥ 𝐵𝑟(𝑀) thus, the input differences of the substitution-diffusion-substitution structures in rounds (1, 2) and (3, 4) will activate a number of s-boxes equal to ( ) (see illustration in figure 3). since in each of these patterns, the number of active s-boxes is no less than 𝐵𝑟(𝑀), the total number of active s-boxes for any input difference will be no less than ( ) × 𝐵𝑟(𝑀).■ fig. 3.3. illustration of difference propagation through four rounds of the block cipher γπθ_spn from theorem 3.2, we derive the following corollary. corollary 3.1. four consecutive rounds of iterative block ciphers using a round function structured like 𝛾𝜋𝜃_𝑆𝑃𝑁 , where the transformation 𝜃 employs mds matrices of size (𝑚 ⋅ 𝑙 ) × (𝑚 ⋅ 𝑙 ) over the field 𝔽 , will have a number of active s-boxes no less than (𝑚 + 1) × (𝑚 ⋅ 𝑙 + 1). indeed, this follows directly from the fact that 𝐵𝑟(𝑀) = (𝑚 ∙ 𝑙′ + 1 ). by applying corollary 3.1, we can recover known results for practical spn block ciphers (see table 2.2). this further confirms the validity of our theorem 3.2and corollary 3.1. table 3.2. lower bounds on the number of active s-boxes over 4 rounds for several spn ciphers cipher 𝒏 𝒎 𝒍′ lower bound on the number of active s-boxes aes 8 4 1 25 kalyna 8 2 4 27 kuneztik 8 1 16 34 remark 3.1. a natural question arises: if we remove the 𝑚-diffuse condition from the shiftrows-type transformation such as 𝜋 in 𝛾𝜋𝜃 − 𝑆𝑃𝑁, will the lower bound on the number of active s-boxes over four rounds be affected? the answer is yes. we examined a specific 82 n.v. long, t.t. luong, n.b. cuong, t.m. phuong copyright ©2025 assa adv. in systems science and appl. (2025) case with aes, where its shiftrows transformation does not satisfy the 𝑚-diffuse property, and the result shows that for its four-round transformation, the number of active s-boxes is always less than 25. thus, the role of the transformation 𝜋 in the generalized spn cipher model is truly significant. its design not only directly impacts the security level but also affects the implementation efficiency of the block cipher algorithm. based on the theoretical foundation presented in this section, in the next section, we propose a modification to the shiftrows layer in aes to obtain a key-dependent dynamic version of aes, which maintains the required level of security without significantly compromising the high-speed implementation capability of the original aes algorithm. 4. proposal of a secure and efficient key-dependent dynamic aes algorithm the original aes algorithm is not only secure but also achieves high execution speed on many platforms. the implementation method using lookup tables for this type of block cipher has become popular and familiar within the cryptographic community. it is regarded as a “musthave property” of a byte-oriented spn block cipher. many dynamic aes algorithm approaches have been proposed, but it seems that they do not maintain the flexible implementation using the lookup tables of the original aes. this may be a limitation in the applicability of the dynamic algorithm. to overcome this issue, we propose a modification in the shiftrows transformation of the original aes to obtain a secure and efficient keydependent dynamic aes variant. as we know, the aes algorithm processes a 128-bit data block divided into 16 bytes arranged into a 4 × 4 two-dimensional state matrix. we name the proposed dynamic aes algorithm aes_dst, which includes the following basic transformations:  the addroundkeys operation is the same as in the original aes  the subbytes operation is the same as in the original aes  the mixcolumns operation is the same as in the original aes  the key schedule is the same as in the original aes the byte permutation consists of two key-dependent transformations: shiftrows and tranbytes. among them, the shiftrows operation is the same as in the original aes, while tranbytes is a transpose of the 4 × 4 data state matrix. these two transformations are selected based on a secret key. the steps performed in the dynamic block cipher algorithm aes_dst are described as in algorithm 4.1. algorithm 4.1. the dynamic block cipher algorithm aes_dst input: a 128-bit input data block (in) is organized into a 4 × 4 state matrix state. the master key (masterkey) has a length of 𝑘 bits ( 𝑘 = 128, 192, 256 bits corresponding to the dynamic aes-128, aes-192, and aes-256 versions with 𝑛 = 10, 12, 14 rounds, respectively). the dynamic key 𝑘𝑑 has a length of 𝑛 bits. output: 128-bit ciphertext block. step 1. from the master key (masterkey) compute the array of round keys (𝑤) through the key schedule. step 2. encryption process – procedure 𝐶𝑖𝑝ℎ𝑒𝑟(𝑖𝑛, 𝑑𝐾𝑒𝑦, 𝑛 , 𝑤): 1. 𝑠𝑡𝑎𝑡𝑒 ← 𝑖𝑛 2. 𝑠𝑡𝑎𝑡𝑒 ← 𝐴𝑑𝑑𝑅𝑜𝑢𝑛𝑑𝐾𝑒𝑦(𝑠𝑡𝑎𝑡𝑒, 𝑤[0. .3]) a word-oriented substitution–permutation network cipher 83 copyright ©2025 assa. adv. in systems science and appl. (2025) 3. for round from 1 to 𝑛 − 1 do 4. 𝑠𝑡𝑎𝑡𝑒 ← 𝑆𝑢𝑏𝐵𝑦𝑡𝑒𝑠(𝑠𝑡𝑎𝑡𝑒) 5. if 𝑘𝑑[𝑟𝑜𝑢𝑛𝑑 − 1] = 1 then 6. 𝑠𝑡𝑎𝑡𝑒 ← 𝑆ℎ𝑖𝑓𝑡𝑅𝑜𝑤𝑠(𝑠𝑡𝑎𝑡𝑒) 7. else 8. 𝑠𝑡𝑎𝑡𝑒 ← 𝑇𝑟𝑎𝑛𝐵𝑦𝑡𝑒𝑠(𝑠𝑡𝑎𝑡𝑒) 9. end if 10. 𝑠𝑡𝑎𝑡𝑒 ← 𝑀𝑖𝑥𝐶𝑜𝑙𝑢𝑚𝑛𝑠(𝑠𝑡𝑎𝑡𝑒) 11. 𝑠𝑡𝑎𝑡𝑒 ← 𝐴𝑑𝑑𝑅𝑜𝑢𝑛𝑑𝐾𝑒𝑦𝑠(𝑠𝑡𝑎𝑡𝑒, 𝑤[4 ⋅ 𝑟𝑜𝑢𝑛𝑑. .4 ⋅ 𝑟𝑜𝑢𝑛𝑑 + 3]) 12. end for 13. 𝑠𝑡𝑎𝑡𝑒 ← 𝑆𝑢𝑏𝐵𝑦𝑡𝑒𝑠(𝑠𝑡𝑎𝑡𝑒) 14. if 𝑘𝑑[𝑛 ] = 1 then 15. 𝑠𝑡𝑎𝑡𝑒 ← 𝑆ℎ𝑖𝑓𝑡𝑅𝑜𝑤𝑠(𝑠𝑡𝑎𝑡𝑒) 16. else 17. 𝑠𝑡𝑎𝑡𝑒 ← 𝑇𝑟𝑎𝑛𝐵𝑦𝑡𝑒𝑠(𝑠𝑡𝑎𝑡𝑒) 18. end if 19. 𝑠𝑡𝑎𝑡𝑒 ← 𝐴𝑑𝑑𝑅𝑜𝑢𝑛𝑑𝐾𝑒𝑦𝑠(𝑠𝑡𝑎𝑡𝑒, 𝑤[4 ⋅ 𝑟 . .4 ⋅ 𝑟 + 3]) 20. return 𝑠𝑡𝑎𝑡𝑒. here, we only describe the procedure for the encryption process; the decryption process can be performed similarly but in reverse order. 5. security evaluation of the dynamic block cipher aes_dst first, we can observe that the tranbytes operation is a transposition of the state matrix. it can be easily verified that it satisfies definition 1, or in other words, it possesses the 𝑚-diffusion property. this means that four encryption rounds of our dynamic aes variant still conform to the results stated in corollary 3.1. figure 4 illustrates the encryption/decryption diagram of the dynamic block cipher aes_dst. in the field of block cipher analysis, several attack techniques have been developed, including differential cryptanalysis [29, 30, 31], linear cryptanalysis [32, 33], and algebraic attacks [31, 34], among others. this section concentrates on two of the most influential and widely studied methods: linear and differential cryptanalysis. these techniques play a key role in assessing the robustness of block cipher designs. linear cryptanalysis seeks to uncover approximate linear correlations between input bits, output bits, and key bits, which can then be exploited to infer parts of the secret key. in contrast, differential cryptanalysis examines how specific differences in input values affect the output, allowing attackers to detect high probability patterns that can aid in key recovery. together, these methods serve as critical benchmarks for measuring the cryptographic strength of modern block ciphers. 5.1. preservation of the wide trail strategy in aes_dst by incorporating key-dependent dynamic byte permutation transformations (including shiftrows and tranbytes) based on the bits of the key 𝑘𝑑, we obtain the dynamic block cipher aes_dst without compromising the wide trail strategy used in the design of the original aes [2]. to clarify this point, we analyze the diffusion level of active bytes as they pass through two consecutive rounds in the aes_dst algorithm. 84 n.v. long, t.t. luong, n.b. cuong, t.m. phuong copyright ©2025 assa adv. in systems science and appl. (2025) fig. 5.1. round structure diagram of the aes_dst encryption/decryption process figures 5.1 and figures 5.2 illustrate the role of the shiftrows transformation in propagating active bytes. input state after subbyte after shiftrow after mixcolumn fig. 5.2. distribution of active bytes during the first round of the aes_dst dynamic block cipher using shiftrows figure 5.1 demonstrates the diffusion behavior during the first round of aes, beginning with a state matrix that contains only one active byte. after the subbytes and shiftrows operations, the state still has a single active byte; however, the mixcolumns transformation, thanks to the diffusion characteristics of the mds matrix, expands this to four active bytes. this number remains unchanged after the addroundkey step. therefore, a single active byte at the start of the round propagates to four active bytes by the end of the first round. active bytes are highlighted as black squares. a word-oriented substitution–permutation network cipher 85 copyright ©2025 assa. adv. in systems science and appl. (2025) input state after subbyte after shiftrow after mixcolumn fig. 5.2. spread of active bytes during the second round of the aes_dst dynamic block cipher using shiftrows figure 5.2 illustrates the key role of shiftrows in distributing active bytes from one column across all four columns of the state matrix. consequently, following the mixcolumns operation, the active bytes spread to cover all 16 bytes in the state matrix. figure 5.3 and figure 5.4 demonstrate how the tranbytes transformation contributes to the diffusion of active bytes. input state after subbyte after tranbytes after mixcolumn fig.5.3. spread of active bytes during the first round of the aes_dst dynamic block cipher using tranbytes input state after subbyte after tranbytes after mixcolumn fig. 5.4. distribution of active bytes during the second round of the aes_dst dynamic block cipher using tranbytes similar to the shiftrows transformation, thanks to the tranbytes transformation, starting from a single active byte, the number of active bytes expands to the maximum of 16 bytes across the entire state matrix after two rounds. from the above illustrations, it is evident that both shiftrows and tranbytes transformations enable the dynamic block cipher aes_dst to adhere to the wide trail strategy used in the design of the original aes. according to the philosophy of dynamic block ciphers proposed by knudsen in 2015 [7], our dynamic aes version not only inherits the security properties of the original aes but, in some respects, also increases the complexity of cryptanalysis. in differential cryptanalysis and its variants, when differential trails depend on the key, the complexity of attacks rises. this is because the attacker must consider all possible cases—on the order of 2 , where 𝑛 is the number of encryption rounds—to determine which differential path the data follows. furthermore, the security of the dynamic block cipher is also supported by vincent rijmen [8], one of the two principal designers of the aes block cipher. we will provide a more detailed analysis of this in the following sections. 5.2. analysis of linear cryptanalysis on the dynamic block cipher aes_dst to perform linear cryptanalysis on the block cipher, one needs to build a linear approximation covering 𝑛 − 1 rounds, along with a corresponding linear expression for the full 𝑛 -round cipher. the procedure involves the following steps: develop a linear approximation spanning 𝒏𝒓 − 𝟏 rounds along with an associated linear equation for the 𝒏𝒓-round block cipher. step 1: generate the linear approximation matrix for the s-box. 86 n.v. long, t.t. luong, n.b. cuong, t.m. phuong copyright ©2025 assa adv. in systems science and appl. (2025) step 2: determine which s-boxes are active (with non-zero input and output masks) in the rounds and assign suitable linear approximations from the matrix created in step 1 to these sboxes. step 3: develop the overall linear equation for the block cipher and assess its deviation from uniform probability. fig. 5.5. the final round of the aes_dst dynamic block cipher aes_dst, the dynamic block cipher, performs 𝑛 rounds on a 128-bit input block and employs 8 × 8 substitution boxes. let 𝐾 represent the secret key for aes_dst, from which the round keys 𝐾 , 𝐾 , … , 𝐾 , 𝐾 are generated. denote by 𝑃 the plaintext and by 𝐶 the ciphertext produced by the aes_dst dynamic block cipher. for the 𝑖-th round, let 𝐴 and 𝐵 be the inputs and outputs of the s-boxes, respectively. the bit 𝐴 , refers to the 𝑗-th bit of 𝐴 , where 𝑖 ranges from 1 to 𝑛 and 𝑗 ranges from 1 to 128. in aes_dst, the permutation is represented by 𝐷 . it is important to note that the linear approximation of aes depends solely on the plaintext, the input bits to the last round (𝑛 -th round), and the bits of the subkeys 𝐾 , 𝐾 , … , 𝐾 . in aes_dst, the permutation 𝐷 —encompassing both the shiftrows/tranbytes and mixcolumns steps—is transformed from a fixed operation into a dynamic one. specifically, unlike the original static shiftrows, this version employs a key-dependent dynamic shiftrows/tranbytes. the shiftrow/tranbytes operation works on a 4 × 4 byte state array. let 𝑆 denote the input state array for shiftrow/tranbytes, with its bytes numbered from 1 to 16 as follows: 𝑆 = 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 the shiftrow transformation works by keeping the first row of the state array intact, while the second row is shifted left by one byte, the third row by two bytes, and the fourth row by three bytes. a word-oriented substitution–permutation network cipher 87 copyright ©2025 assa. adv. in systems science and appl. (2025) hence, shiftrow can be described as a specific permutation of bytes as follows: 𝑆 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 𝑆ℎ𝑖𝑓𝑡𝑅𝑜𝑤(𝑆) 1 6 11 16 5 10 15 4 9 14 3 8 13 2 7 12 the inverse of the shiftrow transformation, denoted as 𝑆ℎ𝑖𝑓𝑡𝑅𝑜𝑤 , can likewise be expressed through a byte permutation as shown below: 𝑆 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 𝑆ℎ𝑖𝑓𝑡𝑅𝑜𝑤 (𝑆) 1 14 11 8 5 2 15 12 9 6 3 16 13 10 7 4 tranbytes functions by rearranging the state matrix, and as such, it can be represented as a byte-level permutation as follows: 𝑆 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 𝑇𝑟𝑎𝑛𝐵𝑦𝑡𝑒𝑠(𝑆) 1 5 9 13 2 6 10 14 3 7 11 15 4 8 12 16 the inverse tranbytes operation, 𝑇𝑟𝑎𝑛𝐵𝑦𝑡𝑒𝑠 , can similarly be expressed as a byte permutation as follows: 𝑆 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 𝑇𝑟𝑎𝑛𝐵𝑦𝑡𝑒𝑠 (𝑆) 1 5 9 13 2 6 10 14 3 7 11 15 4 8 12 16 in the aes_dst dynamic block cipher, a major obstacle for attackers lies in forming a linear approximation over 𝑛 − 1 rounds and deriving a linear expression for the full cipher. while the initial step of cryptanalysis is carried out normally, difficulties arise in the next step when choosing active s-boxes in each round, due to the unknown nature of the permutation. this uncertainty complicates correlating s-boxes between rounds, forcing attackers to select them randomly. even if by chance an attacker succeeds in constructing a linear approximation spanning 𝑛 − 1 rounds and a corresponding expression for the entire aes_dst, the process remains highly challenging. assume that this linear expression takes the following form: 𝐴 , ⊕ 𝐴 , ⊕ … ⊕ 𝐴 , ⊕ ((𝑃 ⊕ 𝑃 ⊕ … ⊕ 𝑃 ) ⊕ 𝐾 , ⊕ … 𝐾 , ⊕ 𝐾 , ⊕ … .⊕ 𝐾 , ⊕ … ⊕ 𝐾 , ) = 0 (2) referring to the last round (see fig. 9), let’s assume the attacker can extract certain bits of the subkey 𝐾 , which correspond to the active s-boxes identified in round 𝑛 . additionally, the attacker can derive some essential bits of 𝐴 . 𝐴 = 𝐷 𝐵 ⊕ 𝐾 (3) using equation (3) as a starting point, the attacker continues by making guesses on the associated key bits of 𝐾 , subsequently trying to derive the required bits of 𝐵 following equation (4). 𝐵 = 𝐷 𝐴 ⊕ 𝐾 (4) in dynamic aes block ciphers, the diffusion-layer permutation 𝐷 varies depending on the key. this means the attacker cannot identify the exact 𝐷 used in the cipher, preventing them from calculating the required bits of 𝐵 based on equation (4). as a result, the attacker is unable to extract the corresponding key bits of 𝐾 , and faces similar obstacles with the other subkeys as well. therefore, under these circumstances, the attacker is limited to obtaining some bits of the subkey 𝐾 and is unable to deduce any further key bits from the remaining subkeys. to perform standard linear cryptanalysis, the attacker needs to determine the specific permutation 𝐷 applied within the dynamic aes block cipher before they can continue with the attack. initially, it is assumed that during the process of forming a linear approximation for the dynamic aes block cipher, the attacker encounters more difficulties in step 2 than with the original aes. because the permutation used in aes_dst varies dynamically and is unknown, the attacker is forced to pick s-boxes randomly across rounds. these selections may lack correlation, which hampers the construction of 𝑛 − 1-round linear approximations and the overall linear expression for the dynamic aes_dst cipher. 88 n.v. long, t.t. luong, n.b. cuong, t.m. phuong copyright ©2025 assa adv. in systems science and appl. (2025) by introducing key-dependent shiftrow/tranbytes operations that make the permutation layer dynamic in the aes_dst block cipher, the difficulty of carrying out linear cryptanalysis rises considerably compared to the conventional static aes cipher. 5.3. analysis of differential cryptanalysis on the dynamic block cipher aes_dst to conduct differential cryptanalysis on the block cipher, it is necessary to develop a differential characteristic spanning 𝑛 − 1 rounds, as well as the associated differential for the full 𝑛 -round cipher. the process for building these is outlined below: develop a differential characteristic covering 𝒏𝒓 − 𝟏 rounds along with the matching differential applicable to the entire 𝒏𝒓 -round block cipher. step 1*: create the differential distribution table for the s-box. step 2*: identify specific active s-boxes in each round (i.e., those exhibiting non-zero differentials) and assign corresponding differentials from the table developed in step 1*. step 3*: calculate the overall differential characteristic of the block cipher and evaluate its probability. it is important to highlight that the block cipher’s differential is linked to a differential characteristic spanning 𝑛 − 1 rounds. this differential is denoted by (∆𝐴, ∆𝐵) , with ∆𝐴 representing the input difference at the plaintext stage, and ∆𝐵 indicating the input difference at the final round. the overall differential characteristic encompasses the individual differentials of the chosen s-boxes across the rounds. aes_dst, the dynamic block cipher, performs 𝑛 rounds on a 128-bit input block and employs 8 × 8 substitution boxes. let 𝐾 represent the secret key for aes_dst, from which the round keys 𝐾 , 𝐾 , … , 𝐾 , 𝐾 are generated. denote by 𝑃 the plaintext and by 𝐶 the ciphertext produced by the aes_dst dynamic block cipher. for the 𝑖-th round, let 𝐴 and 𝐵 be the inputs and outputs of the s-boxes, respectively. the bit 𝐴 , refers to the 𝑗-th bit of 𝐴 , where 𝑖 ranges from 1 to 𝑛 and 𝑗 ranges from 1 to 128. let δ𝐴 = 𝐴 ⊕ 𝐴 and δ𝐵 = 𝐵 ⊕ 𝐵 represent the input and output differences of the s-boxes at round 𝑖. here, ∆𝐴 , denotes the 𝑗-th bit of δ𝐴 , with 1 ≤ 𝑖 ≤ 𝑛 and 1 ≤ 𝑗 ≤ 128. the permutation operation in aes_dst is still represented by 𝐷 . for the aes_dst dynamic cipher, a significant obstacle encountered by attackers lies in forming an 𝑛 − 1-round differential characteristic along with the matching differential for the complete cipher. although step 1* proceeds as usual, step 2*, which involves choosing active s-boxes in various rounds, proves problematic since the permutation is unknown. this uncertainty complicates the task of correlating s-boxes across rounds, forcing the attacker to select these s-boxes randomly. assuming the attacker is fortunate enough, they might still construct a valid differential characteristic for the entire aes_dst cipher. assuming the attacker manages to recover specific bits of the subkey 𝐾 , particularly those associated with the selected active substitution boxes in the final round, they may also be capable of deriving some necessary bits from both inputs 𝐴 and 𝐴 of that round. using equation (3), the cryptanalyst can infer 𝐵 based on equation (4) (𝐵 = 𝐷 𝐴 ⊕ 𝐾 ). following this, the attacker attempts to estimate the key bits of the subkey 𝐾 that influence the output of the round. with these guesses, they reconstruct 𝐵 , and do the same to get 𝐵 . the bitwise difference is then calculated as: δ𝐵 = 𝐵 ⊕ 𝐵 then, if the derived bits of δ𝐵 match those in the differential characteristic previously constructed, the guessed key bits are likely correct for 𝐾 . a word-oriented substitution–permutation network cipher 89 copyright ©2025 assa. adv. in systems science and appl. (2025) in contrast to static designs, the aes_dst cipher employs a key-dependent, dynamically changing permutation 𝐷 . this uncertainty prevents the attacker from identifying the exact permutation applied during encryption. without knowing 𝐷 , they are unable to accurately compute the critical bits of 𝐵 , making it impossible to verify whether the guessed subkey bits for 𝐾 — or any other round keys — are valid. in this case, the attacker can only retrieve a limited number of key bits from the subkey 𝐾 , with no feasible way to uncover additional key bits from the remaining round keys. moreover, as mentioned earlier, developing a differential characteristic for aes_dst proves more challenging than for the standard aes, particularly in step 2*. the core issue stems from the fact that the internal permutation used in aes_dst varies with the key and is not known to the attacker. this forces the attacker to select active s-boxes at random across rounds, reducing the likelihood that those s-boxes are meaningfully connected. as a result, constructing a coherent 𝑛 − 1-round differential characteristic and a complete differential trail becomes nearly impossible in the context of aes_dst. to apply traditional differential cryptanalysis to the aes_dst cipher, an attacker must first identify the specific permutation 𝐷 employed within the cipher. without this information, the attack cannot effectively proceed. by introducing dynamic, key-dependent transformations in the diffusion stage— specifically through the shiftrow and tranbytes operations—the aes_dst cipher substantially raises the difficulty level for differential attacks. this added complexity marks a significant departure from the more predictable structure of standard, static aes, thereby enhancing resistance against such cryptanalytic techniques. 6. evaluation of the randomness of aes_dst this section presents a randomness analysis of the aes_dst dynamic block cipher, based on statistical tests recommended by the nist framework [35]. our evaluation integrates the randomness testing strategy outlined in nist sp 800-22 [35] with the structured plaintext generation approach introduced by sulak [36]. this allows us to investigate how aes_dst behaves under controlled, non-random plaintext conditions across different encryption rounds. specifically, we construct several deterministic input sets—such as low weight (lw), high weight (hw), 1-bit difference (av1), and rotated plaintexts (rot)—as test vectors. we then apply the two-tiered testing strategy from nist sp 800-22 to assess the pseudorandomness of the resulting ciphertexts. notably, the evaluation is limited to test cases optimized for short data sequences, by the guidance from [37]. based on the analysis (table 6.1 and table 6.2), it is evident that employing random keys necessitates at least three rounds of encryption in the dynamic block cipher aes_dst to generate output data exhibiting adequate randomness for the hw, av1, and lw datasets. this implies that for the aes_dst system to maintain essential randomness in its encrypted output, a minimum of three rounds must be executed when random keys are used. additionally, the findings reveal that the randomness level produced by aes_dst matches that of the conventional aes cipher. table 6.1. outcomes of second-level p-value assessments for aes_dst based on statistical tests applied to short data sequences no. of rounds freq. test runs test test for longest run of ones serial test appen. test cusum. test bit autocorr. test byte autocor. test av1 input data 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 2 0.004821 0.000000 0.000000 0.000001 0.000000 0.000001 0.000000 0.023187 3 0.388140 0.690279 0.296774 0.360755 0.113957 0.492432 0.560673 0.009957 4 0.925600 0.526663 0.773360 0.996072 0.965937 0.265690 0.917013 0.909591 5 0.969725 0.359059 0.562529 0.691677 0.590030 0.337997 0.255215 0.770595 6 0.001659 0.357359 0.867525 0.277019 0.118567 0.250200 0.739263 0.051301 90 n.v. long, t.t. luong, n.b. cuong, t.m. phuong copyright ©2025 assa adv. in systems science and appl. (2025) no. of rounds freq. test runs test test for longest run of ones serial test appen. test cusum. test bit autocorr. test byte autocor. test 7 0.715970 0.965837 0.099518 0.856579 0.752517 0.815556 0.335218 0.858150 8 0.915977 0.620556 0.255037 0.102592 0.255590 0.781520 0.087702 0.853995 9 0.677561 0.058897 0.926591 0.799319 0.733049 0.206479 0.023996 0.299263 10 0.190943 0.907337 0.907299 0.506631 0.375749 0.641292 0.409264 0.060240 hw input data 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 2 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 3 0.672962 0.790742 0.956944 0.507715 0.767315 0.557299 0.063931 0.932396 4 0.971935 0.713995 0.569726 0.130933 0.555359 0.712793 0.715195 0.717960 5 0.550656 0.016966 0.930257 0.000669 0.000229 0.193905 0.009111 0.157356 6 0.712010 0.125157 0.269899 0.169880 0.103291 0.611226 0.226911 0.551235 7 0.738873 0.075857 0.790729 0.239091 0.108675 0.953965 0.978852 0.955501 8 0.785150 0.693697 0.305570 0.595033 0.762953 0.777388 0.288855 0.572716 9 0.597013 0.525833 0.287850 0.365215 0.378768 0.837909 0.252725 0.007599 10 0.162613 0.932569 0.722379 0.636367 0.716321 0.316012 0.670659 0.005735 lw input data 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 2 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 3 0.815197 0.936996 0.896293 0.907959 0.950627 0.915199 0.591385 0.256157 4 0.059672 0.760510 0.263700 0.597756 0.657905 0.999708 0.352198 0.628523 5 0.721655 0.928550 0.502976 0.997965 0.995961 0.530596 0.275560 0.769510 6 0.360959 0.275531 0.770137 0.099165 0.095005 0.999728 0.389153 0.370810 7 0.821301 0.850561 0.223176 0.729395 0.728277 0.529035 0.703023 0.110923 8 0.906001 0.922559 0.926138 0.963057 0.931601 0.875985 0.571562 0.035327 9 0.959171 0.162170 0.737297 0.155859 0.356525 0.599980 0.323053 0.565623 10 0.060375 0.297092 0.636063 0.621519 0.537975 0.505325 0.657715 0.266918 rot input data 1 0.357559 0.526258 0.200096 0.750682 0.925820 0.029858 0.526090 0.565327 2 0.595618 0.667512 0.615079 0.635559 0.350689 0.535297 0.951519 0.115702 3 0.779571 0.528596 0.569855 0.506580 0.112316 0.089115 0.093269 0.369525 4 0.690950 0.144490 0.229079 0.297999 0.236025 0.194579 0.215946 0.203690 5 0.199359 0.901740 0.279104 0.979264 0.975541 0.951572 0.769115 0.743401 6 0.355609 0.419073 0.377150 0.647660 0.523603 0.963475 0.929479 0.315552 7 0.057450 0.965519 0.491526 0.749116 0.957566 0.229471 0.961226 0.410902 9 0.298895 0.059195 0.771945 0.507281 0.758229 0.148971 0.509822 0.511561 9 0.751565 0.815415 0.560448 0.877156 0.869469 0.405955 0.789552 0.552502 10 0.207595 0.158116 0.659577 0.752650 0.480489 0.850594 0.124907 0.122742 table 6.2. outcomes of proportion tests for aes_dst conducted on short sequence datasets no. of rouns freq. test runs test test for longest run of ones serial test appen. test cusum. test bit autocorr. test byte autocor. test av1 input data 1 98.98 98.92 99.08 98.99 98.95 99.09 99.29 99.16 2 99.92 99.95 99.07 98.95 98.91 98.98 99.24 99.22 3 99.00 99.96 99.09 99.04 99.95 99.09 99.25 99.22 4 99.99 99.96 99.08 99.01 98.95 99.08 99.25 99.21 5 99.00 98.95 99.09 99.01 98.94 99.08 99.26 99.24 6 99.99 99.94 99.09 99.01 98.94 99.06 99.24 99.21 7 98.98 98.97 99.07 99.01 98.95 99.06 99.25 99.21 8 98.98 99.95 99.08 99.01 98.94 99.06 99.25 99.22 9 98.99 98.95 99.09 99.04 98.95 99.07 99.25 99.22 10 98.99 98.95 99.08 99.00 98.94 99.07 99.25 99.21 hw input data 1 98.74 98.96 99.24 98.65 99.67 98.82 99.01 99.44 2 98.91 99.11 99.29 99.01 99.05 99.04 99.41 99.20 3 98.99 99.94 99.06 99.00 99.94 99.07 99.24 99.22 4 98.98 98.95 99.10 99.02 98.94 99.06 99.25 99.20 5 99.00 98.94 99.10 99.01 98.95 99.08 99.26 99.21 6 98.96 98.95 99.09 98.99 98.91 99.05 99.22 99.21 7 98.99 98.94 99.10 99.02 98.94 99.06 99.25 99.21 8 99.00 98.95 99.09 99.04 98.96 99.06 99.25 99.21 9 99.00 98.96 99.08 99.02 98.95 99.06 99.26 99.21 10 99.00 98.95 99.09 99.02 98.95 99.06 99.25 99.21 lw input data 1 99.14 99.16 99.40 99.12 99.04 99.15 99.46 99.46 2 99.14 99.04 99.18 99.11 99.12 99.06 99.25 99.25 3 99.01 98.94 99.09 99.02 98.95 99.09 99.25 99.21 4 99.00 98.94 99.08 99.01 98.94 99.08 99.24 99.22 5 98.99 98.96 99.08 99.02 98.94 99.06 99.25 99.20 6 99.00 98.96 99.08 99.05 98.96 99.06 99.26 99.22 a word-oriented substitution–permutation network cipher 91 copyright ©2025 assa. adv. in systems science and appl. (2025) 7 98.99 98.95 99.08 99.01 98.94 99.06 99.25 99.22 8 98.99 98.96 99.09 99.04 98.95 99.08 99.26 99.20 9 98.99 98.95 99.08 99.01 98.94 99.07 99.25 99.20 10 98.99 98.95 99.07 99.04 98.95 99.07 99.25 99.22 rot input data 1 98.98 98.95 99.08 99.02 98.94 99.07 99.25 99.21 2 98.99 98.95 99.09 99.02 98.94 99.07 99.25 99.21 3 98.97 98.97 99.09 99.02 98.94 99.06 99.26 99.20 4 98.99 98.96 99.10 99.01 98.94 99.07 99.25 99.22 5 99.00 98.96 99.09 99.04 98.96 99.08 99.26 99.24 6 98.98 98.94 99.09 99.02 98.94 99.06 99.24 99.22 7 98.99 98.95 99.07 99.01 98.94 99.07 99.24 99.21 8 98.99 98.95 99.08 99.01 98.94 99.06 99.25 99.21 9 99.00 98.94 99.09 99.02 98.94 99.06 99.24 99.24 10 98.99 98.94 99.07 99.02 98.94 99.06 99.25 99.24 7. analysis of the implementation efficiency of the dynamic block cipher aes_dst on a software platform unlike previously published versions of dynamic aes block ciphers, our dynamic aes block cipher maintains implementation efficiency on 32-bit software platforms. before analyzing the approach to handling the dynamic byte permutation layer, we briefly review the lookup table implementation technique of the original aes [2]. let 𝑒 denote the data state after the mixcolumns transformation, 𝑎 be the input data state to the subbytes transformation, 𝑏 be the input data state to the shiftrows transformation, and 𝑐 be the input data state to the mixcolumns transformation. then, we have: 𝑒 , 𝑒 , 𝑒 , 𝑒 , = 2 3 1 2 1 1 3 1 1 1 3 1 2 3 1 2 ⊗ 𝑐 , 𝑐 , 𝑐 , 𝑐 , , with 𝑐 , 𝑐 , 𝑐 , 𝑐 , = ⎣ ⎢ ⎢ ⎢ ⎡ 𝑏 , 𝑏 , 𝑏 , 𝑏 , ⎦ ⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎢ ⎡ 𝑆 𝑎 , 𝑆 𝑎 , 𝑆 𝑎 , 𝑆 𝑎 , ⎦ ⎥ ⎥ ⎥ ⎤ , (5) where 𝑆 is the s-box of the original aes, the operation " − " is defined modulo 4, and 𝑗 = 0,1,2,3. thus, we have 𝑒 , 𝑒 , 𝑒 , 𝑒 , = 𝑆 𝑎 , ⊗ 2 1 1 3 ⊕ 𝑆 𝑎 , ⊗ 3 2 1 1 ⊕ 𝑆 𝑎 , ⊗ 1 3 2 1 ⊕ 𝑆 𝑎 , ⊗ 1 1 3 2 (6) then, if we compute the following four tables: 𝑇 [𝑎] = 𝑆[𝑎] ⊗ 2 𝑆[𝑎] 𝑆[𝑎] 𝑆[𝑎] ⊗ 3 , 𝑇 [𝑎] = 𝑆[𝑎] ⊗ 3 𝑆[𝑎] ⊗ 2 𝑆[𝑎] 𝑆[𝑎] , 𝑇 [𝑎] = 𝑆[𝑎] 𝑆[𝑎] ⊗ 3 𝑆[𝑎] ⊗ 2 𝑆[𝑎] , 𝑇 [𝑎] = 𝑆[𝑎] 𝑆[𝑎] 𝑆[𝑎] ⊗ 3 𝑆[𝑎] ⊗ 2 , where 𝑎 ∈ 𝔽 , and the operation " ⊗ " represents multiplication in 𝔽 . each table consists of 256 entries, each of which is 32 bits long. thus, the 𝑗-th column of the state 𝑒 is calculated as follows: 𝑒 = 𝑇 𝑎 , ⊕ 𝑇 𝑎 , ⊕ 𝑇 𝑎 , ⊕ 𝑇 𝑎 , (7) 92 n.v. long, t.t. luong, n.b. cuong, t.m. phuong copyright ©2025 assa adv. in systems science and appl. (2025) now, if we replace shiftrows with tranbytes, then equation (5) becomes: 𝑐 , 𝑐 , 𝑐 , 𝑐 , = ⎣ ⎢ ⎢ ⎢ ⎡ 𝑏 , 𝑏 , 𝑏 , 𝑏 , ⎦ ⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎢ ⎡ 𝑆 𝑎 , 𝑆 𝑎 , 𝑆 𝑎 , 𝑆 𝑎 , ⎦ ⎥ ⎥ ⎥ ⎤ , (8) equation (6) becomes: 𝑒 , 𝑒 , 𝑒 , 𝑒 , = 𝑆 𝑎 , ⊗ 2 1 1 3 ⊕ 𝑆 𝑎 , ⊗ 3 2 1 1 ⊕ 𝑆 𝑎 , ⊗ 1 3 2 1 ⊕ 𝑆 𝑎 , ⊗ 1 1 3 2 , (9) and equation (7) becomes: 𝑒 = 𝑇 𝑎 , ⊕ 𝑇 𝑎 , ⊕ 𝑇 𝑎 , ⊕ 𝑇 𝑎 , (10) from (7) and (10), we see that it is sufficient to use only 4 tables 𝑇 , 𝑗 = 0,1,2,3 to compute the output after the mixcolumns transformation. to eliminate the “if-else” branching in the encryption procedure described in the dynamic aes_dst block cipher algorithm in section 4.1, we calculate the 𝑗-th column in state 𝑒 at round 𝑟 based on the dynamic key bit 𝑘𝑑 as follows: 𝑒 = 𝑇 𝑘𝑑 ⋅ 𝑎 , ⊕ 𝑘𝑑 ⋅ 𝑎 , ⊕ 𝑇 𝑘𝑑 ⋅ 𝑎 , ⊕ 𝑘𝑑 ⋅ 𝑎 , ⊕ 𝑇 𝑘𝑑 ⋅ 𝑎 , ⊕ 𝑘𝑑 ⋅ 𝑎 , ⊕ 𝑇 𝑘𝑑 ⋅ 𝑎 , ⊕ 𝑘𝑑 ⋅ 𝑎 , , (11) where the symbol " ⋅ " denotes multiplication in the decimal system, and 𝑘𝑑 represents the negation of the bit 𝑘𝑑 . in this way, we have eliminated branching in the encryption procedure. moreover, the implementation using precomputed lookup tables can be fully applied to our proposed dynamic aes_dst block cipher. the only difference lies in how the addresses are obtained to access the tables 𝑇 , 𝑗 = 0,1,2,3. to evaluate the execution speed, we implemented the proposed dynamic aes_dst block cipher in c++ using the precomputed lookup table method. below are some statistical results on execution performance along with comparisons to other commonly used block ciphers (compiled and run on the same platform). all tests were conducted on a single-core machine equipped with an intel® core™ i5-7200u cpu @ 2.50ghz (2.471ghz effective), 12gb ram, running windows 10 64-bit, using visual studio 2019 compiler in release mode (x64). the block cipher implementations did not use any assembly instructions and operated in ecb mode. the performance evaluation and speed comparison results are presented in table 7.1. table 7.1. performance evaluation and comparison of the dynamic aes_dst block cipher execution speed no. block cipher block/key size number of rounds lookup table size (encryption + decryption) kbytes encryption/ decryption speed (mb/s) implementation source 1 kalyna 128/128 10 128 1598 oliynykov† 128/256 14 128 1186 2 kuznyechik 128/256 10 128 640 3 aes 128/128 10 8 1696 gladman‡ (on a 32-bit platform) 128/192 12 8 1510 128/256 14 8 1302 4 aes_dst 128/128 10 8 1402 ours 128/192 12 8 1210 † https://github.com/roman-oliynykov/ciphers-speed ‡ http://brg.a2hosted.com//oldsite/cryptography_technology/rijndael/index.php a word-oriented substitution–permutation network cipher 93 copyright ©2025 assa. adv. in systems science and appl. (2025) 128/256 14 8 1087 from table 7.1, it can be observed that with our dynamic approach, the encryption speed of the aes_dst dynamic block cipher decreases only slightly compared to the original aes version (about a 20% reduction). compared to several other block ciphers, the dynamic aes version still maintains a comparable encryption speed. this means that the dynamic aes version we propose can meet the demands of building high-speed cryptographic applications while also enhancing security compared to the original aes block cipher. 8. conclusion the paper provides a different perspective on the security of byte-oriented aes-style block ciphers. to do this, we propose a general spn cipher model. then, we prove using number theory the minimal number of active s-boxes after 4 rounds of encryption in this cipher. our method is generalized and can be applied to evaluate aes-like ciphers such as kalyna, led, kuznyechik, and block ciphers used in hash functions like gost r 34.11-2012, whirlpool, and others. from this analysis, we observe that the role of the byte permutation layer—such as shiftrows in aes—is especially critical. this layer must ensure the 𝑚-diffusion property to preserve the block cipher’s design based on the wide trail strategy. based on these findings, we introduce a key-dependent dynamic variant of the aes block cipher (aes_dst). unlike earlier methods, our approach leverages table lookup implementations without the need for pre-calculated lookup tables. practical tests demonstrate that the dynamic aes cipher’s performance is comparable to well-established block ciphers globally. this dynamic version offers enhanced security over the standard aes while maintaining efficient software implementation. the insights gained here are valuable for the development of secure and adaptable block ciphers, as well as for providing a theoretical basis to assess cryptographic components in current encryption systems. moving forward, we plan to analyze the dynamic aes cipher’s resilience against different types of attacks and estimate its quantum security, particularly considering attacks using grover’s algorithm. acknowledgements funding: no funding was received for conducting this study. financial interests: we declare that we have no financial interests. data availability: data sharing does not apply to this article as no datasets were used, generated, or analyzed during this study declarations competing interests : we declare that we have no competing interests. references [1] shannon, c.e. 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(2010). evaluation of randomness test results for short sequences, proc. of sequences and their applications–seta 2010: 6th international conference (paris, france), pp. 309–319. microsoft word assa word template 2021.doc adv syst sci appl 2022; 02; 85-97 published online at https://ijassa.ipu.ru. environmental assessment of water treatment plants of the republic of ecuador and comparative analysis of water disinfection technologies using the lca method cristian salazar1*, anna i. kurbatova1, milana e. kupriyanova1, kseniya yu. mikhaylichenko1, elena v. savenkova1, alena n. basamykina1 1) peoples’ friendship university of russia (rudn university), moscow, russia e-mail: cristian_salazarf@hotmail.com abstract: in the study, the environmental sustainability of two small water treatment plants with water intake from surface water sources with a high content of organic substances was evaluated. the life cycle assessment was used as a tool to compare two scenarios developed to solve the problem of disinfection by-products (dbps) formation in drinking water. various stages of the obtaining drinking water process were evaluated – from flocculation-coagulation to disinfection. the functional unit was defined as 1 m3 of drinking water produced at a water treatment plant. the proposed scenarios were developed to replace the chlorine disinfection with the uv disinfection of varying intensity (30 mj/cm2 for the first scenario and 186 mj/cm2 for the second scenario), followed by chloroamination. the data were analyzed using the ecoinvent v.3.01 database, modeled and processed in the openlca software. the results showed that at both water treatment plants, the coagulation-flocculation process has the greatest impact on the environment, which is mainly due to the chemical nature of the coagulant. it was revealed that from the point of view of environmental impact, the uv disinfection with an intensity of 30 mj/cm2 is preferable, since the global warming potential (gwp) was 80% less than in the second scenario, while the acidification potential (ap), eutrophication potential (ep), marine aquatic ecotoxicity potential (maep), terrestrial ecotoxicity potential (tetp) and human toxicity potential (htp) were less by 78%, 71%, 72%, 74% and 79%, respectively. keywords: life cycle assessment, water treatment plant, republic of ecuador, uv disinfection 1. introduction the main task of water treatment is to provide consumers with drinking water of proper quality. the treatment includes protection from microorganisms, removal of natural organics and toxic substances, preservation of aesthetic quality, and protection of pipelines from corrosion and re-contamination [1, 2]. drinking water treatment plants in ecuador are usually supplied with surface water (mainly rivers), where water flows from the water catchment area into the water supply network by gravity. according to the economic commission for latin america and caribbean (celac), water purification processes in ecuador are traditional and usually include coagulation, flocculation, precipitation, filtration, and disinfection [3]. these processes require high energy and big amount of chemicals to provide a good quality product for human consumption. however, excessive energy consumption or the use of chemicals, such as coagulants, flocculants, ph stabilizers and disinfectants, not only affect the health of consumers, but also have an impact on the environment [4–6]. this has caused considerable public and industrial interest in the development of new strategies aimed at improving environmental performance throughout the life cycle of water resources management [4, 7], * corresponding author: cristian_salazarf@hotmail.com 86 c.salazar, a.kurbatova, m.kupriyanova, k.mikhaylichenko, e.savenkova, a.basamykina copyright ©2022 assa. adv. in systems science and appl. (2022) as well as obtaining cleaner and more sustainable processes to ensure a better quality of life for the consumer of the final product [8]. according to a study conducted by the national institute of statistics of ecuador, 215 drinking water treatment plants in ecuador use chlorine at the disinfection stage [9]. at these plants, a high dose of chlorine is usually required to meet the desired pathogen inactivation and 0.5 mg/l of disinfectant residue according to the ecuadorian standard 1108 [10]. on the other hand, the high content of natural organics in surface water sources (which is common for reservoirs located in andean ecosystems) [11, 12], combined with a high dosage of chlorine, can lead to a break of the minimum level of disinfection by-products (dbps), therefore, new technologies such as membrane technologies or uv-disinfection are increasingly used in the water management [13]. as usual, the choice of the "best" water treatment system is based primarily on economic and technical constraints. however, the water treatment industry may be responsible for significant global environmental impacts. thus, the complexity of water treatment systems makes it possible to apply broader methods of environmental assessment, such as life cycle analysis regulated by international standards iso 14040 [14] and 14044 [15]. life cycle assessment (lca) is a tool that can be used to obtain information about the environmental impact of water treatment systems. the lca serves to assess global environmental damage potentially caused by a product, process, or service in the "cradle-tograve" approach [16]. consequently, lca offers a more comprehensive approach to quantifying the environmental characteristics of a system or technology to inform management decisions [17]. this method provides a systematic basis for preventing partial optimization, shifts the environmental impact between the stages of the life cycle and helps to identify the main impact factors [18]. the scientific literature on water engineering science includes important studies in which the lca has been used over the past five years. these studies are attracting more and more attention as they are applied to the processes of drinking water treatment plants in developed and developing countries [5, 6, 19, 20]. klopfer and curran (2014) [21] stated the importance of lca-related research for some countries that have not been embedded the idea of life cycle. ecuador is one of these countries. thus, this study represents one of the first lca studies conducted in ecuador on the purification of drinking water. the purpose of this study is to assess the environmental impact of two technological cycles for the drinking water purification, implemented at water treatment plants located in the andean region of ecuador, and to identify environmental "hot spots" of the purification process. the purpose of this study also covers the analysis of the impact of the proposed technologies on the environment as an alternative to chlorine disinfection, improving the understanding of the accommodations of technological solutions while meeting the desired water quality requirements. 2. life cycle assessment of drinking water treatment plants the lca study is carried out in four stages: determination of the purpose and scope of the study, inventory analysis, impact assessment and interpretation of the results. 2.1 definition of the purpose and scope of the study the purpose of this study is to assess the life cycle of 1 m3 of drinking water produced by two water treatment plants located in ecuador. the first water treatment plant is located in latacunga, cotopaxi province, and has a purification capacity of 0.035 m3/s of water coming from the retamales river [22]. the second station is located in the canton of pedro vicente environmental assessment of water treatment plants of the republic of ecuador 87 copyright ©2022 assa. adv. in systems science and appl. (2022) maldonado, pichincha province, and processes an average of 0.030 m3/s of water from the talala river [23]. the physical boundary of the two systems covers the following stages: coagulationflocculation, precipitation, filtration, and disinfection. there are also sub-stages included such as the supply of aluminum sulfate, chlorine gas, caustic soda and polyaluminium chloride (pac). the water supply network is beyond the scope of this study. infrastructure and decommissioning of water treatment plants are not considered, as the results of other studies indicate that the environmental impact is insignificant compared to the operation stage [24, 25]. since this study is not aimed at comparing stations, the results are not subject to mandatory uncertainty analysis at the level of installations as a whole (according to iso standards). fig. 1 shows the purification processes carried out at the latacunga and pedro vicente maldonado water treatment plants. during the purification the flow of water from the catchment area into the water supply network flows by gravity, so there is no energy consumption during sedimentation and filtration. therefore, only coagulation-flocculation and disinfection processes were considered for the assessment. fig. 1. system boundaries of the process at the studied water treatment plants. the chemical reagents used for each process are indicated by a dotted line 2.2 life cycle inventory analysis at this stage, an inventory of the water treatment system was carried out, based not only on information about the electricity consumption and the amount of chemicals needed in the process of preparing drinking water, but also on the equivalent pollution factor. both primary 88 c.salazar, a.kurbatova, m.kupriyanova, k.mikhaylichenko, e.savenkova, a.basamykina copyright ©2022 assa. adv. in systems science and appl. (2022) and secondary sources of information were used for this inventory. the primary source corresponded to the data provided by the technical staff of the studied water treatment plants, who collected information during the 12-month operation period, from january to december 2019. the secondary source corresponded to the ecoinvent v3.1 database [26], designed to provide general bench-mark data on products and processes used at drinking water treatment plants. it should also be noted the main limitation of the study: there are no specific baseline data for ecuador, therefore average technological data for peru were used, since the levels of technology and environmental performance are similar from the point of view of the studied process. table 1 shows the inventory of chemical reagents and the amount of electricity at each of the studied plants. table 1. inventory data of the investigated water treatment plants, expressed in the functional unit (1 m3) of drinking water produced at the stations. process operational data drinking water treatment plant of latacunga city drinking water treatment plant of pedro vicente maldonado canton coagulationflocculation caustic soda (kg) 0.005 0,006 aluminium sulfate (kg) 0.0445 polyaluminium chloride pac (kg) 0.0168 energy consumption (kwh) 0.0022 0.0017 disinfection chlorine gas (kg) 0.0011 0.0017 energy consumption (kwh) 0.0020 0.0024 chemical reagents are supplied by manufacturers located on the territory of ecuador. table 2 shows the logistics data. table 2. transportation of chemical reagents chemicals approximate transportation distance (km) type of transportati on, city drinking water treatment plant of latacunga city drinking water treatment plant of pedro vicente maldonado canton caustic soda 100 130 truck, quito city aluminum sulfate polyaluminium chloride (pac) chlorine gas the openlca v. 10.3 software was used to simulate the systems under study. the cml 2001 method [27] was used to transform input and output streams into impact categories. the exposure categories selected in this study included global warming potential (gwp), acidification potential (ap), eutrophication potential (ep), human toxicity potential (htp), marine aquatic ecotoxicity potential (maep) and terrestrial ecotoxicity potential (tetp). environmental assessment of water treatment plants of the republic of ecuador 89 copyright ©2022 assa. adv. in systems science and appl. (2022) 2.3 impact assessment and interpretation of results table 3 presents the general results obtained by applying environmental impact indicators for the two studied water treatment plants, in comparison with the literature data. table 3. comparison of the environmental impact data obtained with the literature data reference main assumption gwp (kg co2 eq) ap (kg so2 eq) ep (kg po4 eq) htp (kg 1,4-db eq) tetp (kg 1,4-db eq) latacunga water treatment plant distribution network excluded 4.40e-2 5.20e-4 9.26e-5 3.45e-2 1.50e-4 pedro vicente maldonado water treatment plant distribution network excluded 1.68e-2 1.40e-4 3.16e-5 1.26e-2 6.01e-5 ortiz et al. (2016) [5] distribution network excluded 2.21e-2 2.28e-4 5.03e-5 8.04e-3 n/a amores et al. (2013) [19] distribution network excluded 1.77 e-1 6.72e-4 6.87e-5 n/a n/a friedrich and buckley 2002 [24] distribution network excluded 1.73e-1 1.02e-3 6.54e-5 3.31e-3 2.30e-1 friedrich et al. (2012) [4] distribution network excluded 1.85e-1 1.10e-3 7.40e-5 4.09e-3 2.59e-1 saad et al. (2018) [20] distribution network excluded 3.42e-1 1.13e-3 1.06e-4 1.65e-2 2.55e-4 regarding the categories of gwp, ap, ep and htp, the results of this study are either in the range or close to the literary values indicated for the traditional water purification process [4, 5, 19, 20, 24]. however, for tetp, the results of this study are below those given in the literature. differences between the results and the literature data may be related to different operations at the water treatment plant included in the traditional purification process, certain geographical conditions of the place or changes in the configurations of water treatment plants. each process contributes to different impact categories. figures 2 and 3 present the results of the characteristics of lca processes, which show the proportion of each process in different categories of impact. 90 c.salazar, a.kurbatova, m.kupriyanova, k.mikhaylichenko, e.savenkova, a.basamykina copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 2. contribution of caustic soda, aluminum sulfate, chlorine gas, electricity, and transport of chemical reagents to the environmental impact at the latacunga water treatment plant. figure 3. contribution of caustic soda, polyaluminium chloride, chlorine gas, electricity, and transport of chemical reagents to the environmental impact at the pedro vicente maldonado water treatment plant. the exposure profile in the fig. 2 shows that approximately 96% of the total amount of htp, ap and tetp is formed during the coagulation-flocculation process (88% aluminum sulfate and 8% caustic soda). about 98% of the total amount of maep is accounted for by the coagulation-flocculation process (88% aluminum sulfate and 10% caustic soda). the coagulation-flocculation process accounts for 94% of the total gwp (80% aluminum sulfate and 14% caustic soda). according to the results of the fig. 3, at the water treatment plant in the canton of pedro vicente maldonado, approximately 94% of the total amount of htp is formed from pac (65%) and caustic soda (29%). about 93% of the total ep is accounted for by caustic soda (47%) and polyaluminium chloride (46%). also 91% of the ap is formed from polyaluminium chloride (52%) and caustic soda (39%). about 96% of the total tetp is accounted for by polyaluminium chloride (70%) and caustic soda (26%). approximately 96% of all maep is formed during coagulation-flocculation (62% polyaluminium chloride and 34% caustic soda). about 87% of the total gwp is accounted for by the coagulationflocculation process (caustic soda 45% and polyaluminium chloride 42%). environmental assessment of water treatment plants of the republic of ecuador 91 copyright ©2022 assa. adv. in systems science and appl. (2022) at two water treatment plants, the transportation of chemicals is minimal (from 0.3 to 3.2%) in all impact categories. these results are comparable to the literature data. thus, the environmental impact associated with the transportation of chemicals is insignificant [18]. the greatest environmental stress or "trouble spot" of studied systems occurs during coagulation-flocculation process due to the use of coagulating agents and ph stabilizers. some studies report that the greatest contribution to the environmental impact is made by electricity used for cleaning processes [4, 20, 25], while other studies show that the greatest contribution to the environmental impact is made using chemical reagents [5, 6, 28, 29]. in an ideal scenario, water treatment plants involved in providing a sustainable supply of drinking water should reduce the environmental load throughout the process. for example, the use of pac can be reduced using natural coagulants. in addition, it is important to control the water source of water treatment plant since its misuse leads to pollution of the water to be treated. such pollution consists in non-compliance with the standards of chromaticity, turbidity, and suspended solids, and, consequently, the amount of coagulant needed for water purification increases. the best way to reduce an amount of coagulation is to analyze the effectiveness of the pretreatment process designed to improve the physico-chemical characteristics of water entering the water treatment plant. the disinfection process at both water treatment plants has the smallest impact in all categories of exposure (from 0.3 to 2%). although this process poses no danger from an environmental point of view, the use of chlorine gas as a disinfectant is widely discussed because of its possible negative impact on human health [30, 31]. chlorine gas reacts with water to form hypochlorous acid. in the presence of bromine, brominous acid is also formed. hypochlorous and brominous acids form strong oxidizing agents in water and react with a wide variety of compounds, so they are one of the most effective disinfectants. however, these acids also react with natural organics to form dbps [31]. 3. life cycle environmental implication to reduce dbps dbps can be controlled using one of three main strategies: 1) modification of disinfection methods; 2) enhanced removal of dbps precursors; and 3) removal of dbps after formation [32]. in this study, the first strategy of dbps control in the development of alternative scenarios was studied. 3.1 definition of the purpose and scope of the study the two scenarios based on uv-disinfection were identified and evaluated from an environmental point of view. considering the operational capacity of the studied subjects, a reference water flow rate (0.030 m3/s) was used to represent the system. the lca was carried out mainly by data obtained as a result of design calculations based on processes, the ecoinvent database and relevant literature [33, 34]. the first scenario: a combination of uv radiation at a dose of 30 mj/cm2 and disinfection with chlorine is used to achieve the required 4-log inactivation of viruses and 3log inactivation of cryptosporidium and giardia [35]. chloramine is used for residual disinfection, which is achieved by adding anhydrous ammonia to hypochlorite (fig. 4a). the second scenario: a uv dose of 186 mj/cm2 is added to achieve both 4-log virus inactivation and 3-log inactivation of cryptosporidium and giardia [35]. as it was in the first scenario, chloramine also provides residual disinfection (fig. 4b). 92 c.salazar, a.kurbatova, m.kupriyanova, k.mikhaylichenko, e.savenkova, a.basamykina copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 4. schemes of the proposed disinfection processes based on uv-radiation. a) uv dose 30 mj/cm2; b) uv dose 186 mj/cm2 3.2 inventory analysis the type and quantity of materials for the infrastructure of uv disinfection systems were based on the results presented by mo et al. (2018) [33]. in this study, the entire infrastructure was designed for a peak load of 0.033 m3/s and an average concentration of organic carbon of 4.6 mg/l. the infrastructure of the uv disinfection system includes the following components: uv lamps, quartz sleeve, sleeve cleaning system, uv sensor for monitoring of uv radiation, and a control system for the entire uv system. the energy values for existing uv disinfection systems operating at a dose of 30 mj/cm2 were set to the corresponding consumption (0.033 m3/s); electricity indicator was set at 0.041 kwh/m3. this value corresponds to the theoretical values for uv disinfection of drinking water (from 0.01 to 0.05 kwh/m3). disinfection by uv radiation affects the amount of sodium hypochlorite used as the main disinfectant. in the first scenario, hypochlorite is used to achieve 4-log virus inactivation. to calculate the required amount of hypochlorite, a contact timetable provided by the us environmental protection agency was used [36]. at an average temperature of 15 °с with a ph of 6-9, a contact time of 4 min-mg/l is required to achieve 4-log virus inactivation. assuming the contact time is 30 minutes, 0.13 mg/l of free chlorine is required. for residual disinfection, anhydrous ammonia is added to produce chloramine. since chloramine has a low disinfection efficiency, a residual chlorine concentration of 1.0 mg cl2/l is required. considering that the formation of chloramine requires a 5:1 weight ratio of chlorine to ammonia, the amount of hypochlorite required for residual disinfection should be 4.45 mg cl2/l, and the amount of ammonia should be 0.89 mg n/l. in the second scenario, an additional amount of hypochlorite is not required. to completely avoid the formation of dbps, chloramine is used as a residual disinfectant. the processes considered at the operational stage are described in detail in table 4. environmental assessment of water treatment plants of the republic of ecuador 93 copyright ©2022 assa. adv. in systems science and appl. (2022) table 4. the amount of electricity and chemicals required for the proposed disinfection scenarios. the values are given for a functional unit of 1 m3 for 1 year (for a reference water flow rate of 0.03 m3/s) process units quantity (for the first scenario) quantity (for the second scenario) uv energy consumption kwh 39320.45 275243.15 liquid hypochlorite (chlorine) kg 2144.27 1886.16 anhydrous ammonia kg 1022.44 1022.44 3.3 impact assessment and interpretation of results the results of the lca for each impact category for both scenarios are presented in table 5. table 5. results of environmental impact assessment of two disinfection scenarios impact category unit first scenario second scenario difference (%) gwp kg co2 eq 1.73 8.77 80 ap kg so2 eq 107.01 508.76 78 ep kg po4 eq 17.09 59.04 71 maep kg 1,4 db eq 1.15 4.16 72 tetp kg 1,4-db eq 36.63 145.55 74 htp kg 1,4-db eq 1.13 5.46 79 as can be seen from the table 5, the second scenario has higher environmental impacts compared to the first. this is primarily due to the large amount of electricity needed just for uv disinfection to achieve 4-log virus inactivation. this is consistent with the conclusions of cashman et al. (2014) [37] and carré et al. (2017) [38], who reported that the use of traditional uv technology increases the environmental impact during disinfection by increasing the dose of uv radiation, since more electricity is consumed. the global warming potential (gwp) in the first scenario is 80% less than in the second scenario, and the impact of acidification, eutrophication, marine aquatic ecotoxicity, terrestrial ecotoxicity and toxicity to humans in the first scenario is less by 78 %, 71%, 72%, 74% and 79%, respectively. the relative contribution of the infrastructure and the operational phase of the first scenario to various categories of impacts is shown in figure 5. figure 5. the contribution of infrastructure, energy consumption of uv lamps and chemicals to environmental impact of the uv disinfection system at a dose of 30 mj/cm2 94 c.salazar, a.kurbatova, m.kupriyanova, k.mikhaylichenko, e.savenkova, a.basamykina copyright ©2022 assa. adv. in systems science and appl. (2022) according to the results shown in figure 5, the largest impact on global warming, acidification, eutrophication, and marine aquatic ecotoxicity is accounted for by electricity: 66%, 60%, 38% and 34%, respectively. the infrastructure of the uv disinfection system has a minimal impact of global warming and acidification, which ranges from 2 to 4%. on the other hand, the infrastructure of the uv disinfection system has a significant impact in several categories, where it ranges from 40% for terrestrial ecotoxicity to 56% for the category of human toxicity exposure. these results are consistent with carré et al. (2018) [38], where it was indicated that the infrastructure of the uv disinfection system can contribute most to certain categories of environmental impact, depending on the nature of the materials used. however, in other studies, all impacts on the infrastructure of uv systems were insignificant compared to operational impacts [34, 37]. several studies show that the impact on infrastructure becomes increasingly insignificant as energy consumption increases [29] or the complexity of the water treatment system rises (i.e., the number of processes, the amount of chemicals and energy) [25, 37]. the use of uv technology increases the environmental impact during disinfection by increasing electricity consumption but eliminates the formation of disinfection by-products and significantly reduces the use of dangerous chlorine. in general, the negative impact of the uv disinfection system far outweighs the benefits of refusing chlorine, and the use of chlorine or at least partial use of chlorine is environmentally preferable. 4. conclusions this study applied a life cycle environmental assessment methodology to assess the environmental impact of two small drinking water treatment plants located in ecuador, fed from surface waters with a high content of natural organics. two scenarios for solving the problem of disinfection by-products have been developed and compared in terms of life cycle assessment and environmental impact. the results show that the greater environmental damage caused by water treatment plants is mainly explaining by use of coagulants such as aluminum sulfate and polyaluminium chloride. at the latacunga water treatment plant, aluminum sulfate has the largest impacts, while at the pedro vicente maldonado water treatment plant, the contribution of polyaluminium chloride is 65%, 52%, 70% and 62% of the global warming potential, acidification potential, terrestrial ecotoxicity potential and marine aquatic ecotoxicity potential, respectively. it also was revealed that uv disinfection with an intensity of 30 mj/cm2 for an estimated water consumption of 0.030 m3/s has the least impact on the environment compared to the second scenario, where the dose of uv radiation is 186 mj/cm2. for most of the studied criteria, the stage of operation (energy demand of uv lamps, sodium hypochlorite and ammonia) globally has the greatest impact, although for some criteria, such as terrestrial ecotoxicity and human toxicity potential, the main contribution is made by the infrastructure of the uv disinfection system. the environmental impact of the process is only one of the criteria that should be considered when choosing this technology there. in addition, for a comprehensive assessment, it is also necessary to measure operating costs. therefore, for future research, it is recommended to focus on the integration of environmental and economic life cycle assessment. acknowledgements this paper has been supported by the rudn university strategic academic leadership program. conflicts of interest the authors declare no conflict of interest. environmental assessment of water treatment plants of the republic of ecuador 95 copyright ©2022 assa. adv. in systems science and appl. 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(2014) environmental and cost life cycle assessment of disinfection options for municipal drinking water treatment. u.s. epa 2014, no. october, 104. 38. carré, e., beigbeder, j., jauzein, v., junqua, g., lopez‐ferber, m. (2017). life cycle assessment case study: tertiary treatment process options for wastewater reuse. integrated environmental assessment and management, 13, 1113–1121. microsoft word 1385 adv syst sci appl 2023; 01; 1–7 published online at https://ijassa.ipu.ru. comparative analysis of the efficiency of molybdenum trioxide extraction from a spent hydraulic catalyst elena yu. nevskaya1*, olga a. egorova1, sergey p. perekhoda2 1) рeoples’ friendship university of russia (rudn university), moscow, russia 2) baikov institute of metallurgy and materials science, ras, moscow, russia abstract: the technological process uses catalysts containing oxide compounds of molybdenum, cobalt or nickel deposited on an alumina substrate. after a certain time, the activity of the components decreases and the unrecoverable spent catalysts usually become toxic waste. technogenic residues are prohibited for disposal, and it is advisable to use the exhausted catalyst components for subsequent recycling. recycling of industrial waste is a solution to an environmental problem. processing should be organized with the extraction of valuable components, in particular expensive molybdenum oxide (vi). this work is devoted to finding ways to optimize the process of extracting molybdenum from spent hydrotreating catalysts. keywords: hydrotreating catalysts, molybdenum-containing catalysts, kinetic regularities 1. introduction despite the periodic regeneration, the life of the hydrotreating catalysts is short. in the technological process, catalysts are used that represent a system of oxide compounds of molybdenum, cobalt or nickel deposited on an alumina support. after a certain time, the activity of the components decreases and spent catalysts that cannot be restored usually become toxic waste. technogenic residues are prohibited for burial, and it is advisable to use catalyst components that have exhausted their resource for subsequent recycling. disposal or recycling of industrial waste is the solution to an environmental problem. processing should be organized with the extraction of valuable components expensive molybdenum (vi) oxide [10,20]. at present, pyrometallurgical [1,13,25] and hydrometallurgical [2,11,17] methods have been developed, which make it possible to extract molybdenum and associated metals from spent catalysts. hydrometallurgical processes, in comparison with pyrometallurgical ones, pollute the environment less with harmful emissions and, most importantly, allow selective extraction of valuable metals. according to various researchers, the highest degree of extraction of molybdenum by the hydrometallurgical method reaches from 80 to 95% [9,14,18,19]. for "wet" leaching, inorganic strong acids, aqueous solutions of ammonia or soda ash are used. however, the literature provides conflicting information on the carbonate extraction of molybdenum. this work is a continuation of the search for optimal conditions for the selective extraction of molybdenum from spent molybdenum-containing hydrotreating catalysts. the aim of this study is to comparatively study the kinetic regularities of the interaction of solutions of nitric acid (hno3) and sodium carbonate (na2co3) with a molybdenumcontaining catalyst, as well as to determine the optimal conditions for extracting molybdenum from molybdenum (vi) oxide moo3. * corresponding author: nevskaya-eyu@rudn.ru 2 e. yu. nevskaya, a. o. egorova, s. p. perekhoda copyright ©2023 assa adv. in systems science and appl. (2023) 2. experimental part the object of the study was the most common industrial catalyst for hydrotreating diesel fuel, gkd-205. the spent aluminum-nickel-molybdenum (anm) catalyst is cylindrical granules (2.0 × 4.0-7.0 mm) of blue, gray-brown and black color. the active components of the gkd-205 catalyst are oxide compounds of d-metals of variable valence, resistant to poisoning with various poisons: molybdenum, nickel and other elements deposited on a substrate (matrix) of the low-temperature unstable modification γal2o3. durable high-purity alumina al2o3 with optimal acid-base properties forms a specific porous structure of the catalyst and thus ensures the maximum content of molybdenum (vi) oxide moo3. chemical analysis of the gkd-205 brand sample, carried out by atomic emission spectral analysis (aesa) with inductively coupled plasma on an ultima-2 optical emission spectrometer, showed a content of 6-12.2 wt%. molybdenum (vi) oxide moo3; up to 3.14% of the mass. promoter (nickel oxide nio) and up to 88.02% of the mass. aluminum oxide al2o3. by means of x-ray diffraction analysis (x-ray powder diffractometer dron3m, cukαand cokα-radiation) it was found that the spent catalyst matrix consists mainly of two modifications of aluminum oxide α-al2o3 and γ-al2o3. alumina al2o3 is identified on diffraction patterns in the composition of nickel aluminates nial2o4, nial26o40 and sodium na2al2xo3x, which have the closest cubic packing of the spinel type. molybdenum and nickel are found in the oxide forms of molybdenum (vi) trioxide moo3 and nickel (ii) oxide nio, including in the spinel structures al2(moo4)3, nimoo4, na2moo4. ir spectroscopic data in the range of 400-4000 cm-1 were recorded on a specord m-80 spectrometer and an avatar360 esp fourier spectrometer. chemical analysis for the content of molybdenum in the spent hydrotreating catalysts was carried out in accordance with the procedure described in standard 11930.11-79 [15]. the study of the leaching kinetics of the mo-containing catalyst was carried out with solutions of nitric acid and sodium carbonate under isothermal conditions by sampling. a portion of 30 g of the spent anm catalyst, crushed to a particle size of 0.5-1 mm, was loaded into a thermostatically controlled reactor with a volume of 200 cm3. the concentrations of nitric acid hno3 were (mol / l): 1.67, 3.54 and 5.62. the ratio of solid and liquid phases was 1: 5. the concentration of soda ash na2co3 was (mol / l): 0.498, 1.064 and 1.658. the ratio of solid and liquid phases was 1:10. the thermostatic reactor was kept at temperatures of 338 and 368 k. the dissolution time of the catalyst weighed portion was recorded from the moment of loading. the study was carried out in the free-floating mode of particles with continuous stirring of the reaction solution with an electric mixer. the mixer rotations were constant (650 rpm), and this allows us to assume that there are no external diffusion braking in the system. at a certain time, interval, 1.0 ml samples were taken from the reaction mixture using a suction device with a filter attachment. the content of dissolved components in the samples was determined by inductively coupled plasma atomic emission spectral analysis on an ultima-2 optical emission spectrometer with an accuracy of ± 2.0%. 3. results and discussion based on the results of the experimental data, the fraction of dissolved molybdenum trioxide moo3 (α) was calculated according to the equation: α = (1) in this mathematical dependence: comparative analysis of the efficiency of molybdenum trioxide extraction… 3 copyright ©2023 assa. adv. in systems science and appl. (2023) с1 – concentration of molybdenum (vi) in solution, which was determined according to known methods [15-17]; v – volume of solution in the reactor, cm3; m – weight of the spent catalyst, g; с – metal content in the spent catalyst, wt. fraction. to describe the results from the standpoint of formal heterogeneous kinetics [4,7,21], the kinetic parameters were processed using the mathcad 11 software using the nonlinear regression method. for this, the experimental data were presented in the form of the dependence of the degree of conversion of the solute (α) on time (τ) α = f (τ) and described by the equation of the chain mechanism. this equation takes into account the formation of active centers (crystal lattice defects) on the surface of particles during dissolution (fig. 1 (a; b)): α = 1 – exp(a · sh(wi · τ) (2) here: α – the proportion of dissolved molybdenum (vi) oxide; a – a constant proportional to the average number of active centers (the number of crystal lattice defects) on the surface of one particle of the spent catalyst; wi – dissolution rate constant, min-1; τ – time, min. to calculate the values of the constants a and wi, the kinetic curves were rearranged in the coordinates α τ / τ0.5: the degree of conversion of the solute (α) versus the reduced time (τ / τ0.5) (fig. 1). fig. 1. kinetic curves in the coordinates α τ / τ0.5 for nitric acid hno3 from fig. 1 it follows that, considering the error, all the experimental data fall on one curve, which indicates the invariance of the dissolution mechanism at different parameters of the solution. the plotted curve in the coordinates α τ / τ0.5 was superimposed on the grid of theoretical curves of equation (2), which made it possible to determine the constant a of equation (2), equal to 5. the results of processing the experimental data according to equation (2) are presented in table 1, which shows the values of the constant dissolution rate wi. 4 e. yu. nevskaya, a. o. egorova, s. p. perekhoda copyright ©2023 assa adv. in systems science and appl. (2023) table 1.the rate constant of dissolution (wi) in solutions of nitric acid and sodium carbonate, calculated according to equation (2), a = 5 t, k c (hno3), mol / l wi, min-1 c (na2co3), mol / l wi, min-1 338 1,67 0,005 0,498 0,007 3,54 0,008 1,064 0,007 5,62 0,01 1,658 0,0075 368 1,67 0,01 0,498 0,013 3,54 0,01 1,064 0,038 5,62 0,019 1,658 0,088 to determine the order of the dissolution rate of molybdite (moo3) included in the spent catalyst, based on the concentration of nitric acid [hno3] and the concentration of soda ash [na2co3], the experimental data were expressed in coordinates log (wi ) – log (c). the order of the rate of dissolution of molybdite in a nitric acid solution in terms of hno3 concentration is 0.60 ± 0.03. the order of the dissolution rate of molybdenum (iv) oxide in sodium carbonate solution in terms of na2co3 concentration is lower and equal to 0. the obtained values of the reaction orders with respect to the reagent lie in the range from 0 to 2, which allows us to conclude that the processes are proceeding in the kinetic region. the calculation of the activation energy (eact) was carried out according to the arrhenius equation (3): k = a · (3) where k – reaction rate constant, min -1; еаcт – activation energy, j / mol. the activation energy (eact) of the transition to a solution of nitric acid hno3 of molybdenum trioxide in the spent catalyst was 20.00 ± 0.1 kj / mol, which indicates the progress of the process in the intermediate region. for sodium carbonate na2co3 solutions, the activation energy is in the range of 80.00 ÷ 120.00 kj / mol, which confirms the course of the dissolution process in the kinetic region. with an increase in the concentration of mineral acid, the rate of transition of the oxide compound of molybdenum (vi) into the hno3 solution increases. the empirical equations describing the rate of release of the oxide compound of molybdenum (vi) from the aluminum-nickel-molybdenum catalyst into the solution of nitric acid hno3 (4) and into the solution of sodium carbonate na2co3 (5) have the form: wi (moo3) = [hno3]0,60 ± 0,03 · exp(20000/rt) wi (moo3) = [na2co3] · exp(80000/rt) (4) (5) where – the constant of the specific dissolution rate: in nitric acid 0.090 ± 0.005 min-1, and in a solution of soda ash 0.20 ± 0.02 min-1. thus, under these conditions, the rate of extraction of molybdenum by nitric acid solution is minimal. it depends only on the ph value of the solution and does not depend on the concentration of the nitrate anion. based on the data [8,22,23,24] describing the behavior of molybdenum (vi) in electrolyte solutions and the experimental data of this work, we have optimized the scheme [12,23] comparative analysis of the efficiency of molybdenum trioxide extraction… 5 copyright ©2023 assa. adv. in systems science and appl. (2023) showing the distribution of molybdenum-containing ions in electrolyte solutions at different ph values. this makes it possible to derive the dependence of the rate of dissolution of molybdite (moo3) on the concentration of nitric acid [ha] and on temperature, which can be described by a langmuir-type function. in general: wi = · (6) surface concentration is expressed through the concentration of monobasic mineral acid, which coincides with the empirical relationship (4): wi = w0 · · exp(38000/rt) = w0[hno3]0,60 ± 0,03 (7) where w0 – constant of the specific rate of dissolution, min-1. influence of the ph of the medium on the number of acid-base centers on the surface of molybdite was established by the method of ir spectroscopy. thus, the spectroscopic data show the presence in the selected aliquots of a compound of empirical composition moo2oha, the transition of which into the solution limits the rate of dissolution. it is shown that the dissolution of moo3 and the transition of mo6+ ions from the spent catalyst into the solution proceeds in a stepwise dependence on the ph of the solution. it was found that with an increase in the acidity of the solution, the number of active lewis and bronsted weakly basic and weakly acidic centers decreases symbatically with a decrease in the activity of the catalysts. the dissolution of molybdenum (iv) trioxide with an increase in the concentration of mineral acid (hno3) is associated with the formation of poorly soluble compounds of various compositions on the catalyst surface. the rate of dissolution of moo3, considering the influence of the concentration of sodium carbonate solution, was calculated based on the fact that the value of adsorption of na+ and carbonate ions on the surface is determined by the equation: w = w0 · · exp(80000/rt) = w0[na2co3]0 (8) studies of the kinetics of soda decomposition of an aluminum-nickel-molybdenum catalyst for hydrotreating with solutions of different concentrations have confirmed that as the leaching duration and reagent concentration increase, the degree of molybdenum extraction naturally increases. based on the experimental results obtained on the solubility of molybdenum (vi) oxide in solutions with different ph values and kinetic data, the conclusion was clarified that in alkaline media, in comparison with acidic media, the dissolution of molybdenum (vi) oxide occurs deeper at lower concentrations of reagents. acknowledgements this paper has been supported by the rudn university strategic academic leadership program. 6 e. yu. nevskaya, a. o. egorova, s. p. perekhoda copyright ©2023 assa adv. in systems science and appl. 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(2020). modern methods for the regeneration of catalysts used in oil and gas synthesis, the scientific heritage, 50, 41–43. microsoft word 1969.docx adv syst sci appl 2025; 3; 52-73 published online at https://ijassa.ipu.ru. analysis of the thermal behavior of a workspace in the adrar region oudrane abdellatif1*, aour benaoumeur2, bada abdelmalek1,3, kheiri abdelhamid4 1) ahmed draïa university of adrar, adrar, algeria 2) national polytechnic school of oran, oran, algeria 3) university of science and technology of oran, bir el djir, algeria 4) university of lorraine, lorraine, france abstract: this study presents an experimental and numerical investigation of the thermal behavior of a workplace located in the adrar region. temperature variations within the building were analyzed through a detailed modeling approach and validated using data collected during a 20-day experimental campaign conducted in may 2024. all modes of heat transfer were considered, and energy balance equations were formulated for each structural material. the resulting system of equations was solved using an implicit finite-difference scheme combined with the gauss algorithm and an iterative procedure, accounting for the dependence of convective and radiative heat-transfer coefficients on the surrounding temperatures. numerical results indicate that the thermal conductivity of the heavy concrete bricks, together with the local climatic conditions, significantly contributes to indoor temperature increases. comparison of numerical predictions with experimental measurements for two representative days (may 10 and may 18, 2024) demonstrated excellent agreement, with errors not exceeding 4%. the findings provide insights into the thermal performance of public buildings in the adrar region and offer a reliable numerical tool for engineers to select appropriate construction materials. the study also highlights that modern building materials, when used without adequate thermal insulation, can lead to excessive indoor temperatures and uncontrolled energy consumption in this desert environment. keywords: habitable construction, thermal behavior, workplace, heat convection, thermal conductivity, building materials, desert areas, internal temperature. nomenclatures cp specific heat -1-1 .kj.kg e material thickness m fi geometric form factor i electric current intensity ma hr radiation heat exchange coefficient -1-2 .kw.m hconv convection heat exchange coefficient -1 . -2 kw.m hcext external heat exchange coefficient for convection -1 . -2 kw.m l length m mps mass of the south wall kg qccv amount of heat exchanged by convection w qdoor amount of heat exchanged by the door w qwall amount of heat exchanged by the wall w qabsor amount of heat absorbed w qsource amount of heat from the source w qr amount of radiative flux w.m-2 φ heat flux density w.m-2 r electrical resistance ω sps south wall surface m2 tamb room temperature °c tground soil temperature °c tair air temperature °c text external temperature °c tint internal space temperature °c * corresponding author: abdellatif.habadat@gmail.com, +213662830459 oudrane a., aour b., bada a. and kheiri a. 53 copyright ©2025 assa. adv. in systems science and appl. (2025) δt temperature deviation °c tvc temperature of the celestial vault °c tpsi internal south wall temperature °c tpni temperature of the internal north wall °c tpfpi internal temperature of the wall ceiling °c glas u surface thermal transmittance coefficient of glass w.m-2. k-1.j-1 vair air volume m3 vwind average wind speed m. s-1 lettres grecques  absorption coefficient  emissivity coefficient  thermal conductivity -1-1 .kw.m  density kg.m-3 σ stefan-boltzmann constant w. m-2. k-4 nombres adimensionnels gr grashof number pr prandtl number ra rayleigh number abreviation ps south wall pn north wall pe east wall po west wall pfp ceiling wall psi internal south wall pni internal north wall pei internal east wall poi internal west wall pfpi internal ceiling wall pse external south wall pne external north wall pee external eat wall poe external west wall pfpe external ceiling wall htc building heat transfer coefficient dps date palm spikelet dpf recycled date palm fiber dpp petiole of the date palm xps panneau de polystyrène extrudé pcm phase change materials htr hourly temperature reduction hhgr hourly reduction in heat gain eca evaporative air conditioning nv natural ventilation 1. introduction quantifying thermal exchanges between buildings and their surrounding environment at the urban scale requires a detailed understanding of local energy balances across multiple levels, from districts and building clusters down to individual structures and specific building components. the multiscale nature of this problem makes optical sensor technologies particularly relevant, as they enable non-contact spatial measurements that remain effective at various scales. recent advances in sensor technology have enhanced the resolution of detection matrices and improved associated optical systems, allowing for more precise analysis of observed scenes. the luminance measured by these sensors originates from multiple elements within the field of view, from which surface temperatures of the urban environment can be inferred [1,2]. establishing the relationship between measured luminance, the temperature of individual components, and heat transfer through building envelopes requires comprehensive modeling of the underlying physical phenomena [1,3,4]. these mechanisms are highly interdependent, as surface energy balances are strongly influenced by local climatic conditions. a review of existing methods for quantifying thermal exchanges in buildings indicates that their development has been largely shaped by methodological choices, attracting growing interest from the scientific community concerned with energy storage and conservation. several representative studies are highlighted below for illustrative purposes. 54 oudrane a., aour b., bada a. and kheiri a. copyright ©2025 assa adv. in systems science and appl. (2025) kumaraswamy et al. [5] investigated thermal storage in building materials and long-term energy conservation using phase-change materials (pcm). due to their latent heat capacity, pcms can store significant amounts of heat or cold within limited volumes, reducing fluctuations in indoor air temperature and maintaining thermal comfort over extended periods. selecting an appropriate pcm, however, requires careful consideration of local climatic constraints. kaoutari et al. [6] conducted numerical and experimental analyses of coupled heat and moisture transfer in a biobased double-layer wall system, integrating measured material properties into a dynamic model based on the approaches of künzel and mendes. finitedifference methods were employed to solve the coupled transfer processes, with transient simulations representing real winter and summer conditions. comparisons demonstrated that experimental data provided highly accurate solutions. virginia et al. [7] examined uncertainties associated with co-heating tests to evaluate the impact on estimated building heat transfer coefficients (htc), showing that weather conditions, complex building designs, and experimental practices can significantly influence htc estimates. belatrache et al. [8] explored the use of date palm waste as insulating material in arid climates, demonstrating reductions in cooling electricity consumption of up to 64.7% compared to traditional constructions. laidi et al. [9] assessed energy performance of a singlefamily house with a mansard roof in algeria, showing that attics, insulated or not, contribute to reduced cooling demand across various climatic zones. jiang et al. [10] modified clay walls with cement, gravel, sand, and polypropylene fibres, highlighting the strong heat and moisture buffering capacity of the modified walls, reducing indoor temperature and humidity fluctuations. chaoping et al. [11] studied thermal optimization of traditional rural dwellings in northeastern sichuan, comparing four wall renovation strategies through numerical simulations. central insulation combined with extruded polystyrene (xps) panels significantly improved indoor thermal comfort. qudama et al. [12] investigated compacted pcms integrated into building envelopes under warm climates, achieving hourly temperature and heat gain reductions of up to 15.1% and 34.9%, respectively, and lowering indoor temperatures by up to 4 °c. yadeta et al. [13] conducted a field survey of thermal comfort in 104 dwellings, producing adaptive comfort models essential for designing sustainable dwellings in regions with limited energy access. guo et al. [14] performed on-site studies of office buildings in turpan, china, under evaporative cooling and natural ventilation, revealing discrepancies with current adaptive comfort standards and providing guidance for updating design parameters and passive strategies. building on these insights, the present study investigates the thermal behaviour of a workplace in the adrar region of southern algeria. an experimental campaign conducted in may 2024 over 20 days was implemented to validate the numerical results obtained from a thermal model developed for this work. temperature and humidity measurements (testo 175t2, testo 175t3, and testo 174h) included interior surface temperatures of south and north walls, interior roof surface temperature, indoor air temperature, indoor relative humidity, and outdoor air temperature. to achieve the stated objectives, this work is structured as follows. a description of the studied workplace, serving as an office for a university lecturer, is provided in section 1, followed by the climatic characteristics of the adrar region in section 2. section 3 details the measurement instruments and the simplifying assumptions adopted in the experimental protocol. the mathematical modeling of the energy balance is introduced in section 4, and finally, section 5 presents a comprehensive analysis of the results along with a comparison between experimental observations and numerical predictions based on a typical day. oudrane a., aour b., bada a. and kheiri a. 55 copyright ©2025 assa. adv. in systems science and appl. (2025) 2. experimental study element the workspace under study corresponds to the office of a university professor. in the numerical model, the room is idealised as a rectangular control volume, as shown in figure 2.1. the enclosure comprises one external door and one glazed opening. all façades of the office are unobstructed and directly exposed to solar radiation throughout the day, implying that both shortwave and longwave radiative exchanges occur without shading effects. fig. 2.1 view and geometric dimensions of the workplace 2.1. geometrical and physical properties of the workplace the experimental workspace was designed considering both its geometric configuration and physical properties, which serve as input parameters for the thermal model. these characteristics were defined in accordance with civil engineering practices appropriate for desert climates. the external walls are composed of a double-layer masonry construction with a total thickness of 49 cm, incorporating a 5 cm air cavity that functions as thermal insulation. all surfaces are assumed to be homogeneous and isotropic for thermal calculations. table 2.1.1 provides a detailed overview of the geometric dimensions of the experimental element. table 2.1.1. geometric dimensions of the workplace width (m) length (m) height (m) thickness (m) surface (m2) south wall 3.60 3.60 3.20 0.49 12.96 north wall 3.60 3.60 3.20 0.49 12.96 east wall 3.90 3.90 3.20 0.49 15.21 west wall 3.90 3.90 3.20 0.49 15.21 ceiling tile 3.60 3.90 0.20 14.04 floor slab 3.60 3.90 0.22 14.04 table 2.1.2 summarizes the construction materials assigned to each wall of the experimental workspace, along with their relevant thermophysical properties, which are used as input parameters in the numerical thermal model. table 2.1.2 construction materials of the workplace south wall north wall east wall west wall ceiling tile floor slab materials double wall in red brick + exterior coating + air blade + interior coating red brick + exterior coating+ interior coating red brick + exterior coating+ interior coating red brick + exterior coating+ interior coating concrete brick (hourdis) + reinforced concrete + interior coating+ exterior coating granite tiles + heavy concrete building materials with appropriate physical and thermal properties are essential for accurately simulating heat transfer. accordingly, these properties have been carefully determined to enable precise modelling of the various thermal exchanges. table 2.1.3 presents the physical and thermal properties of all materials used in the study: table 2.1.3. physical and thermal properties of building materials [15-18] ρ (kg. m-3) cp (j.kg-1.k-1) λ (w.m-1.k-1) ɛ α red brick 300 800 0.08 0.93 0.60 56 oudrane a., aour b., bada a. and kheiri a. copyright ©2025 assa adv. in systems science and appl. (2025) hourdis brick 1800 1000 1.15 0.97 0.60 reinforced concrete 2400 1000 2 0.97 0.60 heavy concrete 2300 1000 2.71 0.97 0.60 cement + earth + red paint 1959 1.492 0.671 0.95 0.75 plaster 850 936 0.35 0.97 0.21 granite tiles 2200 936 2.1 0.97 0.58 air gap 1.200 1000 0.04 0.40 0.40 glazing 130 720 2.50 0.89 0.08 figure 2.1.1 illustrates the detailed geometric configuration of the external wall section considered in the experimental study. fig. 2.1.1. engineering of the structure of a wall with two walls separated by an air gap 3. description climatic of the adrar region the present study was conducted in adrar, located in the southwest of algeria, covering an area of 424.948 km² with a population of 402,197 inhabitants according to the 2008 census (see figure 3.1). the region possesses significant potential for renewable energy exploitation, particularly solar and wind resources [19-22]. adrar’s climate is classified as hot desert (bwh) according to the köppen system [23,24], characterized by extreme aridity, low and irregular precipitation (annual average <50 mm), and prolonged periods of high ambient temperatures, often exceeding 45 °c during summer months. relative humidity remains very low, typically below 20%, while annual global solar irradiation is exceptionally high, exceeding 2,800 kwh/m²/year. these climatic conditions result in substantial solar and convective heat loads on building envelopes, which are critical parameters for thermal modeling and the assessment of passive cooling strategies in buildings [23,24]. fig. 3.1. geographical location of the study area [22] oudrane a., aour b., bada a. and kheiri a. 57 copyright ©2025 assa. adv. in systems science and appl. (2025) as shown in figure 3.2, adrar exhibits very mild winters and extremely hot, sunny summers. the city experiences a maximum average temperature of 46 °c in july, making it one of the hottest cities in the world. in july 2023, temperatures peaked at 49.9 °c, while in august 2023, a day was recorded with a minimum of 35.5 °c and a maximum of 49.8 °c [25]. fig. 3.2. evolution of the average seasonal temperature range at the adrar region in 2023 [25] 4. temperature measurement devices temperatures were measured and recorded continuously every 24 hours over a period of 20 consecutive days in the workplace using testo devices, as shown in figure 4.1. measurements were performed on the interior surfaces of the south and north walls, the roof, as well as in the indoor air volume, with additional recordings of the outdoor air temperature. the instrumentation provided high-resolution data with a precision of ±0.1 °c for temperature and ±2 % for relative humidity, ensuring accurate input for thermal modeling. fig. 4.1. measuring and recording devices for different temperatures of the workplace 4.1. simplifying assumptions of the experimental process to ensure the proper execution of the experiment, several simplifying assumptions were adopted, which can be summarized as follows:  the outdoor air temperature is assumed to be equivalent to the ground temperature. 58 oudrane a., aour b., bada a. and kheiri a. copyright ©2025 assa adv. in systems science and appl. (2025)  humidity effects are neglected due to the region’s persistent arid conditions.  heat transfer is considered to be unidirectional. figure 4.1.1 illustrates the locations of the measuring devices both inside and outside the workplace. fig. 4.1.1. position of measuring devices inside and outside the workplace 5. mathematical modelling of thermal behaviour 5.1. loads due to the room glazing glass walls are characterized by their absorption and transmission coefficients. accordingly, the heat flux absorbed by a windowpane can be expressed by the following relation [26]: )( intglasglas ttsuq extccv  (5.1.1) in this context, the physical and thermal properties of the glass are considered, as summarized in table 5.1.1. table 5.1.1. physical and thermal properties of the glass of the workplace [27,28]. ) 11 glas ..(  kmw ).( 3 glas  mkg )..( 11 glas  kkgjcp )..( 12 glas  kmwu )( glas cme single glass window 1.05 2300 837 5.6 1 5.2. loads due from the workplace door the door in this workplace influences the internal heat exchange. the approximate amount of heat transferred through the door can be expressed by the following relation [29]:  intdoordoor ttshq extconv  (5.2.1) 5.3. thermal convection loads  convection outside the workplace in this case, the modeling focuses exclusively on heat transfer due to thermal convection through the five walls of the workplace, as described by the following relation [30-32]:  iextiextconviwall ttshq  )()( (5.3.1) the convective heat transfer coefficient on the exterior surfaces of the element is calculated using the following relation [33,34]: wind)( )8,3(7,5 vh extconv  (5.3.2) it was noted that the exterior façade of the south wall is the only surface directly exposed to solar radiation. opaque materials are characterized by an absorption coefficient specific to visible radiation, and the net absorbed flux represents the corresponding heat source. by oudrane a., aour b., bada a. and kheiri a. 59 copyright ©2025 assa. adv. in systems science and appl. (2025) neglecting the portion of flux reflected back to the environment, the heat generated by these radiative exchanges is expressed by the following relation [33]:   n i sourceabsor tfsg i s i qq )( (5.3.3)  convection in the workplace the thermal energy resulting from the heat exchange between the indoor air and the interior surfaces of the workplace walls is calculated using the following relation (equation 5.3.4) [35,36]:   n i amb i walliconvwallconv ttshq ).(. (int))( (5.3.4) natural convection occurs when a temperature gradient induces motion in the air, producing what are referred to as convective flows. heat transfer within a fluid layer is governed both by thermal conduction and by the movement of the fluid. when a thermal gradient is applied across the surface layers, a corresponding density gradient develops [35]. experimentally, it is observed that after a certain period, the fluid begins to move spontaneously, marking the onset of convection. this onset is characterized in table 5.23.1 by a dimensionless parameter known as the rayleigh (ra) number [36]. table 5.3.1. expression of convective transfer coefficients [35,36]. surface description flow regime scope of validity pr.grra  expression vertical wall laminar regime 9 10pr. 4 10 gr   4/1 .42,1 l t vhcon  turbulent regime 9 10pr. gr   3/1 .31,1 l t vhcon  upper surface of a horizontal hot plate or lower surface of a cold plate turbulent regime 9 10pr. 4 10 gr   4/1 .32,1 l t vhcon  laminar regime 9 10pr. gr   3/1 .52,1 l t vhcon  lower surface of a hot plate or upper surface of a cold plate turbulent regime 9 10pr. 4 10 gr   4/1 .59,0 l t vhcon  laminar regime 9 10pr. gr 5.4. thermal conduction loads figure 5.4.1 presents a geometric schematic of the exterior south wall of the study element. it also illustrates the directions of heat transfer and thermal processes occurring within the wall materials. for a linear temperature profile, the heat flux through the wall is calculated using the following relation [37,38]: dx dt sq .. (5.4.1) in steady-state conditions, the heat flux through the wall is conserved and can be expressed using the following relation [39]:      1 int 1 . . . . . . n i i i i i i ext i i i i i s t t q h s t t h s t t e          (5.4.2) 60 oudrane a., aour b., bada a. and kheiri a. copyright ©2025 assa adv. in systems science and appl. (2025) fig. 5.4.1. equivalent electrical diagram of a multilayer wall 5.5. solar radiation loads to consider the optical and geometric properties of the considered surfaces, the form factor f’i+1 can be defined using the following expression [40]: 1 11 1 1 ' 1 111                 i i i i iii i ii s s f f     (5.5.1) the net radiative flux ‘'qr,i i+1’’ between the surfaces at temperatures ‘'ti’' and ‘'ti+1 ’’, is expressed by the following relation [41]:  4 1 4' 11,   iiiiiiir ttfsq  (5.5.2) in linearized form, qr,i+1 is written in expression 5.5.3 as follows:  11,1,   iiiiriir tth (5.5.3) the general expression for the radiative heat transfer coefficient between surfaces i and i+1 is given by the following relation [38-40]:     11 1 1 1 2 1 2 1, 111            i i i i iii i iiii iir s s f tttt h      (5.5.4) 5.6. principle of thermal analysis by the nodal method the nodal method is particularly well suited for simulating the thermal behavior of buildings. it enables a global representation of the entire structure, including fluid circuits, and allows problems of different physical nature to be solved within a unified framework [42,43]. moreover, its computational cost and memory requirements are significantly lower compared to finite element analyses. this approach offers sufficient flexibility to refine the mesh locally when necessary. based on the thermal–electrical analogy, the nodal method consists in discretizing the physical system into a finite number of isothermal elementary volumes (nodes). the thermal response of the system is obtained by solving the set of energy conservation equations, with one governing equation associated to each node. the thermal balance of node i which exchanges heat by conduction with node j, by convection with node k, by radiation with node l, by fluid transport with node m, and interacts oudrane a., aour b., bada a. and kheiri a. 61 copyright ©2025 assa. adv. in systems science and appl. (2025) with heat sources or sinks n, leads to the nonlinear differential equation expressed in relation 5.6.1 [43-45].            4 4 source *. . . . . . . . . . . . . iji j i ik k i i il i i m i ni l n j convection transportrayonnement conduction sdt v cp t t h s t t s f t t m cp t t q dt li                 (5.6.1) the solution of each governing equation yields the temperature of the corresponding node as well as the associated heat fluxes. in the nodal framework, the residential thermal system is represented analogously to an electrical network: the thermal inertia of each node is modeled by a thermal capacitance, conductive and convective exchanges are represented by thermal conductances, imposed temperatures correspond to potential (voltage) nodes, and imposed heat flows are treated as current sources. under this analogy, the global set of coupled nonlinear equations describing the system can be written in the compact matrix form expressed in relation 5.6.2 [45-47].        ( ) ( ) ( ) ( ). . . . .i i cond j j i conv j k i ray l l i fluid m m i source dt c q t t q t t q t t q t t q dt          (5.6.2) 5.7. thermal balance of study element the thermal balance represents an evaluation of all energy inputs and outputs associated with the studied element over a defined time interval [33, 34]. the mathematical model developed for the thermal analysis was validated using experimental measurements, as detailed in the section dedicated to physical results and interpretation. in accordance with the nodal modelling approach, each wall of the structure is treated as an individual node, and a specific thermal balance is established to describe all heat transfer mechanisms acting within that wall. given that the studied enclosure comprises five walls, the resulting system of equations includes ten coupled thermal balance equations, to which an additional equation describing the natural convection of the confined indoor air must be added. relations 5.7.1 and 5.7.2 illustrate representative equations of the global system obtained through numerical modelling in this study [45].  thermal balances of exterior facades of the workplace in this configuration, the global energy balance comprises five independent equations, each associated with one of the external façades of the workplace, which exchange heat with the outdoor environment. each equation explicitly accounts for the local climatic boundary conditions, as well as the thermal and physical properties of the construction materials constituting the corresponding façade. ( ) ( ) ( , ) ( , ) ( ) ( ) ( , ) ( , ) ( ) ( ) . . . . . . . . . . . . . . . . . ps p ps pse ps c ext r vc pse r sol pse ps pse ps ps pn p pn pne pn c ext r vc pne r sol pne pn pne pn pn po p po poe po c ext po po m c t h t t h t h t s t s m c t h t t h t h t s t e m c t h t s t e                                    ( , ) ( , ) ( ) ( ) ( , ) ( , ) ( ) ( ) ( , ) ( , ) . . . . . . . . . . . . . . . . . r vc poe r sol poe po poe pfp p pfp pfpe pfp c ext r vc pfpe r sol pfpe pfp pfpe pfp pfp pe p pe pee pe c ext r vc pee r sol pee pe pe t h t h t m c t h t t h t h t s t e m c t h t t h t h s t e                                 .pe peet                                (5.7.1)  thermal balance of interior facades of the workplace similarly, the internal energy balance of the workplace considered in this study was established. it consists of six governing equations, each describing the heat transfer mechanisms acting on the internal façades of the enclosure. these equations account for 62 oudrane a., aour b., bada a. and kheiri a. copyright ©2025 assa adv. in systems science and appl. (2025) convective, conductive, and radiative exchanges occurring between the indoor air and the corresponding internal surfaces. ( ) ( ) ( , ) ( , ) ( , ) ( , ) ( , ) ( ) ( ) ( , ) ( , ) ( , . . . . . . . . . . . . . . . ps p ps psi ps c int r psi pni r psi poi r psi pei r psi pfpi r psi pp ps ps pn p pn pni pn c int r pni psi r pni pei r pni poi pn pn m c t h t t h t h t h t h t h t s t e m c t h t t h t h t h s t e                              ) ( , ) ( , ) ( ) ( ) ( , ) ( , ) ( , ) ( , ) ( , ) ( ) ( ) . . . . . . . . . . . . . . . . r pni pfpi r pni pp po p po poi po c int r poi psi r poi pni r poi pei r poi pfpi r poi pp po po pfp p pfp pfpi pfp c int pfp pfp t h t h t m c t h t t h t h t h t h t h t s t e m c t h t t s t e                               ( , ) ( , ) ( , ) ( , ) ( , ) ( ) ( ) ( , ) ( , ) ( , ) ( , ) ( , ) . . . . . . . . . . . . . . . r pfpi psi r pfpi pni r pfpi poi r pfpi pei r pfpi pp pe p pe pei pe c int r pei psi r pei poi r pei pni r pei pfpi r pei pp pe pe air h t h t h t h t h t m c t h t t h t h t h t h t h t s t e m                           ( ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ). . . . . . . . . . . . .p air air ps pn po pe pfp pp c psi air c pni air c poi air c pei air c pfpi air c pp air air air air air air air air c t s s s s s s h t h t h t h t h t h t s t s s s s s s                           (5.7.2) processing of input and output data figure 5.8.1 illustrates the detailed computational flowchart employed for the numerical simulation. the diagram details the sequential operations implemented to evaluate the heat transfer mechanisms occurring within and around the workplace over a 24-hour representative cycle. the resulting system of linearized equations obtained from the numerical model was solved using the gauss elimination algorithm. the flowchart also clearly specifies the input parameters required at the initialization stage of the simulation, as well as the output variables produced at each iteration. it is important to highlight that the computation relies on time-averaged boundary and operating conditions, defined over discrete intervals throughout the day. oudrane a., aour b., bada a. and kheiri a. 63 copyright ©2025 assa. adv. in systems science and appl. (2025) fig. 5.8.1. computational flowchart of the numerical heat transfer modeling 6. results and discussion 6.1. validation of the developed numerical model the validation of the numerical model developed in this study, based on data from two representative days, may 10th and may 18th, is presented in figure 6.1.1. the results show a good agreement between the measured and simulated temperatures. validation was performed for the following temperature types: tpn (figure 6.1.1a, may 18th, 2024), tin (figure 6.1.1b, may 18th, 2024; figure 6.1.1c, may 10th, 2024), and text (figure 6.1.1d, may 10th, 2024), with all comparisons yielding satisfactory results. these findings confirm that the developed numerical model is valid and can be reliably used to simulate other thermal quantities, such as the temperature of wall façades and the solar flux incident on both the exterior and interior surfaces of the workplace. the maximum discrepancy between measured and calculated temperatures over the two selected days did not exceed 4%, further demonstrating the model’s accuracy. (a) (b) (c) (d) fig. 6.1.1. comparison between the measured temperature variations and the calculated temperature. 6.2. analysis of experimental data figure 6.2.1 illustrates the temporal evolution of the external text and internal tint temperatures as a function of local time over the course of the experimental period. figure 6.2.1(a) presents the variation of the external temperature text at the study site over seven selected days. it can be observed that the daily external temperature profiles exhibit a similar pattern throughout 64 oudrane a., aour b., bada a. and kheiri a. copyright ©2025 assa adv. in systems science and appl. (2025) the week. during the night, from the early hours until sunrise at around 8:00 a.m., text ranges from 16.3°c (may 1st, 2024) to 31.8°c (may 20th, 2024). following sunrise, the temperature gradually increases, reaching a peak in the late afternoon, approximately 5:00 p.m., with maximum values ranging from 40.31°c on may 1st, 2024, to 49.25°c on may 13th, 2024. thereafter, the temperature gradually decreases and stabilizes within a limited range between 22.9°c (may 1st, 2024) and 31.0°c (may 13th, 2024). the average external temperature over the seven selected days is approximately text (average) = 45.7°c. figure 6.2.1(b) shows the internal temperature tint evolution at the study location as a function of local time. the results indicate that the internal temperature profiles for the seven days follow a similar daily trend. notably, on may 1st and may 2nd, 2024, tint exhibits slightly lower values compared to the other selected days. the maximum internal temperature reached tint = 30.1°c at 6:00 p.m. on may 1st and 2nd, whereas for the remaining days, the peak internal temperature was tint = 33.5°c at approximately 1:00 p.m. these observations indicate that the average internal temperature during early may in this desert region is relatively moderate but shows a gradual increase over the course of the month. (a) (b) fig. 6.2.1. evolution of external and internal temperature as a function of local time for different days of the experiment. 6.3. analysis of typical day may 1st, 2024 figure 6.3.1 illustrates the temporal evolution of the thermal behavior of the study site over a typical day, may 1st, 2024, as a function of local time. figure 6.3.1(a) presents the variations of the internal air temperature tint and relative humidity (rh) throughout the day. it is observed that tint and rh follow a similar trend until approximately 10:00 a.m. from 1:00 p.m. onward, the relative humidity begins to evolve inversely with respect to the internal temperature: as tint increases, rh decreases, reaching a maximum temperature of tint = 30°c at 6:00 p.m. and a minimum relative humidity of rh = 15.5%. figure 6.3.1(b) shows the comparison between the indoor tint and outdoor text air temperatures. both quantities exhibit similar diurnal patterns, with a maximum of t = 41°c at 6:00 p.m. and a minimum of t = 15°c at 8:00 a.m. figure 6.3.1(c) presents the evolution of the internal air temperature tint along with the temperatures of the selected internal wall façades measured during the experiment. the temperature profiles of the walls follow the same general trend as tint, with a minimum of tpfpi = 27.3°c recorded at the lower inner ceiling façade at 6:00 p.m. and a maximum of tpsi = 29.8°c at 10:00 p.m. on the inner south wall façade. a noticeable time lag of approximately δt = 5 h is observed between the peak of the south wall inner façade temperature and the other temperatures tpni, tint and tpfpi. this lag is attributed to the thermal inertia of the building materials, their heat storage capacity, and delayed heat release. figure 6.3.1(d) compares the external temperature text with the temperatures of the internal wall façades selected for measurement. the peak temperature of the inner south wall façade oudrane a., aour b., bada a. and kheiri a. 65 copyright ©2025 assa. adv. in systems science and appl. (2025) tpsi = 29.65°c occurs at 2:00 a.m., showing a time difference of δt = 4 h relative to the other measured temperatures tpni, text and tpfpi. the minimum external temperature of text = 16.3°c is recorded at 7:00 a.m. these observations highlight the significant role of material properties in inducing a time lag (δt) between the internal south wall surface and other measured temperatures, reflecting the impact of thermal inertia and heat storage on the dynamic thermal response of the building. (a) (b) (c) (d) fig. 6.3.1. evolution of the thermal behavior of the element studied as a function of local time during the typical day may 1st, 2024. 6.4. analysis of the typical day may 10th, 2024 figure 6.4.1 illustrates the temporal evolution of the thermal behavior of the study site during a typical day, may 10th, 2024. figure 6.4.1(a) presents the comparison between internal air temperature tint and internal relative humidity (rh) as a function of local time. the results indicate an inverse relationship between tint and rh throughout the day. before noon, the minimum internal temperature of tint = 30.6°c is observed at 8:00 a.m., while the relative humidity reaches a maximum of rh = 16.5%. in the afternoon, this trend reverses: tint peaks at 32.9°c and rh reaches a minimum of 14.9% at 4:00 p.m., confirming the expected inverse correlation between internal temperature and relative humidity. figure 6.4.1(b) compares the internal tint and external text air temperatures throughout the day. both temperatures follow a similar diurnal trend, with a maximum of t = 46.5°c recorded at 4:00 p.m. and a minimum of t = 22°c at 8:00 a.m. a temperature difference of δt = 4°c 66 oudrane a., aour b., bada a. and kheiri a. copyright ©2025 assa adv. in systems science and appl. (2025) is observed between tint and text between 11:00 p.m. and midnight, explaining the higher internal temperature relative to the external environment during this period. figure 6.4.1(c) shows the comparison between the internal temperature tint and the temperatures of the selected internal wall façades tpsi, tpfpi and tpni over the day. a temperature difference of δt = 1.6°c is observed between the north wall inner façade tpni, the roof inner façade tpfpi, and the internal air temperature tint, indicating that the roof inner surface is hotter than the other façades in the afternoon. additionally, the inner south wall façade reaches its peak temperature of tpsi = 33.6°c at 10:00 p.m., exhibiting a time lag of δt = 6 h relative to the internal air temperature tint, due to the thermal inertia of the southern wall. figure 6.4.1(d) presents the comparison between external temperature text and the selected internal wall temperatures tpsi, tpfpi and tpni. the south wall inner façade shows higher temperatures with a clear time lag of δt = 6 h relative to text, tpni and tfpi. the outdoor temperature peaks at text = 45.7°c at 4:00 p.m., while the south wall inner surface reaches its maximum of tpsi = 33.6°c at 10:00 p.m. these observations highlight the influence of construction material properties, as the north wall and roof slab exhibit distinct thermal behavior compared to the south wall, due to differences in thermal inertia and heat storage capacity. (a) (b) (c) (d) fig. 6.4. evolution of the thermal behavior of the element studied as a function of local time during the typical day may 10th, 2024. 6.5. examination of a week during the experience period figure 6.5.1 illustrates the evolution of the thermal behavior at the study site over a 7-day period, from may 1st to may 7th, 2024. figure 6.5.1(a) presents the temporal variations of the oudrane a., aour b., bada a. and kheiri a. 67 copyright ©2025 assa. adv. in systems science and appl. (2025) internal air temperature tint, external air temperature text, and internal relative humidity (rh) as a function of local time. during this period, the maximum external temperature gradually increased, starting from text = 42.7°c on may 1st and reaching text = 50°c on may 7th. in contrast, the internal air temperature remained relatively stable, ranging between tint = 29°c and 33°c over the seven days. meanwhile, the internal relative humidity exhibited a decreasing trend, from a maximum of rh = 16% on may 1st to rh = 13% by may 7th, reflecting the gradual increase in internal temperature. figure 6.5.1(b) shows the comparison between the temperatures of the selected internal wall façades tpsi, tpfpi, tpni and the external air temperature text as a function of local time. the external temperature remains higher than the internal wall temperatures throughout the seven-day period. the north wall inner façade temperature remained relatively stable, between 23°c and 30°c, while the inner south wall and roof inner façade temperatures follow a similar trend, with a mean deviation of δt = 6°c. these observations highlight the influence of wall orientation and material thermal properties on the thermal response of the building envelope. (a) (b) fig. 6.5.1. evolution of temperature and relative humidity of the studied element during 7-days of experiment as a function of local time. 7. analysis and interpretation of computed thermal data 7.1. effect of wall thickness on internal temperature variations figure 6.6.1 presents the temporal evolution of the internal air temperature tint of the workplace as a function of wall thickness over a full diurnal cycle, corresponding to the sun’s trajectory. the results demonstrate a clear inverse correlation between wall thickness and internal temperature, highlighting the thermal damping effect of increased wall mass. for a wall thickness of 0.15 m, the peak internal temperature reaches tint, max = 41°c, whereas for 0.30 m, the peak decreases to tint, max = 36°c. the observed temperature difference (δt = 5°c) is associated with an estimated temporal phase shift of δt ≈ 1 h, indicating a delay in the thermal response of the internal environment due to the walls’ heat storage capacity and thermal inertia. this behavior is consistent with nodal modeling predictions, where each wall is represented as a discrete thermal node with capacitance proportional to its mass and conductance defined by its thermal properties. the phase shift and amplitude attenuation of internal temperature underscore the critical role of wall thickness in modulating diurnal thermal fluctuations and controlling indoor thermal comfort under high solar loading conditions. 68 oudrane a., aour b., bada a. and kheiri a. copyright ©2025 assa adv. in systems science and appl. (2025) fig. 6.6.1. evolution in the temperature of the internal medium is a function of the evolution of the walls thickness. 7.2. seasonal variation of the average external temperature in the year 2024 figure 6.6.2 illustrates the annual evolution of the average outdoor temperature text, avg in the adrar region during 2024. the temperature profile exhibits a distinct pyramidal seasonal pattern, with extreme values observed during summer and winter. peak summer temperatures reach or exceed 50°c, whereas winter minima range between 20°c and 22°c. transitional seasons, spring and autumn, display intermediate average temperatures between 35°c and 40°c. fig. 6.6.2. seasonal variation in the average outdoor temperature on 2024 depending on the year months. these results confirm that the adrar region is among the hottest regions globally, as evidenced by the extreme summer temperatures relative to other seasons. in nodal modeling of building envelopes, these external temperature variations serve as the primary driving forces for conduction, convection, and radiative exchanges, influencing both internal air temperatures and wall surface thermal responses. the extreme summer conditions further underscore the importance of accounting for thermal inertia, phase lag, and material-specific heat capacity in predicting internal thermal comfort and evaluating envelope performance. oudrane a., aour b., bada a. and kheiri a. 69 copyright ©2025 assa. adv. in systems science and appl. (2025) 7.3. seasonal variation in the average global solar flux during the year 2024 figure 6.6.3 shows the seasonal evolution of average solar irradiance for the year 2024. the irradiance exhibits consistent peak values across all seasons, ranging from 1000 w·m⁻² to 11000 w·m⁻², with the maximum occurring in spring (11000 w·m⁻²). these observations highlight the significant seasonal influence of solar radiation on the thermal loading of building envelopes, which is a critical boundary condition for heat transfer simulations and nodal thermal modeling. the high solar loading during peak periods underscores the importance of accounting for material thermal inertia and facade orientation in predicting indoor thermal behavior. fig. 6.6.3. seasonal variation of the average solar flux in 2024 based on the year months. 8. conclusion this study combined experimental measurements and numerical simulations to analyze the thermal behavior of a workplace in adrar. wall temperatures and indoor relative humidity were recorded in may 2024 using testo devices. the developed mathematical model accurately reproduced the experimental results, with an average error below 4%, validating its reliability for predictive thermal simulations. numerical analyses were extended to evaluate the influence of seasonal variations and material properties on internal temperatures over the entire year 2024. the main conclusions from this combined experimental and numerical study are as follows:  the average outdoor temperature at the workplace in may 2024 ranged between 40°c and 45°c.  the average internal temperature was maintained between 30°c and 35°c.  the average internal relative humidity during the same period was approximately 13%.  the temperature difference between the external environment and the interior surfaces of the workplace walls reached ≈6°c. these results indicate that the internal thermal conditions are significantly higher than the ideal thermal comfort range. this can be attributed to extreme solar loading, low ambient humidity, and the thermal properties of the envelope. as a future perspective, practical interventions are proposed to mitigate internal heat gains. notably, the replacement of the existing air-gap insulation with locally available palm residuebased insulation is under investigation, aiming to enhance thermal damping and reduce peak internal temperatures under extreme climatic conditions. 70 oudrane a., aour b., bada a. and kheiri a. copyright ©2025 assa adv. in systems science and appl. 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(2024) experimental and numerical investigations on the thermal and moisture transfer in green dual layer wall for building, case studies in thermal engineering, 53, 103946. doi: 10.1016/j.csite.2023.103946. adv syst sci appl 2022; 03:106–123 published online at https://ijassa.ipu.ru. asymptotic properties of the block-type statistics natalia m. markovich1*, marijus vaičiulis2 1v.a. trapeznikov institute of control sciences russian academy of sciences, moscow, russia, 2vilnius university institute of data science and digital technologies, vilnius university, vilnius, lithuania abstract: extreme value theory is an issue extensively applied in many different fields. one of the central points of this theory is the estimation of a positive extreme value index. in this paper we introduce a new family of block type statistics related to this estimation. a weak consistency of the introduced statistics is proved. a bivariate central limit theorem for newly introduced statistics is derived. we provide the new family of semi-parametric estimators for the positive extreme value index. asymptotic normality of the introduced estimators is proved. it is shown that new estimators have better asymptotic performance comparing with several block-type estimators over the whole range of parameters presented in the second order regular variation condition. an application to the estimation of the positive valued extreme value index for several real data sets is provided. keywords: asymptotic normality, extreme value index, block-type estimator 1. introduction let x1, x2, . . . , xn be a sample of independent identically distributed (i.i.d.) random variables (r.v.s) with an unknown distribution function (d.f.) f . in the present paper we will formulate our assumptions in terms of a quantile type function u associated to f , which is defined by u(t) = { 0, 0 < t ≤ 1 inf {x : f (x) ≥ 1− (1/t)} , t > 1. the main assumption is that u belongs to the class of regularly varying functions at infinity, with γ > 0 (shortly, u ∈ rvγ), i.e., for all x > 0, lim t→∞ u(tx) u(t) = xγ. (1.1) the parameter γ in (1.1) is a positive extreme value index (evi) that is the primary parameter of large extreme events. models satisfying (1.1) are quite common in many application areas such as biostatistics, computer science, finance, insurance and social sciences, among others. to exploit the block approach in the extreme value theory one has to divide the observations x1, . . . , xn into 1 ≤ s ≤ n non-overlapping blocks of size m = [n/s]: bi = {x(i−1)m+1, . . . , xim}, 1 ≤ i ≤ s, ∗corresponding author: nat.markovich@gmail.com asymptotic properties of the block-type statistics 107 where [·] denotes the integer part. let x(i) 1,m ≤ x (i) 2,m ≤ · · · ≤ x (i) m,m denote the order statistics of m observations in the ith block. the aim of the paper is to investigate asymptotic properties, including a weak consistency and an asymptotic normality, of the block-type statistics qn(s, ℓ, r) = 1 s s∑ i=1 1 ℓ ℓ∑ j=1 hr ( x (i) m−j+1,m x (i) m−ℓ,m ) . here, ℓ ∈ n and the family of functions hr, r ∈ r is defined for x > 0 as follows: hr(x) = { (xr − 1) /r, r ̸= 0, ln(x), r = 0. note that the statistics qn(s, ℓ, r) are applicable particularly when only the largest observations {x(i) m−j,m, 0 ≤ j ≤ ℓ, 1 ≤ i ≤ s} are available for the inference. such type of data can be found in [4], where battle deaths in major power wars between 1495 and 1975 were analyzed. for more such data see [10] and references therein. the statistics qn(s, ℓ, r) generalize two families of statistics. firstly, by taking r = 0 we turn back to the estimator for the positive evi, proposed in [18]. secondly, the statistics qn(s, ℓ, r) generalize the statistics qn(s, 1, r), introduced in [17], see also [21]. note that the statistics qn(s, ℓ, r) are scale-free, that is, qn(s, ℓ, r) do not change when observations x1, . . . , xn are replaced by cx1, . . . , cxn with c > 0. the paper is organized as follows. in section 2 our main results are presented. in the next section, we introduce a new family of estimators for positive evis and compare the latter estimators with several known estimators. in section 4 the application to several real data sets is discussed. we finalize with conclusions. the last section contains proofs of the results. 2. main results let d→ and p→ denote the convergence in distribution and in probability, respectively. the equality in distribution will be denoted by d =. theorem 2.1: suppose that x1, . . . , xn are i.i.d. r.v.s with the d.f. f , whose quantile type function u satisfies (1.1) with some γ > 0. let ℓ ∈ n. let r ∈ r be such that γr < 1 holds. let the block number s = sn and the block width m = mn tend to infinity as n → ∞. then qn(s, ℓ, r) p→ λγ(γr), n → ∞, where λγ(t) := γ/(1− t), t < 1. the second order parameter ρ rules the rate of convergence in the first order condition (1.1). moreover, ρ is a non-positive parameter that appears in the limiting relation lim t→∞ 1 a(t) ( u(tx) u(t) − xγ ) = xγhρ(x), (2.2) which we assume to be held for all x > 0. here, a(t) is a measurable function with a constant sign near infinity and which is not identically zero, and a(t) → 0 as t → ∞. it is known that (2.2) implies that |a(t)| ∈ rvρ, see [11]. let z = z(t), t < 1/2 be a gaussian random process with mean 0 and covariance function σ2(t1, t2) := e (z(t1)z(t2)), where σ2(t1, t2) = (1− t1 − t2) −1. copyright © 2022 assa. adv syst sci appl (2022) 108 n. m. markovich, m. vaičiulis theorem 2.2: suppose that x1, . . . , xn are i.i.d. r.v.s with the d.f. f , whose quantile type function u satisfies the second order condition (2.2) with some γ > 0 and ρ ≤ 0. let the block number s = sn and the block width m = mn satisfy the assumptions of theorem 2.1 and, in addition, lim n→∞ √ sa(m) = µ ∈ (−∞,+∞) (2.3) hold. let ℓ ∈ n and ri ∈ r, i = 1, 2 be such that γri < 1/2 holds. then it holds √ s (qn(s, ℓ, r1)− λγ (γr1) , qn(s, ℓ, r2)− λγ (γr2)) d→ ( µν(ℓ, ρ, r1) + λγ (γr1)√ ℓ z(γr1), µν(ℓ, ρ, r2) + λγ (γr2)√ ℓ z(γr2) ) (2.4) as n → ∞, where ν(ℓ, ρ, ri) = γ(1 + ℓ− ρ) γ(1 + ℓ) · 1 (1− γri)(1− γri − ρ) , i = 1, 2, and γ(t) = ∫∞ 0 xt−1 exp{−x}dx, t > 0 is the gamma function. remark 1. if in conditions of theorem 2.2 assumption (2.3) is replaced by lim n→∞ √ sa(m) = ∞, (2.5) then it holds 1 a(m) (qn(s, ℓ, r1)− λγ (γr1)) p→ ν(ℓ, ρ, r1), n → ∞. (2.6) the extension of theorem 2.2 to the multivariate case is quite trivial. thus, theorem 2.2 gives us a lot of possibilities to form asymptotically normal estimators of γ > 0 using statistics qn(s, ℓ, r) with different r. considering the simplest case we propose the following family of the block-type estimators for γ > 0: γ̂(1) n (s, ℓ, r) = { qn(s, ℓ, 0), r = 0, qn(s, ℓ, r) (1 + rqn(s, ℓ, r)) −1 , r ̸= 0. it should be noted that the family of estimators γ̂ (1) n (s, ℓ, r) generalizes the family of estimators given in [18] (for r = 0) and the family of estimators proposed in [17] (for ℓ = 1). we establish the asymptotic normality of γ̂ (1) n (s, ℓ, r) by applying the univariate form of theorem 2.2. corollary 2.3: let the conditions of theorem 2.2 be fulfilled for univariate case. then it holds √ s ( γ̂(1) n (s, ℓ, r)− γ ) d→n ( µν1(ℓ, ρ, γr), γ 2σ2 1(ℓ, γr) ) , n → ∞, where n denotes a normal distribution and ν1(ℓ, ρ, r) = γ(1 + ℓ− ρ) γ(1 + ℓ) · 1−r 1−r− ρ , σ2 1(ℓ, r) = (1−r)2 ℓ(1− 2r) , r < 1/2. taking r = 0 in corollary 2.3 we turn back to theorem 1 in [18]. it should be noted that the asymptotic bias is written in a slightly different form in [18]. another partial case of corollary 2.3 can be found in [17] (see (2.12) therein). there, assuming more simple second order asymptotic condition than (2.3), the asymptotic normality of the estimators 1/γ̂ (1) n (s, 1, r) is obtained. copyright © 2022 assa. adv syst sci appl (2022) asymptotic properties of the block-type statistics 109 3. a new family of the block-type estimators for positive evi let r ̸= 0 satisfy γr < 1, while s = sn and m = mn satisfy the assumptions of theorem 2.1. then qn(s, ℓ, 0)/qn(s, ℓ, r) p→1− γr holds as n → ∞. so, γ̂(2) n (s, ℓ, r) = qn(s, ℓ, r)−qn(s, ℓ, 0) rqn(s, ℓ, r) , r ̸= 0 presents a family of weakly consistent estimators for positive evi. let us define γ̂ (2) n (s, ℓ, 0) := limr→0 γ̂ (2) n (s, ℓ, r). by using the l’hospital’s rule, it follows that γ̂(2) n (s, ℓ, 0) = q̃n(s, ℓ) 2qn(s, ℓ, 0) , where q̃n(s, ℓ) = 1 s s∑ i=1 1 ℓ ℓ∑ j=1 ln2 ( x (i) m−j+1,m x (i) m−ℓ,m ) . an application of theorem 2.2 yields the following result. theorem 3.1: let the conditions of theorem 2.2 be fulfilled with r1 = 0 and r2 = r. then it holds √ s ( γ̂(2) n (s, ℓ, r)− γ ) d→n ( µν2(ℓ, ρ, γr), γ 2σ2 2(ℓ, γr) ) , n → ∞, (3.7) where ν2(ℓ, ρ, r) = γ(1 + ℓ− ρ) γ(1 + ℓ) · 1−r (1−r− ρ)(1− ρ) , σ2 2(ℓ, r) = 2(1−r) ℓ(1− 2r) , r < 1/2. having the asymptotic normality of the introduced estimators γ̂(2) n (s, ℓ, r) we can discuss an optimal choice of s, ℓ and r. we assume further that ρ < 0 and µ ̸= 0. keeping ℓ and r fixed, the limiting mean squared error (mse) for γ̂(2) n (s, ℓ, r) is, approximately, inf s { γ2σ2 2(ℓ, γr) s + a2 (n s ) ν2 2(ℓ, ρ, γr) } , where the sequence s = sn satisfies the assumptions of theorem 2.1. it is known that there exists a positive decreasing function b ∈ rv2ρ−1 such that a2(t) ∼∫∞ t b(x)dx, t → ∞, see here and below, we write an an ∼ cn if an/cn → 1 as n → ∞. let b← denote the inverse function of b. following the lines in [12] we find mse ( γ̂(2) n (s∗2, ℓ, r) ) ∼ 1− 2ρ −2ρ ( ν2 2(ℓ, ρ, γr) ( γ2σ2 2(ℓ, γr) )−2ρ)1/(1−2ρ) b←(1/n) n , (3.8) as n → ∞, where the optimal choice of the number of blocks s∗2 satisfies the relation s∗2 ∼ ( γ2σ2 2(ℓ, γr) ν2 2(ℓ, ρ, γr) )1/(1−2ρ) n b←(1/n) , n → ∞. copyright © 2022 assa. adv syst sci appl (2022) 110 n. m. markovich, m. vaičiulis next, we minimize the right hand side of (3.8) with respect to ℓ and r. for this it is sufficient to minimize the product ν2 2(ℓ, ρ, γr) ( γ2σ2 2(ℓ, γr) )−2ρ = 2−2ρ (1− ρ)2 (φ(ℓ, ρ))2 ϕ(ρ, γr) with respect to ℓ and r. here, we have φ(ℓ, ρ) = ℓργ(ℓ+ 1− ρ) γ(ℓ+ 1) , ϕ(ρ,r) = (1−r)2−2ρ (1−r− ρ)2(1− 2r)−2ρ , r < 1/2. by using the standard technique of derivatives we obtain that the function ϕ(ρ,r) attains its minimum with r∗2 = ρ. whence, the optimal choice of parameter r for γ̂(2) n (s∗2, ℓ, r) has the form r∗2 = ρ/γ. before we proceed, let us introduce the following lemma regarding the function φ(ℓ, ρ). lemma 3.2: for any ℓ ∈ n and ρ < 0, φ(ℓ, ρ) > φ(ℓ+ 1, ρ) holds. by lemma 3.2, for any ℓ ∈ n, the estimator γ̂(2) n (s∗2, ℓ, r ∗ 2) has a bigger asymptotic mse comparing with γ̂ (2) n (s∗2, ℓ+ 1, r∗2). thus, there is no an optimal estimator among estimators γ̂ (2) n (s∗2, ℓ, r ∗ 2), ℓ ∈ n. the same conclusion holds for the estimators γ̂ (1) n (s∗1, ℓ, r ∗ 1), ℓ ∈ n, where s∗1 ∼ ( γ2σ2 1(ℓ, γr) ν2 1(ℓ, ρ, γr) )1/(1−2ρ) n b←(1/n) , n → ∞ and r∗1 = (2γ)−1 ( 2− ρ− (2− 4ρ+ ρ2) 1/2 ) , see [17]. as for mse ( γ̂ (1) n (s∗1, ℓ, r) ) , it has the same asymptotic as in (3.8) with replacing ν2(ℓ, ρ, γr) and σ2 2(ℓ, γr) by ν1(ℓ, ρ, γr) and σ2 1(ℓ, γr), respectively. let us find the limit of the ratio of mse ( γ̂ (i) n (s∗i , ℓ+ 1, r∗i ) ) and mse ( γ̂ (i) n (s∗i , ℓ, r ∗ i ) ) : lim n→∞ mse ( γ̂ (i) n (s∗i , ℓ+ 1, r∗i ) ) mse ( γ̂ (i) n (s∗i , ℓ, r ∗ i ) ) = χ(ℓ, ρ), i = 1, 2, where χ(ℓ, ρ) = (( ℓ ℓ+ 1 )ρ ℓ+ 1− ρ ℓ+ 1 )2/(1−2ρ) . for the well-known heavy-tailed models, like the frechet and the student’s t, condition (2.2) holds with ρ = −1. we represent the numerical approximation of χ(ℓ,−1) for several values of ℓ: χ(1,−1) ≈ 0.825, χ(2,−1) ≈ 0.924, χ(3,−1) ≈ 0.958, χ(4,−1) ≈ 0.973 and χ(5,−1) ≈ 0.981. since the values of χ(ℓ,−1) are close to 1 when ℓ ≥ 3, we recommend to take 3 ≤ ℓ ≤ 5 dealing with real data sets. at the end of this section, we compare the estimators γ̂ (1) n (s∗1, ℓ, r ∗ 1) and γ̂ (2) n (s∗2, ℓ, r ∗ 2). denoting lim n→∞ mse ( γ̂ (1) n (s∗1, ℓ, r ∗ 1) ) mse ( γ̂ (2) n (s∗2, ℓ, r ∗ 2) ) = χ̃(ρ), copyright © 2022 assa. adv syst sci appl (2022) asymptotic properties of the block-type statistics 111 it is not difficult to get that χ̃(ρ) = ( (1− 2ρ)1−1/ρ(1−r∗1) 2−1/ρ 2(1− ρ)(1−r∗1 − ρ)−1/ρ(1− 2r∗1) )−2ρ/(1−2ρ) , where r∗1 = γr∗1. let us note that (χ̃(ρ))(1−2ρ)/(−2ρ) > 1 (3.9) for all −∞ < ρ < 0 (see section 6 for the proof). since (3.9) yields χ̃(ρ) > 1, ρ < 0, we conclude that for any ℓ ∈ n the new estimator γ̂ (2) n (s∗2, ℓ, r ∗ 2) dominates the estimator γ̂ (1) n (s∗1, ℓ, r ∗ 1) in the whole region of parameters {(γ, ρ) : γ > 0, ρ < 0}. whence the conclusion related to the estimators introduced in [17] follows, namely, γ̂ (2) n (s∗2, 1, r ∗ 2) outperforms γ̂(1) n (s∗1, 1, r ∗ 1) in the whole region of parameters {(γ, ρ) : γ > 0, ρ < 0}. comparing the estimators γ̂(1) n (s∗1, ℓ, 0) (proposed in [18]) and γ̂ (2) n (s∗2, ℓ, 0) we have lim n→∞ mse ( γ̂ (1) n (s∗1, ℓ, 0) ) mse ( γ̂ (2) n (s∗2, ℓ, 0) ) = ((1− ρ)2ρ)2/(1−2ρ) . whence we get that the estimator γ̂ (2) n (s∗2, ℓ, 0) has a smaller asymptotic mse in the area {(γ, ρ) : γ > 0,−1 < ρ < 0}, while the estimator γ̂ (1) n (s∗1, ℓ, 0) is better in the area {(γ, ρ) : γ > 0, ρ < −1}. we end this section with comparison of γ̂(2) n (s∗2, ℓ, r ∗ 2) and the hill’s estimator ( [14]) that has the form γ(3) n (k) = 1 k k∑ i=1 ln ( xn−i+1,n xn−k,n ) , where x1,n ≤ · · · ≤ xn,n denote the ascending order statistics of the observations x1, . . . , xn. by [12] we know that the optimal choice of the sample fraction k satisfies the relation k∗ ∼ ( γ2(1− ρ)2 )1/(1−2ρ) n b←(1/n) , n → ∞, while mse ( γ(3) n (k∗) ) ∼ 1− 2ρ −2ρ ( γ−4ρ (1− ρ)2 )1/(1−2ρ) b←(1/n) n , n → ∞ holds. thus, we get immediately lim n→∞ mse ( γ (3) n (k∗) ) mse ( γ̂ (2) n (s∗2, ℓ, r ∗ 2) ) = υ(ℓ, ρ), where υ(ℓ, ρ) = (( ℓ 2 )−2ρ( 1− 2ρ 1− ρ )2−2ρ( γ(1 + ℓ) γ(1 + ℓ− ρ) )2 )1/(1−2ρ) . functions υ(3, ρ) and υ(5, ρ) are shown in fig.3.1. whence one can deduce that for ℓ ≥ 3 there exists ρ̃ = ρ̃(ℓ) < 0 such that the estimator γ̂(2) n (s∗2, ℓ, r ∗ 2) outperforms the hill’s estimator γ(3) n (k∗) within the area (γ, ρ) ∈ (0,∞)× (ρ̃, 0). copyright © 2022 assa. adv syst sci appl (2022) 112 n. m. markovich, m. vaičiulis -1.0 -0.8 -0.6 -0.4 -0.2 ρ 0.90 0.92 0.94 0.96 0.98 1.02 1.04 fig. 3.1. graph of of the functions υ(3, ρ) (grey) and υ(5, ρ) (black) 4. applications we apply the estimators γ̂ (i) n (s, ℓ, 0), i = 1, 2 with ℓ = 1, 3, 5 to real data sets. we analyze absolute log returns of two daily data sets: (i) natural gas prices (dollars usa per cubic feet) between 7 january 1997 and 1 september 2020; (ii) europe brent spot prices (dollars usa per barrel) between 20 may 1987 and 28 august 2020. the data sets contain n = 5389 and n = 8261 non-zero log returns, respectively. let us denote the corresponding absolute log returns by r1,t and r2,t. fig. 1 displays graphs of r1,t (on the left) and r2,t (on the right). 0 1000 2000 3000 4000 5000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 2000 4000 6000 8000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 fig. 4.2. graphs of r1,t, 1 ≤ t ≤ 5389 (on the left) and r2,t, 1 ≤ t ≤ 8261 (on the right) let r(1)i ≥ r (2) i ≥ . . . , i = 1, 2 be the order statistics in the decreasing order. we use qq plots t (i)(n, k) = {( − ln ( j k ) , ln ( r (j) i r (k) i )) , 1 ≤ j ≤ k } , i = 1, 2 provided in [6] for the preliminary analyze. by prop. 4.1 in [6] and assuming that the distribution of the non-zero ri,t belongs to the class rv−1/γ and a sequence k = kn is such that k → ∞, n/k → ∞ as n → ∞, the random sets t (i)(n, s) converge in probability to the half-line t (i) = { (x, γ(i)x) : 0 ≤ x < ∞ } . fig. 2 shows the graph of t (1) (n, k) (on the left) and t (2) (n, k) (on the right) based on the upper 5% of the absolute log returns. the dotted least squares (ls) half-lines are presented in the corresponding graph. by fig. 2, the absolute log returns for both data sets show a classic power law shape, with a slightly worse reflection at the largest values. copyright © 2022 assa. adv syst sci appl (2022) asymptotic properties of the block-type statistics 113 1 2 3 4 5 0.5 1.0 1.5 2.0 1 2 3 4 5 6 0.5 1.0 1.5 2.0 2.5 fig. 4.3. graphs of t (1) ( n(1), n(1)/20 ) (on the left) and t (2) ( n(2), n(2)/20 ) (on the right) with dotted ls half-lines by considering the first data set we provide plots (suggested in [20]) p1 (n, ℓ) = {( θ, γ̂(1) n ([ nθ ] , ℓ, 0 )) , 0 < θ ≤ 0.7 } , p2 (n, ℓ) = {( θ, γ̂(2) n ([ nθ ] , ℓ, 0 )) , 0 < θ ≤ 0.7 } for several ℓ values in fig. 3-5. in practice, a stable region is visually detected in each plot and then an estimate of the parameter γ > 0 is taken as a mean over the stability region (shortly, the sr). the sr and the estimate of γ for each considered value of ℓ is given in tab. 4.1. moreover, the estimate of γ is presented in fig. 3-5 as a dotted line. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 θ 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 θ 0.1 0.2 0.3 0.4 0.5 fig. 4.4. plots p1 (n, 1) (on the left), p2 (n, 1) (on the right) and the estimate of γ (dotted line) the corresponding plots for the second data set are not represented due to their similarity to the plots in fig. 3-5, but the corresponding results obtained by the sr and the estimate of γ are summarized in tab. 4.2. table 4.1. estimation results for the first data set with the size n = 5389 plot p1 (n, ℓ) p2 (n, ℓ) ℓ 1 3 5 1 3 5 sr of θ [0.48,0.6] [0.34,0.56] [0.36,0.54] [0.48,0.6] [0.34,0.56] [0.36,0.54] γ estimate 0.201 0.255 0.278 0.177 0.234 0.258 we end this section with several notings. firstly, the estimates of γ > 0 with ℓ = 1 are in close correspondence for both data sets, see tab. 4.1 and 4.2. from the other hand, by copyright © 2022 assa. adv syst sci appl (2022) 114 n. m. markovich, m. vaičiulis 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 θ 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 θ 0.1 0.2 0.3 0.4 0.5 fig. 4.5. plots p1 (n, 3) (on the left), p2 (n, 3) (on the right) and estimate of γ (dotted line) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 θ 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 θ 0.1 0.2 0.3 0.4 0.5 fig. 4.6. plots p1 (n, 5) (on the left), p2 (n, 5) (on the right) and estimate of γ (dotted line) table 4.2. estimation results for the second data set with the size n = 8261 plot p1 (n, ℓ) p2 (n, ℓ) ℓ 1 3 5 1 3 5 sr of θ [0.32,0.64] [0.32,0.56] [0.34,0.52] [0.32,0.64] [0.32,0.55] [0.34,0.52] γ estimate 0.204 0.228 0.244 0.180 0.212 0.220 comparing the graphs in fig. 1 one can observe that the left graph contains more intermediate size peaks than the right graph. this should be reflected on estimates of γ > 0. such insensitivity of estimators γ̂ (i) n (s, 1, 0), i = 1, 2 in the practical applications is one more reason to use γ̂ (i) n (s, ℓ, 0) with ℓ > 1. secondly, summarizing the results on stability region of parameter θ, we have θ ∈ [1/3, 1/2] for both data sets. this gives that the stability region for block number s is [ [n1/3], [n1/2] ] . consequently, [ [n1/2], [n2/3] ] is stability region for the block width m. keeping in mind that [53891/2] = 73 and [82611/2] = 90, we believe that the block width is large enough in its stability region to take ℓ = 5 for both data sets. finally, we estimate the second order parameter ρ by the estimator proposed in [8] and get ρ̂ = −0.69 and ρ̂ = −0.73 for absolute log returns of the first and second data set, respectively. since the estimator γ̂(2) n (s, 5, 0) is better than the estimator γ̂(1) n (s, 5, 0) when −1 < ρ < 0, we conclude that 0.258 and 0.220 are the estimates of γ for absolute log returns of the data sets under consideration. copyright © 2022 assa. adv syst sci appl (2022) asymptotic properties of the block-type statistics 115 5. conclusions the new family of block-type statistics qn(s, ℓ, r) is introduced. under the classical assumptions on quantile type function u , block number s = sn, block width m = mn and additionally on the parameter r, the weak consistency and the asymptotic normality (the bivariate central limit theorem) of the statistics qn(s, ℓ, r) are proved. the statistics qn(s, ℓ, r) can be used to construct new block-type estimators for positive evi. we list practical reasons for using the block type estimators. firstly, as it is mentioned in introduction, the block type estimators are applicable when only a few largest values are observed within blocks. secondly, due to the recursiveness of the block-type estimators (see, pg. 17 in [15]) they are well suited for the on-line estimation of the positive evi. we proposed the new family of semiparametric estimators γ̂ (2) n (s, ℓ, r) for positive evi. this family is based on statistics qn(s, ℓ, 0) and qn(s, ℓ, r), r ∈ r. hence, the asymptotic normality of γ̂(2) n (s, ℓ, r) is quite simple application of our bivariate central limit theorem for the statistics qn(s, ℓ, r), r ∈ r. the optimal choices s∗2 and r∗2 for s = sn and r are proposed. also, it is proved that for any ℓ ∈ n, the estimator γ̂ (2) n (s∗2, ℓ, r ∗ 2) has a bigger asymptotic mse comparing with γ̂ (2) n (s∗2, ℓ+ 1, r∗2). despite that there is no an optimal estimator among estimators γ̂ (2) n (s∗2, ℓ, r ∗ 2), ℓ ∈ n, we gave some heuristic argument that it is enough to take 3 ≤ ℓ ≤ 5 considering a real data set. moreover, a new family of semiparametric estimators γ̂(1) n (s, ℓ, r) is proposed. it includes several known families of estimators for positive evi. new estimators γ̂(2) n are shown to be better than γ̂ (1) n in the sense of the asymptotic mse. the performance of the estimators γ̂ (1) n (s, ℓ, 0) and γ̂ (2) n (s, ℓ, r) is demonstrated by considering real data sets. we left numerous open questions related to the behavior of new estimators for middle size samples. we admit that quite extensive monte-carlo simulations are needed and we hope to fill this gap in the nearest future. 6. proofs proof of lemma 3.2. the inequality φ(ℓ+ 1, ρ) < φ(ℓ, ρ) is equivalent to −ρ ℓ+ 1 < ( 1 + 1 ℓ )−ρ − 1. (6.10) if ρ = −1, then (6.10) holds obviously. let 0 < −ρ < 1. by using a symmetric form of the bernoulli’s inequality (see, pg.5 in [3]) we obtain that the right hand side of (6.10) is not less than (−ρ/ℓ)(1 + 1/ℓ)−1−ρ. it remains to check that for any ℓ ∈ n the last quantity exceeds −ρ/(ℓ+ 1). let us consider the case −ρ > 1. by applying the symmetric form of the bernoulli’s inequality one more time we get( 1 + 1 ℓ )−ρ − 1 > −ρ ℓ > −ρ ℓ+ 1 , ℓ ∈ n. this completes the proof of lemma 3.2. let y1, y2, . . . , ym be i.i.d. pareto r.v.s with p(y1 ≥ x) = 1/x, x ≥ 1. let yℓ,m, 1 ≤ ℓ ≤ m denote, as usual, the ℓ-th ascending order statistic. if ℓ ∈ n, then by combining example copyright © 2022 assa. adv syst sci appl (2022) 116 n. m. markovich, m. vaičiulis 8.3.3 and theorem 8.4.1 in [1], we get that m−1ym−ℓ,m converges in distribution to some non-degenerate distribution as m → ∞. the next lemma establishes the asymptotic behavior of the moments of m−1ym−ℓ,m. lemma 6.1: let ζ ∈ r satisfy ζ < ℓ+ 1, where ℓ ∈ n. then it holds e ( ym−ℓ,m m )ζ → γ(ℓ− ζ + 1) γ(ℓ+ 1) , m → ∞. (6.11) proof the trivial case ζ = 0 does not require a proof. let ζ > 0. by using (2.2.2) in [1] we derive that the density function of (m−1ym−ℓ,m) ζ is p(x) = m−(ℓ+1)m! ζℓ!(m− ℓ− 1)! x−(ℓ+1)/ζ−1 ( 1− 1 mx1/ζ )m−ℓ−1 , m−ζ ≤ x < ∞. from the above density function we obtain the expectation of (m−1ym−ℓ,m) ζ : e ( ym−ℓ,m m )ζ = m−(ℓ+1)m! ζℓ!(m− ℓ− 1)! ∫ ∞ m−ζ x−(ℓ+1)/ζ ( 1− 1 mx1/ζ )m−ℓ−1 dx = m−ζm! ℓ!(m− ℓ− 1)! b(ℓ− ζ + 1,m− ℓ), where b(a, b) = ∫ 1 0 xa−1(1− x)b−1dx, a > 0, b > 0 is the beta function. the gamma and beta functions are related as b(a, b) = γ(a)γ(b)/γ(a+ b). thus, we have e ( ym−ℓ,m m )ζ = { m−ζγ(m+ 1) γ(m− ζ + 1) } γ(ℓ− ζ + 1) γ(ℓ+ 1) . by using the stirling’s formula for the gamma function one can verify that the quantity in the curly brackets tends to 1 as m → ∞. for the rest case ζ < 0 the proof of (6.11) is similar and thus, it is omitted. lemma 6.1 is proved. proof of the theorem 2.1. we use the potter’s bounds (see, e.g., prop. b.1.9 in [13]). namely, for any ε > 0 and δ > 0 there exists t0 = t0(ε, δ), such that for t ≥ t0 and x ≥ 1, (1− ε)xγ−δ ≤ u(tx) u(t) ≤ (1 + ε)xγ+δ holds. without loss of generality let us assume that 0 < ε < 1. since the function hr is strictly increasing at (0,∞) for any r ∈ r, we get hr ( (1− ε)xγ−δ) ≤ hr ( u(tx) u(t) ) ≤ hr ( (1 + ε)xγ+δ ) , or equivalently, xγrhr ( (1− ε)x−δ ) ≤ hr ( u(tx) u(t) ) − hr (x γ) ≤ xγrhr ( (1 + ε)xδ ) . (6.12) copyright © 2022 assa. adv syst sci appl (2022) asymptotic properties of the block-type statistics 117 keeping in mind u(y (i) j ) d =x (i) j , 1 ≤ j ≤ ℓ, 1 ≤ i ≤ s, (6.13) where y (1) 1 , . . . , y (1) ℓ , . . . , y (s) 1 , . . . , y (s) ℓ are i.i.d. pareto r.v.s with p ( y (1) 1 > x ) = 1/x, x ≥ 1, we have qn(s, ℓ, r) d = 1 s s∑ i=1 1 ℓ ℓ∑ j=1 hr u ( y (i) m−j+1,m ) u ( y (i) m−ℓ,m )  . the right endpoint of the pareto distribution is equal to infinity. thus, for any 1 ≤ i ≤ s, y (i) m−ℓ,m p→+∞ as m → ∞. substituting t = y (i) m−ℓ,m, x = y (i) m−j+1,m y (i) m−ℓ,m , (6.14) into (6.12) we get hr u ( y (i) m−j+1,m ) u ( y (i) m−ℓ,m ) − hr (( y (i) m−j+1,m y (i) m−ℓ,m )γ) ≥ ( y (i) m−j+1,m y (i) m−ℓ,m )γr hr (1− ε) ( y (i) m−j+1,m y (i) m−ℓ,m )−δ , (6.15) hr u ( y (i) m−j+1,m ) u ( y (i) m−ℓ,m ) − hr (( y (i) m−j+1,m y (i) m−j,m )γ) ≤ ( y (i) m−j+1,m y (i) m−ℓ,m )γr hr (1 + ε) ( y (i) m−j+1,m y (i) m−ℓ,m )δ  . (6.16) by summing inequalities (6.15), (6.16) over 1 ≤ j ≤ ℓ and then over 1 ≤ i ≤ s we obtain q̃n(s, ℓ, r)−∆(−) n (s, ℓ, r) ≤ qn(s, ℓ, r) ≤ q̃n(s, ℓ, r) + ∆(+) n (s, ℓ, r), where q̃n(s, ℓ, r) = 1 s s∑ i=1 1 ℓ ℓ∑ j=1 hr (( y (i) m−j+1,m y (i) m−ℓ,m )γ) , ∆(±) n (s, ℓ, r) = 1 s s∑ i=1 1 ℓ ℓ∑ j=1 ( y (i) m−j+1,m y (i) m−ℓ,m )γr hr (1± ε) ( y (i) m−j+1,m y (i) m−ℓ,m )±δ . now it suffices to prove that q̃n(s, ℓ, r) p→ λγ(γr), (6.17) ∆(±) n (s, ℓ, r) p→ 0 (6.18) copyright © 2022 assa. adv syst sci appl (2022) 118 n. m. markovich, m. vaičiulis as n → ∞. the rényi’s representation ( [19]) enables us to write 1 s s∑ i=1 1 ℓ ℓ∑ j=1 g ( y (i) m−j+1,m y (i) m−ℓ,m ) d = 1 s s∑ i=1 1 ℓ ℓ∑ j=1 g ( y (i) j ) , (6.19) where g is any measurable function, see (4.14) in [9]. by applying (6.19) we get q̃n(s, ℓ, r) d = 1 s s∑ i=1 ηi, ηi := 1 ℓ ℓ∑ j=1 hr (( y (i) j )γ) . note that ηi, 1 ≤ i ≤ s is a sequence of i.i.d. r.v.s with eη1 = λγ(γr). thus, by applying the khintchine weak law of large numbers the claim (6.17) follows. let δ > 0 be small enough that such γ(r + δ) < 1 holds. similarly as in (6.17) we find ∆(±) n (s, ℓ, r) p→ { h0(1± ε)± δ, r = 0, hr(1±ε)(1−γr)±γδ/r (1−γ(r±δ))(1−γr) , r ̸= 0. (6.20) let us recall that for any r ∈ r, hr is a continuous function on (0,∞). this, together with hr(1) = 0, gives hr(1± ε) → 0 as ε ↓ 0. so, for any r ∈ r, the right hand side of (6.20) tends to 0 as ε ↓ 0 and δ ↓ 0. thus, the claim (6.18) follows. theorem 2.1 is proved. proof of the theorem 2.2. by using (6.13) one more time we get that for any (θ1, θ2) ∈ r2, θ1 (qn(s, ℓ, r1)− λγ (γr1)) + θ2 (qn(s, ℓ, r2)− λγ (γr2)) d = tn holds, where tn := 1 s s∑ i=1 1 ℓ ℓ∑ j=1 θ1 ( hr1 ( u(y (i) m−j+1,m) u(y (i) m−ℓ,m) ) − λγ (γr1) ) +θ2 ( hr2 ( u(y (i) m−j+1,m) u(y (i) m−ℓ,m) ) − λγ (γr2) ) . by the cramér-wold theorem (see, e.g., pg. 16 in [22]), the claim (2.4) will be proved if we show that √ sntn d→ θ1 ( µν(ℓ, ρ, r1) + λγ (γr1)√ ℓ z(γr1) ) + θ2 ( µν(ℓ, ρ, r2) + λγ (γr2)√ ℓ z(γr2) ) (6.21) as n → ∞. the relation (2.2) is equivalent to lim t→∞ hr ((tx) −γu(tx))− hr (t −γu(t)) (t−γu(t))r a(t) = hρ(x), (6.22) see [16] for details. by applying the drees inequality ( [7], see also prop. 2.1 in [5]), the relation (6.22) implies that there exists the function a0(t), such that a0(t) ∼ (t−γu(t)) r a(t), t → ∞ and for all ε > 0 and δ > 0 there is t0 = t0(δ, ϵ) such that for t ≥ t0 and x ≥ 1,∣∣∣∣hr ((tx) −γu(tx))− hr (t −γu(t)) a0(t) − hρ(x) ∣∣∣∣ ≤ εxρ+δ copyright © 2022 assa. adv syst sci appl (2022) asymptotic properties of the block-type statistics 119 holds, and consequently, it follows hr ( u(tx) u(t) ) ≥ hr (x γ) + xγrhρ(x)a1(t)− εxγr+ρ+δ |a1(t)| , (6.23) hr ( u(tx) u(t) ) ≤ hr (x γ) + xγrhρ(x)a1(t) + εxγr+ρ+δ |a1(t)| , (6.24) where a1(t) = (tγu(t))−r a0(t). we note that a1(t) ∼ a(t) as t → ∞. this, together with |a(t)| ∈ rvρ, implies |a1(t)| ∈ rvρ. next, we apply (6.23), (6.24) with substitutions (6.14). then we get, as in the proof of theorem 2.1, t (1) n + θ1t (2) n (r1) + θ2t (2) n (r2)− ε|θ1|t (3) n (r1)− ε|θ2|t (3) n (r2) ≤ tn, (6.25) t (1) n + θ1t (2) n (r1) + θ2t (2) n (r2) + ε|θ1|t (3) n (r1) + ε|θ2|t (3) n (r2) ≥ tn, (6.26) where t (1) n = 1 s s∑ i=1 1 ℓ ℓ∑ j=1 θ1 ( γhγr1 ( y (i) m−j+1,m y (i) m−ℓ,m ) − λγ (γr1) ) +θ2 ( γhγr2 ( y (i) m−j+1,m y (i) m−ℓ,m ) − λγ (γr2) ) , t (2) n (r) = 1 s s∑ i=1 a1 ( y (i) m−ℓ,m ) υi(r), υi(r) = 1 ℓ ℓ∑ j=1 ( y (i) m−j+1,m y (i) m−ℓ,m )γr hρ ( y (i) m−j+1,m y (i) m−ℓ,m ) , t (3) n (r) = 1 s s∑ i=1 ∣∣∣a1 ( y (i) m−ℓ,m )∣∣∣ 1 ℓ ℓ∑ j=1 ( y (i) m−j+1,m y (i) m−ℓ,m )γr+ρ+δ . let us prove the relation √ snt (1) n d→ θ1 λγ(γr1)√ ℓ z(γr1) + θ2 λγ(γr2)√ ℓ z(γr2), n → ∞. (6.27) by applying (6.19) we get t (1) n d = 1 s s∑ i=1 1 ℓ ℓ∑ j=1 θ1 ( γhγr1 ( y (i) j ) − λγ (γr1) ) + θ2 ( γhγr2 ( y (i) j ) − λγ (γr2) ) . (6.28) the summands over i in the right hand side of (6.28) present a sequence of the i.i.d. zero mean r.v.s. moreover, under assumptions γr1 < 1/2 and γr2 < 1/2 we have e ( 1 ℓ ℓ∑ j=1 θ1 ( γhγr1 ( y (1) j ) − λγ (γr1) ) + θ2 ( γhγr2 ( y (1) j ) − λγ (γr2) ))2 = 1 ℓ ( θ21λ 2 γ(γr1)σ 2(γr1, γr1) + 2θ1θ2λγ(γr1)λγ(γr2)σ 2(γr1, γr2) +θ22λ 2 γ(γr2)σ 2(γr2, γr2) ) . copyright © 2022 assa. adv syst sci appl (2022) 120 n. m. markovich, m. vaičiulis thus, the relation (6.27) follows by applying the lindeberg-lévy central limit theorem (see, e.g., pg. 16 in [22]). we claim that under assumption γr < 1/2, √ snt (2) n (r) p→ µ · ν(ℓ, ρ, r), n → ∞, (6.29) and, assuming additionally, 0 < δ < 1− γr − ρ, √ snt (3) n (r) p→ |µ| 1− γr − ρ− δ , n → ∞. (6.30) the relations (6.29), (6.30), together with (6.27) prove (6.21). let us prove that (6.29) holds. by the definition of the function a, there exists t1, such that the function a1(t) has a constant sign when t ≥ t1. let us recall that |a1| ∈ rvρ holds. by applying the drees inequalities for regularly varying functions (see, e.g., prop. b.1.10 in [13]) we get that for any ε̃ > 0 and δ̃ > 0 there exists t2 = t2(ϵ̃, δ̃), such that for t ≥ t1 ∨ t2 and tx ≥ t1 ∨ t2, it holds −ε̃max { xρ+δ̃, xρ−δ̃ } ≤ a1(tx) a1(t) − xρ ≤ ε̃max { xρ+δ̃, xρ−δ̃ } . we apply the latter inequality with t replaced by m and x replaced by y (i) m−ℓ,m/m. put t (4) n (ρ, r) = 1 s s∑ i=1 ( y (i) m−ℓ,m m )ρ υi(r). as in the proof of theorem 2.1, we get −ε̃a1(m)max { t (4) n (ρ− δ̃, r), t (4) n (ρ+ δ̃, r) } + a1(m)t (4) n (ρ, r) ≤ t (2) n (r),(6.31) ε̃a1(m)max { t (4) n (ρ− δ̃, r), t (4) n (ρ+ δ̃, r) } + a1(m)t (4) n (ρ, r) ≥ t (2) n (r),(6.32) when a1(m) > 0 for m ≥ t1 ∨ t2 and the reverse inequalities to (6.31), (6.32) hold in the case a1(m) < 0 for m ≥ t1 ∨ t2. since assumption (2.3) implies √ sa1(m) → µ as n → ∞, the relation (6.29) will be proved if we show that for any ρ̃ ∈ r satisfying 2ρ̃ < ℓ+ 1, t (4) n (ρ̃, r) p→ ν(ℓ, ρ̃, r), n → ∞. (6.33) as for relation (6.33), it follows from t (4) n (ρ̃, r)− e (y (1) m−ℓ,m m )ρ̃ υ1(r)  p→ 0, (6.34) e (y (1) m−ℓ,m m )ρ̃ υ1(r)  → ν(ℓ, ρ̃, r), n → ∞. (6.35) we will prove e t (4) n (ρ̃, r)− e (y (1) m−ℓ,m m )ρ̃ υ1(r) 2 → 0, n → ∞, (6.36) copyright © 2022 assa. adv syst sci appl (2022) asymptotic properties of the block-type statistics 121 which, in turn, will complete the proof of (6.34), see section 8.3 in [2]. by using the renyi’s representation one can get that for any 1 ≤ i ≤ s, y (i) m−ℓ,m and υi(r) are independent r.v.s. thus, the left hand side of (6.36) is equal to 1 s e ( y (1) m−ℓ,m m )2ρ̃ e (υ1(r)) 2 − e2 ( y (1) m−ℓ,m m )ρ̃ e2 (υ1(r))  ≤ 2 s e ( y (1) m−ℓ,m m )2ρ̃ e (υ1(r)) 2 . by our assumptions s = sn → ∞ holds as n → ∞. this, together with e ( y (1) m−ℓ,m/m )2ρ̃ → γ(ℓ− 2ρ̃+ 1)/γ(ℓ+ 1) as n → ∞ (see lemma 6.1) and e (υ1(r)) 2 < ∞ yields (6.36). to prove that υ1 is square integrable r.v. it is enough to use (6.19) and then to apply a direct integration. it rests to prove (6.35). since y (1) m−ℓ,m and υ1(r) are independent r.v.s we get that the left hand side of (6.35) is equal to e ( y (1) m−ℓ,m/m )ρ̃ e (υ1(r)). now (6.35) follows by applying lemma 6.1 and (6.19) one more time. the proof of (6.30) is similar to the proof of (6.29) and thus, it is omitted. this completes the proof of theorem 2.2. proof of remark 1. with the notation of the previous proof, we have t (1) n /a(m) = √ st (1) n / ( √ sa(m)), where we take θ1 = 1 and θ2 = 0 in the definition of t (1) n . by combining (2.5) and (6.27) we get t (1) n /a(m) p→ 0 as n → ∞. from the proof of theorem 2.2 it follows that t (2) n (r1) p→ ν(ℓ, ρ, r1), n → ∞ and the sequence t (3) n (r1), n = 1, 2, . . . is bounded in probability. thus, keeping in mind inequalities (6.25), (6.26), the relation (2.6) follows. this completes the proof. proof of theorem 3.1. let r ̸= 0. we have γ̂(2) n (s, ℓ, r)− γ = (1− γr) (qn(s, ℓ, r)− λγ(γr))− (qn(s, ℓ, 0)− λγ(0)) rqn(s, ℓ, r) . (6.37) by theorem 2.2, the numerator of the right hand side of (6.37), multiplied by √ sn, converges in distribution to n ( µγr (1− ρ)(1− γr − ρ) · γ(1 + ℓ− ρ) γ(1 + ℓ) , 2γ4r2 ℓ(1− γr)(1− 2γr ) . as the denominator, it tends in probability to rλγ(γr) as n → ∞. now (3.7) follows by applying the slutsky’s lemma (see, e.g., pg. 11 in [22]). let r = 0. the similar argument used in proving (2.4) shows that √ s ( qn(s, ℓ, 0)− γ, q̃(s, ℓ)− 2γ2 ) d→ µγ(1 + ℓ− ρ) (1− ρ)γ(1 + ℓ) ( 1, 2γ(2− ρ) 1− ρ ) + (π1,π2) holds as n → ∞. here, µ is the same as in (2.3), while (π1,π2) is a gaussian random vector with zero means and eπ2 1 = γ2/ℓ, eπ2 2 = 20γ4/ℓ, e (π1π2) = 4γ3/ℓ. now, by using the copyright © 2022 assa. adv syst sci appl (2022) 122 n. m. markovich, m. vaičiulis identity γ̂(2) n − γ = ( q̃(s, ℓ)− 2γ2 ) − 2γ (qn(s, ℓ, 0)− γ) 2qn(s, ℓ, 0) , it is not difficult to verify that (3.7) holds with r = 0. this ends the proof. proof of corollary 2.3. this follows by a similar argument as used in proving theorem 3.1. proof of the inequality (3.9). note that (χ̃(ρ))(1−2ρ)/(−2ρ) can be rewritten as follows (χ̃(ρ))(1−2ρ)/(−2ρ) = ( (χ(ρ) + ρ)2 2 )1−1/ρ( 1 2 + χ(ρ) 2(1− ρ) ) , where χ(ρ) = (ρ2 − 4ρ+ 2) 1/2. it is easy to check that χ(ρ) > √ 2− ρ and, consequently, χ(ρ) > 1− ρ hold for any ρ < 0. thus, (3.9) follows. acknowledgements the reported study was funded by the russian science foundation rsf, project number 22-21-00177 (recipient n.m. markovich, writing-review and editing). references 1. arnold, b.c., balakrishnan, n. & nagaraja, h.n. (1992). a first course in order statistics, new york, wiley. 2. borovkov, a.a. (1968). probability theory. moscow, nauka (in russian). 3. bullen, p.s. (2003). handbook of means and their inequalities. dordrecht/boston/london, kluwer academic publishers. 4. cederman, l.-e., warren, t.c. & sornette, d.(2011). testing clausewitz: nationalism, mass mobilization, and the severity of war, international organization, 65, 605-638. 5. cheng, s. & jiang, c. (2001). the edgerworth expansion for distributions of extreme values, sci. china ser. a, 44, 427-437. 6. das, b. & resnick, s. i. (2008). qq plots, random sets and data from a heavy tailed distribution, stochastic models, 24, 103-132. 7. drees, h. (1998). on smooth statistical tail functions, scand. j. statist., 25, 187-210. 8. fraga alves, m. i., gomes, m. i. & de haan, l. (2003). a new class of semi-parametric estimators of the second order parameter, portugaliae mathematica, 60, 193-214. 9. fraga alves, m.i., gomes, m. i., de haan, l. & neves, c. (2009). mixed moment estimator and location invariant alternatives, extremes, 12, 149-185. 10. ferreira, a. & de haan, l. (2015). on the block maxima method in extreme value theory: pwm estimators, the annals of statistics, 43, 276-298. 11. geluk, j. & de haan, l. (1987). regular variation, extensions and tauberian theorems. in amsterdam: cwi tract 40, center for mathematics and computer science. 12. de haan, l. & peng, l. (1998). comparison of tail index estimators, statist. nederlandica, 52, 60-70. 13. de haan, l. & ferreira, a. (2006). extreme value theory: an introduction. new york, springer. copyright © 2022 assa. adv syst sci appl (2022) asymptotic properties of the block-type statistics 123 14. hill, b. m. (1975). a simple general approach to inference about the tail of a distribution,annals statistics 3, 1163–1174. 15. markovich, n.m. (2007). nonparametric analysis of univariate heavy-tailed data. chichester, wiley. 16. paulauskas, v. & vaičiulis, m. (2013). on the improvement of hill and some others estimators. lith. math. j., 53, 336-355. 17. paulauskas, v. & vaičiulis, m. (2011). several modifications of dpr estimator of the tail index, lith. math. j., 51, 36-50. 18. qi, y. (2010). on the tail index of a heavy tailed distribution. ann. inst. statist. math., 62, 277-289. 19. rényi, a. (1953). on the theory of order statistics, acta math. acad. sci. hungar., 4, 191-231. 20. resnick, s. & stărică, c. (1997). smoothing the hill estimator, adv. in appl. probab., 29, 271-293. 21. vaičiulis, m. (2012). asymptotic properties of generalized dpr statistic. lith. math. j., 52, 95-110. 22. van der vaart, a. w. (1998). asymptotic statistics. cambridge, cambridge university press. copyright © 2022 assa. adv syst sci appl (2022) introduction main results a new family of the block-type estimators for positive evi applications conclusions proofs adv syst sci appl 2021; 01; 22-45 published online at https://ijassa.ipu.ru. design and comparison of plasma h∞ loop shaping and rga-h∞ double decoupling multivariable cascade magnetic control systems for a spherical tokamak yuri mitrishkin1,2*, evgeniia pavlova2, mikhail patrov3 1) faculty of physics, lomonosov moscow state university, moscow, russia e-mail: y_mitrishkin@hotmail.com 2) v. a. trapeznikov institute of control sciences of the russian academy of sciences, moscow, russia e-mail: pavlova@physics.msu.ru 3) ioffe institute of the russian academy of sciences, saint petersburg, russia e-mail: michael.patrov@mail.ioffe.ru abstract: the article aims to present an approach to design and compare cascade h loop shaping and essentially new cascade rga-h double decoupling magnetic control systems for a multivariable dynamical plant, specifically plasma in a vertically elongated tokamak. identification of the present control closed-loops containing a plasma linear model of a relatively high order for the spherical globus-m tokamak (ioffe institute, st petersburg, russia) to derive a low order linear model (without the application of reduction algorithms) as a plant under control is undertaken. a robust h loop shaping method was applied to the identified model to design a plasma position, current, and shape (6 gaps between the first wall and plasma separatrix) multivariable controller. a structural analysis was done to get the most effective structure of the square plant with the 3rd gap eliminated in the feedback and a separate loop for the plasma current control. the methodology of the relative gain array (rga) was applied to this structure to choose the proper correspondences between inputs and outputs (pairing), which brought the plant model closer to a decoupling plant (first decoupling in the open plant model). further, the h adjustment of the control system with the pairing plant and an additional feedback decoupling matrix (second decoupling of the plant model in the feedback) and pi controllers in the feedback gave increased control system accuracy while tracking references. comparison of the two control systems designed has shown that double decoupling gives higher performance accuracy and a less robust stability margin, while the robust loop shaping method allows the stability margin to be increased but gave less accurate control of the gaps. keywords: tokamak, plasma control, identification, structural analysis, decoupling 1. introduction 1.1. state-of-the-art a number of plasma magnetic control systems were designed and applied in simulations and experiments on a set of operating tokamaks with vertically elongated plasmas (jet, asdex upgrade, tcv, diii-d, east, kstar, jt-60u, nstx, mast, and others) and in simulations for iter (international thermonuclear experimental reactor). the survey [21] showed that there is a set of existing approaches which are used for the development of plasma magnetic control systems, for example, robust h∞ control [1, 28], mpc [5], adaptive * corresponding author: y_mitrishkin@hotmail.com mailto:y_mitrishkin@hotmail.com mailto:pavlova@physics.msu.ru mailto:michael.patrov@mail.ioffe.ru mailto:y_mitrishkin@hotmail.com design and comparison of plasma h∞ loop shaping… 23 copyright ©2021 assa. adv. in systems science and appl. (2021) mpc [20], cascade control [10, 27, 35], state control by lmi [32], etc. and allow the desired objectives to be achieved. 1.2. motivation however, multivariable plasma control systems are very far from a satisfactory level concerning their performance (accuracy and speed of response) and robustness (required stability margins) in confining the plasma with thermonuclear reactions inside vacuum vessels of future tokamak-reactors and tokamak power plants with high reliability and survivability. therefore, at present they require further detailed development, like any other technology for fusion, e.g. materials for the first wall and diverter plates. (a) (b) (c) fig. 1. (a) spherical globus-m tokamak, (b) vertical crosssection of globus-m tokamak where pf, cc, hfc, and vfc are poloidal, correction, horizon and vertical field coils, and (c) plasma magnetic configuration with points g1-g6 and directions for gaps control [11] globus-m2 [18] belongs to a new generation of spherical tokamaks and it was launched in 2018. it is the same size as globus-m (fig. 1) [7, 8, 9] (plasma major radius r ≈ 36 cm, minor radius a ≈ 24 cm, elongation ≈ 2), but a new electromagnetic system allows a higher toroidal magnetic field and plasma current to be achieved. in 2018 the first plasma was obtained; discharges with bt = 0.5 t, ip = 0.2 ma, which correspond to the parameters of the last globus-m experimental campaign [2], were demonstrated [17]. the first experimental campaign with the toroidal magnetic field of 0.7 t and the plasma current up to 300 ka was 24 y. mitrishkin, e. pavlova, m. patrov copyright ©2021 assa. adv. in systems science and appl. (2021) carried out in 2019. the increase in toroidal magnetic field and plasma current resulted in an overall improvement of the discharge parameters. significant plasma temperature growth and neutron rate increase was obtained. record stored energy, which is approximately one and a half times higher than in the globus-m experiments, was reached. the toroidal lower hybrid wave launch provided noticeable current drive. preparation for the next experimental campaigns with a higher toroidal magnetic field and plasma current is underway. in future campaigns globus-m2 will operate at designed parameters up to bt = 1 t, ip = 0.5 ma. the experiments will be centred around the reactor-relevant issues: current drive, energy and fast ion confinement, non-inductive plasma current start-up, diverter and sol plasma studies. the poloidal system of the globus-m2 tokamak is unique because it is combined: some of the pf-coils are located inside the toroidal field coil and some are located outside that coil. in most tokamaks, the pf-coils are located completely inside or outside the toroidal field coil [24] or inside the vacuum vessel [21]. the globus-m2 tokamak operates with a diverter plasma configuration similar to iter (france). the power density of the plasma heating in globus-m2 is several times higher than in the larger existing foreign tokamaks: jet (great britain), diii-d (usa), asdex upgrade (germany), tcv (switzerland). tests of the key elements of the diagnostic systems of iter are conducted on the globus-m2 tokamak to support iter. because of that, we intend to design and simulate plasma control systems for the tokamak globus-m, which is one of the advanced spherical tokamaks. 1.3. application of standard control theory methods a set of standard control theory methods was applied to the existing multivariable feedback multi-loop and cascade control system of globus-m to design and find the best control system structures for plasma current and shape control on plasma linear models from the point of view of performance and robustness (h∞ optimisation theory, h∞ decoupling techniques, multilevel control theory, µ-analysis, identification by means of the subspace method, relative gain array approach, singular value frequency responses). the applied control methods of system design and analysis are known, but the results of their application to a plasma in a tokamak are new. the design and simulation of hierarchical plasma control systems for globus-m are done in line with the original methodology developed in [22]. 1.4. results and their novelty obtained for a plasma in a tokamak identification subspace methodology (not standard reduction methods of linear models) [14] was applied to the linear plasma position and pf-currents control systems containing plasma linear models to reduce their high order to an essential lower order so as to design plasma current and shape multivariable control systems. this approach imitated the application of the identification procedure to real control systems used in experiments to get identified linear models for the design of robust controllers which should enable the efficiency of closed-loop control systems. the h robust loop shaping method [16] was applied to the identified (reduced) plasma model to design a plasma shape (6 gaps between the first wall and plasma separatrix and 5 pf-currents) multivariable controller in the presence of tuned pid-controllers inside the plasma position and pf-currents control circuits. at the same time, the normalised coprime factorised plant transfer function with perturbations in factors is shaped by input and output weighting functions, then a robust feedback controller is found to maximise the robust stability margin. the final controller contains the robust controller and weighting functions used. four structures were analysed by singular value frequency responses [34] with the 3rd or 4th gap eliminated in the feedback and a separate loop for the plasma current control to get the most effective structure of the square plant without the 3rd gap. design and comparison of plasma h∞ loop shaping… 25 copyright ©2021 assa. adv. in systems science and appl. (2021) the square plant model obtained was used to design a decentralised multivariable picontroller by means of a new double decoupling: (i) pairing of inputs/outputs in line with the rga matrix [34] and (ii) fixed structure h∞-tuning in the presence of an additional decoupling matrix in the feedback [4]. the two plasma shape control systems designed by the h∞-robust loop shaping approach and by the double decoupling approach were compared using performance and robustness criteria. the first system showed a higher robust stability margin and the second one demonstrated better accuracy while tracking the plasma separatrix. 1.5. paper structure the paper is organised as follows. in section 2, the control problem is formulated using the linear dynamic model of the plasma. in section 3 the identification procedure and in section 4 cascade h loop shaping controller design are represented. section 5 is devoted to input‒output analysis of multi-loop system structures. section 6 discusses loop pairing and introduces the h∞ double decoupling plasma shape control system design and simulation for square plant. the comparison of the two systems is presented in section 7. in conclusion the results obtained are summarised. 2. control problem statement 2.1. parameters of the globus-m tokamak the plasma in the globus-m tokamak (fig. 1, (a)) is the plant of the current investigation [7] with the following parameters: the maximum plasma current ip max = 0.35 mа, the maximum toroidal magnetic field bt max = 0.6 t, the major radius r0 = 0.36 m, the minor radius aр = 0.24 m, the aspect ratio a = r0/ap = 1.5, and the maximum vertical elongation kmax = 2.2. 2.2. plasma magnetic control system of the globus-m/m2 tokamak the plasma control multivariable and cascade system which is used by the globus-m tokamak to control the plasma vertical (z) and horizontal (r) positions (with the diagnostics system to assess the vertical and horizontal position of the plasma), the current in the central solenoid (cs), and the currents in the poloidal field (pf) coils are demonstrated in fig. 2. 2.3. linear model of the globus-m/m2 plasma the linear state space model for the plasma in the tokamak [22, 23], in line with the block diagram of fig. 2, can be written in the general form with concrete matrices: , x ax bu y cx du= + = + where a75×75, b75×8, c39×75, d39×8 are matrices of the model. the outputs of the plant model are as follows: t t p hfc vfc cs pf ccy z r i i i i i i g         =   where δz, δr are plasma vertical and horizontal displacements, δip is the variation of the plasma current, δg are variations of the gaps between the separatrix and the first wall (fig. 1, (c)), δihfc, δivfc, δics, δipf, and δicc are variations of currents in corresponding poloidal field coils (fig. 1, (b)). the inputs of the plant model are as follows:   t hfc vfc cs pf ccu u u u u u    = where δuhfc, δuvfc, δucs, δupf, and δucc are variations of voltages on the hfc, vfc, cs, pf, and cc coils, respectively. the state vector x includes variations of currents in active coils and passive structures (the vacuum vessel) (fig. 1, (b)). 26 y. mitrishkin, e. pavlova, m. patrov copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 2. block diagram of the plasma control system operating in the globus-m/m2: cz, cr, ccs, cpf1, cpf2_top, cpf2_bottom, cpf3, and ccc are analogue controllers, ahfc, avfc, acs, apf1, apf2_top, apf2_bottom, apf3, and acc are actuators. 2.4. aims of the feedback control system design the main control goal is to develop a plasma position (vertical and horizontal), current and shape control system for the globus-m tokamak. the principal challenge is to build the feedback plasma shape control system. currently the plasma shape control on globus-m is carried out by means of program pf-currents, but there is no actual feedback shape control system on the globus-m tokamak (fig. 2). the problem of choosing a particular structure for the control system and the problem of tuning the multivariable controllers in this system are solved in this article. the structure of the control system has a significant effect on the characteristics of the closed-loop system, such as its stability and performance (speed of response and accuracy) as well as its robustness, so it is important to consider the structure in detail. 2.5. controllers to be designed two types of plasma shape controllers: an h∞ controller obtained by means of the loop shaping design procedure [15, 16] and a double decoupling pi-controller obtained by means of the rga [34] and structured h∞ synthesis [4], are to be designed and compared in terms of performance and robustness. these two approaches to the synthesis of control systems allow minimizing a single performance criterion, but the distinctive feature of the second approach is the decoupling of the plant. • h∞ robust control based on open-loop shaping the h∞ loop shaping design procedure consists of two stages: loop shaping and robust stabilisation [15, 16]. during the first stage the singular values of the plant model g0 are shaped by shaping the inputs and outputs of the plant using the preand post-compensators design and comparison of plasma h∞ loop shaping… 27 copyright ©2021 assa. adv. in systems science and appl. (2021) (w1 and w2) in order that the shaped plant model g = w2g0w1 has large minimum singular values at low frequencies and small maximum singular values at high frequencies. at the second stage the robust stabilisation problem is solved for the shaped plant in relation to an h∞-controller k∞ in the feedback. in doing so, a robust stability margin of the closed-loop system is derived as follows: ( ) 1 1 max 1 / inf k i i gk m k   − −      = +    , where 1m n− is a normalised left coprime factorisation of the shaped plant model. the final feedback controller includes the preand post-compensators w1 and w2: k = w1 k∞ w2. • double decoupling with pi-control first decoupling in the open-loop plant model is done using the methodology of the relative gain array (rga) and, as a consequence, the proper pairing between model inputs and outputs, which decreases the interactions between control channels [34]. the loop pairing problem defines the control system structure, i.e., which of the available plant inputs are to be used to control each of the plant outputs. variable pairing is used to establish the one-to-one mapping relationship between manipulated and controlled variables, and then decentralised feedback control loops can be tuned and implemented. the loop pairing decision should be done independently, such that any controller type could be designed after loop pairing. the second decoupling of the plant model in the feedback is done using h fixedstructure adjustment of the control system with the pairing plant model, an additional feedback decoupling matrix, and pi-controllers in the feedback [4]. • comparison of the approaches and plant model in the case of the h∞ robust control system design by the loop shaping approach of mcfarlane and glover [15, 16], the open-loop plant model is shaped by preand postweighting functions to get proper frequency singular values characteristics, and then the shaped plant model is used in the feedback to design the h∞-controller by maximising the robust stability margin of the closed-loop system as the single optimisation criterion. in the case of the fixed-structure h∞ control approach of gahinet and apkarian [4], the structure of the closed-loop control system is given by a designer and then the mixed sensitivity problem is solved by minimisation of the h∞-norm of the stacked transfer matrix of sensitivity, and complementary sensitivity with weighting functions as another optimisation criterion. in both cases of plasma shape control in the globus-m tokamak, the plant model consists of a number of control closed loops of cs&pf-currents, where the references are the inputs (fig. 2) and a plasma equilibrium reconstruction algorithm (not shown in fig. 2) [11] to provide the model outputs, namely gaps between the separatrix and the first wall. the other output values are measured by tokamak diagnostics, specifically the cs&pf-currents, plasma current, horizontal and vertical plasma displacements [22]. 2.6. goals of plasma control the outputs δr, δz, δip, and δg are to be controlled by means of feedback of the plasma control system. the errors of these values to be controlled are the following: , , , , p p j jr r z z i i p g g je r r e r z e r i e r g= − = − = − = − 1,...,6j = . the control problem can be stated as the need to achieve specified accuracy of the control in the finite time intervals: ( )   ( )  1 2, , , , , ,r r z ze t t t t e t t t t     ( )   ( )  3 4, , , , , , p p j ji i g ge t t t t e t t t t     28 y. mitrishkin, e. pavlova, m. patrov copyright ©2021 assa. adv. in systems science and appl. (2021) where 1 4,...,t t are time moments when the proper output values enter into desired bounds determined by positive constants , , and p jr r i g    after transient responses which are caused by the tracking procedure. these time moments 1 4,...,t t are different because of the different plant dynamics along the different channels under control. the time moment t is the moment of finishing the control procedures because plasma discharges in a tokamak are restricted in time. it is important to meet stability requirements in the control system design. stability margins may be specified and obtained by means of the small gain theorem when the result of multiplication of the h∞-norms of the transfer functions of the uncertainty δ and the known part of the system q is less than 1: 1q     . in that case, the µ-analysis gives useful results [28, 36]. 3. identification of plasma position, current, and shape closedloops 3.1. obtaining high order models to be identified to develop the plasma control system, it is often useful to deal with low-dimensional models to design loworder controllers so that these controllers are able to determine the system dynamics during the quasi-stationary phase of plasma discharge with sufficient accuracy. to obtain the simplified, low-order model, one can use a model reduction technique [34] or an identification method [13, 14]. the linear model of the plant under control with a reduced order was produced as a result of the identification of the full (the 75th order) model [11, 31] obtained on the basis of the experimental data of the tokamak globus-m [23]. the outputs of the aforementioned 75-dimensional model were used as the outputs of the real-time globusm tokamak. the order of the model comes from the plasma in a tokamak and actuators which are connected in series with the tokamak (fig. 2). before the beginning of the simulation, the unstable linear model was vertically stabilised by an automatically adjusted pid-controller included in the feedback at a predetermined point of the scenario. the pid-controller was tuned using the pid tuner app in matlab. this app allows one to achieve a balance between performance and robustness while tuning the gains of a controller for a siso plant. then the testing reference signals were fed to the input of the closed loop system (fig. 3(1)). the testing signals were fed to the four control loops as a novel series one by one (fig. 3): (1) vertical and (2) horizontal position control loops, (3) plasma current control loop, (4) gaps between the separatrix and the first wall control loop. all these pidcontrollers were also synthesized using the pid tuner app in matlab. since, as noted above, the numerical experiment was carried out at one point of the scenario, the first half of the plasma discharge was used for identification and the second half was used for verification of the obtained models. there are two types of actuators in the closed loop system, namely, thyristor current inverters [12] and multiphase thyristor rectifiers [29]. the current inverters are complex nonlinear impulse converters based on thyristor bridges working in self-oscillatory mode. there is a proportional relationship on average between the input and output signals of the current inverters [12], so they are modelled with a gain of 250. the 6-pulse rectifier is modelled as a transfer function of the first order with a gain of 2000 and a time constant of 3.3 ms for the purposes of the simulation [29]. design and comparison of plasma h∞ loop shaping… 29 copyright ©2021 assa. adv. in systems science and appl. (2021) pretuned pid vertical position controller testing signal uhfc plasma in globus-m tokamak ipf (5) δg (6) ics δip δr δzcurrent invertor +_ (1) pid vertical position controller testing signal uhfc plasma in globus-m tokamak uvfc ipf (5) δg (6) ics δip δr δzcurrent invertor current invertor (2) pid horizontal position controller pid vertical position controller uhfc plasma in globus-m tokamak rectifier testing signal uvfc ipf (5) δg (6) ics δip δr δzcurrent invertor current invertor +_ p (3) pid horizontal position controller pid vertical position controller uhfc plasma in globus-m tokamak rectifier pid plasma current controller rectifiers (5) uvfc testing signals (6) ipf (5) δg (6) ics δip δr δzcurrent invertor current invertor +_ + _ p p (4) fig. 3. testing and identification of the system control loops: (1) vertical position control loop; (2) horizontal position control loop; (3) plasma current control loop; (4) gaps between the separatrix and the first wall control loop the automatically tuned plasma position and current pid-controllers are presented by the transfer function as follows: 1 ( ) i d pid p f k w k t k s s s s = + + + with the following parameters: 30 y. mitrishkin, e. pavlova, m. patrov copyright ©2021 assa. adv. in systems science and appl. (2021) • vertical position pid-controller: kp = 85, ki = 8507.5, kd = 0.13, tf = 1/29981.5; • horizontal position pid-controller: kp = 544.86, ki = 25716, kd=0, tf = 1/2740.06; • plasma current pid-controller: kp = 0, ki = 131.79, kd = 0, tf = 0, p=1. 3.2. identification of closed-loop control systems the identification was conducted using the system identification toolbox from matlab by means of subspace methodology [13]. as a result of the loop (1) identification, the transfer function of the closed-loop system was derived. the controllers of previously tuned control loops were included in the controllable system during the identification procedure of loops (2)‒(4) (fig. 3). the accuracy of identification of the vertical position control loop (2nd order) and the horizontal position control loop (3rd order) is 89% and 94% respectively, the plasma current control loop (9th order) is 86%, and identification of the multivariate plasma shape control loop (20th order) is ~ 80‒90% for each of the loops. figs. 4 and 5 show the transient processes for the system with the full model of the 75th order and the system with the identified models of the plasma current control loop and multivariable gaps control loop. one can see that on average the components of the transient processes coincide. fig. 4. comparison of transient responses of identified and numerical experimental models for plasma current control loop fig. 5. comparison of transient responses of identified and numerical experimental models for plasma shape (gaps) multivariable control loop the results of the identification of the plasma position and plasma current loops showed that this approach makes it possible to achieve a future understanding of the application of those results to get a plasma model from the closed-loop identified transfer functions when the transfer functions of the actuator and controller are known. in this way, some results of the current inverter linear identification have been obtained in [12]. design and comparison of plasma h∞ loop shaping… 31 copyright ©2021 assa. adv. in systems science and appl. (2021) the identification procedure of the system with the plasma model to determine the gaps in the plasma shape controller design may be used directly in experiments when the plasma equilibrium reconstruction code [23, 25, 26] is connected to the tokamak outputs. 4. robust h∞ loop shaping control system design: 5 inputs – 6 gaps and separate loop for plasma current control the magnetic control system for the plasma position, current, and shape in globus-m was designed as a whole as a complex multivariable multi-loop and cascade system consisting of the following circuits (fig. 6): • two loops for vertical and horizontal plasma positions control coupled via the plant (decentralised control), • scalar cascade circuit with the inner cascade to control the current in the central solenoid and the outer cascade to control the plasma current, • multivariable cascade circuit with the inner cascade to control 5 currents in the poloidal field coils and the outer cascade to control the 6 gaps. pid horizontal position controller pid vertical position controller reference reference uhfc plasma in globus-m tokamak rectifier pid plasma current controller reference rectifiers (5) h∞ shape controller (5×6) uvfc reference ipf (5) δg (6) ics δip δr δzcurrent invertor current invertor +_ +_ +_ +_ +_ +_ p p ∆ w fig. 6. plasma position, current, and shape control system for the globus-m tokamak the robust loop shaping method [15, 16] was used to synthesise the multivariable controller k (which is the 5  6 size transfer matrix) for the identified transfer function of the 20th order between the pf-currents references and gaps. this method is based on the normalised coprime factorisation of the transfer function of the shaped controlled model with position and currents loops containing the plasma model 1 g m n−= (see section 2.2). the robust stability margin of the control system with the robust controller designed is  = 0.36. here n m      is the h∞-norm of two uncertain stable transfer functions, n and m , in the factorisation factors of the perturbed controlled model ( ) ( ) 1 m ng m n −  = +  +  . in doing so, the perturbed transfer function g is shaped by preand post-compensators w1 (s) = i6×6 × 106/s, w2 (s) = i5×5 and a feedback robust controller is found to maximise  . the final controller includes compensators and the robust h∞-controller designed (see section 2.2). 32 y. mitrishkin, e. pavlova, m. patrov copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 7. vertical and horizontal position of the plasma transient responses fig. 8. plasma current and shape transient responses the results of the numerical simulation of the multivariable closed-loop control system are shown in figs. 7 and 8. the following testing reference signals were fed to the input of the system to move the plasma separatrix from the initial position to the final one: 0 m to the vertical position input, 0.01 m to the horizontal position input, 5 ka to the plasma current input, and 0.01 [1 1 -1 -1 0 1] m to the plasma shape inputs (fig. 9). the transient response time of gaps is about 0.02 s. design and comparison of plasma h∞ loop shaping… 33 copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 9. plasma separatrix displacement (the black line is the initial separatrix position, the red line is the final position) along with unit vectors for gap displacements g1-g6 and upper x-point 5. input-output analysis of multi-loop system structures: 5 inputs – 5 gaps, separate and joint plasma current control the objective of this section is to choose the plant structure that will provide an opportunity for decentralised plasma current and shape control. the plasma position stabilisation problem has to be solved first to obtain a stable plant for plasma current and shape control system design. the plasma vertical and horizontal positions were stabilised by pidcontrollers beforehand. two alternative arrangements were considered: with joint and separate controllers for plasma current and shape. the plant for gap displacements δg (between the plasma separatrix and the first wall) control has 5 inputs (δi ref-pf) and 6 outputs (δg) (fig. 6). a loop shaping design procedure using h∞ synthesis was used to build the plasma shape control system (see subsection 3.2) for the plant with 5 inputs and 6 outputs. for the purposes of engineering, it is more convenient to work with a plant that has an equal number of inputs and outputs (squared plant). such a structure provides a possibility for decoupling of the control loops and makes it possible to use simple pid-controllers and gain matrices, which considerably simplifies the control system tuning, but still provides an opportunity to meet the basic design requirements such as speed of response, control bandwidth, robust stability, etc. in order to design effective controllers for the squared plant model, four variants of control systems were designed: the systems with separate control for the plasma current and shape when gap 3 or 4 is eliminated, and the analogous systems with joint control (fig. 10). for objective comparison of these four systems, 6 feedback controllers for the plasma current and shape were synthesised using the h∞ robust loop-shaping method in line with the schemes in fig. 10 (see section 4), and singular value analysis [34] was carried out for these systems. different linear control systems for the same plant model with the same inputs and outputs may be compared in line with the same criterions in time and frequency domains. it does not matter what kind of controllers are used in that comparison in the feedback and what weighting functions have been used in the design. designers are interested to design control systems in these conditions with maximum accuracy, speed of response, and robust stability margins. anyway, one has to minimize the order of the linear controller in continues or discrete time. eventually one has to choose the system which meets mostly the criteria mentioned above. this is basic aim of the designers in this research. the frequency characteristics of the minimum singular values are constructed (fig. 11a) for the four dynamic systems shown in fig. 10, where the plasma current and the gaps 34 y. mitrishkin, e. pavlova, m. patrov copyright ©2021 assa. adv. in systems science and appl. (2021) control loops (the latter connected by dotted lines) were opened by breaking the feedback loop at the plant model output. one generally wants the minimum singular value  of the open loop system (when breaking the loop at the output of the plant model at the point indicated by in fig. 10) to be as large as possible at lower frequencies where the performance of the control system is considered, and the condition number to be small [34]. a large  of the open control system leads to small  of the sensitivity function which links the output disturbance with the plant output. plasma model with position stabilisation loops and separate inner cs&pfcurrents cascades g2 g4 g5 g6 ip g1 gaps controller plasma current controller u y δg (5) δip plasma model with position stabilisation loops and separate inner cs&pfcurrents cascades g2 g3 g5 g6 ip g1 gaps controller plasma current controller u y δg (5) δip (a) (b) plasma model with position stabilisation loops and inner cs&pf currents cascade g2 g4 g5 g6 ip g1 plasma current and gaps controller u y plasma model with position stabilisation loops and inner cs&pf currents cascade g2 g3 g5 g6 ip g1 plasma current and gaps controller u y (c) (d) fig. 10. control systems: (a) separate control, gap 3 is eliminated, (b) separate control, gap 4 is eliminated, (c) joint control, gap 3 is eliminated, (d) joint control, gap 4 is eliminated (a) design and comparison of plasma h∞ loop shaping… 35 copyright ©2021 assa. adv. in systems science and appl. (2021) (b) (c) (d) fig. 11. (a) minimum singular values  and (c) condition numbers for 4 variants of the plant (the graph legend consists of the following components: the number of the gap which was eliminated from the system is the 3rd or 4th, the type of controller for plasma current and shape is separate or joint); (b) a ratio of the minimum singular values  and (d) a ratio of the condition numbers comparing the respective values for the systems with joint and separate control (the graph legend indicates the gap that is eliminated from the system) so   1 ( ) ( ) ( )s s i g s k s − = + is the sensitivity transfer function for the mimo systems given in fig. 10, where ( )g s is the plant model and ( )k s is the controller. to reduce the effects of the output disturbance on the model output, ( )s s should be small over the frequencies of performance interest. to reduce the sensitivity, one requires the maximum singular value of ( )s s to be small while 36 y. mitrishkin, e. pavlova, m. patrov copyright ©2021 assa. adv. in systems science and appl. (2021) ( ) ( )( ) 1 ( ( )) ( ) ( ) 1 ( ( )) l s g s k s s s l s     =   . the frequency range for the globus-m tokamak systems in question is 10‒3000 hz. as  is larger for systems with separate control in the whole frequency range (fig. 11 (a)), one must therefore synthesise two different controllers for current and shape. the condition number is smaller for the system with separate control in almost the whole frequency range (fig. 11 (c)). fig. 11 (b) and (d) show the quantitative difference in the minimum singular values and conditional numbers for the systems with separate and joint control. from the above, it is clear that one should choose the following structure of the system: separate control for the plasma current and gaps and the 3rd gap should be eliminated from consideration and from the feedback of the control system. 6. pairing and h ∞ decoupling plasma shape control system design and simulation for square plant 6.1. pairing of the square plant: 5 inputs – 5 gaps (3rd gap is eliminated), separate plasma current control within the frame of the selected plant model structure, the rga analysis [3, 34] has been applied to the multivariable transfer function g(s) from the references of the inner cascade of pf-currents to the gap’s displacements of the outer cascade. there is an interaction between the controlled and manipulated variables. inevitably, each controlled variable has an impact from all the manipulated variables through a plant. the methodology of the rga allows one to measure the input‒output interactions and choose the best pairing of variables (there are n! possible pairing configurations for the n×n plant). there are two extreme cases that one should consider to build the rga matrix for the system with the given plant (fig. 12): (a) other loops are opened 0,ku k j=  :   0, , . k i ij ij ij j u k j y g g g u =    = =    in this case, only the selected input affects the systems outputs, which allows the effect of a certain input on each output to be measured (fig. 12, a); (b) other loops are closed 0,ky k i=  : 1 0, 1 ˆ, . k i ij ij j y k i ji y g g u g− =    = =        in this case, all inputs affect all system outputs, which allows the effect of all inputs on the selected output to be measured, since the other loops are closed. in fig. 12 (b), the signal from input uj goes not only to the chosen output yi but also to the other outputs and, via the feedback loops, comes to the same output yi carrying the parasitic information of the cross-links. here uj is used to control yi. to derive the rga matrix, one can notice that   0, , k i ij j u k j y y gu g u =    =  =    1 1 0, . k j ji i y k i u u g y g y − − =     =  =     design and comparison of plasma h∞ loop shaping… 37 copyright ©2021 assa. adv. in systems science and appl. (2021) plant u1 =0 uj≠0 uk=0 y1 yk yigij plant u1 uj uk y1≈0 yk≈0 yi≠0 k controller (a) (b) fig. 12. extreme rga configurations: (a) other loops are opened, (b) other loops are closed so the ij-th measure of interactions is   1 . ˆ ij ij ij ji ij g g g g  − = =   the rga matrix is the corresponding matrix of relative gains (where  denotes element-by-element multiplication): 1( ( )) ( ) ( ( ))trga g s g s g s−=  . the central idea around the rga matrix is to assess the effect that a controller of each output variable would have on a specific open loop variable. ijg is the open loop gain between the jth input and ith output, ˆ ijg is the resulting open loop response between the jth input and ith output when all other outputs are under control. so, the smaller the interactions between the chosen channel and the other channels, the closer the ratio of ˆ ij ij g g to 1. there are basic pairing rules that give more effective correspondence between plant outputs and inputs: • input–output pairing with the corresponding rga elements closest to 1 is preferred [3]; • large rga elements should be avoided for input‒output pairing selection [33]; • avoid pairing on negative steady-state rga elements. this condition is closely related to systems with lack of integrity. ( )g s is the transfer function matrix of the open-loop system without a mimo plasma shape controller, which makes it possible to rearrange the inputs and outputs of the system. the rga matrix for the system with the 3rd gap eliminated is as follows: ( ) 7.25 10.83 0.53 3.24 13.67 0.88 1.2 15.03 .2.03 6.06 0.11 0.45 2.55 0.06 0.78 5.53 7.46 23.19 1.25 17.32 rga g −    − − −   = − − −   − − −   − −  0.81 1.73 5.59 8.37 1.33 the initial loop pairing is the following: the 1st output corresponds to the 1st input, and so on: 2-2, 3-3, 4-4, 5-5. after rga matrix analysis in the matrix rga(g) in line with the rules, the new input/output correspondence shown in table 1 was obtained, which is the sequence of matrix entries highlighted in bold type. table 1. rga input/output correspondence outputs 1 2 3 4 5 inputs 3 1 5 2 4 38 y. mitrishkin, e. pavlova, m. patrov copyright ©2021 assa. adv. in systems science and appl. (2021) in accordance with table 1, one can construct a matrix h of the discovered input/output correspondence that may be connected to plant inputs for further controller design: 0 0 1 0 0 1 0 0 0 0 .0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 h        =        6.2. design of plasma shape control system with additional channel decoupling by h fixed structure tuning when the control system structure was analysed and the most effective loop pairing was found, the control problem arose. the structured h∞ synthesis helps one to build a decoupling controller [4, 6]. all control elements were tuned using the looptune command in matlab [4]. this command automatically converts the target bandwidth, performance requirements, and additional design requirements into weighting functions that express the requirements as an h∞ optimisation problem. the optimisation of the h∞ mixed-sensitivity norm s (s is the sensitivity) allows one to tune the controller: ( ) 1 2 ( ) min, ( ) k w s k w i s k  ⎯⎯→ − where w1, w2 are weighting functions. automatic tuning of all feedback loops (picontrollers and decoupling matrix d) for the system with the eliminated 3rd gap and appropriate loop pairing has been done (fig. 13). fig. 13. structural scheme of output-input pairing, multivariable pi-controller and d-decoupling matrix for globus-m when the control system is tuned, the decoupling matrix d ensures that each output of the controlled plant tracks the corresponding reference signal with minimal crosstalk. to tune the control system, including the decoupling matrix d, one first creates a numeric model of the plant. then one creates tunable models of the controller elements and interconnect them to build a controller model. then one uses looptune command to tune the free parameters of the controller model. finally, one has to examine the performance of the tuned system to confirm that the tuned controller yields desirable performance. design and comparison of plasma h∞ loop shaping… 39 copyright ©2021 assa. adv. in systems science and appl. (2021) 7. comparison of the two systems 7.1. performance analysis by means of simulations the systems with two types of controllers, namely, the h∞ controller synthesised by the robust loop shaping method (h∞ loop shaping controller) and the controller synthesised and automatically tuned on the basis of structural analysis, the rga matrix, and structured h∞ synthesis (double rga-h∞ decoupling controller), have been compared. the responses of the systems to the following reference gref = g×[1 1 -1 -1 0 1], 4 , 0 0.005, 0.02, 0.005, t t g t   =     are shown in fig. 14. the minus sign corresponds to the plasma moving to the first wall. the gap tracking errors are presented in fig. 15. the response time is approximately 10 ms and the maximum deviation of gaps is about 0.75 cm for the former system and 0.5 cm for the latter system. the steady-state errors are 0.24 cm for the former system and 0.13 cm for the latter system (fig. 15, table 2). table 2. comparison of the maximum deviation maxd of gaps and the steady-state errors sse gaps h∞ loop shaping controller rga-h∞ decoupling controller maxd , m sse , m maxd , m sse , m g1 0.0031 0.0013 0.0071 0.0001 g2 0.0034 0.0019 0.0045 0 g3 0.005 0.0006 – – g4 0.0027 0.0018 0.002 0 g5 0.007 0.0006 0.002 0 g6 0.0025 0.0025 0.0016 0.0011 fig. 14. reference response of the plasma shape control systems in globus-m 40 y. mitrishkin, e. pavlova, m. patrov copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 15. gap tracking errors tuning the pi-controllers’ parameters and decoupling matrix d by means of the looptune command for the latter system from an unstable initial guess took 25 iterations and about 3 minutes in the matlab environment. figs. 9 and 14 show how the plasma separatrix moved during the simulation. the separatrix is the plasma boundary in the poloidal plane and may be found as the largest closed level line of the poloidal flux [1]. 7.2. robust stability analysis 7.2.1. singular value analysis a plant model additive disturbance ∆ was added into the systems to evaluate the robust stability margins that the two control systems could provide [36]. the structural schemes of the plasma shape control systems in globus-m with additive disturbance are shown in fig. 16. the transfer functions for the known parts of the systems connecting the uncertainty output and input v→u are as follows: (a) ( ) 1 h k i gk − = − + , (b) ( )( ) ( ) 1 h i k d g k d − = − + + + . gk ∆ +_ + + u v gref y gk ∆ +_ + + u v +_ d gref y (a) (b) fig. 16. structural schemes of the plasma shape control systems in globus-m with additive plant model disturbance: (a) system with h∞ controller synthesized by loop shaping method, (b) system with controller synthesized on the basis of structural adjustment from the small gain theorem [36], it is deduced that the robust stability margin of the system can be obtained as follows: design and comparison of plasma h∞ loop shaping… 41 copyright ©2021 assa. adv. in systems science and appl. (2021) 1 h    . the maximum singular values and the corresponding values of the robust stability margins for the given systems are represented in fig. 17 and table 3. however, the h robust loop-shaping controller provides a considerably greater robust stability margin at additive plant uncertainty. compared to the system with the controller synthesised on the basis of structural adjustment, it allows the system with the h∞ controller synthesised by the loop shaping method to ensure a roughly two and a half times higher robust stability margin (table 3). fig. 17. maximum singular values table 3. robust stability margins method type of the system singular value analysis µ-analysis (a) h∞ loop shaping controller 8.91×10-6 1.17 (b) h∞ decoupling controller 3.98×10-6 1.34 the ratio of robust stability margins of system (a) to (b) 2.23 0.87 7.2.2. µ-analysis a lot of dynamic perturbations may occur in different parts of the system. in particular, in our research, we used a multiplicative input uncertainty. however, they can all be pulled out into one perturbation block ∆, represented by the block-diagonal matrix   11 ... : , sr s r idiag i i c    =   where 1 s i i r n = = and n is the dimension of the block ∆. this representation (figs. 6 and 13) of the systems made it possible to test the robust stability of the closed-loop system (w is a weighting function). the frequency response plots of the structured singular values µ [30] ( ) ( ) ( ) 1 : min : det 0m i m   −  =  −  = were examined to determine the system with the highest robust stability margin (fig. 18). from the small gain theorem, one may deduce that the smaller the µ, the higher the robust stability margin. the µ analysis was conducted for m∆ configurations of the systems. for the system with the h∞ controller synthesised by the loop shaping method, the µ peak equals 1.17 and is less than for the system with controller synthesised on the basis of structural analysis, which is equal to 1.34 (table 3). as the structured singular value µ is the direct measure of the robust stability margin of the closed loop system with uncertainties, it was therefore clear that the former system has the higher robust stability margin. 42 y. mitrishkin, e. pavlova, m. patrov copyright ©2021 assa. adv. in systems science and appl. (2021) fig. 18. structured singular values µ for: (a) system with h∞ controller synthesised by loop shaping method; (b) system with controller synthesised on the basis of structural adjustment the interpretation of the robust stability analysis results obtained by the structured singular value µand singular values showed that the conclusions are consistent. 8. conclusion to summarise, the double decoupling approach is proposed for the plasma shape control system design. this novel cascade rga-h double decoupling magnetic control system was compared with the h∞ loop shaping design control system in terms of robustness and performance. the first one deals with structural decisions that must be made before starting the controller design. the system reduction using identification which, in turn, makes it possible to reduce an order of controller, provides the basis for the second one. these two approaches were applied for plasma shape control in the globus-m tokamak. the h∞ loop shaping controller, which uses a combination of loop shaping and robust stabilisation, was designed for the reduced plasma model. identification subspace methodology was applied to linear plasma models to reduce their high order to an essential lower order to design plasma current and shape multivariable control systems. the h∞ double decoupling fixed-structure controller was designed for the plasma shape control linear model, which consists of two feedback loops: the inner loop provides decoupling and the outer loop also provides decoupling and the desired setpoint tracking performance. the first approach results in good robustness (the ratio of the robust stability margins of the latter system to the former system is more than one), whereas the second one made it possible to find the control system structure that ensures better time-domain performance (the maximum deviation of gaps and the steady-state errors are about 50% less for the second control system). in future, it is intended to introduce the control systems designed into experiments on the globus-m2 tokamak [8] on the base of the real time testbed being developed in [19]. acknowledgements the authors thank research associate p.s. korenev (v.a. trapeznikov institute of control sciences of the russian academy of sciences, moscow, russia) for the plasma linear models designed on the basis of the experimental data of the globus-m tokamak. this work was supported by the russian science foundation (rsf) under grant numbers 1719-01022 and 21-79-20180. design and comparison of plasma h∞ loop shaping… 43 copyright ©2021 assa. adv. in systems science and appl. 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(1987). implications of large rga elements on control performance. industrial & engineering chemistry research, 26, 2323–2330. doi: https://doi.org/10.1021/ie00071a025 34. skogestad, s. and postlethwaite, i. (2005). multivariable feedback control. analysis and design. john wiley and sons, ltd, 2nd ed. 35. yuan, q.p. et al. (2013). plasma current, position and shape feedback control on east. nuclear fusion, 53, 043009. doi: https://doi.org/10.1088/0029-5515/53/4/043009 36. zhou, k. and doyle, j.c. (1998). essentials of robust control. prentice hall. https://doi.org/10.1016/j.fusengdes.2016.04.026 https://doi.org/10.1016/0005-1098(93)90175-s https://doi.org/10.1021/ie00071a025 https://doi.org/10.1088/0029-5515/53/4/043009 1. introduction 1.1. state-of-the-art 1.2. motivation 1.3. application of standard control theory methods 1.4. results and their novelty obtained for a plasma in a tokamak 1.5. paper structure 2. control problem statement 2.1. parameters of the globus-m tokamak 2.2. plasma magnetic control system of the globus-m/m2 tokamak 2.3. linear model of the globus-m/m2 plasma 2.4. aims of the feedback control system design 2.5. controllers to be designed 2.6. goals of plasma control 3. identification of plasma position, current, and shape closed-loops 3.1. obtaining high order models to be identified 3.2. identification of closed-loop control systems 4. robust h∞ loop shaping control system design: 5 inputs – 6 gaps and separate loop for plasma current control 5. input-output analysis of multi-loop system structures: 5 inputs – 5 gaps, separate and joint plasma current control 6. pairing and h∞ decoupling plasma shape control system design and simulation for square plant 6.1. pairing of the square plant: 5 inputs – 5 gaps (3rd gap is eliminated), separate plasma current control 6.2. design of plasma shape control system with additional channel decoupling by h( fixed structure tuning 7. comparison of the two systems 7.1. performance analysis by means of simulations 7.2. robust stability analysis 7.2.1. singular value analysis 7.2.2. µ-analysis 8. conclusion acknowledgements adv syst sci appl 2024; 02:94–102 published online at https://ijassa.ipu.ru. practical applicability of the metric approach for a scheduling problem alexander lazarev1*, darya lemtyuzhnikova1,2, ilja kudinov1 1v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia, 2moscow aviation institute, moscow, russia abstract: a given special case of np-complete scheduling problem can be approximated by solving a special case of similar problem with the same precedence graph. we construct a metric space over a set of special cases of this problem and consider the statistical relationship between distance between a pair of special cases of the under consideration and the average error of the approximated solution. sethi, gabow, coffman’s and fujii’s algorithms for this problem are used. it is shown that the absolute and the relative error of the objective function decreases over the density of a graph with a fixed number of jobs. in general case, relative non-zero error value increases with the number of jobs. keywords: scheduling, optimization and control, operations research, makespan 1. introduction nowadays various polynomial algorithms for many combinatoric optimization problems have been developed, but there remain many of them for which the question of existence of polynomial algorithm is still open. at the same time absolute approximate algorithms are known for very few np-hard optimization problems and in most cases only relative error of objective function value estimates are available. therefore, it is important to develop approximate algorithms that allow to obtain solutions of the problem with an acceptable error estimation. the metric approach as one of the effective modern approach is considered in this paper. the main idea of this is following. assume a metric ρ between a pair of instances a, b of some problem: ρ(a,b) ≥ fa(x∗ b)− fa(x∗ a), (1.1) where fa(x) is objective function values of instance a, and the arguments x∗ a and x∗ b are an optimal solution of instances a and b corresponding. then, the metric become an upper bound of the absolute error of using the optimal solution x∗ b of instance b as an approximate solution of instance a. if the searching for an optimal solution of instance a is significantly more time-consuming compared to finding that of b, and we know a rather small upper bound of ρ(a,b), we can use x∗ b as a approximate solution of instance a. in this paper we investigate the feasibility and practicability of applying the metric approach to solve the np-complete scheduling problem of processing jobs on two parallel machines with a partial ordered set of jobs: p2 | prec, pj ∈ {1, 2} | cmax, where processing times of jobs equal 1 or 2. ∗corresponding author: alexander lazarev, jobmath@mail.ru practical applicability of the metric approach... 95 there exist a little of experiments conducted as a proof of method applicability to problems of different types. our research is based on lazarev’s papers [1] and [2], in which the way of metric’s construction for multiple problems is discussed. moreover, we conducted an experiment for a single machine scheduling problem. to fulfill the existing knowledge of feasibility to apply metric approach for various types of problems, we conduct experiments for problem p2 | prec, pj ∈ {1, 2} | cmax. we chose this simplification of the problem p2 | prec, pj = k | cmax for conduction the experiments due to the existence of the variety of polynomial algorithms for the problem p2 | prec, pj = 1 | cmax, to which we can easily apply metric approach. nevertheless, we apply the proof of metric construction for the problem p2 | prec, pj = k | cmax, where k ∈ z+. one of the first papers concerning an algorithm for optimal solution obtaining for problem p2 | prec, pj = 1 | cmax is the work of fujii [4], in which the algorithm based on finding the maximum matching in precedence graph was proposed. the upper bound of the number of operations required to find the optimal sequence is o(n3) operations where n is number of jobs. the coffman [5] and sethi [6] proposed algorithms which, as the fujii’s algorithm, are based on a list of jobs, the list is used to sequentially take elements for distribution on two machines. coffman’s algorithm has found the upper bound is o(n2) operations, and sethi’s work presents two algorithms, one of which makes the labeling for o(e+ n) operations, and the other makes the schedule for o(e+ nα(n)) operations, where e – the number of edges in the precedence graph, and α(n) – ackerman’s function, which slowly grows. the labeling algorithm proposed by sethi is largely based on coffman’s algorithm, but it adds an additional examination of the structure at each level of the graph. gabow’s algorithm [7] works according to the ”high level first” (hlf) principle and requires o(e+ nα(n)) operations. the np-completeness of the problems p2 | prec, pj = 1 | cmax and p2 | prec, pj ∈ {1, 2} | cmax was first proved ullman [8]. furthermore, the bevern in paper [9] proves that the problem p2 | prec, pj ∈ {1, 2} | cmax is w [2]-hard parameterized by the width of the partial order. however it is an open question if pm | prec, pj = 1 | cmax is np-hard for m ⩾ 3. in literature there are references to the work of [10], which gives an algorithm whose complexity is o(n log n) operations for the problem p2 | prec, pj ∈ {1, 2} | cmax, and a heuristic algorithm [11], in the worst case giving error equal to one for the problem p2 | prec, pj ∈ {1, 2} | cmax. in addition, in [12] an algorithm is given that yields the optimal solution for the problem p2 | prec, pj ∈ {1, 2} | cmax using o(n2 log n) operations. hu [13] demonstrates polynomial algorithm for the problem p2 | prec, pj = 1 | cmax, if graph of ordering restrictions is a tree and number of jobs satisfies some conditions. by the way, dolev’s methods [14] lead to polynomial algorithms for the problem p2 | prec, pj = 1 | cmax if the number of machines is fixed and the precedence graph has a certain form. in particular, if the precedence graph contains only in-trees or out-trees, the result leads to linear algorithms for finding an optimal schedule on two and three machines. the structure of the paper is organized as follows. in section 2 the mathematical formulation of the problem p2 | prec, pj ∈ {1, 2} | cmax is given. in section 3 we define the metric we use. in section 4 we describe our motivation and approach for the problem. in section 5 the experiment’s result is given. 2. the problem definition we consider scheduling the problem p2 | prec, pj ∈ {1, 2} | cmax. each instance a has a set of n jobs na = n and set of m machines ma = m . each job j ∈ n can be processed on one machine during pj time units. each machine can process only one job at a time. set copyright © 2024 assa. adv syst sci appl (2024) 96 a. lazarev, d. levtyuzhnikova, i. kudinov n has a partial order ≺ that from j1 ≺ j2 follows j2 can not be started processing until j1 be complete. the partial order is defined by a dag (directed acyclic graph) called a precedence graph. we denote πa = π as a permutation (j1, j2, . . . , jn) over the job set n . in order of constructing a schedule we put each job from π sequentially to the earliest possible place of processing in the schedule. for simplicity, we call π a schedule as well. if saj (π) is a starting time of job j ∈ n processing and ca j (π) = saj (π) + pj is the completion one, the schedule π has the following features: 1. if mj1 = mj2 then [sj1(π), cj1(π)) ∩ [sj2(π), cj2(π)) = ∅ for j1, j2 ∈ n . 2. if j1 ≺ j2 then cj1(π) ≤ sj2(π). the first condition means that a machine cannot process more than one job at a time. the second condition is nothing more than the fulfilltime of precedence relations between jobs. a schedule π of an instance a is feasible if it does not disrupt the conditions above. one schedule π can be feasible for a number of instances with the same precedence graphs. as is usually denoted in scheduling theory, we will denote makespane by cmax(π): cmax(π) = max j∈n cj(π). a schedule π∗ is called an optimal schedule for an instance a of the problem p2 | prec, pj ∈ {1, 2} | cmax if cmax(π ∗) ≤ cmax(π) for each feasible π. as indicated [9], the problem p2 | prec, pj ∈ {1, 2} | cmax is np-complete with respect to the width of the partial order. therefore, this is the reason why approximate solutions of the problem are considered. 3. metric over the set of instances of the problem p2 | prec, pj ∈ {1, 2} | cmax we consider the metric approach application for the problem p2 | prec, pj ∈ {1, 2} | cmax. let π∗ a be an optimal solution of instance a, and π∗ b is that of instance b. using cmax instead of f into (1.1) gives us, the following: ρ(a,b) ≥ ca max(π ∗ b)− ca max(π ∗ a). (3.2) there is a metric for the problem p2 | pj | cmax [3]: ρ(a,b) = ∑ j∈n |paj − pbj |, (3.3) where instances a and b have the same number of jobs n. this is minkowski of order 1. in case of the problem p2 | prec, pj ∈ {1, 2} | cmax there is no known metric for the problem with partial ordered over n . so we believe that for pairs of instances a and b that do not differ otherwise than a partial order over n the metric takes the minimal solution only when the partial order of both a and b is the same. lemma 3.1: let a and b be instances of the problem p2 | prec, pj ∈ {1, 2} | cmax, and π∗ a, π∗ b are optimal solutions of a, b corresponding. then: ca max(π ∗ b)− ca max(π ∗ a) ≤ ∑ j∈n |paj − pbj |. (3.4) copyright © 2024 assa. adv syst sci appl (2024) practical applicability of the metric approach... 97 proof let’s consider the following difference: |ca max(π)− cb max(π)| (3.5) for any a, b, π and give an upper bound of this. now we prove that the maximum of (3.5) reaches on π on which every job is processed on only one machine, so the other one processes no one job. obviously, the decrease value of the target function value is not greater than the difference between sums of processing times of all jobs in terms of each instances a and b: |ca max(π)− cb max(π)| ≤ ∣∣∣∣∣∑ j∈n paj − ∑ j∈n pbj ∣∣∣∣∣ . using the following commonly known inequality: |a+ b| ≤ |a|+ |b|, we obtain |ca max(π)− cb max(π)| ≤ ∣∣∣∣∣∑ j∈n paj − ∑ j∈n pbj ∣∣∣∣∣ ≤ ∑ j∈n |paj − pbj |. (3.6) finally, using (3.6), we get ca max(π ∗ b)− ca max(π ∗ a) = = ( ca max(π ∗ b)− cb max(π ∗ b) ) + ( cb max(π ∗ b)− cb max(π ∗ a) ) + ( cb max(π ∗ a)− ca max(π ∗ a) ) ≤ ≤ ( cb max(π ∗ b)− cb max(π ∗ a) ) + 2 ∑ j∈n |paj − pbj |. that is ( ca max(π ∗ b)− ca max(π ∗ a) ) + ( cb max(π ∗ b)− cb max(π ∗ a) ) ≤ 2 ∑ j∈n |paj − pbj |. from symmetry considerations, we finally obtain the following result: ca max(π ∗ b)− ca max(π ∗ a) ≤ ∑ j∈n |paj − pbj |. theorem 3.1: the function ρ from (3.3) is a metric over the sets of instances of the problem p2 | prec, pj ∈ {1, 2} | cmax. proof that (3.4) corresponds to the axioms of the metric is obvious: 1. ρ(a,b) = 0 ⇐⇒ a = b: ρ(a,b) = ∑ j∈n |paj − pbj | = 0 ⇐⇒ paj = pbj , ∀j ∈ n. copyright © 2024 assa. adv syst sci appl (2024) 98 a. lazarev, d. levtyuzhnikova, i. kudinov 2. ρ(a,b) ≥ 0: ρ(a,b) = ∑ j∈n |paj − pbj | ≥ 0. 3. ρ(a,b) = ρ(b,a): ρ(a,b) = ∑ j∈n |paj − pbj | = ∑ j∈n | − paj + pbj | = ρ(b,a) = 4. ρ(a,c) ≤ ρ(a,b) + ρ(b,c): since |a+ b| ≤ |a|+ |b|, ρ(a,c) =∑ j∈n |paj − pcj | = ∑ j∈n |paj − pbj + pbj − pcj | ≤ ∑ j∈n |paj − pbj |+ ∑ j∈n |pbj − pcj | = ρ(a,b) + ρ(b,c). 4. the polynomial-time approximation scheme as stated above, an instance of the problem p2 | prec, pj = 1 | cmax can be solved by polynomial algorithms. it is possible to convert an instance a of the problem p2 | prec, pj ∈ {1, 2} | cmax to an instance b of the problem p2 | prec, pj = 1 | cmax by setting pj = 1 for all j ∈ n . this leads to a polynomial-time approximation scheme (ptas) that can be seen on fig. 4.1. b π∗ b πa a π∗ a o(n) o(n) o(n) o(2n) fig. 4.1. the ptas scheme for the problem p2 | prec, pj ∈ {1, 2} | cmax. an instance of the problem p2 | prec, pj ∈ {1, 2} | cmax has a set of feasible schedules. but some of them differs from the other only by the order of jobs on each machine, not by the makespan value. since we aim to minimize target function cmax, we take only those schedules that have the minimal value of cmax. no solving instance b algorithm from the ones above must be better than others in general case. indeed, let f∗(a) ⊆ f(a) be a subset of optimal schedules from f(a). since g(a) = g(b), the sets f(a) and f(b) are equal too. but sets f∗(a) and f∗(b) are completely different in general case. it means that the polynomial algorithms one use for the instance b of the problem p2 | prec, pj = 1 | cmax generally just take a random schedule from f(a) as an approximate solution πa for the instance a. 5. computer experiments four polynomial algorithms for the problem p2 | prec, pj = 1 | cmax: fujii’s algorithm [4], coffman’s algorithm [5], sethi’s algorithm [6] and gabow’s algorithm [7] were copyright © 2024 assa. adv syst sci appl (2024) practical applicability of the metric approach... 99 implemented. optimal solution of np -hard problem p2 | prec, pj ∈ {1, 2} | cmax was obtained by glpk solver†. the following indicators were chosen as efficiency evaluation criteria: 1. µ determines the share of cases for which no optimal solution was obtained by the approximate algorithm: µ = k̄ k , where k is the number of generated examples, and k̄ is the one for which the found solution was not optimal, i.e: cmax(πi)− cmax(π ∗ i ) > 0; 2. average relative non-zero error βnre: βnre = 1 k̄ k̄∑ i=1 cmax(πi)− cmax(π ∗ i ) cmax(π∗ i ) ; 3. average absolute non-zero error βnae: βnae = 1 k̄ k̄∑ i=1 ( cmax(πi,a)− cmax(π ∗ i,a) ) ; 4. average relative error βre: βre = 1 k k∑ i=1 cmax(πi)− cmax(π ∗ i ) cmax(π∗ i ) ; 5. average absolute error βae: βae = 1 k k∑ i=1 ( cmax(πi,a)− cmax(π ∗ i,a) ) . there were calculated 1000 of uniformly distributed random instances of the problem p2 | prec, pj = {1, 2} | cmax for instance with number of jobs n ∈ {3, 4, . . . , 13} with the density d = 0.3, where the density is the ratio of the number of edges of the given graph to the one of the complete graph, and 250 of uniformly distributed random instances for each problem for density values d ∈ {0.1, 0.2, . . . , 1.0} for problems with number of jobs n ∈ {3, 4, . . . , 13}. as can be seen from fig. 5.2, the share of cases for which the algorithm obtained a nonoptimal solution, µ grows no more than linearly with increasing the number of vertices (jobs) n. the fluctuations in the graph at odd n are caused by the fact that there are two machines in the problem p2 | prec, pj ∈ {1, 2} | cmax. when a number of jobs n is odd, on average the last job can be put into the schedule, leaving an empty space on another machine. an important result is the dependence of the relative non-zero error βnre and the absolute non-zero error βnae on the number of vertices n. as can be seen from fig. 5.3, respectively, βnre decreases with increasing number of vertices, and βnae grows slowly, which makes it possible to apply the method to graphs with a large number of vertices. the absolute error βnae remains approximately at the same level for all n. this is a consequence of the fact that copyright © 2024 assa. adv syst sci appl (2024) 100 a. lazarev, d. levtyuzhnikova, i. kudinov fig. 5.2. graph of the dependence of the share of cases µ for which the algorithm obtained a non-optimal solution on the number of vertices n. it slowly grows with the growth of n. fluctuations between even and odd values of n is due to the fact that there are two machines in considered problem (a) (b) fig. 5.3. graphs of the dependence of the relative non-zero error βnre (a) and the absolute non-zero error βnae (b) indicators on the number of vertices n at 1000 generations. the cases n ≤ 4 are trivial, and the polynomial algorithms always get an optimal solution the polynomial algorithms absolute errors do not tend to accumulate overtime and ”fixes” each other due to the presence of two machines. but if we start count cases of zero error, the full picture changes. as can be seen from fig. 5.4, the graphs of relative and absolute errors on the number of vertices show a pairwise relationship between the algorithms. during the near-constant graph of βnae, the growth of βae means growth of the rate of errors. all indicators show a decrease in the value of the error when the density of the graph d increases, as look in fig. 5.5. it seems that the less the graph density d, the smaller a space of feasible schedules, and the remain are closer to an optimal one. †https://www.gnu.org/software/glpk/glpk.html copyright © 2024 assa. adv syst sci appl (2024) practical applicability of the metric approach... 101 (a) (b) fig. 5.4. graph of the dependence of the relative error βre (a) and the absolute error βae (b) on the number of vertices n at 1000 generations fig. 5.5. graph of the dependence of the absolute error βae on the density d of the graph at 250 generations 6. conclusions in this paper we consider the possibility of applying the metric approach to solve the np -complete problem p2 | prec, pj ∈ {1, 2} | cmax. it was shown that if the number of restrictions increases, the absolute and relative error of cmax change slowly. besides, the relative non-zero error value decreases, if the number of vertices in the graph grows. in addition, it turned out to be more efficient to use the coffman’s and gabow’s algorithms in finding an approximate solution of this problem by the method discussed in this paper. the obtained average absolute error cmax(πa)− cmax(π ∗ a) seems to be depends only on n and d on sets of all dags of classes considered above. however, this result could not be proved analytically at this moment. acknowledgements copyright © 2024 assa. adv syst sci appl (2024) 102 a. lazarev, d. levtyuzhnikova, i. kudinov the results of this paper partly were obtained within the rsf grant (project no. 22-7110131). we thanks e. bukueva for assistance on in the early stages of the work. references 1. lazarev, a. (2019) scheduling theory. methods and algorithms. moscow, russia: ics ras. 2. lazarev, a., lemtyuzhnikova, d. & werner f. (2021) a metric approach for scheduling problems with minimizing the maximum penalty, applied mathematical modelling, 89, 1163–1176. 3. lazarev, a. & arkhipov, b. (2018) estimation of the absolute error and polynomial solvability for a classical np-hard scheduling problem, doklady mathematics, 97, 262–265. 4. fujii, m., kasami, t. & ninomiya, k. (1969) optimal sequencing of two equivalent processors, siam journal on applied mathematics, 17(4), 784–789. 5. coffman e. g. & graham r. l. (1972) optimal scheduling for two-processor systems, acta informatica, 1(3), 200–213. 6. sethi, r. (1976) scheduling graphs on two processors, siam journal on computing, 5(1), 73–82. 7. gabow, h. n. (1982) an almost-linear algorithm for two-processor scheduling, journal of the acm (jacm), 29(3), 766–780. 8. ullman, j. d. (1975) np-complete scheduling problems, journal of computer and system sciences, 10(3), 384–393. 9. van bevern, r., bredereck, r., bulteau, l., komusiewicz, c., talmon, n., et al. (2016) precedence-constrained scheduling problems parameterized by partial order width, international conference on discrete optimization and operations research – springer, cham, 105–120. 10. nakajima, k., leung, j. & hakimi, s. l. (1981) optimal two processor scheduling of tree precedence constrained tasks with two execution times, performance evaluation, 1(4), 320–330. 11. kaufman, m. t. (1974) an almost-optimal algorithm for the assembly line scheduling problem, ieee transactions on computers, 100(11), 1169–1174. 12. du j. & leung j. y. t. (1989) scheduling tree-structured tasks on two processors to minimize schedule length, siam journal on discrete mathematics, 2(2), 176–196. 13. hu, t. c. (1961) parallel sequencing and assembly line problems, operations research, 9(6), 841–848. 14. dolev, d. & warmuth, m. (1985) scheduling flat graphs, siam journal on computing, 14(3), 638–657. copyright © 2024 assa. adv syst sci appl (2024) introduction the problem definition metric over the set of instances of the problem p2 prec, pj {1, 2} cmax the polynomial-time approximation scheme computer experiments conclusions adv syst sci appl 2023; 01:61–68 published online at https://ijassa.ipu.ru. the problem of controllability with the phase space change irina maximova peoples’ friendship university of russia (rudn university), moscow, russia abstract: in this paper, we consider differential systems of the following structure: at two consecutive time intervals the motion of the object is described by two different systems of differential equations. we study the controllability of the object described by such system from the initial set in one space to the given set in another space through so-called ”transition hypersurface”. the transition of an object from one space to another one is given by a certain reflection. sufficient conditions of the controllability of such differential systems in the problem with phase space change are obtained. approaches to the study of both nonlinear and linear systems are considered. keywords: controllability, reachability set, multiple-valued mapping, support function 1. introduction problems with the change of the phase space are an important class of so-called hybrid (composite) systems, they are characterized by the condition that at different time intervals the motion of the object is described by different differential equations and some links for trajectory mating. the initial source of such problems was the multistage processes of space flight (see [16]). problems with the phase space change arise in various applied problems, including aircraft engineering, robotics, economics, etc. for example, the problem of launching a missile from an underwater object: in this case the dimension of the space does not change, but the environment and conditions of the motion do. the change of the phase spaces can occur in the mathematical modeling of complex dynamic systems, for example, large production complexes, multistage technological processes. the process of controlling a system of chemical reactors, models of the dynamics of metapopulations, and economic systems with a variable structure are also characterized by a sequence of consecutive stages. the dimensionality of dynamic systems used when modelling these processes depends on their state and can change over time, i.e. the decomposition of the complex system is taking place. the possibility of using systems with the variable dimensionality in modeling the dynamics of biological communities was pointed out in the work [15]. during the construction of realization theory, r. kalman proposed to generalize the notion of dynamic system that the dimensionality of its space of states could change over time (see [9]). problems of optimal control of composite systems were studied at different times by, for example, v.g. boltyansky, l.t. ashchepkov, and v.n. rozova. necessary optimality conditions were derived in the general problem of control with intermediate constraints on the trajectory (see [2]). in the paper [7], the necessary optimality conditions for the problem with a phase space change were obtained. the paper [14] considers the problem of optimal control of several objects with sequential mode of operation. the initial state of each subsequent object depends on the final state of ∗corresponding author: maximova-is@rudn.ru 62 i. maximova the previous one. each object is described by a system of ordinary differential equations on the interval of its action. necessary and sufficient optimality conditions were obtained for the problem of optimal control with a criterion of sufficiently general form. necessary optimality conditions for the problem with a change of phase space were obtained by maximova in [11]. the results were generalized to the case of several spaces. in many works devoted to problems of this kind, the optimization issue is mainly studied. meanwhile, typical theorems of the existence of optimal control assume the existence of at least one admissible control that generates a trajectory that satisfies the given boundary conditions, for example, a control that translates a trajectory from one given position to another. the latter problem is the essence of the controllability problem. thus, the problem of controllability is important and relevant in solving optimal control problems. in the paper [3], bargsegyan v.r. considered a mathematical model of control of linear composite systems described at different time intervals by different differential equations and some finite relations for continuity of motion of composite systems. the analytical view of motion of composite systems is constructed, the properties of motion and the geometrical structure of the reachability region were investigated. necessary and sufficient conditions for the complete controllability were formulated. the method of solving the problem of control of composite systems and the method of solving the problem of optimal control were proposed. the monograph [4] is devoted to the control problems of composite linear dynamical systems and systems with multipoint intermediate conditions. particular attention is paid to necessary and sufficient conditions for the complete controllability and observability of composite linear systems, which in the stationary case are comparable to the kalman conditions by their completeness. qualitative properties of controllability and observability of composite systems are revealed. constructive methods for solving control problems of composite systems, systems with undivided multipoint intermediate conditions, with constraints on the values of different parts of the phase vector coordinates at intermediate points in time are proposed. the issue of controllability for some classes of nonlinear systems with a phase space change was studied by the author in [12]. the paper consists of two sections devoted to the cases of nonlinear and linear systems, respectively. an example illustrating this approach is considered. 2. nonlinear systems let x and y be two phase variables: x = (x1, . . . , xn) ∈ x = rn, y = (y1, . . . , ym) ∈ y = rm. the motion of the object is described by the following non-linear systems of differential equations: ẋ = f(t, x(t), u(t)), u(t) ∈ u, t ∈ [0, τ ], x ∈ x, (2.1) ẏ = g(t, y(t), v(t)), v(t) ∈ v, t ∈ [τ, t ], y ∈ y. (2.2) the class of admissible controls consists of all functions u(·) ∈ u = {u(t) ∈ rn | u(·) ∈ l∞[0, τ ]; u(t) ∈ u1 ⊂ rn, t ∈ [0, τ ]}, v(·) ∈ v = {v(t) ∈ rm | v(·) ∈ l∞[τ, t ]; v(t) ∈ v1 ⊂ rm, t ∈ [τ, t ]}, whereu1 ∈ ω(rn), v1 ∈ ω(rm). here ω(rn) and ω(rm) are the sets of all nonempty convex compact subsets of the spaces rn and rm, respectively. the functions f(t, x(t), u(t)), g(t, y(t), v(t)) are such that for systems (2.1) and (2.2) the existence and uniqueness theorem of the cauchy problem is satisfied. solutions of systems (2.1) and (2.2) at t ∈ [0, τ ] and t ∈ [τ, t ] are absolutely continuous functions that satisfy systems (2.1) and (2.2) almost everywhere on [0, τ ] and [τ, t ], respectively. copyright © 2023 assa. adv syst sci appl (2023) the problem of controllability 63 there is an initial set m0 ∈ ω(rn) and a nonintersecting convex ”transition hypersurface” in the space x γ. let’s suppose that τ is the smallest time moment at which the object reaches the hypersurface γ. when an object moving according to the law (2.1) reaches the hypersurface γ, it moves to the space y given by the mapping q : x → y (it is assumed that this mapping translates the convex set into the convex one), and further motion is performed in the space y according to the law (2.2). finally, the terminal set m1 ∈ ω(rm) (not overlapping with the set q(γ)) is given in y . a similar scheme of motion of an object was considered in [7]. the problem is to find the conditions under which the object described by the systems (2.1) and (2.2) will be controllable from m0 to m1. definition 2.1: an object described by systems (2.1) and (2.2) is called controllable from m0 to m1, if there exist allowable controls u(·) and v(·) such that the corresponding solutions of the systems satisfy the boundary conditions x(0) ∈m0, x(τ) ∈ γ and y(τ) = q(x(τ)), y(t ) ∈m1. remark 2.1: for system (2.1), consider the set f(t, x(t), u) consisting of all vectors f(t, x(t), u(t)), where u(t) ∈ u . if x(t) is a trajectory of system (2.1) corresponding to an admissible control u(t), then for almost all t ∈ [0, τ ] the inclusion ẋ(t) ∈ f(t, x(t), u) (2.3) holds true. this leads us to the differential inclusion of ẋ ∈ f(t, x, u). (2.4) solutions of the differential inclusion (2.4) are absolutely continuous functions x(t) defined on the interval [0, τ ] that satisfy the inclusion (2.3) for almost all t ∈ [0, τ ]. so, under rather general assumptions, system (2.1) is equivalent to a differential inclusion (2.4), i.e. for any solution x(·) of the inclusion (2.4) there exists a valid control u(·) such that the function x(·) is the path of system (2.1) with the control u(·). this question is discussed in detail in [8]. taking into account previous remarks, we shall consider differential inclusion (2.4) instead of nonlinear system (2.1). let us denote f(t, x, u) by f (t, x), then the motion of the controlled object is described by the differential inclusion ẋ ∈ f (t, x), t ∈ [0, τ ]. (2.5) similarly, the motion of the controlled object in the space y = rm is described by the differential inclusion ẏ ∈ g(t, y), t ∈ [τ, t ]. (2.6) the motion of an object from the space x to the space y is described above. definition 2.2 ( [6]): a multi-valued mapping f (t, x) is called concave by x on a set m ⊂ x if for any points x1, x2 ∈m and any λ ∈ [0, 1] the condition λf (t, x1) + (1− λ)f (t, x2) ⊂ f (t, λx1 + (1− λ)x2) holds true. we note that the above condition implies the set f (t, x) is convex for every x ∈m (see [6]). the reachability set k(t) for each t ∈ [0, τ ] consists of all points x(t) ∈ rn, where x(t) is the solution of the inclusion (2.5) with initial condition x(0) ∈m0. consider the motion of an object in the space x from the initial set m0 to the transition hypersurface γ. let us suppose that the mapping f (t, x) is concave by x on the reachability copyright © 2023 assa. adv syst sci appl (2023) 64 i. maximova set k(τ) for all t ∈ [0, τ ]. it is known (see [6]) that in this case the family of all solutions on the interval [0, τ ] with initial condition x(0) ∈m0 is a convex set in the space c[0, τ ]. the convexity of the solution family implies that the reachability set k(τ) is convex (the reverse is not true). so, under the above assumptions, the reachability set k(τ) is convex. the intersection of k(τ) with the transition hypersurface γ, we obtain the set k1(τ) = k(τ) ∩ γ. suppose that there exists τ : k1(τ) ̸= ∅. then k1(τ) is convex as intersection of two convex sets. let us transform the set k1(τ) as follows: k2(τ) = q(k1(τ)), where q : x → y. the resulting set k2(τ) is convex due to the properties of mapping q. the set k2(τ) is the initial set when the object moves in the space y into the set m1. in the space y we obtain the following controllability problem: whether the system (2.2) is controllable from the set k2(τ) to the set m1 at the time interval [τ, t ]. let us denote the system reachability set (2.2) from k2(τ) at time t by k3(t ). let us suppose that the mapping g(t, y) is concave on y on the reachability set k3(t ) and k3(t ) is compact. then, for the controllability of the system (2.2) it is sufficient that k3(t ) ∩m1 ̸= ∅ or, in the terms of support functions, the inequality c(k3(t ), ψ) + c(m1,−ψ) ≥ 0 holds true for any ψ ∈ rm (see [5]). thus, the controllability conditions from the set m0 ⊂ x to the set m1 ⊂ y for systems (2.1) and (2.2) can be expressed as the following statement: theorem 2.1: under the above assumptions, for the controllability of an object described by systems (2.1) and (2.2) on the time interval [0, t ] it is sufficient that c(k3(t ), ψ) + c(m1,−ψ) ≥ 0, for any ψ ∈ rm. remark 2.2: for autonomous system (2.2), one can consider the motion of an object in y space in backward time and get the reachability set k4(t ) from the set m1 on the of the transition hypersurface γ. then the controllability condition for the systems (2.1) and (2.2) on the time interval [0, t ] is a non-empty intersection of the sets k4(t ) and k2(τ). 3. linear systems in this part of the paper we consider the case where the motion of the controlled object is described by a linear system of differential equations. let us suppose that systems (2.1) and (2.2) are linear, then we obtain the following problem. 3.1. problem statement let x and y be two phase variables: x = (x1, . . . , xn) ∈ x = rn, y = (y1, . . . , ym) ∈ y = rm. the motion of the object is described by the following linear systems of differential equations: ẋ = ax+ u, u(t) ∈ u, t ∈ [0, τ ], x ∈ x. (3.7) ẏ = by + v, v(t) ∈ v, t ∈ [τ, t ], y ∈ y. (3.8) copyright © 2023 assa. adv syst sci appl (2023) the problem of controllability 65 the class of admissible controls is the sets of functions {u(·) ∈ l∞([0, τ ],rn) | u(t) ∈ u, t ∈ [0, τ ]}, {v(·) ∈ l∞([τ, t ],rm) | v(t) ∈ v, t ∈ [τ, t ]}. in the space x there is an initial set m0 ∈ ω(rn) and a non-overlapping convex the transition hypersurface γ. let τ be the smallest time moment at which the object reaches the hypersurface γ. the motion of the object from one space to another one occurs in the same way as in the first part of the paper. the problem is to find the conditions under which the object described by systems (3.7) and (3.8) is controllable from m0 to m1. 3.2. main result the reachability set k(τ) for system (3.7) is the set of endpoints of trajectories of system (3.7) with initial set m0, corresponding to all possible admissible controls u(·) ∈ u and considered at time τ. due to linearity of system (3.7), the reachability set can be presented in the explicit form: k(τ) = eτam0 + τ∫ 0 e(τ−s)auds, (3.9) here eτam0 is the image of the set m0 under the linear transformation eτa, and the integrand is a multivalued mapping obtained for all s ∈ [0, τ ] as the image of the set u under the linear transformation e(τ−s)a. to find the reachability set with the initial convex set m0, let us first calculate its support function and then reconstruct the set k(τ) by its support function. thus, the support function of the reachability set has the form c(k(τ), ψ) = c(m0, e τa∗ ψ) + τ∫ 0 c(u, esa ∗ ψ)ds. (3.10) since the initial set is convex, the reachability set is also convex (see [5]). then the set k(τ) reconstructed from the support function is intersected at the time τ with the the transition hypersurface γ. by assumption, this yields the convex set k1(τ) = k(τ) ∩ γ. assume that there exists τ such thatk1(τ) ̸= ∅. let us transform the setk1(τ) as follows: k2(τ) = q(k1(τ)), where q : x → y. then the set k2(τ) is convex due to properties of q. the set k2(τ) is the initial set for the system (3.8) when the object moves in space y to the set m1. so, we have formulated the following controllability problem in the space y : whether system (3.8) is controllable from the set k2(τ) to the set m1 on the time interval [τ, t ]. let us define the controllability function φ : rm → r1 by the relation φ(ψ) = c(k2(τ), e (t−τ)b∗ ψ) + c(m1,−ψ) + t−τ∫ 0 c(v, esb ∗ ψ)ds, (3.11) see [5]). according to the controllability theorem [5], an object is controllable on the time segment [τ, t ] from the set k2(τ) to the set m1 if and only if for the controllability function is non-negative, i.e. φ(ψ) ≥ 0 ∀ψ ∈ s, copyright © 2023 assa. adv syst sci appl (2023) 66 i. maximova where s is the unit sphere in rn. in turn, this is equivalent to the condition φ0 = min ψ∈s φ(ψ) ≥ 0. applying this theorem, we obtain the following result: theorem 3.1: under the above assumptions, for the controllability of an object described by systems (3.7) and (3.8) on the interval [0, t ], it is sufficient that the controllability function φ(ψ) = c(k2(τ), e (t−τ)b∗ ψ) + c(m1,−ψ) + t−τ∫ 0 c(v, esb ∗ ψ) ds is non-negative for any ψ ∈ s. 3.3. example let x = r3 and y = r2 and the motion of an object is described by the following systems of equations: ẋ1 = x2 + u1, ẋ2 = −x1 + u2, |u| ≤ 1, u = (u1, u2, u3) ∈ r3, ẋ3 = u3, t ∈ [0, τ ], (3.12) { ẏ1 = y1 + v1, |v| ≤ 1, v = (v1, v2) ∈ r2, ẏ2 = v2, t ∈ [τ, t ]. (3.13) in the space r3, consider the initial set m0 = {(0,−1, 0)} and the transition hypersurface γ = {(x1, x2, x3) ∈ r3|x2 = 0, x3 ≥ 0}. by τ denote the smallest time moment at which the object reaches the hypersurface γ. the mapping that makes the transition from r3 to r2 has the form q(x1, x2, x3) = (x1 + sin τ, x3) = (y1, y2). in the space r2, consider the set m1 = s1(0,−3). the problem is to find conditions for an object described by systems (3.12) and (3.13) be controllable from m0 to m1 on the time interval [0, t ]. denote by k(τ) the reachability set of system (3.12) from the set m0 at time τ . first, consider the motion of the object in the space r3. the support function of the reachability set k(τ) from (3.10) reads c(k(τ), ψ) = −ψ1 sin τ − ψ2 cos τ + τ∥ψ∥, where ∥ψ∥ is the standard euclidean norm of the vector ψ. restoring the set k(τ) by the its support function c(k(τ), ψ), one can easily see that the reachability set k(τ) is the circle of radius τ with the center at the point (− sin τ,− cos τ, 0). the intersection of the reachability set with the transition hypersurface γ occurs at τ > 1 and it is a segment with the endpoints (− sin τ − √ τ 2 − cos2 τ , 0, 0); (− sin τ + √ τ 2 − cos2 τ , 0, 0). we pass to the space r2 under the action of the mapping q. we obtain the set k1(τ), which is the initial for system (3.13) when the object moves in the space r2. the set k1(τ) copyright © 2023 assa. adv syst sci appl (2023) the problem of controllability 67 is the segment with the endpoints (− √ τ 2 − cos2 τ , 0), ( √ τ 2 − cos2 τ , 0). thus, we have the following controllability problem in r2: find such conditions on systems (3.12) and (3.13) that the object described by these systems is controllable from the set k1(τ) to the set m1. the controllability function in this case has the form φ(ψ) = √ τ 2 − cos2 τ · |ψ1|+ ∥ψ∥+ 3ψ2 + t−τ∫ 0 √ ψ2 1 + (sψ1 + ψ2)2 ds, (3.14) where ∥ψ∥ = 1, ψ = (ψ1, ψ2, ψ3). the controllability function (3.14) reaches its minimum at ψ2 = −1, then due to the fact that ∥ψ∥ = 1, ψ1 = ψ3 = 0. this yields min ψ∈s φ(ψ) = t − (2 + τ). therefore, when t > 2 + τ , the object described by systems (3.12) and (3.13) is controllable on the time interval [0, t ] from the set m0 to the set m1. 4. conclusion a special class of controlled differential systems called hybrid or composite is considered. such systems are characterized by the condition that at different time intervals the motion of the object is described by different differential equations and some links for trajectory mating. sufficient conditions of the controllability of such systems from the initial set in one space to the terminal set of the other space are obtained. sufficient controllability conditions for the nonlinear case are obtained using the apparatus of convex analysis, the theory of multivalued mappings, and the control theory. this class of composite systems has not been considered before. an actual application of the proposed approaches and results lies in the field of mathematical biology (see [15]) and mathematical economics with control input, for instance, dynamic market models with continuous, see, e.g., [1, 13]. references 1. arutyunov, a. v. & pavlova, n. g. (2022). equilibrium in market models described by differential equations, differ. equ., 58(9), 1267–1276. 2. ashepkov, l. t. (1981). optimal system control with intermediate conditions, applied mathem. mech., 45(2), 215–222. [in russian] 3. bargsegayn, v. r. (2012). constructive approach to the study of control problems of linear composite systems, control problems, 4, 11–17. 4. bargsegayn, v. r. (2016). controls of composite dynamical systems with multipoint intermediate conditions, moscow, russia: nauka [in russian]. 5. blagodatskikh, v. i. (2001). introduction to optimal control, moscow, russia: visshaya shkola, [in russian]. 6. blagodatskih, v. i. & filippov a. f. (1986). differential inclusions and optimal control, proc. steklov inst. math., 169, 199–259. 7. boltyanskii, v. g. (1983). the optimization problem with phase space change, differ. equ., 19(3), 518–521. 8. borisovich, yu. g., gel’man, b. d., myshkis, a. d. & obukhovskii, v. v. (1999). introduction to the theory of multivalued mappings and differential inclusions, dordrecht, netherlands: kluwer academic publishers. copyright © 2023 assa. adv syst sci appl (2023) 68 i. maximova 9. kalman, r., falb, p., & arbib, m. (1969). topics in mathematical system theory, new york, ny: mcgraw-hill. 10. kramar, v., alchakov, v., & osadchenko, a. (2021). the optimal control of the vessel’s automatic dynamic positioning system under deviation, adv. syst. sci. appl., 21(1), 1–10. 11. maksimova, i. s. (2012). necessary conditions of optimality in the problem with changing phase spaces, rudn reports. series: mathematics. computer science. physics, 4, 5–14. [in russian] 12. maksimova, i. s. & rozova, v. n. (2017). local controllability in the problem with phase space change, rudn reports. series: natural and technical sciences 25(4), 331– 338. [in russian] 13. pavlova, n. g. (2019). study of the continuous-time open dynamic leontief model as a linear dynamical control system, differ. equ., 55(1), 113–119. 14. rozova, v. n. (2006). optimal control of step systems, rudn reports. series: natural and technical sciences, 3, 15–23. [in russian] 15. svirezhev, yu. m. (1987). nonlinear waves, dissipative structures, and catastrophes in ecology, moscow, russia: nauka. [in russian] 16. velichenko, v. v. (1966). on optimal control problems for equations with discontinuous right sides, automation and telemechanics, 7, 20–30. [in russian] copyright © 2023 assa. adv syst sci appl (2023) introduction nonlinear systems linear systems problem statement main result example conclusion microsoft word 1300-article text-6397-1-18-20231013 adv syst sci appl 2023; 03; 164-176 published online at https://ijassa.ipu.ru. semantic image segmentation using a hybrid genetic–cuckoo search algorithm alaa abu srhan1*, mais haj qasem2, hutaf natoureah1, aayat shdaifat1 1) department of basic science, faculty of science, the hashemite university, zarqa, jordan 2)faculty of information technology, computer science department, zarqa university, jordan abstract: image segmentation is the process of dividing a given image into a set of regions or categories. the goal of image segmentation is to change the image representation into a form that is substantially meaningful and easy to analyze. metaheuristic optimization algorithms are widely used algorithms for many applications among them is image segmentation. genetic algorithm (ga) and cuckoo search (cs) algorithm are among the most popular metaheuristic algorithms. in this paper, a hybrid cs and ga (csga) has been used to perform image segmentation and object detection, then compared with other popular algorithms for image segmentation which are fuzzy c-mean (fcm), k-means algorithms, and ga. simulation results of the statistical measures of the performance corroborate that csga outperforms other compared methods. keywords: cuckoo search algorithm, genetic algorithm, image segmentation, fuzzy c-means, kmeans algorithms. 1. introduction image segmentation refers to the process of dividing an image into a set of regions or categories, which correspond to different objects or parts of objects. every pixel in an image is allocated to one of these categories. tasks that use image segmentation aim to represent meaningful areas of the image, such as crops, urban areas, and forests shown in a satellite image [5]. in other analysis tasks, the regions can be sets of border pixels grouped into structures, such as line and circular arc segments in the images of 3d industrial objects [3]. image segmentation aims to understand the image and extraction information from a particular image to accomplish certain tasks. thus, image segmentation has important applications in digital image technology [25]. recently, image segmentation has emerged as one of the hotspots in image processing and computer vision and an important basis for image recognition. image segmentation has two objectives. the first objective is to decompose an image into parts for further analysis. in simple cases, the environment may be controlled sufficiently, such that the segmentation process reliably extracts only the parts that must be analyzed further. the second objective is to perform a change of representation. the pixels of an image must be organized into high-level units that are either substantially meaningful or efficient for further analysis [4]. various techniques for image segmentation have been recognized by scientists and researchers. therefore, several of these techniques are relatively popular, important, and regularly used for image segmentation. image segmentation techniques can be categorized into two main approaches: boundary-based (region delimitation) and region-based methods (pixel clustering). the boundary-based methods focus on images edges while the region-based methods generate enclosed regions [11]. * corresponding author: alaaa@hu.edu.jo 165 semantic image segmentation using a hybrid… copyright ©2023 assa. adv. in systems science and appl. (2023) metaheuristic algorithms are becoming an important component of modern optimization and their general-purpose stochastic procedures are designed to solve complex optimization problems [15]. these algorithms are approximate and often non-deterministic algorithms that guide a search process over the solution space. unlike methods designed specifically for particular types of optimization tasks, metaheuristic algorithms are general purpose algorithms and require no particular knowledge of a problem structure other than the objective function itself [10]. genetic algorithm (ga) and cuckoo search (cs) algorithm [1] are among the most popular metaheuristic optimization algorithms that are used to solve optimization problems. ga is an adaptive heuristic search algorithm based on the evolutionary ideas of natural selection and genetics. by contrast, the cs algorithm is a recently developed metaheuristic optimization algorithm that is used to solve optimization problems. in this study, a hybrid cs and ga (csga) will perform image segmentation [8]. csga combines the advantages of ga and cs. the main disadvantage of ga is that it is easily trapped within a local minimum [16]. accordingly, cs is used to overcome the aforementioned drawback. the use of cs will enable a local search to be performed faster than ga [16]. moreover, cs involves only a single parameter apart from the population size. the proposed algorithm will be compared with the fcm and k-means algorithms and ga. the rest of this paper is organized as follows. section 2 reviews the studies that are closely related to image segmentation techniques in various applications. section 3 presents the methodology of the current research. section 4 presents the experimental results. section 5 presents the conclusion. 2. related work scientists and researchers have developed various image segmentation techniques. image segmentation is used in a wide variety of applications, such as content-based image retrieval, machine vision, medical imaging, object detection, and other related fields. nerurkar [18] presented two of the most efficient and suitable segmentation algorithms for brain tumor detection application. this study detected brain tumor from the (magnetic resonance imaging) mri scans of a brain using two different segmentation algorithms, namely, k-means and region growing techniques. the mri images of a brain were used in the aforementioned experiment and the two algorithms were compared with each other, thereby determining the ideal one based on accuracy. the preceding study concluded that the k-means method is easy and efficient but not as accurate as the region growing method. the region growing method, which proved to be more accurate than the k-means, provided a satisfiable segmentation and proved to be among the most ideal region-based segmentation methods. singh and misra [12] presented an algorithm for image segmentation technique for the automatic detection and classification of plant leaf diseases. they also conducted a survey on different disease classification techniques that can be used for the detection of plant leaf diseases. moreover, these researchers used ga belonging to the evolutionary algorithms that generate solutions for optimization problems. k-means clustering and support vector machine were applied to process the images and obtain a variety of beneficial features needed for subsequent analysis. their experimental results showed the efficiency of their proposed algorithm in recognizing and classifying leaf diseases. another advantage of using their proposed method is that plant diseases can be identified in the early stage. kaur and kaur [28] proposed a new method for the segmentation of medical images to optimize the results of the fcm clustering algorithm by using a hybrid of ga and particle swarm optimization algorithm to optimize the parameters of fcm in medical images. these researchers compared their proposed method, fcm, and kernel fuzzy c-means (kfcm) by using quality parameters, such as the rand index, global consistency error, and variation. their 166 a. abu srhan, m. h. qasem, h.s. natoureah, a. shdaifat copyright ©2023 assa adv. in systems science and appl. (2023) comparison showed that their proposed method is better than the existing one and provided efficient and effective results. zang et al. [2] proposed a novel dna-based ga to learn the kernel intuitionistic fuzzy cmeans (kifcm) clustering for mri image segmentation, particularly using the dna-based ga to determine the optimal number of clusters needed. to determine the optimal number of clusters, these authors attempted to determine the optimal weight exponent (m) by optimizing the kifcm parameters. moreover, they performed an empirical study by comparing their new method with six existing state-of-the-art fuzzy clustering algorithms by using a set of uc irvine machine learning repository (uci) data mining data sets, a set of synthetic mri data, and a set of clinical mri data sets. their experimental result concluded that their new algorithm outperforms the compared algorithms in terms of clustering metrics and computational efficiency. bosch et al. [22] proposed an image segmentation framework that uses color and low-level intensity cues and works on the entire space of the segmentation hypotheses or segmentation volume generated from various choices of parameters. these researchers employed multiple segmentations because they acknowledged that one set of parameters cannot work consistently for different scenes and images. their approach defines a cost function based on two criteria: (1) segments that change constantly and abruptly in the segmentation volume receive larger penalties and (2) segments that do not match with natural image contours should be discouraged. their results concluded that their proposed framework is robust to the choice of segmentation kernel that produces the initial set of hypotheses and capable of outperforming popular segmentation algorithms. wang et al. [24] proposed a novel deep learning-based framework for interactive segmentation by incorporating convolutional neural networks (cnns) into a bounding box and scribble-based segmentation pipeline. they proposed image-specific fine-tuning to make a cnn model adaptive to a specific test image, which can be either unsupervised or supervised, and proposed a weighted loss function by considering network and interaction-based uncertainty for the fine-tuning. these researchers applied their new framework on two applications, namely, 2d segmentation of multiple organs from fetal magnetic resonance (mr) slices and 3d segmentation of brain tumor and the entire brain from various mr sequences. their experimental results showed that the new framework performs well on previously unseen objects. xiong et al. [19] proposed a new algorithm for image segmentation on the basis of realtime color image segmentation method, which is based on color similarity in the rgb color space. these researchers discussed the proposed segmentation method application combined with color sensor in real-time color image segmentation for a cyber physical system (cps) through the application in fire detection. moreover, they summarized a new method in identifying fire in a video based on these characteristics. their experiment result showed that their proposed method in vision-based fire detection and identification in videos was effective. in addition, their results were accurate and can be used in real-time analysis. table 2.1 summarizes a set of the studies on image segmentation in various fields, which contains the techniques used in each study, the application that those techniques is applied for, the metrics used to measure the performance of each study, and "compared with" (in each study the authors compared their proposed technique with other techniques in literatures). table 2.1. research on image segmentation paper techniques application measures compared with [18] k-means and region growing brain tumor accuracy k-means and region growing [12] genetic algorithm plant leaf diseases accuracy k-means and svm [28] genetic algorithm, swarm, and fcm medical images rand index, global consistency error and variation kfcm and fcm 167 semantic image segmentation using a hybrid… copyright ©2023 assa. adv. in systems science and appl. (2023) [2] genetic algorithm, fuzzy c-mean clinic mri image clustering metrics and computational efficiency. gkfcm1, gkfcm2, flicm, kwflicm, mico, and rscfcm. [22] graph-based segmentation human-labeled segmentations bde, pri, voi, and cov ccp, gpb-owt-ucm, selective search, and ms [24] convolutional neural networks 2d, and 3d brain tumor images obstetrician, and radiologist p-net with fcn, u-net, deepmedic, highres3dnet, grabcut, slic-seg, and random walks [19] real-time color image segmentation fire detection average intensity value color similarity and standard deviation according to table 2.1, metaheuristic optimization algorithms are one of the most popular algorithms with promising results for image segmentation. ga is one of the most widely used metaheuristic optimization algorithms for solving optimization problems. its primary drawback is that it is easily trapped within a local minimum [16]. to address this shortcoming, the current study will employ the hydride algorithm of cs and ga to perform local searches faster than [16]. as a result, aside from population size, cs is only single parameter. 3. methodology this study aims to provide an empirical basis for research on image segmentation and boundary detection. accordingly, we applied a hybrid algorithm called csga, which combines the advantages of ga and cs [8], on grayscale images and compared it with fcm, k-means, and ga by using the public berkeley benchmark. this standard uses a human segmented image that provides ground truth boundaries to determine how well this soft boundary map approximates the ground truth boundaries in different algorithms. figure 3.1 illustrates the methodology set up used in this study. the first step involves reading an image taken from an image segmentation benchmark. the second step entails performing segmentation using gacs. the third step involves performing segmentation using other segmentation methods. lastly, the ground truth of the proposed method is compared with other methods using the following evaluation metrics: 1. jaccard similarity coefficient [7]. this metric compares the members of two sets to determine which members are shared and which are distinct. this measure of similarity for two sets of data has a range from 0% to 100%. the higher the percentage, the more similar the two populations. the jaccard equation is as follows: 𝐽(𝑋; 𝑌) = |𝑋 ∩ 𝑌| |𝑋 ∪ 𝑌| (3.1) 2. false positive ratio [27]. this metric involves the probability of falsely rejecting a null hypothesis for a particular test. the false positive ratio is calculated as the ratio between the number of negative events wrongly categorized as positive and the total number of actual negative events: 𝐹𝑃𝑅 = 𝐹𝑃 𝐹𝑃 + 𝑇𝑁 (3.2) where 𝐹𝑃 is the number of false positives, 𝑇𝑁 is the number of true negatives, and 𝑁 is the total number of negatives. 3. false negative ratio [27]. this test result indicates that a condition does not hold, even though it does in reality. the false negative ratio is the conditional probability of a negative test result and the condition being determined is presented by the following equation: 168 a. abu srhan, m. h. qasem, h.s. natoureah, a. shdaifat copyright ©2023 assa adv. in systems science and appl. (2023) 𝐹𝑁𝑅 = 𝐹𝑁 𝑇𝑃 + 𝐹𝑁 (3.3) where 𝐹𝑁 is the number of false negative and 𝑇𝑃 is the number of true positive. fig. 3.1. methodology steps 3.1.cuckoo search yang and deb [27] developed a cs algorithm based on the lévy flight and brood parasitic behaviors. this cs algorithm has been proven to deliver excellent performance in function optimization, engineering design, neural network training, and other continuous target optimization problems, as well as solved the knapsack and nurse-scheduling problems [27]. cuckoo birds have an aggressive reproduction, in which the females hijack and lay their fertilized eggs in other birds’ nests. if the host bird discovers that the eggs do not belong to it, it either throws away or abandons its nest and builds a new one elsewhere [26]. yang and deb [21] explained that the cs algorithm is based on three assumptions. 1. each cuckoo lays one egg at a time and places it in a randomly chosen nest. 2. the best nests with the highest quality of eggs (solutions) carry over to the next generations. 3. the number of available host nests is fixed and a host has a probability 𝑝  (0; 1) of discovering an alien egg. in this case, the host bird either throws out the egg or abandons the nest to build a new one in a different location. the third assumption can be approximated as a fraction: pa of the n nests replaced with new nests with new random solutions at different locations. the lévy flight behavior, rather than a simple random walk behavior, can be used to enhance the performance of cs [20]. the following formula can describe the lévy flight behavior when generating new solutions 𝑥 (𝑡 + 1) for the 𝑖 cuckoo [14]: 𝑥 (𝑡 + 1) = 𝑥 (𝑡) +  ⊙ 𝐿𝑒 𝑣𝑦() (3.4) where   0 is the final size that should be related to the problem of interest scale,  is the step-length, and the product ⊙ refers to an entry-wise multiplication (hadamard product)[9]. the formula that describes the lévy flight behavior in which the step lengths fit a probability distribution is as follows: 𝑙é𝑣𝑦() 𝑡 . (3.5) 169 semantic image segmentation using a hybrid… copyright ©2023 assa. adv. in systems science and appl. (2023) the preceding formula indicates that cuckoo birds’ consecutive jumps or steps mainly form a random walking process that corresponds to a power-law step-length distribution with a heavy tail. figure 3.2 shows the pseudocode of ga for image segmentation [13]. fig. 3.2. cs pseudocode for image segmentation 3.2.genetic algorithm artificial intelligence research within the computer science field produced ga, which is a heuristic search tool designed to mimic the natural process of evolution. this heuristic (or metaheuristic) is commonly used to generate beneficial solutions for optimization and search problems and often employs the natural techniques of evolution, such as inheritance, mutation, selection, and crossover. holland [6] developed the formal theory of ga in the 1970s and continued enhancements in the price and performance value have made ga ideal for many problem-solving optimization methods. ga has been shown to perform well in mixed (i.e., continuous and discrete) combinatorial problems. although ga easily become trapped in the local optima, this algorithm is computationally expensive and a probabilistic one. ga begins with a set of solutions represented by a group of chromosomes called population. a new population can be generated by borrowing solutions from a current population or by applying genetic operators, such as selection, crossover, and mutation to the current population. the new population must be better than the old one [23]. the function of genetic operators warrants considerably detailed attention. the selection operator selects two parent chromosomes from the population based on their fitness to participate in the next operations, crossover, and mutation. these steps are considered important in ga because they have a positive impact on the overall performance [17]. first, parents form new offspring (i.e., children) through crossover probability. thereafter, the mutation operator randomly exchanges alleles, which is similar to what occurs in nature. to work well, ga requires the definition of three important aspects [6]: 1) objective function, 2) genetic representation and its implementation, and 3) genetic operators and their implementation. figure 3.3 shows the pseudocode of ga for image segmentation [13]. input∶ original image=i, 𝐁𝐞𝐠𝐢𝐧: generate initial solutions while (t<max_generation) { generate a cuckoo (i) randomly via levy flights; segment the original input image using the generated cuckoo and evaluate its fitness fi; choose a nest among the host randomly (j); segment the original input image and evaluate its fitness fj; rank the segmented output based on the fitness and find the current best; if (fi>fj) {replace host by the new cuckoo solution;} abandon a fraction (pa) of worse nests [new build new nest at new location via levy flights]; keep the nest with high quality solution; rank the results and find the current best;} end 170 a. abu srhan, m. h. qasem, h.s. natoureah, a. shdaifat copyright ©2023 assa adv. in systems science and appl. (2023) fig. 3.3. ga pseudocode for image segmentation 1.3.csga algorithm the proposed algorithm combines the advantages of ga and cs and overcomes the main disadvantage of ga of easily becoming trapped in the local minima through cs, which performs the local search faster than ga. in addition, cs has only a single parameter, along with population size [8]. csga has the following operations: 1) search new nest operator, 2) abandon operator, and 3) ga operation. csga starts with cs operations and proceeds thereafter to genetic operations. thus, the initial population of ga is not generated randomly but it uses the results of cs. after the genetic operations are completed, the algorithm will start over. figure 3.4 shows the pseudocode of the proposed algorithm: begin: input an image define ga parameters define fitness function generate initial population p0 evaluate population p0 while (stopping ga criteria not satisfied) repeat {calculate fitness for each member reproduction crossover mutation evaluation (muted chromosomes)} end 171 semantic image segmentation using a hybrid… copyright ©2023 assa. adv. in systems science and appl. (2023) fig. 3.4. pseudocode of the proposed algorithm the definition of fitness function is based on the application being used. for image segmentation, it is defined as shown in figure 3.5 [13]: fig. 3.5. fitness function for image segmentation begin input: objective function )(f , image to be segmented; initialize the population of n host nests (cities)) xi ,i=1;2;···;n; optimize initial solutions and saved in the bulletin board. evaluate the fitness of solutions fi; while (t <maxgeneration){ get a cuckoo randomly by le’vy flights; segment the original input image and evaluate its quality/fitness fi; choose a nest among n (say, j) randomly; if fi<fj { replace j by the new solution; } start ga with current population while (t <maxgeneration){ selection: create matting pool production: mutation (flip, swap, slide) evaluate population } host birds abandon pa , )1,0(ap nests, and search pa new nests; refresh the bulletin board and keeping the best solutions (and nests). rank the solutions, and find the best solution. t =t +1; } end giving that (𝑥; 𝑦) = 𝑠𝑖𝑧𝑒 𝑜𝑓 𝑖𝑚𝑎𝑔𝑒 𝑖 = 1: 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑠𝑖𝑧𝑒 𝑗 = 1: 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑐ℎ𝑟𝑜𝑚𝑜𝑠𝑜𝑚𝑒 𝑓𝑖𝑡𝑛𝑒𝑠𝑠(𝑖; 𝑖) = 𝑙𝑜𝑤𝑛𝑢𝑚 ∗ ℎ𝑖𝑔ℎ𝑛𝑢𝑚 ∗ (𝑑1 − 𝑑2) where 𝑑1 = 𝑙𝑜𝑤𝑠𝑢𝑚 𝑙𝑜𝑤𝑛𝑢𝑚 ; 𝑑2 = ℎ𝑖𝑔ℎ𝑠𝑢𝑚 ℎ𝑖𝑔ℎ𝑛𝑢𝑚 ℎ𝑖𝑔ℎ𝑠𝑢𝑚 = ℎ𝑖𝑔ℎ𝑠𝑢𝑚 + 𝑑𝑜𝑢𝑏𝑙𝑒((𝐶(𝑥; 𝑦)) 𝑙𝑜𝑤𝑠𝑢𝑚 = 𝑙𝑜𝑤𝑠𝑢𝑚 + 𝑑𝑜𝑢𝑏𝑙𝑒 (𝑥; 𝑦) where 𝐶(𝑥; 𝑦) = 𝑏(1; 𝑖) 𝑏(1; 𝑖) = 𝑐 ∗ 255 2 𝑐 = 𝑐 + 𝑐ℎ𝑟𝑜𝑚𝑜𝑠𝑜𝑚𝑒(1; 𝑗) ∗ (2 ) 172 a. abu srhan, m. h. qasem, h.s. natoureah, a. shdaifat copyright ©2023 assa adv. in systems science and appl. (2023) 4. implementation and results the current research used the berkeley benchmark, which have collected 12,000 hand-labeled segmentations of 1,000 corel data set images from 30 human subjects. four grayscale images were applied on csga, fcm, k-means, and ga to determine the positive ratio, false negative ratio, and ground truth for each algorithm. various algorithms are simulated using matlab and compared with one another to determine the effectiveness of csga. a set of four images are used to illustrate the usability of the proposed model. segmentation results are shown in figures 4.1, 4.2, 4.3, and 4.4. the evaluation results of the tested images are shown in table 4.1. table 4.1. evaluation results of the tested images csga k-mean fcm ga algorithms image-1 0.8707 0.7527 0.8399 0.6903 false positive ratio 0.9959 0.9963 0.9960 0.9967 false negative ratio 0.6461 0.6302 0.6365 0.6306 jaccard image-2 0.8764 0.6516 0.7332 0.6524 false positive ratio 0.9960 0.9969 0.9967 0.9965 false negative ratio 0.6674 0.5791 0.6639 0.6524 jaccard image-3 0.8816 0.7689 0.8959 0.6574 false positive ratio 0.9958 0.9961 0.8541 0.9968 false negative ratio 0.6601 0.6154 0.6361 0.6451 jaccard image-4 0.9024 0.8347 0.7221 0.7213 false positive ratio 0.9952 0.9883 0.9821 0.9931 false negative ratio 0.7024 0.6800 0.6384 0.6753 jaccard original image ground truth csga fcm k-mean ga fig. 4.1. image-1 segmentation results 173 semantic image segmentation using a hybrid… copyright ©2023 assa. adv. in systems science and appl. (2023) original image ground truth csga fcm k-mean ga fig. 4.2. image-2 segmentation results original image ground truth csga fcm k-mean ga fig. 4.3. image-3 segmentation results 174 a. abu srhan, m. h. qasem, h.s. natoureah, a. shdaifat copyright ©2023 assa adv. in systems science and appl. (2023) original image ground truth csga fcm k-mean ga fig.4.4. image-4 segmentation results the comparison among csga, k-means, fcm, and ga indicates that csga is constantly more efficient and accurate than the other algorithms on the different images (see table 4.4). evidently, csga outperformed the other algorithms, thereby achieving the research goal of the current study. table 4.4. enhancement results algorithms csga over ga csga over fcm csga over kmean image 1 false positive ratio enhancement 18.04% 6.18% 14.90% false negative ratio enhancement 0.08% 0.04% 0.07% jaccard enhancement 1.55% 0.04% 62.10% image 2 false positive ratio enhancement 22.40% 0.08% 8.08% false negative ratio enhancement 0.05% 0.04% 0.02% jaccard enhancement 1.50% 7.33% 1.15% image 3 false positive ratio enhancement 22.42% 11.15% 33.85% false negative ratio enhancement 0.10% 0.07% 14.27% jaccard enhancement 1.50% 2.97% 0.90% image 4 false positive ratio enhancement 18.11% 18.03% 6.77% false negative ratio enhancement 0.21% 1.13% 0.69% jaccard enhancement 2.71% 6.4% 2.24% 5. conclusion image segmentation is a process of dividing a given image into a set of regions or categories. this process aims to understand the image and extraction information from a particular image to accomplish certain tasks. thus, image segmentation has important applications in digital image technology. recently, image segmentation has emerged as one of the hotspots in image processing and computer vision and is an important basis for image recognition. the metaheuristic optimization algorithm is the most popular algorithm for image segmentation. ga and cs algorithm are among the most popular metaheuristic algorithms. in this study, a hybrid csga performed image segmentation to combine the advantages of ga 175 semantic image segmentation using a hybrid… copyright ©2023 assa. adv. in systems science and appl. 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(2017). a kernel-based intuitionistic fuzzy c-means clustering using a dna genetic algorithm for magnetic resonance image segmentation, entropy, 19(11), 578. microsoft word 1393_done adv syst sci appl 2023; 03; 128-151 published online at http://ijassa.ipu.ru. deep feature extraction and weight updated tuned random forest for piper plant species recognition a. pravin1*, c. deepa2 1research scholar, department of computer science, sri ramakrishna college of arts & science, coimbatore, india 2associate professor & head, department of computer science (artificial intelligence & data science), sri ramakrishna college of arts & science, coimbatore, india abstract: recently, identifying plant species has become a significant research area as it is vital for securing biodiversity. plants also possess various medicinal applications. hence, predicting different species of plants is of utmost significance. however, determining plant species through conventional ways is a time-consuming process. that happens due to huge and distinct botanical terms. with the recent evolution of ai (artificial intelligence) based algorithms, researchers have undertaken various attempts to predict plant species. however, most studies averted the consideration of piper plant species which holds huge medicinal benefits. existing research also failed to predict the plant species due to inefficient feature extraction accurately. considering such a pitfall, this study proposes deep cnn (deep convolutional neural network) and inception v3 to extract features to perform all plant classification. in addition, the study proposes deep cnn and vgg16 (visual geometry group16) to extract suitable features for performing piper plant classification. following this, the study considers pca (principle component analysis) for feature fusion as it can reduce noise in data and select relevant features for affording independent and uncorrelated data features. finally, the study proposes wut-rf (weight updated tuned random forest) to classify piper and all plant species. in this process, hyperparameters of rf are tuned with convolutional likelihood weight to attain a high prediction rate. optimal hyperparameter selection and tuning assist in improvising the performance of the proposed classifier. performance analysis of this system about performance metrics exposes its effectiveness in plant species detection. keywords: artificial intelligence, piper plants, deep convolutional neural network, inception v3, vgg16, random forest 1. introduction plants possess several utilities in food, industry, medicine, etc. it also has a huge contribution to protecting the environment. there exists a huge plant species variety, and this count is increasing each year. as plants hold distinct benefits, identifying and classifying them is paramount. among different plants, piper plants have significant applications in medicine, such as asthma, gonorrhoea, diarrhoea, malaria, cough, tumours, etc. (liu, yang, huang, & lin, 2022). comprehending different plant species has become vital for several individuals, like farmers, educators, environmentalists, and other field workers. to describe a plant species, botanists need to be persuaded that the plant varies from other renowned species. accordingly, new plant species are generally determined when a botanist expert revises the nomenclature of the entire species group by revising the taxonomy (anubha pearline, sathiesh kumar, & harini, 2019). due to different botanical terms, such a classification method is tedious and time-consuming. generally, plant species are recognized through their flower, seed, fruit, leaves, etc. however, using the leaf for plant species recognition seems convenient and simple, and researchers have attempted to consider leaves for identifying * corresponding author: pravinresearchscholar@gmail.com deep feature extraction and weight updated tuned random forest… 129 copyright ©2023 assa. adv. in systems science and appl. (2023) plant species. with technological progress, digital images gained significance in various areas. concurrently, with the progress of ai (artificial intelligence), researchers have endeavored to develop an automated system for identifying and classifying plant species that could recognize plants to a certain extent (n. s. kumar, yashwanth, srinivasulu, & rini). by this, an automated plant detection system is suggested that uses ml and computer vision for classifying plant leaf images. texture and color features have been retrieved. following this, the svm classifier has been considered for classification. testing has been performed on the swedish dataset, exposing 93.26% accuracy (kaur & kaur, 2019). further, leaf images are segmented and classified in a complex background through dl based method (yang, zhong, & li, 2020). initially, 2500 images having complex backgrounds were gathered. considered images are fed into mask r-cnn (mask region-based convolutional neural network) (hao et al., 2021)for training the model. subsequently, a training set encompassing 1500 images of 15 species is fed into deep cnn with vgg16 to train the suggested model to classify the leaf. suitable hyperparameters for these methodologies are determined by comparing several parameter integrations. outcomes expose that the average me (misclassification error) of eighty test images is 1.15%. the average accuracy rate is explored to be 91.5%. further, an enhanced cnn structure is suggested for solving the leaf classification issues in small sample cases. metric space quality and selection of supervised samples find the classification rate of the suggested k-nn (k-nearest neighbour) classifier. experimental outcomes reveal that, while training samples are 20, accuracy for classification seems to be high (wang & wang, 2019). furthermore, a model is suggested to analyze the leaf shape that relies on geometrical features and sinuosity coefficients. experimentations undertaken on the leafsnap dataset expose a better classification rate. further, the suggested feature extraction algorithm is improvised by adding the geometrical features of the leaf, and classification rates are explored to be 93% while using rbf and 82% while considering mlp (ganesan, 2021; kala & viriri, 2018). in addition, several visual attributes are used to classify heterogeneous leaves that contrast in shape, surface, and hue. an fsst strategy selects the shape highlights for several leaf classes. subsequently, a progressive method is considered that encompasses preprocessing to normalize the introduction and scaling of several leaves, a hue evaluating stage that includes extracting hue highlights, shape evaluation including the shape-based representation, and surface venture analysis for exposing surface instances of the leaf surface. the individual layer encompasses modules to treat conventional practice and discriminators to pick relevant modules for subsequent preparation. further, nfc grouping is undertaken to exploit likeliness amongst broad leaves for shape and hue layers. finally, the euclidean metric is used to segregate surface components (chaki, parekh, & bhattacharya, 2020). the study [11] also performs piper plant recognition with all plant classifications (pravin & deepa, 2022), considering hyper-parameter-tuned rf (random forest). deep cnn is regarded for extracting features. empirical outcomes reveal the better performance of the suggested system with 0.94 accuracies for all the plant species, while 0.88 accuracies are attained for predicting piper plant while compared with nb (naïve bayes), lr (logistic regression) and svm (support vector machine). conventional works have used different ml and dl-based algorithms to identify pipers and other plants. though such methods have tried to attain better outcomes, most studies have not considered piper plant classification. furthermore, few studies considered piper plant classification but lacked a prediction rate due to ineffective feature extraction. to avoid such negative impacts on accuracy rate, this research aims to propose suitable dl-based feature extraction and ml-based classification with hyperparameter tuning based on the below objectives. the main objective of this study is,  to select relevant features using the proposed deep cnn (deep convolutional neural network) and inception v3 for all plant classification. 130 i. petrov, j.f. doe copyright ©2023 assa adv. in systems science and appl. (2023)  to perform feature extraction through the proposed deep cnn and vgg16 (visual geometry group16) for the piper plant classification.  to classify piper and all plant species through the proposed wut-rf (weight updated tuned random forest) based on hyperparameter tuning with convolutional likelihood weight for attaining a high prediction rate.  to evaluate the proposed system about performance metrics for confirming its effectiveness in predicting piper and all plant species. 1.1. paper organization the paper is organized in the following way, with section ii discussing the conventional works for plant species recognition. section iii discusses the overall proposed system with the proper flow, algorithm, and description, section iv states the results of the proposed system with dataset and performance metrics description, and section v summarizes the overall study with future directions. 2. review of existing work existing researchers have attempted to use different approaches for determining plant species. these methods are discussed in this section with relevant problems. the current study has addressed the solution for better identification of a medical herb (piper vines) through the dbn (deep belief network) network. the suggested system has been implemented on the worldwide availability of the kerala-plants dataset, and various piper vine images have also been collected (pravin & deepa, 2021). the performance of dl has exposed optimizing outcomes in the computerized vision in the current years. thus, the endorsed work has presented a dl technique for classifying and identifying microscopic fragment pictures of the medical herb'simplicia grass’ through cnn (convolutional neutral network), improved by the sift feature removal, denoted as ‘mikrobatx,' that plays an important role in a microscopic categorization of medical herb 'simplicia grass.' crucial simplicia grass features could be extracted by the mikrobatx using microscopic pictures of medical herb leaves. using the mikrobatx dataset, the results of experiments have shown that the recommended model can produce satisfactory accuracy of 89.42% value for the microscopic medical grass simplicia image issues (rahmatulloh & suhendy, 2022). the cnn performance and pre-trained models (vgg19 and vgg16) have been compared for the problem of leaf identification. the dataset considered in the research works contains kerala's indigenous medical herbs. cnn has obtained a classification rate of 95.79%. vgg19 and vgg16 have procured 97.6% and 97.8% accuracy, respectively, which has been better than conventional cnn (paulson & ravishankar, 2020). furthermore, with the transfer learning concept, ten pre-trained networks containing googlenet, alexnet, densenet201, mobilenetv2, inceptionv3, resnet101, resnet50, resnet18, vgg19, and vgg16 have been utilized as a feature extractor. the fp (false positive) rate of 0.1% is less than attained in all the cases(oppong, twum, hayfron-acquah, & missah, 2022). conventional research has also exposed new cases of medical herb dataset termed deepherb dataset containing 2515 leaf pictures from 40 species of indian plants. the efficiency of a dataset has been exposed by comparing it with pre-trained deep cnn structures such as vgg19, vgg16, xception, and inceptivev3. the work has been focused on accepting the transfer learning method on the suggested pre-trained model to undertake feature extraction, and classification has been performed using ann (artificial neutral network) and svm (support vector machine). the hyperparameters of svm have been tuned additionally through the optimization of bayesian to attain a better performance model. the recommended deepherb model studied from ann and xception outperformed by an accuracy of 97.5% (roopashree & anitha, 2021). the current study possesses images of leaves encompassing nine various herbs with 32 various categories of the plantvillage deep feature extraction and weight updated tuned random forest… 131 copyright ©2023 assa. adv. in systems science and appl. (2023) database. besides, cnn has been used to identify a plant leaf. transfer learning has been used for a pre-trained network of alexnet for several amounts of information for network training, and the outcomes have been evaluated with svm and dl categorizers. with 91.15% accuracy, alexnet has performed better than svm, providing 89.69% and 88.96% for circular basis linear kernel and functional kernel, respectively (wagle, 2021). additionally, the current study has recommended a cnn-based process denoted as dleaf. after pre-processing the leaf images, the features have been extracted using three cnn models, specifically fine-tuned alexnet, d-leaf, and pre-trained alexnet. the features were then categorized using five ml methods, specifically ann, svm, cnn, k-nn (knearest neighbour), and nb (naïve bayes). the d-leaf model has accomplished comparative testing of 94.88% accuracy compared to 93.26% (alexnet) and 95.54% (finetuned alexnet) models. the current study has concluded that compared to cmm (conventional morphological method), cnn has been better for retrieving features of herb species (wei tan, chang, abdul-kareem, yap, & yong, 2018). besides, rtsd-net (real-time strawberry detection) has been performed through dnn (deep neural networks) on the embedded system, which has exposed satisfactory performance (zhang et al., 2022). to enhance prediction performance, the research by (dönmez, 2022) has suggested a model for classifying haploid and diploid maize seeds. initially, deep features have been attained from cnn. these features have been selected and fused by varying cnn model combinations. such integrated features were then utilized in traditional classifier methodology training and testing stages. by empirical outcomes, it has been found that the performance rate has been about 96.74%. in addition, crop species and weeds have been classified using textual features of rgb images compared to the svm and dl-based vgg16 model. a feature selection approach, namely relieff, has been employed for selecting suitable features to perform prediction (sunil et al., 2022). performance metrics, namely kappa score, f1-score, and accuracy, have been considered to assess the suggested model's reliability and performance. further, pls (partial least square) regression (goyal & kumar, 2021) has been performed to select suitable features from the retrieved deep feature dataset (saeed et al., 2021). the recommended framework has included three major stages. at the initial stage, deep features have been extracted through a pre-trained vgg19-cnn model (rajesh & bhaskari, 2021). subsequently, a pls-based parallel fusion methodology has been endorsed that integrates the features retrieved from fc (fully connected) layers. the convenient features chosen through pls have been incorporated into the ensemble tree classifier. the suggested system has accomplished accuracy at a rate of 90%. as dl algorithms have exposed better performance, the study (crisóstomo de castro filho et al., 2020) has intended to assess methodologies for detecting rice farming in the southern part of brazil, for which it has used bi-lstm and lstm. comparison has been performed with conventional ml algorithms, namely svm, nb, k-nn, and rf (random forest). outcomes show that lstm and bilstm models have exposed better performance than traditional ml. accordingly, the study (alajrami & abu-naser, 2020) has recommended a solution for assisting individuals in determining the tomato kinds. a model has been constructed through the dl-based cnn model that has been trained and tested. the suggested trained model has been used for predicting the kind of tomato images with the suggested network encompassing 4-max pooling and cnn layers. testing accuracy is 93%. furthermore, a multiscale cnn has been suggested to recognize plant leaves at multiple scales (hu, chen, yang, zhang, & cui, 2018). experimentations have revealed that the suggested model is better than several conventional plant leaf identification algorithms. nevertheless, the research (wäldchen & mäder, 2018) has stated that, despite modern ml algorithms exposing slow performance, in the future, a proliferation of these algorithms will be found to solve the issues of predicting plant species. under this, the study (feng et al., 132 i. petrov, j.f. doe copyright ©2023 assa adv. in systems science and appl. (2023) 2019) has intended to expose the employment of svms and rf in identifying multi-feature crop kinds like spectroscopy, phonological parameters, and vegetation index. training of the sample and verification of accuracy have been undertaken through the use of ml algorithms encompassing svm, maximum likelihood methodologies, and rf. outcomes have been significant. further, for classifying plant species from occluded images of leaves, the research (chaudhury & barron, 2018) has used a dataset comprising varied leaf kinds (m. kumar, gupta, gao, & singh, 2019). initially, 2d-contour points have been indicated as βspline curves. following this, the dce (discrete contour evolution) approach has extracted interest points on such curves. following this, the parameters for similarity transformation have been calculated for an individual open curve. then, distinct open curves were overlaid with inverse similarity curves, and used the frechet metric for computing the measure of match quality and retaining ideal η-matched curves. the function has used global curvature, local curvature, string-cut features, and shape-context descriptors. energy function has been minimized by using a convex and concave relaxation model. experimentations on three accessible leaf image datasets have exposed the better performance of the suggested model. 2.1. problem identification the main issues that have been identified through the evaluation of the above conventional studies are listed below,  only a few studies have endeavored to perform piper plant classification. however, these studies lacked accuracy. for instance, the study (rahmatulloh & suhendy, 2022) has exposed 89.42% accuracy while using mikrobatx.  conventional works have attempted to employ various algorithms for classifying different kinds of plant species. for instance, the research (wagle, 2021) has used alexnet and exposed 91.15% accuracy compared with svm, which affords 89.69% and 88.96% for linear kernel and rbf (radial basis function). the study (wei tan et al., 2018) has exposed 94.88% accuracy while utilizing the d-leaf model. further, the research (saeed et al., 2021) used pls to select features with pre-trained vgg19 and exposed them with 90% accuracy. on the contrary, the study (alajrami & abu-naser, 2020) aimed to classify tomato plants and accomplished a 93% prediction rate using cnns.  suitable feature extraction approaches have to be selected for data pre-processing that could be suitable for the dl model (wagle, 2021).  complex and recently successful dl algorithms could enhance the classifier performance (chaudhury & barron, 2018; sunil et al., 2022).  although the suggested model in (kaur & kaur, 2019) has exposed better performance, it still requires enhancement in executing dl or nn-based methods. in the future, the existing limitation has to be resolved by retrieving suitable cultivated features and executing an enhanced classifier or hybrid classifier. lastly, the plant species have to be identified by considering a real-time dataset. 3. proposed methodology the research attempts to classify all plants and piper plant species based on suitable ml and dl-based feature extraction and classification. though existing studies have tried to perform this prediction, they need to be more accurate. hence, this study intends to resolve the existing pitfalls by proposing methods, and its sequential processes are shown in figure 1 and figure 2. deep feature extraction and weight updated tuned random forest… 133 copyright ©2023 assa. adv. in systems science and appl. (2023) load the data preprocessing feature extraction cnn layer pooling pooling pooling pooling relu dense cnn layer pooling dense pooling pooling relu extracted features principle component analysis train test split classification weight updated tuned random forest trained model prediction phase performance metrics all plants dataset deep cnn inception v3 feature fusion fused features hyperparameter tuning with weighted k – fold cross validation convolutional likelihood weight predicts type of plant figure 1. overall view of the proposed work for all plants classification as described in figure 1, the dataset is initially loaded. following this, pre-processing is performed to eliminate irrelevant and unwanted redundant data to improvise further processing. then, significant features are extracted by deep cnn and inception v3. this phase assists in constructing the proposed model with minimum machine effort and enhances the learning speed and generalization stages in the ml process. extracted features are then fused using pca (principle component analysis). this stage tries to extract highly discriminative information from various input features, thereby avoiding redundant information. the fused features are then fed into the train and test split for classification. in this phase, wut-rf is used for classification, wherein hyperparameter tuning is performed with the convolutional likelihood weight. finally, the kind of plant is predicted by the trained model, and the efficacy of the proposed system is assessed through performance metrics. further, the piper plant classification is performed based on the sequential process, as shown in figure 2. 134 i. petrov, j.f. doe copyright ©2023 assa adv. in systems science and appl. (2023) load the data preprocessing feature extraction cnn layer pooling pooling pooling pooling relu dense cnn layer pooling dense pooling pooling relu extracted features principle component analysis train test split classification weight updated tuned random forest trained model prediction phase performance metrics piper plant dataset deep cnn vgg16 feature fusion fused features hyperparameter tuning with weighted k – fold cross validation convolutional likelihood weight predicts type of piper plant figure 2. overall view of the proposed work for piper plants classification as shown in figure 2, the dataset is initially loaded, and pre-processing is undertaken to eliminate irrelevant data. subsequently, feature extraction is performed by deep cnn and vgg16, wherein relevant features are attained, and then feature fusion is undertaken by pca. lastly, piper plant species classification is accomplished by wut-rf, and the performance of this stage is assessed through performance metrics. the overall flow of the present research is explored in figure 3. load the data preprocessing feature extraction image proposed algorithm extracted features prediction 1 predicts the type of plant if piper plant prediction 2 predicts the type of piper plant display result display result yes no figure 3. overall proposed flow as shown in figure 3, the dataset is loaded, and pre-processing is performed. then, feature extraction is undertaken by the proposed algorithm. based on the extracted features, deep feature extraction and weight updated tuned random forest… 135 copyright ©2023 assa. adv. in systems science and appl. (2023) the type of plant is predicted. for example, if it is a piper plant, then the type of piper plant is predicted. if not, the plant type is displayed as an outcome. 3.1. feature extraction-deep cnn, inception v3, vgg16 feature extraction holds significance in accomplishing better classification performance. the main intention of this process lies in determining informative and compact feature sets for improving classifier efficacy. this study proposes deep cnn and inception v3 for all plant species classification and deep cnn with vgg16 for piper plant classification. 3.1.1. deep cnn (deep convolutional neural network) typically, deep cnns are ffnns (feed forward neural networks) employed for adjusting the parameters encompassing biases and weights of nn to minimize the cost-functional value. the process involved in extracting image features from deep cnn layers is called deep feature extraction. in this case, the extracted features are termed as deep-features. dimension of convolutional layer outcome could be computed by equation.1, (1) input width is given by , and height is represented by for the first convolutional layer. in addition, indicate the width, height, and channels for the convolutional layerkernel filter. an initial max-pooling layer is incorporated to minimize the first convolutional layer result dimensions. moreover, dimensions corresponding to the max-pooling layer result are computed by equation.2, (2) in equation.2, , , indicates the input's width, height, and channels ( ). whereas indicate the width, height, and channels of filter within the max-pooling layer. the outcome of an initial max-pooling layer is fed as input to the subsequent convolutional layer. further, relu activation is utilized on all convolutional layers. that is performed by equation.3, (3) following this, equation.4 denotes the output of neuron ( ) of the initial dense layer. moreover, indicates the inputs of the first dense layer that ranges from (1-512), while represents the resultant overall layers ranging from (1-2048) and is given by, (4) in equation.4, represents the bias value for the jth node. moreover, this layer uses the softmax function for plant leaf classification. softmax ( ) of the ith neuron of the dense layer is given by equation.5, (5) the resultant input image class could be exposed through an equation.6, (6) the resultant value from ( ) indicates the overall piper and all plant leaf classes. 3.1.2. inception v3 inception v3 indicates a pre-trained cnn image recognition framework. it represents a network version already trained upon several million images from the imagenet dataset. this model typically uses several methods to optimize the network to attain better adaptation of the model. as a result, it possesses a deep network compared to the inception-v1 and inception-v2 models. however, its speed is maintained. furthermore, it utilizes auxiliary classifiers and is computationally cheap. due to this advantage, this study considers inception v3 for feature extraction. its overall architecture is shown in figure 4, encompassing different layers of inception v3 with suitable dimensions as considered in this research. 136 i. petrov, j.f. doe copyright ©2023 assa adv. in systems science and appl. (2023) figure 4. inception v3 architecture 3.1.3. vgg16 (visual geometry group16) vgg16 is typically a cnn model comprising 16 convolutional layers. it possesses various benefits, like uniform architecture. similar to alexnet, it possesses convolutions of (3*3) dimensions. it could be trained upon 4gpus for two to three weeks. currently, it is a preferred option to extract features from an image. it finds applicability in several dl-based image classification issues and remains a building block to perform learning and undertakes easy execution. the main merit of this model is that it is optimal for benchmarking, and pretrained networks seem publicly accessible. its overall architecture is depicted in figure 5. deep feature extraction and weight updated tuned random forest… 137 copyright ©2023 assa. adv. in systems science and appl. (2023) figure 5. vgg16 model architecture 3.2. feature fusion-pca (principle component analysis) the present study considers pca for feature fusion. it is an ml algorithm generally used to minimize feature dimensionality while maintaining significant data properties. functioning in data with high dimensions could make classification computationally expensive. it might also lead to low performance in comparison with common feature space. hence, minimizing feature space avoids feature redundancy and affords computational advantages that enhance the classifier's performance. this study selects pca for feature fusion as it could enhance performance at a low cost. other benefits include data noise reduction, feature selection, and the ability to produce uncorrelated and independent data features. the overall mathematical formulation of this process is discussed below, assume represents the actual dataset with size . co-variance matrix ( ) is given as per equation.7, (7) in equation.7, indicates the mean of actual samples given by equation.8, (8) furthermore, eigenvectors are calculated from the alternation of the covariance matrix ( ). then, it is sorted by the eigenvalue in descending order as wherein indicates the eigenvectors. finally, the overall process is given in algorithm-i. 138 i. petrov, j.f. doe copyright ©2023 assa adv. in systems science and appl. (2023) algorithm-i: pca (principal component analysis) step 1: calculate the mean of the feature vector , where represents a pattern , : number of patterns, : feature matrix step 2: determine the covariance matrix // : matrix transposition step 3: calculate the covariance matrix's eigenvalues ( ) and eigenvectors ( . step-4: estimate eigenvectors of high value , number of high valued selected step-5: extract features of low dimension from the raw feature-matrix as where represents the principal component matrix and denotes the feature matrix. as shown in algorithm-i, the mean of the feature vector is initially computed. following this, the covariance matrix is computed. then, eigen values and eigen vectors for the corresponding covariance matrix are calculated. next, eigenvectors are estimated as per step 4. finally, the features with low dimensions are extracted from raw features. 3.3. classificationwut-rf (weight updated tuned random forest) the present study considers rf as it is capable of affording better performance. it could also deal with big data having numerous variables operating in thousands. it could perform automatic dataset balance while a class seems more infrequent than any other data class. this algorithm includes additional randomness to the model while developing trees. during node split, it searches optimal features within random feature subsets rather than for a significant feature. hence, it minimizes overfitting issues in dts (decision trees) and alleviates variance, enhancing accuracy. the process while considering rf for the classification process is depicted in pseudocode-i. pseudocode-i: rf (random forest) deep feature extraction and weight updated tuned random forest… 139 copyright ©2023 assa. adv. in systems science and appl. (2023) in this study, k-fold cross-validation encompasses a dataset split into k partitions. the main notion behind this process is cross-validation with a search algorithm wherein hyperparameters are inputted prior to model training. then, by integrating random search, the model fits individual pairs of varied hyperparameter sets in individual cross-validation. this overall process is depicted in pseudocode-iiwhere require indicates the input to be afforded, and randomsample indicates the function which considers a random set from hyperparameter. pseudocode-ii:k-fold cross-validation with random search through the k-fold cross-validation with random search, optimal hyperparameters are selected. the overall process involved in hyperparameter tuning with weighted cross-fold validation is given by pseudocode-iii, that finally affords the classification of piper and all plant types. in this process, selecting optimal hyperparameters and tuning assists in enhancing the classifier performance. pseudocode-iii: rf hyperparameter tuning with weighted cross fold validation 140 i. petrov, j.f. doe copyright ©2023 assa adv. in systems science and appl. (2023) 4. results and discussion the outcomes procured through the execution of the proposed system are presented in this section with dataset description and performance metrics. deep feature extraction and weight updated tuned random forest… 141 copyright ©2023 assa. adv. in systems science and appl. (2023) 4.1. dataset description dataset considered in this research includes all plant kinds and piper plant kinds. the normal plant kind dataset is depicted in table 1, while the specific piper plant type dataset is shown in table 2. for the normal plant type dataset, different kinds of plants exist, which include piper, peach, apple, strawberry, etc., as shown in table 1, with a total of 1607 images. table 1. all plant dataset class plant classes image count 1 piper plant 172 2 apple plant 260 3 peach plant 180 4 strawberry plant 200 5 cherry plant 160 6 grape plant 130 7 corn plant 151 8 paddy plant 150 9 rice plant 209 10 soya plant 160 11 potato plant 152 12 tomato plant 132 furthermore, for the piper plant dataset, various piper plant kinds are regarded as shown in table 2, namely piper mullesa, piper adunucum, piper argyrites, etc., with 2056 images. table 2. piper plant dataset class piper classes image count 1 piper mullesa 102 2 piper nigrum 100 3 piper adunucum 100 4 piper argyrites 101 5 piper umbellatum 103 6 piper excelsum 107 7 piper parmatum 103 8 piper orantum 102 9 piper porphyrophyllum 102 10 piper sylvaticum 102 11 piper longum 100 12 piper auritum 102 13 pepper bell 180 14 piper sarmentosum 103 15 piper betle 100 moreover, the sample images for all plants, including a piper and other plant kinds, are exposed in figure 6, while the piper plant leaf dataset is depicted in figure 7. 142 i. petrov, j.f. doe copyright ©2023 assa adv. in systems science and appl. (2023) piper plant peach plant apple plant strawberry plant grape plant cherry plant paddy plant soya plant rice plant tomato plant potato plant corn plant figure 6. all plants dataset samples in addition, the samples of the piper plant dataset are explored in figure 7. piper mullesa piper adunucum piper umbellatu piper nigrum piper argyrites piper excelsum piper orantum piper sylvaticum piper parmatum deep feature extraction and weight updated tuned random forest… 143 copyright ©2023 assa. adv. in systems science and appl. (2023) piper porphyrophyllum piper longum piper sarmentosum pepper bell pepper auritum piper beetle figure 7. piper plant dataset samples 4.2. performance metrics the metrics regarded in this study for validating the performance of the proposed system are, i) accuracy it is described as the calculation of overall correct classification. it is indicated by equation.9, (9) ii) precision it is claimed as the correct classification calculation and is represented per equation.10, (10) as represented in equation.9 and equation.10, indicates true positive, is false positive, denotes true negative, and indicates false negative. iii) recall it indicates the proportion of retrieved and relevant images to the proportion of the relevant image. it is given by equation.11, (11) as shown in the equation.11, represents the relevant image, while, denotes the retrieved image. iv) f-measure it is also termed f1-score. it could be claimed as the harmonic mean of recall and precision. it is represented by equation.12, (12) 4.3. experimental results the outcomes that have been attained through the execution of the proposed system are discussed in this section. initially, the image is selected for prediction, as shown in figure 8. 144 i. petrov, j.f. doe copyright ©2023 assa adv. in systems science and appl. (2023) figure 8. selection of image based on the selected image, the plant species are classified as all plants with suitable species, as shown in figure 9. figure 9. plant type classification as shown in figure 9, the paddy leaf is predicted at a rate of 85%, while the soya plant is identified at 35.4%. similarly, the piper plants are also classified by the image taken as input, and the results are depicted in figure 10. figure 10. piper plant type classification as shown in figure 10, the piper longum is predicted at a rate of 41.79%, while piper argyrites is identified at 58.8%. 4.4. performance analysis the performance of the proposed system is evaluated, and the corresponding outcomes are discussed in this section. the accuracy score for piper plant classification is presented in table 3. table 3. accuracy for the piper plant classification accuracy score piper plant 98.82591093 from table 3, it is found that the accuracy score for classifying piper plants is 98.825%. further, the classification performance for the piper plant's performance metrics is tabulated in table 4. deep feature extraction and weight updated tuned random forest… 145 copyright ©2023 assa. adv. in systems science and appl. (2023) table 4. classification performance for piper plants cla sses preci sion re call f1score sup port 0 0.99 1 0.99 276 1 1 0.9 4 0.97 573 2 0.99 0.9 9 0.99 321 3 1 1 1 540 4 1 1 1 321 5 1 1 1 306 6 1 1 1 285 7 0.95 0.9 7 0.96 299 8 0.97 0.9 9 0.98 296 9 0.99 0.9 9 0.99 301 10 0.99 0.9 8 0.99 306 11 1 1 1 252 12 1 1 1 294 13 0.98 0.9 8 0.98 303 14 0.93 1 0.96 267 from table 4, it is exposed that the proposed system's precision, recall, f1-score, and support rate have shown better performance in the range of 0.93-0.99. further, the overall average accuracy for piper plant classification is exposed in figure 11. figure 11. average accuracy rate for classifying piper plants figure 11 shows that the accuracy rate is exposed to be 0.99, while the precision, recall, and f1-score rate is explored to be 0.99.the accuracy score for all plant species classifications is given in table 5. table 5. accuracy for all plant classification accuracy score all plant 97.14975117 from table 5, it is found that the accuracy score for classifying all plants is 97.149%. further, classification performance for all plants about performance metrics is tabulated in table-6. 146 i. petrov, j.f. doe copyright ©2023 assa adv. in systems science and appl. (2023) table 6. classification performance for all plants classes precision recall f1-score support 0 1 1 1 780 1 1 1 1 480 2 0.99 1 1 453 3 1 1 1 390 4 1 1 1 450 5 1 0.99 1 540 6 1 1 1 780 7 1 0.99 1 456 8 1 1 1 627 9 1 1 1 480 10 0.77 0.98 0.86 600 11 0.97 0.7 0.81 595 table 6 shows that the precision, recall, f1-score, and support rate of the proposed system have shown better performance in the range of 0.7-0.99. further, the overall average accuracy for all plant classifications is exposed in figure 12. figure 12. average accuracy rate for classifying all plants figure 12 shows that the accuracy rate is exposed to be 0.97, while the precision, recall, and f1-score rate is explored to be 0.98, 0.97, and 0.97. additionally, test analyses for all plant types are exposed in table 7. table 7. test analysis test-analysis range k-values landis and koch substantial 0.96 fleiss intermediate to good 0.87 cicchetti good 0.94 cramer strong 0.86 matthews moderate 0.86 scott-pi perfect agreement 0.8812 table 7 shows that the proposed system has a higher k-value of 0.96 for the landis and koch test than other test analyses of fleiss exposing 0.87, cramer exploring 0.86, scott-pi showing 0.8812, matthews exposing 0.86 and cicchetti exploring 0.94 as k-value. furthermore, a confusion matrix is procured for the piper plant and all plant classifications. deep feature extraction and weight updated tuned random forest… 147 copyright ©2023 assa. adv. in systems science and appl. (2023) this assists in defining the classifier performance. it also affords information regarding classification errors. finally, it reflects how the classifier needs to be more disordered and clear in making predictions. accordingly, the confusion matrix of piper plant classification is exposed in figure 13. figure 13. confusion matrix of piper plant classification from figure 13, it is exposed that the correct classification rate is found to be high than the misclassification rate. in this case, the correct classifications could be seen in a diagonal. therefore, the proposed system is exposed to be better due to the high rate of correct classification than the misclassification rate. in addition, the confusion matrix of all plant classifications is exposed in figure 14. figure 14. confusion matrix of all plant classification from figure 14, the correct classification rate is higher than the misclassification rate. in such cases, the classifications performed correctly could be seen in diagonal. due to such a high rate of correct classification in comparison to the misclassification rate, the proposed system is found to be better. 4.5. internal comparison the present research has considered a real-time dataset; thus, it is impossible to compare the present study with conventional works. however, the proposed system is internally compared during the implementation phase, and the corresponding outcomes that have been attained are exposed in this section. initially, the comparison was undertaken for piper plant classification. therefore, existing algorithms, namely xgboost, svm, and rf, have been regarded in this case. the attained results are exposed in table 8, with its graphical depiction in figure 15. table 8. analysis of piper plant classification about performance metrics piper plants accuracy precision recall f1-score xgboost 0.56 0.56 0.56 0.57 support vector classifier 0.24 0.24 0.231 0.24 random forest 0.65 0.65 0.64 0.64 proposed algorithm 0.99 0.99 0.99 0.99 148 i. petrov, j.f. doe copyright ©2023 assa adv. in systems science and appl. (2023) figure 15.analysis of piper plant classification table 8 and figure 15 show that existing algorithms like rf have exposed 0.65 as accuracy, while svm has explored 0.24 and xgboost has revealed 0.56. similarly, the precision, recall, and f1-score rate have been different for the considered algorithms. however, the proposed system has explored high classification performance for piper plant classification by exposing 0.99 as the accuracy rate, 0.99 as the precision rate, 0.99 as the recall rate, and 0.99 as the f1-score rate. similarly, the comparison has been undertaken for all plant classifications. again, existing algorithms, namely xgboost, svm, and rf, have been regarded. the attained results are exposed in table 9, with its graphical depiction in figure 16. table-9. analysis of all plant classifications about performance metrics all plants accuracy precision recall f1-score xgboost 0.75 0.75 0.75 0.78 support vector classifier 0.52 0.52 0.53 0.53 random forest 0.8 0.81 0.81 0.81 proposed algorithm 0.97 0.98 0.97 0.97 figure 16.analysis of all plant classification table 9 and figure 16 show that existing algorithms like rf have exposed 0.8 as accuracy, while svm has explored 0.52 and xgboost has revealed 0.75. similarly, the precision, recall, and f1-score rate have been different for the considered algorithms. however, the proposed system has explored high classification performance for all plant classifications by showing 0.97 as the accuracy rate, 0.98 as the precision rate, 0.97 as the recall rate, and 0.97 as the f1-score rate. furthermore, the proposed deep cnn can adjust the parameters to minimize the cost-functional value. furthermore, inception v3 can perform network optimization to adapt the model better, while vgg16 is easy to execute. in addition, pca can reduce data noise and select suitable features for affording uncorrelated and independent data features. lastly, the proposed classifier is capable of minimizing overfitting deep feature extraction and weight updated tuned random forest… 149 copyright ©2023 assa. adv. in systems science and appl. (2023) issues. such advantages have made the proposed system expose better performance in plant species prediction. 5. conclusion the research intended to recognize plant species, including a piper and all plant kinds, by considering suitable ml and dl-based algorithms. to achieve this, the study proposed deep cnn and inception v3 for extracting the features to classify all plant species. at the same time, deep cnn and vgg16 extract relevant features to classify piper plant species. further, for feature fusion, pca was utilized. at the same time, wut-rf was proposed for classification that relied on tuning the hyperparameters of rf with convolutional likelihood weight for obtaining a high prediction rate. finally, a test analysis assessed the performance of the proposed approach. the analysis exposed that the proposed system revealed a higher k-value of 0.96 for the landis and koch test than other test analyses. additionally, a confusion matrix was attained. the matrix found that the proposed system exposed high classification rates for different classes compared to misclassification for classifying piper and all plant types. as this study regarded a real-time dataset, it was impossible to compare with conventional studies. however, the comparison was internally undertaken with existing algorithms like xgboost, svm, and rf. the outcomes found that the proposed system showed a high classification rate for piper plant prediction at a rate of 0.99, while it showed 0.97 for classifying all plant kinds. though better results were attained, there is a scope for future enhancement. in the future, the performance of the proposed models can be evaluated for various agricultural applications like plant disease diagnosis. another dimension involves undertaking experimentations when numerous datasets become publicly accessible. references 1. alajrami, m. a., & abu-naser, s. s. (2020). type of tomato classification using deep learning. international journal of academic pedagogical research (ijapr), 3(12). 2. anubha pearline, s., sathiesh kumar, v., & harini, s. (2019). a study on plant recognition using conventional image processing and deep learning approaches. journal of intelligent & fuzzy systems, 36(3), 1997-2004. 3. chaki, j., parekh, r., & bhattacharya, s. (2020). plant leaf classification using multiple descriptors: a hierarchical approach. journal of king saud university-computer and information sciences, 32(10), 1158-1172. https://doi.org/10.1016/j.jksuci.2018.01.007 4. chaudhury, a., & barron, j. l. (2018). plant species identification from occluded leaf images. ieee/acm transactions on computational biology and bioinformatics, 17(3), 1042-1055. 5. crisóstomo de castro filho, h., abílio de carvalho júnior, o., ferreira de carvalho, o. l., pozzobon de bem, p., et al. 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(2022). real-time strawberry detection using deep neural networks on embedded system (rtsd-net): an edge ai application. computers and electronics in agriculture, 192, 106586. microsoft word review+paper-silky+khurana+-+source+file adv syst sci appl 2025; 02; 23-51 published online at https://ijassa.ipu.ru. emerging perspectives on multiband fractal and hybrid fractal antennas for wireless applications: a comprehensive analysis silky khurana1*, rajni rajni1, yadwinder kumar2 1) shaheed bhagat singh state university, ferozepur, punjab, india 2) punjabi university, punjab, india abstract: in the rapidly evolving field of wireless communications, the demand for compact, efficient, and versatile antenna systems has led to significant interest in multiband hybrid fractal antennas. this paper presents a brief overview to the basic fractal structures and their utilization to develop hybrid fractal patch antenna design. the key interest areas are multiband behaviour, acceptable return loss, voltage standing wave ratio, gain and reduced complexity. many studies have simulated and examined the antenna using the finite-difference time-domain (fdtd) and finite element method (fem) based techniques. various antenna performance parameters such return loss, resonant frequency, bandwidth, and gain are observed and the findings of several articles have been presented. the compact size, multiband/wideband features, and extensive utility in wireless applications make the fractal antenna a promising field of study. these antennas are so popular because of their electrically vast construction, which neatly folds into small spaces. keywords: microstrip patch antenna, fractal structure, hybrid fractals, multiband, return loss, iterated function system (ifs) introduction in the dynamic field of wireless communication, the demand for smaller, more versatile devices that support multiple wireless standards has become increasingly prevalent. in order to meet this need, multiband antennas are essential since they allow the rf front-end to be as small as possible while yet supporting a variety of wireless protocols [1-4]. antenna is an incredible part of telecommunication industry [5-7]. communication devices nowadays are devised to be able to work on different wireless standards thus requiring operational characteristics compatible with multiple frequencies. technological advancements in wireless applications have encouraged the improvisation of antennas to large extent. appreciable enhancement has been witnessed in last few decades in the development of antennas being employed for wireless and multiband applications in communication system [8-10]. multiband antennas are designed to operate across multiple frequency bands, allowing them to support various wireless standards such as gsm, cdma, lte, wi-fi, bluetooth, and more [11-13]. this versatility together with space efficiency is essential for modern devices that need to connect to diverse networks without the need for multiple dedicated antennas. microstrip patch antennas are widely used in current communication systems because of their amazing features, which include ease of manufacture, low cost, tiny size, and light weight [14-16]. *corresponding author: silkykhurana2710@gmail.com 24 s. khurana, rajni, y. kumar copyright ©2025 assa adv. in systems science and appl. (2025) 1.1. microstrip patch antennas microstrip patch antennas have completely changed the wireless sector because of its structural simplicity and ease of production. the demand for microstrip patch antennas has consequently experienced gigantic increase and continues to grow [17-18]. furthermore, they are easily programmable to function in dual or circular polarization, as well as multi-band applications [19-21]. compact, dependable, efficient, and multiband antennas are in increasing demand due to rising data rates and high-speed wireless applications [22-24]. the idea of microstrip as radiator was first proposed by deschamps in 1953 [25]. 1.2. fundamentals of fractal antennas microstrip patch antennas are essentially narrowband devices. the ratio of the operating wavelength to the antenna size has a significant impact on their behaviour [26-28]. this indicates that when the operating frequency is changed, the primary antenna characteristics such as gain, power distribution, impedance, radiation patterns will experience significant fluctuations for a constant antenna size [29-31]. moreover, there is a certain limit on size minimization of an antenna, usually of the order of a quarter wavelength. in this regard, the fractal architecture of antennas can aid in solving the mentioned issue by providing diverse range of geometric forms with astounding characteristics [32-34]. in order to achieve multiband properties [35], which is not possible with a traditional microstrip patch antenna, fractal antenna geometry is mostly employed [36-39]. mandelbrot [40] offered the following definition of fractals: “a fractal is by definition a set for which the hausdorff dimension strictly exceeds the topological dimension”. this definition was later retracted and replaced with: “a fractal is a shape made of parts similar to the whole in some way”. a fractal structure is, thus, one that exhibits symmetry across size, self-similarity at different magnifications, and small-scale replication of the object's overall structure in every section. the self-resemblance and plane-filling properties of fractal geometries are frequently associated with their frequency features, such as their ability to operate at many frequencies in electromagnetic devices [41-42] and by using space-filling and self-similarity features, fractal geometries may generate longer current paths in constrained spaces [43-45]. this improves the antenna's properties, maintaining a smaller antenna size and radiation patterns that are amazingly comparable to those of their bigger counterparts. fractals are mostly used in antenna engineering to broaden the perspective on antenna design and move away from euclidean geometry, which holds that an antenna's closed area is proportionate to its perimeter [46-48],. fractal dimensions are characterized by a fractal similarity parameter d as given in equation (1.1) below [49]. 𝐷 = log(𝐿) log(𝑟) (1.1) where l is total length before performing fractalization & r is number of segment divisions. 1.3. types of fractal geometries sierpinski gasket: sierpinski gasket fractal structure and its properties are described by polish mathematician, wacklaw sierpinski in 1915. it is created by repeatedly eliminating inverted equilateral triangles from an equilateral triangle as shown in fig. 1 below [50]. a triangle from middle of equilateral triangle is removed in first iteration. the sierpinski gasket is referred as self-similar geometry because the left three triangles after first iteration are precise replicas of initial triangle. this antenna's self-similar current distribution demonstrates the multi-band properties [51]. the antenna's multi-band nature may be managed by adjusting its geometry, emerging perspectives on multiband fractal... 25 copyright ©2025 assa adv. in systems science and appl. (2025) and by altering the flare angle, one can modify the antenna's band characteristics [52]. the sierpinski gasket triangle’s form is determined by four factors: the height of the triangle ‘h’, the scaling factor ‘sf’, the flare angle ‘h’, and number of iterations, ‘n’. fig. 1. sierpinski gasket [51] sierpinski carpet: in sierpinski carpet fractal structure, the rectangular patch is used to derive the desired geometry. the center of the primary rectangle is subtracted from the onethird-size rectangle. several iterations of this procedure are performed to get the desired geometry. as shown in fig. 2, this antenna is designed by a series of iterations utilizing the selfsimilarity feature of fractal antenna [53]. fig. 2. generation of sierpinski carpet [53] koch curve: the koch curve is a mathematically proven fractal shape that demonstrates self-similarity at different scales. it gets its name from swedish mathematician helge von koch who developed this geometry. the construction of the koch curve involves iteratively adding smaller segments to a starting line. step by step procedure to construct the koch curve is given below: a) start with a straight-line segment. b) divide the line into three equal parts. c) then triangular shape is formed when centre portion of the straight line is bent with flare angle 60 degree resulting in formation of equilateral triangle [54]. d) repeat the process for each of the four-line segments created in the previous step as shown in fig. 3 below. fig. 3. four iterations of koch fractal [54] minkowski curve: minkowski geometry extends the principles of euclidean geometry into the realm of spacetime [55]. a minkowski fractal is a type of geometric fractal that is 26 s. khurana, rajni, y. kumar copyright ©2025 assa adv. in systems science and appl. (2025) constructed using a recursive process based on the minkowski operation, which involves adding or subtracting geometric shapes at various scales. the resulting structure exhibits selfsimilarity at different levels of magnification as depicted in fig. 4. fig. 4. generation of minkowski fractal up to 2nd iteration [56] cantor set: a typical example of a fractal structure that exhibits self-similarity and is generated through a simple iterative process is a cantor set. we can generate a cantor set by following below mentioned procedure: a) begin with a line segment. this segment represents the "zeroth iteration" of the cantor set. b) divide the line segment into three equal parts. c) out of three parts, remove the middle part of the line segment. d) repeat steps 2 and 3 for each of the remaining line segments (the two outer segments resulting from the previous step). e) continue this process of removing the middle third of each remaining line segment infinitely at each iteration. after performing several iterations, cantor set is designed as shown in fig. 5, which is a fractal with some interesting properties like it is self-similar, meaning that each portion of the set resembles the whole set and it is perfect disconnected, compact metric among the fractal structures [57]. fig. 5. various iterations of cantor set fractal [57] 1.4. iterated function system an iterated function system (ifs) in fractal geometry is used to produce intricate and selfsimilar patterns by repeatedly applying a series of affine transformations [58-59]. fractal antennas, being a subset of fractal geometry, can be designed using ifs techniques to achieve certain desired characteristics such as multiband operation, miniaturization, and wide-angle radiation patterns. by utilizing iterated function systems, fractal antennas can be created with intricate structures and desired electromagnetic properties, making them suitable for various wireless communication and sensing applications. 1.5. design and simulation designing and simulating a fractal antenna involves several steps including understanding fractal geometries, choosing a specific fractal type, using appropriate software for design and simulation, and analysing the performance of the antenna. the brief overview of each process is as follows: emerging perspectives on multiband fractal... 27 copyright ©2025 assa adv. in systems science and appl. (2025) a) choose a fractal geometry: fractal antennas are typically based on self-similar geometric shapes like fractal curves or surfaces. common examples include the koch curve, sierpinski gasket, menger sponge, or various iterations of the mandelbrot set [60]. b) define affine transformations: affine transformations involve translations, rotations, scaling, and shearing. each transformation is represented by a matrix that describes how points in the original fractal pattern are mapped to new locations. these transformations are applied iteratively to generate the fractal antenna structure [61]. c) iterate transformations: apply the chosen set of affine transformations repeatedly to each point in the fractal pattern. as the iterations progress, the antenna structure becomes more complex and self-similar [62]. d) adjust parameters for antenna performance: during the design process, parameters such as the scaling factor, rotation angle, and translation vectors can be adjusted to achieve specific antenna performance characteristics. for instance, tweaking these parameters can optimize the antenna for certain frequency bands or radiation patterns [63]. e) simulation and optimization: once the fractal antenna structure is designed using the ifs method, simulation can be done using various electromagnetic simulation software such as high frequency structure simulator (hfss), cst microwave studio etc. simulation results can help evaluate the antenna's performance metrics like impedance matching, radiation efficiency, gain, and bandwidth. further optimization may be performed to fine-tune the antenna design based on simulation results. 1.6. prototype and testing after finalizing the design, the prototype and testing process of a fractal antenna design involves several stages. first, the design is fabricated using materials such as copper on a suitable substrate, employing methods like pcb etching or photolithography. once the prototype is ready, it undergoes testing with equipment such as a vector network analyzer (vna) for s-parameter measurements, and an anechoic chamber for radiation pattern and gain analysis [64]. the key parameters measured include return loss (s11), bandwidth, radiation pattern, gain, and vswr [65]. these results are then compared with simulation data to identify any discrepancies [66-67] . the process is iterative, involving adjustments to optimize the antenna's performance based on the measurement outcomes. 2. investigations on advancement in fractal antennas j. malik et al. [68] designed a proximity feed, multi-layered, dual band complementary sierpinski fractal antenna construction with bandwidth of 100 and 150 mhz at 3.5 and 5.8 ghz respectively. the overall design scales are 25 x 35 x 1.6 mm3 and substrate used is fr-4. the design is optimized for 3.5 and 5.8 ghz frequencies, respectively, suitable for wlan and wimax applications. the designed antenna is analysed with perfect boundary on fixed integration technique. there are two-levels of excoriation for the stacked equilateral triangle construction. r. ghatak et al. [69] developed a novel and compact cpw fed sierpinski carpet fractal shaped uwb antenna design with hexagonal boundary having compact size of 33 mm x 32 mm. proposed antenna has compact size and omnidirectional radiation pattern. the antenna is incorporating a 'y' shaped slot to reject a band of 5.15-5.825 ghz. the measurement of peak gain varies from 1.25 dbi to 6 dbi. antenna is suitable for uwb application with distortionfree design. the geometry of proposed antenna is shown in fig. 6 below: 28 s. khurana, rajni, y. kumar copyright ©2025 assa adv. in systems science and appl. (2025) fig. 6. sierpinski carpet shaped antenna with hexagonal boundary [69] shrivishal tripathi et al. [70] investigated an ultra-wideband (uwb) monopole antenna based on fractal geometry that resembled minkowski structure as shown in fig. 7. this antenna design makes use of the fractal geometry's numerous resonance properties to produce the wideband phenomena. the fractal geometry uses the space filling feature to give a wider operational bandwidth and a greater effective antenna length. the use of several slots in the ground plane is intended to enhance the reflection coefficient over the whole uwb frequency range. with an optimal diameter of 23.5 x 26.5 mm2, the suggested antenna can operate within the frequency range of 1.4 to 10.4 ghz. the suggested antenna has an almost omnidirectional radiation pattern and correct impedance matching over the whole uwb frequency spectrum. fig. 7. ultra-wideband antenna using minkowski-like fractal geometry [70] avinash kumar et al. [71] studied a reconfigurable microstrip meander line antenna that is suitable for direct-to-home (dth) satellite communication, wi-fi devices, and climatic radar systems by using the switching of currents produced by two pin diodes. the reconfigurability feature of the proposed structure makes it ideal for portable applications due to its compact design. the authors also conducted parametric study of spacing between the meander lines and noted how decreasing the meander line spacing affected the antenna's radiation characteristics, vswr, and reflection coefficient. the proposed structure is represented in fig. 8 below: fig. 8. reconfigurable microstrip meander line antenna [71] mohd h. ramli et al. [72] designed a sierpinski gasket fractal antenna and incorporated slits aiming at compact dimensions, low profile, multiband/broadband. two methods are used emerging perspectives on multiband fractal... 29 copyright ©2025 assa adv. in systems science and appl. (2025) to achieve desired results are adding slits and increasing iterations of fractal antenna as illustrated in fig. 9. the antenna exhibited a return loss of -11.466 db and -8.3543 db at 2.4 ghz and 5 ghz respectively. the proposed antenna design is created in cst software. the design exhibited increased gain when slits are added to sgfa. introduction of dgs in the design is done to increase bandwidth and miniaturize antenna size. fig. 9. sierpinski gasket fractal antenna for multiband applications [72] s. singhal et al. [73] discussed the need for super wideband (swb) antennas in wireless communication systems. various swb antenna structures reported in the literature are reviewed by the author. the author devised a compact hexagonal sierpinski fractal antenna for super wideband applications. the antenna consists of a hexagonal radiation patch with two iterations of sierpinski square slots as mentioned in fig. 10. impedance matching is achieved using a cpw feeding. the antenna exhibits an impedance bandwidth of 3.4-37.4 ghz, with a bandwidth ratio of 11:1. the antenna is omnidirectional, reasonably stable, and has good radiation performance characteristics. there is good agreement between the simulated and experimental results. compared to earlier arrangements, the developed antenna has the benefits of a broader bandwidth and a smaller size. fig. 10. cpw-fed hexagonal sierpinski fractal antenna [73] s. singhal et al. [74] developed a coplanar waveguide fed octagonal super-wideband fractal antenna integrating defected ground structure. the author utilized the fact that fractal geometry allows for compact, multiband, and wideband antenna designs and fractals have convoluted shapes that increase bandwidth and effective radiation. author also studied several superwideband antenna configurations. stable and omnidirectional patterns, among other desirable radiation performance qualities, are achieved. the simulated and experimental results correlate rather well. in comparison to earlier designs, the antenna has a larger bandwidth and a smaller overall size. yogesh k. choukiker et al. [75] devised an antenna for wideband applications by utilizing frequency reconfigurable koch snowflake characteristics. frequency reconfigurability is achieved using rf pin diodes, lumped capacitor, and inductors. the proposed design structure combines the equilateral triangles together with koch fractal design. a recursive process using the self-similar koch fractal structure is used to construct the antenna shape. designed antenna has three operating frequency bands3.34 to 4.52 ghz, 2.2 to 3.4 ghz and 1.45 to 4.1 ghz. 30 s. khurana, rajni, y. kumar copyright ©2025 assa adv. in systems science and appl. (2025) the antenna provides continuous wideband frequency coverage and excellent impedance bandwidth from 1.45 to 4.52 ghz. the antenna's performance is verified by measurements, which show that it produced low cross-polarization, steady, and omnidirectional radiation patterns. the compactness of the developed antenna makes it possible to employ it as an array element. r. hussan m et al. [76] designed a microstrip fed printed monopole antenna and utilized cantor fractal geometry on rectangular patch up to second iteration as shown in fig. 11. the substrate used is having relative permittivity of 4.4 and thickness of 1.6 mm. etching of the monopole radiating element is done on a substrate. antenna is fed with a 50-ohm microstrip line. the em simulator and ie3d are used for modelling and performance assessment, and a parametric analysis of the impact of the radiating element's aspect ratio is also carried out. with -10 db impedance bandwidths in the 1-6 ghz range, dual-band resonant behavior is observed, supporting wireless communication systems like bluetooth, wlan, and its. fig. 11. cantor based monopole antenna with reduced ground plane [76] f. wang et al. [77] proposed and developed a novel minkowski fractal-based method for bandwidth increase and antenna downsizing. sensitivity analysis is used to ascertain how important parameters affect the impedance bandwidth. by measuring fidelity factors, radiation patterns, return loss and transfer function, the author conducted an experimental validation of the theoretical study. asymmetric strips with minkowski fractal shape increase electrical length while decreasing low cut-off frequency. at higher frequencies, improving the fractal direction enhances impedance matching. asymmetric strips and minkowski fractal geometry are used to achieve miniaturization. impedance bandwidth is increased by a triangle notch. c. figueroa-torres et al. [78] proposed a novel fractal antenna for super wideband applications that is based on the sierpinski triangle shape and has a modified ground plane as depicted in fig. 12. the antenna has a 15:1 bandwidth ratio and works between 1.68 and 26 ghz. the antenna is 62 by 64 mm in size and has an average gain of 3 dbi. mobile phones and other portable gadgets can include the innovative design antenna. performance is enhanced by semi-circular sectors in the radiating patch and small gaps on the ground plane. at higher frequencies, the radiation patterns become directed with side lobes. emerging perspectives on multiband fractal... 31 copyright ©2025 assa adv. in systems science and appl. (2025) fig. 12. fractal antenna based on sierpinski structure for super wide-band applications [78] t. benyetho et al. [79] discussed wireless power transmission using microwave energy. the focus is on increasing the efficiency of energy transmission. the author developed a new multiband antenna integrating microstrip patch technology and fractal antenna geometry. the antenna is validated in simulation and measurement are verified in the ism band ranging from 2.45 ghz and 5.8 ghz. its gain is suitable for the mentioned frequency range and its aperture angle is wide. the antenna can be used in a circuit requiring reception of multiple frequencies. the antenna is inexpensive, easy to construct, and appropriate for uses involving wireless power transfer. the design has a small size of 60 x 30 mm2 fabricated on fr4 substrate with dimensions of 60 × 30 mm2. a. reha et al. [80] iterated that antennas with miniaturized design are necessary for restricted environments exhibiting multi-band and broadband characteristics. miniaturization techniques are studied including geometric manipulation and material manipulation. the author presented a coplanar waveguide fed sierpinski curve fractal antenna. a lower resonance frequency is made possible by altering the rectangular geometry of the initial iteration of the sierpinski curve fractal structure. the resonant frequency is further reduced by higher iterations of the fractal structure. thus, a realistic link between the number of iterations and resonance frequencies is established. cadfeko is used to acquire the results, and vector network analyzer is used to confirm them through measurement. binod k. soni et al. [81] designed a minkowskized fractal like antenna using boolean addition of two shapes i.e. isosceles triangle and a square, forming a non-uniform pentagon shaped patch as shown in fig. 13. he also incorporated a parasitic patch imprinted on deformed ground plane. this hybrid fractal structure exhibited multiple resonant frequencies at 0.83 ghz, 2.12 ghz, 1.05 ghz, 3.75 ghz, 1.6 ghz, 3.25 ghz and 5.2 ghz with sufficient bandwidths suitable for gsm, wifi, wimax, wlan services. the author exploited the benefit of larger perimeter provided by minkowski shape structure which aids the current vectors to travel around the antenna structure. fig. 13. minkowskized hybrid fractal antenna [81] thilai thilagam et al. [82] suggested creating slits on the patch surface to create a circular microstrip meander line patch antenna for wireless mobile applications. the author also stacked another hexagonal patch over the meandered circular patch in the final design and achieved the multiband behaviour for frequency range 0.72 ghz to 6 ghz. a gain of maximum value 7.74 dbi is achieved in the proposed structure. the meander technique incorporated in the patch provided increased electrical length and stacking of another patch increased the operational bandwidth of the designed antenna. the antenna design is developed in three stages wherein simple circular patch is designed in first stage. in second stage, circular patch is meandered by 32 s. khurana, rajni, y. kumar copyright ©2025 assa adv. in systems science and appl. (2025) cutting slits on it and finally novelty is introduced in the design by stacking hexagonal patch over the meandered patch. mohd g. siddiqui et al. [83] proposed an a-shaped triangular patch and then used the fractal concept to lengthen the patch's effective length, which produced multiband behavior at frequencies of 11.44 ghz, 13.178 ghz, 15.482 ghz, 19.902 ghz, and 23.529 ghz. this made the patch antenna suitable for use in commercial and x-band, ku band, and k band satellite communication applications. the author exploited the benefits of von koch’s snowflake concept for employing fractal structure on a shaped patch as shown in fig. 14. fig. 14. multi-band triangular microstrip fractal antenna [83] ashwini kumar et al. [84] focuses on radio navigation services for ships using multiband antennas. using an hfss simulation tool, the minkowski fractal antenna is parametrized. to accomplish multiband functioning and decreased size, fractal geometry is applied. with a gain of 2.5 db at 9.4 ghz and 6.3 db at 3 ghz, the proposed antenna functions at both frequencies. antenna radiation and resonance characteristics are measured experimentally. measured outcomes and simulated outcomes accord well. antenna has nearly omni-directional radiation patterns in the xz-plane. to determine the ideal bandwidth in the targeted bands, ground plane width is parametrically optimized. the antenna is tuned to resonant at 9.4 ghz (9.13 ghz to 9.98 ghz) and 3.0 ghz (2.905 ghz to 3.094 ghz), covering frequency ranges for marine radio navigation services with an adequate impedance bandwidth. majed o. dwairi et al. [85] provided a fractal regular slots patch antenna design for use in uwb systems. a uwb monopole rectangular patch antenna serves as the reference antenna. the antenna properties are altered by randomly inserting twenty of the intended fractal slots. following that, four different antenna topologies are examined with respect to a range of factors, including group delay evaluation, radiation patterns, surface current distribution, gain, and impedance bandwidth characteristics. it is analysed that fractal slots create filtering behavior and band rejection characteristics. a good radiation pattern with a maximum gain of more than 4.97 dbi is observed, shifting from a directive-monopole to a somewhat omnidirectional pattern. the group latency varies by less than 4 nsec, however it is generally consistent. because the antenna resonantly operates at wlan ieee802.11a and hiperlan2 systems running in the 5.15-5.825 ghz band and x-band (7.25-7.75 ghz), it is appropriate for uwb applications, satellite applications, and defence systems. hiwa taha sediq et al. [86] exploited the benefits of employing slots to get multiband behaviour. when slots are added to a microstrip antenna patch, discontinuities are included in the electric current's path, providing additional band frequencies. fig. 15 shows the proposed geometry. this technique results in multiple operating frequencies exhibited by proposed antenna making it suitable for mobile and wireless devices. in order to increase the overall length of the current path, the suggested antenna consists of a star monopole patch with a slit gap, a star-shaped slot on the patch as the radiating component, and twin square-shaped slots emerging perspectives on multiband fractal... 33 copyright ©2025 assa adv. in systems science and appl. (2025) on the material as the ground plane. it is easy to use the suggested architecture for ku band, xband, wlan, wimax, and satellite communications applications. fig. 15. multiband monopole fractal antenna for various broadband applications [86] fatima ez-zaki et al. [87] designed a hexagonal shaped microstrip patch antenna. the designed antenna can be suitably used for vehicular communications including lte wimax and v2xdsrc as it covers navigational frequencies of wimax at center frequency of 3.7 ghz and ieee802.11p at center frequency of 5.9 ghz. antenna design includes inset feeding mechanism, cantor fractal slot with partial ground plane. antenna resonates well in the assigned frequency bands for dsrc/ieee802.11p and lte/v2x. the gain of the designed antenna varies from 3.06 to 5.25 db. the antenna covers multiple wireless communication systems such as blind spot detection, wlan, wimax, and dsrc. fabricated antenna shows good results, suitable for wireless applications. a. annou et al. [88] proposed a novel technique to enhance the gain and radiation efficiency by collaborating defected ground structures with koch snowflake fractal design on which unit cell of metamaterials in engraved to achieve triple band operation. a multi-resonance l-dgs antenna with a high gain of 5 db and an efficiency of 99.6% is designed by the author. a new compact double negative metamaterial unit cell and its corresponding circuit are being studied for multi-band (2.8 ghz, 4.1-4.45 ghz, and 5.6 ghz) wireless applications with high miniaturization of 30 x 30 mm2. to enhance the antenna matching, a koch snowflake fractal is inserted along the radiation patch borders. by placing a unique modified csrr unit cell in the center of a rectangular radiating patch, the antenna size is reduced by 63%. the design is then suitable for wlan and wimax applications because the radiation properties and return loss are enhanced through the use of the dgs and koch snowflake fractal techniques. devesh et al. [89] presented a miniaturized hexagonal sierpinski gasket antenna for applications focussing on wireless communication. high gain performance, dual polarization, and multiband operation are attained by the designed antenna. the simulation program hfss v15 is used to analyse the design. since fractal antennas with wideband, multi-band, high gain, and improved radiation properties are in demand, sierpinski gasket fractal structures are employed for multiband operations. with multiband performance demonstrated at 3.46, 8.28, 12.26, 17.21, 23.40, and 26.01 ghz, the design is applicable to 5g and iot applications, as well as satellite and radar communications. the proposed structure is illustrated in fig. 16. 34 s. khurana, rajni, y. kumar copyright ©2025 assa adv. in systems science and appl. (2025) fig. 16. hexagonal sierpinski gasket fractal microstrip antenna [89] ashwini kumar et al. [90] demonstrated, as seen in fig. 17, a modified hilbert curve fractal antenna with a dual-layer substrate supplied by a modified coplanar waveguide cpw feed. to increase bandwidth, the scientists used the dragonfly optimization approach, which is based on swarm intelligence, to improve the shape of the step feed. with a bandwidth of 0.315 ghz (0.68–0.99 ghz), 3.220 ghz (3.98–7.20 ghz), 2.955 ghz (8.043–10.998 ghz), and 1.94 ghz (12.7–14.64 ghz), the antenna offered multiband resonating characteristics. this made it suitable for the ultra high frequency (uhf) band as well as the aeronautical radio navigation service (arns) in c-band, x-band, and ku-band. additionally, the design has parasitic patches that are stimulated via gap coupling. ifs is used to provide geometrical descriptors for the suggested design. fig. 17. hilbert curve fractal antenna [90] ashish kumar et al. [91] investigated the possibility of optimized sierpinski fractal antenna with monolithic microwave integrated circuits for wireless applications in x band and ku band. because of the limited space, the author emphasized the necessity of high index substrate materials for mmics. the limitations of surface waves in high index substrates have been addressed by a novel technique called bulk micro-machining, which produces an air cavity beneath the patch, further creating a low index environment in the patch antenna and leading to a significant improvement in performance parameters as well as compatibility with monolithic microwave integrated circuits, as illustrated in fig. 18. excellent and highly competitive results are obtained at resonant frequencies of 8.29 ghz, 11.93 ghz, and 15.3 ghz by using the ant lion optimization technique. 9.6 ghz has produced the highest gain of 6 db, and 8.29 ghz produces outstanding impedance matching of –26 db. emerging perspectives on multiband fractal... 35 copyright ©2025 assa adv. in systems science and appl. (2025) fig. 18. micro-machined sierpinski carpet fractal antenna [91] lan wang et al. [92] suggested an innovative approach of designing multiband antenna using circle and triangle fractals with microstrip line feed represented in fig. 19. with straightforward geometry, the wideband properties are obtained at frequencies 2.6 ghz, 3.8 ghz, 5.3 ghz with adequate gain ranging between 2.58–3.34 dbi. by modifying the antenna radiating slot, the current direction on the microstrip antenna's metal surface is changed to produce multiband coverage. further the results are enhanced by designing a resonator ring on the ground plane. the antenna model is expandable based on the recommended construction. the antenna may be more fractal outward at lower frequencies and more fractal inward at higher frequencies. fig. 19. multiband antenna using circle and triangle fractals [92] nontuthuko l. nhlengethwa et al. [93] developed a dual band and triple band fractal microstrip patch antenna for wlan and ism wireless applications using a basic sierpinski carpet fractal. in order to obtain multiband behaviour, the author deployed self-similar fractal structures that increases the effective electrical length. as a result, the resonant frequencies of 2.4 ghz, 5.3 ghz and 5.9 ghz are achieved. the suggested antenna is designed to increase its gain and directivity by use of a faulty ground construction that has a reflector plane. the final tbfmpa is showing a maximum directivity of 8.536 dbi at 5.889 ghz and a maximum gain of 2.9 db at 2.385 ghz. fr-4 substrate is used to create the design, which has overall sizes of 57.9 x 53.6 x 1.5 mm3, as seen in fig. 20. fig. 20. sierpinski carpet fractal antenna [93] 36 s. khurana, rajni, y. kumar copyright ©2025 assa adv. in systems science and appl. (2025) aravindraj ezhumalai et al. [57] reviewed various fractal structures using ifs. he suggested that a fractal structure may be designed using a variety of methods, but the most popular method for creating a finite set of construction mappings on an entire metric space is iterated function system. among the fractal geometries produced with this method are the sierpinski carpet and gasket, koch snowflake, deterministic tree/binary tree, gosper island, minkowski island, cantor set, t-square, and a few space filled fractals. he also looked at fractal structures as a virtual combination of capacitor and inductor that produces a wide range of resonant frequencies in fractal patch antennas. he reaffirmed that ifs offers a crucial foundation for the different fractal structures and their modifications that have been mathematically solved. ashwini kumar et al. [94] suggested a novel antenna design for iot applications using minkowski fractal structure. the proposed antenna is miniaturised by 65% using minkowski fractal. to increase the bandwidth, proposed antenna is fed with microstrip step feed. the sub6 ghz band is where the multiband behaviour is obtained, spanning a range of frequencies appropriate for 5g communications, public safety applications, gps, and radar altimeter. after several iterations of inserting slots into the radiating element and partial ground plane, along with parametric optimization, the design is finalized. sourabh raj et al. [95] suggested using the electromagnetic band gap (ebg) structure to create a truncated patch antenna. the fabricated prototype consists of an ebg loaded on the ground plane and a truncated patch with a symmetrical slit. the antenna's optimized volume measures 20 × 15 × 1.57 mm. over the whole covered bandwidth, a good radiation pattern for the e-h field is established. the antenna covers the bandwidths of 24.2 ghz to 27.84 ghz and 33.84 ghz to 36.2 ghz by resonating at 27.2 ghz and 35.04 ghz. the design may be employed in the internet of medical things and for data transfer from rf sensors to healthcare systems because the antenna spans 5g bandwidth. furthermore, data transmission and wireless communication are two more uses for antennas in real-time applications. 3. advances in hybrid fractal antennas azaro et al. [96] designed an antenna that operates in gsm and wifi frequency bands. the design combines a sierpinski-like structure with a meander-like structure. paper focuses on designing miniaturized and multiband antennas for wireless devices. fractal shapes have been found to be effective for antenna development. meander-like structure is added for electrical length compliance. the synthesis problem is solved using a pso algorithm integrated with a method-of-moments simulator. antenna size is about 50% smaller than a quarter-wave monopole. vswr value lower than 3.5 required at input port. atif jamil et al. [97] proposed an antenna for wlan usb dongle with small dimensions and designed a compact koch-meander fractal antenna as depicted in fig. 21. the suggested antenna has high performance attributes together with a large bandwidth. the antenna has the capacity to cover the whole wlan ieee 802.11abg standard band. the antenna's bandwidth is 2.2909 2.553 ghz and 5.1406 5.8737 ghz, with a return loss of -10 db. at 2.41 ghz, a minimum return loss of -28.9 db was attained. the suggested antenna could be a good option for the distribution of wlan usb dongles. emerging perspectives on multiband fractal... 37 copyright ©2025 assa adv. in systems science and appl. (2025) fig. 21. hybrid meander-koch fractal antenna [97] yadwinder kumar et al. [98] explored the use of koch and meander geometry in the creation of a hybrid fractal antenna as depicted in fig. 22 for wireless applications. the small size and multiband behavior are achieved by the application of fractal geometry. the antenna displays quad band behavior at 4.07 ghz, 7.3 ghz, bluetooth, and wlan frequencies because it resonates at four separate frequencies. its radiation pattern is essentially omnidirectional, its structure is planar, and its cost is modest. the author used matlab and the ifs technique to create the hybrid structure. fig. 22. quadband hybrid fractal antenna [98] amandeep singh et al. [99] outlined a wideband antenna design based on a modified sierpinski fractal. the sierpinski fractal antenna's present layout, mentioned in fig. 23, is optimized for broadband behavior. broadband behavior is demonstrated by the developed antenna in the frequency ranges of 21-30ghz and 12.2-13.4ghz. peak gains that are measured lie between 8 and 22 db within specific frequency ranges. high gain is obtained by adjusting the original sierpinski geometry. results from measurements and experiments with a modified coaxial probe fed antenna are validated. the antenna design is examined and refined by the application of the fem electromagnetic field solver. 38 s. khurana, rajni, y. kumar copyright ©2025 assa adv. in systems science and appl. (2025) fig. 23. coaxial probe-fed sierpinski fractal wideband and high gain antenna [99] yadwinder kumar et al. [66] devised a unique hybrid fractal antenna design as shown in fig. 24 utilizing the fractal structures of koch and minkowski. the proposed antenna exhibited resonance at seven frequencies covering important frequency bands such as gps, bluetooth, wlan etc as shown in fig. 25 and thus providing applications in mobile/fixed satellite, aeronautical navigation and other wireless applications. the author utilized iterated function system to formulated the hybrid geometry in hfss. the suggested antenna has acceptable vswr, gain, and return loss characteristics. this antenna's lightweight and small design make it suitable for usage in portable devices. fig. 24. (a) top view (b) side view (c) back view of proposed antenna [66] fig. 25. return loss of compact multiband hybrid fractal antenna showing hepta-band behaviour shrivishal tripathi et al. [100] created a small uwb antenna with better radiation characteristics and wideband utilizing sierpinski fractal geometry. fractal geometry's spacefilling characteristics and self-similarity allow for both miniaturization and wideband features. the effective electrical channel length can be increased by combining the sierpinski fractal with octagonal-shaped geometry. the antenna has a good return loss and almost omnidirectional radiation pattern. the measured features of the designed prototype match simulation, and it exhibits an excellent time domain response with compact dimensions of 25x16 mm2. an analysis of surface current distribution is conducted to comprehend the radiation process. yadwinder kumar et al. [101] proposed combining a meander-like antenna (for lower frequency) with a modified sierpinski gasket to create a hybrid fractal structure. with resonant frequencies of 2.4 ghz, 4.437 ghz, 5.38 ghz, 7.01 ghz, 7.60 ghz, 8.41 ghz, and 9.09 ghz, the suggested radiating structure displays multiband behavior. this makes the developed antenna appropriate for radiolocation, bluetooth, wlan, wi-fi, ism, rfid, 4g/lte, and mobile/fixed satellite services. in order to achieve an effective value of gain at resonating frequencies, as shown in fig. 26, the author modified the original sierpinski gasket structure by emerging perspectives on multiband fractal... 39 copyright ©2025 assa adv. in systems science and appl. (2025) varying various parameters like scale factor and flare angle using the ifs mathematical tool. the modified sierpinski gasket was subsequently paired with a meander-like antenna and a defected ground structure. fig. 26. sierpinski–meander hybrid fractal multiband antenna [101] manisha gupta et al. [102] designed a miniaturized and circularly polarized microstrip fractal antenna using a hexagonal shaped patch for microwave applications in s and x band. the proposed antenna combines the fractal structures of koch and sierpinski, with the koch fractal structure on the edges using a right-angled isosceles triangular construction (fig. 27). the author studied the importance of circular polarization and concluded with hexagonal shaped patch because it is the closest match of circular patch together with defected ground structure to accomplish circular polarization. the proposed structure exhibits multiband behaviour at resonant frequencies 3.1 ghz, 6.9 ghz, 8.4 ghz and 9.1 ghz together with sufficient gain with circular polarized radiation patterns making it suitable for mobile tv, satellite radio, home-based consumer electronics, radars, satellites and military applications. fig. 27. koch fractal-based hexagonal patch antenna [102] devesh et al. [103] integrated both sierpinski gasket and sierpinski carpet to utilize the benefits of fractal antenna which help in achieving miniaturisation of the overall dimensions of antenna as clear from fig. 28. the author also discussed other methods of achieving multiband operations like usage of thick dielectric substrate, slots and notches included in the patch etc and concluded that these techniques lead to larger and bulkier design of antenna. the designer is able to achieve multiband as well as wideband response by creating four triangular slots in square patch structure and repeating the same structure up to four iterations. the resonating frequencies 15.92 ghz, 20.05 ghz, 23.08 ghz and 27.77 ghz as shown in fig. 29 are suitable for k band and ku band wireless applications like satellite communication and radar communication. 40 s. khurana, rajni, y. kumar copyright ©2025 assa adv. in systems science and appl. (2025) fig. 28. modified square sierpinski gasket fractal microstrip antenna [103] fig. 29. return loss of modified square sierpinski gasket fractal microstrip antenna depicting quadband response manisha gupta et al. (2018) [104] designed, fabricated, and measured a hexagonal fractal patch antenna that is inset fed. the antenna has a hexagonal form with edges that include a koch snowflake fractal structure. the frequency range for ultra-wideband characteristics is 3.265 ghz to 8.2 ghz. the antenna is appropriate for multiband and circularly polarized radiations in wireless systems as observed and simulated results correspond well. the design displays radiation patterns that are bidirectional in the e-plane and omnidirectional in the hplane. applications for the prototype antenna include weather radar systems, wi-fi devices, cordless phones, and satellite communication. roman kubacki et al. [105] introduced an ultra-wideband antenna that uses fractal architecture based on planar repetition. through an increase in electrical length and resonant bands, fractal geometry enhanced the antenna's broadband properties and gain. the design of the double-fractal layer antenna allowed a frequency range of 4.1 to 19.4 ghz, with a bandwidth of 15.3 ghz. the highest gain of the designed antenna is 10.9 dbi at 15.2 ghz, with an average observed gain of 6 dbi. the outcomes demonstrated that efficiency increased as frequency increased. the suggested antenna can be applied to communication systems that call for more advanced characteristics. ashish kumar et al. [106] concentrates on adapting the pythagorean fractal tree patch antenna for multiband usage. the antenna is made on a fr4 substrate that has a thickness of 0.8 mm and a dielectric constant of 4.4. with a maximum gain of 4dbi, the final design iteration exhibits multiband characteristics with frequencies spanning from 1.38 ghz to 18.09 ghz. patch antenna bandwidth is improved by thickening the substrate. high index materials and a micromachining technique are used in the integration of patch antenna with mmics to design the patch on the high index material. patch antenna bandwidth issues are solved by using a composite substrate or micromachining, which makes the design appropriate for satellite services related to earth exploration. norun abdul malek et. al [107] suggested the creation of a hybrid koch-minkowski fractal dipole antenna for use in dual band wireless applications. particularly at the low frequency of 900 mhz, which is designated for iot communication for sensors, attenuators, transceivers, emerging perspectives on multiband fractal... 41 copyright ©2025 assa adv. in systems science and appl. (2025) and processors, hybridization led to the shrinking of antenna for iot applications. the goal of the research is to determine how much antenna size may be decreased while still allowing the fractal to resonate at the desired frequency. in this work, the fractal antennas demonstrated dual frequency operation at 900 mhz and 2.6 ghz, with return losses of 18 db and 16 db, respectively. the antenna's compact, lightweight, and thin form (115 x 22.5 x 0.285 mm) makes it a desirable feature for compact wireless applications. mohd gulman siddiqui et al. [108] proposed a fractal antenna for c/x/ku-band applications by utilizing dual fractal approaches based on koch and sierpinski fractals. to lessen the impact of surface currents, the koch fractal idea is applied to the square-shaped patch's outside perimeter and the sierpinski gasket concept is used to its interior segment. three substrates—fr4, rogers ro 4003, and rogers ultralam—have design outcomes that are compared, with a size reduction around 57.2% obtained. with return losses of -38.906 db, 23.58 db, -26.19 db, -34.06 db, and -30.01 db, respectively, the antenna resonates at 5.699 ghz, 10.108 ghz, 11.100 ghz, 15.812 ghz, and 17.621 ghz, demonstrating its suitability for commercial wireless and satellite applications. atif jamil et al. [109] proposed a novel fractal antenna keeping in view the miniaturisation and wideband operation requirements of sophisticated wireless devices. the designed antenna merges meander and koch curve shapes as shown in fig. 30 and it is compared with standard koch curve geometry also. the designer exploited the benefits of combining non-fractal meander line with fractal koch curve using ifs to bring the higher modes together and achieved wider bandwidth. the parametric optimization is also done in cst mws to achieve adjusted thickness of the ring of value of 1.1 mm and width of ground plane of 25 mm. by lengthening the antenna's total electrical length and, consequently, the number of segments that make up the resulting convoluted form, miniaturization is accomplished. fig. 30. wideband hybrid fractal antenna [109] omaima benkhadda et al. [110] developed a sierpinski hexagonal shaped radiating element and incorporated partial ground plane loaded with three rectangular stubs and three rectangular slits. the proposed structure is shown in fig. 31 and it resonates in three frequency ranges of 2.19–4.43 ghz, 4.8–7.76 ghz, and 8.04–11.32 ghz, thus named as tri-wideband fractal antenna suitable for use in variety of wireless technologies such as lte 2300, lte 2500, rfid, bluetooth, 5g spectrum band, wlan, industrial, scientific and medical (ism), worldwide interoperability for microwave access wimax, c-band, and x-band. fig. 31. sierpinski hexagonal shaped antenna [110] 42 s. khurana, rajni, y. kumar copyright ©2025 assa adv. in systems science and appl. (2025) rakhee patil et al. [111] presented the design and optimization of a sierpinski carpet fractal antenna using artificial neural network appropriate for operation between 2 ghz to 8 ghz. the structure is shown in fig. 32. the antenna is simulated using hfss and microstrip line feeding technique is utilized to feed the proposed antenna. in this implementation, the neural network predicts significant parameters, such as s11 and vswr values, based on frequency. a multilayer feed forward artificial neural network is used in the suggested method as an approximation model to determine different antenna parameters. fig. 32. sierpinski carpet fractal antenna [111] n.m. mary sindhuja et al. [112] introduced a novel slotted fractal antenna design with inset fed incorporating βω space filling curve as shown in fig. 33 capable to resonate at multiple frequency bands. the antenna resonating at 1.91 ghz, 3.12 ghz, 5.56 ghz, 10.75 ghz and 13.94 ghz is suitable for various applications such as pcs-1900, rail mobile radio, dcs-imt gap, wcdma, x-band and ku band services. the parametric study by varying the ground plane length is done and optimized performance of antenna is achieved at ground plane length reduced to 50 %. fig. 33. hybrid β ω-indexing based fractal antenna [112] rania hamdy elabd et al. [113] presented a circular ring-shaped fractal antenna incorporating a central plus sign and an outer circular ring with eight smaller rings to enhance the bandwidth for uwb response. the proposed design is demonstrated in fig. 34 below. the author optimized the dimensions of antenna using particle swarm optimization techniques while maintaining the satisfactory return loss response. the antenna covered diverse wireless standards like uwb from 3.1 up to 10.6 ghz, x-band from 8 up to 12.5 ghz, and lower band of ku from 12.5 to 14.5 ghz. the fractal antenna’s bandwidth is increased by mitigating extra capacitive reactance by plus sign which also couples with the partial ground plane. fig. 34. circular ring-shaped uwb fractal antenna [113] sanae attioui et al. [114] presented a novel ultra-wideband (uwb) fractal antenna design that integrates sierpinski carpet and giuseppe peano geometries along with a uniquely emerging perspectives on multiband fractal... 43 copyright ©2025 assa adv. in systems science and appl. (2025) structured ground plane to enhance bandwidth performance. utilizing hfss for simulation, the proposed compact antenna demonstrates effective operation across a broad frequency spectrum of 3.32 to 11.96 ghz, achieving a peak gain of 5.6 db. designed for a variety of wireless applications—including sub-6 ghz 5g, wimax, wlan, c-band, and x-band satellite communications—the antenna's performance was further validated through fabrication on an economical fr4 substrate. the measured results closely align with simulations, confirming the antenna’s wide impedance bandwidth and strong radiation characteristics. its compact form factor and broad usability make it a promising solution for modern uwb communication systems. prasannajeet mohanty et al. [115] introduced a hybrid fractal microstrip antenna design which merged vicsekcross and koch fractal geometries to exploit the strengths of both designs. the vicsekcross structure contributes to antenna miniaturization through its space-filling nature, while the koch curve enhances gain and bandwidth via its self-similar configuration. the vkfa demonstrates a compact, low-cost design fabricated on an fr4 epoxy substrate, operating effectively across 16.92–18.74 ghz with a peak gain of 5.14 dbi at 17.19 ghz. notably, circular polarization is achieved using two l-shaped slots etched 180° apart on the patch, producing orthogonal modes with a 90° phase shift, resulting in right-hand circular polarization and a 3-db axial ratio bandwidth from 17.12 ghz to 18.04 ghz. 4. comparative analysis of hybrid fractal antennas the comparative analysis of various uwb fractal antennas is discussed in this section. the parameters like resonant frequency, gain, operating band, bandwidth and dimension of the fractal antennas are compared and listed in table 1. table 1. comparative analysis of hybrid fractal antennas ref. no. design dimensions substrate type multiband response gain applications [96] sierpinski-like and meander-like structures 40mm × 40mm × 0.8mm arlon ℇr = 3.38 tri-band -gsm and wifi [97] meander-koch fractal antenna 38mm × 10mm fr4 ℇr = 4.34 dual-band 2.281 dbi 2.30 dbi wlan usb dongle [98] koch and meander geometry 55mm × 24mm × 1.6mm fr4 ℇr = 4.4 quad-band 4.3 dbi bluetooth and wlan [99] modified sierpinski fractal antenna 39mm × 40mm × 1.524mm rogers ro4003c ℇr = 3.55 dual-band 21.26 db 8 db broadband satellite receivers, mobile space research activity [66] hybrid koch minkowski fractal antenna 28mm × 15mm × 1.58mm fr4 ℇr = 4.4 hepta-band 4 db gps, bluetooth, wlan [100] sierpinski fractal geometry 25mm × 16mm × 1.6mm fr4 ℇr = 4.4 tri-band - wireless personal area network and wireless body area network [101] modified sierpinski– meander hybrid fractal antenna 54mm × 46mm × 1.6mm fr4 ℇr = 4.4 hepta-band max. 32 db min. 6 db bluetooth, wlan, wifi, ism, rfid, 4g/lte [102] koch fractal-based hexagonal patch antenna 30mm × 35mm × 1.59mm fr4 ℇr = 4.4 penta-band 4.08 db 4.79 db 3.85 db 3.46 db 3.01 db microwave applications, mobile tv and satellite radio 44 s. khurana, rajni, y. kumar copyright ©2025 assa adv. in systems science and appl. (2025) conclusion the development and application of fractal antennas and hybrid fractal antennas represent significant strides in advancing multiband wireless communication systems. through this review, we have explored the evolution of these antennas and their contributions to overcoming the challenges inherent in contemporary wireless communication. the emergence of hybrid fractal antennas, which combine fractal principles with other antenna technologies, has further expanded the capabilities of multiband communication. by leveraging the strengths of different antenna designs, hybrid fractal antennas achieve enhanced performance metrics, including increased bandwidth, improved radiation characteristics, and greater adaptability to evolving communication standards. in the realm of multiband wireless communication, fractal and hybrid fractal antennas have demonstrated their efficacy across various applications, from cellular networks to satellite communication and beyond. their ability to operate across multiple frequency bands facilitates seamless connectivity, robust signal transmission, and [103] modified square sierpinski gasket (mssg) fractal antenna 40mm × 40mm × 1.6mm fr4 ℇr = 4.4 quad-band 6.81 db 7.91 db 7.32 db 8.89 db wireless multi-band communication systems. [104] hexagonal shape with koch snowflake fractal 32mm × 32mm × 1mm fr4 ℇr = 4.4 tri-band 6.8db satellite communication transmissions, some wifi devices, cordless telephones, and weather radar systems [105] minkowski island and crossbar fractal microstrip antennas 27.6mm × 31.8mm × 0.79mm rt/duroid ℇr = 2.2 tri-band 6 dbi communication system [106] modified pythagorean fractal tree patch antenna 40mm × 37mm × 0.8mm fr4 ℇr = 4.4 hexa-band 4 dbi earth exploration satellite services [107] hybrid kochminkowski fractal dipole antenna 115mm × 22.5mm × 0.285mm rt/duroid ℇr = 2.2 dual-band 2.069 db compact wireless applications [108] koch–sierpinski fractal microstrip antenna 60mm × 50mm × 1.6mm fr4 ℇr = 4.4 penta-band - c/x/ku-band applications [109] hybrid meander-koch fractal ring antenna 45mm × 25mm × 1.6mm fr4 ℇr = 4.34 wideband 3 dbi 5 ghz wlan band [110] sierpinski hexagonalshaped fractal antenna 24mm × 30mm × 1.6mm fr4 ℇr = 4.4 tri-band 1.71 dbi 4.61 dbi 4.46 dbi wlan, wimax, ism, lte, rfid, bluetooth [111] sierpinski carpet fractal antenna 35.4mm × 35.4mm × 1.58mm fr4 ℇr = 4.4 hexa-band -wi-fi & wimax [112] hybrid β ω‑indexing based fractal antenna 39.05mm × 32.25mm × 1.6mm fr4 ℇr = 4.4 penta-band 0.5 dbi, 4.32 dbi 14.9 dbi 22.49 dbi 4.51 dbi pcs-1900, rail mobile radio, dcs-imt gap, wcdma, x-band, and ku-band services [113] circular ring-shaped uwb fractal antenna 38mm × 45mm × 1.75mm rogerrt 5880 ℇr = 2.2 penta-band 6 dbi uwb applications emerging perspectives on multiband fractal... 45 copyright ©2025 assa adv. in systems science and appl. 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(2025). circularly polarized hybrid fractal antenna for ku band application, aeu – international journal of electronics and communications, 190, 155641. https://doi.org/10.1016/j.aeue.2024.155641. adv syst sci appl 2024; 02:66–93 published online at https://ijassa.ipu.ru. exploring the relationship between cardiac disease and patterns of 12-lead ecg through neural network: a comprehensive review abu sufiun1, narayan ranjan chakraborty1, shumaiya akter shammi1, sumit kumar banshal2* 1daffodil international university, dhaka, bangladesh 2alliance university, bangalore, india abstract: heart disease is a significant public health concern, affecting a large number of people worldwide every day. with a shortage of qualified cardiologists, particularly in lowincome countries, the diagnosis and management of heart disease can be challenging. the electrocardiogram (ecg) is the primary diagnostic tool for heart disease, but interpreting ecg reports requires the expertise of a qualified cardiologist, making it time-consuming and costly. to address this issue, automated ecg signal interpretation is necessary. hence, this article has made an encyclopedic review of the existing literature. the article includes a demonstration of frequently utilized data sets, tools, and techniques for this domain. therefore, a framework is proposed based on the observation of existing works. the proposed framework aims to improve the analysis of ecg reports for both cardiologists and non-experts. our framework considers the 12-lead ecg, the different types of leads, wave patterns, and their relationship with heart disease. the objective is to produce reliable and accurate results while reducing analysis time. the proposed framework is inherent in improve the diagnosis and management of heart disease by enabling a wider range of healthcare providers and individuals to interpret ecg reports. this could lead to earlier detection and treatment of heart disease, which could improve outcomes and save lives. keywords: cardiac disease, ecg, machine learning, deep learning, heart disease 1. introduction cardiovascular disease (cvd) is a consolidated term for conditions affecting the heart or blood arteries [72]. a science-based organization known as the centers for disease control and prevention (cdc) depicts that the leading cause of death among men, women, and individuals from various racial and ethnic backgrounds in the united states is heart disease. accordingly, the world health organization (who) declares that 17.9 million people die due to heart disease every year. unhealthy and processed food, physical sluggishness, usage of nicotine, and excessive alcohol consumption are the major behaviorally threatening elements for heart disease. these risk factors can lead to elevated levels of blood pressure, blood glucose, blood lipids, and being overweight or obese in individuals [1]. the accumulation of fatty substances known as atheroma in the coronary arteries can cause a blockage or disturbance in the blood flow to the heart muscle, which can lead to the development of coronary heart disease (chd) [101].heart-related illnesses include arrhythmia, myocardial infarction (mi), sometimes known as heart failure, angina, stroke, heart attack, etc. [40,126]. ∗corresponding author: sumitbanshal06@gmail.com exploring the relationship between cardiac disease... 67 the irregularity of the heartbeat known as arrhythmia is linked to a higher risk of blood clots [97, 109, 133]. mi follows when not enough blood reaches a particular area of the heart muscle [59, 101]. the longer it takes for the heart to restore proper blood flow, the greater the harm inflicted on the heart muscle [26]. additionally, coronary artery disease (cad) is the primary cause of heart attacks, while the failure of the heart to effectively pump blood throughout the body is referred to as heart failure, which can result from the heart becoming too stiff or weak [26, 27, 38]. this condition is also known as congestive heart failure (chf) [28, 67]. an ecg is a rapid diagnostic tool that can be utilized to assess the heart’s electrical function and rhythm [114, 150]. in this diagnosis, sensors attached to the skin can detect the electrical impulses that the heart produces with each beat [4, 128]. the signals are recorded by a machine, and a physician evaluates them to determine if there are any irregularities [87]. the 12 ecg leads each reflect a unique 3-d direction of heart action, where leads i, ii, iii, avf, avr, avl, v1, v2, v3, v4, v5, and v6 are the standard ecg leads [19, 23, 91, 106]. however, these leads are classified into two parts: leads i, ii, iii, augmented vector right (avr), augmented vector left (avl), and augmented vector foot (avf) are known as limb leads (figure 1 (a)), and leads v1, v2, v3, v4, v5, and v6 are known as precordial leads (figure 1 (b)) [52, 136, 151]. nowadays, detecting heart problems using ecg has become popular due to its reliability and accurate production of signals [31, 96, 147]. detecting heart disease from ecg signals can be a challenging task for medical professionals due to the time required to understand these signals as well as the expense associated with having qualified experts perform this task. (a) limb leads positions for ecg (b) precordial leads position for ecg fig. 1. leads placement for monitoring ecg signals therefore, the development of an automated system for detecting heart disease from ecg signals may provide a potential solution to this issue. several works have been incorporated to detect various cvds by analyzing ecg signals [20, 99, 124]. a thorough analysis has been conducted on the automated identification of cad through the use of ecg signals [65]. this study employed sixteen entropy measures to detect distinct latent features from ecg signals obtained from patients with cad and healthy individuals. in recent years, various methods such as machine learning (ml), deep learning (dl), and hybrid approaches have been employed for heart disease classification. a review of prior research on the application of dl for ecg diagnosis revealed the use of four standard algorithms: stacked auto-encoders, deep belief network (dbn), convolutional neural network (cnn), and recurrent neural network (rnn) [72]. they conducted a thorough assessment of ecg diagnosis for accomplishing their application, including their advantages and disadvantages. however, most of the research has concentrated on utilizing ecg signals to identify the presence of heart disease [50, 67, 90, 94]. but the working principle of ecg signals and the signal collection procedure of 12 leads of the ecg device are not focal points of the research. therefore, this research aims to incorporate this issue by answering the following research questions: • q1: which data sets are available to analyze heart rate variance? copyright © 2024 assa. adv syst sci appl (2024) 68 a. sufiun, n. r. chakraborty, s. a. shammi, s. k. banshal • q2: what is the importance of the automatic classification of heart diseases, and which approaches are utilized to incorporate this issue? • q3: what is the relation between heart disease and 12 lead ecg mechanisms and how do they help to predict each distinct heart condition? 1.1. inclusion and exclusion criteria in this work, some search strategies are applied to find relevant research in this domain. moreover, this article has analyzed only recent articles to understand the updated and current techniques applied for heart disease detection. in this study, articles that were released between 2019 and 2023 were examined. in addition, we have selected some well-known journals based on ranking and focus on various disease detection, and the medical sector is given more priority to extract the papers. searching criteria: "heart disease" and "ecg" source: ieee transactions on biomedical engineering, heart rhythm, circulation: arrhythmia and electrophysiology, journal of cardiovascular translational research, international journal of cardiology, biomedical signal processing, and control, sensors, pacing and clinical electrophysiology, physiological measurement, journal of medical systems, archives of computational methods in engineering, knowledge-based systems, biocybernetics and biomedical engineering, google scholar year range: 2019-2023 n = 1208 id en tif ic at io n el ig ib ilit y in cl ud ed excluded duplicate records n = 501 excluded records (n = 319) diagnose different disease rather than heart problems full text eligibility n = 198 excluded records (n = 190) core medical-based research, and not related to technology. did not use ecg signals for diagnosis final selection n = 151 excluded records (n = 47) combination of different signals such as ecg and pcg id en tif ic at io n sc re en in g records after removing duplicates n = 707 title and abstract review n = 388 fig. 2. article search strategy the search strategy along with the final list of the articles are illustrated in figure 2. therefore, this study incorporates the popularly utilized data sets and techniques for various cvd. after that, the relation between heart disease and 12 lead ecg mechanisms has also been incorporated in this study. finally, a framework has been developed to suggest an executable approach based on the concomitant literature that is described in the proposed methodology section. 2. frequently used databases data is the fundamental requirement for the detection, analysis, or interpretation of any kind of disease. it is a challenging task to detect disease without any form of information or data. there are several data sets have been built and they are publicly available for disease detection [50, 77, 142, 150]. moreover, some popular data sets are publicly available for the prediction of different heart problems [8, 28, 101]. table 1 illustrates the popular data sets used for the detection of arrhythmia, table 2 refers to the datasets that were used for some copyright © 2024 assa. adv syst sci appl (2024) exploring the relationship between cardiac disease... 69 dangerous disease such as mi, heart failure, etc. and table 3 illustrates the frequently used databases that are utilized for the prediction of several heart problems. table 1. different types of ecg data sets for arrhythmia citation source of data number of disease detectedrecordings [8] mit-bih arrhythmia 47 subjects: 25 males and 22 females, 4000 ecg signal arrhythmia [66] mit-bih arrhythmia n/a arrhythmia [122] mit-bih 47 subjects arrhythmia [28] (mit-bih) arr database, mit-bih normal, sinus rhythm (nsr), and bidmc chf database total 162 records chf, arrhythmia (arr) [95] mit-bih atrial fibrillation database n/a atrial fibrillation (af) [143] mit-bih arrhythmia 29 subjects arrhythmia [97] mit-bih 29 subjects arrhythmia [116] numerical-sultanova, cleveland, ecgphysionet, mit-bih arrhythmia data set, ptb diagnostic ecg database n/a, 1190 people, 18,885 patients, 109446 samples, 14552 samples arrhythmia [83] mit-bih normal sinus rhythm, mitbih arrhythmia, bidmc chf database 18 (5 males, 13 females), 47 (25 males, 22 females), 15 (11 males, 4 females) arrhythmia, chf [38] mit-bih-physionet databases 105 subjects arrhythmia, chf, sudden cardiac death (scd) [22] china physiological signal challenge (cpsc) 2018 data set 6877 recordings 9 categories of arrhythmia [27] mit-bih arr, mitbih nsr, bidmc chf 48 subjects, 18 subjects, 15 subjects chf, arrhythmia [55] ecg data from wearable sensors n/a arrhythmia [109] mit-bih 1800 records arrhythmia [133] mit-bih 48 records 47 from patients arrhythmia [120] mit-bih 47 subjects, 48 recordings arrhythmia [80] mit-bih afdb, cudb, mitdb, mitbih vfdb 23 subjects, 35 subjects, 44 subjects, 22 subjects 6 types of arrhythmia [141] mit-bih 25 subjects af copyright © 2024 assa. adv syst sci appl (2024) 70 a. sufiun, n. r. chakraborty, s. a. shammi, s. k. banshal table 2. a bunch of ecg data sets for prominent heart problems citation source of data number of recordings disease detected [101] ptbdb, mit-bih database 48 records mi [67] rr interval database, bidmc-chf database, nsr-rr, fantasia database, mit-bih, nsr database total 156 subjects heart failure [59] self-developed database (ecg device ‘edan series-3) 1937 patients data covid-19, abnormal heartbeat, mi, previous history of mi, and normal person [53] self-developed 43 patients ihd [104] mit-bih arrhythmia and ptb-ecg databases 360 subjects heart disease [26] ptb diagnostic, bidmc chf, st. petersburg, 236 patient, 15 patient, 7 patient, mi, normal (n), cad, valvular heart disease (vhd) , bundle branch block (bbb), hypertrophic cardiomyopathy (hcm), dilated cardiomyopathy (dcm) [29] ptb database 549 ecg records from 290 subjects mi [138] mit-bih 48 records heartbeats [44] fantasia normal database, european stt database, collected data from ibnal-nafees hospital 40 subjects 40 recordings, 78 subjects 88 recordings, 30 subjects 30 recordings myocardial ischemia [76] cleveland data set 303 records heart disease [93] ptb-xl dataset 21,837 records [152] ptb-xl database 21,837 records cvd [146] st-petersburg, bidmc chf, ptb diagnostic 5 subjects 17 records, 15 subjects 15 records, 52 subjects 80 records cad, chf, mi, normal copyright © 2024 assa. adv syst sci appl (2024) exploring the relationship between cardiac disease... 71 table 3. detecting multiple heart problems using ecg data citation source of data number of recordings disease detected [39] mit-bih, st. petersberg, ptb databases n/a av nodal block (av nb), acute mi, atrial fibrilation (af), cad, earlier mi (emi), healthy, sinus node dysfunction (snd), transient ischemic attack (tia), bbb, cardiomyopathy, dysrhythmia, healthy control, mi, myocarditis, vhd, afib, normal, p, sbr [125] mit/bih-scdh, mit/bih-nsr databases 23 subjects, 18 subjects scd [15] mit-bih arrhythmia database 47 subjects left bundle branch block (lbbb) beat, right bundle branch block (rbbb) beat, pvc beat, ventricular flutter wave beat, nodal (junctional) escape beat, aberrated atrial premature beat, ventricular escape beat, and normal beat [74] mimic-ii 12,000 instances of 942 patients blood pressure (bp) q1: which data sets are available to analyze heart rate variance? one of the most popular data sets regarding heart disease is the massachusetts institute of technology-beth israel hospital (mit-bih) data set [66, 97, 122, 138]. there are various categories of data available in this data set such as the mit-bih arrhythmia data set, mitbih normal sinus rhythm (nsr) data set, mit-bih-physionet databases, mitbih atrial fibrillation database (mit-afdb), mit-bih malignant ventricular ectopy database (mitbih vfdb), mit/bih sudden cardiac death holter (scdh), etc [28,38,80,102,125,143]. among them, the mit-bih arrhythmia data set is the mostly utilized database and this data set is known by several names such as the mit-bih arrhythmia data set, mit-bih arr data set, etc. [15,27,102]. however, it is observed from the existing literature that researchers are more concerned about detecting different types of arrhythmia disease than others [109, 133, 141]. this is why the arrhythmia dataset is popular in this domain for detecting heart problems. additionally, arrhythmia is also referred to as af in some articles because af is a type of arrhythmia [12, 95, 102]. after that, mi, chf, and scd are also predicted in some research using the mit-bih data set [27,38,101]. other heart-related problems such as heart failure, ischemic heart disease (ihd), and abnormal heartbeat are predicted in this field using several popular databases [53, 59, 67]. hence, beth israel deaconess medical center (bidmc) chf data set is employed in some studies to detect heart failure [26–28]. after copyright © 2024 assa. adv syst sci appl (2024) 72 a. sufiun, n. r. chakraborty, s. a. shammi, s. k. banshal that, physikalisch technische bundesanstalt (ptb) diagnostic ecg database is utilized in various literature to detect mi, arrhythmia, etc, [29, 101, 116]. in spite of that, there are some data sets available employed for the detection of heart problems in a few articles. for example, st-petersburg, fantasia database, numerical-sultana, cleveland, creighton university ventricular tachyarrhythmia database (cudb), european st-t database, multiparameter intelligent monitoring in intensive care ii (mimic-ii) waveform database, etc. are utilized in some articles to predict heart disease [44,74,80]. therefore, these data sets are popularly utilized for a combination of detecting several heart diseases. 3. observation of existing approaches q2: what is the importance of the automatic classification of heart diseases, and which approaches are utilized to incorporate this issue? automatic classification of heart disease can help the cardiologist to save their time and they can operate more patients within a short amount of time. not only that, automatic diagnosis of heart problems using ecg signals can also help the patients to acknowledge their condition before affected seriously [8, 101]. therefore, it can also help to diagnose accurately the heart issues since the pre-trained algorithm is trained by the existing database that helps the models to learn the signals specifically. moreover, the available approaches in different domains are depicted in the subsections below. 3.1. deep learning there are several techniques have been utilized to detect different heart problems in many articles such as ml techniques, dl approaches, ensemble methods, hybrid approaches, etc. [8, 29, 143]. among these approaches, some dl algorithms such as cnn, long-short term memory (lstm), cnn-lstm, etc. are widely used in several applications to identify heart illness [28, 66, 122]. cnn is commonly applied in several studies from dl approaches to detect heart diseases [80, 102]. an article introduced a novel neural network architecture based on recent advancements in cnns as a solution to create self-governing systems for diagnosing heart disease using ecg signals [8]. this research employs 1d convolutional layers and the relu activation function, which produces 98.33% accuracy. alternatively, 1d and 2d cnn models with the same activation function are investigated to construct a robust algorithm capable of effectively classifying the ecg signal in the presence of environmental noise [122]. the 1d cnn and 2d cnn have achieved 97.38% and 99.02% accuracy, respectively. another article proposed a method for classifying multiple cardiac illnesses using a one-dimensional cnn with a modified ecg signal as input [39]. they applied their method to three distinct data sets where the st. petersburg data set yielded the best accuracy of 99.71%. moreover, cnn-based hybrid approaches are also popular in this field for classifying heart disease [27, 28, 55, 66, 101, 138]. cnn-lstm is a frequently used algorithm among cnn-based hybrid approaches [50, 67, 95, 143]. an automated detection system is proposed for the detection of mi where cnn, cnn-lstm, and ensemble methods were applied. among them, cnn-lstm and ensemble techniques provided high accuracy of 99.9% [101]. another study suggests an automated diagnosis approach based on deep cnn and lstm architecture (dcnnlstm) for diagnosing chf using ecg signals [67]. this approach has performed similarly to the previous work, 99.52%. in this study, cnn is utilized to extract deep features, while lstm is employed to achieve the goal of detecting chf using the extracted features. however, another cnn-based hybrid approach known as grey wolf optimizer (gwo) artificial bee colony (abc) optimization algorithm (cnngwo-abc) is proposed to detect arrhythmia [55]. the automatic construction of cnn typology using neuro-evolution has been examined in this work. a unique solution based on the abc and the gwo has also been developed. the performance of this algorithm is satisfactory but not excellent as compared to the previous study. it showed 94.27% accuracy which is less than copyright © 2024 assa. adv syst sci appl (2024) exploring the relationship between cardiac disease... 73 the cnn-lstm approaches. another different hybrid strategy is suggested, and it involves a two-stage medical data classification and prediction model [76]. if the results of the initial stage can accurately predict cardiac disease, the second stage may not be necessary. during the first stage, data from medical sensors attached to the patient’s body was categorized, while the second stage involved the classification of ecg images to forecast the likelihood of heart disease. to classify sensor data, a hybrid model using faster r-cnn with se-resnet-101 was used, while for ecg image classification, a hybrid approach utilizing linear discriminant analysis with modified ant lion optimization (hlda-malo) was employed. therefore, the performance of this approach is 98.06% in terms of accuracy. hence, 1d cnn, 2d cnn, and cnn-lstm are commonly used algorithms in this field for detecting various types of heart diseases. in addition, generative adversarial networks (gan) and lstm (ganlstm), convolutional capsule networks, resnet rnns (resrnn), bidirectional long short term memory (bilstm), kernel weight cnn (kwcnn) are also applied in few pieces of literature for heart disease prediction from dl area [22, 26, 104]. the performance of these approaches is good but they do not outperform the other approaches in dl [120,133]. 3.2. machine learning ml based algorithms are also explored in some literature to detect heart illness such as support vector machine (svm), k nearest neighbor (knn), decision tree (dt), etc. [104, 129, 141, 152]. moreover, a deep genetic ensemble of classifiers (dgec) is proposed that consists of three layers where svm is used in every layer [97]. the suggested framework comprises an ensemble of three layers (48 + 4 + 1) consisting of 12 classifiers each from the svm (nu-svc, rbf), knn, pnn, and rbfnn + 4 classifiers from the c-svc and 1 classifier from the c-svc. this method performs with a 99.37% accuracy rate, which is satisfactory. but the effectiveness of the dgec system with additional physiologic signals and the improved method was not examined in this study. however, other svm and fusion svm models are proposed to detect myocardial ischemia, arrhythmia, and chf where they have provided 99.09% and 99.06% accuracy respectively [44, 83]. this study proposes a novel approach for identifying myocardial ischemia using multi-lead long-interval ecg. the method employs choiwilliams time-frequency distribution to detect changes in the st and pr segments of the ecg, which are related to ischemic symptoms, to extract st and pr features [44]. the suggested method is quick, inexpensive, and non-intrusive. moreover, another ml model known as knn has been established to detect mi and it showed 99.96% accuracy by single-channel ecg signal [29]. another study introduced a novel technique for the detection of r-waves and, based on them, the localization of qrs complexes. it was important to evaluate classical classifiers, hence new methods of aggregating ecg signal fragments comprising qrs segments were created. yet, this model’s performance falls short of expectations. it demonstrated a 90.4% accuracy rate for detecting cvd. as a result, using ml algorithms to predict cardiac problems is not widely used. in addition, several different algorithms, including the ridge model, jaya algorithm with red deer algorithm (jrda), ensemble empirical mode decomposition (eemd) with local means (lm) filtering, particle swarm optimization (pso), differential evolution (de), and mdd-net, have been investigated in a few studies [15, 117, 146]. therefore, since ecg signals are one kind of image related data, ml techniques sometimes cannot process them properly and for that reason dl approaches are utilized in this area. 4. correlation between ecg leads and heart diseases q3: what is the relation between heart disease and 12 lead ecg mechanisms and how do they help to predict each distinct heart condition? the 12-lead ecg is vital for detecting and monitoring heart conditions, such as arrhythmia, chd, and electrolyte imbalances [68]. it records the heart’s electrical activity using 10 copyright © 2024 assa. adv syst sci appl (2024) 74 a. sufiun, n. r. chakraborty, s. a. shammi, s. k. banshal electrodes placed on the chest, arms, and legs, generating 12 leads. each lead provides a different view of the heart’s activity and is crucial for identifying specific types of heart disease, such as right ventricular infarction (rvi) in leads v1 and v2, and lateral wall infarction in leads v5 and v6 [127]. the 12-lead ecg plays a crucial role in screening for possible cardiac ischemia and rapidly detecting individuals who may have experienced a heart attack. it’s imperative for healthcare experts to carefully consider the number of leads employed to ensure precise diagnosis and effective treatment [35]. each ecg lead conveys a variety of waveform information, and the ecg waveform itself encompasses several distinct elements, each signifying various phases of the cardiac cycle. these elements encompass the p wave, qrs complex, and t wave, each of which represents distinct waveforms with significant relevance in the interpretation of ecgs. the p wave represents atrial depolarization, the qrs complex represents ventricular depolarization, and the t wave represents ventricular re-polarization. understanding the different types of waves in ecg can help clinicians to diagnose and manage a variety of cardiac conditions. several works have been incorporated for the detection of various heart problems using ecg signals [84, 124, 136]. r to r interval, qrs complex are different portions of an ecg signal and these portions are used for identifying different heart problems [81, 129, 149]. however, the majority of the works utilized rr interval for several heart illnesses such as af, various types of arrhythmias, cad, etc [10, 81]. some works have utilized the qrs complex for incorporating the issue [86, 135, 149]. an article has detected rr interval for af detection using cnn-bilstm [129]. according to earlier clinical investigations, the q, r, and s (qrs complex) are three deflections that reflect a single heartbeat. its timing and structure reveal important details about the heart’s condition. traditional techniques for locating r peaks include wavelet processing, frequency analysis, and digital filters that extract the local maximum value. and r peaks indices have been shown to be important classification indicators for both human and computer-aided categorization. in order to use their model to extract characteristics from pure ecg signals, they would only include r peaks indices in this approach. as a result, they just applied r-r intervals to the original ecg signals in the feature extraction phase to obtain segmentation, and the feature extraction phase will be handled by the model that was used. 0.82 f1 score is achieved by the proposed model in this work. in a different article, the r-peak engzee ecg segmentation technique was used to identify and extract features while recording the position, duration, and quantity of rpeaks [81]. they concentrated on r-r intervals because of the positional invariant nature of cnn layers, the time-dependency of ecg data, and the importance of interval length in ecg interpretation. therefore, cnn architecture can learn the rr interval data rather than the qrs complex. 91.15% accuracy was achieved by the explainable cnn algorithm for the detection of various arrhythmia in this work. on the contrary, the time domain ecg feature based on feed forward neural network (ffnn) and cnn provided 91.5% accuracy for the prediction of arrhythmia using the qrs complex [14]. the only portion of an ecg made up of numerous clustered waves is the qrs complex [30]. the qrs complex consists of q, r, and s waves and signifies ventricular depolarization. after the qrs complex, the t wave denotes ventricular re-polarization. therefore, the qrs complex is utilized for mi detection in research [21]. they stated that a qrs wider than usual is an indication of bbb and ventricular hypertrophy. for that reason, it is easy to recognize mi by increased r wave amplitude, duration, and high voltage qrs. using cnn-bilstm, they achieved 99.62% accuracy. therefore, rr interval and qrs complex both are used for the detection of several heart problems and most of them have utilized the cnn algorithm and cnn-based hybrid algorithms for evaluation purposes the performance is similar to each other for both rr interval and qrs complex. 4.1. p-wave the assessment of p-waves in a 12-lead ecg is a valuable tool for the diagnosis of heart disease [82]. abnormalities in p-wave morphology, duration, and amplitude can copyright © 2024 assa. adv syst sci appl (2024) exploring the relationship between cardiac disease... 75 indicate specific types of heart disease, including atrial enlargement, af, atrial flutter, atrial tachycardia (one kind of arrhythmia), and wpw syndrome. p-wave abnormalities can be detected in leads ii, iii, avf, v1, and v4-6, which are important for the detection of these conditions. the morphology of p-waves in leads ii, iii can detect right atrial enlargement, while leads v1 and v2 can detect left atrial enlargement. hence, irregular p-waves are a hallmark of af. additionally, p wave abnormalities are also associated with other cardiac conditions, such as atrial flutter, atrial tachycardia, and wpw syndrome. to effectively diagnose different heart diseases associated with atrial depolarization abnormalities, it is crucial to conduct a thorough evaluation of p-wave morphology across multiple leads. this is emphasized in the medical literature as well [118]. 4.2. p-r interval the pr interval plays a crucial role in an ecg by showing how electricity travels from the atria to the ventricles of the heart [89]. properly understanding the pr interval’s patterns in a 12-lead ecg is vital for diagnosing heart conditions accurately. specifically, lead ii, lead iii, and avf provide insights into the heart’s lower wall, where pr interval irregularities might signal conduction issues. furthermore, leads v1 to v6 give us a look at the heart’s front, side, and back walls, helping us detect atrial enlargement or fibrillation [68]. remember, the pr interval can be influenced by various heart conditions and medications. this underscores the need for a comprehensive ecg examination to pinpoint the root cause of pr interval variations [88]. often, combining data from multiple leads is necessary for an accurate diagnosis, which is critical for crafting an effective treatment plan [68]. 4.3. qrs complex in a recent research article, we delved into the intricate world of heart disease diagnosis through the examination of the qrs complex within a 12lead ecg system [139]. our focus primarily centered on the significant leads v1 to v6, alongside ii, iii, and avf. the qrs complex itself serves as a mirror to ventricular depolarization, and its fluctuations can serve as critical indicators for a range of cardiac conditions including ventricular hypertrophy, bbb, and mi [62]. when we set our sights on identifying right ventricular hypertrophy, our trusty companions were none other than leads v1 and v2. conversely, left ventricular hypertrophy revealed its presence through the ever-reliable leads v5 and v6. now, for insights into the inferior wall of the heart, leads ii, iii, and avf came into play. these diligent leads held the key to uncovering potential blockages or ischemia through changes in the qrs complex [56]. but that’s not all; the qrs complex’s morphology or shape had a story of its own to tell in the grand scheme of diagnosing heart ailments. a widened qrs complex signaled the presence of a bbb, while a narrow qrs complex hinted at a normal conduction pathway [32]. and let’s not forget about those abnormally deep and wide q waves; they often whispered of a previous mi lurking in the patient’s medical history [105, 108, 134]. it goes without saying that an accurate diagnosis and subsequent treatment plan necessitate a meticulous analysis that takes into account a combination of leads and the distinctive morphology of the qrs complex. 4.4. r-r interval a research paper delves into the effectiveness of a 12-lead ecg system for evaluating the heart’s electrical activity [34]. one crucial element of this system involves the r-r interval waves, which depict the time gap between successive r waves and correlate with ventricular depolarization. variations in the r-r interval can act as signals for various cardiac issues, encompassing tachycardia, bradycardia, and arrhythmias. when dissecting the r-r interval, healthcare professionals typically use lead ii and lead v1 [25], although they may also indicate heart blocks, such as firstdegree av block, second-degree av block, and complete heart block. to precisely pinpoint the heart disease type linked to r-r interval waves, copyright © 2024 assa. adv syst sci appl (2024) 76 a. sufiun, n. r. chakraborty, s. a. shammi, s. k. banshal healthcare providers must meticulously analyze this interval by examining multiple leads. the r-r interval stands as a crucial component of cardiac functioning, facilitating precise diagnosis and treatment of an array of cardiac ailments [6]. in summary, specific leads within a 12-lead ecg system serve a vital role in discerning the heart disease type associated with various waveforms [85]. figure 3 illustrates each waveform’s significant leads. for instance, p waves in leads ii, iii, avl, and v1 can hint at atrial arrhythmias, while q waves in leads i, avl, v5, and v6 may suggest a previous myocardial infarction. t waves in leads v2 to v5, st segment alterations in leads ii, iii, avf, v1 to v6, and u waves in leads v2 to v5 can all point to different cardiac conditions [112]. hence, understanding the importance of each waveform and its corresponding leads plays a pivotal role in identifying the fig. 3. different types of wave-forms prevailing heart disease type and offering suitable treatment. ecgs are invaluable tools for diagnosing various heart conditions, with each type of heart disease manifesting distinct alterations in the different leads of the 12-lead ecg. for instance, cad may produce stsegment depression [131] or t-wave inversion in leads ii, iii, avf, v4-v6, while a heart attack may cause st-segment elevation in leads ii, iii, and avf (inferior mi) or leads v1-v4 (anterior mi). heart failure may exhibit non-specific changes like left ventricular hypertrophy or left bbb [46,49,98]. meanwhile, arrhythmias can produce irregular or abnormal p waves, widened qrs complexes, or absent or abnormal t waves. af may produce an irregular rhythm, absent p waves, and rapid ventricular response. other heart conditions, such as heart valve disease, cardiomyopathy, congenital heart defects, pericarditis, and pulmonary hypertension, also cause different ecg changes [36, 51, 121]. it is important to emphasize that only trained healthcare professionals should interpret ecgs and that ecg changes can vary in different individuals and in different stages of the disease. 5. mapping cardiac territory: anterior, lateral, inferior, and septal leads for heart abnormalities proper placement and interpretation of leads are critical for accurate diagnosis and management of cardiac conditions. anterior wall infarction rarely occurs in isolation and is often associated with infarcts of the septum, lateral wall, or both. the anterior wall is represented by leads v3 and v4 [148]. if both the anterior wall and the septum are affected, the infarct changes will appear in leads v1 to v4, known as an anteroseptal acute mi [61, 107, 145]. in cases where the infarct affects both the anterior and lateral walls (anterolateral ami), changes will appear in v3 to v6 and possibly i and avl. the lateral leads i, avl, v5, v6 are placed on the left side of the chest and are essential in detecting abnormalities in copyright © 2024 assa. adv syst sci appl (2024) exploring the relationship between cardiac disease... 77 (a) cardiac territory i high lateral avr v1 septal v4 anterior ii inferior avl high lateral v2 septal v5 lateral iii inferior avf inferior v3 anterior v6 lateral (b) specification of cardiac territory mapping fig. 4. cardiac territory mapping for early detection of heart abnormalities the left ventricle, such as left ventricular hypertrophy and acute mi [60, 73]. the inferior leads ii, iii, avf are placed on the lower part of the chest and are helpful in detecting abnormalities in the right ventricle and inferior wall of the left ventricle, including rvi [57, 113]. finally, the septal leads v1, v2 are placed on the front of the chest and are crucial in detecting abnormalities in the septum [5], such as septal hypertrophy or septal infarction. the appropriate use and interpretation of these leads shown in figure 4(a) in the 12-lead ecg that can contribute to the accurate diagnosis and management of various cardiac conditions and also the specification for the mapping is illustrated in the figure 4(b). in the field of electrocardiography, specific leads can be used to diagnose and manage different types of mi. the right-sided leads, which include v4r, v5r, and v6r, can show st elevation in a right-side infarct. the posterior leads, v7, v8, and v9, are used to diagnose a posterior acute mi [115]. criteria for rvi include iwmi [2], st segment elevation greater in lead iii than ii, st elevation in v1 (possibly extending to v5 to v6), st depression [144] in v2, and more than 1 mm of st elevation in the right-sided leads (v4r to v6r). most rvis occur in conjunction with inferior wall mi [3]. if st segment elevation is seen in ii, iii, and avf, as well as v1, the most probable explanation is an rvi. the treatment of an rvi is very different from that of a left ventricular infarction, and the diagnostic criteria should be carefully considered in treatment decisions. 6. lead-specific patterns in diagnosing cardiac conditions in general, premature ventricular contractions (pvcs) are best visualized in leads v1 to v3, which are located in the right ventricular outflow tract and the septal region of the heart where pvcs often originate [43, 63, 64, 78, 137]. lead v1 is particularly useful for detecting pvcs because it has a superior view of the right ventricle. pacs (premature atrial contractions) are visualized in lead ii that is one of the most commonly used leads in ecg and can provide valuable information in detecting pacs [9, 11, 37, 100]. pacs are defined as one kind of arrhythmia. additionally, the v1 lead, positioned at the fourth intercostal space on the right side of the sternum, may be helpful in identifying pacs originating from the right atrium. the v2 lead, positioned at the same location on the left side of the sternum, can help identify pacs originating from the left atrium. furthermore, the v4-v6 leads, located on the left side of the chest, can also be useful in detecting pacs originating from the left atrium. rbbb is best visualized in leads v1 and v2, which are located in the right ventricular outflow tract where the right bundle branch is located. rbbb can also be seen in leads v5 and v6, which are located in the left lateral aspect of the heart and may show delayed r-wave progression also help to confirm the diagnosis by showing a ”rabbit ears” pattern in the qrs complex [33, 45, 92]. lbbb is properly envisioned in leads v5 and v6, which are located in the left lateral aspect of the heart where the left bundle branch is located. lbbb is a cardiac condition copyright © 2024 assa. adv syst sci appl (2024) 78 a. sufiun, n. r. chakraborty, s. a. shammi, s. k. banshal characterized by the disruption of the electrical signals that regulate the heart’s pumping function. in the diagnosis of lbbb, v1 and v6 leads are crucial, being the most important on a standard 12-lead ecg. ecg criteria that suggest the presence of lbbb include a qrs duration greater than or equal to 120 ms, broad and monomorphic r waves in leads i, avl, and v6, broad and monomorphic s waves in leads iii and avf, an absence or reduction in the size of q waves in leads v5 and v6, and an rsr’ pattern in v1. these electrocardiographic patterns provide crucial insights into disruptions in the heart’s 17 electrical signals, which are vital for diagnosing various heart conditions [130–134]. medical professionals rely on these criteria to identify conditions like lbbb accurately and offer the appropriate treatment. atrial premature complexes (apcs) become most apparent when looking at leads ii, iii, and avf, which are situated on the lower part of the heart, where the atria reside. apcs might also manifest in other leads [17, 18, 42, 71, 123], like v1 and v2, but they can be trickier to distinguish from other irregularities in those leads. the use of different leads in ecg has been useful in recognizing apbs [24,79,110,140]. lead ii, for example, detects electrical activity between the right arm and left leg and provides a clear image of atria function. similarly, v1 and v2 leads are situated on the right and left sides of the breastbone, and they are capable of detecting atrial premature beats (apbs) from their respective atria. leads v4-v6, which are positioned on the left side of the chest, are also useful in detecting apbs that originate in the left atrium [13, 54, 103]. ventricular ectopic beats (vebs) are abnormal cardiac rhythms that can be diagnosed using an electrocardiogram, a non-invasive diagnostic test that is widely utilized. the ecg gives useful information for predicting vebs [69, 70, 119]. it’s worth mentioning that various ecg leads have a range of sensitivity in identifying vebs. leads v1v3, which are positioned on the chest wall, are more sensitive to detecting vebs coming from the right ventricle, whereas leads v4-v6 are more adept at detecting vebs originating from the left ventricle. lead ii can also identify aberrant electrical activity in the ventricles, making it an effective tool for predicting vebs. a comprehensive study of all ecg leads is required to achieve an accurate diagnosis of vebs. as established in studies, identifying a myocardial infarction (mi) is mainly reliant on specific cardiac areas impacted by decreased blood flow [16, 47, 48, 75]. for example, if the left anterior descending artery (lad), which supplies the anterior wall of the left ventricle, is blocked, you may see st segment elevation, q waves, and t-wave inversion in leads v1v4. conversely, if the blockage occurs in the right coronary artery (rca), which supplies blood to the heart’s inferior wall, leads ii, iii, and avf could display st-segment elevation, q waves, and t-wave inversion. af is best visualize in leads ii, iii, and avf, which are located in the inferior wall of the heart where the atria are located. af can also be seen in other leads, such as v1 and v2, which may show flutter waves or irregular r-r intervals [7, 111]. additionally, leads v5 and v6 may show a rapid ventricular response due to the irregularity of the atrial activity. table 4. best leads for visualization of different heart conditions heart condition best leads for visualization pvcs v1,v2,v3 pacs ii,v1,v2,v4,v5,v6 rbbb v1, v2,v5, v6 lbbb iii,avl,avf,v1,v5,v6 apcs ii, iii, avf,v1, v2 apbs ii,v1,v2,v4,v5,v6 vebs ii,v1,v2,v3,v4,v5,v6 mi ii, iii, avf, v1,v2,v3,v4 af ii, iii, avf,v1, v2, v5, v6 this table 4 presents a comprehensive list of various heart conditions along with the optimal leads for visualizing each of these conditions. the included heart conditions are copyright © 2024 assa. adv syst sci appl (2024) exploring the relationship between cardiac disease... 79 premature ventricular complexes (pvcs), premature atrial complexes (pacs), rbbb, lbbb, apcs, apbs, vebs, mi, and af. by providing the best leads for visualization of each condition, this table can contribute to more accurate diagnoses and effective treatments for these conditions. 7. proposed framework: specific heart disease classification framework based on the existing literature, we have found that the mostly used approach for classifying heart problems using ecg signal is cnn. because, this algorithm is well-known for processing image related data and it is reliable and highest perfomer for predicting heart problems. therefore, a cnn model densenet 201 that is configured using focal loss and adam optimization. the medical sector often deals with imbalanced data sets, where the normal data set exceeds the disease data set. to address this, we adopt focal loss. focal loss is effective for imbalance data set [41]. the adam optimizer performs well with focal loss.the adam technique also works efficiently for the high-dimensional data set [58]. this research aimed to prepare a data set for heart disease prediction. to accomplish this, we combined multiple data sets which are discussed in the data set section. we have also employed a technique to convert one-dimensional ecg signals into two-dimensional ecg images. this conversion aids in reducing the noise of the ecg signals. the conversion is done using ecgkit, where we have transformed the ecg signal waves into image format. next, we split the images into r-r intervals corresponding to one complete cardiac cycle. the resulting images are then stored in separate folders for training and testing, and lead-specific patterns are depicted in figure 5. the ecg wave-to-image generator is used for this conversion, and the heart bit segmentation is accomplished using the ecg-kit with the pantompkins algorithm. finally, we split the data set in 70% for training, 20% for testing and 10% for validation purposes. v1,v2,v3 pvc ii,v1,v2,v4,v5,v6 pac v1,v2,v5,v6 rbb iii,avl,avf,v1,v5,v6 lbb ii, iii, avf,v1,v2 apc ii,v1,v2,v3,v4,v5,v6 veb ii,iii,avf,v1,v2,v3,v4,v5,v6 nor nor pvc pab rbb lbb apc vfw apb ii,v1,v2,v4,v5,v6 apb mi ii, iii, avf,v1,v2,v3,v4 mi ii, iii, avf,v1,v2,v5,v6 af af heartbeat segmentation ecg wave to image generator r r r r r fig. 5. data set preparation copyright © 2024 assa. adv syst sci appl (2024) 80 a. sufiun, n. r. chakraborty, s. a. shammi, s. k. banshal train and test cnn clasifier ecg wave to image generator heartbeat segmentation v1,v2,v3 pvc ii,v1,v2,v4,v5,v6 pac v1,v2,v5,v6 rbb iii,avl,avf,v1,v5,v6 lbb ii,iii,avf,v1,v2 apc ii,v1,v2,v3,v4,v5,v6 veb ii,iii,avf,v1,v2,v3,v4,v5,v6 nor ii,v1,v2,v4,v5,v6 apb ii, iii, avf, v1,v2,v3,v4 mi ii, iii, avf,v1,v2,v5,v6 af nor pac pvc lbb rbb apc apb veb af mi fig. 6. proposed framework utilizing best classifier the ecg-kit is a python-based toolbox that offers a range of tools for the processing and analysis of ecg signals. the toolkit includes functionalities for beat detection, heart rate variability analysis, ecg signal visualization, and ecg signal processing. a noteworthy feature of ecg-kit is its implementation of the pan-tompkins algorithm, a widely used algorithm for detecting the qrs complex in ecg signals. this algorithm utilizes a combination of bandpass filtering, differentiation, squaring, and integration to effectively detect the qrs complex. by leveraging this algorithm, ecg-kit allows users to convert ecg signals into gray scale images, which can be used for further analysis and visualization. in light of the aforementioned background, we suggest a novel dl approach to accurately predict heart disease from ecg signals in real-time scenarios. table 5. ecg report heart disease classification metrics class precision recall f1-score lbb 0.98 0.98 0.98 nor 0.99 0.99 0.99 pac 0.95 0.97 0.96 apc 0.99 0.97 0.98 pvc 0.97 0.98 0.97 rbb 0.99 0.99 0.99 apb 0.96 0.94 0.95 mi 0.97 0.98 0.97 veb 0.98 0.98 0.98 af 0.98 0.97 0.98 accuracy 0.99 macro avg 0.98 0.98 0.98 weighted avg 0.99 0.99 0.99 specifically, our proposed method involves utilizing a cnn architecture densenet-201 to categorize ecg signals into ten distinct classes of heart disease data. to ensure a diverse and comprehensive training data set, we will include unique combinations of lead data for each heart disease class. we use leads v1-v3 for pvcs, leads ii,v1,v2,v4,v5,v6 for pacs, leads v1, v2,v5, v6 for rbbb, leads iii,avl,avf,v1,v5,v6, for lbbb, leads ii, iii, avf,v1, v2 for apcs, leads ii,v1,v2,v4,v5,v6 for apbs, leads ii,v1,v2,v3,v4,v5,v6 for vebs, leads ii, iii, avf, v1-v4 for mi, leads ii, iii, avf,v1, v2, v5, v6 for af, and leads ii, iii, avf, copyright © 2024 assa. adv syst sci appl (2024) exploring the relationship between cardiac disease... 81 v1-v6 for normal (nor). the cnn model will be trained using these segmented images from our proposed data set that precisely classifies each image into its corresponding heart disease class. we will evaluate the proposed model performance using several performance metrics such as precision, accuracy, recall, and f1 score. to demonstrate the effectiveness of our proposed model in real-time scenarios, the proposed model integrates with 12 lead ecg device that produce 12 different types of waveforms. subsequently the ecg signals will be transform into images using an ecg wave to image generator. subsequently, the images will be segmented based on the r-r interval through heartbeats segmentation. moreover, those split ecg images will be processed using the proposed model, and the resulting heart disease predictions will be presented in real-time shown in figure 6. table 6. implemented model comparison with existing works citation dataset used proposed method model architecture accuracy [153] mit-bih high-accuracy ecg heartbeat classification using a cnn 4 convolutional layers and 2 fully connected layers with dropout 99.43% [154] mit-bih deep 2d cnn for ecg arrhythmia classification tae joon jun et al. [2] deep 2d cnn for ecg arrhythmia classification mit-bih arrhythmia database 4 convolutional layers, 2 max-pooling layers, and 3 fully connected layers with dropout and batch normalization 99.05% [155] ptb ecg database deep cnn for automated recognition of mi on ecg signals 5 layers, including convolutional, max pooling, and dropout 99% [156] mit-bih deep cnn 11-layer deep network, normalization using zscore, and smote for imbalance data precision: 98.30% [157] af, vf, st, and normal dataset accurate classification of four ecg patterns using transfer learning transfer learning using pre-trained densenet with 161 convolutional layers and a linear svm 97.23% [122] mit-bih accurate classification of ecg signals using a novel 1d and 2d cnn model novel 1d and 2d cnn model 1d cnn: 97.38% 2d cnn: 99.02% our study leadspecific patterns in diagnosing cardiac conditions dataset neural network for 12lead ecg densenet 201 configured with focal loss and adam optimization 99.57% copyright © 2024 assa. adv syst sci appl (2024) 82 a. sufiun, n. r. chakraborty, s. a. shammi, s. k. banshal however, a major issue encountered in this research was the imbalance in the data set. for example, when considering the lead avf from the 12-lead ecg, it was found that this lead could represent any disease. however, certain classes such as nor, lbb, apc, mi, and af had pictures of avf leads, which were not present in other classes such as pvc, pac, rbb, apb, and veb. this made it difficult for the model to accurately predict diseases that did not have avf lead data. due to the absence of certain types of leads in different types of heart disease classes, the use of 12-lead ecg data as input for the model resulted in data ambiguity. to mitigate this issue, the research team applied a threshold value of 85%. this meant that if the avf signal was determined to be pvc, pac, rbb, apb, or veb with a confidence level below 85%, the prediction would not be made, and the model would discourage misclassification. dealing with unknown data is a challenge in this solution, especially in the sensitive medical sector. a promising result was obtained in our research with densenet 201, achieving an accuracy of 99.57%. the accuracy is assessed using various metrics such as f1 score, precision, and recall shown in table 5. based on the evaluation metrics, the classification model is exhibiting excellent performance. it is achieving high scores for most of the classes, with precision, recall, and f1-score metrics above 0.95 for every class, indicating that the model can accurately classify a substantial portion of instances for each class. furthermore, the precision of our metric is nearly flawless, achieving an impressive score of 0.99. this signifies that the model exhibits exceptional accuracy, effectively categorizing almost every instance with precision. when we examine the macro average of precision, recall, and f1-score, we consistently observe a strong performance at 0.98 across all classes, underscoring the model’s consistent and reliable performance. even the weighted average remains notably high at 0.99, emphasizing the model’s capability to correctly classify instances across a diverse array of classes, maintaining consistently high performance levels. a comparison with the existing models has been incorporated in table 6 that shows the efficiency of our proposed model. in summation, the results presented in this report unequivocally demonstrate the model’s proficiency in accurately classifying instances spanning a wide spectrum of categories, featuring exceptional precision, recall, and f1-score metrics. this compelling performance lends itself to practical utilization in realworld medical applications for the automated classification of ecg reports. 8. conclusion heart disease stands as a pressing global public health concern, with a particularly pronounced impact on low-income countries where the scarcity of qualified cardiologists exacerbates the problem. electrocardiography (ecg) serves as the primary diagnostic tool for heart ailments, yet the process of interpreting ecg reports proves to be both time-consuming and financially burdensome, demanding the expertise of a certified cardiologist. to confront this challenge, the imperative of automated ecg signal interpretation emerges. this article, in response, conducts a comprehensive examination of the existing literature, encompassing popular datasets, tools, and techniques within this domain. noteworthy datasets such as the mit-bih dataset, the ptb database, the bidmc dataset, and the ptb dataset have gained prominence in the realm of heart disease diagnosis. these datasets are available to the public and easily accessible, simplifying their utilization for researchers. furthermore, various methodologies like convolutional neural networks (cnn), long short-term memory (lstm), bidirectional lstm (bilstm), and hybrid models like cnn-lstm and cnnbilstm have emerged as widely adopted approaches to addressing the heart disease detection challenge. in light of these observations, we propose a comprehensive framework that takes into account the 12-lead ecg, diverse lead types, waveform patterns, and their interrelation with heart diseases. this proposed framework holds substantial promise for enhancing the diagnosis and management of heart ailments. it empowers a broader spectrum of healthcare providers and individuals to decipher ecg reports with greater reliability and copyright © 2024 assa. adv syst sci appl (2024) exploring the relationship between cardiac disease... 83 accuracy. this, in turn, facilitates earlier detection and treatment of heart diseases, ultimately leading to improved patient outcomes. furthermore, this study underscores the importance of incorporating various types of leads when developing a cnn model, aiming to simplify the intricacies of detecting unknown patterns in the context of heart disease diagnosis. the proposed framework and 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(2018, october). ecg arrhythmia classification using transfer learning from 2-dimensional deep cnn features. in 2018 ieee biomedical circuits and systems conference (biocas) (pp. 1–4). ieee. copyright © 2024 assa. adv syst sci appl (2024) introduction inclusion and exclusion criteria frequently used databases observation of existing approaches deep learning machine learning correlation between ecg leads and heart diseases p-wave p-r interval qrs complex r-r interval mapping cardiac territory: anterior, lateral, inferior, and septal leads for heart abnormalities lead-specific patterns in diagnosing cardiac conditions proposed framework: specific heart disease classification framework conclusion disclaimer microsoft word 5.1365-article text-5636-1-2-20230113 fin (1) adv syst sci appl 2023; 01; 50-60 published online at https://ijassa.ipu.ru. assessing the fires impact on vegetation cover using remote sensing data: indonesia case study elizaveta grigorets1, anna kurbatova1,2, yaroslav vasyunin3, vasily lobanov1, riri fitri sari4, kseniya mikhaylichenko1, elizaveta anikina1, anastasia kupriyanova1 1) rudn university, moscow, russian federation 2) graphic era (deemed to be) university, dehradun, india 3) paititi research, london, united kingdom 4) university of indonesia, depok, indonesia abstract: wildfires in indonesia have become abnormally frequent due to the human-driven degradation of forest and agricultural lands, as well as climate change. the authors analyze recent studies that provide evidence for an increase in the fire hazard to various ecosystems in indonesia (forests, peatlands, agricultural lands) considering changes in climatic and meteorological parameters of the environment. this work establishes a relationship between burnt areas, measured by moderate resolution imaging spectroradiometer (modis), and the following parameters, retrieved from the reanalysis v5 (era5) ecmwf dataset: monthly precipitation amount, temperature at a height of 2 m above sea level, soil temperature in the upper layer (0 to 7 cm depth), water content in the upper soil layer (0 to 7 cm depth), specific air humidity, zonal wind speed, meridional wind speed, and a standard deviation of precipitation. the authors reveal a correlation and a direct dependence of wildfires on the potential factors influencing the area: air temperature and soil temperature. it is assumed to be associated with the rainfall type, winds (speed, direction, and oscillations), improper land use, and the el niño– southern oscillation. keywords: indonesian region, fires, temperature, precipitation, moisture, climatic zones, remote sensing, modis, era5, google earth engine. 1. introduction according to weekly fire alerts, the peak of the fire season in indonesia typically begins in early august and lasts around 14 weeks. there were 2,303 high-accuracy fire alerts issued by the visible infrared imaging radiometer suite (viirs) from december 2019 to november 2020, and 2205 alerts only in 2020. this value does not exceed the normal compared with previous years going back to 2012. the most fires (32,294 fire alerts) were detected by viirs in 2015 [10]. the fires caused long-term harm to the forestry and agricultural sector, making 0.5% of the gross domestic product (gdp) loss in indonesia in the third quarter of 2019, as estimated by the world bank. in general, it caused reduced gdp rates in indonesia in 2019 and 2020. many factors are contributing to the intensification of wildfires, but it is becoming increasingly apparent that warmer and drier conditions play a large role in exacerbating heat, creating conditions for destructive wildfires. the loss of moisture by the ground cover during evaporation and transpiration occurs under the complex influence of several meteorological factors. the combustibility of forests with high accuracy can be defined only considering all these factors [2]. in addition, there are seasons with different precipitation, air temperature, wind force, and air humidity patterns, which affect the moisture content of the combustible material. among the main factors influencing the intensity of wildfires, there is terrain morphometry (aspect and slope  corresponding author: 5977749@mail.ru. assessing the fires impact on vegetation cover… 51 copyright ©2023 assa adv. in systems science and appl. (2023) steepness), which affects moisture level in the forest floor by influencing the speed of forest combustible materials drying [22]. the fire rate of the ground cover will be different at the same air temperature or the same moisture deficit if this is preceded by a period of dry and hot weather or by a period of low temperatures and precipitation. it matters in what form the precipitation falls, whether it is not intense but prolonged, or has a stormy character. thus, the possibility of the ground cover burning is formed due to the complex effect of many factors, and the influence of these factors must be considered not only by their current state but also by the previous period. the research [14] defines that peat soil with a water content of less than 115% will have a high fire risk, between 115 and 135% – a moderate fire risk, and more than 135% – a low fire risk. active anthropogenic logging actively affects the intensity of forest fires in indonesia [21]. deeper degraded peatlands and mixed agricultural lands are the most vulnerable areas, and anthropogenic activity is suggested to be a strong driver of combustion [26]. the main reasons for deforestation in indonesia are selective logging and conversion to industrial oil palm and pulpwood plantations. industrial plantations were causing over half of indonesia's deforestation until 2016 and then it dropped down to 15%. small-scale farming was a driver for more than onequarter of all deforestation for two years in 2014-2016. also, forest fires converted 20% of forests into grass and shrubland lands in 2015 [24]. improper land use changes the soil water regime, which aggravates soil aridity and makes an ecosystem more vulnerable to fire. one of the impacts resulting from the conversion of peatlands from forest to non-forest lands is drought. it is due to the draining of land for agricultural cultivation purposes [14] because of the increasing bulk density that influences soil moisture. according to [1], the fire frequency is higher in areas where secondary peat swamp forests were transformed into shrubs or plantations (0.15 km−2 × yr−1). in the [29] authors conducted research on the fire frequency in indonesia's two largest peatland regions, sumatra and kalimantan, during 2001-2018. they explored relationships between burning and land-cover types using modis and landsat satellite data. peatlands of these two regions burned five times more than other land covers despite covering less area which could lead to indonesia's peatlands disappearing in the coming decades. moreover, the number of identified hot spots is highly dependent on population density; according to [3] high population density reduces the number and probability of a large number of hot spots, which is associated with quick measures to eliminate them. the groundwater table can be a good indicator for risk zone mapping of peat fire in the peatland area. according to regulations in indonesia, peatland with a groundwater table of more than 40 cm is defined as a degraded peatland area and has a high potential to fire [30]. other analyses have also linked water table depth to a fire risk, prompting the indonesian government to impose regulations regarding drainage based on an allowable water table depth of 40 cm [31]. a strong relationship between burned areas and preceding soil moisture makes the remote sensing of soil moisture a valuable source of information for fire forecasting models and identifying fire-vulnerable locations [5]. the intensity of fires also depends on the type of plant communities and soil profile [26]. the largest carbon loss (94.2 t/ha) occurs on a secondary peat swamp forest and is equivalent to the emission of 345.4 t co2eq. the second-largest carbon loss (36.3 t/ha) occurs in secondary dryland forests followed by forest plantation (18.5 t/ha) and bushes swamp (13.5 t/ha). the most extensive carbon loss on secondary peat swamp forest and forest plantation occurs on the aboveground biomass pool, while secondary dry forest and bushes swamp occur on the deadwood pool. interannual variability in precipitation associated with the southern oscillation of el niño (enso) has been closely linked to fire emissions frequency and duration. from a comparison of fire activities in several areas on both indonesian islands, it is evident that the most severe peat fires occur in the southern part of central kalimantan due to the relatively long dry season (of more than 3 months under enso) [12]. the latest studies show no causal relationship between enso and the intensity of forest fires in indonesia. until 2018, researchers explained fires in 52 e. grigorets et al. copyright ©2023 assa adv. in systems science and appl. (2023) indonesia's peatlands with the enso. however, fires still occurred independently of enso in an area of about 16 × 103 km2 in 2019. therefore, more suitable indicators and methods are required to analyze, evaluate, and forecast peatland fires [13]. fig. 1.1. wildfires effects on forest ecosystems in indonesia [17] 1. study area as the object of study, we chose the plant communities assigned to three climatic zones: broadleaf evergreen forests (in particular, tropical evergreen forests), broadleaf deciduous forests (especially, deciduous monsoon forests), swamp forests, mangrove and nipa palms, alpine vegetation, grassland and cultivated area of indonesia. to better understand the patterns of precipitation preceding and coinciding with the fire season, we based our analysis on a slightly modified division of the three climatic regions (zones) proposed by [15]. the territory of indonesia can be grouped into three climatic zones: – zone 1 – monsoon – has a u-shape. in this case, the rainy season and dry seasons can be clearly differentiated, i.e., november-april for the rainy season, and may-october for the dry season. precipitation rates are not significantly exceeded over time. zone 1 includes east java, bali, nusa tenggara, parts of sulawesi, riau, south sumatra, aceh, all of maluku, parts of east kalimantan, and south kalimantan; – zone 2 – anti-monsoon – has almost the same pattern as the monsoon type, however, the rainfall in the dry season is higher than in the monsoon type. zone 2 includes the areas of sulawesi and papua; – zone 3 – semi-monsoonal – monthly precipitation pattern is very similar to the antimonsoon pattern during rainy seasons and slightly different during dry periods. zone 3 includes most of the areas in sumatra, west java, central java, jakarta, banten, yogyakarta, and northern kalimantan. consequently, the division of the studied area into three clusters allows the most complete tracing of the fire severity on plant formations depending on climatic characteristics. w ild fi re e ff ec ts reduction in population sizes forest edge effects changes in the community structure immigration of exotic species peat subsidence generate changes in soil properties depending on the type of soil amount and quality of fuel duration of the fire temperature changes in soil chemistry: cincentration of nutrients (c, p, ca, mg, na, k) changes the spartial structure and mode of functioning of forest ecosystems assessing the fires impact on vegetation cover… 53 copyright ©2023 assa adv. in systems science and appl. (2023) 2. data and methodology based on information about the distribution of precipitation [15], a detailed mapping of climatic zones was carried out. the source of the initial data was the era5 climate dataset from the european center for medium-range weather forecasts (ecmwf), available since 1981 with a grid spacing of 0.25° (approximately 12 km) [7]. era5 contains total monthly precipitation: accumulated liquid and frozen water, including rain and snow falling on the earth's surface, and does not include fog, dew, or precipitation that evaporates in the atmosphere before reaching the surface. the earth engine cloud platform was used to process the era5 data and to carry out the geospatial data analysis [11]. to identify three climatic zones in indonesia, the authors carried out preliminary data processing and its subsequent clustering using the following code [28]. as part of the preliminary processing, 12 maps were obtained – the average values of total precipitation for each month for the period 2000-2020. as a result of the «reduce» procedure (an ordered procedure for mixing multiple sources of information), these maps were reduced to a raster image consisting of several information channels. each pixel of the generated image contains the following information at a given point for the years 2000-2020: – absolute precipitation minimum – absolute precipitation maximum – month’s ordinal number when an absolute maximum is registered – month’s ordinal number when an absolute minimum is registered – average precipitation – median precipitation – standard deviation of precipitation further, era5 data was used to obtain datasets in geotiff format for each of three zones with the following parameters: – total monthly precipitation – temperature at a height of 2 m above sea level – temperature of the soil in the upper layer (depth from 0 to 7 cm) – water content in the upper soil layer (depth from 0 to 7 cm) – specific air humidity – zonal wind speed – meridian wind speed. obtaining the values of these parameters using the era5 dataset was carried out by directly downloading data from the site [6], converting the data into geotiff format, and obtaining raster images containing the parameter value in each pixel. also, data from the modis instrument, onboard the terra and aqua satellites, was used. modis provides multispectral detection capabilities in 36 spectral bands ranging in wavelength from 400 nm to 14400 nm. each platform delivers daily coverage of the entire globe. it offers advantages over other data sources for studying the occurrence and extent of wildfires by providing two essential fire products: active fire detections and burned area estimates. both are widely used as data sources in many large-scale analyses of wildfire activity and environmental impacts, climate change scenario simulations, and vegetation response projections [16,17,27,32]. the next parameters were used based on modis products: – land cover classification based on mcd12q1 (university of maryland classification, usa) [8,19], including the following land cover types: a) evergreen broadleaf forests (value 2); b) woody savannas (value 8); c) savannas (value 9); d) grasslands (value 10); e) permanent wetlands (value 11); f) croplands (value 12); g) urban and built-up lands (value 13); 54 e. grigorets et al. copyright ©2023 assa adv. in systems science and appl. (2023) h) cropland/natural vegetation mosaics (value 14); – burned area estimations based on mcd64a1 [29,30]; – assessment of net primary production based on mod17a3 [31,32]. in this work, for each of the three zones, the parametric pearson correlation coefficient was calculated using spss statistics software. pearson's correlation criterion is a parametric statistics method that allows determining the presence or absence of a linear relationship between two quantitative indicators and assessing their closeness and statistical significance. the coefficient was calculated between the parameter representing the burnt area of forest fires and the factors affecting the intensity of the pyrogenic factor: humidity, soil temperature (0-7 cm), air temperature (2 m above the ground), horizontal and vertical components of wind speed (at a height of 10 m), and specific air humidity. to determine the closeness of the relationship, the coefficient of determination was calculated. 3. results and discussion firstly, three climatic zones (fig. 3.1) were identified using the commonly used k-means unsupervised classification method [4]. in this case, the iterative algorithm divides a set of pixels into three zones, determining their centers based on the raster data bands, and assigning each pixel the number of the closest zone. fig. 3.1. distribution of zones according to the values of average annual precipitation in indonesia: – green area – monsoon type: pmin=8.5 mm in july, pmax=39 mm in september, pav=20 mm, pmed=19 mm; – yellow area – semi-monsoonal type: pmin=1.2 mm in august, pmax=21.6 mm in october, pav=7.8 mm, pmed=7.5 mm; – pink area – anti-monsoon type: pmin=0.1 mm in october, pmax=18.9 mm in march, pav=7 mm, pmed=7 mm. secondly, for the three zones obtained by clustering by the k-means method, the time series of the fire area's dependence on climatic and meteorological parameters were obtained. the studied parameters were described for each cell with spatial coordinates for each month in the period 20002019. the result of calculating the pearson coefficient showed the highest correlation between the burnt area and air temperature for the monsoon zone, as well as between soil temperature and wind speed. the drainage greatly increases fire risk in peatlands, which lowers the water table, exposing more dry peat to combustion [14]. it can be explained by the fact that during the years of frequent wildfires in northern sumatra in the monsoon zone, the most active fires occurred at the beginning of the dry season when air and soil temperatures are the highest. assessing the fires impact on vegetation cover… 55 copyright ©2023 assa adv. in systems science and appl. (2023) during the period 2000-2015, an average of 75% of all active fires occurring in northern sumatra were in peatlands, with the highest numbers of 87% and 88% in 2005 and 2014, respectively. north sumatra, with approximately 42 000 km2 of peatlands, had the highest proportion of active fires observed on peatlands – 75% – compared to 61% in south sumatra. although south kalimantan has the largest peatland area with over 500 000 km2, the number of active peat fires in this region accounted for 50% of all active fires, which is about 100% of all of indonesia. according to research, it was determined that 120 days of precipitation accumulation before an active fire was the best-correlated time for all fires in south sumatra and kalimantan. in the north of sumatra, this period was less than 30 days. this is partly because the dry season is shorter there. thus, the dry season determines the temperature of the soil and air, which affects the frequency and intensity of fires. a similar trend is observed for the antimonsoon zone, which is due to a combination of the following factors: soil moisture associated with rainfall, improper land use, and the sensitivity of three zones to enso. el niño combined with walker reverse circulation leads to drought in most of indonesia, while la niña events lead to an increase in rainfall. the correlation analysis between the fire area and the temperature at the level of 2 m above the ground showed the highest correlation in zones 2 and 3 for land cover type 2 – evergreen broadleaf forests. it is important to note that for zone 3, in almost 100% of cases, high correlation values are exclusively in this land cover type, while for zone 2, the maximum correlation values are typical for land cover type 8 – woody savannas. zones 1 and 2 also demonstrate less significant correlation values for these land cover types: 9 – savannas, 11 – permanent wetlands, and 14 – cropland/natural vegetation mosaics (fig. 3.2a). a similar situation is for the correlation analysis between the burnt area and the temperature of the soil in the upper layer (depth from 0 to 7 cm), which is associated with a direct dependence of this parameter on the temperature at the level of 2 m above the ground (fig. 3.2b). a) b) fig. 3.2. the distribution of significant correlations for the temperature at a level of 2 m above the ground t (a) and for the temperature of the soil in the upper layer ts (b) by land cover types for zones 1 (z1), 2 (z2), 3 (z3), where: 2 – evergreen broadleaf forests; 8 – woody savannas; 9 – savannas; 10 – grasslands; 11 – permanent wetlands; 12 – croplands; 13 – urban and built-up lands; 14 – cropland/natural vegetation mosaics the correlation analysis between the fire area and the water content in the upper soil layer showed the highest correlation for land cover type 2 – evergreen broadleaf forests – in zone 2, while zone 3 leads in the number of significant correlations for this land cover type (fig.3.3). zone 1, in addition to high correlation values for land cover type 2 – evergreen broadleaf forests, has significant values for land cover type 8 – woody savannas, 14 – cropland/natural vegetation mosaics, 9 – savannas, 12 – croplands, in descending order. 0 20 40 60 80 100 2 8 9 10 11 12 13 14 % land cover type t_z1 t_z2 t_z3 0 20 40 60 80 100 2 8 9 10 11 12 13 14 % land cover type ts_z1 ts_z2 ts_z3 56 e. grigorets et al. copyright ©2023 assa adv. in systems science and appl. (2023) fig. 3.3. distribution of significant correlations for the water content in the upper soil layer sz by land cover types for zones 1 (z1), 2 (z2), 3 (z3), where: 2 – evergreen broadleaf forests; 8 – woody savannas; 9 – savannas; 10 – grasslands; 11 – permanent wetlands; 12 – croplands; 13 – urban and built-up lands; 14 – cropland/natural vegetation mosaics the correlation analysis between the fire area and the specific air humidity (fig. 3.4) showed the highest correlation for zones 2, 3, and land cover type 2 – evergreen broadleaf forests. in zone 1, a significant correlation is for land cover type 2 – evergreen broadleaf forests, as well as for land cover type 8 – woody savannas, 14 cropland/natural vegetation mosaics, 9 – savannas, in descending order. fig. 3.4. distribution of significant correlations for the specific air humidity h by land cover types for zones 1 (z1), 2 (z2), 3 (z3), where: 2 – evergreen broadleaf forests; 8 – woody savannas; 9 – savannas; 10 – grasslands; 11 – permanent wetlands; 12 – croplands; 13 – urban and built-up lands; 14 – cropland/natural vegetation mosaics the correlation analysis between the fire area and the zonal wind speed (fig. 3.5) showed the highest correlation for zone 3 and land cover type 2– evergreen broadleaf forests, then for zone 2 and 1 for the same one. in addition, in zone 1, there is a significant correlation for land cover type 8 – woody savannas. the situation is similar for zone 2. fig. 3.5. distribution of significant correlations for the zonal wind speed windu by land cover types for zones 1 (z1), 2 (z2), 3 (z3), where: 2 – evergreen broadleaf forests; 8 – woody savannas; 9 – savannas; 10 – grasslands; 11 – permanent wetlands; 12 – croplands; 13 – urban and built-up lands; 14 – cropland/natural vegetation mosaics 0 20 40 60 80 100 2 8 9 10 11 12 13 14 % land cover type sw_z1 sw_z2 sw_z3 0 20 40 60 80 100 2 8 9 10 11 12 13 14 % land cover type h_z1 h_z2 h_z3 0 20 40 60 80 100 2 8 9 10 11 12 13 14 % land cover type windu_z1 windu_z2 windu_z3 assessing the fires impact on vegetation cover… 57 copyright ©2023 assa adv. in systems science and appl. (2023) the correlation analysis between the burnt area and the meridian wind speed (fig. 3.6) showed the greatest number of significant correlations in zone 3 for the land cover type 2 – evergreen broadleaf forests. for zone 1, a higher correlation value is for land cover type 8 – woody savannas than for type 2 – evergreen broadleaf forests. fig. 3.6. distribution of significant correlations for the meridian wind speed windv by land cover types for zones 1 (z1), 2 (z2), 3 (z3), where: 2 – evergreen broadleaf forests; 8 – woody savannas; 9 – savannas; 10 – grasslands; 11 – permanent wetlands; 12 – croplands; 13 – urban and built-up lands; 14 – cropland/natural vegetation mosaics thus, the analysis showed that predominantly high correlation values between the burnt area and all parameters in all zones are typical for land cover type 2 – evergreen broadleaf forests, the dominant land cover type in indonesia. at the same time, zone 1 is characterized by the largest scatter of significant correlations for such land cover types as 8 – woody savannas, 14 – cropland/natural vegetation mosaics, 9 – savannas, 12 – croplands, and 11 – permanent wetlands, in descending order. zone 3 has almost 100% high correlation values exclusively for land cover type 2 – evergreen broadleaf forests, which is prevailing for this zone, with some inclusions in the statistics of land cover types 8 – woody savannas and 11 – permanent wetlands. zone 2 is intermediate between zones 1 and 3, therefore, it is characterized by predominantly high correlation values for land cover type 2 – evergreen broadleaf forests – and a slight scatter of high correlation values for land cover types 8 – woody savannas, 14 – cropland/natural vegetation mosaics, 9 – savannas, 12 – croplands, and 11 – permanent wetlands, but in a much lower percentage than for zone 1. 4. conclusion 1. the clustering of terrestrial plant communities in indonesia by the k-means method was carried out based on the data on total precipitation from the era5 monthly climate data set. as a result, three zones were identified: monsoon, semi-monsoon, and anti-monsoon. 2. a database was formed using the google earth engine platform to conduct a multivariate statistical analysis on the following parameters for the period 2000-2020: soil moisture and temperature (0-7 cm), air temperature (2 m above the ground), horizontal and vertical components of wind speed (at 10 m height), specific air humidity, and burnt area. 3. the linear pearson coefficient is calculated between the burnt area and the potential factors influencing the area of the fire for each zone and it was found that the burnt area for all three zones has a direct positive relationship with soil temperature at a depth of 0 to 7 cm. the air temperature affects the fire hazard through the lack of moisture, being the main reason for the loss of the ground cover. on the one hand, the increase in air and soil temperature leads to raising water absorption by plants and increments the ground cover drying. on the other hand, soil heating contributes to an increase in fire hazards. the wind accelerates the evaporation of moisture, promotes more rapid drying of the soil, increases its fire hazard, provides the combustion zones with new portions of oxygen, increases the intensity of combustion at the frontal edge of the fire, leads to the formation of new combustion centers, and, under favorable conditions, can contribute to the transition of strong ground fires to crown fires. 0 20 40 60 80 100 2 8 9 10 11 12 13 14 % land cover type windv_z1 windv_z2 windv_z3 58 e. grigorets et al. copyright ©2023 assa adv. in systems science and appl. (2023) 4. it was revealed that soil temperature is closely related to its moisture. changes in the soil cover and land use of the area significantly affect the runoff characteristics of the drainage basin, which affects the availability of surface and groundwater. therefore, soil moisture is associated with rainfall, improper land use, as well as the sensitivity of all three zones to enso. el niño combined with the walker reverse circulation leads to drought in much of indonesia, while la niña events lead to increased rainfall. 5. in arid conditions, fires are mainly caused by the lack of precipitation, and the additional effect of temperature anomalies does not have a significant impact, despite recent trends in their increase. at the same time, in a humid and warm environment, compared to wet and cool conditions, there is a higher rate of evapotranspiration, which contributes to an increased likelihood of fire. here, high temperatures lead to a deficit in vapor pressure in the atmosphere and evapotranspiration rates, and regardless of changes in rainfall, a warming trend can cause water stress in plants and increase susceptibility to fires. this has important implications for indonesia as atmospheric temperatures rise due to global climate change. acknowledgements this publication has been supported by the rudn university scientific projects grant system, project № 202704-2-074. references [1] adrianto, h. a., spracklen, d. v., arnold, s. r., sitanggang, i. s. & syaufina, l. 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(2004). improvements of the modis terrestrial gross and net primary production global data set, remote sens environ., 95(2), 164–176, doi:10.1016/j.rse.2004.12.011 microsoft word 1398_done adv syst sci appl 2023; 02; 152-163 published online at https://ijassa.ipu.ru. determining sst aerodynamic configuration and power plant parameters under epistemic uncertainty georgy veresnikov1*, igor bashkirov2, sergey gorchakov 2 1) v. a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia 2) central aerohydrodynamic institute named after professor n.e. zhukovsky, zhukovsky, russia abstract: the paper considers the problem of determining parameters of the aerodynamic configuration and power plant of an advanced supersonic passenger transport (sst) at the stage of preliminary aerodynamic design under epistemic uncertainty associated with incomplete information about the initial data. optimization models and algorithms based on them are proposed that operate with the designed sst initial parameters generated by experts on the basis of empirical prediction. such parameters are proposed to be generated within liu’s uncertainty theory as uncertain quantities expressed by uncertainty distribution functions. the use of uncertainty theory will make it possible to formalize and perform aerodynamic design process by replacing the functions that depend on uncertain quantities with their numerical characteristics. such numerical characteristics are effectively interpreted by the decision maker, since they have analogues in probability theory – expected value, quantile, variance. the use of uncertainty theory in solving optimization problems under uncertainty provides low computational costs compared to the theory of probability. the paper discusses the use of numerical methods in the proposed algorithms, since, additionally, it is required to solve the black box function optimization problem. this is due to the lack of simple analytical relations between the sst requirements and the sst aerodynamic configuration and power plant parameters. the adequacy of the developed algorithms is demonstrated by the aerodynamic predictions presented by the pareto fronts of the objective functions, which allow choosing trade-off design solutions. keywords: supersonic passenger transport, aerodynamic design, parametric uncertainty, multiobjective optimization. 1. introduction to determine the parameters of the advanced sst aerodynamic configuration and power plant at the stage of preliminary aerodynamic design is an important task, the successful solution of which largely influences the efficiency of its operation, life cycle cost and competitiveness. a characteristic feature of the early stage of the advanced sst design is considered to be the problem of insufficiency or lack of aerodynamic requirements and initial data. therefore, the use of classical (deterministic) calculation models may be inefficient. to generate the initial data of the preliminary aerodynamic design, experts are involved, operating with subjective estimates of parameter values. therefore, in the sst preliminary aerodynamic design, it is advisable to use mathematical models that make it possible to carry out calculations under parameter uncertainty (indeterminacy). parametric uncertainty is related to the problem of obtaining the exact values of the parameters that are required when using classical deterministic mathematical models. in this case, there are two types of uncertainty reflecting the parameter non-determination aleatory (objective) and epistemic (subjective). within the problem of determining sst parameters (design solutions), aleatory uncertainty occurs when the information about stochastic (random) * corresponding author: veresnikov@mail.ru determining sst aerodynamic configuration and power plant parameters… 153 copyright ©2023 assa. adv. in systems science and appl. (2023) parameters is contained in statistical data, and the parameters are modeled by random variables with the probability distributions determined on the basis of these data (statistical modeling). epistemic uncertainty arises when statistical data are insufficient or not available, and the information about the initial parameters of the sst aerodynamic design is generated by experts. when solving aerodynamic design problems under uncertainty, it is known to use probability theory [1, 2], interval methods [3, 4], possibility theory [5, 6], fuzzy set theory [7, 8], uncertainty theory [9, 10]. a high interest in applying the theories designed to perform "smart"/"soft" computing is usually due to the lack of statistical data. in papers related to solving design problems under uncertainty, with regard to possible constraints, the optimization and verification of the objective function properties, being considered as numerical characteristics of the distribution of non-deterministic model values, are performed. this is explained by the fact that, depending on the theory chosen to represent the aircraft parameters, the result of calculating a function that depends on non-deterministic quantities is a random variable [11], an interval [12], a fuzzy number [13], a possibility value [14], an uncertain quantity [15], which cannot be directly used in optimization calculations. therefore, optimization models are developed, in which the functions that depend on non-deterministic parameters are replaced by their numerical characteristics. the main goals of developing optimization models, in which objective functions and constraints are replaced by their numerical characteristics [16], are to provide: • the least sensitivity of objective functions to possible changes in the optimized and input parameters, i.e. objective robustness; • feasibility of solutions for possible changes in the optimized and input parameters of constraints, i.e. feasibility robustness. the tasks of preliminary aerodynamic design are complicated by the need to solve the black box function optimization problem that occurs when the structure of the objective functions/constraints is hidden or complex. the technique of determining the advanced sst aerodynamic configuration and power plant parameters under uncertainty considered in the paper is based on the deterministic approach to developing the aerodynamic configuration of a supersonic cruise aircraft [17], which includes iterative procedures for numerically solving a system of nonlinear equations. this does not allow using optimization models with analytical relations. traditional methods of solving such problems under uncertainty are based on the use of probability theory. in the general case, the calculation of objective functions and the verification of constraints that depend on a set of random parameters are associated with the calculation of multidimensional integrals. for example, in papers devoted to solving reliability-based design optimization (rbdo) problems [18], in which the main attention is paid to the fulfillment of constraints under parametric uncertainty ensuring robust feasibility, many methods are proposed for calculating a multidimensional integral that determines the probability of fulfilling the constraints dependent on random parameters. the use of statistical modeling methods (monte carlo method) and its modifications, which reduce the required number of calculations to obtain the probability of fulfilling stochastic constraints, does not provide sufficient computational efficiency in many applied problems [19, 20]. in [21], as an alternative approach to solving this problem, it is proposed to calculate the multidimensional integral by numerical methods, but it is noted that, in practice, the possibilities of numerical integration are limited by a relatively small number of measurements (5-6 parameters). a lot of research is devoted to the form (first-order reliability method) [22] and sorm (second-order reliability method) [23] approximation methods, which allow reducing computational costs when calculating the probability of fulfilling stochastic constraints. the sorm usually provides higher accuracy than form, since the first and second order derivatives are used for approximation. when solving problems of preliminary aerodynamic design, it is proposed to use the theory of uncertainty [15] to build optimization models. this approach allows one to determine the parameters of the sst aerodynamic configuration and power plant, when the information about 154 g. veresnikov, i. bashkirov copyright ©2023 assa adv. in systems science and appl. (2023) non-deterministic parameters (uncertain parameters) of the objective functions and constraints is generated by experts. in the theory of uncertainty, based on a strictly described axiomatics, the measure of uncertainty m(•) is introduced as the expert's confidence level that the event • will occur. the uncertain parameters are given by uncertainty distribution functions φξ (x) = m { ξ ≤ x} (φξ -1 is the inverse function), where x is the value of the uncertain variable ξ. appendix provides the fundamental concepts of uncertainty theory used in the paper. it will be shown below that uncertainty theory can be effectively applied in aerodynamic design optimization involving black box function optimization and at the same time provide low computational costs. an example of solving the preliminary aerodynamic design problem is given – the aerodynamic configuration and power plant parametrization, which provides the implementation of the required sst performance with the initial data of non-deterministic nature. 2. methods and algorithms the problem of the sst aerodynamic configuration and power plant parametrization under uncertainty is proposed to be formalized and solved using the flow chart shown in fig. 1. fig. 1. flow chart of the aerodynamic design problem formalization and solution based on input parameters of the black-box model, sets of parameters are formed that are input and optimized for the optimization model with numerical characteristics of objective functions and constraints. the parameters within these sets are divided into deterministic and non-deterministic. for non-deterministic parameters, experts form the uncertainty distribution functions. as a result of determining the objective functions and their numerical characteristics, optimization models are generated that reflect the technical requirements for the designed object. these models can be generally represented as follows [9]: determining sst aerodynamic configuration and power plant parameters… 155 copyright ©2023 assa. adv. in systems science and appl. (2023) ⎩ ⎪⎪ ⎨ ⎪⎪ ⎧ min  x̄' /max d1 f1 x̄' , ξ ,…,dm fm x̄' , ξ , m(gj(x̄ ' , ξ)≤0)≥αgj , j=1, …, p, m((xz + δz) ≥ az) ≥αaz , m((xz + δz) ≤ bz) ≥αbz , z=1, ..., k. where di is the set of numerical characteristics of the objective function fi(x̄ ', ξ), i=1, ..., m; gj(x̄ ', ξ) is the constraint function; x̄'=(x1 ' , …, xk ' ) is the vector of optimized deterministic and uncertain hst (high speed transport) parameters, x1 ' =x1 + δ1, …, xk ' =xk + δk; ξ is the vector of input uncertain parameters; m, p is the number of objective functions and constraints; m(•) is the measure of uncertainty (level of confidence) of the event occurrence •; αgj is the level of the uncertainty measure (level of confidence) given by the decision maker to fulfill the j-th constraint, 0 ≤ αgj ≤1, az and bz are the bounds of the optimized parameters; αaz and αbz are the levels of the measure of uncertainty used to control the bounds by the optimized uncertain parameter xz ' , 0≤αaz ≤1, 0≤αbz ≤1. the generated optimization models are used to synthesize an algorithm that allows one to get a pareto front to analyze and choose trade-off design solutions. the flow chart of this algorithm is shown in fig. 2. fig. 2. generalized flow chart of the synthesized algorithm for solving the aerodynamic optimization problem to test the method for solving the preliminary aerodynamic design problems under uncertainty, the deterministic approach to determining parameters of the aerodynamic 156 g. veresnikov, i. bashkirov copyright ©2023 assa adv. in systems science and appl. (2023) configuration and power plant of a supersonic aircraft [26] was chosen as the basis, the implementation of which is used in the paper as a black-box model. the method is based on a combination of the breguet parameter analytical optimization mcr·(l/d)cr/ce, where l – lift, d – drag, mcr is the cruise mach number, (l/d)cr is the lift-todrag ratio at cruise conditions, ce is the cruise specific fuel consumption coefficient, and the numerical solution of the aerodynamic inverse problem – the transition from the flight performance to the aerodynamic configuration and power plant parameters. the breguet parameter optimization is carried out under the assumptions adopted in the supersonic aircraft aerodynamic calculation: • the number of mcr is assumed to be constant, given in the requirements, • with an increase in cruise altitude hcr 11 km the thrust decreases proportionally to air density, i.e. p(hcr)/p(11 km.) =  (hcr)/ (11 km.), at a constant coefficient of specific fuel consumption ce(hcr) = ce(11 km), • the aerodynamic polar is symmetrical quadratic and is expressed as cd = cd0 +a20cl 2, where cd0 is the drag coefficient at zero lift, a20 is the drag-due-to-lift factor, cl is the lift coefficient, • the calculation corresponds to the conditions of horizontal flight at a constant speed corresponding to the number of mcr: ya=gdes xa=p, where xa the drag force, p the projection of the thrust of the power plant on the velocity axis 0xa, ya the lift force, gdes the design weight of the aircraft, • drag coefficient at zero lift and specified mcr is constant when changing the cruise altitude, • the dependence of the specific fuel consumption coefficient on thrust is assumed to be linear and is expressed as ce(p) = ce1 + ce p (p – p1), where ce1 is the specific fuel consumption coefficient at a given value of thrust p1, ce p is the derivative of the thrust dependence of ce(p). accounting for the conditions of steady horizontal flight makes it is possible to obtain the dependence of (l/d)/ce on thrust normalized to an altitude of h=11 km: (l/d) ce = gdes p11 km ρ(h)/ρ(ce1+ce p(p11 km-p1)) . solution of the system of equations, expressing the conditions of steady horizontal flight: ⎩ ⎪ ⎨ ⎪ ⎧ p(h)= ρ(h)(amcr) 2s 2 (𝐶d0+a20cl 2), gdes= ρ(h)(amcr) 2s 2 cl, where a is the speed of sound, allows one to obtain the relation between the flight altitude and the available thrust, normalized to h = 11 km (engine operation mode): solving the equation ((l/d)/ce)p = 0 with respect to p, it is easy to obtain expression for the optimal thrust corresponding to an altitude of 11 km, which ensures the maximum of the parameter (l/d)/ce: poptim= 4x0+p*+ (4x0-2p*)2-3p*2 6 where p*=p1ce1 ce p . determining sst aerodynamic configuration and power plant parameters… 157 copyright ©2023 assa. adv. in systems science and appl. (2023) substituting this formula into the formula for ((l/d)/ce)cr max, we obtain dependence for the optimal value of (l/d)/ce in the supersonic cruise flight mode: (l/d) ce cr max ≈ cl αqs(poptim-x0) poptim(cemax +ce p(poptim-p1)) the wing area s is determined by solving a system of nonlinear equations relating the flight range, takeoff thrust-to-weight ratio, and maximum subsonic lift-to-drag ratio in accordance with the procedure presented in [26]. the uncertain parameters used in this procedure are:  gtdes consumption in fractions of takeoff weight during descent,  gtalt fuel consumption in fractions of take-off weight during acceleration-climb,  gfload functional load weight (equipment, crew, passengers),  kpp weight characteristic of the power plant,  cf equivalent friction coefficient corresponding to the wetted area,  q1, q2 coefficients in the relation of the specific structural weight (weight of 1 m2 of the structure) to the aircraft volume,  lcl_des climb + descent length. as a result of numerical optimization, the pareto front is determined in the plane of m·k/ce and kmax m < 1, where k means (l/d), fig. 3. fig. 3. pareto front for m·k/ce and kmax m < 1 within the flow charts in fig. 1 and 2, optimization models 1-2 are proposed to solve the presented problem of the sst preliminary design under uncertainty. model1. critical values kmax_cr and br_par=m·k/ce. min  x̄' {e[kmax_cr(x̄ ' , ξ)]}, max  x̄' {e[br_par(x̄' , ξ)]}, model 2. critical values kmax_cr and br_par=m·k/ce min  x̄' {infα[kmax_cr(x̄ ' , ξ)]}, max  x̄' {supα[br_par(x̄' , ξ)]}, 158 g. veresnikov, i. bashkirov copyright ©2023 assa adv. in systems science and appl. (2023) in these optimization models, x̄' = (f, v) is the vector of optimized parameters, f is the configuration factor, v is the aircraft density, ξ = gtdes , gtalt , gfload , kpp, cf, q1, q2, lcl_des) is the vector of input uncertain parameters. the remaining input parameters of the black-box models are considered deterministic and therefore are not specified in the objective functions. using the expression for the expected value in appendix (theorem 1b), the trapezoid method, and the information about the monotonicity type of the objective functions with respect to uncertain parameters, we obtain expressions for approximate calculations e[kmax_cr(x̄ ' , ξ)] and e[br_par(x̄' , ξ)]. e kmax_cr x̄' , ξ = 1 2n [k max_cr (f,  v ,φgtdes -1 i n ,φgtalt -1 i n ,φgfload -1 i n , n-1 i=0 φkpp -1 i n , φcf -1 1i n ,φq1 -1 i n ,φq2 -1 i n ,φlcl_des -1 1i n )+ + kmax_cr(f,  v ,φgtdes -1 i+1 n ,φgtalt -1 i+1 n ,φgfload -1 i+1 n ,φkpp -1 i+1 n , φcf -1 1i+1 n ,φq1 -1 i+1 n ,φq2 -1 i+1 n , φlcl_des -1 1i+1 n ], e br_par(x̄' , ξ) = 1 2n [br_par(f,  v ,φgtdes -1 1i n ,φgtalt -1 1i n ,φgfload -1 1i n , n-1 i=0 φkpp -1 1i n , φcf -1 1i n ,φq1 -1 1i n ,φq2 -1 1i n ,φlcl_des -1 i n + +br_par(f,  v ,φgtdes -1 1i+1 n ,φgtalt -1 1i+1 n ,φgfload -1 1i+1 n ,φkpp -1 1i+1 n , φcf -1 1i+1 n ,φq1 -1 1i+1 n ,φq2 -1 1i+1 n ,φlcl_des -1 i+1 n ], where n is the specified number of parts of the unit interval decomposition. the need for approximate calculations is due to the impossibility of exact calculation of a definite integral, since kmax_cr(x̄ ' , ξ) and br_par(x̄' , ξ) are defined by black-box models. approximate calculations can significantly reduce the overall computational performance of solving preliminary design problems. optimization problems, especially multicriteria ones, require the execution of many similar calculation procedures, which determines the high dependence of computational costs on the method of calculating the numerical characteristics of objective functions and constraints. however, uncertainty theory requires significantly fewer objective function calculations using black-box models than probability theory. using the expressions for critical values in appendix (definition 2 and theorem 1a) and information about the types of monotonicity of objective functions with respect to uncertain parameters, we obtain expressions for infα[kmax_cr(x̄ ' , ξ)] and supα[br_par(x̄' , ξ)]. infα[kmax_cr(x̄ ' , ξ)]= kmax_cr(f,  v ,φgtdes -1 (α),φgtalt -1 (α),φgfload -1 (α), φkcy -1 (α), φcf -1(1-α), φq1 -1 (α),φq2 -1 (α),φlcl_des -1 (1-α)), supα[br_par(x̄' , ξ)]= br_par(f,  v ,φgtdes -1 (α),φgtalt -1 (α),φgfload -1 (α), φkcy -1 (α), φcf -1(α),φq1 -1 (α),φq2 -1 (α),φlcl_des -1 (1-α). where α is the level of confidence. it can be seen that the application of optimization models 1, 2 requires much less calculation of objective functions using black-box models compared to their stochastic analogues. determining sst aerodynamic configuration and power plant parameters… 159 copyright ©2023 assa. adv. in systems science and appl. (2023) the next section presents the results of optimization calculations obtained using the proposed optimization models. 3. results the parameter uncertainty distribution functions in optimization models 1 and 2 were set by experts with piecewise linear functions (6): φ(•)(x)= ⎩ ⎨ ⎧ 0, if x≤x1, αi+ (αi+1-αi)(x-xi) xi+1-xi , if xi≤x≤xi+1, 1≤ i ≤n, 1, if x>xn . where α1, α2,…, αp(•) are the levels of confidence in that (•) ≤ x1, (•) ≤ x2, (•) ≤ 𝑥p(•) , x1 , x2 , …, 𝑥p(•) are the deterministic values of the uncertain parameter (•) considered by an expert, x1 < x2 < … < xp(•) , 0 ≤ α1 ≤ α2 ≤…≤ αp(•) ≤ 1. the use of functions of this type makes it possible to avoid a common error associated with setting parameters by an expert, which, according to physics, can change only in a closed interval, with normal distribution functions having a domain of [-∞; ∞]. in addition, the use of piecewise linear functions simplifies numerical integration when calculating the expected value. an example of the uncertainty distribution function is shown in fig. 4. fig. 4. uncertainty distribution function for the functional load weight as a result of optimization calculations according to models 1-2, pareto fronts were obtained, which reflect trade-off design solutions on determining the sst aerodynamic configuration and power plant parameters under epistemic uncertainty. figures 5 and 6 show the pareto fronts for models 1 and 2. for comparison, each graph shows the pareto front, which was obtained by solving the problem under consideration in a deterministic formulation. 160 g. veresnikov, i. bashkirov copyright ©2023 assa adv. in systems science and appl. (2023) fig. 5. optimization model 1 calculation results fig. 6. optimization model 2 calculation results (α=0,7) figure 4 shows that taking into account the uncertainty of the sst parameters using the expected values in optimization model 1 leads to the pareto front shift to the “worst” range of objective function values (the required growth in the maximum subsonic lift-to-drag ratio to increase the confidence level complicates the aerodynamic design). model 1 provides optimality in the mean, so the function forms of the objective function uncertainty distribution do not matter determining sst aerodynamic configuration and power plant parameters… 161 copyright ©2023 assa. adv. in systems science and appl. (2023) when choosing design solutions using this pareto front. this problem is solved in model 2. the use of critical values makes it possible to obtain design solutions with a given level of confidence (robustness) and leads to a more pronounced pareto front shift to the range of the “worst” objective function values (fig. 5). thus, to ensure the reliability of design solutions, it is preferable to use optimization model 2 with critical values of objective functions. 4. conclusion the paper considers the problem of determining the parameters of the aerodynamic configuration and power plant of a supersonic passenger aircraft under epistemic uncertainty at the preliminary design stage, when some of the initial parameters are set by experts and are nondeterministic. non-determination is the reason for the possibility of implementing parameter values that can lead to a significant deviation of the sst performance from the performance predictions and violation of critical constraints. the paper presents multicri teria optimization models based on uncertainty theory, which provide high computational efficiency when using numerical methods for calculating the numerical characteristics of functions that depend on uncertain quantities. it is shown that the use of the proposed optimization models, which take into account the non-determination of the parameters set by experts, leads to the design solutions that differ from the solutions found using deterministic models. the pareto fronts obtained as a result of optimization calculations for various confidence levels allow a design engineer to determine the aerodynamic configuration and power plant parameters that provide a rational objective function value trade-off. acknowledgements the article is partially prepared in the implementation of the program for the creation and development of the world-class research center “supersonic” for 2020-2025 funded by the ministry of science and higher education of the russian federation (grant agreement of may, 17, 2022 № 075-15-2022-1023). references 1. jaeger l., gogu, c., & segonds, s. 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(2021). second-order reliability methods: a review and comparative study, structural and multidisciplinary optimization, 64, 3233–3263. determining sst aerodynamic configuration and power plant parameters… 163 copyright ©2023 assa. adv. in systems science and appl. (2023) appendix. uncertainty theory: fundamental concepts this section presents the main fundamental concepts of the theory of uncertainty [15], which are used to create a theoretical basis for the parametric synthesis of the hst design solutions under epistemic uncertainty. definition 1 [15]: the uncertainty distribution function of the uncertain variable ξ is the function : r  [0, 1], defined as: φ(x)= m { ξ ≤ x}, where m{•} is the measure of the uncertainty of event occurrence •, defined in [23] as the expert's confidence level that this event will occur. theorem 1 [15]: let the function f(ξ1, ξ2, ..., ξn) be continuous, strictly increasing in ξ1, ξ2, ..., ξm and strictly decreasing in ξm+1, ξm+2, …, ξn. then if ξ1, ξ2, ..., ξn are independent uncertain variables with uncertainty distribution functions φ1, φ2, ..., φn, respectively, then: a) ξ = f(ξ1, ξ2, ..., ξn) is an uncertain variable with the inverse uncertainty: ψ-1(ξ)=f(φ1 -1(α), φ2 -1(α), ..., φm -1(α), φm+1 -1 (1-α), ..., φn -1(1-α)), where φ-1 is the inverse uncertainty distribution. b) expected value: e[ξ]= (f(φ1 -1(α),φ2 -1(α), ..., φm -1(α),φm+1 -1 (1-α), ... 1 0  φn (1 − α))dα. c) variance: v[ξ] = (f(φ (α), φ (α),  . . . , φm (α), φm (1 − α),  . . . 1 0  φn -1(1-α))-e[ξ])2dα. d) for any α  [0, 1]: m{f(ξ1, ξ2, ..., ξn) < 0} > α is equivalent to f(φ1 -1(α),φ2 -1(α), ... ,φm -1(α),φm+1 -1 (1-α), ..., φn -1(1-α))≤0. definition 2 [15]: the quantiles of the uncertain value ξ (critical values) are: supα[ξ] = sup{r | m{ ξ > r} > α}, α  [0, 1], infα[ ξ ] = inf{r | m{ ξ < r} > α}, α  [0, 1]. adv syst sci appl 2024; 01:29–39 published online at https://ijassa.ipu.ru. the piecewise levenberg–marquardt method alexey f. izmailov1*, evgeniy i. uskov2, yan zhibai1 1lomonosov moscow state university, moscow, russia 2derzhavin tambov state university, tambov, russia abstract: we develop sharp local superlinear convergence results for the levenberg–marquardt method applied to constrained piecewise smooth equations, under the local lipschitzian error bound condition for active selections, allowing for nonisolated solutions. we also characterize the level of inexactness allowed when solving subproblems, such that it does not interfere with superlinear convergence rate. applications to constrained reformulations of complementarity systems, using the “min” complementarity function are also discussed. keywords: nonlinear equation, constrained equation, piecewise smooth equation, nonisolated solution, local error bound, levenberg–marquardt method 1. introduction we consider the constrained equation φ(u) = 0, u ∈ p, (1.1) where φ : rp → rq is a given mapping, and p ⊂ rp is a given nonempty closed convex set. let u stand for the solution set of (1.1). the case of p = rp and p = q is the classical setting of an unconstrained nonlinear equation, and assuming that φ is smooth, the fundamental approach for solving such problems is the newton method; see, e.g., [12]. local superlinear convergence of the basic form of the newton method requires a solution in question to be nonsingular, and in particular isolated. the levenberg–marquardt (lm) method for unconstrained nonlinear equations [17,19] (see also [20, section 10.3]) is a classical regularization technique for handling cases when a solution can be singular, and possibly nonisolated, and possibly with p ̸= q. a natural way to approach the lm method is to consider first the (constrained) gauss– newton (gn) method. for the current iterate u ∈ p , the gn method defines the next iterate as u+ v, where v minimizes the (squared) residual of the linearized equation from (1.1) over p − u, i.e., v is a solution of the optimization problem minimize 1 2 ∥φ(u) + φ′(u)v∥2 subject to u+ v ∈ p. (1.2) due to the frank–wolfe theorem [10], this subproblem is always solvable when p is polyhedral, but a solution need not be unique. the potential lack of uniqueness of solutions in the subproblem (1.2) is one of the reasons regularize it. this results in the constrained lm method [2, 15] for (1.1), defining the next ∗corresponding author: izmaf@cs.msu.ru 30 a. f. izmailov, e. i. uskov, y. zhibai iterate as u+ v, where v solves the optimization problem minimize 1 2 ∥φ(u) +g(u)v∥2 + 1 2 σ(u)∥v∥2 subject to u+ v ∈ p, (1.3) with a function σ : p → r+ defining the values of the regularization parameter, and with g(u) being φ′(u), or its suitable substitute when the derivative does not exist or is not available. observe that if σ(u) > 0, and if euclidean norms are used, the objective function of (1.3) is strongly convex quadratic, and hence, this subproblem has the unique solution. moreover, if p is a polyhedral set, then (1.3) is a strongly convex quadratic programming problem. when p = rp, the constrained lm method reduces to the classical (sometimes called unconstrained) lm method with its origins in [17, 19]. for the convergence properties for p = rp in the case of smooth φ and isolated solutions, as well as related references, see [3, theorem 10.2.6]. the cases of nonsmooth (semismooth) φ with isolated solutions was considered in [5]; see also [13, 14]. in this work we deal with local convergence analysis based on weaker assumptions of the local lipschitzian error bound type allowing, in particular, for nonisolated solutions, and originating in the context of lm methods from [21]. moreover, we concentrate on the peculiarities of the case when φ in (1.1) is piecewise smooth, but need not be smooth. as argued in [4], the combination of nonisolated solutions with nonsmoothness is an especially challenging situation. for a recent survey of modern local convergence theories for the lm method, including some applications, the issues of convergence globalization, etc., see [8]. some comments on our notation are in order. the euclidean inner product for u, v ∈ rp is denoted by ⟨u, v⟩, and to avoid any confusion, let ∥ · ∥ stand for the euclidean norm throughout. for a set u ⊂ rp and a point u ∈ rp, dist(x, u) = infv∈u ∥v − u∥, and b(u, δ) = {v ∈ rp | ∥v − u∥ ≤ δ} is the closed ball of radius δ ≥ 0 centered at u. for a given index set h ⊂ {1, . . . , p}, by uj we denote the subvector of u ∈ rp, with components uj , j ∈ j . for a closed convex set u ⊂ rp, by nu(u) we denote the normal cone to u at u, i.e., nu(u) = {v ∈ rp | ⟨v, ũ− u⟩ ≤ 0, ∀ ũ ∈ u} if u ∈ u , and nu(u) = ∅ otherwise. for a sequence {uk} ⊂ rp convergent to some ū ∈ rp, we say that convergence is of qorder θ > 1 if there exists c > 0 such that ∥uk+1 − ū∥ ≤ c∥uk − ū∥θ for all k large enough. such rate of convergence is superlinear, and it is at least quadratic if θ ≥ 2. we say that {uk} converges to ū with r-order θ if there exist c > 0 and a sequence {tk} ⊂ r+ converging to 0 with q-order θ, such that ∥uk+1 − ū∥ ≤ ctk for all k large enough. the rest of this paper is organized as follows. in section 2, we recall the local convergence framework from [9], that will serve as a main tool in our analysis. section 3 contains our new local convergence result for the piecewise smooth case, and section 4 is devoted to the effect of approximate solution of subproblems. finally, in section 5, we discuss applications of the results obtained to constrained reformulations of complementarity systems, using the “min” complementarity function. 2. local convergence framework we start with a discussion of the abstract local convergence framework recently proposed in [9]. for that purpose, it is convenient to consider a problem setting with a single scalar equation: φ(u) = 0, u ∈ p, (2.4) with φ : rp → r+ being a given scalar-valued function, and p ⊂ rp being a given nonempty closed set. obviously, the constrained equation (1.1) can be stated in the form (2.4) with, say, copyright © 2024 assa. adv syst sci appl (2024) the piecewise levenberg–marquardt method 31 φ(u) = ∥φ(u)∥. to that end, there will be no confusion if in this section we will use u for the solution set of (2.4). let ψ : p → p be a given mapping, and consider an abstract iterative process updating the current iterate u ∈ p to the new one of the form ψ(u). the following is a version of [9, theorem 2.1], somehow simplified for our purposes. theorem 2.1: let φ : rp → r+ be a continuous function, p ⊂ rp be a nonempty closed set, ū ∈ u , and assume that φ(u) = o(dist(u, u)) (2.5) as u ∈ p tends to ū. moreover, let ψ : p → p be a mapping such that, with some τ > 1, ψ(u)− u = o(φ(u)) (2.6) and φ(ψ(u)) = o((φ(u))τ ) (2.7) as u ∈ p tends to ū. then, for every δ > 0, and every u0 ∈ p close enough to ū, the sequence {uk} defined by uk+1 = ψ(uk) for all k is contained in b(ū, δ) and converges to some u∗ ∈ u , and the rate of convergence is superlinear with the q-order τ . 3. the piecewise smooth case: local convergence our main focus in this work is on the case when φ in (1.1) is piecewise smooth near a solution ū in question: for a finite collection of selection mappings φ1, . . . , φs : rp → rq which are continuously differentiable near ū, it holds that φ(u) ∈ {φ1(u), . . . , φs(u)} ∀u ∈ rp, and φ is continuous near ū. taking s = 1 recovers the smooth case, i.e. the case when φ is continuously differentiable near ū. according to [11, theorem 2.1], any mapping that is piecewise smooth near some point is necessarily lipschitz-continuous near that point. for every u ∈ rp, define the set a(u) = {j ∈ {1, . . . , s} | φ(u) = φj(u)} (3.8) of indices of all selection mappings active at u. let g be any mapping such that g(u) ∈ {(φj)′(u) | j ∈ a(u)} (3.9) for u ∈ rp close enough to ū. the algorithm with the subproblem (1.3) employing g defined by (3.8) and (3.9), is naturally referred to as the constrained piecewise lm method. observe that with this definition of g, for any u ∈ rp close enough to ū, the subproblem (1.3) can be written in the form minimize 1 2 ∥φj(u) + (φj)′(u)v∥2 + 1 2 σ(u)∥v∥2 subject to u+ v ∈ p, (3.10) with some j ∈ a(u), and this can be seen as the subproblem of the lm method applied to a smooth constrained equation φj(u) = 0, u ∈ p. (3.11) needless to say, the index j can vary from one iteration to another. copyright © 2024 assa. adv syst sci appl (2024) 32 a. f. izmailov, e. i. uskov, y. zhibai we will need to assume the p -property at ū, introduced in [7, p. 434]. it consists of saying that near ū the constraint set p excludes all zeroes of smooth selections active at ū, which are not zeroes of φ. formally it means that uj ⊂ u near ū for all j ∈ a(ū), where uj stands for the solution set of (3.11). the p -property at ū evidently implies that dist(u, u) ≤ dist(u, uj) (3.12) for all u ∈ p close enough to ū. we note that the p -property can be easily guaranteed for reformulations of complementarity systems by means of the “min” (natural residual) complementarity function, see [7] and section 5 below. another assumption employed in theorem 3.1 below is the constrained error bound for each active selection, that is dist(u, uj) = o(∥φj(u)∥) ∀ j ∈ a(ū) (3.13) as u ∈ p tends to ū. under this assumption, the p -property at ū is equivalent to the condition φ(u) = o(∥φj(u)∥) ∀ j ∈ a(ū) as u ∈ p tends to ū. observe also that, employing the inclusion a(u) ⊂ a(ū) for u ∈ rp close enough to ū, one can easily see that the p -property at ū, and the condition (3.13) as u ∈ p tends to ū, imply the constrained error bound dist(u, u) = o(∥φ(u)∥) (3.14) as u ∈ p tends to ū. we proceed with the following characterization of local superlinear convergence of the piecewise lm method, which is the main result of this section. theorem 3.1: let φ : rp → rq be a given mapping, p ⊂ rp a nonempty closed convex set, and ū ∈ u . assume that φ is piecewise smooth near ū, and the derivatives of its smooth selection mappings φ1, . . . , φs : rp → rq are lipschitz-continuous near ū. let the p -property at ū and condition (3.13) be satisfied as u ∈ p tends to ū. moreover, let g : rp → rq×p be a fixed mapping satisfying (3.9), and assume that the function σ : p → r+ defining the regularization parameter satisfies ∥φ(u)∥θ = o(σ(u)), σ(u) = o(∥φ(u)∥θ) (3.15) as u ∈ p tends to ū, with some fixed θ ∈ (0, 2]. then, for every u0 ∈ p , there exists the unique sequence {uk} such that for every k, the displacement uk+1 − uk is a solution of (1.3) with u = uk, and with the additional convention that uk+1 = uk if uk ∈ u . for any δ > 0, if u0 ∈ p is close enough to ū, then this sequence is contained in b(ū, δ) and converges to some u∗ ∈ u , and the rate of convergence superlinear with the q-order min{θ + 1, 2}. proof the first estimate in (3.15) implies that σ(u) > 0 for all u ∈ p \ u close enough to ū, and hence, for such u, the lm subproblem (1.3) has the unique solution v(u). furthermore, for u ∈ rp close to ū, and for j ∈ a(u) ⊂ a(ū) such that g(u) = (φj)′(u) (see (3.9)), we have that φj(u) = φ(u) (see (3.8)). for any such j, (3.15) implies the estimates ∥φj(u)∥θ = o(σ(u)), σ(u) = o(∥φj(u)∥θ) (3.16) as u ∈ p tends to ū, and an iteration of the constrained piecewise lm method can be interpreted as an iteration of the usual constrained lm method for the smooth constrained copyright © 2024 assa. adv syst sci appl (2024) the piecewise levenberg–marquardt method 33 equation (3.11), with the subproblem (3.10), and with the regularization parameter satisfying (3.16). employing the restriction θ ∈ (0, 2], from (3.13) and from the first estimate in (3.16) it follows that (dist(u, uj)) 4 σ(u) = (dist(u, uj)) 2o ( ∥φj(u)∥2 σ(u) ) = o((dist(u, uj)) 2) (3.17) as u ∈ p \ u tends to ū. let ûj stand for a metric projection of u onto uj: ∥u− ûj∥ = dist(u, uj). (3.18) since v(u) is a global solution of (3.10), it holds that ∥φj(u) + (φj)′(u)v(u)∥2 + σ(u)∥v(u)∥2 ≤ ∥φj(u) + (φj)′(u)(ûj − u)∥2 +σ(u)∥ûj − u∥2. (3.19) since j ∈ a(ū), it holds that ū ∈ uj . therefore, evidently, ûj → ū as u→ ū, and from (3.18)–(3.19), applying again the mean-value theorem [12, theorem a.10], we derive that ∥v(u)∥2 ≤ 1 σ(u) ( ∥φj(u) + (φj)′(u)(ûj − u)∥2 + σ(u)∥ûj − u∥2 ) = 1 σ(u) ∥φj(u)− φj(ûj)− (φj)′(u)(u− ûj)∥2 + ∥u− ûj∥2 ≤ 1 σ(u) sup τ∈[0, 1] ∥(φj)′(τu+ (1− τ)ûj)− (φj)′(u)∥2∥u− ûj∥2 + ∥u− ûj∥2 = ∥u− ûj∥2 +o ( ∥u− ûj∥4 σ(u) ) = o((dist(u, uj)) 2), (3.20) where the last estimate is by (3.17). hence, employing again (3.13), v(u) = o(dist(u, uj)) = o(∥φj(u)∥) = o(∥φ(u)∥) (3.21) as u ∈ p \ u tends to ū. furthermore, any u ∈ uj is a (global) solution of the optimization problem minimize 1 2 ∥φj(u)∥2 subject to u ∈ p, and the objective function of this problem is differentiable at u, with the gradient being ((φj)′(u))⊤φj(u). therefore, any such u must satisfy the first-order necessary optimality condition ((φj)′(u))⊤φj(u) +np (u) ∋ 0. (3.22) employing now [2, lemma 1] demonstrating that the constrained error bound implies the upper lipschitzian property of the solutions set of the generalized equation (3.22) subject to the right-hand side perturbations, we obtain that under (3.13), for any solution u of the perturbed generalized equation ((φj)′(u))⊤φj(u) +np (u) ∋ ω, (3.23) copyright © 2024 assa. adv syst sci appl (2024) 34 a. f. izmailov, e. i. uskov, y. zhibai close enough to ū, it holds that dist(u, uj) = o(∥ω∥) as ω → 0. the first-order necessary optimality condition for the subproblem (3.10) has the form ((φj)′(u))⊤(φj(u) + (φj)′(u)v) + σ(u)v +np (u+ v) ∋ 0. (3.24) for any given u ∈ rp, (3.22) is equivalent to saying that ((φj)′(u+ v))⊤φj(u+ v) +np (u+ v) ∋ 0 (3.25) holds with v = 0. then (3.24) can be regarded as a perturbation of the generalized equation (3.25). specifically, if we set ω(u, v) = ((φj)′(u+ v))⊤φj(u+ v)− ((φj)′(u))⊤(φj(u) + (φj)′(u)v)− σ(u)v = ( ((φj)′(u+ v))⊤ − ((φj)′(u))⊤ ) φj(u+ v) +((φj)′(u))⊤(φj(u+ v)− φj(u)− (φj)′(u)v) −σ(u)v, (3.26) then (3.24) can be written in the form ((φj)′(u+ v))⊤φj(u+ v) +np (u+ v) ∋ ω(u, v). from lipschitz-continuity of (φj)′ near ū, and from the second estimate in (3.15) and (3.21), employing again the mean-value theorem [12, theorem a.10], one can readily derive the estimates ω(u, v(u)) = o(σ(u)∥v(u)∥) +o(∥v(u)∥2) = o(∥φ(u)∥θ+1) +o(∥φj(u)∥2) = o(∥φ(u)∥min{θ+1, 2}) (3.27) as u ∈ p \ u tends to ū. summarizing the considerations above, u+ v(u) is a solution of the generalized equation (3.23) with ω = ω(u, v(u)), and it holds that u+ v(u) → ū and ω(u, v(u)) → 0 as u ∈ p \ u tends to ū. therefore, [2, lemma 1] allows to conclude that dist(u+ v(u), uj) = o(ω(u, v(u))) = o(∥φ(u)∥min{θ+1, 2}), where the second estimate is by (3.27). therefore, since φ is lipschitz-continuous near ū, employing the inequality (3.12) (following from the p -property) we derive φ(u+ v(u)) = o(dist(u+ v(u), u)) = o(dist(u+ v(u), uj)) = o(∥φ(u)∥min{θ+1, 2}) (3.28) as u ∈ p \ u tends to ū. estimates (3.21) and (3.28) yield (2.6) and (2.7) in theorem 2.1 with ψ(u) = u+ v(u), φ(u) = ∥φ(u)∥, and τ = min{θ + 1, 2}, while (2.5) is satisfied again because of lipschitz continuity of φ. the needed conclusion now follows by application of theorem 2.1. for the case when θ = 2, the result of theorem 3.1 was essentially obtained in [7, theorems 1, 2]; see also [8, theorem 4.1], where it is emphasized that the convexity assumption on p is actually not needed in that case. on the other hand, in the smooth case, theorem 3.1 is a particular case of [2, theorem 1]. copyright © 2024 assa. adv syst sci appl (2024) the piecewise levenberg–marquardt method 35 as demonstrated in [8, example 3.2], the convergence rate estimate in theorem 3.1 is sharp even in the smooth unconstrained case, and even when the solution ū in question is isolated. this example also shows that the constrained error bound condition (3.14), corresponding to (3.13) in the smooth case, is essential for this analysis. regarding the restriction θ ∈ (0, 2] in theorem 3.1 on the exponent in the regularization parameter, [8, example 3.2] demonstrates that at least the requirement θ < 4 cannot be avoided, even in the smooth case. the possibility to obtain a counterpart of theorem 3.1 for the values θ ∈ (2, 4) seems to remain an open question. the lack of quadratic convergence is claimed in [1, example 4.2] for θ > 3, but this claim relies on some numerical observations only. in the smooth case, superlinear convergence to 0 of the sequence {dist(uk, u)} for θ ∈ (2, 3) was established in [6], with some similar results and extensions in the recent work [22]. 4. the piecewise smooth case: effect of inexactness since solving the constrained lm subproblems exactly can be too expensive, or even impossible, a natural issue consists of characterization of the “level” of controllable inexactness when solving subproblems that does not interfere with local convergence and rate of convergence properties of the lm method established in section 3. yet again, for u ∈ rp close to ū, let j ∈ a(u) ⊂ a(ū) be such that g(u) = (φj)′(u) (see (3.9)); it also necessarily holds that φj(u) = φ(u) (see (3.8)). recall that an iteration of the constrained piecewise lm method can be interpreted as an iteration of the usual constrained lm method for the smooth constrained equation (3.11), with the subproblem (3.10), and with the regularization parameter satisfying (3.16). the first-order necessary optimality condition for (3.10) has the form (3.24), and it is natural to consider the version of the inexact constrained lm method, with inexactness measured by the violation of (3.24). specifically, the process of solving the subproblem (3.10) is terminated once ((φj)′(u))⊤(φj(u) + (φj)′(u)v) + σ(u)v +np (u+ v) ∋ w (4.29) is satisfied with some w ∈ rp smaller than the given tolerance that conforms with the exponent θ in (3.15) (and in (3.16)) as follows: w = o(∥φj(u)∥θ+1) (4.30) as u ∈ p tends to ū. the analysis in [2] relies on the observation that (4.29) is a necessary and sufficient optimality condition for the following convex optimization problem, which is a perturbation of (3.10): minimize 1 2 ∥φj(u) + (φj)′(u)v∥2 + 1 2 σ(u)∥v∥2 − ⟨w, v⟩ subject to u+ v ∈ p. (4.31) then we follow the argument used above to prove (3.21), but taking into account that now the objective function of (4.31) has an extra term involving w. employing again a metric projection ûj of u ∈ p \ u onto uj , for the unique global solution v(u) of (4.31), similarly to (3.19) we obtain that ∥φj(u) + (φj)′(u)v∥2 + σ(u)∥v(u)∥2 − 2⟨w, v(u)⟩ ≤ ∥φj(u) + (φj)′(u)(ûj − u)∥2 +σ(u)∥ûj − u∥2 − 2⟨w, ûj − u⟩. copyright © 2024 assa. adv syst sci appl (2024) 36 a. f. izmailov, e. i. uskov, y. zhibai then, similarly to (3.20) (and in particular, making use of (3.18)), we derive the estimate ∥v(u)∥2 ≤ 1 σ(u) ( ∥φj(u) + (φj)′(u)(ûj − u)∥2 + σ(u)∥ûj − u∥2 +2⟨w, v(u)⟩ − 2⟨w, ûj − u⟩ ) ≤ 2∥w∥ σ(u) (∥v(u)∥+ dist(u, uj)) +o((dist(u, uj)) 2) ≤ o(∥φj(u)∥(∥v(u)∥+ dist(u, uj))) +o((dist(u, uj)) 2) = o(dist(u, uj)∥v(u)∥) +o((dist(u, uj)) 2) (4.32) as u ∈ p \ u tends to ū, where the next-to-the-last estimate is by (4.30) and the first relation in (3.16), and the last one is the lipschitz continuity of φj near ū. evidently, (4.32) implies (3.21) as u ∈ p \ u tends to ū. the remaining part of the argument consists of the reasoning used above to prove theorem 3.1, but with ω(u, v) defined in (3.26) replaced by ω(u, v) + w: taking again into account (4.30), one can see that this modification does not affect (3.27) and the subsequent analysis. this yields theorem 4.1: under the assumptions of theorem 3.1, let the function ψ : p → r+ satisfy ψ(u) = o(∥φ(u)∥θ+1) as u ∈ p tends to ū. then, for every u0 ∈ p , there exists a sequence {uk} such that for every k, the displacement uk+1 − uk is the solution of (4.29) with u = uk, with some w ∈ rp satisfying ∥w∥ ≤ ψ(uk), and with the additional convention that uk+1 = uk if uk ∈ u . for any δ > 0, if u0 ∈ p is close enough to ū, any such sequence is contained in b(ū, δ) and converges to some u∗ ∈ u , and the rate of convergence is superlinear with the q-order min{θ + 1, 2}. for the case when θ = 2, a result related to theorem 4.1 was obtained in [4, theorems 3], but for a somewhat different kind of inexactness. see also [8, section 5] for a discussion of these issues in the smooth case, including establishing the relations between these two kinds of inexactness. the requirement (4.30) on allowed inexactness in theorem 4.1 cannot be relaxed even in the smooth case, as demonstrated by an example in [2, section 5]. 5. application to reformulations of complementarity systems in this section we deal with a nonlinear complementarity system of the form a(x) = 0, b(x) ≥ 0, c(x) ≥ 0, d(x) ≥ 0, ⟨c(x), d(x)⟩ = 0, (5.33) with given smooth mappings a : rn → rl, b : rn → rm, c : rn → rr, d : rn → rr, i.e., possessing lipschitz-continuous derivatives near a solution x̄ of (5.33). various important problem classes can be modeled as (5.33), including the so-called (mixed) complementarity problems, and in particular, karush–kuhn–tucker optimality systems, to mention some. one of the most successful approaches to (5.33) consists of reformulating it as a system of equations by means of the so-called complementarity functions, perhaps the simples (but at the same time prominent) one being the “min” (natural residual) complementarity function: using auxiliary slack variables, system (5.33) can be equivalently written as (1.1) with p = n+ 2r +m, q = l + 3r +m, φ(u) = (a(x), min{y, z}, c(x)− y, d(x)− z, b(x)− µ), u = (x, y, z, µ), (5.34) copyright © 2024 assa. adv syst sci appl (2024) the piecewise levenberg–marquardt method 37 p = rn × rr + × rr + × rm + . (5.35) the mapping φ defined in (5.34) is evidently piecewise smooth. more precisely, for a current iterate uk = (xk, yk, zk, µk) ∈ rp, the smooth selections active at uk have the form φj(u) = (a(x), yic(uk)∩j , zid(uk)∩(i0(uk)\j), c(x)− y, d(x)− z, b(x)− µ), (5.36) where ic(u k) = {i ∈ {1, . . . , r} | yki < zki }, id(u k) = {i ∈ {1, . . . , r} | yki > zki }, i0(u k) = {i ∈ {1, . . . , r} | yki = zki }, and j ⊂ i0(u k). therefore, the iteration of the piecewise lm method consists of choosing such j , setting g(uk) = (φj)′(uk), and solving (1.3) with u = uk for the displacement vk = uk+1 − uk. according to (5.36), for any solution x̄ of system (5.33), and for the corresponding solution ū = (x̄, ȳ, z̄, µ̄) of problem (1.1) with φ and p defined in (5.34) and (5.35), respectively, with ȳ = c(x̄), z̄ = d(x̄), and µ̄ = b(x̄), the smooth selections of φ active at ū have the form φj(u) = (a(x), yic∩j , zid∩(i0\j), c(x)− y, d(x)− z, b(x)− µ), (5.37) where ic = ic(ū) = {i ∈ {1, . . . , r} | ci(x̄) = 0 < di(x̄)}, id = ib(ū) = {i ∈ {1, . . . , r} | ci(x̄) > 0 = di(x̄)}, i0 = i0(ū) = {i ∈ {1, . . . , r} | ci(x̄) = 0 = di(x̄)}, and j ⊂ i0. observe that the nonnegativity conditions on y and z in the definition of p in (5.35) may seem extraneus for this reformulation as they are guaranteed by the equality min{y, z} = 0. however, these constraints appear essential for guaranteing the p property to hold at ū. indeed, if for a given j ⊂ i0, it holds that φj(u) = 0 for some u ∈ p , then according to (5.35), yid∩(i0\j) ≥ 0 and zic∩j ≥ 0, and hence, according to (5.34) and (5.37), it holds that φ(u) = 0. at the same time, if u ∈ rp \ p , then some components of, say, zic∩j can be negative, and then min{yic∩j , zic∩j} = min{0, zic∩j} need not be equal to 0. the remaining ingredient for applicability of theorems 3.1 and 4.1 is the constrained local lipschitzian error bound (3.13) for selections active at ū. for every j ⊂ i0, let xj = { x ∈ rn ∣∣∣∣∣ a(x) = 0, cic∩j(x) = 0, did∩(i0\j)(x) = 0, b(x) ≥ 0, c(x) ≥ 0, d(x) ≥ 0 } . then according to the discussion in [7, section 5] (see, in particular, [7, figure 3]), condition (3.13) is implied by the following: for every j ⊂ i0, it holds that dist(x, xj) = o(∥a(x)∥+ ∥cic∩j(x)∥+ ∥did∩(i0\j)(x)∥+ ∥min{0, b(x)}∥ +∥min{0, cid∩(i0\j)(x)}∥+ ∥min{0, dic∩j(x)}∥) (5.38) as x ∈ rn tends to x̄. condition (5.38) is nothing else but the local lipschitzian error bound for the constraint system a(x) = 0, cic∩j(x) = 0, did∩(i0\j)(x) = 0, b(x) ≥ 0, cid∩(i0\j)(x) ≥ 0, dic∩j(x) ≥ 0, (5.39) copyright © 2024 assa. adv syst sci appl (2024) 38 a. f. izmailov, e. i. uskov, y. zhibai characterizing a branch of the solution set of the complementarity system (5.33) near a given solution x̄. therefore, (3.13) is impled by the so-called piecewise error bound condition for (5.33), that is, the local lipschitzian error bound at x̄ for every branch of the solution set, containing x̄, i.e., for the constraint system (5.39) with every j ⊂ i0. the latter property is implied by the so-called piecewise constrained qualifications at x̄, such as the classical mangasarian–fromovits constraint qualification [18], or the relaxed constant rank constraint qualification [16], for (5.39) with every j ⊂ i0. acknowledgements this work was funded by the russian science foundation grant 24-21-00015 (https: //rscf.ru/en/project/24-21-00015/). references 1. behling, r. (2011) the method and the trajectory of levenberg–marquardt. phd thesis. impa instituto nacional de matemática pura e aplicada, rio de janeiro. https://impa.br/wp-content/uploads/2017/08/tese dout roger behling.pdf 2. behling, r. & fischer, a. (2012) a unified local convergence analysis of inexact constrained levenberg–marquardt methods, optim. lett., 6, 927–940. 3. dennis, j.e. & schnabel, r.b. (1983) numerical methods for unconstrained optimization and nonlinear equations. englewood cliffs, nj: prentice-hall, inc. 4. facchinei, f., fischer, a. & herrich, m. (2013) a family of newton methods for nonsmooth constrained systems with nonisolated solutions, math. methods oper. res., 77, 433–443. 5. facchinei f. & kanzow, c. (1997) a nonsmooth inexact newton method for the solution of large-scale nonlinear complementarity problems, math. program., 76, 493–512. 6. fan, j.-y. & pan, j.-y. (2011) on the convergence rate of the inexact levenberg– marquardt method, j. ind. manag. optim., 7, 199–210. 7. fischer, a., herrich, m., izmailov, a.f. & solodov, m.v. (2016) convergence conditions for newton-type methods applied to complementarity systems with nonisolated solutions, comput. optim. appl., 63, 425–459. 8. fischer, a., izmailov, a.f. & solodov, m.v. (2023) the levenberg–marquardt method: an overview of modern convergence theories and more. (submitted). 9. fischer, a. & strasdat, n. (2023) an extended convergence framework applied to complementarity systems with degenerate and nonisolated solutions, pure appl. func. analys., 8, 1039–1054. 10. frank, m. & wolfe, p. (1956) an algorithm for quadratic programming, naval research logistics quarterly, 3, 95–110. 11. hager, w.w. (1979) lipschitz continuity for constrained processes, siam j. control optim., 17, 321–338. 12. izmailov, a.f. & solodov, m.v. (2014) newton-type methods for optimization and variational problems. cham, germany: springer. 13. kanzow, c. & petra, s. (2004) on a semismooth least squares formulation of complementarity problems with gap reduction, optim. meth. software, 19, 507–525. 14. kanzow, c. & petra, s. (2007) projected filter trust region methods for a semismooth least squares formulation of mixed complementarity problems, optim. meth. software, 22, 713–735. 15. kanzow, c., yamashita, n. & fukushima, m. (2004) levenberg–marquardt methods with strong local convergence properties for solving nonlinear equations with convex copyright © 2024 assa. adv syst sci appl (2024) https://rscf.ru/en/project/24-21-00015/ https://rscf.ru/en/project/24-21-00015/ https://impa.br/wp-content/uploads/2017/08/tese_dout_roger_behling.pdf the piecewise levenberg–marquardt method 39 constraints, j. comput. appl. math., 172, 375–397. 16. kyparisis, j. (1990) sensitivity analysis for nonlinear programs and variational inequalities with nonunique multipliers, math. oper. res., 15, 286–298. 17. levenberg, k. (1944) a method for the solution of certain non-linear problems in least squares, quarterly of appl. math., 2, 164–168. 18. mangasarian, o.l. & fromovitz, s. (1967) the fritz john necessary optimality conditions in the presence of equality and inequality constraints, j. math. anal. appl., 17, 37–47. 19. marquardt, d.w. (1963) an algorithm for least squares estimation of non-linear parameters, siam j., 11, 431–441. 20. nocedal, j. & wright, s.j. (2006) numerical optimization. 2nd edition. new york, ny: springer new york. 21. yamashita, n. & fukushima, m. (2001) on the rate of convergence of the levenberg– marquardt method. in: alefeld, g., chen, x. (eds.), topics in numerical analysis. computing supplementa, vol. 15 (pp. 239–249). vienna, austria: springer. 22. yin, j., jian, j. & ma, g. (2023) a modified inexact levenberg–marquardt method with the descent property for solving nonlinear equations, comput. optim. appl. https://doi.org/10.1007/s10589-023-00513-z. copyright © 2024 assa. adv syst sci appl (2024) https://doi.org/10.1007/s10589-023-00513-z introduction local convergence framework the piecewise smooth case: local convergence the piecewise smooth case: effect of inexactness application to reformulations of complementarity systems microsoft word assa+furb adv syst sci appl 2025; 02; 11-22 published online at https://ijassa.ipu.ru. advances in the development of radiometric measurement system applied to power circuit breaker bernardo a. cezarotto1, sergio h. l. cabral1*, thair i. a. h. mustafa1, ciro a. pitz2, cleiton gili3, hugo a. d. almague1, luiz h. meyer1, marcel v. t. fischer4 1) university of blumenau, blumenau, brazil 2) universidade federal de santa catarina, blumenau, brazil 3) gili energy electronics, presidente nereu, brazil 8) cpfl transmission, porto alegre, brazil abstract: because the importance of the power circuit breaker for the liable operation of a power system added to the increasing demand for continuity in power delivery, non-invasive techniques for evaluating its condition have presented a significant evolution over the last five years. today, the most promising technique is based on a radiometric system, which is based on analyzing the timing of the pole operation through the waveform of the transient signals radiated from the circuit breaker chambers, captured by sensors, and conveyed to a registering & processing unit. even though this may seem to be a simple task it is rather far from being so as it is described and discussed in this work. based on the experience of the authors, represented by several published papers about radiometry applied to power circuit breaker analyses, efforts towards the development of new equipment for replacing the commercially available digital oscilloscope became the milestone in the way to reaching a fully confident radiometric system. although still much has to be done for this, important practical data are given over this work besides an approximate analytical approach for better comprehending the radiometry itself. the key issue is properly guaranteeing the registering of transient signals from the power circuit chambers when any switching is made, which has been shown as being a very challenging task. this work is intended to serve as a basic reference for every radiometric system applied to circuit breakers that may be in development. keywords: radiometry, circuit breaker, trigger, oscilloscope, antennas. 1. introduction because of its fundamental role in guaranteeing the reliability of power system operation based on the endurance of switching current with high amplitudes power circuit breaker is prioritarily chosen as the equipment to be under maintenance. in this case, the main concerns with the power circuit breaker condition are regarding its capacity to properly interrupt any rated shortcircuit current without any arc reignition, which would characterize the circuit breaker to be condemned and thus immediately replaced, representing a rather undesirable situation. for its turn and until today, for this maintenance, the circuit breaker must be periodically deenergized for the application of invasive maintenance techniques, which includes measurement of the contact electrical resistance besides other electrical, mechanical, and cleaning measures [1,2]. in its turn, for the sake of reliability on the power delivery, the time interval elapsed for performing any circuit breaker maintenance must be as short and effective as possible. nevertheless, over the years these maintenance principles have been intensively contested by maintenance personnel of power utilities for being under pressure of increasing demand for power supply continuity. 12 b. a. cezarotto, s. h. l. cabral, t. i. a. h. mustafa, c. a. pitz, c. gili, et al. copyright ©2025 assa adv. in systems science and appl. (2025) alternatively, combined with the mentioned contestation, the increase in the availability of computer tools, as the atp and emtp programs, a new technique for evaluating the actual condition of an operating power circuit breaker was developed, as described in [3]. in its essence, this new technique depends on modeling the electric circuit adjacent to the circuit breaker for studying their behavior under transient conditions of switching operation, which is known as trv-transient recovery voltage study. this sort of study is mainly based on the analysis of the instantaneous voltage developed between circuit breaker poles that should not surpass some pre-defined voltage values defined by some envelope curves [4]. regarding this approach, although trv studies based on computer simulations are accepted by many power utilities as being confident enough for evaluating the actual condition of a power circuit breaker, it is unavoidable to consider the inherent risks that it represents because of the strong dependence on detailed and accurately correct electric circuit modeling, which is not an easy task. nevertheless, it must be emphasized the key advantage of this new method that is its noninvasive nature since there is no need to de-energizing the power circuit breaker. although trv studies are today widely deployed and accepted, because of its drawback of risks due to the possibility of inaccurate circuit modeling the demand for more confident and non-invasive maintenance techniques applied to power circuit breakers contributed to the development of an even newer alternative technique for evaluating the actual condition of an operating power circuit breaker. this newest technique based on radiometry was proposed in [5], by representing a significant change in the approach for evaluating the condition of an operating power circuit breaker maintenance for being based on radiated signals from each of its operations. regarding radiometric measurement applied to the power circuit breaker, the analysis of its condition is based on measuring the time interval elapsed between any two successive poles opening/closing of the three phases, which is to be identifiable on the instantaneous voltage waveform obtained from the captured transient signal radiated from the circuit breaker chambers. for the pole opening the more the value of the mentioned time interval worse the pole contact status because of the evidence of hindering in the arc quenching. therefore, capturing elements as antennas or sensors enable the irradiated signals to be conveyed to recording & processing equipment. in turn, there must be very sensitive coordination between transient signal recording and the circuit breaker switching because of the intrinsically very short time of the transient signal existence. thus, for performing both recording and processing of the captured signal digital oscilloscope is a natural immediate choice as equipment to be adopted because of its usual availability in triggering electrical transient events. therefore, first-generation arrangements of radiometric systems applied to circuit breaker are often centered on a commercially available oscilloscope connected to antennas through coaxial cables. nonetheless, some key drawbacks are immediately identified as reasons for the search for alternative arrangements. namely, the vulnerability of the embedded trigger function when registering transient signals radiated from the circuit breaker chambers and its limited adjust availability for properly registering these signals. moreover, the susceptibility of the oscilloscope to the electromagnetic interference inherently existent in a power substation environment can not only harm the oscilloscope itself as well as it may also cause an undesired trigger. based on this, some efforts have been made by the authors in a search for capturing devices other than antennas in view of optimizing the behavior of the oscilloscope trigger function, as in [6]. instead, after a set of laboratory tests it could be clear that some new registering equipment should replace the commercially available oscilloscope. in the meantime, even a signal analyzer was tested but the mentioned vulnerability remained, which pointed out the need for the development of new and dedicated registering equipment. this new equipment is a second-generation radiometric system applied to the circuit breaker and is under tests running in power substations for adjusting its features in view of fully guaranteeing the registering of transient signals radiated from the circuit breaker chamber during each of its rare and programmed operations. advances in the development of radiometric measurement system… 13 copyright ©2025 assa. adv. in systems science and appl. (2025) thus, by culminating with a presentation of some promising results obtained from the deployment of the radiometry system of the second generation in tests run in a power substation, from now on this work presents a driven description of the basic features of power circuit breaker followed by an analytical approach to help in the comprehension of radiation principles and thus followed by the description of important achievements obtained during the enhancement of the radiometric system. the presented results are intended to be helpful for researchers who may be involved in developing similar radiometric systems. 2. key features of the power circuit breaker the power circuit breaker is a very crucial component of a power system because of its capability in interrupting or applying current with significant amplitude in a reliable way. even though there is a myriad of types of this equipment widely deployed in power systems, this is a sort of equipment to which not many designers are dedicated, in comparison to power transformers or synchronous machines, for example. one of the possible reasons may rely on the fact that because of the inherent presence of an electric arc to be quenched, designing a power circuit breaker requires simultaneous deep knowledge in both mechanical and electrical engineering, with emphasis on plasma dynamics, heat diffusion and radiation and statics & dynamics of metallic structures from the mechanical engineering and electric arc processes, electric field grading and electric charging of dielectric surfaces from the electrical engineering. by the way, these are some reasons for the existence of different types of circuit breakers based on the dielectric material deployed in the chamber in view of arc quenching that may be compressed air, mineral oil, and sf6, among others less usual. in addition, it must be considered the non-linearity in the electrical behavior of the arc, which makes power circuit breaker design to be rather complex. for further details on power-circuit breaker design, some suggested references are [7,9] so, the combination of the described features and the inherent increase in power demand justifies why a power circuit breaker must have its operating condition periodically evaluated. on the other hand, it is hard to define accurately the period for this evaluation because of the inherently stochastic nature of the point of the sinusoidal voltage or current at which the contacts of the power circuit breaker poles will start to unite or separate. this means that arcs with different values of the time interval of existence are decisive for compromising the circuit breaker status. therefore, radiometric measurements that have been shown as being a promising technique for enabling the definition of the right time for the maintenance of the circuit breaker to take place have some barriers to be overcome, yet. this is what is presented and proposed in this work, in which some key achievements in the second generation of radiometric systems applied to circuit breakers are evidenced. 3. radiation principles for understanding the principles of radiometry, in which electromagnetic signals radiated from the operation of a power circuit breaker are captured by antennas located around the arc quenching chambers of a circuit breaker, consider fig. 1, in which it is suggested that the irradiated signal form an operating circuit breaker chamber can be compared to a dipole antenna operation with a sinusoidal injection of signal to be transmitted. 14 b. a. cezarotto, s. h. l. cabral, t. i. a. h. mustafa, c. a. pitz, c. gili, et al. copyright ©2025 assa adv. in systems science and appl. (2025) fig. 1. radiated signal from the chamber of an operating power circuit breaker and its resemblance to a dipole antenna radiating a sinusoidal signal in this case, due to the favorable geometry, spherical coordinates are used to analyze the behavior of magnetic and electric fields that are originated from the current, i, of an elementary section of the antenna, dl, as shown in fig. 2 and adapted from [10]. fig. 2. application of spherical coordinates to analyze the behavior of the dipole antenna the elementary sinusoidal current segment, i dl, will give rise to an angular component of the magnetic field, �⃗̇� , an azimuthal, and a radial component of the electric field, �⃗̇� , and �⃗̇� , respectively. thus, the power density of the irradiated signal, which is the poynting vector, has two components, �̇⃗� , and �̇⃗� . besides the value of its amplitude, which is the power density of the irradiated signal, the poynting vector also allows us to understand how receiver elements can be properly adopted and positioned at a distance from the dipole antenna for being more efficient in capturing the propagating signal. initially, by considering a sinusoidal current flowing through the dipole antenna length segment, in fig. 2, the value of the elementary magnetic field vector, 𝑑�̇⃗�, is 𝑑�̇⃗� = µ 4 𝜋 𝐼 𝑒 𝑅 𝑑�⃗� (1) in which 𝐼 𝑒 is the phasor representation of the current i(t) = 𝐼 𝑠𝑒𝑛 𝜔 𝑡 − ,whereas t is the time, in seconds, r is the radial distance to the elementary segment, in meters, u is the value of the speed of light, in m/s,  is the angular frequency , in rad/s, and µo is the magnetic permeability of the vacuum. in turn, the difference 𝑡 − denotes the establishment of standing waves and the phase difference due to their propagation at the speed of light, at each point of radial distance r from the elementary segment carrying the current. so, considering that the magnetic field, �⃗� is related to the magnetic field vector 𝐴 in the following way advances in the development of radiometric measurement system… 15 copyright ©2025 assa. adv. in systems science and appl. (2025) �⃗� = �⃗� × 𝐴 µ (2) and that because of ampère´s law, only its angular component, , exists, in spherical coordinates, the magnetic field vector will be given by �⃗̇� = 𝐼̇ 𝑠𝑒𝑛𝜃 4 𝜋𝑅 1 𝑅 + 𝑗 𝜔 𝑢 𝑒 𝑑𝑙 𝛼 (3) or, in the time domain, �⃗� (𝑡) = 𝐼 𝑠𝑒𝑛𝜃 4 𝜋 𝑅 𝑠𝑒𝑛 𝜔 𝑡 − 𝑅 𝑢 𝑅 + ω cos 𝜔 𝑡 − 𝑅 𝑢 𝑢 𝑑𝑙 (4) for the electric field, �⃗̇�, it can be obtained from �⃗̇� = 1 𝑗𝜔𝜀 ∇ × �⃗̇� (5) thus, there will be components, r, and , for the electric field, given by �⃗̇� = 𝐼 𝑐𝑜𝑠𝜃 2𝜋𝑅 𝜀 1 𝑢 − 𝑗 1 𝑅𝜔 𝑒 𝑑𝑙 (6) for the radial component which, in the time domain, becomes �⃗� (𝑡) = 𝐼 𝑐𝑜𝑠𝜃 2𝜋𝑅 𝜀 𝑠𝑒𝑛 𝜔 𝑡 − 𝑅 𝑢 𝑢 − 𝑐𝑜𝑠 𝜔 𝑡 − 𝑅 𝑢 𝑅𝜔 𝑑𝑙 (7) while the azimuthal component becomes �⃗̇� = 𝐼 𝑠𝑒𝑛𝜃 4𝜋𝑅𝜀 1 𝑢𝑅 + 𝑗 𝜔 𝑢 − 1 𝜔𝑅 𝑒 𝑑𝑙 (8) which looks like this in the time domain �⃗� (𝑡) = 𝐼 𝑠𝑒𝑛𝜃 4𝜋𝑅𝜀 𝑠𝑒𝑛 𝜔 𝑡 − 𝑅 𝑢 𝑢𝑅 + 𝜔 𝑢 − 1 𝜔𝑅 cos 𝜔 𝑡 − 𝑅 𝑢 𝑑𝑙 (9) thus, for obtaining the expression of the poynting vector, s ⃗(t), the following vector product is to be done 𝑆(𝑡) = �⃗� (𝑡) + �⃗� (𝑡) × �⃗� (𝑡) (10) that becomes 𝑆(𝑡) = (𝐼 𝑑𝑙) 32𝜀 (𝜋𝑅) − 𝑠𝑒𝑛2𝜃 𝑅 𝑠𝑒𝑛 𝜔 𝑡 − 𝑅 𝑢 𝑢 − 𝑐𝑜𝑠 𝜔 𝑡 − 𝑅 𝑢 𝑅𝜔 + (1 − 𝑐𝑜𝑠2𝜃) 𝑠𝑒𝑛 𝜔 𝑡 − 𝑅 𝑢 𝑢𝑅 + 𝜔 𝑢 − 1 𝜔𝑅 cos 𝜔 𝑡 − 𝑅 𝑢 × 𝑠𝑒𝑛 𝜔 𝑡 − 𝑅 𝑢 𝑅 + ω cos 𝜔 𝑡 − 𝑅 𝑢 𝑢 (11) based on (4), (6), and (8), it is noticeable that the amplitude of the components of the magnetic and electric fields that radiate from the current element presents a dependence on the inverse proportion of the radial distance, r. this dependence is proportional to the first, second or third power. on the other hand, from (11) it can be seen that for the power density of radiated signal, the dependence on the inverse of the radial distance is proportional to the second, third, and fourth power. because of this form of dependence, concepts such as "near-field" and "farfield" are defined for a certain frequency range. in turn, for radiometric analysis of a circuit 16 b. a. cezarotto, s. h. l. cabral, t. i. a. h. mustafa, c. a. pitz, c. gili, et al. copyright ©2025 assa adv. in systems science and appl. (2025) breaker, for which the near field condition is considered, these expressions are useful for the choice between a magnetic or an electric field-based antenna for playing the role of a signalcapturing sensor device. nonetheless, it is important to consider that these expressions have limited validity because the circuit breaker chamber itself does attenuate the amplitude of the radiated signal. moreover, the presented analysis is based on a unique value for the angular frequency of the sinusoidal current whereas the electric arc has a range of different frequencies. thus, for more accurate analyses the fourier transform of the electric arc signal in the time domain should be considered and superimposition of effects should be applied. finally, in addition to the dependence on the frequency of the conducted current and the radial distance, for the circuit breaker radiometry, it is also important to consider the value of the azimuth angle, , as it implies in defining the preferred positioning for the sensor element to be used as capturing device. for this purpose and based on fig. 2, by starting with the magnetic field, expressed in (4), it is noted that since its amplitude depends directly proportional to the sine of the azimuth angle, , it is deduced that the azimuthal angle of the preferential positioning of the sensor of the magnetic field has 90 degrees. thus, angles close to 180 and 0 degrees are to be avoided. as for the electric field, the radial component, described in (7), has an amplitude value directly proportional to the cosine of the azimuthal angle, which implies an inversion of the preferential positioning of the electric field sensor, compared to that deduced for the magnetic field. however, it is important to consider that this radial electric field component decreases very quickly with the radial distance from the current element, which makes its importance somewhat secondary. on the other hand, as for the azimuthal component of the electric field, as in (8), this shows a smaller decrease in amplitude as a function of the radial distance and dependence directly proportional to the sine of the azimuthal angle, , which indicates a suggestion of positioning of the electric field sensor similar to the one made for the magnetic field sensor. 4. development in radiometry applied to power circuit breaker as presented in [11], the basic arrangement of a radiometric system applied to a power circuit breaker is shown in fig. 3, in which the transient signal radiated from each of the three-phase chambers of a circuit breaker operation is picked up by the capturing device, converted into a voltage signal, and conveyed to the measurement unit through a coaxial cable in view of being detected and properly registered. this transient signal is superimposed to the emi signal inherently present in the whole environment of a power substation with the predominance of power frequency signal. fig. 3. basic arrangement of a radiometric system – electromagnetic transient signal radiated from the chambers of the power circuit breaker is picked up by the signal capturing device and conveyed by a coaxial cable to the measurement & processing unit advances in the development of radiometric measurement system… 17 copyright ©2025 assa. adv. in systems science and appl. (2025) with the aim of showing how the deployment of a radiometric system to a power circuit breaker can be, fig. 4[11] shows three capacitive antennas aligned to each of the three-phase circuits of a 230 kv power substation yard. it is important to notice that the position of any of the antennas cannot violate any of the required clearance distances when installed. in this case, as a measure for the signal trigger not being subject to the inherent power frequency noise, a fourth antenna was deployed. this antenna was an rf dipole antenna, and it was intended to capture only the high-frequency transient signals from the circuit breaker operation for triggering the signal registering. thus, since it does not capture the power frequency noise interference that is inherently present it prevents such a sort of signal may cause any undesirable triggering. although this antenna is not distinguished in fig. 3, it is a signalcapturing device, in the same way, are the three capacitive antennas. still, concerning fig.3, coaxial cables are not so evidently shown in fig. 4 whereas the workbench oscilloscope adopted as measurement & data processing unit was not shown for being installed in the power substation room. fig. 4. practical deployment of the arrangement of a radiometric system – antennas are dully placed on the ground of a 230 kv power substation yard [11] thus, as also presented in [11], figs. 5 and 6.a to 6.c show the waveform of the registered signal radiated from the power circuit breaker chambers during a closing operation of the power circuit breaker. the graphic of fig. 5 corresponds to the triggering signal picked up by the rf dipole antenna, whereas those of figs. 6.a to 6.c correspond to the signal waveforms of each of the three phases captured by the capacitive antennas. fig. 5. irradiated rf pulses captured by the rf dipole antenna during the closing operation of the circuit breaker 18 b. a. cezarotto, s. h. l. cabral, t. i. a. h. mustafa, c. a. pitz, c. gili, et al. copyright ©2025 assa adv. in systems science and appl. (2025) figs. 6.a to 6.c.signal waveforms of captured by the capacitive coupling antennas during the closing operation of the circuit breaker. phases a, b, and c, respectively the presented waveforms represent what can be typically expected for the closing operation. initially, the signal waveform of fig. 5 which is from the rf dipole antenna is free from any power frequency signal but shows only three high-frequency pulses that correspond to the current interruption in each of the three phases. the first of these pulses triggered the registering of the captured signal. on the other hand, regarding figs. 6.a to 6.c, it is noticeable the occurrence of those same three high-frequency pulses but superimposed to power frequency signal captured by the capacitive antennas. based on the analysis of the behavior of the power frequency signals, it seems that the closure of the poles of phase a is the first one followed by phases c and b, respectively. by the way, this last one presents a signal waveform that is significantly interfered with by signals from phases a and c until it may become steady. in turn, all three phase chambers radiate high-frequency pulses that are mutually picked up by the antennas and each of these pulses is associated with a pre-arcing activity during the displacement of pole contacts [12]. these presented results represent a very successful and stimulating case of data acquisition. nonetheless, in any of the programmed tests in the power substation, the success in triggering signals from each of the rare circuit breaker operations was not plain. at that time, the suspects were on the adopted antennas that would be very sensitive to their correct positioning combined with the stochastic nature of the intensity of the radiated signal because of the dependence of this intensity on the instant of the sinusoidal current or voltage at which the switching operation is made. thus, as presented in [6], an attempt for finding an adequate signal-capturing device was done with a rogowski coil-based ferromagnetic device. for this, a set of tests was performed in the high-voltage laboratory of the university of blumenau with several switching operations of a 35 kv / 800 a, oil chamber, sprecher & schuh ® three-phase mv circuit breaker fed by a low voltage/high current transformer. reasons for this choice were its smaller dimension that facilitates its positioning on the ground of the substation yard, low cost, and likely enhancement in the increase in the amplitude of the picked-up signal for capturing both magnetic and electric radiated fields. advances in the development of radiometric measurement system… 19 copyright ©2025 assa. adv. in systems science and appl. (2025) the main result of that attempt is that the key issue to be addressed was not on the capturing device but on the triggering principles of the commercially available oscilloscope combined with the above-mentioned inherently stochastic nature of the intensity of the radiated signal. fig. 7, adapted from a similar one in [6] help to comprehend one of the many possible reasons for some of the circuit breaker operations not causing the oscilloscope triggering. it shows two superimposed different signals, a and b. both signals are composed of a power frequency sinusoidal signal added to a sort of impulse voltage that is irradiated from the opening of a circuit breaker. the sinusoidal aspect of these signals represents the inherent interfering signal that is in the environment of a power substation. thus, since the aim is in registering the transient voltage, the trigger level of the oscilloscope is set to a value that is greater than that of the sinusoidal amplitude. thus, for signal a the peak of the transient voltage impulse is enough to reach the trigger level set by the operator. therefore, this signal is very likely to cause oscilloscope triggering. on the other hand, for signal b the peak is not enough to reach the trigger level and thus it would very rarely cause the oscilloscope trigger. fig. 7. signal waveforms coupling antennas during the closing operation of the circuit breaker therefore, it became clear that the reason for missing those rare events of circuit breaker operation was not on the signal-capturing device but on the trigger principles adopted by the commercially available oscilloscopes, which is essentially the same one adopted by the signal analyzers. therefore, the only option was to develop new equipment, which represents a milestone in the development of the radiometric system applied to circuit breakers by the research group. as presented in [11], the newly developed equipment for replacing the oscilloscope is based on the time rate of the sampled signal that is obtained from the analogic signal that is picked up by the capturing device and conveyed to it where it is discretized. thus, when in the substation test the operator adjusts the minimum level of triggering with the basis of the time derivative of the signal picked up by the capturing device. moreover, because of the importance of evaluating the time interval elapsed between every two subsequent phases' current interruption or application additional data processing through wavelet transform has been implemented. as an example of how the newly developed equipment can properly replace the commercially available oscilloscope, figs. 8.a to 8.c show the results of the most recent tests performed in a 230 kv power substation. specifically, results are from the closing operation of a power circuit breaker and those same passive capacitive antennas were deployed. 20 b. a. cezarotto, s. h. l. cabral, t. i. a. h. mustafa, c. a. pitz, c. gili, et al. copyright ©2025 assa adv. in systems science and appl. (2025) figs. 8.a to 8.c. signal waveforms of the captured and registered for the closing operation of a circuit breaker. phases a, b, and c, respectively thus, whereas the waveform presentation resembles those from a commercially available oscilloscope, the newly developed equipment also processes the registered data through wavelet transform, which allowed a more precise evaluation of the time interval elapsed between any two subsequent starts of the current application. the advantage of this additional feature is that the wavelet transform of the signal evidences the instant at which the current application of each of the phases starts. for the case of figs. 8.a to 8.c although it may seem identifiable that the phase sequence of the current application is a-c-b, because of the significant distortion in the transient waveform of phase b, which is not always possible to happen. in turn, fig. 9 shows the wavelet transform of that same phase b, on which red circles indicate more precisely the instants at which the current application of each of the indicated phases starts. fig. 9. wavelet transform of the waveform of fig. 8.b with indication of phases sequence of operation in this case, the sequence a-c-b is confirmed, and the measured time interval elapsed between phases a and c could be more precisely evaluated as being 2.47 ms and 3.2 ms between phases c and b, which indicates the circuit breaker to be in good condition. despite the encouraging achievements obtained by the research group of which some have been here presented, much is still to be done in adjusting the newly developed equipment for it to reach the desired condition of liability in triggering the electromagnetic signal radiated from the circuit breaker chamber. thus, because of safety concerns as well as logistics, tests for adjusting the new equipment must be done in situ, which is rarely available, for the reasons given in this work. 5. conclusion the development of a new measurement unit device for replacing the commercially available digital has become the priority in view of fully guaranteeing that the transient irradiated signals advances in the development of radiometric measurement system… 21 copyright ©2025 assa. adv. in systems science and appl. (2025) may be captured during every circuit breaker operation. this remarkable conclusion defines the second generation of radiometric measurement systems applied to circuit breaker of which some important results are presented. besides triggering concerns, this work shows how important can be the addition of items to the newly developed equipment as the real-time processing of registered data. in this case, this processing was through wavelet transform which has shown as being a powerful tool for evidencing the time interval elapsed between phase current application or interruption. the advanced phase of tests performed in power substations with a prototype of newly developed equipment for replacing the formerly adopted digital oscilloscope has their results here shown. acknowledgment the authors acknowledge the electric sector research and development program through the project pd-05785-1902/2019 – de1902: “development of a methodology and device to evaluate the performance of high voltage circuit breakers in substation operation, using a non-invasive technique” regulated by the nacional electric energy agency – aneel, in partnership with cpfl transmission and furb, the university of blumenau, brazil, for the support given to this research. references 1. mihalcea, c., munteanu, f., nemes, c. & baiceanu, f. (2019) lifetime availability analysis of high voltage circuit breakers for maintenance optimization, proc. of 8th international conference on modern power systems (mps) (cluj-napoca, romania), 1–6. 2. promseela, l., suwanasri, c., saribut, s., suwanasri, t. & phadungthin, r. (2020) risk assessment for power circuit breaker by using failure modes, effects and criticality analysis, proc. of international conference on power, energy, and innovations (icpei) (chiangmai, thailand), 149–152. 3. de souza, f. c., cabral, s. h. l., thomas, d. w. p. & hausmann, r. (2021) atp analysis of trv applied to the medium voltage circuit of solar and onshore wind power plants, international journal of electrical power & energy systems, 124, 106364. 4. iec. (2017) high-voltage switchgear and controlgear, standard 62271, part 100: alternating-current circuit breakers, [online]. available: https://webstore.iec.ch/en/publication/62785 5. meier, s. d., moore, p. j. & coventry, p. f. (2011). radiometric timing of high-voltage circuit breaker opening operations, ieee transactions on power delivery, 26(3), 1411– 1417. 6. mustafa, t. i. a. h, cabral, s. h. l., meyer, l. h., almaguer, h. a. d., puchale, l. h. b., et al. (2022) the adoption of ferromagnetic device for signal capturing of radiometric measurement systems, proc. of ieee t&d conference (new orleans, la). 7. flurscheim, c. h. (1982) power circuit breaker theory and design, stevenage, uk: iet. 8. giere, s. (2004) vakuumschalttechnik im hochspannungseinsatz [vacuum switching technology in high-voltage applications], göttingen, germany: cuvillier verlag. [in german] 9. garzon, r. d. (2005) high voltage circuit breakers: design and applications, abingdon-on-thames, uk: taylor & francis. 10. bessonov, l. (1968) applied electricity for engineers, moscow, ussr: mir. 11. mustafa, t. i. a. h, cabral, s. h. l., meyer, l. h., almaguer, h. a. d., pitz, c.a., et al. (2022) a trigger method applied to the development of measurement devices for 22 b. a. cezarotto, s. h. l. cabral, t. i. a. h. mustafa, c. a. pitz, c. gili, et al. copyright ©2025 assa adv. in systems science and appl. (2025) radiometric measurement systems, proc. of international conference on electrical, computer, communications and mechatronics engineering (iceccme) (maldives), 1–5. 12. huang, j., geng, y., jin, q., chen, x.,yan, j.. et al. (2022) the influence of the dispersion in pre-arcing time of vacuum circuit breakers on the controlled switching of capacitor banks, proc. of 6th international conference on electric power equipment – switching technology (icepe-st) (seoul, south korea), 206–211. adv syst sci appl 2025; 1:44–54 published online at https://ijassa.ipu.ru. legendre transformation and its applications natalia pavlova1,2 alexey remizov1* 1moscow institute of physics and technology (state university), dolgoprudnyi, russia 2rudn university, moscow, russia abstract: we present the legendre transformation in a geometric way based on the procedure of the legendrian lift. this approach allows us to understand some interesting properties of it, in particular, the reason for the appearance of singularities of dual curves. also we consider application of the legendre transformation to the clairaut differential equation. finally, we say a few words class of contact transformations and present an infinite group of contact transformations different from the legendre transformation. keywords: legendre transformation, duality, contact transformation, contact structure, pedal curve, singular point, clairaut equation introduction the paper has several goals. first, we present the legendre transformation in a geometric way based on the procedure of the legendrian lift. therefore, we start with the legendre transformation of planar curves and then go to the legendre transformation of functions. as we shall see below, this approach has some important advantages in compare with the standard definition of the legendre transformation of functions, which is thrown in the reader’s face without motivations. moreover, the presented geometric approach allows us to understand the meaning of restrictions in the standard definition of the legendre transformation of functions (for example, the convexity) and the reason for the appearance of singularities of dual curves. second, we consider application of the legendre transformation to ordinary differential equations at the example of the clairaut equation. for this, the geometric interpretation of the legendre transformation is more suitable. finally, we embed the legendre transformation into a class of contact transformations and discuss some examples of contact transformations different from it. in particular, we consider the pedal transformation (apparently the earliest example of contact transformations) and with its aid we obtain an infinite group of contact transformations, which was first constructed by sophus lie. it should be remarked that many of results presented in the paper were previously known, but we presented these results from a more general point of view and explained the relationship between them. we start with a small section presented some notions and results about planar curves and their singularities, which are necessary for understanding the main ideas of the paper. ∗corresponding author: alexey-remizov@yandex.ru legendre transformation and its applications 45 1. curves and singularities let γ be an arbitrary curve on the plane given in the form x = ϕ(t), y = ψ(t), (1.1) where ϕ, ψ are smooth (c∞) functions. further, we shall assume that all considered functions and mappings are smooth (c∞) unless otherwise stated. a point of the curve γ corresponding the parameter t0 is called or critical if ϕ′(t0) = ψ′(t0) = 0. (1.2) otherwise a point is called regular. the curve γ is called regular is all its points are regular. in a neighborhood of every regular point, the curve is diffeomorphic to its tangent line, while for singular points it is not true. example 1.1: in various applications, the following types of singular points often appear: x = ϕ(t) = αtn + o(tn), y = ψ(t) = βtn+1 + o(tn+1), αβ ̸= 0. (1.3) fig. 1.1 presents the graphs of such curves for even and for off n. the singular point of the curve (1.3) with n = 2 is called semicubic casp or simply casp. it is proved that for n = 2 or 3 the germ of any curve (1.3) at zero can be brought to the simplest form x = tn, y = tn+1 by means of an appropriate change of variables (x, y) and a change of the parameter t. however, for n ≥ 4 it is not true [6]. fig. 1.1. the curves (1.3) with even n (left) and odd n (right). the picture is taken from [1]. 2. legendre transformation the geometric definition of the legendre transformation requires the notion of the legendrian lift. we start with the latter notion, which is an important mathematical procedure. it can be applied to curves, functions and many other objects. the legendrian lift appears not only in mathematics and physics, but also in various technical applications and even in the living nature. we shall say a few words about it below. 2.1. legendrian lift the lift or 1-graph of the curve γ given by formula (1.1) is the curve γ : x = ϕ(t), y = ψ(t), p = ψ′(t) ϕ′(t) (2.4) copyright © 2025 assa. adv syst sci appl (2025) 46 n. pavlova, a. remizov in 3-dimensional space with the coordinates (x, y, p). this space denoted by j1 is called the space of 1-jets of functions y(x). in the latter equality in (2.4) the denominator ϕ′(t) may vanish, and we have to discuss the correctness of such operation. if ϕ′(t) = 0, but ψ′(t) ̸= 0, we have p = ∞ at the considered point, that is, the tangent direction of the curve γ is parallel to the y-axis. the dividing by zero disappears, if we interchange the x-axis with the y-axis. the situation is more complicated if the curve γ has a singular point where ϕ′(t) = ψ′(t) = 0. however, in the most of interesting cases the function ψ′(t)/ϕ′(t) has the same one-sided limits, and after an appropriate definition at the point itself, the function ψ′(t)/ϕ′(t) becomes continuous. for example, such a situation takes place for the semicubic parabola, see fig. 2.2. fig. 2.2. legendrian lift of planar curves. remark 2.1: the 1-graph of the curve γ does not depend on the choice of the parametrization the curve γ. according to modern biological concepts going back to the famous work of hubel and wiesel (1959), the brain of mammals performs the legendrian lift of apparent contours of flat images perceived by the retina. there exists an area of the primary cortex of the brain assiciated with vision (v1) that contains neurons united in groups with complex hierarchy (columns, hypercolumn). each column is sensitive to a certain point of the retina and a certain tangent direction at this point. flat images perceived by the retina are lifted in the space of 1-jets, and further work of the brain occurs with the lifted objects. see [13] and the references therein. the only difference between the lift performed by the brain and the lift described above is that the brain uses the third coordinate θ = arctan p instead of p. it is very natural: θ is more appropriate for procedure performed biologically, since it is bounded, while p is not. it is worth observing that this theoretical concept helped to construct some inpainting algorithms that allow to recover corrupted images. in these algorithms, the legendrian lift also plays a crucial role: first, we lift the image, which is considered as a family of contours, in 3-dimensional space and then we act to the lifted object a hypoelliptic diffusion associated with the contact structure. for details, see [4, 5]. 2.2. legendre transformation and duality legendre transformation is an automorphism λ : j1 → j1 given by the formula (x, y, p) 7→ (x, y, p ), where the coordinates are connected by the equalities: x = p, p = x, y + y = xp = xp. (2.5) the legendre transformation can be applied to various objects: curves, functions, differential equations, etc. the objects obtained in this way are called dual of the original, they are often denoted by the same symbols with asterisk. dual objects reflect many important properties of their preimages. copyright © 2025 assa. adv syst sci appl (2025) legendre transformation and its applications 47 consider the application of the legendre transformation to the curve γ given by formula (1.1). it is expressed by the diagram presented in fig. 2.3. fig. 2.3. definition of the legendrian lift of a planar curve. in fig. 2.3, the vertical upward arrow denotes the legendrian lift of the curve γ to the space j1 with coordinates (x, y, p). the horizontal arrow denotes the legendre transformation λ : j1 → j1 given by formula (2.5), the vertical downward arrow denotes the projection π : (x, y, p ) 7→ (x, y ). the composition of these three mappings results in the legendre transformation of the curve γ on the (x, y)-plane to a certain curve γ∗ on the (x, y )-plane, which is called dual to γ. the curve γ∗ dual to a curve γ can have singular points even γ is regular. obviously, they appear due to the projection π : (x, y, p ) 7→ (x, y ). example 2.1: let us find the dual curves for the line γ1 : y = at+ b, the parabola γ2 : y = x2, and the cubic parabola γ3 : y = x3. the line γ1 can be given in the parametric form x = t, y = at+ b. hence p = a and, consequently, we get x = a, y = −b, that is, the dual curve γ∗1 is a single point on the plane. the parabola γ2 can be presented in the parametric form x = t, y = t2. hence p = 2t and, consequently, we get x = 2t and y = t2. thus, the dual curve γ∗2 is the parabola y = x2/4. the cubic parabola γ3 can be presented in the parametric form x = t, y = t3. this yields p = 3t2 and x = 3t2, y = 2t3. this shows that the dual curve γ∗3 is the semicubic parabola with the casp at the origin. the casp corresponds to the inflection point of the initial curve γ3. lemma 2.1: the legendre transformation has the following properties. 1. the curve γ∗ dual to the curve γ of the form (1.1) is regular at all points that correspond to points of the curve γ with non-zero curvature κ(t) ̸= 0. the curve γ∗ has simicubic casp at points that correspond to points of γ where κ(t) = 0, but κ′(t) ̸= 0. 2. assume that the curve γ is the graph of a functions y = f(x) such that df dx (0) = · · · = dnf dxn (0) = 0, dn+1f dxn+1 (0) ̸= 0. (2.6) then the curve γ∗ dual to γ has the form (1.3). 3. the legendre transformation λ is an involution, that is, λ2 is the identity transformation. this is equivalent to the relation λ = λ−1, which yields (γ∗)∗ = γ. the proof is by direct calculation. copyright © 2025 assa. adv syst sci appl (2025) 48 n. pavlova, a. remizov the lemma shows that regular points of γ∗ correspond to points where f ′′(x) ̸= 0, while semicubic casps correspond to points where f ′′(x) = 0, f ′′′(x) ̸= 0, i.e., simple (cubic) inflection points of the function f . example 2.2: in fig. 2.4, the curve dual to the graph of sine is presented. (for better visibility, the scale along the x-axis is increased approximately 20 times.) casps of the dual curve correspond to points x = πn with integer n. fig. 2.4. the curve dual to y = sinx. from the left to right: x runs from 0 to 2π, 4π, 7π, respectively. remark 2.2: it is worth observing that there exist other ways to define dual curves. for instance, in algebraic geometry the definition of dual curves is based on the same construction, where the mapping λ given by formula (2.5) is replaced with the mapping (x, y, p) 7→ (x̄, ȳ , p̄ ), where x̄ = x/y, ȳ = 1/y, p̄ = dȳ /dx̄, (2.7) x, y are defined in (2.5). example 2.3: consider the curve dual to the ellipse (x/a)2 + (y/b)2 = 1. it is easy to see that this curve is the hyperbola y 2 − (ax)2 = b2 in the coordinates (x, y ) and the ellipse (ax̄)2 + (bȳ )2 = 1 in the coordinates (x̄, ȳ ). it is not surprising, because all conic sections (which include ellipses and hyperbolas) are projectively equivalent and the coordinates (x, y ) and (x̄, ȳ ) are connected by a projective transformation (2.7). now consider an important case that the curve γ is the graph of a function y = f(x). it is naturally to ask: whether the dual curve γ∗ is the graph of a certain function y = f ∗(x) as well. if so, the function f ∗ is called the dual function for f or the legendre transformation of f . examples considered above show that the dual curve for the graph of a function is not always the graph of a function. lemma 2.1 gives the following sufficient condition: if the function f is strictly convex (f ′′ > 0) or f is strictly concave (f ′′ < 0) on its domain, then the dual curve is the graph of a function, whence the dual function f ∗ exists. however, the function f ∗(x) can be defined not for allx . indeed, sincex = p, the domain of f ∗ coincides with the range of f ′(x), and f ∗ is defined for x that belong to the range of f ′ only. if the function f is strictly convex, its legendre transformation f ∗ can be defined by the formula f ∗(p) = sup x (xp− f(x)), p ∈ r, (2.8) copyright © 2025 assa. adv syst sci appl (2025) legendre transformation and its applications 49 and if f is strictly concave, its legendre transformation f ∗ can be defined by the formula f ∗(p) = sup x (f(x)− xp), p ∈ r, (2.9) where r denoted the range of f ′. remark 2.3: formulas (2.8) and (2.9) are often used as the definition of the legendre transformation of functions. in our opinion, such a definition has a serious drawback compared to the geometric approach presented above: it is not natural and well motivated. remark 2.4: the legendre transformation can be obviously adapted for hypersurfaces and, consequently, for functions depending on any number of variables. in particular, this way brings us to the same formulas (2.8) and (2.9), where x and p belong to the spaces of the same dimension and xp means the dot product in orthonormal coordinates. 2.3. clairaut differential equation the legendre transformation can be applied to differential equations as well. consider the first-order differential equation f (x, y, p) = 0, p = dy/dx. (2.10) passing to the new variables x, y, p by formula (2.5) and substituting the corresponding expressions in (2.10), we obtain a new first-order differential equation, which is called dual to the initial equation (2.10): f ∗(x, y, p ) = 0, p = dy/dx, (2.11) where f ∗(x, y, p ) = f (p,xp − y,x). in fact, here we have to verify the equality p = dy/dx only, which follows from the equality p = dy/dx and formula (2.5). it more convenient to write the both equalities via differential 1-forms and check that 1-forms pdx − dy and pdx− dy vanish simultaneously: pdx − dy = xdp− d(xp− y) = xdp− d(xp) + dy = −(pdx− dy). from geometric viewpoint, this means that the legendre transformation (2.5) sends contact planes pdx− dy = 0 to contact planes pdx − dy = 0, i.e., it preserves the natural contact structure of the space j1. this implies that the legendre transformation sends integral curves of equation (2.10) to integral curves of its dual equation (2.11). in other words, integral curves of equation (2.11) are dual to integral curves of equation (2.10), and vice versa (since the legendre transformation is involution). this fact can be used for solution of differential equation: if the dual equation can be solved easier that the initial one. the notion of duality also helps to study singularities of implicit differential equations; see [7]. a nice example of application the legendre transformation to differential equation is the clairaut equation, which has the form xp− y = f(p), p = dy/dx, (2.12) where f is an arbitrary function. see, for example, [2, 12]. the legendre transformation sends differential equation (2.12) to the equation y = f(x), which does not contain the derivative. therefore, the legendre transformation sends integral curves of equation (2.12) to points of the (x, y )-plane filling the graph of the function y = f(x). taking into account example 2.1, one can see that integral curves of equation (2.12) are straight lines tangent to a curve dual to the graph y = f(x). copyright © 2025 assa. adv syst sci appl (2025) 50 n. pavlova, a. remizov a direct calculation shows that this curve given by formula x = f ′(p), y = pf ′(p)− f(p) (2.13) is the discriminant curve of equation (2.12) and the envelope of the family of solutions of equation (2.12) – its tangent lines. this gives a simple way to get all solutions of equation (2.12): first, we find the discriminant curve (2.13), which is called singular solution of (2.12). the remaining solutions are tangent lines to the discriminant curve taken at all its points. moreover, it is not hard to see that these tangent lines are given by the formula y = cx− f(c), c = const. (2.14) remark that formula (2.14) can be obtained from equation (2.12) after replacing p with c. example 2.4: in fig. 2.5, integral curves of equation (2.12) with the function f(p) = p3 are presented. in this case, the discriminant curve (2.13) is the semicubic parabola x = 3p2, y = 2p3 (2.15) with the casp at the origin. the family of tangent lines to the curve (2.15) has concentration (patch) near this curve and especially near its cusp. this phenomenon is called a caustic [3]. in physics, a caustic is the envelope of light rays which have been reflected or refracted by a curve (on the plane) or a surface (in the space). fig. 2.5 contains a caustic – the semicubic parabola (2.15). fig. 2.5. integral curves of the clairaut equation xp− y = p3 are tangent lines to the semicubic parabola (2.15). the picture is taken from [10] finally, let us consider the clairaut equation of more general type: f (p, xp− y) = 0, p = dy/dx, (2.16) where f (u, v) is an arbitrary function of two variables. the legendre transformation sends (2.16) to the equation f (x, y ) = 0 without derivative. proceeding in the same way as before, we conclude that integral curves of equation (2.16) are straight lines tangent to the curve dual to f (x, y ) = 0. the equation f (x, y ) = 0 defined the discriminant curves of equation (2.16), which is its singular solution and the envelope of other solutions – tangent lines given by the formula y = ax+ b, f (a,−b) = 0, which is a generalization of (2.14). copyright © 2025 assa. adv syst sci appl (2025) legendre transformation and its applications 51 example 2.5: consider the equation p3 = (xp− y)2, p = dy/dx, (2.17) which has the form (2.16) with f (u, v) = u3 − v2. the curve dual to the curve u3 − v2 = 0 is the cubic parabola y = 4 27 x3, which is the discriminant curve of equation (2.17) and its singular solutions. the family of non-singular solutions of (2.17) consists of all tangent lines to the singular solution, which are given by the formula y = ax+ b, a3 = b2. the latter equation can be written in the form a3 = (ax− y)2, a = const, (2.18) or in the form y = c2x+ c3, where a = c2. formula (2.18) can be obtained from equation (2.17) by replacing the derivative p with the constant c. 3. contact transformations the legendre transformation is not a unique transformation j1 → j1 preserving the contact structure of this space. there exist a large group of diffeomorphisms of the space that also preserve the contact structure, they are called contact transformations. equivalent definition: contact transformations are transformations of curves in the plane in which tangent curves are transformed into tangent curves. let us try to determine condition under which a diffeomorphism (x, y, p) → (x, y, p ) (global or local) preserves the contact structure of the space j1. namely, consider a diffeomorphism (x, y, p) → (x, y, p ) given by the formula x = f (x, y, p), y = g(x, y, p), p = h(x, y, p), (3.19) where the functions f,g satisfy some relation (that we shall find) and the function h is uniquely defined by f,g due to the condition p = dy/dx . therefore, the condition of preservation the contact structure has the form h(x, y, p) = dy dx = gxdx+gydy +gpdp fxdx+ fydy + fpdp = gx + pgy + p′gp fx + pfy + p′fp , (3.20) where p′ = dp/dx. since p = h(x, y, p) does not depend on p′, the right hand side of equality (3.20) does not depend on p′ as well. this yields gx + pgy fx + pfy ≡ gp fp or, equivalently, fp(gx + pgy) ≡ gp(fx + pfy). (3.21) the identity (3.21) connecting the functions f and g is a necessary and sufficient condition for the diffeomorphism (3.19) be a contact transformation. it is not hard to see that the legendre transformation satisfies (3.21). the legendre transform is the most commonly used and the most useful contact transformation, but historically it was not the first one. apparently, the first contact transformation was the so-called pedal transformation, which is defined as follows. we shall copyright © 2025 assa. adv syst sci appl (2025) 52 n. pavlova, a. remizov give another example of contact transformations different from the legendre transform – socalled pedal transformations, which were investigated by colin maclaurin, arthur cayley, and sophus lie. to define pedal transformation, let us fixe an arbitrary pointo on the plane (without loss of generality,o can be the origin). the pointo is called the pole or center of the transformation. let γ be a curve on the plane. then the pedal curve of γ is the locus of points x so that the line ox is perpendicular to the tangent line to γ passing through the point x . example 3.1: the pedal curve of the circle is the limacon (pascal’s snail), the special shape of which is determined by the distance ρ between the pole o and the circle (fig. 3.6). considering ρ as a parameter, we get a family of the pedal curves, some of which have singular points. namely, when the pole o lies on the circle itself (ρ = 0), the pedal curve has the casp at o, this curve is called cardioid (the third figure from the left). fig. 3.6. the pedal curves of the circle (depicted as the dotted line) are limacons, whose shape depends on the distance between the pole o and the circle lemma 3.1: in the cartesian coordinate system centered at o, the pedal transformation is given by the formula p : (x, y, p) 7→ (x̄, ȳ , p̄ ), where x̄ = xy/(1 +x2), ȳ = −y/(1 +x2), p̄ = dȳ /dx̄, (3.22) the relation between x, y, p and x, y, p is given in (2.5). the proof is by direct calculation. in contrast to the legendre transformation λ, the pedal transformation p is not an involution. therefore, one can consider the degrees (compositions) of the pedal transformation. fixing the pole o, consider the cyclic group generated by the elements pn with all integers n: . . . p−3, p−2, p−1, p0, p1, p2, p3 . . . a natural question: is thus group finite or infinite? we will get an answer below. sophus lie suggested a nice way how to extend this discrete grout to a continuous group. for this, consider the polar coordinates (r, ϕ) on the (x, y)-plane and the polar coordinates (r,φ) on the (x̄, ȳ )-plane. as the third coordiante in the space j1 let us take the difference between the angle of inclination of the radius vector to a given point on the plane and the angle of inclination of the tangent to the curve at this point such that the orders of subtraction in the preimage and image are opposite: x = r cosϕ, y = r sinϕ, ψ = ϕ− arctan p, x̄ = r cosφ, ȳ = r sinφ, ψ = arctan p̄ − φ, (3.23) copyright © 2025 assa. adv syst sci appl (2025) legendre transformation and its applications 53 where x, y, p and x̄, ȳ , p̄ are connected via (3.22). it is not hard to check that in the coordinates (3.23), the transformation pn with every integer n is given by the formula r = r| sinψ|n, φ = ϕ− n(ψ + π 2 ), ψ = ψ. (3.24) the obtained formulas show shows that pn ̸= pm for any n ̸= m. therefore, the cyclic group {pn} is infinite. if we consider the same formula (3.24) with real n and the operation pn · pm = pn+m, we obtain a continuous group of contact transformations. this group was constructed by sophus lie. we will not go deep into the theory of contact transformations, referring the interested reader to the specialized literature, which is quite extensive. for example, see [8, 9, 11]. 4. conclusion we presented the legendre transformation in a geometric way, which is based on the notion of the legendrian lift. this approach is more natural than the standard definition for the legendre transformation of functions. indeed, it is applicable to various objects (including functions, curves, differential equations) and it shows that the legendre transformation of functions is a partial case of the legendre transformation of curves. we also considered the legendre transformation to ordinary differential equations at the example of the clairaut equation and discussed its wonderful properties. it was shown that the legendre transformation is a member of a class (group) of contact transformations. we presented several examples of different contact transformations: the pedal transformation and an infinite group of contact transformations. references 1. agakhanova, ya. s., nikachev, a. s., & oblasova, i. n. (2023) on codimension 3 singular points of first-order implicit differential equations, adv. syst. sci. appl., 23(1), 8–21. 2. arnol’d, v. i. (1988) geometrical methods in the theory of ordinary differential equations, berlin: springer. 3. arnol’d, v. i. (1990) singularities of caustics and wave fronts, math appl. (soviet series), 62. kluwer academic publishers. 4. boscain, u., chertovskih, r. a., gauthier, j. p., & remizov, a. o. (2014) hypoelliptic diffusion and human vision: a semidiscrete new twist, siam j. imaging sci., 7(2), 669– 695. 5. boscain, u., chertovskih, r. a., gauthier, j. p., prandi, d., & remizov, a. o. (2018) cortical-inspired image reconstruction via sub-riemannian geometry and hypoelliptic diffusion, esaim, proc. surv., 64, 37–53. 6. bruce, j. w., gaffney t. j. (1982) simple singularities of mappings c, 0 → c2, 0, j. lond. math. soc., ii. ser., 26, 465–474. 7. bruce, j. w., tari, f. (2000) duality and implicit differential equations, nonlinearity, 13:3, 791–811. 8. feld, j. m. (1940) a continuous group of contact transformations containing the generalized pedal transformation, tohoku math. j., 46, 252–260. 9. gizatullin, m. kh.(2008) klein’s conjecture for contact automorphisms of the threedimensional affine space, mich. math. j., 56(1), 89–98. 10. izumiya, s., kurokawa, y. (1994) on systems of clairaut type, kodai math. j., 17(3), 636–643. copyright © 2025 assa. adv syst sci appl (2025) 54 n. pavlova, a. remizov 11. lie, s. (1875) foundations of an invariant theory of contact transformations, math. ann., 8, 215–303. 12. mitrinović, d. s., kečkić, j. d. (1981) variations and generalisations of clairaut’s equations, univ. beograd. publ. elektrotehn. fak. ser. mat. fiz., 716/734, 11–21. 13. petitot, j. (2003) the neurogeometry of pinwheels as a sub-riemannian contact structure, j. physiol. paris, 97, 265–309. copyright © 2025 assa. adv syst sci appl (2025) curves and singularities legendre transformation legendrian lift legendre transformation and duality clairaut differential equation contact transformations conclusion adv syst sci appl 2023; 04:90–103 published online at https://ijassa.ipu.ru. global stability and sensitivity analysis of basic reproduction number of a malaria model adesoye i. abioye1, olumuyiwa j. peter1,2*, festus a. oguntolu3, tawakalt a. ayoola4, asimiyu o. oladapo4 1department of mathematical and computer sciences, university of medical sciences, ondo city, ondo state, nigeria 2department of epidemiology and biostatistics, school of public health, university of medical sciences, ondo city, ondo state, nigeria 3department of mathematics, federal university of technology, minna, niger state, nigeria 4department of mathematical sciences, osun state university, osogbo, osun state, nigeria abstract: this paper explores a mathematical model of malaria, focusing on the basic reproduction number (r0) and employing lyapunov functions to assess the global stability of disease-free and endemic equilibria. sensitivity analysis of key parameters is conducted to evaluate their impact on disease control. the results indicate an active malaria outbreak with decreasing human classes signifying disease progression and increasing mosquito classes suggesting heightened transmission risk. effective control measures, including mosquito control and treatment of infected individuals, are essential to mitigate the outbreak. keywords: malaria, sensitivity, global stability analysis, numerical simulations, lyaponov function 1. introduction malaria is a life threatening tropical disease. it is both preventable and treatable. a case of simple malaria, however, can escalate to a severe form of the disease, which is generally fatal if not treated promptly. this risk of infection is higher in some locations than others, depending on a variety of circumstances, including the species of local mosquitoes. it may also vary according to season, with the danger being greatest in tropical nations during the rainy season [6, 17]. the risk of disease can be lowered by preventing mosquito bites with mosquito nets and insect repellents, or by using mosquito-control techniques such as spraying insecticides. malaria is still frequent in tropical and subtropical nations, despite the fact that it is uncommon in temperate climes. globally, the world health organization estimates that 241 million clinical cases of malaria will occur in 2020, with 627,000 deaths, the majority of whom would be children in africa. the first signs of malaria usually appear after 10-15 days of being bitten by an infected mosquito. fever, headache, and chills are common symptoms, however they could be minor and difficult to distinguish from malaria. in malaria-endemic settings, people who have developed partial immunity may become infected yet exhibit no symptoms [12, 13]. over the years, precise frameworks for comprehending the dynamics of malaria transmission in the human population have been developed using mathematical ∗corresponding author: peterjames4real@gmail.com global stability and sensitivity analysis of basic reproduction number ... 91 models [3–5, 11]. the model developed by [2] ia considered in this study in order to determine the effect of the sensitive parameters to the disease control. further studies on malaria models can be found in [2].the motivation for this study stems from the persistent threat of malaria as a life-threatening disease transmitted by mosquitoes, posing a significant public health challenge for many years. recognizing the gravity of the issue, this research endeavors to explore a sophisticated mathematical model of malaria. by incorporating the fundamental epidemiological indicator, the basic reproduction number (r0), and employing lyapunov functions, we aimed to gain insights into the global stability of disease-free and endemic equilibria. furthermore, the study’s motivation extended to conducting a sensitivity analysis of key parameters, shedding light on their influence in the context of disease control. the findings revealing an active malaria outbreak, characterized by diminishing human populations and burgeoning mosquito populations, underscore the urgency for effective control measures. ultimately, the study’s motivation lies in the imperative need to address this ongoing health crisis through measures such as mosquito control and timely treatment of infected individuals, with the ultimate goal of mitigating the outbreak and safeguarding public health. the paper is structured as follows: section two deals with the models formulation, section three deals with the basic analysis of the model, section four deals with the sensitivity analysis of the model and discussion of result. finally, we concluded the study in section five. 2. model formulation this model developed is an extension of [2] by considering the global stabilities and sensitivity analysis of the model. according to [2], the total population of humans, nh(t) which is known as the host comprises of susceptible class of human (sh(t)), exposed class of human (eh(t)), infectious class of human (ih(t)) and recovered class of human (rh(t)). this can be defined as nh(t) = sh(t) + eh(t) + ih(t) +rh(t) similarly, the total population of mosquitoes, nv(t) which is also known as vector comprises of susceptible class of mosquito (sv(t)), exposed class of mosquito (ev(t)) and infectious class of mosquito (iv(t)). this can be defined as nv(t) = sv(t) + ev(t) + iv(t) moreover, the parameters of human and mosquito populations show the movement of individual at different rates from one class to another. this can be defined below as follows; λh and λv represent the recruitment rates of humans and mosquitoes respectively, σh and σv represent the the probability of transmission of malaria per mosquito bite to susceptible humans and from infected humans to susceptible mosquitoes respectively. µ and η represent natural mortality rates of humans and mosquitoes respectively. also, υ is the developing rate of exposed humans and ψ is the rate of the newborn birth with malaria from the mother’s womb. εh and εv represent the proportion of an antibody produced by human in response to the incidence of infection caused by mosquito and produced by mosquito in response to the incidence of infection caused by human respectively. δh and δv represent the disease-induced mortality rates of humans and mosquitoes respectively. ξ is the biting rate of mosquitoes and α is the developing rate of exposed mosquitoes. γ1 and γ2 represent the rate of loss of immunity in humans and relapse rate of humans respectively. ω is the recovery rate of of humans. 2.1. flow chart of the model the diagram below is the flow chart of the model. copyright © 2023 assa. adv syst sci appl (2023) 92 a.i. abioye, et. al fig. 2.1. flow chart diagram 2.2. differential equations of the model the first order differential equations can be derived from the flow chart model as follows: dsh dt =λh − ξσhivsh 1 + ϵhiv − µsh + γ1rh deh dt = ξσhivsh 1 + ϵhiv − (υ + µ)eh dih dt =υeh − (ω + µ+ δh)ih + ψih + γ2rh drh dt =ωih − (γ1 + γ2 + µ)rh dsv dt =λv − ξσvihsv 1 + ϵvih − ηsv dev dt = ξσvihsv 1 + ϵvih − (α + η)ev div dt =αev − (η + δv)iv (2.1) 3. existence of equilibrium points it is very important to study the equilibrium points of the model (2.1) which comprises of disease free equilibrium (dfe) point and endemic equilibrium point (eep). therefore, to show that the dfe exists by setting the right hand sides of each of the equations in model (2.1) to be zero and substituting eh = ih = rh = ev = iv = 0 to obtain (sh, eh, ih, rh, sv, ev, iv) = ( λh µ , 0, 0, 0, λv η , 0, 0 ) (3.2) also, for the endemic, setting the right hand sides of the equations in model (2.1) to be zero and substituting sh = s∗ h, eh = e∗ h, ih = i∗h, rh = r∗ h, sv = s∗ v , ev = e∗ v and iv = i∗v to copyright © 2023 assa. adv syst sci appl (2023) global stability and sensitivity analysis of basic reproduction number ... 93 obtain s∗ h = (λh + γ1r ∗ h)(1 + ϵhi ∗ v ) µ(1 + ϵhi∗v ) + ξσhi∗v e∗ h = ξσhi ∗ vs ∗ h (1 + ϵhi∗v )(υ + µ) i∗h = υe∗ h + γ2r ∗ h ω + µ+ δh − ψ r∗ h = ωi∗h γ1 + µ+ γ2 s∗ v = λv(1 + ϵvi ∗ h) η(1 + ϵvi∗h) + ξσvi∗h e∗ v = ξσvi ∗ hs ∗ v (1 + ϵvi∗h)(α + η) i∗v = αe∗ v η + δv (3.3) therefore, eq. (3.2) has shown that we can have two equilibriums in the non-negative of r7 + thus, the existence of disease free equilibrium (dfe) point e0 = ( λh µ , 0, 0, 0, λv η , 0, 0 ) and eq. (3.3) has shown the existence of endemic equilibrium point (eep), e1 = (s∗ h, e ∗ h, i ∗ h, r ∗ h, s ∗ v , e ∗ v , i ∗ v ) 3.1. basic reproduction number (r0) this section describes how to derive basic reproduction number r0 of the disease free equilibrium (dfe) using next generation matrix method. according to [18], if f denotes the rate of appearance of new infections and v denotes the rate of transfer of the dfe, then fv−1 is called ”next generation matrix”. therefore f(x) =  ξσhivsh 1 + ϵhiv 0 ξσvihsv 1 + ϵvih 0 0 0 0  and v(x) =  (υ + µ)eh (ω + µ+ δh − ψ)ih − υeh − γ2rh (α + η)ev (η + δv)iv − αev µsh − λh − γ1rh (γ1 + µ+ γ2)rh − ψih ηsv − λv  copyright © 2023 assa. adv syst sci appl (2023) 94 a.i. abioye, et. al solving the above matrices using jacobian matrix at the disease free equilibrium, e0 =( λh µ , 0, 0, 0, λv µ , 0, 0 ) , we obtain f =  0 0 0 ξσhλh µ 0 0 0 0 0 ξσvλv η 0 0 0 0 0 0  and v =  υ + µ 0 0 0 −υ ω + µ+ δh − ψ 0 0 0 0 α + η 0 0 0 −α η + δv  respectively. thus, [18] basic reproduction number r0 means the predominant eigenvalue equivalent to the spectral radius of matrix fv−1. mathematically, r0 = ρ(fv−1) where ρ is the spectral radius. therefore; r0 = √ υξσhλh µ(υ + µ)(ω + µ+ δh − ψ) αξσvλv η(α + η)(η + δv) (3.4) 3.2. global stability analysis of disease-free equilibrium theorem 3.1: if r0 ≤ 1 then the disease free equilibrium e0 given by equation (3.4) is globally asymptotically stable. otherwise, it is unstable. proof let us formed lyapunov function of the type l = υeh (υ + µ)(ω + µ+ δh − ψ) + ih (ω + µ+ δh − ψ) + ηr0ev ξσvλv + η(α + η)r0iv ξσvαλv (3.5) it is easy to identify that the coefficient of each state variables of the above eq. (3.5) are non-negative. thus, differentiating (3.5) as well as substitution model (2.1) to obtain l̇ = υ (υ + µ)(ω + µ+ δh − ψ) ( ξσhivsh 1 + ϵhiv − (υ + µ)eh ) + 1 (ω + µ+ δh − ψ) (υeh − (ω + µ+ δh − ψ)ih + γ2rh)+ ηr0 ξσvλv ( ξσvihsv 1 + ϵvih − (α + η)ev ) + η(α + η)r0 ξσvαλv (αev − (η + δv)iv) (3.6) it is assume that if ϵh = 0 then the saturated incidence ( ξσhivsh 1 + ϵhiv ) becomes bilinear incidence (ξσhivsh). similarly, if ϵv = 0 then the saturated incidence ( ξσvihsv 1 + ϵvih ) becomes bilinear incidence (ξσvihsv) [4, 7, 10, 14]. epidemiologically, it is assumed that, in humans, copyright © 2023 assa. adv syst sci appl (2023) global stability and sensitivity analysis of basic reproduction number ... 95 the rate of antibodies produced against the antigens of human is zero (i.e. ϵh = 0) and in mosquitoes, the rate of antibodies produced against the antigens of mosquitoes is zero (i.e. ϵv = 0). thus, eq. (3.6) becomes l̇ = υ (υ + µ)(ω + µ+ δvh− ψ) (ξσhivsh − (υ + µ)eh) + 1 (ω + µ+ δh − ψ) (υeh − (ω + µ+ δh − ψ)ih + γ2rh) + ηr0 ξσvλv (ξσvihsv − (α + η)ev) + η(α + η)r0 ξσvαλv (αev − (η + δv)iv) l̇ = ( υξσhsh (υ + µ+ δh)(ω + µ+ δh − ψ) − η(α + η)(η + δv)r0 ξσvλvα ) iv + ( ηr0sv λv − 1 ) ih + ( γ2 (ω + µ+ δh − ψ) ) rh ≤ [ υξσhλh µ(υ + µ+ δh)(ω + µ+ δh − ψ) − η(α + η)(η + δv)r0 ξσvλvα ] iv + (r0 − 1) ih = [(√ υσhλhη(α + η)(η + δv) ασvλvµ(υ + µ+ δh)(ω + µ+ δh − ψ) ) iv + ih ] (r0 − 1) if l̇ ≤ 0 on condition that r0 ≤ 1 in addition to l̇ = 0 as long as r0 = 1 or ih = 0 and iv = 0 from the above result. it shows that the supreme invariance set in {(sh, eh, ih, rh, sv, ev, iv) ∈ r7 + : l̇ = 0} is the singleton disease free equilibrium (e0). according to [9], disease free equilibrium (e0) is globally asymptotically stable in r7 +. 3.3. global stability analysis of endemic equilibrium theorem 3.2: if r0 > 1, then equation (2.1) has a unique endemic equilibrium, then e1 = (s∗ h, e∗ h, i∗h, r∗ h, s∗ v , e∗ v , i∗v ) every time r0 > 1 and otherwise there is no endemic equilibrium. proof according to [4, 7, 10, 14], we have defined in eq. (3.3), the existence of eep as e1 = (s∗ h, e ∗ h, i ∗ h, r ∗ h, s ∗ v , e ∗ v , i ∗ v ). it is assume that if ϵh = 0 then the saturated incidence( ξσhivsh 1 + ϵhiv ) becomes bilinear incidence (ξσhivsh). similarly, if ϵv = 0 then the saturated incidence ( ξσvihsv 1 + ϵvih ) becomes bilinear incidence (ξσvihsv) [4,7,10,14]. epidemiologically, it is assumed that, in humans, the rate of antibodies produced against the antigens of human is zero (i.e. ϵh = 0) and in mosquitoes, the rate of antibodies produced against the antigens of mosquitoes is zero (i.e. ϵv = 0). therefore, eq. (3.3) becomes; s∗ h = λh + γ1r ∗ h µ+ ξσhi∗v ; e∗ h = ξσhi ∗ vs ∗ h (υ + µ) ; i∗h = υe∗ h + γ2r ∗ h ω + µ+ δh − ψ ; r∗ h = ωi∗h γ1 + µ+ γ2 ; s∗ v = λv η + β∗ v ; e∗ v = β∗ vs ∗ v (α + η) ; i∗v = αe∗ v η + δv (3.7) copyright © 2023 assa. adv syst sci appl (2023) 96 a.i. abioye, et. al let β∗ h = ξσhi ∗ v and β∗ v = ξσvi ∗ h (3.8) from eq. (3.7) and eq. (3.8), make i∗h subject of formula by substitution method to obtain i∗h = (υλha6 + υγ1ωi ∗ h + a2ωγ2i ∗ h) β ∗ h + a2µωγ2i ∗ h a2a3a6(µ+ β∗ h) (3.9) where a1 = ξ2υλhσhλvσvα, a2 = (υ + µ), a3 = (ω + µ+ δh − ψ), a4 = (α + η), a5 = (η + δv) and a6 = (γ1 + γ2 + µ). similarly, from eq. (3.7) and eq. (3.8), β∗ h can also be written as β∗ h = ξ2λvσhσvαi ∗ h (η + δv)(α + η)(η + ξσvi∗h) (3.10) therefore, eq. (3.9) and eq. (3.10) can now be written as p1i ∗ h + p2 = 0 (3.11) where p1 = (a3η (υγ1ω + a2γ2ω − a2a3a6)r2 0 + ξυλhσv(ωγ2 − a3a6)) and p2 = υλhη (a3a6(r2 0 − 1) + ωγ2) from the above result, eq. (3.11) can be rewritten as i∗h = −p2 p1 ≤ 0 if p2 ≥ 0 at r0 ≤ 1, and there is no endemic equilibrium. furthermore, i∗h = p2 p1 > 0 if p2 < 0 at r0 > 1. thus, ∃ an endemic equilibrium only at r0 > 1. this means that eq. (2.1) has a unique endemic which is non-negative equilibrium anytime r0 > 1. theorem 3.3: if r0 > 1, then e1 which is the endemic equilibrium defined as e1 = (s∗ h, e ∗ h, i ∗ h, r ∗ h, s ∗ v , e ∗ v , i ∗ v ) is globally asymptotically stable in the neighborhood of the region r7 +. proof let us construct goh-volterra type lyapunov function [4, 7, 10, 14]: g =sh − s∗ h − s∗ h ln sh s∗ h + eh − e∗ h − e∗ h ln eh e∗ h + c1 ( ih − i∗h − i∗h ln ih i∗h ) + c2 ( sv − s∗ v − s∗ v ln sv s∗ v ) + c3 ( ev − e∗ v − e∗ v ln ev e∗ v ) + c4 ( iv − i∗v − i∗v ln iv i∗v ) where c1 = ξσhs ∗ hi ∗ v υe∗ h c2 = c3 = σhs ∗ hi ∗ v σvs∗ vi ∗ h and c4 = ξσhs ∗ hi ∗ v αe∗ v . the derivation of g with respect to time is; ġ = ( 1− s∗ h sh ) ṡh + ( 1− e∗ h eh ) ėh + ξσhs ∗ hi ∗ v υe∗ h ( 1− i∗h ih ) i̇h + σhs ∗ hi ∗ v σvs∗ vi ∗ h( 1− s∗ v sv ) ṡv + σhs ∗ hi ∗ v σvs∗ vi ∗ h ( 1− e∗ v ev ) ėv + ξσhs ∗ hi ∗ v αe∗ v ( 1− i∗v iv ) i̇v (3.12) copyright © 2023 assa. adv syst sci appl (2023) global stability and sensitivity analysis of basic reproduction number ... 97 to simplify the eq. (3.3) and consider rh → r∗ h as time, t→ ∞, then we have; λh =ξσhs ∗ hi ∗ v + µs∗ h (υ+µ) = ξσhs ∗ hi ∗ v e∗ h (ω+µ+ δh − ψ) = υe∗ h i∗h λv =ξσvs ∗ vi ∗ h + ηs∗ v (α+η) = ξσvs ∗ vi ∗ h e∗ v (η+δv) = αe∗ v i∗v (3.13) substituting eq. (2.1) and eq. (3.13) into eq. (3.12) when (rh → r∗ h) as time, t→ ∞ to obtain ġ =µs∗ h ( 2− s∗ h sh − sh s∗ h ) + ξσhs ∗ hi ∗ v ( 1− s∗ h sh ) + ξσhs ∗ hi ∗ v ( 1− shive ∗ h s∗ hi ∗ veh ) + ξσhs ∗ hi ∗ v ( 1− i∗heh ihe∗ h ) + ησhs ∗ hi ∗ v σvi∗h ( 2− s∗ v sv − sv s∗ v ) + ξσhs ∗ hi ∗ v ( 1− s∗ v sv ) + ξσhs ∗ hi ∗ v ( 1− svihe ∗ v s∗ vi ∗ hev ) + ξσhs ∗ hi ∗ v ( 1− i∗vev ive∗ v ) (3.14) thus, this can be reduced to ġ =µs∗ h ( 2− s∗ h sh − sh s∗ h ) + ησhs ∗ hi ∗ v σvi∗h ( 2− s∗ v sv − sv s∗ v ) + ξσhs ∗ hi ∗ v( 6− s∗ h sh − i∗heh ihe∗ h − shive ∗ h s∗ hi ∗ veh − s∗ v sv − i∗vev ive∗ v − svihe ∗ v s∗ vi ∗ hev ) (3.15) finally, if arithmetic mean (am) ≥ geometric mean (gm), then the following inequalities satisfy; 2−s ∗ h sh − sh s∗ h ≤ 0, 2− s∗ v sv − sv s∗ v ≤ 0,( 6− s∗ h sh − i∗heh ihe∗ h − shive ∗ h s∗ hi ∗ veh − s∗ v sv − i∗vev ive∗ v − svihe ∗ v s∗ vi ∗ hev ) ≤ 0 (3.16) as a result of this, ġ ≤ 0 for r0 > 1. thus, all the parameters are positive with ġ = 0 as long as sh = s∗ h, eh = e∗ h, ih = i∗h, sv = s∗ v , ev = e∗ v , iv = i∗v . in the meantime, rh → r∗ h as time, t→ ∞ and so [9] the endemic equilibrium (e1) is globally asymptotically stable whenever r0 > 1. 3.4. numerical simulation we conduct a numerical simulation using the developed model to explore the dynamics of malaria within the human population. our simulation utilizes the runge-kutta 4 (r-k 4) scheme integrated into maple 2016, with parameter values specified in table 4.2 and initial conditions as described in reference [2]. figures 3.2(a) to 3.2(g) shows the disease copyright © 2023 assa. adv syst sci appl (2023) 98 a.i. abioye, et. al trend in each population. the results indicate an active malaria outbreak. the decreasing human classes reflect the progression of the disease within the human population, while the increasing mosquito classes suggest that mosquitoes are becoming more infectious, increasing the risk of further transmission to humans. effective control measures, such as mosquito control and treatment of infected individuals, may be needed to mitigate the outbreak. 4. sensitivity analysis 4.1. sensitivity analysis of basic reproduction number r0 parameters this model is of absolute imperative to examine the sensitivity analysis of the parameter standards. sensitivity analysis can be used to determine which of the parameters from basic reproduction number r0 has more influence on the disease spread. according to [7–9,15,16], sensitivity index (s.i) analysis is an essential methodology for virtually all other sensitivity analysis approaches. this is also known as elasticity index. this can be done by using partial derivatives as soon as the variable is a differentiable function of the parameter. the normalized forward sensitivity index (s.i) of a variable or quantity r0 to a parameter α is a ratio of the relative change in the variable or quantity r0 to the relative change in the parameter α. theorem 4.1: the normalized forward sensitivity index (s.i) of a variable or quantity r0 that depends differentiable on a parameter, α, is defined as: xr0 θ = ∂r0 ∂θ × θ r0 (4.17) sensitivity index of the parameters in eq. (3.4) is to use the formula in eq. (3.5). therefore, the sensitivity index (s.i) for the parameters in eq. (3.4) can be written as follows: ∂r0 ∂λh × λh r0 = 1 2 , ∂r0 ∂λv × λv r0 = 1 2 ; ∂r0 ∂δh × δh r0 = − δh 2(ω + µ+ δh − ψ) ; ∂r0 ∂δv × δv r0 = − δv 2(δv + η) ; ∂r0 ∂ψ × ψ r0 = ψ 2(ω + µ+ δh − ψ) ; ∂r0 ∂υ × υ r0 = µ 2(υ + µ) ; ∂r0 ∂µ × µ r0 = 1 2 (2µ (ψ − ω − υ − δh) + υ (ψ − ω − δh)− 3µ2) (υ + µ) (ω + µ+ δh − ψ) ; ∂r0 ∂ξ × ξ r0 = 1; ∂r0 ∂σh × σh r0 = 1 2 ; ∂r0 ∂σv × σv r0 = 1 2 ; ∂r0 ∂α × α r0 = η 2(α + η) ; ∂r0 ∂ω × ω r0 = − ω 2(ω + µ+ δh − ψ) ; ∂r0 ∂η × η r0 = 1 2 ( −3 η2 − 2η (α + δv)− α δv (α + η) (δv + η) ) . (4.18) copyright © 2023 assa. adv syst sci appl (2023) global stability and sensitivity analysis of basic reproduction number ... 99 (a) the graph of susceptible human against time (b) the graph of exposed human against time (c) the graph of infected human against time (d) the graph of recovered human against time (e) the graph of susceptible mosquito against time (f) the graph of exposed mosquito against time (g) the graph of infected mosquito against time fig. 3.2. disease dynamics copyright © 2023 assa. adv syst sci appl (2023) 100 a.i. abioye, et. al 4.2. tables of the model and the sensitivity index table 4.1. the model variables and descriptions variables description sh(t) number of susceptible humans at time t eh(t) number of exposed humans at time t ih(t) number of infectious humans at time t rh(t) number of recovered humans at time t sv(t) number of susceptible mosquitoes at time t ev(t) number of exposed mosquitoes at time t iv(t) number of infectious mosquitoes at time t nh(t) total number of humans population at time t nv(t) total number of mosquitoes population at time t table 4.2. parameters, descriptions values and sources parameter description value source λh recruitment rate of humans 1.2 [2] λv recruitment rate of mosquitoes 0.7 [2] δh disease-induced death rate of humans 0.068 [2] δv disease-induced death rate of mosquitoes 0.001 [2] ψ rate of the newborn birth with infection humans 0.003 [2] υ developing rate of exposed humans 0.05 [2] µ natural death of humans 0.01146 [2] ξ biting rate of mosquitoes 0.12 [2] α developing rate of exposed mosquitoes 0.083 [2] η natural death rate of mosquitoes 0.00083 [2] εh proportion of an antibody produced by human in response to the incidence of infection caused by mosquito 1.0 [2] εv proportion of an antibody produced by mosquito in response to the incidence of infection caused by human 1.0 [2] σh probability of transmission of infection from an infectious mosquitoes to a susceptible humans 0.1 [2] σv probability of transmission of infection from an infectious humans to a susceptible mosquitoes 0.09 [2] ω recovery rate of humans 0.0035 [2] γ1 rate of loss of immunity in humans 0.00017 [2] γ2 relapse rate of humans 0.04 [2] 4.3. graphical representation the diagram below shows the graphical representation of the sensitivity analysis of each parameter present in the basic reproduction number. 4.4. discussion of the results table 4.3 shows the sensitivity index of the parameters by using basic reproduction number (r0) in eq. (3.4) and at the baseline parameter values in table 4.2. the negative sign of sensitivity index shows that, by increasing the value of the parameters, this will reduce the value of r0 but for the positive sign of sensitivity index, it indicates that as parameter increases so also basic reproduction number r0. the most sensitive parameters with respect to the transmission of malaria from table 4.3 is the biting rate of mosquitoes (ξ) with the sensitivity index equals to 1.0000 (s.i = 1.0000). copyright © 2023 assa. adv syst sci appl (2023) global stability and sensitivity analysis of basic reproduction number ... 101 table 4.3. parameters, descriptions and sensitivity indexes of r0 parameter description sensitivity index λh recruitment rate of humans 0.5000 λv recruitment rate of mosquitoes 0.5000 δh disease-induced death of humans 0.4252 δv disease-induced death of mosquitoes 0.2732 ψ rate of the newborn birth with infection humans 0.0188 υ exposed rate of humans to infectious class 0.0932 µ natural death of humans 0.6649 ξ biting rate of mosquitoes 1.0000 α exposed rate of mosquitoes to infectious class 0.00495 η natural death rate of mosquitoes 0.7318 σh probability of transmission of infection from an infectious mosquitoes to a susceptible humans 0.5000 σv probability of transmission of infection from an infectious humans to a susceptible mosquitoes 0.5000 ω recovery rate of humans -0.0219 fig. 4.3. sensitivity index against parameters this shows that if there is 10% increase in the biting rate of mosquitoes, this will increase basic reproduction number (r0) by 10.00% as the respective index for the parameter is one. next parameter to the most sensitive parameter due to the spread of malaria from table 4.3 is the natural death rate of mosquitoes with the sensitivity index equals to 0.7318 (s.i = 0.7318). this shows that increasing the natural death rate of mosquitoes by 10% will result in increases in the value of r0 by 7.318%. similarly, table 4.3 is the natural death rate of humans with the sensitivity index equals to 0.6649 (s.i = 0.6649). this shows that increasing (or decreasing), the natural death rate of humans by 10% will increase or decrease r0 by 6.649%. moreover, the recruitment rates of humans (λh) and mosquitoes (λv) show that probabilities of transmission of infections from infectious mosquitoes to susceptible humans (σh), and from infectious humans to susceptible mosquitoes (σv) with the sensitivity index equal to 0.5000 (s.i = 0.5000). this shows that increasing or decreasing in each of these parameters by 10% increase or decrease in r0 in each of these parameters by 5%. similarly, copyright © 2023 assa. adv syst sci appl (2023) 102 a.i. abioye, et. al table 4.3 is the disease-induced death rate of humans (δh ) with the sensitivity index equals to 0.4252 (s.i = 0.4252). this illustrates that increasing or decreasing the disease-induced death rate of humans by 10% will result in increase or decrease in r0 by 4.252%. likewise, the disease-induced death rate of mosquitoes δv with the sensitivity index is equal to 0.2732 (s.i = 0.2732). this show that increasing or decreasing, disease-induced death rate of mosquitoes by 10% will increase or decrease the value of r0 by 2.732%. on the other hand, the other parameters sensitivity indices are very small which show that they have no impact on r0. 5. conclusion in conclusion, this study delved into a complex mathematical model of malaria, incorporating the crucial basic reproduction number (r0) and employing lyapunov functions to assess the global stability of disease-free and endemic equilibria. sensitivity analysis of key parameters provided insights into their role in disease control. the findings reveal an ongoing malaria outbreak characterized by declining human populations and increasing mosquito populations, signifying elevated transmission risk. the imperative response lies in implementing robust control measures, including mosquito management and prompt treatment of infected individuals, to effectively curb the outbreak and safeguard public health. references 1. abioye, a. i., ibrahim, m. o., peter, o. j., amadiegwu, s. c. & oguntolu, f. a. 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(2002). reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, mathematical biosciences, 180(1–2), 29–48. copyright © 2023 assa. adv syst sci appl (2023) introduction model formulation flow chart of the model differential equations of the model existence of equilibrium points basic reproduction number (r0) global stability analysis of disease-free equilibrium global stability analysis of endemic equilibrium numerical simulation sensitivity analysis sensitivity analysis of basic reproduction number r0 parameters tables of the model and the sensitivity index graphical representation discussion of the results conclusion adv syst sci appl 2022; 04:11–18 published online at https://ijassa.ipu.ru. on a pseudodifferentional calculation in group algebras andronik a. arutyunov1,2* 1v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia 2 moscow institute of physics and technology, moscow, russia abstract: the paper is devoted to the construction of pseudodifferentional operators over group algebras. an algebra of such operators is generated by central derivations. the equipping with an order function of this algebra is also shown. some examples of such operators are given and some further ideas for the development of the new pseudodifferentional calculus are presented. keywords: pseudodifferential operators, group algebras, derivations, pseudodifferential calsulus 1. introduction in a.n. parshin’s paper [12] the following construction was proposed. let d be a derivation of some ring r. then the left module of formal expressions l = ∑ i∈z aid i can be understood as an analog of the pseudo-differential operators algebra. further development of this logic leads us to the shur-sato theory. for the most complete and up-to-date study, see [14]. note also that similar ideas to describe pseudodifferential operators as noncommutative power series in which derivations are substituted as variables have been previously proposed in [9]. in particular, the formalism of the so-called µ-calculus and the corresponding µalgebras was studied. the study of differentials over noncompact spaces is motivated by the author’s earlier work in which the studies of classical and nonlocal pseudodifferential operators over a euclidean space [3, 6]. the study of operators on euclidean spaces has also been carried in [11]. issues related to the calculus of pseudodifferentional operators on abelian groups were considered in [8, 10]. in this paper, we will propose a construction of the algebra of pseudo-differential operators in group algebra, using previously obtained results to describe derivations from papers [2, 4, 5]. using the aforementioned idea from [12] we will take as derivations the central derivations that were studied earlier in [2]. the description of all sorts of derivations as variables is hampered by considerable computational difficulties and the problematic nature of the study of outer derivations (see results of [4] and [5]). in the first section we give a short overview of the concept of central derivations and prove their necessary conditions. especially the formula for the power (see lemma 2.2). in the second section we build firstly the construction for 1-dimensional algebra of pseudodifferentional operators generated by single derivation. then by induction we are constructing the algebra of “large” algebra generated by the cortege of central derivations. ∗corresponding author: andronick.arutyunov@gmail.com 12 a. a. arutyunov in the theorem we prove that this is a well-ordered algebra. we give some examples, the most intresting is the analog of the laplace operator (see examples 3.1 and 3.4). 2. central derivations recall that a derivation d : c[g] → c[g] is a such linear operator that d(uv) = d(u)v + ud(v), ∀u, v ∈ c[g]. central derivations were presented in [2]. it is easy to see that they form a lie algebra, i.e. the commutator of central derivations is a central derivation. central derivations give an example of derivations which are not inner – they cannot be represented as a commutator x → [a, x] for some a ∈ c[g]. in other words, the intersection of the central derivations space and inner derivations space is trivial. moreover, such derivations are, in a sense, fundamentally outer, i.e., they cannot in any sense be approximated by inner derivations. for more details on this, see the paper [4] which shows that central derivations are quasi-outer. recall the definition of central derivations. we are fixing z ∈ z(g) central element, and a homomorphism τ : g → c i.e. τ(uv) = τ(u) + τ(v), u, v ∈ g. and a formula for the action of central derivations on the basis element g ∈ g ⊂ c[g] is dτ,z : g 7→ τ(g)gz. (2.1) sure the following identity holds for central elements z, z1 dτ,z · z1 = dτ,z·z1 . (2.2) from the definition we can get the formula for the derivation commutator. lemma 2.1: the commutator of central derivations is the central derivation such that [dτ1,z1 , dτ2,z2 ] = dτ2(g)τ1(z2)−τ1(g)τ2(z1),z1z2 . (2.3) furthermore the derivations corresponding to the same homomorphism τ commute. proof directly from the formula (2.1) we have the following [dτ1,z1 , dτ2,z2 ](g) = (τ1(gz1)(τ2(g)− τ2(gz1)τ1(g))gz1z2 = = (τ2(g)τ1(z2)− τ2(z1)τ1(g))gz1z2. central elements z1, z2 are fixed so τ1(z2) and τ2(z1) do not depend from the element g. so the mapping (τ2(g)τ1(z2)− τ2(z1)τ1(g)) : g → c is a linear combination of homomorphisms so it as also a homomorphism. the permutability of differentiations corresponding to the same homomorphism follows from the proved formula. we can also calculate the power of the central derivation. lemma 2.2: the formula for the central derivation of the power n ∈ z is following dnτ,z(g) = n−1∏ j=0 (τ(g) + jτ(z))sign(n)gzn. copyright © 2022 assa. adv syst sci appl (2022) on a pseudodifferentional calculation in group algebras 13 note that in the case n = −1 we have d−1 τ,z(g) = 1 τ(g)− τ(z) gz−1. (2.4) proof first we consider the case n > 0. let us fix central derivations for i = 1 . . . n: di : g 7→ τi(g)gzi. (2.5) so the composition is d1d2(g) = d1(τ2(g)gz2) = τ2(g)d1(gz2) = τ2(g)τ1(gz2)gz1z2. that means that for n central derivatons we get the following d1 . . . dn(g) = τn(g)τn−1(gzn−1) . . . τ1(gzn−1 · · · · · z1)gz1 . . . zn. we are calculating the power so τi ≡ τj, zi ≡ zj . hence: dn(g) = τ(g)τ(gz) . . . τ(gzn−1)gzn. (2.6) we can rewrite the last formula in the following way considering that τ is a homomorphism to (c,+) we have that τ(zn) = nτ(z): dn(g) = τ(g)(τ(g) + τ(z)) . . . (τ(g) + (n− 1)τ(z))gzn. (2.7) in other words dn(g) = n−1∏ j=0 (τ(g) + jτ(z))gzn. (2.8) that gives us the statement. the proof for the case n < 0 is simillar. example 2.1: if τ(z) = 0 then we get dn(g) = τn(g)gzn. (2.9) we will use the following coefficient-function θnτ (g) : g → c, θnτ,z(g) := n−1∏ j=0 (τ(g) + jτ(z)) (2.10) this function satisfies the following asymptotic condition proposition 2.1: for n ̸= 0 |θnτ,z(g)| |g|n −→ const , |g| → ∞. (2.11) proof follows from lemma 2.2. copyright © 2022 assa. adv syst sci appl (2022) 14 a. a. arutyunov 3. pseudodifferentional operators over c[g] 3.1. case of one derivation let us consider a laurent polynomial f(x) with coefficients in c f(x) = n∑ k=−∞ fkx k, fk ∈ c. (3.12) for such denoted f with fn ̸= 0 we will say that deg(f) = n . proposition 3.1: for a central derivation d the operator f(d) : c[g] → c[g] is well defined for all g such that θnτ,z(g) ̸= 0. proof almost all coefficients fk in (3.12) are equal to zero. hence, if θnτ,z(g) ̸= 0 then each term in (3.12) is well defined by lemma 2.2. definition 3.1: for fixed n ∈ z and fixed homomorphism τ : g → c the algebra generated by operators f(dτ,∗) from (3.12) and multiplying on central elements will be denoted as ψn τ c[g], ψn τ c[g] = ⟨f(dτ,z,×z⟩, deg f ≤ n, z ∈ z(g). therefore, ψ∞ τ c[g] := ∪n∈zψ n τ c[g]. certainly holds the embedding ψn τ c[g] ⊂ ψn+k τ c[g], for k ∈ n. considering the identity (2.2) we get that each operator a ∈ ψn can be written for the basis element g as a(g) = ∑ i≤n aid i τ,e(g)zi, zi ∈ z(g). so we can define the order for operators in following classic form. definition 3.2: for the operator a ∈ ψ∞ τ c[g] ord(a) := min{n ∈ z|a ∈ ψn τ c[g]}. (3.13) it is well known from the classical pdo-calculation theory that for a good symbolcalculus we need the order function which transforms the space of pseudo-differential operators into a graded algebra. proposition 3.2: ψ∞ τ c[g] is a commutative algebra such that ord(a ◦b) ≤ ord(a) + ord(b); (3.14) proof the commutativity of algebra follows from lemma 2.1. the formula for orders follows from lemma 2.2. example 3.1: let us give some examples of operators from ψ∞ τ c[g] with their orders copyright © 2022 assa. adv syst sci appl (2022) on a pseudodifferentional calculation in group algebras 15 operator n= ∆τ,e(g) := τ 2(g)g 2 ∆τ,z(g) := τ 2(g)gz, z ∈ z(g) 2 idz(g) := gz, z ∈ z(g) 0 ∆−1 τ,z(g) := 1 τ2(g) gz−1 -2 it is easy to see that for such g that τ(g) ̸= 0 holds ∆τ,z ◦∆−1 τ,z = ide. it is also important to note that multiplying operator idz saves the order ∀z ∈ z(g) : idz(ψ n τ c[g]) = ψn τ c[g]. (3.15) we can also show an analogue of the composition formula. proposition 3.3: for z ∈ z(g) – central element of infinite order, and laurant polynomials f, h the following composition formula holds for the central derivation d := dτ,z: ((h ◦ f)(d))(g) = f(h(d))(g) = ∑ i∈z ∑ k+l=i fkhlθ k τ (g)θ l τ (g)gz i. (3.16) proof immediately follows from lemma 2.2 for a laurant polynomial from (3.12) using lemma 2.2 we get that example 3.2: for a central derivation dτ,e for a neutral group element e f(dτ,e)(g) = n∑ k=−∞ fk(τ(g)) kg = f(τ(g))g. (3.17) the situation is arranged similarly for the finite-ordered central elements. example 3.3: let z ∈ z(g) such that zk = e. we want to calculate f(dτ,z). first we have to rewrite f = f 1 + · · ·+ fk where f i(x) = ∑ j∈z fjk+ix jk+i. (3.18) after simple calcultaions we get the following f(dτ,z)(g) = k∑ j=1 f j(τ(g))gzj. (3.19) 3.2. generalized ψ∞(c[g]) algebra if we are working with the pair of central derivations their commutator is also a central derivaion by the formula (2.3). because this commutator is not trivial, the algebra which is generated by all laurant polynimials with the values in central derivations, is not commutative and is arranged more cunningly. we will define a generalized algebra ψ∞ τ1,...,τk (c[g]) by induction. copyright © 2022 assa. adv syst sci appl (2022) 16 a. a. arutyunov fix an algebra ψ∞ τ1,...,τk−1 (c[g]) with given homomorphisms τ1, . . . , τk−1 : g → c and equipped with the order function ord : ψ∞ τ1,...,τk−1 (c[g]) → z such that the following holds ord(a ◦b) ≤ ord(a) + ord(b); ord([a,b]) ≤ ord(a) + ord(b)− 1. then for a homomorphism τk : g → c we will define an algebra ψ∞ τ1,...,τk (c[g]) as ψ∞ τ1,...,τk (c[g]) = ⟨ψ∞ τ1,...,τk−1 (c[g]), dτ,e⟩. the order function is defined as following ord(a ◦ dnτ,e) := ord(a) + n, n ∈ z, (3.20) and naturally continues on linear combinations . so we are ready to formulate and prove the theorem theorem 3.1: algebra ψ∞ τ1,...,τk (c[g]) is well defined and the order function satisfies conditions ord(a ◦b) ≤ ord(a) + ord(b); ord([a,b]) ≤ ord(a) + ord(b)− 1. proof we need to check for ε1,2 = ±1 that the following inequality holds ord([dε1τ1,z1 ◦ d ε2 τ2,z2 ]) < ε1 + ε2. (3.21) if ε1 = ε2 = 1 it immediately follows from 2.1. now let us consider ε1 = 1, ε2 = −1. then we have the following calculation for the meaning of derivations dτ1,z1 , dτ2,z2 on the basis element g: (dτ1,z1 ◦ d−1 τ2,z2 )(g) = dτ1,z1 ( gz−1 2 τ2(g)− τ(z2) ) = τ1(g)− τ1(z2) τ2(g)− τ(z2) gz1z −1 2 = = z−1 2 τ2(g)− τ(z2) (dτ1,z1(g)− τ1(z2)gz1) = (d−1 τ2,z2 ◦ dτ1,z1)(g)− τ1(z2)d −1 τ2,z2 (g)z1. hence we have the following operator view of the commutator:[ dτ1,z1 , d −1 τ2,z2 ] = −τ1(z2)d −1 τ2,z2 × z1. (3.22) considering that τ1(z2) is a constant we get that ord( [ dτ1,z1 , d −1 τ2,z2 ] ) = −1. similarly for ε1 = ε2 = −1 we get that (d−1 τ1,z1 ◦ d−1 τ2,z2 )(g) = d−1 τ1,z1 ( gz−1 2 τ2(g)− τ2(z2) ) = = gz−1 1 z−1 2 (τ2(g)− τ2(z2))(τ1(gz −1 2 )− τ1(z1)) = 1 τ2(g)− τ2(z2) d−1 τ1,z1z2 (g) = = z2 ( d−1 τ2,z2 ◦ d−1 τ1,z1z2 ) (g). copyright © 2022 assa. adv syst sci appl (2022) on a pseudodifferentional calculation in group algebras 17 hence for the commutator we have the following formula: [d−1 τ1,z1 , d−1 τ2,z2 ] = ( d−1 τ2,z2 ◦ d−1 τ1,z1z2 ) (z2 − e). (3.23) the operator on the right hand has the order −2, which is the required result. summarizing lemma 2.1 and formulas (3.22),(3.23) we see that order function defined by (3.20) satisfies conditions of our theorem. we can define the spaces ψn τ1,...,τk (c[g]) with the help of out order function in following way: ψn τ1,...,τk (c[g]) := {a ∈ ψ∞ τ1,...,τk (c[g])|ord(a) ≤ n}. that gives us the structure of filtred algebra in ψ∞ τ1,...,τk (c[g]) by the theorem. also it is easy to see that if, for instance, τk are equal (or can be presented as linear combination as in lemma 2.1) then new algebra is precisely the algebra from the previous step. but it is superfluous variable for the general construction. we can improve example 3.1 for the generalized construction of pseudodifferentional operators for the cortege of homomorphisms {τ1, . . . , τk} example 3.4: let us give some examples of operators from ψ∞ τ1,...,τk c[g] with their orders: operator n= ∆τ1,...,τk,z(g) := (τ 21 (g) + · · ·+ τ 2k (g))gz, z ∈ z(g) 2 ∆−1 τ1,...,τk,z (g) := 1 τ21 (g)+···+τ2k (g) gz−1 -2 we can define the main symbol of the operator a ∈ ψn τ1,...,τk c[g] as an element in factor algebra ψn τ1,...,τk c[g]/ψn−1 τ1,...,τk c[g]. immediately from theorem 3.1 we get that corollary 3.1: main symbols form a commutative algebra. conclusion in conclusion, we would like to note the following. for such a well-defined algebra of pseudo-differential operators, it becomes correct to investigate the index of elliptic operators: operators with invertible main symbol. there is also an opportunity to construct a symbolic calculus. in particular, the question of calculating the parametrix is of interest. the areas of interest seem to be the following • is there a clear formula for the composition? • is there an equipment of the group algebra with the sobolev structure such that an pseudodifferentional operator is bounded as an operator between the corresponding spaces (with respect to the order)? • is there an analogue of the index formula for such operators? another area of activity may also be the transfer of the results obtained to other classes of operators. in particular the case of (σ, τ)−twisted operators. the case of derivations was investigated in [1]. for a more complete overview see [7]. acknowledgements the results in section 1 are obtained under the financial support of the ministry of science and higher education of the russian federation, goszadanie no. 075-00337-20-03, project copyright © 2022 assa. adv syst sci appl (2022) 18 a. a. arutyunov no. 0714-2020-0005. the results in section 2 are obtained under the financial support of the russian science foundation (project no. 20-11-20131). references 1. alekseev, a., arutyunov, a. & silvestrov, s. (2020, august 2) on (σ, τ)derivations of group algebra as category characters. [online] available: https://arxiv.org/abs/2008.00390 2. arutyunov, a. (2020) derivation algebra in noncommutative group algebras, differential equations and dynamical systems. collected papers. proc. steklov inst. math. steklova, 308, 28–41. 3. arutyunov, a. (2015) reduction of nonlocal pseudodifferential operators on a noncompact manifold to classical pseudodifferential operators on a doubledimensional compact manifold, math. notes, 97(4), 502–509. 4. arutyunov, a. & alekseev, a. (2019) cohomology of n-categories and derivations in group algebras, topology and its applications, 275, 107002. 5. arutyunov, a. & mishchenko, a. (2019) a smooth version of johnson’s problem on derivations of group algebras, sb. math., 210(6), 756–782. 6. arutyunov a., & mishchenko, a. (2013) reduction of the calculus of pseudodifferential operators on a noncompact manifold to the calculus on a compact manifold of doubled dimension, math. notes, 94(4), 455–469. 7. hartwig, j., larsson, d. & silvestrov, s. (2006) deformations of lie algebras using σ derivations, j. algebra, 295(2), 314–361. 8. kumar v. & wong m. (2019) c∗ -algebras, h∗-algebras and trace ideals of pseudodifferential operators on locally compact, hausdorff and abelian groups, journal of pseudo-differential operators and applications, 10, 269–283. 9. maslov, v. (1973) operatorniye metodi [operator methods]. moscow, russia: nauka, [in russian]. 10. molahajloo, s. & wong, k. (2015) pseudo-differential operators on finite abelian groups, journal of pseudo-differential operators and applications, 6, 1–9. 11. nicola, f. & rodino, l. (2010) global pseudo-differential calculus on euclidian spaces. basel, switzerland: springer basel ag. 12. parshin, a. (1999) on a ring of formal pseudodifferential operators, algebra. topology. differential equations and their applications, 224, 266–280. 13. shubin, m. (2005) psevdodiffenecialniye operatori i spektralnaya teoriya [pseudodifferential operators and spectral theory]. moscow, russia: dobrosvet, [in russian]. 14. zheglov, a. (2022, may 13) schur-sato theory for quasi-elliptic rings [online] available https://arxiv.org/abs/2205.06790. copyright © 2022 assa. adv syst sci appl (2022) introduction central derivations pseudodifferentional operators over c[g] case of one derivation generalized (c[g]) algebra adv syst sci appl 2020; 04; 113-124 published online at https://ijassa.ipu.ru. development of unified approaches to building neural network and mathematical models based on digital data nailia gabdrakhmanova1*, maria pilgun2 1) peoples friendship university of russia (rudn university), moscow, russia e-mail: gabd-nelli@yandex.ru 2) institute of linguistics russian academy of sciences, moscow, russia e-mail: mpilgun@iling-ran.ru abstract: the paper considers the problem of developing approaches to building mathematical models based on digital data of real objects. the data are in text format and contains information about the behavior of the dynamic system. the information selected from the text data enables building of neural network and mathematical models of the dynamic system. the adequacy of the models is evaluated by analytical and numerical methods. the results are meaningfully interpreted. as a result of the study, it was confirmed that the algorithms and approaches for building mathematical models to solve the considered range of problems using digital data can be unified. the analysis of the obtained solutions showed that the conclusions drawn on the basis of the built mathematical models and the conclusions drawn with the semantic neural network analysis of texts are consistent with each other. therefore, one can talk about the positive results of the models developed. the models developed can be used in solving managerial tasks, planning and situation prediction. keywords: neural network, time series, differential equations, dynamic system, stability, text analysis 1. introduction the purpose of this study is to develop approaches for building mathematical models based on extracting information from large data arrays of various formats. it is assumed that the built neural network and mathematical models will help solve the problems of choosing a managerial decision. for example, the city administration faces such tasks, when choosing a project to be implemented in the city. in such tasks, it becomes necessary to obtain feedback, some evaluation of the attitude of the population to the projects being implemented. the building of mathematical models from digital data, in contrast to mathematical models of technical systems, poses a number of difficulties: the data are in text format; it is impossible to conduct experiments; the problem belongs to the “hard to formalize” class. at present, the progress of neural network technologies allows solving many problems. as shown in this paper, machine learning make it possible to extract hidden information from data of various formats and build mathematical models on their basis. 1.1. data the material for the study was the data of social media, microblogs, blogs, instant messengers, videos, forums and reviews on the construction of the south east chord (sec) * corresponding author: gabd-nelli@yandex.ru mailto:mpilgun@iling-ran.ru 114 n. gabdrakhmanova, m. pilgun copyright ©2020 assa. adv. in systems science and appl. (2020) and nord west chord (nwc) in moscow. the data were collected between 00:00 on july 7, 2019 and 11:59 pm on december 31, 2019 (see table 1.1.). table 1.1. data characteristics parameter messages authors loyalty involvement аudience nord west chord 433 0,9 7 858 6 822 650 south east chord 12 456 7 459 0,1 144 028 17 688 221 1.2. method the study involved several parallel models. with the help of the semantic neural network model, a text analysis was performed; the topic structure and the semantic network were analyzed, as well as the sentiment and the level of aggression; digital conflictogenic zones were identified; and the indices of social stress and well-being were calculated. in parallel with the neural network model, a statistical analysis of experimental data was performed and dynamic models of processes were built. the following models were used for data analysis: a model based on the neural network paradigm of using neural-like elements with time summation of signals [5] and that with differential equations [1]. analysis of solutions for all models showed the consistency of solutions. the chosen model enables automatic semantic ranking of the text data base using several algorithms: an algorithm for forming a homogeneous frequency network of text using an artificial neural network based on neural-like elements with time summation of signals, and an iterative hopfield-like algorithm for ranking network vertices on a scale of “0 -100%” values. in addition, the n-gram network representation is formed by iterative re-weighting at a given number of steps, or based on the ranking process convergence criterion. thus, lexemes are analyzed in the context of syntagmas of a given (n) length on a semantic network formed on the basis of text analysis. the frequency network of the text is built as a set of pairs of words that occur in the sentences of the text. the network vertices are weighted by their frequency of occurrence in the text. the weight of a pair of vertices in the network corresponds to the frequency of occurrence of word pairs in sentences of the text. the neural network approach makes it possible to make a correct analysis of sentiment and aggression and derive indices of social stress and well-being, which are necessary for solving the problems of this study [5; 6]. 2. results and discussion 2.1. semantic neural network model the dynamics of users’ activity (figures 2.2; 2.4) and generated content (figures 2.1; 2.3) regarding the implementation of the sec and nwc projects, enable determination of the communication features and identification of potential digital conflictogenic zones. development of unified approaches to building neural network 115 copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 2.1. dynamics of messages (sec) fig. 2.2. dynamics of users’ activity (sec) fig. 2.3. dynamics of messages (nwc) fig. 2.4. dynamics of users’ activity (nwc) to identify key topics, the user-generated content was analyzed with the allocation of the semantic network, as well as the topic structure, followed by the content analysis. the core of the semantic network composed of nominations with a link weight of 98-100, clearly demonstrates the problems that concern the residents and the complaints of muscovites against these projects (figures 2.5; 2.6.). fig. 2.5. core of the semantic network (sec) 116 n. gabdrakhmanova, m. pilgun copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 2.6. core of the semantic network (nwc) the sec-related content includes a large list of problems that concern the residents: the destruction of the radioactive burial ground of the polymer plant, the possible ecological disaster, radiation; the growth of the number of cancer diseases, the felling of trees, the allrussian environmental protest; the residents are ready to defend their interests in protest actions both in virtual and real space; a sharp negative attitude towards the actions of representatives of the administrative apparatus; expansion of the moscow borders, renovation, construction of other road and transport facilities; violation of work methods, lobbying for laws, deterioration of the transport situation; fake expertise papers. positive and neutral messages are presented in the official content containing a description of the benefits that the residents will receive from improving the transport situation and development of the city’s transport system. the north-west chord-related content includes official messages mainly; the residents almost do not generate any content, which indicates a calm attitude towards the project. key topics are related to coverage of different stages of the construction implementation, as well as information about similar projects; they present the north-west chord as part of the new moscow chord ring. the sentiment analysis made it possible to clusterize the content by sentiment (figure 2.7.). analysis of the level of aggression showed a high conflict potential of messages dedicated to the sec and the absence of indicators of the conflict potential for the content dedicated to the nwc (figure 2.8.). development of unified approaches to building neural network 117 copyright ©2020 assa. adv. in systems science and appl. (2020) fig. 2.7. sentiment-analysis results, sentiment clusterization fig. 2.8. level of aggression in the content 118 n. gabdrakhmanova, m. pilgun copyright ©2020 assa. adv. in systems science and appl. (2020) digital conflictogenic zones were identified taking into account the dynamics of the actors’ activity, the user-generated content, as well as the analysis of textual data that make up the negative cluster. the most indicative criterion was the identification of the presence or absence of aggression; in particular, the presence of strong aggression indicates a high level of negative attitude of local residents towards the sec project. the derivation of the social stress and well-being indices made it possible to calculate the quantitative expression of the residents’ attitude towards the implementation of the sec and nwc urban development projects (figure 2.9.). . fig. 2.9. social stress index and social well-being index 2.2. building of mathematical models this section is dedicated to the building of mathematical models with a focus on the study of the system dynamics (the dynamics of the population’s attitude towards the project p under consideration). the problem was solved on the basis of research data on the attitude of the population to the construction of the south-east (se) chord and the north-west (nw) chord. in the previous section, text data were analyzed after filtering using associative neural networks and visualization programs. further, based on the data selected using the machine learning method, it is required to build mathematical models. to build mathematical models, it is necessary to formalize the problem under consideration. 2.3. formalization of the problem of building a mathematical model given: set w’={w’(t)}, t is within [0,t], where w’(t) is a numerical vector corresponding to the text at time moment t after its vectorization. the elements of the set are divided into clusters according to the criterion of nearness to some feature set. in our problem, this is a relation to a certain project p. in textual analysis, several methods have been developed for solving this problem. for example, one way to determine the proximity of vectors is to calculate the cosine distance using the cosine of the angle between the vectors. the clustering resulted in array , where t is time, w(t) is a numeric vector of messages after filtering by the criterion of nearness to the feature space. at the next step, the set of elements w was clustered in two ways presented below. first. the entire set w is divided into two clusters by the feature of positive and negative attitude towards the project. the resulting clusters with numerical vectors of texts were designated as k1 (‘aggression present’) and k2 (‘no aggression’). “aggression present” means a negative attitude towards project p. “no aggression” means a positive attitude towards project p. development of unified approaches to building neural network 119 copyright ©2020 assa. adv. in systems science and appl. (2020) second. the entire set w is divided into three subclusters according to the degree of negative or positive attitude to project p. we called this feature tonality and identified three degrees of tonality. the constructed subclusters are designated x1 (“negative”), x2 (“neutral”), x3 (“positive”). 2.4. data pre-processing the entire time interval [0; t] is divided into n intervals with the same increment. the following designations are introduced: t is the number of the interval, t=1,2,…,n. the following terms are introduced: k1 (t) is the number of elements in cluster k1 at time t, k2 (t) is the number of elements in cluster k2 at time t, x1 (t) is the number of elements in cluster x1 at time t, x2 (t) is the number of elements in cluster x2 at time t, x3 (t) is the number of elements in cluster x3 at time t. in the models under consideration, it is assumed that an unaccounted individual can join one of the groups or move from one group to another through information received either from the media or through interpersonal communication from a previously informed individual. to build models, the data of all groups are standardized according to formula , where is the estimation of expectation, is the estimation of the mean square deviation, and is the standardized value of х. fragments of standardized data {x1(t)}, {x2(t)}, {x3(t)} , t= are presented in table 2.1 and in the graphs in fig. 2.10. and fig. 2.11. table 2.1. fragment of initial data t x1 x2 x3 1 -1.28 -1.66 -1.13 2 -0.93 -0.57 -0.72 3 -0.34 -0,91 -0.59 4 0.23 1.51 0.28 5 0.09 0.24 0.03 6 -0.65 0.01 -0.96 7 -0.59 -0.22 -0.68 8 -0.37 -0.68 -0.46 9 -0.45 -0.28 -0.85 10 -0.56 0.35 -0.43 11 -0.40 -0.80 -0.30 12 0.22 0.76 -0.11 13 -0.64 -0.80 -0.71 120 n. gabdrakhmanova, m. pilgun copyright ©2020 assa. adv. in systems science and appl. (2020) x3 x2 x11 3 5 7 9 11 13 15 17 19 21 23 25 -4 -2 0 2 4 6 8 10 fig. 2.10. fragment of time series {x1 (t),x2(t),x3(t)} for the sec data xc3 xc2 xc11 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 -3 -2 -1 0 1 2 3 4 5 fig. 2.11. fragment of time series {x1 (t),x2(t),x3(t)} for the nwc data 2.5. model 1. autoregressive time series models data {x1(t)},{ x2(t)}, {x3(t)} , t= are time series. time series models can be built using neural networks [3,4] and methods of mathematical statistics. for our time series, we built autoregressive models. autoregressive models can be used to predict the dynamics of series a few steps ahead. coefficients in time series models indicate the direction in the behavior dynamics. studies of the behavior of time series have shown that the values of the series are affected by such factors as holidays, weekends or weekdays, time of day, etc. to smooth over these factors, the following transformation was used (see fig. 2.12.): according to the following rule: , , . x1 x2 x31 3 5 7 9 11 13 15 17 19 21 23 25 27 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 1,2 fig. 2.12. fragment of transformed time series {x1 (t),x2(t),x3(t)} for the sec data. the time series of the nwc and sec data were studied for stationarity; the coefficients of autocorrelation and partial autocorrelation were found; and estimates of the structural orders were obtained. autoregressive models were built for each time series using the statistica 10 software. development of unified approaches to building neural network 121 copyright ©2020 assa. adv. in systems science and appl. (2020) model 1 results based on the experimental data for the period of n nw and se facilities, autoregressive models of time series {x1 (t)}, {x2(t)},{x3(t)} were built. fig. 2.13 shows trends of time series {x1 (t)}, {x2(t)},{x3(t)} for the se facility of cluster k1. the graphs show that the dynamics of se rows is positive for all rows. to build an autoregressive model, it is required to transform the series so that the series are stationary. therefore, autoregressive models are built according to series {n1(t)},{n2(t)},{n3(t)}. to set the structure of models (p, q, r), estimates of autocorrelation and partial autocorrelation of each time series were calculated. the results of estimating the coefficients of autoregressive models are shown in tables 3, 4. it is important to note that almost all observations for se facilities fell into cluster k1, and for nw – into cluster k2. the analysis of the coefficient estimates shows that for the se facilities the negative attitude towards project p is growing. in models (1), the coefficient before n1 is positive and has a value close to 1, and the coefficient before n3 is negative. the coefficients show that the negative attitude is growing strongly, and the positive attitude is decreasing. models (2) indicate that for the nw facility, the positive attitude to project p grows faster than the negative one. x3 x2 x11 3 5 7 9 11 13 15 17 19 21 23 25 -200 0 200 400 600 800 1000 1200 fig. 2.13. trends of time series {x1 (t)}, {x2(t)},{x3(t)} for the se data. autoregressive models for the se data are given below: ; ; (1) , where . autoregressive models for the nw data are as follows: ; ; (2) , where . 2.6. model 2. modeling with differential equations the dynamics of the process under consideration can be described using the typeepidemiological mathematical model [1,2,7]. in this case, we study the dynamics of the process as some type of epidemiological contact process. there are various kinds of models. the type of models is selected based on the analysis of the interaction of the selected groups in the model. taking into account that there is no contradiction in the problem under consideration, the type of model was chosen using systems of linear differential equations: 122 n. gabdrakhmanova, m. pilgun copyright ©2020 assa. adv. in systems science and appl. (2020) (3) the solution to the system of differential equations (3) is vector x(t)=(x1 (t), x2(t), x3 (t)), then (1) in vector form is written as follows: . the parameters of our system (3) are unknown. the task of building parametric models arises when it is required to identify the value of a parameter from measurement data. various methods have been developed to estimate the parameters of model (3) from experimental points, for example, the nelder-mead method. to solve the parameterization problem, a neural network approach was used. it has been proven that the neural network approach allows obtaining sustainable solutions. this is accounted for as follows. experimental points always have measurement errors. neural network functions are robust against errors in the original data and can be efficiently implemented in software and hardware. neural networks of multi-layer perceptron (mlp) and radial basis (rbf) architectures are suitable for solving the problem. let us explain the essence of the neural network approach using the example of the simplest problem of solving a differential equation: , (*) here is a differential operator, i.e. an algebraic expression containing derivatives of an unknown function u. let us search for an approximate solution (*) in the form of an artificial neural network of a given architecture: here w=(w1,w2,…,wn) is the vector of weights wi=(ci,ai). the basic neuroelement v is set by the choice of the activation function – function f of one real variable. the weights of the neural network – linear input parameters ci and non-linear input parameters ai – are involved in the process of step-by-step training of the network that, in the general case, is built on minimization of some error functional j(w): the point to be emphasized is that what is sought is not the minimum j(w), but point in the space of weights specifying solution . the number is chosen so that the built model is considered sufficiently accurate. after solving the parameterization problem (3), it is necessary to find the solution for (3) and analyze the obtained solutions. then, on the basis of the obtained solutions, the choice of the management decision is made. below is a brief description of the obtained solutions for parametrization problems and solving differential equations using real data of two facilities. outstanding mathematician a.n. kolmogorov [3,7] proposed to abandon the explicit form of functions in the right-hand side of equation (3) in models such as epidemiological contact processes, considering only certain restrictions on the form of these functions. if we follow the idea proposed by a.n. kolmogorov, then neural network mapping models are well suited here. in this case, for a system of differential equations written in general form as development of unified approaches to building neural network 123 copyright ©2020 assa. adv. in systems science and appl. (2020) , the neural network mapping by experimental points will result in f in the form of an expansion in terms of the given basic functions of the neural network (exponential, hyperbolic tangent, identity function). the results of solving this problem are provided in the section below. model 2 results when building models described using a system of differential equations in the form of (3) based on observational data {n1(t)},{n2(t)},{n3(t)}, the parameterization problem was solved. taking into account the stationarity of the transformed series, f(t) was excluded from the right-hand side of (1). numerous numerical experiments have been performed. as a result of experiments on the se data, the following models were built: the locations of the trajectories in the vicinity of the stationary point were studied. the roots of the standard equation were calculated: k1=0,4-0,3i, k2=0,4+0,3i the general solution of the system is obtained: the lyapunov stability parameter has shown that the stationary point for the given system is unstable. conclusion. all models built using differential equations, mathematical statistics and neural networks have their own specifics and their significance in the analysis of system dynamics. we believe that when solving this type of problem, it is necessary to build models using a combination of neural network models together with methods of analytical description of processes. neural network models are not very convenient for analyzing dynamic processes. models built using differential equations (model 2) and regression models (model 1) are more suitable for analyzing the dynamics of processes. however, neural networks are more stable for approximations with distorted data. for example, in problems of parametrization of differential equations and in problems of finding approximate solutions for differential equations, neural networks have shown good results. 3. conclusion algorithms and approaches for building mathematical models to solve the considered range of problems using digital data can be unified. the analysis of the solutions obtained showed that the conclusions drawn on the basis of the built mathematical models and the conclusions drawn using the semantic neural network analysis of texts are consistent with each other. therefore, one can talk about the positive results of the models developed. the models developed can be used in solving managerial tasks, planning and situation prediction. the main conclusions of the solved problem are as follows: • the largest conflictogenic zone was identified in the sec-related data set. the neutral and positive content include official messages mainly, which cause strong rejection of the residents. the user-generated content is characterized by negative evaluativeness and aggression, which makes it possible to predict further escalation of the conflict; • the nwc-related content has no signs of conflictogenity. 124 n. gabdrakhmanova, m. pilgun copyright ©2020 assa. adv. in systems science and appl. (2020) acknowledgements the work was supported by rfbr grant 19-08-00261. references 1. arnold, v.i. (2012) geometric methods in the theory of ordinary differential equations. moscow: icmmo. 2. castillo-ch´avez, c., song, b. (2003). models for the transmission dynamics of fanatic behaviors. in: h.t. banks, c. castillo-ch´avez (eds.), bioterrorism: mathematical modeling applications in homeland security, in: siam frontiers in applied mathematics, vol. 28, siam, philadelphia, 2003, 155–172. 3. gabdrakhmanova n. (2015). neural network models for solving management problems on the main oil pipeline. xvii all-russian scientific and technical conference "neuroinformatics-2015": collection of scientific papers. part 1. m.: niami mephi, 2015. -244, 172-181. 4. gabdrakhmanova n. (2018). forecasting time series using topological data analysis. in itise 2018 international conference on time series and forecasting proceedings of papers 19-21 september 2018 granada (spain), 1367-1374. 5. kharlamov a,a. & pilgun m. (eds.) neuroinformatics and semantic representations. theory and applications. newcastle upon tyne: cambridge scholars publishing. 6. kharlamov, a.a. & pilgun, m. analysis of the situation connotation on the example of assessing the reaction of society: social media data. international journal of future generation communication and networking. (ijfgcn), issn: 2233-7857(print); 22079645(online), nadia, (2020), vol. 13, no. 3, 37-44. doi 10.33832/ijfgcn.2020.13.3.04. 7. kolmogorov, a. n. (1972). qualitative research of population dynamics models. problems of cybernetics. issue 25. 100-106. 1. introduction 1.1. data 1.2. method 2. results and discussion 2.1. semantic neural network model 3. conclusion acknowledgements microsoft word 1147.docx adv syst sci appl 2022; 02; 1-10 published online at https://ijassa.ipu.ru. modeling the detection of moving objects by means of a spatially distributed continuous monitoring system with a dynamic structure vladimir i. goncharenko1,2, azret a. kochkarov3*, danil v. yatskin4, boris v. rumiantsev5 1) moscow aviation institute (national research university), moscow, russia 2) v.a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia. e-mail: goncharenkovi@mai.ru 3) financial university under the government of the russian federation, moscow, russia. e-mail: akochkar@gmail.com 4) moscow institute of physics and technology, moscow, russia. e-mail: danil@frtk.ru 5) moscow state university, moscow, russia. e-mail: rymbor17@gmail.com abstract: the class of monitoring systems, which is equipped with mobile means of detection, is considered. unmanned aerial vehicles are used as detection means. the area of responsibility of the system is a geographic space with counteraction for both detection equipment and objects of observation. an original approach to the development of a simulation model for detecting moving objects of observation in the area of a spatially distributed continuous monitoring system with a structure is proposed. the movement of sensors is displayed along trajectories, which are hamiltonian cycles on the terrain graph. the new approach is to use the approach used to ensure the flexibility of the resulting solutions to the problem of monitoring space and the ability to respond quickly to various factors and other conditions of use based on self-organizing mechanisms. at the same time, both the system itself, designed to solve the tasks of continuous monitoring and the solutions found for specific monitoring tasks and spatially distributed systems, provide continuous monitoring and are resistant to destructive influences of various kinds. the constructed simulation model and the experiments performed using the principles of dynamic detection of detection means in the area of a spatially distributed continuous monitoring system. at each moment of time, the optimal configuration of the parameters is determined, which is used as the most effective solution of the problem from the point of time of detecting an object in the monitoring area of vision. the model does not depend on the type of sensors used in the network and the implementation of the ideological principles embodied in the concept of dynamic systems. based on the results of the experiments carried out, conclusions are drawn about the characteristics of continuous monitoring, which provides a basis for further work to optimize these indicators. keywords: continuous monitoring system, unmanned aerial vehicle, spatially distributed system, monitoring object, mobile detection facility, hamiltonian cycle 1. introduction temporarily distributed continuous monitoring systems demonstrate modern service and support functions of the airborne dynamic detection system, which ensure the solution of problems of maintaining the space-probabilistic-temporal functions (sptf) at the required level [1,2]. sptf related to monitoring of the probability of detecting objects in a given period of time. * corresponding author: akochkar@gmail.com 2 v.i. goncharenko, a.a. kochkarov, d.v. yatskin, b.v. rumiantsev copyright ©2022 assa. adv. in systems science and appl. (2022) the main tasks of the systemic continuous monitoring of the class are prompt and highprecision detection of objects, their further support and timely delivery of the object to the end user [3,4]. the classical approach to solving the problem of monitoring the space of continuous observation over the maximum possible monitoring area. often this problem is solved using stationary systems that are not distributed in nature [5, 6]. the problems of a negative character are presented when solving a general character. to provide more flexibility, network monitoring systems are used [7-10]. however, more attentively, approaches to the organization of the probe, as a rule, are focused on the stationary arrangement of the probe between the devices [11-13]. in this case, a one-time section of the monitoring zone occurs between the subsequent monitoring of each network node of the corresponding section of the zone. this approach is optimal for solving a large class of problems. nevertheless, in recent years, decentralized networks have become more relevant, one of the most important advantages of which is the ability to dynamically change the configuration and adapt the solution to the monitoring problem to any changes in the conditions of the problem, provoked by factors of different nature. thus, it is necessary to develop methods associated with finding solutions to similar problems [14,15]. it is in this ideology that models for the dynamic allocation of means of a spatially distributed continuous monitoring system are developed. an original approach to the development of a simulation model for detecting moving objects of observation in a spatially distributed continuous monitoring system is proposed. the developed simulation model and the conducted experiments use the principles of dynamic detection of detection means in the area of a spatially distributed continuous monitoring system. 2. mathematical model of the dynamic placement as the area of responsibility of the continuous monitoring system in this paper is considered the monitoring area which is bounded by a given square on the plane. obstacles are placed in the monitoring area, represented in the form of parts of the plane bounded by closed curves. neither monitoring objects nor monitoring tools can be placed or moved in these areas. fig. 1 shows the layout of the monitoring objects in a given area of responsibility of the monitoring system. in the general case, mountains, hills, insurmountable for ground and air purposes, water bodies, as well as elements of infrastructure in an urbanized area can be considered as obstacles. in the area of responsibility of the system (monitoring area), monitoring objects can freely move either along random or along specified trajectories, taking into account their technical capabilities. monitoring is carried out by means on the basis of sensors, which at any given time can be located at a certain point in the monitoring area. each monitoring tool has a limited field of view, which in planar projection onto the earth's surface is a circle of radius r. the radius of the field of view of each tool depends on the weather conditions in the monitoring area. also, like the radius r, the speed of the sensor in the area of responsibility of the entire system also depends on the weather conditions. in this work, the failure of monitoring tools is also taken into account. modeling the detection of moving objects by means of a spatially distributed 3 copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 1. general layout of monitoring tools in the area of responsibility of the system to formalize the problem of placing detection equipment, the following designations are used (fig. 1): ω!" – a set of points of the system's area of responsibility, 𝑆!" – the total value of the area of the system's area of responsibility, ω#$% – a set of points of obstacle areas in the area of responsibility of the system, ω& – a set of visibility zone points, 𝑛' – 𝑙-th vertex of the graph by dividing the area of responsibility of the system. assuming that the number of targets detected during the monitoring 𝑡 ∈ [𝑡(; 𝑡)], inversely proportional to mean time < 𝑡 >[!!; !"] detection of one target at a time 𝑡 ∈ [𝑡(; 𝑡)] , the following statement can be obtained: < 𝑡 >[!!; !"] = ∫ 𝑑𝑡. ⋅ 𝑡.𝑝((𝑡.) !" !! , ((1) where 𝑝((𝑡) – probability density of detecting a monitored object at a point in time 𝑡. consider the case when the monitoring object is randomly located in the system's area of responsibility at the moment 𝑡( and does not move in the considered period of time 𝑡 ∈ [𝑡(; 𝑡)]. the randomness of the location of the monitoring object in this case, in the absence of a priori information, should be regarded as uniform. the assumption about the immobility of the monitoring object is valid provided that the speed of its movement is significantly less than the speed of movement of the sensors. (𝑣ом ≪ 𝑣со). under the condition of immobility of the monitoring object, normalization ∫ 𝑑𝑡. ⋅ 𝑝((𝑡.) !" !! = 1 . to minimize the value< 𝑡 >[!!; !"] дresidually minimize the standard deviation σ of the distribution 𝑝((𝑡). the value 𝑝((𝑡) can be represented as 𝑝((𝑡) = 2 2! 7∬ 𝑑𝑥.𝑑𝑦. ⋅ 𝑝((𝑥., 𝑦.)3#$(!) <, (2) 4 v.i. goncharenko, a.a. kochkarov, d.v. yatskin, b.v. rumiantsev copyright ©2022 assa. adv. in systems science and appl. (2022) where ω%6(𝑡) – area of responsibility of the system, "swept" by means of detection at the moment in time 𝑡 ∈ [𝑡(; 𝑡)], 𝑝((𝑥, 𝑦) – density of the probability distribution of the initial location of the monitoring object at the point (𝑥, 𝑦). in the case of a uniform distribution of the initial location of the monitoring object 𝑝((𝑥, 𝑦) = 1/𝑆!" , where 𝑆!" – system area of responsibility. that is why expression (2) can be rewritten as: 𝑝((𝑡) = 27#$(!) 2! ⋅ ( 7#$ . (3) substituting expression (3) into expression (1), obtain: < 𝑡 >[!!; !"] = ( 7%& ⋅ ∫ 𝑑𝑡. ⋅ 𝑡. 27#$(!.) 2!. !" !! . (4) due to the fact that "sweeping" can occur only as long as 𝑆%6 ≤ 𝑆!" (since at 𝑆%6=𝑆!" the monitoring object will be guaranteed to be detected), the task of minimizing the average time of object detection has been reduced to the task of maximizing the "sweeping" speed (shown at fig. 2). fig. 2. schematic representation of the effect of the "sweeping" speed on the length of the integration region in expression (4) in the case when the monitoring object moves in the area of responsibility of the system for a period of time 𝑡 ∈ [𝑡(; 𝑡)] (that is, when the condition 𝑣ом ≪ 𝑣со is not observed), one pass of the detectors may not be enough to detect the monitored objects. in this case, it is necessary to launch an iterative patrol of the system's area of responsibility, in which at each iteration (or, in other words, in a cycle), the system's area of responsibility is swept along the routes (in both directions) that were used above when approaching a stationary target. moreover, the less the period 𝑇8 (in time units) of traversing each point (graph node) of the target zone by monitoring tools, the faster the monitoring object will be detected. this statement clearly demonstrates the limiting case in which the period of the round trip of each modeling the detection of moving objects by means of a spatially distributed 5 copyright ©2022 assa. adv. in systems science and appl. (2022) point 𝑇8 → 0. it should be noted that in view of the fact that the location of the monitoring object in the area of responsibility of the system is modeled by a random process with a uniform distribution over the area of the area of responsibility, none of the points is “selected”. therefore, it is advisable to choose the same period of traversing each point, so 𝑇8 = 𝑐𝑜𝑛𝑠𝑡 for every 𝑘. such a patrol mode can be carried out by a set of n detectors following each other along the constructed route at the same distance from each other. when the weather conditions change and / or the sensors fail, their location should be reconfigured at each subsequent moment in time, taking into account the new route and / or the new number of detection means of the monitoring system. 3. computer modeling computer simulation is carried out in accordance with the following conditions: 1. the area of responsibility of the system is covered by nodes with a radius of the field of view under current weather conditions. such a division is necessary to speed up the operation of the algorithm and is valid provided that during the time between two iterations in time, each monitoring tool has time to move to the next node along the route. at the nodes, a route is built along the terrain graph, which is a hamiltonian cycle along which the detection means will move during the monitoring process. 2. at each iteration, the detector moves along the constructed route by exactly one node. a monitored object is considered detected (registered) if it falls within the scope of at least one detection tool. for the computer experiment, the matlab software package was used. it is proposed to carry out computer modeling of the deployment of mobile detection devices based on sensors in the area of responsibility of the system in the form of four main stages. at the first stage, the input parameters of the task are initialized (the size of the monitoring area, the location of obstacles, the number of detectors, the size of the visibility area of one detector, the number of monitored objects, the speed of the monitored objects). the speed of movement of the detectors is determined by the size of their field of view, since in one iteration the sensor is shifted by a distance of the order of 2𝑅, where 𝑅 – the radius of the visibility area of the detector, which is a circle centered at the sensor location. thus, the average speed of each detection tool over the monitoring time is < 𝑣%9:% >! ≈ 2𝑅 𝑝𝑖𝑥/𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛 (2𝑅 pixel per iteration). at the second stage, a graph of the movement of detectors in the area of responsibility of the system is built. for this, the area of responsibility of the system is covered by the visibility areas of the detection means. the location of the nodes is carried out by means of the gradient descent method, the function of which is the area of the covered area of responsibility of the system. the center of each of these scopes represents the top of the graph. two vertices are connected by edges if the distance between them is < 2𝑅 + 0.2 ⋅ 2𝑅. an example of building a graph is shown in fig. 3. 6 v.i. goncharenko, a.a. kochkarov, d.v. yatskin, b.v. rumiantsev copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 3. left: filling the target area with sensor scopes. right: building a graph of the movement of detectors based on this filling. the red line marks the hamiltonian cycle at the third stage, a hamiltonian cycle of traversing all the vertices of the constructed graph is constructed. on the constructed cycle, detectors are placed with the same distance between two neighboring ones (if the distance is counted along the hamiltonian cycle, see fig. 4). such an arrangement provides an equal traversal period for each vertex of the graph during the monitoring process. fig. 4. left: the location of the sensors on the hamiltonian cycle. right: hamiltonian cycle on the plotted graph the fourth stage is directly devoted to the computer experiment. the number of detected (registered) monitoring objects is calculated for n iterations. at each iteration, the detectors are shifted along the hamiltonian cycle by one vertex. the detected monitoring object is considered registered and cannot be re-registered in the future. the ratio of the number of registered objects of monitoring at the end of monitoring to their initial number determines the probability of object detection (registration) during the monitoring period. monitored objects move in the system's area of responsibility in a random manner with equal probability at each iteration, the offset distance per iteration is initialized by the user and describes the speed of the monitored objects. a frame of an iteration of a computer experiment at a fixed time is shown in fig. 5. modeling the detection of moving objects by means of a spatially distributed 7 copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 5. a frame of an iteration of a computer experiment 4. results and diccussion as a result of the experiment, was built the dependence ;'() ;* (𝑣, 𝑁%9:%) = 𝑃(𝑣, 𝑁%9:%), which is the dependence of the share of registered monitoring objects on their speed and the number of detection tools. the share of registered objects of monitoring can be interpreted as the probability of registering one object of monitoring during the monitoring period. the experimentally obtained dependence 𝑃(𝑣, 𝑁%9:%) with an increase in the number of detectors is illustrated in fig. 6 and 7. fig. 6. dependence of the share of registered targets on the speed of targets and the number of sensors 8 v.i. goncharenko, a.a. kochkarov, d.v. yatskin, b.v. rumiantsev copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 7. dependence of the share of registered objects of monitoring on their speed and the number of detection tools the resulting dependence has characteristic features, among which the following should be noted: 1. there is an area of "stationary" probability, in which for any number of detectors the probability ceases to effectively depend on their speed, since addiction 𝑃(𝑣) this region has the character of small random deviations from the mean value, which we will call the stationary value of the probability 𝑃%!<!. (see fig. 3, left). 2. the dependence of the stationary probability on the number of detection means is approximated with high accuracy by the exponential growth function with saturation (see figs. 6 and 8), i.e. 𝑃%!<!(𝑁%9:%) = 1 − e=>;+(+# (5) 3. for n?@a? = 1 the dependance 𝑃(𝑣) is monotonically decreasing in the region until the stationary probability is reached. therefore, at n?@a? = 1 the maximum probability of registering one monitoring object is achieved at zero target speed 𝑣 = 0. 4. it is also of interest to study the dependence of the threshold point 𝑣!b transition to stationary dependence on the number of detection means. in fig. 6 for 𝑁%9:% = 2,3,4 this point is shifted to the region of bigger speeds. modeling the detection of moving objects by means of a spatially distributed 9 copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 8. dependence of the stationary probability of registration of one monitoring object on the number of detection means 5. conclusion the proposed approach to optimizing the use of a continuous monitoring system makes it possible to efficiently register objects moving randomly in the area of responsibility. systems. within the framework of the budget work, a model that implements the ideology of the proposed approach. the model makes it possible to determine the optimal geometric arrangement, including the trajectory of movement of the means of a spatially distributed continuous monitoring system using the dynamic structure of dynamic targets. for a small number of detectors, the model allows you to find the optimal ratio of target speed to sensor speed. this, in turn, makes it possible to increase such an important indicator as to the probability of registration of one monitoring object. the work investigates statistical dependencies, including: • stationary probability of registration of one monitoring object from the number of detection means; • registered objects of monitoring on their speed and the number of means of detection; • registration of registered targets from speed and number of sensors. the conclusions drawn from the analysis of these dependencies confirm the effectiveness of the proposed model. in the region of stationary probability, an increase in the speed of sensors does not lead to a change in the probability of registering a target object in the singular. this allows us to conclude that it is possible to detect monitoring objects moving at a given speed, which makes it possible to determine such a threshold value for the detection rate when the efficiency of the system (from the point of the probability of object detection) exceeds the maximum values, which makes it possible to determine such a threshold value as optimal for solving detection tasks. acknowledgements this work has been supported by the grants the russian science foundation, rsf № 21-19-00481. 10 v.i. goncharenko, a.a. kochkarov, d.v. yatskin, b.v. rumiantsev copyright ©2022 assa. adv. in systems science and appl. 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(2020) risk management by trend analysis of flight information. russian engineering research, 40, 160–163, https://doi.org/10.3103/s1068798x20020136 microsoft word 1234 adv syst sci appl 2022; 04; 224-232 published online at https://ijassa.ipu.ru. cytokines production in test-volunteers during 120-day confinement in a hermetically sealed chamber sergey a. ponomarev, marina p. rykova, evgeniya n. antropova, anastasiya a. sadova*, ksenia d. orlova, sophia m. shulgina, darya d. vlasova, olga v. kutko, elina a. zhirova, evgeniy a. lysenko, vyacheslav a. schmarov v.a. institute of biomedical problems (ibmp), russian academy of sciences (ras), russian federation, e-mail: cd147@bk.ru abstract: the aim of the current study was to estimate the effect of 120-day confinement in a hermetically sealed chamber with an artificial environment on the ability of cells of innate immunity with pattern-recognition receptors and t-cells of adaptive immunity to synthesize cytokines. the experiment reproducing the conditions of a real spaceflight to the moon was conducted in the ground-based analog facility called nazemnyy eksperimental’nyy kompleks, or nek, within the institute of biomedical problems of the russian academy of sciences in moscow. the study presents the results of the analyses of cytokines production by immunocompetent cells from peripheral blood of six test-volunteers in response to in vitro antigen stimulation. we managed to demonstrate that the confinement led to the decreased lpsinduced cytokines and chemokines production. also, increased th1/th2, th1/th9, and th1/th17 cytokines ratios in response to pha were revealed. keywords: immune system, cytokines, isolation 1. introduction extreme conditions, which people have to inhabit in increasing frequency, cause an extension of functional load (both physical and mental) expressed in the overstrain of almost all physiological systems of the body that ensure adaptation to the extreme environmental factors. unfortunately, the issue of the potential danger of activity in such kind of conditions for a healthy human organism has been poorly studied to date by the traditional medicine, as it focuses mostly on disease states. only in recent decades specialists in aerospace medicine have begun to work on this problem. not coincidently, for this is the spaceflight that causes the most pronounced effects of environmental conditions on human body. in this regard, the data from comprehensive systemic studies of all vital processes that take place in a healthy human organism during space missions and ground-based experiments, enrich the science with the knowledge of the responses to various environmental stimuli. such investigations allow to bring the issues both of normal physiological body characteristics and physiological responses to certain stimuli, and of prediction of alterations in physiological state of organisms’ systems and their early-stage symptoms diagnosis to a close. space flight factors which pose a serious threat on astronauts’ health include increased radioactivity, neuro-psychological stress, intensive physical loads during extravehicular activity, artificial environment and microclimate, overloads (during launch and landing), lack of physical activity (hypodynamia), and, in fact, microgravity. immune system is the first one to be exposed to the extreme conditions of environment. from abundant experience of the studies on astronauts’ immunity functioning, the adaptation to a complex of space flight factors leads to both quantitative and qualitative shifts in the innate immune system and in the adaptive one. among them, the most important cytokines production in test-volunteers during 120-day confinement 225 copyright ©2022 assa. adv. in systems science and appl. (2022) are the following negative alterations: decrease in peripheral blood monocytes and granulocytes expressing pattern recognition receptors (tlr2, tlr4, and tlr6), decline in the functional activities of natural cytotoxicity system and t-cells of adaptive immunity, reduced synthesis of aand g-immunoglobulins (dysgammaglobulinemia) [1–5]. unfortunately, there exist a problem of laboratory of clinical immunology equipment onboard the space station. such a laboratory would not only provide an opportunity to estimate the immune state of astronauts during both preand post-flight periods, but also allow to continuously monitor the state of immunological resistance during the whole mission. undoubtedly, the results of the study of the immune system reactions after orbital flights may form the basis for the development of prospective countermeasures for prevention and correction of immune homeostasis disbalance. however, during the interpretation of these results one should consider the final stage of a space mission as not only a crucial, arduous, and emotionally stressful one, but also as the first trial faced by an astronaut after prolonged stay in the conditions of microgravity. in this regard, the investigation of the principles of immune system adaptation in ground-based model experiments simulating the conditions of long-term manned space missions and of orbital and planetary stations operation is of particular importance. the experiments on long-term confinement in the hermetically sealed chamber held in the ground-based analog facility (nek) within the institute of biomedical problems in moscow are one of the kinds of simulations of basic characteristics of real interplanetary missions. it is worth to note that the obligatory or voluntary lock-down regimes for citizens implemented in the first half of 2020 by the governments of many countries because of the risk of covid-19 expansion became an unexampled social event. many people put themselves in quarantine of their own free will to cease the infection transmission rates. interestingly, in this situation this were ill or potentially ill individuals but the healthy ones, who were in quarantine. undoubtedly, the countermeasures like social distancing and remote working help to stop or at least to decrease the circulation of the virus. however, there increases the risk of immune homeostasis disfunction. apparently, the negative shifts in the functioning of immunity may cause the development not only of allergies, infections and autoimmune disease, but also of malfunctioning in bone, nervous, and cardiac systems, in particular, during the irrational or partial use of countermeasures. thus, even the very small chance of the threatening states and diseases development claims the most rapt attention from the point of view of early diagnostics and treatment. in connection with this, the significance of the studies on the human immunity adaptation to modelled space flight conditions increases. to date, it is generally accepted that the interactions between the cells of immune system and those of different organs and tissues, as well as intercellular cooperation, positive and negative immune regulations are executed by cytokines. the latter are synthesized by immunocompetent cells in response to exogenous stimuli, such as pathogen-associated molecular patterns nearly of all types of microorganisms, chemical and physical damaging agents, as well as to a number of endogenous molecules which concentration rises in many disease states [6-9]. in this case cytokines take part in pathological processes including immunopathology, carcinogenesis, cardiac system disfunction, etc. taking into account the regulatory properties of cytokines the interest to the issue of space flight factors impact on them is of no surprise. cytokine-producing ability is known to inhere in all types of leukocytes, as well as thrombocytes, mast cells, epithelial cells, endotheliocytes, dendric cells, fibroblasts, and cells of nervous system. at the same time, activated monocytes/macrophages and t-lymphocytes being the main producers synthesize and secrete the basic spectrum of cytokines and are by right called their “professional” producers. these very cells play the key role in the maintenance of blood “cytokine background”. in view of aforesaid, the aim of the current study was to estimate the effect of 120-day confinement in the hermetically sealed chamber with an artificial environment on 226 ponomarev s.a. et al. copyright ©2022 assa adv. in systems science and appl. (2022) the ability of cells of pattern-recognition system of innate immunity and of t-cells of adaptive immunity to synthesize cytokines. 2. materials and methods 2.1 general experimental conditions the experiment of 120-day confinement in the hermetically sealed chamber with an artificial environment was conducted in the ibmp ground-based analog facility (nek) with three female and three male participants aged 24-45 years, who received a permission to take part in the experiment from a medical expert commission and signed an informed consent in accordance with the declaration of helsinki. the research program was approved at a meeting of the academic council and verified by the commission on biomedical ethics of ssc rf-ibmp ras (protocol no. 506 dated april 3, 2019). the experiment simulated the conditions of a real space flight to the moon including the following stages: the flight to the satellite and subsequent fly round to search for a landing site; the landing of the four of the crew members for lunar surface operations; the lunar orbital flight and remote control of lunar rover for the preparation of a moon base; the return to the earth. 2.2 sampling the immunological studies were conducted on the fasting samples of venous blood of the six almost healthy test-volunteers, i.e., the crew members of the 120-day confinement experiment “sirius-19”. the blood was sampled in background period (7 days before the experiment), during the experiment (on the 7th, 29th, 54th,71st, 96th, and 120th days of the confinement), and during the recovery period (7 days after the completion of the experiment). the blood was sampled from the cubital vein according to the standard method in aseptic conditions into the greiner bio-one vacuum tubes (austria) containing standard amount of sodium-heparin anticoagulant. 2.3 cytokines analyses cytokine production was determined using the cultured whole blood cells stimulated by the following ligands: lipopolysaccharides (lps) from escherichia coli o127 (sigma, usa) for toll-like receptors (tlr) stimulation (0,1 µg/ml); phytohemagglutinin (pha) fort tlymphocytes stimulation (10 µg/ml). heparinized (20 u/ml) venous blood was dissolved by a factor of five with rpmi-1640 (sigma-aldrich, usa) supplied with l-glutamine (0,3 mg/ml), 5mm hepes, and gentamicin (100 µg/ml), and cultured for 24h or 48h in sterile round-bottom tubes at 37 °c and 5% co2. following cultivation, supernatants were collected and stored at -80 °c until the analyses. in the 24h cultures stimulated with lps the following cytokines were assessed: ifna2, il-10, il-1b, il4, il6, il8, tnf, g-csf, mdc, mcp-1, mcp-3, mip-1, mip-1β, gro, rantes, ip-10; in the 48h cultures stimulated with pha the following cytokines were assessed: ifn, il-2, il-4, il-5, il-6, il-8, il-9, il-10, il-13, il-17а, tnfα, tnfβ, rantes, ip-10, mcp-1, mcp-3, gm-csf, mdc. the cytokines were analyzed using the commercial kit human cytokine/chemokine panel i (millipore, germany) for immunology multiplex assay according to manufacturer’s instructions. the results of the analyses were processed using statistica software (v.10.0) for microsoft windows. the significance of the results was assessed using the wilcoxon signedrank test. cytokines production in test-volunteers during 120-day confinement 227 copyright ©2022 assa. adv. in systems science and appl. (2022) 3. results and discussion during half of the century passed from the beginning of the space exploration by humanity the investigation of immunological reactivity in humans inhabiting and operating onboard a spacecraft has been the main task of space immunology. cytokine profiling in russian crew members of long-term space missions on the international space station demonstrated the pronounced changes in the ability of immunocompetent cells to produce cytokines in vitro upon the return to the earth [1, 2]. for instance, the analysis of one of the key parameters characterizing the state of pattern-recognition receptors system of innate immunity, namely, the production of cytokines by peripheral blood cells in response to lps (the ligand of tlr4), indicated the decrease in the ability of proinflammatory cytokines (il1β, il-6, il-8, tnfα) secretion by cultured monocytes obtained from cosmonauts during the period of early re-adaptation to earth conditions. even more significant changes were observed during the analysis of cytokine-producing ability of peripheral blood lymphocytes upon pha stimulation. in particular, upon the completion of long-term space missions, the shifts towards both the increase and the decrease in cytokines including ifn, il-2, tnfβ, il-4, il-5, il-6, il-10 were detected in the supernatants of cell cultures. it is worth to note that these in vitro investigations of immunocompetent cells’ cytokine-producing ability were conducted using the cultured whole blood cells, since this eliminates the probability of monocytes and granulocytes unwanted activation or death. besides, the cells were cultivated in conditions of natural microenvironment with the preservation of the balance of all humoral factors acting in vivo [10]. based on the above said, we analyzed the ability of cytokine production by immunocompetent cells of test-volunteers in different stages of their stay in conditions of 120-day confinement using the induced cytokine synthesis in response to lps or pha in cultured whole blood cells. during the experiment the significant changes in induced synthesis of some cytokines in response to lps were noted (table 1). thus, we managed to demonstrate a significant decrease in the production of il-1β (on the 54th, 71st, 96th, and 120th days), il-6 (on the 120th day), as well as of tnfα (on the 71st and 120th days) and il-10 (on the 71st and 120th days). interestingly, the decline in lps-induced synthesis of il-1β, tnfα, and il-10 was observed a week past the experiment as well. in addition to the changes described above, a significant decrease in the production of a number of chemokines was also detected: mcp1 (on the 29th day), mcp-3 (on the 29th day), mip-1 (on the 120th day), and ip-10 (on the 29th, 54th, 71th, and 120th days). at the same time, during the experiment the increase in the synthesis and secretion of il-8 (on the 54th day) and mip-1β (on the 29th day) was observed. the concentrations of other cytokines in the supernatants of cell cultures stimulated with lps did not significantly change during the whole experiment, however, there were pronounced individual shifts toward both the increase and the decrease in their concentrations compared to background. table 1. lps-induced cytokine production in whole blood cell cultures from test-volunteers participated in 120-day confinement in a hermetically sealed chamber with an artificial environment. (me; q75-q25) (n=6) cytokines pg/ml days of confinement -7 d 7 d 29 d 54 d 71 d 96 d 120 d +7 d il-1b 886,2; 486,41454,0 351,0; 282,4686,2 498,8; 439,0639,7 366,1*; 232,1402,5 180,8*; 113,7311,8 339,9*; 239,3398,6 190,5*; 181,7215,6 520,8*; 336,6596,8 il-6 1974,0; 1610,52058,5 1481,5; 1218,41793,5 1878,0: 1353,02147,0 1794,0; 1389,01857,0 1442,0; 973,01611,3 2211,0; 1966,82702,8 809,9*; 604,8839,9 2169,0; 1685,32281,5 il-8 1897,0; 1654,34139,3 1863,0; 1488,52729,3 2342,0; 1847,05826,0 2077,0*; 2023,02878,0 3470,0; 2372,34927,8 3756,5; 2734,35363,8 2871,0; 1797,02954,0 3137,5; 2408,56085,0 tnfα 755,4; 681,0615,6; 492,6556,9; 479,8599,7; 416,4408,3*; 295,31385,5; 1129,5506,1*; 335,5573,2*; 428,6228 ponomarev s.a. et al. copyright ©2022 assa adv. in systems science and appl. (2022) 952,1 737,4 713,6 619,4 579,8 1681,3 547,7 725,4 il-4 155,6; 107,5205,7 141,6; 119,6199,4 261,7; 188,4289,8 153,0; 118,3202,7 186,3; 173,9282,3 263,2; 227,8304,5 202,7; 135,6213,8 187,3; 167,8309,0 il-10 630,1; 480,8789,9 291,2; 264,8413,6 665,3; 403,61137,0 353,8; 253,8430,6 189,3*; 137,0255,9 1021,9; 662,51418,8 137,6*; 89,4-402,5 197,5*; 137,7263,0 ifna2 13,7; 12,5-22,3 19,4; 11,7-28,8 18,9; 16,9-20,1 17,8; 12,1-35,0 19,7; 11,2-28,9 24,6; 19,2-36,7 21,9; 21,2-38,9 22,6; 11,6-33,3 g-csf 614,1; 406,0987,3 320,3; 267,5678,6 850,6; 517,81964,0 435,4; 355,9713,0 329,1; 289,1357,6 939,9; 451,41247,3 282,5; 190,8313,6 352,5; 325,1619,8 gro 3363,0; 2613,06756,8 4212,5; 1564,87129,0 3581,0; 2865,07258,0 7438,0; 6616,08041,0 4334,5; 3785,09643,5 10436,0; 6867,011287,0 4891,0; 4266,08520,0 7254,0; 3779,311021,3 mcp-1 3088,5; 1911,33749,0 3987,0; 3561,84311,5 1857,0*; 1222,02765,0 3250,0; 2769,04052,0 4836,5; 3525,55808,5 4499,0; 3948,05983,0 3826,0; 3540,03839,0 3749,0; 2982,84900,8 mcp-3 447,6; 368,8510,5 330,5; 217,7546,6 245,9*; 210,6268,9 344,9; 227,5412,7 470,7; 272,8736,9 607,4; 436,2872,8 329,8; 307,2357,4 482,7; 363,5595,8 mip-1a 5059,0; 4718,05223,0 3442,0; 3108,53917,0 2457,0; 2256,05624,0 5430,0; 3815,05997,0 2793,5; 1527,06358,8 7276,0; 3507,08239,3 2259,0*; 2155,02778,0 4867,0; 3558,86728,0 mip-1b 3258,0; 957,64848,8 2841,5; 1680,84065,8 3765,0*; 1180,04422,0 2701,0; 1628,03618,0 2570,5; 2010,02687,8 3428,5; 1444,56012,5 2436,0; 1329,03994,0 3220,5; 1634,54070,8 mdc 279,5; 156,4416,7 186,7; 154,7261,1 240,6; 146,1306,3 474,9; 235,0557,3 248,9; 185,9301,6 282,4; 238,4484,4 205,4; 157,6326,8 265,6; 145,2394,3 rantes 1325,0; 1053,51682,0 1328,1; 878,81797,5 983,8; 776,71893,0 1256,0; 1047,01475,0 1721,5; 1467,82291,0 2212,0; 1536,82903,8 1323,0; 1224,01929,0 2020,0; 1337,52248,8 ip-10 1084,5; 947,51323,8 465,9; 450,0646,3 300,0*; 268,5443,4 660,3*; 331,1853,4 232,8*; 172,5483,6 341,0; 269,8672,9 392,0*; 391,5405,9 616,3; 450,81121,6 *significant difference compared to background (р<0,05). it is known that lps are complex biopolymers composed of a glycolipid and a polysaccharide characteristic to bacteria; they are the essential structural components of the outer layer of the external membrane of gram-negative bacteria which induce the signal transduction through tlr4. the latter is mostly expressed by myeloid cells, namely, monocytes and macrophages. the activation of tlr4 by lps is accompanied by the signaling via myd88-dependent pathway and leads to the activation of nf-κb and cytokine (tnf-α, il-1, il-6, etc.) and chemokine (mcp-1, mcp-3, gm-csf, etc.) release [11,12,13]. one could suppose that the decrease in the content of cytokines and chemokines such as il1β, il-6, tnfα, mcp-1, mcp-3, mip-1, ip-10 in the supernatants from the lpsstimulated cultures of whole blood cells observed in the experiment was mediated by the decrease in the level in the peripheral blood of monocytes and granulocytes expressing surface cd14/tlr4 complex. however, as it was earlier demonstrated [14], 120-day confinement in a hermetically sealed chamber with an artificial environment led to a significant increase in both absolute and relative counts of tlr4+-monocytes in the peripheral blood. the increase in their percentage was noted on the 29th, 57th, 63rd, 87th, 96th, and 120th days of the experiment. the absolute count of the monocytes rose on the 7th, 29th, and 96th days of the confinement, as well as on the 7th day upon the completion of the experiment. as for granulocytes, their content significantly increased on 57th, 87th, 96th, and 120th days of the experiment compared to background, with the significant changes demonstrated only for absolute counts. on the other hand, the decline in the lps-induced proinflammatory cytokine production could be explained by the increased synthesis of the main regulatory proinflammatory cytokine, namely, il-10, but the data from the experiment demonstrated the decrease in this parameter during the experiment. it is possible, that the cytokines production in test-volunteers during 120-day confinement 229 copyright ©2022 assa. adv. in systems science and appl. (2022) phenomenon observed indicates the changes in the tlr signaling pathways induced either by an inhibitory action of lirls or some similar regulatory receptors, or by the failure of myd88-dependent or myd88-independent intracellular signaling cascade. in general, the data obtained evidence the wavy fluctuations in lps-induced cytokine production in whole blood cell cultures in conditions of the confinement in the hermetically sealed chamber towards the decrease in the supernatant cytokine levels at the final stage of the experiment. during the investigation of cytokine synthesis in response to pha, a polyclonal tlymphocytes activator, a number of peculiar changes was observed (table 2). firstly, a significant decrease in the contents of ifn (on the 7th day), il-6 (on the 29th, 54th, and 71st days), il-8 (on the 71st day), rantes (on the 29th, 54th days), and mdc (on the 120th day) in the supernatants of pha-stimulated cultures was demonstrated. secondly, on the 29th day of the experiment we noted an increase in the production of il-2, il-4, il-17а, and on the 96th day an increase in tnfβ and мср-3 was detected. table 2. pha-induced cytokine production in whole blood cell cultures from test-volunteers participated in 120-day confinement in a hermetically sealed chamber with an artificial environment. (me; q75-q25) (n=6) cytokines pg/ml days of confinement -7 d 7 d 29 d 54 d 71 d 96 d 120 d +7 d ifnγ 2061,5; 833,23329,0 1105,6*; 621,91857,8 3161,5; 1160,85837,0 2702,0; 1168,04171,0 2277,5; 1425,84663,0 2793,5; 1402,54927,8 2570,5; 812,94101,0 4584,5; 2604,96316,3 il-2 231,0; 175,6281,5 197,7; 94,6-350,4 411,8*; 238,7514,4 277,0; 269,9330,4 239,7; 191,7354,6 290,9; 223,1344,9 149,4; 105,6188,0 454,6; 247,6868,5 tnfα 749,7; 459,31482,5 857,6; 511,21062,6 1236,2; 869,22443,5 1293,0; 721,22208,0 965,7; 582,92035,6 1257,5; 694,41795,8 1031,0; 417,81593,0 1644,5; 1264,02602,5 tnfβ 16,5; 11,7-20,3 14,7; 8,1-18,1 41,1; 16,4-77,5 25,5; 22,1-41,7 29,7; 15,5-39,1 18,0*; 13,6-40,3 13,1; 6,6-18,9 33,3; 23,0-71,8 il-4 578,3; 428,9675,1 608,3; 428,5759,3 841,6*; 518,51577,3 795,5; 705,21075,0 617,9; 403,3913,3 661,5; 490,7835,8 452,1; 382,9544,7 836,0; 435,61182,8 il-5 38,0; 25,4-62,1 33,8; 22,7-74,7 85,8; 35,8-133,0 52,4; 49,5-91,8 69,0; 36,7-99,9 67,5; 46,9-70,7 36,7; 31,1-60,8 60,2; 41,0-85,9 il-6 1574,5; 1506,31711,8 1699,0; 1596,02012,8 1133,0*; 794,51284,8 1317,0*; 583,71454,0 1174,5*; 859,11405,8 1458,5; 1008,21466,5 1521,0; 1047,71725,8 1281,9; 684,22105,8 il-13 367,1; 189,7558,9 232,6; 202,3354,9 462,5; 351,2502,7 383,3; 217,2479,7 451,0; 327,2517,4 234,9; 204,1398,7 271,0; 150,1360,1 281,0; 181,6366,8 il-10 1424,4; 438,42583,3 684,9; 523,8926,7 1762,0; 1143,82970,5 1302,0; 1247,01777,0 799,3; 496,01319,2 1150,9; 555,21904,5 554,7; 157,11051,1 610,8; 474,01324,9 il-9 40,4; 30,5-51,6 26,9; 14,1-38,4 53,6; 24,7-62,2 63,4; 36,5-69,8 28,9; 16,1-59,4 23,8; 15,1-37,4 14,8; 11,3-26,6 51,8; 40,6-63,1 il-17a 292,1; 248,1335,1 336,7; 235,6429,7 380,4*; 306,3435,3 336,6; 323,5491,0 255,1; 154,4353,7 315,0; 231,2546,8 259,7; 247,7279,9 237,5; 175,3373,4 il-8 5719,5; 5274,36348,5 5811,5; 4754,36730,8 4919,5; 4238,57701,3 5904,0; 5360,06854,0 5175,5*; 3845,36062,5 5739,5; 4249,88315,3 4487,0; 2689,38489,0 3832,5; 2585,05133,5 gm-csf 195,4; 130,7547,6 166,8; 128,1226,2 457,1; 291,8599,0 236,0; 205,3357,8 358,6; 245,3498,7 284,6; 224,8363,5 259,5; 131,5425,4 251,4; 183,6778,6 mcp-1 4752,0; 2411,85792,5 4277,5; 2845,05665,8 4357,0; 2414,36177,5 4761,0; 3715,07281,0 4699,5; 2183,05160,3 4367,5; 2765,06045,8 3502,5; 2262,84330,5 3869,0; 2102,35050,3 mcp-3 1550,5; 847,82245,5 1421,0; 926,02375,8 1639,0; 890,61926,3 1489,0; 1345,06304,0 1297,5; 818,71563,8 2932,0*; 1533,89231,5 644,0; 566,7988,1 1002,8; 558,81851,0 mdc 1058,4; 490,5996,6; 380,63089,5; 1371,92105,0; 632,82218,5; 489,71435,5; 579,8485,3*; 318,61746,5; 1645,0230 ponomarev s.a. et al. copyright ©2022 assa adv. in systems science and appl. (2022) 2986,8 1931,8 4431,8 3199,0 4325,8 1873,3 1499,0 2280,3 rantes 1921,5; 1382,33050,3 1484,5; 1353,82105,8 1490,5*; 1297,02250,3 1435,0*; 1235,02636,0 1768,0; 1132,92419,0 1758,0; 1238,52738,8 1264,5; 1034,02195,3 2357,5; 1941,42768,3 ip-10 4467,5; 3672,35837,3 4358,5; 3277,35682,8 4371,5; 2892,35610,8 5049,0; 3727,05407,0 3469,5; 3025,34582,0 5290,0; 3538,05714,5 3087,0; 2376,56222,3 3976,0; 2740,56241,0 * significant difference compared to background (р<0,05). during the last decade the researchers focus of the study of t-helper subpopulations (th1, th2, th9, th17). this line of investigation allowed to localize the particular subpopulations among the naïve t-helpers, so-called minor subsets in charge with the secretion of certain cytokines [15]. as we demonstrated in the current work, starting already from the 29th day of the confinement in the hermetically sealed chamber there occurred the disbalance in the production of cytokines by the four types of t-helpers (table 3). these changes were characterized by the high ratios of th1/th2, th1/th9, and th1/th17 cytokines (inf-γ / il-4, inf-γ / il-10, inf-γ / il-9, inf-γ / il-17а) compared to background. at the same time, there were no statistically significant changes in th2/th9, th2/th17, and th9/th17 ratios (il-4/ il-9, il-4/ il-17а, il-10/ il-17а). table 3. the coefficients of the balance of th1-/th2-/th9-/th17-cytokines produced in pha-stimulated cultures of whole blood cells from test-volunteers participated in 120-day confinement in a hermetically sealed chamber with an artificial environment. (me; q75-q25) (n=6) cytokine ratios days of confinement -7 d 7 d 29 d 54 d 71 d 96 d 120 d +7 d ifnγ/il-4 3,22; 1,41-4,31 1,97; 1,14-2,42 3,43; 2,65-3,80 3,45; 3,40-5,06 4,68*; 4,05-5,21 4,05; 3,35-6,00 3,93; 2,19-6,26 7,31; 2,16-16,42 ifnγ/il10 1,33; 1,22-1,54 1,70; 0,98-2,17 1,30; 1,24-1,77 1,59*; 1,52-2,27 3,17; 2,97-4,58 2,31; 2,07-2,83 4,42; 3,68-4,62 4,54; 3,02-6,59 ifnγ/il-9 35,66; 30,7248,00 48,24; 20,8478,01 82,10; 42,46100,60 61,31; 16,7373,97 104,86*; 54,43166,79 133,69*; 86,62190,74 58,60; 42,57220,40 69,61; 43,94117,37 ifnγ/il17а 5,24; 2,8410,20 2,80; 1,7311,47 5,71*; 4,95-12,09 5,40; 3,47-9,88 12,01*; 7,43-21,54 6,20; 4,24-8,33 7,41; 3,13-13,76 15,37; 9,97-22,26 il-4/il-9 15,94; 13,2718,35 26,23; 20,1436,24 27,10; 22,9633,07 17,75; 16,9521,78 25,29; 21,43-33,39 26,88; 19,09-55,36 31,87; 21,5643,01 13,50; 7,20-21,74 il-4/il17а 2,50; 1,32-3,18 1,65; 1,52-3,47 2,87; 1,49-3,85 2,19; 0,91-2,91 2,52; 1,98-4,07 1,54; 1,40-1,63 1,60; 1,35-2,02 3,39; 2,24-4,66 il-10/il17а 3,10; 1,61-8,31 2,14; 1,58-2,75 4,89; 4,27-5,34 3,85; 2,37-3,87 3,02; 2,02-6,50 2,26; 1,80-3,29 1,54; 0,65-3,76 3,49; 2,56-4,26 * significant difference compared to background (р<0,05). these changes indicate not only the alterations in the cytokine producing ability of tlymphocytes but also the appearance of the signs of disbalance in the functioning of cytokines network which results from the increased th1-cytokines secretion. the multidirectional action of the mediators secreted by th1-, th2-, th9-, and th17-cells is known to cause inter-suppression of these populations, due to which the dynamic balance in the functioning of th1-, th2-, th9-, and th17-cells is preserved. the disturbance of this balance can lead to the immune response suppression and the development of immunopathological state [16-18]. stressful experimental conditions simulating the crew activity in “circumlunar orbit” and on the moon surface could not but affect the reserve ability of t-cells to respond to an antigenic stimulus. the study of induced cytokine production by peripheral blood t-lymphocytes also revealed the multidirectional shifts in th1-, th2-, and th17-cytokines secretion. the changes detected indicate the appearance of the sings of disbalance in th1/th2, th1/th9, and th1/th17 types of cytokines which posses both the local effects on monocyte-macrophage and lymphoid cells and the systemic action on distal organs and tissues [19,20]. cytokines production in test-volunteers during 120-day confinement 231 copyright ©2022 assa. adv. in systems science and appl. (2022) thus, the prolonged stay in the hermetically sealed chamber causes an organism reorganize to further function at a new level. taking into consideration the cytokineproducing ability being one of the key parameters characterizing the state of the immune system, the disbalance of antigen-induced synthesis of cytokines can be referred to as a sign of stress in the immunity, and consequently the sign of risk of secondary immunodeficiency development. 4. data availability all data used in this study can be obtained upon reasonable request. 5. funding this work was supported by the russian science foundation, project no. 18-75-10086-p and theme 65.1 of the fundamental research of the ibmp ras 6. conclusion the issue of the mechanisms in charge with the development of the defects in immunological reactivity in response to the effects of the extreme environmental factors on human body is of current importance, for the life of a human today is a continuous sequence of stress and charges. the results of the present study dedicated to immunocompetent cells’ ability to produce cytokines in response to antigens conducted on the test-volunteers participated in the 120-day confinement in nek at ibmp demonstrated that the prolonged exposure to the environmental factors of the hermetically sealed chamber had pronounced effect on the reserve abilities of cells of both innate and adaptive immunity. at this, the noteworthy variability of the results among the participants indicates the individual predisposition to the defects in the immune reactivity in the confinement conditions. the individual differences in the changes of immunocompetent cells activation potential point out that one should consider them when conducting preventive or corrective measures. further accumulation of the data on the peculiarities of the human immune system functioning in conditions of a modelled spaceflight and the subsequent processing of these data and their comparison to those on the state of other systems of the organism will allow to proceed to the diagnosis of a pre-disease state, i.e., the stage when the disease development can be easily prevented without its aggravation in unfavorable environmental conditions. references 1. morukov, v.b., rykova, m.p., antropova, e.n., berendeeva, t.a., ponomarev, s.a., et. al. 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(2016). th1-, th2-, and th17associated cytokine expression in hypopharyngeal carcinoma and clinical significance. eur arch otorhinolaryngol, 273(2), 431-438. 20. jiang, q., yang, g., xiao, f., xie, j., wang, s. (2021). role of th22 cells in the pathogenesis of autoimmune diseases. front immunol., 12, 688066. adv syst sci appl 2023; 04:8–17 published online at https://ijassa.ipu.ru. a state space filtering-based approach for price prediction anton belyakov12, aleksei kurbatskii1*, artur sidorenko1 1moscow school of economics, lomonosov moscow state university, moscow, russia 2 moscow institute of steel and alloys ‘‘misis’’, moscow, russia abstract: we present a method of the forecasting and the data filtering of a linear dynamic system based on the dimension reduction of the space of unobservable states. the method relies on the singular value decomposition of the hankel matrix. the decomposition is used to calculate unknown parameters of the model. the elements of the singular value decomposition are separated into blocks enabling to estimate the initial state and the system matrices and predict the system dynamics and the data filtering by identifying exponential trends and periods of seasonal fluctuations. to illustrate the quality of fitting and the determined periods of an oscillatory system with trends and the white noise, we conducted numerical simulations of such systems. the parameter estimates were obtained with high precision. then, daily electricity price data from the nordpool system from 2016 to 2020 were used to generate in-sample and out-of-sample forecasts. the advantages of the proposed method include the ability to handle ill-conditioned matrices and to determine the periods of oscillatory systems. this is significant due to the presence of seasonality in many economic indicators. in the analyzed daily electricity price data, the method identified the presence of biweekly and monthly seasonality. keywords: state space method, principal components, time series, singular value decomposition, filter, electricity price forecasting 1. introduction the representation of systems in the state space is a common way to describe the dynamics of linear systems. state space models constitute one of the classes of time series models that are successfully applied in management of technical systems and solving system identification problems, as well as in the fields of filtering and forecasting in economics and finance. descriptions of these models in economic and financial contexts can be found in [13,15,17]. these works also discuss parameter estimation methods for such models, including maximum likelihood and bayesian methods. one of the first approaches in this paradigm is the renowned kalman filter, see, e.g., [12]. among the earliest applications of state space methods and the kalman filter, particularly in economics and finance, notable works include [5, 8]. in this work, in addition to filtering, the problem of estimating the hyperparameters of the state space model is considered. in such situations, making forecasts includes preliminary estimation of unknown hyperparameters. in this regard, we rely on and develop the methodology described in [2, 16]. the model used in this work is as follows: there is a linear evolution of hidden states, where observed variables are obtained by projecting from the space of hidden unobservable ∗corresponding author: akurbatskiy@gmail.com a state space filtering-based approach for price prediction 9 variables. algebraically, this concept is expressed as a system of two vector linear equations, where one difference equation describes the change in the hidden state xt over time t, while the other relates the observed parameters yt and the unobservable parameters xt. random factors influence the system, causing the proposed dynamics not to be fully reproduced. random deviations are modeled as mutually independent random variables. to identify the system, it is necessary to estimate the dimension of the vector xt of hidden variables, the initial state x1, and the corresponding representation matrices in state space. the dimension of the hidden state space is a hyperparameter that is either externally specified or determined by cross-validation methods. the approach based on the singular value decomposition of the hankel matrix is used to calculate the unknown parameters of the model [2, 3, 10, 16]. this study focuses on forecasting daily electricity prices from 2016 to 2020, and additionally proposes a data filtering method. on a sample of 1170 observations, it was found that the short-term out-of-sample forecast is comparable to traditional econometric models and to the well-known holt–winters’ filtering method based on the mean absolute error indicator even in a stationary segment of the time series. in addition to modeling on real data and comparing the forecast quality of the proposed method with well-known models, we present several numerical simulations. these simulations demonstrate the specific characteristics of data behavior that the proposed state space model can identify. 2. basics in general, an autoregressive system can be represented in the state space as follows: xt+1 = axt + ξt, yt = cxt + ηt, where t ∈ n, xt is an unobservable k × 1 state vector, x1 is deterministic (but unobservable), yt is an observable l × 1 vector, (ξt, ηt)t∈n is an unobservable white noise, and a and c are state space matrices. it is easily seen that the expected value eyt+1 = catx1. for the sake of simplicity, we set ξt = 0 and ηt = 0 for all t. in such a case, yt+1 = catx1. (2.1) having a sample of size t the hankel matrix h can be written as follows for an odd-sized sample: h =  y1 y2 . . . yn y2 y3 . . . yn+1 ... ... . . . ... yn yn+1 . . . y2n−1  = γ1:nω1:n (2.2) and for an even-sized sample: h =  y1 y2 . . . yn+1 y2 y3 . . . yn+2 ... ... . . . ... yn yn+1 . . . y2n.  = γ1:nω1:(n+1), (2.3) copyright © 2023 assa. adv syst sci appl (2023) 10 a. belyakov, a. kurbatskii, a. sidorenko where n = [ t 2 ] . by virtue of formula (2.1) the hankel matrix is the product of the observability matrix γ1:n =  c ca ... can−1  and the controllability matrix ω1:n = [x1 ax1 . . . an−1x1] for an odd-sized sample, or ω1:(n+1) = [x1 ax1 . . . anx1] for an even-sized sample. therefore, if we find a decomposition of the hankel matrix as a product of two matrices as in (2.2) or (2.3), we can estimate the matrices of the system: ĉ = γ̂1:1, â = ( γ̂1:(n−1) )+ γ̂2:n, x̂1 = ω̂1:1, (2.4) where ( γ̂1:(n−1) )+ is the moore–penrose pseudoinverse of the matrix γ̂1:(n−1) excluding the last block, and the matrix γ̂2:n does not include the first block. the hat sign indicates that these are estimates of matrices γ and ω that will be suggested now. as in [16], we apply the singular value decomposition of the observed matrix ĥ = û ŝv̂ ′, where ŝ is a diagonal matrix of singular values arranged in decreasing order, and û and v̂ are orthogonal matrices. we consider an odd t since the case for even t is similar. these matrices can be split into parts corresponding to, as we assume, the signal and noise: û = [ û1:k û(k+1):n ] , v̂ = [ v̂1:k v̂(k+1):n ] , ŝ = [ ŝ1:k 0 0 ŝ(k+1):n ] , where û1:k and v̂1:k are the matrices consisting of first k columns of matrices û and v̂ correspondingly, while ŝ1:k is the square k × k matrix having in its diagonal first k singular values of ĥ . the estimates for the observability and controllability matrices can be found as follows: γ̂1:n = û1:kŝ 1/2 1:k , ω̂1:n = ŝ 1/2 1:k v̂ ′ 1:k. once the estimates ĉ, â and x̂1 are obtained from (2.4), formula (2.1) can be used for forecasting. it should be noted that matrices c, a and vector x1 are determined, in general, up to a non-degenerate linear transformation with matrix p : c̃ = cp−1, ã = pap−1, x̃1 = px1. but, for forecasting and filtering, it doesn’t matter. however, in practice, formula (2.1) can lead to significant computational errors, as the matrix a is multiplied by itself many times. to find a more accurate estimate, let us express (2.1) as yt = can−1at−tan−1x1 = γn :na t−tωn :n, considering odd t . a suggested estimate for t ≥ t is the following ŷt = γ̂n :nâ t−t ω̂n :n. in addition to the forecasting task, we propose a possible solution to the data filtering problem. let’s consider the case for odd t , as for even t the calculations are similar. it is copyright © 2023 assa. adv syst sci appl (2023) a state space filtering-based approach for price prediction 11 noted that for t ≤ t yt = γk : kω(t−k+1) : (t−k+1), where 1 ≤ k ≤ t. the proposed estimate is set as an average value of estimates with different parameters k in the following way ŷt = 1 min(t, n)−max(0, t+ 1− n) min(t,n)∑ k=max(0,t+1−n) γ̂k : kω̂(t−k+1) : (t−k+1). in order to select the dimension k of the state vector (state space dimension, ssd), one may apply any cross-validation method, e.g. the mean absolute error (mae) of the one-dayahead (or n-days-ahead) forecast in a rolling-sample scheme. 3. modeling 3.1. simulation modeling let us consider three model examples that illustrate the method’s performance on noisy data with some oscillations. 1) oscillatory system with two periods yt = sin(πt/15) + sin(πt/20) + εt/2, where (εt) are independent identically distributed random variables with εt ∼ n (0, 1). a grid with t = 0, . . . , 299 of length 300 was taken, and a trajectory was simulated on it. to estimate the oscillation period, the method described in [2, 3] was applied. according to this method, the spectrum of the matrix log(a) is found. we assume that matrix a does not have multiple eigenvalues. under this condition, oscillation with angular frequency ω will correspond to a pair of complex conjugate numbers z = α + iω and z̄ = α− iω, where the oscillation period is given by π/ω. in the first example, the estimated periods of oscillation were 30.49 and 39.93, while the original periods in the system before noise were 30 and 40 (see fig. 3.1). in the two following examples, the estimated oscillation periods were the same as in the first one, up to rounding accuracy (see fig. 3.2 and 3.3). 0 50 100 150 200 250 300 time −3 −2 −1 0 1 2 3 w av e filtration and forecast of a noisy wave original wave with noise predicted wave filtered wave separation line between in-sample and out-of-sample fig. 3.1. filtration and forecast of a noisy wave. copyright © 2023 assa. adv syst sci appl (2023) 12 a. belyakov, a. kurbatskii, a. sidorenko 0 50 100 150 200 250 300 time −4 −2 0 2 4 6 w av e filtration and foreca t of a noi y wave original wave with noi e predicted wave filtered wave separation line between in-sample and out-of-sample fig. 3.2. filtration and forecast of a noisy wave with exponents. 0 50 100 150 200 250 300 time 0 5 10 15 20 w av e filtration and forecast of a noisy wave original wave with noise predicted wave filtered wave separation line between in-sample and out-of-sample fig. 3.3. filtration and forecast of a noisy wave with exponential trend. 2) oscillatory system with variable amplitude yt = e0.005t sin(πt/15) + e−0.001t sin(πt/20) + εt/2. 3) system with oscillations and exponential trend yt = e0.01t + e0.005t sin(πt/15) + e−0.001t sin(πt/20) + εt/2. 3.2. electricity price modeling for modeling based on real data, we will consider electricity prices. the electricity market is of interest due to the presence of high volatility, multiple seasonality and calendar effects. in [11], it is shown that hourly electricity prices exhibit intraday, weekly, and monthly periodicities, as well as variable volatility. copyright © 2023 assa. adv syst sci appl (2023) a state space filtering-based approach for price prediction 13 we use daily nordpool system data prices† from 2016 to 2020, as considered in the paper [9], because it serves as the unconstrained market clearing reference price for the european nordic region. daily prices were transformed into $/kwh. the sample consists of 1170 observations from january 01, 2016, to june 26, 2020. for out-of-sample forecasting, a rolling window of 22 ∗ 6 days (number of working days in a month multiplied by the number of months) is used (see fig. 3.4). 20 16 -01 -04 20 16 -08 -01 20 17 -02 -27 20 17 -09 -25 20 18 -04 -23 20 18 -11 -19 20 19 -06 -17 20 20 -01 -13 date 0.00 0.02 0.04 0.06 0.08 pr ice daily electricity prices data model in-sample prediction model out-of-sample prediction separation line between in-sample and out-of-sample fig. 3.4. electricity price with the model filtration and 1-day-ahead forecasts with a 6 month floating window. the estimated periods on the interval 2016-01-01 to 2016-06-30 are 11.22 and 22.34. to determine the state space dimension k, we used the mean absolute error for 1, 5, 22 days in a rolling-sample scheme with an interval size for modeling of 226 days (half a year). from the graph, it can be observed that the optimal dimensions for forecasting are k = 5 for daily and weekly predictions (see fig. 3.6 and 3.7). it should be noted that for large dimensions k, out-of-sample errors increase, although in-sample errors decrease. a comprehensive review of literature on electricity price forecasting (epf) methods and their comparison from 2012 to 2022 is presented in the article [14]. notably, during relatively stable times, when conditions for building traditional econometric models are met, such methods work well and demonstrate high predictive characteristics. even considering high volatility, the combined arma-type model and garch-type model are still applied [4, 6, 11]. to make a comparison with arma-type models, for which we selected a stationary series from january 01, 2016, to june 30, 2016. the results of corresponding tests are provided in table 3.1. table 3.1. stationarity tests results test statistic p-value adf −5.3567 4.17× 10−6 kpss 0.2103 > 0.1 df-gls −4.6647 5.38× 10−6 †the nord pool exchange has free data https://www.nordpoolgroup.com/en/market-data1/dayahead/areaprices/sys1/daily/?view=table copyright © 2023 assa. adv syst sci appl (2023) 14 a. belyakov, a. kurbatskii, a. sidorenko besides, an alternative and well-established data filtering method with seasonality, holt– winters exponential smoothing, was considered. fig. 3.5 shows the mean absolute error dependencies for our method for different values of k, as well as for the arma(1,1) model, which proved to be the best model of this class on the training set, and the ets model using holt–winters’ method. one-day-ahead, weekly, and monthly forecasts were made using a rolling-window scheme for all models. over an extended period, the markets often exhibit a complex nonlinear structure as well as natural non-stationarity. an attempt to forecast the price for the next six months is shown in fig. 3.5. it can be seen that our presented model captures the trend, while the arma model and the holt–winters’ method, even though it gives more accurate in-sample forecast, degenerate into a horizontal line. the fact that these models struggle to predict the behavior of such data has long been known, so researchers seek approaches to improve their quality. a survey on subspace methods for estimating linear dynamic models can be found in [1]. 20 15 -12 -31 20 16 -02 -19 20 16 -04 -09 20 16 -05 -29 20 16 -07 -18 20 16 -09 -06 20 16 -10 -26 20 16 -12 -15 date 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 pr ice forecast and filtration of the state space model data state space filtration ets filtration ets forecast arima forecast state space forecast separation line between in-sample and out-of-sample fig. 3.5. state-space filtration and forecast and comparison with arima forecast and holt–winters ets filtration. the periods calculated by the state space model are 11.22 and 22.34 days. fig. 3.6, 3.7 and 3.8 show how the mae of considered method changes depending on the dimension of the state space k. for the short-term forecasts, the optimal value of k is 5, whereas for monthly forecasts, it is equal to 1. the implementation of our forecasting and filtering method in the python programming language can be found at the following link [7]. 4. conclusion methods of forecasting based on the state space representation of a system have a wide range of applications for estimating various dynamic systems. however, in different research contexts and on different types of data, choosing the appropriate method is a non-trivial task, considering the inherent advantages and disadvantages of each method. our proposed forecasting method and an improved filtering method allow working with ill-conditioned system matrices. the ability to identify periods of fluctuations is also of interest for understanding the behavior of cyclical indicators, particularly economic ones. copyright © 2023 assa. adv syst sci appl (2023) a state space filtering-based approach for price prediction 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 hidden dimension 0.002 0.003 0.004 0.005 0.006 m ae electricity mae, 1-day(s)-ahead forecasts, rolling window with length 22 * 6 days state space arima (1, 0, 1) ets fig. 3.6. 1-day-ahead mae over 2016 (6 month window). 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 hidden dimension 0.0025 0.0030 0.0035 0.0040 0.0045 0.0050 0.0055 0.0060 0.0065 m ae electricity mae, 5-day(s)-ahead forecasts, rolling window with length 22 * 6 days state space arima (1, 0, 1) ets fig. 3.7. 5-days-ahead mae over 2016 (6 month window). 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 hidden dimension 0.004 0.006 0.008 0.010 0.012 0.014 0.016 m ae electricity mae, 22-day(s)-ahead forecasts, rolling window with length 22 * 6 days state space arima (1, 0, 1) ets fig. 3.8. 22-days-ahead mae over 2016 (6 month window). to illustrate the method, simulation modeling was conducted on typical types of oscillatory systems, and the forecast quality was evaluated with different horizons for electricity prices. copyright © 2023 assa. adv syst sci appl (2023) 16 a. belyakov, a. kurbatskii, a. sidorenko during the comparison of model forecasts on electricity price data, it was found that the forecasts of the proposed method yield a comparable mean absolute error to the forecasts of traditional econometric models such as arma on stationary series and a well-known seasonal data filtering method like holt–winters’ method. the mean absolute error was calculated for forecasts with a rolling window of 6 months. it is worth noting that the method performs well in handling multiple seasonality and exponential trends inherent in data with such a complex structure as electricity prices. additionally, oscillation periods were calculated, and they turned out to be 11 and 22 days, corresponding to two-week and monthly seasonality. acknowledgements the research was conducted with financial support of the russian science foundation grant no. 19-11-00223. references 1. bauer, d. (2005). estimating linear dynamical systems using subspace methods. econometric theory, 21(1), 181-211. doi:10.1017/s0266466605050127 2. belyakov, a. o., & blazhennova-mikulich, l. yu. (2005) identifikatsiya inertsionnoy matritsy konservativnoy kolebatelnoy sistemy [identification of the inertia matrix of a conservative oscillating system]. vestnik moskovskogo universiteta. seriya 1. matematika. mekhanika, 3, 25–28. 3. belyakov, a. o., & seiranyan, a. p. (2008) determining the moments of inertia of large bodies from vibrations in elastic suspension. mechanics of solids, 43, 205–217. 4. bill´e, a.g., gianfreda, a., del grosso, f., ravazzolo, f. (2023). forecasting electricity prices with expert, linear, and nonlinear models. international journal of forecasting, 39(2), 570-586. https://doi.org/10.1016/j.ijforecast.2022.01.003 5. clark, p. k. (1987) the cyclical component of u. s. economic activity. the quarterly journal of economics, 102(4). https://doi.org/10.2307/1884282 6. girish, g.p., 2016. spot electricity price forecasting in indian electricity market using autoregressive-garch models. energy strategy reviews 11–12, 52–57. https://doi.org/ 10.1016/j.esr.2016.06.005 7. github. (2023, november). subspace method. [online]. available https://github.com/artursidorenko/subspace method 8. fama, e. f., & gibbons, m. r. (1982) inflation, real returns and capital investment. journal of monetary economics, 9(3). https://doi.org/10.1016/0304-3932(82)90021-6 9. fantazzini d., kolodin n. (2020) does the hashrate affect the bitcoin price? journal of risk and financial management, 13(11), 1-–29. 10. kung s. y. (1978) a new identification and model reduction algorithm via singular value decomposition, proc. 12th asilomar conf. on circuits, systems and computer, 705-714. 11. liu, heping & shi, jing. (2013). applying arma–garch approaches to forecasting short-term electricity prices. energy economics, 37, 152-–166. https://doi.org/10.1016/ j.eneco.2013.02.006 12. meinhold, r. j., & singpurwalla, n. d. (1983) understanding the kalman filter. the american statistician, 37(2). https://doi.org/10.2307/2685871 13. mergner, s. (2009) applications of state space models in finance. an empirical analysis of the time-varying relationship between macroeconomics, fundamentals and paneuropean industry portfolios. göttingen: universitätsverlag göttingen. 14. shanglei chai, qiang li, mohammad zoynul abedin, brian m. lucey. (2024) forecasting electricity prices from the state-of-the-art modeling technology and the copyright © 2023 assa. adv syst sci appl (2023) doi:10.1017/s0266466605050127 https://doi.org/ 10.1016/j.ijforecast.2022.01.003 https://doi.org/10.2307/1884282 https://doi.org/10.1016/j.esr.2016.06.005 https://doi.org/10.1016/j.esr.2016.06.005 https://doi.org/10.1016/0304-3932(82)90021-6 https://doi.org/10.1016/j.eneco.2013.02.006 https://doi.org/10.1016/j.eneco.2013.02.006 https://doi.org/10.2307/2685871 a state space filtering-based approach for price prediction 17 price determinant perspectives. research in international business and finance, 67, part a, 102132. https://doi.org/10.1016/j.ribaf.2023.102132 15. triantafyllopoulos, k. (2021) bayesian inference of state space models. kalman filtering and beyond. cham, switzerland: springer. https://doi.org/10.1007/ 978-3-030-76124-0 16. viberg, m. (1995) subspace-based methods for the identification of linear timeinvariant systems. automatica, 31(12), 1835–1851. http://dx.doi.org/10.1109/acc. 2005.1470314 17. zeng, y., & wu, s. (2013) state-space models. applications in economics and finance. new york: springer science & business media. copyright © 2023 assa. adv syst sci appl (2023) https://doi.org/10.1016/j.ribaf.2023.102132 https://doi.org/10.1007/978-3-030-76124-0 https://doi.org/10.1007/978-3-030-76124-0 http://dx.doi.org/10.1109/acc.2005.1470314 http://dx.doi.org/10.1109/acc.2005.1470314 introduction basics modeling simulation modeling electricity price modeling conclusion adv syst sci appl 2023; 04:1–7 published online at https://ijassa.ipu.ru. an exact solution of the hunter–saxton–calogero equation by contact linearization method svetlana mukhina* v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia abstract: in this paper we consider a class of generalized nonlinear hyperbolic partial differential equations of the hunter–saxton–calogero type, which arise in the theory of control of liquid crystals and in the control of unsteady gas flows. we found such conditions that the original equation can be reduced to linear one by contact transformations. the general exact multivalued solutions of the hunter–saxton–calogero equation are found. the obtained solutions are visualized. keywords: contact transformations, cartan form, nematic crystals, exact solution, nonlinear partial differential equation, differential forms. 1. introduction let us consider the generalized nonlinear second-order hunter–saxton–calogero partial differential equation utx = uuxx +g(ux), (1.1) where u(t, x) is an unknown function, t and x are the time and the spatial coordinates, respectively. such equations withg(ux) = κu2x and k = 1 2 arise in the theory of nematic liquid crystals. if, initially, all molecules of a liquid crystal are aligned, then some of them will shift slightly and disorientation will spread throughout the crystal. in this case, the function u(t, x) describes the propagation of weak linear orientation waves in the nematic liquid crystal [1]. the equation with κ ̸= 1 2 is used in hydrodynamics [2], in the geometry of einstein–weyl spaces [3]. the contact equivalence of equation (1.1) and the euler–poisson equation was established for g(ux) = κu2x in [4]. calogero [5], while studying waves in shallow water, found a complex solution of equation (1.1). in this article we present conditions, under which nonlinear equation (1.1) is equivalent to a linear equation with respect to a pseudo-group of contact transformations. this allows us to construct its exact multivalued solutions. these solutions can be used to control the propagation of orientation waves in a nematic crystal. this paper continues the series of articles [6–9] on the application of geometric theory of nonlinear differential equations to constructing their exact solutions. we use the methods developed in [10–12]. ∗corresponding author: ssmukhina@edu.hse.ru 2 s. mukhina 2. geometry of the generalized hunter–saxton–calogero equation let j1 be the 1-jet space of functions on r2 with two independent variables t, x and let t, x, u, p1, p2 be the canonical coordinates on this space. the cartan form κ = du− p1dt− p2dx defines a contact structure on j1 (the so called cartan distribution) c : j1 ∋ θ 7→ c(θ) = kerκθ ⊂ tθj 1. the cartan distribution c is generated by the vector fields ∂ ∂t + p1 ∂ ∂u , ∂ ∂x + p2 ∂ ∂u , ∂ ∂p1 , ∂ ∂p2 . (2.2) a two-dimensional surface γ1 v = { u = v(t, x), p1 = ∂v ∂t , p2 = ∂v ∂x } ⊂ j1 is called a 1-graph of a function v(t, x). let ω2(r2) be the module of differential 2-forms on r2. for an arbitrary differential 2form ω on j1, we can construct the lychagin differential operator ∆ω, which acts by the following rule (see [13]): ∆ω : c∞(r2) → ω2(r2), ∆ω(v) = ω|γ1 v . here ω|γ1 u is a restriction of ω to γ1 v. the equation ∆ω(v) = 0 (2.3) is a second-order differential equation of the monge–ampere class. the restriction of ω to the surface γ1 v vanishes if and only if the function v is a solution of equation (2.3). a surface l ⊂ j1r2 is called a multivalued solution of equation (2.3) if ω|l = 0 and κ|l = 0. equation (1.1) belongs to the class of monge–ampere equations and, therefore, it can be associated with the differential 2-form ω = −2g(p2)dt ∧ dx+ dt ∧ dp1 − dx ∧ dp2 − 2udt ∧ dp2. (2.4) let us introduce a “non-holonomic symplectic structure” ω ∈ ω2(c): ω = dκ|c since the cartan distribution is not completely integrable, this 2-form is defined on vector fields that belong to c only. differential form (2.4) is effective, i.e., ∂u⌋ω = 0 and ω ∧ ω = 0. moreover, it is hyperbolic: ω ∧ ω + ω ∧ ω = 0. (2.5) define the linear operator aω : d(c) → d(c) as follows: aωx⌋ω = x ⌋ω, copyright © 2023 assa. adv syst sci appl (2023) an exact solution of the hunter–saxton–calogero equation 3 where d(c) is a module of vector fields that belong to the cartan distribution c. the operator aω has the following matrix representation in basis (2.2): aω =  1 0 0 0 −2u −1 0 0 0 −2g(p2) 1 2u 2g(p2) 0 0 −1  . its square is scalar: a2 ω = 1, therefore, its eigenvalues are ±1. the eigenvectors define two 2-dimensional characteristic distributions c+ = { x+ = ∂ ∂t − u ∂ ∂x + (−p2u+ p1) ∂ ∂u +g(p2) ∂ ∂p2 , y+ = g(p2) ∂ ∂p1 } , c− = { x− = −u ∂ ∂x − up2 ∂ ∂u +g(p2) ∂ ∂p2 , y− = ∂ ∂x + p2 ∂ ∂u +g(p2) ∂ ∂p1 } . the vector fields x±, y± form a basis of the module d(c±). the first derivatives of distributions c (1) ± = {x±, y±, [x±, y±]} are 3-dimensional. therefore, in the 5-dimensional space j1, they intersect along a 1dimensional distribution l = c (1) + ∩ c(1) − , which is generated by the vector field z = g (p2) ∂ ∂u +g(p2) (g ′′ (p2)− p2) ∂ ∂p1 . at any point a ∈ j1, the tangent space taj1 can be decomposed into a direct sum taj 1 = c+(a)⊕ l(a)⊕ c−(a). denote the distributions c+, l, and c− as p1, p2, and p3, respectively. let dj be the module of vector fields from the distribution pj and let pj : d(j1) → dj be projectors. define the tensors qsj,k ∈ ω2(j1)⊗d(j1) (see [12]): qsj,k(x, y ) := −ps[pjx,pky ], where j, k, s = 1, 2, 3; s ̸= j, k, and skew contraction of two decomposable tensors α⊗ x, β ⊗ y ∈ ω2(j1)⊗d(j1): ⟨α⊗x, β ⊗ y ⟩ = (y ⌋α) ∧ (x⌋β) . this definition is extended to the remaining tensors by linearity. tensor invariants of equation (1.1) have the form: q12,3 = (p2dt ∧ dx+ dt ∧ du)⊗ ( g(p2) 2 ∂ ∂p1 −g (p2) ∂ ∂x +g (p2) p2 ∂ ∂u ) , q31,2 = (p2dt ∧ dx− dt ∧ du+ p1dt ∧ dp2 + p2dx ∧ dp2 − du ∧ dp2) ⊗ ( − (g′′ (p2)− 2) (g (p2)) 2 ∂ ∂p1 ) , q21,1 = (g (p2) dt ∧ dx+ udt ∧ dp2 + dx ∧ dp2)⊗ ( −g (p2) ∂ ∂u + (g′ (p2)− p2)g(p2) ∂ ∂p1 ) , q23,3 = (( g′ (p2) p2 − p22 −g (p2) ) dt ∧ dx+ (−g′ (p2) + p2) dt ∧ du+ dt ∧ dp1 − udt ∧ dp2 ) ⊗ ( g (p2) ∂ ∂u − (g′ (p2)− p2)g (p2) ∂ ∂p1 ) . copyright © 2023 assa. adv syst sci appl (2023) 4 s. mukhina the invariant laplace forms for equation (1.1) are λ+ = 〈 q21,1, q 1 2,3 〉 = −dt ∧ dp2, λ− = 〈 q23,3, q 3 1,2 〉 = −(g′′(p2)− 2)dt ∧ dp2. equation (1.1) satisfies the conditions of contact linearization λ− = 0, λ+ ∧ λ+ = 0, dλ+ = 0 if and only if the function g(p2) has the form g(p2) = p22 + 2k1p2 + k0, where k0, k1 are arbitrary constants. then equation (1.1) has the form utx − uuxx − 2k1ux − u2x − k0 = 0. (2.6) let us construct a linearizing contact transformation. equation (2.6) corresponds to the differential 2-form ω = −2(u2x + 2k1ux + k0)dt ∧ dx+ dt ∧ dut − 2udt ∧ dux − dx ∧ dux. (2.7) we apply the partial legendre transform to this 2-form: φ: (t, x, u, p1, p2) 7→ (t, −p2,−xp2 + u, p1, x) . applying this transformation to differential form (2.7), we obtain a new form ω1 = φ∗(ω) = (2xp2 − 2u)dt ∧ dx+ dt ∧ dp1 + ( 2x2 + 4k1x+ 2k0 ) dt ∧ dp2 − dx ∧ dp2, which corresponds to the linear equation utx + (x2 + k1x+ k0)uxx + xux − u = 0. (2.8) equation (2.8) can be solved by cascade integration method: u(t, x) = ek1t (∫ t t0 f1(τ)e −k1τ cosh ( (τ − t) √ k21 − k0 − arctanh ( x+ k1√ k21 − k0 )) dτ+ +f2 −t− arctanh ( x+k1√ k21−k0 ) √ k21 − k0   √ 2k1x+ x2 + k0 k0 − k21 , (2.9) where f1, f2 are arbitrary functions. note that the legendre transformation maps the multivalued solutions of equation (2.6) to the solutions of equation (2.8). but the inverse legendre transformation maps classical solutions (2.9) to multivalued ones. apply the inverse transformation φ−1 : (t, x, u, p1, p2) 7→ (t, p2,−xp2 + u, p1,−x) to (2.9). let us choose t and p2 as parameters β, α, respectively. then we get general multivalued solution of equation (2.6): copyright © 2023 assa. adv syst sci appl (2023) an exact solution of the hunter–saxton–calogero equation 5 l :  t =β, x =− 1√ −(α2 + 2αk1 + k0)γ ( ek1β ( (α + k1) (∫ β β0 f1(τ)e −k1τ cosh(ψ)dτ ) +(α + k1)f2(η) + γ (∫ β β0 f1(τ)e −k1τ sinh (ψ) dτ + f ′ 2(η) ))) , u =− 1√ −(α2 + 2αk1 + k0)γ (( (αk1 + k0) (∫ β β0 f1(τ)e −k1τ cosh (ψ) dτ ) +(αk1 + k0)f2(η)− α ( γ (∫ β β0 f1(τ)e −k1τ sinh (ψ) dτ ) + f ′ 2(η) )) ek1β ) , ut =e k1β √ α2 + 2αk1 + k0 −γ ( −γ (∫ β β0 f1(τ)e −k1τ sinh (ψ) dτ ) +k1 (∫ β β0 f1(τ)e −k1τ cosh (ψ) dτ ) − f ′ 2(η) + k1f2 ) + f1(β), ux =α, where f1, f2 are arbitrary functions, α, β are parameters, γ = √ k21 − k0, η = − βγ + artanh ( α + k1 γ ) γ  , ψ = ( −arctanh ( α + k1 γ ) + (τ − β)γ ) . to show that l is indeed a multivalued solution, it is enough to check that the restriction of the 2-form ω to it vanishes. 3. visualization let us consider an example of visualization for constructed solution. let k0 = 2, k1 = 0. choose the functions f1(τ) = −τ, f2(η) = −η. then we have t =β, x = β + (arctan (α)α + 1− βα) √ α2 + 1 + α2β α2 + 1 , u = (β + α− arctan (α)) √ α2 + 1− α2 − 1 α2 + 1 . the graph of this solution is shown in fig. 3.1. solution graphs for other f1 and f2 are presented in fig. 3.2 and fig. 3.3. copyright © 2023 assa. adv syst sci appl (2023) 6 s. mukhina fig. 3.1. solution of equation (2.6) with f1(τ) = −τ, f2(η) = −η. fig. 3.2. solution with f1(τ) = τ2, f2(η) = η2 fig. 3.3. solution with f1(τ) = τ3, f2(η) = η3 copyright © 2023 assa. adv syst sci appl (2023) an exact solution of the hunter–saxton–calogero equation 7 acknowledgements this work is supported by the russian science foundation (grant 23-21-00390). references 1. hunter, j. k. & saxton, r. (1991). dynamics of director fields, siam j. appl. math., 51, 1498–1521. 2. golovin, s. v. (2004). group foliation of euler equations in nonstationary rotationally symmetrical case, proc. inst. math. nas of ukraine, 50, 110–117. 3. tod, k. p. (2000). einstein-weil spaces and third order differential equations, j. math. phys., 41, 5572–5581. 4. morozov, o. i. (2007). linearizuemost i integriruemost oboshennogo uravnenia calodgero–hanter–saxton [linearizability and integrability of the generalized calogero–hunter–saxton equation], nauchnii vestnik mgtu ga, 114, 34–42, [in russian]. 5. calogero, f. (1984). a solvable nonlinear wave equation, stud. appl. math., 70, 189– 199. 6. kushner, a. g. & mukhina, s. s. (2022). integration of the deep bed filtration equations, lobachevskii journal of mathematics, 43(10), 73–80. 7. mukhina, s. s. (2023). contact transformations in theory of frontal oil displacement, lobachevskii journal of mathematics, 44(9), 3976–3980. 8. kushner, a. g. (2023). dynamics of evolutionary differential equations with several spatial variables, mathematics, 11(2), 335–346. 9. kushner, a. & sinian, t. (2023). evolutionary systems and flows on solutions spaces of finite type equations, lobachevskii journal of mathematics, 44(9), 3945–3951. 10. krasilshchik, i. s., lychagin, v. v. & vinogradov, a. m. (1986). geometry of jet spaces and nonlinear partial differential equations. new york, ny: gordon and breach. 11. kushner, a. g., lychagin, v. v. & rubtsov, v. n. (2007). contact geometry and nonlinear differential equations. encyclopedia of mathematics and its applications. cambrigde, uk: cambridge university press. 12. kushner, a. g. (2008). a contact linearization problem for monge–ampere equations and laplace invariants, acta appl. math, 101, 177–189. 13. lychagin, v. v. (1979). contact geometry and non-linear second-order differential equations, russian math. surveys, 34(1), 149–180. copyright © 2023 assa. adv syst sci appl (2023) introduction geometry of the generalized hunter–saxton–calogero equation visualization microsoft word 1523-source texts-7016-1-18-20240409 adv syst sci appl 2024; 02; 116-132 published online at https://ijassa.ipu.ru. adjusting the parameters of a pid controller with an asymmetric fuzzy algorithm for an automotive suspension system tuan anh nguyen thuyloi university, hanoi, vietnam abstract: a suspension system plays a crucial role in ensuring the smoothness of a car when traveling on roads. this paper proposes using the active suspension system equipped with a hydraulic actuator to replace a conventional passive suspension. unlike previous publications, this study uses a hybrid proportional-integral-derivative (pid) algorithm, combining a conventional pid controller and a fuzzy solution with two distinct inputs. the coefficients of the pid controller are flexibly adjusted by the fuzzy algorithm. therefore, it can more respond to complex motion conditions. calculations are performed with the matlab-simulink software for four specific cases. according to the research findings, the maximum and root mean square (rms) values of vehicle body acceleration and displacement significantly decrease when applying the hybrid algorithm to the active suspension system. in the last case, these values are only 16.69% and 36.90% when comparing fuzzy pid and passive situations. in general, the smoothness of the sprung mass can be guaranteed in all survey cases. keywords: active suspension system, hybrid pid algorithm, fuzzy pid controller, dynamics vehicle. 1. introduction automotive vibration is an exciting topic that attracts many researchers. car vibrations have an essential influence on the stability and smoothness of the vehicle when traveling. suspension systems are fitted on cars and other vehicles to control their vibration [1]. a conventional suspension system, also known as a passive suspension (mechanical suspension), has three essential components: metal springs, dampers, and lever arms or multi-link bars. this simple suspension system is commonly used on most vehicles today. its cost is relatively low, but its efficiency is not high. therefore, the components of the passive suspension system should be improved to enhance the quality of the system. in [2], yoon et al. came up with the idea of using electromagnetic dampers instead of conventional passive dampers. for this suspension system, the damper's force could be controlled by an electric current applied to electrodes inside the shock absorber. according to zheo et al. [3], spring stiffness could be controlled by utilizing air balloons instead of metal springs. the suspension system performance could be improved by using the above two solutions. however, they only partially met requirements regarding stability and smoothness. pan and sun proposed using the active suspension system to ensure the criteria of road holding and ride comfort [4]. unlike air spring and electromagnetic damping suspension, the active suspension system was equipped with an additional hydraulic actuator that might generate large impact forces, according to kilicaslan [5]. the performance of active suspension systems is often higher than other systems. many studies related to suspension control have been published in a past decade. in [6], jeong et al. designed the linear quadratic (lq) controller for the suspension model of 2 degrees of freedom (dofs) and 7 dofs. this algorithm aimed to minimize the cost function of the controller. algorithm parameters could be determined using various solutions, such as ant adjusting the parameters of a pid controller… 117 copyright ©2024 assa. adv. in systems science and appl. (2024) colony optimization [7] or in-loop solution [8]. a gaussian filter was used to limit the noise from the outside; therefore, it became the lqg controller [9]. when considering the oscillating system to be nonlinear, wang et al. applied a robust predictive control method for a suspension system with a spatial model [10]. to improve the stability of nonlinear control algorithms, li et al. used h nonfragile optimization method, which considered the delay of the hydraulic actuator [11]. using the sliding mode control (smc) algorithm was a suitable solution for nonlinear systems. in [12], ovalle et al. introduced a continuous sliding mode controller for the quarter model. for uncertain systems [13], retrofitting observers was necessary [14, 15]. the smc algorithm might bring high performance to the system, but chattering phenomena still existed when applying this algorithm to oscillating systems. in [16], nguyen showed the in-loop method to define the parameters for the sliding mode controller. this method was implemented based on minimizing the rms values of acceleration and displacement. a newer improvement was shown in [17], where the optimization considered the oscillations' phase difference. in addition, the combination of the smc algorithm and the fuzzy algorithm could help limit chattering in many situations, according to golouje and abtahi [18] and shaer et al. [19]. recently, fuzzy control algorithms have been widely applied in many studies on active suspension control [20, 21]. in [22], mustafa et al. used a fuzzy algorithm with three membership functions of triangular shape. each function had five steps: negative large (nl), negative small (ns), zero (ze), positive small (ps), and positive large (pl). meanwhile, wang et al. used only a simple membership function related to error 𝑒(𝑡) for the fuzzy controller of the suspension system [23]. the membership functions of a fuzzy system could be the same or different, symmetric or asymmetric [24]. this depended on the designer's point of view. fuzzy controllers could be optimized using particle swam optimization (pso) algorithms, which describe the properties of animals such as cuckoos [25], bees [26], etc. in addition, combining fuzzy algorithms and other algorithms could help improve the controller's efficiency [27, 28]. the use of fuzzy algorithms to adjust the parameters of the pid controller is a fascinating idea that has attracted more interest from many researchers. unlike optimization methods like pso [7, 25, 26] or in-loop [8, 16, 17], fuzzy algorithms could generate dynamic changes in different controls instead of just finding a specific optimal value [29]. in [30], nguyen and nguyen used a simple fuzzy algorithm with one input. the input signal was the system's error after it had been derivative twice, while the output signal was the value of the coefficients kp and ki of the pi controller. in [31], khodadadi and hamid ghadiri used a fuzzy controller with two inputs: velocity and acceleration. the membership functions used in [31] were triangular and trapezoidal. fuzzy rules were often chosen symmetrically and described through fuzzy surfaces [32, 33]. constructing fuzzy rules and membership functions significantly depended on the designer's experience. in some cases, the process of determining these rules could be derived from other algorithms [34, 35]. in general, the active suspension system might respond well to changing external conditions once the pid controller parameters were flexibly adjusted by the fuzzy algorithm [36-38]. this was proven through simulation and experiment [37, 39]. based on the above evaluation and analysis, we propose to use the pid algorithm to control the active suspension system. a new fuzzy algorithm dynamically fine-tunes controller parameters; hence, it is called hybrid pid control. fuzzy algorithms in previous studies often only had symmetric or straightforward forms [30-33]. therefore, it could only provide ride comfort under certain conditions. in this work, an asymmetric fuzzy algorithm is designed as an alternative to conventional fuzzy algorithms. this helps to ensure the car's smoothness when oscillating without affecting other problems. this is a new contribution to the paper, demonstrated through simulation results. this paper content is divided into four parts: introduction, mathematical model, result and discussion, and conclusion. 118 t. a. nguyen copyright ©2024 assa adv. in systems science and appl. (2024) 2. mathematical model the mathematical model of the system and the control method are presented in this section. fig. 1. dynamic models. consider a suspension dynamics model, as shown in figure 1. this is called a quarter model, with two masses, 𝑚 and 𝑚 , corresponding to 𝑧 and 𝑧 displacements. applying d'alembert's principle, we can obtain the following equations: 𝐹 − 𝐹 − 𝐹 − 𝐹 = 0, (1) 𝐹 + 𝐹 + 𝐹 + 𝐹 − 𝐹 = 0, (2) where: 𝐹 – inertia force of spung mass, n: 𝐹 = 𝑚 × �̈� ; (3) 𝐹 – inertia force of unsprung mass, n: 𝐹 = 𝑚 × �̈� ; (4) 𝐹 – spring force, n: 𝐹 = 𝐾 × (𝑧 − 𝑧 ); (5) 𝐹 – damping force, n: 𝐹 = 𝐾 × (�̇� − �̇� ); (6) 𝐹 – tire force, n: 𝐹 = 𝐾 × (𝑧 − 𝑧 ); (7) 𝐹 – actuator force, n; 𝐾 – spring coefficient, n/m; 𝐾 – damping coefficient, ns/m; 𝐾 – tire coefficient, n/m; 𝑚 – sprung mass, kg; 𝑚 – unsprung mass, kg; 𝑧 – sprung mass displacement, m; 𝑧 – unsprung mass, m. substituting equations from (3) to (7) into (1) and (2), we get: 𝑚 × �̈� = 𝐾 × (𝑧 − 𝑧 ) + 𝐾 × (�̇� − �̇� ) + 𝐹 (8) 𝑚 × �̈� = 𝐾 × (𝑧 − 𝑧 ) − 𝐾 × (𝑧 − 𝑧 ) − 𝐾 × (�̇� − �̇� ) − 𝐹 (9) the actuating force of the actuator, 𝐹 , is generated by the pressure difference between the chambers inside the hydraulic cylinder (fig. 1). therefore, it is determined by equation (10). 𝐹 = 𝑆 × 𝛥𝑃 (10) adjusting the parameters of a pid controller… 119 copyright ©2024 assa. adv. in systems science and appl. (2024) the variation of fluid pressure between chambers is a complex function that depends on the fluid flow. according to [16], this value is defined as follows: δ𝑃 = 1 𝐶 × 𝐶 × 𝑤 × 𝑥 𝑃 − 𝛥𝑃 × 𝑠𝑔𝑛(𝑥 ) 𝜌 − 𝑉 × 𝛥�̇� 4 × 𝛽 − 𝑆 × (�̇� − �̇� ) (11) the displacement of the servo valve, 𝑥 , plays an essential role in changing the fluid pressure between chambers. this displacement depends on the control signal supplied by the controller, 𝑖(𝑡). 𝑥 = −𝜏 × �̇� + 𝑘 × 𝑖(𝑡) (12) where: 𝑆 – piston cross-sectional area, m2 𝑃 – pressure difference, n/m2 𝐶 – leakage coefficient 𝐶 – discharge coefficient 𝑤 – valve area gradient coefficient 𝜌 – hydraulic oil density, kg/m3 𝑉 – cylinder effective volume, m3 𝑃 – initial pressure, n/m2  – fluid bulk module, n/m2  – time coefficient, s 𝑘 – servo valve coefficient, m/a combining (10), (11), and (12), we get an approximate linear equation (13) (see [33, 40]). �̇� = 𝛼 × 𝑖(𝑡) − 𝛼 × 𝐹 + 𝛼 × (�̇� − �̇� ) (13) with  are the coefficients of equation the (13). let 𝑒(𝑡) be the error between the output signal 𝑦(𝑡) and the reference signal 𝑦 (𝑡). 𝑒(𝑡) = 𝑦 (𝑡) − 𝑦(𝑡). (14) the purpose of the control is to make the value of error approaches zero, that is: |𝑒(𝑡)| ⎯⎯⎯⎯⎯ 0. (15) as suggested above, this study uses a traditional pid control algorithm to eliminate the error 𝑒(𝑡) when the car oscillates. the pid algorithm has three stages corresponding to three functions: proportional, integral, and derivative. therefore, the final control signal is synthesized from these three components. 𝑖(𝑡) = 𝑘 × 𝑒(𝑡) + 𝑘 × 𝑒(𝑡)𝑑𝑡 + 𝑘 × 𝑑𝑒(𝑡) 𝑑𝑡 . (16) in the equation (16), the coefficients 𝑘 , 𝑘 , and 𝑘 play an important role in ensuring the controller's performance. many methods are used to find the optimal values for these coefficients (described in the previous section). however, optimal values are fixed in all cases. in fact, these values are only suitable for some instances, not all. therefore, it is necessary to flexibly tune the controller coefficients to suit each vehicle's movement conditions. in this paper, a fuzzy algorithm is used to adjust the coefficients of the pid controller. unlike previous studies, this fuzzy algorithm has an asymmetrical form. this aims to ensure that vehicle body displacement and acceleration are as minimal as possible. 120 t. a. nguyen copyright ©2024 assa adv. in systems science and appl. (2024) fig. 2. control system diagram. the control system diagram is illustrated in figure 2. according to this description, the values of the coefficients 𝑘 and 𝑘 are flexibly adjusted by the fuzzy algorithm, while the value of the 𝑘 coefficient is unchanged (this is because the 𝑘 coefficient is extremely sensitive, so it should be chosen within a small range). the fuzzy algorithm used in this study has two inputs: vehicle body displacement (the first input) and vehicle body acceleration (the second input). these two parameters can be obtained with sensors on the vehicle's body. they are multiplied by gain coefficients (𝑘 and 𝑘 ) before becoming the formal input of the fuzzy algorithm. figure 3 depicts the membership function of the coefficient 𝑘 , while figure 4 gives information about the membership function of the coefficient 𝑘 . in figure 3, we use trapezoidal membership functions with three levels: pos (positive), neu (neutral), and neg (negative). however, the membership function of the coefficient 𝑘 , which is depicted in figure 4, has seven steps: vlneg (very large negative), lneg (large negative), neg (negative), neu (neural), pos (positive), lpos (large positive), and vlpos (very large positive). although the input ranges of these two membership functions are similar, the output variation is entirely different. for figure 3, the output of the membership function changes only over a small range, while the output of figure 4 changes continuously over a more extensive range. the membership function of the proportional coefficient is designed so that its output value is always high, corresponding to each level. this helps control signal amplification when needed. unlike 𝑘 , the membership function of coefficient 𝑘 is divided into more levels, which is suitable for different oscillation conditions. according to this idea, the system will not operate under minor oscillation conditions. in fact, the design of the membership function depends mainly on the perspective and experience of the researcher. adjusting the parameters of a pid controller… 121 copyright ©2024 assa. adv. in systems science and appl. (2024) fig. 3. membership function (𝑘 coefficient). 122 t. a. nguyen copyright ©2024 assa adv. in systems science and appl. (2024) fig. 4. membership function (𝑘 coefficient). the defuzzification process is performed based on the weighted sum (wtsum) method, according to the equation (17). 𝑦 ′ = 𝛽 𝑦 + 𝛽 𝑦 + ⋯ + 𝛽 𝑦 = 𝑤 𝑦 = 𝛾 𝛷(𝑥) (17) with: 𝑤 = 𝛽 ∑ 𝛽 , 𝛾 = ⎣ ⎢ ⎢ ⎢ ⎡ 𝑦 𝑦 𝑦 . . . 𝑦 ⎦ ⎥ ⎥ ⎥ ⎤ , 𝛷(𝑥) = ⎣ ⎢ ⎢ ⎢ ⎡ 𝑤 (𝑥) 𝑤 (𝑥) 𝑤 (𝑥) . . . 𝑤 (𝑥)⎦ ⎥ ⎥ ⎥ ⎤ . adjusting the parameters of a pid controller… 123 copyright ©2024 assa. adv. in systems science and appl. (2024) fig. 5. fuzzy surface (𝑘 coefficient). fig. 6. fuzzy surface (𝑘 coefficient). fuzzy rules are described in table 1 (𝑘 coefficient) and table 2 (𝑘 coefficient). these rules are plotted in figures 5 and 6. as shown above, the fuzzy rule established in this study has an asymmetric form. this is a novelty of the paper compared to other studies. table 1. fuzzy rules (𝑘 coefficient). 1st input 2nd input output neg neg neg neg neu neg neg pos neu neu neg neg neu neu neu neu pos pos pos neg neu pos neu pos pos pos pos table 2. fuzzy rules (ki coefficient). 1st input 2nd input output 1st input 2nd input output vlneg vlneg vlneg neu pos pos vlneg lneg lneg neu lpos pos vlneg neg lneg neu vlpos lpos 124 t. a. nguyen copyright ©2024 assa adv. in systems science and appl. (2024) vlneg neu neg pos vlneg neg vlneg pos neg pos lneg neu vlneg lpos neu pos neg neu vlneg vlpos neu pos neu pos lneg vlneg lneg pos pos pos lneg lneg lneg pos lpos lpos lneg neg neg pos vlpos lpos lneg neu neg lpos vlneg neu lneg pos neu lpos lneg neu lneg lpos neu lpos neg pos lneg vlpos pos lpos neu pos neg vlneg lneg lpos pos lpos neg lneg neg lpos lpos lpos neg neg neg lpos vlpos vlpos neg neu neu vlpos vlneg neu neg pos neu vlpos lneg pos neg lpos pos vlpos neg pos neg vlpos pos vlpos neu lpos neu vlneg neg vlpos pos lpos neu lneg neg vlpos lpos vlpos neu neg neu vlpos vlpos vlpos neu neu neu the process of calculation and numerical simulation will be carried out after completing the design of the control algorithm for the active suspension system. 3. result and discussion 3.1. condition car vibrations are described in four cases, corresponding to the four types of excitations from the road surface (figure 7). the first and second cases use time-varying cyclic excitation signals. the pulsed excitation signal is used in the third case, while the last case uses a random bump. the first and third cases have low frequencies, while the others have higher frequencies. in addition, these stimuli have large amplitudes except for the first case. fig. 7. road roughness. numerical simulation is carried out with the matlab software. the parameters that are used for the calculation are referenced in table 3. three simulation situations are considered in each case: hybrid pid, traditional pid, and passive. this aims to compare values obtained between situations, including maximum values and rms values. adjusting the parameters of a pid controller… 125 copyright ©2024 assa. adv. in systems science and appl. (2024) table 3. simulation parameters. symbol description value unit ks spring coefficient 43103 n/m kt tire coefficient 3.8103 n/m kd damping coefficient 176103 ns/m ms sprung mass 470 kg mu unsprung mass 48 kg 1 hydraulic actuator coefficient 539103 n3/2/kg1/2m1/2v 2 hydraulic actuator coefficient 1 1/s 3 hydraulic actuator coefficient 551104 n/m 3.2. simulation results the first case: according to figure 7, the road bump in the first case has a sine waveform with a small amplitude and frequency. during the survey period (10 s), the amplitude of the stimulus signal did not exceed 30 mm. the change in body displacement is illustrated in figure 8 with three situations: hybrid pid, traditional pid, and passive. according to these results, the maximum displacement value when the car uses only mechanical suspension is 31.73 mm. the figures for the remaining situations are 16.65 mm and 7.58 mm, respectively, for traditional pid and hybrid pid. an evaluation of the rms value of the displacement is necessary because this is a continuous oscillation. according to this criterion, the rms values of three situations reach 22.19 mm, 11.54 mm, and 5.34 mm, respectively. when using a typical pid controller, the oscillation phase deviates slightly from the passive situation. this phase difference is improved significantly once the fuzzy-pid hybrid algorithm is applied. fig. 8. vehicle body displacement (the first case). the change in body acceleration, in this case, is quite small (figure 9). their maximum values obtained from the simulation are 0.52 m/s2, 0.32 m/s2, and 0.27 m/s2, respectively. these values are obtained at the first stage of the stimulation process. after that, they gradually decrease and oscillate steadily with a smaller amplitude. the phase difference between the signals still exists, but it is not too large. the rms values obtained for all three scenarios are 0.11 m/s2, 0.05 m/s2, and 0.03 m/s2, respectively. 126 t. a. nguyen copyright ©2024 assa adv. in systems science and appl. (2024) fig. 9. vehicle body acceleration (the first case). the second case: in the second case, the road excitation used remains a cyclic sine. however, the amplitude and frequency of the excitation signal are significantly enhanced. according to figure 7, the peak amplitude of the excitation is up to 100 mm, while the frequency is 0.637 hz. therefore, the displacement of the vehicle body changes much more strongly than in the first case (figure 10). according to the results obtained from the calculation process, the peaked displacement of the vehicle body can be up to 133.10 mm, corresponding to the rms value of 76.48 mm. when using the active suspension system controlled by traditional pid algorithm, these values drop sharply to 53.05 mm and 32.26 mm, respectively. in particular, if the hybrid pid algorithm is used, these values are guaranteed to be stable (28.55 mm and 14.94 mm). in terms of applying traditional pid algorithm, the phase difference occurs more strongly, while this is negligible when using the fuzzy algorithm to adjust the controller's parameters. fig. 10. vehicle body displacement (the second case). similar to the first case, the vehicle acceleration in this case also peaks at the first phase of the oscillation. this value may be up to 3.46 m/s2 if the car only uses the passive suspension system. once a modern active suspension system is fitted, the maximum value of acceleration drops sharply to only 2.14 m/s2 and 1.09 m/s2, respectively, for traditional pid and hybrid pid. in subsequent phases, the vehicle body acceleration changes periodically with time, with adjusting the parameters of a pid controller… 127 copyright ©2024 assa. adv. in systems science and appl. (2024) a smaller amplitude. there is a significant phase difference between two situations: passive and traditional pid. this is a significant limitation when using this algorithm to control the suspension system. looking at figure 11 more closely, we can see that the vehicle body's acceleration changes asymmetrically. this is because the fuzzy rule is chosen asymmetrically. this is a novelty in the study because it provides different results than previously published papers. this helps to ensure that the value of acceleration is small and does not cause a phase difference. fig. 11. vehicle body acceleration (the second case). the third case: the third case uses a pulsed excitation signal with a height of 100 mm (figure 7). the wheel is subject to sudden excitation. therefore, the body is raised by a significant amount: 149.95 mm (passive), 57.05 mm (traditional pid), and 21.42 mm (hybrid pid) (figure 12). because these are discontinuous excitation signals, we do not consider their rms values. the pulsed excitation signal causes a considerable acceleration of the vehicle body. this value increases suddenly to 23.76 m/s2 when cars only use passive suspension. this value slightly decreases to 16.88 m/s2 when applying traditional pid algorithm to control the suspension system. in the case of using hybrid pid algorithm, the peak value of acceleration declines sharply, reaching only 7.80 m/s2 (figure 13). these values decrease to zero after the stimulus ends. in two situations where the controller is used, these values may fluctuate briefly before returning to a stable position. 128 t. a. nguyen copyright ©2024 assa adv. in systems science and appl. (2024) fig. 12. vehicle body displacement (the third case). fig. 13. vehicle body acceleration (the third case). the fourth case: when the vehicle is traveling on the road, the wheels often encounter stimuli that take on a random form (figure 7). these stimuli have significant frequencies and amplitudes, which vary continuously over time. according to figure 14, the displacement of the vehicle body changes continuously and follows the stimulus signal of the road surface. the rms values of three situations: passive, traditional pid, and hybrid pid, reach 37.09 mm, 12.15 mm, and 6.19 mm, respectively, while their peak values are 109.37 mm, 38.19 mm, and 19.07 mm. fig. 14. vehicle body displacement (the fourth case). according to the results illustrated in figure 15, the change in body acceleration over time is significant. the maximum acceleration of the body can be up to 13.68 m/s2 if the automobile does not have the active suspension system. in contrast, this value is only 19.07 m/s2 once the car is equipped with the active suspension system controlled by hybrid pid algorithm. in general, the active suspension helps to ensure more ride comfort than conventional mechanical suspension. the vehicle's smoothness can be improved when applying traditional pid control algorithm. however, a significant phase difference occurs. if hybrid pid algorithm is used, the values related to vehicle body displacement and acceleration can adjusting the parameters of a pid controller… 129 copyright ©2024 assa. adv. in systems science and appl. (2024) decrease more sharply. besides, the phase difference is also solved thoroughly. the hybrid algorithm, shown in this study, provides more performance than conventional algorithms. fig. 15. vehicle body acceleration (the fourth case). table 4 summarizes the results obtained from the simulation. table 4. simulation results. hybrid pid traditional pid passive the first case max rms max rms max rms displacement (mm) 7.68 5.34 16.65 11.54 31.73 22.19 acceleration (m/s2) 0.27 0.03 0.32 0.05 0.52 0.11 the second case max rms max rms max rms displacement (mm) 28.55 14.94 53.05 32.26 133.10 76.48 acceleration (m/s2) 1.09 0.56 2.14 1.03 3.46 1.91 the third case max rms max rms max rms displacement (mm) 21.42 57.05 149.96 acceleration (m/s2) 7.80 16.88 23.76 the fourth case max rms max rms max rms displacement (mm) 19.07 6.19 38.19 12.15 109.37 37.09 acceleration (m/s2) 6.30 1.55 10.27 2.76 13.68 4.20 4. conclusion the roughness of the road is the primary source of excitation, which causes car vibrations. active suspension can control car vibrations more effectively than passive suspension. when evaluating vehicle smoothness, the maximum and rms values of body displacement and acceleration should be taken into account. in this study, we design a hybrid pid controller to control the modern suspension system. this is a combination of conventional pid algorithms and fuzzy algorithms, allowing flexible controller parameter adjustment. the paper's results show that body displacement and acceleration values significantly decline when using the hybrid pid algorithm in all four survey cases. in addition, the phase difference is negligible. in general, the smoothness of the vehicle body has been more guaranteed when applying this new algorithm. in this work, some problems still exist and cannot be resolved, including: 1) the change of acceleration is not smooth sufficient (the second case); 2) the convergence of displacement is not fast sufficient 130 t. a. nguyen copyright ©2024 assa adv. in systems science and appl. 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(2021). advance the efficiency of an active suspension system by the sliding mode control algorithm with five state variables, ieee access, 9, 164368–164378, https://doi.org/10.1109/access.2021.3134990 adv syst sci appl 2025; 1:12–21 published online at https://ijassa.ipu.ru. on linear differential equations on the torus and non-standard analysis vladimir p. burskii1* 1moscow institute of physics and technology (state university), dolgoprudnyi, russia abstract: in this paper, we consider periodic boundary value problems for differential equations whose coefficients are trigonometric polynomials. we construct the spaces of generalized functions, where such problems have solutions. in particular, the solvability space of a periodic analogue of the mizohata equation is constructed. we build also a periodic analogue and a generalization of the construction of the nonstandard analysis, where infinitely small are not only functions, but also functional spaces. to show that not all constructions on the torus lead to a simplification in compare with the plane, we consider a periodic analogue of the hypoelliptic differential operator and show that its number-theoretic properties are significant. in particular, it turns out that if a polynomial with integer coefficients is irreducible in the rational field, then the corresponding differential operator is hypoelliptic on the torus. keywords: differential operator on the torus, linear differential equation on the torus, mizohata equation, nonstandard analysis, hypoellipticity 1. introduction periodic boundary value problems for differential equations is a famous object both in mathematical education and research (see, for example, [1], [2], [3]). as lax noted in [4], in the periodic theory of differential equations is free of some technical difficulties that arise in non-periodic theory. this makes it possible to create a more beautiful theory. a study of periodic boundary value problems for linear differential equations brings us to the wonderful world of functions on the torus. since the torus as the product of a finite number of circles, dealing with the torus simplifies studying the behavior of functions of several variables by each variable separately. moreover, the basis elements are eigenfunctions of linear differential operators with constant coefficients. in addition, the topology of the torus allows us to forget about the boundary and the behavior at infinity. this allows us to focus our attention on the only infinity that we face up in this way: the infinite dimension of functional spaces. there are many works devoted to periodic boundary value problems for differential equations considered as equations on the torus. see, for example, the books [1], [2], [3] and the references therein. however, the present paper has no intersection with them. a viewpoint to the structure of the set of infinitesimals presented in this paper is essentially different from those in the non-standard analysis (see, for example, [14]). the first mention of the presented results was published in russian in in hard-to-find publications [15],[16] and it not published in other languages. in the future it would be interesting to consider, in particular, the spectral properties of differential operators on a torus in the spirit of elliptic theory. the author expresses his gratitude to remizov a.o. for his assistance. ∗corresponding author: burskii.vp@phystech.edu linear differential equations on the torus... 13 2. spaces of periodic functions 2.1. spaces of periodic functions the number of variables in this problem is not significant, and without loss of generality we shall consider the case of two variables. it is well known that every fourier series (trigonometric series)∑ k∈z⊕z ake ikx, ∑ k |ak|2 < ∞, where ak ∈ c, k = (k1, k2), x = (x1, x2) ∈ r2, and kx = k1x1 + k1x2, defines a periodic square integrable function on x. thus, such series form the space l2(t 2), where t 2 = r2/z2 is the torus of dimension 2. for every m ∈ z, m ≥ 2, define by hm the space of 2π-periodic complex-valued functions in r2 such that ∥u∥2m = ∫ t 2 u(x)(1−∆)mu(x)dx < ∞, where ∆ = ∂2 ∂x2 1 + ∂2 ∂x2 2 is the laplace operator. all hm are hilbert spaces, namely, the famous sobolev spaces. it is known (see, for example, [1]) that functions exp(ikx), where kx = k1x1 + k2x2, form an orthogonal basis in hm and, consequently, every function f ∈ hm is expandable into the fourier series f = ∑ k∈z⊕z fke ikx converging to f in the topology of hm. further we shall consider trigonometric series with arbitrary real coefficients, not necessarily converging (such series are usually called formal). one can consider hm as the vector space consisting of formal fourier series with the finite norm ∥f∥2m = ∑ k (1 + k · k)m · |fk|2. this formula defines a norm in the space hm with any real m. consider the vector space hm with the topology of the space rz, in which hm is continuously embedded. for m < 0, the space hm is conjugate to the space h−m in the topology of the space h0 = l2(t 2). moreover, can prove a more general statement: proposition 2.1: let e be a barreled vector topological space of 2π-periodic functions continuously embedded in h0. let the system {eikx} be a basis in e, that is, for every v ∈ e there exists a unique sequence {vk} ⊂ e such that ∑ k2≤n vke ikx → v as n → ∞. then the dual space e∗ is naturally isomorphic to a subspace of the space f = { u = ∑ k uke ikx : ∑ vke ikx ∈ e, ⟨u, v⟩ = ∑ ukvk < ∞ ∀v ∈ e } . here the paring ⟨·, ·⟩ gives rise to the duality. copyright © 2025 assa. adv syst sci appl (2025) 14 v. p. burskii proof this statement follows from the known fact that the mackey topology in barrel spaces coincides with the original topology, since it is the strongest among all topologies consistent with duality (see [5]). there are several examples, which we shall use below: example 2.1: the space of infinitely differentiable periodic functions h∞ = ∩ m hm, whose total element has the form u = ∑ uke ikx, k2luk k→∞−→ 0 (∀ l). also the conjugate space (h∞)∗ = h−∞ = ∪ m hm, which is the space of periodic distributions whose fourier coefficients tend to infinity no faster than some power of k2l. example 2.2: the space e0 of 2π-periodic functions u = ∑ uke ikx, ∃δ1 > 0, δ2 > 0 : ∑ e|k1|δ1+|k2|δ2|uk| < ∞, included in the space of periodic real analytic functions. another examples is the conjugate space e∗ 0 , which consists of series ∑ uke ikx whose coefficients uk k→∞−→ ∞ slower than any exponential e|k1|δ1+|k2|δ2 . this space contains the space of hyperfunctions ([6]). example 2.3: the space l1(|k1|!) of functions u = ∑ uke ikx, ∑ k |k1|! |uk| < ∞. also the conjugate space l∗1(|k1|!) of series ∑ k vke ikx, vk = o(|k1|!). let p (x1, x2) be a homogeneous polynomial of degree p with constant coefficients. consider the differential operator p̂ : hm → hm−p generated by the polynomial p : p̂ u = p ( −i ∂ ∂x1 ,−i ∂ ∂x2 ) u. the obvious formula p̂ ( ∑ k∈z⊕z fke ikx ) = ∑ k∈z⊕z p (k1, k2)fke ikx allows us to consider the operator p̂ on the space f of formal trigonometric series. 2.2. solvability of the mizohata equation in [7], g. levy gave an example of a linear differential equation of the first order with infinitely differentiable coefficients that has no solutions in the space of distributions in threedimensional space. developing the ideas of g. levy and p. garabedyan [9], v. v. grushin in [8] gave an example of a first order differential equation with infinity differentiable coefficients that has no solutions in the space of distributions on the plane: ∂u ∂x + ix ∂u ∂y = f(x, y). (2.1) copyright © 2025 assa. adv syst sci appl (2025) linear differential equations on the torus... 15 the operator in the left-hand side of equation (2.1) is one of the mizohata operators, considered in [10]. the function f ∈ c∞ 0 (r2) is even by x, it was constructed by grushin in a special way. consider a periodic modification of equation (2.1): ∂u ∂x + i sinx ∂u ∂y = f̃(x, y), (2.2) where f̃ is 2π-periodic continuation of the function f mentioned above. it can be checked that the grushin’s reasonings are also applicable to equation (2.2). we shall prove that equation (2.2) has a solution in a wider space of generalized functions than the space of schwartz distributions. proposition 2.2: for any even right-hand side f̃ ∈ h−∞ equation (2.2) has a unique periodic solution u(x1, x2) odd in the variable x1, which belongs to the space l∗1(|k1|!). proof let us write the equation (2.2) in the form ∂u ∂x1 + eix1 − e−ix1 2 ∂u ∂x2 = f̃ , which yields k1uk1,k2 + k2 2 (uk1−1,k2 − uk1+1,k2) = fk. (2.3) for a fixed k2 ̸= 0 we obtain the recurrent formula with respect to k1 uk1+1,k2 = 2 k2 (k1uk1,k2 − fk)uk2−1,k2 . (2.4) since the function u(x1, x2) is odd in the variable x1, we have u0,k2 = 0, u−k1,k2 = −uk1,k2 . therefore, the coefficients uk are uniquely determined by (2.4). from (2.4) we have the following estimation: |uk1+1,k2 | < ∑ j=0 (j + 1)!fk1−j < (k1 + 1)! ∑ k fk = c(k1 + 1)! thus, the solution u = ∑ uke ikx belongs to the space l∗1(|k1|!). note that the operations of differentiation and multiplication by a trigonometric polynomial defined formally in the space f , coincide with the analogues operations in the banach space l∗1(|k1|!) defined as usual in spaces of generalized functions through pairing. proposition 2.3: every periodic solution u(x1, x2) of homogeneous equation (2.2) is even in x1 and it is uniquely determined by the functions u0(x2) := ∫ 2π 0 u(x1, x2)dx1 := ⟨u, 1⟩x1 , u1(x2) := ∫ 2π 0 u(x1, x2)e −ix1dx1 := ⟨u, eix1⟩x1 . it belongs to the space l∗1(|k1|!) if u0 and u1 have bounded sequences of coefficients. copyright © 2025 assa. adv syst sci appl (2025) 16 v. p. burskii the proof follows from formula (2.3). here we consider the function u as a formal trigonometric series, and pairing along one coordinate is defined in the standard way: ⟨u, v⟩x1 := ∑ n 〈∑ k ukne ikx1 , ∑ m vmne imx1 〉 einx2 = = ∑ n (∑ m umnv−mn ) einx2 . it is clear that pairing on x1 does not always exist, but if t is a trigonometric polynomial, then the function ⟨t, v⟩x1(x2) exists. 2.3. solvability of general equations now let us consider the general operator l : ∑ |α|≤m tα(x)d α, where tα(x) is a trigonometric polynomial of degree (s1α, s 2 α). it can also be written in the form l = s1∑ n1=−s1 s2∑ n2=−s2 einxpn(d), s1 := max α s1α, s2 := max α s2α. let us assume that the operator l satisfies the following condition: assumption 2.1: for every n the equation pn(x) = 0 has no solutions in integers. then the following generalization of proposition 2.3 is true. proposition 2.4: under assumption 2.1, every formal periodic solution u(x1, x2) of equation lu = 0 is uniquely determined by the functions u02(x2) := ⟨u, 1⟩x1 , u12(x2) := ⟨u, eix1⟩x1 , . . . , us12(x2) := ⟨u, eis1x1⟩x1 , u01(x1) := ⟨u, 1⟩x2 , u11(x1) := ⟨u, eix2⟩x2 , . . . , us21(x1) := ⟨u, eis2x2⟩x2 . if they satisfy the following conditions: ⟨uq1, e ipx1⟩x1 = ⟨up2, e iqx2⟩x2 , ∀ p, q = 1, . . . ,min(s1, s2). the proof is by the direct substitution of formal series into the equation. 2.4. linear sections as objects of non-standard analysis consider the vector space of formal trigonometric series f and define a relation of order u ≤ v in the following way. an element u ∈ f is more regular than v ∈ f or, equivalently, v is more singular than u if there exists an element h ∈ f with bounded positive coefficients hk such that u is the convolution of v and h: u = v ∗ h := ∑ k vkhke ikx. this is obviously equivalent to the condition ∃c > 0 : ∣∣∣un vn ∣∣∣ < c ∀n ∈ z. copyright © 2025 assa. adv syst sci appl (2025) linear differential equations on the torus... 17 similar order relations are used in asymptotic expansions [11]. we shall the following definition: a subspace α ⊂ f is called a linear section of f if from v ∈ α it follows that u ∈ α for every u ≤ v. a trivial example: if v ∈ f , then the set of u ∈ f such that u ≤ v is a linear section of f . it is called the principal linear section of f . all the subspaces considered above are also linear sections. note that the pairing considered in section 1 naturally generates a hausdorff topology on every vector subspace of f (see [5]). therefore, every linear section α can be considered as a complete topological vector space, and the space α∗ consisting of g ∈ f such that ⟨f, g⟩ < ∞ for all f ∈ α is the dual space of α. the set m of all linear sections of f is ordered by the inclusion. for every two sections α and β there exist the linear sections sup(α, β) = α + β, inf(α, β) = α ∩ β defined as minimal (by the inclusion) linear sections that contain respectively α ∪ β or α ∩ β. it is easy to see that the distributivity relations are satisfied: α ∩ (β + γ) = α ∩ β + α ∩ γ, α + (β ∩ γ) = (α + β) ∩ (α + γ). thus, in the set m a certain structure of the distributive lattice with additive and multiplicative identity elements is introduced. if a is an arbitrary set and {δa| a ∈ a} is a family of linear sections, then the supremum of sup δa is minimal linear section that contains all δa. by the zorn lemma, the supremum of any family exists. 2.5. linear sections and solvability of general equations proposition 2.5: 1. the operator l = ∑ |α|≤m tα(x)d α sends any linear section to a linear section an, consequently, it induces a mapping (endomorphism) l̃ in the set m , which preserves the lattice structure. 2. under assumption 2.1, the mapping l̃ is an epimorphism of the lattice m and for every linear section g there exists the maximum β among those linear sections α for which l̃α ≤ g, and β ̸= f . proof since operations of differentiation, multiplication by a scalar, addition, and shifts {un} → {un+k} preserve the relation of order ≤, the mapping l sends every linear section to a linear section and the induced mapping preserves the relation of order and the lattice operations. from assumption 2.1, it follows that every equation lu = einx has a solution un among trigonometric polynomials. further, if f is a formal series and the operator l satisfies condition 2.1, then the equation lu = f = ∑ n fne inx is solvable in the class of formal series. let such a solution be the formal series w = ∑ n,k fnunke ikx, where for each k the sum over n is finite, and the coefficients unk are uniquely defined by condition 2.1. given linear section g, for every f ∈ g consider the set wf of solutions to copyright © 2025 assa. adv syst sci appl (2025) 18 v. p. burskii the equation lu = f and the set of the corresponding principal linear sections. by the zorn lemma, there exists the supremum s ∈ w̃f , which is the desired linear section β. obviously, β does not coincide with f , since otherwise g = f . this completes the proof. remark 2.1: assumption 2.1 can be replaced with be replaced with a weaker condition: ∀m ∈ z2 ∃n : pn(m) ̸= 0. definition 2.1: the linear section β constructed in proposition 2.5 is called a solution of the equation lu = g with linear section g in the right-hand side. for example, from what we proved above, it follows that solution of equation (2.2) with the right-hand side g = hm is the section β ⊂ l∗1(|n1|!). remark 2.2: the term “section” was chosen due to the obvious analogy with dedekind sections when constructing the field of real numbers. note also that the presented construction is consonant with some constructions of non-standard analysis related to the extension of the field of reals: the infinitesimal germs of functions from both a field and an ultrafilter. the set m of linear sections is only partially ordered, but it contains all the germs of sequences as principal sections. in the set m , there exists the operation of convolution, which is associative, commutative and distributive with respect to addition and intersection (however, the inverse exists not for all elements). moreover, in m there exists the conjugation and all linear sections are reflexive spaces. in the set m one can also introduce an associative commutative product, which is not always defined, but covering the product of smooth functions and the product of distributions according to mikusinsky–hirata–ogawa, and, in addition, the inverse element not lying in m as, so to speak, a singular section, containing not all more regular sequences, but all more singular, and so on. the most important thing here is that this set contains not only all functions, but also all functional spaces, which, along with each of their elements, also contain increasingly smooth elements. and now we can consider the question of solving the differential equation in a class of function spaces, if the given right-hand side is a space as it was in statement 5. 3. on the hypoellipticity of differential operators on the torus as we noted above, according to lax’s statement ([4]), in the periodic case it is possible to construct a more beautiful theory. however, the periodic case is not always simpler that the non-periodic one. in this section, we characterize homogeneous differential operators with constant coefficients hypoelliptic in the space of periodic functions on the plane. this is one of the cases when the lax’s statement is quite controversial. let hm, m ≥ 0, be the space of complex functions on the plane, 2π-periodic in the both arguments such that ∥u∥2m = ∫ t 2 u(x)(1−∆)mu(x)dx < ∞, ∆ = ∂2 ∂x2 1 + ∂2 ∂x2 2 . the space h−m is defined as the space dual to hm in the h0-topology. in section 1, we noted that the functions exp(inx), nx = n1x1 + n2x2, copyright © 2025 assa. adv syst sci appl (2025) linear differential equations on the torus... 19 are an orthogonal basis in the hilbert space hm and, consequently, every function f ∈ hm is presented as fourier series f = ∑ n∈z⊕z fne inx converging to f in the topology of hm. thus, one can consider hm, m ∈ r, as the space of formal fourier series with the finite norm ∥f∥2m = ∑ n (1 + n · n)m · |fn|2 (see section 1). these are the famous sobolev spaces. let p (x1, x2) be a homogeneous polynomial of the degree p ≥ 2 with constant coefficients. consider the differential operator l : hm → hm−p acting according to the rule lu = p ( − i ∂ ∂x1 ,−i ∂ ∂x2 ) u. according to [1], define the spaces of infinitely differentiable and generalized functions: h∞ = ∩ m hm, h−∞ = ∪ m hm. definition 3.1: we shall call the operator l hypoelliptic, if for any u ∈ h−∞ the inclusion lu ∈ h∞ implies u ∈ h∞. lemma 3.1: the operator l is hypoelliptic if and only if the exist constants c > 0 and k1 such that |p (n)| > c(n2)k1 ∀n. (3.5) proof assume that there exist constants c > 0 and k1 such that (5) holds true and f = ∑ n fne inx ∈ h∞. then fn/p (n) decreases faster than any power of n. if there exists a sequence of pairs nj such that |p (nj)| → 0 for j → ∞ faster than any power of |n|, then, for example, for the functions f = ∑ j p (nj)ein jx ∈ h∞, but solution ∑ j e injx ∈ h−2, and there is no hypoellipticity. recall that in the case of two variables one of necessary and sufficient conditions of hypoellipticity is the following (see, e.g., [12]): there exist constants c and c such that∣∣∣p (α)(ξ) p (ξ) ∣∣∣ ≤ c |ξ|−|α|c for every multi-index α and any ξ ∈ r2 large enough. for α = p, the latter inequality gives the inequality |p (ξ)| ≥ c|ξ|pc, which coincides with inequality (3.5) on the integer lattice. this means that every operator hypoelliptic on the plane is hypoelliptic on the torus, but not vice versa. proposition 3.1: the operator l is hypoelliptic if and only if for every real root α of the polynomial p (x, 1) copyright © 2025 assa. adv syst sci appl (2025) 20 v. p. burskii there exist constants c > 0 and k such that∣∣∣α− p q ∣∣∣ > c qk (3.6) for every rational p/q sufficiently close to α. proof let u ∈ h−∞, then u ∈ hm with some m. therefore, u = ∑ n une inx, lu = ∑ n unp (n)einx. let f = ∑ n fne inx. it is clear that f ∈ h∞ if and only if |fn| → 0 for n2 → ∞ faster than any power of n2. note that if the operator l is hypoelliptic, then the equation p (n) = 0 has a unique integer solution n = 0. indeed, assume that ν = (ν1, ν2) is another solution, then the function +∞∑ k=−∞ eikνx ∈ h−∞ belongs to the kernel of l, which contradicts the hypoellipticity. therefore, the solution of the equation lu = f can be formally written as u = ∑ n̸=0 fn p (n) einx + c0. the condition f0 = 0 is obviously a condition for the solvability of the equation lu = f . let us apply lemma 3.1. let α be a real root of the polynomial p (x, 1) of multiplicity r. the inequality |p (n)| > cn2k1 is equivalent to∣∣p( n1 n2 , 1 )∣∣ > c(n2)(k1− p 2 ), since p ( n1 n2 , 1 ) tends to zero as n1 n2 tends to one of the roots. thus, we get the inequality∣∣p( n1 n2 , 1 )∣∣ = ∣∣n1 n2 α ∣∣r∣∣p (r) x1 ( n1 n2 + τ ( α− n1 n2 )∣∣ > c(n2)(k1− p 2 ). for n1 n2 close enough to α, we have∣∣n1 n2 − α ∣∣ > cn2k1 > cn2k 2 . a direct calculation gives k = 1 r ( k1 − p 2 ) . on the contrary, if ∣∣n1 n2 − α ∣∣ > cn2k 2 , then ∣∣n1 n2 − α ∣∣ > cn2k(k < 0), whence we obtain |p (n)| > cn2k1 . the proof is complete. remark 3.1: inequality (3.6) is not valid for some transcendental numbers α, for example, ∑∞ ν=1 1 10ν ! (see [13]). on the other hand, the liouville theorem states that for every algebraic number α of degree ν inequality (3.6) holds true with k = ν (see [13]). in particular, we obtain the following proposition 3.2: if a polynomial p with integer (or rational) coefficients is irreducible in the field q, then the operator l is hypoelliptic. copyright © 2025 assa. adv syst sci appl (2025) linear differential equations on the torus... 21 it is easy to see that the hypoelliptic operator l : h∞ → h∞ is reversible. here and below, all spaces are assumed to be quotient by the subspace of constants. the inverse operator l−1 acts from hm into hk with some k. according to the thue–siegel–roth theorem [13], for every algebraic number α of degree r ≥ 2 and any ε > 0 there exists c > 0 such that for every rational number p/q the inequality∣∣∣α− p q ∣∣∣ > c q2+ε . holds true. then, using calculation of the exponents from the proof of proposition 3.2, we obtain the following proposition 3.3: let r be the greatest multiplicity of real roots of the irreducible polynomial p . then for every ε > 0 the operator l−1 acts from hm into hp/2+m−r−ε continuously. remark 3.2: for p = 2, by the liouville theorem, one can put ε = 0. references 1. bers, l., john, f., & schechter, m. (1964) partial differential equations, am. math. soc., 196. 2. bourbaki, n. (1987) topological vector spaces, springer-verlag. 3. burskii, v. p. (1980) on the solvability of the garabedian-grushin equation, in: collection of scientific articles boundary value problems for differential equations, kiev: naukova dumka, 35–39 [in russian]. 4. burskii, v. p. (2018) on differential operators and differential equations on torus, j. samara state techn. univ., ser. phys. math. sci., 22:4, 607–619 [in russian]. 5. davis, m. (1977) applied nonstandard analysis, ny: wiley publication. 6. dezin, a. a. (1987) partial differential equations. an introduction to a general theory of linear boundary value problems, springer-verlag. 7. fedoryuk, m. v. (1977) metod perevala, moscow: nauka [in russian]. 8. garabedian, p. r. (1970) an unsolvable equation, proc. am. math. soc., 25, 207–208. 9. gelfond, a. o. (1960) transcendental and algebraic numbers, new york: dover publications, inc. vii. 10. grushin, v. v. (1971) a differential equation without a solution, math. notes, 10:2, 499–501. 11. hörmander, l. (1963) linear partial differential operators, springer-verlag. 12. hörmander, l. (1983) the analysis of linear partial differential operators ii: differential operators with constant coefficients, springer-verlag. 13. lax, p. d. (1955) on cauchy’s problem for hyperbolic equations and the differentiability of elliptic equations, comm. pure appl. math., 6, 43–59. 14. lewy, h. (1957) an example of a smooth linear partial differential equation without solution, ann. of math., 2:66, 155–158. 15. mizohata, s. (1962) solutions nulles et solution non analytiques, j. math. kyoto unlv., 1, 271–302. 16. ptashnik, b. i. (1984) ill-posed boundary value problems for partial differential equations, kiev: naukova dumka [in russian]. copyright © 2025 assa. adv syst sci appl (2025) introduction spaces of periodic functions spaces of periodic functions solvability of the mizohata equation solvability of general equations linear sections as objects of non-standard analysis linear sections and solvability of general equations on the hypoellipticity of differential operators on the torus adv syst sci appl 2023; 01:35–49 published online at https://ijassa.ipu.ru. piecewise constant functions in dynamic optimization problems olga e. orel1*, evgeny n. orel2 1moscow institute of physics and technology (state university), dolgoprudnyi, russia 2financial university under the government of the russian federation, moscow, russia abstract: we consider a direct method of dynamic optimization to approximate the global extremum. this method is based on splitting the state space into classes (cells) and constructing piecewise constant functions on the partition. such an approach leads to a generalization of the euler polygonal method and makes it possible to use shortest path algorithms on graphs. in the suggested algorithm, for each class we construct a path from an initial point to this class. note that the program remembers only the terminal point of the path, functional value along the path and number of the preceding class. if one found another path with the same boundary conditions but less functional value (for the minimization problem), this path would become the current approximation. we prove that if the partition is sufficiently small, then, by using our method, we obtain the optimal polygon. the suggested approach can also be applied to problems of dynamic optimization with incomplete information (differential games, control of systems with unknown dynamics). in addition, some results of numerical solutions of optimal control problems and differential games are given. keywords: global extremum, optimal control, euler polygonal method, differential games, splitting into classes, piecewise constant function 1. introduction in [11], [12], [9] to solve optimal control problems we used an approach based on construction of piecewise constant functions that are approximations of a bellman function b(x) or a dual function a(x) (i.e., a bellman function with time reversal). in the paper, we show that such an approach is universal and can be applied to various problems of dynamic optimization when we face different types of uncertainty. most general consideration of optimal control problems is given in [18]. to solve a particular problem, we must split the state space into a finite number of subsets, which are called classes, or cells. the space of functions that take constant values on classes is finite dimensional. thus the values of the functions could be computed and stored in computer memory. splitting the space into classes does not mean that we impose the restriction to get into points of the predetermined lattice. our approach is a direct method and is not related to necessary and / or sufficient conditions of extremum. sometimes, the process of search and construction of piecewise constant functions resembles imitation modelling when the scenario of future behavior of the system is played repeatedly. during this process, the behavior of the system is being improved step-by-step and eventually we get a trajectory that approximates the global extremum. to show the relationship between the global extremum and its approximation, we consider some well-known problems, which were investigated earlier (see, for example, [3]). for these ∗corresponding author: orel.oe@phystech.edu 36 o. e. orel, e. n. orel problems, the detailed study of various characteristic of switching surfaces, universal and equivocal surfaces, barriers, etc. was carried out. however, sometimes the solution was not complete. in what follows, we will compare the results achieved by analytic and numerical methods and establish the approximate boundaries of their applications. the suggested approach enables to avoid use of moiseev “elementary operation” due to moving from cell to cell rather than from point to point (for example, vertices of a lattice). the method of piecewise constant functions originates from the euler polygonal method, with which we start. the euler polygonal method plays an important role not only in the applied but also in the pure mathematics [2]: it is used to solve differential equations as well as to derive euler–lagrange equation and to prove the existence theorem in the calculus of variations. in the paper, we consider the application of the euler polygonal method to numerical solution of a variational problem. as a result we can see that the algorithm constructs not only a unique polygonal line, but also a family of such lines that form a discrete central field of curves. each curve consists of segments with constant slopes. in the optimal control problems, by a polygonal line we mean a trajectory with a piecewise constant control. we also consider differential games and survival problems with unknown dynamics. to construct piecewise defined functions, we can choose various approaches, which depend on a particular problem. the initial points at the self-learning stage could be chosen systematically or at random, the process could start with a minorant [14] or with a majorant of the bellman function. one can proceed with the breadth-first or depth-first backtracking search [10], [15]. finally, in the problems with incomplete information on phase limitations, one can use a heuristic function to speed up the process. the algorithms and programs are demonstrated on the base of a pseudocode that is not completely formalized pdl (program design language), where we use elements of pascal. 2. euler field of optimal polygonal lines consider the variational problem j(x(.)) = ∫ t1 t0 l(t, x(t), ẋ(t)dt → min, t0 < t1, x(t0) = x0, x(t1) = x1 (for simplicity, we consider one-dimensional case, however all the results can easily be generalized to the case x ∈ rn). to construct polygonal lines, we divide the time the segment [t0, t1] into m equal parts and pick n points with a fixed step on the ox-axis. thus a regular lattice (τi, xj) appears in the plane. after that we join the nodes of the lattice located at the neighboring verticals by segments and look for the least-cost path on this graph, which has about mn nodes. computation of labels of nodes goes from left to right, so it takes mn2 macro operations to solve the problem. at the same time we construct the function a (τi, xj) that is defined on the nodes of the lattice and is equal to the minimum cost of a polygonal line passing from the initial point to the current node. actually, the algorithm provides not only one trajectory, but rather a discrete central field of euler optimal polygonal lines, starting at (t0, x0) and terminating at all other nodes of the lattice, i.e., the optimal motion is imitated repeatedly. the last circumstance is very important because before we find one globally optimal trajectory, we must pass through all nodes to make sure that there are no regions, at which the cost of steps is too low or even negative. by using euler polygonal lines we significantly simplify the problem and restrict the domain of search. however, as we know, any sufficiently smooth curve can be approximated by such polygonal lines. therefore, if we want to obtain more accurate approximation, we must consider a more fine lattice. copyright © 2023 assa. adv syst sci appl (2023) piecewise constant functions in dynamic optimization problems 37 fig. 2.1. optimal polygonal lines on the standard and distorted euclidean planes the euler polygonal method was verified by the authors on the familiar variational problems, such as the shortest curves on the euclidean and lobachevsky planes, the brachistochrone curve, and the minimal surfaces of revolution [16], [17]. in all cases, visually it would be difficult to distinguish between the fields of curves constructed analytically and numerically. figure 2.1 represents the results of the algorithm for two related problems of searching the shortest paths on the standard and “distorted” euclidean planes. the program has constructed optimal polygonal lines connecting the origin o to the points of the line ac. on the “distorted” plane, the lagrangian function is given by l = { √ 1 + ẋ2, x ≥ 0,√ 1 + ẋ2 ( 1− ( x a )2) , −a ≤ x < 0, where a > 0 is some constant. notice that on the half-plane x ≥ 0 the lagrangian is the same for both problems. thus the segments om to the points m lying above the ot-axis are extremals for the “distorted” plane as well. however, if m is located below the point b, the optimal extremal will not be the standard extremal represented by the segment om , but rather a polygonal line that moves down to a point k, next goes in a horizontal direction, and finally goes up rather than. turning back to the general case, note that picking points xj on the ox-axis, we divide the axis into n+ 1 sets: single-element sets {xj} and all other points of the axis. with this point of view, a (τi, xj) can be considered as a trivial piecewise constant function if we let it be equal to +∞ at the points that are not the nodes of the lattice. nontrivial functions a (t, x) will be considered later. 3. fixed time optimal control problems 3.1. generalized polygonal lines now consider an optimal control problem j ([t0, t1], x(.), u(.)) = ∫ t1 t0 l(t, x(t), u(t))dt → min, dx(t) dt = f(t, x(t), u(t)), (3.1) copyright © 2023 assa. adv syst sci appl (2023) 38 o. e. orel, e. n. orel x(t0) = x0, x(t1) = x1, t0 < t1, x(t) ∈ x, u(t) ∈ u. to solve it numerically, as in calculus of variations, we divide the time segment [t0, t1] into m equal parts by the points t0 = τ0, τ1, τ2, . . . , τi, . . . , τm = t1, τi − τi−1 = ∆τ = t1 − t0 m . when solving variational problems, we have considered piecewise constant velocities. now we will consider controls that are constant on the intervals [τi−1, τi). for convenience, we restrict ourselves to the simple case when the control actions belong to the given finite set v ⊂ u, v = {vj}, j = 1, 2, . . . l. in general, the number l and the set of controls v might vary from point to point. to specify the trajectory γ, emanating from (t0, x0), it suffices to choose a finite sequence of controls ui ∈ v on the intervals [τi−1, τi). we write γ = (ξ0;u1, ξ1; . . . ;uµ−1, ξµ−1;uµ, ξµ) , ui ∈ v, ξi ∈ x, 1 ≤ µ ≤ m, (3.2) where ξ0 = x0 and ξi is a state, at which the system appears at time τi. we will also refer to such trajectories as polygonal lines and to the regions with constant controls as segments. let γ be the set of polygonal lines. the value of the functional j assigned to the polygonal line γ will be called cost and denoted by c(γ). all the polygonal lines, which are constructed by the algorithm, are emanated from a common point (t0, x0). here, just as in the calculus of variations, by considering polygonal lines, we give up the idea to obtain the exact solution, but the finer the step ∆τ and the wider the set v , the closer we get (by functional) to the global extremum. a particular segment is a function x(t) defined over the time interval [τi−1, τi] , 1 ≤ i ≤ m and satisfying the differential equation dx(t) dt = f(t, x(t), vk), vk ∈ v, 1 ≤ k ≤ l (3.3) with constant control vk. the interval has an initial point (τi−1, x (τi−1)), a terminal point (τi, x (τi)), and a cost j ([τi−1, τi], x(.), vj) = ∫ τi τi−1 l(t, x(t), vj)dt. according to the cauchy uniqueness theorem, we can conclude that all these objects are uniquely defined by the number i of the time segment, the number k of the control, and the initial point a = x (τi−1). so for the segment (3.3) we set b (i, k, a) = x (τi) , c (i, k, a) = ∫ τi τi−1 l(t, x(t), vk)dt. (3.4) the complexity of the problem is the following. for µ = m there are lm sequences of the form (3.2). it is important that each polygonal line passes through its own nodes, that is, generally speaking, the polygonal lines have no break points in common, except for the initial one. note that the number of intermediate nodes grows exponentially depending on m in contrast to the linear dependence in variational calculus, where the number of nodes is mn. on the graph with, for example, 3100 ≈ 5 · 1047 vertices, it is almost impossible to find the optimal path. if we try, just as in calculus of variations, to move through the nodes (τi, xj) of the given regular lattice, we will face some difficulties in choosing controls u ∈ v , passing from points (τi−1, xj) to neighboring points (τi, xk), and the moiseev “elementary operation” [7] will sometimes not succeed. copyright © 2023 assa. adv syst sci appl (2023) piecewise constant functions in dynamic optimization problems 39 3.2. splitting into cells we suggest the following algorithm to solve the problem. we should split the set x into a finite number n of subsets x = n⋃ j=1 xj, xj ⋂ xk = ∅, which we refer to as classes, or cells. we recommend that the diameters of the sets xj should be as small as possible. during processing of the algorithm, if there are two points in the same cell at time τi, we must remove less promising one, so that not more than mn nodes are stored in the memory, just as in calculus of variations. if x ∈ xj , we will write x̄ = j. the points x, y ∈ xj of one cell xj will be regarded as equivalent: x ∼ y. as was already mentioned, we can achieve that each cell except for one consists of only one point, and all other points are contained in a common cell. then, just as in variational calculus, there will appear a regular lattice in plane. let (ξ0, ξ1, . . . , ξµ) (3.5) be y-coordinates of break points of a polygonal line (3.2). setting ξ̄i = νi, we get a sequence of integers (ν0, ν1, . . . , νµ) , (3.6) representing the numbers of cells, through which the polygonal line passes. the program is supposed to include procedures to find (exact or approximate) values of the functions b (i, j, a) and c (i, j, a) for arbitrary integers i = 0, 1, . . . ,m, j = 1, . . . , l and a real number a ∈ x (3.4). 3.3. optimization algorithm while processing, the program forms the arrays s[i, j], φ[i, j], w[i, j], a[i, j], i = 1, . . . ,m, j = 1, . . . , n which could be interpreted as follows: σ = s[i, j] is a number of the preceding cell xσ; x = φ[i, j] ∈ xj – is a “best” point of the cell xj , into which the system is got at the moment τi; u = w[i, j] ∈ v is a control bringing the system to the point (τi, φ[i, j]); a[i, j] is a hamilton function of action (the analogue of a bellman function with time reversal). the function a(i, j) represents the estimation of the cost of an optimal polygonal line starting at (t0, x0) and terminating at xj at instant τi. generally, the program consists of four subsequent procedures. the comments are enclosed in braces. program “solution of a fixed time optimal control problem in the class of polygonal lines” {set initial values of action:} a [0, x̄0] := 0; a [i, j] := ∞ for all i, j > 0; fill in the left vertical column t = τ1; fill in the other columns t = τi, i > 1, from left to right; construct the polygonal line by moving backward; end of program now we describe procedures constituting the program. copyright © 2023 assa. adv syst sci appl (2023) 40 o. e. orel, e. n. orel procedure “fill in the left vertical column” for k := 1 to l do begin {sorting all the controls of v } y := b (1, k, x0) ; ν := ȳ; r := c (1, k, x0) ; {next node (τ1, y) and action r = a(1, ȳ)} if r < a [1, ν] then begin {value a [1, ν] is decreased} s [1, ν] := x̄0; {initial class} a [1, ν] := r; {action} φ[1, ν] := y; w[1, ν] := k; {control x̄0 → ν} end; end; end of procedure procedure “fill in the other columns” for i := 2 to m do {processing vertical columns i > 1 from left to right} for j := 1 to n do begin {passing through cells for t = τi−1} x := φ[i− 1, j]; p := a [i− 1, j] ; for k := 1 to l do begin {sorting all the controls of v } y := b (i, k, x) ; ν := ȳ; r := p+ c (i, k, x) ; {next node (τi, y) and action r = a(i, ȳ)} if r < a [i, ν] then begin {value a [i, ν] is decreased} s [i, ν] := j; {preceding class} a [i, ν] := r; {action} φ[i, ν] := y; w[i, ν] := k; {control j → ν} end; end; end; end of procedure procedure “construct γ, starting at (t0, x0) and bringing the system to x1 at t1 = t0 +m ·∆τ” j := x̄1; y := φ[i, j]; if x1 ̸= y then stop; {polygonal line does not exist} u := w[m, j]; j := s [y, u] ; γ := (y, u); {construction starts from the end; now a polygonal line consists of the pair (y, u)} for i := m− 1 by −1 to 1 do begin {backward motion} y := φ[i, j]; u := w[i, j]; j := s [x, u] ; γ := (y, u) ∨ γ; {concatenation: the pair (y, u) is attached to the obtained trajectory γ on the left} end; γ := x0 ∨ γ; end of procedure as a result, we get the sequence of coordinates and controls γ = (ξ0;u1, ξ1; . . . ;um−1, ξm−1;um, ξm) , (3.7) where ξ0 = x0, ξm = x1. 3.4. justification of the algorithm we will show that for a sufficiently fine partition x into cells, the algorithm constructs optimal polygonal lines. fix an arbitrary natural number µ ≤ m and consider a set of polygonal lines (6), consisting of µ segments. there are finitely many such polygonal lines, so the set yµ = {xµ} of their copyright © 2023 assa. adv syst sci appl (2023) piecewise constant functions in dynamic optimization problems 41 right-most nodes (points) is also finite. let ρµ be the minimum distance between pairs of points of the set yµ, and let r = min [rµ|1 ≤ µ ≤ m]. it is clear that r > 0. theorem 3.1: if the diameters of all the sets xj are less than r, then the program constructs the optimal polygonal line from (t0, x0) to (t1, x1), and the cost of this polygonal line is equal to a [m, j] , m = t1 − t0 ∆τ , j = x̄1. proof consider arbitrary natural numbers µ ≤ m and j ≤ n. since the distances between the points of the set yµ are greater than the diameter of the set xj , then the set yµ ∩xj consists of at most one element. denote this element by zµj if it exists. therefore, the polygonal line γ, terminating at or passing through the cell xj at instant τµ, has a node zµj at this set. if yµ ∩xj = ∅, then there are no polygonal line passing through the cell xj at instant τµ. then for this cell we will have a [µ, j] = ∞. now we prove the statement by induction on the time step m. for m = 0 it is trivial because the polygonal line consisting on 0 steps degenerates into the point (t0, x0) and, by construction, a [0, x̄0] := 0. assume the statement is true for µ = m− 1 steps and prove it for the next step m. let (3.7) be an optimal polygonal line. then γ1 = (x0;u1, ξ1; . . . ;um−1, ξm−1) is also an optimal polygonal line, going to (τm−1, ξm−1). by inductive hypothesis, this line or the polygonal line of equal cost would have already been constructed by the algorithm if the point (τm−1, ξm−1) were finite. we have c(γ1) = a [m− 1, j], where j = ξ̄m−1. let x = zm−1,j . analyse what happens when we fill in the vertical column m as the program processes segments emanating from (τm−1, ξm−1). under control um = vk ∈ v the system gets to (τm, ξm). the cost of this step is equal to c (m, k, x). therefore, by inductive hypothesis, we have r = c(γ1) + c (i, k, x) = c(γ). the last equality is valid by virtue of the additivity of a cost functional. as a result, we will definitely get a [ m, ξ̄m ] ≤ c(γ). from the other hand, the strict inequality a [ m, ξ̄m ] < c(γ) is impossible since otherwise the trajectory γ would not be optimal. 3.5. economic example consider the following dynamic optimization problem arisen in economics [19]. the conceptual meaning of the problem is as follows. a firm has received an order for h units of product to be delivered by time t . it seeks a production schedule for filling this order at the specified delivery date at minimum cost, bearing in mind that unit production cost rises linearly with the production rate and that the unit cost of holding inventory per unit time is constant. let x(t) denote the inventory accumulated by time t. then we have x(0) = 0 and must achieve x(t ) = h . the inventory level, at any moment, is the cumulated past production; the rate of change of inventory is the production rate x′(t). thus the firm’s total cost at any moment t is c1 ( x′(t) )2 + c2x(t), where the first term is the total production cost, the product of the unit cost of production and the level of production; the second term is the total cost of holding inventory; and c1 and c2 are positive constants. for simplicity, in what follows we set c1 = c2 = 1. the firm’s copyright © 2023 assa. adv syst sci appl (2023) 42 o. e. orel, e. n. orel fig. 3.2. optimal field of extremals and quasi-optimal field of polygonal lines in the economic problem objective is to determine a production rate u(t) = x′(t) and inventory accumulation x(t) for 0 < t < t such that j = ∫ t 0 [ u2(t) + x(t) ] dt → min, dx(t) dt = u(t), x(0) = 0, x(t ) = h ≥ 0, 0 ≤ u ≤ v, x(t) ≥ 0. to solve this problem, in [19] we have constructed an optimal central field of pontryagin extremals. moreover, the problem was solved numerically with the help of the partition into cells considered above. the results of both investigations is represented in figure 3.2. the demarcation line to the maximum production rate v is shown on the left part of the figure by the dashed line. 4. autonomous problems of optimal control the method of partition into classes for autonomous optimal control problems∫ t 0 l(t, x(t), u(t))dt → min, dx(t) dt = f(x(t), u(t)), x(0) = x0, x(t ) = x1, x(t) ∈ x, u(t) ∈ u, (termination time t > 0 of the process is not fixed) has been considered in [9][11] in details. generally speaking, the state x(t) and the control u(t) are vectors. in frames of a simple version, to solve the problem numerically, we should take a finite set v ⊂ u and a time step ∆τ . thus, the polygonal lines are defined. they consist of segments, i.e., solutions of the differential equation ẋ = f(x, u) on the interval [0,∆τ ] with constant control u ∈ v . now we need to divide x into finite number of classes xi and run the program representing a generalization of any algorithm of finding the shortest path on a graph [9], [11]. copyright © 2023 assa. adv syst sci appl (2023) piecewise constant functions in dynamic optimization problems 43 fig. 4.3. central field of extremals and polygonal lines in the bushaw problem this can be a breadth-first search, such as the dijkstra algorithm (equal price algorithm) or the nillson algorithm (a∗ search algorithm). however, it can be a backtracking depthfirst search [10], [15] or a combination of these algorithms. that machinery could be very important to solve more complicated optimization problems with any lack of information. consider few examples on the subject. 4.1. fel’dbaum-bushaw problem this time-optimal control problem (it is also called an optimal stopping problem) could be stated as follows [20]– [21]: dx dt = ẋ, dẋ dt = u, |u| ≤ 1, x(0) = x0, x(t ) = x1, t → min . figure 4.3 shows the central extremal fields and polygonal lines for the fixed initial point [17]. 4.2. car motion to the parking place the detailed consideration of this problem can be found in [3]. at first glance, it seems that at each particular case the problem could be solved by ruler-and-compass constructions. nevertheless, the authors are not aware of the exact analytic solution of this problem. assuming the speed of a material point (x, y) is constant and the curvature is limited, we can describe the problem as follows: dx dt = cosφ, dy dt = sinφ, dφ dt = u, |u| ≤ 1, x(0) = x0, x(t ) ∈ g, t → min . here by a parking place we mean a point or its neighborhood. note that the state space is three-dimensional, so a point (x, y) in plane corresponds to an entire state set (x, y, φ), where φ is an arbitrary angle. figure 4.4 illustrates the results of numerical solution of the problem for various initial and terminal conditions. in all the cases, we must eventually reach a small square. on the left-hand part of the figure, we are allowed to get to the square at any slope. on the right, we must get to the square in the direction specified by an arrow, precisely, at an angle that is very close to the arrow. thus, in the first case the target set is a two-dimensional neighborhood of a point (x, y), in the second case the target set is a three-dimensional neighborhood of a point copyright © 2023 assa. adv syst sci appl (2023) 44 o. e. orel, e. n. orel fig. 4.4. car motion to the parking place fig. 4.5. finding a way out of the labyrinth (x, y, φ). obviously, on the left-hand part of the figure the task is a little simpler than on the right-hand side. this can be seen also from the form of the “polygonal lines”. figure 4.5 shows the solution of the problem on finding a way out of the labyrinth. the car must get out bypassing all the obstacles. here the region of motion, which is a rectangle of size 8× 16, is partitioned into squares with side 8 10 , and the set of directions — into intervals of length 2π 15 . so we get 10× 20× 30 = 3000 cells. this problem and its solution can be used in robotics and artificial intelligence. therefore, in [9], [11] there was suggested to use the model for testing methods of solving dynamic optimization problems. however, since then no method to solve the problem on finding a way out of the labyrinth has appeared. copyright © 2023 assa. adv syst sci appl (2023) piecewise constant functions in dynamic optimization problems 45 5. differential games now we consider three differential games with two players (see [3]). in all three games, speeds of player p (pursuer) and of player e (evader) are constant. in addition, pursuer p is to be faster than evader e. 5.1. simplest pursuit game in frames of this game, the players make a simple motion. each of them moves with its own constant speed, changing the direction at his own discretion. the game terminates when p captures e (i.e., the distance pe becomes less than a certain prescribed positive quantity). write kinematic equations: ẋ1 = v1 cosφ1, ẏ1 = v1 sinφ1, ẋ2 = v2 cosφ2, ẏ2 = v2 sinφ2, v1 > v2 > 0, where φ1(t) and φ2(t) are controls of the players. for each player, motion along the straight line pe would be optimal, where p moves toward e and e moves away from p . the authors wrote a program, but the game figure is not very informative: one would see only the segments of the straight line pe. the state space is one-dimensional, because the state is completely defined by the distance between the players. in the program, we used the largest distance r (i.e., the distance at which the game terminates with the win for player p ), and the segment [0, r] was, as usual, divided into finite number of cells. 5.2. homicidal chauffeur game a driver attempts to run down a pedestrian at minimum time; the motion is described by the following kinematic equations: ẋ1 = v1 cosφ1, ẏ1 = v1 sinφ1, φ̇1 = v1 r u, ẋ2 = v2 cosφ2, ẏ2 = v2 sinφ2, v1 > v2 > 0. this game was studied by several authors [1], [3], however an exact analytic solution has not been found as yet. figure 5.6 represents some parties constructed by computer. after self-learning, each player acts optimally himself. in this game, the state space is two-dimensional, since it is completely determined by the distance between the players and the angle between the sight line pe and the velocity of p . figure 5.6 shows 8 parties, each terminates with the victory of pursuer p . the trajectory of p is depicted by an ordinary line, whereas the trajectory of e — by a bold line. in addition, along the path of each player, we put markers at equal intervals of time. the markers of pursuer p are depicted by deleted circles, whereas the markers of evader e are represented by solid squares. therefore, at each party, the numbers of circles and squares are the same (sometimes, the last and the first markers are not printed well). the initial location of p is always the origin, but the initial direction of p and the initial coordinates of e are specified at random. in parties 2, 3, 5, 6, and 7 player p captured the opponent easily, since he didn’t need to perform a sharp turn as the initial velocity of p was directed approximately toward e. notice that in parties 1 and 4, player p made a turn maneuver immediately. in the beginning of party 8, evader e caught the tail of p to prohibit him from turning. however p , who has a larger speed, broke away from the pursuit and, having made a loop, aimed for e. to extend the time of the game, evader e has changed the direction, but this has only delayed his defeat. the situation just described is represented in 5.7. 5.3. game of two cars the kinematic equations of players are similar: if speeds satisfy the inequality v1 > v2 > 0, then the following equations are valid: ẋ1 = v1 cosφ1, ẏ1 = v1 sinφ1, φ̇1 = v1 r1 u1, ẋ2 = v2 cosφ2, ẏ2 = v2 sinφ2, φ̇2 = v2 r2 u2. copyright © 2023 assa. adv syst sci appl (2023) 46 o. e. orel, e. n. orel fig. 5.6. several parties of the homicidal chauffeur game fig. 5.7. homicidal chauffeur game. evader e is catching the tail of p fig. 5.8. two parties of the game of two cars the state space is three-dimensional. the coordinates are completely determined by distances between players and angles between the sight line pe and the velocities of players. optimal behaviours of players found by the program in two parties are shown in figure 5.8. limitations of turning radii are the same for both players. in the first party, the motion is to copyright © 2023 assa. adv syst sci appl (2023) piecewise constant functions in dynamic optimization problems 47 fig. 6.9. inverted pendulum on the cart some extent toward each other. in the second game, player e appeared in the back of p and caught his tail, but then he had to change direction. in all three parties, the program involves two steps. at the first step (imitation modelling), for one-step scenarios we used monte–carlo method, whereas at the second step testing was conducted. each player didn’t know actual strategy of the other and produced its own payoff. it was a payoff when both players act optimally. at the initial instant, player p took the payoff to be identical zero, whereas for player e the payoff was infinite. in what follows, the payoff for p was increasing, and for e it was decreasing. as the self-learning progressed, pursuer p was working out his own strategy by minimax principle, whereas evader e — by maximin one. 6. optimization in lack of information dynamic optimization problems are not limited to optimal control models when decisions are made in conditions of complete information, or dynamic games when we have no information about the behavior of the opponent. some other kinds of lack of information are possible. for example, the behavior of the environment could be described by random laws. we will focus on a model, in which the control block has no information about kinematic equations of an object. this is the case for the biological systems. consider a well-known example of holding a pole in an unstable vertical equilibrium [6], [8]. 6.1. problem on plane inverted pendulum the system in this example consists of an inverted pendulum mounted to a motorized cart. the pendulum will simply fall over if the cart isn’t moved to balance it. the objective of the control system is to balance the inverted pendulum by applying a force to the cart that the pendulum is attached to. one wants to maximize the time during which the pendulum is balanced. regarding the cart as a massive body, we get the system of lagrange’s equations of second kind: ẍ = u, l0θ̈ + u cos θ − g sin θ = 0, (6.8) where x is a cart position coordinate; u is an acceleration of the cart, which is a control; θ is a pendulum angle from vertical oy; l0 is a reduced length to pendulum center of mass; g is the acceleration of gravity. in frames of numerical experiments, we choose the following values of parameters and ranges of phase coordinates: g = 980 cm/s2, u = ±1000 cm/s2, l0 = 100 cm, |θ| ≤ 0.2 rad., |x| ≤ 240 cm, ∆τ = 0.02 s. the state space is four-dimensional, the state coordinates are position (x) and velocity (ẋ) of the cart, pendulum angle from vertical (θ) and its angular velocity (θ̇). following copyright © 2023 assa. adv syst sci appl (2023) 48 o. e. orel, e. n. orel michie [6], we divide the state space into 162 classes according to threshold values: in angle ±0.1; ±0.017; 0, in angular velocity ±0.87; in position of the cart ±80, in the velocity of the cart ±50. at each instant of processing, the control block knows only the ordinal number of class, in which the object is located. on the basis of only this consistent information, the block must make a decision: u = 1000 or u = −1000. the cost for the lack of knowledge of dynamics is a doubling of the random access memory: action now depends not only on the state classes, but also on the control ±1000. the algorithms of decision making and self-learning could be different. in the michie program, self-learning is based on the empiric approach. after some learning period, the pendulum eventually fell down though it has been balanced for a long time. the authors used another program that is based on the idea of a priori optimism and the method of forward error correction [13]. as a result, at the end of self-learning, the pendulum was balanced about vertical position forever. 7. conclusion direct methods of dynamic optimization considered in the paper are efficient in the problem on global extremum rather than on local extremum. moreover, this method can be used in the case when searching analytic solution involves enormous time costs. partition of the state set into classes must be such that the corresponding data set could be stored in the computer memory. diameters of classes must be as small as possible. creating piecewise constant functions most of all resembles imitation modelling. as is known, piecewise constant functions are used when it is difficult to obtain the solution in the form of mathematical equations. if assumptions of theorem, which was proved in the paper, are satisfied, the algorithm constructs the optimal curve in the class of polygonal lines. if the assumptions are not satisfied, it is still possible to speak about optimization if we recognize it as a process, at which extra resources of time and memory appear. note that in the simpler problem on searching global extremum of functions of several variables, it is enough to equate the gradient to zero and after that to test critical points. at that case, direct methods are not required. it is shown in textbooks on mathematical analysis. the picture changes drastically and direct methods have become very actual, when we pass to problems of dynamic optimization, since otherwise we have to construct and analyse central field of extremals, and common approaches have not been worked out yet. in the paper, we showed not only relevance and efficiency of direct methods in the dynamic optimization problems of various types, but also demonstrated internal and external similarity of analytic and direct methods. this suggests the idea that in particular cases, combination of both methods would be fruitful. the presented algorithms and programs can be applied to various engineering problems connected with optimal control; see, for instance, the recent papers [4, 5]. references 1. breakwell, j. v., speyer, j. l., & bryson, a. e. 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(2004). optimal control theory and calculus of varianions, moscow, russia: editorial urss. copyright © 2023 assa. adv syst sci appl (2023) introduction euler field of optimal polygonal lines fixed time optimal control problems generalized polygonal lines splitting into cells optimization algorithm justification of the algorithm economic example autonomous problems of optimal control fel'dbaum-bushaw problem car motion to the parking place differential games simplest pursuit game homicidal chauffeur game game of two cars optimization in lack of information problem on plane inverted pendulum conclusion microsoft word 1646-article text-7131-1-4-20240614 adv syst sci appl 2024; 02; 32-39 published online at https://ijassa.ipu.ru. economic development of countries and health technology assessment (hta) in healthcare decision making dmitry meshkov1*, ludeña moreira genesis marley2, elena makeeva1, alexey lobanov1 1) v.a. trapeznikov institute of control science of the russian academy of sciences, moscow, russia e-mail: dmitrymeshkov@mail.ru 2) peoples’ friendship university of russia named after patrice lumumba, moscow, russia e-mail: ludenagenesis9@gmail.com abstract: the research indicated the relationship between the formal criteria of health technology assessment (hta) availability and the economic development of countries considering hta to be an indicator for the effectiveness of the healthcare management process. hta is at the very end of the evidence data generation. it makes possible forecasting medical, economic and social outcomes of healthcare management decisions and is the base for rational use of healthcare resources. effective allocation of healthcare funds leads to an increase in human capital and economic development opportunities followed by the overall healthcare expenditures growth. the number of countries using hta increases with the growth of per capita gdp, reaching a maximum in the countries with 40-50 thousand usd gdp per capita value range and decreases with further growth of this indicator. the low level of economic development, characterized by a low per capita gdp, makes it difficult to implement hta due to strict regulations aimed at expenditures decrease and preventing using effectiveness criteria for health management assessment. nevertheless, a significant number of low-income countries (below 10 thousand usd) per capita gdp are striving to improve the efficiency of health management and are at different stages of the hta creation and implementation into healthcare systems. the opposite countries with high (above 50 thousand usd) per capita gdp are mostly tax haven countries (offshore zones) and as a rule these indicators are not linked with the real economic and industrial development. these countries do not use hta for expert support of healthcare management decisions. regional international cooperation increases the possibility of creating and using hta in both low and high per capita gdp countries. keywords: health technology assessment (hta), healthcare management, healthcare decision making, expert support for decision making, economic development, per capita gross domestic product (gdp), low-middle income countries, high income countries. 1. introduction modern economic models consider healthcare as a tool providing states with labor resources for their economic development. a strong economy, in its turn allows us increasing healthcare expenditures, contributing to further economic growth [1-3]. an important place in this cycle is occupied by the effectiveness of healthcare decision making and control providing different results at the same cost. meanwhile the assessing of the overall effectiveness of healthcare systems in terms of cost-benefit ratio is difficult due to significant number and variety of safety and effectiveness criteria. the reduction to an integral indicator or an artificial restriction cause doubts on the evidence-based nature of the results obtained [4]. however, despite these weaknesses, this approach is currently the only one available for evaluating and comparing the effectiveness of health systems. bloomberg's assessment of the effectiveness of healthcare systems is based on the life expectancy criteria without taking into account its quality, morbidity and prevalence of diseases, without highlighting the * corresponding author: dmitrymeshkov@mail.ru economic development of countries and health technology assessment… 33 copyright ©0000 assa. adv. in systems science and appl. (0000) importance of other factors that shape the health of the population and affect life expectancy, in addition to the health system. expenditures are determined by the share of per capita gross domestic product (gdp) used for health care and medical services [5]. another approach can be based on the healthcare decision making process assessment. the inclusion of comparative effectiveness criteria in the process indicates the focus of healthcare systems to increase efficiency and vice versa, the use of other than effectiveness criteria (like expenditures restrictions) demonstrate the other priorities. in the presented study, effective healthcare management is considered as a process with formal criteria of using modern specific methodology, including situation analysis based on real (evidence-based) data, predictive modeling of possible solutions based on comparative assessment of safety, efficacy, efficiency and effectiveness data [6]. the analysis and the forecast of medical, economic and social results of management decisions are provided by the healthcare technologies assessment (hta) organizations (agencies) in accordance with local legislation and regulatory documents. in majority of cases the decisions are related with budgeting of medical technologies (intervention). the clinical evidence-based data is available at the international databases while economic and social features are countryspecific leading to the necessity of hta agencies establishment at the country level or below (region of the country or hospital) depending on the local situation [7-18]. in the presented study, the existing of specialized organizations (agencies) conducting hta and integrated into the decision-making process in healthcare system is considered as a formal criterion for effective healthcare systems. in accordance with these statements the formal criterion for the availability of hta in the country should be interconnected with the economic development of countries. 2. objectives to assess the relationship between the availability of health technology assessment (hta) organizations (agencies) and their involvement in healthcare decision making process with the economic development of countries. 3. methods the following criteria were applied for assessing of hta at the country level: 1. presence of hta organizations (agencies) in countries; 2. legislative and regulatory documents defining the role of hta agencies and hta procedures in the frames of healthcare systems; 3. any other official documents related to hta (defining the creation of otp organizations at the national level or the use other countries hta data for healthcare decision making, etc.). the who database was used for the primary assessment [19] and then the obtained results were clarified by publications presented in the pubmed and cambridge core databases (243 publications, including reviews) and reviewing of hta agencies websites followed, in some cases, by personal communication. 44 countries met the first 2 criteria: the presence of agencies and a clear definition of their role and interaction with the healthcare decision-making system in accordance with legislation and regulatory procedures. 43 other countries did not use hta for expert support of healthcare decision making but had organizations declaring their participation in hta (without procedures defining their role in healthcare management) or regulatory documents defining the process of hta creation and interaction with the healthcare system in future. the rest 114 of overall 211 countries had neither agencies nor any documents economical level of the countries was carried out by the gross per capita product (gdp per capita) amount according to the world bank [20]. 34 d. meshkov, l. m. g. marley, e. makeeva, a. lobanov copyright ©0000 assa adv. in systems science and appl. (0000) we calculated the number of countries falling within 10 000 usd per capita gdp intervals (fig.1) and next the proportion of countries using hta in their practice (44 countries) or having hta elements (43 countries) without impact on healthcare decision making in these intervals (fig. 2). a correlation analysis (spearman's criterion for heterogeneous statistical data) was performed between the medians of per capita gdp and the share of countries using hta in the same intervals. 4. results the distribution of countries (number of countries in the range) according to gdp per capita is presented on the fig.1. the shares of countries in the same ranges using hta or intending to do it are presented on the fig. 2. the majority of the countries worldwide (124 out of 211) do not use hta in healthcare practice. 44 countries have specialized organizations (hta agencies) and use their expertise for expert support of management decisions in healthcare. 43 countries have separate elements of the hta system or documents regulating the creation of such systems, but do not currently apply them. a significant number of low-income countries do not have hta. just 2 of 125 (bulgaria and serbia) countries in the range of 0 – 9 999 usd per capita gdp use hta in the decision-making system. these countries established hta agencies in accordance with the eu directive but do not have appropriate resources for a full-fledged assessment and thus are forced to use the results obtained by other countries [21-23, 33]. 30 countries in this interval declare their intention to create and use hta in healthcare, but they are at different stages of this process, starting from the early beginning [24] to the creation of full-fledged detailed roadmaps [25, 26]. 32 countries are represented in the range of 10 000-19 999 usd. 8 of these countries use hta and 11 of the just demonstrate their intention to do it. 13 countries are represented in the next interval (20 000 29 000 usd). 6 of them have hta fully implemented into healthcare decision making. all of them are the part of the european region and developing effective healthcare decision making in accordance with the eu directive. 1 country in this range (saudi arabia) is going to establish hta. 6 (european region, south korea and japan) of 12 countries in the 30 000 – 39 999 usd gdp per capita range have already established and used hta. in the next interval (40 000 – 49 999 usd), hta operates in 9 out of 12 countries located in different regions worldwide. 1 country (united arab emirates) is making efforts to establish it. there are just 3 countries in the 50 000 59 999 usd gdp per capita interval. 2 of them use hta and 1 country (iceland) is at the stage of its creation. the next interval (60 000 69 999 usd) is represented by 5 countries, 3 of them have already implemented hta in decision making and 1 is doing it now (singapore), 1 country does not have and is not going to create it (qatar). 7 countries are representing the gdp per capita above 70 000 usd and 5 of them use hta in their practice. for further analyses in accordance with the curve indicating the share of countries with hta the data set was divided into countries in which the use of hta increases with an increase in per capita gdp (left, ascending part of the graph) and countries in which the use of hta decreases with a further increase in income (right part of the graph). the interval from 40 000 to 49 999 usd contained the transition point (the final point for the ascending branch of the chart and the initial one for the descending branch) with maximal number of countries using hta. spearman's rank correlation coefficient for heterogeneous statistical data between the medians of the per capita gdp intervals and the share of countries with hta in these intervals was 0.238 (very weak correlation), but showed a very high (0.9) correlation for the ascending and a very high (0.9) statistically significant inverse economic development of countries and health technology assessment… 35 copyright ©0000 assa. adv. in systems science and appl. (0000) correlation for the descending branches of the graph taken separately. despite the limitations associated with the possibility of using correlation analysis in these conditions, the results obtained suggested the presence of at least two different factors hindering the creation of hta systems in countries with low per capita gdp and in countries with high per capita gdp. an analysis of the publications indicated that low-income countries are characterized by similar features preventing from creation and using of hta: an insufficient number of specialists capable preparing expert opinions on hta and using its results in health practice; weak information infrastructure, making it difficult to obtain national evidence-based data on morbidity and mortality; fragmented health systems; lack of priorities at national levels, making it difficult to form a unified approach to assessment the effectiveness of healthcare and the quality of medical care; the peculiarities of financing and national legislations that excessively strictly regulate the possibility of financing medical technologies according to criteria other than budget savings, as well as the burden on health budgets associated with an increase in morbidity and complicating the possibility of forming flexible and evidencebased decision-making mechanisms [27-30]. accordingly, the increase of economic development reduces the impact of these factors and opens up opportunities for the rational use of health resources based on hta expertise. the share of countries with hta decreases with an increase in per capita gdp in the countries shown on the right side of the graph which indicates a lack of interest or the possibility of establishing a cta in these countries. the assessment of economies of these countries indicated that 10 of 17 countries with per capita gdp above 50 000 usd are the tax haven countries and absolutely all countries with a per capita gdp of more than 80 000 usd (cayman islands, isle of man, ireland, bermuda, switzerland, luxembourg, liechtenstein, monaco) belong to this category or belonged to it until 2023 (cayman islands, bermuda) [31,32]. in this case, financial flows pass through the budgets of countries without real activity, which distorts economic data and leads to artificially inflated gdp per capita figures [33]. in this group of 8 countries with a per capita runway of more than 80 thousand usd, only 4 formally use hta. all these countries (ireland, switzerland, liechtenstein and luxembourg) are a part of the european region and, accordingly, are the subject to the european directive on hta improvement [33]. thus, the use of hta as a criterion for the effectiveness of the healthcare management process has a connection with the real economy. with an underdeveloped economy, countries are usually forced to strictly regulate health care costs, ignoring its effectiveness, reducing the quantity and quality of labor resources and losing the opportunity for economic development. the high per capita gdp in countries primarily over 80 000 usd are not affecting the real economy of these countries due to financial flows passing through offshore zones. despite the difficulties associated with hta establishing and implementation in countries with low and high per capita gdp, the presence of 43 countries with separate elements of hta suggests the creation of full-fledged structures in these countries in the future to ensure an effective process of national health management. international regional cooperation can play a significant role in hta creation using the example of bulgaria, serbia and other countries involved in appropriate regional cooperation. 36 d. meshkov, l. m. g. marley, e. makeeva, a. lobanov copyright ©0000 assa adv. in systems science and appl. (0000) gdp per capita intervals (usd, thousand) fig. 1. the number of countries falling into the per capita gdp intervals of 10 000 usd increments. gdp per capita intervals (usd, thousand) fig. 2. the share of countries using hta in healthcare decision making or having separate hta elements and falling into the per capita gdp intervals of 10 000 usd increments economic development of countries and health technology assessment… 37 copyright ©0000 assa. adv. in systems science and appl. (0000) 5. conclusion 1. the research indicated the relationship between the availability of health technology assessment (hta) organizations (agencies) and their involvement in healthcare decision making process (44 of 211 countries) with the economic development of countries indicated as the amount of per capita gdp. 2. the low level of economic development, characterized by a low per capita gdp makes it difficult to create and implement hta into healthcare decision making due to legislative landscape and preventing from effective healthcare management and leading to a decrease in human capital and loss of opportunities for economic growth in these countries. 3. 43 countries (predominantly with lowand middle-income) are striving to improve the efficiency of health management and are at different stages of hta creation and implementation into healthcare decision making. 4. tax haven countries (offshore zones), despite the high gdp per capita, as a rule do not use hta in healthcare management. obviously, in these countries, gdp does not reflect real economic growth, since the local development is not related to the volume of international cross-cutting financial flows. the exception is the countries of the european region that have created hta agencies in accordance with the european directive 2011/24/eu. 5. international regional cooperation expands opportunities and accelerates the creation and implementation of hta both in lowand high-income countries, as well as provides an opportunity to use the hta-based decision-making process n countries with low expertise or insufficient resources for full-fledged hta. references 1. zahid, h., cuifen, m., zhihao, z. & yingxuan, w. 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(2016). health technology assessment in lowand middle-income countries: a landscape assessment, journal of pharmaceutical health services research, 7(1), 37–42. doi: 10.1111/jphs.12120 30. ey global (2023). el salvador's tax authority annual guide (resolution no. mh.uvi.dgii 006.005/2023) effective for tax year, [online]. available https://www.ey.com/en_gl/tax-alerts/el-salvador-s-tax-authority-publishes-list-of-taxhavens-for-2020 31. tørsløv, t., r., wier, l., s. & zucman, g. (2018). the missing profits of nations, [online]. available http://www.nber.org/papers/w24701 32. damgaard, j., elkjaer, t. & johannesen, n. (2018). piercing the veil, [online]. available https://www.elibrary.imf.org/view/journals/022/0055/002/article-a016en.xml 33. council of the european union, european parliament (2021). regulation (eu) 2021/2282 on health technology assessment and amending directive 2011/24/eu, [online]. available https://www.europeansources.info/record/?p=509515 adv syst sci appl 2022; 04:92–102 published online at https://ijassa.ipu.ru. approximate solutions and estimate of galerkin method for variable third-order operator-differential equation abdel baset i. ahmed engineering mathematics and physics dept, helwan university, cairo, egypt abstract: the paper considers a variable third-order operator-differential equation in a separable hilbert space. under certain assumptions, it is proved that this ode has a unique solution. the proof is based on a classical galerkin discretization of the separable hilbert space in term of certain eigenfunctions. the approximation quality of the galerkin approximations can be controlled in terms of the eigenvalues. we deduce estimates for the convergence rate of the approximate solutions to the exact one. an example provided as application to the investigated method. keywords: galerkin method, operator-differential equation, self-adjoint operator, convergence rate, orthogonal projection 1. introduction an important direction in modern mathematics is the study of operator-differential equations. using the operator method, it is possible to study a wide class of differential equations. the most different types of equations, such as linear and non-linear ordinary differential equations, partial differential equations, integral and integral differential equations can be present in operator form. with the help of the methods of the functional analysis and the theory of operators we can study the question related to the solvability of boundary-values problems and develop algorithms for finding the approximate solutions. it is known that the theory and methodology of operator-differential equations widely used in computational mathematics. among all the differential-operator equations, the most comprehensively studied are the first and the second-order differential-operator equations. in this regard, we can specify the works of a.g. zarubin, p.v. vinogradova [20], a.b.i. ahmed [1] which investigate strong solutions of the cauchy problem for linear and non-linear differential-operator equations of the first order. existence, uniqueness, and continuous dependence of strong solutions of the cauchy problem for various second-order linear differential equations with variable domains proved in the works of d.a. lyakhov [14] and f. e. lomovtsev [13]. the solvability of the third and higher-order linear differential-operator equations were studied in works [3], [4], [8]. from the standpoint of the spectral theory of linear operators in hilbert space and the fourier method for studying the differential-operator equations of the first and higher orders are represented in [6]. through some assumptions, the solvability for differential-operator equations of certain orders with variable coefficients were considered in [1], [2], [3], [7], [8]. note that some of the equations that arise during studying of the process soil and groundwater moisture dynamics, distribution of non-stationary external acoustic waves, ∗corresponding author: abdel2007@yandex.ru approximate solutions and estimate of galerkin method 93 relaxation processes during heat transfer are possible lead to the differential-operator equation of the third order in hilbert space. as a means of proving the theorems of the existence of solutions to non-stationary differential equations, galerkin’s method was used in the works [1], [7], [21] and other works. l.a. kantorovich [9] noted several problems that arise in the general theory of approximate methods for solving operator-differential equations and questions, leading to them, namely: the question of establishing the convergence of the algorithm, the process of studying the fast convergence of an approximate solution, and obtaining effective error estimates for the established approximate solution. the solution of the indicated tasks was devoted to many works. however, this area of research requires further development. when studying galerkin method, special attention is paid to the choice of the basis, since the properties of the basis functions significantly affect the rate of convergence of approximate solution of the differential equation to exact solution. p.e. sobolevsky [19] proposed to choose the eigenfunctions of the differential operator as a basis, which does not depend on time and forms with the investigated operator, so-called, acute angle. this idea was used in [18] for the study of non-stationary differential-operator equations with subordinates operators. at the present time, there is many works on the galerkin method for solving operatordifferential equations of a high order with an arbitrary basis. it should be noted that the dependence of the estimates for the rate of convergence of the behavior of the approximate solution on the type of key elements, properties of operators of the equation and its solution 2. problem statement and auxiliary assertions let h− separable hilbert space over the real scalar field with inner product (., .)h and norm ∥.∥h = ∥.∥ and a separable hilbert space h1 is compactly embedded into h . we denote the hilbert space of all strongly measurable functions f : [0, t ] → h by b2(0, t ;h), where b2(0, t ;h) = { f : ∥f∥b2(0,t ;h) = (∫ t 0 ∥f(t)∥2hdt ) 1 2 < +∞ } . in the space h , we will study the following variable third-order operator-differential equation with zero initial conditions: φ(t) d3u(t) dt3 − s(t)u(t) = f(t), t ∈ [0, t ], (2.1) u(0) = u(t ) = u ′ (0) = 0, (2.2) where u(t) is the required solution and f(t) is a given scalar function. in this work, we introduce some assumptions which concerning the operators φ(t) and s(t) as follows: (i) φ(t) is self-adjoint operator defined only on h1, (φ(t) = φ∗(t) ≥ σ0e, σ0 > 0), e− is the unit operator and σ0− is the spectrum lower bound (σ0 ∈ σ(φ(t)) [1], [8]. φ(t) is a positive definite operator where, ∥u∥h1 = ∥φ(t)u∥h . (ii) φ(t) and s(t) are three-times strongly continuously differentiable operators on the interval [0, t ] [11]. (iii) there exist some positive constants c00, c01, c03 ≥ 0 for all v ∈ h1 such that (φ(t)v, v)h ≥ c00 ∥∥φ1/2(0)v ∥∥2 , (φ ′ (t)v, v)h ≤ −c01 ∥∥φ1/2(0)v ∥∥2 , (φ ′′′ (t)v, v)h ≥ c03 ∥∥φ1/2(0)v ∥∥2 . copyright © 2022 assa. adv syst sci appl (2022) 94 a.b.i. ahmed (iv) s(t) is α−subordinate (with order α : 0 ≤ α < 1) to φ(0) [5], [11] i.e., for any v ∈ h1 there exists a constant k2 ≥ 0 independent on t, v, the s(t) satisfies the following inequality: ∥s(t)v∥ ≤ k2 ∥φ(0)v∥α ∥v∥α−1 . (2.3) (v) there exists a linear operator ψ is similar to φ(0) [16] i.e., ψ is a self-adjoint positive definite operator. moreover, the domain of (ψ) is equal to the domain of (φ(0)) and φ−1(t),ψ−1 : h → h1 are compact in h . both that operators are forming an acute angle in h [19], that is there exists a positive constant k3 independent of the choice v ∈ h1 and t such that ( φ(t)v, ψv ) ≥ k3∥φ(0)v∥∥ψv∥. (2.4) suppose that u(t) and φ(t) have continuous derivatives diu(t) dti , diφ(t) dti , i = 1, 2, 3 respectively in h . moreover, ( φ(i)(t)u, v ) h = ( u,φ(i)(t)v ) h , i = 1, 2, 3 for all u, v ∈ h1 and for almost t ∈ (0, t ) [4] such that w 3 2 (h,h1) = { u(t) ∈ b2(0, t ;h1) : d3 dt3 ( φ(t)u(t) ) ∈ b2(0, t ;h) } , with norm ∥u(t)∥2w 3 2 (h,h1) = ∫ t 0 ( ∥u(t)∥2h1 + ∥∥∥∥ d3 dt3 ( φ(t)u(t) )∥∥∥∥2 h ) dt. define the subspace which contains the strong solution u(t) of problem (2.1)-(2.2) by ◦ w 3 2 (h,h1) = { u(t) ∈ w 3 2 (h,h1) : u(0) = u(t ) = u ′ (0) = 0 } . according to [11], for all t under assumptions (i) and (ii) in the space h and for some constants c1, c2 > 0 that do not depend on t, we have ∥φ(t)φ−1(0)∥h→h ≤ c1, ∥φ(0)φ−1(t)∥h→h ≤ c2. (2.5) 3. galerkin method let e1, . . . , en, . . . be eigenvectors of the spectral problem ψer = λrer, r = 1, . . . , n, . . . , which form complete orthonormal system and λ1, . . . λn, . . . be the corresponding eigenvalues, in which 0 < λ1 ≤ . . . ≤ λn ≤ . . . , λn approaches infinity as n approaches infinity. the galerkin solution (approximate) of (2.1) (2.2) is considered to be in the discrete form un(t) = ∑n i=1 gi(t)ei, i = 1, 2, . . . , n where the functions gi(t) assumed to be unknown and represent the exact solutions to the following cauchy problem: φ(t) d3gj(t) dt3 − n∑ i=1 gj(t) ( s(t)ei, ej ) = ( f(t), ej ) h , (3.6) gj(0) = gj(t ) = g ′ j(0) = 0, j = 1, 2, . . . , n. (3.7) copyright © 2022 assa. adv syst sci appl (2022) approximate solutions and estimate of galerkin method 95 next we denote the orthogonal projection in h onto hn by pn, where hn is the linear span of {e1, e2, . . . , en}. then (3.6) (3.7) is equivalent to the following problem: φ(t) d3un(t) dt3 − pns(t)un(t) = pnf(t), (3.8) un(0) = un(t ) = u ′ n(0) = 0. (3.9) let us establish the unique solvability of problem (2.1) (2.2) and obtain the necessary estimates for the rate of convergence of the approximate solutions by using the galerkin method. now and then, c is used to denote different positive constants independent of t and n. theorem 3.1: under the preceding assumptions, there exists a unique solution un(t) of (3.8)-(3.9) in ◦ w 3 2 (h,h1) for each n. moreover, the sequence {un(t)}n=1,2,... is convergent in w 3 2 (h,h1) and the solution of (2.1) (2.2) is unique. proof look into the cauchy problem: φ(t) d3on(t) dt3 = png(t), (3.10) on(0) = on(t ) = o ′ n(0) = 0. (3.11) if g(t) ∈ b2(0, t ;h), then problem (3.10) (3.11) has a unique solution on(t) ∈ ◦ w 3 2 (h,h1) and hence ∫ t 0 ∥∥∥∥φ(t)d3on(t) dt3 ∥∥∥∥2 dt = ∫ t 0 ∥png(t)∥2dt. from assumption (i) it follows that the operator φ(t) has an inverse φ−1(t) : h → h1. let the operator ( φ(t) d3 dt3 )−1 : b2(0, t ;h) → ◦ w 3 2 (h,h1) be a homeomorphism, then holds the following inequality: ∥∥∥∥∥ ( φ(t) d3 dt3 )−1 ∥∥∥∥∥ b2(0,t ;h)→b2(0,t ;h) ≤ c. (3.12) multiply equation (3.10) by on(t) in h and then integrate with respect to t, where t varies from 0 to t , we get∫ t 0 ( φ(t) d3on(t) dt3 ,on(t) ) dt = −3 2 ∫ t 0 ( dφ(t) dt don(t) dt , don(t) dt ) dt + 1 2 ∫ t 0 ( d3φ(t) dt3 on(t),on(t) ) dt = ∫ t 0 (png(t),on(t))dt. (3.13) as the operator φ(0)φ−1(t) is uniformly bounded. from the heinz inequality [11] yields that the operator φ 1 2 (0)φ− 1 2 (t) is also uniformly bounded. from assumption (iii), it follows that c 2 ∥∥∥φ 1 2 (0)on(t) ∥∥∥2 ≤ ∥png(t)∥b2(0,t ;h) (∫ t 0 ∥on(t)∥2 )1/2 . copyright © 2022 assa. adv syst sci appl (2022) 96 a.b.i. ahmed furthermore, ∥∥∥φ 1 2 (0)on(t) ∥∥∥ ≤ c∥png(t)∥b2(0,t ;h). hence, ∥∥∥∥∥φ 1 2 (0) ( φ(t) d3 dt3 )−1 png(t) ∥∥∥∥∥ ≤ c ∥png(t)∥b2(0,t ;h) . (3.14) then using the following substitution φ(t) d3on(t) dt3 ≡ ωn(t) in (3.8) (3.9), we get ωn(t)− pns(t) ( φ(t) d3 dt3 )−1 ωn(t) = pnf(t). (3.15) in the coming step we need to prove that the operator pns(t) ( φ(t) d3 dt3 )−1 is compact in b2(0, t ;h). according (2.3) and hölder inequality for all ν ∈ h1, we get∥∥∥∥pns(t) ( φ(t) d3 dt3 )−1 ν ∥∥∥∥ b2(0,t ;h) ≤ c ∥∥∥∥pnφ(0) ( φ(t) d3 dt3 )−1 ν ∥∥∥∥α b2(0,t ;h) ∥∥∥∥pn ( φ(t) d3 dt3 )−1 ν ∥∥∥∥1−α b2(0,t ;h) . (3.16) taking into account the uniform boundedness of φ(0)φ−1(t) and inequalities (3.12) (3.16), yields that∥∥∥∥pns(t) ( φ(t) d3 dt3 )−1 ν ∥∥∥∥ b2(0,t ;h) ≤ c ∥∥∥∥pn ( φ(t) d3 dt3 )−1 ν ∥∥∥∥1−α b2(0,t ;h) ∥ν∥αb2(0,t ;h) . (3.17) as a result of the compact embedding of h1 into h and based on the lemma of compactness [15], yields that the space w 3 2 (h,h1) is compactly embedded in b2(0, t ;h). consequently,( φ(t) d3 dt3 )−1 is actually compact in b2(0, t ;h). hence, from inequality (3.17) yields that the operator pns(t) ( φ(t) d3 dt3 )−1 is compact in b2(0, t ;h). the second step, we shall prove that problem (3.15) is resolvable. in (3.15), let the right-hand side be zero, then (3.15) will be equivalent to the following cauchy problem: φ(t) d3un(t) dt3 − pns(t)un(t) = 0, (3.18) un(0) = un(t ) = u ′ n(0) = 0, (3.19) with a solution un(t) ∈ ◦ w 3 2 (h,h1). using assumption (2.3), yields∫ t 0 ∥∥∥∥φ(t)d3un(t) dt3 ∥∥∥∥2 dt ≤ k2 2 ∫ t 0 ∥pnφ(0)un(t)∥2α ∥pnun(t)∥2(1−α)dt ≤ k2 2∥φ(0)φ−1(t)∥2αh→h ∫ t 0 ∥pnφ(t)un(t)∥2α∥pnu(t)n∥2(1−α)dt. copyright © 2022 assa. adv syst sci appl (2022) approximate solutions and estimate of galerkin method 97 thus, ∫ t 0 ∥∥∥∥φ(t)d3un(t) dt3 ∥∥∥∥2 dt ≤ c ∫ t 0 ∥pnφ(t)un(t)∥2α∥pnun(t)∥2(1−α)dt. apply hölder inequality, yields that∫ t 0 ∥∥∥∥φ(t)d3un(t) dt3 ∥∥∥∥2 dt ≤ c (∫ t 0 ∥pnφ(t)un(t)∥2 )α(∫ t 0 ∥pnun(t)∥2dt )(1−α) . consequently, apply the young inequality : ab ≤ εa1/α + (α ε )α/1−α (1− α)b1/1−α, yields that∫ t 0 ∥∥∥∥φ(t)d3un(t) dt3 ∥∥∥∥2 dt ≤ c ( ε ∫ t 0 ∥pnφ(t)un(t)∥2dt+ (α ε ) α 1−α (1− α) ∫ t 0 ∥pnun(t)∥2dt ) . take ε = 1 2c , then ∫ t 0 ∥∥∥∥φ(t)d3un(t) dt3 ∥∥∥∥2 dt ≤ c ∫ t 0 ∥pnun(t)∥2dt. (3.20) whenever we multiply the above-mentioned equation (3.18) by un(t) and integrate the outcome, we get c03 2 ∥φ 1 2 (0)un(t)∥2 ≤ ∫ t 0 ∥un(t)∥∥pns(t)un(t)∥dt ≤ c ∫ s 0 ∥pnφ(0)un(t)∥α∥un(t)∥2−αdt ≤ c (∫ t 0 ∥pnφ(0)un(t)∥2dt )α 2 (∫ t 0 ∥un(t)∥2dt ) 2−α 2 . using (3.20), yields that ∥un(t)∥2 ≤ c ∫ t 0 ∥un(t)∥2dt. (3.21) from the bellman–gronwall theorem [5] it follows that this inequality is possible only in the case when un(t) ≡ 0. thus, by using the the fredholm alternative [17] yields that problem (3.8)-(3.9) has a unique solution un(t) ∈ ◦ w 3 2 (h,h1). let δn(t) = φ(t) d3un(t) dt3 − s(t)un(t)− f(t), follows that ∥δn(t)∥b2(0,t ;h) ≤ ∥(i − pn)f(t)∥b2(0,t ;h) + ∥(i − pn)s(t)un(t)∥b2(0,t ;h). from (3.21), yields that {un(t)} is bounded in w 3 2 (h,h1). since the subspace h1 is compactly embedded in the space h , yields that s(t) : w 3 2 (h,h1) → b2(h, h1) is compact. consequently, {s(t)un(t)} is also compact in b2(h, h1). as in [10], the sequence {i − pn} uniformly converges to zero. therefore, ∥δn(t)∥b2(0,t ;h) → 0 as n → ∞. now, we need to prove that {un(t)} in w 3 2 (h,h1) is a cauchy sequence. copyright © 2022 assa. adv syst sci appl (2022) 98 a.b.i. ahmed from (3.13) we have∥∥∥φ′′′ (0) (un(t)− un+p(t)) ∥∥∥ b2(0,t ;h) ≤ c ∥∥∥∥φ(t)(d3un(t) dt3 − d3un+p(t) dt3 )∥∥∥∥ b2(0,t ;h) ≤ c ( ∥δn(t)∥b2(0,t ;h) + ∥δn+p(t)∥b2(0,t ;h) + ∥s(t)(un(t)− un+p)∥b2(0,t ;h) ) . applying young inequality, we get∥∥∥φ′′′ (0) (un(t)− un+p(t)) ∥∥∥ b2(0,t ;h) ≤ c ( ∥δn(t)∥b2(0,t ;h) + ∥δn+p(t)∥b2(0,t ;h) + ∥(un(t)− un+p)∥b2(0,t ;h) ) . (3.22) in equation (3.8), replace un(t) by un+p(t) and then subtract the resulting equation from equation (3.8), we get the following relation φ(t) ( d3un+p(t) dt3 − d3un(t) dt3 ) − s(t)(un+p − un) = (i − pn)s(t)un(t)− (i − pn)s(t)un+p(t) + (pn+p − pn)f(t). multiply the preceding relation by (un+p(t)− un(t)) and then integrate the outcome. further, employ hölder inequality, yields that∥∥φ1/2(0)(un+p(t)− un(t)) ∥∥2 b2(0,t ;h) ≤ (∥(i − pn)s(t)un(t)∥b2(0,t ;h) + ∥(i − pn)s(t)un+p(t)∥b2(0,t ;h) + ∥(pn+p − pn)f(t)∥b2(0,t ;h))× ∥φ−1/2(0)∥h→h ∥∥φ1/2(0)(un+p(t)− un(t)) ∥∥ b2 . consequently,∥∥φ1/2(0)(un+p(t)− un(t)) ∥∥ b2(0,t ;h) ≤ (∥(i − pn)s(t)un(t)∥b2(0,t ;h) + ∥(i − pn)s(t)un+p(t)∥b2(0,t ;h) + ∥(pn+p − pn)f(t)∥b2(0,t ;h))× ∥φ−1/2(0)∥h→h . from the compactness of {s(t)un(t)} in b2(0, t ;h), yields that the right side of the preceding inequality approaches to zero. therefore, according to (3.22), the {un(t)} in w 3 2 (h,h1) is a cauchy sequence. let {un(t)} → u(t) as n → ∞. then∥∥∥∥−f(t) + φ(t) d3u(t) dt3 − s(t)u(t) ∥∥∥∥ b2(0,t ;h) ≤ ∥∥∥∥φ(t)(d3u(t) dt3 − d3un(t) dt3 ) − s(t) (u(t)− un(t)) ∥∥∥∥ b2(0,t ;h) + ∥δn∥b2(0,t ;h) ≤ c ( φ(0)(∥un(t)− u(t))∥b2(0,t ;h) + δn∥b2(0,t ;h) ) . therefore, there exists a strong solution u(t) ∈ w 3 2 (h,h1) of problem (2.1)-(2.2). now we will show that the solution of problem (2.1)-(2.2) is unique. assume that problem (2.1), (2.2) has two solutions u and û. then φ(t) d3 dt3 (u− û)− s(t) (u− û) = 0. copyright © 2022 assa. adv syst sci appl (2022) approximate solutions and estimate of galerkin method 99 multiply the preceding equation by (u− û) and integrate with respect to t, then we can simply show that ∥u− û∥w 3 2 (h,h1) ≤ 0, which means that the solution of problem (2.1), (2.2) is unique. the theorem is proved. after the question of the solvabilty of problem (2.1) (2.2) is proven by galerkin method, we shall use also the same method to estimate the rate of convergence of the approximate solutions of problem (2.1) (2.2). theorem 3.2: suppose that all of assumptions of theorem 1 be fulfilled. then∥∥φ1/2(0) (un(t)− u(t)) ∥∥ b2(0,t ;h) ≤ cλ −1/2 n+1 . (3.23) proof let un(t) = u(t)− un(t), where u(t) is the exact solution of problem (2.1) (2.2) and un(t) is the approximate solutions of problem (3.8) (3.9). then φ(t) d3un dt3 − s(t)un(t) = (i − pn)(f(t)− s(t)u(t)). multiply the preceding equation by un(t) and hence integrate the outcome taking into account that for each n, we have that un(t) belongs to domain φ(t), we get 3c 2 ∥∥∥φ1/2(0)u ′ n(t) ∥∥∥ b2(0,t ;h) + 3c 2 ∫ t 0 ∥∥φ1/2(0)un(t) ∥∥ ≤ ∫ t o | f(t)((i − pn)un(t))(s(t)un(t), (i − pn)un(t)) | dt. from assumption (iv), we get∥∥φ1/2(0)un(t) ∥∥2 b2(0,t ;h) ≤ c ( ∥f(t)∥b2(0,t ;h) + ∥s(t)un(t)∥b2(0,t ;h) ) × ∥∥φ−1/2(0)ψ1/2 ∥∥ h→h ∥∥ψ−1/2(i − pn)φ 1/2(0)un(t) ∥∥ . consequently, ∥∥φ1/2(0)un(t) ∥∥ b2(0,t ;h) ≤ cλ −1/2 n+1 . the proof is complete. 4. application as an applied example on galerkin method we will investigate an initial-boundary value problem for non-classical higher order differential equations of composite-mixed type with smooth coefficients. the theory of equations of mixed type is one of the most important branches of the theory of non-classical differential equations of mathematical physics.these types of problems have many applications in gas dynamics [12] and in other branches of physics in the rectangle domain q = [0, 1]× [0, t ], consider the initial-boundary value problem χ(x, t) ∂5u(x, t) ∂x2∂t3 + χx(x, t) ∂4u(x, t) ∂x∂t3 − ζ1(x, t) ∂u(x, t) ∂x − ζ0(x, t)u(x, t) = f(x, t), (x, t) ∈ q (4.24) copyright © 2022 assa. adv syst sci appl (2022) 100 a.b.i. ahmed u(x, 0) = ut(x, 0) = u(x, t ) = 0, 0 ≤ x ≤ 1, (4.25) u(0, t) = u0(t), 0 ≤ t ≤ t, (4.26) u(1, t) = u1(t), 0 ≤ t ≤ t. (4.27) where f(x, t) ∈ b2(q), ζi(x, t), ∂ζi(x, t) ∂t ∈ c(q), ui(t) ∈ c3[0, t ], i = 0, 1. assume that χ(x, t) ≥ χ0 > 0, ∂s+1 ∂x∂ts χ(x, t) ∈ c(q), s = 1, 2, 3, and the functions u0(t), u1(t) fulfill the following matching conditions: u0(0) = u0(t ) = u1(0) = u1(t ) = u ′ 0(0) = u ′ 1(0) = 0. by using the substitution v(x, t) = u(x, t)− (1− x)u0(t)− xu1(t), problem (4.24) (4.27) will be equivalent to the following problem: χ(x, t) ∂5v(x, t) ∂x2∂t3 + χx(x, t) ∂4v(x, t) ∂x∂t3 − ζ1(x, t) ∂v(x, t) ∂x − ζ0(x, t)v(x, t) = h(x, t), (x, t) ∈ q (4.28) v(x, 0) = vt(x, 0) = v(x, t ) = 0, 0 ≤ x ≤ 1, (4.29) v(0, t) = v(1, t) = 0, 0 ≤ x ≤ 1, (4.30) where, h(x, t) = f(x, t)− ζ1(x, t)u0(t) + ζ1(x, t)u1(t) + (1− x)ζ0(x, t)u0(t) + xζ0(x, t)u1(t)− (1− x)u ′′′ 0 (t)− xu ′′′ 1 (t). now, we need to redefine some spaces as follows: assign h = b2(0, 1), h1 = w 2 2 (0, 1) ∩ ◦ w 1 2 (0, 1). notice that the w 2 2 (0, 1) is denoted to the vector norm sobolev type space, the subspace ◦ w 1 2 (0, 1) = {v(x) ∈ w 2 2 (0, 1),v(0) = v(1) = 0} and w 3 2 (h,h1) = w 2,3 2 (q). as ∂ ∂x χ(x, t) ∈ c(q), provided that | ∂ ∂x χ(x, t) |≤ χ1, where χ1 is a constant greater than zero. on h1 we identify the following operators φ(t) = ∂ ∂x ( χ(x, t) ∂ ∂x ) −θi, s(t) = ζ1(x, t) ∂ ∂x + (ζ0(x, t)−θ) i, where the constant θ < χ2 1 χ0 and domain s(t) ⊃ domainφ(t). for problem (4.28) (4.30), we take ψ = d2 dx2 . distinctly, it can be seen that the assumptions (i)− (v) are fully satisfied. consequently, simply we can verify that s(t) is α− subordinate to φ(0) with α = 1 2 . copyright © 2022 assa. adv syst sci appl (2022) approximate solutions and estimate of galerkin method 101 in the following step,, we’ll show that the operators φ(t) and ψ fulfill the acute-angle inequality (2.4). for each element y(x) ∈ h1, we have (φ(t)y(x),ψy(x))b2(0,1) ≥ χ0 ∫ 1 0 ( d2y(x) dx2 )2 dx−θ ∫ 1 0 ( dy(x) dx )2 dx+ χ1 ∫ 1 0 | dy(x) dx || d 2y(x) d2x | dx ≥ χ0 ∫ 1 0 ( d2y(x) dx2 )2 dx+ χ1ε 2 ∫ 1 0 ( d2y(x) dx2 )2 dx + χ1 2ε ∫ 1 0 ( dy(x) dx )2 dx−θ ∫ 1 0 ( dy(x) dx )2 dx. we choose ε = χ0 2χ1 . then we get (φ(t)y(x),ψy(x))b2(0,1) ≥ ( χ2 1 χ0 −θ )∫ 1 0 ( dy(x) dx )2 dx ≥ c∥y(x)∥2w 2 2 (0,1) , (4.31) comparing (4.31) with (2.4), we get the fact that φ(t) and ψ verified the acute-angle condition. as well, ψ satisfies the eigenvalue equation ψer(x) = λrer(x), r = 1, 2, . . . , where er(x) = sin rπx√ 2 are eigenfunction solutions, which are normalizable and λr = (rπ)2, r = 1, 2, . . . are the corresponding eigenvalues. all hypotheses of theorems 3.1 and 3.2 are exactly fulfilled. thus, for the galerkin solution of problem (4.28) (4.30) and hence problem (4.24) (4.27), holds the following estimate max 0≤t≤t ∥un(x, t)− u(x, t)∥b2(0,1) ≤ c 1√ n . 5. conclusion for a third-order multi-variable operator differential equations with initial-boundary conditions, the existence and uniqueness theorem of a strong solution was proved using galerkin method. the rate of convergence of the galerkin solution (approximate solutions) to the exact one was estimates. the results of galerkin method enabled us to present an application and prove the solvability of a boundary value problem for non-classical higher order differential equations of composite-mixed type with smooth coefficients. references 1. ahmed, a.b.i. (2021). convergence rate of the galerkin method for boundary value problem for mixed-type operator-differential equations, iaeng international journal of applied mathematics, 51(4), 984–989. 2. ahmed, a.b.i. (2021). existence and uniqueness results for an initial-boundary value problem of parabolic operator-differential equations in a weight space, twms journal of applied and engineering mathematics, 11(3), 628–635. 3. ahmed, a.b.i. (2021). on the solvability of higher-order operator-differential equations in a weighted sobolev space, international journal of applied mathematics, 34(1), 469–478. doi: http://dx.doi.org/10.12732/ijam.v34i1.8 copyright © 2022 assa. adv syst sci appl (2022) 102 a.b.i. ahmed 4. antipin, v.i. (2013). solvability of a boundary value problem for operator-differential equations of mixed type, sib math j, 54, 185–195. 5. beckenbach, f., & bellman, r. (1961) inequalities. springer-verlag, berlin. 6. dezin, a.a. (2000). operator-differential equations,” tr. mat. inst. steklova, 229. 7. dubinskii, yu. a. (1973). on some differential-operator equations of arbitrary order, mat. sb., 90(132-1), 3–22. 8. farid, n., ahmed, a.b.i. & labeeb m.a. (2022). sufficient conditions for regular solvability of an arbitrary order operator-differential equation with initial-boundary conditions, advances in difference equations, 104, 1–14. 9. kantorovich, l.v. (1956). approximate solution of functional equations, uspekhi mat. nauk, 11:6(72), 99–116. 10. krasnosel’skii, m.a., vainikko, g.m., zabreiko, p.p., et al. (1969). approximate solution of operator equations. moscow: nauka. 11. krein, s.g. (1967). lineinye differencialnie uravneniya v banahovih prostranstvah.[linear differential equations in banach spaces]. moscow, russia: nauka, [in russian]. 12. kuzmin, a.g. (1990). neklassicheskie yravneniya smeshannogo tipa i ih prilozheniya k gazovoi dinamike [nonclassical equations of mixed type and its applications to the gas dynamics]. leningrad, ussr: leningrad st. univ, [in russian]. 13. lomovtsev, f.e. (2005). on a stable approximation to boundary value problems for evolution operator-differential equations with variable domains, differ. equ., 41(5), 721–732. 14. lyakhov, d.a. & lomovtsev, f.e. (2010). the method of weak solutions of the auxiliary cauchy problem for investigating the smoothness of solutions of second-order hyperbolic operator-differential equations with variable domains, vestn. belarus. gos. univ, 1(2), 75–82. 15. lyons, j.l. (1973). nekotorie metodi resheniya nelineinih kraevih zadach [some methods for solving nonlinear boundary value problems]. moscow, ussr: mir, [in russian]. 16. mikhlin, s.g. (1970) variacionnie metodi matematicheskoi fiziki [variational methods in mathematical physics]. moscow, ussr: nauka, [in russian]. 17. k. moren (1967). methods of hilbert spaces. poland: national scientific publishers pwn. 18. smagin, v.v. (1997). estimates for the convergence rate of projection and projection difference methods as applied to weakly solvable parabolic equations. mat. sb., 188(3), 143–160. 19. sobolevskii, p. e. (1957). on equations with operators forming an acute angle. dokl. akad. nauk sssr, 116, 754–757. 20. vinogradova, p., & zarubin, a. (2009). projection method for cauchy problem for an operator-differential equation, numer. funct. anal. optim., 30(1-2), 148–167. 21. vishik, m.i. & ladyzhenskaya, o.a. (1956). boundary value problems for partial differential equations and certain classes of operator equations, uspeki. mat. nauk, 11:6(76), 41–97. copyright © 2022 assa. adv syst sci appl (2022) introduction problem statement and auxiliary assertions galerkin method application conclusion microsoft word gundagani murali assa adv syst sci appl 2024; 04; 9-27 published online at https://ijassa.ipu.ru. hall current and ion slip effects on 3d mhd nanofluid flow of eyring-powell fluid with gyrotactic microorganisms murali gundagani1, tan kaun tak2, sivaneasan bala krishnan2, pravin kshirsagar3, venkata madhu javvaji4 1) geethanjali college of engineering and technology, cheeryal, india 2) singapore institute of technology, singapore 3) j.d. college of engineering and management, nagpur, india 4) sreenidhi institute of science and technology, yamnampet, india abstract: this study explores the three-dimensional magnetohydrodynamic (mhd) flow of an eyring-powell nanofluid over a stretching sheet, incorporating the combined effects of hall current, ion slip, gyrotactic microorganisms, thermal radiation, brownian motion, thermophoresis, and key dimensionless parameters such as the schmidt and prandtl numbers. the investigation focuses on the practical applications of such flows, particularly in bioengineering and thermal systems, where magnetic fields and bio-convection play a crucial role. the governing equations are solved numerically using matlab's bvp4c solver, providing detailed insights into velocity profiles, temperature distribution, nanoparticle concentration, and microorganism density. additionally, the study evaluates critical engineering parameters, including skin-friction coefficients, nusselt number, sherwood number, and motile microorganism density profiles. to ensure accuracy, the numerical method is validated against existing literature, confirming the reliability of the results. the findings offer valuable implications for enhancing heat and mass transfer in nanofluid-based technologies, thermal management systems, and bio-convective applications. keywords: hall current; three-dimensional; nanofluid; erying-powell fluid; mhd; hall current; ion slip; stretching sheet; gyrotactic microorganisms. 1. introduction the study of gyrotactic microorganisms in a 3d nano-enhanced eyring-powell fluid flow over a stretching sheet, incorporating magnetohydrodynamics (mhd), hall current, and ion slip effects, is motivated by its relevance to advanced bioengineering and thermal systems. gyrotactic microorganisms play a crucial role in bio-convection, which enhances mixing and heat transfer in nanofluids, making them ideal for applications in biotechnology, medical diagnostics, and energy systems. the inclusion of mhd, hall current, and ion slip effects addresses the influence of magnetic fields, which are pivotal in controlling fluid dynamics in industrial processes like magnetic drug targeting, microfluidic devices, and cooling systems for electronic equipment. this research provides insights into optimizing heat and mass transfer in complex fluid systems, offering potential advancements in biofuel production, wastewater treatment, and thermal management technologies. research on hall and ion slip effects in magnetohydrodynamic (mhd) flows highlights their critical role in altering flow dynamics and heat transfer. the hall effect creates orthogonal electric fields from charged particles in magnetic fields, while ion slip reflects velocity differences between charged particles and neutral fluids. these effects, amplified by stronger magnetic fields, are vital for optimizing mhd systems. extensive research has explored these effects in various contexts. a strong magnetic field applied to an electric field generates an electric potential. this potential is derived from the lorentz force created in charged gases, liquids, or electrically conductive fluids. for propagation to occur, it must go in a linear manner via the electric and magnetic fields. in 10 murali g., tan k. t., sivaneasan b. k., pravin r. k., venkata m. j. copyright ©2024 assa adv. in systems science and appl. (2024) 1879, hall discovered this phenomenon and designated it as the "hall effect" [1]. in some fluids, such as ionized gasses or conductive liquids, the hall effect may alter the fluid's structure during flow due to the influence of magnetic orientation. the primary focus of the present study is the influence of hall currents on the flow patterns of various nanofluids. ions and charged particles traverse electromagnetic field lines, resulting in the electrodynamic phenomena referred to as hall effect and ion-slip. the ion-slip current phenomena may often be disregarded, since ions possess far more mass than electrons and hence exhibit considerably slower movement. these events are very important when looking at the flow features of several problems. das et al. [2] analyzed the hall effects on the spontaneous nanofluid flow with thermal radiation. krishna et al. [3] examined the hall current effects on a water-based spinning nanofluid traversing a porous medium. kumar et al. [4] investigated the influence of spinning and the soret, dufour, and hall currents on magnetohydrodynamic flow in a porous media in their research. bishnoi et al. [5] examined the influence of hall currents on a magnetic nanofluid layer inside a continuous medium subjected to mono-diffusion. they accomplished this by supposing that the nanoparticles at the peripheries are passively organized. in reference [6], swarnalathamma and colleagues examined the influence of hall currents on a thick, incompressible fluid contained inside a porous media. raghunath and mohanaramana [7] examined the soret effect on second-order fluids in rotation inside a porous media influenced by hall currents in their work. ji le et al. [8] investigated the double-diffusive convection of a nanofluid inside a microchannel including hall currents. abbasi et al. examined the stability of electrically conducting nanofluids in the presence of hall currents in their research [9]. the temperature instability of nanofluids under hall currents in the brinkman porous medium with no mass flux of nanoparticles at the edges was talked about in [10]. some of the writers ([11], [12], [13], and [14]) looked into the hall effect in mhd flow issues. these studies [15, 16, 17, 18, 19, and 20] do a great job of explaining what the hall effect means for the flow of mhd nanofluid in a variety of geometrical shapes. a substantial body of additional research ([21]-[30]) further underscores the ongoing interest in understanding hall and ion slip effects on rotating fluid flows past vertical surfaces.the authors conducted numerical analyses on various fluids, as referenced in [32]-[37]. these studies, combined with the extensive research outlined in [38]-[40] and [41]-[44], form a strong basis for the current investigation. this study explores gyrotactic microorganism swimming in an erying-powell nanofluid near a stretching sheet, incorporating hall current, ion slip, magnetic fields, thermal radiation, thermophoresis, and brownian motion. using matlab's "bvp4c," it numerically solves governing equations to analyze heat, mass, momentum, and microorganism density transfer. by examining parameter impacts on temperature, concentration, velocity, and microorganism profiles, the research aims to optimize flow dynamics and enhance design strategies for related applications. 2. modelling of the flow in this current research work, the authors have examined 3-dimensional, eyring-powell fluid with nanofluid particles influenced by thermal radiation, gyrotactic microorganisms, hall current, ion slip, thermophoresis, and brownian motion on a non-conducting stretched sheet with magnetic field. the following assumptions are used for this endeavour: a) the fluid is electrically conducting in the presence of an applied magnetic field b = {0, 0, b0} is applied in the z − direction. b) magnetic reynolds number is assumed very small so that the induced magnetic field is ignored. c) let (u, v, w) be the velocity components along the (x, y, z) directions, respectively. d) flow is precipitated by a surface that is expanding at an exponential rate. hall current and ion slip effects on 3d mhd nanofluid flow… 11 copyright ©2024 assa. adv. in systems science and appl. (2024) e) this means that the joint effects of diffusion are neglected into account when doing the calculations for energy and concentration, respectively. f) when doing the calculation of concentration, the impact of chemical reaction is not taken into consideration. g) in the section on boundary conditions, the influence of the many different slip effects is not taken into consideration. h) the effects of brownian motion and thermophoresis are considered for nanofluid flow. i) the effects of hall current and ion slips are considered in momentum equations. j) the cauchy stress tensor t is written as pi  t (1)  .i ija p j b       (2) where extra stress tensor ij in an eyring-powell fluid model is 11 1 sinh .i i ij j j u u x c x                      (3) here  and c are the characteristics of the eyring-powell fluid, p the pressure and i the identity tensor. considering 3 1 1 1 1 1 1 sinh . . . , . 1 6 i i i i j j j j u u u u c x c x c x c x                        (4) k) the modified generalized formula for ohm’s law is given, which takes into consideration the influences of hall and ion slip at very high magnetic field strengths:      2 e e e e i o o j e v b j b j b b b b                    (5) by applying simplifications, eq. (5) is reduced to        2 22 2 1 1 & 1 1 e e i e i e x o z o e i e e i e u w u w j b j b                                     (6) the radiative heat flux rq is defined as * 4 * 4 3r t q k z         (7) we assume that the temperature variances inside the flow are such that the term 4t can be represented as linear function of temperature. this is accomplished by expanding 4t in a taylor series about a free stream temperature t as follows:    24 4 3 24 6 ............t t t t t t t t          (8) after neglecting higher-order terms in the above equation beyond the first-degree term in  t t , we get 4 3 44 3t t t t   (9) thus substituting eq. (9) in eq. (7), we get 3 * * 16 3r t t q k z        (10) the basic governing equations, continuity, momentum, energy, concentration, and microorganism equations, by applying the boundary layer approximations are 0 u v w x y z          (11) 12 murali g., tan k. t., sivaneasan b. k., pravin r. k., venkata m. j. copyright ©2024 assa adv. in systems science and appl. (2024)     2 22 23 2 2 11 1 2 1 e i eo e i e u wbu u u u u u v w x y z c c z z                                                                        (12)     2 22 23 2 2 11 1 2 1 e e io e i e u wbv v v v v u v w x y z c c z z                                                                        (13) 2 3 *2 2 12 * 2 161 3 t b p d tt t t t t c t t u v w d x y z z z z t z c k z                                                                         (14) 22 2 t b dc c c c t u v w d x y z z t z                                   (15)   * 2 2 c m w b w c u v d x y c c y y y                                    (16) the boundary conditions for this flow are    , , 0, , , 0 0, 0, , , w w w w wu u x ax v v y by w t t c c at z u v t t c c as z                         (17) introducing the following similarity transformations         , , , , , , , , , w w w c c u axf v byg w a f g c c t ta z n u v t t y x                                                 (18) making use of eq. (18), eqs. (12), (13), (14), (15) and (16) take the following form       2 2 2 2 1 2 2 1 1 0 1 e i e e i e f g f f f ff gf f m f m                                (19)       2 2 2 2 2 2 2 1 1 0 1 e e i e i e f g g g g gg fg g m g m                                (20) 24 1 pr pr pr pr 0 3 nbf nbg nb nt nr                      (21) 0nb nbscf nbscg nt          (22) 0n lbfn pen pen pen           (23) the corresponding boundary conditions (10) become                         0 0, 0 0, 0 1, 0 , 0 1, 0 1, 0 1, 0, 0, 0, 0, 0 f g f g n f g n                             (24) where the involved physical parameters are defined as    2 * 12 * 3 *3 2 3 2 1 22 2 , , , , pr , , , 4 1 , , , , , , 2 2 b w t wo b c m m w d c c d t tb b k m nb nt sc nr a t d a t bw na x a y lb pe c c c d d n n                                           (25) quantities of physical interest, the physical parameters of the skin-friction coefficient along x and y directions and local nusselt number are presented as follows: hall current and ion slip effects on 3d mhd nanofluid flow… 13 copyright ©2024 assa. adv. in systems science and appl. (2024)        3 12 1 re 1 0 0 3 wx f x w cfx c f f u               (26)        3 22 1 re 1 0 0 3 wy f y w cfy c g g v               (27)   w w xq nu t t    (28) where   0 w r w y t q q y             4 re 1 0 3xnu nr                 0 re 0ym x b w b w c x yxq sh sh d c c d c c               (29)     1 2 0 re 0w x m w y xd nh nh n d y                  (30) 3.methods the higher order nonlinear ordinary differential equations (odes) presented in equations (19) to (23) and the boundary conditions specified in equation (24) are solved numerically using the bvp4c solver available on the matlab computer platform. the bvp4c solver incorporates the three-stage lobatto iiia runge-kutta method, delivering numerical solutions with fourth-order precision. all subsequent steps have been executed to include the bvp4c solver into our physical model. step-1: new variables are introduced for the system of higher order non-linear odes in eqs. (19)-(23).                         1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 y f y f y f y g y g y g y y y y y n y n                         (31) step-2: reduce the system of higher order non-linear odes in eqs. (19)-(23) to a system of first order non-linear odes using the new variables in eq. (31).                            2 2 2 2 2 2 1 1 2 5 1 3 4 3 2 2 1 ''' 1 3 e i e e i e y y y y y y y m y m f y                             (32)                            2 2 2 2 2 2 2 2 1 5 4 6 1 6 5 5 1 1 6 e e i e i e y y y y y y y m y m g y                              (33)                2 pr 1 8 pr 4 8 pr 8 10 pr 8 4 1 3 nby y nby y nby y nt y nr             (34) 14 murali g., tan k. t., sivaneasan b. k., pravin r. k., venkata m. j. copyright ©2024 assa adv. in systems science and appl. (2024)                       2 1 10 4 10 pr 1 8 pr 4 8 pr 8 10 pr 8 4 1 3 nbscy y nbscy y nby y nby y nby y nt y scnt nr nb                                       (35)                                   2 1 12 10 12 11 pr 1 8 pr 4 8 pr 8 10 pr 811 1 10 4 10 4 1 3 n lby y pey y pey nby y nby y nby y nt yy pe nbscy y nbscy y scnt nb nr                                         (36) step-3: express the boundary conditions in eq. (24) in terms of the new variables in eq. (31).                         1 0, 4 0, 2 1, 5 , 7 1, 9 1, 2 0, 5 0, 7 0, 9 0, 11 1, 12 0, p p p p p p q q q q p q y y y y y y y y y y y y                                                              (37) the subscripts ‘p’ and ‘q’ indicate the location the sheet at 𝜂 = 0, and the location away from the sheet for a specific value of 𝜂. in this study, this location is set at 𝜂 = 3 (for primary velocity profiles), 𝜂 = 4 (for secondary velocity profiles), 𝜂 = 6 (for temperature, concentration and microorganism profiles). step-4: the fourth phase involves using the bvp4c solver to implement the system of first-order non-linear ordinary differential equations specified by equations (32) to (36), together with the boundary conditions indicated in equation (37). step-5: acquire the first and second solutions by using two separate sets of starting estimates sequentially while executing the bvp4c solver. this enables the acquisition of the primary and secondary solutions. if the computed temperature and velocity profiles conform to equation (24), the initial estimates will be deemed acceptable; otherwise, this procedure will be replayed with a new set of initial guesses until the desired outcomes are attained. typically, many attempts are required to get a suitable compilation of first observations. table-1.: comparison of present nusselt number coefficient results with published nusselt number coefficient results of hayat et al. [45] when, βe = βi = ω = sc = nt = nb = pe = lb = 0. m pr nr ε hayat et al. [45] results present results 0.0 0.71 0.5 0.5 0.65496 0.664582750827 0.7 0.62357 0.610058625760 1.5 0.54949 0.539812850855 1.0 0.59586 0.584574450825 1.5 0.78096 0.774707605761 1.0 0.41365 0.420656525663 3.0 0.58315 0.580756545641 0.7 0.57725 0.567775450852 1.5 0.61980 0.608985976591 the authors have compared the present results of nusselt number coefficient with published nusselt number results of hayat et al. [31] in absence of βe, βi, ω, sc, nt, nb, pe, lb. for different values of m, pr, nr, and ε in the table-1. from this table, it is observed that, the comparison results validated the accuracy as well as reliability of our computational approach by demonstrating excellent agreement with earlier studies. hall current and ion slip effects on 3d mhd nanofluid flow… 15 copyright ©2024 assa. adv. in systems science and appl. (2024) 4.results and discussion the purpose of the figures numbered 1 through 19 is to investigate the properties of nondimensional primary and secondary velocity, temperature, concentration, and microorganism profiles across a variety of developmental parameters, including the following: these parameters include the prandtl number (pr), the thermal radiation parameter (nr), the thermophoresis parameter (nt), the brownian motion parameter (nb), the bio-convection lewis number (lb), the peclet number (pe), the microorganism difference parameter (ω), the schmidt number (sc), the erying-powell fluid parameter (ε), the hall current parameter (βi), the ion slip parameter (βe), the stretching sheet parameter (δ), and the magnetic field parameter (m ²). calculated values of three engineering parameters, namely the mass transfer coefficient, the heat transfer rate coefficient, the skin-friction coefficient, and the motile density coefficient, are presented in tables 2, 3, 4, 5, and 6. these values correspond to different values of the developmental parameters that were previously mentioned. it is of utmost importance to acknowledge that, unless otherwise specified in the graphs, the default parameter values that were used throughout the analysis were as follows: m 2 = 0.3, ε = 0.5, δ1 = 0.5, δ2 = 0.5, βe = 0.5, βi = 0.5, δ = 0.1, λ = 0.71, nr = 0.5, nb = 0.3, nt = 0.5, sc = 0.22, lb = 0.5, pe = 0.3, and ω = 0.4. it can be seen in figures 1 and 2 that when the magnetic field parameter increases, the thickness of the boundary layer as well as the amplitudes of the main and secondary velocity profiles decrease. the current phenomenon is the physical outcome that occurs when an electric current travels through a conductive fluid when a magnetic field is present. the velocity of the fluid that is contained inside the boundary layer is therefore slowed down as a result of a drag force. magnetic fields are no longer being used in any way, shape, or form. the main and secondary velocity profiles, as well as the eyringpowell fluid situation, were shown in figures 3 and 4, respectively. according to the findings, there is a correlation between a rise in velocity profiles and an increase in the magnitude of the eyring-powell fluid values. figures 5 and 6 illustrate the impact of the hall parameter (βe) on the velocity profiles of both the main and secondary components. by decreasing the effective electrical conductivity, the hall parameter brings to a reduction in the magnitude of the magnetic resistive lorentz force. because of this, increased fluid dynamics are made possible, which ultimately results in an increase in both the main and secondary velocity profiles, as can be seen in figures 7 and 8. a similar impact is exerted by the ion slip parameter (βi), as may be shown in figures 9 and 10. any rise in the values of the ion slip parameter results in a reduction in the damping or retarding force. the decrease in this damping force results in a rise in both the main and secondary velocity profiles simultaneously. the influence that the stretching ratio parameter has on the secondary velocity profiles is seen in figure 11. it is observed that the secondary velocity profiles increase in proportion to the increasing value of the stretching rate ratio parameter. there is a correlation between the stretching parameter and the flow pressure, which often rises simultaneously. this is seen in figure 12, which shows the temperature of the fluid in proportion to the prandtl number. when the prandtl number is increased, there is a decrease in the temperature gradient that the fluid is experiencing. because of their lower thermal diffusivity, fluids that have a prandtl number that is higher than that of the surrounding fluid exhibit lower temperatures. an illustration of the influence that the thermal radiation parameter, nr, has on temperature profiles may be seen in figure 13. an increase in the thermal radiation parameter results in a decrease in the temperature field. it is possible to see, in figures 14 and 15, respectively, the impact that the brownian motion parameter has on the temperature and concentration curves. according to the findings, an increase in the brownian motion parameter results in a decrease in the concentration profile of nanoparticles and an increase in temperature. brownian motion is the term used to describe the irregular movement of nanoparticles that are suspended in a fluid. this movement occurred as a consequence of interactions with the component particles of the fluid. as thermophoretic activity rises, brownian motion also increases, which leads to an increase in temperature due 16 murali g., tan k. t., sivaneasan b. k., pravin r. k., venkata m. j. copyright ©2024 assa adv. in systems science and appl. (2024) to the kinetic energy that is produced as a consequence. as can be seen in figure 16, the schmidt number, which is a dimensionless quantity that represents the ratio of momentum to mass diffusivity, has an effect on concentration patterns. in the concentration boundary layer, the calculation includes an evaluation of the relative mass and momentum transfer that occurs as a result of diffusion. increasing the schmidt number will result in a decrease in the mass diffusivity of the fluid, which will ultimately lead to a reduction in the concentration. due to the fact that there is a negative connection between mass diffusivity and schmidt number, the concentration boundary layers get thinner as the schmidt number increases. an illustration of the impact that the bioconvection lewis number has on the profiles of microorganisms may be seen in figure 17. the diffusivity of microbial growth diminishes as the lewis number increases owing to the decreasing population of microorganisms. figure 18 illustrates how variations in the peclet number may have an effect on the profiles of microorganisms. for the purpose of evaluating microbial profiles in connection to disease transmission, the peclet number is the major measure that is used. diffusion is the natural movement of particles from regions of higher concentration to regions of lower concentration. as the peclet number increases, the diffusion density of the microorganisms decreases. the pattern shown in figure 19 indicates that the evaluation of the microorganism differential parameter has risen, whilst the value of the microorganism profiles has diminished. fig. 1. the primary velocity profiles for different values of m 2. fig. 2. the secondary velocity profiles for different values of m 2. fig. 3. the primary velocity profiles for different values of ε. hall current and ion slip effects on 3d mhd nanofluid flow… 17 copyright ©2024 assa. adv. in systems science and appl. (2024) fig. 4. the secondary velocity profiles for different values of ε. fig. 5. the primary velocity profiles for different values of βe. fig. 6. the secondary velocity profiles for different values of βe. fig. 7. the primary velocity profiles for different values of βi. 18 murali g., tan k. t., sivaneasan b. k., pravin r. k., venkata m. j. copyright ©2024 assa adv. in systems science and appl. (2024) fig. 8. the secondary velocity profiles for different values of βi. fig. 9. the secondary velocity profiles for different values of δ. fig. 10. the temperature profiles for different values of pr. fig. 11. the temperature profiles for different values of nr. hall current and ion slip effects on 3d mhd nanofluid flow… 19 copyright ©2024 assa. adv. in systems science and appl. (2024) fig. 12. the temperature profiles for different values of nb. fig. 13. the concentration profiles for different values of nb. fig. 14. the temperature profiles for different values of nt. fig. 15. the concentration profiles for different values of nt. 20 murali g., tan k. t., sivaneasan b. k., pravin r. k., venkata m. j. copyright ©2024 assa adv. in systems science and appl. (2024) fig. 16. the concentration profiles for different values of sc. fig. 17. the microorganism profiles for different values of lb. fig. 18. the microorganism profiles for different values of pe. fig. 19. the microorganism profiles for different values of ω. hall current and ion slip effects on 3d mhd nanofluid flow… 21 copyright ©2024 assa. adv. in systems science and appl. (2024) the variations in the values of engineering parameters, including the magnetic field parameter (m 2), the erying-powell fluid parameter (ε), the hall current parameter (βi), the ion slip parameter (βe), the stretching sheet parameter (δ), the prandtl number (pr), the thermal radiation parameter (nr), the thermophoresis parameter (nt), the brownian motion parameter (nb), the bio-convection lewis number (lb), the peclet number (pe), the microorganism difference parameter (ω), and the schmidt number (sc), influence the numerical values of the skin-friction coefficients resulting from primary and secondary velocity profiles in the tables 2 and 3 respectively. the skin-friction coefficients rise with increasing erying-powell fluid parameter (ε), hall current parameter (βi), ion slip parameter (βe), stretching sheet parameter (δ), thermophoresis parameter (nt), and brownian motion parameter (nb), while they decline with increasing magnetic field parameter (m ²), prandtl number (pr), thermal radiation parameter (nr), bio-convection lewis number (lb), peclet number (pe), microorganism difference parameter (ω), and schmidt number (sc). table 4 presents numerical values for the nusselt number, or heat transfer coefficient, with the variable thermal radiation parameter (nr), thermophoresis parameter (nt), and brownian motion parameter (nb) throughout a spectrum of prandtl numbers (pr). the table demonstrates that the heat transfer coefficient progressively rises with elevated thermophoresis (nt) and brownian motion (nb) parameters, while an inverse correlation is observed with increasing prandtl number (pr) and thermal radiation parameter (nr). table 5 examines the influence of the schmidt number (sc), brownian motion parameter (nb), and thermophoresis parameter (nt) on the mass transfer coefficient rate. the mass transfer coefficient is represented by the sherwood number, which corresponds to this table. the table indicates that when the schmidt number (sc) and brownian motion parameter (nb) increase, the mass transfer coefficient diminishes, however the thermophoresis parameter (nt) escalates. the influence of lb, pe and ω on mobile density coefficient is displayed in table6. from this table, it is noticed that the mobile density coefficient is decreasing with rising values of lb, pe and ω. table-2.: computed values of skin-friction coefficient ( cfx ) m 2 ε βi βe δ pr nr nb nt lb pe sc ω cfx 0.3 0.5 0.5 0.5 0.1 0.71 0.5 0.3 0.5 0.5 0.3 0.22 0.4 3.456583525815922 0.5 3.406061268620861 0.7 3.388658682619829 0.8 3.486556276587343 1.2 3.515768202865387 0.8 3.474951668768258 1.2 3.495668082562702 0.7 3.495690226913641 1.0 3.512545858650820 0.2 3.486506133465045 0.3 3.506061156506520 1.00 3.415660441826508 3.00 3.380066805656872 0.8 3.426566252598689 1.2 3.408745636498252 0.6 3.476768467625982 0.9 3.499679336561289 0.7 3.496664206684765 0.9 3.521534510825208 0.8 3.415606252622856 1.3 3.390656309642506 0.5 3.426000109659652 0.7 3.401324843645943 0.30 3.415408658716259 22 murali g., tan k. t., sivaneasan b. k., pravin r. k., venkata m. j. copyright ©2024 assa adv. in systems science and appl. (2024) 0.78 3.370526530927632 0.6 3.425606256702650 0.8 3.406657608764566 table-3.: computed values of skin-friction coefficient ( cfy ) m 2 ε βi βe δ pr nr nb nt lb pe sc ω cfy 0.3 0.5 0.5 0.5 0.1 0.71 0.5 0.3 0.5 0.5 0.3 0.22 0.4 1.352851541825180 0.5 1.326650369060196 0.7 1.306655425952255 0.8 1.376650562106162 1.2 1.390976896576576 0.8 1.386750506916650 1.2 1.402558764556252 0.7 1.396750560165051 1.0 1.413243865483424 0.2 1.386576108650807 0.3 1.406712956641965 1.00 1.315454258528151 3.00 1.296756063590165 0.8 1.326502615760228 1.2 1.306767561057671 0.6 1.375610587618061 0.9 1.396501365196316 0.7 1.385618561672863 0.9 1.405601566583027 0.8 1.315606106059606 1.3 1.285670601656275 0.5 1.330006793647016 0.7 1.315407562576860 0.30 1.305345472383923 0.78 1.284768545982564 0.6 1.314360676452672 0.8 1.296900162572027 table-4.: computed values of rate of heat transfer coefficient pr nb nt nr nu 0.71 0.3 0.5 0.5 1.657056171845669 1.00 1.629752354986598 3.00 1.601407253852805 0.6 1.684502650187268 0.9 1.704558025828522 0.7 1.694524835108735 0.9 1.710764651925378 0.8 1.624645685423892 1.2 1.608576265038681 table-5.: computed values of rate of mass transfer coefficient nb nt sc sh 0.3 0.5 0.22 2.154692359825368 0.6 2.085802530876287 0.9 2.050062352635852 0.7 2.185023654625034 0.9 2.210523854026982 0.30 2.058756087650268 0.78 2.001542083581751 hall current and ion slip effects on 3d mhd nanofluid flow… 23 copyright ©2024 assa. adv. in systems science and appl. (2024) table-6.: computed values of motile density coefficient lb pe ω nh 0.5 0.3 0.4 2.515876257825823 0.8 2.480570257602762 1.3 2.451472085612873 0.5 2.496906962396209 0.7 2.471545585283523 0.6 2.470560162569676 0.8 2.450650276357892 5.conclusions the impacts of the hall current and ion slip current on three-dimensional, incompressible, viscous, electrically conducting, steady, radiating, mhd flow of the nano + powell-erying fluid past a stretching sheet by means of swimming of microorganisms. the concluding remarkable points are represented as of the subsequent.  the study shows that both primary and secondary velocity profiles decrease with an increase in the magnetic field number, while it increases with higher values of the hall parameter, ion slip parameter, powell-erying fluid parameter.  the temperature distribution decreases as the prandtl number and thermal radiation parameter increases but rises with an increase in the thermophoresis parameter and brownian motion parameter.  for concentration, an increase in the thermophoresis parameter enhances the concentration distribution, whereas an increase in the schmidt number and brownian motion parameter decreases it.  finally, the numerical results obtained are ultimately more precise in comparison to the published findings of hayat et al. [45] in absence of βe, βi, ω, sc, nt, nb, pe, lb for different values of m, pr, nr, and ε. acknowldgement this work was conducted as part of my postdoctoral research at the singapore institute of technology (sit). i gratefully acknowledge sit for its support and extend my sincere thanks to my postdoctoral supervisor, dr. tan kaun tak, and the co-pis for their invaluable guidance. i also appreciate the resources provided by the power and energy lab and the stimulating discussions with colleagues. references [1] alamirew, w. d., zergaw, g. a. & gorfie, e. h. (2024). effects of hall, ion slip, viscous dissipation and nonlinear thermal radiation on mhd williamson nanofluid flow past a stretching sheet, international journal of thermofluids, 22, 100646 doi: 10.1016/j.ijft.2024.100646 [2] ramreddy, c., surender, o. & venkata rao, c. 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(2016). effect of hall and ion slip on peristaltic blood flow of eyring powell fluid in a non-uniform porous channel, world journal of modelling and simulation, 12(4), 268-279. [31] babu, n. v. n., murali, g. & bhati, s. m. (2018). casson fluid performance on natural convective dissipative couette flow past an infinite vertically inclined plate filled in porous medium with heat transfer, mhd and hall current effects, international journal of pharmaceutical research, 10(4), 809-819. [32] kirubaharan, d. r., subhashini, a. d. & murali, g. (2024). study of three dimensional casson-nanofluid flow due to a linear porous stretching sheet in the presence of double diffusion effects, advances in systems science and applications, 24(3), 90-103. doi: 10.25728/assa.2024.2024.03.1539 [33] gundagani, m., babu, n. v. n., gadpally, d., bhati, s.m., sanjay, ch, et al. 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(2025). modelling and analysis of three-dimensional maxwell-nanofluid flow over a bi-directional stretching surface in the presence of a magnetic field. international journal for engineering modelling, 38(1), 53-72 doi: 10.31534/engmod.2025.1.ri.04d. hall current and ion slip effects on 3d mhd nanofluid flow… 27 copyright ©2024 assa. adv. in systems science and appl. (2024) [41] singh, n., kaur, j., thakur, p. & murali, g. (2023). structural behaviour of annular isotropic disk made of steel/copper material with gradually varying thickness subjected to internal pressure, structural integrity and life, 23(3), 293-297. [42] sivaiah, s., gundagani, m. & karanamu, m. p. (2012) analysis of heat and mass transfer effects on an isothermal vertical oscillating plate, walailak journal of science and technology, 9(4), 407-415. [43] murali, g., venkata madhu, j., deepa, g., suresh, p. & nagaraju, b. (2025). threedimensional oldroyd-b fluid flow past a stretching surface with magnetic field, nanofluid particles and cattaneo-christov double diffusion effects, johnson matthey technology review, 69(2), 299-320. doi: 10.1595 [44] hayat, t., awais, m. & asghar, s. (2013). radiative effects in a three-dimensional flow of mhd eyring-powell fluid, journal of the egyptian mathematical society, 21, 379-384. doi: 10.1016/j.joems.2013.03.003 adv syst sci appl 2024; 02:19–31 published online at https://ijassa.ipu.ru. piecewise levenberg–marquardt method for generalized nash equilibrium problems alexey izmailov1*, evgeniy uskov2, yan zhibai1 1lomonosov moscow state university, moscow, russia 2 derzhavin tambov state university, tambov, russia abstract: we consider the constrained piecewise levenberg–marquardt method globalized by linesearch, and apply it to “min” reformulations of the optimality systems for generalized nash equilibrium problems. numerical comparison of the performance of this method with some relevant existing alternatives is provided. keywords: generalized nash equilibrium problem, piecewise smooth equation, constrained equation, levenberg–marquardt method, lp-newton method, globalization of convergence 1. introduction this paper aims at application of the globalized constrained piecewise levenberg–marquardt method developed in [24] to “min” reformulations of the optimality systems for generalized nash equilibrium problems (gnep), and comparison with the existing alternatives. in section 2, we recall the statement of a gnep and the related first-order optimality system, and we discuss some possible equivalent reformulations of the latter as constrained systems of equations. section 3 discusses the levenberg–marquardt method for piecewise smooth equations, its linesearch-based globalization, and the related global convergence and superlinear rate of convergence theory. finally, in section 4, we apply this method to the piecewise smooth reformulation of the first-order optimality system for gnep, and compare this algorithm with the usual levenberg–marquardt method for a smooth constrained reformulation, and with the corresponding two versions of the lp-newton method [7] globalized according to [13]. some words about our notation and blanket arrangements are in order. let all the norms be euclidian, unless something different is explicitly specified. for a given y ∈ rm and an index set i ⊂ {1, . . . , m}, we denote by yi the subvector of y with the components yi, i ∈ i . for a sequence {uk} ⊂ rp convergent to some u∗ ∈ rp, the rate of convergence is referred to as superlinear with q-order ν > 1 if there exists c > 0 such that ∥uk+1 − u∗∥ ≤ c∥uk − u∗∥ν for all k large enough. ∗corresponding author: izmaf@cs.msu.ru 20 a. f. izmailov, e. i. uskov, yan zhibai 2. generalized nash equilibrium problem and related constrained equations in a gnep, n players are involved, and each player indexed by ν ∈ {1, . . . , n} controls the variable xν ∈ rnν , and aims at minimizing a smooth objective function fν : rn → r subject to constraints given by a smooth mapping gν : rn → rmν , where n = ∑n ν=1 nν . in particular, both fν and gν may depend not only on xν , but also on the variables of rival players, denoted by x−ν ∈ rn−nν , thus forming the entire vector of variables x = (xν , x−ν) ∈ rn. with this notation, the optimization problem of the ν-th player is written as follows: minimize xν fν(x ν , x−ν) subject to gν(xν , x−ν) ≤ 0. (2.1) the specified problem setting goes back to [30]; recent surveys can be found in [10, 15], including various applications. among other relevant references on theory and numerical methods for gnep are [5, 8, 9, 11, 19–22, 27–29]. gneps form a difficult problem class, in particular, because their solutions are naturally nonisolated. considering x−ν as a parameter, we introduce the lagrangian lν : rnν × rn−nν × rmν → r of (2.1) as lν(x ν , x−ν , λν) = fν(x ν , x−ν) + ⟨λν , gν(xν , x−ν)⟩. then the karush–kuhn–tucker (kkt) system characterizing stationary points and lagrange multipliers of the ν-th player’s optimization problem (2.1) has the form: ∂lν ∂xν (xν , x−ν , λν) = 0, λν ≥ 0, gν(xν , x−ν) ≤ 0, ⟨λν , gν(xν , x−ν)⟩ = 0. concatenating these systems over all players yields the kkt-type system of the gnep: l(x, λ) = 0, λ ≥ 0, g(x) ≤ 0, ⟨λν , gν(x)⟩ = 0, ν = 1, . . . , n, (2.2) where we define λ = (λ1, . . . , λn) ∈ rm, g : rn → rm, g(x) = (g1(x), . . . , gn(x)), with m = ∑n ν=1mν , and l(x, λ) = ( ∂l1 ∂x1 (x1, x−1, λ1), . . . , ∂ln ∂xn (xn , x−n , λn) ) . furthermore, the system (2.2) can be equivalently reformulated as a constrained equation φ(u) = 0, u ∈ p, (2.3) with some mapping φ : rp → rq and a nonempty closed convex set p ⊂ rp. this can be done in many different ways, and we restrict ourselves to the following two adopted, for example, in [13]. in both, p = q = n+ 2m, u = (x, λ, y), with y = (y1, . . . , yn) ∈ rm being a slack variable, and the constraint set is p = rn × rm + × rm + . (2.4) the difference is in how φ is defined. one possibility is to adopt a piecewise smooth “min” reformulation with φ(u) = (l(x, λ), g(x) + y, min{λ, y}) , (2.5) where min is taken componentwise, while another reformulation is smooth, with φ(u) = (l(x, λ), g(x) + y, λ ◦ y) , (2.6) copyright © 2024 assa. adv syst sci appl (2024) piecewise levenberg–marquardt method for gneps 21 employing the hadamard product λ ◦ y = (λ1y1, . . . , λmym). observe that the set p in (2.4) is polyhedral, and moreover, given by simple bounds, and this is the main reason to use slack variables in these reformulations. note also that for φ defined in (2.5), one could actually take p = rn × rm × rm, and (2.3) would still be equivalent to (2.2). however, the nonnegativity constraints on λ and y are needed to ensure the so-called p -property and condition (3.13) that will be essential for the results presented in section 3 below. reformulation employing (2.5) will be of primary interest for us, and to that end, we next recall some terminology related to piecewise smoothness. a mapping φ is refereed to as piecewise smooth if it is continuous, and there exists a finite collection of smooth selection mappings φ1, . . . , φs : rp → rq such that φ(u) ∈ {φ1(u), . . . , φs(u)} ∀u ∈ rp. considering s = 1 recovers the case of a smooth mapping φ. needless to say, a collection of smooth selection mappings corresponding to a given piecewise smooth mapping is not uniquely defined, and we will assume that some such collection is fixed, i.e., φ is defined by a given collection of smooth selection mappings. obviously, the mapping φ defined according to (2.5) is piecewise smooth, with a natural collection of smooth selection mappings φj(u) = ( l(x, λ), g(x) + y, λi(j), y{1, ...,m}\i(j) ) , (2.7) where a one-to-one mapping j 7→ i(j) from {1, . . . , 2m} to the set of all different subsets of {1, . . . , m} is supposed to be fixed. 3. piecewise levenberg–marquardt method getting back to a general piecewise smooth mapping φ, for each u ∈ rp we define the set a(u) = {j ∈ {1, . . . , s} | φ(u) = φj(u)} (3.8) of indices of smooth selection mappings active at u. let g : rp → rq×p be any mapping satisfying g(u) ∈ {(φj)′(u) | j ∈ a(u)} ∀u ∈ rp. (3.9) for a current iterate uk ∈ p , the (constrained) piecewise levenberg–marquardt (lm) method generates the next iterate as uk + vk, where vk is a solution of the problem minimize 1 2 ∥φ(uk) +g(uk)v∥2 + 1 2 σ(uk)∥v∥2 subject to uk + v ∈ p. (3.10) where σ : p → r+ defines the values of the regularization parameter. if σ(uk) > 0, and if p is polyhedral, then (3.10) is a quadratic programming problem with a strongly convex objective function, and in particular this subproblem is uniquely solvable. it follows from (3.8) and (3.9) that the subproblem (3.10) can be written in the form minimize 1 2 ∥φj(uk) + (φj)′(uk)v∥2 + 1 2 σ(uk)∥v∥2 subject to uk + v ∈ p, with some j ∈ a(uk) (that may be different for different k, of course). this means that the iteration of this method coincides with the iteration of the constrained lm method applied to the smooth constrained equation φj(u) = 0, u ∈ p. (3.11) copyright © 2024 assa. adv syst sci appl (2024) 22 a. f. izmailov, e. i. uskov, yan zhibai in the smooth case, the constrained lm method was proposed in [26], and sharp results on its local superlinear convergence were obtained in [1]. the piecewise smooth case was studied in [6, 14], and an overview of these developments can be found in [17]. see also the very recent improvements of the local convergence analysis of the piecewise lm method in [25]. we proceed with an exposition of the globalized piecewise lm method and the related convergence theory developed in [24]. the algorithm below can be regarded as a generalization of the proposal in [18], where the smooth case was considered, and θ = 2 was taken in the algorithm. algorithm 3.1: choose the parameters θ > 0, ε ∈ (0, 1) and κ ∈ (0, 1). choose u0 ∈ p , and set k = 0. 1. if φ(uk) = 0, stop. 2. set σ(uk) = ∥φ(uk)∥θ, and compute vk as the solution of (3.10). if vk = 0, stop. 3. set α = 1. if for j ∈ a(uk) such that g(uk) = (φj)′(uk) (see (3.9)) the inequality ∥φ(uk + αvk)∥2 ≤ ∥φ(uk)∥2 − εσ(uk)α∥vk∥2 (3.12) holds, set αk = α. otherwise replace α by κα and check again the inequality in (3.12), etc., until (3.12) is satisfied, and then set αk = α. 4. set uk+1 = uk + αkv k, increase k by 1 and go to step 1. the key ingredient of the global convergence analysis is the following assumption [24, (1.5)] that appeared before in [13, (4.8)], [16, (32)], and in [3, (0.6)]: ∥φ(u)∥ ≤ ∥φj(u)∥ ∀ j ∈ {1, . . . , s}, ∀u ∈ p. (3.13) theorem 3.1: let φ : rp → rq be a piecewise smooth mapping with continuously differentiable smooth selection mappings φ1, . . . , φs : rp → rq. let p ⊂ rp be a nonempty closed convex set, and assume that (3.13) holds. let g : rp → rq×p be a fixed mapping satisfying (3.9). then algorithm 3.1 uniquely defines the iterates u0, u1, . . ., and either terminates at a point uk ∈ p satisfying ⟨((φj)′(uk))⊤φ(uk), u− uk⟩ ≥ 0 ∀u ∈ p, (3.14) for at least one j ∈ a(uk), or generates an infinite sequence {uk}, and any accumulation point ū of this sequence belongs to p and satisfies ⟨((φj)′(ū))⊤φ(ū), u− ū⟩ ≥ 0 ∀u ∈ p, (3.15) for at least one j ∈ a(ū). we complete this section with a result on asymptotic superlinear convergence rate, and in order to do this, we need the following two assumptions. the p -property at ū, introduced in [14, p. 434], consists of saying that uj ⊂ u near ū for all j ∈ a(ū), where u stands for the solution set of (2.3), while uj stands for the solution set of (3.11). evidently, (3.13) implies the p -property at any solution. another assumption we need is the constrained local lipschitzian error bound for each active selection, that is, ∀ j ∈ a(ū) dist(u, uj) = o(∥φj(u)∥) as u ∈ p tends to ū. (3.16) theorem 3.2: let φ : rp → rq be a given mapping, p ⊂ rp a closed convex set, and ū ∈ u . assume that φ copyright © 2024 assa. adv syst sci appl (2024) piecewise levenberg–marquardt method for gneps 23 is piecewise smooth and the derivatives of its smooth selection mappings φ1, . . . , φs : rp → rq are lipschitz-continuous near ū. let the p -property at ū and condition (3.16) as u ∈ p tends to ū be satisfied. let g : rp → rq×p be a fixed mapping satisfying (3.9). then, if algorithm 3.1 run with θ ∈ (0, 2] generates an iterate close enough to ū, it either terminates with uk ∈ u , or generated an infinite sequence {uk} convergent to some u∗ ∈ u , and the rate of convergence is superlinear with q-order min{θ + 1, 2}. one can easily see that the assumption (3.13), and hence, the p -property at any solution always hold for the gnep kkt-type system reformulation using (2.4) and (2.5). as for the assumption (3.16), some natural sufficient conditions for it were derived in [14, theorems 4, 5], and we adapt them to our problem setting next. proposition 3.1: let fν : rn → r and gν : rn → rmν , ν ∈ {1, . . . , n}, be twice differentiable near x̄ ∈ rn, with their second derivatives being lipschitz-continuous near x̄. assume that ū = (x̄, λ̄) with some λ̄ ∈ rm is a solution of (2.2), where the notation introduces in section 2 for gnep is employed. let φ be defined in (2.5), with its smooth selection mappings φj , j ∈ {1, . . . 2m}, defined according to (2.7), and let p be defined in (2.4). furthermore, define the index set a = {i ∈ {1, . . . , m} | gi(x̄) = 0}, and its partitions a+ = {i ∈ a | ∃ ν ∈ {1, . . . , n} : λ̄ν i > 0}, a0 = a \ a+, and aν + = {i ∈ a | λ̄ν i > 0}, aν 0 = a \ aν +, for every ν ∈ {1, . . . , n}. if for any i ⊂ a0 and any iν ⊂ aν 0 , ν ∈ {1, . . . , n}, the matrices ∂l ∂x (x, λ) ( ∂ga1 +∪i1 ∂x1 (x) )⊤ 0 . . . 0 0 ( ∂ga2 +∪i2 ∂x2 (x) )⊤ . . . 0 ... ... . . . ... 0 . . . . . . ( ∂gan +∪in ∂xn (x) )⊤ g′a+∪i(x) 0  have the same rank for all (x, λ) ∈ rn × rm near (x̄, λ̄), then the piecewise constrained error bound condition (3.16) holds. copyright © 2024 assa. adv syst sci appl (2024) 24 a. f. izmailov, e. i. uskov, yan zhibai corollary 3.1: under the assumptions of proposition 3.1, if the matrix ∂l ∂x (x̄, λ̄) ( ∂ga1 + ∂x1 (x̄) )⊤ 0 . . . 0 0 ( ∂ga2 + ∂x2 (x̄) )⊤ . . . 0 ... ... . . . ... 0 . . . . . . ( ∂gan + ∂xn (x̄) )⊤ g′a(x̄) 0  has full row rank, then the piecewise constrained error bound condition (3.16) holds. the sufficient condition for the piecewise constrained error bound in proposition 3.1 is automatically satisfied if fν is quadratic and gν is affine, for all ν ∈ {1, . . . , n}. observe also that any strict complementarity-like conditions are neither involved in proposition 3.1, nor in its corollary 3.1. 4. numerical results the test set used for the numerical comparisons presented below is a collection of gneps from [5], later also used in [4, 13]. information about these test problems and references to their sources and detailed descriptions are provided in [5]. for each test problem, 20 random starting points were used (the same for all algorithms involved, with random components of x0 distributed uniformly within (0.1, 20), which guarantees that all the functions appearing in all test problems are well-defined at x0. furthermore, λ0 and y0 were chosen according to [13, section 5]. throughout the rest of this section, p is defined according to (2.4). we compared the performance of algorithm 3.1 applied to (2.3) with φ defined in (2.5) (abbreviated below as pwlm) with the following alternatives: • globalized lp-newton method from [13, algorithm 1] supplied with all the performance-improving modification proposed in [13, section 5], and applied to the same instances of (2.3) as pwlm (abbreviated as pwlpn). • the lm method globalized according to algorithm 3.1, but applied to (2.3) with smooth φ defined in (2.6) (abbreviated as lm-had). • globalized lp-newton method, the same as pwlpn, but applied to (2.3) with smooth φ defined in (2.6) (abbreviated as lpn-had). in case of a smooth φ from (2.6), (3.9) implies that g(u) at any u = (x, λ, y) ∈ rp equals φ′(u). for φ from (2.5), the values of a mapping g satisfying (3.9) were computed according to the following rule: for every i ∈ {1, . . . , m}, if λi > yi, then the (n+m+ i)-th row of g(u) is gn+m+i(u) = (0, ei), with ei being the i-th element of the standard basis in rm; otherwise gn+m+i(u) = (ei, 0); the other rows of g(u) are the gradients of the corresponding (smooth) components of φ. the experiments were performed in matlab, with its built-in solver quadprog used for quadratic programming subproblems of pwlm and lm-had, and linprog for copyright © 2024 assa. adv syst sci appl (2024) piecewise levenberg–marquardt method for gneps 25 100 101 0 0.2 0.4 0.6 0.8 1 pwlm 1e-5 pwlm 1e-8 pwlm 1e-10 (a) for “min” reformulation 100 101 0 0.2 0.4 0.6 0.8 1 lm-had 1e-5 lm-had 1e-8 lm-had 1e-10 (b) for hadamard product-based reformulation fig. 4.1. comparison by average iteration counts (the same as evaluations of g) for different values of σ̄. 100 101 0 0.2 0.4 0.6 0.8 1 pwlm 1e-10 pwlpn lm-had 1e-10 lpn-had (a) by average iteration counts 100 101 0 0.2 0.4 0.6 0.8 1 pwlm 1e-10 pwlpn lm-had 1e-10 lpn-had (b) by average evaluations of φ fig. 4.2. comparison of different algorithms. linear programming subproblems of pwlpn and lpn-had, the former run with option ’optimalitytolerance’, 1e-15, and the latter with options ’optimalitytolerance’, 1e-8, ’constrainttolerance’, 1e-8. the parameters in algorithm 3.1: were chosen as follows: θ = 2, ε = 0.001, κ = 0.5. the parameters of the globalized lp-newton methods were chosen according to [13, section 5]. in order to avoid large values of the regularization parameter far from solutions, the rule for it in step 2 of algorithm 3.1 was replaced by σ(uk) = min{σ̄, ∥φ(uk)∥θ}, where several different values of σ̄ > 0 were tied. runs were declared successful when terminated because of ∥φ(uk)∥∞ ≤ 10−6, within 1000 iterations, where for all the algorithms involved, φ defined in (2.5) was used in this stopping test. otherwise, failure was declared, as well as in the cases when the backtracking procedure in step 3 of algorithm 3.1 was producing the trial stepsize parameter copyright © 2024 assa. adv syst sci appl (2024) 26 a. f. izmailov, e. i. uskov, yan zhibai 100 101 0 0.2 0.4 0.6 0.8 1 pwlm 1e-10 pwlpn lm-had 1e-10 lpn-had (a) all test problems 100 101 0 0.2 0.4 0.6 0.8 1 pwlm 1e-10 pwlpn lm-had 1e-10 lpn-had (b) large test problems fig. 4.3. comparison by time. value α ≤ 10−13, and also when ∥((φj)′(uk))⊤φ(uk)∥ ≤ 10−12 was encountered for j ∈ a(uk) such that g(uk) = (φj)′(uk), and in particular, the stationarity condition (3.14) was approximately satisfied. for the lp-newton-based algorithms, similar rules were adopted, but the approximate stationarity test was replaced by |∆(uk)| ≤ 10−12, where the quantity ∆(uk) estimating from above the directional derivative of the merit function is defined according to [13, (2.3)], where γ(uk) is taken the optimal value of the lp-newton subproblem. it turned out that in order to achieve reasonable robustness of the lm-based algorithms, σ̄ has to be taken quite small: larger values lead to long series of short steps far from solutions, and this often ends up with a failure. this happens for lm-had when run for problems a7, heu, tr1b, tr1c, and for pwlm when run for a10e, tr1a. for a10c and a10d, runs of pwlm are generally successful for larger σ̄ as well, but taking smaller values of this parameter reduces the iteration count significantly. our numerical results are presented in the form of performance profiles originally proposed in [2], and later adapted in [23] for the case when multiple starting points are used for every test problem. specifically, for each algorithm a, we present the plot the function πa : [1, +∞) → [0, 1] constructed as follows. let ka τ be the average of some measure of efficiency of algorithm a on problem τ , where the average is taken over successful runs, and let saτ ∈ [0, 1] stand for the portion of successful runs of algorithm a on problem τ . let rτ be the best (say, smallest) value of ka τ among all the algorithms. then for each t ∈ [1, +∞), we set πa(t) = 1 t ∑ τ∈ra(t) saτ , where t the overall number of problems in the test set, and ra(t) is the subset of problems for which the performance of algorithm a is no more than t times worse than that of the best algorithm: ra(t) = {τ ∈ {1, . . . , t} | ka τ ≤ trτ}. in particular, πa(t) for large t is the average portion of successful runs (averaged over all test problems), while πa(1) is the quantity computed in a similar way but using the portions of successful runs only on those problems for which the average performance of the given algorithm over successful runs is the best among all algorithms being tested. in case of a single run for each test problem, this agrees with the original proposal in [2], and in particular, copyright © 2024 assa. adv syst sci appl (2024) piecewise levenberg–marquardt method for gneps 27 table 4.1. average elapsed times (seconds) and percentage of tails of last full steps out of the average iteration counts, over successful runs test pwlm pwlpn lm-had lpn-had time tail time tail time tail time tail a1 0.080 100.0 0.068 100.0 0.016 76.3 0.057 100.0 a2 nan nan 0.044 100.0 nan nan 0.067 100.0 a3 0.003 100.0 0.043 100.0 0.006 100.0 0.066 100.0 a4 0.010 82.8 0.064 98.9 0.011 78.4 0.055 97.1 a5 0.005 100.0 0.042 100.0 0.009 100.0 0.053 100.0 a6 0.014 94.6 0.070 99.7 0.013 90.6 0.067 99.3 a7 0.022 100.0 0.092 100.0 0.018 86.5 0.077 100.0 a8 0.003 100.0 0.025 100.0 0.006 100.0 0.033 100.0 a9a 0.064 75.7 2.508 18.0 0.055 97.7 0.277 70.8 a9b 1.349 35.1 0.612 59.9 0.173 100.0 0.737 77.3 a10a 0.031 49.0 0.101 99.2 0.018 82.3 0.091 98.1 a10b 0.248 75.6 0.466 99.7 0.309 87.3 0.520 95.8 a10c 8.627 10.2 1.612 84.3 nan nan 4.097 81.0 a10d 6.777 36.3 9.110 55.1 2.800 92.3 11.134 60.7 a10e 25.550 37.2 42.698 42.6 nan nan 74.579 34.7 a11 0.004 100.0 0.021 100.0 nan nan 0.029 100.0 a12 0.003 100.0 0.022 100.0 0.003 100.0 0.025 100.0 a13 0.004 100.0 0.027 100.0 0.004 100.0 0.026 100.0 a14 0.010 100.0 0.040 98.9 0.012 81.8 0.049 100.0 a15 0.003 100.0 0.034 100.0 0.004 100.0 0.037 100.0 a16a 0.008 100.0 0.030 100.0 0.004 100.0 0.033 97.9 a16b 0.005 100.0 0.030 100.0 0.004 100.0 0.030 100.0 a16c 0.006 100.0 0.031 100.0 0.005 100.0 0.039 84.0 a16d 0.007 100.0 0.032 100.0 0.007 100.0 0.037 100.0 a17 0.004 100.0 0.024 100.0 0.005 100.0 0.036 100.0 a18 0.007 100.0 0.050 100.0 0.011 94.5 0.053 100.0 harker 0.003 100.0 0.023 100.0 0.006 100.0 0.036 100.0 heu 0.012 100.0 0.053 100.0 0.011 100.0 0.057 100.0 lob 0.068 37.6 nan nan nan nan nan nan ntf1 0.004 100.0 0.022 100.0 0.003 100.0 0.031 100.0 ntf2 0.007 90.5 0.036 99.4 0.006 94.8 0.034 100.0 spam 1380.005 72.7 545.909 95.9 nan nan 166.393 83.7 tr1a 0.059 71.1 0.111 90.4 0.050 94.4 0.071 96.7 tr1b nan nan 1.617 66.8 1.188 43.2 1.636 46.1 tr1c 3.878 26.2 2.245 71.4 2.231 39.7 1.365 62.6 copyright © 2024 assa. adv syst sci appl (2024) 28 a. f. izmailov, e. i. uskov, yan zhibai table 4.2. failures test pwlm pwlpn lm-had lpn-had a1 0 0 0 1 ss a2 20 ss 0 20 (13 sf, 2 il, 5 ss) 0 a3 0 12 sp 1 sf 9 (7 sp, 2 ss) a4 4 (1 sf, 3 ss) 0 4 sf 0 a5 0 0 0 0 a6 2 ss 0 1 sf 0 a7 14 ss 8 sp 0 0 a8 0 0 5 sf 0 a9a 8 (2 sf, 6 ss) 12 il 0 0 a9b 1 il 0 0 0 a10a 0 0 3 sf 0 a10b 0 0 2 (1 sf, 1 ss) 0 a10c 0 0 20 (2 sf, 18 ss) 0 a10d 0 5 (4 il, 1 sp) 19 (8 sf, 11 ss) 2 il a10e 2 (1 sf, 1 ss) 9 il 20 (13 sf, 7 ss) 0 a11 0 0 20 sf 0 a12 0 0 5 sf 0 a13 5 sf 0 0 0 a14 0 0 0 0 a15 0 0 2 sf 0 a16a 1 sf 0 2 sf 0 a16b 2 sf 0 2 sf 0 a16c 1 sf 0 6 sf 0 a16d 0 0 2 sf 0 a17 0 0 2 sf 0 a18 0 0 1 sf 0 harker 0 0 0 0 heu 15 ss 0 6 sf 0 lob 0 20 il 20 ss 20 il ntf1 0 0 4 sf 0 ntf2 0 0 4 sf 0 spam 0 15 sf 20 sf 17 (14 sf, 1 sp, 2 ss) tr1a 16 (9 sf, 7 ss) 2 sp 1 sf 0 tr1b 20 (14 sf, 1 il, 5 ss) 1 sp 12 (9 sf, 3 ss) 0 tr1c 18 (5 sf, 3 il, 10 ss) 4 (3 il, 1 sp) 11 (5 sf, 1 il, 5 ss) 0 πa(t) for large t is the portion of test problems the runs of algorithm a on which are successful, while πa(1) is the portion of problems on which the runs of algorithm a are successful and its efficiency is the best. copyright © 2024 assa. adv syst sci appl (2024) piecewise levenberg–marquardt method for gneps 29 figure 4.1 presents the comparison for three different values of σ̄. figure 4.1a demonstrates the advantages of running pwlm with σ̄ = 10−10, by iteration count, while robustness of all compared variants is nearly the same. in figure 4.1b, the comparison by iteration count is similar, while robustness for this choice of σ̄ is only slightly lower than for the value σ̄ = 10−8. that is why σ̄ = 10−10 was adopted in further experiments. at this point we mention that we have also experimented with different values of the exponent θ in the regularization parameter employed by pwlm. specifically, we tried θ = 1.5 and θ = 1, along with the basic choice θ = 2, but much difference in performance was detected neither by iteration counts not by evaluations of φ. the variant with θ = 1 seriously outperforms the alternatives by the elapsed time, and especially the variant with θ = 2. this phenomenon is explained by the difference in solution times for pwlm subproblems formed with different θ, which shows up mostly for relatively small problems. for large problems (as selected below), the difference in behavior is negligible, by all the criteria, including times, and we kept θ = 2 for the rest of the experiments. in figure 4.2a, we compare the specified versions of pwlm and lm-had with each other, and with pwlpn and lpn-had, with average counts of iterations and evaluations of φ adopted as efficiency measures. pwlm demonstrates robustness somehow lower than that of lp-newton-based methods (thus providing an extra evidence of the use of the latter), but definitely higher than that of lm-had. moreover, the performance of pwlpn and lpn-had is quite similar, while pwlm by far outperforms all the alternatives by efficiency. furthermore, in figure 4.3, we provide the results of comparison by average elapsed time as a measure of efficiency. figure 4.3a is for the entire set of test problems, while in figure 4.3b, only selected large problems were left for comparison, namely, those with more that 100 variables and/or constraints: a9b, a10b, a10c, a10d, a10e, spam, tr1b, tr1c. pwlm outperforms both pwlpn and lpn-had by efficiency on the full test set, although it is somewhat less robust. the picture with robustness remains the same for large problems (and robustness of lm-had appears very low in this case), but now the performance of pwlpn and lpn-had is much improved by efficiency as well, apparently because large linear programming subproblems are solved faster than the quadratic programming subproblems. the detailed information on average times is collected in table 4.1, where the identifiers of large problems are boldfaced. in addition, in that table, we also report on the percentage of tails of last full (with αk = 1) steps of the algorithms in question before successful termination, out of the average iteration counts. the issue of ultimate acceptance of the full step is crucial for superlinear convergence guaranties, like those established for algorithm 3.1 in theorem 3.2. in our experiments, the only cases where the last step was not full were encountered only for lm-had, in 5 runs out of 16 successful for ntf2, and for lpn-had, in 1 run out of 20 successful for a16c. the largest problem in the test set is spam, in which n = 101, n = 2020, m = 4040. it is successfully solved by pwlm, and in a modest number of iterations, while all the other algorithms involved typically fail on it. many failures are caused by inability of quadprog or linprog to solve the corresponding subproblems; for example, this is the case for pwlpn, lm-had, and lpn-had, when run for spam. lm-had also systematically fails for the specified reason for problems a2, a10d, a10e, a11, tr1c. typical failures of other kind are caused by the stepzize parameter becoming too small. problem a2 is also troublesome for pwlm in a different way, with all runs ending up with failures because of a too small stepsize generated, but that was actually caused by convergence to stationary points for an active smooth selection being used, that are not solutions. observe that such scenario is of course not ruled out by theorem 3.1. information about failures and their reasons is summarized in table 4.2 making use of the following abbreviations for failures’ reasons: • sf: solving subproblem failed. • il: failure by iteration limit. • sp: stationary point that is not a solution. copyright © 2024 assa. adv syst sci appl (2024) 30 a. f. izmailov, e. i. uskov, yan zhibai • ss: stepsize too small. at the end, we briefly mention some possibilities to further improve the performance of pwlm for gnep. tuning σ̄ for each problem sometimes gives some positive effect. moreover, alternative rules for σ(·) might be tried, such as σ(u) = σ̄∥φ(u)∥θ/(1 + ∥φ(u)∥θ), or σ(u) = σ̄∥(φ′(u))⊤φ(u)∥θ as proposed in [12], perhaps involving some extra parameters, or some combinations of these rules. the cases when a short step is generated can be handled in a more sophisticated way rather than just declaring failure, like trying some safeguarding steps in such cases. finally, the use of more advanced qp-solvers for subproblems should certainly be helpful, e.g., those applying some pre-processing/scaling. all that will be the subject of future research. acknowledgements the authors thank axel dreves and markus herrich for their assistance with a matlab implementation of the gnep test collection and of the lp-newton method, used in section 4 of this work. this work was funded by the russian science foundation grant 24-21-00015 (https://rscf.ru/en/project/24-21-00015/). references 1. behling, r. & fischer, a. (2012) a unified local convergence analysis of inexact constrained levenberg–marquardt methods, optim. lett., 6, 927–940. 2. dolan, e.d. & more, j.j. (2002) benchmarking optimization software with performance profiles, math. program., 91, 201–213. 3. dorovskikh, d.i., izmailov, a.f. & uskov, e.i. (2024) globalizing convergence of piecewise newton methods, russian universities reports. mathematics, 29, 149–163. 4. dreves, a., facchinei, f., fischer, a. & herrich, m. (2014) a new error bound result for generalized nash equilibrium problems and its algorithmic application, comput. optim. appl., 59, 63–84. 5. dreves, a., facchinei, f., kanzow, c. & sagratella, s. (2011) on the solution of the kkt conditions of generalized nash equilibrium problems, siam j. optim., 21, 1082– 1108. 6. facchinei, f., fischer, a. & herrich, m. (2013) a family of newton methods for nonsmooth constrained systems with nonisolated solutions, math. methods oper. res., 77, 433–443. 7. facchinei, f., fischer, a. & herrich, m. (2014) an lp-newton method: nonsmooth equations, kkt systems, and nonisolated solutions, math. program., 146, 1–36. 8. facchinei, f., fischer, a. & piccialli, v. (2007) on generalized nash games and variational inequalities, oper. res. lett., 35, 159–164. 9. facchinei, f., fischer, a. & piccialli, v. (2009) generalized nash equilibrium problems and newton methods, math. program., 117, 163–194. 10. facchinei, f. & kanzow, c. (2010) generalized nash equilibrium problems, ann. oper. res., 175, 177–211. 11. facchinei, f. & kanzow, c. (2010) penalty methods for the solution of generalized nash equilibrium problems, siam j. optim., 20, 2228–2253. 12. fischer, a. 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(2021) newton-type methods near critical solutions of piecewise smooth nonlinear equations, comput. optim. appl., 80, 587– 615. 17. fischer, a., izmailov, a.f. & solodov, m.v. (2023) the levenberg–marquardt method: an overview of modern convergence theories and more, comput. optim. appl., 2024. 18. fischer, a. & shukla, p.k. (2008) a levenberg–marquardt algorithm for unconstrained multicriteria optimization, oper. res. lett., 36, 643–646. 19. fukushima, m. & pang, j.-s. (2005) quasi-variational inequalities, generalized nash equilibria, and multi-leader-follower games, comput. manag. science, 2, 21–56. 20. harker, p.t. (1991) generalized nash games and quasi-variational inequalities, eur. j. oper. res., 54, 81–94. 21. von heusinger, a., kanzow, c. & fukushima, m. (2012) newton’s method for computing a normalized equilibrium in the generalized nash game through fixed point formulation, math. program., 132, 99–123. 22. izmailov, a.f. & solodov, m.v. (2014) on error bounds and newton-type methods for generalized nash equilibrium problems, comput. optim. appl., 59, 201–218. 23. izmailov, a.f., solodov, m.v. & uskov, e.i. (2015) combining stabilized sqp with the augmented lagrangian algorithm, comput. optim. appl., 62, 405–429. 24. izmailov, a.f., uskov, e.i. & yan zhibai (2024) globalization of convergence of the constrained piecewise levenberg–marquardt method, submitted. 25. izmailov, a.f., uskov, e.i. & yan zhibai (2024) the piecewise levenberg–marquardt method, adv. syst. sci. appl., 24, 29–39. 26. kanzow, c., yamashita, n. & fukushima, m. (2004) levenberg–marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints, j. comput. appl. math., 172, 375–397. 27. kubotam, k. & fukushima, m. (2010) gap function approach to the generalized nash equilibrium problem, j. optim. theory appl., 144, 511–531. 28. kulkarni, a.a. & shanbhag, u.v. 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(1965) existence and uniqueness of equilibrium points for concave n person games, econometrica, 33, 52–534. copyright © 2024 assa. adv syst sci appl (2024) introduction generalized nash equilibrium problem and related constrained equations piecewise levenberg–marquardt method numerical results adv syst sci appl 2022; 03:156–162 published online at https://ijassa.ipu.ru. solvability of control systems with antiperiodic boundary constraints aram v. arutyunov1, zuhra t. zhukovskaya2, sergey e. zhukovskiy2 1v. a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia 2 institute of information transmission problems (kharkevich institute), moscow, russia abstract: in this paper, we consider control systems with implicit dynamics and antiperiodic boundary constraints ont the fixed time interval. we derive sufficient conditions for the existence of admissible control for the systems of this type and obtain estimates of the admissible controls. these results remain considerable in the partial case when the considered dynamics is explicit. keywords: control system, implicit dynamics, antiperiodic boundary constraints, implicit function 1. introduction before proceeding to the formulation of the problem, we introduce the notation used. let rn stand for the n-dimensional real aritheoremetic space with the norm | · | and the inner product ⟨·, ·⟩. for τ > 0, for an absolutely continuous function x : [0, τ ] → rn and for a point t ∈ [0, τ ] at which x is differentiable, we denote by ẋ(t) the derivative of the function x at the point t. denote by ẋ the lebesgue integrable function t 7→ ẋ(t), t ∈ [0, τ ]. everywhere below ln ∞[0, τ ] stands for the set of all lebesgue measurable essentially bounded functions v : [0, τ ] → rn and acn ∞[0, τ ] stands for the set of all absolutely continuous functions x : [0, τ ] → rn such that ẋ ∈ ln ∞[0, τ ]. here and below ∀̇ stands for “for almost all”. let a mapping f : r× rn × rn × rm → rk and a nonempty set u ⊂ rm be given. for every τ > 0 consider the control system f(t, x, ẋ, u(t)) = 0, u(t) ∈ u ∀̇ t ∈ [0, τ ], (1.1) x(0) + x(τ) = 0. (1.2) we will say that a pair of functions (x̄, ū) ∈ acn ∞[0, τ ]× lm ∞[0, τ ] is an admissible process to the problem (1.1), (1.2), if ū(t) ∈ u for almost all t ∈ [0, τ ], the functions x̄(·) is a solution to the differential equation f(t, x, ẋ, ū(t)) = 0 on the segment [0, τ ] and the equality x̄(0) + x̄(τ) = 0 takes place. if a pair of functions (x̄, ū) ∈ acn ∞[0, τ ]× lm ∞[0, τ ] is an admissible process to the problem (1.1), (1.2), then we say that the function x̄ is an admissible trajectory and the function ū is an admissible control. the main goal of our paper is to derive sufficient conditions for the existence of an admissible process to the problem (1.1), (1.2). the proof of the main result is based on the following idea. we pass from the control problem with implicit dynamics to the control problem with explicit dynamics using a nonlocal implicit function theorem; for a function ∗corresponding author: s-e-zhuk@yandex.ru solvability of control systems with antiperiodic boundary constraints 157 ū we consider a mapping φ that corresponds to the initial point x0 the endpoint φ(x0) of a trajectory starting at x0; than we use an analog of brouwer’s fixed point theorem to show the existence of x0 such that x0 = −φ(x0); and prove that the corresponding trajectory is admissible. the idea of reduction of implicit odes (ordinary differential equations) to explicit using implicit function theorems is well known. the usage of nonlocal implicit function theorems to control systems and odes was studied in [1, 3]. the usage of brower’s fixed point theorem and similar results for studying boundary value problems is a standard tool (see, for example, [4]). in this paper, we have combined and modified these known approaches and applied the for studying the problem (1.1), (1.2). 2. main results denote bybn(r) the closed ball in rn centered at zero with the radius r ≥ 0. for an arbitrary linear operator a : rn → rk, denote cova := max{α ≥ 0 : bk(α) ⊂ abn(1)}. it is a straightforward task to ensure that cova > 0 if and only if a is surjective. as is known (see, for example, [2]), that the function cov is continuous. theorem 2.1: let (i) the mapping f be twice continuously differentiable; (ii) ᾱ := inf { covf ′ v(t, x, v, u) : (t, x, v, u) ∈ r1 × rn × rn × u } > 0. given numbers r > 0, α ∈ (0, ᾱ), τ > 0 and functions ū ∈ lm ∞[0, τ ] and w̄(t) := max x∈bn(r) ∣∣∣f(t, x, 0, ū(t))∣∣∣, t ∈ [0, τ ], (2.3) assume that 3 2α τ∫ 0 w̄(s) ds ≤ r. (2.4) then there exists a function x̄ ∈ acn ∞[0, τ ] such that the pair (x̄, ū) is an admissible process to the problem (1.1), (1.2) and |x̄(t)| ≤ r ∀ t ∈ [0, τ ] and |x̄(0)| ≤ 1 2α τ∫ 0 w̄(s) ds. note that the function w̄ in theorem 2.1 is lebesgue integrable. indeed, since f is continuous, then the function (t, u) 7→ max x∈bn(r) ∣∣∣f(t, x, 0, u)∣∣∣, (t, u) ∈ [0, τ ]× u is continuous. moreover, the function ū is lebesgue measurable and essentially bounded. therefore, the function w̄, defined by formula (2.3), is measurable and essentially bounded, hence w̄ is integrable. note that the assumptions (i) and (ii) of theorem 2.1 imply that there exist numbers r > 0, α ∈ (0, ᾱ), τ > 0 and a function ū ∈ lm ∞[0, τ ] such that the inequality (2.4) holds for copyright © 2022 assa. adv syst sci appl (2022) 158 aram v. arutyunov, zuhra t. zhukovskaya, sergey e. zhukovskiy the function w̄ defined by formula (2.3). indeed, take arbitrary numbers r > 0, α ∈ (0, ᾱ) and an arbitrary function ũ ∈ lm ∞([0,+∞)) such that ũ(t) ∈ u for almost all t ≥ 0. since the function τ 7→ τ∫ 0 max x∈bn(r) ∣∣∣f(s, x, 0, ũ(s))∣∣∣ ds, τ ≥ 0 is continuous and vanishes zero at the point τ = 0, then there exists τ > 0 such that 3 2α τ∫ 0 max x∈bn(r) ∣∣∣f(s, x, 0, ũ(s))∣∣∣ ds < r. hence, (2.4) holds for the point τ, for the reduction ū ∈ lm ∞[0, τ ] of ũ to the segment [0, τ ] and for the function w̄ defined by formula (2.3). the regularity assumption (ii) is essential. consider the corresponding example. let n = m = k = 1, f(t, x, ẋ, u) = u(u− 1)ẋ+ x− t2, u = {0, 1}, τ > 0. then ᾱ = 0 and therefore the assumption (ii) fails. moreover, for every measurable essentially bounded function u : [0, τ ] → u, the only solution to the ode f(t, x, ẋ, u(t)) = 0 is the function x(t) = t2. for this function x, we have x(0) + x(τ) > 0. therefore, in this example, there exists no admissible process. now let us derive the sufficient conditions for the existence of an admissible control. the following assertion follows directly from theorem 2.1. corollary 2.1: let the assumptions (i) and (ii) of theorem 2.1 hold, the set u be closed. given numbers r > 0, α ∈ (0, ᾱ), τ > 0 and a function w̄u(t) := min u∈u max x∈bn(r) ∣∣∣f(t, x, 0, u)∣∣∣, t ∈ [0, τ ], assume that 3 2α τ∫ 0 w̄u(s) ds ≤ r. then there exist functions x̄ ∈ acn ∞[0, τ ] and ū ∈ ln ∞[0, τ ] such that the pair (x̄, ū) is an admissible process to the problem (1.1), (1.2) and |x̄(t)| ≤ r ∀ t ∈ [0, τ ] and |x̄(0)| ≤ 1 2α τ∫ 0 w̄u(s) ds. note that both theorem 2.1 and corollary 2.1 remain considerable in the partial case when the dynamics in (1.1) is explicit, i.e. for the problem ẋ = f̃(t, x, u(t)), u(t) ∈ u ∀̇ t ∈ [0, τ ], x(0) + x(τ) = 0. (2.5) in this case the following assertion is valid. let f̃ be twice continuously differentiable and u be closed. given numbers r > 0 and τ > 0 and a function wu(t) := min u∈u max x∈bn(r) ∣∣∣f̃(t, x, u)∣∣∣, t ∈ [0, τ ], copyright © 2022 assa. adv syst sci appl (2022) solvability of control systems with antiperiodic boundary constraints 159 assume that 3 2 τ∫ 0 wu(s) ds < r. then there exist functions x̄ ∈ acn ∞[0, τ ] and ū ∈ ln ∞[0, τ ] such that the pair (x̄, ū) is an admissible process to the problem (2.5) and |x̄(t)| ≤ r ∀ t ∈ [0, τ ], |x̄(0)| ≤ r/3. 3. proof of the main result in the proof of theorem 2.1, along with classical theorems on the properties of solutions to odes and fixed point theorems, we will use the global implicit function theorem from [2]. let us recall it. theorem 3.1: (see [2, theorem 5]) given a positive integer s, an open nonempty set σ ⊂ rs and a twice continuously differentiable mapping f : rn × σ → rk, assume that α0 := inf{covf ′ v(v, σ) > 0 : v ∈ rn, σ ∈ σ} > 0. then there exists a continuously differentiable mapping g : σ× rk → rn such that f (g(y, σ), σ) = y, |g(y, σ)| ≤ |y − f (0, σ)| α0 ∀ y ∈ rk, ∀σ ∈ σ. (3.6) in what follows, we use the following corollary of this theorem. corollary 3.1: let f satisfy the assumptions (i) and (ii) of theorem 2.1. then for any positive α < ᾱ there exists a continuously differentiable mapping g : r1 × rn × u → rn such that f(t, x, g(t, x, u), u) = 0, |g(t, x, u)| ≤ |f(t, x, 0, u)| α ∀ (t, x, u) ∈ r1 × rn × u. (3.7) proof take an arbitrary α < ᾱ. since (i) and (ii) holds, f is sufficiently smooth and the function cov is continuous, then there exists an open neighbourhood u0 ⊂ rm of u such that α ≤ inf { covf ′ v(t, x, v, u) : (t, x, v, u) ∈ r1 × rn × rn × u0 } . (3.8) put σ := r1 × rn × u0. define the mapping f : rn × σ → rk by formula f (v, σ) := f(t, x, v, u) σ := (t, x, u) ∈ σ, v ∈ rn. it is a straightforward task to ensure that f is twice continuously differentiable and f ′ v(v, σ) ≡ f ′ v(t, x, v, u). (3.9) thus, α0 := inf{covf ′ v(v, σ) > 0 : v ∈ rn, σ ∈ σ} (3.9) = (3.9) = inf { covf ′ v(t, x, v, u) : (t, x, v, u) ∈ r1 × rn × rn × u0 } (3.8) ≥ α > 0. (3.10) copyright © 2022 assa. adv syst sci appl (2022) 160 aram v. arutyunov, zuhra t. zhukovskaya, sergey e. zhukovskiy therefore, the mapping f satisfies the assumptions of corollary 3.1. hence, there exists a continuously differentiable mapping g : rk × σ → rn such that (3.6) holds. put g(t, x, u) := g(0, σ), σ = (t, x, u) ∈ r1 × rn × u ⊂ σ. the mapping g is well-defined, since u ⊂ u0. let us show that g is the desired mapping. obviously g is continuously differentiable, since g is. moreover, we have f(t, x, g(t, x, u), u) = f (g(0, σ), σ) (3.6) = 0, |g(t, x, u)| = |g(0, σ)| (3.6) ≤ |f (0, σ)| α0 (3.10) ≤ |f (0, σ)| α = |f(t, x, 0, u)| α for any σ = (t, x, u) ∈ r1 × rn × u. proof of theorem 2.1 take arbitrary numbers r > 0, α ∈ (0, ᾱ), τ > 0, a function ū ∈ lm ∞[0, τ ] and the function w̄ defined by the equality (2.3) such that (2.4) holds. it follows from corollary 3.1 that there exists a continuously differentiable mapping g : r1 × rn × u → rn such that (3.7) holds. put r := 1 2α τ∫ 0 w̄(s) ds. then (2.4) implies 0 ≤ r < r, 1 α τ∫ 0 w̄(s) ds ≤ r− r and − 2r + 1 α τ∫ 0 w̄(s) ds ≤ 0. (3.11) for arbitrary x0 ∈ bn(r), consider the cauchy problem ẋ = g(t, x, ū(t)), t ∈ [0, τ ], x(0) = x0. (3.12) since τ∫ 0 max |x−x0|≤r−r |g(s, x, ū(s))| ds ≤ τ∫ 0 max x∈bn(r) |g(s, x, ū(s))| ds (3.7) ≤ (3.7) ≤ 1 α τ∫ 0 max x∈bn(r) |f(s, x, 0, ū(s))| ds (2.3) = 1 α τ∫ 0 w̄(s) ds (3.11) ≤ r− r, it follows from [6, theorem ii.4.1] that there exists a solution φ(·, x0) : [0, τ ] → bn(r) to the problem (3.12). it follows from [6, theorem ii.4.5] that this solution is unique. it follows from [6, theorem ii.4.11] that the mapping φ(τ, ·) : bn(r) → bn(r) is continuous. define the mapping ψ : bn(r) → rn by formula ψ(x0) := −φ(τ, x0), x0 ∈ bn(r). this mapping is continuous, since φ(τ, ·) is continuous. moreover, the following relation takes place ⟨ψ(x0)− x0, x0⟩ ≤ 0 ∀x0 : |x0| = r. (3.13) copyright © 2022 assa. adv syst sci appl (2022) solvability of control systems with antiperiodic boundary constraints 161 indeed, ψ(x0) = −φ(τ, x0) = −x0 − τ∫ 0 g(s, φ(s, x0), ū(s)) ds (3.14) and, hence, for |x0| = r we obtain ⟨ψ(x0)− x0, x0⟩ (3.14) = −2|x0|2 − 〈 τ∫ 0 g(s, φ(s, x0), ū(s)) ds, x0 〉 ≤ ≤ −2|x0|2 + |x0| τ∫ 0 |g(s, φ(s, x0), 0, ū(s))| ds = −2r2 + r τ∫ 0 |g(s, φ(s, x0), ū(s))| ds (3.7) ≤ (3.7) ≤ −2r2 + r α τ∫ 0 |f(s, φ(s, x0), 0, ū(s))| ds (2.3) ≤ −2r2 + r α τ∫ 0 w̄(s) ds (3.11) ≤ 0. so, it follows from (3.13) that the mapping ψ has a fixed point x̄0 ∈ bn(r), i.e. x̄0 = ψ(x̄0) (see, for example, [5, §1.6]). put x̄(t) := φ(t, x̄0), t ∈ [0, τ ]. let us show that the function x̄ is a desired admissible trajectory. we have f(t, x̄(t), ˙̄x(t), ū(t)) = f(t, x̄(t), φ̇(t, x̄0), ū(t)) = f(t, x̄(t), g(t, x, ū(t)), ū(t)) (3.7) = 0 for almost all t ∈ [0, τ ]. moreover, x̄(0) + x̄(τ) = φ(0, x̄0) + φ(τ, x̄0) = x̄0 − ψ(x̄0) = 0. thus, (x̄, ū) is an admissible process. finally, |x̄(t)| = |φ(t, x̄0)| ≤ r ∀ t ∈ [0, τ ]; |x̄(0)| = |φ̄(0, x̄0)| = |x̄0| ≤ r. therefore, the function x̄ is a desired admissible trajectory. acknowledgements the research is supported by the russian science foundation no 22-11-00042, https://rscf.ru/project/22-11-00042/. references 1. arutyunov, a.v., zhukovskaya, z.t., & zhukovskiy, s.e. (2022) on the existence of admissible positional controls for systems with mixed constraints. 2022 16th international conference on stability and oscillations of nonlinear control systems (pyatnitskiy’s conference), 1–3. 2. arutyunov, a.v. & zhukovskiy, s.e. (2019) application of methods of ordinary differential equations to global inverse function theorems. differ. equations, 55, 437– 448. copyright © 2022 assa. adv syst sci appl (2022) 162 aram v. arutyunov, zuhra t. zhukovskaya, sergey e. zhukovskiy 3. avakov, e.r., arutyunov, a.v., & zhukovskiy, e.s. (2009) covering mappings and their applications to differential equations unsolved for the derivative. differ. equations 45(5), 627–649. 4. krasnosel’skii, m.a. & zabreiko, p.p. (1984) geometrical methods of nonlinear analysis. springer-verlag, berlin. 5. nirenberg, l. (2001) topics in nonlinear functional analysis, courant institute of mathematical sciences. 6. warga, j. (2014) optimal control of differential and functional equations, academic press, new york. copyright © 2022 assa. adv syst sci appl (2022) introduction main results proof of the main result microsoft word 1331_done adv syst sci appl 2023; 02; 79-98 published online at https://ijassa.ipu.ru. analysis of consumer behaviour related to electricity consumption jana héjjová1, jozef bucko1*, emil exenberger2 1) department of applied mathematics and business informatics, faculty of economics, technical university of košice, košice, slovakia 2) henkel slovensko, bratislava, slovakia abstract: almost three-quarters of carbon dioxide emissions originate from households' activities and among the major contributors to climate change is the energy sector. therefore, increased attention is being paid to energy-efficient behaviour in the household segment. over the past decades, a limited number of studies focused on the in-depth understanding of consumer behavior within energy consumption and on comparison of consumers’ own perceptions towards their effectiveness in energy consumption to the reality substantiated by the precisely measurable amounts of energy they have been consuming. this study extends the authors' previous research on changes in consumer behavior, triggered by access to the personalized simplified and userfriendly data related to the energy consumption of a particular household. the focus is on describing the behavior patterns in electricity consumption related to consumers’ age and size of agglomeration they inhabit. the study includes data on 3 years' consumption of over 30 000 households from slovakia. it provides conclusions relevant to the area of the central european region at least, as among the main factors influencing the energy consumption are weather and dwelling conditions. keywords: energy efficiency, disaggregation, behavioral principles, consumer behavior jel code: c12, m31, m37 1. introduction energy production is one of the major factors contributing to climate change as most of the current energy supplies originate from fossil fuels. according to the data from march 2022 [11], 62 % of the world’s electricity have been produced by using fossil fuels. the largest source of electricity is currently coal (36%), which is responsible for a quarter of all energyrelated co2 emissions and therefore significantly threatens humans’ ability to stabilize the climate. combined, clean electricity sources generate 38% of the world’s demand – including hydro (15%), nuclear (10%), wind (7%) and solar (4%) power. it means that fossil power has a downward trend as together wind and solar have generated more than 10% of the world’s electricity, which is twice the share compared to the year 2015 when the paris climate agreement was signed. the urgency of actions towards preventing or at least slowing down climate change also reflects in prioritizing the topic of global energy efficiency improvement by the united nations (un) as one of the three major objectives of their sustainable energy for all (se4all) initiative (un 2020). the aim is to keep the rise in temperature within 2°c by 2050 and 1,5°c by 2100, considering the rapid economic and population growth [29]. and here comes the tricky part, as humans have become heavily dependent on energy consumption to maintain sustainable economic development, which leads to further increase in the world's demand for energy [3, 22]. electricity demand continues to climb, growing by * corresponding author: jozef.bucko@tuke.sk 80 j. héjjová, j.bucko, e. exenberger copyright ©2023 assa adv. in systems science and appl. (2023) 5.4% in 2021 and achieving a new record [11]. due to the complexity of the issue, there is no single universally and effectively applicable policy instrument for environmental protection and sustainable development, which would result in effective mitigation of co2 emissions [15]. there have been numerous initiatives investigating options for changing the course of climate change. along with the initiatives to increase the level of energy produced by renewable resources, also energy conservation and energy efficiency are priorities for governments worldwide. over the past decade, they have motivated intensive research to understand how energy is consumed, and how to translate that knowledge into meaningful information to enable energy consumers to take responsibility over their energy consumption [31] and decrease it or in other words improve their energy efficiency. to achieve the goals, the activities and initiatives need to be targeted properly. as surprising can be perceived the fact that not the biggest production companies, but household sector is a major contributor to greenhouse gas emissions, because of the consumption of a wide range of goods and services. households account for more than 60 % of global greenhouse emissions. to achieve the targets from the paris agreement and to maintain the global average temperature rise below 2 °c plus to further limit it to 1.5 °c above preindustrial levels, is unrealistic without changing the consumption behaviour of household sector [21]. the share of home appliances such as refrigerators, televisions, and air conditioners accounts for around 50 % of the emissions. this effect would be explained as consequence of the regulations on the amounts of co2 emissions produced by the corporate sector. the limits have forced producers to control their emissions and invest in technologies minimizing the impact of their activity on the environment to avoid the fees for exceeding these limits. policymakers in the united states and european union are paying increased attention to residential and commercial buildings as a potential source of major reductions in energy consumption that will help meet that target. the building sectors together account for around 40 % of total energy consumption in these nations and are well understood to have huge potential for cost-effective energy savings that can both reduce greenhouse gas emissions and improve the reliability of the electric grid [20]. over the past 50 years, a limited number of studies have attempted to better understand consumer perceptions of their utility use and their possible effects on consumer decisionmaking. an even smaller number of studies compared consumers' perceptions to actual energy use or savings [19]. this study links up to research on behaviour changes of consumers based on a simple access to disaggregated data related to their energy consumption presented in a transparent and easily understandable version [17]. we focus on describing the behavioural patterns in electricity consumption related to age and size of agglomeration. the study includes consumers from slovakia. as among the main factors influencing the energy consumption is weather and dwelling conditions, the results of the study have the potential to effectively cover the central european region with similar climate conditions and household management style. 2. the current state overview burning the fossil fuels to generate electricity belongs to the major factors causing environmental deterioration. a pleasant ecological environment is not only essential for the good quality of life but supports the economic development of nations as well [34]. among the approaches considered to be relevant in improving the undesirable development is the implementation of policies resulting in changes in consumers' behaviour related to energy consumption. as household energy consumers produce the most of carbon dioxide emissions, they represent the high relevancy target group within activities aimed at increasing energy efficiency. analysis of consumer behaviour related to electricity consumption 81 copyright ©2023 assa. adv. in systems science and appl. (2023) when knowing less is for worse many strategies for realizing potential savings (e.g.: product labelling programs, subsidies for the purchase of efficient devices, behavioural programs that encourage efficient energy use, etc.) rely on the assumption that the buildings’ owners and end users make informed investment and operational decisions [20]. if the precondition of an informed consumer isn't fulfilled, these strategies would probably be ineffective. unfortunately, studies have revealed the presence of several economic, technological, and organizational barriers, which influence the level of consumer awareness and result in poor implementation rates of energy efficiency measures. [18]. as the example could serve the study by attari [1], which confirms differences between consumers' perception and reality. it presents, that consumers tend to overestimate the energy consumed by their low energy-consuming devices and slightly underestimate the amounts used by their most energy-consuming devices. they mostly associate energy savings with reducing the operating time of the devices. understanding consumers' misperceptions can help to design the policies and information campaigns, that will support consumers to make their homes consume less energy. [20]. how to navigate human’s autopilot to changes in energy consumption? households’ energy consumption has a large saving potential estimated up to 27% of their current energy use, which can be saved through more efficient energy use [37]. this is closely related to sustainable changes in consumers' behaviour within energy consumption that can be effectively nudged by respecting behavioural principles in designing the energy efficiency policies and campaigns [7]: a) habits. it is more difficult to change daily routines requiring little or no cognitive effort, which is of significant relevancy in the case of electricity use. b) rewards. the change is conditioned by reward, ideally shortly after the action. in the case of energy efficiency an example is easy access to the data on consumption, which is now available thanks to disaggregation technologies analysing the aggregate household main power measurement in the house and disaggregating this into individual appliances [9]. c) consequences. the consequences of the change are important to us. in this field, no. 1 is an electricity bill. the first-hand output from smart meters is disaggregated data presented in the kilowatt-hours (kwh) consumed. the useful insight for designing the disaggregation report comes from the study by hargreaves [16] examining the communications of energy consumption. it provides the finding that consumers prefer their energy use to be presented in monetary units (e.g. euros) instead of kwh or carbon dioxide emissions. confirming such insight is useful in decision making, whether to put extra effort into transforming the data from disaggregation into monetary units to improve the user experience and the outcome of the steps supporting the behaviour towards energy efficiency. d) involvement. people need to feel involved and effective to make a change. in case of electricity consumption, accessibility of information and complexity can lead to the feeling of helplessness and inaction. the solution is again energy disaggregation, nicknamed the holy grail of energy efficiency as you can hardly improve something, that you cannot measure [14]. currently, there have been applications linked with disaggregation technology, which can produce a user-friendly regular report with the status of consumption in kwh and tips on how to lower it. e) approval. humans' behaviour is strongly influenced by other people's behaviour. an example of the tool enabling common natural influence among consumers is benchmarking similar consumers to present the effective level of consumption and support the feasibility of recommendations. studies on barriers to energy efficiency in sweden and other eu countries within the segment of small and medium enterprises explain that the main obstacles are: lack of time, other business priorities, lack of knowledge, lack of information, lack of staff, high 82 j. héjjová, j.bucko, e. exenberger copyright ©2023 assa adv. in systems science and appl. (2023) investment, etc. [18]. networking platforms could help to overcome these barriers by combining: a) energy audits using behavioural principle of providing the control over the result, and lectures on energy efficiency reflecting the individual needs to feel actively involved in the change, b) consultancy with experts meeting the principle of the authority, c) networking enabling to share the knowledge and providing the social proof [5] in meaning ‘if others do it, i should also do it’, ‘i want to be among energy-efficient ones’ or ‘if it works for others, it will work for me’. friendly data as facilitator of desired changes in behaviour this study provides insight into the consumers' behaviour within electricity consumption based on the data from bidgely technology, which currently belongs to major current disaggregation techniques. the service related to bidgely includes obtaining regular reports on household consumption and recommendations on the household management areas such as space heating, refrigeration, lighting, and always on appliances. disaggregated information itself has two major impacts [14]: • increased consumer engagement, • reduced energy usage. it additionally provides a great number of data, which is a valuable asset for designing effective informational and educational campaigns. disaggregation technology can serve as a tool to reveal customer information that is relevant to optimizing energy efficiency programs of households. according to the study by beckel et al., it is possible to infer 8 of the 18 characteristics of a household with an accuracy between 72 % and 82 %. the characteristics include for example number of inhabitants, the number of adults and children, and employment status (retired, employed, etc.). then with accuracy, around 60 % follow characteristics such as the age of the house, type of the house, or income. [2]. reports on disaggregation can be considered as part of digital marketing, which is most successful when personalized and well-targeted. the mailing list and targeting are considered to be the most important in terms of communication effectiveness. the impact of the mailing list's relevancy on the result of a campaign reaches 40 %, followed by an offer with 30 %, then layout and copy both presented by 15 % impact [4]. therefore, the knowledge of the target group is essential. focus on age and type of agglomeration this study points at the particular areas of households' electricity consumption and investigates how households’ consumption is influenced by their lifestyle shaped by: 1. the age of the property's owner; 2. the size of agglomeration where the property is located. based on the widely-accepted belief that aging is a multi-dimensional process and older people are likely to experience the various aging processes differently due to the wide variability across individuals [33], the age needs to be considered within the complex of factors, as changes in behaviour during a person's life are the results of adaptation and adjustments to life changes, not just aging itself. however, the age is easily accessible input being provided within standard interaction between supplier and consumer with the potential to cluster customers with similar needs into a reasonable number of target groups. examination of consumer responses to marketer-controllable variables (e.g., pricing, sales promotions) of companies in different countries and specific industries, and how the various gerontographic segments respond to these variables, could provide useful guidelines for strategy development, as it has in the united states [25, 26]. another interesting and easily accessible factor influencing the electricity consumers' behaviour is the size of agglomeration, where the dwelling is located [23]. due to the study analysis of consumer behaviour related to electricity consumption 83 copyright ©2023 assa. adv. in systems science and appl. (2023) by buček [5] the size of agglomeration impacts lifestyle in terms of commuting and economic structure. the smaller the village is, the probability of prevailing residential over production character is higher. this effect would be strengthened by the geographic proximity of larger agglomerations. the bigger the city or town is, the services sector becomes more significant. another study by debasish, prusty [8] presents findings that social norms, traditions, caste, and social customs have a greater influence on consumer behaviour in rural areas than in urban ones. their study indicates that rural consumers mostly use friends (45%), tv (17%), and mobile phone retailer (12%) as the source of information. the purchase decision is taken by self-decision (52%), with the help of family (29%) and friends (18%). this knowledge impacts the choice of the channel to effectively deliver the information. the research uses data about customers in slovakia. currently, there is no limit towards volume of energy consumption for households in slovakia. however, among governmental proposals for actions to deal with energy crisis, which has started in the end of 2021 and has been deepened by impact of war in ukraine on the energy market, an idea related to consumption restrictions appeared. the proposal suggested that from the year 2023, 85 % of individual household’s electricity consumption (based on the volume in previous period) would be subsidized by state, billed in prices valid in 2022 and consumption above the limit would be billed in higher prices [35]. this proposal didn’t enter into force at last, but the idea has already appeared. currently the activities leading to decrease in consumption are more of an educative character towards energy efficiency. there are different electricity tariffs available for households. in case of households with the above average electricity consumption, usually over 4 mwh/year, the two tariff products with lower price of electricity during night hours are applied, which means that altering the operating time of selected devices would impact household’s electricity bill. due to the knowledge of the country-of-origin aspect, the country where the product or service is produced impacts consumer behaviour and decision making [24]. when a product or service is complex such as services related to house management, the consumer is more likely to incorporate the country-of-origin aspect into their decisions [36]. therefore, we consider the results of the study to have the potential of effectively covering at least the area of the central european region regarding the common history and presence of several common cultural features. market globalization and the appearance of similarities in consumers' behaviour have been supported in the last few decades due to factors, such as the collapse of communism, the global opening of previously closed markets in india, china, eastern europe, and latin america, worldwide investment and production, increases in world travel and consumer sophistication, advances in telecommunication technologies, and the growth of global media [32]. 3. dataset description and methodology dataset description and hypothesis we examined the datasets ‘ov_data’ and ‘peer_insight’, which contain the data from smart meters of electricity consumers. the smart meter measures the progress of consumers' consumption electricity continuously within various time intervals. bidgely application recognizes the consumption of individual appliances separately due to their specific electric stamp. bidgely records these data on each individual consumer into above listed datasets. the datasets include consumers meeting the legislative criteria for an obligatory instalment of smart meters, which has been proceeded in slovakia sequentially since 2014. the consumers with smart meters have been offered a free application bidgely and the datasets include the ones, who have activated it. the data have been anonymized and have the following structure: 84 j. héjjová, j.bucko, e. exenberger copyright ©2023 assa adv. in systems science and appl. (2023) ov_data contains attributes: ov (char), eic (char), bidgely_start_date (date), bidgely_end_date (date), consumption (int), age (num), postcode (char), town (char). peer_insight contains attributes: ov (character), etag (character), id_bidgely (char), id_user (char), id_appliance (char), type (char), start_date (date), end_date (date), last_modified (date), actual_usage_kwh (num), benchmark_usage_kwh (num), peer_radius_km (num), peer_user_count (int), size_bytes (int). we merged the datasets in the r-studio environment using the "ov" attribute, which is common for both datasets. it provided comprehensive information about customers and their consumption trends for individual appliances during the monitored period. the datasets examined, contain parameters that were not relevant for our research, therefore they were removed before processing as well as records with incomplete and extreme values. the output contained data of 33 095 electricity consumers within the household segment in slovakia. the data analysed include the electricity consumption of the consumers during the period from june 2018 till august 2021. these consumers are primarily households using electricity for heating or equipped with electric water boilers. this study examines customer behaviour related to electricity use in the following particular areas of the consumption: • total consumption, • lighting, • water heating, • air conditioning, • space heating, • refrigeration, • always on appliances. it focuses on the behaviour of households divided into categories due to their age as well as the size of agglomeration, where the property is located and compares their consumption. the dataset contains over 3 mils. records on the electricity consumption in the above-listed categories. our study links to research, which examined whether the increase of individuals' control over electricity consumption based on the access to personalized energy disaggregation reports leads to the changes in their consumption patterns [17]. the results confirmed the hypothesis that consumers' access to customized analysis of their electricity consumption, influences their behaviour and has an impact on the decrease of the electricity consumption. another conclusion has been, that there are differences in the behaviour of consumers related to energy efficiency due to the demographic factor of age. this study dealt with the households’ total consumptions, which in this case has been categorized into particular areas of electricity consumption to get more detailed picture needed to support the effectiveness of energy efficiency initiatives. the motivation of our research was to provide in depth information on electricity consumption patterns of individual customers due to the demographic factor of age and the size of agglomeration, where the consumer lives. we established the following hypothesis and subsequently the research question. the study examines the hypothesis: hypothesis 1 (h1): there is a statistically significant difference in electricity consumption within particular areas due to the demographic factor of age. following the proof of the h1, the study continues to examine the consumption in relation to the size of agglomeration, where the property is located. within the framework of the established hypothesis, we will analyze the correlation between the electricity consumption expressed in units of kwh and age. to analyse the correlation between these numerical variables, we use the pearson correlation coefficient test [28]. analysis of consumer behaviour related to electricity consumption 85 copyright ©2023 assa. adv. in systems science and appl. (2023) hypothesis h0: "there is no statistically significant difference in electricity consumption within particular areas due to the demographic factor of age.", we will test at a significance level of alpha = 0.01. if the p-value for a given consumption category is lower than 0.01, then h0 will be rejected, which means that there is a statistical relationship between age and consumption within the given consumption category. if a statistical relationship between consumption and age will be proven, we will determine the nature of this relationship based on the pearson correlation coefficient (r), which will take on values from -1 to 1. a positive pearson correlation coefficient means positive linear relationship; a negative value means negative linear relationship. the higher the absolute value of the pearson correlation coefficient, the higher the linear relationship between the variables. based on results, the analysis of consumers’ average electricity consumption will follow. the analysis will focus on comparing consumption in three areas:  analysing electricity consumption in particular categories;  analysing consumers' consumption related to age;  analysing consumers' consumption related to the size of the agglomeration. methodology of analysing electricity consumption in particular categories the analysis compares consumers' monthly average electricity use within the categories of total consumption, lighting, water heating, air conditioning, space heating, refrigeration, and always on appliances. even the average monthly consumption differs within the year because of the weather, which increases electricity consumption mainly due to the exterior temperatures (heating in case of low temperatures, air-conditioning in case of high exterior temperatures), for the purpose of our study, this aspect has been considered irrelevant as the study focuses on analyzing and comparing differences among different demography groups, not on electricity consumption trend in particular months throughout the year. methodology for analysing consumers’ consumption related to age following our previous research providing an outcome, that consumers' behaviour within electricity consumption differs by age, we've analysed how age impacts the consumption in particular areas. we divided consumers into 3 age categories: a) 18 34 years b) 35 – 54 years c) 55 – 70 years the categorization has been realized based on the following:  the commodity consumption is usually billed to the property owner, which is restricted by the age of 18 years, when a person is considered an adult.  due to the study of different age groups' behaviour related to media consumption, this categorization reflects the lifestyle of the particular groups and the way they acquire information. it influences the selection of effective tools and relevant content to deliver messages related to education in energy efficiency, which is among the practical implications of our research.  our categorization also reflects the current society's lifecycle by considering the age when people start managing households of their own. in european union states, it is between 17,5 to 32,4 years [12].  the insight listed above, impacts interval of the third category from 55 to 70 years. we assume that at this point of our lifecycle decreases the number of households' permanent residents, which influences the household management including the activities related to energy consumption. it is established on the data on the european union population starting a family in the age between 26 to 31 years [12] combined with the fact that the average age of leaving parental household is 26 years. 86 j. héjjová, j.bucko, e. exenberger copyright ©2023 assa adv. in systems science and appl. (2023)  consumers aged above 70 years were not included in the dataset analysed as the decision-making process related to household management is realized with the strong influence of children or grandchildren of the property's owner. therefore, online communication related to energy efficiency is targeted to people aged under 70. the reason also is, that the genuine dataset included consumers over 100 or 110 years, where we assume that the owner is deceased and the formal transfer of property hasn't been completed, the decision-maker is a relative. methodology for analyzing consumers’ consumption related to the size of the agglomeration to categorize the agglomerations examined in the research, the degree of urbanisation (degurba) was applied, as a classification method developed to facilitate international comparisons. the method is acknowledged by the un statistical commission [10]. degurba classification indicates the character of an area and is based on the share of the local population living in urban clusters and urban centres. it classifies local administrative units into three types of areas [13]: cities (densely populated areas), towns and suburbs (intermediate density areas), and rural areas (thinly populated areas). 1. city is defined as an urban centre consisting of contiguous grid cells of 1 km2 that have a density of at least 1500 inhabitants per km2. the contiguous cells should have a total population of at least 50,000. gaps in these urban centres are filled and edges are smoothed. 2. town and semi-dense areas are an urban cluster, that consists of contiguous grid cells with a density of at least 300 inhabitants per km2. the contiguous cells should have a total population of at least 5000. 3. rural area is thinly populated area outside urban clusters. based on the above-described approach, agglomerations included in the research were divided into 3 following categories: 1. rural size (no. of inhabitants) up to 5000 2. town size (no. of inhabitants) 5001 – 50 000 3. city size (no. of inhabitants) over 50 000 4. results this research has focused on analysing customer behaviour related to electricity use in individual areas of electricity consumption: total consumption, lighting, water heating, air conditioning, space heating, refrigeration, and always on. the 1st part of the study has analysed the relationship between consumers' behaviour within energy efficiency in particular consumption categories and the demography factor of age. we attempted to confirm hypothesis 1 stating, that there are differences in the behaviour of consumers related to energy efficiency within particular consumption categories due to the demographic factor of age. the null hypothesis: there is no dependency between the variable of age and the amount of electricity consumption, was measured by employing the linear relation between the two scale variables. the pearson correlation coefficient was used [28] and according to its values presented in table, the null hypothesis was rejected and the alternative hypothesis on the dependency between variables was confirmed. analysis of consumer behaviour related to electricity consumption 87 copyright ©2023 assa. adv. in systems science and appl. (2023) table 4.1. test of dependency between consumptions within categories and age consumption category pearson correlation coefficient p-value alpha hypothesis result space heating -0.11266 <0.001 0.01 rejected consumption depends on age refrigeration 0.09867 <0.001 0.01 rejected consumption depends on age air conditioning -0.07248 4.16e-67 0.01 rejected consumption depends on age lighting 0.05971 <0.001 0.01 rejected consumption depends on age always on 0.01983 1.96e-3 0.01 rejected consumption depends on age water heating 1.46e-01 0.01 not rejected consumption does not depend on age pool 1.22e-01 0.01 not rejected consumption does not depend on age the results show the negative linear relationship between age and amount of electricity consumption in the categories of space heating and air conditioning, which are among 3 categories with the highest average level of consumption. space heating is the first, followed by refrigeration and air conditioning. water heating and pool are on a very similar level and there is not dependence on age. these two categories differ in terms of consumers' perception of necessity. water heating is in central european climate conditions closer to basic needs in comparison to air conditioning, which is considered to be an appliance providing extra comfort. the tendency of spending less electricity in the categories with the highest average level of consumption suggests higher cost sensitivity with the increasing age of consumers. the results on categories related to physical presence in the property such as lightning or always on show the positive linear relationship between age and the amount of electricity consumption. it reflects the more active lifestyle of younger categories, on the other hand, lightning and always on belong to the categories with the lowest average consumption and therefore lowest impact on the cost of electricity consumption. in the 2nd part of the study, we've attempted to extract more detailed insight into consumers' behaviour, which would support the potential of energy efficiency initiatives by improved targeting. therefore, we've compared the areas with more and less significant differences in consumption behaviour. consumption areas, where obvious differences in consumption have occurred among age and location categories, are dependent on the amount of time that residents spend in the household. to achieve a higher tangibility of the results, the differences in selected cases are presented in euros. as appliances' consumption varies based on their energy class or model, the middle values were applied. the price in the table is the maximum commodity price for the households in slovakia for the year 2022, set by the state regulatory authority. the price in the table does not include the fixed cost related to electricity supply e.g. distribution fees. table 4.2. the average electricity consumption of appliances and price in slovakia for 2022 appliance average electricity consumption (kwh)* cost to use per hour (eur) electricity cost per kwh (commodity cost)** fridge with freezer 0,3 0,0183 washing machine 2 0,122 dishwasher 1,25 0,07625 lcd tv 0,16 0,00976 0,061 kwh air conditioning 3,5 0,2135 led bulb 60 w 0,01 0,00061 water heater 2 0,122 we have visualized the results through the following figures exported from pivot tables. 88 j. héjjová, j.bucko, e. exenberger copyright ©2023 assa adv. in systems science and appl. (2023) total consumption average agglomeration category age category rural area town city average value young 410,8296727 338,611025 383,832007 377,7575681 middle 472,7231584 439,700794 473,22596 461,8833039 older 463,2863979 436,822543 445,503892 448,537611 average value 448,9464097 405,044787 434,187286 429,3928277 fig. 4.1. total consumption (kwh) consumers in rural areas have the highest average electricity consumption, followed by consumers in the cities. the lowest level of average electricity consumption is observed within the group living in the towns.  this order applies to categories of young and older consumers. in the category of middle-aged consumers, consumption in rural areas and cities reaches a similar level.  the highest consumption in all the types of agglomeration relates to middle-aged consumers (35 – 54 years). lighting consumption average agglomeration category age category rural area town city average value young 27,78205703 23,39657635 23,56821858 24,91561732 middle 38,2736304 36,68005287 34,96266095 36,63878141 older 39,60238301 37,98430817 36,71322331 38,0999715 average value 35,21935681 32,68697913 31,74803428 33,21812341 analysis of consumer behaviour related to electricity consumption 89 copyright ©2023 assa. adv. in systems science and appl. (2023) fig. 4.2. lighting consumption (kwh)  electricity consumption on lighting is dependent on the consumer's physical presence on the property.  in each type of agglomeration, the biggest electricity consumption related to lighting is in the category of consumers aged 55 to 70 years (older), which establishes the dependency on a lifestyle of this category, which spends more time at their homes.  in all age categories, consumption in rural areas is higher than in towns and cities. the difference reaches 3 to 4 kwh in average monthly consumption when comparing rural areas to cities. to increase the tangibility of the result, the following example would serve: o led bulb consumption: 0,01 kwh o using lamp with 3 lightbulbs: 0,03 kwh o 3 – 4 kwh difference in monthly consumption represents 100 hours of lighting monthly or approximately 3 hours daily.  this supports the claim that different living habits and behavioural routines impact energy consumption. consumers in bigger agglomerations have easier access to a wider spectre of out-of-home activities, which impacts the amount of their time spent on the property. there is a significant difference between the group of "young" consumers in comparison to categories of "middle" and "older" consumers. it is represented by the amount of 10 – 14 kwh in monthly average consumption, which equals approximately 250 to 500 hours monthly (based on the example above). water heating consumption average agglomeration category age category rural area town city average value young 124,5838846 123,0391415 141,5799019 129,7343093 middle 123,6553438 130,463145 128,8536727 127,6573872 older 126,9154341 126,821314 134,834856 129,523868 average value 125,0515542 126,7745335 135,0894769 128,9718548 90 j. héjjová, j.bucko, e. exenberger copyright ©2023 assa adv. in systems science and appl. (2023) fig. 4.3. water heating consumption (kwh)  the average electricity consumption for water heating is on a similar level. in rural areas and towns. it is higher by 7 kwh per month in the cities in comparison to rural areas and towns, but this difference represents heating of approximately 120 liters of hot water which equals, for example, one bathtub, therefore does not reflect the significant difference in consumption behaviour. air conditioning average agglomeration category age category rural area town city average value young 112,372785 111,3322813 109,3141469 111,0064044 middle 123,3632023 127,2176276 133,2868927 127,9559075 older 113,377211 121,3758597 126,5448923 120,4326543 average value 116,3710661 119,9752562 123,0486439 119,7983221 fig. 4.4. air conditioning consumption (kwh) analysis of consumer behaviour related to electricity consumption 91 copyright ©2023 assa. adv. in systems science and appl. (2023)  the consumption in bigger agglomerations is higher as the temperature there can be up to approximately 2 or 3 times higher than in smaller agglomerations. also due to air quality, the air ventilation through opening the windows would probably be less intensive.  the highest consumption occurs in the category of middle-aged customers. compared to other age categories, it represents the extent of 7 to almost 25 kwh monthly, which equals 2 to 7 hours of air-conditioning usage. the category of young consumers shows the lowest consumption related to air-conditioning with the most obvious differences in towns and cities reaching a difference of 10 to 20 kwh per month.  this category is related to the amount of time spent at home – in the house, flat; and the results display a similar but less significant trend as the lighting category, where younger consumers in bigger agglomerations trend towards out of home lifestyle.  electricity consumption within the category also relates to the amount of money the person is able or willing to invest into her/his comfort and how intense is the discomfort (worse air quality in the bigger city vs. cleaner air and lower temperature in the rural areas). the willingness to spend more on comfort appears to be strongest in the category of middleaged consumers, who seem to spend more time at home than younger consumers and are probably less cost-sensitive than the category of older consumers. space heating consumption average agglomeration category age category rural area town city average value young 301,1342769 270,8408179 363,2576857 311,7442602 middle 259,378239 262,5976718 325,4578095 282,4779067 older 210,3373863 212,3748125 267,4037899 230,0386629 average value 256,9499674 248,6044341 318,7064284 274,7536099 fig. 4.5. space heating consumption (kwh)  significant differences can be observed within different age categories, where there is a trend of younger consumers spending more energy. the difference between young and older consumers regardless of the size of agglomeration reaches approximately 80 kwh per 92 j. héjjová, j.bucko, e. exenberger copyright ©2023 assa adv. in systems science and appl. (2023) month, which means more than 900 kwh yearly equaling approximately 55 euros yearly (based on max. electricity price in slovakia for households 0,061 eur/kwh set for year 2022).  as for the agglomeration categories, the most significant difference appears in the cities, where the average monthly consumption on space heating is approximately 60 kwh higher in comparison to rural areas and towns. the effect can be caused by a stronger tendency to combine electricity with wood as a fuel for space heating.  this category is not related to the physical presence of a consumer in the household as it is in the case of for example lighting or air conditioning. consumption related to space heating is more dependent on the financial effect on the consumer and her/his comfort. "i want to be warm, and i can pay for it". the group of people preferring lower temperatures at home due to the positive health effects or as a part of their lifestyle that includes winterization is still a minority. another aspect of higher consumption within the young category would be the presence of small children and providing them with a comfortable environment.  the consumption in the category of middle-aged consumers is closer to young consumers, they spend monthly approximately 30 kwh less than young customers. in comparison to older consumers, the middle-aged category spends monthly approximately 50 kwh more.  in case of the middle-aged customers, the incentive towards this type of behaviour can be higher motivation for financial effectivity than in the category of younger consumers. the reason for prioritizing the financial effectiveness can come from the principle of not spending more than needed, but not at any cost as there is probably a lower willingness to decrease own comfort as in the case of older customers, where the energy bill needs to be controlled due to lower-income related to the retirement period. refrigeration consumption average agglomeration category age category rural area town city average value young 32,86679155 29,99534424 31,44946436 31,43720005 middle 37,40352933 35,70179346 34,35879422 35,82137234 older 39,85576864 38,22623818 37,2992759 38,46042757 average value 36,70869651 34,64112529 34,36917816 35,23966665 fig. 4.6. refrigeration consumption (kwh) analysis of consumer behaviour related to electricity consumption 93 copyright ©2023 assa. adv. in systems science and appl. (2023)  the biggest difference appears between categories of older consumers in rural areas vs. young consumers in the towns. it is 10 kwh per month, which exceeds the yearly consumption of one a category refrigerator including the freezing section (approx. 110 kwh).  the overall results indicate that: o these consumers have inefficient appliances (lower energy class); o at least 2 appliances – separate refrigerator and freezer (e.g. highest consumption occurs in the category of older-aged consumers in rural areas. the reason could be the intention to store special meet purchased from farmers, which are sold in bigger amounts, or to store the bigger amount of food and decrease the frequency of shopping, which influences the need for mobility and traveling to the nearby cities and costs some amount of time.); o their behaviour in the area of electricity consumption related to refrigeration is ineffective, probably due to low knowledge of efficient habits. always – on average agglomeration category age category rural area town city average value young 56,67039872 52,71637142 55,31790336 54,90155783 middle 76,11951268 79,89100522 89,40651329 81,80567706 older 80,72345299 87,25934023 98,02581712 88,66953678 average value 71,17112147 73,28890562 80,91674459 75,12559056 fig. 4.7. always – on consumption (kwh).  older aged consumers spend the most electricity within this category. the reason is probably they spend more time at home, which is observed in the category of lighting as well.  older consumers spend over 30 kwh more than the young-aged category, and middle-aged have the consumption higher by 25 kwh more than the young. at the consumption level of for example lcd tv, which is 0,16 kwh, the 25-kwh difference represents over 150 hours of watching tv.  from the comfort point of view, the older people seem to prefer always on or entertainment appliances to higher temperatures and they tend to focus their energy-saving 94 j. héjjová, j.bucko, e. exenberger copyright ©2023 assa adv. in systems science and appl. (2023) activity more on heating, which generates multiple times higher costs than always on appliances.  from the agglomeration size point of view, the level of consumption is similar in the category of young consumers. more obvious differences appear in middle-aged and older consumers, where the consumption of middle-aged consumers and older consumers rises with the size of agglomeration. even though the category is related to consumers' presence in the household, it doesn't show the trends in lighting and air conditioning, where the consumption is lower in sizeable agglomerations with more opportunities for out of home activities. the reason would be, that the always on category includes more appliances therefore the average consumption is impacted by the number of always on appliances present on the property. 5. limitations our current research has been accompanied by a few limitations, that need to be taken into account in the application of the results:  the research confirms different patterns in behaviour in consumption related to age and size agglomeration, but further research to confirm the triggers causing differences in electricity consumption patterns is needed to reveal various motivators, which would be addressed to activate these groups of consumers in decreasing the level of consumption.  for the further use of these results, we would like to point out, that we have worked with average values, which do not reflect the seasonality of energy consumption. but still, it hasn't been relevant in our study focusing on the long-term trend.  the level of energy consumption and efficiency is a complex topic influenced by a group of factors. we focused on the impact of a tool based on behavioural principles and our study reflects the insight that energy consumption is highly influenced by climate as well as dwelling conditions, therefore we recommend the practical implementation of the results mainly in the central european region.  we are also aware that the intrapersonal factor of "age" itself should not be considered a sufficient parameter to evaluate or predict customers' behaviour within household management, and consequently to improve targeting. but it is an easily accessible and precise type of data about consumers. 6. conclusion and practical implications among the most aggressive originators of environmental deterioration is electricity production by burning fossil fuels. cutting down electricity consumption is a tough challenge during the period of the human race's existence, when electricity has become a candidate with the potential to revolutionize items in maslow's pyramid. a relevant group of energy consumers are households as they are the source of more than half of the carbon dioxide emissions. therefore, numerous initiatives have been established to address the problem. along with the initiatives to increase the level of energy produced by renewable resources, energy conservation, and energy efficiency are priorities for governments worldwide. they have motivated intensive research over the past decade into understanding how energy is consumed, and how to translate the knowledge into meaningful information to enable energy consumers to take responsibility over their energy consumption [31]. strategies for realizing potential savings rely on informed decisions and rational actions of the consumers. unfortunately, studies reveal existence of several economic, technological, and organizational barriers, that influence the level of consumer awareness and knowledge. therefore, such strategies result in poor implementation rates of energy efficiency measures. [18]. analysis of consumer behaviour related to electricity consumption 95 copyright ©2023 assa. adv. in systems science and appl. (2023) to achieve the desired result, which is a sustainable behavioural change leading to permanent decrease in electricity consumption, a mastered mix of relevant information in an easily perceivable shape combined with effective tools grabbing the attention of households needs to be created. as targeting is the most influential factor, when it comes to the success of the communication campaigns, knowledge of consumers' desires or barriers to performing desired behaviour and actions is a topic of high relevancy. the household segment is especially challenging due to its size and variability in habits or behaviour within household management. energy consumption is influenced by the characteristics as for example the number of occupants and their age; type of occupants as consumption of households where there are no children or where couples work, consume energy differently than households with small children or older people. these characteristics multiply the level of difficulty in attempts to understand the energy consumers' behaviour and consequently increase the difficulty in addressing them successfully in an attempt to improve their energy efficiency. the first part of this study concludes that age provides numerous insights into household lifestyle resulting in differences in electricity consumption. the outputs show both positive and negative linear relationships between age and amount of electricity consumption in the examined categories. the tendency of spending less electricity in the categories with the highest average level of consumption suggests higher cost sensitivity with the increasing age of consumers. the results on categories related to physical presence in the property such as lightning or always on, show the positive linear relationship between age and the amount of electricity consumption. it reflects the more active lifestyle of younger categories, on the other hand, lightning and always on belong to the categories with the lowest average consumption and therefore lowest impact on the cost of electricity consumption. the second part of the study pointed at extracting more detailed insight into consumers' behaviour, which would support the potential of energy efficiency initiatives by improved targeting. it specifically dealt with the place of living as one of the causes of different behaviour in electricity consumption combined with the factor of aging as a reflection of lifestyle. the results have revealed possible significant inefficiency in the area of refrigeration, showed non-essential differences in the areas of air conditioning and water heating, and provided detail on relevant differences in consumption within lighting, always on, and space heating. disaggregation provides valuable insights for the improvement of energy efficiency campaigns and makes them applicable to the mass market as they scale to thousands or millions of customers with little additional effort. energy efficiency appears to be a contraproductive activity to utility companies at first sight. but on the other hand, it reflects the already ongoing trend of digitalization and decentralization, which enables the expansion of the peer-to-peer (p2p) model within the energy market and makes end-users less dependent on utility suppliers. this probably leads to the same impact as airbnb or uber models have had on the accommodation or taxi sector, by creating a highly competitive way of satisfying related customer needs by cutting down costs, especially intermediary ones. technologies allow this trend to enter the energy sector and develop pressure on energy companies' transformation and force them to search for new sources of income reflecting the needs of customers. one way is to transfer the income from the electricity supply to other energyrelated services and products. among the key preconditions of success is image and reputation, therefore proving it via consultancy and advice on energy efficiency within households appears as an effective option. [30]. more detailed information on customers also supports the improvement of customer retention, which is becoming more relevant in a liberalized energy market [30]. utilities benefit from customer insights, which can they reveal using the proposed system to estimate household characteristics. the same information that helps to make energy-saving campaigns more effective, can help market products and services more successfully. the latter includes 96 j. héjjová, j.bucko, e. exenberger copyright ©2023 assa adv. in systems science and appl. 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(2016). understanding household energy consumption behavior: the contribution of energy big data analytics. renewable and sustainable energy reviews, 56, 810–819. adv syst sci appl 2023; 04:31–39 published online at https://ijassa.ipu.ru. study of a redundant residue number system for single error correction viktor kuchukov1*, mikhail babenko2, safwat al-galda3 1north-caucasus center for mathematical research, north-caucasus federal university, stavropol, russia 2north-caucasus federal university, stavropol, russia 3university of misan, amarah, iraq abstract: one approach to reliability is to use a redundant residue number system. in general, two redundant moduli are required to detect and correct a single error. this paper considers an approach to error correction using a single redundant modulo, which allows a significant reduction in the hardware used, but at a significant cost in computational speed. the use of an approximate method based on the chinese remainder theorem allows the speed of computation to be increased by eliminating the computationally complex operation of taking the remainder from the division by the range of the residue number system. a method based on the approximate method with one redundant modulo is proposed. modelling of the considered methods on asics in rtl and physical synthesis environment cadence genus synthesis solution is carried out. keywords: residue number system, error correction, chinese remainder theorem, applicationspecific integrated circuit 1. introduction the problem of reliability and guaranteed validity of the results obtained is relevant for remote control systems where autonomy and fault tolerance are important, e.g. on railways and oil and gas pipelines. it is known that for data transmission channels the probability of single errors significantly exceeds the probability of errors of higher multiplicity (generalised binomial law of error distribution), and for digital devices the law of error distribution is close to uniform [1, 2]. most correction codes, especially block binary codes, which include parity check codes, have two groups of digits — information and control digits. the information group contains digits that represent the numerical value of the coded value, and the control group contains digits that are redundant and additionally introduced for the purpose of detecting and correcting possible distortions [3]. since there is no possibility of composing the control part of the result by the control parts of the arithmetic operation components, the possibility of controlling the correctness of arithmetic operations is excluded. this non-arithmetic character of special position codes prevents their use in calculating machines, for which the control of arithmetic operations is of particular importance. one of the examples of arithmetic codes are an codes [4,5], but these codes allow only addition and subtraction operations. another approach to the construction of arithmetic codes is the residue number system (rns). further, the paper is organized as follows. section 2 considers a redundant residue number system and shows the basic properties of error correction in rns. section 3 considers the ∗corresponding author: vkuchukov@ncfu.ru 32 v. kuchukov, m. babenko, s. al-galda residue number system with one redundant modulo for single error correction. section 4 shows the application of the approximate method based on the chinese remainder theorem for error correction in rns. section 5 is devoted to the application of the approximate method to rns with one redundant modulo. section 6 presents the modelling of the considered methods. 2. redundant residue number system one of the most promising representations of numbers in parallel processing is the use of non-positional number systems, such as the residue number system. if a set of positive integers p1, p2, . . . , pn, called moduli of the system, is given, then a residue number system is a system in which a positive integer is represented as a set of residues on the chosen moduli x = (x1, x2, . . . , xn), where xi = x mod pi for i = 1, 2, . . . , n [3]. if the moduli satisfy the condition p1 < p2 < . . . < pn, then the system is called ordered. it is known from number theory that if the moduli pi are mutually prime, then the representation of the number x = (x1, x2, . . . , xn) < p = p1 · p2 · . . . · pn is singular, where p is the dynamic range of the number representation. the peculiarity of the residue number system is the possibility of performing addition, subtraction and multiplication operations in parallel and independently for each of the modulo [6]. if we add to the residue number system with moduli {p1, p2, . . . , pn} and dynamic range p = ∏n i=1 pi, which is often called the working range, an additional modulo pn+1 > pi, i = 1, n, then the full range of the system is p = p · pn+1. in the case of operations on numbers in the range [0, p ), if the result of the operation is less than p , it is correct, otherwise it is incorrect. thus, the introduction of the redundant modulo pn+1 allows us to detect calculation errors. the introduction of constraints on the redundant modulo or the introduction of additional moduli allows not only the detection but also the correction of errors [7, 8]. let’s take as an example the rns {3, 5, 7, 8}, where p4 = 8 is the control modulo. then the working range p = 105, the full range p = 840. take the number x = (1, 2, 3, 4) = 52, since x < p , there is no error. suppose there is an error on the second modulo and the number x = (1, 3, 3, 4) = 388, and since x > p , the error can be detected. one of the methods of locating errors in modular code is the method of projections. the projection xi of a number x = (x1, x2, . . . , xn+1) onto the modulo pi is the number obtained by crossing out the digit xi in the representation of x . if in an ordered residue number system the projection xi of a number x = (x1, x2, . . . , xi, . . . , xn, xn+1) on the modulo of pi satisfies the condition xi > p pn+1 , then the digit xi is correct if only one error is possible [3]. the introduction of only one control modulo generally fails to locate the error. to correct the error, we can use a method based on the chinese remainder theorem (crt), by which the number x can be obtained from the formula x = ∣∣∣∣∣ n∑ i=1 pi · xi · ∣∣p−1 i ∣∣ pi ∣∣∣∣∣ p , (2.1) where p is the dynamic range, pi = p/pi, ∣∣p−1 i ∣∣ pi is the multiplicative inversion of pi modulo pi, and the operator |x|pi denotes the remainder of the division of x by pi, i.e. x mod pi [9]. copyright © 2023 assa. adv syst sci appl (2023) study of a rrns for single error correction 33 consider the above number x = (1, 3, 3, 4) = 388, for which the projections are x1 = (3, 3, 4) = 108, x2 = (1, 3, 4) = 52, x3 = (1, 3, 4) = 28, x4 = (1, 3, 3) = 73. we can see that the projections x2, x3, x4 are within the working range and the error cannot be corrected. let’s take a rns with two redundant moduli {3, 5, 7, 11, 13} with working range p = 105 and full range p = 15015. let’s introduce an error in the number x = (1, 2, 3, 8, 0) = 52 on the second modulo x = (1, 3, 3, 8, 0) = 6058. then the projections x1 = 1053, x2 = 52, x3 = 1768, x4 = 598, x5 = 283. from here we can see that only the projection x2 falls within the working range and then the remainder of x2 = 52 modulo p2 = 5 is 2 and the corrected number x = (1, 2, 3, 8, 0). 3. correction of an error with one redundant modulo the paper [10] proposes an approach to error correction with one redundant modulo. given a residue number system with a control modulo p1 < p2 < p3 < . . . < pn < pn+1, it is assumed that the control modulo is reliable and cannot contain an error. if pn+1 > pn · pn−1, then algorithm 1 is executed. algorithm 1. number recovery based on the method of projections and crt require: x ′ = ( x′ 1, x ′ 2, . . . , x ′ n, x ′ n+1 ) ensure: x data in memory: {p1, p2, . . . , pn+1}, p = ∏n i=1 pi, p = pn · p wi = ∣∣∣p−1 i ∣∣∣ pi · p i, where p i = p/pi, for all i ∈ [1, n+ 1] wi,j = ∣∣∣p−1 i,j ∣∣∣ pj · p i,j , where p i,j = p i/pj , for all i ∈ [1, n+ 1] and j ̸= i 1: s = 0 2: for i = 1 to n+ 1 do 3: s = s + x′ i · wi 4: end for 5: s = s mod p 6: if s < p then 7: x = s 8: return x 9: else 10: for i = 1 to n+ 1 do 11: si = 0 12: for j = 1 to n+ 1 do 13: if i ̸= j then 14: si = si + x′ i · wi,j 15: end if 16: end for 17: x = si mod p i 18: if x < p then 19: return x 20: end if 21: end for 22: end if this algorithm allows a working modulo error to be corrected. however, the error in the control modulo is not always corrected exactly. let’s take the rns {3, 5, 7, 37} and copyright © 2023 assa. adv syst sci appl (2023) 34 v. kuchukov, m. babenko, s. al-galda the number x = (1, 2, 3, 15) = 52 and introduce the error on the control modulo x = (1, 2, 3, 17) = 1312. algorithm 1 gives the number x ′ = (2, 2, 3, 17) = 17. however, the number of projections required is reduced if the reliability of the calculations on the control modulo is ensured. 4. approximate method based on the chinese remainder theorem formula (2.1) and algorithm 1 use the operation of finding the remainder of the division by a large modulo p. this operation is computationally expensive. one way to improve the efficiency of this operation is to use an approximate method based on crt. to detect and correct a single error in the rns, we consider the addition of two redundant moduli pn+1 and pn+2. in [11, 12] a fractional approximate representation of numbers based on crt is proposed. let us divide (2.1) by p and we get x p = ∣∣∣∣∣∣∣ n+2∑ i=1 xi · ∣∣∣p−1 i ∣∣∣ pi pi ∣∣∣∣∣∣∣ 1 = ∣∣∣∣∣ n+2∑ i=1 xi · ki ∣∣∣∣∣ 1 , (4.2) where ki = ∣∣∣p−1 i ∣∣∣ pi pi are constants of the chosen system, and the operation |x|1 means taking the fractional part of the number x. in this case, the value of the expression (4.2) is in the interval [0, 1). taking the fractional part of a number is much simpler than finding the remainder of the division by the full range, but in hardware implementation the coefficients ki can rarely be represented as finite fractions and there is a question of rounding accuracy. to solve this problem, the fractional coefficients ki are multiplied by 2n , where n is the number of binary digits after the decimal point, providing the necessary level of accuracy of the calculations, each number obtained is rounded up to an integer, and then all calculations are performed modulo 2n . finding the remainder by this modulo is solved in hardware by truncation, which is a trivial task. for the calculations we can use the estimation proposed in [13]: n = ⌈ log2 ( p · n+2∑ i=1 (pi − 1) )⌉ . in this case, the working range p is represented in the redundant rns and it is obvious that p = (0, . . . , 0, πn+1, πn+2), where πn+1 = p mod pn+1, πn+2 = p mod pn+2. thus, when calculating projections, it is necessary to find relative values for both x and p . let’s look at a similar example in rns with two redundant moduli {3, 5, 7, 11, 13} and the number x = (1, 3, 3, 8, 0) = 6058. according to formula (4.2), the coefficients ki are k1 = 1/3, k2 = 2/5, k3 = 5/7, k4 = 1/11, k5 = 6/13, rounded to the nearest n = 19. to reduce the record, we will use fractional representation. then the relative value for x is 466/1155 and for p is 1/143. since 466/1155 > 1/143, the number x contains an error. for projection modulo p1, the coefficients ki are k1 = 1/5, k2 = 1/7, k3 = 3/11, k4 = 5/13. then the relative value for x1 is 81/385 and for p is 3/143. since 81/385 > 3/143, the number x1 contains an error. for projection modulo p2, the coefficients ki are k1 = 2/3, k2 = 4/7, k3 = 5/11, k4 = 4/13. then the relative value for x2 is 4/231 and for p is 5/143. since 4/231 < 5/143, the number x2 is error free. the other projections show that the number contains an error. copyright © 2023 assa. adv syst sci appl (2023) study of a rrns for single error correction 35 to recover the number we need to multiply the value of 4/231 by the full range of the projection, i.e. by p/p2 = 3003, we get 52, which corresponds to the number without error. 5. application of the approximate method to error correction with one redundant modulo let us consider the application of the approximate method to algorithm 1. the disadvantage of this algorithm is the need to find the remainder by the large modulo. let us introduce algorithm 2, which allows us to correct the working modulo error in rns with one redundant modulo pn+1 > pn · pn−1. algorithm 2. number recovery based on the projection method and approximate crt require: x ′ = ( x′ 1, x ′ 2, . . . , x ′ n, x ′ n+1 ) ensure: x data in memory: {p1, p2, . . . , pn+1}, p = ∏n i=1 pi, p = pn · p , π = p mod pn+1 p i = p/pi, wi = ∣∣∣p−1 i ∣∣∣ pi pi , i ∈ [1, n+ 1] p i,j = p i/pj , wi,j = ∣∣∣p−1 i,j ∣∣∣ pj pj , i ∈ [1, n], j ∈ [1, n+ 1], i ̸= j 1: x = ∣∣∑n+1 i=1 x′ i · wi ∣∣ 1 2: w = |π · wn+1|1 3: if x < w then 4: return x · p 5: else 6: for i = 1 to n do 7: xi = ∣∣∣∑n+1 j=1,j ̸=i x ′ j · wi,j ∣∣∣ 1 8: wi = |π · wi,n+1|1 9: if xi < wi then 10: return xi · p i 11: end if 12: end for 13: end if let’s look at an example of how algorithm 2 works for rns {3, 5, 7, 37} and the number x = (1, 2, 3, 17) = 1312. the data stored in memory: p = 105, p = 3885, π = 31, p1 = 1295, p2 = 777, p3 = 555, p4 = 105, p1,2 = p2,1 = 259, p1,3 = p3,1 = 185, p1,4 = p4,1 = 35, p2,3 = p3,2 = 111, p2,4 = p4,2 = 21, p3,4 = p4,3 = 15, copyright © 2023 assa. adv syst sci appl (2023) 36 v. kuchukov, m. babenko, s. al-galda w1 = 2 3 , w2 = 3 5 , w3 = 4 7 , w4 = 6 37 , w1,2 = 4 5 , w1,3 = 5 7 , w1,4 = 18 37 , w2,1 = 1 3 , w2,3 = 6 7 , w2,4 = 30 37 , w3,1 = 2 3 , w3,2 = 1 5 , w3,4 = 5 37 . then x = ∣∣∣∣1 · 23 + 3 · 3 5 + 3 · 4 7 + 15 · 6 37 ∣∣∣∣ 1 = 2383 3885 , w = ∣∣∣∣31 · 6 37 ∣∣∣∣ 1 = 1 37 . since 2383/3885 > 1/37, the number contains an error. let’s construct the projections. then the first projection x1 = ∣∣∣∣3 · 45 + 3 · 5 7 + 15 · 18 37 ∣∣∣∣ 1 = 1088 1295 , w1 = ∣∣∣∣31 · 1837 ∣∣∣∣ 1 = 3 37 . since 1088/3885 > 3/37, the number contains an error. the second projection x2 = ∣∣∣∣1 · 13 + 3 · 6 7 + 15 · 30 37 ∣∣∣∣ 1 = 52 777 , w2 = ∣∣∣∣31 · 3037 ∣∣∣∣ 1 = 5 37 . since 52/777 < 5/37, the number does not contain an error. the third projection x3 = ∣∣∣∣1 · 23 + 3 · 1 5 + 15 · 5 37 ∣∣∣∣ 1 = 373 555 , w3 = ∣∣∣∣31 · 5 37 ∣∣∣∣ 1 = 7 37 . since 373/555 > 7/37, the number contains an error. so the error was in the second modulo. to recover the number, it is necessary to multiply 52/777 by p2 = 777, so the correct number is 52. this approach can also contain an error only on working moduli. 6. modelling single error correction simulations were performed on an asic in the rtl and physical synthesis cadence genus synthesis solution environment using the osu018 stdcells library. the metrics measured were the time for the signal to travel through the circuit (picoseconds, ps), the power required (watts, w) and the area used (square microns, µm2). rns sets with one and two redundant moduli with 4-6 working moduli covering the 8, 16, 24 and 32 bit ranges shown in table 6.1 are selected for the simulation. tables 6.2–6.4 show the simulation results for the signal propagation time through the circuit (picoseconds, ps), the area used (square microns, µm2) and the power required (watts, w). algorithm 1 with one redundant modulo, denoted as i in the tables, an approximate method based on crt with two redundant moduli, denoted as ii, an approximate method based on crt with one redundant modulo, denoted as iii, are chosen for the simulations. tables 6.2–6.4 show that the approximate method with two moduli is on average 0.43% faster, uses 16.96% more area and has a 25.44% higher power consumption compared to the approximate method with a one redundant modulo. the single redundant modulo method is on average 180% slower, uses 23% more power but 69% less area. copyright © 2023 assa. adv syst sci appl (2023) study of a rrns for single error correction 37 table 6.1. sets of moduli for modelling number of working moduli working range size, bits 8 16 24 32 one redundant modulo 4 {3, 5, 7, 11, 79} {13, 17, 19, 23, 439} {59, 61, 67, 71, 4759} {251, 257, 263, 269, 70753} 5 {2, 3, 5, 7, 11, 79} {5, 7, 11, 13, 17, 223} {23, 29, 31, 37, 41, 1523} {79, 83, 89, 97, 101, 9803} 6 {2, 3, 5, 7, 11, 13, 149} {3, 5, 7, 11, 13, 17, 223} {11, 13, 17, 19, 23, 29, 673} {31, 37, 41, 43, 47, 53, 2503} two redundant moduli 4 {3, 5, 7, 11, 13, 17} {13, 17, 19, 23, 29, 31} {59, 61, 67, 71, 73, 79} {251, 257, 263, 269, 271, 277} 5 {2, 3, 5, 7, 11, 13, 17} {5, 7, 11, 13, 17, 19, 23} {23, 29, 31, 37, 41, 43, 47} {79, 83, 89, 97, 101, 103, 107} 6 {2, 3, 5, 7, 11, 13, 17, 19} {3, 5, 7, 11, 13, 17, 19, 23} {11, 13, 17, 19, 23, 29, 31, 37} {31, 37, 41, 43, 47, 53, 59, 61} table 6.2. results of the simulation of the signal propagation time through the circuit, ps number of working moduli algorithm working range size, bits 8 16 24 32 4 i 25097 39429 84673 119623 ii 11369 15876 21509 31825 iii 11476 15707 24436 29211 5 i 22903 36698 76512 111360 ii 12424 15646 23134 27172 iii 12374 16354 22028 27055 6 i 29077 41193 66569 66711 iii 15867 17340 22252 27402 iv 15542 18717 21620 26845 table 6.3. results of the simulation of the used area, µm2 number of working moduli algorithm working range size, bits 8 16 24 32 4 i 60759 132332 319916 538922 ii 186419 393166 816822 1470043 iii 148011 314708 684626 1206436 5 i 62136 131767 287343 493900 ii 258575 422888 877680 1448269 iii 200183 347296 715083 1329741 6 i 94223 155115 282315 399128 ii 444720 566191 916218 1393492 iii 343388 473257 783562 1281555 copyright © 2023 assa. adv syst sci appl (2023) 38 v. kuchukov, m. babenko, s. al-galda table 6.4. results of the simulation of the required power, w number of working moduli algorithm working range size, bits 8 16 24 32 4 i 4,82e-01 1,24e+00 3,37e+00 6,25e+00 ii 1,92e-01 8,36e-01 2,67e+00 7,03e+00 iii 1,42e-01 6,12e-01 1,98e+00 5,45e+00 5 i 4,75e-01 1,29e+00 3,11e+00 5,68e+00 ii 2,96e-01 8,73e-01 2,91e+00 7,26e+00 iii 2,14e-01 5,94e-01 2,27e+00 6,27e+00 6 i 8,24e-01 1,57e+00 2,95e+00 4,51e+00 ii 8,59e-01 1,37e+00 3,32e+00 6,95e+00 iii 5,32e-01 9,80e-01 2,56e+00 5,60e+00 7. conclusion from the simulation it can be seen that the method using a single redundant modulo reduces the area used, but the running time of the algorithm increases significantly. using the approximate method improves the performance significantly, but has a larger area used. adapting the approximate method based on crt to the single redundant modulo method can significantly improve the computational speed compared to the single redundant modulo method and reduce the area used compared to the approximate method with two redundant moduli. however, the disadvantage of this method is the limitation imposed by the redundant modulo. one possible application of this approach is distributed data storage systems, such as the one described in russian patent 2780148. in this case, in order to ensure reliability, the residue of the redundant modulo may be stored in the user’s storage, while the residues of the working moduli are distributed to cloud storage. thus, in case of an error or non-receipt of one of the file parts from the distributed environment, the original data can be restored in the manner described above. a direction for further research may be to develop a single redundant modulo error correction method where the error on the control modulo can also be corrected. acknowledgements this work was supported by the russian science foundation 19-71-10033, https://rscf.ru/project/19-71-10033/. references 1. tolstyakov, v. s. 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(1983) an improved residue number system digital-to-analog converter, ieee transactions on circuits and systems, 30(12), 903–907. 12. cherviakov, n. i. (2011). metody, algoritmy i tekhnicheskaya realizaciya osnovnyh problemnyh operacij, vypolnyaemyh v sisteme ostatochnyh klassov [methods, algorithms and technical implementation of the main problem operations performed in the residue number system], infokommunikacionnye tekhnologii, 9(4), 4–12. [in russian] 13. cherviakov, n. i., babenko, m. g., lyakhov, p. a. & lavrinenko, i. n. (2013) priblizhennyj metod opredeleniya znaka chisla v sisteme ostatochnyh klassov i ego tekhnicheskaya realizaciya [approximate method for determining the sign of a number in the residue number system and its technical implementation]. nauchno-tekhnicheskie vedomosti sankt-peterburgskogo gosudarstvennogo politekhnicheskogo universiteta. informatika. telekommunikacii. upravlenie. 4(176), 131–141. [in russian] copyright © 2023 assa. adv syst sci appl (2023) introduction redundant residue number system correction of an error with one redundant modulo approximate method based on the chinese remainder theorem application of the approximate method to error correction with one redundant modulo modelling conclusion microsoft word article+text,+kibnbatouta+-article+assa-revised+version adv syst sci appl 2025; 1; 30-43 published online at https://ijassa.ipu.ru. green industry: energy efficiency improvement of steam generation in a moroccan plastic recycling industry kawtar ibn batouta*, sarah aouhassi, khalifa mansouri école normale supérieure de l'enseignement technique, mohammedia, casablanca, morocco abstract: steam systems in industry represent a substantial share of industrial energy consumption. therefore, improving the energy efficiency of these systems is necessary to reduce fossil fuel consumption and greenhouse gas emissions in the context of energy transition. this improvement not only leads to environmental benefits but also enhances financial gains, thereby increasing industrial competitiveness. the objective of this study is to identify energy-saving opportunities in a plastic recycling plant in morocco while adopting a system approach based on thermodynamics fundamentals. after conducting a benchmark analysis based on a scoping tool questionnaire, the steam generation section is diagnosed to quantify major losses. subsequently, an action plan is suggested for more efficient thermal energy use. the assessments focus on energy loss reductions along with waste heat recovery. the implementation of these actions results in a potential saving of 22%, achieving an energy gain of 5574 gj/y, a financial benefit of 96 884 $/y, and a reduction in co2 emissions by 332.15 tons annually. the global payback period does not exceed one year. the study conducted encourages the adoption of costeffective energy efficiency measures by industry stakeholders, in line with the sustainable development goals. furthermore, it benefits all industries utilizing boilers in their steam production systems, regardless of the diversity of industrial systems. keywords: steam systems; energy consumption; energy efficiency; greenhouse gas emissions; energy-saving; industry; energy transition; sustainable development nomenclature: p: pressure in bar gauge (bar) or bar absolute (bara) t: temperature in °c h: specific enthalpy in kj/kg ṁ: mass flow in t/h β: blowdown rate in kg/h e: energy in gj hhv: fuel higher heating value in mj/kg x: thermodynamic quality (mass basis) η: efficiency σ: savings λ: loss k: operating cost in $ q̇: heat transfer rate ẇ: power 1. introduction the industrial sector consumes nearly 40% of global energy. relying on fossil fuels, it is responsible for 20% of total co2 emissions [12]. upon closely examining energy use in this sector, half of the consumed energy is used for process heating, primarily through steam production systems, which represent 30% of the final energy use in industry [21]. indeed, steam is an adaptable energy transfer medium, effectively *corresponding author: ibnbatouta.kawtar@gmail.com green industry: energy efficiency improvement… 31 copyright ©2025 assa adv. in systems science and appl. (2025) serving both process heating and power generation with exceptional flexibility. steam is generated in the boiler, a pressure vessel that heats water by convection heat transfer of flue gas resulting from burning the fuel. to improve the boiler and the overall system efficiency, it is necessary to reduce boiler losses and recover the resulting waste energy [2]. consequently, fuel consumption and ghg emissions would be reduced. fig. 1 details the diagram of a fire-tube boiler, highlighting the various energy losses. the most significant are, in descending order, stack loss, blowdown loss and shell loss. in scientific literature, several studies have focused on steam system optimization [20]. the work of [14] determines a method to compute the optimal excess air value for fuel combustion in the boiler’s burner, as this value impacts the stack loss and the boiler efficiency. in the case study of [17], boiler energy efficiency has been improved using a variable speed drive in the fan motor supplying the boiler combustion air. the authors in [7] propose energy-saving measures for steam generation in the us industry, leading to an 18% economic potential of total boiler energy use. the research of [5] proposes a closed tank system to eliminate energy losses, improving the coefficient of efficiency by about 6%. in morocco, some case studies have addressed energy efficiency in industrial sectors, focusing on either the electrical [3], [4], [6] or thermal [1], [13] aspects. the thermal side has included other sides such as furnaces. however, rare case studies have addressed the boiler performances [16] and have yet to be dedicated to the optimization of the overall industrial steam system. additionally, it has been identified that there needs to be more energy efficiency case studies in the country that use benchmarking tools, as well as limited diversity in the studied industrial sectors within the context of energy transition [11]. the present study contributes to filling these gaps by adopting a system approach to modeling and improving the energy efficiency of a steam system in a plastic recycling industry located in morocco. this approach results in significantly greater energy and cost savings compared to an analysis conducted at the component level. this work covers diagnostic, metering and optimization research categories of energy efficiency [10]. first, the actual steam system is benchmarked using a scoping tool questionnaire. the energy loss investigation was conducted in the lowest-rated part of the steam system, the boiler house, to propose energy-saving actions. these opportunities aim at improving thermal energy efficiency by both reducing and recovering the major boiler losses, namely stack and blowdown losses. each action is quantified in terms of economic and energy saving, along with greenhouse gas (ghg) emissions reduction. although industrial systems are highly diversified, they share the common elements of steam production systems, making this case study useful for various industrial sectors as well. fig. 1. energy losses of a fire-tube boiler adapted from [8] 32 k. ibn batouta, s. aouhassi, k. mansouri copyright ©2025 assa adv. in systems science and appl. (2025) 2. methods 2.1. process & steam system description 2.1.1. process description in this recycling industry, polyethylene terephthalate (pet), sourced from discarded plastic bottles, undergoes three main production stages: washing, extrusion, and finishing (see fig. 2). in the washing line, pet bottles are shredded, screened, and manually sorted. after grinding with cold water, the material passes through a series of separators and flotation units to remove labels and residues. it is then filtered, hot-washed at 90°c using steam via heat exchangers, rinsed, and screened again to eliminate small particles and colored fragments. in the extrusion line, the cleaned pet is first dried in ovens to remove moisture before entering extruders, where it is molded into specific diameters. the product is stabilized in boxes using direct steam injection to ensure material integrity and precise temperature control. in the finishing line, the extruded material is cooled and solidified using steam-heated water to ensure consistent fiber quality. this process prepares the recycled pet (rpet) for its final application in the textile industry. fig. 2. process description 2.1.2. steam system description a typical steam system consists of four main areas, namely generation, distribution, end-use, and recovery. the generation part takes place in the boiler room, where the combustion process transfers energy from the fuel to the water, converting it into steam. the boiler used in this system is a fire-tube boiler constructed in 2020. one inherent efficiency advantage of this type over the water-tube type is that shell loss is minimal. propane is used as the fuel, and the makeup water is treated with softeners. the boiler room is located next to the entrance of the first production line (washing line). the generated steam, produced at a flow rate of 2 t/h at 6 bar, is then distributed through pipes and valves to the end-use processes, either with direct injection or through heat exchangers. finally, the recovery of the formed condensate allows it to be reused in the boiler room for steam generation; the condensate recovery rate of the studied system is 80%. in this article, a systems approach is adopted, as it leads to significantly higher energy and cost savings compared to a component-level analysis. however, the energy efficiency improvement actions are focused on the steam generation part. fig. 3 illustrates the steam system areas. green industry: energy efficiency improvement… 33 copyright ©2025 assa adv. in systems science and appl. (2025) fig. 3. steam system description 2.2. data collection the first step in data collection was conducting an interview with the plant’s technical director based on a questionnaire of 26 questions divided according to four levels of steam system management. this questionnaire is called the steam system scoping tool (ssst), developed by the united states department of energy [18]. its role is to benchmark the studied system against best practices in industries that use steam in their processes. next, a site visit was conducted to closely observe the different parts of the steam system and to complete and confirm the questionnaire responses. the subsequent phase was a measurement campaign focused on the steam generation part using portable measuring devices, namely the conductivity meter and the combustion analyzer, whose specifications are in table 1. the measured data included the conductivities of the feedwater and blowdown water, as well as the oxygen content and temperature of the combustion gases in the stack. data collection also included recording the data displayed on the plant's on-site equipment. table 1. measurement instruments technical specification item general specifications conductivity and tds meter measure norm: en 50581:2012 accuracy: ±2% combustion analyzer temperature accuracy: ±0,5 °c o₂ in flue gas accuracy: ±0,2% 2.3. analysis tools 2.3.1. mass and energy balance the laws of mass and energy balance are used for the calculation in this study. in the control volume (fig. 4), the mass conservation law states that mass is neither created nor destroyed eq. (2.1). �̇� = �̇� (2.1) in the energy conservation law, energy can neither be created nor destroyed in the control volume (fig.4). it can only be changed from one form to another eq. (2.2). �̇� + �̇� × ℎ + �̇� = �̇� × ℎ (2.2) 34 k. ibn batouta, s. aouhassi, k. mansouri copyright ©2025 assa adv. in systems science and appl. (2025) fig. 4. mass and energy balance in a steady state steady flow 2.3.2. measur software in order to generate steam system models, the free online tool named manufacturing energy assessment software for utility reduction (measur) has been used, developed by the u.s. department of energy's advanced manufacturing office (amo) [22]. measur provides a suite of analytical tools and resources that assist in identifying opportunities for energy savings, improving efficiency, and reducing utility costs. it includes features such as system-level assessments, detailed energy modeling, and simulation capabilities for various manufacturing processes. 3. results and discussion 3.1. benchmarking results an extensive interview with the plant's energy manager, combined with an on-site visit that focused specifically on the steam system, produced results on the ssst, which are summarized in table 2. the total obtained score is 32.1%, which is relatively low compared to industries operating with steam systems on an international scale. the ssst results are given as supplementary material n°1. the results of this preliminary evaluation have facilitated the prioritization of the steam sections to address for proposing potential energy-saving actions. this article will concentrate on the area that received the lowest score (25%), namely “boiler plant operating practices”. table 2. summary of steam scoping tool results steam section possible score plant score % score boiler plant operating practices 80 20 25,0% steam system profiling 90 24 26,7% steam system operating practices 140 49 35,0% distribution, end use, recovery practices 30 16 53,3% total scoping tool questionnaire score 340 109 32,1% 3.2. collected data the data is either measured using portable measuring devices or collected from the factory's on-site equipment. all measurements are in metric units, based on the international system of units (si). table 3 summarizes the collected data. green industry: energy efficiency improvement… 35 copyright ©2025 assa adv. in systems science and appl. (2025) table 3. data collection values parameter value source plant operating hours 4000 h/y plant data fuel type gplpropane (c3h8) plant data fuel hhv 50 mj/kg plant data fuel emission rate 60 kg co2/gj plant data make-up water temperature 20 °c in site instrumentation feedwater conductivity 1500 µs/cm measured blowdown conductivity 9000 µs/cm measured % o2 in flue gas 17% measured flue gas temperature 151°c measured steam pressure 6 barsaturated steam in site instrumentation steam flow 2 t/h in site instrumentation condensate recovery rate 80% in site instrumentation condensate return temperature 65°c in site instrumentation 3.3. data analysis first, we calculate the boiler combustion efficiency using the stack loss calculator in measur. this data is influenced by the percentage of oxygen in the combustion gases (17%) and the temperature of these gases (151°c), both measured by the combustion analyzer at the stack. fig. 5 shows the obtained boiler combustion efficiency of 74.3% with a stack loss (𝜆 ) of 25.7% (eq. 3.1). 𝜂 = 100% − 𝜆 (3.1) fig. 5. stack loss calculator in measur-baseline boiler stack loss is typically the largest source of efficiency loss during boiler operation. various factors influence this loss, but the primary contributors are the flue gas temperature and the amount of excess air. in practice, achieving a perfect stoichiometric mixture of fuel and air is nearly impossible, which is why excess air is required to ensure complete combustion. while excess air is essential for complete combustion, too much of it can reduce efficiency, as additional energy is needed to heat the unnecessary air. therefore, the key to optimizing boiler performance is finding the right balance – enough excess air to guarantee combustion without incurring significant thermal losses. in the case of a fire-tube boiler using propane as fuel, like the one under study, the optimal percentage of excess air typically falls between 5% and 10%. this corresponds to a flue gas oxygen concentration of 1% to 2%, as calculated in eq. (3.2) [9]. % 𝑒𝑥𝑐𝑒𝑠𝑠 𝑎𝑖𝑟 = % 𝑂 21 − % 𝑂 (3.2) 36 k. ibn batouta, s. aouhassi, k. mansouri copyright ©2025 assa adv. in systems science and appl. (2025) the analysis conducted here relies on combustion principles, with the primary inputs or measured data being the flue gas exit temperature, ambient temperature, and flue gas oxygen content. the evaluation of combustion efficiency, based on stack losses, is divided into two main components: the effect of flue gas temperature at a constant oxygen percentage (17% in the reference model) and the effect of excess air at a constant flue gas temperature (151°c in the reference model). as shown in fig. 6, the sensitivity analysis reveals a clear trend: boiler combustion efficiency decreases as the oxygen content in the flue gas increases and as the flue gas temperature rises. fig. 6. combustion efficiency analysis the indirect method for efficiency calculation focuses on each kind of boiler losses [15], mainly stack (λstack) and blowdown (λblowdown) losses. while the direct method uses steam, feedwater and fuel properties, referring to the amount of thermal energy absorbed by feedwater to make steam [19]. the fuel flow is calculated based on the previous boiler combustion efficiency using the formula for direct efficiency calculation eq. (3.3): green industry: energy efficiency improvement… 37 copyright ©2025 assa adv. in systems science and appl. (2025) 𝜂 = �̇� × ℎ − ℎ �̇� × 𝐻𝐻𝑉 (3.3) where  ṁ (steam flow) = 2 t/h;  ℎ (𝑃 = 7 bara; 𝑋 = 1) = 2762.83 kj/kg (x refers to the quality of steam (saturated));  ℎ (𝑃 = 1 bara; 𝑇 = 100°c) = 419 kj/kg (bara is the absolute pressure);  ṁ = 130 kg/h;  𝐻𝐻𝑉 = 50 mj/kg. the boiler efficiency 𝜂 , calculated by the direct method, is then 72%. the blowdown rate can be calculated in eq. (3.4) or using measur calculator: 𝛽 = 𝛾 1 − 𝛾 (3.4) where 𝛾 = [18]. the obtained value is a 20% blowdown rate. the blowdown flow, from energy and mass balance, is calculated in eq. (3.5): �̇� = 𝛽 1 − 𝛽 × �̇� = 500 kg/h. (3.5) eq. (3.6) is used to quantify the boiler energy losses from blowdown. however, eq. (3.7) is the adequate formula to calculate the overall system losses due to blowdown. indeed, the water discharged during the purging process is replaced by makeup water at ambient conditions from a system approach rather than feedwater when calculating efficiency at the boiler level. the results show that these losses at the boiler level reach 557 gj/year and at the system level reach 1227 gj/year. this latter value corresponds to an annual financial loss of $20860. �̇� = �̇� × ℎ − ℎ (3.6) �̇� = �̇� × ℎ − ℎ (3.7) where ℎ (𝑃 = 1 𝑏𝑎𝑟𝑎; 𝑇 = 20 °c) = 84.01 kj/kg. at this stage, the percentage of energy losses due to blowdown (λblowdown) relative to fuel consumption is calculated by eq. (3.8) and eq. (3.9). this value is 2.2%. the blowdown rate of the studied boiler is 20%, which is significantly higher than the normal values typically observed in similar boilers. this elevated blowdown rate is primarily attributed to the poor quality of the makeup water used in the system, which has a high conductivity of 1500 µs/cm. additionally, the blowdown is not conducted on a regular basis, further contributing to the high conductivity of the blowdown water. for comparison, high-quality water systems often have a blowdown rate of less than 1%, while lowquality water systems may exhibit blowdown rates as high as 10%. thus, the 20% blowdown rate observed in our case far exceeds these typical ranges. this highlights the challenges faced by our system due to suboptimal water quality and inconsistent blowdown practices. 𝜆 = �̇� �̇� (3.8) �̇� = �̇� × 𝐻𝐻𝑉 (3.9) 𝜂 = 100 − 𝜆 − 𝜆 − 𝜆 (3.10) the most significant boiler losses are stack and blowdown losses, while other losses, such as shell loss, are neglected. the boiler efficiency calculated using the indirect method (eq. 3.10) is 72.1%, which is very close to the 72% efficiency obtained using the direct method. the slight difference may be attributed to 38 k. ibn batouta, s. aouhassi, k. mansouri copyright ©2025 assa adv. in systems science and appl. (2025) uncertainties in the measurement instruments. this efficiency is slightly lower than the combustion efficiency since it accounts for blowdown energy losses, but it is more accurate. the accuracy of the results is intrinsically linked to the quality of the input data. in our study, we took care to ensure the reliability of the data by verifying and cross-referencing it with historical plant records and manufacturer specifications, which provided an additional layer of validation. regarding the calculation of boiler efficiency using the indirect method, although this approach is more complex, it is generally more precise. this is because the potential errors, despite being relatively larger in a proportional sense, only affect the measurement of losses, which represent a fraction of the total energy introduced. for instance, when measuring the oxygen content in the flue gas using a combustion analyzer, the equipment accuracy is specified as ±0.2%. in practical terms, this means that if the measured oxygen percentage is 17%, the true value could lie between 16.966% and 17.034%. this small variation in oxygen measurement results in a slight fluctuation in calculated combustion efficiency, ranging from 74.1% to 74.3%. such precision underscores the importance of both accurate measurement tools and the indirect method in obtaining reliable efficiency results, while also highlighting that minor deviations in data can influence the outcomes. 3.4. reference model the measur model depicts the steam system under study, as shown in fig. 7. it provides a detailed overview of the various components of the steam system, starting from steam generation to condensate recovery, encompassing steam distribution and its process end-use. the thermodynamic model is given as a supplementary material n°2. table 4 summarizes the key steam system results. the total operational cost, including fuel and water consumption, is 446 664 $/y. for its part, co2 emissions amounts to 1552.35 t/y. fig. 7. baseline model in measur table 4. steam system summary fuel data water and steam data boiler fuel use 26 050.94 gj/y make-up water required 3453.21 m3 boiler fuel cost 442 866 $/y make-up water cost 3799 $/y emissions from fuel 1552.35 t co2/y steam generated 2 t/h total operation cost 446 664 $/y marginal steam cost 62.38 $/y green industry: energy efficiency improvement… 39 copyright ©2025 assa adv. in systems science and appl. (2025) 3.5. energy saving opportunities 3.5.1. stack loss improvement excess air and temperature of exhaust gases are two major factors impacting the energy efficiency of the boiler. in this section, the focus will be on these two parameters to enhance energy efficiency. the evaluation of energy, financial, and environmental savings will be examined.  excess air control the actual value of 17% oxygen in flue gas suggests an excess air level of around 700-800%, which is far beyond normal operational parameters. in this improvement action, we suggest installing an automatic o2 trim control. the reduction of % o2 from 17% to a typical value of 2% will improve the boiler efficiency from 74.3% to 86.7%, as shown in the calculator in fig. 8. fig. 8. stack loss calculation – o2 reduction the annual potential savings from this action, calculated from eq. (3.11), are 63 340 $, corresponding to 3725.8 gj, with an emissions reduction of 222 t co2. this represents a saving of 14% relative to the plant thermal energy bill. 𝜎 = 𝐾 × 1 − 𝜂 𝜂 (3.11) where 𝐾 is the total fuel operating cost.  stack loss recovery the proposed economizer will decrease the flue gas temperature from 151 to 105°c. using this heat exchanger to preheat feedwater will reduce the thermal loss in the stack from 25.7% to 19.5%, improving the boiler efficiency from 74.3 to 80.5% (see fig. 9). fig. 9. stack loss calculation – waste heat recovery the annual savings from this energy-saving action amount to 34 109 $/y, representing 2006.4 gj/y, with emissions reductions of about 120 t co2/y. from an energy bill perspective, a saving of 8% is expected. 40 k. ibn batouta, s. aouhassi, k. mansouri copyright ©2025 assa adv. in systems science and appl. (2025) 3.5.2. blowdown loss management as calculated in the data analysis section 3.3, the studied steam system has a relatively high blowdown rate of 20%. reducing this rate or recovering the blowdown thermal energy will certainly reduce fuel consumption. these two actions are investigated in this stage.  blowdown rate reduction the reduction of the blowdown rate from the actual value to a fair value of 1% requires close monitoring of makeup water chemical treatment. the water treatment station based on softeners must be periodically checked, and the conductivities monitored. additionally, it is suggested to install an automatic blowdown controller for intermittent bottom blowdown. this modification results in a financial saving of 29 301 $/y, an energy conservation of 1599 gj/y and an emissions reduction of 95 t co2/y. this is embodied by a 7% saving on fuel consumption.  blowdown energy recovery in this action, it is suggested that blowdown heat recovery equipment be installed to preheat makeup water. the potential savings are 27 210 $/y, 1593 gj/y, 95 t co2/y and a 6% saving reported to the fuel bill. fig. 10 outlines the impact of the two modifications regarding blowdown management in comparison to the reference model. fig. 10. baseline vs. blowdown reduction and recovery models 3.5.3. combined opportunities scenarios the combination of the four proposed improvement actions leads to an energy savings of 22%, resulting in an annual decrease in fuel consumption by 111.5 tons and a reduction in co2 emissions by 332.15 tons and a financial saving of 96 884$/y. the details of the achieved savings are summarized in table 5. table. 5. savings from actions combination baseline assessments combination % saving energy input (gj/y) 26 050.9 20 476.9 21.4% fuel usage (t/y) 521 409.5 fuel cost ($/y) 442 866 348 108 annual emissions (tonne co2) 1 552.35 1 220.2 make-up water cost ($/y) 3 799 1 673 56% annual cost ($) 446 664 349 781 21.7% the sankey diagrams in fig. 11-a and fig. 11-b illustrate the energy flows before and after adopting the combined actions scenario. the primary energy losses in steam generation – specifically stack losses and purge losses – were originally 24.5% and 5.3%, respectively, at the system level. these losses have now been significantly reduced to 11.4% for stack losses and 0.04% for purge losses. additionally, the green industry: energy efficiency improvement… 41 copyright ©2025 assa adv. in systems science and appl. (2025) end-use process has improved its energy utilization efficiency, using 71.8% of the supplied energy compared to just 56.1% in the reference model. fig.11-a. sankey diagram of energy losses-baseline fig.11-b. sankey diagram of energy losses-actions combination conclusion this paper presented the roadmap and methodology of a steam system optimization carried out for a moroccan plastic recycling industry. the benchmarking study disclosed that the “boiler plant operating practices” area needs the most attention in efficiency improvement, as it is the lowest-rated zone (25%). the diagnostic of the current situation in the boiler house revealed many thermal energy losses at the system level, namely the stack (25.4%) and the blowdown losses (5.3%). these percentages, reported to the fuel cost, present a financial loss of 135 960 $/y. to mitigate the impact of these losses, four energy-saving opportunities were presented. their objectives are to reduce and recover energy losses. two actions focus on reducing stack losses: decreasing excess air in combustion with an automatic air trim controller and preheating feedwater using an economizer in the chimney. the other two actions address blowdown management: reducing the blowdown rate and recovering heat from it. this involves using an automatic blowdown system, enhancing makeup water treatment, and installing blowdown recovery equipment to preheat makeup water. the combination of the precited actions leads to 22% potential savings, resulting in an energy gain of 5574 gj/y, financial benefit of 96 884 $/y and co2 emissions reduction of 332.15 t/y. a preliminary cost analysis has shown that the implementation of these actions has a payback period of less than one year. the current modeling approach provides a solid foundation for analyzing and improving the energy efficiency of the steam system. while the developed model allows for the adjustment of system parameters online, facilitating the assessment of their impact on energy savings, it currently lacks real-time simulation capabilities. our study has primarily focused on optimizing the system using static data and specific operational scenarios. however, we recognize the potential value of incorporating dynamic modeling to account for real-time fluctuations in system behavior, and this represents a promising area for future 42 k. ibn batouta, s. aouhassi, k. mansouri copyright ©2025 assa adv. in systems science and appl. (2025) research. specifically, we aim to explore the integration of dynamic simulation tools that would enable continuous monitoring and real-time adjustment of system parameters, potentially uncovering additional opportunities for optimization. moreover, incorporating more advanced mathematical models, performing sensitivity analyses, and regularly validating and calibrating the models – along with employing sophisticated simulation tools – could significantly enhance both the accuracy and efficacy of the modeling approach. these improvements would not only increase the reliability of the outcomes but also identify further possibilities for energy savings and system optimization. the motivation behind this case study is to demonstrate to industrialists that significant energy savings can be achieved by improving the energy efficiency of the existing system without the need for capitalintensive investments to change the current technologies. even though this study adopts a systemic approach, it has focused solely on improving the steam generation aspect in its improvement actions. in continuation of this work, we intend to address other parts of the steam system, namely distribution, enduse, and recovery. additionally, we plan to conduct a financial analysis of the energy-saving opportunities to demonstrate their cost-effectiveness. acknowledgement this study was conducted as part of a training program for experts in steam system optimization in morocco, organized by the united nations industrial development organization (unido) under the "accelerating energy efficiency in industry in morocco" (aeeim) program. we extend our sincere gratitude to the anonymous host plant for welcoming us and allowing us to conduct this case study on their premises. their cooperation and support were invaluable in enabling us to carry out this assessment and contribute to advancing energy efficiency practices in the industry. references 1. balli, l., atifi, a., hlimi, m., hamri, b. & achenani, y. 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(2015) efficiency studies of combination tube boilers, alexandria engineering journal, 55(1), 193–202, doi: 10.1016/j.aej.2015.12.007 20. tirumala srinivas, g. (2017) efficiency of a coal fired boiler in a typical thermal power plant, american journal of mechanical and industrial engineering, 2(1), 32, doi: 10.11648/j.ajmie.20170201.15 21. unido (2023) energy efficiency potentials in industrial steam systems in china development of a steam systems energy efficiency cost curve, [online]. available: https://www.unido.org/sites/default/files/201509/ee_potentials_steam_systems_china__0.pdf 22. us doe (2024) measur | department of energy, [online]. available: https://www.energy.gov/eere/iedo/measur microsoft word manuscript adv syst sci appl 2024; 03; 11-30 published online at https://ijassa.ipu.ru. does political instability affect business efficiency of russian firms? han-sol lee1, aleksei kurbatskii2*, nikita artamonov3 1patrice lumumba rudn university, moscow, russia 2lomonosov moscow state university, moscow, russia 3mgimo university, moscow russia abstract. this study delves into the impact of political instability on the business efficiency (proxied by growth rate of roa) of 3,706 russian firms for the period 2019-2022. the aim of the study is to reveal the impact of governance quality on the managerial efficiency of russian companies during recent years with a particular concern of the on-going political instability: the russia-ukraine conflict. the results from the binary probit regression analysis allow us to substantiate that the probability of roa growth and governance quality are inversely related in the recent years. however, when the governance indicators are interacted with the year dummy 2022, its effect becomes positive on the probability of growth of roa. this implies that due to the ongoing russia-ukraine conflict the improved governance quality starts to exert impacts on the business efficiency of russian firms. the overall impact of governance indicators is stronger for smes than for large companies in russia, indicating that large companies have better capacity to withstand external shocks and instabilities than smes. key words: political economy, governance, micro-data analysis, russia, russia-ukraine conflict 1. introduction in recent years, the world has become more turbulent in many ways, and russia holds one of the central places in the ongoing geopolitical and economic processes. some researchers believe that russia has suffered significant damage due to the severe sanctions, brain drain, and withdrawal of companies from the russian market [6]. conclusions about the significant impact of sanctions on gdp were made during the period of the first sanctions after march 2014 in the works [21, 47]. another camp of economists believes that despite certain limitations russia has the opportunity to once again become the leader of the global economic system [11, 28]. regardless of these polar points of view, the issue of effective governance is more acute than ever for the country's leadership and business [34, 46]. do not forget that russia in the 20th century was repeatedly the epicenter of political instability (participation in two world wars, the revolution, the collapse of the ussr). at the same time, extraordinary measures were often required to stabilize the situation, and the economy often had a mobilizational character [33]. the current political situation in russia and the world is fundamentally different from previous turbulent periods, which gives additional relevance to the study on assessing the impact of public administration on economic indicators. this study will analyse microdata of russian companies, thus, an attempt will be * corresponding author: akurbatskiy@gmail.com 12 h.-s. lee, a. kurbatskii, n. artamonov copyright ©2024 assa adv. in systems science and appl. (2024) made to reflect the influence of management factors in unstable political and economic situations on russian business (proxied by growth rate of roa). it should be noted that the current unstable situations in the economy and political life of countries do not entail the consequences that could have been expected 20-30 years ago. this is largely due to digitalization, which has entered all spheres of life [7]. even with such a vivid phenomenon as brain drain during periods of instability in the country, we now cannot draw the unambiguous conclusions that were obvious to most studies in the 20th century [10]. the current situation in russia's economy and politics was preceded by several, though not so comprehensive, crisis periods. the work [44] is devoted to the analysis of factors determining the efficiency of enterprises in countries with unstable economies using the example of the russian manufacturing industry for the period 2012-2016. we note one of the authors' conclusions that an increase in interest rates can have a negative effect on the profitability of companies, and in russia they are currently at a high level. for enterprises from various industries, the multidirectional effects of various variables, in particular, internal financial factors, are highlighted. an analysis of the activities of companies in three industries during the turbulent period of 2013-2017 in russia showed that an aggressive strategy has a positive effect on efficiency. it also surpasses the conservative one during the crisis [5]. this indirectly indicates that external public administration, if it is not very strict, may not play a primary role in the performance of firms. many studies have been devoted to the effectiveness of government policy to support enterprises during the covid19 pandemic, as well as various aspects of improving economic policy in crisis and post-crisis periods [40, 50]. the paper [20] discusses the features of a turbulent business environment and examines various effective strategies that companies can use to adapt to the new realities that occurred during and after the covid19 pandemic. our research fully covers these periods and additionally includes the challenging year 2022. the purpose of this study is to reveal the impact of governance quality on the managerial efficiency of russian firms during the post-covid19 periods with special concern for the ongoing russiaukraine conflict. although, multiple papers delve into the significant role of governance in russia [15, 27, 37, 48], the changing role of governance under this ongoing political instability has not yet been investigated. remaining sections of the paper is composed of as follows. in section 2, previous studies will be studied and hypotheses will be developed. section 3 presents data and methodology. section 4 describes regression results. section 5 compares our results with the previous studies and provides policy implications. section 6 concludes the study. 2. literature review 2.1. the impact of the governance on the economy governance measures how well the government creates and enforces regulations, as w ell as provides services and is concerned with agents' achievements in conducting the objectives of principals, rather than the aims set by leaders [19]. however, as rothste in [42] has pointed out, strictly separating governance from the executive capabilities from the normative goals that government is intended to serve is rather difficult. but, at the same time, he criticizes the world governance indicators (wgi) for containing an excessive number of criteria that reflect normative policy preferences. even that, wgi is the most widely adopted as a proxy to the governance quality in empirical st udies due to its comprehensiveness, large coverage of countries and periods and reliab ilities of sources. however, as pointed out by rothstein [42] and fukuyama [19], the does political instability affect business efficiency… 13 high scores in wgi do not fully guarantee that the government showed a better performance in the aspect of outcomes but rather mostly do that it follows well the standards that are commonly considered good, when to interpret the regression results, it should be taken into account. as can be expected, the significance of the good governance and its positive impact on the economy were empirically demonstrated in a plethora of studies [8, 12, 14, 17, 22, 24, 32]. the results of these studies support the importance of implementing good governance policies, such as citizen participation, transparency, systemized regulatory bodies and law, fight against corruption and so forth, to promote sustainable economic growth. some studies demonstrate the significance of a specific factor of the good governance. for instance, [41] demonstrated that stronger democratic institutions influence governance by limiting the actions of corrupt officials and thereby democracy is a significant determinant of growth, but only to the extent that it is associated with better governance. [9] revealed that corruption has a significant negative impact on gross domestic product (gdp) per capita. however, interestingly, some heterogeneities are found in previous researches. the study by [23] examined the relationship between governance and economic growth in twelve asian countries classified as "free", "partly free" and "non-free" according to a country’s level of democracy. the paper found that, overall, various aspects of good governance led to more significant economic growth in “non-free” countries compared to “free” and “partly free” countries. in particular, [2] demonstrated that although the relationship between the good governance and economic growth relationship are positive, but its significance varies depending on the level of development of each country during the crisis-period: its significance is strongly shown in highly developed nations but not in medium and less developed nations. on the other hand, some studies demonstrated that the bad governance can serve as a chance to gain benefits. [18] revealed that transnational companies (with own strategic resources and competencies) in the nigerian oil industry sometimes even benefit from political risk. [36] demonstrated that political stability and regulatory quality boost economic growth, while government effectiveness negatively influences on it and the role of anti-corruption in it is ambiguous. [49] revealed that solid regulatory control and political stability have a positive association with the economic growth but its significance depends on the income level of the country. while, the study added that in some cases corruption can play a positive role because it bypasses ineffective rules and bureaucratic bottlenecks. in a study of [1], it is turned out that political instability has a negative impact on economic growth as hypothesized, while, unlike the general notion, democracy also can have a negative effect on economic growth. in addition, there are very limited studies in this type of topic, which is dedicated to the russian economy. [25] empirically measured the relation between efficient public administration and the russian economic growth. the study revealed that the association between regional governance system performance indicators and socioeconomic territorial growth is absent, while higher maintenance expenditures of the local executive authorities (e.g., the wage of civil servants, the number of civil servants, etc.) can exert significant effects on the socioeconomic growth in russia. [31] revealed a non-linear relationship between the democracy and economic growth of russian regions: regions with either high or low democracy outperform those with intermediate. on the other hand, the topic of governance in russia has been extensively studied [15, 27, 37, 48], but few papers have directly measured its impact on the economy like the two aforementioned. from the review of previous studies several research gaps were identified. first of all, considering the significance of geo-political factors in russia, some papers have addressed the governance issue but it is rare to find studies that measure its effects on the economy. secondly, most studies investigated the macro-economic impact, although it is interesting and worthwhile 14 h.-s. lee, a. kurbatskii, n. artamonov copyright ©2024 assa adv. in systems science and appl. (2024) to see its effects at the micro-level. thirdly, in spite of the fact that the impact of the quality governance on the economy can vary depending on the geopolitical stability, most of previous studies have not considered such potential heterogeneity. to elaborate on this point, suppose that the impact of the governance quality on the economy turned out to be insignificant when there is no geopolitical conflict. however, during the era when the external instability is sharply enhanced due to factors such as global conflicts, economic crisis, and others, the systematic governance can be highly important and relieve the negative effects from the external instability on the economy. our study will provide new insights in the following two strands: (1) to empirically prove the impact of the governance factors on the business efficiency of the russian firms by employing the micro-level data; (2) to estimate the heterogeneity depending on the political instability. 2.2. hypothesis development based on the above literature review, research hypotheses can be formulated. first of all, the relationship between the quality of governance and economic (profit) growth of the country (firms) is rather vague. companies can maximize profits under well-established governance and low-risk environments. however, in some cases, they may utilize the lenient governmental systems and consider it as an additional business opportunity for existing companies. to elaborate on this point, entrepreneurship is suppressed under the poor governance due to a high risk of opening a business and which will reduce the market entry of new firms and competitions as well accordingly. thereby, it is difficult to say that a quality governance always leads to better firms’ performance. meanwhile, the drawbacks from the government systems can be beneficial to existing companies in some cases. if the former effects are larger, then the relationship between the two will be positive, while if the latter effects are larger, then it will be negative. hypothesis 1 (h1): the impact of governance quality on business efficiency (proxied by the probability of roa growth) of russian firms is uncertain. however, during the period that the social instabilities are sharply enhanced due to international political (e.g., war, conflict, etc.) or economic (e.g., financial crisis, inflation, currency depreciation, etc.) issues, the risk of conducting business increases remarkably for existing companies because the nation’s consumer confidence can be weakened and the household’s purchasing power can be reduced due to related financial policies such as interest rate, inflation rate, exchange rate, and so forth. during such unstable era, there is low motivations of the market entry of new firms (especially foreign firms) regardless of the governance quality but the consumptions for existing firms may decrease. in this case, the risk may be way larger than the benefits that firms can enjoy from the lenient governmental systems and firms may prefer more stable governmental environments than before because it can lead to the promotion of the nation’s consumer confidence. thereby, we can conjecture that: hypothesis 2 (h2): the interaction term between the governance quality indicators and the year 2022 variable is positively correlated with business efficiency (proxied by the probability of roa growth) of russian firms. 3. data and methodology for our study, binary probit model is adopted. the dependent variable is either 1 or 0. the calculation for the probability of y=1 can be made as follows: 𝑃𝑟𝑜𝑏(𝑌 = 1|𝑥) = φ(𝑥 𝛽) (1) here x is a vector of independent variables, while 𝛽 is a a vector of coefficients. φ is a distribution function of standard gaussian distribution. in the same way, the probability of y=1 does political instability affect business efficiency… 15 can be rendered as follows [29]: 𝑃𝑟𝑜𝑏(𝑌 = 0|𝑥) = 1 − φ(𝑥 𝛽) (2) the probit model of our study is follows: 𝑃𝑟𝑜𝑏 𝐺𝑟𝑤 = φ(𝛽 + 𝛽 𝐺𝑜𝑣𝑒𝑟𝑛𝑎𝑛𝑐𝑒 + 𝛽 𝐺𝑜𝑣𝑒𝑟𝑛𝑎𝑛𝑐𝑒 ∗ 2022 + 𝛽 𝐶𝑜𝑛𝑡𝑟𝑜𝑙 ) (3) the description of variables and its sources are presented in table 1. in the baseline model, 𝐺𝑟𝑤_𝑅𝑂𝐴 = 1, 𝑖𝑓 𝑔𝑟𝑜𝑤𝑡ℎ 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑅𝑂𝐴 > 0 0, 𝑖𝑓 𝑔𝑟𝑜𝑤𝑡ℎ 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑅𝑂𝐴 ≤ 0 while, in the model for the robustness checks, an alternative threshold is used as follows: 𝐺𝑟𝑤_𝑅𝑂𝐴 = 1, 𝑖𝑓 𝑔𝑟𝑜𝑤𝑡ℎ 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑅𝑂𝐴 ≥ 0.5 0, 𝑖𝑓 𝑔𝑟𝑜𝑤𝑡ℎ 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑅𝑂𝐴 < 0.5 the key variable is: 𝐺𝑜𝑣𝑒𝑟𝑛𝑎𝑛𝑐𝑒 ∈ {𝑉𝐴, 𝑃𝐴, 𝐺𝐸, 𝑅𝑄, 𝑅𝐿, 𝐶𝐶, 𝐶𝑙𝑢𝑠𝑡𝑒𝑟1, 𝐶𝑙𝑢𝑠𝑡𝑒𝑟2, 𝐶𝑙𝑢𝑠𝑡𝑒𝑟3, 𝐴𝑙𝑙}. 𝐺𝑜𝑣𝑒𝑟𝑛𝑎𝑛𝑐𝑒 ∗ 2022 is an interaction term between governance indicators and the year 2022 to see the effect of the russo-ukrainian conflict. due to multicollinearity, it is not advisable to include six governance variables in one model. in general, there are three ways to build a model using wgi datasets. first, consider one variable at a time and create 6 models [17, 23]. second, create 3 clusters of variables [3, 4] and generate 3 models. to simply use the average value of six indicators and make 1 model [4, 16, 30]. in our study, all three approaches are adopted. control variables are as follows: 𝐶𝑜𝑛𝑡𝑟𝑜𝑙 ∈ {𝐺𝑟𝑤_𝑒𝑥𝑝𝑒𝑛𝑠𝑒, 𝐿𝑛(𝑎𝑔𝑒)} . for the robustness checks, models with alternative control variables (which are grw_gdp, openness, ln(bop), and fdi_ratio) as well as a logit estimator are also constructed. in addition, the issue of heterogeneity depending on the company size is explored. table 1. variable definitions and data sources notation definition source i companies in russia spark 𝑡 year (2019-2022) spark 𝐺𝑟𝑤_𝑅𝑂𝐴 growth rate of roa of company i in year t, binary dependent variable (1 or 0) spark 𝑉𝐴 voice and accountability (va) of russia in year t in the interval [-2.5, +2.5] the world governance indicators (wgi) 𝑃𝐴 political stability and absence of violence/terrorism (pa) of russia in year t in the interval [-2.5, +2.5] wgi 𝐺𝐸 government effectiveness (ge) of russia in year t in the interval [-2.5, +2.5] wgi 𝑅𝑄 regulatory quality (rq) of russia in year t in the interval [-2.5, +2.5] wgi 𝑅𝐿 rule of law (rl) of russia in year t in the interval [-2.5, +2.5] wgi 𝐶𝐶 control of corruption (cc) of russia in year t in the interval [-2.5, +2.5] wgi 𝐶𝑙𝑢𝑠𝑡𝑒𝑟1 the average of va and pa of russia in year t in the interval [-2.5, +2.5] wgi 𝐶𝑙𝑢𝑠𝑡𝑒𝑟2 the average of ge and rq of russia in year t in the interval [-2.5, +2.5] wgi 𝐶𝑙𝑢𝑠𝑡𝑒𝑟3 the average rl and cc in the interval [-2.5, +2.5] wgi 16 h.-s. lee, a. kurbatskii, n. artamonov copyright ©2024 assa adv. in systems science and appl. (2024) 𝐴𝑙𝑙 the average va, pa, ge, rq, rl and cc of russia in year t in the interval [-2.5, +2.5] wgi 2022 year dummy (2022=1, otherwise 0) 𝐺𝑟𝑤_𝑒𝑥𝑝𝑒𝑛𝑠𝑒 the growth rate of the sum of the commercial and management expenses of company i in year t spark 𝐿𝑛(𝑎𝑔𝑒) the age of company i in year t spark 𝐺𝑟𝑤_𝐺𝐷𝑃 the gdp growth rate of russia in year t the world development indicators (wdi) 𝑂𝑝𝑒𝑛𝑛𝑒𝑠𝑠 trade openness of russia in year t wdi 𝐿𝑛(𝐵𝑜𝑃) balance of payment in trade of russia in year t wdi 𝐹𝐷𝐼_𝑟𝑎𝑡𝑖𝑜 fdi inflows/gdp ratio of russia in year t wdi 𝜖 error term note: before calculating roa and expenses in a growth term, the current values in rubles are converted to constant values in ruble by applying the gdp deflator of russia (base year=2016) from the world development indicators (wdi). figure 1 illustrates the dynamics of governance quality in russia for the period 2018-2022. in 2019, the average of governance indicators was the highest because 4 out of 6 indicators increased. particularly, government effectiveness (ge) was above than the 0 in this year. however, due to the russia-ukraine conflict since the early 2022, all the governance indicators decreased from 2021 to 2022 to the level below than before. especially, the downward slope of regulatory quality (rq) was the steepest, while control of corruption (cc) showed the most minimal decreases. -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 2018 2019 2020 2021 2022 va pa ge rq rl cc figure 1. the world governance indicators in russia figure 2 depicts the mean of roa dynamics of russian companies for the period 20182022. in 2019, the mean of roa hit the bottom and its patterns are retrieved as it had continuously increased till 2021. in 2022, its roa increased while the slope of the rising curve became flatter. this pattern is commonly shown without the heterogeneity in the firm size. throughout the study period, the mean of roa was higher in smes than large companies. although the slope of the rising curve of roa became flatter both in large and smes, the does political instability affect business efficiency… 17 degree of flatness is more severe in smes. it seems that the russia-ukraine conflict somewhat applied the brakes of the rising tendencies of roa, and influences much strongly on smes than large companies. figure 2. the mean of roa of companies in russia 4. results table 2 presents the results of the baseline regressions. va is positively correlated with the probability of the growth of roa at the 1% significant level. however, when interacted with the year 2022 dummy variable, its coefficient becomes negative. it implies that during the recent years citizens freedom in politics, expressions, media, and so forth enhances the probability of firms’ business efficiency but its positive impact turns negative when interacted with the year 2022. to take into account the factor of political instability in 2022, to restrict the voices of citizens to a certain level is beneficial to firms’ efficient business management. on the other hand, it is revealed that pa, ge, rq, rl and cc are negatively associated with the probability of the growth of roa at the 1% significant level. however, when it is interacted with the year 2022 dummy variable, its coefficient becomes positive. it is likely that during the recent years, russian firms take advantages even from the vulnerability of society and systems, such as political conflict/violence, inferiority of public service, inability of government’s policy implementation and vulnerability of law and corruption, to increase the efficiency of their business. however, as the political instability and uncertainty becomes deepened, a situation and status of firms has changed, and as a result, having a bit more stable and quality governance than before leads to increase in the probability of firms’ business efficiency. the same results are shown in clustered variables (i.e., cluster 1, cluster2, cluster3, and all). to interpret the results intuitively, eq.(4) is rendered as follows from the table 2: 𝑃𝑟𝑜𝑏(𝐺𝑟𝑤 ) = 𝛷 −1.64 − 2.13 ⋅ 𝐴𝑙𝑙 + 0.76 ⋅ (𝐴𝑙𝑙 ⋅ 2022) + 0.02 ⋅ 𝐺𝑟𝑤 + 0.09 ⋅ 𝐿𝑛(𝑎𝑔𝑒) (4) at here, the coefficient of the interaction term of governance and the year 2022 (all*2022) 0 0,05 0,1 0,15 0,2 0,25 2018 2019 2020 2021 2022 all companies large companies smes 18 h.-s. lee, a. kurbatskii, n. artamonov copyright ©2024 assa adv. in systems science and appl. (2024) is positive, while its absolute value |+ 0.76| is less than the absolute value of the coefficient of the governance indicator (all) |−2.13|. it indicates that the negative correlation between governance indicators and the probability of roa growth is reduced during the year 2022, while its correlation still remains negative. the results are consistent in models with a different threshold of roa, logit estimator, and alternative control variables [see table a1, a2 and a3]1. to check the heterogeneity, observations are divided by company size. table 3 represents regression results of large companies, while table 4 depicts that of smes. the results are following that of the baseline regressions in general. however, there are points that require o give a special attention. first of all, rq and rq*2022 turned out to be insignificant for large companies. second, the absolute value of the coefficient is larger in smes than large companies. this implies that smes are much influenced by external governance factors than large companies. it seems that large companies have better capacities to mitigate external shocks and instabilities than smes through their internal systems and these systems acts as a cushion against shocks, while smes absorb shocks more completely without a shock-resistant mediator. 1 models for robustness checks are presented in appendix. does political instability affect business efficiency… 19 table 2. baseline regressions (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) constant 5.22*** -1.23*** -0.31*** -1.18*** -1.49*** -2.26*** -2.99*** -0.65*** -2.18*** -1.64*** (0.70) (0.12) (0.05) (0.21) (0.21) (0.22) (0.36) (0.07) (0.23) (0.18) va 4.97*** (0.64) va*2022 -0.62*** (0.09) pa -1.59*** (0.18) pa*2022 0.51*** (0.06) ge -1.01*** (0.12) ge*2022 0.90*** (0.11) rq -1.91*** (0.41) rq*2022 1.10*** (0.23) rl -1.55*** (0.25) rl*2022 0.51*** (0.08) cc -2.30*** (0.25) cc*2022 0.24*** (0.03) cluster1 -3.20*** (0.42) cluster1*2022 0.67*** (0.09) cluster2 -1.46*** (0.20) cluster2*2022 1.01*** (0.13) cluster3 -2.31*** (0.27) cluster3*2022 0.52*** (0.06) all -2.13*** (0.27) all*2022 0.76*** (0.10) grw_expense 0.02*** 0.02*** 0.02*** 0.02*** 0.02*** 0.02*** 0.02*** 0.02*** 0.02*** 0.02*** (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) ln(age) 0.10*** 0.09*** 0.09*** 0.10*** 0.10*** 0.09*** 0.09*** 0.09*** 0.09*** 0.09*** (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) mcfadden r-squared 5x10-3 6x10-3 5x10-3 3x10-3 4x10-3 6x10-3 5x10-3 5x10-3 6x10-3 5x10-3 n 14824 14824 14824 14824 14824 14824 14824 14824 14824 14824 note: standard errors are in parentheses (*: p<0.1, **: p<0.05, ***: p<0.01) 20 h.-s. lee, a. kurbatskii, n. artamonov copyright ©2024 assa adv. in systems science and appl. (2024) table 3. heterogeneity checks: the impact of governance indicators on roa of large companies (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) constant 6.14*** -0.93*** -0.08*** -0.65 -0.99** -2.02*** -2.39*** -0.36** -1.78*** -1.24*** (1.37) (0.24) (0.11) (0.40) (0.40) (0.43) (0.70) (0.15) (0.45) (0.36) va 5.60*** (1.24) va*2022 -0.75*** (0.17) pa -1.47*** (0.35) pa*2022 0.42*** (0.12) ge -0.89*** (0.24) ge*2022 0.73*** (0.22) rq -1.27 (0.79) rq*2022 0.69 (0.45) rl -1.20** (0.49) rl*2022 0.36** (0.16) cc -2.29*** (0.48) cc*2022 0.18*** (0.06) cluster1 -2.76*** (0.81) cluster1*2022 0.54*** (0.17) cluster2 -1.24*** (0.38) cluster2*2022 0.81*** (0.26) cluster3 -2.10*** (0.53) cluster3*2022 0.43*** (0.12) all -1.86*** (0.52) all*2022 0.62*** (0.19) grw_expense -7x104 -9x105 -4x104 -1x103 -1x103 1x1043 -5x104 -6x104 -2x104 -4x104 (0.01) (0.01) 0.01 (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) ln(age) 0.02 0.01 0.01 0.02 0.02 0.01 0.01 0.02 0.01 0.01 (0.04) (0.04) (0.04) (0.04) (0.04) (0.04) (0.04) (0.04) (0.04) (0.04) n 3916 3916 3916 3916 3916 3916 3916 3916 3916 3916 note: standard errors are in parentheses (*: p<0.1, **: p<0.05, ***: p<0.01) does political instability affect business efficiency… 21 table 4. heterogeneity checks: the impact of governance indicators on roa of smes (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) constant 5.06*** -1.41*** -0.42*** -1.45*** -1.76*** -2.47*** -3.39*** -0.80*** -2.46*** -1.89*** (0.82) (0.14) (0.06) (0.24) (0.24) (0.26) (0.43) (0.09) (0.27) (0.22) va 4.92*** (0.75) va*2022 -0.61*** (0.10) pa -1.72*** (0.21) pa*2022 0.56*** (0.07) ge -1.10*** (0.15) ge*2022 1.01*** (0.13) rq -2.25*** (0.48) rq*2022 1.30*** (0.27) rl -1.76*** (0.29) rl*2022 0.59*** (0.09) cc -2.42*** (0.29) cc*2022 0.26*** (0.04) cluster1 -3.54*** (0.49) cluster1*2022 0.76*** (0.10) cluster2 -1.62*** (0.23) cluster2*2022 1.13*** (0.16) cluster3 -2.51*** (0.32) cluster3*2022 0.57*** (0.07) all -2.34*** (0.32) all*2022 0.85*** (0.11) grw_expense 0.08*** 0.09*** 0.09*** 0.08*** 0.08*** 0.09*** 0.09*** 0.09*** 0.09*** 0.09*** (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) ln(age) 0.14*** 0.13*** 0.13*** 0.13*** 0.13*** 0.13*** 0.13*** 0.13*** 0.13*** 0.13*** (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) n 10908 10908 10908 10908 10908 10908 10908 10908 10908 10908 note: standard errors are in parentheses (*: p<0.1, **: p<0.05, ***: p<0.01) 22 h.-s. lee, a. kurbatskii, n. artamonov copyright ©2024 assa adv. in systems science and appl. (2024) 5. discussion from the regression results, two main findings are rendered. first of all, the relationship between the quality of governance and the probability of roa growth in russia is negative. this implies that companies were more likely to experience roa growth during the period when wgi scores for russia are low. these results are contradicting previous studies like [8, 12, 14, 17, 22, 24, 32]. however, some previous studies support the idea that lenient governance can act as an opportunity to obtain benefits the other way around [18, 49]. one possible explanation of the results is that business in russia were prepared to a certain extent. it is worth noting that the first international sanctions on russian business were introduced back in 2014 [13], when the russia-ukraine conflict actually began. it was then that import substitution tasks were set in many industries, and counter-sanctions were adopted [43]. also in 2018, after the election of the president of russia, national projects were announced, one of which is called “small and medium-sized entrepreneurship and support for individual entrepreneurial initiative.” the government is well aware of the priority tasks facing the country, but their implementation will require the development of appropriate institutions. in fact, in our opinion, institutions of catch-up development are needed to build coordination between the government and business in russia [38]. companies, and not only russian ones, will have to adapt to the current situation and the consequences of the russia-ukraine conflict. at the same time, it is not so much the date of the end of hostilities that is unpredictable and extremely important for business, but how world trade will be change and what restrictions will be lifted from russia [39]. the results of the work [35] provide an interesting look at the capabilities of firms during external shocks. it is noted that new firms can find themselves in favourable conditions due to arbitration opportunities, and existing firms are better able to focus on innovation. apparently, the activity and adaptability of business in russia makes it possible to avoid serious shocks in the economy, which were assumed by the west when introducing various kinds of restrictions. it is impossible not to note that the results are based on wgi indicators, the limits of applicability of which [45], especially for russia, may be considerably worse than desired. it is worth paying attention to the fact is that as criticized by [42] and [19], the composition of wgi indicators is largely dependent on whether a nation adheres well to the normative values of good governance rather than whether a nation achieves outcomes effectively. this implies that there is a chance that when russia did not follow the norms of governance which is perceived good in general, adversely, the real efficiency and outcomes of governmental administrations were improved. in addition, as hypothesized, the interaction term between governance and the year 2022 variable turned out to be positive. it implies that during the politically unstable period, firms performed better under the improved governance quality. the year 2022 moderated the negative effect of the governance quality on the probability of roa growth. as explained previously in the hypothesis development, during the politically unstable period, the good governance may enhance the consumer confidence, while firms are reluctant to enter in the market due to excessively high risks. during the politically stable period, bad governance can play a chance for existing companies as it prevents the entry of new companies. however, during the politically crisis period, the firms’ motivation of entry in the market is reduced already regardless of the quality of governance. at this point, the quality of governance becomes a factor in determining consumer confidence. although, the overall effects of governance quality in the year 2022 on the russian firms were negative, if its time spans are enlarged, it can turn into be positive even. one thing that we need to focus on is that the ongoing russia-ukraine conflict made foreign companies highly careful when to enter in or return does political instability affect business efficiency… 23 back to the russian market. thereby, we can expect that the future competition in the market will be reduced even when the conflictual situation ends. in this sense, we can somehow expect that the overall effect of governance quality can turn into positive in russia in the long-term as it will enhance the consumer confidence and increase efficiency of the market. however, our conclusions cannot be considered exceptional, discussions on similar issues have been ongoing on for quite a long time [26]. in our opinion, only the course of further events and the availability of necessary new data will allow us to see how long-term the identified trends are. 6. conclusion this study explores the impact of governance quality and political instability on the b usiness efficiency of russian companies. probit models are constructed based on panel data of 3,706 russian firms for the period 2019-2022. the aim of the study is to rev eal the impact of governance quality on the managerial efficiency of russian compani es during the recent years with a particular concern of the on-going political instabilit y: the russia-ukraine conflict. the results from the binary probit regression analysis a llow us to substantiate that the probability of roa growth and governance quality are inversely related in recent years. however, when the governance indicators are interact ed with the 2022 dummy variable, its effect becomes positive on the probability of r oa growth. this implies that during the recent years, russian firms have taken advantages from the vulnerability of society and systems, such as political conflict/violence, inferiority of public service, inability of government’s policy formulation/implementation and vulnerability of law and corruption, to increase the efficiency of their business. however, as the political instability and uncertainty becomes deepened, a situation and status of firms has changed, and as a result, having an improved governance leads to increase in the probability of firms’ business efficiency. the overall impact of governance indicators is stronger for smes than larges companies in russia and it shows that large companies have better capacities to relieve external shocks and instabilities than smes. as pointed out previously, wgi indicators are normative and may not fully describe the quality of governance in some countries. thus, we can conjecture that there is a chance that when russia did not adhere to the norms of governance which is perceived good in general (thus, marked low in the wgi), the efficiency and outcomes of governmental administrations may have improved. on the other hand, it could also be true that some companies obtain more business chances under the weak governmental system. thus, the interpretation of the negative relationship between the roa growth probability and governance quality leaves larges rooms for interpretations and discussions. while, during the politically unstable period, the good governance may enhance the consumer confidence, while firms are loath to enter in the market due to excessively high risks. during the politically unstable period, the firms’ motivation of entry in the market is reduced already regardless of the quality of governance. at this point, the quality of governance is a factor only to determine the consumer confidence not a market competition. thus, we can cautiously conjecture that although, the overall effects of governance quality in 2022 on russian firms were negative, if its time span is extended, it can turn into be positive even considering the expected reductions in future foreign direct investments and following low competitions at the russian market. however, our conclusions cannot be considered exceptional, and only the course of further events and the availability of new necessary data will allow us to determine how long-term the identified trends are. 24 h.-s. lee, a. kurbatskii, n. artamonov copyright ©2024 assa adv. in systems science and appl. 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(2024) appendix table a1. robustness checks: alternative threshold of roa growth (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) constant 3.78*** -1.52*** -0.79*** -1.39*** -1.64*** -2.37*** -2.82*** -1.05*** -2.25*** -1.81*** (0.75) (0.13) (0.05) (0.22) (0.22) (0.24) (0.39) (0.08) 0.25 (0.20) va 4.11*** (0.68) va*2022 -0.54*** (0.09) pa -1.25*** (0.19) pa*2022 0.37*** (0.06) ge -0.77*** (0.13) ge*2022 0.66*** (0.12) rq -1.31*** (0.43) rq*2022 0.74*** (0.24) rl -1.12*** (0.27) rl*2022 0.35*** (0.09) cc -1.86*** (0.27) cc*2022 0.16*** (0.03) cluster1 -2.42*** (0.45) cluster1*2022 0.49*** (0.09) cluster2 -1.10*** (0.21) cluster2*2022 0.74*** (0.14) cluster3 -1.79*** (0.29) cluster3*2022 0.38*** (0.07) all -1.62*** (0.29) all*2022 0.56*** (0.10) grw_expense 0.02*** 0.02*** 0.02*** 0.02*** 0.02*** 0.02*** 0.02*** 0.02*** 0.02*** 0.02*** (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) ln(age) 0.06*** 0.05** 0.05** 0.05*** 0.05*** 0.05** 0.05** 0.05** 0.05** 0.05** (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) n 14824 14824 14824 14824 14824 14824 14824 14824 14824 14824 note: standard errors are in parentheses (*: p<0.1, **: p<0.05, *** p<0.01); the threshold of grw_roa is ≥ 0.5 in these models. does political instability affect business efficiency… 29 table a2. robustness checks: alternative logit model (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) constant 8.42*** -2.03*** -0.53*** -1.96*** -2.45*** -3.71*** -4.92*** -1.08*** -3.60*** -2.71*** (1.13) (0.20) (0.08) (0.33) (0.33) (0.36) (0.58) (0.12) (0.38) (0.30) va 8.05*** (1.02) va*2022 -1.01*** (0.14) pa -2.60*** (0.29) pa*2022 0.83*** (0.10) ge -1.65*** (0.20) ge*2022 1.47*** (0.18) rq -3.13*** (0.65) rq*2022 1.79*** (0.37) rl -2.53*** (0.40) rl*2022 0.83*** (0.13) cc -3.75*** (0.39) cc*2022 0.38*** (0.05) cluster1 -5.24*** (0.67) cluster1*2022 1.10*** (0.14) cluster2 -2.38*** (0.31) cluster2*2022 1.65*** (0.21) cluster3 -3.78*** (0.44) cluster3*2022 0.84*** (0.10) all -3.47*** (0.43) all*2022 1.24*** (0.15) grw_expense 0.08*** 0.09*** 0.08*** 0.07*** 0.08*** 0.09*** 0.08*** 0.08*** 0.09*** 0.08*** (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) ln(age) 0.17*** 0.16*** 0.16*** 0.17*** 0.16*** 0.16*** 0.16*** 0.16*** 0.16*** 0.16*** (0.03) (0.03) (0.03) (0.03) (0.03) (0.03) (0.03) (0.03) (0.03) (0.03) n 14824 14824 14824 14824 14824 14824 14824 14824 14824 14824 note: standard errors are in parentheses (*: p<0.1, **: p<0.05, *** p<0.01) 30 h.-s. lee, a. kurbatskii, n. artamonov copyright ©2024 assa adv. in systems science and appl. (2024) table a3. robustness checks: alternative control variables threshold: grw_roa>0 threshold: grw_roa≥0.5 (1) (2) (3) (4) (5) (6) (7) (8) constant -1.69*** -0.14 3.83*** -1.44*** -1.92*** -0.54 2.99*** -1.70*** (0.19) (0.32) (1.08) (0.18) (0.20) (0.34) (1.14) (0.20) all -2.51*** -2.52*** -2.31*** -2.39*** -1.93*** -1.95*** -1.76*** -1.83*** (0.27) (0.28) (0.27) (0.27) (0.30) (0.30) (0.29) (0.29) all*2022 0.81*** 1.04*** 0.69*** 1.13*** 0.59*** 0.80*** 0.49*** 0.88*** (0.10) (0.11) (0.10) (0.12) (0.10) (0.12) (0.10) (0.13) grw_gdp -0.90*** -0.80*** (0.18) (0.19) openness -3.15*** -2.80*** (0.64) (0.67) ln(bop) -0.21*** -0.19*** (0.04) (0.05) fdi_ratio -8.80*** -7.83*** (1.78) (1.87) n 14824 14824 14824 14824 14824 14824 14824 14824 note: standard errors are in parentheses (*: p<0.1, **: p<0.05, *** p<0.01) adv syst sci appl 2023; 03:139–152 published online at https://ijassa.ipu.ru. accurate indoor positioning system based on visible light asmaa hadjer saboundji*, mokhtar keche, mohammed dahmani laboratoire signaux et images, université des sciences et de la technologie d’oran mohamed boudiaf (usto-mb), el m’naouar-bir el djir-oran, algeria abstract: besides its use in high-rate wireless communications, visible light has recently emerged as an interesting positioning technology, which is referred to as visible light positioning (vlp). its main advantage is its high positioning accuracy. like with other positioning technologies, many techniques may be used with vlp, such as received signal strength (rss), (difference) time of arrival ((d)toa), angle of arrival (aoa), etc. in this paper, we investigate the use of the rss technique for vlp. two new methods are proposed and their performances are compared experimentally to those of three methods from the literature. both of the proposed methods are based on polynomial fitting. the obtained results show that the second one has a positioning accuracy that is comparable to that of the best method among these three reference methods, while being simpler. it should be highlighted that with only standard low-cost commercial components relationship with, we could obtain a very high positioning accuracy, with a mean localization error in the order of 1 cm. keywords: indoor positioning, visible light positioning, rss, machine learning, neural networks 1. introduction the localization of people or objects has become an essential part of the deployment of services in many fields, such as retail, logistics, urban planning, leisure, etc. the success of global positioning systems (gps) and the mobile revolution have forever changed our relation to technology, both in business and in our personal lives. gps cannot be used inside buildings because of the high-power attenuation, the multipath phenomena, and the low signal penetration [20]. many technologies have been proposed for indoor location, such as: wifi [28], bluetooth [24], radio frequency identification (rfid) [9], zigbee [5], and visible light [18]. the latter technology relies on the use of a light-emitting diode (led), which is a promising light source, due to its low energy consumption. besides this, compared to traditional light sources, leds have many advantages, including brightness efficiency and long service life. in addition to lighting purposes, leds can also be easily modulated at relatively high data rates for data communications [18] . such a technology is called visible light communication (vlc). they also can be used for visible light positioning (vlp). vlp is considered as an alternative solution for indoor positioning since it does not suffer from radio or electromagnetic interference. it has potential advantages such as cost reduction, durability, and environmental friendliness, but above all, from the point of view of localization, systems based on visible light have the capacity to estimate the position of an object with high precision [36]. there are several vl-based positioning methods, which can be classified into two categories according to the type of receiver [2]: the photodiode-based methods and the camera-based methods. ∗corresponding author: asmaahadjar.saboundji@univ-usto.dz 140 a.h. saboundji, m. keche, m. dahmani camera based vlp refers to the use of an image sensor as a terminal for the positioning receiver; it is preferred in commerce and industry because of its high positioning accuracy [6]. in a vlp system, a camera is used to capture led images. then, the position of the receiver is determined based on the relationship between the known 3d coordinates of the leds in a real-world plane and their 2d coordinates in the image plane [11]. a lot of work has been done both for leds identification in the image plane and positioning. a camera based positioning system, which uses triple, or double leds is proposed in [7]; it achieves a centimeter accuracy. another real-time vlp method, which uses double leds, with a lightweight image processing algorithm, is proposed in [19]; it achieves a 3.93 cm positioning accuracy. zhang et al. proposed in [33] a single led vlp method, which exploits the property of circular projection and uses the geometric characteristics of the led image to determine the position and orientation of the receiver with respect to the led lamps reference plane. the average positioning error is reported to be 17.52 cm. in [16], an angleof-arrival localization algorithm that uses a camera and three or more leds is proposed; it can achieve decimeter-level accuracy. although vlp-based camera localization methods can achieve decimeter or even centimeter accuracy, they require expensive sensors and complex image processors [23]. on the other hand, pd-based positioning methods have a wide range of applications in the indoor positioning field due to the high sensitivity of a pd to the light, its low price, and the possibility of high reception bandwidth. pd-based positioning systems are generally based on multilateration or triangulation, which requires the calculation of the distance or angle between the emitter and the receiver. many techniques have been developed for this purpose, such as the time of arrival (toa), the time difference of arrival (tdoa), angle of arrival (aoa), and received signal strength (rss). toa or tdoa is an important technique in localization, which is used in gps [17]. it requires highly synchronized hardware, which increases the implementation costs [14]. in toa, the absolute travel time of a wireless signal from the transmitter to the receiver is first estimated [3]. knowing the speed of propagation of light, the toa measurement allows measuring the distances between the transmitters and the receiver, which are then applied to a trilateration method to determine the position of the receiver. this makes the toa technique difficult to implement because it requires precise synchronization between the transmitters and the receiver. therefore, to avoid the need for precise time synchronization, indoor location systems generally opt for tdoa rather than toa, because tdoa requires synchronization between transmitters only, which is easier to achieve [21]. however, while tdoa methods could be an appropriate solution with some technologies, they are not an optimal choice for economic led indoor positioning, because the differences in the traveling times of the signals are too short, due to the short distances between the transmitters and the receiver, which requires highly accurate and synchronized clocks to estimate them. the angle of arrival (aoa) is defined as the angle between the propagation path of a wave (its incidence) and a reference direction [22]. geometrically, aoa localization consists of finding a point in 3d space so as to minimize the sum of the distances between this point and the lines drawn between the led transmitters and the receiver [4,25,27]. aoa technique can provide high accuracy with an expensive sensor array, such that the measured angles are not limited by the receiver’s field of view. therefore, rss-based methods are the most common for indoor vlp, since they do not require synchronization, which reduces the cost and complexity of implementation [26]. in addition, rss values are easy to obtain using a photodetector and a simple power measurement algorithm [37]. many research works have been carried out for rss based indoor vlp. jung et al, [13] proposed a localization technique using the received signal strength ratio (rssr), which is the relative ratio of optical powers detected between each led and optical receiver. three equations could be obtained using the force ratio between the received signals, and the position of the terminal is located by solving these equations. in [34]a positioning system, by varying the transmission power, was proposed in order to estimate the distances to several light sources, however, this leads to unstable lighting. yang et al [29] also copyright © 2023 assa. adv syst sci appl (2023) accurate indoor positioning system based on visible light 141 presented a localization method, which uses a single transmitter and multiple receivers, whose center is determined using their rss and the relative positions between them. an average positioning error of 0.65 cm was reported. however, the implementation of the method is not straightforward and requires equipping the mobile device with three receivers that are deployed in fixed positions, relatively to each other. in [31], a positioning error lower than 3 cm has been reported in 3-d localization, however, besides the light sensor, an extra accelerometer or gyro-sensor is required by the system, which increases the positioning algorithm complexity. yang [30] proposed a 3-d indoor positioning system using aoa and rss with a single transmitter and multiple optical receivers. they showed that the positioning error is less than six cm. [12] puts forward an indoor 3-d visible light fingerprint positioning system based on the fusion of a k-means algorithm and a rf algorithm. the reported average 3-d location error is 4.45 cm. in [8], chin-wei hsu et al. proposed to subdivide the entire positioning zone into several small sub-zones, and in each sub-zone, they applied a positioning algorithm that directly estimates the mobile’s position from the rss. the two coordinates of this position are estimated separately and each of them is expressed as a quadratic polynomial of the powers received from four leds. the coefficients of this polynomial are determined during a training phase by what they called “machine learning”, which in fact uses the steepest descent algorithm to find the solution. artificial neural networks (ann) have also been proposed for estimating the position of a receiver directly from the rss. for example, the authors in [10] trained a typical back-propagation ann with the rss received from four leds at different points on the receiver plane to learn the relation rss-position, then the trained ann is used to retrieve the position of the receiver from the four rss during the operational phase. this paper, proposes two new visible light-based positioning methods. the performances of these methods are assessed experimentally and compared to that of some other methods from the specialized literature. the rest of this paper is organized as follows: the system model for indoor positioning is described in section. 2. section. 3 presents the proposed two vlp methods. in section 4, the experimental results are presented and discussed. finally, in section 5, some conclusions are drawn. 2. indoor positioning channel model as shown in figure 2.1, a typical indoor scenario with four leds and a photodetector is considered in this paper. in rss-based vlp, the position of the mobile device (receiver) is determined by a location algorithm, from the strength of the signals received from leds. the led is often considered as a lambertian source [1, 15]. therefore, when only the lineof-sight (los) channel is considered, the distance between a led and the receiver can be estimated by measuring the received signal power, which can be expressed by: pr = { (m+1)a cosm(ϕ) cos(θ) 2πd2 t (θ)g(θ)pt 0 ≤ θ ≤ fov 0 θ > fov (2.1) where • pt is the transmitted power, • a is the physical area of the detector in the pd, • ϕ and θ are, respectively, the irradiance angle and the incidence angle, as shown in figure 3.2, • t (θ) is the gain of the optical filter, • g(θ) is the gain of the optical concentrator, • fov is the field of view at the pd, • d is the distance between the led and receiver, • m is the lambertian mode number related to the semi-angle at half-power of the led, f1/2, by: copyright © 2023 assa. adv syst sci appl (2023) 142 a.h. saboundji, m. keche, m. dahmani m = − ln 2 ln(cosf1/2) (2.2) led 1 led 2 led 3 led 4 (0,100 cm,165 cm) (100 cm,100 cm,165 cm) (100 cm,0 ,165 cm) (0,0 ,165 cm) 16 5 cm 100 cm 10 0 cm d1 d 2 d3 d4 receiver fig. 2.1. scenario considered for visible light positioning 3. the vlp proposed methods in the following, we describe the two proposed vlp methods. both methods are based on polynomial fitting. in the first method, the relation distance-rss is expressed by a polynomial h led photodetector d fov fig. 3.2. parameters of the optical wireless channel whose coefficients are determined by a polynomial fitting method, during a training phase. copyright © 2023 assa. adv syst sci appl (2023) accurate indoor positioning system based on visible light 143 this relation is then used during the operational phase to retrieve the distances between the receiver and at least three emitters (sources), from the powers of the signals received from these emitters. once the distances are determined, they can be used to estimate the position of the receiver, by applying a multilateration method. in the second method, the coordinates of the receiver are directly expressed as a function of the received powers. this function is assumed to be a polynomial, whose parameters are determined using polynomial surface fitting. 3.1. the first proposed method as we have seen in the previous section, the rss is related to the distance between the emitter and the receiver. a usual approach is to model the relation distance-power, during an offline training phase, then use this relation to retrieve the distance from the rss, during the operational phase. once the distances between a receiver and at least three emitters, or an emitter and at least three receivers are estimated, trilateration can be applied to estimate the position of the mobile device. many models have been proposed in the related literature to express the received powerdistance relation. for example, the following relation was proposed in [35] . p i r/p i 0 = h dni (3.3) where: • p i r is the power received form led i, • p i 0 is a reference power, i.e. the power received form led i at a given reference point, • h is the vertical distance between the emitters and the receiver, • di is the distance between led i and the receiver, • n is a parameter whose value can be determined experimentally by applying a non-linear fitting to a set of training measurements. fig. 3.3. experimental test bed in the following, this fitting will be denoted the simplified lambertian fitting. for a better fitting, we assume in this paper that the relation distance-power is polynomial. we have investigated the use of polynomial curve fitting, which is probably one of the simplest copyright © 2023 assa. adv syst sci appl (2023) 144 a.h. saboundji, m. keche, m. dahmani and most used data fitting techniques. in this fitting, the relation distance-power is expressed as follows: d = n∑ k=0 ckp k (3.4) where d is the distance, p is the power, ck is the kth coefficient of the polynomial and n is its degree. to determine the coefficients ck, we first carried out a measurement campaign during which the powers received from a given led were measured at different points located on a square grid. then, knowing the distances of these points to that led, we applied a polynomial fitting method to obtain the coefficients ck. this is achieved offline during what is called a training phase. during the operational phase, relation 3.4 is used to estimate the distances of the receiver to the sources from the powers of the signals received from these sources. these distances are then inputted into a multilateration method to estimate the position of the receiver. this method solves the following system of non-linear equations: (x− x1) 2 + (y − y1) 2 = d21 (x− x2) 2 + (y − y2) 2 = d22 ... (x− xn) 2 + (y − yn) 2 = d2n (3.5) where n is the number of sources, (x, y) are the coordinates of the receiver, (xi, yi) are the coordinates of the ith source, and di is the distance between the receiver and the ith source. by using linearization and the least squares estimation, it can be shown that the position may be obtained as follows [32]: [ x y ] = 1 2 (ata)−1atb (3.6) where: a = (x2 − x1) (y2 − y1) ... ... (xn − x1) (yn − y1)  (3.7) b = x2 2 − x2 1 + y22 − y22 + d22 − d21 ... x2 n − x2 1 + y2n − y21 + d2n − d21  (3.8) 3.2. the second proposed method in this method, each of the two coordinates of the receiver is modeled separately as a polynomial of the received powers whose coefficients are determined by using surface polynomial fitting. in order to use available methods for this purpose the coordinate x or y is expressed as a function of only two received powers. if we denote these two powers by p1 and p2, then the coordinate x is approximated by: x̂ = c0,0 + c1,0p1 + c0,1p2 + · · ·+ c(i, j)p i 1p j 2 + · · · (3.9) a similar relation is used for the y coordinate. notice that since only two powers are used in relation 3.9, the solution is not unique. in fact, there are two solutions that correspond to the intersections of two circles, centered, respectively on the first and second emitters with radii equal, respectively, to the distances between the receiver and these two emitters. copyright © 2023 assa. adv syst sci appl (2023) accurate indoor positioning system based on visible light 145 generation of the signal driver circuit led locating algorithm detection photodetector sound card (acquisition) v l c c h a n n e l arduino pc fig. 3.4. diagram of the constructed experimental vlp system. therefore, for each couple of powers (emitters), two couples of relations should be established, one for the points that represent the first solution, and one for the points that represent the second solution. this implies that the training set should be split into two nonoverlapping sub-sets; the first one is used for the first relation and the second one for the second relation. if n emitters are used, then in total 2c2n couples of relations should be used and thus the same number of solutions is obtained. since only half of these solutions are valid, a pruning method can be applied to exclude the outsiders and average the remaining solutions to get an estimate of the true position. 4. experimental results and discussion to evaluate the performances of the proposed vlp methods, many experiments were carried out using a test bed that has been built up for this purpose. before presenting the obtained results and discussing them, we first present the experimental environment. 4.1. experimental environment and parameters a photo of the constructed test bed is shown in figure 3.3. it consists of four low-consumption commercial led lamps that are fixed on a ceiling at emplacements, as indicated in figure 2.1, an emitter circuit, and a receiver circuit (see figure 3.4). the emitter circuit is composed of an arduino card that generates four signals; each signal controls the lightening of a led lamp through a driver circuit. in order to discriminate the signals received from the leds, the lights emitted by these leds are modulated at different frequencies, which are respectively, 800 hz, 900 hz, 1000 hz, and 1100 hz. the receiver includes a photo resistance, which is displaced horizontally at a distance of 165 cm from the ceiling, and a pc. the received signal is acquired by the sound card of the pc at a sampling frequency equal to 96 khz. the digitized signal is then applied to a detection algorithm that uses the fft to compute the powers of the four frequencies, i.e. the powers of the signals received from the four leds. these powers are then applied to a locating algorithm to determine the position of the mobile, i.e. the photo resistance. the following five locating algorithms were implemented and tested: • the two proposed in the present paper, • the one proposed in [8] and named “machine learning (ml)method”, copyright © 2023 assa. adv syst sci appl (2023) 146 a.h. saboundji, m. keche, m. dahmani normalized received power 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 d is ta n c e ( c m ) 160 170 180 190 200 210 220 all data fitting data polynomial fitting simplified lambertien fitting fig. 4.5. comparison between the polynomial fitting and the simplified lambertian fitting • the locating algorithm proposed in [10], which is based on neural networks, • the one that uses multilateration and the simplified lambertian fitting (slf) to estimate the distances from the powers [35]. x 0 20 40 60 80 100 y 0 20 40 60 80 100 120 estimated position vs. real position real position estimated position fig. 4.6. training results for the second proposed method (surface fitting) all of the five methods cited above require learning. for this purpose, a training phase was conducted during which 121 normalized measurements were collected and stored in a file. each measurement consists of the four powers received from the four leds at a given point, which is located on a uniform square grid that covers the (100cm x 100cm) positioning zone, whose origin, (0cm, 0cm), is taken to be the projection of the center of the first led lamp. to test the generalization of the learning, two test files were used. the first test file contains 121 normalized power measurements collected at the same points as during the training phase, whereas the second test file stores 121 normalized power copyright © 2023 assa. adv syst sci appl (2023) accurate indoor positioning system based on visible light 147 x (cm) 0 20 40 60 80 100 y ( c m ) 0 20 40 60 80 100 120 estimated position / real position real position estimated position fig. 4.7. training results for the machine learning method measurements, collected at points situated on a uniform square grid whose origin is (5 cm, 5 cm). notice that the measurements in the learning file and the two test files were acquired at different dates. 4.2. experimental results analysis first, let us present the results of the training. figure 4.5 compares the polynomial fitting used in the first proposed method and the simplified lambertian fitting proposed in [35]. for the polynomial fitting, a degree six was found to be sufficient. it should be mentioned that only the 11 measurements lying along the diagonal that connect the points (0 cm, 0 cm) and (100 cm, 100 cm) were used for both fittings. from this figure, it can be observed that the polynomial fitting is more precise. x (cm) -20 0 20 40 60 80 100 120 y (c m ) -20 0 20 40 60 80 100 120 estimated position / real position real position estimated position fig. 4.8. training results for the ann method copyright © 2023 assa. adv syst sci appl (2023) 148 a.h. saboundji, m. keche, m. dahmani error (cm) 0 1 2 3 4 5 6 7 8 9 p r( e rr o r< x ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 empirical cdf 1st proposed method 2nd proposed method simp.lamb.fitting mach.learn.method ann method fig. 4.9. cdf of the positioning errors obtained with the five vlp methods, using test file 1 for the proposed surface fitting method, the machine learning method [8] and the ann method [10], the whole 121 measurements were used for training. for the surface fitting method, a polynomial of degree 3 in both variables was found to give the best results. for the machine learning method, the least squares method was employed to compute the parameters of the model. for the ann method, a network with 10 hidden layers was trained using the backpropagation algorithm, with bayesian regulation. the learning results for these three methods are presented in figures 4.6, 4.7, and 4.8, where the true positions and the estimated ones are plotted together. these figures show that the ann has the lowest training table 4.1. position errors’ statistics obtained with the five vlp methods, using test file 1 method errors (cm) min mean 90% max ml [8] 0.12 2.27 3.50 8.47 ann [10] 0.05 0.43 0.78 1.15 slf [35] 0.30 3.13 4.90 8.00 first proposed method 0.02 2.06 3.37 4.39 second proposed method 0.10 0.91 1.47 3.42 error, followed by the surface fitting method, followed by the machine learning method. the obtained mean position errors are respectively, 0.24 cm, 0.79 cm, and 2.23 cm. the results obtained by the five implemented vlp methods with test file 1 are summarized in table 4.1 and figure 4.9, whereas those obtained with test file 2 are summarized in table 4.2 and figure 4.10. in the tables, some statistics of positioning errors are reported, while the figures display the cumulative distribution function of these errors. the positioning error is defined by: e = √ (xt − xe)2 + (yt − ye)2 (4.10) where (xt, yt) and (xe, ye) denote, respectively, the true position and the estimated position. from these results, it can be observed that to different extents the generalization in the five compared vlp methods is good enough. the best results were obtained with the ann method, whereas the worst results were obtained with the method that uses the simplified copyright © 2023 assa. adv syst sci appl (2023) accurate indoor positioning system based on visible light 149 table 4.2. position errors’ statistics obtained with the five vlp methods, using test file 2 method errors (cm) min mean 90% max ml [8] 0.23 2.81 5.35 8.72 ann [10] 0.04 1.29 2.70 4.07 slf [35] 0.16 3.25 5.34 9.65 first proposed method 0.06 2.49 4.47 10.90 second proposed method 0.11 1.55 2.89 6.06 lambertian fitting to model the relation distance-rss and the multilateration to retrieve the position from the estimated distances. the second proposed method has a performance that is slightly inferior to that of the ann method, but it has the advantage of being simpler. error (cm) 0 2 4 6 8 10 12 p r( e rr o r< x ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 empirical cdf 1st proposed method 2nd proposed method simp.lamb.fitting mach.learn.method ann method fig. 4.10. cdf of the positioning errors obtained with the five vlp methods, using test file 2 from figures 4.9 and 4.10, it can be observed that large errors may occur with the simplified lambertian method and the machine learning method. it is the case also of the first proposed method with the second test file. however, we have observed that in this case these errors may be reduced if all the measurements are used for the training, instead of using only those taken at the 11 points lying on the diagonal (the maximum error is reduced from 10.90 cm to 8.21 cm). this is not the case for the simplified lambertian method. 5. conclusion in this paper, we have proposed two new rss-based vlp methods and compared their performances to that of three other methods from the literature. in the first method, we have proposed to replace the simplified lambertian fitting used in [35] with an improved polynomial fitting. we have shown experimentally that this improvement in the fitting results in an improvement in the positioning accuracy. in the second method, we have proposed to estimate the coordinates of the mobile separately and directly from the received powers. we have applied surface fitting to learn the relation coordinates-received power. since this type of fitting is limited to two variables, we have devised a technique that combines several coordinates estimates, each estimate being based on two powers among the four received copyright © 2023 assa. adv syst sci appl (2023) 150 a.h. saboundji, m. keche, m. dahmani powers, taking into account the existence of two possible estimates for each couple of powers. we have shown that the performance of this method is comparable to that of the best method, which is the ann method, while being simpler. it should be noted that by the same way we have validated experimentally the ann method, which has been evaluated only by simulations in the original paper where it has been presented. it should also be highlighted that given that only standard low cost commercial components were used in our experimentation, the obtained positioning accuracy is very interesting. references 1. cossu, g., presi, m., corsini, r., choudhury, p., khalid, a. m., et. al. 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(2018). a survey of positioning systems using visible led lights. ieee communications surveys & tutorials, 20(3), 1963–1988. copyright © 2023 assa. adv syst sci appl (2023) introduction indoor positioning channel model the vlp proposed methods the first proposed method the second proposed method experimental results and discussion experimental environment and parameters experimental results analysis conclusion adv syst sci appl 2022; 04:19–30 published online at https://ijassa.ipu.ru. spacecraft emergency situations automated scenarios design within the framework of fdir concept and dynamical neural networks technique alexey tuturov1, olga andrianova1,2*, ernest sleptsov, evgeniy yurkevich1, lidia kryukova1 1v. a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia 2tecnologico de monterrey-campus guadalajara, research institute of advanced materials for sustainable manufacturing, guadalajara, mexico abstract: the space complex is considered as a socio-cyberphysical system (scps) in order to point out that the human factor is one of the most important ones in the analysis of emergency situations using spacecraft telemetry in the conditions of the current level of software development, information and technical resources of the space complex to ensure its functional stability. functional stability of the system means its functionality under the influence of destructive factors of the external or internal environment. the concept of fault detection, isolation, and recovery (fdir) is described, its advantages and drawbacks are pointed out in the context of the current problem. based on the fdir concept, decision support systems will be developed in future. in the framework of fdir, spacecraft telemetry data is proposed to be used to design the monitor, the task of which is to identify and register emergency situations, based on modern data processing tools, namely, dynamical neural networks (dnns). keywords: fdir concept, dynamical neural networks, telemetry, socio-cyberphysical system 1. introduction a space complex may be considered as a socio-cyberphysical system (scps) due to the increased influence of the human factor on the functional stability of the system in the context of modern trends in increasing software and hardware implementations in space technology. functional stability is understood as the ability to preserve and/or restore the performance of system functions, as well as to adapt information communication channels when changing system elements under the influence of destructive factors of the external and/or internal environment (korolev, 2018). thus, along with technical, informational, and software processes and resources, a set of technical experts can be represented as human resources, taking into account the importance of their daily participation in ensuring the functional stability of the entire system. increasing of personnel interaction efficiency with software and hardware is carried out, among other things, by developing decision support systems. in this paper, a new concept for operability diagnostics of a spacecraft using telemetry data from its service and target systems is proposed and presented. based on this methodology, decision support systems are planned to be developed in future. one of the modern methods of ensuring functional stability is the application of the concept of fault detection, isolation, and recovery (fdir). the main tasks of fdir are the automatic switch of the spacecraft to a safe mode of operation, logging, and localization ∗corresponding author: andrianovaog@gmail.com 20 a. tuturov, o. andrianova, e. sleptsov, e. yurkevich, l. kryukova of emergency situations. this methodology makes it possible to increase the reliability of the target application without using excess reserves, therefore it is widely used in the rocket and space industry (akhmetov, makarov, & sollogub, 2014; eickhoff, 2012). to construct and implement the mentioned concept above during the system design and after the start of the flight mission, it is necessary to classify the nature of possible faults. to do this, methods of probability theory, reliability theory, and expert evaluation are currently used in engineering practice. another way is to process the telemetry information coming from the spacecraft, to analyse and classify it, using classical and modern concepts of data processing: the method of support vectors (azevedo, ambrosio, & vieira, 2012), self-learning kohonen maps (talalaev & fralenko, 2018), artificial neural networks, one of the extensions of which are dynamical neural networks (dnns) (a. s. poznyak, oria, & poznyak, 2019), which will be used further in the current research. dnns have the aim of approximating uncertain systems represented by differential or difference equations or measuring (telemetry) information. in case of dynamical representations, both unknown parameters in the right-hand side of the system or its unknown form may be recovered, meanwhile, if the system is represented by time series, omissions or errors in the input sequence can be restored due to the flexible structure of dnns. the main difference between dynamical neural networks and well-known data processing methods is time-dependence of their coefficients (weighting matrices), which allows them to be used in approximation problems for dynamical plants. dnns have proven to be efficient approximate representations of different classes of uncertain models (a. poznyak, sanchez, & yu, 2001; wu, 1998; t. poznyak, chairez, & poznyak, 2019). they can also use their internal memory to process arbitrary sequences of inputs. when analyzing time series, this property is important not only to recover the missing data, but also to detect non-standard behavior, that can lead to faults and emergency situations. the following main contributions of this study are: 1. the space complex is considered as a scps and the concept of its functional stability is introduced and explained; 2. the telemetry data of the spacecraft is analyzed from the point of view of detecting and classifying anomalies; 3. in the fdir concept, dnns are proposed to be used as a monitor on telemetry data. the rest of the paper is organised as follows. in section 2, components of the space complex are described as well as a scps and its functional stability are defined. section 3 analyzes the spacecraft telemetry data, introducing two types of anomalies and emergency engine start, that can occur the system. in section 4, basics of fdir concept are given and its modification for the current problem is discussed. section 5 introduces theoretical background for dnn identification, using telemetry data. section 6 presents some concluding remarks and ideas for future work. 2. the space complex as a socio-cyberphysical system this section describes the main components of the space complex and their functions, as well as definitions of functional stability and telemetry. the necessity of transition from a cyberphysical system to a socio-cyberphysical one is justified when describing the space complex and ensuring its functioning. the structure, characteristics, and operational standards of the equipment for a particular spacecraft are directly related to its purpose as part of the space complex (sc) defined as a set of interconnected orbital and ground-based technical devices that provide solutions to various tasks based on the use of outer space. copyright © 2022 assa. adv syst sci appl (2022) spacecraft emergency situations automated scenarios design 21 out of all the variety of sc components, only the following will be used in the current research: • satellite constellation (scon) as a set of spacecrafts located on the orbits in accordance with the ballistic structure and united by the commonality of the problems to be solved as part of the sc; • ground control complex (gcc) as a set of technical devices and structures designed to control the functioning of the spacecraft of scon from the moment of launching to orbit; • mission control center (mcc) as structures with technical systems and technological tools of command-software, telemetry, and ballistic-navigation support, external information exchanges, trunk and special communications, mapping, designed to ensure the activities of service personnel for the formation, transmission, reception, processing, storage, documentation of information during the continuous process of spacecraft flight control; • payload data management center (pdmc) as a set of interconnected ground-based technical devices with software designed to provide the customer and its consumers with target information obtained on the basis of space data (piro et al., 2018). due to the concept of telemetry, the space complex may be represented and described as a single system. telemetry is defined as a field of science and technology that deals with the development and operation of automated devices complex that ensures the receipt, conversion, transmission via a communication channel, reception, processing, and registration of measuring information and information about events in order to monitor at a distance the state and functioning of technical and biological systems of various objects and the study of natural phenomena. thus, there is necessarily a telemetry system as a part of any spacecraft, considered as an automated system. a typical model of a telemetry system functionating is shown in figure 2.1. fig. 2.1. hierarchical model of a telemetry system using ccsds (the consultative committee for space data systems) recommendations copyright © 2022 assa. adv syst sci appl (2022) 22 a. tuturov, o. andrianova, e. sleptsov, e. yurkevich, l. kryukova functional stability is understood as a necessary condition for ensuring the fulfillment of the sc and its components objectives. for correct operation of the concept of functional stability, the sc should be considered as a complex object, represented by a cyberphysical system (cps), meaning an intelligent system based on engineering interacting networks of physical and computational components. taking into account the interaction of the scon ballistic structure features and the geography of ground-based communications (gc), the unity of the information space for a particular spacecraft and the entire sc is possible only within a time-limited communication session over a physical connection via a radio signal, by establishing a communication channel. usually, the communication channel between the spacecraft and the gc is considered as the basic level when presenting the sc as a cps. with this approach, all resources and processes in the cps belong to one of the categories: information, software or technical. however, in this paper, this functionality is proposed to be extended by adding the category “personnel”, designed to reflect the knowledge, experience, intuition, and ability to improvise of the personnel teams of the gcc, mcc, and pdmc in the form of expert assessments described by weight functions. thus, in this paper it is proposed to consider a sc as a scps. the idea of considering a human (social) factor in automatic control systems is not new, associated with increasingly complex control and navigation tasks. for example, a concept of control subject has been introduced by v. lefebvre (lefebvre, 1977) and developed by v.e. lepskiy (lepskiy, 1998), d.a. novikov (d.a. & e., 2012). according to this new general statement of control problems, control subject and control object should always be considered as integral parts. in (sleptsov & andrianova, 2021), the control subject’s goals are described as follows: “control subjects task is to ensure the required state of the system from its point of view, taking into account information about external disturbances. naturally, the control system functioning this way is closed-loop as control object and control subject form a closed cycle.” consideration of sc as a scps will make it possible to introduce into the control process a human factor not as a statistical disturbance, but as one of the vectors determining the development of sc. to solve this problem, methods of scenario tree construction are used to develop emergency situation scenarios. however, this method requires a sufficient amount of accumulated event statistics, i.e. recorded facts of qualitative or quantitative changes in the parameters of the internal and/or external environment of the spacecraft, which is directly related to the level of wear of the spacecraft systems components and contradicts the trend of high functional stability of the spacecraft for short periods of its operation. thus, in this paper, the scenario tree method will not be directly applied. 3. spacecraft telemetry as an information environment for constructing the scenarios consider the communication channel between the spacecraft and the gcc as the only way for a single information space existence between the spacecraft and the sc. characteristics of the communication channel under this consideration act as boundary conditions, and the main limitation of information exchange is the duration of the communication channel stable existence, mainly due to the mutual radio visibility of the spacecraft and the gss. figure 3.2 shows the telemetry data of an element of the spacecraft orientation and stabilization (sos) system for 14 days: from 305 to 319. the sos data are selected because the functional stability of the entire spacecraft directly depends on the operation of this system, as well as the quality of its target tasks: in this example, the telemetry information from the earth remote sensing satellite is used. the color of plots in figure 3.2 shows the change of day in the considered interval. from the color change, it can be seen that the oscillation of the plots is cyclical: the data of the first days differ qualitatively from the last by the presence of a shift equal to about 17 minutes, copyright © 2022 assa. adv syst sci appl (2022) spacecraft emergency situations automated scenarios design 23 fig. 3.2. telemetry data from the element of sos system fig. 3.3. telemetry data from the magnetometer: x axis which corresponds to the natural precession of the orbit for this spacecraft. this pattern can also be traced for other elements. thus, figures 3.3, 3.4, and 3.5 show telemetry data from one magnetometer along three axes, but the tendency to cyclicity persists here, despite the difference in the nature of the recorded characteristic. in each of figures 3.2–3.5, a characteristic area of data dominance is clearly visible, and it is not difficult to find mathematical functions to describe its boundaries. denote this statistical area by the “norm” for the time period of observations under consideration. two types of anomalies in relation to the “norm” can be distinguished. copyright © 2022 assa. adv syst sci appl (2022) 24 a. tuturov, o. andrianova, e. sleptsov, e. yurkevich, l. kryukova fig. 3.4. telemetry data from the magnetometer: y axis fig. 3.5. telemetry data from the magnetometer: z axis the first type of anomalies are long straight lines in the near-horizontal plane of the plot. these anomalies are nothing but gaps in the data. these gaps indicate the presence of problems in the recording equipment on board of the spacecraft and/or in the data transmission over the communication channel. in any case, the analysis of possible reasons for the presence of these gaps is the prerogative of specialists and experts, and not automatic tools, due to the lack of data on which it is possible to make a mathematically sound forecast about the cause of this phenomenon. the second type of anomalies are vertical bursts that go beyond the boundaries of the “norm”. these situations can be either single values as in figure 3.3, or blue indications copyright © 2022 assa. adv syst sci appl (2022) spacecraft emergency situations automated scenarios design 25 similar in nature in figure 3.2. in the future, the anomalies of the second type will be called events, i.e. the occurrence of conditions or disturbances. fig. 3.6. example of telemetry with an emergency engine start figure 3.6 shows a fragment of telemetry, which represents the emergency activation of the spacecraft engine (orange color), which can be classified as an emergency situation. the sequence of events in the telemetry of various spacecraft systems leading to an emergency situation will be called an emergency scenario. the development of automated software tools for localization and relief of the emergency situation consequences, followed by the recovery of the spacecraft operability, is one of the main tasks, solved in the framework of the fdir concept. 4. description of fdir methodology the fdir concept is based on the principles that emergency situations of varying criticality occur with a certain probability. the probability and criticality of the occurrence of an emergency situation is determined from previous experience, the results of game-theoretic modeling, expert knowledge. the parrying of emergency situations depends on its criticality and can be carried out with the help of predefined scenarios developed on the basis of expert opinion and experience, see figure 4.7. the fdir also provides for the possibility of parrying single emergency situations. this is achieved through the use of watchdog timers. a watchdog timer is an electronic timer that is used to detect and recover faults. during normal operation, this timer is constantly restarted to prevent the entire system from restarting, while in case of a software or hardware fault, this timer stops restarting and initiates system restart. this concept is based on the following principles (akhmetov et al., 2014; eickhoff, 2012; wander & förstner, 2013; salarkaleji & dayyani, 2013): copyright © 2022 assa. adv syst sci appl (2022) 26 a. tuturov, o. andrianova, e. sleptsov, e. yurkevich, l. kryukova fig. 4.7. levels of fdir • design a precise hierarchy of types of faults, determination of the level of software and hardware that ensures the elimination of every fault; • construction and implementation of algorithms for automatic transfer of the system to the safe mode; • exploitation of autonomous onboard radio command and information-telemetry systems in the design of the spacecraft, providing reliable control and guaranteed receiving of telemetry information; • ensuring the protection of the issuance and realization of potentially dangerous commands; • providing the possibility of full or partial cleaning of the onboard ram and restoring the original state of the onboard software; • identification of the faults using explicit and implicit characteristics; • usage of flexible configuration of software and hardware in system design. the safe mode of operation of the spacecraft assumes: • automatic transfer of the spacecraft to the orientation mode, which provides guaranteed power from solar panels, disconnection of the electrical load associated with the intended use and with other energy-consuming processes; • ensuring the operation of the thermal regime system, onboard control system; • use of redundancy: functional and/or substitution; • the ability to restore onboard software using a new configuration of onboard hardware and software, which provides a mode of restoring normal functioning. fdir has several levels of settings: 1. the settings that are embedded in rom during the preparation of the spacecraft on the ground; 2. software settings in ram that are implemented and changed during flight tests; copyright © 2022 assa. adv syst sci appl (2022) spacecraft emergency situations automated scenarios design 27 3. flexible and adaptive settings, laid down according to certain cycles: year, month, decade. the advantages of this concept include: 1. adaptability and flexibility of spacecraft systems; 2. autonomous and prompt elimination of emergency situations or automatic transfer to safe mode for further actions of the mcc; 3. the fdir concept implies the use of the onboard computer resource, and, accordingly, the analysis of telemetry information on board, which, on the one hand, reduces the load on the mcc, allows for the analysis of information promptly and allows you to make decisions during a communication session with the earth, and on the other hand, selectively record the most interesting fragments of telemetry information for long periods of time. accordingly, the disadvantage is the complication of the onboard hardware and software complex, which requires: 1. complication of initial design; 2. increased requirements on the reliability of the electronic component base; 3. increased requirements on the quality of the program code. as an implementation of the fdir concept, the presence of the automatic reconfiguration (aro) mechanism in the onboard software can be considered. the essence of the work of this software mechanism is that each working element of all spacecraft systems is expertly assigned with weight values, which are summed up in the aro indicator. when the value of the aro indicator due to faults in the operation of the spacecraft elements falls below the expert-defined value, the procedure for changing the orientation mode of the spacecraft is automatically started, aimed at stabilizing the functioning of the spacecraft. the telemetry data of the aro software mechanism is shown in figure 4.8. fig. 4.8. telemetry data from aro the fdir concept involves the use of several execution units: a monitor, a script block, and a logging block. they can be both software and hardware. later in this article, the design of the software monitor will be described. monitor’s task is to identify and register emergency situations. copyright © 2022 assa. adv syst sci appl (2022) 28 a. tuturov, o. andrianova, e. sleptsov, e. yurkevich, l. kryukova the main tools for monitoring used in the aerospace industry are various types of kalman filters (pontuschka & da fonseca, 2014; lu, eykeren, kampen, chu, & yu, n.d.; zolghadri, 2017), and also time series (troiano, tipaldi, hoping, de pasquale, & bruenjes, 2012). time series have no predictive power. kalman filter makes it possible to predict, but it can diverge. dnns that will be described in the next section have predictive ability, and their convergence is proved using lyapunov methods. the speed of their convergence depends on the selection of parameters. 5. dnn identification for electro-mechanical part of the spacecraft system dnns have the aim of approximating the right-hand side of uncertain systems represented by dynamical equations. for continuous-time systems, differential neural networks have proven to be efficient approximate representations of different classes of uncertain models (a. poznyak et al., 2001; wu, 1998; t. poznyak et al., 2019), meanwhile up to now recurrent neural networks with time-dependent weighting matrices have not been used much to represent discrete-time systems. measurements from the spacecraft sensors are received discretely, which is why in the development of this study it is planned to consider the electro-mechanical part of the measured data as a discrete-time system of the “black box” type, i.e. with unknown motion laws, and based on the available sensors data to approximate this system using recurrent neural networks. generalization of existing methods to discrete-time systems is not a difficult task, since the mathematical basis for the approximate representation of systems using dnn is the lyapunov stability theory, which is sufficiently developed for both continuous and discrete-time cases. move on to the description of the system under consideration. 5.1. problem description in figure 5.9, the spacecraft is represented by an unknown electro-mechanical model, which has the following input and output signals: αi, being the spacecraft rotation angle; ωi, being the angular velocity of the spacecraft; ωm i , being the angular velocity of the wheel; ui = ui ( ωm i (kt ) ) , being the control signal; ξi, being the norm-bounded disturbances, i = 1, 3. fig. 5.9. spacecraft as a black box representation dynamical representation of the system, given in figure 5.9, may be proposed in the following form: copyright © 2022 assa. adv syst sci appl (2022) spacecraft emergency situations automated scenarios design 29 αi(kt ) = fi ( αi((k − 1)t ), ωi((k − 1)t ), ωm i ((k − 1)t ) ) + ξi(kt ), (5.1) ωi(kt ) = gi ( αi((k − 1)t ), ωi((k − 1)t ), ωm i ((k − 1)t ) ) + (5.2) ui ( ωm i ((k − 1)t ) ) + ξi(kt ), ωm i (kt ) = hi ( αi((k − 1)t ), ωi((k − 1)t ), ωm i ((k − 1)t ) ) (5.3) where functions fi ( αi, ωi, ω m i ) , gi ( αi, ωi, ω m i ) , and hi ( αi, ωi, ω m i ) are unknown and should be approximated; k is a non-negative integer value; t > 0 is a time step. 5.2. dynamical (recurrent) neural network identifier denoting the state vector of system (5.1)–(5.3) by x = [ αi, ωi, ω m i ] , supposing that the righthand side of the system is unknown or partly known, represent system (5.1)–(5.3) in the form x(kt ) = ax((k − 1)t ) +w ∗ 1 σ1(x((k − 1)t ))+ w ∗ 2 σ1(x((k − 1)t ))ui ( ωm i (k − 1)t ) ) + f̃(x((k − 1)t )) where a ∈ r9×9 is a hurwitz matrix, w ∗ 1 ∈ r9×9 and w ∗ 2 ∈ r9×m are time-invariant (unknown) weighting matrices (m can be chosen), σ1(x((k − 1)t )) ∈ r3×1 and σ2(x((k − 1)t )) ∈ rm×1 are activation functions (for example, sigmoidal ones). dnn identifier takes the following form: x̂(kt ) = ax̂((k − 1)t ) +w1(kt )σ1(x̂((k − 1)t ))+ w2(kt )σ2(x̂((k − 1)t ))ui ( ωm i (k − 1)t ) ) . by means of the lyapunov stability theory for discrete systems and using the concept of ultimate boundedness, difference equations for the weighting matrices may be found in the form wi(kt ) = σi(x((k − 1)t ), x̂((k − 1)t ), u(ωm i (k − 1)t )), i = 1, 2. the values of w1(kt ) and w2(kt ), obtained from the solution of the minization problem, connected with lyapunov inequality, starting from some moment k0t do not or slightly change, that is why these values may be fixed and used further for solving the prediction problem. the main task of prediction is not only to restore the future dynamics of the system using previous states and fixed weighting matrices, but also to detect possible faults in the system in the future and prevent them. 6. conclusion the purpose of this work was to explain the necessity of representation of sc in the form of scps and, in this context, to describe the tools that will be used for detection and localization of anomalies and emergency situations in the system. for this purpose, the experience of controlling the spacecraft scon was used, in particular, the analysis of telemetry and the application of the fdir concept, as an example of the introduction of expert participation of the mcc, gcc, and pdmc teams in controlling the spacecraft during the communication session. in order to achieve the goal in the mentioned operating conditions of the spacecraft, it was proposed to consider the use of dnn as a mechanism for automated analysis of telemetry for constructing emergency scenarios, which in the future will become the basis for creating a decision support system that allows increasing the functional stability of the entire sc and saving the limited time of the spacecraft communication session for detection, localization, and recovery of the consequences of an emergency situation on board the spacecraft by creating an automated archive of emergency scenarios. copyright © 2022 assa. adv syst sci appl (2022) 30 a. tuturov, o. andrianova, e. sleptsov, e. yurkevich, l. kryukova references akhmetov, r., makarov, v., & sollogub, a. (2014). problems of unmanned spacecraft reengineering in abnormal flight situations and their knowledge-based solutions. vestnik of the samara state aerospace university, 07, 9. azevedo, d. r., ambrosio, a. m., & vieira, m. (2012). applying data mining for detecting anomalies in satellites. in 2012 ninth european dependable computing conference (p. 212-217). d.a., n., & e., r. (2012). foundations of control methodology. advances in systems science and applications, 12, 238-257. eickhoff, j. (2012). onboard computers, onboard software and satellite operations.. korolev, a. (2018). functional stability of navigation information systems. izvestiâ vysših učebnyh zavedenij. priborostroenie, 61, 559-565. lefebvre, v. (1977). the structure of awareness : toward a symbolic language of human reflexion.. lepskiy, v. (1998). the concept of subject-oriented computerization of management activities (in russian). institute of psychology ras, moscow. lu, p., eykeren, l. v., kampen, e.-j. v., chu, q. p., & yu, b. (n.d.). adaptive hybrid unscented kalman filter for aircraft sensor fault detection, isolation and reconstruction. in aiaa guidance, navigation, and control conference. piro, a., casella, d., pinori, s., diciolo, l., cappelletti, c., graziani, f., & battistini, s. (2018, 05). hybris: analysis and design of a hyper-spectral cubesat mission for multiple remote sensing applications and earth observation synergies.. pontuschka, m., & da fonseca, i. (2014). fdir for the imu component of aocs systems. mathematical problems in engineering, 09, 1-9. poznyak, a., sanchez, e., & yu, w. (2001). differential neural networks for robust nonlinear control (identification, state estimation and trajectory tracking). world scientific. poznyak, a. s., oria, i. c., & poznyak, t. (2019). a survey on artificial neural networks application for identification and control in environmental engineering: biological and chemical systems with uncertain models. annu. rev. control., 48, 250-272. poznyak, t., chairez, i., & poznyak, a. (2019). ozonation and biodegradation in environmental engineering: dynamic neural network approach (1st ed.). elsevier. salarkaleji, f., & dayyani, a. (2013). a survey on fault detection, isolation and recovery (fdir) module in satellite onboard software. in 2013 6th international conference on recent advances in space technologies (rast) (p. 545-548). sleptsov, e., & andrianova, o. (2021). control theory concepts: analysis and design, control and command, control subject, model reduction. ifac-papersonline, 54, 204-208. talalaev, a., & fralenko, v. (2018, 01). fault-tolerant system for organizing highperformance computing for solving data processing problems. program systems: theory and applications, 9, 85-108. troiano, l., tipaldi, m., hoping, m., de pasquale, d., & bruenjes, b. (2012, 10). satellite fdir practices using timed failure propagation graphs. in (vol. 11). wander, a., & förstner, r. (2013). innovative fault detection, isolation and recovery strategies on-board spacecraft: state of the art and research challenges.. wu, j. (1998). symmetric functional differential equations and neural networks with memory. transactions of the american mathematical society, 350(12), 4799–4838. zolghadri, a. (2017, 11). the challenge of advanced model-based fdir for real-world flightcritical applications. engineering applications of artificial intelligence, 68. copyright © 2022 assa. adv syst sci appl (2022) introduction the space complex as a socio-cyberphysical system spacecraft telemetry as an information environment for constructing the scenarios description of fdir methodology dnn identification for electro-mechanical part of the spacecraft system problem description dynamical (recurrent) neural network identifier conclusion references adv syst sci appl 2024; 01:163–168 published online at https://ijassa.ipu.ru. exact solutions of the groundwater filtration equation via dynamics tao sinian lomonosov moscow state university, moscow, russia abstract: the paper is devoted to the application of finite dimensional dynamics of evolutionary partial differential equations to the boussinesq filtration equation. the boussinesq equation is a second order nonlinear equation with three independent variables: time and two spatial coordinates. it describes a shape of the groundwater free surface as it flows through a porous medium under the influence of gravity. this paper proposes a method for constructing exact solutions of the equation, based on the method of finite dimensional dynamics. an example of the evolution of the free surface of groundwater is given. keywords: boussinesq equation, filtration, groundwater, finite dimensional dynamics, completely integrable distributions, symmetry 1. introduction the boussinesq equation [2] ∂u ∂t = k ∂ ∂x ( (h(x, y) + u) ∂u ∂x ) + k ∂ ∂y ( (h(x, y) + u) ∂u ∂y ) (1.1) describes a two-dimensional non-stationary filtration of groundwater in a thin layer of porous media caused by gravity. a porous medium is a layer of permeable material (for example, sand or clay). the layer is limited below by a surface that does not allow water to pass through (for example, granite), and above by the surface of the earth. if, as a result of any physical processes (operation of artesian wells, drainage or heavy rainfall), the water level in any place of the layer changes, then under the influence of gravity, the liquid begins to move, leveling its free surface. in equation (1.1) t is time, x, y are spatial coordinates in the fixed horizontal plane π, and k is a constant, depending on the physical properties of water and porous media. without loss of generality we can suppose that k = 1. the functions h(x, y) and u(t, x, y) show the distances from the plane π to the underlying surface and to the free surface of the water at a point (x, y) respectively at a moment t. the porous medium in this case is a thin layer of permeable material (for example, sand or clay). this layer is limited below by a surface that does not allow water to pass through (for example, granite), and above by the surface of the earth. if, as a result of any physical processes (the operation of artesian wells or precipitation), the water level in any place of the layer changes, then under the influence of gravity, the liquid begins to move, leveling its free surface. the model is built under the following assumptions: ∗corresponding author: 86taosinian@gmail.com 164 tao sinian – water is incompressible and has a constant density; – the soil is homogeneous; – the underlying surface is smooth; – groundwater does not reach the surface of the earth anywhere; – the pressure on the free surface of the liquid is constant. 2. evolutionary equations and their flows equations (1.1) is a nonlinear evolutionary equation. the idea of the method that we implement in this work is as follows (see [14]). let φ be the generating function of an infinitesimal symmetry of a differential equation e . the evolutionary equation ∂u ∂t = φ ( x, u, ∂u ∂x , ∂2u ∂x2 , . . . ) (2.2) generates the flow φt on the solution set of e . therefore, knowing some solution v(x) of e , we can construct a one-parameter family of solutions vt = ( φ−1 t )∗ (v(x)). to do this we need to solve the cauchy problem u(0, x) = v(x) for equation (2.2). then the function u(t, x) = vt is a solution of evolutionary equation (2.2). the method of (finite dimensional) dynamics is based precisely on this idea. if for a given function φ we can find a finite type equation e , such that this function generates a symmetry, then we can find a finite dimensional family of solutions of equation (2.2). for equations with one spatial variable, the method of finite dimensional dynamics was proposed in [3,11] and it was further developed in [1,5,8]. for equations with several spatial variables, the method was proposed in [4]. in [6] it was applied to equations with two spatial variables, and in [10] it was extended to general systems of evolutionary equations. the practical application of the finite dimensional dynamics method involves cumbersome symbolic calculations in jet spaces. to overcome these difficulties, we used a symbolic computation system maple (see details in [9, 13]). this article is a direct continuation of our article [6], so we refer readers to it for basic definitions and explanations (see also [7]). 3. dynamics of the boussinesq equation we look for dynamics of equation (1.1) in the form of a second-order overdetermined system of differential equations  ∂2v ∂x2 = p (x, y), ∂2v ∂x∂y = q(x, y), ∂2v ∂y2 = r(x, y), (3.3) where p,q,r are smooth functions. let j1 = j1(r2) be the space of 1-jets with canonical coordinates x1 = x, x2 = y, v, p1, p2. system (3.3) defines the two-dimensional distribution p : a ∋ j1 7→ p(a) = 2⋂ i=0 kerωi,a ⊂ taj 1 (3.4) copyright © 2024 assa. adv syst sci appl (2024) exact solutions of the groundwater filtration equation 165 on this space. here differential 1-forms ω0 = dv − p1dx− p2dy, ω1 = dp1 − p (x, y)dx−q(x, y)dy, ω2 = dp2 −q(x, y)dx−r(x, y)dy. the distribution integrability condition has the form p = ∂2a ∂x2 , q = ∂2a ∂x∂y , r = ∂2a ∂y2 , where a = a(x, y) are some smooth function. then the general solution of system (3.3) is v(x, y) = a(x, y) + c1x+ c2y + c0, (3.5) where c1, c2, c3 are arbitrary constants. rewriting the right-hand side of equation (1.1) in 2-jet space coordinates and restricting it to the distribution p we get the function φ̄ = (hx + p1)p1 + (hy + p2)p2 + (h + v)(axx + ayy). construct the vector field s̄ = sφ̄ = φ̄ ∂ ∂v +d1(φ̄) ∂ ∂p1 +d2(φ̄) ∂ ∂p2 , (3.6) where d1 = ∂ ∂x + p1 ∂ ∂v + axx ∂ ∂p1 + axy ∂ ∂p2 , d2 = ∂ ∂y + p2 ∂ ∂v + axy ∂ ∂p1 + ayy ∂ ∂p2 . note that [d1,d2] = 0. the vector field is an infinitesimal symmetry of the distribution p if and only if (see [6])  d2 1(φ̄)− s̄(axx) = 0, d2 2(φ̄)− s̄(ayy) = 0, d1d2(φ̄)− s̄(axy) = 0. (3.7) equations (3.7) are governing. they link functions h and a. the solvability condition of system (3.7) imposes restrictions on the function h . for example, for quadratic functions h functions a should be quadratic, too. so, to construct a solution of equation (1.1), we need: 1. find a solution a of system (3.7) for a given function h; 2. construct vector field (3.6) and find its flow φt; 3. act on solutions of system (3.3) by the flow φt. 4. quadratic function let the function h be quadratic: h(x, y) = h20x 2 + 2h11xy + h02y 2 + h10x+ h01y + h00. (4.8) copyright © 2024 assa. adv syst sci appl (2024) 166 tao sinian substituting (4.8) into (3.7), we see that system (3.7) contains equations that do not depend on the coefficients hij . solving them, we find that the function a is also quadratic: a(x, y) = a20x 2 + 2a11xy + a02y 2. (4.9) here we discard the linear part since it do not affect equation (3.3). the remaining equations of system (3.7) provide restrictions on the coefficients of (4.8) and (4.9). note that for an arbitrary quadratic function h the functions a(x, y) = −h(x, y) and a(x, y) = 0 are solutions of (3.7). 5. example: evolution of the free surface of groundwater let us find solutions of equation (1.1) when h = 2xy. then we can choose a(x, y) = −1 3 x2 − 4 3 xy − 1 3 y2. by formula (3.5) we get v(x, y) = −1 3 x2 − 4 3 xy − 1 3 y2 + c1x+ c2y + c0. (5.10) then φ̄ = (2y + p1)p1 − 8 3 xy − 4 3 v + (2x+ p2)p2 and vector field (3.6) takes the form s̄ = ( 2p1y − 8 3 xy − 4 3 v + 2p2x+ p21 + p22 ) ∂ ∂v − ( 4y − 2 3 p2 + 8 3 p1 + 8 3 x ) ∂ ∂p1 − ( 2 3 p1 + 4x+ 8 3 p2 + 8 3 y ) ∂ ∂p2 . the translation along this vector field is φt :  x 7→x, y 7→y, v 7→[−(1/3(x2 + y2 + 4xy)) exp((4/3)t) − (1/3(x− y+))(x− y − (3/2)p1 + (3/2)p2) exp(−(2/3)t) − (3/8)(y + x+ (1/2)p2 + (1/2)p1) 2 exp(−(16/3)t) − (1/12)(x− y − (3/2)p1 + (3/2)p2) 2 exp(−(8/3)t) + (y + x+ (1/2)p2 + (1/2)p1)(y + x) exp(−2t) + (1/8)x2 − (1/3((21/8)p1 − (15/8)p2 + (9/4)y))x+ (1/8)y2− (1/3((21/8)p2 − (15/8)p1))y + (9/32)p22 + (9/32)p21 − (3/16)p2p1 + v] exp((−4/3)t), p1 7→ − (2/3(x+ 2y)) exp(2t) exp((10/3)t) exp(−(16/3)t) + (1/6(6y + 6x+ 3p2 + 3p1)) exp(−(10/3)t) − (1/3) exp(−2t)(x− y − (3/2)p1 + (3/2)p2), p2 7→ − (4/3)x− (2/3)y + ((y + x+ (1/2)(p2 + p1))) exp(−(10/3)t) + (1/6) exp(−2t)(−2y + 2x+ 3p2 − 3p1). copyright © 2024 assa. adv syst sci appl (2024) exact solutions of the groundwater filtration equation 167 applying the transformation φ−1 t to (5.10) we get the following 3-parameter solutions family of boussinesq equation (1.1): u(t, x, y) =− 1 3 x2 − 4 3 xy + 1 2 x exp ( −10 3 t ) c1 + 1 2 exp(−2t)c3x − 1 2 exp(−2t)c1x+ 1 2 x exp ( −10 3 t ) c3 − 1 3 y2 + 1 2 exp(−2t)c1y − 1 2 exp(−2t)c3y + 1 2 exp ( −10 3 t ) yc3 + 1 2 exp ( −10 3 t ) yc1 + exp ( −4 3 t ) c2 − 3 16 exp(−4t)c2 3 + 9 32 exp ( −4 3 t ) c2 3 − 3 32 exp ( −20 3 t ) c2 1 − 3 16 exp(−4t)c2 1 + 9 32 exp ( −4 3 t ) c2 1 − 3 32 exp ( −20 3 t ) c2 3 + 3 8 exp(−4t)c3c1 − 3 16 exp ( −4 3 t ) c3c1 − 3 16 ) exp ( −20 3 t ) c3c1. (5.11) a direct check shows that this is indeed a solution of equation (1.1) when h(x, y) = 2xy. graphs of function (5.11) whit c1 = 1, c2 = c3 = 0 for various time values are presented in fig. 5.1. fig. 5.1. evolution of the free surface of groundwater at t = 0, 0.1, 0.3. another example of the use of differential geometric methods for constructing exact solutions of nonlinear partial differential equations is proposed in [12]. the author expresses gratitude to alexei g. kushner for stating the problem and useful discussions. copyright © 2024 assa. adv syst sci appl (2024) 168 tao sinian references 1. akhmetzyanov, a. v., kushner, a. g. & lychagin, v. v. (2017) attractors in models of porous media flow, doklady. mathematics, 472(6), 627–630. 2. boussinesq, j. (1904) recherches théoriques sur lécoulement des nappes deau infiltrées dans le sol et sur le débit des sources [theoretical research on the flow of water tables infiltrated into the ground and on the flow of springs], journal de mathématiques pures et appliquées 5e série, 10, 5–78, [in french]. 3. kruglikov, b. s. & lychagina, o. v. (2005) finite dimensional dynamics for kolmogorov – petrovsky – piskunov equation, lobachevskii journal of mathematics, 19, 13–28. 4. kushner, a. g. (2023) dynamics of evolutionary differential equations with several spatial variables, mathematics, 11(2), 335–346. 5. kushner, a. g. & gorinov, a. a. (2020) dynamics of evolutionary pde systems, lobachevskii journal of mathematics, 41(12), 2448–2457. 6. kushner, a. g. & kushner, e. n. & tao s. (2022) dynamics of evolutionary equations with 1+2 independent variables, advances in systems science and applications, 22(4), 1–10. 7. kushner, a. g., lychagin, v. v. & rubtsov, v. n. (2007) contact geometry and nonlinear differential equations. cambridge, uk: cambridge university press. 8. kushner, a. g. & matviichuk, r. i. (2020) exact solutions of the burgers – huxley equation via dynamics. journal of geometry and physics, 151, 103615. 9. kushner, a. g. & matviichuk, r. i. (2021) finite dimensional dynamics of evolutionary equations with maple, in: m. ulan, e. schneider (eds.), differential geometry, differential equations, and mathematical physics. tutorials, schools, and workshops in the mathematical sciences, springer nature switzerland ag, 123–149. 10. kushner, a. g. & sinian, t. (2023) evolutionary systems and flows on solutions spaces of finite type equations, lobachevskii journal of mathematics, 44(9), 3945– 3951. 11. lychagin, v. v. & lychagina, o. v. (2007) finite dimensional dynamics for evolutionary equations. nonlinear dyn., 48, 29–48. 12. mukhina, s. (2023) an exact solution of the hunter – saxton – calogero equation by contact linearization method, advances in systems science and applications, 23(04), 1–7. 13. tychkov, s. n. (2019) introduction to symbolic computations in differential geometry with maple. in: kycia, r. a., ułan, m., & schneider, e. (eds.) nonlinear pdes, their geometry, and applications (pp. 157–182). zurich, switzerland: springer nature switzerland ag. 14. vinogradov, a. m. (1984) local symmetries and conservation laws, acta appl. math., 3, 21–78. copyright © 2024 assa. adv syst sci appl (2024) introduction evolutionary equations and their flows dynamics of the boussinesq equation quadratic function example: evolution of the free surface of groundwater microsoft word 1418-article text-6611-1-18-20240109 dynamic approach to the energy functioning of a built space… 77 dynamic approach to the energy functioning of a built space and associated carbon footprint: application to a school complex located in ouagadougou daouda sawadogo1, ousmane coulibaly1, xavier chesneau2, belkacem zeghmati2 1) laboratoire de physique et de chimie de l'environnement (lpce), department of physics, université joseph ki-zerbo, ouagadougou, burkina faso 2) multidisciplinary modeling and simulation laboratory (lamps), mechanics and energetics group (gme), université perpignan via domitia, perpignan, france abstract: we propose in this study a dynamic simulation of the energy functioning of a built space in order to deduce an evaluation of the associated carbon footprint. a simulation and modeling tool is implemented based on several methodological references. we first use a systemic approach to describe the structural and functional aspects of the systems that make up the built space and the energy flux that we observe. this simulation is done in particular in a hierarchical way to allow a relevant analysis of flux from adequate data processing. the tool is also based on a typological approach around the notions of typical days, typical structural and functional configurations at different scales and angles of observation. the main purpose of this study is to present the correlation or weighting functions that will enable us to move from typical days to months of the year. we present results at different scales of observation in time and space. these results are explained in terms of energy consumption and in terms of carbon footprint based on emission factors of the energy mix of the waemu territory, more specifically burkina faso. keywords: energy balance, carbon balance, built space, consumption item, systemic approach, typological approach. 1. introduction the sustained growth in energy demand, the dependence on hydrocarbons and the fluctuation of their prices, as well as environmental and budgetary constraints have prompted many countries to promote energy efficiency [1]. the major challenge is to reduce energy consumption in the buildings and in particular the problems of greenhouse gas emissions in an environment of crisis and pollution which are getting worse and worse. to do this, our architectural practices, our equipment methods and our consumption habits must necessarily be reviewed. in france, the built-up space sector more particularly buildings, is the second most emitting sector of greenhouse gases (ghg) [2]. in this article we propose a detailed and dynamic approach for evaluating the energy consumption of a built space. we deduce the associated carbon footprint from the energy mix emission factors in a given country. for this we are developing a systemic and typological description tool of a built space. this tool will provide information on the structural and functional characteristics of the subsystems that make up the built space at different scales of observation. we consider on a typical day an operating scenario explaining the energy consumption by consumption item for each functional space. then from the weighting 1 corresponding author: daoudasawadogo2017@gmail.com 78 d. sawadogo, o. coulibaly, x. chesneau, b. zeghmati coefficients, we determine the energy consumption at the scale of the month and the year. we also work on typical functional spaces according to their structural characteristics in order to develop our results at the level of the entire built space. typological approaches will make it possible to implement data acquisition that is both detailed and the least tedious possible. the systemic approach will make it possible to conduct an analysis at different scales of observation in space and time [3]. it is observed that electrical energy is the most used in most built spaces. the evaluation of electrical energy consumption is complex [4]. therefore, the application of systemic tools and models seems to be a relevant, even essential, approach to decision-making in such a complex system. many works on the analysis of physical and environmental phenomena have been carried out using this new approach. we can cite: r. cantin et al. [5] conducted studies on thermal comfort in buildings. they considered a systemic approach to thermal comfort for a new characterization of different adaptive factors and mechanisms observed in situ. this systemic approach allows the representation of the dynamics of comfort and an elaboration of the structural characteristics of a dynamic model. the results show how a systemic approach contributes to a new analysis of thermal comfort allowing modeling likely to give rise to a dynamic simulation, then néria isabelle rakotonarivo [6] carried out a dynamic analysis of the carbon balance of the energy consumed in any built space. for this, it has designed a tool to assess, through a dynamic approach at different time scales and at different operating levels, the carbon footprint of the energy consumed by built spaces during their construction and operating phases. this evaluation is based on a systemic description of the different entities making up the built space and of the energy and material fluxes. the typological approach, frequent in the modeling of the thermal behavior of the building, refers to the classification of the building according to various characteristics. several works have been carried out using the topological approach for the description of energy consumption systems. we can cite: kavgic et al. [7] who simulated the operation in dynamic thermal regime (std: dynamic thermal simulation) of a representative building by a topological approach. energy consumption can be estimated at the scale of a territory by multiplying the energy consumption of the representative building of a class by the number of buildings of this class listed on the territory studied, and this, for each class defined in the model. however, the hypotheses of modeling and definition of the typological classes (number, precision, availability of data, influence of the discriminating parameters on the model) are to be dealt with subsequently swan et al. [8] carry out a modeling of the final energy consumption in the residential sector. in the energy field, they consider that this approach, based on the typological approach, allows the separation of consumption according to energy use. life cycle assessment (lca) is a standardized method of assessing the environmental impact of a product, system or activity throughout its life cycle, from cradle -to-grave. it includes all the consecutive stages from resource extraction, through transformation and use, to end of life, possibly with recycling [9] . several studies have been carried out using life cycle analysis (lca). for example: marie-lise pannier et al. [10] who investigated the identification of influential life cycle assessment (lca) uncertain parameters with buildings. this work, carried out with the lca tool novaequer, has highlighted the influence of the building's lifespan, the electricity production mix and the major contributors to the energy efficiency of the envelope. the results show a better understanding of the effect of influential parameters which contributes to make the use of lca more reliable for the energy performance of buildings. clyde zhengdao et al. [11] have shown for a life cycle assessment (lca) that a systematic and comprehensive assessment of dynamic approach to the energy functioning of a built space… 79 the energy of buildings is essential to maintain the sustainability of projects. life cycle assessment based on the energy of buildings is increasingly favored by researchers. 2. methodological approach 2.1. systemic approach fig. 1. systemic description of a built space the systemic approach allows us to implement a hierarchical and organic description of the different subsystems constituting the built spaces (eb) and the energy fluxes associated with them around entities that we will call: buildings (bât), functional spaces (ef), consumer station (pc) and components (c). the explanation of the energy fluxes will be done at the level of each of these sub-systems which will constitute as many scales of observation [12,13]. 2.2. typological approach fig. 2. typological approach 80 d. sawadogo, o. coulibaly, x. chesneau, b. zeghmati the typological approach will be considered in relation to several descriptive indicators: energy typologies, activity sector typologies, built space typologies and associated subsystems, spatial, structural and functional configuration typologies the types of reference observation day. these typological approaches will in particular make it possible to deduce detailed energy consumption fluxes from reduced data [14]. 2.3. normative framework our approach will also be based on iso standards around life cycle assessment considerations, particularly concerning the assessment of carbon footprints.  series of iso standards 14040 to 14044: specifies the principles and the framework applicable to carrying out life cycle assessments.  series of iso 14000 standards which designates all the standards relating to environmental management.  iso 14001 standard: defines a series of specific requirements for the implementation of an environmental management system within an organization, regardless of its size and field of activity.  iso 9001 standard: the iso 9001 standard defines a series of requirements concerning the implementation of a quality management system in an organization, whatever its size and sector of activity. its application is complementary to the iso 14001 standards. we also take into account a set of regulatory documents, referenced for the territory of application of the tool implemented: regional code for energy efficiency in buildings (ceeb) of uemoa [15,16]. 3. built space functioning simulation tool 3.1. architecture of the simulation tool the tool is produced within the framework of a java programming language controlling the navigation in each subsystem of the built space around several phases:  a phase of description of the study framework and the environment of the built space;  a phase of description of the observed subsystems;  a data acquisition phase;  a phase of energy consumption calculation;  a phase of analysis and presentation of results. in this article, we have illustrated the implementation of the simulation tool on the built space type "school complex" located on the territory of the west african economic and monetary union (waemu) more precisely in burkina faso in the city of ouagadougou in saaba. 3.1.1. description of the study framework and the environment of the built space it will initially be a question of specifying the types of energy considered, the sectors of activity and the environment of the built space according to various scales of observation of the territory considered in a systemic approach. identification of the type of energy: fuel, electricity, heat, or a combination of these sources. with the aim of influencing energy consumption and greenhouse gas emissions. identification of sector of activity: residential, commercial, industrial, agricultural. with the aim of affecting energy demand, equipment and system requirements. territory identification: urban or rural areas. with the aim of affecting climate, weather conditions and building regulations. dynamic approach to the energy functioning of a built space… 81 identification of the urban or rural agglomeration: to influence population density, traffic and available infrastructures. identification of the activity zone: residential, commercial, industrial, agricultural. the objective is to determine how the building is to be used, and what equipment and systems are required. building space system identification: encompasses the building structure, its mechanical and electrical systems, and its equipment. the aim is to determine the building's maintenance and repair requirements. building space subsystem identification: this is a set of building space system components that work together for a specific function. with the importance of assigning specific maintenance and repair requirements to each subsystem. this description provides a clear visualization of the building identification process, enabling effective communication with stakeholders and guaranteeing correct follow-up of all stages. 3.1.2. spatial description of the built space we identify in this description phase, the subsystems constituting the built space, as well as the components of each energy consumption item: 1. identification of the built space subsystem; 2. identification of the building; 3. identification of the functional space; 4. identification of consumption items; 5. component identification. before the official result of our school complex is published, we will present the validation of our study model. we have chosen to validate ('in situ' validation) our simulation tool for analysing energy flows, i.e. we are going to compare the response of our systems analysis application with the annual results of the electricity bills of a university campus. the design of our systemic model stems from a typological model that should be validated beforehand to avoid modeling errors. in view of the figure above, the university residence of patte d’oie has an electricity consumption of 124,387 kwh per year. it is close to the average annual value indicated by the electricity bills, which is 125,224 kwh with a maximum relative difference observed which does not exceed 1%. this discrepancy can be attributed to the lack of fictitious loads that enter and leave the site (mobile phones, pcs, etc.) brought by students who come to study or eat in the city. this assessment, depending on the rate of equipment, the time of use and the type of device, is therefore linked to the level of income of the residents. the distribution can be deduced from the fig. 3: fig. 3. annual electricity consumption of the analysis tool and the three-year average electricity consumption of the university city of patte d’oie 82 d. sawadogo, o. coulibaly, x. chesneau, b. zeghmati the pages of the tool providing these two description phases are presented more concretely around navigation sequences. this identification phase lists precisely the data reduced to the different scales of observation presented. these are structural data around in particular the dimensions and capacities linked to the subsystems of the built space and global functional data around in particular the sociological and economic characteristics [17]. at the level of the components or devices of a consumption station, the reduced data are their number and their power. the consumption items are clearly specified for a given built space in relation to the activities carried out there. the components are very variable; it is practical to define them in a drop-down menu which will allow them to be identified a priori. two scenarios can be programmed at the level of the functional spaces. we can, either consider a detailed data acquisition, or consider a reduced data acquisition by having previously defined typical functional spaces. 3.1.3. data acquisition phase this stage of description leads us, starting from scenario of use, to specify the characteristics of operation of each component of the whole of the stations of consumption for all the functional spaces of the buildings constituting the built space studied [18]. the pages of the tool providing this description phase for a typical day of operation, around the durations of use of the devices linked to the consumption item described, allowing the calculation of energy consumption for each hour of the typical day. then we will define the weighting coefficients making it possible to deduce the consumption for each month of the year and their summary on the scale of the year. several data acquisition strategies can be implemented to inform the simulation and analysis tool. we can proceed according to the following possibilities:  reference to statistical databases;  online data acquisition from bms or smart grid device;  use of experimental database;  introduction of simulated databases from building thermal or physical behavior software;  reference to seemingly relevant databases. the choices that will be made depend on the study contexts. in the present study we will rely on a priori data acquisitions and the use of experimental databases. 3.1.4. calculation phase calculation on a typical day we use the systemic approach calculation method to deduce the energy consumption associated with the hourly use profiles of the devices as part of an observation on a typical day: (1) (2) (3) dynamic approach to the energy functioning of a built space… 83 with: : total daily duration of use; : duration of use of the device at each hour of the day; : total hourly consumption of a component; : installed power of the device defined in the previous phase; : total daily consumption. these calculations are carried out at the level of components and consumer items directly during the first stage of data acquisition. calculation over a month and a year during the second stage of data acquisition, by integrating the monthly usage scenarios and the weighting coefficients making it possible to go from a typical day for a given month to typical days for all the months of the year, we continue the calculation for the other time scales over the year. we deduce the monthly and annual consumption from the following chart: fig. 4. flowchart for calculating consumption on a monthly and annual scale energy signature the energy signatures (es) make it possible to observe energy consumption (ec) by consumption item (pc), by functional space (ef) and by building in the built space (eb). these calculations are made at the level of functional spaces, buildings and built space. they are found from the following calculations carried out on the scale of the hour, the day, the month and the year. (4) (5) (6) 84 d. sawadogo, o. coulibaly, x. chesneau, b. zeghmati (7) carbon footprint the carbon balances will be deduced from the energy consumption based on the emission factors of the energy mix produced in a given place. the carbon footprint (bc) of the built space will therefore be determined using the following formula: (8) with: bc: carbon footprint expressed in kg eq co2 ce: energy consumption expressed in kwh ef: emission factor expressed in kg eq co2 /kwh the energy consumption here represents the consumption of each building constituting the built space. burkina faso's emission factor is 0.469 kg eq co2 /kwh. [19]. 3.2. presentation of results and analyzes 3.2.1. description of the built space we have chosen to carry out our calculations on a school complex with a typical construction using the materials most used in construction in ouagadougou. it is a multi-zone structure of 490 m². this site was modeled to determine the weighting coefficients. we were led to define 14 thermal zones within the complex. 3.2.2. determination of weighting coefficients the illuminance is measured using a luxmeter. table 1 shows the average measured and calculated illuminance values for some typical locations. the average illuminance values in the different locations do not respect the recommended average illuminance level. nevertheless, the led tubes used have a good luminous efficiency compared with the best current values which are between 80 and 150 lm/w. however, there are possible improvements with respect to lamp management. from all the above, we will take the monthly weighting coefficient for lighting equal 1. for certain consumption items, it is possible to obtain the weighting coefficients at the level of the consumption items by similarity for example. we also note that for some consumption items, the weighting coefficient is equal to 1. table 1. illuminance evaluation results typical place mean illuminance measured (lux) theoretical average illumination (lux) luminous efficacy (lm/w) bedrooms 186.2 243 92 study rooms 193.5 174 92 offices 432.6 464 92 the weighting coefficients for the air conditioning are taken from the doctoral thesis of coulibaly [20]. the graph shows the sensible loads obtained in the building. this study was carried out by taking the materials most used in construction in the uemoa zone, followed by meteorological data for the city of ouagadougou over a full year in hourly time steps. these metrological data include in particular temperature, humility, radiation, wind. dynamic approach to the energy functioning of a built space… 85 fig. 5. the sensitive loads of a building after determining the building loads, we deduce the correlation coefficients for the "air conditioning or comfort" consumption item. the correlation coefficients in relation to the sensitive loads are given for each month of the year. the results obtained by considering the month of april as the reference month are presented in table 2: table 2. weighting coefficients by item of consumption in the year month/pc jan feb mar apr may jul jul aug sep oct nov dec lighting 1 1 1 1 1 1 1 1 1 1 1 1 air conditioning 0.36 0.53 0.87 1.00 0.98 0.73 0.64 0.55 0.59 0.73 0.54 0.36 office automation 1 1 1 1 1 1 1 1 1 1 1 1 multimedia 1 1 1 1 1 1 1 1 1 1 1 1 these correlations or weighting coefficients are then integrated into our java simulation tool at the level of the corresponding consumption items mentioned above. then, we release the numerical and graphical results which will be presented following this publication. the graphical results will be given with the excel tool. 3.2.3. output of results from the simulation tool our simulation tool allows several digital outputs presenting consumption curves at the scale of the day, month or year at the level of components and consumption items, energy signatures per consumption item at the level of functional spaces, buildings and built space, energy signatures by consumption item and by functional space at the level of buildings and built spaces. we have chosen some of its digital outputs followed by the plotting of possible curves in the following figures. 3.2.3.1. energy flux by consumption item the energy fluxes by consumption item are observed at the level of a functional space, a building or the built space. at this stage we can have daily (hour by hour) or annual (month by month) consumption curves for each component and each consumption item of the built space. we then deduce for a given functional space, a first level of energy signature by consumption station on the scale of the day, the month or the year. table 3 shows the daily and monthly energy balance for january 2022 for the 6th classroom. table 3. electricity consumption by consumption item for the "6th class" functional area. daily consumption (kwh) monthly consumption (kwh) monthly carbon footprint (kwh) lighting 2.60 67.60 31.70 air conditioning 3.37 31.59 14.82 office automatique 1.62 42.12 19.75 86 d. sawadogo, o. coulibaly, x. chesneau, b. zeghmati sécurité 0.24 7.44 3.49 total 7.83 148.75 69.76 table 3 shows that the daily consumption of the "air conditioning" consumption item in the "6th class" functional area is higher than that of the other consumption items. this is justified by its power and operating time, the monthly consumption of the "lighting" consumption item outweighs that of the other consumption items in the same functional area. this is justified by its wattage, usage factor and weighting coefficient and the "lighting" consumption item ejects a higher monthly carbon footprint than the other consumption items. this is justified by the fact that the "lighting" consumption item admits a higher monthly consumption than the other consumption items. fig. 6. daily consumption (hour by hour) of the “6th grade” functional space fig. 6 shows a peak in power consumption from 8 pm to 12 pm followed by a constant until 11 a.m. around 12 noon. electricity consumption falls until 1pm, then rises until 2 pm and remains constant until 6 pm. this can be explained by the fact that the teaching courses started at 8 am. and 3 pm. and all the equipment was switched on, and by the fact that the students went downstairs to have lunch. for constant consumption from 1 am. to 6 pm. and 7 pm. to midnight, it's the safety device that's on only. 3.2.3.2. energy flow by functional space the same types of graphic outputs can be produced by functional space at the level of each building or the built space at the scale of the month or the year. by way of illustration, we present for example the signature of the monthly energy consumption for the month april 2022 by functional space, from which we deduce the daily and monthly consumption curves (fig. 7) and (fig. 8). fig. 7. daily consumption per functional space in the “education” building dynamic approach to the energy functioning of a built space… 87 fig. 8. monthly consumption per functional space in the “education” building the daily consumption of the "education" building is 65.48 kwh. fig. 7 shows the results obtained with the analysis tool. fig. 8 shows consumption trends for the "education" building over the month of april. we remind you that it is evaluated on the basis of usage coefficients and monthly weighting factors. monthly consumption was 1743.15 kw. high consumption is observed in the "supervisors' office" functional area. this consumption is responsible for the emission of 817.54 kg eq co2. 3.2.3.3. energy flux per building in the built-up area we can also present signatures of energy consumption or greenhouse gas emissions within the built space by building also at different time scales fig. 9, 10 and 11 and an annual consumption curve for built space (fig. 12). fig. 9. daily consumption per building of built space "level r+2" fig. 10. monthly consumption per building of built space "level r+2" 88 d. sawadogo, o. coulibaly, x. chesneau, b. zeghmati fig. 11. monthly carbon footprint per building of the “r+2 level” built space fig. 12. annual consumption of built space "level r+2" the tool shows the daily and monthly consumption of the "r+2 level" building area. in all, the building space consumes a total of 158 kwh per day and 4435.07 kwh per month. the greenhouse gas (ghg) emission is 2080.05 kg eq co2. in fig. 9, 10 and 11, consumption in the "education" building is higher than in the bedrooms of students. this is confirmed by the appliances and monthly usage factors. we note that electricity consumption in the "r+2 level" building area is higher in october than in the other months (fig. 12). during the hottest months of the year (march, april and may), consumption is higher than in the other months. this is logical, as during theses months, the population uses more means to cool buildings, such as air-conditioning or fans; in addition, consumption drops sharply as school staff and students are on vacation during the month of july; august and september. consumption peaks at around 4130.04 kwh in october. electricity consumption then falls to around 3053.31 kwh in december. conclusion at the end of our work, we have a tool for simulating and analysing detailed information at different scales of observation in time and space. data acquisition is initially done in hourly time steps. this allows a fine dynamic approach to changes in energy fluxes over time. successive integrations based on weighted correlation laws make it possible to visualize these fluxes at the scale of the month and the year. beyond this dynamic representation, several types of signature can be observed. at this stage 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(2010). description and systemic modeling of energy working built spaces, proc. of the 7th international conference indoor air quality, ventilation and energy conservation in buildings, (syracuse, new york). 18. vorger, e. (2014). étude de l'influence des occupants sur la performance énergétique des logements par le biais d'une modélisation stochastique globale [study of the influence of occupants on the energy performance of dwellings using global stochastic modelling]. proc. of conférence ibpsa france-arras (paris, france), 33–50, [in french]. 19. l’agence internationale de l’energie (2013). émissions de co2 dues à la combustion des carburants fournissant des données pour 2011210 [international energy agency (2013). co2 emissions from fuel combustion providing data for 2011210] [online]. available: https://fr.globalpetrolprices.com/, [in french]. 20. coulibaly, o. (2011). contribution à l’élaboration d’une réglementation thermique et énergétique des bâtiments au burkina faso : données de base multiparamétriques et modélisation thermo-aéraulique sous codyba et trnsys [contribution to the development of thermal and energy regulations for buildings in burkina faso : multiparameter baseline data and thermo-aerodynamic modelling using codyba and trnsys]. ph.d. thesis, université joseph ki-zerbo, [in french]. adv syst sci appl 0000; 00; 1-4 published online at http://ijassa.ipu.ru. optimization of a hybrid system (pv-fuel cell) of energy production for an isolated site touhami ghaitaoui1*, sliman laribi1 , essama ahmed ghaitaoui2, touhami drissi 1, ibrahim ben omar1, youcef halali1 1) 1 laboratoire de développement durable et informatique (lddi) , faculté des sciences et de la technologie, université d’adrar, adrar, algeria. 2) iceps laboratory ,djillali liabes university of sidi bel-abbes, sidi bel-abbes, algeria e-mail: touhami.eln@gmail.com abstract: this recently, there has been a shift in autonomous micro-grid systems, moving away from reliance on fossil fuels. instead, these systems are now integrating existing renewable energies in specific areas to promote sustainability in the energy sector. one promising method for energy storage and transmission is through the use of hydrogen and solar energy, especially in remote regions where hybrid techniques can meet the required energy demands. our research focuses on electrifying a agricultural site located in the sbaa region, specifically in adrar city, southwest algeria. we propose the implementation of an independent small electric network that combines various renewable energy sources (pv/full cell). to ensure cost-effectiveness, we aim to optimize and control the system's components, carefully sizing them through a meticulous selection process. this study evaluates the technical and economic feasibility of renewable energy systems, with the goal of reducing the total net cost, energy expenses, and unfulfilled load, while also minimizing co2 emissions. the simulation program homer pro is used to conduct this analysis. overall, the proposed system demonstrates positive results in electrifying remote areas while maintaining a conservative cost of electrical energy, ultimately establishing sustainable regions. keywords: agricultural site , electrolyzer, fuel cell, homer pro software, hydrogen tank, pv energy, power to hydrogen to power, renewable energy, stand-alone micro grid. 1. introduction only diesel generators can power algeria's irrigation systems, rural areas, and arid regions without access to the electricity grid. an irrigation system powered by diesel is a major contributor to climate change, the depletion of fossil fuel reserves, rising operating costs, and diesel-related maintenance issues, all of which prompt the search for alternative energy sources for the system [1]. as of now, sun powered pv-based water system framework is considered as an elective water system arrangement. it demonstrates the financial, ecological and specialized feasibility contrasted with the diesel-based water system framework [2]. presently, battery capacity framework is excluded from sun oriented water system framework in algeria because of significant expense and upkeep issues. however, a small amount of water is stored in the water storage tank that is included. what's more, the significance of energy stockpiling ought to have been considered in the midst of appeal and awful climate [3]. without a doubt, on account of long haul energy capacity, it is desirable over utilize the hydrogen framework which is more proficient, productive and solid [4]. in the second half of the 21st century, the production of hydrogen from water will gradually replace fossil fuels and become the primary energy carrier [5]. when the term "renewable energy" is used, water is typically regarded as a sustainable and clean source for the production of hydrogen [6]. mailto:touhami.eln@gmail.com 2 i. petrov, j.f. doe copyright ©0000 assa adv. in systems science and appl. (0000) by achieving a balance between the demand for water and the supply of power, energy storage is an effective solution to the storage and daylighting issues. among power capacity innovations, electrochemical energy stockpiling has gotten more consideration because of its security, dependability, lattice similarity and high proficiency [7]. due to its high calorific value, energy density, and emission-free or near-emission-free nature, hydrogen generation, which is based on the electrolysis of water, is one of the most widely used chemical energy storage methods [6]. the hydrogen storage tank's outflow is low in this storage system. the hydrogen stockpiling framework can likewise be utilized for dependable and nonstop capacity on the grounds that its energy conservativeness is high and easy to introduce and move anyplace [8]. mechanically created nations have previously carried out hydrogen power module innovation in their vehicle area [9] and in the water system area utilizing an inverter [10]. in order to power rural homes, a feasibility study of a hydrogen fuel system for ac electrification in algeria's rural areas was conducted in [11]. additionally, a number of works in the literature on hybrid systems presented two or three different kinds of renewable energy sources for design and planning [12-13]. instead of putting pressure on algeria's economy, the hydrogen fuel cell irrigation system is an alternative way to meet the current demands of agriculture. in algeria, particularly in the southern desert regions, irrigation is typically required for 10 to 13 hours per day for cultivation, while solar radiation is available for 8 to 10 hours. this alternative solution will be profitable and beneficial to the environment. the load is powered by approximately half of the solar radiation. any storage system can reduce the size of the solar panel and the amount of land needed to install it by using 50% of the radiation that is not being used. as a result, farmers can cultivate more land. in this postulation, a substituting current siphon is chosen to plan and recreate the half breed water system framework (power device/pv) in the space of sbaa adrar algeria. this study is carried out by working with an optimization tool: the simulation software homer (hybrid optimization of multiple energy resources) is an economic optimization model, which minimizes the cost of energy production to satisfy the final energy demand. energy based on current net cost. thus, homer-based hydrogen fuel cell system for irrigation is analyzed and designed as a renewable green energy technology by reducing energy storage requirements, improving efficiency and reliability. 2. methods and materials the stand-alone hybrid system shown in fig. 2.1 consists of several basic components, including photovoltaic cells, fuel cell, electrolyser, hydrogen tank, inverter and battery. the photovoltaic cell, power device and electrolyzer are associated through a dc transport. the transformer connects the ac output to the load while the dc output connects to the dc bus. photovoltaics use solar energy to create electricity during the day, whereas hydrogen systems generate power both during the day and at night. the fuel cell serves as a backup power source for the hybrid system, ensuring that electricity remains available in the case of breakdowns or power outages. word template for assa manuscript 3 copyright ©0000 assa. adv. in systems science and appl. (0000) fig. 2.1. model of a hybrid pv/fuel cell power plant 3. description of study area this work was done with homer pro version 3.14.2. fig. 3.1 shows the location of the agricultural zone of sbaa in the willaya of adrar algeria. fig. 3.1.geographical coordinates of the agricultural region sbaa (adrar ،algeria) the agricultural zone of sbaa, located in the wilaya of adrar in algeria, is an important region for agriculture in the region. it is characterized by a desert climate and arid geographical conditions. despite these challenges, the area benefits from groundwater resources and adapted agricultural techniques that allow the practice of agriculture fig. 3.2. fig. 3.2. real photo of the agricultural region concerned by the study 4 i. petrov, j.f. doe copyright ©0000 assa adv. in systems science and appl. (0000) 3.1. resources of the study area solar radiation and mean annual temperature are shown in fig. 3.3 and 3.4, respectively, based on data from the nasa surface (weather and sun) webpage for the research region. fig. 3.3. global monthly data on horizontal irradiance for the sbaa adrar agricultural region fig. 3.4. monthly average temperature data of the sbaa adrar agricultural region 3.1. concepts and components of micro-grids the average load indicated in fig. 3.5 has been entered into the program, which uses this data to estimate the remainder of the information over the course of a year. this first stage of the process shown that the agricultural region sbaa adrar consumes 275.04 kw, a figure that must be addressed in the planning and design of the mr, which includes a pv system, a fuel cell, an electrolyzer, a hydrogen tank, and a storage system. the system is not linked to a conventional network. the project's lifespan is set at 25 years. table 3.1. electrical characteristics of pivoted system the pivots pump power (w) power of 5 motors (kw) rotation time (h) 1 44741.99 1.1 13h number of pivots total irrigation area (ha) total load (kw) 6 180 275.04 word template for assa manuscript 5 copyright ©0000 assa. adv. in systems science and appl. (0000) fig. 3.5. monthly average temperature data of the sbaa adrar agricultural region 4. architecture of the system table 4.1 shows the overall input data for the hybrid system employed, including the cost of volume, capital, replacement, and maintenance, as well as the lifetime. table 4.1. architecture of the system [11] component size capital cost ($)/kw replacement cost ($) o&m cost ($/yr) lifetime pv 1 kw 783.3 783.3 10.00 25 years fuel cell 1 kw 3000.00 3000.00 0.10 40000 h electrolyzer 1 kw 2000.00 2000.00 0 15 years hydrogen tank 1 kg 1500.00 1500.00 30 25 years converter 1kw 800.00 750.00 1 15 years 5. results and discussion using homer pro, the suggested scenario for upgrading the small standalone hybrid power system shown in fig. 5.1, which employs 100% renewable energy, was evaluated for system performance and optimum sensitivity. the conceivable combinations were distinguished based on the given imperatives and inputs, taking into account their capital expenditures (capex), operating expenses (opex), net present cost (npc), and cost of energy (coe). fig. 5.1. monthly average temperature data of the sbaa adrar agricultural region 6 i. petrov, j.f. doe copyright ©0000 assa adv. in systems science and appl. (0000) 5.1. results of simulation the simulation results were obtained using homer pro software, which allowed for an hourly analysis of different configurations based on size and cost parameters. table 5.1 provides an overview of these configurations. following the simulations, a set of groups was established to analyze and discuss the results obtained from each group. these groups likely focused on specific aspects or variations in the system design, such as different component sizes or cost considerations. table 5.1. optimization results 5.2. pv panel / p2h2p system the pv/p2h2p panel system combines solar pv panels with a fuel cell to evaluate the technical and economic viability of a 100% renewable energy system under the 2023 electricity tariff. the optimized system design includes a fuel cell capacity of 250 kw, an electrolyzer capacity of 60 kw, a hydrogen tank capacity of 30 kw, a photovoltaic capacity of 507 kw, and a converter capacity of 41.5 kw. the estimated cost of energy (coe) for the system is 0.509$/kwh, with a net present cost (npc) of 765,306.00$. the power supply in this system is derived from both the photovoltaic module and the fuel cell. on-site power generation is predominantly sourced from the solar pv panels, accounting for 87.5% of the monthly power generation, while the fuel cell contributes 12.5%, as depicted in fig. 5.2. the system achieves 100% renewable energy utilization due to the favorable solar potential and economic competitiveness of solar pv panels. fig. 5.2 presents the energy production from both the photovoltaic panels and fuel cells, along with their contribution to energy generation and consumption over a one-year period. the study area experiences nearly 5 months (march, april, may, september, october) with similar climate conditions, characterized by temperatures ranging from 25 to 35°c. for the remaining months, the temperature exceed 45°c. table 5.3 provides a summary of the annual electricity production and usage for the pv/p2h2p system at the site. the total capital cost of the system, considering replacement, operation, and capital maintenance costs for one year and over a 25-year period, is estimated at $619,319.68, as depicted in fig. 5.3. table 5.2. financial characteristics of optimized pv/p2h2p systems. pv (kw) fuel cell (kw) electrolyzer (kw) hydrogen tank (kg) converte r npc (ud) coe (usd/kwh) 507 250 60 30 41.5 765,305.84 0.590 word template for assa manuscript 7 copyright ©0000 assa. adv. in systems science and appl. (0000) table 5.3. yearly power generation and utilize of on-site grid/pv frameworks. pv production (kwh/an) fc production (kwh/an) demand (kwh/an) total use (kwh/an) 40.581 114542 4885008 4445412 fig. 5.2. monthly electric production for the pv / p2h2p system. fig. 5.3. the hybrid energy system's current total net cost however, the cost of running and maintaining this technology is substantially higher, thus we conclude that the element determining the overall cost of the system is the inclusion of the fuel tank, with the current net cost of the system anticipated over 25 years being 765,305.84. $. 6. conclusions the objective of this work is to evaluate the feasibility of a stand-alone hybrid renewable energy system capable of meeting energy demand. a stand-alone power system can avoid the need to build an expensive grid in rural areas. the use of solar energy and hydrogen has considerable advantages as the primary power sources for this system. however, stand-alone photovoltaic systems have their own limitations, including their due high cost and low efficiency. simulation of this system using homer software revealed that it has the lowest net present cost (npc) and lowest cost of electricity (coe) over the life of the project. 25 years. the proposed and evaluated model can be reproduced under similar load profiles and with available renewable energy resources, but it must be re-evaluated if the input data changes. given the high feed-in tariffs and capital costs, it is essential to determine the life cycle cost of the system, i.e. the total net present cost (tnpc), in order to obtain a optimal energy hybrid system. it is important to observe the actual effects on the power system and its economic parameters, as well as variations in sensitive variables that influence the simulation results. in this study, two 100% renewable energy scenarios were defined, modeled, simulated and compared for a stand-alone solution. from simulations carried out with homer pro, the main conclusions are as follows: the evaluation of the two scenarios showed that the scenario based on the hybrid pv-p2h2p-battery combination (scenario 2) is the most promising approach for the selected stand-alone micro grid site in the agricultural region of sbaa in adrar, in algeria .the hybrid system can further reduce the amount of excess electricity compared to a battery-only storage system. the hybrid system is able to extend the autonomy of the system compared to a battery storage system by using the stored 8 i. petrov, j.f. doe copyright ©0000 assa adv. in systems science and appl. (0000) hydrogen. hydrogen production from renewable energy sources could be a long-term energy storage solution and provide an energy carrier for various uses, with excellent environmental characteristics. future work aims to analyze the technical and economic advantages of integrating these new ways of using hydrogen. references [1] d. k. yadav, sai pranith girimaji and t.s. bhatti, “optimal hybrid power system design using homer,” power electronics (iicpe), ieee 5th india international conference, pp. 1– 6, dec 2012 [2] m. rezwan khan, prospect of solar pv based irrigation in rural bangladesh: a comparative study with diesel based irrigation system developments in renewable energy technology x. han, x. cui, x. hang, t. yan, z. qu and c. wang, “distributed generation system design based on energy storage converter technology,” 10th world congress on intelligent control and automation, 2012,wcica 2012 [3] o.v. marchenko, s.v. solomin, the future energy: hydrogen versus electricity, international journal of hydrogen energy 40 (2015) 38013805, doi: 10.1016/j.ijhydene.2015.01.132 [4] bockris jo’m, the hydrogen economy: its history. international journal hydrogen energy 2013; 38: 2579e88. [5] y. li, d.w. chen, m. liu, r.z. wang; life cycle cost and sensitivity analysis of a hydrogen system using low-price electricity in china, international journal of hydrogen energy 42 (2017) pp. 1899-1911 [6] mahlia tmi, saktisahdan tj, jannifar a, hasan mh, matseelar hsc. a review of available methods anddevelopment on energy storage; technology update. renew sustain energy rev 2014; 33(2):532-45 [7] a. gargoom, abu mohammad osman haruni, m. e. haque and m. negnevitsky, "hybrid stand-alone power systems with hydrogen energy storage for isolated communities," ieee pes t&d 2010, new orleans, la, usa, 2010, pp. 1-6. doi: 10.1109/tdc.2010.5484293 [8] dieter weigl, m. henz, inayati and h. saidi, "converted batterypowered electric motorcycle and hydrogen fuel cell-powered electric motorcycle in south east asia: development and performance test," proceedings of the joint international conference on electric vehicular technology and industrial, mechanical, electrical and chemical engineering (icevt & imece), surakarta, indonesia, 2015, pp. 1-4. doi: 10.1109/ icevtimece.2015.7496637 [9] h. rezk and m. shoyama, "techno-economic optimum sizing of standalone photovoltaic/fuel cell renewable system for irrigation water pumping applications," power and energy (pecon), 2014 ieee international conference on, kuching, 2014, pp. 182-186. doi: 10.1109/pecon.2014.7062437 [10] k. k. hossain and t. jamal, "solar pvhydrogen fuel cell system for electrification of a remote village in bangladesh," 2015 international conference on advances in electrical engineering (icaee), dhaka, 2015,pp.22-25. doi: 10.1109/icaee.2015.7506787 [11] ghennou, s., nasser, t.m., benmoussa, d., chellali, b. (2022). energy optimization of a purely renewable autonomous micro-grid to supply a tourist region. european journal of electrical engineering, [12] s. ashok, “optimised model for community-based hybrid energy system”, renewable energy 32 (2007) pp. 1155–1164 [13] s. m. shamim reza and n. i. sarkar, "design and performance analysis of a directlycoupled solar photovoltaic irrigation pump system at gaibandha, bangladesh," green energy and technology (icget), 2015 3rd international conference on,dhaka, 2015, pp. 1-6. doi: 10.1109/icget.2015.7315116 microsoft word 1674.docx adv syst sci appl 2025; 3; 40-51 published online at https://ijassa.ipu.ru. the 2023 dengue outbreak in bangladesh: serotypes, causes, and impact md yasin ali parh1,2, mst sharmin akter sumy1,2, md. murad hossain3*, nayeem bin saifuddin4, jannatul ferdousi elma5, hamid zarei6, md nazmul hasan7 1) university of louisville, louisville, ky, usa 2) department of statistics, islamic university, kushtia, bangladesh 3) gopalganj science and technology university, gopalganj, bangladesh 4) dhaka nursing college, dhaka, bangladesh 5) kumudini nursing college, mirzapur, tangail, bangladesh 6) university of louisville, louisville, ky, usa 7) university of saskatchewan, saskatoon, sk, canada abstract: the rapid spread of the mosquito-borne fever, dengue, is a growing public health concern in bangladesh. the nation experienced its worst dengue outbreak in 2023. this study aims to present an overview of the 2023 dengue situation in bangladesh. we utilized monthly reported data on dengue cases and dengue-related deaths obtained from the bangladesh institute of epidemiology, disease control and research (iedcr). keywords: dengue outbreaks, cases, deaths, cfr, public health crisis. 1. introduction dengue virus (denv) infection remains a significant global public health problem that is brought on by four closely related serotypes of the virus (denv 14) and it is most commonly found in urban and semiurban areas of tropical and subtropical climates [33]. according to the world health organization (who), dengue fever is now endemic in more than 100 nations [8]. in 2023, and as of the end of july, over three million cases and over 1500 dengue-related deaths have been reported globally [12]. furthermore, every year, an estimated 400 million individuals become infected with the dengue virus, 100 million become sick, and 21,000 die as a result of dengue [8]. a minor fever to the potentially fatal dengue hemorrhagic fever (dhf) or dengue shock syndrome (dss) can be brought on by any of the four serotypes of denv infection [23]. infection with one serotype confers lifetime immunity to that serotype but does not provide long-term protection against the other three serotypes [28, 21]. dengue has a wide spectrum of clinical signs and is usually accompanied by hematological changes such as leukopenia and thrombocytopenia in mild cases and vascular alterations, including vasodilation and increased permeability. denv infection contributes to plasma leakage, hemorrhage, or organ impairment, such as liver damage [22, 34]. evidence suggests that severe dengue patients have systolic and diastolic cardiac dysfunction as a result of the septum and right ventricular wall injury [35]. while clinical manifestations are typically modest, a potentially fatal complication known as severe dengue can arise and is the leading cause of hospitalization and mortality of patients [26, 32]. many factors are contributing to increase the dengue transmission intensity. increased population, international travel, worldwide trade, uncontrolled urbanization, mosquito * corresponding author: murad.stat@gstu.edu.bd word template for assa manuscript 41 copyright ©2025 assa. adv. in systems science and appl. (2025) migration, climate and environmental changes, human host immunity, and virus genotype are regarded to be key reasons for increasing the intensity of dengue virus transmission [7,9,13,20,24,27,31]. the el nin˜o weather phenomenon, anticipated to intensify the already rising temperatures in asia, is also projected to heighten the prevalence of mosquito-borne diseases such as dengue [4]. dengue fever outbreak control is difficult in tropical settings, where rising temperatures can make longer the length of the mosquito season and provide more time for dengue transmission. bangladesh, like other southeast asian (se) countries, with a population of approximately 165 million, has become a perfect environment for the dengue vector, resulting in an increase in dengue transmission since the first outbreak in 2000 due to its proximity to the equator and subtropical/tropical temperature [25]. dengue fever has spread to 64 districts in bangladesh, and public health experts believe that the situation is worse now and posing a substantial risk to a large number of individuals. between 2012 and 2019, the most dengue cases were reported in dhaka, bangladesh’s capital and most populous city, with a population of almost 16 million people [5, 29]. last year the number of dengue infections has soared more, especially in the capital, dhaka. hospitals are overwhelmed by the outbreak. the dengue outbreak in bangladesh has taken a concerning turn this year, with an alarming spike in cases and fatalities. according to a directorate general of health services (dghs) survey of 50 dengue patients, women are more likely than men to die from dengue (62% versus 38%, respectively). additionally, it demonstrates that dengue killed the majority of individuals between the ages of 19 and 50 [18]. there is evidence of a potential gender and age-specific difference in morbidity and death, which may point to more serious infections in specific age groups. even though only a few hospitals were reported in the majority of the research, and the sample sizes were modest, these studies may nevertheless offer useful data on the distribution of age, sex, and other demographic characteristics. a safe and effective dengue vaccine must be available and should have strong and long-lived protection against all four serotypes to avoid making a person particularly vulnerable to dengue disease. furthermore, community awareness and health care prevention are critically important in disease prevention [3]. inadequate health care infrastructure, lack of outbreak planning, and awareness of dengue infection may result in public health crises if suitable prevention techniques are not taken ahead of time. it is an urgent need to reveal the reasons for mosquito behavior changes, possible reasons for dengue outbreaks, and gender and age-specific differences in morbidity. as a result, the purpose of this study was to investigate the prevalence of dengue and precise accounting of the number of cases, mortality rates, dominating serotypes, epidemiology, and clinical presentation of dengue virus in order to present a comprehensive picture of the dengue pandemic in bangladesh, which could benefit in the diagnosis and control the future outbreaks. 1. methods 1.1. data collection data was compiled from the national database, which had released the confirmed dengue cases and related deaths through a government press release from the directorate general of health services (dghs), ministry of health and family welfare, bangladesh. the website https://old.dghs.gov.bd/index. php/bd/, from which we collected all the released data. the dataset included all officially reported dengue cases and dengue-related deaths compiled by dghs/iedcr from january 1, 2023 to december 31, 2023. monthly reports covering this full one-year period were extracted and used for all analyses. 42 parh m. y. a., et al. copyright ©2025 assa adv. in systems science and appl. (2025) 1.2. data source and validation all dengue-related data were retrieved from the directorate general of health services (dghs) official press releases and the bangladesh institute of epidemiology, disease control and research (iedcr). these data undergo internal verification at dghs before public release, including validation of case counts reported from government and private hospitals. to ensure reliability, we cross-checked the monthly cumulative data with daily situation reports published by dghs and confirmed consistency of case and death counts across releases. we also screened the dataset for anomalies, duplicate entries, inconsistencies in cumulative curves, and missing values; no significant missingness was identified. 1.3. statistical analysis we conducted descriptive statistical analyses to summarize dengue cases, deaths, and case fatality rates (cfr) by month, age group, gender, and geographic location. categorical variables (e.g., gender differences in cfr and case distribution across regions) were compared using the chi-square test. when comparing continuous variables, we used independent sample t-tests or the mann–whitney u test when normality assumptions were not met. age-specific and gender-specific cfr differences were evaluated with proportion tests. time-series trends (weekly and monthly fluctuations) were assessed through graphical visualization and percentage change calculations. all analyses were performed using standard statistical software (e.g., r or spss), and a p-value < 0.05 was considered statistically significant. 2. results and analysis in this section we will discuss our results and analysis through a subsection of brief history of dengue outbreak in bangladesh 2.1. a brief history of dengue outbreak in bangladesh dengue was first recorded in the 1960s in bangladesh (east pakistan) and was known as “dacca fever” [30]. from that time dengue fever has become an endemic disease, infecting thousands of people and lowering people’s quality of life. dengue fever occurrences documented every year since 2000 with new cases being reported every day. in 2019, bangladesh saw the greatest number of dengue infections (101,354 cases) and 179 people died [17]. due to excessive rainfall, water-logging, flooding, temperature rises, and odd shifts in the country’s conventional seasons, bangladesh’s climate conditions are becoming more suitable for the transmission of dengue and other vector-borne diseases such as malaria and chikungunya [2]. an abnormally lengthy wet season, and resistance to pesticides which are used by local authorities considered major drivers of that outbreak [19]. based on accessible hospital-based surveillance data, seven outbreaks with over 5,000 hospitalized cases were recorded in dhaka city between 2000 and 2022. a significant segment of bangladesh’s populace opts to reside in urban areas due to the potential for improved earnings, access to quality education, and better overall amenities. the dengue outbreak in dhaka in 2019 marked the most severe instance since dengue outbreaks first appeared in bangladesh in the late 1990s [16]. this epidemic was followed by a significant outbreak in 2021 (28,429 cases and 85 deaths), the real size of which is thought to be hidden by the covid-19 pandemic (delta onslaught). another large outbreak occurred (61,732 cases, 281 deaths) in 2022. up to 2021, dhaka remained the epicenter of all large outbreaks. later on, all the districts in bangladesh experienced a significant outbreak with a large number of cases and death rates. even though dengue is common in bangladesh, the most recent outbreak (2023) is exceptional in terms of the number of cases and deaths, seasonality, and the abrupt early upsurge. fig. 1 presents a clear picture of dengue-related cases from 2014 to 2023. the majority of confirmed dengue fever cases occurred in the latter half of the year, as depicted by the graph. word template for assa manuscript 43 copyright ©2025 assa. adv. in systems science and appl. (2025) as of august 3, in 2023, the death toll exceeded that of previous years, and by august 21, the number of hospitalizations had surpassed the highest recorded during the 2019 outbreak [11]. in september 2023, 79,598 confirmed cases of dengue fever were the highest monthly total on record. also, in september 2023, 25.0% of cases and 23.2% of deaths of total cases and deaths of this year were reported. the reported number of dengue cases and deaths this year so far is the highest ever recorded in the country. the subsequent highest dengue confirmation was in august 2019 (confirmed cases 52,636). in 2019, 101,354 cases of dengue fever were confirmed, with nearly 98% of cases occurring between july and december. it’s noteworthy that bangladesh’s death rate has somewhat grown in recent years. similar to past years, the epidemic commenced during the summer months (april–may), proliferated, and escalated across the entire nation during the monsoon season (july–august) [33]. fig. 1. year and month-wise dengue reporting in 2023 fig. 2 provides a distinct visualization of the weekly variations in dengue cases throughout the year, 2023. in dhaka division, the surge in cases began during epidemic week 17 (23-29 april 2023, 27 new cases in dhaka and 14 cases outside of dhaka), while all eight divisions experienced an uptick from epidemic week 26 (25 june to 1 july 2023, 1409, cases) onward. fig. 2. weekly fluctuation of dengue cases in 2023 fig. 3 shows the monthly dengue cfr of 2019 and from 2021 to 2023. the picture shows that although the death rate in april of 2023 is low compared to the death rate in april of 2019, and for other months of 2023, death rates are higher compared to the other months of different years. also, the figure shows that the start of may has the highest concentration of denguereported deaths for all different years. in 2023, although the cfr was very high in february but march onwards, it has just reached a steady level with a high cfr for the next several months. it indicates that last year’s outbreak is exceptional due to the high caseload, high fatality rate, seasonality, and abrupt early spike. the dengue outbreak is getting worse due to altered weather patterns brought on by climate change. less rainfall does not always stop the spread of dengue, even in bangladesh when the monsoon seasons end, and the weather turns dry. the mix of temperature, humidity, and rainfall determines the ideal circumstances for aedes mosquitoes to flourish. 44 parh m. y. a., et al. copyright ©2025 assa adv. in systems science and appl. (2025) fig. 3. year and month-wise case fatality rates (cfr) of 2019, and from 2021 to 2023 analyses of data from inpatient records and geographic coordinates of residence locations between 01 january and 31 december of 2023 highlighted two important findings that require further epidemic investigation: first, approximately half of the cases (n=110,008; 34% of confirmed dengue cases) and deaths (n=980; 57% of dengue-related deaths) were recorded in the capital city dhaka raising concern about the high density of denv patients in the proximity of the hospitals (fig. 4(a)). dhaka bears the brunt of the outbreak and serves as the epicenter, with over half of the reported cases originating in this mega-city [6]. fig. 4. (a) case fatality rate cfr (%) of patients living in dhaka city and outside of dhaka city (b) gender wise case fatality rate (%) (c) percentage of gender wise cases in dhaka city (d) percentage of gender wise cases in outside of dhaka city second, although the percentage of female patients in dhaka and outside of dhaka city are less compared to male patients, the overall cfr for female patients (cfr: 0.8) is higher all over the country compared to male patients (cfr: 0.4) (figure 4(b)). the percentage of gender wise dengue cases in dhaka and outside of dhaka are shown in figure 4(c & d). a notable prevalence of males was observed, constituting 60% of cases in both dhaka and outside of dhaka city. women may have preferred to stay at home for longer periods and delay visiting a doctor, these could be the possible reasons for high cfr. in every study, the percentage of male cases was approximately twice as high as that of female cases and the male-to-female ratio reached 2.7 [1]. hospital-based studies reveal that male patients are more common in developing nations like bangladesh, this disparity may be the result of case selection bias. another factor can be the cultural fact, as asian women are typically wearing long dresses. gender wise dengue cases for different age groups in dhaka and outside of dhaka are shown in fig. 5 the distribution of age among individuals infected outside of dhaka city reveals that the majority of cases (46%) occurred in the 16 to 35 years age group, whereas the age group above 71 years constituted the smallest proportion (< 2%). in terms of gender, males word template for assa manuscript 45 copyright ©2025 assa. adv. in systems science and appl. (2025) exhibited higher infection rates than females across all age groups outside of dhaka city. likewise, the breakdown of age among infected individuals in dhaka city shows that the largest percentage of cases (50%) occurred in the 16 to 35 years age group, while the age group above 71 years constituted the smallest fraction (< 2%). regarding gender, males demonstrated elevated infection rates compared to females across all age groups in dhaka city. fig. 5. gender wise dengue cases in dhaka and outside of dhaka we discovered indications of a potential age-specific variation in cfr nationwide, suggesting more severe infections in older age groups of male patients (≥ 76 years with cf r : 2.9) compared to female patients (≥ 76 years with cf r : 1.8). this information is illustrated in table 1 and fig. 6. the older age group recorded a higher cfr compared to the younger age group. in 2023, more than 57% of confirmed cases were adults, with the greatest age range for cases being between 16 and 40 years old. approximately 29% of all dengue cases in bangladesh are observed in children and youth under the age of 20. children frequently engage in outdoor activities without wearing long-sleeved clothing, and a significant number do not utilize mosquito nets during sleep at night. children under the age of five, in particular, face an elevated risk of experiencing severe symptoms such as dehydration and shock resulting from dengue. table 1.1. distribution and predominant denv serotypes in different districts of bangladesh age groups number of female cases number of male cases number of female deaths number of male deaths 𝐶𝐹𝑅 = 𝐷𝑒𝑎𝑡ℎ 𝐶𝑎𝑠𝑒𝑠 × 100 𝑓𝑜𝑟 𝐹𝑒𝑚𝑎𝑙𝑒 𝐶𝐹𝑅 = 𝐷𝑒𝑎𝑡ℎ 𝐶𝑎𝑠𝑒𝑠 × 100 𝑓𝑜𝑟 𝑀𝑎𝑙𝑒 0-5 6854 9165 35 31 0.51 0.34 6-10 6990 9219 37 21 0.53 0.23 11-15 7630 14299 23 21 0.3 0.15 16-20 12930 25865 44 56 0.34 0.22 21-25 15625 29718 82 43 0.52 0.14 26-30 15985 25299 90 58 0.56 0.23 31-35 13015 18045 116 39 0.89 0.22 36-40 12465 15419 107 58 0.86 0.38 41-45 9375 11142 79 54 0.84 0.48 46-50 8623 9309 84 55 0.97 0.59 51-55 6183 7676 57 55 0.92 0.72 56-60 5602 6463 78 56 1.39 0.87 61-65 3334 4827 51 67 1.53 1.39 66-70 2155 3161 38 49 1.76 1.55 71-75 802 1481 21 28 2.62 1.89 76-80 588 878 8 18 1.36 2.05 >80 413 644 10 26 2.42 4.04 males between 21 to 25 years have been seen to be more affected whereas more female patients are affected at age between 26 to 30 years old. in 2023, more than 57% of deaths were adults aged 21-50 years old. people 30 years old accounted for 44% of deaths in 2023. on further analysis, the overall cfr is higher in females than in males (male cfr: 0.4, female cfr: 0.7). females have around three times higher cfr than males among those aged 16-40 years (male cfr: 0.2, female cfr: 0.6). in the majority 46 parh m. y. a., et al. copyright ©2025 assa adv. in systems science and appl. (2025) of age categories, women exhibited a higher number of reported deaths compared to men. this discrepancy could be attributed to gender related differences in health-seeking patterns, suggesting that men may have greater accessibility to healthcare resources, while women of working age might seek healthcare less frequently than their male counterparts. in bangladesh, dengue primarily impacted young adults. more than 80% of infections during the initial outbreak in 2000 were adults (> 18 years old), with the greatest age range for cases being between 18 and 33 [25]. in addition, the majority (62%) of confirmed cases in the 2002 outbreak were in the 16–30 age range, with a mean age of 29 [16]. additionally, the majority of cases in 2016 (21-40; 55%), 2018 (15-29; 65%), and 2019 (21-40; 50%) outbreaks were older adolescents and young adults [16]. during the 2019 outbreak, a hospital-based investigation indicated that the majority (46.1%) of the kids were in the 10 to 14 age range, with a mean age of 8.8 years [19]. fig. 6. gender wise cfr (%) and total cases for different age groups in 2023 according to the data presented in fig. 7, the current dengue surveillance of government and non-government autonomous hospitals reveals that throughout the year 2023, cfr is consistently higher in government autonomous hospitals compared to their non-government counterparts. government hospitals are free of charge and impoverished people tend to visit more. private hospitals typically charge far more for supportive care, which is beyond reach for most bangladeshis. by july 25, cases requiring hospitalization were documented in all districts, and the count of hospitalizations outside dhaka city exceeded that within the capital on august 14. government hospital visit to impoverished households has a disproportionate impact on high cfr. government hospitals have insufficient risk management, lack of necessary care, testing kits, blood bags, platelet kits, staff, and beds, and the surveillance mechanisms are not up to the task of handling the growing dengue hazards. fig. 7. case fatality rate (cfr%) in government and non-government autonomous hospitals in bangladesh, 2023 word template for assa manuscript 47 copyright ©2025 assa. adv. in systems science and appl. (2025) fig. 8 reported the percentage of the top 10 districts across the country. dhaka city corporation stands out as the most severely affected area within dhaka division, contributing to 35.3% of cases and 57.8% of deaths. dhaka city recorded 63% (n = 41,321) of the total cases and was one of the most affected regions in 2022. in 2023, dhaka city experienced its highest temperatures in six decades, characterized by an unusually hot spring and subsequent heavy monsoon rains. even small amounts of stagnant water create an ideal breeding environment for mosquitoes, and the rainy season is still ongoing. other impacted districts include chattogram (4.42% of cases), manikganj (4.03% cases), barishal (4.24% cases), faridpur (2.34% cases), gazipur (2.26% cases), lakshmipur (2.16% cases), khulna (2.13% cases), patuakhali (2.36% cases), and pirojpur (2.29% cases). fig. 8. top 10 affected districts in 2023 the important question is whether bangladesh is ready to combat widespread outbreaks of dengue. bangladesh is currently experiencing its biggest dengue fever outbreak on record, with overflowing patients in hospitals and an increasing number of illnesses and deaths. the recent outbreaks are found to be dominated by serotypes denv-2 and denv-3, which have played a major role in increasing clinical presentation severity. the expansion of dengue outbreaks and denv transmission in many geographic locations of bangladesh is impending due to the fast and unplanned urbanization, environmental suitability of vector adaption, unconsciousness of people, lack of healthcare facilities, and so on. the dengue outbreak in bangladesh in 2023 serves as a reminder to pay attention to controlling the dengue epidemic in bangladesh. in the context of the pandemic our recommendations are as follows, (a) cross-infection with different variants increases the risk of complications and fatality, even though a patient does become immune to a particular variant after contracting it. since the signs of secondary infections change, they suffer shock syndrome or severe dengue, and more deaths are being caused by delayed hospital admissions. older people and people with neutralizing antibodies to 2 denv serotypes have a higher likelihood of secondary or tertiary dengue infection and that contributes disproportionately more to denv transmission [14]. it is necessary to conduct public health campaigns to educate the population about the risks of cross-infection and the importance of seeking medical care promptly. emphasize the changing signs and symptoms of secondary infections to encourage early recognition and intervention. (b)enhancements should be made to the community’s efforts to eradicate mosquito breeding areas, which are spearheaded by local leaders and council members as well as voluntary mosquito control groups. from the pattern of this year’s hit february is the best time to begin mosquito control measures in bangladesh. although dengue vaccinations are finally on the market [10], in order to fully understand their potential to prevent denv transmission and lower dengue-related morbidity and mortality, field research and controlled trials are needed to assess the vaccines’ efficacy and safety. (c) bangladesh’s main city of dhaka serves as the focal hub for the management of specialized health care. according to recent data, almost 44% of patients admitted to dhaka’s hospitals came from outside the city [15]. there is a need to expand specialized 48 parh m. y. a., et al. copyright ©2025 assa adv. in systems science and appl. (2025) healthcare services to other regions of bangladesh. this can help alleviate the burden on dhaka’s healthcare infrastructure and ensure better access to care for people living in rural areas or smaller towns. in dhaka, there was a severe lack of electrolytes in both hospitals and pharmacies. strengthen the supply chain for essential medical supplies, including electrolytes, to ensure consistent availability in hospitals and pharmacies in dhaka. (d)community education, community involvement in risk reduction, and the fortification of preventative measures are all necessary at the community level. people should sleep under insecticide-treated bed nets. the cost of bed nets and insecticides should be reduced. the complete picture of bangladesh’s dengue outbreak is not entirely captured by this constrained surveillance system. it is necessary to conduct proactive nationwide geographic-based surveillance that gathers clinical, demographic, and socioeconomic data to better understand the characteristics of denv transmission hot-spots as well as the disparities in morbidity and mortality related to age and gender. 2.2. gender differences in morbidity and mortality the higher case fatality rate (cfr) among women observed in 2023 may reflect several underlying socio-cultural and economic factors that influence health outcomes in bangladesh. evidence suggests that women often delay seeking medical attention due to household responsibilities, reduced mobility, and cultural expectations that prioritize caregiving roles over personal health needs. financial dependency on male family members may further restrict timely access to healthcare, particularly in rural areas where out-of-pocket expenses represent a significant barrier. in addition, sociocultural norms may prevent women from traveling alone to healthcare facilities, contributing to delayed diagnosis and treatment. these factors collectively may explain the higher cfr observed among women, despite their lower proportion of total reported cases. biological differences in immune response and comorbidities may also play a role, although additional research is needed to confirm this. 2.3. potential drivers behind the 2023 dengue surge the unprecedented increase in dengue incidence in 2023 is likely the result of multiple interacting factors. climate change has altered bangladesh’s temperature and rainfall patterns, creating highly favorable environmental conditions for aedes mosquito breeding. the country experienced unusually high temperatures early in the year, followed by heavy monsoon rainfall, both of which extend the mosquito breeding season and increase vector density. urbanization also played a substantial role: rapid, unplanned expansion of densely populated cities such as dhaka has led to overcrowding, poor waste management, and stagnant water accumulation—ideal conditions for aedes proliferation. another major driver of the outbreak was the circulation of more virulent and highly transmissible denv serotypes, particularly denv-2 and denv-3, which have historically been linked to more severe clinical outcomes. serotype switching and secondary infections may have contributed to the severity of the outbreak, as individuals previously exposed to one serotype are at increased risk of severe disease upon infection with another. additionally, increased human mobility, population density, and inadequate vector control activities during critical pre-monsoon months further amplified the outbreak’s magnitude. 3. conclusions and future cautions in 2023, bangladesh experienced the highest dengue-related mortality ever recorded, marking a critical point in the nation’s public health history. over the past two decades, there has been a cyclical shift in the circulating denv serotypes, with all four serotypes (denv-1 to denv4) detected during recent outbreaks, contributing to increased disease severity. the epidemiological pattern of dengue has shifted toward more frequent and severe outbreaks since the first major event in 2000. several factors contribute to the rising burden of dengue in word template for assa manuscript 49 copyright ©2025 assa. adv. in systems science and appl. (2025) bangladesh, including high population density, rapid and unplanned urbanization, inadequate housing and waste management systems, and favorable climatic conditions for mosquito breeding. these conditions underscore the urgency for both immediate and sustained interventions. to effectively reduce the risk of future outbreaks, year-round efforts to eliminate mosquito breeding sites and destroy aedes habitats must be intensified. strong government policies that support vector control, urban planning, and waste management are essential. furthermore, community engagement plays a vital role in prevention strategies. raising public awareness through community healthcare workers, local campaigns, social media, and television can empower citizens to take proactive measures at the household and neighborhood levels. lastly, continuous surveillance to detect the circulating serotypes, monitor clinical manifestations, and understand underlying risk factors is crucial for informed public health planning. a coordinated effort involving government bodies, healthcare institutions, and the general public is imperative to control dengue and minimize its impact in the coming years. declarations the authors declare that they have no competing interests. author contribution declaration myap and msas contributed to the design of the study, analyzed data, and drafted the manuscript. mmh, nbs, jfe, and hz interpreted results, contributed to the reviewing of the manuscript, and took the responsibility to check the accuracy of the data analysis. myap, msas, and nbs provided a critical review of the manuscript. funding statement the research presented in this article was conducted without the support of any external funding from institutions or organizations. all expenses associated with this study were covered by the authors. acknowledgements i would like to express my sincere gratitude to the bangladesh institute of epidemiology, disease control and research (iedcr) for providing access to the monthly reported data on dengue cases and dengue-related deaths used in this study. i also acknowledge the efforts of local public health authorities and healthcare workers whose contributions in data reporting were essential. although this study was not supported by any funding, the cooperation of these institutions greatly facilitated the completion of my research. references 1. ahmad, f. u., paul, s. k., aung, m. s., et al. 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(2012) cardiac function in vietnamese patients with different dengue severity grades, crit care med, 40(2), 477–483. adv syst sci appl 2022; 03:124–134 published online at https://ijassa.ipu.ru. the model of the production side of the russian economy: the updated version nikolay pilnik, stanislav radionov*, financial research institute of the ministry of finance of the russian federation, moscow, russia abstract: in this paper we propose the new version of the model of the production side of the russian economy presented in [25]. we add several new features such as more general production function, more tractable human capital variable and the method of covid-19 pandemic shock description. the new version of the model is capable of reproducing the wide set of main russian macroeconomic indicators with sufficiently high accuracy even during the covid-19 pandemic. keywords: dynamic optimization, complementary slackness conditions, production of gdp 1. introduction and literature review in [25] we presented the model of production side of the economy, which represents the aggregate description of all economic agents who produce the added value. this aggregated agent, which can be called producer, constitutes the important part of economy and, together with consumer, government etc., forms the basic structure of general equilibrium description of the economy. the model presented in [25] is based on the solution of the optimization problem of the agent who maximizes his discounted profit flow under technologic, demographic and financial constraints in the spirit of the standard microeconomic description of the firm. the model, however, contains a wider set of economic instruments available to producer – it is connected not only to labor and capital markets, but also to the banking system via loans in both national and foreign currencies. several heuristic methods such as relaxation of complementary slackness conditions are used to derive the model which is capable of reproducing the russian macroeconomic variables with sufficiently high precision. in this paper we add several new features to the model. first, we use more general constant elasticity of substitution (ces) production function instead of cobb-douglas production function. ces production function was introduced in [4] and is widely used both in theoretic and applied works dedicated to the description of the production side of economy both on macroand microeconomic levels (see [20], [21], [9], [14], [10], [22], [2], [7], [5] and the special issue of journal of macroeconomics in june 2008 for examples and discussion). the modelling of gross domestic product using the production functions of different kinds (see for example [11], [12], [15], [18]) can be mentioned separately due to the closeness of these works to ours. the cobb-douglas production function, which is a special case of ces, is used more commonly than ces due to its analytic tractability. in particular, cobbdouglas production functions are ubiquitous in dsge models (see for example [28], [29], [8]). however, one of the properties of the cobb-douglas production function is that it implies the constant shares of labor and capital contributions to production, which is not confirmed on the russian macroeconomic data. hence we decided to switch to more sophisticated ces production function. ∗corresponding author: saradionov@edu.hse.ru the model of the production side of the russian economy: the updated version 125 second, in the new version of the model we use the human development index calculated by the united nations (see [27]) as a measure of human capital. this index is calculated based on such indicators as gross national income per capita, literacy rate, education level. note that these indicators are essentially out of control of the production side of economy, so this variable is considered as exogenous in the model. third, we introduce the mechanism which exogenously restricts the ability to hire the labor force. it us used to explain the dynamics of employment during the covid-19 pandemic. it is worth mentioning that the impact of covid-19 pandemic and the related government measures stimulated the great volume of research. notable examples include [13], [3], [19], [1], [6], [16] among many others. in our model we use the covid-19 stringency index introduced in [17], which is a numeric indicator of strictness of covidrelated measures, as a factor which puts a pressure on employment rate. in the rest of the paper we describe the new version of the model and analyze its ability to reproduce the russian macroeconomic variables. 2. the model: statement as in [25], we consider the model of the whole sector of the economy, which generates the added value, as a single agent. we formulate the optimization problem, solve it and then, using several heuristics, we reduce the solution to the dynamic model. as in [25], the investments are divided into two parts: investments in the maintenance of fixed assets ju(t) and investments in building up fixed assets jm(t). the formation of production capital (fixed assets) m(t) is described by the following relations: d dt m (t) = jm (t)− δam (t)m (t) , (2.1) 0 ≤ jm (t) . (2.2) where δam (t) is the coefficient of depreciation rate, which is assumed to be constant and equal to 0.00275. the equation (2.1) allows to calculate the series jm (t) based on statistics on the level of fixed assets and depreciation. the balance of investments j (t) = jm (t) + ju (t) , (2.3) where j (t) is the overall level of investment (gross fixed capital formation), allows to calculate the investments in the maintenance of fixed assets ju (t). the balance can also be written in nominal terms pj (t) = pm (t) jm (t) + pu (t) ju (t) , (2.4) where pj (t) is the overall level of investments (gross fixed capital formation) in the current prices, pm (t) and pu (t) are the deflators of the corresponding elements of investments. in this version of the model, however, we assume them being equal. we introduce the human capital per capita denoted by h(t). as was noted in the introduction, in this version of the model we consider it as an exogenous variable. next, we introduce the total amount of payments made by the producer to its employees, which is called wage and mixed income in the system of national accounts. we denote this variables by w (t). as was noted in the introduction, we assume that the production function of the agent we consider has a constant elasticity of substitution. namely, we assume the following structure of the production function: copyright © 2022 assa. adv syst sci appl (2022) 126 n. pilnik, s. radionov y (t) = a ( α (um (t)m (t))θ + (1− α) (uh (t)h (t)rmax (t)) θ ) 1 θ , (2.5) where um (t) and uh (t) are the utilization rates, respectively, of fixed assets and human capital. we define these indicators in the following non-linear way: um (t) = ( ju (t) m (t) )b , uh (t) = ( r (t) rmax (t) )a . (2.6) we substitute (2.6) into (2.5) and also substitute a with several normalization coefficients in order to avoid the dimensionality problem. as a result, the production function takes the form y (t) y0 = [ α ( ju(t) j0 )bρ( m(t) m0 )(1−b)ρ + (1− α) ( h(t) h0 )ρ( r(t) r0 )bρ( rmax(t) rmax0 )(1−b)ρ ] 1 ρ . (2.7) the volume of production 0 ≤ y (t) and the volume of products sold 0 ≤ yp (t) may vary, which leads to the introduction of stock 0 ≤ z (t) in the model. its change can be calculated as d dt z (t) = y (t)− yp (t) . (2.8) denote the loans attracted by the producer in national currency (rubles) by l (t). the average terms for which loans are attracted will be denoted by (βl (t)) −1. the variable βl (t) is the inverse duration and interpreted as the average frequency of the loans return. then the dynamics of ruble loans is defined by the equation d dt l (t) = k (t)− βl (t)l (t) , (2.9) where 0 ≤ k (t) is the flow of newly attracted loans and rl (t) is the effective interest rate on loans. producer also attracts loans in foreign currency, which are nominated in dollars in the russian statistical data. we denote the corresponding value by vl (t). the dynamics of foreign currency loans is described by the equation similar to (2.9): d dt vl (t) = vk (t)− βvl (t) vl (t) . (2.10) here βvl (t) is the reverse durations of currency loans, newly attracted foreign currency loans are denoted by 0 ≤ vk (t) and the effective interest rate on foreign currency loans is denoted by rvl (t). the taxes producer pays can be aggregared into four groups: value added tax, labor taxes, property taxes and income taxes. the rates of these aggregated taxes are denoted by τy (t), τr (t), τtm (t), τpr (t) correspondingly. for servicing operations related to loans and investments, the producer uses current account n (t). in this version of the model it is assumed that its volumes are proportional to ruble and foreign currency loans and fixed assets. these proportions are defined by the coefficients νl, νvl, νm: n (t) ≥ νll (t) + νvlwvl (t) vl (t) + νmpm (t)m (t) , (2.11) where wvl (t) denotes the exchange rate. copyright © 2022 assa. adv syst sci appl (2022) the model of the production side of the russian economy: the updated version 127 other expenses of the producer are denoted by oc o (t) and are considered as the exogenous variable. we also denote the profit after taxation by pr (t). the financial balance of the producer can be written as d dt n (t) = k (t)− βl (t)l (t)− rl (t)l (t)− (1 + τpr (t)) pr (t)+ wvl (t) (vk (t)− βvl (t) vl (t)− rvl (t) vl (t))−oc o (t) + (1− τy (t)) py (t)yp (t)− pj (t) ju (t)− pm (t) jm (t)− τtm (t) ptm (t)m (t)− (1 + τr (t))wr (t)r (t) . (2.12) the above relationships represent the limitations imposed on the producer’s ability to choose the values of its planned variables (controls): ju (t) , jm (t) , k (t) , l (t) ,m (t) , n (t) ,pr (t) , r (t) ,yp (t) , z (t) , vk (t) , vl (t) , y (t) . (2.13) according to the principle of rational expectations, when planning its control variables, the producer can rely on an accurate forecast of information variables: h (t) ,oc o (t) , rmax (t) , βl (t) , βvl (t) , δah (t) , δam (t) , pj (t) , pm (t) , ptm (t) , py (t) , rl (t) , rvl (t) , τh (t) , τpr (t) , τr (t) , τtm (t) , τy (t) , wh (t) , wr (t) , wvl (t) . as in [25], the goal of the producer in the model is to maximize the total discounted utility from the profit after taxation. we assume that the discount rate in the utility function is equal to the gdp deflator: ∫ t t0 e−∆ t 1− η ( pr (t) py (t) )1−η dt. (2.14) the problem is supplemented with terminal condition, which can be interpreted as a growth condition for some linear combination of the phase variables: ω (t0 ) γ ≤ ω (t ) , (2.15) where ω (t) = al (t)l (t) + am (t)m (t) + an (t)n (t) + az (t)z (t) + avl (t) vl (t). this is the analogue of no ponzi condition written in terms of the producer’s own capital which is a difference between its assets and liabilities. as in [25], we derive the solution of the maximization problem of the function (2.14) under constraints (2.1) – (2.12) and the terminal condition (2.15) with respect to the variables (2.13) using the lagrange method. we obtain the system of sufficient optimality conditions, which can be divided into several groups: equations for primal variables, differential equations for dual variables, complementary slackness conditions and the terminal condition. this system is then tranformed to the more tractable form using the method described in detail in [26]. first, we transform it to the discrete time by replacing derivatives with increments (backward increments are used for direct variables and forward increments are used for dual variables). second, some direct variables are replaced by their values in the previous period. third, differential equations for the dual variables are replaced by expressions of corresponding separatrices. forth, the complementary slackness conditions are replaced by their more regular approximations (the procedure is called the relaxation of complementary slackness conditions). after all these transformations, we proceed to the dynamic system which can also be presented by several groups of expressions. copyright © 2022 assa. adv syst sci appl (2022) 128 n. pilnik, s. radionov the first group consists of several expressions defining growth rates of different exogenous variables (in different forms) and two more auxiliary variables: gpm (t) = pm (t)− pm (t− 1) pm (t− 1) , gwh (t) = wh (t)− wh (t− 1) wh (t− 1) , gpy (t) = py (t)− py (t− 1) py (t− 1) , gtaupr (t) = τpr (t)− τpr (t− 1) τpr (t− 1) + 1 , gtauh (t) = τh (t)− τh (t− 1) 1 + τh (t− 1) , grmax (t) = rmax (t)−rmax (t− 1) rmax (t− 1) , gttauy (t) = −τy (t)− τy (t− 1) −τy (t− 1) + 1 , rr(t) = ( h(t− 1) h0 ) θ −bθ+1 ( y (t) y 0 )− θ −bθ+1 ( rmax(t) rmax0 ) (1−b)θ −bθ+1 × × ( by (t) py(t) (1− α) (1− τy(t)) wr(t)r0 (1 + τr(t)) ) bθ −bθ+1 , rr2(t) = −pj(t) ju(t)− (1 + τr(t))wr(t)r(t)−oco(t)+ + y (t) (1− τy(t)) py(t)− pr(t) (τpr(t) + 1) . the second group consists of expressions defining “real” variables such as gdp and its components included in the model, fixed capital stock and the number of employed. real gdp in the model is defined as ( y (t) y 0 ) (1−b)θ −bθ+1 = α ( m(t− 1) m0 ) (1−b)θ −bθ+1 ( αbpy(t) (1− τy(t)) pj(t) j0 ) bθ −bθ+1 + + (1− α) ( rmax(t) rmax0 ) (1−b)θ −bθ+1 ( h(t− 1) h0 ) θ −bθ+1 ( bpy(t) (1− α) (1− τy(t)) wr(t)r0 (1 + τr(t)) ) bθ −bθ+1 . the number of employed r (t) and the level of investments in the maintenance of fixed assets ju (t) are adjusted at this stage to better take into account the particularities of russian macroeconomic dynamics during the recent years. the expression which defines the number of employed persons derived as first order optimality condition is denoted by r∗(t). it is adjusted to better take into account the situation of partial employment. this leads to the following expressions: r∗(t) = ( h(t− 1) h0 ) θ −bθ+1 ( rmax(t) rmax0 ) (1−b)θ −bθ+1 r0 ( y (t) y 0 )− θ −bθ+1 × × ( by (t) py(t) (1− α) (1− τy(t)) wr(t)r0 (1 + τr(t)) ) 1 −bθ+1 , r (t) = arrmax (t) + brr ∗ (t) , similarly, the expression which defines the investments in the maintenance of fixed assets obtained from first order optimization conditions is denoted by ju∗(t). it is adjusted to take into account the exogenous constraint related to the measures introduced to limit the copyright © 2022 assa. adv syst sci appl (2022) the model of the production side of the russian economy: the updated version 129 covid-19 spread. as was mentioned in the introduction, we use the covid-19 stringency index denoted by covid(t) to measure the strictness of the introduced measures. the resulting expressions are the following: ju∗(t) = ( y (t) y 0 )− θ −bθ+1 ( m(t− 1) m0 ) (1−b)θ −bθ+1 j0 ( αby (t) py(t) (1− τy(t)) pj(t) j0 ) 1 −bθ+1 , ju (t) = min{m(t− 1) (mm (1− covid(t))) 1 b , ju∗(t)}. the profit of producer pr(t), investments in building up fixed assets jm(t) and the change of stocks z(t) are defined by the following relations: pr(t) = ( −∆− gtaupr(t) + ρ(t) η + ( −1 η + 1 ) gpy(t) ) pr(t− 1) + pr(t− 1) , jm(t) = b2 ( δam(t) + −δam(t)− τtm(t) + gpm(t)− rr(t)α(−1+b)(τy(t)−1)y (t)py(t) m(t−1)pm(t) δam(t)num+ 1 ) m(t− 1)− b2gpm(t)m(t− 1)− a2 ((δam(t)num+ 1) ρ(t) + δam(t) + τtm(t)− gpm(t))m(t− 1)+ + rr(t)α (−1 + b) (τy(t)− 1)y (t) py(t) pm(t) + cc2rr2(t) pm(t) , z(t) = z(t− 1) + (b5 − a5 (−gpy(t)− gttauy(t) + ρ(t)))z(t− 1) + cc5rr2(t) py(t) (1− τy(t)) . the expression defining the stock of the fixed capital m(t) is also adjusted – we introduce its smoothed version denoted by msm(t). this smoothed variable is compared with the statistical values of m(t). m (t) = jm (t)− δam (t)m (t− 1) +m (t− 1) , msm(t) = κmsm(t− 1) + (1− κ)m(t). the last group are the equations defining financial variables, both stocks and flows. vk(t) = b3 ( βvl(t)− rvl(t) + gwvl(t) βvl(t)nuvl − 1 − gwvl(t) ) vl(t− 1)− a3 ((βvl(t)nuvl − 1) ρ(t) + rvl(t) + gwvl(t)) vl(t− 1)− cc3rr2(t) wvl(t) , k(t) = ( b4 ( − rl(t) η(t)nul − 1 + η(t) ) − a4 ((η(t)nul − 1) ρ(t) + rl(t)) ) l(t− 1)+ + (−1 + cc1 + cc2 + cc3 + cc5)rr2(t), n (t) = νll (t− 1) + νvlwvl (t) vl (t− 1) + νmpm (t)m (t− 1) , l (t) = k (t)− βl (t)l (t− 1) + l (t− 1) , vl (t) = vk (t)− βvl (t) vl (t− 1) + vl (t− 1) . copyright © 2022 assa. adv syst sci appl (2022) 130 n. pilnik, s. radionov 3. the model: identification and results the presented version of the model is identified on the russian macroeconomic data from 2005q1 to 2021q4. the sources of the data are essentially the same as in [25] – rosstat data is used for gdp and its components, capital stock etc., bank of russia data is used for the variables related to the interactions with the banking sector, and the federal treasury data is used for the taxation data. the seasonality identification procedure described in [23] is applied where necessary. the new data sources include the covid-19 stringency database [17] and the human capital development database [27]. the first dataset is available on a daily basis, and we use the quarterly average value. the second dataset is available on a yearly basis, and we extrapolate the given values for the quarters. we identify the parameters of the model in two ways. the first one is the standard dynamic system approach used in [25]. we calculate the growth rates of eight variables (y , r, m , l, vl, n , pr, w ) to the corresponding quarter of the previous year. the function of errors is the sum of squares of the discrepancies between the statistics and the model data with some weights. we put the weight 25 for the gdp as the most important variable and 1 for all other variables. using the command lsqnonlin of the matlab optimization package, we find the set of model parameters which minimizes this function of errors. the accuracy of data fit is shown on the plots below. as we can see, the model performs rather well even during the covid-19 pandemic. (a) gdp (b) number of employed (c) fixed assets (d) ruble loans copyright © 2022 assa. adv syst sci appl (2022) the model of the production side of the russian economy: the updated version 131 (e) foreign currency loans (f) current accounts (g) profit (h) wages mean average percentage errors of forecasts are shown in the table 3.1. table 3.1. mean absolute percent errors of the model variables, dynamic system y r m l vl n pr w 1.77 1.01 8.38 8.91 8.82 12.45 5.13 1.13 the second method of the parameters identification we apply is based on the idea of multistep forecasting proposed in [24]. in this framework the parameters of the model are identified in such a way that the error of forecasts for a given length of forecasting is minimized. we consider the forecasts up to 6 quarters ahead and again using the command lsqnonlin of the matlab optimization package find a set of parameters which minimizes the sum of squares of forecast errors. as for the previous case, we minimize the total error of growth rates of the modelled variables compared to their values 4 quarters ago. mean absolute percentage errors of the forecasts for a given variable and a given number of quarters ahead is presented in the table 3.2. for the comparison, we also present mean absolute percentage errors of the simple autoregressive ar(1) models. as we can see, for the variables vl, pr, w the model performs better than ar(1) for the whole period, for r, l, n the model performs better or about the same for long-term forecasts, for m the model and ar(1) perform about the same, and for y the model performs worse than ar(1). copyright © 2022 assa. adv syst sci appl (2022) 132 n. pilnik, s. radionov table 3.2. mean absolute percent errors of the model variables, multistep forecasting 1 2 3 4 5 6 y model 3.58 3.41 3.30 3.31 3.21 3.09 y ar(1) 1.12 1.58 1.80 2.17 2.29 2.35 r model 0.89 0.87 0.89 0.87 0.83 0.78 r ar(1) 0.48 0.65 0.72 0.85 0.83 0.91 m model 0.08 0.16 0.23 0.31 0.37 0.42 m ar(1) 0.08 0.16 0.23 0.30 0.35 0.40 l model 2.66 3.57 4.49 5.21 5.50 5.63 l ar(1) 2.12 3.35 4.25 5.11 5.71 6.29 vl model 3.06 3.72 3.90 4.05 3.81 4.72 vl ar(1) 3.43 4.74 6.21 7.27 8.20 8.78 n model 10.48 10.18 10.06 10.10 9.80 9.97 n ar(1) 4.30 6.57 8.48 10.24 10.17 9.82 pr model 3.21 3.99 4.71 5.86 5.63 5.45 pr ar(1) 4.43 5.81 6.42 7.88 8.39 9.08 w model 1.03 0.99 1.00 0.98 0.93 0.87 w ar(1) 1.36 1.77 2.32 3.00 3.25 3.74 4. conclusion we propose the new version of the model of the production side of the russian economy. introduction of several new features such as more complex production function and the exogenous covid-19-related restrictions allowed to reproduce the wide set of macroeconomic variables with rather high accuracy even during the covid-19 pandemic. one of the possible future directions of research is the introduction of sanctions-related restrictions the russian economy currently faces into the model. acknowledgements this work is supported by the russian science foundation under grant 21-18-00482. references [1] d. acemoglu, v. chernozhukov, i. werning, and m. d. whinston. optimal targeted lockdowns in a multigroup sir model. american economic review: insights, 3(4):487– 502, 2021. 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[29] f. smets and r. wouters. shocks and frictions in us business cycles: a bayesian dsge approach. american economic review, 97(3):586–606, 2007. copyright © 2022 assa. adv syst sci appl (2022) introduction and literature review the model: statement the model: identification and results conclusion microsoft word 1420-article text-6914-1-18-20240322 adv syst sci appl 2024; 01; 114-128 published online at https://ijassa.ipu.ru. multistate system performance analysis incorporating human error, mathematical modeling and reliability approach amit kumar1*, pardeep kumar2, majid forghani-elahabad3 1symbiosis institute of technology, symbiosis international (deemed university) (siu), lavale, pune, maharashtra, india 2lovely professional university, chaheru, phagwara, punjab, india 3federal university of abc, santo andré, são paulo, brazil abstract: this paper proposes a novel framework for analyzing the performance of a sugar mill (wahid sugar mill, situated in punjab, india) with human errors, mathematical modeling, and reliability approach. the proposed framework considers the complex interactions between the different components of the sugar mill and the potential impact of human error on the same. the performance of the considered system is affected by the different components failures and unplanned outages. considering these facts, a mathematical model is developed for the sugar mill to calculate different reliability measures such as availability, reliability, and mttf. the mathematical modeling aspect of the framework utilizes markov chains to model the stochastic behavior of the sugar mill. the authors also perform sensitivity analysis to identify the impact of different components' failure on the performance of the same. the results of the paper demonstrate the potential of the proposed framework in providing valuable insights into the performance of the sugar mill under different scenarios, including the impact of human error. keywords: reliability measures; unplanned outages; sensitivity analysis; sugar mill. 1. introduction in this age of science and technology, heavy and automated machines are being used to enhance production in industries, but at the same pace, it increases the complexity of the associated industry. so, to maintain the desired production of a system, the maintenance team takes good care of different components of the same. it can be done by obtaining the various performance measures of the system as well as by keeping the proper track of the maintenance of the components. hence, keeping all these things into consideration, in the present study, we consider a sugar mill in punjab, india, a complex system comprising many components like an unloader, conveyor, cutter, crusher, bagasse carrying machine, and boiler. failure of any of these components may lead the whole system to a degraded or failed state. these failures may be mechanical, electrical or due to human operators. human operators sometimes give wrong commands to the machines, which may cause the failure of costly components or machines. sometimes, humans do not pay proper attention to their work because they feel tired or bored inside the mill. sometimes failures also occur due to power outages, corrosion, manufacturing defects and wear out, natural calamities like earthquakes or tornadoes, etc. these system failures can’t be avoided (expect natural calamities) but can be mitigated with proper repair and maintenance or using the redundancy in the system. nandhini and padmavathy [10] analyzed the year-wise production of sugarcane and described the reason for the production change in the sugar cane industry from 2000-2010. zhao and li [20] presented the effects of climate change on sugar cane production and described the strategies to mitigate the effects of climate change. performance analysis of the sugar mill taking various parameters into consideration was presented by [12]. sharma and vishwakarma [16] utilized the genetic multistate system performance analysis… 115 copyright ©2024 assa. adv. in systems science and appl. (2024) algorithm technique to analyze the performance of the feeding system in the sugar industry. zaidi [19] examined the transient and steady-state behavior of the feeding system within the sugar industry. dahiya et al. [4] employed a fuzzy reliability approach to develop a mathematical model for the a-pan crystallization system in the sugar industry. navyata et al. [11] evaluated the reliability, availability, and mttf of a dual-channel logic communication system using the boolean function technique. tewari and kumar [17] presented the availability analysis of the milling system in the rice milling plant. bansal et al. [2] applied the boolean function technique for the evaluation of reliability parameters for a milk powder plant manufacturing plant. li [9] discussed the system's comparison between active and standby redundancy. some industrial systems take rest after working for some specific amount of time, such as an industrial system that works under a cost-free warranty and rest policy were presented by kumar and kumar [8]. reliability and sensitivity analysis of a thermal power plant were presented [13]. k-out-of-n: the f/g system has gained popularity in the industrial system and is used for improving the reliability of the system. ram and kumar [14] presented a study on the performance analysis of an industrial system using a 2-out-of-3: f configuration considering human error. in a separate work, ram and manglik [15] investigated a system with parallel redundancy with human error, partial failure, and catastrophic failure and evaluated the various reliability parameters. in the literature, it has been observed that repair facilities may not always be available with the system. a new approach that repairmen can take multiple vacations when there is no product for repair was presented by [18]. kalaiarasi et al. [6] analyzed a system consisting of four components with a human error rate with the help of markov modeling. haggag [5] presented the profit analysis and availability of 3-out-of-4 systems under preventive maintenance. chatterjee and nath [3] presented a case study on an indian railway passenger reservation system using a smart computing application. aly et al. [1] presented the ram analysis of a 3-out-of-4 system and identified the most critical components that affect the system's performance. markov modeling is a highly effective tool for analyzing different ofsystems' performance, as it focuses on the potential states a system can assume throughout its functioning. in the initial state, referred to as the perfect state, all system components are in optimal working condition. as time passes, components of the system start to degrade, and the performance of the system reduces significantly. if proper maintenance is done at the right time it saves the system from major failures. if this component maintenance is not done at the right time, then the system is bound to fail, which may cost a lot to the organization. there can be many possible failure states in the markov model. but after the failure of the component system is repaired and failed components are either repaired or replaced to bring the system back to the good working state. kumar and kumar [7] used markov modeling to analyze the performance of automatic ticket vending machines for the same performance analysis. in the above studies, it has been observed that elite researchers investigated many industrial systems through different techniques for finding their various system measures. also, specific authors were investigating some of the components of sugar mill. but no one has ever tried to investigate a sugar mill as a whole for performance analysis by taking human error into consideration, and also sensitivity analysis of the sugar mill plant regarding its components failure/repair has never been performed. hence, in the present study, authors have investigated a sugar mill by taking its various important components along with a human operator. the next section briefly describes the sugar mill's components, which have been taken into consideration for the reliability analysis of the system. 1. system description the description of the components of the sugar mill is as follows 116 a. kumar, p. kumar, m. forghani-elahabad copyright ©2024 assa adv. in systems science and appl. (2024) component-a: unloader is represented by component a. basically; it is used to unload the cane from the means of transport. in the present study, two unloaders in parallel configuration have been taken into consideration. if one of them fails, then the sugar mill goes into the degraded state. component-b: the conveyor is represented by component b. once the cane is unloaded, then it is kept on the conveyor for further process. failure of the conveyer results in the failure of the whole system. component-c: cutter is represented as component c. basically, it used to cut the cane into specific size of pieces. component-d: component d represents the crushing system, which serves the purpose of cane crushing and juice extraction. component-e: bagasse carrying system is represented by component e. after the juice extraction from canes, bagasse is used as a fuel in the sugar mill. it is used in the heatgenerating system of the mill. the bagasse carrying machine is used to carry the bagasse to the heat-generating system. component-f: the boiler is represented by component f. it is used to generate heat in the various stages of production in the sugar mill. optimizing the boiler's performance can significantly improve the reliability of the sugar mill. in this study, two boilers are taken to enhance the overall production of the mill. in the event of a single boiler failure, the sugar mill operates in a degraded state, leading to reduced production. however, complete failure occurs only when both boilers cease to function. the interconnection of these components (flow diagram) in the sugar mill is represented in the following fig. 2.1. fig. 2.1: configuration of the system 2. assumptions the following assumptions are associated with this model.  initially, the system is in good condition and all the components are working with full efficiency.  system components can be in working, partially failed, or in a failed state.  a repair facility is always available.  failure and repair rates have been taken as constant and follow exponential distribution.  human operators always available to operate the system. bagasse carrying machine boiler boiler a unloader unloader conveyor cutter crusher b c d e f multistate system performance analysis… 117 copyright ©2024 assa. adv. in systems science and appl. (2024)  the raw material is always available for production. 3. nomenclature the description of the various nomenclature and states, which followed throughout the manuscript, is given in the following table 4.1. table 4.1. nomenclature t signifies that the system is in a satisfactory condition. indicates that the system is in a compromised state. shows that the system has experienced a failure. time frame. s laplace transformation variable. 𝑃 (𝑡) likelihood of the system being in a state 𝑆 at instant t (𝑖 = 0,1,2,3, . . . ,27). 𝑃 (𝑡) laplace transform of 𝑃 (𝑡). 𝛼 failure rate of 𝑖 ℎ component of the system. 𝛼 human error failure rate. 𝛽 repair rate of 𝑖 ℎ component of the system. 𝛽 human error repair rate. 𝑆 good state: all components of system are operating properly and in optimal working condition. 𝑆 degraded state: the state in which the first unloader experiences a failure. 𝑆 failed state: state in which the second unloader fails after the failure of the first unloader. 𝑆 failed state: state in which the conveyer of the system fails. 𝑆 failed state: state in which the cutter of the system fails. 𝑆 failed state: state in which the crusher of the system fails. 𝑆 failed state: state in which the bagasse carrying machine of the system fails. 𝑆 failed state: a state in which the system fails due to human error. 𝑆 degraded state: state in which the first boiler of the system fails. 𝑆 failed state: state in which the conveyer of the system fails after the failure of the first boiler. 𝑆 failed state: state in which the cutter of the system fails after the failure of the first boiler. 𝑆 failed state: state in which the crusher of the system fails after the failure of the first boiler. 𝑆 failed state: state in which the bagasse carrying machine of the system fails after the failure of the first boiler. 𝑆 failed state: state in which the second boiler fails after the failure of the first boiler. 𝑆 failed state: a state in which the system fails due to human error after the failure of the first boiler. 𝑆 degraded state: state in which the first unloader and first boiler fail. 𝑆 failed state: state in which the second unloader fails after the failure of the first unloader and first boiler. 𝑆 failed state: the state in which the conveyer fails after the failure of the first unloader and first boiler. 𝑆 failed state: state in which the cutter fails after the failure of the first unloader and first boiler. 𝑆 failed state: state in which the crusher fails after the failure of the first unloader and first boiler. 𝑆 failed state: the state in which the bagasse carrying machine fails after the failure of the first unloader and first boiler. 𝑆 failed state: state in which second boiler fails after the the first unloader and first boiler fails. 𝑆 failed state: state in which the system fails due to human error after the failure of the first unloader and first boiler. 𝑆 failed state: state in which the conveyer fails after the failure of the first unloader. 𝑆 failed state: state in which the cutter fails after the failure of the first unloader. 𝑆 failed state: state in which the crusher fails after the failure of the first unloader. 𝑆 failed state: the state in which the bagasse carrying machine fails after the failure of the first unloader. 𝑆 failed state: state in which the system fails due to human error after the failure of the first unloader. 118 a. kumar, p. kumar, m. forghani-elahabad copyright ©2024 assa adv. in systems science and appl. (2024) 4. mathematical modeling of the sugar mill plant based on the system analysis, the authors would have developed a mathematical model to represent the behavior of the sugar mill system. this model would take into account the various parameters and variables that influence the reliability of the system, such as failure rates, repair times, maintenance policies, and external factors like human error. critically analyzing the probability of various failure/repair of the components of the sugar mill during its production, different possible states and their interconnection are identified and represented in the following state transition diagram (fig. 4.2). all the possible states𝑆 : 𝑖 = 0,1,2, . . . ,27are shown in the diagram below. for a better understanding of these states table 4.1 is given previously. the chapman-kolmogorov differential equations are developed from the state transition diagram in the interval (𝑡, 𝑡 + 𝛥𝑡) as follows. fig. 4.2: state transition diagram 5. formulation and solution of the model the state transition diagram for the considered process (fig. 4.2) shows the transition between the ith state to the jth state in a small time interval 𝛥𝑡, which can be represented by the following set of chapmankolmogorov differential equations, with the aid of markov birth-death process, as follows. multistate system performance analysis… 119 copyright ©2024 assa. adv. in systems science and appl. (2024) 𝑑 𝑑𝑡 + 𝛼 + 𝛼 𝑃 (𝑡) = 𝛽 𝑃 (𝑡) + 𝛽 𝑃 (𝑡) + 𝛽 𝑃 (𝑡) + 𝛽 𝑃 (𝑡 ) (5.1) 𝑑 𝑑𝑡 + 𝛽 + 𝛼 + 𝛼 𝑃 (𝑡) = 𝛼 𝑃 (𝑡) + 𝛽 𝑃 (𝑡) + 𝛽 𝑃 (𝑡) + 𝛽 𝑃 (𝑡) + 𝛽 𝑃 ( )(𝑡) (5.2) 𝑑 𝑑𝑡 + 𝛽 𝑃 (𝑡) = 𝛼 𝑃 (𝑡); (5.3) 𝑑 𝑑𝑡 + 𝛽 𝑃 (𝑡) = 𝛼 𝑃 (𝑡), 𝑖 = 2,3,4,5,6,7; (5.4) 𝑑 𝑑𝑡 + 𝛽 𝑃 (𝑡) = 𝛼 𝑃 (𝑡); (5.5) 𝑑 𝑑𝑡 + 𝛼 + 𝛽 + 𝛼 𝑃 (𝑡) = 𝛽 𝑃 (𝑡) + 𝛽 𝑃 (𝑡) + 𝛽 𝑃 (𝑡) + 𝛽 𝑃 (𝑡) + 𝛽 𝑃 (𝑡) + 𝛽 𝑃 (𝑡) + 𝛽 𝑃 (𝑡 ) + 𝛼 𝑃 (𝑡 ); (5.6) 𝑑 𝑑𝑡 + 𝛽 𝑃 (𝑡) = 𝛼 𝑃 (𝑡), 𝑖 = 2,3,4,5,6; (5.7) 𝑑 𝑑𝑡 + 𝛽 𝑃 (𝑡) = 𝛼 𝑃 (𝑡); (5.8) 𝑑 𝑑𝑡 + 𝛼 + 𝛽 + 𝛽 + 𝛼 𝑃 (𝑡) = 𝛽 𝑃 (𝑡 ) + 𝛼 𝑃 (𝑡 ) + 𝛼 𝑃 (𝑡 ) + 𝛽 𝑃 (𝑡) ; (5.9) 𝑑 𝑑𝑡 + 𝛽 𝑃 (𝑡) = 𝛼 𝑃 (𝑡), 𝑖 = 1,2,3,4,5,6; (5.10) 𝑑 𝑑𝑡 + 𝛽 𝑃 (𝑡) = 𝛼 𝑃 (𝑡); (5.11) 𝑑 𝑑𝑡 + 𝛽 𝑃 (𝑡) = 𝛼 𝑃 (𝑡), 𝑖 = 2,3,4,5; (5.12) 𝑑 𝑑𝑡 + 𝛽 𝑃 (𝑡) = 𝛼 𝑃 (𝑡). (5.13) initial condition 𝑃 (𝑡) = 1, 𝑡 = 0 and 𝑖 = 0; 0, otherwise. (5.14) on taking the laplace transformation in equation (5.1)–(5.14), we get 𝑠 + 𝛼 + 𝛼 𝑃 (𝑠) = 1 + 𝛽 𝑃 (𝑠) + 𝛽 𝑃 (𝑠) + 𝛽 𝑃 (𝑠) + 𝛽 𝑃 (𝑠); (5.15) 120 a. kumar, p. kumar, m. forghani-elahabad copyright ©2024 assa adv. in systems science and appl. (2024) 𝑠 + 𝛽 + 𝛼 + 𝛼 𝑃 (𝑠) = 𝛼 𝑃 (𝑠) + 𝛽 𝑃 (𝑠) + 𝛽 𝑃 (𝑠) + 𝛽 𝑃 (𝑠) + 𝛽 𝑃 ( )(𝑠) ; (5.16) [𝑠 + 𝛽 ]𝑃 (𝑠) = 𝛼 𝑃 (𝑠); (5.17) [𝑠 + 𝛽 ]𝑃 (𝑠) = 𝛼 𝑃 (𝑠), 𝑖 = 2,3,4,5,6,7; (5.18) [𝑠 + 𝛽 ]𝑃 (𝑠) = 𝛼 𝑃 (𝑠); (5.19) 𝑠 + 𝛼 + 𝛽 + 𝛼 𝑃 (𝑠) = 𝛽 𝑃 (𝑠) + 𝛽 𝑃 (𝑠) + 𝛽 𝑃 (𝑠) + 𝛽 𝑃 (𝑠) + 𝛽 𝑃 (𝑠) + 𝛽 𝑃 (𝑠) + 𝛽 𝑃 (𝑠 ) + 𝛼 𝑃 (𝑠); (5.20) [𝑠 + 𝛽 ]𝑃 (𝑠) = 𝛼 𝑃 (𝑠), 𝑖 = 2,3,4,5,6; (5.21) [𝑠 + 𝛽 ]𝑃 (𝑠) = 𝛼 𝑃 (𝑠); (5.22) 𝑠 + 𝛼 + 𝛽 + 𝛽 + 𝛼 𝑃 (𝑠) = 𝛽 𝑃 (𝑠) + 𝛼 𝑃 (𝑠) + 𝛼 𝑃 (𝑠) + 𝛽 𝑃 (𝑠) ; (5.23) [𝑠 + 𝛽 ]𝑃 (𝑠) = 𝛼 𝑃 (𝑠), 𝑖 = 1,2,3,4,5,6; (5.24) [𝑠 + 𝛽 ]𝑃 (𝑠) = 𝛼 𝑃 (𝑠); (5.25) [𝑠 + 𝛽 ]𝑃 (𝑠) = 𝛼 𝑃 (𝑠), 𝑖 = 2,3,4,5; (5.26) [𝑠 + 𝛽 ]𝑃 (𝑠) = 𝛼 𝑃 (𝑠). (5.27) initial condition 𝑃 (𝑠) = 1, 𝑠 = 0 and 𝑖 = 0; 0, otherwise. (5.28) in order to find the various performance indicators of the considered system, the authors solve the above set of equations and find the various state probabilities 𝑃 (𝑠); 𝑖 = 0,1, . . . ,27 for the sugar mill. equation (5.29 and equation (5.30) give the up state (working) and down states (failed state) probability of the sugar mill. 𝑃 (𝑠) = 𝑃 (𝑠) + 𝑃 (𝑠) + 𝑃 (𝑠) + 𝑃 (𝑠), (5.29) 𝑃 (𝑠) = 𝑃 (𝑠) − 𝑃 (𝑠). (5.30) the following state probabilities are obtained when solving the above set of equations using initial and boundary conditions. 𝑃 (𝑠) = 1 𝐻 , (5.31) 𝑃 (𝑠) = 𝛼 𝐻 + 𝛽 𝛼 𝛼 𝐻 𝐻 𝐻 𝑃 (𝑠), (5.32) 𝑃 (𝑠) = 𝛽 𝛽 𝛼 𝛼 𝐻 𝐻 𝐻 + 𝛼 𝐻 + 𝛽 𝛼 𝛼 𝐻 𝐻 𝐻 𝑃 (𝑠), (5.33) 𝑃 (𝑠) = 𝛼 𝛽 𝛼 𝐻 𝐻 𝐻 + 𝛽 𝛽 𝛼 𝛼 𝐻 𝐻 𝐻 + 𝛼 𝛼 𝐻 𝐻 + 𝛼 𝛼 𝐻 𝐻 + 𝛼 𝛽 𝛼 𝐻 𝐻 𝐻 𝑃 (𝑠), (5.34) multistate system performance analysis… 121 copyright ©2024 assa. adv. in systems science and appl. (2024) where 𝐻 = 𝑠 + 𝛼 + 𝛼 − 𝛽 𝛼 𝐻 − 2𝛽 𝛽 𝛼 𝛼 𝐻 𝐻 𝐻 − 𝛽 𝛼 𝑠 + 𝛽 − 𝛽 𝛼 𝑠 + 𝛽 − 𝛽 𝛼 𝐻 − 𝛽 𝛽 𝛼 𝛼 𝐻 𝐻 𝐻 ; (5.35) 𝐻 = 𝑠 + 𝛽 + 𝛼 + 𝛼 − 𝛽 𝛼 𝑠 + 𝛽 − 𝛽 𝛼 𝑠 + 𝛽 − 𝛽 𝛼 𝐻 − 𝛽 𝛽 𝛼 𝛼 𝐻 𝐻 , (5.36) 𝐻 = 𝑠 + 𝛽 + 𝛼 + 𝛼 − 𝛽 𝛼 𝐻 − 𝛽 𝛼 𝑠 + 𝛽 − 𝛽 𝛼 𝑠 + 𝛽 , (5.37) 𝐻 = 𝑠 + 𝛽 + 𝛽 + 𝛼 + 𝛼 − 𝛽 𝛼 𝑠 + 𝛽 − 𝛽 𝛼 𝑠 + 𝛽 . (5.38) 6. numerical calculation and evaluation of different reliability measures 6.1. availability system availability is a performance metric used to assess the operational effectiveness of a system while accounting for appropriate maintenance practices. to calculate the timedependent availability of the system, substitute the numerical values of different failure and repair parameters as 𝛼 = 0.01 , 𝛼 = 0.02 , 𝛼 = 0.03 , 𝛼 = 0.015 , 𝛼 = 0.028 , 𝛼 = 0.04, 𝛽 = 1, 𝛽 = 1, 𝛽 = 1, 𝛽 = 1, 𝛽 = 1, 𝛽 = 1, 𝛽 = 1 in equation (5.29) and apply the inverse laplace transform. the resulting equation (6.1) provides the time-dependent availability of the sugar mill. 𝐴(𝑡) = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ −0.00096837955𝑒 . − 0.0001949522158𝑒 . + 0.1073957516𝑒 . + 0.0001833977962𝑒 . − 0.09037726155𝑒 . − 0.002608553632𝑒 . − 0.1570307560𝑒 . − 0.01092951713𝑒 . + 0.002032653966𝑒 . + 0.002724385526𝑒 . + 0.0004439964442𝑒 . − 0.0009123318364𝑒 . − 0.0009921283752𝑒 . + 0.8938297613𝑒 . . ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ . (6.1) by fitting the time unit t in equation (6.1), the time-dependent availability of the sugar mill can be observed. table 6.1 and the corresponding fig. 6.1 display the availability values for the sugar mill. table 6.1. behaviors of availability of the sugar mill with time time unit (t) availability 𝐴(𝑡) 0 1.00000 1 0.90691 2 0.90425 3 0.89853 4 0.89626 122 a. kumar, p. kumar, m. forghani-elahabad copyright ©2024 assa adv. in systems science and appl. (2024) 5 0.89560 6 0.89553 7 0.89567 8 0.89588 9 0.89611 10 0.89636 fig. 6.1. behaviour of availability of the sugar mill with time 6.2. reliability reliability refers to the probability of a system successfully carrying out its intended task within specified operating conditions for a given duration. to determine the reliability of the sugar mill, the various failure rates as 𝛼 = 0.05, 𝛼 = 0.01, 𝛼 = 0.02, 𝛼 = 0.03, 𝛼 = 0.015, 𝛼 = 0.028, 𝛼 = 0.04 while the repair rate is set to zero in equation (5.29). the resulting reliability of the sugar mill is obtained as given in equation (6.2). 𝑅(𝑡) = [0.00020000(5000 + 390𝑡 + 7𝑡 )𝑒 . ] (6.2) the behavior of time-dependent reliability of sugar mill can be obtained by varying time unit t in (6.2). table 6.2 and corresponding fig. 6.2 represent the reliability of the sugar mill. table 6.2. behavior of reliability of the system with time unit time unit (t) reliability r(t) 0 1.00000 1 0.88994 2 0.78962 3 0..69866 4 0.61661 5 0.54290 6 0.47695 7 0.41815 8 0.36589 9 0.31959 10 0.27868 0 2 4 6 8 10 0.88 0.90 0.92 0.94 0.96 0.98 1.00 a va lia bi lit y time(t) multistate system performance analysis… 123 copyright ©2024 assa. adv. in systems science and appl. (2024) fig. 6.2. behavior of reliability of the system with time unit 6.3. mean time to failure (mttf) mathematically, the mttf of a system is calculated as given in equation (6.3). 𝑀𝑇𝑇𝐹 = 𝑅(𝑡)𝑑𝑡 = lim → 𝑅(𝑠) (6.3) now, using equation (6.2) in (6.3), authors obtained the mttf of the considered system as: 𝑀𝑇𝑇𝐹 = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 1 𝛼 + 𝛼 + 𝛼 + 𝛼 + 𝛼 + 𝛼 + 𝛼 + 𝛼 + 𝛼 (𝛼 + 𝛼 + 𝛼 + 𝛼 + 𝛼 + 𝛼 + 𝛼 ) + 2𝛼 𝛼 (𝛼 + 𝛼 + 𝛼 + 𝛼 + 𝛼 + 𝛼 + 𝛼 ) ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ . (6.4) varying failure rates one by one from 0.01 to 0.09 with an interval of 0.01 and fixing other failure rates, table 6.3 and fig. 6.3 are obtained for mttf of the considered system. table 6.3. mttf of the sugar mill with various failure rates variations in failure rates mttf with respect to failure rates 𝛼 𝛼 𝛼 𝛼 𝛼 𝛼 𝛼 0.01 8.31561 7.66484 8.25048 8.92729 7.94742 7.86005 9.71726 0.02 8.20020 7.15360 7.66484 8.25048 7.40081 7.76656 8.92729 0.03 8.04273 6.70381 7.15360 7.66484 6.92172 7.63667 8.25048 0.04 7.86050 6.30529 6.70381 7.15360 6.49869 7.48393 7.66484 0.05 7.66484 5.94995 6.30529 6.70381 6.12266 7.31754 7.15360 0.06 7.46321 5.63130 5.94995 6.30529 5.78639 7.14403 6.70381 0.07 7.26054 5.34404 5.63130 5.94995 5.48403 6.96762 6.30529 0.08 7.06005 5.08387 5.34404 5.63130 5.21081 6.79138 5.94995 0.09 6.86383 4.84719 5.08387 5.34404 4.96279 6.61736 5.63130 0 2 4 6 8 10 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 r el ia bi lit y r (t ) time(t) 124 a. kumar, p. kumar, m. forghani-elahabad copyright ©2024 assa adv. in systems science and appl. (2024) fig. 6.3. mttf w.r.t. failure rates 6.4. sensitivity analysis of mttf the objective of the sensitivity analysis is to determine the input variables which affect the system performance most. here authors perform the sensitivity analysis on the mttf of the sugar mill. table 6.4 shows the change in the meantime to failure mttf of the system resulting from changes in parameters𝛼 ,𝛼 ,𝛼 ,𝛼 ,𝛼 ,𝛼 ,𝛼 . the same is depicted in fig. 6.4. table 6.4: sensitivity of the mttf variation in failure rates sensitivity with respect to mttf 𝜕(𝑀𝑇𝑇𝐹) 𝜕𝛼 𝜕(𝑀𝑇𝑇𝐹) 𝜕𝛼 𝜕(𝑀𝑇𝑇𝐹) 𝜕𝛼 𝜕(𝑀𝑇𝑇𝐹) 𝜕𝛼 𝜕(𝑀𝑇𝑇𝐹) 𝜕𝛼 𝜕(𝑀𝑇𝑇𝐹) 𝜕𝛼 𝜕(𝑀𝑇𝑇𝐹) 𝜕𝛼 0.01 -8.64993 -54.60011 -62.80532 -72.91918 -58.49511 -6.93044 -85.55885 0.02 -13.99608 -47.86119 -54.60011 -62.80532 -51.06985 -11.43978 -72.91918 0.03 -17.21471 -42.26548 -47.86119 -54.60011 -44.93705 -14.31633 -62.80532 0.04 -19.04569 -37.57298 -42.26548 -47.86119 -39.81881 -16.08062 -54.60011 0.05 -19.96413 -33.60267 -37.57298 -42.26548 -35.50705 -17.08201 -47.86119 0.06 -20.28051 -30.21609 -33.60267 -37.57298 -31.84373 -17.55829 -42.26548 0.07 -20.20068 -27.30604 -30.21609 -33.60267 -28.70721 -17.67328 -37.57298 0.08 -19.86272 -24.78852 -27.30604 -30.21609 -26.00272 -17.54093 -33.60267 0.09 -19.36012 -22.59706 -24.78852 -27.30604 -23.65563 -17.24117 -30.21609 0.02 0.04 0.06 0.08 0.10 5 6 7 8 9 10 he 6 5 4 3 2 1 m t t f variation in failure rates multistate system performance analysis… 125 copyright ©2024 assa. adv. in systems science and appl. (2024) fig. 6.4. sensitivity of the mttf 6.5. estimated profit from the sugar mill the estimated profit of the sugar mill in the interval [0, 𝑡) is calculated by using 𝐸 (𝑡) = 𝐾 𝑃 (𝑡)𝑑𝑡 − 𝐾 𝑡 (6.5) equation (6.5) will provide the expected profit from the sugar mill for the various revenue and service cost. using equation (5.29) in equation (6.5), authors obtain the profit function for the considered sugar mill as follows. 𝐸 (𝑡) = 𝐾 ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ 0.0006559165477𝑒 . + 0.0001335207330𝑒 . − 0.09612524081𝑒 . − 0.0002118925909𝑒 . + 0.03883492317𝑒 . + 0.001710050295𝑒 . + 0.06576538233𝑒 . + 0.004545896979𝑒 . − 0.002916565831𝑒 . − 0.003943782037𝑒 . − 0.005142893452𝑒 . + 0.001068197741𝑒 . + 0.001454181879𝑒 . + 3161.198061𝑒 . − 3161.208517 ⎭ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎫ − 𝐾 𝑡. (6.6) now vary the service cost 𝐾 = 0.1,0.2,0.3,0.4,0.5 , fix revenue 𝐾 as one, and vary time unit t in (6.6). table 6.5 and corresponding fig. 6.5 are obtained as follows. 0.02 0.04 0.06 0.08 0.10 -90 -80 -70 -60 -50 -40 -30 -20 -10 5 3 he 6 4 3 1 se ns iti vi ty o f m t t f variation of failure rates 126 a. kumar, p. kumar, m. forghani-elahabad copyright ©2024 assa adv. in systems science and appl. (2024) table 6.5. expected profit of the system time unit (t) expected profit from the system 𝐾 = 0.1 𝐾 = 0.2 𝐾 = 0.3 𝐾 = 0.4 𝐾 = 0.5 0 0 0 0 0 0 1 0.76029 0.66029 0.56029 0.46029 0.36029 2 1.56722 1.36722 1.16722 0.96722 0.76722 3 2.36828 2.06828 1.76828 1.46828 1.16828 4 3.16547 2.76547 2.36547 1.96547 1.56547 5 3.96133 3.46133 2.96133 2.46133 1.96133 6 4.75687 4.15687 3.55687 2.95687 2.35687 7 5.55246 4.85246 4.15246 3.45246 2.75246 8 6.34824 5.54824 4.74824 3.94824 3.14824 9 7.14424 6.24424 5.34424 4.44424 3.54424 10 7.94048 6.94048 5.94048 4.94048 3.94048 fig. 6.5. expected profit of the sugar mill vs. time unit t 4. result discussion in this paper, the authors have developed a mathematical model based on the working of a sugar mill to determine the various reliability measures of the sugar mill. for this, the six components of the plant have been taken into consideration. after analyzing the system mathematically, the following results are obtained. the behavior of sugar mill availability is shown in fig. 6.1. it is observed that system availability decreases very slowly as time passes. also, the availability of sugar mill at ten units of time is 0.89636. fig. 6.2 reflects the reliability of the sugar mill concerning time unit. it is found that the reliability of the sugar mill at ten units of time is 0.27868. this means that with the passage of time unit system’s unit reliability is also decreasing. this may be due to the system component's aging, corrosion, stress, etc. fig. 6.3 shows the nature of mttf of the sugar mill concerning variation in failure rates. it reflects that the mttf of the sugar mill with respect to the failure rate of the unloader is the highest. so, the performance of the sugar mill is less effected by the failure of the unloader. despite increasing the failure rate of the unloader mttf is higher as compared to other components of the system. the graph of sugar mill sensitivity for its mttf is shown in fig. 6.4, it is observed that the system’s mttf is highly sensitive with respect to human error. as the rate of human error increases, it adversely affects the system mttf. fig. 6.5 shows the behavior of expected profit from the sugar mill. it shows 0 2 4 6 8 10 0 2 4 6 8 5.02 k 4.02 k 3.02 k 2.02 k 1.02 k e xp ec te d pr of it time unit (t) multistate system performance analysis… 127 copyright ©2024 assa. adv. in systems science and appl. (2024) that the expected profit of the sugar mill decreases as the service cost increases. hence, to optimize the profit function, the management needs to give more attention to the maintenance policy. conclusion in this paper, we utilized the concept of mathematical modeling and the markov model to obtain the various reliability measures of the “wahid sugar mill” situated in punjab, india. from the above discussion presented in section 7, the authors conclude that the mill needs to pay more attention to human error along with other failures. failures can be mitigated by employing skilled labor and imparting them training from time to time. it asserts that this research is beneficial for the management of the same for improving productivity. acknowledgements the authors are really grateful to the managing director (r.p. dubey) and the engineers of the wahid sugar mill who guided us to carry out this research. they always supported us whenever we needed their guidance. the last author also thanks cnpq (grant 306940/2020-5). references 1. aly, m. f., afefy, i. h., abdel-magied, r. k. & elhalim, e. k. 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(2015). climate change and sugarcane production: potential impact and mitigation strategies, international journal of agronomy, 2015, 1–10. adv syst sci appl 2025; 1:108–119 published online at https://ijassa.ipu.ru. grading structure for derivations of group algebras andronick arutyunov*1, igor zhiltsov 1v.a. trapeznikov institute of control sciences russian academy of sciences, moscow, russia abstract: in this paper we give a way of equipping the derivation algebra of a group algebra with the structure of a graded algebra. the derived group is used as the grading group. for the proof, the identification of the derivation with the characters of the adjoint action groupoid is used. these results also allow us to obtain the analogous structure of a graded algebra for outer derivations. a non-trivial graduation is obtained for all groups that are not perfect. keywords: derivations, bimodule, outer derivations, graded algebras calculation of derivations in a group algebra is a well-known problem. present work elaborates results from articles [1–3] focused on studying derivations in terms of characters of adjoint action gruppoid. an important result of this research are handy formulas for quick calculation of derivations. these articles explore derivations’ link to combinatorial properties of the group. among applications note use in coding theory (see [4, 5]), novikov algebras (see recent work [6]) and more general constructions, like (σ, τ)−derivations (see [7]). 1. introdution aim of the present work is grading the derivation algebra by identifying derivations and characters on a certain groupoid (all the necessary definitions are given in section 2). the main result of the paper follows. let n be a fixed normal subgroup in g such that g/n is abelian. theorem 1.1. if |g/n | > 1, der is graded with g/n , that is der = ⊕ k∈g/n derk, ∀k, l ∈ g/n : [derk,derl] ⊂ derkl. here derk is a subalgebra of derivation whose characters’ support is localised entirely in one coset an = k. the structure of the work follows. section 2 provides main definitions and propositions. section 3 describes the construction of grading and contains the main result and its proof. section 4 provides an example of grading for g equal to discrete heisenberg group along with an example of localising central derivations for such groups g that g is not a stem group. fix an infinite finitely-generated group g for the rest of the text. ∗corresponding author: andronick.arutyunov@gmail.com grading structure for derivations of group algebras 109 2. preliminaries recall that group algebra c[g] is an algebra of formal finite sums of type (a1, . . . , an ∈ c, g1, . . . , gn ∈ g) a1g1 + · · ·+ angn we define derivation as a linear operator d that satisfies the leibniz rule (for all a, b ∈ c[g]) d(ab) = d(a) · b+ a · d(b) derivations over a group algebra form a lie algebra with respect to commutator. we will denote this algebra as der or der(c[g]). 2.1. characters we use the technique of characters following [1–3]. definition 2.1. for a given group g consider a small groupoid γ: 1. objects (obj) — elements of g, 2. arrows (hom) — pairs of elements of g. for an arrow (u, v) its source s(u, v) is given by v−1u, and its target t(u, v) — by uv−1 (hom(a, b) denotes a set of all arrows for which the source is a and target is b), 3. consider two arrowsφ = (u2, v2) ∈ hom(b, c), ψ = (u1, v1) ∈ hom(a, b) (we will call a(n ordered) pair of arrows φ, ψ such that s(φ) = t(ψ) composable). the composition for these two arrows is given by: (u2, v2) ◦ (u1, v1) := (v2u1, v2v1) the formula for composition does not comprise u2 since if we have a pair of composable arrows (u2, v2), (u1, v1), u2 can be expressed in terms of u1, v1, v2. the reader may consider this as an exercise. γ is the groupoid of group’s inner action on itself. fix an element a of g. define following symbols: definition 2.2. • [a] = {xax−1 : x ∈ g} is a’s conjugacy class in g, • gg := {[g] : g ∈ g} • γ[a] is γ’s subgroupoid, informally, a connected component in γ, given by: obj(γ[a]) := [a] = {x ∈ obj : x ∈ [a]} hom(γ[a]) := {(u, v) ∈ hom : u, v ∈ [a]} definition 2.3. a character on γ is a function χ : hom→ c, such that: • (composition) for each pair of composable arrows φ, ψ: χ(φ ◦ ψ) = χ(φ) + χ(ψ) • (locally finite) ∀y ∈ g there is a finite set of x ∈ g, such that χ(x, y) ̸= 0. holds the following decomposition: lemma 2.1. (decomposition) γ = ⊔ [a]∈gg γ[a] copyright © 2025 assa. adv syst sci appl (2025) 110 a. arutyunov, i. zhiltsov remark. although characters being locally finite may seem as an alien and a bit too technical detail, it is deliberately placed in the definition to stress that we will not consider ”non-locally finite characters”. the reasons will become clear, among the rest, in theorem 2.1. we will need the following statement: statement 2.1. let (u, v) = φ ∈ hom, a ∈ g. then the following statements are equivalent φ ∈ hom(γ[a]), s(φ) ∈ [a], t(φ) ∈ [a]. it is proved by direct calculation. 2.2. connection between characters and derivations the following theorem motivates to consider (locally finite) characters when studying derivations. informally, theorem 2.1 shows that characters may be seen as a generalization of linear operator’s matrix. theorem 2.1 (derivation formula and derivation character, [1, section 2]). for each derivation d there exists a unique character χ such that for each x ∈ g holds d(x) = ∑ k∈g χ(k, x)k (2.1) consider d, χ from theorem 2.1. we will say that character χ gives derivation d (derivation d is given by character χ; we will omit words ”derivation” and ”character”). for a derivation d let χd be a character such that χd gives d. theorem 2.1 implies: corollary 2.1. let d, ∂ be derivations given by characters χd, χ∂ correspondingly, then d+ ∂ be given by χd + χ∂ . definition 2.4. let d be given by α, ∂ be given by β. then {α, β} is the character that gives [d, ∂]. statement 2.2 (”matrix” product, [2, proposition 2.4]). let α, β be characters. then {α, β} satisfies (a, b ∈ g) {α, β}(a, b) = ∑ k∈g α(a, k)β(k, b)− β(a, k)α(k, b) two examples of derivations follow. example 2.1 will be needed to prove theorem 1.1. let a ∈ g. recall that derivation da is called inner if for any x ∈ c[g] da(x) = [x, a] = xa− ax example 2.1 (character of inner derivation [3, proposition 3]). let a ∈ g. then character χa given by formula χa(φ) =  1, a = s(φ), −1, a = t(φ), 0, otherwise. (2.2) gives da(x) = [x, a]. example 2.2. another possible example of derivations are central derivations. we will call derivation d central if there exists such central element z ∈ z(g) and homomorphism τ : g→ (c,+) such that for all basis elements g ∈ g: d(g) = τ(g)gz copyright © 2025 assa. adv syst sci appl (2025) grading structure for derivations of group algebras 111 such an operator is indeed a derivation, see [3, proposition 4]. [3, proposition 5] shows that non-trivial central derivations are not inner. moreover, [3, proposition 6] shows that central derivations form a lie subalgebra in der(g). definition 2.5. for a given character χ we define support of χ as following supp χ = {φ ∈ hom : χ(φ) ̸= 0} for the given subset m ⊂ g denote by derm the set of derivations d such that for character χ that gives d: supp χ ⊂m . example 2.3. recall character χa from example 2.1 (where a ∈ g.) its support is easily calculated supp χa = {φ : s(φ) = a} ∪ {ψ : t(ψ) = a} by statement 2.1, we can localize supp χa in a single conjugacy class a (we will need such a localisation later in theorem 1.1) supp χa ⊂ γ[a] 2.3. applying decomposition lemma 2.1 establishes decomposition of groupoid γ. the current section presents decompositions for (locally finite) characters and derivations. the following two lemmas are equivalent. we prove the first one. lemma 2.2. let χ be a character. then there exists finitely many a1, . . . , an ∈ g such that supp χ ≤ n⋃ k=1 γ[ak] lemma 2.3 (derivation decomposition). then for each u ∈ g such that χu is a character, holds decomposition d = ∑ [u]∈gg du, and the set {[u] ∈ gg : ∃x ∈ g : du(x) ̸= 0} is finite. proof for lemma 2.2 let g = ⟨x | r⟩, where x =: {x1, . . . , xk} is finite (g is assumed to be finitely-generated throughout the text). consider (u, v) ∈ γ. let n = n(v) be minimal nonnegative integer such that ∃y0, . . . , yn ∈ x ∪x−1 : v = y0y1 . . . yn 1. show that ∃z0, . . . , zn ∈ g : (z0, y0) ◦ · · · ◦ (zn, yn) = (u, v) (2.3) subproof induction by n = n(v). base: n = 0 — z0 = u. step: consider z0 = uv−1y0. then (z, y0), (y −1 0 u, y−1 0 v) are composable since s(z0, y0) = y−1 0 z0 = y−1 0 uv−1y0 = = y−1 0 u(y−1 0 v)−1 = t(y−1 0 u, y−1 0 v) moreover, (z0, y0) ◦ (y−1 0 u, y−1 0 v) = (u, v) notice that y−1 0 v = y1 . . . yn, thus n(y−1 0 v) < n = n(v). therefore, applying the step of induction, get eq. (2.3). □ copyright © 2025 assa. adv syst sci appl (2025) 112 a. arutyunov, i. zhiltsov 2. since χ is locally finite, the set b = (g× (x ∪x−1)) ∩ supp χ is finite. by ?? for each arrow φ there exists a unique element a ∈ g such that φ ∈ hom(γ[a]); thus, there exists a finite seta = {a1, . . . , an} such that for any a /∈ a:b ∩hom(γ[a]) = ø. thus, by item 1 for any a /∈ a: supp χ ∩hom(γ[a]) = ø. therefore: supp χ ≤ n⋃ k=1 γ[ak] ■ a very nice alternative proof for lemma 2.2 was submitted in an anonymous review. alternative proof for lemma 2.2 let d be the derivation given by character χ. let p be a union of conjugacy classes such that supp χ ⊂ ⋃ g∈p γ[g] =: u the following statements are equivalent: • (x, y) ∈ hom(u), • y−1x = s(x, y) ∈ p consider an element y ∈ g such that y−1d(y) ∈ ⟨p ⟩ here ⟨x⟩ forx ⊂ g denotes a set of all finite sums a1x1 + · · ·+ anxn such that a1, . . . , an ∈ c and x1, . . . , xn ∈ x . let’s calculate y−1d(y) by statement 2.2. y−1d(y) = y−1 ∑ x∈g χ(x, y)x = ∑ x∈g χ(x, y)y−1x = ∑ x∈g χ(x, y)s(x, y) ∈ ⟨p ⟩ therefore, for each x such that χ(x, y) ̸= 0 : y−1x ∈ p . consider a set h = { y : y−1d(y) ∈ ⟨p ⟩ } as a simple exercise, check h’s being a subgroup in g. to summarize, if y is in subgroup h ≤ g (that is y−1d(y) ∈ ⟨p ⟩) then for each x such that χ(x, y) ̸= 0 : y−1x ∈ p . to finish the proof let’s choose such p that p is a union of a finite number of conjugacy classes and h = g. to achieve this, consider a finite generating set s for g and a finite subset m ⊂ g such that for any s ∈ s there exist complex numbers am,m ∈m such that s−1d(s) = ∑ m∈m amm informally, calculate all s−1d(s), s ∈ s, which are finite sums, and store all elements in g present in at least one of finite sums. copyright © 2025 assa. adv syst sci appl (2025) grading structure for derivations of group algebras 113 since m is finite, p = ⋃ m∈m [m] is a union of finite number of conjugacy classes. moreover, for such p s ⊂ h thus, g = h . all in all, there exists a union p of a finite number of conjugacy classes such that supp χ ⊂ ⋃ g∈p γ[g] =: u ■ let d be the derivation given by character χ, and χu(φ) := { χ(φ), φ ∈ hom(γ[u]), 0, otherwise . we will denote the derivation given by χu as du. 3. constructing graded algebra grading with abelian quotients definition 3.1. let a be an abelian group, e be a neutral element in a, a be a lie algebra, a can be expressed as a direct sum a = ⊕ n∈a an, such that ∀n, l ∈ a : [an,al] ⊂ anl such that ae ̸= a. then a is called graded (with a). the direct sum def 3.1 is called a’s grading with a. notice that trivial gradings are excluded. definition 3.2. a commutator subgroup of group g is g′ := {xyx−1y−1 : x, y ∈ g} recall a few well-known definitions and statements that will be required below. statement 3.1. for any group g a subgroup g′ is normal, g′ ◁ g let n be a fixed normal subgroup in g such that g/n is abelian. conceptually, the latter condition derives from the following def 3.1, however it is also needed for more technical, yet crucial details, like lemma 3.1. statement 3.2. g′ ⊂ n . lemma 3.1. let a ∈ n . then [a] ⊂ an . proof a calculation for any a, t ∈ g proves lemma: tat−1n = tat−1a−1na = na = an the first and the last equivalences hold since n is normal; the second equivalence holds by statement 3.2 since g/n is abelian (tat−1a−1 ∈ g′ ⊂ n .) ■ copyright © 2025 assa. adv syst sci appl (2025) 114 a. arutyunov, i. zhiltsov note that lemma 3.1 would have been false if we did not requireg/n to be abelian. consider g = s4, n = v4, a = (12) for counterexample. lemma 3.1 motivates the following symbols: γan = ⋃ k∈an γ[k], deran = { d ∈ der : supp χd ⊂ γan } . lemma 3.2. let a, b ∈ g, d is given by character α : supp α ⊂ γan , ∂ is given by character β : supp β ⊂ γbn . then supp {α, β} ⊂ γabn . proof statement 2.2 implies: {α, β}(h, g) = ∑ k∈g α(h, k)β(k, g)− β(h, k)α(k, g) consider an arrow (h, g) : {α, β}(h, g) ̸= 0. there exists k ∈ g:[ α(h, k)β(k, g) ̸= 0 β(h, k)α(k, g) ̸= 0 { α(h, k) ̸= 0 β(k, g) ̸= 0{ β(h, k) ̸= 0 α(k, g) ̸= 0 (3.4) expressing eq. (3.4) in terms of supp : { (h, k) ∈ supp α ⊂ hom(γan) (k, g) ∈ supp β ⊂ hom(γbn){ (k, g) ∈ supp α ⊂ hom(γan) (h, k) ∈ supp β ⊂ hom(γbn) (3.5) by statement 2.1, an arrow φ belongs to γ[x] ⊂ γxn iff its target t(φ) belongs to [x] ⊂ xn . thus, eq. (3.5) implies:  { hk−1 = u ∈ an kg−1 = v ∈ bn{ kg−1 = u ∈ an hk−1 = v ∈ bn (3.6) multiplying the equations in eq. (3.6):[ hg−1 = uv hg−1 = vu , for u ∈ an, v ∈ bn copyright © 2025 assa. adv syst sci appl (2025) grading structure for derivations of group algebras 115 by definition, t(h, g) = hg−1. by statement 2.1:[ (h, g) ∈ hom(γ[uv]) (h, g) ∈ hom(γ[vu]) since [uv] = [u · vu · u−1] = [vu] and uv ∈ abn , then by lemma 3.1: [uv] ⊂ uvn = abn thus: (h, g) ∈ γ[uv] ⊂ γabn , for u ∈ an, v ∈ bn all in all, {α, β}(h, g) ̸= 0 ⇒ (h, g) ∈ γabn , therefore supp {α, β} ⊂ γabn . ■ theorem 1.1. if |g/n | > 1, der is graded with g/n , that is der = ⊕ k∈g/n derk, ∀k, l ∈ g/n : [derk,derl] ⊂ derkl. proof consider the sum ∑ k∈g/n derk. 1. first, show that ∑ k∈g/n derk is equal to der. • ∑ k∈g/n derk ⊂ der — since der is closed under finite sums. • now we show an opposite inclusion. consider an arbitrary d ∈ der. lemmas 2.3 and 3.1 imply (see lemma 2.3 for definition of du) d = ∑ [u]⊂g du = ∑ k∈g/n ( ∑ [u]⊂k du ) consider for given k ∈ g/n sk := ∑ [u]⊂k du ∈ derk by lemma 2.3, there is only a finite number of such [u] ⊂ g that du is not constant 0. thus, there exists an integer n and k1, . . . , kn ∈ g/n such that for any k ∈ g/n, k ̸= k1, . . . , kn : sk is constant 0. thus, d = n∑ i=1 ski thus, der ⊂ ∑ k∈g/n derk copyright © 2025 assa. adv syst sci appl (2025) 116 a. arutyunov, i. zhiltsov • all in all, der = ∑ k∈g/n derk. 2. ∑ k∈g/n derk is direct. subproof let dk ∈ derk be given by χk. suppose that∑ k∈g/n dk ≡ 0 since constant 0 is a derivation given by constant character (equal to 0), by corollary 2.1 for any φ ∈ hom ∑ k∈g/n χk(φ) = 0 since supp χk, supp χl are disjoint (for k ̸= l), for each φ ∈ hom exists no more than one k ∈ g/n such that χk(φ) ̸= 0. thus, for each φ ∈ hom and for each k ∈ g/n : χk(φ) = 0, and for each k ∈ g/n : dk ≡ 0. therefore, ∑ k∈g/n derk = ⊕ k∈g/n derk is direct by definition of direct sum. □ 3. we established that der = ⊕ k∈g/n derk since there exists such m ∈ g that mn ̸= n . thus, by examples 2.1 and 2.3 the inner derivation [x,m] is given by a character χm such that supp χm ⊂ [m] ⊂ mn by lemma 3.1. thus, der ̸= dern (i.e. the grading is not trivial.) finally, check that (∀k, l ∈ g/n ) ∀k, l ∈ g/n : [derk,derl] ⊂ derkl proof for item 3 let dk ∈ derk, dl ∈ derl. let character χk give dk, character χl give dl. thus, by definition of derk,derl supp χk ≤ γk, supp χl ≤ γl therefore, by lemma 3.2 supp {χk, χl} ≤ γkl finally, [dk, dl] ∈ derkl □ ■ copyright © 2025 assa. adv syst sci appl (2025) grading structure for derivations of group algebras 117 example 3.1. let g be a perfect group (|g/g′| = 1), that is g′ = g. theorem 1.1 yields a trivial grading(which we do not regard as a grading in this text) for der(c[g]) since |g/g′| = 1. and vice versa: corollary 3.1. ifg is not a perfect group (g ̸= g′) then der admits a (non-trivial) grading with g/g′. example 3.2. let g be a knot group (i.e. let there exist some knot k such that g is the knot group ofk). it is well-known that in this caseg/g′ = z, thereforeder(g) admits a grading with z. 4. examples discrete heisenberg group consider discrete heisenberg group (a group of 3× 3 upper unitriangular matrices with integer entries). following [3], we use this group as a handy example since it admits easy calculations. definition 4.1. consider a group of integer unitriangular matrices with respect to matrix multiplication: h = {( 1 a c 0 1 b 0 0 1 ) : a, b, c ∈ z } ( 1 a c 0 1 b 0 0 1 )( 1 x z 0 1 y 0 0 1 ) := ( 1 a+ x c+ z + ay 0 1 b+ y 0 0 1 ) since all the matrices in h have determinant 1, the inverse is well-defined and given by( 1 a c 0 1 b 0 0 1 )−1 = ( 1 −a ab− c 0 1 −b 0 0 1 ) our goal is to grade der(h). definition 4.2. the centre of g is z(g) = {z ∈ g : ∀g ∈ g : gz = zg} the follwing statements are well-known and trivial. statement 4.1. h ′ = z(h) = {( 1 0 a 0 1 0 0 0 1 ) : a ∈ z } statement 4.2. h/h ′ ≃ z⊕ z let ψ : z⊕ z → h/h ′ be an isomorphism. recall the symbols: γan = ⋃ k∈an γ[k], deran = { d ∈ der : supp χd ⊂ γan } . copyright © 2025 assa. adv syst sci appl (2025) 118 a. arutyunov, i. zhiltsov define: der(i,j) := derψ(i,j) corollary 4.1 (from statement 4.2, theorem 1.1). der(h) is graded with z⊕ z, that is der(h) = ⊕ (i,j)∈z⊕z der(i,j) ∀(i, j), (k, l) ∈ z⊕ z : [der(i,j),der(k,l)] ⊂ der(i+k,j+l) example 4.1. statement 4.1 implies: if d is given by such χ that supp χ ≤ γ[z] for z ∈ z(h), then d ∈ der(0,0). definition 4.3. g is a stem group if z(g) ≤ g′ example 4.1 can be generalised under the assumption that g is a stem group. note that h is a stem group since h ′ = z(h) by statement 4.1. see [8] for more details on (finite, which is not our case) stem groups. proposition 4.1. let g be a stem group. if d is given by such χ that supp χ ≤ γ[z] for z ∈ z(g), then d ∈ derg′ . central derivations were introduced in [3] as operators given on a group algebra generators g ∈ g with the homomorphism τ : g→ (c,+) and the central element z ∈ z(g) by formula dτ,z : g 7→ τ(g)gz. recall that [3, proposition 6] shows that central derivations form a lie subalgebra in der(g) (denote as zder(g)). proposition 4.2. let g be not a stem group. then there is an ”induced” nontrivial grading of zder(g) with g′. note that if g is a stem group, the grading is trivial, i.e. not a grading at all in our terms. another example of an induced grading follows. let a ∈ g. recall that derivation da is called inner if for any x ∈ c[g] da(x) = [x, a] = xa− ax let innerder be the set of all derivations of the form (a1, . . . , an ∈ c; y1, . . . , yn ∈ g) g ∋ x 7→ a1[x, y1] + · · ·+ an[x, yn] = [x, a1y1 + · · ·+ anyn] a direct calculations shows that innerder is an ideal in der (i.e. for any d ∈ innerder and for any ∂ ∈ der : [d, ∂], [∂, d] ∈ innerder; recall that [d, ∂](x) = d(∂(x))− ∂(d(x)).) therefore, there exists a factor-algebra outerder := der/innerder. corollary 4.2. if g is not a stem group, there is an induced (non-trivial) grading of outerder with g′ of the form outerder = ⊕ k∈g/g′ derk/innerderk, where innerderk := derk ∩ innerder copyright © 2025 assa. adv 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(1940) the classification of prime-power groups, journal für die reine und angewandte mathematik, 1940, 130-141, doi: 10.1515/crll.1940.182.130 copyright © 2025 assa. adv syst sci appl (2025) introdution preliminaries characters connection between characters and derivations applying decomposition constructing graded algebra examples microsoft word 1243 article text, copyedited.doc adv syst sci appl 2022; 03; 84-95 published online at https://ijassa.ipu.ru. improving health through healthcare technologies assessment development: further perspectives for hta tools development dmitry meshkov1, oyvind melien2, sergey cherkasov1, alexey lobanov1, anna fedyaeva1 1) v.a. trapeznikov institute of control science of the russian academy of sciences, moscow, russia e-mail: dmitrymeshkov@mail.ru 2) norwegian institute for public health, oslo, norway e-mail: oyvind.melien@fhi.no abstract: the hta progress requires analyses of existing and forthcoming needs as well as the development of tools meeting these needs. the proposed research presents hierarchical scheme for hta development created based on environmental scanning and analytic hierarchy process (ahp). the scheme presents the overall prognosis of hta development indicating the key blocks of needs and relevant tools. each of these tools are considered to be exclusively important for hta development. local registries are considered to be the priority for further hta improving meeting more needs and providing less risks comparing to other tools. keywords: health technology assessment, hta development, patients registries, prognosis, forecast, environmental scanning, analytic hierarchy process. 1. introduction healthcare technologies assessment (hta) is at the very end of the data generation chain assessing the value of medical intervention and allowing putting together the entire set of data on the risks and benefits associated with its use. consistent data generation includes information on the effectiveness of individual clinical trials and systematic reviews that allow assessing the full range of clinical effects associated with the use of a specific technology or treatment of nosology, and summarizing them in the form of clinical guidelines in accordance with who requirements. following these stages, a comprehensive clinical and economic assessment of medical technology makes it possible to assess not only medical, but also economic and social consequences of the intervention. [20] since the eighties of the last century, organizations specializing in complex clinical and economic assessment of medical technologies (or medical interventions) have been established in countries. these organizations carry out their activities at the level of an individual health facility, a region within the country, at the level of a country or even a group of countries. they may have different lists of tasks to be solved and methods used. the existing progress of research as well as changes in the epidemiological, economic and social environment present new requirements for health technologies assessment (hta), which is the base for the rational use of medical technologies. hta was initially established in high income countries but lowand middle-income countries also understand the need for rational approach in healthcare technologies applications and consider it as the basis for effective healthcare systems, achieving social stability and providing labor resources for economic development. in line with this the member states of world health organization approved the resolution “health intervention and technology assessment in support of universal health improving health through healthcare technologies assessment development 85 coverage» at the world health assembly in may 2014 (wha67/2014/rec/1 (who.int) wha 67.23). initially, there were different approaches regarding assessment and allocating or redistributing resources for the implementation of innovative healthcare interventions in highand low-middle-income countries but currently, there is a process of harmonization of the requirements and methodology of hta in both highand low-middle-income countries. there is also a revision of the tasks facing hta as well as, the development of new methods of evaluation and interaction with decision-making systems. in addition, a legislative regulation for hta at a regional level has been approved recently (regulation (eu) 2021/2282 of the european parliament and of the council of 15 december 2021 on health technology assessment and amending directive 2011/24/eu). however, the process of modern hta creation is not finished yet [12, 13]. the harmonization of hta objectives and methodology allows us to assess the key principles of development of hta as a tool ensuring global health [2, 4, 8, 9, 10, 16, 19, 24, 25]. 2. objectives the objectives of this research were: 1. to create a comprehensive hierarchical structural scheme presenting the key information and technological blocks and reflecting the forecast of the hta development; 2. to highlight and determine the most relevant areas of development of practical tools meeting existing and promising hta needs which can be considered the objectives for further research and development of practical tools. 3. methods two key methods were used to conduct the study. environmental scanning was used to search and analyze data on the current state and prospects of hta development. as a result, key information messages were identified and then structured into a hierarchical scheme using the analytic hierarchy process (ahp) by 11 experts specialized in hta and healthcare management. environmental scanning was based on the systematic review of literature (pubmed and embase; 353 relevant publications) and other modes of communication including available english-language websites of organizations assessing healthcare technologies and the search for "grey" literature using google search as well personal messages [18, 21]. the determined emerging issues presented both the specific topics and their general environment making possible identifying important emerging issues that may constitute either obstacles or opportunities. [22, 17, 32,] the obtained data was used for analytic hierarchy process (ahp)which is a structured technique for organizing and analyzing complex decisions using accurate approach to quantifying the weights of decision criteria estimating the relative magnitudes of factors through pair-wise comparisons. first step in ahp was the decomposing of decision problem into a hierarchy of more easily comprehended sub-problems, each of which can be analyzed independently. nest step included systematical evaluation of various elements of hierarchical structure by comparing them to each other two at a time, with respect to their impact on an element above them in the hierarchy. in making the comparisons, the experts assessed both concrete data about the elements and their judgments about the elements' relative meaning and importance based on “environmental scanning” data. these numerical values reflected the number of links with higher hierarchical elements and the capability of the lower element to meet needs of these higher elements. thus, the higher numerical weight was delivered to the element providing the most significant impact on the whole hierarchical structure and represented the ability to achieve developing hta needs [7, 26, 27, 29]. 86 d. meshkov, o. melien, s. cherkasov, a. lobanov, a. fedyaeva 4. results the hierarchical structure of the main information blocks for hta development is shown in fig. 1. it demonstrates the following hierarchal levels: key needs and requirements for hta; addressing of these needs to more specific questions and participants; possible solutions and practical tools; threats and conflicts that may arise when using the proposed tools for implementing solutions and also the description of potential conflicts resolution. the key needs included relevance, reliability and timelines for the interventions availability and were addressed in more detailed subjects. relevance of expert opinion with local and regional peculiarities and the needs of the local healthcare systems. it is can be addressed to availability and use of local data reflecting the morbidity, mortality, epidemiological and demographic data as well as the other features of the territory or administrative unit including the results of local clinical trials and local real world data evidence (rwd/rwe). the expert opinion must be relevant to the interests of various stakeholders in addition to healthcare authorities and patients, that can somehow be involved in the implementation of management decisions and are associated with the outcomes of their implementation. a separate need concerns the compliance of expert opinions with formalized requests, regulatory documents and standard operating procedures of the local healthcare systems. reliability of expert opinion ensured by the quality of the source data, analysis methods and evaluation criteria. the quality and reliability of source data means appropriate amount of evidence including disease burden, reports on international and local clinical and academic trials, systematic reviews and clinical recommendations (global evidence) related to safety, efficacy, efficiency and effectiveness. reliable methods and criteria assure that practical results of regulatory decision and redistribution of resources will match the expert forecast both in the same criteria and figures. timelines are the dates when the medical technologies in question become available to those who need them. modern medical technologies can significantly add years of life and improve its quality. ensuring early availability means added years of quality life (qaly) for patients. it is provided by various schemes of early access of patients to therapy before market authorization, by shortening of market authorization timelines and market entry of the medicines or by using more effective scenarios providing medications to patients after market authorization. shortening the timelines for reimbursement has also an impact on ensuring accessibility through the interest of the industry. on the other hand, since the industry is one of the stakeholders in the evaluation of healthcare technologies and its interests should be taken into account together with the interests of other participants. identified solutions included stakeholders and patients’ involvement, access to local, regional and global research and disease burden data, data and expertise sharing, pre-launch access for patients, early access to clinical trials data and faster data processing. early hta was also reviewed as a solution shortening timelines for reimbursement. the development and humanization of society lead to an increase in social groups that have their own interests in ensuring the availability of medical technology to patients. often these stakeholder groups have conflicting interests, which, however, should be taken into account when making a decision. stakeholders' needs can be met by their involvement in the hta process. the involvement of patients is presented in most publications as a separate task, which is due to their obvious exclusive role in the assessment and decision-making process. a necessary element ensuring effective involvement of stakeholders is access to local information on the clinical, economic and social burden of the disease, as well as possible solutions and their outcomes is necessary for the effective participation of various stakeholders in the process of analysis, consensus finding and decision-making. improving health through healthcare technologies assessment development 87 88 d. meshkov, o. melien, s. cherkasov, a. lobanov, a. fedyaeva local laws and regulations are the background on which the assessment takes place and the results should correspond to this legislation. at the same time, however, there is also feedback, which consists in the influence of stakeholders, especially patients, who, based on local data on the burden of the disease, have the opportunity to change legislation in accordance with social needs. global evidence consists of the data that is received and stored at the local, regional and global levels which sometimes have differences in quantity and quality of data. global databases, such as cochrane, g-i-n and others, accumulate and process with all available data providing this way the most complete and evidence-based information on diseases and treatments. local data are included in global data as an integral part, but they are also of some interest, reflecting local specifics and, obviously, should be taken into account when conducting local assessments. regional databases allow to expand the sample and increase the reliability by combining similar local evidence based on epidemiological, medical, economic, cultural, linguistic community of countries in the region. regional cooperation serves also as an intermediate link between local and global collaboration making possible accelerating accumulation and increasing accessibility of evidence for global cooperation. methodology and criteria of assessment are rather related blocks, and their development is more intensive and efficient in case of data and expertise sharing making possible to spend as much resources as possible but to have an opportunity to use the entire results of cooperation. early access to patients can be achieved by pre-launch participation in clinical studies or any other similar activities. to some extend it is also linked to patients’ data making possible to speed up the process of screening and enrollment of eligible patients into studies. authorization timelines in terms of hta are closely related with early access to clinical trials data as well as faster data proceeding. it also makes possible to use early hta technology and finally to reduce the time for reimbursement. section tools is of the very interest for the development of health technology assessment, since it allows you to identify the main areas of research, as well as to rank them by selecting the most effective, promising and relevant. communicational and informational platforms the first thing that experts face when assessing various outcomes and developing predictive scenarios is the need to combine multidisciplinary information and compress it into several verbal statements that can be used to formulate regulatory documents that determine the allocation of resources. information platforms serve as a tool for information transformation, structuring the studied information into blocks, which, in turn, are connected by certain relationships and algorithms. clinical trial was reviewed as an example of such platform. data generation consists of thousands of data flows between blocks of information represented by dozens of clinical indicators obtained at various stages of screening, treatment and follow-up of hundreds or thousands of patients observing in dozens of clinical centers. upper levels of information blocks include, but are not limited to, comprehensive information on individual patients (case report form), on clinical centers (investigator file), aggregating information on individual safety and efficacy indicators for all included patients or stratified groups, for example, by age or by presence of concomitant diseases. obtaining and processing of all this information require qualified personnel, appropriate equipment for measurements, recording, transmission, accumulation and processing of multidisciplinary data resulted in expert analysis and a conclusion that fits just few statements. these activities require coordinated action of various specialists including physicians, laboratory specialists, pharmacists, data management specialists, medical writers and others. hta requires also assessment of economic and social outcomes alongside with the clinical improving health through healthcare technologies assessment development 89 copyright ©2022 assa. adv. in systems science and appl. (2022) ones generating the need for interdisciplinary interactions between specialists and stakeholders with conflicting interests developing mutually acceptable solution. the concept of "informational platform" also includes blocks of information and transparent algorithms making possible communicating specialists and stakeholders with different background and areas of interests. as a result, further hta development requires a hierarchical complex of information and communication platforms assessing data, ranging from monitoring and forecasting of public health, preclinical, clinical and clinical-economic research to social outcomes forecasting and resulting in prognostic scenarios on various issues related to public health management. modern information and technology platforms have a complex system of connections, and a change in one of the elements of this complex leads to a change in other elements. every intervention, from the simplest to the most complex, has an impact on the overall system, and the overall system affects every intervention [30]. further developing of the hierarchical structure and interactions between these platforms as well as the existence of regulatory and legislative bodies determining these elements and the degree of their impact on the system as a whole, we get a classic "blockchain" structure which is rather plausibly describes the interaction of elements of the overall public health management information platform. [3]. the creation and further development of these information and communication platforms requires the appropriate development of digital infrastructure and digital technologies capable of proceeding with big data and identifying links and dependencies between health indicators and multidisciplinary factors forming these indicators. the main risks identified by experts in connection with the creation of communication platforms are the possibility of incomplete or asymmetric representation of stakeholders and their unpreparedness, primarily of patients as the main stakeholders. accordingly, the resolution of these conflicts is associated with transparency of the assessment decision making process, and conflict-of-interest management mechanisms and training of key stakeholders. bioethics is of great importance for creation of communication platform considering it as a science and technology that provides solutions to interdisciplinary and interdepartmental issues related to human health in conditions of conflicting interests. assessment criteria the allocation of resources in healthcare is a rather complex process that ensures a transparent and rational procedures resulting in clear decisions. as a result, various criteria have to be considered within the framework of hta and simplified to a one-dimensional solution – to "include or not include" technology in the reimbursement lists or similar documents [1, 6, 11]. these are criteria that often conflict with each other [15]. insufficient awareness of the value elements considered throughout the decision-making continuum, as well as the lack of transparency in decision-making, can potentially create tension among stakeholders [31]. as a result, a comprehensive "value assessment approach" is increasingly being used in the assessment of healthcare technologies, which requires a clear list of criteria to be taken into account [23]. in addition, it is a generally accepted principle that decisionmaking in the field of healthcare should be based on evidence-based scientific data [26], which call for the need for a scientifically based method of evaluating more than one aspect of the problem of decision-making at the same time. usually the country data, even obtained on the basis of an integrated approach are almost never complete and can only reflect the main trends in the development of the local hta, which imposes certain restrictions on the results of any analysis in this area. as a rule, countries use a set of methods even if only some of these methods are fixed by law. the multicriterial approach leads to attempts creating the universal tool assessing multiple criteria at once and providing clear conclusion on the matter. 90 d. meshkov, o. melien, s. cherkasov, a. lobanov, a. fedyaeva copyright ©0000 assa adv. in systems science and appl. (0000) the development of evaluation criteria (and, accordingly, evaluation methods) face the need to combine local data and global data based on local legislation, taking into account numerous interdisciplinary factors, forming a forecast of medical, economic and social outcomes and the possibility of a simple conclusion acceptable with local regulatory procedures. the key risks and potential conflict are also linked to stakeholder’s unpreparedness and asymmetric representation of particular stakeholders needs. this may include, for example, the creation of assessment criteria aimed at saving money when the position of payers is dominant or, conversely, the criteria which may cause unjustified expenses when the position of industry is dominated. conflict resolution is related to the transparence of criteria and procedures, education of stakeholders and especially patients, conflict of interest management and the organization of communication platforms based on bioethics principles. local registries patient registers are a special case of information platforms allowing receiving and analyzing data on a specific disease. the term local implies that the register collects limited information that is not integrated into larger unified registers, and serves to accumulate and analyze information on individual diseases at its own level (hospitals, cities, countries). local registers are the most effective and reliable tools for obtaining rwd/rwe information about the number of patients, health determinants and other factors related to the occurrence of the disease, the course of the disease, treatment features and outcomes. depending on the registries structure and the information entered, they can be a source of data on the economic and social consequences of the disease. local registries are the basis for regional and, potentially, global registries that provide comprehensive information on diseases. there are two main directions for the further development of registries. the first one is linked to increasing number of parameters including information about social health determinants as well as economic and social outcomes that can be very helpful for hta and healthcare decision making. the second direction is linked to merging of registries databases wherever possible and increasing the sample size of the studied population. combining of both makes possible increasing the reliability of expertise and relevance to stakeholders and healthcare authorities’ needs. an opportunity to speed up patient’s enrollment in clinical trials and to use the existing patients’ databases for early access schemes and iterative studies can be considered as additional advantage of such development of registries. difficulties and conflicts are primarily related to the insufficient number of available local registers (and appropriate institutions capable creating such registries) and their characteristics. the majority of registers were created to address specific issues and included only patients who received the studied therapy for specific diseases or by medical institutions for their specific tasks and research programs. the infrastructure and financing of registers need a major reassessment that they can meet harmonization requirements and could be used beyond these existing tasks. the lack of qualified personnel with sufficient skills ensuring harmonization and extension of the of registers functions are also an issue. accordingly, the resolution of these problems is associated with the reassessment of infrastructure and funding of registries and education of qualified staff as well as creation of specific tools that allow obtaining the necessary data, processing them and presenting real time assessments and. the creation of such tools is closely connected with cloud information technologies that provide optimal opportunities for obtaining and processing information at the current level of science and technology development, as well as the harmonization of registers at various levels (local, regional and global) and the integration of the entire complex of rwd/rwe in hta process and decision making in healthcare. improving health through healthcare technologies assessment development 91 copyright ©2022 assa. adv. in systems science and appl. (2022) regional and global registries merging and harmonization of registries beyond the countries was appointed as a special task due importance of the global access to clinical data. [14] local legislation provides country-special requirements on data transfer but there are some opportunities in case of existing agreements of legislative uniformity in groups of countries [5]. as a rule, these countries have similar historical linguistic and cultural background as well as exiting economic and healthcare cooperation. thus, regional registries were considered as a separate tool and intermediate stage on the global data flow significantly facilitating the task of transparency and integrity of clinical, economic and social data on disease burden and treatment effects. the development of such registries also needs to merge and combine the data from the smaller and subordinated ones as well as the expanding the list of indicators including different diseases data and social health determinants, clinical, economic and social outcomes. the amount of interdisciplinary data and need for processing previously blinded (personal data) or generalized information (due do local legislations) requires implementation of specialized digital capable collecting and processing with this data in realtime frames. difficulties and solutions associated with local and global registries are the same as for the local ones: restructuring and re-targeting registers to more complex tasks and the corresponding reallocation of resources and information structures, staff training and creation of appropriate tools (including cloud-based technologies) for the registries maintenance and data analysis in real time frames. regional and global cooperation hta development reviewed in this study concerns only the one block of the information platform for expert support of healthcare decision-making. [6]. the main objective of the platform is to ensure the continues and appropriate transformation of the multidisciplinary data flow. international cooperation allows to merge local data and to use a win-win strategy in methodology development and achieving global core competencies when each party invests as much resources as it can, but has the opportunity to use the complete results of the research or education. each party invests as much resources as it can, but has the opportunity to use the complete results of the research or education programs. regional structuring of cooperation may significantly increase efficiency of this cooperation. common cultural, linguistic, economic, educational features as well as the traditions of scientific activity facilitate and accelerate the exchange of information. consistent collaboration on local level followed by regional and global cooperation and the counterflow of expertise from global to regional and local levels considered to be more effective comparing to the flat structureless global cooperation. as a positive example, we can consider guidance international network (https://g-i-n.net), which unites 7 regional associations (gin africa, gin arab, gin asia, gin australia & new zealand (anz), gin iberoamerica, gin nordic, gin north america (na)). difficulties related with regional and global cooperation are similar to those ones related with registries. the main are the lack of appropriate organizations and staff. these issues can be solved by creation of relevant infrastructure and development of the personnel global core competencies. facilitating and/or ensuring early access to interventions this block consists of different methods aimed at proving earlier access for more patients and thus increasing the global value outcomes for global health. initially they were reviewed 92 d. meshkov, o. melien, s. cherkasov, a. lobanov, a. fedyaeva copyright ©0000 assa adv. in systems science and appl. (0000) as different methods with similar characteristics including: supplementary role in current healthcare decision making; developmental status or opportunities for improvements and the requirement for better implementation in decision making process. managed entry agreements (mea) are complementing the established decision-making schemes for medicines budgeting. the application of meas requires the development of more advanced algorithms and procedures for their application, which is associated with certain difficulties caused by differences in legislation and application experience even in countries of the same region (central and western europe). nevertheless, these agreements are considered as one of the promising areas of expert support for decision-making on reimbursement (conditional or unconditional) of medicines, which increases their accessibility for patients and, in some cases, provides earlier access. adaptive pathways or medicines adaptive pathways for patients (mapps) is an approach aimed at maximizing the positive impact of new medicines on public health, achieved by balancing the need for timely patient access through phased (iterative) implementation using both existing tools and a more flexible application of the existing regulatory framework. thanks to adaptive pathways, a drug can follow an "iterative" process in which it can be initially approved for only a small group of patients (based on limited scientific evidence), and then, when more evidence is collected, the drug can be approved for extended group of patients. in addition, a drug may be first approved based on surrogate endpoints, which should subsequently be tested with more clinically important outcome endpoints predicting and horizon scanning incorporated into the model of proactive public health management and allowing the necessary set of activities to be formed in advance, increasing the efficiency of bringing medical technologies to market and creating decision support tools that allow adequate and timely allocation of necessary resources. currently used for particular tasks and investigation of competitive landscape for interventions in shortand medium-term forecast. early hta provides algorithms for early access to and analysis of study treatment outcomes as well as early awareness of stakeholders and decision makers reducing the time for hta process and allocating funding for intervention. there were five main risks closely linked one to another and with the early access methodology: these schemes were not deeply incorporated into regulatory process or were under development or required additional tools (databases or predictive models, etc.). there was also shortage of qualified staff and organizations (local, regional or global). resolution of these issues was linked with infrastructure and funding for establishing of appropriate institution and personnel competencies as well as for the development of comparative effectiveness methodology and interactive tools for decision making (to summarize comprehensive analyses of big data and different prognostic scenarios in a friendly and assessable format). selected tools for prioritization comprehensive hta development requires improvement implementation of all mutually linked blocks. nevertheless, some of these blocks can provide more impact on the rest and trigger their development or can cause risks and additional tools required for conflict resolution. the analytic hierarchy process (ahp) made possible to range the importance of the proposed hta tools and their efficacy. the importance was considered to be aligned with number of addressed needs which can solved by the tool. the efficacy was considered as a ratio between the number resolved needs and the number of associated risks. the number of links between specific tools the addressed need is shown in fig. 2. local registries were considered to be the most important tool allowing increasing awareness and involvement of patients into hta process as well as obtaining data on disease burden and treatment outcomes necessary for local regulatory decisions and also generating data for regional and global cooperation in comprehensive hta area. improving health through healthcare technologies assessment development 93 copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 2. the number of addressed needs to be solved by perspective tools development the relative efficacy of the perspective tools is presented on fig. 3. it indicates that development of local registries can be the most effective step indicating that meeting of 5 needs will cause 3 potential risks (ratio is 1,7) requiring 4 additional activities for conflict resolution. fig. 3. the ratio between the number resolved needs and the number of associated risks (efficacy of the perspective tool development) 94 d. meshkov, o. melien, s. cherkasov, a. lobanov, a. fedyaeva copyright ©0000 assa adv. in systems science and appl. 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(2010) environmental scanning: an application of information literacy skills at the workplace. journal of information science, 36(6), 719–732. adv syst sci appl 2024; 03:1–10 published online at https://ijassa.ipu.ru. on the compatibility of metric and linear order petr klimov1, richik sengupta2,3* 1moscow institute of physics and technology, moscow, russia 2skolkovo institute of science and technology, moscow, russia 3derzhavin tambov state university, tambov, russia abstract: there is a classic question of introducing some reasonable order on the real plane r2. we consider this problem through the prism of the relation between metric proximity and linear order proximity, introducing a way to determine the compatibility relation between metric and linearly ordered spaces, and study the basic properties of this connection. for r and its metric subspaces, the order on the line is compatible with the metric according to the definition we introduce. it turns out that for rn for n > 1, as well as for other similar metric spaces in which there is a subspace isometric to an open ball in rn, it is impossible to introduce an order that is compatible with the metric. at the same time, we also introduce a family of spaces called discrete curved lines, which are not generally isometric to the subspaces of r, but for which it is possible to introduce a linear order compatible with the natural metric on them. for them we prove sufficient conditions for the compatibility of the metric with the order. using them, we construct spaces with non-trivial properties in which the order is compatible with the metric. in particular, we show that although on all of rn no order is compatible with the metric, in any normed space one can introduce non-trivial metric subspaces compatible with some order (in the case of infinite-dimensional normed spaces, this subspace can be constructed to be non-embeddable in any finite-dimensional subspace) keywords: ordered metric spaces, metric spaces, linear order. 1. introduction one of the natural questions one might have is how one can reasonably introduce an order on a real plane r2 in general given that there is a standard order on the real line r. a common way is to introduce lexicographic order, and there are other ways to introduce an order on a plane, but all of these methods lose some of the natural properties that order on a straight line possessed. lexicographic order is very inconsistent with our sense of nearness. according to the lexicographic order, it turns out that (a, b) ≤ (a, b+ ϵ) ≤ (a+ δ, b), as arbitrarily small we do not take δ > 0 and as arbitrarily large we do not take ϵ > 0. this is very inconsistent with our understanding that in some ways the point (a, b) is much closer to (a+ δ, b) than to (a, b+ ϵ). what gives rise to this feeling? this understanding is based on the notion of metric proximity by the natural classical euclidean metric on r2. ∗corresponding author: r.sengupta@skoltech.ru 2 p. klimov, r. sengupta this gives rise to a natural way to impose the requirement to impose compatibility in the definitions of metric proximity [2–4, 6, 8] and linear proximity [1, 5, 10, 11], which we do in our article. it turns out that there are no linear spaces compatible with the classical metric on rn for n > 1 and similar metric spaces with linear orders according to our definition. further we introduce a series of non-trivial metric spaces that are compatible with respect to the order but are not isometric to subsets of the real line. 2. basic definitions let (x, d,≤) be a set equipped with a metric d and a non-strict linear order denoted by ≤. let us introduce what it means for a metric to be compatible with an order. definition 2.1: we will call the order ≤ on x compatible with the metric d on the left if d(x, y) ≤ d(x, z) for all x, y, z ∈ x : x ≤ y ≤ z. definition 2.2: we will call the order ≤ on x compatible with the metric d on the right if d(y, z) ≤ d(x, z) for all x, y, z ∈ x : x ≤ y ≤ z. definition 2.3: we will say that the order of x is compatible with the metric d if it is compatible on the right and left sides. remark 2.1: further, we will say both ”the order is compatible with the metric” and ”the metric is compatible with the order”. in other words, this property connects both of these structures symmetrically. the spaces where this property holds will be called ”order-compatible metric spaces.” in this article, we will focus on studying the compatibility of certain metrics with order in the sense of definition 2.3. definition 2.4: if x is a vector space, we will call the order ≤ on x compatible with the norm || · || : x → r if it is compatible with the metric d(x, y) = ||x− y|| induced by it. remark 2.2: it is easy to see that the metric d in the space x is compatible with the order ≤ if and only if d(x, z) ≥ max{d(y, z), d(x, y)} ∀x, y, z ∈ x : x ≤ y ≤ z. obviously, r is an order-compatible metric space. in addition, there are other examples of order-compatible metric spaces. a trivial example is a discrete metric space with an arbitrary linear order. the following lemma follows from the definition of compatibility. lemma 2.1: in the metric space (x, d), let the metric d be compatible with the order ≤. then the induced metric on any subset of x will be compatible with the induced order. recall that on a linearly ordered set (x,<) with strict order <, we can define an order topology [7] as one with a pre-base of ”open rays” {x | a < x}, {x | x < b}, a, b ∈ x. we might assume that the definition of compatibility between metric spaces and linear orders reduces to an order topology. we will show that this is not the case. remark 2.3: metrizable order topologies and metric spaces in which the metric is compatible with copyright © 2024 assa. adv syst sci appl (2024) on the compatibility of metric and linear order 3 the order do not ”naturally” reduce to each other. it is not difficult to see that we can unambiguously obtain an order topology from an order and an order from an order topology. unlike defining an order topology, an order can be compatible with different metrics, including from the point of view of the generated topology, and this metric can be compatible with different orders. for example, the discrete metric and the euclidean metric on a straight line are compatible with the standard linear order on a straight line, but they generate different topologies. further, a discrete metric on any set is compatible with any order on that set. 3. incompatibility with order of classic multidimensional spaces in this section, we will construct examples of metric spaces where it is impossible to define any linear order that is compatible with the metric. we will demonstrate for a broad range of multidimensional normed spaces that it is not possible to establish any linear order that is compatible with their norm. proposition 3.1: for any metric space with at most three points, you can always create an order that is compatible with the metric. it turns out that this is no longer always true for four points. theorem 3.1: let there be four distinct points x, y, z, t in the metric space m such that min{d(x,z), d(y, t )} ≥ max{d(y, z), d(x, y ), d(x,t ), d(t, z)}. then there does not exist an order that is compatible with the metric. proof let us assume that in m an order-compatible metric can be introduced. since d(x,z) ≥ max{d(y, z), d(x, y )} by the definition of compatibility, we have x ≤ y ≤ z or z ≤ y ≤ x. without loss of generality, we will assume that x ≤ y ≤ z. therefore, since d(x,z) ≥ max{d(x,t ), d(t, z)}, we have that x ≤ t ≤ z. since d(y, t ) ≥ max{d(y,x), d(t,x)}, we have t ≤ x ≤ y or y ≤ x ≤ t. it follows that x = t or x = y. hence, we have obtained a contradiction. this construction can be used further. to do so, we need the following definitions: definition 3.1: let the real numbers r, x and y with r > 0 be given. we will call (r, x, y)1 system a set of points on the plane {x, y, z, t}, x = (r + x, r + y), y = (−r + x, r + y), z = (−r + x,−r + y), x = (r + x,−r + y). we will call (r, x, y)∞ system a set of points on the plane {x, y, z, t}, x = (r + x, y), y = (x, r + y), z = (−r + x, y), t = (x,−r + y). lemma 3.1: on the (r, x, y)k-system, it is impossible to introduce an order compatible with the p-norm induced on the metric plane, for 1) k = 1 and 1 ≤ p < ∞; 2) k = ∞ and 1 < p ≤ ∞. proof the proof follows directly from checking the condition of theorem 3.1. copyright © 2024 assa. adv syst sci appl (2024) 4 p. klimov, r. sengupta x y z t fig. 3.1. example of (r, 0, 0)∞ systems s = {x,y, z, t} on the plane around the origin. it is easy to see that the diameters of the resulting rhombus with vertices from s in the 2 norm are larger than the sides, which shows that the condition of the theorem 3.1 is true. the definition and lemma above allow us to obtain the following useful lemma, which can be used to prove that the metric is incompatible with the order for a number of spaces: lemma 3.2: on an open ball br(x, y) in r2, for all 1 < p ≤ ∞, it is impossible to introduce an order that is compatible with the metric induced from the p-norm. proof it is easy to see that for any x, y ∈ r, r > 0, 1 ≤ p ≤ ∞ we have ( r 3 , x, y)1 ⊂ br(x, y) and ( r 2 , x, y)∞ ⊂ br(x, y). theorem 3.2: there is no metric-compatible order for any metric space that contains a subset that is isometric to an open ball in the plane with p-norm, 1 ≤ p ≤ ∞ . in particular, there is no metric-compatible order in the following spaces: 1. rn with the p norm for n > 1, 1 ≤ p ≤ ∞; 2. ℓp for 1 ≤ p ≤ ∞; 3. lp[a, b] for [a, b] ⊂ r, 1 ≤ p ≤ ∞; 4. lp(a), where a is a measurable space that contains two disjoint subsets a1 and a2 with a finite nonzero measure, 1 ≤ p ≤ ∞; 5. an open ball in any of the spaces listed above. proof from the statement of point (5), the statements of points (1)-(4) follow by lemma 2.1. we prove point (5) for each of these spaces. to do this, we isometrically embed an open ball from r2 in the corresponding space and use the lemma 3.2. 1. let x ∈ rn. we can embed isometrically br(0, 0) into br(x) with the p norm: f(y1, y2) = (x1 + y1, x2 + y2, x3, . . . , xn). 2. let x ∈ ℓp. we can isometrically embed br(0, 0) with the p-norm into br(x) with the ℓp norm: f(y1, y2) = (x1 + y1, x2 + y2, x3, . . .). 3. we can embed r2 isometrically with the p norm into lp[a, b]: copyright © 2024 assa. adv syst sci appl (2024) on the compatibility of metric and linear order 5 f(y1, y2) = y1 p√ b−a 2 i [a,a+ (b−a) 2 ] + y2 p√ b−a 2 i [a+ (b−a) 2 ,b] . we show the isometricity of f. for (y1, y2) and (y′1, y ′ 2) : d((y1, y2), (y ′ 1, y ′ 2)) = p √ |y1 − y′1|p + |y2 − y′2|p = = p √√√√√ b∫ a 2 b− a ( |y1 − y′1|pi[a,a+ (b−a) 2 ] + |y2 − y′2|pi[a+ (b−a) 2 ,b] ) (z)dz = = d(f(y1, y2), f(y ′ 1, y ′ 2)) and f is injective. for any br(x) ⊂ lp[a, b] , we can restrict f to some open ball lying in f−1(br(x)), thus constructing an isometric embedding of the open ball from r2 into br(x). 4. it is enough to repeat the reasoning of point 3 for f(y1, y2) = y1 p√ µ(a1) ia1 + y2 p√ µ(a2) ia2 . 4. constructions of order-compatible metric spaces in the previous section, we provided numerous examples of spaces that are not compatible with any linear order. these spaces were primarily created by naturally embedding certain subdomains of rn into them, meaning they were ”locally multidimensional” and ”closely resembled” a multidimensional real space. at the same time, only degenerate finite metric spaces (consisting of no more than 3 points), discrete metric spaces, and r fit the conditions for compatibility with some order. let us construct an example of a more nontrivial space with such properties obtained by a certain deformation of a countable subset of points on a line. to begin with, we recall the definition of a uniformly discrete set [9]. definition 4.1: if (m,d) is a metric space, and a ⊂ m , then the packing radius r of the set a is defined as: r = 1 2 inf x,y∈a d(x, y). definition 4.2: a set is called uniformly discrete if it has a nonzero covering radius. now we introduce the definition of a discrete curved line, which will be a model example of a set for many constructions in this section. definition 4.3: let (m,d) be a metric space. let d be a countable uniformly discrete subset of r with covering radius r. denote by x the image of an isometric embedding of d into m. let ux ∈ m be chosen for any x ∈ x such that there exists c > 0 such that for all x ∈ x it is true that d(x, ux) ≤ c. let ε = inf{c| ∀x ∈ x d(x, ux) ≤ δ}. we refer to a set {ux}x∈x with a metric induced from m and an order induced from x (meaning that ux1 ≤ ux2 if x1 < x2) as a discrete curved line. alternatively, if we want to highlight the parameters, we can call it a (r, ε)-discrete curved line. let’s try to consider the motivation for such a definition. it comes down to two simple concepts: copyright © 2024 assa. adv syst sci appl (2024) 6 p. klimov, r. sengupta ≥ 2r ε ε ε f u d x u e fig. 4.2. the concept of a discrete curved line: we start by selecting a set of points in d ⊂ r. these points are uniformly distributed and have a covering radius of r, meaning that no two points are closer than 2r. we then embed this set into a new metric space, creating a new space called x . next, we define a subset of u by shifting the elements of x by no more than ε. this allows us to create a curved line structure within the space u . 1. we select a specific subset of points from a straight line (embedded in another metric space). these points are chosen such that they are a sufficient distance apart from each other, for example, using sets like n or z, where all points are at least 1 unit apart. 2. we perform a small shift on each point, relative to the distance between them. since the shift is minimal, we can expect that after the shift, for any three ordered points x ≤ y ≤ z, the distances d′(x, z), d′(y, y′), and d′(y′, z′) will remain larger than the original distances d(x, y), d(y, z), and d(x, z), respectively. this is because all pairwise distances change slightly compared to the significant difference between d(x, z) and the distances d(x, y) and d(y, z). a discrete curved straight line allows us to make any desired shifts relative to the original location of points from the perspective of the geometry of the space where the embedding occurs. now we will prove the general theorem on the necessary and sufficient conditions for the compatibility of the metric and order for discrete curved lines. theorem 4.1: (sufficient condition for the compatibility of the metric with the order for a discrete curved line) let u be an (r, ε)-discrete curved line in the metric space (m,d). if ε ≤ r 2 , then the metric on u is compatible with the order on u . copyright © 2024 assa. adv syst sci appl (2024) on the compatibility of metric and linear order 7 proof according to the definition 4.3 for u , there exists x , which is a uniformly discrete subset of r with a covering radius of r. the set u is defined as {ux}x∈x , where d(x, ux) < ε. consider three points: ux1 ≤ ux2 ≤ ux3 (which implies x1 ≤ x2 ≤ x3) then, d(ux1 , ux2) is less than or equal to the sum of three terms: d(x1, x2), d(ux1 , x1), and d(ux2 , x2). this can be shown using the triangle inequality and the definition of a uniformly discrete line. thus, we have d(ux1 , ux2) ≤ d(x1, x2) + d(ux1 , x1) + d(ux2 , x2) ≤ 2r + 2ε. it is not difficult to see that on x (since x is isometric to a subset of the real line with covering radius r) d(x1, x3) = d(x1, x2) + d(x2, x3) ≥ 4r. similar to the reasoning above, we have: d(x1, x3) ≤ d(ux1 , ux3) + d(ux1 , x1) + d(ux3 , x3) ≤ d(ux1 , ux3) + 2ε from which it follows that d(ux1 , ux3) ≥ d(x1, x3)− 2ε ≥ 4r − 2ε. it is easy to see that if ε ≤ r 2 then d(ux1 , ux2) ≤ 2r + 2ε ≤ 4r − 2ε ≤ d(ux1 , ux3). similarly, for the case where d(ux2 , ux3) ≤ d(ux1 , ux3), the metric is compatible with the order. as we can see, for a small enough value of ε, a discrete curved line’s metric is compatible with its order. this is an example of a space with a non-trivial metric that is compatible with a certain order. on the other hand, if ε is increased relative to r, the order compatibility property may be violated. example 4.1: (left shift of a point of the stretched natural number line) consider the set of natural numbers scaled by 2r as a subset of r with the standard metric. let x = 2rn. move the point 4r to the left by an amount ε > 2r. let u = {ux}x∈x , where ux = { 4r − ε, if x = 4r; x, otherwise. thus, we obtain a (r, ε)-discrete curved line, where the order is not compatible with the metric. generally, when we create various examples of discrete curved lines, we observe that if we increase the parameter ε compared to r, there will be more instances where the metric is incompatible with the order. it would be natural to assume that for large ε, the metric is guaranteed to cease to be compatible with the order. however, we will demonstrate a counterexample. example 4.2: (the set of natural numbers scaled and shifted upwards.) let u = 2rne1 + εe2 = {(2rn, ε)}n∈n ⊂ r2. it is not difficult to see that this set is (r, ε)discrete curved line. however, the order is compatible with the metric, no matter how small or large ε we choose. this is because all metric properties and order properties remain unchanged under this isometric shift. thus, we see the following picture for the (r, ε)-discrete curved line: 1) for a small ε relative to r (it is enough if ε ≤ r 2 according to theorem 4.1 ), the metric is compatible with the order. 2) if we exceed a certain limit (following the example of 4.1, it is enough to have ε > 2r), then we observe discrete curved lines where the metric is not compatible with the order. however, for arbitrarily large ε there are also cases where the metric is compatible with the order, as demonstrated by the example 4.2. copyright © 2024 assa. adv syst sci appl (2024) 8 p. klimov, r. sengupta this leads us to the question: what is the highest possible value of ε that ensures that the order is compatible with the metric? problem 4.1: what is the minimum m(r) such that any (r, ε)-discrete curved line with ε ≤ m(r) is guaranteed to have a compatible metric with the order? according to theorem 4.1 and the 4.1 example, it is true that m(r) is between r 2 and 2r. of course, the exact conditions on ε in relation to r for making the metrics compatible with the order may change depending on the specific metric spaces we consider, their classes, or the methods we use to implement the isometric embedding and select new points. one might think that all types of order-compatible discrete curved lines, in terms of their metric and general properties, are exactly the same as an ordinary lines. this is because they are created by simply shifting certain points on a line. at the same time, we can select discrete curved lines that can significantly differ from a straight line in terms of certain properties. this allows us to explore non-trivial examples of spaces with a metric that is compatible with an order. lemma 4.1: for any infinite-dimensional normed space v over r, there exists a countable subset of u compatible with some order that does not does not belong to any finite-dimensional subspace, meaning that dim span(u) = ∞. proof let us consider a specific subset of the basis vectors: {e1, . . . }. consider an (r, r 2 )-discrete curved line u in the vector space v . let u be a set of points that can be represented as un = 2nre1 + r 2 en, where en is the n-th basis vector. it is easy to see that u is an (r, r 2 )-discrete curved line with a specific structure. it is obtained by shifting up along the nth coordinate of the embedding of natural numbers, which are scaled by 2r. this scaled set is then embedded on the first coordinate axis. by the theorem 4.1, we know that the metric in this curved line is compatible with the order and vectors in u are linearly independent, since for each unique ui in u , there is a corresponding unique ei in the basis expansion. remark 4.1: theorem 4.3 shows that there are nontrivial examples of metric spaces that are not isometric to any subset of the real line, but in which the order is compatible with the metric. theorem 4.1 gives sufficient conditions for the discrete curved line to be compatible with the order. at the same time, it is possible to find examples of spaces where the metric is compatible with the order through specific directed deformations of individual points. sometimes we can use general conditions for the entire line, but in general it is preferred to have a more precise method for evaluating the allowed changes at a specific point. theorem 4.2: (pointwise sufficient condition for compatibility of the metric with an order for a discrete curved line) let us consider a discrete curved line u that is obtained from a set x . we can assign numbers to the elements of x according to their order on the real number line using integers. if there is an upper or lower bound, we can use positive or negative integers accordingly. let d(uxi , xi) = εi. then, u can be considered a order-compatible metric space if the following condition holds for all i: εi ≤ min j<i, k>i max (2r(k − i)− 2εj − εk, 2r(i− j)− 2εk − εj). proof let us take an arbitrary point xi and consider arbitrary points xj , xk, xj ≤ xi ≤ xk. copyright © 2024 assa. adv syst sci appl (2024) on the compatibility of metric and linear order 9 then d(uxi , uxj ) ≤ d(xi, xj) + d(uxi , xi) + d(uxj , xj) ≤ 2(i− j)r + εi + εj (since on the straight line between xi and xj there is exactly j − i times 2r ). similarly d(uxj , uxk ) ≤ 2(k − i)r + εi + εk. similar to the above d(xj, xk) ≤ d(uxj , uxk ) + d(uxj , xj) + d(uxk , xk) ≤ d(uxj , uxk ) + εj + εk which implies that d(uxj , uxk ) ≥ d(xj, xk)− εj − εk ≥ (k − j)2r − εj − εk. then if (k − j)2r − εj − εk ≥ max (2(i− j)r + εi + εj, 2(k − i)r + εi + εk), then if εi ≤ max (2r(k − i)− 2εj − εk, 2r(i− j)− 2εk − εj) our metric will be compatible with the order for the triple uxj , uxi , uxk . if we select an arbitrary value of i and then choose any two other values, j and k, where j is less than i and k is greater than i, and keep in mind that εi is bounded from above, we obtain the conditions of the lemma. all the examples of order-compatible discrete curved lines in normed spaces had the property that when we projected them onto the coordinate axis passing through their starting points x , the order of the points was the same as it was on the original curved line itself. in other words, the points were simply ordered based on their coordinate along the original straight line. fig. 4.3. an example of a discrete curved line where the metric is compatible with the order. since we make a small shift, the projection preserves the order on the original set. let us show that this is not always the case. example 4.3: consider the set of natural numbers x = ne1 embedded in the plane with the euclidean metric. then the set u = {ux}x∈x , where ux = { (0.5, 1.5), if x = (2, 0); x, otherwise will be a discrete curved line with the metric compatible with the order, but the projection of u on the x axis (on which x lies) has an order that does not coincide with the order on this axis as a straight line. proof to check compatibility with the order, note that the only modified point is u(2,0). it is easy to see that if we take the points ux1 = (1, 0), ux2 = (0.5, 1.5), ux3 = (3, 0), the compatibility is satisfied (checked by calculating distances). the remaining cases follow from theorem 4.1 and the fact that εi = 0 for all i ̸= 2. 5. acknowledgements the research of the second author was supported by the russian science foundation (project no. 23-11-20020, https://rscf.ru/en/project/23-11-20020/). copyright © 2024 assa. adv syst sci appl (2024) 10 p. klimov, r. sengupta references 1. arutyunov, a. v., zhukovskiy, e. s., & zhukovskiy, 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(2020) some questions of the analysis of mappings of metric and partially ordered spaces, russian universities reports. mathematics, 25(132), 345–358. doi: 10.20310/2686-9667-2020-25-132-345358. copyright © 2024 assa. adv syst sci appl (2024) introduction basic definitions incompatibility with order of classic multidimensional spaces constructions of order-compatible metric spaces acknowledgements microsoft word khachatryan_2023_new adv syst sci appl 2024; 01; 82-94 published online at https://ijassa.ipu.ru. synchronization of inbound and outbound flows at stations in the model of freight transportation organization nerses khachatryan* state academic university for the humanities, moscow, russia abstract: this article presents a model for organizing railway freight transportation between two node stations based on the interaction of neighboring stations depending on their technical capabilities and the demand for freight transportation. the main objective of such interaction is to synchronize the inbound and outbound flow at the stations, specifically reducing the degree of inconsistency between the receipt and dispatch of goods at the stations. this characteristic represents the imbalance between the volume of incoming and outgoing goods at the stations per unit of time and is described by a nonnegative function bounded above by unity. its dynamics are described by a system of differential equations containing a set of parameters characterizing the infrastructure of the stations, the mode of distribution of goods from the final node station, and the demand for freight transportation. the range of parameter variations for which the specified system has a solution has been determined, as well as such a set of these parameters for which the task of synchronizing the inbound and outbound flow at the stations is best solved. it has been found that regardless of the initial value, starting from a certain point in time, the degree of inconsistency between the receipt and dispatch of goods at all stations except the initial node station becomes zero. keywords: freight transportation organization, flow synchronization, system of differential equations, model parameters, stationary solutions, stability of solutions. 1. introduction one of the largest foundational sectors of any country's economy is transportation. it provides geographical connectivity across the nation's territories and coordinates the functioning of all sectors of the economy. transportation creates the conditions for the effective operation of a country, and its development is a vital component of economic modernization. additionally, transportation contributes to the development of international economic relations, the exploration of new economic regions, and ensures the country's defense capabilities. two major groups of mathematical models for transportation systems can be distinguished: 1. the first group focuses on modeling transportation networks and their utilization. it includes models for calculating correspondences, such as the gravity model [63], entropy model [28, 52], models within the competing centers framework [21], as well as models for flow distribution within the network [57, 47, 3]. 2. the second group involves modeling the dynamics of transportation flow [56]. it comprises fundamental classes of dynamic models: macroscopic (hydrodynamic), kinetic (gas dynamic), and microscopic models. macroscopic models [17, 43, 34, 24] describe averaged characteristics of transportation flow and are sometimes referred to as hydrodynamic models because they liken the flow itself to the movement of compressible fluid. macroscopic diagrams, which illustrate the relationships between performance parameters such as traffic density, traffic flow, and vehicle speed, are used to represent traffic states and system configurations [16, 26, 12]. microscopic * corresponding author: nerses-khachatryan@yandex.ru synchronization of inbound and outbound flows at stations in the model… 83 copyright ©2024 assa adv. in systems science and appl. (2024) models explicitly describe the movement of each individual vehicle. they provide a more detailed description of movement on specific segments of the transportation network but require significantly greater computational resources for practical implementation. the first microscopic models were introduced in the 1950s [51, 56]. examples of such models include car-following models [25, 7], optimal velocity models [2, 59], the treiber model [60], as well as cellular automaton models [15, 14]. kinetic models occupy an intermediate position between macroscopic and microscopic models. in kinetic models, traffic flow is characterized by the density distribution of vehicles in phase space, and the dynamics of the phase density are described by kinetic equations. these equations result from averaging the effects of interactions among individual vehicles [30, 50]. it is worth noting that the models mentioned above are most suitable for studying automobile traffic. for a vast country like russia, however, railway transportation plays a pivotal role. it ensures reliable and cost-effective delivery of goods, especially when rapid transportation of large volumes of cargo is required. publications related to railway logistics can be categorized into three main groups, depending on the types of problems under investigation. the first group deals with the design of railway infrastructure [32, 42, 20]. the second group focuses on the management of locomotives and wagons. depending on regulatory and market characteristics, different regions may have their own models to account for specific factors. for instance, the work of r. fukasawa et al. [22] presents a model used by one of the largest railway operators in latin america. another example is the study by a. chezelli et al. [13], which examines multiple optimization models for freight delivery by swiss federal railways' cargo express service. several publications focus on models designed to address the specifics of the freight transport market in italy [49, 11]. some works present cost-minimization models for transporting goods across multiple european countries via railway [1, 33]. there are also models created for the russian railway transport market [54, 45, 6]. the third group addresses railway planning tasks, which traditionally centered on scheduling freight train movements [44, 10, 46]. in recent years, publications in this group have been complemented by studies that apply macroscopic traffic theory to describe processes in railway transportation. n. weik's work [61], for example, provides a theoretical analysis of traffic flow properties on unidirectional railway lines. it constructs macroscopic fundamental diagrams and demonstrates how they can be used to determine flow regimes and various phases of train movement, which is valuable for system design and operational planning. another area of research that has been actively developing in recent years is related to predicting delays in railway systems. trains in this system follow predefined schedules, which allows for the efficient use of routes and tracks. deviations from such planned operations manifest as delays and can reduce the system's efficiency. minor delays are often absorbed by built-in buffers and thus do not impact larger scales [63, 18]. however, logistical disruptions, often caused by external factors such as weather, occasionally lead to congestion or even large-scale stoppages with negative consequences for society and the economy [48, 9]. most models that study delays are based on railway system schedules and typically use trains as agents that can incur delays [27, 23, 29]. in contrast, in the work by dekker et al. [19], delays are treated as variables associated not with trains but with nodes (stations) and edges of the railway network, which remain in place. the spread of delays between these nodes does not necessarily have to be described in terms of discrete trains and events but can be based solely on common (or even systemic) quantities, such as network topology and schedules. the authors draw an analogy with hydrodynamics: while delays are traditionally considered lagrangian particles (i.e., following trains like a fluid carrying particles), they suggest considering delays from an eulerian perspective (i.e., defining incoming and outgoing delays in a fixed spatial system). they refer to this representation of delays as diffusion-like spreading. when examining the microscale, it is expected that this unconventional approach to delay handling may be less accurate than more detailed models, but on a large scale, the performance of such a model increases. the model contains only basic schedule information 84 n. khachatryan copyright ©2024 assa adv. in systems science and appl. (2024) (e.g., train frequencies and travel times), and all the model's information is embedded in a single matrix, facilitating the analysis of system properties. another crucial issue in railway planning is the study of freight transportation modes and their corresponding cargo flows within a dynamic system that describes the transportation process as the interaction of key elements of railway infrastructure, primarily stations. this problem has been addressed in the works of l.a. beklaryan and n.k. khachatryan [4, 5, 35–41]. they present dynamic models in which the organization of freight transportation involves forming cargo flows based on station interactions. the rules of station interaction depend on the nature of demand for freight transportation. in cases of consistently high demand for freight transportation, the focus is on utilizing the station's technical capabilities to their fullest. in the absence of consistently high demand for freight transportation, the primary goal of station interaction is to synchronize incoming and outgoing flows, allowing for more efficient freight transportation, minimizing delays, and ensuring a smooth flow of cargo. the work [41] describes a model of organizing freight transportation between major node stations when there is no consistently high demand for freight transportation. it explores the relationship between the degree of mismatch between cargo arrival and departure at stations and the parameters that characterize the demand for freight transportation, the technical capacity of stations, and the degree of its utilization. it assumes that the stations are identical, meaning that the specified parameters are the same for all stations. this article is dedicated to advancing this model and examines the case where stations have different characteristics. 2. problem statement the movement of cargo on a segment of the railway network between two terminal stations, connected by multiple intermediate stations, is being considered. the primary characteristic of station 𝑖 at time 𝑡, where 𝑖 ∈ {0, 1, . . , 𝑚, 𝑚 + 1}, is the degree of mismatch between cargo reception and dispatch 𝑧 (𝑡), which varies in the range [0, 1]. the precise definition is provided in the work [41]. the technical capacity of station 𝑖 is determined by the maximum permissible increase in the degree of mismatch between cargo reception and dispatch per unit of time and is defined by a nonnegative decreasing function 𝜑 (𝑧) defined on the interval [0, 1], satisfying the condition 𝜑 (1) = 0. the initial node station (𝑖 = 0) receives cargo based on the demand for transportation within its technical capacity and dispatches it to the next station within its technical capacity. each of the intermediate stations (𝑖 = 1, 2, … 𝑚) receives cargo within its technical capacity and dispatches it within the technical capacity of the next station. the final node station (𝑖 = 𝑚 + 1) receives cargo within its technical capacity and distributes it according to a specific regime. the dynamics of the degrees of mismatch between cargo reception and dispatch at the stations are described by the following system of differential equations �̇� (𝑡) = min 𝑑 , 𝜑 𝑧 (𝑡) − 𝜆 𝜑 𝑧 (𝑡) , 𝑡 ∈ [𝑡 , +∞); (2.1) �̇� (𝑡) = 𝜆 𝜑 𝑧 (𝑡) − 𝜆 𝜑 𝑧 (𝑡) , 𝑖 = 1, 2, … , 𝑚, 𝑡 ∈ [𝑡 , +∞); (2.2) �̇� (𝑡) = 𝜆 𝜑 𝑧 (𝑡) − 𝑑 , 𝑡 ∈ [𝑡 , +∞); (2.3) 0 ≤ 𝑧 (𝑡) ≤ 1 , 𝑖 = 0, 1, … , 𝑚 + 1, 𝑡 ∈ [𝑡 , +∞). (2.4) here 𝑑 > 0, 0 < 𝜆 ≤ 1, 𝑑 > 0 are model parameters: 𝑑 is a characteristic of the demand for transportation; 𝜆 is a characteristic of the degree of utilization of the technical potential of station number 𝑖; 𝑑 is a characteristic of the distribution mode of goods from the final node station. next, consider the function 𝜑 (𝑧), that defines the technical potential of station number 𝑖, of the following form synchronization of inbound and outbound flows at stations in the model… 85 copyright ©2024 assa adv. in systems science and appl. (2024) 𝜑 (𝑧) = 𝑎 (1 − 𝑧), 𝑎 > 0. (2.5) the parameter 𝑎 > 0, which is involved in defining the function 𝜑 (𝑧), represents the capability characteristic of station number 𝑖 in increasing the flow of goods. parameter 𝑑 , which is a characteristic of the demand for transportation and is involved in equation (1), is represented as follows: 𝑑 = 𝜇𝑎 , 0 < 𝜇 ≤ 1. (2.6) let's rewrite the system (2.1)–(2.4), where the function 𝜑 (𝑧) is defined according to (2.5), and the parameter 𝑑 is defined according to (2.6). �̇� (𝑡) = min 𝜇𝑎 , 𝑎 1 − 𝑧 (𝑡) − 𝜆 𝑎 1 − 𝑧 (𝑡) , 𝑡 ∈ [𝑡 , +∞), (2.7) �̇� (𝑡) = 𝜆 𝑎 1 − 𝑧 (𝑡) − 𝜆 𝑎 1 − 𝑧 (𝑡) , 𝑖 = 1, 2, … , 𝑚, 𝑡 ∈ [𝑡 , +∞), (2.8) �̇� (𝑡) = 𝜆 𝑎 1 − 𝑧 (𝑡) − 𝑑 , 𝑡 ∈ [𝑡 , +∞), (2.9) 0 ≤ 𝑧 (𝑡) ≤ 1 , 𝑖 = 0, 1, … , 𝑚 + 1, 𝑡 ∈ [𝑡 , +∞). (2.10) here 𝜇, 𝑎 , 𝜆 , 𝑑 are model parameters: 𝜇 (0 < 𝜇 ≤ 1) is a characteristic of the demand range for transportation that can be satisfied with the existing technical capacity of the stations; 𝑎 (𝑎 > 0) is a characteristic of the ability of station number 𝑖 to increase freight flow 𝑖; 𝜆 (0 < 𝜆 ≤ 1) is a characteristic of the degree of utilization of the technical capacity of station number 𝑖; 𝑑 (𝑑 > 0) is a characteristic of the distribution mode of goods from the final node station. let's outline the main research tasks: determine the ranges of variation of the parameters 𝜇, 𝑎 , 𝜆 , 𝑑 , for which the cargo transportation system can operate smoothly, i.e., the system (2.7)–(2.10) has a solution. for a given value of the demand characteristic for cargo transportation (parameter 𝜇) establish the most acceptable achievable levels of inconsistency between receiving and sending goods at all stations, by managing the values of the following characteristics: the capabilities of stations to increase freight flow (parameter 𝑎 ), the degree of utilization of the technical capacity of stations (parameter 𝜆 ) and the mode of distribution of goods from the final node station (parameter 𝑑 ). 3. investigation of solutions of the system (2.7)–(2.10) studying the solution set of the system (2.7)–(2.10), let's begin by investigating all the solutions of the system of differential equations (2.7)–(2.9). first and foremost, let's identify the stationary solutions of the system (2.7)–(2.9). by direct examination, we can verify the validity of the following statement. statement 3.1: the system (2.7)–(2.9) for any values of the parameters 0 < 𝜇 ≤ 1, 𝑎 > 0, 0 < 𝜆 ≤ 1, 𝑖 = 0, 1, . . . , 𝑚 + 1, 𝑑 > 0 such that 𝑑 ≤ 𝜇𝑎 has stationary solutions: 𝑧 (⋅) ≡ 1 − 𝑑 𝑎 , 𝑧 (⋅) ≡ 1 − 𝑑 𝜆 𝑎 , 𝑖 = 1, … , 𝑚 + 1, 𝑤ℎ𝑒𝑛 𝑑 < 𝜇𝑎 (3.1) 𝑧 (⋅) ≤ 1 − 𝜇, 𝑧 (⋅) ≡ 1 − 𝑑 𝜆 𝑎 , 𝑖 = 1, … , 𝑚 + 1, 𝑤ℎ𝑒𝑛 𝑑 = 𝜇𝑎 (3.2) 86 n. khachatryan copyright ©2024 assa adv. in systems science and appl. (2024) for 𝑑𝑚+1 > 𝜇𝑎0 the system (2.7)–(2.9) has no stationary solutions. let's proceed to investigate the remaining solutions of the system (2.7)–(2.9). theorem 3.1: any solution of the system (2.7)–(2.9) with 𝑑 < 𝜇𝑎 eventually converges to a stationary trajectory (3.1), and when 𝑑 = 𝜇𝑎 it converges to one of the stationary trajectories (3.2). for 𝑑 > 𝜇𝑎 the coordinates 𝑧 (⋅), 𝑖 = 1, . . . , 𝑚 + 1 of the solution of the system (2.7)–(2.9) eventually converge to the stationary mode described in (3.1), while the function 𝑧 (⋅) decreases linearly. proof. let's find the general solution of the system (2.7)–(2.9). we will start with the last equation, rewritten as follows �̇� (𝑡) + 𝜆 𝑎 𝑧 (𝑡) = 𝜆 𝑎 − 𝑑 . (3.3) the linear equation (3.3) has the following general solution 𝑧 (𝑡) = 1 − 𝑑 𝜆 𝑎 + 𝑐 𝑒 . (3.4) using this, let's find the solution to the penultimate equation of the system (2.7)–(2.9). it is easy to verify that if 𝜆 𝑎 ≠ 𝜆 𝑎 , then it has the following form 𝑧 (𝑡) = 1 − 𝑑 𝜆 𝑎 + 𝜆 𝑎 𝑐 𝜆 𝑎 − 𝜆 𝑎 𝑒 + 𝑐 𝑒 . (3.5) otherwise (𝜆 𝑎 = 𝜆 𝑎 ) the solution will be as follows 𝑧 (𝑡) = 1 − 𝑑 𝜆 𝑎 + 𝜆 𝑎 𝑐 𝑡𝑒 + 𝑐 𝑒 . (3.6) similarly, we can find solutions to all the other equations of the system (2.7)–(2.9), except for the initial one. if all 𝜆 𝑎 , 𝑖 = 1, 2, … , 𝑚 + 1 are pairwise distinct, then we obtain the following solutions: 𝑧 (𝑡) = 1 − 𝑑 𝜆 𝑎 + 𝜆 𝑎 𝜆 𝑎 𝑐 (𝜆 𝑎 − 𝜆 𝑎 )(𝜆 𝑎 − 𝜆 𝑎 ) 𝑒 + 𝜆 𝑎 𝑐 𝜆 𝑎 − 𝜆 𝑎 𝑒 + 𝑐 𝑒 ; … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … synchronization of inbound and outbound flows at stations in the model… 87 copyright ©2024 assa adv. in systems science and appl. (2024) 𝑧 (𝑡) = 1 − 𝑑 𝜆 𝑎 + 𝜆 𝑎 𝜆 𝑎 … 𝜆 𝑎 𝑐 𝑒 (𝜆 𝑎 − 𝜆 𝑎 )(𝜆 𝑎 − 𝜆 𝑎 ) … (𝜆 𝑎 − 𝜆 𝑎 ) + 𝜆 𝑎 𝜆 𝑎 … 𝜆 𝑎 𝑐 𝑒 (𝜆 𝑎 − 𝜆 𝑎 )(𝜆 𝑎 − 𝜆 𝑎 ) … (𝜆 𝑎 − 𝜆 𝑎 ) + 𝜆 𝑎 𝜆 𝑎 … 𝜆 𝑎 𝑐 𝑒 (𝜆 𝑎 − 𝜆 𝑎 )(𝜆 𝑎 − 𝜆 𝑎 ) … (𝜆 𝑎 − 𝜆 𝑎 ) + … + 𝜆 𝑎 𝑐 𝑒 (𝜆 𝑎 − 𝜆 𝑎 ) + 𝑐 𝑒 ; (3.7) … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 𝑧1(𝑡) = 1 − 𝑑𝑚+1 𝜆1𝑎1 + 𝜆2𝑎2𝜆3𝑎3 … 𝜆𝑚+1𝑎𝑚+1𝑐𝑚+1𝑒−𝜆𝑚+1𝑎𝑚+1𝑡 (𝜆𝑚𝑎𝑚 − 𝜆𝑚+1𝑎𝑚+1)(𝜆𝑚−1𝑎𝑚−1 − 𝜆𝑚+1𝑎𝑚+1) … (𝜆1𝑎1 − 𝜆𝑚+1𝑎𝑚+1) + 𝜆2𝑎2𝜆3𝑎3 … 𝜆𝑚𝑎𝑚𝑐𝑚𝑒−𝜆𝑚𝑎𝑚𝑡 (𝜆𝑚−1𝑎𝑚−1 − 𝜆𝑚𝑎𝑚)(𝜆𝑚−2𝑎𝑚−2 − 𝜆𝑚𝑎𝑚) … (𝜆1𝑎1 − 𝜆𝑚𝑎𝑚) + 𝜆2𝑎2𝜆3𝑎3 … 𝜆𝑚−1𝑎𝑚−1𝑐𝑚−1𝑒−𝜆𝑚−1𝑎𝑚−1𝑡 (𝜆𝑚−2𝑎𝑚−2 − 𝜆𝑚−1𝑎𝑚−1)(𝜆𝑚−3𝑎𝑚−3 − 𝜆𝑚−1𝑎𝑚−1) … (𝜆1𝑎1 − 𝜆𝑚−1𝑎𝑚−1) + … + 𝜆2𝑎2𝑐2𝑒−𝜆2𝑎2𝑡 (𝜆1𝑎1 − 𝜆2𝑎2) + 𝑐1𝑒−𝜆1𝑎1𝑡. if 𝜆 𝑎 = 𝜆 𝑎 = ⋯ 𝜆 𝑎 , then the solutions take the following form: 𝑧 (𝑡) = 1 − 𝑑 𝜆 𝑎 + 𝑒 𝜆 𝑎 𝑐 2 𝑡 + 𝑐 𝑡 + 𝑐 , … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 𝑧 (𝑡) = 1 − 𝑑 𝜆 𝑎 + 𝑒 𝜆 𝑎 𝑐 (𝑘 + 1)! 𝑡 + 𝑐 𝑡 + ⋯ + 𝑐 𝑡 + 𝑐 , … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 𝑧 (𝑡) = 1 − 𝑑 𝜆 𝑎 + 𝑒 𝜆 𝑎 𝑐 𝑚! 𝑡 + 𝑐 𝑡 + ⋯ + 𝑐 𝑡 + 𝑐 . (3.8) in all other cases, the solutions have the following form 𝑧 (𝑡) = 1 − 𝑑 𝜆 𝑎 + 𝐺 (𝑡), (3.9) where 𝐺 (𝑡) is the sum of functions of the form 𝑏 𝑒 and 𝑐 𝑒 𝑡𝑘, 𝑝 = 𝑖, … , 𝑚 + 1; 𝑗 = 𝑖, … , 𝑚; 𝑘 = 1, … , 𝑚 − 𝑖 + 1 (3.10) from (3.4)–(3.10), it follows that lim 𝑡→+∞ 𝑧𝑖(𝑡) = 1 − 𝑑𝑚+1 𝜆𝑖𝑎𝑖 , 𝑖 = 1, 2, … , 𝑚. we now need to solve the first equation of the system (2.7)–(2.9). to do this, let's rewrite it in the following form: 88 n. khachatryan copyright ©2024 assa adv. in systems science and appl. (2024) �̇� (𝑡) = 𝜇𝑎 − 𝜆 𝑎 1 − 𝑧 (𝑡) , if 𝑧 (𝑡) < 1 − 𝜇, 𝑡 ∈ [𝑡 , +∞), 𝑎 1 − 𝑧 (𝑡) − 𝜆 𝑎 1 − 𝑧 (𝑡) , if 𝑧 (𝑡) ≥ 1 − 𝜇 , 𝑡 ∈ [𝑡 , + ∞). (3.11) let's consider the following two equations �̇� (𝑡) = 𝜇𝑎 − 𝜆 𝑎 1 − 𝑧 (𝑡) , 𝑡 ∈ 𝑡, +∞ , (3.12) �̇�0(𝑡) = 𝑎0 1 − 𝑧0(𝑡) − 𝜆1𝑎1 1 − 𝑧1(𝑡) , 𝑡 ∈ 𝑡, +∞ , (3.13) where 𝑡 ≥ 𝑡0. using the expressions for 𝑧1(𝑡), we obtain the solutions to equations (3.12) and (3.13). they can be represented as follows 𝑧 (𝑡) = (𝜇𝑎 − 𝑑 )𝑡 + 𝐹 (𝑡) + 𝑐 where 𝐹 (𝑡) ∈ с 𝑡, +∞ , lim → 𝐹 (𝑡) = 0, (3.14) 𝑧 (𝑡) = 1 − 𝑑 𝑎 + 𝐹 (𝑡) where 𝐹 (𝑡) ∈ с 𝑡, +∞ , lim → 𝐹 (𝑡) = 0. (3.15) using these solutions, let's investigate the asymptotic behavior of the solution to equation (3.11). when 𝑑 < 𝜇𝑎 the asymptotics of the solution to equation (3.11) is determined by relation (3.15), i.e., lim → 𝑧 (𝑡) = 1 − . when 𝑑 > 𝜇𝑎 the asymptotics of the solution to equation (3.11) is determined by relation (3.14), i.e., 𝑧 (⋅) decreases linearly with lim → 𝑧 (𝑡) = − ∞. however, if 𝑑 = 𝜇𝑎 then the asymptotics of the solution to equation (3.11) may be determined by either relation (3.14) or relation (3.15), depending on the initial conditions. in other words, either lim → 𝑧 (𝑡) = с , where с < 1 − 𝜇 or lim → 𝑧 (𝑡) = 1 − . ∎ 4. investigation of solutions of the system (2.7)–(2.10) let's proceed to study the solutions of the system (2.7)–(2.9) that satisfy the constraints (2.10). lemma 4.1: for all parameter values 𝑎 > 0, 0 < 𝜆 ≤ 1, 𝑖 = 1, 2, … , 𝑚 + 1, 𝑑 > 0 satisfying the condition 𝜆 𝑎 ≥ 𝜆 𝑎 ≥ ⋯ 𝜆 𝑎 ≥ 𝜆 𝑎 ≥ 𝑑 the components 𝑧 (⋅), 𝑧 (⋅), … 𝑧 (⋅) of any solution of the system (2.7)–(2.10) that satisfy the constraints (2.10) at the initial moment in time will continue to satisfy them at subsequent moments in time. proof. let's begin by examining the last component of the solution to the system (2.7)–(2.9), namely 𝑧 (⋅). it takes the form (3.4), where с is determined by the condition 1 − 𝑑 𝜆 𝑎 + 𝑐 𝑒 = 𝑧 , where 0 ≤ 𝑧 ≤ 1, i.e. 𝑐 = 𝑑 𝜆 𝑎 − 1 + 𝑧 𝑒 , where 0 ≤ 𝑧 ≤ 1 (4.1) synchronization of inbound and outbound flows at stations in the model… 89 copyright ©2024 assa adv. in systems science and appl. (2024) from (4.1), it follows that 𝑑 𝜆 𝑎 − 1 𝑒 ≤ 𝑐 ≤ 𝑑 𝜆 𝑎 𝑒 . using this estimate for с and the expression (3.4), we obtain an estimate for 𝑧 (⋅). it will take the following form 1 − 𝑑 𝜆 𝑎 − 1 − 𝑑 𝜆 𝑎 𝑒 ( ) ≤ 𝑧 (𝑡) ≤ 1 − 𝑑 𝜆 𝑎 1 − 𝑒 ( ) (4.2) from (4.2), it follows that when the condition 𝑑 ≤ 𝜆 𝑎 is met, the inequality holds 0 ≤ 𝑧 (𝑡) ≤ 1, 𝑡 ∈ [𝑡 , +∞). (4.3) let's demonstrate that similar inequalities to (4.3) hold for the other components of the solution to the system (2.7)–(2.9). we'll start with the component 𝑧 (⋅). to do this, consider the equation (2.8) for 𝑖 = 𝑚: �̇� (𝑡) = 𝜆𝑚𝑎𝑚 1 − 𝑧𝑚(𝑡) − 𝜆𝑚+1𝑎𝑚+1 1 − 𝑧𝑚+1(𝑡) , 𝑡 ∈ [𝑡0, +∞). let's demonstrate that the function 𝑧 (⋅) cannot take a value greater than 1. indeed, otherwise, due to the continuity of the function 𝑧 (⋅) there must exist a point 𝑡∗ > 𝑡 , such that 𝑧 ( 𝑡∗) = 1. in such a case �̇�𝑚( 𝑡∗) = −𝜆𝑚+1𝑎𝑚+1 1 − 𝑧𝑚+1( 𝑡∗) and due to inequality (4.3), it follows that �̇� ( 𝑡∗) ≤ 0, i.e. 𝑧 (⋅) is upper-bounded by one. let's now show that the function 𝑧 (⋅) cannot take a value less than 0. indeed, otherwise, due to the continuity of the function 𝑧 (⋅) there must exist a poin 𝑡∗∗ > 𝑡 , such that 𝑧 ( 𝑡∗∗) = 0. then �̇�𝑚( 𝑡∗∗) = 𝜆𝑚𝑎𝑚 − 𝜆𝑚+1𝑎𝑚+1 1 − 𝑧𝑚+1( 𝑡∗∗) and when the condition 𝑑 ≤ 𝜆 𝑎 ≤ 𝜆 𝑎 is satisfied, the inequality �̇� ( 𝑡∗∗) ≥ 0, holds, i.e., 𝑧 (⋅) is lower-bounded by zero. similarly, the validity of all the other inequalities (2.10) can be proven, except for the first one (related to the function 𝑧 (⋅)).∎ let's formulate a similar lemma for the zeroth component of the solution to the system (2.7)– (2.9). lemma 4.2: for all parameter values 0 < 𝜇 ≤ 1, 𝑎 > 0, 𝑖 = 0, 1, … 𝑚 + 1, satisfying the condition 𝑎 ≤ 𝑎 , there exists 𝜆 (𝜇, 𝑎 , 𝑎 , 𝑧 (𝑡 ), 𝑧 (𝑡 ), . . . , 𝑧 (𝑡 )), 𝜇 ≤ 𝜆 ≤ 1 such that for all 𝜆 in the interval 𝜇, 𝜆 , 0 < 𝜆 ≤ 1, 𝑖 = 2, 3, … , 𝑚 + 1, 𝑑 > 0, satisfying the condition 𝜆 𝑎 ≥ 𝜆 𝑎 ≥ ⋯ 𝜆 𝑎 ≥ 𝜆 𝑎 ≥ 𝑑 the zeroth component 𝑧 (⋅) of any solution to the system (2.7)–(2.9) that satisfies the constraint (2.10) at the initial moment in time will continue to satisfy it at subsequent moments in time. proof. similarly to the other components, let's show that the function 𝑧 (⋅) cannot take a value greater than 1. indeed, otherwise, due to the continuity of the function 𝑧 (⋅) there must exist a point �̃� > 𝑡 , such that 𝑧 (�̃�) = 1. then, from (2.7), it follows that �̇�0(𝑡) = 𝜆1𝑎1(𝑧1(𝑡) − 1), 90 n. khachatryan copyright ©2024 assa adv. in systems science and appl. (2024) i.e., according to lemma 1, �̇� (�̃�) ≤ 0. this means that the function 𝑧 (⋅) is upper-bounded by one. now, let's estimate the function 𝑧 (⋅) from below. to do this, we'll investigate the behavior of its derivative as 𝑧 (⋅) → 0 +. according to (2.7), it is described by the equation �̇�0(𝑡) = 𝜇𝑎0 − 𝜆1𝑎1 1 − 𝑧1(𝑡) . let's analyze the inequality 𝜇𝑎 − 𝜆 𝑎 (1 − 𝑧 (𝑡)) ≥ 0. we will show that for any 0 < 𝜇 ≤ 1, 𝑡 ∈ [𝑡 , +∞) there exists a range of 𝜆 values on the half-open interval (0, 1], for which this inequality holds. to do this, rewrite it as 𝑧1(𝑡) ≥ 1 − 𝜇𝑎0 𝜆1𝑎1 . (4.4) according to lemma 4.1, 0 ≤ 𝑧 (𝑡) ≤ 1, 𝑡 ∈ [𝑡 , +∞). (4.5) from (4.5), it follows that for any 𝜇, satisfying 0 < 𝜇 ≤ 1, and for any 𝑎 , 𝑎 , satisfying 𝑎 ≤ 𝑎 there exists 𝜆 , 𝜇 ≤ 𝜆 ≤ 1 such that for any value of 𝜆 from the interval 𝜇, 𝜆 the inequality (4.4) will hold for all 𝑡 ∈ [𝑡 , +∞), i.e. �̇� (𝑡) ≥ 0 as 𝑧 (𝑡) → 0 +. this demonstrates the lower bound of the function 𝑧 (⋅) as 0. it is evident that 𝜆 depends on the parameters 𝜇, 𝑎 , 𝑎 , as well as the initial conditions, so we denote it as 𝜆 (𝜇, 𝑎 , 𝑎 , 𝑧 (𝑡 ), 𝑧 (𝑡 ), . . . , 𝑧 (𝑡 )). ∎ let's state the main result of this study. theorem 4.1: for any initial values 0 ≤ 𝑧 (𝑡 ) ≤ 1, parameters 0 < 𝜇 ≤ 1, 𝑎 > 0, 𝑖 = 0, 1, . . . , 𝑚 + 1, satisfying the condition 𝑎 ≤ 𝑎 , there exists 𝜆 (𝜇, 𝑎 , 𝑎 , 𝑧 (𝑡 ), 𝑧 (𝑡 ), . . . , 𝑧 (𝑡 )), 𝜇 ≤ 𝜆 ≤ 1 such that for all 𝜆 in the interval 𝜇, 𝜆 , 0 < 𝜆 ≤ 1, 𝑖 = 2, 3, … , 𝑚 + 1, 𝑑 > 0, satisfying the condition 𝜆 𝑎 ≥ 𝜆 𝑎 ≥ ⋯ 𝜆 𝑎 ≥ 𝜆 𝑎 ≥ 𝑑 the solution to the system (2.7)–(2.10) exists and converges to either the stationary solution (3.1) (when 𝑑 < 𝜇𝑎 ) or to one of the stationary solutions (3.2), which is the same for all 𝑑 and 𝑎 (when 𝑑 = 𝜇𝑎 ). proof. the proof directly follows from theorem 3.1, lemma 4.1, and lemma 4.2. ∎ corollary 1: the system of differential equations (2.7)–(2.10) has a globally stable stationary solution (3.1) and a family of stable solutions of the form (3.2). proof. the proof directly follows from theorem 4.1. ∎ corollary 2: for any initial values 0 ≤ 𝑧 (𝑡 ) ≤ 1, parameters 𝑎 > 0 , 𝑖 = 0, 1, . . . , 𝑚 + 1, satisfying the condition 𝑎 ≥ 𝑎 , 0 < 𝜇 ≤ 1, �̅� = 𝜇, 𝑖 = 1, . . . , 𝑚 + 1 and 𝑑 = 𝜇𝑎 the solution to the system (2.7)–(2.10) exists and converges to one of the stationary solutions of the form 𝑧 (⋅) ≤ 1 − 𝜇, 𝑧 (⋅) ≡ 0, 𝑖 = 1, . . . , 𝑚 + 1. proof. synchronization of inbound and outbound flows at stations in the model… 91 copyright ©2024 assa adv. in systems science and appl. (2024) the proof directly follows from theorems 3.1 and 4.1. ∎ let's move on to the interpretation of the main results obtained in this study. before doing that, let's remind ourselves that the research on this model, described by the system (2.7)–(2.10), boils down to solving two fundamental tasks. the first task is to determine the ranges of model parameter variations within which the mentioned system has a solution (ensuring uninterrupted freight flow). theorem 4.1 defines these specified ranges. the second task is to determine the most acceptable achievable levels of inconsistency between cargo reception and dispatch on all stations for a given value of the freight demand characteristic using control parameters. the solution to this task is presented in corollary 2. according to it, for any values of inconsistency levels between cargo reception and dispatch at the initial time, if the capabilities of all stations to increase freight flow, starting from the first one, are not less than the capabilities of the zero station (initial node station), it is always possible to activate station potentials in such a way (by selecting values for the parameter 𝜆 ) and align the cargo distribution mode with the final node station's demand for freight transport (by selecting the value for the parameter 𝑑 ), such that the degree of inconsistency at all stations except the initial node station will gradually become zero over time. the corresponding characteristic at the initial node station will depend on both the freight demand characteristic and the initial conditions 5. conclusion a dynamic model for organizing cargo transportation has been investigated on a segment of the railway network, which represents a railway line between two node stations. this model is represented by a system of differential equations that describe the dynamics of the discrepancy between cargo reception and dispatch at the stations. it includes a set of parameters that define the characteristics of cargo transportation demand, the technical capabilities of the stations, the extent of their utilization, as well as the mode of cargo distribution from the node station. ranges of parameter variations have been determined within which the cargo transportation system can operate smoothly without interruptions. for each station, the most suitable level of utilizing its technical potential has been identified, allowing for a smooth and efficient flow of cargo and the ability to respond to changes in cargo transportation demand. acknowledgements the article was prepared at the state academic university for the humanities as part of the state assignment of the ministry of science and higher education of the russian federation (topic no. fznf-2023-0004 "digitalization and formation of modern information society: cognitive, economic, political and legal aspects"). references 1. andersen, j. & christiansen, m. 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(2018). the influence of buffer time distributions in delay propagation modelling of railway networks, journal of rail transport planning & management, 8(3–4), 220–232. adv syst sci appl 2023; 04:60–75 published online at https://ijassa.ipu.ru. bayesian estimation and prediction from a mixture of weibull and gompertz distributions abdulqader al-dugin1,2*, mostafa mohie el-din1, amr sadek1 1department of mathematics, al-azhar university, cairo, egypt 2department of mathematics, abyan university, abyan, yemen abstract: we study different methods for estimation the parameters of a mixture of weibull and gompertz distributions as a lifetime model, based on a complete sample. maximum likelihood estimation and bayes estimation under informative and non-informative priors have been obtained using the symmetric squared error (se) loss function, the asymmetric linear exponential (linex) loss function and general entropy (ge) loss function. also, we discuss two-sample bayesian prediction intervals of the proposed model. for the illustration of the developing results, some computation results for the proposed model is presented. keywords: mixture model; loss function; weibull; gompertz; maximum likelihood estimation; bayesian estimation and prediction 1. introduction mixtures models have received great attention from analysts in the recent years due to their important role in life testing and reliability. in many applications, mixture models are used to analyze random duration in possibly heterogeneous populations, statistical analysis and machine learning such as modeling, classification, and survival analysis. attention has been paid by some authors to the finite mixtures to discuss lifetime distributions, [see, everitt and hand (1981),titterington et al. (1985), mclachlan and basford (1988), lindsay (1995), mclachlan and peel (2000)]. also, mixture distributions have been considered extensively by several researchers using both classical and bayesian techniques, for example, shawky and bakoban (2009), abu-zinadah (2010), erisoglu et al.(2011), feroze and aslam (2014), daniyal and rajab (2015), elshahat and mahmoud (2016) and mahmoud et al.(2017). the weibull distribution has been widely used in modeling of lifetime event data; this is due to the variety of shapes of the probability density function (pdf) based on its parameters, mohie el-din et al. (2018) presented a new study on progressive-stress accelerated life testing for power generalized weibull distribution under progressive type-ii censoring. the weibull distribution has been shown to be useful for lifetime modeling and data analysis in applied engineering sciences, and also the static stress accelerated life test of the generalized weibull distribution has been studied under step-wise type-ii [see, mohie el-din et al.(2019), murthy et al.(2003)]. the gompertz distribution is used to model human survival and mortality times and actuarial tables. it has many real-life applications, particularly in medical and actuarial studies, and for statistical inference of the gompertz distribution on the basis of typeii hybrid progressively censored data [see, mohie el-din et al. (2017)]. the gompertz distribution is also used as a survival model in reliability. it has an increased risk rate for the ∗corresponding author: abdulqader@azhar.edu.eg bayesian estimation and prediction 61 life of the systems. because of its complex shape, it has not received enough attention in the past. however, recently, this distribution has received a lot of attention from demographers and actuaries, and several studies have been presented in this field, including statistical inference and prediction of the gompertz distribution based on multiply typei censored data. mohie el-din and abumoussa,(2018). the characterization for gompertz distribution based on general progressively type-ii right censored order statistics is done by mohie el-din,et al.(2017). pollard and valkovics (1992) were the first to deal with the gompertz distribution thoroughly. however, their results are true only in cases where the initial level of mortality is very close to zero. willemse and koppelaar (2000) proposed a new formulation of epistemic elicitation of gompertz’s law of mortality. if x has the same units and the same rates of variation of two distributions, then a random variable x is said to have a mixture distribution. let a random variable x is to have a mixture of two components weibull and gompertz distribution, with the probability density function (pdf) is given by: f(x) = 2∑ j=1 pjfj(x); ; j = 1, 2, (1.1) where f1(x) = α1θx θ−1e−α1x θ , x > 0, α1 > 0, θ > 0, f2(x) = α2e x−α2(e x−1) , x > 0, α2 > 0. the mixing proportions pj, are such that 0 ≤ pj≤ 1, 2∑ j=1 pj = 1. the corresponding cumulative distribution function(cdf) and reliability function, receptively are given by: f (x) = 2∑ j=1 pjfj(x), j = 1, 2, (1.2) where f1(x) = 1− e−α1x θ , f2(x) = 1− e−α2(e x−1), and r(x) = 2∑ j=1 pjrj(x), j = 1, 2, (1.3) where r1(x) = e−α1x θ , r2(x) = e−α2(e x−1). the objective of this work is to apply the bayesian procedure to estimate the parameters and obtain two sample prediction bounds for future observations from the proposed model, based on complete sample. the rest of this paper is organized as follows: in section 2, we obtain maximum likelihood estimators of the proposed parameters. the bayesian estimation is discussed in section 3. bayesian prediction presented in section 4. simulation study and real data presented in section 5 to compare the performance of different estimation methods of the parameters. finally, conclusions are presented in section 6. 2. maximum likelihood estimation in this section, we present maximum likelihood estimators (mles) for the unknown parameters α1, α2 and p based on complete sample. suppose a sample of n units are put on operation in life testing experiment and that the test is terminated if all n items taken copyright © 2023 assa. adv syst sci appl (2023) 62 a. al-dugin, m. m. mohie el-din, a. sadek from this population have failed. after the n units have failed each item can be attributed to the appropriate sub-population. thus if the n units have failed during the interval (0, x(n)); r1 from the first sub-population and r2 from the second sub-population. let xij denote the failure of the jth unit that belongs to the ith sub-population and xij ≤ x(n); j = 1, 2, ...., ri; i = 1, 2;n = r1 + r2 , where x(n) denotes the failure time of the nth unit. for a two-component mixture model, the likelihood function is given by: l(α1, α2, θ, p |x) = n! [ r1∏ j=1 p1 f 1(x1j)][ r2∏ j=1 p2 f 2(x2j)], (2.4) where f1(x1j) = α1θx θ−1 1j e−α1x θ 1j , x1j > 0, α1 > 0, θ > 0, f2(x2j) = α2e x 2j−α2(e x2j−1), x2j > 0, α2 > 0, then, l(α1, α2, θ, p|x) ∝ r1∏ j=1 p1 α1 θ x θ−1 1j e−α1x θ 1j r2∏ j=2 p2 α2 e x2j−α2(e x2j−1). (2.5) put p1 = p, p2 = 1− p, and assuming that the parameter θ is known, the likelihood function (2.5) reduces to l(α1, α2, p|x, θ) ∝ 2∏ i=1 (piαi) ri e −α1 r1∑ j=1 x θ 1j e −α2 r2∑ j=1 (ex2j−1) . (2.6) thus, the log-likelihood function of parameters α1, α2 and p are given by: lnl = lnl(p, α1, α2|x) ∝ 2∑ i=1 {ri ln pi+ri lnαi} − α1 r1∑ j=1 x θ 1j − α2 r2∑ j=1 (ex2j−1). (2.7) taking derivatives with respect to α1, α2 and p in equation (2.7), and equating by zero, the maximum likelihood estimators of the three parameters are obtained as follows α̂1 = r1 r1∑ j=1 xθ ij , α̂2 = r2 r2∑ j=1 (ex2j − 1) , p̂ = r1 r1 + r2 . 3. bayesian estimation in this section, we derive bayesian estimators of the parameters α1, α2 and p of the considered model by using various priors based on different symmetric and asymmetric loss functions. 3.1. loss function in decision theory, the loss criterion is specified in order to obtain the best estimator. three loss functions are proposed, symmetric squared error loss function (se) and asymmetric (linex and general entropy) loss functions, as follows: copyright © 2023 assa. adv syst sci appl (2023) bayesian estimation and prediction 63 • squared error loss function (es): a simple, and very common loss function is defined by l1(θ̂ , θ) = c (θ̂ − θ)2, where c is constant which is symmetrical in nature and gives equal weight to overestimation as well as underestimation. however, in real applications, estimation of reliability and failure rate functions, an overestimate is more serious than the underestimate. the use of symmetric loss function might be inappropriate as has been recognized by basu and ebrahimi (1991). • linear exponential loss function (linex): one of the most commonly used asymmetric loss functions, introduced by varian (1975) under the assumption that the minimal loss occurs at θ̂ = θ, it can be expressed as l2(θ̂ , θ) ∝ e−q( θ̂ −θ ) − q( θ̂ − θ)− 1, q ̸= 0, where q determines the shape of the loss function. it is used in both overestimation and underestimation, if q > 0 means overestimation and if q < 0, means underestimation but in a situation where q = 0, the linex loss function is almost symmetric and approaches squared error loss function. under the above loss function, the bayes estimator θ̂linex of θ can be obtained as θ̂linex = −1 q ln [e (e−qθ|x)], provided that the expected value with respect to the posterior function of θ, e(e−qθ|x) exists and is finite. • general entropy loss function (ge): another commonly asymmetric loss function is the modified linex loss function called a general entropy loss function proposed by calabria and pulcini (1996). l3(θ̂, θ) ∝ ( θ̂ θ )h − h ln ( θ̂ θ ) − 1, h ̸= 0, which has a minimum at θ̂ = θ. also, this loss function used by several authors, in the original form having the shape parameter h = 1, for h > 0, a positive error has a more effect than a negative error. in this case, the bayes estimate of θ is given by: θ̂ge = [ e ( θ−h|x )]− 1 h , provided that the expected value with respect to the posterior function of θ, e ( θ−h|x ) exists and is finite. 3.2. the posterior distribution under the informative prior assume the prior distribution of the parameters α1, α2 and p are α1 ∼ γ(a1, b1) , α2 ∼ γ(a2, b2), and p ∼ β(c, d) for the mixing parameter, pi; i = 1, 2 where p1 = p and p2 = 1− p. assuming, now, the independence of parameters, the joint distribution prior for α1, α2 and p is: π(α1, α2 , p)= π1(α1)π2(α2)π3(p), (3.8) copyright © 2023 assa. adv syst sci appl (2023) 64 a. al-dugin, m. m. mohie el-din, a. sadek where πi(αi) = baii γ(ai) αai−1 i e−biαi , αi > 0, ai, bi > 0, i = 1, 2, π3(p) = γ(c+ d) γ(c)γ(d) pc−1(1− p)d−1, 0 < p < 1, c, d > 0. then, the joint prior distribution of α1, α2 and p can be written as follows π (α1, α2 , p) ∝ [ 2∏ i=1 α ai−1 i e−biαi ] × p c−1 ( 1− p) d−1. (3.9) it follows from (2.6) and (3.9), that the joint posterior density function of α1, α2 and p is given by g (α1, α2, p|x)= k−1 1 pr1+c−1 (1− p)r2+d−1 α r1+a1−1 1 α r2+a2−1 2 e−α1ϕ1 e −α2ϕ2 , (3.10) where, k1 is the normalizing constant given by k1 = β (r1+c , r2+d) γ (r1+a1) γ (r2+a2) (ϕ1) r1+a1 (ϕ2) r2+a2 , with ϕ1 = b1+ ∑r1 j=1x θ 1j , and ϕ2 = b2+ ∑r2 j=1(e x2j−1). 3.2.1. bayes estimator under squared error loss function (se) the bayes estimators of α1, α2 and p based on the squared error loss function are given by: α̂1se = k−1 1 β(r1+c , r2+d) γ(r1+a1+1) γ (r2+a2) (ϕ1) r1+a1+1 (ϕ2)r2+a2 , α̂2 se = k−1 1 β(r1+c , r2+d) γ (r1+a1) γ(r2+a2+1) (ϕ1)r1+a1(ϕ2)r2+a2+1 , p̂ se = k−1 1 β(r1+c+ 1 , r2+d ) γ (r1+a1) γ (r2+a2) (ϕ1)r1+a1(ϕ2)r2+a2 . 3.2.2. bayes estimator under linex loss function (linex) the bayes estimators of α1, α2 and p based on the linex loss function are given by: α̂1linex = −1 q ln [ k−1 1 β(r1+c , r2+d ) γ (r1+a1) γ (r2+a2) (ϕ1+q)r1+a1(ϕ2)r2+a2 ] , α̂2linex = −1 q ln [ k−1 1 β (r1+c , r2+d) γ (r1+a1) γ (r2+a2) (ϕ1)r1+a1(ϕ2+q)r2+a2 ] , p̂ linex = −1 q ln [ k−1 1 γ (r1+a1) γ (r2 + a2) (ϕ1)r1+a1(ϕ2)r2+a2 ∞∑ k=0 (−q)k k! β(r1+c+ k , r2+d) ] . copyright © 2023 assa. adv syst sci appl (2023) bayesian estimation and prediction 65 3.2.3. bayes estimator under general entropy loss function (ge) the bayes estimator of α1, α2 and p based on general entropy loss function are given by: α̂1ge = [ k−1 1 β (r1+c , r2+d) γ(r1+a1−h )γ (r2+a2) (ϕ1)r1+a1−h(ϕ2)r2+a2 ]− 1 h , α̂2ge = [ k−1 1 β (r1+c, r2+d) γ (r1+a1) γ(r2+a2−h) (ϕ1) r1+a1 (ϕ2) r2+a2−h ]− 1 h , p̂ge= [ k−1 1 β(r1+c− h , r2+d) γ (r1+a1) γ (r2+a2) (ϕ1)r1+a1(ϕ2)r2+a2 ]− 1 h . 3.3. the posterior distribution under the non-informative prior jeffery’s prior is a formal rule for obtaining a non-informative prior. the non-informative priors are recommended when there is no formal prior information about the parameters. this is defined as the distribution of the parameters proportional to the square root of the determinants of the fisher information matrix. the prior distribution for the mixing parameter p, i.e., p ∼ uniform(0,1). assuming independence, the joint prior distribution is given by: πi(αi) ∝ 1 ai , αi > 0, i = 1, 2, π3(p) = 1, 0 < p < 1. (3.11) then, the joint prior distribution of α1, α2 and p can be written as follows π (α1, α2, p)∝ 1 α1α2 , α1, α2> 0 , 0 < p < 1. (3.12) it follows from ( 2.6) and ( 3.12) that, the joint posterior density function of α1, α2 and p is given by g (p, α1, α2|x) ∝ k−1 2 pr1 (1− p)r2 αr1−1 1 αr2−1 2 e−α1ϕ ∗ 1 e−α2ϕ ∗ 2 , (3.13) where, k2 is the normalizing constant given by k2 = β(r1+1 , r2+1 ) γ (r1) γ (r2) (ϕ∗ 1) r1 (ϕ∗ 2) r2 , with ϕ∗ 1= ∑r1 j=1x θ 1j and ϕ∗ 2= ∑r2 j=1(e x2j−1). 3.3.1. bayes estimator under squared error loss function (se) the bayes estimators of α1, α2 and p based on the squared error loss function are given by: α̂1sen = k−1 2 β(r1+1, r2+1) γ(r1+1)γ (r2) (ϕ∗ 1) r1+1 (ϕ∗ 2) r2 , α̂2 sen = k−1 2 β(r1+1 , r2+1) γ (r1) γ (r2+1) (ϕ∗ 1) r1 (ϕ∗ 2) r2+1 , p̂sen = k−1 2 β(r1+2 , r2+1) γ (r1) γ (r2) (ϕ∗ 1) r1 (ϕ∗ 2) r2 . 3.3.2. bayes estimator under linex loss function (linex) copyright © 2023 assa. adv syst sci appl (2023) 66 a. al-dugin, m. m. mohie el-din, a. sadek the bayes estimators of α1, α2 and p based on the linex loss function are given by: α̂1linexn = −1 q ln [ k−1 2 β (r1+1, r2+1) γ(r1)γ (r2) (ϕ∗ 1+q)r1 (ϕ∗ 2) r2 ] , α̂2linexn = −1 q ln [ k−1 2 β (r1+1 , r2+1) γ (r1) γ(r2) (ϕ∗ 1) r1 (ϕ∗ 2+q)r2 ] , p̂linexn= −1 q ln [ k−1 2 γ(r1)γ(r2) (ϕ∗ 1) r1(ϕ∗ 2) r2 ∞∑ k=0 (−q)k k! β(r1+k + 1 , r2+1) ] . 3.3.3. bayes estimator under general entropy loss function (ge) the bayes estimators of α1, α2 and p based on general entropy loss function are given by: α̂1gen = [ k−1 2 β(r1+1, r2+1) γ(r1−h)γ(r2) (ϕ∗ 1) r1−h(ϕ∗ 2) r2 ]− 1 h , α̂2gen = [ k−1 2 β(r1+1 , r2+1) γ(r1)γ(r2−h) (ϕ∗ 1) r1(ϕ∗ 2) r2−h ]− 1 h , p̂gen = [ k−1 2 β(r1 − h+ 1, r2 + 1) γ(r1)γ(r2) (ϕ∗ 1) r1(ϕ∗ 2) r2 ]− 1 h . 4. bayesian prediction in this section, the bayesian two-sample prediction of a future order statistics sample considered on the basis of observed data used in the informative and non-informative prior. a random sample of size m of future observation, independent sample of size n, is drawn from the same population with eq(1.1) . therefore ys represents the sth ordered statistic in the future sample of size m, 1 ⩽ s ⩽ m. the sth order statistic in a sample of size m represents the life length of a (m− s+ 1) out of m system. the distribution function of ys , the ordered future sample is given by (see, arnold et al. (1992) and jaheen (2003)), fys (ys|α1, α2, p) = m∑ l=s ( m l ) [fx(ys|α1, α2, p)] l [1− fx (ys|α1, α2, p) ] m−l = m∑ l=s l∑ j1=0 ( m l )( l j1 ) (−1)j1 [r(ys)] m−l+j1 , (4.14) where fx (ys|α1, α2, p) = 1−r(ys) is the distribution function of the mixture model and r(ys) is the reliability function of the mixture model after replacing x by ys. using the binomial expansion for [r(ys)] m−l+j1 as follows: [r(ys)] m−l+j1= [ p1 e −α1y θ s+p2 e −α2(e ys−1) ]m−l+j1 , = m−l+j1∑ j2=0 ( m− l + j1 j2 ) pδ11 pj22 e−α1δ1y θ s e−α2j2(e ys−1). (4.15) therefore, we get copyright © 2023 assa. adv syst sci appl (2023) bayesian estimation and prediction 67 fys (ys|α1, α2, p) = m∑ l=s l∑ j1=0 m−l+j1∑ j2=0 ( m l )( l j1 )( m− l + j1 j2 ) (−1)j1 pδ11 pj22 (e−α1y θ s )δ1 (e−α2(e ys−1))j2 , (4.16) where δ1 = m− l + j1 − j2. f ∗(ys|x) = ∫ 1 0 ∫ ∞ 0 ∫ ∞ 0 f (ys|α1, α2, p) g (α1, α2, p|x) dα1dα2dp, (4.17) where g (α1, α2, p|x) is the joint posterior density for parameters α1, α2 and p and f (ys|α1, α2, p) is the pdf of sth component in a future sample. therefore, bayesian prediction density of ys for a given value v, can be obtained as: p [ys ≥ v|x] = ∞∫ v f ∗(ys|x)dys =1− ∫ 1 0 ∫ ∞ 0 ∫ ∞ 0 fys (v|α1, α2, p) g (α1, α2, p|x) dα1dα2dp, (4.18) substitution of (3.9) and (4.16) in (4.18), we get bayes predictive distribution bounds with value v for ys in case of informative as p [ys ≥ v |x] = 1− k−1 1 ∑ b β (δ1+δ2, δ3) γ (r1+a1) γ (r2+a2) (ϕ∗∗ 1 )r1+a1 (ϕ∗∗ 2 )r2+a2 . where δ2 = r1 + c, δ3 = r2 + d+ j2. substitution of (3.11) and (4.16) in (4.18), we get bayes predictive distribution bounds with value v for ys in case of noninformative as p [ys ≥ v |x] = 1− k−1 2 ∑ b β (δ∗1+1 , δ∗2+1) γ (r1) γ (r2) (ϕ∗∗∗ 1 )r1 (ϕ∗∗∗ 2 )r2 . where∑ = m∑ l=s l∑ j1=0 m−l+j1∑ j2 , b = ( m l )( l j1 )( m−l+j1 j2 ) (−1)j1 δ∗1 = δ1 + r1 , δ∗2 = j2 + r2 ϕ∗∗ 1 = δ1v θ + ϕ1 , ϕ∗∗ 2 = j2(e v − 1) + ϕ2 ϕ∗∗∗ 1 = δ1v θ + ϕ∗ 1 , ϕ∗∗∗ 1 = j2(e v − 1) + ϕ∗ 2 a 100 γ% prediction interval for ys is given by p [l(x) < ys < u(x)] = γ, where l(x) and u(x) are obtained respectively by solving the following two equations: p [ys > l(x)] = 1 + γ 2 and p [ys > u(x)] = 1− γ 2 . (4.19) copyright © 2023 assa. adv syst sci appl (2023) 68 a. al-dugin, m. m. mohie el-din, a. sadek 5. simulation study and real data in this section, we present some simulation results to compare the performance of various estimates by using the monte carlo simulation method from the mixture of weibull and gompertz distributions. the following steps were considered: 1. for the parameter we have considered (α1, α2, p) = (0.2, 0.1, 0.7) along with θ is known and θ = 0.61. the values of q and h are (0.5, -0.5) for linex and general entropy loss functions. the method of choosing the hyper parameters values introduced in ahmadi et al.(2020). let (a1 = 0.03, a2 = 0.04, b1 = 0.2, b2 = 0.35, c = 96.23, and d = 41.167) for informative prior. in case of non-informative prior, we take (a1 = a2 = b1 = b2 = 0; c = d = 1). in all these cases samples of size n = 20, 40, 60, 80, 100 and 150, are generated. 2. generate a uniform random number u from the interval(0,1). 3. if u ⩽ p the observation has been randomly taken from first sub-population and if u > p then the observation have been taken from the second sub-population. 4. the obtained results for maximum likelihood estimates and bayes estimates are calculated. 5. bayesian prediction for the future observations of ys are obtained by solving numerically, equations (4.6) with γ = 0.95. 6. the above steps are repeated 1000 times, and the average of the estimates are calculated and presented in the tables (5.1)-(5.3). also the average lower and average upper interval of ys when s = m 2 is even or m+ 1 2 is odd for different sample size n and different future sample size m, simulated coverage probability and average interval lengths are obtained in tables (5.4)-(5.5). all results were obtained by using mathematica 10. copyright © 2023 assa. adv syst sci appl (2023) bayesian estimation and prediction 69 table 5.1. average estimates and corresponding mse of the parameter α1 = 0.2 based on informative and noninformative prior n mle bayes se linex ge informative non-informative informative non-informative informative non-informative q=-0.5 q=0.5 q=-0.5 q=0.5 h=-0.5 h=0.5 h=-0.5 h=0.5 20 0.217318 0.217035 0.21770 0.217996 0.216086 0.218285 0.216363 0.213051 0.205008 0.21332 0.205247 (0.00451) (0.00446) (0.00475) (0.00458) (0.00435) (0.00463) (0.0044) (0.00418) (0.00372) (0.00423) (0.00376) 40 0.209797 0.209695 0.21531 0.210106 0.209287 0.210209 0.209388 0.207823 0.204061 0.207922 0.204154 (0.00176) (0.00175) (0.00273) (0.00177) (0.00173) (0.00178) (0.0017) (0.00169) (0.00158) (0.00169) (0.00159) 60 0.205851 0.205791 0.20781 0.206052 0.205531 0.206113 0.205591 0.204558 0.202084 0.204617 0.202141 (0.00107) (0.00107) (0.00116) (0.00108) (0.00106) (0.00108) (0.0011) (0.00105) (0.00101) (0.00105) (0.00101) 80 0.202925 0.202884 0.20450 0.203072 0.202696 0.203113 0.202737 0.2019757 0.200154 0.202016 0.200194 (0.00076) (0.00076) (0.00098) (0.00077) (0.00076) (0.00077) (0.0008) (0.00075) (0.00073) (0.00075) (0.00074) 100 0.202708 0.202675 0.20293 0.202825 0.202526 0.202858 0.202558 0.201949 0.200494 0.201981 0.200525 (0.00063) (0.00062) (0.00067) (0.00063) (0.00062) (0.00063) (0.0006) (0.00062) (0.0006) (0.00062) (0.00061) 150 0.201981 0.200088 0.20288 0.202024 0.20209 0.202079 0.201883 0.200249 0.200195 0.198195 0.200575 (0.00041) (0.00078) (0.00052) (0.00055) (0.00042) (0.00041) (0.0004) (0.00064) (0.00044) (0.00106) (0.0004) table 5.2. average estimates and corresponding mse of the parameter α2 = 0.1 based on informative and noninformative prior n mle bayes se linex ge informative non-informative informative non-informative informative non-informative q=-0.5 q=0.5 q=-0.5 q=0.5 h=-0.5 h=0.5 h=-0.5 h=0.5 20 0.122528 0.121892 0.12360 0.123059 0.120817 0.123818 0.121374 0.116016 0.103891 0.116521 0.104117 (0.00532) (0.004878) (0.00641) (0.00532) (0.00454) (0.00594) (0.00488) (0.00401) (0.00273) (0.00432) (0.00287) 40 0.111112 0.11105 0.11245 0.111384 0.110721 0.111449 0.11078 0.10856 0.103519 0.10861 0.103544 (0.00183) (0.001781) (0.00289) (0.00183) (0.00173) (0.00188) (0.00178) (0.00163) (0.00139) (0.00168) (0.00142) 60 0.108024 0.108021 0.10953 0.108201 0.107841 0.108205 0.107844 0.106479 0.103373 0.106479 0.103365 (0.00079) (0.00079) (0.00095) (0.0008) (0.00078) (0.0008) (0.00079) (0.00074) (0.00067) (0.00075) (0.00067) 80 0.103682 0.103692 0.10633 0.103813 0.103572 0.103803 0.103562 0.102581 0.100348 0.10257 0.100332 (0.00043) (0.000433) (0.00079) (0.00044) (0.00043) (0.00044) (0.00043) (0.00042) (0.00039) (0.00042) (0.00039) 100 0.103475 0.103483 0.10521 0.103578 0.103389 0.103569 0.10338 0.102607 0.100849 0.102598 0.100837 (0.00041) (0.000411) (0.00052) (0.00041) (0.00041) (0.00041) (0.00041) (0.0004) (0.00038) (0.0004) (0.00038) 150 0.101673 0.101554 0.10172 0.101613 0.101495 0.101733 0.101614 0.100986 0.0998459 0.101101 0.099953 (0.00027) (0.000266) (0.00029) (0.00027) (0.00027) (0.00027) (0.00027) (0.00026) (0.00026) (0.00027) (0.00026) table 5.3. average estimates and corresponding mse of the parameter p = 0.7 based on informative and noninformative prior n mle bayes se linex ge informative non-informative informative non-informative informative non-informative q=-0.5 q=0.5 q=-0.5 q=0.5 h=-0.5 h=0.5 h=-0.5 h=0.5 20 0.69115 0.699258 0.69973 0.699589 0.6989326 0.676418 0.671834 0.69878 0.367819 0.670447 0.662696 (0.01073) (0.000173) (0.00947) (0.00017) (0.00017) (0.0093) (0.00967) (0.00017) (0.00018) (0.009988) (0.01083) 40 0.7034 0.701061 0.70132 0.701354 0.700767 0.694919 0.692503 0.70064 0.699792 0.691894 0.68816 (0.0052) (0.00026) (0.00474) (0.00027) (0.00026) (0.00471) (0.0048) (0.00027) (0.00027) (0.00483) (0.00502) 60 0.697767 0.699586 0.69986 0.69985 0.699321 0.692219 0.690552 0.699205 0.698441 0.690144 0.687615 (0.00346) (0.00032) (0.00331) (0.00032) (0.00032) (0.00329) (0.0033) (0.00032) (0.00032) (0.00336) (0.00347) 80 0.700113 0.700282 0.70053 0.700521 0.700042 0.695861 0.6946 0.699937 0.699244 0.694303 0.692422 (0.00272) (0.00037) (0.00261) (0.00037) (0.00037) (0.0026) (0.0026) (0.00037) (0.00037) (0.00264) (0.0027) 100 0.69908 0.699832 0.69991 0.700052 0.699612 0.695685 0.694666 0.699516 0.698881 0.69443 0.692921 (0.00203) (0.00036) (0.00197) (0.00036) (0.00036) (0.00197) (0.002) (0.00036) (0.00036) (0.00199) (0.00203) 150 0.700313 0.699868 0.69998 0.699949 0.699786 0.69802 0.697335 0.699506 0.698982 0.697181 0.69618 (0.00136) (0.00035) (0.00132) (0.00035) (0.00035) (0.00132) (0.0013) (0.00035) (0.00035) (0.00133) (0.00135) copyright © 2023 assa. adv syst sci appl (2023) 70 a. al-dugin, m. m. mohie el-din, a. sadek table 5.4. the 95% bayesian prediction bounds, length of bayesian prediction and their simulated coverage probability for ys based informative prior (n,m) y1 y4 ym (lower, upper) length coverage (lower, upper) length coverage (lower, upper) length coverage (15,8) (0.000303,0.4654) 0.465106 0.934 (0.103358,1.96123) 1.85788 0.934 (0.620589,34.0201) 33.3995 0.977 (20,8) (0.000275,0.45017) 0.449896 0.936 (0.106456,1.90021) 1.79375 0.935 (0.760848,32.4311) 31.6703 0.967 (30,8) (0.000265,0.44291) 0.442654 0.945 (0.109328,1.86521) 1.75588 0.945 (0.463508,30.5226) 30.0591 0.981 (50,8) (n,m) y1 y5 ym (lower, upper) length coverage (lower, upper) length coverage (lower, upper) length coverage (15,10) (0.000216,0.36387) 0.363661 0.946 (0.130549,1.82091) 1.69036 0.933 (0.879757,39.4524) 38.5727 0.958 (20,10) (0.0002006,0.3515) 0.351314 0.941 (0.137363,1.78166) 1.6443 0.944 1.05329,34.2686) 33.2153 0.957 (30,10) (0.000190,0.34323) 0.343049 0.951 (0.141384,1.72623) 1.58484 0.936 (0.936194,32.7176) 31.7814 0.963 (50,10) (0.000174,0.33177) 0.331604 0.946 (0.145217,1.69538) 1.55017 0.937 (0.779346,34.6254) 33.8461 0.983 (n,m) y1 y6 ym (lower, upper) length coverage (lower, upper) length coverage (lower, upper) length coverage (15,12) (0.000165,0.29767) 0.297512 0.935 (0.1511,1.69131) 1.54021 0.929 (1.06404,39.1433) 38.0793 0.968 (20,12) (0.000154,0.28254) 0.282393 0.945 (0.150129,1.61944) 1.46931 0.936 (0.900551,38.0184) 37.1179 0.965 (30,12) (0.000139,0.27243) 0.272296 0.954 (0.165389,1.63105) 1.46566 0.946 (0.861137,34.6709) 33.8098 0.978 (50,12) (0.000129,0.26469) 0.264562 0.947 (0.169259,1.56466) 1.3954 0.947 (0.949135,33.8375) 32.8884 0.978 (n,m) y1 y7 ym (lower, upper) length coverage (lower, upper) length coverage (lower, upper) length coverage (15,13) (0.000152,0.27705) 0.277058 0.938 (0.193216,1.9467) 1.75349 0.936 (0.964702,42.2606) 41.2959 0.973 (20,13) (0.000141,0.26195) 0.26181 0.941 (0.208467,2.03536) 1.82689 0.935 (1.17018,39.2434) 38.0732 0.976 (30,13) (0.0001234,0.24464) 0.244524 0.945 (0.210109,1.90317) 1.69306 0.931 (1.17348,35.6211) 34.4476 0.966 (50,13) (0.0001143,0.23821) 0.238096 0.957 (0.211075,1.81535) 1.60427 0.958 (1.18585,33.1661) 31.9803 0.982 table 5.5. the 95% bayesian prediction bounds,length of bayesian prediction and their simulated coverage probability for ys based non informative prior (n,m) y1 y4 ym (lower, upper) length coverage (lower, upper) length coverage (lower, upper) length coverage (15,8) (0.000248,0.49451) 0.49427 0.953 (0.111943,2.86765) 2.75571 0.948 (0.779858,33.9345) 33.1547 0.955 (20,8) (0.0002421,0.46972) 0.469481 0.951 (0.102928,2.46754) 2.36461 0.949 (0.558534,36.6563) 36.0978 0.988 (30,8) (0.000240,0.46191) 0.461669 0.963 (0.106012,2.21552) 2.10951 0.961 (0.537214,32.6924) 32.1552 0.979 (50,8) (0.000235,0.45122) 0.450991 0.948 (0.108414,2.01425) 1.90583 0.952 (0.183484,30.7059) 30.5224 0.988 (n,m) y1 y5 ym (lower, upper) length coverage (lower, upper) length coverage (lower, upper) length coverage (15,10) (0.000187,0.38934) 0.389161 0.954 (0.147564,2.78801) 2.64045 0.948 (1.14207,45.6324) 44.4903 0.949 (20,10) (0.000177,0.36678) 0.366609 0.951 (0.141181,2.34187) 2.20069 0.947 (1.11316,37.6697) 36.5566 0.961 (30,10) (0.000165,0.34543) 0.345268 0.957 (0.1373962.01525) 1.87786 0.953 (1.05137,32.9594) 31.908 0.968 (50,10) (0.0001621,0.33803) 0.337875 0.947 (0.143411,1.86337) 1.71996 0.953 (1.06816,39.8492) 38.7811 0.973 (n,m) y1 y6 ym (lower, upper) length coverage (lower, upper) length coverage (lower, upper) length coverage (15,12) (0.0001403,0.3082) 0.30812 0.961 (0.16975,2.64656) 2.47681 0.923 (1.31578,45.0772) 43.7614 0.953 (20,12) (0.0001340,0.291961) 0.291827 0.952 (0.173968,2.29415) 2.12018 0.939 (1.17472,40.9827) 39.808 0.967 (30,12) (0.0001238,0.275178) 0.275055 0.947 (0.179458,2.0297) 1.85024 0.961 (1.35481,38.1587) 36.8039 0.968 (50,12) (0.0001214,0.265905) 0.265784 0.962 (0.171414,1.74186) 1.57044 0.959 (1.33663,33.6541) 32.3175 0.974 (n,m) y1 y7 ym (lower, upper) length coverage (lower, upper) length coverage (lower, upper) length coverage (15,13) (0.0001372,0.28479) 0.284626 0.941 (0.205341,2.78688) 2.58154 0.939 (1.01779,49.069) 48.0512 0.983 (20,13) (0.000191,0.266334) 0.266213 0.949 (0.206602,2.48416) 2.27756 0.956 (1.37849,43.4506) 42.0721 0.967 (30,13) (0.0001098,0.24637) 0.246269 0.956 (0.217119,2.22986) 2.01274 0.961 (1.1813,37.8901) 36.7088 0.976 (50,13) (0.000125,0.23432) 0.234541 0.954 (0.224957,1.99452) 1.76956 0.958 (1.4746,33.8247) 32.3501 0.958 copyright © 2023 assa. adv syst sci appl (2023) bayesian estimation and prediction 71 5.1. numerical example now, we presents a numerical example to illustrate the methodology for the proposed estimates based on real data. consider the following data set is from kotz and johhnson (1983) and represents the survival times (in years) after diagnosis of 43 patient with certain kind of leukemia. the data has been classified into two sets using probabilistic mixing weights for p=0.7, which produced r1 = 30 and r2 = 13,, as follows: population-i population-ii 0.019, 0.159, 0.636, 0.748, 1.175, 1.206, 1.282, 1.356, 1.362, 1.458, 1.564, 1.586, 1.592, 1.781, 1.923, 1.959, 2.134, 2.548, 2.652, 2.951, 3.038, 3.655, 3.754, 4.690, 4.888, 5.143, 5.167, 5.603, 6.192, 6.874 6.655, 0.129, 0.485, 2.466, 0.203, 4.203, 2.413, 1.219, 1.219, 0.781, 0.869, 3.6, 5.633 the estimated and prediction results are presented in tables [(5.6)-(5.11)]. table 5.6. average estimates corresponding to real data set in case informative prior parameter mle bayes loss function se linex ge q=0.5 q=-0.5 q=1 q=-1 h=0.5 h=-0.5 h=1 h=-1 α1 0.241776 0.241112 0.240629 0.241597 0.240148 0.242086 0.235093 0.239111 0.233075 0.241112 α2 0.010971 0.0109709 0.0109686 0.0109733 0.0109663 0.0109756 0.0103402 0.010762 0.0101271 0.0109709 p 0.697674 0.65625 0.655097 0.657399 0.653939 0.658542 0.650806 0.654462 0.648936 0.65625 table 5.7. average estimates corresponding to real data set in case noninformative prior parameter mle bayes loss function se linex ge q=0.5 q=-0.5 q=1 q=-1 h=0.5 h=-0.5 h=1 h=-1 α1 0.241776 0.241776 0.24129 0.242264 0.240807 0.242755 0.23574 0.239769 0.233716 0.241776 α2 0.010971 0.0109713 0.010969 0.0109736 0.0109667 0.010976 0.0103405 0.0107624 0.0101274 0.0109713 p 0.697674 0.688889 0.687721 0.690051 0.686547 0.691206 0.683628 0.687163 0.681818 0.688889 copyright © 2023 assa. adv syst sci appl (2023) 72 a. al-dugin, m. m. mohie el-din, a. sadek table 5.8. bayesian prediction bounds ys, length of the bayesian prediction corresponding 90% in case informative prior for the real data set (n,m) y1 y4 ym (lower, upper) length (lower, upper) length (lower, upper) length (63,8) (0.0897561,2.00536) 1.9156039 (1.23083,4.29157) 3.06074 (4.01878,9.16364) 5.14486 (n,m) y1 y5 ym (lower, upper) length (lower, upper) length (lower, upper) length (63,10) (0.0759191,1.69281) 1.6168909 (1.38585,4.2249) 2.83905 (4.29473,9.51146) 5.21673 (n,m) y1 y6 ym (lower, upper) length (lower, upper) length (lower, upper) length (63,12) (0.0662061,1.4729) 1.4066939 (1.50481,4.17095) 2.66614 (4.49234,9.79406) 5.30172 (n,m) y1 y7 ym (lower, upper) length (lower, upper) length (lower, upper) length (63,13) (0.0623424,1.38559) 1.3232476 (1.72943,4.32339) 2.59396 (4.57268,9.9177) 5.34502 table 5.9. bayesian prediction bounds ys, length of the bayesian prediction corresponding 90% in case non-informative prior for the real data set (n,m) y1 y4 ym (lower, upper) length (lower, upper) length (lower, upper) length (63,8) (0.0870625,1.93105) 1.8439875 (1.18344,4.27051) 3.08707 (3.97367,9.4282) 5.45453 (n,m) y1 y5 ym (lower, upper) length (lower, upper) length (lower, upper) length (63,10) (0.0736783,1.62945) 1.5557717 (1.33145,4.19961) 2.86816 (4.27084,9.78129) 5.51045 (n,m) y1 y6 ym (lower, upper) length (lower, upper) length (lower, upper) length (63,12) (0.0642801,1.41804) 1.3537599 (1.44492,4.14233) 2.69741 (4.4837,10.0682) 5.5845 (n,m) y1 y7 ym (lower, upper) length (lower, upper) length (lower, upper) length (63,13) (0.0605408,1.33419) 1.2736492 (1.65938,4.30482) 2.64544 (4.57015,,10.1938) 5.62365 table 5.10. bayesian prediction bounds ys, length of the bayesian prediction corresponding 95% in case informative prior for the real data set (n,m) y1 y4 ym (lower, upper) length (lower, upper) length (lower, upper) length (63,8) (0.0528,2.36052) 2.30772 (1.03033,4.54519) 3.51486 (3.69989,10.2809) 6.58101 (n,m) y1 y5 ym (lower, upper) length (lower, upper) length (lower, upper) length (63,10) (0.0446395,1.99962) 1.9549805 (1.18738,4.46186) 3.27448 (4.01896,10.6249) 6.60594 (n,m) y1 y6 ym (lower, upper) length (lower, upper) length (lower, upper) length (63,12) (0.0389124,1.74192) 1.7030076 (1.31001,4.39521) 3.0852 (4.24416,10.9047) 6.66054 (n,m) y1 y7 ym (lower, upper) length (lower, upper) length (lower, upper) length (63,13) (0.0366346,1.63903) 1.6023954 (1.52286,4.53485) 3.01199 (4.33457,11.0273) 6.69273 copyright © 2023 assa. adv syst sci appl (2023) bayesian estimation and prediction 73 table 5.11. bayesian prediction bounds ys, length of the bayesian prediction corresponding 95% in case non-informative prior for the real data set (n,m) y1 y4 ym (lower, upper) length (lower, upper) length (lower, upper) length (63,8) (0.0513029,2.27814) 2.2268371 (0.991506,4.54152) 3.550014 (3.63099,10.5643) 6.93331 (n,m) y1 y5 ym (lower, upper) length (lower, upper) length (lower, upper) length (63,10) (0.0434,1.92644) 1.88304 (1.14161,4.45264) 3.31103 (3.97304,10.9139) 6.94086 (n,m) y1 y6 ym (lower, upper) length (lower, upper) length (lower, upper) length (63,12) (0.0378516,1.67744) 1.6395884 (1.25864,4.38166) 3.12302 (4.21589,11.1984) 6.98251 (n,m) y1 y7 ym (lower, upper) length (lower, upper) length (lower, upper) length (63,13) (0.0356443,1.57834) 1.5426957 (1.46169,4.53063) 3.06894 (4.31345,11.323) 7.00955 6. conclusion in this paper, we have addressed the estimation and prediction problems of the mixture of weibull and gompertz distributions. different estimators of the parameters are obtained using maximum likelihood and bayesian methods, under the informative and non-informative prior distributions, we conclude. 1. the bayes estimates perform better under informative prior that non-informative prior for all different loss functions. 2. the mean squared error of maximum likelihood estimates and bayesian estimates of the proposed parameters 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(1988). mixture models: inferences and applications to clustering. new york, ny: marcel dekker. 16. mclachlan, g. j. & peel, d. (2000). finite mixture models. new york, ny: john wiley & sons. 17. mohie el-din, m. m., abd el-raheem, a. m. & abd el-azeem, s. o. (2018). on progressive-stress accelerated life testing for the power generalized weibull distribution under progressive type-ii censoring, journal of statistics applications & probability letters, 5(3), 131–143. 18. mohie el-din, m. m., abd el-raheem, a. m. & abd el-azeem, s. o. (2019). inference for a constant-stress accelerated life testing for power generalized weibull distribution under progressive type-ii censoring, journal of statistics applications & probability, 8(3), 201–216. 19. mohie el-din, m. m., abdel-aty, y. & abu-moussa, m. h. (2017). statistical inference for the gompertz distribution based on type-ii progressively hybrid censored data, communications in statisticssimulation and computation, 46(8), 6242–6260. 20. mohie el-din, m. m. & abu-moussa, m.h. (2018). statistical inference and prediction for the gompertz distribution based on multiply type-i censored data, journal of the egyptian mathematical society, 26(2), 218–234. 21. mohie el-din, m.m., sadek, a., mohie el-din, marwa m. and sharawy, a.m., (2017). characterization for gompertz distribution based on general progressively type-ii right censored order statistics, international journal of advanced statistics and probability (ijasp), 5(1), 52–56. 22. murthy, d. n. p., xie, m. & jiang, r. (2003). weibull models. new york, ny : john wiley & sons. 23. pollard, j. & valkovics, e. (1992). the gompertz distribution and its applications, genus, 48(3–4), 15–29. copyright © 2023 assa. adv syst sci appl (2023) bayesian estimation and prediction 75 24. shawky, a. i. & bakoban, r. a. (2009). on finite mixture of two component exponentiated gamma distribution, journal applied sciences research, 5(10), 1351– 1369. 25. titterington, d. m., smith, a. f. m. & makov, u. e. (1985). statistical analysis of finite mixture distributions. new york, ny: john wiley & sons. 26. varian, h. r. (1975). a bayesian approach to real estate assessment. in: stephen, e.f & zellner, a.(eds). studies in bayesian econometrics and statistics in honor of leonard j. savage (pp. 195–208). amsterdam, nl: north holland. 27. willemse, w. & koppelaar, h. (2000). knowledge elicitation of gompertz’ law of morality, scandinavian actuarial journal, 94, 168–180. copyright © 2023 assa. adv syst sci appl (2023) introduction maximum likelihood estimation bayesian estimation loss function the posterior distribution under the informative prior bayes estimator under squared error loss function (se) bayes estimator under linex loss function (linex) bayes estimator under general entropy loss function (ge) the posterior distribution under the non-informative prior bayes estimator under squared error loss function (se) bayes estimator under linex loss function (linex) bayes estimator under general entropy loss function (ge) bayesian prediction simulation study and real data numerical example conclusion microsoft word assa_1241-article text-4841-1-4-20220512 adv syst sci appl 2022; 04; 212-223 published online at http://ijassa.ipu.ru. computer program to automate the determination of the size of sanitary protection zones (spzs) and surveillance zones for the storage of sources of ionizing radiation in places of burial / storage of radioactive waste petr istratov1, yury minaev1, elena zarubina1, petr zolotariov1 1) medical university reaviz, moscow, russia abstract: the rapid development of nuclear power and industry, has led to the need to address the issues of waste disposal, which are sources of ionizing radiation and can be hazardous to human health. in this regard, the development of automated systems for preliminary assessment of the place of future disposal of radioactive waste (rw) and the width of the sanitary protection zone (spz), to avoid falling outside residential areas, is an urgent task. the purpose of this study was to develop algorithms for solving the task of processing and analyzing information for modeling the operation of the radioactive waste disposal system, as well as to test the performance of these algorithms in practice. on the basis of artificial intelligence (neural network) the "computer program for automation of determination of dimensions of sanitary-protective zones (spz) and surveillance zones for storage of sources of ionizing radiation in places of burial/storage of radioactive wastes" was developed. the main advantage of the developed neural network is that it can identify non-obvious attributes for determining the size of sanitary protection zones (spz). these algorithms do not rely on the existing knowledge about the influence of the analyzed indicators, which makes them objective and allows to use them to create models of the state of storage of radiation sources. keywords: neural networks, algorithm, sanitary protection zone. 1. introduction rapid development of nuclear power and industry has led to the necessity of solving the issues of waste disposal, which are sources of ionizing radiation and may pose a danger to human health. all issues related to the organization of measures to protect the population living in the vicinity of such facilities are strictly regulated, however, when choosing a place for disposal of radioactive waste one always has to take into account, in addition to the class of waste, the combined effect of various factors both natural and climatic (wind rose, amount of precipitation, soil nature, natural radioactive background of the area, presence or absence of forests, their nature and much more) and anthropogenic nature (for example, presence of industrial waste). in this connection, development of automated systems for preliminary assessment of the place of future disposal of radioactive waste (rw) and the width of the sanitary protection zone (spz), in order to avoid getting outside the zones of residential development, is an urgent task. the purpose of this study was to develop algorithms for solving the problem of information processing and analysis to simulate the operation of the radioactive waste disposal system, as well as to test the performance of these algorithms in practice. in general case the problem to be solved refers to the problem of classification. this is the main and very extensive group of medical and biological problems. the answer in them is a class the choice of one variant from a predetermined set of variants. for our problem, p. istratov, y. minaev, e. zarubina, p. zolotariov 213 copyright ©2022 assa. adv. in systems science and appl. (2022) classification is defined as a binary problem (elementary classification) in this case, the set of possible answers consists of two options (classes), with the first option determining the size of the obtained sanitary protection zone as sufficient, and the second option as not adequate enough to solve the problem. 2. objectives the scientific goal of this work consisted of solving the following main tasks: 1. choose the form of ai implementation for the developed program; 2. analyze and select the most significant indicators for the system development; 3. create an expert system based on the selected indicators; 4. test the system operation under conditions of real estimation of the swd width in the area of the raw disposal site. 3. methods several stages of development of "computer program for automation of determining the size of sanitary protection zones (spz) and surveillance zones for storage of sources of ionizing radiation in places of burial / storage of radioactive waste" can be distinguished in creation of a self-training neural network, some of which coincide with the stages of creation of traditional systems. problem formulation: the same as for the traditional systems plus the choice of the optimal neural network structure and training methods (for most tasks the structure and methods are standard). 1. collection of training data. a set of examples for training the network, each of which represents an array of input data and its corresponding response known in advance. 2. creating and training a neural network. if the task is within the standard scheme (in most cases), you don't need to do any statistical calculations, but programming work, too. if the task is non-standard, you need to adapt the structure of a neural network and the method of estimation calculation in training. training of a neural network in most standard cases is an automatic process, which only after its completion requires a specialist to evaluate the results. of course, it may often require correction, creation of additional networks with other parameters, etc., but it is always possible to evaluate the system performance at any stage of training by testing a control sample. developing methodology of neural network expert systems, we proceeded from the possibility to develop the most individualized (designed for one particular user-specialist) systems by this specialist. of course, nothing prevents to combine in one system individual experience of several experts. the absence of "mathematical" stages realizes such possibilities. the subject specialist is able to set the task himself; moreover, no one but him can do it better. the subject specialist must also collect the material. schemes of problem setting, ways of data representation and ways of response production by a neural network are designed in such a way that most tasks in many fields fit into these standard schemes. so with well-designed neural network software tools and neural network documentation, most specialists are able to develop not very complex neural network applications on their own. 3. interface creation. the same as for traditional expert systems. 4. debugging and testing. this stage mainly includes debugging of the program, since testing is often carried out during the training of networks. 5. post-training. 214 program to automate the determination of the size of sanitary protection zones copyright ©0000 assa adv. in systems science and appl. (0000) this stage is typical only for learning systems. when creating neuroexpert programs it is quite seldom possible to collect enough data for good network training at once. so when creating a neural network researchers determine the best parameters of networks and do the initial training. subsequently users train the system in conditions of real work and real data, transferring the experience to it. moreover, the fundamental difference of methodology of neural network systems creation from traditional ones is exactly, that system is never created at once ready and never completely finished, continuing to accumulate experience in the process of operation. let's look at the structure and functioning of an individual neuron. each connection from neuron to neuron is called a synapse. fig. 1 shows a neuron with a group of synapses connecting the neuron either to other neurons or to the outside world. to consider how a neuron works, it does not matter whether the signal comes to the neuron from the outside world or from another neuron, and it does not matter where the signal is sent from the neuron. in full-connected networks the output signal is sent to all other neurons. s y n a p s e 1 synapse 2 synapse 3 adder converter synapse 1 fig. 1. diagram of a neuron a neuron consists of two functional units: the input adder and the neuron itself, or the transducer. a neuron functions as follows: at the current moment signals from other neurons and/or from the external world are sent to it via input synapses (there are 3 on the figure). each synapse has a parameter called synapse weight, which represents some number. a signal passing through a synapse is multiplied by the weight of that synapse. depending on the weight, the signal can be amplified (weight modulus > 1) or attenuated (weight modulus < 1) in amplitude. the signals from all the synapses leading to a given neuron are received by the adder. the summator sums all incoming signals and sends one number, the resulting sum, to the neuron itself (transducer). the value of this number will depend both on the values of initial signals and synapse weights. the neuron, which receives this number, transforms it according to its function resulting in another number, and sends it to all other neurons via the "axon" through the corresponding synapses. subsequent neurons perform the same operations with the received signals, the only difference being that, firstly, the weights of their synapses may already be different, and secondly, other neurons may have another type of the transformation function. in the neural networks we construct, all neurons have the same function. this function, called the characteristic function, has the form: f (x) = x/(c+x), (1.0) where x is the signal coming from the adder, c is a constant called the neuron characteristic. experimentally we got that the optimal range of the characteristic for solving p. istratov, y. minaev, e. zarubina, p. zolotariov 215 copyright ©2022 assa. adv. in systems science and appl. (2022) the vast majority of problems is from 0.1 to 0.8. plots of the characteristic function for both cases are shown in fig. 2. the choice of such a function is due to the fact that it is smooth and continuous over the entire range of variables x, the range of values is always limited. -1 -0,8 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8 1 -1 -0,5 0 0,5 1 fig. 2. diagram of the characteristic function training the back propagation network includes the following operations [24]: 1. select the next training pair from the training set; feed the input vector to the input of the network. 2. compute the output of the network. 3. calculate the difference between the output of the network and the required output (target vector of the training pair). 4. adjust the weights of the network to minimize the error. 5. repeat steps 1 to 4 for each vector of the training set, until the error on the whole set reaches an acceptable level. the operations in steps 1 and 2 are similar to those in the already trained network, i.e., you feed the input vector and calculate the resulting output. the computation is performed in layers. in fig. 3.3 first the outputs of neurons of layer j are computed, then they are used as inputs of layer k, the outputs of neurons of layer k are computed, which form the output vector of the network. in step 3, each of the outputs of the network, which in fig. 3 are labeled out, is subtracted from the corresponding component of the target vector to get an error. this error is used in step 4 to correct the weights of the network, with the sign and magnitude of the weight changes determined by the learning algorithm (see below). after a sufficient number of repetitions of these four steps, the difference between the actual outputs and the target outputs should decrease to an acceptable value, and the network is said to be trained. now the network is used for recognition and the weights are unchanged. steps 1 and 2 can be looked at as a "forward pass" as the signal propagates through the network from input to output. steps 3, 4 constitute the "reverse pass", here the calculated error signal propagates back through the network and is used to adjust the weights. these two passes will now be detailed and expressed in a more mathematical form. forward passage. steps 1 and 2 can be expressed in vector form as follows: input vector x is fed and the output is vector y. the vector pair input-target x and t is taken from the training set. calculations are performed on vector x to get output vector y. as we have seen, calculations in multilayer networks are performed layer by layer, starting from the layer closest to the input. the net value of each neuron of the first layer is calculated as a weighted sum of the neuron's inputs. then the activation function f "compresses" the net and gives the out value for each neuron in that layer. when the 216 program to automate the determination of the size of sanitary protection zones copyright ©0000 assa adv. in systems science and appl. (0000) output set of a layer is obtained, it is the input set for the next layer. the process is repeated layer by layer until the final set of outputs of the network is obtained. this process can be expressed in condensed form using vector notation. the weights between neurons can be thought of as a matrix w. for example, the weight from neuron 8 in layer 2 to neuron 5 in layer 3 is denoted by w8,5. then the net vector of layer n can be expressed not as the sum of products, but as the product of x and w. in vector notation, n = xw. component application of the function f to the net-vector n yields the output vector o. thus, for this layer the computational process is described by the following expression: о = f(xw) (2.1) the output vector of one layer is the input vector for the next layer, so calculating the outputs of the last layer requires applying equation (2.3) to each layer from the network input to its output. reverse pass. adjusting the weights of the output layer. since each neuron in the output layer has a target value, weights can easily be adjusted using a modified delta rule. the inner layers are called "hidden layers", for their outputs there are no target values for comparison. therefore, the training becomes more complicated. neuron in the hidden layer neuron in the outpyt layer tuning factor fig. 3. adjusting the weight in the output layer fig. 3 shows the learning process for one weight from neuron p in hidden layer j to neuron q in output layer k. the output of neuron of layer k, subtracted from the target value (target), gives an error signal. it is multiplied by the derivative of the compressive function [out(1 out)] calculated for this neuron of layer k, thus giving the value δ. δ = out(1 – out)(target – out) (2.2) then δ is multiplied by the out value of neuron j, from which the weight in question comes out. this product is in turn multiplied by the learning rate coefficient η (usually from p. istratov, y. minaev, e. zarubina, p. zolotariov 217 copyright ©2022 assa. adv. in systems science and appl. (2022) 0.01 to 1.0), and the result is added to the weight. the same procedure is performed for each weight from the hidden layer neuron to the neuron in the output layer. the following equations illustrate this calculation: δwpq,k = η δq,k out (2.3) wpq,k(n+1) = wpq,k(n) + δwpq,k (2.4) where wpq,k(n) is the value of weight from neuron p in the hidden layer to neuron q in the output layer at step n (before correction); note that index k refers to the layer where this weight ends, i.e, according to the convention adopted in this book, with which it is combined; wpq,k(n+1) is the value of the weight at step n + 1 (after correction); δq,k is the value δ for neuron q, in the output layer k; outp,j is the value out for neuron p in the hidden layer j. adjusting the weights of the hidden layer. consider one neuron in the hidden layer that precedes the output layer. as it passes forward, this neuron transmits its output to neurons in the output layer through the weights that connect them. during learning, these weights function in reverse order, passing the value δ from the output layer back to the hidden layer. each of these weights is multiplied by the value of δ of the neuron to which it is connected in the output layer. the value δ required for a hidden layer neuron is obtained by summing all such products and multiplying by the derivative of the compressive function (see fig. 4):         q kpq,kq,jp,jp,kq, δ)out(1outδ w (2.5) when the value of δ is obtained, the weights feeding the first hidden level can be adjusted using equations (2.5) and (2.6), where the indices are modified according to the layer. p r e l i m i n a r y l a y e r ( i ) hi dden layer ( j) output layer ( k) neuron p preli mi nary layer (i) fig. 4. setting the weight in the hidden layer for each neuron in a given hidden layer, δ must be calculated and all the weights associated with this layer adjusted. this process is repeated layer by layer towards the input, until all weights are adjusted. using vector notations, the error back propagation operation can be written in a much more compact way. let us denote the set of values δ of the output layer by dk and the set of 218 program to automate the determination of the size of sanitary protection zones copyright ©0000 assa adv. in systems science and appl. (0000) weights of the output layer as array wk. to obtain dj, the δ vector of the output layer, the following two operations are sufficient: 1.multiply the o-vector of the output layer dk by the transpose matrix of weights w'k connecting the hidden layer to the output layer. 2.multiply each component of the resulting product by the derivative of the compressive function of the corresponding neuron in the hidden layer. in the symbolic notation dj = dkw’k $[0j $(i – 0j)], (2.6) where the operator $ denotes the component product of vectors, oj is the output vector of layer j, and i is a vector whose components are all equal to 1. adding a neural bias. in many cases it is desirable to endow each neuron with a learning bias. this allows to shift the origin of the logistic function, giving the effect similar to the adjustment of perseptron neuron threshold, and leads to the acceleration of the learning process. this feature can be easily introduced into the learning algorithm by adding a weight to each neuron, attached to +1. this weight is trained the same way as all other weights, except that the signal applied to it is always equal to +1, not the output of the previous layer neuron. pulse. the literature [24] describes a learning acceleration method for the backpropagation algorithm that also increases the stability of the process. this method, called pulse, consists in adding to the weight correction a term proportional to the magnitude of the previous weight change. once a correction occurs, it is "remembered" and serves to modify all subsequent corrections. the correction equations are modified as follows: δwpq,k(n+1)= η δq,k outp,j + δwpq,k(n) (2.7) wpq,k(n+1) = wpq,k(n) + δwpq,k(n+1) (2.8) where (α – is the momentum coefficient, usually set around 0.9. using the momentum method, the network tends to follow the bottom of narrow gullies of the error surface (if any), rather than moving from slope to slope. this method seems to work well on some problems, but has a weak or even negative effect on others. there is a similar method based on exponential smoothing, which may have an advantage in a number of applications. δwpq,k(n+1)= (1-) δq,k outp,j + δwpq,k(n) (2.9) then the change in weight is calculated wpq,k(n+1) = wpq,k(n) + ηδwpq,k(n+1), (2.10) where α the smoothing coefficient varies and ranges from 0.0 to 1.0. if equal to 1.0, the new correction is ignored and the previous correction is repeated. in the region between 0 and 1, the weight correction is smoothed by a value proportional to . still, η is the learning rate coefficient, serving to control the average weight change. for the development of the program was used (type of implementing computer) pc (personal computer). the programming language was c++ builder xe8. operating system used: windows xp, vista, windows 7, windows 8, windows 10. for system training there was created a database (certificate of database registration no. 2021622507 dated 09.11.2021) for storage and issuing of operative and reference information about condition of ionizing radiation sources (in places of radioactive waste disposal/storage) and toxic pollution at their quantitative content which is/are insignificantly exceeding operating hygienic norms/levels. the description of the database provides a list of information fields for a comparative analysis of toxic air pollution in the analyzed conditions of the residential area and residential area of comparison settlements. for each item, 32 measurements were made (presented in the database). p. istratov, y. minaev, e. zarubina, p. zolotariov 219 copyright ©2022 assa. adv. in systems science and appl. (2022) indicators were analyzed in the computer program: natural background radioactivity (values were analyzed); character of soil (variants); presence of forests (variants); amount of precipitation; radionuclides (excess of specific activity in the analyzed soil samples); content of chemical pollutants in the air of the residential area (normal/exceeding the average daily mpc). 4. results for the development and preparation of computer implementation of the proposed program, neural networks were chosen as a variant of implementation of artificial intelligence systems. any expert system, including the one under development, should conditionally consist of four blocks: an interface with the user, a knowledge base, a computing block, a block of explanations, allowing the user to trace the "course of reasoning" of the system in a particular case. the connecting element between these blocks is the method by which the expert system, in response to the user's request, produces a result (conclusion). the paper proposes a classification of such methods into three main groups: 1) methods of logical rules "in pure form", when the formalization of the rules of obtaining the result is carried out by the expert; 2) the same methods, but formalization of rules is carried out by the researcher, observing the work of the specialist from outside; 3) methods based on the principle "look and learn". creating even simple expert systems based on methods 1 and 2 is a complex task, primarily because it requires the joint work of specialists of different profiles. traditional expert systems based on knowledge bases and logical rules require quite a lot of time and money for creation. creation of expert system can be divided into several stages. 1. task setting: defining the objectives of the expert system, the set of input data and the form of presentation of the answer. 2. data collection: a set of representative material for statistical research and its structuring division into subgroups according to various attributes. 3. statistical processing: identification of patterns linking the input data to the answer calculation of averages and relative values, their comparison, correlation, regression, factor analysis, etc. 4. creating a knowledge base: drawing up logical rules by which the expert system should work. 5. algorithm programming: transferring logical rules to a programming language. 6. creation of system interface: development of means of interaction of the system with the user data input forms, response output, etc. 7. debugging and testing: checking the program operation and testing in real conditions. when creating logical expert systems, the 3rd, 4th and 5th stages take the most time and require the joint work of both subject specialists and programmers and mathematicians. despite the emergence of computer tools for designing expert systems, the main work is entrusted to specialists. several serious problems arise in this case. the first one is that when solving complex real-world problems (medicine, biology), the number of logical rules increases significantly. often there is such a complex system of interrelations between them that it simply cannot be comprehended. breaking a problem into blocks does not always help either: firstly, it is not always easy to do, and secondly, when breaking it down, some connections may sometimes get lost. 220 program to automate the determination of the size of sanitary protection zones copyright ©0000 assa adv. in systems science and appl. (0000) the second and even more serious problem is that it is not always possible to express the computational process by logical rules. this may be due both to the complexity of the problem itself, and to the peculiarities of the subject specialist's activity. this is especially evident in medicine, where the decision-making process largely relies on the intuition and experience of a physician who is not an expert in his own thinking. in all of these cases, it is said that the task is not amenable to algorithmization. moreover, even if the creators manage to develop an algorithm, there is never a sufficient guarantee that it will work correctly in real conditions, and this can be checked only after all the work on creating the system has been completed. in the creation of the self-learning system developed by the authors (certificate of registration of computer software №2021681088 from 02.12.2021), you can identify several stages, some of which coincide with the stages of creating traditional systems. 1. problem statement. the same as for the traditional systems plus the choice of the optimal structure of a neural network and training methods (for the majority of problems the structure and methods are standard). \ 2. gathering of training data. a set of examples for training the network, each of which presents an array of input data and a corresponding response known in advance (database registration certificate № 2021622507, 09.11.2021). collection of data for further training of the network was carried out during 3 years for all selected parameters in conditions of different radiation background, chemical pollution, natural-climatic factors and their combinations, which entered into a single database. 3. neural network creation and training. this stage does not require any statistical calculations. the training of the developed neural network was an automatic process, which only after its completion required the participation of a specialist to evaluate the results and compare them with the existing database. schemes of problem statement, ways of data representation and ways of response production by the neural network are designed in such a way that most of the problems fit into these standard schemes. 4. creation of interface of this program is the same as for traditional expert systems. 5. debugging and testing. the stage included mainly debugging of program work, as testing was carried out in the process of network training. the available advantages of this neural network expert system are shown when solving hard-to-algorithm problems: 1. a neural network makes decisions on the basis of experience, which it acquires independently. "independently" in this case means that a user of expert system does not need to establish relationships between input data and necessary decision, spending time on various statistical processing, selection of mathematical apparatus, creation and checking of mathematical models. 2. the decision made by a neural network is not categorical. the network gives out the decision together with a degree of confidence in it, that leaves to the user an opportunity to critically estimate its answer. 3. a neural network allows to model a decision-making situation. 4. a neural network gives an answer very quickly (fractions of a second), which allows to use them in various dynamic systems, which require immediate decision-making. it should be emphasized that application of implicit algorithms does not contradict and does not cancel use of formal methods, but can be complemented by them if necessary. the program has additional functions: saving the results of the work; printing the results of the forecast; clearing the values of the indicators to be filled in. the selection of indicators was carried out in various combinations taking into account the results obtained after training of artificial neural networks and evaluation of the results of their work (fig. 5). p. istratov, y. minaev, e. zarubina, p. zolotariov 221 copyright ©2022 assa. adv. in systems science and appl. (2022) feature name meaning score natural radioactivity background soil characteristics forests downfall radionuclides chemical contamination per year slay soil greenwood downfall per year cl clear calculati on about pri nt exi t fig. 5. main screen of the computer program subsequent tests of the neural network showed that the quality of its work for an objective assessment was 80% of the correct solutions, which confirmed the possibility of using ai for mathematical modeling of the assessment of harm of personnel work in the conditions of sanitary protection zones, as well as calculating the minimum distances from the raw object to the residential development in a particular area (fig. 6). expert system forecast model prediction expert system conclusion estimated size of the sanitary protection zone 16 km close fig. 6. results screen 5. conclusion the developed computer program: "automation of determination of dimensions of sanitary protection zones (spz) and surveillance zones for storage of sources of ionizing radiation in places of burial / storage of radioactive waste" was intended to automate the process of calculations of dimensions of sanitary protection zones (spz) and surveillance zones for storage of sources of ionizing radiation in places of burial / storage of radioactive waste based on mathematical regression models. the main advantage of the developed neural network is that it can identify non-obvious attributes for determining the size of sanitary protection zones (spz). these algorithms do not rely on the available knowledge about the influence of the analyzed indicators, which makes them objective and allows to use them to create models of the state of storage of radiation sources. 222 program to automate the determination of the size of sanitary protection zones copyright ©0000 assa adv. in systems science and appl. 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(1989) neurocomputer technique: theory and practice. new york, ny: van nostrand reinhold co. microsoft word 1359-article text-6029-1-6-20230530 adv syst sci appl 2023; 02; 115-127 published online at https://ijassa.ipu.ru. evaluating the efficacy of the sliding mode algorithm for the active suspension system tuan anh nguyen* thuyloi university, hanoi, vietnam abstract: the active suspension system is installed in the car to improve the smoothness and comfortable ride while driving. in this paper, a quarter-dynamic model is utilized to describe the car's vibrations. the sliding mode control algorithm is proposed to control the operation of the suspension system. the problem's input signal is the excitation from the road surface, while the output signal includes values related to the acceleration and displacement of the vehicle body. in addition, the interaction between the wheel and the road surface, which is expressed through the change of dynamic force, is also considered. the numerical simulation is performed with a sinusoidal bump excitation signal. according to this result, the car body displacement was greatly reduced when using an active suspension controlled by the sliding mode method. the fluctuation of the unsprung mass is insignificant, so it does not affect the car's smoothness. this helps increase wheel stability and holding. the effect that the sliding mode algorithm brings to this study is positive. keywords: sliding mode control; vehicle vibration; vehicle dynamic; matlab simulink 1. introduction in recent years, car vibration has become an important and researched topic. there are many causes of car vibration [1], such as wind and external forces. besides, the engine and transmission system operation also cause the car to oscillate [2]. however, these factors only affect a small part of the vehicle's vibration. the main cause of car vibration is the stimulus from the road surface [3]. in fact, the road surface is not flat, so it causes the wheels to oscillate. the vibrations from the wheels are transmitted to the car body through the suspension system, which causes the body to vibrate. vibrations in the car make passengers feel tired and uncomfort [4]. in addition, the goods' quality and the car's life may be reduced if the oscillation occurs continuously over a long period. the suspension system on a car plays an important role in ensuring the smoothness of the vehicle [5]. the suspension system quenches much of the car's vibrations, while the rest depends on the tires and other factors. traditional suspension consists of only three main components: the lever arm, the spring, and the damper. some other components, such as the rubber, stabilizer bar, multi-link bar, etc., can also be considered components of the suspension system [6]. only springs and dampers are considered for studies related to vehicle vibration and ride comfort [7], meanwhile remaining components, such as the lever arm, stabilizer bar, etc., can be mentioned in the studies of vehicle oscillation and stability [8]. a suspension system using springs and dampers with constant stiffness (k = const, c = const) is called passive suspension. today, many modern suspension systems have been used to improve the car's smoothness when traveling. modern suspension systems can either change the spring stiffness (by using pneumatic springs) [9] or change the damping stiffness (by using electromagnetic dampers) [10]. the stiffness of these elements can be changed dynamically to improve the car's smoothness. although suspension stiffness can be changed based on using the semi-active suspension (damper control) or air suspension (air spring * corresponding author: anhngtu@tlu.edu.vn 116 tuan anh nguyen copyright ©2023 assa adv. in systems science and appl. (2023) control), the system response has not yet reached the ideal state. therefore, the solution of retrofitting a hydraulic actuator to the suspension system is a perfect fit. the hydraulic actuator can apply force to the vehicle body to suppress unwanted vibrations. the suspension using a separate hydraulic actuator is known as the active suspension [11]. many papers related to active suspension control have been published in recent years. in [12], khodadadi and ghadiri introduced the pid algorithm (proportional integral derivative) to control the suspension system. this algorithm has three coefficients: kp, ki, and kd for amplification, integration, and differential stages, respectively. these coefficients are tuned by a fuzzy algorithm that includes three states: input, output, and processing. the pid algorithm is quite simple and widely used in industry; however, it is only suitable for systems with one input and one output. if the system has a lot of output and/or input, it is necessary to use a different algorithm instead. in [13], nguyen et al. applied the lqr (linear quadratic regulator) algorithm to control the quarter suspension model. according to nguyen et al., determining the controller's coefficients involves four specific steps. the determination of these values is based on the point of view of selecting the minimum rms (root mean square) value of the outputs. an additional filter is necessary if the system is affected by interference signals from the outside. usually, the gaussian filter can be combined with the lq algorithm to become the lqg controller [14]. the above control algorithms are perfectly suitable for linear oscillation systems. however, the actual oscillation of the car is nonlinear, so more complex control algorithms need to be applied. in [15], qin et al. proposed using raisc (road adaptive intelligent suspension control) for the quarter dynamics model. according to qin et al., the structure of the raisc controller is divided into two phases: offline and online. the offline phase assumes the task of determining the optimal values of the controller based on the pso (particle swarm optimization) method, while the online phase performs the task of generating control signals for the system. in [16], huang et al. proposed a new adaptive robust control law for the suspension system. this rule can satisfy the uncertainty conditions of the nonlinear system. this controller is synthesized from three component signals. the first signal is received from the system's output, while the other two must undergo filtering and adaptive tuning. for complex nonlinear algorithms, it is common to design indirect rather than direct controllers [16, 17]. once the dynamic model of the actuator (direct control) is considered, the algorithm becomes much more complex. therefore, the model of the actuator should be linearized to simplify the control problem. in [18], nguyen used a quarter model with 5 state variables (the fifth variable is the influence of the actuator). the actuator coefficients are described in the equation (26), and they depend on many factors such as the piston cross-sectional area, fluid pressure, hydraulic cylinder flow, fluid viscosity, fluid leaks, etc. [19]. in [20], nguyen and nguyen proposed using the osmc solution (optimal sliding mode control) for an active suspension system. the model parameters are optimally selected by the loop algorithm. besides, there are many other intelligent control algorithms utilized to control the suspension system, such as type-2 fuzzy [21], fuzzy-pi [22], type-2 fractional order fuzzy [23], fuzzy smc [24], neural network [25], etc. overall, the efficiency of these algorithms is positive. however, the controller design process is quite complicated. the author proposes using the smc method for a quarter suspension model in this study. because the actuator is considered, the controller designed in this study is a direct controller. previous studies often only considered the car's smoothness when designing the controller, but ignored the car's stability. in this paper, both issues related to smoothness and stability are considered. the paper's content is divided into 4 sections, including an introduction section, a model section, a simulation section, and a conclusion section. specific content corresponding to each section is presented in the below sections. evaluating the efficacy of the sliding mode algorithm… 117 copyright ©2023 assa. adv. in systems science and appl. (2023) 2. model 2.1. sliding mode control theory considering a nonlinear control object with y(t) output and u(t) input signals. the nth derivative of the output signal is a function that depends on the component derivation signals and input signals.         1, , ,...,n ny t f y y y y u t   . (1) assume that the function f(.) is undefined and bounded, i.e.:  || . ||f    . (2) let xi be the state variables of the model:   1 2 1 ... n n x y x y x y      . (3) the model of the object is returned in the form of a system of state equations as follows:     1i ix x x f x u t       1, 1i n  (4) where  1 2 ... t nx x x x . assuming that the set point signal is zero, that is, h(t) = 0 (figure 1). if the nonlinear object (4) is restricted by condition (3), there is always a signal response controller that does not depend on the function f(.) of the model. so, the control signal can be rewritten as follows:      u t k sgn s  . (5) fig. 1. system control. here: s(e): sliding surface (when s(e) = 0), e(t): error signal,                2 1 0 1 2 2... n n ns e b e t b e t b e t b e t e t         , (6)        e t h t y t y t    . (7) the bi coefficients of the slip surface need to be chosen appropriately, so that (8) is a hurwitz polynomial. once this condition is met, the state variables tend to zero after a specific time (9).   1 2 2 1 0 1 2 2... n n np b b b b            . (8)  lim 0 t x t   . (9) because:   1 1i i e x x e       . (10) 118 tuan anh nguyen copyright ©2023 assa adv. in systems science and appl. (2023) therefore, the equation (9) is able to write in the following form:  lim 0i t t e    (11) where t is a finite time point. if the equation s(e) = 0 has coefficients bi satisfying the condition of the hurwitz polynomial (8), the sliding surface s(e) tends towards zero, i.e.:   0s e s  . (12) the equation (12) is called the sliding condition of the controller. from (4), (6), and (12), we have:       1 2 2 1 2 2 0 0 0 ( ) n n n i i i i n i i i i i s e b e b x x b x f x u t                     . (13) the value of (13) is positive if s(e) is less than zero, and vice versa if s(e) is greater than zero. the control signal u(t) can be rewritten by combining (2) and (13) as follows:         2 2 0 2 2 0 ; 0 ; 0 n i i i n i i i u t b x s e u t b x s e                      . (14) the control signal u(t), which is determined by the equation (14), does not depend on (4). therefore, it is considered a robust controller. if condition (2) is not satisfied, it is necessary to define an upper limit of the function f(.), i.e.:     ,f x g x x  . (15) then condition (13) becomes:             2 2 0 2 2 0 ; 0 ; 0 n i i i n i i i u t b x g x s e u t b x g x s e                    . (16) 2.2. suspension control model considering the active suspension model (figure 2). the model consists of m1 (the sprung mass) and m2 (the unsprung mass). the spring is described by its k stiffness, and the damper is described by its c coefficient. the damping component of the tire is ignored, so only the elastic characteristic of the tire (kt) is considered. a hydraulic actuator is considered when using an active suspension. the force that is produced by the actuator is considered a state variable of the system. evaluating the efficacy of the sliding mode algorithm… 119 copyright ©2023 assa. adv. in systems science and appl. (2023) fig. 2. a quarter dynamic model. applying d'alembert's principle, the car's oscillation is described by equations (17) and (18), respectively, with two degrees of freedom, x1 and x2. 1m k c a if f f f   , (17) 2m kt k c a if f f f f    . (18) where: the inertia force: 1 1 1 m if m x  , (19) 2 2 2 m if m x  . (20) the elastic force of spring and tire:  2 1kf k x x  , (21)   2kt tf k r t x  . (22) the shock absorber resistance force:  2 1cf c x x   . (23) combining equations from (17) to (23), we get:    2 1 2 1 3 1 1k x x c x x x m x       , (24)       2 2 1 2 1 3 2 2tk r t x k x x c x x x m x         . (25) where: r(t): excitation signal from road surface, x3: acting force of the actuator, x3 = fa, according to [26], the state variable x5 can be approximated by the equation (26).    3 1 2 3 3 2 1x u t x x x        . (26) with: u(t): the output voltage signal of the controller, i: actuator's coefficients. let's: 4 1x x  , (27) 5 2x x  . (28) 120 tuan anh nguyen copyright ©2023 assa adv. in systems science and appl. (2023) taking the derivative of the variables x4 and x5, we get:  4 1 2 3 4 5 1 1 x kx kx x cx cx m       , (29)   5 1 2 3 4 5 2 1 tx kx k k x x cx cx m       . (30) the output signal of the simulation problem is the sprung mass displacement.   1y t x . (31) the model's higher-order derivative signal has the form of the equation (32) (n = 5):       1 2 1 2 1 2 2 2 5 1 2 3 3 4 1 2 1 2 1 2 1 2 2 2 3 5 1 2 1 1 t t t t k kk kc x c x m m m m k kc c c c y x k x u t m m m m m m m m c c k k x m m                                                              . (32) the sliding surface is an important component of the sm controller. according to (6), the sliding surface is written in the following form:      3 4 0 1 2 3 4s e e e e e e          . (33) the control signal is written as a function depending on the sliding surface of the controller.             5 4 4 5 1 1 0 i i s i i i i i i i u t c y a x e t rsgn e t                    . (34) where: ai, i, c, and r are coefficients. after the control algorithm has been established, the simulation will be performed. 3. simulation in this work, the numerical simulation method is used to investigate car vibration. the simulation process is done in the matlab/simulink environment. a sinusoidal bump excitation is utilized (figure 3). this is a common type of excitation signal that is frequently used in many simulation problems related to suspension control. the bump from the road surface is the input to the oscillation problem, while the output of the problem includes changes in displacement, acceleration, dynamic force, etc. these results are represented in the time domain corresponding to the simulation time. the maximum and rms values of the results obtained are used for comparison and evaluation. the following mathematical function represents the excitation signal from the road:    0 2h t h sin ft   . (35) there are three scenarios covered when simulating, including: +) a car using traditional suspension system (none) +) a car using linear control suspension (pid) +) a car using nonlinear control suspension (sm) evaluating the efficacy of the sliding mode algorithm… 121 copyright ©2023 assa. adv. in systems science and appl. (2023) fig. 3. bump on the road. the technical parameters of the simulation vehicle are shown in table 1. table 1. the simulation vehicle parameters. symbol unit value description k n/m 40500 coefficient spring kt n/m 174000 coefficient tire c ns/m 3150 coefficient damper m2 kg 36 unsprung mass m1 kg 410 sprung mass the road excitation signal used is sinusoidal. this is a low-frequency cyclic variable signal that is often used in problems related to control in suspension systems. the change in the vehicle body displacement over time is shown in figure 4. the vehicle body oscillation is greatest when using a traditional suspension system without control. in contrast, car vibrations are smaller if an active suspension system is used. the vertical displacement of the sprung mass is 70.83 (mm), 23.36 (mm), and 10.03 (mm) respectively, corresponding to three investigated situations: none, pid, and sm. the rms of values related to vehicle vibrations is also commonly used to assess car smoothness. in this case, the rms value of displacement corresponding to the situations is: 45.81 (mm) – none, 16.54 (mm) – pid, and 7.07 (mm) – sm. 122 tuan anh nguyen copyright ©2023 assa adv. in systems science and appl. (2023) fig. 4. vertical displacement (sprung mass). when the displacement of the vehicle body changes, the value of the sprung mass acceleration will change accordingly. the change in vehicle body acceleration is shown in figure 5. in the first stage, the change in acceleration is the largest; these values can be up to 2.52 (m/s2), 1.82 (m/s2), and 1.08 (m/s2), corresponding to the three investigated cases. this value changes cyclically in later stages and reaches a more stable state. because this is a continuous periodic oscillation, the rms indicator should be used. the result of the acceleration value when calculating according to the rms criterion is 1.28 (m/s2) – none, 0.44 (m/s2) – pid, and 0.19 (m/s2) – sm, respectively. according to this result, the car body's rms acceleration when using the active suspension system directed by the sm method is only 14.84% compared to the car using the traditional suspension system. meanwhile, the rms value of acceleration when the pid algorithm is used is 34.38% compared to the first situation. this leads to the conclusion that the sm control algorithm enhances the stability and smoothness of the sprung mass. evaluating the efficacy of the sliding mode algorithm… 123 copyright ©2023 assa. adv. in systems science and appl. (2023) fig. 5. vertical acceleration (sprung mass). the vehicle body oscillation can be further improved using the active suspension system with the sm mentioned above algorithm. however, it is still necessary to investigate the oscillations of the unsprung mass. besides, the change in the dynamic load of the wheel is of great significance to assessing the car's stability when moving. figure 6 shows the change in the unsprung mass's vertical displacement during the simulation period. this value tends to be asymptotic to the excitation signal from the road (figure 3). the difference in values in all three situations is very small, namely 54.84 (mm), 51.57 (mm), and 50.82 (mm), which correspond to the maximum values; 38.51 (mm), 36.68 (mm), and 36.15 (mm), which correspond to the rms values. fig. 6. vertical displacement (unsprung mass). 124 tuan anh nguyen copyright ©2023 assa adv. in systems science and appl. (2023) for the acceleration of the unsprung mass, the difference exists in the first period (figure 7). the acceleration in the sm case is larger than the other two; however, this only happens for a very small amount of time. in the subsequent phases of the oscillation, the difference between the values is very small. for the car using the active suspension controlled by the sm solution, "chattering" occurred when calculating the acceleration of the unsprung mass. the unsprung mass acceleration signal vibrates at the phase transition's beginning. this vibration is not large, so it does not affect the smoothness and stability of a car. fig. 7. vertical acceleration (unsprung mass). in order to improve the ride comfort of the automobile, the actuator of the active suspension system needs to generate actuating forces that affect both the sprung and the unsprung mass. if the impact force is too large, the change in dynamic load will also be larger. this increases the risk of wheels separating from the road and can lead to vehicle instability. therefore, the change of dynamic load at the wheel is one of the very important factors to help evaluate the automobile's stability. this change is shown in figure 8. as a result of figure 8, the value of the dynamic force changes continuously based on the excitation signal from the road. the biggest change belongs to the situation where cars only use conventional suspension systems. the dynamic load on wheels was reduced from 4286.97 (n) to 3381.09 (n). meanwhile, the minimum dynamic load values of the other two situations are 3384.33 (n) and 3596.45 (n), respectively, for pid and sm situations. the rms value of the dynamic load did not change too much, reaching 4331.92 (n), 4295.78 (n), and 4290.47 (n), respectively. evaluating the efficacy of the sliding mode algorithm… 125 copyright ©2023 assa. adv. in systems science and appl. (2023) fig. 8. dynamic load. the change in the dynamic load is expressed as a percentage, as shown in figure 9. this graph depicts the change in figure 8 as a percentage to all three scenarios. during the first oscillation phase, the "none" situation variation can be as close as 25%. in subsequent oscillation phases, the value of "none" does not exceed 17.1%, while the values of "pid" and "sm" are only about 6.8% and 3.3%, respectively. this result has shown that the actuation force of the actuator when a car uses the active suspension system does not harm the vehicle's stability. fig. 9. dynamic load change. 126 tuan anh nguyen copyright ©2023 assa adv. in systems science and appl. (2023) 4. conclusion irritation of the road is the main cause of car vibration. the suspension is utilized to quench these oscillations. the active suspension system with hydraulic actuators is more efficient than the conventional one. as a result, the car's smoothness and stability have improved. in this study, the author proposes to use the smc solution to control the active suspension system in cars. a quarter-dynamic model that incorporates hydraulic actuators is used to describe vehicle vibrations. the numerical simulation is conducted in the matlab/simulink environment in the time domain. according to the research findings, the car body displacement and acceleration values are strongly reduced when utilizing the active suspension system directed by the sm algorithm. in addition, the oscillation of the unsprung mass is not large (equivalent to vehicles using passive suspension). so, the comfort and smoothness of a car are improved. besides, the change in dynamic load at the wheel when using active suspension is smaller than traditional suspension. this helps to enhance the vehicle's stability when oscillating. the sm algorithm can help the active suspension operate more steadily; however, the phenomenon of "chattering" still exists. in some cases, "chattering" can have a negative effect on the smoothness of the car (usually with high-frequency nonlinear oscillations). in the near future, studies on reducing the "chattering" phenomenon when applying the sm algorithm to the suspension system will be carried out. references [1] guo, h., et al. (2019). research on improvement of truck vibration based on systematic g8d method. shock and vibration, 2019, 1–11, https://doi.org/10.1155/2019/1416340 [2] lee, y., lee, s., choi, h., and min, k. (2019). analysis of vibration on an engine block caused by combustion in a diesel engine. international journal of automotive technology, 20(1), 187-195, doi: https://doi.org/10.1007/s12239-019-0018-7 [3] agostinacchio, m., ciampa, d., & olita, s. (2014). the vibrations induced by surface irregularities in road pavements – a matlab approach, european transport research review, 6, 267–275, https://doi.org/10.1007/s12544-013-0127-8 [4] asua, e., et al. (2022). analysis of the motion sickness and the lack of comfort in car passengers. applied sciences, 12(8), 3717, https://doi.org/10.3390/app12083717 [5] moradi, s. m., akbari, a., & mirzaei, m. (2019). an offline lmi-based robust model predictive control of vehicle active suspension system with parameter uncertainty, transactions of the institute of measurement and control, 41(6), 1699–1711, https://doi.org/10.1177/0142331218787599 [6] nguyen, d. n., nguyen, t. a., and dang, n. d. (2022). a complex rollover dynamics model with active stabilizer bar controlled by the fuzzy algorithm, heliyon, 8(11), e11715, https://doi.org/10.1016/j.heliyon.2022.e11715 [7] omar, m., el-kassaby, m. m., & abdelghaffar, w. (2017). a universal suspension test rig for electrohydraulic active and passive automotive suspension system, alexandria engineering journal, 56, 359–370, http://doi.org/10.1016/j.aej.2017.01.024 [8] nguyen, t. a. (2021). preventing the rollover phenomenon of the vehicle by using the hydraulic stabilizer bar controlled by a two-input fuzzy controller, ieee access, 9, 129168-129177, http://doi.org/10.1109/access.2021.3114023 [9] nazemian, h., & masih-tehrani, m. (2020). development of an optimized game controller for energy saving in a novel interconnected air suspension system, proceedings of the institution of mechanical engineers, part d: journal of automobile engineering, 234(13), 3068–3080, http://doi.org/10.1177/0954407020927147 [10] ma, x., wong, p. k., & zhao, j. (2019). practical multi-objective control for automotive semi-active suspension system with nonlinear hydraulic adjustable evaluating the efficacy of the sliding mode algorithm… 127 copyright ©2023 assa. adv. in systems science and appl. (2023) damper, mechanical systems and signal processing, 117, 667–688, https://doi.org/10.1016/j.ymssp.2018.08.022 [11] kilicaslan, s. (2018). control of active suspension system considering nonlinear actuator dynamics, nonlinear dynamics, 91, 1383–1394, https://doi.org/10.1007/s11071-017-3951-x [12] khodadadi, h., & ghadiri, h. 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(2022). a novel hybrid control algorithm sliding mode-pid for the active suspension system with state multivariable, complexity, 2022(30), 1–14, https://doi.org/10.1155/2022/9527384 adv syst sci appl 2023; 03:27–35 published online at https://ijassa.ipu.ru. deep-learning-based tracing for satellite telemetry ilya nachevsky1, olga andrianova1*, isaac chairez2 1v.a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia 2tecnologico de monterrey, institute of advanced materials for sustainable manufacturing, guadalajara, jalisco, mexico abstract: the mechanical section of the telemetry data from the scientific satellite “lomonosov”, (such as yaw, pitch, and roll angles of the spacecraft’s main axes, along with its programmed and measured velocities) is pre-processed (alignment, reflection, and binarization) and used for anomaly behavior propagation. the main goal of this study is to estimate possible abnormal behavior of the system in the future and to help to restore normal behavior during a limited communication session with a spacecraft. the system model uses a recurrent architecture approach, namely tracing methodology, considering time shifts in the target data sequence. a deep learning strategy is used to model the abnormal behavior using the onboard collected mechanical information as inputs. the results are compared with the onboard anomaly detection system (aro) data. the reproduction of the obtained information shows better performance compared to traditional estimation techniques, using binary cross-entropy and receiver operating characteristic curve (rocauc) as comparison criterion. future model modifications, which can improve its quality, are discussed at end of the study. keywords: trajectory tracing, anomaly propagation, telemetry analysis, recurrent model, time-distributed 1. introduction telemetry data from the scientific satellite “lomonosov” could have anomaly behavior, which must be caught during a limited communication session with a spacecraft or propagated based on the previous values of the estimated data, i.e., the preliminary session. space companies have onboard systems for catching anomalies in data. still, unfortunately, they are far from ideal and often provide information about a critical anomaly in data after the fact, so it is crucial to develop anomaly propagation systems based on the current data, considering the dynamic capture of the satellite motion. the problem of restoring dynamic dependencies in the various data types is significant and widely studied in artificial intelligence and data analysis. for example, a recent paper (tuturov, andrianova, sleptsov, yurkevich, & kryukova, 2022) has already described an approach to identifying and predicting anomalies, based on the previous experience of the system, using differential neural networks (poznyak, oria, & poznyak, 2019), whose workability is strictly proven by stability theory methods. there also exists a scope of techniques to detect anomalies (zeng, jin, xu, chen, & zhang, 2022; luo & nagarajan, 2018), that aims to get deviant behavior from a system dynamics. the first approach uses parametric causality, and double-criteria drift streaming peaks over threshold (dcdspot) (chen, lu, fang, et al., 2014) in a space satellite system. in contrast, the second approach employs an autoencoder architecture in a wireless sensor networks (wsn) (ddos ∗corresponding author: andrianovaog@gmail.com 28 i. nachevsky, o. andrianova, i. chairez attacs) (nishio et al., 2022). another approach uses a cnn-based technique to detect the anomalies of zero-day attacks (hairab, elsayed, jurcut, & azer, 2022). this paper aims to propagate deviant or abnormal behavior, which means that the device operates under disturbed conditions or has a defect in the control or mechanical systems. the cases of abnormal system functioning can be described as sophisticated linked moments that can be detected in telemetry data, and the nature of which can be tracked in some patterns of event sequence. in contrast to the papers mentioned above, the main goal is to detect anomalies in the data. this paper intends to propagate target anomaly events, i.e., to estimate the eventual presence of anomaly behavior for future moments. the main contributions of this study are: 1. tracing methodology is implemented to propagate anomaly behavior in telemetry data; 2. the developed architecture considers time shifts in target data sequence in the same manner as the onboard anomaly detection system; 3. propagation results of the new technique show better characteristics than the onboard system. the rest of the paper is organized as follows. in section 2, telemetry data is described, and the main problem is stated. section 3 deals with the preprocessing of the measured data. in section 4, model design, including model structure and training and validation processes, is presented. simulation results are available in section 5. section 6 contains some concluding remarks and ideas for future work. 2. telemetry data description and problem statement 2.1. system under consideration the system under study is given by telemetry data from the scientific satellite “lomonosov”. from a variety of different technical data. the spacecraft orientation and stabilization (sos) system has been selected because the functional stability (korolev, 2018) of the entire spacecraft directly depends on the operation of this system. it can be helpful for anomalies and emergency detection, localization, and further prediction after designing a mathematical model of the considered processes. the onboard software’s existing automatic reconfiguration (aro) mechanism can be considered a baseline for anomaly detection. to imagine the type of data aro mechanism provides (see fig. 2.1), this figure shows three triggers of the anomaly detection system in 160 seconds. in aro indicator, weight values, expertly assigned to each functional element of all spacecraft systems, are summed up. when the value of the aro indicator leads to faults in the operation of the spacecraft elements and it falls below the expert-defined value, then the procedure for changing the orientation mode of the spacecraft is automatically started. this procedure aims at stabilizing the spacecraft operation. the following mechanical data, provided by sos system, are chosen to form the state vector x for further processing and model design: x = [ α υ ϑ ] where α ∈ r3 stands for the main spacecraft axes angles (yaw, pitch, and roll); υ ∈ r4 and ϑ ∈ r4 are the vectors of the measured and programmed angular velocities of the four wheels, respectively. for example, fig. 2.2 depicts the dynamics of the roll angle within 160 seconds, while fig. 2.3 shows the programmed angular velocity of the first wheel. some problems happened due to the specifics of data reading from sensors, processing, and writing in log files. the troubles contain time shifts and data absents. for example, in this telemetry report, the most extended shift is approximately 1.875 seconds (15 system beat). one system beat is 0.125 seconds. copyright © 2023 assa. adv syst sci appl (2023) deep-learning-based tracing for satellite telemetry 29 fig. 2.1. aro data fig. 2.2. roll angle fig. 2.3. programmed angular velocity of the first wheel copyright © 2023 assa. adv syst sci appl (2023) 30 i. nachevsky, o. andrianova, i. chairez 2.2. problem statement the main problem is to propagate anomalies in telemetry data and make anomaly detection better than the existing aro response. in the presented system, the state vector has nonlinear nature. so, the tracer architecture, which can catch sophisticated connections between vector elements and time dependence due to its recurrent nature (piech et al., 2015), can be successfully used for model design. another problem that needs to be solved in this study is to convert raw telemetry data to a binarized form (where “1” stands for an anomalous value of the element, and “0” indicates the absence of the anomaly at the current time) of each state vector element using statistical methods, such as lowess aligner, anomaly detector in the form of quantile barrier (zeng et al., 2022). 3. preprocessing in statistics, the term lowess refers to “locally weighted scatterplot smoothing”. as one of the standard approaches to produce a smooth curve that fits the data points in a scatterplot, this method is applied to the raw telemetry data. the results of its operation for the programmed angular velocity of the first wheel are shown in fig. 3.4. fig. 3.4. lowess algorithm applied to the programmed angular velocity of the first wheel further, negative values are reflected in the positive area, as the module of lowess stands for the barrier of the quantile barrier function qbar (|lowess(x)|) where qbar(z) = { 1 if qx(0.95) < z, 0 if qx(0.95) ≥ z, and qx is the quantile by sampling the state vector. fig. 3.5 demonstrates data processing results, namely alignment, reflection, and binarization for the programmed angular velocity of the first wheel data. the binarized plot has 100 values by ordinate axis for the reasons of clarity. the binarized data for each element of the state vector is recorded with the sequence of labels from 0 to 10 (the spacecraft main axes angles, measured and programmed angular velocities of the wheels). in this form, the dataset is used for further network training. copyright © 2023 assa. adv syst sci appl (2023) deep-learning-based tracing for satellite telemetry 31 fig. 3.5. lowess algorithm applied to the programmed angular velocity of the first wheel 4. model design the recurrent architecture approach has already been used for propagation sequence behavior (piech et al., 2015). in this work, a similar methodology, namely tracing, is applied. the sequence of events has information about sophisticated linked moments, which are causes of abnormal behavior. 4.1. model structure fig. 4.6. model architecture fig. 4.6 depicts the first part of information coded on the embedding layer (khrulkov, hrinchuk, mirvakhabova, & oseledets, 2019). the approach helps to represent data in the new form and catch knowledge of relation from parts of the sequence. the second one is the recurrent neural network (rnn)-based layer considering a long-short term memory (lstm) implementation. the layer can remember information and gather it through all query items. this approach helps to get specific dependencies between events or even a chain of events. after getting a response from the layer, we need to make a solution about the nature of the sequence by compressing the sparse answer space. the third part is a fully connected layer, which compresses the lstm response and helps predict the sequence’s nature by items. by the way, we make a propagation about the next step of the sequence with analysis from different sources (telemetry from each sensor). the learning strategy is copyright © 2023 assa. adv syst sci appl (2023) 32 i. nachevsky, o. andrianova, i. chairez based on the optimization by adam method (kingma & ba, 2017) through back-propagation through-time mechanism (rumelhart, hinton, & williams, 1986). 4.2. training and validation process the above-described architecture is trained with satellite telemetry data. fig. 4.7 and fig. 4.8 demonstrate training and validation processes. rocauc is used as a metric such as in related work (piech et al., 2015), which gives clearer quality in case of unbalanced classes. binary cross-entropy (bce loss function) is chosen as a loss function, which shows a dissimilarity measure between two distributions and is based on kullback-leibler divergence (kullback-leibler divergence, n.d.). thus, minimizing the loss function means that the loss value between the target labels and the estimated labels becomes smaller: j(x∗, x̂∗) = {j1, . . . , jn}⊤, jk = − [x̂∗ k · log x∗ k + (1− x̂∗ k) · log(1− x∗ k)] , k ∈ {1, . . . , n}, j(x∗, x̂∗) → min ∼ rocauc(x∗, x̂∗) → max where j stands for the loss function, x∗ is a target value, and x̂∗ is an estimated value. fig. 4.7. rocauc train/test curves fig. 4.8. bce loss train/test curves copyright © 2023 assa. adv syst sci appl (2023) deep-learning-based tracing for satellite telemetry 33 5. simulation results fig. 5.9 demonstrates results (smoothed values) of tracing, reflecting the potential appearance of anomaly events. each time point has information about the previous sequence, and the new time point has a prediction for each info source (sensors). the approach is robust concerning time shifts and information disappearing because the data batches have been shuffled in a training period, and the neural network can predict such events based on only one element in sequence (dictionary of known keys axes, measured and programmed wheel velocities). the quality of the onboard aro system is 54% by auc metrics, while the presented neural network in a pre-trained condition has a quality of about 82% by auc metrics. this value of above the results presented before considering the implementation of traditional regressors and static neural networks. intending to simplify access to the proposed model, the authors of this manuscript prepared a digital repository where the software used to develop the presented results is included. these programs can be found in github repository. fig. 5.9. tracing result with comparison to strict data copyright © 2023 assa. adv syst sci appl (2023) https://github.com/aristarx-lintter/deep_satellite_telemetry_tracing 34 i. nachevsky, o. andrianova, i. chairez 6. discussion this study describes the propagation of anomalies in satellite telemetry data. the suggested and tested method to improve deviant system behavior propagation can be further used for onboard implementation due to its lightweight architecture. the system can be fine-tuned using new information. in this case, the system with the tracing neural network core is able to predict with actual information about the target satellite device. it could be implemented by fine-tuning the control center and pushing new weights to the satellite board (the model weight is about 700kb). another approach time-distributed crnn (luna-álvarez, mújica-vargas, matuz-cruz, vianney kinani, & ramos-dı́az, 2020) can also be applied to make a solution to this problem. the idea regarding applying convolution layers to sum the information can get a better result than the current architecture approach (tracing with the embedding layer). however, in this way, a change of the 2d convolutions to 1d is suggested, or the information should be represented in a bit new form to feed the data in a recurrent layer. the quality of the problem solution can also be improved when transformer architecture is used (vaswani et al., 2017). the transformer architecture can represent the latent space more complexly in comparison to the approach from this paper, which, consequently, will lead to the ability to separate distributions and to make the high dissimilarity between them. it’s worth mentioning that the loss function can be picked the same, while the data representation is needed to be changed in a straight way of queries, keys, and values matrices. acknowledgements the paper was prepared with the partial financial support of the tecnologico de monterrey, institute of advanced materials for sustainable manufacturing under the grant challengebased research funding program 2022 number i006-iamsm004-c4-t2-t. references chen, b., lu, g., fang, h. z., et al. (2014). method of satellite anomaly detection based on least squares support vector machine [j]. computer measurement & control, 2022(3), 690–696. hairab, b. i., elsayed, m. s., jurcut, a. d., & azer, m. a. (2022). anomaly detection based on cnn and regularization techniques against zero-day attacks in iot networks. ieee access, 10, 98427–98440. khrulkov, v., hrinchuk, o., mirvakhabova, l., & oseledets, i. v. (2019). tensorized embedding layers for efficient model compression. corr, abs/1901.10787. retrieved from http://arxiv.org/abs/1901.10787 kingma, d. p., & ba, j. (2017). adam: a method for stochastic optimization. korolev, a. (2018). functional stability of navigation information systems. izvestiâ vysših učebnyh zavedenij. priborostroenie, 61, 559-565. kullback-leibler divergence. (n.d.). http://hanj.cs.illinois.edu/cs412/bk3/ kl-divergence.pdf. (accessed: 2023-04-02) luna-álvarez, a., mújica-vargas, d., matuz-cruz, m., vianney kinani, j. m., & ramosdı́az, e. (2020). self-driving through a time-distributed convolutional recurrent neural network. in 2020 17th international conference on electrical engineering, computing science and automatic control (cce) (p. 1-6). doi: 10.1109/cce50788.2020.9299181 luo, t., & nagarajan, s. g. (2018). distributed anomaly detection using autoencoder neural networks in wsn for iot. ieee icc. doi: arxiv:1506.05908v1 copyright © 2023 assa. adv syst sci appl (2023) http://arxiv.org/abs/1901.10787 http://hanj.cs.illinois.edu/cs412/bk3/kl-divergence.pdf http://hanj.cs.illinois.edu/cs412/bk3/kl-divergence.pdf deep-learning-based tracing for satellite telemetry 35 nishio, t., nakahara, m., okui, n., kubota, a., kobayashi, y., sugiyama, k., & shinkuma, r. 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(2022). spacecraft emergency situations automated scenarios design within the framework of fdir concept and dynamical neural networks technique. advances in systems science and applications, 4, 19-30. vaswani, a., shazeer, n., parmar, n., uszkoreit, j., jones, l., gomez, a. n., . . . polosukhin, i. (2017). attention is all you need. corr, abs/1706.03762. retrieved from http://arxiv.org/abs/1706.03762 zeng, z., jin, g., xu, c., chen, s., & zhang, l. (2022). spacecraft telemetry anomaly detection based on parametric causality and double-criteria drift streaming peaks over threshold. applied sciences, 12(4). retrieved from https://www.mdpi.com/ 2076-3417/12/4/1803 doi: 10.3390/app12041803 copyright © 2023 assa. adv syst sci appl (2023) http://arxiv.org/abs/1706.03762 https://www.mdpi.com/2076-3417/12/4/1803 https://www.mdpi.com/2076-3417/12/4/1803 introduction telemetry data description and problem statement system under consideration problem statement preprocessing model design model structure training and validation process simulation results discussion references adv syst sci appl 2022; 04:193–201 published online at https://ijassa.ipu.ru. extending obstacle map of autonomous vehicles based on network model of local positioning alexander abdulov* v. a. trapeznikov institute of control sciences, moscow, russia abstract: at present, an unmanned autonomous vehicle (av) to provide accurate navigation during motion depends on gps. the search and study for the alternative methods of av localization is the demand to implement smart city concepts because in real-world conditions, the gps signal may either be absent, or its accuracy may be insufficient to trajectory following and perform maneuvers. for collision avoidance to raise av safety, the network model of local positioning is proposed. control systems based on obtained local maps partially solve the problem of a safety motion. the presented simulation results confirm the effectiveness of the described approach to improving traffic safety in uncontrolled dynamic environments. keywords: autonomous vehicle, local positioning, collision avoidance, simulation modeling 1. introduction an unmanned autonomous vehicle (av) for safety motion applies optimal control rules enabling one to avoid collisions. due to that, avs form models of observed obstacles, accounting for dynamic characteristics of moving objects (including other vehicles). the accuracy of determining obstacle model parameters is directly concerned with the probability of avoiding a collision [1]. one should point out that in real-time, it is not always possible to estimate the position and dynamics of other road traffic participants accurately directly using only onboard computational tools. as a rule, all road traffic participants are interested in minimizing the collision number, so it is reasonable for them to exchange helpful information with each other under the availability of communication channels. then the communication can become a key factor in searching for safe motion paths. let us consider a case of motion in a “crowd” with a possibility of avs communicating with each other by the wireless network. the avs are equipped with rangefinders enabling them to evaluate distances to obstacles and wireless communication modules to receive and transmit messages. suppose the radius of action of the wireless communication module is comparable with the av dimension and its speed characteristics to receive helpful information from other participants from the network in advance. 2. problem statement all avs have the possibility to build their local map of observed objects, transmit it to other network participants, and receive their maps of obstacles. for the two-dimensional case, a local map of obstacles (see fig. 2.1a) which is represented by the data type mapmsg: < id,g = {gi}, u = (uj) >, where gi = (d, a, s) – sets of relative positions of directly ∗corresponding author: aabdulov@asmon.ru 194 a. abdulov (a) local map of obstacles (b) principle of a safe path choosing fig. 2.1. simple model of the mobile autonomous vehicle. observed objects (distant at the distance d and deviated on the angle a from observer heading), and s – approximate dimensions of the obstacle (since they are unknown, they should be estimated). note that it is quite difficult to focus on the dimensions estimated by rangefinders in practice. therefore, this information is rather needed for the subsequent evasion from unidentified objects, which in theory appear to be stationary objects with some size. in turn, u is the state vector of the av (dimension, planned path in local coordinate system, and other useful information). it is advisable to use wireless network ip addresses or mac addresses of network devices as unique identifiers id. all examples are used simple integer identifiers. there are n observable objects for each av in the general case, and m received messages from the network. so, a problem arises to determine message sender location at the local map of obstacles. choosing safety path control (see fig. 2.1b) can be implemented by modifications of dwa [2], teb [3], or mpc [4]. 3. algorithm description partially this uncertainty can be decreased by solving for each av the local positioning problem, which is to determine a relative location and parameters of motion of different objects. a feature of the problem is the absence of common orient beacons for all road traffic participants. groups of avs with communication channels can apply network models of local positioning based on known approaches: triangulation, trilateration, and navigation numeration [5]. the proposed method is applicable, taking into account the following statement: the distance between the participants of the movement observing each other is the same. the complete cycle of the algorithm for obtaining an extended local obstacle map is shown in fig. 3.2. let’s consider it in more detail. 3.1. rangefinder data capture first of all, it is necessary to capture readings from the rangefinders installed on board, these can be circular or multi-beam lidars, depth or stereo cameras, etc. copyright © 2022 assa. adv syst sci appl (2022) extending obstacle map of autonomous vehicles 195 fig. 3.2. algorithm cycle to obtaining an extended local obstacle map. 3.2. create self local map of obstacles then create self local obstacle map gself . here, rangefinder readings are used as input, but they should be prepared. it is required to isolate individual objects (obstacles) based on raw data gi: determine the distance di, the angleai of deviation from its course on it and additionally evaluate its dimensions si. this is a very important stage, especially in practice when the sensor readings are noisy. the prepared map gself should be packed into a message msgself ∈ mapmsg (supplemented with information uself about its dimensions and the estimated trajectory of movement relative to the current position), and sent to the network so that other participants can also use it. 3.3. database of local maps self message msgself =< self,gself , uself > and others {msgid} received from the network are stored on board in a special database for further operational use. outdated records from the database can be automatically deleted. in this case, the component g of messages in the database will be represented in the form of a directed tree g(v,e). so each local obstacle map consists of a root node id ∈ v and its associated terminal vertices {v = (id, ai)|d(v) = 1} ⊂ v as “star” topology. all nodes of this graph have the attribute s (object dimensions), and edges e = (id, (id, a)) ∈ e with attribute d (distance of the obstacle). examples of such “stars” are shown in fig. 3.3. 3.4. edge clustering by its lengths from all relevant messages in the database edges ei ∈ e(gself ∪gid) are extracted to perform clustering based on their lengths d with an unknown number of clusters. methods like dbscan [6] are suitable for this task. note: when clustering edges, one can take into account not only their lengths, but also the dimensions of the nodes connected by them. through this clustering, many sets ck ⊂ e of similar edges are generated. if any two edges ei, ej ∈ ck are from the same cluster, then the corresponding nodes (av) are potential candidates for docking into a combination. in fig. 3.3 edges from the same cluster have the same color. copyright © 2022 assa. adv syst sci appl (2022) 196 a. abdulov 3.5. many possible docking combinations for each cluster with multiple nodes (i.e. |ck| > 1), its docking edges ei ∈ ck can be represented by nodes of a connected graph gk = g(ck, ck × ck) of possible permutations within a given cluster. joints within such a graph of permutations are mutually exclusive. let gk!! denote their number. note: for a fully connected graph (upper bound), the number of permutations of the choice of all possible docking edges will be equal to the double factorial (3.1) of the number of nodes. if the number of nodes n is even, then take the nearest less odd: n = n− 1. n!! = n! 2 n−1 2 · (n−1 2 )! . (3.1) if n = 9 then n!! = 945. therefore, even with a small number of candidates, the total number of permutations may turn out to be quite large. the total number of permutations of all combinations is (3.2). ∏ |ck|>1 gk!! (3.2) taking into account the specifics of the problem, the number of permutations can be significantly reduced to improve performance, for calculations on board. first, take fig. 3.3. examples of docking items. copyright © 2022 assa. adv syst sci appl (2022) extending obstacle map of autonomous vehicles 197 the possible docking with self av edges obtained directly. further, may be use not all combinations available on the network, and append the list of possible connections only twice (k1 = 2). 1. docking observed by self av, because for successful docking the avs must be in fov (field of view) of each other. 2. control docks for the appended list in the previous iteration. in the process of forming possible combinations, invalid ones should be eliminated immediately, for example, based on limited fov. the proposed docking may add to the participants’ fov some objects that they did not initially observe – these should be excluded. there are also cases when, in false positives, the identified node goes beyond the radius of the wireless network signal – it shouldn’t be there. another way to reduce the number of permutations is the probabilistic approach. the more samples in the cluster (more candidates), the more likely it is to misplace the nodes. given the mutually exclusiveness of the docking, only a few of the most probable combinations can be selected. for simulation experiments, the value of the corresponding parameter k2 = 3 was chosen (the algorithm still reliable). 3.6. optimal extending of the local obstacle map of the many admissible combinations of docking, the most suitable one will be selected in accordance with the given criterion of optimality, which will be further used for navigation. as a result, its own extended local obstacle map ĝself = (gi) is a certain combination (ordered set) of docking permutations connecting all currently relevant “stars” into a single map. as an optimality criterion when choosing the most suitable combination of joints used: |{v ∈ v (gself )|d(v) = 1}| |e(gself ∪ {gid})| → min (3.3) here, the denominator is the total number of joining edges, and the numerator is the number of remaining terminal vertices. also have ideas for using weighting factors to take into account the importance of joining certain nodes when calculating the combination scores. note: the resulting extended local obstacle map can also be sent to the network as the initial one, especially if there is high confidence in its reliability. an example of the result is shown in fig. 3.4. the negative identifiers mean that this node is a common obstacle for several observers, but has never been identified on the network. the presence of such common obstacles, one might say, increases the reliability of the docking. the optimality criterion (3.3) does not guarantee perfect results and “false positives” are quite acceptable. 4. simulation modeling experiments have been implemented on the specially developed program simulator to evaluate the efficiency of the communication factor and collisions minimization. the av model with a given frequency of the main cycle builds its local map, sends data to the network, receives messages from other ones, analyzes the situation, and selects maximally safe control approaching the destination based on the extended obstacle map. simulation scene is the corridor 10 m (see fig. 4.5). the intensity of the av model generator is 0.5 2.5 (units per second). there are 2 generators on the scene. av models move in counterflows: one group (from generator 1, cyan) travels towards generator 2 (magenta) and vice versa. for each intensity, the experiment lasted 10 min. simulation time, statistics were collected every dt = 0.1 s. during this time, two generators are released on the stage somewhere from 600 (0.5 hz) to 3000 (2.5 hz) av models. there are no side walls in this experiment to eliminate the congestion effect. the parameters of the av models at: copyright © 2022 assa. adv syst sci appl (2022) 198 a. abdulov fig. 3.4. example of an extended obstacle map. • linear velocities: 1.5± 0.5 m/s; • speed of rotations: 180± 60 deg/s; • dimensions (radii): 0.15± 0.05 m; • wireless network signal range (radii): 2.5± 0.5 m; • rangefinder maximum distance: 1.5± 0.5 m; • field of view (sector): 180± 60 deg. series of experiments: • t1 everything is known about those who fall into direct observation (uses known dimensions and planned trajectories); • t2 direct observation (information from the network is not available, all observed obstacles are considered as static); • t3 the proposed approach based on an extended obstacle map (uses information from the network for those network participants whose position has been established, including false positives, all other nodes are considered as static obstacles). copyright © 2022 assa. adv syst sci appl (2022) extending obstacle map of autonomous vehicles 199 5. experimental results. during the experiments, the following average indicators were evaluated: • count of hits (collisions). • delay on the way. delay = exitt ime− startt ime mint ime − 1 if it is possible to reach the goal in the minimum possible time mintime, then the delay is 0, and if 1, then the delay relative to the minimum possible time is 100%. the simulation results (see fig. 5.6) have confirmed the efficiency of the proposed approach: the developed model enables one to considerably restrict the number of collisions under motion over non-regulated counter flows. such a model of local positioning enables one to increase the awareness of the road traffic participants without a common regulation center. this approach will work both in networks with centralized common access points and in distributed self-organizing mesh networks [7]. in addition, a number of subtasks appear to reduce the number of permutations and develop a more reliable criterion of optimality to reduce the number of false positives, so that the method continues to work not only in conditions of artificial noisy readings of rangefinders, but also in reality, where there are a lot of other inaccuracies, for example, in the the experiments did not take into account the delays in data transmission over the network. in addition, in connection with the combinatorial problem, the question remains about the minimum sufficient number of successfully identified participants to ensure safe movement in clusters. at high intensities, even with k1 = 2 and k2 = 3, lengthy calculations had to be performed at the stage of enumerating combinations. fig. 4.5. scene example of the developed simulator: (a) move area; (b) ideal extended local map; (c) proposed method in scene; (d) extended map by the proposed method with normed angles. copyright © 2022 assa. adv syst sci appl (2022) 200 a. abdulov fig. 5.6. experimental results. 6. conclusion the smart city strategically considers transport development. the transport structure becomes intelligent. the tendency to apply dynamic and multi-modal information is appreciable in control of the city logistics. big data are sampled from avs sensors, safety cameras, rfid (radio frequency identification) labels, etc. the avs take a particular place in the transport infrastructure, which requires the development of technologies: artificial intelligence, telecommunication systems, cyber-physical interfaces, informationand cybersecurity [1, 5, 8]. the “smart city” approach, oriented to safety, is to react not only to existing but also to occurring vulnerabilities, account dangers arising under developing new technologies. du to that, safety management systems [9–12] based on determining risks of different kinds (cybersecurity, efficiency, and reliability of software and hardware tools, etc.), will enable one to reveal vulnerabilities, dangerous factors, and to develop corresponding compensation measures, what, in turn, will increase the smart city cii safety. applying the network models of local positioning as additional method localization to organize the avs traffic in the smart city will enable one to increase motion safety even in the case of the absence of gps. in the case of successfully extending the local map of obstacles by comparing relative mutual locations, i.e., if for the av will be correspondence between network addresses of message senders with their location on the map has been established, then it can able to use data received from the network. the simulation results show that the proposed approach allows to reduce the number of collisions when driving in counterflow traffic on the basis of the considered algorithm for extending the local obstacle map. acknowledgements section 3 of the reported study was funded by rfbr, project number 19-29-06044. references 1. abdulov, a. & abramenkov, a. (2018) collision avoidance by communication for autonomous mobile robots in crowd. in 2018 eleventh international copyright © 2022 assa. adv syst sci appl (2022) extending obstacle map of autonomous vehicles 201 conference “management of large-scale system development” (mlsd), 1–4, https://dx.doi.org/10.1109/mlsd.2018.8551804. 2. fox, d., burgard, w., & thrun, s. (1997) the dynamic window approach to collision avoidance. ieee robotics automation magazine, 4(1), 23–33, https://dx.doi.org/10.1109/100.580977. 3. rösmann, c., hoffmann, f., & bertram, t. (2017) integrated online trajectory planning and optimization in distinctive topologies. robotics and autonomous systems, 88, 142– 153, https://dx.doi.org/https://doi.org/10.1016/j.robot.2016.11.007. 4. li, j., ran, m., wang, h., & xie, l. (2019) mpc-based unified trajectory planning and tracking control approach for automated guided vehicles. in 2019 ieee 15th international conference on control and automation (icca), 374–380, https://dx.doi.org/10.1109/icca.2019.8899955. 5. abdulov, a., abramenkov, a., & shevlyakov, a. (2018) visual odometry approaches to autonomous navigation for multicopter model in virtual indoor environment. advances in systems science and applications, 18(3), 17–28, https://dx.doi.org/10.25728/assa.2018.18.3.583. 6. khan, k., rehman, s. u., aziz, k., fong, s., & sarasvady, s. (2014) dbscan: past, present and future. in the fifth international conference on the applications of digital information and web technologies (icadiwt), 232–238, https://dx.doi.org/10.1109/icadiwt.2014.6814687. 7. guss, s. (2016) private wireless mesh networks. mathematical structures and modeling, 4, 102–115. 8. promyslov, v., sakrutina, e., & meshcheryakov, r. (2019) coherence criterion for security architecture of digital control system. in 2019 international russian automation conference (rusautocon), 1–5, https://dx.doi.org/10.1109/rusautocon.2019.8867615. 9. li, c.-y., wang, j.-h., zhi, y.-r., wang, z.-r., & gong, j.-h. (2018) simulation of the chlorination process safety management system based on system dynamics approach. procedia engineering, 211, 332–342, 2017 8th international conference on fire science and fire protection engineering (icfsfpe 2017), https://dx.doi.org/10.1016/j.proeng.2017.12.020. 10. kalashnikov, a. & sakrutina, e. (2019) safety management system and significant plants of critical information infrastructure. ifac-papersonline, 52, 1391–1396, 9th ifac conference on manufacturing modelling, management and control mim 2019, https://dx.doi.org/10.1016/j.ifacol.2019.11.393. 11. sakrutina, e. & kalashnikov, a. 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(2021) unmanned vehicles: safety management systems and safety functions. communications in computer and information science, 1396, 112–121, https://dx.doi.org/10.1007/978-981-16-1483-5. copyright © 2022 assa. adv syst sci appl (2022) https://dx.doi.org/10.1109/mlsd.2018.8551804 https://dx.doi.org/10.1109/100.580977 https://dx.doi.org/https://doi.org/10.1016/j.robot.2016.11.007 https://dx.doi.org/10.1109/icca.2019.8899955 https://dx.doi.org/10.25728/assa.2018.18.3.583 https://dx.doi.org/10.1109/icadiwt.2014.6814687 https://dx.doi.org/10.1109/rusautocon.2019.8867615 https://dx.doi.org/10.1016/j.proeng.2017.12.020 https://dx.doi.org/10.1016/j.ifacol.2019.11.393 https://dx.doi.org/10.1109/mlsd49919.2020.9247693 https://dx.doi.org/10.1007/978-981-16-1483-5 introduction problem statement algorithm description rangefinder data capture create self local map of obstacles database of local maps edge clustering by its lengths many possible docking combinations optimal extending of the local obstacle map simulation modeling experimental results. conclusion microsoft word 2025 09 26 meshkov assa adv syst sci appl 2025; 02; 81-91 published online at https://ijassa.ipu.ru. ensuring the sustainability of health care management expert support and decision making in modern conditions dmitry meshkov1, sergey cherkasov1, alexey lobanov1, anna dzebisashvili1, polina lysova1, viktor glebov1, yana matkovskaya1,2, yuri sidelnikov1, 3, anatoly mikhalsky1, dmitry stefanocvsky4, taisia dozortseva1, olga rodionova1, elena makeeva1, ludeña moreira genesis marley5 1) v.a. trapeznikov institute of control science of the russian academy of sciences, moscow, russia 2) financial university under the government of the russian federation, moscow, russia 3) moscow aviation institute (national research university), moscow, russia 4) the state university of management, moscow, russia 5) peoples’ friendship university of russia named after patrice lumumba, moscow, russia abstract: current technological and environmental changes requiring new approaches for healthcare data generation and assessment as well as decision-making support. the analysis performed using delphi method indicated that sustainability of healthcare decision-making by using a number of approaches based on existing opportunities for proactive risk-based assessment of population health and healthcare interventions. non-numerical qualitative expert opinions are of the same value as quantitive data while the balance between them and the methodology of mutual transformation form an integrated evidence-based support for the sustainable data generation and healthcare decision-making. qualitative expert opinions make it also possible to decrease the asymmetry and lack of quantitative data between model studies on animals and the characteristics of the human body. the balance between qualitative expert assessment and the digital evidence changes during the lifecycle of the intervention or possibly process (like pandemic). sustainability of healthcare decision-making in changing environment depends on availability and methods of qualitative expert support. currently, there are no generally accepted methods for the final qualitative expert assessment, which makes it possible to reduce the diversity of interdisciplinary big data to a transparent and understandable assessment and these methodology and methods are to be developed. automated systems for early detection of emergent cases based on medical statistics data are also of great importance and capable increasing sustainability of healthcare management and this information should be linked to preprepared action plans to avoid covid-19 lessons in future. keywords: healthcare management, healthcare decision making, expert support for decision making, public health emergencies and contingency planning. 1. introduction healthcare systems around the world experience difficulties meeting any deviations from the standard situation. this applies both to negative changes like the emergence of new diseases as well as the positive ones like the development of new interventions. the fight against the pandemic has shown enough examples of positive and effective actions that have reduced the burden of the disease. however, it is necessary to consider and 1 corresponding author: dmitrymeshkov@mail.ru 82 d. meshkov, s. cherkasov, a. lobanov, a. dzebisashvili, p. lysova, et al. copyright ©2025 assa adv. in systems science and appl. (2025) analyze negative experiences in order to avoid them in the future. healthcare systems could not identify appropriately react the early signs and description of covid-19 pandemic as well as performing effective management and clinical decisions. first personal messages from healthcare practitioners indicated an increase in deaths from sars as early as march 2019. next the actual signs of sars-cov-2 circulation in northern italy (lombardy) were officially detected as early as september-october 2019 and in france in novemberdecember 2019 [1]. a description of the upcoming pandemic as a threat to global health and a list of countermeasures were presented by the hopkins institute in october 2019 but they did not lead to a noticeable response from the health systems in any country. [2]. the ineffective management decisions and the irrational use of available resources, as well as the distortion of medical statistics, became apparent within a few months of the outbreak of the pandemic. [3,4]. the treatment of patients infected with covid-19 was accompanied by clinically unjustified use of antibiotics, which are not only useless for viral infection, but can cause immune suppression and worsen the condition of such patients. additionally, antibiotics lead to the emergence and spread of resistant bacterial strains, contributing to the spread and complicating bacterial infections [5-7]. even a few years after the outbreak of the pandemic, confronting the clinical, economic and social burden of the disease requires innovative thinking and the development of strategies aimed at improving health efficiency, at the level of policy decisions and national health management, including a complete revolution in the entire system of medical care [8-13]. the development and introduction of medical interventions is expected to have a positive impact on health systems contributing the increase of safety, efficacy, efficiency and effectiveness. however, insufficient accessibility and irrational use of medical technologies occur in different countries, regardless of the payment model or healthcare system, and in various clinical institutions, from rural clinics to specialized hospitals. [14] health systems in lowand middle-income countries face additional challenges that hinder innovation. such problems include insufficient awareness of barriers and accessibility factors reducing the possibility and effectiveness of innovations. unsuccessful implementation of innovations, in turn, requires additional resources and affects the possibility of future implementations and as well as decrease the accessibility of medical care in other clinical areas [15]. the development of interventions is also experiencing difficulties. over the past 60 years, the pharmaceutical industry has provided the world with more than 1,220 new drugs that have played an important role in improving the global health and increasing life expectancy by 2 months per year [16]. however, the existing r&d model that fueled this success now is providing a set of symptoms has often been a harbinger of major failures in other industries [17]. the experience of these industries shows that the latter can withstand such upheavals based on the accidental appearance of several products, the socalled «black swans», and the pharmaceutical industry has a chance to «rethink the culture of functioning in connection with these modern conditions. [18] thus, the current state of health management systems at the global and national levels is unable to respond both the creation of new health interventions and technologies as well as the new threats. modern conditions require healthcare systems to be ready functioning in changing conditions related to the risks of infectious diseases caused by new pathogens, the emergence of new medical interventions, organizational and information technologies. contemporary technology development makes it possible to ensure proactive risk-based public health management [19]. the term technology should in this case must be understood as the entire complex of substances, devices and activities performed by qualified personnel, including diagnosis, treatment, safety and prevention of adverse events including and supporting the intervention itself. [20]. ignoring a comprehensive concept of technology in healthcare ensuring the sustainability of health care management expert support… 83 copyright ©2025 assa. adv. in systems science and appl. (2025) decision making leads to an irrational expenditure of resources and a negative result of its application [15]. the use of separate elements of this model in healthcare decision making make it possible to increase clinical efficacy, social benefits and economic effectiveness providing additional benefits of integrated development to the countries that have already implemented this approach [21]. these facts indicate that the complex consisting of public health monitoring, expert support for healthcare decision making and healthcare management itself, despite the available technological and information capabilities, is in practice a fragile structure that does not meet both the challenges of global health risks and technological capabilities, and thus requiring the development and implementation of data generation management and utilization principles meeting the modern challenges and ensuring the sustainability of healthcare management in modern conditions. 2. objectives development of key statements and recommendations ensuring the sustainability of health care management expert support and decision making in modern conditions. the following tasks were accomplished for the purpose: 1. evaluation of existing approaches to data management and creation of predictive scenarios providing information about both the status and risks to population health and the development of medical technologies, and capable providing proactive risk-based management within a specific existing administrative and legal system (country, region) in modern conditions. 2. analysis of possible problems and gaps related to these approaches and caused by the peculiarities of the formation and transformation of interdisciplinary multiformat numerical evidence data into qualitative non-numerical expert opinions that are used for healthcare decision making. 3. analysis of the of digital data transformation into non-numerical qualitative conclusions over time in the process of data generation during preclinical and clinical studies as well as during the commercial use. 4. assessment of the impact of this consequent data transformation on sustainability and fragility of the decision-making process in healthcare. 5. assessment of availability of the generally accepted methods of expert assessment making possible justifying decisions based on interdisciplinary multi-format data under the conditions of stakeholders conflicting interests. 6. assessing the opportunities of early detection of population health changers and alerts using automated systems or in any other way. 7. assessing potential activities and countermeasures regarding healthcare threats with of uncertainty elements that allowing us to quickly respond to any risks and avoiding inappropriate actions typical of the initial stage of the covid-19 pandemic. 3. methods modified delphi investigation was used to for the investigation [22]. the initial information was obtained using the environmental scanning procedure abased on the review of literature (pubmed and embase; 467 articles and reviews selected) and other modes of communication including personal messages [23-25]. based on the environmental scanning data, we formulated the provisions and presented them to a group of analysts for 4 iterations of discussion before the final 7 provisions were agreed by all analysts upon essence and wording. the group of analysts consisted of 12 people, clinical specialists specialized in the field of health management (4), biology (3), statistics (1), management theory (2), economics (1), information technology (2). after 84 d. meshkov, s. cherkasov, a. lobanov, a. dzebisashvili, p. lysova, et al. copyright ©2025 assa adv. in systems science and appl. (2025) several iterations, 7 main provisions were formulated, approved by all participants of the group of analysts. 4. results a data management model for proactive risk management in healthcare is most appropriate approach and the tool capable increasing the sustainability of healthcare systems. the model consists of elements or blocks related to a specific evidence-based cluster of information and the links between these elements indicating the sequential data flows. almost all of these elements (population health and health interventions monitoring, disease burden assessment, pre-clinical and clinical studies, systematic reviews, health technology assessment and others) exist and can be used in almost any country, but they are not fully used in any country in the world [19]. using or not using of these elements as well as the extent of the use depend on the level of economic development of countries and, in turn, the evidence-based healthcare decision-making affects the economic and social development of countries via human resources [21]. countries with low level of economic development (gdp per capita) are characterized by a lack of priorities at national levels, making it difficult to form a unified approach to assessing the effectiveness and the quality of medical care; strict legislation aimed at reducing costs without taking into account their effectiveness and making it difficult to form flexible and evidence-based decision-making mechanisms; a weak information infrastructure that makes it difficult to obtain national evidence-based data on morbidity and mortality; and an low number of specialists capable of preparing expert opinions. all these issues impact an opportunity for effective management in healthcare and leading to an increase in morbidity and treatment costs. [26-29]. the model can be used for development of national healthcare systems forecasting depending on existing economic development of these countries and the use of a set of evidence-based methods in its healthcare. it allows you also to identify problem areas and develop recommendations for improving evidence-based decision making. the model also indicates patterns of sequential transformation of data and their transition from quantitative digital estimates to nonnumerical qualitative expert opinions. such transitions take place both the elements (information blocks) of the model and at the final level of interaction with regulatory authorities. the need to use non-numerical methods of estimation and forecasting serves as a natural separation of the information clusters of the model, within which quantitative digital probabilistic estimation is possible. non-numerical qualitative expert opinions and the methodology of mutual transformation of qualitative data into quantitative and vice versa are of the same importance as numerical research data and form all together an integrated evidencebased support for the healthcare decision making. the results of a comprehensive assessment and prognostic scenarios that are submitted to regulatory authorities depend also on non-numerical qualitative assessments assuming numerical data obtained during pre-clinical and clinical studies and creating hypotheses for the next numerical quantitative step of the integrated research. the formation of evidencebased data for expert support of management decisions in healthcare at the example of medical intervention consist of several stages including pre-clinical (pathogenesis and mechanism of action data) and clinical studies (testing the hypothesis of comparability of animal and human data during the phase i; testing the hypothesis about the therapeutic effect based on surrogate indicators during phase ii and testing the hypothesis about the therapeutic effect based on clinical indicators during phase iii studies) [30]. the safety and efficiency data obtained at this stage is sufficient to obtain market authorization. the inclusion of the intervention into reimbursement lists in many countries is associated with obtaining additional data on effectiveness, economic and social outcomes as ensuring the sustainability of health care management expert support… 85 copyright ©2025 assa. adv. in systems science and appl. (2025) part of the health technology assessment. in addition, as preand clinical data accumulates, this information is summarized and structured in the form of clinical guidelines and systematic reviews. the values of digital quantitative assessments and qualitative non-digital expert opinions for each element of the model are changing in the process of the medical technology global value pack creation. the value of qualitative assessment is the highest at the decision-making stage. there is no real-world data (rwd) obtained at this stage as well as the correlations between rwd and controlled trials. moreover, the decision must be based on a number of other interfering technological, economic, administrative, psychological factors which cannot be expressed in a single numeric format [19,31]. qualitative expert opinions make it possible to decrease the asymmetry and lack of quantitative data between model studies on animals and the characteristics of the human body. animal-based research makes possible to simulate the majority of pathologic conditions providing a detail about the mechanisms of pathogenesis and treatment which cannot be performed in human subjects due to ethical or technological barriers [30]. at the same time, we can obtain statistically reliable data on human physiology and treatment outcomes which can be different from the animal one. as a result, we have a distribution of our knowledge about the problem and the possibility of solving it, illustrated in figure 1. the only way to reduce this uncertainty at the decision-making stage may be a qualitative expert opinion filling the gap between human and animal information clusters, ensuring the continuity of data flow formation and allowing generalizing conclusions to be drawn. the expert opinion, along with regulatory procedures, determines management decisions on the use of medical intervention, which can be expressed in the format of authorization on the market or inclusion in reimbursement lists [32-34]. the value of qualitative expert assessment decreases as digital evidence accumulates, but any changes in the decision-making system increase its importance. the dynamics of this process is illustrated in fig. 1 using the example of the life cycle of a medical intervention. as preclinical studies progress, data is generated on the mechanisms of pathogenesis and the intervention mechanism of action the in conditions which are close to the real-world impacts, but in models that differ from humans. the results of clinical trials provide safety and efficacy data about medical intervention in humans, but under controlled conditions that differ from real-world conditions. market authorization of the intervention is made on the basis of regulatory procedures and expert opinion, which compensates for the lack of data on the intervention effects in humans under the real-world conditions. after market authorization, this medical technology begins to be used in people in real conditions. after the market authorization real-world data (rwd) is accumulated in digital format, which is more evidence-based from the point of view of regulatory authorities and replaces expert assessment. with any external changes, positive (the emergence of a new medical technology, competitive or vice versa, increasing the effectiveness of the considered one) or negative (a new infection) and the formation of gaps in digital evidence, the value of expert assessment increases. changes in regulatory procedures also require reassessment of the intervention and may increase the value of qualitative expert review. any change increases the degree of uncertainty, and in conditions of uncertainty, increases the role of the expert capable providing and implementing management decisions. [35-37]. sustainability of healthcare decision-making in changing environment depends on availability and methods of qualitative expert support the long period without pandemics or other changes leads to a reduction of the expert support value and a simplification of the decision-making mechanism. under these conditions, the effectiveness of management decisions depends more on reducing costs within an existing and well-established mechanism and less on the response to external influences and the emergence of «lean management» is justified. on the other hand, the development of information and healthcare technologies leads to continuous changes as well as the increase of 86 d. meshkov, s. cherkasov, a. lobanov, a. dzebisashvili, p. lysova, et al. copyright ©2025 assa adv. in systems science and appl. (2025) uncertainty identifying the challenge the healthcare decision-making adaptation to new conditions and the emergence of «change management». theoretically, the alternating approaches of lean management and change management could have allowed us forming a model of cyclical evolution of healthcare systems, but the uncertainty associated with the emergence of technological innovations and appearance of new pathogens forces us to be critical of this possibility, as well as the successful coexistence of lean and change management. obviously, the model of the healthcare system will be successful, in which standard processes are effectively managed as an element taken from lean management, but there are sufficient resources and developed action plans in case of even unlikely or uncertain events, as the covid-19 pandemic seemed before it began. the model of proactive risk-based expert support for management decisions in healthcare supported by appropriate methods of qualitative expert review are the most suitable for existing situation [2, 16-18]. currently, there are no generally accepted methods for the final qualitative expert assessment, which makes it possible to reduce the diversity of interdisciplinary big data to a transparent and understandable assessment. it is quite easy to translate quantitative research results into qualitative conclusions, but the process of translating qualitative data into quantitative ones is extremely difficult. the need for qualitative peer review eliminating gaps between the structural elements of the model representing digital interdisciplinary data in different formats, as well as between the integrated results of the assessment and the regulatory authorities deciding on the reallocation of resources, is critical in any healthcare system. currently, there are no generally accepted standardized methods for that purpose. multi-criteria decision-making (mcdm) is most often proposed for this purpose and mentioned in specific publications but it has has significant limitations for biological and social systems [38-42]. automated systems for early detection of emergent cases based on medical statistics capable increasing sustainability of healthcare management. the automatic system based on the international classification of diseases (icd-10) coding and capable providing real-time data was presented at the who conference in 2015. it had been already created and tested in a number of locations but did not meet any interest after that [43]. the information on the increase in deaths from sars would have been available in march 2019 or earlier, i.e. more than 6 months before the officially declared covid-19 pandemic in case of existence of such a system in at least one country. an alternative is automated search systems for specialized publications or information presented at web-sites. since the ontological model of a potential epidemic was formulated in october 2019, and publications date back to the same date this approach would make it possible to draw conclusions about the risks to public health 3 months before the official start of the pandemic and take preventive measures with the appropriate capacity of health systems [1,2]. action plans are needed for the most unpredictable and uncertain cases related to public health. paradoxically, commercial companies had an action plan for various issues (iso 31000 and/or others), while healthcare systems around the world took actions that did not correspond to the situation during the pandemic. obviously, humanity is facing more than one epidemic. the probability of these events are high but we do not know when such an epidemic will begin (uncertainty). it is useless and costly to prepare infection hospitals and vaccines for an unknown pathogen in advance. but you can have an action plan prepared in advance, in which, at a minimum, it will be written that it is not effective and safe to use antibiotics for patients with acute viral infection [5-7]. ensuring the sustainability of health care management expert support… 87 copyright ©2025 assa. adv. in systems science and appl. (2025) fig. 1. illustration of the changes in values of qualitative (expert assessment) and quantitative (preclinical, clinical studies, data from real clinical practice) assessments during the early and late phases of the intervention lifecycle. 5. conclusion a data management model for proactive risk management in healthcare is most appropriate approach and the tool capable increasing the sustainability of healthcare systems. nonnumerical qualitative expert opinions and the methodology of mutual transformation of qualitative data into quantitative and vice versa are of the same importance as numerical research data and form all together an integrated evidence-based support for the healthcare decision making. qualitative expert opinions make it possible to decrease the asymmetry and lack of quantitative data between model studies on animals and the characteristics of the human body. the value of qualitative expert assessment decreases as digital evidence accumulates, but any changes in the decision-making system increase its importance. sustainability of healthcare decision-making in changing environment depends on availability and methods of qualitative expert support. currently, there are no generally accepted methods for the final qualitative expert assessment, which makes it possible to reduce the diversity of interdisciplinary big data to a transparent and understandable assessment. automated systems for early detection of emergent cases based on medical statistics capable increasing sustainability of healthcare management. action plans are needed for the most unpredictable and uncertain cases related to public health. references 1. reis, j., faou, a. l., buguet, a., sandner, g. & spencer, p. 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(2015). activities of russian who-fic collaboration centre, who-family of international classifications network annual meeting (17–23 october 2015, manchester, united kingdom), c216. microsoft word 1301_final adv syst sci appl 2022; 04; 79-91 published online at http://ijassa.ipu.ru. bifurcation in the model of cargo transportation organization nerses k. khachatryan1* 1) state academic university for the humanities, moscow, russia e-mail: nerses-khachatryan@yandex.ru abstract: this article is devoted to the study of the model of cargo transportation organization between two nodal stations. the main characteristic of an arbitrary station is the degree of inconsistency between receiving and sending cargo, which is the difference between the volume of incoming and outgoing cargo per unit of time. the initial node station accepts goods depending on the demand for them within its technical potential, which is determined by the maximum allowable increase in the degree of inconsistency between the reception and dispatch of goods per unit of time. the movement of goods from one station to another is carried out within the framework of their technical potentials. the distribution of goods from the final node station is carried out in a certain mode. such a model is described by a system of differential equations with a number of parameters that define the characteristics of the demand for cargo transportation, the degree of use of the technical potential of the stations and the mode of cargo distribution from the final node station. when changing the parameters of the model, a bifurcation effect occurs and multiple stationary solutions appear, among which the most acceptable from the point of view of economic feasibility are identified. keywords: cargo transportation organization model, differential equations, stationary solutions, model parameters, stability 1. introduction the mathematical models used for the analysis of transport networks are diverse in terms of the tasks to be solved, the mathematical apparatus, the data used and the degree of detail of the description of traffic. based on the functional role of models, i.e. on the tasks for which they are used, three main classes can be conditionally distinguished: predictive, simulation and optimization models [24]. predictive models allow to determine what will be the traffic flows in the network with known geometry and properties of the transport network. the forecast of the load of the transport network includes the calculation of the average characteristics of traffic, such as the volume of inter-district movements, the intensity of the flow, the distribution of vehicles along the paths, etc. with the help of such models, it is possible to predict the consequences of changes in the transport network or in the placement of objects. simulation modeling aims to reproduce all the details of the movement, while the averaged value of the flows and the distribution along the paths are considered known and serve as the initial data for these models. thus, flow forecasting and simulation modeling are complementary directions [9]. the third class of models is aimed at optimizing the functioning of transport networks. with their help the problems of optimization of transport routes, development of the optimal network configuration, etc. are solved [10, 23, 26, 27]. according to the method of describing traffic flows, all models of transport networks can be divided into classes: models-analogues, models-following the leader, models-probabilistic. * corresponding author: nerses-khachatryan@yandex.ru 80 n.k. khachatryan copyright ©0000 assa adv. in systems science and appl. (0000) in analog models, the movement of a vehicle is likened to some kind of physical flow (hydroand gas-dynamic models) [6, 7, 9]. in models of following the leader, it is important to assume that there is a connection between the movement of the driven and the main vehicle [5]. in probabilistic models, the traffic flow is considered as a result of the interaction of vehicles on the elements of the transport network. due to the rigid nature of network restrictions and the mass nature of traffic in the traffic flow, there are clear patterns of formation of queues, intervals, loads on the road lanes, etc. these patterns are significantly stochastic [25]. one of the most popular modes of transport for cargo transportation in russia is rail. publications devoted to railway logistics can be divided into the following main groups according to the type of tasks studied: 1) the tasks of designing the infrastructure of the railway network; 2) tasks of managing the fleet of locomotives and wagons; 3) tasks of railway planning. in the first group we can highlight the works [8, 11, 19, 22]. the task of managing the fleet of locomotives and wagons is described in [4]. the third group, in particular, is represented by the tasks of forming the schedule of freight trains and the organization of freight flows [20, 21]. one of the approaches to the organization of cargo traffic is described in [1, 2, 12-16]. they present macroscopic dynamic models in which the process of organizing railway freight transportation is the formation of cargo traffic based on the interaction of neighboring stations. by their functional role, they are predictive, because they allow predicting the dynamics of loading of the stations and flows arising in the railway network, at the set procedure of the organization of a cargo flow. in the way of describing traffic flows, they are close to the models of following the leader, if the flow on a particular section of the railway network is identified with the congestion of the respective stations. at the same time, these models have a significant difference from the models of following the leader, which means that each station interacts not with one, but with the two nearest neighboring stations. in [17] and [18], another approach to the organization of cargo traffic is presented, in which the process of its formation is determined by the demand for cargo transportation and the degree of use of the technical potential of stations. this work is devoted to the study of the modification of the model described in [17], and unlike it, it is initially aimed at finding practically feasible modes of cargo transportation. such models are characterized by the global stability of a stationary solution, which is sometimes economically impractical. at the same time, when the model parameters change, a bifurcation effect occurs and multiple stationary solutions appear, among which there are economically feasible ones. thus, in this model, the presence of bifurcation makes it possible to obtain solutions that describe the optimal functioning of the system. 2. problem statement consider the movement of goods on a section of the railway network between two nodal stations connected by a set of intermediate stations. denoting the number of intermediate stations by 𝑚 , we get the following set of station numbers {0,1, . . , 𝑚, 𝑚 + 1}, where 0 – is the number of the initial node station, and, а 𝑚 + 1 – is the number of the final node station. denote by 𝑛 the number of roads at the station with the number 𝑖. we assume that all paths are used with the same degree of efficiency. consider discrete moments of time 𝑡 , 𝑡 , 𝑡 , . . . ; 𝑡 = 𝑡 + δ𝑡, 𝑘 = 1,2, . .. let 𝑉 (𝑡 ) is the volume of cargo received on the 𝑗th road of the 𝑖th station for a period of time [𝑡 , 𝑡 ], and 𝑉 (𝑡 ) is the volume of cargo sent from the 𝑖th road of the 𝑖th station for a period of time [𝑡 , 𝑡 ]. denote bifurcation in the model of cargo transportation organization 81 copyright ©0000 assa. adv. in systems science and appl. (0000) 𝑥 (𝑡 ) = 𝑉 (𝑡 ) − 𝑉 (𝑡 ) 𝑉 (𝑡 ) , 𝑖𝑓 𝑉 (𝑡 ) > 𝑉 (𝑡 ) 0, 𝑖𝑓 𝑉 (𝑡 ) ≤ 𝑉 (𝑡 ). it's obvious that 0 ≤ 𝑥 (𝑡 ) ≤ 1, 𝑖 = 0,1, . . . , 𝑚 + 1; 𝑗 = 1,2, . . . , 𝑛 and characterize the degree of inconsistency between receiving and sending on the 𝑗th road of the 𝑖th station at time 𝑡 . denote 𝑧 (𝑡 ) = 1 𝑛 𝑥 (𝑡 ). it is also obvious that 0 ≤ 𝑧 (𝑡 ) ≤ 1 and characterizes the degree of inconsistency between receiving and sending at the 𝑖th station at time 𝑡 . the technical potential of the station is determined by the maximum allowable increase in the degree of inconsistency between the reception and dispatch of goods per unit of time and is given by a non-negative decreasing function ( )z defined on the segment [0,1] and satisfying the condition 𝜑(1) = 0. the initial node station accepts cargo depending on the demand for transportation within its technical potential and sends it to the next station within its technical potential. each of the intermediate stations receives cargo within its technical potential and sends it within the technical potential of the next station. the final node station accepts cargo within its technical potential and distributes it in a certain mode. taking into account the above, we will write down a system of finite-difference equations describing the change in the degree of inconsistency between the reception and dispatch of goods at stations. 𝑧 (𝑡 ) = 𝑧 (𝑡 ) + min 𝑑 , 𝜑 𝑧 (𝑡 ) − 𝜆𝜑 𝑧 (𝑡 ) δ𝑡, 𝑘 = 1, 2, 3 … (2.1) 𝑧 (𝑡 ) = 𝑧 (𝑡 ) + 𝜆𝜑 𝑧 (𝑡 ) − 𝜆𝜑 𝑧 (𝑡 ) δ𝑡, 𝑖 = 1, 𝑚, 𝑘 = 1, 2, 3 … (2.2) 𝑧 (𝑡 ) = 𝑧 (𝑡 ) + 𝜆𝜑 𝑧 (𝑡 ) − 𝑑 δ𝑡, 𝑘 = 1, 2, 3 … (2.3) 0 ≤ 𝑧 (𝑡 ) ≤ 1, 𝑖 = 0, 1, … , 𝑚 + 1, 𝑘 = 0, 1, 2, … (2.4) here 𝑑 > 0, 0 < 𝜆 ≤ 1, 𝑑 > 0 are the model parameters: 𝑑 is a characteristics of the demand for transportation; 𝜆 is a characteristics of the degree of use of the technical potential of the stations; 𝑑 is a characteristics of the cargo distribution mode from the final node station. let's move on to the continuous analog of the system of discrete-difference equations (2.1)–(2.4), presented below �̇� (𝑡) = min(𝑑 , 𝜑(𝑧 (𝑡))) − 𝜆𝜑(𝑧 (𝑡)), 𝑡 ∈ [𝑡 , +∞), (2.5) �̇� (𝑡) = 𝜆[𝜑(𝑧 (𝑡)) − 𝜑(𝑧 (𝑡))], 𝑖 = 1,2, . . . , 𝑚, 𝑡 ∈ [𝑡 , +∞), (2.6) �̇� (𝑡) = 𝜆𝜑(𝑧 (𝑡)) − 𝑑 , 𝑡 ∈ [𝑡 , +∞), (2.7) 0 ≤ 𝑧 (𝑡) ≤ 1 , 𝑖 = 0, 1, . . ., 𝑚 + 1, 𝑡 ∈ [𝑡 , +∞). (2.8) next, consider the function that sets the technical potential of the stations, of the following type 𝜑(𝑧) = 𝑎(1 − 𝑧), 𝑎 > 0. (2.9) 82 n.k. khachatryan copyright ©0000 assa adv. in systems science and appl. (0000) the parameter 𝑎 > 0 , which participates in the definition of the function 𝜑(𝑧) , is a characteristic of the ability of stations to increase cargo traffic. since 𝜑(𝑧) ≤ a for all 0 ≤ 𝑧 ≤ 1 then the parameter 𝑑 , which is a characteristic of the demand for transportation and is involved in equation (1.1), can be represented as follows: 𝑑 = 𝜇𝑎, 0 < 𝜇 ≤ 1. (2.10) the parameter 𝑑 , which is a characteristic of the cargo distribution mode from the final node station, is represented as follows: 𝑑 = 𝛾𝑎, 𝛾 > 0. (2.11) let's rewrite the system (2.5)-(2.8), in which the function 𝜑(𝑧) is defined according to (2.9), and the parameters 𝑑 and 𝑑 are defined according to (2.10) and (2.11), respectively. �̇� (𝑡) = min(𝜇𝑎, 𝑎(1 − 𝑧 (𝑡))) − 𝜆𝑎(1 − 𝑧 (𝑡)), 𝑡 ∈ [𝑡 , +∞), (2.12) �̇� (𝑡) = 𝜆𝑎 (𝑧 (𝑡) − 𝑧 (𝑡)), 𝑖 = 1,2, . . . , 𝑚, 𝑡 ∈ [𝑡 , +∞), (2.13) �̇� (𝑡) = 𝜆𝑎(1 − 𝑧 (𝑡)) − 𝛾𝑎 , 𝑡 ∈ [𝑡 , +∞), (2.14) 0 ≤ 𝑧 (𝑡) ≤ 1 , 𝑖 = 0,1, . . . , 𝑚 + 1, 𝑡 ∈ [𝑡 , +∞). (2.15) here 𝜇, 𝑎, 𝜆, 𝛾 are the model parameters: 𝜇 (0 < 𝜇 ≤ 1) is a characteristics of the range of demand for transportation, which can be satisfied with the existing technical potential of the stations; 𝑎 (𝑎 > 0) is a characteristic of the ability of stations to increase cargo traffic; 𝜆 (0 < 𝜆 ≤ 1) is a characteristics of the degree of use of the technical potential of the stations; 𝛾 (𝛾 > 0) is a characteristics of the cargo distribution mode from the final node station. here are the main objectives of the study: to determine the ranges of change of parameters 𝜇, 𝑎, 𝜆, 𝛾, in which the system of cargo transportation can function smoothly, i.e. system (2.12)-(2.15) has a solution, describe the qualitative behavior of solutions depending on the parameters. for a given value of the demand characteristics for cargo transportation (parameter 𝜇) set the most acceptable achievable levels of the degree of inconsistency between the reception and dispatch of goods at all stations, by controlling the values of the following characteristics: the ability of stations to increase cargo traffic (parameter 𝑎), the degree of use of the technical potential of stations (parameter 𝜆) and the mode of cargo distribution from the final node station (parameter 𝛾). 3. investigation of system solutions (2.12)-(2.15) the study of the set of solutions of the system (2.12)–(2.15) begins with the study of all solutions of the system of differential equations (2.12)–(2.14). first of all, we will highlight the stationary solutions of the system (2.12)–(2.14). with the help of direct verification, you can verify the validity of the following statement proposition 3.1: system (2.12)–(2.14) for any parameter values 0 < 𝜇 ≤ 1, 𝑎 > 0, 0 < 𝜆 ≤ 1, 𝛾 > 0 such that 𝛾 ≤ 𝜇 has stationary solutions: 𝑧 (. ) ≡ 1 − 𝛾, 𝑧 (. ) ≡ 1 − , 𝑖 = 1, . . . , 𝑚 + 1, for 𝛾 < 𝜇; (3.1) 𝑧 (. ) ≤ 1 − 𝜇, 𝑧 (. ) ≡ 1 − , 𝑖 = 1, . . . , 𝑚 + 1, for 𝛾 = 𝜇. (3.2) bifurcation in the model of cargo transportation organization 83 copyright ©0000 assa. adv. in systems science and appl. (0000) for 𝛾 > 𝜇 the system (2.12)–(2.14) has no stationary solutions. let's proceed to the study of the remaining solutions of the system (2.12)–(2.14). theorem 3.1: an arbitrary solution of the system (2.12)–(2.14) at 𝛾 < 𝜇 eventually goes to a stationary solution (3.1), and at 𝛾 = 𝜇 to one of the stationary solutions (3.2). for 𝛾 > 𝜇 coordinates 𝑧 (. ), 𝑖 = 1, . . ., 𝑚 + 1 of solutions of the system (2.12)–(2.14) eventually enter the stationary mode specified in (3.1), and the function 𝑧 (. ) becomes linearly decreasing. proof. let is find the general solution of the system (2.12)–(2.14). let is start with the last equation, which contains one variable (𝑧 ). let 's rewrite it in the following form �̇� (𝑡) + 𝜆𝑎𝑧 (𝑡) = (𝜆 − 𝛾)𝑎. (3.3) it is not difficult to verify that linear equation (3.3) has the following general solution 𝑧 (𝑡) = 1 − + 𝑐 𝑒 . (3.4) substituting the expression for 𝑧 from (3.4) into the penultimate equation of the system (2.12)–(2.14), we find its general solution. it has the following form 𝑧 (𝑡) = 1 − + 𝑒 (𝜆𝑎𝑐 𝑡 + 𝑐 ). (3.5) similarly, we will find general solutions to all other equations of the system (2.12)-(2.14) except the initial one: 𝑧 (𝑡) = 1 − + 𝑒 ( 𝑡 + 𝑐 𝑡 + 𝑐 ); … 𝑧 (𝑡) = 1 − + 𝑒 ( ( )! 𝑡 + 𝑐 𝑡 + ⋯ + 𝑐 𝑡 + 𝑐 ); (3.6) 𝑧 (𝑡) = 1 − + 𝑒 ( ! 𝑡 + 𝑐 𝑡 + ⋯ + 𝑐 𝑡 + 𝑐 ). it follows from (3.5) and (3.6) that lim → 𝑧 (𝑡) = 1 − , 𝑖 = 1, 2, … , 𝑚. let's move on to solving the first equation of the system (2.12)-(2.14). let 's rewrite it in the following form �̇� (𝑡) = 𝜇𝑎 − 𝜆𝑎 1 − 𝑧 (𝑡) , if 𝑧 (𝑡) < 1 − 𝜇, 𝑡 ∈ [𝑡 , +∞), 𝑎 1 − 𝑧 (𝑡) − 𝜆𝑎 1 − 𝑧 (𝑡) , if 𝑧 (𝑡) ≥ 1 − 𝜇 , 𝑡 ∈ [𝑡 , + ∞). (3.7) consider the following two equations �̇� (𝑡) = 𝜇𝑎 − 𝜆𝑎 1 − 𝑧 (𝑡) , 𝑡 ∈ 𝑡, +∞ , (3.8) �̇� (𝑡) = 𝑎 1 − 𝑧 (𝑡) − 𝜆𝑎 1 − 𝑧 (𝑡) , 𝑡 ∈ 𝑡, +∞ , (3.9) where 𝑡 ≥ 𝑡 . using the expression for 𝑧 (𝑡) from (3.6), we obtain the solution of equations (3.8) and (3.9). they can be represented as follows 𝑧 (𝑡) = 𝑎(𝜇 − 𝛾)𝑡 + 𝐹 (𝑡) + 𝑐 , where 𝐹 (𝑡) ∈ с 𝑡, +∞ , lim → 𝐹 (𝑡) = 0, (3.10) 𝑧 (𝑡) = 1 − 𝛾 + 𝐹 (𝑡), where 𝐹 (𝑡) ∈ с 𝑡, +∞ , lim → 𝐹 (𝑡) = 0 (3.11) these solutions allow us to investigate the asymptotic behavior of the solution of equation (3.7). it is easy to see that when 𝛾 < 𝜇 the asymptotics of the solution of equation (3.7) is 84 n.k. khachatryan copyright ©0000 assa adv. in systems science and appl. (0000) determined by the relation (3.11), i.e. lim → 𝑧 (𝑡) = 1 − 𝛾 . for 𝛾 > 𝜇 the asymptotics of equation (3.7) is determined by the relation (3.10), i.e. 𝑧 (. ) decreases linearly and lim → 𝑧 (𝑡) = − ∞ . for 𝛾 = 𝜇 the asymptotics of equation (3.7), depending on the initial conditions, can be determined by both equation (3.10) and equation (3.11), i.e. either lim → 𝑧 (𝑡) = с , where с < 1 − 𝜇 or lim → 𝑧 (𝑡) = 1 − 𝜇. □ let us proceed to the study of solutions of system (2.12)-(2.14) satisfying constraints (2.15). lemma 3.1: for all values of parameters 𝑎 > 0 , 𝛾 > 0 and 𝜆 , 𝛾 ≤ 𝜆 ≤ 1 components 𝑧 (. ), 𝑧 (. ), … 𝑧 (. ) of an arbitrary solution of the system (2.12)-(2.14) satisfying the constraints (2.15) at the initial moment of time will satisfy them at subsequent moments of time. proof. let's start by considering the last component of the solution of the system (2.12)-(2.14), i.e. 𝑧 (. ). it has the form (3.4), where с is determined from the condition 1 − + 𝑐 𝑒 = 𝑧 , where 0 ≤ 𝑧 ≤ 1, i.e. 𝑐 = ( − 1 + 𝑧 )𝑒 , where 0 ≤ 𝑧 ≤ 1. (3.12) from (3.12) follows ( − 1)𝑒 ≤ 𝑐 ≤ 𝑒 . using this estimate for 𝑐 and expression (3.4), we get an estimate for 𝑧 (. ). it will take the following form (1 − )(1 − 𝑒 𝑒 ) ≤ 𝑧 (𝑡) ≤ 1 − (1 − 𝑒 𝑒 ). (3.13) it follows from (3.13) that for all 𝛾 ≤ 𝜆 there is an inequality 0 ≤ 𝑧 (𝑡) ≤ 1, 𝑡 ∈ [𝑡 , +∞) . (3.14) we show that for the other components of the solution of the system (2.12)-(2.14), inequalities similar to inequality (3.14) are fulfilled. let's start with the component 𝑧 (. ). to do this, consider equation (2.13) for 𝑖 = 𝑚: �̇� (𝑡) = 𝜆𝑎 𝑧 (𝑡) − 𝑧 (𝑡) , 𝑡 ∈ [𝑡 , +∞). we show that the function 𝑧 (. ) cannot take a value greater than 1. indeed, otherwise, due to the continuity of the function 𝑧 (. ) there must be a point 𝑡∗ > 𝑡 , such that 𝑧 ( 𝑡∗) = 1. then it follows from (3.14) that �̇� ( 𝑡∗) ≤ 0. similarly, the function 𝑧 (. ) cannot take a value less than 0. thus, it is proved that for all 𝛾 ≤ 𝜆 function 𝑧 (. ) also satisfies an inequality similar to inequality (3.14). similarly, the satisfiability of all other inequalities is proved. □ we formulate a similar lemma for the zero component of the solution of system (2.12)(2.14). lemma 3.2: for all values of parameters 𝑎 > 0, 𝛾 > 0 and 𝜇, 𝛾 ≤ 𝜇 ≤ 1 there exists 𝜆(𝜇, 𝑧 (𝑡 ), 𝑧 (𝑡 ), . . ., 𝑧 (𝑡 )), 𝜇 ≤ 𝜆 ≤ 1 such that for any the value of the parameter 𝜆 from the segment [𝜇, 𝜆] the zero component 𝑧 (. ) of an arbitrary solution of the system (2.12)-(2.14) satisfying the constraint (2.15) at the initial moment of time, satisfies it at subsequent moments of time. bifurcation in the model of cargo transportation organization 85 copyright ©0000 assa. adv. in systems science and appl. (0000) proof. as for the other components, we show that the function 𝑧 (. ) cannot take a value greater than 1. indeed, otherwise, due to the continuity of the function 𝑧 (. ) there must be a point 𝑡∗∗ > 𝑡 , such that 𝑧 ( 𝑡∗∗) = 1. then it follows from (2.12) that �̇� ( 𝑡∗∗) = 𝜆𝑎(𝑧 ( 𝑡∗∗) − 1), that is, according to lemma 1, �̇� ( 𝑡∗∗) ≤ 0. let's move on to evaluating the function 𝑧 (. ) from below. to do this, we investigate the behavior of its derivative at 𝑧 (. ) → 0 +. according to (2.12), it is described by the equation �̇� (𝑡) = 𝑎(𝜇 − 𝜆(1 − 𝑧 (𝑡))). let 's investigate the inequality 𝜇 − 𝜆(1 − 𝑧 (𝑡)) ≥ 0. let 's rewrite it in the form 𝑧 (𝑡) ≥ 1 − . (3.15) according to lemma 1, for arbitrary 𝛾 > 0, 𝜆 satisfying the condition 𝛾 ≤ 𝜆 ≤ 1 there is an inequality 0 ≤ 𝑧 (𝑡) ≤ 1, 𝑡 ∈ [𝑡 , +∞) . (3.16) it follows from (3.16) that an arbitrary μ satisfying the condition 𝛾 ≤ 𝜇 ≤ 1 exists 𝜆, 𝜇 ≤ 𝜆 ≤ 1 such that for any value of the parameter 𝜆 from the segment [𝜇, 𝜆] inequality (3.15) will hold for all 𝑡 ∈ [𝑡 , +∞), i.e. �̇� (𝑡) ≥ 0 for 𝑧 (𝑡) → 0 +, which shows that the function 𝑧 (. ) is limited from below by the value 0. obviously 𝜆 depends on both 𝜇 , and initial conditions, so we denote it 𝜆(𝜇, 𝑧 (𝑡 ), 𝑧 (𝑡 ), . . ., 𝑧 (𝑡 )). □ we formulate the main result of this paragraph. theorem 3.2: for any initial values 0 ≤ 𝑧 (𝑡 ) ≤ 1, 𝑖 = 0, 1, . . ., 𝑚 + 1, parameters 𝑎 > 0, 𝛾 > 0 and 𝜇, 𝛾 ≤ 𝜇 ≤ 1 there exists 𝜆(𝜇, 𝑧 (𝑡 ), 𝑧 (𝑡 ), . . ., 𝑧 (𝑡 )) , 𝜇 ≤ 𝜆 ≤ 1 such that for any value of the parameter 𝜆 from the segment [𝜇, 𝜆] the solution of the system (2.12)(2.15) exists and converges to the stationary solution (3.1) (for 𝛾 < 𝜇 ) or to one of the stationary solutions (3.2), the same for all γ and a (for 𝛾 = 𝜇). proof. the proof follows directly from theorem 1, lemma 1 and lemma 2. □ corollary 3.1: the system of differential equations (2.12)–(2.15) has a globally stable stationary solution (3.1) and a family of stable solutions of the form (3.2). proof. the proof follows directly from theorem 2. □ 4. examples of solutions of the system (2.12)–(2.15) let's consider some examples of solutions of the system (2.12)–(2.15) that clearly demonstrate the results of the previous paragraph. in all the examples below, the number of stations is 10 (the initial node station, eight intermediate stations, the final node station), the values of parameters, 𝜇, 𝑎 and the initial conditions are fixed and take the following values: 𝜇 = 0.6, 𝑎 = 1.5 (4.1) 𝑧 (𝑡 ) = 0.4, 𝑧 (𝑡 ) = 0.2, 𝑧 (𝑡 ) = 0.1, 𝑧 (𝑡 ) = 0.4, 𝑧 (𝑡 ) = 0.7, 𝑧 (𝑡 ) = 0.1, 𝑧 (𝑡 ) = 0.2, 𝑧 (𝑡 ) = 0.3, 𝑧 (𝑡 ) = 0.5, 𝑧 (𝑡 ) = 0.3. (4.2) 86 n.k. khachatryan copyright ©0000 assa adv. in systems science and appl. (0000) according to theorem 3.2, for any initial values satisfying condition (2.15), parameters 𝑎 > 0, 𝛾 > 0 and 𝜇, 𝛾 ≤ 𝜇 ≤ 1 there exists 𝜆 (𝜇 ≤ 𝜆 ≤ 1) depending on the initial conditions and the parameter 𝜇, such that for all 𝜆 from the segment [𝜇, 𝜆] the solution of the system (2.12)–(2.15) exists. calculations have shown that for the above initial conditions (4.2) and parameter 𝜇 (4.1), 𝜆 takes the value equal to 0.898 (𝜆 = 0.898). the nature of the solution of the system (2.12)–(2.15) depends on the value of the parameter 𝛾. depending on whether it is less than the value of the parameter 𝜇 or equal to it, there are two types of solutions of the system (2.12)–(2.15). let's start with the first case (𝛾 < 𝜇). in fig. 4.1, fig. 4.2 and fig. 4.3. graphs of solutions of the system (2.12)–(2.15) with a fixed value of parameter 𝛾 (𝛾 = 0.5) and three different values of parameter 𝜆 are given: two at the ends of the segment [𝜇, 𝜆] and one at an internal point. fig. 4.1. graph of the solution of the system (2.12)–(2.15) (𝛾 = 0.5, 𝜇 = 𝜆 = 0.6). fig. 4.2. graph of the solution of the system (2.12)–(2.15) (𝛾 = 0.5, 𝜇 = 0.6, 𝜆 = 0.85) bifurcation in the model of cargo transportation organization 87 copyright ©0000 assa. adv. in systems science and appl. (0000) fig. 4.3. graph of the solution of the system (2.12)–(2.15) (𝛾 = 0.5, 𝜇 = 0.6, 𝜆 = 𝜆 = 0.898) note that due to the global stability of the stationary solution (3.1), the asymptotic behavior of the solutions shown in figures 4.1–4.3 does not change when the initial values change. these figures clearly show that the steady – state mode for the zero component of the solution of the system (2.12)–(2.15) does not depend on the parameter 𝜆, and for the remaining components, an increase in the parameter 𝜆 leads to their asymptotic increase. recall that the components of the solution of the system (2.12)–(2.15) determine the dynamics of the degree of inconsistency between the reception and dispatch of goods at stations. therefore, in this case, the question of choosing a parameter 𝜆 is uniquely determined, its value should be taken equal to the value of the parameter 𝜇. let's move on to the second case (𝛾 = 𝜇). fig. 4.4, fig. 4.5 and fig. 4.6 show graphs of solutions of the system (2.12)–(2.15) at 𝛾 = 0.6 and the same values of parameter 𝜆, as for the previous case. fig. 4.4. graph of the solution of the system (2.12)–(2.15) 𝛾 = 𝜆 = 𝜇 = 0.6 88 n.k. khachatryan copyright ©0000 assa adv. in systems science and appl. (0000) fig. 4.5. graph of the solution of the system (2.12)–(2.15) 𝛾 = 𝜇 = 0.6, 𝜆 = 0.85 fig. 4.6. graph of the solution of the system (2.12)–(2.15) 𝛾 = 𝜇 = 0.6, 𝜆 = 𝜆 = 0.898 comparing figures 4.4, 4.5 and 4.6, it can be seen that an increase in the value of parameter 𝜆 leads to an asymptotic decrease in the zero component of the solution of the system (2.12)– (2.15), in contrast to the first case. the remaining components of the solution, as in the first case, increase asymptotically with an increase in the value of the parameter 𝜆. this means that in this case, by controlling the parameter 𝜆 you can set the desired degree of inconsistency between the reception and dispatch of goods at all stations (including at the zero station). if we take 𝜆 equal to 𝜇 , then over time the degree of inconsistency between the reception and dispatch of goods at all stations except zero will become zero, and at zero station 1 − 𝜇. if the value of 𝜇 is close to one, then this choice of the parameter 𝜆 will be optimal. otherwise, everything will depend on the specific value of the parameter 𝜇, as well as on how important it is in a given situation to reduce the degree of inconsistency between receiving and sending goods at the initial node station by increasing this characteristic at other stations. comparing figures 4.1 and 4.4, figures 4.2 and 4.5, as well as figures 4.3 and 4.6, it can be concluded that in terms of minimizing the degree of inconsistency between the reception and dispatch of goods at stations, it is advisable to take the parameter 𝛾 equal to the parameter 𝜇. this means that the mode of cargo distribution from the final node station must be coordinated with the characteristics of the demand for transportation. bifurcation in the model of cargo transportation organization 89 copyright ©0000 assa. adv. in systems science and appl. (0000) 5. investigation of the attraction area of stationary solutions of the system (2.12)–(2.15) of the form (3.2) according to consequence 1, the system of differential equations (2.12)–(2.15) has a globally stable stationary solution (3.1) and a family of stable solutions of the form (3.2). we select the following stationary solution from this family 𝑧 (. ) ≡ 1 − 𝜇, 𝑧 (. ) ≡ 1 − , 𝑖 = 1, . . . , 𝑚 + 1. (5.1) as numerical experiments have shown, this stationary solution is in contrast to other stationary solutions of the form (3.2) presented below 𝑧 (. ) < 1 − 𝜇, 𝑧 (. ) ≡ 1 − , 𝑖 = 1, . . . , 𝑚 + 1 (5.2) and being locally stable, has a certain area of attraction. denote с(𝜇, 𝜆) = ( )( / ) . the analysis of a large number of numerical experiments made it possible to describe the region of attraction of the stationary solution (5.1). the results of this analysis are given in the following proposition. proposition 5.1: for any parameter values 0 < 𝜇 ≤ 1, 𝑎 > 0, 𝜇 ≤ 𝜆 ≤ 𝜆 , 𝛾 = 𝜇 the solution of the system (2.12)–(2.15) converges to the stationary solution (5.1) if the initial values satisfy the condition 𝑧 (𝑡 ) ≥ �̃�(𝜇, 𝜆), 𝑖 = 0, 1, . . . , 𝑚 + 1. otherwise, i.e. if the condition𝑧 (𝑡 ) ≤ �̃�(𝜇, 𝜆), 𝑖 = 0, 1, . . . , 𝑚 + 1 and ∃ 𝚤̅ ∈ {0, 1, . . . ., 𝑚 + 1} such that 𝑧 ̅(𝑡 ) < �̃�(𝜇, 𝜆) is met, then the solution of the system (2.12)–(2.15) converges to one of the stationary solutions (5.2). □ 6. conclusion this article presents a model of organization of cargo transportation between two nodal stations, described by a system of differential equations with a number of parameters that define the characteristics of demand for cargo transportation, the degree of use of the technical potential of the stations and the mode of cargo distribution from the final nodal station. the ranges of parameter changes under which the cargo transportation system can function smoothly are determined. for a given value of the demand characteristics for cargo transportation, by controlling the degree of use of the technical potential of the stations and the mode of cargo distribution from the final node station, the most acceptable achievable levels of the degree of inconsistency between the reception and dispatch of goods at all stations are established. references [1] beklaryan l.a. & khachatryan n.k. 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[optimization of planning and control of transport systems]. moscow: transport [in russian]. adv syst sci appl 2023; 03:153–163 published online at https://ijassa.ipu.ru. butler group direct decomposition classification with applications to parallel algorithms ekaterina blagoveshchenskaya*, ilya mikulik emperor alexander i st. petersburg state transport university, st. petersburg, russia abstract: the graphical approach to the classification problem of butler group direct decompositions is used to preserve the indecomposability property of some rigid subgroups in all possible direct decompositions of the group itself. the group class under consideration as well as torsion-free abelian groups as a whole admits non-isomorphic direct decompositions. the proof of decomposition existence with predicted properties is one of the investigation streams. until now the related results concerned only the ranks of indecomposable summands. now the way of controlling the other properties of group decompositions is suggested. all the results in this direction are closely connected with the algorithm parallelization. the special feature of the results presented is that they give the method of constructing certain dependence graphs as the subgraphs of the algorithm graph to be given in a parallel form preserving the corresponding fragments. such dependence subgraphs can define the data relations in parallel computations, which reflect various conditions of parallelism. keywords: butler groups, direct decompositions, parallel algorithms 1. introduction we consider a class of torsion-free abelian groups which is a subclass of the so-called almost completely decomposable groups. the latter belong to the wider class of butler groups which are epimorphic images of finite rank completely decomposable groups, see [1, theorem 2.4.19]. recall the main notions. the monographs [1] [4] serve as the main references and we adopt their standard notations throughout this paper. any torsion-free abelian group the can be considered as a subgroup of a direct sum of copies of the rationals q. if the minimal possible number of the summands is finite then it is called the rank of x , which is denoted by rkx (in comparison, there exist abelian groups of infinite ranks). a cd-group (completely decomposable group) is a direct sum of rank-one groups (subgroups of the rationals). an acd-group x (almost completely decomposable group) is a torsion-free abelian group of finite rank, that contains a completely decomposable group v with finite x/v . if in addition x/v is a cyclic group then x is called a crq-group (i.e. an acd-group with cyclic regulator quotient). any acd-group x has a distinguished completely decomposable fully invariant subgroup r(x). this group r(x) is called the regulator of x , and [x : r(x)] is the regulator index of x . its regulator exponent is the exponent e =: exp x/r(x) of the regulator quotient x/r(x). in general, for an abelian group generated by a system of elements we use the symbol ⟨. . . ⟩. as usual, v ⊂ x means that v is a subgroup of x and v∗ = {g ∈ x : there is n ∈ ∗corresponding author: blagoveschenskaya@pgups.ru, kblag2002@yahoo.com 154 ekaterina blagoveshchenskaya, ilya mikulik n with ng ∈ v } denotes the purification of v in x . the type of an element g ∈ x denoted by tpx g can be determined as the isomorphism class of the rational group τ , which is isomorphic to ⟨g⟩∗ in x and contains the group of integers z. then we may say that the element g is of type τ , z ⊂ τ ⊂ q. furthermore, tpx g coincides with the group type, tpx , if x is a homogeneous group. we will write τ(p) = ∞ or pτ = τ if 1/pn belongs to τ for any natural number n (p is a prime). following standard definitions also x∗(τ) = ∑ σ>τ x(σ) and x♯(τ) is the purification of x∗(τ) in x . a type τ is critical, a member of the set tcr(x) of critical types of x , if x(τ)/x♯(τ) ̸= 0, see [1, p. 37, definition 2.4.6 ]. any direct sum of rank-one groups of a fixed type τ is called (τ−) homogeneous cd-group, i.e. completely decomposable group, whose elements are of type τ . we are concentrated on the so-called block-rigid crq-groups x of ring type and their direct decompositions. this means that tcr(x) is an antichain and consists only of idempotent types (i.e. the ones which are types of idempotent rational groups, or in other words, can be represented by characteristics, consisting only of 0′s and ∞′s, see [1, p. 13], [3, section 85]). the regulator a = r(x) decomposes uniquely a = ⊕ τ∈tcr(x) aτ into its τ homogeneous components aτ = a(τ) = x(τ), which are pure in x (that is na ∈ aτ with natural n and a ∈ x implies a ∈ aτ ). if τ ̸∈ tcr(x)) then aτ = 0. if rkaτ = 1 for all τ ∈ tcr(x), then a and x are called rigid groups. torsion-free abelian groups admit non-isomorphic direct decompositions. traditionally, direct decomposition classification is based on the near-isomorphism equivalence, which preserves their properties in detail. here we introduce a very natural weaker equivalence with the one goal to preserve only the main properties of abelian group direct decompositions. 2. direct decomposition theory of crq-groups: basic results near-isomorphism is an equivalence, which is weaker than isomorphism and traditionally used for classification of groups of this class, see [1, definition 9.1.2, theorem 9.1.4, 5], [2, theorem 7.16]: definition 2.1: let g and h be torsion-free abelian groups of finite rank. then g and h are called nearly isomorphic (in symbols g ∼=nr h) if and only if for any prime q there are monomorphisms ηq : g −→ h and ξq : h −→ g such that h/ηq(g) and g/ξq(h) are finite groups and |h/ηq(g)| and q as well as |g/ξq(h)| and q are relatively prime. this equivalence preserves decomposability properties of torsion-free abelian groups of finite rank: theorem 2.1 (12.9 (b), p. 144, [2]): let x and y be nearly isomorphic torsion-free abelian groups of finite rank and x = x1 ⊕x2. then there exists a decomposition y = y1 ⊕ y2 with y1 ∼=nr x1 and y2 ∼=nr x2. we concentrate on a block-rigid crq-group x of ring type with the regulator a = r(x) and cyclic regulator quotient x/a with e = exp x/a = |x/a|. the completely decomposable group a is a direct sum of its τ -homogeneous components aτ = a(τ) with aτ ∼= nττ (a direct sum of nτ copies of τ , z ⊂ τ ⊂ q), that is a = ⊕ τ∈tcr(a) a(τ), and n = rkx = rka = ∑ τ∈tcr(a) nτ . choose a generator b+ a of x/a, then eb = ∑ τ∈t vτ , vτ ∈ aτ , and mτ = mτ (x) = |vτ | = |vτ + ea|, (2.1) clearly e = lcmτ∈tcr(x) mτ . it is shown in [5, lemma 2.2] that mτ (x) do not depend on the choice of element b and serve as invariants of group x . moreover, they are the same for nearly isomorphic groups: copyright © 2023 assa. adv syst sci appl (2023) butler group direct decomposition classification 155 theorem 2.2 (near–isomorphism criterion, theorem 2.4, [5]): let x and y be block-rigid crq-groups. then x ∼=nr y if and only if r(x) ∼= r(y ) and mτ (x) = mτ (y ) for all types τ . we always put mτ (x) = 1 if τ /∈ tcr(x)). it is easy to see that for any prime divisor p of e there exist at least two members of tcr(x), say τ and σ, such that gcd(mτ ,mσ) is divisible by p (if only one mτ is divisible by p then p|aτ which contradicts the purity of aτ in a = r(x)). we list the additional restrictions on the groups x under consideration: s1. e is a square-free natural number; s2. for any prime divisor p of e there exist exactly two invariants, say mτ and mσ, which are divisible by p; s3. mτ (x) ̸= 1 for any τ ∈ tcr(x). a special decomposition, called the main decomposition in [1, theorem 13.1.6] and [5, theorem 3.5], always exists for almost completely decomposable groups, according to main decomposition theorem (see [1, theorem 9.2.7]). for the groups under consideration it is described in theorem 2.3 (main decomposition, theorem 3.5, [5]): let x be a block–rigid crq-group. then there exists a decomposition x = x0 ⊕ a′ such that a′ is completely decomposable, x0 is a rigid crq-group and τ ∈ tcr(x0) if and only if mτ (x0) = mτ (x) > 1. the group x0 is unique up to near isomorphism and a′ is unique up to isomorphism. such a main decomposition is not unique but uniquely determined up to near isomorphism. the rigid summand x0 from any main decomposition is called a main summand of x . since direct decompositions into indecomposable summands are of great importance the special role belongs to the following theorem 2.4 (indecomposability criterion, theorem 3.7, [5]): let x be a block–rigid crq–group. then x is directly indecomposable if and only if x is rigid, and there is no non–trivial partition tcr(x) = t1 ∪ t2 such that gcd(mσ(x),mτ (x)) = 1 whenever σ ∈ t1 and τ ∈ t2. the next theorem describes all the decompositions of a block-rigid crq-group into direct sum of indecomposable summands up to near isomorphism. theorem 2.5 (decomposability criterion, theorem 3.3, [5]): let x be a block-rigid crq-group. if x = x1 ⊕x2 ⊕ . . .⊕xt ⊕xt+1 with completely decomposable xt+1 (rkxt+1 ⩾ 0) and rigid indecomposable crq-groups xi with mτi = mτ (xi), i = 1, . . . , t, then, for all types τ , mτ (x) = ∏t i=1 mτi is a factorization such that: d1. the integers mτi and mσj are relatively prime whenever i ̸= j; d2. |{i : mτi > 1}| ≤ rk(x(τ)); d3. for any i = 1, . . . , t there is no non–trivial partition tcr(xi) = t i 1 ∪ t i 2 such that gcd(mσi,mτi) = 1 whenever σ ∈ t i 1 and τ ∈ t i 2. conversely, if mτ (x) = ∏t i=1mτ i is a factorization of the mτ (x) such that the decomposability conditions d1, d2 and d3 are satisfied, then there is a decomposition x = x1 ⊕x2 ⊕ · · · ⊕xt ⊕xt+1 such that xi are rigid indecomposable crq-groups, tcr(xi) = {ρ : mρi > 1} and mτ (xi) = mτi for i = 1, . . . , t, and xt+1 is a completely decomposable group (may be it is 0). remark 2.1: i. if the condition d3 is excluded, the theorem remains true, but the rigid direct summands of x are not necessarily indecomposable. copyright © 2023 assa. adv syst sci appl (2023) 156 ekaterina blagoveshchenskaya, ilya mikulik ii. let x = x1 ⊕x2 ⊕ · · · ⊕xv be a decomposition into indecomposable summands with tcr(x1) ∩ tcr(x2) = {τ1, . . . , τs} ≠ ∅. then there exists a decomposition x ∼= x1,2 ⊕ τ1 ⊕ . . .⊕ τs ⊕x3 ⊕ . . .⊕xv with the indecomposable group x1,2 satisfying the conditions: tcr(x1,2) = tcr(x1) ∪ tcr(x2), rkx1,2 ∼= rkx1 + rkx2 − s, mτ (x1,2) = mτ (x1)mτ (x2). iii. let x = x1 ⊕x2 and x = x ′ 1 ⊕x ′ 2 be two decompositions with indecomposable summands x1, x ′ 1 and let tcr(x1) ∩ tcr(x ′ 1) = {τ1, . . . , τs} ≠ ∅. then there exists a decomposition x = x1,2 ⊕x ′′ 2 having the indecomposable summand x1,2, which satisfies the conditions: tcr(x1,2) = tcr(x1) ∪ tcr(x ′ 1), rkx1,2 ∼= rkx1 + rkx ′ 1 − s, mτ (x1,2) = mτ (x1)mτ (x ′ 1). iv. let a = r(x) = ⊕ τ∈tcr(x) aτ and let x = x1 ⊕x2 ⊕ · · · ⊕xv be a decomposition into indecomposable summands. if p | gcd(mτ (xi),mσ(xi)) ̸= 1 for some i and rk(aτ ) = 1 then σ ∈ tcr(xi) with p | exp(xi/r(xi)) . 3. classification of block-rigid crq-groups: general approach let x be a block-rigid crq-group of ring type having the regulator a = r(x) and the cyclic regulator quotient x/a with e = exp x/a = |x/a|. denote by p = p(x) the set of all prime divisors of e = expx/a of a crq-group x . the completely decomposable group a is a direct sum of its τ -homogeneous components aτ = a(τ) with aτ ∼= nττ (a direct sum of nτ copies of τ , z ⊂ τ ⊂ q), that is a = ⊕ τ∈tcr(a) a(τ), and n = rkx = rka =∑ τ∈tcr(a) nτ . for any acd-group x (not only for that with a cyclic regulator quotient) its regulator a is a fully invariant subgroup. then for any group decomposition x = x1 ⊕x2 there exists the corresponding regulator decomposition a = a1 ⊕ a2 such that x/a ∼= x1/a1 ⊕x2/a2. then e = e1e2 with ei = exp xi/ai, i = 1, 2. let us introduce the definition of the new very natural equivalence on the class of acd-groups which is stronger than quasi-isomorphism but weaker than near isomorphism and preserves only the regulators and regulator quotients of groups and their direct decompositions. definition 3.1: let x and y be acd-groups. then x and y are called factor-identical groups (in symbols x ∼=fi y ) if r(x) ∼= r(y ) and x/r(x) ∼= y/r(y ). definition 3.2: let x and y be acd-groups. then x and y are called strongly factor-identical groups (in symbols x ∼=sfi y ) if and only if x ∼=fi y and for any decomposition x = x1 ⊕x2 ⊕ . . .⊕xt into indecomposable summands there also exists a decomposition y = y1 ⊕ y2 ⊕ . . .⊕ yt with indecomposable summands such that xi ∼=fi yi for any i = 1, . . . , t. remark 3.1: i. let x and y be nearly isomorphic acd-groups. then x and y are strongly factor-identical. ii. let x and y be factor-identical indecomposable acd-groups. then x and y are strongly factor-identical. iii. let x and y be factor-identical indecomposable acd-groups of rank 2. then x and y are strongly factor-identical, moreover, x ∼=nr y . copyright © 2023 assa. adv syst sci appl (2023) butler group direct decomposition classification 157 iv. let x and y be factor-identical acd-groups and x = x1 ⊕x2, y = y1 ⊕ y2 with x1 ∼=fi y1 . then x2 ∼=fi y2. v. let x and y be strongly factor-identical acd-groups and x = x1 ⊕x2, y = y1 ⊕ y2 with x1 ∼=sfi y1 . then x2 ∼=sfi y2. we are interested in crq-groups x and y which are strongly factor-identical but not nearly isomorphic. without loss of generality assume that a completely decomposable group a serves as the regulator for the both groups and x/a ∼= y/a with e = exp x/a = exp y/a. for a crq-group x define the set of its near-isomorphism invariants: mx = {mτ (x), τ ∈ tcr(a)}. for any prime divisor p of e denote tp(x) = {τ ∈ tcr(a) : p | mτ (x)}, (3.2) by the condition, |tp(x)| = 2. definition 3.3: let x be a crq-group. a subset m ′ of mx is called a type-connected set if and only if for any non-trivial partition m ′ = m ′ 1 ∪m ′ 2 with disjoint m ′ 1 and m ′ 2 there exist mτ (x) ∈ m ′ 1 and mσ(x) ∈ m ′ 2 which are not relatively prime. if any mτ (x) ∈ (mx \m ′) is not p-divisible for a prime p ∈ p , then a type-connected set m ′ is called a cover for the factor p in x and it is denoted by m ′(p). in general, this typeconnected set is not uniquely determined. in particular, the two members mτ (x), mσ(x) of the set mx , which are divisible by p, form the minimal cover for p, exactly, {mτ (x),mσ(x)} is the minimal cover for p if and only if tp(x) = {τ, σ}, see (3.2). for a prime p ∈ p define t x p , the set of all the primes such that for each q ∈ t x p there exists a cover m ′(p, q) for the factors p and q simultaneously. if m ′(p, q) is not empty, it is not uniquely determined in general. we have the set union p = ⋃ p∈p t x p (3.3) as there exists the minimal cover for each p. the members of the same set t x p are equivalent in accordance with the equivalence relation introduced on the set p as being members of the same tp. indeed, this relation is trivially reflexive because p ∈ t x p . it is obviously symmetric which means that q ∈ t x p implies p ∈ t x q . the transitivity follows the fact that p ∈ t x q , q ∈ t x r imply r ∈ t x q by symmetry, therefore, p and r belong to the same set t x q . then (3.3) is a union of pairwise disjoint or completely coinciding subsets. fix a set px of primes from p such that p = ⋃ p∈px t x p (3.4) consists of only pairwise disjoint sets. for a crq-group x denote mx p = {mτ (x) : there exists q ∈ t x p such that q | mτ (x)}. clearly, mx = ⋃ p∈px mx p (3.5) copyright © 2023 assa. adv syst sci appl (2023) 158 ekaterina blagoveshchenskaya, ilya mikulik is a uniquely determined union of pairwise disjoint type-connected subsets mx p , which will be called a canonical decomposition of the set of the near-isomorphism invariants of a crqgroup x . note that mx p is the maximal cover for each q ∈ t x p , therefore, any two members of different sets mp x are relatively prime. we also need rx p = {τ : mτ (x) ∈ mp x}. denote exp = lcm{mτ (x) : τ ∈ rx p }. clearly, e = ∏ p∈px exp (3.6) with gcd(exp , e x q ) = 1, and tcr(x) = ⋃ p∈px rx p with rx p ∩rx q = ∅ if p /∈ t x q (or p ̸= q if we take the numbers from the set px). we also have that any q ∈ t x p divides exp . 4. direct decomposition classification of block-rigid factoridentical crq-groups since factor-identical crq-groups x and y have isomorphic regulators we assume that a ring type block-rigid completely decomposable group a serves as the regulator for the both groups. furthemore, according to the definition 3.1 we have that e = |x/a| = |y/a| and p is the set of all prime divisors of e. we start with rigid crq-groups x . theorem 4.1: let x and y be factor-identical rigid crq-groups. then x and y are strongly factor-identical if and only if t x p = t y p and rx p = ry p for any p ∈ p . proof as x is a rigid crq-group, it follows from theorem 2.4 that their exists the only one decomposition x = ⊕ p∈px xp into indecomposable summands xp which satisfy the conditions rx p = tcr(x p) and ep = expxp/r(xp) = lcm{mτ (x) : τ ∈ rx p }. note that xp is allowed to be a rank-one group isomorphic to some τ ∈ tcr(x) with mτ (x p) = 1 and ep = 1. let y = ⊕ p∈py y p be a decomposition into indecomposable summands. evidently, crqgroups x and y are strongly factor-identical if and only if the corresponding indecomposable summands of their main decompositions are factor-identical (after a suitable reordering if necessary). it takes place if and only if r(xp) ∼= r(y p) and x/r(xp) ∼= y/r(y p) for any p. since the regulators of xp and y p are rigid completely decomposable groups and the factor-groups over the regulators are cyclic groups with square-free orders ep, these conditions are equivalent to the following, rx p = ry p and t x p = t y p for any p ∈ p as required. for a block-rigid crq-group we define the set p ′(x) = {q ∈ p : q|mτ (x) if and only if rkaτ ⩾ 2} (4.7) which satisfies the condition: if q ∈ p ′(x) and tq(x) = {τ, σ} then rkaτ ⩾ 2 and rkaσ ⩾ 2, see (3.2). denote p ′′(x) = p \ p ′(x), then p = p ′(x) ∪ p ′′(x), (4.8) copyright © 2023 assa. adv syst sci appl (2023) butler group direct decomposition classification 159 with p ′′(x) = {q ∈ p : rkaσ < 2 or rkaσ < 2 with tq(x) = {τ, σ}}. (4.9) introduce the subset p 0(x) of p ′′(x): p 0(x) = {q ∈ p : rkaσ = 1 and rkaσ = 1 with tq(x) = {τ, σ}}. (4.10) note that p 0(x) = p and consists of all prime divisors of |x/r(x)| if x is a rigid crqgroup. we also introduce the subset t ′(x) of tcr(x) which consists of all the types τ satisfying rkaτ = 1. we use the so-called primary factor representation x = ∑ p∈p xp (4.11) for uniquely determined fully invariant crq-groups xp having the same regulator a and the regulator quotients xp/a ∼= z/pz. for any subset p̃ ⊂ p define a fully invariant subgroup of x as xp̃ = ∑ p∈p̃ xp, (4.12) then x = xp ′(x) +xp ′′(x). theorem 4.2 (criterion of strong factor-identity for crq-groups): let x and y be factor-identical block-rigid crq-groups with p ′ = p ′(x). then x and y are strongly factor-identical if and only if the following conditions hold: (1) p ′ = p ′(y ); (2) xp ′ ∼=nr yp ′; (3) xp ′′ ∼=sfi yp ′′ . proof without loss of generality we assume that the groups x and y have the same regulator a = r(x) = r(y ) and the same regulator exponent e = |x/a| = |y/a|, see definition 3.1. if x is indecomposable then it is rigid which means x = xp 0(x) and the theorem is trivial. in general we apply the decomposability criterion for crq-groups, see theorem 2.5. assume that x and y are strongly factor-identical. it is immediate from (4.11) that xp = xp 0 ⊕ a′p is the only one possible decomposition up to near isomorphism, which is the main decomposition, whose main summand xp 0 is an indecomposable crq-group of rank 2 with tcr(x p 0 ) = tp(x) = {τ, σ} and expxp 0/r(xp 0 ) = p. i. if p ∈ p ′(x) then x = v p 0 ⊕ v p with v p 0 ∼=nr x p 0 . it corresponds to the following factorizations of the invariants mτ (x) = mτ (v p 0 )mτ (v p): 1. tcr(v p 0 ) = {τ, σ} and mτ (v p 0 ) = mσ(v p 0 ) = p; 2. tcr(v p) = tcr(x), mτ (v p) = mτ (x) p , mσ(v p) = mσ(x) p and mρ(v p) = mρ(x) if ρ /∈ tp(x). this means that for any p ∈ p ′(x) the main summand xp 0 of xp is nearly isomorphic to an indecomposable summand of x . then y has an indecomposable summand of rank 2, which is also nearly isomorphic to xp 0 for the prime p ∈ p , see remark 3.1(iii). therefore, copyright © 2023 assa. adv syst sci appl (2023) 160 ekaterina blagoveshchenskaya, ilya mikulik p ∈ p ′(y ), and, by symmetry, we conclude that p ′ = p ′(x) = p ′(y ) with xp ′ ∼=nr yp ′ by theorem 2.2. then p ′′(x) = p ′′(y ). denote p ′′ = p ′′(x) = p ′′(y ). ii. let xp ′′ = z ⊕g with indecomposable z, see (4.9, 4.12). clearly, z is a rigid crq-group and z/r(z) ∼= ( ⊕ τ∈tcr(z) aτ ) x ∗ / ⊕ τ∈tcr(z) aτ by remark 2.1(iii). there exists a decomposition x = w0 ⊕w with w0 ∼=cr z by theorem 2.5. it corresponds to the following factorizations of the invariants mτ (x) = mτ (w0)mτ (w ): 1. tcr(w0) = tcr(z) and mτ (w0) = mτ (z) by theorem 2.2; 2. tcr(w ) = tcr(x) \ (t ′(x) ∩ tcr(z)), mρ(w ) = mρ(x) mρ(z) for any ρ ∈ tcr(w ). (traditionally we set mτ (z) = 1 if τ ∈ tcr(x) \ tcr(z)). since x and y are supposed to be strongly factor-identical, y has an indecomposable summand, say v0, which is factor-identical to z. let y = v0 ⊕ v . we conclude from remark 3.1(ii,v) that w ∼=sfi v with p ′′(w ) = p ′′(v ) = p ′′(x) \ p 0(z), and the induction on the number of indecomposable summands leads to the conclusion that xp ′′ ∼=sfi yp ′′ . conversely, assume that p ′(x) = p ′(y ). this implies p ′′(x) = p ′′(y ). for p ′ = p ′(x) and p ′′ = p ′′(x) we have xp ′ ∼=nr yp ′ and xp ′′ ∼=sfi yp ′′ by the condition. let x = x1 ⊕x2 with indecomposable x1. then xp ′ = x ′ 1 ⊕x ′ 2 and xp ′′ = x ′′ 1 ⊕ x ′′ 2 with x ′ i = xp ′ ∩xi, x ′′ i = xp ′′ ∩xi, i = 1, 2, as xp ′ and xp ′′ are fully invariant in x . denote e1 = expx ′ 1/r(x ′ 1) and e2 = expx ′′ 1 /r(x ′′ 1 ) with tcr(x1) = tcr(x ′ 1) ∪ tcr(x ′′ 1 ) and e1e2 = |x1/r(x1)|. it follows from e1 | expxp ′/a and e2 | expxp ′′/a that gcd(e1, e2) = 1. there exist direct summands y ′ 1 and y ′′ 1 of y such that x ′ 1 ∼=nr y ′ 1 and x ′′ 1 ∼=sfi y ′′ 1 by the condition, and we apply remark 2.1(iii) to get that y = y1 ⊕ y2 with indecomposable y1 ∼=sfi x1 satisfying expy ′ 1/r(y ′ 1) = e1e2. we apply the same approach to the groups x2 and y2 by remark 3.1(v), which is actually an induction on the number of indecomposable summands leading to the required conclusion x ∼=sfi y . now we need to obtain the coditions for crq-groups to be strongly factor-identical if their regulator homogeneous components satisfy the condition: if gcd(mτ ,mσ) ̸= 1 then aτ or aσ is a rank-one group (in particular, rk(aτ ⊕ aσ) = 2) . it takes place if p(x) = p ′′(x), and we fix this condition. let p ∗(x) = p ′′(x) \ p 0(x), then x = xp 0(x) +xp ∗(x), see (4.12). denote p 0 = p 0(x), p ∗ = p ∗(x), v = xp 0 and w = xp ∗ , see (4.12). we have that xp ′′ = v +w and consider the canonical decomposition of the the near-isomorphism invariant set of the rigid crq-group v : mv = ⋃ p∈pv mv p into pair-wise disjoint subsets with pv ⊂ p 0, see (3.5). applying (3.6) to the group v we have that evp = lcm{mτ (v ) : τ ∈ rv p ⊂ tcr(v ) ⊂ tcr(x)} with rv p = {τ : mτ (v ) ∈ mp v }. we also introduce the numbers e′ x p = lcm{mτ (x) : τ ∈ rv p }. copyright © 2023 assa. adv syst sci appl (2023) butler group direct decomposition classification 161 it is important that e = |x/a| = ∏ p∈pv e′xp by the condition p(x) = p ′′(x). denote m ′ x = {mτ (x) : gcd(mτ (x), e′ x p ) ̸= 1} and r′x p = {τ ∈ tcr(x) : gcd(mτ (x), e′xp ) ̸= 1} = {τ ∈ tcr(x) : mτ (x) ∈ m ′ x}. let x ′ p = ( ⊕ τ∈r′x p aτ ) x ∗ . by construction, the sets of types r′x p = tcr(x ′ p) are not supposed to be pair-wise disjoint and tcr(x) = ⋃ p∈pv r′x p . therefore, x = ∑ p∈pv x ′ p. denote t x′ p = {q ∈ p : q|e′xp }, we see that p = ⋃ p∈pv t x′ p . recall that the set pv can be chosen differently as a subset of p , see (3.4). in fact, each number q determines one of the sets {t x′ p : p ∈ pv }, exactly, that one, which contains this prime number q. without loss of generality, instead of pv we take the set qx which can have prime divisors of e′xp evp , that is qx ⊂ p(x) = p ′′(x). then we write p = ⋃ p∈qx t x′ p and x = ∑ p∈qx x ′ p with tcr(x) = ⋃ p∈qx r′x p . theorem 4.3 (criterion of strong factor-identity for special crq-groups): let x and y be factor-identical block-rigid crq-groups with the regulator a and p ′′ = p(x) = p(y ) such that x = xp ′′ and y = yp ′′ . then x and y are strongly factor-identical if and only if there exists a subset q of p ′′ such that the groups x ′ p and y ′ p are factor-identical for any p ∈ q. proof each group x ′ p has the only one decomposition, which is its main decomposition x ′ p = x ′ 0p ⊕ a′ p into the rigid indecomposable group x ′ 0p, having the critical typeset tcr(x ′ 0p) = tcr(x ′ p), and a completely decomposable group a′ p. then x ′ p ∼=fi y ′ p implies x ′ 0p ∼=fi y ′ 0p, that is x ′ p and y ′ p are strongly factor-identical for any p ∈ q by the condition, see theorem 4.1. the necessity of this condition follows from the fact that, by construction, the groups x and y have direct decompositions with indecomposable summands nearly isomorphic to x ′ 0p and y ′ 0p accordingly, see theorem 2.5. the sufficiency is obtained on the basis of the same direct decompositions by remark 2.1(ii, iii). 5. algorithms of butler group direct decompositions as a parallel programming model let us recall some links between crq-group direct decomposition theory and the theory of graphs from [6] [7]. we need a graph γ, which is an r-colorable graph, that is each vertex can be assigned one of the r colors so that no two adjacent vertices are of the same color, see [8, 14.1]. as for the crq-group x associated with graph γ, it will be characterized by some restriction on the invariants mτ = mτ (x). recall that for any prime divisor p of e there exist two members of tcr(a), say τ and σ, such that gcd(mτ ,mσ) is divisible by p (if only one mτ is divisible by p then p|aτ which contradicts the purity of τaτ in a). we constructed a crq-group x of rank n satisfying the special previously listed conditions (s1, s2, s3). recall the following definition 5.1 (definition 3.9, [5]): let x be a block–rigid crq–group. the frame of x is a graph f (x) whose vertices are the copyright © 2023 assa. adv syst sci appl (2023) 162 ekaterina blagoveshchenskaya, ilya mikulik elements of tcr(x) and two vertices σ and τ are joined by the edges designated by ”p” for each prime divisor p of gcd(mσ(x),mτ (x)). note that if x is rigid then the frame vertex number coincides with rkx and it an indecomposable group if and only if its frame is connected, see theorem 2.4. passing to the graphical illustration of the main decomposition of x by graph γ, we denote r = |tcr(x)| and arrange the elements of its critical typeset in an arbitrary way, tcr(x) = {τ1, . . . , τr}. then a = aτ1 ⊕ aτ2 ⊕ . . .⊕ aτr . let us distribute n vertices of γ on the r radiuses τ1, . . . , τr of some concentric circles so that the circle of the smallest radius would be filled with the vertices of the frame f (x), and each radius τi would contain ni = rkaτi vertices, i = 1, . . . , r. recall that all the prime divisors p of e are naturally assigned to the edges of the frame, and, therefore, f (x) is located on the circle of the smallest radius, see definition 5.1. by construction, any two adjacent (connected by an edge) vertices of graph γ never share the same radius. thus, graph γ is an r-colorable graph (with only vertices of different colors connected by edges). the graph γ is the union of the frame f (x) and the set of isolated vertices. we call γ the main graph of group x since its main decomposition x = y ⊕ a′ can be easily reconstructed from γ up to near isomorphism by visualization y as f (x) (condition s3.) and identifying completely decomposable a′ with the set of all isolated vertices of γ. the decomposability criterion (theorem 2.5) states that all the other decompositions x = x1 ⊕x2 ⊕ . . .⊕xt ⊕xt+1 up to near-isomorphism can be seen as the special transformations of graph γ, which will be called ”admissible transformations”. they can be obtained by moving the edges so that the edge p of the new graph γ′ would join (another) pair of vertices, but at the same radiuses as those in γ and under the important restriction that no pair of vertices of the same connected component share the same radius (see theorem 2.2). then the vertex number of each component coincides with the rank of the corresponding indecomposable summand. moreover, the connected components are the frames of rigid indecomposable crq-groups x1, x2, . . . , xt, whose critical types and, therefore, the regulators are determined by their vertex radiuses, while the prime divisors of e = expx/a are interpreted as the edges allowing us to calculate the invariants mτ (xi), i = 1, . . . , t, as their partial products. meantime, the isolated vertices correspond to completely decomposable group xt+1. thus, admissible transformations of γ lead to the new set of connected components forming a graph with the same number of vertices of each color and the same number of edges connecting the vertices of any two different colors as those in γ. the graphical approach to direct decompositions of the considered groups opens some new prospects in parallel programming modeling. the main graph of an almost completely decomposable group after the choosing homogeneous component numeration becomes an oriented graph of a sequential algorithm. the group direct decompositions are in the correspondence with the set of possible parallel decompositions of the algorithm into a number of threads. we can apply them to modify the parallel random access machine (pram) using the parallel programming with hints, called semi-implicit parallelism, see [9][10]. more exactly, the special choice of the group homogeneous component ranks preserves the fixed graph fragments associated with the so-called maximal stable subgroups. we call the subgroups x ′ 0p with p ∈ q the maximal stable subgroups of x as each of them is a maximal indecomposable subgroup always serving up to near isomorphism as a fully invariant subgroup of one of the indecomposable summands for any possible direct decomposition of x . then the theorem 4.3 says that crq-groups x and y with the same regulator a and p ′′ = p(x) = p(y ) are strongly factor-identical if only if the sets of their maximal stable subgroups consist of the factor-identical groups. copyright © 2023 assa. adv syst sci appl (2023) butler group direct decomposition classification 163 this meets the programmer’s instruction to save these parts, which correspond to the maximal stable subgroups, for the separated execution. it is a way of creating a universal parallel programming for solving the tasks, which are of the same kind but differ in the particular segments. note, that it is also the way to save the cores involved for the other processes during the time of the special segment fulfillment. thus, the direct decomposition graph of an acd-group with some predicted properties can be applied to the parallel programming solution of the tasks with the special segments needing the particular attention of the programmer. it can be also used as one of the first steps within the general approach to the task solution under consideration. acknowledgements the research was supported by the russian science foundation (project no. 22-2100267). references 1. mader, a. 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(2016). torsion-free abelian groups, graphs and algorithms, proc. of international conference ”mathematics and informatics”, moscow, russia. 8. bondy, j.a. & murty, u.s.r. (1976). graph theory with applications. amsterdam, netherlands: elsevier. 9. ding, c. (2011). parallel programming by hints. proc. of the acm international conference on object oriented programming systems languages and applications, cascais , portugal, 13–14, https://doi.org/10.1145/2048147.2048154 10. helm, d., et al. (2020). a programming model for semi-implicit parallelization of static analyses. proc. of the 29th acm sigsoft international simposium on software testing and analyses, los angeles, ca, 428–439, https://doi.org/10.1145/3395363.3397367 copyright © 2023 assa. adv syst sci appl (2023) introduction direct decomposition theory of crq-groups: basic results classification of block-rigid crq-groups: general approach direct decomposition classification of block-rigid factor-identical crq-groups algorithms of butler group direct decompositions as a parallel programming model microsoft word article+text,+krasnovdv+utkinav+assa+(1) adv syst sci appl 2024; 04; 57-65 published online at https://ijassa.ipu.ru. estimating the unmeasured output of nonlinear siso system under uncertainty dmitry v. krasnov, anton v. utkin* v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia abstract: nonlinear single-channel control plant with affine unmached external disturbances are considered. their mathematical model has the so-called triangular (based on the composition of the arguments of the functions in each equation) “input-output” form, which has the following structural properties. if there are no external disturbances, then this form is controllable and observable in terms of output. if external disturbances are smooth, then the control plant model is representable in the canonical “input-output” form with respect to mixed variables (functions of state variables, external disturbances and their derivatives). in the new mixed variable basis, the system remains controllable, observable, and has the same relative degree as the original system. if the output variable is measured, then an observer of mixed variables can be constructed based on the canonical system and, using suitable feedback, ensure the desired behavior of the output variable. in this case, there is no need to detail external disturbances and design individual observers for them, which greatly simplifies the structure of the controller. this paper discusses the problem of implementing the specified synthesis procedure in the case when the output variable is not measured. motivating examples are given and conditions are formulated under which it is possible to first restore the output variable from the measurement of other state variables invariantly with respect to the action of external disturbances. methods for estimating external disturbances using observers with piecewise linear corrective actions, which do not require the introduction of dynamic disturbance models, constitute the methodological basis for solving this problem. using the example of various mathematical models of single-link manipulators, the fulfillment/failure of the conditions necessary to restore an unmeasured output variable is demonstrated. the proposed approach can be extended to multichannel nonlinear control plants without loss of generality. keywords: nonlinear siso system, unmatched disturbances, observability, reduced-order state observer, piecewise linear function with saturation, single-link manipulator. 1. introduction the problem of constructing the reduced-order state observers is considered for nonlinear minimal-phase systems with one input and one output (siso), represented in a triangular (by the composition of the function arguments in each equation) “input-output” form. this form is convenient for synthesis and feedback when solving the tracking problem, and for the state observer for estimating unmeasured variables, when only the output (controlled) variable is measured, and all parametric uncertainties and external disturbances are match, i.e. act on the input in the last equation [12, 15, 17–19]. in this paper, we study a case little studied in theory, but common in practice, when the output (controlled) variable of the tracking system for one reason or another cannot be measured [5], the set of sensors is not complete, and the parametric and external disturbances are unmatched. the problem of observing non-measurable (including controlled) variables is considered in a narrow setting, when the identification of unknown parameters and the construction of external disturbance generators are not provided or are not possible. our goal is to formulate for a triangular “input-output” system the conditions necessary for constructing a physically * corresponding author: utkin-av@rambler.ru 58 d.v. krasnov, a.v. utkin copyright ©0000 assa adv. in systems science and appl. (0000) realizable observer, which depend not only on the structural properties of observability, but also on the channels of action of parametric and external disturbances. the scientific novelty lies in the identification of a practically significant class of siso systems with unmatched disturbances, where the problem of estimating unmeasured state variables is feasible without additional identification of the existing parametric and external disturbances. the reducedorder state observer of a special structure with piecewise linear control actions provides a solution to this problem [8, 9, 13], if the differential equations of the measured variables in the control plant model do not contain parametric and external disturbances. then, the observer is constructed as a copy of these equations, where unmeasured state variables are treated as undefined bounded inputs and estimated with the observer's corrective actions. in this case, the uncertainty of differential equations of unmeasured state variables is allowed. the proposed approach, which provides a given estimation accuracy in a finite time, is fundamentally different from the traditional methods for constructing reduced observers based on differential equations of unmeasured variables [3, 4, 6, 7, 14], which in a narrow setting can be implemented only if the entire control plant model is fully defined. the paper is structured as follows. design options for a state and disturbance observer with piecewise linear correcting actions for a second-order system are discussed in section 2. the main result is presented in section 3. the conditions are formulated necessary for constructing a physically realizable observer for estimating the controlled but unmeasured variable of the n-th order siso system. as an example, the equations of motion of singlelink manipulators with a rigid and elastic type of articulation with the shaft of a dc motor are considered. 2. motivating examples methods for estimating external disturbances in terms of their act on the control plant are the methodological basis for this study. these methods provide the given accuracy of the estimating signals when certain conditions are met [8, 9, 13]. to explain them, we will consider a triangular siso system “input-output” of the second order ,)(),( ),(),( 22122 12111 butxxfx txxfx       (2.1) ,0,0 ),( 2 211    b x xxf (2.2) where 2t 21 ),( rxxxx  is the state vector, x is an open bounded area of change of state variables in the control process, )(1 tх is controlled output, ),( 21 ххfi are functionally and parametrically defined expressions satisfying the lipschitz conditions, ri are external disturbances, ru is the control (input), which is assumed to be a known function of time, for simplicity b is assumed to be a known constant. satisfaction of conditions (2.2) means that system (2.1) is controllable and observable with respect to the output in the absence of external disturbances. all internal and external signals in system (2.1) are assumed to be bounded together with their derivatives in the control process, in particular: ,)(,),(,),( ],,0[,)(,)( 2121 xtxfxxffxxf tttt iidt d ii iiii     (2.3) where t is regulation time, iiii ff  ,,, are known constants, in this section everywhere 2,1i . estimating the unmeasured output of nonlinear siso system under uncertainty 59 copyright ©0000 assa. adv. in systems science and appl. (0000) let us consider various variants of measurements in system (2.1) and the corresponding conditions for the physical realizability of observers of unmeasured state variables and external disturbances. example 2.1: if state variables )(),( 21 txtx are measured in system (2.1), then we can obtain estimates of external disturbance s using two first-order autonomous observers: ,),( ,),( 22122 12111 vbuxxfz vxxfz     (2.4) where rzi  is the state vector, rv i are corrective actions of the observer. taking into account (2.1), (2.4), the system with respect to observation errors rzx iii  takes the following form .2,1,)(  ivt iii  (2.5) synthesis of observer (2.4) consists in choosing the parameters of piecewise linear corrective actions        ,/1, ,/1),sign( )(sat iiiii iiii iiii llm lm lmv    (2.6) where ,0const, ii lm so as to stabilize the observation errors and their derivatives with a given accuracy .,)()()(,)( 0 ttttvttt iiiiiii    (2.7) we can obtain inequalities for the choice of observer parameters that provide (2.7) using the second lyapunov method [8]. taking into account the measurements in systems (2.4), (2.5), we set the initial values ./1)0(0)0()0()0( iiiii lxz   then we can provide ii lt /1)(  during the entire control time ],0[ tt by choosing the amplitudes of corrective actions (2.6) in the form .0)())sign((  iiiiiiiiiii mmm   (2.8) in the linear zone ii l/1 , system (2.5) and its derivatives take the form .)( ,)( iiiii iiiii lmt lmt       the state variables )(),( tt ii   of this virtual system will converge to the given domains (2.7) in a finite time ttt  00 0: if the conditions 2,1,if0)( ))(()/( ;if0)( ))(()/(     ilm lmtml lm lmtml iiiiiii iiiiiiiiiii iiiiiii iiiiiiiiiii        (2.9) are met. the simultaneous fulfillment of conditions (2.8)–(2.9) 60 d.v. krasnov, a.v. utkin copyright ©0000 assa adv. in systems science and appl. (0000) 2,1,,max 1 ,           i m lm i i i i i iii  (2.10) ensures the solution of the formulated estimation problem (2.7). example 2.2: if in system (2.1)–(2.2) )(1 tx is measured, and ,0)(1 t then, similarly to (2.4), we can reconstruct )(2 tx with a first-order observer based on the first equation of system (2.1) in the form ., ),(sat, 111111 111111 vfzx lmvvz       (2.11) then, after analyzing virtual systems 1112111 111211111 11211111 ),( ,),(:/1 );sign(),(:/1       lmxxf lmxxfl mxxfl dt d    we obtain inequalities for choosing the parameters of the observer (2.11) similarly to (2.8)– (2.10) ,,max 1 , 1 1 1 1 1 111         ff m lfm  (2.12) providing 11211111 ),()(,)(  vxxftt   , then ,),,()(~)(),( 1111221211 tttvxhtxtxvxxf  (2.13) where ),( 111 vxh is the solution of the equation 1211 ),( vxxf  with respect to 2x , which exists due to (2.2). thus, we applied the technique for estimating external disturbances, presented in the first case, to estimate the unmeasured state variable )(2 tx using the reduced observer (2.11), constructed on the basis of the differential equation of the measured variable, which does not depend on the external disturbance. at the same time, we did not explicitly use the second equation of system (2.1), which depends on the disturbance, so identification 0)(2 t is not needed to solve this problem. the main limitation of this approach is that at the design stage it is necessary to obtain the estimates 11, ff required to tune the observer (2.12), taking into account the specific control law and the admissible range of initial values of the state variables. note that in this case, the standard reduced observer, which is constructed on the basis of the differential equation of an unmeasured variable, is physically unrealizable in a narrow setting, since the second equation of system (2.1) is under the influence of an external uncontrolled disturbance [6, 7]. note that due to (2.13) we can reconstruct both )(2 tx and 0)(2 t using a full-order observer .~~),~(sat~ ,~),(, 2222222 2212211 zxlmv vbuzxfzvz     (2.14) taking into account (2.1), (2.14), the system with respect to observation errors rzx iii  takes the form estimating the unmeasured output of nonlinear siso system under uncertainty 61 copyright ©0000 assa. adv. in systems science and appl. (0000) ),~(~),(),( ),(),( 2222122122 112111   vzxfxxf vxxf     (2.15) where for 1tt  the inequalities 222122212 11211111112222 ),(),( ,),(),(~~   lzxfzxf lvxhvxhxx    (2.16) are met, 212, ll are well-known lipschitz constants. then, under the conditions similar to (2.10), taking into account (2.16), the estimation )()(,)( 2222 tvtztx   is provided in a finite time .: 212 tttt  the conditions for observability of the controlled variable, regardless of external disturbances, have the following form: if in system (2.1) )(2 tx is measured, and 0)(2 t , ,0/),( 1212  xxxf then we can reconstruct )(1 tx using a first-order observer similar to (2.11) but based on the second equation of system (2.1), and provide 2212 ),( vxxf  ).,()(~)( 22211 хvhtxtx  in addition, we can obtain estimates of both )(1 tx and )(1 t using a full-order observer, similar to (2.14). example 2.3: if in system (2.1) only )(1 tx is measured and ,0)(1 t then in a narrow setting, i.e. without introducing a dynamic model of )(1 t , it is impossible to solve the problem of separately estimating the variable )(2 tх and external disturbances. similarly, if only )(2 tx is measured, and 0)(2 t , then it is impossible to solve the problem of estimating separately the variable )(1 tх and external disturbances in a narrow setting. in the next section, the presented results are used to formalize the observability conditions for the output controlled variable of a single-channel arbitrary order system. 3. main result 3.1. general case of triangular siso system let us consider a minimum-phase nonlinear multidimensional siso system, represented in a triangular affine form “input-output” ,),()( ;1,1),,...,,,()...,,,( 21121 butxxfx nitxxxxxxfx nnn iiiii        (3.1) where n n rxxxx  t 1 )...,,( is the state vector, rtx )(1 is controlled output, ru is the control, )...,,,( 121 ii xxxf are functionally and parametrically defined expressions satisfying the lipschitz conditions, b is a known constant. all uncertainties of the control plant model and external disturbances are concentrated in expressions i , in some equations of system (3.1) these terms are absent. it is assumed that in the control process all internal and external signals are bounded similarly to (2.3). the fulfillment of the conditions 0,,1,1,0 )...,,( 1 11      bxxni x xxf i ii (3.2) means that system (3.1) is observable with respect to the output in the absence of disturbances, and the output is controllable with respect to the input. in a narrow setting, when the identification of unknown parameters and the construction 62 d.v. krasnov, a.v. utkin copyright ©0000 assa adv. in systems science and appl. (0000) of external disturbance generators are not provided or not possible, there are two options for synthesizing feedback that provides tracking of the output variable )(1 tx of a reference signal )(tg . the first variant [9]: if conditions 1,1,,   nicf in ii  and nсtg )( are satisfied, system (3.1) is representable in the canonical basis of mixed variables (functions of state variables, external influences and their derivatives), for the estimation of which it is enough to measure )(1 tx и ),(tg at the same time, it is possible to provide )()(lim 1 tgtx t   using dynamic feedback. the second variant [1]: if the functions )(),( tgti  are not smooth, then when measuring )(...,),(1 txtx n and )(tg , it is possible to provide a given accuracy of the tracking error using static feedback. with an incomplete set of sensors, the possibility of estimating the state variables necessary for feedback synthesis depends on the observability structure of system (3.1) and the absence of disturbances 0)( tj in specific j -th equations. let the first variant of feedback synthesis be used for system (3.1)–(3.2), but the controlled variable )(1 tx is not measured. let us formulate the conditions for its observability regardless of disturbances, based on the motivating examples from section 2. in what follows, a narrow statement of the problem is considered by default, i.e., without additional identification of existing uncertainties. theorem 3.1: if in system (3.1)–(3.2) there is i -th equation ( ni ...,,2 ) such that: 1. ,0)( ti ,0/)...,,( 111   xxxf ii ;xx 2. )(1 txi is measured ( 0)(1  txn ); 3. )(txi is measured or observed independently of disturbances (see example 2.2); 4. )(...,),( 12 txtx i either are not arguments of the function if , or are measured, or are observed regardless of disturbances, then for estimating )(1 tx it is possible to construct a physically realizable reduced observer. proof the conditions formulated in the lemma are dictated by the triangular form of the considered system (3.1)–(3.2). in the simplest case, when )(txi ( ni ...,,2 ) and all available arguments of the function if are measured (except for )(1 tx ), ,0)( ti then the estimate of )(1 tx can be obtained using the corrective action of the first-order observer constructed on the basis of the i -th equation of system (3.1) similarly to (2.11)–(2.13): )....,,,()(~)( )...,,( ,, ),(sat, 1211 11       iii iii iiiiii iiiiii xxvhtxtx vxxf vfzx lmvvz     (3.3) note that if ni  , then the first expression (3.3) will take the form buvz nn  , but the following expressions (3.3) will not change. if )(txi and/or the available arguments )(...,),( 12 txtx i of the function if necessary for restoring )(1 tx (3.3) are not measured, then the conditions for their observability are similar to those formulated in the lemma. for example, if in system (3.1)–(3.2) )(2 tх is measured, then under conditions ,0)( tj ,...,,3,2 kj  1 ik , based on the j th equations, we can construct a physically realizable observer for estimating )(...,),( 13 txtx k similarly to (2.14)– estimating the unmeasured output of nonlinear siso system under uncertainty 63 copyright ©0000 assa. adv. in systems science and appl. (0000) (2.16). if in system (3.1)–(3.2) )(3 tх is measured, then under conditions ,0)( tj ,...,,3 kj  1 ik , based on the j th equations, we can construct a physically realizable observer for estimating ),(...,),( 14 txtx k if at the same time )(2 tx is measured or is not an argument of functions jf , etc. □ if the conditions of theorem 3.1 are satisfied and an estimate )(~ 1 tx can be obtained, then for the second variant of the synthesis of the tracking system, the conditions for constructing physically realizable observers for the remaining unmeasured variables with respect to the measured (observed) variables are verified similarly. for example, if ,0)(1 t then there is the possibility of estimating )(2 tx ; if at the same time 0)(2 t , then and )(3 tx etc. 3.2. application to single link manipulator as an example, let us consider two systems of the form (3.1)–(3.2), which describe the equations of motion of a single-link manipulator [16] with a rigid joint type with a dc motor shaft ubxcxcx txcxcxcx xx    3332323 3232221212 21 ),(sin ,     (3.4) and with an elastic joint type , ,)( , ),(sin)( , 5554545 54544413434 43 12213212 21 ubxaxax xaxaxxax xx txaxxax xx            (3.5) where )(1 tх is the angular position of the pendulum (controlled output), an overline means that this parameter is not exactly known, .0)( t as we can see, the conditions of the theorem 3.1 are not satisfied in system (3.4). if the controlled variable )(1 tx is not measured, then in a narrow formulation it is impossible to obtain its estimate from measurements of ),(),( 32 txtx this is possible only in the absence (or identification) of uncertainties .,,, 232221 ccc thus, in system (3.4), the first variant of the tracking system synthesis is realized only when measuring .1x for the second variant of the synthesis of the tracking system, the minimum set of sensors is measurements ),(),( 31 txtx since, based on the first equation of system (3.4), it is possible to construct a reduced observer for estimating ),(2 tx and it is not necessary to identify the existing uncertainties to solve the observation problem. in system (3.5), the conditions of the lemma are satisfied for 4i and measurements ).(),(),( 543 txtxtx then the corrective action of the first-order observer constructed on the basis of the fourth equation of system (21) will give the estimate ).(1 tx sensors )(),( 53 txtx are the minimum set for estimating ).(1 tx in this case, estimates of both )(1 tx and )(4 tx can be obtained using a second-order observer constructed on the basis of the third and fourth equations of system (3.5) [10, 11]. sensors )(),( 53 txtx are the minimum set for the second variant of the synthesis of the tracking system. in this case, the third-order observer is constructed on the basis of the 64 d.v. krasnov, a.v. utkin copyright ©0000 assa adv. in systems science and appl. (0000) unperturbed subsystem 21 54544413434 43 ,)( , xx xaxaxxax xx       (3.6) in the form of 11 454513434 343 ,)( , vz vxazxaz vzz       allows one to obtain the estimates ).(),(),( 214 txtxtx if )(),( 31 txtx are measured, then the observer constructed on the basis of subsystem (3.6) in the form of 11 444413434 343 ,)( , vz vzaxxaz vzz       allows one to obtain the estimates ).(),(),( 254 txtxtx an example of constructing a reduced-order observer for a multilink manipulator whose movements are described by matrix equations of the type (3.4) is presented in the work [11]. simulation modeling of closed systems (3.4), (3.5) with different sets of sensors and with the corresponding reduced observers confirmed their performance [2, 9, 10]. 4. conclusion the control plant model is often presented in the canonical “input-output” form in order to make it convenient to perform mathematical analysis and synthesis of the tracking system. however, the canonical form completely depersonalizes the control plant model and hides its features. this paper shows that if the output variable is not measured in tracking systems operating under parametric and external disturbances, then it is advisable to use the triangular shape “input-output” to solve the problem of its observation. at the same time, the use of the technique for estimating external disturbances for estimating unmeasured state variables makes it possible to construct robust reduced-order observers. acknowledgements research partially supported by rsf, project no. 24-21-20009. references 1. antipov, a.s., krasnova, s.a., & utkin, v.a (2021). methods of ensuring invariance with respect to external disturbances: overview and new advances, mathematics, 9(23), 3140. 2. antipov, a.s. & krasnov, d.v. 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(2022). continuous output feedback sliding mode control for underactuated flexible-joint robot, j. franklin inst., 359, 7847–7865. adv syst sci appl 2024; 02:40–53 published online at https://ijassa.ipu.ru. stability analysis of switched positive persidskii systems with distributed and unbounded delays alexander aleksandrov*, natalya andriyanova saint petersburg state university, saint petersburg, russia abstract: switched positive persidskii systems with distributed and unbounded delays are studied. right-hand sides of these systems are linear combinations of nonlinearities of a sector type. special constructions of diagonal lyapunov–krasovskii functionals are proposed and conditions are derived under which the absolute stability of the considered systems can be proved with the aid of such functionals. the developed approaches are applied to the stability analysis of a mechanical system with switched nonlinear positional forces and to a problem of mobile agent deployment. results of numerical simulations are presented confirming the theoretical conclusions. keywords: switched positive system, sector nonlinearity, lyapunov–krasovskii functional, distributed and unbounded delays, mechanical system, formation control 1. introduction studying the stability of nonlinear time-delay systems is an actual problem of the contemporary control theory due to wide applications of these systems (see, e.g., [1, 2] and the bibliography therein). moreover, it is worth noticing that, in numerous models of practical systems, state variables are restricted to be nonnegative [3, 4]. therefore, an important class of time-delay systems is that of positive systems. methods for stability analysis of positive systems with delay are well developed for linear systems [3,4]. in the nonlinear case, the basic approaches are the comparison method and the lyapunov direct method [5–8]. however, under the constructing comparison systems, usually it is required that nonlinearities satisfy linear estimates (see [9]). as regards to the lyapunov direct method, its application to time-delay systems is based on the using lyapunov–razumikhin functions or lyapunov– krasovskii functionals [1, 5]. at the same time, a general constructive technique to finding such functions and functionals is still lacking. this problem becomes even more difficult if the model under consideration needs to take into account uncertainties and switching of operation modes. as a result, stability conditions are well investigated only for special classes of nonlinear time-delay systems, for instance, for homogeneous systems, lurie control systems, etc. [5, 10–13]. one of the interesting and important such classes is that of persidskii systems [3]. righthand sides of these systems are represented as linear combinations of separable nonlinearities satisfying sector conditions. persidskii systems are widely used for modeling automatic control systems, neural networks, opinion dynamics, digital filters, etc. [3, 14]. first, stability conditions for delay-free persidskii systems were obtained by barbashin via constructing a lyapunov function in the form of linear combination of integrals from nonlinearities [15]. in [16], linear lyapunov functions were proposed for the stability analysis ∗corresponding author: a.u.aleksandrov@spbu.ru stability analysis of switched positive persidskii systems 41 of positive delay-free persidskii systems. in a number of subsequent works, various canonical forms of lyapunov functions were used to derive stability conditions for both switched and nonswitched persidskii systems (see [3, 17, 18] and the references therein). stability of switched positive persidskii systems with constant delay was studied in [19] with the aid of a special construction of lyapunov–krasovskii functionals. in [8, 9], positive persidskii systems with distributed delay were investigated. in recent paper [14], generalized persidskii systems with constant delays were considered and conditions of input-to-state stability formulated in terms of lmis were derived. in [20], an approach to constructing lyapunov–krasovskii functionals for switched positive persidskii systems with constant delay was developed. the proposed functionals depend on a positive tuning parameter. it was shown that, via an appropriate parameter choice, less conservative stability conditions can be obtained compared to known ones. the objective of the present paper is an extension of the above approach to systems with distributed and unbounded delays. in addition, we will apply our results to the stability analysis of a mechanical system with nonlinear positional forces and to a problem of formation control. through the paper we will use the following notation: • r is the field of real numbers, rn and rn×n are the n-dimensional euclidean space and the vector space of n× n matrices, respectively. • let ∥ · ∥ be the euclidean norm of a vector. • a matrix c ∈ rn×n is called nonnegative if all its entries are nonnegative. • a matrix c ∈ rn×n is called metzler if all its off-diagonal entries are nonnegative. • the identity matrix is denoted by i . • inequalities for vectors are understood componentwise. 2. statement of the problem consider the following switched persidskii system with distributed delay: ẋ(t) = pσφ(x(t)) +qσ ∫ t t−τ f(ξ − t)φ(x(ξ))dξ. (1) here x(t) ∈ rn, φ(x) is a separable vector function, i.e., φ(x) = (φ1(x1), . . . , φn(xn)) ⊤, where the functions φi(xi), which are said to be admissible nonlinearities, are continuous and locally lipschitz for |xi| < ∆ (0 < ∆ ≤ +∞) and satisfy the sector constraints xiφi(xi) > 0 for xi ̸= 0, i = 1, . . . , n, τ = const > 0, f(ζ) is nonnegative and continuous for ζ ∈ [−τ, 0] scalar kernel, σ = σ(t) is a piecewise constant function defining the switching law, σ(t) : [0,+∞) 7→ {1, . . . , n}, ps, qs are constant matrices, s = 1, . . . , n . according to the standard assumption [21], we will consider the case where the function σ(t) admits only finitely many discontinuities on every bounded interval. such switching laws will be called admissible. initial functions for solutions of (1) are chosen from the space c([−τ, 0], rn) of continuous functions θ(ξ) : [−τ, 0] 7→ rn with the uniform norm ∥θ∥τ = max ξ∈[−τ,0] ∥θ(ξ)∥. for a solution x(t), xt denotes the restriction of this solution to the interval [t− τ, t], i.e., xt : ξ 7→ x(t+ ξ) for ξ ∈ [−τ, 0]. assumption 2.1: the matrices p1, . . . , pn are metzler and the matrices q1, . . . , qn are nonnegative. remark 2.1: under assumption 2.1, the system (1) is positive (see [6]). copyright © 2024 assa. adv syst sci appl (2024) 42 a. aleksandrov, n. andriyanova from the continuity of φ(x) and the sector restrictions it follows that φ(0) = 0. hence, there exists the zero solution of (1). definition 2.1: the system (1) is called absolutely stable if its zero solution is asymptotically stable for any admissible switching signal and any admissible nonlinearities. to derive the absolute stability conditions, we will use a special construction of diagonal lyapunov–krasovskii functional. this functional is a counterpart of that proposed in [20] for switched positive persidskii systems with constant delays. furthermore, we will study the problem of absolute stability for a switched positive persidskii system with unbounded delay. finally, we will provide applications of the developed approaches to the stability analysis of a mechanical system with switched nonlinear positional forces and to a problem of mobile agent deployment on a line segment. 3. absolute stability conditions for the system with distributed delay construct a lyapunov–krasovskii functional candidate for (1) as follows: v1(xt) = n∑ i=1 αi ∫ xi(t) 0 φν i (u)du+ n∑ i=1 βi ∫ t t−τ ∫ ξ−t −τ f(ζ)dζφν+1 i (xi(ξ))dξ, (2) where αi, βi are positive coefficients, ν is a positive rational number with odd numerator and denominator. we will look for conditions under which the absolute stability of (1) can be proven with the aid of such a functional. theorem 3.1: let assumption 2.1 be fulfilled. if there exist a positive rational number ν with odd numerator and denominator, positive vectors λ, η and numbers ω1, ω2 such that νω1 + ω2 < 0, (3) ( ps + ∫ 0 −τ f(ζ)dζqs ) λ ≤ ω1λ, s = 1, . . . , n, ( ps + ∫ 0 −τ f(ζ)dζqr )⊤ η ≤ ω2η, s, r = 1, . . . , n, then one can choose coefficients αi, βi for which the functional (2) guarantees the absolute stability of the system (1). proof differentiating v1(xt) along the solutions of (1), we obtain v̇1 = n∑ i,j=1 αip (σ) ij φν i (xi(t))φj(xj(t)) + n∑ i,j=1 αiq (σ) ij φν i (xi(t)) ∫ t t−τ f(ξ − t)φj(xj(ξ))dξ − n∑ i=1 βi ∫ t t−τ f(ξ − t)φν+1 i (xi(ξ))dξ + f n∑ i=1 βiφ ν+1 i (xi(t)), copyright © 2024 assa. adv syst sci appl (2024) stability analysis of switched positive persidskii systems 43 where f = ∫ 0 −τ f(ζ)dζ , p(σ)ij and q (σ) ij are entries of the matrices pσ and qσ, respectively. let αi = ηi/λ ν i , i = 1, . . . , n, (4) yi(t) = φi(xi(t))/λi, i = 1, . . . , n. (5) here λi and ηi are components of the vectors λ and η, respectively. then the derivative of the functional can be rewritten in the form v̇1 = n∑ i,j=1 ηiλjp (σ) ij yνi (t)yj(t) + n∑ i,j=1 ηiλjq (σ) ij yνi (t) ∫ t t−τ f(ξ − t)yj(ξ)dξ − n∑ i=1 βiλ ν+1 i ∫ t t−τ f(ξ − t)yν+1 i (ξ)dξ + f n∑ i=1 βiλ ν+1 i yν+1 i (t). using the young inequality [1], we arrive at the estimate v̇1 ≤ ν ν + 1 n∑ i=1 ηiy ν+1 i (t) n∑ j=1 ( p (σ) ij + fq (σ) ij ) λj + n∑ i=1 λiy ν+1 i (t) ( 1 ν + 1 n∑ j=1 ηjp (σ) ji + fβiλ ν i ) + 1 ν + 1 n∑ i=1 λi ∫ t t−τ f(ξ − t)yν+1 i (ξ)dξ n∑ j=1 ηjq (σ) ji − n∑ i=1 βiλ ν+1 i ∫ t t−τ f(ξ − t)yν+1 i (ξ)dξ. choose the coefficients βi as follows: βi = 1 (ν + 1)λν i ( max r=1,...,n n∑ j=1 ηjq (r) ji + δ ) , (6) i = 1, . . . , n, where δ is a positive parameter. if δf < −(νω1 + ω2)/2, then v̇1 ≤ νω1 + ω2 2(ν + 1) n∑ i=1 ηiλiy ν+1 i (t)− δ ν + 1 n∑ i=1 λi ∫ t t−τ f(ξ − t)yν+1 i (ξ)dξ. hence (see [1]), the zero solution of the system (1) is asymptotically stable. this completes the proof. remark 3.1: it is worth mentioning that, under assumption 2.1, the conditions of theorem 3.1 are fulfilled iff one of the following conditions is satisfied: (a) there exists a vector λ > 0 such that (ps + fqs)λ < 0, s = 1, . . . , n, (7) copyright © 2024 assa. adv syst sci appl (2024) 44 a. aleksandrov, n. andriyanova (b) there exists a vector η > 0 such that (ps + fqr) ⊤η < 0, s, r = 1, . . . , n. really, in the case (a) the conditions of theorem 3.1 hold for sufficiently large values of ν, whereas in the case (b) they hold for sufficiently small values of ν. however, in some problems, e.g., estimation of attraction domain or the convergence rate of solutions, analysis of the impact of external perturbations, etc., to derive less conservative results, it is useful to have opportunity for an appropriate choice of the parameter ν (see, for instance, [20]). the presented statement of theorem 3.1 permits us to obtain the domain of admissible values for this parameter. 4. absolute stability conditions for the system with unbounded delay next, consider the system ẋ(t) = pσφ(x(t)) +qσ ∫ t 0 g(t− ξ)φ(x(ξ))dξ, (8) where g(ζ) is nonnegative and continuous for ζ ≥ 0 scalar kernel and the remaining notation is the same as for (1). thus, we will study switched persidskii system with unbounded delay. it should be noted that models with unbounded delays are widely used in various applications such as pid controller design, population dynamics, networked control, social science, etc. (see [1, 5, 22, 23]). every solution x(t) of (8) is defined by an initial time instant t0 ≥ 0 and an initial function θ(ξ) ∈ c([0, t0], r n), where c([0, t0], r n) is the space of continuous functions θ(ξ) : [0, t0] 7→ rn with the uniform norm ∥θ∥[0,t0] = max ξ∈[0,t0] ∥θ(ξ)∥. let xt be the restriction of a solution x(t) of (8) to the interval [0, t], i.e., xt : ξ 7→ x(t+ ξ) for ξ ∈ [−t, 0]. remark 4.1: it is known [7] that, under assumption 2.1, the system (8) is positive. definition 4.1: the system (8) is called absolutely stable if its zero solution is asymptotically stable for any admissible switching signal and any admissible nonlinearities. to obtain absolute stability conditions for (8) we impose additional constraints on the kernel g(ζ). assumption 4.1: let ∫ +∞ 0 g(ζ)dζ < +∞. assumption 4.2: let ∫ +∞ 0 g(ζ)dζ < +∞, where g(ζ) = ∫ +∞ ζ g(ξ)dξ. remark 4.2: in particular, assumptions 4.1 and 4.2 are satisfied for the function g(ζ) = exp(−cζ), where c = const > 0. it is worth mentioning that pid-controllers with exponential kernels are widely used in the modern control theory (see [1]). copyright © 2024 assa. adv syst sci appl (2024) stability analysis of switched positive persidskii systems 45 we will choose a lyapunov–krasovskii functional candidate for (8) in the form v2(xt) = n∑ i=1 αi ∫ xi(t) 0 φν i (u)du+ n∑ i=1 βi ∫ t 0 g(t− ξ)φν+1 i (xi(ξ))dξ, (9) where αi, βi are positive coefficients, ν is a positive rational number with odd numerator and denominator. theorem 4.1: let assumptions 2.1, 4.1, 4.2 be fulfilled. if there exist a positive rational number ν with odd numerator and denominator, positive vectors λ, η and numbers ω1, ω2 such that (ps +g(0)qs)λ ≤ ω1λ, s = 1, . . . , n, (10) (ps +g(0)qr) ⊤ η ≤ ω2η, s, r = 1, . . . , n, (11) and the inequality (3) hold, then one can choose coefficients αi, βi for which the functional (9) guarantees the absolute stability of the system (8). proof consider the derivative of the functional (9) along the solutions of (8). we obtain v̇2 = n∑ i,j=1 αip (σ) ij φν i (xi(t))φj(xj(t)) + n∑ i,j=1 αiq (σ) ij φν i (xi(t)) ∫ t 0 g(t− ξ)φj(xj(ξ))dξ − n∑ i=1 βi ∫ t 0 g(t− ξ)φν+1 i (xi(ξ))dξ +g(0) n∑ i=1 βiφ ν+1 i (xi(t)). define the coefficients αi and βi by the formulae (4) and (6), where λi and ηi are components of the positive vectors λ and η satisfying the inequalities (10) and (11), respectively. similarly to the proof of theorem 3.1, using the substitution (5) and the young inequality, we arrive at the estimate v̇2 ≤ −c1 n∑ i=1 yν+1 i (t)− c2 n∑ i=1 ∫ t 0 g(t− ξ)yν+1 i (ξ)dξ, where c1 > 0, c2 > 0. hence (see [5]), the zero solution of the system (8) is asymptotically stable. this completes the proof. it is worth noticing that, regarding theorem 4.1, a remark similar to remark 3.1 can be formulated. 5. applications consider two applications of the proposed approaches. copyright © 2024 assa. adv syst sci appl (2024) 46 a. aleksandrov, n. andriyanova 5.1. stability analysis of a mechanical system let motions of a mechanical system be defined by the equations q̈(t) + haq̇(t) +bσφ(q(t)) + cσ ∫ t 0 g(t− ξ)φ(q(ξ))dξ = 0. (12) here q(t), q̇(t) ∈ rn are vectors of generalized coordinates and generalized velocities, respectively, φ(q) = (φ1(q1), . . . , φn(qn)) ⊤, where the functions φi(qi) are continuous for |qi| < ∆ (0 < ∆ ≤ +∞) and satisfy the sector constraints qiφi(qi) > 0 for qi ̸= 0, i = 1, . . . , n, g(ζ) is nonnegative and continuous for ζ ≥ 0 scalar kernel, σ = σ(t) is an admissible switching law, σ(t) : [0,+∞) 7→ {1, . . . , n}, a,bs, cs are constant matrices, s = 1, . . . , n , h is a positive parameter. it is worth noting that the term cσ ∫ t 0 g(t− ξ)φ(q(ξ))dξ can be treated as integral part of a pid-controller [5, 22]. every solution q(t) of (12) is defined by an initial time instant t0 ≥ 0 and an initial function θ(ξ) ∈ c1([0, t0], r n), where c1([0, t0], r n) is the space of continuously differentiable functions θ(ξ) : [0, t0] 7→ rn with the uniform norm ∥θ∥[0,t0] = max ξ∈[0,t0] ( ∥θ(ξ)∥+ ∥θ̇(ξ)∥ ) . the system (12) admits the equilibrium position q = q̇ = 0. (13) to derive stability conditions for (13), we will use a special approach based on the decomposition method. this approach was first proposed in [24] for stability analysis of linear delay-free mechanical systems and it was subsequently extended to some classes of linear and nonlinear mechanical systems with constant delay [12, 25]. consider the following auxiliary subsystems: ẋ(t) = −a−1 ( bσφ(x(t)) + cσ ∫ t 0 g(t− ξ)φ(x(ξ))dξ ) , (14) ẏ(t) = −ay(t). (15) assumption 5.1: the matrices ps = −a−1bs, s = 1, . . . , n, (16) are metzler and the matrices qs = −a−1cs, s = 1, . . . , n, (17) are nonnegative. assumption 5.2: the system (15) is asymptotically stable. assumption 5.3: vector function φ(q) satisfies the lipschitz condition for ∥q∥ < ∆ with a constant l > 0. theorem 5.1: let the matrices ps, qs be defined by the formulae (16), (17), respectively, and assumptions 4.1, 4.2, 5.1–5.3 be fulfilled. if one of the following conditions is satisfied: (a) there exists a vector λ > 0 such that (ps +g(0)qs)λ < 0, s = 1, . . . , n, copyright © 2024 assa. adv syst sci appl (2024) stability analysis of switched positive persidskii systems 47 (b) there exists a vector η > 0 such that (ps +g(0)qr) ⊤η < 0, s, r = 1, . . . , n, then one can find a number h̄ > 0 such that the equilibrium position (13) is asymptotically stable for h ≥ h̄, for any admissible switching law and any admissible nonlinearity φ(q). proof define new variables as follows: x(t) = q(t) + 1 h a−1q̇(t), y(t) = q̇(t). (18) then hẋ(t) = pσφ(x(t)) +qσ ∫ t 0 g(t− ξ)φ(x(ξ))dξ +pσ ( φ ( x(t)− 1 h a−1y(t) ) − φ(x(t)) ) +qσ ∫ t 0 g(t− ξ) ( φ ( x(ξ)− 1 h a−1y(ξ) ) − φ(x(ξ)) ) dξ, ẏ(t) = −hay(t) + pσφ ( x(t)− 1 h a−1y(t) ) +qσ ∫ t 0 g(t− ξ)φ ( x(ξ)− 1 h a−1y(ξ) ) dξ. (19) it is worth mentioning that (19) can be considered as a complex system describing the interaction of subsystems (14) and (15). under the conditions of theorem 5.1, there exist a positive rational number ν with odd numerator and denominator and positive numbers αi, βi, i = 1, . . . , n, such that w (t, xt) ≤ −c1∥φ(x(t))∥ν+1 − c2 ∫ t 0 g(t− ξ)∥φ(x(ξ))∥ν+1dξ, where w (t, xt) is the derivative of the functional (9) along the solutions of the subsystem (14) and c1, c2 are positive coefficients. it is known (see [26]) that assumption 5.2 implies the existence of a lyapunov function v3(y) with the following properties: (i) v3(y) is continuously differentiable for y ∈ rn; (ii) v3(y) is positive definite; (iii) v3(y) is a homogeneous of the order ν + 1 with respect to the standard dilation function; (iv) the derivative of v3(y) along the solutions of the subsystem (15) is negative definite. construct a lyapunov–krasovskii functional candidate for (19) in the form ṽ (xt, yt) = h n∑ i=1 αi ∫ xi(t) 0 φν i (u)du+ n∑ i=1 βi ∫ t 0 g(t− ξ)φν+1 i (xi(ξ))dξ +v3(y(t)) + ∫ t 0 g(t− ξ)∥y(ξ)∥ν+1dξ. copyright © 2024 assa. adv syst sci appl (2024) 48 a. aleksandrov, n. andriyanova differentiating this functional along the solutions of (19) and using the lipschitz condition for φ(q), we obtain ˙̃ v ≤ −c1∥φ(x(t))∥ν+1 − c2 ∫ t 0 g(t− ξ)∥φ(x(ξ))∥ν+1dξ −(hc3 −g(0))∥y(t)∥ν+1 − ∫ t 0 g(t− ξ)∥y(ξ)∥ν+1dξ + c4 h ∥φ(x(t))∥ν ( ∥y(t)∥+ ∫ t 0 g(t− ξ)∥y(ξ)∥dξ ) +c5∥y(t)∥ν ( ∥φ(x(t))∥+ ∫ t 0 g(t− ξ)∥φ(x(ξ))∥dξ + 1 h ∥y(t)∥+ 1 h ∫ t 0 g(t− ξ)∥y(ξ)∥dξ ) , where c3, c4, c5 are positive constants. with the aid of the young inequality, it is easy to prove the existence of a number h̄ > 0 such that ˙̃ v ≤ −1 2 ( c1∥φ(x(t))∥ν+1 + hc3∥y(t)∥ν+1 + ∫ t 0 g(t− ξ) ( c2∥φ(x(ξ))∥ν+1 + ∥y(ξ)∥ν+1 ) dξ ) for h ≥ h̄. this implies the asymptotic stability of the zero solution of (19), see [5]. taking into account the properties of the transformation (18), we obtain that, for such values of h, the equilibrium position (13) of the system (12) is asymptotically stable, as well. this completes the proof. 5.2. a problem of mobile agent deployment let a group of n mobile agents on a line be given. the agents are interpreted as numbered points with coordinates zi(t) ∈ r, i = 1, . . . , n, and the agent dynamics is modeled by the first-order integrators żi(t) = ui, i = 1, . . . , n, where ui is a control input. assume that a segment [a, b] of the line is given. consider the problem of synthesis of a decentralized control protocol that provides a prescribed agent deployment on the segment. in [27], this problem was solved for the case of the equidistant agent distribution under the condition that each agent receives information about the distances between itself and its nearest left and right neighbors. it should be noted that neighbors are understood in terms of agent numbers. in [28, 29], the result of [27] was extended to the case where each agent receives information about the distances between itself and several its left neighbors and several its right neighbors (not necessarily nearest neighbors). furthermore, the effect of communication delay and switching of network topology (replacing chosen neighbors by the other ones) was investigated [28,29]. linear control protocols were designed for which neither constant delay values nor switching disturb the agent convergence to the equidistant distribution. in addition, the problem of nonlinearly-uniform (uniform with respect to a given nonlinear function) agent deployment was studied [29]. it is worth mentioning that such a problem is important for coverage control of mobile sensors, where a cost function is introduced to evaluate how copyright © 2024 assa. adv syst sci appl (2024) stability analysis of switched positive persidskii systems 49 well a given curve or domain is covered by the sensor network, and for synchronization of processes relatively to certain functions of phase coordinates [30, 31]. in [29], nonlinear control protocols were proposed and robustness of these protocols with respect to constant communication delays and network topology switching was proved. however, it should be noted that, in various formation control models, instead of discrete delays, distributed ones are used (see [1,32]). in particular, an example of such a model is the traffic flow dynamics [32]. therefore, in the present subsection, we will consider the problem of nonuniform agent deployment under distributed delay in the communication channels. let a continuous, locally lipschitz and strictly increasing for z ∈ (−∞,+∞) scalar function ϱ(z) be given. our objective is to design a decentralized control protocol providing the convergence of agents to the positions z̄i for which the corresponding points ϱ(z̄i) are uniformly distributed on the segment [ϱ(a), ϱ(b)]. hence, ϱ(z̄i) = ϱ(a) + i n+ 1 (ϱ(b)− ϱ(a)), i = 1, . . . , n. denote z̄ = (z̄1, . . . , z̄n) ⊤. in what follows, the points a and b will be interpreted as static agents, i.e., z0(t) = a, zn+1(t) = b for t ∈ [0,+∞). we will consider the scenario where each agent receives information from some its neighbors, and connections between agents can be switched on and off at any time instant. let σ(t) : [0,+∞) 7→ {1, . . . , n} be an admissible switching law defining the operating order of communication topologies, ξ(σ(t)) il and ξ (σ(t)) ir be the sets of indices of left and right neighbors, respectively, from which the ith agent receives information at the instant t, and ξ (σ(t)) i = ξ (σ(t)) il ∪ ξ (σ(t)) ir , i = 1, . . . , n. assumption 5.4: let ξ(s) il ̸= ∅ and ξ (s) ir ̸= ∅ for i = 1, . . . , n, s = 1, . . . , n . assumption 5.5: each ith agent at each time instant knows the values ϱ(zi(t))− c ∫ t t−τ f(ξ − t)ϱ(xj(ξ))dξ for j ∈ ξ (σ(t)) i , where τ = const > 0, f(ζ) is a nonnegative and continuous for ζ ∈ [−τ, 0] kernel, ∫ τ 0 f(ζ)dζ > 0, c = 1/ ∫ τ 0 f(ζ)dζ . assumption 5.6: each agent at each time instant knows how many agents are located between itself and agents from which the signals are received. define the coefficients a(s)ij by the formulae a (s) ij = δ (s) i (i− j)m (s) il for j ∈ ξ (s) il , a (s) ij = δ (s) i (j − i)m (s) ir for j ∈ ξ (s) ir , where m (s) il and m (s) ir are the numbers of elements of the sets ξ(s) il and ξ (s) ir , respectively, and δ (s) i =  1 m (s) il ∑ j∈ξ(s) il 1 i− j + 1 m (s) ir ∑ j∈ξ(s) ir 1 j − i  −1 , i = 1, . . . , n, s = 1, . . . , n. copyright © 2024 assa. adv syst sci appl (2024) 50 a. aleksandrov, n. andriyanova construct a control protocol as follows: ui = ∑ j∈ξ(σ(t)) i a (σ(t)) ij ( c ∫ t t−τ f(ξ − t)ϱ(zj(ξ))dξ − ϱ(zi(t)) ) , i = 1, . . . , n. then the corresponding closed-loop system takes the form żi(t) = ∑ j∈ξ(σ(t)) i a (σ(t)) ij ( c ∫ t t−τ f(ξ − t)ϱ(zj(ξ))dξ − ϱ(zi(t)) ) , i = 1, . . . , n. (20) theorem 5.2: let assumptions 5.4–5.6 be fulfilled. then the system (20) admits the equilibrium position z̄ that is globally asymptotically stable for any admissible function ϱ(z) and any admissible switching law. proof it is easy to verify that z̃ is an equilibrium position for (20). hence, the system (20) can be rewritten in the form żi(t) = ϱ(z̄i)− ϱ(zi(t)) + c ∑ j∈ξ(σ(t)) i , 1≤j≤n a (σ(t)) ij ∫ t t−τ f(ξ − t) (ϱ(zj(ξ))− ϱ(z̄j)) dξ, i = 1, . . . , n. let xi(t) = zi(t)− z̄i, i = 1, . . . , n. then ẋi(t) = −φi(xi(t)) + c ∑ j∈ξ(σ(t)) i , 1≤j≤n a (σ(t)) ij ∫ t t−τ f(ξ − t)φ(xj(ξ))dξ, i = 1, . . . , n. (21) here φi(xi) = ϱ(xi + z̄i)− ϱ(z̄i). thus, we arrive at the system of the form (1) where ps = −i and the components q (s) ij of the matrices qs are defined as follows: q(s)ij = ca (s) ij for j ∈ ξ (s) i , and q (s) ij = 0 for j /∈ ξ (s) i , i, j = 1, . . . , n, s = 1, . . . , n . it is known (see the proof of theorem 1 in [29]), that, for such matrices ps and qs, the corresponding inequality system (7) admits a positive solution. according to remark 3.1 we obtain that, if ν is sufficiently large, then there exists a functional of the form (2) guaranteeing the asymptotic stability of the zero solution of (21). it is worth noticing that in this case the asymptotic stability is global [5]. this implies the global asymptotic stability of the equilibrium position z̄ of the system (20). the proof is completed. 6. numerical example for simulation, consider a group consisting of ten agents. let [a, b] = [1, 6], ϱ(z) = arctan z, τ = 1, f(ζ) = e−ζ , σ(t) = 1 for t ∈ [10k, 10k + 5) and σ(t) = 2 for t ∈ [10k + 5, 10(k + 1)), k = 0, 1, 2, . . .. assume that initial functions for agents are constant on the interval [−1, 0]. their values and sets of agent neighbors are presented in table 6.1. in fig. 6.1, the dependence of agent coordinates on time is given. the simulation results demonstrate the convergence of agents to the prescribed distribution. copyright © 2024 assa. adv syst sci appl (2024) stability analysis of switched positive persidskii systems 51 table 6.1. network topologies and agents initial positions agent number initial position ξ (1) i ξ (2) i 1 0.41 {0, 4} {0, 5} 2 -0.39 {0, 5} {0, 6} 3 0.4 {0, 6} {0, 7} 4 -0.3 {1, 7} {0, 8} 5 -0.22 {2, 8} {1, 9} 6 0.8 {3, 9} {2, 10} 7 0.3 {4, 10} {3, 11} 8 -0.5 {5, 11} {4, 11} 9 -0.6 {6, 11} {5, 11} 10 1.7 {7, 11} {6, 11} 0 100 200 300 400 t 0 1 2 3 4 5 6 fig. 6.1. the agent time history. 7. conclusion in the present contribution, original constructions of diagonal lyapunov–krasovskii functionals for switched positive persidskii systems with distributed and unbounded delays are proposed. with the aid of these functionals, new absolute stability conditions for considered systems are derived. the obtained results are used for the stability analysis of a mechanical system with switched nonlinear positional forces and for the design of decentralized protocols ensuring a prescribed mobile agent deployment on a line segment. it is worth noticing that the proofs of theorems 3.1 and 4.1 provide us constructive algorithms for finding parameters of the functionals. an interesting direction for further research is an application of the developed approaches in stability analysis of generalized lotka–volterra models of population dynamics. acknowledgements the research was supported by the grant of russian science foundation, n 24-21-00091, https://rscf.ru/en/project/24-21-00091/ copyright © 2024 assa. adv syst sci appl (2024) 52 a. aleksandrov, n. andriyanova references 1. fridman, e. (2014) introduction to time-delay systems: analysis and control. basel: birkhauser. 2. kharitonov, v. (2013) time-delay systems. lyapunov functionals and matrices. boston: birkhäuser. 3. kazkurewicz, e. & bhaya, a. (1999) matrix diagonal stability in systems and computation. boston: birkhauser. 4. farina, l. & rinaldi, s. (2000) positive linear systems: theory and applications. new york: wiley. 5. kolmanovskii, v. & myshkis, a. (1999) applied theory of functional differential equations. dordrecht: springer science & busines media. 6. ngoc, p. (2013) stability of positive differential systems with delay. ieee trans. automat. control, 58, 203–209. 7. ngoc, p., naito, t., shin, j., & murakami, s. (2008) on stability and robust stability of positive linear volterra equations. siam j. control optim., 47, 975–996. 8. aleksandrov, a. (2018) construction of the lyapunov–krasovskii functionals for some classes of positive delay systems. siberian math. j., 59, 753–762. 9. shen, j. & chen, s. (2017) stability and l∞-gain analysis for a class of nonlinear positive systems with mixed delays. int. j. robust nonlinear control, 27, 39–49. 10. liao, x. & yu, p. (2008) absolute stability of nonlinear control systems. new york, heidelberg: springer science & business media. 11. feyzmahdavian, h., charalambous, t., & johansson, m. (2014) asymptotic stability and decay rates of homogeneous positive systems with bounded and unbounded delays. siam j. control. optim., 52, 2623–2650. 12. aleksandrov, a. & efimov, d. (2021) stability analysis of switched homogeneous timedelay systems under synchronous and asynchronous commutation. nonlinear analysis: hybrid systems, 42, 101090. 13. rajagopal, j. & balamurugan, j. 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(2018) design of robust stabilizing pi/pid controller for time delay interval process plants using particle swarm optimization. advances in systems science and applications, 18, 92–120, https://dx.doi.org/doi:10.25728/assa.2018.18.4.536. copyright © 2024 assa. adv syst sci appl (2024) https://dx.doi.org/doi.org/10.25728/assa.2021.21.1.1044 https://dx.doi.org/doi: 10.25728/assa.2018.18.4.536 stability analysis of switched positive persidskii systems 53 23. priyanka, e., maheswari, c., ponnibala, m., & thangavel, s. (2019) scada based remote monitoring and control of pressure & flow in fluid transport system using imc-pid controller. advances in systems science and applications, 19, 140–162, https://dx.doi.org/doi.org/10.25728/assa.2019.19.3.676. 24. zubov, v. (1970) analytical dynamics of gyroscopic systems (in russian). leningrad: sudostroenie. 25. aleksandrov, a., aleksandrova, e., & zhabko, a. 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(2007) stability of traffic flow behavior with distributed delays modeling the memory effects of the drivers. siam j. appl. math., 68, 738–759. copyright © 2024 assa. adv syst sci appl (2024) https://dx.doi.org/doi.org/10.25728/assa.2019.19.3.676 introduction statement of the problem absolute stability conditions for the system with distributed delay absolute stability conditions for the system with unbounded delay applications stability analysis of a mechanical system a problem of mobile agent deployment numerical example conclusion adv syst sci appl 2025; 02:51–57 published online at https://ijassa.ipu.ru. multivalued solutions of the cauchy problem for one quasilinear system elena n. kushner bauman moscow state technical university, moscow, russia abstract: the article proposes a generalization of the method of characteristics to quasilinear systems of evolutionary equations with one spatial variable. such systems are a special case of hydrodynamic type systems. it is shown that in some cases self-intersection of solution graphs is possible. an example of a system of two equations is considered in detail. keywords: hydrodynamic type systems, multivalued solutions, jet spaces, caustics, selfintersection, cauchy problem 1. introduction consider the following first order quasilinear system of m equations ∂u1 ∂t + a(u1, . . . , um) ∂u1 ∂x = 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . ∂um ∂t + a(u1, . . . , um) ∂um ∂x = 0. (1.1) here t is time, x is a spatial coordinate, the function a ∈ c∞(rm). this is a special case of a hydrodynamic type systems and a generalization of the scalar euler–hopf equation ut + uux = 0. (1.2) it is well known that solutions of equation (1.2) become discontinuous at some moment of time even for a smooth initial condition. a similar phenomenon occurs for the considered system. in addition, we will show that for such systems the effect of self-intersection of the graphs of their solutions is possible. consider the following cauchy problem for equation (1.1) ui|t=0 = u i(x), i = 1, . . . ,m. (1.3) assume that the initial functions u i belong to the class c∞. 2. multivalued solutions let j1 = j1(2,m) be the 1-jet space with canonical coordinates (see [1, 2]) t, x, u1 0,0, u 1 1,0, u 1 0,1, . . . , u m 0,0, u m 1,0, u m 0,1; i = 1, . . . ,m. ∗corresponding author: ekushner@ro.ru 52 e. kushner the cartan distribution c is generated by the differential 1-forms κi = dui 0,0 − ui 1,0dt− ui 0,1dx, i.e. c(θ) = m⋂ i=1 kerκi |θ ⊂ tθj 1. system (1.1) generates a 2(m+ 1)-dimensional submanifold e = { ui 1,0 + aui 0,1 = 0 ∣∣ i = 1, . . . ,m } ⊂ j1. a two-dimensional integral manifold of the cartan distribution lying in e is called a multivalued solution of system (1.1). let us show a method for constructing multivalued solutions to cauchy problem (1.3). define the following vector field on the 0-jet space j0(2,m): x = − ∂ ∂t − a(u1 0,0, . . . , u m 0,0) ∂ ∂x . its prolongation to j1(2,m) is x(1) = x + m∑ j=1 ( m∑ i=1 ∂a ∂ui 0,0 ui 1,0 ) uj 0,1 ∂ ∂uj 1,0 + m∑ j=1 ( m∑ i=1 ∂a ∂ui 0,0 ui 0,1 ) uj 0,1 ∂ ∂uj 0,1 . lemma 2.1: the vector field x(1) is tangent to e . proof the lie derivative of the function fk = uk 1,0 + auk 0,1 is x(1)(fk) = ( m∑ i=1 ∂a ∂ui 0,0 ui 1,0 ) uk 0,1 + ( m∑ i=1 ∂a ∂ui 0,0 ui 0,1 ) auk 0,1. therefore, the restriction x(1)(fk)|e = 0 for k = 1, . . . ,m. 3. cauchy data the initial data (1.3) define a curve in the 1-jet space: k : { t = 0, x = χ, ui 0,0 = u i(χ), ui 0,1 = du i dχ , ui 1,0 = −ã du i dχ ∣∣∣∣ i = 1, . . . ,m } , where ã = a(u1(χ)), . . . , um(χ)). this curve is called cauchy curve. let φτ be a flow of x(1). since the vector field x(1) is transversal to the hyperplane t = 0, we see that x(1) a /∈ tak for any point , i.e. trajectories of the vector field intersect the cauchy curve. so, for sufficiently small τ , the union of φτ (k) forms a surface l: l = ⋃ τ φτ (k). (3.4) copyright © 2025 assa. adv syst sci appl (2025) multivalued solutions of the cauchy problem... 53 theorem 3.1: surface (3.4) is a multivalued solution of equation (1.1). proof due to lemma 2.1, the vector field x(1) is tangent to e . then l ⊂ e . it remains to prove that l is an integral manifold of the cartan distribution. the cauchy curve is an integral curve of the cartan distribution. since the vector field x(1) is contact, the curve φτ (k) is an integral for the cartan distribution, too. that is taφτ (k) ⊂ c(a) for any point a ∈ k. the value of the 1-form κi on the vector field x(1) is κi(x (1)) = ui 1,0 + aui 0,1 for i = 1, . . . ,m. therefore, the restriction κi(x (1)|e) = κi(x (1))|e = 0 and the integral curves of the vector field x(1)|e are integral for the cartan distribution as well. let a ∈ l be an arbitrary point. the tangent space tal is the direct sum tal = taφτ (k)⊕ rx(1) a ⊂ c(a). therefore, l is a 2-dimensional integral manifold of the cartan distribution. 4. example consider the system { ut + uux = 0, vt + uvx = 0 (4.5) with initial data u|t=0 = 1 1 + x2 , v|t=0 = 1 1 + (x− 1)2 . then the vector field x = − ∂ ∂t − u0,0 ∂ ∂x and its prolongation is x(1) = − ∂ ∂t − u0,0 ∂ ∂x + u1,0u0,1 ∂ ∂u1,0 + u2 0,1 ∂ ∂u0,1 + u1,0v0,1 ∂ ∂v1,0 + u0,1v0,1 ∂ ∂v0,1 . the corresponding flow is φτ :  t 7→ t− τ, x 7→ x− u0,0τ, u0,0 7→ u0,0, v0,0 7→ v0,0, u1,0 7→ u1,0 1− τu0,1 u0,1 7→ u0,1 1− τu0,1 v1,0 7→ u1,0v0,1τ − v1,0u0,1τ + v1,0 1− τu0,1 , v0,1 7→ v0,1 1− τu0,1 . copyright © 2025 assa. adv syst sci appl (2025) 54 e. kushner the cauchy curve is k :  t = 0, x− χ = 0, u0,0 − 1 1 + χ2 = 0, v0,0 − 1 1 + (χ− 1)2 = 0, u1,0 − 2χ (1 + χ2)3 = 0, v1,0 − 2χ− 2 (1 + χ2)(1 + (χ− 1)2)2 = 0, u0,1 + 2χ (1 + χ2)2 = 0, v0,1 + 2χ− 2 (1 + (χ− 1)2)2 = 0. shifting the curve k along the vector field x(1), we obtain the multivalued solution to the cauchy problem: lk :  t = τ, x = τ + χ+ χ3 1 + χ2 , u0,0 = 1 1 + χ2 , v0,0 = 1 2 + χ2 − 2χ , u1,0 = 2χ (1 + χ2)(1 + 2χ2 + χ4 − 2τχ) , u0,1 = − 2χ 1 + 2χ2 + χ4 − 2τχ , v1,0 = 2χ3 − 2χ2 + 2χ− 2 (2 + χ2 − 2χ)2(1 + 2χ2 + χ4 − 2τχ) , v0,1 = − 2(χ− 1))(1 + χ2)2 (2 + χ2 − 2χ)2(1 + 2χ2 + χ4 − 2τχ) . the projections of this solution to 3-dimensional spaces t, x, u and t, x, v are shown in fig. 4.1. let us construct caustics. for this goal, we introduce two planes π and γ with coordinates t, x and τ, χ respectively and construct the mapping π : γ −→ π, where π : t = τ, x = χ3 + χ+ τ 1 + χ2 . copyright © 2025 assa. adv syst sci appl (2025) multivalued solutions of the cauchy problem... 55 fig. 4.1. projections lu k and lv k of the multivalued solution lk to the spaces t, x, u and t, x, v respectively. the jacobi matrix of π is jπ =  1 0 1 1 + χ2 1 + 2χ2 + χ4 − 2χτ (1 + χ2)2  . a caustic σ is defined by the critical points of this mapping, that is, the points at which the determinant of the jacobi matrix vanishes (see fig. 4.2): σ = {1 + 2χ2 + χ4 − 2χτ = 0} ⊂ γ. fig. 4.2. the caustics σ on the plane γ (left) and its projection to the plane π (right). copyright © 2025 assa. adv syst sci appl (2025) 56 e. kushner fig. 4.3. birth of the self-intersection curve: sections of the surface lv k at t = 0, t = 2, t = 5 respectively. as can be seen from fig. 4.1, the surface lv k :  t = τ, x = τ + χ+ χ3 1 + χ2 , v0,0 = 1 2 + χ2 − 2χ , (4.6) has self-intersection along some curve. let us find this curve. coordinates a = a(τ) and b = b(τ) of the points of this curve are solutions of the system τ + a+ a3 1 + a2 = τ + b+ b3 1 + b2 , 1 2 + a2 − 2a = 1 2 + b2 − 2b . its real solutions are a = 1∓ (2τ − 4)1/4, b = 1± (2τ − 4)1/4 we see that self-intersection occurs only when τ ≥ 2. so, we can choose χ = 1 + (2τ − 4)1/4. substituting this value to (4.6), we obtain a parametric representation of the self-intersection curve:  t = τ, x = τ + (2τ − 4)1/4 + 1 + ((2τ − 4)1/4 + 1)3 1 + ((2τ − 4)1/4 + 1)2 , v0,0 = 1√ 2τ − 4 + 1 . 5. conclusion the proposed method can be extended to a wider class of evolutionary systems. note that another approach to evolutionary systems is proposed in [3–5]. that method is based on the finite-dimensional dynamics. copyright © 2025 assa. adv syst sci appl (2025) multivalued solutions of the cauchy problem... 57 fig. 4.4. projection of the self-intersection curve to the plane x, v references 1. krasilshchik, i. s., lychagin, v. v. & vinogradov, a. m. (1986) geometry of jet spaces and nonlinear partial differential equations. new york, ny: gordon and breach. 2. kushner, a. g., lychagin, v. v. & rubtsov, v. n. (2007) contact geometry and nonlinear differential equations. cambridge, uk: cambridge university press. 3. gorinov, a. a. & kushner, a. g. (2020) dynamics of evolutionary pde systems, lobachevskii j. math., 41, 2448–2457. 4. kushner, a. g. & matviichuk, r. i. (2021) dynamics and exact solutions of nonevolutionary partial differential equations, differ. geom. appl., 76, 101761. 5. kushner, a. g. & sinian, t. (2024) evolutionary systems and flows on solutions spaces of finite type equations, lobachevskii j. math., 44(9), 3945–3951. copyright © 2025 assa. adv syst sci appl (2025) introduction multivalued solutions cauchy data example conclusion microsoft word 1441-article text-6891-1-18-20240313 adv syst sci appl 2024; 01; 95-113 published online at https://ijassa.ipu.ru. game-theoretic models of quality management in organizations under corruption makka agieva1, movlatkhan agieva1, guennady ougolnitsky*2, 1) ingush state university, nazran, russia 2) i.i. vorovich institute for mathematics, mechanics, and computer science, southern federal university, rostov-on-don, russia abstract: we consider game theoretic models of the quality management in an organizational system under corruption. a graph theoretic formalization of the process approach in quality management is proposed. in a static case, a problem of quality management in a two-level organizational system under corruption is formalized as an inverse stackelberg game with an additional viability condition. it is solved by means of germeier theorem. in a three-level model we analyze how the principal can constraint corruption by penalties. the conditions under which the principal rather supports corruption than constraints it are determined. in a dynamic case, a quality indicator is considered as a state variable which changes in a discrete time due to an equation of dynamics. a two-level model of the type "supervisor-agent" is studied first. it is assumed that a quality requirement is obligatory for the agent but he can weaken this requirement in exchange for a bribe to the supervisor. then we build and analyze a three-level model by adding the principal that can charge penalties to other players if a bribe is found. the game theoretic models are investigated numerically by means of simulation modeling. keywords: corruption; game theory; quality control; organizational systems; simulation modeling 1. introduction a problem of quality management is very actual for any organization. a well known concept of quality management based on a process approach is used worldwide. there are official iso 9000 standards that describe and regulate the quality management. however, employees are active agents whose behavior is strategic. particularly, the employees can propose bribes to their supervisors if they want to weaken too strong quality requirements, and then a corporative corruption arises. there is a big stream of the mathematical modeling of corruption. particularly, some papers are devoted to the corruption in hierarchical organizations. several important in-sights and recommendations are received. the authors' concept of corruption modeling is based on two ideas. first, corruption is treated as a feedback on bribe in an organizational control system. second, a struggle with corruption is successful if some requirements of sustainable development (for example, quality requirements) are satisfied for this system. therefore, the most adequate model of the corporative corruption is an inverse stackelberg game with phase constraints. in a static case, an inverse stackelberg game has a special additional constraint that plays a role of the phase constraint. for the solution of inverse stackelberg games we use germeier theorem. in a static setting it allows for an analytical solution, and in a dynamic setting some algorithms of simulation modeling are used. a different software on java and python programming languages is used for numerical calculations. the contribution of this paper is the following: *corresponding author: gaugolnickiy@sfedu.ru 96 t.y. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2024 assa adv. in systems science and appl. (2024)  a problem of quality management in a two-level organizational system under corruption is formalized as a static inverse stackelberg game with an additional requirement, and its solution is found by means of germeier theorem;  in a three-level organizational system a similar model is analyzed analytically and numerically. the conditions under which the principal supports corruption are determined;  in a dynamic setting, a problem of quality management in two-level and three-level organizational systems under corruption is formalized as a dynamic inverse stackelberg game with phase constraints;  a method of qualitatively representative scenarios in simulation modeling is used for the solution of the mentioned dynamic inverse stackelberg games. in section 2, a literature review on the topic of the paper is provided. in section 3, we consider static models of quality management under corruption in two-level and three-level organizations. these models are inverse stackelberg games. in section 4, we consider dynamic models of the same type, also in two-level and three-level organizational systems. in section 5, concluding remarks are presented. 2. literature review a concept of quality management was introduced by w. edwards deming [22] and developed by many other authors [29,59]. they proposed several principles of quality management, specified their implementation in industrial organizations, and used statistical methods to improve their efficiency. we develop this approach in our concept of sustainable management in organizations [25,52]. most papers concerned modeling of corruption are based on gary becker's idea [11] that struggle with any crime makes sense (is economically rational) if benefits from pre-venting the crime are greater than the respective costs. in application to corruption this idea was developed by susan rose-ackerman [54,55] and many other authors [58]. the main reviews are presented in [3,28]. political corruption, especially in elections, is considered in [1,45– 47]. cultural aspects of corruption are touched in [30], its psychological aspects are considered in [10]. other papers are devoted to the corruption in economics and state administration. so called "games of inspection" are described in [21], tax and other audits are studied in [18,19,56]. influence of competition to corruption is studied in [46,57], compensation of market failures in [22], corruption in natural resources exploitation in [12]. in [20] they analyze a connection between corruption and shadow economics, in [36–38] – a "petty" corruption in licensing. in [28] the authors propose methods of struggle with corruption, in [65,66] they reveal the role of collusion. many papers [7–9,17,43,44] are devoted to the corruption in hierarchical organizations. as a rule, all mentioned models use the techniques of static games in normal form or multistage games. essentially less number of papers is dedicated to the dynamic models of corruption based on optimal control models or differential games. some examples can be found in the manual [27], and papers [14–16,42,68,70]. thus, in [13] they have shown that an evolution of corruption in economics can result in the «revolution of honesty» due to which the system control vector moves to another state. in the model [16] situations exist in which national economies with the same development parameters are on different corruption levels. the paper [23] is devoted to the cyclicity in political corruption when struggle with corruption interchanges with its silent support on the macroeconomic level. organization of inspections as an evolutionary game is presented in [33,34], and the same approach is used in [39] for description of the struggle with illegal logging. in [35] they use mean-field games that describe an interaction of a very big number of rational agents, including the case of impact of a special "main" player. dynamic model of general coordination and private interests 97 copyright ©2024 assa. adv. in systems science and appl. (2024) an important domain of description of the strategic behavior of active agents is auction models [4,41]. two big groups can be differentiated there: models of collusion between the participants of an auction and models of collusion between an auctioneer and some participants. the data of specific studies are presented in [2,30,60], and the normative documents are given in [61-64]. in the last time the attention is attracted by the models of sustainable procurement that consider social and environmental aspects of the auction proposals [53,67]. the authors' concept of modeling corruption is presented in [25,68]. olsder [49,50] has presented a review of inverse stackelberg games. an original concept of hierarchical games belongs to germeier [24]. he proposed a classification of hierarchical games based on a principle of guaranteed payoff of the leader, and proved a very useful theorem about a solution of inverse stackelberg games in a static setting. in a dynamic setting, this approach was developed by kononenko [26]. we proposed the respective numerical algorithms for the case of several agents [69]. these algorithms are based on the method of qualitatively representative scenarios in simulation modeling [51] which permits to reduce enumeration essentially. 3. static models 3.1. quality management without corruption we consider an organizational-economic system as a basic object. this system is a controlled one because its state depends on a production strategy. the system state is described by a set of characteristics that include costs and quality indicators. an economic agent exercises control. namely, the agent chooses a strategy that maximizes his payoff function (his profit on a period). so, we receive an optimization problem for the agent in the form: 𝑝(𝑞) − 𝑟 → max, (𝑞, 𝑟) ∈ 𝑆, (3.1.1) where 𝑞 = (𝑞 , 𝑞 , … , 𝑞 ) is a vector of quality indicators; 𝑟 – production cost; 𝑝(𝑞) – income from the sale of a product with quality indicators 𝑞; 𝑆 – a set of feasible production strategies. a function p is continuous, and s is a compact set. the quality indicators can reflect absolutely different characteristics of the product. such indicators can describe consumption quality or characterize production (for example, its environmental impact). suppose that all indicators take their values on the segment [0,1], and a greater value corresponds to a better quality. this model is a static one, and the income and cost are calculated for the whole report period. any solutions are made in the beginning of the period. an element of the set 𝑆 is a pair(𝑞, 𝑟) that corresponds to such a production strategy which provides a vector of quality indicators 𝑞 and cost 𝑟. besides, for any pair of strategies: (𝑞 , 𝑟 ) ∈ 𝑆, (𝑞 , 𝑟 ) ∈ 𝑆, (𝑞 > 𝑞 ) => (𝑟 > 𝑟 ); (𝑞 > 𝑞 ) ≝ ∀𝑖 ∈ {1,2, … , 𝑚}(𝑞 ≥ 𝑞 ), ∃𝑗: 𝑞 ≥ 𝑞 . we can write 𝑆 = 𝑃𝑂 𝑓(𝑃) , where 𝑃 = {(𝑝 , 𝑝 , … , 𝑝 )} is a set of controlled parameters; 𝑓– a standard production function; 𝑃𝑂 – a function that makes the set paretooptimal. assume that for any element from 𝑆 we can determine its preimage in 𝑃, i.e., such values of the controlled parameters that provide given values 𝑟 and 𝑞. thus, instead of the solution of the problem on the whole set 𝑃it is sufficient to solve it on the set 𝑆 and apply to the solution the function 𝑓 . a solution of the problem (3.1.1) can be considered both as a pair(𝑞, 𝑟) and the respective set of parameters(𝑝 , 𝑝 , … , 𝑝 ). a viability condition in the static setting means that all essential indicators of the system activity take their values from a given range. for an industrial organization such indicators 98 t.y. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2024 assa adv. in systems science and appl. (2024) are, for example, a quality of production, an adequate wage, a satisfaction of some environmental protection requirements, and so on. for the quality indicators 𝑞 these conditions can be written as 0 ≤ 𝑞 ≤ 1, 𝑖 = 1, … , 𝑚. the greater are these indicators, the higher is the production quality. therefore, a viability condition (an admittance) for an indicator 𝑖can be written as: 𝑞 ≥ 𝑎 , 𝑎 ∈ [0,1], 𝑖 = 1, … , 𝑚. (3.1.2) the value 𝑎 determines a minimal admissible value of the respective indicator. if there are no constraints for an indicator then 𝑎 = 0. thus, we go from the problem (3.1.1) to the problem 𝑝(𝑞) − 𝑟 → 𝑚𝑎𝑥, (𝑞, 𝑟) ∈ 𝑆 ∩ 𝐴, 𝐴 = [𝑎 , 1] × [𝑎 , 1] × [𝑎 , 1] × [0, ∞). (3.1.3) as viability conditions narrow down a domain 𝑆 , a maximal payoff found with consideration with these conditions will be less or the same. this fact motivates to neglect a viability condition to increase a payoff. according to the iso 9000 international standards, any activity should be considered as a technological process. in any organization these processes have a complex interaction that form a system of processes [22]. let us consider an industrial system as a treed = (x, u), directed to the root, where x is a set of processes (vertices), u is a set of flows between these processes (arcs). fig. 3.1.1. an example of a tree-like industrial structure the root of the tree corresponds to the last production process: its output goes to a consumer. the processes can reflect arbitrary types of activity. besides a production, it can be an intermediate quality control, a preparation of reports or other documents, a maintenance of a ready production. a production flow from one process to another is characterized by two quantities: a vector of quality indicators 𝑞 and a production cost 𝑟. an agent's objective is a choice of the optimal set of strategies that provides a maximal payoff. this set distributes the strategies among the processes. in general case a process can have several input flows. then characteristics of the output flow depend on all these variables. an initial process has no input flows. an output of the last process goes to the consumer, and its characteristics determine the agent's payoff. this description of an organizational-economic system allows to find a pareto-optimal set of strategies 𝑆 step by step, from one process to another. here we use a dynamic programming principle. the java platform was used for writing the application, and the netbeans 6.9 package served as a development environment. the program builds a set 𝑆 and finds the strategies that provide a maximal payoff with consideration of viability conditions or without it. 3.2. two-level system under corruption it is not always advantageous for an agent to satisfy the viability conditions. thus, the principal who establish these conditions must control their satisfaction. usually, the principal is a legislative body, a local control agency or a certification organization. so, the principal dynamic model of general coordination and private interests 99 copyright ©2024 assa. adv. in systems science and appl. (2024) cannot control the viability conditions personally and delegates the control functions to special supervisors. consider the following regulation pattern. an organizational-economic system is a controlled object. the first control subject is an agent (a governing body of the organization) who chooses a production strategy. the model is a static one, and the agent knows which vector of quality indicators 𝑞 and cost 𝑟 corresponds to each strategy 𝑝 . also, the agent knows the viability conditions established by the principal (the latter does not participate in the model explicitly). the second control subject is a supervisor who is responsible for the satisfaction of viability conditions. thus, we receive a two-level hierarchically organized control subsystem "supervisor-agent". let us assume first that there is only one quality indicator, and the supervisor has a complete and certain information about its value. if the viability conditions are satisfied then any actions are required, otherwise the supervisor should fix the situation. besides, the supervisor can charge a penalty to the agent. the penalty value can depend on the difference between normative and actual values of the indicator. ideally, the supervisor is an impartial arbitrator who honestly does her job. her reward is fixed by the principal in advance, and the possible penalty goes completely to the principal. however, a real situation may differ. suppose that the supervisor can extend the domain of viability by the controlled indicator. naturally, she does it for a reward (bribe) from the agent. the check-up procedure for an𝑖-th indicator is the following. 1. the principal reports an admittance range that determines the viability condition. suppose that an𝑖-th component of this vector is equal to𝑎 . 2. the agent chooses a control strategy. after this, the value of the𝑖-th quality indicator is equal to𝑞 , and the cost is equal to𝑟 . the viability condition may be satisfied or not. 3. the supervisor checks the value of the𝑖-th indicator and exposes𝑞 . 4. if𝑞 < 𝑎 then the supervisor reports to the agent a corruption function𝑎(𝑏). this function defines a dependence of the bound of admittance𝑎on a bribe value𝑏. 5. the agent pays to the supervisor a bribe𝑏 and uses a production strategy that provides a new extended admittance condition𝑞 ≥ 𝑎(𝑏). in this case the agent has additional cost connected with some production changes and, probably, pays a penalty. suppose that the agent does not know when a check-up will be performed and which indicator will be checked. thus, in a two-level system the corruption concerns only steps 3-5. a mathematical formalization of this situation is a complicated version of the model (3.1.3). the agent's payoff function𝑔 is his profit, or a difference between an income from new quality indicators and a sum of new production cost, bribe, and penalty. thus, themodelhastheform 𝑔 (𝑎, 𝑏, 𝑞, 𝑟) = 𝑝(𝑞) − 𝑟 − 𝑏 − 𝑑(𝑞, 𝑞 ) → max, (3.2.1) 𝑏 ≥ 0, (𝑞, 𝑟) ∈ 𝑆, 𝑞 ≥ 𝑎(𝑏), 𝑖 = 1, … , 𝑚; where𝑖 – index of the controlled indicator; 𝑏 – bribe value;𝑎(𝑏) – a new bound of admittance by the indicator𝑖; 𝑞 – a new vector of quality indicators; 𝑟 – a new production cost; 𝑆 – a compact set of admissible strategies; 𝑑 –a penalty function for the transfer from an old strategy that provides the quality𝑞 to a new strategy. it includes additional cost and possible penalty for violation of the viability condition. this function is supposed to be given and proportional to the distance between 𝑞 and 𝑞 , for example, 𝑑(𝑞, 𝑞 ) = 𝑘‖𝑞 − 𝑞 ‖, 𝑘 > 0. the control variables are a new production strategy and a bribe. the domain of production strategy is an initial domain of feasible strategies with an additional condition of admittance by the checked 𝑖-th indicator. the domain of bribe is a semi-interval [0, +∞) that is specified by the function 𝑎(𝑏). the supervisor's payoff function also corresponds to her profit. we do not consider a fixed reward, therefore the function coincides with the bribe value: 100 t.y. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2024 assa adv. in systems science and appl. (2024) 𝑔 (𝑎, 𝑏) = 𝑏 → max, 𝑎(𝑏) ∈ 𝐴. (3.2.2) the supervisor chooses 𝑎(𝑏). this function must satisfy the following constraints that define the admissible domainа: 𝐴: 𝑎(0) = 𝑎 , 𝑞 ≤ 𝑎(𝑏) ≤ 𝑎 . (3.2.3) the first condition determines a capture strategy of the supervisor: if there is no bribe then an initial viability condition is satisfied. the second condition means that the supervisor can extend an admittance bound up to the current value of the checked indicator. an additional extension has no reason. greater values of the bribe correspond to not less admittance domain. therefore, 𝑎(𝑏)is a monotonous non-increasing function. thus, we receive the model (3.2.1)–(3.2.3). this is a game theoretic model because the interests of players are different. a greater bribe is advantageous to the supervisor and nonadvantageous to the agent, and the payoff function of each player depends on strategies of both players. this game is a hierarchical one because the supervisor makes the first move when she reports to agent a response function 𝑎(𝑏) . given 𝑎(𝑏)the agent chooses his bribe 𝑏and control variables 𝑝. the supervisor anticipates a best response of the non-benevolent agent and maximizes her guaranteed payoff. thus, we receive an inverse stackelberg game [49,50], or germeier game of the type г [24,26]. however, the agent's payoff function is too complicated. let us transform this function. a choice of the production strategy by the agent has no impact to the supervisor's payoff function. in the same time, this choice depends on the admittance bound. let us introduce the function 𝑢(𝑎) = max ( , )∈ , 𝑝(𝑞) − 𝑟 − 𝑑(𝑞, 𝑞 ) . (3.2.4) this function solves a problem of profit maximization for a given value of the admittance bound𝑎. the agent can calculate this function for each 𝑎 without any information about the supervisor's strategy. as evident from the formula (3.2.4), this function is non-increasing on the segment [0,1] and attains its maximum in the point𝑞 because a further reduction of the quality is non– advantageous to the agent. moreover, even small deviations of 𝑎 from 𝑞 imply some changes in the current production strategy that incurs additional cost. therefore, ∀𝜀 > 0 ∃𝛿 > 0: 𝑢(𝑞 ) − 𝑢(𝑞 + 𝜀) > 𝛿. thus, we can rewrite the model (3.2.2)–(3.2.4) in the form 𝑔 (𝑎, 𝑏) = 𝑏 → max ( ) , 𝑔 (𝑎, 𝑏) = 𝑢(𝑎) − 𝑏 → max, (3.2.5) 𝑏 ≥ 0, 𝑞 ≤ 𝑎 ≤ 𝑎 , 𝑎(0) = 𝑎 . this model is a germeier game of the type г [64], or an inverse stackelberg game. we can generalize the model for the case of several quality indicators. the supervisor's strategy and the function 𝑢 become multidimensional. however, the idea of solution and the final result are the same, and the supervisor's strategy also includes two important points: the first one corresponds to the current system state, and the second one corresponds to the viability condition. for solution of the game (3.2.5) we use germeier theorem [64]. in this case the supervisor's payoff function does not depend directly on the admittance bound. therefore, any valueof 𝑎 may serve as her dominant strategy. it is natural to take 𝑎 = 𝑎 because it corresponds to the law requirements: 𝑎 (𝑏) = 𝑎 , 𝑔 (𝑎 (𝑏), 𝑏) = max ∈[ , ] 𝑔 (𝑎, 𝑏) = 𝑏. the supervisor's punishment strategy consists in the compulsion of the agent to provide the viability condition established by the principal: 𝑎 (𝑏) = 𝑎 . dynamic model of general coordination and private interests 101 copyright ©2024 assa. adv. in systems science and appl. (2024) if the supervisor punishes the agent, then his payoff is equal to 𝑔 (𝑎 (𝑏), 𝑏) = 𝑢(𝑎 ) − 𝑏. the maximal agent's payoff in the case of his punishment is equal to: 𝐿 = max ∈[ , ) 𝑔 (𝑎 , 𝑏) = 𝑢(𝑎 ). in this case the agent changes a production strategy to provide an initial admittance range for the checked quality indicator. thus, there is no reason to pay a bribe, and the optimal agent's strategy is 𝐸 = {0}. let us introduce a set: 𝐷 = {(𝑎, 𝑏): 𝑢(𝑎) − 𝑏 > 𝑢(𝑎 )}. (3.2.6) a stepwise form of the bound of this set corresponds to the form of the function 𝑢(𝑎). the value 𝑏∗is a maximal bribe which the agent is agree to pay for 𝑎 = 𝑞 . in this case to pay a bribe is advantageous for the agent. then, we find 𝐾 = sup ( , )∈ 𝑔 (𝑎, 𝑏) = 𝑏∗. (3.2.7) (𝑥 , 𝑥 ): 𝑔 (𝑥 , 𝑥 ) ≥ 𝐾 − 𝜀. 𝑥 = 𝑞 , 𝑥 = 𝑏∗ − 𝜀. 𝐾 = max ∈[ , ] 𝑔 (𝑎, 0) = 0. we have 𝐾 > 𝐾 , therefore, the 𝜀-optimal supervisor's strategy is 𝑎 (𝑏) = 𝑞 , 𝑏 = 𝑏∗ − 𝜀, 𝑎 , 𝑏 < 𝑏∗ − 𝜀. (3.2.8) it remains to calculate𝑏∗. for this purpose, let us build the set𝐷 in the coordinates 𝑏, 𝑢(𝑎). itisseenthat 𝑏∗ = 𝑢(𝑞 ) − 𝑢(𝑎 ). (3.2.9) thus, the solution of the game is the pair 𝑎 (𝑏), 𝑢(𝑞 ) − 𝑢(𝑎 ) , and the respective values of payoff functions are equal to: 𝑔∗ = 𝑢(𝑞 ) − 𝑢(𝑎 ) − 𝜀, (3.2.10) 𝑔∗ = 𝑢(𝑎 ) + 𝜀. now consider the supervisor's optimal strategy with incomplete information: 𝑎 (𝑏) = 𝑞 , 𝑏 = 𝑢(𝑞 ) − 𝑢(𝑎 ) − 𝜀, 𝑎 , 𝑏 < 𝑢(𝑞 ) − 𝑢(𝑎 ) − 𝜀. (3.2.11) it has an essential shortage, namely, the supervisor must know the function 𝑢(𝑎). as it follows from (3.2.4), this function has a complex structure and depends on three other functions. meanwhile, a check-up gives to the supervisor a certain information only about an actual value of the quality indicator. however, the supervisor may not know a complete form of the function𝑢. to construct an optimal strategy, it is sufficient to have estimates for two points. suppose that the supervisor has a sufficient information to evaluate a distribution of the values 𝑢(𝑞 )and 𝑢(𝑎 ) (denote them by 𝑢 and𝑢 ). 𝑢 ∈ 𝑁(𝜇 , 𝜎 ), 𝑢 𝜖𝑁(𝜇 , 𝜎 ). in this case 𝑏∗ ∈ 𝑁 𝜇 − 𝜇 , 𝜎 + 𝜎 = 𝑁(𝜇, 𝜎). (3.2.12) the supervisor chooses a value𝑏 as an estimate for 𝑏∗and builds a function: 𝑎 (𝑏) = 𝑞 , 𝑏 ≥ 𝑏 , 𝑎 , 𝑏 < 𝑏 . (3.2.13) after that the supervisor reports her strategy to the agent, two variants are possible. 1. a point with coordinates (𝑞 , 𝑏 )belongs to the domain 𝐷. in this case the agent gives a bribe𝑏 , and the supervisor closes her eyes to the violation. respectively, the payoffs are equal to: 102 t.y. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2024 assa adv. in systems science and appl. (2024) 𝑔∗ = 𝑢(𝑞 ) − 𝑢(𝑎 ) − 𝑏 , 𝑔∗ = 𝑏 . 2. a point with coordinates (𝑞 , 𝑏 ) does not belong to the domain 𝐷. the agent does not pay a bribe and provide the condition of viability. the supervisor receives nothing: 𝑔∗ = 𝑢(𝑞 ) − 𝑢(𝑎 ), 𝑔∗ = 0. as the supervisor knows a distribution of𝑏∗ , it is possible to calculate the following probability: 𝑃(𝑏∗ > 𝑏 ) = 1 𝜎√2𝜋 𝑒 ( ) 𝑑𝑠 . (3.2.14) in this case we can modify the supervisor's payoff function as follows: 𝑓(𝑏 ) = 𝑏 , 𝑏 < 𝑏∗, 0, 𝑏 ≥ 𝑏∗, (3.2.15) 𝐸𝑓(𝑏 ) = √ ∫ 𝑒 ( ) 𝑑𝑠 → max. (3.2.16) the supervisor maximizes an expectation of her payoff. the problem (3.2.16) was solved numerically by means of the maple package for different parameters of distribution of 𝑏∗. the supervisor's payoff function and its derivative for 𝑏∗ ∈ 𝑁(50,15)are presented in fig. 3.2.2. fig. 3.2.2. an example of the graphs of the supervisor's payoff function and its derivative for normal distribution besides, it is possible to calculate numerically a value of 𝑏 that is optimal for the supervisor. in the considered example 𝑏∗ = 38.7. however, an expectation of the payoff is less than a required bribe: 𝐸𝑓(𝑏∗) = 29.97. using the log-normal distribution as a distribution rule with the same median and variance implied an increasing of the optimal for the supervisor values of bribe and payoff expectation (43.26 and 41.75 respectively) (fig. 3.2.3). dynamic model of general coordination and private interests 103 copyright ©2024 assa. adv. in systems science and appl. (2024) fig. 3.2.3. an example of the graphs of the supervisor's payoff function and its derivative for log-normal distribution in the considered two-level model of corruption the principal does not interfere in the supervisor's activity in any way. respectively, any corruption interactions remain unpunished. that's why an optimal strategy of the supervisor is a maximal violation of the viability conditions in exchange of the maximal possible bribe from the agent. it is natural that the principal is not satisfied and tries to fix the situation. the most evident way to do it is to charge penalties both on a bribe-giver (agent) and on a bribe-taker (supervisor). let us consider a modified model where the principal still participates implicitly. this model is based on the model (3.2.16) where the supervisor does not know an exact value of 𝑏∗. the supervisor's payoff function is defined by the formula (3.2.15) and is equal to the bribe or to zero if the agent does not pay a bribe. however, in the latter case the agent may report to the principal a fact of extortion. then the principal charges to the supervisor a penalty as a function of the required bribe and a new admittance bound. let us first consider a case when the penalty depends only on a bribe. in this case an admittance bound still does not impact the supervisor's payoff, a punishment strategy and a domain 𝐷 do not change, and therefore an optimal strategy remains the same. however, the value of 𝑏 is found with consideration of a penalty. the supervisor's payoff is equal to: 𝑓(𝑏 ) = 𝑏 , 𝑏 < 𝑏∗, −𝑥(𝑏 ), 𝑏 ≥ 𝑏∗. (3.2.17) the objective is to maximize a payoff expectation. this expectation does not depend on 𝑏 because it is predetermined by supervisor's strategy and a value of 𝑏∗: 𝐸𝑓(𝑏 ) = 𝑏𝑃(𝑏 ∗> 𝑏 ) − 𝑥(𝑏 ) 1 − 𝑃(𝑏 ∗> 𝑏 ) → max. (3.2.18) a penalty function 𝑥(𝑏 )is determined by the principal and is known to the supervisor. it is established officially and is known in advance. the simplest example is a constant function, i.e., a fixed penalty that does not depend on bribe. certainly, the penalty may be proportional to the bribe or be a piecewise continuous function. in the latter case the domain segment is subdivided on several intervals, and the values of penalty are different on the intervals. in general case the principal tries to choose such penalty function that the supervisor's payoff expectation is negative (for all𝑏 or for the values from a domain). in this case the corruption becomes non-advantageous for the supervisor. 104 t.y. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2024 assa adv. in systems science and appl. (2024) denote by 𝐹 ∗(𝑏 ) = 𝑃(𝑏∗ ≤ 𝑏 ) a distribution function of 𝑏∗ . then for given 𝑏 it is necessary for the principal that an inequality holds: 𝑏 1 − 𝐹 ∗(𝑏 ) − 𝑥(𝑏 )𝐹 ∗(𝑏 ) ≤ 0, 𝑜𝑟𝑥(𝑏 ) ≥ 𝑏 1 − 𝐹 ∗(𝑏 ) 𝐹 ∗(𝑏 ) . (3.2.19) the main shortage of the model is that the supervisor's payoff depends directly on her information about a real value of 𝑏∗ . therefore, if a variance of the respective random variable tends to zero, or the supervisor is able to evaluate quite accurate the agent's losses from correction the viability violations then the penalty increases. in the limit case when the supervisor knows exactly the value of 𝑏∗, even a maximal possible penalty is useless because a probability of bribe becomes equal to one. at last, let us consider a case when the penalty function depends on the admittance bound. to solve the problem in this setting it is necessary to use the following formulation: 𝐾 = sup ( , )∈ 𝑏𝑃(𝑏∗ > 𝑏) − 𝑥(𝑎, 𝑏) 1 − 𝑃(𝑏∗ > 𝑏) . (3.2.20) this solution is less trivial and depends directly on the form of the function 𝑥(𝑎, 𝑏). in this case the supervisor can decrease an admittance bound not up to the current value of the indicator but to an intermediate value. however, a general trend and the main shortage are the same: if the supervisor is well informed about𝑏∗ then no penalty will help to overcome corruption. 3.3. three-level system under corruption the respective model has the form 𝑔 (𝑎, 𝑏, 𝑐) = 𝑐 + 𝑧(𝑎) − 𝑙 (𝑎, 𝑏, 𝑐) → min, , , , 𝑔 (𝑎, 𝑏, 𝑐) = 𝑏 − 𝑙 (𝑎, 𝑏, 𝑐) → max, ( ) (3.3.1) 𝑔 (𝑎, 𝑏, 𝑐) = 𝑢(𝑎) − 𝑏 − 𝑙 (𝑎, 𝑏, 𝑐) → max. here 𝑐 isthe principal's anti-corruption cost; 𝑧(𝑎) – the principal's losses from the viability violation (a non-decreasing function); 𝑧(𝑎 ) = 0; 𝑙 (𝑎, 𝑏, 𝑐) – a penalty charged to the supervisor; 𝑙 (𝑎, 𝑏, 𝑐) – a penalty charged to the agent; 𝑙∗, 𝑙∗– functions that show which part of a penalty has an economic utility for the principal. in some cases, penalties may be non-material, for example, a prohibition to hold an office or even an imprisonment. such penalties punish the briber but have no economic utility for the principal. in the absence of corruption the penalty functions are equal to zero, as well as in the absence of the principal's cost: 𝑙 (𝑎 , 0, 𝑐) = 𝑙 (𝑎, 𝑏, 0) = 0. in general case, the principal strategies are𝑙 (𝑎, 𝑏, 𝑐),𝑙 (𝑎, 𝑏, 𝑐), 𝑙∗, 𝑙 ∗ and 𝑐. however, in real life the principal can control only the variable 𝑐, and we will stick to this interpretation. introduce the following penalty functions: 𝑙 (a, b, c) = 𝑝(a, c) ∙ 𝑠 (𝑎, 𝑏), 𝑖 ∈ {1,2}. (3.3.2) here 𝑝(𝑎, 𝑐) is a probability of detection a fact of corruption. the probability of detection increases with augmentation of anti-corruption cost and is decreasing of an admittance bound 𝑎. denote 𝑠 (𝑎, 𝑏) – a penalty, 𝑠 (𝑎 , 0) = 0, 𝑝(𝑎, 0) = 0; (3.3.3) 𝑔 (𝑎, 𝑏, 𝑐) = 𝑐 + 𝑧(𝑎) − 𝑙 (𝑎, 𝑏, 𝑐) − 𝑙 (𝑎, 𝑏, 𝑐) → min, 𝑔 (𝑎, 𝑏, 𝑐) = 𝑏 − 𝑝(𝑎, 𝑐)𝑠 (𝑏) → max, ( ) 𝑔 (𝑎, 𝑏, 𝑐) = 𝑢(𝑎) − 𝑏 − 𝑝(𝑎, 𝑐)𝑠 (𝑎, 𝑏) → max, (3.3.4) an algorithm of solution of the problem (3.3.4) is the following. dynamic model of general coordination and private interests 105 copyright ©2024 assa. adv. in systems science and appl. (2024) 1. given a fixed value of 𝑐 the supervisor and the agent find a solution of the inverse stackelberg game (germeier game г ) – a value of bribe and an admittance bound. this solution depends on the parameter с. 2. the principal chooses the value of that minimizes its payoff function. consider as an example a pure financial penalty. the penalty is equal to the upper bound of the respective intervals. then the payoff functions have the form: 𝑠 (𝑏) = 50𝑏, 𝑏 ∈ [0,25000] 60𝑏, 𝑏 ∈ [25000,150000] 90𝑏, 𝑏 ∈ [150000,1000000] 100𝑏, 𝑏 ∈ (1000000, +∞). (3.3.5) �̃� (𝑏) = 30𝑏, 𝑏 ∈ [0,25000] 40𝑏, 𝑏 ∈ [25000,150000] 80𝑏, 𝑏 ∈ [150000,1000000] 90𝑏, 𝑏 ∈ (1000000, +∞). (3.3.6) 𝑠 (𝑎, 𝑏) = �̃� (𝑏) − 𝑢(𝑎) + 𝑢(𝑎 ) (3.3.7) additional summands in the second function mean that in the case of detection the agent should provide an initial viability condition. these summands are not included in the principal's payoff function: 𝑙 ∗ (a, b) = 𝑝(a, c) ∙ 𝑠 (𝑏); (3.3.8) 𝑙 ∗ (a, b) = 𝑝(a, c) ∙ �̃� (𝑏). denote: 𝑠∗(𝑏) = 80𝑏, 𝑏 ∈ [0,25000] 100𝑏, 𝑏 ∈ [25000,150000] 170𝑏, 𝑏 ∈ [150000,1000000] 190𝑏, 𝑏 ∈ (1000000, +∞). (3.3.9) the problem takes the form: 𝑔 (𝑎, 𝑏, 𝑐) = 𝑐 + 𝑧(𝑎) + 𝑝(𝑎, 𝑐)𝑠 ∗ (𝑏) → min, (3.3.10) 𝑔 (𝑎, 𝑏, 𝑐) = 𝑏 − 𝑝(𝑎, 𝑐)𝑠 (𝑏) → max, ( ) 𝑔 (𝑎, 𝑏, 𝑐) = 𝑢(𝑎) − 𝑏 − 𝑝(𝑎, 𝑐)𝑠 (𝑎, 𝑏) → max. let us solve the problem (3.3.10) according to the proposed algorithm. assume that 𝑝(𝑎, 𝑐) = 𝑝(𝑐) because this simplifies the solution of the game г essentially. as before, the punishment strategy is: 𝑎𝐻(𝑏) = 𝑎 . the difference from the model (3.2.5) begins from the definition of the domain 𝐷 . 𝐷 = {(𝑎, 𝑏): 𝑔 (𝑎, 𝑏, 𝑐) > 𝑢(𝑎 )}. (3.3.11) 𝑢(𝑎) − 𝑏 − 𝑝(c)𝑠 (𝑎, 𝑏) − 𝑢(𝑎 ) > 0 ⇔ 1 − 𝑝(c) 𝑢(𝑎) − 𝑢(𝑎 ) − 𝑏 − 𝑝(c)�̃� (𝑏) > 0 (3.3.12) the domain that corresponds to the inequality (3.3.12) is similar to the domain 𝐷 from (3.3.6). let us find a maximal value of 𝑏 that belongs to the domain: 𝑏∗ = min 1 − 𝑝(c) 𝑢(𝑎) − 𝑢(𝑎 ) 1 + 𝑝(c)𝑘 ; 𝑒 . (3.3.13) here 𝑘 is a coefficient at 𝑏 in the 𝑖-th row of the formula (3.3.13), 𝑒 – the right bound of the interval in the 𝑖-th row. 𝑏∗ = max ∈{ , , , } (𝑏∗) . (3.3.14) it is evident that 𝑏∗ > 0 because each of 𝑏∗ is positive. in fact, the supervisor solves the following problem: 𝑔 (𝑏, 𝑐) = 𝑏 − 𝑝(𝑐)𝑠 (𝑏) → 𝑚𝑎𝑥,   0 ≤ 𝑏 ≤ 𝑏 ∗ −𝜀, (3.3.15) 106 t.y. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2024 assa adv. in systems science and appl. (2024) 𝑏 = argmax ∗ 𝑏 − 𝑝(c)𝑠 (𝑏) . (3.3.16) as 𝑠 (0) = 0 then 𝑔 ≥ 0, and the equality holds when the bribe is equal to zero. suppose that in this case the supervisor sets an admittance bound equal to 𝑎 because she has no motivation to change the viability condition. thus, an 𝜀-optimal supervisor's strategy is the function: 𝑎 (𝑏) = 𝑞 , 𝑏 = 𝑏 , 𝑏 ≠ 0, 𝑎 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (3.3.17) the only agent's best response to this function is: 𝑏 = 𝑏 . go to the second step of the algorithm. find such a value of 𝑐 that minimizes the principal's payoff function: 𝑔 (𝑎, 𝑏, 𝑐) = 𝑐 + 𝑧(𝑎) − 𝑠 ∗ (𝑏) → min. (3.3.18) optimal for the game "supervisor-agent" values 𝑎and 𝑏 are expressed by 𝑐: 𝑔 (𝑎(𝑐), 𝑏(𝑐), 𝑐) = 𝑐 + 𝑧(𝑞 ) − 𝑝(c)s ∗ (𝑏 (𝑐)), 𝑏 (𝑐) > 0, 𝑐, 𝑏 (𝑐) = 0. (3.3.19) notice that 𝑏 (0) > 0, i.e. 𝑔 (0) = 𝑧(𝑞 ). then, 𝑝(𝑐)is a non-decreasing function. suppose that it has a non-decreasing inverse function 𝑐(𝑝)in the sense that 𝑐(𝑝) = min{𝑐: 𝑝(c) > 𝑝}. (3.3.20) as it is seen from (3.3.13)–(3.3.14), when с increases the value of 𝑏∗decreases. the value of 𝑏 also decreases because, first, a penalty increases and, second, an interval of taking the maximum decreases. it is simple to check that 𝑏 = 0 <=> 𝑏 − 𝑝(c)𝑠 (𝑏) ≤ 0, ∀b > 0 <=> 𝑝(c) ≥ ( ) , ∀b > 0. in turn, the last inequality holds when 𝑝(𝑐) ≥ 𝑏 𝑏 ∙ min ∈{ , , , } 𝑘 = 1 50 . (3.3.21) here𝑘 is a coefficient at 𝑏 in the 𝑖-th row of the supervisor's penalty function. for big values of 𝑐 the supervisor will also refuse from taking a bribe. however, the principal's payoff function will increase. therefore, if 𝑐∗ is the principal's optimal strategy then 𝑐∗ ∈ 0, 𝑐 1 50 . (3.3.22) if a struggle with corruption is the main objective of the principal then 𝑐∗ = 𝑐 is its optimal strategy. it remains to answer the following question: may it be advantageous to the principal to support a certain level of corruption that minimizes its cost? otherwise, is there an internal point of maximum on the segment (3.3.22) that does not coincide with its right bound? in the point с(1/50) the principal's payoff function has a break. so, let us first estimate 𝑔 (𝑎(𝑐), 𝑏(𝑐), 𝑐) when с → с(1/50) − 0 . then 𝑎(𝑐) = 𝑞 . denote for simplicity 𝑈 = 𝑢(𝑎) − 𝑢(𝑞 ). 𝑏 = argmax 0 ≤ 𝑏 < 𝑏∗ − 𝜀 (b − 𝑝(c)𝑠 (𝑏) ) = 𝑚𝑖𝑛{𝑏∗ − 𝜀; 𝑒 }; 𝑏∗ = max ∈{ , , , } min 1 − 𝑝(𝑐) 𝑢(𝑞 ) − 𝑢(𝑎 ) 1 + 𝑝(𝑐)𝑘 ; 𝑒 = dynamic model of general coordination and private interests 107 copyright ©2024 assa. adv. in systems science and appl. (2024) = 𝑚𝑎𝑥 ∈{ , , , } 𝑚𝑖𝑛 𝑈 1 + 𝑘 ; 𝑒 = = 𝑚𝑎𝑥 𝑚𝑖𝑛 𝑒 , 49 80 𝑈 , 𝑚𝑖𝑛 𝑒 , 49 90 𝑈 , 𝑚𝑖𝑛 𝑒 , 49 130 𝑈 , 49 140 𝑈 . 𝑏 =𝑚𝑖𝑛{𝑏∗ − 𝜀; 𝑒 }= 𝑚𝑖𝑛 𝑒 , 49 80 𝑈 − 𝜀. the principal's payoff is equal to 𝑔 (𝑎, 𝑏, 𝑐) = c − 0 + z(𝑞 ) − ∙ 𝑚𝑖𝑛 𝑒 , 𝑈 − 𝜀 . as 𝑐 is non-decreasing function then in the case when z(𝑞 ) < 80 50 ∙ 𝑚𝑖𝑛 𝑒 , 49 80 𝑈 , it is advantageous to the principal to support a non-zero level of corruption. if 𝑧 > 40000 then simple bribes are not advantageous to the principal. let us consider the case 𝑝(𝑐) ∈ 0, 1 100 , 𝑏 = argmax 0 ≤ 𝑏 < 𝑏∗ − 𝜀 (b − 𝑝(c)𝑠 (𝑏) = 𝑏∗ − 𝜀. 𝑏∗ = max ∈{ , , , } min 1 − 𝑝(𝑐) 𝑢(𝑞 ) − 𝑢(𝑎 ) 1 + 𝑝(𝑐)𝑘 ; 𝑒 = = 𝑚𝑎𝑥 ∈{ , , , } 𝑚𝑖𝑛 𝑈 1 + 𝑘 ; 𝑒 = = 𝑚𝑎𝑥 𝑚𝑖𝑛 𝑒 , 99 130 𝑈 , 𝑚𝑖𝑛 𝑒 , 99 140 𝑈 , 𝑚𝑖𝑛 𝑒 , 99 180 𝑈 , 99 190 𝑈 . though the calculations are not finished it is already clear that 99 190 𝑈 > 𝑒 <=> 𝑈 > 1.919.192 => 𝑏 = 99 190 𝑈. 𝑔 (𝑎, 𝑏, 𝑐) ≤ c 1 100 + z(𝑞 ) − 99 100 ∙ 𝑈. thus, if 𝑈 > 1919192 and 𝑧(𝑞 ) < (99/100)𝑈 then for the principal it is more advantageous to support a maximally possible level of corruption than to get rid of it completely. more accurate evaluations of the parameter 𝑐 require a specific information about 𝑧 and 𝑈, and functions с(𝑝)or 𝑝(с). an additional penalty in the case of detection encourages corruption. for example, if a function 𝑈contains compensation of the principal's losses equal to 𝑧(𝑞 ) then a probability that the principal will not fight with corruption essentially increases. besides, only very large corporations with immense profits can allow themselves such big bribes. therefore, one of the ways of fighting with a big corruption is a small business development. 4. dynamic models let us consider dynamic modifications of the proposed models. 108 t.y. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2024 assa adv. in systems science and appl. (2024) agent's production function 𝑝(𝑞) reflects his income in dependence on the quality indicator 𝑞. a part of the income 𝐼 is invested to the quality support, and the other part is his profit. the agent must provide a viability condition 𝑞 ≥ 𝑎 that defines quality requirements. however, it is a supervisor who monitors the viability condition and can weaken it subject to the function 𝑎(𝑏)in exchange to a bribe 𝑏 from the agent. this process is implemented during a finite period of time 𝑇 with a discrete step 𝑡. a dynamical problem of control has the following form: 𝐺 = 𝑔 = 𝑏 → max, (4.1) 𝑎(𝑏) ∈ 𝐴; 𝐴 = {𝑎: 𝑎(0) = 𝑎 ; 0 ≤ 𝑎(𝑏) ≤ 𝑎 }; (4.2) 𝐺 = 𝑔 = (1 − 𝑏 − 𝐼 )𝑝(𝑞 ) → max, (4.3) 𝑏 ≥ 0; 𝐼 ≥ 0; 𝑏 + 𝐼 ≤ 1; (4.4) 𝑞 ≥ 𝑎(𝑏 ); (4.5) 𝑞 = 𝐼 𝑝(𝑞 ) − 𝜇𝑞 , 𝑞 = 𝑞 ; 𝑡 = 0,1,2, … , 𝑇. (4.6) here𝐺 , 𝐺 are total payoffs of the supervisor and the agent respectively for the period 𝑇; 𝑔 , 𝑔 – their current payoffs on the step 𝑡; 𝑝(𝑞 ) – the agent's production function on the step𝑡; 𝑏 –a share of bribe ("kickback") in the agent's payoff on the step 𝑡; 𝐼 – a share of investments to the quality support in the agent's payoff on the step 𝑡; 𝑎(𝑏 ) – the supervisor's corruption function on the step 𝑡 ; 𝑎 – an initial threshold value of the quality indicator providing the viability; 𝜇 –a coefficient of quality decreasing in the absence of investments; 𝑞 – an initial value of the quality indicator. a difference game (4.1)–(4.6) with phase constraints (4.5) has the following information structure. at each step of the discrete time 𝑡 = 1,2, … , 𝑇: (1) given the function 𝑎(𝑏) the agent chooses the control values 𝑏 , 𝐼 ; (2) the values of 𝑎(𝑏 ), 𝑝(𝑞 ), 𝑞 are calculated; (3) if 𝑞 ≥ 𝑎(𝑏 )then 𝑡: = 𝑡 + 1 else go to step (1). for identification of the model (4.1)–(4.6) we use the following hypotheses. the supervisor's corruption function describes capture [61] that is reflected by the conditions (4.2). the function 𝑎(𝑏) decreases monotonously on the segment [0,1] . in the absence of corruption(𝑏 = 0)its value coincides with an initial threshold value 𝑎 , and in case of the maximal kickback(𝑏 = 1)the supervisor does not check the viability condition at all. the function 𝑝(𝑞) has standard properties of a production function: it increases monotonously in its domain, it is equal to zero if the quality is equal to zero, it has a positive derivative and a non-positive second derivative. so, it is natural to use a power function. notice that in the case of corruption the value of quality may become less than a threshold value 𝑎 that defines viability, and the value of income decreases. however, to attain a value 𝑎(𝑏) < 𝑎 of the quality indicator 𝑎 less share of investments 𝐼 may be sufficient. thus, in dependence on the correlation of values of the model parameters and the form of functions 𝑎(𝑏), 𝑝(𝑞)the agent should decide whether to give a bribe and how big it should be. respectively, a model analysis allows to expose the conditions that make corruption nonadvantageous. now add to the model the principal that can charge penalties on the supervisor and on the agent if their corruption is detected. the quantities of respective penalties 𝑠 (𝑎), 𝑠 (𝑎)are functions of the threshold value of quality𝑎determined subject to corruption 𝑎(𝑏) . the functions are supposed to be given. however, to catch a briber the principal should exercise some efforts. the quantity of these efforts𝑧is the principal's control variable that serves as an argument of two functions: probability of corruption detection 𝑃(𝑧) and the respective cost 𝐶(𝑧) . in the case of corruption (𝑎(𝑏) < 𝑎 ) the principal is charged a conditional dynamic model of general coordination and private interests 109 copyright ©2024 assa. adv. in systems science and appl. (2024) penalty (an estimate of its losses) proportional to the deviation 𝑎 − 𝑎 with a large coefficient 𝑀. this dynamical control problem has the following form: 𝐺 = 𝑔 = [𝑠 (𝑎) + 𝑠 (𝑎)]𝑃(𝑧) − 𝐶(𝑧) − 𝑀(𝑎 − 𝑎) → max, (4.7) 𝑧 ≥ 0; (4.8) 𝐺 = 𝑔 = 𝑏 − 𝑠 (𝑎)𝑃(𝑧) → max, (4.9) 𝑎(𝑏) ∈ 𝐴; 𝐴 = {𝑎: 𝑎(0) = 𝑎 ; 0 ≤ 𝑎(𝑏) ≤ 𝑎 }; (4.10) 𝐺 = 𝑔 = [(1 − 𝑏 − 𝐼 )𝑝(𝑞 ) − 𝑠 (𝑎)𝑃(𝑧)] → max, (4.11) 𝑏 ≥ 0; 𝐼 ≥ 0; 𝑏 + 𝐼 ≤ 1; (4.12) 𝑞 ≥ 𝑎(𝑏 ); (4.13) 𝑞 = 𝐼 𝑝(𝑞 ) − 𝜇𝑞 , 𝑞 = 𝑞 ; 𝑡 = 0,1,2, … , 𝑇. (4.14) in comparison with the model (4.1)–(4.6) we added the following denotations: 𝐺 – a total payoff of the principal for the period 𝑇; 𝑔 – its current payoff at the step 𝑡; 𝑠 (𝑎), 𝑠 (𝑎) – penalty functions; 𝑧 – a quantity of the principal efforts in detecting corruption; 𝐶(𝑧) – a function of the principal's respective cost; 𝑃(𝑧) – a probability of detection; 𝑀 – a conditional penalty coefficient. the information structure of the difference game (4.7)–(4.14) with phase constraints (4.13) remains the same, and the supervisor and the agent know the payoff functions𝑠 (𝑎), 𝑠 (𝑎)and a quantity of the principal efforts in detecting corruption 𝑧. a function of cost of the corruption detection 𝐶(𝑧)is increasing and convex, 𝐶(0) = 0. a function of the corruption detection 𝑃(𝑧) is increasing and concave, 𝑃(0) = 0 . penalty functions of the supervisor and the agent 𝑠 (𝑎), 𝑠 (𝑎), as in the case of the principal, are linearly proportional to the deviation 𝑎 − 𝑎. an analytical investigation of the dynamical control problems (4.1)–(4.6), (4.7)–(4.14) seems very complicated. thus, we used a method of qualitatively representative scenarios in simulation modeling [51]. the results are similar to the static case. 5. conclusion mathematical modeling of the social-economic systems is a very complicated and ambitious task. it is absolutely evident in the case of organizational corruption where a problem of model identification is especially difficult. however, there is a big stream of literature on mathematical modeling of corruption. even simplified mathematical models increase our understanding of this phenomenon and allow for some practical recommendations on its control. in this paper we build and investigated game theoretic models of quality management under corruption. a quality management is characterized by viability conditions. we considered static and dynamic inverse stackelberg games in two-level (supervisor-agent) and three-level (principal-supervisor-agent) organizations. some results are received analytically using germeier theorem, and other are obtained by numerical calculations on the base of simulation modeling. in some cases, we revealed conditions under which the corruption is not advantageous for the players. in the considered models the principal does not participate in a corruption activity. moreover, it has anti-corruption cost and charges penalties to the bribers, and ideally these penalties compensate a damage from violation of the viability conditions. however, for 110 t.y. anopchenko, o.i. gorbaneva, e.i. lazareva, a.d. murzin, g.a. ougolnitsky copyright ©2024 assa adv. in systems science and appl. 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(2005). corruption, political competition and environmental policy, j. of env. econ. and management, 49, 516–535. adv syst sci appl 2024; 01:20–28 published online at https://ijassa.ipu.ru. methods of estimation and processing of aerospace information in image contouring problem andrey a. galyaev*, alexander v. dobrovidov, pavel v. lysenko, leonid m. berlin, maria a. kolygo v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia abstract: the article proposes approaches to the solution of the segmentation problem of real satellite images based on random markov vector fields and new statistical textural characteristics for gray-level co-occurrence matrix. the authors introduce a novel image textural characteristic called statistical complexity that successfully distinguishes deterministic structures on naturally or artificially noised images. software modules have been developed and satellite image examples are processed in order to determine the contours of the land structures and the presence of static and dynamic objects on the ocean surface. a comparison of two segmentation methods on real images has been made, which showed the prospects of their use in automatic image processing. keywords: earth’s surface monitoring, image segmentation and contouring problem, aerospace information processing 1. introduction aircraft and space monitoring of the earth’s surface using optical and radio devices is the main data source for building electronic maps for various purposes. in the problem of constructing electronic terrain maps based on data obtained by aerospace carriers, the task of constructing contours of objects of natural and artificial nature often arises [1, 16, 19, 22]. similar texture analysis tasks emerge for oil spill detection on the ocean surface [23], as well as for classifying atmospheric phenomena and cloud types [5]. another problem is associated with type determination of natural and artificial waves on the ocean surface [9,17], as well as finding archaeological or sunken objects [12]. some segmentation problems include the shoreline of lakes and seas, the border of forests and ice fields, etc. others include the boundaries of buildings, roads, transmission lines, etc. [21] at the same time, one of the main tasks of creating electronic maps is the task of outlining objects for their further vectorization and compressed representation. the totality of measurements of various physical quantities on the sea surface, deep flat sections and flat sections in the atmosphere generate measurement fields, the structure of which varies depending on various processes occurring both under the sea surface and above the surface [6, 14]. such digitized data fields are the source material for making decisions about the presence of various anomalies [20] in the observed field implementations and highlighting their shape. however, direct handling of physical models is extremely inconvenient, since they are described, as a rule, by complex nonlinear equations [24]. the procedure of contouring is often reduced to operator delineating with an electronic pencil along the visible boundaries of objects. it is proposed to replace this heavy unproductive work ∗corresponding author: galaev@ipu.ru processing of aerospace information in image contouring problem 21 with an automatic contouring procedure, which will delineate most of the visible objects in the image. a small part of the remaining ”difficult” objects can be handled by operator. to solve the problem of image segmentation, the article suggests two approaches: the one based on the use of a spatial interaction model (spm) and the second based on new textural characteristic called statistical complexity. such models can describe some piecewise homogeneous spatial structures, where homogeneous areas are called textures. the main feature of the spm is that they allow taking into account the intra-textural dependence of observations. this circumstance turns out to be decisive, since, for example, experience with radar images has shown that it is statistical relationships that are the characteristic features by which it is possible to distinguish objects in noisy images. the usage of statistical complexity provides an advantage compared to other image textural characteristics for the detection problem of the deterministic signal component [2], [3]. the application of information models developed in this research and their modification to the characteristics of real images make it possible to obtain stable boundaries of objects and reliably distinguish their contours. the current article describes the results of segmentation algorithms on noisy images based on markov’s field and novel statistical feature called statistical complexity. 2. information model of spatial interaction to develop an automatic contouring procedure, an effective information approach is proposed, in which a data field is considered as an implementation of a random field, where each point of the field is statistically related to the surrounding points according to statistical laws. a set of field points with the same (or similar statistical characteristics) form a field segment. therefore, the task of dividing the implementation of the observed field (hereinafter the image) into areas with different statistical characteristics is called the image segmentation task. 2.1. markov model approach the basis of the spm is a set of conditional distributions of the values of the observed field, defined at each point of the image at the specified values of the field at neighboring points p {xij|xkl, (k, l) ∈ sij}, (2.1) where sij is the set of neighboring points around a point (i, j) called a pattern, i.e. sij = {(k, l) : (k, l) ̸= (i, j)}. another way to define the field model is given by the coupling equation xij = φ(sx ij, ξij), (2.2) where φ is a given borel function, {ξij} is a field of mutually independent random variables, and sx ij = {xkl : (k, l) ∈ sij} is a set of observations whose indices belong to the template. when constructing algorithms for statistical image analysis, linear auto-regressive models of two-dimensional fields are most often used xij = µ+ ∑ (k,l)∈sij akl(xi−k,j−l − µ) + bξij, (2.3) where µ is a mean value, akl and b are some constant model values. depending on the type of template, there are causal, semi-causal and non-causal models. it turns out that only causal models allow us to obtain convenient recurrent image segmentation procedures consistent with a certain (for example, scanning) method of obtaining image data. it should be noted that models (2.2) and (2.3) are also used to generate random fields with copyright © 2024 assa. adv syst sci appl (2024) 22 a.a. galyaev, a.v. dobrovidov, p.v. lysenko, l.m. berlin, m.a. kolygo specified statistical properties in model experiments, and in this case only the causal model allows generating the next field value from those already calculated up to this point. another class of commonly used models includes markov models that generate fields with a finite number of states θij . for these models , textures are described directly by a set of conditional probabilities p {θij|θkl, (k, l) ∈ sij}. the mottled structure of the image (for example, a sea surface) containing anomalous areas, can be described as a composition of various textures generated by the spm. the approach to texture composition proposed here allows us to achieve the following important results: 1. take into account the relationship of all observations in the image field; 2. make the border between homogeneous textures blurred, which directly corresponds to the actual data obtained; 3. get a spotty structure of arbitrary random shape; 4. to build an informational mathematical model, which is a constructive basis for the synthesis of recurrent image segmentation procedures. a texture composition with the specified properties is constructed as follows. first, a markov field {θij} is generated based on a given system of conditional probabilities. the composition of textures is obtained using the equation xij = µ(θij) + ∑ (k,l)∈sij akl(θij)(xi−k,j−l − µ(θij)) + b(θij)ξij, (2.4) in which the coefficients are controlled by a markov field {θij}. now this image is the source for its segmentation into areas with the same statistical properties. model (2.4) allows one to use optimal recurrent algorithms developed by the authors for segmentation, while making a minimal error. however, to do this, it is necessary to use all the statistical characteristics of both the markov field and the coefficients of equation (2.4). however, in practice, it is impossible to obtain the characteristics of an unobservable markov field. therefore, a modified cumulative sum algorithm is proposed here for segmentation, using various sample statistics obtained directly from the noise image of the type. therefore, the linear structure of the solution is caused by a row-by-row method of calculating statistics. there are several ways to get rid of this. in this paper, a nonlinear smoothing filter with windows of various sizes is proposed. in addition to the likelihood ratio statistics, one can use simpler statistics in the form of a square of the second difference: ∆ij = (xij + xi−1j−1 − xi−1j − xij−1). (2.5) studies have shown that such statistics are sensitive to changes in texture structure, but not sensitive to changes in mean and variance. therefore, it is good to apply it to those images in which the average does not change, and the correlation properties change from one segment to another. at the same time, the advantage of statistics is that it does not require the evaluation of any parameters. such nonparametric statistics often turn out to be the most suitable for real situations. to segment real images using likelihood ratio statistics, it is necessary to be able to calculate sample correlation characteristics obtained from homogeneous sections of this image. a white rectangle describing it is built around this contour, defining the minimum and maximum coordinates. an example of such a contour is shown in fig. 2.1. according to the data inside this contour, the program automatically calculates the sample mean, variance and correlation characteristics according to a given algorithm. for clarity, the program plots correlation functions, allowing the operator to assess by eye the degree of difference in characteristics in different areas of the image. copyright © 2024 assa. adv syst sci appl (2024) processing of aerospace information in image contouring problem 23 fig. 2.1. real image. correlation characteristics using the yule-walker equations are recalculated into the coefficients of equation (2.3), and the latter, in turn, are substituted into the likelihood ratio statistics. the result of image segmentation is shown in fig. 2.2. of particular interest is the fig. 2.2. segmentation result. possibility of applying the developed algorithms and programs to radar images of the sea surface. the sensitivity of the algorithms to small changes in the surface structure allows detecting and detecting the form of anomalies on the windy surface caused by various natural and man-made sources [9, 17, 24]. 2.2. statistical and information approaches there is an alternative approach for image texture analysis which is based on statistical characteristics of the image. this approach uses different so called textural features based on gray-level co-occurrence matrix. gray-level co-occurrence matrix (glcm) method have been first described by julesz [13], where gray level co-occurrence matrix as the frequency of the appearance of individual pairs of values in a specific image fragment was introduced. the calculations of the element of normalized symmetrical glcm p(i, j) are based on the analysis of quantized levels of brightness i p(i, j) = |{(n,m), (k, l) : i(n,m) = i, i(k, l) = j}|, (2.6) copyright © 2024 assa. adv syst sci appl (2024) 24 a.a. galyaev, a.v. dobrovidov, p.v. lysenko, l.m. berlin, m.a. kolygo where i(n,m) are the values of the brightness matrix elements with coordinates (n,m), i, j are the quantized levels of brightness (i, j = 1, . . . , n ), p(i, j) is the frequency value of the pair having index (i, j). according this matrix statistical features concerning certain texture might be calculated. most of these functions were suggested by haralick et al. [10]. table 2.1 contains known popular textural features [4,15] as well as two new ones, which are introduced in this article: so called statistical complexities csq and ctv [2, 3, 18]. no. textural feature formula 1 energy ∑ i,j p(i, j)2 2 contrast ∑ i,j p(i, j)(i− j)2 3 correlation ∑ i,j p(i, j) (i− µ)(j − µ) σ2 4 inertia ∑ i,j p(i, j)(i, j)2 5 entropy − ∑ i,j p(i, j) log2 p(i, j) (0 if p(i, j) = 0) 6 complexity csq ( − ∑ i,j p(i, j) log2 p(i, j) )(∑ i,j (p(i, j)− pw(i, j)) 2 ) 7 complexity ctv ( − ∑ i,j p(i, j) log2 p(i, j) )(∑ i,j |p(i, j)− pw(i, j)| )2 table 2.1. most common features used in textural image analysis. explanations for table 2.1: pw(i, j) – the normalized glcm for the brightness levels matrix i distributed according to the normal distribution for white noise simulation; µ is the mean glcm (which is an estimate of the intensity of all pixels in the relationship that contributed to the glcm), calculated as: µ = ∑ i,j ip(i, j); σ2 is the variance of the intensities of all reference pixels in the relationship contributing to the glcm is calculated as: σ2 = ∑ i,j p(i, j)(i− µ)2. 3. modelling and comparison of constructed methods two methods described in article are compared for different satellite image examples in this section. fast python implementation of glcm has been used from [11] and modified for complexities functions’ computation (items 6 and 7 in table 2.1). markov field algorithm has been implemented by authors. 3.1. satellite images contouring the first example is devoted solely to image contouring problem itself. two satellite images are used. both are taken from esa ”earth from space image collection” gallery: the first one copyright © 2024 assa. adv syst sci appl (2024) processing of aerospace information in image contouring problem 25 is esa sentinel-2 picture of faroe islands [7]; the second one is esa sentinel-1 picture of lofoten archipelago in norway [8]. these pictures have been chosen due to their extensive map ruggedness and contour line complexity as a perfect material for contouring algorithms validation. two methods of textural image analysis are applied to each picture, which is illustrated in figures 3.3 and 3.4. it can be seen in figure 3.3 that image contouring problem is solved successfully with both methods in the first example. 0 200 400 600 800 1000 0 200 400 600 800 1000 1200 (a) original picture. 0 200 400 600 800 1000 0 200 400 600 800 1000 1200 (b) processed with markov field algorithm picture. 0 200 400 600 800 1000 0 200 400 600 800 1000 1200 (c) processed with glcm complexity algorithm picture. fig. 3.3. the image of faroe islands by copernicus sentinel-2 mission. 0 500 1000 1500 2000 0 500 1000 1500 2000 2500 3000 (a) original picture. 0 500 1000 1500 2000 0 500 1000 1500 2000 2500 3000 (b) processed with markov field algorithm picture. 0 500 1000 1500 2000 0 500 1000 1500 2000 2500 3000 (c) processed with glcm complexity algorithm picture. fig. 3.4. the image of the archipelago of lofoten by copernicus sentinel-1 mission. copyright © 2024 assa. adv syst sci appl (2024) 26 a.a. galyaev, a.v. dobrovidov, p.v. lysenko, l.m. berlin, m.a. kolygo figure 3.4 contains a much more difficult image due to windy sea surface. it can be seen that glcm complexity approach performs better than markov field method, denoising slightly more of clouds and sea surface. 3.2. ship detection second example studies a more practical problem of object detection on satellite images. the first and second picture is taken from [17] for research purposes. figure 3.5 demonstrates the image of many anchored static ships. as one can see, both algorithms perform satisfactorily in this example, although markov field algorithm loses a little more pixels of ships. 0 200 400 600 800 1000 0 200 400 600 800 1000 (a) original picture. 0 200 400 600 800 1000 0 200 400 600 800 1000 (b) processed with markov field algorithm picture. 0 200 400 600 800 1000 0 200 400 600 800 1000 (c) processed with glcm complexity algorithm picture. fig. 3.5. anchored cargo ships from [17]. figure 3.6 explores a more challenging case of moving ship. in this situation there can be also a problem of ship wake detection as studied in [24]. as one can see from figures 3.6b and 3.6c, both algorithms successfully detect a moving object but second one allows also to detect a wake of the ship, which can be useful in different applications. 0 200 400 600 800 1000 0 200 400 600 800 1000 (a) original picture. 0 200 400 600 800 1000 0 200 400 600 800 1000 (b) processed with markov field algorithm picture. 0 200 400 600 800 1000 0 200 400 600 800 1000 (c) processed with glcm complexity algorithm picture. fig. 3.6. the internal waves generated by a cargo ship in the pearl river estuary in china from [17]. copyright © 2024 assa. adv syst sci appl (2024) processing of aerospace information in image contouring problem 27 4. conclusion the article proposes a solution to the problem of segmentation of real satellite images based on random markov vector fields and on the basis of new textural features. the authors propose to use two types of statistical complexity that distinguish deterministic structures well on naturally or artificially noised images. on the example of several satellite images the two presented methods are compared and conclusions are drawn about their applicability depending on the imaging conditions and complexity of the depicted scene as well as the prospects of their use in automatic image processing are discussed. future work can be devoted to implementation of classification algorithms based on textural features introduced above. acknowledgements the work was supported by the russian science foundation under grant no 23-19-00134. references 1. benecki, p., kawulok, m., kostrzewa, d. & skonieczny, l. (2018) evaluating superresolution reconstruction of satellite images, acta astronautica, 153, 15–25. 2. berlin, l., galyaev, a. & lysenko, p. (2023) comparison of information criteria for detection of useful signals in noisy environments, sensors, 23(4). 3. berlin, l., galyaev, a. & lysenko, p. (2023) statistical complexity as a criterion for the useful signal detection problem, automation and remote control, 84(7). 4. bharati, m. h., liu, j. j. & macgregor, j. f. (2004) image texture analysis: methods and comparisons, chemometrics and intelligent laboratory systems, 72(1), 57–71. 5. christodoulou, c. i., michaelides, s. c. & pattichis, c. s. (2003) multifeature texture analysis for the classification of clouds in satellite imagery, ieee transactions on geoscience and remote sensing, 41(11), 2662–2668. 6. ermakov, s. a., lavrova, o. yu., kapustin, i. a., ermoshkin, a. v., molkov, a. a. & danilicheva, o. a. (2018) o grebenchatoi strukture granits slikov na morskoi poverhnosti [on the “comb” structure of the edges of slicks on the sea surface], sovremennye problemy distantsionnogo zondirovaniya zemli iz kosmosa, 15(7), 208– 217, [in russian]. 7. esa (2018). faroe islands, modified copernicus sentinel data, [online]. available: https://www.esa.int/esa multimedia/images/2020/01/faroe islands. 8. esa (2020). lofoten, norway, modified copernicus sentinel data, [online]. available: https://www.esa.int/esa multimedia/images/2022/03/lofoten norway 9. graziano, m. d. (2020) preliminary results of ship detection technique by wake pattern recognition in sar images, remote sensing, 12(18), 2869. 10. haralick, r. m. (1979) statistical and structural approaches to texture, proceedings of the ieee, 67(5), 786–804. 11. izumi, t. (2018). fast gray-level co-occurrence matrix by numpy, [online]. available: https://github.com/tzm030329/glcm. 12. jahjah, m. & ulivieri, c. (2010) automatic archaeological feature extraction from satellite vhr images, acta astronautica, 66(9), 1302–1310. 13. julesz, b. (1962) visual pattern discrimination, ire transactions on information theory, 8(2), 84–92. 14. keeler, r. n., bondur, v. g. & gibson c. h. (2005) optical satellite imagery detection of internal wave effects from a submerged turbulent outfall in the stratified ocean: optical satellite imagery detection, geophysical research letters, 32(12). copyright © 2024 assa. adv syst sci appl (2024) https://www.esa.int/esa_multimedia/images/2020/01/faroe_islands https://www.esa.int/esa_multimedia/images/2022/03/lofoten_norway https://github.com/tzm030329/glcm 28 a.a. galyaev, a.v. dobrovidov, p.v. lysenko, l.m. berlin, m.a. kolygo 15. kiema, j. b. k. (2002) texture analysis and data fusion in the extraction of topographic objects from satellite imagery, international journal of remote sensing, 23(4), 767– 776. 16. kupidura, p. (2019) the comparison of different methods of texture analysis for their efficacy for land use classification in satellite imagery. remote sensing, 11(10), 1233. 17. liu, y. & deng, r. (2018) ship wakes in optical images, journal of atmospheric and oceanic technology, 35(8), 1633–1648. 18. lópez-ruiz, r., mancini, h. l. & calbet, x. (1995) a statistical measure of complexity, physics letters a, 209(5), 321–326. 19. napolano, g., vela, c., nocerino, a., opromolla, r. & grassi, m. (2023) a multi-sensor optical relative navigation system for small satellite servicing, acta astronautica, 207, 167–192. 20. pan, x., li, n., yang, l., huang, z., chen, j., wu, z. & zheng, g. (2023) anomalybased ship detection using sp feature-space learning with false-alarm control in sea-surface sar images, remote sensing, 15(13), 3258. 21. pesaresi, m. (2000) texture analysis for urban pattern recognition using fine-resolution panchromatic satellite imagery, geographical and environmental modelling, 4(1), 43–63. 22. sun, c. & wen, z. (2021) research progress in remote sensing image texture analysis, journal of physics: conference series, 1961(1), 012028. 23. sun, z., sun, s., zhao, j., ai, b. & yang, q. (2022) detection of massive oil spills in sun glint optical imagery through super-pixel segmentation, journal of marine science and engineering, 10(11), 1630. 24. wang, j., guo, l., wei, y. & chai, s. (2023) study on ship kelvin wake detection in numerically simulated sar images, remote sensing, 15(4), 1089. copyright © 2024 assa. adv syst sci appl (2024) introduction information model of spatial interaction markov model approach statistical and information approaches modelling and comparison of constructed methods satellite images contouring ship detection conclusion adv syst sci appl 2024; 04:82–93 published online at https://ijassa.ipu.ru. adaptive disturbance rejection control of 2d crane based on drem with instrumental variables vladislav petrov1, anton glushchenko2*, konstantin lastochkin2 1stary oskol technological institute n.a. a.a.ugarov (branch) nust “misis”, stary oskol, russia 2v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia abstract: a problem of control of uncertain 2d overhead crane affected by disturbances with unknown model is considered. an adaptive controller is proposed to solve the above-mentioned problem, which main salient feature is that it ensures asymptotic convergence of errors for both trolley position and payload swing angle not to a bounded set, but to zero without requirement of a priori knowledge on bounds of the system parameters and disturbance. first of all, the crane is represented in the feedback linearizable form via coordinate transformations under mild assumptions. then such representation is parameterized in the form of perturbed linear regression equation, which parameters are exactly identified via application of procedure, which has been recently proposed by the authors and is based on the dynamic regressor extension and mixing method and instrumental variables approach. obtained parameters estimates are substituted into derived ideal control law, and asymptotic stability of the proposed adaptive control system is rigorously proved. all theoretical results are validated via numerical experiments. keywords: 2d crane, parameter uncertainty, asymptotic stability, dynamic regressor extension and mixing, instrumental variables 1. introduction overhead cranes are widely used to manipulate heavy objects at different enterprises and construction cites. they usually consist of a cart moving on a line/plane and a hook suspended to the cart via cables. on the one hand, as one of the main goals of any enterprise is to increase productivity, then cranes are expected to move the payload from one position to another in minimal time. on the other hand, such plants are underactuated, thus cart movement, as well as exogenous disturbances, cause payload swing, which deteriorates positioning and overall crane performance. therefore, better productivity can be achieved via application of control systems that ensure high control quality under above-mentioned conditions [7, 22]. in this study we restrict our attention to 2d single pendulum cranes. problem of such plants control has been paid considerable attention by the control community during several past decades, and main obtained results are condensed in [7]. as sway reduction is of high priority, open-loop [16] and collocated control [11] strategies are not considered in this study, and further only non-collocated ones are of interest. plenty of closed-loop control methods have been exploited for crane systems, e.g., gainscheduling [10], feedback linearization [19], singular perturbation analysis [21], predictive control [5, 12]. all these methods are based on at least one of the following restrictive assumptions: the system parameters are known, disturbance-free case is considered. ∗corresponding author: aiglush@ipu.ru adaptive disturbance rejection control of 2d crane... 83 at present, as far as parameter uncertainty and perturbations are concerned, the crane control problem is still a fairly open topic. without being exhaustive, further some techniques to cope with the above-mentioned case are considered. in [14] a new passivity-based control law is elucidated, which includes both actuated cart and underactuated payload positions, and the asymptotic stability is proved via application of lyapunov and lasalle’s invariance theorem, but the robustness to disturbances is shown only via numerical experiments. parameter uncertainty and unknown perturbations are coped with simultaneously using sliding mode control (smc) technique [4] in conjunction with sigmoid function to avoid chattering [1] and adaptive control [22] to relax assumption of a priori known bounds of the system parameters and disturbance. particularly in [22], using a coordinate transformation, a new sliding surface is proposed, and uncertainty is parametrized as a linear regression equation, which parameters are then estimated (exactly, as no disturbances were included into initial model description, while friction was also parametrized as a linear regression). then obtained uncertainty estimate is added to the smc-based control law with a negative sign. robustness to perturbations is again shown only via simulation. in [9], using another coordinate transformation, the original system is represented as a chain of integrators with unmatched uncertainty. then controller is designed with the help of backstepping approach for disturbance free case. this result has been further improved in [20], [23], where the transformation from [9] is augmented with an additional assumption to represent the crane model in the feedback linearizable form with matched uncertainty. then smcand backstepping-based controllers with finite-time and extended observers of bounded disturbance were designed. one of the main drawbacks of such designs from [20], [23] is the requirement to know the system parameters for control signal calculation. the fact that the overhead crane model can be represented in feedback linearizable form means that other methods apart from smc and backstepping can be applied to solve the control problem under consideration, particularly, adaptive control methods [15]. considering them, firstly, an ideal control law is derived, which guarantees the control goal achievement under the assumption that the system parameters are known. secondly, using certainty equivalence principle, the control law unknown parameters are substituted with their dynamic estimates obtained with the help of adaptive laws from initial system parametrizations represented in the form of linear regression equations. this principle has already been mentioned in this section when approach from [22] was described. to ensure convergence of cart and payload position errors to zero, exact estimates of the system parameters are required, otherwise only uniform ultimate boundedness of mentioned errors can be shown. but existing online continuous-time parameter estimation laws provide zero identification error only if disturbance-free case is considered (like in [22]) or the external perturbations are vanishing to zero or independent with the regressor of the system, which are far away from practical scenarios [2, 8, 18]. in [6] the authors have recently proposed a new method of online asymptotic identification of perturbed linear systems, which is based on dynamic regressor extension and mixing procedure [3] and instrumental variables approach [13] and ensures online exact asymptotic estimation of the unknown parameters of linear systems in the perturbed case even if the perturbation and regressor of the system are dependent. such method became an inspiration for this study, as it allows one to overcome the hindrances mentioned in the previous paragraph. therefore, in order to overcome the above-mentioned limitations of the existing methods for the crane control, in this study we apply the approach from [6] to propose a control method, which exhibits simultaneously the following remarkable properties: p1) it ensures asymptotic convergence to zero of both cart/payload position and parametric errors in case when non-vanishing bounded disturbance with unknown model affects the overhead crane, copyright © 2024 assa. adv syst sci appl (2024) 84 v. petrov, a. glushchenko, k. lastochkin p2) it does not require prior knowledge of system real/nominal parameters and disturbance or their bounds, p3) it does not use smc approach and free from complexity of the backstepping-based control laws. the remaining parts of this paper are organized as follows. the rigorous problem statement is given in in section ii. main result is elucidated in section iii. section iv presents the results of the numerical experiments followed by the conclusions in section v. finally, proofs are postponed to appendix. notation. further the following notation is used: |.| is the absolute value, ∥.∥ is the suitable norm of (.), in×n = in is an identity n× n matrix, 0n×n is a zero n× n matrix, 0n stands for a zero vector of length n, det{.} stands for a matrix determinant, adj{.} represents an adjoint matrix. we also use the fact that for all (possibly singular) n× n matrices m the following holds: adj{m}m = det{m}in×n and say that f ∈ lq if q √ t∫ t0 |f (s)|qds <∞ for all t ≥ t0. 2. problem statement a mathematical model of a 2d overhead crane is considered: (m +m) q̈1+mlq̈2cos (q2)−mlq̇22sin (q2) = τ + τd, ml2q̈2 +mlq̈1cos (q2) +mglsin (q2) = 0, (2.1) where q1 ∈ r stands for a trolley position, q2 ∈ r denote a payload swing angle, τ ∈ r is a control signal, τd ∈ r is an unknown external perturbation. the masses of the trolley m and payload m are unknown, the gravity constant g and the cable length l are known. the following assumptions are adopted for the system (2.1). assumption 1. the payload swing angle q2 is mechanically bounded, i.e., q2 ∈ ( −π 2 , π 2 ) . assumption 2. it holds that l g cos(q2) q̇22 ≪ 1. assumption 3. it holds that τ̇d (t) ∈ lp ∩ l∞ for 1 ≤ p <∞. the control goal for (2.1) is formulated as simultaneous stabilization of the payload swing angle q2 at zero and asymptotic tracking of the reference position signal by the trolley. considering the states x1 = q1, x2 = q̇1, x3 = q2, x4 = q̇2, this goal can be written as follows: lim t→∞ ∥x (t)− xd∥ = 0, (2.2) where xd = [x1d 0 0 0]t. 3. main result the main result of the study is divided into three parts. in the first one, using [9] and assumptions 1 and 2, the system is transformed into a feedback linearizable form. in the second part, an ideal control law is derived that allows one to achieve the stated goal in case that all plant parameters are known. in the third part, a procedure from [6] is used to form an estimation law for exact identification of parameters of the overhead crane affected by perturbations, and then an adaptive control system is designed that ensures the goal (2.2) achievement in case of unknown crane parameters. copyright © 2024 assa. adv syst sci appl (2024) adaptive disturbance rejection control of 2d crane... 85 3.1. system parametrization in this subsection transformations from [9] are described to convert the system into a feedback linearizable form. the term q̈1 is expressed from the second equation of (2.1): q̈1 = −ml2q̈2 −mglsin (q2) mlcos (q2) = −gtan (q2)− lq̈2sec (q2) . (3.3) equation (3.3) is substituted into (2.1), and after series of simple transformations we have: −l [m +msin2 (q2)] q̈2 − (m +m) gsin (q2)−mlq̇22sin (q2) cos (q2) = = cos (q2) [τ + τd] . (3.4) the leftand right-hand sides of (3.4) is divided by −l [m +msin2 (q2)] to obtain: q̈2 = cos (q2) −l (m +msin2 (q2)) [ τ + τd + (m +m) gtan (q2) +mlq̇22sin (q2) ] . (3.5) the following notation is introduced: θt = [m m], ϕt (t) = [−l −lsin2 (q2)], ωt (t) = [gtan (q2) lq̇22sin (q2) + gtan (q2)], (3.6) in order to rewrite (3.5) in a simpler form: q̈2 = cos (q2) ϕt (t) θ [ τ + τd + ωt (t) θ ] . (3.7) equations (3.3) and (3.7) are written in a state space form: ẋ1 = x2, ẋ2 = −gtan (x3)− lusec (x3) , ẋ3 = x4, ẋ4 = u, u = cos(x3) ϕt(t)θ [ τ + τd + ωt (t) θ ] . (3.8) in accordance with [9], the following coordinate transformation t : r4 7→ r4 is introduced: p = t (x) = x1 + χ (x3) x2 + lx4 cos(x3) x3 x4 , (3.9) where χ (x3) = l x3∫ 0 1 cos(s) ds = l · log ( 1+tan(x3 2 ) 1−tan(x3 2 ) ) . then, owing to d dt [χ (x3)] = l ẋ3 ( tan2 ( x3 2 ) + 1 )( tan2 ( x3 2 ) − 1 ) = l x4 cos (x3) , (3.10) copyright © 2024 assa. adv syst sci appl (2024) 86 v. petrov, a. glushchenko, k. lastochkin it is written with the help of the new coordinates that: ṗ1 = p2, ṗ2 = −gtan (p3)− lusec (p3) + l u+p24tan(p3) cos(p3) = tan (p3) ( −g + lp24 cos(p3) ) , ṗ3 = p4, ṗ4 = u. (3.11) then, following [9], another one coordinate transformation s: r4 7→ r4 is introduced: ξ = s (p) =  p1 p2 −gtan (p3) −g (1 + tan2 (p3)) p4 , (3.12) which allows one to rewrite (3.11) as follows: ξ̇1 = ξ2, ξ̇2 = ξ3 − lξ3ξ24 (g2+ξ23) 3 2 , ξ̇3 = ξ4, ξ̇4 = v , (3.13) where the following notation is used: ξ3 − lξ3ξ24 (g2+ξ23) 3 2 = −gtan (p3) + lgtan(p3)g2(1+tan2(p3)) 2 p24 (g2+g2tan2(p3)) 3 2 = −gtan (p3) + l tan(p3)p24 cos(p3) , v = −g (tan2 (p3) + 1) [u + 2p24tan (p3)] = = g (ξ) [ cos(x3) ϕt(t)θ [ τ + τd + ωt (t) θ ] + f (ξ) ] , f (ξ) = 2 (tan (atan (−g−1ξ3))) ( ξ4 −g(1+tan2(atan(−g−1ξ3))) )2 , g (ξ) = −g (tan2 (atan (−g−1ξ3)) + 1) . (3.14) if assumptions 1 and 2 are met, then it holds for the second term from the second equation of (3.13) that: lξ24 (g2 + ξ23) 3 2 = lg2(1 + tan2 (p3)) 2 p24 (g2 + g2tan2 (p3)) 3 2 = lp24 g cos (p3) = l g q̇22 cos (q2) ≪ 1, (3.15) which guarantees that ζ3 − lζ3ζ 2 4 (g2 + ζ23 ) 3 2 ∆ = ζ3 and allows one to rewrite (3.13) as: ξ̇ (t) = a0ξ (t) + b0g (ξ) [ cos (x3) ϕt (t) θ [ τ (t) + τd (t) + ωt (t) θ ] + f (ξ) ] , (3.16) where a0 = [ 04 i3 01×3 ] , b0 = [ 03 1 ] . copyright © 2024 assa. adv syst sci appl (2024) adaptive disturbance rejection control of 2d crane... 87 therefore, in case assumptions 1 and 2 are met, the original problem (2.2) is reduced to tracking of the following reference signal by the new coordinates ξ: ξd = s (t (xd)) = x1d00 0  in such a way that lim t→∞ ∥ξ (t)− ξd∥ = 0. (3.17) since, under assumption 1, the transformations (3.9) and (3.12) are diffeomorphisms, the objectives (2.2) and (3.17) are equivalent. 3.2. ideal control law derivation to define the required control quality for the system (3.16), the following reference model is introduced: ξ̇ref (t) = akξref (t) + b0e t 1kx1d, (3.18) where k ∈ rn is a parameter vector, which defines a hurwitz matrix ak = a0 + b0k t. then the tracking error ξ̃ = ξ (t)− ξref (t) is described by: ˙̃ξ (t) = ak ξ̃ (t) + b0g (ξ) [ cos(x3) ϕt(t)θ [ τ (t) + τd (t) + ωt (t) θ ] + f (ξ) ] − −b0 ( ktξ (t) + et1kx1d ) . (3.19) to compensate for all perturbations in equation (3.19), the control law is chosen as: τ (t) = ϕt(t)θ cos(x3) τb (t)− τdf (t)− ωt (t) θ, τb (t) = 1 g(ξ) ( ktξ + et1kx1d ) − f (ξ) , (3.20) where τdf (t) denotes the filtered disturbance, τb (t) is a baseline component of the control signal. in order to obtain the filtered version of τd (t), the stable first order filter k s+k [.] , k > 0 is applied to the leftand right-hand sides of the first equation from (2.1): (m +m) k s+k [q̈1] +ml k s+k [q̈2cos (q2)]−ml k s+k [ q̇22sin (q2) ] = k s+k [τ ] + k s+k [τd] , (3.21) having applied the swapping lemma: k s+k [xy] = x k s+k [y]− k s+k [ ẋ k s+k [y] ] , we have from (3.21) that: τdf (t) = φt (t) θ − z (t) , φt (t) = [ ks s+k [x2] ks s+k [x2] + lφ(x) ] , z (t) = k s+k [τ ] , (3.22) where τdf (t) : = k s+k [τd (t)] , φ(x) = cos (x3) ks s+k [x4] + k s+k [ sin (x3) ( x4 ks s+k [x4]− x24 )] . copyright © 2024 assa. adv syst sci appl (2024) 88 v. petrov, a. glushchenko, k. lastochkin the substitution of (3.22) into (3.20) yields the final equation for the ideal control law, which is applicable in case all overhead crane parameters are known: τ (t) = ϕt(t)θ cos(x3) τb (t)− ωt (t) θ + z (t) , (3.23) where ω (t) = ω (t) + φ (t). if the crane parameters are unknown, they are required to be estimated to design an adaptive control system. 3.3. adaptive control system design the unknown parameters are proposed to be identified using the estimation law proposed in [6]. following the procedure from [6], an instrumental variable is introduced on the basis of the reference model (3.18) states: ζiv (t) = [ ks s+k [x2ref ] ks s+k [x2ref ] + lφ(xref ) ] (3.24) where xref (t) = x1ref (t)x2ref (t) x3ref (t) x4ref (t)  = si ( t i (ξref ) ) . using the instrumental variable (3.24), the regression equation (3.22) is extended with the help of a filter with averaging and a sliding window filter: ϑ̇ (t) = ζiv (t) z (t)− ζiv (t− t ) z (t− t ) , ϑ (t0) = 0, ψ̇ (t) = ζiv (t)φ t (t)− ζiv (t− t )φt (t− t ) , ψ (t0) = 02×2, (3.25) ẏ (t) = − 1 f (t) ḟ (t) (y (t)− ϑ (t)) , y (t0) = 0, φ̇ (t) = − 1 f (t) ḟ (t) (φ (t)− ψ (t)) , φ (t0) = 02×2, ḟ (t) = ptp−1, f (t0) = f0 > 0, (3.26) where t > 0 is a sliding window width, p ⩾ 1, f0 ⩾ tp0 stand for the parameters of the filter with averaging. considering that θ = const and applying the filters (3.25) and (3.26), the following regression equation is obtained [6, proposition 4]: y (t) = φ (t) θ +w (t) , (3.27) where the disturbance satisfies following equations: ẇ (t) = − 1 f (t) ḟ (t) (w (t)− ε (t)) , w (t0) = 0, ε̇ (t) = ζiv (t)w (t)− ζiv (t− t )w (t− t ) , ε (t0) = 0, (3.28) where, in order to follow notation from [6], we introduce w (t) : = k s+k [τd (t)]. multiplication of (3.27) by adj {φ (t)} yields a set of scalar regression equations: y (t) = ∆ (t) θ +w (t) , (3.29) copyright © 2024 assa. adv syst sci appl (2024) adaptive disturbance rejection control of 2d crane... 89 where y (t) : = adj { φ(t) 1+∥φ(t)∥ } y (t) , ∆(t) : = det { φ(t) 1+∥φ(t)∥ } , w (t) : = adj { φ(t) 1+∥φ(t)∥ } w (t) . we are in position to prove the following proposition for the disturbance from (3.29). proposition 3.1: if the following inequality holds: ∀t ⩾ t0 ∣∣∣∣∣∣ t∫ t0 ζi (s)w (s) ds ∣∣∣∣∣∣ ⩽ c <∞ ∀i = {1,2} , (3.30) then i) wi ∈ ll for all l ∈ (1, ∞), ii) there exists a scalar cw > 0 such that |wi (t)| ⩽ cw ḟ (t) f (t) <∞. proof proof of proposition 3.1 coincides with the one for proposition 5 from [6]. the second lyapunov method, the perturbation properties from proposition 3.1, and the results from [6] jointly motivate introduction of the following adaptive control law: τ (t) = ϕt(t)θ̂(t) cos(x3) τb (t)− ωt (t) θ̂ (t) + z (t) , (3.31) where ˙̂ θ (t)=−γ∆(t) ( ∆(t) θ̂ (t)− y (t) ) , (3.32) where γ > 0 are adaptive gains. the properties of the closed-loop adaptive control system are described in the following theorem. theorem 3.1: let assumptions 1-3 be met and inequality (3.30) hold, then the goal (2.2) is achieved. proof proof of theorem 3.1 is postponed to appendix. owing to the chosen structure of the reference model, boundedness (asymptotic stability) of the error ξ̃ leads to boundedness (asymptotic stability) of the error ξ (t)− ξd. thus, the control law (3.31) + (3.32) ensures that the goal (2.2) is achieved even when the overhead crane parameters are unknown. 4. numerical experiments the overhead crane model parameters have been picked from the state standard of the russian federation ’electric single girder overhead cranes’ and had the following values: m = 800, m = 4660, l = 1, g = 9.81. (4.33) the disturbance was defined as: τd (t) = 1000e−0.001(t−t0)sin (0.2πt) + 500, (4.34) copyright © 2024 assa. adv syst sci appl (2024) 90 v. petrov, a. glushchenko, k. lastochkin where the exponential multiplier formally allows one to meet assumption 3. the reference signal was formed using the following function: r (t) = 10sign {sin (0.05π)} , ẋ1d (t) = sat|+2 −2 {100 (r (t)− x1d (t))} . (4.35) the parameters of the control law (3.23) were chosen as follows: θ̂(t0) = [0.5m 0.1m]t, kt = [−16 −32 −24 −8], k = 1, p = 1, t = 10, γ = 104. (4.36) figure 1 shows behavior of the unknown parameter estimates. fig. 4.1. behavior of θ̂(t). figure 2 presents behavior of x1, x1d, ξ̃1, x3, lξ3ξ24 (g2+ξ23) 3 2 and f , ff . fig. 4.2. behavior of x1, x1d, ξ̃1, x3, lξ3ξ 2 4 (g2+ξ23) 3 2 and f , ff . copyright © 2024 assa. adv syst sci appl (2024) adaptive disturbance rejection control of 2d crane... 91 the results of conducted experiments allow one to conclude that the proposed control system ensures goal (2.2) achievement under conditions of parametric uncertainty. owing to the estimation of unknown parameters of the system, compensation of the disturbance is ensured without oscillations of the payload swing angle. the above-given transients of lξ3ξ24 (g2+ξ23) 3 2 allow one to conclude that assumption 2 and inequality (15) are met. 5. conclusion in this study a new adaptive control system for perturbed uncertain overhead crane is proposed, which, adopting non-strict assumptions, simultaneously (i) ensures asymptotic convergence to zero of both cart/payload position and parametric errors in case when nonvanishing bounded disturbance with unknown model affects the overhead crane, (ii) does not require prior knowledge of system real/nominal parameters and disturbance or their bounds, (iii) does not use smc approach and free from complexity of the backstepping-based control laws. conducted numerical experiments fully validated all theoretical results. acknowledgements the first author was financially supported by the ministry of science and higher education of the russian federation (state assignment fsme-2023-0014). a. theorem 3.1 proof proof proof is divided into two steps. the first one is to show asymptotic convergence of parametric error θ̃ (t) = θ − θ̂ (t) to zero. in the second one we prove convergence of the tracking error ξ̃ (t) to zero. step 1. the differential equation for the parametric error is written as follows: ˙̃θ (t) = γ∆(t) ( ∆(t) θ̂ (t)− y (t) ) = − γ∆2 (t) θ̃ (t)−γ∆(t)w (t) . (a.37) following theorem 2 from [6], in case w ∈ l2 and ∆ /∈ l2 we have lim t→∞ θ̃ (t) = 0, θ̃ (t) ∈ l∞. step 2. the control law (3.31) is substituted into (3.19) to obtain: ˙̃ξ (t) = ak ξ̃ (t)− b0 ( ktξ (t) + et 1kx1d ) + +b0g (ξ) [ cos(x3) ϕt(t)θ [ ϕt(t)θ̂(t) cos(x3) τb (t) + τ̃d (t) + ωt (t) θ̃ (t) ] + f (ξ) ] = = ak ξ̃ (t) + b0 g(ξ)cos(x3) ϕt(t)θ [ −ϕt(t)θ̃(t) cos(x3) τb (t) + τ̃d (t) + ωt (t) θ̃ (t) ] (a.38) where τ̃d (t) = τd (t)− τdf (t). copyright © 2024 assa. adv syst sci appl (2024) 92 v. petrov, a. glushchenko, k. lastochkin then error equations can be written in the following cascade form: ˙̃ξ (t) = f1 ( ξ̃, θ̃, τ̃d, t ) , ˙̃θ (t) = f2 ( θ̃, t ) , ˙̃τ d (t) = f3 (τ̃d, t) , (a.39) where f1 ( ξ̃, θ̃, τ̃d, t ) : = ak ξ̃ (t)+ +b0 g(ξ)cos(x3) ϕt(t)θ [ −ϕt(t)θ̃(t) cos(x3) ( 1 g(ξ̃+ξref) ( ktξ̃ (t)+ktξref (t)+e t 1kx1d ) −f ( ξ̃+ξref )) + +τ̃d (t) + ωt (t) θ̃ (t) ] , f2 ( θ̃, t ) : = −γ∆2 (t) θ̃ (t)−γ∆(t)w (t) , f3 (τ̃d, t) : = −kτ̃d + τ̇d (t) . according to step 1, when the premises of the theorem under consideration are met, we have lim t→∞ θ̃ (t) = 0, and since τ̇d (t) ∈ lp ∩ l∞, 1 ⩽ p <∞, then, owing to lemma 2.19 from [15, p. 83], it is also true that lim t→∞ τ̃d (t) = 0. then, as sup t⩾t0 sup ∥θ̃∥⩽θ̃max ∥∥∥∇θ̃f2 ( t, θ̃ )∥∥∥ <∞ (because of normalisation in (3.29)), following theorem 3.2 from [17], it is concluded that the error ξ̃ (t) converges asymptotically to zero, which was to be proved. references 1. antipov, a., & krasnova, s. 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(2024) eso-based and ftdo-based anti-swing control for overhead cranes with external disturbance, measurement and control, 1–10. early access. https://doi.org/10.1177/00202940241246608 copyright © 2024 assa. adv syst sci appl (2024) introduction problem statement main result system parametrization ideal control law derivation adaptive control system design numerical experiments conclusion theorem 3.1 proof microsoft word 1197 article text, copyedited.doc adv syst sci appl 2022; 02; 34-45 published online at https://ijassa.ipu.ru. improvement of the traffic control on complex crossroads via sending randomized recommendations to drivers andrey m. valuev* mechanical engineering research institute named after a.a. blagonravov, russian academy of sciences, moscow, russia e-mail: valuev.online@gmail.com abstract: the efficient use of signalized crossroads, especially intersections of multi-lane highways playing a great role in the urban traffic, demands not only the adequate control by traffic lights regulation but rational adaptation of the drivers’ totality to current traffic organization and control as well. the latter results from their complex structure and traffic organization on them when some lanes within an intersection split and/or merge with the other ones and their passage is regulated individually. besides, changes of lane counts on entrance and exit roads is typical. the lack of visible indicators helping drivers in their error-free rational lane choice may be overcome by the traffic control system endowed with the function of the advicetick system for the drivers’ totality. the proposed way to elaborate proper recommendations is based on calculation of the optimal distribution of vehicles between possible routes according to the current intensity of entering traffic flows in all passage directions obtained from the monitoring data. to approach this ideal distribution it is proposed to send impersonal recommendations to drivers of approaching vehicles depending on their desired passage directions; these recommendations must randomly change in time according to the determined distribution since some admissible directions must be splitted between certain routes in the calculated proportion. the proposed control method can significantly reduce improper lane choice by drivers, including their principal errors not allowing them to reach the needed road and so increase the intersection capacity utilization and reduce traffic delays. keywords: road traffic control, traffic organization, multi-lane highway, road intersection, traffic flows, driver behavior, simulation 1. introduction road junctions, especially junctions of multi-lane roads, play a very significant role in the urban traffic and, if used inefficiently, they can practically block the traffic in large segments of road networks. only a small part of them are designed as multi-layer road constructions including bridges and/or tunnels. most junctions are single-layer intersections and in the cases of the medium or high traffic they are regulated by the traffic lights. for the reason of safety the organization of the traffic through a certain intersection foresees the separations of routes between phases of the traffic light cycle (tlc). routes’ crossing on each phase is forbidden and they try to exclude or minimize routes’ merging as well. each phase of the traffic light cycle serves a certain set of passage directions with individual traffic intensities of each of them. so the determination of the tlc phases must provide the possibility for all admitted directions to pass through the intersection with an average intensity of the entering traffic flow. however, even the best adaptation of the traffic light regulation parameters to the intensity of incoming flows and their distribution between the permitted directions of the * corresponding author: valuev.online@gmail.com improvement of traffic control on a complex crossroads via sending 35 copyright ©2022 assa. adv. in systems science and appl. (2022) intersection passage (both factors express locally the social demand for relocation within the city) may be insufficient for the most efficient implementation of the demand. this situation takes place if some directions may be passed by more than one route. then the efficient use of the intersection throughput require the proper distribution of vehicles on these directions between these routes that may not be automatically reached by the drivers’ totality selforganization because of the lack of visible indicators for their choices. it is especially difficult when some lanes within the intersection are branched into several lanes or, conversely, merge with other ones. to cope with the situation, it is advisable to endow the traffic control system with the function of an intelligent advisor for drivers, the specific implementation of which at a given time and at a given intersection should be integrated with monitoring the local traffic situation and determining the parameters of the traffic light cycle on it. in this case, recommendations should be issued to the registered users of the intelligent driver assistance system, who, in order to receive the advice, must announce the direction they need to travel through the intersection. in order to approximate the distribution of vehicles by lanes to the most effective, these recommendations should be issued to each user individually, however based not on their personal data (the system only needs to know the type of each vehicle) but on statistical considerations. 2. background this work is related to works on adaptive traffic control systems at the intersections. works of this kind have been known since the 1970s [3] and continue to develop [1, 7, 10, 12, 19]. in theoretical terms, such works are based on some general idea of the road traffic at the intersection, which can be of one of the types: 1) empirical macroscopic, considering “portions” of traffic at the intersection as a whole, or 2) microscopic, implying reproduction of the individual vehicles’ movement, which, the latter having the form of random streams, cellular automata or dynamical systems [4]. this work is based on the latter model type— the integral characteristics of the passage of vehicle flows through the intersection are supposed to be determined by calculating the trajectories of individual vehicles in the traffic flows [15, 16]. a number of works, both scientific and practical, are devoted to the structure of the intersection [5, 6, 9, 13, 14]. its detailed presentation is necessary for the application of the methodology proposed in this article. finally, a number of new works is devoted to the opportunities offered by the information interaction both between the drivers in a vehicle group and between the drivers and the traffic control system. the concepts of advanced driver assistance systems (adas), adaptive cruise control (acc) and even connected autonomous vehicles (cav) are being developed either individually or in combination [2, 11, 17, 18, 21]. some features of the proposed systems have already been implemented in some advanced models of cars. it also solves the emerging problem of data transmission in communications between drivers of individual vehicles (v2v) and between drivers and the traffic control infrastructure or vice versa (v2i) [20]. on the other hand, the emerging opportunities to improve vehicle handling should lead to both the increased safety of their passengers and the improvement in the road situation in general. the latter mean better conditions for meeting social needs for movement in the city. we focus on the new possibilities of v2i systems, but also note the usefulness of information interaction between drivers of neighboring vehicles for its more complete implementation. 36 a.m. valuev copyright ©2022 assa. adv. in systems science and appl. (2022) 3. problems of individual and social route choice for drivers of vehicles on an intersection of multi-lane roads the vehicle driver in a big city faces a lot of various forms of traffic organization, especially on highway intersections. this fact may be seen from fig. 3.1 showing the principal schemes of 15 from the 59 main junctions in the moscow urban road network [8]. all these structures as well as 44 others are different. fig. 3.1. principal schemes of main road junctions in moscow [8] besides, these principal schemes do not cover the entire set of variants for choosing by the drivers approximating these intersections and passing them since many passage directions are splitted into several lanes some of which branch and so serve more than one direction. the difference between the real traffic organization of a complex road junction and its representation in the internet or on a printed map is demonstrated by fig. 3.2 and 3.3. markings on the road cover show the permitted directions of movement but it is often too late for a driver to make choice when seeing them. improvement of traffic control on a complex crossroads via sending 37 copyright ©2022 assa. adv. in systems science and appl. (2022) fig. 3.2. principal scheme of the road junction on serpukhovskaya square fig. 3.3. actual traffic organization of the road junction on serpukhovskaya square in addition, typical traffic organization involves changing the number of lanes on the entry and exit roads when approaching the intersection. all the above mentioned makes sufficient difficulties for the drivers in their choices of paths. this choice is especially difficult if some passage directions are served by more than one lane. in this case each driver must answer the question what entry queue to join for passing the intersection in the minimal time. the actual organization of traffic determines the choice of drivers (on each entry road) and the reasons for the particular choice. the latter can be: first of all, its correctness for the desired direction of movement; then the expectation of a faster crossing, and finally 38 a.m. valuev copyright ©2022 assa. adv. in systems science and appl. (2022) maximum safety and convenience. the first reason is relevant for every multi-lane entry road, the others only if there is more than one road for the desired travel direction. drivers who do not regularly pass a certain intersection of multi-lane roads may need assistance in choosing the correct lane in all cases. however, we only focus on cases where there is more than one route for some travel directions. the totality of drivers’ choices for the given tlc parameters determines the efficiency of the intersection throughput use that may be characterized by average delays in its passage. such a choice is relatively simple only if the entrance lanes are strictly separated between directions, for every direction they are geometrically similar, and there are no merging points or at each tlc phase they are passed in the only one direction. in this case maintaining approximately equal lengths of queues in front of the entrance stop lines for a group of lanes serving the same direction leads to close values of delays for all these lanes and corresponding routes. thus drivers can focus on the lengths of queues and — in the aggregate — ensure their alignment. in this case self-organization of the local driver community is useful for them all, but even then the choice of an individual driver is random. in other cases there are no visible landmarks for adapting the totality of drivers to the local road situation to achieve traffic efficiency. for example, in the typical case when one lane allows only driving straight ahead and the neighboring one — straight ahead and to the right, and the duration of the green phases for these lanes, as is often the case, is not the same. in these complex cases that are typical for intersections of multi-lane highways it is very unlikely that a joint choice of drivers can provide favorable conditions for traffic through an intersection. an intelligent traffic management system should help drivers make choices in order to form an aggregate choice that will benefit most of them. 4. proposed approach 4.1 methodology the aim of our research is to work out the concept of an advisory system for drivers affecting their behavior in the way that must improve the traffic passage through a definite signalized intersection. to reach the goal, the way of quantitative prognosis of traffic indices must be proposed linking it with the cumulative choice of the drivers’ totality. besides, we must determine the informational basis of the projected system. in fact, the required indices of the traffic passage through a definite intersection depend on: 1. the static traffic organization of the intersection, i.e., the set of the permitted passage directions connecting some pairs of the entry and exit roads and the system of the routes through the intersection area corresponding to these directions; 2. the dynamic traffic organization of the intersection, or, in other words, the structural control, consisting in the separation of these roads among phases of the traffic lights cycle (a so-called phase-wise traffic separation scheme — tss); 3. the parametric control of the intersection, i.e., the duration of each phase of the traffic light cycle; 4. the actual (or desired) average traffic intensity for each passage direction; it must be emphasized that the road traffic has the stochastic nature, so these intensities oscillate. to elaborate the qualitative and quantitative attributes of the set of messages for drivers, the traffic control system must obtain and take into account the following information: 1. representation of the existing traffic organization at the crossroads area on two levels. the first one means the set of pairs (an entry road, an exit road) that define the permitted passage directions. the second one is the detailed representation of the intersection including neighboring segments of entry and exit roads in the form of the oriented graph of all permitted routes through the crossroads area. its arcs are axes of road lanes’ segments and improvement of traffic control on a complex crossroads via sending 39 copyright ©2022 assa. adv. in systems science and appl. (2022) nodes are of three types: 1) points of intersection of arcs, 2) stop lines on entries to the intersection or within it, 3) conditional borders of the intersection area on entry and exit roads. notice that these data entirely define alternatives in the intersection passage for vehicles entering it by every road. every possible route on this graph connecting a pair of entry and exit points may be used as a route through the intersection. 2. separation of the routes between tlc phases in the form of tss that is introduced to eliminate the danger of vehicles collision in the simultaneous passage of points of crossing and diminish this danger in the simultaneous passage of points of merging. 3. the average throughput of each active route on each tss phase as a function of its duration must be established either from the traffic monitoring treatment or the computer simulation on the verified model. 4. the characteristic of current traffic flows to which the system of drivers’ informing is being adapted is the existing (desired) intensity of traffic flows on each passage direction. the implementation of the proposed idea principally consists in three items: 1. for known values of duration of each phase the ideal distribution among the proper lanes the vehicles incoming by each entry road for each passage direction . it must be performed by the solution of the below optimization problems. 2. if this ideal distribution is, nevertheless, unsatisfactory to meet the existing traffic demand, recommendations are made how to choose more efficient values of tlc phases’ durations. 3. finally, the way of formation of the stream of messages addressed individually (but impersonally) to drivers of approaching vehicles is being defined. it is aimed to the best attainable correspondence between the real distribution of vehicles along the routes and the ideal one. 4.2. determination of dependencies that characterize quantitatively the route passage the separation of passage directions between tlc phases minimizes the interaction between traffic flows on different directions, although not entirely. typical case of the vehicular traffic on a route during one green phase is the movement of a chain of vehicles (cluster) with the minimal safe distance between subsequent vehicles. each cluster primarily consists of cars that formed an immobile queue up to the moment of the green phase beginning; if the queue is relatively short, extra cars join the cluster that does not change the cluster structure. nevertheless, quantitative indices of the clusters change from one cycle to another one because of many random factors, such as the distribution of vehicles’ types as well as psycho-physiological types of vehicles’ drivers in the cluster. an example of fluctuations in counts of passed vehicles on a definite lane for 5 subsequent cycles is shown in table 4.1. table 4.1. observation data of the route passage by traffic flows during several cycles cycle number the number of vehicles that crossed the stop line, for time intervals ( ) 0–10 sec. 10–20 sec. 20–30 sec. 0–30 sec. 1 5 5 4 14 2 3 4 3 10 3 3 5 3 11 4 3 5 4 12 5 4 4 4 12 such data, although in much greater amount, enable us to define the required dependences. the maximum average number of vehicles passing a definite i-th route for a given time interval is a definite value with a certain composition of the traffic flow on it (i.e., a certain distribution of types of vehicles in the traffic flow) and some other )(tn i )(max tn i 40 a.m. valuev copyright ©2022 assa. adv. in systems science and appl. (2022) influencing factors (whether, illumination). based only on the data presented in table 4.1, we may define as 3.6 for t=10 s, as 8.2 for t=20 s and as 11.8 for t=30 s. for intermediate values of t the required values of may be assessed by interpolation and for t>30 s by extrapolation. however, their assessment by simulation on a model with correctly identified parameters seems preferable. if the i-th route branches and, thus, serves multiple passage directions then depends on the proportion of vehicles’ separation between the passage directions, i.e., on the values of qij/qi where qij is the traffic flow intensity for the direction j and qi is the sum of them. for more concise notation this dependence may be written as for the overwhelming majority of signalized intersections each route branches one time only into two ones; so only one value of the type qij/qi determines and we may choose the fraction of the number of vehicles moving in the straight direction after the branching point (fstr i) for this role. to establish the values of we must split all obtained data of (4.1) into classes with close values of it looks very likely that the data set of observations may be insufficient for a reliable estimate and simulation must be used. the introduced dependences are related to the distribution of vehicles entering the intersection along one road. it is sufficient if the current tss does not involve any active merge point that is the most common traffic organization. the case of active merging points is more complicated and demands further investigations. 4.3 calculation of the optimal distribution of vehicles among the routes according to the known intensity and structure of input traffic flows the problem under consideration is solved in a certain context: for a definite tss and given durations of tlc phases. the duration of the tlc and its r-th phase are denoted further, respectively, by ttlc and tp r. let ri be the set of tlc phases on which the i-th route is passed. let the set of passage directions for vehicles entering the crossroads area along the entrance road k be jk and for the passage direction j the set of routes ikj is used. the set of routes for all is denoted by ik. let qpd j be the actual or desired traffic flow intensity for the passage direction our aim for each entrance road k is to distribute all qpd j, in time and space, namely between routes from ikj and green phases for these routes in the optimal way, i.e., to determine the optimal values of where (4.2) such a distribution is feasible if the throughput of each route is not smaller than the intensity of vehicles’ appearance in its beginning, so (4.3) may be treated as the throughput reserve indicator for the passage direction so the target value is the total reserve indicator for all passage directions for which (4.4) )(max tn i )(max tn i ,pd ij )(max tn i }).,{,( pmax idiji jjqtn î }),{,( pmax idiji jjqtn î inmax )()()( turnstr tntntn iii += ).(/)(strstr tntnf iii = kjjî .kjjî ,kjjî ijrq ,pd j ii rr ijr qq kj i =å å î î .kjjî ,1}),{,(/)( pd ppmaxtlc i rr jj idijrriijr i i jjqtnqt s-=îå å î î ,0³is .kiiî is .kiiî k0s ,0ki ss ³ .kiiî improvement of traffic control on a complex crossroads via sending 41 copyright ©2022 assa. adv. in systems science and appl. (2022) the introduced optimization problem consists in maximizing under restrictions (4.2)–(4.4). the problem in question is a nonlinear programming problem. it must be emphasized that dependences expressing average maximal numbers of passing vehicles are not integer-valued due to the stochastic nature of the traffic and are represented by smooth functions. the case when the condition (4.5) is satisfied for all directions is favorable and does not require changes of durations of the tlc phases. otherwise, the problem of these rational changes emerges: to increase the phase duration causing the violation of (4) for some entrance roads; it demands decreasing the rest ones. if we are sure that the calculated optimal distributions of vehicles between routes are attainable, then it is reasonable to determine them and phase durations simultaneously. thus the following problem may be solved: to maximize under restrictions (4.2), (4.3) and (4.6) for all k=1,…, k. to illustrate the proposed solution, a specific example is analyzed, namely two highly curved parallel lanes shown in the rightmost part of fig. 3.3; they implement the turn from the inside of the garden ring to the outside. according to the existing traffic organization the traffic flow from valovaya street is separated between these lanes that are passed simultaneously. the following table provides the necessary data for the case in question. two opposite factors, namely, a greater curvature and a smaller length of the left lane, determine the specific form of dependences obtained from observational data. table 4.2. dependences for two parallel curves lane the average number of vehicles that crossed the stop line during the specified green phase duration 5 s 10 s 15 s 20 s 25 s 30 s left 1.29 3.14 4.71 6.29 7.71 8.99 right 1.71 3.71 5.43 7.02 8.01 9.00 it follows from (4.2)–(4.4) that in this case the optimal distribution of vehicles between the lines must satisfy the condition (4.7) the necessary availability of the throughput reserve requires satisfying the condition (4.8) then we have in (4.7). so if and then the maximal incoming flow may be not greater than 10.14/30=0.338 vehicles per second. the temporal headway is slightly less 3 s (its minimal value for the most intense single-lane traffic flow is about 2 s). most importantly, appearing vehicles must be distributed between the left and right lanes when entering the intersection in the proportion 4.71: 5.43, or 46.4% to the left and 53.6% to the right. in reality, the distribution of appearing vehicles between lanes changes randomly. certainly, the optimal proportion cannot be achieved with deterministic messages. for the k0s }),{,( pmax idiji jjqtn î 00 ³ks 0s ,0ss ³i .kiiî ,kjjî )(max tn i .1)(/)(/ 11p2max21tlc1p1max11tlc s-== tnqttnqt .)()( 1pdtlc1p2max1p1max qttntn ³+ 0³is 151p =t ,30tlc =t 42 a.m. valuev copyright ©2022 assa. adv. in systems science and appl. (2022) case with practically the same amounts of vehicles should join both queues but it is only a specific situation. if the straight-line flow is divided between the left and right lanes, and the flow on the right passes only along the right lane, then the conditions must be satisfied (4.9) (4.10) (4.11) and the optimization criterion may be expressed as maximization of 4.3. generation of messages for drivers it follows from the above that the best situation for the crossroads passage in all directions is when each entry queue in each lane is a sequence of clusters corresponding to subsequent green phases for this lane, with each cluster consisting of the calculated numbers of vehicles. besides, for branching lanes there must be a definite share of vehicles for each possible direction in every cluster. however, the exact match between the reality and the calculated deterministic distributions is unattainable because of the stochastic nature of the urban vehicular traffic. e.g., in (1) it is supposed to distribute vehicles that must drive up to the intersection for one cycle for passing the j-th direction. in fact, it is only a mean value. nevertheless, the calculated distribution may be obtained on average through randomly generated messages. problems (1)–(3) yield values that admit their direct utilization for forming the stream of messages. they are the shares of routes implementing a certain passage direction (in the general case, for a certain tlc phase). if the j-th direction is passed on the single phase r (the most common case) along two routes, 1 and 2, then these shares are constant in time and equal to and , respectively. then the recommendations to go along the 1st and the 2nd routes must be issued with probabilities and it is important that each driver would receive the message only one time; besides, the message must be sent and received well in advance to let enough time for maneuvers aimed at changing lanes. just the same situation takes place if all routes for a definite direction are passed on two successive phases; in this case the two phases must be united in (2). sending messages in the form: “if you drive straight then choose the second lane on the left; if you turn left then…” the advisory system does not need any information from individual drivers. in other cases the share of routes differs for phases. then values of route shares used in messages must change in time. we suppose that the most reliable way to define moments of switching is to link them with the count of vehicles after the previous switching. we assume that after appearing the average number of vehicles passing the intersection in the definite direction for one future phase that next vehicles will pass the intersection in the next phase and use for them respective values of shares. then, a minimal feedback from drivers is required: everyone must inform whether he intends to go straight or to turn left or right. в остальных случаях доли маршрутов для направления проезда перекрестка различается по фазам. тогда значения долей каждого маршрута в приходящем потоке, используемые при формировании сообщений, должны меняться во времени. для этого нужно разделить последовательности подъезжающих тс на порции, соответствующие количеству тс, прошедших на предыдущей фазе и по этой последней фазе определять требуемые доли для новой порции тс. потребуется также минимальная обратная связь от водителей: каждый должен сообщить, намерен ли он ехать прямо или повернуть налево или направо. s301p =t ,1))/(,(/)( 22221121p2max2221tlc s-=++* qqqtnqqt ,02 ³s ,1)(/ 11p1max11tlc s-=tnqt ,01 ³s ,1pd2111 qqq =+ ,2pd22 qq = }.,min{ 21 ss jqttlc jjr qqq pd11 /= jjr qqq pd22 /= 1q .2q improvement of traffic control on a complex crossroads via sending 43 copyright ©2022 assa. adv. in systems science and appl. (2022) in other cases, the shares of routes for some directions of the intersection passage differ in cycles and phases. then the values of the shares of each route in the incoming stream, used in the formation of messages, must change over time. to do this, it is necessary to divide the sequences of approaching vehicles into portions corresponding to the number of vehicles that passed in the previous phase and determine the required shares for a new portion of the vehicle from this last phase. minimal feedback from drivers will also be required: everyone must indicate whether he intends to go straight or turn left or right. 5. existing limitations for the application of the proposed methodology and ways to overcome them although there are no principal obstacles to implement a test version of the proposed advicetick system, the initial number of its users may be very small and its effect as well, so it would be difficult to demonstrate practically the benefits of its implementation. on the other hand, full-scale experiments at a certain intersection can be carried out with vehicles whose drivers have agreed to participate in the experiment.. video fixation of the experiment must convince other drivers to follow the recommendations of the proposed advice-tick system. to achieve the real usefulness of the proposed approach, its implementation on its initial stage requires both some perfections of its methodology and notably extra means of monitoring and information exchange. the technical aim is to maintain the actual distributions of vehicles between routes for each direction close to the optimal ones. it is necessary to establish in real time: 1) the share of drivers who receive messages and use them; 2) the actual distributions between the routes of vehicles whose drivers do not use the proposed advice-tick system. then the identification of vehicles in the intersection area as driven by the users and non-users is required. 6. conclusion the urban road traffic is a complex phenomenon including technical, informational, ecological and socio-psychological aspects, but often it does not meet the present day requirements. their significance for all these aspects is great. so efforts to improve it are reasonable. the present paper puts forward both a new concept and a formal method for its implementation, and seems promising in some aspects. based on modern achievements in control theory, traffic modeling and infocommunication technologies, it can be implemented in the near future. for the effective use of the intersection throughput, it is necessary both to develop an adequate control by the traffic lights regulation and to achieve rational behavior of drivers for the current organization and traffic control. the proposed approach is theoretically developed for the case when the traffic organization at the intersection excludes active points of route merging. this is a more common case, but merging routes is also used somewhere. besides, a driver having intention to change the lane may not have a success, especially in the case of heavy traffic. however, the congested traffic causes so many difficulties that only a system of measures at many levels can overcome or rather mitigate them. at any rate, these aspects require consideration in further research. 44 a.m. valuev copyright ©2022 assa. adv. in systems science and appl. 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(2013). integrated traffic information service system for public travel based on smart phones applications: a case in china, international journal of intelligent systems and applications, 5(12), 72–80, https://doi.org/10.5815/ijisa.2013.12.0 21. zohdy, i. h., & rakha, h. a. (2016). intersection management via vehicle connectivity: the intersection cooperative adaptive cruise control system concept, journal of intelligent transportation systems, 20(1), 17-32, https://doi.org/10.1080/15472450.2014.889918 microsoft word 1513-source texts-7008-1-6-20240404 adv syst sci appl 2024; 02; 133-165 published online at https://ijassa.ipu.ru. text classification technologies in document categorization systems. a survey alla kravets, dmitriy semenochkin volgograd state technical university, volgograd, russia abstract: this paper presents a literature review from 2013 to 2022 on technologies and datasets used in the field of text classification. the review covered 110 sources from 5 scientific databases, the main criterion for inclusion was the presence of an experimental part involving a classifier or other technologies related to the classification process. the study reviewed the classification process and highlighted three main stages of text classification: data preparation, classifier training, and evaluation of results. using kitchenham's systematic literature review methodology, scholarly articles dealing with text classification problems were collected and analyzed. a sample of 243 articles was obtained, and after screening, a resulting sample of 110 articles was obtained. guided by the two research questions posed, this sample was analyzed and the results of the analysis were presented in graphical format. for each of the identified stages of classification, the frequency of use of the main technologies used in a particular stage was analyzed. each technology was reviewed within its respective source. in addition, considerable attention was given to analyzing the different datasets used for text classification, with a particular focus on the less frequently used ones. an analysis of the frequency of use of datasets concluded that researchers often use proven and popular datasets to demonstrate the effectiveness of their method. datasets are less frequently used to solve localized text classification problems. one notable trend identified in the analysis is the increasing prevalence of deep learning technologies in text classification. these technologies, including neural networks, recurrent neural networks (rnns), convolutional neural networks (cnns), transformers, and attention mechanisms, have gained considerable popularity among researchers. this study provides valuable insights into the evolution of text classification by shedding light on a variety of technologies, approaches, and datasets used by researchers. as text classification continues to evolve and diversify, this review can be a valuable resource for scholars and practitioners in the field, providing. keywords: text classification, text classifiers, deep learning, classification methods, datasets, text representation, feature extraction 1. introduction in today's information-driven world, the exponential growth of textual data has posed significant challenges in managing, organizing, and extracting meaningful insights from this sea of information. the need to efficiently categorize text documents has never been more crucial. text classification plays a pivotal role in addressing this challenge, facilitating applications ranging from information retrieval and sentiment analysis to spam filtering and content recommendation. in the era of big data, it's not just the quantity of data that's overwhelming; it's also the diversity of sources and formats. with the advent of social media, blogs, research papers, and other digital platforms, text data now spans a wide spectrum of genres, languages, and modalities. this evolution necessitates the continuous development of novel methods and approaches to effectively classify this expanding corpus of textual information [12]. text classification is one of the most fundamental challenges in the field of natural language processing (nlp). its significance is underlined by the fact that it often serves [70] [98] as a prerequisite for more complex nlp tasks such as machine translation, text summarization, and question-answering systems [7]. 134 a. kravets, d. semenochkin copyright ©2024 assa adv. in systems science and appl. (2024) text classification modules are components of numerous real-world applications, and they entail not only the core classifier responsible for mapping input data to specific classes but also a pipeline composed of sequential steps. there is no consensus on a universally accepted set of constituent stages of this pipeline, as researchers emphasize different aspects. for instance, within the work of [113], the text classification pipeline encompasses stages such as data collection, data analysis for labeling, feature construction and weighting, feature selection and projection, training of a classification model, and solution evaluation. in a related vein, [38] delineates a slightly distinct yet conceptually akin series of phases, including preprocessing, text representation, training of a classification model, and evaluation. conversely, [62] propounds a distinct framework, encompassing document representation, feature selection, training of a classification model, and evaluation of a text classifier. within the framework of this comprehensive literature review, we aim to harmonize these disparate views by aggregating all phases relevant to text preparation and formalizing this central stage as «data preprocessing». at the same time, we recognize the intrinsic importance of the stages of training a classification model and conducting evaluations. it is noteworthy that a significant proportion of the review studies in text classification have a relatively narrow focus, often focusing exclusively on the intrinsic properties of text classification models or delving into individual components within the classification process. for example, the studies by [147] and [159] are particularly specialized, closely examining the intricacies of classification algorithms and providing valuable insights into their merits and limitations. conversely, the work of [104] omits any specific mention of the algorithms used for classification, drawing attention to other facets of the process. of particular interest, [126] and [1] devote their attention to the area of feature selection methods, highlighting techniques for optimizing classification results. in stark contrast, the study by [76] abstains from any discourse on the underlying data and datasets used in their research. the main objective of this study is to surpass the typical limitations seen in previous reviews on text classification. our goal is to provide a thorough and extensive examination of the entire text classification pipeline, instead of limiting our investigation solely to classification algorithms or individual pipeline stages. this review will evaluate the technologies used for the process of text classification, including data preparation, processing, training classification models, and subsequent evaluations. furthermore, this review will accentuate the role of datasets in the domain of text classification. our objective is to conduct a systematic evaluation of the technologies used for the text classification process. this systematic literature review's main contributions can be outlined as follows:  analyzing 110 scientific articles published between 2013 and 2022, this review extends existing research.  by following established literature review protocols, it guarantees a trustworthy and replicable examination.  the review catalogues and evaluates technologies used in the analyzed articles, including classification algorithms, feature engineering and text representation techniques.  the review provides detailed listings of datasets used in the articles. this paper is organized as follows: section 2 summarizes the architecture of systems for text classification and provides information about the main elements of the text classification process, presents a description of data preparation, existing classification methods and classification evaluation. the methodology of the review, including the research questions, is detailed in section 3. section 4 presents the analysis of the main findings according to the research questions. the outcome of the entire review, taking into account the opinions of the authors, can be seen in section 5. text classification technologies in document categorization systems. a survey 135 copyright ©2024 assa. adv. in systems science and appl. (2024) 2. background this background section introduces the main concepts directly related to the topic of this review: section 2.1 discusses the basic concept and architecture of the text document classification process. section 2.2 presents information on existing technologies in the field of text data preparation. section 2.3 presents the main types of architectures and algorithms for text classification. 2.1. architecture of text classification systems text classification, is a fundamental task in the field of nlp and text mining. its primary objective is to automatically categorize or assign labels to a given text document into predefined categories or classes based on its content [75]. a machine learning model or algorithm typically performs this classification. here are the key components and concepts related to text document classification [42]:  text classification involves assigning a document to one or more predetermined categories.  the process typically requires a dataset (d) of labeled training documents (d1, d2, ..., dn), each with corresponding class labels (c1, c2, ..., cn). these labels serve as the ground truth to train a classification model.  the aim of text classification is to create a model for assigning the correct class label to a new, unclassified document (d) automatically. this model learns from training data patterns and features that enable predictions about the class of new documents.  there are two primary types of text classification: single and multi-label. in single label classification, each document is assigned to only one class. it is assumed to belong exclusively to a single category. a document in multi-label classification can belong to multiple classes at the same time. for example, a news article may be classified as both «politics» and «sports» if it covers topics from both categories. a text classification pipeline is a structured sequence of processes designed to transform unstructured text data into useful information. as stated in the introduction we use tripartite division encompassing preparation, classifier training, and result evaluation. the initial and fundamental step of the text classification pipeline involves preparing the text [150], which is an essential process for transforming unorganized textual data into an easily comprehendible and analyzable format. the importance of this phase cannot be overstated since the effectiveness and importance of the consequent analysis and model training are directly dependent on the meticulousness of text preparation. the second phase of the text classification pipeline is dedicated to classifier training. a diverse array of machine learning algorithms, is contemplated in accordance with the nature of the text classification task. armed with feature-extracted and labeled data, the selected classifier undergoes training to discern underlying patterns and relationships within the data. the ultimate objective is to endow the model with the ability to make accurate predictions on novel, unlabeled textual inputs. the culmination of the text classification pipeline lies in result evaluation, a quintessential phase to gauge the efficacy and generalizability of the trained classifier. evaluation metrics such as precision, recall, f1-score, and accuracy are employed to assess the model's performance [38]. these metrics provide a comprehensive view of the classifier's ability to correctly categorize text inputs. figure 2.1 shows a graphical representation of the text classification process. 136 a. kravets, d. semenochkin copyright ©2024 assa adv. in systems science and appl. (2024) fig. 2.1. stages of text classification process 2.2. data preparation the text preparation phase in a text classification pipeline is a crucial step that involves transforming raw, unstructured textual data into a structured and analyzable format. this phase encompasses several key components: text preprocessing, text representation and feature selection. 2.2.1 text preprocessing the text preparation phase in a text classification pipeline is a crucial step that in-volves transforming raw, unstructured textual data into a structured and analyzable format. this phase encompasses several key components: text preprocessing, text representation and feature selection. text preprocessing is the initial step in text preparation. its objective is to clean and standardize the raw text data to ensure consistency and remove unnecessary noise. research papers ([150] [43] [115]) agree on important aspects of text preprocessing, such as stopword removal, tokenization, stemming and lemmatization. stopwords are common words like «the», «and», «in», which appear frequently in text but typically do not carry significant information for classification tasks. removing stopwords reduces dimensionality and can improve model performance. tokenization involves breaking down the text into individual units, typically words or subwords, known as tokens. the process of tokenization plays a fundamental role in text analysis and classification. it allows the model to treat each token as a separate entity. tokenization can be achieved through: after tokenization, stemming and lemmatization are employed to reduce words to their base or root forms. the purpose of these processes is to standardize variations of words and reduce dimensionality. stemming is a more aggressive approach that involves removing prefixes and suffixes from words to reduce them to their root form [57]. for example, «running» and «runner» would both be stemmed to «run». lemmatization, on the other hand, is a more sophisticated technique that considers the context and grammar of words. it converts words to their base or dictionary forms, known as lemmas. for instance, «ran» would be lemmatized to «run» and «better» would be lemmatized to «good». lemmatization often results in more meaningful and linguistically accurate base forms. 2.2.2 text representation to enable mathematical operations on text data, it is often transformed into numerical vectors. this process, known as word representation, is essential for machine learning models. two text classification technologies in document categorization systems. a survey 137 copyright ©2024 assa. adv. in systems science and appl. (2024) common methods for word representation are n-gram based models and word embedding models. n-gram based models represent a classical approach to text analysis, particularly in the context of representing words and sequences of words [38]. these models extract linguistic information by analyzing the frequency and co-occurrence of contiguous sequences of n words, where n represents the desired context window size. the tf-idf score for a term t in a document d within a corpus is calculated using two key components: term frequency (tf) and inverse document frequency (idf) [65]. the tf component measures how frequently a term appears in a document, while the idf component quantifies the importance of the term across the entire corpus. the overall tf-idf score is a product of these two components: tf-idf(t, d, d) = tf(t, d) × idf(t, d) (2.1) where, tf(t, d) represents the term frequency of term t in document d , which is often computed as the raw count of term occurrences or using normalized variants such as the sublinear or logarithmic tf. idf(t, d) represents the inverse document frequency of term t in the corpus d, calculated as: idf(t, d) = log n n (2.2) where n is the total number of documents in the corpus, n is the number of documents in which term t appears at least once. word embedding models represent a paradigm shift in word representation by encoding words as dense vectors in continuous vector spaces [38]. this approach has gained prominence due to its ability to capture semantic relationships between words, enabling advanced natural language understanding. this dense representation enables the capture of semantic information. word embeddings can capture semantic relationships between words. for example, vector arithmetic operations like «king – man + woman» can yield the vector representation of «queen», illustrating the ability to infer word relationships [15]. word embedding models, such as word2vec [112], glove [125], and fasttext [17], map words to continuous vector spaces where words with similar meanings have similar vector representations. word2vec is a word embedding technique that maps words from a large text corpus into continuous vector representations in a high-dimensional vector space [112]. the central idea behind word2vec is to learn distributed representations of words such that semantically similar words are mapped to nearby points in the vector space. word2vec operates under two primary architectures: continuous bag of words (cbow) and skip-gram. glove is a text representation model that aims to capture semantic relationships between words in a corpus by learning distributed word vectors [125]. it operates on the principle that word vectors should encode word co-occurrence information effectively. the core idea behind glove is to factorize a word co-occurrence matrix to learn word embeddings. fasttext is a text representation model developed by facebook ai research [17], designed to handle text classification efficiently. it extends the idea of word2vec by incorporating subword information. instead of representing words as fixed vectors, fasttext represents words as bags of character n-grams. 2.2.2 feature selection feature selection is the final step in the text preparation phase. in document classification, textual data is transformed into high-dimensional vectors through basic representation models, as previously described in section 2.2.2. while these high-dimensional vector representations capture the essence of the documents, they also introduce several inherent challenges – curse of dimensionality, sparse data and overfitting. to mitigate these challenges, feature selection techniques are employed to reduce the dimensionality of vector representations while preserving essential information [153]. feature selection aims to identify and retain the most relevant features (terms or words) while discarding less informative ones. 138 a. kravets, d. semenochkin copyright ©2024 assa adv. in systems science and appl. (2024) source [77] defines three categories of feature selection methods: filter, wrapper, and embedded. filter methods like chi-squared test (chi), information gain (ig) and mutual information (mi) [83] are independent of specific machine learning algorithms. they evaluate features based on statistical metrics, such as chi-squared tests, information gain, or correlation coefficients, without considering the classification model. selected features are typically based on their intrinsic relevance to the target class or document category. the chi algorithm is a statistical technique designed to assess the independence between a given feature (term) and class labels, effectively quantifying the significance of individual terms in a classification context. the chi algorithm estimates a chi-squared statistic for each feature, which is formulated as: χ = ∑ o − e e (2.3) where, o represents the observed frequency of feature i in class j , and e denotes the expected frequency under the assumption of independence. a high chi value indicates strong dependence between the feature and class labels, rendering it a valuable metric for feature selection. ig method, deeply rooted in information theory, plays a pivotal role in the selection of relevant features for text classification. ig quantifies the reduction in uncertainty (entropy) regarding class labels achieved by incorporating a particular feature into the classification model. essentially, it measures how much information a feature contributes to class discrimination. a higher ig signifies that the feature imparts substantial insights, enhancing the model's capability to distinguish between various classes. mi measures the degree of dependency between a feature and class labels by assessing the shared information. features with high mi values are considered informative as they exhibit a strong association with specific classes. mi, as a feature selection technique, aids in identifying features that contribute significantly to classification accuracy by quantifying the information shared between features and class labels. embedded methods incorporate feature selection into the model training process itself [83]. these methods optimize feature selection as an integral part of the learning algorithm. examples include regularization techniques like l1 regularization (lasso) [83] used in logistic regression and decision trees with built-in feature selection. embedded methods aim to strike a balance between filter and wrapper methods, considering feature relevance and model performance simultaneously. 2.3 classifier training in this section, we adopt a division of classification algorithms proposed by [7] to categorize them into two distinct groups: basic machine learning and deep learning. 2.3.1 basic machine learning basic machine learning approaches are grounded in statistical principles and have been foundational in text classification tasks. these algorithms employ features derived from text data to make predictions. examples of algorithms in this category include naive bayes (nb), svm, random forest (rf), logistic regression (lr) and k-nearest neighbors (k-nn). svm [18] is a powerful and versatile machine learning algorithm commonly employed in text classification tasks. svm operates by finding an optimal hyperplane that best separates data points belonging to different classes while maximizing the margin between them. in the context of text classification, documents are typically represented as high-dimensional feature vectors, where each dimension corresponds to a term or word. nb [161] is a probabilistic classification algorithm widely used in text classification tasks. it operates on the principles of bayes' theorem and conditional probability. one of the fundamental assumptions of nb is the «naive» assumption, which assumes that features (words or terms in the case of text classification) are conditionally independent, given the class text classification technologies in document categorization systems. a survey 139 copyright ©2024 assa. adv. in systems science and appl. (2024) label. in text classification, nb assigns class labels to documents based on the likelihood of observing certain terms given each class. the class with the highest probability is assigned as the predicted class label. knn [60] is a non-parametric classification algorithm used in text classification tasks. it operates based on the principle of proximity, where documents are assigned a class label based on the majority class among their k-nearest neighbors in the feature space. in text classification, knn treats documents as points in a high-dimensional space, with each dimension representing a term or word. when classifying a new document, knn identifies the k most similar documents from the training dataset, typically based on distance metrics like euclidean distance or cosine similarity. the class label is then determined by a majority vote among these neighbors. lr [39] linear classification algorithm for text classification. it models the relationship between the binary or categorical dependent variable (class labels in text classification) and one or more independent variables (features or terms in text documents) using a logistic function. in text classification, lr assigns class probabilities to documents based on their features. it models the probability that a document belongs to a particular class given its feature values. rf [56] is an ensemble learning algorithm used in text classification tasks. it combines multiple decision trees to improve classification accuracy and robustness. rf constructs an ensemble of decision trees, where each tree is trained on a bootstrapped subset of the training data and a random subset of features. the final class prediction is determined by aggregating the predictions of individual trees, often through majority voting. 2.3.2 deep learning deep learning represents a transformative paradigm in text classification, driven by the advancement of neural network architectures. deep learning models, with their ability to automatically learn hierarchical representations, have excelled in capturing complex textual patterns. cnns, initially designed for image processing, have been adapted effectively for nlp tasks [58]. in text classification, cnns utilize convolutional filters of varying sizes to scan and extract local features, such as n-grams, from the input text. these features are then processed hierarchically through multiple layers of the network to capture higher-level abstractions. this approach can effectively identify patterns and representations in text data, making it valuable for various text classification tasks. rnns, specifically long short-term memory (lstm) [48] and gated recurrent unit (gru) [25] variants, are adept at modeling sequential data. rnns are recurrent in nature, allowing them to process sequences of data by maintaining internal states and utilizing past information to influence future predictions [52]. they are particularly effective in tasks that involve sequential dependencies, such as text generation and sentiment analysis. lstm comprises recurrent cells with memory gates that enable the retention of crucial information over extended sequences. it tackles the vanishing gradient problem that plagues traditional rnns, making it well-suited for tasks where context from distant words is essential. the lstm cell consists of three primary gates: the input gate, the forget gate, and the output gate. these gates regulate the flow of information through the cell, allowing it to capture longrange dependencies in the text. grus, like rnns, are recurrent networks, but they incorporate gating mechanisms that enable better handling of long-range dependencies and mitigate the vanishing gradient problem. this makes them suitable for text classification tasks that involve capturing contextual information. transformer-based models [151], exemplified by the bert (bidirectional encoder representations from transformers) architecture, have revolutionized text classification. bert and its variants leverage attention mechanisms to capture contextual information, 140 a. kravets, d. semenochkin copyright ©2024 assa adv. in systems science and appl. (2024) achieving state-of-the-art results in various nlp tasks, including document classification and question answering. 3. systematic literature review the primary objective of this literature review is to systematically collect, analyze, and synthesize scholarly articles within the realm of text classification, specifically focusing on the diverse technologies employed in this domain. this comprehensive exploration aims to provide readers with a comprehensive understanding of the evolving landscape of text classification systems and the pivotal role of technological components therein. this review adheres to the well-established procedures outlined in kitchenham's systematic literature review protocol [74]. 3.1 research questions in pursuit of a comprehensive analysis, we have formulated two key research questions to guide this literature review: r1. what components have been used in classification systems? this primary research question directs our attention toward the diverse components and technologies embedded within text classification systems. we aim to categorize and explain the different components of these systems, including preprocessing techniques, feature selection methods, and the classification algorithms. r2. what data (dataset) was used to classify the text in the works under study? our secondary research question examines the basic datasets employed in the analyzed works. we endeavor to identify and describe the datasets used for these classification systems' training and evaluation. 3.2 constructing a literature review the first critical step in constructing this literature review was the selection of electronic databases housing scholarly articles. to ensure comprehensive coverage of relevant research, five prominent databases were chosen as primary sources: ieee xplore, acm digital library, springer, sciencedirect, and google scholar. these databases were selected due to their extensive repositories of academic publications across various domains, including artificial intelligence and nlp. given the focus of this review on text classification and the associated technologies, a comprehensive list of relevant keywords was compiled. these keywords include terms such as “text classification techniques”, “document classification techniques”, “text categorization”, “document categorization”, “text classification”, and “text classification applications”. leveraging the compiled keywords, a structured query was formulated for each of the selected electronic databases. an example query used is: “(((text or texts or document or documents) and (categorization or classification or categorize or classify or categorizing or classifying or classifier)) and (techniques or applications))”. the period for the selection of articles spanning from 2013 to 2022 was strategically chosen to encapsulate a decade of scholarly research. this timeframe not only reflects a substantial period of academic investigation but also coincides with the exponential growth of research related to artificial intelligence, a prevailing trend in contemporary academia. 3.3 literature selection process the literature selection process went through several stages to guarantee the inclusion of both relevant and informative sources. at the outset, a search of designated electronic databases generated a substantial number of results – 9435. following this, a methodical selection process was performed to enhance the quality of this population. the process involved text classification technologies in document categorization systems. a survey 141 copyright ©2024 assa. adv. in systems science and appl. (2024) assessing titles and keywords for relevance to the review's objectives while also removing duplicates. ultimately, 243 sources were identified as potentially relevant. to further narrow down the selection, we manually analyzed the abstracts and introductions of 243 sources based on a defined criterion. the presence of a classifier and a real-world experiment in the articles was the key inclusion criterion to ensure that the selected sources made a significant empirical contribution to the objectives of the review, rather than just being review articles. 110 research articles were selected for in-depth analysis during this literature review. table 3.1 displays the respective number of articles for each scholarly database. table 3.1. destitution articles for each scholarly database database potentially relevant results relevant results ieee xplore 68 37 science direct 62 25 acm digital library 47 19 springer 37 16 google scholar 29 13 4. literature review results in the results section, we present a comprehensive analysis of the 110 sources obtained from a thorough examination of five electronic databases. section 4.1 pro-vides an overarching view of the literature review's outcomes, including the distribution of sources by publication year and classification of each source based on re-search focus. section 4.2 delves into the outcomes of the investigation pertaining to the first research question, which centers on the technological aspects of text classification. this subsection is further divided into three sub-sections, each dedicated to elucidating the findings related to specific stages of the text classification process: data preparation, classifier training, and result evaluation. these sub-sections offer detailed insights into the myriad techniques and methods employed at each stage of the text classification pipeline. lastly, section 4.3 concentrates on the outcomes of the study concerning the second research question, focusing on the foundational datasets used in text classification research. 4.1 literature search results a graphical representation (fig. 4.1) depicting the distribution of sources by publication year within the selected 10-year research period (2013-2022) has been compiled. notably, this diagram reveals a significant surge in articles related to text classification in the year 2019, marking the pinnacle of research interest within the chosen timeframe. however, it's noteworthy that subsequent years display a gradual decline in scholarly output, a trend that can be attributed, at least in part, to the global impact of the coronavirus pandemic. as elucidated in section 3, the deliberate selection of a 10-year span aimed to spotlight the burgeoning interest in methods and approaches to text classification rooted in neural networks, particularly deep learning models. to scrutinize this evolution, a year-wise analysis was conducted, distinguishing between sources focusing on basic machine learning algorithms and those centered on deep learning. 142 a. kravets, d. semenochkin copyright ©2024 assa adv. in systems science and appl. (2024) fig. 4.1. publishing summary from 2013 to 2022 figure 4.2 provides a compelling narrative. it indicates that the emergence of interest in neural networks began around 2015, steadily gaining momentum. by 2019, deep learning and neural networks had reached their zenith, signifying a transformative shift in the field. notably, the data suggests that this enthusiasm remains unabated, with deep learning approaches increasingly supplanting basic algorithms as the preferred choice for text classification tasks. this trajectory underscores the enduring relevance and potency of neural networks in text classification research. fig. 4.2. distribution of articles by type of classification algorithm from 2013 to 2022 although the research theme consistently relates to text classification, it is important to note that various aspects of this field are often studied. however, all reviewed studies share an experimental component that involves creating and assessing a text classifier. the objectives of the research can be expressed by either focusing on the text classification problem or conducting a specialized study on the critical preparatory steps, including data preprocessing, text representation, and feature selection. the figure 4.3 reflects these nuances: most studies (83.64%) concentrate on the central issue of text classification or classifier development. a few works ([54] [55] [67] [89] [162] [96]) tackle text representation. in addition, several sources ([120] [171] [59] [14] [139] [90] [124] [149] [79] [129] [142] [130]) are dedicated to investigating feature selection techniques in-depth. text classification technologies in document categorization systems. a survey 143 copyright ©2024 assa. adv. in systems science and appl. (2024) fig. 4.3. distribution of all articles by subject of study 4.2 rq1 results 4.2.1 text representation while researching the sources, we compiled a list of text representation algorithms utilized in text classification tasks. to present our findings, table 4.1 displays the frequency of usage for particular text representation schemes. among the total number of references, word2vec and tf-idf are the two main methods, with respect to the total number of text representation algorithms in use, they are used in 48 percent of them. table 4.1. text representation algorithms algorithm percentage references tf-idf 24.5% (27 of 110) [161] [56] [67] [89] [171] [14] [124] [149] [142] [130] [72] [87] [102] [157] [156] [107] [170] [165] [111] [24] [158] [22] [19] [8] [128] [13] [160] word2vec 23.6% (27 of 110) [54] [67] [96] [72] [168] [163] [87] [127] [137] [82] [103] [102] [157] [11] [140] [156] [27] [85] [53] [135] [174] [47] [36] [45] [154] [169] bow 10.9% (12 of 110) [56] [162] [14] [8] [103] [11] [3] [44] [28] [92] [69] [21] glove 6.3% (7 of 110) [54] [102] [103] [176] [177] [81] [152] fasttext 2.7% (3 of 110) [103] [53] [166] doc2vec 2.7% (3 of 110) [30] [157] [92] in the following, we briefly outline the text representation algorithms highlighted by the review. in 6.3% of the sources, the bow text representation model is used. the bow model treats each document as an unordered set of words and represents it as a sparse vector, where each dimension corresponds to a unique term in the entire corpus. doc2vec, also known as paragraph vectors [80], extends the concept of word embeddings to entire documents. it learns fixed-length vector representations for documents by treating them as continuous pieces of text. doc2vec employs a neural network architecture that combines word vectors with a document-specific vector. source [96] employs an algorithm founded upon word2vec, known as towe (taskoriented word embedding), tailored explicitly for text classification tasks. it introduces a function-aware component, strategically accentuating the functional attributes of words within the embedding space. 144 a. kravets, d. semenochkin copyright ©2024 assa adv. in systems science and appl. (2024) source [89] presents an enhanced tf-idf algorithm that emphasizes the concentration of feature words within specific categories, the dispersion degree within a class, and the strength of association between a feature word and a class. source [142] introduces the tax2vec approach, which integrates semantic information from taxonomies to construct enhanced semantic features for improving the performance of short text classification. this novel algorithm leverages a labeled or unlabeled corpus of documents and a taxonomy of words. source [162] introduces a novel approach termed the attribute of network extended to bag-of-words (aebow) model. aebow seeks to enhance traditional bow representations by incorporating attribute weights from a network structure. 4.2.2 feature selection in the corpus of research articles reviewed, it is noteworthy that only a minority, specifically 15.45%, explicitly incorporate feature selection techniques within their text classification methodologies ([171] [59] [14] [139] [124] [149] [79] [129] [142] [130] [158] [3] [152] [51] [5] [6] [2]). among the feature selection methods encountered, the chi-square (chi) algorithm emerges as the most prevalent and widely adopted ([171] [124] [149] [129] [152] [51] [6]). information gain stands as the second most prevalent feature selection algorithm within the scope of this review ([139] [129] [142] [51] [5]). sources [129] and [142] utilize mi as a feature selection algorithm. sources [129] and [51] employ document frequency (df) as a feature selection algorithm. df assesses the importance of a feature by quantifying the number of documents in which it appears. source [59] utilizes normalized difference measure (ndm) as a feature selection algorithm. ndm quantifies the dissimilarity between term distributions in different classes. source [14] employs genetic algorithms (ga) as a feature selection technique. gas are optimization algorithms inspired by the process of natural selection. they iteratively evolve a population of feature subsets to maximize classification performance. moreover, it is noteworthy that source [139] introduces a novel adaptation of the information gain algorithm, termed maximizing global information gain (mgig). this innovative approach substantially mitigates computational complexity, reducing it from o(vk2) to o(vk), where v signifies the number of terms and k represents the number of terms. source [149] employs a trio of feature selection algorithms: poisson distribution, relative discrimination criterion, and discriminative features selection (dfss). poisson distribution models feature occurrence in documents as a poisson process [119]. features with a significantly different poisson distribution across classes are selected. relative discrimination criterion evaluates feature importance based on the relative entropy between class-conditional feature distributions [131]. features that exhibit higher relative discrimination are chosen. source [79] employs the firefly algorithm as a feature selection technique. the firefly algorithm is a nature-inspired optimization algorithm that models the flashing behavior of fireflies to solve optimization problems [164]. in the context of feature selection, it operates by iteratively selecting a subset of features that maximize a specific objective function. source [129] employs an array of feature selection algorithms, including odd ratio and ngl correlation coefficient, in addition to chi, ig, and mi: source [158] employs singular value decomposition (svd) as a feature selection algorithm. svd is a dimensionality reduction technique that factors the term-document matrix into singular values, providing a low-rank approximation [31]. it effectively captures the most significant latent semantic structures within the data. source [2] employs pearson correlation coefficients as a feature selection algorithm. pearson correlation measures the linear relationship between two variables. in feature selection, it quantifies the degree of association between individual features and class labels. text classification technologies in document categorization systems. a survey 145 copyright ©2024 assa. adv. in systems science and appl. (2024) 4.2.3 text classification in line with the classification made in section 2.3 of this review, we will present the findings regarding text classification research by distinguishing between basic machine learning algorithms and deep learning algorithms. to provide an overview of the findings regarding basic machine learning algorithms, a table 4.2 has been generated. table 4.2. basic machine learning algorithms algorithm percentage references svm 29% (32 of 110) [55] [67] [89] [120] [171] [59] [14] [139] [149] [142] [72] [87] [157] [107] [111] [158] [13] [160] [103] [11] [3] [44] [28] [69] [51] [2] [88] [33] [68] [71] [16] [46] nb 20% (22 of 110) [161] [67] [120] [59] [14] [139] [124] [129] [157] [111] [158] [13] [3] [44] [28] [51] [2] [68] [71] [46] [122] [155] [141] knn 13,6% (15 of 110) [162] [120] [14] [129] [157] [170] [111] [24] [13] [92] [2] [88] [71] [46] [116] rf 8,1% (9 of 110) [56] [120] [13] [44] [69] [68] [46] [122] [136] lr 7,2% (8 of 110) [120] [102] [157] [22] [13] [21] [68] [46] sources [89] and [87] uses a linearsvm a variant of svm designed specifically for linearly separable data, where classes can be separated by a hyperplane [37]. linearsvm employs a linear kernel function, which simplifies the classification boundary into a linear equation. source [88] introduces an innovative approach called svm-knn for text classification, which combines two distinct algorithms: svm and knn. the svm-knn algorithm aims to improve classification performance by leveraging the strengths of both svm and knn. source [33] introduces cbssvm, a method that combines svm with a learning strategy known as center-based similarity space learning [32]. sources [44] and [122] employ a variant of naive bayes known as nb network for text classification [35]. nb network extends the basic nb algorithm by introducing a network structure, which allows for capturing dependencies among features (words) in the data. sources [161] and [46] used a multi-nominal naive bayes (multi-nb). it is a specialized variant of naive bayes [106] designed for dealing with categorical data, which includes text data where the features represent the frequency or presence of terms. source [155] introduce a nbtree a hybrid classification algorithm that combines the principles of decision trees with naive bayes classifiers. in this approach, a decision tree is constructed to make decisions based on the values of certain features. source [56] introduces an rf-based algorithm called semantics aware random forest (sarf) for text classification. sarf aims to enhance the performance of rf by considering semantics and the relevance of features in the classification process. 4.2.4 deep learning algorithms in analyzing the results of the literature review in section 4.1, it was found that deep learning models have attracted considerable attention in the field of text classification. the results of the research on classifiers related to deep learning are shown in table 4.3. table 4.3. deep learning algorithms algorithm percentage references lstm 23.6% (26 of 110) [54] [130] [102] [163] [137] [82] [103] [85] [53] [135] [174] [176] [81] [152] [141] [50] [175] [95] [114] [94] [138] [91] [23] [4] cnn 20% (22 of 110) [90] [19] [160] [82] [103] [11] [47] [45] [154] [169] [5] [68] [23] [20] [99] [100] [93] [173] [84] [134] [73] [10] [86] 146 a. kravets, d. semenochkin copyright ©2024 assa adv. in systems science and appl. (2024) rnn 10,9% (12 of 110) [22] [8] [127] [103] [53] [45] [177] [4] [20] [99] [78] [132] transformer 8,1% (9 of 110) [22] [103] [138] [23] [145] [41] [108] [61] in the area of deep learning, it should be noted that the use of lstm models has become one of the most prominent topics. lstm, although technically an rnn model, has been widespread enough to merit a separate category because of its unique architecture and usefulness for analyzing sequential data. lstm and its derived models represent the most commonly used classification algorithms among the sources reviewed. in addition, along with lstm-based models, cnn models are the second most popular. rnn-based models and transformer-based models are not the least popular. other learning models include graph neural networks (gnns) ([165] [128] [29]) and hierarchical attention networks (hans) ([142] [22] [127] [47] [36]). sources ([177] [4] [132]) use gru architecture for text classification. sources [130] [137] [152] [95] [114] [23] [4] employ the conventional lstm architecture as a fundamental component of their text classification models. sources [54], [103] [102] [27] [176] [135] employ bidirectional lstm (bi-lstm) models for text classification. bi-lstm, an extension of standard lstm, offers the ability to capture contextual information from both past and future states of a word within a sequence. bi-lstm operates by employing two lstm layers—one that processes the sequence from the beginning to the end (forward lstm), and another that processes the sequence in reverse (backward lstm). source [91] employs an innovative bilstm model called attention-based bi-directional lstm with a convolutional layer (ac-bilstm). in ac-bilstm, the convolutional layer extracts higher-level phrase representations from word embedding vectors, and bilstm is used to access representations of preceding and following contexts. source [82] employs a bi-lstm-based model known as bi-lstm-cnn. in this architecture, the bi-lstm component captures contextual information from both directions, and this information is then fused through a cnn. source [85] introduces a model, known as bidirectional attention-based lstm or ablstm, combines deep learning techniques with regular expression-based classification strategies to effectively categorize medical texts. source [174] employs a hybrid model called brcan (bidirectional recurrent attentionbased convolutional neural network), which combines bi-lstm with a cnn, attention mechanisms, and word2vec embeddings. source [50] introduces the universal language model fine-tuning (ulmfit) architecture, an extension of the awd-lstm model. this model, based on awd-lstm, represents a state-of-the-art approach to text classification. furthermore, source [110] presents a modification to the lstm model, focusing on a set of highly effective regularization strategies that can be seamlessly integrated into existing lstm implementations. source [141] and source [138] also leverage the ulmfit model. additionally, source [175] introduces the c-lstm architecture, which combines the strengths of cnns and rnns to enhance text classification. source [81] similarly employs a model inspired by the c-lstm architecture for text classification, further emphasizing the efficacy of this combined cnn and lstm approach. sources [47] and [100] uses a similar architecture, also combining cnn and rnn. source [163] introduces the ranklstm classification model, which operates through a unified learning-ranking process. in ranklstm, the arrangement of document labels is transformed based on a semantic tree, ensuring compatibility and consistency with the sequential learning lstm framework. to address the challenge of variable label numbers within each document, ranklstm employs connectionist temporal classification (ctc) techniques [40], effectively mitigating the problem of error propagation. source [94] proposes an asp-mtl (adversarial shared-private model for multi-task learning) that incorporates lstm layers. this architecture features a shared recurrent neural text classification technologies in document categorization systems. a survey 147 copyright ©2024 assa. adv. in systems science and appl. (2024) layer that engages in adversarial learning against a trained multi-layer perceptron, actively preventing it from making precise predictions regarding task types. moreover, sources [22] and [127] employ bidirectional gru models with an attention mechanism. bidirectional grus process input sequences in both forward and backward directions, enhancing their ability to capture contextual information. the attention mechanism allows the model to focus on important words or phrases within a sentence. source [53] introduces the hierarchical attention-based recurrent neural network (harnn), which leverages a hierarchical attention mechanism to model dependencies between different levels of a hierarchical structure. this top-down approach facilitates the capture of associations between texts and the hierarchical structure, making it effective for tasks that involve multi-level information. source [8] presents the dal (deep active learning) algorithm, which combines deep learning, based on rnn architecture, with active learning. this fusion of deep learning with active learning techniques enhances the model's performance while minimizing the need for extensive labeled data. source [45] employs a bidirectional recurrent neural network (brnn). brnns have the advantage of processing variable-length sequences and maintaining access to both past and future information, which can be valuable for text classification tasks where contextual understanding is crucial. source [5] implements a single-layer multisize filters convolutional neural network (smfcnn), which incorporates multiple filters of different sizes applied to various windows of text. source [86] presents a convolutional spatial attention neural network (sva-cnn) that utilizes a carefully designed training set of multiview representations, along with a combination of heterogeneous attention mechanisms and cnn-based operations. sva-cnn automatically extracts and weights multiple granular and fine-grained representations, enhancing its performance in capturing complex textual features. most of these sources rely on the bert (bidirectional encoder representations from transformers) ([22] [103] [138] [145] [108] [61]) model for text classification. bert is a pretrained transformer-based model that learns contextual word representations by considering the surrounding words in both directions (left and right) of a given word. this bidirectional context makes bert particularly powerful in understanding context and semantics in text data. source [41] introduces the multi-scale self-attention and multi-scale transformer, which combines multi-scale knowledge with self-attention mechanisms. this approach allows the model to extract rich and robust features from text data at different scales, enhancing its ability to understand context and semantics. source [61] presents lightxml, which adopts an end-to-end training approach with dynamic negative label sampling. lightxml employs generative cooperative networks to recall and rank labels, with the label recalling part generating negative and positive labels, and the label ranking part distinguishing positive labels. source [165] introduces text gcn (graph convolutional network), a method that leverages gnns for text classification. it constructs a single large graph from an entire corpus, where words and documents serve as nodes. text gcn utilizes graph convolutional operations to capture the structural dependencies and relationships between words and documents within this graph [128]. source [29] introduces a text classification model based on gnns known as hypergat (hypergraph attention networks). hypergat is designed to capture high-order word interactions within a hypergraph representation of text data. it employs a dual attention mechanism to highlight crucial contextual information, enabling the learning of highly expressive text representations. 148 a. kravets, d. semenochkin copyright ©2024 assa adv. in systems science and appl. (2024) source [36] introduces an architecture called hcan (hierarchical convolutional attention networks), which combines the strengths of self-attention, capable of capturing linguistic relationships over long sequences like rnns, with the efficiency of training found in cnns. 4.2.5 evaluation during the analysis of the reviewed sources, the metrics employed by authors to evaluate the effectiveness of their text classification solutions were identified. seven metrics were found to be used, with varying frequencies, and their usage are displayed in the figure 4.4. the most commonly employed metrics among researchers were precision, accuracy, recall, and fmeasure. fig. 4.4. classification evaluation metrics f-measure was utilized in 46% (51 of 110) percent of the analyzed sources. the f-measure, also known as the f1 score, is a metric that combines both precision and recall into a single value, providing a balanced assessment of a classification model's performance. it is particularly useful when dealing with imbalanced datasets and is calculated as the harmonic mean of precision and recall. furthermore, researchers often reported micro, macro, and weighted averages of the f1 score to account for variations in class distribution. precision was employed in 43% (48 of 110) percent of the sources. precision measures the accuracy of positive predictions made by a classifier, indicating the proportion of true positive predictions out of all positive predictions. it focuses on the precision of the model's positive predictions. accuracy was utilized in 40% (44 of 110) percent of the sources. accuracy is a straightforward metric that measures the ratio of correctly predicted instances to the total instances in the dataset. it provides an overall assessment of a classifier's correctness. recall, also known as sensitivity or true positive rate, was employed in 34% (38 of 110) percent of the sources. recall measures the proportion of true positive predictions out of all actual positive instances in the dataset. it emphasizes the classifier's ability to correctly identify positive instances. in contrast, fewer sources utilized metrics like error rate ([102] [82] [135] [177] [51] [141] [50] [114] [94] [145]) and area under the curve (auc) ([90] [72] [11] [53] [3] [2]). error rate is a simple metric used to measure the accuracy of a classification model. it represents the proportion of incorrectly classified instances in a dataset. auc is a metric used to evaluate the performance of binary classification models, particularly for tasks involving the calculation of receiver operating characteristic (roc) curves. text classification technologies in document categorization systems. a survey 149 copyright ©2024 assa. adv. in systems science and appl. (2024) 4.3 rq2 results during the literature review, all datasets used by the researchers in their studies were organized and documented. these datasets have been compiled and are presented in accompanying tables 4.4 and 4.5. table 4.4 presents the distribution by domain. table 4.4. dataset domain distribution domain number of datasets number of references news 12 65 reviews 10 59 text 8 15 medicine 10 15 web content 5 12 questions 2 10 academic papers 5 7 legal 2 7 articles 1 4 documents 2 2 sms 1 2 the table 4.5 shows the top 10 most popular datasets by number of mentions. table 4.5. 10 most popular dataset name domain source percentage references 20 news group news [157] 15,5% (17 of 110) [161] [162] [59] [14] [139] [130] [107] [158] [165] [81] [176] [33] [136] [78] [108] [29] reuters news [158] 13.6% (15 of 110) [55] [162] [59] [14] [139] [124] [149] [129] [130] [107] [165] [81] [44] [136] [29] imdb reviews [159] 11.8% (13 of 110) [96] [135] [45] [141] [50] [145] [95] [114] [94] [91] [23] [108] [105] mr reviews [160] 10% (11 of 110) [96] [165] [81] [154] [176] [94] [91] [173] [84] [86] [29] sst reviews [161] 9% (10 of 110) [162] [96] [45] [154] [176] [175] [95] [91] [23] [78] ag news news [162] 8.1% (9 of 110) [102] [135] [154] [50] [4] [100] [86] [145] [66] amazon reviews [164] 8.1% (9 of 110) [54] [162] [36] [33] [93] [86] [108] [61] [66] dbpedia text [165] 7.2% (8 of 110) [135] [16] [50] [4] [100] [145] [108] [66] sogou news news [162] 6.3% (7 of 110) [140] [27] [174] [100] [132] [145] [66] yelp reviews [166] 6.3% (7 of 110) [174] [36] [50] [100] [145] [66] [109] however, for the sake of brevity and focused analysis, we will primarily delve into the datasets that do not fall within the realm of the top 10 most frequently employed datasets in the field of text classification. while the top 10 datasets provide valuable insights into the foundational and widely recognized benchmarks for text classification, our objective is to shed light on datasets that might not receive as much attention but are nonetheless significant for specific domains, research angles, or unique challenges. by examining these lesser-known datasets, we aim to broaden the scope of our analysis and capture a more comprehensive understanding of the diverse applications and contexts of text classification in the academic literature. the eur-lex ([54] [22] [138] [93] [132] [61]) text collection is a collection of documents about european union law [97]. it contains many several different types of documents, including treaties, legislation, case-law and legislative proposals, which are indexed according to several orthogonal categorization schemes to allow for multiple search facilities. the most important categorization is provided by the eurovoc descriptors, which is a topic hierarchy with almost 4000 categories regarding different aspects of european law. trec ([154] [176] [177] [50] [91]) question dataset involves classifying a question into 6 question types [146]. trec divides all questions into 6 categories, including 505 locations, 150 a. kravets, d. semenochkin copyright ©2024 assa adv. in systems science and appl. (2024) human, entity, abbreviation, description and numeric. the training dataset contains 5452 labelled questions while the testing dataset contains 500 questions. thucnews ([8] [160] [82] [27]) corpus includes total of 10 news categories and total of 740,000 chinese news texts, all in utf-8 plain text format. it selects news from the ten categories of sports, finance, real estate, home, education, technology, fashion, politics, games and entertainment as experimental data. the webkb dataset ([162] [139] [44] [136]) contains 4,199 documents. each document contains around 10,358 characters that are divided into around 2,918 words. the documents are webpages collected by the world-wide knowledge base project of the cmu text-learning group. these pages were manually classified into seven different classes: student, faculty, staff, department, course, project, and other. the yahoo! answers dataset ([174] [100] [145] [66]) consists of 10 categories including society and culture, science and mathematics, health, education and reference, computer and internet, sports, business and finance, entertainment and music, family and relationships, and politics and government. documents include the title of the question, the background of the question and the best answer. there are 140,000 training samples and 5,000 test samples [172]. three sources use datasets consisting of material from scholarly articles from the arxiv website. source [54] uses a variation of the dataset consisting of 55,840 abstracts of computer science papers from 54 categories. source [109] uses a variation of the dataset that contains 1 million abstracts of scientific papers, including the title and abstract of the article. each paper has one primary category, e.g., cs.ai, stat.ml, etc., used as a class label. source [135] uses a variation of the dataset consisting of 230,105 abstracts from 53 categories. medline ([122] [136] [134]) and the ohsumed ([165] [128] [29]) corpus is from the medline database [118], which is a bibliographic database of important medical literature maintained by the national library of medicine. elec ([135] [114]) consists of reviews of electronics products on amazon and contains 200,000 reviews [64]. mimic-iii ([103] [152]) dataset is an openly available dataset developed by the mit lab for computational physiology [63]. it comprises clinical notes, demographics, vital signs, laboratory tests and other data associated with 40,000 critical care patients. 2 sources use the new york times news articles dataset. source [109] independently constructs a dataset of 13,081 news articles using the new york times api. this news corpus covers 5 supercategories and 25 subcategories. source [47] uses the nytimes, a large-scale corpus containing nearly every article published in the new york times between january 01, 1987 and june 19th, 2007 [148], selecting articles with the news tag. rotten tomatoes ([114] [91]) consists of short snippets of movie reviews, for sentiment classification [121]. rotten tomatoes web site pages where reviews marked with «fresh» are labeled as positive, and reviews marked with «rotten» are labeled as negative the sms spam collection ([44] [46]) dataset is a public set of sms labeled messages that have been collected for mobile phone spam research [143]. this dataset contains 5,574 english, real, and non-encoded messages, tagged as being legitimate (ham) or spam. each sms message contains around 82 characters that are divided into around 19 words. sources [88] and [20] use datasets based on information from sohu, one of china's major websites. source [20] uses a dataset that comes from the news data of 18 channels including domestic, international, sports, society and entertainment from june 2012 to july 2012, and provides url and body information. source [88] uses a dataset from sohu laboratory. dataset uses in [13] is titled as «covid fake news dataset» developed by [123] and published on coronavirus disease research community-covid-19. dataset contains 10202 covid fake news shared all over social media platforms including facebook, instagram posts, and news on social media with the keywords covid-19, coronavirus, and pandemic. text classification technologies in document categorization systems. a survey 151 copyright ©2024 assa. adv. in systems science and appl. (2024) 5. discussion this section presents the discussion on all the rqs stated in section 3. 5.1 what components have been used in classification systems? based on the comprehensive analysis conducted in this literature review, we can confidently assert that text classification has firmly established itself as a prominent and evolving problem within the domain of nlp. researchers continue to devote considerable attention to this area, driven by its increasing significance and the diverse applications it encompasses. one notable trend observed is the shift towards adopting cutting-edge technologies and novel approaches to address text classification challenges. as revealed in section 4.1, more and more researchers are applying neural network and deep learning technologies. this transformative trend extends beyond using deep learning solely as a classification tool; it significantly influences various facets of text classification. for instance, word representation algorithms like word2vec and glove, which are fundamental to text classification, rely on neural networks to learn word embeddings, thereby enhancing the quality of text representations. traditional text preprocessing techniques such as bag of words (bow) and tf-idf, which have historically underpinned text classification, still maintain their relevance, representing 44 percent of all analyzed sources. researchers, however, are continually innovating within this space, often infusing these established techniques with neural network architectures ([89] [162] [142] [107]), thereby optimizing text preprocessing for modern classification tasks. feature selection for text classification remains an active research area, with researchers employing a variety of statistical algorithms in their classification pipelines ([171] [139] [79] [129]). feature selection plays a pivotal role in enhancing the efficiency and effectiveness of classifiers. while text representation and preprocessing are essential, classifiers themselves are the focal point of many studies. there is a great interest in recurrent neural networks, namely lstm according to the results of the study, it is on the basis of this variant of rnn that researchers most often proposed new methods, such as ranklstm [163], awd-lstm [50], c-lstm [175], crnn [100], asp-mtl [94], ac-bilstm [91], bi-lstm-cnn [82] and ablstm [85]. it is also worth noting that the use of multilayer networks gives researchers the opportunity to cross layers in different architectures, such as ac-bilstm, the convolutional layer extracts the higher-level phrase representations from the word embedding vectors and bilstm is used to access both the preceding and succeeding context representations. bi-lstm-cnn and na-cnn-lstm [99] present an architecture that integrates cnns and rnns. the attention mechanism has also gained prominence in text classification ([54] [85] [53] [174] [36] [91] [73] [41] [29] [86]). attention mechanisms enable models to focus on specific parts of input sequences, enhancing their ability to capture contextual information effectively. another pivotal development is the increasing utilization of transformer models in text classification ([103] [22] [138] [23] [145] [41] [108] [61]). the ascendancy of transformer models, epitomized by gpt and its variants, signifies the transformative potential of largescale language models. researchers recognize that these models are at the forefront of nlp, offering unmatched capabilities for understanding and generating human-like text. at present, the chatgpt language model is the center of attention of the world community [34]. despite the growing prevalence of neural network-based classification algorithms, basic algorithms like naive bayes and svm continue to hold their ground 50 percent of sources mention just basic algorithms. these basic methods offer several advantages, including simplicity, interpretability, and efficiency, which make them well-suited for certain classification tasks. furthermore, for datasets with limited training examples, basic algorithms can often deliver competitive performance without the computational demands associated with deep learning [13]. 152 a. kravets, d. semenochkin copyright ©2024 assa adv. in systems science and appl. (2024) in general, it can be observed that the evolving landscape of text classification is characterized by the fusion of deep learning, neural networks, basic methods and innovative approaches. researchers are constantly pushing the boundaries of what is possible, driven by the need to develop efficient, accurate, and context-aware text classification solutions that meet the requirements of modern applications. the prevalence of deep learning, attention mechanisms, and transformational models emphasizes their key role in shaping the future of text classification. 5.2 what data (dataset) was used to classify the text in the works under study? analyzing datasets used in text classification studies reveals interesting trends and preferences of researchers. similar to source 11, we have given the 10 most popular datasets and also categorized them by domain. among the most frequently used dataset types, news sets (20 news group, reuters, ag news), covering areas such as sports, finance, health and politics, stand out, occurring in 65 of the 110 sources examined. these datasets are commonly used to solve multi-class categorization problems. in addition, datasets containing reviews, such as imdb, mr, sst and amazon, enjoy considerable popularity, mainly for binary categorization tasks where reviews are categorized as positive or negative. however, this study also sheds light on less frequently employed datasets, often tailored to address specific domains or niche problems. one such example is the medicine field 10 datasets were identified in the literature review and only 3 of them were mentioned 2 or more times (medline, ohsumed, mimic-iii), the remaining studies used unique datasets ([127] [21] [152] [173]). similarly, datasets like eur-lex find their niche in legal domain applications ([54] [22] [138] [93] [61] [132]). another noteworthy trend is the use of question-and-answer datasets like trec and yahoo! answers, featured in 10 out of 110 sources, particularly for multiclass classification, which often includes question categorization. in summary, while popular datasets in news and reviews remain at the forefront of text classification research, the emergence of domain-specific datasets and specialized corpora highlights the evolving landscape of text classification, catering to a broader range of applications and research inquiries. 6. conclusion this paper presents a comprehensive literature review spanning the years 2013 to 2022, focusing on the technologies and datasets employed in the field of text classification. the review encompassed a total of 110 sources, all of which contained experimental data related to text classification processes. the study emphasized three fundamental stages in text classification: data preparation, classifier training, and result evaluation. detailed statistics were compiled concerning the use of text representation and feature selection algorithms, as well as the various text classifiers adopted by researchers. each technology integrated into the classification pipeline was meticulously examined within its respective source. additionally, significant attention was dedicated to analyzing the diverse datasets utilized for text classification, with a particular emphasis on the less frequently employed ones. one prominent trend gleaned from this analysis is the increasing prevalence of deep learning-based technologies in the field. this study offers valuable insights into the evolving landscape of text classification, shedding light on the diverse array of technologies, approaches, and datasets employed by researchers. references 1. abiodun, e. o., alabdulatif, a., abiodun, o. i., alawida, m., alabdulatif, a., et al. 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(2019). efficient processing of gru based on word embedding for text classification. international journal on informatics visualization, 3(4). https://doi.org/10.30630/joiv.3.4.289 adv syst sci appl 2022; 04:51–64 published online at https://ijassa.ipu.ru. oscillating and proper solutions of singular quasi-linear differential equations natalya g. pavlova1,2,3 alexey o. remizov2* 1v.a. trapeznikov institute of control sciences, russian academy of sciences, moscow, russia 2moscow institute of physics and technology (state university), dolgoprudnyi, russia 3rudn university, moscow, russia abstract: we study the behaviour of solutions of quasi-linear differential equations of the second order at their singular points, where the coefficient of the second-order derivative vanishes. we consider solutions entering a singular point either with definite tangential direction (proper solutions) or without definite tangential direction (oscillating solutions). equations of this type appear in many problems arising in analysis, geometry, dynamical systems theory, and physics. first, we prove that a generic equation of the considered type has no oscillating solutions. then we concentrate on proper solutions, which can enter a singular point in admissible tangential directions only. great attention is paid to second-order differential equations, whose right-hand sides are cubic polynomials by the first-order derivative. we obtain local representations for solutions of such equations in a form similar to newton–puiseux series – series with fractional exponents (and, in a special case, with logarithmic terms). keywords: differential equations, singular points, normal forms, oscillating solutions 1. introduction 1.1. a brief look at quasi-linear equations the study of singular quasi-linear differential equations goes back to the middle of the xix century. one of the first examples is the equation xmy′ = f(x, y) (1.1) named after briot and bouquet (french mathematicians of the 19th century), who carried out a detailed study of such equations for analytic functions f ; the main achievement of this study was representation of solutions in the form of power series with fractional exponents. in that time, the theory developed by briot and bouquet seemed so significant that one of the problems at the famous prize competition (1885) sponsored by oscar ii†, was to develop a comparable theory for non-linear equations of the general form f (x, y, y′) = 0. in the same year, poincaré published his third memoir “on curves defined by differential equations”, where he gave a start to the geometric approach in the theory of differential equations.‡ ∗corresponding author: alexey-remizov@yandex.ru †king of sweden (1872–1907) and norway (1872–1905). in his youth he studied at uppsala university, where he distinguished himself in mathematics. ‡it is worth observing that poincaré was awarded oscar ii prize for solution of another problem. 52 n. g. pavlova, a. o. remizov later on, the interest to singular quasi-linear differential equations was motivated either by pure theoretical aspects or by various applications. for example, equations ∆(x, y)y′′ = m(x, y, y′) (1.2) appear in many problems arising in analysis, geometry, dynamical systems theory, and physics. here one can name the bessel equation and the gaussian hypergeometric equation, to name a few. we also notice equations appearing in two recent works: a generalized ginzburglandau model for liquid crystals [8] and newton’s aerodynamic problem [16]. definition 1.1: a point q0 = (x0, y0) is called a regular point of equation (1.2) if ∆(x0, y0) ̸= 0, and it is called a singular point of equation (1.2) if ∆(x0, y0) = 0. among equations (1.2), a great attention is paid to the case that the right-hand side is a cubic polynomial in y′, that is, m(x, y, y′) = 3∑ i=0 µi(x, y)(y ′)i. (1.3) an attention to equations (1.2), (1.3) is motivated by their role in physics and geometry, for instance, the description of various geometric structures (geodesic flows in affine or projective connection, etc.). equations of this class were studies by sophus lie, a. tresse, j. liouville, e. cartan, etc. among others, an important case is the levi–civita connection, which is generated by the metric tensor ds2 = a(x, y) dx2 + 2b(x, y) dxdy + c(x, y) dy2. (1.4) geodesics in the levi–civita connection are solutions of equation (1.2), (1.3), where ∆ = ac− b2 and the coefficients of the cubic polynomial m are polynomials on the coefficients a, b, c and their first-order derivatives: µ0 = a(ay − 2bx) + axb, µ1 = b(3ay − 2bx) + axc− 2acx, µ2 = b(2by − 3cx) + 2ayc− acy, µ3 = c(2by − cx)− bcy. (1.5) here ∆ vanishes at points where the quadratic form (1.4) degenerates. in riemannian geometry, such is not the case, since the inequality ∆ > 0 holds true everywhere. however, metrics (quadratic forms) with varying signature generically appear on surfaces embedded into pseudo-euclidean spaces: the metric induced on a surface into pseudo-euclidean space degenerates at those points where the surface tangents the light cone of the ambient space. singularities of the geodesic equation appearing at degenerate points of the metric (1.4) are studied in the series of papers [22] – [19]. 1.2. phenomenon of oscillating solutions a. f. filippov showed [5] that a system of ordinary differential equations f (x, y, p) = 0, p = dy/dx, (1.6) where x ∈ r1, y = (y1, . . . , yn) ∈ rn, p = (p1, . . . , pn), and f : r2n+1 → rn is a smooth (by smooth we mean c∞, if the otherwise is not stated) vector function, may have copyright © 2022 assa. adv syst sci appl (2022) oscillating and proper solutions 53 solutions without definite tangential direction at a certain point. this motivates the following terminology. a solution y(x) of system (1.6) is called oscillating at a point (x0, y0), if y(x) is a vector function differentiable on the interval (x0, x0 + δ) or (x0 − δ, x0), δ > 0, such that y(x) → y0 but y′(x) has neither finite not infinite limit as x → x0. a solution y(x) is called proper at a point (x0, y0), if y′(x) has limit (finite or infinite) as x → x0. let j1 be the 1-jet space of smooth functions y(x), with the coordinates (x, y, p). filippov showed that if for a point t0 ∈ j1 there exists t ′ 0 ∈ j1 such that x0 = x′ 0, y0 = y′0, p0 ̸= p′0, and the matrix fp degenerates at t ′ 0, then besides a unique proper solution passing through (x0, y0) with the tangential direction p0, system (1.6) may have oscillating solutions, which pass through (x0, y0) and even have definite tangential direction p0 at (x0, y0). the following example is taken from [5]. consider solutions of the system{ p1(1− p21 − p22) + 8xy1 + 4y2 = 0, p2(1− p21 − p22) + 8xy2 − 4y1 = 0, (1.7) that pass through the origin with the tangential direction pi = 0, that is, satisfy the conditions yi(0) = y′i(0) = 0, i = 1, 2. the matrix fp is non-degenerate at the point t0 = 0, and system (1.7) has a unique proper solution yi(x) ≡ 0, which obviously satisfies the required conditions. however, system (1.7) has an infinite number of oscillating solutions given by the formula y1(x) = x2 cos(x−1 + c), y2(x) = x2 sin(x−1 + c), c = const, for x ̸= 0, and yi(0) = 0. the both derivatives y′i(x) are zero at x = 0, but the limits y′i(x) as x → 0 do not exist. on the other hand, filippov showed that systems with oscillating solutions are a sort of exception. given a point q0 = (x0, y0) ∈ rn+1, we define the set q(q0) = {t = (q0, p) ∈ j1 : f (t ) = 0, det(fp(t )) = 0}. theorem 1.1 ( [5]): assume that at least one of the following conditions holds true: 1. p0 /∈ coq(q0), where co denotes the convex hull, 2. the set q(q0) is at most countable (i.e., countable, or finite, or empty). then system (1.6) has no solution oscillating at q0 with tangential direction p0 at q0. generically, the condition 2 holds true at all points, and system (1.6) has no oscillating solutions. although theorem 1.1 has a very general character, it gives no information for some special classes of systems. for instance, it gives only a trivial result for quasi-linear implicit systems a(x, y) p = b(x, y), p = dy/dx, (1.8) where a is a n× n matrix depending on (x, y), b is a vector function. it is worth observing that system (1.8) are of great interest due to various applications, especially in electrical circuit theory. see, e.g., the papers [27] – [21] and the references therein). theorem 1.1 guarantees the absence of oscillating solution at all points where deta ̸= 0, which obviously follows from the fact that then system (1.8) is locally equivalent to the explicit system p = a−1(x, y)b(x, y). however, it says nothing about points where the matrix a degenerates, since q(q0) = {q0} × rn if deta(q0) = 0, and the both conditions in theorem 1.1 fail. 1.3. the aim, scope and structure of the paper the main emphasis of this paper is on oscillating solutions of quasi-linear second-order differential equations (1.2). it should be remarked that in all papers mentioned above (as well copyright © 2022 assa. adv syst sci appl (2022) 54 n. g. pavlova, a. o. remizov as in other works known to us) only proper geodesics were considered, while the possibility of oscillating geodesics is not studied yet. in the present paper, we shall fill this gap. the paper is organized as follows. in section 2, we establish a necessary condition for quasi-linear equation (1.2) to have an oscillating solution – theorem 2.1, which plays the same role as that played by theorem 1.1 for non-linear systems (1.6). this condition shows that a generic equation (1.2) with m analytic in p has no oscillating solutions, but proper solutions only. in particular, equation (1.2) with a cubic polynomial (1.3) that describes geodesics in the levi–civita connection generated by the metric tensor (1.4) has no oscillating solutions if ∆, d∆ do not vanish simultaneously. in section 3, we briefly describe proper solutions of equation (1.2) with the right-hand side (1.3) entering its generic singular points. proper solutions cannot enter singular points in arbitrary tangential directions, but only at so-called admissible directions p that correspond to real roots of the cubic polynomial m in p. to every admissible direction p, where p is a primer root of m , corresponds either a unique smooth solution or an infinite family of solutions with the common tangential direction and power singularity. theorem 3.3 gives the local representation of such solutions in a form similar to newton–puiseux series. this can be considered as a far development of the results obtained by poincaré and his predecessors (briot, bouquet, fuchs, etc.) for differential equations (1.1). however, in studying secondorder equations we encounter a special difficulty that does not appear in the case of the first order equations: vector fields with non-isolated singular points. a systematic study of such vector fields begins in the end of xx century, the first known works are [25, 28]. a brief survey of results on the local classification of such fields can be found in [24] (appendix). in section 4, we present a natural generalization of the results obtained before for quasilinear differential equations of higher orders. finally, we should remark the closely related papers [9] – [29] and especially [26]. 2. oscillating solutions in this section, we shall deal with differential equation (1.2). following the tradition to denote derivatives by single letters going back to gaspard monge, we write this equation in the following form: ∆(x, y) dp dx = m(x, y, p), p = dy/dx. (2.9) the functions ∆(x, y), m(x, y, p) are supposed to be smooth. denote the locus of singular points of equation (2.9) by γ. generically, the set γ = {(x, y) : ∆(x, y) = 0} is a curve on the (x, y)-plane, which is called singular curve or degenerate curve of the equation. we shall assume this in what follows. consider the initial value problem for equation (2.9) with the initial condition y(x0) = y0. if the point q0 = (x0, y0) is regular, for every p0 this problem has a unique solution satisfying the additional condition p(x0) = p0, which is defined and smooth on a real interval including the point x0, and it has no solutions without lim p(x) as x → x0. the situation becomes different if q0 is singular (see the examples below). this makes sense to refine the notion of solutions for the case of singular points. throughout this paper, we use the following notation: if i is an open real interval, then i denotes the closed interval (segment) with the same endpoints. definition 2.1: let q0 = (x0, y0) ∈ γ. a solution y(x) of equation (2.9) enters the point q0 if the function y(x) satisfies the following conditions: copyright © 2022 assa. adv syst sci appl (2022) oscillating and proper solutions 55 1. it is continuous on a non-empty segment iε with endpoints x0 and x0 + ε, where ε can be either positive or negative, y(x0) = y0. 2. the function y(x) is differentiable and it satisfies (2.9) on the open interval iε. 3. the graph of y(x) has no common points with γ except for q0. if in addition to the above conditions, the derivative p(x) has a (finite or infinite) limit as x → x0, the solution y(x) is called proper. otherwise the solution y(x) is called oscillating at the point q0. see fig. 2.1. definition 2.2: a solution y(x) is called passing through the point q0 if the function y(x) is differentiable and it satisfies (2.9) at all points of an open interval i that contains x0, y(x0) = y0, and the graph of y(x), x ∈ i , intersects γ at the point q0 only. the significance of the given definitions will become clear in the examples presented below. obviously, every solution passing through q0 is the union of two solutions entering q0. it is worth observing that solutions from definitions 2.1, 2.2 are also called one-sided or twosided solutions, respectively (for example, in [26]). in this paper, we prefer the terminology given above, which shows the geometric flavour of these notions. example 2.1: the parabolas y = αx2, α = const, are integral curves of the second-order equation 2y dp dx = p2, p = dy/dx, passing through the origin. however, the graphs of functions defined by the formula f(x) = { α1x 2, x ≤ 0, α2x 2, x > 0, are also integral curves, and all such functions are solution of the given equation passing through the origin. to avoid such ambiguity, it is sufficient to use the notion of a solution entering a point defined above. then solutions entering the origin are the branches of the parabolas y = αx2, α = const, and only them. in the previous example, all solutions are proper. now we give two examples of equations (2.9) that possess oscillating solutions. example 2.2: the equation x4 dp dx = 2x3p− (2x2 + 1)y, p = dy/dx, has the family of solutions defined by the formula y = x2(α cosx−1 + β sinx−1) for x ̸= 0, α, β = const, (2.10) and zero for x = 0. these functions are differentiable on the whole real axis, but their derivatives are not continuous at the origin except for α = β = 0. according to the given definitions, formula (2.10) gives a family of solutions passing through the origin with the tangential direction p = 0. for α = β = 0, we have the regular solution y = 0, while for all others α, β the solutions (2.10) are oscillating at the origin. example 2.3: the equation x2 dp dx = xp− 2y, p = dy/dx, copyright © 2022 assa. adv syst sci appl (2022) 56 n. g. pavlova, a. o. remizov has the family of solutions y = x(α cos ln |x|+ β sin ln |x|), α, β = const, (2.11) entering the origin. except for α = β = 0, all these solutions are oscillating at the origin and (unlike the previous example) they have no definite tangential directions at the origin. moreover, using formula (2.11), one can construct an equation of the form (2.9) with oscillating solution, whose right-hand side is a polynomial of arbitrary degree in p. for instance, one can see that the function (2.11) with α = β = 1 is a solution of the first-order equations fi(x, y, p) = 0, where f2 = (xp)2 − 2xyp+ 2(y2 − x2), f3 = (xp)3 + y(xp)2 − 2xp(x2 + 2y2) + 6y(y2 − x2). therefore, y = x(cos ln |x|+ sin ln |x|) is an oscillating solution of the second-order equations x2 dp dx = xp− 2y + fi(x, y, p), p = dy/dx. we presented several examples demonstrating the existence of oscillating solutions. the following theorem shows that oscillating solutions do not appear generically. theorem 2.1: let q0 ∈ γ and m(q0, p) be an analytic function not identically zero. then equation (2.9) has no oscillating solutions entering q0. proof assume that the above conditions hold true, but equation (2.9) has an oscillating solution y(x) that enters q0 without definite tangential direction. by the definition, y(x) is defined on a segment iε = [x0, x0 + ε] and ∆(x, y(x)) ̸= 0 at all its inner points: x0 < x < x0 + ε. without loss of generality, assume that ∆(x, y(x)) > 0 for all x such that x0 < x < x0 + ε (otherwise we multiply the both sides of (2.9) by −1). from the absence of the limit y′(x) as x → x0 it follows that there exist two sequences x′ n → x0 + 0 and x′′ n → x0 + 0 such that p(x′ n) → p′ and p(x′′ n) → p′′, p′ ̸= p′′. for definiteness, assume that p′ < p′′. since the function y′(x) is continuous on the interval x0 < x < x0 + ε, for every p∗ ∈ (p′, p′′) there exist two sequences ξn → x0 + 0 and ξ′n → x0 + 0 such that lim n→∞ p(ξn) = lim n→∞ p(ξ′n) = p∗, dp dx (ξn) > 0, dp dx (ξ′n) < 0. (2.12) see fig. 2.1 (right). substituting the solution y(x) into (2.9), at the points x = ξn and x = ξ′n we have the equalities ∆(ξn, y(ξn)) dp dx (ξn) = m(ξn, y(ξn), p(ξn)), (2.13) ∆(ξ′n, y(ξ ′ n)) dp dx (ξ′n) = m(ξ′n, y(ξ ′ n), p(ξ ′ n)). (2.14) copyright © 2022 assa. adv syst sci appl (2022) oscillating and proper solutions 57 the right-hand sides of both equalities (2.13), (2.14) have the finite limit m(x0, y0, p∗), whence their left-hand sides have the same limit m(x0, y0, p∗). on the other hand, ∆(ξn, y(ξn)) and ∆(ξ′n, y(ξ ′ n)) are positive for every n, and from (2.12) it follows that the left-hand sides of equalities (2.13), (2.14) have different signs. therefore, their limit m(x0, y0, p∗) = 0. thus, we proved that m(q0, p∗) = 0 for every p∗ ∈ (p′, p′′). since the function m(q0, p) is analytic in p, this implies m(q0, p) = 0 for every p. this completes the proof. y x p x n¢n * p fig. 2.1. an oscillating solution entering the origin (on the left) and its derivative (on the right). corollary 2.1: let q0 be a singular point of equation (1.2), (1.3). if the coefficients µ0, . . . , µ3 of the cubic polynomial m do not vanish at q0 simultaneously, there are no oscillating solutions entering q0. moreover, in the case of the geodesic equation, d∆(q0) = 0 is a necessary condition for the existence of an oscillating solution entering q0. proof the first statement is obvious, let us prove the second one. the geodesic equation for the metric (1.4) has the form (1.2), (1.3), where ∆ = ac− b2 and µ0, . . . , µ3 are expressed through a, b, c by formula (1.5). by theorem 2.1, for the nonexistence of solutions oscillating at q0 ∈ γ, it is sufficient to prove that if all coefficients µ0, . . . , µ3 simultaneously vanish at q0, then d∆(q0) = 0. let us assume on the contrary that µi(q0) = 0 for all i and d∆(q0) ̸= 0. to simplify the calculations, we choose local coordinates centered at q0 ∈ γ such that b(q0) = 0, which is always possible to attain by an appropriate linear transformation. then from the equalities ∆(q0) = 0, d∆(q0) ̸= 0 it follows that |a(q0)|+ |c(q0)| ≠ 0. without loss of generality one can assume that a(q0) ̸= 0 and c(q0) = 0. then from the equalities µi(q0) = 0 and (1.5) it follows cx(q0) = cy(q0) = 0. finally, from the equalities b(q0) = c(q0) = cx(q0) = cy(q0) = 0 it follows d∆(q0) = 0. the obtained contradiction completes the proof. 3. second-order equations cubic in the first-order derivative in this section, we shall briefly describe generic singularities of equation (1.2), whose righthand side has the form (1.3). here we shall assume that ∆, µ0, . . . , µ3 are smooth functions not connected with each other, in contrast with the geodesic equation, where all these functions are expressed through a, b, c and their derivatives. according to the genericity copyright © 2022 assa. adv syst sci appl (2022) 58 n. g. pavlova, a. o. remizov assumption, we shall assume that the coefficients µ0, . . . , µ3 of the cubic polynomial (1.3) do not vanish simultaneously and the set of singular points γ is a regular curve, which locally separates the (x, y)-plane into two domains. then equation (1.2) has no oscillating solutions, and we consider only proper solutions passing through or entering a point q0 ∈ γ. further we shall omit the word “proper”. theorem 3.1: solutions of equation (1.2) can enter a point q0 ∈ γ at directions p that corresponds to the real roots of m(q0, p) only. the proof trivially follows from the qualitative analysis of the vector field ẋ = ∆(q), ẏ = p∆(q), ṗ = m(q, p) (3.15) generated by equation (1.2) in the 1-jets space. given point q0 ∈ γ, we define the admissible directions p that correspond to singular points (q0, p) of the vector field (3.15), i.e., satisfy the conditions ∆(q0) = 0, m(q0, p) = 0. the field (3.15) is a partial case of vector fields ẋ = v, ẏ = w, żi = aiv + biw, i = 1, . . . , n, (3.16) where x ∈ r1, y ∈ r1 and z = (z1, . . . , zn) ∈ rn. here v, w and ai, bi are smooth functions on the variables (x, y, z). the field (3.16) can be invariantly defined as follows: all its components belong to the ideal i generated by two of them (for instance, v and w). obviously, the set of singular points of this field is given by two equations v = 0, w = 0. therefore, the spectrum of its linear part at every singular point is (λ1, λ2, 0, . . . , 0), where the eigenvalues λ1,2 (either real or complex) continuously depend on the point. if reλ1,2 ̸= 0, then the center manifold w c of the field (3.16) coincides with the set of its singular point, and the reduction principle (see [1] or [7]) states that the germ of (3.16) is topologically equivalent to ẋ = σ1x, ẏ = σ2y, żi = 0, i = 1, . . . , n, where σi = sign (reλi). we recall that the topological equivalence of two vector fields means that there exists a homeomorphism that sends the phase portrait of the first field to the phase portrait of the second field. smooth local normal forms of vector fields (3.16) are also known; see., e.g. [6] or [24] (appendix). here we present only one result of smooth classification. let q0 ∈ γ and m(q0, p∗) = 0, i.e., p∗ be an admissible direction at q0. the number and the behaviour of solutions of equation (1.2) that enter q0 with the direction p∗ is determined by the ratio of the non-zero eigenvalues of the field (3.15): λ1(q0, p∗) = (∆x + p∆y)(q0, p∗), λ2(q0, p∗) = mp(q0, p∗). further we shall always assume that λ1,2(q0, p∗) ̸= 0. define the value λ = λ2(q0, p∗) : λ1(q0, p∗) and the set n−1 = {1/n, n ∈ n}. copyright © 2022 assa. adv syst sci appl (2022) oscillating and proper solutions 59 theorem 3.2: if λ > 0, the germ of (3.16) is c∞-smoothly equivalent to ξ̇ = a1(ζ)ξ, η̇ = a2(ζ)η, ζ̇j = 0, λ /∈ n ∪ n−1, ξ̇ = a1(ζ)ξ, η̇ = a2(ζ)η + b2(ζ)ξ n, ζ̇j = 0, λ ∈ n ∪ n−1\{1}, ξ̇ = a1(ζ)ξ + b1(ζ)η, η̇ = a2(ζ)η + b2(ζ)ξ, ζ̇j = 0, λ = 1. using theorem 3.2, one can get the following result: theorem 3.3: if λ < 0, then equation (1.2) has only one solutions passing though the point q0 with the tangential direction p∗. if λ > 0, then equation (1.2) has an infinite number of solutions entering the point q0 with the direction p∗. in appropriate local coordinates centered at q0, these solutions have one of two following forms: y = f (x, c|x|λ), if λ /∈ n, y = f (x, xλ(c+ ε ln |x|)), ε ∈ {0, 1}, if λ ∈ n, where f is a smooth function, c = const. the proof of this theorem is quite similar to those for analogous theorem for the equation of geodesics in signature varying metrics (see [22] – [19]) although it has a much more general character. indeed, for a generic cubic polynomial m(q0, p), there are four (up to rearrangements of its real roots pi) possible cases: c1. 1 admissible direction p0 with λ(q0, p0) > 0, c2. 3 admissible directions p0, p1, p2 with λ(q0, p0) > 0 and λ(q0, pi) < 0 for i = 1, 2, c3. 1 admissible direction p0 with λ(q0, p0) < 0, c4. 3 admissible directions p0, p1, p2 with λ(q0, p0) < 0 and λ(q0, pi) > 0 for i = 1, 2. here negative λ corresponds to a unique solution passing through q0 with given tangential direction, while positive λ corresponds to an infinite family of solutions passing through q0 with given tangential direction. for equations of geodesics only the cases c1 and c2 can be realized. example 3.1: consider the differential equation x dp dx = αp(p2 − 1), α ̸= 0, (3.17) whose singular points fill the curve γ = {x = 0}. at every q0 ∈ γ the cubic polynomial m(p) = αp(p2 − 1) has three prime roots: p0 = 0 and p1,2 = ±1. here λ(q0, 0) = −α and λ(q0,±1) = 2α. therefore, if α > 0 then equation (3.17) has a single solution passing through the point q0 with the tangential direction p0 = 0 (the bold line in fig. 3.2, left) and two infinite families of solutions entering q0 with the directions p1,2 = ±1 (solid lines in fig. 3.2, left). if α < 0 then equation (3.17) has an infinite family of solutions entering the point q0 with the tangential direction p0 = 0 (solid lines in fig. 3.2, right) and two single solutions passing through the point q0 with the directions p1,2 = ±1 (the bold line in fig. 3.2, right). copyright © 2022 assa. adv syst sci appl (2022) 60 n. g. pavlova, a. o. remizov fig. 3.2. solutions of equation (3.17). on the left: α > 0. on the right: α < 0. example 3.2: consider the differential equation x dp dx = αp(p2 + 1), α ̸= 0. (3.18) whose singular points fill the curve γ = {x = 0}. at every q0 ∈ γ the cubic polynomial m(p) = αp(p2 + 1) has only one real root p0 = 0, whose λ(q0, 0) = α. therefore, if α > 0 then equation (3.18) has an infinite family of solutions entering the point q0 with the tangential direction p0 = 0 (solid lines in fig. 3.3, left). if α < 0 then equation (3.18) has a single solution entering the point q0 with the tangential direction p0 = 0 (the bold line in fig. 3.3, right). fig. 3.3. solutions of equation (3.18). on the left: α > 0. on the right: α < 0. example 3.3: three more examples taken from [18] present families of geodesics in signature varying metrics entering a point q0 (or passing through q0) where the metrics degenerates. in the series [22] – [19], it is shown that for equations of geodesics only the cases c1 and c2 can be realized. three such examples are presented in fig. 3.4. on the left and center: one admissible direction p0 = ∞ with λ > 0, an infinite family of geodesics entering q0. on the right: three admissible directions p0 = ∞ with λ > 0 and p1,2 with λ < 0, which give an infinite family of geodesics entering q0 with the direction p0 and two single geodesics (bold lines) passing trough q0 with the directions p1,2. copyright © 2022 assa. adv syst sci appl (2022) oscillating and proper solutions 61 fig. 3.4. three families of geodesics in degenerate metrics with varying signature. 4. generalization for higher order equations here we present some generalizations of the previous results. consider the quasi-linear differential equation of the order n ≥ 2 using the notations similar to (2.9): ∆(x, ȳ) dp dx = m(x, ȳ, p), ȳ = (y(0), . . . , y(n−2)), p = y(n−1), (4.19) where y(i) denotes the i-th derivative of y, and ∆(x, ȳ), m(x, ȳ, p) are smooth functions. it is important to emphasize that the coefficient ∆ standing with the higher derivative depends on x, y(0), . . . , y(n−2) and it does not depend on y(n−1). this plays an important role for the further study. denote the locus of singular points of equation (4.19) by γ i.e., γ = {(x, ȳ) : ∆(x, ȳ) = 0}. generically, γ is a hypersurface in the (x, ȳ)-space. we start with the following definitions that generalize definitions 2.1 and 2.2: definition 4.1: let q0 = (x0, ȳ0) ∈ γ. a solution y(x) of equation (4.19) enters the point q0 if the following conditions hold true: 1. the vector-function ȳ(x) is continuous on a non-empty segment iε with endpoints x0 and x0 + ε, where ε can be either positive or negative, ȳ(x0) = ȳ0. 2. the vector-function ȳ(x) is differentiable and it satisfies (4.19) on the open interval iε. 3. the graph of ȳ(x) has no common points with γ except for q0. if in addition to the above conditions, the function p(x) has a (finite or infinite) limit as x → x0, the solution y(x) is called proper. otherwise the solution y(x) is called oscillating at the point q0. definition 4.2: a solution y(x) is called passing through the point q0 if the vector-function ȳ(x) is differentiable and it satisfies (4.19) at all points of an open interval i that contains x0, ȳ(x0) = ȳ0, and the graph of ȳ(x), x ∈ i , intersects the hypersurface γ at the point q0 only. finally, we introduce one more notion. if y(x) is a solution of equation (4.19), the corresponding vector-function ȳ(x) we shall call an integral curve of this equation. theorem 4.1: let q0 ∈ γ and m(q0, p) be an analytic function not identically zero. then equation (4.19) has no oscillating solutions entering q0. the proof of this theorem repeats the proof of theorem 2.1. copyright © 2022 assa. adv syst sci appl (2022) 62 n. g. pavlova, a. o. remizov further we shall assume that the conditions of theorem 4.1 hold true. moreover, we assume that γ is a regular hypersurface, which locally separates the (x, ȳ)-plane into two domains denoted by d and d′. then equation (4.19) has no oscillating solutions, and we shall study its proper solutions (omitting the adjective proper). similarly to what was done in section 3, equation (4.19) generates a direction field in the (n− 1)-jet space jn−1 with the coordinates (x, ȳ, p). this direction field is determined by the vector field ẋ = ∆(x, ȳ), ẏ(i) = y(i+1)∆(x, ȳ), ẏ(n−2) = p∆(x, ȳ), ṗ = m(x, ȳ, p). (4.20) the projection π : jn−1 → jn−2 (along the p-direction called vertical) of an integral curve of the field (4.20) different from a straight vertical line, is an integral curve of equation (4.19). here we generalize the main statements of section 3 except for those of them that are connected with the direction p = ∞. the case p = ∞ is excluded from consideration, since it cannot be reduced to a finite value similarly to section 3 and it requires a special detail research for various types of equations. theorem 4.2: solutions of equation (4.19) can enter a singular point q0 ∈ γ only with admissible values of p that correspond to zeros of the function m(q0, p). the proof of this theorem repeats the proof of theorem 3.1. definition 4.3: we shall call a singular point q0 generic, if d∆(q0) ̸= 0, the function m(q0, p) has a finite number of zeros, all prime, and the corresponding admissible value pi determines the direction (y(1), . . . , y(n−2), p) in the (x, )̄-space that is transversal to the hypersurface γ, i.e., ∆x + y(1)∆y(0) + y(2)∆y(1) + · · ·+ p∆y(n−2) ̸= 0. singular points of the field (4.20) have the form (q, pi), where q ∈ γ and pi is one of zeros of the function m(q, p). the spectrum of the linear part of the field (4.20) at its singular point (q0, pi) has the form (0, . . . , 0, λ1, λ2), where λ1 = ∆x + y(1)∆y(0) + y(2)∆y(1) + · · ·+ p∆y(n−2) , λ2 = mp evaluated at (q0, pi). if q0 ∈ γ is generic, then the both eigenvalues λ1,2 are non-zero. denote λ(q0, pi) = λ2(q0, pi)/λ1(q0, pi). the sign of λ(q0, pi) determines the number of solutions entering the point q0 with the given admissible direction pi and their rough local properties: theorem 4.3: let q0 ∈ γ be a generic singular point of equation (4.19) and pi be an admissible value of p, that is, m(q0, pi) = 0. 1. if λ < 0, then the equation has a c∞-smooth solution passing through q0 and it has no other solutions entering q0 with the value pi. 2. if λ > 0, then the equation has an infinite number of solutions entering q0 with the admissible value pi. moreover, the family of corresponding integral curves contains an infinite number of curves entering q0 from the domain d and an infinite number of curves entering q0 from the domain d′. there exist local coordinates centered at q0 such that solutions mentioned above have one of two forms: y = f (x, c|x|λ), if λ /∈ n, y = f (x, xn(c+ ε ln |x|)), if λ = n ∈ n, copyright © 2022 assa. adv syst sci appl (2022) oscillating and proper solutions 63 where ε is 0 or 1, f is a smooth function on two variables, c = const. moreover, in the case that equation (4.19) is analytic, in the above statements c∞ can be replaced with cω. then the function f is analytic, and the latter two formulas give newton– puiseux series for solutions. theorem 4.3 is similar to theorem 3.3, the main difference is that now we do not bring the hypersurface γ in the (x, ȳ)-space to a simple form (for instance, the hyperplane x = 0). in general, it is impossible, since the coordinates of the (x, ȳ)-space with n > 2 are not independent variables. the proof of theorem 4.3 repeats the proof of theorem 3.3 with obvious changes. 5. conclusion first, we established a necessary condition for quasi-linear equations (1.2) to have an oscillating solution, which shows that generically such equations have no oscillating solution. however, the proof of this result (theorem 2.1) is not appropriate for quasi-linear systems (1.8), and the question of the existence of oscillating solution for generic systems (1.8) is still open. the interest to this question is motivated, in particular, by the studying of the dynamics generated by singular lagrangians, which goes back to the works of p.a.m. dirac and p. g. bergmann (middle of 20th century). second, we described the phenomenon of admissible directions for proper solutions of equations (1.2), (1.3) and considered the case that such directions correspond prime roots of the cubic polynomial m . the case of double roots was previously studied in the partial case if equation (1.2), (1.3) is the geodesic equation generated by a pseudo-riemannian metric with varying signature; see the survey [18] or the original works [19, 22, 24]. the study of double roots for equation (1.2) with a generic cubic polynomial (1.3) is more complicated and it is not done yet. acknowledgements the research is supported by the russian foundation for basic research (project 2001-00610) and by the ministry of science and higher education of the russian federation (goszadanie n. 075-00337-20-03, project 0714-2020-0005). references 1. arnol’d, v. i. & ilyashenko, yu. s. 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(2021) mathematical modeling of color printing devices: color gamut visualization and colorimetric measurements regularization, adv. syst. sci. appl., 21:2, 117–132. copyright © 2022 assa. adv syst sci appl (2022) introduction a brief look at quasi-linear equations phenomenon of oscillating solutions the aim, scope and structure of the paper oscillating solutions second-order equations cubic in the first-order derivative generalization for higher order equations conclusion microsoft word 1567-article text-7289-1-18-20240805 adv syst sci appl 2024; 04; 28-43 published online at https://ijassa.ipu.ru. modeling of consensus in bilateral trade negotiations olga v. maksimova1,2*, iosif z. aronov3 1) federal state budgetary institution yu. a. izrael institute of global climate and ecology, moscow, russia 2) university of science and technology misis, moscow, russia 3) university of the ministry of foreign affairs of the russian federation, moscow, russia abstract: any trade transaction is preceded by negotiations between at least two parties, under which a sale and purchase are made. this work studies the factors that influence the time it takes for the parties to reach mutual agreement in the presence of coalitions. de groot's model of the process of achieving consensus based on regular markov chains is presented in the work. it is built on the assumption that participants in the negotiation process exchange opinions and can influence the point of view of other members of the negotiating parties during the discussion. the main advantage of such a model is the ability to identify and study the key characteristics of the negotiating parties. the obtained modeling results helped to formulate practical recommendations for making an agreed-upon decision within the framework of trade negotiations. analysis of the current model of face-to-face negotiations between the two parties creates softer conditions for the composition of the group, and the negotiations themselves contribute to the development of new ways to take into account the interests of both sides. keywords: trade negotiations, consensus, coalitions, regular markov chains, number of negotiations. 1. introduction 1.1. historical aspect of the development of trade negotiations and their role in modern economic development trade negotiations are one of the oldest types of human activity aimed at solving problems when organizing cooperation or resolving conflict situations in the trade process. they have the same ancient history as conflicts and wars and were used in practice long before the formation of legal procedures. the first information about the existence of trade relations can be found in ancient times. already three and a half millennia bc, the first king of assyria-babylonia from the sumerian dynasty established trade relations with the north and south from his capital, sirtella. the prosperity of today's civilization owes much to international and national trade. any trade transaction is preceded by negotiations between at least two parties, under which a sale and purchase are made. there are many historical examples of such negotiations, from antiquity to the present day. the first attempts to describe the negotiation process are associated with the name francois de callières (1645–1717), a french diplomat and advisor to french king louis xiv. callières created the concept of diplomacy based on the absolute priority of negotiations between equal partners over force and violence, manifested in its various forms in interstate relations. he defined the goals, objectives, forms, and methods of the negotiation process and *corresponding author: o-maximova@yandex.ru modeling of consensus in bilateral trade negotiations 29 copyright ©2024 assa adv. in systems science and appl. (2024) other aspects of professional diplomacy [6]. although his book was devoted to diplomatic negotiations, it can rightfully (with some reservations) be attributed to the trade sphere. the heart of a trade deal is negotiation. modern researchers of the negotiation process focus on such aspects of this activity as communications [21, 26], strategy and tactics of negotiations [27], fair equilibrium prices [24, 28], the best choice from alternatives [5], etc. at the same time, as a rule, the dynamics of the negotiation process and the issues of time to reach agreement between the parties (consensus) are on the periphery of the research. the trade negotiation process is not continuous. as a rule, it consists of several rounds (stages). their number, in fact, determines the time it takes to reach consensus. it should be emphasized that the time of the negotiation process until the parties reach an agreement can be considered an integral indicator characterizing the effectiveness of negotiations. an increase in the time of the negotiation process may indicate poor selection of negotiators, a lack of flexibility in negotiations, excessive ambition, etc. thus, the analysis of the negotiation process in this regard is a relevant task that meets the ideas of globalization. during the first round of negotiations, the parties get to know each other and determine their starting positions, which need to be brought closer together during subsequent rounds. further, we will assume that two parties participate in the negotiation process. this assumption does not in any way detract from the value of the study, since most trade negotiations are bilateral. as noted above, the main function of negotiations is to reach an agreement, without which a trade deal cannot take place. joint activity with the other party focused mainly on finding a mutually acceptable solution is the most important characteristic of the negotiation process and, in this sense, is opposed to the unilateral actions of the participants. it should be noted that many researchers generally view negotiations as a process of finding a joint solution to a problem with the other side. in order to emphasize the importance of this point, r. fisher and w. urey introduced the special term “batna”, which is an abbreviation for the english “best alternative to negotiated agreement”, as one of the main elements of the negotiation process [10]. negotiations between the two parties will not take place if at least one of the participants has batna, i.e., an alternative that he considers to be more advantageous than the negotiated solution. if, when discussing the initial position, it turns out that the parties’ proposals do not have any common ground – in other words, there is no common negotiation space – then such negotiations are unlikely to end in agreement [30]. moreover, if, for some reason, for example, due to an incorrect assessment of the situation by one of the parties, the negotiations end in mutual agreement, in the future they will be perceived as unfair or dishonest. therefore, the authors of the present study proceed from the fact that there is a negotiation space (points of intersection of the parties’ proposals) that makes it possible to achieve a convergence of positions during the negotiation process. it should be emphasized that the presence of negotiation space and the absence of batna are necessary conditions for achieving consensus but are clearly insufficient. even the absence of batna does not guarantee the achievement of agreement in trade negotiations, which may be due, for example, to the psychophysical characteristics of the specialists participating in the discussion, the complexity of the subject of discussion, and other factors. thus, during the negotiation process, two coalitions are formed (according to the number of parties), which leads to a deadlock and a conflict between the parties. in the theory of conflictology, there are works devoted to the search for consensus in coalitions [14, 16, 19, 20]. these works examined the issues of resolving conflicts within specific cases, which does not allow us to draw general conclusions related to the achievement of agreement by the parties during negotiations. 30 o. v. maksimova, i. z. aronov copyright ©2024 assa adv. in systems science and appl. (2024) the purpose of this work is to study the factors that influence the time it takes for the parties to reach mutual agreement (consensus) in the presence of coalitions in bilateral trade negotiations. 1.2. the role of consensus as a method of agreement in trade negotiations an insoluble situation with coalitions can be changed in different ways: to come up with some compromise from one side or from both coalitions (parties), or to invite an external expert to moderate negotiations. in the first case, the main idea of “removing” the conflict and achieving consensus is associated with the proposal to relax demands (i.e., making a concession), which one party to the negotiations makes to the other, and in the second case, the main idea is associated with the streamlining of the negotiation process by the moderator (an external expert). if there is no search for a compromise between the coalitions (parties) or the absence of external intervention, negotiations may drag on for years and consensus will not be achieved. consensus research raises many questions related to ensuring the reproducibility of research results. an analysis by a collaboration of evidence-based psychologists found that out of 100 original experimental studies in the field of social psychology, only some groups were able to reproduce no more than 39 experiments [1]. thus, it is not possible to organize socio-psychological studies of the nature of consensus for large groups. the complexity of formalizing the negotiation procedure itself leads to the search for fundamentally new approaches to solving the problem of finding consensus. the phenomenon of reaching consensus can be analyzed on the basis of a modeling methodology that is not limited by the number of group participants. the relevance of building such models is related to the need to study factors that influence the time to reach consensus in trade negotiations in order to increase their efficiency and reduce time costs. de groot's model of the process of achieving consensus based on regular markov chains presented in the work showed its efficiency in studies of general consensus in a group [2−4, 31]. the model is built on the assumption that discussion participants exchange opinions and can influence the opinions of other group members during the discussion. the main advantages of this model are the ability to identify and study the key characteristics of the participants in the negotiation process and develop practical recommendations to avoid delaying the process of making an agreed-upon decision. recently, this model has been applied in various domains such as network automation management [7], negotiation processes [23], and social media management [8]. 2. methods and materials 2.1. description of the theoretical consensus building model based on regular markov chains let us describe a model of consensus in the process of negotiation based on regular markov chains [3]. 1. let some question concerning the parameter ɵ (not necessarily real) in a group be discussed. it is assumed that the value of the parameter ɵ itself is not known/not fully known. each of the negotiators holds his initial opinion about ɵ. thus, n – the number of members of the group participating in the discussion; s(0) = 𝑆 , 𝑆 ,…,𝑆 – the vector of the initial opinions of these members, where 𝑆 is the initial opinion of the i-th member, 𝑖 = 1, … , 𝑛 . for example, in the structure of russian exports to china a significant share is occupied by pulp and paper products (paper, cardboard). thus, in the process of trade negotiations, the quality indicators of these products, prices, delivery times and other characteristics related to the import of modeling of consensus in bilateral trade negotiations 31 copyright ©2024 assa adv. in systems science and appl. (2024) goods are discussed. therefore, the vector s(0) describes the initial values of any characteristic along which the sides of negotiations must agree. note that the opinion 𝑆 of each member (i = 1, …, n) can vary depending on that member's degree of confidence in the opinions of other, as well as the degree of selfconfidence. group members exchange their opinions about the vector s(0) during a group meeting. therefore, each of them can either change their opinion or keep it the same at any stage of the negotiation. 2. let's set the probability of trust of the i-th participant to the opinion of the j-th participant by a strict inequality 0 < 𝑝 < 1 (i = 1, …, n; j = 1, …, n, 𝑖 ≠ 𝑗). this is because consensus often requires flexibility in decision-making as well as active participation. if i = j, let 𝑝 be the probability that the i-th member trusts himself, and we also suppose 0 < 𝑝 < 1. also, these values can be explained as levels of authoritarianism: the higher the pii value, the higher the authoritarianism of the group member. the issues related to authoritarianism are described quite well in the literature, which allows us to introduce an appropriate scale (authoritarianism – a socio-psychological characteristic of a person reflecting his/her desire to maximally subordinate partners in interaction and communication to his/her influence). this is important for practical guidance on negotiation processes. let's form a quadratic confidence matrix consisting of probabilities p = (𝑝 ) (i = 1, …, n; j = 1, …, n). the resulting matrix p is stochastic (the condition ∀𝑖 ∈ 1, 𝑛 ∑ 𝑝 = 1 is satisfied). 3. the vector of opinions of the members of the negotiation process at the first stage can be calculated by the formula st(1) = p∙st(0) = (𝑆 , 𝑆 ,…,𝑆 )t, where st(∙) is a column vector of dimension 𝑛 × 1.and after the k-th step, we get the vector st(k) = (𝑆 , 𝑆 ,…,𝑆 )t = p∙st(k – 1) = pk·st(0). 4. the iterative process stops at the m-th step when all rows of the matrix pm become equal with a given accuracy. in this case, we consider that group cohesion has been achieved, i.e. consensus has been reached. mathematically, this means that the matrix p has reached the final matrix f after being raised to degree m. since the final matrix f does not change in subsequent iterations, the vector of negotiators' opinions st(m) = pm·st(0) = 𝑆 , 𝑆 ,…,𝑆 t. thus, after m iterations a consensus emerges. comment. in markov chain theory, a necessary and sufficient condition for the convergence of the matrix p to the final matrix f any vector of initial opinions is the regularity of the matrix p [13]. that is, for the regularity of the matrix it is sufficient for the sums of probabilities on the rows of the matrix p to be equal to 1, with the strict inequality 0 < pij < 1 being satisfied for any probabilities pij [17] (theorem 4.1.2). 2.2. description of the theoretical consensus building model in two coalitions as a special case of the consensus model let us consider the theoretical model of consensus using the example of having two coalitions for a group of five people. for such a case, the confidence matrix p can have, for example, the following form: p = ⎝ ⎜ ⎛ 0.7 0.3 0 0.5 0.5 0 0 0 0 0 0 0 0.5 0.1 0.3 0 0 0 0 0.5 0.7 0.4 0 0.2 0.3⎠ ⎟ ⎞ . thus, we consider two coalitions, the first involving two negotiators and the second involving three negotiators. 32 o. v. maksimova, i. z. aronov copyright ©2024 assa adv. in systems science and appl. (2024) in the coalition model, the regularity condition of the trust matrix p is violated, so in this situation consensus cannot be reached for any 𝑛 ≥ 2 [13]. in other words, the condition 0 < pij < 1 for any rows is violated. matrices of this type and their corresponding markov chains are decomposable. if a compromise solution is reached, it will allow to get rid of the existing coalitions. such solutions can be different. for example, it is possible to make a concession by members of one coalition to the second coalition. mathematically, this corresponds to redistributing the i-negotiator's probabilities of trust pij for the members of his coalition by the probabilities pij for the members of the competing coalition. the probabilities pij are redistributed through additional reasoning in the deliberation process. a model of such concessions is described in the work of the social psychologist homans [15]. the article considers the scheme of functioning of small groups on the basis of utility exchange. we can talk about two variants of concessions:  unilateral concessions, where members of one coalition redistribute their probabilities of trust to members of the other coalition;  bilateral concessions, where members of one coalition redistribute their confidence probabilities to members of the second coalition and, conversely. we will find the time it takes to reach consensus in the presence of unilateral concessions in different types of coalitions. unilateral concessions in the negotiation process have been examined in a number of papers, such as [20]. however, they did not analyze the factors affecting the time until the final decision is reached. 2.3. elements of experiment planning for the analysis of the theoretical model of coalitions it is known that as the size of a group increases, interactions between its members become less frequent and formal, which leads to a weakening sense of belonging, and this is an obstacle to the formation of group cohesion [18]. based on this premise, it is obvious that the group of negotiators should not be too large so that direct interaction between its members can be ensured. let us take this fact into account further when setting up the experiment. for the general case of consensus without coalitions, it has been revealed in a number of papers that the time of consensus depends primarily on the authoritarian number of group members and their size [2−4]. there are many combinations of size, concessions, and the authoritarianism of negotiators. in order to control this variability, experimental planning is applied [10]. 1) let’s build a full-scale experiment with an orthogonal plane. in such an experiment, all combinations of factor level are realized, which allows us to independently assess the influence of each factor and each pair interaction [10]. 2) in order to draw stable conclusions about the average number of conversations, the experimental plan will include the simulation of 100 parallel experiments per line [9]. each series of parallel experiments that correspond to one line of the plan includes information about the reaction. in addition, parallel experiments protect us from errors and allow us to evaluate reproducibility errors. 3) the homogeneity of experiment variances is tested by the cochrane g-test at 0.05 [10]: 𝐺(𝑣 − 1, 𝑙 − 1) = σ σ + σ + ⋯ + σ where σ(∙) – experimental (∙)-line variation, 𝑣 – number of parallel experiments in one line. if the obtained 𝐺 is less than the 𝐺 , then we can recognize the homogeneity of the variances of parallel experiments at a given level of significance. in this case, we can proceed to building a model. cochrane test for checking the homogeneity of variances works for normally distributed variables. for each line of the plan we checked the modeling of consensus in bilateral trade negotiations 33 copyright ©2024 assa adv. in systems science and appl. (2024) consistency of the obtained data with the normal distribution by pearson's criterion (the significance level for the test is assumed to be 0.05). 4) regression analysis of the experimental results was performed in the context of excel16 and statistica-15 with a significance level of α = 5 percent. 3. consensus model in two mono-authoritarian coalitions 3.1. description of the experiment with unilateral concession and its theoretical results let each coalition consist of members (negotiators) whose authoritarian levels are close to each other, i.e., their corresponding probabilities vary within ± 0.005 of some predetermined probability p. in this case, we will assume that negotiations are conducted by monoauthoritarian coalitions. assumptions and definitions build a model of unilateral concessions when there are two mono-authoritarian coalitions in a group of 20 negotiators. other works have shown that such a number of negotiators in a group is optimal in terms of the number of rounds of negotiations, all other things being equal [31]. the first stage. the response choice. the response will be the average number of rounds m before consensus is reached, i.e. the average time to reach consensus. from now on, we’ll count the number of rounds. the second stage. the factors of choice. after a preliminary analysis, the following factors were selected: x0 – coalition impact index i – is the quotient of the first coalition to the second coalition; x1 – average authoritarianism of the first coalition negotiators; x2 – average authoritarianism of the second coalition negotiators; x3 – unilateral concession that members of the first coalition make to the second coalition. the third stage. the factor-levels choice. preliminary analysis showed that the constructed mathematical model is nonlinear. since we do not assume to go out of the framework of quadratic polynomial it was not assumed that two or three factor levels are sufficient in such models. the choice of these factor levels was made after research. after preliminary analyses, the levels were selected (table 3.1). table 3.1. factors and levels of the first experiment factors factor levels factors x0 x1 x2 x3 lower «–1» 0.25 0.005 0.005 10 medium «0» 1 0.05 0.05 50 upper «+1» 4 0.895 0.895 75 let us describe the levels of each factor. factor x0. level “–1” corresponds to two coalitions with the number of members n1 = 4 and n2 = 16 (i = ¼), level “0” corresponds to two coalitions with the number of members n1 = n2= 10 (i = 1) and level “+ 1” corresponds to two coalitions with the number of members n1 = 16 and n2 = 4 (i = 4). factor x1. level “–1” corresponds to the authoritarianism of the members of the first coalition that makes a concession, from 0 to 0.01 (with an average value of 0.005), level “0” corresponds to authoritarianism from 0.045 to 0.055 (with an average value of 0.05), level “+ 1” corresponds to the authoritarianism from 0.89 to 0.90 (with an average value of 0.895). factor x2. level “–1” corresponds to the authoritarianism of the members of the second coalition that accept a concession from the first coalition, from 0 to 0.01 (with an average 34 o. v. maksimova, i. z. aronov copyright ©2024 assa adv. in systems science and appl. (2024) value of 0.005), level “0” corresponds to authoritarianism from 0.045 to 0.055 (with an average value of 0.05), level “+1” corresponds to the authoritarianism from 0.89 to 0.90 (with an average value of 0.895). consider the remarks for the factors x1 and x2. level «–1» is chosen so as to ensure consideration of the group of negotiators with low authoritarianism. for a group of negotiators with low authoritarianism, we will have that ∀ 𝑖 = 1, 𝑛 𝑝 < . based on this estimate, for groups of 20 negotiators, an upper boundary is provided 𝑝 < 0.005. authoritarianism varied in the range of ± 0.005 from the average level. if 𝑝 > , the group ceases to be a low authoritarian group because its members can trust themselves more than others in the group [3]. in a group with low authoritarian members, we can get the following situation for some 𝑖 = 1, 𝑛: ∀𝑗 = 1, 𝑛; 𝑗 ≠ 𝑖 𝑝 < 𝑝 , i.e., there is a negotiator whose level of confidence in his opinion is less than the level of confidence in any other. level «0» is chosen to ensure that a group of negotiators with an upper bound on authoritarianism is considered pii = 0.005. authoritarianism was varied in the range of ± 0.005 from the average level. level «+1» level is chosen to ensure that the group of negotiators with high levels of authoritarianism is taken into account: ∀𝑖 = 1, 𝑛 ∃𝑗 = 1, 𝑛; 𝑝 > 𝑝 , 𝑗 ≠ 𝑖 i.e., there is at least one member whose level of confidence in his opinion exceeds the level of confidence in any other member of the group. note that for a group with authoritarianisms 𝑝 = 1 we obtain a decomposable matrix 1 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ 1 . it is shown in markov chain theory that this matrix does not converge to the final matrix [17]. therefore, the level «+1»is chosen close to the 𝑝 = 1. authoritarianism varied in the range of ± 0,005 from the average level. factor x3. level “–1” corresponds to a concession of y = 10 percent, level “0” corresponds to a concession of y = 50 percent, and level “+1” corresponds to a concession of y = 75 percent. the fourth stage. the planning matrix choice. the orthogonal design of the full-factor experiment included 81 lines and 4 factors in total. preliminary analysis showed that the model would not be linear, so we will include pairwise interactions. the fifth stage. the experiment. for each factor level, the modelling of elements pii of matrix p was carried out using a uniform distribution law. it was taken into account that the sum of probabilities in each row should equal 1, i.e. that the matrix p should be stochastic. next, all probabilities in the rows of matrix p for the first coalition were recalculated in proportion to the amount of concession it made to the second coalition. the probabilities of the second coalition accepting the concession remained the same. for each line, the average m is calculated. then, for all lines, the homogeneity of dispersion is checked by cochran's g-test. the homogeneity of variances was confirmed at a level of only 15 percent. let us analyze the variability of each experimental line at the end of the chapter. the following regression model was obtained: 𝑚 = 36.7 + 20.0 ∙ 𝑋2 − 15.5 ∙ 𝑋3 + 10.9 ∙ 𝑋2 ∙ 𝑋3 (3.1) modeling of consensus in bilateral trade negotiations 35 copyright ©2024 assa adv. in systems science and appl. (2024) the plan matrix is orthogonal, which allows us to interpret the model coefficients straightly. the model (1) is adequate according to fisher's criterion, the determination shown is high enough, which also demonstrates the good quality of the model (r2 = 0.723, f(3, 77) = 66.853, pvalue = 2.2∙10-21 < 0.05). fig. 3.1 shows the spline of the model along the experimental points derived from the factors x2 и x3 included into the models. it confirms the results of the preliminary studies, which showed that the model is nonlinear with respect to the included factors x2 and x3. a joint increase in factors x2 and x3 leads to an increase in the value of m (fig. 3.1). let us carry out a theoretical analysis of the resulting model (3.1) and further give its interpretation. the sixth stage. results. result 1. using the step-by-step inclusion of variables, confirmed the inclusion gave the following results:  a statistically significant relationship between the output variable mand the regressors x2, x3 and x2x3, while the strongest one was revealed with the regressor x2;  a statistically insignificant relationship between the output variable m and the regressors x0, x1 and all remaining pairwise interactions. the structure of the final matrix in each such case is remarkable: the rows corresponding to the first coalition consist of zeros, as shown in the example from section 2.2. also, based on the results of the experiment, it is important to note one non-random fact of the model’s behavior, consistent with the studies for the general group without coalitions [17]: if x2 = −1, then for fixed values of x3 the average number of rounds m will be slightly greater than for the level x2 = 0, as shown in fig. 3.1. fig. 3.1. spline based on experimental data (x2 – authoritarianism of the coalition accepting the concession, x3 – concession, m – average number of rounds) result 2. the regression graph (3.1) resulting from the experiment is presented in fig. 3.2. let us fix each factor in the experiment in turn and analyze the dependencies (fig. 3.3, fig. 3.4). 1. when x2 is fixed, we obtain the following section equations (fig. 3.3): x2 = −1, 𝑚 = 16,7 − 26.4 ∙ 𝑋3, x2 = 0, 𝑚 = 36.7 − 15.5 ∙ 𝑋3, x2 = 1, 𝑚 = 56.7 − 4.6 ∙ 𝑋3. 2. when x3 is fixed, we obtain the following section equations (fig. 3.4): x3 = −1, 𝑚 = 52.2 + 9.1 ∙ 𝑋2, x3 = 0, 𝑚 = 36.7 + 20.0 ∙ 𝑋2, 36 o. v. maksimova, i. z. aronov copyright ©2024 assa adv. in systems science and appl. (2024) x3 = 1, 𝑚 = 21.2 + 30.9 ∙ 𝑋2. when fixing x2:  the largest slope angle is given by the first equation for x2 = −1, which records high sensitivity to changes in factor x3: the decrease of x3 by 1 unit implies an average increase of m response for 26−27 rounds;  the smallest slope angle is given by the last equation for x2 = 1, which indicates low sensitivity to changes in factor x3: the increase of x3 by 1 unit implies an average decrease of m response for only 4−5 rounds.  when fixing x3:  the largest slope angle is given by the last equation for x3 = 1, which records high sensitivity to changes in factor x2; the increase of x2 by 1 unit implies an average increase of m for 30−31rounds;  the smallest slope angle is given by the last equation for x3 = −1, which records low sensitivity to changes in the factor x2; the increase of x2 by 1 unit implies an average increase of m for only 9−10 rounds. fig. 3.2. regression model based on experimental data (x2 – authoritarianism of the coalition accepting the concession, x3 – concession, m – average number of rounds) result 3. in figures 3.3 and 3.4 and in the sections of the regression model, it can be clearly seen that the greatest slope is reached when x3 is fixed, which indicates the sensitivity of the model, particularly to changes in factor x2. it is also clearly evident that the effect of factor x3 becomes less as it tends to x2 = 1 (the sections in fig. 3.4 converge as x2 → 1, the slope angle at x2 = 1 in the sections displayed in fig. 3.3 is the smallest among the others). fig. 3.3. sections of the regression model when factor x2 is fixed 0 10 20 30 40 50 60 70 80 -1 -0.5 0 0.5 1 m (r eg r) x3 x2 = -1 x2 = 0 x2 = 1 modeling of consensus in bilateral trade negotiations 37 copyright ©2024 assa adv. in systems science and appl. (2024) fig. 3.3. sections of the regression model when factor x3 is fixed result 4. the highest values of m = 55-66 according to experimental data are observed at x2 = 1, which is confirmed by the model within the experimental region: x2 = 1, 𝑚 = 56.7 − 4.6 ∙ 𝑋3; for different 𝑋3 ∈ [−1, 1], the values according to the model will be 𝑚 = 56.7 ± 4.6. the model also shows high values for x2 = 0 when x3 = −1, which is also evident from the experimental data. note that outside the experimental area, for any values of x3 when factor x2 increases (i.e., as the authoritarianism of the group members accepting the concession tends to 1), we can obtain an arbitrarily large number of rounds since, as noted earlier, we can arrive to decomposable matrix. result 5. the smallest value m = 5−13 according to experimental data is observed at x3 = 1 (table 3.2) and at x3 = 0, if x2 ≠ 1, which is confirmed by the model within the experimental area: x3 = 1, 𝑚 = 21.2 + 30.9 ∙ 𝑋2; level of values m = 5−13 in fig. 3.3 is achieved only for x2 = 0 and x2 = −1 at a high value of x3, in fig. 3.4 is achieved only for x3 = 0 and x3 = 1 at low values of x2. 3.2. variability in the number of negotiations in the experiment and particular cases of the model the experiment examined the effect of various factors on the average number of negotiation rounds conducted before consensus is reached. it was found that there is different variability in the number of negotiations under different conditions. variability allows us to understand the limits of variation in the number of negotiations in coalitions, as there is no complete homogeneity within a coalition. therefore, it characterizes the constancy of the number of steps taken to reach consensus. let us compile a table where we display the main range of variances together with the corresponding levels of factors (table 3.2). result 6. from the analysis of table 3.2 it is clear that, on average, the tests are subject to the greatest variation when factor x0 = 1 (moreover, when factor x2 = 1, then the level of factor x3 is not important). the smallest variation is most often observed in the tests when: a) x0 = −1, b) x0 = 1 when x2≠ 1 и x3≠1, c) x0 = 0 when x2≠ 1. note that in the case of x0 = −1 there are no tests with high or medium variation, and that the factor x1 does not affect the variation of the tests. 0 10 20 30 40 50 60 70 80 -1 -0.5 0 0.5 1 m ( re gr ) x2 x3 = -1 x3 = 0 x3 = 1 38 o. v. maksimova, i. z. aronov copyright ©2024 assa adv. in systems science and appl. (2024) an additional analysis of variance showed that factor x0 has a statistically significant effect on the variation (fisher's criterion: f = 33.602 , f = f , (2. 78) = 3.114 and f > f ) [9]. let us consider special cases of the model for each level of factor x0. the obtained regression partial dependencies under conditions of homogeneous variances (at a significance level of 5 percent) have the form: x0 = −1. 𝑚 = 37.5 + 21.4 ∙ 𝑋2 − 16.9 ∙ 𝑋3 + 12.6 ∙ 𝑋2 ∙ 𝑋3. x0 = 0.𝑚 = 35.6 + 22.6 ∙ 𝑋2 − 14.0 ∙ 𝑋3 + 10.4 ∙ 𝑋2 ∙ 𝑋3. x0 = 1.𝑚 = 37.0 + 16.1 ∙ 𝑋2 − 15.7 ∙ 𝑋3 + 9.7 ∙ 𝑋2 ∙ 𝑋3. for the level x0 = 0 at x2 = x3 = 0, the response value 𝑚 is the smallest among all other levels of x0, which is consistent with the simulation results and is essential in interpretation (see section 3.3). in general, the coefficients in the equations are close and reflect the same behavior of the dependent variable. therefore, to obtain general conclusions when interpreting the model, you can use the general model (3.1) obtained in section 3.1. table 3.2. ranges of variances of parallel tests of the experiment variation levels factors x0 x1 x2 x3 variation lower − 1 any any any 0−2 0 any 0; − 1 any 1 any 0 0; − 1 medium 0 any − 1 any 2−10 1 any 0 1 upper 1 any 1 any 10−32 − 1 1 3.3. experiment with a bilateral concession with a bilateral concession, it is expected that the negotiation process should move faster. but how fast? and will the process always be faster? in order to find out, we will conduct a shortened experiment with two equal coalitions with the same level of authoritarianism of members. let us briefly describe it. factors x1, x2. level “–1” corresponds to the authoritarianism of coalition members from 0 to 0.01 (with an average value of 0.005), level “0” corresponds to authoritarianism from 0.045 to 0.055 (with an average value of 0.05), level “+1” corresponds to the authoritarianism of coalition members from 0.89 to 0.90 (with an average value of 0.895). consider the remarks for the factors x1 and x2 in 3.2. factor x3. level “–1” corresponds to a bilateral concession of y = 10 percent, level “0” corresponds to y = 50 percent and level “+1” corresponds to y = 75 percent. the full-factor experiment orthogonal plan included 9 lines and 2 factors in total. for each level, the modeling of elements 𝑝 of matrix p was carried out using the uniform law of distribution under given conditions so that the sum of probabilities within each row equals 1. next, all probabilities in the rows of matrix p for both coalitions (sides) were recalculated in proportion to the concessions they make for each other. for each line, the average m is calculated. result 7. it is possible to identify cases not only when we observe a decrease in the number of rounds but also cases where their number may increase (highlighted in bold in table 3). these cases correspond to the factor levels: x1, x2 = −1 or x1, x2 = 0 and x3 = 1. the largest decrease in the average number of negotiation rounds m is observed for the factor modeling of consensus in bilateral trade negotiations 39 copyright ©2024 assa adv. in systems science and appl. (2024) levels: x1, x2 = 1 and x3 = 0 or x3 = 1. note that a bilateral small concession (x3 = −1 or x3 = 0) led in all cases to a decrease in m. table 3.3. results of a full factorial experiment for mono-authoritarian coalitions with bilateral concessions x1, x2 x3 x1 x2 x3 −1 −1 47.6 22.8 52 −1 0 8.1 4.0 50 −1 1 5.9 7.3 rise 0 −1 47.5 22.8 52 0 0 8 4.0 50 0 1 5.6 7.2 rise 1 −1 66.1 33.3 50 1 0 65.6 9.0 86 1 1 65.8 7.0 89 the modeling revealed that variability m decreased on average by 2 times compared to the variability with unilateral concession. this shortened experiment served the purpose of identifying possible cases of reduction/increase in m. the experiment does not include the analysis of coalitions with different authoritarian levels of negotiators and different numbers of group members with mutual concessions. this requires further detailed research in accordance with the stated objectives. 3.4. interpretation of the model of mono-authoritarian coalitions interpretation of results 1 and 3. the most significant factor that influences the number of rounds of negotiations to reach consensus is the authoritarianism of the coalition accepting the concession. if the coalition accepting the concession consists only of non-authoritarian members, then the average number of rounds will be slightly higher than in the case when the members of this coalition have authoritarian levels close to 1/n. since the coalition accepting the concession ultimately forms a consensus decision, it is very important that this group includes specialists who are ready to listen to others and at the same time have their own opinion; that is, they are not absolutely non-authoritarian or absolutely authoritarian. it was revealed that with average values of concession and average authoritarianism of the coalition accepting the concession, the smallest number of agreements will be in the case when the coalitions are equal in number. this corresponds to general recommendations for negotiations [27]. the influence of the authoritarianism of the coalition that moves towards rapprochement, on the total number of rounds of negotiations is statistically insignificant. the size of the coalition also does not have a significant effect on the value of m. interpretation of result 2. a high sensitivity to the size of the concession was revealed for a coalition with nonauthoritarian members that accepts this concession: as the concession increases, the number of rounds quickly decreases. it was revealed that with an increase in concessions from 10 percent to 75 percent, the number of rounds can be reduced by 5 or more times. low sensitivity to the size of the concession was revealed for the authoritarian coalition accepting this concession: with an increase in the concession, the number of rounds is reduced slightly. thus, the size of the concession plays a significant role in reducing the number of rounds only for non-authoritarian coalitions that accept the concession. interpretation of result 4. 40 o. v. maksimova, i. z. aronov copyright ©2024 assa adv. in systems science and appl. (2024) the highest value of the number m was found for authoritarian coalitions accepting a concession. the size of this authoritarian coalition does not significantly affect the average number of rounds in this case. the negative role of authoritarianism among the group members was also noted in the general case, i.e., during negotiations within one group without the presence of coalitions [31]. note that by replacing the authoritarian members of the coalition accepting the concession with non-authoritarian members, it is possible to reduce the negotiation time for a 50 percent concession by 4 or more times, and for a 75 percent concession by 10 or more times. interpretation of result 5. the smallest value of the number of rounds of the negotiation process within the experiment was revealed at the highest value of the concession, but only if it is accepted by a non-authoritarian coalition (all other things being equal). this finding is consistent with result 2: for an authoritarian coalition accepting a concession, the size of the concession has little effect on the number of negotiation rounds. interpretation of result 6. the greatest variation in average negotiation time occurs when a large coalition makes any significant concession to an authoritarian small coalition. the smallest variation in average negotiation time is observed when: – a small coalition makes any concession; – a larger coalition makes a concession to a coalition that consists of non-authoritarian members (increasing concessions in this case entails an increase in the variation in the average number of negotiation rounds). thus, in the first case, we can unexpectedly observe both a small and a large number of rounds of the negotiation process; in the second case, low variability in the number m is observed. the variation in the number m is not affected by the authoritarianism of the coalition that makes the concession. interpretation of result 7. with a bilateral concession, both cases of reduction in the number m and cases of its increase were identified. the average number of negotiation rounds increases when coalitions consist of members with low or close to 1/n authoritarianism and make a large concession. thus, the transfer of responsibility for the decision made from one side to the other leads to an increase in the time of the negotiation process. the average number of rounds of the negotiation process is reduced to the maximum when authoritarian coalitions decide to make a mutual concession. note that a small bilateral concession leads to a reduction in the negotiation process. 4. interpretation of the results for the trade negotiation process. findings the obtained modeling results make it possible to develop general recommendations for the negotiation process of the two parties to avoid delay in the negotiation process. in what follows, we will call a coalition (side) authoritarian if all its members have high authoritarianism (𝑝 ≥ 0.9), and non-authoritarian if its members have low authoritarianism (𝑝 < 1/𝑛). the values of authoritarianism 𝑝 can be obtained by expert assessment of the components developed by f. fiedler and later tested in many laboratory experiments [12]. let us consider the main recommendations arising from the present study. 1. without a concession in the negotiations, the parties (if there are contradictions) will not be able to reach a consensus. at least one of the parties must make a concession. 2. it is advisable (other things being equal) for the parties to be equal in number. 3. negotiators on both sides should not all be completely non-authoritarian or authoritarian. modeling of consensus in bilateral trade negotiations 41 copyright ©2024 assa adv. in systems science and appl. (2024) 4. if condition 3 is met, then, from the point of view of reducing negotiation time, it is better if both parties make a concession. if condition 1 is not met, then reaching a consensus is possible only by replacing the authoritarian members of one side with specialists who are ready to make a concession. note that resolving the conflict by replacing negotiators in order to eliminate coalitions does not always lead to the expected consensus [29]. a striking example of this situation is the conflict in panama during the construction of a hydroelectric power station between two coalitions: indigenous peoples and the company generadora del istmo s. a. (genisa). the replacement of members of one of the coalitions did not lead to the development of a consensus decision. therefore, a concession serves as a primary factor in developing a solution during negotiations. next, we assume that condition 1 is satisfied, but the following condition is not fulfilled: a) condition 2. achieving consensus will be faster if one of the possibilities is implemented:  any concession is made by the minority side, and the other side will not be absolutely authoritarian,  a larger coalition makes a concession, while the party accepting the concession consists of non-authoritarian members (increasing concessions in this case entails an increase in the variation in the average number of negotiation rounds). b) condition 3. reaching consensus can be achieved by replacing absolutely authoritarian members or absolutely non-authoritarian members of one side with members with average authoritarianism. analysis of the current model of face-to-face negotiations between the two parties creates softer conditions for the composition of the group, and the negotiations themselves contribute to the development of new ways to take into account the interests of both sides. 5. conclusion issues of achieving consensus, which is based, as a rule, on the possibility and ability of negotiators to compromise, currently have been poorly studied; the complexity of formalizing the negotiation procedure itself leads to the search for fundamentally new approaches to constructing mathematical models. the relevance of developing such models for studying the interaction between experts in groups and the factors influencing the time to reach consensus is determined by the possibility of delaying the negotiation process. the model considered in the work is built on the assumption that participants in the negotiation process exchange opinions and can influence the point of view of other members of the negotiating parties during the discussion. the main advantage of such a model is the ability to identify and study the key characteristics of the negotiating parties. it helped to formulate practical recommendations for making an agreed-upon decision within the framework of trade negotiations. despite these advantages, the considered model does not reveal the problem of correlating consensus and truth, when the deliberate formation of a group of “ignoramuses” leads to a consensus that is far from the truth [25, 22]. therefore, the problem of approaching the truth of a collective decision deserves special attention and analysis of the experience of using consensus procedures and is beyond the scope of the current study. acknowledgements put here your acknowledgements and grant funding. references 1. aarts, a. a., anderson, j. e., anderson, c. j., attridge, p. r., attwood, a., et al. 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(2019). control of consensus convergence in technical committees of standardization on the basis of regular markov chains model, international journal of systems assurance engineering and management, 10(1), 29–36, doi: 10.1007/s13198-019-00765-1 microsoft word molostvov-3 adv syst sci appl 2025; 1; 1-11 published online at https://ijassa.ipu.ru. optimization of multi-currency deposit structure by two indicators (income and risk) under uncertainty vitaly s. molostvov* international center of decision choice and analysis of hse university, russia, moscow abstract. a two-criteria vector optimization problem – finding pareto-optimal solutions in linear systems with interval uncertainty of coefficients – is considered. the problem of resource allocation to multiple activities is investigated. the uncertainty-adjusted income is a bilinear function, linear by strategy under fixed uncertainty and by uncertain parameters under fixed strategy. guaranteed income is a linear function of variables and guaranteed risk is a piecewise linear function. finding the optimal guaranteed risk is reduced to a linear programming problem by piecewise linear programming methods. to solve the two-criteria problem of the optimal allocation of financial resource on three currency deposits, the parameterization of the pareto set by the value of the guaranteed income criterion is applied. thus, the construction of a representative subset of pareto-optimal solutions is reduced to solving a finite number of linear programming problems. the results can be used in analyzing the problems of financial management under conditions of incomplete information. keywords: deposit diversification, bicriteria optimization, incomplete information, minimax regret solution. 1. introduction decision-making tasks in most cases are complicated by the factor of incomplete information. we can distinguish some typical situations by the nature of information available to the decision maker (dm) or to the researcher of the problem, the developer of appropriate models. a deterministic version in which the values of all model parameters are known completely and accurately. a stochastic variant in which some parameters of the model are not known exactly, but the stochastic characteristics of these non-deterministic parameters are specified. a case of substantial uncertainty, when neither exact values nor any stochastic characteristics are known for some model parameters. in this case, the uncertain parameters are known only to the precision of some known set. the first case is most fully developed both in theory and in terms of applications. the stochastic variant in optimization problems is usually reduced to the deterministic case by considering the corresponding deterministic problem for the mean values and variance of the optimized indicators. in problems with substantial uncertainty, the following situation is typical: only the boundaries of possible values for uncertain parameters are known, often – two-sided ranges. the principle of the best-guaranteed result (wald principle [1]) is fruitful here. it is applied either to the initial target indicator or to some secondary indicators, for example, to the risk function according to savage [2]. this risk can be interpreted as a function of losses due to ignorance – incomplete information. savage optimization and other approaches in optimization problems under uncertainty were considered in [3-6]. in particular, the paper [3] studied the case of "mixed type" uncertainty, in which statistical distributions are known for nondeterministic parameters of the problem, but some characteristics of these distributions (e.g., mathematical distribution or dispersion) are uncertain parameters with known ranges of possible but unknown in advance values. the works [5, 6] * corresponding author: molostvov@list.ru v.s. molostvov 2 copyright ©2025 assa adv. in systems science and appl. (2025) consider in detail the problems of optimal distribution of deposits in three currencies according to one criterion (guaranteed income or guaranteed (minimal) savage risk), algorithms for calculating the optimal deposit structure were obtained, and multivariate calculations were carried out. this paper considers a two-criteria variant of the optimization of a three-currency deposit, in which both of the mentioned criteria are used simultaneously. 2. statement of the problem decision-making problems are characterized by two features. first, for some uncontrollable parameters only the limits of their possible values are known (they are known with the accuracy of some set). second, in order to assess the quality of decisions, it is necessary to consider simultaneously several quality indicators – criteria. for example, in the problems of rational use of investments, one of the main criteria is income. at the same time, the uncertainty of the future economic environment entails uncertainty of expected financial results. this leads to the need to consider risks, understood as the difference between the desired or expected results and the results obtained. in the field of single-criteria optimization of income under uncertainty, the most convincing and elaborated, in our opinion, is the wald principle [1] – the principle of the best-guaranteed result. to measure risk, the risk function (loss function) according to savage [2] is often used, i.e. the difference between the best result (at a known ahead value of uncertainty) and the actual result at a particular real strategy. applying the principle of the best-guaranteed outcome to the risk function leads to the concept of guaranteed risk and optimal guaranteed risk. the problem of multi-currency deposit optimization under uncertainty in terms of each these indicators (income and risk) separately was considered in [5, 6]. this paper develops a two-criteria approach to the above problem. the generally recognized concept of pareto (or slater) optimality is used. guaranteed income (according to wald) and guaranteed risk (according to savage) are chosen as criteria. the pareto principle identifies a set of efficient solutions (synonyms – non-dominated, nonimprovable, efficient, rational), but the dm must choose a specific solution. there are many approaches, recommendations and procedures proposed in the vast literature on multicriteria optimization regarding the choice of a single solution. some of them, despite their apparent logical simplicity (e.g., minimization of linear convolution of criteria with given weight "importance coefficients"), have been subjected to reasonable criticism. one of the possible solutions here is to visualize the whole set of pareto-optimal solutions or at least a sufficiently representative discrete subset of this set and leave the choice of a single solution to the discretion of the dm. it is desirable that the parameterization of the set of paretooptimal solutions be transparent and understandable for the dm. the following property of two-criteria problems is used below. if the constraint "no worse than p= const" is imposed on one criterion and optimization is performed on the second criterion, then the obtained solution (if it exists) is pareto optimal or, in the extreme case, slater optimal. in case the obtained solution is unique in terms of the value of the second criterion, it is also pareto optimal [7]. if the minimum and maximum values of the criteria are finite (in our deposit problem it is so), then the procedure with a "sufficiently detailed" partitioning by the parameter p will give a "sufficiently detailed" representation of the set of optimal solutions. within the framework of this approach, we propose an algorithm for constructing a subset of pareto-optimal solutions for the problem of optimal deposit diversification. 3. bilinear resource allocation problems under uncertainty consider the problem of allocating a unit of some resource to 𝑚 types of activity. the profitability of the latter depends on uncertain factors. let the initial efficiency indicator (income) be given by a bilinear function of the form 3 optimization of deposit structure by income and risk under uncertainty copyright ©2025 assa adv. in systems science and appl. (2025) 𝑓(𝑥, 𝑦) = 𝑐 𝑥 𝑦 , 𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌, 𝑎𝑙𝑙 𝑐 ≥ 0. (1) here, the vector 𝑥 = (𝑥 , … , 𝑥 ) is the dm's strategy of allocating the resource to m activities, 𝑋 = {𝑥 = (𝑥 , … , 𝑥 ): 𝑥 ≥ 0 (𝑖 = 1, … , 𝑚), 𝑥 + ⋯ + 𝑥 = 1} (2) is the set of dm's strategies whose components are non-negative and satisfy the budget constraint (canonical simplex in 𝑚-dimensional space), 𝑌 = 𝑦 = (𝑦 , … , 𝑦 ): 0 ≤ 𝑎 ≤ 𝑦 ≤ 𝑏 , (𝑗 = 1, … , 𝑛) (3) is the set of possible states of uncertainty (a parallelepiped in 𝑛-dimensional space given by interval boundaries of uncertain factors). the coefficients 𝑐 have the sense of income at full utilization of a resource unit only for the i-th activity under conditions when the uncertainty vector has the form 𝑦 = (0, … , 𝑦 = 1, … ,0) the upper and lower bounds of the possible values of the uncertain parameters 𝑎 , 𝑏 (j= 1,...,n) are assumed to be known. by virtue of compactness of the sets 𝑋 and 𝑌 and continuity of the function𝑓(𝑥, 𝑦), minima and maxima occurring further exist [9]. the problem is to find strategies that are either waldor savage-optimal. in the first case, it is about maximizing guaranteed income 𝑓[𝑥] = min ∈ 𝑓(𝑥, 𝑦) = min ∈ 𝑐 𝑥 𝑦 (4) on the set of the strategies 𝑋 (risk is excluded). let us rewrite the guaranteed income function in the form 𝑓[𝑥] = min ∈ 𝑓(𝑥, 𝑦) = min ∈ 𝑐 𝑦 𝑥 . (5) since by assumption 𝑐 ≥ 0 , the minima of the expressions in the last brackets in formula (5) are reached at 𝑦 = 𝑎 (j=1,...,n). therefore, taking into account the non-negativity of variables 𝑥 , the guaranteed income is a linear function 𝑓[𝑥] = min ∈ 𝑐 𝑎 𝑥 = 𝑑 𝑥 , (6) where 𝑑 = ∑ 𝑐 𝑎 . let 𝑑 = max 𝑑 . then the optimal (maximal) guaranteed revenue 𝑓 = d is attained for any dm strategy 𝑥 with arbitrary full allocation of the unit of resource to the activities with 𝑑 = 𝑑. the problem of savage risk optimality is more complicated. here the basic concept of risk function (regret function or loss function), defined by the formula φ(𝑥, 𝑦) = max ∈ 𝑓 (𝑧, 𝑦) − 𝑓(𝑥, 𝑦). (7) it means the loss of income due to incomplete information as the difference between the best outcome that could have been obtained with a known forward uncertainty value and the actual outcome when the strategy 𝑥 is chosen. the concept of the best-guaranteed outcome can also be applied to this secondary indicator according to the following definition. v.s. molostvov 4 copyright ©2025 assa adv. in systems science and appl. (2025) definition 1: an admissible solution 𝑥 ∈ 𝑋 is called a risk-guaranteed optimal solution if it delivers a minimum guaranteed risk: min ∈ max ∈ φ(𝑥, 𝑦) = max ∈ φ(𝑥 , 𝑦) = φ , (8) where max ∈ φ(𝑥, 𝑦) is the guaranteed risk to the dm when he uses the strategy 𝑥 ∈ 𝑋. let us call the value 𝛷 the optimal guaranteed risk. for brevity, we will also speak in this case of risk and savage-optimal solutions. the above definition applies to arbitrary income function𝑓(𝑥, 𝑦), the set of dm’s strategies 𝑋 and the set of possible values of uncertainty 𝑌. the considered bilinear income functions of the form (1) are linear by 𝑥 at each fixed 𝑦 and linear by 𝑦 at each fixed 𝑥, hence continuous over the set of variables. the sets of strategies 𝑋 and uncertainties 𝑌 are polyhedra, hence compact. therefore, all maxima and minima in (8) are attained and the optimal solution and the optimal risk exist. the guaranteed risk 𝛷[𝑥] = max ∈ 𝛷(𝑥, 𝑦) depends continuously on 𝑥. 4. savage guaranteed risk optimization and goal programming preliminarily consider the case where the dm has a finite number of strategies and the uncertainty can take a finite number of states: 𝑋 = {𝑢 , … , 𝑢 }, 𝑌 = {𝑠 , … , 𝑠 }. (9) then the income is defined by the income (𝑘 × 𝑙)-matrix 𝐶 = 𝑐 , whose element 𝑐 is equal to the income of the dm when he chooses a strategy 𝑢 and the uncertainty state 𝑠 realizes. in this case, finding the optimal guaranteed income and optimal guaranteed risk does not cause computational difficulties – it is reduced to choosing maximal and minimal values from finite sets of numbers. thus, finding a wald solution consists in finding a maximin strategy for matrix game in pure strategies with a payoff matrix 𝐶. the dm's use of mixed strategies can improve his expected payoff. the task of calculating the savage solution can be interpreted in terms of goal programming [8]. let us associate to each uncertainty state 𝑠 j-th criterion, the values of which are specified by the j-th column of the matrix c . consider the corresponding multi-criteria optimization problem with the set 𝑋 of admissible strategies of the dm. as a target (or ideal, utopian) point in the ldimensional space of criteria, let us assume a point – a set of maximum values of income at a known uncertainty state: 𝑐∗ = max 𝑐 , … , max 𝑐 . (10) the risk function in this case is (𝑘 × 𝑙) − matrix 𝑅 = 𝑚𝑎𝑥 𝑐 − 𝑐 . (11) the application of the savage minimax approach to the risk matrix formally coincides with the search for an admissible strategy that gives the value of the vector criterion that is as close as possible to the target point in the uniform chebyshev metric. consequently, the search for a savage-optimal solution can be interpreted as a goal-programming problem. in the main considered continuous case (1) (3), the construction of the risk function (1) in explicit form is much more complicated. the analogy with goal programming is preserved if we consider the corresponding multi-criteria problem with an infinite continuous set of criteria {𝑓(𝑥, 𝑦)} ∈ , in which each possible value of uncertainty corresponds to its own criterion function. however, this analogy has little computational value here. 5 optimization of deposit structure by income and risk under uncertainty copyright ©2025 assa adv. in systems science and appl. (2025) 5. optimization of guaranteed risk and linear programming let us consider step-by-step the process of constructing the optimal risk-guaranteed solution in problem (1) (3), (8) in accordance with the definition in formula (8). to construct the risk function, we first calculate the function 𝑓[𝑦] = max ∈ 𝑓(𝑥, 𝑦) (12) of the best results for the dm at each known value of uncertainty y∈ 𝑌 . due to the compactness of the sets 𝑋 and 𝑌 and the continuity of the function 𝑓(𝑥, 𝑦) the function (12) is continuous on the set y. moreover, since 𝑋 is a polyhedron, by virtue of the known extremal property of linear functions on a polyhedron, at any 𝑦 ∈ 𝑌 the maximum in (12) is also reached at one of the vertices of this polyhedron. therefore, the function 𝑓[𝑦] is a piecewise linear function of the form 𝑓[𝑦] = max 𝑓 𝑥( ), 𝑦 = max 𝑐 𝑥 ( ) 𝑦 , 𝑦 ∈ 𝑌, (13) where 𝑥( ), … , 𝑥( ) is the set of vertices of the polyhedron 𝑋. the risk function (3) will also be at each 𝑥 ∈ 𝑋 a piecewise linear convex function on the argument y as the sum of linear and piecewise linear functions. the next step is to find the guaranteed risk φ[𝑥] = max ∈ φ(𝑥, 𝑦) . (14) the piecewise linear property for 𝛷[𝑥] is preserved given that y is a polyhedron. the form of the function 𝛷[𝑥] depends on the configuration of the set y of possible uncertainty values. in the case of bilateral constraints, it is found explicitly in specific cases (see, for example, the next section). suppose that an explicit form of the guaranteed risk function 𝛷[𝑥] is established: φ[𝑥] = max 𝑙 (𝑥), (15) where 𝑙 (𝑥) (𝑖 = 1, … , 𝐿) are linear functions. then the problem of minimization of the piecewise linear function of guaranteed risk can be [9] reduced to the following linear programming (lp) problem: 𝑧 → min 𝑥 ∈ 𝑋 𝑧 − 𝑙 (𝑥) ≥ 0 (𝑘 = 1, … , 𝐿) (16) with m+1 variable and l additional constraints. the optimal value of the variable 𝑧 is equal to the optimal guaranteed risk; the minimum point of this problem determines the optimal allocation of a unit of resource by activity. the solution of the lp problem (16) can be performed by the simplex method, including the regular means of the "solution search" section in excel, while the main difficulty is to construct the guaranteed risk function φ[𝑥] an alternative method of solving the original problem is to find the minimum of the function φ[𝑥] by investigating its extremal properties on separate parts of the boundary and inside the set 𝑋. this approach was used in [4, 5] in the case of three types of activity. we considered the problem of distributing a monetary unit resource over deposits in three types of currencies with uncertain future rates. the advantage of the method is to obtain the solution in analytical form, which made it possible to perform multivariate calculations in excel and present their results in tabular and graphical form. however, the capabilities of the method are limited by low dimensions of the problems. on the other hand, the use of piecewise linear programming methods allows to deal with problems of higher dimensionality, but there remains the need to pre-calculate the guaranteed risk v.s. molostvov 6 copyright ©2025 assa adv. in systems science and appl. (2025) function. in addition, this method is less explicit and it is more difficult to perform multivariate calculations. therefore, a combination of these two approaches seems promising. 5. solving the problem of optimal resource allocation for three types of deposits the problem of the optimal structure of a multi-currency deposit [4] is of great practical interest. let a unit of money in the currency 𝑉 be distributed over three deposits in the currencies 𝑉 , 𝑉 ,𝑉 with known interest rates 𝑑 , 𝑑 , 𝑑 . one of the currencies, 𝑉 , is used to calculate the total income at the end of the period of deposits: 𝑓(𝑥, 𝑦) = 1 + 𝑑 𝐾 𝑥 𝑦 + 1 + 𝑑 𝐾 𝑥 𝑦 + (1 + 𝑑 )𝑥 , (17) where 𝑥 is the amount of currency 𝑉 , converted at the initial rate 𝐾 to currency 𝑉 (𝑖 = 1,2), 𝑥 is the remaining resource to deposit in currency 𝑉 . the initial rates 𝐾 , 𝐾 are assumed to be known, while 𝑦 , 𝑦 are the unknown rates of currencies 𝑉 , 𝑉 relative to currency 𝑉 at the end of the deposit period. regarding uncertain rates 𝑦 , 𝑦 we assume that they can take any value from the given intervals 𝑦 ∈ [𝑎 , 𝑏 ], (𝑖 = 1,2). (18) this problem is a special case of the problem (1)–(3): 𝑚 = 𝑛 = 3; 𝑐 = 0, 𝑖𝑓 𝑖 ≠ 𝑗; 𝑐 = 1 + 𝑑 𝐾 (𝑖 = 1,2), 𝑐 = (1 + 𝑑 ), 𝐾 = 1 (19) and the set of acceptable strategies of the dm and the set of possible states of uncertainty are given by linear constraints 𝑥 + 𝑥 + 𝑥 = 1, 𝑥 ≥ 0 (𝑖 = 1,2,3), 𝑎 ≤ 𝑦 ≤ 𝑏 (𝑗 = 1,2), 𝑦 = 1. (20) the dimensionality of the problem can be reduced by excluding the variable 𝑥 from the constraints and the target function, using the constraint-equality 𝑥 = 1 − 𝑥 − 𝑥 . we obtain a problem with two variables (𝑥 , 𝑥 ) as dm’s decisions and two uncertain exchange rates (𝑦 , 𝑦 ): 𝑓(𝑥 , 𝑥 , 𝑦 , 𝑦 ) = 1 + 𝑑 𝐾 𝑥 𝑦 + 1 + 𝑑 𝐾 𝑥 𝑦 + (1 + 𝑑 )(1 − 𝑥 − 𝑥 ), (21) 𝑋 = 𝑥 + 𝑥 ≤ 1 𝑥 , 𝑥 ≥ 0 , (22) 𝑌 = 𝑎 ≤ 𝑦 ≤ 𝑏 𝑎 ≤ 𝑦 ≤ 𝑏 . (23) in [5] an explicit form of the piecewise linear risk function was obtained. in the same article, the expression for the guaranteed risk was established: φ[𝑥] = max {𝛼 𝑥 + 𝛼 𝑥 , 𝛽 (1 − 𝑥 ) + 𝛼 𝑥 , 𝛽 (1 − 𝑥 ) + 𝛼 𝑥 }, 𝑥 ∈ 𝑋. (24) here, the coefficients of the linear functions forming the piecewise linear function (24) are determined by the initial parameters of the problem: 𝛼 = (1 + 𝑑 ) − 1 + 𝑑 𝐾 𝑎 (𝑖 = 1,2), (25) 7 optimization of deposit structure by income and risk under uncertainty copyright ©2025 assa adv. in systems science and appl. (2025) 𝛽 = 1 + 𝑑 𝐾 𝑏 − (1 + 𝑑 ) (𝑖 = 1,2). (26) in accordance with the results of the previous section, the optimal risk-guaranteed solution can be found from the following linear programming problem: ⎩ ⎪ ⎨ ⎪ ⎧ 𝑧 ⟶ min, 𝑥 + 𝑥 ≤ 1, 𝑥 , 𝑥 ≥ 0, 𝑧 − 𝛼 𝑥 − 𝛼 𝑥 ≥ 0, 𝑧 + 𝛽 𝑥 − 𝛼 𝑥 ≥ 𝛽 , 𝑧 − 𝛼 𝑥 + 𝛽 𝑥 ≥ 𝛽 . (27) the optimal value 𝑧 equals the optimal guaranteed risk, the minimum point of 𝑥∗ = (𝑥∗, 𝑥∗) together with 𝑥∗ = 1 − 𝑥∗ − 𝑥∗ sets the optimal allocation of a unit of resource 𝑉 to the three contributions. in [6] the method of direct solution calculation was used through the analysis of extreme properties of the guaranteed risk function on the boundary and interior parts of the set 𝑋. the corresponding algorithm and its implementation in excel were used for multivariate oneand twoparameter calculations in the problem with three types of currencies. in parallel, "pointwise" control calculations by the method of piecewise linear programming were carried out selectively, in which the methods gave matching results. 6. the multi-criteria approach: preliminary information consider a general two-criteria optimization problem: 𝐺 = ⟨𝑋, 𝑔 (𝑥), 𝑔 (𝑥)⟩, (28) where 𝑋 is the set of admissible solutions (alternatives), 𝑔 (𝑥) and 𝑔 (𝑥) are the target functions to be maximized by choosing the alternative 𝑥 ∈ 𝑋. suppose that the set 𝑋 is closed and bounded (compact), and the functions 𝑔 (𝑥) and 𝑔 (𝑥) are continuous. then there are finite limits (partial maxima) of the target functions. definition 1: an admissible solution 𝑥 ∈ 𝑋 is called a pareto-improvement for an admissible solution 𝑥 ∈ 𝑋 , if 𝑔 (𝑥 ) ≥ 𝑔 (𝑥) (𝑖 = 1,2) and at least one inequality is strong. definition 2: an admissible solution 𝑥 ∈ 𝑋 is called pareto optimal (p-optimal) if no pareto-improvement exists for it. we denote the set of all pareto optimal solutions by 𝑋 , the set of corresponding optimal estimates 𝑔(𝑥) = (𝑔 (𝑥), 𝑔 (𝑥)) by 𝑌 . this set forms the so-called northeast boundary of the set of all estimates 𝑌 = 𝐺(𝑋). similarly to the above, the notion of slater optimality is defined and sets 𝑋 , 𝑌 are introduced. by slater-improvement here we mean the improvement of all criteria at once. from pareto optimality follows slater optimality: 𝑋 ⊇ 𝑋 , 𝑌 ⊇ 𝑌 . the converse, generally speaking, is not true. note that the sets 𝑌 and 𝑌 , 𝑋 and 𝑋 in many problems coincide or differ insignificantly. as for the choice of the final optimal solution, there is a wide range of recommendations and procedures, significant part of which consists in "scalarization" of the problem, i.e., in replacing the vector criterion by a single scalar criterion. as such, linear convolutions, nonlinear monotonic v.s. molostvov 8 copyright ©2025 assa adv. in systems science and appl. (2025) convolutions (in particular, leontief functions), and the lexicographic criterion are used. various iterative procedures, such as the method of successive concessions and others, are also proposed. one approach is to make available to the dm the entire set of por s-optimal estimates or at least some sufficiently representative subset, a kind of 𝜀-network. as a rule, the pareto frontier has a complex structure. if it is possible to construct it in a visual form and with a transparent for the dm algorithm of construction, it facilitates the task of choosing the final decision. let us use the following method of explicitly constructing the set s. let us fix the minimum acceptable level of one criterion by the constraint 𝑔 (𝑥) ≥ 𝐶 and solve the single-criteria optimization problem 𝑔 (𝑥) → max 𝑥 ∈ 𝑋 𝑔 (𝑥) ≥ с . (29) prorosition: let 𝐶 ≤ 𝑚𝑎𝑥 ∈ 𝑔 (𝑥) and 𝑥∗ be a solution of the problem (29). then 𝑥∗ is slater-optimal. proof. indeed, let 𝑌 = 𝑔(𝑋) = {𝑔(𝑥)|𝑥 ∈ 𝑋} be the set of all achievable criterion values, and 𝑋 = {𝑥 ∈ 𝑋|𝑔 (𝑥) ≥ 𝐶}. the alternatives from the set 𝑋\𝑋 cannot be slater-improvements for 𝑥∗ , since they have 𝑔 (𝑥) < 𝐶 ≤ 𝑔 (𝑥∗). on the other hand, for all alternatives in the set𝑋 , by definition the solution 𝑥∗, 𝑔 (𝑥) ≤ 𝑔 (𝑥∗) = max ∈ 𝑔 (𝑥) . hence, 𝑋 doesn’t contain be a slater-improvement for 𝑥∗. since 𝑋 = (𝑋\𝑋 ) ∪ 𝑋 , the proposition is proved. remark: in nondegenerate cases, the solution 𝑥∗ and the estimate 𝑔∗(𝑥) are pareto optimal well. for this, in particular, it is sufficient that the maximum in problem (29) is unique in the value of the function 𝑔 (𝑥) [7]. 8. algorithm for constructing the set of pareto-optimal solutions let us return to the problem of the optimal deposit structure given two criteria: guaranteed income and guaranteed risk (section 5). the formula for the guaranteed income in accordance with expressions (21) and (23) takes the form: 𝑓[𝑥] = 1 + 𝑑 𝐾 𝑎 𝑥 + 1 + 𝑑 𝐾 𝑎 𝑥 + (1 + 𝑑 )(1 − 𝑥 − 𝑥 ) ⇒ max. (30) or 𝑓[𝑥] = 1 + 𝑑 𝐾 𝑎 − (1 + 𝑑 ) 𝑥 + 1 + 𝑑 𝐾 𝑎 − (1 + 𝑑 ) 𝑥 + (1 + 𝑑 ) ⇒ max. (31) taking into account definitions (25) we finally have: 𝑓[𝑥] = −𝛼 𝑥 − 𝛼 𝑥 + (1 + 𝑑 ) ⇒ max. (31) secondary coefficients 𝛼 (𝑖 = 1,2) in (31) depend on initial parameters and have the following meaningful interpretation. the value 𝛼 specifies the difference between the direct income of a unit of currency𝑉 and the income when using this unit to buy currencyм at the beginning of the period at the rate 𝐾 and selling the total (1 + 𝑑 ) at the minimum rate 𝑎 at the end of the period. the expression for the guaranteed risk is given above in (24): 𝛷[𝑥] = max{𝛼 𝑥 + 𝛼 𝑥 , 𝛽 (1 − 𝑥 ) + 𝛼 𝑥 , 𝛽 (1 − 𝑥 ) + 𝛼 𝑥 } ⇒ min. (31) 9 optimization of deposit structure by income and risk under uncertainty copyright ©2025 assa adv. in systems science and appl. (2025) the coefficient 𝛽 (𝛽 ) specifies the difference between the return on a unit of currency𝑉 , deposited via currency 𝑉 (𝑉 ), and the direct return on currency𝑉 at the highest final rate 𝑉 (𝑉 ). consider the vector maximization problem ⟨𝑋, 𝑓[𝑥], −𝛷[𝑥]⟩, (32) in which risk minimization is replaced by risk maximization with minus sign, the set of admissible alternatives is given by conditions (22). let us apply to it the ideas from the previous section. as it was shown above, the minimization of guaranteed risk can be performed by solving the lp problem (27). let us supplement it with a restriction on the value of the first criterion – guaranteed income: ⎩ ⎪⎪ ⎨ ⎪⎪ ⎧ 𝑧 ⟶ min, 𝑥 + 𝑥 ≤ 1, 𝑥 , 𝑥 ≥ 0, 𝑧 − 𝛼 𝑥 − 𝛼 𝑥 ≥ 0, 𝑧 + 𝛽 𝑥 − 𝛼 𝑥 ≥ 𝛽 , 𝑧 − 𝛼 𝑥 + 𝛽 𝑥 ≥ 𝛽 , 𝑓[𝑥] = −𝛼 𝑥 − 𝛼 𝑥 + (1 + 𝑑 ) ≥ 𝐶. (33) the parameter 𝐶 has the meaning of the minimum acceptable guaranteed income. potentially, the solution of the lp problem (33) at all values of 𝐶 from 𝐶 = max ∈ 𝑓(𝑥) to 𝐶 = min ∈ 𝑓(𝑥) will give the whole set 𝑋 of pareto-optimal solutions (more precisely, we obtain the set of all slater-optimal solutions 𝑋 ⊇ 𝑋 ). the interval for the parameter 𝐶 can be reduced until the moment of reaching the extreme value of the second criterion, because further reduction of this parameter will only lead to deterioration (reduction) of the guaranteed income without improving the value of the guaranteed risk. therefore, new neither slater-optimal nor pareto-optimal solutions will be obtained. in practice, of course, one has to consider a discrete finite set of values of the parameter 𝐶 . for example, we can use the sequence {𝐶 }: 𝐶 = 𝐶 (1 − ∆ ), 𝑘 = 0,1,2, … 𝑘 , (34) where 𝑘 is defined from the condition φ 𝑥 = min ∈ φ[𝑥], {𝑥 } is the sequence of solutions of problem (33). it follows from the results of section 5 that 𝐶 = max ∈ 𝑓[𝑥] = max(−𝛼 , −𝛼 ) + (1 + 𝑑 ), (35) 𝐶 = min ∈ 𝑓[𝑥] = min(−𝛼 , −𝛼 ) + (1 + 𝑑 ), (36) and min ∈ φ[𝑥] is found from the solution of the single-criteria lp problem (27) about the minimum guaranteed risk. the result of solving the problem (33) (𝑥( ), 𝑧( )) has a transparent and understandable interpretation for the dm. let the dm agrees to reduce the guaranteed income by no more than 100∆ % of the maximum value. under this condition, the solution 𝑥( ) guarantees the risk of no more than φ[𝑥( )] = 𝑧( ). the sequence of points {𝑥( )} = {(𝑥 ( ) , 𝑥 ( ) )} generates a sequence of values of the vector criteria {(f[𝑥( )], φ[𝑥( )])} . recall that the volume of currency 𝑥 = 1 − 𝑥 −𝑥 . this information, presented in tabular and graphical form, can be used by the dm, along with other considerations available to him, to make the final decision on the deposit structure. v.s. molostvov 10 copyright ©2025 assa adv. in systems science and appl. (2025) conclusion the problems of decision making under complete information for linear systems are sufficiently developed both in theory and in applications. however, as a rule, in practice, decision making occurs under uncertainty of a number of parameters. in economic research, these are usually future values of prices, exchange rates, demand, terms of transactions, etc. the two most commonly used performance indicators in this situation are income and savage risk [2], understood as loss of income due to incomplete information. the maximizing (for income) or minimizing (for risk) strategies of dm are considered as the optimal solution in accordance with the principle of the best-guaranteed result [1]. the possibility of obtaining optimal solutions in linear systems with interval uncertainty of coefficients by methods of piecewise linear programming was considered within the framework of the above approach. the problem of resource allocation by several types of activity was studied. the income given uncertain factors is a bilinear function, linear on strategy under fixed uncertainty and on uncertain parameters under fixed strategy. the sets of admissible strategies and possible values of uncertainty are assumed to be polyhedra. because of these properties, the guaranteed return and risk are piecewise linear functions defined on the polyhedron of strategies. this allowed us to apply the piecewise linear programming method and reduce the problem of finding the optimal guaranteed risk to some linear programming problem. the obtained results are applied to the solution of the problem of optimal allocation of financial resource on three currency deposits. as a rule, the higher the return on investment, the higher the risk. the attempt to find a compromise between these two indicators leads to the vector optimization problem and the concept of pareto optimality. we propose a method for solving the two-criterion problem of optimal allocation of financial resource to three currency deposits, which uses the parameterization of the pareto set by the value of the guaranteed income criterion. thus, the construction of a representative subset of pareto-optimal solutions is reduced to solving a finite number of linear programming problems. the obtained information, presented in tabular and graphical form, can be used by the dm along with other available considerations to make the final decision on the allocation of financial resources. the results can be used in analyzing financial management problems under conditions of incomplete information. acknowledgements this article is an output of a research project «decision-making in socio-economic, political and financial spheres» implemented as part of the basic research program at the national research university higher school of economics (hse university). the work was partially supported by the international center of decision choice and analysis of the national research university higher school of economics. the author is grateful to the reviewers for their thorough and in-depth study of the paper, constructive comments, and valuable suggestions, which improved the paper significantly. references. 1. wald, a. 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(1990). introduction to minimax. new york: dover. microsoft word 1456-article text-6596-1-18-20240108 adv syst sci appl 2023; 04; 18-30 published online at https://ijassa.ipu.ru. optimizing frequency stability in distributed power grids through advanced power flow control with 25mw photovoltaic integration ali abderrazak tadjeddine*, mohammed sofiane bendelhoum, iliace arbaoui, ridha iliace bendjillali, mohamed alami lseter laboratory, nour bachir university centre, el-bayadh, algeria abstract: this study addresses the challenges associated with integrating photovoltaic (pv) systems into the western algerian power grid, primarily focusing on mitigating frequency fluctuations induced by variable electrical load profiles. these fluctuations can have adverse effects on power quality and grid stability. to address this challenge, the study introduces the benefits of frequency stability analysis and optimal power flow (opf) control within distributed and adaptable power networks. the primary goal is to enhance the integration of pv systems by ensuring the dynamic stability of the network, optimizing solar energy utilization through the maximum power point tracking (mppt) technique, and employing var compensation methods. these strategies are designed to facilitate reliable and economically viable pv integration into the power grid, particularly in the saida-naama region. through the implementation of opf control, var compensation techniques, and pv integration, this research achieves a notable reduction in power losses, ranging from 10% to 25% within a single day. furthermore, the utilization of reactive power control, employing naama's static var compensator (svc) for the transmission network, along with localized compensatory measures for the distribution network, effectively maintains voltage levels within acceptable parameters. these combined efforts provide critical support for the advancement of pv integration, the transition towards variable renewable energy (vre) sources, and the promotion of sustainable power generation practices within the region. keywords: static var compensator, transient stability, photovoltaic integration, mppt, variable renewable energy, power grid, optimal power flow 1. introduction in the realm of electrical system operations, operators encounter notable challenges associated with maintaining voltage stability and managing frequency fluctuations, especially during periods of high load. the preservation of stable voltage and frequency levels stands as a critical pillar in guaranteeing the dependable and efficient functioning of power grids. variations in these parameters can result in adverse consequences, such as equipment malfunctions, diminished power quality, and potential interruptions in power supply. consequently, dispatchers bear the vital responsibility of adeptly overseeing voltage stability and frequency control to uphold the reliability of the system [15]. a pivotal factor influencing voltage stability is the conduct of nonlinear electrical loads under demanding transient conditions. nonlinear loads exhibit intricate characteristics that can affect overall system performance. these loads have the potential to introduce harmonics, distort the voltage waveform, and exhibit dynamic behaviors that can further complicate voltage stability. an accurate understanding and representation of the behavior of these nonlinear loads assume great significance in designing control equations and devising effective operational strategies [1]. * corresponding author: atadj1@gmail.com optimizing frequency stability in distributed power grids… 19 copyright ©2023 assa. adv. in systems science and appl. (2023) static load models offer a simplified portrayal of load behavior within electrical systems, primarily focusing on static temporal attributes such as fixed voltages, constant powers, and constant impedances. while these models provide a valuable foundational framework, they may fall short in capturing the full complexity and dynamic nature of actual load behavior. in scenarios where traditional static load models prove inadequate in faithfully representing load characteristics, the adoption of alternative dynamic load models becomes imperative. dynamic load models, such as frequency-dependent load models, offer a more detailed and precise comprehension of load characteristics in electrical systems [9, 11]. by accounting for the dynamic responses of loads to fluctuations in frequency, a comprehensive assessment of load interactions and their impact on system performance can be achieved. this knowledge assumes a pivotal role in safeguarding the reliability, stability, and efficient operation of electrical grids. the general expression of the impedance z, current i, and power p, namely zip model can be represented as follows: ⎩ ⎪ ⎨ ⎪ ⎧𝑃 = 𝑃 𝑎 𝑉 𝑉 + 𝑏 𝑉 𝑉 + 𝑐 1 − 𝑘 δ𝑓 q = 𝑄 𝑎 𝑉 𝑉 + 𝑏 𝑉 𝑉 + 𝑐 1 − 𝑘 δ𝑓 (1) in these equations: 𝑃 and 𝑄 represent the active and reactive powers of the load for the operating voltage v, 𝑃 and 𝑄 denote the active and reactive powers of the load for the nominal voltage 𝑉 , 𝑎 , 𝑏 , 𝑐 , 𝑎 , 𝑏 , and 𝑐 are the coefficients of the zip model, 𝑘 is the sensitivity parameter of the active power, 𝑘 is the sensitivity parameter of the reactive power, 𝛥𝑓 represents the frequency deviation in p.u. furthermore, frequency fluctuations can occur due to imbalances between the power generation and load demand in the electrical system. these fluctuations can lead to synchronization issues among devices connected to the grid, potentially causing operational challenges and compromising system stability. dispatchers must closely monitor and regulate the system's frequency to ensure its operation within an acceptable range. effective frequency control mechanisms and feedback systems are necessary to maintain the desired frequency levels [12, 10]. in the modelling and control equations of electrical systems, it is crucial to consider the impact of nonlinear loads and their behaviour under critical transient conditions. these models help capture the complex characteristics of various electrical devices and their interactions within the system. by incorporating these models, dispatchers can gain a deeper understanding of the system dynamics, enabling them to devise robust control strategies to mitigate voltage instability and frequency fluctuation [16, 17]. the power generation facilities need to be designed to operate under exceptional conditions for limited durations. the operating durations are specified in table 1 below, indicating the minimum times an electricity generation unit must be able to remain connected to the grid at different frequencies deviating from the nominal value [3, 6-8]. moreover, it is essential to recognize that frequency fluctuations can arise from disparities between power generation and load demand within the electrical system. these fluctuations have the potential to trigger synchronization issues among grid-connected devices, which, in turn, can give rise to operational challenges and jeopardize overall system stability. dispatchers bear the responsibility of vigilant monitoring and control over the system's frequency to ensure it operates within an acceptable range [13, 18]. when it comes to modeling and establishing control equations for electrical systems, it is paramount to account for the influence of nonlinear loads and their responses under critical transient conditions. these models serve as invaluable tools for capturing the intricate characteristics of diverse electrical devices and their intricate interactions within the system. 20 a. tadjeddine, m. bendelhoum, i. arbaoui, r. bendjillali, m. alami copyright ©2023 assa adv. in systems science and appl. (2023) by incorporating these models into their analyses, dispatchers can attain a more profound understanding of system dynamics, empowering them to formulate robust control strategies aimed at mitigating voltage instability and addressing frequency fluctuations [8, 2]. furthermore, it is imperative that power generation facilities are designed to operate under exceptional conditions for specific durations. the power generation facility's ability to remain connected and synchronized with the electricity transmission network. even during rapid frequency variations of up to 1 hz per second, is pivotal [19, 4]. this capability ensures the generation unit's ongoing reliable operation and the provision of a stable electricity supply, even when confronted with substantial frequency fluctuations. maintaining coupling and synchronization under such rapid variations is essential for safeguarding safety and system stability during dynamic conditions. frequency stability analysis plays an indispensable role in understanding and mitigating these fluctuations, thereby ensuring the optimal performance of power systems [10, 20]. it encompasses the examination of the dynamic behavior of grid frequency and the identification of factors contributing to its oscillations [14]. through a comprehensive analysis of frequency stability, power system operators and engineers can devise effective control strategies to maintain the frequency within acceptable limits. several factors contribute to the attainment of such an impressive effective energy production rate: 1.1. solar resource availability the ample solar energy resources within the esc region play an instrumental role in maximizing energy production. these regions, characterized by high solar irradiation levels and minimal shading or obstructions, allow pv panels to receive abundant sunlight, resulting in heightened energy generation. 1.2. efficient pv system design efficient pv system design involves a thorough assessment of these factors to maximize energy production and ensure the long-term performance and economic viability of the pv installation. by paying careful attention to these details, the esc pv park can harness the full potential of solar energy resources for sustainable electricity generation. 1.3. effective operations and maintenance routine maintenance and continuous monitoring of the pv park are essential for ensuring optimal system performance. this includes tasks such as panel cleaning, fault inspections, and addressing technical issues promptly to minimize energy losses and maximize production [6]. 1.4. grid integration and power management the integration of the esc pv park into the grid and the application of efficient power management systems are vital for optimizing its output. this may involve the use of technologies like grid-tied inverters, energy storage systems, and advanced control algorithms to regulate power flow, maintain grid stability, and minimize energy curtailment. this signifies the effective utilization of available solar resources, maximizing the pv park’s output, profitability, and contributing to greener and more sustainable energy generation. frequency stability analysis assumes a critical role in the successful integration of pv systems into power grids. by ensuring frequency stability, we can effectively address the challenges posed by fluctuations in renewable energy generation and promote the reliable and efficient operation of electrical grids. the subsequent sections of this paper will delve deeper into the control measures and strategies employed in frequency stability analysis, presenting their potential benefits and practical applications [7, 14]. frequency monitoring and control are pivotal in maintaining the electrical grid within predetermined frequency ranges. the balance between power generation and demand influences grid frequency; excessive generation raises frequency, while excess demand lowers optimizing frequency stability in distributed power grids… 21 copyright ©2023 assa. adv. in systems science and appl. (2023) it [5, 12]. this study aims to explore the advantages of frequency stability analysis in the context of pv integration. by comprehending the factors influencing frequency variations and implementing control measures, we can enhance pv system integration, optimize performance, and reduce disruptions to the grid. this paper centers its attention on elucidating the significance of frequency stability analysis in the context of integrating pv systems into power grids, with a specific focus on the case of western algeria, as illustrated in fig. 1.1(a) below. its objective is to shed light on the challenges arising from frequency fluctuations resulting from variable electrical load profiles and underscore the criticality of preserving frequency stability to ensure reliable and efficient grid operations. (a) (b) fig. 1.1. sample figure with caption, (a) geographical photovoltaic park in the esc region, (b) average temperature on may 21, 2023 in esc region, algeria through an in-depth exploration of these facets, this study endeavors to offer valuable insights and practical methodologies to assist dispatchers and system operators in effectively managing voltage stability, frequency control, and energy management within electrical systems. by adeptly addressing these challenges, power grids can operate with heightened reliability, ensure optimal power quality, and provide uninterrupted electricity supply to consumers. in fig. 1.1(b), the graphical representation exhibits a line graph with markers denoting the maximum and minimum average temperatures recorded on may 21, 2023, in el-abiodh sidi cheikh (esc) region, algeria, at 6:12 am, a marker positioned at 17.09°c signifies the minimum temperature, while at 4:10 pm; a marker located at 28.52°c represents the maximum temperature. this graph visually captures the temperature variations throughout the day. indeed, by harnessing these factors and implementing best practices in the deployment and operation of pv systems, it becomes feasible to achieve a remarkably high effective energy production rate of 64.75%. the fig. 1.2. illustrates daily load profiles on pq buses, displaying typical electricity demand patterns with peak hours and lower demand periods. fig. 1.2. evolution of the daily load profile 22 a. tadjeddine, m. bendelhoum, i. arbaoui, r. bendjillali, m. alami copyright ©2023 assa adv. in systems science and appl. (2023) 2. materials and mathematical formulation the production generators in the network supply the required active and reactive powers based on the demand profile while ensuring that the generated reactive power remains within specified limits. the complex power at node 𝑖 can be represented by: 𝑆̅ = 𝑃 + 𝑗𝑄 𝑆̅ = 𝑃 − 𝑃 − 𝑃 + 𝑗 𝑄 − 𝑄 − 𝑄 (2) where 𝑃 , 𝑄 : active and reactive power at node 𝑖; 𝑃 , 𝑄 : generation active and reactive power at node 𝑖; 𝑃 ,𝑄 : active and reactive load power at node 𝑖; 𝑃 ,𝑄 : active and reactive power transmitted. by properly managing the active and reactive power flow in the network, the production generators can ensure that the demand is met while maintaining the required voltage and power factor levels within acceptable limits. this balance between generation and consumption is crucial for operation of system. modelling of the powers transmitted via the transmission line by: 𝑆̅ = |𝑉 | . 𝑌∗ − 𝑉 . 𝑉∗. 𝑌∗ + |𝑉 | . 𝑌∗ 𝑆̅ = 𝑉 . 𝑌∗ − 𝑉 . 𝑉∗. 𝑌∗ + 𝑉 . 𝑌∗ (3) where 𝑆̅ , 𝑆̅ are apparent powers transited from node 𝑖 𝑡𝑜 𝑗 & 𝑗 𝑡𝑜 𝑖 respectively. active and reactive losses are involved during power transmission. ⎩ ⎪ ⎨ ⎪ ⎧𝑆̅ = 𝑆̅ = (𝑆̅ + 𝑆̅ ) 𝑃 = 𝑅 𝑆̅ 𝑄 = 𝐼𝑚𝑎𝑔 𝑆̅ (4) where 𝑆̅ , 𝑃 , 𝑄 are total apparent, active and reactive power lost in the network. the complex voltage and the active and reactive powers, for each pv bus and pq bus, we have the following equation can be written as follows: ⎩ ⎪ ⎨ ⎪ ⎧ ∆𝑃 = 𝑃 − 𝑃 = 0 ∆𝑄 = 𝑄 − 𝑄 = 0 ∆𝑃 = 𝑃 − 𝑉 ∑ 𝑉 (g cos θ + jb sin 𝜃 ) = 0 ∆𝑄 = 𝑄 − 𝑉 ∑ 𝑉 (g sin θ − jb cos 𝜃 ) = 0 (5) where 𝜃 = 𝜃 − 𝜃 is the transport angle between buses 𝑖 and 𝑗. at the core of particle swarm optimization (pso) is a swarm of particles, each representing a potential solution. these particles move through the solution space in search of the best solution. they adjust their positions based on their current velocity and their memory of the best solution they've encountered so far. additionally, they are influenced by the best solutions found by their neighbors within the swarm. pso is used in optimization problems, particularly in cases where the objective function is complex and lacks a simple mathematical expression. it has been applied in various fields, including engineering, economics, and machine learning. pso's ability to explore solution spaces efficiently and its simplicity make it a popular choice for solving optimization problems. 2.1. power grid presentation figure 2.1. presents the single-line diagram illustrating the topology of the saida-naama grid region. within this diagram, we can observe the naama power plant and the esc photovoltaic park, serving as key generation sources. additionally, the static var compensator optimizing frequency stability in distributed power grids… 23 copyright ©2023 assa. adv. in systems science and appl. (2023) is featured as a device responsible for controlling the transmission network, thereby ensuring both voltage stability and power factor regulation. the distribution network is connected to various consumer categories representing electrical loads, collectively contributing to the overall power demand. in terms of power generation, the system comprises a 180 mw gas turbine power plant situated in the naama region and a 25𝑀𝑊𝑝 photovoltaic park located in the el-abiodh sidi cheikh region of algeria. the electrical loads within the network predominantly fall into two categories: residential and industrial consumers. the transmission network operates at 220 kv, while the distribution network functions at 60 kv. fig. 2.1. single-line topology diagram of the saida-naama grid under normal operating conditions, the power transit limits for transmission lines and autotransformers are conventionally set at 80% of their maximum capacity. this choice of power transit limits, below the maximum capacity, serves to establish a safety margin, ensuring that these components operate within a secure and reliable range. these limits take into account factors such as fluctuations in load demand, variations in environmental conditions, and the potential occurrence of unforeseen contingencies. this approach helps uphold the system's security and minimizes the risks associated with equipment failures or overloads. 2.2. automatic active power generation control a frequency deviation step response characterizes how a power system's frequency reacts when subjected to a sudden change in power generation or load demand. when such a step change occurs, the frequency temporarily deviates from its nominal value before stabilizing again. this behavior typically follows a specific pattern as shown in fig. 2.2. with the following steps, initial deviation and transient response. the duration of the transient response depends on the system's inherent inertia and the speed of the control mechanisms in place. figure 2.2 illustrates a block diagram of the closed-loop control system, where the frequency deviation is continuously monitored and processed by the pi (proportional-integral) controller. the control signal generated by the pi controller adjusts the generator set points through the governor control system. the power system model provides feedback on the actual frequency, closing the control loop. this mw load frequency control (lfc) model with a pi controller ensures frequency stability by dynamically adjusting the generator outputs based on the frequency deviation, maintaining the power system within acceptable limits. for a load change of ∆𝑃 (𝑠) = 0.2 𝑝. 𝑢, if you set the integral controller gain 𝑘 to 4, the closed-loop transfer function relating the load change to the frequency deviation ∆𝛺(𝑠) can be determined based on the pi controller settings. 24 a. tadjeddine, m. bendelhoum, i. arbaoui, r. bendjillali, m. alami copyright ©2023 assa adv. in systems science and appl. (2023) fig. 2.2. blocks diagram of active/load power frequency control model with pid controller key parameters include: turbine time constant 𝜏 = 0.5𝑠 governor time constant 𝜏 = 0.2𝑠 generator inertia constant 𝐻 = 5𝑠 governor speed regulation 𝑅 = 0.05 𝑃. 𝑢 load change factor 𝐷 = 0.67 the turbine is designed to generate electricity at a constant frequency of 50hz, with a rated output of 180mw. however, a load change of 67.74mw (∆𝑃 = 0.376 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡) occurs. this results in a steady-state frequency deviation of 0.91hz 𝛥𝑓 = |(−0.0182)(50)|when a 67.74mw load change is applied. ⎩ ⎪ ⎨ ⎪ ⎧ ∆ω(𝑠) −∆𝑃 (𝑠) = 𝑇(𝑠) = 𝑠 1 + 𝜏 𝑠 (1 + 𝜏 𝑠) 𝑠(𝐷 + 2𝐻𝑠) 1 + 𝜏 𝑠 (1 + 𝜏 𝑠) + 𝑘 + 𝑠 𝑅 𝑇(𝑠) = 𝑠 + 7𝑠 + 10𝑠 50𝑠 + 353.35𝑠 + 523.45𝑠 + 1033.5𝑠 + 200 (6) 3. results and discussion the simulation results in this study underwent comprehensive analysis and were presented using multiple approaches, including the examination of small signal stability, dynamic voltage behaviour in the time domain, and the utilization of specific resolution algorithms. notably, the study employed the fast decoupled load flow (fdlf) method and the multiobjective optimal power flow (mo-opf) method for in-depth analysis. these analyses were conducted on the electrical network comprising the 220kv transmission system and the 60kv distribution system within the saida-naama region situated in western algeria (as depicted in fig. 2.1.). the software tools used for conducting the analysis and visualizing the results were matlab 2021a and etap 2019. these advanced software platforms provided the necessary computational capabilities to carry out the complex analysis required to evaluate the integration stability of photovoltaic systems in the region effectively. 3.1. voltages profiles results fig. 3.1. demonstrates the impact of integrating the photovoltaic park in the esc region on the dynamic voltage profile throughout the study day of may 21, 2023. it also includes the mw pv production curve. optimizing frequency stability in distributed power grids… 25 copyright ©2023 assa. adv. in systems science and appl. (2023) fig. 3.1. impact of pv integrating dynamic voltage profile in the esc region fig. 3.2. displays the voltage evolution scenario during the study day for maximum loads, comparing the cases with and without reactive power compensation. fig. 3.2. voltage profile during load maximum conditions in both cases: with & without compensation fig. 3.3. illustrates the voltage evolution scenario throughout the study day for minimum loads, comparing the cases with and without reactive power compensation (voltage sag regulation). fig. 3.3. voltage profile during load minimum conditions in both cases: with & without compensation 3.2. frequency control results fig. 3.4. depicts the frequency evolution in grid during the study day of may 21, 2023, incorporating the correction of fluctuations based on the model pi-agc we previously proposed. 26 a. tadjeddine, m. bendelhoum, i. arbaoui, r. bendjillali, m. alami copyright ©2023 assa adv. in systems science and appl. (2023) fig. 3.4. frequency evolution in grid with pi regulation. 3.3. active power production control fig. 3.5. illustrates the evolution of active power production by the naama gas turbine (gt) and the power demand from the electrical grid (pg) without photovoltaic introduction of esc park’s. fig. 3.5. mw production from gas turbine and power grid without pv fig. 3.6. illustrates the evolution of active power production by the naama gas turbine (gt) and the power demand from the electrical grid (pg) without photovoltaic introduction of esc park’s, during a study day on may 21, 2023. fig. 3.6. mw production from gas turbine and power grid with pv the decrease in curves fig. 3.1. and fig. 3.6 between 10h and 18h present the power active evolution production of photovoltaic park (esc_pv). 3.4. total active losses evolution fig. 3.7. displays a comparison between the evolutions of power losses in mw, when considering the presence of photovoltaic energy with var compensation. fig. 3.7. comparison between evolutions of mw power losses 3.5. multi-objective optimization algorithm fig. 3.8. presents the optimal voltages attained by employing the multi-objective-opf with a optimization algorithm, namely mo-opf. optimizing frequency stability in distributed power grids… 27 copyright ©2023 assa. adv. in systems science and appl. (2023) fig. 3.8. optimisation voltage profiles of load max and min with moopf methods with svc and pf control table 3.1 displays the daily power losses and reduction rates for the maximum and minimum load scenarios on may 21, 2023. table 3.1 daily power losses and reduction rates for may 21, 2023 load 𝑀𝑊 𝑀𝑣𝑎𝑟 𝑀𝑊 𝑀𝑣𝑎𝑟 power losses power flow 8.369 -31.542 2.292 -14.976 power losses optimal power flow 6.300 -31.092 2.064 -9.456 mw reduction % 24.72% 10% the integration of photovoltaic energy into an electrical system has several notable impacts on various system parameters: active power 3.1: the generation of photovoltaic energy is contingent on factors such as solar radiation, weather conditions, and time of day. consequently, the active power output from photovoltaic sources can vary. when integrated into the electrical grid, this energy contributes to the overall available active power. injection bus voltage 3.2: integrating photovoltaic energy can positively influence voltage levels within the electrical system. however, maximizing electrical loads can potentially lead to voltage degradation at certain buses, necessitating a thorough examination of local and global reactive power compensation. integration power factor 3.3: the power factor, indicating the efficiency with which a load utilizes supplied energy, may experience fluctuations due to variations in solar energy production. system frequency 3.4: substantial fluctuations in solar energy production can indirectly influence the system's frequency. when photovoltaic energy production diminishes suddenly, such as during passing clouds or adverse weather conditions, an increase in grid load may lead to a minor frequency reduction. these frequency variations are typically modest and can be mitigated through grid regulation mechanisms. the study employs an algorithm that simultaneously addresses various constraints and objectives to optimize power flow. key objectives encompass minimizing mw and mvar power losses, swing bus mw power, shunt var devices, and achieving a flat voltage profile. implementing this algorithm resulted in a notable reduction of power losses, with a 24.72% reduction observed for maximum load conditions and a 10% reduction for minimum load scenarios, all while maintaining optimal voltage adjustments. 5. conclusion this paper presents a comprehensive study that focuses on addressing the static and dynamic integration stability of photovoltaic (pv) systems in the saida-naama region of western algeria. the central challenge addressed in this research pertains to the complex issue of 28 a. tadjeddine, m. bendelhoum, i. arbaoui, r. bendjillali, m. alami copyright ©2023 assa adv. in systems science and appl. (2023) frequency fluctuations, arising from a combination of variations in load curves and electrical production. recognizing the critical nature of this challenge, the study introduces an innovative approach that integrates frequency stability analysis, optimal power flow control, and reactive power compensation within a flexible distributed power grid. the primary objective of this research is to enhance the overall energy quality and stability of the pv park located in el-abiodh-sidi-cheikh (esc) by ensuring dynamic stability across the network. this objective is achieved through the implementation of maximum power point tracking (mppt) techniques, the maximization of solar resource utilization, and the compensation for reactive power variations. the results obtained from this study demonstrate substantial progress in improving the integration of photovoltaic systems. by utilizing optimal power flow control in conjunction with var compensation and enhanced photovoltaic integration, the research reveals significant reductions in active power losses, with reductions of up to 24.72% observed, contingent on the specific load scenarios. furthermore, the study addresses reactive power control through the application of naama's static var compensator (svc) for the transmission network, along with localized compensations for the distribution network. this approach effectively maintains voltage levels well within acceptable thresholds. these findings are of significant importance as they lay the foundation for the integration of photovoltaic systems by increasing the production capacity of the esc photovoltaic station beyond its current 25mwp capacity. moreover, the study provides insights into implementing additional pv stations throughout the region. these advancements contribute significantly to strengthening ongoing efforts toward transitioning to sustainable energy sources and the utilization of green energy. ultimately, this research delivers invaluable knowledge to the field, offering a pathway to overcome the challenges associated with the integration of photovoltaic systems into algeria's national interconnection network, commonly referred to as the rin. references 1. abdollahi, r., pisu, p., & schenwald, d. 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(2023). optimal power flow analysis in distributed grid connected photovoltaic systems, smart grids for smart cities, 1, 131– 142, doi: 10.1002/9781119872108.ch6. 18. tokumitsu, k., amano, h., & kawabe, k. (2021). improved load frequency controller for reduction of both area control error and automatic frequency restoration reserve energy cost, proc. of 2021 ieee madrid powertech (madrid, spain), doi: 10.1109/powertech46648.2021.9494880. 19. wang, p., jiang, q., li, b., liu, t., et. al. (2023). ultra-low frequency oscillation analysis considering thermal-hydro power proportion, international journal of electrical power & energy systems, 148, 108919, doi: 10.1016/j.ijepes.2022.108919. 20. young, j., wilson, d. g., robinet, r. d., & weaver, w. (2023). supervisory optimal control with information flow for wind and solar resilient integration onto an electric power grid. proc. of 22nd wind and solar integration workshop (wiw 2023) (helsinki, finland). doi: 10.1049/icp.2023.2809. 30 a. tadjeddine, m. bendelhoum, i. arbaoui, r. bendjillali, m. alami copyright ©2023 assa adv. in systems science and appl. (2023) appendix table a.1, provided below, outlines these operating durations, delineating the minimum times an electricity generation unit must remain connected to the grid while experiencing variations in frequency away from the nominal value. 21. table a.1 minimum frequency range and operating durations without frequency range operating duration 47 hz 47.5 hz 10 seconds 47.5 hz 48 hz 20 seconds 48 hz 52 hz unlimited 52 hz 52.5 hz 10 seconds adv syst sci appl 2024; 04:66–81 published online at https://ijassa.ipu.ru. on the external estimation of the limit reachable set for the linear discrete-time system based on support hyperplanes anastasiya simkina* moscow aviation institute, moscow, russia abstract: a linear discrete-time system with geometric constraints on control is considered. for the given system, the problem of constructing the limit reachable set is solved. it is proven that it is a cylinder oriented along certain elements of the real jordan basis. statements about the structure of the supporting hyperplane to the convex section of this cylinder are formulated and proven. necessary and sufficient conditions for the boundedness of the limit reachable set are determined. keywords: discrete-time system, limit reachable set, support hyperplane, convex set, polyhedral approximation 1. introduction the classical controllability condition for linear dynamic systems assumes unbounded control actions. however, when considering practical problems, constraints often arise that are determined by various technical characteristics of the control system under consideration. as a result, even with an infinite number of steps, it is not always possible to transfer the system from a given initial state to a certain fixed final state. this fact makes it relevant to study not only the issues of reachability and controllability of various dynamic systems but also to develop methods for constructing and evaluating the limit reachable and controllable sets for an arbitrary control system. on the other hand, these sets can be applied to solving synthesis problems for discrete-time systems [7]. currently, two main directions can be distinguished in this area: the study of individual states for controllability [3–5, 12] and geometric methods for constructing reachable and controllable sets [?, 1,15,16]. thus, when studying nonlinear systems, it is possible to obtain only general properties of controllable sets [3] or their estimates [15, 16]. for the case of linear dynamic equations, it is possible to construct more constructive results for various classes of systems: periodic [11], switchable [4], with positive control [1]. the most rigorous results are formulated for the case of compact and convex constraints on control values [?,7], even allowing the description of limit reachable and controllable sets [2, 5, 12]. analogous problems were also studied in the works of kurzhanskiy a.b. [13–15]. in [8], for linear discrete-time systems with scalar control, which is subject to a first-order summary constraint, it is shown that in the case of stable systems, it is possible to explicitly find the limit reachable set, which is a convex polyhedron symmetric about zero. for higher-order constraints, the description of the limit reachable and null-controllable sets is obtained using supporting halfspaces [9]. the structure of the article is as follows. section 2 is devoted to the problem statement. section 3 describes the procedure for decomposing the original system into subsystems of smaller dimensions, allowing the problem to be simplified. section 4 develops a method ∗corresponding author: abv1998@yandex.ru on the external estimation of the limit reachable set... 67 for the external evaluation of limit reachable sets. the resulting estimates are cylinders or polyhedra, whose borders are determined by supporting hyperplanes and half-spaces. based on the developed methods, section 5 constructs the limit reachable set for the system of stabilizing the glucose level in blood plasma. 2. problem statement consider linear stationary system with discrete time and geometric constraints on the control (a,u): x(k + 1) = ax(k) + u(k), x(0) = 0, u(k) ∈ u , k ∈ n ∪ {0}, (2.1) where x(k) ∈ rn is the state vector, u(k) ∈ rn is the control action at step k, u ⊂ rn denotes the set of admissible control values defining the geometric constraints, and a ∈ rn×n is the system matrix. it is assumed that u is a convex compact set and 0 ∈ intu . we define the limit reachable set y∞ of the system (2.1) as the set of states to which the system (a,u) can be transferred from the origin in any finite number of steps: y∞ = { x ∈ rn : x = n−1∑ k=0 an−k−1u(k), u(0), . . . , u(n − 1) ∈ u , n ∈ n } . (2.2) it is required to investigate the properties and construct estimates of the set (2.2). as an internal estimate of the limit reachable set, the reachable set in a finite number of steps y(n) can be considered, where y(n) is the set of terminal states to which the system (a,u) can be transferred from 0 in n ∈ n steps. a description of y(n) is known [7, lemma 1]. from (2.2) the following representation for y∞ follows: y∞ = ∞⋃ n=0 y(n). it is known that the set (2.2) is convex [18, theorem 1]. this determines the range of problems addressed in this article. it is necessary to determine the necessary and sufficient conditions for the boundedness of the limit reachable set y∞, to investigate its closedness property, and, in the case of boundedness, to construct a description in terms of supporting hyperplanes, which is guaranteed for any convex set [17, theorem 18.8]. in many respects, the reachable sets share properties with the null-controllable sets, which were studied in [2]. for this reason, some of their properties will be analogous. however, there are also fundamental differences, which will be revealed in the subsequent sections. 3. the general structure of reachable sets the article [19] demonstrates that the limit reachable and null-controllable sets for systems with scalar control are cylinders. their orientation is determined by the elements of the real jordan basis and the eigenvectors of the matrix a. a similar result was obtained in [2, lemmas 2–4] for 0-controllable sets for systems with vector control. we will use a similar approach to describe the limit reachable sets (2.2) for the system (2.1). let h1, . . . , hn ∈ rn denote the real jordan basis of the matrix a, which is a set of linearly independent eigenvectors and generalized eigenvectors, in which basis a is described by its real jordan form, that is the following decomposition [6] holds for the matrix s = copyright © 2024 assa. adv syst sci appl (2024) 68 a.v. simkina (h1, . . . , hn) ∈ rn×n: a = s j1 . . . o ... . . . ... o . . . jm s−1, ji ∈ rni×ni , i = 1,m, n1 + . . .+ nm = n, (3.3) where the jordan blocks j1, . . . , jm have one of the following two forms: j =  λ 1 · · · 0 0 λ . . . 0 ... ... . . . 1 0 0 · · · λ  ∈ rñ×ñ, (3.4) j =  raφ i · · · o o raφ . . . o ... ... . . . i o o · · · raφ  ∈ r2ñ×2ñ, aφ = ( cosφ sinφ − sinφ cosφ ) , i = ( 1 0 0 1 ) . (3.5) let p ∈ rñ×n denote the projection matrix onto an ñ-dimensional subspace: p = 0 · · · 0 1 · · · 0 ... ... ... . . . ... 0 · · · 0 0 · · · 1  ñ×n . we describe the structure of the limit reachable set in the following theorem. theorem 3.1: let the order of h1, . . . , hn in the decomposition (3.3) be chosen such that the representation holds a = s ( a⩾1 o o a<1 ) s−1, where a⩾1 ∈ r(n−ñ)×(n−ñ) consists of jordan blocks of the form (3.4) and (3.5) corresponding to eigenvalues with modulus not less than 1, and a<1 ∈ rñ×ñ consists of jordan blocks corresponding to eigenvalues with modulus less than 1. let ỹ∞ ⊂ rñ be the limit reachable set (2.2) for the system (a<1, ps−1u). then the relation holds y∞ = s ( rn−ñ × ỹ∞ ) . proof let us denote ũ = ps−1u . first, we will demonstrate that the theorem holds for the case when the matrix a coincides with its real jordan form, i.e. s is the identity matrix. since u is bounded, there exists a convex set v ∈ r(n−ñ) such that u ⊂ v × ũ . (3.6) consider the dynamic equations (2.1) for the system (a,v × ũ) and an arbitrary control sequence {u(k)}n−1 k=0 ⊂ v × ũ . for all n ∈ n the following representation holds: x(n) = an−1u(0) + an−2u(1) + . . .+ u(n − 1) = copyright © 2024 assa. adv syst sci appl (2024) on the external estimation of the limit reachable set... 69 =  n−1∑ k=0 an−k−1 ⩾1 v(k) n−1∑ k=0 an−k−1 <1 ũ(k)  = ( y(n) x̃(n) ) , where v(0), . . . , v(n − 1) ∈ v , ũ(0), . . . , ũ(n − 1) ∈ ũ , y(n) ∈ rn−ñ, x̃(n) ∈ rñ. therefore, if we denote by ŷ∞ ⊂ rn the limit reachable set for the system (a,v × ũ), then due to the representation (2.2), for any x(n) ∈ ŷ∞ the inclusion x̃(n) ∈ ỹ∞ must also hold. we obtain ŷ∞ ⊂ rn−ñ × ỹ∞. on the other hand, from (3.6) it follows that y∞ ⊂ ŷ∞. finally, we obtain: y∞ ⊂ rn−ñ × ỹ∞. (3.7) let’s prove the reverse inclusion. to do this, we will choose an arbitrary x = ( yt, x̃t )t ∈ rn−ñ × ỹ∞. from the inclusion 0 ∈ intu it follows that there exists v′ > 0 such that [−v′; v′]n−ñ × {0} ⊂ u . therefore, for all n ∈ n the following control will satisfy the constraints of the system (2.1): u(k) = ( v(k) 0 ) , v(k) ∈ [−v′; v′]n−ñ, k = 0, n − 1. since x̃ ∈ ỹ∞, by definition (2.2), there exist m ∈ n and ũ(n), . . . , ũ(n +m − 1) ∈ ũ such that x̃ = n+m−1∑ k=n an+m−k−1 <1 ũ(k). additionally, considering the definition of ũ , it is possible to choose v(n), . . . , v(n +m − 1) ∈ rn−ñ such that u(k) = ( v(k) ũ(k) ) ∈ u , k = 0,m − 1. thus, the control sequence {u(k)}n+m−1 k=0 is admissible for the system (a,u) and, with a zero initial state, leads to the following relations: x(n +m) =  m+n−1∑ k=0 am+n−k−1 ⩾1 v(k) m+n−1∑ k=0 am+n−k−1 <1 ũ(k)  = =  n−1∑ k=0 an+m−k−1 ⩾1 v(k) + n+m−1∑ k=n an+m−k−1 ⩾1 v(k) n−1∑ k=0 an+m−k−1 <1 ũ(k) + n+m−1∑ k=n an+m−k−1 <1 ũ(k)  = am ⩾1 n−1∑ k=0 an−k−1 ⩾1 v(k) + y′ x̃  , where y′ = n+m−1∑ k=n an+m−k−1 ⩾1 v(k). copyright © 2024 assa. adv syst sci appl (2024) 70 a.v. simkina therefore, the equality x(n +m) = x can be obtained by demonstrating that for some n ∈ n, it is possible to choose v(0), . . . , v(n − 1) ∈ [−v′; v′]n−ñ that satisfy the following relations: y = am ⩾1 n−1∑ k=0 an−k−1 ⩾1 v(k) + y′, a−m ⩾1 (y − y′) = n−1∑ k=0 an−k−1 ⩾1 v(k). considering (2.2), is equivalent to the inclusion a−m ⩾1 (y − y′) ∈ y ′ ∞, where y ′ ∞ ⊂ rn−ñ is the limit reachable set for the system ( a⩾1, [−v′; v′]n−ñ ) . for further reasoning, we will use known analytical representations of reachable and 0controllable sets for discrete-time linear systems. it is known [20, lemma 1] that for any n ∈ n the 0-controllable set of the system ( a−1 ⩾1, a −1 ⩾1[−v′; v′]n−ñ ) in n steps has the form: x ′(n) = − n∑ k=1 ( a−1 ⩾1 )−k ( a−1 ⩾1[−v′; v′]n−ñ ) = − n∑ k=1 ak−1 ⩾1 [−v′; v′]n−ñ. on the other hand, according to [7, lemma 1], the reachable set in n steps for the system (a⩾1, [−v′; v′]n1) has the form: y ′(n) = n−1∑ k=0 ak ⩾1[−v′; v′]n−ñ. thus, the 0-controllable set of the system ( a−1 ⩾1, a −1 ⩾1[−v′; v′]n−ñ ) and the reachable set of the system (a⩾1, [−v′; v′]n1) over the same number of steps coincide, and consequently, their limit set analogs x ′ ∞ and y ′ ∞ also coincide. however, according to [2, lemma 3] x ′ ∞ = rn−ñ, since all eigenvalues of a−1 ⩾1 do not exceed 1 in magnitude, being reciprocals of the eigenvalues of the matrix a⩾1 [6]. this trivially implies the inclusion: a−m ⩾1 (y − y′) ∈ y ′ ∞ = rn−ñ, from which it follows that the equality x(n +m) = x is admissible, and as a consequence, x ∈ y∞, leading to rn−ñ × ỹ∞ ⊂ y∞. (3.8) together, (3.7) and (3.8) for the case s = i define the identity: y∞ = rn−ñ × ỹ∞. (3.9) let us consider the case where s ∈ rn×n is an arbitrary non-singular matrix. from (3.9), it follows that for the system (s−1as, s−1u), the limit reachable set y0 ∞ has the form: y0 ∞ = rn−ñ × ỹ∞. taking into account the decomposition (3.3) and for n ∈ n from the dynamic equations we obtain: x(n) = n−1∑ k=0 an−k−1u(k) = n−1∑ k=0 s ( an−k−1 ⩾1 o o an−k−1 <1 ) s−1u(k) = copyright © 2024 assa. adv syst sci appl (2024) on the external estimation of the limit reachable set... 71 = s n−1∑ k=0 ( an−k−1 ⩾1 o o an−k−1 <1 ) s−1u(k). from this, considering (2.2), it follows that x ∈ y∞ if and only if s−1x ∈ y0 ∞, i.e. y∞ = sy0 ∞. theorem 3.1 is proven. for the limit null-controllable sets, it is also guaranteed that they are open and convex [2, theorem 1]. however, when considering the limit reachable sets, general statements about their openness or closedness do not seem possible. example 3.1: consider a two-dimensional system (a,u) with parameters defined as follows: a = ( 0 1 0 0 ) , u = [−1; 1]× [−1; 1]. given that a2 = o, for n ⩾ 2 we have x(n) = u(n − 1) + au(n − 2). therefore, by definition (2.2) the inclusions x ∈ y∞ and x ∈ u + au are equivalent. we can derive the following representation from this: y∞ = u + au = conv {( 1 1 ) , ( −1 1 ) , ( 1 −1 ) , ( −1 −1 )} + conv {( 1 0 ) , ( −1 0 )} = = conv {( 2 1 ) , ( −2 1 ) , ( 2 −1 ) , ( −2 −1 )} . finally, y∞ is closed. example 3.2: consider a two-dimensional system (a,u) with parameters defined as follows: a = ( 0 0 0 1 2 ) , u = [−1; 1]× [−1; 1]. for any n ∈ n and control sequence {u(k)}n−1 k=0 ⊂ u it holds that x(n) = u(n − 1) + au(n − 2) + . . .+ an−1u(0) =  u1(n − 1) n−1∑ k=0 1 2k u2(n − k − 1)  , |x1(n)| ⩽ 1, |x2(n)| ⩽ n−1∑ k=0 1 2k < 2. according to (2.2) y∞ ⊂ [−1; 1]× (−2; 2). on the other hand, for any x = (x1, x2) t ∈ [−1; 1]× (−2; 2) there exists n ∈ n such that |x2| ⩽ 2− 1 2n−1 . copyright © 2024 assa. adv syst sci appl (2024) 72 a.v. simkina consequently, for all k = 0, n − 1 the following control actions will be admissible u2(k) = x2 2− 2−n+1 ∈ [−1; 1], u1(k) = x1 ∈ [−1; 1]. then x(n) =  x1 n−1∑ k=0 1 2k x2 2−2−n+1  =  x1 x2 2−2−n+1 n−1∑ k=0 1 2k  = ( x1 x2 2−2−n+1 ( 2− 2−n+1 )) = ( x1 x2 ) . we have the inclusion [−1; 1]× (−2; 2) ⊂ y∞. finally, the representation holds y∞ = [−1; 1]× (−2; 2), meaning that the limit reachable set y∞ in this example is neither closed nor open. example 3.3: let us consider the case where n = 1 and in the system (a,u) a = 1 2 , u = [−1; 1]. by repeating the reasoning from example 3.2 but applying it only to the single coordinate, we obtain: y∞ = (−2; 2). thus, the limit reachable set y∞ in this example is open. 4. construction of estimates of limit null-controllable sets although theorem 3.1 guarantees that the set (2.2) is a cylinder, it does not determine the structure of the cross-section of this cylinder. essentially, this statement only makes it possible to reduce the original problem of constructing the limit set to an analogous problem for a system of smaller dimension, but it does not solve it completely. for this reason, in this section, we will consider a method for constructing polyhedral external estimates of the set y∞, based on the apparatus of supporting half-spaces and properties of convex sets. however, theorem 3.1 allows us to assume, without any loss of generality, that all eigenvalues of the matrix a are strictly less than 1. indeed, in the opposite case, the original problem is equivalent to the problem of constructing the set ỹ∞ for the subsystem (a<1, ps−1u), which satisfies the eigenvalue constraint. therefore, we will henceforth assume that the spectral radius of the matrix a is strictly less than 1, i.e., the system (2.1) is stable. we will use the previously proven property of the convexity of y∞ [18, theorem 1]. as it is known, every closed and convex set is the intersection of its supporting half-spaces [17, theorem 18.8]. although, as demonstrated in examples 3.1–3.3, the set (2.2) is not necessarily either closed or open. by means of supporting half-spaces, it is possible to obtain an external estimate of its closure, i.e., to reconstruct its structure up to boundary points. theorem 4.1: the set y∞ in the system (a,u) is bounded if and only if the spectral radius of the matrix a ∈ rn×n is less than 1. in this case, when the set is bounded, for each p ∈ rn \ {0} the following representation holds for the support function: s (p,y∞) △ = sup x∈y∞ (p, x) = ∞∑ k=0 max uk∈u ( (ak)tp, uk ) . copyright © 2024 assa. adv syst sci appl (2024) on the external estimation of the limit reachable set... 73 proof let the spectral radius of a be at least 1. then, in the condition of theorem 3.1, the inequality n > ñ holds, and as a consequence, y∞ is not bounded. now consider the reverse situation. note that from the condition 0 ∈ u for all k ∈ n ∪ {0} it follows that: max uk∈u ( (ak)tp, uk ) ⩾ 0. then for any p ∈ rn \ {0} due to (2.2), the following chain of equalities holds: sup x∈y∞ (p, x) = sup n∈n∪{0} uk∈u k=0,n−1 n∑ k=0 ( p,akuk ) = sup n∈n∪{0} n∑ k=0 max uk∈u ( p,akuk ) = ∞∑ k=0 max uk∈u ( p,akuk ) . the convergence of the last series follows from the fact that all eigenvalues of the matrix a are strictly less than 1 in magnitude, boundedness of the set u and classical inequality: |(p,akv)| ⩽ ∥p∥∥ak∥∥v∥. thus, we conclude that: s(p,y∞) = ∞∑ k=0 sup uk∈u ( (ak)tp, uk ) < ∞. the boundedness of y∞ follows from the boundedness of the support function for the basis vectors and their opposites. theorem 4.1 is proven. the challenge in applying theorem 4.1 lies in the need to compute the exact value of the series. for an arbitrary choice of the support vector p ∈ rn \ {0} solving this problem is difficult. however, if p is chosen to be an element of the real jordan basis of a, it is possible to obtain the exact structure of the supporting half-space. corollary 4.1: consider the system (2.1) where all eigenvalues of the matrix a have magnitudes less than 1, and the decomposition (3.3) holds. let j ∈ rñ×ñ be a jordan block of the form (3.4), located in the rows numbered n0 + 1, . . . , n0 + ñ. let p ∈ rn \ {0} be chosen such that for some i = 1, ñ, the following equality holds: stp = (0, . . . , 0︸ ︷︷ ︸ n0+i−1 , 1, 0, . . . , 0)t. additionally, the following notations are introduced vn0+j = max v∈s−1u vn0+j, vn0+j = min v∈s−1u vn0+j, j = 1, ñ. then for the set (2.2), the following inclusion holds y∞ ⊂ { x ∈ rn : xn0+i ⩽ (p, x) ⩽ xn0+i } , where: 1. if λ ≥ 0, then xn0+i = ñ−i∑ j=0 vn0+i+j (1− λ)j+1 , copyright © 2024 assa. adv syst sci appl (2024) 74 a.v. simkina xn0+i = ñ−i∑ j=0 vn0+i+j (1− λ)j+1 ; 2. if λ < 0, then xn0+i = ñ−i∑ j=0 ( vn0+i+j + vn0+i+j 2(1 + |λ|)j+1 − vn0+i+j − vn0+i+j 2(1− |λ|)j+1 ) , xn0+i = ñ−i∑ j=0 ( vn0+i+j + vn0+i+j 2(1 + |λ|)j+1 + vn0+i+j − vn0+i+j 2(1− |λ|)j+1 ) . proof by the given condition, for the chosen vector p, it holds that stp = (0, . . . , 0, 1, 0, . . . , 0)t ∈ rn, where 1 corresponds to the (n0 + i)-th coordinate of the vector stp. let k ⩾ ñ− 1. then, as known [6, section 3.2.5] jk =  λk c1 kλ k−1 c2 kλ k−2 . . . c ñ−1 k λk−ñ+1 0 λk c1 kλ k−1 . . . c ñ−2 k λk−ñ+2 ... ... ... . . . ... 0 0 0 . . . λk  , where cj k denotes the binomial coefficient: cj k = k! (k − j)!j! . taking into account (3.3), (ak)tp = (s−1)t diag ( jk 1 , . . . , j k m )t stp = = (s−1)t(0, . . . , 0︸ ︷︷ ︸ n0+i−1 , λk, c1 kλ k−1, . . . , c ñ−i k λk−ñ+i︸ ︷︷ ︸ ñ−i+1 , 0, . . . , 0︸ ︷︷ ︸ n−n0−ñ )t, ( (ak)tp, u ) = λkvn0+i + c1 kλ k−1vn0+i+1 + c2 kλ k−2vn0+i+2 + . . .+ c ñ−i k λk−ñ+ivn0+ñ = = ñ−i∑ j=0 cj kλ k−jvn0+j+i, where s−1u = v. let k < ñ− 1. then jk =  λk c1 kλ k−1 c2 kλ k−2 . . . ck kλ k−k 0 . . . 0 0 λk c1 kλ k−1 . . . ck−1 k λk−(k−1) ck kλ k−k . . . 0 ... ... ... . . . ... ... . . . ... 0 0 0 . . . 0 0 . . . λk  . taking into account (3.3), (ak)tp = (s−1)t diag ( jk 1 , . . . , j k m )t stp = copyright © 2024 assa. adv syst sci appl (2024) on the external estimation of the limit reachable set... 75 = (s−1)t(0, . . . , 0︸ ︷︷ ︸ n0+i−1 , λk, c1 kλ k−1, . . . , ck−1 k λ, 1︸ ︷︷ ︸ k+1 , 0, . . . , 0︸ ︷︷ ︸ n−n0−i−k )t, ( (ak)tp, u ) = λkvn0+i + c1 kλ k−1vn0+i+1 + c2 kλ k−2vn0+i+2 + . . .+ ck kλ k−kvn0+i+k = = k∑ j=0 cj kλ k−jvn0+j+i. thus, ( (ak)tp, u ) = min{k,ñ−i}∑ j=0 cj kλ k−jvn0+j+i, sup u∈u ( (ak)tp, u ) = sup v∈s−1u min{k,ñ−i}∑ j=0 cj kλ k−jvn0+j+i ⩽ ⩽ min{k,ñ−i}∑ j=0 cj k sup vn0+j+i∈[vn0+j+i;vn0+j+i] λk−jvn0+j+i = =  min{k,ñ−i}∑ j=0 cj kλ k−jvn0+j+i, λ ⩾ 0, min{k,ñ−i}∑ j=0 cj k|λ| k−j max { (−1)k−jvn0+j+i, (−1)k−jvn0+j+i } , λ < 0. on the other hand, inf u∈u ( (ak)tp, u ) = inf v∈s−1u min{k,ñ−i}∑ j=0 cj kλ k−jvn0+j+i ⩾ ⩾ min{k,ñ−i}∑ j=0 cj k inf vn0+j+i∈[vn0+j+i;vn0+j+i] λk−jvn0+j+i = =  min{k,ñ−i}∑ j=0 cj kλ k−jvn0+j+i, λ ⩾ 0, min{k,ñ−i}∑ j=0 cj k|λ| k−j min { (−1)k−jvn0+j+i, (−1)k−jvn0+j+i } , λ < 0, taking into account theorem 4.1, for any x ∈ y∞ we obtain for λ ⩾ 0 (p, x) ⩽ ∞∑ k=0 min{k,ñ−i}∑ j=0 cj kλ k−jvn0+j+i = ñ−i∑ j=0 vn0+j+i ∞∑ k=j cj kλ k−j = ñ−i∑ j=0 vn0+j+i (1− λ)j+1 = xn0+i, copyright © 2024 assa. adv syst sci appl (2024) 76 a.v. simkina and we obtain for λ < 0 (p, x) ⩽ min{k,ñ−i}∑ j=0 ∞∑ k=0 max { (−1)k−jvn0+j+i, (−1)k−jvn0+j+i } cj k+j|λ| k−j = = ñ−i∑ j=0 ∞∑ k=0 max{(−1)kvn0+j+i; (−1)kvn0+j+i}cj k+j|λ| k = = ñ−i∑ j=0 ( ∞∑ k=0 vn0+j+ic j 2k+j|λ| 2k − ∞∑ k=0 vn0+j+ic j 2k+j+1|λ| 2k+1 ) = = ñ−i∑ j=0 ( vn0+j+i ∞∑ k=0 cj 2k+j|λ| 2k − vn0+j+i ∞∑ k=0 cj 2k+j+1|λ| 2k+1 ) = = ñ−i∑ j=0 ( vn0+j+i 2 ( 1 (|λ|+ 1)j+1 + 1 (1− |λ|)j+1 ) + vn0+j+i 2 ( 1 (|λ|+ 1)j+1 − 1 (1− |λ|)j+1 )) = = ñ−i∑ j=0 ( vn0+j+i + vn0+j+i 2(1 + |λ|)j+1 + vn0+j+i − vn0+j+i 2(1− |λ|)j+1 ) = xn0+i. similarly, from theorem 4.1, it can be derived since max uk∈u ( −(ak)tp, uk ) = −min uk∈u ( (ak)tp, uk ) that (p, x) ⩾ xn0+i. corollary 4.1 is fully proven. for the case of a jordan block of size 3.5, it is possible to construct supporting hyperplanes for an infinite number of vectors p. therefore, the corresponding external estimate of the set y∞ will be a cylinder. corollary 4.2: consider the system (2.1) where all eigenvalues of the matrix a have magnitudes less than 1, and the decomposition (3.3) holds. let j ∈ r2ñ×2ñ be a jordan block of the form (3.5), located in the rows numbered n0 + 1, . . . , n0 + 2ñ. let p ∈ rn \ {0} be chosen such that for some i = 1, ñ, the following equality holds: stp = (0, . . . , 0︸ ︷︷ ︸ n0+2i−2 , p̃1, p̃2, 0, . . . , 0) t, p̃11 + p̃22 = 1. additionally, the following notations are introduced rn0+j = max v∈s−1u √ v2n0+2j−1 + v2n0+2j, j = 1, ñ. then the following inclusion holds for the set (2.2) y∞ ⊂ { x ∈ rn : x2 n0+2i−1 + x2 n0+2i ⩽ rn0+j 2 } , rn0+j = ñ−i∑ j=0 rn0+i+j (1− r)j+1 . copyright © 2024 assa. adv syst sci appl (2024) on the external estimation of the limit reachable set... 77 proof by the given condition, for the chosen vector p, it holds that stp = (0, . . . , 0, p̃1, p̃2, 0, . . . , 0) t ∈ rn, where the two-dimensional subvector p̃ corresponds to the (n0 + 2i− 1)-th and (n0 + 2i)-th coordinates of the vector p. let k ⩾ ñ− 1. then jk =  rkakφ c1 kr k−1a(k−1)φ . . . c ñ−1 k rk−ñ+1a(k−ñ+1)φ 0 rkakφ . . . c ñ−2 k rk−ñ+2a(k−ñ+2)φ ... ... . . . ... 0 0 . . . rkakφ  , taking into account (3.3), (ak)tp = (s−1)t diag ( jk 1 , . . . , j k m )t stp = = (s−1)t ( 0, . . . , 0︸ ︷︷ ︸ n0+2i−2 , c0 kr k−0 ( at (k−0)φp̃ )t , . . . , c ñ−i k rk−ñ+i ( at (k−ñ+i)φp̃ )t , 0, . . . , 0 )t . in case k < ñ− 1 jk =  rkakφ c1 kr k−1a(k−1)φ . . . ck k r k−ka(k−k)φ 0 . . . 0 0 rkakφ . . . ck−1 k rk−(k−1)a(k−(k−1))φ ck k r k−ka(k−k)φ . . . 0 ... ... . . . ... ... . . . ... 0 0 . . . 0 0 . . . rkakφ  , taking into account (3.3), (ak)tp = (s−1)t diag ( jk 1 , . . . , j k m )t stp = = (s−1)t ( 0, . . . , 0︸ ︷︷ ︸ n0+2i−2 , c0 kr k−0 ( at (k−0)φp̃ )t , . . . , ck−1 k rk−(k−1) ( at (k−(k−1))φp̃ )t , p̃t, 0, . . . , 0 )t . let us denote for v ∈ s−1u by vj ∈ r2, j = 1, ñ its two-dimensional subvectors by: vj = (vn0+2j−1, vn0+2j) t. note that by the assumptions, ∥vj∥ ⩽ rn0+j . also, let u = sv ∈ u . we obtain for k ⩾ ñ− 1( (ak)tp, u ) = rk(a−kφp̃, v i) + c1 kr k−1(a−(k−1)φp̃, v i+1) + c ñ−i k rk−ñ+i(a−(k−ñ+i)φp̃, v ñ) ⩽ ⩽ rk∥a−kφp̃∥∥vi∥+ c1 kr k−1∥a−(k−1)φp̃∥∥vi+1∥+ . . .+ c ñ−i k rk−ñ+i∥a−(k−ñ+i)φp̃∥∥vñ∥ ⩽ ⩽ rkrn0+i + c1 kr k−1rn0+i+1 + . . .+ c ñ−i k rk−ñ+irn0+ñ = ñ−i∑ j=0 cj kr k−jrn0+i+j. for k < ñ− 1( (ak)tp, u ) = rk(a−kφp̃, v i) + c1 kr k−1(a−(k−1)φp̃, v i+1) + ck kr k−k(a−(k−k)φp̃, v i+k) ⩽ ⩽ rk∥a−kφp̃∥∥vi∥+ c1 kr k−1∥a−(k−1)φp̃∥∥vi+1∥+ . . .+ ck kr k−k∥a−(k−k)φp̃∥∥vi+k∥ ⩽ rkrn0+i + c1 kr k−1rn0+i+1 + . . .+ ck kr k−krn0+i+k = k∑ j=0 cj kr k−jrn0+i+j. copyright © 2024 assa. adv syst sci appl (2024) 78 a.v. simkina due to the arbitrariness of the choice of u ∈ u we obtain max u∈u ( (ak)tp, u ) ⩽ min{k,ñ−i}∑ j=0 cj kr k−jrn0+i+j, according to theorem 4.1, for any x ∈ y∞, the following relations hold: (p, x) ⩽ ∞∑ k=0 min{k,ñ−i}∑ j=0 cj kr k−jrn0+i+j = n−i∑ j=0 ∞∑ k=0 cj k+jr krn0+i+j = = n−i∑ j=0 rn0+i+j (1− r)j+1 = rn0+i. taking into account the choice of p, we obtain the inclusion: y∞ ⊂ ⋂ p̃21+p̃22=1 { x ∈ rn : (p, x) ⩽ rn0+i } = { x ∈ rn : √ x2 n0+2i−1 + x2 n0+2i ⩽ rn0+i } . corollary 4.2 is proven. using corollaries 4.1 and 4.2, it is possible to estimate from above the sought set y∞ in certain specific directions determined by the real jordan basis. the problem] of constructing supporting hyperplanes is reduced to computing the spectrum of the system matrix. additionally, the more general theorem 4.1 allows for refining the computed external estimates if necessary. 5. blood plasma glucose stabilization system we will demonstrate the effectiveness of the developed methods using the example of constructing the limit reachable set for a stabilization glucose and insulin levels in blood plasma system. for clarity and the ability to visualize the computed sets, we will consider the simplest mathematical model by bergman [10], described by a three-dimensional system: ġ(t) = −q1g(t)−x(t)(g(t) +gb) + v1(t), ẋ(t) = −q2x(t) + q3i(t), i̇(t) = −m(i(t) + ib) + v2(t) vi , (5.10) where g(t) and i(t) represent the deviations of the glucose and free insulin concentrations in blood plasma from their normal values gb and ib, respectively. x(t) is a term accounting for the delay in the metabolism processes of free insulin. the coefficient m denotes the partial elimination (removal from the body), vi is the volume of distribution of insulin in tissues, v1(t) and v2(t) are the rates of external glucose and insulin input, respectively, and q1, q2, q3 are auxiliary numerical parameters. the following numerical values of the parameters, obtained experimentally in [10], are used for the calculations,: q1 = 0.028, q2 = 0.025, q3 = 0.000013, vi = 12 l, m = 5 54 min−1, gb = 4.5 mmol/l, ib = 15 nmol/l. copyright © 2024 assa. adv syst sci appl (2024) on the external estimation of the limit reachable set... 79 by performing linearization and subsequent discretization with a step size of δt = 10 min assuming the control inputs v1(t) and v2(t) are piecewise constant, we obtain the following finite-difference relations of the form (2.1) with the parameters: ad = ( 0.7557 −34.5225 −0.0018 0 0.7788 0.000073 0 0 0.3961 ) , ud = conv {( −30.61 −0.1418 −0.93 ) , ( −28.03 −0.1418 −0.93 ) , ( −30.62 −0.1423 2.11 ) , ( −28.04 −0.1423 2.11 ) ,(−28.03 0.1421 −0.93 ) , ( 30.62 0.1421 −0.93 ) , ( 28.02 0.1416 2.11 ) , ( 30.61 0.1416 2.11 )} . note that for the system (ad,ud) the condition 0 ∈ intud is not satisfied because dimud = 2. for this reason, we will double the quantization step and transite to an equivalent system in terms of constructing the limit reachable sets with the following parameters: if we linearize (5.10) and put v1(t) = v1k, v2(t) = v2k for t = [kδ; (k + 1)δ] for some δ > 0, we can solve the resulting system of linear differential equations explicitly. then for x(k) = (g(kδ), x(kδ), i(kδ))t the finite-difference system relations will be correct when selecting: a = a2 d = ( 0.5712 −52.9824 −0.0046 0 0.6065 0.0001 0 0 0.1569 ) , u = u + adu . the eigenvalues of the matrix a are λ1 = 0.5712, λ2 = 0.6065 and λ3 = 0.1569, which satisfy the conditions of corollary 4.1. we will use this result to construct an external polyhedral estimate of the set y∞. we will compute the vectors p1, p2, p3 from corollary 4.1: p1 = ( (1, 0, 0)s−1 )t = (1, 1500, 0.3)t, p2 = ( (0, 1, 0)s−1 )t = (0, 1500, 0.28)t, p3 = ( (0, 0, 1)s−1 )t = (0, 0, 1)t. additionally, the following equalities hold min v∈s−1u v1 = −997.88, min v∈s−1u v2 = −963.25, min v∈s−1u v3 = −1.54, max v∈s−1u v1 = 998.18, max v∈s−1u v2 = 963.53, max v∈s−1u v3 = 3.506. ultimately, we obtain the following estimate for the limit reachable set: y∞ ⊂ ŷ∞ = {x ∈ r3 : − 997.88 ⩽ (p1, x) ⩽ 998.18}∩ ∩{x ∈ r3 : − 963.25 ⩽ (p2, x) ⩽ 963.53} ∩ {x ∈ r3 : − 1.54 ⩽ (p3, x) ⩽ 3.506}. the set ŷ∞ is graphically represented in figure 5.1. copyright © 2024 assa. adv syst sci appl (2024) 80 a.v. simkina fig. 5.1. external estimate y∞ (by lines) and reachable set y(11) (by surface) for discretized system (5.10). conclusion the article investigates various properties of the limit reachable sets for linear stationary discrete-time systems with geometric constraints on the control. specifically, it is proven that these sets are cylinders oriented 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(2003) reachability and controllability of discretetime systems under control actions bounded in magnitude and norm, autom. remote control, 64(12), 1844–1857. 20. ibragimov, d. n. & sirotin, a. n. (2017) on the problem of operation speed for the class of linear infinite-dimensional discrete-time systems with bounded control autom. remote control, 78(10), 1731–1756. copyright © 2024 assa. adv syst sci appl (2024) introduction problem statement the general structure of reachable sets construction of estimates of limit null-controllable sets blood plasma glucose stabilization system adv syst sci appl 2024; 02:1–18 published online at https://ijassa.ipu.ru. correlations of the sojourn times of subtasks in fork-join queueing systems with m |m |1-type subsystems anastasia v. gorbunova1*, alexey v. lebedev2 1v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia 2lomonosov moscow state university, moscow, russia abstract: the article analyzes the relationship between the residence times of subtasks in the fork-join subsystems of the queuing system. when arriving the fork-join system, the task is divided into subtasks, each of which is serviced in its own subsystem, the task is considered serviced after the completion of all subtasks that originally comprised it servicing. there is a dependence between the residence times of subtasks in subsystems, which affects the main performance indicators of the system, for example, the response times, which greatly complicates their analysis. the paper examines the characteristics of the existing dependence. in particular, with the help of generating functions and the laplace-stieltjes transformation, exact expressions for the pearson and spearman correlation coefficients are obtained. in addition, using a combination of several methods, including the nelder-mead optimization method, the estimation of the kendall correlation coefficient was obtained, and the model for the response time estimation based on the resulting correlation coefficients was described. despite the many works on the study of fork-join queuing systems, there are practically no articles devoted to the correlation analysis of the temporal metrics of the model. therefore, this article can become one of the first in this area, laying the foundation for the study of the correlation dependence and its influence on the performance parameters of the model, not only for the classical case of m |m |1 subsystems, but also for more complex architectures of this system. keywords: fork-join queueing system, system with parallel service of tasks, correlation coefficient, pearson correlation, spearman correlation, kendall correlation, generating function, meta-gaussian model, simulation modeling. 1. introduction the article studies the classical fork-join queuing system (qs) with m |m |1-type subsystems. upon arriving the system, the task is divided into several subtasks (the number of which is equal to the number of subsystems), each of which rises in turn for service. after servicing, the subtasks are again combined into a whole task, and the collected task leaves the system. thus, the residence time of the entire task (the response time) is determined by the maximum of the residence times of its constituent subtasks. the described mechanism for the functioning of the fork-join qs is suitable for modeling multitasking processes of various nature. due to the fact that the division of a complex combined task into several subtasks is one of the ways to save time and resources, the analysis of fork-join qs is still an urgent task. first of all, if we talk about the practical application of this model, it is worth mentioning the processes occurring in information and computing systems. for example, we can talk ∗corresponding author: avgorbunova@list.ru 2 a.v. gorbunova, a.v. lebedev about computing platforms that support data-intensive applications, and in general about various high-performance environments for big data analysis, which are based on parallel or distributed structures (mapreduce technology) [12,21,27,35]. there are also many examples of parallel structures in manufacturing systems, such as the process of assembling multi-item orders in warehouses of one or more suppliers [6, 19, 31]. in the field of healthcare, when patients are admitted or discharged, medical studies are carried out, which are distinguished by their duration [2, 15, 20]. in the banking sector, consideration of a loan application can be carried out by several departments simultaneously, which also leads to parallelization of this procedure [17]. in all systems described, the response time plays a key role. this characteristic is critical and quite often appears in agreements on the quality of services provided. therefore, predicting both the average response time and the quantiles of this random variable allows one to make a qualitative prediction of one of the most important indicators of system performance. the main reason for the complexity of the analysis of fork-join qs is the existing dependence between the times subtasks spend in subsystems. this dependence arises due to the commonality of the moment when subtasks appear in the system itself, since they are constituent elements of one task, which was divided into parts at the time of arriving into the system. the dependence between the sojourn times of subtasks is a distinctive feature of the fork-join qs with k subsystems m |m |1 (the analyzed type of qs in this particular case) from k parallel qs m |m |1. therefore, it is of particular interest to estimate the correlation coefficients between the residence times of subtasks. it is also important to estimate correlation coefficients because this expands the rather limited range of known methods for analyzing system response time. in particular, the article presents one of the variants of the model based on the known value of the spearman correlation coefficient, which makes it possible to evaluate various response time characteristics by using the example of mathematical expectation based on the meta-gaussian model. this model is not the only one possible in this case, but it serves as a clear example of the viability of the proposed approach for assessing the characteristics of a fork-join queueing system using correlation coefficients. despite the fact that the fork-join qs variant with m |m |1 subsystems is considered the easiest to study, accurate results were obtained only for the average response time in the case with two subsystems (k = 2) [23]. for k > 2, only approximations of the mean response time and its variance of varying degrees of accuracy [32–34] were obtained. there are also known publications in which estimates of the time moments of the task’s stay in the qs, as well as estimates of high-level percentiles (for example, the 99th percentile) of the response time distribution in the case of more complex types of qs subsystems [4,25,26,29] . by more complex types of subsystems we mean qs with non-poisson input flows and non-exponential distribution of service time. along with the classical analytical methods of queuing theory, fork-join systems have been studied empirically, using elements of the theory of order statistics and even using machine learning methods [7,9,36]. various generalizations and modifications of fork-join systems [3, 28, 30] are also considered. we also note that the authors previously studied the dependence of the maximum residual service times in two subsystems for a fork-join system with an infinite number of servers and general service times, for which the copulas and blomquist coefficients [8] were found. as for the evaluation of the correlation coefficients between the residence times of subtasks, there are practically no studies in this direction, and the authors could not find publications in which such an analysis would be carried out even in the case of subsystems of the m |m |1 type. in [10], the authors studied systems with pareto service and obtained empirical formulas for the pearson, spearman and kendall correlation coefficients (along with mean and standard deviation estimates) that give a good fit over a wide range of parameters. in the same work, it was possible to derive not estimates of the pearson and copyright © 2024 assa. adv syst sci appl (2024) correlations of the sojourn times of subtasks in fork-join queueing systems 3 spearman correlation coefficients, but exact formulas. an approximation for the kendall correlation coefficient is also obtained. the article provides the detailed description of the approach to the derivation of expressions for the correlation coefficients between the residence times of a pair of subtasks in the respective subsystems. by itself, the approach for deriving pearson and spearman correlations is classical and is based on the theory of generating functions and laplacestieltjes transformations, while considering a system for which the number of subsystems is two (k = 2). to evaluate the kendall correlation, the combined approach is used, which includes the nelder-mead optimization method and graphical analysis. an interesting phenomenon has been discovered: at low load, the pearson correlation coefficient is greater, and at high load, spearman’s. in this case, the kendall correlation coefficient is always less than both of them. it should be noted that the value of the correlation coefficients between any pair of subtasks is not affected by the number of subsystems, i. e. the value of the correlation coefficient does not depend on k and will be the same for any paired combination of subtasks from two different subsystems with fixed input flow and service parameters, and hence also true for all k > 2. the article is organized as follows. section 2 describes the mathematical model of forkjoin qs. sections 3–5 present a description of approaches to finding each of the three pearson, spearman and kendall correlation coefficients, respectively, with elements of numerical analysis using a specific example of numerical data to confirm the obtained analytical result comparing with the results of simulation modeling in the python software environment. in section 6, the limiting two-dimensional distribution for the normalized residence times of subtasks under high load is obtained and all three correlation coefficients are compared. section 7 describes the meta-gaussian model for the average response time estimation using the spearman correlation coefficient, also the numerical example is presented. the conclusion (section 8) summarizes some results and outlines ways for further research. 2. mathematical model of fork-join qs consider a classical fork-join queuing system with k ⩾ 2 subsystems (fig. 2.1). tasks enter the system according to a poisson process with a rate λ > 0. upon arriving the system, a task is instantly divided into k subtasks, each of which is queued for service to one of k subsystems or immediately starts to be serviced if the queue is empty. all subsystems have one server for servicing subtasks and buffer of unlimited capacity. the service time on each server has an exponential distribution with the parameter µ > 0. thus, the subsystems are k . . . fork point join point ... ... ... λ λ λ λ µ µ µ fig. 2.1. fork-join model of a queuing system with k subsystems of type mλ|mµ|1. identical qss of type mλ|mµ|1. a task is considered serviced only after all its parts, i. e. all its constituent subtasks, have been serviced. next, we turn to the consideration of the case when k = 2, since the total number of subsystems has no effect on the pairwise dependence of the times of sojourn in subsystems. copyright © 2024 assa. adv syst sci appl (2024) 4 a.v. gorbunova, a.v. lebedev consequently, the random residence time of a task in the qs (or the response time) of the r system is the maximum of two random residence times of the subtasks ξi, i = 1.2, in the corresponding subsystems r = max{ξ1, ξ2}. the functioning of the system is described by the markov process x(t) = {x1(t), x2(t)}, where xi(t) is the number of subtasks in the i-th subsystem, i = 1, 2. then the set of process states can be written as follows x = {(i, j), i ⩾ 0, j ⩾ 0}. further, let pij be the stationary probability that the first subsystem contains i customers and the second subsystem has j customers. the system ergodicity condition is standard and common for both queuing subsystems, i. e. ρ = λ/µ < 1. now we introduce a generating function for the number of subtasks in the system p (z, w) = ∞∑ i=0 ∞∑ j=0 ziwjpij. (2.1) according to [5] it will take the form p (z, w) = n(z, w) q(z, w) , where for λ = 1 (which, without loss of generality, we will assume everywhere below) n(z, w) = µz(w − 1)p (z, 0) + µw(z − 1)p (0, w), q(z, w) = (1 + 2µ)zw − µw − µz − z2w2, p (z, 0) = (µ− 1)3/2 µ(µ− z)1/2 = (1− ρ)3/2 (1− ρz)1/2 , p (0, w) = (1− ρ)3/2 (1− ρw)1/2 , then p (z, w) = z(w − 1)p (z, 0) + w(z − 1)p (0, w) (2 + ρ)zw − w − z − ρz2w2 . (2.2) in the following sections, we will describe the sequence of actions required to calculate the correlation coefficients of several types. 3. pearson correlation coefficient the dependence between the residence times of subtasks in the fork-join subsystems of a system with parallel servicing of tasks arises due to the common moments of tasks arrival in these subsystems. up or down fluctuations of the input flow (according to the number of arrivals for some time) lead to an increase or decrease in the length of queues in subsystems and, accordingly, an increase or decrease in the residence time of subtasks of one task in subsystems. data visualization makes it possible to demonstrate the dependence of the sojourn times in subsystems. so, in fig. 3.2 shows the values of ξ1 and ξ2 obtained by simulation in the case of a fork-join qs and for the case of two parallel qss m |m |1 with identical values of the parameters λ and µ. the number of pairs of points (ξ1, ξ2) is the same in both cases and is one million. of course, such a number of points is not enough to obtain a good estimate of the correlation coefficient, but it is quite acceptable to illustrate the presence of a difference in the behavior of random variables for two options for the functioning of systems. from the theory it is clear and clearly visible that for independent residence times (fig. 2b) the joint density level lines have the form x1 + x2 = const (since partial distributions copyright © 2024 assa. adv syst sci appl (2024) correlations of the sojourn times of subtasks in fork-join queueing systems 5 are exponential), while for dependent residence times (fig. 2a) it can be seen that the lines have a bulge from the origin, which is the greater, the greater the load. this reflects a greater likelihood of jointly larger values than with independence. note that visually the nature of the dependence is not similar to the classical case in statistics, when there is a functional dependence of quantities (linear or monotonic), on which random noise is superimposed. therefore, the question arises of how the known correlation coefficients will be able to capture (reflect) this dependence. a) b) fig. 3.2. illustration of presence/absence of dependence between random variables ξ1 and ξ2 at ρ = 0.8 in case of a) fork-join qs with two subsystems m |m |1; b) two parallel functioning qs m |m |1. to calculate the pearson correlation coefficient, we will use the classical tools from queuing theory. consider the laplace-stieltjes transformation (lst) for the residence times of subtasks in subsystems φ(s, t) = ∞∫ 0 ∞∫ 0 e−sxe−tyv(x, y)dxdy, (3.3) where v(x, y) = ∂2v (x, y) ∂x∂y is the two-dimensional distribution density of the residence times of subtasks in subsystems, v (x, y) = p (ξ1 < x, ξ2 < y). next, we carry out the standard arguments. if, when a task enters the system, it encounters i subtasks in the first qs and j subtasks in the second qs, then the waiting time in the queue of each of its two related elements in the corresponding subsystems will consist of the sum of the random service times of all preceding them in queue i or j of subtasks, respectively, as well as the service time of the newly arrived element itself. taking into account the fact that the service time of one subtask has an exponential distribution with the parameter µ, then the sum consisting of (i+ 1) and (j + 1) of such random variables will have the erlang distribution with the distribution function ei+1(x) and ej+1(x). then the two-dimensional distribution function of the residence time of subtasks in subsystems will take the form v (x, y) = ∞∑ i=0 ∞∑ j=0 ei+1(x)ej+1(y)pij. (3.4) copyright © 2024 assa. adv syst sci appl (2024) 6 a.v. gorbunova, a.v. lebedev using the properties of the lst, taking into account the expression (3.4), we obtain that φ(s, t) = ∞∑ i=0 ∞∑ j=0 εi+1(s)εj+1(t)pij = ∞∑ i=0 ∞∑ j=0 ( µ µ+ s )i+1( µ µ+ t )j+1 pij, where εi+1(s) and εj+1(t) — lst for ei+1(x). thus, the lst of the residence times of subtasks in subsystems (3.3) can be represented as φ(s, t) = p ( µ µ+ s , µ µ+ t ) · µ µ+ s · µ µ+ t , (3.5) where p (·, ·) is the generating function from (2.1). taking into account λ = 1, we have φ(s, t) = p ( 1 1 + ρs , 1 1 + ρt ) · 1 1 + ρs 1 1 + ρt . (3.6) the pearson correlation coefficient between the residence times of subtasks in subsystems is determined by the following expression rp = e[ξ1 · ξ2]− e[ξ1]e[ξ2]√ v ar[ξ1]v ar[ξ2] . (3.7) taking into account the fact that, as is known, the residence time of an task in a system of type mλ|mµ|1 has an exponential distribution with the parameter (µ− λ), i. e. e[ξ1] = e[ξ2] = 1 µ− λ , v ar[ξ1] = v ar[ξ2] = 1 (µ− λ)2 , the expression (3.6) with λ = 1 is converted to the form rp = (µ− 1)2e[ξ1 · ξ2]− 1 = µ2(1− ρ)2e[ξ1 · ξ2]− 1. (3.8) therefore, to determine the correlation coefficient, it is necessary to calculate e[ξ1 · ξ2]. this can be done with lst φ(s, t) e[ξ1 · ξ2] = ∞∫ 0 ∞∫ 0 xy · v(x, y)dxdy = = ∂2φ(s, t) ∂s∂t ∣∣∣∣ s=0,t=0 = ∂2 [ p ( 1 1+ρs , 1 1+ρt ) · 1 1+ρs 1 1+ρt ] ∂s∂t ∣∣∣∣ s=0,t=0 , while p ( 1 1 + ρs , 1 1 + ρt ) · 1 1 + ρs 1 1 + ρt = = (1− ρ)3/2 ( t √ ρs+1√ ρs−ρ+1 + s √ ρt+1√ ρt−ρ+1 ) ρ2s2t+ ρ2st2 − ρ2st+ ρs2 + 2ρst− ρs+ ρt2 − ρt+ s+ t . if we introduce the following notation f (s) = (1− ρ)3/2 √ 1 + ρs√ 1 + ρs− ρ , copyright © 2024 assa. adv syst sci appl (2024) correlations of the sojourn times of subtasks in fork-join queueing systems 7 g(s, t) = ρ2s2t+ ρ2st2 − ρ2st+ ρs2 + 2ρst− ρs+ ρt2 − ρt+ s+ t, then the formula can be written in a more compact form p ( 1 1 + ρs , 1 1 + ρt ) · 1 1 + ρs 1 1 + ρt = tf (s) + sf (t) g(s, t) . taking derivatives will lead to the following expression ∂2 ∂s∂t [ tf (s) + sf (t) g(s, t) ] = f(s, t) g(s, t) , where f(s, t) = g2(s, t)[f ′(s) + f ′(t)] +g′ s(s, t)g ′ t(s, t)[2tf (s) + 2sf (t)]− −g′ s(s, t)g(s, t)[f (s) + sf ′(t)]−g′ t(s, t)g(s, t)[f (t) + tf ′(s)]− −g′′ st(s, t)g(s, t)[tf (s) + sf (t)], g(s, t) = g3(s, t). next, we need to calculate ∂2φ(s, t) ∂s∂t ∣∣∣∣ s=0,t=0 = lim s→0 t→0 ∂2φ(s, t) ∂s∂t = lim s→0 t→0 f(s, t) g(s, t) . moreover, note that the functions f(s, t) and g(s, t) are infinitesimal of the third order as (s, t) → (0, 0), i. e. lim s→0 t→0 f(s, t) = lim s→0 t→0 g(s, t) = 0, f ′ s(0, 0) = f ′ t(0, 0) = g′s(0, 0) = g′t(0, 0) = 0, f ′′ ss(0, 0) = f ′′ tt(0, 0) = f ′′ st(0, 0) = g′′ss(0, 0) = g′′tt(0, 0) = g′′st(0, 0) = 0, f (3) sn1 tn2 (0, 0) ̸= 0, g (3) sn1 tn2 (0, 0) ̸= 0, n1 + n2 = 3. therefore, for the existence of a double limit at the point (0, 0), it is necessary and sufficient that the following equality [13, 14] to be true: f (3) sn1 tn2 (0, 0) g (3) sn1 tn2 (0, 0) = m, m ̸= 0,m ̸= ±∞, n1 + n2 = 3. (3.9) and the double limit itself will be equal to m from (3.9) lim s→0 t→0 f(s, t) g(s, t) = m. after appropriate calculations, we get that f (3) sn1 tn2 (0, 0) = 3(1− ρ)ρ2(4ρ− ρ2 + 8) 4 , g (3) sn1 tn2 (0, 0) = 6(1− ρ)3, n1 + n2 = 3. thus, we have ∂2φ(s, t) ∂s∂t ∣∣∣∣ s=0,t=0 = lim s→0 t→0 ∂2φ(s, t) ∂s∂t = f (3) sn1 tn2 (0, 0) g (3) sn1 tn2 (0, 0) = ρ2(4ρ− ρ2 + 8) 8(1− ρ)2 , copyright © 2024 assa. adv syst sci appl (2024) 8 a.v. gorbunova, a.v. lebedev then rp = µ2(1− ρ)2 ρ2(4ρ− ρ2 + 8) 8(1− ρ)2 − 1 = ρ(4− ρ) 8 . (3.10) the correctness of the (3.10) formula for the pearson correlation coefficient between the residence times of any two subtasks in the corresponding subsystems is confirmed by a numerical experiment. with the help of simulation for the values λ = 1, ρ ∈ [0.1, 0.9] with a step of 0.05, the values of rp were calculated. the results of the comparison with the analytical expression (3.10) are shown in 3.3. small deviations in the range of ρ values close to unity 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.00 0.20 0.40 0.60 0.80 1.00 rp ρ simulation analytics fig. 3.3. pearson correlation coefficient rp are explained as follows. an increase in the system load factor requires a significant increase in the duration of the run within one launch of the simulation model, since an increase in the load factor leads to an increase in the correlation of the data used to build a point estimate of the quantity under study. the features of the simulation fork-join qs and the construction of confidence intervals of the obtained estimates can be found in the work [11]. an increase in the duration of the run, in turn, leads to a significant increase in the time spent on the numerical experiment, which is not rational in this case, since the data match up to the third digit after the decimal point, and the module of the maximum relative error does not exceed 0.62%. based on the obtained results, we can conclude that the dependence between the residence times ξ1 and ξ2 in the subsystems is well reflected by the pearson correlation coefficient, and it increases quadratically (with deceleration) with the load ρ. 4. spearman correlation coefficient let’s analyze the spearman’s correlation coefficient in order to get a complete picture of the relationship between the random residence times of subtasks ξ1 and ξ2 in subsystems. in statistics, the spearman correlation coefficient is calculated as the pearson correlation coefficient applied to the ranks of observations (in ascending order) as a measure of the strength of a monotonic dependence (increasing or decreasing) between random variables, usually to test hypotheses about their independence. from the point of view of probability theory, for continuous random variables, the spearman correlation coefficient is conveniently defined as follows. let random variables x1 and x2 have distribution functions f1 and f2, we consider u1 = f1(x1) and u2 = f2(u2), then the spearman correlation coefficient x1 and x2 can be expressed as the pearson correlation coefficient of u1 and u2 random variables [24, p. 170]: rs(x1, x2) = rp(u1, u2). copyright © 2024 assa. adv syst sci appl (2024) correlations of the sojourn times of subtasks in fork-join queueing systems 9 the coefficient rs, as well as rp, takes values on the interval [−1, 1]. to determine the coefficient rs, let’s make some preparatory calculations. consider φ(s, t) from (3.5), where s and t are multiplied by (µ− λ), then taking into account that ρ = λ/µ, we get φ((µ− λ)s, (µ− λ)t) = p ( µ µ+ (µ− λ)s , µ µ+ (µ− λ)t ) · µ µ+ (µ− λ)s · µ µ+ (µ− λ)t = = p ( 1 1 + (1− ρ)s , 1 1 + (1− ρ)t ) · 1 1 + (1− ρ)s · 1 1 + (1− ρ)t . now let λ = 1, then we will substitute the corresponding expression for the generating function. we will get φ((µ− λ)s, (µ− λ)t) = = 1 1+(1−ρ)s ( 1 1+(1−ρ)t − 1 ) · (1−ρ)3/2 (1− ρ 1+(1−ρ)s )1/2 + 1 1+(1−ρ)t ( 1 1+(1−ρ)s − 1 ) · (1−ρ)3/2 (1− ρ 1+(1−ρ)t )1/2 (2 + ρ) 1 1+(1−ρ)s · 1 1+(1−ρ)t − 1 1+(1−ρ)s − 1 1+(1−ρ)t − ρ (1+(1−ρ)s)2(1+(1−ρ)t)2 · · 1 1 + (1− ρ)s · 1 1 + (1− ρ)t = a(ρ) b(ρ) · 1 1 + (1− ρ)s · 1 1 + (1− ρ)t . simplify the numerator a(ρ) a(ρ) = 1 1 + (1− ρ)s · 1− 1− (1− ρ)t 1 + (1− ρ)t · (1− ρ)3/2( 1+(1−ρ)s−ρ 1+(1−ρ)s )1/2+ + 1 1 + (1− ρ)t · 1− 1− (1− ρ)s 1 + (1− ρ)s · (1− ρ)3/2(1+(1−ρ)t−ρ 1+(1−ρ)t )1/2 = = −(1− ρ)2 (1 + (1− ρ)s)(1 + (1− ρ)t) · t(1 + (1− ρ)s)1/2 (s+ 1)1/2 + + −(1− ρ)2 (1 + (1− ρ)s)(1 + (1− ρ)t) · s(1 + (1− ρ)t)1/2 (t+ 1)1/2 = = −(1− ρ)2 · c(ρ) (1 + (1− ρ)s)(1 + (1− ρ)t) . (4.11) simplify the denominator b(ρ) with an additional factor b(ρ) · (1 + (1− ρ)s)(1 + (1− ρ)t) = = 2 + ρ− (1 + (1− ρ)s)− (1 + (1− ρ)t)− ρ (1 + (1− ρ)s)(1 + (1− ρ)t) = = ρ− (1− ρ)(s+ t)− ρ (1 + (1− ρ)s)(1 + (1− ρ)t) = ρ(1− ρ)(s+ t) + ρ(1− ρ)2st− (1− ρ)(s+ t)− (1− ρ)2(s+ t)2 − (1− ρ)3(s+ t)st (1 + (1− ρ)s)(1 + (1− ρ)t) = = −(1− ρ)2 · ( (s+ t)− ρst+ (s+ t)2 + (1− ρ)(s+ t)st ) (1 + (1− ρ)s)(1 + (1− ρ)t) = copyright © 2024 assa. adv syst sci appl (2024) 10 a.v. gorbunova, a.v. lebedev = −(1− ρ)2 ·d(ρ) (1 + (1− ρ)s)(1 + (1− ρ)t) (4.12) then φ((µ− λ)s, (µ− λ)t) = c(ρ) d(ρ) , where c and d are defined in (4.11) and (4.12), i. e. φ((µ− λ)s, (µ− λ)t) = t(1 + (1− ρ)s)1/2(s+ 1)−1/2 + s(1 + (1− ρ)t)1/2(t+ 1)−1/2 (s+ t)− ρst+ (s+ t)2 + (1− ρ)(s+ t)st . (4.13) recall that the sojourn times of subtasks in the fork-join subsystems of the qs have an exponential distribution with the parameter µ− λ, i. e., ξi ∼ exp(µ− λ). consider random variables ui = f (ξi), where fξi(x) = 1− e−(µ−λ)x, x > 0, which will have uniform distribution on the interval [0, 1], i. e., ui ∼ r[0, 1], i = 1, 2. then rs — spearman’s correlation coefficient for random variables ξ1 and ξ2 will be pearson’s correlation coefficient for random variables u1 and u2 [24, p. 170], i. e. rs = e[u1 · u2]− e[u1]e[u2]√ v ar[u1] · v ar[u2] . compute e[u1 · u2]: e[u1 · u2] = e[(1− e−(µ−λ)ξ1)(1− e−(µ−λ)ξ2)] = = 1− e[e−(µ−λ)ξ1 ]− e[e−(µ−λ)ξ2 ] + e[e−(µ−λ)ξ1e−(µ−λ)ξ2 ] since ηi = (µ− λ)ξi ∼ exp(1), i. e., pηi(x) = e−x, x > 0, we have e[e−(µ−λ)ξi ] = e[e−ηi ] = ∞∫ 0 e−xe−xdx = 1 2 . then we get e[u1u2] = 1− 1 2 − 1 2 + e[e−(µ−λ)ξ1e−(µ−λ)ξ2 ] = e[e−(µ−λ)ξ1e−(µ−λ)ξ2 ] = = ∞∫ 0 ∞∫ 0 e−(µ−λ)xe−(µ−λ)ypξ1ξ2(x, y)dxdy = = ∞∫ 0 ∞∫ 0 e−(µ−λ)sxe−(µ−λ)sypξ1ξ2(x, y) dxdy ∣∣∣∣ s=1,t=1 = = φ((µ− λ)s, (µ− λ)t) ∣∣∣∣ s=1,t=1 . now, substituting s = 1 and t = 1 into the formula (4.13), we obtain e[u1 · u2] = √ 2 √ 2− ρ 8− 3ρ . copyright © 2024 assa. adv syst sci appl (2024) correlations of the sojourn times of subtasks in fork-join queueing systems 11 and given that e[ui] = 1/2, and v ar[ui] = 1/12, i = 1.2, we can write the spearman correlation coefficient in a final form rs = √ 2 √ 2−ρ 8−3ρ − 1 2 · 1 2 1 12 = 12 √ 2 √ 2− ρ 8− 3ρ − 3. (4.14) simulation modeling confirms the correctness of the (4.14) formula for the spearman 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.00 0.20 0.40 0.60 0.80 1.00 rs ρ simulation analytics fig. 4.4. spearman correlation coefficient rs coefficient (fig. 4.4). note that the spearman correlation coefficient also increases with load in a non-linear manner, which however is visually closer to linear than for the pearson correlation coefficient. 5. kendall correlation coefficient in this section, we construct an approximation for the kendall correlation coefficient, since it has not yet been possible to derive an exact formula for it. the kendall correlation coefficient, like the spearman correlation coefficient, is a rank correlation coefficient. it also evaluates the nature of the monotonic dependence between random variables and the closeness of this connection [1, 18]. statistically, in the sample, the proportion of pairs of observations of random vectors (from two random variables as components) is estimated, in which the components have the same order (monotonic character), i. e., for example, one component increases with the growth of the other, or vice versa, decreases with the growth of another. the kendall correlation coefficient is estimated as the difference between the proportions of pairs of vectors for which the orders are the same and for which they differ. the formula for it can be written as follows r̂k = 1− 4 n(n − 1) n−1∑ i=1 n∑ j=i+1 1{[ξ1i < ξ1j] ̸= [ξ2i < ξ2j]}, (5.15) where 1{·} is the event indicator function {·}, and (ξ1i, ξ2i), 1 ≤ i ≤ n , is random sample of n vectors from random variables ξ1 and ξ2. from the point of view of probability theory, the kendall correlation coefficient of the random variables x and y is defined as rk = e sign(x1 −x2)(y1 − y2), copyright © 2024 assa. adv syst sci appl (2024) 12 a.v. gorbunova, a.v. lebedev where (x1, y1), (x2, y2) are independent random vectors distributed as (x, y ). to approximate the kendall correlation coefficient, we use a combination of several methods. first, we will carry out a graphical analysis of the data obtained using simulation modeling. after plotting the dependence of rk on the system load ρ, you can see that, as in the case of rs, this dependence is close to linear in appearance (compare fig. 4.4 and fig. 5.5), although in fact, as follows from the (4.14) formula, this is not the case. assume for simplicity that rk depends on ρ quadratically, i. e. rk ≈ ρ(c1 + c2ρ). now, to find the unknown coefficients, we use the nelder-mead optimization method. in the process of optimization with respect to c, we will minimize the modulus of the relative approximation error ape = ∣∣∣∣rk − r̂k rk ∣∣∣∣ · 100% between data generated by simulation (rk — “true” values of kendall’s correlation coefficient) and predictions calculated by the proposed analytical formula (r̂k — kendall’s correlation coefficient estimate), with the initial values of the coefficients estimated from the graph. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.20 0.40 0.60 0.80 1.00 rk ρ simulation analytics fig. 5.5. kendall correlation coefficient rk table 5.1. errors in approximating the kendall correlation coefficient rk by (5.16) for ρ ∈ {0.10, 0.15, ..., 0.90} estimated error types characteristic max ape, % min ape, % mape, % rk 0.92828 0.09158 0.43138 as a result of optimization in the python software environment, we obtain the values of the coefficients c1 ≈ 0.25134, c2 ≈ 0.02517. therefore, the final expression for rk will look like rk ≈ ρ(0.25134 + 0.02517ρ). (5.16) in the 5.5 figure and in the 5.1 table, you can compare the results of simulation modeling of the kendall coefficient with the results of calculations using the analytical formula (5.16). as can be seen, the approximation error is not at all large and does not exceed 1 copyright © 2024 assa. adv syst sci appl (2024) correlations of the sojourn times of subtasks in fork-join queueing systems 13 as for the values themselves, then, for example, rk ≈ 0.2 is true for ρ = 0.8, and this means that for about 60% of pairs of observations of the vectors (ξ1, ξ2) from the entire set of their the order (character of monotonicity) coincides, but for about 40% of the pairs (ξ1, ξ2) it does not. in this case, each vector corresponds to a task and consists of the sojourn times of its subtasks. thus, the nature of the dependence of the spearman and kendall correlation coefficients on the load in both cases is close to linear, but the kendall coefficient takes smaller values. 6. limiting two-dimensional distribution and ratios of correlation coefficients from the obtained results, it can be seen that all correlation coefficients have some limits at high load (ρ → 1), and these limits are different from both 0 and 1, which suggests that they have meaningful meaning. note that from the formula (4.13) one can obtain the limit lim ρ→1 φ((µ− λ)s, (µ− λ)t) = t(s+ 1)−1/2 + s(t+ 1)−1/2 s+ t− st+ (s+ t)2 , (6.17) and this is the lst of some two-dimensional distribution. namely, this is the limit distribution of normalized sojourn times η1 = (µ− λ)ξ1, η2 = (µ− λ)ξ2, whose joint distribution is described by the lst φ((µ− λ)s, (µ− λ)t), with the random variables η1 and η2 individually are always equally distributed (they are standard exponentials), and the dependence between them is determined by the load of ρ. all limit values of the correlation coefficients are thus the values of the correlation coefficients for the limit distribution (6.17). let’s write them out explicitly†: rp = 3 8 = 0.375, rs = 12 √ 2 5 − 3 ≈ 0.394, rk ≈ 0.276. in the figure 6.6 we also present a joint graph of all three coefficients (with the addition of limit values). as can be seen from the figure, the lines of the pearson and spearman correlation coefficients intersect at one point (except zero). the coordinates of the intersection point can be found by equating the expressions (3.10) and (4.14). after simplification, we obtain the equation 9ρ6 − 120ρ5 + 160ρ4 + 2752ρ3 − 6080ρ2 + 3072ρ = 0. the numerical solution of this equation allows us to find the only root that belongs to the interval ρ ∈ (0, 1), namely ρ ≈ 0.803146. thus, the intersection point has the following approximate coordinates: (0.803146, 0.320943). to the left of it is more pearson’s correlation coefficient, and to the right — spearman’s. 7. meta-gaussian model the calculation of the residence time correlation coefficients is not only of academic interest, but can also be useful for assessing system performance. indeed, when it comes to selecting an †the value of rk is an estimate according to the formula (5.16). copyright © 2024 assa. adv syst sci appl (2024) 14 a.v. gorbunova, a.v. lebedev 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 r ρ pearson correlation spearman correlation kendall correlation 0.375 0.394 0.276 fig. 6.6. coefficients of correlation and their limit values approximate model of the dependence of residence times, such a model can be parameterized by one of the correlation coefficients. next, we will consider the meta-gaussian model based on reducing an arbitrary distribution to a multivariate normal one. such models are used in financial mathematics [22, ch. 5, 9], hydrology [16], etc. at the same time, we do not claim that this model is optimal in this case, but we present it only as an example and plan to pass on to more accurate models in the future. let the number of subsystems be k ≥ 2, and the random variables ξi, 1 ≤ i ≤ k, as before, be the residence times of subtasks from one task. all of them have an exponential distribution with parameter (µ− λ). let’s put ζi = φ−1 ( 1− e−(µ−λ)ξi ) , 1 ≤ i ≤ k, where φ−1 is the inverse function of the standard normal distribution. then the random variables ζi, 1 ≤ i ≤ k, have the standard normal distribution and ξi = − 1 µ− λ ln (1− φ(ζi)) , 1 ≤ i ≤ k. suppose that ζi, 1 ≤ i ≤ k, have a joint multivariate normal distribution. then, due to the symmetry of the system, any pair of quantities ζi and ζj , i ̸= j, has the same pearson correlation coefficient, which we denote as r. in addition, this pair has the same spearman correlation coefficient rs as the original pair ξi and ξj , since the spearman correlation coefficient is preserved under continuous monotonically increasing transformations of random variables. for a multivariate normal distribution [22, theorem 5.36, p. 215] it is true, that: rs = 6 π arcsin r 2 , where we can find r = 2 sin πrs 6 , (7.18) with known rs, calculated by using the formula (4.14). a set of random variables ζi with the required distributions and correlations can be obtained using the formulas: ζi = √ rε0 + √ 1− rεi, 1 ≤ i ≤ k, copyright © 2024 assa. adv syst sci appl (2024) correlations of the sojourn times of subtasks in fork-join queueing systems 15 where εi, 0 ≤ i ≤ k, are independent standard normal random variables. so for the response time rk = max{ξ1, . . . , ξk} we get the estimate r̂k = − 1 µ− λ ln ( 1− φ( √ rε0 + √ 1− rmax{ε1, . . . , εk}) ) . (7.19) using formulas (4.14), (7.18) and (7.19), a simulation was carried out for the values λ = 1, ρ ∈ [0.1, 0.9] with steps of 0.05 and k from 3 to 20, in order to estimate the average response time in comparison with the results of the simulation of a fork-join queuing system carried out by the authors earlier. the errors of the obtained approximations are presented in the table 7.2. table 7.2. errors in approximations of the average response time using the formula (7.19) (meta-gaussian model) for values k = 3, ..., 20 and ρ ∈ {0.10, 0.15, ..., 0.90} evaluated types of errors characteristic max ape, % min ape, % mape, % rk 4.18222 0.29829 2.46825 note that maxape turns out to be close to the error of the classical nelson-tantavi formula [23] on a given set of parameter values (about 4%). the model can be easily used to estimate not only the average response time, but also variance, quantiles, etc. the disadvantages of the meta-gaussian model include the fact that it requires simulation rather than provides an explicit formula, but this simulation is much simpler and faster than the simulation of the original fork-join queuing system. of course, more advanced models, which parameterized by correlation coefficients, are possible. if the model uses parameterization for other reasons, then it seems desirable that its predictions of the values of the correlation coefficients do not diverge too much from the actual ones. 8. conclusion the article presents exact analytical expressions for the pearson and spearman correlation coefficients between the residence times of subtasks in the fork-join subsystems of qs. these formulas were obtained using the classical method of generating functions and laplacestieltjes transformations. the authors could not find sources where such studies were carried out, perhaps one of the reasons is the cumbersomeness of the calculations necessary to derive these formulas. an approximate expression was obtained for the kendall correlation coefficient, the approximation accuracy of which is quite high. to derive the kendall correlation estimate, graphical analysis and the nelder-mead optimization method were used. the results obtained in the framework of a numerical experiment were compared with the data of simulation modeling, which confirmed their correctness. it is shown that all coefficients increase with increasing load, with pearson’s correlation coefficient quadratic (with slowdown), and spearman and kendall’s in a more complex nonlinear manner (but close to linear). at a high load, they approach some limiting values due to the properties of the limiting two-dimensional distribution of the normalized residence times of subtasks. the coefficients were compared according to how well they capture the dependence (they take larger values). it has been established that the kendall coefficient is copyright © 2024 assa. adv syst sci appl (2024) 16 a.v. gorbunova, a.v. lebedev the worst, and as for the rest, there is a critical load value, below which the pearson coefficient is better, and above it is the spearman coefficient. formulas for correlation coefficients make it possible to meaningfully describe the existing relationship between the random variables of the residence times of subtasks, which in turn may allow further more accurate analysis of the response time characteristics of the entire system for the case when the number of subsystems is more than two, since for such systems only approximations of the average response time and its variance have been obtained with varying degrees of accuracy. as a simple example along this path, the paper presents the meta-gaussian model. to describe the dependence and evaluate the characteristics, it is also planned to use the modern theory of copulas, with the fitting of copulas to the available data. references 1. agresti, a. 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(2022). application of machine learning methods to solving problems of queuing theory, communications in computer and information science, 1605, 304–316. https://doi.org/10.1007/978-3-031-09331-9 24 copyright © 2024 assa. adv syst sci appl (2024) introduction mathematical model of fork-join qs pearson correlation coefficient spearman correlation coefficient kendall correlation coefficient limiting two-dimensional distribution and ratios of correlation coefficients meta-gaussian model conclusion adv syst sci appl 2023; 03:66–90 published online at https://ijassa.ipu.ru. covid-19 spread modeling incorporating suggestive optimal control strategies under uncertainty p.k. santra1*, d. pal2, g.s. mahapatra3, h. alrabaiah4,5 1abada nsup school, howrah, west bengal, india 2chandrahati dilip kumar high school, chandrahati, west bengal, india 3department of mathematics, national institute of technology puducherry, karaikal, india 4college of engineering, al ain university, al ain, united arab emirates 5department of mathematics, tafila technical university, tafila, jordan abstract: in the present paper, we have provided a five-compartmental epidemic model in an interval environment to analyze the spread of covid-19 infection in india. the proposed model divides the entire population of india into five classes. they are susceptible, exposed, asymptomatic, symptomatic, and recovered classes. under some suppositions, the crisp model is constructed and converted to an imprecise model by the interval number. we introduced a parametric functional form of an interval number to study the imprecise epidemiological model. the main objective of this study is to develop an epidemiological model in an imprecise environment and to try to understand the dynamics of the epidemic model of covid-19 infection spread in india. we also presented the covid-19 model with two controls to effectively control covid-19 disease in india. finally, a numerical simulation is carried out considering that the model parameters are imprecise. the numerical results show that our proposed imprecise model is reliable from a practical point of view. keywords: covid-19 infection, basic reproduction number, stability, optimal control, interval number 1. introduction the objective of mathematical epidemiology is to find the factors behind the occurrence of a disease. today, mathematical modeling has played a vital role in understanding epidemiological prototypes of diseases. it also predicts the consequences of the foreword of public health interventions to manage the spread of diseases. in recent times, many researchers and scientists [1–3] have attracted attention to epidemiological systems and have paid great attention to their corresponding research. in an epidemiological system, the transmission of disease from an infected person to a susceptible individual occurs by a mechanism known as the incidence rate, that is, a function describing the above mechanism through different forms. among them, homogeneous transmission is widely used. this transmission follows the law of mass action. it is also notable that the cost of the treatment strategy to control epidemic diseases is very high. in some cases, proper treatment is only available and inexpensive. therefore, it is very important to provide the most costeffective control strategies to prevent the spread of epidemic diseases. several researchers and scientists [4–6] are trying their best to provide appropriate optimal control strategies for ∗corresponding author: prasunsantra5@gmail.com covid-19 spread modeling under uncertainty 67 epidemic diseases to protect the human community. their efforts predict some cost-effective strategies to restrict the extent of epidemic diseases and minimize the cost of the control program. recently, coronavirus turned out to be a pandemic disease spreading throughout the world and created a panic situation in many countries. covid-19 infection is a large family of viruses that cause disease like ordinary cold in the primary stage. the latter can create a big problem, such as acute respiratory syndrome. this virus also showed the ability to cause serious health problems among a certain group of individuals, including the elderly population and patients with cardiovascular disease and diabetes. however, the nature of this epidemiology is still changing [7]. on 11 february 2020, this disease was termed coronavirus disease 2019 (covid-19). the virus behind this disease was declared severe acute respiratory syndrome coronavirus 2 (sars-cov-2) [8]. this virus has been shown to transform from human to human [9]; as a result, the covid-19 disease spreads day by day around the world [10–14]. recently, india has become one of the hardest affected country in asia with covid19 endemic due to its very high population density. the number of covid positive cases increases progressively, the indian government has monitor the succession of this outburst and have taken different public health actions together with social distancing procedures in real time. the figure of positive covid-19 infection started to increase from 4th march, 2020. as on 8th may, 2020, a total of 59690 confirmed covid cases together with 17887 recovered and 1986 deaths in india [15]. the indian government has taken different precaution measures [16–18] to maintain the social distance [19] among the large population of india. to date, there are no specific drugs for covid-19 infection. mostly, doctors recommended different treatments via medications to covid-19 patients depending on their symptoms. as it is not possible to invented anti-covid therapeutic treatment, it should be restricted via apposite precautionary measures like quarantine mechanisms [21,22], individual safeguard from the infected individual by using social distancing [20], etc. as the covid-19 virus spread very quickly throughout the world, so various mathematical models depending on the covid-19 outbreak [23–27] have been performed already. recntly, cadoni and gaeta [28] presented covid-19 infection in sir framework. zhang et al. [29] developed covid-19 dynamics via fractional derivative. higazy [30] described covid-19 virulent disease by using sidarthe model. wu et al. [31] studied a seir model to understand the dynamics behind the spread of covid-19 virus world wise. read et al. [32] developed a covid-19 seir model based on poisson-distributed daily time augmentation. pal et al. [33] explored a covid-19 based seqir model to understand the situation of disease in india. since the covid-19 virus has sinister harmful health upshots and pessimistic social impact, a remarkable quantity of awareness and research work are desired on covid-19 infection. recently, many researchers [34–36] concentrate their research to develop epidemic models in uncertain / imprecise environments. from their observations, they feel that in reality data cannot be collected precisely due to some unavoidable reasons such as fluctuation of environment, changes in the nature of the virus, ambiguous death rate (like co-morbidity death), uncertainty about the number of undetected infections, human errors, availability of proper information, etc. this paper introduces a five-compartmental covid-19 infection model in interval environment. we consider parameters of the model that are imprecise in nature and represented by interval numbers. first we formulate the crisp model then we formulate the imprecise model by considering the parameters are interval in nature. we have also introduced two treatment controls in our proposed model under impriseness. the entire paper is organized into several sections and subsections. section 1 presents the introduction to the paper, which includes a description of the problem and its origin. section 2 presents the material & methods which are divided into model variables and parameters, some useful basic mathematical concepts, and the mathematical formulation of the model. copyright © 2023 assa. adv syst sci appl (2023) 68 p.k. santra et al. the model derivation and preliminaries are explained in section 2. the basic properties of our proposed model structure are discussed in section 3. in section 4, we introduce the concept of the basic reproduction number r0 [37]. next, we are dealing with two types of equilibrium points in the system, namely disease-free equilibrium (dfe) e0, and endemic equilibrium e1. it is clear that covid-19 infection is not only a community health trouble [38] but also a great social and monetary shock for developing countries like india. therefore, it is a very essential concern to control [39–42] the spread of covid-19 infection in india by adopting an optimal control policy. in section 5, we have formulated the covid-19 epidemic model with control treatment. this section provides a procedure for finding an optimal control [43] u(t) that increases the recovery rate and minimizes the cost associated with treatment. the analytical results obtained in the previous sections are numerically verified in section 6 with the help of realistic values of the model parameters using matlab. lastly, a general conclusion about our proposed model structure is provided in section 7. 2. materials & methods 2.1. variables and parameters involved in the model following variables and parameters are used to develop the covid-19 model: variable description n(t) : total populations at time t s(t) : susceptible people at the time t e(t) : exposed populations at the time t ia(t) : infective people in the asymptomatic phase at the time t is(t) : infective populations in the symptomatic phase at the time t r(t) : recovered populations at the time t parameter description λ : recruitment rate. α : transmission coefficient due to asymptomatic class (ia). β : transmission coefficient due to symptomatic class (is). µ : natural death rate. ρ : incubation period. (1− σ) : fraction of symptomatically infected people (is). σ : fraction of asymptotically infected people (ia). (1− k)r : transfer rate from the asymptomatic phase (ia) to symptomatic phase (is). kr : recover rate from the asymptomatic class (ia). m : recover rate from symptomatic class (is). d : death rate from covid-19 in symptomatic class (is). copyright © 2023 assa. adv syst sci appl (2023) covid-19 spread modeling under uncertainty 69 2.2. some useful definitions definition 2.1: interval number: a closed interval number a = [al, ar] in the set of real numbers (r) is defined by a = [al, ar] = {x : al ≤ x ≤ ar, x ∈ r} , where al, and ar are the left and right bound of the interval, respectively. definition 2.2: interval valued function: the interval valued function [44, 45] of [al, ar] can be expressed as f(p) = a1−p l apr for p ∈ [0, 1]. arithmetic operations on interval numbers using the concept of interval valued functions are given in appendix c. 2.3. covid-19 pandemic model formulation to develop a mathematical model of covid-19 infection spread in india. the following assumptions and notations are considered throughout the paper. the total human population of india is split up into five mutually exclusive compartments, specifically, susceptible class (s), exposed class (e), infective class in asymptomatic phase (ia), infective class in symptomatic phase (is) and recovered class (r). therefore, total population of india n(t) = s(t) + e(t) + is(t) + ia(t) +r(t). assumptions: 1. the susceptible population (s) is collected of non-infected persons by covid-19 still now but can be infected via get in touch with both types of infective (asymptomatic (ia) as well as symptomatic (is) phase). 2. the exposed population (e) is collected of infected persons by covid-19 but not infectious. 3. the asymptomatic phase (ia) population is collected of covid-19 infection persons without any symptoms (but capable of infecting). 4. the infective symptomatic (is) population is collected of persons with difficulties and diverse symptoms of covid-19 and need treatment. 5. the recovered (r) population from the disease by treatment or otherwise. 6. the susceptible persons become infected by ample contact with contagious (asymptomatic or symptomatic) persons, and enter into the exposed class. 7. one fraction of asymptomatic infective goes into symptomatic phase and one portion becomes recovered. 8. one fraction of symptomatic contagious goes into recovered class and other portion dies out due to disease related death. 9. all class of population is decreased by natural death. the flow of individuals from one class to another class is presented in figure 2.1. copyright © 2023 assa. adv syst sci appl (2023) 70 p.k. santra et al. fig. 2.1. flow diagram of individuals of covid-19 infection 2.4. crisp model the proposed covid-19 infection is presented in a crisp environment based on the above stated assumptions through the following set of ordinary differential equations: ds dt = λ− (αia + βis)s − µs de dt = (αia + βis)s − (ρ+ µ)e dia dt = σρe − (r + µ)ia (2.1) dis dt = (1− σ)ρe + (1− k)ria − (m+ µ+ d)is dr dt = kria +mis − µr with their initial conditions : s(0) > 0;e(0) ≥ 0; ia(0) ≥ 0; is(0) ≥ 0;r(0) ≥ 0. (2.2) 2.5. imprecise model with interval coefficient most of the models have been considered in the crips environment, but the data may not be recorded or collected exactly owing to several occasions in reality. therefore, the model parameters are not to be considered as constants. however, it may be considered as imprecise in nature. to handle with this type of impreciseness, we take the parameters as interval numbers. therefore, for an imprecise coefficient, we present the epidemic model with an interval counterpart of coefficients, i.e., λ̂, α̂, β̂, µ̂, ρ̂, r̂, m̂ and d̂. then the covid-19 copyright © 2023 assa. adv syst sci appl (2023) covid-19 spread modeling under uncertainty 71 infection model (2.1) with interval valued parameters modified to the following form: ds dt = λ̂− ( α̂ia + β̂is ) s − µ̂s de dt = ( α̂ia + β̂is ) s − (ρ̂+ µ̂)e dia dt = σρ̂e − (r̂ + µ̂)ia (2.3) dis dt = (1− σ)ρ̂e + (1− k)r̂ia − (m̂+ µ̂+ d̂)is dr dt = kr̂ia + m̂is − µ̂r with their initial conditions of equation (2.2), where λ̂ ∈ [λl,λu], α̂ ∈ [αl, αu], β̂ ∈ [βl, βu], µ̂ ∈ [µl, µu], ρ̂ ∈ [ρl, ρu], r̂ ∈ [rl, ru], m̂ ∈ [ml,mu] and d̂ ∈ [dl, du]. also λl > 0, αl > 0, βl > 0, µl > 0, ρl > 0, rl > 0, ml > 0, dl > 0. using the parametric functional form of the interval number, the pandemic system (2.3) in the parametric form is as follows: ds dt = λ1−p l λp u − { α1−p l αp uia + β1−p l βp uis } s − µ1−p l µp us de dt = { α1−p l αp uia + β1−p l βp uis } s − { ρ1−p l ρpu + µ1−p l µp u } e dia dt = σρ1−p l ρpue − { r1−p l rpu + µ1−p l µp u } ia (2.4) dis dt = (1− σ)ρ1−p l ρpue + (1− k)r1−p l rpuia − { m1−p l mp u + µ1−p l µp u + d1−p l dpu } is dr dt = kr1−p l rpuia +m1−p l mp uis − µ1−p l µp ur for p ∈ [0,1]. the above system formulation (2.4) can be rewritten as: ds dt = λ1−p l λp u − { α1−p l αp uia + β1−p l βp uis } s − µ1−p l µp us de dt = { α1−p l αp uia + β1−p l βp uis } s − ae dia dt = σρ1−p l ρpue −bia (2.5) dis dt = (1− σ)ρ1−p l ρpue + (1− k)r1−p l rpuia − cis dr dt = kr1−p l rpuia +m1−p l mp uis − µ1−p l µp ur where a = [ ρ1−p l ρpu + µ1−p l µp u ] ; b = [ r1−p l rpu + µ1−p l µu) p ] ; c = [(ml) 1−p(mu) p + (µl) 1−p(µu) p + (dl) 1−p(du) p] and p ∈ [0,1], with initial conditions as stated in equation (2.2). copyright © 2023 assa. adv syst sci appl (2023) 72 p.k. santra et al. 3. theoretical study of the model 3.1. positivity of the solutions theorem 3.1: each solution of the system (2.5) under conditions (2.2) exists in the interval [0,∞) and initial conditions (2.2) are satisfied for all t ≥ 0. proof in view of the fact that the right-hand side of the parametric pandemic system (2.5) is completely continuous and locally lipschitzian on c, the solution (s(t), e(t), ia(t), is(t), r(t)) of (2.5) with initial conditions (2.2) exists and is unique on [0, κ), where 0 < κ < +∞. first, we show that s(t) > 0, ∀ t ∈ [0, κ). otherwise, ∃ a t ∈ (0, κ) such that s(t1) = 0, · s(t1) ≤ 0 and s(t) > 0 for all t ∈ [0, t1). hence, there must have e(t) ≥ 0 for all t ∈ [0, t1). if this statement is not true, then there exists a t2 ∈ (0, t1) such that e(t2) = 0, · e(t2) < 0 and e(t) ≥ 0 on [0, t2). we claim that ia(t) ≥ 0, ∀ t ∈ [0, t2). if this is not true, then ∃ a t3 ∈ (0, t2) such that ia(t3) = 0, · ia(t3) < 0 and ia(t) ≥ 0 on [0, t3). now, from the third equation of (2.5), we have dia(t3) dt = σ(ρl) 1−p(ρu) pe(t3)−bia(t3) = σ(ρl) 1−p(ρu) pe(t3) ≥ 0, which is a contradiction with · ia(t3) < 0. so ia(t) ≥ 0, ∀ t ∈ [0, t2). next, we assert that is(t) ≥ 0, ∀ t ∈ [0, t2) . if this is not correct, then ∃ a t4 ∈ (0, t2) such that is(t4) = 0, · is(t4) < 0 and is(t) ≥ 0 on [0, t4). now, from the fourth equation of (2.5), we have dis(t4) dt = (1− σ)(ρl) 1−p(ρu) pe(t4) + (1− k)(rl) 1−p(ru) pia(t4)− cis(t4) = (1− σ)(ρl) 1−p(ρu) pe(t4) + (1− k)(rl) 1−p(ru) pia(t4) ≥ 0, which contradict the consideration · is(t4) < 0, therefore is(t) ≥ 0, ∀ t ∈ [0, t2). now, using the second equation of (2.5), we have de(t2) dt = ( (αl) 1−p(αu) pia(t2) + (βl) 1−p(βu) pis(t2) ) s(t2)− ae(t2) = ( (αl) 1−p(αu) pia(t2) + (βl) 1−p(βu) pis(t2) ) s(t2) ≥ 0 which is a contradiction with · e(t2) < 0, therefore e(t) ≥ 0, ∀ t ∈ [0, t1). hence, e(t), ia(t), is(t) ≥ 0, ∀ t ∈ [0, t1). next, we claim that r(t) ≥ 0, ∀ t ∈ [0, t1). if this is not true, then ∃ a t5 ∈ (0, t1) such that r(t5) = 0, · r(t5) < 0 and r(t) ≥ 0 on [0, t5) . from the last equation of (2.5), we have dr(t5) dt = k(rl) 1−p(ru) pia(t5) + (ml) 1−p(mu) pis(t5)− (µl) 1−p(µu) pr(t5) = k(rl) 1−p(ru) pia(t5) + (ml) 1−p(mu) pis(t5) ≥ 0, which contradict the consideration · r(t5) < 0, therefore r(t) ≥ 0, ∀ t ∈ [0, t1). copyright © 2023 assa. adv syst sci appl (2023) covid-19 spread modeling under uncertainty 73 again, from the first equation of (2.5) follows that ds(t1) dt = λ1−p l λp u − ( α1−p l αp uia(t1) + β1−p l βp uis(t1) ) s(t1)− µ1−p l µp us(t1) = λ1−p l λp u > 0, this shows that s(t) > 0, ∀ t ∈ [0, κ). based on the steps discussed previously, it observes that e(t) ≥ 0, ia(t) ≥ 0, is(t) ≥ 0, r(t) ≥ 0 for all t ∈ [0, κ), where 0 < κ < +∞. this completes the proof. 3.2. invariant region theorem 3.2: the feasible region µ defined by µ = { (s(t), e(t), ia(t), is(t), r(t)) ∈ r6 + : 0 < n ≤ λ1−p l λp u ϑ } where ϑ = min {(µl) 1−p(µu) p, (µl) 1−p(µu) p + (dl) 1−p(du) p}, with intial conditions (2.2) is positively invariant. proof by adding the equations of the system (2.5) we obtain dn dt = (λl) 1−p(λu) p − (µl) 1−p(µu) p(s + e + ia + is +r)− (dl) 1−p(du) pis therefore, dn dt + ϑn = λ1−p l λp u − (µ1−p l µp u − ϑ)(s + e + ia +r)− (µ1−p l µp u + d1−p l dpu − ϑ)is ≤ (λl) 1−p(λu) p (3.6) where ϑ = min {(µl) 1−p(µu) p, (µl) 1−p(µu) p + (dl) 1−p(du) p}. the solution n(t) of the differential equation (3.6) has the following property, 0 < n(t) ≤ n(0) exp(−ϑt) + (λl) 1−p(λu) p ϑ (1− exp(−ϑt)), where n(0) represents the sum of the initial values of the variables. as t → ∞, we have 0 < n(t) ≤ (λl) 1−p(λu)p ϑ . moreover, if n(0) ≤ (λl) 1−p(λu)p ϑ then also n(t) ≤ (λl) 1−p(λu)p ϑ for all t. this means that (λl) 1−p(λu)p ϑ is the upper bound of n . on the other hand, if n(0) > (λl) 1−p(λu)p ϑ , then n(t) will decrease to (λl) 1−p(λu)p ϑ . this means that if n(0) > (λl) 1−p(λu)p ϑ , then the solution (s(t), e(t), ia(t), is(t), r(t)) enters µ or approaches it asymptotically. hence, it is positively invariant under the flow induced by the systems (2.5) with initial conditions (2.2). thus in µ the model (2.5) with initial conditions (2.2) is well-posed both mathematically and epidemiologically. therefore, the study of the dynamics of the proposed covid-19 system in µ is mathematically sufficient. 3.3. existence of equilibria and basic reproduction number we will study the existence and stability behavior of the system (2.5) at different equilibrium points. the equilibrium points of the proposed system are as follows: copyright © 2023 assa. adv syst sci appl (2023) 74 p.k. santra et al. (i) disease-free equilibrium (dfe): e0( (λl) 1−p(λu)p (µl)1−p(µu)p , 0, 0, 0, 0), (ii) endemic equilibrium: e1(s ∗, e∗, i∗a, i ∗ s, r ∗). the basic reproduction number (brn): the basic reproduction number [46] is defined as ”the number of new infective individuals produced by a single infective individual during his or her effective infectious period when introduced into susceptible populations”. now, the brn of covid-19 system (2.5) will be derived using the next generation matrix method [47]. let z = (e(t), ia(t), is(t), r(t), s(t))t , the proposed pandemic system (2.5) can be written in the following form: dz dt = 𭟋(z)− υ(z) where 𭟋(z) =  ((αl) 1−p(αu) pia + (βl) 1−p(βu) pis)s 0 0 0 0  and υ(z) =  ae −σ(ρl) 1−p(ρu) pe +bia −(1− σ)(ρl) 1−p(ρu) pe − (1− k)(rl) 1−p(ru) pia + cis −k(rl) 1−p(ru) pia − (ml) 1−p(mu) pis + (µl) 1−p(µu) pr −(λl) 1−p(λu) p + ((αl) 1−p(αu) pia + (βl) 1−p(βu) pis + (µl) 1−p(µu) p)s  the jacobian matrices of 𭟋(z) and υ(z) at e0 are as follows: d𭟋(e0) = [ f3×3 0 0 0 0 0 0 0 0 ] and dυ(e0) =  v3×3 0 0 0 −kr −(ml) 1−p(mu) p (µl) 1−p(µu) p 0 0 (λl) 1−p(λu)pα (µl)1−p(µu)p (λl) 1−p(λu)pβ (µl)1−p(µu)p 0 (µl) 1−p(µu) p  where f = 0 (λl) 1−p(λu)pα (µl)1−p(µu)p (λl) 1−p(λu)pβ (µl)1−p(µu)p 0 0 0 0 0 0  and v = [ a 0 0 −σ(ρl) 1−p(ρu) p b 0 −(1− σ)(ρl) 1−p(ρu) p −(1− k)r c ] now, fv −1 is the next generation matrix of (2.5), consequently the spectral radius of the matrix fv −1 is denoted and defined by [47]: r ( fv −1 ) = λ1−p l λp u µ1−p l µp u [α1−p l αp uσρ 1−p l ρpuc + β1−p l βp uρ 1−p l ρpu{σ(1− k)r1−p l rpu +b(1− σ)}] abc copyright © 2023 assa. adv syst sci appl (2023) covid-19 spread modeling under uncertainty 75 according to ref. [47], the brn of system (2.5) is r0 = r ( fv −1 ) = λ1−p l λp u µ1−p l µp u [α1−p l αp uσρ 1−p l ρpuc + β1−p l βp uρ 1−p l ρpu{σ(1− k)r1−p l rpu +b(1− σ)}] abc notice that (λl) 1−p(λu)p (µl)1−p(µu)p is the number of susceptibles at the dfe. existence of endemic equilibrium e1(s ∗, e∗, i∗a, i ∗ s, r ∗): here, we analyze the existence of nontrivial endemic equilibrium e1(s ∗, e∗, i∗a, i ∗ s, r ∗) of the system (2.5). to find the endemic equilibrium of pandemic model (2.5), we consider the following: s(t) > 0, e(t) > 0, ia(t) > 0, is(t) > 0, r(t) > 0 and ds dt = 0, de dt = 0, dia dt = 0, dis dt = 0, dr dt = 0 (3.7) the solution of the equations of system (3.7) is: s∗ = abc (αl)1−p(αu)pσ(ρl)1−p(ρu)pc+(βl)1−p(βu)p(ρl)1−p(ρu)p{σ(1−k)(rl)1−p(ru)p+b(1−σ)} = (λl) 1−p(λu)p (µl)1−p(µu)p 1 r0 , e∗ = bi∗a σ(ρl)1−p(ρu)p , i∗a = (λl) 1−p(λu)pσ(ρl) 1−p(ρu)p(1− 1 r0 ) ab , i∗s = {σ(1−k)(rl) 1−p(ru)p+b(1−σ)}i∗a σc , r∗ = [k(rl) 1−p(ru)pσc+(ml) 1−p(mu)p{σ(1−k)(rl) 1−p(ru)p+b(1−σ)}]i∗a σ(µl)1−p(µu)pc , hence, i∗a has positive solution iff r0 > 1. summarizing the above discussions, we arrive at the following result. theorem 3.3: the system (2.5) has a dfe e0( λ1−p l λp u µ1−p l µp u , 0, 0, 0, 0) for all parameter values. if r0 > 1, the system (2.5) also asserts a unique endemic equilibrium e1(s ∗, e∗, i∗a, i ∗ s, r ∗). 3.4. stability behavior here, we examine the condition of stability of the pandemic system (2.5) in two different equilibrium points e0 and e1. for the stability of dfe e0( (λl) 1−p(λu)p (µl)1−p(µu)p , 0, 0, 0, 0) we consider the theorems given below theorem 3.4: the dfe e0 of the system (2.5) is locally asymptotically stable if r0 < 1. proof detailed proof of this theorem is given in appendix a. theorem 3.5: the dfe e0( (λl) 1−p(λu)p (µl)1−p(µu)p , 0, 0, 0, 0) is globally asymptotically stable for the system (2.5) if r0 < 1 and became unstable if r0 > 1. proof the covid-19 system (2.5) rewrite as dx dt = f (x, v ) dv dt = g(x, v ), such that g(x, 0) = 0 copyright © 2023 assa. adv syst sci appl (2023) 76 p.k. santra et al. where x = (s,r) ∈ r2 denotes compartments of the number of uninfected individuals, v = (e(t), ia(t), is(t)) ∈ r4 is the number of infected individuals compartments, and e0( (λl) 1−p(λu)p (µl)1−p(µu)p , 0, 0, 0, 0) is the dfe of the system (2.5). the global stability of the dfe is guaranteed if satisfying the following two conditions: (i) for dx dt = f (x, 0), x∗ is globally asymptotically stable, (ii) g(x, v ) = bv − ĝ(x, v ), ĝ(x, v ) ≥ 0 for (x, v ) ∈ ω where b = dvg(x∗, 0) is a metzler matrix and ω is the positively invariant set of the system(2.5). following castillo-chavez et al. [48], we investigate the aforementioned conditions for pandemic system (2.5), f (x, 0) = [ (λl) 1−p(λu) p − (µl) 1−p(µu) ps 0 ] b =  −a (λl) 1−p(λu)pα (µl)1−p(µu)p (λl) 1−p(λu)pβ (µl)1−p(µu)p σ(ρl) 1−p(ρu) p −b 0 (1− σ)(ρl) 1−p(ρu) p (1− k)(rl) 1−p(ru) p −c  and ĝ(x, v ) = [ 0 0 0 ] as the off-diagonal elements of b are non negative, and ĝ(x, v ) ≥ 0, hence the disease-free equilibrium e0( (λl) 1−p(λu)p (µl)1−p(µu)p , 0, 0, 0, 0) is globally asymptotically stable if r0 < 1 and became unstable if r0 > 1. theorem 3.6: the endemic equilibrium point e1(s ∗, e∗, i∗a, i ∗ s, r ∗) of the system (2.5) is locally asymptotically stable if r0 > 1, b1b2 −b3 > 0 and b1b2b3 −b2 1b4 −b2 3 > 0 proof detailed proof of this theorem is given in appendix b. 4. optimal control for covid-19 model the most important motive of studying infectious diseases is to control the transmission of the disease and finally stop the infection from the population. transmission of the disease and finally stop the infection from the population. in the present scenario, many developed countries are working hard to discover the medicine or vaccine of covid-19. as vaccine of covid-19 has not yet been discovered, we have to control this pandemic by suitable controlling strategies like the use of proper drugs, etc. this section presents an optimal control problem relative to the covid-19 epidemic model (2.5) with two control variables u1(t) and u2(t), as it is essential to construct an optimal control problem in such a way that the total amount of drugs can be minimized. the first control u1(t) represents the recovery rate of the infectious asymptomatic individuals with treatment. the second control u2(t) represents the recovery rate of the infectious symptomatic individuals with treatment. therefore, the proposed covid-19 epidemic model with two controls becomes: copyright © 2023 assa. adv syst sci appl (2023) covid-19 spread modeling under uncertainty 77 ds dt = λ− ( (αl) 1−p(αu) pia + (βl) 1−p(βu) pis ) s − (µl) 1−p(µu) ps de dt = ( (αl) 1−p(αu) pia + (βl) 1−p(βu) pis ) s − ((ρl) 1−p(ρu) p + (µl) 1−p(µu) p)e dia dt = σ(ρl) 1−p(ρu) pe − (µ+ (µl) 1−p(µu) p + u1)ia (4.8) dis dt = (1− σ)(ρl) 1−p(ρu) pe + µia − (u2 + (µl) 1−p(µu) p + (dl) 1−p(du) p)is dr dt = u1ia + u2is − (µl) 1−p(µu) pr satisfying, s(0) = s0, e(0) = e0, ia(0) = ia0 , is(0) = is0 , r(0) = r0 (4.9) here µ stands for the rate of transfer from the asymptomatic phase to the symptomatic phase of the infective individual, and all other system parameters remain unchanged as described earlier. let µ̂ be the interval counterparts of µ, where µ̂ ∈ [µl, µu]. parametric form of interval valued parameter µ̂ is (µl) 1−p(µu) p for p ∈ [0,1]. the objective functional [50] is defined as j(u1(t), u2(t)) = tf∫ 0 [g1e + iae +g3is + k1 2 u2 1 + k2 2 u2 2]dt (4.10) where g1,g2,g3,k1 2 and k2 2 are positive constants. the square of the control variable reflects the severity of the side effects of the treatment. we have to minimize the objective functional j(u1(t), u2(t)) given in (4.8) so that the covid infected individuals and the cost of treatment can be minimized. therefore, we look for an optimal control (u∗ 1, u ∗ 2) such that j(u∗ 1, u ∗ 2) = min {j(u1, u2) : (u1, u2) ∈ u} (4.11) where admissible control set u = {(u1, u2) : uiis measurable, 0 ≤ ui ≤ 1, t ∈ [0, tf ], fori = 1, 2}. 4.1. existence of an optimal control theorem 4.1: an optimal control (u∗ 1, u ∗ 2) exists for j(u∗ 1, u ∗ 2) = min {j(u1, u2) : (u1, u2) ∈ u} for the control system (4.8) with initial conditions (4.9). proof this theorem can be proved by using the results of the existence of an optimal control pair from ref. [51]. the system of equation (4.8) is bounded from above by a linear system. the boundedness of solutions of system (4.8) for a finite time interval is used to prove the existence of an optimal control. we first check the following properties before using of the theorem from ref. [51]: (i) set of state variables and their control is non-empty, (ii) u is convex and closed, (iii) the right-hand side of (4.8) is bounded by a linear function in the state control, (iv) the integrand of the objective functional is convex on u , (v) there exist constants c1, c2 > 0 and q > 1 such that the integrand of (4.10) satisfies g1e +g2ia +g3is + k1 2 u2 1 + k2 2 u2 2 ≥ c1(u 2 1 + u2 2) q 2 − c2. copyright © 2023 assa. adv syst sci appl (2023) 78 p.k. santra et al. we use a result by lukes ( [52] theorem 9.2.1) to verify the condition (i) for the system (4.8) with bounded coefficients. the control set u is convex and closed by definition, which gives the condition (ii). the right-hand side of the state system (4.8) satisfies condition (iii) as the state solutions are a priori bounded. the integrand in the objective functional, g1e +g2ia +g3is + k1 2 u2 1 + k2 2 u2 2, is clearly convex on u , which gives condition (iv). finally, because the state variables are bounded, then there are c1, c2 > 0 and q > 1 satisfying g1e +g2ia +g3is + k1 2 u2 1 + k2 2 u2 2 ≥ c1(u 2 1 + u2 2) q 2 − c2 based on the above conditions, there exists an optimal control (u∗ 1, u ∗ 2) such that j(u∗ 1, u ∗ 2) = min {j(u1, u2) : (u1, u2) ∈ u} 4.2. characterization of the optimal control the pontryagin’s maximum principle [53] is used to derive the necessary conditions for the optimal control pair. the necessary conditions for optimum control are obtained by means of pontryagin’s maximum principle. theorem 4.2: there exists an optimal control (u∗ 1, u ∗ 2) and corresponding solutions s ∗ , e ∗ , ia ∗ , is ∗ , r ∗ that minimizes j of (4.10) over u . the explicit optimal controls are connected to the existence of continuous specific functions λi(t), for i = 1, 2, 3, 4, 5, the solutions of the following adjoint system: dλ1 dt = (λ1 − λ2) ( (αl) 1−p(αu) pia + (βl) 1−p(βu) pis ) + λ1(µl) 1−p(µu) p dλ2 dt = −g1 + λ2((ρl) 1−p(ρu) p + (µl) 1−p(µu) p)− (λ3 − λ4)σ(ρl) 1−p(ρu) p − λ4(ρl) 1−p(ρu) p dλ3 dt = −g2 + (λ1 − λ2)α 1−p l αp us + (λ3 − λ5)u1(t) + (λ3 − λ4)(µl) 1−p(µu) p − λ3(µl) 1−p(µu) p dλ4 dt = −g3 + (λ1 − λ2)(βl) 1−p(βu) ps + (λ4 − λ5)u2(t) + λ4(µl) 1−p(µu) p dλ5 dt = λ5(µl) 1−p(µu) p on the conditions of transversality: λi(tf ) = 0, for i = 1, 2, 3, 4, 5. in addition, the following property holds: u∗ 1 = min{max{0, (λ3 − λ5)ia ∗ k1 }, 1}, and u∗ 2 = min{max{0, (λ4 − λ5)is ∗ k2 }, 1}. proof the hamiltonian is defined as follows: ĥ = g1e +g2ia +g3is + k1 2 u2 1 + k2 2 u2 2 + λ1[λ 1−p l λp u − (α1−p l αp uia + β1−p l βp uis)s(4.12) −µ1−p l µp us] + λ2[(α 1−p l αp uia + β1−p l βp uis)s − (ρ1−p l ρpu + µ1−p l µp u)e] +λ3[σρ 1−p l ρpue − (µ1−p l µp u + µ1−p l µp u + u1)ia] + λ4[(1− σ)ρ1−p l ρpue + µ1−p l µp uia −(u2 + µ1−p l µp u + d1−p l dpu)is] + λ5[u1ia + u2is − (µl) 1−p(µu) pr] copyright © 2023 assa. adv syst sci appl (2023) covid-19 spread modeling under uncertainty 79 where λi (i = 1, 2, 3, 4, 5, 6) are the adjoint functions to be determined suitably. the form of the equations of adjoint and transversality conditions are the standard consequences from pontryagin’s maximum principle [53]. the adjoint system for the proposed covid-19 system can be obtained as follows: dλ1 dt = −∂ĥ ∂s = (λ1 − λ2) ((αl) 1−p(αu) pia + (βl) 1−p(βu) pis) + λ1(µl) 1−p(µu) p dλ2 dt = −∂ĥ ∂e = −g1 + λ2 ( ρ1−p l ρpu + µ1−p l µp u ) − (λ3 − λ4)σ(ρl) 1−p(ρu) p − λ4(ρl) 1−p(ρu) p dλ3 dt = − ∂ĥ ∂ia = −g2 + (λ1 − λ2)α 1−p l αp us + (λ3 − λ5)u1(t) + (λ3 − λ4)µ 1−p l µp u + λ3µ 1−p l µp u dλ4 dt = − ∂ĥ ∂is = −g3 + (λ1 − λ2)β 1−p l βp us + (λ4 − λ5)u2(t) + λ4((µl) 1−p(µu) p + (dl) 1−p(du) p) dλ5 dt = −∂ĥ ∂r = λ5(µl) 1−p(µu) p (4.13) the transversality conditions (or boundary conditions) are λi(tf ) = 0, i = 1, 2, 3, 4, 5 (4.14) using the optimality condition, we have ∂ĥ ∂u1 = k1u ∗ 1 − (λ3 − λ5)i ∗ a = 0 at u1 = u∗ 1(t) ⇒ u∗ 1(t) = (λ3−λ5)i ∗ a k1 ∂ĥ ∂u2 = k2u ∗ 2 − (λ4 − λ5)i ∗ s = 0 at u2 = u∗ 2(t) ⇒ u∗ 2(t) = (λ4−λ5)i ∗ s k2 (4.15) again, from the bounds for the control u1(t), we get u∗ 1 =  (λ3−λ5)i ∗ a k1 , if 0 ≤ (λ3−λ5)i ∗ a k1 ≤ 1 0, if (λ3−λ5)i ∗ a k1 ≤ 0 1, if (λ3−λ5)i ∗ a k1 ≥ 1 in compact form: u∗ 1 = min{max{0, (λ3 − λ5)i ∗ a k1 }, 1} (4.16) from the bounds for the control u2(t), we get u∗ 2 =  (λ4−λ5)i ∗ s k2 , if 0 ≤ (λ4−λ5)i ∗ s k2 ≤ 1 0, if (λ4−λ5)i ∗ s k2 ≤ 0 1, if (λ4−λ5)i ∗ s k2 ≥ 1 the above can be written in dense form: u∗ 2 = min{max{0, (λ4 − λ5)i ∗ s k2 }, 1} (4.17) copyright © 2023 assa. adv syst sci appl (2023) 80 p.k. santra et al. using (4.16), we obtain the following optimality system: ds dt = (λl) 1−p(λu) p − ( (αl) 1−p(αu) pia + (βl) 1−p(βu) pis ) s − (µl) 1−p(µu) ps (4.18) de dt = ( (αl) 1−p(αu) pia + (βl) 1−p(βu) pis ) s − ( (ρl) 1−p(ρu) p + (µl) 1−p(µu) p ) e dia dt = σ(ρl) 1−p(ρu) pe − ((µl) 1−p(µu) p + (µl) 1−p(µu) p)ia −min{max{0, (λ3 − λ5)i ∗ a k1 }, 1}ia dis dt = (1− σ)ρ1−p l ρpue + µ1−p l µp uia −min{max{0, (λ4 − λ5)i ∗ s k2 }, 1}is − (µ1−p l µp u + d1−p l dpu)is dr dt = min{max{0, (λ3 − λ5)i ∗ a k1 }, 1}ia +min{max{0, (λ4 − λ5)i ∗ s k2 }, 1}is − (µl) 1−p(µu) pr dλ1 dt = (λ1 − λ2) ( (αl) 1−p(αu) pia + (βl) 1−p(βu) pis ) + λ1(µl) 1−p(µu) p dλ2 dt = −g1 + λ2 ( (ρl) 1−p(ρu) p + (µl) 1−p(µu) p ) − (λ3 − λ4)σ(ρl) 1−p(ρu) p − λ4(ρl) 1−p(ρu) p dλ3 dt = −g2 + (λ1 − λ2)α 1−p l αp us + (λ3 − λ5)min{max{0, (λ3 − λ5)i ∗ a k1 }, 1}+ (λ3 − λ4)µ 1−p l µp u + λ3µ 1−p l µp u dλ4 dt = −g3 + (λ1 − λ2)β 1−p l βp us + (λ4 − λ5)min{max{0, (λ4 − λ5)i ∗ s k2 }, 1}+ λ4(µ 1−p l µp u + d1−p l dpu) dλ5 dt = λ5(µl) 1−p(µu) p subject to the following conditions: s(0) = s0, e(0) = e0, ia(0) = ia0, is(0) = is0, r(0) = r0 and λ1(tf ) = 0, λ2(tf ) = 0, λ3(tf ) = 0, λ4(tf ) = 0, λ5(tf ) = 0 this completes the proof. 5. numerical simulations in this section, we analyze our mathematical model in different environment through some simulation works. the simulation works are done by using matlab r2008a software package with the use of the most versatile ordinary differential equation solver ode45. simulations of crips covid-19 model: to understand the system dynamics of our proposed crisp model more clearly, we present the following graphical presentation based on table 5.1 and table 5.2. table 5.1. model parameters for covid-19 system parameters values references λ 60000 assumed α 0.0000000001 fitted β 0.0000000001 fitted µ 0.00002 assumed ρ 0.2 fitted r 0.1 fitted m 0.07 fitted d 0.0018 fitted σ 0.64 fitted k 0.7 fitted copyright © 2023 assa. adv syst sci appl (2023) covid-19 spread modeling under uncertainty 81 table 5.2. preliminary population density for covid-19 mode s(0) 1200000000 e(0) 200000 ia(0) 200000 is(0) 220114 r(0) 347979 using the parameter values given in table 5.1, the endemic equilibrium point is given by (713600234.09, 228324.60, 291846.24, 350560.38, 2251062473.25) and correspoinding values of r0 is given by 4.1999 > 1. therefore, all population of our system is all persist, i.e., the population of our model system converges to the corresponding endemic equilibrium point (713600234.09, 228324.60, 291846.24, 350560.38, 2251062473.25). also, since r0 > 1, indicates the presence of infection in the system. using the parameter values given in table 5.1 for different initial conditions given in table 5.2, the dynamics of the model is presented in fig. 5.2 and fig. 5.3. 0 50 100 150 200 250 300 350 400 450 500 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10 7 time a c ti v e i n fe c te d ( i s ) fig. 5.2. time series plot of the active infected population (is) for the parameter values given in table 1 and the initial conditions given in table 2. 0 100 200 300 400 500 2 4 6 8 10 12 x 10 8 time s u s c e p ti b le ( s ) 0 100 200 300 400 500 0 1 2 3 4 x 10 7 time e x p o s e d ( e ) 0 100 200 300 400 500 0 1 2 3 4 5 x 10 7 time a s y m p to m a ti c ( i a ) 0 100 200 300 400 500 0 2 4 6 8 10 x 10 8 time r e c o v e re d ( r ) i ii iviii fig. 5.3. time series plot of (i) susceptible population, (ii) exposed population, (iii) infective population asymptomatic phase, (iv) recovered population based on parameter values given in table 1 and initial conditions given in table 2. from fig. 5.2, it is observed that the pick of height is nearly equal to 4.7× 107. moreover, after reaching the height pick of the infection, the active infective curve is gradually decreased as the time progress. furthermore, the curve of exposed population (e), the infective population in asymptomatic phase (ia) shows the similar behavior as infective (is) curve for different with respect to time t (see fig. 5.3). on the contrary, susceptible (s) population curve gradually decreases and gradually increases as the time progress. imprecise model to verify our analytical findings, we set the values of the parameters as interval numbers given in table 5.1, and the initial population density is given in table 5.2, respectively. based on table 5.1, we calculate the disease-free equilibrium, endemic equilibrium, and r0 for different values of p ∈ [0, 1] which are given in table 5.3. from table 5.3, we observe that e0 and e1 exist for all values of p. it is also interesting to observe that the value of r0 is continuously increases with the increasing value of p. it copyright © 2023 assa. adv syst sci appl (2023) 82 p.k. santra et al. table 5.3. disease-free equilibrium (e0), endemic equilibrium (e1) and r0 p disease-free equilibrium endemic equilibrium r0 0.0 (2947368421.05, 0, 0, 0, 0) (778798079.86, 216835.35, 266284.03, 321556.78, 2138994795.46) 3.7845 0.1 (2957243264.96, 0, 0, 0, 0) (765297598.12, 219172.65, 271318.04, 327289.35, 2161811456.07) 3.8642 0.2 (2967151193.49, 0, 0, 0, 0) (752031614.29, 221489.81, 276390.65, 333055.41, 2184422467.99) 3.9455 0.3 (2977092317.47, 0, 0, 0, 0) (738996051.42, 223787.25, 281502.46, 338855.54, 2206831952.79) 4.0286 0.4 (2987066748.13, 0, 0, 0, 0) (726186903.48, 226065.38, 286654.12, 344690.33, 2229043960.98) 4.1134 0.5 (2997074597.06, 0, 0, 0, 0) (713600234.09, 228324.60, 291846.24, 350560.38, 2251062473.25) 4.1999 0.6 (3007115976.22, 0, 0, 0, 0) (701232175.32, 230565.32, 297079.47, 356466.27, 2272891401.66) 4.2883 0.7 (3017190997.95, 0, 0, 0, 0) (689078926.54, 232787.93, 302354.44, 362408.60, 2294534590.87) 4.3786 0.8 (3027299774.97, 0, 0, 0, 0) (677136753.17, 234992.80, 307671.80, 368387.99, 2315995819.22) 4.4707 0.9 (3037442420.37, 0, 0, 0, 0) (665401985.62, 237180.32, 313032.22, 374405.02, 2337278799.96) 4.5648 1.0 (3047619047.62, 0, 0, 0, 0) (653871018.08, 239350.87, 318436.33, 380460.31, 2358387182.31) 4.6609 is also interesting to see that brn of our proposed number is gradually increasing with the increasing value of p. table 5.3 also indicates that brn r0 is always much greater than 1 for any values of p. hence, the infection spread very quickly. therefore, government should take proper policy to reduce the value of r0 below or near to 1. again, we see from table 5.3 that our model system’s population converges to the corresponding endemic equilibrium points e1 for all values of p (by our analytical results stated in theorem 6). graphical presentation of brn r0 with respect to is presented through fig. 5.4. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 p r 0 fig. 5.4. reproduction number r0 with respect to p this figure shows that as the parameter p increases (0 to 1) brn is also increases i.e., speeding rate of infection are increased as the parameter p increases. therefore, from table 5.3 and fig. 5.4, we observe that parametric model is essential to study the behavior of the covid model. the dynamical behavior of our proposed covid-19 model with respect time for different values of p is represented by fig. 5.5 and fig. 5.6, respectively. 0 50 100 150 200 250 300 350 400 450 500 0 1 2 3 4 5 6 7 x 10 7 time a c ti v e i n fe c te d ( i s ) p=0.0 p=0.2 p=0.4 p=0.6 p=0.8 p=1.0 p=0.5 fig. 5.5. dynamical behavior with time for different values of p for the active infected population from fig. 5.5, we observe that the pick of height occurs when p = 1 and which is nearly equal to 6× 107. it is also noticed that when p is gradually decreases form 1 then the number of height active individual is also decreases. for p = 0 the infective curve attains its pick level copyright © 2023 assa. adv syst sci appl (2023) covid-19 spread modeling under uncertainty 83 0 100 200 300 400 500 2 4 6 8 10 12 x 10 8 time s u s c e p ti b le ( s ) 0 100 200 300 400 500 0 1 2 3 4 5 x 10 7 time e x p o s e d ( e ) 0 100 200 300 400 500 0 2 4 6 x 10 7 time a s y m p to m a ti c ( i a ) 0 100 200 300 400 500 0 2 4 6 8 10 x 10 8 time r e c o v e re d ( r ) p=0.0 p=0.2 p=0.4 p=0.6 p=0.8 p=1.0 p=0.5 i ii iviii fig. 5.6. dynamical behavior with time for exposed (e), susceptible (s), asymptomatic infected (ia), recovered (r) population curve for different values of p at nearly equal to 3× 107. therefore, for our covid-19 parametric model the total active number of infected individuals lies in the interval [3× 107, 6× 107]. again, after reaching the height pick of the infection the active infective curve is gradually decreases with time for any values of p. moreover, the curve of population e and ia shows the similar behavior as infective (is) curve for different values of p with respect to time t. on the contrary, susceptible (s) population curve shows an opposite behavior for different values of p. in this figure, we observe that as p decreases from 1 the susceptible curve gradually increases. finally, we observed that as p increases from 0 the recovered population curve (r) is also increases, i.e., the number of recovered population gradually increased. now, we solve the imprecise optimality system numerically and the obtained results are plotted graphically. the proposed optimality system is a two point boundary value system with divided boundary conditions at times t = 0 and t = tf . here, we have solved this twopoint boundary value optimality problem for tf = 25 in the time unit day at which the treatment is stopped. as in the objective functional (4.10), the different populations and control functions are different scales, we are balancing the objective functional (4.10) by opting the weight constants g1, g2 , g3, k1 and k2 given in table 5.4. table 5.4. values of the weight constants weight constant value g1 0.1 g2 0.2 g3 0.2 k1 300000 k2 300000 the numerical simulation of the control problem is performed using the parameter values and weight constants specified in table 5.1 and table 5.4, respectively. here, we search for two optimal control functions u1 and u2, minimize the objective functional (j) provided by (4.10). the graph of optimal control for u1 and u2 is presented in fig. 7 for different p. 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 time u 1 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 time u 2 p=0.0 p=1.0 p=0.5 p=0.0 p=1.0 p=0.5 iii fig. 5.7. optimal control of two controls, i.e., treatment control (u 1 and u 2 ) using the parameter values given in table 3 with g1 = 0.1, g2 = 0.2, g3 = 0.2, k1 = 300000, k2 = 300000 for different values of p. copyright © 2023 assa. adv syst sci appl (2023) 84 p.k. santra et al. from the optimal control diagrams (fig. 5.7), one could conclude that the total effort should be given in both controls at the start of the disease. the system’s dynamical behavior under control and without control is presented in fig. 5.8 for different p. 0 5 10 15 20 25 1.1985 1.199 1.1995 1.2 1.2005 1.201 x 10 9 time p o p u la ti o n s 0 5 10 15 20 25 0 2 4 6 x 10 5 time p o p u la ti o n e 0 5 10 15 20 25 0 2 4 6 x 10 5 time p o p u la ti o n i a 0 5 10 15 20 25 0 2 4 6 x 10 5 time p o p u la ti o n i s p=0.5 p=0.5 p=0.0 p=0.0 p=1.0 p=1.0 i iii iv ii fig. 5.8. time series plots of (i) susceptible population, (ii) exposed population, (iii) infective population in asymptomatic phase, (iv) infective population in symptomatic phase with two controls and without control using the parameter values given in table 5.1 and the values of the weight constants given in table 5.4. the solid lines of fig. 5.8 are presented as the population with control, and the dotted lines are presented as the population without control. fig. 5.8 shows that using control, exposed class (e), asymptomatic class (ia) and symptomatic class (is) are decreasing. this study indicates that the two controls can control the disease successfully. in fig. 5.9 and fig. 5.10, we have presented the comparative study without control (u1 = u2 = 0) through the time series diagrams of every population class, also for only first treatment control (u1 ̸= 0, u2 = 0) and only second treatment control (u1 = 0, u2 ̸= 0) for different values of p. 0 5 10 15 20 25 1.1985 1.199 1.1995 1.2 1.2005 x 10 9 time p o p u la ti o n s 0 5 10 15 20 25 1 2 3 4 5 6 x 10 5 time p o p u la ti o n e 0 5 10 15 20 25 0 1 2 3 4 5 6 x 10 5 time p o p u la ti o n i a 0 5 10 15 20 25 1 2 3 4 5 6 x 10 5 time p o p u la ti o n i s p=0.5 p=0.5 p=0.0 p=0.0 p=1.0 p=1.0 iii i ii iv fig. 5.9. time series plots of (i) susceptible, (ii) exposed, (iii) infected in asymptomatic phase, (iv) infected in symptomatic phase populations with u1 control and without control using input values given in table 5.1 and table 5.4. 0 5 10 15 20 25 1.1985 1.199 1.1995 1.2 1.2005 x 10 9 time p o p u la ti o n s 0 5 10 15 20 25 1 2 3 4 5 6 x 10 5 time p o p u la ti o n e 0 5 10 15 20 25 1 2 3 4 5 6 x 10 5 time p o p u la ti o n i a 0 5 10 15 20 25 0 2 4 6 x 10 5 time p o p u la ti o n i s p=0.5 p=0.5 p=0.0 p=0.0 p=1.0 p=1.0 i ii iviii fig. 5.10. time series plots of (i) susceptible, (ii) exposed, (iii) asymptomatic phase of infected, (iv) symptomatic phase of infected populations with u2 control and without control using the parameter values given in table 5.1 and the values of the weight constants given in table 5.4. the solid lines of fig. 5.9 are presented as the population with u1 control and the dotted lines are presented as the population without control. in fig. 5.10, the solid lines are presented as the population with u2 control and dotted lines are presented as the population without copyright © 2023 assa. adv syst sci appl (2023) covid-19 spread modeling under uncertainty 85 control. these figures clearly show that using both controls together is more effective to control the disease compared to without any control and single controls. from the above observations, it can conclude that both the treatment controls together yield a relatively noticed result on the controlling of the disease. 6. conclusions this paper considered a pandemic disease in recent years, the so-called covid-19 infection. this study developed a five-compartmental epidemic model and investigated the dynamical behavior of this model. most of the covid-19 infection models are generally based on the assumption that the model’s parameters are precisely known. however, the scenario is different since it is impossible to know all the parameter values specifically. in this paper, we developed a method to discuss the dynamical behavior of covid-19 epidemic model with imprecise parameters by considering that the model’s coefficients are ambiguous for the lack of precise numerical information. by using the concept of the next-generation matrix method, we have found r0 = λ1−p l λp u µ1−p l µp u [α1−p l αp uσρ 1−p l ρpuc + β1−p l βp uρ 1−p l ρpu{σ(1− k)r1−p l rpu +b(1− σ)}] abc as brn of the system, which helps us to determine the dynamical behavior of the system for all p ∈ [0, 1]. the system (2.5) is locally asymptotically stable in the dfe e0 when r0 < 1 for all p ∈ [0, 1]. the endemic equilibrium e1 exists when r0 > 1 and p ∈ [0, 1] and the system becomes unstable at e0 but under some conditions locally asymptotically stable at e1. the most important part of this study is to establish an optimal control for the pandemic model to minimize the asymptomatic and symptomatic populations and also to minimize the cost of treatment under impreciseness. we have considered two types of treatment as two controls to reduce the spread of the disease. the first treatment control u1, and the second treatment control u2 are designed in such a way that they minimize the objective functional as given in equation (4.10). the main mathematical findings for the dynamical behavior of the covid-19 pandemic are also numerically verified using matlab. a comparison between the dynamical behavior of every population with three different control options is presented graphically for different values of p in fig. 5.7 to fig. 5.10, which clearly shows the effectiveness of the treatment control. in addition, it is precisely shown that optimal control is much more effective in reducing the number of asymptomatic and symptomatic individuals, implying that treatment can successfully control the spread of the disease. furthermore, it was noticed that both treatments are essential at the beginning of the outbreak to prevent the spread of the disease. thus, our analytical and numerical studies show that the optimal control policy is a beneficial technique to reduce the spread of covid-19 infection. the proposed pandemic model is based on the effects of covid-19 infection in a population. in most of the covid-19 mathematical models, the model parameters are taken as constants. however, in reality, it is not true, as the parameters depend on the environmental conditions; therefore, they are not constant. to avoid these difficulties, in our present model we consider that the parameters of the models are imprecise and present them by interval number. then, using the parametric functional form of the interval number, we study the dynamical behavior of the imprecise model. it is also true that a time lag between susceptible individuals may be infected, hence, the present model can be extended by incorporating the time lag into the system, which is kept for future work consideration. controlling covid-19 spread is now a challenging and significant issue. therefore, predicting and identifying 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(1986). the mathematical theory of optimal processes. london, uk: gordon and breach science. copyright © 2023 assa. adv syst sci appl (2023) covid-19 spread modeling under uncertainty 89 appendix a: the proof of theorem-6 the variational matrix of system (2.5) at dfe e0 is given by me0 =  −µ1−p l µp u 0 −α1−p l αp uλ 1−p l λp u µ1−p l µu)p −β1−p l βp uλ 1−p l λp u µ1−p l µp u 0 0 −a α1−p l αp uλ 1−p l λp u µ1−p l µp u β1−p l βp uλ 1−p l λp u µ1−p l µp u 0 0 σρ1−p l ρpu −b 0 0 0 (1− σ)ρ1−p l ρpu (1− k)r1−p l rpu −c 0 0 0 kr1−p l rpu m1−p l mp u −µ1−p l µp u  therefore, eigenvalues of the characteristic equation of me0 are −µ1−p l µp u, −µ1−p l µp u and the solution of the cubic equation, p (λ) ≡ λ3 + a1λ 2 + a2λ+ a3 = 0 (a.1) where a1 = a+b + c, a2 = ab(1−r0) + ac(1−r0) +bc + λ1−p l λp u µ1−p l µp u[ ρ1−p l ρpuβ 1−p l βp u (1− σ)b + (1− k)σr1−p l rpu c + σρ1−p l ρpu α1−p l αp uc + (1− k)β1−p l βp ur 1−p l rpu b ] and a3 = abc(1−r0) now, it is easily noted that a1 > 0, a2 > 0, a3 > 0 if r0 < 1. after some simplifications, we get a1a2 −a3 = (a2b +ab2 +a2c +ac2 +abc)(1−r0) + (a+b + c)( bc + λ1−p l λp uρ 1−p l ρpu µ1−p l µp u [ β1−p l βp u (1− σ)b + σ(1− k)r1−p l rpu c + σ α1−p l αp uc + (1− k)β1−p l βp ur 1−p l rpu b ]) here, we can notice that, if r0 < 1 then a1a2 − a3 > 0. therefore, by the routh–hurwitz routh–hurwitz criterion [49] it follows that p (λ) = 0 has negative real roots if r0 < 1.this completes the proof. appendix b: the proof of theorem-8 the variational matrix of system (2.5) at e1(s ∗, e∗, i∗a, i ∗ s, r ∗) is given by, me1 =  b11 0 b13 b14 0 b21 b22 b23 b24 0 0 b32 b33 0 0 0 b42 b43 b44 0 0 0 b53 b54 b55  where, b11 = − ( α1−p l αp ui ∗ a + β1−p l βp ui ∗ s ) − µ1−p l µp u, b13 = −α1−p l αp us ∗, b14 = −β1−p l βp us ∗, b21 = ( α1−p l αp ui ∗ a + β1−p l βp ui ∗ s ) , b22 = −a, b23 = α1−p l αp us ∗, b24 = β1−p l βp us ∗, b32 = copyright © 2023 assa. adv syst sci appl (2023) 90 p.k. santra et al. σρ1−p l ρpu, b33 = −b, b42 = (1− σ)ρ1−p l ρpu, b43 = (1− k)r1−p l rpu, b44 = −c, b53 = kr1−p l rpu, b54 = m1−p l mp u, b55 = −µ1−p l µp u. therefore, the eigenvalues of the characteristic equation of me1 are −µ1−p l µp u and the solution of the equation, q (λ) ≡ λ4 +b1λ 3 +b2λ 2 +b3λ+b4 = 0 (b.1) where b1 = −(b11 + b22 + b33 + b44), b2 = b11b22 + b11b33 + b11b44 + b22b44 + b33b44 + b22b33 − b23b32 − b24b42, b3 = −b11b22b44 − b11b33b44 − b11b22b33 − b22b33b44 − b24b32b43 − b21b13b32 − b21b14b42 + b11b23b32 + b11b24b42 + b23b32b44 + b24b42b33, and b4 = b11b22b33b44 + b11b24b32b43 + b21b13b32b44 + b21b14b33b42 − b11b23b32b44 − b11b24b33b42 − b14b21b32b43. by the routh–hurwitz criterion [49] it follows that q (λ) = 0 has negative real roots if bi > 0 (i = 1, 2, 3, 4), d1 = b1 > 0, d2 = ∣∣∣∣b1 b3 1 b2 ∣∣∣∣ = b1b2 −b3 > 0, d3 =∣∣∣∣∣b1 b3 0 1 b2 b4 0 b1 b3 ∣∣∣∣∣ = b1b2b3 −b2 1b4 −b2 3 > 0. therefore, the system (2.5) shows local asymptotic stability at e1 when r0 > 1, bi > 0(i = 1, 2, 3, 4), b1b2 −b3 > 0 and b1b2b3 −b2 1b4 −b2 3 > 0. this completes the proof. appendix c: arithmetic operations on interval numbers using the concept of interval-valued functions [45] between a = [a, a] and b = [ b, b ] for p ∈ [0, 1] are as follows: addition: a+b = [a, a] + [ b, b ] = [ a+ b, a+ b ] , then the interval-valued function for a+b is h (p) = c (1−p) l cpu where cl = a+ b > 0 and cu = a+ b. subtraction: a−b = [a, a]− [ b, b ] = [ a− b, a− b ] , then the interval-valued function for a−b is h (p) = c (1−p) l cpu where cl = a− b > 0 and cu = a− b. scalar multiplication: εa = ε [a, a] = { [εa, εa] if ε ≥ 0 [αa, εa] if ε < 0 , provided a > 0 , then the interval-valued function of εa is h (p) = c (1−p) l cpu if ε ≥ 0 and h (p) = −d (1−p) l dpu if ε < 0, where c l = εa, c u = εa, dl = |ε|a and du = |ε|a. copyright © 2023 assa. adv syst sci appl (2023) introduction materials & methods variables and parameters involved in the model some useful definitions covid-19 pandemic model formulation crisp model imprecise model with interval coefficient theoretical study of the model positivity of the solutions invariant region existence of equilibria and basic reproduction number stability behavior optimal control for covid-19 model existence of an optimal control characterization of the optimal control numerical simulations conclusions adv syst sci appl 2023; 03:36–47 published online at https://ijassa.ipu.ru. feedback design in linear control problems as an optimization problem mikhail khlebnikov* v. a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia abstract: we provide and discuss a new approach to the design of linear control systems based on the optimization viewpoint. three basic classes of control problems are analyzed: a) static state and output feedback for linear quadratic regulator problem; b) rejection of nonrandom bounded exogenous disturbances via static linear feedback; c) the same rejection via dynamic output feedback using an observer. these three problems are considered as optimization ones with feedback gains as matrix variables. the iterative algorithms for its solution are formulated in a uniform way, and the explicit expressions for gradients of the cost functions are provided. the gradient method exhibits its efficiency for test examples, including double pendulum. keywords: linear systems, output feedback, optimization, gradient method, lyapunov equation, linear quadratic regulator problem, external disturbances 1. introduction recently, the approach to linear control systems from the point of view of optimization has become very popular. so, in the classical problem of the linear quadratic controller, one can consider the linear feedback matrix as a variable and reduce the problem to minimizing the performance index with respect to this variable; this approach goes back to the works of r. kalman in the middle of the last century. one of the main objectives of this paper is to discuss the possibility of a unified and systematic approach to a number of problems that arise in linear control theory as matrix optimization problems and trace it on the example of three classical control problems. in addition to the linear quadratic problem, from the same positions, the paper considers the problem of rejecting nonrandom bounded external disturbances by constructing a static linear output feedback, as well as using dynamic output feedback with observer. for each of these problems, we state the corresponding problems of nonconvex matrix optimization, and the iterative algorithms for its solution are formulated in a uniform way. herewith, the relations for calculating the gradients of the corresponding cost functions are written out in closed form. from now on, | · | is the euclidean norm of a vector, ∥ · ∥ is the spectral norm of a matrix, ∥ · ∥f is the frobenius norm of a matrix, and ⟨·, ·⟩ is the frobenius inner product of matrices. 2. linear quadratic regulator recall the new approaches to the classical problem of linear quadratic control. it can be considered as an optimization problem, where the variable is the feedback matrix, and the ∗corresponding author: khlebnik@ipu.ru feedback design in linear control problems as an optimization problem 37 integral quadratic performance index of the transient process is minimized. the gradient of such function (for the state feedback) was written out in the seminal work [5] by kalman, and for output feedback in the paper [6] by levin and athans. since then, the iterative gradienttype optimization methods have been used repeatedly (see [7]), but the justification for such methods has appeared only recently in [2–4, 8, 12]. namely, consider the linear stationary continuous-time control system: ẋ = ax+bu, y = cx, with state x(t) ∈ rn, output y(t) ∈ rl, and control input u(t) ∈ rp. the infinite-horizon linear quadratic performance criterion is given by f(k) = e ∞∫ 0 ( xt(t)qx(t) + ut(t)ru(t) ) dt, where the expectation is taken over the distribution of an initial condition x(0) with zero mean and covariance matrix σ, and the quadratic cost is parameterized by positive-definite matrices q, r. design of the appropriate static feedback u(t) = −ky(t) with constant gain matrix k ∈ rl×p can be reformulated as the optimization problem f(k) = trxς subject to the constraint (a−bkc)tx +x(a−bkc) + ctktrkc +q = 0 for the matrix variables k ∈ rl×p and x = xt ∈ rn×n. the properties of the corresponding function were studied in [3]: it turns out to be smooth but non-convex, see fig. 2.1. moreover, it is defined on a possibly disconnected and nonconvex domain, see fig. 2.2. fig. 2.1 fig. 2.2 nevertheless, it was possible to construct a gradient method with a special choice of step length, which converges to the optimal solution in the case of state feedback and to a stationary point in the case of output feedback. now we formulate the following algorithm to minimize f(k). copyright © 2023 assa. adv syst sci appl (2023) 38 m. khlebnikov algorithm 1. 1. set the parameters ε > 0, t1 > 0, 0 < τ < 1, and the initial stabilizing approximation k0. 2. at the jth iteration, the value kj is given. compute aj = a−bkjc, solve the lyapunov equations at j x +xaj +q+kt j rkj = 0 and ajy + y at j + σ = 0 and find the matrices x and y respectively; compute the gradient hj = ∇f(kj) from the relation 1 2 ∇f(k) = (rk −btx)y. if ∥hj∥ ≤ ε, then kj is taken as an approximate solution. 3. solve the lyapunov equation at j x ′ +x ′aj +mt∇f(kj) + ( ∇f(kj) )t m = 0 and find its solution x ′. compute the value ∇2f(kj)[hj, hj] = 2 〈 r∇f(kj)y,∇f(kj) 〉 − 4 〈 btx ′y,∇f(kj) 〉 and set the trial step as γ = min { t1, ∥∇f(kj)∥2f ∇2f(kj)[hj, hj] } . 4. make one step of the gradient method kj+1 = kj − γjhj. the step length γj > 0 is selected by splitting γ until the following conditions are met: a. kj+1 is a stabilizing controller; b. f(kj+1) ≤ f(kj)− τγj∥hj∥2. go to step 2. 3. suppression of external disturbances: static feedback in [9], a similar approach is applied for the first time to the systems subjected to external disturbances. the problem of suppressing bounded external disturbances (peak-to-peak gain minimization) is formulated as follows. consider the linear time-invariant control system ẋ = ax+bu+dw, x(0) = x0, y = c1x, z = c2x+b1u, (3.1) with state x(t) ∈ rn, measured output y(t) ∈ rl, controlled output z(t) ∈ rr, control u(t) ∈ rp, and t-measurable exogenous disturbance w(t) ∈ rm bounded at every moment of time: |w(t)| ≤ 1 for all t ≥ 0. copyright © 2023 assa. adv syst sci appl (2023) feedback design in linear control problems as an optimization problem 39 the problem is to design a stabilizing control in the form of state feedback (if available) u = kx or output feedback u = ky to reduce the “peak” output z(t), i.e. the value maxt |z(t)|. the exact solution of such a problem is difficult, but it is possible to minimize the upper bound of this quantity using the concept of invariant ellipsoid. we recall the following definition 3.1: ellipsoid e = { x ∈ rn : xtp−1x ≤ 1 } , p ≻ 0, (3.2) centered at the origin is called invariant for the dynamical system if the condition x(0) ∈ e implies x(t) ∈ e for all times t ≥ 0 and all admissible external disturbances. in other words, any system trajectory starting from a point lying in the invariant ellipsoid, with all admissible external disturbances acting on the system, will be in this ellipsoid at any time. it is easy to see that if e is an invariant ellipsoid with the certain matrix p , then the linear output z = cx of the system for x0 ∈ e belongs to the so-called bounding ellipsoid ez = {z ∈ rn : zt(cpct)−1z ≤ 1}. this approach based on the technique of invariant ellipsoids was first used in the monograph [1]; a detailed exposition of this technique can be found in the book [11]. in this case, the state feedback design is reduced to a parametric problem of semidefinite programming with the help of special changes of variables; there exist convenient numerical methods for solving such problems. however, for the output feedback design, such reduction is fundamentally impossible. the initial problem of the output feedback design for system (3.1) rejecting the external disturbances can be reduced to the following matrix optimization problem: min f(k,α), f(k,α) = trc2pct 2 + ρ∥k∥2f , (3.3) subject to the constraint( a+bkc1 + α 2 i ) p + p ( a+bkc1 + α 2 i )t + 1 α ddt = 0 with respect to the matrix variables p = pt ∈ rn×n, k ∈ rp×n, and the scalar parameter α > 0. the first component in (3.3) defines the size of the bounding ellipsoid with respect to the trace criterion, and the second one represents the penalty for using large control values (wherein, the coefficient ρ > 0 regulates its importance). the presence of the second component avoids the appearance of large values of the gain matrix. within the framework of the proposed optimization approach, a gradient method for finding static output feedback is written out and its justification is given, see [9]. we formulate the following algorithm to minimize f(k,α). algorithm 2. 1. set the parameters ε > 0, γ > 0, 0 < τ < 1 and the initial stabilizing approximation k0. compute the value α0 = σ(a+bk0c1). 2. at the jth iteration, the values kj and αj are given. compute aj = a+bkjc1; solve the lyapunov equations( aj + αj 2 i ) p + p ( aj + αj 2 i )t + 1 αj ddt = 0 copyright © 2023 assa. adv syst sci appl (2023) 40 m. khlebnikov and ( aj + αj 2 i )t y + y ( aj + αj 2 i ) + ct 2 c2 = 0, and find the matrices p and y respectively. compute the gradient hj = ∇kf(kj, αj) from the relation 1 2 ∇kf(k,α) = bty pct 1 + ρk. if ∥hj∥ ≤ ε, then kj is taken as an approximate solution. 3. make one step of the gradient method kj+1 = kj − γjhj. the step length γj > 0 is selected by splitting γ until the following conditions are met: a. kj+1 is a stabilizing controller; b. f(kj+1) ≤ f(kj)− τγj∥hj∥2. 4. for the resulting kj+1 compute aj+1 = a+bkj+1c1 and the matrices p , y and x as the solution of the lyapunov equations( aj+1 + αj 2 i ) p + p ( aj+1 + αj 2 i )t + 1 αj ddt = 0, ( aj+1 + αj 2 i )t y + y ( aj+1 + αj 2 i ) + ct 2 c2 = 0, and ( aj+1 + αj 2 i ) x +x ( aj+1 + αj 2 i )t + p − 1 α2 j ddt = 0 respectively. 5. solve the problem of minimizing f(kj+1, α) with respect to α via the newton method: αj+1 = αj − ∇αf(kj+1, αj) ∇2 ααf(kj+1, αj) where ∇αf(k,α) = try ( p − 1 α2 ddt ) , ∇2 ααf(k,α) = 2 try ( x + 1 α3 ddt ) , and obtain αj+1. go to step 2. the method converges in the following sense. theorem 3.1: in algorithm 2, only a finite number of subdivisions γj is realized at each iteration, the function f(kj) monotonically decreases, and the gradient tends to zero lim j→∞ ∥hj∥ = 0 at the rate of geometric progression. copyright © 2023 assa. adv syst sci appl (2023) feedback design in linear control problems as an optimization problem 41 4. output dynamic feedback consider a linear time-invariant control system in continuous time: ẋ = ax+bu+dw, x(0) = x0, y = c1x+d1w, z = c2x, (4.4) with state x(t) ∈ rn, observed output y(t) ∈ rl, optimized output z(t) ∈ rr, control u(t) ∈ rp and bounded at every moment of time exogenous disturbance w(t) ∈ rm: |w(t)| ≤ 1 for all t ≥ 0. assume that the measurements of the state x are not available, and the information about the system is provided by its output y only. the goal is to find the minimal (in a certain sense) ellipsoid containing the optimized output z. within the framework of the proposed approach, it was possible to write down a gradient method for constructing the feedback using the dynamic controller u = kx̂, where x̂ is an observer described by the linear differential equation ˙̂x = ax̂+bu+ l(y − c1x̂), x̂(0) = 0, including the discrepancy between the output y and its forecast c1x̂ (here l ∈ rn×l is the observer’s matrix) and provide its justification. namely, for this setup the initial problem is reduced to the minimization problem f(k,l, α) = tr (c2 0)p (c2 0)t + ρk∥k∥2f + ρl∥l∥2f (4.5) subject to the constraint( a0 +m1kn1 +m2ln2 + α 2 i ) p + p ( a0 +m1kn1 +m2ln2 + α 2 i )t + 1 α ( d d − ld1 )( d d − ld1 )t = 0, where a0 = ( a 0 0 a ) , m1 = ( b 0 ) , n1 = (i −i) , m2 = ( 0 i ) , n2 = (0 −c1) , with respect to the matrix variables p = pt ∈ r2n×2n, k ∈ rp×n, l ∈ rn×l, and the scalar parameter α > 0. in (4.5), in addition to the component that determines the size of the bounding ellipsoid with respect to the trace criterion, the penalties ρk and ρl are introduced for the size of the controller and observer matrices. we formulate the following algorithm to minimize f(k,l, α), see the details in [10]. copyright © 2023 assa. adv syst sci appl (2023) 42 m. khlebnikov algorithm 3. 1. set the parameters ε > 0, γk , γl > 0, 0 < τk , τl < 1, and the initial stabilizing approximation (k0, l0). compute the value α0 = σ(a0 +m1k0n1 +m2l0n2). 2. at the jth iteration, the values kj , lj , and αj are given. compute aj = a0 + m1kjn1 +m2ljn2 and find the matrices p and y as the solutions of the lyapunov equations( aj + αj 2 i ) p + p ( aj + αj 2 i )t + 1 αj ( d d − ljd1 )( d d − ljd1 )t = 0 and ( aj + αj 2 i )t y + y ( aj + αj 2 i ) + (c2 0)t (c2 0) = 0 respectively. 3. compute the gradient hk j = ∇kf(kj, lj, αj) from the relation 1 2 ∇kf(k,l, α) = mt 1 y pnt 1 + ρkk. if ∥hk j ∥ ≤ ε, then kj is taken as an approximate solution. 4. make one step of the gradient method for k: kj+1 = kj − γk j hk j . the step length γk j > 0 is selected by splitting γk until the following conditions hold: a. kj+1 stabilizes the matrix a0 +m1kn1 +m2ljn2; b. f(kj+1) ≤ f(kj)− τkγ k j ∥hk j ∥2. 5. having kj+1, compute the gradient hl j = ∇lf(kj+1, lj, αj) from the relation 1 2 ∇lf(k,l, α) = mt 2 y pnt 2 − 1 α (0 i)y ( d d − ld1 ) dt 1 + ρll. if ∥hl j ∥ ≤ ε, then lj is taken as an approximate solution. 6. make one step of the gradient method for l: lj+1 = lj − γl j h l j , the step lengths γl j > 0 is selected by splitting γl until the following conditions hold: a. lj+1 stabilizes the matrix a0 +m1kj+1n1 +m2ln2; b. f(lj+1) ≤ f(lj)− τlγ l j ∥hl j ∥2. 7. for the resulting kj+1, lj+1 compute aj+1 = a0 +m1kj+1n1 +m2lj+1n2 and the matrices p , y and x as the solutions of the lyapunov equations( aj+1 + αj 2 i ) p + p ( aj+1 + αj 2 i )t + 1 αj ( d d − lj+1d1 )( d d − lj+1d1 )t = 0, copyright © 2023 assa. adv syst sci appl (2023) feedback design in linear control problems as an optimization problem 43 ( aj+1 + αj 2 i )t y + y ( aj+1 + αj 2 i ) + (c2 0)t (c2 0) = 0, and( aj+1 + αj 2 i ) x +x ( aj+1 + αj 2 i )t + p − 1 α2 j ( d d − lj+1d1 )( d d − lj+1d1 )t = 0 respectively. 8. solve the problem of minimizing f(kj+1, lj+1, α) with respect to α via the newton method: αj+1 = αj − ∇αf(kj+1, lj+1, αj) ∇2 ααf(kj+1, lj+1, αj) , where ∇αf(k,l, α) = try [ p − 1 α2 ( d d − ld1 )( d d − ld1 )t] , ∇2 ααf(k,l, α) = 2 try [ x + 1 α3 ( d d − ld1 )( d d − ld1 )t] , and get αj+1. go to step 2. 5. examples example 1. consider a double mathematical pendulum consisting of two weightless rods of length l1 and l2, at the ends of which weights of masses m1 and m2 are fixed. the system moves in a viscous medium with a drag coefficient γ, in a vertical plane xy, and the position of the pendulum is determined by the angles φ1 and φ2 of the deviation of the rods from the vertical, see fig. 5.3. @ @@•a a aa• �u �w m1 ϕ1 m2 ϕ2 -x ?y l1 l2 fig. 5.3 the lower body is affected by a bounded external disturbance |w| ≤ 1, to compensate for which the control action u is applied to the upper body. copyright © 2023 assa. adv syst sci appl (2023) 44 m. khlebnikov introducing auxiliary variables φ3 = φ̇1, φ4 = φ̇2, we arrive at a linearized system φ̇1 = φ3, φ̇2 = φ4, φ̇3 = − ( 1 + m2 m1 ) g l1 φ1 + m2 m1 g l1 φ2 − γ m1 φ3 + 1 m1 u, φ̇4 = ( 1 + m2 m1 ) g l2 φ1 − ( 1 + m2 m1 ) g l2 φ2 − γ m2 φ4 + 1 m2 w. at m1 = m2 = 1, l1 = l2 = g, γ = 0.15, the matrices of system (3.1) have the form a =  0 0 1 0 0 0 0 1 −2 1 −0.15 0 2 −2 0 −0.15  , b = 0 0 1 0  , d = 0 0 0 1  . as the observed output, we choose y = ( φ1 φ2 ) and as a regulated output we take the vector z = ( φ̇1 φ̇2 ) , that is c1 = ( 1 0 0 0 0 1 0 0 ) , c2 = ( 0 0 1 0 0 0 0 1 ) . let us also ρ = 0.1. since the open-loop system is stable, we choose k0 = (0 0) as an initial approximation for the controller. calculations carried out in accordance with algorithm 2 led to the following results. the dynamics of the criterion f(k) = trc2pct 2 + ρ∥k∥2 is shown in fig. 5.4. the process ended at the 9th step by finding the stabilizing static output feedback k∗ = (0.0050 −0.9024) ; wherein f(k∗) = 48.9381. copyright © 2023 assa. adv syst sci appl (2023) feedback design in linear control problems as an optimization problem 45 1 2 3 4 5 6 7 8 9 48 50 52 54 56 58 60 62 64 66 68 fig. 5.4 at the same time, we obtain the matrix( 2.0663 −0.0516 −0.0516 46.7904 ) of the corresponding bounding ellipse for the regulated output z. example 2. consider the system from example 1 again and construct the dynamic feedback for its observed output. since the matrix a of the system is stable, it would seem natural to choose as initial approximations zero, but the point k = l = 0 is a saddle point for the minimized function. therefore, as an initial approximation for the controller, we take the matrix k0 = (0 0 0 0) , and as an initial approximation for the observer, we generate a certain admissible matrix l0 =  0.0826 −0.0346 0.7379 0.6160 0.1141 0.4720 −0.9572 0.1446  . let ρk = ρl = 0.1. according to algorithm 3, the optimization procedure led to the controller matrix k∗ = (0.8219 −0.0402 −1.3024 0.4692) , the observer matrix l∗ =  0.7944 −0.0845 0.6222 0.9323 0.8885 −0.0391 −0.4905 0.8054  , copyright © 2023 assa. adv syst sci appl (2023) 46 m. khlebnikov and the corresponding bounding ellipse for the regulated output z with the matrix( 1.2117 0.1784 0.1784 2.5120 ) . 0 10 20 30 40 50 60 0 5 10 15 20 25 30 35 40 fig. 5.5 the dynamics of the criterion f(k,l) = tr (c2 0)p (c2 0)t + ρk∥k∥2f + ρl∥l∥2f is shown in fig. 5.5. 6. conclusion we consider and discuss a new approach to control design in linear systems. it is based on reducing the original problem to an optimization problem in the feedback matrix variable (by state or by output). further, the corresponding problems are solved by the gradient method; its convergence is theoretically substantiated for a number of important cases. numerous examples demonstrate the effectiveness of this approach. in future research, the author plans to extend the proposed approach to the pi/pid controller design and filtering problems. references 1. boyd, s., el ghaoui, l., feron, e. & balakrishnan, v. (1994) linear matrix inequalities in systems and control theory. philadelphia, pa: siam studies in applied mathematics. 2. bu, j., mesbahi, a., fazel, m. & mesbahi, m. (2019) lqr through the lens of first order methods: discrete-time case, arxiv: 1907.08921, [online]. available: https://arxiv.org/abs/1907.08921 copyright © 2023 assa. adv syst sci appl (2023) feedback design in linear control problems as an optimization problem 47 3. fatkhullin, i. & polyak, b. (2021) optimizing static linear feedback: gradient method, siam journal on control and optimization, 59(5), 3887–3911. 4. fazel, m., ge, r., kakade, s. & mesbahi, m. (2018) global convergence of policy gradient methods for the linear quadratic regulator. proc. 35th international conference on machine learning, stockholm, sweden, 1467–1476. 5. kalman, r. e. (1960) contributions to the theory of optimal control, boletin de la sociedad matematica mexicana, 5(1), 102–119. 6. levine, w. & athans, m. (1970) on the determination of the optimal constant output feedback gains for linear multivariable systems, ieee transactions on automatic control, 15(1), 44–48. 7. mäkilä, p. m. & toivonen, h. t. (1987) computational methods for parametric lq problems – a survey, ieee transactions on automatic control, 32(8), 658–671. 8. mohammadi, h., zare, a., soltanolkotabi, m. & jovanović, m. r. (2019) global exponential convergence of gradient methods over the nonconvex landscape of the linear quadratic regulator, proc. 2019 ieee 58th conference on decision and control, nice, france, 7474–7479. 9. polyak, b. t. & khlebnikov, m. v. (2021) static controller synthesis for peak-to-peak gain minimization as an optimization problem, automation and remote control, 82(9), 1530–1553. 10. polyak, b. t. & khlebnikov, m. v. (2022) observer-aided output feedback synthesis as an optimization problem, automation and remote control, 83(3), 303–324. 11. polyak, b. t., khlebnikov, m. v. & shcherbakov, p. s. (2014) upravlenie lineinymi sistemami pri vneshnikh vozmushcheniyakh: tekhnika lineinykh matrichnykh neravenstv [control of linear systems under exogenous disturbances: technique of linear matrix inequalities]. moscow, russia: lenand [in russian]. 12. zhang, k., hu, b. & başar t. (2020) policy optimization for h2 linear control with h∞ robustness guarantee: implicit regularization and global convergence, arxiv: 1910.09496, [online]. available: https://arxiv.org/abs/1910.09496 copyright © 2023 assa. adv syst sci appl (2023) introduction linear quadratic regulator suppression of external disturbances: static feedback output dynamic feedback examples conclusion adv syst sci appl 2024; 01:40–57 published online at https://ijassa.ipu.ru. a cauchy type problem for causal functional inclusions in banach spaces garik petrosyan*, maria soroka voronezh state university, voronezh, russia abstract: in this paper, we study a cauchy type problem in banach spaces for various classes of functional inclusions with causal multioperators. based on the topological degree theory for condensing multimaps, we prove a global theorem on the existence of trajectories for systems governed by functional inclusions. as an application, we obtain generalizations of existence theorems for a cauchy type problem for semilinear second order differential inclusions of this type and semilinear differential inclusions of the fractional order 1 < q < 2. keywords: causal multioperator, functional inclusion, cauchy type problem, differential inclusion, fractional derivative, measure of non-compactness, fixed point, topological degree, condensing multioperator 1. introduction recently, the attention of many researchers (see [2], [19], [20], [23] and references therein) has been attracted to generalizations of differential equations and inclusions, namely to the class of functional equations and inclusions with causal operators. the term of a causal or volterra operator in the sense of a.n. tikhonov (see [29]), was used in mathematical physics to solve problems of differential equations, integro-differential equations, functionaldifferential equations with finite or infinite delay, integral equations of volterra type, functional equations of neutral type, etc. (see, for example, [6]). the papers [5], [7], [8], [13] among others are devoted to the study of equations and inclusions with causal multioperators of various types, theorems on the existence of solutions, description of qualitative properties of solutions and various applications. in recent decades, the interest to the theory of fractional-order differential equations has significantly increased, thanks to applications in various branches of applied mathematics, physics, engineering, biology, economics, etc. (see, for example, monographs [17], [27] papers [1], [4], [9], [21], etc.). the boundary value problems of various types for fractional differential equations and inclusions were considered in the works [10], [15], [16], [24], [25]. in this paper, we study a cauchy type problem in banach spaces for various classes of functional inclusions with causal multioperators. based on the topological degree theory for condensing multimaps, we prove a global theorem on the existence of trajectories for systems governed by functional inclusions. as an application, we obtain generalizations of existence theorems for solutions of a cauchy type problem for second order semilinear differential inclusions and semilinear differential inclusions of the fractional order 1 < q < 2. in this paper we use standard notation, the symbol “⊸” denotes a multimap. ∗corresponding author: garikpetrosyan@yandex.ru a cauchy type problem... 41 2. preliminaries 2.1. measures of noncompactness we denote by e a banach space and introduce the following notation: • p (e) = {a ⊆ e : a ̸= ∅} is the collection of all non-empty subsets of e ; • pb(e) = {a ∈ p (e) : a is bounded} ; • pv(e) = {a ∈ p (e) : a is convex} ; • c(e) = {a ∈ p (e) : a is closed} ; • cv(e) = pv(e) ∩ c(e); • k(e) = {a ∈ p (e) : a is compact} ; • kv(e) = pv(e) ∩k(e). definition 2.1: (see [3]). let (a,≥) be a partially ordered set. a function β : pb(e) → a is called the measure of noncompactness (mnc) in e if for each ω ∈ pb(e) we have: β(coω) = β(ω), where coω denotes the closure of the convex hull of ω. a measure of noncompactness β is called: 1) monotone, if for each ω0,ω1 ∈ pb(e), from ω0 ⊆ ω1 it follows β(ω0) ≤ β(ω1); 2) nonsingular, if for each a ∈ e and each ω ∈ pb(e) we have β({a} ∪ ω) = β(ω). if a is a cone in e , the mnc β is called: 3) regular, if β(ω) = 0 is equivalent to the relative compactness of ω ∈ pb(e); 4) real, if a is the set of all real numbers r with the natural ordering. as the example of a real mnc satisfying all above properties, we can consider the hausdorff mnc χ(ω): χ(ω) = inf{ε > 0, for which ω has a finite ε-net in e }. as other examples, consider the measures of noncompactness defined in the space of continuous functions c([a, b]; e) with values in the banach space e : (1) the modulus of fiber noncompactness: φ(ω) = sup t∈[a,b] χe(ω(t)), where χe is the hausdorff mnc in e and ω(t) = {y(t) : y ∈ ω}; (2) the fading modulus of fiber noncompactness: γ(ω) = sup t∈[a,b] e−ltχe(ω(t)), where l > 0 is a given number; (3) the modulus of equicontinuity: modc (ω) = lim δ→0 sup y∈ω max |t1−t2|≤δ ∥y (t1)− y (t2)∥ . these measures of noncompactness satisfy all the above properties, except for the regularity. copyright © 2024 assa. adv syst sci appl (2024) 42 g. petrosyan, m. soroka 2.2. multivalued maps let x be a metric space and y be a normed space. let us recall some notations (see, for example, [14], [22]). definition 2.2: a multivalued map (multimap) f : x → p (y ) is said to be upper semicontinuous (u.s.c.) at a point x ∈ x, if for every open set v ⊂ y such that f(x) ⊂ v, there exists a neighborhood u(x) of x such that f(u(x)) ⊂ v. definition 2.3: a multivalued map f : x → p (y ) is called closed if its graph gf = {(x, y) : x ∈ x, y ∈ f(x)} is a closed subset of x × y. definition 2.4: a multivalued map f : x → p (y ) is called quasicompact if its restriction to each compact subset a ⊂ x is compact. definition 2.5: for a given p ≥ 1, a multifunction g : [0, t ] → k(y ) is called: • lp–integrable if it admits an lp–bochner integrable selection, i.e., there exists a function g ∈ lp ([0, t ];y ) such that g(t) ∈ g(t) for a.e. t ∈ [0, t ]; • lp–integrably bounded if there exists a function ξ ∈ lp([0, t ]) such that ∥g(t)∥y ≤ ξ(t) for a.e. t ∈ [0, t ]. definition 2.6: a multimap f : x ⊆ e → k(e) is called condensing with respect to a mnc β (or β– condensing) if for each bounded set ω ⊆ x which is not relatively compact, we have: β(f (ω)) ̸≥ β(ω). let d ⊂ e be a non-empty closed convex subset, v be a non-empty bounded open subset of d, β is a monotone nonsingular mnc in e and f : v → kv (d) be a u.s.c. β-condensing map such that x /∈ f (x) for all x ∈ ∂v , where v and ∂v denote the closure and the boundary of the set v in the relative topology of d. in such a setting, the (relative) topological degree degd ( i−f , v ) of the corresponding vector field i−f , satisfying the standard properties is defined (see, for example, [14], [22]). in particular, the condition degd ( i−f , v ) ̸= 0 implies that the fixed points set fixf = {x : x ∈ f(x)} is a nonempty subset of v. application of topological degree theory leads to the following fixed point principles, which will be used in the what follows. theorem 2.1: ( [14], corollary 3.3.1). let m be a convex closed bounded subset of e and f : m → kv(m) be a β–condensing multimap, where β is a monotone nonsingular mnc in e . then the fixed point set fixf is non-empty. copyright © 2024 assa. adv syst sci appl (2024) a cauchy type problem... 43 theorem 2.2: ( [14], theorem 3.3.4). let v ⊂ d be a bounded open neighborhood of a point a ∈ v and f : v → kv(d) be a u.s.c. β-condensing multimap, where β is a monotone nonsingular mnc in e , satisfying the boundary condition x− a /∈ λ(f(x)− a) for all x ∈ ∂v and 0 < λ ≤ 1. then fixf ̸= ∅ is a non-empty compact set. 2.3. family of cosine operator functions definition 2.7: (see [12], [18], [28]) a family of bounded operators {c(t)}t∈r in a banach space e is called a strongly continuous family of cosine operator functions if: (1) c(0) = i; (2) c(s+ t) + c(s− t) = 2c(s)c(t) for all t, s ∈ r; (3) t→ c(t)x is continuous for all x ∈ e . the family of strongly continuous sine operator functions associated with the family of cosine operator functions {c(t)}t∈r is the family of operators {s(t)}t∈r such that s(t)x = ∫ t 0 c(s)xds, x ∈ e , t ∈ r. the operator a generates a family of cosine operator functions {c(t)}t∈r if: ax = d2 dt2 c(t)x ∣∣∣ t=0 , for all x ∈ d(a) for which the last expression is defined. 2.4. causal multioperators let e be a separable banach space. by lp ([0, t ];e) , 1 ≤ p ≤ ∞, we denote the banach space of all bochner summable functions f : [0, t ] → e with the usual norm. for each subset n ⊂ lp ([0, t ];e) and τ ∈ (0, t ) we define restriction n on [0, τ ] as n |[0,τ ]= {f |[0,τ ]: f ∈ n}. definition 2.8: a multivalued map q : c ([0, t ];e) ⊸ lp ([0, t ];e) is said to be a causal multioperator, if for each τ ∈ (0, t ) and for every u, v ∈ c ([0, t ];e) the condition u |[0,τ ]= v |[0,τ ] implies that q(u) |[0,τ ]= q(v) |[0,τ ] . let us give some examples of causal multioperators. example 2.1: we assume that the multimap f : [0, t ]× e → kv (e) satisfies the following conditions: (f1) for each ϕ ∈ e the multifunction f (·, ϕ) : [0, t ] → kv (e) admits a measurable selection; (f2) for a.e. t ∈ [0, t ] the multifunction f (t, ·) : e → kv (e) is u.s.c.; (f3) there exists a function α ∈ lp +[0, t ], 1 ≤ p ≤ ∞, such that ∥f (t, ϕ)∥e := sup{∥z∥e : z ∈ f (t, ϕ)} ≤ α(t)(1 + ∥ϕ∥e) for a.e. t ∈ [0, t ] and ϕ ∈ e. copyright © 2024 assa. adv syst sci appl (2024) 44 g. petrosyan, m. soroka from the above conditions (f1)− (f3) it follows that the multimap pf : c([0;t ];e) → p (lp([0, t ];e)), given in the following way pf (x) = {f ∈ lp([0, t ];e) : f(t) ∈ f (t, x(t)) for a.e. t ∈ [0, t ]} is well defined (see, for example, [14], [22]). it is clear that the multioperator pf is causal. example 2.2: let f : [0, t ]× e → kv(e) be a multimap satisfying conditions (f1)− (f3) from example 2.1. suppose that {k(t, s) : 0 ≤ s ≤ t ≤ t} is a continuous (with respect to the corresponding norm) family of bounded linear operators in e and m ∈ l1([0, t ];e) is a given function. consider the volterra integral multioperator v : c ([0, t ];e) ⊸ l1 ([0, t ];e) defined as v(u)(t) = m(t) + ∫ t 0 k(t, s)f (s, u)ds, i.e., v(u) = {y ∈ l1 ([0, t ];e) : y(t) = m(t) + ∫ t 0 k(t, s)f(s)ds : f ∈ pf (u)}. it is also clear that the multioperator v is causal. 3. cauchy type problem for functional inclusions with the causal multioperators we will assume that the causal operator q : c ([0, t ];e) → c (lp ([0, t ];e)) satisfies the following conditions: (q1) q is weakly closed in the following sense: conditions {un}∞n=1 ⊂ c ([0, t ];e) , {fn}∞n=1 ⊂ lp ([0, t ];e) , 1 ≤ p ≤ ∞, fn ∈ q(un), n ≥ 1, un → u0, fn l1 ⇀ f0 imply f0 ∈ q(u0); (q2) there exists a function α ∈ l∞ + ([0, t ]) such that ∥q (u) (t) ∥e ≤ α (t) (1 + ∥u(t)∥e) , for a.e. t ∈ [0, t ], for all u ∈ c([0, t ];e); (q3) there exists a function ω : [0, t ]× r+ → r+ such that (ω1) for all x ∈ r+ : ω(·, x) ∈ lp +([0, t ]), 1 ≤ p ≤ ∞, ; (ω2) for a.e. t ∈ [0, t ] a function ω(t, ·) : r+ → r+ is continuous, nondecreasing and quasihomogeneous in the sense that ω(t, λx) ≤ λω(t, x) for all x ∈ r+ and λ ≥ 0; (ω3) for each bounded set ∆ ⊂ c ([0, t ];e) we have χ (q (∆) (t)) ≤ ω (t, φ (∆(s))) for a.e. t ∈ [0, t ], where the set ∆(s) = {y(s) : y ∈ ∆} ⊂ e and φ is the modulus of fiber noncompactness in c ([0, t ];e) . note that the condition (ω2) means that ω(t, 0) = 0 for a.e. t ∈ [0, t ] and as an example of such a function we can consider ω(t, x) = k(t) · x, where k(·) ∈ lp +([0, t ]). consider a linear operator s : lp([0, t ];e) → c([0, t ];e), which is causal in the sense that for every τ ∈ (0, t ] and f, g ∈ lp([0, t ];e) condition f(t) = g(t) for a.e. t ∈ [0, τ ] implies (sf) (t) = (sg) (t) for all t ∈ [0, τ ]. following [14], we impose the next conditions on operator s : copyright © 2024 assa. adv syst sci appl (2024) a cauchy type problem... 45 (s1) for 1 ≤ p <∞ there exists d ≥ 0 such that ∥sf(t)− sg(t)∥pe ≤ d ∫ t 0 ∥f(s)− g(s)∥peds for all f, g ∈ lp([0, t ];e) and 0 ≤ t ≤ t ; if p = ∞ then there exists d1 ≥ 0 such that ∥sf(t)− sg(t)∥e ≤ d1 ∫ t 0 ∥f(s)− g(s)∥eds for all f, g ∈ l∞([0, t ];e) and 0 ≤ t ≤ t. (s2) for an arbitrary compact set k ⊂ e and a sequence {fn}∞n=1 ⊂ lp ([0, t ];e) , 1 ≤ p ≤ ∞, such that {fn(t)}∞n=1 ⊂ k for all t ∈ [0, t ] the weak convergence fn l1 ⇀ f0 implies sfn → sf0 in c([0, t ];e). also we suppose that s satisfies the relation: (s3) (sf) (0) = 0 for each function f ∈ lp([0, t ];e). notice, that condition (s1) implies that the operator s satisfies the lipschitz condition: (s1′) ∥sf − sg∥c ≤ d∥f − g∥l1 . consider the following important examples. (i) let a closed linear operator a : d (a) ⊂ e → e generates a family of strongly continuous cosine operator functions {cos(at)}t≥0. the operator l : l1([0, t ];e) → c([0, t ];e) defined as lf(t) = ∫ t 0 sin(at) f(s)ds is a special case of the operator s. note that taking a = 0 we obtain, as a special case, the usual integral operator li : l1([0, t ];e) → c([0, t ];e), lif(t) = ∫ t 0 f(s)ds. it is easy to verify the following statement. lemma 3.1: the operator l satisfies conditions (s1)− (s3). (ii) consider the operator functions g(t) = ∫ ∞ 0 ξq(θ)c(t qθ)dθ, k(t) = ∫ t 0 g(s)ds, t (t) = q ∫ ∞ 0 θξq(θ)s(t qθ)dθ, where ξq(θ) = θ−1− 1 q q ψq(θ −1/q), ψq(θ) = ∞∑ n=1 (−1)n−1θ−qn−1γ(nq + 1) πn! sin(nπq), θ ∈ r+. lemma 3.2: (see [30]). the operator functions g,k and t possess the following properties: copyright © 2024 assa. adv syst sci appl (2024) 46 g. petrosyan, m. soroka 1) for each t ∈ [0, t ], g(t),k(t) and t (t) are linear bounded operators, more precisely, for each x ∈ e we have ∥g(t)x∥e ≤m ∥x∥e , ∥k(t)x∥e ≤m ∥x∥e t, ∥t (t)x∥e ≤ m γ(2q) ∥x∥e t q, where m = sup{∥c(t)∥; t ∈ [0, t ]}. 2) the operator functions g(·),k(·) and tq−1t (·) are strongly continuous, i.e., functions t ∈ [0, t ] → g(t)x, t ∈ [0, t ] → k(t)x and t ∈ [0, t ] → tq−1t (t)x are continuous for each x ∈ e. by using lemma 3.2 it is easy to see that the operator functions g,k and t are also satisfy the conditions (s1)− (s3). (iii) let a : d(a) ⊂ e → e be a closed linear operator e generating a family of strongly continuous cosine operator functions {c(t)}t≥0 . the operator g : lp([0, t ];e) → c([0, t ];e), p ≥ 1, defined as gf(t) = ∫ t 0 (t− s)q−1t (t− s)f(s)ds, 1 < q ≤ 2, is a special case of the operator s. lemma 3.3: the operator g satisfies conditions (s1)− (s3). proof (s1) let 1 ≤ p <∞. by using the holder inequality, we get: ∥g(f)(t)−g(g)(t)∥e ≤ ∫ t 0 (t− s)q−1 ∥t (t− s)(f(s)− g(s))∥e ds ≤ ≤ m γ(2q) [∫ t 0 (t− s) (2q−1)p p−1 ds ] p−1 p [∫ t 0 ∥f(s)− g(s)∥pe ds ] 1 p . then ∥g(f)(t)−g(g)(t)∥pe ≤ d ∫ t 0 ∥f(s)− g(s)∥pe ds, where d = [ p− 1 2qp− 1 ]p−1 mpt 2qp−1 γp(2q) . for p = ∞, we have: ∥g(f)(t)−g(g)(t)∥e ≤ ∫ t 0 (t− s)q−1 ∥t (t− s)(f(s)− g(s))∥e ds ≤ ≤ mt q−1 γ(2q) ∫ t 0 ∥f(s)− g(s)∥e ds. then ∥g(f)(t)−g(g)(t)∥e ≤ d1 ∫ t 0 ∥f(s)− g(s)∥e ds, copyright © 2024 assa. adv syst sci appl (2024) a cauchy type problem... 47 where d1 = mt q−1 γ(2q) . (s2) letk ⊂ e be a compact set and a sequence {fn}∞n=1 ⊂ lp ([0, t ];e) , 1 ≤ p ≤ ∞, such that {fn(t)}∞n=1 ⊂ k for all t ∈ [0, t ] and fn l1 ⇀ f0. applying lemma 3.2, we obtain: χ ({ g (fn) (t) }) ≤ ∫ t 0 (t− s)q−1χe({t (t− s) fn})ds = 0. this means that the sequence {g (fn) (t)}∞n=1 ⊂ e is relatively compact for each t ∈ [0, t ]. from the other side, if we take ϵ > 0 and t1, t2 ∈ [0, t ] such that 0 < t1 < t2 ≤ t , then for each fn, we have: ∥∥∥∥∥g (fn) (t2)−g (fn) (t1) ∥∥∥∥∥ e = ∥∥∥∥∥ ∫ t2 0 (t2 − s)q−1 t (t2 − s) fn(s)ds− ∫ t1 0 (t1 − s)q−1 t (t1 − s) fn(s)ds ∥∥∥∥∥ e ≤ ∥∥∥∥∥ ∫ t2 t1 (t2 − s)q−1 t (t2 − s) fn(s)ds ∥∥∥∥∥ e + + ∥∥∥∥∥ ∫ t1 0 ( (t2 − s)q−1 t (t2 − s)− (t1 − s)q−1 t (t1 − s) ) fn(s)ds ∥∥∥∥∥ e = z1 + z2, where z1 = ∥∥∥∥∥ ∫ t2 t1 (t2 − s)q−1 t (t2 − s) fn(s)ds ∥∥∥∥∥ e , z2 = ∥∥∥∥∥ ∫ t1 0 ( (t2 − s)q−1 t (t2 − s)− (t1 − s)q−1 t (t1 − s) ) fn(s)ds ∥∥∥∥∥ e . since {fn(t)}∞n=1 ⊂ k then there exists a constant n > 0, such that ∥fn(t)∥e ≤ n. by using lemma 3.2 and condition (f2), for each ϵ > 0 we can choose δ1 > 0 such that |t2 − t1| < δ1, implies the following estimate: z1 ≤ mn(t2 − t1) 2q γ (2q + 1) < ϵ 2 . taking into account that the family of operators tq−1t (t) is strongly continuous, for each ϵ > 0 we can choose γ = γ(ϵ) > 0 and δ2 > 0 such that the inequality |t2 − t1| < δ2 implies∥∥(t2 − s)q−1 t (t2 − s)x− (t1 − s)q−1 t (t1 − s)x ∥∥ e < γ, x ∈ e, we get the following estimate: z2 ≤ γt < ϵ 2 . copyright © 2024 assa. adv syst sci appl (2024) 48 g. petrosyan, m. soroka therefore, for each ϵ > 0 we may choose δ = min {δ1, δ2} such that∥∥∥∥∥g (fn) (t2)−g (fn) (t1) ∥∥∥∥∥ e ≤ z1 + z2 < ϵ 2 + ϵ 2 = ϵ. so, the sequence {g (fn)} is equicontinuous. by the arzela-ascoli theorem, we conclude that the sequence {g (fn)} ⊂ c ([0, t ] ;e) is relatively compact. from (s1) it follows that the operator g is linear and bounded, therefore it is continuous with respect to the topology of weak sequential convergence, hence from the weak convergence fn ⇀ f0 in l1([0, t ];e) implies g(fn)⇀ g(f0). since the sequence {g(fn)} is relatively compact, we conclude that g(fn) → g(f0) in c([0, t ];e). the satisfaction of the condition (s3) is obvious. consider a system governed by a functional inclusion with causal operators q and s, of the following form: x(t) ∈ g(t)x0 +k(t)x1 + s ◦ q(x), t ∈ [0, t ] (3.1) x(0) = x0, x′(0) = x1, (3.2) definition 3.1: a function x ∈ c([0, t ];e) is called a mild solution to problem (3.1)-(3.2), if it satisfies the conditions: (1) the function x ∈ c([0, t ];e) and it satisfies inclusion (3.1); (2) x(0) = x0, x ′(0) = x1. consider the multioperator γ : c([0, t ];e) ⊸ c([0, t ];e) defined as γ(x) = {x ∈ c([0, t ];e) : x(t) = g(t)x0 +k(t)x1 + s ◦ q(x)}. it is clear that if the function x is a fixed point of the multioperator γ, then x is a solution to the problem (3.1) (3.2). therefore, our goal is to prove the existence of a fixed point of the multioperator γ. definition 3.2: a sequence of functions {ξn} ⊂ lp([0, t ];e) is called lp–semicompact if it is lp–integrably bounded, i.e., ∥ξn(t)∥e ≤ v(t) for a.e. t ∈ [0, t ] and for all n = 1, 2, ..., where v ∈ lp([0, t ]), and the set {ξn(t)} is relatively compact in e for a.e. t ∈ [0, t ]. lemma 3.4: (see. [14], proposition 4.2.1.). every lp–semicompact sequence is weakly compact in l1([0, t ];e). we need the following properties of the multioperator s ◦ q. since for every 1 < p ≤ ∞ : lp([0, t ];e) ⊂ l1([0, t ];e), we can formulate a modification of theorem 5.1.1 from [14] in the following form. lemma 3.5: let s : lp([0, t ];e) → c([0, t ];e) be an operator satisfying the conditions (s1) and (s2). then for every lp-semicompact sequence {fn}∞n=1 ⊂ lp([0, t ];e), the sequence {sfn}∞n=1 is relatively compact in c([0, t ];e) and, moreover, the weak convergence fn l1 ⇀ f0 implies that sfn → sf0 in c([0, t ];e). copyright © 2024 assa. adv syst sci appl (2024) a cauchy type problem... 49 theorem 3.1: (see [2]). let a multioperator q satisfy the conditions (q1)–(q3) and a operator s satisfy (s1), (s2). then the composition s ◦ q : c([0, t ];e) ⊸ c([0, t ];e) is a u.s.c. multimap with compact values. let us find conditions for the multioperator s ◦ q to be condensing with respect to a corresponding mnc. for this we need the following statements. lemma 3.6: (see [2]) let a sequence of functions {fn}∞n=1 ⊂ lp([0, t ];e) be lp-integrally bounded and there exists a function υ ∈ lp +([0, t ]) such that χ ({fn (t)}∞n=1) ≤ υ(t) for a.e. t ∈ [0, t ]. if an operator s satisfies the conditions (s1) and (s2), then for 1 ≤ p <∞ we have χ ({sfn (t)}∞n=1) ≤ ( 4pd ∫ t 0 υp(s)ds )1/p , and for p = ∞ χ ({sfn (t)}∞n=1) ≤ 2d1 ∫ t 0 υ(s)ds, where d,d1 are the constants from condition (s1). consider the measure of noncompactness ν in the space c([0, t ];e) with values in the cone r2 +. on a bounded subset of ω ⊂ c([0, t ];e) we define the values of ν as follows: ν(ω) = (γ (ω) ,modc (ω)) , where modc is the modulus of equicontinuity, γ is the fading modulus of the fiber noncompactness γ(ω) = sup t∈[0,t ] e−ltχ(ω(t)). the constant l > 0 is chosen so that max{q1, q2} < 1, where q1 = sup t∈[0,t ] ( 4d1/p (∫ t 0 e−pl(t−s)ωp (s, 1) ds )1/p ) , q2 = sup t∈[0,t ] ( 2d1 ∫ t 0 e−l(t−s)ω (s, 1) ds ) , where the constants d,d1 are from the condition (s1), ω is a function from the condition (q3). it is easy to see that the mnc ν is monotone, nonsingular, and algebraically semi-additive. it follows from the arzela–ascoli theorem that it is also regular. theorem 3.2: let a causal multioperator q : c([0, t ];e) ⊸ lp ([0, t ];e) satisfy the conditions (q2) and (q3) and for a causal operator s : lp ([0, t ];e) → c ([0, t ];e) the conditions (s1)–(s3) be satisfied. then the multioperator γ is ν-condensing. copyright © 2024 assa. adv syst sci appl (2024) 50 g. petrosyan, m. soroka proof by lemma 3.2, it suffices to prove the assertion of the theorem for the multioperator s ◦ q. let ω ⊂ c([0, t ];e) be a bounded set such that ν (s ◦ q (ω)) ≥ ν (ω) . (3.3) let us show that the set ω is relatively compact. inequality (3.3) means that γ({s ◦ q (ω)}) ≥ γ(ω). (3.4) applying the condition (q3) and by using the properties of the function ω, we obtain for a.e. t ∈ [0, t ] χ ({f(t) : f ∈ q (ω)}) ≤ ω ( t, sup s∈[0,t] χ ({y(s) : y ∈ ω}) ) = ω (t, φ ({y : y ∈ ω})) = ω ( t, elte−ltφ ({y : y ∈ ω}) ) ≤ ω ( t, eltγ (ω) ) ≤ ω ( t, elt ) · γ (ω) . first, we consider the case 1 ≤ p <∞. by lemma 3.6 we have for each t ∈ [0, t ] : χ ({sf(t) : f ∈ q (ω)}) ≤ ( 4pd ∫ t 0 ωp ( s, els ) ds · γp (ω) )1/p ≤ 4d1/p (∫ t 0 eplsωp (s, 1) ds )1/p · γ (ω) . inequality (3.4) and the last inequality imply the following γ(ω) ≤ sup t∈[0,t ] ( 4d1/p (∫ t 0 e−pl(t−s)ωp (s, 1) ds )1/p ) γ (ω) = q1 · γ (ω) . therefore γ (ω) = 0, and φ (ω) = 0 for all t ∈ [0, t ]. let us turn to the case p = ∞. by lemma 3.6 we have for each t ∈ [0, t ] : χ ({sf(t) : f ∈ q (ω)}) ≤ 2d1 ∫ t 0 ω ( s, els ) ds · γ (ω) ≤ 2d1 ∫ t 0 elsω (s, 1) ds · γ (ω) ; inequality (3.4) and the last inequality imply the following γ(ω) ≤ sup t∈[0,t ] ( 2d1 ∫ t 0 e−l(t−s)ω (s, 1) ds ) γ (ω) = q2 · γ (ω) . therefore γ (ω) = 0, and φ (ω) = 0 copyright © 2024 assa. adv syst sci appl (2024) a cauchy type problem... 51 for each t ∈ [0, t ]. now we will show that the set ω is equicontinuous. we take sequences {yn}∞n=1 ⊂ ω, n ≥ 1, and {fn}∞n=1, fn ∈ q(yn). from the conditions (q2) and (q3) it follows that the sequence {fn}∞n=1 is lp-semicompact, and therefore by lemma 3.5 the sequence {sfn}∞n=1 is relatively compact. hence modc({sfn}∞n=1) = 0. thus ν (s ◦ q (ω)) = (0, 0), but then it follows from the inequality (3.3) that ν(ω) = (0, 0), and the last expression yields that the set ω is relatively compact. to prove the main theorem of the paper, we need the following statements, known as the gronwall bellman lemma and the generalized gronwall bellman lemma. lemma 3.7: let v(t) and f(t) be nonnegative continuous functions on the segment [a, b], moreover v(t) ≤ c+ ∫ t a f(s)v(s)ds, t ∈ [a, b], where c is a positive constant. then for each t ∈ [a, b] the inequality v(t) ≤ ce ∫ t a f(s)ds, holds. lemma 3.8: let h(t), u(t) and v(t) be nonnegative functions integrable on [a, b] satisfying the inequality: v(t) ≤ u(t) + ∫ t a h(s)v(s)ds, t ∈ [a, b]. then the following inequality holds: v(t) ≤ u(t) + ∫ t a e ∫ t a h(θ)dθh(s)u(s)ds, t ∈ [a, b]. theorem 3.3: let a causal multioperator q : c([0, t ];e) → cv(lp([0, t ];e)), 1 ≤ p ≤ ∞, satisfy the conditions (q1)–(q3) and a linear causal operator s : lp([0, t ];e) → c([0, t ];e) satisfy the conditions (s1)–(s3). then the set σ of all solutions to problem (3.1)-(3.2) is a nonempty compact set. proof let us show that the set of all solutions x ∈ c([0, t ];e) of a one-parameter inclusion x ∈ λγ(x), λ ∈ [0, 1], (3.5) is a priori bounded. we divide the proof into three cases: p = 1, 1 < p <∞, p = ∞. let p = 1, if a function x ∈ c([0, t ];e) satisfies condition (3.5). then by lemma 3.2 for each t ∈ [0, t ], we have the following estimates: copyright © 2024 assa. adv syst sci appl (2024) 52 g. petrosyan, m. soroka ∥x(t)∥e ≤ ∥g(t)x0∥e + ∥k(t)x1∥e +d ∫ t 0 ∥f(s)∥eds ≤ m∥x0∥e +mt∥x1∥e +d ∫ t 0 ∥f(s)∥eds ≤m∥x0∥e +mt∥x1∥e +d ∫ t 0 ∥f(s)∥eds, where f ∈ q(x) and, therefore, by condition (q2) : ∥f(s)∥e ≤ α(s)(1 + ∥x(s)∥e). then ∥x(t)∥e ≤m∥x0∥e +mt∥x1∥e +d ∫ t 0 α(s)(1 + ∥x(s)∥e)ds ≤ m∥x0∥e +mt∥x1∥e +d∥α∥l∞t +d∥α∥l∞ ∫ t 0 ∥x(s)∥eds. the last expression is a non-decreasing function of t, so we get ∥x(t)∥e ≤m∥x0∥e +mt∥x1∥e +d∥α∥l∞t + ∫ t 0 d∥α∥l∞∥x(t)∥eds. this means that the function v(t) = ∥x(t)∥e satisfies the inequality v(t) ≤m∥x0∥e +mt∥x1∥e +d∥α∥l∞t + ∫ t 0 d∥α∥l∞v(s)ds. applying lemma 3.7, we obtain the required a priori boundedness: v(t) = ∥x(t)∥e ≤ ued∥α∥l∞ = γ1, where u =m∥x0∥e +mt∥x1∥e +d∥α∥l∞t. then ∥x∥c = supt∈[0,t ] ∥x(t)∥e ≤ γ1. for the case 1 < p <∞, we have ∥x(t)∥e ≤m∥x0∥e +mt∥x1∥e + ( d ∫ t 0 ∥f(s)∥peds )1/p ≤m∥x0∥e +mt∥x1∥e + ( d ∫ t 0 αp(s)(1 + ∥x(s)∥e)pds )1/p ≤m∥x0∥e +mt∥x1∥e + ( d ∫ t 0 αp(s)ds+d ∫ t 0 αp(s)∥x(s)∥peds )1/p ≤ ≤m∥x0∥e +mt∥x1∥e + ( d ∫ t 0 αp(s)ds )1/p + ( d ∫ t 0 αp(s)∥x(s)∥peds )1/p . let us introduce the following notation: c0 =m∥x0∥e +mt∥x1∥e +d1/p∥α∥lp , h(s) = d1/pα(s). copyright © 2024 assa. adv syst sci appl (2024) a cauchy type problem... 53 then we get: ∥x(t)∥e ≤ c0 + (∫ t 0 hp(s)∥x(s)∥peds )1/p . let v(t) = ∥x(t)∥pe, then from the last inequality we obtain the estimate: v(t) ≤ 2pcp0 + 2p ∫ t 0 hp(s)v(s)ds. now applying lemma 3.8 to the last inequality, we get v(t) = ∥x(t)∥pe ≤ 2pcp0 ( 1 + ∫ t 0 e2 p ∫ t 0 hp(θ)dθhp(s)ds ) . then we have the final estimate for 1 < p <∞ : ∥x(t)∥e ≤ 2c0 p √ 1 + ∫ t 0 e2 p ∫ t 0 hp(θ)dθhp(s)ds = γ2. then ∥x∥c = supt∈[0,t ] ∥x(t)∥e ≤ γ2. for the case p = ∞, in the same way as in the case p = 1, the following estimate holds: ∥x∥c ≤ u1e d1∥α∥l∞ = γ3, where u1 =m∥x0∥e +mt∥x1∥e +d1∥α∥l∞t. now, if we take r ≥ max{γ1, γ2, γ3}, then we can guarantee that the set v ⊂ c([0, t ];e), given as v = {x ∈ c([0, t ];e) : ∥x∥c < r}, contains all solutions of inclusion (3.5). thus, the multioperator γ satisfies on ∂v the condition of theorem 2.2 with a = 0, hence the set of its fixed points is non-empty and compact. 4. cauchy type problems for semilinear differential inclusions 4.1. a cauchy type problem for a second order semilinear differential inclusions consider the following system governed by a differential inclusion in a separable banach space e : y′′ (t) ∈ ay (t) + f (t, y(t)) , t ∈ [0, t ] , (4.6) y(0) = y0, y′(0) = y1, (4.7) where f : [0, t ]× e → kv(e) is a multivalued map, y0, y1 ∈ e are given functions. suppose that (a) a : d (a) ⊂ e → e generates a family of strongly continuous cosine operator functions {cos(at)}t≥0. a multimap f : [0, t ]× e → kv(e) is such that: (f1) for each ψ ∈ e the multifunction f (·, ψ) : [0, t ] → kv (e) admits a measurable selection; copyright © 2024 assa. adv syst sci appl (2024) 54 g. petrosyan, m. soroka (f2) for a.e. t ∈ [0, t ] the multimap f (t, ·) : e → kv (e) is upper semicontinuous (u.s.c.); (f3) there exists a function α ∈ l∞ + [0, t ] such that ∥f (t, ψ)∥e := sup{∥z∥e : z ∈ f (t, ψ)} ≤ α(t)(1 + ∥ψ∥e) for a.e. t ∈ [0, t ] and for all ψ ∈ e; (f4) there exists a function ωf : [0, t ]× r+ → r+ satisfying the conditions (ω1)-(ω3) such that for each bounded set ω ⊂ c([0, t ];e) we have χ (f (t,ω)) ≤ ωf (t, φ (ω)) for a.e. t ∈ [0, t ]. notice that when q = 2 : g(t) = cos(at), k(t) = sin(at), t (t) = sin(at) t . in accordance with [11], a function y ∈ c([0, t ];e) is a mild solution to problem (4.6)-(4.7), if it can be represented in the form: y(t) = cos(at)y0 + sin(at)y1 + ∫ t 0 sin(a(t− s))f(s)ds, where f ∈ pf (y), pf is a superposition multioperator (see example 2.1). the fact that the superposition multioperator pf : c([0, t ];e) ⊸ l1([0, t ];e) satisfies the condition (q1) can be verified by lemma 5.1.1 from [14]. conditions (q2) and (q3) for pf follow from (f3) and (f4), respectively. taking into account lemma 3.1, we can consider relation (4.6) as a special case of functional inclusion (3.1) with q = pf , and s = l is the cauchy operator. as a direct consequence of theorem 3.3, we obtain the following result. theorem 4.1: suppose that the conditions (a), (f1)–(f4), hold true. then the set of solutions to problem (4.6)-(4.7) is a non-empty compact subset of the space c([0, t ];e). 4.2. a cauchy type problem for fractional semilinear differential inclusions let us recall the notion of the caputo fractional derivative. definition 4.1: the caputo fractional derivative of the order q ∈ (1, 2) of a function g ∈ c2([0, t ];e) is the function cdq 0g of the following form: cdq 0g(t) = 1 γ(2− q) ∫ t 0 (t− s)1−qg′′(s) ds, where γ is euler’s gamma-function γ(q) = ∫ ∞ 0 xq−1e−xdx. consider the following system governed by a functional differential inclusion in a separable banach space e : cdq 0y (t) ∈ ay (t) + f (t, y(t)) , t ∈ [0, t ] , (4.8) y (0) = y0, y′(0) = y1. (4.9) suppose that copyright © 2024 assa. adv syst sci appl (2024) a cauchy type problem... 55 (a) a : d (a) ⊂ e → e is a closed linear operator in e generating a family of strongly continuous cosine operator functions {c(t)}t≥0 . assume also that a multimap f : [0, t ]× e → kv(e) satisfies the conditions (f1)– (f4) from section 4.1. a function y ∈ c([0, t ];e) is a mild solution to problem (4.8)–(4.9), if it can be presented in the form: y(t) = g(t)y0 +k(t)y1 + ∫ t 0 (t− s)q−1t (t− s)f(s)ds, where f ∈ pf (y). the fact that the superposition multioperator pf : c([0, t ];e) ⊸ lp([0, t ];e), p > 1/q satisfies the condition (q1) can be verified as in the paper [26]. conditions (q2) and (q3) for pf follow from (f3) and (f4), respectively. taking into account lemma 3.3, we can consider the relation (4.8) as a special case of functional inclusion (3.1) with q = pf , and s = g is the cauchy type operator. as a direct consequence of theorem 3.3, we obtain the following result. theorem 4.2: suppose that the conditions (a), (f1)–(f4) hold true. then the set of solutions to problem (4.8)-(4.9) is a non-empty compact subset of the space c([0, t ];e). acknowledgements the work was supported by the russian science foundation (project no. 23-71-10026). references 1. afanasova, m., liou, y. ch., obukhoskii, v., petrosyan, g. (2019). on controllability for a system governed by a fractional-order semilinear functional differential inclusion in a banach space, journal of nonlinear and convex analysis, 20(9), 1919–1935. 2. afanasova, m., obukhovskii, v., petrosyan g. (2021). a controllability problem for causal functional inclusions with an infinite delay and impulse conditions, advances in systems science and applications, 21(3), 40–62. 3. ahmerov, r.r., kamenskii, m.i., potapov, a.s., rodkina, a.e., sadovskii, b.n. (1992). measures of noncompactness and condensing operators. boston–basel– berlin, birkhauser. 4. appell, j., lopez, b., sadarangani, k. (2018). existence and uniqueness of solutions for a nonlinear fractional initial value problem involving caputo derivatives, j. nonlinear var. anal., 2, 25–33. 5. bulgakov, a.i., maximov, v.p. (1981). functional and functional-differential inclusions with volterra operators, differential equations, 17(8), 1362–1374. 6. corduneanu, c. (2002). functional equations with causal operators. stability and control: theory, methods and applications. london, uk: taylor and francis. 7. drici, z., mcrae, f.a., vasundhara devi, j. (2005). differential equations with causal operators in a banach space, nonlinear anal., 62(2), 301–313. 8. drici, z., mcrae, f.a., vasundhara devi, j. (2006). monotone iterative technique for periodic boundary value problems with causal operators, nonlinear anal., 64(6), 1271– 1277. 9. gomoyunov, m.i. (2018). fractional derivatives of convex lyapunov functions and control problems in fractional order systems. fract. calc. appl. anal., 21, 1238–1261. copyright © 2024 assa. adv syst sci appl (2024) 56 g. petrosyan, m. soroka 10. gomoyunov, m.i. (2019). approximation of fractional order conflict-controlled systems, prog. fract. differ. appl., 5, 143–155. 11. hartman, p. (1964). ordinary differential equations. new york, ny: john wiley and sons. 12. he, j.w., liang, y., ahmad, b., zhou, y. (2019). nonlocal fractional evolution inclusions of order α ∈ (1, 2), mathematics, 7(2), 1–17. 13. jankowski, t. (2008). boundary value problems with causal operators, nonlinear anal., 68(12), 3625–3632. 14. kamenskii, m., obukhovskii, v., zecca, p. (2001). condensing multivalued maps and semilinear differential inclusions in banach spaces. berlin, germany: walter de gruyter. 15. kamenskii, m., obukhoskii, v., petrosyan, g., yao, j.-c. (2019). on a periodic boundary value problem for a fractional order semilinear functional differential inclusions in a banach space, mathematics, 7(12), 5–19. 16. kamenskii, m., obukhoskii, v., petrosyan, g., yao, j.-c. (2021). on the existence of a unique solution for a class of fractional differential inclusions in a hilbert space, mathematics, 9(2), 136–154. 17. kilbas, a.a., srivastava, h.m., trujillo, j.j. (2006). theory and applications of fractional differential equations. amsterdam, netherlands: north-holland mathematics studies, elsevier science b.v. 18. kostin, v.a., kostin, d.v., kostin, a.v. (2019). operator cosine functions and boundary value problems, reports of the academy of sciences, 486(5), 531–536. 19. kulmanakova, m.m., ulyanova, e.l. (2019). o razreshimosti kauzalnih funkcionalnih vkluchenii s beskonechnim zapazdivaniem [on the solvability of causal functional inclusions with infinite delay]. tambov university reports. series: natural and technical sciences, 24(127), 293–315, [in russian]. 20. lupulescu, v. (2008). causal functional differential equations in banach spaces, nonlinear anal., 69(12), 4787–4795. 21. mainardi, f. (1994). on the initial value problem for the fractional diffusion-wave equation. in rionero, s. and ruggeri, t., eds., waves and stability in continuous media, world scientific, singapore, 246–251. 22. obukhovskii, v., gel’man, b. (2020). multivalued maps and differential inclusions. elements of theory and applications. hackensack, nj: world scientific publishing co. pte. ltd. 23. obukhovskii, v., zecca, p. (2011). on certain classes of functional inclusions with causal operators in banach spaces, nonlinear anal., 74(8), 2765–2777. 24. petrosyan, g. (2021). antiperiodic boundary value problem for a semilinear differential equation of fractional order, the bulletin of irkutsk state university. series: mathematics, 34, 51–66. 25. petrosyan, g. (2020). on antiperiodic boundary value problem for a semilinear differential inclusion of fractional order with a deviating argument in a banach space, ufa mathematical journal, 12(3), 69–80. 26. petrosyan, g.g. (2015). teorema o slaboi zamknutosti superpozicionnogo multioperatora [a theorem on the weak closure of superposition multioperator], tambov university reports. series: natural and technical sciences, 20(5), 1355–1358, [in russian]. 27. podlubny, i. (1999). fractional differential equations. san diego, ca: academic press. 28. travis, c.c., webb, g.f. (1978). cosine families and abstract nonlinear second order differential equations, acta math.hungar, 32, 75–96. 29. tychonoff, a.n. (1938). sur les equations fonctionnelles de volterra et leurs applications a certains problemes de la physique mathematique, bulletin de i’universite d’etat de moscou. ser. internationale. section a. mathematiques et mecanique, 1(8), 1–25, [in french]. copyright © 2024 assa. adv syst sci appl (2024) a cauchy type problem... 57 30. zhou, y., he, j.w. (2019). new results on controllability of fractional evolution systems with order α ∈ (1, 2), evol. eq. control theory, 10(3), 491–509. copyright © 2024 assa. adv syst sci appl (2024) introduction preliminaries measures of noncompactness multivalued maps family of cosine operator functions causal multioperators cauchy type problem for functional inclusions with the causal multioperators cauchy type problems for semilinear differential inclusions a cauchy type problem for a second order semilinear differential inclusions a cauchy type problem for fractional semilinear differential inclusions microsoft word 1539-article text-7336-1-18-20240902 adv syst sci appl 2024; 03; 90-103 published online at https://ijassa.ipu.ru. study of three dimensional casson-nanofluid flow due to a linear porous stretching sheet in the presence of double diffusion effects d. r. kirubaharan1, a. d. subhashini1, g. murali2* 1department of mathematics, prist university, thanjavur, india; 2department of mathematics, geethanjali college of engineering and technology, cheeryal, india. abstract: in this work, the bidirectional flow of magnetohydrodynamics (mhd) with slip effects, a magnetic field, and the inclusion of casson and nanofluid particles is investigated. a stretched sheet that is porous and linear in three dimensions creates this flow. similarity transformations are used to translate the fundamental boundary layer model into a system of ordinary differential equations. after that, the equations are solved using the runge-kutta technique along with shooting technique. the investigation looks at how several parameters, including temperature, concentration, and main and secondary velocities, affect how flow variables change over time. additionally, it looks at the impact of engineering parameters including mass transfer coefficients, skin-friction coefficients, and heat transfer rates. a combination of tabular and graphical displays is used in this investigation. in the end, the validity of the present model is confirmed by contrasting the numerical values derived from the findings that have been published with the available data. this instruction give you guidelines for preparing papers for assa publication. please use this document as a template to prepare your manuscript for submission. keywords: multiple slip effects; mhd; three dimensional; nanofluid; casson fluid; porous medium; linear stretching sheet; runge-kutta method 1. introduction the fundamentals of nanofluids, slip effects, and their combined consequences in diverse applications are covered in this introduction. slip at fluid-solid contacts introduces a distinctive boundary condition that substantially influences general flow characteristics. concentration, thermal, and velocity slip are among the numerous types of slip effects in fluid dynamics that distinguish them from the typical no-slip situation at the solid-fluid interface. in-depth comprehension of the interactions between numerous slide effects and their impact on fluid flow is required as complexity increases. the flux of boundary layers is significantly influenced by thermal slip conditions, which has an impact on both industrial and natural processes. these processes encompass the extrusion of plastic and metal, as well as the refrigeration and drying processes employed in the production of paper and textiles. these characteristics have an impact on a variety of technical applications, including oil recovery, heat exchangers, catalytic reactors with rectangular ducts, chemical vapor deposition on surfaces, and chemical reactions in reactor chambers. flow models in porous media are applicable in a variety of biological contexts, including brain tissue diffusion, tissue bio-heat transfer during genesis, blood flow in cancer, and bioconvection. authors mentioned in the literature [1]-[5] developed a model that considers slip effects on time-dependent magnetohydrodynamic (mhd) heat and mass transfer through a porous stretching sheet. the model shows that slip parameter values increase and the fluid's velocity near the boundary layer region decreases. other studies have focused on the multiple slip effects on mhd axisymmetric buoyant nanofluid flow above a stretching sheet, the presence of magnetic nanoparticles, the numerous slip effects on erratic viscoelastic nanofluid flow across a permeable stretched sheet with radiation, the thickness and velocity of the slip layer in * corresponding author: prof.murali20@gmail.com study of three dimensional casson-nanofluid flow… 91 copyright ©2024 assa. adv. in systems science and appl. (2024) concentrated suspensions, the influence of absorption and boundary slip in a channel, entropy analysis in a 3-d non-linear radiative hybrid nanofluid flow between two parallel stretched permeable sheets with slip velocities, and the darcy-forchheimer up/down flow of entropy optimized radiative nanofluids with second-order slip, nonuniform source/sink, and shape effects.authors in the references section [6]-[10] et al. studied multiple slip and cross diffusion on mhd carreau-casson fluid over a slandering sheet with non-uniform heat source/sink. asmat et al. examined thermal analysis in electrically conducting fluid with varying slips and radiation. das et al. examined absorption and boundary slip in a channel. nayak et al. investigated 3-d non-linear radiative hybrid nanofluid flow between two parallel stretched permeable sheets with slip velocities. samantaray et al. studied darcy-forchheimer up/down flow of entropy optimized radiative nanofluids. in geophysical systems and chemical engineering, cross-diffusion-that which results from instantaneous heat and mass transfer in a flowing fluid-is seen in the soret and dufour effects. thermal diffusion-sometimes known as the soret effect-describes specifically the mass transfer resulting from a temperature difference. the energy flow arising from changes in concentration is known as the dufour effect. the soret effect results from the separation of big, heavy molecules and small, light ones brought about by temperature differential. many times, this effect is considered when many chemical species exist in a chemical environment with a notable temperature gradient. many researchers drew interest in the soret and dufour effects on various fluid flow systems. references [11]-[15]cited in the section have studied steady free convective flow with soret and dufour effects using heat radiation and a magnetic field on porous materials. shojaei et al. examined the dufour and soret effect in a convective stretchable cylinder, hayat et al. studied the influence of soret and dufour effects on mhd casson fluid flow, pal and mondal analyzed the impact of heat-generating and nonlinear thermal radiation on steady-state flow through a non-isothermal wedge, and bekezhanova and goncharova examined the properties of liquid flows undergoing evaporation. authors mentioned in the references section [16]-[20]studied the impact of soret and dufour effects on forced convection flow towards a moveable thin needle using buongiorno's nanofluid model. jawad et al. analyzed boundary layer mhd darcy-forchheimer radiative nanofluid flow using marangoni convection and also studied fluid flow dynamics in soret-dufour collisions. das and majumdar investigated the magnetohydrodynamic flow of a casson fluid on an inclined permeable plate, considering the soret effect, thermal radiation, viscous dissipation, and joule heating. raja sekhar et al. studied the movement of a magnetohydrodynamic casson nanofluid across a non-linear inclined surface.the works mentioned in the literature [21]–[25] had a significant impact on understanding the nature of the reported work.works which were done by the authors [27]-[32] have shown the solutions for the problems. the works mentioned in the literature [33]–[34] had a significant impact on understanding the nature of the reported work. motivated by the above research, this current research is a follow-up to the “mhd threedimensional casson fluid flow past a porous linearly stretching sheet” by nadeem et al. [26]. the dynamics of the boundary layer of a nanofluid moving through a stretched sheet in a magnetic field is studied here by taking multiple slips (thermal and concentration slips), dufour and soret effects into account. the set of regulating parameters determines how the governing equations are transformed into nonlinear differential equations. these non-linear differential equations are then numerically solved using the runge-kutta algorithm, along with shooting technique. a considerable physical quantity may be numerically calculated using tables, such as the local skin-friction coefficient, local nusselt and sherwood numbers, and so on. this non-linear stretching sheet has designs and structures that are ideal for the mechanical, aerospace, civil, and marine industries. 2. flow governing equations this work investigates the three-dimensional, continuously flowing, electrically conductive, noncompressible casson-nanofluid around a linearly stretched sheet. the work uses numerical techniques to identify answers while accounting for the effects of soret, dufour, porous media, multiple slips, 92 d. r. kirubaharan, a. d. subhashini, g. murali copyright ©2024 assa. adv. in systems science and appl. (2024) thermophoresis, and brownian motion. the geometry and physical coordinate system of this flow problem are shown in figure 1. fig. 1. geometric representation of the movement of casson and nanofluid particles a) let (u, v, w) be the velocity components along the (x, y, z) directions, respectively. b) flow is generated due to an exponentially stretching surface. c) a uniform magnetic field of strength b0 is applied in the z − direction. d) magnetic reynolds number is assumed very small so that the induced magnetic field is ignored. e) the effects of soret and dufour are considered in concentration and energy equations respectively. f) the effects of multiple slip effects are considered in boundary conditions section. g) the characteristics of brownian motion and thermophoresis are accounted. h) the rheological equation for a non-newtonian fluid is defined as, * o    (1) eq. (1) can be expanded for casson fluid as, 2 , & 2 , 2 2 y y ij b ij c b ij c c p p e e                         (2) where ij jie e  with ije is the  , th i j component of the fluid deformation rate and 2b yp     is the yield stress of the casson fluid. for this flow, the governing boundary layer equations can be written as continuity equation: 0 u v w x y z          (3) momentum equation: 22 2 * 1 1 obu u u u u v w u u x y z z k                                                (4) 22 2 * 1 1 obv v v v u v w v v x y z z k                                               (5) equation of thermal energy: study of three dimensional casson-nanofluid flow… 93 copyright ©2024 assa. adv. in systems science and appl. (2024) 22 2 12 2 m tt b s p d kdt t t t t c t c u v w d x y z z z z t z c c y                                                                  (6) equation of species concentration: 22 2 2 2 m tt b m d kdc c c c t t u v w d x y z z t z t y                                            (7) the boundary conditions for this flow are     1 2, , , 0 0, 0, , w w w w t c u u x ax v v y by t t l c c l at z y y u v t t c c as z                               (8) introducing the following similarity transformations              , , , , , , w w c c u a x y f v b x y g c c t ta w a f g z t t                                     (9) making use of eq. (9), equation of continuity is identically satisfied and eqs. (4), (5), (6) and (7) take the following form  21 1 0f ff gf f m k f                  (10)  21 1 0g fg gg g m k g                  (11) 2pr pr pr pr pr 0f g nb nt du                  (12) 0nb nbscf nbscg nt scnbsr             (13) the corresponding boundary conditions (8) become                         0 0, 0 0, 0 1, 0 , 0 1 0 , 0 1 0 0, 0, 0, 0 f g f g f g                                (14) where the involved physical parameters are defined as           2 1 1* 2 , , , ( )1 , , , , pr , , , , b w t wo m t w b m w m t w s p d c c d t tb m nb nt a t d k t t a sc k sr l d ak t c c d k c c a b du l c c t t a                                            (15) engineering quantities quantities of physical interest, the physical parameters of the skin-friction coefficient along x and y directions and local nusselt number are presented as follows:    2 1 re 1 0wx x x x w cf cf f u             (16)    2 1 re 1 0wy y y y w cf cf g u             (17)        0 re 0w z x w w t x xq z nu nu t t t t                    (18) 94 d. r. kirubaharan, a. d. subhashini, g. murali copyright ©2024 assa. adv. in systems science and appl. (2024)        0 re 0ym x b w b w c x yxq sh sh d c c d c c                (19) where   re x x u x x   is local reynolds number based on the stretching velocity uw (x) and   re w y v y y   is local reynolds number based on the stretching velocity vw (y). 3. solution by r-k method along with shooting technique using the runge-kutta method alongside the shooting approach allows for the solution of non-linear governing equations with partial derivatives. this approach produces more accurate outcomes when compared to alternative numerical methods. similarity transformations are employed to convert the controlling partial differential equations into ordinary differential equations. through the use of additional variables, the non-linear equations are converted into linear equations. introducing additional variables into a higher-order differential equation to convert it into a linear equation. 1 2 3 4 5 6 7 8 9 10, , , , , , , , ,f f f f f f f g f g f g f f f f                  (20) the equations (13)-(15) are transformed to following first order ode.     2 1 3 4 3 2 2 3 2 3 2 1 6 4 6 5 5 6 5 6 2 8 1 8 4 8 8 10 8 10 1 10 4 10 8 8 10 , , 1 1 , , 1 1 pr pr pr pr pr , f f f f f m k f f f f f f f f f m k f f f f f f f f f nbf f nt f du f nb scf f nbscf f nt f scnbsrf f nb                                                   (21) using eq. (21), the corresponding boundary conditions (16) are                         1 4 2 5 7 8 9 10 2 5 7 9 0 0, 0 0, 0 1, 0 , 0 1 0 , 0 1 0 0, 0, 0, 0 f f f f f f f f f f f f                      (22) the approximate solutions are numerically resulted in graphical representation using matlab bvp4c programming to present the physical significance of non-dimensional parameters. the iterative procedure is repetitive till we obtained results correct up to the preferred precision of 10-6 level. study of three dimensional casson-nanofluid flow… 95 copyright ©2024 assa. adv. in systems science and appl. (2024) 3.1. comparison of present results with published results table 1 valuation of present skin-friction results with published results of nadeem et al. [26] when sr = nb = pr = sc = nt = du = λ = δ = 0, and β → ∞. for comparison of present results with the published results, the authors have compared the present skin-friction results with published results of nadeem et al. [26] for variations of m at β → ∞ in table 1 when sr = nb = pr = sc = nt = du = λ = δ = 0. this table exposes a good agreement between the current findings and those obtained by nadeem et al. [26], as previously stated. 4. results and discussion on a casson-nanofluid stream over a linearly stretching sheet, an analysis of three-dimensional, steady flow was performed. due to the combination of soret and dufour, an electrically charged fluid is investigated in the presence of magnetic field and multiple slip effects. an approximate numerical approach called runge kutta method along with shooting technique is used to handle the altered equations created from the said model. examine the effects of various dynamic parameters of the contours of our system on velocity, and temperature, and concentration profiles. also, the study of variables for different values through the graphs. for this study, we have taken on the settings of engineering parameters for our study as m = 0.5, k = 0.3, β = 0.5, pr = 0.71, nb = 0.3, nt = 0.5, du = 0.5, sc = 0.22, sr = 0.5, ω = 0.5, δ = 0.5, and λ = 0.5. the figures, 2 and 3, clearly demonstrate that when the magnetic field parameter increases, both the boundary layer thickness and the amplitude of the main and secondary velocity profiles decrease. physically, the current phenomenon occurs when a magnetic field causes the flow of electric current in a conductive fluid. this results in the fluid inside the boundary layer encountering a drag force, which decelerates the fluid's velocity. therefore, the use of magnetic fields is terminated. the figures 4 and 5 demonstrate that the primary and secondary velocity profiles inside the boundary layer diminish as the porosity parameter rises. in addition, figures 4 and 5 illustrate that an increase in the permeability parameter leads to a reduction in the thickness of the boundary layer. the influence of the non-newtonian casson fluid parameter on the primary and secondary velocity profiles may be seen in figures 6 and 7. it is shown that the resistance to the flow of fluid is caused by an elevation in the non-newtonian casson fluid parameter. in figures 6 and 7, it can be shown that when the non-newtonian casson fluid parameter has bigger values, the amplitude of the main and secondary velocity profiles, as well as the thickness of the boundary layer, decrease. it is observed that when we increase the non-newtonian casson fluid parameter indefinitely, the present events obviously transition to a newtonian fluid. figure 8 illustrates the impact of the stretching ratio parameter on the secondary velocity profiles. the secondary velocity profiles increase as the value of the stretching rate ratio parameter grows. generally, increasing the stretching parameter leads to a rise in flow pressure. the fluid's temperature in relation to the prandtl number is shown in figure 9. as the prandtl number increases, the temperature differential in the fluid decreases. lower thermal diffusivity in fluids with a higher prandtl number results in a lower temperature. 96 d. r. kirubaharan, a. d. subhashini, g. murali copyright ©2024 assa. adv. in systems science and appl. (2024) the temperature and the concentration profiles are both influenced by the brownian motion parameter, as figures 10 and 11 respectively demonstrate. figure 10 illustrates how increasing the brownian motion parameter results in improved temperature profiles. as the viscous force decreases and the brownian motion parameter increases, the brownian diffusion coefficient increases. as a result, both temperature and the thermal boundary layer's thickness vary. these effects might be better understood by looking at figure 10. it is evident from figure 11 that as the brownian motion parameter increases, the concentration falls. the impact of the thermophoresis parameter on the temperature and concentration curves is seen in figures 12 and 13, respectively. the thermophoresis parameter is defined by the viscous force and the heat diffusion coefficient. as its inversely proportional nature implies, a decrease in the viscous force occurs with an increase in the thermophoresis parameter. the thermophoresis value indicates that the temperature and nanoparticle concentration have increased due to a decrease in viscous force and an increase in the thermal diffusion coefficient. figure 14 illustrates how the fluid's temperature profile rises as the number of dufour increases. this phenomenon may be explained by describing the release of two chemically insensitive fluids at room temperature inside a system, which increases the system's internal temperature differential. figure 15 shows how the number of soret affects concentration profiles, indicating that an increase in soret number results in an increase in liquid concentration. the soret effect occurs when an irreversible process generates a temperature gradient in a concentration field. it has the ability to increase the circulatory system's concentration flow. the original reason is shown in figure 15. in figure 16, you can see how the thermal slip measure changes the temperature curve. as the slip limit gets bigger, the temperature goes down. when the thermal slip parameter goes up, the thickness of the thermal boundary layer goes down, even though there isn't much heat transfer from the layer to the fluid. as the concentration slip parameter goes up, the concentration curve goes down, as shown in figure 17. figure 18 illustrates the impact of schmidt number on concentration patterns. the schmidt number is a dimensionless quantity that denotes the ratio of momentum to mass diffusivity. by diffusion in the concentration (species) boundary layer, the comparative magnitudes of momentum and mass transfer are determined by the calculation. the concentration will decrease as a result of the fluid's lower mass diffusivity, which is a consequence of a higher schmidt number. the schmidt number and mass diffusivity are inversely proportional, indicating that concentration boundary layers with higher schmidt numbers are thinner. fig. 2. influence of m on  f  velocity profiles fig. 3. influence of m on  g  velocity profiles study of three dimensional casson-nanofluid flow… 97 copyright ©2024 assa. adv. in systems science and appl. (2024) fig. 4. influence of k on  f  velocity profiles fig. 5. influence of k on  g  velocity profiles fig. 6. influence of β on  f  velocity profiles fig. 7. influence of β on  g  velocity profiles fig. 8. influence of ω on  g  velocity profiles fig. 9. influence of pr on    temperature profiles fig. 10. influence of nb on    temperature profiles fig. 11. influence of nb on    concentration profiles 98 d. r. kirubaharan, a. d. subhashini, g. murali copyright ©2024 assa. adv. in systems science and appl. (2024) fig. 12. influence of nt on    temperature profiles fig. 13. influence of nt on    concentration profiles fig. 14. influence of du on    temperature profiles fig. 15. influence of sr on    concentration profiles fig. 16. influence of δ on    temperature profiles fig. 17. influence of λ on    concentration profiles fig. 18. influence of sc on    concentration profiles adv syst sci appl 2024; 03; 90-103 published online at https://ijassa.ipu.ru. table 2 skin-friction coefficient  1 1 0f        numerical values m k β ω pr nb nt du δ sc sr λ  1 1 0f        0.5 0.3 0.5 0.5 0.71 0.3 0.5 0.5 0.5 0.22 0.5 0.5 1.8563629652 0.8 1.8158282823 1.2 1.7956738784 0.5 1.8067859839 0.8 1.7842383084 0.7 1.8256764870 0.9 1.7934574573 0.8 1.8858288221 1.2 1.9168756287 1.00 1.8067762857 3.00 1.7745654726 0.5 1.8945652975 0.7 1.9245629259 0.8 1.9035482582 1.2 1.9354665928 1.0 1.9156760876 1.5 1.9565876878 0.7 1.8257106508 0.9 1.8065865818 0.30 1.8046496591 0.78 1.7945635972 1.0 1.8846658658 1.5 1.9354345082 0.8 1.8258582830 1.2 1.8067730304 the effects of thermophoresis parameter (nt = 0.5, 0.8, 1.2), brownian motion parameter (nb = 0.3, 0.5, 0.7), soret number (sr = 0.5, 1.0, 1.5), concentration slip parameter (λ = 0.5, 0.8, 1.2), and schmidt number (sc = 0.22, 0.30, 0.60). on rate of mass transfer coefficient or in terms sherwood number coefficient are discussed in table-4. from this table, it is observed that the rate of mass transfer coefficient is increasing with increasing values of thermophoresis parameter (nt = 0.5, 0.8, 1.2) and decreasing with increasing values of brownian motion parameter (nb = 0.3, 0.5, 0.7), soret number (sr = 0.5, 1.0, 1.5), concentration slip parameter (λ = 0.5, 0.8, 1.2), and schmidt number (sc = 0.22, 0.30, 0.60). table 2 shows the numerical values of skin-friction coefficient due to primary velocity profiles for variations in values of the engineering parameters such as, magnetic field parameter (m = 0.5, 0.8, 1.2), permeability parameter (k = 0.3, 0.5, 0.8), casson fluid parameter (β = 0.5, 0.7, 0.9), prandtl number (pr = 0.71, 1.0, 3.0), thermophoresis parameter (nt = 0.5, 0.8, 1.2), brownian motion parameter (nb = 0.3, 0.5, 0.7), dufour number (du = 0.5, 1.0, 1.5), soret number (sr = 0.5, 1.0, 1.5), thermal slip parameter (δ = 0.5, 0.7, 0.9), concentration slip parameter (λ = 0.5, 0.8, 1.2), stretching sheet parameter (ω = 0.5, 0.8, 1.2) and schmidt number (sc = 0.22, 0.30, 0.60). from this table, it is observed that the skin-friction coefficient is increasing with rising values of stretching sheet parameter (ω = 0.5, 0.8, 1.2), thermophoresis parameter (nt = 0.5, 0.8, 1.2), brownian motion parameter (nb = 0.3, 0.5, 0.7), dufour number (du = 0.5, 1.0, 1.5), soret number (sr = 0.5, 1.0, 1.5), while it is decreasing with increasing values of magnetic field parameter (m = 0.5, 0.8, 1.2), permeability parameter (k = 0.3, 0.5, 0.8), casson fluid parameter(β = 0.5, 0.7, 0.9), prandtl number (pr=0.71,1.0,3.0),thermal slip parameter (δ = 0.5, 0.7, 0.9),concentration slip parameter(λ = 0.5, 0.8, 1.2), and schmidt number (sc = 0.22, 0.30, 0.60). 100 d. r. kirubaharan, a. d. subhashini, g. murali copyright ©2024 assa. adv. in systems science and appl. (2024) table-3.: skin-friction coefficient  1 1 0g        numerical values m k β ω pr nb nt du δ sc sr λ  1 1 0g        0.5 0.3 0.5 0.5 0.71 0.3 0.5 0.5 0.5 0.22 0.5 0.5 0.4366750857 0.8 0.3965760101 1.2 0.3656671086 0.5 0.4167403701 0.8 0.3856750385 0.7 0.4067685989 0.9 0.3746502354 0.8 0.4645462482 1.2 0.4845881772 1.00 0.3976868198 3.00 0.3567777081 0.5 0.4756165023 0.7 0.5056650185 0.8 0.4806547108 1.2 0.5141769232 1.0 0.4967603643 1.5 0.5256897128 0.7 0.4067832591 0.9 0.3856740633 0.30 0.4063634014 0.78 0.3677461449 1.0 0.4612576708 1.5 0.5034906340 0.8 0.3965760137 1.2 0.3518576825 the numerical values of rate of heat transfer coefficient in terms of nusselt number are displayed in table-3 for different values of prandtl number (pr = 0.71, 1.0, 3.0), thermophoresis parameter (nt = 0.5, 0.8, 1.2), brownian motion parameter (nb = 0.3, 0.5, 0.7), dufour number (du = 0.5, 1.0, 1.5), and thermal slip parameter (δ = 0.5, 0.7, 0.9). from this table, it is observed that the rate of heat transfer coefficient is gradually rising with increasing values of thermophoresis parameter (nt = 0.5, 0.8, 1.2), brownian motion parameter (nb = 0.3, 0.5, 0.7), dufour number (du = 0.5, 1.0, 1.5), while the reverse effect is observed in increasing values of prandtl number (pr = 0.71, 1.0, 3.0) and thermal slip parameter (δ = 0.5, 0.7, 0.9). study of three dimensional casson-nanofluid flow… 101 copyright ©2024 assa. adv. in systems science and appl. (2024) table-4.: rate of heat transfer coefficient  0  numerical values pr nb nt du δ  0 0.71 0.3 0.5 0.5 0.5 0.8755277123 1.00 0.8356746803 3.00 0.8015465292 0.5 0.9056751045 0.7 0.9354859826 0.8 0.9105965752 1.2 0.9456756037 1.0 0.9216576501 1.5 0.9454658522 0.7 0.8465710456 0.9 0.8216587815  table-5.: rate of mass transfer coefficient  0 numerical values nb nt sc sr λ  0 0.3 0.5 0.22 0.5 0.5 0.9156736034 0.5 0.8719546129 0.7 0.8356761852 0.8 0.9365781087 1.2 0.9512539732 0.30 0.8656708544 0.78 0.8154043804 1.0 0.9367165760 1.5 0.9554598181 0.8 0.8845881953 1.2 0.8656746818 5. conclusions in the current research work, the joint effects of soret and dufour, brownian motion on threedimensional, steady, incompressible, electrically conducting, viscous, casson-nanofluid flow towards a stretching sheet in presence of porous medium, magnetic field and multiple slip effects is investigated. for this investigation, the resultant governing flow equations are solved by using rungekutta method along with shooting technique. the aspire of this research article is to examine the impacts of m,β,k, ω, du, sr,pr, δ, sc, nb, nt, λ on flow variables such as velocity, temperature, and concentration profiles as well as skin-friction, rate of heat and mass transfer coefficients are studied through graphically and tabular forms. from this investigation, the main findings are:  an increase in the parameters of the magnetic field, casson fluid, and porous material leads to a decrease in the main velocity profiles.  as the stretching sheet parameter grows, the secondary velocity profiles similarly increase. however, the magnetic field, casson fluid, and porous medium parameters exhibit the reverse tendency.  the temperature profile of the nanofluid demonstrates enhanced performance when the thermophoresis, dufour effect, and brownian motion are increased. furthermore, it exhibits a diminishing trend when the prandtl number and thermal slip parameter are large.  the concentration profiles drop as the schmidt number, concentration slip parameter, and brownian motion parameters increase, and increase as the thermophoresis and soret parameters increase.  by using a constraining scenario, the numerical results obtained are ultimately more precise in comparison to the published findings of nadeem et al. 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(2010). slip velocity and slip layer thickness in flow of concentrated suspensions, appl. polym. sci., 70, 515–522. microsoft word 8._done_1251-article text-5754-1-18-20230405 adv syst sci appl 2023; 01; 69-90 published online at https://ijassa.ipu.ru. a novel method for predicting technology trends based on processing multiple data sources nguyen thanh viet1,2**, alla kravets1,3 1) volgograd state technical university, volgograd, russia 2) pham van dong university, quang ngai, vietnam 3) dubna state university, dubna, moscow region, russia abstract: in order to gain competing capability in conditions of quickly scientific changes, it is crucial to track the evolution of existing technologies and to explore promising and emerging technologies. moreover, numerous previous studies showed that sudden changes in r&d and patents are actually correlated with great variations in the market profit of firms. for this reason, if stock prices of an enterprise keep uptrend, then the technologies developed by considered one will be likely to become promising innovations in future. in this paper, we proposed a method to predict technology trends based on processing multiple data sources by mining web news, forecasting stock price trends of high-tech companies, and patent clustering analysis. different from other studies, our proposed method promotes an idea of predicting technology trends by forecasting stock price trend using univariate and multivariate data preparation approaches, with the utilization of bayesian optimization for exploring best hyperparameters of machine/deep learning models, also a new method for patent analysis. besides, a program system was created for analyzing word burst detection, predicting trend of stock prices, and analyzing patent applications. after collecting patents of samsung electronics co ltd, as a case study, clustering analysis is implemented on extracted noun phrases to explore technology trends developed by the company. these technology trends have recently been confirmed by domain experts in their corresponding published articles. the obtained forecast precision is about 93.8%, which proves that the proposed method gains positive reliability. keywords: technology forecast, web news, stock price, samsung, bayesian optimization, machine learning, deep learning, patent analysis, keyword extraction, clustering analysis. 1. relationship between innovation and stock prices the development of emerging technologies not only has affected current industrial production but also has generated promising manufacturing opportunities that impact significantly on social and economic factors. exploring upcoming renovation tendencies of emerging technologies prematurely is essential for governments, research and development institutes, and industrial companies in managing strategies to achieve dominant advantages in business competitiveness. moreover, numerous policy makers and organization managers are always conscious of the importance of forecasting the upcoming innovation tendencies of emerging technologies for their stable developments [18], [48]. on the other hand, over the industrial revolutions with speedy transformations in societies, capital assets are invested in positions as long as there are promising economic benefits. technologies are forming various domains of the modern life. they enhance manufacture speed, and facilitate numerous aspects of life more effective. a domain that is being revolutionized by the progresses of technologies is the stock and financial market. there are different ways of how technologies have affected and making the present shape of the financial markets, and the future state as well. * corresponging author: vietqn1987@gmail.com 70 n. t. viet, a. kravets copyright ©2023 assa adv. in systems science and appl. (2023) more specifically, in paper [54] 73 considered companies published declarations of blockchain application in period 2016-2019 were chosen as research objects, while the shortterm event analysis method was adopted to examine quantitatively the influence of blockchain technology applications on market achievement of these companies. the results revealed that after their declarations of blockchain implementation in business, considered companies attained more obvious unusual profits, which showed that the financial market responded positively to the blockchain application. concurrently, the event analysis method helped identify the effect of emerging technologies application on the company's stock values. for instance, a few authors examined empirically the effect of 3d printing application and cloud computing implementation on the stock revenue amount [19]. if stock prices indicate belief about future profits, then we can specify a correlation between innovation, which in case of successful implementation may impact positively on growth and profit of firms, and stock prices. in fact, there is a close relation between the volatility of stock prices and market shares when technology is still unpredictable through premature industrial evolution. in other words, the volatility of stock prices that reflects the uncertainty and risk attaches to technological innovation [25]. furthermore, another group of studies that combines innovation to stock price refers to the relationship between patents and market values. the argument is that if patent analysis statistics hold facts about perspectives of technology opportunities, then they will be related to present shifts in market values, which are affected by the beliefs about future growing. some authors investigate the correlation between the amount of granted patent applications of a firm, an estimation of the firm’s expenditure in invented activities (r&d), and a measure of its inventive returns, i.e., stock market price. their results showed that sudden shifts in r&d and patents are really correlated with substantial shifts in the firm market value. for instance, vitt et al. [49] investigated the relation between the patenting efforts of high-technology companies and the movement of their stock price. also, the authors demonstrated correlation between the amount of patent applications along with the variety of their respective patent categories, and the market-adjusted stock revenues. besides, the relation between those patent activity measures and volatility of the market-adjusted stock revenues, were confirmed completely. herein, utilizing the moving window method, the scholars presented numerous lagged terms of patenting activity features in order to generate fitted models. the final results confirmed the essential effect of patenting efforts on stock price change direction and its significant volatility and drift statistics. moreover, senarathne and wei [35] stated that patent applications, backward patent citations, and issued patents affect trading volume to some extent, proposing that each of these parameters could be considered as absorptive capacity parameter when new information flow is correlated with economic innovation. likewise, experimental confirmation with stock price revealed that firms succeeded in investing in comprehensive innovation attained higher stock incomes. the paper [12] proposed a model that made clear the correlation between extent of firm innovation (or absorptive capacity) and stock incomes, the change direction of which presents prospect of firm’s growth and profitability. the final results showed that the premise of ambiguityaversion is absolutely significant in specifying higher incomes with comprehensive innovation presence, and the particular specification of desirable benefit formed the magnitude of the returns. lee [20] proved that the stock price of a company utilizing a certain technology would fluctuate along with the life cycle of the considered technology. concretely, the author selected companies that typically supply augmented reality and were registered in korea kosdaq market. then these companies were grouped depending on the detailed technologies that make augmented reality. the author adopted the event study method to estimate the stock incomes in comparison with a predefined benchmark. the result stated a novel method for predicting technology trends… 71 copyright ©2023 assa. adv. in systems science and appl. (2023) that in the “peak of inflated expectations” phase, the portfolios of all listed company utilizing augmented reality mostly presented greater incomes than the benchmark. additionally, the paper [11] investigated the correlation between patent activity of the top corporate international r&d investors and their market values. the research exploited a collection of over 1,250 public multinational corporations and their intellectual property rights – trademarks and patents – registered between 2005 and 2012. finally, the study allows us to differentiate the effects of increasing market value from additional intellectual property rights. in [5] the experimental evidence recommended that stock incomes in the emerging technology conditions showed high return volatility. the article analyzed the movement, time series attributes of the relationship between daily log returns and the extent of volatility disseminations from the emergent technologies conditions to the spanish market portfolio, the spanish banking sector, and the financial market of the eu countries. on the other hand, recent studies [46], [47] present the current progress and existing problems of analyzing, forecasting technological trends and state that web news is an essential data source for investigating overall public recognition of emergent technologies. web news, which points out the latest changes of everyday life and publishes analyses throughout the world, has the attribute of real time update and becomes significant for exploring the upcoming technological development tendencies. after utilizing burst detection algorithm and clustering of burst terms, we obtained some prominent enterprises and main technology trends that occurred from january 2016 to january 2020 [46]. subsequently, based on the above discussions, the authors putted forward a hypothesis: if we are able to predict the uptrend stock price of some high-tech enterprises, then the technologies developed by the enterprise will be most likely to become promising innovations in the future. hence, we propose a method to predict technology trends by aggregating following steps: mining web news to extract significant high-technology enterprises, forecasting stock price trends of selected ones, and patent analysis of considered enterprises with uptrend stock price. the general process of the method is depicted in figure 1. figure 1. overall workflow of proposed method for technology prediction 72 n. t. viet, a. kravets copyright ©2023 assa adv. in systems science and appl. (2023) 2. reviewing studies on stock price prediction stocks (shares) are financial product distinguished by flexible trading, great income and much uncertainty, which are preferred by a lot of investors. they can gain large revenues by correctly forecasting stock price movements. stock price forecast is not new but a challenging and complicated problem, which is paid much attention from both data scientists and economists. in particular, if the goal is predicting the exact price values, then we consider it as regression problem, while if the goal is predicting the price direction, i.e., moving down or up, then we consider it as classification problem. most of existing studies are investigating the daily forecast (5 or 10 days ahead) and only a few of them are investigating the forecast during one day, e.g., 15-min or hourly forecast. with the aim of creating powerful forecast model, both linear and machine learning methods have been adopted since the past few decades. the primary attempts included fundamental analysis and technical analysis. the fundamental analysis estimates stock price depending on its native financial value, while the technical analysis considers mainly trends and charts. besides, some works utilized technical indicators as additional input features for machine and deep learning models. thereafter linear models were presented for solving stock price forecast, which mainly comprise arima (autoregressive integrated moving average) [13] and garch (generalized autoregressive conditional heteroskedasticity) [6]. nowadays, machine learning models are also utilized for predicting stock price with the success, e.g., linear regression, gradient boosting, support vector machine, etc. [53], [31]. recently various deep learning models have been recommended for solving this problem with great achievements [56], [10]. in terms of powerful capability processing big data and recognizing nonlinear correlations between input features and prediction output, for many years deep learning models have been providing better results than both linear and machine learning models on the problem of stock price forecast [15]. some recent results suggest that the effectiveness of lstm (long short-term memory) is better than non-time series and traditional machine learning models. for instance, in [29] nguyen and yoon proposed a new effective framework, called deep transfer with related stock information, which made use of transfer learning and deep neural network. to do so, a base model adopting lstm was trained with large dataset obtained from numerous stocks to initialize optimal training parameters. then, this base model is finetuned by utilizing a small dataset from target stock along with numerous input features (derived from the correlation between stocks) to increase accuracy. moreover, vo et al. [50] presented a deep responsible investment portfolio model and compared the accuracy of lstm, bidirectional lstm (bilstm) and gated recurrent unit (gru) on the stock price forecast problem. authors stated that the bilstm model reading the data backward once more would help enhance forecast performance, especially in predicting sequential data like financial time series. besides, giang et al. [41] putted forward a method adopting lstm along with deep concern in historical data of stocks and their nearest neighbors by similarity. empirical results on 4 active stocks in the united state and 3 stocks in the vietnamese stock market confirmed that the proposed model outperformed other methods in comparison (linear regression, random forest, convolutional neural network – cnn, and vanilla lstm). chen et al. [8] proposed a trend predictive model (tpm) relying on an encoder-decoder framework that forecasts the stock price movement tendency correctly. this model included two stages: in the first stage, a dual feature extraction method basing on numerous time periods was adopted to attain important information from the stock market. differing from existing methods which extract significant features only at certain time points, the authors applied the cnn and plr method to obtain both spatial short-term and temporal long-term features. then, in the second stage of the tpm, an encoder-decoder framework basing on a novel method for predicting technology trends… 73 copyright ©2023 assa. adv. in systems science and appl. (2023) dual attention mechanism was utilized to pick out appropriate dual features and forecast the stock price movement tendency. in study [22], by employing an eight-trigram feature engineering scheme of the candlestick samples between days, lin et al. constructed a new ensemble machine learning framework for daily stock movement forecast, integrating normal candlestick charting with the most recent artificial intelligence (ai) methods. more specifically, various machine/deep learning methods were also adopted to forecast the closing stock price movement. accordingly, this proposed framework helps select proper machine learning forecast method for all patterns basing on the trained outputs. additionally, the extensive surveys of jiang [16] and li [21] give latest reviews of recent works on deep learning models for stock price prediction. authors not only categorize and introduce numerous data sources, many neural network models, and frequent used metrics for evaluation, but also the reproducibility and implementation. in summary, econometric and machine learning models were the most frequently adopted methods in stock price forecast problem. nevertheless, econometric models do not provide effective accuracy for nonlinear time series problems, while common machine learning models disregard much hidden knowledge evolving through times. deep learning models can process effectively multi-period and time series data [41]. concurrently, combining various deep learning models together has recently provided better performance than single model and is becoming the dominant approach for solving stock price forecast problems [16], [55]. 3. proposed method and empirical results of stock price prediction in the course of the study, a program system was developed for analyzing the word burst detection in web news, predicting the trend of stock prices, and patent analysis. the information-functional diagram of the system is shown in figure 2. this program system consists of 3 modules: – word frequency analysis module; – stock price forecast module; – patent analysis module. the program code is written in python, with support of popular machine/deep learning libraries such as scikit-learn, keras, pandas, matplotlib, seaborn, numpy, spacy, nltk, sentence transformer, etc. all empirical runs were carried out in the interactive google colab environment. 74 n. t. viet, a. kravets copyright ©2023 assa adv. in systems science and appl. (2023) figure 2. information-functional diagram of the program system wherein the module for analyzing the words frequency in web news were described specifically in our previous article [46]. this paper describes the functions of the stock price forecast module and patent analysis module. in this manner, our recent studies [46], [47] state that web news is a crucial data source for investigating social recognition of emergent technologies. also, after utilizing burst detection algorithm on web news data and clustering of burst terms, we obtained some main technology trends that occurred from january 2016 to january 2020. moreover, some of high-technology enterprises appeared among the explored burst terms is: samsung, huawei, intel, uber, tesla, apple, and facebook. as a case study, the proposed method is applied to predict the stock price trend of samsung electronics co ltd. 3.1. raw data available stock price data of samsung (stock code 005930.ks) is downloaded from yahoo finance [33] in the period from january, 2000 to december, 2021 for further prediction model. the objective of this section is predicting the stock price overall trend (upward or downward) of samsung in the year 2022 by adopting numerous machine learning and deep learning models. the raw data consists of all financial attributes that exist in a stock market, e.g., open/ close/ adjusted close/ high/ low prices, and trading volume. the downloaded dataset of samsung stock price is demonstrated in figure 3. a novel method for predicting technology trends… 75 copyright ©2023 assa. adv. in systems science and appl. (2023) figure 3. samsung stock price (currency in korean won) dataset from 2000 to 2021 3.2. data preprocessing and preparation there are 2 common approaches to prepare data for predicting stock price: univariate and multivariate data. for univariate input data we utilize autocorrelation to calculate how past stock prices affect their upcoming prices. correlogram (acf plot) is used to help determine whether our stock price series is presenting autocorrelation in overall, also at which point we can capture the pattern of values in the series for better future prediction. by observing acf plot (figure 4) derived from statsmodels python paсkage, we set lag window as 50, which means basing on adjusted close price data of previous 50 days, the model needs to predict the adjusted close price in the 51th day (the next day). in univariate data preparation approach only the adjusted close price column is used, while all remaining columns (open/ close/ high/ low prices, trading volume) are eliminated. this feature column is utilized as both input features (e.g., the historical prices in a look-back window) and prediction target (e.g., the adjusted close price of the next day). figure 4. correlogram (max 60 lags) of samsung stock price with respect to multivariate data approach, horizon time window is defined as the future time length needed for prediction by the model. most of previous works consider long-term forecast horizon time windows, e.g., 7 days or 10 days [16], while fewer studies examine shorter horizon time windows such as 15 minutes or 1 day. however in this study, for multivariate data approach, we consider the horizon time window as 250, because it is the average number of working days within a year (the stock market is closed at weekends and on holidays). to do so, input data is financial data within a day (open/ adjusted close/ high/ low prices, trading volume), the developed model needs to predict the adjusted close price in 76 n. t. viet, a. kravets copyright ©2023 assa adv. in systems science and appl. (2023) the next 250th day. the difference between 2 data preparation approaches is illustrated in figure 5 below. figure 5. univariate and multivariate data preparation methods additionally the resulting dataset is divided into a training set with 90% of the observations that are used to train the model (herein, the training set also contains 10% of the data for validation), and the remaining 10% for testing the model. on the other hand, relating to input features of various scale values, feature normalization preprocess is utilized to assure that many machine/deep learning models can be trained effectively with performance and speed improvement. hereby feature normalization is namely the process of rescaling the input feature values between 0 and 1. 3.3. adopted machine/deep learning models for prediction furthermore, generally machine/deep learning models include two types of parameters (not for learning in training process): model design variables and hyperparameters. the model design variables consist of type of optimizer, loss function, and activation function, while hyperparameters are used to control the learning process. they may include number of layers, number of neurons, kernel size, dropout rate, and learning rate, etc. thus, in order to select effective hyperparameters of the models, the bayesian optimization (bo) is adopted with the utilization of scikit-optimize [34] and keras-tuner [30] libraries. the purpose of bayesian optimization [52] is to find and adjust the hyperparameters of a given model for effective performance on the validation dataset. this optimization method is popularly considered to be superior to grid or random searches, because bo utilizes gaussian processes to estimate an efficient probability distribution model of objective function. hence, bo method can help identify better hyperparameter settings than a grid or random search within larger variable search space and fewer iteration cycles. besides, in this paper the authors employ popular machine learning models for regression, e.g., extreme gradient boosting (xgb), gradient boosting regression (gbr), support vector regression (svr), and k-nearest neighbor (knn). on the other hand, these following models also provide high evaluation accuracy. a novel method for predicting technology trends… 77 copyright ©2023 assa. adv. in systems science and appl. (2023) 1. linear regression (with elastic net regularization [37]). linear regression is a classical linear model that fit the relation between the forecast target and the input variables, in which model parameters can be identified by the constrained least squares. while fitting linear regression model, the elastic net is a regularized regression method that combines linearly the l1 and l2 penalties of the lasso and ridge methods. in this study the search space of model hyperparameters is following: alpha {0.01, 0.02, …, 0.99, 1}, l1_ratio is a random real number in range [0, 1]. 2. feedforward neural network (ffnn). it is the most basic kind of ann (artificial neural network) where connections between the neuron nodes do not create cycle. in this network, the information flow passes only in one forward direction from the input nodes, through the hidden nodes and to the output nodes [40]. we define the search space of hyperparameters as following: number of layers {1, 2, 3}, number of neurons {100, 150, 200, 250, 300} for univariate data approach and {200, 250, 300, 350, 400} for multivariate data, dropout rate {0.01, 0.1}. 3. convolutional neural network (cnn). the model is mainly regarded for processing 2d pictures, where each group of neurons named filter executes convolution operations to various subsequent regions of the input picture. because the neurons share the same weight values, the amount of trained parameters is reduced compared to the densely connected ffnn. afterwards max or average pooling operations can be employed for multiple times to reduce the original dimensions, until the final output is concatenated to a dense layer. by applying the convolutional and pooling operations to a single dimension, the model 1d cnn [9] is often used for time series prediction and classification. the search space of 1d cnn hyperparameters is following: number of filters {150, 200, 250, 300, 350, 400} (univariate data) and {300, 350, …, 750, 800} (multivariate data); kernel size {2, 3, 4, 5} (univariate data) and {2, 3, 4} (multivariate data); dropout rate {0.01, 0.1}. 4. recurrent neural network (rnn). differing from ffnn, rnn is an ann wherein links between the nodes generate cycles along a time sequence, which helps express time changing attributes. nevertheless, standard rnn is affected by the vanishing and exploding gradient issue in training, namely, the gradients of node weights become diminishing or extending when the back propagation calculation is executed through multiple times. lstm network [24] is rnn that resolved the vanishing/exploding gradient problems, where the hidden layers are replaced by recurrent gates. lstm model has been utilized widely for many recent years in text analysis, emotion analysis, and speech recognition tasks, since the network restores its own memory of time sequences and can improve forecast accuracy. thus in this paper we apply the search space of lstm hyperparameters as following: for univariate data: number of units in the first layer {50, 100, 150, 200}, number of units in the second layer {50, 100, 150, 200}, for multivariate data: number of units {200, 250, …, 500}; dropout rate {0.01, 0.1}. 5. cnn~lstm hybrid model. in feature engineering stage cnn network is often utilized to extract the most significant features, while lstm has ability to memorize the time sequences with popular usage in time series analysis. according to these special characteristics of cnn and lstm, a hybrid model based on cnn~lstm is also constructed for solving stock price forecast. moreover, the search space of hyperparameters in this hybrid model is a combination of search spaces in the abovementioned cnn and lstm models. 3.4 evaluation metrics in this study mae, rmse, and r2 are used as metrics to evaluate and compare the performance of the abovementioned forecast models. mae (mean absolute error) computes the sum of absolute differences between true and predicted values, as shown in formula (3.1). rmse (root mean square error) firstly calculates the average of sum of square magnitude of 78 n. t. viet, a. kravets copyright ©2023 assa adv. in systems science and appl. (2023) residuals caused by estimating predicted values, and then it gets the root value, as shown in formula (3.2). mae and rmse are measurements of closeness between the predicted and the actual values, which show the reliability of forecast model. additionally, r2 metric computes the linear correlation between 2 variables in comparison and helps eliminate the impact of dimensions on regression tasks, as shown in formula (3.3). we need to explore the best model that provides small mae, rmse, and high r2. these 3 metrics are defined as follows: (3.1) (3.2) (3.3) where regarding target variable of sample i, is the true value, is the predicted value; is the mean of true value of all samples; and m is the amount of samples in the dataset. the following subsection presents results derived from different data preparation approaches and machine/deep learning models. 3.5. empirical testing results 3.5.1. univariate data approach in this univariate data approach, basing on adjusted close price data of previous 50 days, we have to explore the model that best predicts the adjusted close price in the 51th day. figure 6 demonstrates evaluation metrics of various adopted models. all positive r2 values are also multiplied by 1000 for better visualization. figure 6. evaluation metrics of various adopted models (univariate data) from the figure 6 we can define that deep learning models, i.e., ffnn, cnn, lstm, and cnn~lstm, provide quite good forecast quality (low mae, rmse, and high r2), as demonstrated by scatter plot in figure 7a. basic purpose of scatter plot is displaying and observing relation between two numerical variables (true and predicted adjusted close price). the closer the points are to a diagonal (red dashed) line, the more precise forecast stock prices are. a novel method for predicting technology trends… 79 copyright ©2023 assa. adv. in systems science and appl. (2023) (a) (b) figure 7. evaluation of prediction quality: a) scatter plot between truth values and predictions of the 4 best models; b) structure of the best adopted model cnn~lstm among them, cnn~lstm model attains the best evaluation measures, i.e., rmse = 1,283.49; mae = 969.05; r2 = 0.991. the final tuned hyperparameters are: filters = 350, kernels = 4, number of units = 200, dropout rate = 0.01. also, the structure of the best model cnn~lstm is presented in figure 7b. 3.5.2. multivariate data approach in multivariate data approach, we set horizon time window as 250 (the average number of working days within a year), as input data is financial data within a day (open/ adjusted close/ high/ low prices, trading volume), we need to select the model that best predicts adjusted close price in the next 250th day. figure 8 shows evaluation scores of various adopted models. figure 8. evaluation scores of various adopted models (multivariate data) from the above figure, we can identify the best model as elastic net, which provides the most optimal forecast testing metrics with respect to other models, i.e., rmse = 14,622.6; mae = 11,393; r2 = –1.546. in this case, the test score is not as good as in univariate data approach, because the model needs to predict 250 days ahead with the financial input of one day; while in univariate data approach the model only has to forecast the stock price in the next day, basing on values of 50 previous days. additionally, if we decrease the horizon time window, we can obtain better evaluation scores, for instance in case of time window as 50, 80 n. t. viet, a. kravets copyright ©2023 assa adv. in systems science and appl. (2023) then r2 increases to 0.679, while rmse = 8,613.5; mae = 6,761. however, as mentioned above, the goal of current research is to forecast the general upward/downward trend of stock prices, not the exact values. thus, with utilization of the selected elastic net model, the upward trend is predicted quite precisely with test data, as shown in figure 9 below. figure 9. comparison between true and predicted values by the best model elastic net 3.6. samsung stock price prediction for 2022 from previous subsections, the best models for univariate and multivariate data have been chosen. accordingly, for univariate data approach, the hybrid model cnn~lstm gets the last 50 adjusted close prices in 2021, and predicts the price in the next day. this derived value is appended to the end of input vector, which forms the input data for the next prediction phase. this forecast iteration is repeated 250 times. for the sake of clarity, the forecast algorithm from univariate data is illustrated in figure 10. figure 10. sequential forecast algorithm from univariate data in case of multivariate data, the best model elastic net gets the last 250 stock prices data in the dataset, each sample in this set provides predicted value in the next 250th day. thus, with the input stock price data of 250 days from 29/12/2020 to 30/12/2021, the output predicted values will be approximately in date range of 2022. the predicted values of year 2022, derived from the 2 best models are demonstrated in figure 11. moreover, in order to check the trend created by the two models, authors also try to calculate another forecast values: it is the combination of these 2 lines, which means every new forecast value equals linear combination of univariate and multivariate forecast values with the coefficients 0.718 and 0.068, as shown in formula (3.4). these numbers are extracted after fitting linear regression model with true samsung stock prices of the first 35 days in 2022. the new forecast values are also shown in the figure 11. herein, the formula of a novel method for predicting technology trends… 81 copyright ©2023 assa. adv. in systems science and appl. (2023) new forecast future values (yfut) basing on predicted values by multivariate (ymult) and univariate data (yuni) approaches is following: yfut = 0.718ymult + 0.068yuni (3.4) figure 11. samsung stock price prediction for 2022 additionally, in order to check the general trend of samsung stock prices in 2022 in comparison with the last 2 years, the authors depict all the last true and predicted values together in one figure 12. by observing this figure, we can conclude that generally stock prices of samsung will keep the upward trend in some time periods of 2022. figure 12. the general trend of samsung stock price 4. samsung patent analysis for inferring technology trends in 2022 arguments in section 1 have already stated that: if stock prices of an enterprise stay uptrend, then the technologies developed by the considered one will be likely to become promising innovations in future. after observing the predicted values, derived from proposed models, with high confidence we suppose that generally stock price of samsung will keep the upward trend in some time periods of 2022, especially in comparison with 2020. for this reason, in this section we collect patent applications of samsung, which were filed from 2020 to 2021, in order to predict technology trends in 2022. the metadata of samsung patent applications was downloaded from the famous platform the lens [3]. the filter query for obtaining samsung patents is: earliestprioritydate = (2020-01-01 2021-12-31), grouped by simple families, document type = (excl design_right), owner name exact = (samsung electronics co ltd). that means the earliest priority date (the 82 n. t. viet, a. kravets copyright ©2023 assa adv. in systems science and appl. (2023) earliest filing date in a family of patent applications) is from 01/01/2020 to 31/12/2021; and design right patents were also excluded. at the time of writing this paper (march, 2022), the query returns 1678 patent applications; accordingly, we can synthesize the most frequent cpc classification codes of samsung patents, which are presented in figure 13. figure 13. top cpc classification codes of developing technologies by samsung among them, the top 5 cpc classification codes are: 1. g06n3/08: computing settings based on biological models, which rely on physical entities operated by simulated intelligence to reproduce smart life arrangements; 2. h01l27: devices made up of a majority of semiconductor or solid-state components formed in common substrate; 3. h01l25: assemblies made up of a majority of individual semiconductor or solid state devices; 4. g06n3/0454: computing settings established from biological models adopting integration of many neural networks; 5. g06f3: input settings for transferring data to be processed by the computer; output settings for transferring data from processing unit to output unit. on the other hand, in order to understand technologies developed by samsung, we devise a new method for analyzing patents with words clustering (figure 14). figure 14. method for analyzing patents with words clustering herein the stage “extracting all possible noun phrases” consists of 3 steps:  part-of-speech tagging by utilizing spacy library [14];  noun phrase identification: the longest possible noun phrases sequence of many consecutive words within a sentence such that the last word in the sequence is a noun and each of the other words is either a noun or an adjective; a novel method for predicting technology trends… 83 copyright ©2023 assa. adv. in systems science and appl. (2023)  eliminating plural forms by utilizing natural language toolkit (nltk) platform [28]. the pseudo-code of noun phrase extraction algorithm is presented in figure 15. figure 15. pseudo-code of noun phrases extraction algorithm then the authors adopt sentencetransformers python framework [36] in the stage “converting extracted phrases to embedding vectors” to convert extracted phrases into embedding vectors of 768 dimensions. in the stage “ranking extracted phrases by relevance and diversity” we follow the method maximal marginal relevance, proposed by maarten grootendorst [17]. the idea of this method is minimizing redundancy (appearance of similar phrases) and maximizing the diversity of results in text summarization tasks (reducing data while increasing knowledge). concretely, the method starts by selecting the phrase that is the most similar to the document. then, it iteratively selects new promising phrases that are both similar to the document and different from the previously selected phrases by the diversity parameter, which is set to 0.3. herein authors select the most 3 meaningful phrases from each patent application, which consists of title and abstract, in the output of the stage “extracting top meaningful phrases from patent documents”. at the final result set, after eliminating duplicates we get 3496 unique noun phrases from 1678 patent applications of samsung in 2020-2021. then the silhouette score [38] is used to identify the optimal number of clusters, which is ranged from 80 to 120 (as a case study). figure 16 shows that 112 clusters give the best silhouette score. 84 n. t. viet, a. kravets copyright ©2023 assa adv. in systems science and appl. (2023) figure 16. silhouette analysis for optimal cluster number for this reason, by using the k-means algorithm we divide 3496 extracted phrases into 112 clusters, herein 15 out of these 112 clusters are demonstrated in figure 17 (only some prominent phrases of each cluster are shown). additionally, the principle component analysis (pca) method is adopted to convert these phrase vectors (768 dimensions) into 2d vectors for visualization. figure 17. visualization of 15 phrase clusters extracted from samsung patent applications subsequently, from the obtained clusters we have inferred 16 technological sectors, which are considered as technology trends in 2022. also, domain experts have recently proved these trends by their published articles, which are placed in corresponding following phrase clusters: 1) semiconductor industry, storage and computing integration, ic packaging (phrases: 3d semiconductor memory device, semiconductor package device, memory storage device, on-chip storage, semiconductor chip, semiconductor memory device, oxide semiconductor material, integrated circuit semiconductor device, in-memory operation unit, in-memory processing, processor-in-memory, in-memory computation, chip-stacked semiconductor package, package-on-package device, etc.): this can remove the delay and power consumption of data transfer, hundreds of times more efficient ai computing, hence it is especially appropriate for neural networks [4], [43]. a novel method for predicting technology trends… 85 copyright ©2023 assa. adv. in systems science and appl. (2023) 2) wireless technology, 5g network (phrases: cellular wireless communication system, wireless communication apparatus, wireless power transmitter, wireless charging method, mmwave antenna module, pre-5th-generation, 5th generation, etc.): 5g construction focuses on independent networking and millimeter-wave [4]. 3) electronic design automation – eda (phrases: semiconductor chip design, first chip structure, first eda tools, first electronic device, etc.): ai will not only be the final formation of eda development in future, but also the central solution to enhance the performance of the chip design in the upcoming years [4]. 4) autonomous driving technology (phrases: autonomous driving vehicle, automatic driving device, driving method, smart car, lidar sensor, etc.): in the future the car will certainly be an electromechanical intelligence. the current subsystems are integrated as much as possible will be a trend [4], [1], [45]. 5) bio-sensing-ppg (photoplethysmography) technology (phrases: photoplethysmography sensor, wearable device, ear wearable device, blood pressure, systolic blood pressure, mean arterial pressure, hemoglobin index, cardiac output, blood vessel, pulmonary health parameter, blood compound concentration value, etc.): with upgrading algorithm, wearable devices are more and more likely to represent ppg-based bio-sensing functioning, which can determine not only heart rate, but also other parameters like body hydration, blood lipid, blood glucose, and blood oxygen levels in the future [45]. 6) metaverse – eye tracking and 3d sensing (phrases: vr image, screen mirroring, virtual 3d graphic information, 3d navigation information, reconstructed 3d point cloud, eye tracking sensor, object recognition method, sensor-detected movement, trigger recognition, target tracking method, near-eye display apparatus, position sensor, etc.) [45]. 7) proximity sensor (phrases: first proximity sensor, second proximity sensor, photoelectric device, photoelectric conversion, etc.): the proximity sensor market was evaluated at 3790.5 million usd in 2020 and is predicted to achieve 5761.12 million usd by 2026 and expand at 7.3% cagr over the period 2021-2026 [45], [32]. 8) neuromorphic computing technology (phrases: neural network-implementing neuromorphic device, neuromorphic method, processor-implemented neural network, neural network accelerator, deep convolutional neural network, etc.): the prominent advantages of neuromorphic computing in comparison with usual procedures are fast executive speed, energy efficient, minimizing local failures, and the capacity to process and learn simply. the obvious potentials for ai is the greatest driving force for investing in neuromorphic computing [39]. 9) lithium-ion batteries – libs (phrases: lithium battery, inorganic lithium-ion conductor, lithium-air battery, lithium garnet, battery pack uptime, rechargeable battery, etc.). libs have been one of the leading energy storage solutions recently. the market share and application fields of libs have been increasing swiftly and continuously exhibiting a stable rising trend [23], [44]. 10) biometric industry (phrases: biometric authentication smart card, biometric authentication sensor, biometric authentication data, virtual enrollment fingerprint, fake fingerprint detection, fingerprint sensor, fingerprint verification, blended fingerprint image, etc.) [7]. 11) metal-air battery (phrases: metal reflective layer, metal air battery, conductive metal oxide, lithium-containing metal oxide, metal-semiconductor oxide, bump metal layer, pad metal pattern, metal interconnect pattern, etc.): the worldwide metal-air battery market size was evaluated at 540.6 million usd in 2021 and is forecasted to reach 1303.5 million usd with a 13.4% cagr by 2028 [26]. 12) natural language processing (phrases: speech signal processing method, natural language understanding, automatic speech recognition, audio processing block, voice assistant service, expressive text-to-speech system, voice recognition system): the 86 n. t. viet, a. kravets copyright ©2023 assa adv. in systems science and appl. (2023) worldwide natural language processing market is forecasted to reach at 18.78% cagr during period 2017-2023 [27]. 13) time series intelligence software (phrases: forecasting model, threshold time range, retrosynthesis prediction model, attention-based sequence-to-sequence model, timing relationship indication, input prediction time, time-series data prediction method): [42]. 14) video streaming (phrases: video quality assessment method, hdr video, streaming server, interactive streaming service, video-based point cloud compression data, video bitstream, video stream, etc.) [2]. 15) water purifier and wastewater treatment technology (phrases: purifying filter, automatic water supply device, water level sensor, wet cleaning process, chemical mechanical polishing, air purification device, water purifier body, water purifying filter, clean water flow path, fluorine-containing wastewater, etc.) [51]. 16) plasma technology (phrases: plasma electrode, plasma processing system, local plasma cleaning process, plasma treatment apparatus, plasma generator, microwave plasma source, etc.). accordingly, 15 out of these 16 technology clusters have been approved and predicted to be quite significant in the near future, hence the evaluation for prediction accuracy (precision) can be estimated as 15/16 (~93.8%). 5. conclusion prediction of technological development trends impact essentially on making strategic decisions. the early realization of potential emerging and promising technological trends helps enhance the market position and competitiveness of companies. for this reason, if innovation-oriented companies disregard emerging and promising technologies, they will not exploit the entire potentials and advantages of their own technologies or products. a domain that is being revolutionized by the progresses of technologies is the stock and financial market. there are different ways of how technologies have affected and made the present shape of the financial markets, and the future state as well. concretely, there is a strong relation between the volatility of stock prices and market shares when technology is still unpredictable through premature industrial evolution, while numerous previous studies showed that sudden shifts in r&d and patents are really correlated with substantial shifts in the firm market value. likewise, experimental confirmation with stock prices revealed that firms succeeded in investing in comprehensive innovation attained higher stock incomes. for this reason, if stock prices of an enterprise keep uptrend in some time periods in comparison with the last few years, then the technologies developed by considered one will be likely to become promising innovations in the future. relying on this assumption, also existing progress and problems stated in the previous studies [46], [47], in this paper we proposed a novel method to predict promising technology trends based on processing multiple data sources by aggregating following steps: mining web news to extract significant high-technology enterprises, forecast stock price trends of selected ones, and patent clustering analysis. different from other studies, our proposed method promotes an idea of predicting technology trends by forecasting stock price trend, with the utilization of bayesian optimization for exploring best hyperparameters of machine/deep learning models, also a new method of patent analysis. moreover, authors combined using univariate and multivariate data preparation approaches for predicting stock price trend. after collecting patent applications of samsung company (as a case study) and extracting most significant noun phrases of patent applications (consisting of titles and abstracts), the clustering method of noun phrases is utilized to explore technology trends developed by the enterprise. these technology trends have recently been confirmed by domain experts in their corresponding published articles. thus, the obtained forecast accuracy (precision) is about 93.8%, which proves that the proposed method has gained positive reliability. a novel method for predicting technology trends… 87 copyright ©2023 assa. adv. in systems science and appl. (2023) further work will focus on improving the evaluation metrics (rmse, mae, r2) by combining other relevant data sources (sentiment analysis in social networks, technological news, technical financial indicators), in order to enhance stock price prediction models using univariate and multivariate data approaches. acknowledgments the reported study was funded by rfbr, projects no. 19-07-01200 and 20-37-90092. references 1. 5 trends to watch in 2022: autonomous vehicles. (2022). [online] available https://www.gtlaw.com/en/insights/2022/1/published-articles/5-trends-to-watch-in-2022autonomous-vehicles 2. 9 streaming trends for 2022/2023: top predictions you should be thinking about. (2022). [online] available https://financesonline.com/streaming-trends/ 3. about the lens. (2022). [online] available https://about.lens.org/ 4. top 10 technology trends in the global semiconductor industry in 2022. (2022). 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(2021). prediction model for stock price trend based on recurrent neural network, journal of ambient intelligence and humanized computing, 12(1), 745–753 https://doi.org/10.1007/s12652-020-02057-0 microsoft word morozov+mv adv syst sci appl 2025; 02; 75-80 published online at https://ijassa.ipu.ru. stability criteria for periodic selector-linear difference inclusions mikhail morozov v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia abstract: the paper considers periodic selector-linear difference inclusions. lyapunov functions from the parametric class of homogeneous forms of even degree are constructed. these functions establish necessary and sufficient conditions of asymptotic stability and can be used in the development of numerical methods for investigating the stability of systems equivalent to the considered difference inclusions. using piecewise linear lyapunov functions, an algebraic criterion of asymptotic stability is obtained. an example of a mechanical system leading to periodic selector-linear difference inclusion is considered. keywords: periodic selector-linear difference inclusions, asymptotic stability, lyapunov functions, algebraic criterion of asymptotic stability 1. introduction. statement of the problem the study of discrete control systems in a number of cases leads to difference inclusions. in [9] a brief review of publications on this topic is given, see for example [1-3,5]. in [9] for periodic difference inclusions a necessary and sufficient condition of uniform asymptotic stability in the form of some limit relation is obtained on the basis of the variational approach. in [10], for periodic selector-linear difference inclusions, a class of time-periodic lyapunov functions of quasi-quadratic form, as well as parametric classes of piecewise quadratic and piecewise linear lyapunov functions were distinguished. with the help of these functions the necessary and sufficient conditions for asymptotic stability were obtained. this paper is a continuation of [9,10] and is devoted to obtaining new asymptotic stability criteria for periodic selector-linear difference inclusions based on the method of lyapunov functions. consider periodic selector-linear difference inclusion ),,()1( xsfsx  , ,...,1,0 nrxs  (1.1) where the set-valued map nn rrf 1: has the form  ,)()( ,)(:),( ssbxsbyyxsf  here ),(s )()( sns  )number natural a is ,...,1,0 ( ns  is a convex, compact set of real )( nn matrices .b the sequence of vectors  ,)(sx satisfying for all ,...1,0s inclusion (1.1), is the solution of inclusion (1.1). the definitions of asymptotic stability, uniform asymptotic stability and uniform exponential stability of inclusion (1.1) are given in [9]. the equivalence of these properties for inclusion (1.1) is proved there. keeping this in mind, further we will speak about asymptotic stability of inclusion (1.1). the problem is to construct stability criteria for inclusion (1.1) using the discrete analogue of the direct lyapunov method [4]. 76 m. morozov copyright ©2025 assa adv. in systems science and appl. (2025) 2. results in [10] parametric classes of piecewise quadratic ,),(max),( 2 1 xslxsv j mj m   )()( slnsl jj  (2.1) and piecewise linear ,),(max),( 1 xslxsv j mj m   )()( slnsl jj  (2.2) lyapunov functions were considered. in (2.1) and further we denote by , a scalar product of vectors. in [10] it was proved that for asymptotic stability of inclusion (1.1) it is necessary and sufficient that for some integer nm  there exists periodic on s (period n ) lyapunov function (2.1) or (2.2) satisfying the condition ,)( mnsrankl  ))(),...,(()( 1 slslsl m (2.3) and the inequality ),(),1(max ),( xsvysv mm xsfy   (2.4) for all ,nrx 0s and some ).10(  consider periodic on s lyapunov functions from the class of homogeneous forms of degree r2 ( hereafter r is natural) ,),(),( 1 2 ,    m j rj rm xslxsv ),()( slnsl jj  ,...1,0s (2.5) the condition of positive definiteness of function (2.5), as well as for functions (2.1), (2.2), is condition (2.3). if this condition is fulfilled, the function ),( , xsv rm will be strictly convex on nrx for every .0s theorem 2.1: inclusion (1.1) is asymptotically stable iff there exists lyapunov function (2.5) periodic in s (of period n ), its vectors )),()(( )( slnslsl jjj  mj ,1 satisfy for all 0s condition (2.3), for the function inequality (2.4) is satisfied for some 1r . proof. sufficiency. the sufficiency of the conditions of theorem 2.1 is proved by using inequality (2.4) and estimates 0 ,),( 12 2 2 , 2 1   r rm r xxsvx for the positively homogeneous strictly convex function (2.5), which were obtained in lemma [8]. necessity. to prove the necessity, we will use the statement of theorem 3.2 in [10], according to which for asymptotically stable inclusion (1.1) there exists a piecewise quadratic lyapunov function (2.1) satisfying (2.3) and the inequality ),(),1(max 1 ),( xsvysv mm xsfy   (2.6) for some 1 ).10( 1   we construct the function ),( , xsv rm by choosing as vectors ),(sl j mj ,1 in (2.5) the vectors defining the lyapunov function ).,( xsvm since the inequalities are true     m j rjrjrj mj xslmxslxsl 1 2 mj1 22 1 ,),( max ),(),(max then the estimates ),,(),(),( , xsmvxsvxsv r mrm r m  ,nrx ,...1,0s (2.7) the inequality follows from (2.6) and (2.7) stability criteria for periodic selector-linear difference inclusions 77 copyright ©2025 assa. adv. in systems science and appl. (2025) ),(),1(max ,1, ),( xsvmysv rm r rm xsfy   . choosing a positive integer ,0rr  where ,1)]ln/([ln 10  mr we obtain that the inequality ),(),1(max ,2, ),( xsvysv rmrm xsfy   is satisfied for the constructed function ),( , xsv rm , since for 0rr  the number 2 12  m satisfies the condition .10 2   the periodicity of the function ),( xsvm implies the fulfillment of the equality ).,(),( , , xsvxnsv rmrm  theorem 2.1 is proved.  lyapunov function (2.5) can be represented in the form ,)()(),( 1 ,    rn i iirm xsxsv  ),()( sns ii   ,,1 ,...,1,0 rnis  (2.8) where  ,1 ),( ri nix all possible elementary monomials of degree r2    n j r rnrji m n m i cnrmxxx nii 1 2 121 )2 ,...)(( 1 – the total number of such monomials. in accordance with the discrete analog of the direct lyapunov method [4], the construction of lyapunov function (2.8) for difference inclusion (1.1) is reduced to the search of the parameter vector ,,1 )),(( ri nix   defining the solution of the set of inequalities ,0 ,0),())(,1(),,( ,,  xxsvxsbsvxs rmrmb  ).()( ssb  (2.9) equation (2.9) specifies the conditions of strict monotonic decreasing of the function ),(, xsv rm on the solutions of inclusion (1.1). considering (2.8) and (2.9) we obtain the inequalities ,0 ,0))()())(()1((),,( 1   xxsxsbsxs pn i iiiib  ).()( ssb  (2.10) let i ni xx   1 max be cubic norm of vector .x consider the problem of mathematical programming ),,,(max max max min )()(110 xsb ssbxnsg      .1: 1 2          rn i ig  (2.11) using the scheme of the proof of theorem 1 in [11], it can be shown that for existence of lyapunov function (2.8), satisfying condition (2.10), it is necessary and sufficient that the solution of the problem (2.11) satisfies the inequality .0 to solve the minimax problem (2.11), known methods of numerical solution can be used. the periodic components of the vector , ,,1 ),( ri nix  can be searched, for example, in the form of fourier series segments [11]. through     m j ij mi dsd 1 1 max)( denote the cubic norm of the square matrix m jiijdd 1,)(  of order .m with lyapunov functions (2.2) we obtain the criterion of asymptotic stability of inclusion (1) in algebraic form. theorem 2.2: for asymptotic stability of inclusion (1.1) it is necessary and sufficient that the following conditions are satisfied: 1. for some 1p there exist periodic -)( nn matrices )(sbi ( ),()( sbnsb ii  pi ,1 ), satisfying the condition )),(),...,(()( 1 sbsbcos p ,...1,0s 78 m. morozov copyright ©2025 assa adv. in systems science and appl. (2025) 2. there exist a number ,nm  periodic -)( mn matrix )(sl ))()(( slnsl  of rank ,n and periodic -)( mm  matrices )(sdi )),()(( sdnsd ii  ,,1 pi  satisfying the conditions 1)(   sdi ( ,,1 pi  ,...1,0s ), such that the matrix relations are satisfied ),()()1()( sdslslsb ii  ,,1 pi  ,...1,0s (2.12) in (2.12) and further the dash denotes the transpose operation. proof. necessity. by theorem 3.3 [10], for asymptotically stable inclusion (1.1), there exists piecewise linear lyapunov function (2.2) satisfying condition (2.3) and inequality (2.4) for some ).10(  it follows from theorem 20.4 [12] that any non-empty closed bounded set can be approximated by a polyhedral convex set. thus, for the set )(s there exists a natural number 1p and such periodic -)( nn matrices )(sbi ( ),()( sbnsb ii  pi ,1 ), that )),(),...,(()( 1 sbsbcos p and the sets )(s and ))(),...,(( 1 sbsbco p will differ from each other as little as possible. also, the vectors ,)()( xsbsy ii  ,,1 pi  0s will be as close as desired to the elements of the set ),( xsf in (1.1). therefore, from (2.2) and (2.4) follow the inequalities ,),(max)(,),1(max 11 xslxsbxsl j mj i f mf    ,,1 pi  ,...,1,0s ,nrx hence the inequalities ,),(max)(,),1()( 1 xsqlxsbxslsb j mj j f i   ,nrx (2.13) for all sif and , ,,1( mf  ,,1 pi  ,...).1,0s applying lemma [7] to (2.13) we obtain that for each i and f ,,1( mf  ),1 pi  there exist periodic (period )n functions ),(si fj ,,1 mj  ,...,1,0s such that ,)()()1()( 1    m j ji fj f i slsqslsb     m j i fj s 1 .1)( introducing the periodic functions ),()( sqsd i fj i fj  we come to the following equations ,)()()1()( 1    m j ji fj f i slsdslsb ,,1 mf  ,,1 pi  ,...1,0s (2.14) since    m j i fj qsd 1 )( for any sif и , ,,1( mf  ,,1 pi  ,...),1,0s then each of the -)( mm  matrices ,))(()( , m jf i fji sdsd  ,,1 pi  satisfies the condition 1)(   qsdi for all ,...1,0s in matrix notation, vector equations (2.14) are equivalent to (2.12). the periodicity of the matrices )(sl and )(sdi in (2.12) follows from the periodicity of the vectors ),(sl f mf ,1 in (2.2). the necessity is proved. sufficiency. according to the conditions of the theorem )),(),...,(()( 1 sbsbcos p therefore, any vector ),( xsfy in (1.1) can be represented as ,)()( 1    p i ii xsbsy  ,0)( si ,,1 pi  .1)( 1    p i i s determining by the matrix )(sl in (2.12), satisfying condition (2.3), lyapunov function (2.2), we obtain ))()(,1(max),1(max 1),( sxsbsvysv im pi m xsfy   = (2.15) stability criteria for periodic selector-linear difference inclusions 79 copyright ©2025 assa. adv. in systems science and appl. (2025) .)(),1()(maxmax 11 sxslsb f i mfpi   since matrix equalities (2.12) are equivalent to vector equalities (2.14), for any ,f ,i s ( ,,1 mf  ,,1 pi  ,...1,0s ) the following relations are true      m j m j i fjm ji fj m j ji fj f i sdxsvxslsdxslsdxslsb 1 11 .)(),(),()(),()(),1()( hence .,1 ),,()(),1()(max 1 pixsvsdxslsb mi f i mf   therefore, the equality is valid ),,(),1()(maxmax 11 xsvxslsb m f i mfpi   (2.16) where .1)(maxmax 110   sdi pins  it follows from (2.15) and (2.16) that the function ),( xsvm satisfies inequality (2.4). therefore, by theorem 3.3 [9] inclusion (1.1) will be asymptotically stable. theorem 2.2 is proved.  4. example consider a pendulum of length l , whose suspension axis makes vertical harmonic oscillations with small amplitude  and frequency . the differential equation of motion of the pendulum is [6] ,0sin1 2 22 2        zt gl g dt zd   (3.1) where g is acceleration of free fall. as for a pendulum with a fixed pendulum axis, the vertical is the equilibrium position. but in contrast to the case of a pendulum with a fixed axis, this equilibrium position can be either stable or unstable, depending on the value of . equation (3.1) is a special case of an equation of the more general form [6] ,0)( 2 2  ztbf dt zd (3.2) where )(tf is a periodic function of time (with period 0t ), and , ib ],[ 21 bbi  is a certain parameter. just as in the examples in [9,10], it can be shown that equation (3.1) can be represented as a second-order system of differential equations with periodic coefficients, depending on the parameter .ib the discrete analog of this system is equivalent to inclusion (1.1). 4. conclusion for periodic selector-linear difference inclusion (1.1) the asymptotic stability criterion is obtained using lyapunov functions (2.5). such functions can be used in the development of numerical methods for investigating the stability of systems equivalent to difference inclusion (1.1). it is shown that the construction of lyapunov functions (2.5) reduces to the solution of the minimax problem (2.11). the algebraic criterion of asymptotic stability is established using piecewise linear lyapunov functions. the example of a mechanical problem leading to the consideration of a periodic difference inclusion is given. 80 m. morozov copyright ©2025 assa adv. in systems science and appl. 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(1970). convex analysis. new jersey, nj: princeton university press. adv syst sci appl 2025; 02:58–74 published online at https://ijassa.ipu.ru. a cauchy problem for functional inclusions with causal operators in banach spaces garik petrosyan*, maria soroka voronezh state university, voronezh, russia abstract: in this paper, we consider the cauchy problem for functional inclusions containing the sum of n causal single-valued operators and a multivalued causal operator in banach spaces. the peculiarity of single-valued operators is that, starting from the second term, each operator is represented by an integral with a variable upper limit of the previous term. such functional inclusions generalize the cauchy problem for semilinear differential equations and inclusions of arbitrary order n, as well as a cauchy-type problem in the case of inclusions and equations of fractional order not exceeding n. to solve the problem, we will apply the theory of topological degree for multivalued condensing mappings. to prove the existence of a solution, we will construct a resolving multivalued operator in the space of continuous functions corresponding to the problem. based on the properties of the resolving operator, we will prove a theorem on the existence of solutions to this problem. keywords: causal multioperator, functional inclusion, cauchy type problem, differential inclusion, fractional derivative, measure of non-compactness, fixed point, topological degree, condensing multioperator 1. introduction the study of boundary value problems for differential equations and inclusions is a complex and extremely important section of modern mathematical science, which has numerous applications and currently attracts the attention of many scientists. many physical, economic, biological and engineering problems, primarily related to the flow of processes in dynamical systems, lead to the need for new research in the field of differential equations and inclusions in abstract spaces. in turn, the development of the theory of differential inclusions is due to the fact that they are a convenient and natural tool for describing control systems of various classes, systems with discontinuous characteristics studied in various sections of optimal control theory, mathematical physics, radio physics, acoustics, etc. one of the best tools for studying this kind of problem, provide multivalued and nonlinear analysis methods that stand out as very powerful, efficient and useful. recently, the attention of many researchers (see [1], [2], [3], [4] and references therein) has been attracted to generalizations of differential equations and inclusions, namely to the class of functional equations and inclusions with causal operators. the term of a causal or volterra operator in the sense of a.n. tikhonov (see [5]), was used in mathematical physics to solve problems of differential equations, integro-differential equations, functional-differential equations with a finite or infinite delay, integral equations of volterra type, functional equations of a neutral type, etc. (see, for example, [6]). the papers [7], [8], [9], [10], [11] among others are devoted to the study of equations and inclusions with causal operators of ∗corresponding author: garikpetrosyan@yandex.ru a cauchy problem for functional inclusions with causal operators... 59 various types, theorems on the existence of solutions, description of qualitative properties of solutions and various applications. in this paper we study a cauchy type problem in banach spaces for various classes of functional inclusions with causal operators. basing on the topological degree theory for condensing multimaps we prove a global theorem on the existence of trajectories for systems governed by functional inclusions. as an application, we obtain generalizations of existence theorems for solutions of a cauchy type problem for semilinear differential inclusions an arbitrary order n and semilinear differential inclusions of a fractional order n− 1 < q < n. 2. preliminaries 2.1. measures of noncompactness we denote a banach space by e and introduce the following notation: • p (e) = {a ⊆ e : a ̸= ∅} is the collection of all non-empty subsets of e ; • pb(e) = {a ∈ p (e) : a is bounded} ; • pv(e) = {a ∈ p (e) : a is convex} ; • c(e) = {a ∈ p (e) : a is closed} ; • cv(e) = pv(e) ∩ c(e); • k(e) = {a ∈ p (e) : a is compact} ; • kv(e) = pv(e) ∩k(e). definition 2.1: (see [12]). let (a,≥) be a partially ordered set. a function β : pb(e) → a is called the measure of noncompactness (mnc) in e if for each ω ∈ pb(e) we have: β(coω) = β(ω), where coω denotes the closure of the convex hull of ω. a measure of noncompactness β is called: 1) monotone, if for each ω0,ω1 ∈ pb(e), from ω0 ⊆ ω1 follows β(ω0) ≤ β(ω1); 2) nonsingular, if for each a ∈ e and each ω ∈ pb(e) we have β({a} ∪ ω) = β(ω). if a is a cone in a banach space, the mnc β is called: 3) regular, if β(ω) = 0 is equivalent to the relative compactness of ω ∈ pb(e); 4) real, if a is the set of all real numbers r with the natural ordering. as the example of a real mnc obeying all above properties, we can consider the hausdorff mnc χ(ω): χ(ω) = inf{ε > 0, for which ω has a finite ε-net in e }. as other examples, consider the measures of noncompactness defined in the space of continuous functions c([a, b]; e) with values in the banach space e : (1) the modulus of fiber noncompactness: φ(ω) = sup t∈[a,b] χe(ω(t)), where χe is the hausdorff mnc in e and ω(t) = {y(t) : y ∈ ω}; (2) the fading modulus of fiber noncompactness: γ(ω) = sup t∈[a,b] e−ltχe(ω(t)), where l > 0 is a given number; copyright © 2025 assa. adv syst sci appl (2025) 60 g. petrosyan, m. soroka (3) the modulus of equicontinuity: modc (ω) = lim δ→0 sup y∈ω max |t1−t2|≤δ ∥y (t1)− y (t2)∥ . these measures of noncompactness satisfy all the above properties, except for the regularity. 2.2. a family of cosine operator functions definition 2.2: (see [13]). a family of bounded operators {c(t)}t∈r in a banach spacee is called a strongly continuous family of cosine operator functions if: (1) c(0) = i; (2) c(s+ t) + c(s− t) = 2c(s)c(t) for all t, s ∈ r; (3) t→ c(t)x is continuous for all x ∈ e. the family of strongly continuous sine operator functions associated with the family of cosine operator functions {c(t)}t∈r is the family of operators {s(t)}t∈r such that s(t)x = ∫ t 0 c(s)xds, x ∈ e, t ∈ r. the operator a is a generator a family of cosine operator functions {c(t)}t∈r if: ax = d2 dt2 c(t)x|t=0, for all x ∈ d(a) for which the last expression is well defined. 2.3. multivalued maps let x be a metric space and y be a normed space. let us recall some notations (see, for example, [14], [15]). definition 2.3: a multivalued map (multimap) f : x → p (y ) is said to be upper semicontinuous (u.s.c.) at a point x ∈ x, if for every open set v ⊂ y such that f(x) ⊂ v, there exists a neighborhood u(x) of x such that f(u(x)) ⊂ v. definition 2.4: a multivalued map f : x → p (y ) is called closed if its graph gf = {(x, y) : x ∈ x, y ∈ f(x)} is a closed subset of x × y. definition 2.5: a multivalued map f : x → p (y ) is called quasicompact if its restriction to each compact subset a ⊂ x is compact. definition 2.6: for a given p ≥ 1, a multifunction g : [0, t ] → k(y ) is called: • lp–integrable if it admits an lp–bochner integrable selection, i.e., there exists a function g ∈ lp ([0, t ];y ) such that g(t) ∈ g(t) for a.e. t ∈ [0, t ]; • lp–integrably bounded if there exists a function ξ ∈ lp([0, t ]) such that ∥g(t)∥y ≤ ξ(t) for a.e. t ∈ [0, t ]. copyright © 2025 assa. adv syst sci appl (2025) a cauchy problem for functional inclusions with causal operators... 61 definition 2.7: a multimap f : x ⊆ e → k(e) is called condensing with respect to a mnc β (or β– condensing) if for each bounded set ω ⊆ x which is not relatively compact, we have: β(f(ω)) ̸≥ β(ω). let d ⊂ e be a non-empty closed convex subset, v be a non-empty bounded open subset of d, β is a monotone nonsingular mnc in e and f : v → kv (d) be a u.s.c. β-condensing map such that x /∈ f (x) for all x ∈ ∂v , where v and ∂v denote the closure and the boundary of the set v in the relative topology of d. in such a setting, the (relative) topological degree degd ( i−f , v ) of the corresponding vector field i−f , satisfying the standard properties is well defined (see, for example, [14], [15]). in particular, the condition degd ( i−f , v ) ̸= 0 implies that the fixed points set fixf = {x : x ∈ f(x)} is a nonempty subset of v. application of topological degree theory leads to the following fixed point principles, which will be used in the that follows. theorem 2.1: (see [14], corollary 3.3.1). let m be a convex closed bounded subset of e and f : m → kv(m) a β–condensing multimap, where β is a monotone nonsingular mnc in e . then the fixed point set fixf is non-empty. theorem 2.2: (see [14], theorem 3.3.4). let v ⊂ d be a bounded open neighborhood of a point a ∈ v and f : v → kv(d) a u.s.c. β-condensing multimap, where β is a monotone nonsingular mnc in e , satisfying the boundary condition x− a /∈ λ(f(x)− a) for all x ∈ ∂v and 0 < λ ≤ 1. then fixf ̸= ∅ is a non-empty compact set. 2.4. causal multioperators let e be a separable banach space. by lp ([0, t ];e) , 1 ≤ p ≤ ∞, we denote the banach space of all bochner summable functions f : [0, t ] → e with the usual norm. for each subset n ⊂ lp ([0, t ];e) and τ ∈ (0, t ) we define restriction n on [0, τ ] as n |[0,τ ]= {f |[0,τ ]: f ∈ n}. definition 2.8: a multivalued map q : c ([0, t ];e) ⊸ lp ([0, t ];e) is said to be a causal multioperator, if for each τ ∈ (0, t ] and for every u, v ∈ c ([0, t ];e) the condition u |[0,τ ]= v |[0,τ ] implies that q(u) |[0,τ ]= q(v) |[0,τ ] . let us give examples of causal multioperators. example 2.1: we assume that the multimap f : [0, t ]× e → kv (e) satisfies the following conditions: (f1) for each ϕ ∈ e the multifunction f (·, ϕ) : [0, t ] → kv (e) admits a measurable selection; copyright © 2025 assa. adv syst sci appl (2025) 62 g. petrosyan, m. soroka (f2) for a.e. t ∈ [0, t ] the multifunction f (t, ·) : e → kv (e) is u.s.c.; (f3) there exists a function α ∈ lp +[0, t ], 1 ≤ p ≤ ∞, such that ∥f (t, ϕ)∥e := sup{∥z∥e : z ∈ f (t, ϕ)} ≤ α(t)(1 + ∥ϕ∥e) for a.e. t ∈ [0, t ] and ϕ ∈ e. from above conditions (f1)− (f3) it follows that the multimap pf : c([0;t ];e) → p (lp([0, t ];e)), given in the following way pf (x) = {f ∈ lp([0, t ];e) : f(t) ∈ f (t, x(t)) for a.e. t ∈ [0, t ]} is well defined (see, for example, [14], [15]). it is clear that the multioperator pf is causal. example 2.2: let f : [0, t ]× e → kv(e) be a multimap satisfying conditions (f1)− (f3) from example 2.1. suppose that {k(t, s) : 0 ≤ s ≤ t ≤ t} is a continuous (with respect to the corresponding norm) family of bounded linear operators in e and m ∈ l1([0, t ];e) is a given function. consider the volterra integral multioperator v : c ([0, t ];e) ⊸ l1 ([0, t ];e) defined as v(u)(t) = m(t) + ∫ t 0 k(t, s)f (s, u)ds, i.e., v(u) = { y ∈ l1 ([0, t ];e) : y(t) = m(t) + ∫ t 0 k(t, s)f(s)ds : f ∈ pf (u) } . it is also clear that the multioperator v is causal. 3. cauchy type problem for functional inclusions with the causal operators we will assume that the causal operator q : c ([0, t ];e) → c (lp ([0, t ];e)) satisfies the following conditions: (q1) the operator q is weakly closed in the following sense: conditions {un}∞n=1 ⊂ c ([0, t ];e) , {fn}∞n=1 ⊂ lp ([0, t ];e) , 1 ≤ p ≤ ∞, fn ∈ q(un), n ≥ 1, un → u0, fn l1 ⇀ f0 implies f0 ∈ q(u0); (q2) there exists a function α ∈ l∞ + ([0, t ]) such that ∥q (u) (t) ∥e ≤ α (t) (1 + ∥u(t)∥e) , for a.e. t ∈ [0, t ], for all u ∈ c([0, t ];e); (q3) there exists a function ω : [0, t ]× r+ → r+ such that (ω1) for all x ∈ r+ : ω(·, x) ∈ lp +([0, t ]), 1 ≤ p ≤ ∞; (ω2) for a.e. t ∈ [0, t ] a function ω(t, ·) : r+ → r+ is continuous, nondecreasing and quasihomogeneous in the sense that ω(t, λx) ≤ λω(t, x) for all x ∈ r+ and λ ≥ 0; (ω3) for each bounded set ∆ ⊂ c ([0, t ];e) we have χ (q (∆) (t)) ≤ ω ( t, sup s∈[0,t] φ (∆(s)) ) for a.e. t ∈ [0, t ], where the set ∆(s) = {y(s) : y ∈ ∆} ⊂ e and φ is the modulus of fiber noncompactness in c ([0, t ];e) . copyright © 2025 assa. adv syst sci appl (2025) a cauchy problem for functional inclusions with causal operators... 63 note that the condition (ω2) means that ω(t, 0) = 0 for a.e. t ∈ [0, t ] and as an example of such a function we can consider ω(t, x) = k(t) · x, where k(·) ∈ lp +([0, t ]). consider a linear operator s : lp([0, t ];e) → c([0, t ];e), which is causal in the sense that for every τ ∈ (0, t ] and f, g ∈ lp([0, t ];e) condition f(t) = g(t) for a.e. t ∈ [0, τ ] implies (sf) (t) = (sg) (t) for all t ∈ [0, τ ]. following [14], we impose the next conditions on operator s : (s1) for 1 ≤ p <∞ there exist d ≥ 0 such that ∥sf(t)− sg(t)∥pe ≤ d ∫ t 0 ∥f(s)− g(s)∥peds for all f, g ∈ lp([0, t ];e) and 0 ≤ t ≤ t ; if p = ∞ there exist d1 ≥ 0 such that ∥sf(t)− sg(t)∥e ≤ d1 ∫ t 0 ∥f(s)− g(s)∥eds for all f, g ∈ l∞([0, t ];e) and 0 ≤ t ≤ t. (s2) for an arbitrary compact set k ⊂ e and a sequence {fn}∞n=1 ⊂ lp ([0, t ];e) , 1 ≤ p ≤ ∞, such that {fn(t)}∞n=1 ⊂ k for all t ∈ [0, t ] the weak convergence fn l1 ⇀ f0 implies sfn → sf0 in c([0, t ];e). also we suppose that s satisfies the relation: (s3) (sf) (0) = 0 for each function f ∈ lp([0, t ];e). notice, that condition (s1) implies that the operator s satisfies the lipschitz condition: (s1′) ∥sf − sg∥c ≤ d∥f − g∥l1 . consider following important examples of special cases of the operator s. (i) the operator m1 : l 1([0, t ];e) → c([0, t ];e) defined as m1f(t) = ∫ t 0 ea(t−s)f(s)ds, where a closed linear operator a : d (a) ⊂ e → e is the infinitesimal generator of a uniformly bounded c0-semigroup {eat}t≥0. remark 3.1: the laplace transform for a real-valued function f l[f ](λ) = ∫ ∞ 0 e−λsf(s)ds, λ ∈ r, λ > 0, is a special cases of the operator s. notice, that the inverse laplace transform is a causal operator also and satisfy conditions (s1)− (s3). (ii) the operator m2 : l 1([0, t ];e) → c([0, t ];e) defined as m2f(t) = ∫ t 0 (t− s)sh(at) f(s)ds, where a closed linear operator a : d (a) ⊂ e → e is a generator of a uniformly bounded family of strongly continuous cosine operator functions {ch(at)}t≥0. copyright © 2025 assa. adv syst sci appl (2025) 64 g. petrosyan, m. soroka (iii) the operator l1 : l p([0, t ];e) → c([0, t ];e), p > 1/q, defined as l1f(t) = ∫ t 0 (t− s)q−1t1(t− s)f(s)ds, 0 < q < 1, where t1(t) = q ∫ ∞ 0 θξq(θ)u(t qθ)dθ, ξq(θ) = 1 q θ−1− 1 qψq(θ −1/q), ψq(θ) = 1 π ∞∑ n=1 (−1)n−1θ−qn−1γ(nq + 1) n! sin(nπq), θ ∈ r+, anda : d(a) → e is a closed linear operator in e generating a uniformly bounded c0semigroup {u(t)}t≥0 . (iv) the operator l2 : l p([0, t ];e) → c([0, t ];e), p ≥ 1, defined as l2f(t) = ∫ t 0 (t− s)q−1t2(t− s)f(s)ds, 0 < q ≤ 1, where t2(t) = q ∫ ∞ 0 θξq(θ)s(t qθ)dθ, ξq(θ) = 1 q θ−1− 1 qψq(θ −1/q), ψq(θ) = 1 π ∞∑ n=1 (−1)n−1θ−qn−1γ(nq + 1) n! sin(nπq), θ ∈ r+. and a : d(a) ⊂ e → e is a closed linear operator in e generating a uniformly bounded family of strongly continuous cosine operator functions {c(t)}t≥0 . let a closed linear operator a : d (a) ⊂ e → e be a generator of a uniformly bounded family operator functions {g0(t)}t≥0. we suppose the following conditions: (g1) for each t ∈ [0, t ], g0(·) are linear bounded operators, more precisely, for each x ∈ e ∥g0(t)x∥e ≤m ∥x∥e where m = sup{∥g0(t)∥; t ∈ [0,+∞)}. (g2) the operator function g0(·) are strongly continuous, i.e., functions t ∈ [0, t ] → g0(t)x are continuous for each x ∈ e. consider the operator functions gi, i = 1, 2, ..., n− 1 defined as gi(t) = ∫ t 0 gi−1 (s)ds. lemma 3.1: under conditions (g1), (g2) the operator functions gi, (i = 1, · · · , n− 1) possess the following properties: 1) for each t ∈ [0, t ], gi are linear bounded operators, more precisely, for each x ∈ e we have ∥gi(t)x∥e ≤m ∥x∥e ti i! ; copyright © 2025 assa. adv syst sci appl (2025) a cauchy problem for functional inclusions with causal operators... 65 2) the operator functions tigi(·) are strongly continuous, i.e., functions t ∈ [0, t ] → tigi(t)x are continuous for each x ∈ e. proof 1) for every fixed t ≥ 0 and for each x ∈ e, we prove the statement using the principle of mathematical induction. show it is true for i = 1. by using condition (g1) we have ∥g1(t)x∥e ≤ ∫ t 0 ∥g0(s)x∥e ds ≤m∥x∥e t. let us assume that ∥gk(t)x∥e ≤m ∥x∥e tk k! is true. now, we prove that it is true for i = k + 1. ∥gk+1(t)x∥e ≤ ∫ t 0 ∥gk(s)x∥e ds ≤m∥x∥e 1 k! ∫ t 0 sk ds =m ∥x∥e tk+1 (k + 1)! . 2) for any x ∈ e, i = 1, · · · , n− 1, and any 0 ≤ t1 ≤ t2 we have ∥gi(t1)x− gi(t2)x∥e ≤ ∫ t2 t1 ∥gi−1(s)x∥eds ≤∫ t2 t1 m ∥x∥e si−1 (i− 1)! ds ≤m ∥x∥e (t2 − t1) i i! . then if t2 → t1 we have ∥gi(t1)x− gi(t2)x∥e → 0 for every x ∈ e. by using lemma 3.1 it is easy to see that the operator functions gi, i = 0, 1, · · · , n, are satisfy conditions (s1)− (s3). consider a system governed by a functional inclusion with causal operators q and s, of the following form: x(t) ∈ g0(t)x0 + g1(t)x1 + · · ·+ gn−1(t)xn−1 + s ◦ q(x)(t), t ∈ [0, t ] (3.1) x(0) = x0, x′(0) = x1, · · · , x(n−1)(0) = xn−1. (3.2) definition 3.1: a function x ∈ c([0, t ];e) is called a mild solution to problem (3.1)-(3.2), if it satisfies inclusion (3.1) and condition (3.2). consider the multioperator γ : c([0, t ];e) ⊸ c([0, t ];e) defined as γ(x) = {x ∈ c([0, t ];e) : x(t) = g0(t)x0 + g1(t)x1 + · · ·+ gn−1(t)xn−1 + s ◦ q(x)}. it is clear that if the function x is a fixed point of the multioperator γ, then x is a solution to the problem (3.1) (3.2), therefore, our goal is to prove the existence of fixed point of the multioperator γ. copyright © 2025 assa. adv syst sci appl (2025) 66 g. petrosyan, m. soroka definition 3.2: a sequence of functions {ξn} ⊂ lp([0, t ];e) is called lp–semicompact if it is lp–integrably bounded, i.e., ∥ξn(t)∥e ≤ v(t) for a.e. t ∈ [0, t ] and for all n = 1, 2, ..., where v ∈ lp([0, t ]), and the set {ξn(t)} is relatively compact in e for a.e. t ∈ [0, t ]. lemma 3.2: (see. [14], proposition 4.2.1.). every lp–semicompact sequence is weakly compact in l1([0, t ];e). we need the following properties of the multioperator s ◦ q. since for every 1 < p ≤ ∞ : lp([0, t ];e) ⊂ l1([0, t ];e), we can formulate a modification of theorem 5.1.1 from [14] in the following form. lemma 3.3: let s : lp([0, t ];e) → c([0, t ];e) be an operator satisfying conditions (s1) and (s2). then for every lp-semicompact sequence {fn}∞n=1 ⊂ lp([0, t ];e), the sequence {sfn}∞n=1 is relatively compact in c([0, t ];e) and, moreover, the weak convergence fn l1 ⇀ f0 implies that sfn → sf0 in c([0, t ];e). theorem 3.1: (see [4]). let the multioperator q satisfy conditions (q1)–(q3) and the operator s satisfy (s1), (s2). then the composition s ◦ q : c([0, t ];e) ⊸ c([0, t ];e) is a u.s.c. multimap with compact values. let us proceed to finding conditions under which the multioperator s ◦ q will be condensing with respect to a corresponding mnc. for this we need the following statements. lemma 3.4: (see [4]). let a sequence of functions {fn}∞n=1 ⊂ lp([0, t ];e) be lp-integrally bounded and there exist a function υ ∈ lp +([0, t ]) such that χ ({fn (t)}∞n=1) ≤ υ(t) for a.e. t ∈ [0, t ]. if an operator s satisfies conditions (s1) and (s2), then for 1 ≤ p <∞ we have χ ({sfn (t)}∞n=1) ≤ ( 4pd ∫ t 0 υp(s)ds )1/p , and for p = ∞ χ ({sfn (t)}∞n=1) ≤ 2d1 ∫ t 0 υ(s)ds, where d,d1 are the constants from condition (s1). consider the measure of noncompactness ν in the space c([0, t ];e) with values in the cone r2 +. on a bounded subset of ω ⊂ c([0, t ];e) we define the values of ν as follows: ν(ω) = (γ (ω) ,modc (ω)) , where modc is the modulus of equicontinuity, γ is the fading modulus of fiber noncompactness γ(ω) = sup t∈[0,t ] e−ltχ(ω(t)). copyright © 2025 assa. adv syst sci appl (2025) a cauchy problem for functional inclusions with causal operators... 67 the constant l > 0 is chosen so that max{q1, q2} < 1, where q1 = sup t∈[0,t ] ( 4d1/p (∫ t 0 e−pl(t−s)ωp (s, 1) ds )1/p ) , q2 = sup t∈[0,t ] ( 2d1 ∫ t 0 e−l(t−s)ω (s, 1) ds ) , where the constants d,d1 are from condition (s1), ω is a function from condition (q3). it is easy to see that the mnc ν is monotone, nonsingular, and algebraically semi-additive. it follows from the arzela-ascoli theorem that it is also regular. theorem 3.2: let a causal multioperator q : c([0, t ];e) ⊸ lp ([0, t ];e) satisfy conditions (q2) and (q3) and for a causal operator s : lp ([0, t ];e) → c ([0, t ];e) the conditions (s1)–(s3) be satisfied. then the multioperator γ is ν-condensing. proof by lemma 3.1 it is suffices to prove the assertion of the theorem for the multioperator s ◦ q. let ω ⊂ c([0, t ];e) be a bounded set such that ν (s ◦ q (ω)) ≥ ν (ω) . (3.3) let us show that the set ω is relatively compact. inequality (3.3) means that γ({s ◦ q (ω)}) ≥ γ(ω). (3.4) applying the condition (q3) and by using the properties of the function ω, we obtain for a.e. t ∈ [0, t ] χ ({f(t) : f ∈ q (ω)}) ≤ ω ( t, sup s∈[0,t] φ ({y(s) : y ∈ ω}) ) = ω (t, φ ({y : y ∈ ω})) = ω ( t, elte−ltφ ({y : y ∈ ω}) ) ≤ ω ( t, eltγ (ω) ) ≤ ω ( t, elt ) · γ (ω) . at first, we consider the case 1 ≤ p <∞. by lemma 3.4 we have for each t ∈ [0, t ] : χ ({sf(t) : f ∈ q (ω)}) ≤ ( 4pd ∫ t 0 ωp ( s, els ) ds · γp (ω) )1/p ≤ 4d1/p (∫ t 0 eplsωp (s, 1) ds )1/p · γ (ω) . inequality (3.4) and the last inequality imply the following estimate γ(ω) ≤ sup t∈[0,t ] ( 4d1/p (∫ t 0 e−pl(t−s)ωp (s, 1) ds )1/p ) γ (ω) = q1 · γ (ω) , therefore γ (ω) = 0, copyright © 2025 assa. adv syst sci appl (2025) 68 g. petrosyan, m. soroka thus φ (ω) = 0 for all t ∈ [0, t ]. let us turn to the case p = ∞. by lemma 3.4 we have for each t ∈ [0, t ] : χ ({s(f)(t) : f ∈ q (ω)}) ≤ 2d1 ∫ t 0 ω ( s, els ) ds · γ (ω) ≤ 2d1 ∫ t 0 elsω (s, 1) ds · γ (ω) ; inequality (3.4) and the last inequality imply the following γ(ω) ≤ sup t∈[0,t ] ( 2d1 ∫ t 0 e−l(t−s)ω (s, 1) ds ) γ (ω) = q2 · γ (ω) , therefore γ (ω) = 0, thus φ (ω) = 0 for each t ∈ [0, t ]. now we will show that the set ω is equicontinuous. we take sequences {yn}∞n=1 ⊂ ω, n ≥ 1, and {fn}∞n=1, fn ∈ q(yn). from conditions (q2) and (q3) it follows that the sequence {fn}∞n=1 is lp-semicompact, and therefore by lemma 3.3 the sequence {sfn}∞n=1 is relatively compact. hence modc({sfn}∞n=1) = 0. thus ν (s ◦ q (ω)) = (0, 0), but then it follows from the inequality (3.3) that ν(ω) = (0, 0), and the last expression yields that the set ω is relatively compact. to prove the main theorem of the paper, we need the following statements, known as the gronwall bellman lemma and the generalized gronwall bellman lemma (see [16]). lemma 3.5: let v(t) and f(t) be nonnegative continuous functions on the segment [a, b], moreover v(t) ≤ c+ ∫ t a f(s)v(s)ds, t ∈ [a, b], where c is a positive constant. then for each t ∈ [a, b] the inequality v(t) ≤ ce ∫ t a f(s)ds, holds. lemma 3.6: let h(t), u(t) and v(t) be nonnegative functions integrable on [a, b] satisfying the inequality: v(t) ≤ u(t) + ∫ t a h(s)v(s)ds, t ∈ [a, b]. then the following inequality holds: v(t) ≤ u(t) + ∫ t a e ∫ t a h(θ)dθh(s)u(s)ds, t ∈ [a, b]. copyright © 2025 assa. adv syst sci appl (2025) a cauchy problem for functional inclusions with causal operators... 69 theorem 3.3: let a causal multioperator q : c([0, t ];e) → cv(lp([0, t ];e)), 1 ≤ p ≤ ∞, satisfy conditions (q1)–(q3) and a linear causal operator s : lp([0, t ];e) → c([0, t ];e) satisfy conditions (s1)–(s3). then the set σ of all solutions to problem (3.1)-(3.2) is a non-empty compact set. proof let us show that the set of all solutions x ∈ c([0, t ];e) of a one-parameter inclusion x ∈ λγ(x), λ ∈ [0, 1], (3.5) is a priori bounded. we divide the proof into three cases: p = 1, 1 < p <∞, p = ∞. let p = 1, if a function x ∈ c([0, t ];e) satisfies condition (3.5), then by lemma 3.1 for each t ∈ [0, t ], we have the following estimates: ∥x(t)∥e ≤ ∥g0(t)x0∥e + ∥g1(t)x1∥e + · · ·+ ∥gn−1(t)xn−1∥e +d ∫ t 0 ∥f(s)∥eds ≤ m∥x0∥e +mt∥x1∥e + · · ·+m tn−1 (n− 1)! ∥xn−1∥e +d ∫ t 0 ∥f(s)∥eds ≤ m∥x0∥e +mt∥x1∥e + · · ·+m t n−1 (n− 1)! ∥xn−1∥e +d ∫ t 0 ∥f(s)∥eds, where f ∈ q(x) and, therefore, by condition (q2) : ∥f(s)∥e ≤ α(s)(1 + ∥x(s)∥e), we have ∥x(t)∥e ≤m∥x0∥e +mt∥x1∥e + · · ·+m t n−1 (n− 1)! ∥xn−1∥e+ d ∫ t 0 α(s)(1 + ∥x(s)∥e)ds ≤m∥x0∥e +mt∥x1∥e + · · ·+m t n−1 (n− 1)! ∥xn−1∥e+ d∥α∥l∞t +d∥α∥l∞ ∫ t 0 ∥x(s)∥eds. the last expression is a non-decreasing function of t, so we get ∥x(t)∥e ≤m∥x0∥e +mt∥x1∥e + · · ·+m t n−1 (n− 1)! ∥xn−1∥e+ d∥α∥l∞t + ∫ t 0 d∥α∥l∞∥x(s)∥eds. this means that the function v(t) = ∥x(t)∥e satisfies the estimate v(t) ≤m∥x0∥e +mt∥x1∥e + · · ·+m t n−1 (n− 1)! ∥xn−1∥e+ d∥α∥l∞t + ∫ t 0 d∥α∥l∞v(s)ds. copyright © 2025 assa. adv syst sci appl (2025) 70 g. petrosyan, m. soroka applying lemma 3.5, we obtain the required a priori boundedness: v(t) = ∥x(t)∥e ≤ ued∥α∥l∞ = γ1, where u =m∥x0∥e +mt∥x1∥e + · · ·+m t n−1 (n− 1)! ∥xn−1∥e + ∥xn∥e +d∥α∥l∞t. then ∥x∥c = supt∈[0,t ] ∥x(t)∥e ≤ γ1. for the case 1 < p <∞, we have ∥x(t)∥e ≤m∥x0∥e +mt∥x1∥e + · · ·+m t n−1 (n− 1)! ∥xn−1∥e + ( d ∫ t 0 ∥f(s)∥peds ) 1 p ≤ m∥x0∥e +mt∥x1∥e + · · ·+m t n−1 (n− 1)! ∥xn−1∥e + ( d ∫ t 0 αp(s)(1 + ∥x(s)∥e)pds ) 1 p ≤ m∥x0∥e +mt∥x1∥e + · · ·+m t n−1 (n− 1)! ∥xn−1∥e+( d ∫ t 0 αp(s)ds+d ∫ t 0 αp(s)∥x(s)∥peds ) 1 p ≤ m∥x0∥e +mt∥x1∥e + · · ·+m t n−1 (n− 1)! ∥xn−1∥e+( d ∫ t 0 αp(s)ds )1/p + ( d ∫ t 0 αp(s)∥x(s)∥peds ) 1 p . let us introduce the following notation: c0 =m∥x0∥e +mt∥x1∥e + · · ·+m t n−1 (n− 1)! ∥xn−1∥e +d1/p∥α∥lp , h(s) = d1/pα(s). then we get: ∥x(t)∥e ≤ c0 + (∫ t 0 hp(s)∥x(s)∥peds )1/p . let v(t) = ∥x(t)∥pe, then from the last inequality we obtain the estimate: v(t) ≤ 2pcp0 + 2p ∫ t 0 hp(s)v(s)ds. now applying lemma 3.6 to the last inequality, we get v(t) = ∥x(t)∥pe ≤ 2pcp0 ( 1 + ∫ t 0 e2 p ∫ t 0 hp(θ)dθhp(s)ds ) . then we have the final estimate for 1 < p <∞ : ∥x(t)∥e ≤ 2c0 p √ 1 + ∫ t 0 e2 p ∫ t 0 hp(θ)dθhp(s)ds = γ2. copyright © 2025 assa. adv syst sci appl (2025) a cauchy problem for functional inclusions with causal operators... 71 then ∥x∥c = supt∈[0,t ] ∥x(t)∥e ≤ γ2. for the case p = ∞, in the same way as in the case p = 1, the following estimate holds: ∥x∥c ≤ u1e d1∥α∥l∞ = γ3, where u1 =m∥x0∥e +mt∥x1∥e + · · ·+m t n−1 (n− 1)! ∥xn−1∥e +d1∥α∥l∞t. now, if we take r ≥ max{γ1, γ2, γ3}, we can guarantee that the set v ⊂ c([0, t ];e), given as v = {x ∈ c([0, t ];e) : ∥x∥c < r}, contains all solutions of inclusion (3.5). thus, the multioperator γ satisfies on ∂v the condition of theorem 2.2 with a = 0, hence the set of its fixed points is nonempty and compact. 4. examples we will need the following notion (see [17, 18]). definition 4.1: the gerasimov-caputo fractional derivative of an order q ≥ 0 of a function g ∈ cn([0, t ];e) is the function cdq 0g of the following form: cdq 0g(t) = 1 γ(n− q) ∫ t 0 (t− s)n−q−1g(n)(s) ds, n = [q] + 1, where γ is the euler’s gamma-function γ(q) = ∫ ∞ 0 xq−1e−xdx. notice, that the last definition includes the case, when q = n ∈ n. consider the following system governed by a differential inclusion in a separable banach space e : cdq 0y (t) ∈ ay (t) + f (t, y(t)) , t ∈ [0, t ] , n− 1 < q ≤ n, (4.6) y (0) = y0, y′(0) = y1, ..., y(n−1)(0) = yn−1. (4.7) let a multimap f : [0, t ]× e → kv(e) is such that: (f1) for each ψ ∈ c([0, t ];e) the multifunction f (·, ψ) : [0, t ] → kv (e) admits a measurable selection; (f2) for a.e. t ∈ [0, t ] the multimap f (t, ·) : e → kv (e) is u.s.c.; (f3) there exists a function α ∈ l∞ + [0, t ] such that ∥f (t, ψ)∥e := sup{∥z∥e : z ∈ f (t, ψ)} ≤ α(t)(1 + ∥ψ∥e) for a.e. t ∈ [0, t ] and for all ψ ∈ e; (f4) there exists a function ωf : [0, t ]× r+ → r+ satisfying the conditions (ω1)-(ω3) such that for each bounded set ω ⊂ c([0, t ];e) we have χ (f (t,ω)) ≤ ωf (t, φ (ω)) for a.e. t ∈ [0, t ]. copyright © 2025 assa. adv syst sci appl (2025) 72 g. petrosyan, m. soroka the fact that the superposition multioperator pf : c([0, t ];e) ⊸ lp([0, t ];e), p > 1/q, satisfies condition (q1) can be verified as in paper [19]. conditions (q2) and (q3) for pf follow from (f3) and (f4), respectively. if 0 < q ≤ 1 we will suppose that (ai) a : d (a) ⊂ e → e is a linear closed operator generating a uniformly bounded c0semigroup {u(t)}t≥0. then g0(t) = ∫ ∞ 0 ξq(θ)u(t qθ)dθ, and a function y ∈ c([0, t ];e) is a mild solution to problem (4.6)–(4.7), if it can be presented in the form: y(t) = g0(t)y0 + ∫ t 0 (t− s)q−1t1(t− s)f(s)ds, where f ∈ pf (y). we can consider the relation (4.6) as a special case of functional inclusion (3.1) with q = pf , and the cauchy type operator s = l1. as a direct consequence of theorem 3.3, we obtain the following result. theorem 4.1: suppose that conditions (ai), (f1)–(f4) hold and 0 < q ≤ 1. then the set of solutions to problem (4.6)-(4.7) is a non-empty compact subset of the space c([0, t ];e). now, let 1 < q ≤ 2. we will suppose that (aii) a : d (a) ⊂ e → e be a closed linear operator in e generating a uniformly bounded family of strongly continuous cosine operator functions {c(t)}t≥0 . then g0(t) = ∫ ∞ 0 ξq(θ)c(t q/2θ)dθ, g1(t) = ∫ t 0 g0(s)ds, and a function y ∈ c([0, t ];e) is a mild solution to problem (4.6)–(4.7), if it can be presented in the form: y(t) = g0(t)y0 + g1(t)y1 + ∫ t 0 (t− s)q/2−1t2(t− s)f(s)ds, where f ∈ pf (y). we can consider the relation (4.6) as a special case of functional inclusion (3.1) with q = pf , and the cauchy type operator s = l2. as a direct consequence of theorem 3.3, we obtain the following result. theorem 4.2: suppose that conditions (aii), (f1)–(f4) hold and 1 < q ≤ 2. then the set of solutions to problem (4.6)-(4.7) is a non-empty compact subset of the space c([0, t ];e). finally, in the general case n− 1 < q < n, we will suppose that (aiii) a : d (a) ⊂ e → e generating a uniformly bounded family operator functions {g0(t)}t≥0 such that (g01) g0(0) = i; copyright © 2025 assa. adv syst sci appl (2025) a cauchy problem for functional inclusions with causal operators... 73 (g02) the operator function g0(·) is strongly continuous, i.e., the function t→ g0(t)x, t ≥ 0 is continuous for each x ∈ e. then, the operator functions gi, i = 1, 2, ..., n− 1, will be define as gi(t) = ∫ t 0 gi−1 (s)ds. a function y ∈ c([0, t ];e), is a mild solution to problem (4.6)–(4.7), if it can be presented in the form: y(t) = g0(t)y0 + g1(t)y1 + · · ·+ gn−1(t)yn−1 + l−1 [ (λq − a)−1l [f ] (λ) ] (t), where f ∈ pf (y), l is a laplace transform and λq ∈ ρ(a). we can consider the relation (4.6) as a special case of functional inclusion (3.1) with q = pf , and the cauchy type operator s = l−1 [(λq − a)−1l [f ] (λ)] . as a direct consequence of theorem 3.3, we obtain the following result. theorem 4.3: suppose that conditions (aiii), (f1)–(f4) hold. then the set of solutions to problem (4.6)(4.7) is a non-empty compact subset of the space c([0, t ];e). acknowledgements the work was supported by the russian science foundation (project no. 23-71-10026). references 1. lupulescu, v. (2008). causal functional differential equations in banach spaces, nonlinear anal., 69(12), 4787–4795. 2. obukhovskii, v. & zecca, p. (2011). on certain classes of functional inclusions with causal operators in banach spaces, nonlinear anal., 74(8), 2765–2777. 3. kulmanakova, m. m. & ulyanova, e. l. (2019). o razreshimosti kauzalnih funkcionalnih vkluchenii s beskonechnim zapazdivaniem [on the solvability of causal functional inclusions with infinite delay]. tambov university reports. series: natural and technical sciences, 24(127), 293–315, [in russian]. 4. afanasova, m., obukhovskii, v., petrosyan g. (2021). a controllability problem for causal functional inclusions with an infinite delay and impulse conditions, advances in systems science and applications, 21(3), 40–62. 5. tychonoff, a. n. (1938). sur les equations fonctionnelles de volterra et leurs applications a certains problemes de la physique mathematique, bulletin de i’universite d’etat de moscou. ser. internationale. section a. mathematiques et mecanique, 1(8), 1–25, [in french]. 6. corduneanu, c. (2002). functional equations with causal operators. stability and control: theory, methods and applications. london, uk: taylor and francis. 7. bulgakov, a. i. & maximov, v. p. (1981). functional and functional-differential inclusions with volterra operators, differential equations, 17(8), 1362–1374. 8. drici, z., mcrae, f. a. & vasundhara devi, j. (2005). differential equations with causal operators in a banach space, nonlinear anal., 62(2), 301–313. 9. drici, z., mcrae, f. a. & vasundhara devi, j. (2006). monotone iterative technique for periodic boundary value problems with causal operators, nonlinear anal., 64(6), 1271–1277. copyright © 2025 assa. adv syst sci appl (2025) 74 g. petrosyan, m. soroka 10. jankowski, t. (2008). boundary value problems with causal operators, nonlinear anal., 68(12), 3625–3632. 11. petrosyan g. & soroka m. (2024). a cauchy type problem for causal functional inclusions in banach spaces, advances in systems science and applications, 24(1), 40–57. 12. ahmerov, r. r., kamenskii, m. i., potapov, a. s., rodkina, a. e. & sadovskii, b. n. (1992). measures of noncompactness and condensing operators. berlin, germany: birkhauser. 13. sova, m. (1966). cosine operator functions, rozprawy mat., 49. 14. kamenskii, m., obukhovskii, v. & zecca, p. (2001). condensing multivalued maps and semilinear differential inclusions in banach spaces. berlin, germany: walter de gruyter. 15. obukhovskii, v. & gel’man, b. (2020). multivalued maps and differential inclusions. elements of theory and applications. hackensack, nj: world scientific publishing co. pte. ltd. 16. hartman, p. (1964). ordinary differential equations. new york, ny: john wiley and sons. 17. kilbas, a. a., srivastava, h. m. & trujillo, j. j. (2006). theory and applications of fractional differential equations. amsterdam, the netherlands: north-holland mathematics studies. 18. podlubny, i. (1999). fractional differential equations. san diego, ca: academic press. 19. petrosyan, g. g. (2015). teorema o slaboi zamknutosti superpozicionnogo multioperatora [a theorem on the weak closure of superposition multioperator], tambov university reports. series: natural and technical sciences, 20(5), 1355–1358, [in russian]. copyright © 2025 assa. adv syst sci appl (2025) introduction preliminaries measures of noncompactness a family of cosine operator functions multivalued maps causal multioperators cauchy type problem for functional inclusions with the causal operators examples microsoft word real-time flight expariment_1 adv syst sci appl 2023; 03; 127-138 published online at https://ijassa.ipu.ru. quality control of a real-time flight experiment using neural networks vasiliy akhrameev1, eugeniy tsvetkov1, alexander pashchenko2* 1) moscow institute of physics and technology, moscow, russia 2) v.a. trapeznikov institute of control sciences of ras, moscow, russia abstract: the paper presents some investigations on automatic processing of flight information. the aim of this investigation is to reduce the cost and timing required for aviation equipment testing. the algorithms are developed for recognizing various test modes as well as evaluating the correctness of their implementation. while developing the algorithms a multilayer perceptron and a kohonen network were used as basics. the results of experiments are presented when applying these algorithms for real testing process. keywords: automatic processing of flight information, test modes, flight experiment, multilayer perceptron, kohonen network 1. introduction one of the ways to reduce the time and cost of flight tests is the use of automatic flight information processing tools. this is especially important, since the amount of information collected during the flight experiment is growing more and more and requires a lot of time and human resources for secondary data processing. modern methods of quantifying the stability and controllability of an aircraft are based on a comprehensive analysis of the records of its movement obtained in flight when the pilot performs special control maneuvers [1], [2], which are one of the main elements of the flight task of a test pilot. for qualitative and quantitative assessment of the dynamic stability and controllability of the aircraft, the following test control modes are usually performed: • impulse deviations of control surfaces; • stepwise deviations of control surfaces; • two-way deviations of control surfaces. to carry out secondary processing of experimental data in real time, it is necessary to recognize the test modes and, based on the recognition results, include certain secondary processing algorithms. this task should be solved completely automatically and as secretly as possible from the operator who solves the main tasks of controlling the flight experiment. the main feature of the flight modes during the flight experiment is their determinacy, since the flight task is negotiated in advance before departure, the required values of the deviation of the aircraft controls are specified. typically, a test pilot is specifically trained to perform the modes. then the problem of recognition is reduced to the well-known problem of coherent reception in statistical radio engineering 2. problem statement in order to conduct secondary processing of the results of the flight experiment, it is required to develop an algorithm for recognizing test modes and assessing the correctness of its implementation. * corresponding author: paschenko_alex@mail.ru 128 akhrameev v., tsvetkov e., pashchenko a. copyright ©2023 assa adv. in systems science and appl. (2023) the work sets the following tasks: 1) development of an algorithm for finding flight modes corresponding to a given reference mode, based on the use of neural networks of two types: a multilayer perceptron and a kohonen network; 2) development of an algorithm for training networks; 3) application of the resulting algorithm for real-time recognition; 4) investigation of the recognition system for the stability of decision-making when submitting noisy and highly distorted records. 3. main tasks of recognition system construction each recognition system is adapted to recognize only this type of objects or phenomena. let's consider the main tasks that arise in the process of designing and building recognition systems. the first task is to study in detail the objects that will be recognized by the system being created. the main purpose of this task is to find out the features of the studied objects and find out which features unite and which distinguish objects from each other. the second task is to divide the entire set of recognizable objects or phenomena into separate classes. to do this, you need to choose the correct classification method based on the requirements that apply to the recognition system itself. the main criterion here is a set of decisions made when recognizing unknown objects and phenomena. in the third task, the goal is to compile a dictionary of features used both for a priori description of the class and for classification of each unknown object submitted to the recognition system input. the dictionary can only include features about which information can be obtained sufficient to describe classes in the language of these features. it is impractical to include all the signs in the dictionary, even if the information about them is received in full. when designing a dictionary in the language of features, classes are described and after selecting recognition algorithms, the informativeness of each feature is evaluated [3]. thus, it is possible to weed out the least useful signs. after that, taking into account the restrictions imposed on the creation of technical means of obtaining information, a final decision is made on the composition of the dictionary of features for the operation of the recognition system. the fourth task involves the description of classes of objects based on the resulting dictionary of features. the task of describing classes is as follows. let the dictionary contain an ordered set of features 𝑋 , . . . , 𝑋 . this set of parameters is a vector 𝑋 = {𝑋 , . . . , 𝑋 } defined in the feature space of the recognition system. each point of this space represents recognizable objects. the whole set of objects is divided into classes 𝛺 , . . . , 𝛺 , each class in the feature space corresponds to a region 𝐷 , 𝑖 = 1, . . . , 𝑚, so that a point in the feature space representing an object that has features 𝑋 , . . . , 𝑋 and belongs to the class 𝛺 belongs to the region 𝐷 . it is also required to construct separating functions 𝐹 (𝑋 , . . . , 𝑋 ), 𝑖 = 1, . . . , 𝑚 , having the following property – if the object belongs to the class 𝛺 , then the value of 𝐹 (𝑋 , . . . , 𝑋 ) should be the largest. the separating function 𝐹 must take the largest value for all objects 𝑋 of the class 𝛺 . 𝐹 (𝑋 ) > 𝐹 (𝑋 ), 𝑞, 𝑔 = 1, … , 𝑚, 𝑞 ≠ 𝑔. (1) thus, in the feature space of the recognition system, the boundary of the partition, called the decisive boundary between the regions 𝐷 corresponding to the classes 𝛺 , is expressed by the equation 𝐹 (𝑋) − 𝐹 (𝑋) = 0 (2) quality control of a real-time flight experiment… 129 copyright ©2023 assa. adv. in systems science and appl. (2023) the processing of information about recognized objects and the description of classes is a very large and time–consuming part in the task of creating a recognition system associated with a detailed study of the features of objects. the fifth task is to develop recognition algorithms that determine the class to which the recognized object belongs. recognition methods are based on comparing some measure of proximity or similarity of the classified object with each class. thus, if some measure of the proximity of a given object to any class 𝛺 exceeds the measure of proximity to other classes, then a decision is made on whether this object belongs to the class 𝛺 the sixth task is understood as the development of additional methods for controlling the recognition system to optimize the functioning and maximize the selected criterion for the quality of the system. the selection of criteria by which the effectiveness of the system will be evaluated is the last seventh task. performance indicators, such as recognition probability, decision-making time and required costs, are determined based on experimental studies of the recognition system using physical and mathematical models. these seven tasks fully describe the stages of the recognition system development. in this paper, the recognition system is based on the operation of two types of neural networks: a multilayer perceptron and a kohonen network [3]. 4. relevance of the choice of neural networks for creating a recognition system artificial neural network (ins) is a mathematical model, as well as its software or hardware implementation, built on the principle of organization and functioning of biological neural networks – networks of nerve cells of a living organism. this concept arose when studying the processes occurring in the brain, and when trying to simulate these processes. the first such attempt was the neural networks of w. mcculloch and w. pitts [4], [5]. when performing a flight experiment, each maneuver corresponds to a certain input and output image. obtaining the characteristics of a system from a set of input and output signals is a nonlinear task. as described above, algorithms based on the use of multilayer perceptrons are suitable for such tasks. kohonen networks are used to solve the recognition problem. kohonen networks are a type of networks that learn "without a teacher". the network consists of one layer of adjustable weights (fig. 1) and functions in the spirit of a strategy according to which the winner takes everything, i.e. only one neuron is excited, the remaining outputs of the layer are suppressed, the outputs of the network are: 𝑦 = 𝑤 𝑥 , ∃𝑘: 𝑦 = 1, 𝑦 = 0. (3) fig. 1. kohonen network 130 akhrameev v., tsvetkov e., pashchenko a. copyright ©2023 assa adv. in systems science and appl. (2023) the kohonen network classifies input vectors into groups of similar ones, adjusting the weights so that input images belonging to the same class will activate the same output neuron 𝑦 . it is essential to note that all input vectors {𝑥 } must be normalized. at each training step, the output neuron with the maximum value is selected. its weights are adjusted as follows: 𝑤 = 𝑤 + 𝜇(𝑥 − 𝑤 ) (4) where 𝑤 – is the new weight value. adjusting the weights in a similar way reduces to minimizing the difference between the input vector and the vector of weights of the selected neuron. at the end of training, neurons corresponding to two close classes can be at a significant "distance" from each other. in order for them to be located side by side, you can use a learning algorithm that changes the weights of not only the winning neuron, but also the weights of neighboring neurons. we introduce a function having a maximum equal to one at zero and decreasing to zero at infinity. a typical example of such a function satisfying the requirements is the gaussian function: in this work, a network with output neurons arranged in a row is used. therefore, the algorithm can be simplified to a discrete model in which, in addition to the winning neuron, the weights of only two neighboring neurons change: 𝑤 ± = 𝑤 ± + 0.1 × 𝜇 𝑥 − 𝑤 ± (5) thus, the output neurons corresponding to one class of images will be adjacent. compared to a multilayer perceptron, the kohonen network is resistant to noise of small amplitude, which is always present in real data obtained in a flight experiment. as a result, the flight mode search algorithm uses a combination of two networks. the first model, kohonen's neural network, is used to compress the input vector. the second network, the multilayer perceptron, determines how much this mode corresponds to a certain reference image. 5. results of mathematical modeling 5.1. setting up a numerical experiment in this work, a computer simulation of the giving of elevators was carried out. the longitudinal channel of the aircraft movement in this case is described by the equations 𝐼 �̇� = 𝑀 𝛼 + 𝑀 𝜔 + 𝑀 в𝛿 , (6) �̇� = 𝜔 − 𝑛 − cos 𝛼 𝑔/𝑉, (7) 𝑛 = 𝐶 𝛼 + 𝐶 𝛿 𝑞𝑆/𝑚𝑔. (8) in all simulations, the time values of four parameters were recorded: angle of attack – 𝛼, angular velocity – 𝜔 , vertical acceleration – 𝑛 , elevator deflection – 𝛿 . the resulting recordings were also overlaid with noise with a relative amplitude of 5%. fig. 2 shows an example of the result of computer simulation of the parameters of the longitudinal motion of the aircraft in the test mode when performing an altitude control. quality control of a real-time flight experiment… 131 copyright ©2023 assa. adv. in systems science and appl. (2023) fig. 2. results of computer simulation of longitudinal motion 5.2. description of recognition algorithms and numerical experiment a characteristic record of the flight experiment when performing a control maneuver with the elevator cottages in the time interval is shown in fig. 3. fig. 3. cottages by the elevator to train kohonen neural networks, whose task is to recognize the beginning and end of the test mode, it is necessary to allocate separate intervals at which the elevator is given (fig. 4). fig. 4. intervals at which the steering wheel is given the resulting set of intervals is modeled by reference modes, which can be divided into two groups. in the first group there are modes equivalent to the one shown in fig. 5a, in the second group – the one shown in fig. 5b. 132 akhrameev v., tsvetkov e., pashchenko a. copyright ©2023 assa adv. in systems science and appl. (2023) a) the first type of standard b) the second type of standard fig. 5. reference modes of driving also, it is worth adding intervals to the set of training examples in which there are no images of the control maneuver (fig. 6). fig. 6. intervals in which there are no images of the control maneuver a similar division into sections of the beginning and end of the test mode is performed for all flight parameters that will be used for recognition. it is important to note that the dimension of the input vector of the kohonen network is set constant, and the number of points on the interval where the image of the control maneuver is present can vary greatly depending on the duration of the mode and the frequency of sensor polling. therefore, before submitting a vector to the network input, it is necessary to interpolate this vector into a vector of a given dimension that coincides with the dimension of the network inputs. in this paper, all kohonen networks used have 50 input and 10 output neurons. since the image record may contain areas with negative values, the minimum value is subtracted from all values in the interval. all intervals are normalized to one and the kohonen network training stage is started. after training, the output neurons of the network are divided into 3 groups. in the first group there are neurons with the maximum value for the mode shown in fig. 5a, in the second group there are neurons with the maximum value for the mode shown in fig. 5b. the remaining neurons form a third group. the values of these neurons are maximal at the intervals shown in fig. 6. table 1 below shows examples of values for the output layer of neurons after the images depicted earlier are fed into the input layer of the trained network. table 1 examples of values for the output layer of neurons fig. 5а 0.1338 0.1155 0.1115 0.2016 0.5873 0.7659 0.8697 0.9336 0.7043 0.7527 fig. 5b 0.9017 0.9701 0.9864 0.8886 0.5594 0.4452 0.2485 0.1331 0.3088 0.2726 fig. 6 0.4887 0.4241 0.4120 0.6095 0.8218 0.9092 0.8090 0.7291 0.6543 0.6737 quality control of a real-time flight experiment… 133 copyright ©2023 assa. adv. in systems science and appl. (2023) as can be seen, in the first case, neurons are mainly excited at numbers 7–10, in the second at numbers 1–4. this result allows us to form a rule for searching and selecting flight recording sites at the primary stage of the algorithm. at this stage, a flight recording section is fed to the input of the kohonen neural network, in which the experimental mode is presumably located. this section is selected by a "sliding" window moving along the record. the size of the window depends on the type of maneuver performed, and varies within the specified limits. these limits must be set by the operator on the basis of the flight task set before the flight. if any of the neurons takes a value greater than a certain critical value and corresponds to a group of neurons responding to the beginning or end of the mode, then the recognition system stores the corresponding values of the start or end time of the submitted interval. further, the recognition system, based on the information received about the start and end time of the found mode, allocates entire intervals at which the maneuver was performed. an example of such a vertical overload recording interval is shown in fig. 7. fig. 7. recording of vertical overload the arrays obtained earlier already contain the time values corresponding to the beginning and end of the maneuver shown in fig. 7, so the following algorithm can be applied. denote the previously obtained arrays as 𝑅𝑒𝑔1(𝑗, 𝑖) and 𝑅𝑒𝑔2(𝑗, 𝑖), where the array 𝑅𝑒𝑔1 contains time intervals corresponding to the class of images shown in fig. 5, the array 𝑅𝑒𝑔2 contains the corresponding values related to the class of images shown in fig. 6. each 𝑗-th element of the arrays contains 2 values. we are interested in the value 𝑅𝑒𝑔1(𝑗, [𝑡 , 𝑡 ]) and 𝑅𝑒𝑔2(𝑗, [𝑡 , 𝑡 ]) , provided 𝑅𝑒𝑔1(𝑗, 𝑡𝑒𝑛𝑑) − 𝑅𝑒𝑔2 𝑘, 𝑡𝑏𝑒𝑔𝑖𝑛 < 𝑇 (9) where 𝑇 is the characteristic time between the execution of the two parts of the desired mode. the found modes should then be evaluated for the quality of performance by the test pilot. in this paper, a multilayer perceptron is used for this purpose, since the principle of operation and training of this type of neural networks allows us to introduce a fairly free method for evaluating the mode. as mentioned earlier, kohonen networks are excellent for compressing data obtained during a flight experiment. the principle of operation of this network, based on the method of selecting the main components, allows you to separate all vectors entering the network by components that distinguish one mode from another. to train the network, you can use both mode cuts from real flight recordings and simulated ones. in this paper, the second option was used. to create a training set, 2000 modes were modeled, differing in quality when performing elevator handlebars. an example of a correctly executed mode of giving the steering wheel in time is shown in fig. 8. 134 akhrameev v., tsvetkov e., pashchenko a. copyright ©2023 assa adv. in systems science and appl. (2023) fig. 8. correctly executed driving mode if the mode is performed qualitatively, then sharp changes in the angular velocity are observed (fig. 9). fig. 9. changes in angular velocity ωz with poor-quality execution of the mode (fig. 10), a smooth change in the angular velocity occurs (fig. 11). fig. 10. poorly executed mode quality control of a real-time flight experiment… 135 copyright ©2023 assa. adv. in systems science and appl. (2023) fig. 11. smooth change of angular velocity in this numerical experiment, a kohonen compression network with an input vector dimension of 100 points was trained, and the output vector had a dimension of 30 points. in other words, as a result of training, the network created an internal structure of 30 code vectors, a measure of proximity to which the input vector is encoded. each vector supplied to the input of the compression network is smoothed by the moving average method. let there be an original vector of the mode image 𝑦 = {𝑦 , . . . , 𝑦 }, then the smoothed vector 𝑦 = {𝑦 , . . . , 𝑦 } is obtained according to the following rule 𝑦𝑦 = 𝑦 , (10) 𝑦𝑦 = (𝑦 + 𝑦 + 𝑦 ) 3 , (11) 𝑦𝑦 = (𝑦 + 𝑦 + 𝑦 + 𝑦 + 𝑦 ) 5 , (12) 𝑦𝑦 = (𝑦 + 𝑦 + 𝑦 + 𝑦 + 𝑦 ) 5 , (13) having received a working kohonen compression network, you can start training a multilayer perceptron. the following architecture was chosen for the perceptron: 4 layers of neurons with an activation function – a logistic sigmoid, 30 neurons on the input layer, 40 neurons on the first hidden layer, 10 on the second hidden layer and 2 neurons on the output layer. the training takes place according to the following algorithm: a compressed image of the simulated mode is fed to the input, as a result of the network operation, two output neurons output values from 0 to 1. the difference between the output values of the network and the evaluation of the quality of the execution of the mode specified during the simulation is calculated. next, the weights of the neural network are adjusted by the method of back propagation of the error. training continues until the total quadratic error is less than some value specified by the operator. in this paper, this value was equal to 0.25. once again, we will fully form the algorithm of the recognition system. initially, a record of the flight in which the test mode is performed is fed to the input of the system. a search window with a changing size moves throughout the flight record within the limits set in advance. each image that enters the window is smoothed and interpolated into a vector with a dimension equal to 50 points. the resulting vector is fed to the input of the primary kohonen recognition network. if any of the neurons of the control group defined at the training stage takes a value greater than 0.9, the time values corresponding to this section of the flight are recorded in the array. this creates a slicing of the entire mode into sections where a highquality maneuver is presumably located. further, these sections are encoded by the second kohonen compression network and the outputs of this network are fed to the input of a multilayer perceptron, which determines the quality of the recognized mode. the evaluation 136 akhrameev v., tsvetkov e., pashchenko a. copyright ©2023 assa adv. in systems science and appl. (2023) of the mode varies from 0 to 1, where 1 is a qualitatively executed mode, 0 – this mode is not suitable for determining the characteristics of the aircraft. consider in detail the work of the primary kohonen recognition network with sequential mode recording. let a window capturing 100 points move according to the record shown in fig. 12. fig. 12. an example of a flight recording of a change in the angle of attack based on the results of the training, it was determined that when the beginning and end of the angle of attack recording are applied during the test mode, the maximum values are taken by 2 and 10 neurons, respectively. fig. 13 shows the time values of these neurons along the entire recording of the test maneuver. fig. 13. values of neurons along the recording of the test maneuver as you can see, the neurons take a value greater than 0.9 in those parts of the recording where the beginning and end of the mode are located. the remaining neurons of the kohonen network are excited similarly, when other flight parameters are applied. the probability of recognition was estimated for noise with a maximum relative amplitude from 1% to 50% with different recording mode integrity up to 60% of the duration, with a 20% cut off from the beginning and end of the recording. after conducting 1000 tests for each mode, the probability of recognition was estimated as the ratio of the number of correct recognitions to the total number of tests. fig. 14–15 below shows the dependences of the recognition probability on the noise amplitude according to the test results for various specified distortion parameters. the probability of recognition varies from 0 to 1. quality control of a real-time flight experiment… 137 copyright ©2023 assa. adv. in systems science and appl. (2023) fig. 14. test result when adding noise from 0% to 50% fig. 15. test result when cropping the image. the image was cropped to 20% at both ends. it can be seen from the results that the algorithm used recognizes the modes with sufficient probability with large distortions. at the current time, the relative amplitude of the noise in the telemetry data is approximately 5%. with such a measurement error, the probability of recognizing the mode is more than 80%, even when only 60% of the recording is submitted. the probability of recognition is 50% with a maximum amplitude of relative error greater than 50%. now let's consider the complex operation of the recognition system with the sequential submission of an entire record of a simulated test flight, in order to verify the operation of the system in real time. fig. 16 shows a graph of changes in the vertical overload of the aircraft during flight. this record was processed by the recognition system, and three modes were found satisfying the specified execution conditions. the fourth mode was not recognized, because the steering wheel was not correctly shifted on it. fig. 16. allocation of recognized modes by the processing system red strokes (see fig. 16) the system marked the beginning of the recognized modes, and green marked their end. as you can see, all the qualitatively executed modes were recognized. the recognition system rated the first mode 0.42260, the second 0.9063, and the third 0.7609. when performing the first mode, it moved smoothly, which does not meet the quality requirements of the test mode, the second mode meets the quality requirements of the test mode. when performing the third mode at the end, the handle moved too smoothly. fig. 17 shows a graph of the change in the angular velocity 𝑤 . 138 akhrameev v., tsvetkov e., pashchenko a. copyright ©2023 assa adv. in systems science and appl. (2023) fig. 17. change in the angular velocity of the pitch 6. conclusion in this paper the major types of neural networks and methods of their training in reference modes were studied. two types of networks were used: a multilayer perceptron and a kohonen network. the multilayer perceptron in this work is used to obtain a final conclusion about the validity of the performed mode. the kohonen network is used both to determine the intervals of the estimated finding of modes close to the reference, and to compress the input image for the perceptron. the neural networks used in this work and trained on reference images showed excellent results in recognizing modes in the data obtained by the flight experiment. the described network learning algorithms can be used to work with images of other modes. an algorithm has been developed for recognizing the flight experiment mode using the described networks on the example of data on the deviation of the elevator, the magnitude of the overload and the angle of attack when performing the "dacha pitch" maneuver. this algorithm can be used to recognize other flight modes. the operation of the system has been tested when simulating real-time data acquisition. the results obtained show that the system recognizes all qualitatively executed modes and their assessment corresponds to reality. the system has shown sufficient stability of operation with a relative noise of up to 30% references 1. kotik, m. g., pavlov, a. v., pashkovsky, i. m., sardanovsky, yu. s., & shchitaev, n. g. (1965). letnye ispitaniya samoletov [flight tests of aircraft]. russia, moscow: mashinostroenie, [in russian]. 2. pashkovsky, i. m. (1975). ustoichivost’ i upravlyaemost’ samoleta [stability and controllability of the aircraft]. moscow, russia: mashinostroenie, [in russian]. 3. gorelik a. l., & skripkin, v. a. (1977). metody raspoznavaniya [recognition methods]. moscow, russia: vischaya shkola, [in russian]. 4. aksenov, s. v., & novoseltsev, v. b. (2006). organizaciya i ispolzovanie neironnyh setei (metody i tehnologii) [organization and use of neural networks (methods and technologies)]. tomsk, russia: izdatelstvo ntl, [in russian]. 5. haikin, s. (2006). neironnie seti: polniy kurs, 2-e izdanie [neural networks: a complete course, 2nd edition]. moscow, russia: i.d. williams, [in russian]. microsoft word article+text,+kornienko adv syst sci appl 2024; 03; 73-89 published online at https://ijassa.ipu.ru. the introduction of cloud storage as one of the ways to save the organization's data management resource dmitriy kornienko1*, alexander nikulin1, nikita kornienko2 1bunin yelets state university, yelets, russia 2national research university higher school of economics, moscow, russia abstract: the article discusses the use of cloud technologies in enterprise data management, provides examples of the use of cloud technologies. the analysis of the problems of implementation and functioning of cloud data storages in organizations has been carried out. based on the analysis, the technological and functional problems of implementing cloud data storages are highlighted. the attention is focused on unsolvable problems of building distributed data transmission systems. the practical implementation of cloud storage and data exchange processes through it has been carried out. a comparative analysis of the proposed methods with existing ones is demonstrated and their effectiveness is proved. keywords: cloud technologies, data, distributed storage system, protocols. 1. introduction improving information technology occupies an important place among the many new directions in the development of organizations. it is aimed at the development of the information environment, which involves the introduction and effective use of new information services. one of the promising directions of development of modern information technologies is cloud technologies. cloud technologies are understood as technologies of distributed data processing, in which computer resources and capacities are provided to the user as an internet service. let us analyze the essence and main characteristics of cloud technologies in order to substantiate the possibility and expediency of their application in organizations. cloud computing is a model for providing ubiquitous and convenient network access (on an as-needed basis) to a shared pool of configurable computing resources (e.g., networks, servers, storage, applications, and services) that can be quickly provisioned and released with minimal management effort and need interaction with the service provider. in cloud computing, data is permanently stored on virtual servers located in the cloud, and is also temporarily cached on the client side on computers, laptops, netbooks, mobile devices, etc. one of three basic models is used to build a cloud: software as a service, platform as a service, infrastructure as a service. the cloud is not the internet itself, but the entire set of hardware and software that ensures the processing and execution of client requests. cloud computing is a new paradigm that involves distributed and remote processing and storage of data. the essence of cloud technologies is as follows: you can not have any programs on your computer, but only have access to the internet. remote access to data in the cloud can be found anywhere on the planet where there is access to the internet. a cloud data storage system, or data storage as a service is an abstract concept that corresponds to a data storage system, which can be administered on demand using a special interface [1-3]. this interface abstracts the location of the system, so it does not matter whether it is local, remote, or hybrid. cloud storage infrastructures form new architectures that support different levels of service on top of a potentially large group of users and * corresponding author: dmkornienko@mail.ru 74 d. kornienko, a. nikulin, n. kornienko copyright ©2024 assa adv. in systems science and appl. (2024) geographically distributed storage devices. it is important that clients can control and manage how their data are stored as well as how their money is spent. numerous cloud service providers offer management tools that provide users with enhanced control over costs. storage efficiency is an important characteristic of a cloud storage infrastructure, especially given its emphasis on overall savings. to make the storage system more efficient, you need to store more data. a common solution is to reduce the amount of source data so that they occupy less physical space. there are two ways to achieve this goal: compression, i.e., packaging data by encoding them using different representations and deduplication, i.e., excluding all duplicate data [4; 5]. although both methods are useful, compression involves processing (transcoding data to and from the infrastructure), and deduplication involves calculating signatures to search for duplicates. saving is one of the most important features of cloud data storage. this involves purchasing storage devices, their power supply, repair, as well as storage management. considering cloud storage from this standpoint (including a service level agreement and the increased storage efficiency), it can be beneficial for certain usage models. a storage access api (application programming interface) is an essential component of the services. many applications require access to storage services using an api that is optimised for a particular storage system, either on their own hardware or cloud-based. the amazon s3 api cloud storage system provides developers with an sdk (software development kit) for .net and java, as well as libraries for additional platforms and languages. these interfaces usually use representational state transfer (rest) protocols and/or simple object access protocol (soap) [6-8]. when considering the architecture, it is necessary to take into account its operating parameters. they are understood as various characteristics of the architecture, taking into account the cost, performance, remote access, etc. the architecture of cloud data storage is primarily the delivery of data storage resources on-demand in a highly scalable and multiagent environment. in general, the cloud storage architecture is an external interface that provides an api to access storage devices. in traditional storage systems, this is the scsi (small computer system interface) protocol, but the cloud requires new ones. among them, you can find external protocols of web services, file protocols and even more traditional external interfaces (internet scsi, iscsi, etc.). middleware i.e. the data storage logic is behind the external interface. this middleware functions as data replication and data reduction, according to traditional data placement algorithms taking into account geographical location. finally, the internal interface organises the physical storage of data. this can be an internal protocol that functions specifically or a traditional server with physical disks. 2. theoretical overview the rapid increase in the bandwidth of computer networks allows the development of numerous applications that provide for intensive data processing [1]. these new applications can perform tasks ranging from mass data transmission (sdss (sloan digital sky surve) and electronic very long baseline interferometry) to high-bandwidth interactive systems (geowall) [2]. however, different applications have different requirements for data transmission services [3]. for example, the geowall application may require a long-term speed variation of data transmission, whereas data should be transmitted at the maximum possible speed in private networks for the sdss application [4]. however, the current internet system is designed to provide support for a variety of different types of applications. this internet design philosophy affects fundamentally the development of transport protocols. on the internet, most of the traffic is generated by tcp (transmission control protocol), but there are applications, which are not provided with a sufficient level of efficiency by tcp. in the introduction of cloud storage… 75 copyright ©2024 assa. adv. in systems science and appl. (2024) the context of high-performance computing, tcp is well known for its low efficiency and network fair sharing with threshing [5; 6]. modifications to the network stack of the protocol kernel (for example, new variants of tcp) usually require several years for standardisation, implementation and widespread deployment. indeed, since the advent of tcp, about three decades ago, only four versions of this protocol have been widely deployed, namely tahoe, reno, newreno, and sack [7]. although today the data transmission speed is 1 gbit / s and higher for a growing number of networks, a wide bandwidth remains an urgent problem for web applications because of the limitations of existing network transport protocols. the limitations of the implemented network transport protocols are one of the main reasons why it is so difficult to scale applications with intensive use of network connections from local clusters to global networks [8]. the transmission control protocol (tcp) has been successfully used for decades as the main transport layer protocol of the network protocol stack. however, it has recently been shown that tcp loses performance when used for high-speed wide area networks, especially for geographically remote networks. tcp uses the additional increase/multiplicative decrease congestion control algorithm, which cannot provide all the available bandwidth and has sufficiently large latency in the case of large packet loss in highperformance networks [9]. computer network researchers labour at new transport protocols and congestion avoidance algorithms to support next-generation high-speed networks. many studies, including the tcp variants (fast, bic, scalable, and highspeed) and xcp showed higher performance when modelling them [10]. however, the practical use of these protocols is still very limited in real applications due to the difficulties of their deployment, installation and hardware restrictions [11]. network users usually turn to application-level solutions when they need to transfer a large bulk of data, among which udp-based (user datagram protocol) protocols are very popular, for example, sabul, udt (data transfer protocol), tsunami, rbudp (reliable blast udp), fobs and gtp [12]. such udp-based protocols provide much better algorithm strength and are easy to install. however, despite the simplicity of deployment at the user level, the protocols are quite difficult to configure in the kernel to make them as efficient as possible [13]. since the user-level deployment cannot change the kernel code, there may be additional context switches and copying process of memory sections between the user level and the kernel one. at high data transfer rates, these operations are very sensitive to processor load and protocol performance [14]. 3. materials and methods for the practical application of the developed methods, it is necessary to design and develop a distributed data storage system. the specifics of modern data storage are in their transition to distributed systems. therefore, a necessary condition is to ensure the redundancy of critical network nodes. it is also an important condition that network nodes have information about other nodes and their data. in the conditions of distributed operations, their necessary functionality is to ensure timely identification of an inoperable data exchange/storage node and their extraction from the operation. also, after the node is restored, it should join the data exchange/storage process as soon as possible. designing such a system requires a preliminary analysis of the requirements and planning its internal processes. the software of the system must ensure complete functionality and have the means to organise all the required processes of processing, transmitting and storing data in all regulated modes of operation. the system software must be universal; functionally sufficient; reliable; adaptive; upgradeable and scalable; have an intuitive user interface; protected from external influences; able to record all actions of software users. the software 76 d. kornienko, a. nikulin, n. kornienko copyright ©2024 assa adv. in systems science and appl. (2024) should be developed using the principles of structural and modular programming. each of the system tasks should be as independent as possible from the others. quality control of developed software tools should be provided by testing and conducting a trial operation. the system architecture should be based on the principles of high-load and fault-tolerant systems. it must meet the following basic requirements: the system should support total scaling-out (both data storage nodes and satellites to improve the efficiency of data access); the system should be able to duplicate all critical network nodes and data storage to ensure system fault tolerance. the system should include the following functional components: 1. data storage is a component that provides data storage for solving the following tasks: data storage; data access; accounting for user actions. 2. satellite is a component that provides fast information and hardware interaction of the system with its users. 3. adaptive dns (domain name system) server is a component that provides logistics of requests from a user to the system. general requirements for the system reliability: 1. the software and hardware complex should function around the clock, in a continuous mode, except for force majeure circumstances. 2. the databases should be backed up regularly (at least once a day). it is necessary to have at least two backups of all data. backups should be stored in physically remote locations. 3. failures and malfunctions of workstations and network devices should not result in data destruction and affect the performance of the whole system. 4. failure of one of the subsystems should not shut down other subsystems, i.e., all other subsystems must function. 5. a planned stop or failure of the information resource should not lead to a software failure. 6. incorrect user actions should not result in an emergency. 7. errors of technical personnel should be minimised, including by clearly delineating access rights to the system, as well as logging system events. the reliability of an information system is understood as the general system ability to preserve its main specified properties in time. with this understanding, the software should: be resistant to erroneous actions of users (errors in the actions of personnel should not lead to failures (failures) in the operation of the information system software); provide guaranteed control of incoming and outgoing information; ensure rapid recovery after failure (failures). the software is developed on the basis of common operating systems, programming tools and dbmss (database management systems). the system should use standard solutions based on standard protocols and interaction interfaces, which provide for the possibility of interfacing and collaboration of hardware and software from various manufacturers, as well as for interfacing with information systems of other organisations. all hardware and software solutions used in the system design must comply with the requirements of national standards or international standards (if applicable). the hardware and software used as part of the is must be certified or otherwise documented by a supplier company confirming their compliance with the specifications. unified solutions are preferred due to the great social significance of the project, and the tight deadlines for commissioning. such solutions should have the following properties: access to the system should be provided via the global internet; modularity (component solution); be able to integrate with external automated systems. the system must provide the following functions: registration of system clients; creation of a company cloud record in the system; creation of an administrator account for the company; administrative part; registration of system users for the company; editing user profiles; providing access to data via protocols: http (hypertext transfer protocol); https (hypertext transfer protocol the introduction of cloud storage… 77 copyright ©2024 assa. adv. in systems science and appl. (2024) secure); ftp (file transfer protocol); ftps (file transfer protocol + ssl); sftp (secure file transfer protocol); error handling; storage errors; errors satellites; script errors; communication interruptions; heavy load; registration of user actions. 4. results and discussion since the list of necessary functions and the relationships between them have been analysed, it is advisable to present them in the form of system use cases (fig. 1, 2). the system should provide effective information exchange between internal components. information exchange should be carried out between the system components using local computer networks and global data transmission networks [15; 16]. appropriate specified information exchange protocols should determine the composition, structure, volume and frequency of transmission. information exchange protocols should provide measures to exclude the possibility of unauthorised access to data. there should also be means for monitoring the transmitted input/output data and means for monitoring information in databases. the requirements for information exchange between the system components should be determined at the development stage, based on the capabilities to deploy the platform [17; 18]. fig. 1. use cases of the system from the point of view of a user and a client of the system 78 d. kornienko, a. nikulin, n. kornienko copyright ©2024 assa adv. in systems science and appl. (2024) fig. 2. options for using the system from the point of view of the system owner the overall architecture of the system consists of an adaptive dns server, several data containers (dcs) and many satellites (fig. 3). each data container should include (fig. 4): two firewalls; two application servers; two storage units with disks. such an architecture is necessary to ensure the fault tolerance of the system in case of loss of a physical network node [19; 20]. at the first stage, requests come to the firewall level, which is displayed with the same ip (internet protocol) address with indexes 0 and 1 for external nodes. if a device with 0 exits, then all requests are readdressed to a device with 1. fig. 3. general architecture of the system the introduction of cloud storage… 79 copyright ©2024 assa. adv. in systems science and appl. (2024) after a firewall receives a user request, the request gets in turn to the available application server, the availability status of which is constantly monitored at the first level. after the hardware connection, the two storage units are connected at the hardware level as a single storage unit. also, at the software level, data synchronisation is configured between the units to ensure data backup. since the speed of processors and the amount of ram increase extremely quickly, the application servers cannot make full use of them without complex mathematical algorithms and are limited by the number of users, therefore, it is advisable to use containers (virtualisation at the operating system level). then, structurally, the application server will consist of several containers and a load balancer (fig. 5). fig. 4. single storage general architecture fig. 5. application server general structure there are several software tools aimed at monitoring the availability of services in containers, and keepalived is the most popular of them. their operational disadvantage is that they cannot group services when used in our architecture. since the data accessibility of different users from different companies via ftp is one of the key requirements (each company has its own hostname), one needs different access ip address for each company (ftp specifics). an ftp server can support the simultaneous use of many ip addresses. in this case, keepalived will try to create connections and check availability for each ip address, and the load will increase exponentially on the server to check the availability of containers if there are a lot of them. to solve this problem, a system has been designed that completely replaces the keepalived one with the following features: – grouping ip addresses and services as one service (in case of unavailability, this service ceases to be available at all specified ip. and vice versa, the services become available again at all ip addresses if available); – storing statistics for all ip addresses and services, taking into account the number of requests; the data transmitted; the data received; inoperability; restoration of operability. 80 d. kornienko, a. nikulin, n. kornienko copyright ©2024 assa adv. in systems science and appl. (2024) architecturally, a satellite corresponds to the architecture of an application data storage server. it does not require a large and reliable data warehouse, since it acts exclusively as a "proxy server" between clients and their data. in addition, there is no need to completely duplicate it, since it does not contain critical data and all input requests will be automatically resent to other nearby satellites. modern cloud data warehouses must provide both the most rational data flow and a significant increase in its rate, i.e., acceleration of source-consumer data transmission and processing. first, data flows are analysed to solve these problems, when designing an information system: – all the links of the data processing and storage system, starting with initial information, its gradual transformation and formation of final data sent to the managed system as a reporting command and other information. at the same time, the role is determined of each system element in the cloud storage and recorded in the data processing scheme, as well as their structure and functions are specified; – a data communication diagram of all system elements between themselves and the external environment. the diagram may contain information about specific forms of data links and indicate their quantitative and temporal characteristics; – find primary (output) system data. an information flow analysis is the most important stage in the deployment of an existing data storage system, which will meet the design objectives. studying information flows gives a general idea of how a system functions and is the first step in analysing the effective design of a highly loaded data processing, storage and transmission system. further study of data flows allows identifying the elements of the information display of objects, their relationships, as well as the structure and dynamics of data flows. the movement of data, accompanied by appropriate information flows, is the basis for ensuring the operation of a cloud data warehouse in the system. there is a continuous movement of information flows between all the elements of the system functioning in the data warehouse environment, that provide the receipt of information data necessary for the analysis and transmission of client data. the main elements of the system are (fig. 6): 1) a data warehouse, which is one of the main elements of the system, where client data are physically and long-term stored (fig. 7); 2) a data access satellite, which is a data access element in the system that is located as close as possible to users (data consumers), and may be used as temporary storage (caching) (fig. 8); 3) a dns server is an element of the system that receives dynamic updates from its other elements and provides data users with reliable and up-to-date information. fig. 6. main system elements interaction diagram the introduction of cloud storage… 81 copyright ©2024 assa. adv. in systems science and appl. (2024) fig. 7. satellite deployment diagram fig. 8. storage deployment diagram a dns server is used to update information about the location of available system elements and their availability. it acts as a key element of the information interaction of other elements of the system. to do this, each element of the system sends information about itself with a certain frequency and is authorized on the dns server using a tcp connection (fig. 6). in turn, other elements of the system exchange information using a data protocol based on udt (fig. 9). data warehouses, as an element of a data management system, actively consume information and, accordingly, are the recipients and senders of data flows. a satellite, as an element of data consumption, also actively consumes information resources in a two-way direction. for a more detailed analysis of data flows, it is necessary to consider each element of the system in the context of data and subsystems of this element. 82 d. kornienko, a. nikulin, n. kornienko copyright ©2024 assa adv. in systems science and appl. (2024) fig. 9. storage units and satellites interaction diagram (self-engineered) to maximise the use of system resources, each physical server will be divided into the most independent units (containers). the containers are an operating system-level virtualization system intended for running multiple isolated instances of the linux operating system on a single computer. they do not use virtual machines but create a virtual environment with its own process space and network stack. all the containers use a single os kernel to maximise efficiency. to ensure the operation of a system consisting of several containers, two types are used as one system (fig. 7, 8): a load balancer container and an application container. the load balancer acts like a proxy server that checks the availability of containers and redirects the received requests to an available container for processing them. the application container is a full-fledged copy of the software application that processes user requests and returns the result. the data storage container (fig. 10) receives and executes requests exclusively using the udt protocol, for which independent udt server and udt client processes are running in the container. using this protocol, the container receives data from other data storage or from clients via satellites and sends data to other data storage and clients using data access satellites. fig. 10. storage container deployment diagram the satellite application container (fig. 11) contains: 1. a udt server and udt client for data exchange with other elements of the global data warehouse system. 2. a web server for users’ easy access to data. 3. the proftpd server is designed to expand the functionality of data access using ftp, ftps, sftp. the introduction of cloud storage… 83 copyright ©2024 assa. adv. in systems science and appl. (2024) having considered the functionality of the system as its separate business processes, one will get: 1. search for the nearest data access satellite. 2. uploading data to the storage. 3. access to data. 4. data replication. the search for the nearest data access satellite is shown in figure 12. fig. 11. satellite container deployment diagram fig. 12. nearest satellite searching sequence diagram when a client needs to access the data, the software application addresses the dns client of the operating system with a request to get the ip address of the system using the domain name. in turn, the dns client uses a network of dns servers to access the ns servers of the system and, using the client's ip addresses, it calculates the optimal data access satellite that is "closest" (the minimum number of steps and the maximum speed) to it. fig. 13. data loading sequence diagram 84 d. kornienko, a. nikulin, n. kornienko copyright ©2024 assa adv. in systems science and appl. (2024) data is loaded to the storage according to fig. 13. after receiving the ip addresses of the nearest satellite, the client logs in to the service and only can access the data after that. when the client needs to download a document, he or she sends the document to the satellite using tcp and a software application. the satellite, in turn, caches the data and sends them to the receiving storage directly, or through the nearest storage using udt. fig. 14. data access sequence diagram figure 14 demonstrates data access. if it is necessary to receive the document, the user sends a request to the satellite using tcp protocol a software application to receive the document. the satellite, in turn, receives the document from the storage directly or through another storage using udt. the document is temporarily cached on the satellite and sent to the client (tcp). there are two approaches to data exchange in the global network: serial data transmission i.e. data transmission using one communication channel; parallel data transmission i.e. data transmission using several communication channels. speed is the advantage of parallel data transmission. this method is used in computer peripherals for data exchange in data buses. the quality and conductivity of conductors is the main disadvantage of this approach. with different properties of the conductors, the bits in the data transmission may arrive with a delay, which leads to significant errors. in turn, serial data transmission is less dependent on conductors, since data are transmitted using one communication channel, but it is much slower compared to parallel data transmission. this approach is the most common in global networks. the introduction of cloud storage… 85 copyright ©2024 assa. adv. in systems science and appl. (2024) data transmission methods are also divided into synchronous data transmission; asynchronous data transmission. asynchronous exchange is the most common form of serial communication, which involves the transmission of a data packet that contains information about the beginning and the end of data transmission, information for error control and data themselves. since the architecture of the cloud data storage assumes both simultaneous data transmission in different directions and receiving data from different data sources, asynchronous serial data transmission was chosen. an asynchronous approach to the system design was also chosen for efficient resource allocation. a system data transmission module can be represented as a system of states and transitions (fig. 15). when the process is started, a data channel is created, which switches to the "waiting for events" status. since this communication channel provides for both data transmission and data reception, this communication channel responds to the following events: 1. input connection: during this event, the system needs to authenticate the client and add this connection to the connection pool. 2. input data: get the data, decrypt and perform the actions provided in this data package. if there is no data for the communication channel to read, and data packets are not received, this packet is placed in the queue of data packets that are not received, and the system waits for the possibility of receiving data through this communication channel. 3. data transmission envisages checking a connection with a remote client, forming data packets, encrypting them and transmitting them over a communication channel. if it is impossible to transmit data immediately, the data packets are moved to the packet queue and wait for the possibility of transmission (when the output communication channel is free). 4. connection error: in this case, the system evaluates an error and makes a decision. if the system receives a false packet, or an unexpected one, the system "asks" the client to retransmit the data. if a larger system failure has occurred, such as a data encryption synchronization failure, the system breaks the connection with this client, and the client recreates the connection and re-sends the data for which the server did not send confirmation. since client data is stored in the cloud data storage as a file structure, data transfer in the system is reduced to file transfer. the transfer of an individual file can be represented as the transfer of individual data packets (fig. 16). fig. 15. data channel status diagram 86 d. kornienko, a. nikulin, n. kornienko copyright ©2024 assa adv. in systems science and appl. (2024) during the transmission of any amount of data, the system generates data transfer packets and adds them to the data transfer queue. after that, the system performs several functions: 1. checking for a connection to the data destination. 2. if there is no connection: creating a connection with the destination; authentication at the destination; 3. sequential encryption and transmission of data packets (from the queue) over this communication channel. 4. if the output cache is full, the system goes into waiting for other events. fig. 16. if the output cache is full, the system goes into waiting for other events. after successfully sending a data packet, the system notices the packet as sent in the data transfer queue. after the client successfully receives the data packet, it sends a confirmation packet. the server part deletes the packet from the data transfer queue after receiving this packet. in case of disconnection with the data receiver, the system automatically changes the status of the sent data packets to new ones for retransmission of data (fig. 17). the logical model of the data transmission system can be represented in the form of a class diagram in the cloud data storage (fig. 18). it is obvious from the class diagram that the conceptual model of the data transfer system is similar in the data warehouse and the satellite. this approach simplifies the construction of systems of this nature. it is also obvious from this structure that "connection" is an abstract class and can be used by protocols such as tcp or udt to transport data. this approach provides significant advantages in cases with a single limited communication channel, which is very relevant for the current state of communication. fig. 17. data packet transmission status diagram the introduction of cloud storage… 87 copyright ©2024 assa. adv. in systems science and appl. (2024) fig. 18. system class diagram if it is necessary to transfer data (a file), the system performs several simple actions: 1. creating an instance of the "file" class. 2. adding it to the "source file queue". 3. getting a link to the connections (create it, if needed). 4. creating instances of the "packet" class within the free window size for the connection. 5. if the connection is ready to receive a "packet" from the "outgoing packet queue", encryption and transmission over the communication channel. 6. receiving confirmation of the end of the file. 7. deleting the instance of the "file" class. on the far side of the connection, the recipient of the data (file) performs the following actions: 1. getting a data packet with information about the data (file) and decrypting it 2. creating an instance of "file". 3. adding it to the "input file queue". 4. creating a physical file. 5. receiving data packets, decrypting and filling the physical file with them, sending confirmation of receiving data packets. 6. getting a file completion package. 7. closing the file. 8. deleting the instance of the "file" class. 9. sending confirmation of receipt of the full file. the process of receiving a data packet can be divided into two stages: – getting the packet header; – getting the main body of the packet. the message header contains information about: 1) the communication channel for multiplex data transmission of several files at one time; 2) the size of the data packet body; 88 d. kornienko, a. nikulin, n. kornienko copyright ©2024 assa adv. in systems science and appl. (2024) 3) the command is a type of data that is transmitted in the packet body (creating a directory, a file, data from the file, confirming receipt of the packet...); additional information may also be contained, depending on the command. 5. conclusion a session-level protocol for such networks was developed taking into account the difficulties and disadvantages of the existing state of large-volume data transmission through highperformance regionally distributed networks, which were identified during the study of the problem. the new session layer protocol is compatible with the standard osi tcp and udp stack protocols. in addition, it is adapted for modern networks, i.e., it has the functionality for efficient use of network bandwidth, regardless of its characteristics. when using such a protocol, the transmission delays between different types of networks do not increase. thus, the protocol ensures efficient use of the resources of the extreme nodes of the network. in the matter of security, it supports data encryption, i.e. it can allow the connection of encryption protocols. the efficiency of the udp-based protocol was optimised and it was shown that these ideas could be used to deploy effective and practical applications based on udp. for example, using the udt protocol environment (based on udp) can easily support various congestion control algorithms, for example, high-speed tcp or explosive rbudp. the use of this approach made it possible to increase the performance of both the data transfer element and the system as a whole, in contrast to the methods proposed in [21; 22]. the use of multi-channel communication allowed simultaneous data transmission over one communication channel, and the universalization of the architecture allowed the use of various data exchange protocols on different parts of the network, depending on efficiency. the authors would like to thank the management of bunin yelets state university for financial support of this study. references 1. cong, w., zheng, y., zhang, z., kang, q., & wang, x. 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(2021) the single page application architecture when developing secure web services, journal of physics: conference series, 2091(1), 012065, doi: 10.1088/1742-6596/2091/1/012065 microsoft word kozlovad_adv_20241028_edited adv syst sci appl 2024; 04; 44-56 published online at https://ijassa.ipu.ru. possibilities of assessing information security risks using fuzzy logic and econometrics methods alexander kozlov1, nikolai noga v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia abstract. this paper provides a brief overview and analysis of the most commonly used methods for assessing information security risks in various complex systems. these methods involve creating specific models that help assess, manage, and predict the possible occurrence of adverse situations related to information security. these methods help to make decisions aimed at minimizing the potential damage that could occur because of external attacks on information resources or other threats that exploit existing vulnerabilities. however, they provide an acceptable result in risk assessment in conditions where it is possible to quantify the parameters on which the risk depends. in conditions of high uncertainty, for example, when determining the dependence of risk on subjective factors, the use of these methods can lead to high errors. as a rule, risk assessment is associated with a high degree of uncertainty of parameter values and their mutual influence on the information security risk level. the method proposed by the authors, based on the combined use of fuzzy logic and regression analysis, makes it possible to assess the information security risk in conditions of uncertainty in complex information systems with a network structure. this method also allows you to identify the parameters that most influence the risk level, which in turn enables you to protect information resources while optimizing the cost of implementing an effective protection system. using this method and basing on the predicted risk level values, allows you to plan events to improve the level of information system protection in both the short and long term. keywords: information security, information leakage, multifactorial risk assessment, fuzzy logic, econometrics, production rules, regression equation, multiple determination coefficient, subjective factors, uncertainty conditions. introduction currently, the global digitalization of the economy and the rapid development of digital technologies have led to the fact that the digital world has penetrated into all areas of human activity. now a person can no longer refuse digital services. moreover, these services, in addition to new opportunities, carry new risks and threats. it is difficult for an untrained person to understand all the nuances of the digital world. attackers are ready to take advantage of this, always striving to benefit themselves in any unclear situation. the situation regarding countering hacker attacks on any information resources of companies, as well as on critical information infrastructures, has become extremely complicated. in particular, such attacks intensified during the global pandemic, when the staff of many companies had to switch to remote operation. the activity of such attacks has also increased because of the aggravation of the political situation in the world. confidence in software and hardware manufactured abroad has significantly decreased. thus, it became necessary to solve the problem of switching to domestic hardware and software. both large companies and small businesses face these challenges. we should note that, in the first half of 2024, the number of data breaches worldwide decreased by 19.2% compared to the same period in 2023. however, in russia, during this time, there was an increase of 10.1% in the number of information breaches compared to the first half of 2023. additionally, an all 1 corresponding author: alex4590alex@yandex.ru possibilities of assessing information security risks… 45 copyright ©2024 assa adv. in systems science and appl.(2024) time high in the number of compromised personal data records was recorded in russia during this period. overall, approximately 1 billion personal data records were compromised in the first half of 2024, which is a 33.8% increase from the first half of 2023 [1]. it is worth noting that the nature of attacks has also changed. currently, attacks are mainly aimed at disrupting various systems of critical infrastructure, leading to complete failure and maximum damage. additionally, these attacks try to achieve political objectives by creating a media impact. although, in the past, attacks on companies were mainly carried out for the purposes of industrial espionage and theft of personal and financial data. in order to protect the company's business and ensure its efficiency and continuity, it is important to assess both current and potential risks associated with its future activities. risk refers to the impact of uncertain events on the achievement of goals [2]. therefore, risk assessment involves considering these uncertainties. there can be a wide range of risk factors. typically, general factors are identified for a particular class of systems, but specific ones can also be considered, which are unique to each individual system. it is also possible to observe the impact of various factors over time, meaning that risk assessments you should perform regularly and iteratively, using up-to-date information. this process includes risk identification, analysis, and comparative evaluation [2]. based on the above, it follows that the methodology used for risk assessment should be straightforward and allow for dealing with various uncertainties. in this article, the authors propose a solution to the problem of information security risk assessment using fuzzy logic and regression analysis together. these methods allow for solving this problem under conditions of uncertainty in the dependence of various parameters within complex information structures. additionally, these methods allow for the identification of parameters that have a significant impact on information security risks, as well as those that you can ignore in a given situation. this information you can use to plan measures for improving the information resource protection system, reducing potential damage, and minimizing potential risks in both the short and long term. literature review as mentioned above, the focus of attacks on organizations and their information resources has changed. therefore, it is necessary to look for extraordinary solutions to identify information security risks in operated information systems. based on these solutions, methodologies for assessing, analyzing, and managing information security risks have been developed. for example, well-established but still relevant methodologies include coras (a risk and threat modeling and system analysis tool), octaves (for assessing critical threats, assets, and vulnerabilities), cramm (a central computer and telecommunications agency analysis and management risk method), vulture 2006 (an integrated risk analysis and management tool for digital security information systems), riskwatch, and the methodology from microsoft [3-4]. a number of works have also been dedicated to identifying security threats. for example, work [5] introduces the recursive internetwork architecture and carries out a security risk assessment to identify threats at runtime and verify the correctness of built-in security tools. it also examines mitigation measures designed to combat attacks related to these threats. paper [6] presents a methodology for assessing information security risks when using cloud technologies in corporate information systems. this methodology uses fuzzy logic methods and takes into account the level of control over information resources. also, the assessment of security risks in cloud computing is considered in [7] based on a two-stage procedure for quantifying the information security risk of cloud computing, aimed at calculating the coefficient of countering possible attacks and comparing the amount of damage with the value of the organization's assets. work [8] shows that with each distributed access implemented 46 a. kozlov, n. noga copyright ©2024 assa adv. in systems science and appl.(2024) during data exchange. there is a security risk that is assessed using security metrics such as the average cost of failure and the multidimensional cost of failure. in all the above–mentioned works, risk assessment is the assessment of the threats, as well as the vulnerabilities that these threats are implemented through, and, finally, the damage resulting from their implementation. risk analysis and management include the construction of models that adequately show the occurrence of adverse conditions, taking into account various parameters that characterize these risks. in addition, these models help decision-making in order to reduce the potential damage that attacks on information resources can caused or the implementation of other threats through existing vulnerabilities, as well as the development of preventive measures to increase the security of the informatization object. the above methods provide an acceptable result for risk assessment in situations where it is possible to quantify the parameters that affect risk. however, in conditions of high uncertainty, such as when determining the dependence of risk on subjective factors [9] that may affect business goals, including information security, these methods may not be sufficient. factors such as the level of wages and professionalism can affect the outcome of a project, and incorrect management decisions based on incomplete information can lead to negative consequences such as increased costs, delays, or even the loss of the business due to information leakage. methods based on fuzzy logic have been shown to be effective for risk assessment in these situations, as they allow for the consideration of uncertain parameters and their impact on risk [10]. these methods can help identify potential risks and develop strategies to mitigate them, ensuring that businesses can achieve their goals while minimizing the risk of negative outcomes. in [11], a methodology is proposed that allows assessing risk under conditions of uncertainty regarding the interrelation between factors such as the level of threats, presence of vulnerabilities, potential damage, amount of various resources, costs for creating and maintaining a system, and partial loss of control over one's information resources when using cloud-based structures. in [12], the authors propose a risk analysis process combining qualitative and quantitative methods. the focus is on developing a methodology for assessing and analyzing risks and vulnerabilities in the context of security risk management. the authors carry out the risk assessment in four stages. at the first stage, a mathematical model is built in accordance with the results of risk identification. at the second stage, basic information or data available as a result of expert assessments are obtained. a mathematical method is chosen to quantify the information. at the third stage, a suitable model and analysis methods are selected. at the last stage, the risk level is determined in accordance with certain criteria. at the same time, the focus of the study is on the analysis of risk assessment using fuzzy logic. in [13], to assess cyber threats to digital instrumentation and i/o systems, the authors proposed a risk assessment method based on multi-fuzzy systems that assesses the risk of cybersecurity depending on the following risk factors: the overall capabilities of the attacker, the probability of success of the attack and the consequences of the attack. this method involves the operation of three fuzzy inference system. the first system evaluates the overall capabilities of the attacking attacker, the second system evaluates the overall probability of success of the attack, and the third system evaluates the risk level based on the impact of the attack and the results of the first two systems. in [14], a technique was proposed to solve the problem of detecting a set of critical nodes by fragmenting the graph as a result of removing a set of vertices with a certain power in such a way that the residual graph has minimal pairwise connectivity at a user-defined power value. traditional optimization algorithms are unable to find the optimal set of vertices in large graphs with thousands or millions of nodes due to the high computational costs associated with them. the authors used the within-method method in a greedy algorithm to quickly identify a set of possibilities of assessing information security risks… 47 copyright ©2024 assa adv. in systems science and appl.(2024) critical vertices. the proposed algorithm can be easily extended to vertexand edge-weighted variants to detect a critical node. in [15], a method is presented that allows us to estimate the predicted value of information security risk for complex systems and its confidence interval, using regression analysis and fuzzy logic techniques, in terms of how risk depends on various factors such as the cost of resources and the threat level. this method is used to implement a risk assessment process under uncertainty. the use of a fuzzy model in [16] allowed for a more flexible processing of inaccurate information security risk factors, and enabled us to switch to a quantitative representation of qualitative characteristics. the fuzzy model and methods proposed in this work can be used to assess both specific types of information security risks for an erp system (enterprise resource planning), as well as the general information security risk for an erp. at the same time, the fuzzy logic toolbox in the matlab system [17] was used to implement the methodology described in [11] and [16]. the paper [18] provides a detailed description of the information risk analysis algorithm, as well as the specifics of the implementation of the information risk assessment stage in the overall analysis process. the authors proposed a technique based on a neuro-fuzzy network that ensures the adequacy of the assessment and applicability to non-numeric input data using the example of the implementation of a neuro-fuzzy network in the matlab environment. materials and methods in this paper, we propose a methodology based on fuzzy logic and regression analysis for determining the factors that significantly influence information security risk. this methodology allows us to identify a set of critical parameters that need to be taken into account when evaluating information security risks, as well as a list of less significant parameters that can be ignored. additionally, this approach enables us to address the problem of identifying critical nodes in networked information systems [19]. it is clear that an increase in the number of parameters under consideration and their ranges of values can significantly complicate the construction of a table of production rules. the proposed methodology makes it possible to solve the problem of managing the security of a network information system with a large number of parameters, as well as with a different number of ranges of values for each parameter included in the consideration. according to our methodology, the risk level we can represent as a function of n parameters. ),...,,( 21 nppprr  (3.1) where ,,...,1, nipi  – parameters that affect the risk level. it is assumed that the quantitative values of all these parameters are in the range [0; 1], and the qualitative ones take values, for example, low, medium and high. the goal of this task is to identify the linguistic variables that have the most significant impact on the value of a specific output variable, which in turn determines the level of risk. in order to accomplish this, we will develop a multiple regression model. the ranges of values for these variables will be established based on expert assessments, such as those obtained through the delphi method or other group-based techniques [20-22]. we will use mathematical statistical methods to process the estimates provided by the experts. the authors of this paper did not intend to provide a detailed description of the process for processing expert estimates, such as assigning relative weights to factors considered during the evaluation process. a detailed description of this procedure you can find in [21]. for simplifying the coordination of results from expert assessments in this example, we will assume that experts provided estimates that were close in value and the concordance coefficient was close to one. 48 a. kozlov, n. noga copyright ©2024 assa adv. in systems science and appl.(2024) it is proposed to use fuzzy logic methods to determine the risk level [11, 15, and 19]. this makes it possible to conduct a multifactorial risk assessment. the following features we should consider when using these methods:  the impact of the parameters under consideration can be very uncertain due to their possible influence on each other, i.e. there is no clarity on the issue of the mutual correlation of these parameters and, accordingly, their possible redundancy;  the accuracy of the assessment depends on the quality of the formation of production rules;  and how well and in detail are the terms of variables described , indicating the ranges of their values for each linguistic variable. to calculate the coefficients of the regression equation, we use the least squares method (ols) however, since the obtained coefficients of the equation are incomparable, in order to be able to compare them [23] and arrange the parameters according to the degree of influence on the risk level, we build the regression equation on a standardized scale. as a result, it is possible to rank coefficients according to the degree of influence on the risk level and, having determined redundant parameters, i.e. parameters that have little effect on the risk level, exclude these parameters from the equation. the solution scheme provides that we perform the task in several stages. step 1. conducting a survey of the informatization facility according to the results of the survey of the informatization object, a set of n factors (parameters) is determined, on which the information security of this object (the operated information system) depends. in relation to information systems, risk factors we can divide into four main types, which include: 1. economic parameters: the value of the asset (information resource), the level of dependence of the main production on the information infrastructure, the level of costs for the information component in general and means of ensuring cybersecurity in particular, the level of potential damage and others. 2. technical parameters, which include: the level of vulnerabilities in the hardware and software complex of the operated system; the level of technological independence from imported developments (the ratio of the number of software and hardware developed and manufactured in russia to the total number of tools used at the informatization facility), the level of technical protection of information and others. 3. organizational parameters: the level of isolation of the facility from external systems, the level of use involved external specialists, the organization level of the information system protection (availability and execution of instructions and regulations, availability of access controls, availability of security administrators, availability of contractors) and others. 4. subjective parameters: the employees and administrators qualification level, the employee’s salary level, the level of "turnover" of personnel and others [9]. according to the results of the examination of the informatization object, other risk factors may be established. the combination of such factors will represent the result of the risk identification. step 2. creating a fuzzy knowledge base from the set of n parameters obtained at the first step, m parameters (from five to eight) are selected, representing all four of the above groups, on which, according to experts, the information security of the object depends largely. if we take into account all n parameters, then the table of production rules may take an unacceptably large size, which will be quite difficult to process. the selected parameters (linguistic variables) are normalized so that their values are within the range from 0 to 1. all input linguistic variables pi are evaluated each on their own scale, both at a qualitative level ("low", "medium" and "high") and in quantitative form in the range from 0 to 1. the boundaries of the terms of these linguistic variables are set by expert assessments, in tabular possibilities of assessing information security risks… 49 copyright ©2024 assa adv. in systems science and appl.(2024) form. similarly, we assign values for the output parameter of the risk level r: "low", "acceptable", "high" and "critical". then we form a table of production rules, each row of which is a combination of the values of the input parameters. in this case, each line corresponds to a certain value of the output parameter. moreover, we do this for any possible combination of values of the selected parameters. further, the averaged quantitative values are correlated with the qualitative values of the variables [19]. we get many data for the possibility of using regression analysis methods. step 3. building a regression equation using matlab, ms excel or other software products with options for working with regressions, we build a linear model of multiple regressions from (3.1) with explanatory variables introduced into consideration for simplicity of presentation:    m i ii pkar 1 0  (3.2) where a0 and miki ,...,1,  , incomparable coefficients that can be found using ols. next, from equation (3.2), we proceed to the equation on a standardized scale and find standardized coefficients. step 4. analysis of the result obtained we build the coefficients in ascending order to determine the variables on which the information security risk depends largely. in addition, we examine the variables for their dependence on each other. to assess the quality of the constructed model, one of the important indicators in the construction of regression is calculated the coefficient of multiple determination to assess the joint influence of variables. the higher it is, the greater the impact of the selected variables on the risk level r. with a low value of the multiple determination coefficient (less than 0.6), variables that have little effect on the risk value with the lowest value of the ki coefficients are excluded from consideration. we return to step 2, where instead of the excluded parameters, we select other parameters from the set n and repeat the above steps until the value of the multiple determination coefficient is at least 0.8. we can complete the iterative process when we make sure that any new combination of parameters does not significantly increase the coefficient of multiple determinations. in this case, we will assume that the parameters that maximally affect the value of the information security risk level of the information object in question have been found. applying further fisher's f-criterion and student's t-criterion [23], one can also verify the statistical significance of both the regression coefficients and the resulting regression equation as a whole at a certain level of significance. after determining the parameters that have the maximum impact on the level of information security risk, we implement the stage of comparative risk assessment. we compare the value of the risk level obtained with the above parameters with its acceptable value. if necessary, we develop a set of measures aimed at ensuring such values of the established parameters, at which the value of the risk level lies within acceptable limits. it should be borne in mind that the cost of providing the required values of the desired parameters should not exceed the possible losses from the implementation of potential threats. example as an illustration of the proposed methodology, let us consider an example of determining the factors that most affect the level of risk in the network information infrastructure. we select 50 a. kozlov, n. noga copyright ©2024 assa adv. in systems science and appl.(2024) five linguistic variables from the four groups indicated above in the first step. these are the following factors:  the value level of the information asset of the network node,  the level of wear of technical means in the node,  node load level,  the level of organization of node protection,  the qualification level of the node staff. of the five indicators selected above, three indicators increase the level of risk with increasing values, and two indicators (the level of organization of node protection and the qualification level of the node staff), on the contrary, reduce the level of risk. production rules are much easier to form when the indicators are unidirectional, i.e. the risk level increases (decreases) with an increase (decrease) in the values of each of the considered indicators. in order to comply with these conditions, while leaving the essence of the indicators unchanged, we will change the level of organization of node protection to the level of node vulnerability, and the level of staff qualification to the level of staff incompetence. that is, we are now considering the following factors:  the value level of the information asset of the network node – x1,  the level of wear of technical means in the node – x2,  node load level – x3,  the vulnerability level of the node – x4,  the level of incompetence of the node staff – x5. now, increasing the value of any of the selected indicators leads to an increase in the level of risk. we believe that the values of all these variables take both quantitative values in the range [0; 1] and qualitative values (for example, low, medium, high). now the risk level from (3.2) we can represent as follows: 5 0 1 i i i r a k x      (3.3) where is an error that includes the influence of factors unaccounted for in this equation, as well as random errors and measurement features. to determine the coefficients in (3.3), we use ols. as mentioned above, the coefficients obtained are incomparable. in order to be able to compare coefficients and build parameters according to the degree of influence on risk, a regression equation is constructed on a standardized scale [23]: 5 1 ir i x i t l t   (3.4) , 1,...,5,il i  standardized coefficients and , , 1,...,5 ir xt t i  standardized variables such that , , 0, 1, 1, ..., 5. i i r xi i i i r x r x t t r x x xr r t t t t i            the boundaries of the terms of the above-mentioned linguistic variables, as indicated above, are set based on expert assessments, in tabular form. we should note that the influence of the above-mentioned factors might be quite uncertain due to their possible influence on each other. in this example, we will assume that the ranges of terms values of the above linguistic variables with the replacement of qualitative values with the corresponding averaged quantitative values correspond to those shown in tables 3.1 to 3.5. possibilities of assessing information security risks… 51 copyright ©2024 assa adv. in systems science and appl.(2024) table 3.1. the value level of an information resource (asset) in a node (x1). level (fuzzy variable) the ratio of the cost of information stored in the node to the cost of all information in the system the boundaries of the term "asset value level" the average value of the term 1 low up to 10% 0-0.11 0.05 2 middle from 10% to 40% 0.07-0.45 0.21 3 high more than40% 0.4-1.00 0.7 table 3.2. the level of wear of technical means (x2) level (fuzzy variable) the ratio of the age of the equipment to the maximum service life (8 years) the boundaries of the term “technical means wear level” the average value of the term 1 low the equipment has been in operation for less than 2 years 0-0.21 0.1 2 middle the equipment is operated from 2 to 6 years 0.15-0.65 0.4 3 high the equipment has been in operation for more than 6 years, there are signs of physical and moral deterioration 0.6-1.0 0.8 table 3.3. node load level (x3) level (fuzzy variable) the ratio of the volume of passing information to the maximum throughput of the node's communication equipment the boundaries of the term "node load level" the average value of the term 1 low there is a large margin for node throughput 0-0.28 0.14 2 middle the node works stably and can withstand peak loads 0.2-0.6 0.4 3 high the node is loaded and may not withstand peak load 0.55-1.0 0.725 table 3.4. the vulnerability level of the node to security threats (x4) level (fuzzy variable) availability of an information security system boundaries of the term “node vulnerability level” the average value of the term 1 low the node is protected by certified an information security system, and constantly monitored for employee actions and external attacks. 0-0.31 0.15 2 middle additional protective equipment has been installed at key positions of the node, and periodic monitoring is carried out. 0.2-0.7 0.45 3 high there is only an information security system, which is usually included in the hardware and software 0.6-1.0 0.8 table 3.5. the level of incompetence of security staff (x5) level (fuzzy variable) work experience in the field of information security and data protection the boundaries of the term "the level of incompetence of employees" the average value of the term 1 low work experience of more than 10 years 0-0.25 0.125 2 middle work experience from 2 to 10 years 0.2-0.7 0.45 3 high work experience up to 2 years 0.6-1.0 0.8 to form the production rules, we will also need the following table 3.6 of changes in the level of risk. 52 a. kozlov, n. noga copyright ©2024 assa adv. in systems science and appl.(2024) table 3.6. the output variable is risk (r) level measures taken the boundaries of the term "risk" the average value of the term 1 insignificant ignoring the risk 0-0.2 0.1 2 acceptable the operation of the system is possible, it is necessary to plan additional measures to protect the node. 0.16-0.5 0,33 3 high only limited operation of the node is possible, urgent measures must be taken to reduce the risk 0.45-0.65 0,55 4 critical the node impossible to work 0.6-1.0 0,8 the task of creating a fuzzy knowledge base at the risk assessment stage we propose to solve using the matlab software package (fuzzy logic option) [17]. thus, we obtain the production rules in the form of table 3.7. table 3.7. production rules x1 x2 x3 x4 x5 r 1 low low low low low insignificant 2 low low low low middle insignificant … 100 middle low high low high high … 242 high high high high middle critical 243 high high high high high critical next, using the averaged values from tables 3.1-3.6, we obtain from table 3.7 of the production rules a fuzzy knowledge base in quantitative form in table 3.8. table 3.8 fuzzy knowledge base, production rules x1 x2 x3 x4 x5 r 1 0,05 0,1 0,14 0,15 0,125 0,1 2 0,05 0,1 0,14 0,15 0,45 0,1 … 100 0,21 0,1 0,725 0,15 0,8 0,55 … 242 0,7 0,8 0,725 0,8 0,45 0,8 243 0,7 0,8 0,725 0,8 0,8 0,8 the values of the standardized coefficients are found using the matlab system [17] (regression option) or ms excel. thus, equation (3.4) takes the following form: 1 2 3 4 5 0,3529 0, 2169 0, 2588 0, 2954 0,1791 .r x x x x xt t t t t t     (3.5) now we calculate the coefficient of multiple determination to assess the combined effect of these five parameters on the value of the risk level 1 2 3 4 5 2 0,826rx x x x xm  this means that 82.6% of the risk level value we can explain by the parameters included in the regression equation and gives an assessment of the quality of the constructed model. to verify the statistical significance of the obtained regression equation as a whole and the regression coefficients at a certain level of significance, you can use fisher's f-criterion and student's t-criterion. ranking in ascending order of standardized coefficients from (3.5) shows that in the given example, the parameter the level of incompetence of the node staff – has the least influence on the overall risk value, and the value of information assets has the greatest influence. possibilities of assessing information security risks… 53 copyright ©2024 assa adv. in systems science and appl.(2024) results and discussion based on the results of our example, we can conclude that in order to reduce the risk of information security of an information system, first, you should pay attention to the proper distribution of stored information assets between network nodes. having established a list of parameters on which the level of information security risk of the node mainly depends, the next step is to rank the nodes of a complex network by criticality based on these parameters, which will identify the most critical nodes of the network [19]. node ranking can be carried out using multi-criteria optimization methods, for example, the method of borda counts or the method of determining the number of elements in the upper and lower boundary sets [24-25]. as a result of conducting an information security risk assessment using the proposed methodology, the following decisions can be made:  no additional measures are required, an acceptable level of risk is provided;  urgent measures are required to eliminate critical vulnerabilities;  to continue the risk analysis taking into account additional parameters, both external and internal;  adjust the goals to reduce the level of risk and costs. in addition, in the case of any of the above decisions, it is necessary to take into account the need for constant monitoring of the risk level, since any change in the external or internal conditions of the information infrastructure may affect its significance. the authors note that one of the results of using this technique is the formation of a knowledge base using production rules. at the same time, instead of values in qualitative form (values, as indicated above, "low", "medium", "high", etc.), data are formed in quantitative form, as in [19. 26] by averaging the values of terms of linguistic variables. based on the knowledge base obtained, using regression analysis methods, it is possible to build the regression model in such a way as to minimize the number of explanatory variables that significantly affect the value of the output variable "risk level". at the same time, it is possible to study the dependence of explanatory variables on each other, i.e. their mutual correlation. moreover, according to the results of the study, the proposed methodology makes it possible not to take into account variables that have little effect on the risk level. the formed knowledge base simplifies significantly the monitoring of the risk level, since this requires making changes to it only according to those parameters whose values have changed during the period under review. the proposed methodology has been tested in solving the problem of determining critical nodes in information systems with a complex network structure [19], as well as the problem of assessing insurance risks in the insurance of information systems, resources and their supporting infrastructures [26]. in both cases, the methodology showed a good result in assessing risk under conditions of uncertainty. this technique has a certain versatility and you can use it not only to solve problems of assessing information security risks, but also to solve other problems in conditions of uncertainty. thus, in [27], using the methods of multicriteria optimization used in the methodology, the parameters that most affect the demographic situation in various regions were determined. conclusion the methodology we have considered, which combines the use of fuzzy logic and regression analysis techniques to assess information security risk, enables us to calculate predicted values of risk levels in conditions of uncertainty, taking into account various factors that may influence risk, including subjective factors such as the level of employee qualification, staff turnover within the company, and others. 54 a. kozlov, n. noga copyright ©2024 assa adv. in systems science and appl.(2024) this technique provides an opportunity to determine the parameters that most affect the value of the risk level, which allows companies to pay special attention to measures to counter threats in the most dangerous areas of activity and minimize the cost of measures to protect their information resources. the proposed methodology you can apply to any complex network structure in both public and private companies with an extensive network of regional divisions (branches). at the same time, this technique allows not only to determine the level of risk, but also, in the case of additional multicriteria optimization methods, to find critical nodes and evaluate the effectiveness of branches in companies with a distributed structure [19, 28]. this technique you can also apply in the field of cyber insurance or risk insurance in the field of information technology [26]. its application expands the possibilities of using information risk insurance tools in various insurance companies and makes it relatively easy to determine the level of insurance risk in conditions of uncertainty and greatly facilitate the underwriting procedure. a similar technique based on the combined use of regression analysis and ranking methods [27], you can use to conduct a comparative analysis of the demographic situation in various regions of russia with the determination of indicators that have the greatest impact on the demography of the region. this makes it possible for regional authorities to make longand medium-term forecasts for the formation of a policy in the field of demography. we should note that this indicates some university of the proposed methodology. the versatility of the methodology allow you to apply it in many areas of management from small companies and enterprises to the public sector, i.e. where it is required to assess various risks under conditions of uncertainty. currently, different companies increase using artificial neural networks in economics and business [29]. neural networks are used, among other things, to increase the level of attacks detection on computer networks, as well as to determine the level of information security risk [30-33]. world statistics show that attackers have recently been massively organizing attacks on neural networks. therefore, the creation and operation of such networks is also associated with certain risks, most of which lie in the area of uncertainty. the authors believe that the proposed methodology allows for the assessment of information security risk when using neural networks and artificial intelligence systems based on them. references 1. information leaks in the world and russia for the first half of 2024: an analytical report, [online]. available https://www.infowatch.ru/analytics/analitika/utechki-informatsiiv-mire-i-rossii-za-pervoye-polugodiye-dve-tysyachi-dvadtsat-chetvertogo-goda.pdf [in russian] 2. gost r iso 31000-2019. 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(2022). multi-criteria decision-making tasks: theory and methods of analysis: university textbook. moscow, russia: yurayt, [in russian]. 26. kozlov, a. d. & noga, n. l. (2023). methodology for determining the parameters that most affect insurance risks in the field of information technology, insurance business, 12, 17-26, [in russian]. 27. kozlov, a. d. & noga, n. l. (2023). the comparative assessment methodology of the demographic situation in the regions, proc. of the 16th international conference “management of large-scale system development (mlsd)” (moscow, russia). 56 a. kozlov, n. noga copyright ©2024 assa adv. in systems science and appl.(2024) 28. kozlov, a. d. & noga, n. l. (2022). the selection of the comparative evaluation method of the effectiveness of the insurance company branches, proc. of the 15th international conference “management of large-scale system development (mlsd)”, (moscow, russia). 29. gorodnova, n. v. 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(2022). risk assessment using a neuro-fuzzy system, information and security, 25(4), 583-592, [in russian]. adv syst sci appl 2023; 03:16–26 published online at https://ijassa.ipu.ru. newton method vs. semismooth newton method for singular solutions of nonlinear complementarity problems alexey izmailov1*, evgeniy uskov2, yan zhibai1 1lomonosov moscow state university, moscow, russia 2derzhavin tambov state university, tambov, russia abstract: among the most successful techniques for solving nonlinear complementarity problems is the one consisting of reformulation of the problem in question as a system of nonlinear equations, by means of the so-called complementarity functions. different complementarity functions lead to nonlinear systems with different smoothness and regularity properties, thus allowing for application of different classed of numerical methods. in this paper we compare the newton method for the smooth reformulation with the semismooth newton method for the reformulation relying on the nonsmooth fischer–burmeister complementarity function, with a special emphasis on the cases when the solution in question violates the strict complementarity condition. keywords: nonlinear complementarity problem, complementarity function, strict complementarity, singular solution, newton method, semismooth newton method, extrapolation 1. introduction we consider the nonlinear complementarity problem (ncp) u ≥ 0, f (u) ≥ 0, ⟨u, f (u)⟩ = 0, (1.1) where f : rp → rp is a given smooth mapping. here and throughout the paper, ⟨·, ·⟩ stands for the euclidian inner product. accordingly, by ∥ · ∥ we will denote the euclidian norm. among the most successful approaches to numerical solution of ncp (1.1) relies on a reformulation of this problem as an equivalent system of equations. recall that a function ψ : r× r → r is referred to as a complementarity function if ψ(a, b) = 0 ⇔ a ≥ 0, b ≥ 0, ab = 0. with any such function at hand, one can readily observe that (1.1) is equivalent to the equation φ(u) = 0 (1.2) with φ : rp × rp, φ(u) = ψ(u, f (u)), (1.3) where ψ is applied componentwise. known complementarity functions are plenty [17, 18], and their instances may possess different smoothness properties. therefore, the specific choice of such a function naturally ∗corresponding author: izmaf@cs.msu.ru newton method vs. semismooth newton method 17 restricts the class of numerical methods applicable to the corresponding instance of equation (1.2). the rest of the paper is organized as follows. in section 2, we consider a smooth complementarity function, allowing for the basic newton method to be applied to the corresponding equation (1.2). we discuss the local convergence and rate-of-convergence properties, as well the linesearch technique for globalization of convergence, and the extrapolation technique for potential acceleration of convergence to solutions violating strict complementarity. section 3 deals with the fischer–burmeister complementarity function, in which case the corresponding equation (1.2) can be tackled by the semismooth newton method. local convergence, rate-of-convergence, and globalization of convergence are discussed. section 4 presents the results of a numerical comparison of the two methods on a collection of ncps with solutions violating strict complementarity. 2. smooth complementarity function and newton method consider, for example, a complementarity function defined by ψ(a, b) = 2ab− (min{0, a+ b})2. (2.1) this function is smooth, and assuming that f is differentiable at u ∈ rp, the corresponding mapping φ defined in (1.3) is also differentiable at u, with the rows of the jacobian φ′(u) given by φ′ i(u) = 2uif ′ i (u) + 2fi(u)e i − 2min{0, ui + fi(u)}(f ′ i (u) + ei), i = 1, . . . , p, (2.2) where e1, . . . , ep is the standard basis in rp. moreover, assuming that f is differentiable on a neighborhood of some point in rp, with its derivative f ′ being lipschitz-continuous on this neighborhood, the same property is possessed by φ′. under these smoothness assumptions, one can apply the basic newton method to equation (1.2) with φ defined according to (1.3), (2.1): for a given iterate uk ∈ rp, if f is differentiable at uk, one can compute the next iterate as uk+1 = uk + vk, where vk is a solution of the linear equation φ(uk) + φ′(uk)v = 0; (2.3) see, e.g., [15, section 2.1.1]. let ū be a solution of ncp (1.1). assume that f is differentiable near ū, with its derivative f ′ being continuous at ū, and that φ′(ū) with rows defined in (2.2) is nonsingular. then, according to [15, theorem 2.2], for any u0 ∈ rp close enough to ū, the newton method specified above uniquely defines a sequence {uk}, and this sequence converges to ū superlinearly, or even quadratically if f ′ is lipschitz-continuous near ū. convergence of the newton method can be globalized, e.g., by linesearch techniques. the following is essentially [7, algorithm 2.1]. algorithm 2.1: define the mapping φ according to (1.3), (2.1). choose the parameters σ ∈ (0, 1) and θ ∈ (0, 1). choose u0 ∈ rp, and set k = 0. 1. if φ(uk) = 0, stop with success. 2. compute vk ∈ rp as a solution of (2.3). if such vk cannot be found, stop with a failure. 3. set τ = 1. if the inequality ∥φ(uk + τvk)∥ ≤ (1− στ)∥φ(uk)∥ (2.4) is satisfied, set τk = τ . otherwise, replace τ by θτ , check inequality (2.4) again, etc., until (2.4) becomes valid. copyright © 2023 assa. adv syst sci appl (2023) 18 a. izmailov, e. uskov, y. zhibai 4. set uk+1 = uk + τkv k. 5. increase k by 1 and go to step 1. observe that if φ(uk) ̸= 0, then the function ∥φ(·)∥ is differentiable at uk, and its gradient is (φ′(uk))⊤φ(uk)∥φ(uk)∥. then the inner product of this gradient and vk solving (2.3) evidently equals −∥φ(uk)∥ < 0, while the stepsize test (2.4) is nothing else but the classical armijo inequality; see, e.g., [15, (2.62)], and step 3 of the algorithm is the standard armijo backtracking procedure. in practical implementations, and for global convergence analysis, instead of declaring failure when vk cannot be found in step 2, one can supply algorithm 2.1 with some safeguarding tools; see, for example, [7, algorithm 3.1]. according to [7, theorem 3.1], any accumulation point ū of any sequence {uk} generated by such enhanced algorithm 2.1 satisfies (φ′(ū))⊤φ(ū) = 0, and in particular, can be guaranteed to be a solution of ncp (1.1) under some reasonable additional assumptions. however, in the numerical experiments reported in section 4, we employ algorithm 2.1 as it is, without any additional safeguards, and this indeed leads to some failures on step 2. moreover, assuming that φ′(ū) is nonsingular at a solution ū of ncp (1.1), and employing the argument in [15, theorem 5.4], one can easily see that (2.4) is satisfied with α = 1 provided uk is close enough to ū. therefore, the algorithm asymptotically accepts the full step, and inherits the local convergence and rate of convergence properties of the basic newton method. observe, however, that if ū violates that strict complementarity condition ū+ f (ū) > 0, which means that there exist i ∈ {1, . . . , p} such that ūi = fi(ū) = 0. then (2.2) yields φ′ i(ū) = 0, and hence, φ′(ū) is necessarily singular, not allowing to apply the theory outlined above. the strict complementarity condition is regarded as rather restrictive in the literature, and therefore, one should be ready to face singularity of a solution ū of (1.2) when the latter is obtained as a reformulation of ncp (1.1), employing the smooth complementarity function (2.1) (and actually, any other smooth complementarity function). there exists quite a rich literature on the behavior of newton-type methods near singular solutions of nonlinear equations; we mention only some references most tightly related to the contents of this work: [7–11, 13]. apart from characterizing local linear convergence from asymptotically dense domains starlike with respect to such solution, some acceleration techniques were developed, such as the so-called extrapolation [9, 11]. in its simplest form, this procedure amounts to generating, in parallel with the main sequence {uk} of the newton method, an auxiliary sequence {ûk} obtained by doubling the newton step: for each k set ûk+1 = uk + 2vk. (2.5) this specific procedure is suggested by the convergence pattern of the newton method to singular solutions of certain wide classes. according to [11, theorem 4.1], under the appropriate assumptions, the sequence {ûk} generated this way converges to ū linearly with the asymptotic ratio of 1/4 (instead of 1/2 for the sequence {uk} generated by the newton method), from all points in the domain of convergence of newtonian sequences. the main iterative sequences {uk} are not affected by this acceleration technique in any way, and generating the auxiliary sequence {ûk} is very cheap. therefore, this acceleration technique is is fully practical, and can be easily incorporated into the globalized algorithm 2.1 [7]. that said, in all the references cited above in connection with newton-type methods near singular solutions it is assumed that φ is al least twice differentiable, which is not the case for the mapping defined by (1.3), (2.1). to the best of our knowledge, the current paper is the first one where the properties in question will be assessed for equations with locally lipchitzian first derivatives but perhaps without second derivatives. copyright © 2023 assa. adv syst sci appl (2023) newton method vs. semismooth newton method 19 3. fischer–burmeister complementarity function and semismooth newton method if one does not want to deal with potential singularity of solutions (as in section 2), a natural thing to do is to employ different (nonsmooth) complementarity functions. one of the most successful and widely used is the so-called fischer–burmeister complementarity function defined by ψ(a, b) = a+ b− √ a2 + b2. (3.1) this function is not differentiable at (0, 0), and the corresponding mapping φ defined in (1.3) need not be differentiable at u ∈ rp with ui = fi(u) = 0 for at least one i ∈ {1, . . . , p}, no matter how smooth is f . in particular, φ need not be differentiable at a solution violating strict complementarity. at the same time, if f is differentiable near u ∈ rp, with f ′ being continuous at u, then φ is lipschitz continuous at u, and even semismooth at u; see [15, proposition 3.8] therefore, even though the newton method is in general not applicable to equation (1.2) with φ defined according to (1.3), (3.1), one can make use of the so-called semismooth newton method [15, section 2.4.1], which is the iterative procedure specified above for newton method, but with the iteration equation (2.3) replaced by φ(uk) + jkv = 0, (3.2) where jk is some element of ∂bφ(uk), here, for any u ∈ rp, ∂bφ(u) = {j ∈ rp×p | ∃ {uj} ⊂ sφ : {uj} → u, {φ(uj)} → j}, is the so-called b-differential of φ at u, with sφ standing for the set of all points where φ is differentiable; see, e.g., [15, section 1.4.1]. if ū is a solution of ncp (1.1), and all matrices in ∂bφ(ū) are nonsingular, then, according to [15, theorem 2.42], for any u0 ∈ rp close enough to ū, the semismooth newton method uniquely defines a sequence {uk}, and this sequence converges to ū superlinearly, or even quadratically under further smoothness assumptions on f . the upper estimate of ∂bφ(u) for φ defined according to (1.3), (3.1) is given in [15, proposition 3.11]: the rows of any matrix j ∈ ∂bφ(u) satisfy the equalities ji =  αif ′ i (u) + βie i if ui = fi(u) = 0, f ′ i (u) + ei − fi(u)f ′ i (u) + uie i√ u2i + (fi(u))2 if ui ̸= 0 or fi(u) ̸= 0, i = 1, . . . , p, (3.3) with some (αi, βi) ∈ s for i = 1, . . . , p such that ui = fi(u) = 0, where s = {(a, b) ∈ r2 | (a− 1)2 + (b− 1)2 = 1}. this upper estimate serves as the basis for practical procedures allowing to compute an element jk ∈ ∂bφ(u k), to be used in (3.2); see, e.g., [15, p. 161]. however, in section 4, we employ the upper estimate above directly, i.e., we pick up jk in (3.2) as some matrix with rows satisfying (3.3) for u = uk. this choice is not fully clean theoretically, but it is somehow practically justified, in particular, by the natural expectation that the case when uki = fi(u k) = 0 for some i ∈ {1, . . . , p} will not be encountered along the iteration sequences {uk}, and hence, φ will differentiable at uk, and the upper estimate of ∂bφ(uk) given be (3.3) will reduce to {φ′(uk)}, for all k. as for globalization of convergence, we next present a counterpart of algorithm 2.1 for the semismooth newton method. copyright © 2023 assa. adv syst sci appl (2023) 20 a. izmailov, e. uskov, y. zhibai algorithm 3.1: define the mapping φ according to (1.3), (3.1). choose the parameters σ ∈ (0, 1) and θ ∈ (0, 1). choose u0 ∈ rp, and set k = 0. 1. if φ(uk) = 0, stop with success. 2. compute some jk ∈ ∂bφ(u k). compute vk ∈ rp as a solution of (3.2). if such vk cannot be found, stop with a failure. 3. set τ = 1. if the inequality (2.4) is satisfied, set τk = τ . otherwise, replace τ by θτ , check inequality (2.4) again, etc., until (2.4) becomes valid. 4. set uk+1 = uk + τkv k. 5. increase k by 1 and go to step 1. the use of the stepsize test (2.4) in this context can be explained as follows. the function ∥φ(·)∥ with φ defined according to (1.3), (3.1) need not be differential at uk, even if φ(uk) ̸= 0. that said, according to [15, proposition 5.5], the function φ : rp → r, φ(u) = ∥φ(u)∥2, is differentiable at every point u ∈ rp where f is differentiable, and its gradient satisfies φ′(u) = 2j⊤φ(u) ∀ j ∈ ∂φ(u), (3.4) where ∂φ(u) is clarke’s generalized jacobian of φ at ū, defined as the convex hull of ∂bφ(u); see, e.g., [15, section 1.4.1]. therefore, for vk solving (3.2) it holds that ⟨φ′(uk), vk⟩ = 2⟨φ(uk), jkvk⟩ = −2∥φ(uk)∥2 = −2φ(uk). (3.5) hence, ∥φ(uk + τvk)∥2 = φ(uk + tvk) = φ(uk) + τ⟨φ′(uk), vk⟩+ o(τ) = ∥φ(uk)∥2 − 2τ∥φ(uk)∥2 + o(τ) = (1− 2τ + o(τ))∥φ(uk)∥2 as t→ 0, implying that (2.4) is satisfied for all τ > 0 small enough, for any fixed σ ∈ (0, 1). observe further that employing (3.5), condition (2.4) can be equivalently written in the form φ(uk + τvk) ≤ (1− 2στ + σ2τ 2)φ(uk) = φ(uk) + στ(1− στ/2)⟨φ′(uk), vk⟩. (3.6) since we are not aware of any existing global convergence theories for the linesearch semismooth newton method with the specific stepsize test (2.4), we next address this issue for the following safeguarded version of algorithm 3.1. algorithm 3.2: define the mapping φ according to (1.3), (3.1). choose the parameters c > 0, q > 0, σ ∈ (0, 1) and θ ∈ (0, 1). choose u0 ∈ rp, and set k = 0. 1. if φ(uk) = 0, stop with success. 2. compute some jk ∈ ∂bφ(u k). compute vk ∈ rp as a solution of (3.2). if such vk cannot be found, or violates ∥vk∥ ≤ max{c, 1/∥φ(uk)∥q}, (3.7) go to step 4. 3. set τ = 1. if the inequality (2.4) is satisfied, set τk = τ . otherwise, replace τ by θτ , check inequality (2.4) again, etc., until (2.4) becomes valid. go to step 6 4. set vk = −2j⊤ k φ(u k). if vk = 0, stop. copyright © 2023 assa. adv syst sci appl (2023) newton method vs. semismooth newton method 21 5. set τ = 1. if the inequality φ(uk + τvk) ≤ φ(uk)− στ∥vk∥2 (3.8) is satisfied, set τk = τ . otherwise, replace τ by θτ , check (3.8), etc., until (3.8) becomes valid. 6. set uk+1 = uk + τkv k. 7. increase k by 1 and go to step 1. theorem 3.1: let f : rp → rp be continuously differentiable. then, for any starting point u0 ∈ rp, algorithm 3.2 either terminates with some iterate uk satisfying j⊤φ(uk) = 0 ∀ j ∈ ∂φ(uk), (3.9) or generates an infinite sequence {uk} such that every accumulation point ū of this sequence satisfies j⊤φ(ū) = 0 ∀ j ∈ ∂φ(ū), (3.10) where φ according to (1.3), (3.1). moreover, if for some ū there exists an infinite subsequence {ukj} convergent to ū such that the semismooth newton direction is used by algorithm 3.2 for all indices k = kj (i.e., for all j, for k = kj equation (3.2) is solvable, and the computed solution satisfies (3.7)), then {φ(uk)} → 0 (3.11) as k → ∞, and in particular, all accumulation points of {uk} are solutions of the ncp (1.1). proof if for a current iterate uk it holds that φ(uk) = 0, the algorithm terminates, and (3.9) is evidently satisfied in this case. otherwise, if the algorithm accepts the semismooth newton direction vk, then the discussion before algorithm 3.1 implies that (2.4) is always satisfied after a finite number of backtrackings, i.e., the linesearch procedure at step 3 of algorithm 3.2 terminates with some τk > 0. if the newton direction is not used, then, according to (3.4), at step 4 the algorithm picks up vk = −φ′(uk). in this case, the linesearch procedure at step 5 of the algorithm with the stepsize test (3.8) terminates finitely with some τk > 0 (as it is the standard armijo linesearch procedure for a function which is smooth at uk, and in the direction the direction of the negative gradient of this function at uk) if φ′(uk) ̸= 0, while otherwise, the algorithm terminates with (3.9) being satisfied again, the latter following from (3.4). therefore, algorithm 3.2 either terminates with some iterate uk satisfying (3.9), or generates an infinite sequence {uk}, and it remains to analyze the latter case. observe that the sequence {∥φ(uk)∥} is always monotonically non-increasing, no matter which search directions are used. but this sequence is bounded below (by zero), and hence converges. consider first the case when there exists an infinite subsequence {ukj} convergent to some ū, and such that the semismooth newton direction is accepted by algorithm 3.2 at every point of this subsequence. if τ̄ = lim sup j→∞ τkj > 0, (3.12) then (2.4) implies that for infinitely many indices kj , the residual ∥φ(ukj)∥ is reduced at least linearly (by a factor of at least (1− στ̄/2)), and hence, {φ(ukj)} → 0 as j → ∞. since {∥φ(uk)∥} is convergent, we conclude that (3.11) holds. copyright © 2023 assa. adv syst sci appl (2023) 22 a. izmailov, e. uskov, y. zhibai on the other hand, if (3.12) is not valid, then lim j→∞ τkj = 0, (3.13) implying that for each j large enough, the initial trial stepsize value had been reduced at least once, i.e., the value τ = τkj/θ > τkj does not satisfy (2.4). since (2.4) is equivalent to (3.6), it then holds that φ(ukj + τkjv kj/θ)− φ(ukj) τkj/θ > σ(1− στkj/2)⟨φ′(uk), vk⟩. (3.14) observe that according to [15, proposition 5.5], φ is continuously differentiable on rp, and employing the mean-value theorem [15, theorem a.10(a)], we then obtain from (3.14) that for each j there exists ωkj ∈ [0, 1] such that ⟨φ(ukj + ωkjτkjv kj/θ), vkj⟩ > σ(1− στkj/2)⟨φ′(uk), vk⟩. (3.15) if the sequence {vkj} of the semismooth newton directions is unbounded, condition (3.7) implies that lim inf j→∞ ∥φ(ukj)∥ = 0. in view of convergence of the sequence {∥φ(uk)∥}, this again implies (3.11). therefore, it remains to consider the case when {vkj} is bounded. taking a further subsequence, if necessary, we may assume that {vkj} converges to some ṽ, and therefore, by (3.5), ⟨φ′(ū), ṽ⟩ = −2φ′(ū). then by (3.13), passing onto the limit in (3.15) as j → ∞, we obtain that −φ(ū) ≥ −σφ(ū), which is only possible when φ(ū) = 0. it remains to consider the case when there exists an infinite subsequence {ukj} convergent to some ū, and such that the semismooth newton direction is not accepted by algorithm 3.2 at every point of this subsequence. in this case, the iterates {ukj+1} are generated by the gradient steps with armijo linesearch for the continuously differentiable merit function φ. then by standard argument (see, e.g., [2, proposition 1.2.3]), it follows that φ′(ū) = 0, and according to (3.4), this yields (3.10). 4. numerical results in this section we provide some numerical comparisons of the algorithms discussed above for a collection of small ncps taken from [19], and for some other examples of ncps with solutions violating strict complementarity, taken from various sources. for the former, we use the identifiers of the problems adopted in [19], while the latter are cited as they appear in the related references. for each of these test problems, a solution of interest ū, violating strict complementarity, is known; this information is provided in table 3.1. the algorithms being tested are abbreviated as follows: • nm is algorithm 2.1. • nm-ep (for “extrapolation”) is algorithm 2.1, generating also an auxiliary sequence {ûk} according to (2.5). copyright © 2023 assa. adv syst sci appl (2023) newton method vs. semismooth newton method 23 100 101 0 0.2 0.4 0.6 0.8 1 nm nm-ep snm snm-ep fig. 3.1. single runs from “standard” starting points. 100 101 0 0.2 0.4 0.6 0.8 1 nm nm-ep snm snm-ep fig. 3.2. multiple runs from random starting points. copyright © 2023 assa. adv syst sci appl (2023) 24 a. izmailov, e. uskov, y. zhibai • snm is algorithm 3.1. • snm-ep is algorithm 3.1, generating also an auxiliary sequence {ûk} according to (2.5). all algorithms were run with with parameter values σ = 0.01 and θ = 0.5. runs of nm and snm were terminate with success declared once an iterate uk was generated satisfying ∥φ(uk)∥ ≤ 10−11. for nm-ep and snm-ep, for k = 1, 2, . . ., we first generated ûk according to (2.5), and terminated the run with success if ∥φ(ûk)∥ ≤ 10−11. (4.1) convergence to the solution of interest ū was declared when, at successful termination, we had ∥uk − ū∥ ≤ 10−3 (with uk replaced by ûk for nm-ep and snm-ep when termination happened because of (4.1)). if successful termination did not occur after 50 iterations, or the backtracking procedures in steps 3 of algorithms 2.1 and 3.1 produced a trial value τ such that τ∥vk∥ ≤ 10−10, the process was terminated with failure declaring that the step became too short to proceed. all problems in [19] are supplied with “recommended” starting points. for each of the other test problems being used, we have also selected some starting point, trying to keep our choices reasonably generic. information on “standard” starting points u0 and solutions of interest ū is collected in table 3.1. in the first part of the experiments, for each test problem, we have performed a single run of each algorithm using the specified “standard” starting point. the results (iteration counts until successful termination) are reported in table 3.2; cases of failure are shown as “–”. information in table 3.2 is further summarized in figure 3.1 in the form of performance profile, proposed in [4]. the value of any function whose graph is presented in the figure at a point t of the horizontal axis corresponds to the portion of test problems for with the result (in our case, the iteration count) of the algorithm associated with this graph was no more than t times worse than the best result among all the algorithms being tested (the result of a failure is regarded infinitely many times worse than any result). in the second part of the experiments, for each test problem, we have performed 1000 runs of each algorithm from randomly generated starting points with uniform distribution in the cubic neighborhood of the solution of interest, with the half-edge of the cube equal to 1. in table 3.3, we report the percentage of successful runs, the average iteration count over successful runs, and the average distance to the solution of interest over cases when convergence to such solution was detected (separated by slashes). the cases when there were no successful runs are still marked by “–”. the corresponding performance profile in figure 3.2 is an adaptation of the original proposal in [4] to the case of multiple runs for each test problem, suggested in [16]. the failures for dis64 and [1, example 3.1] happen at steps 2 of both algorithms 2.1 and 3.1 because the jacobians of the corresponding φ at u0 are singular, and the iteration equation cannot be solved. generally, this is a typical reason of failures encountered in these experiments. there are also some rare cases when failure was declared because of too short steps (for quarquad and doubleknot), or when the iteration limit was achieved. another observation is that the full step of nm was always asymptotically accepted in these experiments, while for snm, there were runs with the last step before successful termination using τk < 1. looking at the results of the experiments, one can conclude that nm is definitely outperformed in terms of efficiency by any of the concurrent algorithms, and the reason for copyright © 2023 assa. adv syst sci appl (2023) newton method vs. semismooth newton method 25 that must be precisely the slow (linear, at best) convergence to so-called critical solutions, established in [9–11, 13] for newton-type methods in the twice differentiable case. this might give an impression that the use of the fischer–burmeister complementarity function and semismooth newton-type methods is definitely preferable with respect to the smooth complementarity function and usual newton-type methods. indeed, it appears that attraction to any special (“critical”) singular solutions and the related special convergence pattern show up in a much less persistent way for nonsmooth equations. nevertheless, passing from snm to snm-ep makes the efficiency somewhat higher, even though there currently exists no any theory supporting this behavior. observe, however, that the impact of extrapolation on the newton method is even more evident: nm-ep outperforms snm in terms of efficiency, and is demonstrated performance similar to that of snm-ep. all in all, nm-ep demonstrates a concurrent behavior when compared with snm and snm-ep, even though nm evidently suffers from the negative effect of critical solutions on its convergence rate. 5. concluding remarks in this paper, we provided a numerical comparison of two approaches to numerical solution of nonlinear complementarity problems, one employing a smooth complementarity function, while another one making use of the (nonsmooth) fischer–burmeister complementarity function. the results obtained demonstrate that even in the presence of solutions violating strict complementarity, the first approach may still be fully competitive if equipped with the simplest extrapolation procedure intended for acceleration of convergence to such solutions. theoretical justification of the effects demonstrated in this work will be a subject of future research. acknowledgements this research was supported by the russian science foundation grant 23-11-20020 (https://rscf.ru/en/project/23-11-20020/). references 1. arutyunov, a.v. & izmailov, a.f. (2018) stability of possibly nonisolated solutions of constrained equations with applications to complementarity and equilibrium problems. set-valued var. anal., 26, 327–352. 2. bertsekas, d.p. (1999) nonlinear programming. 2nd edition. belmont: athena. 3. daryina, a.n. & izmailov, a.f. (2004) on identification of active indices in mixed complementarity problems. in: bereznev, v.a. (ed.), questions of modelling and analysis in decision making problems. ccas, moscow, 72–87. 4. dolan, e.d. & more, j.j. (2002) benchmarking optimization software with performance profiles. math. program., 91, 201–213. 5. fischer, a., izmailov, a.f. & jelitte, m. (2021) newton-type methods near critical solutions of piecewise smooth nonlinear equations. comput. optim. appl., 80, 587– 615. 6. fischer, a., izmailov, a.f. & solodov, m.v. (2019) local attractors of newton-type methods for constrained equations and complementarity problems with nonisolated solutions. j. optim. theory appl., 180, 140–169. 7. fischer, a., izmailov, a.f. & solodov, m.v. (2021) accelerating convergence of the globalized newton method to critical solutions of nonlinear equations. comput. optim. appl., 78, 273–286. copyright © 2023 assa. adv syst sci appl (2023) https://rscf.ru/en/project/23-11-20020/ 26 a. izmailov, e. uskov, y. zhibai 8. fischer, a., izmailov, a.f. & solodov, m.v. (2021) unit stepsize for the newton method close to critical solutions. math. program., 187, 697–721. 9. griewank, a. (1980) analysis and modification of mewton’s method at singularities. phd thesis. australian national university, canberra. 10. griewank, a. (1980) starlike domains of convergence for newton’s method at singularities. numer. math., 35, 95–111. 11. griewank, a. (1985) on solving nonlinear equations with simple singularities or nearly singular solutions. siam rev., 27, 537–563. 12. izmailov, a.f. & kurennoy, a.s. (2014) on regularity conditions for complementarity problems. comput. optim. appl., 57, 667–684. 13. izmailov, a.f. kurennoy, a.s. & solodov, m.v. (2018) critical solutions of nonlinear equations: local attraction for newton-type methods. math. program., 167, 355–379. 14. izmailov, a.f. & solodov, m.v. (2001) error bounds for 2-regular mappings with lipschitzian derivatives and their applications. math. program., 89, 413–435. 15. izmailov, a.f. & solodov, m.v. (2014) newton-type methods for optimization and variational problems. springer series in operations research and financial engineering. cham: springer. 16. izmailov, a.f., solodov, m.v. & uskov, e.i. (2015) combining stabilized sqp with the augmented lagrangian algorithm comput. optim. appl., 62, 405–429. 17. kanzow, c. (1994) some equation-based methods for the nonlinear complementarity problem. optim. methods software, 3, 327–340. 18. mangasarian, o.l. (1976) equivalence of the complementarity problem to a system of nonlinear equations. siam j. appl. math., 31, 89–92. 19. oberlin, c. & wright, s.j. (2009) an accelerated newton method for equations with semismooth jacobians and nonlinear complementarity problems. math. program., 117, 355–386. copyright © 2023 assa. adv syst sci appl (2023) newton method vs. semismooth newton method 27 test problems u0 ū quarp 0.9 1 dis61 (0.2, 0.85) (0, ( √ 5− 1)/2) quarquad, 1 (0.1, 0.9) (0, 1) quarquad, 2 (0.9, 0.9) (1, 0) affknot1 (0.9, 0.1) (0, 1) affknot2 (0.5, 0.5) (0, 1) quadknot (0.5, 0.5) (0, 1) munson4 (0, 0) (1, 1) dis64 (2, 4) (0, 0) ne-hard (10, 1, 10) (0, 0, √ 200) doubleknot (0.5, 0.5, 0.5, 0.5) (1, 0, 0, 1) quad1, 1 (0.9, -0.1) (1, 0) quad1, 2 (0.9, 0.1) (1, 0) quad2, 1 (-1, -1) (0, 0) quad2, 2 (1, 1) (0, 0) quarn 0.9 1 [1, example 3.1] (-0.5, -0.9) (0, 0) [1, example 3.2] (-0.5, -0.5) (0, 0) [1, example 3.3], 1 (-0.9, 0.5) (0, 0) [1, example 3.3], 2 (0.9, -0.9) (1, 0) [1, example 3.4] (1.9, -0.5) (1, 0) [3, example 6] (0.9, 0.1) (0, 0) [3, example 7] (0.9, 0.1) (0, 0) [3, example 8] (1.9, 0.9) (0, 1) [3, example 9] (0.9, 0.9) (0, 0) [5, example 1] -0.9 0 [5, example 3], 1 (0.9, 0.1) (0, 1) [5, example 3], 2 (-0.9, 0.5) (0, 0) [6, example 4.2] (0.9, 0.9) (0, 0) [6, example 4.3] (0.9, 0.9) (0, 0) [12, example 3.4] 0.9 0 [12, example 3.5] (0.9, 0.9) (0, 0) [14, example 1] (0.9, 0.9) (0, 1) [14, example 2] (0.9, 0.1) (0, 0) table 3.1. “standard” starting points and solutions. copyright © 2023 assa. adv syst sci appl (2023) 28 a. izmailov, e. uskov, y. zhibai test problems nm nm-ep snm snm-ep quarp 15 14 15 13 dis61 19 10 18 12 quarquad, 1 16 8 4 4 quarquad, 2 18 17 17 15 affknot1 20 2 6 6 affknot2 18 1 1 1 quadknot 18 9 16 11 munson4 19 10 20 8 dis64 – – – – ne-hard 25 16 7 7 doubleknot 21 11 8 8 quad1 15 5 12 12 quad2 20 9 18 18 quarn 15 14 15 13 [1, example 3.1] – – – – [1, example 3.2] 18 9 4 4 [1, example 3.3], 1 20 11 7 7 [1, example 3.3], 2 19 5 5 5 [1, example 3.4] 20 11 18 6 [3, example 6] 19 1 1 1 [3, example 7] 19 1 1 1 [3, example 8] 20 11 18 13 [3, example 9] 19 1 1 1 [5, example 1] 19 8 17 17 [5, example 3], 1 21 16 5 5 [5, example 3], 2 19 10 18 18 [6, example 4.2] 19 1 1 1 [6, example 4.3] 19 9 5 5 [12, example 3.4] 19 1 1 1 [12, example 3.5] 20 1 1 1 [14, example 1] 19 7 15 4 [14, example 2] 19 1 1 1 table 3.2. single runs from “standard” starting points. copyright © 2023 assa. adv syst sci appl (2023) newton method vs. semismooth newton method 29 test problems nm nm-ep snm snm-ep quarp 100/18/5.1e-04 100/16/6.7e-04 100/16/4.9e-04 100/15/6.9e-04 dis61 100/17/1.7e-06 100/9/1.2e-06 100/15/2.4e-06 100/11/5.9e-09 quarquad, 1 100/20/6.7e-07 100/13/4.0e-07 100/10/1.7e-13 100/10/1.2e-13 quarquad, 2 97/20/5.6e-04 97/17/6.2e-04 100/17/5.6e-04 100/16/7.1e-04 affknot1 100/11/4.9e-06 100/5/2.8e-07 94/5/5.1e-05 94/5/5.2e-05 affknot2 100/19/4.5e-06 100/9/5.7e-13 100/5/2.7e-13 100/5/3.3e-13 quadknot 100/19/2.1e-06 100/9/1.3e-06 49/17/2.2e-06 68/10/1.3e-07 munson4 100/19/1.7e-06 100/9/1.5e-06 100/18/2.5e-06 100/7/1.4e-06 dis64 100/19/1.7e-06 100/1/5.8e-17 100/1/2.5e-16 100/1/2.9e-16 ne-hard 100/22/2.1e-06 100/15/2.4e-07 100/6/5.6e-06 100/6/7.7e-06 doubleknot 88/19/2.6e-06 88/10/6.6e-13 70/5/4.4e-13 71/5/4.3e-13 quad1 100/19/7.2e-05 100/13/7.0e-05 100/16/2.3e-06 100/16/2.2e-06 quad2 100/19/6.3e-05 100/12/7.0e-05 100/16/2.3e-06 100/16/2.2e-06 quarn 100/19/1.0e-04 100/18/7.3e-04 100/19/6.2e-04 100/18/5.3e-04 [1, example 3.1] – – – – [1, example 3.2] 50/18/2.3e-06 51/8/8.4e-13 53/4/5.7e-13 50/4/5.5e-13 [1, example 3.3], 1 100/10/2.8e-06 100/4/3.4e-7 100/7/1.0e-10 100/5/8.9e-07 [1, example 3.3], 2 100/5/2.3e-06 100/2/3.0e-05 100/10/7.3e-06 100/6/8.2e-06 [1, example 3.4] 100/17/7.0e-05 100/11/6.0e-05 100/15/3.3e-06 100/7/3.9e-06 [3, example 6] 100/19/1.9e-06 100/1/2.2e-16 100/1/1.8e-16 100/1/1.7e-16 [3, example 7] 100/18/2.8e-06 100/1/2.7e-16 100/1/2.1e-16 100/1/2.0e-16 [3, example 8] 100/17/1.9e-06 100/8/1.3e-06 100/15/2.5e-06 100/11/2.2e-12 [3, example 9] 100/19/2.2e-06 100/1/6.0e-16 100/1/1.7e-16 100/1/1.7e-16 [5, example 1] 100/19/7.2e-05 100/12/7.0e-05 100/16/2.3e-06 100/16/2.2e-06 [5, example 3], 1 100/19/6.9e-05 100/13/3.8e-09 100/10/4.3e-13 100/10/5.4e-13 [5, example 3], 2 100/19/6.9e-05 100/13/6.2e-05 100/15/2.2e-06 100/15/2.2e-06 [6, example 4.2] 100/18/2.1e-06 100/1/1.4e-16 100/1/6.5e-15 100/1/7.4e-15 [6, example 4.3] 100/19/1.6e-06 100/8/1.3e-10 100/5/5.3e-13 100/5/4.7e-13 [12, example 3.4] 100/18/1.6e-06 100/1/6.3e-18 100/1/6.6e-17 100/1/6.6e-17 [12, example 3.5] 100/19/1.6e-06 100/1/8.4e-16 100/1/3.5e-15 100/1/3.6e-15 [14, example 1] 100/19/1.7e-06 100/9/1.2e-06 100/18/2.3e-06 100/6/1.3e-06 [14, example 2] 100/18/2.0e-06 100/1/7.2e-16 100/1/9.5e-16 100/1/5.3e-16 table 3.3. multiple runs from random starting points. copyright © 2023 assa. adv syst sci appl (2023) introduction smooth complementarity function and newton method fischer–burmeister complementarity function and semismooth newton method numerical results concluding remarks adv syst sci appl 2023; 02:99–114 published online at https://ijassa.ipu.ru. on estimating the characteristics of a fork-join queueing system with poisson input and exponential service times anastasia v. gorbunova1*, alexey v. lebedev2 1v.a. trapeznikov institute of control sciences of russian academy of sciences, moscow, russia 2lomonosov moscow state university, moscow, russia abstract: the paper studies the classical fork-join queueing system with m |m |1 subsystems. the analysis of this system is still relevant due to the lack of exact solutions for assessing its performance characteristics if the number of subsystems exceeds two. in addition, the fork-join system is a mathematical model of parallel or distributed computing systems that have become widespread as one of the most effective methods for processing big data. an approach based on graphical analysis, non-linear regression, and the use of the nelder-mead optimization method is proposed to estimate the mathematical expectation and dispersion of the response time of a fork-join system. as a result, the authors managed to modify the known approximations and significantly (many times) improve their approximation quality. the paper also examines the quality of the experimental data of simulation modeling used to estimate the approximation error of the obtained expressions. as a rule, this issue remains outside the scope of ongoing research in the field of this topic due to the complexity of such an analysis. and sometimes, it is due to the underestimation of the importance of this issue. the article proposes an approach to finding confidence intervals for simulation results. it provides an algorithm for their construction and also gives some recommendations. keywords: fork-join queueing system, parallel service, parallel computing, average response time, response time variance, simulation. 1. introduction the fork-join queueing system (qs) is a system with parallel service where the task is divided into a fixed number k of smaller components (subtasks). then each of the subtasks enters the queue for service in one of the subsystems in the system with an infinite storage capacity and one server. after each subtasks, initially constituting one task, is serviced, the task will be considered fully serviced and will be able to leave the system. the described system is a mathematical model of many real-life and similarly functioning physical systems. the most striking example of it is data-intensive systems. it is because dividing a large task into subtasks with their subsequent parallel processing saves time [1,2]. the study of the characteristics of such systems is currently relevant due to the breadth of the so-called “big data” phenomenon and, accordingly, the emergence of an impressive number of services that provide services for processing and analyzing big data [3–5]. studies of fork-join systems have been conducted for a relatively long time. however, the exact result for such an essential characteristic of any qs as its average response time was obtained only in the relatively simplest case. namely, when the number of subsystems is k = 2, the input flow is poissonian, and the service time on servers is exponential (in other ∗corresponding author: avgorbunova@list.ru 100 a.v. gorbunova, a.v. lebedev words, subsytems are markov queues of type m |m |1) [6]. if the number of subsystems is more than two, then only approximations of varying degrees of accuracy [6–9] are known. the reason for the complexity of the analysis of the described system is the presence of dependencies between the sojourn times of subtasks in subsystems. this is the main difference between the described system and the set k of parallel functioning qs. in this case, the average response time of the entire system will be determined by the longest sojourn time in the corresponding subsystem of one of its subtasks. thus, the average response time of a fork-join qs is essentially the mathematical expectation of the maximum of k dependent random variables of sojourn times in subsystems, i.e., the k-th order statistic in the sequence of specified random variables. therefore, elements of the theory of order statistics [10] were used along with the methods of queuing theory to approximate the average response time. however, even in this case, the obtained estimates are approximate since the theory of order statistics focuses more on the study of samples of independent random variables. in addition, if we are not talking about the poisson input flow and exponential service times, it is quite difficult to determine the distribution of the sojourn time of the task in the qs. in more recent publications in the field of this topic, an approach is applied to the analysis of fork-join systems using machine learning methods, in particular, artificial neural networks or multiple non-linear regression [11, 12]. as a result, approximations of the mathematical expectation and higher order moments of the response time were obtained for rather complex fork-join qs structures, for example, with the pareto distribution for service time. details of the approach based on machine learning methods can be found in [13]. there are other approaches to deriving an estimate of the average response time of fork-join systems. many of them are described in the review [14]. you can also highlight [1, 15–17] among recent works. this article proposes another approach to estimating the response time of a fork-join qs. this approach is based on well-known approximations for the mathematical expectation and dispersion of the sojourn time of the task in the qs. using graphical analysis, as well as using the nelder-mead optimization method, the authors managed to modify the known approximations for fork-join qs with k subsystems m |m |1 and significantly (several times) improve their quality of the approximation. analytical formulas were obtained using a limited set of experimental data; however, as shown by a numerical experiment, they are quite accurate for large values of k. in addition, the proposed approach can be used to estimate the response time of a fork-join qs not only with subsystems m |m |1 but also for more complex variants of distributions of input flow and service time. the vast majority of solutions for fork-join qs are approximate. therefore, the question of the reliability of the experimental data used to assess the quality of the approximation of the obtained expressions comes to the fore. as a rule, numerical data can only be obtained through simulation. the values obtained using the simulation are considered reference or true. however, it is clear that in order for the simulation data to become the most accurate, a large number of realizations of the estimated random variable are required within one run of the simulation model. at the same time, determining a sufficient number of realizations and confidence intervals of the obtained estimates remains a rather difficult question due to the correlation of these data (more on this will be discussed in the corresponding section). the article proposes an approach to finding confidence intervals, provides an algorithm for their construction, as well some recommendations. the algorithm is considered in the example of a fork-join qs with subsystems of the form m |m |1; however, it can also be extended to the case with more complex qs as subsystems of the fork-join system. so, the article is organized as follows. the second section describes an approach to constructing estimates for the mathematical expectation and variance of the response time, presents the obtained estimates in the form of analytical expressions, as well as the results of a numerical experiment; the third section raises the question of the correct organization of simulation modeling, evaluates the correlation between realizations of the random value copyright © 2023 assa. adv syst sci appl (2023) assa latex template 101 of the response time (in within the framework of the simulation), on the basis of which a confidence interval is built using the proposed algorithm. 2. evaluation of the main performance characteristics of fork-join qs in this section, we present new estimates for such important characteristics of a fork-join qs with k subsystems m |m |1 as its average response time and variance. the estimates proposed by the authors are based on the modification of known approximations obtained earlier. experimental data, which we obtained by modeling, were needed to correct these estimates. in particular, using a simulation model written in the python software environment, the values of the mathematical expectation and dispersion of the response time were calculated for a constant arrival rate λ = 1 and a changing service rate value µ. namely, the load factor ρ = λ/µ took values from 0.1 to 0.9 inclusive with a step of 0.05, and the number of subsystems k varied from 3 to 20 inclusive (because for k = 2 the exact value of the average response time). the number of tests (r.v. implementations) within one run of the simulation model varied from 5 million in the case of a low system load to 10 million in the case of a high system load. next, we describe the process of deriving new estimates based on the results of the simulation and what it leads to. 2.1. average response time let e[rk ] and √ var[rk] denote the mean and standard deviation of the response time in a fork-join qs with k subsystems of type mλ|mµ|1. according to the approximate nelsontantawi formula for estimating the average response time [6], which gives the smallest relative error compared to other known [11] formulas, which does not exceed 5% for k ⩽ 32, we have e[rk ] ≈ e[rk ]nt = [ hk h2 + 4 11 ( 1− hk h2 ) ρ ] 12− ρ 8 1 µ− λ , (2.1) where hk = ∑k i=1 1/i is the partial sum of the harmonic series. since the dependence of the average response time on the parameters ρ, (µ− λ) and (hk/h2 − 1) is observed in the (2.1) formula, it is natural to suggest that our correction depends on from the same settings. in particular, suppose that the improved estimate for the average response time is e[rk ] ≈ ( hk h2 − 1 ) ρ µ− λ · µ̃+ e[rk ]nt , (2.2) where we assume that the nelson-tantawi estimate is sharp for k = 2 and asymptotically sharp for ρ → 0, while the scale factor 1/(µ− λ) is preserved. next, to refine µ̃, plot the modified mean response time expression as a function of ρ and (hk/h2 − 1). in this case, by the modified expression, we mean the relation µ̃ = (e[rk ]− e[rk ]nt )(µ− λ) (hk/h2 − 1)ρ , (2.3) the calculation of the right side of which is possible due to the results of simulation modeling for e[rk ] and calculations using the formula (2.1) for e[rk ]nt . in the figures 2.1 and 2.2 one can catch the linear dependence of µ̃ on ρ and hk/h2 − 1 (though not very strict). thus, the analysis of graphs allows us to propose an expression of copyright © 2023 assa. adv syst sci appl (2023) 102 a.v. gorbunova, a.v. lebedev 0.00 0.05 0.10 0.15 0.20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 µ ρ fig. 2.1. dependence µ̃ from (2.3) on ρ. 0.00 0.05 0.10 0.15 0.20 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 µ hk/h2 – 1 fig. 2.2. dependence µ̃ from (2.3) on (hk/h2 − 1). the form µ̃ = µ̃(ρ,hk) ≈ c1 − c2 ( hk h2 − 1 ) + c3ρ, (2.4) i.e. e[rk ] ≈ ( hk h2 − 1 ) ρ µ− λ · ( c1 − c2 ( hk h2 − 1 ) + c3ρ ) + e[rk ]nt . (2.5) we use the nelder-mead optimization method [18, 19] to determine the values of the coefficients ci, i = 1, 2, 3. the idea of using this method is to minimize the maximum value of the modulus of the relative error of approximation of the average response time by the copyright © 2023 assa. adv syst sci appl (2023) assa latex template 103 expression (2.5) to its “true” values calculated using simulation max ∣∣∣∣ ê[rk ]− e[rk ] e[rk ] · 100% ∣∣∣∣ → min . (2.6) as a result of minimizing the error (2.6) on the entire set of data obtained through simulation, the optimal values of the coefficients c1, c2 and c3 are determined, which best reflect the dependence (2.5) within the chosen optimization method. as a result, after applying the nelder-mead optimization method to (2.6) as a function of several ci variables in the python programming environment, we get c1 ≈ 0.087197, c2 ≈ 0.070236, c3 ≈ 0.09638. (2.7) in this case, for the above values λ = 1, ρ ∈ [0.1, 0.9] with step 0.05 and k = 3, ..., 20, the following holds maxape ≈ 0.343207%, minape ≈ 0.001150%, mape ≈ 0.142565%, while for nelson-tantawi formula (2.1) with the same values of λ, ρ and k the following is true maxape ≈ 3.944700%, minape ≈ 0.000628%, mape ≈ 1.322934%. thus, our correction improves maxape by a factor of 11.5 and mape by a factor of 9.3. the results of the approximation of the average response time using the (2.5) formula in comparison with the simulation results are shown in the figure. 0 5 10 15 20 25 0 5 10 15 20 25 f o rm u la (5 ) simulation fig. 2.3. average response time. the efficiency of the expression (2.5) is not limited to the maximum value of k = 20. this formula is expected to be good for a larger number of subsystems due to the built-in logic of the proposed approach to approximating the average response time. in particular, for k = 100 and values ρ ∈ [0.1, 0.9] with a step of 0.05 we have the following approximation errors maxape ≈ 2.666589%, minape ≈ 0.024029%, mape ≈ 0.642155%. this is better than the nelson-tantawi formula, which gives maxape ≈ 2.741596%, minape ≈ 0.100878%, mape ≈ 1.084311%. copyright © 2023 assa. adv syst sci appl (2023) 104 a.v. gorbunova, a.v. lebedev for the value of k = 1000, due to the saving of computational and time resources of simulation modeling, we limited ourselves to checking the quality of the approximation under low load ρ = {0.1, 0.2, 0.3, 0.4, 0.5}, since with increasing values it does not only k, but also ρ, it is necessary to significantly increase the number of tests (realizations of r.v.) within one run of the simulation model. as a result of the approximation error, the formulas (2.5) amounted to maxape ≈ 1.485234%, minape ≈ 0.010079%, mape ≈ 0.491061%, and by the nelson-tantawi formula, we have maxape ≈ 3.389362%, minape ≈ 1.090899%, mape ≈ 2.530636%. note that if we add one more term to the (2.4) formula, i.e. µ̃ ≈ c1 − c2 ( hk h2 − 1 ) + c3ρ+ c4ρ ( hk h2 − 1 ) , (2.8) and recalculate the optimal coefficients after substituting (2.8) into (2.2), which turn out to be equal to c1 ≈ 0.150293, c2 ≈ 0.152088, c3 ≈ 0.012684, c4 ≈ 0.105121, then we obtain a further significant improvement in the estimates for k from 3 to 20, namely, maxape ≈ 0.252453%, minape ≈ 0.000495%, mape ≈ 0.105624%. for k = 100 we get maxape ≈ 1.548824%, minape ≈ 0.467368%, mape ≈ 1.185327%. an increase in mape may indicate the influence of a small systematic error, which becomes noticeable at large k. it is possible that adjusting the coefficients (while keeping the general formulas) considering the simulation for k > 20 can further improve the approximation. 2.2. response time standard deviation in [20] a formula was proposed to estimate the variance of the fork-join qs response time with k subsystems mλ|mµ|1 in the general case of different service rates µi, 1 ≤ i ≤ k. in the case of a single service rate µ, it takes the form var[rk ] ≈ 2 (µ− λ)2 k∑ i=1 ( k i ) (−1)i−1 1 i2 − ( 1 µ− λ k∑ i=1 1 i )2 , (2.9) which in turn simplifies to var[rk ] ≈ qk (µ− λ)2 , where qk = k∑ i=1 1 i2 . we note that the above formula, by its construction, proceeds from the independence of the sojourn times in subsystems, i. e., we are talking about the dispersion of the maximum of independent exponential random variables. copyright © 2023 assa. adv syst sci appl (2023) assa latex template 105 suppose that our estimate will depend, as well as the (2.9) expression itself, on the variance of the average response time in qs m |m |1, i.e. on 1/(µ− λ)2, and on the partial sum of the inverse square series. let us put var[rk ] ≈ qk − 1 (µ− λ)2 · σ̃ + 1 (µ− λ)2 , (2.10) where we assume that the estimate is sharp for k = 1. next, to specify the expression for σ̃, we construct and analyze plots of the modified variance versus ρ, (qk − 1), and (hk − 1). under the modified variance expression, we mean the following: σ̃ = var[rk ](µ− λ)2 − 1 qk − 1 , (2.11) where the values of var[rk ] were obtained by simulation. 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0.35 0.4 0.45 0.5 0.55 0.6 σ qk – 1 fig. 2.4. dependence σ̃ from (2.11) on (qk − 1). 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0.65 1.15 1.65 2.15 2.65 σ hk – 1 fig. 2.5. dependence σ̃ from (2.11) on (hk − 1). copyright © 2023 assa. adv syst sci appl (2023) 106 a.v. gorbunova, a.v. lebedev 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 σ ρ fig. 2.6. dependence σ̃ from (2.11) on ρ. in figures 2.4 and 2.5 at fixed values of ρ, σ̃ is observed to depend quadratically on (qk − 1) and on (hk − 1), respectively. however, the analysis of the experimental data for large k showed that the dependence on (hk − 1) is closer to the truth since the actual values of the variance grow without limit with the growth of k. otherwise, the dispersion values would quickly stabilize since, as is known, qk has a finite limit at k → ∞ equal to π2/6. further, in the figure 2.6 there is an explicit linear dependence on ρ with a slope depending on k, and the straight lines pass through the point (0, 1). so let us assume that σ̃ = σ̃(ρ,hk) = 1 + ρ ( c1 + c2(hk − 1) + c3(hk − 1)2 ) (2.12) and correspondingly, var[rk ] ≈ qk − 1 (µ− λ)2 · ( 1 + ρ ( c1 + c2(hk − 1) + c3(hk − 1)2 )) + 1 (µ− λ)2 . (2.13) then, similarly, using the nelder-mead method, we solve the optimization problem of minimizing the maximum approximation error of the formula (2.13) max ∣∣∣∣v̂ar[rk ]− var[rk ] var[rk ] · 100% ∣∣∣∣ → min . (2.14) as a result, we obtain the following values of the coefficients ci c1 ≈ −0.113658, c2 ≈ 0.339780, c3 ≈ 0.053745. (2.15) the figure 2.7 shows the results obtained using the (2.13) formula and simulation modeling to illustrate the quality of the obtained approximation. for the standard deviation of the response time for λ = 1, ρ ∈ {0.1, 0.15, 0.20, ..., 0.90} and k = 3, ..., 20, we have maxape ≈ 0.564247%, minape ≈ 0.000755%, mape ≈ 0.188812%, while for the formula (2.9) for the same values λ, ρ and k is true maxape ≈ 14.475134%, minape ≈ 0.173239%, mape ≈ 6.055298%. thus, our estimate improves maxape by a factor of 25.7 and mape by a factor of 32.1. copyright © 2023 assa. adv syst sci appl (2023) assa latex template 107 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 f o rm u la (1 3 ) simulation fig. 2.7. the standard deviation of the response time. as in the case of the mathematical expectation of the response time, the formula for the standard deviation of the response time turns out to be good not only for the maximum value of k = 20, but also for a much larger number of subsystems. thus, for k = 100 and values ρ ∈ [0.1, 0.9] with a step of 0.05 we have the following approximation errors maxape ≈ 1.211417%, minape ≈ 0.094622%, mape ≈ 0.724898%. for the value k = 1000, for the same reasons as in the case of the formula for the mathematical expectation, we present the results of the approximation error only under conditions of low system load ρ = {0.1, 0.2, 0.3, 0.4, 0.5} maxape ≈ 1.029874%, minape ≈ 0.203567%, mape ≈ 0.592974%. as a result, a very good approximation takes place. 3. features of fork-join qs simulation the fork-join queuing system is one of the systems that is difficult to study. exact analytical results are known only for a small number of particular cases. in most works on this topic, many approximate methods, including numerical algorithms, have been developed to analyze the performance characteristics of fork-join qs. the effectiveness of a particular method can be determined by various criteria. the most significant among them are economy and high accuracy. efficiency is understood as the required amount of resources: the less computer memory is required, or the shorter the time of the algorithm itself, the better. regarding the accuracy of the method, data for comparison are required here. the only possible tool for obtaining them in this context can only be simulation modeling. thus, the assessment of the accuracy of new methods for studying fork-join qs directly depends on the accuracy of the values obtained as a result of simulation modeling. simulation modeling is based on the monte carlo method, which consists of multiple reproductions of the random process of the system functioning and further statistical processing of the results obtained. as a result of the fork-join qs simulation, we can get a data set or a vector, which is a sequence of realizations of a random value of the time the request stays in the system. further, using these data, it is necessary to construct a point estimate copyright © 2023 assa. adv syst sci appl (2023) 108 a.v. gorbunova, a.v. lebedev of the average response time and the corresponding confidence interval. however, the main difficulty in constructing a confidence interval is that the data obtained are correlated. in particular, it is clear that the sojourn time of the n-th task in the system will depend on the sojourn time of the previous one, i.e. (n− 1)-th task, the sojourn time of which, in turn, depends on the sojourn time of the (n− 2)-th task, and so on. therefore, the question of a sufficient number of realizations of the studied quantity during the duration of one run of the simulation model is quite relevant. of course, in practice, it is possible to experimentally determine the required size of the sequence, gradually increasing the number of its elements with each run of the model and observing how the value of the sample average response time changes. if these changes become insignificant, the experiment is stopped, and the final result is chosen. despite the fact that the described approach to determining the duration of the run of the simulation model is the most common, it requires a lot of time, but it does not guarantee the result due to the lack of knowledge about the confidence level of the obtained values. therefore, this paper proposes an approach to constructing confidence intervals for the average response time of a fork-join qs under the conditions of correlation of simulation data. an estimate of the correlation between realizations of the random value of the fork-join response time of the ri system during the duration of one run, which are at a distance of n, n = 1, 2, ... is determined by the expression r̂(n) = ê[riri+n]− ê[r]2 v̂ar[r] , (3.16) where ê[riri+n] = 1 n − n n−n∑ i=1 ri ·ri+n, σ̂2 = v̂ar[r] = ê[r2]− ê[r]2, r̂ = ê[r] = 1 n n∑ i=1 ri, ê[r2] = 1 n n∑ i=1 r2 i . next, we calculate the variance of the response time estimate obtained using simulation modeling var[r̂] = var [ 1 n n∑ i=1 ri ] = 1 n2 var [ n∑ i=1 ri ] = = 1 n2 ( n∑ i=1 var[ri] + 2 n−1∑ i=1 n∑ j=i+1 cov(ri, rj) ) = = σ2 n ( 1 + 2 n n−1∑ i=1 n∑ j=i+1 corr(ri, rj) ) = σ2 n ( 1 + 2∆n ) , where ∆n = 1 n n−1∑ i=1 n∑ j=i+1 corr(ri, rj) = 1 n n−1∑ n=1 (n − n)r(n), r(n) is the correlation coefficient between a pair of elements (response times) separated from each other (according to the numbers of serviced tasks) by a distance of n. copyright © 2023 assa. adv syst sci appl (2023) assa latex template 109 then ∆n = 1 n n−1∑ n=1 (n − n)r(n) = n−1∑ n=1 ( 1− n n ) r(n) −−−→ n→∞ ∞∑ n=1 r(n) = ∆. the question arises of how to estimate the sum of an infinite series ∆, while in practice, we can estimate only a finite number of autocorrelations r(n). in the dissertation [21] it was proposed to use for this the exponential asymptotics r(n) ∼ ae−bn, n → ∞, (3.17) observed for the types of qs considered there. our calculations also show good agreement with the exponential law of decreasing autocorrelations (for large ρ). for clarity, the figure 3.8 shows a graph of the dependence of sample autocorrelations calculated by the formula (3.16) on the step length n at ρ = 0.7. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 r( n ) n k=2 k=10 k=20 fig. 3.8. correlation coefficient between the sojourn times of i-th and (i+ n)-th tasks in the system, ρ = 0.7, n = 1, 2, ...100. we can now estimate ∆ by calculating the values of r̂(n) using simulation for a finite number of steps, i.e. for n = 1, 2, ...,m, ∆ = +∞∑ n=1 r(n) = m∑ n=1 r(n) + +∞∑ n=m+1 r(n), (3.18) taking into account the fact that due to (3.17) it is true +∞∑ n=m+1 r(n) ∼ a e−b(m+1) 1− e−b , m → ∞. (3.19) two situations are possible. we sum r̂(n) as long as it is true that r̂(n) ⩾ 2/ √ n , otherwise we calculate the values of r̂(n) statistically not significant. in the first case, the number copyright © 2023 assa. adv syst sci appl (2023) 110 a.v. gorbunova, a.v. lebedev of obtained elements r̂(n), n = 1, 2, ...,m, may be sufficient for summation, i.e., r̂(m) < 2/ sqrtn . otherwise, when r̂(m) ⩾ 2/ √ n , we will need to estimate summation residuals for n from (m+ 1) to +∞ from (3.19). this can be done by building a regression model. we add the resulting estimate for the tail of the sum to the sum over n from 1 to m. as a result of the described actions, we will calculate some estimate ∆̂ for ∆ from (3.18). thus, for specific values of the parameters of the system under consideration, for example, for a certain value of the load factor ρ and the number of subsystems k: 1. choose some value m; 2. based on the simulation results for a large number of implementations n , we calculate the set of values r̂(n) using the formula (3.16), n = 1, 2, ...m; 3. sum the elements of r̂(n) until r̂(n) < 2/ √ n , n = 1, 2, ...m; 4. if r̂(m) ⩾ 2/ √ n 1) we build a regression model of the form (3.17), which is reduced to a linear one by taking the logarithm of both parts of the expression: ln r(n) = ln a− bn+ ε (ε — random error of the model); 2) estimate the parameters a and b of the constructed regression model using the least squares method; 3) calculate the estimate of the summation tail by the formula (3.19), substituting into it the values of the obtained estimates of the regression parameters a and b, as well as m; 4) calculate the estimate ∆ according to the formula (3.18), substituting the corresponding values into it ∆̂ = m∑ n=1 r̂(n) + â e−b̂(m+1) 1− e−b̂ ; 5. if r̂(n∗ + 1) < 2/ √ n , and n∗ ⩽ m, we get the estimate ∆̂ = n∗∑ n=1 r̂(n). due to the asymptotic normality of the estimate of the mean, we can build asymptotic confidence intervals for arbitrary levels of reliability, for example, by applying the “three sigma” rule. then with probability p = 0.997, it will be true e[r] = ê[r]± 3σê[r] ≈ ê[r]± 3σ√ n √ 1 + 2∆n ≈ ê[r]± 3σ̂√ n √ 1 + 2∆̂. (3.20) as you can see, compared to the case of independent random variables, the number of trials n during one model run will be required √ 1 + 2∆n times more due to the existing correlation between the data. next, we construct an estimate for ∆ for a fork-join qs with k subsystems of type m |m |1 in the form of some analytic expression depending on ρ and k. to do this, using simulation for various combinations of pairs of values k = 2, ..., 20 and ρ = 0.1, 0.2, ...0.9, according to the algorithm described above, we calculate the values ∆̂ = ∆̂(k, ρ). the values of m were chosen depending on ρ, namely m = 10, 20, 30, 40, 50, 50, 100, 250, 1000. note that here we start the simulation with k = 2, since for this characteristic, even in this case, there are still no exact results, just as there were no approximate ones, i.e. research is completely new. copyright © 2023 assa. adv syst sci appl (2023) assa latex template 111 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k=10 k=11 k=12 k=13 fig. 3.9. dependence [∆̂/(2ρ/(1− ρ2))− 1] on ρ. now let us define a specific type of functional dependence by analyzing the results of constructing various graphs. in particular, the plot of ∆̂/(2ρ/(1− ρ2))− 1 as a function of ρ resembles a bunch of straight lines with different slopes passing through the point (0, 0), taking into account fluctuations caused by random errors (fig. 3.9). therefore, it remains to find a formula that describes these lines. taking into account the approximations obtained in earlier publications for the average response time of a fork-join system in which there is a partial sum of the harmonic series, it is natural to assume that ∆ will similarly depend on k not directly but through hk . with respect to hk , a dependence close to linear was also observed. therefore, suppose that the lines can be described by the formula c · ρ(hk − 1). finally, we get an expression for the ∆ estimate of the following form: ∆(k, ρ) ≈ 2ρ (1− ρ)2 ( 1 + c · ρ(hk − 1) ) , (3.21) where c is some coefficient whose value is to be found. the search for the c coefficient is carried out using the nelder-mead optimization method. thus, we get c ≈ 0.186722. to verify the quality of the obtained approximation, we construct the corresponding graph (fig. 3.10) and analyze the approximation errors. let ape = ∆̂−∆ ∆ · 100%, then we get maxape ≈ 5.27449%, minape ≈ 0.013941%, where the average value of the modules of relative approximation errors mape ≈ 0.835525%, copyright © 2023 assa. adv syst sci appl (2023) 112 a.v. gorbunova, a.v. lebedev 0 50 100 150 200 250 0 50 100 150 200 250 f o rm u la (2 1 ) simulation fig. 3.10. ∆ coefficient: comparison of simulation results with the formula (3.21). which indicates the good quality of the resulting expression. in this case, we can assume that the formula (3.21) can be successfully used to construct confidence intervals not only in the region 2 ⩽ k ⩽ 20, but also for large values of k. we also note that the values of ∆ observed in the studied range of parameters reach approximately 252, which means, in this case, an increase in the required number of trials n by more than 22 times compared to the case of independent trials (to achieve the same accuracy of the estimate average response time), and this effect cannot be neglected. the algorithm presented above can be used to build confidence intervals of the form (3.20) to estimate the average response time not only in the case of subsystems of the m |m |1 type but also in more complex cases, for example, for subsystems of the type g|g|1 and others. as regards the method of constructing a functional dependence for the coefficient ∆, more laborious work is possible here on the selection of the type of dependence in each specific case, i.e., for specific distributions of input flow and service times. in particular, dependence on hk naturally arises for an exponential distribution, and for example, in the case of distributions with heavy (power-law) tails (for example, as in [12]), dependence on some degree of k will be natural. however, the general approach to constructing an estimate for ∆, described by the example of deriving the (3.21) expression, can be applied to other types of qs fork-join subsystems. 4. conclusion the paper presents an approach to obtaining more accurate than known estimates for the expectation and response time variance of a fork-join system with k subsystems m |m |1. as shown by the numerical experiment, the obtained analytical formulas are valid for a wide range of values of the parameters included in them. the considered approach can be used to construct good quality estimates in more complex cases. for example, when the subsystems are queuing systems of the form g|g|1, which can be the subject of further research. in addition, the article describes the essential aspects of fork-join qs simulation and the evaluation of its results, which is an integral part of the organization of a numerical experiment to check the quality of approximation of any approximation. an algorithm for copyright © 2023 assa. adv syst sci appl (2023) assa latex template 113 constructing confidence intervals for sample estimates obtained by simulating the functioning of a fork-join qs is presented, and some recommendations are also given. references 1. nguyen, m., alesawi, s., li, n., che, h. & jiang, h. (2020). a black-box forkjoin latency prediction model for data-intensive applications, ieee transactions on parallel and distributed systems, 31(9), 1983-2000. 2. peng, p., soljanin, e. & whiting, p. (2022). diversity/parallelism trade-off in distributed systems with redundancy ieee transactions on information theory, 68(2), 1279-1295. 3. samyukta sethuraman (2022). analysis of fork-join systems: network of queues with precedence constraints. boca-raton, fl: crc press. 4. de oliveira, d. c. m., liu, j. & pacitti, e. (2019). data-intensive workflow management: for clouds and data-intensive and scalable computing environments. san rafael, ca: morgan & claypool publishers. 5. khalid, m. & yousaf, m. m. (2021). a comparative analysis of big data frameworks: an adoption perspective, applied sciences, 11(22), 11033. 6. nelson, r. & tantawi, a. n. (1988). approximate analysis of fork/join synchronization in parallel queues, ieee transactions on computers, 37(6), 739–743. 7. varki, e., merchant, a. & chen, h. (2002). the m/m/1 fork-join queue with variable subtasks, [online]. available: http://www.cs.unh.edu/ varki/publication/2002nov-open.pdf. 8. varma, s. & makowski, a. m. (1994). interpolation approximations for symmetric fork-join queues, performance evaluation, 20, 245–265. 9. kemper, b. & mandjes, m. (2012). mean sojourn time in two-queue fork-join systems: bounds and approximations, or spectrum, 34, 723–742. 10. david, h. a. & nadaraja, h. n. (2003). order statistics. hoboken, nj: john wiley & sons. 11. gorbunova, a. v. & vishnevsky, v. m. (2020). estimating the response time of a cloud computing system with the help of neural networks, advances in systems science and applications, 20(3), 105–112. 12. gorbunova, a. v. & lebedev, a. v. (2022). response time estimate for a fork-join system with pareto distributed service time as a model of a cloud computing system using neural networks, communications in computer and information science, 1552, 318–332. 13. vishnevsky, v. m. & gorbunova, a. v. (2022). application of machine learning methods to solving problems of queuing theory, communications in computer and information science, 1605, 304–316. 14. thomasian, a. (2014). analysis of fork/join and related queueing systems, acm computing surveys (csur), 47(2), 17:1–17:71. 15. qiu, z., perez, j. f. & harrison, p. g. (2015). beyond the mean in fork-join queues: efficient approximation for response-time tails, performance evaluation, 91, 99–116. 16. wang, w., harchol-balter, m., jiang, h., scheller-wolf, a. & srikant, r. (2019). delay asymptotics and bounds for multitask parallel jobs, queueing syst., 91, 207–239. 17. gorbunova, a. v. & lebedev, a. v. (2020). bivariate distributions of maximum remaining service times in fork-join infinite-server queues, problems of information transmission, 56(1), 73–90. 18. nelder, j. a. & mead, r. (1965). a simplex method for function minimization, computer journal, 7, 308–313. 19. kochenderfer, m. j. & wheeler, t. a. (2019). algorithms for optimization. cambridge, uk: mit press. copyright © 2023 assa. adv syst sci appl (2023) 114 a.v. gorbunova, a.v. lebedev 20. gorbunova, a., zaryadov, i, matushenko, s., sopin, e. (2016). the estimation of probability characteristics of cloud computing systems with splitting of requests, communications in computer and information science, 678, 418–429. 21. zaharenkova, e. (2019). analitiko-statisticheskie metody rascheta i optimizacii sistem i setej massovogo obsluzhivanija so stepennymi hvostami raspredelenij [analytical and statistical methods for calculating and optimizing queuing systems and networks with power tails of distributions]. ph. d. thesis, omsk, [in russian]. copyright © 2023 assa. adv syst sci appl (2023) introduction evaluation of the main performance characteristics of fork-join qs average response time response time standard deviation features of fork-join qs simulation conclusion microsoft word 3._done_1376 adv syst sci appl 2023; 01; 22-34 published online at http://ijassa.ipu.ru. the disease centered multimorbidity model at the example of type 2 diabetes mellitus dmitry meshkov1, alexey lobanov1, larisa danilova2, sergey cherkasov1, alexander shiroky1, anna fedyaeva1, irina moroz3, elena makeeva1 1) v.a. trapeznikov institute of control science of the russian academy of sciences, moscow, russia 2) belarusian medical academy of postgraduate education, minsk, belarus 3) belarusian state medical university, minsk, belarus abstract: multimorbidity is a condition when few diseases (multimorbidity pattern) are diagnosed in same patient. according to the disease centered model the pattern is formed around one “marker” disease at the expense of other diseases which have non-random relationships with the “marker” disease and between each other. the totality and probabilistic connections between pairs of diseases determine the specifics of the pattern and indicate the possibilities for managing individual and population health as a result of interactions of all components of the the pattern. algorithms have been developed assessing the degree of non-random connections between pairs of diseases in the pattern and establishing a hierarchical relationship between both “marker” and other diseases as well as between pairs of non-marker diseases. the calculation of non-random statistical relationships between pairs of diseases included in the t2dm (type 2 diabetes mellitus) multimorbidity pattern proved the consistency of the data obtained with the current clinical description of t2dm pathogenesis and complications. keywords: multimorbidity, multimorbidity pattern, type 2 diabetes mellitus, expert support, decision making, patient-centered healthcare, cumulative life course impairment, proactive risk management, healthcare. 1. introduction multimorbidity is defined as the presence of two or more diseases in one person [ошибка! источник ссылки не найден.]. attention to multimorbidity has been caused by an increase in life expectancy and the proportion of elderly people suffering from a number of conditions worldwide. the multimorbidity prevalence in older adults varies from 40 to 99% depending on population studied. major consequences of multimorbidity are disability and functional decline, poor quality of life, high health care costs and complex pharmacological regimes. currently, there is a transition in multimorbidity perception from a set of separate random diseases to systematic approach assessing multimorbidity pattern as a special condition requiring appropriate clinical and healthcare management decisions. the viability of this approach is the consequence of the appropriate diseases inclusion and exclusion criteria into multimorbidity patterns as well as the assessment of relationships between these diseases [12 7]. some of the approaches aimed at association or, conversely, the division of a set of diseases into patterns are associated with the diseases grouping according to various formal characteristics. this approach is based mainly on the reassessment of statistical data and is applicable for disease burden calculation. at the same time, it is extremely difficult to use it for justifying clinical and managerial decisions. the opposite approaches are based on accumulated clinical experience proving the causal relationship of pathogenetic mechanisms of a group of  corresponding author: dmitrymeshkov@mail.ru the disease centered multimorbidity model… 23 copyright ©2023 assa adv. in systems science and appl. (2023) diseases and are the basis for evidence-based solutions in healthcare, but they do not allow us to fully identify and evaluate all the links between diseases forming multimorbid pattern. the complexity and heterogeneity of patients with multimorbidity when using traditional clinical guidelines focused on a single disease quite often complicate and sometimes lead to inadequate clinical solutions like polypragmasia [12, 22, 5]. cumulative life course impairment (clci) and proactive risk management in healthcare approaches are based on both statistical big data and clinical evidence, with the purpose of economical effectiveness of high-cost innovative interventions and treatment of high-cost patients [2, 13]. successful solution of the multimorbidity challenge need the concept (description of the basic principles, elements and interrelations between them) realized in form of a mathematical model and practical tools for expert support of healthcare decision making capable combining both the big data processing and the ability to take into account clinical evidence of non-random and causal clinical relationships of diseases. [19]. the group of diabetes mellitus diseases can be considered as the most suitable object for testing of the concept itself and the practical tools as well. this group of conditions includes t1dm (insulin dependent) and t2dm (insulin-non-dependent) with different pathogenesis but similar clinical manifestations. a number of t2dm associations have common hereditary and pathogenetic interrelations (obesity, atherosclerosis, hypothyroidism, coronary heart disease, vascular catastrophes, a number of oncological and neurodegenerative conditions as well as diabetic complications themselves). the t2dm associated diseases influence medical, social and economic disease burden impacting healthcare budgets in many countries what makes t2dm and associated diseases remaining the optimal model for assessing multimorbidity and improving therapeutic approaches as well as health economics decisions [ошибка! источник ссылки не найден., 20 24]. 2. the objective for the study development of multimorbidity model aimed at effective population health management and applicable for practical use in expert support of healthcare decision making. the following tasks have been solved to achieve the objective: 1. creation of the multimorbidity concept considering it is a set of provisions and recommendations describing the elements and interrelations between them and aimed at improving of population health and reducing the disease burden. 2. selection or creation algorithms for evaluating non-random association in pairs of diseases allowing creation a model of a multimorbid pattern that includes a hierarchical list of diseases with the highest values of association with a marker disease and relationships, including also links between non-marker diseases. 3. practical assessment of concept and the model applicability for expert support of healthcare decision making on the example of t2dm in females considering the condition is a socially significant one with a sufficiently known pathogenesis. 2. methods 2.1. disease centered multimorbidity pattern concept the first task was accomplished using a combination of environmental scanning (200+ publications found in international databases pubmed and embase as well as personal messages from experts) [15, 9] with analytic hierarchy process (ahp) [4, 18 ] modified for development and assessment of promising models of healthcare management [14]. the ahp was divided into 2 stages. the objective for the first stage was to formulate the necessary criteria without taking into account whether they were realistic or not. at the second stage the experts had to find the solutions for realization of these criteria and to formulate the tasks for mathematical analyses and 24 d. meshkov et al. copyright ©2023 assa adv. in systems science and appl. (2023) modelling of multimorbidity patterns. the expert council consisted of 8 members including 3 experts in healthcare management (and healthcare practitioners as well), 2 endocrinologists specialized in diabetes and 3 mathematicians. 2.2. algorithms for non-random interrelations assessment between pairs of diseases the requirements for the mathematical analysis and modelling were formulated during the second stage of ahp. associative rules machine learning was selected to create algorithms determining the probability of random and non-random connections between pairs of diseases [1]. 2.3. concept and the model assessment at the example of t2dm type 2 diabetes mellitus (t2dm) and associated diseases patients were the optimal model for assessing the multimorbidity patterns and improving personalized therapeutic approaches as well as health management decisions. female patients were selected due to gender differences in pathogenesis and complications of the disease [ошибка! источник ссылки не найден., 7 24] according to the ahp requirements the standard statistical reporting database provided by local healthcare administration was used. the database contained the data collected during 1 year period about 392 215 visits and diagnosis of 28153 female patients at the age from 30 to 61 years including 1977 patients with t2dm condition. 3. results and dicussion 3.1. disease centered multimorbidity pattern concept the final results of the ahp are shown at the table 1. the disease-centered multimorbidity pattern is formed around one disease chosen as the basis of the pattern. this disease is grouped together with other diseases that have a non-random connection with it. any disease represented in the international classification of diseases (icd-10/icd-11) can act as a pattern-forming one and it is convenient to use this condition for the pattern identification, so next we use the term "marker" disease or condition to identify the pattern. the totality of such patterns forms the structure of morbidity and prevalence of diseases in selected population, but from practical point of view, the most important patterns are formed around marker diseases that have a greater medical, social and economic disease burden. the value of a non-random connection with other diseases is regarded as a higher probability of their joint manifestation in the same patients. the direction of the connection indicates the probability of appearance of a new disease under the existing one indicating the presence of causal relationships between pairs of diseases. links from non-marker disease to the marker one may reflect the pathogenesis of the marker disease and also the links from the marker to non-marker ones reflect the pathogenesis of complications. links between nonmarker conditions can also provide additional information about interrelations within the pattern. in addition to connections between pairs of diseases there can be also assessment of risk factors and social health determinants influence on the pattern providing new opportunities for disease management. non-random connections may indicate the presence of causal relationships between pattern elements, although experimental clinical confirmation is necessary to substantiate them. the direction of non-random connections provides additional information about possible causeand-effect relationships in pairs of diseases. the hierarchical list of diseases included in the pattern is determined mainly by the value of their non-random association with the "marker" disease. based on the above information it is necessary to go through the following stages to form the multimorbidity pattern: identification of the "marker" disease jointly by clinical specialists and health care organizers, calculation of non-random statistical relationships of the "marker" disease and other conditions (or risk factors and social health determinants depending of the objective of the assessment) based on available standard medical statistics data and in-depth clinical analysis of the identified patterns by medical specialists. the disease centered multimorbidity model… 25 copyright ©2023 assa adv. in systems science and appl. (2023) table 1. the requirements for multimorbidity concept (ahp fist stage results) key needs breaking points requirements for the concept the ability to reduce the disease burden medical (safety, efficacy and efficiency) appropriate target setting based on achievable clinical needs. selection of clinically significant (marker) disease treatment as the primary objective. improving the effectiveness of treatment of other diseases included in the pattern is a secondary task. identification of non-random links between conditions in the pattern which can be the signs of similar or interrelated pathogenic mechanisms for better interventions allocation. economical (effectiveness) economic disease burden is a criteria for marker condition selection. social impact and bioecthical compliance social impact of the disease is a criteria for "marker" condition selection. compliance with existing global and local healthcare standards clinical guidelines matching treatment of single diseases included into pattern is based on existing guidelines but it does not exclude preparing of the new guidelines taking into consideration relationship between diseases in the pattern. regional and local country regulations compliance marker disease centered patterns using and accomplishing the existing regulations based on separate disease treatment. applicable or can be adjusted to any healthcare infrastructure marker disease centered pattern using the existing diagnostics and treatment facilities as well as patients’ journey. financing and reimbursement assessing healthcare financing and reimbursement programs criteria for the identification of a marker disease healthcare professionals core competencies focusing the concept on existing healthcare professionals’ specialization by highlighting the most applicable condition in the multimorbidity pattern. required information for multimorbidity pattern assessment existing medical and statistical data using for creation and assessment of multimorbidity pattern. 3.2. algorithms for non-random interrelations assessment between pairs of diseases the algorithm for finding strong connections between pairs of diseases was created using association rules. let the set 𝑂 = {𝑜 , 𝑜 , . . 𝑜 } is a set of diseases. each element of the set represents a separate unique disease found in the available data and takes the value 1 or 0. the following family of sets 𝑃 = {𝑝 , 𝑝 , . . 𝑝 }, is a set of patients and consists of m ordered non-empty sets of non-zero elements from o. each element of p corresponds to a unique patient from the dataset. it means that, for each patient 𝑝 ∈ 𝑃, we have a list of his diagnoses o, without taking into account the number of times he was diagnosed. next, we introduce the concept of "rules". in this case, the rule will be introduced as an implication of the following form: 𝑋 ⇒ 𝑌, where x, y – are sets consisting of elements from o. in the future, in accordance with established tradition, this implication will be written as: 𝑋 → 𝑌. in our case, the sets x and y consist of disjoint sets of unique diseases, and we say, "if x, then y". for example, the set x consists of diseases {i, j, k}, and y of {l}, where i, j, k, l are unique elements of the set o. and we will say: "rule: if {i, j, k}, then {l}". actually, “rule” may not contain practical sense. it’ just a form to fix directed pair of two sets. the rules with no sense and meaningful ones the characteristics were separated on the base of calculated values. the rules were 26 d. meshkov et al. copyright ©2023 assa adv. in systems science and appl. (2023) considered to be valid in case characteristics satisfied high values. below there is a description of the associative rules stable concepts and the definition of characteristics. antecedents – is the conditional designation of the first set of x objects. abbreviated a. consequences – the conditional designation of the second set of y objects. abbreviated c. let z be a set of variables of interest, 𝑡 is a set containing z, then: support – coefficient denoting the ratio of the number of sets containing the set x to the total number of available sets. in fact, this is the frequency of the presence of set x in the studied set of sets p. 𝑠𝑢𝑝𝑝(𝑋) = {𝑡 ∈ 𝑃, 𝑍 ⊆ 𝑡} |𝑃| based on this definition, we introduce 𝑠𝑢𝑝𝑝(𝐴 → 𝐶)as the frequency of parallel encounters of objects from a and c. 𝑠𝑢𝑝𝑝(𝐴 → 𝐶) = 𝑠𝑢𝑝𝑝(𝐴 ∪ 𝐶) 𝑠𝑢𝑝𝑝(𝐴) (by definition) is the frequency of encounters of objects from antecedent among the entire sample of sets; and 𝑠𝑢𝑝𝑝(𝐶) is the frequency of encounters of objects from consequence. based on the definition of 𝑠𝑢𝑝𝑝(𝑋), it takes values from 0 to 1, 𝐸(𝑠𝑢𝑝𝑝) ∈ ℚ. the greater this value, the more often the selected group of objects occurs among all sets at the same time. 𝒄𝒐𝒏𝒇(𝑨 → 𝑪) – directional characteristic – the ratio between 𝑠𝑢𝑝𝑝(𝐴−> 𝐶) and the frequency of a. 𝑐𝑜𝑛𝑓(𝐴 → 𝐶) = 𝑠𝑢𝑝𝑝(𝐴 → 𝐶) 𝑠𝑢𝑝𝑝(𝐴) in fact, 𝑐𝑜𝑛𝑓(𝐴 → 𝐶) replaces the concept of conditional probability, i.e., the value will denote the conditional probability of being in the set of objects from c, provided there are objects from a. 𝑐𝑜𝑛𝑓(𝐴 → 𝐶) takes values from 0 to 1, 𝐸(𝑐𝑜𝑛𝑓) ∈ ℚ. lift – is a coefficient describing the ratio of the dependence of antecedents and consequences to their independence. otherwise, it is the ratio of the validity of the rule when both sets of objects meet together to the validity of the rule when a set of consequences is met. based on the definition, this characteristic is symmetrical. 𝑙𝑖𝑓𝑡 = 𝑠𝑢𝑝𝑝(𝐴 → 𝐶) 𝑠𝑢𝑝𝑝(𝐴) ⨯ 𝑠𝑢𝑝𝑝(𝐶) = 𝑐𝑜𝑛𝑓(𝐴 → 𝐶) 𝑠𝑢𝑝𝑝(𝐶) = 𝑐𝑜𝑛𝑓(𝐶 → 𝐴) 𝑠𝑢𝑝𝑝(𝐴) it takes values from 0 to infinity, 𝐸(𝑙𝑖𝑓𝑡) ∈ ℚ. if it is equal to 1, then a and c are independent. the high values (higher that 1) indicate the more dependence. the smaller values (below 1), indicate counter–dependence. 𝒄𝒐𝒏𝒗(𝑨 → 𝑪) – a high value here means that consequences strongly depends on antecedents. definition of this characteristic is: 𝑐𝑜𝑛𝑣(𝐴 → 𝐶) = 1 − 𝑠𝑢𝑝𝑝(𝐶) 1 − 𝑐𝑜𝑛𝑓(𝐴 → 𝐶) the conviction values from 0 to infinity, 𝐸(𝑐𝑜𝑛𝑣) ∈ ℚ. it is equal to 1 in case the elements are independent. value below 1 indicates counter-dependence. the values higher than 1 indicates non the disease centered multimorbidity model… 27 copyright ©2023 assa adv. in systems science and appl. (2023) random dependence. the higher the value of the indicator, the greater the probability of a nonrandom association of the appearance of another disease in the presence of a marker condition. the highest value (infinity) was obtained when the probability of t2dm diagnosis was estimated in patients with a previously diagnosed marker t2dm condition. clinical interpretation of these symbols is presented in the table 2. table 2. clinical interpretation of the most significant final symbols. symbol range clinical interpretation support [0, 1] frequency of the disease occurrence in the data set support a (antecedent) [0, 1] the frequency of the marker disease in the data set (t2dm in the example below) support c (consequent) [0, 1] frequency of another disease in a-c pair (another disease in t2dm patients in the example below) confidence (а –> с) [0, 1] the conditional probability that disease c will randomly or nonrandomly occur in the presence of disease a (directional value). this indicator is of interest for assessing the burden of disease in multi-morbidity, since it allows us to estimate the probability of the manifestation of condition c in the presence of condition a and, accordingly, to estimate the total burden of disease a and c in the presence of disease a conviction (а –> с) [0 ∞] indicates the probability of non-random appearance of the disease c under the condition of the disease a (directional value). high values of this indicator can be a sign that there are common pathogenetic mechanisms or external factors (risk factors or social health determinants) contributing to the development of both conditions. if the value is 1, the connection is regarded as random. from 0 to 1 – there is a counter-dependence, i.e., a decrease in the probability of developing disease c in the presence of disease a. 3.3. concept and the model assessment at the example of t2dm the conviction (probability of non-random appearance) of diseases under the condition of t2dm is presented in graph format on fig. 1 making visible the relationship between t2dm and other diseases designated by the icd-10 codes. arrows indicate the direction of the connection and their thickness shows its value. there are arrows indicating also the directional relationship between other than t2dm conditions and the probability of their feedback influence on t2dm. the values themselves are presented on fig. 2. fig. 3 represents, on the contrary, the conditional probability of the appearance of t2dm in the presence of other (non-marker) diseases. the convenience magnitude of these connections is presented fig. 4 representing a hierarchical list and interactions of the 10 most significant (highest conviction values) diseases that can contribute to the development of t2dm. these pictures show a limited number of diseases (10) with the highest conviction values with the marker disease (t2dm). the highest conviction value on fig. 4 is infinite (non-random relationship between t2dm and t2dm) but, on the histogram, it is artificially limited. the clinical meaning is quite obvious if there have been a t2dm patient visit earlier the patient will continue to visit the doctor with t2dm forming a non-random statistical relationship within the same diagnosis. there are few comments below showing the possibility of using of indicated links between diseases in the pattern as a basis for further pathogenesis pathways evaluation by clinicians. t2dm can influence on the developing of i11.9 (hypertensive disease with predominant heart damage without (congestive) heart failure) (fig. 1). this condition is also affected by other 28 d. meshkov et al. copyright ©2023 assa adv. in systems science and appl. (2023) fig. 1. conditional probability of non-random association (conviction) between existing t2dm (e11) and other diseases (9 diseases with the highest values of the code) in women (graph format). the conviction value is demonstrated by the hierarchical order of diseases (the highest is on the left) and the thickness of the arrows (the highest is the thickest). diseases caused by dm2 and the development of hypertensive disease with predominant heart damage can be considered as the possible result of direct and indirect effects of a complex of diseases included in the t2dm multimorbid pattern [8]. the example of proliferative retinopathy (h35.2) show that a number of non-marker diseases can be on the one hand a condition for the development of t2dm and on the other hand the t2dm can be a condition for the development of these diseases. proliferative retinopathy (h35.2) is included in the hierarchical list of diseases that are likely to develop under the existing t2dm (fig. 1 and 2) and also in the list of conditions pre-existing t2dm (fig. 3 and 4). in turn, the nonrandom combination of t2dm and proliferative retinopathy diseases can have quite a strong feedback effect on t2dm, demonstrated by the thickest arrow on fig. 1. thus, there is a complex of two diseases: proliferative retinopathy and t2dm and the development of each of the disease non-randomly depend on the other. the presence of isolated closed non random connections between separate diseases included in the pattern may be an example of its stratification probably related with some pathogenetic features [ошибка! источник ссылки не найден., 20 -24]. predicting of t2dm by numerous t1dm conditions (10.0; 10.5; 10.6) on fig. 3 and 4 can be explained as the result of diagnosis clarification and change in the same patients during period of data collection. it can be assumed that the expected stratification of t2dm into 5 or more subtypes will lead to the formation of similar statistical relationships that will need to be taken into account in the clinical interpretation of calculations [ошибка! источник ссылки не найден.]. thus, the evaluation on the example of t2dm of the proposed disease-centered multimorbidity concept and the algorithms for assessing of non-random connections between pairs of diseases in t2dm multimorbidity pattern allowed us to obtain data that do not contradict the modern the disease centered multimorbidity model… 29 copyright ©2023 assa adv. in systems science and appl. (2023) fig. 2. value of the conditional probability of non-random association (conviction) between existing t2dm (e11) and other diseases (9 diseases with the highest values) in women. 0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 h40.1 (primary open-angle glaucoma) i11.0 (hypertensive disease with predominant heart damage with heart failure) h35.2 (other proliferative retinopathy) h35.8 (other specified retinal disorders) i20.8 (other forms of angina pectoris) i67.8 (other specified lesions of the cerebral vessels) i67.2 (cerebral atherosclerosis) h35.0 (background retinopathy and retinal vascular changes) i11.9 (hypertensive disease with predominant heart damage without heart failure) 30 d. meshkov et al. copyright ©2023 assa adv. in systems science and appl. (2023) fig. 3. conditional probability of non-random association (conviction) between existing diseases (9 diseases with the highest values of the code) and t2dm (e11) and other in women (graph format). the conviction value is demonstrated by the hierarchical order of diseases (the highest is on the left) and the thickness of the arrows (the highest is the thickest). understanding of the t2dm pathogenesis and, moreover, are in accordance with the latest research and forecasts in this area. moreover, the proposed evaluation indicated a number of observations which can be taken as a basis for further evaluation of pathogenesis pathways. it is necessary to remember that the examples demonstrate just statistical assessment and hypothesis requiring clinical evidence. 4. conclusions the research made it possible to formulate the disease-centered multimorbidity concept aimed at improving the efficiency of public health management in modern conditions. multimorbidity traditionally defined as 2 or more diseases presented in the same patients, and is represented by multimorbidity patterns forming all together the morbidity, disease prevalence and mortality in population. in accordance the disease-centered multimorbidity concept, the pattern of multimorbidity is formed around one marker disease at the expense of other diseases that have a non-random connection with the marker disease and between each other. the totality and probabilistic connections between the pairs of diseases determine the specifics of the pattern and indicate the possibilities for managing individual and population health as a result of the interaction of all components of the system forming the pattern. calculation algorithms have been developed to assess the degree of non-randomness of connections between pairs of diseases in the pattern of multimorbidity, allowing to build a hierarchical relationship between both marker and other diseases, and between pairs of non-marker diseases. the calculation of non-random statistical relationships between pairs of diseases included in the t2dm multimorbidity pattern proved the consistency of the data obtained with the modern understanding of t2dm pathogenesis and the disease centered multimorbidity model… 31 copyright ©2023 assa adv. in systems science and appl. (2023) fig. 4. value of the conditional probability of non-random association (conviction) between existing diseases (9 diseases with the highest values) and t2dm (e11) in women. 0 2 4 6 8 10 12 e66.2 (extreme obesity accompanied by alveolar hypoventilation) e10.6 (t1dm with other specified complications) s43.1 (dislocation of the acromioclavicular joint) n03.2 (diffuse membranous glomerulonephritis) e10.0 (t1dm with coma) h35.2 (other proliferative retinopathy) e10.5 (t1dm with peripheral circulatory disorders) i73.8 (other specified peripheral vascular diseases) o24.1 (pre-existing t2dm) 32 d. meshkov et al. copyright ©2023 assa adv. in systems science and appl. (2023) complications. at the same time, it should be taken into account that the presented algorithms forming a pattern model around the selected marker disease represent just statistical relationships between pairs of diseases included in the pattern and cannot serve as a basis for clinical decisions by themselves. the same remark applies to defining the boundaries of the pattern that should include the most significant diseases in terms of pathogenesis and disease burden. implementation of the proposed model into practice does not require changes in any healthcare infrastructure and financing as well as of the core competences of healthcare practitioners. at the same time, it provides additional opportunities for better and faster understanding of diseases pathogenesis and expert support of healthcare decision making. a hierarchical list of diseases that have a non-random relationship with a marker disease allows clarifying the structure and the necessary data for creating or improving patient registers. clarification and identification of common pathogenetic mechanisms of the development of diseases included in the multimedia pattern allows the use of high-cost target therapy for the treatment of a complex of diseases instead of administering numerous interventions to each of the diseases. such approach makes possible decreasing poligragmasia and increasing the effectiveness healthcare expenditures for high-cost patients. references 1. agrawal, r., imieliński, t. & swami, a. 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(2016). multimorbidity and mortality in older adults: a systematic review and meta-analysis. arch gerontol geriatr., 67, 130–138. 18. saaty, t. l. & peniwati, k. (2008). group decision making: drawing out and reconciling differences. pittsburgh, pennsylvania: rws publications. 19. šendelj, r. (2020). information technology and information management in healthcare. stud health technol inform., 274, 139–158. 20. skou, s.t., et al. (2022). multimorbidity. nat rev dis primers, 8, 48. 21. teck, j. (2022). diabetes-associated comorbidities. prim care, 49(2), 275–286. 22. valderas, j.m., et al. (2009). defining comorbidity: implications for understanding health and health services. ann fam med., 7(4), 357–363. 23. van den akker, m., buntinx, f. & knottnerus, j. (1996). comorbidity or multimorbidity: what’s in a name? a review of literature. eur j gen pract., 2: 65–70. 24. vetrano, d.l., et al. (2018). an international perspective on chronic multimorbidity: approaching the elephant in the room. j gerontol a biol sci med sci, 11, 73(10), 1350–1356. adv syst sci appl 2025; 3:25–39 published online at https://ijassa.ipu.ru. a mathematical model for reducing divorce cases according to social indicators for morocco: optimal control approach sakkoum ayoub, hamza toufga, wafae chahid, mustapha lhous hassan ii university of casablanca, casablanca, morocco abstract: in this research, we discuss a mathematical model for a person’s social status with a focus on marital status. in most societies, the marital status of men and women can be categorized as follows: unmarried refers to a young man or woman of legal marriageable age who is neither married nor engaged. the second category is engagement, a stage that precedes marriage, often characterized by acquaintance and typically of short duration. the third status is marriage, which is customarily conducted according to the traditions and customs of each religion or culture. the next category is separation, referring to married couples who are experiencing difficulties and are living apart but are not legally divorced. following that is divorce, which denotes a legal dissolution of marriage. the final status is widowed. our research aims to address and reduce the phenomenon of divorce, which has become widespread, with more than 50 percent of marriages reportedly lasting less than a year. we identify two control strategies that help minimize the number of divorced individuals while increasing the number of married individuals. the optimal control problem is formulated and analyzed by applying a discrete version of pontryagin’s maximum principle. numerical simulations are performed to validate the theoretical results. keywords: the marital status model, optimal control, pontryagin’s maximum principle. 1. introduction divorce is one of the social-psychological problems that has become increasingly prevalent in societies worldwide, especially in arab communities. it results in many negative effects, including family disintegration, the spread of hostility, the emergence of children’s adverse behaviors, and their reluctance toward marriage in the future. additionally, divorce can lead to psychological disorders that may cause behavioral deviations later in life. marriage, as we can see now, confronts many challenges. marriage’s sacramental essence is increasingly changing, and divorce is being integrated into the legal system through legislation. the legal measures have now made it possible for an unhappy couple to seek a way out of the dead lock in the wedding lock. this has brought about dynamic changes in the social environment. divorce, desertion, and separation are frequent occurrences in a modern family, whereas they were a rare phenomenon in traditional society. marriages are not always successful, as some end in discord. divorce is the final indication of a marriage breakdown. it is the legal method of dissolving a marriage. the conjugal relationship is the central bond uniting the family in any society. when this bond is broken, the family is automatically broken. the existence of family groups as a functioning unit depends upon the continuation of many personal relationships. when this bond breaks down, the family organisation breaks down. ∗corresponding author: abauth1@gmail.com 26 s. ayoub, h. toufga, w. chahid, m. lhous the practice of divorce has existed for centuries. divorce is described as a court-ordered legal dissolution of a marriage. divorce can also be defined as the legal separation of two people impacted by a court’s judgment or decree, either completely terminating the marriage relationship or suspending its consequences as far as the parties’ cohabitation is concerned. thus, from the two definitions, it may be established that a divorce can be said to be validly granted when it is made by a judgement or decree of a court with the aim of dissolving the marriage completely or suspending its effects, where these effects are the rights and obligations acquired by and imposed on the couple in their cohabitation as a result of the marriage contract. divorce is the only recognised procedure for annulling a marriage contract. people who are in a relationship before marriage are one of the compartments we discussed in our mathematical model, a man and a woman who are in a relationship can be considered two mature people in the dating or courtship stage. in some countries and cultures, two people in a relationship mean that they practice family life as a married couple and can have children as long as the marriage contract exists between them. in some other countries, the union of two people or their sexual relations outside of marriage is considered a crime punishable by law. as for the model we are discussing, we consider only two related people who are engaged or in a state of acquaintance before marriage, according to moroccan law, which criminalizes relationships outside the institution of marriage. the high commission for planning (hcp) [11], a national statistical institution, has included social indicators for morocco: 2023 edition, exciting and indicative digital data on the development of many aspects of personal status issues for moroccans, especially regarding issues of marriage and divorce. annually, hcp addresses the state of the population structure (demography) in detail. since the family code came into effect, the number has increased from 236.574 during the year 2004 to a total of 275.978 marriages concluded in 2019. however, there was a significant decline in 2020 until it reached 194.318, and after that there was a slight increase until it reached 251.847 in 2022. 50% of marriages in morocco end in divorce. the morocco divorce rate increased from 45.01% in 2017 to 48.83% in 2018, then 50.34% in 2019 to 55.17% in 2020 during the covid-19 pandemic. the year 2021 marked a slight drop in this rate, which fell to 51.18% and back again in 2022 to 74.95%. in numbers, divorce cases registered in moroccan courts were 107,136 in 2017, followed by an increase of 7.75% the next year with 115,436 cases. the year 2019 registered 129,417 cases, which dropped to 105,471 in 2020. in our mathematical system, we discuss a mathematical model of family dynamics based on marital status. people in society are considered to be classified into six classes: virgin s, engagement c, married m, separated p, divorced d, widowed w. many research themes close to the subject of the individual’s status in society have been discussed, for example, law, religion, and statistics [1, 7–9, 18, 21, 24]. m.lhous in [12] has talked about a mathematical model of monogamous marriage. and in [13], m.lhous, by using a multi-region discretetime model, developed a mathematical model of the family status in various areas, as well as the outcome of region interconnectivity on marital status. a.sakkoum [22] has talked about a discrete mathematical model of islamic polygamy. optimal control theory is widely utilized as accessible and successful tool for managers to create and simulate control schemes, see [2,23]. we use an optimal control strategy to reduce divorces and increase the number of spouses at a low cost. the first two controls (u1, u2) are introduced to decrease the number of divorces using reconciliation, advice, seminars, and classes within the framework of the accompanying relationship to solve the problems for which the spouses want to divorce, explain to adults the positive effects of marriage on the psychological and social equilibrium of the individual and society, and provide clarifications of the psychological problems that divorce entails for children and all members of both families. the second control (u3, u4) lies in issuing laws that make divorce not easy. is chosen for the individuals who started the divorce process. this control is regarded as a timeconsuming and expensive legal processes. there are several approaches for computing the copyright © 2025 assa. adv syst sci appl (2025) a mathematical model for reducing divorce cases... 27 fig. 1.1. divorce rate in morocco between 2017 and 2022 [11]. best control for a given mathematical model. pontryagin’s maximum principle [19] calculates the optimal control for a system under a specified constraint. the paper is structured as follows. in section 2, we explore equations of social status. in section 3, we create positivity of solutions and invariant area. in section 4, free equilibrium and basic reproduction number analysis are conducted, proving the local stability of the free equilibrium point and the global stability. in section 5, we present some findings on the existence of optimum control and use pontryagin’s maximal principle to analyse control strategies and deduce the essential conditions for optimal control. the numerical simulations are reported in section 6. finally, we give conclusions in section 7. 2. mathematical model in this social system, we create a mathematical model to minimise divorces and separations. the notations and their descriptions are shown below. s(t): number of unmarried persons reach marriageable age. c(t): number of people who are in a relationship before marriage. m(t): number of people who are married. p(t): the number of persons who are separated but not divorced. d(t): the number of divorced individuals. w(t): number of people who are widowed. λ is the percentage of people who are recruited. who remain unmarried when they reach marriageable age. this individual got into a couple without marriage at a rate of α1. α2 is the rate at which individuals in a couple become single. this individual got married at a rate of β1 and engaged at a rate of β2. r1 is the rate of married individuals who become widowed. r2 is the rate of widows who become married. r3 represents the proportion of widows in relationships. the unhappy couple moved past their disagreements and remarried at a rate of λ1. λ2 rate of broken marriage individuals but not divorced. δ1 rate of married people getting divorced. δ2 rate of divorced individuals who become married. ρ1 is the rate of failed marriages ending in permanent divorce. the rate at which divorced individuals enter the engaged population is ρ2, while the natural death rate is µ. furthermore, we assume that members blend in a uniform manner, meaning they have the same degree of interaction, and that factors such as sex, colour, and social standing have no bearing on the likelihood of divorce. table 1 depicts and describes the model’s state variables, while table 2 provides copyright © 2025 assa. adv syst sci appl (2025) 28 s. ayoub, h. toufga, w. chahid, m. lhous fig. 2.2. the schematic illustration of the scmpdw model. a description of the model’s parameters. figure 1 displays the model’s compartmental flow diagram. in light of the aforementioned presumptions, the differential equation system that governs the model is as follows: ds dt = λ− (α1 + β1 + µ)s + α2c (2.1) dc dt = α1s − (α2 + β2 + µ)c + r3w + ρ2d (2.2) dm dt = β1s + β2c − (r1 + µ+ λ2d)m + δ2d + λ1p + r2w (2.3) dp dt = λ2md − (λ1 + ρ1 + µ)p (2.4) dd dt = ρ1p − (δ2 + ρ2 + µ)d (2.5) dw dt = r1m − (r2 + r3 + µ)w (2.6) with positive initial conditions given by s0 = s(0) ⩾ 0, c0 = c(0) ⩾ 0,m0 = m(0) ⩾ 0, p0 = p (0) ⩾ 0, (2.7) d0 = d(0) ⩾ 0 and w0 = w (0) ⩾ 0. all parameters of the system are considered to be positive at all times t. variable state description s(t) number of single adults at time t. c(t) number of engaged without marriage adults at time t. m(t) number of married adults at time t. p (t) number of broken adults at time t. d(t) number of divorced individuals at time t. w (t) number of widowed at time t. n(t) total number of population at time t. table 2.1. model’s (2.1)-(2.6) variable states. copyright © 2025 assa. adv syst sci appl (2025) a mathematical model for reducing divorce cases... 29 parameter description λ recruitment percentage of adults at the age of single. α1 average percentage of single adults who got engaged. α2 average percentage of engaged adults who got single. β1 average percentage of single adults who got married. β2 average percentage of engaged adults who got married. r1 average percentage of married adults who become widowed. r2 average percentage of widowed who got married. r3 average percentage of widowed who got engaged. λ1 percentage of broken adults who renew their previous marriage. λ2 the contact percentage of divorced individuals with married individuals. δ2 percentage of divorced individuals who got married. ρ1 percentage of broken adults who got divorced. ρ2 percentage of divorced individuals who got engaged. µ natural death percentage of adults. table 2.2. description of model parameters (2.1)-(2.6). 2.1. positivity of solutions a marriage model system must demonstrate that every state variable is always non negative in order to be considered socially realistic. theorem 2.1: for all solutions of system (2.1)-(2.6) with intial conditions (2.7) the functions s(t), c(t), m(t), p (t), d(t), w (t) are nonnegative for every t ≥ 0. proof we start our investigation of system (2.1)-(2.6) by looking at a few basic model characteristics. firstly, we will show that any solution of (2.1)-(2.6), initiated from a nonnegative initial condition in r6 +, will remain nonnegative. specifically, we can see that ds dt |s=0 = λ + α2c > 0for allc ⩾ 0 dc dt |c=0 = α1s + r3w + ρ2d ⩾ 0 for all s,w,d ⩾ 0 dm dt |m=0 = β1s + β2c + δ2d + λ1p + r2w ⩾ 0 for all s, c,d, p,w ⩾ 0 dp dt |p=0 = λ2md ⩾ 0 for all m,d ⩾ 0 dd dt |d=0 = ρ1p ⩾ 0 for all p ⩾ 0 dw dt |w=0 = r1m ⩾ 0 for all m ⩾ 0 this proves that r6 + is positively invariant with respect to system (2.1)-(2.6), meaning that any solution of (2.1)-(2.6) will remain in r6 + for all times. which proves that any solution of (2.1)-(2.6) is nonnegative. 2.2. invariant region theorem 2.2: for all t > 0 solutions of system (2.1)-(2.6) with intial conditions (2.7) are contained in the copyright © 2025 assa. adv syst sci appl (2025) 30 s. ayoub, h. toufga, w. chahid, m. lhous region ω ⊂ r6 +, characterized by ω = {(s(t), c(t),m(t), p (t), d(t),m(t)) ∈ r6 + : n(t) ≤ λ µ }. (2.8) proof the sum of all equations in the model system yields dn(t) dt = ds(t) dt + dc(t) dt + dm(t) dt + dp (t) dt + dd(t) dt + dw (t) dt . the change in the overall population is defined as dn dt = λ− µn. the answer of inequation is n(t) = λ µ − ( λ µ −n0)e −µt, where n0 = n(0). using the theorem of birkhoff-rota [4], we observe that, if n0 < λ µ , then n −→ λ µ asymptotically as t −→ ∞ in ω and the total population size n −→ λ µ , which means that 0 ⩽ n ⩽ λ µ and the solutions are bounded. as a result, all of the model’s possible solutions converge in the region ω [10]. 3. analysis of free equilibrium, basic reproduction number r0 fundamentally, the divorce–free equilibrium can be obtained by taking the equation of the system with s = c = m = p = d = m = 0 into consideration, we arrive at: m0 = λ[β1n2 + β2α1]n6 [n1n2 − α1α2][α1(r1 + µ)n6 + β1r1r3 − α1r1r2 − r1r3] s0 = λ n1 + α2 n1 c0 c0 = 1 α1β2 + β1n2 [α1r1 + α1µ+ β1r1r3 n6 − α1r1r3 n6 ]m0, w0 = r1 n6 m0. where, n1 = α1 + β1 + µ, n2 = α2 + β2 + µ and n6 = r2 + r3 + µ. therefore, system (2.1)-(2.6) has a unique free equilibrium state: e0 = (s0, c0,m0, 0, 0,w0) (3.9) copyright © 2025 assa. adv syst sci appl (2025) a mathematical model for reducing divorce cases... 31 3.1. basic reproduction number r0 to calculate the fundamental r0, we employed the next generation matrix approach of (2.1)(2.6), where f is the matrix of fresh divorce or separation terms and v is the matrix of transition terms. the matrices f and v are calculated using the coefficients of p and d in the system’s fourth and fifth equations. equations of the model are re-written, beginning with newly infected classes: dp dt = λ2md − (λ1 + ρ1 + µ)p (3.10) dd dt = ρ1p − (δ2 + ρ2 + µ)d (3.11) f= [ 0 λ2m0 0 0 ] and v= [ (λ1 + ρ1 + µ) 0 −ρ1 (δ2 + ρ2 + µ) ] r0 (spectral radius of fv −1) must be taken into consideration. thus, the reproduction number r0 can be obtained as follows: r0 = ρ(fv −1) = λ2m0 (λ1 + ρ1 + µ)(δ2 + ρ2 + µ) (3.12) 3.2. local stability of free equilibrium point theorem 3.1: e0 is locally asymptotically stable if r0 < 1. whereas, e0 is unstable if r0 > 1 . proof consider the model system (2.1)-(2.6), at the equilibrium point (e0), the jacobian becomes:  x1 α2 0 0 0 0 α1 x2 0 0 ρ2 r3 β1 β2 x3 λ1 λ2m0 + δ2 r2 0 0 0 x4 λ2m0 0 0 0 0 ρ1 x5 0 0 0 r1 0 0 x6  (3.13) with: x1 = −(α1 + β1 + µ), x2 = −(α2 + β2 + µ), x3 = −(α1 + µ), x4 = −(λ1 + ρ1 + µ), x5 = −(δ2 + ρ2 + µ) and x6 = −(r3 + r2 + µ). three eigenvalues of (13) are n1 = −(α1 + β1 + µ), n2 = −(α2 + β2 + µ), n3 = −(α1 + µ), and n6 = −(r3 + r2 + µ), while the remaining two eigenvalues are obtained from the 2 by 2 matrix a = ( −(λ1 + ρ1 + µ) λ2m0 ρ1 −(δ2 + ρ2 + µ) ) (3.14) copyright © 2025 assa. adv syst sci appl (2025) 32 s. ayoub, h. toufga, w. chahid, m. lhous if the routh-hurwitz [15] condition is met, the eigenvalues of matrix a will be real and negative. applying the routh-hurwitz condition: ⋆ tr(a) < 0, ⋆ det(a) > 0. tr(a) = −(λ1 + ρ1 + µ)− (δ2 + ρ2 + µ) = − ( (λ1 + ρ1 + µ) + (δ2 + ρ2 + µ) ) . therefore, we have tr(a) < 0. det(a) = (λ1 + ρ1 + µ)(δ2 + ρ2 + µ)− ρ1λ2m0 = (λ1 + ρ1 + µ)(δ2 + ρ2 + µ)[1− ρ1λ2m0 (λ1 + ρ1 + µ)(δ2 + ρ2 + µ) ] = (λ1 + ρ1 + µ)(δ2 + ρ2 + µ)[1− ρ1r0, ] thus, det(a) > 0 if, r0 < 1 ρ1 . using theorem 2 of (2.1)-(2.6), we deduce that the free equilibrium point is locally asymptotically stable. 3.3. global stability of free equilibrium point we create a lyapunov function [14] to demonstrate the free equilibrium point’s global stability. we define the following: l = p λ1 + ρ1 + µ +d. (3.15) we will show that dl dt ⩽ 0 for all t ⩾ 0. we have: dl dt = d dt p λ1 + ρ1 + µ + dd dt = λ2md − (λ1 + ρ1 + µ)p λ1 + ρ1 + µ + ( ρ1p − (δ2 + ρ2 + µ)d ) = ( λ2m λ1 + ρ1 + µ − (δ2 + ρ2 + µ) ) d − (1− ρ1)p = (δ2 + ρ2 + µ)[ λ2m (λ1 + ρ1 + µ)(δ2 + ρ2 + µ) − 1]d − (1− ρ1)p = (δ2 + ρ2 + µ)[r0 − 1]d − (1− ρ1)p. thus, if r0 < 1, dl dt is negative. the biggest compact invariant set in ω is the singleton set e0. according to lasalle’s invariance principle [3], e0 is asymptotically stable in ω. 4. the optimal control problem as well, we recognize morocco, like other countries, suffers until this moment from the phenomenon of divorce. divorce is found in all age groups, young and old. we suggest a copyright © 2025 assa. adv syst sci appl (2025) a mathematical model for reducing divorce cases... 33 strategy to control and decrease the rate of divorces, first u1 and u2 to advise and find ways of reconciliation before falling into divorce. the second control u3 and u4 lies in issuing laws that make divorce not easy. the controlled model corresponding to (2.1)-(2.6) has the form: ds dt = λ− (α1 + β1 + µ)s + α2c (4.16) dc dt = α1s − (α2 + (1 + u1)β2 + µ)c + r3w + ρ2d (4.17) dm dt = β1s + (1 + u1)β2c − (r1 + µ+ λ2d)m + (1 + u4)δ2d + (1 + u3)λ1p + r2w (4.18) dp dt = λ2md − ((1 + u3)λ1 + (1− u2)ρ1 + µ)p (4.19) dd dt = (1− u2)ρ1p − ((1 + u4)δ2 + ρ2 + µ)d (4.20) dw dt = r1m − (r2 + r3 + µ)w, (4.21) and the challenge is to minimize the objective functional j(u1, u2, u3, u4) = ad(t ) +bp (t )− cm(t )+∫ t 0 (ad(t) +bp (t)− cm(t) + i 2 u2 1 + j 2 u2 2 + k 2 u2 3 + l 2 u2 4)dt (4.22) where u1, u2, u3, u4 are measurable functions satisfying 0 ⩽ ui,min ⩽ ui ⩽ ui,max ⩽ 1, 1 ⩽ i ⩽ 4 j(u∗ 1, u ∗ 2, u ∗ 3, u ∗ 4) = min{j(u1, u2, u3, u4)}. (4.23) 4.1. the optimal control: existence we initial demonstrate the existence of the system’s solutions, followed by the presence of optimal control. theorem 4.1: the optimal control problem given by (4.16)-(4.21), (4.23) has a solution (u∗ 1, u ∗ 2, u ∗ 3, u ∗ 4). proof we shall employ fleming and rishel [6] to demonstrate the existence of the optimal control. • as a result, the collection of controls and their respective state variables is not zero. we’ll utilize a simplified version of the existence result. • the control space u , is convex and closed by definition. • j(u1, u2, u3, u4) is convex in u . • the main sides of system equations are continuous, limited higher by a sum of bounded control and state, and can be represented as a linear function of u1, u2, u3, and u4, with coefficients varying with time and state. • the integrand in the objective functional, a1d(t) + a2p (t)− a3m(t) + i 2 u2 1 + j 2 u2 2 + k 2 u2 3 + l 2 u2 4 is clearly convex on u . copyright © 2025 assa. adv syst sci appl (2025) 34 s. ayoub, h. toufga, w. chahid, m. lhous • it rests to show that there exist constants ζ1, ζ2, ζ3, ζ4, ζ5 > 0 and ζ such that a1d(t) + a2p (t)− a3m(t) + i 2 u2 1 + j 2 u2 2 + k 2 u2 3 + l 2 u2 4 ⩾ ζ1 + ζ2|u1|ζ + ζ3|u2|ζ + ζ4|u3|ζ + ζ5|u4|ζ , where ζ1 = inft∈[0,t ]{a1d(t) + a2p (t)− a3m(t)}, ζ2 = i , ζ3 = j , ζ4 = k and ζ5 = l and ζ = 2. the state variables are being bounded (n(t) ⩽ λ µ and n(t) = s(t) + c(t) +m(t) + p (t) +d(t) +m(t), ∀t ⩾ 0), then, from fleming and rishel [6], we conclude that there exists an optimal control of (4.23). 4.2. the optimal control characterization at the same time by using pontryagin’s maximum principle [19], we derive the necessary conditions for our optimal control. for this purpose we define the hamiltonian as h = a1d(t) + a2p (t)− a3m(t)) + i 2 u2 1 + j 2 u2 2 + k 2 u2 3 + l 2 u2 4 + 5∑ i=1 ςifi(s,c,m,p,d,w ). where fi denotes the right side of the difference equation for the ith state variable at the time step. theorem 4.2: considering optimal controls u∗ 1, u ∗ 2, u ∗ 3, u ∗ 4 and their respective solutions s, c, m, p, d, and w in the appropriate state system (2.1)-(2.6), there are adjoint variables. ς1, ς2, ς3, ς4, ς5, and ς6 meet ς ′ 1 = ς1(α1 + β1 + µ)− ς2α1 − ς3β1 ς ′ 2 = −ς1α2 + ς2(α2 + (1 + u1)β2 + µ)− ς3(1 + u1)β2 ς ′ 3 = a3 + ς3[(r1 + µ+ λ2d]− ς4λ2d − ς6r1 ς ′ 4 = −a2 − ς3(1 + u3)λ1 + ς4[(1 + u3)λ1 + (1− u2)ρ1 + µ]− ς5(1− u2)ρ1 ς ′ 5 = −a− ς2ρ2 + ς3[λ2m − (1 + u4)δ2]− ς4λ2m + ς5[(1 + u4)δ2 + ρ2 + µ]} ς ′ 6 = −ς2r3 − ς3r2 + ς6(r2 + r3 + µ). with the transversality conditions at time tf , ς1(tf ) = 0, ς2(tf ) = 0, ς3(tf ) = a3, ς4(tf ) = −a2, ς5(tf ) = −a1, ς6(tf ) = 0. furthermore, for t ∈ [0, tf ], the optimal controls u∗ 1, u ∗ 2 and v∗1 are given by u∗ 1 = min[1,max(0, (ς2 − ς3)β2c i )], u∗ 2 = min[1,max(0, (ς5 − ς4)ρ1p j )], u∗ 3 = min[1,max(0, (ς4 − ς3)λ1p k )], u∗ 4 = min[1,max(0, (ς5 − ς3)δ2d l )]. copyright © 2025 assa. adv syst sci appl (2025) a mathematical model for reducing divorce cases... 35 proof we derive the necessary conditions for our optimal control. for this purpose, the hamiltonian is described as below. h = a1d(t) + a2p (t)− a3m(t) + i 2 u2 1 + j 2 u2 2 + k 2 u2 3 + l 2 u2 4 + 5∑ i=1 ςifi(s,c,m,p,d,w ). where f1(s,c,m,p,d,w ) = λ− (α1 + β1 + µ)s + α2c, f2(s,c,m,p,d,w ) = α1s − (α2 + (1 + u1)β2 + µ)c + r3w + ρ2d, f3(s,c,m,p,d,w ) = β1s + (1 + u1)β2c − (r1 + µ+ λ2d)m + (1 + u4)δ2d + (1 + u3)λ1p + r2w, f4(s,c,m,p,d,w ) = λ2md − ((1 + u3)λ1 + (1− u2)ρ1 + µ)p, f5(s,c,m,p,d,w ) = (1− u2)ρ1p − ((1 + u4)δ2 + ρ2 + µ)d, f6(s,c,m,p,d,w ) = r1m − (r2 + r3 + µ)w. using pontryagin’s maximum principle [19, 20], the adjoint equations and transversality requirements for t ∈ [0, tf ] can be derived as follows: ς ′ 1 = −∂h ∂s = −{−ς1(α1 + β1 + µ) + ς2α1 + ς3β1} = ς1(α1 + β1 + µ)− ς2α1 − ς3β1 ς ′ 2 = −∂h ∂c = −{ς1α2 − ς2(α2 + (1 + u1)β2 + µ) + ς3(1 + u1)β2} = −ς1α2 + ς2(α2 + (1 + u1)β2 + µ)− ς3(1 + u1)β2 ς ′ 3 = − ∂h ∂m = −{−a3 − ς3[(r1 + µ+ λ2d] + ς4λ2d + ς6r1} = a3 + ς3[(r1 + µ+ λ2d]− ς4λ2d − ς6r1 ς ′ 4 = −∂h ∂p = −{a2 + ς3(1 + u3)λ1 − ς4[(1 + u3)λ1 + (1− u2)ρ1 + µ] + ς5(1− u2)ρ1} = −a2 − ς3(1 + u3)λ1 + ς4[(1 + u3)λ1 + (1− u2)ρ1 + µ]− ς5(1− u2)ρ1 ς ′ 5 = −∂h ∂d = −{a1 + ς2ρ2 − ς3[λ2m − (1 + u4)δ2] + ς4λ2m − ς5[(1 + u4)δ2 + ρ2 + µ]} = −a− ς2ρ2 + ς3[λ2m − (1 + u4)δ2]− ς4λ2m + ς5[(1 + u4)δ2 + ρ2 + µ]} ς ′ 6 = − ∂h ∂w = −{ς2r3 + ς3r2 − ς6(r2 + r3 + µ)} = −ς2r3 − ς3r2 + ς6(r2 + r3 + µ) copyright © 2025 assa. adv syst sci appl (2025) 36 s. ayoub, h. toufga, w. chahid, m. lhous for t ∈ [0, tf ], the optimal controls u∗ 1, u ∗ 2, u∗ 3 and u∗ 4 can be solved from the optimality condition ∂h ∂u1 = iu1 − β2cς2 − β2cς3 = 0 ∂h ∂u2 = ju2 + ρ1pς4 − ρ2pς5 = 0 ∂h ∂u3 = ku3 + λ2pς3 − λ2pς4 = 0 ∂h ∂u4 = lu4 + δ2dς3 − δ2dς5 = 0 then, we have (ς2 − ς3)β2c i , (ς5 − ς4)ρ1p j , (ς4 − ς3)λ1p k and (ς5 − ς3)δ2d l . by the bounds in u of the controls, we deduce that u∗ 1, u ∗ 2, u ∗ 3 and u∗ 4 are given the form in the theorem. 5. numerical simulation our reasons for participating in this investigation was to assess the rate of controls, the number of divorces, and the rate of increase in marriages affect each other. some numerical simulation results are given in this section mainly to demonstrate the numerical results achieved. the ” social indicators for morocco-2023 edition ” document included ” exciting and indicative ” digital data on the development of many aspects of personal status issues for moroccans. especially with regard to issues of marriage, divorce, fertility rates, and childbearing. the values of the model parameters chosen for the simulation are given by document ” social indicators for morocco-2023 edition ” [11]. s(0) = 184052, c(0) = 153694,m(0) = 258470, p (0) = 183441, d(0) = 68145, w (0) = 81542,λ = 82319, α1 = 0.0158, α2 = 0.11, β1 = 0.0003889, β2 = 0.00656, r1 = 0.364, r2 = 0.314, r3 = 0.00684, λ1 = 0.119, λ2 = 0.000025, δ2 = 0.0000219, rho1 = 0.000142, ρ2 = 0.0000275, µ = 0.0017. copyright © 2025 assa. adv syst sci appl (2025) a mathematical model for reducing divorce cases... 37 the following two figures show developments in the number of marriage contracts and divorces in morocco in 2022. the development of the number of marriages in morocco in one year, as we notice that starting from the first day, the development is noticeable day after day. without controls: on the first day, we launched with more than 200,000 marriage contracts, but more than 50 percent do not last more than a few weeks. after that, we notice a slow increase in the number of marriages, as we moved from 61,228 marriage contracts until we reached 170,349 successful marriage contracts a year later. these results remain much lower than the numbers we started with. as for divorce cases in morocco, we note that we started with 68,145 on the first day, and with a significant increase, we reached more than 97,000 in one year, and this is an abnormal situation. with controls: after using the control, we notice a good development from 258,470 marriage contracts on the first day until we reached more than 570,000 marriage contracts per year. after applying the control, it appears that there is an interesting decline in the number of divorces, as the difference was good. we started from 68,145 divorce cases and reached 36,000 divorce cases. this means the continuation of marriage and the decline in divorce. the following figure represent the changes in the number of separated people, in one year. the changes occurring in this chart show that before implementing the control, we started on the first day with 183441 and after a year had passed, the number of the separated people reached more than 340000. this situation is not reassuring, and this is due to the fact that most of the cases separation reach to divorce, and this is what we reduce from it. after applying the control, we notice that the graph shows an increase in the number of cases copyright © 2025 assa. adv syst sci appl (2025) 38 s. ayoub, h. toufga, w. chahid, m. lhous separated people but not in such a large way that after a year it reached an average of 210000 cases, and most of them do not reach divorce. this is due to the control that we applied on two levels. 6. conclusion a mathematical model has been developed as a system to examine the social status of people in all societies. on the one hand, it has been proven that the systems are stable locally and globally without symptoms at the equilibrium point. on the other hand, the two controls are applied by reducing the number of divorce cases as well as increasing the number of marriage cases after we developed controls that gave effectiveness and encouraging results. finally, numerical simulations were used to evaluate the departments’ results, which showed that providing counseling and implementing an awareness campaign, as well as the administrative complexities and severe financial and societal impacts of divorce, can help reduce divorce cases. references 1. al-qurashi, m., 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(1962). the mathematical theory of optimal processes, new york, ny: crc press. 20. b.v, pontryagin’s maximum principle, mathematics in science and engineering, 1962. 21. syamsir, s. & kasbawati, t. (2022). stability analysis of divorce dynamics models, jurnal matematika statistika dan komputasi, 17(2), 267–279. 22. sakkoum, a., lhous, m., rachik, m. & tridane, a. discrete mathematical modeling and optimal control of the marital status: islamic polygamous marriage model case, mathematical modeling and computing, 10(3), 748–763. 23. toufga, h., sakkoum, a., benahmadi, l. & lhous, m. analysis of the dynamics and optimal control of cutaneous lieshmania during human immigration, iranian journal of numerical analysis and optimization, 15(1), 311–345, doi: 10.22067/ijnao.2024.88709.1466. 24. tessemal, h., haruna, i., shaibu, o. & endeshaw, k. a mathematical model analysis of marriage divorce, communications in mathematical biology and neuroscience, 20, 2052–2541. copyright © 2025 assa. adv syst sci appl (2025) introduction mathematical model positivity of solutions invariant region analysis of free equilibrium, basic reproduction number r0 basic reproduction number r0 local stability of free equilibrium point global stability of free equilibrium point the optimal control problem the optimal control: existence the optimal control characterization numerical simulation conclusion adv syst sci appl 2021; 04:130–144 published online at https://ijassa.ipu.ru. equilibrium in market models with known elasticities aram arutyunov1, alexander kotyukov2*, natalia pavlova1,2,3 1 rudn university, moscow, russia 2 v.a. trapeznikov institute of control sciences of ras, moscow, russia 3 moscow institute of physics and technology, moscow, russia abstract: closed and open market models, in which the supply and demand functions are restored by their price elasticities, are studied. for the closed market model criteria on the existence of equilibrium is obtained as the corollary of existence theorems for the solutions to systems of linear equations and inequalities. the results of the covering maps theory, namely existence theorems of a coincidence point, are applied to obtain sufficient conditions on the existence of an equilibrium in open market model and to develop search algorithm of an equilibrium in this model. numerical experiments illustrating the obtained results are conducted. keywords: demand, supply, elasticity, market model, economical equilibrium 1. introduction the concept of equilibrium plays an important role in the research of market models. basically it is a situation on the market in which none of the participants is interested in the change of their state. let us explain the meaning of equilibrium. assume that the market consists of the producers and the consumers. the producers tend to maximize their own income by manufacturing some goods. the consumers spend their budget to purchase these goods satisfying their needs. the total volume of goods produced is called the supply. the total volume of goods needed is called the demand. it is obvious that insufficient amount of goods on the market can lead to unfavorable consequences such as hunger, epidemic or even death depending on the market described. therefore, there must be enough goods on the market to satisfy the needs of the consumers. on the other hand, if some goods produced is not sold to the consumers, the producers suffer an obvious income loss. hence, they must produce enough goods to satisfy the needs of the consumers, but not more than this required amount. from these considerations we obtain that the amount of goods produced must be equal to the amount of goods needed. in other words, the value of the supply must be equal to the value of the demand. this situation on the market is called an equilibrium. the concept of equilibrium allows us to determine the best prices for the producers and the consumers to set in the modeled region. indeed, with these prices (which are called equilibrium prices) producers obtain maximal profit since they sell all the goods produced, and consumers obtain the goods needed in the full amount. however, the mathematical approach to this concept remained poor due to the state of the progress of mathematics at the time. ∗corresponding author: amkotyukov@mail.ru equilibrium in market models with known elasticities 131 the advancement in the theory of covering mappings and coincidence points (see, e.g., [1], [2]) makes the deep research into the equilibrium theory possible. this mathematical field allows us to make a research on the equilibrium even in the nonlinear market models. for example, in [5]– [7] theorems on the existence of coincidence points for mappings of metric spaces are applied to obtain sufficient conditions on the existence of an equilibrium in the market models. in [3], [4], [7] the mappings of supply and demand are determined as solutions of extremal problems. the supply function can also be obtained as the realization of some production model (see, e.g., [5], [13]). here we restore the supply and demand function using the concept of elasticitiy. elasticities are the values that connect supply and demand volumes with parameters obtained from the statistical data (e.g., prices, income, transport costs etc.). that allows us to determine the supply and demand functions that describe real market of some region. using equilibrium theory, we can improve the economical situation in the region. in this paper, we investigate the existence of equilibrium in open and closed market models. here, we used price elasticity coefficients of goods presented on the market to obtain the explicit forms of the supply and demand functions. for these functions, we obtained a criteria for the existence of equilibrium in the closed market model. also for the open market model we obtained sufficient conditions using corollaries of coincidence points theorems. we also developed a search algorithm of an equilibrium for the open market model and conducted a numerical experiment. this algorithm is based on pattern search method and illustrate the effectiveness of the obtained sufficient conditions. this method is chosen due to its simplicity and visibility. the obtained results show that equilibrium is found in the case when the model satisfies the sufficient conditions with moderate precision. 2. market models this section provides sample equations, figures, and tables. 2.1. closed market model consider a market of n goods with uniform prices set: p = (p1, . . . , pn), pi ∈ [c1i, c2i], where vectors c1, c2 set the natural restrictions on these prices. suppose that we have n× n matrices e = (eij)i,j=1,n, ẽ = (ẽij)i,j=1,n (eij, ẽij ∈ r for all i, j = 1, n, vectors c̄1 = (c11, ..., c1n), c̄2 = (c21, ..., c2n) ∈ rn + with 0 < c1i < c2i for all i = 1, n, a vector p̄∗ = (p∗1, ..., p ∗ n) ∈ rn + such that c1i ≤ p∗i ≤ c2i for all i = 1, n, and vectors d̄∗ = (d∗1, ..., d ∗ n), s̄∗ = (s∗1 , ..., s ∗ n) ∈ rn +. we call σc = (e , ẽ , d̄∗, s̄∗, p̄∗, c̄1, c̄2) (2.1) a closed market model (the goods presented on the market are manufactured by the producers only). the set of all σc is denoted by σc. the parameters of the model have the following sense. elements of e are the price elasticities of demand: eij = ∂di ∂pj pj di , i, j = 1, n. (2.2) here pi is the price of the ith good, di = di(p1, ..., pn) is a total demand on the ith good, eij is the elasticity of demand on the ith good for the price of the jth good, i, j = 1, n. copyright © 2021 assa. adv syst sci appl (2021) 132 a. arutyunov, a. kotyukov, n. pavlova elements of ẽ are the price elasticities of supply: ẽij = ∂si ∂pj pj si , i, j = 1, n. (2.3) here si = (si(p1, ..., pn) is a total supply of the ith good, ẽij is the elasticity of supply on the ith good for the price of the jth good, i, j = 1, n. components of the vectors c̄1 and c̄2 generate natural conditions on the prices of the goods: c1i ≤ pi ≤ c2i, i = 1, n. component of the vector d̄∗ = d̄(p∗1, ..., p ∗ n) is the demand on the corresponding good and component of the vector s̄∗ = s̄(p∗1, ..., p ∗ n) is the supply of the corresponding good for the prices set p̄∗ = (p∗1, ..., p ∗ n). parameters (e , ẽ , d̄∗, s̄∗, p̄∗, c̄1, c̄2) uniquely define the function of demand d : [c11; c21]× ...× [c1n; c2n]→ rn +, d(p1, ..., pn) = (d1(p1, ..., pn), ..., dn(p1, ..., pn)) (2.4) and the function of supply s : [c11; c21]× ...× [c1n; c2n]→ rn +, s(p1, ..., pn) = (s1(p1, ..., pn), ..., sn(p1, ..., pn)) (2.5) solving the system of partial differential equations (2.2) we obtain the explicit form of the demand function: di(p1, ..., pn) = d∗i n∏ j=1 (p∗j) −eijp eij j . (2.6) similarly, solving the system of partial differential equations (2.3) we get the explicit form of the supply function: si(p1, ..., pn) = s∗i n∏ j=1 (p∗j) −ẽijp ẽij j . (2.7) definition 2.1: a vector p̄ ∈ [c11, c21]× ...× [c1n, c2n] is called an equilibrium price vector (an equilibrium), in the model σc if d(p̄) = s(p̄). 2.2. open market model suppose that we have n× n matrices e = (eij)i,j=1,n, ẽ = (ẽij)i,j=1,n (eij, ẽij ∈ r for all i, j = 1, n, vectors c̄1 = (c11, ..., c1n), c̄2 = (c21, ..., c2n) ∈ rn + with 0 < c1i < c2i for all i = 1, n, a vector p̄∗ = (p∗1, ..., p ∗ n) ∈ rn + such that c1i ≤ p∗i ≤ c2i for all i = 1, n, and vectors d̄∗ = (d∗1, ..., d ∗ n), s̄∗ = (s∗1 , ..., s ∗ n) ∈ rn +. moreover, we suppose that a vector ā = (a1, ..., an) ∈ rn +, such that there exists at least one number i = 1, n : ai > 0, is given. we call σo = (e , ẽ , d̄∗, s̄∗, p̄∗, c̄1, c̄2, ā) (2.8) an open market model (at least one kind of the goods is imported externally). the set of all σo is denoted by σo. the parameters e , ẽ , d̄∗, s̄∗, p̄∗, c̄1, c̄2 of the model (2.8) have the same sense as the ones for the model (2.1). the component of vector āwith the number i = 1, n equals to the volume of imported ith good. copyright © 2021 assa. adv syst sci appl (2021) equilibrium in market models with known elasticities 133 the parameters (e , ẽ , d̄∗, s̄∗, p̄∗, c̄1, c̄2, ā) uniquely define the demand function (2.4) and supply function (2.5). these functions are defined by (2.6) and (2.7) respectively. definition 2.2: a vector p̄ ∈ [c11, c21]× ...× [c1n, c2n] is called an equilibrium price vector (an equilibrium), in the model σo if d(p̄) = s(p̄) + ā. 3. equilibrium in market models the following theorems (criteria on the existense of an equilibrium) are extenstions of the results obtained in [8] for the closed market model. theorem 3.1: in closed market model σc ∈ σc there exists an equilibrium vector iff for any vectors λ̄ = (λ1, . . . , λn) ∈ rn + and x̄ = (x1, . . . , xn) ∈ rn the following conditions hold: n∑ i,j=1 λiaijxj + n∑ i=1 λibi + n∑ i=1 λixi = 0, n∑ i=1 λi ln c1i c2i ≤ 0 with aij = ẽij − eij, i, j = 1, n, bi = ln d∗i s∗i + n∑ k=1 (ẽik − eik) ln p∗k, i = 1, n. proof follows from alexandrov theorem and fan ky theorem on the consistency of linear inequality system (see, e.g., [9]) and the fact that for closed market model σc ∈ σc the condition on the existence of an equilibrium is equivalent to the condition on the existence of the solution to the following system of equations (which is linear by ln pj, j = 1, n): n∑ j=1 (ẽij − eij) ln pj = ln d∗i s∗i + n∑ j=1 (ẽij − eij) ln p∗j , i = 1, n, satisfying inequalities: c1j ≤ pj ≤ c2j ∀j = 1, n. the following theorem is another criteria on the existence of equilibrium. its conditions (unlike theorem 3.1) can be easily verified. to do that, one must simply calculate the finite number of determinants. theorem 3.2: for closed market model σc ∈ σc there exists an equilibrium vector iff there exist such natural numbers ik ≤ n, jl ≤ n, k, l = 1, r with r = rang ( ẽ − e ) and square matrix a = {aikjl}, aikjl = ẽikjl − eikjl with rang r such that the following conditions hold: copyright © 2021 assa. adv syst sci appl (2021) 134 a. arutyunov, a. kotyukov, n. pavlova deta 6= 0, det  a bi1 ... bir aij1 · · · aijr bi  = 0 ∀i = 1, n, det  a bi1 ... bir δij1 · · · δijr ln c1i  ≤ 0, det  a bi1 ... bir δij1 · · · δijr ln c2i  ≥ 0 ∀i = 1, n with bi = ln d∗i s∗i + n∑ k=1 (ẽik − eik) ln p∗k, i = 1, n. and δij is kronecker delta. proof the vector p̄ = (p1, ..., pn) is an equilibrium prices vector in model σc iff it is the solution to the following system of equations and inequalities n∑ j=1 (ẽij − eij) ln pj = ln d∗i s∗i + n∑ j=1 (ẽij − eij) ln p∗j , pi ≥ c1i, pi ≤ c2i, i = 1, n. (3.9) the system (3.9) is consistent iff the following system is consistent n∑ j=1 aijxi = bi, xi ≥ c1i, xi ≤ c2i, i = 1, n. (3.10) here c1i = ln c1i, i = 1, n;c2i = ln c2i, i = 1, n. applying theorem 1.3 from [10] to the system of linear equations and inequalities (3.10) we obtain the conditions of the theorem. 3.1. equilibrium in open market models let us introduce the following notation: ᾱ(σ) = [ max i=1,n (( s∗i n∏ j=1 (p∗j) −ẽij min { c |ẽij | 1j , c −|ẽij | 2j })−1 × n∑ k=1 c2k − c1k 2 c2k|ẽ−1ki | )]−1 , β̄(σ) = max i=1,n (( d∗i n∏ j=1 (p∗j) −eij max { c |eij | 2j , c −|eij | 1j }) × n∑ k=1 ck2 − c1k 2c1k |eik| ) , copyright © 2021 assa. adv syst sci appl (2021) equilibrium in market models with known elasticities 135 γ̄(σ) = max i=1,n |si(c̃) + ai −di(c̃)| , where e−1ij is the element of the inverse matrix e−1 to ẽ , c̃ = c1+c2 2 . theorem 3.3: let the parameters of the open market model σo ∈ σo satisfy the conditions: • β̄(σ) < ᾱ(σ); • γ̄(σ) < ᾱ(σ)− β̄(σ). then there exists an equilibrium prices vector p̄ ∈ rn + with c̄1 ≤ p̄ ≤ c̄2. to prove this theorem, we need the following definitions and results from the theory of covering maps. let us formulate them. let (x, ρx) and (y, ρy ) be the metric spaces and ψ and φ be the maps from x to y . by bx(x, r) denote a closed ball with the center at point x and radius r in the space x . analogously we define by (x, r). definition 3.1: (see [11]). a map ψ : x → y is called α-covering if ψ(bx(x, r)) ⊇ by (ψ(x), αr)∀x ∈ x, ∀r > 0. definition 3.2: (see [11]). a map ψ is called metrically κ-regular if ∀x0 ∈ x, y ∈ y ∃ ∈ x : ψ(x) = y and ρx(x, x0) ≤ κρy (y,ψ(x0)). proposition 3.1: a map ψ is α-covering iff ψ is 1/α-regular. note that the maps ψ and φ are, obviously, surjective. it is easy to show that from the properties of metric regularity we can obtain the following proposition. proposition 3.2: let σo ∈ σo, s,d : rn + → rn be such that s is α-covering and d is lipschitz continuous with lipschitz constant β, ∃p∗, p∗∗ ∈ rn + : s(p∗) = d(p∗), s(p∗∗) = d(p∗). if s(p∗) = s(p∗∗), then p∗ = p∗∗. proof indeed, if s is α-covering, then s is 1/α-regular and ρx(p∗, p∗∗) ≤ 1 α ρy (s(p∗), s(p∗∗)) = 0, which leads to p∗ = p∗∗. theorem 3.4: (see theorem 1 from [11]) let the space x be complete, x0 ∈ x,α > 0, r > 0. let the map ψ : x → y be closed and α-covering on bx(x0, r). then for any nonnegative β < α and any map φ : bx(x0, r)→ y satisfying lipschitz condition with the constant β and such that ρy (ψ(x0),φ(x0)) ≤ (α− β)r, there exists a coincidence point ξ ∈ x for the maps ψ,φ, i.e., ψ(ξ) = φ(ξ), such that ρy (x0, ξ) ≤ ρy (ψ(x0),φ(x0)) α− β . copyright © 2021 assa. adv syst sci appl (2021) 136 a. arutyunov, a. kotyukov, n. pavlova proof proof of theorem 3.3. introduce the following notation. by cov(s|m) denote the supremum of all α > 0 such that s is α-covering on m . by lip(d|m) denote the infimum of all β ≥ 0 such that d satisfies lipschitz condition with a constant β. then lip(d|m) = sup p∈intm ∥∥∥∥∂d∂p (p) ∥∥∥∥ . in spaces rn +,rn define the norms by ‖x‖1 = 2 max j=1,n |xj| c2j − c1j ∀x = (x1, ..., xn) ∈ rn, ‖y‖2 = max j=1,n |yj|∀y = (y1, ..., yn) ∈ rn. let x = rn +, y = rn. put m = bx(c̃, 1). it is obvious that m = [c11, c21]× ...× [c1n, c2n]. consider the metric spaces (x, ρy ) and (y, ρy ) with the metrics ρx , ρy defined by the norms ‖ · ‖1, ‖ · ‖2 correspondingly. it is obvious that rn + is not complete, but we only need that bx(c̃, 1) is complete. let us estimate lip(d|m). to do that, estimate ∥∥∥∂d ∂p (p) ∥∥∥ first. from (2.2) it follows that∥∥∥∥∂d∂p (p) ∥∥∥∥ = d∗ieik pk n∏ j=1 (p∗j) −eijp eij j therefore, ∥∥∥∥dddp ∥∥∥∥ = max ‖x‖=1 ∥∥∥∥dddp x ∥∥∥∥ = max ‖x‖=1 max i=1,n n∑ k=1 ∣∣∣∣∂di ∂pk xk ∣∣∣∣ ≤ ≤ max ‖x‖=1 max i=1,n n∑ k=1 d∗i |eik| pk |xk| n∏ j=1 (p∗j) −eijp eij j ≤ ≤ max i=1,n ( n∏ j=1 (p∗j) −eijp eij j d∗i ) n∑ k=1 c2k − c1k 2pk |eik| ≤ ≤ max i=1,n d∗i ( n∏ j=1 (p∗j) −eij max { c eij 2j , c eij 1j }) × × n∑ k=1 ck2 − c1k 2c1k |eik| = β̄(σ). now we estimate cov(s|m). according to proposition 3.1, if the map s is α-covering, the inverse map s−1 is 1/α-lipschitz continuous. we obtain the estimate using this proposition. firstly we find (∂s/∂p)−1. by (2.3) we have ∂si ∂pk (p) = ẽiks ∗ i pk n∏ j=1 (p∗j) −ẽijp ẽij j copyright © 2021 assa. adv syst sci appl (2021) equilibrium in market models with known elasticities 137 therefore, det ∂s(p) ∂p = n∏ i=1 ( s∗i p −1 i n∏ j=1 (p∗j) −ẽijp ẽij j ) det ẽ . by sik, ẽik denote a cofactor to element ∂s/∂pk, ẽik of ∂s/∂p, ẽ correspondingly. thus: sik =  n∏ m=1 m 6=k p−1m  n∏ l=1 l 6=i ( s∗l n∏ j=1 (p∗j) −ẽljp ẽlj j ) ẽik. hence, the element of inverse matrix (∂s/∂p)−1:( ∂s(p) ∂p )−1 ki = ẽ−1ki p−1k s∗i n∏ j=1 (p∗j) −ẽijp ẽij j , where ẽ−1ki is the element of inverse matrix to ẽ . now we estimate the lipschitz constant of (∂d/∂p)−1:∥∥∥∥∥ ( ∂s(p) ∂p )−1∥∥∥∥∥ = max ‖x‖1=1 max i=1,n n∑ k=1 ∥∥∥∥∥ ( ∂si(p) ∂pk )−1∥∥∥∥∥ ≤ ≤ max ‖x‖1=1 max i=1,n n∑ k=1 pk ∣∣∣ẽ−1ki xk ∣∣∣ s∗i n∏ j=1 (p∗j) −ẽij min { c ẽij 1j , c ẽij 2j } ≤ ≤ max i=1,n ( s∗i n∏ j=1 (p∗j) −ẽij min { c ẽij 1j , c ẽij 2j })−1 × × n∑ k=1 ∣∣∣ẽ−1ki ∣∣∣ c2k − c1k 2 c2k = 1 ᾱ(σ) . by the conditions of theorem and inequalities cov(s|m) ≥ ᾱ(σ), lip(d|m) ≤ β̄(σ) there exist positive numbers α, β such that β̄(σ) < β < α < ᾱ(σ), s is α-covering on m, d is β-lipschitz continuous on m. since ρy (s(c̃+ a,d(c̃)) = γ̄(σ), from condition 2 of the theorem it follows that ρy (s(c̃+ a,d(c̃)) < α− β. therefore, there exists a vector p̄ ∈ x such that d(p̄) = s(p̄) + a and ρy (p̄, c̃) ≤ ρy (s(c̃) + a,d(c̃)) α− β . from the last inequality it follows that p ∈ intm , since m = bx(c̃, 1) and ρy(s(c̃) + a,d(c̃)) < α− β. hence, c1j < pj < c2j, j = 1, n. theorem 3.5: let the following conditions be satisfied for the model σo ∈ σo: copyright © 2021 assa. adv syst sci appl (2021) 138 a. arutyunov, a. kotyukov, n. pavlova • ᾱ(σ) > 2β̄(σ); • ∃p̃ = (p̃1, . . . , p̃n) ∈ rn +: s(p̃1, . . . , p̃n) = d(p∗1, . . . , p ∗ n), where (p∗1, . . . , p ∗ n) is an equilibrium in model σo. then in the model σo there exists an equilibrium prices vector not equal (p∗1, . . . , p ∗ n). proof follows from theorem 1 in [12]. the results similar to theorems 3.3–3.5 for dynamic market models may be obtained, applying the results of [13]– [16] besides existence theorems for coincidence points. in such models the supply and demand functions depend not only on prices on the goods, but on price change rates. in turn, the question of the existence for an equilibrium can be considered as the question of existence for the solution of a system of differential equations. 4. iteration process define the norms in the spaces rn + and rn: ‖x‖x = 2 max j=1,n |xj| c2j − c1j ∀x = (x1, ..., xn) ∈ rn, ‖y‖y = max j=1,n |yj| ∀y = (y1, ..., yn) ∈ rn, and consider metric spaces (rn +, ρx), (rn, ρy ) with the metrics ρx and ρy generated by the norms ‖ · ‖x and ‖ · ‖y correspondingly. fix an arbitrary δ > 0. from definition (3.1) we obtain that ρy (s(x′), d(x)) ≤ δρy (s(x), d(x)), (4.11) ρx(x′, x) ≤ α−1ρy (s(x), d(x)). (4.12) based on these inequalities we can construct the following iteration process. let s,d : x → y , s be α-covering and d be lipschitz continuous with the constant β. fix an arbitrary p0 ∈ x and a sequence of nonnegative numbers {δi}: β + α lim i→∞ δi < α. (4.13) then there exists a sequence {pi}, i = 0, 1, ... such that ρy (s(pi+1), d(pi)) ≤ δiρy (s(pi), d(pi)), (4.14) ρx(pi+1, pi) ≤ α−1ρy (s(pi), d(pi)). (4.15) theorem 4.1 (a. arutyunov, see [1]): let the space x be complete, s : x → y be α-covering, gphs = {(x, y) ∈ x × y : y = s(x)} be closed and d : x → y satisfy lipschitz condition with a constant β < α and, moreover, β + α lim i→∞ δi < α. (4.16) then ∀p0 ∈ x there exists a sequence {pi} satisfying (4.14), (4.15) ∀i, and any such sequence converges to some coincidence point ξ = ξ(x0) with ρx(ξ, p0) ≤ α−1 ( 1 + ∞∑ j=1 i∏ j=1 ( δi + β α )) ρy (s(p0), d(p0)). copyright © 2021 assa. adv syst sci appl (2021) equilibrium in market models with known elasticities 139 we construct search algorithm on the base of iteration process (4.14), (4.15) and theorem 4.1. the search of a new point on every step of this iteration process is based on direct search method. namely, we search the new point of the iteration process in the form: pi+1 = pi + h = (p1i + h1, ..., pni + hn), (4.17) where h = (h1, ..., hn) is the step that needs to be defined on every iteration. the search radius, i.e., the maximal value of every coordinate of the step, can be found using (4.15). indeed, from (4.15) we obtain that: max i=1,n |hi| ci1 − ci2 6 α−1 2 max i=1,n |si(p)−di(p)|, where s(p) = (s1(p), ..., sn(p)), d(p) = (d1(p), ..., dn(p)). then: |hi| ci1 − ci2 6 α−1 2 max i=1,n |si(p)−di(p)|. hence we obtain the estimate for the search radius: |hi| 6 ci1 − ci2 2α max i=1,n |si(p)−di(p)|. (4.18) next we must define the elements of the sequence {δi}. from (4.16) it follows that: lim i→∞ δi < 1− β α . (4.19) here we put: δi = δ = 1− β α − ε, (4.20) where ε > 0 is an arbitrary small positive number (process error). constructed sequence obviously satisfies (4.19). now we describe the step of the iteration process. let pi be the point obtained on ith iteration. define the search radius by considering an equality in (4.18) and put p̃i+1 = (p1i + h1, p2i, ..., pni), i.e., take the initial point with ¡¡incremented¿¿ first coordinate. for this point we check the condition (4.14). if it is satisfied, we put pi+1 = p̃i+1 but we continue the search process by decreasing h1. if we find another point satisfying (4.14), we compare it with the previous one by the value at ρ2(s(·), d(pi)) and take the one with the minimal value. then we decrease h1 and continue the search while h1 > ε. after we complete the search by the first coordinate in positive direction we put p̃i+1 = (p1i − h1, p2i, ..., pni), and conduct this search for this point. once we finish this search (i.e. h1 < ε), we ¡¡increment¿¿ the second coordinate and conduct this search for it etc. when we complete the search for the last coordinate we finish the iteration and obtain the point pi+1. numerical experiments show that on the first step of the iteration process δi chosen above can violate this process. condition (4.19) allows us to redefine δi to satisfy (4.14), from which we obtain that max p∈uh(pi) ρy (s(p), d(pi)) ≤ δiρy (s(pi), d(pi)), copyright © 2021 assa. adv syst sci appl (2021) 140 a. arutyunov, a. kotyukov, n. pavlova where uh(pi) = {x ∈ rn + : |pij − xj| ≤ hj, j = 1, n}. from the last inequality we get the lower estimate for δi: δi ≥ (ρy (s(pk), d(pk)))−1 max p∈uh(pk) ρy (s(p), d(pk)). it is possible to define δi as shown above on each iteration but this can significantly decrease the iteration process. 5. conclusion the open and closed market models are studied. the existence of an equilibrium is investigated. for the closed market model we obtained necessary and sufficient conditions for the existence of an equilibrium price vector. to do that, we used corollaries of consistence theorems for the systems of linear equations and inequalities. these conditions can be easily verified numerically. for the open market model we obtained sufficient existence conditions. to do that, we used corollaries of the theorems on coincidence points for covering and lipschitz continuous mappings. these conditions can be also verified numerically. also we obtained the conditions for the uniqueness and nonuniqueness of equilibrium in the open and closed market models. they are easily obtained as corollaries from coincidence points theorems. all these results can be applied to investigate the power of the set of equilibrium price vectors and to find them numerically. acknowledgements this paper was written with the financial support of rfbr (grant no. 19-01-00080, 2001-00610). theorem 3.3 is obtained by the first author under financial support of russian science foundation (project no. 20-11-20131). references 1. arutyunov, a.v. 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(2016) non-degenerate necessary optimality conditions for the optimal control problem with equality-type state constaints, journal of global optimization, 64(4), 623–647. copyright © 2021 assa. adv syst sci appl (2021) 142 a. arutyunov, a. kotyukov, n. pavlova a pp e n d ix .n u m e r ic a l e x pr e r im e n t r e su lt s ta bl e 5. 1. n um er ic al re su lts (n = 1) . a e ẽ c 1 c 2 d ∗ s ∗ p ∗ e xa ct so lu tio n a pp ro xi m at e so lu tio n d iff er en ce 12 ,0 49 67 25 7 0, 49 84 97 05 0, 31 75 12 80 3, 42 51 30 70 3, 90 65 44 50 52 ,3 29 66 27 0 39 ,7 80 66 12 0 3, 66 58 37 60 3, 53 06 37 18 3, 53 07 15 53 0, 00 00 78 35 23 ,9 46 10 55 9 0, 37 99 69 09 0, 87 86 58 00 1, 08 46 55 00 2, 26 08 07 60 62 ,5 57 06 23 0 42 ,4 19 23 65 0 1, 67 27 31 30 1, 10 99 18 76 1, 11 00 59 81 0, 00 01 41 05 75 ,3 90 41 16 9 0, 61 29 43 72 0, 94 26 46 90 2, 00 81 69 40 2, 23 83 04 10 82 ,0 56 52 56 0 7, 06 13 41 90 2, 12 32 36 75 2, 14 25 03 95 2, 14 24 84 99 0, 00 00 18 96 3, 46 47 92 08 0, 84 91 81 28 0, 98 56 63 00 1, 17 15 36 60 1, 86 12 72 60 14 ,1 80 57 21 0 10 ,3 13 52 89 0 1, 51 64 04 60 1, 21 93 94 22 1, 21 96 33 61 0, 00 02 39 39 12 ,0 49 67 25 4 0, 49 84 97 05 1, 31 75 12 80 3, 42 51 30 70 3, 90 65 44 50 52 ,3 29 66 27 0 39 ,7 80 66 12 0 3, 66 58 37 60 3, 73 46 44 90 3, 73 46 49 99 0, 00 00 05 09 45 ,7 59 60 18 8 0, 21 80 86 95 1, 36 43 36 50 1, 68 36 60 00 3, 10 19 71 30 83 ,7 68 17 07 0 36 ,6 72 68 59 0 2, 39 28 15 65 2, 49 14 02 47 2, 49 13 66 71 0, 00 00 35 76 57 ,9 20 58 65 8 0, 29 80 26 99 1, 39 38 11 70 5, 47 84 52 30 6, 84 55 39 20 98 ,1 63 66 61 0 37 ,7 83 17 68 0 6, 16 19 95 75 6, 76 01 78 29 6, 76 01 31 04 0, 00 00 47 25 52 ,0 17 25 02 3 1, 78 50 43 26 0, 74 30 21 60 1, 31 06 19 30 1, 41 11 85 60 83 ,4 28 42 96 0 27 ,7 29 30 28 0 4, 00 61 36 80 1, 32 13 04 17 1, 32 13 83 81 0, 00 00 19 28 45 ,8 26 52 17 4 1, 48 40 95 65 0, 97 46 88 80 3, 82 62 40 60 4, 18 60 33 00 61 ,0 37 24 68 0 15 ,9 49 39 75 0 2, 94 41 30 05 4, 04 54 58 93 4, 04 54 39 65 0, 00 00 40 24 25 ,4 09 51 76 1 1, 11 90 65 93 0, 25 28 51 10 2, 84 52 80 20 3, 04 29 79 90 97 ,1 41 77 20 0 74 ,4 14 36 64 0 4, 27 68 34 30 3, 03 15 88 98 3, 03 15 48 74 0, 00 00 11 91 24 ,6 69 58 65 5 1, 35 44 84 68 0, 81 66 33 60 3, 81 63 91 70 4, 73 72 76 90 80 ,8 88 50 35 0 61 ,8 26 36 12 0 7, 17 07 06 85 4, 66 85 27 86 4, 66 85 15 95 0, 00 03 75 21 8, 24 24 59 60 1, 74 15 68 54 1, 94 87 13 80 1, 51 48 66 20 1, 92 54 00 30 10 ,9 46 97 02 0 2, 62 02 12 60 6, 90 77 53 25 1, 70 97 24 80 1, 70 97 35 03 0, 00 00 03 48 59 ,3 89 03 65 3 1, 97 22 58 93 1, 92 99 66 70 6, 59 08 41 10 7, 22 46 65 40 76 ,1 39 83 21 0 20 ,3 01 44 89 0 5, 10 37 47 60 7, 12 53 61 38 7, 12 53 57 90 0, 00 03 58 18 25 ,9 81 30 49 0 1, 06 15 89 55 1, 96 27 97 30 4, 74 29 31 00 5, 46 45 64 20 45 ,8 81 08 57 0 20 ,3 65 32 23 0 3, 88 47 47 45 5, 41 41 73 42 5, 41 38 15 24 0, 00 02 70 41 4, 17 41 79 59 1, 18 72 77 26 1, 12 76 44 60 3, 24 92 54 90 4, 52 02 40 00 86 ,8 19 87 70 0 82 ,2 83 27 37 0 0, 75 43 47 15 3, 74 61 59 23 3, 74 64 29 64 0, 00 06 71 30 4, 86 89 45 08 0, 45 99 18 70 -0 ,6 65 85 27 0 7, 67 71 90 70 8, 79 74 88 20 9, 24 30 15 45 4, 38 45 89 11 8, 23 73 39 45 8, 24 94 31 89 8, 24 94 31 35 0, 00 00 00 54 8, 49 60 13 14 1, 80 79 54 21 -1 ,4 85 82 48 0 1, 25 34 82 80 1, 37 77 86 50 8, 12 00 03 01 0, 12 08 65 98 1, 31 56 34 65 1, 35 90 74 73 1, 35 90 74 84 0, 00 00 00 11 0, 35 32 40 02 2, 73 43 24 05 -2 ,7 00 23 78 0 0, 27 42 14 80 8, 02 55 73 70 0, 60 35 63 16 4, 57 84 65 68 4, 14 98 94 25 6, 26 31 73 87 6, 26 31 73 74 0, 00 00 00 13 4, 29 33 12 68 3, 94 83 31 92 -3 ,4 95 45 33 0 6, 40 13 22 30 7, 26 92 56 00 9, 98 52 49 81 4, 81 04 50 34 6, 83 52 89 15 6, 72 75 93 15 6, 72 75 93 30 0, 00 00 00 15 copyright © 2021 assa. adv syst sci appl (2021) equilibrium in market models with known elasticities 143 ta bl e 5. 2. n um er ic al re su lts (n = 2) . a e ẽ c 1 c 2 d ∗ s ∗ p ∗ e xa ct so lu tio n a pp ro xi m at e so lu tio n d iff er en ce (1 0 − 4 ) 5, 44 92 46 34 0. 18 90 0. 88 20 0. 78 92 0. 20 51 6, 15 66 38 00 11 ,8 27 80 96 0 8, 12 07 83 80 0, 97 49 50 70 8, 99 22 23 80 7, 45 51 09 45 7, 45 50 90 20 0, 00 00 19 25 1, 50 26 96 78 0. 69 93 0. 96 98 0. 03 57 0. 86 06 3, 73 11 10 20 13 ,4 09 70 39 0 7, 41 03 93 50 4, 25 43 88 20 8, 57 04 07 05 6, 24 36 31 97 6, 24 36 31 97 6, 08 64 05 69 2. 27 65 1. 37 60 0. 81 56 0. 24 02 0, 77 74 84 50 0, 93 91 76 60 6, 40 65 20 17 6, 25 21 56 23 0, 85 83 30 55 0, 93 91 72 82 0, 93 89 80 26 0, 00 02 71 11 5, 29 11 89 33 4. 52 51 7. 37 93 0. 03 05 0. 97 65 5, 38 25 48 20 5, 45 60 87 70 3, 79 43 71 35 8, 70 64 45 23 5, 41 93 17 95 5, 45 60 70 10 5, 45 58 79 26 7, 84 59 66 91 6. 13 38 8. 32 64 0. 91 73 0. 31 66 7, 42 23 43 60 7, 50 77 17 10 8, 47 62 63 15 6, 92 24 53 70 7, 46 50 30 35 7, 50 77 10 74 7, 50 77 01 23 0, 00 00 14 74 2, 77 77 15 60 9. 25 62 6. 30 13 0. 66 89 0. 69 81 2, 18 74 84 60 2, 34 25 98 40 2, 37 61 39 70 5, 24 03 01 45 2, 26 50 41 50 2, 34 25 80 83 2, 34 25 69 57 0, 93 75 43 08 6. 22 04 4. 73 21 0. 98 47 0. 16 37 7, 34 77 19 70 7, 45 62 99 40 7, 56 45 49 29 9, 52 37 66 09 7, 40 20 09 55 7, 45 62 72 36 7, 45 61 14 10 0, 00 01 98 82 3, 03 70 09 92 8. 44 00 5. 62 22 0. 31 53 0. 70 35 4, 77 91 20 10 4, 87 65 29 30 5, 34 61 82 34 9, 61 66 58 66 4, 82 78 24 70 4, 87 64 83 42 4, 87 63 63 07 32 ,6 44 72 26 0 89 .9 77 7 22 .7 80 0 9. 67 36 6. 45 68 87 ,4 42 05 80 0 87 ,4 83 79 64 0 36 ,5 58 47 46 8 81 ,6 56 30 91 9 87 ,4 62 92 72 0 87 ,4 83 79 04 5 87 ,4 83 67 33 1 0, 00 02 81 63 1, 97 42 33 17 43 .3 59 5 23 .2 55 9 6. 97 41 6. 63 99 4, 94 66 53 80 5, 03 38 54 10 51 ,9 18 10 77 8 80 ,3 05 65 52 0 4, 99 02 53 95 5, 03 38 53 05 5, 03 35 96 94 6, 62 85 81 47 83 .4 10 5 37 .7 34 4 5. 96 30 0. 33 56 94 ,7 63 27 61 0 94 ,8 75 13 01 0 40 ,5 95 97 35 9 81 ,5 09 84 57 6 94 ,8 19 20 31 0 94 ,8 75 07 15 0 94 ,8 75 04 71 1 0, 00 00 70 92 75 ,8 26 74 31 4 40 .7 93 5 73 .9 58 4 0. 98 47 8. 39 60 44 ,1 84 89 15 0 44 ,3 11 03 04 0 99 ,7 88 15 74 4 93 ,8 29 97 67 7 44 ,2 47 96 09 5 44 ,3 11 00 34 0 44 ,3 10 93 68 1 0, 93 75 43 08 6. 22 04 4. 73 21 0. 98 47 0. 16 37 7, 34 77 19 70 7, 45 62 99 40 7, 56 45 49 29 9, 52 37 66 09 7, 40 20 09 55 7, 45 62 72 36 7, 45 60 67 59 0, 00 02 61 15 3, 03 70 09 92 8. 44 00 5. 62 22 0. 31 53 0. 70 35 4, 77 91 20 10 4, 87 65 29 30 5, 34 61 82 34 9, 61 66 58 66 4, 82 78 24 70 4, 87 64 83 42 4, 87 63 21 34 2, 60 77 08 48 4. 01 64 9. 47 26 0. 31 68 0. 42 43 7, 55 57 16 60 7, 56 82 48 90 6, 04 56 77 02 9, 75 81 69 92 7, 56 19 82 75 7, 56 82 45 33 7, 56 82 12 05 0, 00 00 66 78 5, 41 95 22 09 3. 87 16 1. 42 76 0. 79 69 0. 63 16 5, 03 93 81 00 5, 06 07 73 20 5, 25 34 99 15 8, 17 91 71 63 5, 05 00 77 10 5, 06 07 65 88 5, 06 07 07 98 7, 58 08 78 11 2. 81 64 1. 45 88 0. 71 91 0. 22 19 3, 12 70 26 60 3, 32 65 84 70 9, 14 18 50 23 9, 46 44 01 26 3, 22 68 05 65 3, 32 65 67 08 3, 32 65 84 67 0, 00 00 61 68 0, 60 92 50 99 4. 16 22 6. 04 16 0. 29 73 0. 68 45 9, 16 04 05 30 9, 23 57 00 10 7, 33 79 82 84 8, 85 95 19 59 9, 19 80 52 70 9, 23 56 87 40 9, 23 56 28 28 0, 37 63 45 16 0. 96 67 0. 57 79 -0 .5 49 2 -0 .1 57 6 5, 03 02 07 50 5, 30 80 02 40 9, 06 97 82 03 8, 77 30 20 14 5, 16 91 04 95 5, 09 26 56 85 5, 08 56 82 69 0, 00 69 74 16 0, 37 63 45 16 0. 45 16 0. 22 16 -0 .8 89 3 -0 .6 16 6 5, 03 02 07 50 5, 30 80 02 40 9, 06 97 82 03 8, 77 30 20 14 5, 16 91 04 95 5, 20 89 78 41 5, 30 17 36 41 0, 37 63 45 16 0. 96 67 0. 57 79 -0 .5 49 2 -0 .1 57 6 5, 03 02 07 50 5, 30 80 02 40 9, 06 97 82 03 8, 77 30 20 14 5, 16 91 04 95 5, 08 54 88 70 5, 08 56 82 69 0, 00 01 93 99 0, 37 63 45 16 0. 45 16 0. 22 16 -0 .8 89 3 -0 .6 16 6 5, 03 02 07 50 5, 30 80 02 40 9, 06 97 82 03 8, 77 30 20 14 5, 16 91 04 95 5, 30 18 90 90 5, 30 17 36 41 5, 34 46 57 76 0. 08 06 0. 91 13 -0 .2 61 3 -0 .7 45 8 5, 89 70 07 00 6, 01 06 39 50 9, 77 38 45 69 4, 51 11 80 83 5, 95 38 23 25 5, 98 00 72 40 5, 98 01 63 42 0, 00 00 91 02 5, 34 46 57 76 0. 41 83 0. 21 31 -0 .7 01 8 -0 .2 88 8 5, 89 70 07 00 6, 01 06 39 50 9, 77 38 45 69 4, 51 11 80 83 5, 95 38 23 25 6, 01 06 39 50 6, 01 06 39 48 4, 43 20 44 23 0. 32 71 0. 38 22 -0 .9 38 8 -0 .0 42 8 7, 05 48 84 90 7, 14 02 09 00 5, 94 09 92 36 1, 49 78 93 01 7, 09 75 46 95 7, 07 48 50 70 7, 07 48 52 27 0, 00 00 01 57 4, 43 20 44 23 0. 05 78 0. 65 83 -0 .0 55 6 -0 .6 05 9 7, 05 48 84 90 7, 14 02 09 00 5, 94 09 92 36 1, 49 78 93 01 7, 09 75 46 95 7, 09 22 56 90 7, 09 22 73 83 copyright © 2021 assa. adv syst sci appl (2021) 144 a. arutyunov, a. kotyukov, n. pavlova ta bl e 5. 3. n um er ic al re su lts (n = 3) . a e ẽ c 1 c 2 d ∗ s ∗ p ∗ e xa ct so lu tio n a pp ro xi m at e so lu tio n d iff er en ce 0, 00 06 71 64 -0 ,5 23 1 0, 21 73 0, 96 89 0, 55 78 -0 ,7 80 4 0, 54 95 0, 00 09 08 00 0, 00 15 87 80 0, 00 08 13 57 0, 00 05 03 46 0, 00 12 47 90 0, 00 12 83 20 0, 00 14 04 21 0, 00 01 21 01 0, 00 01 12 86 0, 72 10 -0 ,1 72 6 0, 98 37 0, 69 00 -0 ,3 33 0 0, 28 00 0, 00 05 04 00 0, 00 09 08 60 0, 00 08 67 91 0, 00 08 75 20 0, 00 07 06 30 0, 00 09 08 60 0, 00 07 69 05 0, 00 02 74 98 0, 00 00 0, 62 94 -0 ,9 42 3 -0 ,6 84 8 -0 ,4 30 6 0, 26 99 0, 00 02 05 00 0, 00 09 78 10 0, 00 06 58 43 0, 00 05 11 56 0, 00 05 91 55 0, 00 08 88 50 0, 00 08 35 23 0, 00 01 07 50 0, 22 74 0, 85 85 -0 ,3 71 0 0, 07 15 0, 84 18 0, 60 96 0, 00 00 22 80 0, 00 05 24 10 0, 00 09 35 67 0, 00 03 90 17 0, 00 02 73 45 0, 00 01 60 20 0, 00 04 59 87 0, 00 02 99 67 0, 00 09 57 15 -0 ,7 54 7 -0 ,5 59 8 -0 ,2 74 8 0, 60 20 0, 77 08 0, 96 75 0, 00 00 48 70 0, 00 04 08 10 0, 00 09 23 20 0, 00 08 66 16 0, 00 02 28 40 0, 00 01 35 30 0, 00 01 44 92 0, 00 05 39 50 0, 00 00 -0 ,2 46 3 0, 04 60 0, 70 82 0, 90 49 -0 ,7 07 5 0, 00 00 62 60 0, 00 07 87 40 0, 00 07 71 32 0, 00 01 36 70 0, 00 04 25 00 0, 00 06 20 40 0, 00 07 49 18 0, 00 00 39 63 -0 ,5 02 6 0, 43 13 0, 76 71 0, 57 29 -0 ,9 93 4 0, 38 43 0, 00 00 20 70 0, 00 00 45 80 0, 00 00 58 73 0, 00 00 05 03 0, 00 00 33 25 0, 00 00 45 80 0, 00 00 45 80 0, 00 00 00 00 0, 00 00 71 97 0, 12 15 0, 12 76 -0 ,0 76 8 -0 ,8 32 8 0, 62 71 0, 88 35 0, 00 00 15 50 0, 00 01 11 90 0, 00 00 58 27 0, 00 00 87 90 0, 00 00 63 70 0, 00 00 15 60 0, 00 00 15 50 0, 00 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13 -0 ,4 71 5 0, 57 01 0, 00 00 39 90 0, 00 01 34 70 0, 00 00 48 45 0, 00 00 22 84 0, 00 00 87 30 0, 00 01 34 70 0, 00 01 34 50 0, 00 00 00 20 0, 00 00 32 30 -0 ,0 20 8 0, 77 08 0, 18 09 -0 ,7 56 5 -0 ,2 71 6 -0 ,4 08 5 0, 00 00 76 80 0, 00 01 34 10 0, 00 00 09 04 0, 00 00 59 78 0, 00 01 05 45 0, 00 01 34 10 0, 00 01 34 10 0, 00 00 87 44 0, 00 00 -0 ,8 70 8 0, 86 00 -0 ,4 45 5 0, 64 63 0, 54 18 0, 00 00 51 60 0, 00 01 23 80 0, 00 00 58 10 0, 00 00 48 79 0, 00 00 87 70 0, 00 01 51 60 0, 00 01 10 73 copyright © 2021 assa. adv syst sci appl (2021) introduction market models closed market model open market model equilibrium in market models equilibrium in open market models iteration process conclusion